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+{"text":"\\section{Introduction}\n\nNeural Networks (NN's) have become in the last years a very effective\ninstrument for solving many difficult problems in the field of Signal\nProcessing due to their properties like non-linear dynamics, adaptability,\nself-organization and high speed computational capability (see for example \n\\cite{Luo97} and the papers therein quoted).\n\nAim of this paper is to show the feasibility of the use of NN's to solve\ndifficult problems of signal processing regarding the so called VIRGO\nproject. Gravitational Waves (GW's) are travelling perturbations of the\nspace-time predicted by the theory of \\ General Relativity, emitted when\nmassive systems are accelerated. Up to now, there is only an indirect\nevidence of their existence, obtained by the observations of the binary\npulsar system PSR 1913+16. Moreover, the direct detection of GW's is not\nonly a relevant test of General Relativity, but the start of a new picture\nof the Universe. In fact, GW's carry complementary information with respect\nto electromagnetic and optical waves, since the GW's are practically not\nabsorbed by the matter.\n\nThe aim of the Virgo experiment is the direct detection of gravitational\nwaves and, in joint operation with other similar detectors, to perform\ngravitational waves astronomical observations. In particular, the VIRGO\nproject is designed for broadband detection from $10Hz$ to $10kHz$. The\nprinciple of the detector is shown in figure~1.\n\n A $3$ $km$ arm-length Michelson\ninterferometer with suspended mirrors (test masses) is used. The phase\ndifference $\\Delta \\phi $ between the two arms is amplified using\nFabry-Perot cavities of Finesse $50$ in each arm. Aiming for detection\nsensitivity of $3.10E-23\\frac{1}{Hz}\\quad at\\quad 100Hz$, VIRGO is a very\ndelicate experimental challenge because of the competition between various\nsources of noise and the very small expected signal. In fact, the\ninterferometer will be tuned on the dark fringe, and then the signal to\nnoise ratio will be mainly limited, in the above defined range of\nsensitivity, by residual seismic noise, thermal noise of the suspensions\nphoton counting noise (shot noise). In figure 2 the overall sensitivity of\nthe apparatus is shown. \nIn this figure it is easy to see the contribution of\nthe different noise sources to the global noise.\n\nIn this context we use a Multi-Layer Perceptron (MLP) NN with\nthe back-propagation learning algorithm to model and identify the noise in the\nsystem, because we experimentally found that FIR NN's and Elman NN's did not work in a satisfying manner.\n\nBoth the FIR \\cite{Tsoi91} and Elman \\cite{Elman90} models proved to be very\nsensible to overfitting and were not stable.  Furthermore the Elman network\nrequired a great number of hidden units, while the FIR network required a\ngreat number of delay terms. Instead, the MLP proved succesfull and easy to\ntrain because we used the Bayesian learning paradigm.\n\n\\section{Neural networks for time-domain system identification}\n\nNN's are massively parallel, distributed processing systems. They\nare composed of a large number of processing elements (called nodes, units\nor neurons) which operate in parallel. Scalars (called weights) are\nassociated to the connections between units and determine the strength of\nthe connections themselves. Computational capability is due to the\nconnections between the units and to their collective behaviour.\nFurthermore, information representation is distributed, i.e. no single unit\nis charged with specific information.\nNN's are well-known for their universal approximation capability\n\\cite{Hornik89}.\n \nSystem identification consists in finding the input-output mapping of a\ndynamic system. A discrete-time Multi-Input Multi-Output (MIMO) system (or a\ncontinuous-time sampled-data system) can be described with a set of\nnon-linear difference equations of the form ({\\em input-state-output}\nrepresentation): \n\\begin{equation}\n\\left\\{ \n\\begin{array}{ll}\n{\\bf z}(n+1)=F({\\bf z}(n),{\\bf u}(n)) &  \\\\ \n{\\bf y}(n)=G({\\bf z}(n)) & \n\\end{array}\n\\right.  \\label{eq:iso}\n\\end{equation}\nwhere ${\\bf z}\\in R^{L}$ is the state vector of the system, ${\\bf u}\\in\nR^{M} $ is the input vector and ${\\bf y}\\in R^{N}$ is the output vector. \nSince we\ncan not always access the state vector of the system, therefore we can use an\ninput-output representation of the form: \n\\begin{eqnarray}\n{\\bf y}(n) &=&{\\cal F}[{\\bf y}(n-1),{\\bf y}(n-2),\\ldots ,{\\bf y}(n-n_{y}), \n\\nonumber \\\\\n&&{\\bf u}(n-n_{d}),\\ldots ,{\\bf u}(n-n_{u}-n_{d}+1)]  \\label{eq:io}\n\\end{eqnarray}\nwhere $n_{u}$ and $n_{y}$ are the maximum lags of the input and output\nvectors, respectively, and $n_{d}$ is the pure time-delay (or dead-time) in\nthe system. This is called an ARX model (autoregressive with exogenous\ninputs,) and it can be shown that a wide class of discrete-time systems can\nbe put in this form~\\cite{Chen89}. To build a model of the form~(\\ref{eq:io}%\n), we must therefore obtain an estimation of the functional ${\\cal F}[\\cdot\n] $, which generally is nonlinear.\n\nGiven a set of input-output pairs, a neural network can be built \\cite{Luo97}\nwhich approximates the desired functional ${\\cal F}[\\cdot ]$. Such a network\nhas $n_{y}N+(n_{u}-n_{d})M$ inputs and $N$ outputs (see figure 3). \n\nA\ndifficulty in this approach arises from the fact that generally we do not\nhave information about the model order (i.e. the maximum lags to take into\naccount) unless we have some insight into the system physics. Furthermore,\nthe system is non-linear. Recently \\cite{He93} a method has been proposed for\ndetermining the so-called {\\em embedding dimension} of nonlinear dynamical\nsystems,\n when the input-output pairs are affected by very low noise.\nFurthermore, the lags can be determined by evaluating the \\emph{average mutual\ninformation} (AMI)\\cite{Abarbanel96}. Such methodologies, although not always\nsuccessful, can be nevertheless used as a starting point in model design.\n\n\\section{Virgo}\n \nIn the VIRGO data analysis, the most difficult problem is the gravitational\nsignal extraction from the noise due to the intrinsic weakness of the\ngravitational waves, to the very poor signal-to-noise ratio and to their not\nwell known expected templates. Furthermore, the Virgo detector is not yet\noperational, and the noise sources analyzed are purely theoretical models\n(often stationary noises), not based on experimental data. Therefore, we\nexpect a great difference between the theoretical noise models and the\nexperimental ones. As a consequence, it is very important to study and to\ntest algorithms for signal extraction that are not only very good in signal\nextraction from the theoretical noise, but also very adaptable to the real\noperational conditions of Virgo.\n\nFor this reason, we decided the following strategy for the study, the\ndefinition and the tests of algorithms for gravitational data analysis. The\nstrategy consists of the following independent research lines.\n\nThe first line starts from the definition of the expected theoretical noise\nmodels. Then a signal is added to the Virgo noise generated and the\nalgorithm is used for the extraction of the signal of known and unknown\nshape from this noise at different levels of signal-to-noise ratio. This\nwill allow us to make a number of data analysis controlled experiments to\ncharacterize the algorithms.\n\nThe second line starts from the real measured environmental noise (acoustic,\nelectromagnetic, ...) and tries to identify the noise added to a theoretical\nsignal. In this way we can test the same algorithms in a real case when the\nnoise is not under control.\n\nUsing this strategy, at the end, when in a couple of years Virgo will be\nready for the first test of data analysis, the procedure will be moved to\nthe real system, being sure to find small differences from theory and\nreality after having acquired a large experience in the field.\n\n\\section{A neural network-based model of the Virgo system}\n\nAs we have seen in the introduction, the Virgo interferometer can be\ncharacterized by a sensitivity curve, which expresses the capability of the\nsystem to filter undesired influences from the environment, and which could\nspoil the detection of gravitational waves (such a noise is generally called\nseismic noise). The sensitivity curve has the following expression: \n\\begin{equation}\nS(f)=\\left\\{ \n\\begin{array}{ll}\n\\frac{S_{1}}{f^{5}}+\\frac{S_{2}}{f}+S_{3}\\left[ 1+\\left( \\frac{f}{f_{k}}\n\\right) ^{2}\\right] +S_{\\nu } & \\quad f\\geq f_{\\textrm{min}} \\\\ \nS(f_{\\textrm{min}}) & \\quad f<f_{\\textrm{min}}\n\\end{array}\n\\right.   \\label{eq:exprsenscurve}\n\\end{equation}\nwhere:\n\n\\begin{itemize}\n\\item  $f_{\\textrm{min}}=4Hz$\n\n\\item  $f_{k}=500Hz$ is the shot noise cut-off frequency\n\n\\item  $S_{1}=1.8\\cdot 10^{-36}$ is the pendulum mode\n\n\\item  $S_{2}=0.33\\cdot 10^{-42}$ is the mirror mode\n\n\\item  $S_{3}=3.24\\cdot 10^{-46}$ is the shot noise\n\\end{itemize}\n\nThe contribution $S_{\\nu }(f)$ of violin resonances is given by: \n\\begin{equation}\nS_{\\nu }(f)=\\sum_{i}^{n}\\frac{1}{i^{4}}\\frac{f_{i}^{(c)}}{f}\\frac{C_{c}\\phi\n_{i}^{2}}{\\left[ \\left( \\frac{f}{if_{i}^{(c)}}\\right) ^{2}-1\\right]\n^{2}+\\phi _{i}^{2}}+\\frac{1}{i^{4}}\\frac{f_{i}^{(f)}}{f}\\frac{C_{f}\\phi\n_{i}^{2}}{\\left[ \\left( \\frac{f}{if_{i}^{(f)}}\\right) ^{2}-1\\right]\n^{2}+\\phi _{i}^{2}}  \\label{eq:violinres}\n\\end{equation}\nwhere the different masses of close and far mirrors are taken into account:\n\n\\begin{itemize}\n\\item  $f_{i}^{(c)}=i\\cdot 327Hz$\n\n\\item  $f_{i}^{(f)}=i\\cdot 308.6Hz$\n\n\\item  $C_{c}=3.22\\cdot 10^{-40}$\n\n\\item  $C_{f}=2.82\\cdot 10^{-40}$\n\n\\item  $\\phi _{i}^{2}=10^{-7}$\n\\end{itemize}\n\nNote that we used\na simplified curve for our simulations, in which we neglected the resonances (see\nfigure 4).\n\nSamples of the sensitivity curve $\\{S_{i}\\}_{i}$ can be obtained by\nevaluating the expression (\\ref{eq:exprsenscurve}) at a set of frequencies $%\n\\{f_{i}\\}_{i}$, $f_{i}\\in \\lbrack 10,10000]$. The samples of the sensitivity\ncurve allow us to obtain the system transfer function (in the frequency\ndomain), $N(j2\\pi f)$, such that: \n\\begin{equation}\n|N(j\\omega )|^{2}=S(j\\omega )  \\label{eq:tranfun}\n\\end{equation}\n>From this, by means of an inverse Discrete Fourier Transform, samples of the\nsystem transfer function (in the time domain) can be obtained. Our aim is to\nbuild a model of the system transfer function (\\ref{eq:tranfun}).\n\nAssuming that the interferometer input noise is a zero mean Gaussian\nprocess, by feeding it to the system (i.e. filtering it through the system\ntransfer function) we obtain a coloured noise. The so obtained white\nnoise-colored noise pairs can then be used to train an MLP, as shown in\nfigure 3.\n\n\\subsection{Experimental Results}\n\nThe first step in building an ARX model is the model order determination. To\ndetermine suitable lags which describe the system dynamics, we used the\nAMI criterion \\cite{Abarbanel96}. \nThis can be seen as a generalization of the autocorrelation function, used to\ndetermine lags in linear systems. A strong property of the AMI statistic is\nthat it takes into account the non-linearities in the system. Usually, the\nlag is chosen as the first minimum of the AMI function. The result is reported\nin figure~5, in which the first minimum is at 1.\nTo find how many samples are necessary to unfold the (unknown) state-space of\nthe model (the so called \\emph{embedding dimension} \\cite{Abarbanel96}) we used\nthe method of \\cite{He93}, the Lipschitz decomposition. The result of the\nsearch is reported in\n figure~6.\n\nFrom the figure we can see that,\nstarting from lag three, the order\n index decreases very slowly, and so we can\nderive that the width of the input window is at\n least three. In order to test\nthe NN's capability in solving the problem, we\nchose a width of 5, both for\ninput and output (i.e. $n_{y}=n_{u}=5$). In\n this way, we obtained a NN with a\nsimple structure. Furthermore, some\n preliminary experiments showed that the\nsystem dead-time is $n_{d}=0$; this\n gives the best description of the system\ndynamics.\n\nAnother fundamental issue is the NN complexity, i.e. the number\nof units in\n the hidden layers of the NN. Usually the determination of the\nnetwork complexity is critical because of the risks of overfitting.\nSince \nthe NN was trained following a Bayesian framework, then overfitting was of no\nconcern; so we directed our search for a model with the minimum possible\ncomplexity. In our case, we found a hidden layer with 6 $tanh$ units is\noptimal.\n\nThe Bayesian learning framework (see \\cite{MacKay94} and \\cite{Neal94}) allows\nthe use of a \\emph{distribution} of NN's,\nthat is, the model is a realization of a random vector whose\ncomponents are the NN weights. The so obtained NN is the {\\em most probable}\ngiven the data used to train it. This approach avoids the bias of the\ncross-validatory techniques commonly used in practice to reduce model\noverfitting \\cite{Bishop96}. To allow for a smooth mapping which does not\nproduce overfitting, several regularization parameters (also called\n\\emph{hyperparameters}) embedded in the error criterion have been used:\n\\begin{itemize}\n\\item one for each set of connections out of each input node,\n\\item one for the set of connections from hidden to output units,\n\\item one for each of the bias connections,\n\\item one for the data-contribution to the error criterion.\n\\end{itemize}\nUsually, the hyperparameters of the first three kinds are called \\emph{alphas},\nwhile the last is called a \\emph{beta}.\n \nThe approach followed in the\napplication of the Bayesian framework is the ``exact integration'' scheme,\nwhere we sample from the analytical form of the distribution of the network\nweights. This can be done if we assume an analytic form for the prior\nhyperparameters distribution, in particular a distribution uniform on a\nlogarithmic scale: \n\\begin{displaymath}\np(\\ln \\alpha _i) = \\frac{1}{\\alpha _i}.\n\\end{displaymath}\nThis distribution is \\emph{non-informative}, i.e. it embodies our complete\nlack of knowledge about the values the hyperparameters should take.\n\nThe chosen model was trained using a sequence of little less than a million of\npatterns (we sampled the system at 4096Hz for 240s) normalized to zero mean\nand unity variance with a \\emph{whitening} process. Note that the\n input-output\npairs were processed through discrete integrators to obtain\n pattern-target\npairs, as shown in figure 3. The NN was then tested on a 120s long sequence.\n\n \nThe NN was trained for $370$ epochs, with the hyperparameters being\nupdated every $15$ epochs. A close look at the $\\alpha $ hyperparameters\nshows that all the inputs are relevant for the model (they are of the same\nmagnitude; note that this further confirms the pre-processing analysis). The\n$\\beta $ hyperparameter shows that the data contribution to the error is very\nsmall (as we would expect, since the data are synthetic).\n\nThe simulations were made using the MATLAB$^{\\copyright}$\nlanguage, the Netlab Toolbox~\\cite{Bishopetal96} and other software designed\nby us.\n \nIn figure~7, the PSDs of the target and the predicted time series are shown; in the lower \npart of the figure is reported the PSDs of the prediction residuals.\n\nIn figure~8, the PSD of a 100Hz sine wine added to the noise is shown, with the signal \nextracted by the network. As can be seen, the network recognizes the frequency of the\nsine wave with the maximum precision allowed by the residuals.\n\n\\section{Conclusion}\n\n\\bigskip\n\nIn this paper we have shown some preliminary tests on the use of NN's for\nsignal processing in the VIRGO Project. Some observations can be elicited\nfrom the experimental results:\n\n\\begin{itemize}\n\\item  In evaluating the Power Spectral Densities (PSDs), we made the\nhypothesis that the system is sampled at $4096Hz$. It is only a work\nhypothesis, but it shows how the network reproduces the system dynamics up\nto $2048Hz$. Note that the PSDs are nearly the same also if we were near\nthe Nyquist frequency.\n \n\n\\item  The PSD of the residuals shows a nearly-white spectrum, which is\nindex of the model goodness (see \\cite{Ljung95}).\n\n\\end{itemize}\n\nThe next steps in the research are: \n\n\\begin{description}\n\\item[-]  to increase the system model order and to test if there are\nsignificant differences in prediction;\n\n\\item[-]  to test the models with a greater number of samples to obtain a\nbetter estimate of the system dynamics;\n\n\\item[-]  to model the noise inside the system model to improve the system\nperformance and to allow a multi-step ahead prediction (i.e. an output-error\nmodel)\n\\end{description}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA substantial fraction of main sequence stars, up to 50\\%, are known\nto exist in systems with multiple components.\nThe importance of close binaries on stellar evolution has been\nrecognized lately, but their effects on the close circumstellar\nenvironment is under active study. \nThe influence of a companion on the region near to the surface of\nAsymptotic Giant Branch (AGB) stars may play a pivotal role in the\nformation of aspherical Planetary Nebulae (PNe).  \nObservations of molecules in the winds of a sample of AGB stars\n\\citep{Ramstedt2014,Ramstedt2017,Ramstedt2018,Doan2017,Decin2020} show\na variety of non spherical shapes which are \nattributed to the interactions with companion stars.\nSimilar shaping might affect the episodes of mass loss leading to the\nformation of PNe.\n\nSiO masers are usually found within the pulsating atmospheres of AGB\nstars at about 2 stellar radii, at about the distance at which\nsilicate and other dust forms in the outflows \n\\citep{Karovicova2013,Gobrecht2016,Hoefner2019}.\nThese masers generally appear in the form of a ring centered on the\nphotosphere and reveal motions in the inner envelope\n\\citep{Boboltz1997,Hollis1997b,Cotton2004,Cotton2006,Ragland2008}.\nSingle dish monitoring of semi-regular and Mira AGB stars by\n\\cite{Gomez-Garrido2020} show large and rapid variations of the SiO\nmasers in the semi-regular giants RX Boo and RT Vir and slower\nvariations in several Miras.\n\nOne of the current unknowns in stellar evolution is the shaping\nmechanism from asymptotic giant branch (AGB) stars to planetary\nnebulae. \nOn the one hand, both recent optical spectro--polarimetry\nand radio\/submm maser observations indicate unexpectedly high magnetic\nfields near or at the stellar surface \\citep{Lebre2014,Vlemmings2018}\nand references therein. \nOn the other hand, ALMA observations indicate the presence of\ncompanions around stars that were previously believed to be isolated\ne.g. the iconic ALMA observations of R Sculptoris;\n\\citep{Maercker2012}.  \nWhether binarity or magnetic fields dominate the evolutionary shaping\nmechanism is a matter of current debate. \n\nOne of the best places to study how the out-flowing material is shaped\nis at, or near, the stellar surface. \nLong baseline ALMA observations of the surfaces of AGB stars Mira and\nW Hya reveal hot-spots that likely influence and cause asymmetry in the\nnear--circumstellar environment \\citep{Vlemmings2015,Vlemmings2017}.\nTowards binary system Mira AB, we have also recently discovered signs\nof the influence of the white dwarf Mira B at only a few stellar radii\nfrom AGB star Mira A (Humphreys et al., in prep). \nIn this case, using ALMA long baseline data we found that SiO masers\ntrace a portion of a bubble wall formed by the interacting Mira AB\nwinds, also seen as a spiral plume in SiO thermal gas.\nIn this paper, we present observations of continuum and SiO maser\nlines using the JVLA of Mira and two other AGB stars with known\ncompanions, W Aql and R Aqr. \nOur aims are to study the effect of the companions on the\nobserved maser spot distribution. \n\n\\cite{Cotton2004,Cotton2006} \ngive a sequence of VLBA images of the SiO masers in the atmosphere of Mira.\nThese appear as partial rings with diameters in the range 60 to 75\nmas.\nThe diameter of the $\\nu$=1,J=1-0 maser ring tends to be slightly larger\nthan that of the $\\nu$=2,J=1-0 masers.\n\\cite{Mennesson2002} give a uniform disk diameter of the photosphere in\nK band of 28.79 $\\pm$ 0.10 mas.\n\\cite{Wittkowski2016} give a diameter of 28.5 $\\pm$ 1.5 mas at 2.25\n$\\mu$m.\n\\cite{Perrin2020} give the photospheric diameter in H band of 21.1 mas.\nThe distance to Mira is given by \\cite{Hannif1995} as 110 $\\pm$ 9\npc. at which distance 1'' corresponds to 110 AU.\nThe Hipparcos distance \\citep{vanLeeuwen2007} is 92$\\pm$11 pc.\n\nW Aquilae (W Aql)  is a binary S--type AGB star sometimes showing SiO\nmaser emission. \nSiO maser emission observed towards W Aql \\citep{Nakashima2007} \ndisappeared for a time around 2011 \\citep{Ramstedt2012}, which\ncould be due to disruption by an additional companion (the known\ncompanion lies at about 0.46 arcseconds from the S--type AGB star,\nabout 180 AU at a distance to W Aql of 395 pc). \nPeak SiO maser emission can reach 21 Jy (single--dish;\n\\cite{Ramstedt2012}).  \n\nR Aquarii (R Aqr) is a  symbiotic binary and a well--known host of\nmillimeter and submillimeter SiO masers \\citep{Boboltz1997,\nHollis1997b,Gray1998,Cotton2004,Cotton2006,Ragland2008}.\n\\cite{Cotton2004,Cotton2006} give a sequence of VLBA images\nof the SiO masers in the atmosphere of R Aqr with ring diameters of 31\nto 33 mas.\nInfrared measurements of the photospheric size show a range of values\nbetween 11.2 and 18.4 mas \\citep{Tuthill2000,Mennesson2002,Ragland2008,Wittkowski2016}.\n\n\nThe maser emission is variable but peak values in single--dish\nobservations lie in the range up to 300 Jy. \nR Aqr consists of a Mira variable and an accreting, hot companion\nwith a remarkable jet outflow. \nHST observations of the jet shows considerable change over a several\nyear timescale \\citep{Hollis1997a}. \nIn 2014 the stars were separated by 45 mas \\citep{Schmid2017} or 9.8 AU\nat the distance to R Aqr of 218 pc \\citep{Min2014}.\nALMA observations of \\cite{Bujarrabal2018} detect both stars with a\nbridge of material joining them, likely material flowing from the AGB\nprimary to the accretion disk around the WD secondary.\n\\cite{Cotton2004} show an unusual rotating ring of masers in January\n2001. \nThis ring was interpreted as arising in a rotating equatorial disk of\nmaterial. \n\n\n\\section{Observations}\nThe observations were made on the JVLA near 7 mm wavelength during 2019\nAugust 12 from 06:00:00 UT to 13:00:00 UT under project code 19A-220\nin the most extended (``A'') configuration. \nThe target stars (astrometric\/bandpass calibrators) were W Aquilae\n(J1939-1525), R Aquarii (J2348-1631) and Mira, AKA omicron Ceti,\n(J0217+0144). \nThe pulsation phase for W Aquilae was 0.27, for R Aquarii was 0.21 and\n0.79 for Mira.\nThe photometric calibrator was 3C48.\n\nThe observing sequence consisted of cycling between one of the targets (7 min)\nand its calibrator (2 min) for an hour and then proceeding to another\ntarget.\nReference pointing was used with a pointing measurement made on the\ncalibrator at the beginning of each sequence.\nThe integration time was 2 seconds and only the parallel hand data\n(RR,LL) were recorded.\n\nThe masers observed were the SiO $\\nu$=2,J=1-0 and\n$\\nu$=1,J=1-0  transitions at 42.820587 and 43.122027 GHz\nrespectively.\nEach transition was covered by 256 channels of 62.5 kHz\nbandwidth($\\sim$0.9 km\/s after Hanning). \nThe center LSR velocity of the observation was 47 km\/sec for Mira and\n-22 km\/sec for R Aqr and W Aql for which the true systemic velocity is\napproximately -26 and -18 km\/sec respectively.\nThe continuum observations consisted of 58 spectral windows in the\nrange of 41.0 to 48.7 GHz using 7424 $\\times$ 1 MHz channels; spectral\nregions containing the masers were not included.\n\n\\section{Calibration}\nThe conditions of these observations, Summertime, high frequency and\nhigh resolution render the proper calibration of the data more\ndifficult. \nIn order to allow accurate comparison of the continuum and line images\nthey must be aligned both photometrically and especially\nastrometrically.\nThe relatively poor phase coherence of the data make this more difficult.\nPhotometric calibration is complicated by the relatively poor\nsensitivity and strong resolution of the photometric calibrator by the\n$\\sim$45 mas resolution of the given data.\nFortunately, the masers in two of the targets are quite\nstrong and allow relative phase calibration following the general\napproach of \\cite{RM90,RM97,RM07}, \nadapted to the wide-band nature of the present data.\n\nData were translated from the archive format to AIPS format and were\nprocessed in the Obit package \\citep{OBIT}.\nHanning smoothing was used on the line data to\nsuppress the artifacts resulting from the interaction of the strong,\nnarrow maser signals and the finite delay range of the correlator.\n\n\\subsection{Amplitude \\label{ampcal}}\nThe traditional amplitude calibration scheme for connected element\ninterferometers is to compare the gain solutions of the astrometric\nand photometric calibrators to infer the flux density of the\nastrometric calibrator.\nThis works well in the high signal-to-noise regime with well modeled\nand only marginally resolved calibrators but can produce poor results\nwith low SNR data with strongly resolved photometric calibrators\nas is the case here.\nAll of the centimeter wavelength stable photometric calibration\nsources, including 3C48 used in this work, are strongly resolved at\nthe $\\sim$45 mas resolution of this data.\n\nWe have adopted an alternate approach of using an initial, assumed\nflux density for the astrometric calibrator and apply this calibration\nto the photometric calibrator.\nThe ratio of the apparent integrated flux density of the derived image\nof the photometric calibrator to its true flux density gives the\nfactor needed to correct the assumed flux density of the astrometric\ncalibrator. \nAstrometric calibrators are generally physically very small sources,\nhence variable, but are well modeled by a point at the current\nresolution. \nThis approach has the advantage that a super accurate structural model\nof the photometric calibrator is not needed but only an accurate total\nflux density.\nThe adopted flux density of 3C48 at 40.98 GHz was 0.611 Jy and at 48.6\nGHz was 0.506 Jy \\citep{Perley}.\nThe systematic error in the amplitude scale is estimated to be 20\\%.\n\n\\subsection{Phase}\nThe phase calibration approach of \\cite{RM90,RM97} is to\nuse the phase calibration derived from a self calibration of a strong,\nsimple maser to calibrate the phase of the continuum data.\nThis can work well in the present case of AGB stars which may have very\nstrong, non-thermal masers but very weak thermal emission from the\nphotosphere. \n\nThe original scheme of \\cite{RM90,RM97}  was appropriate for\na narrow band system for which a standard group delay calibration was\nadequate and did not need to be determined from the data-set in\nquestion.\nThe continuum bandwidth of the current data is sufficiently wide that\ncorrections to the group delay need to be determined  and applied to\nthe data. \n\nThe ``phase'' corrections derived from the self calibration of a maser\nare largely due to tropospheric refraction, hence are narrow band\nsamples of a group delay function and this needs to be taken into\naccount in transferring the maser calibration to the continuum data\nwhich extends for several GHz from the frequency of the maser.\nFurthermore, the signal from a given maser spot is very narrow band\nand inadequate to measure the relevant group delay.\nThe calibration method used is similar to that described by\n\\cite{Matthews2015,Matthews2018}.\n\nWe have adopted the following scheme.\n\\begin{enumerate}\n\\item {\\bf System temperature calibration}.\nAmplitude corrections were determined using program SYGain from the\non--line calibration measurements. \n\\item {\\bf Group delay calibration}.\nThe continuum observations of the phase reference sources were used to\ndetermine the group delay error (program Calib). \nThese calibrations were transferred to the maser line data and used to\ncalibrate both data sets.\n\\item {\\bf Bandpass calibration}.\nThe observations of the phase reference sources were used to derive a\nbandpass response function for each of the line and continuum data for\neach target star using program BPass.\n\\item {\\bf External calibration}.\nThe amplitude calibration scheme described in Section \\ref{ampcal} was\nused to derive the flux densities of the astrometric calibrators.\nThese were then used to obtain complex gain corrections (program\nCalib) which were applied to the data.\n\\item {\\bf Self calibrate a  maser}.\nA strong, simple maser appearing in a few line channels was imaged\nby SCMap for each target using self calibration.\nAfter several cycles of phase only calibration, an amplitude and phase\nself calibration was done.\n\\item {\\bf Transfer phase to continuum data}.\nThe time sequences of gain corrections from the maser self calibration\nwere transferred to the continuum data extrapolating phase in\nfrequency using frequency scaling. \nThis is done with program SnCpy.\n\\item {\\bf Doppler corrections}.\nProgram CVel was used to correct the spectroscopic data for the\nEarth's motion.  \n\\item {\\bf Image}.\nThe continuum data were imaged with no further calibration producing a\nwide-band image using program MFImage.\nChannels in which maser emission might appear were excluded from the\ncontinuum imaging. \nA spectral cube was imaged (via Imager) from the maser data for each\ntransition using the calibration from the self calibration.\n\\end{enumerate}\n\nUse of self calibration to set the coordinates of the derived\nimages results in a large uncertainty of the absolute positions.\nHowever, the resulting images should have accurate relative positions\nof both masers and continuum emission with respect to the reference\nmaser. \n\n\\section{Results}\nNo maser or continuum emission was detected from W Aql but strong\nmasers were detected in Mira and R Aqr and were used to calibrate the\ncontinuum. \nFrequency channels with maser emission are identified from long term, scalar\naveraged, visibility spectra from individual baselines.  \nThe mean channel (0.9 km sec$^{-1}$ ) visibility amplitude on a given\nbaseline for W Aql is 18 mJy and 5 times this should be easily\ndetectable, a plausible upper limit to a maser peak is 90 mJy. \nPimpanuwat et al. (in prep.) report that ALMA observations of the\nATOMIUM\n\\footnote{https:\/\/fys.kuleuven.be\/ster\/research-projects\/aerosol\/atomium\/atomium}\nproject detected thermal but not maser emissions of several 1.2 mm SiO\nJ=5-4 and J=6-5 transitions in W Aql during June and July 2019.\n\nMany maser features are well separated in velocity and are quite small;\ncentroids can be measured to a small fraction of the resolution.\nThe maser rings can therefore be defined with an accuracy much\nbetter than the size of the stellar photosphere.\nThe noise contribution to the position uncertainty of an unresolved,\nisolated component is $\\sim \\theta_{FWHM}\/(2\\times {\\rm SNR})$ where\n$\\theta_{FWHM}$ is the full width at half maximum of the resolution and\n${\\rm SNR}$ is the signal-to-noise ratio.\nSince the spatial and frequency resolution of this data is such that\nmasers overlap in space and velocity, a list of ``masers'' was derived\nby fitting one or two Gaussians to the image in alternate channels in\neach of the transitions.\n\n\\subsection{Mira}\nThe off--source channel noise in the maser cubes is 3.5 and 3.0 mJy\nbeam$^{-1}$ for $\\nu$=1,J=1-0 and $\\nu$=2,J=1-0 respectively and the\nCLEAN restoring beams were  51 $\\times$ 45 mas at position angle\n41$^\\circ$ and 50 $\\times$ 40 mas at position angle -27$^\\circ$.\nThe integrated spectra of the SiO masers in Mira are shown in Figure\n\\ref{FigMiraSpec}.\n\nThe locations of the masers are plotted on the gray scale of the\ncontinuum in Figure \\ref{FigMira}.\nThe velocities and relative strengths of the masers are shown in\nFigure \\ref{FigMira_Vel}.\nThe SNR of the masers seen in Figure \\ref{FigMiraSpec} exceeds 100 so\nthe uncertainty due to noise of the maser locations is less than 0.25\nmas, much less than the maser ring size of 60 - 70 mas\n\\citep{Cotton2004} or the photospheric size of 28.79 mas\n\\citep{Mennesson2002}. \nThe velocity ranges of the masers in the two transitions isn't the\nsame but this is also the case at several epochs shown in\n\\cite{Cotton2006} and is common in the single dish spectra of\n\\cite{Gomez-Garrido2020}. \nThe continuum of the secondary, Mira B, is clearly detected but no SiO\nwas detected except at the location of Mira A.\nNo continuum was detected away from the two stars.\n\nThe off-source RMS in the continuum image is 28 $\\mu$Jy beam$^{-1}$,\nand the CLEAN restoring beam is 41 $\\times$ 39 mas at position angle\n-79$^\\circ$. \nThe peak of Mira B is 434 $\\pm$ 3 mas (53 $\\pm$ 0.3 AU) from Mira A at\nan orientation of 95$^\\circ$ from north through east.\nThe peak flux density of Mira A is 2.7 $\\pm$ 0.6 mJy.\nThe total flux density of Mira A is 5.0 $\\pm$ 1.0 mJy \nwith a deconvolved Gaussian size (FWHM) of 39.4 $\\pm$\n1.5 mas $\\times$ 32.9 $\\pm$ 1.5 mas at position angle -56.1$^\\circ$\n$\\pm$ 4$^\\circ$.  \n\nMira A being only moderately resolved and its profile, hence\narea, being not well constrained by the present data, two possible\nextremes are a uniform disk and a Gaussian.\nWe directly use the  factor of 1.6 in \\citep{Pearson1999} to convert the\nGaussian size to an equivalent uniform disk size. \nWe thus get for this extreme possibility a uniform disk diameter\nof 63.0$\\times$52.6 mas.\nThe brightness temperature of Mira A calculating its area\nassuming this uniform disk and a total flux density of 5.0 mJy is 1300 $\\pm$\n270 K.\nThe brightness temperature calculating its area assuming a \nGaussian profile with the flux density of 5.0 mJy is 2320 $\\pm$ 480K.\nThe true brightness temperature is likely between 1300 and 2300 K.\nOur total flux density was obtained by an intergral over the\nimage pixels and its error is dominated by the 20\\% estimated\nuncertainty in the flux density scale. \nThe errors on the Gaussian size include the effects of correlated\nnoise in the image \\citep{Condon97}.\nThe true uncertainty in the brightness temperature is dominated by the\nuncertainty in the effective area of the source.\n\n\n\n\nThe peak flux density of Mira B is 0.62 $\\pm$ 0.14 mJy with an\nintegrated value of 0.64 $\\pm$ 0.14 mJy.\nThis component is at most marginally resolved with an upper limit\nof 20 mas.\n\\begin{figure}\n  \\includegraphics[width=2.7in,angle=-90]{Mira.Spec.ps}\n  \\caption{Spectrum of Mira SiO masers, ``+'' is $\\nu$=1,J=1-0, and\n    ``x'' is  $\\nu$=2,J=1-0.}\n  \\label{FigMiraSpec}%\n\\end{figure}\n\\begin{figure}\n  \\includegraphics[width=3in]{Mira_stars.ps}\n  \\caption{Continuum reverse gray-scale of Mira with ``+'' marking\n    the locations of $\\nu$=1,J=1-0 SiO masers and ``x'' for\n    $\\nu$=2,J=1-0 masers.\n    A scalebar for the grayscale in $\\mu$Jy Beam$^{-1}$ is given at the top.\n    The resolution is shown in the box in the lower left corner.}\n  \\label{FigMira}%\n\\end{figure}\n\\begin{figure}\n  \\includegraphics[width=3.5in]{Mira_color_vel_crop.eps}\n  \\caption{Color coded velocities of SiO masers surrounding Mira A\n    with ``+'' marking  the locations of $\\nu$=1,J=1-0 SiO masers and ``x'' for\n    $\\nu$=2,J=1-0 masers.\n    Velocities are indicated by the color with a scale bar to the right.\n    The size of each symbol is proportional to the $\\sqrt{\\quad}$ of the\n    peak flux density.} \n  \\label{FigMira_Vel}%\n\\end{figure}\n\n\\subsection{R Aquarii}\nThe off--source channel noise in the maser cubes is 4.4 and 3.9 mJy\nbeam$^{-1}$ for $\\nu$=1,J=1-0 and $\\nu$=2,J=1-0 respectively and the\nCLEAN restoring beams were  59 $\\times$ 41 mas at position angle\n-4$^\\circ$.\nThe integrated spectra of the SiO masers in R Aqr are shown in Figure\n\\ref{FigR_AqrSpec}.\n\nThe locations of the masers are plotted on the gray scale image of the \ncontinuum in Figure \\ref{FigR_Aqr}.\nThe velocities and relative strengths of the masers are shown in\nFigure \\ref{FigR_Aqr_Vel}.\nAs seen from Figure \\ref{FigR_AqrSpec}, the SNR of the masers exceed\n100 so the uncertainty of the maser locations due to image noise is\nless than 0.3 mas.\n\nThe off--source RMS in the continuum image is 36 $\\mu$Jy beam$^{-1}$\nand the CLEAN restoring beam is 50 $\\times$ 36 mas at position angle\n-1$^\\circ$. \nThe continuum component in which the masers are imbedded has a peak\nflux density of 6.1 $\\pm$ 1.2\nmJy and an integrated flux density of 10.8 $\\pm$ 2.2\nmJy and a deconvolved Gaussian size of 58.9 $\\pm$ 1.6 mas $\\times$\n16.9 $\\pm$ 1.8 mas at a position angle of 17$^\\circ$ $\\pm$ 1$^\\circ$.\nThe Gaussian brightness temperature of this component is 6,550 $\\pm$\n1490 K. \nFor the equivalent uniform disk, the brightness temperature is 3,690\n $\\pm$ 840 K.\nThe integrated flux density of the emission visible in Figure\n\\ref{FigR_Aqr} is 40 mJy.\nNo continuum or masers were detected outside of the region shown in\nFigure \\ref{FigR_Aqr}.\n\\begin{figure}\n  \\includegraphics[width=2.7in,angle=-90]{R_Aqr.Spec.ps}\n  \\caption{Spectrum of R Aquarii SiO masers, ``+'' is $\\nu$=1,J=1-0, and\n    ``x'' is  $\\nu$=2,J=1-0.}\n  \\label{FigR_AqrSpec}%\n\\end{figure}\n\\begin{figure}\n  \\includegraphics[width=3in]{R_Aqr_stars_label.ps}\n  \\caption{Continuum reverse grayscale of R Aqr with ``+'' marking\n    the locations of $\\nu$=1,J=1-0 SiO masers and ``x'' for\n    $\\nu$=2,J=1-0 masers.\n    A scalebar for the grayscale in $\\mu$Jy Beam$^{-1}$ is given at\n    the top.\n    Labels mark features approximately corresponding to those in\n    \\cite{Schmid2017}. \n    The circle shows the location of the secondary as estimated from the\n    orbit of \\cite{Bujarrabal2018}.\n    The resolution is shown in the box in the lower left corner}\n  \\label{FigR_Aqr}%\n\\end{figure}\n\\begin{figure}\n  \\includegraphics[width=3.5in]{R_Aqr_color_vel_crop.eps}\n  \\caption{Color coded velocities of SiO masers surrounding R Aqr with ``+'' marking\n    the locations of $\\nu$=1,J=1-0 SiO masers and ``x'' for\n    $\\nu$=2,J=1-0 masers.\n    Velocities are indicated by the color with a scale bar to the right.\n    The size of each symbol is proportional to the $\\sqrt{\\quad}$ of the\n    peak flux density.} \n  \\label{FigR_Aqr_Vel}%\n\\end{figure}\n\\section{Discussion}\n\\subsection{Mira}\nThe continuum emission from Mira A is slightly elongated 40$^\\circ$\nfrom the direction of Mira B. \nOtherwise there is no hint in the present results of an interaction.\nWhile not well defined, the maser ring is basically round.\nNumerous measurements of the 7 mm SiO masers are available in the\nliterature. \nThe maser rings seen at much higher resolution in \\cite{Cotton2004}\nand \\cite{Cotton2006} are even less well defined but also show little\nhint of an interaction.\nWhile there are variations in maser velocity around the ring seen in\nFigure \\ref{FigMira_Vel}, this is commonly seen in AGB maser rings and\nshows no obvious effect of the companion. \nSiO masers tracing the interaction of the AGB star with a close\ncompanion was reported in  $\\pi^1$ Gru \\citep{Homan2020}\nalthough with a closer companion (6 AU) than Mira B is to A. \n\n\\cite{RM07} give a uniform disk size for Mira A of 54 $\\pm$ 4 mas\n$\\times$ 50 $\\pm$ 4 mas at position angle -30$^\\circ$ $\\pm$ 50 with a\nflux density of 4.8 $\\pm$ 0.2 mJy and a brightness temperature of 1680 \n$\\pm$ 250 K in 2000 October.\nThis is somewhat smaller than our uniform disk equivalent of\n63.0$\\times$52.6 mas. \nThe \\cite{RM07} flux density of 4.8 mJy is in good agreement with our\nvalue of 5.0 mJy. \nOur larger size results in a lower estimated uniform disk brightness\ntemperature, 1300 $\\pm$ 270 K at 45 GHz; however,\nthis difference is within the errors.\nThe alignment of the SiO  maser ring with Mira A in \\cite{RM07}\ngenerally agrees with that in Figure \\ref{FigMira}; centered on the\nphotosphere. \nThey do not detect Mira B.\n\nObservations of Mira A in 2014 February by \\cite{Matthews2015} give an\nelliptical Gaussian fit at 46 GHz of 37.5 $\\pm$ 2.1 $\\times$ 31.7\n$\\pm$ 2.0 mas at position angle 147$^\\circ$ $\\pm$ 6$^\\circ$ in excellent\nagreement with our 2019 August results.\nHowever, the flux density quoted by \\cite{Matthews2015}, 7.6 $\\pm$ 1.5\nmJy, is 1.5 $\\sigma$ higher than our 5.0 mJy and their uniform\ndisk brightness temperature of 2110 $\\pm$ 440 K is correspondingly\nhigher than our 1300 $\\pm$ 270 K. \nThe difference between the current results and those of\n\\cite{Matthews2015} is 1.5 $\\sigma$.\n\\cite{Matthews2018} presents evidence for variations in the sizes and\nshapes of several other Mira variables so the differences between our\nresults and those of \\cite{RM07} and \\cite{Matthews2015} may simply\nreflect changes in Mira A.\n\nThe observations of \\cite{Matthews2015} detected Mira B and give a\ndeconvolved Gaussian size at 46 GHz of 18 $\\pm$ 5 $\\times$  1 $\\pm$ 6\nmas at position angle 172$^\\circ$ $\\pm$ 30$^\\circ$ with a flux density of\n0.97 $\\pm$ 0.2 mJy.\nThis is in good agreement with our upper limit of 20 mas\nalthough the star is at most marginally resolved in both cases.\nSince Mira B is thought to be a white dwarf, any resolution of this\nobject is very unlikely to be of the star itself but more likely an\naccretion disk around it.\n\nWhile we detect no emission away from Mira A and B, \\cite{Wong2016}\nshow ALMA observations of a plume of molecular gas in an arc from\nMira A up to 3'' in the direction away from Mira B but do not\nassociate it with an effect by Mira B. \nThey also report that the SiO density drops sharply past 4 stellar radii.\n\nALMA observations by \\cite{Kaminski2016} of gas phase AlO, the\nprecursor molecule to Al$_2$O$_3$ in aluminum dust, shows that the\ndensity peaks at about 2R$\\star$. \nThis is near the size of the radio photosphere and the SiO maser\nring seen in Figure \\ref{FigMira}.\n\n\\newpage\n\\subsection{R Aquarii}\nThe orbit of the secondary star in R Aqr has been the subject of\ndebate for a number of years.\n\\cite{Hollis1997b} suggest that an extended feature seen in a VLA 7 mm\nimage corresponded to the secondary.\nHowever, \\cite{Schmid2017} dispute this interpretation and on the\nbasis of high spatial resolution H$_\\alpha$ imaging suggest another.\nA spectroscopic orbit has been determined by\n\\cite{Gromadzki2009} who derive an eccentric orbit ($e$=0.25) with a\n43.6 year period but cannot determine where on the sky the secondary\nappears. \nDetection of the secondary star by \\cite{Bujarrabal2018} allowed them\nto update the orbit of \\cite{Gromadzki2009} giving an estimate of\nthe location at the epoch of the observations presented here.\nThe region of emission enclosing the stars is shown in Figure\n\\ref{FigR_Aqr_Close} with the estimated location of the white dwarf\nshown as a circle.\nAs for Mira, Figure \\ref{FigR_Aqr_Vel} shows no clear signature of an\neffect of the secondary on the velocity structure of the ring.\n\nHowever, ALMA observations by \\cite{Bujarrabal2018} show plumes of CO\ngas which they intreprete as arising from the outflow from the AGB star\nbeing focused into its orbital plane by the white dwarf companion\nmaximizing the accretion onto the secondary.\nThey also detect continuum emission between the stars with a\nspectral index suggesting dust.\nSuch structure would be obscured at 7 mm wavelength in 2019 by the\ninner jet emission seen in Figure \\ref{FigR_Aqr_Close}.\n\nThe continuum emission seen in the ALMA 0.8 mm observations are\nnot expected  to be the same as the 7 mm continuum emission as the\nformer is dominated by the photosphere and the dust forming around\nthe star and the latter is from free-free emission from ionized \nmaterial in the circumstellar envelop.\n\\begin{figure}\n  \\includegraphics[width=3.0in]{R_Aqr_Close.eps}\n  \\caption{Like Figure \\ref{FigR_Aqr} but a closeup of the emission\n    enclosing the stars and a scalebar labeled in mJy beam$^{-1}$\nThe circle shows the location of the secondary as estimated from the\norbit of \\cite{Bujarrabal2018}.}\n  \\label{FigR_Aqr_Close}%\n\\end{figure}\n\nThis system contains a very extended and complex jet to the\nnorth--east and south--west which is presumed to arise from the\nsecondary star \\citep{Hollis1997a,Hollis1997b,Schmid2017}.\nThe jet material is quite visible in the H$_\\alpha$ images presented\nin \\cite{Schmid2017} which also include the potential secondary.\nThe 7 mm image in Figure \\ref{FigR_Aqr} is strikingly similar to\nseveral of the H$_\\alpha$ features south-west of the AGB star shown in\n\\cite{Schmid2017}, especially their features C, A$_{\\rm SW}$, \nB$_{\\rm SW}$ and C$_{\\rm SW}$ and are approximately indicated in Figure\n\\ref{FigR_Aqr}. \nThese features in Figure \\ref{FigR_Aqr}, with the partial exception of\nC, are presumably related to the jet.\nThis correspondance is puzzling given the 5 years between the\nobservations and the very short recombination times of these features\nestimated by \\cite{Schmid2017}.\nThis suggests either longer recombination times or a continuing source\nof ionization.\n\nThe continuum feature associated with the SiO masers in Figures\n\\ref{FigR_Aqr} and \\ref{FigR_Aqr_Close} is elongated in the general\ndirection of the jet and is not centered on the maser ring but rather\non the companion white dwarf. \nThis suggests that this feature also encloses the secondary and some\nof its emission is from the jet.\nThe Gaussian brightness temperature of this feature, 6,550  $\\pm$ 970 K\nis greatly in excess of that expected for the photosphere of an AGB\nstar further suggesting that much of the emission is from the\naccretion disk\/jet which is nearly aligned on the sky with the AGB star.\n\n\nThe circumstellar masers seen in Figures \\ref{FigR_Aqr} and\n\\ref{FigR_Aqr_Close} show little evidence of being affected by the jet\nor the secondary star. \nThe VLBI image of the masers in R Aqr shown in \\cite{Ragland2008}\nshow masers streaming off in the direction of the north--east jet in\n2006 September but those in \\cite{Cotton2004,Cotton2006} do not.\n\n\n  The continuum emission see in Figures \\ref{FigR_Aqr} and\n\\ref{FigR_Aqr_Close} is similar in shape and orientation to that\nreported by \\cite{Hollis1997b} from 1996, a half orbital period earlier,\nbut with a different offset from the SiO masers marking the atmosphere\nof the AGB star.\nThis suggests that in both cases, the emission is dominated by the jet\nfrom the accretion disk around the white dwarf and that the image of\n\\cite{Hollis1997b} does show the location of the secondary.\n\nUnfortunately, the poor absolute astrometry resulting from the\natmospheric instability in the current data prevent them from being very\nuseful quantitatively in refining the orbit.\n\nThe distributions of masers in AGB atmospheres are extremely\nvariable as they track conditions suitable for maser amplification\nrather than following concentrations of material.  \nObservations several months apart, as shown in the references for VLBI\nobservations usually show very different distributions.  \nThe frequent monitoring of the Mira TX Cam \\citep{Gonidakis2013}\nshows very turbulent behavior. \n\nThe nature of the SiO maser distribution in January 2001 from\n\\cite{Cotton2004} was of a very different character than at much later\nepochs in showing a high degree of symmetry across the ring.  \nIn April 2001 the general velocity pattern of the masers was\nconsistent with a general rotation but without much evidence for an\nequatorial disk. \n\\cite{Hollis2001} claim some evidence for differential rotation in Dec\n2000 as well as earlier with a pole of the rotation  at a position\nangle of 150$^\\circ$.\nBy 2006 \\citep{Ragland2008} the velocity structure was incompatible\nwith the differential rotation of 2000-2001 and the 2019 velocity\nstructure in Figure \\ref{FigR_Aqr_Vel} is also incompatible.  \nThe 2000-2001 event may have been a transient feature although the\norientation of the apparent pole is difficult to explain by an\ninteraction with either the gravity of, or the jet coming from, the\nsecondary. \nThe jet is closer to the direction of the pole of the putative\nrotation rather than orthogonal to it and the orbit of\n\\cite{Bujarrabal2018} shows a substantial separation between the two\nstars in 2001. \n\n\n\\section{Conclusions}\nObservations of 7 mm continuum and SiO masers in three symbotic AGB\nstars are reported. \nThe continuum data were phase referenced to the masers allowing\nsensitive imaging and accurate registration.\nTechniques for photometric calibration and this phase referencing were\ndeveloped to exploit these data.\n\nNo masers were detected in W Aquilae which prevented a sensitive\ncontinuum image from being made.\nBoth Mira and R Aquarii have strong masers which allows deep\ncontinuum imaging.\nThe image of Mira shows the masers well centered on the photosphere of\nMira A and the white dwarf Mira B was well detected.\nNo clear evidence of an interaction between the white dwarf and the\nenvelope of Mira A was found.\n\nThe inner portion of the jet in R Aquarii is visible in the continuum\nimage and the radio emission around the AGB star is extended in the\ndirection of the jet.\nThe continuum emission is centered on the location of the white dwarf\nestimated from the orbit of \\cite{Bujarrabal2018} rather than on the\nAGB star as indicated by the SiO masers.\nThat, plus the high brightness temperature of this feature, indicates\nthat it is dominated by emission from the accretion disk + jets\nassociated with the white dwarf.\n\\cite{Bujarrabal2018} also shows gas apparently flowing from the AGB star\nonto the secondary.\nThe alignment of the continuum peak of R Aqr with the predicted\nlocation from the orbit of the secondary by \\cite{Bujarrabal2018}\nappears to support their orbit.\n\n\\begin{acknowledgements}\nWe thank B. Pimpanuwat\/ATOMIUM consortium for the J=5-4 and J=6-5\nmeasurements.\nThis work made use of the AAVSO International Database contributed by\nobservers worldwide. \n\n\\end{acknowledgements}\n\\vspace{5mm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nFluctuations, especially those of conserved charges, are promising \nobservables for the analysis of primordial thermodynamics \nin relativistic heavy-ion collisions \\cite{Stephanov:1999zu,\nAsakawa:2000wh,Jeon:2000wg}.\nIn particular, non-Gaussian fluctuations characterized by \nhigher order cumulants acquire much attention recently\n\\cite{Ejiri:2005wq,Stephanov:2008qz,Asakawa:2009aj}.\nActive studies have been carried out experimentally\n\\cite{STAR,ALICE,Adamczyk:2013dal,Adamczyk:2014fia,Adare:2015aqk,Luo:2015ewa}\nand by lattice QCD numerical simulations\n\\cite{Ding:2015ona,Borsanyi:2015axp}.\nSee for reviews \\cite{Koch:2008ia,Asakawa:2015ybt} and\nlatest progress \\cite{Luo:2015doi,Nahrgang:2016ayr}.\n\nExperimentally, the fluctuations are measured \nby the event-by-event analysis.\nIn this analysis,\nthe number of particles arriving at some range\nof detectors are counted in each event.\nThe event-by-event distribution characterized by the histogram\nof the particle number is called the event-by-event fluctuations.\nThe cumulants of the particle number are constructed from \nthe histogram.\n\nIn real experiments, however, particle numbers in each \nevent cannot be measured accurately, because the detectors\ncan measure particles with some probability called efficiency\nwhich is less than unity \\cite{Abelev:2008ab}.\nDue to the imperfect measurement, the event-by-event histogram \nand accordingly the cumulants constructed from the histogram \nare modified in a nontrivial way.\n\nThe effect of the efficiency on cumulants can be understood if it is\nassumed that the efficiency for individual particles are uncorrelated, \ni.e. the probability to observe different particles in an event \nis not correlated with one another.\nIn this case, the probability distribution \nof experimentally-observed particle numbers can be\nrelated to the original one without efficiency loss \nusing binomial distribution function \\cite{Kitazawa:2012at}.\nIn this paper, we call this relation the binomial model.\nThe binomial model enables us to relate the cumulants of \nthe original and observed particle number distributions.\nIt has been recognized that \nthe imperfect efficiency can significantly modify the values of \nthe cumulants especially for higher order ones \n\\cite{Kitazawa:2012at,Bzdak:2012ab,Luo:2014rea}.\nThe efficiency correction in experimental analyses thus is \nan important procedure to obtain the relevant values of the \ncumulants.\nThe binomial model is then extended to the case with \nmultiple values of local efficiencies for different particle\nspecies and phase spaces \\cite{Bzdak:2013pha,Luo:2014rea}.\n\nIn Refs.~\\cite{Bzdak:2013pha,Luo:2014rea}, the formulas for \nthe efficiency correction with multiple efficiencies \nare obtained using factorial moments.\nThese formulas consist of $M^m$ factorial moments \nfor $m$-th order cumulants with $M$ different efficiencies.\nIn practical analyses for efficiency correction, therefore, \nthe numerical cost for the efficiency correction increases rapidly \nas $M$ becomes larger.\nThe number of efficiencies $M$ thus is limited to small values\nin the present experimental analyses \\cite{private}.\n\nIn the present study, we derive a set of formulas \nwhich relate the cumulants of original and observed \ndistribution functions with multiple efficiencies.\nThey are derived by a straightforward extension \nof the method used in Ref.~\\cite{Kitazawa:2012at}.\nIn these formulas, the original cumulants before the efficiency\nloss are represented by the mixed cumulants of observed \nparticles.\nThe number of the cumulants in these relations does not depend on $M$.\nThey thus would enable one to carry out the efficiency correction \nmore effectively for large $M$ and higher order cumulants.\nEven the realistic $p_T$-dependent efficiency \\cite{Abelev:2008ab} \nwould be treated with these formulas.\n\nThis paper is organized as follows.\nIn Sec.~\\ref{sec:main}, we first present our main results,\ni.e. the formulas to relate the original and observed cumulants.\nTheir derivation are then discussed in later sections.\nSection~\\ref{sec:basic} is devoted to reviews on \nthe definition and properties of cumulants.\nWe then present the derivation for single-variable \ndistribution functions in Sec.~\\ref{sec:single}, as a simple\nillustration of the full derivation addressed in Sec.~\\ref{sec:mult}.\nSection~\\ref{sec:summary} is devoted to discussions\nand a short summary.\n\n\\section{Main result}\n\\label{sec:main}\n\nIn this section, we clarify the problem considered in this paper \nand show the answer, which is the main result of this paper.\n\n\\subsection{Problem}\n\\label{sec:main1}\n\nWe consider a probability distribution function\n\\begin{align}\nP(N_1,N_2,\\cdots,N_M) = P(\\vec{N}),\n\\label{eq:P}\n\\end{align}\nfor $M$ integer stochastic variables, $N_1,~N_2,\\cdots,N_M$,\nwith $\\vec{N}=(N_1,N_2,\\cdots,N_M)$ being the vectorical \nrepresentation of the stochastic variables.\nIn the following we call $N_i$ the number\nof $i$-th particle in an event.\nThe purpose of the present study is to obtain the cumulants of\na linear combination of $N_i$,\n\\begin{align}\nQ = \\sum_{i=1}^M a_i N_i ,\n\\label{eq:Q}\n\\end{align}\nwith $a_i$ being numerical numbers.\n\nIf $P(\\vec{N})$ is obtained directly in an experiment \nas an event-by-event histogram, the cumulants of $Q$ can, \nof course, be constructed straightforwardly from the histogram.\nIn real experiments, however, the detectors \nmiss the measurement of particles with some efficiency.\nThe number of the $i$-th particles observed by the detector, \n$n_i$, is thus different from and smaller than $N_i$.\nThe probability distribution function \nof observed particle numbers obtained by the imperfect experiment\n\\begin{align}\n\\tilde{P}(n_1,n_2,\\cdots,n_M) = \\tilde{P}(\\vec{n}),\n\\label{eq:tildeP}\n\\end{align}\ndiffers from Eq.~(\\ref{eq:P}).\n\nIf the efficiencies for the observation of individual particles \nare assumed to be independent with one another,\none can relate $\\tilde{P}(\\vec{n})$ and $P(\\vec{N})$ in a simple form.\nIn this case, the probability distribution of $n_i$ for a fixed $N_i$ \nobeys the binomial distribution function \n\\begin{align}\nB_{p,N}(n) = \\frac{N!}{n!(N-n)!} p^n (1-p)^n ,\n\\label{eq:binomial}\n\\end{align}\nwith $N=N_i$ and $p=\\epsilon_i$ being the efficiency of \nthe $i$-th particle.\nTherefore, $P(\\vec{N})$ and $\\tilde{P}(\\vec{n})$ are related \nwith each other as \\cite{Kitazawa:2012at,Bzdak:2013pha,Luo:2014rea}\n\\begin{align}\n\\tilde{P}(\\vec{n}) = \\sum_{N_1,\\cdots,N_M}\nP(\\vec{N}) \\bigg( \\prod_i B_{\\epsilon_i,N_i}(n_i)\\bigg).\n\\label{eq:tildeP=PB}\n\\end{align}\nIn this paper we refer to Eq.~(\\ref{eq:tildeP=PB}) \nas the binomial model.\nThe purpose of the present study is to represent the cumulants of \n$Q$ using those of $\\tilde{P}(\\vec{n})$ in the binomial model \nin a compact form.\n\n\\subsection{Answer}\n\\label{sec:main2}\n\nThe cumulants of $Q$ up to fourth order are given by\n\\begin{align}\n\\langle Q \\rangle_{\\rm c} \n=& \\llangle q_{(1)} \\rrangle_{\\rm c} ,\n\\label{eq:Q1}\n\\\\\n\\langle Q^2 \\rangle_{\\rm c} \n=& \\llangle q_{(1)}^2 \\rrangle_{\\rm c} - \\llangle q_{(2)} \\rrangle_{\\rm c} ,\n\\label{eq:Q2}\n\\\\\n\\langle Q^3 \\rangle_{\\rm c} \n=& \\llangle q_{(1)}^3 \\rrangle_{\\rm c} - 3 \\llangle q_{(2)} q_{(1)} \\rrangle_{\\rm c} \n+ \\llangle 3 q_{(2,1|2)} - q_{(3)} \\rrangle_{\\rm c} ,\n\\label{eq:Q3}\n\\\\\n\\langle Q^4 \\rangle_{\\rm c} \n=& \\llangle q_{(1)}^4 \\rrangle_{\\rm c} - 6 \\llangle q_{(2)} q_{(1)}^2 \\rrangle_{\\rm c} \n+ 12 \\llangle q_{(2,1|2)} q_{(1)} \\rrangle_{\\rm c} \n\\nonumber \\\\\n&\n+ 6 \\llangle q_{(1,1|2)} q_{(2)} \\rrangle_{\\rm c} \n-4 \\llangle q_{(3)} q_{(1)} \\rrangle_{\\rm c} - 3 \\llangle q_{(2)}^2 \\rrangle_{\\rm c} \n\\nonumber \\\\\n&\n+ \\llangle -18 q_{(2,1,1|2,2)} + 6 q_{(2,1,1|3)} + 4 q_{(3,1|2)} \n\\nonumber \\\\\n&\n+ 3 q_{(2,2|2)} - q_{(4)} \\rrangle_{\\rm c} ,\n\\label{eq:Q4}\n\\end{align}\nwhere the cumulants $\\langle\\cdot\\rangle_{\\rm c}$ and \n$\\llangle\\cdot\\rrangle_{\\rm c}$ are taken \nfor $P(\\vec{N})$ and $\\tilde{P}(\\vec{n})$, respectively.\n$q_{(\\cdots)}$ are linear combinations of $n_i$\ndefined by\n\\begin{align}\nq_{(s)} \n=& \\sum_{i=1}^M c_{(s)}^{(i)} n_i,\n\\label{eq:q(s)}\n\\\\\nq_{(s_1,\\cdots,s_j|t_1,\\cdots,t_k)} \n=& \\sum_{i=1}^M c_{(s_1,\\cdots,s_j|t_1,\\cdots,t_k)}^{(i)} n_i.\n\\label{eq:q(sstt)}\n\\end{align}\nThe coefficients $c_{(\\cdots)}^{(i)}$ are numerical numbers \nwhich depend on $a_i$ and $\\epsilon_i$ defined by\n\\begin{align}\nc_{(s)}^{(i)} =& \\tilde{a}_i^s \\tilde\\xi_s^{(i)} ,\n\\\\\nc_{(s_1,\\cdots,s_j|t_1,\\cdots,t_k)}^{(i)}\n=& \\tilde{a}_i^{s_1+\\cdots+s_j} \\tilde\\xi_{s_1}^{(i)} \\cdots \\tilde\\xi_{s_j}^{(i)} \n\\tilde\\xi_{t_1}^{(i)} \\cdots \\tilde\\xi_{t_k}^{(i)} ,\n\\end{align}\nwith \n$\\tilde{a}_i = a_i\/\\epsilon_i$,\n$\\tilde{\\xi}_m^{(i)}=\\xi_m^{(i)}\/\\epsilon_i$, and \n$\\xi_m^{(i)}=\\xi_m(\\epsilon_i)$ being coefficients of \nthe binomial cumulants with probability $\\epsilon_i$;\nexplicit forms of $\\xi_m$ up to sixth order are given \nin Eqs.~(\\ref{eq:xi1}) -- (\\ref{eq:xi6}).\n\nWhen one applies Eqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}) to the \nefficiency correction in event-by-event analyses, \nthe required procedure is as follows:\n\\begin{enumerate}\n\\item\nCalculate $q_{(\\cdots)}$ in each event from $n_i$ \nobserved experimentally in the event.\n\\item\nUsing the event-by-event distribution of $q_{(\\cdots)}$,\nobtain the (mixed) cumulants of $q_{(\\cdots)}$ \nwhich appear in Eqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}).\n\\end{enumerate}\nRemark that the numerical cost for the latter procedure \ndoes not depend on $M$.\n\nIn Ref.~\\cite{Kitazawa:2012at}, \nthe relation of the cumulants for the ``net-particle number'' \nwith common efficiencies for particles and anti-particles, \nrespectively, are derived.\nThis result is reproduced by substituting $M=2$, $a_1=1$ and $a_2=-1$ \nin Eqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}).\n\nIn the rest of this paper,\nwe derive Eqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}).\n\n\\section{Cumulants}\n\\label{sec:basic}\n\nIn this section we first summarize properties of cumulants \nrequired for the derivation of Eqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}).\n\n\n\\subsection{Definition}\n\\label{sec:cumulants}\n\nThe cumulants are defined from the cumulant generating function.\nFor the probability distribution function Eq.~(\\ref{eq:P}),\nthe generating function is defined by\n\\begin{align}\nK(\\theta_1,\\cdots,\\theta_M)\n= \\ln \\bigg[ \\sum_{N_1,\\cdots,N_M}\nP(\\vec{N}) \\exp( \\sum_{i=1}^M N_i \\theta_i ) \\bigg].\n\\label{eq:K}\n\\end{align}\nThe cumulants are then defined by the derivatives of \nEq.~(\\ref{eq:K}) as \n\\begin{align}\n\\langle N_m^s \\rangle_{\\rm c}\n= \\frac{\\partial^s}{\\partial \\theta_m^s} K(\\vec{\\theta})|_{\\vec{\\theta}=0}\n\\equiv \\partial_m^s K(\\vec{0}),\n\\label{eq:cumulant}\n\\end{align}\nand \n\\begin{align}\n\\langle N_{m_1}^{s_1} \\cdots N_{m_j}^{s_j}  \\rangle_{\\rm c}\n= \\partial_{m_1}^{s_1} \\cdots \\partial_{m_j}^{s_j} K(\\vec{0}).\n\\end{align}\nFrom these definitions, one immediately obtains \nthe ``distributive property'' of the cumulants such as \n\\begin{align}\n\\lefteqn{ \\langle (a_1 N_1 + a_2 N_2 )^2 \\rangle_{\\rm c} }\n\\nonumber \\\\\n&= a_1^2 \\langle N_1^2 \\rangle_{\\rm c}\n+ 2 a_1 a_2 \\langle N_1 N_2 \\rangle_{\\rm c}\n+ a_2^2 \\langle N_2^2 \\rangle_{\\rm c} .\n\\end{align}\nOther properties of cumulants, such as their relation\nwith moments, are found in Ref.~\\cite{Asakawa:2015ybt}.\n\n\\subsection{Cumulant expansion}\n\\label{sec:exp}\n\nConsider a (non-homogeneous) linear function of $\\vec{N}$, \n\\begin{align}\nL(\\vec{N}) = d_0 + \\sum_i d_{i=1}^M N_i,\n\\end{align}\nwith numerical numbers $d_i$.\nThe cumulants of $L(\\vec{N})$ are given by\n\\begin{align}\n\\langle (L(\\vec{N}))^m \\rangle_{\\rm c}\n= \\frac{\\partial^m}{\\partial \\bar\\theta^m} K_L(\\bar\\theta) |_{\\bar\\theta=0},\n\\label{eq:<L^m>}\n\\end{align}\nwith \n\\begin{align}\nK_L(\\bar\\theta) \n= \\ln \\sum_{N_1,\\cdots,N_{M}} P(\\vec{N}) e^{\\bar\\theta L(\\vec{N})}\n= \\ln \\langle e^{\\bar\\theta L(\\vec{N})} \\rangle .\n\\label{eq:K_Lln}\n\\end{align}\nFrom Eq.~(\\ref{eq:<L^m>}), the generating function \n$K_L(\\bar\\theta)$ is expanded as \n\\begin{align}\nK_L(\\bar\\theta) = \\sum_{m=1}^\\infty \\frac{\\bar\\theta^m}{m!} \n\\langle (L(\\vec{N}))^m \\rangle_{\\rm c}.\n\\label{eq:K_Lexp}\n\\end{align}\nNote that the sum starts from $m=1$ because $K_L(0)=0$ which is \nensured from the fundamental property of probability \n$\\sum_{\\vec{N}} P(\\vec{N})=1$.\n\nBy substituting $\\bar\\theta=1$ \nin Eqs.~(\\ref{eq:K_Lln}) and (\\ref{eq:K_Lexp}) we obtain,\n\\begin{align}\n\\ln \\langle e^{L(\\vec{N})} \\rangle\n= \\sum_{m=1}^\\infty \\frac1{m!} \n\\langle (L(\\vec{N}))^m \\rangle_{\\rm c}.\n\\label{eq:cumulantexp}\n\\end{align}\nEquation~(\\ref{eq:cumulantexp}) is referred to as \nthe cumulant expansion \\cite{Asakawa:2015ybt}, \nand plays a crucial role in the following derivations.\n\n\\subsection{Binomial distribution function}\n\nThe cumulant generating function of the binomial distribution \nfunction $B_{p,N}(n)$ is given by\n\\begin{align}\nk_N(\\theta) = \\ln \\sum_n e^{\\theta n} B_{p,N}(n).\n\\label{eq:k_N}\n\\end{align}\nBy taking derivatives, the $m$-th order cumulant is given by \n$\\langle n^m \\rangle_{\\rm c,binomial} = \\xi_m(p) N$ with \n\\begin{align}\n\\xi_m(p) = \\frac1N \\frac{\\partial^m}{\\partial \\theta^m} k_N(0).\n\\label{eq:xi(p)}\n\\end{align}\nExplicit forms of $\\xi_m(p)$ up to sixth order are given by\n\\begin{align}\n\\xi_1(p) =& p,\n\\label{eq:xi1}\n\\\\\n\\xi_2(p) =& p(1-p),\n\\\\\n\\xi_3(p) =& p(1-p)(1-2p),\n\\\\\n\\xi_4(p) =& p(1-p)(1-6p+6p^2),\n\\\\\n\\xi_5(p) =& p(1-p)(1-2p)(1-12p+12p^2),\n\\\\\n\\xi_6(p) =& p(1-p)(1-30p+150p^2-240p^3+120p^4).\n\\label{eq:xi6}\n\\end{align}\nUsing $\\xi_m(p)$, Eq.~(\\ref{eq:k_N}) is written as \n\\begin{align}\nk_N(\\theta) = \\sum_m \\frac{\\theta^m}{m!} \\xi_m(p) N, \n\\label{eq:ktheta}\n\\end{align}\nwhich shows that $k_N(\\theta)$ is proportional to $N$.\nIn this study, we fully make use of this property of $k_N(\\theta)$.\n\n\n\\section{Single variable case}\n\\label{sec:single}\n\nBefore the full derivation of Eqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}),\nin this section we first deal with a simplified problem \nwith $M=1$, as this analysis would become a good \nexercise for the full derivation addressed in the next section.\n\nIn this section, we consider probability distribution functions\n$P(N)$ and $\\tilde{P}(n)$ for single stochastic variables\n$N$ and $n$, respectively, which are related with each other as\n\\begin{align}\n\\tilde{P}(n) = \\sum_N P(N) B_{p,N}(n).\n\\label{eq:Psingle}\n\\end{align}\nThe cumulant generating function of $\\tilde{P}(n)$\nis calculated to be\n\\begin{align}\n\\tilde{K}(\\theta) \n&= \\ln \\sum_n e^{\\theta n} \\tilde{P}(n)\n\\nonumber \\\\\n&= \\ln \\sum_n e^{\\theta n} \\sum_N P(N) B_{p,N}(n)\n\\nonumber \\\\\n&= \\ln \\sum_N P(N) \\sum_n e^{\\theta n} B_{p,N}(n)\n\\nonumber \\\\\n&= \\ln \\sum_N P(N) e^{k_N(\\theta)}\n\\nonumber \\\\\n&= \\ln \\langle e^{k_N(\\theta)} \\rangle,\n\\label{eq:tildeKsingle}\n\\end{align}\nwhere in the fourth equality we used Eq.~(\\ref{eq:k_N}).\nThe expectation value in the last line is taken for $P(N)$.\n\nUsing the fact that $k_N(\\theta)$ is linear in $N$,\nEq.~(\\ref{eq:tildeKsingle}) is expressed by the cumulant \nexpansion Eq.~(\\ref{eq:cumulantexp}) as \n\\begin{align}\n\\tilde{K}(\\theta) \n= \\sum_m \\frac1{m!} \\langle (k_N(\\theta))^m \\rangle_{\\rm c},\n\\label{eq:Kk}\n\\end{align}\nwhile $\\tilde{K}(\\theta)$ also defines \nthe cumulants of $\\tilde{P}(n)$ by their derivatives \nwith $\\theta=0$,\n\\begin{align}\n\\llangle n^m \\rrangle_{\\rm c} = \\partial^n \\tilde{K} ,\n\\label{eq:<n^m>c}\n\\end{align}\nwhere $\\partial=\\partial\/\\partial\\theta$ and \nwe introduced the following notations:\n\\begin{enumerate}\n\\item\nThe cumulants of single and double brackets are taken \nfor $P(\\vec{N})$ and $\\tilde{P}(\\vec{n})$, respectively.\n\\item\nWhen the argument of generating functions $\\tilde{K}(\\theta)$ and\n$k_N(\\theta)$ is suppressed, it is understood that $\\theta=0$ is\nsubstituted.\n\\end{enumerate}\nThese notations are used throughout this paper.\n\nFrom Eqs.~(\\ref{eq:Kk}) and (\\ref{eq:<n^m>c}), \nthe $j$-th order cumulant $\\llangle n^j \\rrangle_{\\rm c}$\nis represented by the cumulant expansion \n\\begin{align}\n\\llangle n^j \\rrangle_{\\rm c} \n= \\partial^j \\sum_m \\frac1{m!} \\langle k_N^m \\rangle_{\\rm c}.\n\\label{eq:<n^m>c=<>}\n\\end{align}\nWe now represent the right-hand side of Eq.~(\\ref{eq:<n^m>c=<>}) \nby the cumulants of $N$.\nTo proceed this analysis, there are two convenient rules.\n\\begin{description}\n\\item[Rule 1]\n$\\theta$ derivatives on $\\langle k_N^m \\rangle_{\\rm c}$ act on \n$k_N$'s as if $\\langle \\cdot \\rangle_{\\rm c}$ were a standard bracket;\nfor example,\n\\begin{align}\n\\partial \\langle k^m \\rangle_{\\rm c}\n=& m \\langle k^{m-1} (\\partial k) \\rangle_{\\rm c} .\n\\\\\n\\partial^2 \\langle k^m \\rangle_{\\rm c}\n=& m(m-1) \\langle k^{m-2} (\\partial k)^2 \\rangle_{\\rm c}\n+ m \\langle k^{m-1} (\\partial^2 k) \\rangle_{\\rm c}.\n\\end{align}\nThis is because $k_N(\\theta)$ is proportional to $N$ and only the \ncoefficient in front of $N$ depends on $\\theta$.\n\\item[Rule 2]\nBy substituting $\\theta=0$, \nall $k_N(\\theta)$'s which do not receive $\\theta$ derivative \nvanish. Therefore, all $k(\\theta)$ must receive \nat least one differentiation so that the term gives nonzero \ncontribution to $\\langle n^j \\rangle_{\\rm c}$ in Eq.~(\\ref{eq:<n^m>c=<>}).\nThis immediately means that the $m$-th order term in \nEq.~(\\ref{eq:<n^m>c=<>}) can affect $\\langle n^j \\rangle_{\\rm c}$ \nonly if $m\\le j$.\n\\end{description}\n\nTo see the manipulation for the right-hand side of\nEq.~(\\ref{eq:<n^m>c=<>}) with these rules, let us consider the $j=3$ case.\nIn this case, the terms with $m=1,~2$, and $3$ have nonvanishing\ncontribution because of Rule 2.\nThese terms are calculated to be\n\\begin{align}\n\\partial^3 \\langle k_N \\rangle_{\\rm c}\n=& \\langle \\partial^3 k_N \\rangle_{\\rm c}\n= \\xi_3 \\langle N \\rangle_{\\rm c},\n\\\\\n\\frac12 \\partial^3 \\langle k_N^2 \\rangle_{\\rm c}\n=& 3 \\langle (\\partial^2 k_N)(\\partial k_N)  \\rangle_{\\rm c}\n= 3 \\xi_2 \\xi_1 \\langle N^2  \\rangle_{\\rm c},\n\\\\\n\\frac1{3!} \\langle \\partial^3 k_N^3 \\rangle_{\\rm c}\n=& \\langle (\\partial k_N )^3 \\rangle_{\\rm c}\n= \\xi_1^3 \\langle N^3 \\rangle_{\\rm c},\n\\end{align}\nwhere all terms with $k_N$ without derivatives are neglected\nfrom Rule 2.\nSubstituting them in Eq.~(\\ref{eq:<n^m>c=<>}) we obtain\n\\begin{align}\n\\llangle n^3 \\rrangle_{\\rm c}\n=& \\partial^3 \\tilde{K}\n\\nonumber \\\\\n=& \\partial^3 \\langle k_N \\rangle_{\\rm c}\n+ \\frac12 \\partial^3 \\langle k_N^2 \\rangle_{\\rm c}\n+ \\frac1{3!} \\partial^3 \\langle k_N^3 \\rangle_{\\rm c}\n\\nonumber \\\\\n=& \\xi_3 \\langle N \\rangle_{\\rm c}\n+ 3 \\xi_2 \\xi_1 \\langle N^2 \\rangle_{\\rm c}\n+ \\xi_1^3 \\langle N^3 \\rangle_{\\rm c}.\n\\end{align}\nSimilar manipulations up to $6$-th order lead to \n\\begin{widetext}\n\\begin{align}\n\\llangle n \\rrangle_{\\rm c} \n=& \\xi_1 \\langle N \\rangle_{\\rm c} ,\n\\label{eq:<n1>c}\n\\\\\n\\llangle n^2 \\rrangle_{\\rm c} \n=& \\xi_2 \\langle N \\rangle_{\\rm c} + \\xi_1^2 \\langle N^2 \\rangle_{\\rm c} ,\n\\\\\n\\llangle n^3 \\rrangle_{\\rm c} \n=& \\xi_3 \\langle N \\rangle_{\\rm c} + 3 \\xi_2 \\xi_1 \\langle N^2 \\rangle_{\\rm c} \n+ \\xi_1^3 \\langle N^3 \\rangle_{\\rm c} ,\n\\\\\n\\llangle n^4 \\rrangle_{\\rm c} \n=& \\xi_4 \\langle N \\rangle_{\\rm c} \n+ (4 \\xi_3 \\xi_1 + 3 \\xi_2^2 ) \\langle N^2 \\rangle_{\\rm c} \n+ 6 \\xi_2 \\xi_1^2 \\langle N^3 \\rangle_{\\rm c} \n+ \\xi_1^4 \\langle N^4 \\rangle_{\\rm c} ,\n\\\\\n\\llangle n^5 \\rrangle_{\\rm c} \n=& \\xi_5 \\langle N \\rangle_{\\rm c} \n+ (5 \\xi_4 \\xi_1 + 10 \\xi_3 \\xi_2 ) \\langle N^2 \\rangle_{\\rm c} \n+ (10 \\xi_3 \\xi_1^2 + 15 \\xi_2^2 \\xi_1 ) \\langle N^3 \\rangle_{\\rm c} \n+ 10 \\xi_2 \\xi_1^3 \\langle N^4 \\rangle_{\\rm c} \n+ \\xi_1^5 \\langle N^5 \\rangle_{\\rm c} ,\n\\\\\n\\llangle n^6 \\rrangle_{\\rm c} \n=& \\xi_6 \\langle N \\rangle_{\\rm c} \n+ (6 \\xi_5 \\xi_1 + 15 \\xi_4 \\xi_2 + 10 \\xi_3^2 ) \\langle N^2 \\rangle_{\\rm c} \n+ (15 \\xi_4 \\xi_1^2 + 60 \\xi_3 \\xi_2 \\xi_1 + 15 \\xi_2^3 ) \\langle N^3 \\rangle_{\\rm c} ,\n\\nonumber \\\\\n&\n+ (20 \\xi_3 \\xi_1^3 + 45 \\xi_2^2 \\xi_1^2 ) \\langle N^4 \\rangle_{\\rm c} \n+ 15 \\xi_2 \\xi_1^4 \\langle N^5 \\rangle_{\\rm c} \n+ \\xi_1^6 \\langle N^6 \\rangle_{\\rm c} .\n\\label{eq:<n6>c}\n\\end{align}\n\\end{widetext}\nThese are the formulas which represent the cumulants of \nobserved particles numbers, $\\llangle n^m \\rrangle_{\\rm c}$, \nusing the original ones $\\langle N^m \\rangle_{\\rm c}$.\n\nFor the efficiency correction,\nwe have to represent $\\langle N^m \\rangle_{\\rm c}$\nusing $\\llangle n^m \\rrangle_{\\rm c}$.\nThese relations are most straightforwardly obtained by \nrepresenting Eqs.~(\\ref{eq:<n1>c}) -- (\\ref{eq:<n6>c})\nin a matrix form,\n\\begin{align}\n\\vec{V}_n =& \\mathbb{M} \\vec{V}_N,\n\\label{eq:V_n=MV}\n\\end{align}\nwith \n\\begin{align}\n\\vec{V}_n = ( \\llangle n \\rrangle_{\\rm c} , \\cdots , \\llangle n^m \\rrangle_{\\rm c} )^T,\n\\quad\n\\vec{V}_N = ( \\langle N \\rangle_{\\rm c} , \\cdots ,\n\\langle N^m \\rangle_{\\rm c} )^T,\n\\nonumber \n\\end{align}\nand taking the inverse.\nWe note that the matrix $\\mathbb{M}$ in Eq.~(\\ref{eq:V_n=MV}) \nis lower triangular.\nAccordingly, the inverse of $\\mathbb{M}$ is also lower triangular.\nThe $m$-th order cumulant $\\langle N^m \\rangle_{\\rm c}$ thus is \nrepresented \nby $\\llangle n^l \\rrangle_{\\rm c}$ with $l\\le m$.\nThe results correspond to a special case of\nEqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}) with $M=1$ and $a_1=1$.\nThe explicit forms up to fourth order are found \nin Ref.~\\cite{Asakawa:2015ybt}.\n\n\n\\section{Multi-variable case}\n\\label{sec:mult}\n\nNext, we derive Eqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}).\nWe start from the cumulant generating function of\n$\\tilde{P}(\\vec{n})$ in Eq.~(\\ref{eq:tildeP}),\n\\begin{align}\n\\tilde{K}(\\vec{\\theta}) \n=& \\sum_{n_1,\\cdots,n_M} \\tilde{P}(\\vec{n}) \n\\exp ( \\sum_i \\theta_i n_i ) \n\\nonumber \\\\\n=& \\ln \\sum_{N_1,\\cdots,N_M} P(\\vec{N}) \n  \\sum_{n_1,\\cdots,n_M} \\prod_i \\big( e^{\\theta_i n_i} B_{\\epsilon_i,N_i}(n_i) \\big)\n\\nonumber \\\\\n=& \\ln \\sum_{N_1,\\cdots,N_M} P(\\vec{N}) \n  \\prod_i \\big( \\sum_{n_i} e^{\\theta_i n_i} B_{\\epsilon_i,N_i}(n_i)\\big) \n\\nonumber \\\\\n=& \\ln \\sum_{N_1,\\cdots,N_M} P(\\vec{N}) e^{\\kappa(\\vec{\\theta})} \n\\nonumber \\\\\n=& \\ln \\langle e^{\\kappa(\\vec{\\theta})} \\rangle_{\\rm c},\n\\end{align}\nwhere\n\\begin{align}\n\\kappa(\\vec{\\theta}) = \\sum_{i=1}^M k_{N_i} (\\theta_i),\n\\label{eq:kappa}\n\\end{align}\nwith $k_{N_i}(\\theta_i)$ being the cumulant generating function \nof $B_{\\epsilon_i,N_i}(n_i)$ defined in Eq.~(\\ref{eq:k_N}).\nIn the following, we use the notations 1 and 2 introduced in \nthe previous section.\n\nFrom the definition, it is clear \nthat $\\kappa$ is a linear function of $N_i$.\nDerivatives of $\\kappa(\\vec{\\theta})$ for $\\theta_i=0$ are given by\n\\begin{align}\n\\partial_i^m \\kappa = \\xi_m^{(i)} N_i ,\n\\label{eq:dkappa=xiN}\n\\end{align}\nwith $\\xi_m^{(i)} = \\xi_m(\\epsilon_i)$, while \nderivatives of $\\kappa(\\vec{\\theta})$ with \ndifferent $\\theta_i$'s vanish, i.e.\n\\begin{align}\n\\partial_j^m \\partial_k^l \\kappa = 0 ,\n\\label{eq:ddk}\n\\end{align}\nfor all $j\\ne k$ and nonzero $m$ and $l$, and so forth,\nwhich is trivial from Eq.~(\\ref{eq:kappa}).\nWe also note that  $\\kappa(\\vec{0})=0$.\n\nBecause of the linearity of $\\kappa(\\vec{\\theta})$ on $\\vec{N}$,\n$\\tilde{K}(\\vec{\\theta})$ can be written in the \ncumulant expansion as \n\\begin{align}\n\\tilde{K}(\\vec{\\theta}) = \\sum_m \\frac1{m!} \n\\langle (\\kappa(\\vec{\\theta}))^m \\rangle_{\\rm c} .\n\\label{eq:Kkappa}\n\\end{align}\nEquations~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}) are obtained by \ntaking appropriate derivatives of Eq.~(\\ref{eq:Kkappa})\nsimilarly to the previous section.\nIn this manipulation, one can again apply the Rules 1 and 2 \nintroduced in the previous section because of \nthe linearity of $\\kappa(\\vec{\\theta})$.\n\nIn order to obtain the cumulants of $Q$ defined in Eq.~(\\ref{eq:Q}), \nwe apply a differential operator,\n\\begin{align}\nD = \\sum_i \\tilde{a}_i \\partial_i ,\n\\end{align}\nto both sides in Eq.~(\\ref{eq:Kkappa}),\nwith $\\tilde{a}_i=a_i\/\\epsilon_i$.\nThe left-hand side is then given by\n\\begin{align}\nD^m \\tilde{K} = \\llangle ( \\sum_i \\tilde{a}_i n_i)^m \\rrangle_{\\rm c}\n= \\llangle q_{(1)}^m \\rrangle_{\\rm c},\n\\label{eq:DmKq}\n\\end{align}\nwhere $q_{(1)}$ is defined in Eq.~(\\ref{eq:q(s)}).\n\nNext, we see derivatives of the right-hand side order by order.\nFor the first derivative, only the first term in Eq.~(\\ref{eq:Kkappa})\nhas nonzero contribution and we obtain\n\\begin{align}\nD \\tilde{K} = \n\\langle D \\kappa(\\vec{\\theta}) \\rangle_{\\rm c}\n= \\langle Q \\rangle_{\\rm c} .\n\\label{eq:Dkappa}\n\\end{align}\nFrom Eq.~(\\ref{eq:DmKq}) with $m=1$ and Eq.~(\\ref{eq:Dkappa}),\nwe obtain Eq.~(\\ref{eq:Q1}).\n\nThe second derivative is calculated to be\n\\begin{align}\nD^2\\tilde{K} \n= D^2 \\big( \\langle \\kappa \\rangle_{\\rm c} + \n\\frac12 \\langle \\kappa^2 \\rangle_{\\rm c} \\big)\n= \\langle D^2 \\kappa \\rangle_{\\rm c} + \\langle ( D\\kappa )^2 \\rangle_{\\rm c},\n\\label{eq:D^2tildeK}\n\\end{align}\nwhere we used Rules 1 and 2.\nThe second term in the far right-hand side \nis $\\langle Q^2 \\rangle_{\\rm c}$ as a special case of the relation\n\\begin{align}\n\\langle ( D\\kappa )^m \\rangle_{\\rm c} = \\langle Q^m \\rangle_{\\rm c}.\n\\label{eq:Dkappa^m}\n\\end{align}\nThe first term, on the other hand, needs a further manipulation.\nFor this calculation, we note the relation\n\\begin{align}\nD^m \\kappa \n=& \\big( \\sum_i \\tilde{a}_i \\partial_i \\big)^m \\kappa \n= \\sum_i \\tilde{a}_i^m \\partial_i^m \\kappa \n= \\sum_i \\tilde{a}_i^m \\xi_m^{(i)} N_i\n\\nonumber \\\\\n=& \\sum_i \\tilde{a}_i^m \\tilde\\xi_m^{(i)} \\partial_i \\kappa \n= D_{(m)} \\kappa,\n\\label{eq:D2kappa}\n\\end{align}\nwhere we have defined a differential operator \n\\begin{align}\nD_{(s)} = \\sum_i \\tilde{a}^s \\tilde\\xi_s^{(i)} \\partial_i,\n\\end{align}\nwith $\\tilde\\xi_s^{(i)} = \\xi_s^{(i)} \/\\xi_1^{(i)}$.\nNote that $D=D_{(1)}$.\nIn the second equality in Eq.~(\\ref{eq:D2kappa}) we have used\nEq.~(\\ref{eq:ddk}).\nWe then take $D_{(2)}$ derivative of $\\tilde{K}$ as\n\\begin{align}\nD_{(2)} \\tilde{K} = \\langle D_{(2)} \\kappa \\rangle_{\\rm c} .\n\\label{eq:D_2tildeK}\n\\end{align}\nSubstituting Eqs.~(\\ref{eq:D2kappa}) and (\\ref{eq:D_2tildeK}) \nin Eq.~(\\ref{eq:D^2tildeK}) and \n\\begin{align}\nD_{(m)} \\tilde{K} = \\llangle q_{(m)} \\rrangle_{\\rm c},\n\\label{eq:D(m)K}\n\\end{align}\nwe obtain Eq.~(\\ref{eq:Q2}).\n\nTo extend the calculation to third and higher orders,\nwe need another differential operator\n\\begin{align}\nD_{(s_1,\\cdots,s_j|t_1,\\cdots,t_k)} \n= \\sum_i c_{(s_1,\\cdots,s_j|t_1,\\cdots,t_k)}^{(i)} \\partial_i,\n\\label{eq:D(sstt)}\n\\end{align}\nwhich appears in differentiations \n\\begin{align}\nD_{(s_1)} D_{(s_2)} \\kappa =& D_{(s_1,s_2|2)} \\kappa ,\n\\\\\nD_{(s_1)} D_{(s_2)} D_{(s_3)} \\kappa =& D_{(s_1,s_2,s_3|3)} \\kappa ,\n\\end{align}\nand\n\\begin{align}\nD_{(\\vec{s}_1|\\vec{t}_1)} D_{(\\vec{s}_2|\\vec{t}_2)} \\kappa\n&= D_{(\\vec{s}_1,\\vec{s}_2|\\vec{t}_1,\\vec{t}_2,2)} \\kappa,\n\\\\\nD_{(\\vec{s}_1|\\vec{t}_1)} D_{(\\vec{s}_2|\\vec{t}_2)} D_{(\\vec{s}_3|\\vec{t}_3)} \\kappa\n&= D_{(\\vec{s}_1,\\vec{s}_2,\\vec{s}_3|\\vec{t}_1,\\vec{t}_2,\\vec{t}_3,3)} \\kappa,\n\\end{align}\nand etc., where the vectorical representations for subscripts are\nunderstood.\nOn the other hand, the operations of these operators on $\\tilde{K}$ give \n\\begin{align}\nD_{(\\vec{s}|\\vec{t})} \\tilde{K}\n&= \\llangle q_{(\\vec{s}|\\vec{t})} \\rrangle_{\\rm c},\n\\\\\nD_{(\\vec{s}_1|\\vec{t}_1)} D_{(\\vec{s}_2|\\vec{t}_2)} \\tilde{K}\n&= \\llangle q_{(\\vec{s}_1|\\vec{t}_1)} q_{(\\vec{s}_2|\\vec{t}_2)} \\rrangle_{\\rm c},\n\\end{align}\nand so forth.\n\nFor third order, all equations required to obtain \n$\\langle Q^3 \\rangle_{\\rm c}$ are\n\\begin{align}\nD^3 \\tilde{K}\n=& \\llangle q_{(1)}^3 \\rrangle_{\\rm c}\n\\nonumber \\\\\n=& \\langle D_{(3)} \\kappa \\rangle_{\\rm c}\n+ 3 \\langle (D_{(2)}\\kappa) (D_{(1)}\\kappa) \\rangle_{\\rm c}\n\\nonumber \\\\\n& + \\langle (D_{(1)}\\kappa)^3 \\rangle_{\\rm c},\n\\label{eq:D^3tildeK1}\n\\\\\nD_{(2)} D_{(1)} \\tilde{K}\n=& \\llangle q_{(2)} q_{(1)} \\rrangle_{\\rm c}\n\\nonumber \\\\\n=& \\langle D_{(2,1|2)}\\kappa \\rangle_{\\rm c}\n+ \\langle (D_{(2)}\\kappa) (D_{(1)}\\kappa) \\rangle_{\\rm c},\n\\label{eq:D^3tildeK2}\n\\\\\nD_{(2,1|2)} \\tilde{K}\n=& \\llangle q_{(2,1|2)} \\rrangle_{\\rm c}\n= \\langle D_{(2,1|2)}\\kappa \\rangle_{\\rm c}\n\\label{eq:D^3tildeK3}\n\\\\\nD_{(3)} \\tilde{K} =& \\llangle q_{(3)} \\rrangle_{\\rm c}\n= \n\\langle D_{(3)}\\kappa \\rangle_{\\rm c}.\n\\label{eq:D^3tildeK4}\n\\end{align}\nUsing these results with Eq.~(\\ref{eq:Dkappa^m}),\nwe obtain Eq.~(\\ref{eq:Q3}).\nNote that the calculation to obtain Eq.~(\\ref{eq:Q3})\nfrom Eqs.~(\\ref{eq:D^3tildeK1}) -- (\\ref{eq:D^3tildeK4})\nis straightforwardly carried out by representing \nthese equations in a matrix form,\n\\begin{align}\n\\llangle V_q  \\rrangle_{\\rm c} \n= \\mathbb{M}_3 \\langle V_\\kappa \\rangle_{\\rm c},\n\\label{eq:M3}\n\\end{align}\nwith \n\\begin{align}\nV_q =& ( q_{(1)}^3 , q_{(2)} q_{(1)} , q_{(2,1|2)} , q_{(3)} )^T,\n\\\\\nV_\\kappa =& ( (D_{(1)}\\kappa)^3 , (D_{(2)}\\kappa) (D_{(1)}\\kappa) ,\nD_{(2,1|2)}\\kappa , D_{(3)}\\kappa )^T,\n\\end{align}\nand taking the inverse of $\\mathbb{M}_3$.\n\nSimilarly, the $m$-th order relation for $m\\ge4$\nis obtained by the following procedures:\n\\begin{enumerate}\n\\item\nCalculate $D^m \\tilde{K}$. \nThe result contains $\\langle (D_{(1)}\\kappa)^m \\rangle_{\\rm c} \n= \\langle Q^m \\rangle_{\\rm c}$.\n\\item\nThe remaining terms in the above result consists of \nderivatives of $\\kappa$.\nCalculate the derivatives of $\\tilde{K}$ with the same differential\noperator. For example, when one obtains\n$\\langle ( D_{(2)}\\kappa )( D_{(1)}\\kappa )^2 \\rangle_{\\rm c}$\nin the first process, calculate $D_{(2)} D_{(1)}^2 \\tilde{K}$.\nThe result contains the original term (in the example,\n$\\langle ( D_{(2)}\\kappa )( D_{(1)}\\kappa )^2 \\rangle_{\\rm c}$).\n\\item\nRepeat this procedure until all derivatives of $\\kappa$ \nare represented by derivatives of $\\tilde{K}$.\n\\item\nUnify them in a matrix form like Eq.~(\\ref{eq:M3}), \nand take the inverse.\n\\end{enumerate}\nFor fourth order, Eq.~(\\ref{eq:Q4}) is obtained by \ncalculating the following $11$ \ndifferentiations of $\\tilde{K}$:\n\\begin{align}\n& D^4 ,~D_{(3)}D_{(1)},~ D_{(2)}^2,~\nD_{(3,1|2)} ,~ D_{(2,2|2)} \n\\nonumber \\\\\n& D_{(2)} D_{(1)}^2 ,~ D_{(2,1|2)} D_{(1)} ,~ D_{(1,1|2)} D_{(2)} ,\n\\nonumber \\\\\n& D_{(2,1,1|2,2)} ,~ D_{(2,1,1|3)} ,~ D_{(4)}.\n\\end{align}\n\n\n\\section{Discussions}\n\\label{sec:summary}\n\nIn this paper we have presented the formulas representing\nthe cumulants of $Q$ up to fourth order.\nThese results can straightforwardly be extended to much higher\norders and to mixed cumulants, though the calculation\nbecomes more lengthy as the order becomes higher.\n\nIn this paper, we considered the binomial model Eq.~(\\ref{eq:binomial}).\nOnly a property of the binomial distribution function $B_{p,N}(n)$\nused in the derivations of Eqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4}) is \nthe fact that the cumulant generating function $k_N(\\theta)$ is\nproportional to $N$.\nTherefore, $B_{p,N}(n)$ can be replaced by other distribution \nfunctions satisfying this condition by replacing the values \nof $\\xi_i$.\n\n\nIn this paper we discussed the efficiency correction assuming the\nbinomial model Eq.~(\\ref{eq:binomial}).\nAs discussed already, \nthis model can be justified when the efficiencies for \nthe individual particles can be regarded independent.\nIn real detectors, however, efficiencies of individual particles\ncan be correlated.\nThe effect of such correlations are discussed recently \\cite{efficiency},\nand it is suggested that a small correlation can give rise to large\ndiscrepancy of the reconstructed values especially for higher orders.\nThe efficiency corrected cumulants based on the binomial model \nhave to be interpreted with this caveat.\n\nIn typical detectors in heavy ion collisions,\nneutrons cannot be observed.\nBecause of this problem, the proton number cumulants are \nexperimentally analyzed and compared with \nthe theoretical studies on the baryon number cumulants.\nAs suggested in Refs.~\\cite{Kitazawa:2011wh,Kitazawa:2012at},\nthe reconstruction of the baryon number cumulants is possible \nin the binomial model, since the measurement of protons among \nnucleons can be regarded as the $50\\%$ efficiency loss.\nIn this case, the assumption of the independence of efficiencies\nis well justified for high-energy collisions because of the \nisospin randomization \\cite{Kitazawa:2011wh,Kitazawa:2012at}.\nIt is an important subject to analyze the baryon number\ncumulants experimentally in this method, and compare \ntheir values directly with theoretical studies.\n\nIn real experiments, particle misidentifications and secondary\nparticles also affect the event-by-event analysis \\cite{Ono:2013rma}.\nThese effects are another issue which has to be taken care of\nbesides the problem of the efficiency correction.\n\n\nIn this paper, we derived the relations Eqs.~(\\ref{eq:Q1}) -- (\\ref{eq:Q4})\nwhich relate the ``original'' and ``observed'' cumulants of\nparticle numbers in event-by-event analyses.\nIn these formula, the cumulants of original particle numbers\nare represented by the mixed cumulants of observed particles.\nThe number of cumulants does not depend on the number of\ndifferent efficiencies, $M$.\nThese formulas thus would effectively be applied to practical \nanalyses of efficiency correction of the cumulants \nwith large $M$.\n\n\n\\section*{Acknowledgment}\n\nThe author thanks ShinIchi Esumi for inviting him\nto ``The second CiRfSE Workshop'' held at Tsukuba University\non Jan. 18-19, 2016, and for his hospitality during his stay.\nThe author also acknowledges fruitful discussion during \nthis stay with S.~Esumi, Xiaofeng Luo, Hiroshi Masui, \nToshihiro Nonaka, and Tetsuro Sugiura.\nThis work is supported in part by JSPS KAKENHI Grant\nNumber 25800148.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\tSwarms of low-cost agents such as small aerial robots may pose risk to safety-critical infrastructure such as government facilities, airports, and military bases. Interception strategies \\cite{chen2017multiplayer, coon2017control} against these threats may not be feasible or desirable in an urban environment due to posing greater risks to humans and the surrounding infrastructure. Under the assumption of risk-averse and self-interested adversarial agents (attackers) that tend to move away from the defending agents (defenders) and from other dynamic objects, herding can be used as an indirect way of guiding the attackers to a safe area.  \n\t\n\tIn our recent work \\cite{chipade2019swarmherding,chipade2020swarmherding}, we developed a herding algorithm, called `StringNet Herding', to herd a swarm of adversarial attackers away from a safety-critical (protected) area. A closed formation (`StringNet') of defending agents connected by string barriers is formed around a swarm of attackers staying together to confine their motion within given bounds, and guide them to a safe area. However, the assumption that the attackers will stay together in a connectivity region, and they will react to the defenders collectively as a single swarm while attacking the protected area, can be quite conservative in practice.\n\t\t\n\tIn this paper, we build upon our earlier work on `StringNet Herding' \\cite{chipade2020swarmherding} and study the problem of defending a safety-critical (protected) area from adversarial agents that \\textit{may} or \\textit{may not} stay together. We propose a `Multi-Swarm StringNet Herding' approach that uses clustering-based defender assignment, and the `StringNet Herding' method to herd the adversarial attackers to known safe areas.\n\t\n\t\\subsubsection{Related work}\n\tSeveral approaches have been proposed to solve the problem of herding. Some examples are: the $n$-wavefront algorithm \\cite{gade2015herding,paranjape2018robotic}, where the motion of the birds on the boundary of the flock is influenced based on the locations of the airport and the safe area; herding via formation control based on a potential-field approach \\cite{pierson2018controlling}; biologically-inspired \"wall\" and \"encirclement\" methods that dolphins use to capture a school of fish \\cite{haque2011biologically}; an RRT approach that finds a motion plan for the agents while maintaining a cage of potentials around the sheep \\cite{varava2017herding}; sequential switching among the chased targets \\cite{licitra2017single}.\n\tIn general, the above approaches suffer from one or more of the following: 1) dependence on knowing the analytical modeling of the attackers' motion, 2) lack of modeling of the adversarial agents' intent to reach or attack a certain protected area, 3) simplified motion and environment models. The proposed `StringNet Herding' approach relaxes the first and the third issue above, and takes into account the second one for control design.\n\n\t\n\t\n\n\n\t\n\n\t\n   \n    Clustering of data points is a popular machine learning technique \\cite{xu2015comprehensive}. \n\n\tThere are various categories of clustering algorithms: 1) partition based (K-means \\cite{macqueen1967some}), 2) hierarachy based (BIRCH \\cite{zhang1996birch}), 3) density based (DBSCAN \\cite{ester1996density}), 4) stream based (STREAM \\cite{o2002streaming}), 6) graph theory based (CLICK \\cite{sharan2000click}). Spatial proximity of the agents is crucial for the problem at hand so our focus will be mostly on the density based approaches in this paper.\n\n\n\tAssignment problems have also been studied extensively \\cite{burkard2012assignment}. In this paper, we are interested in a generalized assignment problem (GAP) \\cite{oncan2007survey}, in which there are more number of objects than knapsacks to be filled. GAP is known to be NP-hard but there are approximation algorithms to solve an arbitrary instance of GAP \\cite{oncan2007survey}.\n\t \n\t\\subsubsection{Overview of the proposed approach}\n\t\n\t\n\tThe proposed approach involves: 1) identification of the clusters (swarms) of the attackers that stay together, 2) distribution and assignment of the defenders to each of the identified swarms of the attackers, 3) use of `StringNet Herding' approach by the defenders to herd each identified swarm of attackers to the closest safe area. \n\t\n\tMore specifically, we use the ``Density based Spatial Clustering of Application with Noise (DBSCAN)\" algorithm \\cite{ester1996density} to identify the swarms of the attackers in which the attackers stay in a close proximity of the other attackers in the same swarm. We then formulate a generalized assignment problem with additional constraints on the connectivity of the defenders to find which defender should go against which swarm of attackers and herd it to one of the safe areas. This connectivity constrained generalized assignment problem (C2GAP) is modeled as a mixed integer quadratically constrained program (MIQCP) to obtain an optimal assignment solution. We also provide a hierarchical algorithm to find the assignment quickly, which along with the MIQCP formulation is the major contribution of this paper.\n\t\n\n\t\n\t\n\n\n\t\n\n\t\n\n\t\n\t\n\t\n\n\n\t\n\t\\subsubsection{Structure of the paper}\n\tSection \\ref{sec:math_model} describes the mathematical modeling and problem statement. The StringNet herding approach is briefly discussed in Section \\ref{sec:herding}. The approach on clustering and the defenders-to-attackers assignment for multiple-swarm herding is discussed in Section \\ref{sec:multi_swarm_herding}. \n\n\tSimulations and conclusions are provided in Section \\ref{sec:simulations} and \\ref{sec:conclusions}, respectively.\n\t\n\t\\section{Modeling and Problem Statement}\\label{sec:math_model}\n\t\\textit{Notation}: The set of integers greater than 0 is denoted by $\\mathbb{Z}_{>0}$. Vectors and matrices are denoted by small and capital bold letters, respectively (e.g., $\\textbf{r}$, $\\textbf{P}$). $\\norm{.}$ denotes the Euclidean norm of its argument. $\\abs{.}$ denotes the absolute value of a scalar, and cardinality if the argument is a set. $n!$ is a factorial of $n$.\n\n\t\n\t\\begin{comment}\n\t$\\bm{\\sigma}_{\\ell}: \\mathbb{R}^2\\rightarrow \\mathbb{R}^2$ is a saturation function defined as: \n\t\\begin{equation}\n\t\\bm{\\sigma}_{\\ell}(\\mathbf{g}) = \n\t\\begin{cases}\n\t\\mathbf{g} & \\text{if } \\norm{\\mathbf{g}} < u_{m{\\ell}}\\\\\n\tu_{m{\\ell}} \\frac{\\mathbf{g}}{\\norm{\\mathbf{g}}} & \\text{otherwise}\n\t\\end{cases}\n\t\\end{equation}\n\tfor ${\\ell}=\\{a,d,\\bar{d}_h,\\bar{d}_h, r_s,r_h\\}$ such that $u_{mr_h}<u_{mr_s}<u_{md}$ and $u_{m\\bar{d}_s}=u_{md}-u_{mr_s}$,$u_{m\\bar{d}_h}=u_{md}-u_{mr_h}$.\n\t\\end{comment} \n\t\n\tWe consider $N_a$ attackers $\\mathcal{A}_i$, $i \\in I_a= \\{1,2,...,N_a\\}$, and $N_d$ defenders $\\mathcal{D}_j$, $j \\in I_d= \\{1,2,...,N_d\\}$, operating in a 2D environment $\\mathcal{W} \\subseteq \\mathbb{R}^2$ that contains a protected area $\\mathcal{P} \\subset \\mathcal{W}$, defined as $\\mathcal{P}=\\{\\textbf{r} \\in \\mathbb{R}^2 \\;| \\; \\norm{\\textbf{r}-\\textbf{r}_p}\\le \\rho_p\\}$, and $N_s$ safe areas $\\mathcal{S}_{m} \\subset \\mathcal{W}$, defined as $\\mathcal{S}_{m}=\\{\\textbf{r} \\in \\mathbb{R}^2 \\; | \\; \\norm{\\textbf{r}-\\textbf{r}_{sm}}\\le \\rho_{sm}\\}$, for all $m \\in I_s=\\{1,2,...,N_s\\}$, where $(\\textbf r_p, \\rho_p)$ and $(\\textbf r_{sm}, \\rho_{sm})$ are the centers and radii of the corresponding areas, respectively. The number of defenders is no less than that of attackers, i.e., $N_d \\ge N_a$. The agents $\\mathcal{A}_i$ and $\\mathcal{D}_j$ are modeled as discs of radii $\\rho_a$ and $\\rho_d\\le \\rho_a$, respectively and move under double integrator (DI) dynamics with quadratic drag: \n\t\\begin{subequations} \\label{eq:attackDyn1}\n\t\\begin{align}\n\t\\dot{\\textbf{r}}_{ai}\n\t=\\textbf{v}_{ai}, \\quad \\quad \n\t\\dot{\\textbf{v}}_{ai}\n\t=\\textbf{u}_{ai}-C_{D} \\norm{\\textbf{v}_{ai}}\\textbf{v}_{ai};\\\\\n\t\\dot{\\textbf{r}}_{dj}\n\t=\\textbf{v}_{dj}, \\quad \\quad \n\t\\dot{\\textbf{v}}_{dj}\n\t=\\textbf{u}_{dj}-C_{D} \\norm{\\textbf{v}_{dj}}\\textbf{v}_{dj};\\\\\n\t\\norm{\\mathbf{u}_{ai}} \\le \\bar{u}_a, \\quad\n\t\\norm{\\mathbf{u}_{dj}} \\le \\bar{u}_d; \n\t\\end{align}\n\t\\end{subequations}\n\twhere $C_D$ is the drag coefficient, $\\mathbf{r}_{ai}=[x_{ai}\\; y_{ai}]^T$ and $\\mathbf{r}_{dj}=[x_{dj}\\; y_{dj}]^T$ are the position vectors of $\\mathcal{A}_i$ and $\\mathcal{D}_j$, respectively; $\\mathbf{v}_{ai}=[v_{x_{ai}}\\; v_{y_{ai}}]^T$, $\\mathbf{v}_{dj}=[v_{x_{dj}}\\; v_{y_{dj}}]^T$ are the velocity vectors, respectively, and $\\mathbf{u}_{ai}=[u_{x_{ai}}\\; u_{y_{ai}}]^T$, $\\mathbf{u}_{dj}=[u_{x_{dj}}\\; u_{y_{dj}}]^T$ are the accelerations (the control inputs), respectively.\t\n\n\n\t The defenders are assumed to be faster than the attackers, i.e., $\\bar{u}_a < \\bar{u}_d$.\n\tThis model poses a speed bound on each player with limited acceleration control, i.e., $v_{ai}=\\norm{\\mathbf{v}_{ai}}<\\bar{v}_a=\\sqrt{\\frac{\\bar{u}_a}{C_d}}$ and $v_{dj}=\\norm{\\mathbf{v}_{dj}}<\\bar{v}_d=\\sqrt{\\frac{\\bar{u}_d}{C_d}}$. \n\n\tWe assume that every defender $\\mathcal{D}_j$ senses the position $\\textbf r_{ai}$ and velocity $\\textbf{v}_{ai}$ of the attacker $\\mathcal{A}_i$ when $\\mathcal{A}_i$ is inside a circular sensing-zone $\\mathcal{Z}_d^s=\\{\\mathbf r \\in \\mathbb{R}^2 |\\; \\norm{\\textbf{r}-\\textbf{r}_p} \\le \\rho_d^s\\}$. Each attacker $\\mathcal{A}_i$ has a similar local sensing zone $\\mathcal{Z}_{ai}^s=\\{\\textbf{r} \\in \\mathbb{R}^2 \\;| \\; \\norm{\\textbf{r}-\\textbf{r}_{ai}}\\le \\rho_{ai}^s \\}$ inside which they sense defenders' positions and velocities.\n\n\n\n\n\t\n\n\t\n\t\n\tThe attackers aim to reach the protected area $\\mathcal{P}$. The attackers may use flocking controllers \\cite{dai2014flocking} to stay together, or they may choose to split into different smaller swarms \\cite{goel2019leader,raghuwaiya2016formation}. The defenders aim to herd each of these attackers to one of the safe areas in $\\mathcal{S}=\\{\\mathcal{S}_1,\\mathcal{S}_2,...,\\mathcal{S}_{N_s}\\}$ before they reach $\\mathcal{P}$. We consider the following problems.\n\t\n\t\\begin{problem}[Swarm Identification] \\label{prob:1}\n\tIdentify the swarms $\\{\\mathcal{A}_{c_1},\\mathcal{A}_{c_2},...,\\mathcal{A}_{c_{N_{ac}}}\\}$ of the attackers for some unknown $N_{ac}\\ge 1$ such that attackers in the same swarm $\\mathcal{A}_{c_k}$, and only them, satisfy prescribed conditions on spatial proximity.\n\t\\end{problem}\n\t\\begin{problem}[Multi-Swarm Herding]\n\tFind subgroups $\\{\\mathcal{D}_{c_1},\\mathcal{D}_{c_2},...,\\mathcal{D}_{c_{N_{ac}}}\\}$ of the defenders and their assignment to the attackers' swarms identified in Problem \\ref{prob:1}, such that all the defenders in the same subgroup are connected via string barriers to enclose and herd the assigned attacker's swarm.\n\t\\end{problem}\n\t\n\n\n\n\t\n\t\n\t\n\t\n\t\\begin{comment}\n\t\\begin{lemma} \\label{lem:potential_fun}\n\tPotential function $V_{i}^{j}$ has a unique minima at $R_{i}^{j}=\\bar{R}_{i}^{j}+R_{i}^{j,m}$.\n\t\\end{lemma}\n\t\n\t\\begin{proof}\n\tLet $\\Delta R=R_{i}^{j}-R_{i}^{j,m}$. The derivative of $V_{i}^{j}$:\n\t\\begin{equation*}\n\t\\pdydx{V_{i}^{j}}{R_{i}^{j}} =\\frac{(\\Delta R)^2-(\\bar{R}_{i}^{j})^2}{(\\Delta R)^2+(\\bar{R}_{i}^{j})^2}\\frac{1}{(\\Delta R)}\n\t\\end{equation*}\n\t\\begin{equation}\\label{eq:dV_by_dR}\n\t\\pdydx{V_{i}^{j}}{R_{i}^{j}}=0 \\implies \\Delta R = \\pm \\bar{R}_{i}^{j}\n\t\\end{equation}\n\t\n\tHowever $V_{i}^{j}$ is not defined for $\\Delta R = - \\bar{R}_{i}^{j}$. The second derivative of $V_{i}^{j}$ at $\\Delta R = + \\bar{R}_{i}^{j}$ is \n\t\\begin{equation} \\label{eq:d2V_by_dR2}\n\t\\left. \\frac{\\partial^2{V_{i}^{j}}}{\\partial (R_{i}^{j})^2} \\right |_{\\Delta R} = \\left. \\frac{(\\bar{R}_{i}^{j})^4+4(\\bar{R}_{i}^{j})^2\\Delta R^2 -\\Delta R^4 }{\\Delta R^2\\left(\\bar{R}_{i}^{j})^2+\\Delta R^2 \\right)^2} \\right |_{\\Delta R} >0\n\t\\end{equation}\n\tThis means ${\\Delta R}=\\bar{R}_{i}^{j}$ (i.e. $R_{i}^{j}=\\bar{R}_{i}^{j}+R_{i}^{j,m}$) is a local minima of $V_{i}^{j}$. Since it is the only point in the domain of $V_{i}^{j}$ satisfying~\\eqref{eq:dV_by_dR} and \\eqref{eq:d2V_by_dR2} it is also the unique minima of $V_{i}^{j}$.\n\t\\end{proof}\n\t\\end{comment}\n\n\n\n\n\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\t\n\n\n\t\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\t\n\n\n\n\n\n\n\n\n\n\n\t\n\n\n\n\n\t\t\n\t\t\n\t\\section{Herding a Single Swarm of Attackers}\\label{sec:herding}\n\tTo herd a swarm of attackers to $\\mathcal{S}$, we use `StringNet Herding', developed in\\cite{chipade2020swarmherding}. StringNet is a closed net of strings formed by the defenders as shown in Fig. \\ref{fig:clusterAssignment}. The strings are realized as impenetrable and extendable line barriers (e.g., spring-loaded pulley and a rope or other similar mechanism \\cite{mirjan2016building}) that prevent attackers from passing through them. The extendable string barrier allows free relative motion of the two defenders connected  by the string. The string barrier can have a maximum length of $\\bar{R}_{s}$. If the string barrier were to be physical one, then it can be established between two defenders $\\mathcal{D}_j$ and $\\mathcal{D}_{j'}$ only when they are close to each other and have almost same velocity, i.e., $\\norm{\\mathbf{r}_{dj}-\\mathbf{r}_{dj'}}\\le \\underline{R}_s<\\bar{R}_s$ and $\\norm{\\mathbf{v}_{dj}-\\mathbf{v}_{dj'}}\\le \\epsilon$, where $\\underline{R}_s$ and $\\epsilon$ are small numbers. The underlying graph structure for the two different ``StringNet'' formations defined for a subset of defenders $\\mathcal{D}'=\\{\\mathcal{D}_j \\;|\\; j \\in I_d'\\}$, where $I_d' \\subseteq I_d$, are defined as follows:\n\t\n\t\\begin{definition}[Closed-StringNet] \\label{def:closed_StringNet} The Closed-StringNet $\\mathcal{G}^{s}_{cl}(I_d')= (\\mathcal{V}^s_{cl}(I_d'),$ $\\mathcal{E}^s_{cl}(I_d'))$ is a cycle graph consisting of: 1) a subset of defenders as the vertices, $\\mathcal{V}^s_{cl}(I_d')=\\{\\mathcal{D}_j \\;|\\; j \\in I_d'\\}$, 2) a set of edges, $\\mathcal{E}^s_{cl}(I_d')=\\{(\\mathcal{D}_j,\\mathcal{D}_{j'}) \\in \\mathcal{V}^s_{cl}(I_d') \\times \\mathcal{V}^s_{cl}(I_d') | \\mathcal{D}_j \\overset{s} \\longleftrightarrow \\mathcal{D}_{j'} \\}$, where the operator $\\overset{s} \\longleftrightarrow$ denotes an impenetrable line barrier between the defenders.\n\t\\end{definition}\n\n\\begin{definition}[Open-StringNet] The Open-StringNet $\\mathcal{G}^{s}_{op}(I_d')= (\\mathcal{V}^s_{op}(I_d'),$ $\\mathcal{E}^s_{op}(I_d'))$ is a path graph consisting of: 1) a set of vertices, $\\mathcal{V}^s_{op}(I_d')$ and 2) a set of edges, $\\mathcal{E}^s_{op}(I_d')$, similar to that in Definition \\ref{def:closed_StringNet}.\n\\end{definition}\n\t\n\tThe StringNet herding consists of four phases: 1) gathering, 2) seeking, 3) enclosing, and 4) herding to a safe area. These phases are discussed as follows.\n\t\n\t\\subsection{Gathering}\\label{sec:gatheting}\n     We assume that the attackers start as single swarm that stays together, however, they may start splitting into smaller groups as they sense the defenders in their path. The aim of the defenders is to converge to an open formation $\\mathscr{F}_d^g$ centered at the gathering center $\\mathbf{r}_{df^g}$ located on the\n\texpected path of the attackers, where the expected path is defined as the shortest path of\n\tthe attackers to the protected area, before the attackers reach $\\mathbf{r}_{df^g}$. Let $\\mathscr{R}_d(N_a):\\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0}$ be the resource allocation function that outputs the number of the defenders that can be assigned to the given $N_a$ attackers. The open formation $\\mathscr{F}_d^g$ is characterized by\n\tthe positions $\\bm{\\xi}_l^g$, for all \n\n\t$l \\in I_{d{c_0}}= \\{1,2,...,\\mathscr{R}_d(N_a)\\}$, as shown in Fig.~\\ref{fig:clusterAssignment}. Once\n\tthe defenders arrive at these positions, the defenders get\n\tconnected by strings as follows: the defender\n\tat $\\bm{\\xi}_l^g$ gets connected to the defender at $\\bm{\\xi}_{l+1}^g$ for all\n\t$l \\in \\{1,2,...,\\mathscr{R}_d(N_a)-1\\}$ (see Fig.~\\ref{fig:clusterAssignment}). The formation $\\mathscr{F}_d^g$ is chosen to be a straight line formation as opposed to a semicircular formation\\footnote{Completing a circular formation starting from a semicircular formation of the same radius is faster. However, the semicircular formation, for a given length constraint on the string barrier ($\\bar{R}_s$), creates smaller blockage to the attackers as compared to the line formation. It is a trade-off between speed and effectiveness.} chosen in \\cite{chipade2020swarmherding} to allow for the largest blockage in the path of the attackers. The angle made by the normal to the line joining $\\bm{\\xi}_1^g$ and $\\bm{\\xi}_{N_d}^g$ (clockwise from $\\bm{\\xi}_1^g$, see Fig.~\\ref{fig:clusterAssignment}) is the orientation of the formation. The formation $\\mathscr{F}_d^g$ is chosen such that its orientation is toward the attackers on their expected path (defined above), see the blue formation in Fig~\\ref{fig:clusterAssignment}.\n\tThe desired positions $\\bm{\\xi}_l^g$ on $\\mathscr{F}_d^g$ centered at the gathering center $\\mathbf{r}_{df^g}$ are:\n\t\\begin{equation} \n\t\\arraycolsep=1.4pt\n\t\\begin{array}{ll}\n\t\\bm{\\xi}_{l}^g=\\mathbf{r}_{df^g} + \\hat{R}_{l} \\hat{\\mathbf{o}} (\\theta_{df^g}+\\frac{\\pi}{2}), \\quad \\text{for all } l \\in I_{d{c_0}};\n\t\\end{array}\n\t\\end{equation}\n     where $\\hat{R}_{l}= \\hat{R}_{d}^{d,g} \\left(\\frac{N_d-2l+1}{2} \\right)$, $\\hat{\\mathbf{o}}(\\theta)=[ \\cos(\\theta), \\; \\sin(\\theta) ]^T$ is the unit vector making an angle $\\theta$ with $x$-axis,\n\t $\\theta_{df^g}=\\theta_{a_{cm}}^*+\\pi$, where $\\theta_{a_{cm}}^*$ is the angle made by the line segment joining the attackers' center of mass (ACoM) to the center of the protected area (the shortest path from the initial position of ACoM to $\\mathcal{P}$) with $x$-axis. These positions are static, i.e., $\\dot{\\bm{\\xi}}_{l}^g=\\ddot{\\bm{\\xi}}_{l}^g=\\mathbf{0}$. The gathering center  $\\mathbf{r}_{df^g} = \\rho_{df}^g \\hat{\\mathbf{o}}(\\theta_{df^g}) $ is such that $\\rho_{df}^g> \\rho_p$.\n\t\n\t \n\t\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=.8\\linewidth,trim={5.3cm 1.2cm 4.4cm 3.15cm},clip]{clusterAssignment.eps}\n\t\\caption{Assignment of defenders to the attackers' swarms}\n\t\\label{fig:clusterAssignment}\n    \\end{figure}\n    We define the defender-goal assignment as:\n\t \\begin{definition}[Defender-Goal Assignment]\n\t A bijective mapping $\\bm{\\beta}_0:\\{1,2,...,\\mathscr{R}_d(N_a)\\} \\rightarrow I_d$ such that the defender $\\mathcal{D}_{\\bm{\\beta}_0(l)}$ is assigned to go to the goal $\\bm{\\xi}_{l}^g$.\n\t \\end{definition}\n\tAs discussed in \\cite{chipade2020swarmherding}, we design a time-optimal motion plan so that the defenders converge to the formation $\\mathscr{F}_d^g$ as early as possible. Given initial positions for the $N_d$ defenders, and desired goal positions on the formation $\\mathscr{F}_d^g$, we recursively solve a mixed integer quadratic program (MIQP) using bisection method to find: 1) the best gathering center, if feasible, and 2) the best defender-goal assignment.\n\t\n\tThe MIQP finds the best defender-goal assignment by using: 1) the time information of the time-optimal trajectories obtained for each defender to go from its initial position to any goal position $\\bm{\\xi}_l^g$ under bounded acceleration \\cite{chipade2020swarmherding}, and 2) the information of collision of all pairs of the time-optimal trajectories. The bisection method is then used to find the best gathering center by comparing the maximum time for the defenders obtained from the MIQP and the minimum time required by the attackers to reach the gathering center.\n\t\n\t\n\t\\subsection{Seeking}\n\tAfter the defenders accomplish gathering, suppose a group of defenders $\\mathcal{D}_{c_k}=\\{\\mathcal{D}_j |j \\in I_{dc_k}\\}$, $I_{dc_k} \\subseteq I_d$, is tasked to herd a swarm of attackers $\\mathcal{A}_{c_k}=\\{\\mathcal{A}_i |i \\in I_{ac_k}\\}$, $I_{ac_k} \\subseteq I_a$, the details are discussed later in Section \\ref{sec:multi_swarm_herding}. Let\n\t$\\bm{\\beta_{k}}:\\{1,2,...,|\\mathcal{D}_{c_k}|\\}\\rightarrow I_{dc_k}$ be the mapping that gives the indexing order of the defenders in $\\mathcal{D}_{c_k}$ on the Open-StringNet line formation $\\mathscr{F}_{dc_k}^{s}$ (similar to $\\mathscr{F}_{d}^{g}$).\n\tIn the seeking phase, the defenders in $\\mathcal{D}_{c_k}$ maintain the line formation $\\mathscr{F}_{dc_k}^{s}$ and try to get closer to the swarm of attackers $\\mathcal{A}_{c_k}$ by using state-feedback, finite-time convergent, bounded control laws as discussed in \\cite{chipade2020swarmherding}. The control actions as derived in \\cite{chipade2020swarmherding} for the defenders in $\\mathcal{D}_{c_k}$ are modified to incorporate collision avoidance from the other StringNet formations by $\\mathcal{D}_{c_{k'}}$, for $k'\\neq k$.\n\t\n\t\\subsection{Enclosing: Closed-StringNet formation}\nOnce the Open-StringNet formation reaches close to the attackers' formation, the enclosing phase begins in which the defenders start enclosing the attackers by moving to their desired positions on the enclosing formations while staying connected to their neighbors. We choose two formations for this phase that the defenders sequentially achieve: 1) Semi-circular Open-StringNet formation ($\\mathscr{F}_{dc_k}^{e_{op}}$), 2) Circular Closed-StringNet formation ($\\mathscr{F}_{dc_k}^{e_{cl}}$). When the defenders directly try to converge to a circular formation from a line formation during this phase, the defenders at the either end of the Open-StringNet formation will start coming closer to each other reducing the length of the overall barrier in the attackers' path significantly. This is because the desired positions of these terminal defenders in the circular formation would be very close to each other on the opposite side of the circular formation (see Fig.~\\ref{fig:clusterAssignment}) and collision avoidance part of the controller is only active locally near the circle of maximum radius $\\bar{\\rho}_{ac_k}$ around the swarm $\\mathcal{A}_{c_k}$. So the defenders would first converge to a semi-circular formation and would converge to a circular formation after the former is achieved.\n\t\n\tThe desired position $\\bm{\\xi}_{c_k,l}^{e_{op}}$ on the Open-StringNet formation $\\mathscr{F}_{dc_k}^{e_{op}}$ (Fig.~\\ref{fig:clusterAssignment}) is chosen on the circle with radius $\\rho_{sn_k}$ centered at $\\mathbf{r}_{\\hat{ac}_k}$ as:\n\t\\begin{equation} \\label{eq:goal_in_OpenStringNet}\n\t\\arraycolsep=1.4pt\n\t\\begin{array}{ll}\n\t\\bm{\\xi}_{c_k,l}^{e_{op}}=\\mathbf{r}_{\\hat{ac}_k} + \\rho_{sn_k} \\hat{\\mathbf{o}}(\\theta_{l}) \\text{, where }\n\t\\theta_{l} = \\theta_{df_k}^{e*}+\\frac{\\pi}{2}+\\frac{\\pi(l-1)}{|\\mathcal{D}_{c_k}|-1},\n\t\\end{array}\n\t\\end{equation}\n\tfor all $l \\in \\{1,2,...,|\\mathcal{D}_{c_k}|\\}$, where $\\theta_{df_k}^{e*}=\\theta_{df_k}^{s*}$. The center $\\mathbf{r}_{\\hat{ac}_k}=\\mathbf{r}_{ac_k}+\\tilde{\\mathbf{r}}_{\\hat{ac}_k}$, where $\\tilde{\\mathbf{r}}_{\\hat{ac}_k}$ is the position of the centroid of the convex hull of the position coordinates of the attackers in $\\mathcal{A}_{c_k}$ relative to the center of mass $\\mathbf{r}_{ac_k}=\\sum_{i \\in I_{ac_k}} \\frac{\\mathbf{r}_{ai}}{|\\mathcal{A}_{c_k}|}$ of $\\mathcal{A}_{c_k}$ at the latest time when the swarm $\\mathcal{A}_{c_k}$ was identified. \n\tThe radius $\\rho_{sn_k}$ should satisfy, $\\bar{\\rho}_{ac_k} +b_d < \\rho_{sn_k}$, where $\\bar{\\rho}_{ac_k}$ is maximum radius of swarm $\\mathcal{A}_{c_k}$. The parameter $b_d$ is the tracking error for the defenders in this phase \\cite{chipade2020swarmherding}. \n\t\n\tSimilarly, the desired positions $\\bm{\\xi}_{c_k,l}^{e_{cl}}$ on the Closed-StringNet formation $\\mathscr{F}_{dc_k}^{e_{cl}}$ same as in Eq.~\\ref{eq:goal_in_OpenStringNet} with $\n\t\\theta_{l} = \\theta_{df_k}^{e*}+\\frac{\\pi(2l-1)}{|\\mathcal{D}_{c_k}|}$,\n\tfor all $l \\in \\{1,2,...,|\\mathcal{D}_{c_k}|\\}$.\n\tBoth the formations move with the same velocity as that of the attackers' center of mass, i.e., $\\dot{\\bm{\\xi}}_{c_k,l}^{e_{op}}=\\dot{\\bm{\\xi}}_{c_k,l}^{e_{cl}}=\\dot{\\mathbf{r}}_{ac_k}$. \n\t\n\tThe defenders $\\mathcal{D}_{c_k}$ first track the desired goal positions $\\bm{\\xi}_{c_k,l}^{e_{op}}$ by using the finite-time convergent, bounded control actions given in \\cite{chipade2020swarmherding}. Once the defender $\\mathcal{D}_{\\bm{\\beta}_{k}(1)}$ and $\\mathcal{D}_{\\bm{\\beta}_{k}(|\\mathcal{D}_{c_k}|)}$\n reach within a distance of $b_d$ from $\\bm{\\xi}_{c_k,1}^{e_{op}}$ and $\\bm{\\xi}_{c_k,|\\mathcal{D}_{c_k}|}^{e_{op}}$, i.e., $\\norm{\\mathbf{r}_{d\\bm{\\beta}_{k}(1)}-\\bm{\\xi}_{c_k,1}^{e_{op}}}<b_d$ and $\\norm{\\mathbf{r}_{d\\bm{\\beta}_{k}(|\\mathcal{D}_{c_k}|)}-\\bm{\\xi}_{c_k,|\\mathcal{D}_{c_k}|}^{e_{op}}}<b_d$, respectively, the desired goal positions are changed from $\\bm{\\xi}_{c_k,l}^{e_{op}}$ to $\\bm{\\xi}_{c_k,l}^{e_{cl}}$\n for all $l \\in \\{1,2,...,|\\mathcal{D}_{c_k}|\\}$.\nThe StringNet is achieved when $\\norm{\\mathbf{r}_{d\\bm{\\beta}_{k}(l)}-\\mathbf{\\bm{\\xi}}_{c_k,l}^{e_{cl}}} \\le b_d$ for all $l \\in \\{1,2,...,|\\mathcal{D}_{c_k}|\\}$ during this phase.\n\t\\subsection{Herding: moving the Closed-StringNet to safe area}\n\tOnce a group of defenders $\\mathcal{D}_{c_k}=\\{\\mathcal{D}_j| j \\in I_{dc_k} \\}$, for $I_{dc_k}\\subseteq I_d$, forms a StringNet around a swarm of attackers, they move while tracking a desired rigid closed circular formation $\\mathscr{F}_{dc_k}^h$ centered at a virtual agent $\\mathbf{r}_{df_k^h}$ as discussed in \\cite{chipade2020swarmherding}. The swarm is herded to the closest safe area $S_{\\varsigma(k)}$, where $\\varsigma(k)=\\displaystyle \\argmin_{m \\in I_s} \\norm{\\mathbf{r}_{df_k^h}-\\mathbf{r}_{sm }}$.\n\t\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\t\n\t\\section{Multi-Swarm Herding}\\label{sec:multi_swarm_herding}\n\n We consider that the attackers split into smaller groups as they sense the defenders in their path, to maximize the chance of at least some attackers reaching the protected area by circumnavigating the oncoming defenders. \n To respond to such strategic movements of the attackers, the defenders need to collaborate intelligently. In the approach presented in this paper, the defenders first identify the spatial clusters of the attackers. Then, the defenders distribute themselves into smaller connected groups, subsets of defenders that have already established an Open-StringNet formation, in order to herd these different spatial clusters (swarms) of the attackers to safe areas. In the next subsections, we discuss the clustering and the defender to swarm assignment algorithms. \n\n    \n\\subsection{Identifying Swarms of the Attackers} \\label{sec:clustering}\nIn order to identify the spatially distributed clusters (swarms) of the attackers, the defenders utilize the Density Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm \\cite{ester1996density}. Given a set of points, DBSCAN algorithm finds clusters of high density points (points with many nearby neighbors), and marks the points as outliers if they lie alone in low-density regions (whose nearest neighbors are too far away). DBSCAN algorithm can identify clusters of any shape in the data and requires two parameters that define the density of the points in the clusters: 1) $\\varepsilon_{nb}$ (radius of the neighborhood of a point), 2) $m_{pts}$ (minimum number of points in $\\varepsilon_{nb}$-neighborhood of a point). In general, attackers can split into formations with varied range of densities making the choice of the parameters $\\varepsilon_{nb}$ and $m_{pts}$ challenging.\nVariants of the DBSCAN algorithm, such as OPTICS \\cite{ankerst1999optics}, can find clusters of varying density, however, they are more time consuming. To keep computational demands low, we use the DBSCAN algorithm with fixed parameters $\\varepsilon_{nb}$ and $m_{pts}$, which quickly yields useful clustering information about the attackers satisfying a specified connectivity constraints. \n\nThe neighborhood of an attacker is defined using weighted distance between two attackers: $d(\\mathbf{x}_{ai},\\mathbf{x}_{ai'})=\\sqrt{(\\mathbf{x}_{ai}-\\mathbf{x}_{ai'})^T \\mathbf{M} (\\mathbf{x}_{ai}-\\mathbf{x}_{ai'})}$, where $\\mathbf{x}_{ai}=[\\mathbf{r}_{ai}^T,\\mathbf{v}_{ai}^T]^T$ and $\\mathbf{M}$ is a weighing matrix defined as $\\mathbf{M}=diag([1,1,\\varphi, \\varphi])$, where $\\varphi$ weights relative velocity against relative position. We choose $\\varphi<1$ because relative position is more important in a spatial cluster than the velocity alignment at a given time instance. The $\\varepsilon_{nb}$-neighborhood of an attacker $\\mathcal{A}_i$ is then defined as the set of points $\\mathbf{x} \\in \\mathbb{R}^4$ such that $d(\\mathbf{x}_{ai},\\mathbf{x})<\\varepsilon_{nb}$.\n\nThe largest circle inscribed in the largest Closed-StrignNet formation formed by the $N_d$ defenders has radius $\\bar{\\rho}_{ac} =\\frac{\\bar{R}_s}{2} \\cot(\\frac{\\pi}{N_d})$. Maximum radius of any cluster with $N_a$ points identified by DBSCAN algorithm with parameters $\\varepsilon_{nb}$ and $m_{pts}$ is $\\frac{\\varepsilon_{nb}(N_a-1)}{m_{pts}-1}$. If all of the attackers were to be a single swarm enclosed inside the region with radius $\\bar{\\rho}_{ac}$ then we would require $\\varepsilon_{nb}$ to be greater than $\\frac{\\bar{\\rho}_{ac} (m_{pts}-1) }{N_a-1}$ in order identify them as a single cluster. So we choose $\\varepsilon_{nb}=\\frac{\\bar{\\rho}_{ac} (m_{pts}-1) }{N_a-1}$ and since we want to identify even clusters with as low as 3 agents we need to choose $m_{pts}=3$. With this parameters for DBSCAN algorithm, we have:\n\n\\begin{lemma}\nLet $\\{\\mathcal{A}_{c_1},\\mathcal{A}_{c_2},...,\\mathcal{A}_{c_{N_{ac}}}\\}$ be the clusters identified by DBSCAN algorithm with $\\varepsilon_{nb} =  \\frac{\\bar{\\rho}_{ac} }{N_a-1} \\floor{\\frac{m_{pts}}{2}}$. For all $k \\in I_{ac}=\\{1,2,...,N_{ac}\\}$, we have  $\\rho_{ac_k}=\\max_{i \\in I_{ac_k}} \\norm{\\mathbf{r}_{ai}-\\mathbf{r}_{\\hat{ac}_k}} \\le \\frac{\\bar{R}_s}{2}\\cot\\left(\\frac{\\pi}{|\\mathcal{A}_{c_k}|}\\right)$, if $|\\mathcal{A}_{c_k}|>3$ and $N_a=N_d$.\n\\end{lemma}\n\nAs the number of attackers increases, the computational cost for DBSCAN becomes higher and looses its practical usefulness. Furthermore, the knowledge of the clusters is only required by the defenders when a swarm of attackers does not satisfy the assumed constraint on its connectivity radius. So the DBSCAN algorithm is run only for swarms of attackers $\\mathcal{A}_{c_k}$ for some $k \\in I_{ac}$ whenever the connectivity constraint is violated by them i.e., when the radius of the swarm of attackers $\\mathcal{A}_{c_k}$ defined as $\\rho_{ac_k}=\\max_{i \\in I_{ac_k}} \\norm{\\mathbf{r}_{ai}-\\mathbf{r}_{\\hat{ac}_k}}$ exceeds the value $\\bar{\\rho}_{ac_k}=\\frac{\\bar{R}_s}{2}\\cot\\left(\\frac{\\pi}{N_d}\\right) \\frac{|\\mathcal{A}_{c_k}|-1}{N_a-1}$.\n\n\t\\subsection{Defender Assignment to the Swarms of Attackers}\\label{sec:defender_cluster_assignment}\n\tAs the initial swarm of attackers splits into smaller swarms, the defenders must distribute themselves into smaller groups and assign the attackers' swarms (clusters) to these groups in order to enclose these swarms and subsequently herd them to the closest safe area. Let $\\mathcal{A}_c=\\{\\mathcal{A}_{c_1},\\mathcal{A}_{c_2},\\dots, \\mathcal{A}_{c_{N_{ac}}}\\}$ be a set of swarms of the attackers after a split event has happened at time $t_{se}$. We assume that none of the swarms in $\\mathcal{A}_c$ is a singular one (i.e., a swarm with less than three agents), $|\\mathcal{A}_{c_k}|>2$ for all $k \\in I_{ac}=\\{1,2,...,N_{ac}\\}$. We formally define the defender to attackers' swarm assignment as: \n\t \\begin{definition}[Defender-Swarm Assignment]\n\t\tA set $\\bm{\\beta}$ $=\\{\\bm{\\beta}_1,\\bm{\\beta}_2,...\\bm{\\beta}_{N_{ac}} \\}$ of mappings $\\bm{\\beta}_k: \\{1,2,...,$ $\\mathscr{R}_d(|\\mathcal{A}_{c_k}|)\\} \\rightarrow I_d$, where $\\bm{\\beta}_k$ gives the indices of the defenders assigned to the swarm $\\mathcal{A}_{c_k}$ for all $k \\in I_{ac}$.\n\t\\end{definition}\n\t\n\tWe consider an optimization problem to find the best defender-swarm assignment as:\n\n\t\\begin{equation} \\label{eq:swarm_assignment_problem}\n\t\\begin{array}{ll} \n\\bm{\\beta}^\\star = \t\\text{argmin} & \\displaystyle \\sum_{k=1}^{N_{ac}} \\sum_{j'=1}^{\\mathscr{R}_d(|\\mathcal{A}_{c_k}|)} \\norm{\\mathbf{r}_{\\hat{ac}_k}-\\mathbf{r}_{d\\bm{\\beta}_k (j')}}\\\\\n\t\\text{Subject to} &\n\t (\\mathcal{D}_{\\bm{\\beta}_k(j')},\\mathcal{D}_{\\bm{\\beta}_k(j'-1)}) \\in \\mathcal{E}^s_{op}(I_d), \\\\\n\t&\\forall j' \\in \\{2, ..., \\mathscr{R}_d(|\\mathcal{A}_{c_k}|)\\},\\forall k \\in I_{ac}.\n\t\\end{array}\t\n\t\\end{equation}\n\tThe optimization cost is the sum of distances of the defenders from the centers of the attackers' swarms to which they are assigned. This ensures that the collective effort needed by all the defenders is minimized when enclosing the swarms of the attackers. The constraints in Eq.~\\eqref{eq:swarm_assignment_problem} require that all the defenders that are assigned to a particular swarm of the attackers are neighbors of each other, are already connected to each other via string barriers and the underlying graph is an Open-StringNet. Assuming $N_d=N_a$, we choose $\\mathscr{R}_d(|\\mathcal{A}_{c_k}|)=|\\mathcal{A}_{c_k}|$, i.e., the number of defenders assigned to a swarm $\\mathcal{A}_{c_k}$ is equal to the number of attackers in $\\mathcal{A}_{c_k}$. This is to ensure that there are adequate number of defenders to go after each attacker in the event the attackers in swarm $\\mathcal{A}_{c_k}$ disintegrate into singular swarms\\footnote{In this case, herding may not be the most economical way of defense. How to handle the situations with singular swarms is out of the scope of this paper and will be studied in the future work.}. \n\t\n\tThis assignment problem is closely related to generalized assignment problem (GAP) \\cite{oncan2007survey}, in which $n$ objects are to be filled in $m$ knapsacks $(n\\ge m)$.  This problem is modeled as a GAP with additional constraints on the objects (defenders) that are assigned to a given knapsack (attackers' swarm). We call this constrained assignment problem as connectivity constrained generalized assignment problem (C2GAP) and provide a mixed integer quadratically constrained program\t(MIQCP) to find the optimal assignment as: \n\n\n\t\\setlength{\\abovedisplayskip}{10pt}\n\t\\begin{subequations} \\label{eq:defender_swarm_assign_MIQCP}\n\t\\begin{align}\n\t\\text{Minimize } &  J=  \\textstyle \\sum_{k=1}^{N_{ac}} \\sum_{j=1}^{N_d} \\norm{\\mathbf{r}_{\\hat{ac}_k}-\\mathbf{r}_{dj}}\\delta_{jk} \\label{eq:MIQCP_cost}\\\\\n\t\\vspace{2mm}\n\t\\text{Subject to } & \\scriptstyle \\sum_{k \\in I_{ac}} \\delta_{jk}=1, \\quad \\forall j \\in I_d; \\label{eq:MIQP_constraint_1}\\\\\n\t\\vspace{2mm}\n    & \\scriptstyle \\sum_{j \\in I_{d}} \\delta_{jk}=\\mathscr{R}_d(|\\mathcal{A}_{c_k}|), \\quad \\forall k \\in I_{ac}; \\label{eq:MIQP_constraint_2}\\\\\n\t\\vspace{2mm}\n\t&\\scriptstyle \\sum_{j \\in \\tilde{I}_d} \\delta_{jk}\\delta_{(j+1)k}\\ge \\mathscr{R}_d(|\\mathcal{A}_{c_k}|)-1, \\quad \\forall k \\in I_{ac}; \\label{eq:MIQP_constraint_3}\\\\\n\t& \\scriptstyle \\sum_{k \\in I_{ac}} \\sum_{j \\in {I_d}} \\delta_{jk}=\\mathscr{R}_d(N_a); \\label{eq:MIQP_constraint_4}\\\\\n\t& \\scriptstyle \\delta_{jk}\\in \\{0,1\\}, \\quad \\forall j \\in I_d, k \\in I_{ac};\n\t\\end{align}\n\t\\end{subequations}\n\twhere $\\tilde{I}_d=I_d-\\{N_d\\}$, $\\delta_{jk}$ is a decision variable which is equal to 1 when the defender $\\mathcal{D}_j$ is assigned to the swarm $\\mathcal{A}_{c_k}$ and 0 otherwise. The constraints \\eqref{eq:MIQP_constraint_1} ensure that each defender is assigned to exactly one swarm of the attackers, the  capacity constraints \\eqref{eq:MIQP_constraint_2} ensure that for all $k \\in I_{ac}$ swarm $\\mathcal{A}_{c_k}$ has exactly $\\mathscr{R}_d(|\\mathcal{A}_{c_k}|)$ defenders assigned to it, the quadratic constraints \\eqref{eq:MIQP_constraint_3} ensure that all the defenders assigned to swarm $\\mathcal{A}_{c_k}$ are connected together with an underlying Open-StringNet for all $k \\in I_{ac}$ and the constraint \\eqref{eq:MIQP_constraint_4} ensures that all the $\\mathscr{R}_d(N_a)$ defenders are assigned to the attackers' swarms. This MIQCP can be solved using a MIP solver Gurobi \\cite{gurobi}.\n\n  As shown in an instance of the defender-swarm assignment in Fig.~\\ref{fig:clusterAssignment}, the defenders at $\\bm{\\xi}_{l}^g$ for $l \\in  \\{1,2,...,5\\}$ are assigned to swarm $\\mathcal{A}_{c_2}$ and those at $\\bm{\\xi}_{l}^g$ for $l \\in \\{6,7,...,10\\}$ are assigned to swarm $\\mathcal{A}_{c_1}$.\n\n\\subsection{Hierarchical Approach to defender-swarm assignment}\nFinding the optimal defender-swarm assignment by solving the MIQCP discussed above may not be real-time implementable for a large number of agents $(>100)$. In this section, we develop a computationally efficient hierarchical approach to find defender-swarm assignment. A large dimensional assignment problem is split into smaller, low-dimensional assignment problems that can be solved optimally and quickly. Algorithm \\ref{alg:defender_cluster_assignment} provides the steps to reduce the problem of size $N_{ac}$ to smaller problems of size smaller than or equal $\\underline{N}_{ac}(<N_{ac})$.\n\\begin{algorithm}[h]\n\t\t\\caption{Defender-Swarm Assignment}\n\t\t\\label{alg:defender_cluster_assignment}\n\t\n\t\n\t\n\t\t\\SetKwFunction{assignHierarchical}{assignHierarchical}\n\t\t\\SetKwFunction{assignMIQCP}{assignMIQCP}\n\t\t\\SetKwFunction{splitEqual}{splitEqual}\n\t\t\\SetKwProg{Fn}{Function}{:}{}\n        \\Fn{\\assignHierarchical{$\\mathscr{A}, \\mathscr{D}$}}{\n\t\t\\eIf{$\\mathscr{A}.N_{ac}>\\underline{N}_{ac}$}{[$\\mathscr{A}^l,\\mathscr{D}^l,\\mathscr{A}^r,\\mathscr{D}^r$]=\\splitEqual($\\mathscr{A},\\mathscr{D})$;\\\\\n\t\t\\eIf{$\\mathscr{A}^l.N_{ac}>\\underline{N}_{ac}$}{\n\t\t$\\bm{\\beta}^l=$\\assignHierarchical($\\mathscr{A}^l, \\mathscr{D}^l$);}\n\t\t{$\\bm{\\beta}^l=$\\assignMIQCP($\\mathscr{A}^l,\\mathscr{D}^l$);}\n\t\t\\eIf{$\\mathscr{A}^r.N_{ac}>\\underline{N}_{ac}$}{\n\t\t$\\bm{\\beta}^r=$\\assignHierarchical($\\mathscr{A}^r, \\mathscr{D}^r$);}\n\t\t{$\\bm{\\beta}^r=$\\assignMIQCP($\\mathscr{A}^r, \\mathscr{D}^r$);}\n\t\t$\\bm{\\beta}=\\{\\bm{\\beta}^l,\\bm{\\beta}^r\\};$\n\t\t}\n\t\t{$\\bm{\\beta}$=\\assignMIQCP($\\mathbf{r}_{ac}$,$\\mathbf{r}_d$);}\n\t\t\\Return{$\\bm{\\beta} =\\{\\bm{\\beta}_1,\\bm{\\beta}_2,...,\\bm{\\beta}_{N_{ac}}\\}$}\n\t\t}\n\t\t\\textbf{End Function}\n\t\\end{algorithm}\n\tIn Algorithm \\ref{alg:defender_cluster_assignment}, $\\mathscr{A}$ is a data structure that stores the information of: centers of the attackers' swarms $\\mathbf{r}_{ac}=[\\mathbf{r}_{\\hat{ac}_1}, \\mathbf{r}_{\\hat{ac}_2},...,\\mathbf{r}_{\\hat{ac}_{N_{ac}}}]$, numbers of the attackers in each swarm $\\mathbf{n}_{ac}=[|\\mathcal{A}_{c_1}|,|\\mathcal{A}_{c_2}|,...,|\\mathcal{A}_{c_{N_{ac}}}|]$, total number of attackers $N_a$; and $\\mathscr{D}$ is a data structure that stores the information of: defenders' positions $\\mathbf{r}_{d}=\\{\\mathbf{r}_{dj}| j \\in I_d'\\}$, and the goal assignment $\\bm{\\beta}$. \\splitEqual function splits the attackers into two groups $\\mathscr{A}^l$ and $\\mathscr{A}^r$ of roughly equal number of attackers and the defenders into two groups $\\mathscr{D}^l$ and $\\mathscr{D}^r$. The split is performed based on the angles $\\psi_{k}$ made by relative vectors $\\mathbf{r}_{\\hat{ac}_k}-\\mathbf{r}_{dc}$, for all $k \\in I_{ac}$, with the vector $\\mathbf{r}_{\\hat{ac}_k}-\\mathbf{r}_{dc}$ where $\\mathbf{r}_{dc}$ is the center of $\\mathbf{r}_d$. We first arrange these angles $\\psi_k$ in descending order. The first few clusters in the arranged list with roughly half the total number of attackers become the left group $\\mathscr{A}^l$ and the rest become the right group $\\mathscr{A}^r$. Similarly, the left group $\\mathscr{D}^l$ is formed by the first $\\mathscr{A}^l.N_a$ defenders as per the assignment $\\bm{\\beta}$ and the rest defenders form the right group $\\mathscr{D}^r$. We assign the defenders in $\\mathscr{D}^l$ only to the swarms in $\\mathscr{A}^l$ and those in $\\mathscr{D}^r$ only to the swarms in $\\mathscr{A}^r$. By doing so we may or may not obtain an assignment that minimizes the cost in \\eqref{eq:MIQCP_cost} but we reduce the computation time significantly and obtain a reasonably good assignment quickly. As in Algorithm \\ref{alg:defender_cluster_assignment}, the process of splitting is done recursively until the number of attackers' swarms is smaller than a pre-specified number $\\underline{N}_{ac}$. The function \\assignMIQCP finds the defender-swarm assignment by solving \\eqref{eq:defender_swarm_assign_MIQCP}. As shown in Figure~\\ref{fig:runTimeAssign}, the average computation time over a number of cluster configurations and initial conditions for the hierarchical approach to assignment is significantly smaller than that of the MIQCP formulation and also the cost of the hierarchical algorithm is very close to the optimal cost (MIQCP), see Fig.~\\ref{fig:assignCostError}.\n\t\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=.98\\linewidth,trim={.6cm 0cm .4cm .75cm},clip]{runTimeAssign.eps}\n\t\\caption{Run-time for assignment algorithms}\n\t\\label{fig:runTimeAssign}\n    \\end{figure}\n    \t\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=.98\\linewidth,trim={.6cm 0cm .4cm .75cm},clip]{assignCostError.eps}\n\t\\caption{\\% Error in the costs of the assignment algorithms}\n\t\\label{fig:assignCostError}\n    \\end{figure}\n\n\n\n\n\n\n\t\\section{Simulations}\\label{sec:simulations} \n\n\tWe provide a simulation of 18 defenders herding 18 attackers to $\\mathcal{S}$ with bounded control inputs. Figure~\\ref{fig:multiSwarmHerd} shows the snapshots of the paths taken by all agents. The positions and paths of the defenders are shown in blue color, and that of the attackers in red. The string-barriers between the defenders are shown as wide solid blue lines with white dashes in them.\n\t\n\tSnapshot 1 shows the paths during the gathering phase. As observed the defenders are able to gather at a location on the shortest path of the attackers to the protected area before the attacker reach there. Five attackers are already separated from the rest thirteen in reaction to the incoming defenders in their path. The defenders have identified two swarms of the attackers $\\mathcal{A}_{c_1}$ and $\\mathcal{A}_{c_2}$ at the end of the gathering phase and assign two subgroups $\\mathcal{D}_{c_1}$ and $\\mathcal{D}_{c_2}$ of the defenders to $\\mathcal{A}_{c_1}$ and $\\mathcal{A}_{c_2}$ using Algorithm \\ref{alg:defender_cluster_assignment}. As shown in snapshot 2, $\\mathcal{D}_{c_1}$ and $\\mathcal{D}_{c_2}$ seek $\\mathcal{A}_{c_1}$ and $\\mathcal{A}_{c_2}$, but the attackers in swarm $\\mathcal{A}_{c_2}$ further start splitting and the defenders identify this newly formed $\\mathcal{A}_{c_2}$ and $\\mathcal{A}_{c_3}$ at time $t=120.11 s$. The group $\\mathcal{D}_{c_2}$ is then split into two subgroups $\\mathcal{D}_{c_2}$ and $\\mathcal{D}_{c_3}$ of appropriate sizes and assigned to the new swarms $\\mathcal{A}_{c_2}$ and $\\mathcal{A}_{c_3}$ using Algorithm \\ref{alg:defender_cluster_assignment}. \n\t\n\tSnapshot 3 shows how the 3 subgroups of the defenders are able to enclose the the identified 3 swarms of the attackers by forming Closed-StringNets around them. Snapshot 4 shows how all the three enclosed swarms of the attackers are taken to the respective closest safe areas while each defenders' group ensures collision avoidance from other defenders' groups. Additional simulations can be found at \\href{https:\/\/drive.google.com\/drive\/folders\/11qJxjlxR_AWbc4vicIRchZpvCaByCJGP?usp=sharing}{\/drive\/video}.\n\t\\begin{figure*}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[h]{0.48\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth,trim={.1cm 0.2cm 2.cm .3cm},clip]{figures\/multiSwarmHerd1.eps}\n\t\t\n\t\t\t\\label{fig:multiSwarmHerd1}\n\t\t\\end{subfigure}\t\n\t\t\\begin{subfigure}[h]{0.48\\textwidth}\n\t\t\\includegraphics[width=1\\linewidth,trim={.9cm 0.2cm 2.cm .65cm},clip]{figures\/multiSwarmHerd2.eps}\n\t\n\t\t\\label{fig:multiSwarmHerd2}\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}[h]{0.48\\textwidth}\n\t\t\\includegraphics[width=1\\linewidth,trim={.9cm 0.2cm 2cm .85cm},clip]{figures\/multiSwarmHerd3.eps}\n\t\n\t\t\\label{fig:multiSwarmHerd3}\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}[h]{0.48\\textwidth}\n\t\t\\includegraphics[width=1\\linewidth,trim={.9cm 0.2cm 2cm 1.25cm},clip]{figures\/multiSwarmHerd4.eps}\n\t\n\t\t\\label{fig:multiSwarmHerd4}\n\t\t\\end{subfigure}\n\t\t\\caption{Snapshots of the paths of the agents during Multi-Swarm StringNet Herding}\n\t\t\\label{fig:multiSwarmHerd}\n\t\\end{figure*}\n\n\n\t\n\t\\vspace{-1mm}\n\t\\section{Conclusions} \\label{sec:conclusions}\n\tWe proposed a clustering-based, connectivity-constrained assignment algorithm that distributes and assigns groups of defenders against swarms of the attackers, to herd them to the closest safe area using `StringNet Herding' approach. We also provide a heuristic for the defender-swarm assignment based on the optimal MIQCP that finds the assignment quickly. Simulations show how this proposed method improves the original 'StringNet Herding' method and enables the defenders herd all the attackers to safe areas even though the attackers start splitting into smaller swarms in reaction to the defenders.\n\t\n\n\t\n\t\n\n\n\n\t \n\n\n\n\n\n\n\t\n\n\t\n\t\n\t\n\n\t\n\t\n\t\n\n\n\t\n\n\t\n\n\n\t\\vspace{-2mm}\n\t\\bibliographystyle{IEEEtran}\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nWithin the field of stochastic thermodynamics~\\cite{Seifert2012} systems with interacting degrees of freedom play an important role since they offer case studies for the interplay between thermodynamics and information theory~\\cite{Parrondo2015}.\n\nFor such systems there are diverse setups. For example, systems with degrees of freedom which are inaccessible to the experimenter show deviations in the fluctuation relation for the entropy production~\\cite{Mehl2012}. Further, there is a wide variety of measurement-feedback setups in which a stochastic system is measured and subsequently controlled by a feedback controller. Additionally, sensing is a manifestation of such a coupled joint system. Here, a stochastic system is influenced by an external stochastic process. Usually the task of the sensor is to measure the external process.\n\nSystems with interacting degrees of freedom have conveniently been modeled using bipartite Markov processes. For such systems a splitting of the second law of thermodynamics has been achieved such that individual second laws applicable to each one of the subsystems retain the influence of the other through information theoretic terms like information flow~\\cite{Horowitz2014} or learning rate~\\cite{Hartich2014}. Recently, Crooks and Still~\\cite{Crooks2016} obtained marginal fluctuation relations applicable to one of two subsystems within a bipartite setup. In these the influence of the other subsystem is encoded in a transfer-entropy~\\cite{Schreiber2000} term.\n\nMeasurement-feedback systems have been studied by Sagawa and Ueda~\\cite{Sagawa2010,Sagawa2012}, Horowitz and Vaikuntanathan~\\cite{Horowitz2010} and Ponmurugan~\\cite{Ponmurugan2010}. They established fluctuation theorems applicable to such systems. Various sensor setups have been studied. Within the context of stochastic thermodynamics, the main focus is on the relation between the sensor's information about the external environment and its energy dissipation~\\cite{Mehta2012,Lan2012,Barato2014,Sartori2014,Bo2015}.\n\nHowever, it is instructive to study these setups from a common perspective and find fluctuation relations which, when evaluated for these special cases, recover the known results. This also offers a formalism to reliably derive fluctuation relations applicable to one of several interacting subsystems.\n\nThe aim of this paper is to (1) put to use the concept of detached path probabilities and detached entropy production for bipartite Markov chains. Both have been introduced as preliminary quantities for further evaluation in~\\cite{Crooks2016}; (2) elaborate on a series of special cases that have been studied separately before, namely measurement-feedback systems, sensors and hidden Markov models and show that fluctuation theorems involving the detached entropy production can recover known results; and (3) show how to use our formalism to confirm model parameters for hidden Markov models.\n\n\n\\section{Detached path probabilities}\nWe consider a two-variate Markov chain $(x_{0:T},y_{0:T})$ with \n\\begin{subequations}\n\\begin{align}\\label{defx}\n x_{0:T} &= \\{x_0,x_1,...,x_T\\}\\eqqcolon\\{x_0,\\mathbf{x}\\}\\\\\\label{defy}\n y_{0:T} &= \\{y_0,y_1,...,y_T\\}\\eqqcolon\\{y_0,\\mathbf{y}\\}.\n\\end{align}\n\\end{subequations}\nThe process is assumed to be bipartite as sketched in Fig.~\\ref{fig_setupJointProcess} such that only one subsystem changes its state at a time \\cite{Hartich2014,Horowitz2014,Crooks2016}. A simple example is given by two discretized coupled Langevin equations with independent noise sources. \n\nThe stochastic dynamics depend on external protocols $\\mathbf{u}=\\{u_1,...,u_T\\}$ and $\\mathbf{v}=\\{v_0,...,v_{T-1}\\}$ that influence the individual transition probabilities $p_x(x_t|x_{t-1},y_t;u_t)$ and $p_y(y_t|x_{t-1},y_{t-1};v_{t-1})$, respectively. For the joint probability of the entire sequence of states, starting from an initial condition $(x_0,y_0)$, we hence have\n\\begin{multline}\\label{defjointprob}\n p(\\mathbf{x},\\mathbf{y}|x_0,y_0;\\mathbf{u},\\mathbf{v}) = p_y(y_1|x_0,y_0;v_0)\\,p_x(x_1|x_0,y_1;u_1)...\\\\\n            ...\\times p_y(y_T|x_{T-1},y_{T-1};v_{T-1})\\,p_x(x_T|x_{T-1},y_T;u_T).\n\\end{multline}\nTogether with the initial distribution $p_0(x_0,y_0)$ this probability uniquely determines the entire process.\n\n\n\\begin{figure}[ht]\n \\centering\n\\includegraphics[width = \\linewidth]{1.pdf}\n \\caption{Setup of a joint trajectory. The subsystems $x$ and $y$ update one after the other.}\n \\label{fig_setupJointProcess}\n\\end{figure}\n\nFrom the structure of the transition probabilities as well as from Fig.~\\ref{fig_setupJointProcess} it is clear that the stochastic variables $x$ and $y$ influence each other. Nevertheless, as observed by Crooks and Still~\\cite{Crooks2016}, it is instructive to split the joint probability \\eqref{defjointprob} according to \n\\begin{align}\n  p(\\mathbf{x},\\mathbf{y}&|x_0,y_0;\\mathbf{u},\\mathbf{v})= q_x(\\mathbf{x}|x_0; \\mathbf{y}, \\mathbf{u}) \\, q_y(\\mathbf{y}|y_0;x_0,\\mathbf{x},\\mathbf{v}),  \\label{eqn_decompositionJointProbability}\n\\end{align}\nwhere \n\\begin{subequations}\n\\begin{align}\\label{eqn_xTrajectoryProb}\n  q_x(\\mathbf{x}|x_0;\\mathbf{y},\\mathbf{u}) &\\coloneqq \\prod\\limits_{t=1}^{T} p_x(x_t|x_{t-1},y_t;u_t)\\\\\n  q_y(\\mathbf{y}|y_0;x_0,\\mathbf{x},\\mathbf{v}) &\\coloneqq \\prod\\limits_{t=1}^{T} p_y(y_t|x_{t-1},y_{t-1};v_{t-1}). \\;  \\label{eqn_yTrajectoryProb}\n\\end{align}\n\\end{subequations}\nA related concept in information theory goes under the name of \\emph{causal conditioning} and is applicable even to non-Markovian processes \\cite{Marko1973, Massey1990, Jiao2013}.\n\nWe call $q_x$ and $q_y$ the \\emph{detached} path probabilities of the individual subsystems. They are normalized according to\n\\begin{align}\n 1 &= \\int d\\mathbf{x}\\,q_x(\\mathbf{x}|x_0; \\mathbf{y}, \\mathbf{u})\\label{eqn_detachedNormalized}\\\\\n&= \\int dx_1\\,p_x(x_1|x_0,y_1;u_1) \\, ... \\int dx_T\\,p_x(x_T|x_{T-1},y_T;u_t)\\nonumber\n\\end{align}\nand similarly for $q_y(\\mathbf{y}|y_0;\\mathbf{x},x_0,\\mathbf{v})$. The decomposition of the joint probability into a product of detached probabilities according to Eq.~\\eqref{eqn_decompositionJointProbability} can be represented graphically as a splitting of the joint process into two sub-processes in which the state of the other process enters as an additional time-dependent protocol. This is depicted in Fig.~\\ref{fig_splitJointProcess}.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=\\linewidth]{2.pdf}\n \\caption{Decomposition of the joint trajectory into two sub-trajectories with the other process acting as a time-dependent protocol.}\n \\label{fig_splitJointProcess}\n\\end{figure}\n\nIt is important to distinguish the detached path probabilities from the \\emph{marginal} and the \\emph{conditional} probabilities of the individual sequences. The marginal probability of the sequence of $x$-values is defined by \n\\begin{align}\\label{defmargprobx}\np_x(\\mathbf{x}|x_0;\\mathbf{u},\\mathbf{v})\\coloneqq \\int dy_0d\\mathbf{y} \\, p_0(y_0|x_0)\\,p(\\mathbf{x},\\mathbf{y}|x_0,y_0;\\mathbf{u},\\mathbf{v}),\n\\end{align}\nwhere $p_0(y_0|x_0) = p_0(x_0,y_0)\/\\int dy_0\\,p_0(x_0,y_0)$. Hence, in contrast to $q_x(\\mathbf{x}|x_0; \\mathbf{y}, \\mathbf{u})$, all dependence on $y$ is averaged out. Similarly, \n\\begin{align}\\label{defmargproby}\np_y(\\mathbf{y}|y_0;\\mathbf{u},\\mathbf{v})\\coloneqq \\int dx_0d\\mathbf{x} \\, p_0(x_0|y_0)\\, p(\\mathbf{x},\\mathbf{y}|x_0,y_0;\\mathbf{u},\\mathbf{v})\n\\end{align}\nwith no trace left of the $x$-dynamics. \n\nThe conditional probability of $\\mathbf{x}$ given the $y$-sequence, on the other hand, is defined by:\n\\begin{align}\n p_x(\\mathbf{x}|x_0, &y_0,\\mathbf{y}; \\mathbf{u},\\mathbf{v})\n= \\frac{p(\\mathbf{x},\\mathbf{y}|x_0,y_0;\\mathbf{u},\\mathbf{v})}{\\int d\\mathbf{x}\\,p(\\mathbf{x},\\mathbf{y}|x_0,y_0;\\mathbf{u},\\mathbf{v})}\\nonumber\\\\\n &=\\frac{q_x(\\mathbf{x}|x_0;\\mathbf{y},\\mathbf{u})\\, q_y(\\mathbf{y}|y_0;\\mathbf{x},x_0,\\mathbf{v})}{\\int d\\mathbf{x}\\, q_x(\\mathbf{x}|x_0;\\mathbf{y},\\mathbf{u})\\, q_y(\\mathbf{y}|y_0;\\mathbf{x},x_0,\\mathbf{v})}.\\label{diffqp}\n\\end{align}\nAgain analogous results hold for \n\\begin{equation}\n p_y(\\mathbf{y}|x_0, \\mathbf{x},y_0; \\mathbf{u},\\mathbf{v}) = \\frac{p(\\mathbf{x},\\mathbf{y}|x_0,y_0;\\mathbf{u},\\mathbf{v})}{\\int d\\mathbf{y}\\,p(\\mathbf{x},\\mathbf{y}|x_0,y_0;\\mathbf{u},\\mathbf{v})}.\\label{defcondproby} \n\\end{equation}\nThe conditional probability $p_x(\\mathbf{x}|x_0, y_0,\\mathbf{y}; \\mathbf{u},\\mathbf{v})$ depends on the particular $y$-values considered. Yet, it is different from the detached probability $q_x(\\mathbf{x}|x_0; \\mathbf{y}, \\mathbf{u})$ since the latter ignores the feedback from $x$ to $y$. Formally, this implies that we cannot evaluate the integral in the denominator in Eq.~\\eqref{diffqp}. More intuitively, the \nfeedback between the subsystems implicates that a conditioning on future values of one trajectory constrains the evolution of the other one. It is precisely this effect that is eliminated in the definition of the detached path probabilities.\n\n\n\\section{Entropy production and fluctuation relations}\n\n\\subsection{Reverse process}\nEntropies together with the respective fluctuation theorems play a pivotal role in stochastic thermodynamics. Their definition generally involves a conjugate process \\cite{Seifert2012}.\n\nAlthough there is some freedom in choosing the conjugate process, here we will always take the time-reversed process driven by the time-reversed protocols $\\mathbf{\\bar{u}}\\coloneqq\\{ u_T, u_{T-1},..., u_1\\}$ and $\\mathbf{\\bar{v}}\\coloneqq\\{ v_{T-1}, v_{T-2},..., v_0\\}$ as the conjugate one. For simplicity and to lighten the notation, we assume that the protocol values $u$ and $v$ as well as the state variables $x$ and $y$ are even under time-reversal, i.e. $\\bar u_t = u_t$, $\\bar v_t = v_t$, $\\bar x_t = x_t$ and $\\bar y_t = y_t$ for all $t=0,...,T$. \n\nThe stochastic dynamics of the reversed process are characterized by a joint probability of the form \\eqref{defjointprob} with $\\mathbf{u}$ und $\\mathbf{v}$ replaced by $\\mathbf{\\bar{u}}$ and $\\mathbf{\\bar{v}}$, respectively. We will need the path probabilities of the reversed process in particular for the time-reversed sequences of $x$ and $y$ states that we denote by\n\\begin{subequations} \n\\begin{align}\\label{defbarx}\n x_{T:0} &= \\{x_T,x_{T-1},...,x_0\\}\\eqqcolon\\{x_T,\\mathbf{\\bar{x}}\\}\\\\\\label{defbary}\n y_{T:0} &= \\{y_T,y_{T-1},...,y_0\\}\\eqqcolon\\{y_T,\\mathbf{\\bar{y}}\\}.\n\\end{align}\n\\end{subequations}\nThe joint probability of the reversed trajectory starting at $(x_T,y_T)$ under the reversed process is therefore given by\n\\begin{align}\n \\bar p(\\mathbf{\\bar{x}},&\\mathbf{\\bar{y}}|x_T,y_T;\\mathbf{\\bar{u}},\\mathbf{\\bar{v}})\\nonumber\\\\\n &= p_x(x_{T-1}|x_T,y_T,u_T)\\,p_y(y_{T-1}|x_{T-1},y_T;v_{T-1}) ...\\nonumber\\\\\n &\\qquad\\qquad \\times p_x(x_0|x_1,y_1; u_1)\\,p_y(y_0|x_0,y_1; v_0).\\label{defjointprobrev}\n\\end{align}\nThis is shown graphically in Fig.~\\ref{fig_reverseJointProcess}. Note that, compared to Fig.~\\ref{fig_setupJointProcess}, horizontal components of arrows (indicating time) are reversed whereas vertical ones (indicating causal influence) remain unchanged \\cite{Crooks2016}.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width = \\linewidth]{3.pdf}\n \\caption{Reverse joint trajectory.}\n \\label{fig_reverseJointProcess}\n\\end{figure}\n\nAs initial distribution of the reverse process we take the final distribution of the forward process: \n\\begin{align}\n p_T(x_T,y_T) \\coloneqq p(x_T,y_T|x_0,y_0;\\mathbf{u},\\mathbf{v}).\n\\end{align}\n\nAnalogous to Eq.~\\eqref{eqn_decompositionJointProbability}, we decompose the joint probability of the reversed process into the respective detached probabilities~\\cite{FootnoteReverseDecomposition}:\n\\begin{align}\n  \\bar p(\\mathbf{\\bar{x}},\\mathbf{\\bar{y}}|x_T,y_T;\\mathbf{\\bar{u}},\\mathbf{\\bar{v}})= \\bar{q}_x(\\mathbf{\\bar{x}}|x_T; \\mathbf{\\bar{y}}, y_T,\\mathbf{\\bar{u}}) \\, \\bar{q}_y(\\mathbf{\\bar{y}}|y_T;\\mathbf{\\bar{x}},\\mathbf{\\bar{v}}),\\label{eqn_decompositionReverseJointProbability}\n\\end{align}\nwhere now \n\\begin{subequations}\n\\begin{align}\\label{eqn_xReversedTrajectoryProb}\n  \\bar{q}_x(\\mathbf{\\bar{x}}&|x_T;\\mathbf{\\bar{y}},y_T,\\mathbf{\\bar{u}}) \\coloneqq \\prod\\limits_{t=1}^{T} p_x(x_{t-1}|x_t,y_t; u_t)\\\\\n  \\bar{q}_y(\\mathbf{\\bar{y}}&|y_T;\\mathbf{\\bar{x}},\\mathbf{\\bar{v}}) \\coloneqq \\prod\\limits_{t=1}^{T} p_y(y_{t-1}|x_{t-1},y_t; v_{t-1}) \\; . \\label{eqn_yReversedTrajectoryProb} \n\\end{align}\n\\end{subequations}\n\n\\subsection{Joint entropy production}\n\nWith the specification of the reverse process, we may define the corresponding entropy productions. It is natural to consider the \\emph{joint} entropy production \n\\begin{align}\n \\sigma_{xy} \\coloneqq \n     \\ln \\frac{p_0(x_0,y_0)\\,p(\\mathbf{x},\\mathbf{y}|x_0,y_0;\\mathbf{u},\\mathbf{v})}\n                {p_T(x_T,y_T)\\,\\bar p(\\mathbf{\\bar{x}},\\mathbf{\\bar{y}}|x_T,y_T;\\mathbf{\\bar{u}},\\mathbf{\\bar{v}})}\\label{eqn_defJointEntropy}\\; .\n\\end{align} \nAs usual $\\sigma_{xy}$ quantifies the relative surprise to observe the time-reversed trajectory \n$\\{x_T,\\mathbf{\\bar{x}},y_T,\\mathbf{\\bar{y}}\\}$ under the influence of the time-reversed protocols $\\{\\mathbf{\\bar{u}},\\mathbf{\\bar{v}}\\}$ if the forward trajectory $\\left\\{x_0,\\mathbf{x},y_0,\\mathbf{y}\\right\\}$ was seen under the protocols $\\left\\{\\mathbf{u},\\mathbf{v}\\right\\}$. \n\nTo avoid clutter, we will from now on suppress the dependence on the protocols $\\mathbf{u}$ and $\\mathbf{v}$ and their reversed versions and show it only where it matters.\n\nThe joint entropy production fulfills an integral fluctuation theorem (IFT):\n\\begin{align}\n \\left\\langle e^{-\\sigma_{xy}}\\right\\rangle &= \\int d\\mathbf{x} d\\mathbf{y} dx_0 dy_0 \\,e^{-\\sigma_{xy}}\\, p_0(x_0,y_0)\\,p(\\mathbf{x},\\mathbf{y}|x_0,y_0)\\nonumber\\\\\n  &= \\int d\\mathbf{\\bar{x}} d\\mathbf{\\bar{y}} dx_T dy_T \\, p_T(x_T,y_T)\\,\\bar p(\\mathbf{\\bar{x}},\\mathbf{\\bar{y}}|x_T,y_T)\\nonumber\\\\\n &=1.\\label{eqn_jointIFT}\n\\end{align}\n\n\\subsection{Detached entropy production}\n\nBuilding on the detached path probabilities we may define the \\emph{detached} entropy productions\n\\begin{subequations}\n\\begin{align}\\label{appentx}\n \\tilde\\sigma_x &\\coloneqq \\ln\\frac{p_0(x_0|y_0)\\,q_x(\\mathbf{x}|x_0;\\mathbf{y})}{p_T(x_T|y_T)\\,\\bar{q}_x(\\mathbf{\\bar{x}}|x_T;y_T,\\mathbf{\\bar{y}})}\n            \\\\\\mathrm{and}\\nonumber &\\\\\\label{appenty}\n \\tilde\\sigma_y &\\coloneqq \\ln\\frac{p_0(y_0|x_0)\\,q_y(\\mathbf{y}|y_0;x_0,\\mathbf{x})}{p_T(y_T|x_T)\\,\\bar{q}_y(\\mathbf{\\bar{y}}|y_T;\\mathbf{\\bar{x}})}\\; , \n\\end{align}\n\\end{subequations}\nwhere the initial conditions may be calculated from $p_0(x_0,y_0)$ and $p_T(x_T,y_T)$, respectively.\n\nUsing~Eq.~\\eqref{appentx}, we obtain:\n\\begin{align}\n \\left\\langle e^{-\\tilde\\sigma_{x}}\\right\\rangle &= \\int d\\mathbf{x} d\\mathbf{y} dx_0 dy_0 \\,\\frac{p_T(x_T|y_T)\\,\\bar q_x(\\mathbf{\\bar{x}}|x_T;y_T,\\mathbf{\\bar{y}})}{p_0(x_0|y_0)\\,q_x(\\mathbf{x}|x_0;\\mathbf{y})}\\nonumber\\\\\n &\\qquad\\qquad \\times p_0(x_0,y_0)\\,p(\\mathbf{x},\\mathbf{y}|x_0,y_0)\\\\\n &= \\gamma,\\label{eqn_detachedIFTx}\n\\end{align}\nwhere, with Eq.~\\eqref{eqn_decompositionJointProbability},\n\\begin{align}\n \\gamma &\\coloneqq \\int d\\mathbf{x} d\\mathbf{y} dx_0 dy_0 \\, \\left[ p_T(x_T|y_T)\\,\\bar q_x(\\mathbf{\\bar{x}}|x_T;y_T,\\mathbf{\\bar{y}})\\right] \\nonumber\\\\\n&\\qquad\\qquad \\times \\left[ p_0(y_0)\\,q_y(\\mathbf{y}|y_0;x_0,\\mathbf{x}) \\right] \\label{eqn_defGamma}\n\\end{align}\nis a parameter related to feedback from $x$ to $y$. In general $\\gamma \\neq 1$. Only if there is no feedback from $x$ to $y$, $q_y(\\mathbf{y}|y_0;x_0,\\mathbf{x}) =  p_y(\\mathbf{y}|y_0)$, we find\n\\begin{align}\n \\gamma &= \\int d\\mathbf{y} dy_0 \\, p_0(y_0)\\,p_y(\\mathbf{y}|y_0) \\nonumber\\\\\n &\\qquad \\times \\int d\\mathbf{\\bar{x}} dx_T\\, p_T(x_T|y_T)\\,\\bar q_x(\\mathbf{\\bar{x}}|x_T;y_T,\\mathbf{\\bar{y}}) = 1 \\label{eqn_gammaEqual1}.\n\\end{align}\nIn general, therefore, the detached entropy productions do not fulfill IFTs of the usual type.\n\n\n\\subsection{Relations between entropy productions}\n\nThe entropy productions defined above are not independent of each other. With the help of the mutual information \\cite{Cover2006}\n\\begin{align}\n i(x,y)&\\coloneqq \\ln\\frac{p(x|y)}{p(x)} = \\ln\\frac{p(y|x)}{p(y)}\\label{eqn_defMutualInfo}\n\\end{align} \nwe may disentangle the initial distribution according to \n\\begin{equation}\n p_0(x_0,y_0)=p_0(x_0|y_0)\\,p_0(y_0|x_0)\\,e^{-i_0(x_0,y_0)}\n\\end{equation} \nand analogously for $p_T(x_T,y_T)$. We then find from Eqs. \\eqref{eqn_decompositionJointProbability}, \\eqref{eqn_decompositionReverseJointProbability}, \\eqref{eqn_defJointEntropy}, \\eqref{appentx}, and \\eqref{appenty}\n\\begin{equation}\\label{eqn_splitJointEntropy}\n \\sigma_{xy}=\\tilde\\sigma_x+\\tilde\\sigma_y+\\Delta i\n\\end{equation} \nwhere \n\\begin{equation}\\label{eqn_defDeltaI}\n \\Delta i =i_T(x_T,y_T)-i_0(x_0,y_0)\\; .\n\\end{equation} \n\nThe IFT~\\eqref{eqn_jointIFT} for the joint entropy production as well as the IFT~\\eqref{eqn_detachedIFTx} for the detached entropy production together with the relation~\\eqref{eqn_splitJointEntropy} between the entropy productions are the main general results of this study. We now specify the setup depicted in Fig.~\\ref{fig_setupJointProcess} to various particular situations and elucidate the relation between the IFTs \\eqref{eqn_jointIFT} and \\eqref{eqn_detachedIFTx} and results obtained in previous studies of these systems.\n\n\\section{Special Cases}\n\n\\subsection{Measurement-feedback systems}\nA measurement-feedback system consists of a system in contact with a thermal reservoir and a controller which measures the state of the system and uses that information to influence it. Most prominent examples of this type are Maxwell's demons and information engines~\\cite{Parrondo2015}.\n\nWe adapt our framework to this situation by describing the system by $x$ and the controller by $y$ (see Fig.~\\ref{fig_measFBSetup}). Since subsystem $x$ is coupled to a thermal reservoir at constant inverse temperature $\\beta$, the detached entropy production $\\tilde\\sigma_x$ of Eq.~\\eqref{appenty} has, due to detailed balance, a clear thermodynamic interpretation~\\cite{Seifert2005}:\n\\begin{align}\n \\tilde\\sigma_x &= \\ln\\frac{p_0(x_0|y_0)}{p_T(x_T|y_T)}\\nonumber\\\\\n&\\;\\;\\;+\\ln\\frac{p_x(x_1|x_0,y_1;u_1)\\,...\\,p_x(x_T|x_{T-1},y_T;u_T)}{p_x(x_{T-1}|x_T,y_T;\\bar u_T)\\,...\\,p_x(x_0|x_{1},y_1; \\bar u_1)}\\\\\n&= \\ln\\frac{p_0(x_0|y_0)}{p_T(x_T|y_T)}\\nonumber\\\\\n&\\;\\;\\;-\\beta\\left[ H_x(x_1;y_1,u_1) - H_x(x_0;y_1,u_1) + ... \\right.\\nonumber\\\\\n&\\qquad\\;\\;\\;\\; \\left. + H_x(x_T;y_T,u_T)-H_x(x_{T-1};y_T,u_T) \\right]\\\\\\label{eq:h1}\n&\\eqqcolon \\Delta s_{x|y} -\\beta\\, Q_x.\n\\end{align}\nHere $H_x(x;y,u)$ is the Hamiltonian governing the $x$-dynamics, $\\Delta s_{x|y}$ is the change in conditional system entropy, and $Q_x$ denotes the heat that the system exchanges with the reservoir.\n\nThe evolution of the $y$-subsystem is solely determined by the sequential measurements it takes of the state of the $x$-subsystem:\n\\begin{align}\n p_y(y_{t+1}|x_t,y_t;v_t) = p_y(y_{t+1}|x_t;v_t).\n\\end{align}\nGraphically, this is expressed by the absence of horizontal arrows in the $y$-trajectory in Fig.~\\ref{fig_measFBSetup}. Moreover, the noise in the controller is \\emph{measurement} noise and has no direct thermodynamic interpretation. The role of the protocol $v_t$ can be understood as influence on the measurement procedure, e.g. controlling its variance.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width = \\linewidth]{4.pdf}\n \\caption{Setup of a measurement-feedback system. Top: the system $x$ is under feedback control from the controller $y$. In every timestep, $y$ represents a measurement of the current state of $x$ which is then used to influence its dynamics. Bottom: time-reversed process.}\n \\label{fig_measFBSetup}\n\\end{figure}\n\nThe detached entropy production $\\tilde\\sigma_y$ of Eq.~\\eqref{appenty} then simplifies to:\n\\begin{align}\n \\tilde\\sigma_y &= \\ln\\frac{p_0(y_0|x_0)\\,\\prod\\limits_{t=1}^T p_y(y_t|x_{t-1};v_{t-1})}{p_T(y_T|x_T)\\,\\prod\\limits_{t=1}^T p_y(y_{t-1}|x_{t-1};v_{t-1})}\\\\\n&=\\ln\\frac{p_0(y_0|x_0)}{p_T(y_T|x_T)} + \\sum\\limits_{t=1}^T \\ln\\frac{p(y_t|x_{t-1})}{p(y_{t-1}|x_{t-1})}\\\\\n&= \\sum\\limits_{t=1}^T \\ln \\frac{p(y_t|x_{t-1})}{p(y_t|x_t)}\n\\end{align}\nwhich upon using Eq.~\\eqref{eqn_defMutualInfo} becomes:\n\\begin{align}\n \\tilde\\sigma_y &= -\\sum\\limits_{t=1}^T \\ln\\frac{p(y_t|x_t)}{p(y_t)} - \\ln\\frac{p(y_t|x_{t-1})}{p(y_t)}\\\\\n&= -\\sum\\limits_{t=1}^T i(x_{t},y_t) - i(x_{t-1},y_{t})\\\\\n&\\eqqcolon -i_x.\\label{eqn_detachedEntMeasFB}\n\\end{align}\nHere $i_x$ denotes the \\emph{information flow}~\\cite{Horowitz2014}, which is the change in mutual information between $x$ and  $y$ that is caused by the changes in $x$ only. Conversely, \n\\begin{equation}\\label{eqn_infoFlowy}\n i_y:=\\sum\\limits_{t=1}^T i(x_{t-1},y_{t}) - i(x_{t-1},y_{t-1})\n\\end{equation} \nis the part of the mutual information that is solely due to the $y$-dynamics. Clearly,\n\\begin{align}\n i_x + i_y = \\Delta i.\\label{eqn_relationInfoFlows}\n\\end{align}\nFrom Eqs.~\\eqref{eqn_splitJointEntropy}, \\eqref{eq:h1}, and \\eqref{eqn_detachedEntMeasFB}  the joint entropy production~\\eqref{eqn_defJointEntropy} reads:\n\\begin{align}\n \\sigma_{xy} &= \\Delta s_{x|y} -\\beta Q_x-i_x+\\Delta i\\\\\n&= \\Delta s_{x|y} -\\beta Q_x + i_y\\; .\n\\end{align}\n\nThe joint IFT~\\eqref{eqn_jointIFT} acquires the form\n\\begin{align}\n \\left\\langle e^{-\\Delta s_{x|y} + \\beta Q_x - i_y} \\right\\rangle = 1, \\label{eqn_jointIFTMeasFB}\n\\end{align}\nwhich is a special version of the \\emph{generalized Jarzynski equality}~\\cite{Sagawa2010,Ponmurugan2010,Horowitz2010,Sagawa2012} for measurement-feedback processes: the entropy production of the feedback-controlled system needs to be augmented by an information quantity to fulfill the fluctuation theorem. \n\nWithout this additional term the right-hand-side of the fluctuation relation for the entropy production $\\tilde\\sigma_x$ of the system alone deviates from unity, cf. Eq.~\\eqref{eqn_detachedIFTx}:\n\\begin{align}\n \\left\\langle e^{-\\Delta s_{x|y} - \\beta Q_x} \\right\\rangle = \\gamma. \n\\end{align}\nThe parameter $\\gamma$ defined in Eq.\\eqref{eqn_defGamma} transforms to:\n\\begin{align}\n  \\gamma &\\coloneqq \\iint d\\mathbf{x} d\\mathbf{y} dx_0 dy_0 \\, \\left[ p_T(x_T|y_T)\\,\\bar q_x(\\mathbf{\\bar{x}}|x_T;y_T,\\mathbf{\\bar{y}})\\right] \\nonumber\\\\\n&\\qquad\\qquad \\times \\left[ p_0(y_0)\\,\\prod\\limits_{t=1}^T p_y(y_{t}|x_{t-1}; v_{t-1}) \\right],\n\\end{align}\nand coincides with the ``efficacy parameter'' found by Sagawa and Ueda~\\cite{Sagawa2012}. In our case, the time-reversed measurements are similar to the forward measurements because of the assumption that all variables are even under time-reversal. \n\n\n\\subsection{Sensors}\nAnother setup involving the interplay between thermodynamics and information is sensing~\\cite{Parrondo2015}. Accordingly, it has already been studied extensively within the framework of stochastic thermodynamics~\\cite{Lan2012,Mehta2012,Still2012,Barato2013,Sartori2014,Barato2014,Ito2015,Bo2015,Hartich2016}. Here, a system is tasked with measuring an external signal. In the case of a biomolecular sensor, the external signal might be a chemical concentration, the pH, or osmotic pressure. The key ingredient is the fact that these signals are by themselves random and may be modeled by a stochastic process. We adapt our general setup to fit this situation by eliminating the feedback from $x$ to $y$, see Fig.~\\ref{fig_sensorSetup}. Consequently, the $x$-subsystem acts as the sensor that measures the external process $y$.\n\nNote that this is the biologically relevant interpretation of a \\emph{sensor} since any molecular reaction system retains at least some memory of its past and is, in that sense, not an optimal sensor.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width = 1\\linewidth]{5.pdf}\n \\caption{Setup of a sensor. Top: the sensor $x$ is influenced by the environment $y$ which is modeled as a stochastic process. Bottom: time-reversed process.}\n \\label{fig_sensorSetup}\n\\end{figure}\n\nAs in the previous example, the noise in the $x$-subsystem is of thermal nature and we have, cf. Eq.~\\eqref{eq:h1}\n\\begin{align}\\label{eqn_detachedEntSensor}\n \\tilde\\sigma_x = \\Delta s_{x|y} - \\beta Q_x.\n\\end{align}\n\nSince $y$ remains unaffected by $x$, the detached path probability $q_y$~[Eq.~\\eqref{eqn_yTrajectoryProb}] equals the marginal path probability $p_y$~[Eq.~\\eqref{defcondproby}]:\n\\begin{align}\n q_y(\\mathbf{y}|y_0;x_0,\\mathbf{x}) = p_y(\\mathbf{y}|y_0)\\label{eqn_sensorDetachedyEqualMarginaly}\n\\end{align}\nand similarly for the reverse process:\n\\begin{align}\n \\bar q_y(\\mathbf{\\bar{y}}|y_T;\\mathbf{\\bar{x}}) = \\bar p_y(\\mathbf{\\bar{y}}|y_T),\n\\end{align}\nwhere $p_y(\\mathbf{\\bar{y}}|y_T)$ is the reverse marginal probability of the $y$-process.\n\nThis implies that, apart from the initial and final states, the detached entropy production $\\tilde\\sigma_y$ does not depend on the specific trajectory traversed by $x$ and is therefore related to the usual entropy production $\\sigma_y$ of the $y$-process. Using Eqs.~\\eqref{eqn_defMutualInfo} and~\\eqref{eqn_defDeltaI} we find:\n\\begin{align}\n \\tilde\\sigma_y &= \\ln\\frac{p_0(y_0|x_0)\\,p_y(\\mathbf{y}|y_0)}{p_T(y_T|x_T)\\,\\bar p_y(\\mathbf{\\bar{y}}|y_T)}\\\\\n&= \\ln\\frac{p_0(y_0)\\,p_y(\\mathbf{y}|y_0)}{p_T(y_T)\\,\\bar p_y(\\mathbf{\\bar{y}}|y_T)} -\\Delta i\\\\\n&\\eqqcolon \\sigma_y -\\Delta i.\\label{eqn_sensorDetachedEPyEqualMarginaly}\n\\end{align}\n\nWith Eqs.~\\eqref{eqn_splitJointEntropy} and~\\eqref{eqn_detachedEntSensor} the joint IFT now reads\n\\begin{align}\n \\left\\langle e^{-\\Delta s_{x|y} + \\beta Q_x -\\sigma_y } \\right\\rangle = 1.\n\\end{align}\nJensen's inequality implies \n\\begin{align}\\label{eq:h3}\n \\left\\langle \\Delta s_{x|y} \\right\\rangle - \\beta \\left\\langle Q \\right\\rangle \\geq - \\left\\langle \\sigma_y \\right\\rangle,\n\\end{align}\nwhich means that the average dissipation of the sensor is bounded from below by the negative entropy production of the external process. This relation has been shown in the steady state and for entropy rates in~\\cite{Hartich2014}.\n\nSince there is no feedback from $x$ to $y$ in this setup, the right-hand-side of the detached IFT given in Eq.~\\eqref{eqn_detachedIFTx} equals 1, as we have shown in Eq.~\\eqref{eqn_gammaEqual1}:\n\\begin{align}\n\\left\\langle e^{-\\Delta s_{x|y}+ \\beta Q_x} \\right\\rangle = 1.\\label{eqn_detachedIFTSensor}\n\\end{align}\nFor the sensor the usual fluctuation theorem thus holds if the conditional system entropy is used in the definition of the entropy production. This measure of dissipation has been applied to a nonequilibrium sensor in~\\cite{Still2012}. If one wants to retain the usual definition of marginal system entropy change $\\Delta s_x$, Eq.~\\eqref{eqn_detachedIFTSensor} implicates a lower bound on the system's dissipation:\n\\begin{align}\n \\left\\langle \\Delta s_x \\right\\rangle - \\beta \\left\\langle Q \\right\\rangle \\geq \\left\\langle \\Delta i \\right\\rangle,\n\\end{align}\nwhere we have used: $s_{x|y} = s_x - i(x,y)$. The average total sensor dissipation is therefore bounded from below by the change in mutual information it managed to build up during the process. In the context of sensory adaptation, this bound has been obtained and further analyzed by Sartori \\emph{et al.}~\\cite{Sartori2014}. We showed here that it can be deduced from a specialization of the detached fluctuation theorem.\n\n\n\n\\subsection{Hidden Markov models}\nHidden Markov models~\\cite{Bishop2006} are a tool used, e.g., in machine learning to model sequential data coming from a Markov chain that is not directly accessible. Instead, one observes only measurements of its hidden states. Bechhoefer~\\cite{Bechhoefer2015} has used the formalism of hidden Markov models to clarify feedback schemes in stochastic thermodynamics. Phrased in our setup the hidden Markov model appears as a special case of a sensor that has no memory of its past, cf. Fig.~\\ref{fig_hmmSetup}.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width = \\linewidth]{6.pdf}\n \\caption{Setup of a hidden Markov model. Top: $y$ is a stochastic process that can only be observed through noisy measurements $x$. Bottom: time-reversed process.}\n \\label{fig_hmmSetup}\n\\end{figure}\n\nCompared to the sensor, the $y$-dynamics remain unmodified. However, since the $x$-dynamics consist of measurements, the detached entropy production $\\tilde\\sigma_x$ simplifies in much the same way as the $y$-entropy production in the setup of a measurement-feedback system:\n\\begin{align}\n p_x(x_t|x_{t-1},y_t;u_t) = p_x(x_{t}|y_t; u_t).\n\\end{align}\nWe obtain, with Eqs.~\\eqref{appentx}, \\eqref{eqn_defMutualInfo}, and~\\eqref{eqn_infoFlowy}\n\\begin{align}\n \\tilde\\sigma_x &= \\ln\\frac{p_0(x_0|y_0)}{p_T(x_T|y_T)}+ \\sum\\limits_{t=1}^T \\ln\\frac{p_x(x_t|y_t;u_t)}{p_x(x_{t-1}|y_t;u_t)}\\\\\n&= \\sum\\limits_{t=1}^T \\ln\\frac{p(x_{t-1}|y_{t-1})}{p(x_{t-1}|y_t)}\\\\\n&= \\sum\\limits_{t=1}^T i(x_{t-1},y_{t-1}) - i(x_{t-1},y_t)\\\\\n&= -i_y,\\label{eqn_HMMDetachedEPx}\n\\end{align}\nwhere, as before, $i_y$ is the information flow due to the $y$-dynamics.\n\nDue to the lack of feedback from $x$ to $y$, the apparent entropy production $\\tilde\\sigma_y$ equals the marginal entropy production minus the change in mutual information as in Eq.~\\eqref{eqn_sensorDetachedEPyEqualMarginaly}.\n\nThe joint entropy production of Eq.~\\eqref{eqn_splitJointEntropy} is therefore given by:\n\\begin{align}\n \\sigma_{xy} = -i_y + \\sigma_y.\n\\end{align}\n\nThus the joint IFT reads:\n\\begin{align}\n\\left\\langle e^{-\\sigma_y + i_y} \\right\\rangle = 1,\n\\end{align}\nwhich is similar to the joint IFT for measurement-feedback processes~[Eq.\\eqref{eqn_jointIFTMeasFB}]. In this case the roles of $x$ and $y$ are exchanged and the entropy production $\\sigma_y$ needs not necessarily have a thermodynamic interpretation. \n\nThe detached IFT~\\eqref{eqn_detachedIFTx} yields:\n\\begin{align}\\label{eqn_detachedIFTHMM}\n\\left\\langle e^{i_y} \\right\\rangle  = 1.\n\\end{align}\n\nThis is an integral fluctuation theorem involving an informational quantity characterizing the correlation between hidden ($y$) and observed ($x$) process. It allows us to directly assess the direction of the information flow in the process. It implies $\\langle i_y \\rangle \\leq 0$, meaning that on average information flows from the hidden process to the observations. This holds for arbitrary initial conditions. The larger the information flow, the more predictive the current observation is of the future hidden state.\n\nIn real-world applications, however, the computation of the information flow $i_y$ requires the knowledge of the observed sequence as well as the hidden sequence, which by definition is not accessible. To evaluate Eq.~\\eqref{eqn_detachedIFTHMM} when only the observed sequence is available, one first needs to average over all possible hidden sequences:\n\\begin{align}\n \\left\\langle e^{i_y} \\right\\rangle = \\left\\langle \\left\\langle e^{i_y} \\right\\rangle_{p(y_0,\\mathbf{y}|x_0,\\mathbf{x})} \\right\\rangle_{p_x(x_0,\\mathbf{x})} = 1. \\label{eqn_HMMModelVerification}\n\\end{align}\nIn this way the detached IFT can be used for model verification given a sufficiently large data set of $x$-trajectories as we demonstrate in the following. For this we provide the auxiliary entropy production $\\bar\\sigma_x[x_0,\\mathbf{x}]$ simply by redefining the quantity in the average in Eq.~\\eqref{eqn_HMMModelVerification}:\n\\begin{align}\n e^{-\\bar\\sigma_x[x_0,\\mathbf{x}]} &\\coloneqq \\left\\langle e^{i_y[x_0,y_0,\\mathbf{x},\\mathbf{y}]} \\right\\rangle_{p(y_0,\\mathbf{y}|x_0,\\mathbf{x})}\\\\\n&= \\iint dy_0 d\\mathbf{y} \\; p(y_0,\\mathbf{y}|x_0,\\mathbf{x}) \\, \\prod\\limits_{t=1}^{T} \\frac{p(x_{t-1}|y_{t})}{p(x_{t-1}|y_{t-1})},\\label{eqn_defAuxEnt}\n\\end{align}\nwhere $p(y_0,\\mathbf{y}|x_0,\\mathbf{x})$ is the \\textit{posterior distribution} of the entire sequence $\\{y_0,\\mathbf{y}\\}$ of hidden states given the entire sequence $\\{x_0,\\mathbf{x}\\}$ of observed states. The details of how this distribution can be calculated are given in the appendix.\n\nNow we can assign to each observed trajectory $\\{x_0,\\mathbf{x}\\}$ an auxiliary entropy production $\\bar\\sigma_x[x_0,\\mathbf{x}]$. From Eq.~\\eqref{eqn_HMMModelVerification} we infer that this entropy production fulfills an IFT:\n\\begin{align}\\label{eqn_IFTAuxEnt}\n \\left\\langle e^{-\\bar \\sigma_x[x_0,\\mathbf{x}]} \\right\\rangle_{p_x(x_0,\\mathbf{x})} = 1.\n\\end{align}\n\nThus, if we are given a set of trajectories and we want to infer whether the data have been generated from some specific model $\\{p_y(y_{t+1}|y_t), p_x(x_t|y_t), p_0(y_0)\\}$, we can calculate the auxiliary entropy production $\\bar \\sigma_x[x_0,\\mathbf{x}]$ for each trajectory and verify that the IFT in Eq.~\\eqref{eqn_IFTAuxEnt} holds. If it does not, the model is not correct.\n\n\\subsubsection*{Example}\n\nTo illustrate this procedure, we consider a hidden Markov model with two hidden and two observed states each labeled with 0 and 1. The transition probabilities for the hidden Markov chain are given by the transition matrix $\\tilde{\\mathbb{T}}_y$:\n\\begin{align}\n\\tilde{\\mathbb{T}}_y &\\coloneqq \\begin{pmatrix}\np_y(y_{t+1} = 0|y_t = 0) & p_y(y_{t+1} = 0|y_t = 1) \\\\\np_y(y_{t+1} = 1|y_t = 0) & p_y(y_{t+1} = 1|y_t = 1)\n\\end{pmatrix}\\\\ \n&= \\begin{pmatrix}\n1-a & b \\\\\na & 1-b\n\\end{pmatrix}.\n\\end{align}\n\nThe observed sequence is uniquely determined by the hidden sequence. For each time step the following holds:\n\\begin{align}\n p_x(x_t = y_t|y_t) = 1-\\epsilon\\qquad p_x(x_t = 1 - y_t|y_t) = \\epsilon.\n\\end{align}\nThus each measurement is wrong with probability $\\epsilon$. The transition probabilities for the observed chain is given by the transition matrix $\\tilde T_x$:\n\\begin{align}\n \\tilde{\\mathbb{T}}_x &\\coloneqq \\begin{pmatrix}\np_x(x_{t} = 0|y_t = 0) & p_x(x_{t} = 0|y_t = 1) \\\\\np_x(y_{t} = 1|y_t = 0) & p_x(x_{t} = 1|y_t = 1)\n\\end{pmatrix} \\\\\n&= \\begin{pmatrix}\n1-\\epsilon & \\epsilon \\\\\n\\epsilon & 1-\\epsilon\n\\end{pmatrix}.\n\\end{align}\n\nThe full bipartite process is composed of the four states $(x,y) = (0,0), (0,1), (1,0)$, and $(1,1)$ in that order. It is described by the transition matrices $\\mathbb{T}_y$ and $\\mathbb{T}_x$ for the $y$- and $x$-step separately:\n\\begin{align}\n\\mathbb{T}_y &\\coloneqq \\mathbb{P}(x_t,y_{t+1}|x_t,y_t) = \\begin{pmatrix}\n1-a & b & 0 & 0 \\\\\na & 1-b & 0 &0 \\\\\n0 & 0 & 1-a & b \\\\\n0 & 0 & a & 1-b\n\\end{pmatrix}\\\\\n\\mathbb{T}_x &\\coloneqq \\mathbb{P}(x_{t+1},y_{t+1}|x_t,y_{t+1})\\nonumber\\\\ \n&= \\begin{pmatrix}\n1-\\epsilon & 0 & 1-\\epsilon & 0 \\\\\n0 & \\epsilon & 0 & \\epsilon \\\\\n\\epsilon & 0 & \\epsilon & 0 \\\\\n0 & 1-\\epsilon & 0 & 1-\\epsilon\n\\end{pmatrix}\n\\end{align}\n\nWith the initial condition $p_0(y_0) \\eqqcolon \\begin{pmatrix}\n  p_y^0\\\\\n  1-p_y^0\n\\end{pmatrix} $, it follows:\n\\begin{align}\np_0(x_0,y_0) &= \\begin{pmatrix}\n1-\\epsilon & 0 \\\\\n0 & \\epsilon  \\\\\n\\epsilon & 0  \\\\\n0 & 1-\\epsilon\n\\end{pmatrix}\\, p_0(y_0)\\\\\n&= \\begin{pmatrix}\n    (1-\\epsilon)\\,p_y^0\\\\\n    \\epsilon\\, (1-p_y^0)\\\\\n    \\epsilon\\,p_y^0\\\\\n    (1-\\epsilon)\\, (1-p_y^0)\n   \\end{pmatrix}.\n\\end{align}\n\nFrom this we can calculate all joint probabilities:\n\\begin{align}\n p(x_t,y_t) = \\left( \\mathbb{T}_x \\mathbb{T}_y \\right)^t \\, p_0(x_0,y_0).\n\\end{align}\n\nTo evaluate the information flow [Eq.~\\eqref{eqn_HMMDetachedEPx}], we need the joint probabilities $p(x_{t-1},y_t)$. These can be obtained from $p(x_{t-1},y_{t-1})$:\n\\begin{align}\n p(x_{t-1},y_t) = \\mathbb{T}_y\\,p(x_{t-1},y_{t-1}).\n\\end{align}\n\nWe demonstrate our formalism by generating $N$ trajectories of length $T$ with model parameters $a$, $b$, $p_0$ and $\\epsilon$. We then calculate the auxiliary entropy production using Eq.~\\eqref{eqn_defAuxEnt} for each trajectory and plot the left-hand-side of Eq.~\\eqref{eqn_IFTAuxEnt} when evaluated with some other parameters $a'$, $b'$, $p_0'$ and $\\epsilon'$. Figure.~\\ref{fig_HMMConv} shows the result for a specific set of parameters.\n\n\\begin{figure}[ht]\n \\centering\n\\includegraphics[width = \\linewidth]{7}\n \\caption{Convergence of the IFT for the auxiliary entropy production for our specific hidden Markov model. The trajectories were generated using the following parameters: $T = 5$, $a= 0.3$, $b= 0.2$, $p_0= 0.1$, and $\\epsilon = 0.2$. We analyzed the IFT using the generating parameters (black line\/ third from the top) and three sets of parameters in which one model parameter differs from the generating parameter (other lines). In the interest of clarity, error bars are only shown for larger sample sizes.}\n \\label{fig_HMMConv}\n\\end{figure}\n\nOne recognizes that only the correct set of parameters ensures convergence to one. This means that the IFT for the auxiliary entropy production can indeed be used to confirm the model parameters. \n\nIn this specific hidden Markov model the IFT is more sensitive to the parameter $\\epsilon$ governing the dynamics of the observed sequence than to the parameters $a$ and $b$. \n\n\n\\section{Discussion}\\label{sec:discussion}\nWe have shown that within the field of stochastic thermodynamics measurement-feedback systems, sensors and hidden Markov models are related because they appear as special cases of a joint bipartite Markov chain.\n\nThis general joint bipartite Markov chain models a system with two interacting degrees of freedom. For such a case the influence of both subsystems as well as their interaction on the total entropy balance have to be considered. However, one can define other entropy productions like the detached entropy production which can be assigned a physical meaning.\n\nFrom the study of the special cases we may deduce the following: whenever a subsystem is coupled to a thermal reservoir and the other subsystem influences its internal energy, its stochasticity is of thermal nature. Consequently, the detached entropy production equals the usual sum of system entropy change plus contribution of exchanged heat. On the other hand, when the influence is purely \\emph{informational}, meaning it is due to a measurement, the detached entropy production takes the form of an information flow. This also holds true when the measurement noise is of thermal nature because it is not included in the energy budget of the joint system.\n\nWe point out that the reverse process from which we derived the entropy production has a special structure because the subsystem updates swap order under time inversion. Generally, this fact is not important when the fluctuation theorems only involve forward quantities. It nonetheless plays a role if the parameter $\\gamma$ in the detached IFT in Eq.~\\eqref{eqn_detachedIFTx} differs from 0, since in that case $\\gamma$ depends explicitly on the reverse detached probability.\n\nThe dissection of the joint entropy production into detached entropy productions is similar to the partitioning of the joint entropy production rates into two sub-entropy productions with information flow~\\cite{Horowitz2014}. Our formalism reveals the path probabilities (and reverse processes) from which one can define such sub-entropy productions.\n\nAdditionally, there is another way to decompose the joint entropy production which is close in spirit. Instead of detached entropy productions one may use \\emph{marginal} entropy productions which are defined solely based on the marginal path probabilities. For example for $x$ one obtains:\n\\begin{align}\n \\sigma_x \\coloneqq \\ln \\frac{p_0(x_0)\\,p_x(\\mathbf{x}|x_0)}{p_T(x_T)\\,\\bar p_x(\\mathbf{\\bar{x}}|x_T)}.\n\\end{align}\nThe consequences of this separation have been studied by Crooks and Still~\\cite{Crooks2016}. One finds that contrary to detached entropy production plus information flow, one obtains marginal entropy production plus transfer entropy. Both approaches seem equally valid. A comparison of the second-law inequalities these and other information measures provide has been made by Horowitz and Sandberg~\\cite{Horowitz2014a}.\n\nMore than that, the decomposition of the joint process into two sub-processes is similar to the approach of causal conditioning within the information theory community. It is used to define directed information~\\cite{Marko1973, Massey1990, Jiao2013} which is closely related to transfer entropy and has proven useful in the study of information thermodynamics~\\cite{Ito2013, Hirono2015, Vinkler2016, Tanaka2017}.\n\nThe application of fluctuation theorems to validate the underlying model for data stemming from a stochastic process is a promising approach. It has already proven successful when estimating drift and diffusion coefficients in the Markov analysis of turbulent flows~\\cite{Nickelsen2013, Reinke2016}.\n\nOur findings point in two directions for future research. Firstly, it seems valuable to follow up on the study of how fluctuation theorems can be put to use to infer model parameters in, e.g., hidden Markov Models. It might be possible to use the fluctuation relation as a cost function in parameter learning or to infer the number of hidden parameters best describing the observed data.\n\nSecondly, the question of how to appropriately describe systems which are strongly coupled to a thermal environment has recently gained attention~\\cite{Seifert2016,Jarzynski2017,Miller2017}. A key issue is the partitioning into system and environment. Due to the strong coupling, perturbations of the environment due to the system may feed back into the future evolution of the system, thus violating the Markov assumption. We propose the detached path probabilities as a method to circumvent this problem and demonstrated that meaningful entropy measures can be derived from these probabilities.\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper we used the detached path probabilities from which we defined the detached entropy production. This is a quantity which helps to analyze the stochastic thermodynamics of systems with interacting degrees of freedom.\n\nWe showed that one can understand measurement-feedback systems, sensors and hidden Markov models as special cases of one joint bipartite Markov chain. When applied to the special cases, the fluctuation relations involving the detached entropy production recover useful relations which have been found separately before.\n\nEspecially for hidden Markov models such a fluctuation relation can be used to confirm that model parameters have been learned correctly.\n \n\\begin{acknowledgments}\nWe thank Marcel Kahlen for a critical reading of the manuscript.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nLenticular, or S0, galaxies make up some 25\\% of large galaxies in the\nlocal Universe (Dressler 1980), so understanding how\nthey form must constitute a significant element of any explanation of\ngalaxy evolution.  Their location at the crossroads between\nellipticals and spirals in Hubble's tuning-fork diagram underlines\ntheir importance in attempts to develop a unified understanding of\ngalaxy evolution, but also means that it is not even clear to which of\nthese classes of galaxy they are more closely related.\n\nOne often-cited piece of evidence comes from the fact that the\nproportion of S0s is substantially smaller in distant ($z\\sim0.5$) clusters\nthan in nearby ones, while spirals show the opposite trend (Dressler\net al.\\ 1997), strongly suggesting a transformation from\none to the other.  However, even if this scenario is accepted, it does\nnot answer the question as to whether S0s are more closely related to\nspirals or ellipticals, which is intimately connected to the mechanism\nof transformation.  If the transformation simply involves a spiral\ngalaxy losing its gas content through ram pressure stripping (Gunn \\&\nGott 1972) or ``strangulation'' (Larson et al.\\ 1980), \nso ceasing star formation and fading into an S0, then\nclearly S0s and spirals are closely related.  However, it is also\npossible that mergers can cause such a transformation: while\nequal-mass mergers between spirals create elliptical galaxies, more\nminor mergers can heat the original disk of a spiral and trigger a\nbrief burst of star formation, using up the residual gas and leaving\nan S0.  In such a merger scenario, the mechanism for creating an S0 is\nmuch more closely related to that for the formation of ellipticals.\n\nClues to which mechanism is responsible are to be found in the\n``archaeological record'' that can be extracted from spectral\nobservations of nearby S0s.  In particular, the present-day stellar\ndynamics should reflect the system's origins, with the gentle gas\nstripping of a spiral resulting in stellar dynamics very similar to\nthe progenitor spiral, while the merger process will heat the stars,\nresulting in kinematics more dominated by random motions, akin to an\nelliptical.  In addition, the absorption line strengths can be\ninterpreted through stellar population synthesis to learn about the\nmetallicity and star formation histories of these systems.  Even more\ninterestingly, these dynamical and stellar properties can be compared\nto see if a consistent picture can be constructed for the formation of\neach system. I present here some recent \nevidence suggesting that such a consistent picture is indeed emerging. \n\n\n\\section{Evidence from the Tully-Fisher relation}\n\nCombining published data with high-quality VLT\/FORS spectroscopy of sample\nof Fornax S0s (Bedregal et al.\\ 2006a)  we have carried out a combined\nstudy of the Tully-Fisher relation and the stellar populations of these\ngalaxies.  Despite the relatively small sample and the considerable\ntechnical challenges involved in determining the true rotation velocity $V_{\\rm\nrot}$ from absorption line spectra of galaxies with significant non-rotational\nsupport (see Mathieu et al.\\ 2002), some very interesting results arise.\nS0s lie systematically below the spiral galaxy Tully-Fisher relation in both\nthe optical and near-infrared (Figure~1). If S0s are the descendants of spiral\ngalaxies, this offset can be naturally interpreted as arising from the\nluminosity evolution of spiral galaxies that have faded since ceasing star\nformation. Moreover, the amount of fading implied by the offset of individual\nS0s from the spiral relation seems to correlate with the luminosity-weighted\nage of their stellar population, particularly at their centres (Figure~2). \nThis correlation suggests a scenario in which the star formation clock stopped\nwhen gas was stripped out from a spiral galaxy and it began to fade into an S0.\nThe stronger correlation at small radii indicates a final last-gasp burst of\nstar formation in this region. See Bedregal, Arag\\'on-Salamanca  \\& Merrifield\n(2006b) for details. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=3.3in,width=4.0in,angle=0]{Aragon-Salamanca_fig1.ps}\n\\end{center}\n\\caption{$B$-band Tully-Fisher relation (TFR) for \n  S0 galaxies using\n  different samples from the literature (open symbols) and our VLT Fornax   \n  data (filled circles). \n  The solid and dashed lines show two independent determinations of \n  the TFR relation for local spirals. On average (dotted line),\n  S0s are $\\sim3$ times fainter\n  than spirals at similar rotation velocities \n  (Bedregal, Arag\\'on-Salamanca \\& Merrifield 2006b). \n       }\n\\label{fig:fig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=1.65in,angle=0]{Aragon-Salamanca_fig2.eps}\n\\end{center}\n\\caption{\n  For our VLT Fornax data we plot the\n  shift in magnitudes from the\n  $B$-band spiral TFR versus the stellar population age at the galaxy\n  centre (left panel), at $1\\,R_e$ (middle panel) and at $2\\,R_e$ (right\n  panel). The lines show models for fading spirals. \n  Note that the correlation is strongest for the central stellar \n  populations of the galaxies, suggesting that the last episode of star\n  formation took place there (Bedregal, Arag\\'on-Salamanca \\& Merrifield 2006b).  \n    }\n\\label{fig:fig2}\n\\end{figure}\n\n\n\n\n\\section{Evidence from the globular cluster populations}\n\nEntirely consistent and independent evidence comes from our recent\nstudies of the properties of the globular cluster (GC) systems and stellar\npopulations of SOs (Arag\\'n-Salamanca, Bedregal \\& Merrifield 2006; Barr et\nal.\\ 2007).  If interactions with the intra-cluster medium are responsible  for\nthe transformation of spirals into S0s, the number of globular clusters in\nthese galaxies  will not be affected. That is probably not true if more violent\nmechanisms such as galaxy-galaxy interactions are the culprit (see, e.g.,\nAshman \\& Zepf 1998). If we assume that the number of globular clusters remains\nconstant, the GC specific frequency  ($S_N\\propto\\,$number of GCs per unit\n$V$-band Luminosity) would increase due to the fading of the galaxy. On\naverage, the GC specific frequency is a factor $\\sim 3$ larger for S0s than it\nis for spirals (Arag\\'on-Salamanca et al. 2006),  meaning that\nin the process S0s become, on average,  $\\sim 3$ times fainter than their\nparent spiral.  Furthermore, in this scenario the amount of fading (or increase\nin GC specific frequency) should grow with the time elapsed  since the star\nformation ceased, i.e., with the luminosity-weighted age of the S0 stellar\npopulation.  Figure~3 shows that this is indeed the case, adding considerable\nweight to the conclusions reached from our Tully-Fisher studies. \n\n\n\n\n\\section{Additional evidence from the stellar populations and dynamics} \n\nIn Bedregal et al.\\ (2007) we show that the central  absorption-line indices in\nS0 galaxies correlate well with the central velocity dispersions in accordance\nwith what previous studies found for elliptical galaxies.  However, when these\nline indices are converted into stellar population properties, we find that the\nobserved correlations seem to be driven by systematic age and alpha-element\nabundance variations, and not changes in overall metallicity as is usually\nassumed for ellipticals. These correlations become even tighter when the\nmaximum circular velocity is used instead of the central velocity dispersion.  \nThis improvement in correlations is interesting because the maximum rotation\nvelocity is a better proxy for the S0's dynamical mass than its central\nvelocity dispersion. Finally,  the  $\\alpha$-element over-abundance seems to be\ncorrelated with dynamical mass, while the absorption-line-derived ages also\ncorrelate with these over-abundances. These correlations imply that the most\nmassive S0s have the shortest star-formation timescales and the oldest stellar\npopulations,  suggesting that mass plays a large role in dictating the life\nhistories of S0s.\n\n\n\n\n\n\n\n\\section{Conclusions}\n\n\nThe stellar populations, dynamics and globular clusters of S0s  provide\nevidence consistent with these galaxies being the descendants of fading spirals\nwhose star formation ceased.  However, caution is needed since significant\nproblems could still exist with this picture (see, e.g., Christlein \\&\nZabludoff 2004; Boselli \\& Gavazzi 2006). Moreover, the  number of \ngalaxies studied\nhere is still small, and  it would be highly desirable to extend this kind of\nstudies to much larger samples covering a broad range of galaxy masses and\nenvironments. \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=2.9in,angle=0]{Aragon-Salamanca_fig3.eps}\n\\end{center}\n\\caption{\nLog$_{10}$ of the luminosity-weighted ages is Gyr  \n  vs.\\ the globular cluster specific frequency\n  ($S_N$) of S0s. The line shows the evolution \n  expected for a\n  fading galaxy according to the stellar population models of \n  Bruzual \\& Charlot\n  (2003). The correlation between the fading of the galaxies \n  (or increase in $S_N$) and the spectroscopically-determined\n  age of their stellar populations is clearly consistent with the predictions of\n  a simple fading model.   \n  Note that the $S_N$ value for NGC3115B\n  is very unreliable and almost certainly \n  severely overestimated due\n  to contamination from the GC systems of neighbouring galaxies. \n  See Barr et al.\\ (2007) for details.    \n    }\n\\label{fig:fig3}\n\\end{figure}\n\n\\begin{acknowledgments}\n\nI thank A.G.\\ Bedregal,  M.\\ Merrifield, J.M.\\ Barr, B.\\ Milvang-Jensen,  S.P.\\\nBamford and N. Cardiel for allowing me to discuss here results obtained with\ntheir help.  \n\n\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSuper-resolution fluorescent microscopy has transformed many domains of biology.\nTo date there are two far-field classes of techniques that lead to fluorescence-based microscopy with a resolution far beyond the Rayleigh diffraction limit \\cite{ref1}. The first class  is generically referred to as super-resolved ensemble fluorophore microscopy and the second as super-resolved single fluorophore microscopy. \n\nThe first class of techniques can be implemented either by stimulated emission depletion (STED) of fluorescence from all molecules in a sample except those in a small region of the imaged biological sample or by structured illumination microscopy (SIM). STED builds on the deterministic transitions that either switch fluorescence on or off to reduce the emission volume \\cite{ref4,ref5,ref6,STED}. In SIM, interference patterns used in sample illumination lead to a twofold\ngain in resolution \\cite{ref2,ref3,SIM}. \n\n\nThe second class of techniques is based on the a prior knowledge that the measurements at a given time are from single fluorescent molecules that are separated from each other by distances larger than the Rayleigh diffraction limit. This information is used to super-localize single molecules in an image, which means finding the position of each molecule to\na precision better than the Rayleigh diffraction limit. Super-resolved single fluorophore microscopy relies on the stochastic switching of fluorophores in a time sequence to localize single molecules.  It can be implemented either by photo-activated localization microscopy (PALM) \\cite{ref7,PALM} or by stochastic optical reconstruction microscopy (STORM) \\cite{ref8,ref9}. \n\n\nA major disadvantage of these two classes of techniques is that they  suffer from the trade-off between the spatial and temporal resolutions, which makes live cell imaging quite challenging. On one hand, super-resolved single fluorophore microscopy techniques require hundreds of thousands of exposures. This is because in every frame, the diffraction-limited image of each\nemitter must be well separated from its neighbours, to enable the identification of its exact\nposition. This inevitably leads to a long acquisition cycle, typically on the order of several minutes. Consequently, fast dynamics cannot be captured by these techniques. On the other hand, SIM techniques require only tens of frames (thus they are with high temporal resolution). But, their spatial resolution enhancement is limited by a factor of two. \n\nMost previous works on enhancing the temporal resolution focused on improving the localization accuracy in PALM\/STORM. Some of them (such as CS-STORM \\cite{CS-STORM} and SPARCOM\\cite{ref11}) used compressive sensing (CS) recovery algorithms to reduce the number of measurements, but any PALM\/STORM-based techniques inevitably suffers from the trade-off challenge. The trade-off originates from the fact that stochastic single molecule switching activates only a small part of the solution in each frame. \n\nTo better explain this, let us describe the mathematical problem for PALM\/STORM. We activate the fluorescent solution by stochastic switching with $T$ number of frames. Denote by $\\rho_1,\\rho_2,\\cdots,\\rho_T$ the sparse distribution of the activated fluorescent molecules (point scatterers). Then we collect the corresponding measured images $Y_1,Y_2,\\cdots,Y_T$. Hence we have\n$$\nS \\rho_t :=h\\circledast \\rho_t = Y_t, \\quad t=1,2,\\cdots,T,\n$$\nwhere $h$ is a blurring kernel and $\\circledast$ is the convolution product. \nIn CS-STORM, we apply compressive sensing to the following deconvolution problem for reconstructing the unknown $\\rho_t$:\n\\begin{equation} \\label{l10}\n \\min_{\\rho_t} \\| \\rho_t \\|_1 \\quad \\mbox{subject to}\\quad \\rho_t \\geq 0 \\mbox{ and } \\| S \\rho_t - Y_t\\|_2\\leq \\sigma,\n\\end{equation}\nwhere $\\sigma$ is the noise level. \nThen the super-resolved image can be obtained by\n$$\n \\rho = \\sum_{t=1}^T \\rho_t.\n$$\nWhen the density of activated molecules in each single frame is small, then in average the  point scatterers are well separated. Then it is easy to localize them by deconvolution procedure. But the lower the density of molecules, the higher the number of frames $T$. This is the spatio-temporal resolution trade-off of PALM\/STORM-based approaches. \n\n\nIn \\cite{beam2022}, a novel imaging modality called  Brownian Excitation Amplitude Modulation microscopy (BEAM) is introduced, which is based on speckle imaging and compressive sensing.  On one hand, it reduces significantly the number of exposures by exposing the most part of the solution at each frame to the illumination pattern.  On the other hand, it  involves multiple incoherent illuminations of the biological sample and achieves super-resolution\nmicroscopy across both space and time from a sequence of diffraction-limited images and can capture fast dynamics of biological samples. Hence, BEAM outperforms the PALM\/STORM-based techniques. Their  two key ingredients are spatial sparsity and temporal incoherence.  BEAM combines the sparsity of the point scatterers and  the incoherence between the illumination patterns in different frames. \n\n\nThere are some related works to BEAM. The Blind-SIM\\cite{blindSIM} and RIM\\cite{RIM} use random speckle modulations but compressive sensing was not exploited there and so the spatial resolution enhancement is limited by a factor of two (they also require a large number of measurements). The Joint Sparse Recovery approach in \\cite{JSR-Ye} uses both random speckles and compressive sensing. But their inverse problem is formulated in MMV (multiple measurement vectors) form whose sensing matrix has no incoherence, which is not optimal for CS, and hence requires a large number of measurements.   \n\nLet us now briefly describe the inverse problem in BEAM. \nSuppose we have multiple speckle patterns $I_1,I_2,\\cdots,I_T$ illuminating the sparse fluorescent solution  and then collect the corresponding measured images $Y_1,Y_2,\\cdots,Y_T$. Then we have\n$$\nA_t \\rho :=h\\circledast (I_t \\rho) = Y_t, \\quad t=1,2,\\cdots,T,\n$$\nwhere, as before, $h$ is a blurring kernel. We apply compressive sensing to reconstruct the unknown $\\rho$ with estimated speckle patterns $I_t$ \\cite{candes2014towards, denoyelle2017support, duval2015exact, morgenshtern2016super, morgenshtern2020super}:\n\\begin{equation} \\label{l1}\n \\min_{\\rho} \\| \\rho \\|_1 \\quad \\mbox{subject to}\\quad \\rho \\geq 0 \\mbox{ and } \\| A \\rho - Y\\|_2\\leq \\sigma,\n\\end{equation}\nwhere $\\sigma$ is the noise level, $A =(A_t)_{t=1,\\ldots,T}$ is the sensing matrix, and $Y=(Y_1,\\ldots,Y_T)^\\top$ (with $\\top$ denoting the transpose). \nNotice that the columns of $A$ have a high degree of incoherence coming from the Brownian motion of the speckle patterns $I_t$. This incoherence in the sensing matrix is an optimal feature for compressive sensing to work properly. The sparsity prior in BEAM enhances the spatial resolution (beyond SIM's two-fold enhancement), and at the same time, the required number of measurements  stays small since our sensing matrix satisfies CS requirement (incoherence). \nTo the best of our knowledge, BEAM is the first compressive imaging approach satisfying the incoherence requirement, which is the key to overcome the trade-off barrier between the spatial and temporal resolutions.\n\nBEAM can be then seen as the first experimental realization of spatio-temporal sparsity-based super-resolved imaging, where threefold resolution enhancement can be achieved by applying compressive sensing over only  few frames. \nMotivated by BEAM, our aim in this paper is to pioneer the mathematical foundation of spatio-temporal sparsity-based super-resolution. We consider mathematical models similar to (\\ref{l1}) \nbut tackle instead the sparsest solution ($l_0$ pseudo-norm minimizer) under the measurement constraints. The sparsest solution is usually the one targeted in sparsity-based imaging and also in the general compressive sensing theory (using tractable convex $l_1$-minimization). Moreover, we  consider that the values of the illumination patterns may not be known. Our main results (Theorems \\ref{thm:l0normrecovery0} and \\ref{thm:twodl0normrecovery0}) consist in deriving lower bounds for the resolution enhancement in both the one- and two-dimensional cases. More precisely, we estimate the minimal separation distance for stable recovery of point scatterers from multi-illumination incoherent data. Our estimations reveal the dependence of the resolution enhancement on the cut-off frequency of the imaging system, the signal-to-noise ratio, the sparsity of the point scatterers, and more importantly on the incoherence of the illumination patterns. Our theory highlights the importance of incoherence in the illumination patterns and theoretically demonstrates the possibility of achieving super-resolution for sparsity-based multi-illumination imaging using very few frames. \n\n\nIt is worth emphasizing that there are many mathematical theories for estimating the stability of super-resolution in the single measurement case. To our knowledge, the first work  was by Donoho \\cite{donoho1992superresolution}. He considered a grid setting where a discrete measure is supported on a lattice (spacing by $\\Delta$) and regularized by a so-called  \"Rayleigh index\" $d$. He demonstrated that the minimax error for the  recovery of the strength of the scatterer is bounded by $SRF^{\\alpha}\\sigma$ ($2d-1\\leq \\alpha \\leq 2d+1$) with $\\sigma$ being the noise level and the super-resolution factor $SRF = 1\/({\\Omega \\Delta})$. Here, $\\Omega$ is the cut-off frequency. Donoho's results emphasized the importance of sparsity (encoded in the Rayleigh index) in the super-resolution problem. In \\cite{demanet2015recoverability}, the authors considered $n$-sparse scatterers supported on a grid and obtained sharper bounds ($\\alpha=2n-1$) using an estimate of the minimum singular value for the measurement matrix. The case of multi-clumps was considered in \\cite{li2021stable, batenkov2020conditioning} and similar minimax error estimations were derived. See also other related works for the understanding of resolution limit from the perceptive of sample complexity \\cite{moitra2015super,chen2020algorithmic}. In \\cite{akinshin2015accuracy, batenkov2019super}, the authors considered the minimax error for recovering off-the-grid point scatterers. Based on an analysis of the \"prony-type system\", they derived bounds for both strength and location reconstructions of the point scatterers. More precisely, they showed that  for $\\sigma \\lessapprox (SRF)^{-2p+1}$ where $p$ is the number of point scatterers in a cluster, the minimax error for the strength and the location recoveries scale respectively as $(SRF)^{2p-1}\\sigma$, $(SRF)^{2p-2} {\\sigma}\/{\\Omega}$. Moreover, for the isolated non-cluster point scatterer, the corresponding minimax error for the strength and the location recoveries scale respectively as $\\sigma$ and ${\\sigma}\/{\\Omega}$.\n\nDue to the popularity of sparse modeling and compressive sensing, many sparsity-promoting algorithms were proposed to address the super-resolution problem. In the groundbreaking work of Cand\\`es and Fernandez-Granda \\cite{candes2014towards}, it was demonstrated that off-the-grid sources can be exactly recovered from their low-pass Fourier coefficients by total variation minimization under a minimum separation condition. Other sparsity promoting methods include the BLASSO algorithm \\cite{azais2015spike, duval2015exact, poon2019} and the atomic norm minimization method \\cite{tang2013compressed, tang2014near}. These two algorithms were proved to be able to stably recover the sources under a minimum separation condition or a non-degeneracy condition. The resolution of these convex algorithms are limited by a distance of the order of the Rayleigh diffraction limit \\cite{tang2015resolution, da2020stable} for recovering general signed point scatterers. But for the case of positive sources \\cite{morgenshtern2016super, morgenshtern2020super, denoyelle2017support}, there is no such limitation on the resolution and the performance of these algorithms could be nearly optimal. \n\nMore recently, to analyze the resolution for recovering multiple point scatterers, in \\cite{liu2021mathematicaloned, liu2021mathematicalhighd, liu2021theorylse} the authors defined \"computational resolution limits\" which characterize the minimum required distance between point scatterers so that their number and locations can be stably resolved under certain noise level. By developing a non-linear approximation theory in a so-called Vandermonde space, they derived bounds for computational resolution limits for a deconvolution problem \\cite{liu2021mathematicaloned} and a line spectral problem \\cite{liu2021theorylse} (equivalent to the super-resolution problem considered here). In particular, they showed in \\cite{liu2021theorylse} that the computational resolution limit for number and location recovery should be respectively $\\frac{C_{\\mathrm{num}}}{\\Omega}(\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{2n-2}}$ and  $\\frac{C_{\\mathrm{supp}}}{\\Omega}(\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{2n-1}}$ where $C_{\\mathrm{num}}, C_{\\mathrm{supp}}$ are constants and $m_{\\min}$ is the minimum strength of the point scatterers. Their results demonstrate that when the point scatterers are separated larger than $\\frac{C_{\\mathrm{supp}}}{\\Omega}(\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{2n-1}}$, we can stably recover the scatterer locations. Conversely, when the point scatterers are separated by a distance less than $O(\\frac{C_{\\mathrm{supp}}}{\\Omega}(\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{2n-1}})$, stably recovering the scatterer locations is impossible in the worst case. This resolution limit indicates that super-resolution is possible for the single measurement case but requires very high signal-to-noise ratio (according to the exponent $\\frac{1}{2n-1}$). This explains why it is so hard to achieve super-resolution by single illumination. Therefore, we have to resort to multiple illuminations in order to super-resolve point scatterers. \n\nAs we have seen, the mathematics behind resolution limit for single illumination imaging is towards to be fully understood. Nevertheless, the multiple illumination case  still lacks or even is without any mathematical foundation. Thus, our paper serves as a first step towards understanding the resolution limit (or performance) of multi-illumination imaging. We consider both the one- and two-dimensional cases. Our results demonstrate that the resolution for the multiple illumination imaging problem in the one-dimensional case is less than \n\\[\n\\frac{2.2e\\pi }{\\Omega }\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}, \n\\]\nwhere $\\sigma_{\\infty, \\min}(I)$ is defined by (\\ref{sigmadef}). In two dimensions, the resolution limit is multiplied by $(n +1) (n+2)$ when the $n$ point scatterers are assumed to be in a disk of radius $n \\pi\/\\Omega$. \n\nOur paper is organized in the following way.\nSection \\ref{sect2} formulates the minimization problem for recovering point scatterers from multi-illumination data. Sections \\ref{sect3} and \\ref{section:twodcase} present the main results on the spatio-temporal super-resolution in respectively the one- and two-dimensional case and a detailed discussion on their significance. \nSection \\ref{section:approxinvandermonde}  introduces the main technique (namely the approximation theory in Vandermonde space) that is used to show the main results of this paper. \nIn Section \\ref{section:proofofthml0normrecover}, Theorem \\ref{thm:l0normrecovery0}\nis proved. Section \\ref{section:prooftwodl0normrecovery} is devoted to the proof of Theorem \\ref{thm:twodl0normrecovery0}.\nFinally, the appendix provides some lemmas and inequalities\nthat are used in the paper.\n\n\n\\section{Resolution in the one-dimensional case} \\label{sect3}\n\n\\subsection{Problem setting} \\label{sect2}\nLet $\\Omega >0$ be the cut-off frequency. For a smooth function $f$ supported in $[-\\Omega, \\Omega]$, let $$||f||_2 = \\frac{1}{2\\Omega} \\int_{-\\Omega}^{\\Omega}|f(\\omega)|^2 d\\omega \\quad \\mbox{ and } \\quad ||f||_\\infty =\\max_{\\omega \\in [-\\Omega, \\Omega]}|f(\\omega)|.$$  For $\\Lambda >0$, we define the warped-around distance for $x,y\\in \\mathbb R$ by\n\\begin{equation}\\label{equ:warpedarounddistance}\n\\Big|x-y\\Big|_{\\Lambda} = \\min_{k\\in \\mathbb{Z}} \\babs{x-y-k\\Lambda}. \n\\end{equation}  \n\nLet $\\displaystyle \\mu=\\sum_{j=1}^{n}a_{j}\\delta_{y_j}$  be a discrete measure, \nwhere $y_j \\in \\mathbb R,j=1,\\cdots,n$, represent the locations of the point scatterers  and $a_j\\in \\mathbb C, j=1,\\cdots,n,$ their strengths. We set\n\\begin{equation} \\label{mmin}\nm_{\\min}=\\min_{j=1,\\cdots,n}|a_j|, \\quad d_{\\min}=\\min_{p\\neq j}| y_p-y_j|.\n\\end{equation}\nWe assume that the point scatterers are illuminated by some illumination pattern $I_t$ for each time step $t \\in \\mathbb{N},  1\\leq t\\leq T$, where $T$ is the total number of frames. Then   $I_t \\mu$ is given by \n\\[\nI_t \\mu = \\sum_{j=1}^n I_t(y_j)a_j\\delta_{y_j},\\ t=1, \\cdots, T.\n\\]\nThe available measurements are the noisy Fourier data of  $I_t \\mu$ in a bounded interval. More precisely, they are given by \n\\begin{equation}\\label{equ:multimodelsetting1}\n\\mathbf Y_t(\\omega) = \\mathcal F [I_t \\mu] (\\omega) + \\mathbf W_t(\\omega)= \\sum_{j=1}^{n}I_t(y_j)a_j e^{i y_j \\omega} + \\mathbf W_t(\\omega), \\quad 1\\leq t\\leq T, \\ \\omega \\in [-\\Omega, \\Omega], \n\\end{equation}\nwhere $\\mathcal F[I_t \\mu]$ denotes the Fourier transform of $I_t \\mu$ and $\\vect W_t(\\omega)$ is the noise. We assume that $||\\mathbf W_t||_2<\\sigma$ with $\\sigma$ being the noise level. Recall that $\\pi\/\\Omega$ is the Rayleigh resolution limit.\n\n\nThe inverse problem we are concerned with is to recover the sparsest measure that could generate these diffraction-limited images $\\vect Y_t$'s under certain illuminations. In modern imaging techniques, there are three different cases of interest:\n\\begin{itemize}\n\t\\item The illumination patterns are exactly known, such as in SIM and STORM;\n\t\\item The illumination patterns are unknown but can be approximated, such as in BEAM;\n\t\\item The illumination patterns are completely unknown.\n\\end{itemize}\nIn this paper, we consider reconstructing the point scatterers as the sparsest solution  under the measurement constraint for all these three cases. More specifically, when the illumination patterns are exactly known, we consider the following $l_0$-minimization problem:\n\\begin{equation}\\label{prob:l0minimization}\n\\min_{\\rho} ||\\rho||_{0} \\quad \\text{subject to} \\quad ||\\mathcal F[I_t \\rho] -Y_t||_2< \\sigma, \\quad 1\\leq t\\leq T,\n\\end{equation}\t\nwhere $||\\rho||_{0}$ is the number of Dirac masses representing the discrete measure $\\rho$. When the illumination patterns are not exactly known but could be approximated, we consider the $l_0$-minimization problem:\n\\begin{equation}\\label{prob:l0minimization1}\n\\min_{\\rho} ||\\rho||_{0} \\quad \\text{subject to} \\quad ||\\mathcal F[\\hat I_t \\rho] -Y_t||_2< \\sigma, \\quad 1\\leq t\\leq T,\n\\end{equation}\t\nwhere $\\hat I_t$ is an approximation of each $I_t$ so that the feasible set contains some discrete measures with  $n$ supports. When the illumination patterns are completely unknown, we consider the following $l_0$-minimization problem:\n\\begin{equation}\\label{prob:l0minimization2}\n\\min_{\\rho} ||\\rho||_{0} \\quad \\text{subject to the existence of $\\hat I_t$'s such that}\\  ||\\mathcal F[\\hat I_t \\rho] -Y_t||_2< \\sigma, \\quad 1\\leq t\\leq T.\n\\end{equation} \nOur main result in the next section gives an estimation of the resolution of these sparsity recovery problems in the one-dimensional case. \n\n\n\n\\subsection{Main results}\nWe first introduce the illumination matrix as\n\\begin{equation}\\label{equ:illuminationpattern1}\nI = \\begin{pmatrix}\nI_1(y_1)&\\cdots&I_1(y_n)\\\\\n\\vdots&\\vdots&\\vdots\\\\\nI_T(y_1)&\\cdots&I_T(y_n)\\\\\n\\end{pmatrix}.\n\\end{equation}\nThen we define, for a $m\\times k$ matrix $A$, $\\sigma_{\\infty, \\min}(A)$ by\n\\begin{equation} \\label{sigmadef}\n\\sigma_{\\infty, \\min}(A) = \\min_{x\\in \\mathbb C^k, \\mathbb ||x||_{\\infty}\\geq 1} ||Ax||_{\\infty}.\n\\end{equation}\nIt is easy to see that $\\sigma_{\\infty, \\min}(A)$  characterizes the correlation between the columns of $A$. \n\nWe have the following result on the stability of problems (\\ref{prob:l0minimization}), (\\ref{prob:l0minimization1}), and (\\ref{prob:l0minimization2}). Its proof is given in Section \\ref{section:proofofthml0normrecover}.\n\\begin{thm}\\label{thm:l0normrecovery0}\n\tSuppose that $\\mu= \\sum_{j=1}^n a_j \\delta_{y_j}$ and the following separation condition holds:\n\t\\begin{equation}\\label{equ:sepaconditionl0normrecovery}\n\td_{\\min} := \\min_{p\\neq j}\\babs{y_p-y_j}_{\\frac{n\\pi}{\\Omega}}\\geq \\frac{2.2e\\pi }{\\Omega }\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}},\n\t\\end{equation}\n\twith  $\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\leq 1$.\nHere, $m_{\\min}$ is defined in (\\ref{mmin}) and \t$\\frac{\\sigma}{m_{\\min}}$ is the noise-to-signal ratio. \n\t Then any solution to (\\ref{prob:l0minimization}), (\\ref{prob:l0minimization1}), or (\\ref{prob:l0minimization2}) contains exactly $n$ point scatterers. Moreover, for $\\rho = \\sum_{j=1}^n \\hat a_j \\delta_{\\hat y_j}$ being the corresponding solution, after reordering the $\\hat y_j$'s, we have \n\t\\begin{equation}\n\t\\Big|\\hat y_j-y_j\\Big|_{\\frac{n\\pi}{\\Omega}}<\\frac{d_{\\min}}{2},\n\t\\end{equation} \n\tand \n\t\\begin{equation}\n \\Big|\\hat y_j-y_j\\Big|_{\\frac{n\\pi}{\\Omega}}< \\frac{C(n)}{\\Omega}SRF^{n-1}\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}, \\quad 1\\leq j\\leq n,\n\t\\end{equation}\n\twhere $C(n)=2\\sqrt{2 \\pi} ne^{n}$ and $SRF = \\frac{\\pi}{\\Omega d_{\\min}}$ is the super-resolution factor.\n\\end{thm}\n\n\n\\begin{remark}\n In this paper, for simplicity, we assume that the measurements are for all $\\omega \\in [-\\Omega, \\Omega]$. Nevertheless, our results can be easily extended to the discrete sampling case, for example, when the measurements are taken at $M$ evenly spaced points $\\omega_l\\in [-\\Omega, \\Omega]$ with $M\\geq  n$. The minimum number of sampling points at each single frame is only $n$, which shows that the sparsity recovery can reduce significantly the number of measurements. \n Moreover,  if we consider that the point scatterers (as well as the solution of (\\ref{prob:l0minimization})) are supported in an interval of length of several Rayleigh resolution limits, then the warped-around distance in Theorem \\ref{thm:l0normrecovery0} can be replaced by the Euclidean distance (with only a slight modification of the results). Under this scenario, by utilizing the projection trick introduced in \\cite{liu2021mathematicalhighd}, our results can also be extended to multi-dimensional spaces. \n\\end{remark}\n\n\n\\begin{remark}\nFor the case when $n=2$, the minimal separation distance in Theorem \\ref{thm:l0normrecovery0} \n\\[\n\\frac{2.2e\\pi }{\\Omega }\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{2}},\n\\]\napplies for any $k$-dimensional spaces. It means that for multi-illumination imaging in general $k$-dimensional space, the two-point resolution \\cite{shahram2004imaging, shahram2005resolvability, shahram2004statistical, chen2020algorithmic,den1997resolution} of sparsity recoveries like (\\ref{prob:l0minimization}), (\\ref{prob:l0minimization1}), or (\\ref{prob:l0minimization2}) is less than $\\frac{2.2e\\pi }{\\Omega }\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{2}}$. \n\\end{remark}\n\n\n\\begin{remark}\nNote that the stability result in Theorem \\ref{thm:l0normrecovery0} holds for any algorithm that can recover the sparsest solution (solution with $n$ point scatterers). Thus it also helps to understand the performance of other sparsity-promoting algorithms, such as the $l_1$-minimization that is frequently used in the sparsity-based super-resolution. Also, our results can be generalized to the multi-clump case, where the resolution is related to the sparsity of the point scatterers in each clump rather than the total number of point scatterers. This can explain the fact that we can achieve super-resolution imaging even in the case where we have tens or hundreds of point scatterers.  \n\\end{remark}\n\n\n\\begin{remark}\nNote also that our results can be extended to other kinds of imaging systems with different point spread functions. For example, let the point spread function be $f$. In the presence of an additive noise $w(t)$, \nthe measurement in the time-domain is \n$$\nf \\circledast \\mu (t) + w(t) = \\sum_{j=1}^n  a_j f (t-y_j) + w(t).\n$$\nBy taking the Fourier transform, we obtain\n$$\n\\mathcal F y (\\omega) = \\mathcal F f(\\omega) \\mathcal F \\mu (\\omega) + \\mathcal F w (\\omega) = \\mathcal F f(\\omega) \\big(\\sum_{j=1}^n a_j e^{i y_j \\omega} \\big) + \\mathcal F w (\\omega) .\n$$ \nSuppose that $| \\mathcal F f(\\omega)| >0$ at the sampling points. Then our results can be easily extended to the case when the point spread function is $f$. \n\\end{remark}\n\n\nTheorem \\ref{thm:l0normrecovery0} demonstrates that when the point scatterers are separated by the distance $d_{\\min}$ in (\\ref{equ:sepaconditionl0normrecovery}), we can stably recover the scatterer locations. Under the minimal separation condition, each of the recovered locations is in a neighborhood of the ground truth and the deviation of them from the ground truth is also estimated. Thus the resolution of our sparsity-promoting algorithms for the multi-illumination data is less than\n\\[\n\\frac{2.2e\\pi }{\\Omega }\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}.\n\\]\nBased on this formula of the resolution limit, we demonstrate that the incoherence (encoded in $\\sigma_{\\infty, \\min}(I)$) between the illumination patterns (or columns in illumination matrix (\\ref{equ:illuminationpattern1})) is crucial to the sparsity-based spatio-temporal super-resolution. More precisely, applying any sparsity-promoting algorithm for images from illumination patterns with high degree of incoherence can achieve desired super-resolution, even when only a small number of frames are provided, which yields high spatio-temporal resolution. This is the most important contribution of our paper. \n\n\n We remark that our result can even serve as a way to estimate explicitly the resolution for the multi-illumination imaging when we could know or estimate the incoherence of the illumination patterns and the signal-to-noise ratio. We present a simple example as follows that calculates explicitly the resolution limit of our sparsity recovery problem by the estimation (\\ref{equ:sepaconditionl0normrecovery}). We leave the other detailed discussions on Theorem \\ref{thm:l0normrecovery0} to the following three subsections.\n\n\\begin{example}\nWe consider two point scatterers that are illuminated by two illumination patterns. Suppose for instance that the illumination matrix is given by\n\\[\nI = \\begin{pmatrix}\n1&0.7\\\\\n0.7&1\n\\end{pmatrix}.\n\\]  \nSuppose also that the noise level is $\\sigma = 10^{-3}$ and the noise-to-signal ratio is $\\frac{\\sigma}{m_{\\min}}=10^{-3}$. By Lemma \\ref{lem:sigmainftyminestimate1}, $\\sigma_{\\infty, \\min}(I)$, defined in (\\ref{sigmadef}), is equal to $0.3$. Hence, by Theorem \\ref{thm:l0normrecovery0}, the resolution limit $d_{\\min}$ in solving problem (\\ref{prob:l0minimization}) (\\ref{prob:l0minimization1}), or (\\ref{prob:l0minimization2}) is smaller than \n\\[\n\\frac{2.2e\\pi }{\\Omega }\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}} \\approx 0.34 \\frac{\\pi}{\\Omega},\n\\]\nwhere $\\frac{\\pi}{\\Omega}$, as said before,  is the classical Rayleigh resolution limit. This shows that even with only two illuminations with mild degree of incoherence, there is a threefold resolution improvement.\n\n\\end{example}\n\n\n\n\\subsection{Discussion of $\\sigma_{\\infty, \\min}(I)$ and the effect of multiple illumination}\n\n\n\n\\subsubsection{Adding the same illumination pattern will not enhance the resolution}\nLet \n\\[\nI = \\begin{pmatrix}\nI_1(y_1)&\\cdots&I_1(y_n)\\\\\n\\vdots&\\vdots&\\vdots\\\\\nI_{T-1}(y_1)&\\cdots&I_{T-1}(y_n)\\\\\nI_T(y_1)&\\cdots&I_T(y_n)\n\\end{pmatrix}, \\quad \\hat I = \\begin{pmatrix}\nI_1(y_1)&\\cdots&I_1(y_n)\\\\\n\\vdots&\\vdots&\\vdots\\\\\nI_T(y_1)&\\cdots&I_T(y_n)\\\\\nI_{T+1}(y_1)& \\cdots & I_{T+1}(y_n)\n\\end{pmatrix}\n\\]\nwith $I_{T+1} = I_{T}$. \nBy the definition of $\\sigma_{\\infty, \\min}$, it is clear that $\\sigma_{\\infty, \\min}(\\hat I) =\\sigma_{\\infty, \\min}(I)$. Thus, adding the same illumination pattern can not increase the resolution in Theorem \\ref{thm:l0normrecovery0}. This is consistent with our observation that multiple illuminations with different patterns are key for spatio-temporal super-resolution. \n\n\\subsubsection{The incoherence between the illumination patterns is crucial}\nThe value of $\\sigma_{\\infty, \\min}(I)$ is related to the correlation between the columns of the illumination matrix $I$. In particular, we have the following rough estimation of  $\\sigma_{\\infty, \\min}(I)$:\n\\begin{equation}\\label{equ:sigmainftyminestimate1}\n\\sigma_{\\infty, \\min}(I)\\geq \\frac{\\sigma_{\\min}(I)}{\\sqrt{T}},\n\\end{equation}\nwhere $\\sigma_{\\min}(I)$ is the minimum singular value of $I$. This clearly illustrates that the correlation between the columns of $I$ is crucial to $\\sigma_{\\infty, \\min}(I)$. The correlation between columns of $I$ is related to the incoherence of the illumination patterns. Thus, we should employ illumination patterns with high degree of incoherence in order to increase $\\sigma_{\\infty, \\min}(I)$, and consequently, obtain a significant resolution enhancement. \n\n\n\\subsection{Comparison with the single illumination case}\nIn this subsection, we compare the resolution in the single illumination case (i.e., the single measurement case) with that in the multiple illumination case, whereby we illustrate the effect of multiple illuminations in enhancing the resolution. \n\nIn \\cite{liu2021theorylse}, the authors estimate the so-called computational resolution limit for the line spectral estimation problem of the single measurement case. The line spectral estimation problem is to estimate the locations of some line spectra from the Fourier data (in a bounded domain) of one of their linear combination. So the line spectral problem is equivalent to the super-resolution problem considered here. The results in \\cite{liu2021theorylse} show that, for the single measurement case, when the point scatterers are separated by\n\\[\n\\tau = \\frac{c_0}{\\Omega} (\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{2n-1}},\n\\]\nfor some positive constant $c_0$,  there exists a discrete measure $\\mu = \\sum_{j=1}^n a_j \\delta_{y_j}$ with $n$ point scatterers located at $\\{-\\tau, -2\\tau, -n\\tau\\}$ and another discrete measure $\\hat \\mu = \\sum_{j=1}^n \\hat a_j \\delta_{\\hat y_j}$ with $n$ point scatterers located at  $\\{0,\\tau,\\cdots, (n-1)\\tau\\}$ such that\n\\[\n||\\mathcal F[\\hat \\mu]-\\mathcal F[\\mu]||_{\\infty}< \\sigma,\n\\]\nand either $\\min_{1\\leq j\\leq n}|a_j|= m_{\\min}$ or $\\min_{1\\leq j\\leq n}|\\hat a_j|= m_{\\min}$. \n\nBy the definition of $||\\cdot||_{\\infty}$ and $||\\cdot||_{2}$, we also have \n\\[\n||\\mathcal F[\\hat \\mu]-\\mathcal F[\\mu]||_{2}< \\sigma. \n\\]\nThis result demonstrates that when the point scatterers are separated by $\\frac{c_0}{\\Omega} (\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{2n-1}}$, the solution of the $l_0$-minimization problem in the single measurement case\n\\begin{equation}\\label{prob:singlel0minimization}\n\\min_{\\rho} ||\\rho||_{0} \\quad \\text{subject to} \\quad ||\\mathcal F[\\rho] -Y||_2< \\sigma, \n\\end{equation}\t\nis not stable. In particular, the recovered point scatterers by (\\ref{prob:singlel0minimization}) may be located in an interval completely disjoint from that of the ground truth. \n\nTherefore, for the single measurement case, when the scatterers are separated by $O(\\frac{(\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{2n-1}}}{\\Omega})$, the \n$l_0$-minimization may be unstable. However, for the multiple illumination case, when the point scatterers are separated by $O(\\frac{(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{n}}}{\\Omega})$, the $l_0$-minimization (\\ref{prob:l0minimization}) is still stable. \n\nSuppose we have illumination patterns such that $\\frac{1}{\\sigma_{\\infty, \\min}(I)}$ is of constant order, the resolution now is of order $O(\\frac{(\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{n}}}{\\Omega})$. Compared with the resolution in the single measurement case, say of order $O(\\frac{(\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{2n-1}}}{\\Omega})$, this clearly shows a significant enhancement and illustrates the effect of multiple illuminations in improving the resolution. \n\n\\subsection{Lower bound for the resolution of multi-illumination imaging}\nBy Theorem \\ref{thm:l0normrecovery0}, when we have desired illumination patterns with high degree of incoherence so that $\\sigma_{\\infty, \\min}(I)$ is of order one, the resolution of the sparsity recovery is expected to be less than $\\frac{c_0}{\\Omega}(\\frac{\\sigma}{m_{\\min}})^{\\frac{1}{n}}$ for some positive constant $c_0$. We next demonstrate that this resolution order is the best we can obtain if the illumination patterns are unknown. More precisely, we have the following proposition whose proof is given in Appendix \\ref{section:proofofsupportlowerbound}. \n\n\n\\begin{prop}\\label{prop:multisupportlowerboundthm1}\n\tGiven $n \\geq 2$, $\\sigma, m_{\\min}$ with $\\frac{\\sigma}{m_{\\min}}\\leq 1$,  and unknown illumination pattern $I_t$ with $|I_t(y)|\\leq 1, y\\in \\mathbb R, 1\\leq t\\leq T$, let $\\tau$ be given by\n\t\\begin{equation}\\label{equ:multisupportlowerboundsepadis2}\n\t\\tau = \\frac{0.043}{\\Omega}\\Big(\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}.\n\t\\end{equation}\n   Then there exist  $\\mu=\\sum_{j=1}^{n}a_j\\delta_{y_j}$ with $n$ supports at $\\big\\{-\\tau, -2\\tau,\\ldots, -n\\tau \\big\\}$ and $|a_j| =  m_{\\min}, 1\\leq j\\leq n$, and $\\rho=\\sum_{j=1}^{n-1}\\hat a_j \\delta_{\\hat y_j}$ with $n$ supports at  $\\big\\{0, \\tau,\\cdots, (n-1)\\tau\\big\\}$, such that\n\t\\[\n\t\\text{there exist $\\hat I_t$'s so that } \\ ||\\mathcal F [\\hat I_t \\rho]-\\mathcal F[I_t \\mu]||_{2}< \\sigma, \\ t=1,\\cdots, T.\n\t\\]\n\\end{prop} \n\n\\section{Resolution in the two-dimensional case}\\label{section:twodcase}\n\\subsection{Problem setting}\nLet $\\Omega >0$ be the cut-off frequency. For a smooth function $f:\\mathbb R^2\\rightarrow \\mathbb R$ supported on $||\\vect \\omega ||_2 \\leq \\Omega $, let \n\\[\n ||f||_\\infty =\\max_{||\\vect \\omega||_2 \\leq \\Omega }|f(\\vect \\omega)|.\n\\]  \nLet $\\displaystyle \\mu=\\sum_{j=1}^{n}a_{j}\\delta_{\\vect y_j}$  be a discrete measure, \nwhere $\\vect y_j \\in \\mathbb R^2,j=1,\\cdots,n$, represent the locations of the point scatterers  and $a_j\\in \\mathbb C, j=1,\\cdots,n,$ their strengths. We set\n\\begin{equation} \\label{twodmmin}\nm_{\\min}=\\min_{j=1,\\cdots,n}|a_j|, \\quad d_{\\min}=\\min_{p\\neq j}||\\vect y_p - \\vect  y_j||_2 .\n\\end{equation}\nAgain, we assume that the point scatterers are illuminated by some illumination pattern $I_t$ for each time step $t \\in \\mathbb{N},  1\\leq t\\leq T$, where $T$ is the total number of illumination patterns. Then  $I_t \\mu$ is \n\\[\nI_t \\mu = \\sum_{j=1}^n I_t(\\vect y_j)a_j\\delta_{\\vect y_j},\\quad t=1, \\cdots, T.\n\\]\nIn the time-domain, the measurements are\n\\[\nA_t\\mu:= h \\circledast (I_t \\mu) , \\quad  t= 1, 2, \\cdots, T, \n\\]\nwhere $h$ is a blurring kernel in $\\mathbb R^2$. Thus, in the Fourier-domain, the available measurements are given by  \n\\begin{equation}\\label{equ:twodmultimodelsetting1}\n\\mathbf Y_t(\\vect \\omega) = \\mathcal F [I_t \\mu] (\\vect \\omega) + \\mathbf W_t(\\vect \\omega)= \\sum_{j=1}^{n}I_t(\\vect y_j)a_j e^{i \\vect y_j \\cdot \\vect \\omega} + \\mathbf W_t(\\vect \\omega), \\ 1\\leq t\\leq T, ||\\vect \\omega ||_2 \\leq \\Omega,\n\\end{equation}\nwhere $\\mathcal F[I_t \\mu]$ denotes the Fourier transform of $I_t \\mu$ and $\\vect W_t(\\vect \\omega)$ is the noise. We assume that $||\\mathbf W_t||_{\\infty}<\\sigma$ with $\\sigma$ being the noise level. \n\nWe consider reconstructing the point scatterers as the sparsest solution (solution to the $l_0$-minimization problem) under the measurement constraints for the three cases of illumination patterns that are discussed in Section \\ref{sect2}. With a slight abuse  of notation, we also denote by $\\mathcal F[\\rho]$ the function $\\mathcal F[\\rho](\\vect \\omega), ||\\vect \\omega||_2\\leq \\Omega$. In this section, we suppose that the point scatterers are located in a disk $\\mathcal O$ with radius of several Rayleigh resolution limits. Then we consider the following optimization problems. When the illumination patterns are exactly known, we consider the following $l_0$-minimization problem:\n\\begin{equation}\\label{prob:twodl0minimization}\n\\min_{\\rho \\ \\text{supported in $\\mathcal O$}} ||\\rho||_{0} \\quad \\text{subject to} \\ ||\\mathcal F[I_t \\rho] -\\vect Y_t||_{\\infty}< \\sigma, \\quad 1\\leq t\\leq T,\n\\end{equation}\t\nwhere $||\\rho||_{0}$ is the number of Dirac masses representing the discrete measure $\\rho$. When the illumination patterns are not exactly known but could be approximated, we consider the $l_0$-minimization problem\n\\begin{equation}\\label{prob:twodl0minimization1}\n\\min_{\\rho \\ \\text{supported in $\\mathcal O$}} ||\\rho||_{0} \\quad \\text{subject to} \\ ||\\mathcal F[\\hat I_t \\rho] - \\vect Y_t||_{\\infty}< \\sigma, \\quad 1\\leq t\\leq T,\n\\end{equation}\t\nwhere $\\hat I_t$ is an approximation of each $I_t$ so that the feasible set contains some measures with $n$ supports. When the illumination patterns are completely unknown, we consider the following $l_0$-minimization problem: \n\\begin{equation}\\label{prob:twodl0minimization2}\n\\min_{\\rho \\ \\text{supported in $\\mathcal O$}} ||\\rho||_{0} \\quad\n\\text{subject to the existence of $\\hat I_t$'s such that}\\  ||\\mathcal F[\\hat I_t \\rho] - \\vect Y_t||_{\\infty}< \\sigma,\\ 1\\leq t\\leq T.\n\\end{equation} \nOur main result in the following  subsection gives an estimation of the resolution of these two-dimensional sparsity recovery problems. \n\n\\subsection{Main results for the stability of sparsity recoveries in two dimensions}\nThe illumination matrix in the two-dimensional case is\n\\begin{equation}\\label{equ:twodilluminationpattern1}\nI = \\begin{pmatrix}\nI_1(\\vect y_1)&\\cdots&I_1(\\vect y_n)\\\\\n\\vdots&\\vdots&\\vdots\\\\\nI_T(\\vect y_1)&\\cdots&I_T(\\vect y_n)\\\\\n\\end{pmatrix}.\n\\end{equation}\n We have the following theorem on the stability of problems (\\ref{prob:twodl0minimization}), (\\ref{prob:twodl0minimization1}), and (\\ref{prob:twodl0minimization2}). Its proof is given in Section \\ref{section:prooftwodl0normrecovery}.\n \n \n\\begin{thm}\\label{thm:twodl0normrecovery0}\nLet $n\\geq 2$ and let the disk $\\mathcal O$ be of radius  $\\frac{c_0n\\pi}{\\Omega}$ with $c_0\\geq 1$. Let $\\vect Y_t$'s be the measurements that are generated by an $n$-sparse measure $\\mu=\\sum_{j=1}^{n}a_j \\delta_{\\vect y_j}, \\vect y_j \\in \\mathcal O$ in the two-dimensional space. Assume that\n\\begin{equation}\\label{equ:highdsupportlimithm0equ0}\nd_{\\min}:=\\min_{p\\neq j}\\Big|\\Big|\\mathbf y_p-\\mathbf y_j\\Big|\\Big|_2\\geq \\frac{2.2c_0e\\pi(n+2)(n+1)}{\\Omega }\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}. \n\\end{equation}\nHere, $I$ is the matrix in (\\ref{equ:twodilluminationpattern1}), $m_{\\min}$ is defined in (\\ref{twodmmin}) and $\\frac{\\sigma}{m_{\\min}}$ is the noise-to-signal ratio. \nThen any solution to (\\ref{prob:twodl0minimization}), (\\ref{prob:twodl0minimization1}), and (\\ref{prob:twodl0minimization2}) contains exactly $n$ point scatterers. Moreover, for $\\rho = \\sum_{j=1}^n \\hat a_j \\delta_{\\hat {\\mathbf y}_j}$ being the corresponding solution, after reordering the $\\hat {\\mathbf y}_j$'s, we have \n\\begin{equation}\n\\btwonorm{\\hat {\\mathbf y}_j- \\vect y_j}<\\frac{d_{\\min}}{2},\n\\end{equation} \nand \n\\begin{equation}\n\\btwonorm{\\hat {\\mathbf y}_j-\\vect y_j} < \\frac{C(n)}{\\Omega}SRF^{n-1}\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}, \\quad 1\\leq j\\leq n,\n\\end{equation}\nwhere $C(n)=(n+1)^n(n+2)^n\\sqrt{2\\pi}nc_0^{n-1}e^{n}$ and $SRF = \\frac{\\pi}{\\Omega d_{\\min}}$ is the super-resolution factor.\n\\end{thm}\n\n\nTheorem \\ref{thm:twodl0normrecovery0} is the two-dimensional analogue of Theorem \\ref{thm:l0normrecovery0}. It reveals the dependence of the resolution of two-dimensional sparsity recoveries on the cut-off frequency of the imaging system, the signal-to-noise ratio, the sparsity of point scatters, and the incoherence of illumination patterns. It highlights the importance of multiple illumination patterns with high degree of incoherence in achieving two-dimensional spatio-temporal super-resolution. \n\n\\section{Non-linear approximation theory in Vandermonde space}\\label{section:approxinvandermonde}\nIn this section, we present the main technique that is used in the proofs of the main results of the paper, namely the approximation theory in Vandermonde space. This theory was first introduced in \\cite{liu2021mathematicaloned, liu2021theorylse}.\n Instead of considering the non-linear approximation problem there, we consider a different approximation problem, which is relevant to the stability analysis of (\\ref{prob:l0minimization}). More specifically,  for  $s \\in \\mathbb{N}, s \\geq 1,$ and $z\\in \\mathbb C$, we define the complex Vandermonde-vector\n\\begin{equation}\\label{equ:multiphiformula}\n\\phi_s(z)=(1,z,\\cdots,z^s)^\\top.\n\\end{equation}\n\nThroughout this paper, for a complex matrix $A$, we denote $A^\\top$ its transpose and $A^*$ its conjugate transpose. \n\nWe consider the following non-linear problem:  \n\\begin{equation}\\label{equ:multinon-linearapproxproblem1}\n\\min_{\\hat \\theta_j \\in \\mathbb R, j=1,\\cdots,k}\\max_{t=1, \\cdots, T} \\min_{\\hat a_{j,t}\\in \\mathbb{C}, j=1,\\cdots,k}\\Big|\\Big|\\sum_{j=1}^k \\hat a_{j,t}\\phi_s(e^{i\\hat \\theta_j})-v_t\\Big|\\Big|_2,\n\\end{equation}\nwhere $v_t=\\sum_{j=1}^{k+1}a_{j,t}\\phi_s(e^{i\\theta_j})$ is given with $\\theta_j$'s being real numbers. \nWe shall derive a lower bound for the optimal value of the minimization problem for the case when $s=k$. The main results are presented in Section \\ref{section:mainresultsapproxinvandermonde}. \n\n\\subsection{Notation and Preliminaries}\nWe first introduce some notation and preliminaries. We denote for $k \\in \\mathbb{N}, k\\geq 1$, \n\\begin{equation}\\label{equ:multizetaxiformula1} \n\\zeta(k)= \\left\\{\n\\begin{array}{cc}\n(\\frac{k-1}{2}!)^2,& \\text{$k$ is odd,}\\\\\n(\\frac{k}{2})!(\\frac{k-2}{2})!,& \\text{$k$ is even,}\n\\end{array} \n\\right. \\ \\xi(k)=\\left\\{\n\\begin{array}{cc}\n1\/ 2,  & k=1,\\\\\n\\frac{(\\frac{k-1}{2})!(\\frac{k-3}{2})!}{4},& \\text{$k$ is odd,\\,\\,$ k\\geq 3$,}\\\\\n\\frac{(\\frac{k-2}{2}!)^2}{4},& \\text{$k$ is even}.\n\\end{array} \n\\right.\t\n\\end{equation}\nWe also define for  $p, q \\in \\mathbb{N}, p,q \\geq 1$, and $z_1, \\cdots, z_p, \\hat z_1, \\cdots, \\hat z_q \\in \\mathbb C$, the following vector in $\\mathbb{R}^p$:\n\\begin{equation}\\label{equ:multieta}\n\\eta_{p,q}(z_1,\\cdots,z_{p}, \\hat z_1,\\cdots,\\hat z_q)=\\left(\\begin{array}{c}\n|(z_1-\\hat z_1)|\\cdots|(z_1-\\hat z_q)|\\\\\n|(z_2-\\hat z_1)|\\cdots|(z_2-\\hat z_q)|\\\\\n\\vdots\\\\\n|(z_{p}-\\hat z_1)|\\cdots|(z_{p}-\\hat  z_q)|\n\\end{array}\\right).\n\\end{equation}\n\nWe present two auxiliary lemmas that are helpful for deriving our main results. These lemmas are slightly different from the ones in \\cite[ Section III]{liu2021theorylse}. Thus, we employ different techniques for proving them. Their proofs  are presented in Appendix \\ref{section:proofofproductloweraandstable}. \n\n\\begin{lem}\\label{lem:multimultiproductlowerbound1}\n\tFor $\\theta_j \\in \\mathbb R, j=1, \\cdots, k+1$, assume that $\\min_{p\\neq j}|\\theta_p-\\theta_j|_{2\\pi}=\\theta_{\\min}$. Then, for any $\\hat \\theta_1,\\cdots, \\hat \\theta_k\\in \\mathbb R$, we have the following estimate: \t\n\t\\[\n\t||\\eta_{k+1,k}(e^{i\\theta_1},\\cdots,e^{i\\theta_{k+1}},e^{i \\hat \\theta_1},\\cdots,e^{i\\hat \\theta_k})||_{\\infty}\\geq \\xi(k)(\\frac{2 \\theta_{\\min}}{\\pi})^k.  \n\t\\] \n\\end{lem}\n\n\\medskip\n\\begin{lem}\\label{lem:multistablemultiproduct0}\nLet $\\epsilon >0$. For $\\theta_j, \\hat \\theta_j \\in \\mathbb R, j=1, \\cdots, k$, assume that\n\t\\begin{equation}\\label{equ:stablemultiproductlemma1equ1}\n\t||\\eta_{k,k}(e^{i\\theta_1},\\cdots,e^{i\\theta_k}, e^{i\\hat \\theta_1},\\cdots, e^{i\\hat \\theta_k})||_{\\infty}< (\\frac{2}{\\pi})^{k}\\epsilon,\n\t\\end{equation}\n\twhere $\\eta_{k,k}$ is defined as in (\\ref{equ:multieta}), and that\n\t\\begin{equation}\\label{equ:stablemultiproductlemma1equ2}\n\t\\theta_{\\min} =\\min_{q\\neq j}|\\theta_q-\\theta_j|_{2\\pi}\\geq \\Big(\\frac{4\\epsilon}{\\lambda(k)}\\Big)^{\\frac{1}{k}},\n\t\\end{equation} \n\twhere \n\t\\begin{equation}\\label{equ:lambda1}\n\t\\lambda(k)=\\left\\{\n\t\\begin{array}{ll}\n\t1,  & k=2,\\\\\n\t\\xi(k-2),& k\\geq 3.\n\t\\end{array} \n\t\\right.\t\n\t\\end{equation}\n\tThen, after reordering the $\\hat \\theta_j$'s, we have\n\t\\begin{equation}\\label{equ:stablemultiproductlemma1equ4}\n\t|\\hat \\theta_j -\\theta_j|_{2\\pi}< \\frac{\\theta_{\\min}}{2},  \\quad j=1,\\cdots,k,\n\t\\end{equation}\n\tand moreover,\n\t\\begin{equation}\\label{equ:stablemultiproductlemma1equ5}\n\t|\\hat \\theta_j -\\theta_j|_{2\\pi}< \\frac{2^{k-1}\\epsilon}{(k-2)!(\\theta_{\\min})^{k-1}}, \\quad j=1,\\cdots, k.\n\t\\end{equation}\n\\end{lem}\n\n\n\n\n\\subsection{Main results on the approximation theory in Vandermonde space}\\label{section:mainresultsapproxinvandermonde}\nBefore presenting a lower bound for problem (\\ref{equ:multinon-linearapproxproblem1}), we introduce a basic approximation result in Vandermonde space. This result was first derived in \\cite{liu2021theorylse}. \n\n\\begin{thm}\\label{thm:multispaceapprolowerbound0}\n\tLet $k\\geq 1$. For fixed $\\hat \\theta_1,\\cdots, \\hat \\theta_k\\in \\mathbb{R}$, denote $\\hat A= \\big(\\phi_{k}(e^{i\\hat \\theta_1}),\\cdots, \\phi_{k}(e^{i\\hat \\theta_k})\\big)$, where  the $\\phi_{k}(e^{i\\hat \\theta_j})$'s  are defined as in (\\ref{equ:multiphiformula}). Let $V$ be the $k$-dimensional complex space spanned by the column vectors of $\\hat A$ and let $V^\\perp$ be the one-dimensional orthogonal complement of $V$ in $\\mathbb{C}^{k+1}$. Denote by $P_{V^{\\perp}}$ the orthogonal projection onto $V^{\\perp}$ in $\\mathbb{C}^{k+1}$. Then, we have\n\t\\begin{equation*}\n\t\\min_{\\hat a\\in \\mathbb C^{k}}||\\hat A\\hat a-\\phi_{k}(e^{i\\theta})||_2=||P_{V^{\\perp}}\\big(\\phi_{k}(e^{i\\theta})\\big)||_2 = |v^*\\phi_{k}(e^{i\\theta}) |\\geq \\frac{1}{2^k}|\\Pi_{j=1}^k(e^{i\\theta}-e^{i\\hat \\theta_j})|,\n\t\\end{equation*}\t\n\twhere $v$ is a unit vector in $V^\\perp$ and $v^*$ is its conjugate transpose. \t\n\\end{thm}\n\n\\medskip\nWe then have the following result for non-linear approximation (\\ref{equ:multinon-linearapproxproblem1}) in Vandermonde space.\n\\begin{thm}\\label{thm:multispaceapprolowerbound1}\n\tLet $k\\geq 1$ and $\\theta_j\\in\\mathbb R, 1\\leq j\\leq k+1,$ be $k+1$ distinct points with $\\theta_{\\min}=\\min_{p\\neq j}|\\theta_p-\\theta_j|_{2\\pi}>0$. For $q\\leq k$, let $\\hat \\alpha_t(q)=(\\hat a_{1,t},\\cdots, \\hat a_{q,t})^\\top$, $\\alpha_t=(a_{1,t},\\cdots, a_{k+1, t})^\\top$ and\n\t\\[\n\t\\hat A(q)= \\big(\\phi_{k}(e^{i\\hat \\theta_1}),\\cdots, \\phi_{k}(e^{i\\hat \\theta_q})\\big), \\quad A= \\big(\\phi_{k}(e^{i\\theta_1}),\\cdots, \\phi_{k}(e^{i\\theta_{k+1}})\\big),\n\t\\]\n\twhere $\\phi_{k}(z)$ is defined as in (\\ref{equ:multiphiformula}). Then, for any $\\ \\hat \\theta_1, \\cdots,  \\hat \\theta_q\\in \\mathbb{R}$,\n\t\\begin{equation*}\n\t\\max_{t=1,\\cdots,T}\\min_{\\hat \\alpha_t(q)\\in \\mathbb C^q}||\\hat A(q)\\hat \\alpha_t(q)-A \\alpha_t||_2\\geq  \\frac{\\sigma_{\\infty, \\min}(B)\\xi(k)(\\theta_{\\min})^{k}}{\\pi^{k}},\n\t\\end{equation*}\n\twhere \n\t\\begin{equation}\\label{equ:multispaceapprolowerbound1equ1}\n\tB=\\left(\\begin{array}{cccc}\n\ta_{1,1}&a_{2,1}&\\cdots&a_{k+1,1}\\\\\n\t\\vdots&\\vdots&\\vdots&\\vdots\\\\\n\ta_{1,T}&a_{2,T}&\\cdots&a_{k+1, T}\n\t\\end{array}\\right).\n\t\\end{equation}\n\\end{thm}\n\\begin{proof}\n\\textbf{Step 1.} \nNote that, for any  $\\hat \\theta_1, \\cdots, \\hat \\theta_q, \\cdots, \\hat \\theta_k\\in \\mathbb R$, if $q<k$, then \n\\[\n\\min_{\\hat a_t(q)\\in \\mathbb C^q}||\\hat A(q)\\hat \\alpha_t(q)-A\\alpha_t||_2\\geq \\min_{\\hat \\alpha_t(k)\\in \\mathbb C^k}||\\hat A(k)\\hat \\alpha_t(k)-A\\alpha_t||_2, \\ 1\\leq t\\leq T.\n\\]\nSo, we only need to consider the case when $q=k$. We shall verify that, for any $k$ distinct points $\\hat \\theta_1,\\cdots,\\hat \\theta_{k}\\in \\mathbb R$, we have \n\\begin{equation}\\label{equ:multispaceapproxlowerboundequ-3}\n\\max_{t=1,\\cdots,T}\\min_{\\hat \\alpha_t\\in \\mathbb{C}^k}||\\hat A(k)\\hat \\alpha_t(k)-A\\alpha_t||_2\\geq \\frac{\\sigma_{\\infty, \\min}(B)\\xi(k)(\\theta_{\\min})^{k}}{\\pi^{k}}.\n\\end{equation}\nLet us then fix $(\\hat \\theta_1,\\cdots,\\hat \\theta_{k})$ in our subsequent arguments.\\\\\n\n\\textbf{Step 2.}  Let $V$ be the complex space spanned by the column vectors of $\\hat A(k)$ and let $V^\\perp$ be the orthogonal complement of $V$ in $\\mathbb C^{k+1}$. It is clear that $V^\\perp$ is a one-dimensional complex space. We let $v$ be a unit vector in $V^{\\perp}$ and denote by $P_{V^{\\perp}}$ the orthogonal projection onto $V^{\\perp}$ in $\\mathbb C^{k+1}$. Note that $\\|P_{V^{\\perp}}u\\|_{2} =  |v^* u| $ for $u \\in \\mathbb C^{k+1}$, where\n$v^*$ is the conjugate transpose of $v$. We have\n\\begin{align}\\label{equ:multispaceapproxlowerboundequ-2}\n\\min_{\\hat \\alpha_t\\in \\mathbb C^k}||\\hat A(k)\\hat \\alpha_t-A\\alpha_t||_2 = ||P_{V^{\\perp}}(A\\alpha_{t})||_2= |v^*A\\alpha_{t}|= |\\sum_{j=1}^{k+1}a_{j,t}v^*\\phi_{k}(e^{i\\theta_j})| =|\\beta_{t}|,\n\\end{align} \nwhere $\\beta_{t}= \\sum_{j=1}^{k+1} a_{j,t}v^*\\phi_{k}(e^{i\\theta_j}), \\ t = 1,\\cdots, T$. Denote by $\\beta=(\\beta_{1}, \\beta_{2},\\cdots, \\beta_{T})^\\top$. Thus, we only need to estimate the lower bound of $||\\beta||_{\\infty}$. By (\\ref{equ:multispaceapproxlowerboundequ-2}), we have $\\beta= B \\hat \\eta$,\nwhere $B$ is given by (\\ref{equ:multispaceapprolowerbound1equ1}) and \n$\\hat \\eta = (v^*\\phi_{k}(e^{i\\theta_1}), v^*\\phi_{k}(e^{i\\theta_2}), \\cdots, v^*\\phi_{k}(e^{i\\theta_{k+1}}))^\\top$. By the definition of $\\sigma_{\\infty, \\min}(B)$, we have \n\\[\n||\\beta||_{\\infty}\\geq \\sigma_{\\infty, \\min}(B)||\\hat \\eta||_{\\infty}.\n\\]\nOn the other hand, by Theorem \\ref{thm:multispaceapprolowerbound0}, we obtain that\n\\begin{equation*}\\label{equ:multispaceapproxlowerbound1equ2}\n||\\hat \\eta||_{\\infty} \\geq \\frac{1}{2^k}||\\eta_{k+1, k}(e^{i\\theta_1}, \\cdots, e^{i\\theta_{k+1}},  e^{i\\hat \\theta_1}, \\cdots, e^{i\\hat \\theta_k} )||_{\\infty},\n\\end{equation*}\nwhere $\\eta_{k+1,k}$ is defined by (\\ref{equ:multieta}). Combining this with Lemma \\ref{lem:multimultiproductlowerbound1}, we get $$||\\hat \\eta||_{\\infty} \\geq \\frac{1}{2^k}\\xi(k)(\\frac{2 \\theta_{\\min}}{\\pi})^k.$$ \nIt then follows that \n\\begin{align*}\n||\\beta||_{\\infty}\\geq \\frac{\\sigma_{\\infty, \\min}(B)\\xi(k)(\\theta_{\\min})^{k}}{\\pi^{k}},\n\\end{align*}\nwhich proves (\\ref{equ:multispaceapproxlowerboundequ-3}) and hence the theorem.  \\end{proof}\n\n\n\\medskip\n\\begin{thm}\\label{thm:multispaceapproxlowerbound2}\n\tLet $k\\geq 2$ and $\\theta_j \\in \\mathbb R, j=1, \\cdots, k,$ be $k$ different points with $$\\theta_{\\min}=\\min_{p\\neq j}|\\theta_p-\\theta_j|_{2\\pi}>0.$$ Assume that there are $k$ distinct points $\\hat \\theta_1,\\cdots,\\hat \\theta_k\\in \\mathbb R$ satisfying\n\t\\[ \\max_{t=1, \\cdots, T} ||\\hat A\\hat \\alpha_t-A \\alpha_t||_2< \\sigma, \\]\n\twhere\n\t$\\hat \\alpha_t=(\\hat a_{1,t},\\cdots, \\hat a_{k,t})^\\top$, $\\alpha_t=(a_{1,t},\\cdots, a_{k,t})^\\top$ and\n\t\\[\n\t\\hat A= \\big(\\phi_{k}(e^{i \\hat \\theta_1}),\\cdots, \\phi_{k}(e^{i \\hat \\theta_k})\\big), \\quad A= \\big(\\phi_{k}(e^{i \\theta_1}),\\cdots, \\phi_{k}(e^{i \\theta_{k}})\\big).\n\t\\]\n\tThen\n\t\\[\n\t||\\eta_{k,k}(e^{i \\theta_1},\\cdots,e^{i \\theta_k},e^{i \\hat \\theta_1},\\cdots,e^{i \\hat \\theta_k})||_{\\infty}<\\frac{2^{k}}{\\sigma_{\\infty, \\min}(B)}\\sigma,\n\t\\]\n\twhere\n\t\\begin{equation}\\label{equ:multispaceapproxlowerbound2equ1}\n\tB=\\left(\\begin{array}{cccc}\n\ta_{1,1}&a_{2,1}&\\cdots&a_{k,1}\\\\\n\t\\vdots&\\vdots&\\vdots&\\vdots\\\\\n\ta_{1,T}&a_{2,T}&\\cdots&a_{k, T}\n\t\\end{array}\\right).\n\t\\end{equation}\n\\end{thm}\n\\begin{proof} Let $V$ be the complex space spanned by the column vectors of $\\hat A$ and let  $V^\\perp$ be the orthogonal complement of $V$ in $\\mathbb C^{k+1}$.  Let $v$ be a unit vector in $V^{\\perp}$ and denote by $P_{V^{\\perp}}$ the orthogonal projection onto $V^{\\perp}$ in $\\mathbb C^{k+1}$. Similarly to {Step 2} in the proof of Theorem \\ref{thm:multispaceapprolowerbound1}, we obtain that\n\\begin{align}\\label{equ:multispaceapproxlowerbound2equ2}\n\\min_{\\hat \\alpha_t\\in \\mathbb C^k}||\\hat A\\hat \\alpha_t-A\\alpha_t||_2 = ||P_{V^{\\perp}}(A\\alpha_{t})||_2= |v^*A\\alpha_{t}|= |\\sum_{j=1}^{k}a_{j,t}v^*\\phi_{k}(e^{i\\theta_j})| =|\\beta_{t}|,\n\\end{align} \nwhere $\\beta_{t}= \\sum_{j=1}^{k} a_{j,t}v^*\\phi_{k}(e^{i\\theta_j}), \\ t = 1,\\cdots, T$. Denote by $\\beta=(\\beta_{1}, \\beta_{2},\\cdots, \\beta_{T})^\\top$, we have $\\beta= B \\hat \\eta$,\nwhere $B$ is given by (\\ref{equ:multispaceapproxlowerbound2equ1}) and \n$\\hat \\eta = (v^*\\phi_{k}(e^{i\\theta_1}), v^*\\phi_{k}(e^{i\\theta_2}), \\cdots, v^*\\phi_{k}(e^{i\\theta_{k}}))^\\top$. By the definition of $\\sigma_{\\infty, \\min}(B)$, we arrive at\n\\[\n||\\beta||_{\\infty}\\geq \\sigma_{\\infty, \\min}(B)||\\hat \\eta||_{\\infty}.\n\\]\nOn the other hand, by Theorem \\ref{thm:multispaceapprolowerbound0}, we get\n\\begin{equation*}\\label{equ:multispaceapproxlowerbound2equ3}\n||\\hat \\eta||_{\\infty} \\geq \\frac{1}{2^k}||\\eta_{k, k}(e^{i\\theta_1}, \\cdots, e^{i\\theta_{k}},  e^{i\\hat \\theta_1}, \\cdots, e^{i\\hat \\theta_k} )||_{\\infty},\n\\end{equation*}\nand hence  the theorem is proved. \\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{thm:l0normrecovery0}}\\label{section:proofofthml0normrecover}\n\nThe proof of Theorem \\ref{thm:l0normrecovery0} is divided into four steps. \n\n\\textbf{Step 1.} We only prove the theorem for problem (\\ref{prob:l0minimization}) and the other two cases can be proved in the same manner. We first prove that the solution to (\\ref{prob:l0minimization}) is a discrete measure corresponding to at least $n$ point scatterers. For $\\rho = \\sum_{j=1}^{k}\\hat a_j \\delta_{\\hat y_j}$ and $\\mu = \\sum_{j=1}^n a_j \\delta_{y_{j}}$, we set $\\hat \\mu_t = I_t \\rho = \\sum_{j=1}^k\\hat a_{j,t} \\delta_{\\hat y_j}$ and $\\mu_{t} = \\sum_{j=1}^n I_t(y_j)a_j \\delta_{y_{j}}$. We shall prove that if $k<n$, then for any $\\hat y_j\\in \\mathbb R, \\hat a_{j,t} \\in \\mathbb C, j=1,\\cdots,k, t=1,\\cdots,T$, \n\\begin{equation}\\label{equ:multinumberresultequ0}\n\\max_{t=1, \\cdots, T}||\\mathcal F[\\hat \\mu_t]-\\mathcal F[\\mu_t]||_{2}> 2\\sigma.\n\\end{equation}\nFor ease of presentation, we fix $\\hat y_j, \\hat a_{j,t}$'s in the subsequent arguments. In view of $||\\vect W_t||_2<\\sigma$, from (\\ref{equ:multinumberresultequ0}) we further have \n\\begin{equation}\n\\max_{t=1, \\cdots, T} ||\\mathcal F[\\hat \\mu_t]-\\vect Y_t||_{2}>\\sigma,\n\\end{equation} \nwhereby any solution corresponding to only $k<n$ point scatterers cannot be a solution to (\\ref{prob:l0minimization}). We now begin our proof. Let $h=\\frac{2\\Omega}{n}$ and $g(\\omega) = \\mathcal F[\\hat \\mu_t](\\omega)-\\mathcal F[\\mu_t](\\omega)$ for each $t$, we have \n\\begin{align*}\n&\\frac{1}{2\\Omega} \\int_{-\\Omega}^\\Omega |g(\\omega)|^2 d\\omega = \\frac{1}{2\\Omega} \\sum_{j=1}^n\\int_{-\\Omega+(j-1)h}^{-\\Omega+jh}|g(\\omega)|^2d\\omega \\\\\n=& \\frac{1}{2\\Omega} \\int_{0}^{h}\\sum_{j=1}^n|g(\\omega+(j-1)h-\\Omega)|^2d\\omega.\n\\end{align*}\nWith this decomposition in hand, in order to prove (\\ref{equ:multinumberresultequ0}) we shall prove in the following steps that for all $\\omega$ in $[0,h]$, \n\\begin{equation}\\label{equ:multinumberresultequ2}\n\\max_{t=1, \\cdots, T}\\frac{1}{n}\\sum_{j=1}^n|\\mathcal F[\\hat \\mu_t](\\omega+(j-1)h-\\Omega)-\\mathcal F[\\mu_t](\\omega+(j-1)h-\\Omega)|^2>4\\sigma^2.\n\\end{equation}\n Without loss of generality, we only show (\\ref{equ:multinumberresultequ2}) for $\\omega =0$. For $k<n$, we consider \n\\begin{equation}\\label{equ:multinumberresultequ1}\n\\left(\\mathcal F [\\hat \\mu_t](\\omega_1), \\mathcal F [\\hat \\mu_t](\\omega_2), \\cdots,\\mathcal F [\\hat \\mu_t](\\omega_{n-1})\\right)^\\top -\\left(\\mathcal F [\\mu_t](\\omega_1), \\mathcal F  [\\mu_t](\\omega_2), \\cdots,\\mathcal F [\\mu_t](\\omega_{n-1})\\right)^\\top,\n\\end{equation}\nwhere $\\omega_j=(j-1)h-\\Omega, \\ j= 1, \\cdots, n-1$. We write (\\ref{equ:multinumberresultequ1}) as \n\\begin{equation*}\n\\hat \\Phi \\hat \\alpha_t- \\Phi \\alpha_t, \n\\end{equation*}\nwhere $\\hat \\alpha_t= (\\hat a_{1,t},\\cdots, \\hat a_{k,t})^\\top$, $\\alpha_t=(I_t(y_1)a_1, \\cdots, I_t(y_n)a_n)^\\top$ and  \n\\begin{equation*}\n\\hat \\Phi= \\left(\n\\begin{array}{ccc}\ne^{i\\hat y_1\\omega_1}&\\cdots& e^{i\\hat y_k\\omega_1}\\\\\ne^{i\\hat y_1\\omega_{2}} &\\cdots& e^{i\\hat y_k\\omega_{2}}\\\\\n\\vdots&\\vdots&\\vdots\\\\\ne^{i\\hat y_1\\omega_{n-1}}&\\cdots& e^{i\\hat y_k \\omega_{n-1}}\\\\\n\\end{array}\n\\right), \\quad \\Phi=\\left(\n\\begin{array}{ccc}\ne^{i y_1\\omega_1}&\\cdots& e^{i y_n\\omega_1}\\\\\ne^{i y_1\\omega_{2}} &\\cdots& e^{i y_n\\omega_{2}}\\\\\n\\vdots&\\vdots&\\vdots\\\\\ne^{i y_1\\omega_{n-1}}&\\cdots& e^{i y_n \\omega_{n-1}}\n\\end{array}\n\\right).\n\\end{equation*}\n\nWe shall prove that the following estimate holds:\n\\begin{equation}\\label{equ:multinumberresultequ3}\n\\max_{t=1, \\cdots, T} \\frac{1}{\\sqrt{n}}||\\hat \\Phi \\hat \\alpha_t- \\Phi \\alpha_t||_2> 2\\sigma,\n\\end{equation}\n and consequently arrive at (\\ref{equ:multinumberresultequ2}). \n\n\n\\textbf{Step 2.} We let $\\theta_j = \ty_j\\frac{2\\Omega}{n}, j=1,\\cdots,n$ and $\\hat \\theta_j = \\hat y_j\\frac{2\\Omega}{n}$.  From the following decompositions:  \n\\begin{equation}\\label{equ:multimatrixdecomposition1}\n\\begin{aligned}\n&\\hat \\Phi=\\big(\\phi_{n-1}(e^{i \\hat \\theta_1}), \\cdots,\\phi_{n-1}(e^{i\\hat \\theta_k} ) \\big)\\text{diag}(e^{-i\\hat y_1\\Omega},\\cdots,e^{-i\\hat y_k\\Omega}),\\\\\n&\\Phi=\\big(\\phi_{n-1}(e^{i \\theta_1}), \\cdots,\\phi_{n-1}(e^{i \\theta_n})\\big)\\text{diag}(e^{-i y_1\\Omega},\\cdots,e^{-iy_n\\Omega}),\n\\end{aligned}\n\\end{equation}\nwhere $\\phi(\\cdot)$ is defined as in (\\ref{equ:multiphiformula}), we readily obtain that\n\\begin{align} \\label{equ:multi1111}\n\\max_{t=1, \\cdots, T} ||\\hat \\Phi \\hat \\alpha_t-\\Phi \\alpha_t||_2= \\max_{t=1, \\cdots, T} ||\\hat D \\hat \\gamma_t-D \\tilde{\\gamma}_t||_2,\n\\end{align}\nwhere $\\hat \\gamma_{t} = (\\hat a_{1,t}e^{-i\\hat y_1\\Omega},\\cdots, \\hat a_{k,t}e^{-i\\hat y_k\\Omega})^\\top, \\gamma_t=(I_{t}(y_1)a_1e^{-iy_1\\Omega},\\cdots, I_{t}(y_n)a_{n}e^{-iy_n\\Omega})^\\top$, \\\\\n$\\hat D=\\big(\\phi_{n-1}(e^{i \\hat \\theta_1}), \\cdots,\\phi_{n-1}(e^{i \\hat \\theta_k} ) \\big)$ and $D=\\big(\\phi_{n-1}(e^{i \\theta_1}), \\cdots,\\phi_{n-1}(e^{i \\theta_n})\\big)$.\nWe consider $I$ in (\\ref{equ:illuminationpattern1}) and denote $B= I \\text{diag}(a_1e^{-iy_1\\Omega},\\cdots, a_{n}e^{-iy_n\\Omega})$. Applying Theorem \\ref{thm:multispaceapprolowerbound1}, we get \n\\begin{equation*}\n\\max_{t=1, \\cdots, T} ||\\hat D \\hat \\gamma_t-D \\gamma_t||_2\\geq   \\frac{\\sigma_{\\infty, \\min}(B)\\xi(n-1)(\\theta_{\\min})^{n-1}}{\\pi^{n-1}}, \n\\end{equation*}\nwhere $\\theta_{\\min}=\\min_{j\\neq p}|\\theta_j-\\theta_p|_{2\\pi}$.\nOn the other hand, by the definition of $\\sigma_{\\infty, \\min}$, we have\n\\begin{equation}\\label{equ:multinumberupperboundequ2}\n\\begin{aligned}\n&\\sigma_{\\infty, \\min}(I)m_{\\min} = \\min_{||\\alpha||_{\\infty}\\geq m_{\\min}}||I \\alpha||_{\\infty}\\\\\n \\leq & \\min_{||\\alpha||_{\\infty}\\geq 1}||I\\text{diag}(a_1e^{-iy_1\\Omega},\\cdots, a_{n}e^{-iy_n\\Omega})\\alpha||_{\\infty} \\quad (\\text{by $m_{\\min} =\\min_{1\\leq j\\leq n}|a_j|$})\\\\\n  =& \\sigma_{\\infty, \\min}(B).\n\\end{aligned}\n\\end{equation}\nThus,\n\\begin{equation*}\n\\max_{t=1, \\cdots, T} ||\\hat D \\hat \\gamma_t-D \\gamma_t||_2\\geq   \\frac{m_{\\min}\\sigma_{\\infty, \\min}(I)\\xi(n-1)(\\theta_{\\min})^{n-1}}{\\pi^{n-1}}. \n\\end{equation*}\nBy (\\ref{equ:multi1111}), it follows that \n\\[\n\\max_{t=1, \\cdots, T} ||\\hat \\Phi \\hat \\alpha_t-\\Phi \\alpha_t||_2 \\geq   \\frac{m_{\\min}\\sigma_{\\infty, \\min}(I)\\xi(n-1)(\\theta_{\\min})^{n-1}}{\\pi^{n-1}}. \n\\]\nOn the other hand, recall that $d_{\\min}= \\min_{j\\neq p}|y_j-y_p|_{\\frac{n\\pi}{\\Omega}}$. Using the relation $\\theta_j = y_j\\frac{2\\Omega}{n}$, we have $\\theta_{\\min}=\\frac{2\\Omega}{n}d_{\\min}$. Then the separation condition (\\ref{equ:sepaconditionl0normrecovery}) and $\\frac{1}{\\sigma_{\\infty, \\min}(I) \\frac{\\sigma}{m_{\\min}}} \\leq 1$ imply that\n\\begin{equation*}\n\\theta_{\\min}\\geq \\frac{4.4 e\\pi }{n}\\Big(\\frac{1}{\\sigma_{\\infty, \\min}( I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}} \\geq \\frac{4.4 e\\pi }{n}\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n-1}}> \\pi \\Big(\\frac{2\\sqrt{n}}{\\xi(n-1)\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n-1}},\n\\end{equation*}\nwhere here we have used Lemma \\ref{lem:multinumbercalculate1} for deriving the last inequality. Therefore, \n\\[\n\\max_{t=1, \\cdots, T} ||\\hat \\Phi \\hat \\alpha_t-\\Phi \\alpha_t||_2 > 2\\sqrt{n}\\sigma,\n\\]\nwhence  (\\ref{equ:multinumberresultequ3}) is proved.\n\n\\textbf{Step 3.} By the above results, the solution of (\\ref{prob:l0minimization}) corresponds exactly to $n$ point scatterers. Suppose that the solution is $\\rho = \\sum_{j=1}^n \\hat a_j \\delta_{\\hat y_j}$ and $\\hat \\mu_{t} = I_t \\rho = \\sum_{j=1}^n \\hat a_{j,t} \\delta_{\\hat y_j}$.  We now prove the stability of the location recovery. Similarly to {Step 1}, using the constraints in (\\ref{prob:l0minimization}) \n\\[\n||\\mathcal F[\\hat \\mu_t] - \\vect Y_t||_2 < \\sigma, \\quad 1\\leq t\\leq T,\n\\]\nwe can derive that \n\\[\n\\frac{1}{2\\Omega} \\int_{0}^{h}\\max_{t=1, \\cdots, T} \\sum_{j=1}^n|\\mathcal F[\\hat \\mu_t](\\omega+(j-1)h-\\Omega)-\\mathcal F[\\mu_t](\\omega+(j-1)h-\\Omega)|^2d\\omega < 4\\sigma^2.\n\\]\nHence, there exists $\\omega_0 \\in [0,h]$ ($h = \\frac{2\\Omega}{n}$) such that \n\\begin{equation}\\label{equ:multisupportresultequ2}\n\\max_{t=1, \\cdots, T}\\frac{1}{n}\\sum_{j=1}^n|\\mathcal F[\\hat \\mu_t](\\omega_0+(j-1)h-\\Omega)-\\mathcal F[\\mu_t](\\omega_0+(j-1)h-\\Omega)|^2< 4 \\sigma^2.\n\\end{equation}\nWithout loss of generality, we suppose that $\\omega_0 = 0$ and consider \n\\begin{equation*}\n\\left(\\mathcal F [\\hat \\mu_t](\\omega_1), \\mathcal F [\\hat \\mu_t](\\omega_{2}), \\cdots,\\mathcal F [\\hat \\mu_t](\\omega_{n})\\right)^\\top -\\left(\\mathcal F [\\mu_t](\\omega_1), \\mathcal F [\\mu_t](\\omega_{2}), \\cdots,\\mathcal F [\\mu_t](\\omega_{n})\\right)^\\top =\\hat \\Phi \\hat \\alpha_t- \\Phi \\alpha_t, \n\\end{equation*}\nwhere $\\omega_j = (j-1)h-\\Omega$, $\\hat \\alpha_t= (\\hat a_{1,t},\\cdots, \\hat a_{n,t})^\\top$, $\\alpha_t=(I_t(y_1)a_1, \\cdots, I_{t}(y_n)a_n)^\\top$ and \n\\begin{equation*}\n\\hat \\Phi= \\left(\n\\begin{array}{ccc}\ne^{i\\hat y_1\\omega_1}&\\cdots& e^{i\\hat y_n\\omega_1}\\\\\ne^{i\\hat y_1\\omega_{2}} &\\cdots& e^{i\\hat y_n\\omega_{2}}\\\\\n\\vdots&\\vdots&\\vdots\\\\\ne^{i\\hat y_1\\omega_{n}}&\\cdots& e^{i\\hat y_n \\omega_{n}}\\\\\n\\end{array}\n\\right), \\quad \\Phi=\\left(\n\\begin{array}{ccc}\ne^{i y_1\\omega_1}&\\cdots& e^{i y_n\\omega_1}\\\\\ne^{i y_1\\omega_{2}} &\\cdots& e^{i y_n\\omega_{2}}\\\\\n\\vdots&\\vdots&\\vdots\\\\\ne^{i y_1\\omega_{n}}&\\cdots& e^{i y_n \\omega_{n}}\\\\\n\\end{array}\n\\right).\n\\end{equation*}\nBy (\\ref{equ:multisupportresultequ2}), it is clear that \n\\[\n\\max_{t=1, \\cdots, T}||\\hat \\Phi \\hat \\alpha_t- \\Phi\\alpha_t||_{2}<2\\sqrt{n}\\sigma.\n\\]\nNote that \n\\begin{align}\\label{equ:multiupperboundsupportlimithm1equ1}\n\\max_{t=1, \\cdots, T}||\\hat \\Phi \\hat \\alpha_t-\\Phi \\alpha_t||_2= \\max_{t=1, \\cdots, T}||\\hat D \\hat \\gamma_t -D \\gamma_t||_2,\n\\end{align}\nwhere $\\hat \\gamma_{t}=(\\hat a_{1,t}e^{-i\\hat y_1\\Omega},\\cdots, \\hat a_{n,t}e^{-i\\hat y_n\\Omega})^\\top, \\gamma_t=(I_1(y_1)a_1e^{-iy_1\\Omega},\\cdots, I_n(y_n)a_{n}e^{-iy_n\\Omega})^\\top$,\\\\\n $\\hat D=\\big(\\phi_{n}(e^{i \\hat \\theta_1}),\\cdots,\\phi_{n}(e^{i \\hat \\theta_n})\\big)$ and $D=\\big(\\phi_{n}(e^{i \\theta_1}),\\cdots,\\phi_{n}(e^{i \\theta_n})\\big)$. Thus,\n\\begin{equation}\\label{equ:multisupportupperboundequ1}\n\\max_{t=1, \\cdots, T}||\\hat D \\hat \\gamma_t-D \\gamma_t||_2 < 2\\sqrt{n}\\sigma. \n\\end{equation}\nWe can apply Theorem \\ref{thm:multispaceapproxlowerbound2} to get\n\\begin{align}\\label{equ:multiupperboundsupportlimithm1equ3}\n||\\eta_{n,n}(e^{i \\theta_1},\\cdots,e^{i \\theta_n},e^{i \\hat \\theta_1},\\cdots,e^{i \\hat \\theta_n})||_{\\infty}<\\frac{2^{n+1}\\sqrt{n}\\sigma}{\\sigma_{\\infty, \\min}(B)}, \n\\end{align}\nwhere $\\eta_{n,n}$ is defined by (\\ref{equ:multieta}) and $B= I\\text{diag}(a_1e^{-iy_1\\Omega},\\cdots, a_{n}e^{-iy_n\\Omega})$. By (\\ref{equ:multinumberupperboundequ2}),\nit follows that \n\\begin{align*}\n\\sigma_{\\infty, \\min}(I)m_{\\min} \\leq \\sigma_{\\infty, \\min}(B).\n\\end{align*}\nThus, we have\n\\begin{align}\\label{equ:multisupportupperboundequ2}\n||\\eta_{n,n}(e^{i \\theta_1},\\cdots,e^{i \\theta_n},e^{i \\hat \\theta_1},\\cdots,e^{i \\hat \\theta_n})||_{\\infty}<\\frac{2^{n+1}}{\\sigma_{\\infty, \\min} (I)}\\frac{\\sigma}{m_{\\min}}. \n\\end{align}\n\n\\textbf{Step 4.}\nWe apply Lemma \\ref{lem:multistablemultiproduct0} to estimate $|\\hat \\theta_j -\\theta_j|_{2\\pi}$'s. For this purpose, let  $\\epsilon = \\frac{2\\sqrt{n}\\pi^{n}}{\\sigma_{\\infty, \\min}(I)} \\frac{\\sigma}{m_{\\min}}$. It is clear that \n$||\\eta_{n ,n}||_{\\infty}<(\\frac{2}{\\pi})^n\\epsilon$ and we only need to check the following condition:\n\\begin{equation}\\label{equ:multiupperboundsupportlimithm1equ4}\n\\theta_{\\min}\\geq \\Big(\\frac{4\\epsilon}{\\lambda(n)}\\Big)^{\\frac{1}{n}}, \\quad \\mbox{or equivalently}\\,\\,\\, (\\theta_{\\min})^n \\geq \\frac{4\\epsilon}{\\lambda(n)}.\n\\end{equation}\nIndeed, by $\\theta_{\\min} = \\frac{2\\Omega}{n} d_{\\min}$ and the separation condition (\\ref{equ:sepaconditionl0normrecovery}), \n\\begin{equation}\\label{equ:multiupperboundsupportlimithm1equ-1}\n\\theta_{\\min}\\geq  \\frac{4.4\\pi e}{n}\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}\\geq   \\Big(\\frac{8\\sqrt{n}\\pi^n}{\\lambda(n)}\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}.\n\\end{equation}\nHere, we have used Lemma \\ref{lem:multisupportcalculate1} for deriving the last inequality. Then, we get (\\ref{equ:multiupperboundsupportlimithm1equ4}). Therefore, we can apply Lemma \\ref{lem:multistablemultiproduct0} to get that, after reordering $\\hat \\theta_j$'s,\n\\begin{equation} \\label{equ:multiupperboundsupportlimithm1equ7}\n\\Big|\\hat \\theta_{j}-\\theta_j\\Big|_{2\\pi}< \\frac{\\theta_{\\min}}{2}, \\text{ and } \\Big|\\hat \\theta_{j}-\\theta_j\\Big|_{2\\pi}< \\frac{2^n\\sqrt{n}\\pi^{n}}{(n-2)!(\\theta_{\\min})^{n-1}}  \\frac{1}{\\sigma_{\\infty, \\min}(I)} \\frac{\\sigma}{m_{\\min}},\\ j=1,\\cdots,n.\n\\end{equation}\nFinally, we estimate $|\\hat y_j - y_j|_{\\frac{n\\pi}{\\Omega}}$. Since $|\\hat \\theta_{j}-\\theta_j|_{2\\pi}< \\frac{\\theta_{\\min}}{2}$, we have after reordering the $\\hat y_j's$,\n$$|\\hat y_j-y_j|_{\\frac{n\\pi}{\\Omega}}< \\frac{d_{\\min}}{2}.$$\nOn the other hand,  $\\Big|\\hat y_j-y_j\\Big|_{\\frac{n\\pi}{\\Omega}}= \\frac{n}{2\\Omega}\\Big|\\hat \\theta_j -\\theta_j\\Big|_{2\\pi}$. \t\nCombining (\\ref{equ:multiupperboundsupportlimithm1equ7}) and (\\ref{equ:stirlingformula}), a direct calculation shows that\n\\begin{align*}\n\\Big|\\hat y_j-y_j\\Big|_{\\frac{n\\pi}{\\Omega}}< \\frac{C(n)}{\\Omega} (\\frac{\\pi}{\\Omega d_{\\min}})^{n-1} \\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}, \n\\end{align*}\nwhere $C(n)=2\\sqrt{2}ne^{n}\\sqrt{\\pi}$.\n\n\\section{Proof of Theorem \\ref{thm:twodl0normrecovery0}}\\label{section:prooftwodl0normrecovery}\n\\subsection{Number and location recoveries in one-dimensional case}\nWe first introduce some results for the number and location recoveries of the one-dimensional case, which will help us to derive the stability results for two-dimensional super-resolution. Unlike Theorem \\ref{thm:l0normrecovery0}, the stability results here consider Euclidean distance between point scatterers.  \n\nFor source $\\mu = \\sum_{j=1}^n a_j \\delta_{y_j}$ and illumination patterns $I_t$'s, the measurements are\n\\begin{equation}\\label{equ:onedmultimodelsetting1}\n\\mathbf Y_t(\\omega) = \\mathcal F [I_t \\mu] (\\omega) + \\mathbf W_t(\\omega)= \\sum_{j=1}^{n}I_t(y_j)a_j e^{i y_j \\omega} + \\mathbf W_t(\\omega), \\quad 1\\leq t\\leq T, \\ \\omega \\in [-\\Omega, \\Omega], \n\\end{equation}\nwhere $\\mathcal F[I_t \\mu]$ denotes the Fourier transform of $I_t \\mu$ and $\\vect W_t(\\omega)$ is the noise with $||\\mathbf W_t||_{\\infty}<\\sigma$. \n\n\n\\begin{thm}\\label{thm:onednumberbound}\nSuppose the measurements $\\vect Y_t$'s in (\\ref{equ:onedmultimodelsetting1}) are generated from $\\mu= \\sum_{j=1}^n a_j \\delta_{y_j}, y_j\\in \\mathbb R$ where $y_j$'s are in an interval $\\mathcal O$ of length $\\frac{c_0n\\pi}{\\Omega}$ with $c_0\\geq 1$ and satisfy\n\\begin{equation}\\label{equ:onedsepaconditionnumber}\nd_{\\min} := \\min_{p\\neq j}\\babs{y_p-y_j}\\geq \\frac{4.4c_0e\\pi }{\\Omega }\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}\n\\end{equation}\nwith  $\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\leq 1$. Then there is no $k<n$ locations $\\hat y_j\\in \\mathbb R, j=1, \\cdots,k$ such that there exists $\\hat \\mu_t = \\sum_{j=1}^k \\hat a_{j,t} \\delta_{\\hat y_j}$'s so that \n\\[\n||\\mathcal F[\\hat \\mu_t] - \\vect Y_t||_{\\infty}< \\sigma, \\quad t=1, \\cdots, T. \n\\] \n\\end{thm}\n\\begin{proof} \nLet $\\mu_t = \\sum_{j=1}^n a_j I_t(y_j)\\delta_{y_j}$. Similar to Step 1 and Step 2 in the proof of Theorem \\ref{thm:l0normrecovery0}, we only need to prove that if $k<n$, then for any $\\hat y_j\\in \\mathbb R, \\hat a_{j,t} \\in \\mathbb C, j=1,\\cdots,k, t=1,\\cdots,T$, \n\\begin{equation}\\label{equ:twodmultinumberresultequ0}\n\\max_{t=1, \\cdots, T}||\\mathcal F[\\hat \\mu_t]-\\mathcal F[\\mu_t]||_{\\infty}> 2\\sigma.\n\\end{equation}\nSpecifically, for $k<n$, we consider \n\\begin{equation}\\label{equ:twodmultinumberresultequ1}\n\\left(\\mathcal F [\\hat \\mu_t](\\omega_1), \\mathcal F [\\hat \\mu_t](\\omega_2), \\cdots,\\mathcal F [\\hat \\mu_t](\\omega_{n-1})\\right)^\\top -\\left(\\mathcal F [\\mu_t](\\omega_1), \\mathcal F  [\\mu_t](\\omega_2), \\cdots,\\mathcal F [\\mu_t](\\omega_{n-1})\\right)^\\top,\n\\end{equation}\nwhere $\\omega_j=(j-1)h-\\Omega, \\ j= 1, \\cdots, n-1$ with $h = \\frac{\\Omega}{c_0n}$. We write (\\ref{equ:twodmultinumberresultequ1}) as \n\\begin{equation*}\n\\hat \\Phi \\hat \\alpha_t- \\Phi \\alpha_t, \n\\end{equation*}\nwhere $\\hat \\alpha_t= (\\hat a_{1,t},\\cdots, \\hat a_{k,t})^\\top$, $\\alpha_t=(I_t(y_1)a_1, \\cdots, I_t(y_n)a_n)^\\top$ and  \n\\begin{equation*}\n\\hat \\Phi= \\left(\n\\begin{array}{ccc}\ne^{i\\hat y_1\\omega_1}&\\cdots& e^{i\\hat y_k\\omega_1}\\\\\ne^{i\\hat y_1\\omega_{2}} &\\cdots& e^{i\\hat y_k\\omega_{2}}\\\\\n\\vdots&\\vdots&\\vdots\\\\\ne^{i\\hat y_1\\omega_{n-1}}&\\cdots& e^{i\\hat y_k \\omega_{n-1}}\\\\\n\\end{array}\n\\right), \\quad \\Phi=\\left(\n\\begin{array}{ccc}\ne^{i y_1\\omega_1}&\\cdots& e^{i y_n\\omega_1}\\\\\ne^{i y_1\\omega_{2}} &\\cdots& e^{i y_n\\omega_{2}}\\\\\n\\vdots&\\vdots&\\vdots\\\\\ne^{i y_1\\omega_{n-1}}&\\cdots& e^{i y_n \\omega_{n-1}}\n\\end{array}\n\\right).\n\\end{equation*}\nWe shall prove that the following estimate holds:\n\\begin{equation}\\label{equ:onedmultinumberresultequ3}\n\\max_{t=1, \\cdots, T} \\frac{1}{\\sqrt{n}}||\\hat \\Phi \\hat \\alpha_t- \\Phi \\alpha_t||_2> 2\\sigma,\n\\end{equation}\nand consequently it yields (\\ref{equ:twodmultinumberresultequ0}). Let $\\theta_j = y_j h =  y_j\\frac{\\Omega}{c_0n}$ and $\\hat \\theta_j = \\hat y_j h = \\hat y_j\\frac{\\Omega}{c_0n}$. Similar to Step 2 in the proof of Theorem \\ref{thm:l0normrecovery0}, we can have\n\\[\n\\max_{t=1, \\cdots, T} ||\\hat \\Phi \\hat \\alpha_t-\\Phi \\alpha_t||_2 \\geq   \\frac{m_{\\min}\\sigma_{\\infty, \\min}(I)\\xi(n-1)(\\theta_{\\min})^{n-1}}{\\pi^{n-1}},\n\\]\nwhere $\\theta_{\\min} = \\min_{p\\neq j}|\\theta_j-\\theta_p|_{2\\pi}$. Because $y_j$'s are in an interval of length $\\frac{c_0n\\pi}{\\Omega}$, by $\\theta_j = y_jh$ we have $\\theta_{\\min} = \\min_{p\\neq j}|\\theta_j-\\theta_p|_{2\\pi}=d_{\\min}\\frac{\\Omega}{c_0n}$. Then the separation condition (\\ref{equ:onedsepaconditionnumber}) and $\\frac{1}{\\sigma_{\\infty, \\min}(I) \\frac{\\sigma}{m_{\\min}}} \\leq 1$ imply that\n\\begin{equation*}\n\\theta_{\\min}\\geq \\frac{4.4 e\\pi }{n}\\Big(\\frac{1}{\\sigma_{\\infty, \\min}( I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}} \\geq \\frac{4.4 e\\pi }{n}\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n-1}}> \\pi \\Big(\\frac{2\\sqrt{n}}{\\xi(n-1)\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n-1}},\n\\end{equation*}\nwhere here we have used Lemma \\ref{lem:multinumbercalculate1} for deriving the last inequality. Therefore, \n\\[\n\\max_{t=1, \\cdots, T} ||\\hat \\Phi \\hat \\alpha_t-\\Phi \\alpha_t||_2 > 2\\sqrt{n}\\sigma,\n\\]\nwhence we prove (\\ref{equ:onedmultinumberresultequ3}).\n\\end{proof}\n\n\\begin{thm}\\label{thm:onedsupportbound}\nSuppose that the measurements $\\vect Y_t$'s in (\\ref{equ:onedmultimodelsetting1}) are generated from $\\mu= \\sum_{j=1}^n a_j \\delta_{y_j}, y_j\\in \\mathbb R$, where $y_j$'s are in an interval $\\mathcal O$ of length $\\frac{c_0n\\pi}{\\Omega}$ with $c_0\\geq 1$ and satisfy\n\\begin{equation}\\label{equ:onedsepaconditionsupport}\nd_{\\min} := \\min_{p\\neq j}\\babs{y_p-y_j}\\geq \\frac{4.4c_0e\\pi }{\\Omega }\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}\n\\end{equation}\nwith  $\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\leq 1$. Moreover, for $\\hat \\mu_t = \\sum_{j=1}^n \\hat a_{j,t} \\delta_{\\hat y_j}, \\hat y_j \\in \\mathcal O$ satisfying $||\\mathcal F[\\hat \\mu_t] - \\vect Y_t||_{\\infty}< \\sigma, t=1, \\cdots, T$, after reordering the $\\hat y_j$'s, we have \n\t\\begin{equation}\n\t\\Big|\\hat y_j-y_j\\Big|<\\frac{d_{\\min}}{2},\n\t\\end{equation} \n\tand \n\t\\begin{equation}\n\t\\Big|\\hat y_j-y_j\\Big| < \\frac{C(n)}{\\Omega}SRF^{n-1}\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}, \\quad 1\\leq j\\leq n,\n\t\\end{equation}\n\twhere $C(n)=2^n\\sqrt{2\\pi}nc_0^{n-1}e^{n}$ and $SRF = \\frac{\\pi}{\\Omega d_{\\min}}$ is the super-resolution factor.\n\\end{thm}\n\\begin{proof}\nLet $\\mu_t = \\sum_{j=1}^n a_j I_t(y_j)\\delta_{y_j}$. Similar to the proof of Theorem \\ref{thm:l0normrecovery0}, we consider \n\\begin{equation*}\n\\left(\\mathcal F [\\hat \\mu_t](\\omega_1), \\mathcal F [\\hat \\mu_t](\\omega_{2}), \\cdots,\\mathcal F [\\hat \\mu_t](\\omega_{n})\\right)^\\top -\\left(\\mathcal F [\\mu_t](\\omega_1), \\mathcal F [\\mu_t](\\omega_{2}), \\cdots,\\mathcal F [\\mu_t](\\omega_{n})\\right)^\\top =\\hat \\Phi \\hat \\alpha_t- \\Phi \\alpha_t, \n\\end{equation*}\nwhere $\\omega_j = (j-1)h-\\Omega$ with $h = \\frac{\\Omega}{c_0n}$, $\\hat \\alpha_t= (\\hat a_{1,t},\\cdots, \\hat a_{n,t})^\\top$, $\\alpha_t=(I_t(y_1)a_1, \\cdots, I_{t}(y_n)a_n)^\\top$ and \n\\begin{equation*}\n\\hat \\Phi= \\left(\n\\begin{array}{ccc}\ne^{i\\hat y_1\\omega_1}&\\cdots& e^{i\\hat y_n\\omega_1}\\\\\ne^{i\\hat y_1\\omega_{2}} &\\cdots& e^{i\\hat y_n\\omega_{2}}\\\\\n\\vdots&\\vdots&\\vdots\\\\\ne^{i\\hat y_1\\omega_{n}}&\\cdots& e^{i\\hat y_n \\omega_{n}}\\\\\n\\end{array}\n\\right), \\quad \\Phi=\\left(\n\\begin{array}{ccc}\ne^{i y_1\\omega_1}&\\cdots& e^{i y_n\\omega_1}\\\\\ne^{i y_1\\omega_{2}} &\\cdots& e^{i y_n\\omega_{2}}\\\\\n\\vdots&\\vdots&\\vdots\\\\\ne^{i y_1\\omega_{n}}&\\cdots& e^{i y_n \\omega_{n}}\\\\\n\\end{array}\n\\right).\n\\end{equation*}\nBy the constraint on the noise, it is clear that \n\\[\n\\max_{t=1, \\cdots, T}||\\hat \\Phi \\hat \\alpha_t- \\Phi\\alpha_t||_{2}<2\\sqrt{n}\\sigma.\n\\]\nLet $\\theta_j = y_j h$ and $\\hat \\theta_j = \\hat y_j h.$\nSimilar to the proof of Theorem \\ref{thm:l0normrecovery0}, we can prove that, after reordering $\\hat \\theta_j$'s,\n\\begin{equation} \\label{equ:onedmultiupperboundsupportlimithm1equ7}\n\\Big|\\hat \\theta_{j}-\\theta_j\\Big|_{2\\pi}< \\frac{\\theta_{\\min}}{2}, \\text{ and } \\Big|\\hat \\theta_{j}-\\theta_j\\Big|_{2\\pi}< \\frac{2^n\\sqrt{n}\\pi^{n}}{(n-2)!(\\theta_{\\min})^{n-1}}  \\frac{1}{\\sigma_{\\infty, \\min}(I)} \\frac{\\sigma}{m_{\\min}},\\ j=1,\\cdots,n.\n\\end{equation}\nFinally, we estimate $|\\hat y_j - y_j|$. Since $|\\hat \\theta_{j}-\\theta_j|_{2\\pi}< \\frac{\\theta_{\\min}}{2}$ and $\\hat y_j$'s, $y_j$'s are in $\\mathcal O$, we have after reordering the $\\hat y_j$'s,\n$$|\\hat y_j-y_j|< \\frac{d_{\\min}}{2}.$$\nOn the other hand,  $\\Big|\\hat y_j-y_j\\Big| = \\frac{nc_0}{\\Omega}\\Big|\\hat \\theta_j -\\theta_j\\Big|_{2\\pi}$. \t\nCombining (\\ref{equ:onedmultiupperboundsupportlimithm1equ7}) and (\\ref{equ:stirlingformula}), a direct calculation shows that\n\\begin{align*}\n\\Big|\\hat y_j-y_j\\Big|< \\frac{C(n)}{\\Omega} (\\frac{\\pi}{\\Omega d_{\\min}})^{n-1} \\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}, \n\\end{align*}\nwhere $C(n)=2^n\\sqrt{2\\pi}nc_0^{n-1}e^{n}$.\n\\end{proof}\n\n\n\\subsection{Projection lemmas}\nNext we introduce two auxiliary lemmas whose ideas are from \\cite{liu2021mathematicalhighd}. We introduce some notation. For $0<\\theta\\leq \\frac{\\pi}{2}$ and $N=\\lfloor \\frac{\\pi}{\\theta} \\rfloor$, we denote the unit vectors in $\\mathbb R^2$ by\n\\begin{align}\\label{equ:vectorlist1}\n\\vect v(\\tau \\theta)= \\big(\\cos(\\tau \\theta), \\sin(\\tau \\theta)\\big)^T, \\quad 1\\leq \\tau \\leq N.\n\\end{align}\nIt is obvious that there are $N$ different unit vectors of the form (\\ref{equ:vectorlist1}). \n\nFor a vector $\\vect v \\in \\mathbb R^2$, we denote $\\mathcal P_{\\vect v}$ the projection to the one-dimensional space spanned by $\\vect v$. We have the following lemmas. \n\n\\begin{lem}\\label{lem:highdsupportproject1}\n\tLet $n\\geq 2$ and $\\vect{y}_1, \\cdots,  \\vect{y}_n$ be $n$ different points in $\\mathbb R^2$. Let $\\ d_{\\min}=\\min_{p\\neq j}||\\vect{y}_p-\\vect{y}_j||_2$ and $\\Delta = \\frac{\\pi}{(n+2)(n+1)}$. Then there exist  $n+1$ unit vectors $\\vect{v}_q$'s such that $0 \\leq \\vect v_p \\cdot \\vect v_j \\leq \\cos(2\\Delta)$ for $p\\neq j$ and\n\t\\begin{equation}\\label{equ:highdprojectionlower3}\n\t\\min_{p\\neq j, 1\\leq p, j \\leq n}||\\mathcal P_{\\vect v_q}(\\vect{y}_p)-\\mathcal P_{\\vect v_q}(\\vect{y}_j)||_2\\geq \\frac{2\\Delta d_{\\min}}{\\pi},\\quad  q=1,\\cdots, n+1. \n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\tNote that there are at most $\\frac{n(n-1)}{2}$ different vectors of the form $\\vect u_{pj}= \\vect y_p -\\vect y_j, p<j$. For each $\\vect u_{pj}$, consider the set $N(\\vect{u}_{pj}, \\Delta)= \\Big\\{ \\vect v \\Big| ||\\vect v||_2=1, \\vect v \\in \\mathbb R^2, |\\vect v \\cdot \\vect{u}| <||\\vect{u}||_2 \\sin\\Delta \\Big\\}$. Let $\\theta = 2\\Delta$ and introduce the vectors $\\vect v(\\tau \\theta)$ as in (\\ref{equ:vectorlist1}). It is clear that $0\\leq \\vect v(\\tau_1 \\theta) \\cdot \\vect v(\\tau_2 \\theta) \\leq \\cos(\\theta)$ for $\\tau_1\\neq \\tau_2$. Thus if $|\\vect v(\\tau_1 \\theta)\\cdot \\vect{u}| <||\\vect{u}||_2 \\sin\\Delta$, then, for other $\\tau_2 \\neq \\tau_1$, we have $|\\vect v(\\tau_2 \\theta) \\cdot \\vect{u}| \\geq ||\\vect{u}||_2 \\sin\\Delta$. We can derive that each set $N(\\vect{u}_{pj}, \\Delta)$ contains \n\tat most one of the vectors $\\vect v(\\tau \\theta)$'s. As a result, $\\cup_{p<j, 1\\leq j, p\\leq n}N(\\vect{u}_{pj}, \\Delta)$ contains at most $\\frac{n(n-1)}{2}$ vectors of the form $\\vect v(\\tau \\theta)$. \n\t\n\tNext recall that there are $N$ different vectors of the form in (\\ref{equ:vectorlist1}), where $N = \\lfloor \\frac{\\pi}{\\theta} \\rfloor $. Since \n\t\\[\n\t\\theta=2 \\Delta = \\frac{2\\pi}{(n+2)(n+1)}, \n\t\\]\n\twe have\n\t\\begin{align*}\n\tN\\geq \\lfloor \\frac{\\pi}{\\theta} \\rfloor = \\frac{(n+2)(n+1)}{2} > \\frac{(n+1)^2}{2}.\n\t\\end{align*}\n\tNote that $\\frac{(n+1)^2}{2} - \\frac{n(n-1)}{2} = n+1$, we can find $n+1$ vectors of the form $\\vect v(\\tau \\theta)$ that are not contained in the set  $\\cup_{p<j, 1\\leq j, p\\leq n} N(\\vect u_{pj}, \\Delta)$. That is, we can find $n+1$ unit vectors, say, $\\vect v_q$, $1\\leq q \\leq n+1$, which satisfy (\\ref{equ:highdprojectionlower3}).\n\\end{proof}\n\n\n\\begin{lem}\\label{lem:highdsupportproject2}\nFor a vector $\\vect{u}\\in \\mathbb R^2$, and two unit vectors $\\vect{v}_1, \\vect{v}_2\\in \\mathbb R^2$ satisfying $0 \\leq \\vect v_1 \\cdot \\vect v_2 \\leq \\cos(\\theta)$, we have \n\\begin{equation}\\label{equ:highdprojectionlower4}\n|\\vect v_1\\cdot \\vect u|^2+|\\vect v_2\\cdot \\vect u|^2  \\geq (1-\\cos(\\theta))||\\vect u||_2^2.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nFor $\\vect{u}\\in \\mathbb R^2$, and two unit vectors $\\vect{v}_1, \\vect{v}_2\\in \\mathbb R^2$ satisfying $0 \\leq \\vect v_1 \\cdot \\vect v_2 \\leq \\cos \\theta$, we have \n\\begin{equation}\\label{equ:projectionlower3}\n\\btwonorm{(\\vect{v}_1\\cdot \\vect u,\\  \\vect{v}_2\\cdot \\vect u)^T}^2\n= \\btwonorm{\\begin{pmatrix}\n\t\\vect{v}_1^T\\\\\n\t\\vect{v}_2^T\n\t\\end{pmatrix}\\cdot \\vect u}^2 \\geq \\sigma_{\\min}^2(\\begin{pmatrix}\n\\vect{v}_1^T\\\\\n\\vect{v}_2^T\n\\end{pmatrix})||\\vect u||_2^2 \\geq (1-\\cos \\theta)||\\vect u||_2^2,\n\\end{equation}\nwhere the last inequality follows from calculating $\\sigma_{\\min}(\\begin{pmatrix}\n\\vect{v}_1^T\\\\\n\\vect{v}_2^T\n\\end{pmatrix})$.\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:twodl0normrecovery0}}\n\\begin{proof}\nWe only prove the theorem for problem (\\ref{prob:twodl0minimization}). The other cases can be proved in a similar manner. Let the measurements $\\vect Y_t$'s be generated by $\\mu=\\sum_{j=1}^n a_j \\delta_{\\mathbf y_j}, \\vect y_j \\in \\mathcal O$ satisfying the minimum separation condition \n\\begin{equation}\\label{equ:twodsupportlimithm0equ-2}\nd_{\\min}^{(2)}:=\\min_{p\\neq j, 1\\leq p, j\\leq n}\\btwonorm{\\mathbf y_p- \\mathbf y_j} \\geq \\frac{2.2c_0 e \\pi (n+2)(n+1)}{\\Omega}\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}.\n\\end{equation}\nLet $\\Delta = \\frac{\\pi}{(n+2)(n+1)}$. By Lemma \\ref{lem:highdsupportproject1}, there exist $n+1$ unit vectors $\\vect v_q$'s so that $$0\\leq \\vect v_p \\cdot \\vect v_j \\leq \\cos 2\\Delta, 1\\leq p<j \\leq n,$$ and for each $q$, \n\\begin{equation}\\label{equ:proofhighdsupport1equ0}\n\\min_{p\\neq j}\\btwonorm{\\mathcal P_{\\vect v_q}(\\vect{y}_p)-\\mathcal P_{\\vect v_q}(\\vect{y}_j)}\\geq  d_{\\min}^{(1)},\n\\end{equation}\nwhere\n\\begin{align}\\label{equ:twodsupportlimithm0equ-1}\nd_{\\min}^{(1)} = \\min_{p\\neq j}\\btwonorm{\\vect y_p-\\vect y_j} \\frac{2\\Delta}{\\pi}=\\frac{2d_{\\min}^{(2)}}{(n+2)(n+1)} \\geq \\frac{4.4c_0e \\pi }{\\Omega}\\Big(\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}\\Big)^{\\frac{1}{n}}. \n\\end{align}\nBecause the projected point scatterers in these one-dimensional subspaces are separated by a distance beyond (\\ref{equ:twodsupportlimithm0equ-1}), by the constraints on the measurements and Theorem \\ref{thm:onednumberbound}, any solution to (\\ref{prob:twodl0minimization}) must contain $n$ point scatterers. Assume that $\\rho=\\sum_{j=1}^n \\hat a_j \\delta_{\\mathbf {\\hat y}_j}, \\mathbf {\\hat y}_j\\in \\mathcal O$ is a solution to (\\ref{prob:twodl0minimization}). Then we consider $$\\mu_t = I_t \\mu = \\sum_{j=1}^{n} a_j I_t(\\vect y_j) \\delta_{\\vect{ y}_j}, \\quad \\hat \\mu_t = I_t \\rho = \\sum_{j=1}^{n} \\hat a_{j,t} \\delta_{\\mathbf {\\hat y}_j},$$ and the projected measures \n$$\\sum_{j=1}^n a_jI_t(\\vect y_j) \\delta_{\\mathcal P_{\\vect v_q}(\\mathbf{y}_j)}, \\quad \\sum_{j=1}^n a_{j,t} \\delta_{\\mathcal P_{\\vect v_q}(\\mathbf {\\hat y}_j)}$$ in the one-dimensional subspace spanned by $\\vect v_q$. We also consider the corresponding measurements $\\vect Y_t(\\vect \\omega)$, $\\vect \\omega$ in the subspace spanned by $\\vect v_q$. It is clear that for each $q$, $\\mathcal P_{\\vect v_q}(\\mathbf{y}_j)$'s are in an interval (i.e., in a one-dimensional subspace) of length $ \\frac{c_0n\\pi}{\\Omega}$ and the separation condition (\\ref{equ:twodsupportlimithm0equ-1}) is satisfied. On the other hand, the measurement constraints in (\\ref{prob:twodl0minimization}) still hold for the measurements $\\vect Y_t(\\vect \\omega)$ in that subspace. By Theorem \\ref{thm:onedsupportbound}, we can conclude that for each $q$, we have a permutation $\\tau_q$ of $\\big\\{1,\\cdots, n \\big\\}$ so that\n\\begin{equation}\\label{equ:proofhighdsupport1equ2}\n\\btwonorm{\\mathcal P_{\\vect v_q}(\\mathbf{\\hat y}_{\\tau_q(j)})- \\mathcal P_{\\vect v_q}(\\mathbf{y}_j)}< \\frac{C(n)}{\\Omega}\\Big(\\frac{\\pi}{d_{\\min}^{(1)}\\Omega}\\Big)^{n-1}\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}, \\quad 1\\leq j\\leq n,\n\\end{equation}\nwhere $C(n) = 2^n\\sqrt{2\\pi}nc_0^{n-1}e^{n}$. \n\nNote that, for fixed $j$ in (\\ref{equ:proofhighdsupport1equ2}), we have $(n+1)$ different $\\tau_q(j)$'s, while $\\mathbf{\\hat y}_p$'s take at most $n$ values. Therefore, by the pigeonhole principle, for each fixed $\\vect y_j$, we can find two different $q$'s, say, $q_1$ and $q_2$, such that $\\mathbf{\\hat y}_{\\tau_{q_1}(j)} =\\mathbf{\\hat y}_{\\tau_{q_2}(j)} = \\mathbf{\\hat y}_{p_j}$ for some $p_j$. \nSince $0\\leq \\vect v_{q_1} \\cdot \\vect v_{q_2} \\leq \\cos 2\\Delta$,  we can apply Lemma \\ref{lem:highdsupportproject2} to get\n\\[\n\\btwonorm{\\mathbf {\\hat y}_{p_j} - \\vect y_j}< \\frac{\\sqrt{2}}{\\sqrt{1-\\cos(2\\Delta)}} \\frac{C(n)}{\\Omega}\\Big(\\frac{\\pi}{d_{\\min}^{(1)}\\Omega}\\Big)^{n-1}\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}, \\quad 1\\leq j\\leq n.\n\\]\nUsing the inequality \n$$ 1-\\cos 2\\Delta\\geq \\frac{8}{\\pi^2}\\Delta^2\\geq \\frac{8}{(n+2)^2(n+1)^2},$$ we further obtain\n\\begin{equation}\\label{equ:proofhighdsupport1equ3}\n\\btwonorm{\\mathbf {\\hat y}_{p_j} - \\vect y_j}< \\frac{(n+2)(n+1)C(n)}{2\\Omega}\\Big(\\frac{\\pi}{d_{\\min}^{(1)}\\Omega}\\Big)^{n-1}\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}, \\quad 1\\leq j\\leq n.\n\\end{equation}\nCombining together (\\ref{equ:twodsupportlimithm0equ-2}) and (\\ref{equ:twodsupportlimithm0equ-1}), we can verify by a direct calculation that\n\\[\n\\btwonorm{\\mathbf {\\hat y}_{p_j} - \\vect y_j}< \\frac{1}{2} d_{\\min}^{(2)}, \\quad  j =1, \\cdots, n. \n\\]\nThus we can reorder the $\\mathbf {\\hat y}_{j}$'s so that \n\\[\n\\btwonorm{\\mathbf {\\hat y}_{j} - \\vect y_j}< \\frac{1}{2} d_{\\min}^{(2)}, \\quad j=1, \\cdots, n. \n\\]\nBy (\\ref{equ:proofhighdsupport1equ3}), we obtain that \n\\[\n\\btwonorm{\\mathbf {\\hat y}_{j} - \\vect y_j} < \\frac{((n+2)(n+1)\/2)^nC(n)}{\\Omega}\\Big(\\frac{\\pi}{d_{\\min}^{(2)}\\Omega}\\Big)^{n-1}\\frac{1}{\\sigma_{\\infty, \\min}(I)}\\frac{\\sigma}{m_{\\min}}, \\quad 1\\leq j\\leq n,\n\\]\nwhich follows from (\\ref{equ:proofhighdsupport1equ3}) and the fact that $d_{\\min}^{(1)}=\\frac{2d_{\\min}^{(2)}}{(n+2)(n+1)}$. \n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn recent years compelling evidence for neutrino oscillations has been found in experiments with atmospheric~\\cite{SKatm}, solar~\\cite{sol,SKsolar,SNO1,SNO2,SNO3,SNOsalt}, reactor~\\cite{KamLAND} and long-baseline accelerator neutrinos~\\cite{K2K,MINOS}. Two mass squared differences, $\\Delta m^2_{ji} \\equiv m_j^2 - m_i^2$, have been measured with good accuracy, their present best fit values being $|\\Delta m^2_{31} |= 2.5 \\times 10^{-3} \\ \\mathrm{eV}^2$ and $|\\Delta m^2_{21} |= 8.0 \\times 10^{-5} \\ \\mathrm{eV}^2$. \nIn addition, explaining the experimental data in terms of neutrino oscillations requires two large mixing angles in the leptonic mixing matrix $U$. Their best fit values are:\n $\\sin^2 \\theta_{12} = 0.30$ and $\\sin^2 2 \\theta_{23} =1$, see Refs.~\\cite{thomas,newfit,concha}.  \n Despite the remarkable recent progress in our understanding of neutrino physics, fundamental questions remain unanswered. It is crucial to establish the nature of neutrinos - whether they are Dirac or Majorana particles, the neutrino mass ordering, the absolute neutrino mass scale, the value of the unknown mixing angle $\\theta_{13}$, the presence or absence of CP violation in the leptonic sector, and the precise values of the already known oscillation parameters. This information will help shed light on the physics beyond the Standard Model responsible for neutrino masses and for the leptonic mixing structure.\n \nIn order to achieve these goals, very sensitive neutrino experiments will be required. In particular, long baseline oscillation experiments are expected to play an important role in providing precision measurements of the neutrino oscillation parameters, the CP-violating phase $\\delta$, and a determination of the neutrino mass ordering.\n\nNeutrino Factories~\\cite{geer}, in which a neutrino beam is generated from muons decaying within the straight sections of a storage ring, have been studied extensively in the past, and have been shown to be sensitive tools for studying neutrino oscillation physics~\\cite{geer,nf1,nf2,nf4,nf5,nf6,silver,study1-physics,nf8,study2,yo,nf9,GeerMenaPascoli,hubernew}. In a Neutrino Factory far detector, the experimental signature for the so called \\emph{golden channel}~\\cite{nf5} is the presence of a \\emph{wrong-sign} muon~\\cite{geer,nf1}, i.e. a muon with opposite sign to the muons stored in the neutrino factory. Wrong-sign muons result from $\\nu_{e} \\rightarrow \\nu_{\\mu}$ oscillations, and can be used to measure the unknown mixing angle $\\theta_{13}$, determine the neutrino mass hierarchy, and search for CP violation in the neutrino sector. This physics program requires the detection of charged current (CC) muon-neutrino interactions, and the measurement of the sign of the produced muon. If the interacting neutrinos have energies of more than a few GeV, standard neutrino detector technology, based on large magnetized sampling calorimeters, can be used to measure wrong-sign muons with high efficiency and very low backgrounds. This has been shown to work for Neutrino Factories with energies of about 20 GeV or greater~\\cite{nf5,ISS-Detector Report,study1-physics}. \n\nLower energy Neutrino Factories~\\cite{GeerMenaPascoli}, which store muons with energies $<10$~GeV, require a detector technology that can detect lower energy muons. Recently ideas have emerged for a Neutrino Factory detector based on a fully active calorimeter within a potentially affordable large volume magnet. These ideas encourage consideration of low energy Neutrino Factories. Initial studies~\\cite{GeerMenaPascoli}, based on a first guess for the performance of a low energy Neutrino Factory detector, suggested that a Neutrino Factory with an energy of about 4 GeV would enable very precise measurements of the neutrino mixing parameters. \nIn the present paper, we consider in more detail the expected performance of a low energy Neutrino Factory detector, and update and extend our previous studies to include a more realistic detector model and a more comprehensive study of systematic effects. \nIn particular we exploit the low energy threshold of the detector and\nmake a very conservative estimate for its energy resolution which, together with the broad spectrum of the neutrino factory beam, facilitates the elimination of degeneracies~\\cite{FL96,MN01,BMWdeg,deg}. It is well known that even a very precise measurement of the appearance probability for neutrinos and antineutrinos at a fixed $L\/E$ allows different solutions for $(\\theta_{13}, \\mathrm{sign} (\\Delta m^2_{13}), \\delta)$, weakening severely the sensitivity to these parameters. Many strategies have been advocated to resolve this issue which in general involve another detector~\\cite{MN97,silver,BMW02off,SN1,twodetect,SN2,T2kk} or the combination with another experiment~\\cite{otherexp1,BMW02,HLW02,MNP03,otherexp,mp2,HMS05,Choubey,yo,nf9,huber2,lastmine}.\nUsing the energy dependence of the signal in the low energy Neutrino Factory, \nwe find that a 4 GeV Neutrino Factory can unambiguously determine all of the neutrino oscillation parameters with good precision provided $\\sin^2 2\\theta_{13}>$ few $\\cdot 10^{-3}$. Hence a low energy Neutrino Factory would be a precision tool for both large and small $\\theta_{13}$.\n  \nIn Section II we describe the design for the low-threshold detector and its performance. In Section III, we discuss in detail the physics reach of the proposed setup. We first consider the disappearance $\\nu_\\mu$ signal in order to determine precisely the value of the atmospheric mass squared difference and, possibly, the type of hierarchy even for $\\theta_{13}=0$.\nThen, we consider the appearance signals $\\nu_e \\rightarrow \\nu_\\mu$ and $\\bar{\\nu}_e \\rightarrow \\bar{\\nu}_\\mu$, which depend on $\\theta_{13}$, $\\delta$ and the type of neutrino mass ordering. \nWe perform a detailed numerical simulation and discuss the sensitivity of the low-energy Neutrino Factory to these parameters.\nIn Section IV, we draw our conclusions.\n\n\\section{Detector design and performance}\n\n\nA totally active scintillator detector (TASD) has been proposed for a Neutrino Factory, and results from a first study of its expected performance are  described in the recent International Scoping Study Report~\\cite{ISS-Detector Report}. Using a TASD for neutrino physics is not new.  Examples are KamLAND~\\cite{KamLAND}, which has been operating for several years, and the proposed NO$\\nu$A detector~\\cite{Nova}, which is a $15-18$~Kton liquid scintillator detector that will operate off-axis to the NuMI beam line~\\cite{Numi} at Fermilab. Note that, unlike KamLAND or NO$\\nu$A, the TASD we are investigating for the low energy Neutrino Factory is magnetized and has a segmentation that is approximately 10 times that of \\mbox{NO$\\nu$A}. Magnetization of such a large volume ($>30,000$~m$^3$) is the main technical challenge in designing a TASD for a Neutrino Factory, although R\\&D to reduce the detector cost (driven in part by the large channel count, $7.5 \\times 10^6$) is also needed. \n\nThe Neutrino Factory TASD we are considering consists of long plastic scintillator bars with a triangular cross-section arranged in planes which\n make x and y measurements (we plan to also consider an x-u-v readout scheme).  Optimization of the cell cross section still needs further study since a true triangular cross section results in tracking anomalies at the corners of the triangle.  The scintillator bars have a length of $15$~m and\nthe triangular cross-section has a base of $3$~cm and a height of $1.5$~cm. We have considered a design using liquid as in \\mbox{NO$\\nu$A}, but, compared to \\mbox{NO$\\nu$A}, the cell size is small (\\mbox{NO$\\nu$A}  uses a $4\\times 6$~cm$^2$ cell) and the non-active component due to the PVC extrusions that hold the liquid becomes quite large (in \\mbox{NO$\\nu$A}, the scintillator is approximately $70\\%$ of the detector mass). Our design is\nan extrapolation of the MINER$\\nu$A experiment~\\cite{minerva_www} which in turn was an extrapolation of the D0\npreshower detectors~\\cite{D0}.  We are considering a detector mass of approximately $35$~Kton (dimensions $15 \\times 15 \\times 150$~m).\nWe believe that an air-core solenoid can produce the field required ($0.5$~Tesla) to do the physics.\n\nAs was mentioned above, magnetizing the large detector volume presents the main technical challenge for a Neutrino Factory TASD.  Conventional\nroom temperature magnets are ruled out due to their prohibitive power consumption, and conventional superconducting magnets are believed to be too expensive, due to the cost of the enormous cryostats needed in a conventional superconducting magnet design. In order to eliminate \nthe cryostat, we have investigated a concept based on the  superconducting transmission line (STL) that was \ndeveloped for the Very Large Hadron Collider superferric magnets~\\cite{VLHC}. The solenoid windings now consist of this superconducting cable which is confined \nin its own cryostat (Fig.~\\ref{fig:STL}).  Each solenoid ($10$ required for the full detector) consists of $150$ turns and requires $7500$ m of cable. There is \nno large vacuum vessel and thus no large vacuum loads which make the cryostats for large conventional superconducting magnets very expensive. \n\nThe Neutrino Factory TASD response has been simulated with GEANT4 version 8.1 (Fig.~\\ref{fig:tasd}).  The GEANT4 model of the detector included each of the individual scintillator bars, but did not include edge effects on light collection, or the effects of a central wavelength shifting fiber. A uniform 0.5 Tesla magnetic field was simulated.\n \nSamples of isolated muons in the range of momentum between $100$~MeV$\/c$ and $15$~GeV$\/c$ were simulated to allow the determination of the momentum resolution and charge identification capabilities. The NUANCE~\\cite{Nuance} event generator was also used to simulate 1 million $\\nu_e$ and 1 million $\\nu_\\mu$ interactions. Events were generated in $50$ mono-energetic neutrino energy bins between $100$~MeV and $5$~GeV. The results that follow only have one thousand events processed through the GEANT4 simulation and reconstruction.\n \nThe detector response was simulated assuming a light yield consistent with MINER$\\nu$A measurements and current photo detector performance~\\cite{Nova}. In addition, a 2 photo-electron energy resolution was added through Gaussian smearing to ensure that the energy resolution used in the following physics analysis would be a worst-case estimate. Since  a complete pattern recognition algorithm was beyond the scope of our study, for our analysis the Monte Carlo information was used to aid in pattern recognition. All digitised hits from a given simulated particle where the reconstructed signal was above 0.5 photo electrons were collected. When using the isolated particles, hits in neighboring x and y planes were used to determine the 3 dimensional position of the particle. The position resolution was found to be approximately $4.5$~mm RMS with a central Gaussian of width $2.5$~mm~\\footnote{At this stage, the simulation does not take into account light collection inefficiencies in the corners of the base of the triangle.}. These space points were then passed to the RecPack Kalman track fitting package~\\cite{recpack}. \n \nFor each collection of points, the track fit was performed with an assumed positive and negative charge. The momentum resolution and charge misidentification rates were determined by studying the fitted track in each case which had the better $\\chi^2$ per degree of freedom. Figure~\\ref{fig:tasd_mom} shows the momentum resolution as a function of muon momentum. The tracker achieves a resolution of better than $10\\%$ over the momentum range studied. Figure~\\ref{fig:Track}(a) shows the efficiency for reconstructing positive muons as a function of the initial muon momentum. The detector becomes fully efficient above $400$~MeV.\n \nThe charge mis-identification rate was determined by counting the rate at which the track fit with the incorrect charge had a better $\\chi^2$ per degree of freedom than that with the correct charge. Figure~\\ref{fig:Track}(b) shows the charge mis-identification rate as a function of the initial muon momentum.\n \nThe neutrino interactions were also reconstructed using the aid of the Monte Carlo information for pattern recognition. In an attempt to produce some of the effects of a real pattern recognition algorithm on the detector performance, only every fourth hit was collected for track fitting. Tracks were only fit if $10$ such hits were found from a given particle. The Monte Carlo positions were smeared (Gaussian smearing using the $4.5$~mm RMS determined previously) and passed to the Kalman track fit. The reconstruction returned:\n\\begin{itemize}\n\\item The total momentum vector of all fitted tracks,\n\\item The momentum vector of the muon (muon ID from MC truth),\n\\item The reconstructed and truth energy sum of all the hits that were not in a particle that was fitted, and \n\\item The reconstructed energy sum of all hits in the event.\n\\end{itemize}\n\nThe $\\nu_{\\mu}$ CC event reconstruction efficiency as a function of neutrino energy is shown in Fig.~\\ref{fig:NuMuCC}(a).  The fraction of\n$\\nu_{\\mu}$ CC events with a reconstructed muon is shown in Fig.~\\ref{fig:NuMuCC}(b).  In this figure the bands represent the limits of the \nstatistical errors for this analysis.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=6.5in]{STL.eps}\n\\end{center}\n\\caption[]{\\textit{Diagram of Superconducting Transmission Line design.}}\n\\label{fig:STL}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=4in]{TASD_sch.eps}\n\\end{center}\n\\caption[]{\\textit{Schematic of Totally Active Scintillator Detector.}}\n\\label{fig:tasd}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=4in]{pres.eps}\n\\end{center}\n\\caption[]{\\textit{Momentum resolution as a function of the muon momentum.}}\n\\label{fig:tasd_mom}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{ll}\n\\includegraphics[width=3in]{fig5a.eps}&\\hskip 0.cm\n\\includegraphics[width=3in]{fig5b.eps}\\\\\n\\hskip 4.truecm\n{\\small (a)}            &\n\\hskip 4.truecm\n{\\small (b)}       \\\\  \n\\end{tabular}\n\\end{center}\n\\caption{\\textit{(a) Efficiency for reconstructing positive muons. (b)\nMuon charge mis-identification rate as a function of the initial muon momentum.}}\n\\label{fig:Track}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{ll}\n\\includegraphics[width=3in]{eff_vs_e.eps}&\\hskip 0.cm\n\\includegraphics[width=3in]{numu_frac.eps}\\\\\n\\hskip 4.truecm\n{\\small (a)}            &\n\\hskip 4.truecm\n{\\small (b)}       \\\\  \n\\end{tabular}\n\\end{center}\n\\caption{\\textit{(a) Reconstruction efficiency of NuMu CC events as a function of neutrino interaction energy. (b) Fraction of NuMu CC events with a reconstructed muon.}}\n\\label{fig:NuMuCC}\n\\end{figure}\n\nBased on these initial Neutrino Factory TASD studies, in our phenomenological analysis we assume the detector has an effective threshold for measuring muon neutrino CC events at $E_\\nu = 500$~MeV, above which it has an energy independent efficiency of $73\\%$. The $73\\%$ efficiency is primarily driven by the neutrino interaction kinematics, not by the detector tracking efficiency.  No charge-ID criterion is applied here. The charge misidentification rate information is used as input into the effect of backgrounds on the analysis.\n\nWe note that, to fully understand the backgrounds in the TASD requires a simulation that includes neutrino interactions and a full event reconstruction.  Although this is beyond the scope of the present study, a consideration of backgrounds in the well studied Magnetized Fe-Scintillator detector proposed for the high-energy Neutrino Factory~\\cite{nf5} and The International Scoping Study for a Neutrino Factory~\\cite{ISS-Detector Report} motivates the $10^{-3}$ background (contamination) assumption used in this paper for the TASD. Before kinematic cuts, the main backgrounds for the Fe-Scintillator detector are muon charge mis-ID, charm decay, pion and kaon decay, and are all of comparable order: $1-5\\times 10^{-4}$.  For the TASD at a low energy Neutrino Factory the muon charge mis-ID rate (Fig.~\\ref{fig:Track}(b)) and the charm decay background is suppressed (at the level $4-8\\times 10^{-5}$) due to the low energy beam.  Pion and Kaon decay in flight become the main background concerns at the $1-5\\times 10^{-4}$ level.  A figure of merit for comparing the TASD to a conventional Magnetized Fe-Scintillator detector is the ratio of their respective particle decay lengths to interaction lengths.  For TASD the ratio is about $1$; for the Magnetized Fe-Scintillator detector it is approximately $8$.  So naively we can conclude that the decay background in TASD will be 10 times worse than in the conventional detector ignoring any kinematic or topological cuts.  However the TASD will have vastly superior kink detection to identify decay-in-flight.  For example we will typically have $40$ hits on the pion track before decay.  In addition TASD will have continuous $dE\/dx$ measurements along the track and better overall energy resolution.  We believe that these properties will allow us to control backgrounds to the $10^{-3}$ level or better.\n\n\n\n\\section{Physics reach of the low energy Neutrino Factory}\n\nWe have previously mentioned that, by exploiting the energy dependence of the signal, it is possible to extract from the measurements the correct values of $\\theta_{13}$ and $\\delta$, and eliminate the additional solutions arising from discrete ambiguities. In the present study, we include the detector simulation results described in the previous section, which suggests a lower energy threshold (500~MeV) than previously assumed~\\cite{GeerMenaPascoli}, and an energy resolution $dE\/E=30\\%$~\\footnote{We have assumed a very conservative $dE\/E=30\\%$ because at this time the\nsimulation work has not yet produced a number for the TASD.  Based on \\mbox{NO$\\nu$A} \nresults, we expect the TASD $dE\/E$ to be better than $6\\%$ at 2~GeV.}. Above threshold, the detector efficiency for muon neutrino CC events is taken to be 73\\%.\n\nIn the following we consider the representative baseline $L=1480$~km, which corresponds to the distance from Fermilab to the Henderson mine. However, we believe that the TASD will not require operation deep underground in order to remove backgrounds. Results are similar for other baselines in the 1200--1500~km range. The results are presented for the high-statistics scenario described in~\\cite{GeerMenaPascoli} as well as for a more aggressive scenario which improves the statistics of the old high-statistics scenario by a factor of three, to quantify the benefits of increased detector sizes and\/or stored-muon luminosities. The high-statistics scenario corresponds to $1 \\times 10^{23}$~Kton-decays (10 years of data taking, with $5 \\times 10^{20}$ useful muon decays of each sign per year, and a detector fiducial mass times efficiency of 20~Kt). The more aggressive scenario corresponds to $3 \\times 10^{23}$~Kton-decays (which could correspond, for instance, to 10 years of data taking, with $1 \\times 10^{21}$ useful muon decays of each sign per year, and a detector fiducial mass times efficiency of 30~Kt).\n\nTable~\\ref{tab:tab1} shows the number of CC muon events expected in the two scenarios explored here for, respectively, the positive and negative muons stored in the Neutrino Factory. Notice that, in the absence of oscillations, there would be a few times $10^4$~$\\nu_e$ CC interactions, which would allow a search for $\\nu_e \\rightarrow \\nu_\\mu$ oscillations with probabilities below $10^{-4}$.\n\\begin{table}[thb]\n\\centering\n\\begin{tabular}{||c|c||c|c||c|c||}\n\\hline \\hline\n\\multicolumn{2}{||c||}{$E_{\\mu^{\\mp}}=$} & \n\\multicolumn{2}{c||} {$\\mu^{+}$} &  \n\\multicolumn{2}{c||}{$\\mu^{-}$}\\\\\n\\cline{3-6}\n\\multicolumn{2}{||c||}{$4.12$ GeV}& $N_{\\bar{\\nu}_\\mu}\/10^3$ & $N_{\\nu_e}\/10^3$  & \n$N_{\\nu_\\mu}\/10^3$ &$N_{\\bar{\\nu}_e}\/10^3$ \\\\  \n\\hline\\hline\nstatistics& $1$& $13$ & $22$ & $25$ & $11$\\\\\n\\cline{2-6}\n$(10^{23})$ Kt-decays\n   & $3$& $39$ & $66$ & $77$ & $34$\\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{\\it{ \nNeutrino and antineutrino charged currents interaction rates \nfor L = 1480~km, for the $10^{23}$~Kton-decays and the $3 \\times 10^{23}$~Kton-decays-statistics scenarios.}} \n\\label{tab:tab1}\n\\end{table}\n\nAll numerical results reported in the next subsections have been obtained with the exact formulae for the oscillation probabilities. Unless specified otherwise, we take the following central values for the remaining oscillation parameters: $\\sin^{2}\\theta_{12}=0.29$, $\\Delta m^2_{21} =  8 \\times 10^{-5}$ eV$^2$,  $|\\Delta m^2_{31}| = 2.5 \\times 10^{-3}$ eV$^2$ and  $\\theta_{23}=40^\\circ$. We show in Tables~\\ref{tab:tab2} and \\ref{tab:tab3}, for two representative values of $\\theta_{13}=1^\\circ$ and $8^\\circ$, and the CP phase $\\delta=0^\\circ, 90^\\circ, 180^\\circ$ and $270^\\circ$, the number of \\emph{wrong-sign} muon events in the two scenarios explored here, for, respectively, the positive and negative muons stored in the Neutrino Factory, for normal (inverted) hierarchy.\n\\begin{table}[bht]\n\\centering\n\\begin{tabular}{||c|c||c||c||}\n\\hline\\hline\nstatistics (Kt-decays) & $\\delta(^{o})$ & $\\mu^{+}$ stored (wrong-sign: $\\mu^-$)  &$\\mu^{-}$ stored (wrong-sign: $\\mu^{+}$) \\\\\n\\hline\\hline\n     & 0 & 880 (340) & 180 (520)\\\\\n\\cline{2-4}\n$1 \\times 10^{23}$     \n     & 90 & 1230 (505) & 90 (330)\\\\\n\\cline{2-4}\n     & 180 & 1000 (340) & 170 (440)\\\\\n\\cline{2-4}\n     & 270 & 645 (175) & 260 (625) \\\\\n\\hline\\hline  \n       & 0 &2640 (1020)& 540 (1550)\\\\ \n\\cline{2-4}\n$3 \\times 10^{23}$     \n     & 90 & 3700 (1520) &270 (990)\\\\\n\\cline{2-4}\n     & 180 &2990 (1020) & 510 (1310)\\\\\n\\cline{2-4}\n     & 270 &1930 (520) &780 (1870) \\\\\n\\hline\\hline \n\\end{tabular}\n\\caption{\\it \n{Wrong sign muon event rates for normal (inverted) hierarchy, assuming $\\nu_e\\to \\nu_\\mu$ ($\\bar{\\nu}_e \\to \\bar{\\nu}_\\mu$)\noscillations in a 20~Kt fiducial volume detector, for a L = 1480~km baseline. \nWe assume here $\\theta_{13}=8^{o}$, i.e. $\\sin^2 2\\theta_{13}\\simeq 0.076$. \nWe present the results for several possible values of the CP-violating phase $\\delta$  for both the two scenarios.}}\n\\label{tab:tab2}\n\\end{table} \n\\begin{table}[bht]\n\\centering\n\\begin{tabular}{||c|c||c||c||}\n\\hline\\hline\nstatistics (Kt-decays)& $\\delta(^{o})$ & $\\mu^{+}$ stored (wrong-sign: $\\mu^-$)  &$\\mu^{-}$ stored (wrong-sign: $\\mu^{+}$) \\\\\n\\hline\\hline\n     & 0 & 54 (50) &  27 (37) \\\\\n\\cline{2-4}\n$1 \\times 10^{23}$     \n     & 90 & 100 (70) & 13 (10)\\\\\n\\cline{2-4}\n     & 180 & 67 (50) & 70 (25)\\\\\n\\cline{2-4}\n     & 270 & 22 (30) & 37 (50) \\\\\n\\hline\\hline  \n       & 0 &160 (150)& 80 (110)\\\\ \n\\cline{2-4}\n$3 \\times 10^{23}$     \n     & 90 & 300 (210)&40 (30)\\\\\n\\cline{2-4}\n     & 180 &200 (150)& 230 (250)\\\\\n\\cline{2-4}\n     & 270 &65 (90) &110 (150)\\\\\n\\hline\\hline \n\\end{tabular}\n\\caption{\\it \n{As Table~\\protect\\ref{tab:tab2} but for  $\\theta_{13}=1^{o}$, i.e. $\\sin^2 2\\theta_{13}\\simeq 0.001$. \n}}\n\\label{tab:tab3}\n\\end{table} \n\n\nFor our analysis, we use the following $\\chi^{2}$ definition\n\\begin{equation}\n\\chi^2 = \\sum_{i,j} \\sum_{p,p'} \\; (n_{i,p} - N_{i,p}) C_{i,p:,j,p'}^{-1} (n_{j,p'} - N_{j,p'})\\,,\n\\end{equation}\n where $N_{i,\\pm}$ is the predicted number of muons for\n a certain oscillation hypothesis, $n_{i,p}$ are the simulated ``data'' from a Gaussian or Poisson smearing and $C$ is the $2 N_{bin} \\times 2 N_{bin}$ covariance matrix given by:\n\\begin{equation}\nC_{i,p:,j,p'}^{-1}\\equiv \\delta_{ij}\\delta_{pp'}(\\delta n_{i,p})^2 \n\\end{equation}\nwhere $(\\delta n_{i,p}) = \\sqrt{n_{i,p} + (f_{sys}\\cdot n_{i,p})^2}$\ncontains both statistical and a $2\\%$ overall systematic error ($f_{sys}=0.02$).\n\n\\subsection{Exploring the disappearance channel}\n\nConsider first the disappearance channels, already considered in the context of Neutrino Factories~\\cite{nf4,dis} and carefully explored in Ref.~\\cite{Stef}.\nIn Ref.~\\cite{GeerMenaPascoli} it was shown that, with its high statistics and good energy resolution, a low energy neutrino factory can be used to precisely determine the atmospheric neutrino oscillation parameters, $\\theta_{23}$ and $\\Delta m^2_{31}$. In particular, for an exposure of $3 \\times 10^{22}$~Kton-decays for each muon sign, and allowing for a $2\\%$ systematic uncertainty, it was shown that: (i) \nMaximal mixing in the 23-sector could be excluded at $99\\%$ CL if $\\sin^2 \\theta_{23}<0.48$ ($\\theta_{23}<43.8^\\circ$), independently of the value of $\\theta_{13}$, and (ii) For a large value of $\\theta_{13}$, i.e. $\\theta_{13}>8^\\circ$, the $\\theta_{23}$-octant degeneracy would be resolved at the $99\\%$ CL for $\\sin^2 \\theta_{23}<0.44$ ($\\theta_{23} < 41.5^\\circ$). \nIn our present study, the good energy resolution of the TASD provides sensitivity to the oscillatory pattern of the disappearance signal that is comparable to, and somewhat better than, we previously assumed. \n\nIn Fig.~\\ref{fig:dis} we show the $68\\%$, $90\\%$ and $95\\%$~CL contours (for 2 d.o.f) resulting from the fits to the measured energy dependent $\\nu_\\mu$ and $\\overline{\\nu}_{\\mu}$ CC rates at $L = 1480$ km. Results correspond to $1 \\times 10^{23}$~Kton-decays, and are shown for $\\Delta m^2_{31}=2.5 \\times 10^{-3}$ eV~$^2$ and two simulated values of $\\sin^2 \\theta_{23}$ (= $0.4$ and $0.44$). For $\\theta_{13}=0$, $P_{\\nu_\\mu \\to \\nu_\\mu} (\\theta_{23})=  P_{\\nu_\\mu \\to \\nu_\\mu} (\\pi\/2 -\\theta_{23})$, i.e. the disappearance channel is symmetric under $\\theta_{23}\\to \\pi\/2-\\theta_{23}$. However, when a rather large non-vanishing value of $\\theta_{13}$ is switched on, a $\\theta_{23}$ asymmetry appears in the $P_{\\nu_\\mu \\to \\nu_\\mu}$. Notice that the asymmetry grows with increasing $\\theta_{13}$ and the four-fold degeneracy in the atmospheric neutrino parameters is resolved more easily. We conclude that, using only the $\\nu_\\mu$-disappearance data, the uncertainty on $\\Delta m^2_{31}$ could be reduced down to the $1\\%-2\\%$ level. In principle, the $\\nu_e$ disappearance channel could also be used, which is sensitive to $\\theta_{13}$ and matter effects. However, charge discrimination for electrons has not yet been adequately studied to determine the relevant TASD performance parameters.\n\n\nThe extremely good determination of the atmospheric mass squared difference opens the possibility \nto determine the mass hierarchy by exploiting the effects of the solar mass squared difference on the $\\nu_\\mu$ disappearance probability, even for negligible values of $\\theta_{13}$.\nThis strategy was studied in detail in Refs.~\\cite{deGouvea:2005hk,deGouvea:2005mi,Minakata:2006gq}.  The vacuum $\\nu_\\mu \\to \\nu_\\mu$ oscillation probability is given b\n\\begin{equation}\nP(\\nu_\\mu \\to \\nu_\\mu) = 1 - 4 | U_{\\mu 1}|^2 | U_{\\mu 2}|^2 \\sin^2 \\frac{\\Delta m^2_{12} L}{4E} - 4 | U_{\\mu 1}|^2 | U_{\\mu 3}|^2 \\sin^2 \\frac{\\Delta m^2_{13} L}{4E} - 4 | U_{\\mu 2}|^2 | U_{\\mu 3}|^2 \\sin^2 \\frac{\\Delta m^2_{23} L}{4E} ~,\n\\end{equation}\nwhere the usual notation is used for the mass squared \ndifferences $\\Delta m^2_{ij}$ and for the elements of the leptonic mixing matrix $U$.\nIn the following we take $\\theta_{13}=0$.\nThe oscillation probabilities depend on whether\n$|\\Delta m^2_{13}| > |\\Delta m^2_{23}|$ (normal hierarchy) or $|\\Delta m^2_{13}| < |\\Delta m^2_{23}|$ (inverted hierarchy).\nPrecisely measured disappearance probabilities can distinguish between normal and inverted hierarchies if there is sensitivity to effects driven by both $|\\Delta m^2_{13}|$ and  $\\Delta m^2_{12}$.\nThis requires the atmospheric mass squared difference to be measured at different $L\/E$ with a precision of better than $|\\Delta m^2_{21}| \/ |\\Delta m^2_{31}| \\sim 0.026$. In fact, it was pointed out in Ref.~\\cite{deGouvea:2005hk} that, for a fixed $L\/E$, the disappearance\nprobabilities for the normal and inverted hierarchies are \nthe same if $|\\Delta m^2_{13}|$ is substituted with\n$-|\\Delta m^2_{13}| +  \\Delta m^2_{12} + \\frac{4 E}{L} \\arctan \\Big(\n\\cos {2 \\theta_{12} } \\tan \\frac{  \\Delta m^2_{12} L}{4E} \\Big)$.\nIn order to break this degeneracy it is necessary to measure the atmospheric mass squared difference at different energies and at distances for which the oscillations driven by the solar term are non negligible.\nIn our setup, if we assume a $0\\%$ ($2\\%$) overall systematic error, we find that the hierarchy can be measured at the $1 \\sigma$ level ($1 \\sigma$ level) for the $10^{23}$~Kton-decays case, while for the $3 \\times 10^{23}$~Kton-decays scenario it can be determined at $4 \\sigma$ level ($2 \\sigma$ level). Note that the systematic errors play a crucial role. It is in principle possible to reduce the impact of the systematics errors using the ratios of the number of events at the near and far detectors:\n\\begin{equation}\n{\\cal R} (E)  = \\frac{\\frac{N_{\\mathrm N} (\\nu_\\mu) }{N_{\\mathrm N} (\\bar{\\nu}_e) }}{\\frac{N_{\\mathrm F} (\\nu_\\mu) }{N_{\\mathrm F} (\\bar{\\nu}_e) }}~,\n\\end{equation}\nwhere $N_{\\mathrm N (F)} (\\nu_\\mu [\\bar{\\nu}_e]) $ refer to the number of $\\nu_\\mu \\ [\\bar{\\nu}_e]$ events in the near (far) detector for a fixed energy $E$. \nVery good energy resolution is required for such cancellations to be effective.\nIn this case, a low energy Neutrino Factory \ncan give important information on the type of hierarchy \neven if $\\theta_{13}=0$.\n\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{ll}\n\\includegraphics[width=3in]{dis_new_8.ps}&\\hskip 0.cm\n\\includegraphics[width=3in]{dis_new_3.ps}\\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{$68\\%$, $90\\%$ and $95\\%$ (2 d.o.f)  CL contours resulting from the fits at $L = 1480$ km assuming two central values for $\\sin^2 \\theta_{23}=0.4$ and $0.44$ and  $\\Delta m^2_{31}=2.5 \\times 10^{-3}$ eV~$^2$. In the left (right) panel, $\\theta_{13}=8^\\circ$ ($3^\\circ$). The statistics considered for both simulations corresponds to $1\\times 10^{23}$~Kton-decays.  Only disappearance data have been used to perform these plots.} }  \\label{fig:dis}\n\\end{figure}\n\n\n\n\\subsection{Simultaneous fits to $\\theta_{13}$ and $\\delta$}\n\nNext, we study the extraction of the unknown parameters $\\theta_{13}$ and $\\delta$, using the \\emph{golden channel} ($\\nu_e (\\bar{\\nu}_e) \\to \\nu_\\mu (\\bar{\\nu}_\\mu)$).\nWe start by considering a neutrino factory scenario with $1 \\times 10^{23}$~Kton-decays. We find that, for values of $\\theta_{13}>2^\\circ$, the sign degeneracy is resolved at the $95\\%$~C.L. Note that for $\\theta_{13} >4^\\circ$ the octant degeneracy has already been resolved using the disappearance data. \n\nFigure~\\ref{fig:fig2} shows, for a fit to the simulated data at a baseline $L=1480$ km, the $68\\%$, $90\\%$ and $95\\%$~CL contours in the ($\\theta_{13}, \\delta$)-plane. Results are shown for background levels set to zero (left panel) and $10^{-3}$ (right panel) for the $10^{23}$~Kton-decays scenario. The four sets of contours correspond to four simulated test points in the ($\\theta_{13}, \\delta$)-plane, which are depicted by a star. The simulations are for the normal mass hierarchy and $\\theta_{23}$ in the first octant ($\\sin^2 \\theta_{23} = 0.41$ which corresponds to $\\theta_{23}=40^\\circ$). Our analysis includes the study of the discrete degeneracies. That is, we have fitted the data assuming both the right and wrong hierarchies, and the right and wrong choices for the $\\theta_{23}$ octant. If present, the additional solutions associated to the $\\theta_{23}$ octant ambiguity are shown as dotted contours.\n\nNotice from Fig.~\\ref{fig:fig2} that the sign ambiguity is resolved at the $95\\%$ CL in the $10^{23}$~Kton-decays scenario. Additional solutions associated to the wrong choice of the $\\theta_{23}$ octant are still present in the $10^{23}$ Kton-decays scenario, but notice that the presence of these additional solutions does not interfere with a measurement of the CP violating phase $\\delta$ and $\\theta_{13}$, since the locations of the  fake solutions in the ($\\theta_{13}$, $\\delta$) plane, are almost the same as the correct locations.\n\nThe effect of the background can be easily understood in terms of the statistics presented in Tables~\\ref{tab:tab2} and \\ref{tab:tab3}. For small values of $\\theta_{13}$, the addition of the background has a larger impact for $\\delta \\sim -90^\\circ$, since for that value of the CP phase the statistics are dominated by the antineutrino channel, which suffers from a larger background (from $\\nu_\\mu$'s) than the neutrino channel (from $\\bar{\\nu}_\\mu$'s). For a background level smaller than $\\sim 10^{-4}$, the results are indistinguishable from the zero background case. \n\nWe illustrate the corresponding results for the improved scenario of $3\\times 10^{23}$~Kton-decays in Fig.~\\ref{fig:fig1}. Note that the higher statistics allow us to consider a smaller value for $\\mbox{$\\theta_{13}$}=1^\\circ$. The additional solutions arising from the wrong choice for the neutrino mass hierarchy or $\\theta_{23}$ octant are not present at the $95\\%$ CL. Furthermore, the addition of a background level of $10^{-3}$ does not significantly affect the resolution of the degeneracies, and has only an impact on the CP violation measurement.\n\nThe performance of the \\emph{low energy neutrino factory} in the two high statistics scenarios explored here is unique. The sign($\\Delta m_{31}^2$) can be determined at the $95\\%$~CL in the $10^{23}$~Kton-decays ($3 \\times 10^{23}$~Kton-decays) scenario if $\\theta_{13}>2^\\circ$ ($>1^\\circ$) for all values of the CP phase $\\delta$. The $\\theta_{23}$-octant ambiguity can be removed at the $95\\%$ CL down to roughly $\\theta_{13}>0.5-1.0^\\circ$  for the representative  choice of $\\sin^2 \\theta_{23}=0.41$, independently of the value of $\\delta$, except for some intermediate values of $\\theta_{13}\\sim 2^\\circ$, for which the $\\theta_{23}$ degeneracy is still present for some values of the CP violating phase $\\delta$. Resolving the $\\theta_{23}$-octant degeneracy therefore is easier for small values of $\\theta_{13}<2^\\circ$. This is due to the fact that, as explored in Ref.~\\cite{GeerMenaPascoli}, the $\\theta_{23}$-octant degeneracy is resolved using the information from the low energy bins, which are sensitive to the solar term. For the setup described in this paper, the solar term starts to be important if $\\theta_{13}<2^\\circ$. However, notice that the presence of the $\\theta_{23}$ octant ambiguity at $\\theta_{13}\\sim 2^\\circ$ will not interfere with the extraction of $\\theta_{13}$ and $\\delta$, since the locations of the degenerate (fake) solutions almost coincide with the positions of ``true'', nature solutions.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{ll}\n\\includegraphics[width=3in]{2_hl_nback_new.ps}&\\hskip 0.cm\n\\includegraphics[width=3in]{2_hl_back_new.ps}\\\\\n\\hskip 2.truecm\n{\\small}            &\n\\hskip 2.truecm\n{\\small}       \\\\  \n\\end{tabular}\n\\end{center}\n\\caption{\\textit{$68\\%$, $90\\%$ and $95\\%$ (2 d.o.f) CL contours resulting from the fits at $L = 1480$ km assuming four central values for $\\delta=0^{\\circ}$, $90^{\\circ}$, $-90^{\\circ}$ and $180^{\\circ}$ and  $\\mbox{$\\theta_{13}$}=2^\\circ$ without backgrounds (left panel) and with a background level of $10^{-3}$ (right panel). The additional $\\theta_{23}$ octant solutions are depicted in dotted blue. The statistics considered for both simulations corresponds to $10^{23}$~Kton-decays.}}\n\\label{fig:fig2}\n\\end{figure}\n\\begin{figure}[h]\n\n\\begin{center}\n\n\\begin{tabular}{ll}\n\n\\includegraphics[width=3in]{1_hhl_nback_new.ps}&\\hskip 0.cm\n\\includegraphics[width=3in]{1_hhl_back_new.ps}\\\\\n\\hskip 2.truecm\n{\\small}            &\n\\hskip 2.truecm\n{\\small}       \\\\  \n\\end{tabular}\n\\end{center}\n\\caption{\\textit{$68\\%$, $90\\%$ and $95\\%$ (2 d.o.f) CL contours resulting from the fits at $L = 1480$ km assuming four central values for $\\delta=0^{\\circ}$, $90^{\\circ}$, $-90^{\\circ}$ and $180^{\\circ}$ and  $\\mbox{$\\theta_{13}$}=1^\\circ$ without backgrounds (left panel) and with a background level of $10^{-3}$ (right panel). The statistics considered for both simulations corresponds to $3\\times 10^{23}$~Kton-decays.}}\n\\label{fig:fig1}\n\n\\end{figure}\n\n\nIn Figs.~\\ref{fig:hier} and \\ref{fig:cp} we summarize, for the $10^{23}$ and $3\\times 10^{23}$~Kton decays scenarios, the physics reach for a TASD detector located 1480 km from a low energy neutrino factory. The analysis takes into account the impact of both the intrinsic and discrete degeneracies. Figure~\\ref{fig:hier} shows the region in the ($\\sin^2 2 \\theta_{13}$, ``fraction of $\\delta$'') plane for which the mass hierarchy can be resolved at the $95\\%$ CL (1 d.o.f). Contours are shown for zero background, and for when a background level of $10^{-3}$ is included in the analysis. Note that, with a background level of $\\simeq 10^{-3}$, the hierarchy can be determined in both scenarios if $\\sin^2 2 \\theta_{13}>$ few $10^{-3}$ (i.e. $\\theta_{13}> 2-3^\\circ$) for all values of the CP violating phase $\\delta$. For a background level smaller than $\\sim 10^{-4}$, the results are indistinguishable from the zero background case.\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=3.5in]{hierarchy_new_v1.ps}\n\\end{center}\n\\caption[]{\\textit{$95\\%$ CL (1 d.o.f) hierarchy resolution assuming that the far detector is located at a distance of $1480$ km at the Henderson mine. The solid (dotted) red curves depict the results assuming  $1\\times 10^{23}$~Kton-decays ($3\\times 10^{23}$~Kton-decays) without backgrounds.  The long-dashed (short-dashed) black curves depict the results assuming  $1\\times 10^{23}$~Kton-decays ($3\\times 10^{23}$~Kton-decays) with a background level of $10^{-3}$. }}\n\\label{fig:hier}\n\\end{figure}\n\nFigure~\\ref{fig:cp} shows the region in the ($\\sin^2 2 \\theta_{13}$, $\\delta$) plane for which a given (non-zero) CP violating value of the CP-phase $\\delta$ can be distinguished at the $95\\%$ CL (1 d.o.f) from the CP conserving case, i.e. $\\delta =0, \\pm 180^\\circ$. The results are given for the two statistics scenarios studied here.  Note that, even in the presence of a $10^{-3}$ background level, the CP violating phase $\\delta$ could be measured with a $95\\%$ CL precision of better than $20^\\circ$ in the $10^{23}$~Kton-decays ($3\\times 10^{23}$~kton-deacys) luminosity scenario if $\\sin^2 2 \\theta_{13}>0.01$ ($\\sin^2 2 \\theta_{13}>0.002$).\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=3.5in]{cp_new_v1.ps}\n\\end{center}\n\\caption[]{\\textit{$95\\%$ CL (1 d.o.f) CP Violation  extraction assuming that the far detector is located at a distance of $1480$. The solid (dotted) red curves depict the results assuming  $1\\times 10^{23}$~Kton-decays ($3\\times 10^{23}$~Kton-decays) without backgrounds. The long-dashed (short-dashed) black curves depict the results assuming  $1\\times 10^{23}$~Kton-decays ($3\\times 10^{23}$~Kton-decays) with a background level of $10^{-3}$.}}\n\\label{fig:cp}\n\\end{figure}\n\n\n\\section{Summary and Conclusions}\n\nWe have studied the physics reach of a \\emph{low energy neutrino factory}, first presented in Ref.~\\cite{GeerMenaPascoli}, in which the stored muons have an energy of $4.12$~GeV. The simulated detector performance is based upon a\nmagnetized Totally Active Scintillator Detector. Our simulations suggest this detector will have a threshold for measuring muon neutrino CC interactions of about 500 MeV and an energy independent efficiency of about 73\\% above threshold. We have assumed a conservative energy resolution of 30\\% for the detector.\n\nIn our analysis, we consider the representative baseline of $1480$ Km, divide the simulated observed neutrino event spectrum into 9 energy bins above the 500~MeV threshold, and exploit both the disappearance ($\\nu_\\mu \\to \\nu_\\mu$)  and the \\emph{golden} ($\\nu_e \\to \\nu_\\mu$) channels by measuring CC events tagged by ``right-sign'' and ``wrong-sign'' muons.  The results can be easily generalized to other baselines in the 1200--1500~km range. We have investigated the dependence of the physics sensitivity on statistics by considering a high statistics scenario corresponding to  $1 \\times 10^{23}$~Kton-decays for each muon sign, and a more aggressive scenario corresponding to $3 \\times 10^{23}$~Kton-decays for each muon sign. We have also explored the impact of backgrounds to the wrong-sign muon signal by considering background levels of zero and $10^{-3}$.\n\nWe find that, based only on the disappearance channel, maximal atmospheric neutrino mixing can be excluded at $95\\%$ CL if $\\sin^2 \\theta_{23}<0.44$ ($\\theta_{23}<41.5^\\circ$). The atmospheric mass difference could be measured with a precision of $1\\%-2\\%$, opening the possibility of determining the neutrino mass hierarchy even if $\\theta_{13}=0$, provided systematic uncertainties can be controlled. Neglecting systematic uncertainties, the mass  hierarchy could be determined at the $1 \\sigma$ level ($4 \\sigma$ level) in the $1 \\times 10^{23}$~Kton-decays ($3 \\times 10^{23}$~Kton-decays) statistics scenario.\n\nThe rich oscillation pattern of the $\\nu_e\\to \\nu_\\mu$ ($\\bar\\nu_e \\to \\bar\\nu_\\mu$) appearance channels at energies between $0.5$ and $4$ GeV for baselines $\\mathcal{O}(1000)$~km facilitates an elimination of the degenerate solutions.  If the atmospheric mixing angle is not maximal, for the representative choice of $\\sin^2 \\theta_{23}=0.4$, the octant in which $\\theta_{23}$ lies could be extracted at the $95\\%$ CL in both scenarios if $\\theta_{13}> 0.5-1^\\circ$, for all values of the CP violating phase $\\delta$, except for some intermediate values of the mixing angle $\\theta_{13}$ in which the fake solutions's location coincides with the true's solution position and therefore the presence of these fake solutions does not interfere with the extraction of $\\delta_{CP}$ and $\\theta_{13}$.\n\nIn the $10^{23}$ kton-decays scenario, if the background level is $\\sim 10^{-3}$ ($10^{-4}$), the neutrino mass hierarchy could be determined at the $95\\%$ CL, and the CP violating phase $\\delta$ could be measured with a $95\\%$ CL precision of better than $20^\\circ$, if $\\sin^2 2 \\theta_{13}>0.01$ ($\\sim 0.006$). With a factor of three improvement in the former statistics, the numbers quoted above are $\\sin^2 2 \\theta_{13}= 0.005$ and $\\sin^2 2 \\theta_{13}=0.002$, for background levels of $10^{-3}$ and $10^{-4}$, respectively. In our analysis we have included a $2\\%$ systematic error on all measured event rates.\n\n\nIn summary, the low statistics low energy Neutrino Factory scenario we have described, with a background level of $10^{-3}$, for both large and very small values of $\\theta_{13}$ would be able to eliminate ambiguous solutions, determine $\\theta_{13}$, the mass hierarchy, and search for CP violation. Higher statistics and lower backgrounds would further improve the sensitivity, and may enable the mass hierarchy to be determined even if $\\theta_{13}=0$.\n\n\n\\vspace{1cm}\n\\section*{Acknowledgments} \nThis work was supported in part by the European Programme ``The Quest for\nUnification''  contract MRTN-CT-2004-503369, and by the Fermi National Accelerator Laboratory, which is operated by the Fermi Research Association, under contract No. DE-AC02-76CH03000 with the U.S. Department of Energy. SP acknowledges the support of CARE,\ncontract number RII3-CT-2003-506395.\nOM and SP would like to thank the Theoretical Physics Department at Fermilab for hospitality and support. \n\\section*{Appendix} \nAll detector concepts for the Neutrino Factory (NF) require a magnetic field in order to determine the sign of muon (or possibly the electron) produced in the neutrino interaction. For the baseline detector, this is done with magnetized iron. Technically this is very straightforward, although for the $100$~Kton baseline detector does present challenges because of its size. The cost of this magnetic solution is felt to be manageable. Magnetic solutions for the TASD become much more problematic.  The solution that we propose is to use the Superconducting Transmission Line developed for the VLHC~\\cite{VLHC}, Fig.~\\ref{fig:STL}, as windings for very large solenoids that form a magnetic cavern (see Fig.~\\ref{fig:mag_cav}) for the detector.  The Superconducting Transmission Line (STL) consists of a superconducting cable inside a cryo-pipe cooled by supercritical liquid helium at $4.5-6.0$~ K placed inside a co-axial cryostat. It consists of a perforated Invar tube, a copper stabilized superconducting cable, an Invar helium pipe, the cold pipe support system, a thermal shield covered by multilayer superinsulation, and the vacuum shell. One of the possible STL designs developed for the VLHC is shown in Fig.~\\ref{fig:mag_cav} within the main text.  The STL is designed to carry a current of $100$~kA at $6.5$~K in a magnetic field up to $1$~T. This provides a $50\\%$ current margin with respect to the required current in order to reach a field of $0.5$~T.  This operating margin can compensate for temperature variations, mechanical or other perturbations in the system.\n\n\nThe solenoid windings now consist of this superconducting cable which is confined in its own cryostat.  Each solenoid consists of $150$ turns and requires $\\sim 7500$~ m of cable. There is no large vacuum vessel and access to the detectors can be made through the winding support cylinder since the STL does not need to be close-packed in order to reach an acceptable field. We have performed a simulation of the Magnetic Cavern concept using STL solenoids and the results are shown in Fig.~\\ref{fig:stl_sol}.  With the iron end-walls ($1$~m thick), the average field in the XZ plane is approximately $0.58$~T at an excitation current of $50$~kA. This figure shows the field uniformity in the X-Z plane which is better than $\\pm 2\\% $ throughout the majority of the volume with approximately $20\\%$ variations near the end-irons. Figure~\\ref{fig:app3} shows the on-axis $B$ field as a function of position along the $z$ axis (in meters).\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=6in]{Mag_cav_A.1.eps3}\n\\end{center}\n\\caption[]{\\textit{Simulation results for magnetic cavern design.}}\n\\label{fig:mag_cav}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=6in]{Field_map_A.2.eps3}\n\\end{center}\n\\caption[]{\\textit{STL Solenoid Magnetic Cavern Field Uniformity in XZ plane.}}\n\\label{fig:stl_sol}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=6in]{BvZ2_A.3.eps3}\n\\end{center}\n\\caption[]{\\textit{On-axis $B$ field in $T$ as a function of position along the $z$ axis (m).}}\n\\label{fig:app3}\n\\end{figure}\nWe have not yet been able to do a detailed costing of the magnetic cavern.  The STL costs can be estimated quite accurately ($30\\%$) from the VLHC work and current SC cable costs and are believed to be $\\$50$M.  The total magnetic cavern cost is estimated to be less than $\\$150$M.  This is to be compared to a fully loaded cost savings of the Low-Energy Neutrino Factory (compared to the $50$~GeV design) as indicated in Ref.~\\cite{ISS-Detector Report}.\n\n\n\n\n  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and instructions for participants}\nOverall, using the ASCL is straight-forward, but how it interacts with other resources, such as ADS, may not be apparent to new or casual users, and certainly there are tips and tricks for using it more efficiently and effectively. This tutorial was developed to teach new users of the ASCL how this resource works. Prospective attendees were provided with instructions, via the ADASS website, as to what they needed to have to participate in the tutorial (a computer with a browser and Internet access) and what knowledge they were assumed to already have (rudimentary familiarity with using ADS for searching for articles\/resources; what a bibcode is; how to use a general search engine such as Google). They were also asked to bookmark several URLs in their browser, including the ASCL home page (\\url{https:\/\/ascl.net}), the ADS home page (\\url{https:\/\/ui.adsabs.harvard.edu\/}, and the Google search page (\\url{https:\/\/www.google.com}).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics [scale=0.5]{T01_f1.eps}\n    \\caption{Poll results: Familiarity with the ASCL}\n    \\label{fig:ASCLfamiliarity}\n\\end{figure}\n\nUpon entering the virtual space in which the tutorial was given, participants were asked to answer a poll to gauge their starting familiarity with the ASCL; Figure \\ref{fig:ASCLfamiliarity} shows that most attendees had little experience with the resource.\n\n\\section{Tutorial outline}\nThe tutorial covered:\n\\begin{itemize}\n    \\item what the ASCL is and components of an ASCL entry\n    \\item common and alternate ways to bring up ASCL records\n    \\item how to find software using different methods and tools\n    \\item how citation tracking and preferred citation work\n    \\item how to find a code's preferred citation (where one exists)\n    \\item how to create a metadata file that informs others how to cite your code \n    \\item the best place(s) to put preferred citation information\n\\end{itemize}\n\n\\section{Hands-on activities}\nTutorial participants were encouraged to follow along and mirror the presenter's actions as different tasks, such as bringing up ASCL records or finding software in ADS, were demonstrated.  Four specific hands-on activities provided practice with the ASCL, and with using ADS and Google, too, for information relating to the ASCL, reinforcing the training. Figures \\ref{fig:Searching1} and \\ref{fig:Searching2} show the searching activities. Readers are encouraged to try these exercises to gauge their familiarity with the ASCL and even with ADS. The full set of slides is available on the ASCL (\\url{https:\/\/tinyurl.com\/ASCLtutorial}).\n\n\\begin{figure}\n     \\centering\n     \\includegraphics [scale=0.47]{T01_f3.eps}\n     \\caption{Hands-on Activity \\#1}\n     \\label{fig:Searching1}\n\\end{figure}\n\n\\begin{figure}\n     \\centering\n     \\includegraphics [scale=0.47]{T01_f4.eps}\n     \\caption{Hands-on Activity \\#2}\n     \\label{fig:Searching2}\n\\end{figure}\n\n\n\n\\section{Questions, answers, and comments}\nAttendees were encouraged to post questions and comments to the Q\\&A and chat areas provided by the virtual tool used to present the tutorial, and to unmute themselves to ask questions or make comments directly. Questions were varied, but three topics were of particular interest: keywords, citing software, and submitting software to ASCL. The ASCL currently has keywords only for NASA missions and HITS software. Participants asked, \\textit{``Is there a guideline on list of keywords?''}, \\textit{``Do you have a taxonomy for  your keywords?''}, \\textit{``Can we suggest more?''}, and \\textit{``Can we suggest keywords to you for our own codes?''} I was glad to learn there is so much interest in the ASCL providing more keywords! How to do that, and what keywords to use, will be discussed with the community at a later date. Regarding citation, one attendee asked, \\textit{``What would you define as a good method?''}. Criteria for inclusion in the ASCL came up in questions on whether there is a way for a new code to be added to the resource before research that describes or uses the software is published (yes, there is), and what is considered a refereed resource, with the point made that SPIE proceedings, for example, may not be refereed in the traditional sense. \n\nIn addition, one participant mentioned, \\textit{``I'm really interested in these new sort options, I didn't realise ADS had added those.''} This was not surprising to me; I include information on ADS searching when I show people how to use the ASCL because so many do not know about using, for example, doctype to find just software. \n\n\\section{Using every minute}\nFour additional optional topics were made available that could be discussed if time permitted; participants were asked to express which of these was of the most interest to them in a second poll, as shown in Figure \\ref{fig:LastTopic}. As a result, the differences between and similarities of ASCL and Zenodo was the final topic of the tutorial.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics [scale=0.5]{T01_f2.eps}\n    \\caption{Poll results: Final topic for discussion}\n    \\label{fig:LastTopic}\n\\end{figure}\n\n\n\\section{Summary}\nThis tutorial was intended for people relatively new to the ASCL; polling the audience before the start of the session showed that it was on-target as to who would attend. Only 11\\% of the participants had any significant experience with the resource. I thank the Heidelberg Institute for Theoretical Studies, Michigan Technological University, and the University of Maryland College Park for support, the ADASS POC for selecting this tutorial, the participants for coming, and software authors everywhere for providing the computational methods on which research depends.\n\n\n\n\n\n\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTraditionally, convex optimization problems have been divided into two main classes: the class of smooth problems and the class of non-smooth problems \\cite{polyakintroduction}. However, introducing an intermediate class of problems with convex differentiable objectives with H$\\ddot{\\text{o}}$lder continuous gradient allows us to view the classes of smooth and non-smooth convex optimization problems as two extreme cases of this intermediate class. \n\nThe first optimal methods for this class were introduced in \\cite{nemirovski1985optimal}. However, both these procedures and some others presented later had a serious drawback: they required too much information about the objective (for example, the degree of the objective function's smoothness or the distance from the initial point to the solution) to be used efficiently. \n\nIn \\cite{nesterov2015universal} the Universal Fast Gradient Method is presented. It is optimal for the class of problems with convex differentiable objectives with H$\\ddot{\\text{o}}$lder continuous gradient, has a low iteration cost, and does not involve any parameters dependent on the objective.\n\nSome minimization methods allow for the use of an exact line search procedure. A classic example of such a method is the steepest descent method, which is a version of the gradient descent method in which on each iteration instead of performing a step of fixed length in the direction of the negative gradient the objective function is minimized along said direction. Although this does not improve the worst-case convergence rate, such line search procedures often perform very well in practice. The aim of this work was to construct a universal method which allowed for the use of an exact line search procedure. By combining the core idea of Nesterov's Universal Fast Gradient Method with the framework described by Allen-Zhu et al. in \\cite{allen2014linear}, such a method was devised. As far as it is known to the authors of this paper, our work contains the first example of such a method, although a method utilising exact line search for solving minimization problems with convex Lipschitz continuous objectives was recently constructed by Drori et al \\cite{drori2018efficient}. Their work also contains an example of a universal method which uses an exact three dimension subspace minimization on each iteration. Our numerical experiments indicate that the exact line search step does indeed demonstrate great performance on some non-smooth problems. Note that in the well-known Shor's type methods with variable metric for non-smooth convex optimization problems line search is performed not in the direction of the negative gradient. These methods also require quadratic memory  \\cite{polyakintroduction}.  \n\nThe paper is organized as follows. Firstly, we define the intermediate class of problems which we refer to above, set the problem and give other definition used later in this paper. Secondly, we define Nesterov's Universal Fast Gradient Method, which we will be using as a benchmark in our numerical experiments. In \\textbf{Section 2} we present our Universal Linear Coupling Method, prove its convergence and equip it with a stopping criterion. \\textbf{Section 3} contains notes on how to implement the line search procedure and how its accuracy affects the method's convergence. Finally, \\textbf{Section 4} is dedicated to the results of our numerical experiments.\n\n\n\\subsection{Preliminaries}\n\nOne of the conditions often used in convergence analysis of numerical optimization methods is $L$-smoothness.\n\n\\begin{definition}\nA  function $f:\\ \\mathbb{R}^n\\rightarrow \\mathbb{R}^m$ is called Lipschitz continuous with constant $L$ if\n\n\\[\\|f(x)-f(y)\\|\\leq L\\|x-y\\|\\quad \\forall x,y\\in\\mathbb{R}^n.\\]\n\n\\end{definition}\n\n\\begin{definition}\nA differentiable function $f:\\ \\mathbb{R}^n\\rightarrow \\mathbb{R}^m$ is called $L$-smooth if its gradient is Lipschitz continuous with constant $L$:\n\\[\\|\\nabla f(x)-\\nabla f(y)\\|\\leq L\\|x-y\\|\\quad \\forall x,y\\in\\mathbb{R}^n.\\]\n\\end{definition}\n\nWe will be using the following natural generalisation of Lipschitz continuity.\n\n\\begin{definition}\nA  function $f:\\ \\mathbb{R}^n\\rightarrow \\mathbb{R}^m$ satisfies the H$\\ddot{\\text{o}}$lder condition (or is H$\\ddot{\\text{o}}$lder continuous) if there exist constants $\\nu\\in[0,1]$ and $M_\\nu\\geqslant 0$, such that \\[\\|f(x)-f(y)\\| \\leqslant M_\\nu\\|x-y\\|^\\nu\\quad \\forall\\ x,y\\in\\mathbb{R}^n.\\]\n\\end{definition} \n\nThe constant $\\nu$ in this definition is called the exponent of the H$\\ddot{\\text{o}}$lder condition. H$\\ddot{\\text{o}}$lder continuity coincides with Lipschitz continuity if $\\nu=1$. On the other hand, H$\\ddot{\\text{o}}$lder continuity with $\\nu=0$ is just boundedness. If a function is differentiable and its gradient is H$\\ddot{\\text{o}}$lder continuous, then exponent $\\nu$ is a measure of the function's smoothness.\n\n\nThroughout this paper we will be working with the problem \\[f(x)\\rightarrow \\min_{x\\in\\mathbb{R}^n}, \\] where $f(x)$ is a convex differentiable function and its gradient satisfies the H$\\ddot{\\text{o}}$lder condition for some $\\nu\\in[0,1]$ with some constant $M_{\\nu}$. We denote some solution to this problem as $x^\\ast$.\n\nLet us define Bregman divergence $V_x(y)$ as follows:\n\n\\[V_x(y)=\\omega(y)-\\langle\\nabla\\omega(x),y-x\\rangle-\\omega(x), \\] where $\\omega(x)$ is a 1-strongly convex function. $\\omega$ is also called a distance generating function. By definition, \n\n\\[V_x(y)\\geqslant \\frac{1}{2}\\|y-x\\|^2.\\]\n\n\n\\subsection{Universal Method}\n\nIn \\cite{Devolder2014} it is shown that the notion of inexact oracle allows one to apply some methods of smooth convex optimization to non-smooth problems. The following lemma plays a key role in this:\n\n\\begin{lemma}\nLet function $f$ be differentiable and have H$\\ddot{o}$lder continuous gradient. Then for any $\\delta>0$ we have \\[f(y)\\leqslant f(x)+\\langle\\nabla f(x), y-x\\rangle+\\frac{M}{2}\\|y-x\\|^2+\\frac{\\delta}{2},\\] where \\[M=M\\left(\\delta,\\nu, M_\\nu\\right)=\\left[\\frac{1-\\nu}{1+\\nu}\\frac{M_\\nu}{\\delta}\\right]^{\\frac{1-\\nu}{1+\\nu}}M_\\nu.\\]\n\n\\end{lemma}\nThe exact values $\\left(f(x),\\nabla f(x)\\right)$ of a differentiable function $f$ with H$\\ddot{\\text{o}}$lder continuous gradient allow us to obtain an upper bound similar to the one obtained by using inexact information for a differentiable and L-smooth function. This allows one to apply methods reliant on the usage of an inexact oracle for L-smooth objectives to optimize objectives with H$\\ddot{\\text{o}}$lder continuos gradient.\n\nHowever, knowledge of the parameters $\\nu$ and $M_\\nu$ from the definition of H$\\ddot{\\text{o}}$lder continuity is still required to apply such an approach. In \\cite{nesterov2015universal} a line search procedure was used to estimate the needed parameters similarly to how the constant of $L$-smoothness is estimated in adaptive methods. For a general norm on $\\mathbb{R}^n$ and a corresponding Bregman divergence $V_x(y)$ the Universal Fast Gradient Method may be written as follows.\n\\newpage\n\\begin{algorithm}\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n\t\n    \\caption{UFGM($f$, $L_0$, $x_0$, $\\varepsilon$, $T$)}\n    \\Input{$f$ a differentiable convex function with H$\\ddot{\\text{o}}$lder continuous gradient;\n    initial value of the \"inexact\" Lipschitz continuity constant $L_0$;\n    initial point $x_0$;\n    accuracy $\\varepsilon$;\n    number of iterations $T$.}\n    $y_0\\gets x_0$, $z_0\\gets x_0$, $\\alpha_0 \\gets 0$, $\\psi_0(x)\\gets V_{x_0}(x)$\\\\\n  \\For{$k=0$ to $T-1$}{\n  \t$L_{k+1}\\gets\\frac{L_{k}}{2}$\\\\\n    \\While{True}{\n    $v_k=\\operatornamewithlimits{argmin}\\limits_{x\\in \\mathbb{R}^n} \\psi_k(x)$\\\\\n    $\\alpha_{k+1}\\gets\\frac{1}{2L_{k+1}}+\\sqrt{\\frac{1}{4L^2_{k+1}}+\\alpha^2_k\\frac{L_k}{L_{k+1}}}$\\\\\n    $\\tau_k\\gets\\frac{1}{\\alpha_{k+1}L_{k+1}}$\\\\\n    $x_{k+1}\\gets\\tau_kv_k+(1-\\tau_k)y_k$\\\\\n    $z_{k+1}\\gets \\operatornamewithlimits{argmin}\\limits_{z\\in \\mathbb{R}^n} \\alpha_{k+1}\\langle\\nabla f(x_{k+1}), z-v_k\\rangle +V_{v_k}(z)$\\\\\n    $y_{k+1}\\gets\\tau_kz_{k+1}+(1-\\tau_k)y_k$\\\\\n    \\If{$f(y_{k+1})\\leqslant f(x_{k+1})+\\langle\\nabla f(x_{k+1}),y_{k+1}-x_{k+1}\\rangle+\\frac{L_{k+1}}{2}\\|y_{k+1}-x_{k+1}\\|^2+\\frac{\\tau_k\\varepsilon}{2}$}{\\textbf{break}}\n    \\Else{$L_{k+1}\\gets 2L_{k+1}$}\n  \t}\n    $\\psi_{k+1}(x)\\gets\\psi_k(x)+\\alpha_{k+1}\\left[f(x_{k+1})+\\langle\\nabla f(x_{k+1}),x-x_{k+1}\\rangle\\right]$  \n  }\n  \\Return{$y_T$}\n  \n\\end{algorithm}\n\n\nThe above method does not require a priori knowledge of the smoothness parameter $\\nu$ or the corresponding $M_\\nu$. The following theorem gives the convergence rate of the above algorithm:\n\n\\begin{theorem}\nLet f be a differentiable convex function with H$\\ddot{\\text{o}}$lder continuous gradient with some exponent $\\nu$ and $M_\\nu<\\infty$. Let $L_0\\leqslant M(\\varepsilon,\\nu,M_\\nu)$. Then\n\\[f(y_k)-f(x^\\ast) \\leq \\left[\\frac{2^{2+4\\nu}M_\\nu^2}{\\varepsilon^{1-\\nu}k^{1+3\\nu}}\\right]^{\\frac{1}{1+\\nu}} +\\frac{\\varepsilon}{2}.\\]\n\n\\end{theorem} \n\nWhat follows is that one may obtain an $\\varepsilon$-accurate solution in \\[k\\leqslant \\inf_{\\nu\\in[0,1]} \\left[\\left(\\frac{2^\\frac{3+5\\nu}{2}M_\\nu}{\\varepsilon}\\right)^\\frac{2}{1+3\\nu} \\left(\\frac{1}{2}\\|x_0 -x^\\ast\\|^2\\right)^{\\frac{1+\\nu}{1+3\\nu}}\\right]\\] iterations. If the problem admits multiple solutions, then $x^\\ast$ may be considered to be the solution minimizing $\\frac{1}{2}\\| x_0 - x^\\ast \\|^2$. As shown in \\cite{nemirovskii1983problem}, this is optimal up to a multiplicative constant independent of the accuracy, the initial point, and the objective function.\n\n\\section{Universal Linear Coupling Method}\n\nWe are now ready to present our universal method based on the linear coupling method proposed by Allen-Zhu et al. \\cite{allen2014linear} The Linear Coupling framework is chosen as a basis for our method because it allows for the usage of an exact line search step, which is our goal. The original linear coupling method utilizes gradient and mirror descent steps to guarantee optimal convergence rate for convex objectives. However, it is clear from the convergence analysis of said method that the gradient step is only used to obtain a lower bound on the decrease of the objective during this step. This means that any procedure capable of guaranteeing at least such a decrease may be utilized instead. Since in the unconstrained Euclidean setting the gradient step is always performed in the direction of the negative of the gradient, one may use the steepest descent method instead. This idea combined with the idea of Nesterov's universal method allows us to modify the Linear Coupling method in the following way:\n\n\\begin{algorithm}\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n\t\n    \\caption{ULCM($f$, $L_0$, $x_0$, $\\varepsilon$, $T$)}\n    \\Input{$f$ a differentiable convex function with H$\\ddot{\\text{o}}$lder continuous gradient;\n    initial value of the \"inexact\" Lipschitz continuity constant $L_0$;\n    initial point $x_0$;\n    accuracy $\\varepsilon$;\n    number of iterations $T$.}\n    $y_0 \\gets x_0$, $z_0 \\gets x_0$, $\\alpha_0 \\gets 0$\\\\\n  \\For{$k=0$ to $T-1$}{\n  \t$L_{k+1}\\gets\\frac{L_{k}}{2}$\\\\\n    \\While{True}{\n  \t$\\alpha_{k+1}\\gets\\frac{1}{2L_{k+1}}+\\sqrt{\\frac{1}{4L^2_{k+1}}+\\alpha^2_k\\frac{L_k}{L_{k+1}}}$\\\\\n    $\\tau_k\\gets\\frac{1}{\\alpha_{k+1}L_{k+1}}$\\\\\n    $x_{k+1}\\gets\\tau_kz_k+(1-\\tau_k)y_k$\\\\\n    $h_{k+1}\\gets\\operatornamewithlimits{argmin}\\limits_{h\\geqslant 0} f(x_{k+1}-h\\nabla f(x_{k+1}))$\\\\\n    $y_{k+1}\\gets x_{k+1}-h_{k+1}\\nabla f(x_{k+1})$\\\\\n    $z_{k+1}\\gets z_k-\\alpha_{k+1}\\nabla f(x_{k+1})$\\\\\n    \\If{$\\langle \\alpha_{k+1}\\nabla f(x_{k+1}),z_k-z_{k+1}\\rangle-\\frac{1}{2}\\|z_k-z_{k+1}\\|^2\\leq \\alpha^2_{k+1}L_{k+1}(f(x_{k+1})-f(y_{k+1})+\\frac{\\tau_k\\varepsilon}{2})$}{\\textbf{break}}\n    \\Else{$L_{k+1}\\gets 2L_{k+1}$}\n  \t}\n \n  }\n  \\Return{$y_T$}\n\\end{algorithm}\n\\newpage\n\n\nAs far as it is known to the authors of this paper, this is the first universal method of non-smooth optimization utilizing steepest descent steps.\n \nFrom this point onwards $L_k$ will always denote the value obtained at the end of a full iteration of the \"for\" loop.\n\nWe shall now show that the above algorithm is well-defined. To be more precise, we shall prove that the if-condition inside the while loop is satisfied after a finite number of iterations for any $k$. \n\n\\begin{lemma}\n$f(x)$ is a convex differentiable function and its gradient satisfies the H$\\ddot{\\text{o}}$lder condition for some $\\nu\\in[0,1]$ with some constant $M_{\\nu}$. Then for all steps k of above algorithm\n\n\\[\\alpha_{k+1}\\langle\\nabla f(x_{k+1}),z_k-z_{k+1}\\rangle-\\frac{1}{2}\\|z_k-z_{k+1}\\|^2\\leqslant \\alpha^2_{k+1}L_{k+1}\\left(f(x_{k+1})-f(y_{k+1})+\\frac{\\tau_k\\varepsilon}{2}\\right), \\] for all $L_{k+1}$ satisfying\n\n\\[L_{k+1}\\geqslant M(\\tau_k\\varepsilon,\\nu,M_\\nu)=\\left[\\frac{1-\\nu}{1+\\nu}\\frac{M_\\nu}{\\tau_k\\varepsilon}\\right]^{\\frac{1-\\nu}{1+\\nu}}M_\\nu.\\]\n\n\\end{lemma}\n\\begin{proof}\n\n\\begin{align*}\n\\alpha_{k+1}\\langle\\nabla f(x_{k+1}),z_k-z_{k+1}\\rangle-\\frac{1}{2}\\|z_k-z_{k+1}\\|^2\\\\\\leqslant\\frac{\\alpha^2_{k+1}}{2}\\|\\nabla f(x_{k+1})\\|^2&\\leqslant M\\alpha_{k+1}^2\\left(f(x_{k+1})-f(y_{k+1})+\\frac{\\tau_k\\varepsilon}{2})\\right)\n\\end{align*}\n\nHere the first inequality follows from the fact that $\\|\\alpha_{k+1}\\nabla f(x_{k+1})-(z_k-z_{k+1})\\|^2\\geq 0$. To get the last inequality we will use \\textsc{Lemma 1.1} with $\\delta=\\tau_k\\varepsilon$ and $x=x_{k+1}$, $y=x_{k+1}-\\beta\\nabla f(x_{k+1})$: \n\n\\begin{align*}\nf(y)&\\leqslant f(x_{k+1})+\\langle\\nabla f(x_{k+1}), -\\beta\\nabla f(x_{k+1})\\rangle+\\frac{\\beta^2M}{2}\\|\\nabla f(x_{k+1})\\|^2+\\frac{\\tau_k\\varepsilon}{2}\\\\ &= f(x_{k+1})-\\beta\\|\\nabla f(x_{k+1})\\|^2+\\frac{\\beta^2M}{2}\\|\\nabla f(x_{k+1})\\|^2+\\frac{\\tau_k\\varepsilon}{2}.\n\\end{align*}\nMinimising the right-hand side over $\\beta\\in\\mathbb{R}$, we get $\\beta=\\frac{1}{M}$. This results in the following guarantee:\n\n\\[f(y)-f(x_{k+1})\\leqslant -\\frac{\\|\\nabla f(x_{k+1})\\|^2}{2M}+\\frac{\\tau_k\\varepsilon}{2}. \\]\n\nIn our algorithm \\begin{align*}\ny_{k+1}= x_{k+1}-h_{k+1}\\nabla f(x_{k+1}),\\\\h_{k+1}=\\operatornamewithlimits{argmin}\\limits_{h\\geqslant 0} f(x_{k+1}-h\\nabla f(x_{k+1})),\n\\end{align*} so \\[f(y_{k+1})-f(x_{k+1})\\leq f(y)-f(x_{k+1})\\leq -\\frac{\\|\\nabla f(x_{k+1})\\|^2}{2M}+\\frac{\\tau_k\\varepsilon}{2}.\\]\n\n\\end{proof}\n\\subsection{Comparison with the UFGM method}\n\nNote that in the case of Euclidean norm and $V_x(y)=\\frac{1}{2}\\|x-y\\|^2$, in the UFGM algorithm the mirror descent step \\[z_{k+1}\\gets \\operatornamewithlimits{argmin}\\limits_{z\\in \\mathbb{R}^n} \\alpha_{k+1}\\langle\\nabla f(x_{k+1}), z-v_k\\rangle +V_{v_k}(z)\\] may be rewritten as \\[z_{k+1}\\gets v_k-\\alpha_{k+1}\\nabla f(x_{k+1}).\\] Moreover, in the case of the Euclidean norm the sequence $\\{v_k\\}$ turns out to be identical to the sequence $\\{z_k\\}$. Now by direct substitution of $z_{k+1}$ and by using $(1-\\tau_k)y_k=x_{k+1}-\\tau_kv_k$ we get that \\[y_{k+1}=\\tau_k(z_k-\\alpha_{k+1}\\nabla f(x_{k+1}))+(1-\\tau_k)y_k=x_{k+1}-\\frac{1}{L_{k+1}}\\nabla f(x_{k+1}).\\] This means that the two methods are not just very similar, but are practically identical. The only difference between them is the usage of exact line search instead of a fixed-length gradient descent step.\n\n\\subsection{Convergence Analysis}\nTo ascertain the convergence of the above algorithm we will require the following lemmas:\n\n\\begin{lemma}\nFor any $u\\in\\mathbb{R}^n$\n\n\\[\\alpha_{k+1}\\langle\\nabla f(x_{k+1}),z_k-u\\rangle\\leqslant\\alpha_{k+1}^2L_{k+1}\\left(f(x_{k+1})-f(y_{k+1})+\\frac{\\tau_k\\varepsilon}{2}\\right)+\\frac{1}{2}\\|z_k-u\\|^2-\\frac{1}{2}\\|z_{k+1}-u\\|^2. \\]\n\\end{lemma}\n\n\\begin{proof}\n\\begin{align*}\n\\alpha_{k+1}&\\langle\\nabla f(x_{k+1}),z_k-u\\rangle = \\alpha_{k+1}\\langle\\nabla f(x_{k+1}),z_k-z_{k+1}\\rangle+\\alpha_{k+1}\\langle\\nabla f(x_{k+1}),z_{k+1}-u\\rangle \\\\\n&\\stackrel{\\scriptsize{\\circled{1}}}{=} \\alpha_{k+1}\\langle\\nabla f(x_{k+1}),z_k-z_{k+1}\\rangle+\\langle z_k-z_{k+1} ,z_{k+1}-u\\rangle\\\\\n&\\stackrel{\\scriptsize{\\circled{2}}}{=}\\alpha_{k+1}\\langle\\nabla f(x_{k+1}),z_k-z_{k+1}\\rangle+\\frac{1}{2}\\|z_k-u\\|^2-\\frac{1}{2}\\|z_{k+1}-u\\|^2-\\frac{1}{2}\\|z_k-z_{k+1}\\|^2\\\\\n&\\stackrel{\\scriptsize{\\circled{3}}}{\\leqslant} \\alpha_{k+1}^2L_{k+1}\\left(f(x_{k+1})-f(y_{k+1})+\\frac{\\tau_k\\varepsilon}{2}\\right)+\\frac{1}{2}\\|z_k-u\\|^2-\\frac{1}{2}\\|z_{k+1}-u\\|^2.\n\\end{align*}\n\nHere, $\\circled{1}$ is due to \\[z_{k+1}=\\operatornamewithlimits{argmin}\\limits_{z\\in\\mathbb{R}^n} \\langle\\alpha_{k+1}\\nabla f(x_{k+1}),z\\rangle+\\frac{1}{2}\\|z_k-z\\|^2,\\] which implies \\[\\nabla \\left(\\frac{1}{2}\\|z_k-z\\|^2+\\langle\\alpha_{k+1}\\nabla f(x_{k+1}),z\\rangle\\right)\\bigg\\rvert_{z=z_{k+1}}=0.\\] $\\circled{2}$ follows from the triangle equality of Bregman divergence \n\\[\\langle -\\nabla V_x(y),y-u\\rangle = V_x(u)-V_y(u)-V_x(y),\\] which takes the following form when $V_x(y)=\\frac{1}{2}\\|x-y\\|^2$: \\[\\langle x-y,y-u\\rangle=\\frac{1}{2}\\|x-u\\|^2-\\frac{1}{2}\\|y-u\\|^2-\\frac{1}{2}\\|x-y\\|^2\\]\nFinally, $\\circled{3}$ is due to our choice of $L_{k+1}.$\n\n\\end{proof}\n\n\\begin{lemma}\nFor any $u\\in\\mathbb{R}^n$\n\n\\begin{align*}\n\\alpha_{k+1}^2L_{k+1}f(y_{k+1})-&\\left(\\alpha^2_{k+1}L_{k+1}-\\alpha_{k+1}\\right)f(y_k)+\\\\&\n\\left(\\frac{1}{2}\\|z_{k+1}-u\\|^2-\\frac{1}{2}\\|z_k-u\\|^2\\right)-\\frac{\\alpha_{k+1}\\varepsilon}{2}\\leqslant\\alpha_{k+1}f(u).\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nWe deduce the following sequence of relations:\n\\begin{align*}\n&\\alpha_{k+1}(f(x_{k+1})-f(u))\\leqslant \\alpha_{k+1}\\langle \\nabla f(x_{k+1}), x_{k+1}-u\\rangle\\\\&=\\alpha_{k+1}\\langle \\nabla f(x_{k+1}), x_{k+1}-z_k\\rangle+\\alpha_{k+1}\\langle \\nabla f(x_{k+1}), z_k-u\\rangle\\\\\n&\\stackrel{\\scriptsize{\\circled{1}}}{=} \\frac{(1-\\tau_k)\\alpha_{k+1}}{\\tau_k}\\langle \\nabla f(x_{k+1}), y_k-x_{k+1}\\rangle+\\alpha_{k+1}\\langle \\nabla f(x_{k+1}), z_k-u\\rangle\\\\\n&\\stackrel{\\scriptsize{\\circled{2}}}{\\leqslant} \\frac{(1-\\tau_k)\\alpha_{k+1}}{\\tau_k}(f(y_k)-f(x_{k+1}))+\\alpha^2_{k+1}L_{k+1}\\left(f(x_{k+1})-f(y_{k+1})+\\frac{\\tau_k\\varepsilon}{2}\\right)\\\\&+\\frac{1}{2}\\|z_{k}-u\\|^2-\\frac{1}{2}\\|z_{k+1}-u\\|^2\n\\stackrel{\\scriptsize{\\circled{3}}}{=} (\\alpha^2_{k+1}L_{k+1}-\\alpha_{k+1})f(y_k)-\\alpha_{k+1}^2L_{k+1}f(y_{k+1})\\\\&+\\alpha_{k+1}f(x_{k+1})+\\left(\\frac{1}{2}\\|z_{k}-u\\|^2-\\frac{1}{2}\\|z_{k+1}-u\\|^2\\right)+\\frac{\\alpha_{k+1}\\varepsilon}{2}.\n\\end{align*} Here, $\\circled{1}$ uses the fact that our choice of $x_{k+1}$ satisfies $\\tau_k(x_{k+1}-z_k)=(1-\\tau_k)(y_k-x_{k+1})$. $\\circled{2}$ is by convexity of $f(\\cdot)$ and \\textsc{Lemma 2.2}, while $\\circled{3}$ uses the choice of $\\tau_k=\\frac{1}{\\alpha_{k+1}L_{k+1}}$.\n\\end{proof}\n\nWe are now ready to begin our proof of the method's convergence.\n\n\n\\begin{theorem}\nLet $f(x)$ be a convex, differentiable function such that its gradient satisfies the H$\\ddot{\\text{o}}$lder condition for some $\\nu\\in[0,1]$ with some finite $M_{\\nu}$. Let $L_0$ also satisfy \n\\[L_0\\leqslant \\inf_{\\nu\\in[0,1]}4\\left[\\frac{1-\\nu}{1+\\nu}\\frac{M_\\nu}{\\varepsilon}\\right]^{\\frac{1-\\nu}{1+\\nu}}M_\\nu.\\] Then ULCM($f$, $L_0$, $x_0$, $\\varepsilon$, $T$) outputs $y_T$ such that $f(y_T)-f(x^\\ast)\\leqslant\\varepsilon$ in the number of iterations\n\n\\[T\\leqslant\\inf_{\\nu\\in[0,1]}\\ \\left[\\frac{1-\\nu}{1+\\nu}\\right]^\\frac{1-\\nu}{1+3\\nu}\\left[\\frac{2^\\frac{3+5\\nu}{2}M_\\nu}{\\varepsilon}\\right]^\\frac{2}{1+3\\nu}\\Theta^\\frac{1+\\nu}{1+3\\nu},\\]where $\\Theta$ is any upper bound on $\\frac{1}{2}\\|x_0-x^\\ast\\|^2$.\n\\end{theorem}\n\\begin{proof}\nNote that our choice of $\\alpha_{k+1}$ satisfies\n\\[\\alpha^2_{k+1}L_{k+1}-\\alpha_{k+1}=\\alpha^2_{k}L_k,\\addtag\\] which allows us to telescope \\textsc{Lemma 2.3}. Summing up \\textsc{Lemma 2.3} for $k=0,1,\\ldots, T-1$ and $u=x^\\ast$, we obtain\n\n\\[\\alpha_{T}^2L_{T}f(y_T)+\\left(\\frac{1}{2}\\|z_T-x^\\ast\\|^2-\\frac{1}{2}\\|z_0-x^\\ast\\|^2\\right)\\leqslant\\sum_{k=1}^T\\alpha_kf(x^\\*) +\\sum_{k=1}^T\\frac{\\alpha_k\\varepsilon}{2}.\\] By using $(1)$ we get that $\\sum\\limits_{k=1}^T\\alpha_k=\\alpha^2_TL_T$. We also notice that $\\frac{1}{2}\\|z_t-x^\\ast\\|^2\\geqslant 0$ and $\\frac{1}{2}\\|z_0-x^\\ast\\|^2\\leqslant \\Theta$. Therefore, \n\n\\[f(y_T)-f(x^\\ast)\\leqslant \\frac{\\Theta}{\\alpha^2_TL_T}+\\frac{\\varepsilon}{2}.\\]\n\nNote that our process of calculating $L_k$ guarantees that if the step $L_{k+1}\\gets 2L_{k+1}$ of the algorithm was executed at least once for some $k$, then for that $k$\n\n\\[L_{k+1}\\leqslant 2\\left[\\frac{1-\\nu}{1+\\nu}\\frac{M_\\nu}{\\varepsilon\\tau_k}\\right]^{\\frac{1-\\nu}{1+\\nu}}M_\\nu. \\addtag\\]\n\nAssume that $L_n\\leq 2\\left[\\frac{1-\\nu}{1+\\nu}\\frac{M_\\nu}{\\varepsilon\\tau_{n-1}}\\right]^{\\frac{1-\\nu}{1+\\nu}}M_\\nu$ and $L_{n+1}=\\frac{L_n}{2}$ for some $n\\geq 1$. Then \\[\\frac{1}{\\tau_n}=\\alpha_{n+1}L_{n+1}=\\frac{1}{2}+\\sqrt{\\frac{1}{4}+\\alpha_n^2L_nL_{n+1}}\\geq\\frac{1}{2}+\\sqrt{\\frac{1}{4}+\\frac{\\alpha_n^2L_n^2}{2}}\\geq\\frac{1}{\\sqrt{2}\\tau_{n-1}}.\\]\n\n\\[L_{n+1}=\\frac{L_n}{2}\\leq \\left[\\frac{1-\\nu}{1+\\nu}\\frac{\\sqrt{2}M_\\nu}{\\varepsilon\\tau_{n}}\\right]^{\\frac{1-\\nu}{1+\\nu}}M_\\nu\\leq2\\left[\\frac{1-\\nu}{1+\\nu}\\frac{M_\\nu}{\\varepsilon\\tau_{n}}\\right]^{\\frac{1-\\nu}{1+\\nu}}M_\\nu.\\] This shows that even if we don't execute the step $L_{k+1}\\gets 2L_{k+1}$, (2) remains true as long as it held true on the previous iteration. All of the above proves that the assumption about $L_0$ in the statement of the theorem implies that (2) is true for all $k=0,\\ldots T-1$. \n\nDenote $A_k=\\alpha^2_kL_k$. We may now proceed to attain a lower bound on $A_T$.\n\n\\[\\frac{\\alpha^2_k}{A_k}=\\frac{1}{L_k}\\geqslant\\frac{1}{2M_\\nu}\\left[\\frac{1+\\nu}{1-\\nu}\\frac{\\varepsilon}{M_\\nu}\\right]^{\\frac{1-\\nu}{1+\\nu}}\\left[\\frac{\\alpha_k}{A_k}\\right]^{\\frac{1-\\nu}{1+\\nu}}\\]\n\n\\[\\alpha_k\\geqslant\\frac{1}{2^\\frac{1+\\nu}{1+3\\nu}M_\\nu^\\frac{2}{1+3\\nu}}\\left[\\frac{1+\\nu}{1-\\nu}\\varepsilon\\right]^\\frac{1-\\nu}{1+3\\nu}A_k^\\frac{2\\nu}{1+3\\nu}.\\]\n\nDenote $\\gamma=\\frac{1+\\nu}{1+3\\nu}\\geqslant \\frac{1}{2}.$ Since $A_{k+1}=A_k+\\alpha_{k+1},$ \n\n\\[A^\\gamma_{k+1}-A^\\gamma_{k+1}\\geqslant \\frac{A_{k+1}-A_{k}}{A_{k+1}^{1-\\gamma}+A_k^{1-\\gamma}}\\geqslant\\frac{\\alpha_{k+1}}{2A_{k+1}^{1-\\gamma}}\\geqslant\\frac{1}{2^\\frac{2+4\\nu}{1+3\\nu}M_\\nu^\\frac{2}{1+3\\nu}}\\left[\\frac{1+\\nu}{1-\\nu}\\varepsilon\\right]^\\frac{1-\\nu}{1+3\\nu}. \\addtag\\]Now we telescope (3) for $k=0,\\ldots,T-1$ and  get\n\n\\[A_T\\geqslant\\left[\\frac{1+\\nu}{1-\\nu}\\right]^\\frac{1-\\nu}{1+\\nu}\\frac{T^\\frac{1+3\\nu}{1+\\nu}\\varepsilon^\\frac{1-\\nu}{1+\\nu}}{2^\\frac{2+4\\nu}{1+\\nu}M_\\nu^\\frac{2}{1+\\nu}}.\\] This allows us to estimate the number of iterations necessary to achieve error no more than $\\varepsilon$. However, beforehand we shall note that this estimate heavily depends on $\\nu$. By allowing $M_\\nu$ to be infinite, we make the gradient of any differentiable function satisfy the H$\\ddot{\\text{o}}$lder condition for all $\\nu\\in[0,1]$. This in turn allows to easily select the most appropriate estimate:\n\n\\[T\\leqslant\\inf_{\\nu\\in[0,1]}\\ \\left[\\frac{1-\\nu}{1+\\nu}\\right]^\\frac{1-\\nu}{1+3\\nu}\\left[\\frac{2^\\frac{3+5\\nu}{2}M_\\nu}{\\varepsilon}\\right]^\\frac{2}{1+3\\nu}\\Theta^\\frac{1+\\nu}{1+3\\nu}.\\]\n\nNote that since the solution $x^\\ast$ was arbitrary, $x^\\ast$ may now be considered to be the solution which minimizes $\\frac{1}{2}\\|x_0-x^\\ast\\|^2$.\n\n\\end{proof}\n\\subsection{Stopping criterion}\n\nIn \\cite{anikin2017dual} it is shown that the original version of the Linear Coupling Method may be equipped with a stopping criterion. By using similar techniques, we are now going to show that our universal modification of said method may also be equipped with a calculable stopping criterion.\n\nBy ignoring the first inequality in the proof of \\textsc{Lemma 2.3}, we get that for all $u\\in\\mathbb{R}^n$ (remember that $A_k=\\alpha^2_k L_k$)\n\\begin{align*}\nA_{k+1} f(y_{k+1})-A_k f(y_k)&+\\frac{1}{2}\\|z_{k+1}-u\\|^2-\\frac{1}{2}\\|z_k-u\\|^2-\\frac{\\alpha_{k+1}\\varepsilon}{2}\\\\&\\leq \\alpha_{k+1}\\left(f(x_{k+1})+\\langle\\nabla f(x_{k+1}),u-x_{k+1}\\rangle\\right).\n\\end{align*}\n Summing up for $k=0,\\ldots,m-1$, we obtain \n\n\\[f(y_m)\\leq \\frac{\\varepsilon}{2}+\\frac{1}{A_m}\\min_{u\\in\\mathbb{R}^n}\\left\\lbrace\\frac{1}{2}\\|z_0-u\\|^2+\\sum_{i=1}^m\\alpha_{i}\\left(f(x_{i})+\\langle\\nabla f(x_{i}),u-x_{i}\\rangle\\right)\\right\\rbrace.\\]\n\nDenote \\[l_m(u)=\\sum_{i=1}^m\\left[ \\alpha_{i}\\left(f(x_{i})+\\langle\\nabla f(x_{i}),u-x_{i}\\rangle\\right)\\right]\\] and\n\n\\[\\hat{f}_m=\\min_{u:\\ \\frac{1}{2}\\|z_0-u\\|^2\\leq\\Theta} \\frac{1}{A_m} l_m(u).\\]\n\nThen by using strong duality one may see that\n\n\\begin{align*}\n\\hat{f}_m&=\\min_{u\\in \\mathbb{R}^n}\\max_{\\lambda\\geq 0}\\left\\lbrace\\frac{1}{A_m} l_m(u)+\\lambda\\left(\\frac{1}{2}\\|z_0-u\\|^2-\\Theta\\right)\\right\\rbrace\\\\&=\\max_{\\lambda\\geq 0} \\min_{u\\in \\mathbb{R}^n}\\left\\lbrace\\frac{1}{A_m} l_m(u)+\\lambda\\left(\\frac{1}{2}\\|z_0-u\\|^2-\\Theta\\right)\\right\\rbrace.\n\\end{align*}\n\nBy setting $\\lambda=\\frac{1}{A_m}$, we get that \n\n\\[\\hat{f}_m\\geq\\frac{1}{A_m}\\min_{u\\in\\mathbb{R}^n}\\left\\lbrace\\frac{1}{2}\\|z_0-u\\|^2+\\sum_{i=1}^m\\alpha_{i}\\left(f(x_{i})+\\langle\\nabla f(x_{i}),u-x_{i}\\rangle\\right)\\right\\rbrace-\\frac{\\Theta}{A_m}.\\]\n\nThen $f(y_m)-\\hat{f}_m\\leq\\frac{\\varepsilon}{2}+\\frac{\\Theta}{A_m}.$ This means that our method is primal-dual. By the convexity of $f$ we also get that $f(x^\\ast)\\geq\\hat{f}_m$, so $f(y_m)-f(x^\\ast)\\le f(y_m)-\\hat{f}_m\\leq\\varepsilon$ may be used as an implementable stopping criterion. Of course, an estimate of $\\Theta$ is required to compute $\\hat{f}_m$. Overestimating $\\Theta$ may lead to performing an excessive amount of iterations, while underestimating it invalidates the criterion completely. However, the stopping criterion requires an estimate of only one unknown parameter, which is also not used in the algorithm's definition. On the other hand, three unknown parameters ($\\nu, M_\\nu, \\Theta$) need to be estimated to calculate the upper bound on the number of iterations required to get an $\\varepsilon$-accurate solution\n\\section{Line search}\n\nDuring all of the previous analysis we assumed that $\\forall x\\in\\mathbb{R}^n\\ f(x),\\ \\nabla f(x)$, the steepest descent step, and the mirror descent step may be calculated exactly. However, in relation to the steepest descent step this assumption is not critical for the method's convergence.\n\nFor any convex function of one real argument defined on a segment of the form $[a,b]$ of length $l=b-a$ a point $y$ such that \\[\\|y-\\operatornamewithlimits{argmin}_{x\\in[a,b]}f(x)\\|\\leqslant \\varepsilon \\] may be found in $O(\\log\\frac{l}{\\varepsilon})$ function value calculations by using the bisection method. However, to perform an exact line search in our algorithm one needs to localize the solution first. To do that we propose the following simple procedure:\n\n\n\\begin{algorithm}\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n\t\n    \\caption{Localize(f,$l_0$)}\n    \\Input{$f(x)$ -- convex function defined on $[0,+\\infty)$; initial segment length $l_0$.}\n    \\Output{$l$ such that $\\operatornamewithlimits{argmin}\\limits_{x\\in[0,+\\infty)} f(x) \\in [0,l]$}\n    $l\\gets l_0$\\\\\n    \\While{$f(2l)\\leqslant f(l)$}{\n  \t\t$l\\gets2l$\n    }\n  \\Return{$l$}\n\\end{algorithm}\n\n\nLet us estimate the accuracy with which the steepest descent must be performed to guarantee our method's convergence. \nLet's say we want to get a solution with accuracy of $\\varepsilon+\\delta$, where $\\delta$ is the term resulting from the inaccuracy of the steepest descent step. To do that we need to slightly modify our algorithm: \n\\newpage\n\\begin{algorithm}\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n\t\n    \\caption{$\\delta$-ULCM($f$, $L_0$, $x_0$, $\\varepsilon$, $\\delta$, $T$)}\n    \\Input{$f$ a differentiable convex function with H$\\ddot{\\text{o}}$lder continuous gradient;\n    initial value of the \"inexact\" Lipschitz continuity constant $L_0$;\n    initial point $x_0$;\n    accuracy $\\varepsilon$;\n    line search accuracy $\\delta$;\n    number of iterations $T$.}\n    $y_0 \\gets x_0$, $z_0 \\gets x_0$, $\\alpha_0 \\gets 0$\\\\\n  \\For{$k=0 \\to T-1$}{\n  \t$L_{k+1}\\gets\\frac{L_{k}}{2}$\\\\\n    \\While{True}{\n  \t$\\alpha_{k+1}\\gets\\frac{1}{2L_{k+1}}+\\sqrt{\\frac{1}{4L^2_{k+1}}+\\alpha^2_k\\frac{L_k}{L_{k+1}}}$\\\\\n    $\\tau_k\\gets\\frac{1}{\\alpha_{k+1}L_{k+1}}$\\\\\n    $x_{k+1}\\gets\\tau_kz_k+(1-\\tau_k)y_k$\\\\\n    Choose $y_{k+1}$ such that $f(y_{k+1})\\leq \\operatornamewithlimits{argmin}\\limits_{h\\geq 0} f(x_{k+1}-h\\nabla f(x_{k+1}))+\\frac{\\tau_k\\delta}{2}$\\\\\n  $z_{k+1}\\gets\\operatornamewithlimits{argmin}\\limits_{z\\in\\mathbb{R}^n}\\ \\langle\\alpha_{k+1}\\nabla f(x_{k+1}),z-z_k\\rangle +\\frac{1}{2}\\|z_k-z\\|^2$\\\\\n    \\If{$\\langle \\alpha_{k+1}\\nabla f(x_{k+1}),z_k-z_{k+1}\\rangle-\\frac{1}{2}\\|z_k-z_{k+1}\\|^2\\leq \\alpha^2_{k+1}L_{k+1}(f(x_{k+1})-f(y_{k+1})+\\frac{\\tau_k\\varepsilon}{2})$}{\\textbf{break}}\n    \\Else{$L_{k+1}\\gets 2L_{k+1}$}\n  \t}\n \n  }\n  \\Return{$y_T$}\n\\end{algorithm}\n\n\\begin{theorem}\nLet $f(x)$ be a convex, differentiable function such that its gradient satisfies the H$\\ddot{\\text{o}}$lder condition for some $\\nu\\in[0,1]$ with some finite $M_{\\nu}$. Let $L_0$ also satisfy \n\\[L_0\\leqslant \\inf_{\\nu\\in[0,1]}4\\left[\\frac{1-\\nu}{1+\\nu}\\frac{M_\\nu}{\\varepsilon}\\right]^{\\frac{1-\\nu}{1+\\nu}}M_\\nu.\\] Then $\\delta$-ULCM($f$, $L_0$, $x_0$, $\\varepsilon$, $\\delta$, $T$) outputs $y_T$ such that $f(y_T)-f(x^\\ast)\\leqslant\\varepsilon+\\delta$ in the number of iterations\n\n\\[T\\leqslant\\inf_{\\nu\\in[0,1]}\\ \\left[\\frac{1-\\nu}{1+\\nu}\\right]^\\frac{1-\\nu}{1+3\\nu}\\left[\\frac{2^\\frac{3+5\\nu}{2}M_\\nu}{\\varepsilon}\\right]^\\frac{2}{1+3\\nu}\\Theta^\\frac{1+\\nu}{1+3\\nu},\\]where $\\Theta$ is any upper bound on $\\frac{1}{2}\\|x_0-x^\\ast\\|^2$.\n\\end{theorem}\n \nThis immediately follows from the proof of $\\textsc{Theorem 2.4}$. To see that, note that if for some $L_{k+1}$ and the exact solution of the line search problem $\\hat{y}_{k+1}$\n\n\\[\\langle \\alpha_{k+1}\\nabla f(x_{k+1}),z_k-z_{k+1}\\rangle-V_{z_k}(z_{k+1})\\leq \\alpha^2_{k+1}L_{k+1}\\left(f(x_{k+1})-f(\\hat{y}_{k+1})+\\frac{\\tau_k\\varepsilon}{2}\\right)\\] holds true, then by definition of $y_{k+1}$ we have\n\\[\\langle \\alpha_{k+1}\\nabla f(x_{k+1}),z_k-z_{k+1}\\rangle-V_{z_k}(z_{k+1})\\leq \\alpha^2_{k+1}L_{k+1}\\left(f(x_{k+1})-f(y_{k+1})+\\frac{\\tau_k(\\varepsilon+\\delta)}{2}\\right).\\] This leads to an analogue of \\textsc{Lemma 2.1}. Then by proceeding with the proof the same way it was done in \\textsc{Theorem 2.4}, one gets the desired result.\n\n\\subsection{Simplified function evaluation during line search}\n\nAs noted in \\cite{SESOP}, for objectives of particular form the steepest descent step may be performed significantly faster.\n\nConsider a function of the form \n\n\\[f(x)=\\phi(\\bm{\\rm{A}}x)+\\psi(x),\\] where $x\\in \\mathbb{R}^n$, $\\bm{\\rm{A}}\\in \\mathbb{R}^{n\\times n}$.\n\nIf $n$ is sufficiently large, the computation of $\\bm{\\rm{A}}x$ may be the most time-consuming operation during computation of $f(x)$. However, if we are performing the steepest descent step, we can be sure that $x$ is of the form $x_k+\\alpha\\nabla f(x_k)$. Then\n\n\\[\\bm{\\rm{A}}x=\\bm{\\rm{A}}x_k+\\alpha \\bm{\\rm{A}}\\nabla f(x_k)=v_0+\\alpha v_1.\\] This shows that one may calculate the two points $v_0$ and $v_1$ just once at the beginning of a steepest descent step. \n\nIf $\\psi(y)$ and $\\phi(y)$ with $y$ known may be calculated in $\\mathcal{O}(n)$ arithmetic operations, then this representation of $\\bm{\\rm{A}}x$ allows us to evaluate $f(x)$ in $\\mathcal{O}(n)$ arithmetic operations after performing matrix multiplication, which requires $\\mathcal{O}(n^2)$ arithmetic operations, only twice. This may significantly decrease the cost of one steepest descent step.\n\n\\section{Numerical experiments}\n\nThe proposed methods were implemented in C$++$ and tested using the\nmodern versions of GCC, clang and ICC (Intel C Compiler) on both GNU\/Linux,\nMac OS X and Microsoft Windows operating systems. The source code is available at \\url{http:\/\/github.com\/htower\/ulcm}.\n\nFor the presented computational experiments we have also implemented a variant of the conjugate gradients method proposed by Y.~Nesterov in \\cite{nesterov1989book}, which we denote as NCG. The method has high numerical stability and a number of interesting properties. In particular, it lacks a restart procedure. This results in an increased iteration complexity relatively to \"classic\" conjugate gradient methods, which may be attributed to the necessity of solving two line search problems at each iteration. Details are presented in Algorithm \\ref{alg_ncg} and Figure \\ref{fig_ncg}.\n\n\\begin{algorithm}\n  \\SetKwInOut{Input}{Input}\n  \\SetKwInOut{Output}{Output}\n\n  \\caption{NCG($f$, $x_0$, $\\delta$, $T$)}\n  \\Input{$f$ a differentiable convex function with H$\\ddot{\\text{o}}$lder continuous gradient;\n  initial point $x_0$;\n  line search accuracy $\\delta$;\n  number of iterations $T$.}\n  $y_{-2} \\gets x_0$,\n  $y_{-1} \\gets x_0$,\n  $y_{ 0} \\gets x_0$ \\\\\n  \\For{$k = 0$ to $T-1$}\n  {\n    $\\alpha_k \\gets \\operatornamewithlimits{argmin}\\limits_{\\alpha \\in \\mathbb{R}} f(x_k + \\alpha (y_{k-2} - x_k))$\\\\\n    $y_k=x_k+\\alpha_k(y_{k-2} - x_k)$\\\\\n    $\\beta_k \\gets \\operatornamewithlimits{argmin}\\limits_{\\beta\\ge 0} f(y_k - \\beta \\nabla f(y_k))$\\\\\n    $x_{k+1}=y_k-\\beta_k\\nabla f(y_k)$\n  }\n  \\Return{$x_T$}\n  \\label{alg_ncg}\n\\end{algorithm}\n\n\\begin{figure}\n\\includegraphics[]{pics\/cg_nesterov.pdf}\n\\caption{Illustration of the NCG method.}\n\\label{fig_ncg}\n\\end{figure}\n\nThe behaviour of the proposed methods was investigated by a series of numerical experiments on different smooth and non-smooth optimization problems. For all experiments we set the starting point $x_0$ to $10 \\cdot e$, where $e = (1,...,1)$, and the precision value $\\varepsilon = 10^{-4}$. The methods were interrupted as soon as the objective function's value became lower than $f(x^*) + 5\\varepsilon = f(x^*) + 5\\times 10^{-4}$. The dimensionality of the problem was up to $10^6$ .\n\nFirstly, we considered the following smooth (quadratic) problem:\n\\begin{equation}\n  f(x) = \\sum_{i=1}^{n} i x_i^2 .\n\\label{eq_problem_s}\n\\end{equation}\nThis function is $L$-smooth, but the parameter $L$ depends on the number of dimensions $n$ linearly. This minimization problem can be solved analytically, the optimal value $f(x^\\ast)$ is equal to $0$. The results of our experiments are presented in Table~\\ref{tbl_s} and Figure~\\ref{fig_s36}.\n\\begin{table}\n\\begin{tabular}{r|l||r|r|r|r|r|r}\n\\multirow{2}{26mm}{$n$, problem size} & \\multirow{2}{10mm}{$f(x_0)$} & \\multicolumn{2}{c|}{UFGM} & \\multicolumn{2}{c|}{ULCM} & \\multicolumn{2}{c}{NCG} \\\\\n & & iterations & t, sec. & iterations & t, sec. & iterations & t, sec. \\\\ \\hline\n$10^3$ & $5 \\cdot 10^7$    &   743 & 0.035 &   722 & 0.035 &  121 & 0.004 \\\\\n$10^4$ & $5 \\cdot 10^9$    &  3230 & 1.429 &  3459 & 3.233 &  385 & 0.079 \\\\\n$10^5$ & $5 \\cdot 10^{11}$ & 15231 & 141.2 & 18053 & 372.6 & 1217 & 2.796 \\\\\n$10^6$ & $5 \\cdot 10^{13}$ & 73185 & 6857 & 84117 & 22373 & 3850 & 98.40 \\\\\n\\end{tabular}\n\\caption{Method's complexity for the smooth problems.}\n\\label{tbl_s}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[]{pics\/s3.pdf}\n\\includegraphics[]{pics\/s6.pdf}\n\\caption{Methods convergence for the smooth problems with $n = 10^3$ (top) and $n = 10^6$ (bottom). The solid line stands for the UFGM method, the dotted line stands for the ULCM method, the dashed line stands for the NCG method.}\n\\label{fig_s36}\n\\end{figure}\n\nNext, we consider the following non-smooth problem:\n\\begin{equation}\n  f(x) = \\max_{i=1,...,n} x_i + \\frac{\\mu}{2} \\| x \\|_2^2.\n\\label{eq_problem_ns}\\end{equation}\n\nIn our experiments $\\mu=0.1$.\nThough this function is differentiable almost everywhere. Though it does not have globally H$\\ddot{\\text{o}}$lder continuous gradients, the gradient satisfies the H$\\ddot{\\text{o}}$lder continuity condition on any bounded set. \n\nThis minimization problem can be solved analytically, the optimal value $f(x^\\ast)$ is equal to $-\\frac{1}{2\\mu n}=-\\frac{5}{n}$.\n\nThe gradient (subgradient, in case $f$  is not differentiable at $x$) can be evaluated as\n\\[\n  \\nabla f(x) = \\mu x + z(x), \\quad\n  z(x) = (0,...0,1,0,...0) ,\n\\]\nwhere $1$ is located at position $k = \\operatornamewithlimits{argmin}\\limits_{i=1,...,n} x_i .$\n\nThe results are shown in Table~\\ref{tbl_ns} and Figure~\\ref{fig_ns3_ns6}.\n\n\\begin{table}\n\\begin{tabular}{r|l||r|r|r|r}\n\\multirow{2}{26mm}{$n$, problem size} & \\multirow{2}{10mm}{$f(x_0)$} & \\multicolumn{2}{c|}{UFGM} & \\multicolumn{2}{c}{ULCM} \\\\\n & & iterations & t, sec. & iterations & t, sec. \\\\ \\hline\n$10^3$ & $1 \\cdot 10^4$ &  535795 &  17.48 & 1376 & 0.175 \\\\\n$10^4$ & $1 \\cdot 10^5$ &  706870 &  233.8 & 6930 & 6.059 \\\\\n$10^5$ & $1 \\cdot 10^6$ & 1751285 &   4713 & 6950 & 34.18 \\\\\n$10^6$ & $1 \\cdot 10^7$ & 4341186 & 165435 & 6977 & 575.1 \\\\\n\\end{tabular}\n\\caption{Method's complexity for the non-smooth problems.}\n\\label{tbl_ns}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[]{pics\/ns3.pdf}\n\\includegraphics[]{pics\/ns6.pdf}\n\\caption{Methods convergence for the non-smooth problems with $n = 10^3$ (top) and $n = 10^6$ (bottom).}\n\\label{fig_ns3_ns6}\n\\end{figure}\n\nNote that in our particular case, since the ULCM and UFGM methods become identical if the steepest descent of the ULCM methods is replaced with a gradient descent step with step length $\\frac{1}{L_{k+1}}$, all the differences in actual performance may be attributed to the line search procedure.\n\nThe results of our experiments may be summarized as follows:\n\\begin{enumerate}\n  \\item For the smooth problems (\\ref{eq_problem_s}) the NCG method showed best performance. Its convergence rate significantly exceeds the convergence rates of UFGM and ULCM methods by up to two orders of magnitude. Although the ULCM method took less iterations to converge, it was slower (about 3 times) in terms of running time. \n  \\item For the non-smooth problems (\\ref{eq_problem_ns}) the situation is opposite. In that case the ULCM method significantly outperformed UFGM, both in terms of required iterations and elapsed time. In the case of $10^6$ arguments our method converged about 300 times faster.\n\\end{enumerate}\n\n\\section*{Conclusions}\nIn this paper we propose the first primal-dual method of non-smooth convex optimization with auxiliary line search. Practical experiments show that this method significantly outperforms Nesterov's Universal Fast Gradient Method \\cite{nesterov2015universal}. Moreover, we prove that the presented method is also optimal for all the problems with intermediate level of smoothness. The advantage of such an approach is that one can generalize it to stochastic programming using mini-batches \\cite{gasnikov2016universal} and to gradient-free methods \\cite{dvurechensky2017randomized}.  \n\n\\section*{Acknowledgements}\n\nThe authors would like to thank Boris Polyak and Yurii Nesterov for helpful comments.\n\n\\section*{Funding}\nThis work was partially supported by RNF 17-11-01027.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nMotivated by applications in a wide array of fields ranging from sociology to systems biology and most closely related to this work, in probabilistic combinatorics and statistical physics, the last few years have witnessed an explosion in both network models as well as interacting particle systems on these models. In this context, the two major themes of this work are as follows:\n\n\n\\begin{enumeratea}\n\\item {\\bf Connectivity, percolation and critical random graphs:} A fundamental question in this general area is understanding connectivity properties of the network model, including the time and nature of emergence of the giant component.  Writing $[n] = \\set{1,2,\\ldots, n}$ for the vertex set, most of these models have a parameter $t$ (related to the edge density) and a model dependent critical time $t_c$ such that for $t< t_c$ (subcritical regime), there exists no giant component (size of the largest component $|\\cC_1(t)| = o_P(n)$), while for $t> t_c$ (supercritical regime), the size of the largest component scales like $f(t)n$ with $f(t) > 0$ and is model dependent. Behavior in the so-called ``critical regime'' (i.e., when $t = t_c$) is the main content of this paper.\n To prime the reader let us informally describe the types of results closest in spirit to this work. We defer precise definitions of technical aspects (e.g., definitions of the limiting objects, the underlying topology etc.) to Section \\ref{sec:notation} and precise statements of related results to Section \\ref{sec:disc}.\nThe prototypical example of the ``critical'' phenomenon is the Erd\\H{o}s-R\\'enyi  random graph at criticality which is constructed as follows: Fix a parameter $\\lambda\\in \\bR$ and vertex set $[n]$ and let $\\ERRG(n,\\lambda)$ be the random graph obtained by placing each of the ${n\\choose 2}$ possible edges independently with probability $n^{-1} + \\lambda n^{-4\/3}$.\nMaximal component sizes in $\\ERRG(n,\\lambda)$ were studied extensively in \\cite{bollobas1984evolution,luczak1990component,janson1993birth,aldous-crit,nachmias-peres-diameter}. The scaling limit of the maximal components of $\\ERRG(n,\\lambda)$ when viewed as metric spaces was identified in \\cite{BBG-12}. \nIt is believed that a large class of random discrete structures, in the critical regime, belong to the ``Erd\\H{o}s-R\\'enyi  universality class.''\nSoon after the work  \\cite{BBG-12}, an abstract universality principle was developed in \\cite{SBSSXW14,bhamidi-broutin-sen-wang} which was used to establish Erd\\H{o}s-R\\'enyi  type scaling limits for a wide array of critical random graph models including the configuration model and various models of inhomogeneous random graphs.\nIt is strongly believed that the components of critical percolation on high-dimensional tori \\cite{hofstad-sapozhnikov,heydenreich-hofstad-1,heydenreich-hofstad-2}, and the hypercube \\cite{hofstad-nachmias} also share the Erd\\H{o}s-R\\'enyi  scaling limit, but these problems are open at this point.\n\n\n\\item {\\bf Vacant set left by random walk on graphs:} The second main theme is the area of random interlacements and percolative properties of the vacant set of random walks on finite graphs, see e.g. \\cite{sznitman2010vacant}. See \\cite{vcerny2012random} for a recent survey most closely related to this paper, and \\cite{drewitz-rath-sapozhnikov} for an introduction to random interlacements. This question was initially posed by Hilhorst who wanted to understand the geometry of crystals affected by corrosion. The precise mathematical model is as follows: consider a finite  graph on $[n]$ vertices (and to fix ideas assumed connected) which represents the crystalline structure of the object of interest. Now suppose a ``corrosive particle'' wanders through the structure via a simple random walk $\\set{X_t:t\\geq 0}$ (started from say a uniformly chosen vertex),  marking each vertex it visits as ``corroded'' (this marking does not affect the dynamics of the walk). For a fixed parameter $u\\geq 0$, define the vacant set as the set of all vertices that have not been ``corroded'' (i.e., not visited by the walk) by time $u n$,\n\\begin{equation}\n\\label{eqn:vac-set-def}\n\t\\mathcal{V}}\\newcommand{\\cW}{\\mathcal{W}}\\newcommand{\\cX}{\\mathcal{X}^u = [n]\\setminus \\big\\{X_t: 0 \\leq t \\leq n u\\big\\}.\n\\end{equation}\nWhen $u$ is ``small'' one expects that only a small fraction of the vertices have been visited by the corrosive particle and thus the maximal connected component $\\cC_1(u)$ of the non-corroded set $\\mathcal{V}}\\newcommand{\\cW}{\\mathcal{W}}\\newcommand{\\cX}{\\mathcal{X}^u$ has a large connected component of size $\\cC_1(u) = \\Theta_P(n)$, while if $u$ increases beyond a ``critical point'' $u_{\\star}$ then the corrosion in the crystal has spread far enough that the maximal connected component in $\\cC_1(u) = o_P(n)$. The ``critical'' $u=u_{\\star}$ regime and in particular the fractal properties of connected components in this regime are of great interest.\n\n\\end{enumeratea}\n\n\n\\subsection{Organization of the paper}\n\\label{sec:org}\nIn the remaining subsections of the introduction, we describe the random graph models considered in this paper and give an informal description of our results. Section \\ref{sec:res} contains precise statements of our main results. We have deferred major definitions to Section \\ref{sec:not}. We discuss the relevance of this work and connections to existing results in Section \\ref{sec:disc}.\nIn Section \\ref{sec:prf-prelim} we define some constructs used in the proof.\nWe start Section \\ref{sec:proof-plane-tree} with the statement of Lemma \\ref{lem:plane-trees} that contains the main technical estimates related to the uniform measure on plane trees with a prescribed children sequence. This lemma forms the crucial work horse in the rest of the proofs. The proof of this lemma occupies the rest of Section \\ref{sec:proof-plane-tree}.\nSection \\ref{sec:proof-conn} and Section \\ref{sec:proof-main-deg-vac-thm} contain the proofs of our main results.\n\n\n \\subsection{Random graph models}\n \\label{sec:rg-models}\n\n  Fix a collection of $n$ vertices labeled by $[n] := \\set{1,2,\\ldots, n}$ and an associated degree sequence $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(n)} = (d_v: v\\in [n])$ where $\\ell_n := \\sum_{v\\in [n]}d_v$ is assumed even. There are two natural constructions resulting in generating a random graph on the above vertex set with the prescribed degree sequence.\n\n \\begin{enumeratea}\n \t\\item {\\bf Uniform distributed simple graph:} Let $\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ denote the space of all simple graphs on $[n]$-labeled vertices with degree sequence $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}$. Let $\\pr_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ denote the uniform distribution on this space and write $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ for the random graph with distribution $\\pr_{n, \\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$.\n\t\\item {\\bf Configuration model \\cite{Boll-book,molloy1995critical,bender1978asymptotic}:} Recall that a multigraph is a graph where we allow multiple edges and self-loops. Write $\\overline\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ for the space of all multigraphs on vertex set $[n]$ with prescribed degree sequence $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}$.\nWrite $\\CM_{n}(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$ for the random multigraph\nconstructed sequentially as follows: Equip each vertex $v\\in [n]$ with $d_v$ half-edges or stubs. Pick two half-edges uniformly from the set of half-edges that have not yet been paired, and pair them to form a full edge. Repeat till all half-edges have been paired. Write $\\overline\\pr_{n, \\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ for the law of $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$.\n \\end{enumeratea}\n\n\n\\subsection{Informal description of our contribution}\nThis work has four major contributions which we now informally describe:\n\\begin{enumeratea}\n\t\\item  We provide an explicit algorithm (Lemma \\ref{lem:alternate-construction}) for sampling a uniform random graph with given degree sequence by first sampling a planar tree with a modified degree sequence via an appropriately defined tilt with respect to the uniform distribution on the space of trees with this modified degree sequence. This allows us to derive scaling limits for the uniform distribution on the space of simple connected graphs with degree sequence satisfying regularity conditions including a finite number of surplus edges (Theorem \\ref{prop:condition-on-connectivity}).\n\t\\item We then use this result to derive scaling limits for the critical regime of both the configuration model as well as the uniform random graph model with prescribed degree sequence (Theorem \\ref{thm:graphs-given-degree-scaling}). This is the strongest metric space scaling result for graphs with given degree sequence under optimal assumptions. This result improves work of \\cite{bhamidi-broutin-sen-wang} which was obtained as a consequence of a general universality principle but under stronger assumptions (exponential moment condition on the degree sequence). The technique used in this paper is also completely disjoint from \\cite{bhamidi-broutin-sen-wang}.\n\t\\item Write $C(n, n+k)$ for the number of {\\bf connected graphs} with $n$ labeled vertices and $n+k$ edges. Deriving asymptotics for $C(n, n+k)$ for fixed $k$ as $n\\to\\infty$ has inspired a large body work both in the combinatorics community \\cite{wright1977number,bender1990asymptotic,spencer-count} as well as in the probability community \\cite{aldous-crit,janson2007brownian,takacs1991bernoulli}. Extending such results to count {\\bf connected graphs} with {\\bf prescribed degree} sequence seems beyond the ken of existing techniques.   As a consequence of our proof technique, we derive asymptotics for such counts (Theorem \\ref{thm:number-of-connected-graphs}). \n\\item As a substantive application, we answer a question raised by \\v{C}ern{\\'y} and Teixeira (\\cite[Remark 6.1 (2)]{cerny-teixeira} (also see the work of Sznitman, e.g. \\cite[Remark 4.5]{sznitman2011lower}) by obtaining the metric space scaling limit of the vacant set left by random walk on random regular graphs. This is the first result about continuum scaling limit of maximal components in the critical regime for this model.  The eventual hope, albeit not addressed in this paper is as follows: Consider spatial systems such as the $d$-dimensional lattice or perhaps more relevant to this paper, asymptotics for the vacant set left by random walks on  the $d$-dimensional torus $(\\bZ\/n\\bZ)^d$ in the large $n\\to\\infty$ network limit.  As described in \\cite{sznitman2011lower}, the basic intuition is that for high enough dimensions $d$, the corresponding objects should behave similar to what one sees in the context of $2d$-regular random graphs.\n\\end{enumeratea}\n\n\n\n\n\\section{Main results}\n\\label{sec:res}\nWe will now describe our main results. To motivate the interested reader, we will eschew slight efficiency in the order of the statement of the results and first show the implications in the context of the main substantive application in Section \\ref{sec:res-vac}.  We will then come back to the general results in Section \\ref{sec:abs-res}. We first fix a convention that we will follow throughout this paper.\n\n\\medskip\n\n\\noindent{\\bf Convention.} All our metric measure spaces will be probability spaces. For any metric measure space $\\mvX=(X,d,\\mu)$ and $\\alpha>0$, $\\alpha\\mvX$ will denote the metric measure space $(X,\\alpha d,\\mu)$, i.e, the space where the metric has been multiplied by $\\alpha$ and the (probability) measure $\\mu$ has remained unchanged. Precise definitions of metric space convergence including the Gromov-Hausdorff-Prokhorov (GHP) topology are deferred to Section \\ref{sec:not}.\n\\subsection{Geometry of vacant sets left by random walk}\n\\label{sec:res-vac}\nFix $r\\geq 3$ and $n\\geq 1$. Here and throughout we assume $nr$ is even.\nRecall the definitions of $\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$, $\\pr_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$, $\\overline\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$, and $\\overline\\pr_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ from Section \\ref{sec:rg-models}.\nLet $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)}=(r,r,\\ldots,r)$, and\ndefine\n\\[\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,r}:=\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)}},\\ \\ \\text{ and }\\ \\ \\overline\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,r}:=\\overline\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)}}.\\]\nAnalogously write $\\pr_{n,r}$ (resp. $\\overline\\pr_{n,r}$) for $\\pr_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)}}$ (resp. $\\overline\\pr_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)}}$).\nLet $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,r}\\sim \\pr_{n,r}$.\nFor any (multi)graph $G$, write $P^G$ for the distribution of a simple random walk $\\set{X_t: t\\geq 0}$ on $G$ with the initial state $X_0$ chosen uniformly at random. Recall the definition of the vacant set from \\eqref{eqn:vac-set-def}. Define,\n\\begin{equation}\n\\label{eqn:ustar-def}\n\tu_{\\star}=\\frac{r(r-1)\\ln(r-1)}{(r-2)^2}\n\\end{equation}\n\nWrite $\\cC_1(u)$ for the maximal connected component in $\\mathcal{V}}\\newcommand{\\cW}{\\mathcal{W}}\\newcommand{\\cX}{\\mathcal{X}^u$.  Then the following was shown in \\cite{vcerny2011giant}: With $\\pr_{n,r}$-probability converging to one as $n\\to\\infty$,\n\\begin{enumeratea}\n\t\\item given any $u< u_{\\star}$ and $\\sigma > 0$, there exist strictly positive constants $\\rho, c> 0$ depending only on $u,\\sigma,r$ such that\n\t\\[P^{\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,r}}\\big(\\big|\\cC_{\\scriptscriptstyle(1)}(u)\\big|\\geq \\rho n\\big)\\geq 1-cn^{-\\sigma};\\]\n\t\\item for any fixed $u> u_{\\star}$ and $\\sigma > 0$, there exists $\\rho^\\prime >0$ depending only on $u, \\sigma, r$ such that\n\t\\[P^{\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,r}}\\big(\\big|\\cC_{\\scriptscriptstyle(1)}(u)\\big|\\geq \\rho^\\prime \\log(n)\\big)\\leq cn^{-\\sigma}.\\]\n\\end{enumeratea}\nFigures \\ref{fig:test1} and \\ref{fig:test2} display the connectivity structure of the vacant set just above and just below $u_{\\star}$ respectively where the underlying graph is a $r=4$-regular random graph on $n=50,000$ vertices. The maximal component has been colored red, the second largest component blue, and all other components have been colored cyan.\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[trim=5.5cm 5cm 3.8cm 4.5cm, clip=true, angle=0, scale=0.9]{Ghigh_50000-4-5-purp.pdf}\n  \\captionof{figure}{Connectivity structure at $u=u_{\\star}+0.5$.}\n  \\label{fig:test1}\n\\end{minipage}%\n\\begin{minipage}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[trim=5.7cm 5.5cm 3.6cm 4cm, clip=true, angle=0, scale=0.9]{Glow_50000-4-5-purp.pdf}\n  \\captionof{figure}{Connectivity structure at $u=u_{\\star}-0.5$. }\n  \\label{fig:test2}\n\\end{minipage}\n\\end{figure}\n\nThe main aim of this section is the study of the annealed measures:\n\\[\\vP_{n,r}(\\cdot) := \\frac{1}{|\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,r}|} \\sum_{G\\in \\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,r}} P^G(\\cdot)\\ \\ \\text{ and }\\ \\\n\\overline\\vP_{n,r}(\\cdot) := \\sum_{G\\in\\overline\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,r}}\\overline\\pr_{n,r}\\big(G\\big) P^G(\\cdot).\\]\nBuilding on the work of Cooper and Frieze \\cite{cooper-frieze}, \\v{C}ern{\\'y} and Teixeira in \\cite[Theorem 1.1]{cerny-teixeira} showed the following for the above annealed distribution: Let $\\set{u_n:n\\geq 1}$ be any sequence such that there exists fixed $\\beta < \\infty$ such that, $n^{1\/3}|u_n - u_{\\star}|\\leq \\beta$ for all large $n$. Then given any $\\varepsilon> 0$, $\\exists~ A:= A(\\varepsilon,r,\\beta) > 0$ such that for all $n$ large,\n\\begin{equation}\n\\label{eqn:cerny-tiex}\n\t\\vP_{n,r}\\left(A^{-1} n^{2\/3} \\leq |\\cC_{\\scriptscriptstyle(1)}|\\leq An^{2\/3}\\right)\\geq 1-\\varepsilon.\n\\end{equation}\n\n\nThey also showed that analogous results hold for $\\overline\\vP_{n,r}(\\cdot)$.  The $n^{2\/3}$ scaling of the maximal component size suggests that the critical behavior for this model resembles that of the critical Erd\\H{o}s-R\\'enyi  random graph. Our first main result stated below confirms this assertion.\n\n\\begin{thm}[Scaling limit of the vacant set]\\label{thm:vacant-set-scaling}\nLet $r\\geq 3$.\n\\begin{enumeratei}\n\\item\nLet $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,r}\\sim\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}_{n,r}$ and $u_\\star$ be as in \\eqref{eqn:ustar-def}.\nRun a simple random walk on $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,r}$ up to time $nu_n$ starting from a uniformly chosen vertex, where\n\\begin{align}\\label{eqn:40}\nn^{1\/3}(u_{\\star}-u_n)\\to a_0\\in\\bR.\n\\end{align}\nLet $\\cC_{(j)}$ be $j$-th largest component of the subgraph of $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,r}$ induced by the vacant set $\\mathcal{V}}\\newcommand{\\cW}{\\mathcal{W}}\\newcommand{\\cX}{\\mathcal{X}^{u_n}$. Endow $\\cC_{(j)}$ with the graph distance and the uniform probability measure on its vertices.\nThen there exists a sequence $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{\\mathrm{vac}}(a_0)=\\big(M_1^{\\mathrm{vac}}(a_0),M_2^{\\mathrm{vac}}(a_0),\\ldots\\big)$ of random compact metric measure spaces such that under the annealed measure $\\vP_{n,r}$,\n\\[\\frac{1}{n^{1\/3}}\\big(\\cC_{(1)}, \\cC_{(2)},\\ldots\\big)\\stackrel{\\mathrm{w}}{\\longrightarrow}\n\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{\\mathrm{vac}}(a_0)=\\big(M_1^{\\mathrm{vac}}(a_0),M_2^{\\mathrm{vac}}(a_0),\\ldots\\big)\\]\nwith respect to product topology induced by GHP distance (see Section \\ref{sec:gh-mc} for definition) on each coordinate.\n\\item\nThe conclusion in part (i) continues to hold with the same limiting sequence $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{\\mathrm{vac}}(a_0)$ if we replace $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,r}$ by $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)})$ and $\\vP_{n,r}$ by the corresponding annealed measure $\\overline\\vP_{n,r}$.\n\\end{enumeratei}\n\\end{thm}\n\n\\begin{rem}\nA complete description of the limiting spaces appearing in Theorem \\ref{thm:vacant-set-scaling}\nrequires certain definitions and is thus deferred to Section \\ref{sec:not}. The limiting object $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{\\mathrm{vac}}(a_0)$ is explicitly defined in Construction \\ref{constr:M-vac}.   The connection between the scaling limit of the critical Erd\\H{o}s-R\\'enyi  random graph $\\ERRG(n,\\lambda)$\nand the limiting spaces in the results stated in this section is also explained in Section \\ref{sec:not}.\n\\end{rem}\n\nThe above result deals with the vacant set left by random walk (VSRW) on the random $r$-regular graph. We in fact conjecture that for the corresponding problem on random graphs with general prescribed degree sequence, one has analogous results with a {\\bf universality} phenomenon under moment conditions on the degree sequence.\n\\begin{conj}\\label{conj:general-degree-vacant}\nLet $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f} = \\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(n)}=(d_1,\\ldots,d_n)$ be a degree sequence, and let $D_n$ denote the degree of a vertex chosen uniformly from $[n]$.\nAssume that as $n\\to\\infty$,  $D_n\\stackrel{\\mathrm{w}}{\\longrightarrow} D$ with $\\E(D^2) < \\infty$ and $\\E(D_n^2)\\to \\E(D^2)$. Further assume\n\\[\\nu := \\frac{\\E[D(D-1)]}{\\E[D]} > 1 \\mbox{ and } \\pr(D \\geq 3) > 0.\\]\nConsider the VSRW on $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n, \\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ or $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$ at level $u$. We conjecture that the following hold:\n\\begin{enumeratea}\n\\item There exists a (model dependent) critical point $u_{\\star}$ such that for $u< u_{\\star}$, size of the maximal component $|\\cC_{\\scriptscriptstyle(1)}(u)| = \\Theta_P(n)$ whilst for $u > u_{\\star}$,  $|\\cC_{\\scriptscriptstyle(1)}| = o_P(1)$.\n\n\\item If $~\\E(D^3) < \\infty$ and $\\E(D_n^3)\\to E(D^3)$, then for $u_n$ satisfying\n\t\\[\\lim_{n\\to\\infty} n^{1\/3} (u_{\\star} - u_n)=a_0\\]\nfor some $a_0\\in \\bR$, the connectivity structure of VSRW at level $u_n$ with edges in the maximal components rescaled by $n^{-1\/3}$ satisfy results analogous to Theorem \\ref{thm:vacant-set-scaling}.\n\t\n\\item Let $p_k:=\\pr(D = k)$, $k=0, 1, \\ldots$. Assume that there exists $C>0$ and $\\tau\\in (3,4)$ such that\n\\[p_k \\sim C k^{-\\tau}\\ \\ \\text{ as }\\ \\ k\\to\\infty.\\]\n(In particular, $\\E[D^2]<\\infty$, but $\\E[D^3]=\\infty$.)\nThen for any $a_0\\in\\bR$, there exists a sequence\n$\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{\\mathrm{vac}}_{\\tau}(a_0)=\\big(M^{\\mathrm{vac}}_{\\tau, 1}(a_0),M^{\\mathrm{vac}}_{\\tau, 2}(a_0),\\ldots\\big)$\nof random metric measure spaces such that under the annealed measure $\\vP_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ (or $\\overline\\vP_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$),\n\\[n^{-\\frac{\\tau-3}{\\tau -1}}\\big(\\cC_{(1)}(u_n), \\cC_{(2)}(u_n),\\ldots\\big)\\stackrel{\\mathrm{w}}{\\longrightarrow}\n\t\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{\\mathrm{vac}}_{\\tau}(a_0),\\]\nfor any sequence $u_n$ satisfying\n\\[\\lim_{n\\to\\infty} n^{(\\tau-3)\/(\\tau -1)} (u_{\\star} - u_n) \\to a_0.\\]\n \\end{enumeratea}\n\n\n\n\\end{conj}\n\n\\begin{rem}\n\t\\label{rem:clar}\n\tAt this point, we owe the reader two clarifications regarding the conjecture. First we need to clarify the phrase ``results analogous to Theorem \\ref{thm:vacant-set-scaling}'' in (b). Second we need to give some details on the limit objects $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{\\mathrm{vac}}_{\\tau}(a_0)$ in (c).\nBoth of these clarifications are deferred to the discussion Section \\ref{sec:disc} \\eqref{disc:vacant-set}.\n\\end{rem}\n\n\\subsection{Scaling limits of random graphs with prescribed degrees}\n\\label{sec:abs-res}\nThis section describes our main results on graphs with prescribed degree sequence. The first result describes maximal component structure for critical random graphs under appropriate assumptions. For each $n\\geq 1$ let $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}=\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(n)}=(d_v : v\\in[n])$ be a degree sequence with vertex set $[n]$. We will work with degree sequences that satisfy the following assumption.\n\n\\begin{ass}\n\t\\label{ass:cm-deg}\n\tLet $D_n$ be a random variable with distribution given by\n\\[\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(D_n=i\\big)=\\frac{1}{n}\\#\\big\\{j\\ :\\ d_j=i\\big\\},\\]\ni.e., $D_n$ has the law of the degree of a vertex selected uniformly at random from $[n]$. Assume the following hold as $n\\to\\infty$:\n\\begin{enumeratei}\n\\item There exists a limiting random variable $D$ with $\\pr(D = 1)> 0$ such that $D_n \\stackrel{\\mathrm{w}}{\\longrightarrow} D$.\n\\item Convergence of third moments (and hence all lower moments):\n\\[\\E\\big[D_n^3\\big]:= \\frac{1}{n} \\sum_{v\\in [n]} d_v^3 \\to \\E\\big[D^3\\big]<\\infty. \\]\n\\item We are in the critical scaling window, i.e., there exists $\\lambda \\in \\bR$ such that\n\\[\\nu_n:= \\frac{\\sum_{v\\in [n]} d_v(d_v-1)}{\\sum_{v\\in [n]} d_v} = 1+\\frac{\\lambda}{n^{1\/3}} + o(n^{-1\/3}). \\]\nIn particular, $\\E[D^2]=2\\E[D]$.\n\\end{enumeratei}\n\\end{ass}\n\nRecall that $\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ denotes the space of all simple graphs on $n$ vertices with degree sequence $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}$ and $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ is uniformly distributed over $\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$.\n\n\n\n\n\\begin{thm}[Scaling limit of graphs with given degree sequence under optimal assumptions]\\label{thm:graphs-given-degree-scaling}\nSuppose $\\set{\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(n)}:n\\geq 1}$ satisfy Assumption \\ref{ass:cm-deg} with limiting random variable $D$.\n\\begin{enumeratei}\n\\item Let $\\cC_{(j)}$ be the $j$-th largest component of $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$. Endow $\\cC_{(j)}$ with the graph distance and the uniform probability measure on its vertices.  Then there exists a sequence $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^D(\\lambda)=(M_1^D(\\lambda), M_2^D(\\lambda),\\ldots)$ of (random) compact metric measure spaces such that\n\\[\\frac{1}{n^{1\/3}}\\big(\\cC_{(1)}, \\cC_{(2)},\\ldots\\big)\\stackrel{\\mathrm{w}}{\\longrightarrow}\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^D(\\lambda)\\]\nwith respect to product topology induced by GHP distance on each coordinate.\n\\item The conclusion of part (i) continues to hold with the same limiting sequence $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^D(\\lambda)$ if we replace $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ by $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$.\n\\end{enumeratei}\n\\end{thm}\n\\begin{rem}\n\tThe limit objects $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^D(\\lambda)$ are described explicitly in Construction \\ref{constr:M-D}.\n\\end{rem}\n\n\nThe main ingredient in proving the above result is the following result about the uniform distribution on the space of all {\\bf connected} simple graphs with a prescribed degree sequence.\nFor each fixed $\\widetilde m\\geq 1$, let $\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}=\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(\\widetilde m)}=(\\widetilde d_1,\\hdots,\\widetilde d_{\\widetilde m})$ be a given degree sequence. Consider the following assumption on $\\set{\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(\\widetilde m)}: \\widetilde m\\geq 1}$:\n\\begin{ass}\\label{ass:degree}\\hfill\n\\begin{enumeratei}\n\\item\\label{it:1}\n$\\widetilde d_j\\ge 1$ for $1\\le j\\le\\widetilde m$, and $\\widetilde d_1=1$.\n\\item\\label{it:2}\nThere exists a pmf $(\\tilde p_1,\\tilde p_2,\\ldots)$ with\n\\[\\widetilde p_1>0,\\quad  \\sum_{i\\ge 1}i\\widetilde p_i=2,\\quad\\text{and }\\sum_{i\\ge 1}i^2\\widetilde p_i<\\infty\\]\nsuch that\n\\[\\frac{1}{\\widetilde m}\\#\\set{j: \\widetilde d_j=i}\\to\\widetilde p_i\\ \\text{ for }\\ i\\ge 1,\\ \\text{ and }\\\n\\frac{1}{\\widetilde m}\\sum_{i\\ge 1}\\widetilde d_i^2\\to\\sum_{i\\ge 1}i^2\\widetilde p_i.\\]\nIn particular, $\\max_{1\\le j\\le \\widetilde m}\\widetilde d_j=o(\\sqrt{\\widetilde m})$.\n\\end{enumeratei}\n\\end{ass}\n\\begin{rem}\\label{rem:d1=1-irrelevant}\nWe make two observations about the above set of assumptions.\n\\begin{enumeratei}\n\\item\nThe assumption $\\widetilde d_1=1$ makes the notation in the proofs simpler. It has no other special relevance. Indeed,\nsince $\\widetilde p_1>0$, a positive proportion of vertices have degree one when $\\widetilde m$ is large. Thus, we can always consider the vertex that has the smallest label among all vertices that have degree one.\n\\item We will work with connected graphs with fixed complexity, i.e., for all $\\widetilde m\\geq 1$, the degree sequence $\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(\\widetilde m)}$ will satisfy $\\sum_{j\\in[\\widetilde m]}\\widetilde d_j=2(\\widetilde m-1)+2k$ for some fixed $k\\geq 0$.\n    Hence in this case, the assumption $\\sum_{i\\geq 1}i\\widetilde p_i=2$ is redundant as it follows from the other assumptions.\n\\end{enumeratei}\n\\end{rem}\nLet $\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}^{\\con}$ be the set of all {\\bf connected}, simple, labeled (by $[\\widetilde m]$) graphs with degree sequence $\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}$ where the vertex labeled $j$ has degree $\\tilde d_j$.\n\n\\begin{thm}[Scaling limit of {\\bf connected} graphs with given degree sequence under optimal assumptions]\\label{prop:condition-on-connectivity}\nConsider a sequence of degree sequences $\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(\\widetilde m)}=(\\widetilde d_1,\\hdots,\\widetilde d_{\\widetilde m})$ satisfying Assumption \\ref{ass:degree}. In addition, assume that for all $\\widetilde m$,\n\\[\\sum_{j\\in[\\widetilde m]}\\widetilde d_j=2(\\widetilde m-1)+2k\\]\nfor some (fixed) nonnegative integer $k$.\nSample $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}^{\\con}$ uniformly from $\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{\\widetilde{\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}}^{\\con}$, and endow it with the graph distance and the uniform probability measure on vertices. Then there exists a random compact metric measure space $M^{\\scriptscriptstyle(k)}$ such that\n\\[\\frac{1}{\\sqrt{\\widetilde m}}\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}^{\\con}\\stackrel{\\mathrm{w}}{\\longrightarrow} \\frac{1}{\\sigma}M^{\\scriptscriptstyle(k)}\\]\nin the GHP sense, where $\\sigma^2=\\sum_{i\\ge 1}i^2\\tilde p_i-4$ is the asymptotic variance.\n\\end{thm}\n\n\\begin{rem}\n\tThe limit object $M^{\\scriptscriptstyle(k)}$ is described explicitly in Construction \\ref{constr:M-k}.\n\\end{rem}\nOur next result concerns enumeration of connected graphs with prescribed degrees.\n\\begin{thm}[Asymptotic number of connected graphs with given degree sequence when the complexity is fixed]\\label{thm:number-of-connected-graphs}\nConsider a sequence of degree sequences $\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(\\widetilde m)}=(\\widetilde d_1,\\hdots,\\widetilde d_{\\widetilde m})$ satisfying Assumption \\ref{ass:degree}. Assume further that for all $\\widetilde m$,\n\\[\\sum_{j\\in[\\widetilde m]}\\widetilde d_j=2(\\widetilde m-1)+2k\\]\nfor some (fixed) nonnegative integer $k$. Let $\\sigma^2=\\sum_{i\\ge 1}i^2\\tilde p_i-4$ be the asymptotic variance. Then\n\\[\\lim_{\\widetilde m}\\ \\frac{\\big|\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}^{\\con}\\big|\\times\n\\prod_{i=1}^{\\widetilde m}\\big(\\widetilde d_i-1\\big)!\\times\n\\widetilde m^{k\/2}}{\\big(\\widetilde m+2k-2\\big)!}=\\frac{\\sigma^k}{k!}\\E\\bigg[\\bigg(\\int_0^1\\ve(x)dx\\bigg)^k\\bigg],\\]\nwhere $\\ve(x)$, $x\\in[0,1]$, is a standard Brownian excursion.\n\\end{thm}\n\\begin{rem}\\label{rem:wright}\nLet $C(n,n+k)$ denote the number of connected graphs with $n$ labeled vertices and $n+k$ edges. Wright \\cite{wright1977number} showed that for any fixed $k\\geq -1$,\n\\[C(n, n+k)\\sim\\rho_k n^{n+(3k-1)\/2}\\qquad \\text{as }\\ n\\to\\infty,\\]\nwhere the constants $\\rho_k$ satisfy a certain recurrence relation. Spencer \\cite{spencer-count} proved a connection between this purely combinatorial result and a probabilistic object by showing that\n\\[\\rho_k=\\frac{1}{(k+1)!}\\E\\bigg[\\bigg(\\int_0^1\\ve(x)dx\\bigg)^{k+1}\\bigg],\\ \\ k\\geq -1,\\]\nwhere $\\ve(x)$, $x\\in[0,1]$, is a standard Brownian excursion. Theorem \\ref{thm:number-of-connected-graphs} proves the analogue of this result for connected graphs when the degree sequence is fixed.\n\\end{rem}\n\n\\section{Definitions and limit objects }\n\\label{sec:not}\nThis section contains basic constructs required to state our main results.\n\n\\subsection{Notation and conventions}\\label{sec:notation}\n\n\nFor any set $A$, we write $|A|$ for its cardinality and $\\mathds{1}\\set{A}$ for the associated indicator function. Given two intervals $A, B \\subset \\bR$, we write $C(A,B)$ for the space of continuous functions $f: A \\to B$, equipped with the $L_\\infty$-norm $\\|f\\|_\\infty := \\sup_{x \\in A}|f(x)|$. We write $D(A,B)$ for the space of RCLL (right-continuous-left-limit) functions $f: A \\to B$, equipped with the Skorohod topology. We use the standard Landau notation of $o(\\cdot)$, $O(\\cdot)$ and the corresponding \\emph{order in probability} notation $o_P(\\cdot)$ and $O_P(\\cdot)$.\n\n We use $\\stackrel{\\mathrm{P}}{\\longrightarrow}$, $\\stackrel{\\mathrm{w}}{\\longrightarrow}$ and $\\stackrel{\\mathrm{a.e.}}{\\longrightarrow}$ to denote convergence in probability, weak convergence and almost-sure convergence. We say a sequence of events $E_n$, $n \\in \\bN$, occur \\emph{with high probability} if $\\pr(E_n) \\to 1$ as $n \\to \\infty$.\n\n\n\\subsection{Gromov-Hausdorff-Prokhorov metric}\n\\label{sec:gh-mc}\n\nWe mainly follow \\cite{EJP2116,AddBroGolMie13,metric-geometry-book}. All metric spaces under consideration will be compact. Let us recall the Gromov-Hausdorff distance $d_{\\GH}$ between metric spaces.  Fix two metric spaces $X_1 = (X_1,d_1)$ and $X_2 = (X_2, d_2)$. For a subset $C\\subseteq X_1 \\times X_2$, the distortion of $C$ is defined as\n\\begin{equation}\n\t\\label{eqn:def-distortion}\n\t\\dis(C):= \\sup \\set{|d_1(x_1,y_1) - d_2(x_2, y_2)|: (x_1,x_2) , (y_1,y_2) \\in C}.\n\\end{equation}\nA correspondence $C$ between $X_1$ and $X_2$ is a measurable subset of $X_1 \\times X_2$ such that for every $x_1 \\in X_1$, there exists at least one $x_2 \\in X_2$ such that $(x_1,x_2) \\in C$ and vice-versa. The Gromov-Hausdorff distance between the two metric spaces  $(X_1,d_1)$ and $(X_2, d_2)$ is defined as\n\\begin{equation}\n\\label{eqn:dgh}\n\td_{\\GH}(X_1, X_2) = \\frac{1}{2}\\inf \\set{\\dis(C): C \\mbox{ is a correspondence between } X_1 \\mbox{ and } X_2}.\n\\end{equation}\n\nSuppose $(X_1, d_1)$ and $(X_2, d_2)$ are two metric spaces and $p_1\\in X_1$, and $p_2\\in X_2$. Then the {\\it pointed Gromov-Hausdorff distance} between $\\mvX_1:=(X_1, d_1, p_1)$ and $\\mvX_2:=(X_2, d_2, p_2)$ is given by\n\\begin{align}\n\\label{eqn:dgh-pointed}\n\td_{\\GH}^{\\point}(\\mvX_1, \\mvX_2) = \\frac{1}{2}\\inf \\set{\\dis(C): C \\mbox{ is a correspondence between }X_1 \\mbox{ and } X_2\\mbox{ and }(p_1, p_2)\\in C}.\n\\end{align}\n\n\n\nWe will use the Gromov-Hausdorff-Prokhorov distance that also keeps track of associated measures on the corresponding metric spaces. A compact metric measure space $(X, d , \\mu)$ is a compact metric space $(X,d)$ with an associated finite measure $\\mu$ on the Borel sigma algebra $\\cB(X)$.  Given two compact metric measure spaces $(X_1, d_1, \\mu_1)$ and $(X_2,d_2, \\mu_2)$ and a measure $\\pi$ on the product space $X_1\\times X_2$, the discrepancy of $\\pi$ with respect to $\\mu_1$ and $\\mu_2$ is defined as\n\n\\begin{equation}\n\t\\label{eqn:def-discrepancy}\n\tD(\\pi;\\mu_1, \\mu_2):= ||\\mu_1-\\pi_1|| + ||\\mu_2-\\pi_2||\n\\end{equation}\nwhere $\\pi_1, \\pi_2$ are the marginals of $\\pi$ and $||\\cdot||$ denotes the total variation of signed measures. Then define the metric $d_{\\GHP}$ between $X_1$ and $X_2$ is defined\n\\begin{equation}\n\\label{eqn:dghp}\n\td_{\\GHP}(X_1, X_2):= \\inf\\bigg\\{ \\max\\bigg(\\frac{1}{2} \\dis(C),~D(\\pi;\\mu_1,\\mu_2),~\\pi(C^c)\\bigg) \\bigg\\},\n\\end{equation}\nwhere the infimum is taken over all correspondences $C$ and measures $\\pi$ on $X_1 \\times X_2$. As mentioned in the introduction, unless otherwise stated typically the associated measures will be \\emph{probability} measures.\n\nSimilar to \\eqref{eqn:dgh-pointed}, we can define a ``{\\it pointed Gromov-Hausdorff-Prokhorov distance}'' $d_{\\GHP}^{\\point}$ between two metric measure spaces $X_1$ and $X_2$ having two distinguished points $p_1$ and $p_2$ respectively by taking the infimum in \\eqref{eqn:dghp} over all correspondences $C$ and measures $\\pi$ on $X_1 \\times X_2$ such that $(p_1, p_2)\\in C$.\n\n\nNow let $\\mathfrak{S}$ be the space of all compact metric measure spaces.  The function $d_{\\GHP}$ is a pseudometric on $\\mathfrak{S}$, and defines an equivalence relation $X \\sim Y \\Leftrightarrow d_{\\GHP}(X,Y) = 0$ on $\\mathfrak{S}$. Let $\\bar \\mathfrak{S} := \\mathfrak{S} \/ \\sim $ be the space of isometry equivalent classes of compact metric measure spaces and $\\bar d_{\\GHP}$ be the induced metric. Then by \\cite{EJP2116}, $(\\bar \\mathfrak{S}, \\bar d_{\\GHP})$ is a complete separable metric space. To ease notation, we will continue to use $(\\mathfrak{S}, d_{\\GHP})$ instead of $(\\bar \\mathfrak{S}, \\bar d_{\\GHP})$ and $X = (X, d, \\mu)$ to denote both the metric space and the corresponding equivalence class.\n\nSince we will be interested in not just one metric space but an\ninfinite sequence of metric spaces, the relevant space will be $\\mathfrak{S}^{\\bN}$ equipped with the product topology inherited from $d_{\\GHP}$.\n\n\\subsection{Gromov-weak topology}\\label{sec:gromov-weak}\nHere we mainly follow \\cite{Winter-gromov-weak}. Introduce an equivalence relation on the space of complete and separable metric spaces that are equipped with a probability measure on the associated Borel $\\sigma$-algebra by declaring two such spaces $(X_1, d_1, \\mu_1)$ and $(X_2, d_2, \\mu_2)$ to be equivalent when there exists an isometry $\\psi:\\mathrm{support}(\\mu_1)\\to\\mathrm{support}(\\mu_2)$ such that $\\mu_2=\\psi_{\\ast}\\mu_1:=\\mu_1\\circ\\psi^{-1}$, i.e., the push-forward of $\\mu_1$ under $\\psi$ is $\\mu_2$. Write $\\mathfrak{S}_{*}$ for the associated space of equivalence classes. As before, we will often ease notation by not distinguishing between a metric space and its equivalence class.\n\nFix $l\\geq 2$, and a complete separable metric space $(X, d)$. Then given a collection of points $\\vx:=(x_1, x_2, \\ldots, x_l)\\in X^l$, let $\\vD(\\vx):= (d(x_i, x_j))_{i,j\\in [l]}$ denote the symmetric matrix of pairwise distances between the collection of points. A function $\\Phi\\colon \\mathfrak{S}_* \\to \\bR$ is called a polynomial of degree $l$ if there exists a bounded continuous function $\\phi\\colon \\bR_+^{l^2}\\to \\bR$ such that\n\\begin{equation}\\label{eqn:polynomial-func-def}\n\t\\Phi((X,d,\\mu)):= \\int  \\phi(\\vD(\\vx)) d\\mu^{\\otimes l}(\\vx).\n\\end{equation}\nHere $\\mu^{\\otimes l}$ is the $l$-fold product measure of $\\mu$. Let $\\boldsymbol{\\Pi}}\\newcommand{\\mvrho}{\\boldsymbol{\\rho}}\\newcommand{\\mvsigma}{\\boldsymbol{\\sigma}$ denote the space of all polynomials on $\\mathfrak{S}_*$.\n\n\\begin{defn}[Gromov-weak topology]\n\t\\label{def:gromov-weak}\n\tA sequence $(X_n, d_n, \\mu_n)_{n\\geq 1} \\in \\mathfrak{S}_*$ is said to converge to $(X, d, \\mu) \\in \\mathfrak{S}_*$ in the Gromov-weak topology if and only if $\\Phi((X_n, d_n, \\mu_n))\\to \\Phi((X, d, \\mu))$ for all $\\Phi\\in \\boldsymbol{\\Pi}}\\newcommand{\\mvrho}{\\boldsymbol{\\rho}}\\newcommand{\\mvsigma}{\\boldsymbol{\\sigma}$.\n\\end{defn}\nIn \\cite[Theorem 1]{Winter-gromov-weak} it is shown that $\\mathfrak{S}_*$ is a Polish space under the Gromov-weak topology. It is also shown that, in fact, this topology can be completely metrized using the so-called Gromov-Prokhorov metric.\n\n\\subsection{Spaces of trees with edge lengths, leaf weights and root-to-leaf measures}\n\\label{sec:space-of-trees}\nThe rest of this section largely follows \\cite{bhamidi-hofstad-sen}.  In the proof of the main results we need the following two spaces built on top of the space of discrete trees. The first space $\\vT_{IJ}$ was formulated in \\cite{aldous-pitman-edge-lengths,aldous-pitman-entrance} where it was used to study trees spanning a finite number of random points sampled from an inhomogeneous continuum random tree (ICRT). A more general space $\\vT_{IJ}^*$ was used in the proofs in \\cite{bhamidi-hofstad-sen}.\nThe index $I$ in $\\vT_{IJ}$ and $\\vT_{IJ}^*$ is needed for the purpose of keeping track of the number of marked ``hubs,'' i.e., vertices of high (or infinite) degrees in such trees (see \\cite{aldous-pitman-edge-lengths,aldous-pitman-entrance,bhamidi-hofstad-sen} for a proper definition). For our purpose it will suffice to consider the case $I=0$. So we only define the space $\\vT_{J}:=\\vT_{0J}$ and $\\vT_{J}^*:=\\vT_{0J}^*$.\n\\medskip\n\n\\noindent{\\bf The space $\\vT_{J}$:} Fix $J\\geq 1$. Let $\\vT_{J}$ be the space of trees having the following properties:\n\\begin{enumeratea}\n\t\\item There are exactly $J$ leaves labeled $1+, \\ldots, J+$, and the tree is rooted at another labeled vertex $0+$.\n\n\t\\item Every edge $e$ has a strictly positive edge length $l_e$.\n\\end{enumeratea}\nA tree $\\vt\\in \\vT_{J}$ can be viewed as being composed of two parts:\\\\\n(1) $\\shape(\\vt)$ describing the shape of the tree (including the labels of leaves) but ignoring edge lengths. The set of all possible shapes $\\vT_{J}^{\\shape}$ is obviously finite for fixed $J$.\\\\\n(2) The edge lengths $\\vl(\\vt):= (l_e:e\\in \\vt)$. Consider the product topology on $\\vT_{J}$ consisting of the discrete topology on $\\vT_{J}^{\\shape}$ and the product topology on $\\bR^d$.\n\n\\medskip\n\n\\noindent{\\bf The space $\\vT_{J}^*$:} We will need a slightly more general space. Along with the two attributes above in $\\vT_{J}$, the trees in this space have the following two additional properties. Let $\\cL(\\vt):= \\set{1+, \\ldots, J+}$ denote the collection of non-root leaves in $\\vt$. Then every leaf $v\\in \\cL(\\vt) $ has the following attributes:\n\n\\begin{enumeratea}\n\t\\item[(d)] {\\bf Leaf weights:} A nonnegative number $A(v)$. Write $\\vA(\\vt):=(A(v): v\\in \\cL(\\vt))$.\n\t\\item[(e)] {\\bf Root-to-leaf measures:} A probability measure $\\nu_{\\vt,v}$ on the path $[0+,v]$ connecting the root and the leaf $v$. Here the path is viewed as a line segment pointed at $0+$ and has the usual Euclidean topology. Write $\\boldsymbol{\\nu}(\\vt):= (\\nu_{\\vt,v}: v\\in \\cL(\\vt))$ for this collection of probability measures.\n\\end{enumeratea}\nIn addition to the topology on $\\vT_{J}$, the space $\\vT_{J}^*$ with these additional two attributes inherits the product topology on $\\bR^{J}$ owing to leaf weights and $(d_{\\GHP}^{\\point})^J$ owing to the root-to-leaf measures.\n\nAdditionally, we include a special element $\\partial$ in $\\vT_{J}^*$. This will be useful in the proofs as we will view any rooted tree that does not have exactly $J$ distinct leaves as $\\partial$, which will allow us to work entirely in the space $\\vT_{J}^*$.\n\\subsection{Scaling limits of component sizes at criticality}\n\\label{sec:erdos-scaling-limit}\nThe starting point for establishing the metric space scaling limit is understanding the behavior of the component sizes.\nWe first set up some notation. Fix parameters $\\alpha, \\eta, \\beta > 0$, and write $\\mvmu = (\\alpha, \\eta, \\beta)\\in \\bR_+^3$. Let $\\set{B(s):s\\geq 0}$ be a standard Brownian motion. For $\\lambda \\in \\bR$, define\n\\begin{equation}\n\\label{eqn:bm-lamb-kapp-def}\nW^{\\mvmu,\\lambda}(s):= \\frac{\\sqrt{\\eta}}{\\alpha} B(s)+\\lambda s - \\frac{\\eta s^2}{2\\alpha^3},\\qquad s\\geq 0.\n\\end{equation}\nWrite $\\overline{W}^{\\mvmu,\\lambda}$  for the process reflected at zero:\n\\begin{equation}\n\\label{eqn:reflected-pro-def}\n\t\\overline{W}^{\\mvmu,\\lambda}(s) := W^{\\mvmu,\\lambda}(s) -\\min_{0\\le u\\le s} W^{\\mvmu,\\lambda}(u),   \\qquad s\\geq 0.\n\\end{equation}\n Consider the metric space,\n\\begin{equation}\n\\label{eqn:ldown-def}\n\tl^2_{\\downarrow}:= \\bigg\\{\\vx= (x_i:i\\geq 1): x_1\\geq  x_2 \\geq \\ldots \\geq 0, \\sum_{i=1}^\\infty x_i^2 < \\infty\\bigg\\},\n\\end{equation}\nequipped with the natural metric inherited from $l^2$.\nIt was shown by Aldous in \\cite{aldous-crit} that the excursions of $\\overline{W}^{\\mvmu,\\lambda}$ from zero can be arranged in decreasing order of lengths as\n\\begin{equation}\n\\label{eqn:mvxi-def}\n\t\\mvxi^{\\mvmu}(\\lambda) = \\big(|\\gamma_{\\scriptscriptstyle(i)}^{\\mvmu}(\\lambda)|: i \\geq 1\\big),\n\\end{equation}\nwhere $|\\gamma_{\\scriptscriptstyle(i)}^{\\mvmu}(\\lambda)|$ is the length of the $i$-th largest excursion $\\gamma_{\\scriptscriptstyle(i)}^{\\mvmu}(\\lambda)$, and further $\\mvxi^{\\mvmu}(\\lambda) \\in l^2_{\\downarrow}$. Let $\\mathcal{P}}\\newcommand{\\cQ}{\\mathcal{Q}}\\newcommand{\\cR}{\\mathcal{R}_\\beta$ be a rate $\\beta$ Poisson process $\\bR_+^2$ independent of $W^{\\mvmu,\\lambda}(\\cdot)$.\nFor each $i\\geq 1$, write $N_{\\scriptscriptstyle(i)}^{\\mvmu}(\\lambda)$ for the number of points of $\\mathcal{P}}\\newcommand{\\cQ}{\\mathcal{Q}}\\newcommand{\\cR}{\\mathcal{R}_\\beta$ that fall under the excursion $\\gamma_{\\scriptscriptstyle(i)}^{\\mvmu}(\\lambda)$.\n\n\nAldous in \\cite{aldous-crit} studied the maximal components of the Erd\\H{o}s-R\\'enyi  random graph in the critical regime and  proved a remarkable result that says that the sizes of the maximal components scaled by $n^{-2\/3}$ and the number of surplus edges in the maximal components of $\\ERRG(n^{-1}+\\lambda n^{-4\/3})$ converge jointly in distribution to\n$\\big(\\big(|\\gamma_{\\scriptscriptstyle(i)}^{\\mvmu_{\\er}}(\\lambda)|, N_{\\scriptscriptstyle(i)}^{\\mvmu_{\\er}}(\\lambda)\\big): i\\geq 1 \\big)$,\nwhere $\\mvmu_{\\er} = (1,1,1)$.\nThis result has since been generalized to a number of other random graph models. In the context of graphs with given degree sequence,  Nachmias and Peres \\cite{nachmias-peres-random-regular} studied critical percolation on random regular graphs;  Riordan \\cite{riordan2012phase} analyzed the configuration model with bounded degrees; Joseph \\cite{joseph2014component} considered i.i.d. degrees.\nThe strongest possible such result under minimal assumptions was obtained in \\cite{dhara-hofstad-leeuwaarden-sen}. We will state a weaker version of this result next.\n\n\\begin{thm}[\\cite{dhara-hofstad-leeuwaarden-sen}]\\label{thm:cm-component-sizes}\nConsider a degree sequence $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}=\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(n)}$ satisfying Assumption \\ref{ass:cm-deg} with the limiting random variable $D$ and define $\\sigma_r:=\\E[D^r]$, $r=1,2,3$.\nWrite $\\cC_{\\scriptscriptstyle(i)}(\\lambda)$ for the $i$-th largest connected component of $\\ \\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$ (or $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$). Let\n\\[\nN_{\\scriptscriptstyle(i)}^n(\\lambda) := \\#\\text{ edges in } \\cC_{\\scriptscriptstyle(i)}^n(\\lambda) - (|\\cC_{\\scriptscriptstyle(i)}(\\lambda)| -1)\n\\]\ndenote the number of surplus edges in $\\cC_{\\scriptscriptstyle(i)}(\\lambda)$. Then as $n \\to \\infty$,\n\\[\\bigg(\\big(n^{-2\/3}|\\cC_{\\scriptscriptstyle(i)}^n(\\lambda)|, N_{\\scriptscriptstyle(i)}^n(\\lambda)\\big): i\\geq 1\\bigg)\\stackrel{\\mathrm{w}}{\\longrightarrow} \\mvZ^D(\\lambda):= \\big(\\big(|\\gamma_{\\scriptscriptstyle(i)}^{\\mvmu_{D}}(\\lambda)|, N_{\\scriptscriptstyle(i)}^{\\mvmu_{D}}(\\lambda)\\big): i\\geq 1 \\big)\n\\]\nwith respect to product topology. Here $\\mvmu_{D} = (\\alpha_D, \\eta_D, \\beta_D)$ is given by\n\\begin{equation*}\n\\alpha_D = \\sigma_1,\\ \\\n\\eta_D = \\sigma_3\\sigma_1-\\sigma_2^2,\\ \\text{ and }\\\n\\beta_D = 1\/\\sigma_1.\n\\end{equation*}\n\\end{thm}\nThis result, in a stronger form, can be found in \\cite[Theorem 2 and Remark 5]{dhara-hofstad-leeuwaarden-sen}.\nWe will use this result in the next section to describe the limiting metric measure spaces arising in Section \\ref{sec:res}.\n\n\n\n\n\n\\subsection{The limiting metric measure spaces}\n\nA compact metric space $(X,d)$ is called a \\emph{real tree} \\cite{legall-survey,evans-book} if between every two points there is a unique geodesic such that this path is also the only non self-intersecting path between the two points. Functions encoding excursions from zero can be used to construct such metric spaces via a simple procedure. We describe this construction next.\n\nFor $0 < a < b <\\infty$, an \\emph{excursion} on $[a,b]$ is a continuous function $h \\in C([a,b], \\bR)$ with $h(a)=0=h(b)$ and $h(t) > 0$ for $t \\in (a,b)$. The length of such an excursion is $b-a$. For $l \\in(0,\\infty)$, let $\\cE_l$ be the space of all excursions on the interval $[0,l]$. Given an excursion $h \\in \\cE_l$, one can construct a real tree as follows. Define the pseudo-metric $d_h$ on $[0,l]$:\n\\begin{equation}\n\\label{eqn:d-pseudo}\n\td_h(s,t):= h(s) + h(t) - 2 \\inf_{u \\in [s,t]}h(u), \\; \\mbox{ for } s,t  \\in [0,l].\n\\end{equation}\nDefine the equivalence relation $s \\sim t \\Leftrightarrow d_h(s,t) = 0$. Let $[0,l]\/\\sim$ denote the corresponding quotient space and consider the metric space $\\cT_h:= ([0,l]\/\\sim, \\bar d_h)$, where $\\bar d_h$ is the metric on the equivalence classes induced by $d_h$. Then $\\cT_h$ is a real tree (\\cite{legall-survey,evans-book}).\nLet $q_h:[0,l] \\to \\cT_h$ be the canonical projection and write $\\mu_{\\cT_h}$ for the push-forward of the Lebesgue measure on $[0,l]$ onto $\\cT_h$ via $q_h$. Further, we assume that $\\cT_h$ is rooted at $\\rho := q_h(0)$.  Equipped with $\\mu_{\\cT_h}$, $\\cT_h$ is now a rooted compact metric measure space. Note that by construction, for any $x\\in \\cT_h$, the function $h$ is constant on $q_h^{-1}(x)$. Thus for each $x\\in [0,l]$, we write $\\mathrm{ht}(x) = h(q_h^{-1}(x))$ for the height of this vertex.\n\nThe Brownian continuum random tree defined below is a fundamental object in the literature of random real trees.\n\n\n\\begin{defn}[Aldous's continuum random tree \\cite{Aldo91a}]\\label{def:aldous-crt}\n\tLet $\\ve$ be a standard Brownian excursion on $[0,1]$. Construct the random compact real tree $\\cT_{2\\ve}$ as in \\eqref{eqn:d-pseudo} with $h=2\\ve$. The associated measure $\\mu_{\\cT_{2\\ve}}$ is supported on the collection of leaves of $\\cT_{2\\ve}$ almost surely.\n\\end{defn}\nWrite $\\nu$ for the law of a standard Brownian excursion on the space of excursions on $[0,1]$ namely $\\cE_1$.\nFor $k\\geq 0$, let $\\widetilde \\ve_{(k)}$ be a random excursion with distribution $\\tilde \\nu_{k}$ given via the following Radon-Nikodym density with respect to $\\nu$:\n\\begin{equation}\\label{eqn:tilde-nu-k-def}\n\\frac{d \\tilde \\nu_{k} }{d\\nu}(h)\n= \\frac{\\left[\\int_0^{1}h(u) du\\right]^k}{\\E\\left[\\left(\\int_0^{1} \\ve(u)du\\right)^k\\right]}\n= \\frac{\\big(\\int_{\\cT_h}\\mathrm{ht}(x)\\mu_{\\cT_h}(dx)\\big)^k}{\\E\\big[\\int_{\\cT_{\\ve}}\\mathrm{ht}(x)\\mu_{\\cT_{\\ve}}(dx)\\big]^k}, \\qquad h\\in \\cE_{1}.\n\t\\end{equation}\n\n\\begin{constr}[The space $M^{\\scriptscriptstyle(k)}$]\\label{constr:M-k}\nFix $k\\geq 0$.\n\\begin{enumeratea}\n\\item Let $\\widetilde \\ve_{(k)}$ be as above, and write $\\cT^\\star=\\cT_{2\\widetilde\\ve_{(k)}}$. Let $\\mu_{\\cT^\\star}$ denote the associated measure.\n\\item Conditional on $\\cT^\\star$, sample $k$ leaves $\\set{x_i: 1\\leq i\\leq k}$ in an i.i.d. fashion from $\\cT^\\star$ with density proportional to $\\mathrm{ht}(x) \\mu_{\\cT^\\star}(dx)$.\n\\item Conditional on the two steps above, for each of the sampled leaves $x_i$, sample a point $y_i$ uniformly at random on the line $[\\rho, x_i]$. Identify $x_i$ and $y_i$, i.e., introduce the equivalence relation $x_i\\sim y_i$, $1\\leq i\\leq k$, and form the quotient space $\\cT_{\\star}\/\\sim$.\n\\end{enumeratea}\nSet $M^{\\scriptscriptstyle(k)}$ to be the resultant (compact) random metric measure space.\n\\end{constr}\n\nNext recall the definition of $\\mvZ^D(\\lambda)$ from Theorem \\ref{thm:cm-component-sizes}.\n\\begin{constr}[The sequence $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^D(\\lambda)$]\\label{constr:M-D}\n\\hfill\n\\begin{enumeratea}\n\\item Sample $\\mvZ^D(\\lambda)=\\big(\\big(|\\gamma_{\\scriptscriptstyle(i)}^{\\mvmu_{D}}(\\lambda)|, N_{\\scriptscriptstyle(i)}^{\\mvmu_{D}}(\\lambda)\\big)\\ :\\ i\\geq 1\\big)$.\nFor simplicity, write\n\\[\\xi_i=|\\gamma_{\\scriptscriptstyle(i)}^{\\mvmu_{D}}(\\lambda)|,\\ \\text{ and }\\ N_i=N_{\\scriptscriptstyle(i)}^{\\mvmu_{D}}(\\lambda).\\]\n\\item Conditional on $\\mvZ^D(\\lambda)$, construct the spaces $S_i:=M^{(N_i)}$ independently for $i\\geq 1$.\n\\end{enumeratea}\nSet\n\\[\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^D(\\lambda)=\\big(M_1^D(\\lambda), M_2^D(\\lambda),\\ldots\\big),\\ \\text{ where }\\ M_i^D(\\lambda)=\\frac{\\alpha_D\\sqrt{\\xi_i}}{\\sqrt{\\eta_D}}\\cdot S_i,\\quad\\ i\\geq 1.\\]\n\\end{constr}\n\nNote that the sequence $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^D(\\lambda)$ of limiting spaces depends only on the first three moments of the random variable $D$ (which is also true for $\\mvZ^D(\\lambda)$--the scaling limit of the component sizes and the number of surplus edges).\n\nFinally, let $a_0$ be as in theorem \\ref{thm:vacant-set-scaling}. Define\n\\begin{align}\\label{eqn:lambda-vac-def}\n\\lambda_{\\mathrm{vac}}=\\frac{a_0(r-2)^2}{r(r-1)},\\ \\text{ and }\\ p_{\\mathrm{vac}}=\\exp\\bigg(-\\frac{r\\ln(r-1)}{(r-2)}\\bigg),\n\\end{align}\nand let $D_{\\mathrm{vac}}$ be the mixture random variable\n\\begin{align}\\label{eqn:D-vac-def}\nD_{\\mathrm{vac}}=(1-p_{\\mathrm{vac}})\\cdot\\delta_0+p_{\\mathrm{vac}}\\cdot\\mathrm{Binomial}\\bigg(r,\\ \\frac{1}{r-1}\\bigg).\n\\end{align}\n\\begin{constr}[The sequence $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{\\mathrm{vac}}(a_0)$]\\label{constr:M-vac}\nSet\n\\[\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{\\mathrm{vac}}(a_0):=\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{D_{\\mathrm{vac}}}\\big(\\lambda_{\\mathrm{vac}}\\big).\\]\n\\end{constr}\n\n\n\\begin{rem}\\label{rem:erdos-renyi}\nThe Erd\\H{o}s-R\\'enyi  scaling limit identified in \\cite{BBG-12} can be recovered by taking the limiting random variable to be $D_{\\er}\\sim\\mathrm{Poisson}(1)$, i.e., the scaling limit of $\\ERRG(n^{-1}+\\lambda n^{-4\/3})$ (after rescaling the graph distance by $n^{-1\/3}$) is given by\n\\[\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}_{\\er}(\\lambda):=\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{D_{\\er}}(\\lambda).\\]\n(Note that in this case, $\\alpha_{D_{\\er}}=\\eta_{D_{\\er}}=\\beta_{D_{\\er}}=1$.) The result for $\\ERRG(n^{-1}+\\lambda n^{-4\/3})$ can be obtained from our results as a special case of Theorem \\ref{thm:graphs-given-degree-scaling} by observing the following two facts:\n\\begin{enumeratei}\n\\item The (random) degree sequence of $\\ERRG(n^{-1}+\\lambda n^{-4\/3})$ satisfies Assumption \\ref{ass:cm-deg} with limiting random variable $D_{\\er}$.\n\\item Conditional on the event where the degree sequence equals $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}$, $\\ERRG(n^{-1}+\\lambda n^{-4\/3})$ is uniformly distributed over $\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$.\n\\end{enumeratei}\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:disc}\n\nHere we briefly discuss related work, the relevance of the work in this paper and possible extensions and questions raised by this work.\n\n\\begin{enumeratea}\n\t\\item {\\bf Graphs with prescribed degree distribution:} Graphs with prescribed degree sequence have played an integral part in probabilistic combinatorics over the last decade and have also been  heavily used in the applied fields including epidemic modeling \\cite{britton2007graphs,newman2002spread,keeling2005networks} community detection and clustering \\cite{fortunato2010community} and so on. In the context of this paper, the critical point for existence of a giant component was established in \\cite{molloy1995critical}. When the degree sequence results in trees, under suitable assumptions on the degree sequence, Broutin and Marckert in \\cite{broutin-marckert} showed that these trees appropriately normalized converge to Aldous's continuum random tree; this result will show up in a number of our proofs.\n\\item {\\bf Critical random graphs: } In the context of continuum scaling limits of maximal components in the critical regime, the only other result for the configuration model was derived in \\cite{bhamidi-broutin-sen-wang}; here using completely different techniques, critical percolation on the {\\bf supercritical} regime of the configuration model where the degree distribution has exponential tails was studied. Associated dynamic versions of this model were constructed and coupled appropriately to Aldous's multiplicative coalescent. A general universality principle also derived in the same paper then resulted in the scaling limits of maximal components at critical percolation. The techniques in that paper, however, do not extend to this work. Here we need start directly with a {\\bf critical} prescribed degree sequence; the proof techniques in this paper are completely different and use a combinatorial description of the uniform distribution on the space of {\\bf connected} simple graphs with a prescribed degree sequence.\n\n\t\\item \\label{disc:vacant-set} {\\bf Vacant sets and random interlacements on general random graphs:} With regards to vacant sets, as stated Theorem \\ref{thm:vacant-set-scaling} applies to random regular graphs. However as elucidated in Conjecture \\ref{conj:general-degree-vacant}, we believe that analogous results hold for the VSRW problem on $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n, \\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ or $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$ constructed using general degree sequence satisfying the hypothesis of Conjecture \\ref{conj:general-degree-vacant}. Let us now address the two clarifications described in Remark \\ref{rem:clar}. Assuming one can establish the critical point for VSRW for such graphs,\n\t\\begin{enumeratei}\n\t\t\\item Finite third moment: we conjecture that one can construct a (model dependent) random variable $D^*_{\\mathrm{vac}}$ (analogous to \\eqref{eqn:D-vac-def} for the random regular graph) and $\\lambda_{\\mathrm{vac}}^*$ a function of both the distribution of $D$ and $a_0$ (analogous to \\eqref{eqn:lambda-vac-def}) such that the maximal connected components in the critical regime with edges rescaled by $n^{-1\/3}$ converge to $\\boldsymbol{M}}\\newcommand{\\mvN}{\\boldsymbol{N}}\\newcommand{\\mvO}{\\boldsymbol{O}^{D_{\\mathrm{vac}}^*}(\\lambda_{\\mathrm{vac}}^*)$. This explicates the ``universality'' phenomenon we expect in this regime.\n\t\t\\item Infinite third moment regime: In \\cite{bhamidi-hofstad-sen}, various results related to scaling limits of maximal components in Aldous's multiplicative coalescent were established in terms of tilted inhomogeneous continuum random trees. One ramification of these  results (\\cite[Theorem 1.2]{bhamidi-hofstad-sen}) is the continuum scaling limits of the maximal components in the critical regime of the so-called Norros-Reitu model where the driving weight sequence is assumed to have heavy tails with exponent $\\tau\\in (3,4)$. A full description of this random graph model as well as the corresponding limits is beyond the scope of this paper, we refer the interested reader to \\cite{bhamidi-hofstad-sen}. We conjecture that the maximal components in the critical regime for the VSRW model are described by the limit objects derived in \\cite{bhamidi-hofstad-sen}.\n\t\\end{enumeratei}\n\t\n\tLet us now say a few words on how one can go about proving the above conjecture (at least in the finite third moment setting).  As will become evident from the proofs, the result follows owing to the following three ingredients \\begin{inparaenumi}\n\t\t\\item Theorem \\ref{thm:graphs-given-degree-scaling};\n\t\t\\item a result of Cooper and Frieze \\cite{cooper-frieze} which expresses the annealed measure for the vacant set problem in terms of the random graphs with prescribed degree sequence;\n\t\t\\item refined bounds on the degree sequence of the vacant in the critical scaling window derived in \\cite{cerny-teixeira}.\n\t\\end{inparaenumi}\n\tParts (i) and (ii) continue to hold for the vacant set problem for random walks on general graphs with prescribed degree sequence. Thus to extend our results to the vacant set problem for general graphs, all one needs is an extension of the refined bounds in (iii) to random walks on general graphs.\n\t\n\n\n\t\\item {\\bf Proof techniques:} The techniques used in this paper differ from the standard techniques used to show convergence of such random discrete objects to limiting random tree like metric spaces. One standard technique (used in \\cite{BBG-12,SBSSXW14}) is to construct an exploration process of the discrete object of interest that converges to the exploration process of a continuum random tree (see \\cite{legall-survey,evans-book} for beautiful treatments), and encode the ``surplus'' edges as a random point process falling under the exploration, and show that this point process converges to a Poisson point process in the limit.\nIn this work, we use a different technique that requires less work. We first prove convergence of the object of interest in the Gromov-weak topology, essentially showing that for each fixed $k\\geq 2$, the distance matrix constructed from $k$ randomly sampled vertices converges in distribution to the distance matrix constructed from $k$ points appropriately sampled from the limiting structure. This result, coupled with a global lower mass bound implies via general theory \\cite{athreya-lohr-winter} that convergence occurs in the stronger Gromov-Hausdorff-Prokhorov sense. In the context of critical random graphs, this technique was first used in \\cite{bhamidi-hofstad-sen} to analyze the so-called rank-one critical inhomogeneous random graph.\n\\end{enumeratea}\n\n\n\n\\section{Preliminary constructions}\n\\label{sec:prf-prelim}\n\nIn this section we set up important notation related to plane trees that will be used throughout the proof.\n\n\n\\subsection{Plane trees: Basic functionals and exploration}\n\\label{sec:plane-tree-def}\nThroughout the sequel we will let $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$ denote a {\\bf plane rooted} tree. We use $\\rho$ to denote the root and think of this as the original progenitor of the tree so that every vertex other than the root has a unique parent. Write $\\cL(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$ for the set of leaves of  $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$. For each non-root vertex $u\\in \\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$, let $\\overleftarrow{u}$ denote the parent of $u$. Let $[\\rho, u]$ (resp. $[\\rho, u)$) denote the ancestral line of $u$ including (resp. excluding) $u$. Thus $[\\rho, u)=[\\rho,\\overleftarrow{u}]$. Using the planar embedding, any plane tree $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$ can be explored in a depth-first manner (our convention is to go to the ``leftmost'' child first). Let $\\prec_{\\mathrm{DF}}$ be the linear order on vertices of a plane tree induced by a depth-first exploration starting from the root, i.e., $x\\prec_{\\mathrm{DF}} y$ if $x$ is explored before $y$ in a depth-first search of the plane tree.\n\n\\begin{defn}[Admissible pairs of leaves]\n\t\\label{def:admissible}\n\tFor leaves $u,v\\in\\cL(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$, we say that the {\\bf ordered} pair $(u,v)$ is {\\bf admissible} if\n\t\\[\\overleftarrow{\\overleftarrow{v}}\\in[\\rho, \\overleftarrow{u}),\\ \\text{ and }\\\n\t\\overleftarrow{u}\\prec_{\\mathrm{DF}}\\overleftarrow{v}.\\]\n\tLet $\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$ denote the set of admissible pairs of $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$.\n\\end{defn}\nSee Figure \\ref{fig:admiss} for an example of an admissible pair.\n\\begin{figure}[htbp]\n\t\\centering\n\t\\pgfmathsetmacro{\\nodebasesize}{1}\n\t\n\t\t\\begin{tikzpicture}[scale=.25,  iron\/.style={circle, minimum size=6mm, inner sep=0pt, ball color=red!20}, wat\/.style= {circle, inner sep=3pt, ball color=green!20}\n]\n\t\t\n\t\n\t\n\t\t   \n\t\n\t\n\t\t    \\node (1) [iron] at (0,0) {$\\rho$};\n\t\t    \\node (11) [iron] at (-4,5) {$\\overleftarrow{\\overleftarrow{v}}$};\n\t\t    \\node (12) [iron] at (4,5) {};\n\t\t    \\node (111) [iron] at (-8,10) {$\\overleftarrow{u}$};\n\t\t\t\\node (112) [iron] at (1,10) {$\\overleftarrow{v}$};\n\t\t\t\\node (1111) [wat] at (-12,15) {${u}$};\n\t\t\t\\node (1121) [wat] at (5,15) {${v}$};\n\t\t   \n\t\t   \n\t\t   \n\t\t   \n    \t\n\t\n\t\t\n\t\t\n\t        \\draw[blue, very thick] (1)-- (11) -- (111) -- (1111);\n\t\t\n\t        \\draw[blue, very thick] (1) -- (12);\n\t\t\n\t        \\draw[blue, very thick] (11) -- (112) -- (1121);\n\t\t\n\t\n\t\n\n\t\n\t\t\\end{tikzpicture}\n\t\\caption{An example of an admissible pair of leaves.  }\n\t\\label{fig:admiss}\n\\end{figure}\n\tWe introduce a linear order $\\ll$ on $\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$ as follows: $(u_1,v_1)\\ll (u_2,v_2)$ if either $\\overleftarrow{u}_1\\prec_{\\mathrm{DF}}\\overleftarrow{u}_2$ or if $\\overleftarrow{u}_1=\\overleftarrow{u}_2$ and $\\overleftarrow{v}_1\\prec_{\\mathrm{DF}}\\overleftarrow{v}_2$. For $u\\in\\cL(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$, define\n\\begin{equation}\n\\label{eqn:ftu-atu-def}\n\t\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, u):=\\set{v\\in\\cL(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})\\ :\\ (u,v)\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})},\\ \\text{ and }\\\n\tf_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}}(u):=|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, u)|.\n\\end{equation}\nNote that\n\\begin{equation}\n\\label{eqn:At=sum-ftu}\n|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|=\\sum_{u\\in\\cL(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})} f_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}}(u).\n\\end{equation}\nFor any plane tree $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$ and a vertex $u$, define\n\\[\\mvB_1(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},u)=\\set{v\\ :\\ \\overleftarrow{v}\\in[\\rho,u)},\\ \\text{ and }\\\n\\mvB_2(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},u)=\\set{v\\ :\\ \\overleftarrow{\\overleftarrow{v}}\\in[\\rho,u)},\\]\nwhere $\\rho$ is the root of $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$. Thus for any $u\\in\\cL(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$,\n\\begin{equation}\n\\label{eqn:ft-lessb2}\n\tf_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}}(u)=|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, u)|\\le|\\mvB_2(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, u)|.\n\\end{equation}\nFor any plane tree $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$ and a vertex $u$, define\n\\[\\mvB_1^-(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},u)=\\set{v\\ :\\ \\overleftarrow{v}\\in[\\rho,u),\\ u\\prec_{\\mathrm{DF}} v},\\ \\text{ and }\\\n\\mvB_1^+(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},u)=\\set{v\\ :\\ \\overleftarrow{v}\\in[\\rho,u),\\ v\\prec_{\\mathrm{DF}} u}.\\]\nSo if our convention is to explore the children of a vertex from left to right in a depth-first search, then\n$\\mvB_1^{-}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},u)$ (resp. $\\mvB_1^{+}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},u)$) is the collection of vertices that are at distance one from the path $[\\rho,u)$ and lie on the right (resp. left) side of $[\\rho,u)$.\n\n\n\n\nNow fix $k\\ge 1$. Define\n\\[\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})=\\bigg\\{\\big\\{(u_1,v_1),\\ldots,(u_k,v_k)\\big\\}\\ \\mid\\ (u_j,v_j)\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})\\text{ and }u_1,v_1,\\ldots,u_k,v_k\\text{ are }2k\\text{ distinct leaves}\\bigg\\}.\\]\nLet $\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$ be the collection of all such ordered $k$-tuples of admissible pairs. Clearly\n\\[\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_1(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})=\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}),\\ |\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|=k!\\times|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|,\\ \\text{ and }|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|\\le |\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|^k.\\]\nFor later use, define $\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})^k = \\otimes_{i=1}^k \\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$ be the $k$-fold Cartesian product of $\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$.   For a plane tree $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$ and a vertex $u$, let $\\mathrm{Anc}^{\\scriptscriptstyle (1)}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},u)$ be the plane subtree (with the root and $u$ marked) whose vertex set is given by\n\\[V=\\big\\{v\\ \\big|\\ v\\in[\\rho, u],\\ \\text{ or }\\ \\overleftarrow{v}\\in[\\rho, u)\\big\\}.\\]\nLet $\\mathrm{Anc}^{\\scriptscriptstyle(2)}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, u)$ be the plane subtree (with the root and $u$ marked) whose vertex set is given by\n\\[V^{\\scriptscriptstyle(2)}=\\big\\{v\\ \\big|\\ v\\in[\\rho, u],\\ \\text{ or }\\ \\overleftarrow{v}\\in[\\rho, u),\\ \\text{ or }\\\n\\overleftarrow{\\overleftarrow{v}}\\in[\\rho, u)\\big\\}.\\]\nGiven a plane tree $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$, write $\\mvc(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}) = (c_v(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}): v\\in \\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$, where $c_v(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$ is the number of children of $v$ in $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$. Further write $\\mvs(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}) = (s_i(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}):i\\geq 0)$ for the {\\bf empirical children distribution (ECD)} of $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$. Namely,\n \t\\[s_i(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}):= \\set{v\\in \\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}: c_v(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}) = i}, \\qquad i\\geq 0.\\]\nGiven a sequence of integers $\\mvs = (s_i:i\\geq 0)$, we say that the sequence is a \\emph{tenable} ECD for a tree if there exists a finite plane tree $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$ with $\\mvs(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}) = \\mvs$. It is easy to check $\\mvs$ is tenable if and only if $s_i \\geq 0$ for all $i$ with $s_0 \\geq 1 $, and\n\\[\\sum_{i\\ge 0} s_i = 1+ \\sum_{i\\ge 0} i s_i<\\infty.\\]\nGiven a tenable ECD $\\mvs$, let  $\\bT_{\\mvs}$ denote the set of all plane trees having ECD $\\mvs$.\n\nIf $\\mvf$ is a finite forest of plane trees, define the ECD $\\mvs(\\mvf)$ analogously. Note that for any sequence of integers $\\mvs=(s_i: i\\ge 0)$ satisfying\n\\[s_i\\ge 0,\\ \\ \\sum_i i s_i<\\infty,\\ \\text{ and }\\ k(\\mvs):=\\sum_i s_i-\\sum_i i s_i\\geq 1,\\]\nthere exists a forest with ECD $\\mvs$. Such a forest has exactly $k(\\mvs)$ many trees.\nGiven such a sequence $\\mvs$, let $\\bF_{\\mvs}$ denote the set of all plane forests with {\\bf ranked roots} having ECD $\\mvs$. Thus each forest in $\\bF_{\\mvs}$ comes with an ordering of the roots so it makes sense talking about the ``first'' tree of the forest, the ``second'' tree etc.\n\n\n\nFinally fix $k\\geq 1$ and let $\\bT_{\\mvs}^{\\scriptscriptstyle(k)}$ denote the set of all pairs $(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, \\mvx)$, where $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}$ and $\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$. For a plane tree $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$ and $\\mvx=\\big\\{(u_1,v_1),\\ldots,(u_k,v_k)\\big\\}\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$, let $\\cI(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, \\mvx)$ be the rooted space obtained by adding an edge between $\\overleftarrow{u}_j$ and $\\overleftarrow{v}_j$, and deleting $u_j,v_j$ and the two edges incident to them for $j=1,\\ldots,k$. (See Figure \\ref{fig:admiss-ident} for an example of this.) We endow the space $\\cI(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, \\mvx)$ with the graph distance and the uniform probability measure on all vertices. Similarly if $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}^{\\mathrm{lab}}$ is a labeled plane tree and $\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}^\\mathrm{lab})$, then $\\cI(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}^\\mathrm{lab}, \\mvx)$ is the {\\bf labeled} graph obtained by following the same construction and retaining the labels.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\pgfmathsetmacro{\\nodebasesize}{1}\n\t\n\t\t\\begin{tikzpicture}[scale=.25,  iron\/.style={circle, minimum size=6mm, inner sep=0pt, ball color=red!20}, wat\/.style= {circle, inner sep=3pt, ball color=green!20}\n]\n\t\t\n\t\n\t\n\t\t   \n\t\n\t\n\t\t    \\node (1) [iron] at (0,0) {$\\rho$};\n\t\t    \\node (11) [iron] at (-4,5) {$\\overleftarrow{\\overleftarrow{v}}$};\n\t\t    \\node (12) [iron] at (4,5) {};\n\t\t    \\node (111) [iron] at (-8,10) {$\\overleftarrow{u}$};\n\t\t\t\\node (112) [iron] at (1,10) {$\\overleftarrow{v}$};\n\t\t   \n\t\t   \n\t\t   \n\t\t   \n    \t\n\t\n\t\t\n\t\t\n\t        \\draw[blue, very thick] (1)-- (11) -- (111);\n\t\t\n\t        \\draw[blue, very thick] (1) -- (12);\n\t\t\n\t        \\draw[blue, very thick] (11) -- (112);\n\t\t\t\\draw[red, very thick] (111) -- (112);\n\t\t\n\t\n\t\n\n\t\n\t\t\\end{tikzpicture}\n\t\\caption{An example of the operation $\\cI$ applied on the tree $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$ and admissible pair $(u,v)$ in Figure \\ref{fig:admiss}.  }\n\t\\label{fig:admiss-ident}\n\\end{figure}\nFinally for a plane tree $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$ and $\\mvx=\\big((u_1,v_1),\\ldots,(u_k,v_k)\\big)\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})^k$, let $\\cQ(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, \\mvx)$ be the space obtained by identifying $u_j$ and $\\overleftarrow{\\overleftarrow{v}}_j$ for $1\\leq j\\leq k$.\nWe endow the space $\\cQ(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}, \\mvx)$ with the graph distance and the push-forward of the uniform probability measure on $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$.\n\n\n\n\n\n\n\n\\section{Properties of plane trees}\\label{sec:proof-plane-tree}\nWe start by describing the setting and assumptions. Assume that for each $m\\geq 1$, $\\mvs^{\\scriptscriptstyle(m)}=(s_i: i\\ge 0)$ (where $\\sum s_i=m$) is a tenable ECD for a tree. Analogous to Assumption \\ref{ass:degree}, we make the following assumption on $\\set{\\mvs^{\\scriptscriptstyle(m)}: m\\geq 1}$:\n\n\\begin{ass}\\label{ass:ecd}\nThere exists a pmf $(p_0, p_1,\\ldots)$ with\n\\[ p_0>0,\\quad  \\sum_{i\\ge 1}i p_i=1,\\quad\\text{and }\\sum_{i\\ge 1}i^2 p_i<\\infty\\]\nsuch that\n\\[\\frac{s_i}{m}\\to p_i\\ \\text{ for }\\ i\\ge 0,\\ \\text{ and }\\\n\\frac{1}{m}\\sum_{i\\ge 0}i^2 s_i\\to\\sum_{i\\ge 1}i^2 p_i.\\]\nIn particular, $\\Delta_m:=\\max\\set{i: s_i\\neq 0}=o(\\sqrt{m})$.\n\nWe will write $\\sigma^2 = \\sum_i i^2 p_i - 1$ for the variance associated with the pmf $(p_0, p_1,\\ldots)$.\n\\end{ass}\nThen the following was shown in \\cite{broutin-marckert}.\n\\begin{theorem}[{\\cite[Theorem 1]{broutin-marckert}}]\n\\label{thm:broutin-marckert}\nLet $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}$ be a uniform element of $\\bT_{\\mvs^{\\scriptscriptstyle(m)}}$ endowed with the with the {\\bf uniform probability measure on $m$ vertices} and viewed as a metric measure space. Under Assumption \\ref{ass:ecd}, as $m\\to\\infty$,\n\\[\\frac{\\sigma}{\\sqrt{m}}\\cT_{\\mvs^{\\scriptscriptstyle(m)}}\\stackrel{\\mathrm{w}}{\\longrightarrow}\\cT_{2\\ve}\\]\nin the GHP sense (see Definition \\ref{def:aldous-crt}).\n\\end{theorem}\n\n\\begin{rem}\nIn \\cite[Theorem 1]{broutin-marckert}, the convergence is stated to hold in the Gromov-Hausdorff sense. However, it is easy to see that the proof in fact implies convergence in the GHP sense.\n\\end{rem}\n\n\nThe following technical lemma collects all the ingredients necessary for proving our main results.\n\\begin{lem}\\label{lem:plane-trees}\nLet $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}$ be a uniform plane tree with ECD $\\mvs^{\\scriptscriptstyle(m)}$. Under Assumption \\ref{ass:ecd} the following assertions hold.\n\\begin{enumeratei}\n\\item\\label{lem:leaves-counting-measure}\nFor each $k\\ge 1$, we can construct independent random vectors $(U_m^{(1)}, V_m^{(1)}),\\ldots,(U_m^{(k)}, V_m^{(k)})$ such that $U_m^{(j)}$, $j=1,\\ldots,k$ have uniform distribution on the $s_0$ leaves and $V_m^{(j)}$, $j=1,\\ldots,k$ have uniform distribution on the $m$ vertices, and\n\\[m^{-1\/2}d_{\\cT_{\\mvs^{\\scriptscriptstyle(m)}}}\\big(U_m^{(j)}, V_m^{(j)}\\big)\\stackrel{\\mathrm{P}}{\\longrightarrow} 0\\ \\text{ for }j=1,\\ldots,k.\\]\nIn particular,\n\\[d_{\\GHP}\\big(\\frac{1}{\\sqrt{m}}\\cT_{\\mvs^{\\scriptscriptstyle(m)}}, \\frac{1}{\\sqrt{m}}\\cT_{\\mvs^{\\scriptscriptstyle(m)}}^{\\cL}\\big)\\stackrel{\\mathrm{P}}{\\longrightarrow} 0,\\]\nwhere $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}^{\\cL}$ denotes the metric measure space obtained when the underlying tree is endowed with the uniform probability measure on the set of leaves $\\cL(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})$. (Recall that the measure on the space $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}$ is the uniform probability measure on {\\bf all} vertices.)\n\\item\\label{lem:uniform-integrability}\nRecall the definition of an admissible pair of leaves (Definition \\ref{def:admissible}) and of the function $f_{\\cT_{\\mvs^{\\scriptscriptstyle(m)}}}$ (Equation \\eqref{eqn:ftu-atu-def}). Let $U_m$ be uniformly distributed over $\\cL(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})$. Then for every $k\\ge 1$,\n\\[\\sup_m\\ \\E\\left(\\frac{|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})|}{s_0\\sqrt{m}}\\right)^k\n\\le \\sup_m\\ \\E\\left(\\frac{f_{\\cT_{\\mvs^{\\scriptscriptstyle(m)}}}(U_m)}{\\sqrt{m}}\\right)^k\n<\\infty.\\]\n\\item\\label{lem:independent-sampling}\nFor every $k\\ge 1$,\n\\[\\frac{1}{m^{3k\/2}}\\bigg(\\big|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})\\big|^k-\\big|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})\\big|\\bigg)\\stackrel{\\mathrm{P}}{\\longrightarrow} 0.\\]\n\\item\\label{lem:finite-dim-convergence}\nLet $k\\ge 0$ and $\\ell\\ge 1$.\nConsider independent random variables $U_m^{(i)}$, $i=1,\\ldots, k$ and $V_m^{(j)}$, $j=1,\\ldots,\\ell$, where $U_m^{(i)}$, $i=1,\\ldots,k$ have uniform distribution on the $s_0$ leaves of $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}$, and $V_m^{(j)}$, $j=1,\\ldots,\\ell$ have uniform distribution on the $m$ vertices.\n\nIf the subtree of $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}$ spanned by the root $\\rho$ and $U_m^{(i)}$, $i=1,\\ldots, k$ and $V_m^{(j)}$, $j=1,\\ldots,\\ell$ does not have $(k+\\ell)$ distinct leaves, then set $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}(\\mvU, \\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X})=\\partial$.\nOtherwise set $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}(\\mvU, \\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X})$ to be the subtree of $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}$ spanned by the root $\\rho$ and $U_m^{(i)}$, $i=1,\\ldots, k$ and $V_m^{(j)}$, $j=1,\\ldots,\\ell$, and view it as an element of $\\ \\vT_{k+\\ell}^*$ as in Section \\ref{sec:space-of-trees} via the following prescription:\nFor $1\\leq i\\leq k$, attach the ``leaf value'' $m^{-1\/2}f_{\\cT_{\\mvs^{\\scriptscriptstyle(m)}}}(U_m^{(i)})$ to $U_m^{(i)}$.\nEndow $[\\rho, U_m^{(i)}]$ with a probability measure by assigning mass $p_x^{\\scriptscriptstyle(i)}$ to each $x\\in[\\rho, U_m^{(i)})$, where\n  \\[p_x^{\\scriptscriptstyle(i)}:=\\frac{\\#\\bigg\\{v\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}\\big(\\cT_{\\mvs^{\\scriptscriptstyle(m)}}, U_m^{(i)}\\big)\\ \\ \\bigg| \\ \\\n  \\overleftarrow{\\overleftarrow{v}}=x\\bigg\\}}{f_{\\cT_{\\mvs^{\\scriptscriptstyle(m)}}}(U_m^{(i)})}.\\]\n(The leaf values and root-to-leaf measures attached to $V_m^{(j)}$, $1\\le j\\le\\ell$, are irrelevant in our proof and can be taken to be zero and $\\delta_{\\{\\rho\\}}$ respectively.)\n\nThen\n\\[\\frac{1}{\\sqrt{m}}\\cT_{\\mvs^{\\scriptscriptstyle(m)}}(\\mvU, \\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X})\\stackrel{\\mathrm{w}}{\\longrightarrow} \\frac{1}{\\sigma}\\cT_{k,\\ell},\\]\nwhere $\\cT_{k,\\ell}$ is the random element of $\\ \\vT_{k+\\ell}^*$ constructed as follows: The shape of $\\cT_{k,\\ell}$ is that of the subtree of $\\cT_{2\\ve}$ spanned by $(k+\\ell)$ points $x_1,\\ldots,x_{k+\\ell}$ sampled independently according to the mass measure $\\mu_{\\cT_{2\\ve}}$. The leaf weight attached to $x_i$ is $p_0\\sigma\\cdot\\mathrm{ht}(x_i)\/2$, $i=1,\\ldots,k$, and the measure on $[\\rho,x_i]$ is the normalized line measure.\n\\item\\label{lem:conditional-expectation}\nThe following joint convergence holds:\n\\[\\bigg(\\frac{1}{\\sqrt{m}}\\cT_{\\mvs^{\\scriptscriptstyle(m)}}, \\frac{|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})|}{s_0\\sqrt{m}}\\bigg)\n\\stackrel{\\mathrm{w}}{\\longrightarrow}\\bigg(\\frac{1}{\\sigma}\\cT_{2\\ve},\\frac{p_0\\sigma}{2}\\int_{\\cT_{2\\ve}}\\mathrm{ht}(x)\\ \\mu_{\\cT_{2\\ve}}(dx)\\bigg)\\]\nwith respect to product topology induced by GHP topology on the first coordinate and Euclidean topology on the second coordinate.\n\\end{enumeratei}\n\\end{lem}\n\\subsection{Proof of Lemma \\ref{lem:plane-trees}\\eqref{lem:leaves-counting-measure}}\nConsider $m$ vertices labeled $1,\\ldots,m$. Let $\\bT_{\\mvs}^{\\mathrm{lab}}$ be the set of plane trees where vertices labeled $1,\\ldots,s_0$ are leaves, vertices labeled $s_0+1,\\ldots, s_0+s_1$ have one child,..., vertices labeled $(m-s_{\\Delta}+1),\\ldots, m$ have $\\Delta$ many children. For convenience, denote by $c_j$ the number of children of the vertex labeled $j$. We will now describe a way of generating a tree uniformly distributed over $\\bT_{\\mvs}^{\\mathrm{lab}}$.\n\nLet $\\pi$ be a uniform permutation on $[m]$. Let\n\\begin{align}\\label{eqn:F}\nS(j):=\\sum_{i=1}^j (c_{\\pi(i)}-1),\\ j=1,\\ldots,m,\\ \\text{ and }\\ F(x) = S(\\lfloor m x \\rfloor), \\quad 0\\leq x\\leq 1.\n\\end{align}\nExtend the definition of $\\pi$ periodically by letting $\\pi(j):=\\pi(j \\mod m+1)$.\nLet $i_0$ denote the location of the first minima of $\\set{S(j):1\\leq j\\leq m}$ and consider the Vervaat transform w.r.t. this location:\n\\begin{align}\\label{eqn:F-exc}\nS^{\\exec}(j) = \\sum_{i=1}^{j} \\big(c_{\\pi(i_0 + i)}-1\\big),\\quad 1\\leq j\\leq m,\\ \\text{ and\\ set }\nF^{\\exec}(x) = S^{\\exec}(\\lfloor mx \\rfloor), \\quad 0\\leq x\\leq 1.\n\\end{align}\nLet $\\cT_{\\mvs}^{\\mathrm{lab}}$ be the labeled tree whose {\\L}ukasiewicz walk is $F^{\\exec}(\\lfloor mx\\rfloor),\\ x\\in[0,1]$. Note that $\\cT_{\\mvs}^{\\mathrm{lab}}$ can be constructed sequentially from $F^{\\exec}$, and further every interval $((i-1)\/m, i\/m]$, $1\\leq i\\leq m$ corresponds to a unique vertex in the exploration process. Let $\\cT_{\\mvs}^{\\mathrm{lab}-}$ be the tree obtained by removing all labels from $\\cT_{\\mvs}^{\\mathrm{lab}}$.\n\\begin{lem}\\label{lem:generating-tree-from-permutation}\nLet $\\cT_{\\mvs}^{\\mathrm{lab}}$ and $\\cT_{\\mvs}^{\\mathrm{lab}-}$ be as above.\n\\begin{enumeratea}\n\\item $\\cT_{\\mvs}^{\\mathrm{lab}}\\sim\\mathrm{Unif}(\\bT_{\\mvs}^{\\mathrm{lab}})$.\n\\item $\\cT_{\\mvs}^{\\mathrm{lab}-}\\stackrel{\\mathrm{d}}{=}\\cT_{\\mvs}$, i.e., $\\cT_{\\mvs}^{\\mathrm{lab}-}\\sim\\mathrm{Unif}(\\bT_{\\mvs})$.\n\\end{enumeratea}\n\\end{lem}\n\\noindent{\\bf Proof:} Each of the $(m-1)!$ rotation classes of the $m!$ permutations gives rise to a unique $F^{\\exec}$. Since there is a bijection between $\\bT_{\\mvs}^{\\mathrm{lab}}$ and the possible realizations of $F^{\\exec}$,\n\\[\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}(\\cT_{\\mvs}^{\\mathrm{lab}}=\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}^\\mathrm{lab})=\\frac{1}{(m-1)!}=\\frac{1}{|\\bT_{\\mvs}^{\\mathrm{lab}}|}\\ \\text{ for any }\\ \\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}^\\mathrm{lab}\\in\\bT_{\\mvs}^{\\mathrm{lab}}.\\]\nThis implies\n\\[\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\cT_{\\mvs}^{\\mathrm{lab}-}=\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\big)=\\frac{\\prod_{i\\ge 0}s_i!}{(m-1)!}=\\frac{1}{|\\bT_{\\mvs}|}\\ \\text{ for any }\\ \\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}.\\]\n \\hfill $\\blacksquare$\n\n\nWe now state a useful concentration inequality.\n\\begin{lem}\\label{lem:concentration-uniform-permutation}\nThere exist universal constants $c_1, c_2>0$ such that for any $m\\geq 1$ and probability vector $\\mvp:=(p_1,\\ldots,p_m)$,\n\\[\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\max_{j\\in[m]}\\bigg|\\sum_{i=1}^j p_{\\pi(i)}-\\frac{j}{m}\\bigg|\\geq x\\sigma(\\mvp)\\bigg)\n\\leq\\exp\\big(-c_1 x\\log\\log x\\big),\\ \\text{ for }\\ x\\geq c_2,\\]\nwhere $\\pi$ is a uniform permutation on $[m]$, and $\\sigma(\\mvp):=\\sqrt{p_1^2+\\ldots+p_m^2}$.\n\\end{lem}\n\n\n\\noindent{\\bf Proof:}\nThe result is essentially contained in \\cite[Lemma 4.9]{bhamidi-hofstad-sen}, and we only outline how to extract the result from its proof.\nWe can work with $\\pi$ generated in the following way: let $X_1,\\ldots,X_m$ be i.i.d. Unif$[0,1]$, and set $\\pi(i)=j_i$, where $X_{j_1}<\\ldots<X_{j_m}$. Write $X_{(i)}=X_{j_i}$. Then\n\\begin{align}\\label{eqn:20}\n\\max_{j\\in[m]}\\bigg(\\sum_{i=1}^j p_{\\pi(i)}-\\frac{j}{m}\\bigg)\n\\leq\\max_{j\\in[m]}\\bigg(\\sum_{i=1}^j p_{\\pi(i)}-X_{(j)}\\bigg)+\\max_{j\\in[m]}\\bigg|X_{(j)}-\\frac{j}{m}\\bigg|.\n\\end{align}\nBy the DKW inequality \\cite{massart-DKW},\n\\begin{equation}\n\\label{eqn:dkw}\n\\pr\\bigg(\\max_{j\\in [m]} \\bigg|X_{\\scriptscriptstyle(j)}-\\frac{j}{m}\\bigg| \\geq \\sigma(\\mathbf{p}}\\newcommand{\\vq}{\\mathbf{q}}\\newcommand{\\vr}{\\mathbf{r}) x\\bigg)\n\\leq 2\\exp\\left(-2m\\cdot\\left(\\sigma(\\mathbf{p}}\\newcommand{\\vq}{\\mathbf{q}}\\newcommand{\\vr}{\\mathbf{r})x\\right)^2\\right)\n\\leq 2\\exp\\left(-2x^2\\right),\n\\end{equation}\nwhere the last step uses the inequality: $m\\sigma(\\mvp)^2\\geq(\\sum_i p_i)^2=1$.\nFrom \\cite[Equations (4.14) and (4.15)]{bhamidi-hofstad-sen} we have\n\\begin{equation}\\label{eqn:21}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\max_{j\\in[m]}\\bigg(\\sum_{i=1}^j p_{\\pi(i)}-X_{(j)}\\bigg)\\geq x\\sigma(\\mvp)\\bigg)\n\\leq\\exp\\big(-c_3 x\\log\\log x\\big),\\ \\text{ for }\\ x\\geq c_4.\n\\end{equation}\nCombining \\eqref{eqn:20}, \\eqref{eqn:dkw}, and \\eqref{eqn:21}, it follows that\n\\begin{equation}\\label{eqn:22}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\max_{j\\in[m]}\\bigg(\\sum_{i=1}^j p_{\\pi(i)}-\\frac{j}{m}\\bigg)\\geq x\\sigma(\\mvp)\\bigg)\n\\leq\\exp\\big(-c_5 x\\log\\log x\\big),\\ \\text{ for }\\ x\\geq c_6.\n\\end{equation}\nNow note that\n\\begin{align*}\n\\max_{j\\in[m]}\\bigg(\\frac{j}{m}-\\sum_{i=1}^j p_{\\pi(i)}\\bigg)\n\\leq\\max_{j\\in[m]}\\bigg(X_{(j)}-\\sum_{i=1}^j p_{\\pi(i)}\\bigg)+\\max_{j\\in[m]}\\bigg|X_{(j)}-\\frac{j}{m}\\bigg|.\n\\end{align*}\nFurther, the arguments used in the proof of \\cite[Equation (4.15)]{bhamidi-hofstad-sen} can be used to prove a tail bound similar to \\eqref{eqn:21} for  $\\max_{j\\in[m]}\\big(X_{(j)}-\\sum_{i=1}^j p_{\\pi(i)}\\big)$. Combining this observation with \\eqref{eqn:22} yields the desired result.\n \\hfill $\\blacksquare$\n\n\n\n\\begin{lem}\\label{lem:leaves-counting-measure-labeled-tree}\nFor each $m\\geq 1$, let $\\mvp=\\mvp^{\\scriptscriptstyle(m)}=(p_1^{\\scriptscriptstyle(m)},\\ldots, p_m^{\\scriptscriptstyle(m)})$ be a probability vector such that $p_{\\max}\\to 0$ as $m\\to\\infty$. Then for each $k\\ge 1$, we can construct independent random vectors $(U_m^{(1)}, V_m^{(1)}),\\ldots,(U_m^{(k)}, V_m^{(k)})$ on $\\cT_{\\mvs^{\\scriptscriptstyle(m)}}^\\mathrm{lab}$ such that $U_m^{(j)}$, $j=1,\\ldots,k$ are distributed according $\\mvp$ and $V_m^{(j)}$, $j=1,\\ldots,k$ have uniform distribution on the $m$ vertices, and\n\\[m^{-1\/2}d_{\\cT_{\\mvs^{\\scriptscriptstyle(m)}}^\\mathrm{lab}}\\big(U_m^{(j)}, V_m^{(j)}\\big)\\stackrel{\\mathrm{P}}{\\longrightarrow} 0\\ \\text{ for }j=1,\\ldots,k.\\]\n\\end{lem}\n\n\n\\noindent{\\bf Proof:}\nRecall from around \\eqref{eqn:F} the permutation $\\pi$ used to construct $\\cT_{\\mvs}^\\mathrm{lab}$, and set\n\\[ G(x)=\\sum_{i=0}^{\\lfloor mx\\rfloor} p_{\\pi(i+i_0+1)},\\ \\ x\\in[0,1],\\]\nwhere as before $i_0$ is the location of the first minima of $\\set{S(j): j\\in [m]}$.\nLet $\\sigma(\\mvp)$ be as in Lemma \\ref{lem:concentration-uniform-permutation}. Then $\\sigma(\\mvp)\\geq p_{\\max}\\geq 1\/m$.\nHence Lemma \\ref{lem:concentration-uniform-permutation} implies\n\\begin{align}\\label{eqn:9}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\sup_{x\\in[0,1]}\\bigg|G(x)-x\\bigg|\\geq y\\sigma(\\mvp)\\bigg)\\le\\exp\\big(-C_1 y\\log\\log y\\big),\\ \\ y\\ge C_2.\n\\end{align}\nSince $\\sigma(\\mathbf{p}}\\newcommand{\\vq}{\\mathbf{q}}\\newcommand{\\vr}{\\mathbf{r})\\leq\\sqrt{p_{\\max}}\\to 0$ by assumption, we get, in particular,\n\\begin{align}\\label{eqn:10}\n\\sup_{u\\in[0,1]}|G^{-1}(u)-u|\\stackrel{\\mathrm{P}}{\\longrightarrow} 0.\n\\end{align}\nLet $X\\sim\\mathrm{Unif}[0, 1]$. Let $V_m^{\\scriptscriptstyle(1)}$ (resp. $U_m^{\\scriptscriptstyle(1)}$) be the point on $\\cT_{\\mvs}^\\mathrm{lab}$ corresponding to $X$ (resp. $G^{-1}(X)$). Then $V_m^{\\scriptscriptstyle(1)}$ has uniform distribution on the $m$ vertices, and $U_m^{(1)}$ is distributed according to $\\mvp$.\nLet $H_m$ be the height function of $\\cT_{\\mvs}^\\mathrm{lab}$. Then\n\\[\\bigg|d_{\\cT_{\\mvs}^\\mathrm{lab}}\\big(U_m^{(1)}, V_m^{(1)}\\big)-\n\\bigg(H_m\\big(G^{-1}(X)\\big)+H_m(X)-2\\min\\displaystyle^\\star H_m(u)\\bigg)\\bigg|\\le 2,\\]\nwhere $\\min^\\star$ denotes minimum taken over the interval $[G^{-1}(X)\\wedge X, G^{-1}(X)\\vee X]$. Now \\cite[Proposition 5 and Lemma 7]{broutin-marckert} imply that the random function $m^{-1\/2}H_m(\\cdot)$ converges weakly to the continuous function $2\\ve\/\\sigma$ under the usual Skorohod topology on $D[0,1]$. Since $|G^{-1}(X)-X|\\stackrel{\\mathrm{P}}{\\longrightarrow} 0$, we conclude that\n\\[m^{-1\/2} d_{\\cT_{\\mvs}^\\mathrm{lab}}\\big(U_m^{(1)}, V_m^{(1)}\\big)\\stackrel{\\mathrm{P}}{\\longrightarrow} 0.\\]\nTaking $k$ independent copies of $X$ and repeating the same argument completes the proof. \\hfill $\\blacksquare$\n\n\\noindent{\\bf Completing the proof of Lemma \\ref{lem:plane-trees}\\eqref{lem:leaves-counting-measure}:}\nBy Lemma \\ref{lem:generating-tree-from-permutation}, $\\cT_{\\mvs}\\stackrel{\\mathrm{d}}{=}\\cT_{\\mvs}^{\\mathrm{lab}-}$. Thus the claim follows by an application of Lemma \\ref{lem:leaves-counting-measure-labeled-tree} with the choice\n\\[p_i=\\frac{\\mathds{1}\\set{c_i=0}}{s_0}. \\]\n \\hfill $\\blacksquare$\n\\subsection{Proof of Lemma \\ref{lem:plane-trees}\\eqref{lem:uniform-integrability}}\nWe start with the following set of estimates.\n\\begin{lem}\\label{lem:moments}\nLet $\\tilde\\mvs$ be a tenable ECD for a forest of plane trees, and let\n\\[\\tilde z=\\sum_i \\tilde s_i-\\sum_i i\\tilde s_i,\\ \\ \\tilde\\Delta=\\max_i\\set{i:\\tilde s_i\\neq 0},\\ \\text{ and }\\ \\tilde m:=\\sum_i\\tilde s_i,\\]\ni.e., $\\tilde z$ is the number of roots, $\\tilde\\Delta$ is the maximum number of children, and $\\tilde m$ is the total number of vertices in any forest in $\\bF_{\\tilde\\mvs}$. Sample a forest uniformly from $\\bF_{\\tilde\\mvs}$, and let $X_j$ denote the number of children of the root of the $j$-th tree, $1\\le j\\le \\tilde z$. Then for any $\\sigma_1,\\ldots,\\sigma_r\\ge 1$ and $1\\le j_1<\\ldots<j_r\\le\\tilde z$,\n\\begin{align}\\label{eqn:23}\n\\E\\bigg[X_{j_1}^{\\sigma_1}\\times\\ldots\\times X_{j_r}^{\\sigma_r}\\bigg]\n\\le r 2^r\\bigg(\\sum_i\\frac{i^2\\tilde s_i}{\\tilde m}\\bigg)^r\n\\bigg(1+\\frac{\\tilde\\Delta}{\\tilde z}\\bigg)\n(\\tilde\\Delta)^{\\sigma_1+\\ldots+\\sigma_r-r},\n\\end{align}\nwhenever $r\\le\\tilde m\/2$. As a consequence the sum of children of all roots in the randomly sampled forest satisfies the moment bound\n\\begin{align}\\label{eqn:24}\n\\E\\big[X_1+\\ldots+X_{\\tilde z}\\big]^k\\le K_{k}\\bigg(1+\\frac{\\tilde\\Delta}{\\tilde z}\\bigg)\n\\sum_{r=1}^k\\tilde z^r\\bigg(\\sum_i\\frac{i^2\\tilde s_i}{\\tilde m}\\bigg)^r\\tilde\\Delta^{k-r}\n\\end{align}\nwhenever $r\\le\\tilde m\/2$, and where $K_k$ is a constant depending only on $k$.\n\\end{lem}\n\\noindent{\\bf Proof:} Let $\\delta_i=(0,\\ldots,0,1,0,\\ldots)$ with $1$ at the $i$-th position.\nRecall (see, e.g., \\cite[Equation 6.19]{pitman-book}) that\n\\begin{align}\\label{eqn:pitman}\n\\big|\\bF_{\\tilde\\mvs}\\big|=\\frac{\\tilde z(\\tilde m-1)!}{\\prod_{i\\ge 0}\\tilde s_i!}.\n\\end{align}\nBy exchangeability, it is enough to consider $X_1,\\ldots,X_r$. Since there is a bijection between $\\bF_{\\tilde\\mvs-\\delta_{i_1}-\\delta_{i_2}}$ and the set of forests in $\\bF_{\\tilde\\mvs}$ for which $X_1=i_1$ and $X_2=i_2$,\n\\begin{align}\\label{eqn:25}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(X_1=i_1,X_2=i_2\\big)=\\frac{|\\bF_{\\tilde\\mvs-\\delta_{i_1}-\\delta_{i_2}}|}{|\\bF_{\\tilde\\mvs}|}.\n\\end{align}\nWhen $i_1\\neq i_2$,\n\\begin{align*}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(X_1=i_1,X_2=i_2\\big)\n=\\frac{(\\tilde z-2+i_1+i_2)\\tilde s_{i_1}\\tilde s_{i_2}}{\\tilde z(\\tilde m-1)(\\tilde m-2)}\n\\le 4\\bigg(1+\\frac{i_1+i_2}{\\tilde z}\\bigg)\\frac{\\tilde s_{i_1}\\tilde s_{i_2}}{\\tilde m^2}.\n\\end{align*}\nWhen $i_1=i_2=i$,\n\\begin{align*}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(X_1=i,X_2=i\\big)\n=\\frac{(\\tilde z-2+2i)\\tilde s_{i}(\\tilde s_{i}-1)}{\\tilde z(\\tilde m-1)(\\tilde m-2)}\n\\le 4\\bigg(1+\\frac{2i}{\\tilde z}\\bigg)\\frac{\\tilde s_{i}^2}{\\tilde m^2}.\n\\end{align*}\nIn general,\n\\begin{align*}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(X_1=i_1,\\ldots,X_r=i_r\\big)\n&\\le 2^r\\bigg(1+\\frac{i_1+\\ldots+i_r}{\\tilde z}\\bigg)\\frac{\\tilde s_{i_1}\\times\\ldots\\times\\tilde s_{i_r}}{\\tilde m^r}\\\\\n&\\le r2^r\\bigg(1+\\frac{\\tilde\\Delta}{\\tilde z}\\bigg)\\frac{\\tilde s_{i_1}\\times\\ldots\\times\\tilde s_{i_r}}{\\tilde m^r}.\n\\end{align*}\nNote that\n\\[\\sum_{i\\ge 1}i^{\\sigma_j}\\frac{\\tilde s_{i}}{\\tilde m}\n\\le\\tilde\\Delta^{(\\sigma_j-2)\\vee 0}\\sum_{i\\ge 1}i^2\\frac{\\tilde s_{i}}{\\tilde m}\n\\le\\tilde\\Delta^{(\\sigma_j-1)}\\sum_{i\\ge 1}i^2\\frac{\\tilde s_{i}}{\\tilde m}.\\]\nCombining the above, we get\n\\[\\E\\bigg[X_1^{\\sigma_1}\\times\\ldots\\times X_r^{\\sigma_r}\\bigg]\n\\le r 2^r\\bigg(1+\\frac{\\tilde\\Delta}{\\tilde z}\\bigg)\\bigg(\\sum_i\\frac{i^2\\tilde s_i}{\\tilde m}\\bigg)^r\n\\tilde\\Delta^{\\sum_{j=1}^r(\\sigma_j-1)},\n\\]\nwhich proves \\eqref{eqn:23}. The bound in \\eqref{eqn:24} follows by a direct expansion. \\hfill $\\blacksquare$\n\n\\medskip\n\nFrom \\eqref{eqn:At=sum-ftu} and \\eqref{eqn:ft-lessb2} it is clear that\n\\begin{align}\\label{eqn:12}\n\\E\\left(\\frac{|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})|}{s_0\\sqrt{m}}\\right)^k\n&=\\E\\bigg[\\E\\bigg(\\frac{1}{\\sqrt{m}}f_{\\cT_{\\mvs^{\\scriptscriptstyle(m)}}}(U_m)\\bigg|\\cT_{\\mvs^{\\scriptscriptstyle(m)}}\\bigg)\\bigg]^k\\notag\\\\\n&\\le\\E\\left(\\frac{f_{\\cT_{\\mvs^{\\scriptscriptstyle(m)}}}(U_m)^k}{m^{k\/2}}\\right)\n\\le\\E\\left(\\frac{|\\mvB_2(\\cT_{\\mvs}, U_m)|^k}{m^{k\/2}}\\right).\n\\end{align}\nLet $\\mvs_{1}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})=(s_{1,i}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}),\\ i\\geq 0)$ be the ECD corresponding to vertices in $[\\rho, U_m]$ when $\\mathrm{Anc}^{\\scriptscriptstyle (1)}(\\cT_{\\mvs}, U_m)=\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$, i.e., $s_{1,i}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$ is the number of vertices in $[\\rho, U_m]$ with $i$ many children when $\\mathrm{Anc}^{\\scriptscriptstyle (1)}(\\cT_{\\mvs}, U_m)=\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}$. Let\n\\[\\tilde\\mvs(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}):=\\mvs-\\mvs_1(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}).\\]\nLet $\\cF$ be the (random) plane forest with ranked roots obtained by deleting the vertices of $[\\rho, U_m]$ and the edges incident to them, rooting the resulting trees at the vertices in $\\mvB_1(\\cT_{\\mvs}, U_m)$, and ranking the roots using the depth-first order.\nNote that there is a bijection between the realizations of $(\\cT_{\\mvs}, U_m)$ and of $\\big(\\mathrm{Anc}^{\\scriptscriptstyle (1)}(\\cT_{\\mvs}, U_m), \\cF\\big)$. Thus\n\\begin{align}\\label{eqn:forest}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\cF=\\mvf,\\ \\mathrm{Anc}^{\\scriptscriptstyle (1)}(\\cT_{\\mvs}, U_m)=\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\big)=\\frac{1}{s_0|\\bT_{\\mvs}|},\\ \\text{ for all }\\ \\mvf\\in\\bF_{\\tilde\\mvs(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})}.\n\\end{align}\nHence\n\\[\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\mathrm{Anc}^{\\scriptscriptstyle (1)}(\\cT_{\\mvs}, U_m)=\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\big)=\\frac{|\\bF_{\\tilde\\mvs(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})}|}{s_0|\\bT_{\\mvs}|}.\n\\]\nWriting $\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}}=(\\cdot\\ |\\ \\mathrm{Anc}^{\\scriptscriptstyle (1)}(\\cT_{\\mvs}, U_m)=\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$, it follows that\n\\begin{equation}\\label{eqn:30}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}}\\big(\\cF=\\mvf\\big)=\\frac{1}{|\\bF_{\\tilde\\mvs(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})}|}\\ \\text{ for any }\\ \\mvf\\in\\bF_{\\tilde\\mvs(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})}.\n\\end{equation}\nDefine the conditional expectation operator $\\E_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}}$ in an analogous fashion. Let\n\\[E:=\\big\\{\\mathrm{ht}(U_m)< m\/2\\big\\}.\\]\nBy \\cite[Theorem 1]{addario-berry},\n\\begin{align}\\label{eqn:13}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}(E^c)\\le 7\\exp(-cm).\n\\end{align}\nLet $\\tilde z, \\tilde m, \\tilde\\Delta$  be as in Lemma \\ref{lem:moments} with $\\tilde\\mvs=\\tilde\\mvs(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})$. Then $\\tilde m\\ge m\/2$ on the event $E$, $\\tilde s_i\\le s_i$, and $\\tilde\\Delta\\le\\Delta$. Thus it follows from Lemma \\ref{lem:moments} that\n\\begin{align*}\n\\mathds{1}_E\\E_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}}\\bigg[|\\mvB_2(\\cT_{\\mvs}, U_m)|^k\\bigg]\n\\le K\\sum_{r=1}^k\\bigg(\\tilde z^r\\Delta^{k-r}+\\tilde z^{r-1}\\Delta^{k-r+1}\\bigg).\n\\end{align*}\nHence\n\\begin{align}\\label{eqn:14}\n\\E\\bigg[\\mathds{1}_E|\\mvB_2(\\cT_{\\mvs}, U_m)|^k\\bigg]\n\\le K\\sum_{r=1}^k\\bigg(\\Delta^{k-r}\\E\\big[\\big|\\mvB_1(\\cT_{\\mvs}, U_m)\\big|^r\\big]\n+\\Delta^{k-r+1}\\E\\big[\\big|\\mvB_1(\\cT_{\\mvs}, U_m)\\big|^{r-1}\\big]\\bigg).\n\\end{align}\nLet $S^{\\exec}$ be as in \\eqref{eqn:F-exc}. Letting\n\\[\\mvp:=(p_1,\\ldots, p_m),\\ \\text{ where }\\ p_i=\\frac{c_i}{m-1},\\ i=1,\\ldots,m,\\]\nan application of Lemma \\ref{lem:concentration-uniform-permutation} shows that\n\\[\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\max_{j\\in[m]}\\bigg|\\sum_{i=1}^j p_{\\pi(i+i_0)}-\\frac{j}{m}\\bigg|\\geq x\\sigma(\\mvp)\\bigg)\n\\leq\\exp\\big(-c_1 x\\log\\log x\\big),\\ \\text{ for }\\ x\\geq c_2,\\]\nwhere $i_0$ has the same meaning as in \\eqref{eqn:F-exc}. Since\n\\[\\bigg|\\sum_{i=1}^j \\frac{\\big(c_{\\pi(i+i_0)}-1\\big)}{m-1}-\\bigg(\\sum_{i=1}^j p_{\\pi(i+i_0)}-\\frac{j}{m}\\bigg)\\bigg|\n=\\frac{j}{m(m-1)}\\leq\\frac{1}{m-1},\\]\nwe conclude that\n\\begin{align*}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\max_{j\\in[m]}\\big|S^{\\exec}(j)\\big|\\ge y\\sqrt{m}\\bigg)\\le\\exp\\big(-cy\\log\\log y\\big),\\ \\ y\\ge c'.\n\\end{align*}\nThis implies that the same tail bound holds for $\\mvB_1^-(U_m)$. Since $\\mvB_1^+(U_m)\\stackrel{\\mathrm{d}}{=}\\mvB_1^-(U_m)$ and $\\mvB_1=\\mvB_1^-+\\mvB_1^+$,\n\\begin{align}\\label{eqn:15}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\big|\\mvB_1(\\cT_{\\mvs}, U_m)\\big|\\ge y\\sqrt{m}\\bigg)\\le\\exp\\big(-cy\\log\\log y\\big),\\ \\ y\\ge c'.\n\\end{align}\nCombining \\eqref{eqn:14} and \\eqref{eqn:15}, we conclude that\n\\[\\sup_m\\frac{1}{m^{k\/2}}\\E\\bigg[\\mathds{1}_E|\\mvB_2(\\cT_{\\mvs}, U_m)|^k\\bigg]<\\infty.\\]\nFrom \\eqref{eqn:13}, it follows that\n\\[\\frac{1}{m^{k\/2}}\\E\\bigg[\\mathds{1}_E^c|\\mvB_2(\\cT_{\\mvs}, U_m)|^k\\bigg]\\le 7m^{k\/2}\\exp(-cm).\\]\nWe complete the proof of Lemma \\ref{lem:plane-trees}\\eqref{lem:uniform-integrability} by combining these observations with \\eqref{eqn:12}.  \\hfill $\\blacksquare$\n\n\\subsection{Proof of Lemma \\ref{lem:plane-trees}\\eqref{lem:independent-sampling}}\nNote first that Lemma \\ref{lem:plane-trees}\\eqref{lem:uniform-integrability} shows that $|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})|\/m^{3\/2}$ is tight. Now\nclearly\n\\begin{align}\\label{eqn:16}\n&\\big|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})\\big|^k-\\big|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})\\big|\\notag\\\\\n&\\hskip10pt\n=\\#\\bigg\\{\\big((u_1,v_1),\\ldots,(u_k,v_k)\\big)\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})^k\\ \\big|\\ \\set{u_i,v_i}\\cap\\set{u_j,v_j}\\neq\\emptyset\\text{ for some } i\\neq j\\bigg\\}\\notag\\\\\n&\\hskip20pt\n\\le k^2\\big|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})\\big|^{k-2}(R_{11}+2R_{12}+R_{22}),\n\\end{align}\nwhere\n\\[R_{11}=\\#\\bigg\\{\\big((u_1,v_1),(u_2,v_2)\\big)\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})^2\\ \\big|\\ u_1=u_2\\bigg\\},\\\nR_{12}=\\#\\bigg\\{\\big((u_1,v_1),(u_2,v_2)\\big)\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})^2\\ \\big|\\ u_1=v_2\\bigg\\},\\]\nand\n\\[R_{22}=\\#\\bigg\\{\\big((u_1,v_1),(u_2,v_2)\\big)\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})^2\\ \\big|\\ v_1=v_2\\bigg\\}.\\]\nNow\n\\begin{align*}\nR_{11}\n&=\\sum_{u\\in\\cL(\\cT_{\\mvs})}\\sum_{v_1\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs},u)}\n\\#\\bigg\\{v_2\\ \\big|\\ (u,v_2)\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})\\bigg\\}\\\\\n&\\hskip20pt\\le m\\sum_{u\\in\\cL(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})}f_{\\cT_{\\mvs^{\\scriptscriptstyle(m)}}}(u)\n=m\\big|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})\\big|\n=mO_{P}(m^{3\/2})=o_P(m^3).\n\\end{align*}\nA similar argument shows that $R_{12}+R_{22}=o_P(m^3)$. Combined with \\eqref{eqn:16}, this gives\n\\[\\big|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})\\big|^k-\\big|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\cT_{\\mvs^{\\scriptscriptstyle(m)}})\\big|\n=O_P\\big(m^{3(k-2)\/2}\\big)o_P(m^3)=o_P(m^{3k\/2}),\\]\nas desired.  \\hfill $\\blacksquare$\n\n\n\n\\subsection{Proof of Lemma \\ref{lem:plane-trees}\\eqref{lem:finite-dim-convergence}}\nWe will make use of the following lemma.\n\\begin{lem}\\label{lem:line-measure}\nLet $U_m$ be uniformly distributed on the $s_0$ leaves. Then\n\\[\\frac{1}{\\sqrt{m}}\\ \\max_{k\\le\\mathrm{ht}(U_m)}\n\\Bigg|\\#\\bigg\\{v\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}\\big(\\cT_{\\mvs^{\\scriptscriptstyle(m)}}, U_m\\big)\\ \\bigg| \\\n\\mathrm{ht}\\big(\\overleftarrow{\\overleftarrow{v}}\\big)\\le k\\bigg\\}-\\frac{p_0\\sigma^2 k}{2}\\Bigg|\\stackrel{\\mathrm{P}}{\\longrightarrow} 0.\\]\n\\end{lem}\n\nWe first prove Lemma \\ref{lem:plane-trees}\\eqref{lem:finite-dim-convergence} assuming Lemma \\ref{lem:line-measure}.\n\n\\noindent{\\bf Completing the proof of Lemma \\ref{lem:plane-trees}\\eqref{lem:finite-dim-convergence}:}\nTheorem \\ref{thm:broutin-marckert} shows that the shape of the subtree of $m^{-1\/2}\\cT_{\\mvs}$ spanned by $(k+\\ell)$ vertices sampled independently and uniformly from $[m]$ converges to the shape of $\\sigma^{-1}\\cT_{k,\\ell}$. By Lemma \\ref{lem:plane-trees}\\eqref{lem:leaves-counting-measure}, the same conclusion holds if the first $k$ vertices are sampled independently and uniformly from the $s_0$ leaves, and the other $\\ell$ vertices are sampled independently and uniformly from $[m]$. Convergence of the root-to-leaf measures and the leaf values is a consequence of Lemma \\ref{lem:line-measure}.\n \\hfill $\\blacksquare$\n\nThe rest of this section is devoted to the proof of Lemma \\ref{lem:line-measure}.\nWe start with the following lemma.\n\\begin{lem}\\label{lem:concentration}\nLet $\\tilde\\mvs=\\tilde\\mvs_{\\kappa}$ be a sequence of ECDs indexed by $\\kappa$. We will suppress $\\kappa$ in the notation most of the time. Let $\\tilde m,\\ \\tilde\\Delta,\\ \\tilde z$ and $X_1,\\ldots, X_{\\tilde z}$ be as in Lemma \\ref{lem:moments}. Let $f:\\bZ_{\\ge 0}\\to\\bR_{\\ge 0}$. Assume that\n\\begin{enumeratei}\n\\item $\\tilde z\\to\\infty$;\n\\item there exists $a>0$ such that $\\sum_{i\\ge 0}f(i)\\tilde s_i\/\\tilde m\\to a$;\n\\item $\\sup_{\\kappa}\\sum_{i\\ge 0}f^2(i)\\tilde s_i\/\\tilde m<\\infty$;\n\\item $\\tilde\\Delta=o(\\tilde z)$; and\n\\item $\\max_{1\\le i\\le\\tilde\\Delta}f(i)=o(\\tilde z)$.\n\\end{enumeratei}\nThen\n\\[\\frac{1}{\\tilde z}\\cdot\\max_{1\\le j\\le\\tilde z}\\big|\\sum_{i=1}^j f(X_i)-aj\\big|\\stackrel{\\mathrm{P}}{\\longrightarrow} 0.\n\\]\n\\end{lem}\n\n\\noindent{\\bf Proof:}\nAn argument similar to the one used in \\eqref{eqn:25} gives\n\\[\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(X_1=i\\big)\n=\\frac{(\\tilde z-1+i)\\tilde s_{i}}{\\tilde z(\\tilde m-1)}.\n\\]\nHence\n\\begin{align}\\label{eqn:26}\n\\E\\bigg[\\sum_{i\\ge 1}^{\\tilde z}\\frac{f(X_i)}{\\tilde z}\\bigg]\n=\\E\\big[f(X_1)\\big]=\\sum_{i=1}^{\\tilde\\Delta}\n\\bigg(\\frac{(\\tilde z-1+i)\\tilde m}{\\tilde z(\\tilde m-1)}\\bigg)\\frac{f(i)\\tilde s_{i}}{\\tilde m}\\to a.\n\\end{align}\nSimilarly, using \\eqref{eqn:25}, a direct computation shows that\n$\\cov\\big(f(X_1), f(X_2)\\big)\\to 0,$\nwhich in turn implies\n\\begin{align}\\label{eqn:27}\n\\var\\bigg[\\sum_{i\\ge 1}^{\\tilde z}\\frac{f(X_i)}{\\tilde z}\\bigg]\\to 0.\n\\end{align}\nCombining \\eqref{eqn:26} and \\eqref{eqn:27}, we see that\n\\begin{align}\\label{eqn:29}\n\\sum_{i\\ge 1}^{\\tilde z}f(X_i)\/\\tilde z\\stackrel{\\mathrm{P}}{\\longrightarrow} a.\n\\end{align}\n\n\nLet $\\hat S=(\\hat S_0,\\hat S_1,\\ldots)$ denote the empirical distribution of $X_1,\\ldots, X_{\\tilde z}$.\nSince\n\\begin{align*}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(X_1=i_1,\\ldots, X_{\\tilde z}=i_{\\tilde z}\\big)=\\frac{|\\bF_{\\mvs-\\hat\\mvs}|}{|\\bF_{\\mvs}|}\n\\end{align*}\nfor any $(i_1,\\ldots,i_{\\tilde z})$ with empirical distribution $\\hat\\mvs$,\n\\begin{align}\\label{eqn:17}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(X_1=i_1,\\ldots, X_{\\tilde z}=i_{\\tilde z}\\ \\big|\\ \\hat S=\\hat\\mvs\\big)=\\frac{\\prod_{i\\geq 0}\\hat s_i!}{\\tilde z!}\n\\end{align}\nDefine $y_1,\\ldots, y_{\\tilde z}$ as follows:\n\\begin{align*}\ny_1=\\ldots=y_{\\hat S_0}=0,\\ \\ y_{\\hat S_0+1}=\\ldots=y_{\\hat S_0+\\hat S_1}=1,\\ldots.\n\\end{align*}\nThen conditional on $\\hat S$, the distribution \\eqref{eqn:17} can be generated by uniformly permuting $y_1,\\ldots,y_{\\tilde z}$ and removing the $y$ labels. Set\n\\[\\hat\\mvp:=(\\hat p_1,\\ldots,\\hat p_{\\tilde z}),\\ \\text{ where }\\\n\\hat p_i=\\frac{f(y_i)}{\\sum_{j=1}^{\\tilde z}f(y_j)}.\\]\nFrom Lemma \\ref{lem:concentration-uniform-permutation}, for a uniform permutation $\\pi$ (independent of $\\hat S$) on $\\tilde z$ elements and $\\varepsilon>0$,\n\\begin{align}\\label{eqn:28}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\max_{1\\le j\\le\\tilde z}\\bigg|\\sum_{i=1}^j \\hat p_{\\pi(i)}-\\frac{j}{\\tilde z}\\bigg|\\ge\\varepsilon\\ \\bigg|\\ \\hat S\\bigg)\n\\le \\exp\\bigg(-c_1\\bigg(\\frac{\\varepsilon}{\\sigma(\\hat\\mvp)}\\bigg)\\log\\log\\bigg(\\frac{\\varepsilon}{\\sigma(\\hat\\mvp)}\\bigg)\\bigg)\\\n\\text{ on }\\ \\big\\{c_2\\sigma(\\hat\\mvp)\\leq\\varepsilon\\big\\}.\n\\end{align}\nSince\n\\[\\sigma(\\hat\\mvp)^2\\le \\hat p_{\\max}\n=\\frac{\\tilde z}{\\sum_{1\\le i\\le\\tilde z}f(X_i)}\\times\\max_{1\\le i\\le\\tilde\\Delta}\\frac{f(i)}{\\tilde z}\n\\stackrel{\\mathrm{P}}{\\longrightarrow} 0,\\]\nwe conclude from \\eqref{eqn:28} that\n\\[\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\max_{1\\le j\\le\\tilde z}\\bigg|\n\\frac{\\sum_{i=1}^j f(X_i)}{\\sum_{k=1}^{\\tilde z}f(X_k)}-\\frac{j}{\\tilde z}\\bigg|\\ge\\varepsilon\\bigg)\\to 0,\n\\]\nwhich combined with \\eqref{eqn:29} yields the claim.\n \\hfill $\\blacksquare$\n\n\\medskip\n\n\n\\noindent{\\bf Proof of Lemma \\ref{lem:line-measure}:}\nIt follows from \\cite[Proposition 5]{broutin-marckert} that\n\\begin{align}\\label{eqn:31}\n\\frac{1}{\\sqrt{m}}\\ \\max_{k\\le\\mathrm{ht}(U_m)}\\bigg|\\#\\bigg\\{v\\ \\big|\\\n v\\in\\mvB_1^-(\\cT_{\\mvs}, U_m),\\ \\mathrm{ht}\\big(\\overleftarrow{v}\\big)\\le k\\bigg\\}-\\frac{\\sigma^2 k}{2}\\bigg|\\stackrel{\\mathrm{P}}{\\longrightarrow} 0,\n\\end{align}\nand a similar statement is true with $\\mvB_1^+$ by symmetry.\nNow recall from \\eqref{eqn:30} that conditional on $\\mathrm{Anc}^{\\scriptscriptstyle (1)}(\\cT_{\\mvs},U_m)$, the forest $\\cF$ (defined around \\eqref{eqn:forest}) is uniformly distributed over $\\bF_{\\tilde\\mvs}$, where $\\tilde\\mvs$ is the ECD of the vertices in $[\\rho, U_m]^c$.\nWe make the following observations about the forest $\\cF$:\n\\begin{enumeratei}\n\\item The ECD $\\tilde\\mvs$ of $\\cF$ satisfies\n\\[\ns_i-\\mathrm{ht}(U_m)\\leq\\tilde s_i\\leq s_i,\n\\]\nwhere $\\mathrm{ht}(U_m)=\\Theta_P(\\sqrt{m})$ by Theorem \\ref{thm:broutin-marckert}.\n\\item Similarly, the number of vertices $\\tilde m$ (say) in $\\cF$ satisfies\n\\[\nm-\\mathrm{ht}(U_m)\\leq\\tilde m\\leq m.\n\\]\n\\item \\eqref{eqn:31} and its analogue for $\\mvB_1^+$ combined with the fact $\\mathrm{ht}(U_m)=\\Theta_P(\\sqrt{m})$ shows that the number of roots of $\\cF$, namely $|\\mvB(\\cT_{\\mvs}, U_m)|$ satisfies\n\\[|\\mvB(\\cT_{\\mvs}, U_m)|=\\Theta_P(\\sqrt{m}).\\]\n\\end{enumeratei}\nThus, when $\\mvs$ satisfies Assumption \\ref{ass:ecd}, the ECD of $\\cF$ and the function $f(i)=i$ satisfy the assumptions of Lemma \\ref{lem:concentration} with $a=1$. This combined with \\eqref{eqn:31} gives\n\\begin{align}\\label{eqn:32}\n\\frac{1}{\\sqrt{m}}\\ \\max_{k\\le\\mathrm{ht}(U_m)}\\bigg|\\#\\bigg\\{v\\ \\big|\\\n \\overleftarrow{v}\\in\\mvB_1^-(\\cT_{\\mvs}, U_m),\\ \\mathrm{ht}\\big(\\overleftarrow{\\overleftarrow{v}}\\big)\\le k\\bigg\\}\n -\\frac{\\sigma^2 k}{2}\\bigg|\\stackrel{\\mathrm{P}}{\\longrightarrow} 0,\n\\end{align}\nand a similar statement is true with $\\mvB_1^+$.\n\n\nLet $\\cF^{\\scriptscriptstyle(2)}$ be the plane forest with ranked roots obtained by deleting the vertices of $\\mathrm{Anc}^{\\scriptscriptstyle (1)}(\\cT_{\\mvs},U_m)$ and the edges incident to them, rooting the resulting trees at the vertices of $\\mvB_2(\\cT_{\\mvs}, U_m)$, and ranking them in the depth-first order.\nThen conditional on $\\mathrm{Anc}^{\\scriptscriptstyle(2)}(\\cT_{\\mvs},U_m)$, $\\cF^{\\scriptscriptstyle(2)}$ is again uniformly distributed over the set of plane forests with ranked roots with the remaining children sequence.\nFurther, reasoning similar to above shows that the ECD of $\\cF^{\\scriptscriptstyle(2)}$ and the function $f(i)=\\mathds{1}\\set{i=0}$ satisfies the assumptions of Lemma \\ref{lem:concentration} with $a=p_0$.\n\nApplying Lemma \\ref{lem:concentration} to the forest $\\cF^{\\scriptscriptstyle(2)}$ and the function $f(i)=\\mathds{1}\\set{i=0}$, and combining this with \\eqref{eqn:32} yields the claim in Lemma \\ref{lem:line-measure}.\n \\hfill $\\blacksquare$\n\n\\subsection{Proof of Lemma \\ref{lem:plane-trees}\\eqref{lem:conditional-expectation}}\nFirst recall from Lemma \\ref{lem:plane-trees} \\eqref{lem:leaves-counting-measure} that $\\cT_{\\mvs}^{\\cL}$ denoted the metric measure space obtained when the underlying tree is endowed with the uniform probability measure on $\\cL(\\cT_{\\mvs})$.  Let $U_m^{\\scriptscriptstyle(i)}$, $1\\leq i\\leq k$, and $x_1,\\ldots,x_k$ be as in Lemma \\ref{lem:plane-trees}\\eqref{lem:finite-dim-convergence}.\nLemma \\ref{lem:plane-trees}\\eqref{lem:leaves-counting-measure} together with Theorem \\ref{thm:broutin-marckert} shows that for all $k\\geq 1$,\n\\begin{align*}\n&\\left(\\frac{1}{\\sqrt{m}}\\cT_{\\mvs},\\\n\\frac{1}{\\sqrt{m}}\\cT_{\\mvs}^{\\cL},\\\n\\frac{1}{k\\sqrt{m}}\\bigg(\\mathrm{ht}(U_m^{\\scriptscriptstyle(1)})+\\ldots+\\mathrm{ht}(U_m^{\\scriptscriptstyle(k)})\\bigg)\\right)\\\\\n&\\hskip30pt\n\\stackrel{\\mathrm{w}}{\\longrightarrow}\\left(\\frac{1}{\\sigma}\\cT_{2\\ve},\\\n\\frac{1}{\\sigma}\\cT_{2\\ve},\\\n\\frac{1}{k\\sigma}\\bigg(\\mathrm{ht}(x_1)+\\ldots+\\mathrm{ht}(x_k)\\bigg)\n\\right)\n\\end{align*}\nwith respect to product topology induced by GHP topology on the first two coordinates and Euclidean topology on $\\bR$ on the third coordinate.\nThus by Lemma \\ref{lem:line-measure},\n\\begin{align}\\label{eqn:33}\n\\left(\\frac{1}{\\sqrt{m}}\\cT_{\\mvs},\\\n\\frac{1}{k\\sqrt{m}}\\bigg(f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(1)})+\\ldots+f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(k)})\\bigg)\\right)\n\\stackrel{\\mathrm{w}}{\\longrightarrow}\\left(\\frac{1}{\\sigma}\\cT_{2\\ve},\\\n\\frac{p_0\\sigma}{2k}\\bigg(\\mathrm{ht}(x_1)+\\ldots+\\mathrm{ht}(x_k)\\bigg)\n\\right).\n\\end{align}\nNow for any $\\varepsilon>0$ and $k\\geq 1$,\n\\begin{align*}\n&\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\bigg|\n\\frac{1}{k\\sqrt{m}}\\bigg(f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(1)})+\\ldots+f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(k)})\\bigg)-\\frac{|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs})|}{s_0\\sqrt{m}}\n\\bigg|\\geq\\varepsilon\n\\bigg)\\\\\n&\\hskip20pt\n=\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\bigg|\n\\frac{1}{k\\sqrt{m}}\\bigg(f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(1)})+\\ldots+f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(k)})\\bigg)-\n\\frac{\\E\\big(f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(1)})\\big|\\cT_{\\mvs}\\big)}{\\sqrt{m}}\n\\bigg|\\geq\\varepsilon\n\\bigg)\\\\\n&\\hskip40pt\n\\leq\\frac{1}{\\varepsilon^2 k m}\\E\\bigg[\\var\\bigg(f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(1)})\\big|\\cT_{\\mvs}\\bigg)\\bigg]\n\\leq\\frac{1}{\\varepsilon^2 k m}\\E\\bigg[f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(1)})^2\\bigg]\\leq\\frac{C}{\\varepsilon^2k},\n\\end{align*}\nwhere the first equality holds because of \\eqref{eqn:At=sum-ftu} and the last step uses Lemma \\ref{lem:plane-trees}\\eqref{lem:uniform-integrability}.\nBy a similar argument, we can show that\n\\begin{align*}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\bigg(\\bigg|\n\\frac{1}{k}\\big(\\mathrm{ht}(x_1)+\\ldots+\\mathrm{ht}(x_k)\\big)-\n\\int_{\\cT_{2\\ve}}\\mathrm{ht}(x)\\mu_{\\cT_{2\\ve}}(dx)\\bigg|\n\\geq\\varepsilon\n\\bigg)\\leq\\frac{C}{\\varepsilon^2k}.\n\\end{align*}\nThese observations combined with \\eqref{eqn:33} yield\n\\[\\bigg(\\frac{1}{\\sqrt{m}}\\cT_{\\mvs}, \\frac{|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs})|}{s_0\\sqrt{m}}\\bigg)\n\\stackrel{\\mathrm{w}}{\\longrightarrow}\\bigg(\\frac{1}{\\sigma}\\cT_{2\\ve},\\frac{p_0\\sigma}{2}\\int_{\\cT_{2\\ve}}\\mathrm{ht}(x)\\ \\mu_{\\cT_{2\\ve}}(dx)\\bigg),\\]\nwhich is the desired result.  \\hfill $\\blacksquare$\n\n\n\n\\section{Asymptotics for connected graphs with given degree sequence}\n\\label{sec:proof-conn}\nThe aim of this section is to prove Theorems \\ref{prop:condition-on-connectivity} and \\ref{thm:number-of-connected-graphs}.\nWe will start with a construction of the uniform measure on the space of simple connected graphs with a prescribed degree sequence with some fixed number $k$ of surplus edges (Lemma \\ref{lem:alternate-construction}). Using this lemma together with the technical lemma \\ref{lem:plane-trees}, we will then complete the proofs of the above two theorems.\n\n\n\n\\subsection{Construction of connected graphs with given degree sequence}\nLet $\\widetilde{\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}} = \\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(\\widetilde m)}$ be as in Theorem \\ref{prop:condition-on-connectivity}, and recall that $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{\\widetilde{\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}}^{\\con}$ represents a random connected graph with degree sequence $\\widetilde{\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ sampled uniformly from $\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{\\widetilde{\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}}^{\\con}$. Recall also that under Assumption \\ref{ass:degree},  $\\widetilde d_1 = 1$. Consider the remaining vertices $\\set{2,\\ldots, \\widetilde m}$, and form the children sequence $\\mvc = (c_j: 2\\leq j \\leq (\\widetilde m+2k))$ via\n\\begin{equation}\\label{eqn:children}\n\\mvc:=\\big(\\widetilde d_2-1,\\ldots,\\widetilde d_{\\widetilde m}-1, 0,\\ldots,0\\big)\\ \\ \\ (\\text{with } 2k\\text{ zeros at the end}).\n\\end{equation}\nBy the hypothesis of Theorem \\ref{prop:condition-on-connectivity},\n\\begin{align}\\label{eqn:18}\n\\sum_{j=2}^{\\widetilde m+2k} c_j = (\\widetilde m-1+2k) - 1,\n\\end{align}\nand thus $\\mvc$ represents a valid children sequence for a tree on\n\\begin{align}\\label{eqn:def-m}\nm:=\\widetilde m-1+2k\n\\end{align}\nvertices.\nLet $\\mvs=\\mvs^{\\scriptscriptstyle(m)}=(s_0, s_1,\\ldots)$ be the empirical distribution of $\\mvc$, i.e.,\n\\begin{align}\\label{eqn:35}\ns_i=\\#\\big\\{j\\ :\\ c_j=i\\big\\}.\n\\end{align}\n\nSample $(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX)$ from $\\bT_{\\mvs}^{\\scriptscriptstyle(k)}$ uniformly. Assume that $\\widetilde\\mvX=\\{(u_1, v_1),\\ldots, (u_k,v_k)\\}$, where\n\\[(u_1, v_1)\\ll\\ldots\\ll (u_k, v_k).\\]\nLabel $u_j$ as $\\widetilde m+2j-1$ and $v_j$ as $\\widetilde m+2j$, $1\\leq j\\leq k$. Label the other $\\widetilde m-1$ vertices of $\\widetilde\\cT_{\\mvs}$ uniformly using labels $2,\\ldots, \\widetilde m$ so that in the resulting labeled plane tree $j$ has $\\widetilde d_j-1$ many children. (Thus there are $(s_0-2k)!\\times\\prod_{i\\geq 1} s_i!$ many ways of obtaining such a labeling of $\\widetilde\\cT_{\\mvs}$). Call this labeled, plane tree $\\widetilde\\cT_{\\mvs}^\\mathrm{lab}$. Construct the graph $\\cI(\\widetilde\\cT_{\\mvs}^\\mathrm{lab}, \\widetilde\\mvX)$, attach a vertex labeled $1$ to the root, and then forget about the planar order and the root. Let $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}$ be the resulting graph.\n\\begin{lem}\\label{lem:alternate-construction}\nLet $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}$ be the random graph resulting from the above construction. Then  $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}\\sim\\Unif(\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}^{\\con})$, i.e., $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}\\stackrel{\\mathrm{d}}{=}\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}^{\\con}$.\n\\end{lem}\n\\noindent{\\bf Proof:} Fix a graph $G\\in\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{\\widetilde{\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}}^{\\con}$. Root the graph at the only neighbor of $1$ (recall that $\\tilde d_1=1$), and remove the vertex $1$ and the edge incident to it. Suppose $H$ is the resulting rooted, labeled graph. We can construct a labeled plane tree from $H$ in the following way:\n\\begin{enumeratei}\n\\item Call the root $u_1$. Set the status of all its neighbors as ``discovered,'' and set the status of $u_1$ as ``explored.'' Shuffle all its neighbors uniformly and go to the ``leftmost'' neighbor and call it $u_2$.\n\\item When we are at $u_k$ ($k\\ge 2$), search for all its neighbors (other than $u_{k-1}$) in the graph at that time. If none of these neighbors have been discovered previously, then shuffle them uniformly, set their status as ``discovered,'' set the status of $u_k$ as ``explored,'' and go to the leftmost neighbor and call it $u_{k+1}$.\n\n    If some of these neighbors have been previously discovered, then these edges create surplus. Suppose we have found $\\ell_0$ many surplus edges before exploring $u_k$, and at $u_k$ we found $\\ell_1$ many new surplus edges $e_1,\\ldots,e_{\\ell_1}$.\n    Assume that $e_j=(u_k, y_j)$ and $y_1\\prec_{\\mathrm{DF}}\\ldots\\prec_{\\mathrm{DF}}y_{\\ell_1}$. For $j=1,\\ldots,\\ell_1$, delete the edge $e_j$, and create two leaves labeled $\\widetilde m+2\\ell_0+2j-1$ and $\\widetilde m+2\\ell_0+2j$, where $u_k=\\overleftarrow{\\widetilde m+2\\ell_0+2j-1}$ (i.e., $u_k$ is the {\\bf parent} of leaf labeled $\\widetilde m+2\\ell_0+2j-1$) and similarly $y_j=\\overleftarrow{\\widetilde m+2\\ell_0+2j}$.\n    Shuffle the neighbors of $u_k$ uniformly (including the newly created leaves), set their status as ``discovered,'' set the status of $u_k$ as ``explored,'' and move to the leftmost neighbor and call it $u_{k+1}$. (Note that we do {\\bf not} set the status of $\\widetilde m+2\\ell_0+2j$, $j=1,\\ldots,\\ell_1$ as discovered.)\n\n    If $u_k$ has no neighbors other than $u_{k-1}$, then go to the next (in the depth-first order) discovered but unexplored vertex and call it $u_{k+1}$.\n\\end{enumeratei}\nLet $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}^{\\mathrm{lab}}$ be the resulting labeled plane tree and set\n$\\mvx=\\{(\\widetilde m+1,\\widetilde m+2),\\ldots,(\\widetilde m+2k-1,\\widetilde m+2k)\\}$.\nNote that $(\\widetilde m+2\\ell_0+2j-1, \\widetilde m+2\\ell_0+2j)$ is an admissible pair for $1\\leq j\\leq k$. Note also that the children sequence of $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}^{\\mathrm{lab}}$ is always $\\mvc$ defined in \\eqref{eqn:children}. Thus $(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}^\\mathrm{lab},\\mvx)$ is a (labeled) element of $\\bT_{\\mvs}^{(k)}$. Let $\\mathrm{DF}(H)$ be the set of all labeled elements of $\\bT_{\\mvs}^{(k)}$ one can obtain in this way. Then\n\\[\\big|\\mathrm{DF}(H)\\big|=\\prod_{j=2}^{\\widetilde m}(\\tilde d_j-1)!.\\]\n\nNow\n\\[\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big((\\widetilde\\cT_{\\mvs}^\\mathrm{lab}, \\widetilde\\mvX)=(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx)\\big)=\\frac{1}{|\\bT_{\\mvs}^{\\scriptscriptstyle(k)}|}\n\\times\\frac{1}{(s_0-2k)!\\times\\prod_{i\\geq 1} s_i!}\n\\ \\text{ for every }\\ (\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx)\\in\\mathrm{DF}(H).\\]\nThus\n\\begin{align}\\label{eqn:34}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\cI(\\widetilde\\cT_{\\mvs}^\\mathrm{lab}, \\widetilde\\mvX)=H\\big)\n&=\\sum_{(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx)\\in\\mathrm{DF}(H)}\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big((\\widetilde\\cT_{\\mvs}^\\mathrm{lab}, \\widetilde\\mvX)=(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx)\\big)\\notag\\\\\n&=\\frac{\\prod_{j=2}^{\\widetilde m}(\\tilde d_j-1)!}{|\\bT_{\\mvs}^{\\scriptscriptstyle(k)}|}\\times\\frac{1}{(s_0-2k)!\\times\\prod_{i\\geq 1} s_i!}.\n\\end{align}\nSince this probability is constant and the map from $G$ to $H$ is a bijection, we get the desired result. \\hfill $\\blacksquare$\n\\subsection{Proof of Theorem \\ref{prop:condition-on-connectivity}}\nLet $\\mvs=\\mvs^{\\scriptscriptstyle(m)}$ be as in \\eqref{eqn:35}.\nNote that when $\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}^{\\scriptscriptstyle(\\widetilde m)}$ satisfies Assumption \\ref{ass:degree}, $\\mvs^{\\scriptscriptstyle(m)}$ satisfies Assumption \\ref{ass:ecd} with limiting p.m.f. $(p_i : i\\geq 0)$, where\n\\[p_i:=\\widetilde p_{i+1},\\ \\ i=0,1,\\ldots.\\]\nIn view of Lemma \\ref{lem:alternate-construction}, it is enough to prove the result for $\\widetilde m^{-1\/2}\\cI(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX)$. Recall the definition of $\\cQ(\\cdot, \\cdot)$ from Section \\ref{sec:plane-tree-def}. Then\n\\[d_{\\GHP}\\bigg(\\frac{1}{\\sqrt{\\widetilde m}}\\cI(\\widetilde\\cT_{\\mvs},\n\\widetilde\\mvX),\\frac{1}{\\sqrt{\\widetilde m}}\\cQ(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX)\\bigg)\\leq \\frac{5k}{\\sqrt{\\widetilde m}}.\\]\nThus it is enough to prove the the result for the space $m^{-1\/2}\\cQ(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX)$ (recall the definition of $m$ from \\eqref{eqn:def-m}).\n\nRecall the various notions of weak convergence on the space of metric measure spaces defined in Section \\ref{sec:notation}. We will first prove convergence of $m^{-1\/2}\\cQ(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX)$ in the Gromov-weak topology by making use of a technique from \\cite{bhamidi-hofstad-sen} and then strengthen it to convergence in the GHP sense. Let $\\Phi$ and $\\phi$ be as in \\eqref{eqn:polynomial-func-def}. Then\n\\begin{align}\\label{eqn:1}\n\\E\\left(\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX\\big)\\bigg)\\right)\n&=\\frac{\\sum_{(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx)\\in\\bT_{\\mvs}^{\\scriptscriptstyle(k)}}\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx\\big)\\bigg)}{|\\bT_{\\mvs}^{\\scriptscriptstyle(k)}|}\\notag\\\\\n&=\\frac{\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}\\sum_{\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})}\n\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx\\big)\\bigg)\\big\/\\big(|\\bT_{\\mvs}|\\cdot s_0^k m^{k\/2}\\big)}{\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|\\big\/\\big(|\\bT_{\\mvs}|\\cdot s_0^k m^{k\/2}\\big)}\\notag\\\\\n&=\\frac{\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}\\sum_{\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})}\n\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx\\big)\\bigg)\\big\/\\big(|\\bT_{\\mvs}|\\cdot s_0^k m^{k\/2}\\big)}{\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|\\big\/\\big(|\\bT_{\\mvs}|\\cdot s_0^k m^{k\/2}\\big)}.\n\\end{align}\n\nNext,\n\\begin{align}\\label{eqn:4}\n\\frac{\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}\\sum_{\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})}\n\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx\\big)\\bigg)}{|\\bT_{\\mvs}|\\cdot s_0^k m^{k\/2}}\n&=\\E\\bigg[\\sum_{\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\cT_{\\mvs})}\n\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\cT_{\\mvs},\\mvx\\big)\\bigg)\\frac{1}{s_0^km^{k\/2}}\\bigg]\\\\\n&=\\E\\bigg[\\sum_{\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs})^k}\n\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\cT_{\\mvs},\\mvx\\big)\\bigg)\\frac{1}{s_0^km^{k\/2}}\\bigg]+o(1),\\notag\n\\end{align}\nwhere the second equality follows from Lemma \\ref{lem:plane-trees} \\eqref{lem:uniform-integrability} and \\eqref{lem:independent-sampling}. Writing\n\\[\\mvx=\\big((u_1,y_1),\\ldots,(u_k,y_k)\\big), \\\n\\sum\\displaystyle_1=\\sum_{\\substack{u_1\\in\\cL(\\cT_{\\mvs})\\\\ \\vdots\\\\ u_k\\in\\cL(\\cT_{\\mvs})}},\n\\ \\text{ and }\\\n\\sum\\displaystyle_2=\\sum_{\\substack{y_1\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs},u_1)\\\\ \\vdots\\\\ y_k\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs},u_k)}},\n\\]\nwe note that\n\\begin{align}\\label{eqn:5}\n\\frac{1}{s_0^km^{k\/2}}\\sum_{\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}(\\cT_{\\mvs})^k}\n\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\cT_{\\mvs},\\mvx\\big)\\bigg)\n&=\\frac{1}{s_0^km^{k\/2}}\n\\sum\\displaystyle_1\\sum\\displaystyle_2\\ \\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\cT_{\\mvs},\\mvx\\big)\\bigg)\\\\\n&=\n\\sum\\displaystyle_1\\bigg(\\frac{1}{s_0^k}\\bigg)\\prod_{i=1}^k \\bigg(\\frac{f_{\\cT_{\\mvs}}(u_i)}{\\sqrt{m}}\\bigg)\n\\sum\\displaystyle_2  \\prod_{i=1}^k \\bigg(\\frac{1}{f_{\\cT_{\\mvs}}(u_i)}\\bigg)\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\cT_{\\mvs},\\mvx\\big)\\bigg).\\notag\n\\end{align}\nLet $\\mvU=(U_m^{\\scriptscriptstyle(i)}: 1\\leq i\\leq k),\\ \\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X}=(V_m^{\\scriptscriptstyle(j)}: 1\\leq j\\leq \\ell)$, and $\\cT_{\\mvs}(\\mvU,\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X})$ be as in Lemma \\ref{lem:plane-trees}\\eqref{lem:finite-dim-convergence}. We now recall some constructs from \\cite{bhamidi-hofstad-sen}.\nRecall the space $\\vT_{J}^*$ from Section \\ref{sec:space-of-trees}. Let $\\vt$ be an element in $\\vT_{k+\\ell}^*$. Denote its root by $\\rho$ and its leaves by\n\\begin{equation}\n\\label{eqn:leaves-def}\n\\vz_{k,k+\\ell}:= (z_1, z_2, \\ldots,z_{k}, z_{k+1}, \\ldots, z_{k+\\ell}).\n\\end{equation}\nAlso recall that for each $i$, there is a probability measure $\\nu_{\\vt,i }(\\cdot)$ on the path  $[\\rho, z_i]$ for $1\\leq i\\leq k+\\ell$. For $1\\leq i\\leq k$, sample $y_i$ according to the distribution  $\\nu_{\\vt,i}(\\cdot)$ independently for different $i$ and identify $z_i$ with $y_i$. Let $\\vt'$ denote the (random) space thus obtained, and let $d_{\\vt'}$ denote the induced metric on $\\vt'$. Define the function $g^{\\scriptscriptstyle(k)}_\\phi:\\vT_{k+\\ell}^*\\to \\bR$ by\n\\begin{align}\n\\label{eqn:gphi-def}\ng^{\\scriptscriptstyle(k)}_{\\phi}(\\vt):=\n\\left\\{\n\\begin{array}{l}\n\\E\\left[\\phi\\left(d_{\\vt'}(z_i, z_j): k+1\\leq i\\leq k+\\ell\\right)\\right],\\text{ if }\\vt\\neq\\partial,\\\\\n0,\\text{ if }\\vt=\\partial.\n\\end{array}\n\\right.\n\\end{align}\nIn words, we look at the expectation of $\\phi$ applied to the pairwise distances between the last $\\ell$ leaves after sampling $y_i$ on the path $[\\rho, z_i]$ for $1\\leq i\\leq k$ and identifying $z_i$ with $y_i$. Note that here the expectation is only taken over the choices of $y_i$.\n\n\n\nWrite $d_{\\cQ}$ for the induced metric on the space $m^{-1\/2}\\cQ\\big(\\cT_{\\mvs},\\mvx\\big)$,\nand let $\\sum_3=\\sum_{v_1,\\ldots,v_{\\ell}\\in [m]}$. Then\n\\[\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\cT_{\\mvs},\\mvx\\big)\\bigg)\n=\\sum\\displaystyle_3\\frac{1}{m^{\\ell}}\\phi\\bigg(d_{\\cQ}(v_i, v_j): 1\\leq i<j\\leq\\ell\\bigg).\\]\nWriting $\\mvu=(u_1,\\ldots, u_k)$ and $\\mvv=(v_1,\\ldots, v_{\\ell})$, we immediately see that\n\\[\n\\bigg|\\sum\\displaystyle_2\\prod_{i=1}^k\n\\bigg(\\frac{1}{f_{\\cT_{\\mvs}}(u_i)}\\bigg)\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\cT_{\\mvs},\\mvx\\big)\\bigg)\n-\\sum\\displaystyle_3\\frac{1}{m^{\\ell}}\ng_{\\phi}^{\\scriptscriptstyle(k)}\\bigg(\\frac{1}{\\sqrt{m}}\\cT_{\\mvs}(\\mvu,\\mvv)\\bigg)\\bigg|\n\\leq \\|\\phi\\|_{\\infty}\\pr\\big(\\cT_{\\mvs}(\\mvu,\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X})=\\partial\\big|\\ \\cT_{\\mvs}\\big).\n\\]\nSince $\\pr\\big(\\cT_{\\mvs}(\\mvU,\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X})=\\partial\\big)\\to 0$ (by Lemma \\ref{lem:plane-trees}\\eqref{lem:finite-dim-convergence}),\nand $\\prod_{i=1}^k \\big(f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(i)})\/\\sqrt{m}\\big)$ is uniformly integrable (by Lemma \\ref{lem:plane-trees}\\eqref{lem:uniform-integrability}),\ncombining the last display with \\eqref{eqn:4} and \\eqref{eqn:5},\n\\begin{align}\\label{eqn:6}\n&\\frac{\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}\\sum_{\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})}\n\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx\\big)\\bigg)}{|\\bT_{\\mvs}|\\cdot s_0^k m^{k\/2}}\n=\\E\\bigg[\\prod_{i=1}^k \\bigg(\\frac{f_{\\cT_{\\mvs}}(U_m^{\\scriptscriptstyle(i)})}{\\sqrt{m}}\\bigg)\ng_{\\phi}^{\\scriptscriptstyle(k)}\\bigg(\\frac{1}{\\sqrt{m}}\\cT_{\\mvs}(\\mvU,\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X})\\bigg)\\bigg]+o(1).\n\\end{align}\nSince the functional $g_{\\phi}^{\\scriptscriptstyle(k)}$ is continuous on the space $\\vT_{k+\\ell}^*$ \\cite[Proposition 4.25]{bhamidi-hofstad-sen}, combining \\eqref{eqn:6} with Lemma \\ref{lem:plane-trees}\\eqref{lem:finite-dim-convergence} and using uniform integrability\n(Lemma \\ref{lem:plane-trees}\\eqref{lem:uniform-integrability}),\nwe get\n\\begin{align}\\label{eqn:7}\n&\\frac{\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}\\sum_{\\mvx\\in\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})}\n\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v},\\mvx\\big)\\bigg)}{|\\bT_{\\mvs}|\\cdot s_0^k m^{k\/2}}\n\\\\\n&\\hskip40pt\\to\\left(\\frac{p_0\\sigma}{2}\\right)^k\n\\E\\bigg[\\int_{x_1\\in\\cT_{2\\ve}}\\ldots\\int_{x_{k+\\ell}\\in\\cT_{2\\ve}}\\mu_{\\cT_{2\\ve}}^{\\otimes k+\\ell}(dx_1\\ldots dx_{k+\\ell})\\prod_{i=1}^k\\mathrm{ht}(x_i)\\cdot\ng_{\\phi}^{\\scriptscriptstyle(k)}\\bigg(\\frac{1}{\\sigma}\\cT_{k,\\ell}\\bigg)\\bigg].\\notag\n\\end{align}\nTaking $\\Phi\\equiv 1$ in \\eqref{eqn:7}, we get\n\\begin{align}\\label{eqn:3}\n\\frac{\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|}{|\\bT_{\\mvs}|\\cdot s_0^k m^{k\/2}}\n\\to \\left(\\frac{p_0\\sigma}{2}\\right)^k\\E\\bigg[\\bigg(\\int_{\\cT_{2\\ve}}\\height(x)\\ \\mu_{\\cT_{2\\ve}}(dx)\\bigg)^k\\bigg].\n\\end{align}\n\nCombining the above, we conclude that\n\\begin{align*\n\\E\\left[\\Phi\\bigg(\\frac{1}{\\sqrt{m}}\\cQ\\big(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX\\big)\\bigg)\\right]\n&\\to\\frac{\\E\\bigg[\\int_{x_1}\\ldots\\int_{x_{k+\\ell}}\\mu_{\\cT_{2\\ve}}^{\\otimes k+\\ell}(dx_1\\ldots dx_{k+\\ell})\\prod_{i=1}^k\\mathrm{ht}(x_i)\\cdot\ng_{\\phi}^{\\scriptscriptstyle(k)}\\bigg(\\frac{1}{\\sigma}\\cT_{k,\\ell}\\bigg)\\bigg]}{\\E\\bigg[\\bigg(\\int_{\\cT_{2\\ve}}\\mathrm{ht}(x)\\ \\mu_{\\cT_{2\\ve}}(dx)\\bigg)^k\\bigg]},\\\\\n&=\\E\\bigg[\\Phi\\bigg(\\frac{1}{\\sigma}M^{\\scriptscriptstyle(k)}\\bigg)\\bigg].\n\\end{align*}\nThis shows that\n\\begin{align}\\label{eqn:19}\n\\frac{1}{\\sqrt{m}}\\cQ\\big(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX\\big)\n\\stackrel{\\mathrm{w}}{\\longrightarrow}\\frac{1}{\\sigma}M^{\\scriptscriptstyle(k)}\n\\end{align}\nin the Gromov-weak sense.\n\nWe now want to lift it to GHP convergence. To this end, for any metric measure space $(X,d,\\mu)$ and $\\delta>0$, define\n\\[\\kappa_{\\delta}(X)=\\kappa_{\\delta}(X,d,\\mu):=\\inf_{x\\in X}\\bigg\\{\\mu\\big\\{y\\ :\\ d(y,x)\\le\\delta\\big\\}\\bigg\\}.\\]\nFrom the definition of $(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX)$ (given right below \\eqref{eqn:18}), it is clear that\n\\begin{align*}\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\widetilde\\cT_{\\mvs}=\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\big)\n=\\frac{|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|}{|\\bT_{\\mvs}^{\\scriptscriptstyle(k)}|}=\\frac{|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|}{\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}'\\in\\bT_{\\mvs}}|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}')|}\n\\end{align*}\nfor any $\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}$. Hence for any bounded continuous (w.r.t. GHP topology) $h$,\n\\[\\E\\bigg[h\\bigg(\\frac{1}{\\sqrt{m}}\\widetilde\\cT_{\\mvs}\\bigg)\\bigg]\n=\\frac{\\E\\bigg[h\\bigg(\\frac{1}{\\sqrt{m}}\\cT_{\\mvs}\\bigg)\\cdot \\big|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\cT_{\\mvs})\\big|s_0^{-k} m^{-k\/2}\\bigg]}{\\E\\big[|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\cT_{\\mvs})|s_0^{-k}m^{-k\/2}\\big]}.\\]\nUsing Lemma \\ref{lem:plane-trees}\\eqref{lem:independent-sampling}, and Lemma \\ref{lem:plane-trees}\\eqref{lem:conditional-expectation}\ntogether with uniform integrability (\\ref{lem:plane-trees}\\eqref{lem:uniform-integrability}), we conclude that\n\\[\\E\\bigg[h\\bigg(\\frac{1}{\\sqrt{m}}\\widetilde\\cT_{\\mvs}\\bigg)\\bigg]\n\\to\\E\\bigg[h\\bigg(\\frac{1}{\\sigma}\\cT_{2\\widetilde{\\ve}_{(k)}}\\bigg)\\bigg],\n\\]\nwhere $\\widetilde{\\ve}_{(k)}$ is as defined before \\eqref{eqn:tilde-nu-k-def}. Hence $m^{-1\/2}\\widetilde\\cT_{\\mvs}\\stackrel{\\mathrm{w}}{\\longrightarrow}\\sigma^{-1}\\cT_{2\\widetilde{\\ve}_{(k)}}$ in the GHP sense, and in particular, for each $\\delta>0$, $1\/\\kappa_{\\delta}\\big(m^{-1\/2}\\widetilde\\cT_{\\mvs}\\big)$, $m\\geq 1$ is a tight sequence of random variables.\nThis immediately implies that $1\/\\kappa_{\\delta}\\big(m^{-1\/2}\\cQ(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX)\\big)$, $m\\geq 1$ is also a tight sequence of random variables for each $\\delta>0$. Combining this with \\eqref{eqn:19} and \\cite[Theorem 6.1]{athreya-lohr-winter}, we see that\n\\begin{align*}\n\\frac{1}{\\sqrt{m}}\\cQ\\big(\\widetilde\\cT_{\\mvs}, \\widetilde\\mvX\\big)\n\\stackrel{\\mathrm{w}}{\\longrightarrow}\\frac{1}{\\sigma}M^{\\scriptscriptstyle(k)}\n\\end{align*}\nin the GHP sense. This concludes the proof of Theorem \\ref{prop:condition-on-connectivity}.  \\hfill $\\blacksquare$\n\n\\subsection{Proof of Theorem \\ref{thm:number-of-connected-graphs}}\\label{sec:proof-cor-number-of-connected-graphs}\nRecall the relation between $m$ and $\\widetilde m$ from \\eqref{eqn:def-m}.\nNow it follows from \\eqref{eqn:34} that\n\\[\\big|\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}^{\\con}\\big|=\n\\frac{\\big|\\bT_{\\mvs}^{\\scriptscriptstyle(k)}\\big|\\times(s_0-2k)!\\times\\prod_{i\\geq 1} s_i!}{\\prod_{j=1}^{\\widetilde m}(\\widetilde d_j-1)!}.\n\\]\nFurther,\n\\[\n\\big|\\bT_{\\mvs}^{\\scriptscriptstyle(k)}\\big|=\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|\n=\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|\/k!.\n\\]\nIt thus follows that\n\\begin{align*}\n\\big|\\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{\\widetilde\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}^{\\con}\\big|\n&=\\frac{s_0^k m^{k\/2}\\times\\big|\\bT_{\\mvs}\\big|\\times (s_0-2k)!\\times\\prod_{i\\geq 1} s_i!}{\\prod_{j=1}^{\\widetilde m}(\\widetilde d_j-1)!\\times k!}\n\\times\\sum_{\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v}\\in\\bT_{\\mvs}}\\frac{|\\boldsymbol{A}}\\newcommand{\\mvB}{\\boldsymbol{B}}\\newcommand{\\mvC}{\\boldsymbol{C}_k^{\\ord}(\\boldsymbol{t}}\\newcommand{\\mvu}{\\boldsymbol{u}}\\newcommand{\\mvv}{\\boldsymbol{v})|}{\\big|\\bT_{\\mvs}\\big|s_0^k m^{k\/2}}\\\\\n&\\sim \\frac{s_0^k m^{k\/2}\\times(\\widetilde m+2k-2)!\\times (s_0-2k)!}{s_0!\\times\\prod_{j=1}^{\\widetilde m}(\\widetilde d_j-1)!\\times k!}\\times\n\\left(\\frac{p_0\\sigma}{2}\\right)^k\\E\\bigg[\\bigg(\\int_0^1 2\\ve(x)dx\\bigg)^k\\bigg],\n\\end{align*}\nwhere the last step uses \\eqref{eqn:3} and the expression for $|\\bT_{\\mvs}|$ from \\eqref{eqn:pitman}.\nUsing the relations $m\/\\widetilde m\\sim 1$ and $s_0!\/(s_0-2k)!\\sim s_0^k(mp_0)^{k}$, a simple rearrangement of terms completes the proof.\n \\hfill $\\blacksquare$\n\n\n\n\n\n\\section{Proof of Theorems \\ref{thm:vacant-set-scaling} and \\ref{thm:graphs-given-degree-scaling}}\\label{sec:proof-main-deg-vac-thm}\n\nWe start with the distribution of the configuration model.\n\\begin{lem}[\\cite{Hofs13}, Proposition 7.7]\\label{lem:CM-distribution}\nLet $G$ be a multigraph on vertex set $[n]$ in which there are $x_{ij}$ many edges between $i$ and $j$, $1\\leq i<j\\leq n$, and vertex $i$ has $x_{ii}$ many loops. Let\n$\\overline d_i=x_{ii}+\\sum_{j=1}^n x_{ij}$\nbe the total degree of $i$ (note that a loop contributes two to the degree). Let\n\\[\\overline\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}=(\\overline d_1,\\ldots,\\overline d_n),\\ \\text{ and }\\ \\ell_n=\\sum_{i=1}^n\\overline d_i.\\]\nThen\n\\[\\overline\\pr_{n,\\overline\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}\\big(G\\big)=\n\\frac{1}{(\\ell_n-1)!!}\\times\n\\frac{\\prod_{i\\in[n]}\\overline d_i!}{\\prod_{i\\in[n]}2^{x_{ii}}\\prod_{1\\leq i\\leq j\\leq n}x_{ij}!}.\\]\n\\end{lem}\nWe now state two fundamental results about the configuration model and uniform simple graphs with prescribed degree.\n\n\\begin{enumeratea}\n\t\\item From Lemma \\ref{lem:CM-distribution} (see also \\cite{Boll-book,mcdiarmid1998concentration}), it follows that conditional on being simple, the configuration model has the same distribution as $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$, i.e.,\n\t\\begin{equation}\n\t\\label{eqn:cm-conditional-uniform}\n\t\t\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}) \\in \\cdot\\ \\big|\\ \\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}) \\in \\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n, \\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}\\big):= \\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}\\in \\cdot\\big).\n\t\\end{equation}\n    \\item By \\cite[Theorem 1.1]{janson2009probability}, under Assumption \\ref{ass:cm-deg}, there exists a constant $c > 0$ such that the probability that $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$ is simple satisfies\n\t\\begin{equation}\n\t\\label{eqn:cm-simple-0}\n\t\t\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}) \\in \\mathbb{G}}\\newcommand{\\bH}{\\mathbb{H}}\\newcommand{\\bI}{\\mathbb{I}_{n, \\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}\\big) \\to c, \\qquad \\text{ as } n\\to\\infty.\n\t\\end{equation}\n\\end{enumeratea}\n\nThis connection between $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$ and $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ is a very useful tool as it enables one to prove certain results about the uniform simple graph with given degrees by first obtaining a similar result for the configuration model, and then using \\eqref{eqn:cm-conditional-uniform} and \\eqref{eqn:cm-simple-0} to deduce the same for the simple graph.\n\n\nFor any nonnegative random variable $X$ with $\\E X>0$, define the corresponding size biased random variable $X^\\circ$ with distribution given by\n\\[\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(X^\\circ\\leq x\\big)=\\frac{\\E\\big[X\\mathds{1}_{X\\leq x}\\big]}{\\E\\big[X\\big]},\\ \\ x\\in[0,\\infty).\\]\n\n\n\\begin{prop}\\label{prop:CM-facts}\nAssume that $\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}$ satisfies Assumption \\ref{ass:cm-deg} with limiting random variable $D$, and let $D^\\circ$ denote the corresponding size-biased random variable. Let\n\\[p_i^\\circ:=\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(D^\\circ=i\\big)=\\frac{i \\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(D=i\\big)}{\\E\\big[D\\big]},\\ \\ i=1, 2, \\ldots.\\]\n\\begin{enumeratei}\n\\item\nLet $\\cC_{\\scriptscriptstyle(k)}$ be the $k$-th largest component of $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$. Then the following hold for each $k\\geq 1$:\n\\begin{align}\n&\\hskip28pt\\frac{1}{|\\cC_{\\scriptscriptstyle(k)}|}\\sum_{v\\in \\cC_{\\scriptscriptstyle(k)}} d_v^2\\stackrel{\\mathrm{P}}{\\longrightarrow} \\sum_{i\\ge 1}i^2 p_i^{\\circ}<\\infty;\\label{eqn:cm-sum-square} \\\\\n&\\hskip52pt\\pr\\big(\\cC_{\\scriptscriptstyle(k)} \\text{ is simple}\\big)\\to 1;\\label{eqn:cm-simple}\\\\\n&\\frac{1}{|\\cC_{\\scriptscriptstyle(k)}|}\\#\\set{v\\in \\cC_{\\scriptscriptstyle(k)}: d_v=i}\\stackrel{\\mathrm{P}}{\\longrightarrow}  p^{\\circ}_i\\ \\text{ for }\\ i\\ge 1.\\label{eqn:cm-degree}\n\\end{align}\n\\item Further \\eqref{eqn:cm-sum-square}\nand \\eqref{eqn:cm-degree} continue to hold if we replace $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$ by $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$.\n\\end{enumeratei}\n\\end{prop}\n\\noindent {\\bf Proof:}\nGiven a sequence $a_1,\\ldots,a_{\\ell}$ of positive real numbers, the (random) size-biased permutation $\\pi(1),\\ldots, \\pi(\\ell)$ can be obtained as follows:\n\\[\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\pi(1)=i\\big)=\\frac{a_i}{\\sum_{j=1}^{\\ell}a_j},\\ \\text{ and }\\\n\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\pi(k)=i\\ \\big|\\ \\mathcal{S}}\\newcommand{\\cT}{\\mathcal{T}}\\newcommand{\\cU}{\\mathcal{U}_{k-1}\\big)=\\frac{a_i}{\\sum_{j\\notin\\mathcal{S}}\\newcommand{\\cT}{\\mathcal{T}}\\newcommand{\\cU}{\\mathcal{U}_{k-1}}a_j},\\ \\ k=2,\\ldots,\\ell,\\]\nwhere $\\mathcal{S}}\\newcommand{\\cT}{\\mathcal{T}}\\newcommand{\\cU}{\\mathcal{U}_k=\\{\\pi(1),\\ldots,\\pi(k)\\}$.\nIt is a standard fact that the random graph $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$ can be explored in a depth-first way so that the vertices appear as a size-biased permutation, where vertex $i$ has size $d_i$; see \\cite[Section 5.1]{dhara-hofstad-leeuwaarden-sen} or \\cite{riordan2012phase}.\nIt further follows from \\cite[Lemma 15]{dhara-hofstad-leeuwaarden-sen} and Theorem \\ref{thm:cm-component-sizes} that for every $\\varepsilon>0$,\nthere exists $T_{\\varepsilon}>0$ such that\n\\begin{align}\\label{eqn:36}\n\\limsup_n\\ \\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(\\cC_{(k)}\\text{ is explored by time }T_{\\varepsilon}n^{2\/3}\\big)\\geq 1-\\varepsilon.\n\\end{align}\n\\cite[Lemma 5]{dhara-hofstad-leeuwaarden-sen} shows that for every $T>0$,\n\\begin{align}\\label{eqn:37}\n\\sup_{0\\leq u\\leq T}\\bigg|\\frac{1}{n^{2\/3}}\\sum_{i=1}^{\\lfloor un^{2\/3}\\rfloor}d_{\\pi(i)}^2-\\frac{\\sigma_3 u}{\\sigma_1}\\bigg|\\stackrel{\\mathrm{P}}{\\longrightarrow} 0,\n\\end{align}\nwhere $\\sigma_r=\\E[D^r]$, $r=1,2,3$. Combining \\eqref{eqn:36} and \\eqref{eqn:37}, we get\n\\[\\frac{1}{n^{2\/3}}\\bigg(\\sum_{v\\in \\cC_{\\scriptscriptstyle(k)}} d_v^2-\\frac{\\sigma_3}{\\sigma_1}\\big|\\cC_{(k)}\\big|\\bigg)\\stackrel{\\mathrm{P}}{\\longrightarrow} 0.\\]\nSince $\\sigma_3\/\\sigma_1=\\E\\big[{D^\\circ}^2\\big]=\\sum_{i\\geq 1}i^2 p_i^\\circ$, the last display together with Theorem \\ref{thm:cm-component-sizes} yields \\eqref{eqn:cm-sum-square}.\n\n\n\\eqref{eqn:cm-simple} follows from \\cite[Section 5.3]{dhara-hofstad-leeuwaarden-sen} or by following verbatim the proof of this exact result but under slightly different moment assumptions in \\cite[Equation 7.6]{dhara-hofstad-leeuwaarden-sen-2}. \\eqref{eqn:cm-degree} follows from \\cite[Equation 6.4]{dhara-hofstad-leeuwaarden-sen}.\n\nFinally, part (ii) follows from (i) by an application of \\eqref{eqn:cm-conditional-uniform} and \\eqref{eqn:cm-simple-0}.\n \\hfill $\\blacksquare$\n\n\n\n\n\n\n\n\n\n\\vskip12pt\n\n\\noindent{\\bf Proof of Theorem \\ref{thm:graphs-given-degree-scaling}(i).}\\ \\\nNote that $\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}\\big(D=1\\big)>0$ under Assumption \\ref{ass:cm-deg}. Hence $p_1^\\circ>0$.\nFurther, under Assumption \\ref{ass:cm-deg},\n\\[\\sum_{i\\geq 1}i p_i^\\circ=\\E\\big[D^\\circ\\big]=\\E D^2\/\\E D=2.\\]\nHence, by Proposition \\ref{prop:CM-facts} (ii), for every $k\\geq 1$, $(d_v\\ :\\ v\\in\\cC_{(k)})$ satisfies Assumption \\ref{ass:degree} (after a possible relabeling of vertices) with limiting p.m.f. $(p_1^\\circ, p_2^\\circ,\\ldots)$.\n\nLet $\\mathcal{P}}\\newcommand{\\cQ}{\\mathcal{Q}}\\newcommand{\\cR}{\\mathcal{R}$ denote the partition of $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}$ into different components. Then conditional on $\\mathcal{P}}\\newcommand{\\cQ}{\\mathcal{Q}}\\newcommand{\\cR}{\\mathcal{R}$, each component is uniformly distributed over the set of simple, connected graphs with the degrees prescribed by the partition $\\mathcal{P}}\\newcommand{\\cQ}{\\mathcal{Q}}\\newcommand{\\cR}{\\mathcal{R}$. Further, different components are conditionally independent.\nWe thus conclude using Theorem \\ref{thm:cm-component-sizes} and Theorem \\ref{prop:condition-on-connectivity} that for every $k\\geq 1$,\n\\[n^{-2\/3}\\big(|\\cC_{(1)}|,\\ldots,|\\cC_{(k)}|\\big)\n\\stackrel{\\mathrm{w}}{\\longrightarrow}\\big(|\\gamma_{\\scriptscriptstyle(1)}^{\\mvmu_{D}}(\\lambda)|,\\ldots,|\\gamma_{\\scriptscriptstyle(k)}^{\\mvmu_{D}}(\\lambda)|\\big)\\]\njointly with\n\\[\\bigg(\\frac{1}{\\sqrt{|\\cC_{(1)}|}}\\cC_{(1)},\\ldots, \\frac{1}{\\sqrt{|\\cC_{(k)}|}}\\cC_{(k)}\\bigg)\n\\stackrel{\\mathrm{w}}{\\longrightarrow}\\frac{\\alpha_D}{\\sqrt{\\eta_D}}\\big(S_1,\\ldots,S_k\\big)\\]\nin the GHP sense, where $S_i$ are as in Construction \\ref{constr:M-D}, and $\\eta_D$, and $\\alpha_D$ are as in Theorem \\ref{thm:cm-component-sizes}. (Here we have used the fact $\\sum_{i\\geq 1}i^2 p_i^\\circ-4=\\eta_D\/\\alpha_D^2$.)\nCombining the two yields the result.\n \\hfill $\\blacksquare$\n\n\n\\vskip12pt\n\n\\noindent{\\bf Proof of Theorem \\ref{thm:graphs-given-degree-scaling}(ii).}\\ \\\nBy Proposition \\ref{prop:CM-facts} (i), for every $k\\geq 1$, $(d_v\\ :\\ v\\in\\cC_{(k)})$ satisfies Assumption \\ref{ass:degree} (after a possible relabeling of vertices) with limiting p.m.f. $(p_1^\\circ, p_2^\\circ,\\ldots)$.\nAs before, let $\\mathcal{P}}\\newcommand{\\cQ}{\\mathcal{Q}}\\newcommand{\\cR}{\\mathcal{R}$ denote the partition of $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f})$ into different components. For each $k\\geq 1$, define the event\n\\[E_k:=\\big\\{\\cC_{(j)}\\ \\text{ is simple for all }\\ 1\\leq j\\leq k\\big\\}.\\]\nThen note that by Lemma \\ref{lem:CM-distribution}, conditional on the event $E_k\\cap\\{\\mathcal{P}}\\newcommand{\\cQ}{\\mathcal{Q}}\\newcommand{\\cR}{\\mathcal{R}=P\\}$, $\\cC_{(j)}$, $j\\geq 1$ are independent, and for each $i\\leq k$, $\\cC_{(i)}$ is uniformly distributed over the set of simple, connected graphs with the degrees prescribed by the partition $P$. Since $\\mathbb{P}}\\newcommand{\\bQ}{\\mathbb{Q}}\\newcommand{\\bR}{\\mathbb{R}(E_k^c)\\to 0$ by \\eqref{eqn:cm-simple}, the result follows by imitating the argument used in the proof of Theorem \\ref{thm:graphs-given-degree-scaling}(i). \\hfill $\\blacksquare$\n\n\n\n\n\n\\vskip12pt\n\n\n\\noindent{\\bf Proof of Theorem \\ref{thm:vacant-set-scaling}(ii).}\\ \\\nFor every $u\\geq 0$, let $\\mathcal{V}}\\newcommand{\\cW}{\\mathcal{W}}\\newcommand{\\cX}{\\mathcal{X}^u$ denote the vacant set left by a random walk on $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)})$ run up to time $nu$.\nLet $\\cE^u$ be the set of all edges of $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)})$ both of whose endpoints are in $\\mathcal{V}}\\newcommand{\\cW}{\\mathcal{W}}\\newcommand{\\cX}{\\mathcal{X}^u$, i.e.,\n\\[\\cE^u:=\\big\\{\\{v_1,v_2\\}\\in\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)}) :\\ v_1, v_2\\in\\mathcal{V}}\\newcommand{\\cW}{\\mathcal{W}}\\newcommand{\\cX}{\\mathcal{X}^u\\big\\}.\\]\nDefine the vacant graph $\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X}^u$ by $\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X}^u:=([n],\\cE^u)$, and let $\\vD^u:=\\big(\\vD^u(j)\\ :\\ j\\in[n]\\big)$ be the degree sequence of $\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X}^u$.\nThen, by \\cite[Proposition 3.1]{cerny-teixeira}, for any collection $A$ of multigraphs on $[n]$,\n\\begin{align}\\label{eqn:38}\n\\overline\\vP_{n,r}\\big(\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X}^u\\in A\\big)\n=\\sum_{\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}\\overline\\vP_{n,r}\\big(\\vD^u=\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}\\big)\\times\\overline\\pr_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}\\big(A\\big).\n\\end{align}\nIn words, the vacant graph $\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X}^u$ can be generated in two steps: (i) sample the degree sequence $\\vD^u$ under the annealed measure $\\overline\\vP_{n,r}$, and then (ii) construct a configuration model with this degree sequence.\n\nLet $\\mvs^{u}=\\big(s_0^{u},\\ldots, s_r^{u}\\big)$ denote the empirical distribution of $\\vD^{u}$. Then, by \\cite[Equation 6.1]{cerny-teixeira},\nfor every $\\varepsilon>0$,\n\\begin{align*}\n\\overline\\vP_{n,r}\\bigg(\\bigg|\\frac{1}{n}s_i^{u_{\\star}}-\\pr\\big(D_{\\mathrm{vac}}=i\\big)\\bigg|\\geq\\varepsilon\\bigg)\n\\to 0,\\ \\ 0\\leq i\\leq r,\n\\end{align*}\nwhere $u_{\\star}$ is as in \\eqref{eqn:ustar-def}, and $D_{\\mathrm{vac}}$ is as in \\eqref{eqn:D-vac-def}. The simple observation\n\\[\\big|s_i^{u_{\\star}}-s_i^{u_n}\\big|\\leq (|a_0|+1)n^{2\/3}(r+1)\\]\nfor large $n$ when $u_n$ satisfies \\eqref{eqn:40} leads to\n\\begin{align}\\label{eqn:39}\n\\overline\\vP_{n,r}\\bigg(\\bigg|\\frac{1}{n}s_i^{u_n}-\\pr\\big(D_{\\mathrm{vac}}=i\\big)\\bigg|\\geq\\varepsilon\\bigg)\n\\to 0,\\ \\ 0\\leq i\\leq r.\n\\end{align}\nFurther, by \\cite[Equation 6.4]{cerny-teixeira},\n\\begin{align}\\label{eqn:41}\n\\overline\\vP_{n,r}\\bigg(n^{1\/3}\n\\bigg|\\frac{\\sum_{i=0}^r (i^2-2i)s_i^{u_n}}{\\sum_{i=0}^r i s_i^{u_n}}\n-\\lambda_{\\mathrm{vac}}\\bigg|\\geq\\varepsilon\\bigg)\\to 0,\n\\end{align}\nwhere $\\lambda_{\\mathrm{vac}}$ is as in \\eqref{eqn:lambda-vac-def}. Combining \\eqref{eqn:39} and \\eqref{eqn:41}, we see that\nthe degree sequence $\\vD^{u_n}$ satisfies Assumption \\ref{ass:cm-deg} with limiting random variable $D_{\\mathrm{vac}}$ and $\\lambda=\\lambda_{\\mathrm{vac}}$.\nIn view of \\eqref{eqn:38}, an application of Theorem \\ref{thm:graphs-given-degree-scaling}(ii) completes the proof. \\hfill $\\blacksquare$\n\n\\vskip12pt\n\n\n\n\\noindent{\\bf Proof of Theorem \\ref{thm:vacant-set-scaling}(i).}\\ \\\nLet $\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X}^u,\\ \\vD^u$, and $\\mvs^u$ be as in the proof of Theorem \\ref{thm:vacant-set-scaling}(ii), but with $\\mathcal{G}}\\newcommand{\\cH}{\\mathcal{H}}\\newcommand{\\cI}{\\mathcal{I}_{n,r}$ as the underlying graph (instead of $\\CM_n(\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}_r^{\\scriptscriptstyle(n)})$). By \\cite[Lemma 7]{cooper-frieze}, the analogue of \\eqref{eqn:38} is true in this case, i.e.,\n\\[\\vP_{n,r}\\big(\\boldsymbol{V}}\\newcommand{\\mvW}{\\boldsymbol{W}}\\newcommand{\\mvX}{\\boldsymbol{X}^u\\in A\\big)\n=\\sum_{\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}\\vP_{n,r}\\big(\\vD^u=\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}\\big)\\times\\pr_{n,\\mathbf{d}}\\newcommand{\\ve}{\\mathbf{e}}\\newcommand{\\vf}{\\mathbf{f}}\\big(A\\big),\\]\nfor any collection $A$ of simple graphs on $[n]$. Further, using \\eqref{eqn:cm-conditional-uniform} and \\eqref{eqn:cm-simple-0},\nwe conclude that  \\eqref{eqn:39} and \\eqref{eqn:41} continue to hold in this case. We complete the proof by an application of Theorem \\ref{thm:graphs-given-degree-scaling}(i). \\hfill $\\blacksquare$\n\n\\section*{Acknowledgements}\nThe authors thank Christina Goldschmidt and Remco van der Hofstad for many helpful discussions, and Bal\\'{a}zs R\\'{a}th for useful comments on a preliminary version of the paper.\nSB has been partially supported by NSF-DMS grants 1105581, 1310002, 160683, 161307 and SES grant 1357622.\nSS has been supported in part by EPSRC grant EP\/J019496\/1, a CRM-ISM fellowship, and the Netherlands Organization for Scientific Research (NWO) through the Gravitation Networks grant 024.002.003.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sect:intro}\n\nThe aim for theoretical precision predictions for the LHC requires next-to-next-to-leading order (NNLO) calculations for\na number of processes.\nIf one goes beyond the simplest $2 \\rightarrow 2$-processes,\nconsidering a $2 \\rightarrow (n-2)$-process with possibly $n>4$, one is in particular interested\nin methods which allow for automation.\nNumerical methods \nlike numerical loop integration \\cite{Soper:1998ye,Soper:1999xk,Nagy:2003qn,Gong:2008ww,Assadsolimani:2009cz,Assadsolimani:2010ka,Becker:2010ng,Becker:2011vg,Becker:2012aq,Becker:2012nk,Becker:2012bi,Goetz:2014lla,Seth:2016hmv}\ncombined with loop-tree duality \\cite{Catani:2008xa,Bierenbaum:2010cy,Bierenbaum:2012th,Buchta:2014dfa,Hernandez-Pinto:2015ysa,Buchta:2015wna,Sborlini:2016gbr,Driencourt-Mangin:2017gop,Driencourt-Mangin:2019aix,Runkel:2019yrs}\nor methods based on numerical unitarity \\cite{Ita:2015tya,Abreu:2017idw,Abreu:2017xsl,Abreu:2017hqn,Abreu:2018jgq}\nare a promising path for this approach.\n\nStarting from two-loops, there are Feynman integrals with higher powers of the propagators.\nThey arise from self-energy insertions on internal lines.\nAn example is shown in fig.~\\ref{fig_1}.\nNote that the contributions we are concerned about are not an artefact of a gauge choice.\nFor gauge theories, we will use Feynman gauge throughout this paper.\nIn Feynman gauge the Feynman propagator has just simple poles.\nIn an analytic calculation Feynman integrals with higher powers of the propagators are not a problem.\nThey are reduced by integration-by-parts identities to master integrals.\nThe master integrals are then calculated analytically.\nIt is possible that the set of master integrals itself contains integrals with raised propagators.\n\nThe situation is different for numerical approaches.\nIn this paper we focus on numerical loop integration \\cite{Soper:1998ye,Soper:1999xk,Nagy:2003qn,Gong:2008ww,Assadsolimani:2009cz,Assadsolimani:2010ka,Becker:2010ng,Becker:2011vg,Becker:2012aq,Becker:2012nk,Becker:2012bi,Goetz:2014lla,Seth:2016hmv}\nin combination with loop-tree duality \\cite{Catani:2008xa,Bierenbaum:2010cy,Bierenbaum:2012th,Buchta:2014dfa,Hernandez-Pinto:2015ysa,Buchta:2015wna,Sborlini:2016gbr,Driencourt-Mangin:2017gop,Driencourt-Mangin:2019aix,Runkel:2019yrs}.\nLet us mention that our results have also implications for methods\nbased on numerical unitarity \\cite{Ita:2015tya,Abreu:2017idw,Abreu:2017xsl,Abreu:2017hqn,Abreu:2018jgq}.\nRaised propagators have been considered previously in\n\\cite{Bierenbaum:2012th,Sogaard:2014ila,Abreu:2017idw}.\nWithin these numerical approaches one is interested in\nthe residue when a raised propagator goes on-shell.\nIf $f(z)$ is a function of a complex variable $z$, which has a pole of order $\\nu$ at $z_0$, the standard formula for the \nresidue at $z_0$ is given\nby\n\\begin{eqnarray}\n\\label{residue_example_one_dim}\n \\mathrm{res}\\left(f,z_0\\right)\n & = &\n \\frac{1}{\\left(\\nu-1\\right)!}\n \\left.\n \\left(\\frac{d}{dz}\\right)^{\\nu-1}\n \\left[ \\left(z-z_0\\right)^\\nu f\\left(z\\right) \\right]\n \\right|_{z=z_0}.\n\\end{eqnarray}\nWe may think of the variable $z$ as being the energy flowing through the raised propagator.\nFor $\\nu > 1$ we have a derivative acting on all $z$-dependent quantities in the diagram.\nAlthough this can be done, it is process-dependent and not very well suited for automation.\n(Eq.~(\\ref{residue_example_one_dim}) is a simple univariate example, the generalisation to the multivariate case\nis discussed in ref.~\\cite{Sogaard:2014ila}. The computation of the residue in the multivariate case with higher powers of the propagators\nis based on Gr\\\"obner bases.)\nAlternatively, ref.~\\cite{Bierenbaum:2012th} proposes to reduce Feynman integrals with raised propagators through integration-by-parts\nidentities to Feynman integrals without raised propagators.\nThis is possible, but again it is process-dependent and therefore not very well suited for automation.\n\nAlthough our focus is on the loop-tree duality method, where we cut an $l$-loop contribution exactly $l$-times,\nlet us briefly comment on the numerical unitarity method.\nHere, one writes the $l$-loop amplitude as a linear combination of (known) master integrals with (unknown) coefficients.\nThe coefficients are determined by cutting the internal propagators, starting with the maximal cut and working down\nthe hierarchy.\nThe method exploits the fact that the integrand of a loop amplitude factorises on leading poles into products of tree amplitudes.\nHowever, if higher powers of the propagators are present, one needs in addition to the coefficient of the leading poles\nalso the coefficients of the subleading poles, where the above mentioned factorisation property no longer holds.\nRef.~\\cite{Abreu:2017idw} presents a numerical method to extract the coefficients of the subleading poles by considering equations obtained from cutting less propagators.\n\nLet us now return to the loop-tree duality method.\nWe would like to isolate the complication into a small process-independent part.\nIf we only look at the left diagram of fig.~\\ref{fig_1} there is nothing we can do.\nHowever, we may look at the set of all diagrams corresponding to a self-energy insertion on a specific internal line.\nAt two-loops and in $\\phi^3$-theory there are two diagrams, as shown in fig.~\\ref{fig_1}:\nThe left diagram of fig.~\\ref{fig_1}, which we already discussed, and the right diagram of fig.~\\ref{fig_1}, corresponding\nto the counterterm from renormalisation.\nIn the on-shell scheme the counterterm is basically the Taylor expansion to second order around the on-shell value of the self-energy.\nThus, if we would perform the one-loop calculation of the self-energy analytically and combine it with the counterterm, \nwe would obtain a transcendental function, which vanishes quadratically in the on-shell limit.\nThis will cancel the double pole and the residue will vanish.\nThis is fine, but has the drawback that we introduced transcendental functions from an analytic one-loop calculation.\nWe would like to work entirely with rational functions, as we do in the numerical approach.\nIt is therefore natural to ask, if there exists an integral representation for the counterterm, such that the residue\nvanishes already at the integrand level.\nThis is the topic of this paper and we show that such an integral representation for the counterterm exists.\nSuch integral representations are not unique.\nThere is quite some freedom to construct an integral representation, only the integral, the UV-behaviour and the on-shell behaviour \nis fixed by the requirement that the counterterm should be a proper counterterm, local in loop momentum space \nand leading to a vanishing residue.\nA sufficient condition for the last condition to hold is that the sum of the integrands for the self-energy vanishes quadratically as\nthe external momentum of the self-energy goes on-shell.\nThus\n\\begin{eqnarray}\n\\label{eq_condition}\n \\lim\\limits_{k^2\\rightarrow m^2} \\left(\\mbox{Self-energy integrand}\\right)\n & = &\n {\\mathcal O}\\left( \\left(E-E^\\flat\\right)^2 \\right),\n\\end{eqnarray}\nwhere $E^\\flat$ denotes the on-shell value of the energy flowing through the raised propagator, \nThis condition will cancel the double pole (and the single pole)\nfrom the propagators, resulting in a vanishing residue.\nFor gauge theories it is sufficient to require eq.~(\\ref{eq_condition}) only up to gauge terms, which vanish when contracted\ninto gauge-invariant quantities.\n\nIn this paper we construct counterterms with the property given in eq.~(\\ref{eq_condition}).\nThus, the main result of this paper is that when summed over all relevant diagrams (including counterterms from renormalisation)\nresidues due to higher poles from self-energy insertions on internal lines can be made to vanish at the integrand level.\n\nLet us mention that the counterterms we construct have higher powers of the propagators \nin the self-energy parts.\nAt first sight, this may seem like nothing has been gained: \nWe removed higher powers of the propagators in one part,\nbut introduced new higher powers of the propagators in another part.\nThe essential point is that we removed the higher powers of the propagators from the process-dependent part\nand isolated the higher powers of the propagators in a universal process-independent part.\nThe derivatives for the residues may therefore be calculated once and for all.\n\nOne final remark: Although the explicit results for the counterterms\nfor $\\phi^3$-theory and QCD presented in this paper are for stable particles, where the masses are real,\nthis assumption is not essential.\nWe may allow complex masses.\nWe only require that the renormalised propagator has a pole at the renormalised mass with residue $1$.\nOur method has a straightforward extension towards the complex mass scheme \\cite{Denner:2005fg}.\n\nThis paper is organised as follows:\nIn the next section we consider a simple toy example from complex analysis.\nIn section~\\ref{sect:scalar_theory} we present our argument in detail for the case of a scalar $\\phi^3$-theory.\nAll essential features are already in there.\nIn section~\\ref{sect:QCD} we specialise to the case of quantum chromodynamics, treating spin $1\/2$-fermions and massless spin $1$-gauge bosons.\nFinally, our conclusions are contained in section~\\ref{sect:conclusions}.\nAppendix~\\ref{sect:Feynman_rules} lists the Feynman rules for the scalar $\\phi^3$-theory.\n\n\\section{A toy example}\n\\label{sect:toy}\n\nLet us first look at a toy example and consider the polynomials\n\\begin{eqnarray}\n f_2 \\; = \\; z_2,\n \\;\\;\\;\\;\\;\\;\n f_1 \\; = \\; z_1 + \\frac{1}{2} z_2 + 1,\n \\;\\;\\;\\;\\;\\;\n f_6 \\; = \\; z_1 - \\frac{1}{2} z_2 - 1\n\\end{eqnarray}\nin two complex variables $z_1$ and $z_2$.\nWe are interested in the local residues (i.e. two-fold residues in $z_1$ and $z_2$) of the rational function\n\\begin{eqnarray}\n R \n & = &\n \\frac{1}{f_2^2 f_1 f_6}.\n\\end{eqnarray}\nThe local residues are at\n\\begin{eqnarray}\n \\left(z_1,z_2\\right)\n & \\in &\n \\left\\{\n \\; \\left(-1,0\\right),\n \\; \\left(1,0\\right),\n \\; \\left(0,-2\\right) \\;\n \\right\\}.\n\\end{eqnarray}\nThe location of the residues is shown in the left drawing of fig.~\\ref{fig_toy}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.0]{fig_toy}\n\\hspace*{20mm}\n\\includegraphics[scale=1.0]{fig_toy_CT}\n\\end{center}\n\\caption{\nThe left figure shows the location of the residues in the $(z_1,z_2)$-plane of the rational function $R$,\nthe right figure shows the location of the residues of the rational function $R_{\\mathrm{CT}}$.\n}\n\\label{fig_toy}\n\\end{figure}\nWe are in particular interested in the local residues at $P_1=(-1,0)$ and $P_2=(1,0)$, where we have a double pole from $f_2^2$. We have\n\\begin{eqnarray}\n \\mathrm{res}\\left(R,P_1\\right)\n \\; = \\;\n \\frac{1}{4},\n & &\n \\mathrm{res}\\left(R,P_2\\right)\n \\; = \\;\n - \\frac{1}{4}.\n\\end{eqnarray}\nLet us now define\n\\begin{eqnarray}\n f_1^\\flat \\; = \\; z_1 + 1,\n \\;\\;\\;\\;\\;\\;\n f_6^\\flat \\; = \\; z_1 - 1\n\\end{eqnarray}\nand consider rational functions with poles only along $f_2$, $f_1^\\flat$ and $f_6^\\flat$, i.e. rational functions\nof the form\n\\begin{eqnarray}\n \\frac{P\\left(z_1,z_2\\right)}{f_2^{\\nu_2} \\left(f_1^\\flat\\right)^{\\nu_1} \\left(f_6^\\flat\\right)^{\\nu_6}},\n\\end{eqnarray}\nwith $\\nu_1,\\nu_2,\\nu_6 \\in {\\mathbb N}$ and $P(z_1,z_2)$ a polynomial in $z_1$ and $z_2$.\nThese functions have local residues only at the two points $P_1=(-1,0)$ and $P_2=(1,0)$\n(this is shown in the right picture of fig.~\\ref{fig_toy}), and we are interested in a function\n$R_{\\mathrm{CT}}$ which cancels the residues of $R$ at $P_1$ and $P_2$.\nLet us first note that the function\n\\begin{eqnarray}\n R_{\\mathrm{try}}\n & = &\n \\frac{1}{f_2^2 f_1^\\flat f_6^\\flat}\n \\;\\; = \\;\\;\n \\frac{1}{z_2^2 \\left(z_1+1\\right) \\left(z_1-1\\right)}\n\\end{eqnarray}\nhas no residues at $P_1$ or $P_2$:\n\\begin{eqnarray}\n \\mathrm{res}\\left(R_{\\mathrm{try}},P_1\\right)\n \\; = \\;\n 0,\n & &\n \\mathrm{res}\\left(R_{\\mathrm{try}},P_2\\right)\n \\; = \\;\n 0,\n\\end{eqnarray}\nsince $R_{\\mathrm{try}}$ does not have a single pole in $z_2$.\nHowever, expanding $R$ to second order in $z_2$ does the job:\n\\begin{eqnarray}\n R_{\\mathrm{CT}}\n & = &\n -\n \\frac{1}{f_2^2 f_1^\\flat f_6^\\flat}\n \\left( 1 - \\frac{z_2}{2 f_1^\\flat} + \\frac{z_2}{2 f_6^\\flat} \\right)\n\\end{eqnarray}\nWe have\n\\begin{eqnarray}\n \\mathrm{res}\\left(R_{\\mathrm{CT}},P_1\\right)\n \\; = \\;\n - \\frac{1}{4},\n & &\n \\mathrm{res}\\left(R_{\\mathrm{CT}},P_2\\right)\n \\; = \\;\n \\frac{1}{4}.\n\\end{eqnarray}\nThus\n\\begin{eqnarray}\n \\mathrm{res}\\left(R+R_{\\mathrm{CT}},P_1\\right)\n \\; = \\; \n \\mathrm{res}\\left(R+R_{\\mathrm{CT}},P_2\\right)\n & = & 0,\n\\end{eqnarray}\nand the residues at $P_1$ or $P_2$ cancel in the sum.\n\nThe analogy with quantum field theory is as follows: We may think of $z_1$ and $z_2$ as two energy\nvariables, $f_1$, $f_2$ and $f_6$ as propagators and of $f_1^\\flat$ and $f_6^\\flat$ as the on-shell projections\nof $f_1$ and $f_6$, respectively, as $f_2$ goes on-shell.\n\n\n\\section{The method for a scalar theory}\n\\label{sect:scalar_theory}\n\nLet us now discuss a simple quantum field theory.\nWe consider a massive $\\phi^3$-theory.\nThe Lagrangian in renormalised quantities is given by\n\\begin{eqnarray}\n {\\mathcal L} \n & = &\n \\frac{1}{2} \\left( \\partial_\\mu \\phi \\right) \\left( \\partial^\\mu \\phi \\right)\n - \\frac{1}{2} m^2 \\phi^2\n + \\frac{1}{3!} \\lambda^{(D)} \\phi^3\n +  {\\mathcal L}_{\\mathrm{CT}}.\n\\end{eqnarray}\nUnder renormalisation we have\n\\begin{eqnarray}\n \\phi_0 = Z_\\phi^{\\frac{1}{2}} \\phi,\n \\;\\;\\;\\;\\;\\;\n \\lambda_0 = Z_\\lambda \\; \\lambda^{(D)},\n \\;\\;\\;\\;\\;\\;\n m_0 = Z_m m,\n\\end{eqnarray}\nwhere we denote bare quantities with a subscript ``0''.\nWe work in dimensional regularisation and set $D=4-2\\varepsilon$.\nWe further set\n\\begin{eqnarray}\n\\label{def_lambda_D}\n \\lambda^{(D)}\n & = &\n \\mu^\\varepsilon S_\\varepsilon^{-\\frac{1}{2}} \\lambda.\n\\end{eqnarray}\nThe arbitrary scale $\\mu$ is introduced to keep the mass dimension of the renormalised coupling $\\lambda$ equal to one.\nThe factor $S_\\varepsilon = (4\\pi)^\\varepsilon \\exp(-\\varepsilon \\gamma_E)$ absorbs artefacts of dimensional regularisation \n(logarithms of $4\\pi$ and Euler's constant $\\gamma_E$).\nThe Lagrangian for the counterterms is given by\n\\begin{eqnarray}\n {\\mathcal L}_{\\mathrm{CT}}\n & = &\n - \\frac{1}{2} \\left(Z_\\phi-1\\right) \\phi \\Box \\phi \n - \\frac{1}{2} \\left(Z_\\phi Z_m^2 -1 \\right) m^2 \\phi^2\n + \\frac{1}{3!} \\left(Z_\\phi^{\\frac{3}{2}} Z_\\lambda - 1 \\right) \\lambda^{(D)} \\phi^3.\n\\end{eqnarray}\nThe Feynman rules for the scalar $\\phi^3$-theory are listed in appendix~\\ref{sect:Feynman_rules}.\nFor the perturbative expansion of the renormalisation constants we write\n\\begin{eqnarray}\n Z_a\n & = &\n 1 + \\sum_{n=1}^\\infty Z_a^{(n)} \\left( \\frac{\\lambda^2}{\\left(4\\pi\\right)^2} \\right)^n,\n \\;\\;\\;\\;\\;\\;\n a \\in \\{ \\phi, m, \\lambda \\}.\n\\end{eqnarray}\nWe will need $Z_m^{(1)}$ and $Z_\\phi^{(1)}$.\nIn the on-shell scheme these renormalisation constants are given by\n\\begin{eqnarray}\n\\label{renormalisation_constants_phi_cubed}\n Z_m^{(1)}\n & = & \n \\frac{1}{4m^2} B_0\\left(m^2,m^2,m^2\\right),\n \\nonumber \\\\\n Z_\\phi^{(1)}\n & = & \n \\frac{2-\\varepsilon}{6m^2} B_0\\left(m^2,m^2,m^2\\right) - \\frac{1-\\varepsilon}{3m^4} A_0\\left(m^2\\right).\n\\end{eqnarray}\nThe scalar one-loop integrals $A_0$ and $B_0$ are defined by\n\\begin{eqnarray}\n A_0\\left(m^2\\right)\n & = &\n 16 \\pi^2\n  S_\\varepsilon^{-1} \\mu^{2\\varepsilon} \n \\int \\frac{d^Dk}{(2 \\pi)^D i } \\; \n \\frac{1}{k^2-m^2},\n \\nonumber \\\\\n B_0\\left(p^2,m_1^2,m_2^2\\right)\n & = &\n 16 \\pi^2\n  S_\\varepsilon^{-1} \\mu^{2\\varepsilon} \n \\int \\frac{d^Dk}{(2 \\pi)^D i } \\; \n \\frac{1}{\\left[\\left(k+\\frac{1}{2}p\\right)^2-m_1^2\\right]\\left[\\left(k-\\frac{1}{2}p\\right)^2-m_2^2\\right]}.\n\\end{eqnarray}\nIn this paper we are concerned with diagrams like the one shown in the left picture of fig.~\\ref{fig_1}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.0]{fig_oneloop}\n\\hspace*{20mm}\n\\includegraphics[scale=1.0]{fig_CT}\n\\end{center}\n\\caption{\nThe left figure shows a self-energy insertion on an internal line.\nThe same momentum is flowing through the two red lines, resulting in a propagator raised to power two.\nA self-energy insertion on an internal line is always accompanied by a counterterm, shown in the right figure.\n}\n\\label{fig_1}\n\\end{figure}\nIn fig.~\\ref{fig_2} we show our choice for the labelling of the propagators and the orientation of the momenta.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.0]{fig_labelled_propagators}\n\\hspace*{20mm}\n\\includegraphics[scale=1.0]{fig_labelled_momenta}\n\\end{center}\n\\caption{\nThe labelling of the propagators (left figure) and the labelling of the momenta (right figure).\n}\n\\label{fig_2}\n\\end{figure}\nWith\n\\begin{eqnarray}\n D_j & = & k_j^2 - m^2 + i \\delta\n\\end{eqnarray}\nwe have for this diagram\n\\begin{eqnarray}\n I_{\\mathrm{twoloop}}\n & = &\n \\frac{i \\lambda^6}{2}\n \\mu^{4\\varepsilon} S_\\varepsilon^{-2}\n \\int \\frac{d^Dk_1}{\\left(2\\pi\\right)^D}\n \\int \\frac{d^Dk_2}{\\left(2\\pi\\right)^D}\n \\frac{1}{D_1 D_2^2 D_3 D_4 D_5 D_6},\n\\end{eqnarray}\nwhere we ignored a prefactor $\\mu^{2\\varepsilon} S_\\varepsilon^{-1}$, accompanying also the Born amplitude.\nWe see that $D_2$ is raised to the power two.\nWithin the loop-tree duality method we take residues in the energy integrations $E_1$ and $E_2$.\nThe residues are classified by the set of spanning trees for our diagram.\nWe may denote a spanning tree by the propagators we remove to get a tree diagram.\nThe set of spanning trees for our two-loop diagram is given by\n\\begin{eqnarray}\n \\left\\{\n \\;\n  \\left(1,2\\right),\n \\;\n  \\left(1,3\\right),\n \\;\n  \\left(1,4\\right),\n \\;\n  \\left(1,5\\right),\n \\;\n  \\left(1,6\\right),\n \\;\n  \\left(2,6\\right),\n \\;\n  \\left(3,6\\right),\n \\;\n  \\left(4,6\\right),\n \\;\n  \\left(5,6\\right)\n \\;\n \\right\\}.\n\\end{eqnarray}\nEach spanning tree defines also a cut graph.\nFor a cut graph, we don't remove internal edges but cut them into half-edges.\nThe half-edges become additional external lines of the cut graph.\nIn fig.~\\ref{fig_3} we show a few examples of cut graphs obtained from spanning trees. \nProblematic are the cuts $(1,2)$ and $(2,6)$.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.0]{fig_cut_13}\n\\hspace*{20mm}\n\\includegraphics[scale=1.0]{fig_cut_12}\n\\hspace*{20mm}\n\\includegraphics[scale=1.0]{fig_cut_16}\n\\end{center}\n\\caption{\nVarious cuts of the two-loop diagram, which correspond to spanning trees.\nThe cut $(1,3)$ (left diagram) is unproblematic.\nThe cut $(1,2)$ (middle diagram) requires the residue of a doubled propagator.\nThe right diagram shows the cut $(1,6)$.\n}\n\\label{fig_3}\n\\end{figure}\nAs $D_2$ occurs quadratically, taking the residue for $D_2=0$ forces us to compute a derivative.\n\nIn this paper we would like to point out, that the left diagram of fig.~\\ref{fig_1} always comes \nin combination with a counterterm, shown in the right picture of fig.~\\ref{fig_1}.\nThe contribution from the counterterm is \n\\begin{eqnarray}\n I_{\\mathrm{twoloop},\\mathrm{CT}}\n & = &\n -\n \\frac{\\lambda^6}{\\left(4\\pi\\right)^2}\n \\mu^{2\\varepsilon} S_\\varepsilon^{-1}\n \\int \\frac{d^Dk_2}{\\left(2\\pi\\right)^D}\n \\frac{\\left[Z_\\phi^{(1)} k_2^2 - \\left(Z_\\phi^{(1)} + 2 Z_m^{(1)}\\right) m^2 \\right]}{D_2^2 D_3 D_4 D_5}.\n\\end{eqnarray}\nLet us write\n\\begin{eqnarray}\n I_{\\mathrm{twoloop}}\n & = &\n i \\lambda^6\n \\mu^{4\\varepsilon} S_\\varepsilon^{-2}\n \\int \\frac{d^Dk_1}{\\left(2\\pi\\right)^D}\n \\int \\frac{d^Dk_2}{\\left(2\\pi\\right)^D}\n R_{\\mathrm{twoloop}}\\left(k_1,k_2\\right),\n \\nonumber \\\\\n R_{\\mathrm{twoloop}}\\left(k_1,k_2\\right)\n & = &\n \\frac{1}{2 D_1 D_2^2 D_3 D_4 D_5 D_6}.\n\\end{eqnarray}\n$R_{\\mathrm{twoloop}}(k_1,k_2)$ is a rational function in $k_1$ and $k_2$.\nWithin the numerical method one writes \n$I_{\\mathrm{twoloop},\\mathrm{CT}}$ also as a two-loop integral:\n\\begin{eqnarray}\n\\label{twoloop_CT}\n I_{\\mathrm{twoloop},\\mathrm{CT}}\n & = &\n i \\lambda^6\n \\mu^{4\\varepsilon} S_\\varepsilon^{-2}\n \\int \\frac{d^Dk_1}{\\left(2\\pi\\right)^D}\n \\int \\frac{d^Dk_2}{\\left(2\\pi\\right)^D}\n R_{\\mathrm{twoloop},\\mathrm{CT}}\\left(k_1,k_2\\right).\n\\end{eqnarray}\nWe may now ask the question if there exists a function $R_{\\mathrm{twoloop},\\mathrm{CT}}(k_1,k_2)$, rational\nin the energies $E_1$ and $E_2$, such that\n\\begin{enumerate}\n\\item $R_{\\mathrm{twoloop},\\mathrm{CT}}(k_1,k_2)$ satisfies eq.~(\\ref{twoloop_CT}),\n\\item the sum of $R_{\\mathrm{twoloop}}$ and $R_{\\mathrm{twoloop},\\mathrm{CT}}$ falls off \nfor $|k_1|\\rightarrow \\infty$ as $|k_1|^{-5}$, i.e.\n\\begin{eqnarray}\n \\lim\\limits_{|k_1|\\rightarrow \\infty} \\left( R_{\\mathrm{twoloop}}\\left(k_1,k_2\\right) + R_{\\mathrm{twoloop},\\mathrm{CT}}\\left(k_1,k_2\\right) \\right)\n & = & {\\mathcal O}\\left(|k_1|^{-5}\\right),\n\\end{eqnarray}\n\\item the sum of $R_{\\mathrm{twoloop}}$ and $R_{\\mathrm{twoloop},\\mathrm{CT}}$\nvanishes quadratically as $k_2$ goes on-shell, i.e.\n\\begin{eqnarray}\n \\lim\\limits_{k_2^2 \\rightarrow m^2} \\left( R_{\\mathrm{twoloop}}\\left(k_1,k_2\\right) + R_{\\mathrm{twoloop},\\mathrm{CT}}\\left(k_1,k_2\\right) \\right)\n & = & {\\mathcal O}\\left( \\left(E_2-E_2^\\flat\\right)^2 \\right),\n\\end{eqnarray}\n\\item $R_{\\mathrm{twoloop},\\mathrm{CT}}(k_1,k_2)$ is independent of the energy $E_2$.\n\\end{enumerate}\nThe first two requirements are just the statement that $R_{\\mathrm{twoloop},\\mathrm{CT}}$ is \na local counterterm at the integrand level \nfor the ultraviolet sub-divergence given by the self-energy sub-graph.\nRequirement 3 is the new condition which we would like to enforce and ensures that the residue\nfrom $D_2 \\rightarrow 0$ will vanish.\nCondition 4 is an additional technical requirement and ensures that \n$I_{\\mathrm{twoloop},\\mathrm{CT}}$ does not receive contributions from the cut $(1,6)$.\nThis cut is shown\nin the right diagram of fig.~\\ref{fig_3}.\n\nLet us point out that all conditions laid out above\nrefer only to the self-energy sub-diagram, not to the full diagram.\nThe conditions are therefore universal process-independent conditions.\n\nLet us now look at the self-energy.\nIt is convenient to adopt a slightly different notation for the momenta, shown in fig.~\\ref{fig_4}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.0]{fig_selfenergy}\n\\end{center}\n\\caption{\nThe labelling of the momenta for the one-loop self-energy.\n}\n\\label{fig_4}\n\\end{figure}\nFor the (bare) one-loop self-energy\nwe have\n\\begin{eqnarray}\n -i \\Sigma_{\\mathrm{oneloop}}\n & = &\n \\lambda^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1}\n \\int \\frac{d^Dk}{\\left(2\\pi\\right)^D}\n R_{\\mathrm{oneloop}},\n \\;\\;\\;\\;\\;\\;\n R_{\\mathrm{oneloop}}\n \\; = \\;\n \\frac{1}{2 D_1 D_2},\n \\nonumber \\\\\n & &\n D_1\n \\; = \\; \\left(k+\\frac{1}{2}p\\right)^2 - m^2,\n \\;\\;\\;\\;\\;\\;\n D_2\n \\; = \\; \\left(k-\\frac{1}{2}p\\right)^2 - m^2.\n\\end{eqnarray}\nGiven $p=(E,\\vec{p})$ we define $p^\\flat$ by\n\\begin{eqnarray}\n\\label{def_p_flat}\n p^\\flat & = & \\left( \\mathrm{sign}\\left(E\\right) \\sqrt{\\vec{p}^2+m^2} ,\\vec{p}\\right).\n\\end{eqnarray}\nThe momentum $p^\\flat$ is on-shell\n\\begin{eqnarray}\n \\left(p^\\flat\\right)^2 & = & m^2,\n\\end{eqnarray}\nand does not depend on $E$ (apart from the sign).\nWith $n=(1,\\vec{0})$ we may write $p^\\flat$ equally as\n\\begin{eqnarray}\n p^\\flat\n & = &\n p - c n,\n \\;\\;\\;\\;\\;\\;\n c \\; = \\;\n \\frac{1}{2n^2} \\left( 2 p \\cdot n - \\mathrm{sign}\\left(2 p \\cdot n\\right) \\sqrt{ \\left(2 p \\cdot n\\right)^2 - 4 n^2 \\left(p^2-m^2\\right)} \\right).\n\\end{eqnarray}\nFor the counterterm we write\n\\begin{eqnarray}\n -i \\Sigma_{\\mathrm{oneloop},\\mathrm{CT}}\n & = &\n \\lambda^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1}\n \\int \\frac{d^Dk}{\\left(2\\pi\\right)^D}\n R_{\\mathrm{oneloop},\\mathrm{CT}}.\n\\end{eqnarray}\nWe require that the only poles of $R_{\\mathrm{oneloop},\\mathrm{CT}}$ originate from\n\\begin{eqnarray}\n D_1^\\flat \n \\; = \\;\n \\left(k + \\frac{1}{2}p^\\flat\\right)^2 - m^2,\n \\;\\;\\;\\;\\;\\;\n D_2^\\flat \n \\; = \\;\n \\left(k - \\frac{1}{2}p^\\flat\\right)^2 - m^2.\n\\end{eqnarray}\n$D_1^\\flat$ and $D_2^\\flat$ are the images of $D_1$ and $D_2$ under the map $p \\rightarrow p^\\flat$.\nA possible choice for $R_{\\mathrm{oneloop},\\mathrm{CT}}$ is given by\n\\begin{eqnarray}\n\\label{counterterm_phi_cubed}\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n & = &\n - \\frac{1}{2 D_1^\\flat D_2^\\flat}\n \\left[ 1 - \\frac{4 k \\cdot \\left(p-p^\\flat\\right) + p^2-m^2}{4 D_1^\\flat} + \\frac{4 k \\cdot \\left(p-p^\\flat\\right) - p^2+m^2}{4 D_2^\\flat} \\right]\n \\nonumber \\\\\n & &\n + \\frac{\\left(p-p^\\flat\\right)^2}{8 m^2} \n   \\left( \\frac{2}{D_1^\\flat D_2^\\flat} - \\frac{1}{\\left(D_1^\\flat\\right)^2} - \\frac{1}{\\left(D_2^\\flat\\right)^2} \\right).\n\\end{eqnarray}\nThe first line is the expansion of $R_{\\mathrm{oneloop}}$ around the on-shell kinematics, such that\nthe difference between the first line and $R_{\\mathrm{oneloop}}$ is of order ${\\mathcal O}((p^2-m^2)^2)$.\nThe first line gives also a local UV-counterterm, such that\nthe difference between the first line and $R_{\\mathrm{oneloop}}$ is of order ${\\mathcal O}(|k|^{-5})$ or better.\nThus, we see that the first line satisfies conditions~(2) and (3).\nCondition~(4) is trivially satisfied due to our definition of $p^\\flat$ in eq.~(\\ref{def_p_flat}).\nIt remains to satisfy condition~(1).\nThis is the job of the term in the second line.\nThis term ensures that\n\\begin{eqnarray}\n \\lambda^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1}\n \\int \\frac{d^Dk}{\\left(2\\pi\\right)^D}\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n & = &\n \\frac{\\lambda^2}{\\left(4\\pi\\right)^2}\n \\;\n i \\left[ Z_\\phi^{(1)} p^2 - \\left( Z_\\phi^{(1)} + 2 Z_m^{(1)} \\right) m^2 \\right],\n\\end{eqnarray}\nwhere $Z_\\phi^{(1)}$ and $Z_m^{(1)}$ have been defined in eq.~(\\ref{renormalisation_constants_phi_cubed}).\nAt the same time, the term on the second line does not spoil the on-shell limit nor the UV-limit.\nTo see this, we note that\n\\begin{eqnarray}\n \\left(p-p^\\flat\\right)^2\n\\end{eqnarray}\nvanishes quadratically in the on-shell limit.\nSecondly, the combination\n\\begin{eqnarray}\n \\frac{2}{D_1^\\flat D_2^\\flat} - \\frac{1}{\\left(D_1^\\flat\\right)^2} - \\frac{1}{\\left(D_2^\\flat\\right)^2}\n\\end{eqnarray}\nfalls off as $\\mathcal{O}(|k|^{-6})$ in the UV-limit.\n\nThe counterterm $R_{\\mathrm{oneloop},\\mathrm{CT}}$ is a rational function in the energy variable $E_k$.\nAn inspection of eq.~(\\ref{counterterm_phi_cubed}) shows that $R_{\\mathrm{oneloop},\\mathrm{CT}}$ has double poles in the variable $E_k$.\nThis is however unproblematic, as it occurs in a universal building block. The residues can be calculated once and for all.\nAs an example we consider the residue at\n\\begin{eqnarray}\n E_{k,D_1^\\flat} & = & - \\frac{1}{2} E_{p^\\flat} + \\sqrt{\\left(\\vec{k}+\\frac{1}{2}\\vec{p}\\right)^2+m^2}.\n\\end{eqnarray}\nAs an abbreviation we set\n\\begin{eqnarray}\n E_1 & = &  \\sqrt{\\left(\\vec{k}+\\frac{1}{2}\\vec{p}\\right)^2+m^2}.\n\\end{eqnarray}\nWe find\n\\begin{eqnarray}\n \\mathrm{res}\\left(R_{\\mathrm{oneloop},\\mathrm{CT}}, E_k = E_{k,D_1^\\flat}\\right)\n & = &\n- \\frac{1}{4 E_1 D_2^\\flat}\n + \\frac{\\left(E_p-E_{p^\\flat}\\right)^2}{8 E_1 m^2 D_2^\\flat}\n + \\frac{\\left(E_p-E_{p^\\flat}\\right)^2}{32 E_1^3 m^2}\n - \\frac{\\left(E_p-E_{p^\\flat}\\right)^2}{32 E_1^3 D_2^\\flat}\n \\nonumber \\\\\n & &\n - \\frac{\\left(E_1-E_{p^\\flat}\\right)\\left(E_p-E_{p^\\flat}\\right)}{2 E_1 \\left(D_2^\\flat\\right)^2}\n + \\frac{E_{p^\\flat} \\left(E_p-E_{p^\\flat}\\right)^2}{16 E_1^2 \\left(D_2^\\flat\\right)^2},\n\\end{eqnarray}\nwhere $D_2^\\flat$ is understood to be evaluated at $k=(E_{k,D_1^\\flat},\\vec{k})$.\n\nThe rational function $R_{\\mathrm{oneloop}}$ has a corresponding residue at\n\\begin{eqnarray}\n E_{k,D_1} & = & - \\frac{1}{2} E_{p} + \\sqrt{\\left(\\vec{k}+\\frac{1}{2}\\vec{p}\\right)^2+m^2}.\n\\end{eqnarray}\n$R_{\\mathrm{oneloop}}$ has only single poles and the residue is given by\n\\begin{eqnarray}\n \\mathrm{res}\\left(R_{\\mathrm{oneloop}}, E_k = E_{k,D_1}\\right)\n & = &\n \\frac{1}{4 E_1 D_2},\n\\end{eqnarray}\nwhere $D_2$ is understood to be evaluated at $k=(E_{k,D_1},\\vec{k})$.\nFor the sum of the two residues we have\n\\begin{eqnarray}\n\\lefteqn{\n \\mathrm{res}\\left(R_{\\mathrm{oneloop}}, E_k = E_{k,D_1}\\right)\n +\n \\mathrm{res}\\left(R_{\\mathrm{oneloop},\\mathrm{CT}}, E_k = E_{k,D_1^\\flat}\\right)\n = } & &\n \\nonumber \\\\\n & &\n \\frac{1}{4 E_1}\n \\left[\n \\frac{1}{D_2}\n - \\frac{1}{D_2^\\flat}\n - \\frac{2 \\left(E_1-E_{p^\\flat}\\right)\\left(E_p-E_{p^\\flat}\\right)}{\\left(D_2^\\flat\\right)^2}\n \\right]\n \\nonumber \\\\\n & &\n + \\frac{\\left(E_p-E_{p^\\flat}\\right)^2}{8 E_1 m^2 D_2^\\flat}\n + \\frac{\\left(E_p-E_{p^\\flat}\\right)^2}{32 E_1^3 m^2}\n - \\frac{\\left(E_p-E_{p^\\flat}\\right)^2}{32 E_1^3 D_2^\\flat}\n + \\frac{E_{p^\\flat} \\left(E_p-E_{p^\\flat}\\right)^2}{16 E_1^2 \\left(D_2^\\flat\\right)^2}.\n\\end{eqnarray}\nWe note that the term in the square bracket vanishes also quadratically in the on-shell limit.\nIn technical terms we have for $D_2$ evaluated at $k=(E_{k,D_1},\\vec{k})$ and for $D_2^\\flat$ evaluated at $k=(E_{k,D_1^\\flat},\\vec{k})$:\n\\begin{eqnarray}\n \\frac{1}{D_2}\n - \\frac{1}{D_2^\\flat}\n - \\frac{2 \\left(E_1-E_{p^\\flat}\\right)\\left(E_p-E_{p^\\flat}\\right)}{\\left(D_2^\\flat\\right)^2}\n & = &\n {\\mathcal O}\\left( \\left(E_p-E_{p^\\flat}\\right)^2 \\right).\n\\end{eqnarray}\nLet us now go back to fig.~\\ref{fig_1}.\nWe combine the two-loop diagram (left diagram in fig.~\\ref{fig_1})\nwith the one-loop diagram with a counterterm insertion (right diagram in fig.~\\ref{fig_1}).\nFor the latter we derived a two-loop integral representation.\nWe may evaluate the sum of the two-loop integrals by taking residues in the two energy integrations.\nOur construction ensures that there is no residue from the cut $(1,2)$ (middle diagram of fig.~\\ref{fig_3}).\nThere are of course residues from an unproblematic cut like $(1,3)$ (left diagram of fig.~\\ref{fig_3}).\nFinally, let us note that the residue for the cut $(1,6)$ (right diagram of fig.~\\ref{fig_3})\nreceives only a contribution from the genuine two-loop diagram, but not from the diagram with the counterterm insertion.\nBy construction, the integral representation of the counterterm is independent of the energy flowing through\nthe outer loop, therefore there is no residue in this energy variable.\n\n\\section{QCD}\n\\label{sect:QCD}\n\nLet us now consider QCD with $N_f$ massless quarks and $N_Q$ massive quarks.\nIt is sufficient to discuss the case where all massive quarks have the same mass $m$.\nWe denote the renormalisation constant for the gluon field by $Z_3$, \nthe one for a massless quark field by $Z_2$ and the one for a massive quark field by $Z_{2,Q}$.\nThe renormalisation constant for the heavy quark mass $m$ is denoted by $Z_m$. \nFor the renormalisation constants we write\n\\begin{eqnarray}\n Z_a\n & = &\n 1 + \\sum_{n=1}^\\infty Z_a^{(n)} \\left( \\frac{\\alpha_s}{4\\pi} \\right)^n.\n\\end{eqnarray}\nWe will need the one-loop renormalisation constants. For $Z_3^{(1)}$ we write\n\\begin{eqnarray}\n Z_3^{(1)}\n & = &\n Z_{3,l}^{(1)}\n + \n Z_{3,Q}^{(1)},\n\\end{eqnarray}\nseparating the contributions from the massless particles in the loop ($Z_{3,l}^{(1)}$) \nfrom the contribution of the massive quark in the loop ($Z_{3,Q}^{(1)}$).\nIn the on-shell scheme we have\n\\begin{eqnarray}\n Z_2^{(1)}\n & = &\n 0,\n \\nonumber \\\\\n Z_{2,Q}^{(1)}\n & = &\n - \\left(3-2\\varepsilon\\right) C_F B_0\\left(m^2,m^2,0\\right),\n \\nonumber \\\\\n Z_m^{(1)}\n & = &\n - \\left(3-2\\varepsilon\\right) C_F B_0\\left(m^2,m^2,0\\right),\n \\nonumber \\\\\n Z_{3,l}^{(1)}\n & = &\n 0,\n \\nonumber \\\\\n Z_{3,Q}^{(1)}\n & = &\n - \\frac{4}{3} T_R N_Q B_0\\left(0,m^2,m^2\\right).\n\\end{eqnarray}\nThe self-energies are diagonal in colour space. \nWe suppress the Kronecker delta's in colour space.\n\n\\subsection{Light quarks}\n\nIn this paragraph we set\n\\begin{eqnarray}\n D_1^\\flat \n \\; = \\;\n \\left(k + \\frac{1}{2}p^\\flat\\right)^2,\n \\;\\;\\;\\;\\;\\;\n D_2^\\flat \n \\; = \\;\n \\left(k - \\frac{1}{2}p^\\flat\\right)^2.\n\\end{eqnarray}\nThe self-energy for a massless quark is given by\n\\begin{eqnarray}\n - i \\Sigma_{\\mathrm{oneloop}}\n & = & \n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1} \n \\int \\frac{d^Dk}{(2\\pi)^D} \n R_{\\mathrm{oneloop}},\n \\;\\;\\;\\;\\;\\;\n R_{\\mathrm{oneloop}}\n \\; = \\;\n C_F\n \\frac{2 \\left(1-\\varepsilon\\right) \\left( \\slashed{k} + \\frac{1}{2} \\slashed{p} \\right)}{D_1 D_2}.\n\\end{eqnarray}\nFor the counterterm we write\n\\begin{eqnarray}\n - i \\Sigma_{\\mathrm{oneloop},\\mathrm{CT}}\n & = & \n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1} \n \\int \\frac{d^Dk}{(2\\pi)^D} \n R_{\\mathrm{oneloop},\\mathrm{CT}}.\n\\end{eqnarray}\nA possible choice for $R_{\\mathrm{oneloop},\\mathrm{CT}}$ is given by\n\\begin{eqnarray}\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n & = &\n - C_F \\frac{2 \\left(1-\\varepsilon\\right) \\left( \\slashed{k} + \\frac{1}{2} \\slashed{p} \\right)}{D_1^\\flat D_2^\\flat}\n \\left[ 1 - \\frac{4 k \\cdot \\left(p-p^\\flat\\right) + p^2}{4 D_1^\\flat} + \\frac{4 k \\cdot \\left(p-p^\\flat\\right) - p^2}{4 D_2^\\flat} \\right].\n \\;\\;\\;\n\\end{eqnarray}\nIntegration is in this case particularly simple. All integrals are scaleless integrals, which vanish in dimensional regularisation. Therefore\n\\begin{eqnarray}\n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1}\n \\int \\frac{d^Dk}{\\left(2\\pi\\right)^D}\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n & = &\n \\frac{\\alpha_s}{4\\pi} \n \\;\n i Z_2^{(1)} \\slashed{p}\n \\; = \\; 0.\n\\end{eqnarray}\n\n\\subsection{Heavy quarks}\n\nIn this paragraph we set\n\\begin{eqnarray}\n D_1^\\flat \n \\; = \\;\n \\left(k + \\frac{1}{2}p^\\flat\\right)^2 - m^2,\n \\;\\;\\;\\;\\;\\;\n D_2^\\flat \n \\; = \\;\n \\left(k - \\frac{1}{2}p^\\flat\\right)^2.\n\\end{eqnarray}\nThe self-energy for a massive quark is given by\n\\begin{eqnarray}\n - i \\Sigma_{\\mathrm{oneloop}}\n & = & \n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1} \n \\int \\frac{d^Dk}{(2\\pi)^D} \n R_{\\mathrm{oneloop}},\n \\nonumber \\\\\n R_{\\mathrm{oneloop}}\n & = &\n C_F\n \\frac{2 \\left(1-\\varepsilon\\right) \\left( \\slashed{k} + \\frac{1}{2} \\slashed{p} \\right) - 4 \\left(1-\\frac{1}{2} \\varepsilon \\right) m}{D_1 D_2}.\n\\end{eqnarray}\nFor the counterterm we write\n\\begin{eqnarray}\n - i \\Sigma_{\\mathrm{oneloop},\\mathrm{CT}}\n & = & \n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1} \n \\int \\frac{d^Dk}{(2\\pi)^D} \n R_{\\mathrm{oneloop},\\mathrm{CT}}.\n\\end{eqnarray}\nA possible choice for $R_{\\mathrm{oneloop},\\mathrm{CT}}$ is given by\n\\begin{eqnarray}\n\\label{counterterm_heavy_quark}\n\\lefteqn{\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n = \n C_F \\left\\{\n - \\frac{2 \\left(1-\\varepsilon\\right) \\left( \\slashed{k} + \\frac{1}{2} \\slashed{p}^\\flat \\right) - 4 \\left(1-\\frac{1}{2} \\varepsilon \\right) m}{D_1^\\flat D_2^\\flat}\n \\left[ 1 - \\frac{4 k \\cdot \\left(p-p^\\flat\\right) + p^2 - m^2}{4 D_1^\\flat} \n \\right. \\right.\n } & & \\nonumber \\\\\n & &\n \\left.\n + \\frac{4 k \\cdot \\left(p-p^\\flat\\right) - p^2 + m^2}{4 D_2^\\flat} \\right]\n - \\frac{\\left(1-\\varepsilon\\right) \\left( \\slashed{p} - \\slashed{p}^\\flat\\right)}{D_1^\\flat D_2^\\flat}\n \\nonumber \\\\\n & &\n - \\frac{1}{4} \\left( \\slashed{p}^\\flat - m \\right) \\left(p^2-m^2\\right) \\frac{D_1^\\flat-D_2^\\flat+4m^2}{\\left(D_1^\\flat\\right)^2 \\left(D_2^\\flat\\right)^2}\n + \\frac{m \\left(p-p^\\flat\\right)^2}{4 m^2} \n   \\frac{\\left(D_1^\\flat-D_2^\\flat\\right)\\left(D_1^\\flat-D_2^\\flat+2m^2\\right)}{\\left(D_1^\\flat\\right)^2 \\left(D_2^\\flat\\right)^2}\n \\nonumber \\\\\n & &\n + \\frac{\\left( \\slashed{p}^\\flat - m \\right) \\left[ p^\\flat \\cdot \\left(p-p^\\flat\\right) \\right]}{m^2}\n   \\frac{\\left(D_1^\\flat-D_2^\\flat\\right)\\left(D_1^\\flat-D_2^\\flat+\\frac{3}{2}m^2\\right)}\n        {\\left(D_1^\\flat\\right)^2 \\left(D_2^\\flat\\right)^2}\n - \\frac{\\varepsilon m \\left(p-p^\\flat\\right)^2}{2 \\left(D_1^\\flat\\right)^2 D_2^\\flat}\n \\nonumber \\\\\n & &\n \\left.\n + \\frac{\\left[ 2 \\left( \\slashed{p} - m \\right) m^2 - m \\left(p^2-m^2\\right) \\right]}{2m^2}  \n   \\frac{\\left(D_1^\\flat-D_2^\\flat\\right)\\left(2 D_1^\\flat+D_2^\\flat\\right)}{\\left(D_1^\\flat\\right)^2 \\left(D_2^\\flat\\right)^2}\n \\right\\}.\n\\end{eqnarray}\nThe terms in the first two lines approximate $R_{\\mathrm{oneloop}}$ in the on-shell and in the ultraviolet\nlimit.\nThe terms in the third to fifth line ensure that the integration of $R_{\\mathrm{oneloop},\\mathrm{CT}}$\ngives the desired result.\nWe have\n\\begin{eqnarray}\n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1}\n \\int \\frac{d^Dk}{\\left(2\\pi\\right)^D}\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n & = &\n \\frac{\\alpha_s}{4\\pi} \n \\;\n i \\left[ Z_2^{(1)} \\slashed{p} - \\left( Z_2^{(1)} + Z_m^{(1)} \\right) m \\right].\n\\end{eqnarray}\nThe terms in the third to fifth line vanish\nin the on-shell and in the ultraviolet\nlimit.\nFor example, the last term in eq.~(\\ref{counterterm_heavy_quark})\nfalls of like ${\\mathcal O}(|k|^{-5})$ in the UV-limit.\nFor the on-shell limit we note that\n\\begin{eqnarray}\n 2 \\left( \\slashed{p} - m \\right) m^2 - m \\left(p^2-m^2\\right)\n & = &\n - m \\left( \\slashed{p} - m \\right) \\left( \\slashed{p} - m \\right).\n\\end{eqnarray}\n\n\\subsubsection{The $\\overline{\\mathrm{MS}}$-scheme}\n\nFor the mass of a heavy quark, the $\\overline{\\mathrm{MS}}$-scheme and the on-shell scheme are two popular\nrenormalisation schemes.\nIn this paragraph, we comment on the $\\overline{\\mathrm{MS}}$-scheme.\nIn the previous section we constructed an integral representation\n$R_{\\mathrm{oneloop},\\mathrm{CT}}$ in the on-shell scheme with the property\nthat\n\\begin{eqnarray}\n \\lim\\limits_{k^2\\rightarrow m^2} \\left(R_{\\mathrm{oneloop}}+R_{\\mathrm{oneloop},\\mathrm{CT}}\\right)\n & = &\n {\\mathcal O}\\left( \\left(E-E^\\flat\\right)^2 \\right).\n\\end{eqnarray}\nThis is not possible in the $\\overline{\\mathrm{MS}}$-scheme.\nTo see this, let us perform a finite renormalisation from the on-shell mass to the $\\overline{\\mathrm{MS}}$-mass.\nThis amounts to adding the term\n\\begin{eqnarray}\n\\lefteqn{\n \\frac{\\alpha_s}{4\\pi} \n \\;\n i \\left\\{ \n    \\left[ Z_2^{(1)} \\slashed{p} - \\left( Z_2^{(1)} + Z_{m,\\overline{\\mathrm{MS}}}^{(1)} \\right) m \\right]\n    -\n    \\left[ Z_2^{(1)} \\slashed{p} - \\left( Z_2^{(1)} + Z_m^{(1)} \\right) m \\right]\n   \\right\\}\n = }\n \\nonumber \\\\\n & = & \n \\frac{\\alpha_s}{4\\pi} \n \\;\n i \\left(\n         Z_m^{(1)} - Z_{m,\\overline{\\mathrm{MS}}}^{(1)}\n   \\right) m\n \\; = \\;\n \\frac{\\alpha_s}{4\\pi} \n \\;\n i C_F \\left( -4 + 3 \\ln\\frac{m^2}{\\mu^2} \\right) m\n + {\\mathcal O}\\left(\\varepsilon\\right),\n\\end{eqnarray}\nwhere we used\n\\begin{eqnarray}\n Z_{m,\\overline{\\mathrm{MS}}}^{(1)}\n & = &\n - \\frac{3 C_F}{\\varepsilon}.\n\\end{eqnarray}\nThe term from the finite renormalisation is a non-zero constant in the on-shell limit and hence does not vanish\nquadratically in the on-shell limit.\nNeither can there be an integral representation, which vanishes quadratically in the on-shell limit.\n\n\n\\subsection{Gluons}\n\nWe now consider the gluon self-energy.\nLet us first briefly discuss what happens in an analytic calculation.\nWe denote by $-i\\Pi^{\\mu\\nu}_{\\mathrm{oneloop}}$ the one-loop contribution to the gluon self-energy \nand by $-i\\Pi^{\\mu\\nu}_{\\mathrm{oneloop},\\mathrm{CT}}$ the contribution from the counterterm.\nThe self-energy is transverse and we may write\n\\begin{eqnarray}\n -i \\Pi^{\\mu\\nu}_{\\mathrm{oneloop}}\n & = &\n  i \\left( p^2 g^{\\mu\\nu} - p^\\mu p^\\nu \\right) \\Pi_{\\mathrm{oneloop}}\\left(p^2\\right),\n\\end{eqnarray}\nwith a scalar function $\\Pi_{\\mathrm{oneloop}}(p^2)$.\nWe expand $\\Pi_{\\mathrm{oneloop}}(p^2)$ around $p^2=0$:\n\\begin{eqnarray}\n \\Pi_{\\mathrm{oneloop}}\\left(p^2\\right)\n & = &\n \\Pi_{\\mathrm{oneloop}}\\left(0\\right)\n + {\\mathcal O}\\left(p^2\\right).\n\\end{eqnarray}\nThis defines $Z_3^{(1)}$:\n\\begin{eqnarray}\n \\frac{\\alpha_s}{4\\pi} \\; Z_3^{(1)} & = & \\Pi_{\\mathrm{oneloop}}\\left(0\\right).\n\\end{eqnarray}\nThus\n\\begin{eqnarray}\n -i \\left( \\Pi^{\\mu\\nu}_{\\mathrm{oneloop}} + \\Pi^{\\mu\\nu}_{\\mathrm{oneloop},\\mathrm{CT}} \\right)\n & = &\n  i \\left( p^2 g^{\\mu\\nu} - p^\\mu p^\\nu \\right) \\cdot {\\mathcal O}\\left(p^2\\right).\n\\end{eqnarray}\nWe see that the term proportional to $g^{\\mu\\nu}$ has a factor $(p^2)^2$ and will cancel a double pole from the propagators.\nOn the other hand, the term proportional to $p^\\mu p^\\nu$ comes only with a single factor $p^2$, leaving a residue from a single pole.\nHowever, this term is proportional to $p^\\mu p^\\nu$. We may neglect the contribution from this residue if we contract\nthis term into quantities, which vanish when contracted with an on-shell momentum $p^\\mu$ or $p^\\nu$.\n\nFor the gluon self-energy we distinguish the case of massless particles in the loop and the case of a massive quark loop.\n\n\\subsubsection{Contributions from massless particles}\n\nIn this paragraph we set\n\\begin{eqnarray}\n D_1^\\flat \n \\; = \\;\n \\left(k + \\frac{1}{2}p^\\flat\\right)^2,\n \\;\\;\\;\\;\\;\\;\n D_2^\\flat \n \\; = \\;\n \\left(k - \\frac{1}{2}p^\\flat\\right)^2.\n\\end{eqnarray}\nThe contribution to the gluon self-energy from massless particles is given by\n\\begin{eqnarray}\n - i \\Pi^{\\mu\\nu}_{\\mathrm{oneloop}}\n & = & \n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1} \n \\int \\frac{d^Dk}{(2\\pi)^D} \n R_{\\mathrm{oneloop}},\n \\nonumber \\\\\n R_{\\mathrm{oneloop}}\n & = &\n \\left\\{ - 2 C_A \\left[ -p^2 g^{\\mu\\nu} + p^\\mu p^\\nu - 2 \\left( 1 - \\varepsilon \\right) k^\\mu k^\\nu \n                      + \\frac{1}{2} \\left( 1 - \\varepsilon \\right) g^{\\mu\\nu} \\left( D_1 + D_2 \\right) \\right]\n \\right. \\nonumber \\\\\n & & \\left.\n       - 2 T_R N_f \\left[ p^2 g^{\\mu\\nu} - p^\\mu p^\\nu + 4 k^\\mu k^\\nu \n                          - g^{\\mu\\nu} \\left( D_1 + D_2 \\right) \\right] \\right\\}\n \\frac{1}{D_1 D_2}.\n\\end{eqnarray}\nFor the counterterm we write\n\\begin{eqnarray}\n - i \\Pi^{\\mu\\nu}_{\\mathrm{oneloop},\\mathrm{CT}}\n & = & \n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1} \n \\int \\frac{d^Dk}{(2\\pi)^D} \n R_{\\mathrm{oneloop},\\mathrm{CT}}.\n\\end{eqnarray}\nA possible choice for $R_{\\mathrm{oneloop},\\mathrm{CT}}$ is given by\n\\begin{eqnarray}\n\\lefteqn{\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n = \n \\left\\{ 2 C_A \\left[ -p^2 g^{\\mu\\nu} + p^\\mu p^\\nu - 2 \\left( 1 - \\varepsilon \\right) k^\\mu k^\\nu \n                      + \\frac{1}{2} \\left( 1 - \\varepsilon \\right) g^{\\mu\\nu} \\left( D_1 + D_2 \\right) \\right]\n \\right. } & &\n \\nonumber \\\\\n & & \\left.\n       + 2 T_R N_f \\left[ p^2 g^{\\mu\\nu} - p^\\mu p^\\nu + 4 k^\\mu k^\\nu \n                          - g^{\\mu\\nu} \\left( D_1 + D_2 \\right) \\right] \\right\\}\n \\frac{1}{D_1^\\flat D_2^\\flat}\n \\left\\{ 1 - \\frac{4 k \\cdot \\left(p-p^\\flat\\right) + p^2}{4 D_1^\\flat} \n \\right. \\nonumber \\\\\n & & \\left. \n + \\frac{4 k \\cdot \\left(p-p^\\flat\\right) - p^2}{4 D_2^\\flat} \n + \\left[ k \\cdot \\left(p-p^\\flat\\right) \\right]^2 \\left( \\frac{1}{\\left(D_1^\\flat\\right)^2} + \\frac{1}{\\left(D_2^\\flat\\right)^2} - \\frac{1}{D_1^\\flat D_2^\\flat}\\right)\n \\right\\}.\n\\end{eqnarray}\nIntegration yields\n\\begin{eqnarray}\n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1}\n \\int \\frac{d^Dk}{\\left(2\\pi\\right)^D}\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n & = &\n \\frac{\\alpha_s}{4\\pi} \n \\;\n i Z_{3,l}^{(1)} \\left( - g^{\\mu\\nu} p^2 + p^\\mu p^\\nu \\right)\n \\; = \\; \n 0.\n\\end{eqnarray}\n\n\n\\subsubsection{Contributions from a massive quark}\n\nIn this paragraph we set\n\\begin{eqnarray}\n D_1^\\flat \n \\; = \\;\n \\left(k + \\frac{1}{2}p^\\flat\\right)^2 - m^2,\n \\;\\;\\;\\;\\;\\;\n D_2^\\flat \n \\; = \\;\n \\left(k - \\frac{1}{2}p^\\flat\\right)^2 - m^2.\n\\end{eqnarray}\nThe contribution to the gluon self-energy from massive quarks is given by\n\\begin{eqnarray}\n - i \\Pi^{\\mu\\nu}_{\\mathrm{oneloop}}\n & = & \n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1} \n \\int \\frac{d^Dk}{(2\\pi)^D} \n R_{\\mathrm{oneloop}},\n \\nonumber \\\\\n R_{\\mathrm{oneloop}}\n & = &\n - 2 T_R N_Q\n \\left[\n        p^2 g^{\\mu\\nu} - p^\\mu p^\\nu + 4 k^\\mu k^\\nu - g^{\\mu\\nu} \\left( D_1 + D_2 \\right) \n \\right]\n \\frac{1}{D_1 D_2}.\n\\end{eqnarray}\nFor the counterterm we write\n\\begin{eqnarray}\n - i \\Pi^{\\mu\\nu}_{\\mathrm{oneloop},\\mathrm{CT}}\n & = & \n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1} \n \\int \\frac{d^Dk}{(2\\pi)^D} \n R_{\\mathrm{oneloop},\\mathrm{CT}}.\n\\end{eqnarray}\nA possible choice for $R_{\\mathrm{oneloop},\\mathrm{CT}}$ is given by\n\\begin{eqnarray}\n\\lefteqn{\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n = \n T_R N_Q\n \\left\\{\n  \\frac{2 \\left( p^2 g^{\\mu\\nu} - p^\\mu p^\\nu \\right)}{D_1^\\flat D_2^\\flat}\n  \\left[ 1 - \\frac{4 k \\cdot \\left(p-p^\\flat\\right) + p^2}{4 D_1^\\flat} + \\frac{4 k \\cdot \\left(p-p^\\flat\\right) - p^2}{4 D_2^\\flat} \\right]\n \\right.\n } & & \n \\nonumber \\\\\n & &\n \\left.\n + \n  \\frac{\\left[ 8 k^\\mu k^\\nu-2g^{\\mu\\nu} \\left( D_1^\\flat + D_2^\\flat \\right) \\right]}{D_1^\\flat D_2^\\flat}\n  \\left[ \n        1 - \\frac{4 k \\cdot \\left(p-p^\\flat\\right) + p^2}{4 D_1^\\flat} + \\frac{4 k \\cdot \\left(p-p^\\flat\\right) - p^2}{4 D_2^\\flat} \n \\right. \\right.\n \\nonumber \\\\\n & &\n \\left.\\left.\n        + \\left( k \\cdot \\left(p-p^\\flat\\right) \\right)^2 \\left( \\frac{1}{\\left(D_1^\\flat\\right)^2} + \\frac{1}{\\left(D_2^\\flat\\right)^2} - \\frac{1}{D_1^\\flat D_2^\\flat}\\right)\n  \\right]\n - \\frac{p^2 g^{\\mu\\nu}}{D_1^\\flat D_2^\\flat}\n \\right.\n \\nonumber \\\\\n & &\n \\left.\n - \\frac{3}{14} \\frac{\\left( p^\\flat \\cdot \\left(p-p^\\flat\\right) \\right)^2 p^\\flat{}^\\mu p^\\flat{}^\\nu }{\\left(D_1^\\flat\\right)^2 \\left(D_2^\\flat\\right)^2}\n + \\left[\n          \\left( \\frac{1}{3} p^\\flat \\cdot \\left(p-p^\\flat\\right) - \\frac{p^2}{2} \\right) \\left(p-p^\\flat\\right)^\\mu \\left(p-p^\\flat\\right)^\\nu\n \\right. \\right.\n \\nonumber \\\\\n & & \n \\left. \\left.\n        + \\left( \\frac{2}{15} p^\\flat \\cdot \\left(p-p^\\flat\\right)  - \\frac{p^2}{2} \\right) \n          \\left( \\left(p-p^\\flat\\right)^\\mu p^\\flat{}^\\nu + p^\\flat{}^\\mu \\left(p-p^\\flat\\right)^\\nu \\right) \n        - \\frac{1}{6} \\left(p-p^\\flat\\right)^2 p^\\flat{}^\\mu p^\\flat{}^\\nu\n \\right. \\right.\n \\nonumber \\\\\n & & \n \\left. \\left.\n        + \\frac{2}{5} \\left(p^\\flat \\cdot \\left(p-p^\\flat\\right)\\right)^2 g^{\\mu\\nu}\n        + \\frac{1}{6} \\left( \\left(p-p^\\flat\\right)^2 + 2p^2 \\right) p^2 g^{\\mu\\nu}\n        - \\frac{4}{15} p^2 p^\\flat{}^\\mu p^\\flat{}^\\nu\n   \\right] \n   \\frac{D_1^\\flat+D_2^\\flat}{\\left(D_1^\\flat\\right)^2 \\left(D_2^\\flat\\right)^2}\n \\right\\}.\n \\nonumber \\\\\n\\end{eqnarray}\nThe terms in the first three lines approximate $R_{\\mathrm{oneloop}}$ in the on-shell and in the ultraviolet\nlimit.\nThe terms in the fourth to sixth line ensure that the integration of $R_{\\mathrm{oneloop},\\mathrm{CT}}$\ngives the desired result.\nWe have\n\\begin{eqnarray}\n g^2 \\mu^{2\\varepsilon} S_\\varepsilon^{-1}\n \\int \\frac{d^Dk}{\\left(2\\pi\\right)^D}\n R_{\\mathrm{oneloop},\\mathrm{CT}}\n & = &\n \\frac{\\alpha_s}{4\\pi} \n \\;\n i Z_{3,Q}^{(1)} \\left( - g^{\\mu\\nu} p^2 + p^\\mu p^\\nu \\right).\n\\end{eqnarray}\nLet us note that the last term\n\\begin{eqnarray}\n   - \\frac{4}{15} T_R N_Q p^2 p^\\flat{}^\\mu p^\\flat{}^\\nu\n   \\frac{D_1^\\flat+D_2^\\flat}{\\left(D_1^\\flat\\right)^2 \\left(D_2^\\flat\\right)^2}\n\\end{eqnarray}\nonly vanishes linearly in the on-shell limit.\nIt is however proportional to $p^\\flat{}^\\mu p^\\flat{}^\\nu$ and will give a vanishing contribution when contracted\ninto quantities, which vanish when contracted with $p^\\flat{}^\\mu$ or $p^\\flat{}^\\nu$.\n\n\n\n\\section{Conclusions}\n\\label{sect:conclusions}\n\nIn this paper we showed that residues (or cuts) from raised propagators can be made to vanish for renormalised quantities in the on-shell scheme.\nThis is a significant simplification for numerical methods at two-loops and beyond.\nWe achieve this by constructing an integral representation for the ultraviolet counterterms in the on-shell scheme.\nWe worked out these counterterms explicitly for $\\phi^3$-theory and QCD.\n\n\\subsection*{Acknowledgements}\n\nThis work has been supported by the \nCluster of Excellence ``Precision Physics, Fundamental Interactions, and Structure of Matter'' \n(PRISMA+ EXC 2118\/1) funded by the German Research Foundation (DFG) \nwithin the German Excellence Strategy (Project ID 39083149).\n\n\n\n\\begin{appendix}\n\n\\section{Feynman rules}\n\\label{sect:Feynman_rules}\n\nIn this appendix we list the Feynman rules for $\\phi^3$-theory.\nThe Feynman rule for the propagator is\n\\begin{eqnarray}\n\\begin{picture}(85,20)(0,5)\n \\Line(70,10)(20,10)\n\\end{picture} \n & = &\n \\frac{i}{p^2-m^2+i\\delta},\n \\nonumber \\\\\n\\end{eqnarray}\nwith $\\delta$ an infinitesimal small positive number.\nThe vertex is given by\n\\begin{eqnarray}\n\\begin{picture}(100,35)(0,50)\n\\Vertex(50,50){2}\n\\Line(50,50)(80,50)\n\\Line(50,50)(29,71)\n\\Line(29,29)(50,50)\n\\end{picture}\n \\;\\; = \\;\\;\n i \\lambda^{(D)}.\n \\\\ \\nonumber\n\\end{eqnarray}\nThe coupling $\\lambda^{(D)}$ is defined in eq.~(\\ref{def_lambda_D}).\nThe Feynman rules for the counterterms are\n\\begin{eqnarray}\n\\begin{picture}(85,20)(0,5)\n \\Line(70,10)(20,10)\n \\Line(40,5)(50,15)\n \\Line(40,15)(50,5)\n\\end{picture} \n & = &\n i \\left[ \\left(Z_\\phi-1\\right) p^2 - \\left( Z_\\phi Z_m^2 - 1 \\right) m^2 \\right],\n \\nonumber \\\\\n\\begin{picture}(100,35)(0,50)\n\\Line(56,50)(80,50)\n\\Line(43,57)(29,71)\n\\Line(29,29)(43,43)\n\\Line(45,45)(55,55)\n\\Line(45,55)(55,45)\n\\end{picture}\n& = &\n i \\left(Z_\\phi^{\\frac{3}{2}} Z_\\lambda-1\\right) \\lambda^{(D)}.\n \\\\ \\nonumber\n\\end{eqnarray}\n\n\\end{appendix}\n\n{\\footnotesize\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{}} \n\n\\begin{document}\n\\normalem\n\n\\title{Rotational Resonances and Regge Trajectories in Lightly Doped Antiferromagnets}\n\n\\author{A. Bohrdt}\n\\email[Corresponding author email: ]{annabelle.bohrdt@tum.de}\n\\affiliation{Department of Physics and Institute for Advanced Study, Technical University of Munich, 85748 Garching, Germany}\n\\affiliation{Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\\\"unchen, Germany}\n\\address{ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA}\n\\affiliation{Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA}\n\n\\author{E. Demler}\n\\affiliation{Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA}\n\n\\author{F. Grusdt}\n\\affiliation{Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-Universit\\\"at M\\\"unchen, Theresienstr. 37, M\\\"unchen D-80333, Germany}\n\\affiliation{Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\\\"unchen, Germany}\n\n\n\n\\pacs{}\n\n\n\\date{\\today}\n\n\\begin{abstract}\nUnderstanding the nature of charge carriers in doped Mott insulators holds the key to unravelling puzzling properties of strongly correlated electron systems, including cuprate superconductors. Several theoretical models suggested that dopants can be understood as bound states of partons, the analogues of quarks in high-energy physics. However, direct signatures of spinon-chargon bound states are lacking, both in experiment and theory. Here we numerically identify long-lived rotational resonances at low doping, which directly reveal the microscopic structure of spinon-chargon bound states. Similar to Regge trajectories reflecting the quark structure of mesons, we establish a linear dependence of the rotational energy on the super-exchange coupling. Rotational excitations are strongly suppressed in standard angle-resolved photo-emission (ARPES) spectra, but we propose a multi-photon rotational extension of ARPES where they have strong spectral weight. Our findings suggest that multi-photon spectroscopy experiments should provide new insights into emergent universal features of strongly correlated electron systems.\n\\end{abstract}\n\n\\maketitle\n\n\n\\section{Introduction}\nOur understanding of strongly correlated quantum matter often involves new emergent structures. For example, emergent gauge fields play a central role for understanding quantum spin liquids \\cite{Wen2004}, and spin-charge separation in the one-dimensional Hubbard model can be related to the fractionalization of fermions into deconfined spinons and chargons \\cite{Giamarchi2003,Kim1996,Vijayan2020}. The fate of those partons in dimensions higher than one remains unresolved. Theoretically and experimentally, one faces similar problems as in high-energy physics: the mathematical models are too challenging to solve and signatures of parton formation are often indirect or buried in complex observables. In this article we draw analogies to high energy physics and report on unambiguous signatures for parton structures in the two-dimensional (2D) $t-J$ model.\n\nIn quantum chromodynamics it is well established that directly observable nucleons are not the most elementary constituents of matter. The quark model introduced more than fifty years ago explains the larger class of mesons and baryons as composite objects consisting of two or three valence quarks. A smoking gun demonstration of the quark model was its ability to explain many additional resonances observed in collider experiments as ro-vibrational excitations of the fundamental parton configurations. In the quark model, many heavy mesons are thus understood as excited states of the fundamental mesons: they contain the same quark content but realize a higher vibrational state or have non-zero orbital angular momentum \\cite{Micu1969,Olive2014}. \n\nA hallmark signature of rotational mesons comes from analysis of their mass, which can be related to the excitation energy of the pair of quarks relative to the lowest irrotational energy state. In a simplistic model, a meson can be described as a rigid line-like object with constant energy density per unit length, also known as ``string tension''. The two nearly massless quarks are located at the respective ends of this line and carry the (flavor) quantum numbers of the system. The relativistic expression for the energy of a rotating meson of this type scales linearly with the string tension and with the square root of its angular momentum \\cite{Greensite2003}. The latter relation, known as Regge trajectory, can be directly probed in collider experiments and has been observed experimentally \\cite{Bali2001}. It provides a strong indication that the observed mesons are bound states of partons. \n\n\\begin{figure\n\\centering\n\\epsfig{file=FIG1.pdf, width=0.48\\textwidth}\n\\caption{\\textbf{Rotational meson resonances.} Bound states of spinons and chargons in a $C_4$-symmetric doped 2D AFM Mott insulators feature characteristic ro-vibrational excitations. (a) In an effective microscopic theory, the string with the light chargon (gray) can rotate around the heavy spinon (blue). (b) To detect rotational resonances, we propose a multi-photon ARPES scheme. Following the creation of a hole by a first photon, a second photon couples to lattice vibrations and excites rotational modes with $C_4$ angular momentum $m_4=0,1,2,3$. \n(c) Energy distribution curves for rotational ARPES spectra at the nodal point $\\vec{k}= \\vec{\\pi}\/2$ with different angular momenta, from top to bottom: $m_4=2,1,0$. The lowest (dash-dotted) curve corresponds to the usual ARPES spectrum with $m_4=0$. All spectra are normalized by their total area, $\\int d\\omega~ A_{{\\rm (rot)}}^{(m)}$. The lowest mesonic resonances (ground state $\\mathsf{1S}$, vibrational $\\mathsf{2S}$ and rotational $\\mathsf{1P,1D,1F}$ excited states) correspond to long-lived excited states. We performed time-dependent DMRG simulations for a $t-J$ model on a $4 \\times 40$ cylinder, with $t=3 J$. The shaded areas correspond to toy model calculations (see methods \\ref{MethodsC}) where we introduced small energy shifts and broadening as fit parameters. \n}\n\\label{figRotSetup}\n\\end{figure}\n\nAn idea almost as old as the problem of high-$T_c$ superconductivity itself comprises that strongly correlated electrons in these systems may be ruled by similar principles as high-energy physics \\cite{Lee2006}. In analogy with quark confinement, B\\'eran et al. suggested a description of a hole doped into a 2D antiferromagnet (AFM) in terms of a composite quasiparticle, consisting of two partons -- a chargon, carrying the charge quantum number and a spinon, carrying the spin quantum number -- bound together by ``an interaction obeying a string law'' \\cite{Beran1996}. However, finding direct experimental or theoretical signatures for such structure has proven to be difficult. In angle resolved photo-emission spectra (ARPES) no sign of rotational resonances has been seen, and the nature of a possible first vibrational excitation is debated \\cite{Dagotto1990,Leung1995,Mishchenko2001,Manousakis2007,Bohrdt2020}. \n\nDiscerning the nature of charge carriers in lightly doped Mott insulators should provide a major boost to understanding properties of the underdoped cuprates and elucidating the origin of the pseudogap (PG) phase. In particular, it should provide a basis for constructing a consistent description of transport \\cite{Badoux2016} and spectroscopy \\cite{Shen2005} experiments. Several theoretical proposals involve emergent structures of partons, starting on a single dopant level \\cite{Bulaevskii1968,Beran1996,Trugman1988,Manousakis2007}, to effective theories of the PG phase involving exotic composite Fermi liquids \\cite{Anderson2007,Baskaran2007} and including fractionalized Fermi liquids \\cite{Senthil2003} where deconfined spinons and chargons form electron-like bound states \\cite{Punk2015PNASS,Sachdev2016,Zhang2020}. These scenarios may also explain the sudden and pronounced change of ARPES spectra \\cite{Chen2019} and the carrier density \\cite{Badoux2016} observed in cuprates around $p^* = 19 \\%$ doping, as being related to an unbinding transition of spinons and chargons.\n\nHere we provide strong numerical evidence that charge carriers in a lightly doped 2D AFM Mott insulators are comprised of partons, which are bound to each other, and exhibit telltale rotational excitations following Regge-like trajectories. We show that these rotational excitations are strongly suppressed in standard ARPES measurements and propose a multi-photon extension of ARPES imparting $C_4$-angular momentum into the system and allowing to access rotational excitations experimentally in solids \\cite{Damascelli2003} or using ultracold atoms \\cite{Bohrdt2018,Brown2019}. Our numerical DMRG simulations of the rotational ARPES spectra in the $t-J$ model, see Fig.~\\ref{figRotSetup}, reveal narrow quasiparticle peaks at low excitation energies, which we interpret as a striking proof of the parton picture. Moreover, we describe the rotational resonances by a microscopic spinon-chargon toy model which explains the observed features without any free fit parameters.\n\n\n\n\\section{Rotational ARPES spectrum}\nIn traditional ARPES the spectral function $A(\\vec{k},\\omega) = - \\pi^{-1} {\\rm Im} \\mathcal{G}(\\vec{k},\\omega)$ is measured which reveals information about the one-hole Green's function $\\mathcal{G}(\\vec{j},t) = \\theta(t) \\sum_\\sigma \\bra{\\Psi_0} \\hat{c}^\\dagger_{\\vec{j},\\sigma}(t) \\c_{\\vec{0},\\sigma}(0) \\ket{\\Psi_0}$; the latter describes how a fermion $\\c_{\\vec{j},\\sigma}$ with spin $\\sigma$ is removed from the initial state $\\ket{\\Psi_0}$ and leads to a hole propagating through the system. In the 2D Fermi-Hubbard model, believed to describe lightly doped copper oxides \\cite{Lee2006}, a long-lived quasiparticle peak is found in $A(\\vec{k},\\omega)$ \\cite{Wells1995,Ronning2005,Graf2007} which describes how a hole interacting with magnetic fluctuations forms a spin- or magnetic polaron \\cite{Kane1989,Sachdev1989,Dagotto1990,Martinez1991,Liu1992,Koepsell2019} and moves through the surrounding AFM.\n\nOur goal is to search for long-lived rotational excitations in the one-hole spectrum, which provide a direct route to reveal the composite nature of charge carriers in the Hubbard model. To couple to rotationally excited states one must impart discrete $C_4$ angular momentum into the system. However, the Green's function $\\mathcal{G}(\\vec{k},\\omega)$ respects the symmetries of the underlying Hamiltonian: In this case we are particularly interested in the discrete rotational $C_4$ symmetry of the Hubbard model, which is unbroken in the undoped parent AFM $\\ket{\\Psi_0}$. Hence, for $C_4$ invariant momenta (C4IM) in the magnetic Brillouin zone (MBZ) no angular momentum transfer is allowed and rotational excitations have no weight in the traditional ARPES spectrum $A(\\vec{k},\\omega)$. \n\nFor non-C4IM, lattice effects can in principle impart $C_4$ angular momentum into the system. However since the Green's function couples to the center-of-mass momentum $\\vec{k}$ of the hole, the spectral weight of rotational states is still expected to be strongly suppressed if the effective masses of the two supposed partons are significantly different. In this limit, the lighter parton rotates around the heavier parton which carries most of the linear momentum $\\vec{k}$, thus suppressing couplings of $\\vec{k}$ to the \\emph{relative} angular momentum of the two partons. We confirm this intuition for an analytically solvable toy-model in one dimension (see supplements \\ref{SuppMat1Dmes}). The Hubbard model in cuprates, with super-exchange coupling $J \\approx t \/ 3$, is in such a regime where significantly different parton masses $\\simeq 1\/J$ and $\\simeq 1\/t$ are expected. \n\nTo allow significant overlap with possible rotational excitations, we devise a rotational extension of ARPES where angular momentum is directly imparted into the system, even at C4IM. The simplest term creating an excitation with discrete angular momentum $m_4=0,1,2,3$, spin $\\sigma$, charge one, and total momentum $\\vec{k}$ is given by\n\\begin{equation}\n\\hat{R}_{m_4,\\sigma}(\\vec{k}) =  \\sum_{\\vec{j}} \\frac{e^{- i \\vec{k} \\cdot \\vec{j}}}{\\sqrt{V}}  \\sum_{\\vec{i}: \\ij} e^{i m_4 \\varphi_{\\vec{i} - \\vec{j}}}  \\sum_{\\sigma'} \\hat{c}^\\dagger_{\\vec{j},\\sigma'} \\c_{\\vec{i},\\sigma'} \\c_{\\vec{j},\\sigma},\n\\label{eqDefRmk}\n\\end{equation}\nwith $\\varphi_{\\vec{r}} = {\\rm arg}(\\vec{r})$ the polar angle of $\\vec{r}$. The action of this operator on a product N\\'eel state, $\\hat{R}_{m_4,\\sigma} \\ket{{\\rm N}}$, is illustrated in Fig.~\\ref{figRotSetup} (b): Here the second and third fermion operators in $\\hat{R}_{m_4,\\sigma}$ create a string-like excitation with $C_4$ angular momentum $m_4$ and non-zero overlap to the rotational states predicted for a hole in an Ising AFM \\cite{Grusdt2018PRX}.\n\nInstead of the usual Green's function, we consider the rotational Green's function\n\\begin{equation}\n\\mathcal{G}_{\\rm rot}^{(m_4)}(\\vec{k},t) = \\theta(t) \\sum_\\sigma \\bra{\\Psi_0} \\hat{R}^\\dagger_{m_4,\\sigma}(\\vec{k},t) \\hat{R}_{m_4,\\sigma}(\\vec{k},0) \\ket{\\Psi_0},\n\\end{equation}\nwhich we calculate by time-dependent DMRG (see methods \\ref{MethodsB}) \\cite{Paeckel2019a,Kjall2013,Zaletel2015}. The corresponding rotational spectrum, $- \\pi^{-1} {\\rm Im} \\mathcal{G}_{\\rm rot}^{(m_4)}(\\vec{k},\\omega)$, in Lehmann representation is\n\\begin{equation}\n A_{\\rm rot}^{(m_4)}(\\vec{k},\\omega) =  \\sum_{\\sigma, n>0} \\delta \\l \\omega-E_n+E_0 \\r  | \\bra{\\Psi_n} \\hat{R}_{m_4,\\sigma}(\\vec{k})  \\ket{\\Psi_0} |^2,\n \\label{eqDefArot}\n\\end{equation}\nwhere $\\ket{\\Psi_0}$ ($E_0$) is the correlated ground state (energy) and $\\ket{\\Psi_n}$ ($E_n$) for $n>0$ are the eigenstates (eigenenergies) with an added hole. Hence, if long-lived rotational excitations exist, they manifest in pronounced quasiparticle peaks in the rotational ARPES spectrum in Eq.~\\eqref{eqDefArot}. For $m_4=0$ the same selection rules apply as for the conventional ARPES spectrum and the same states contribute, but with modified spectral weights. \n\nThe rotational spectrum can be experimentally measured using a multi-photon extension of ARPES. We propose to use one set of beams for lattice modulation, which imparts angular momentum into the system by coupling to specific phonon modes in solids \\cite{Devereaux1994} or directly by modulating the optical potential with appropriate phases in ultracold atoms \\cite{Bloch2008}. The other beam is the usual ARPES beam which creates the hole excitation. Details of our scheme are provided in the methods \\ref{MethodsA}.\n\n\n\\section{Rotational resonances}\nNow we present our numerical results obtained for one hole doped into a 2D AFM Mott insulator. Specifically, we considered the $t-J$ model on extended four-leg cylinders and for $t\/J=3$, the experimentally most relevant value for cuprates (see methods \\ref{MethodsB}). In Fig.~\\ref{figRotSetup} (c) we show numerically obtained spectra (energy distribution curves) at the nodal point $\\vec{k}= \\vec{\\pi}\/2$, with $\\vec{\\pi} = (\\pi,\\pi)$. For $m_4=0$ (red line, second from bottom) the rotational spectrum shows the same quasiparticle peak as the conventional ARPES spectrum (black, bottom line), at the same energy. This peak, labeled $\\mathsf{1S}$, corresponds to the magnetic polaron ground state \\cite{Bohrdt2020}. A possible excited state is also visible at $m_4=0$, which we label $\\mathsf{2S}$ and which has previously been argued to correspond to the first vibrational excitation of the magnetic polaron \\cite{Dagotto1990,Leung1995,Mishchenko2001,Manousakis2007,Bohrdt2020}. The $\\mathsf{2S}$ state has a reduced spectral weight in the rotational ARPES spectrum, where it is difficult to identify at all. \n\nMuch clearer indications for long-lived excitations of magnetic polarons can be found in the non-trivial rotational ARPES spectra with $m_4 \\neq 0$. For $m_4=2$ (top, blue curve in Fig.~\\ref{figRotSetup}) we find a pronounced quasiparticle peak corresponding to an excitation energy $\\Delta E \\sim 1.7 J$. Remarkably, no significant spectral weight appears below this energy, in particular we find zero spectral weight at the polaron ground state energy. We note that this is not simply a consequence of selection rules: Firstly, the nodal point does not correspond to a C4IM, not even in the reduced MBZ. Secondly, the AFM has gapless magnon modes which should in principle allow to carry away angular momentum and allow an excited magnetic polaron to decay to its ground state. Based on these observations, we identify the resonance found at $m_4=2$ with a $\\mathsf{1D}$ excited state of the magnetic polaron. \n\nSimilarly, the rotational spectrum with $m_4=1$ features a pronounced peak at a slightly higher excitation energy $\\Delta E \\sim 2.3 J$ above the ground state (second from top, turquoise curve in Fig.~\\ref{figRotSetup}). In this case we find weak hybridization with the $\\mathsf{1S}$ state, as indicated by a small quasiparticle peak at the ground state energy. Based on its quantum numbers, we identify the new excited state as $\\mathsf{1P}$. By applying a combination of time-reversal and inversion symmetry, it follows that the $m_4=1$ and $m_4=3$ rotational spectra coincide exactly. Hence the $\\mathsf{1P}$ state is associated with a degenerate $\\mathsf{1F}$ state at $m_4=3$. \n\nOur observation of long-lived quasiparticle peaks in the rotational spectrum provides a direct indication that mobile holes in lightly doped AFM Mott insulators have a discrete internal structure. To understand our reasoning, consider a theoretical model of magnetic polarons without a rigid internal structure. In this case, the action of the operator $\\hat{R}_{m_4,\\sigma}(\\vec{k})$ in the rotational Green's function would generically be expected to have two separate effects: The first fermion operator $\\c_{\\vec{j},\\sigma}$ in Eq.~\\eqref{eqDefRmk} creates a mobile hole with a large overlap to the structureless magnetic polaron. The subsequent pair of fermion operators, $\\sum_{\\sigma'} \\hat{c}^\\dagger_{\\vec{j},\\sigma'} \\c_{\\vec{i},\\sigma'}$ in Eq.~\\eqref{eqDefRmk}, then couples to the surrounding spins and creates separate magnon excitations. In this case one would expect $A_{\\rm rot}^{(m_4)}(\\vec{k},\\omega)$ to become a convolution of a polaron and a magnon contribution, possibly renormalized weakly by interaction effects. This would lead to a broad and mostly featureless spectrum -- in stark contrast with our numerical findings in Fig.~\\ref{figRotSetup} (c). \n\n\n\\section{Regge-like trajectories}\nB\\'eran et al. \\cite{Beran1996} have suggested that mobile holes in an AFM Mott insulator can be described as mesonic bound states of two strongly interacting partons, a light chargon and a heavy spinon, see also \\cite{Laughlin1997,Grusdt2018PRX}. In that case the operators $\\hat{R}_{m_4,\\sigma}(\\vec{k})$ should create rotational excitations, which explains the peaks we found in the rotational spectra in Fig.~\\ref{figRotSetup}. Now we study how the excitation energies $\\Delta E$ of these rotational peaks, as well as the first vibrational peak, depend on the underlying coupling strength $J\/t$ in the system, which gives further insights into the nature of the bound state. \n\nIn Fig.~\\ref{figReggeTraj} we numerically extracted the positions of the peaks from frequency cuts $A_{\\rm rot}^{(m_4)}(\\vec{\\pi}\/2,\\omega)$ of rotational spectra at the nodal point, for different values of $J\/t$. We find that the positions of the rotational peaks scale linearly with the spin exchange \n\\begin{equation}\n\\Delta E_{\\rm rot} \\simeq J,\n\\label{eqErotScaling}\n\\end{equation}\nwhereas the gap to the vibrational excitation $\\mathsf{2S}$ has a characteristic power-law dependence on $t$ and $J$ \\cite{Bohrdt2020}, \n\\begin{equation}\n\\Delta E_{\\rm vib} \\simeq t^{1\/3} J^{2\/3}.\n\\label{eqEvibScaling}\n\\end{equation} \n\nThese scaling behaviors can be explained by a simplistic meson model~\\cite{Grusdt2018PRX}: In this model the two partons are connected by a line-like object on the square lattice with constant energy density $dE\/d\\ell$. This string tension must be proportional to the spin exchange energy $dE\/d\\ell \\propto J$ to obtain the observed scaling laws in Eqs.~\\eqref{eqErotScaling}, \\eqref{eqEvibScaling}.\n\nSince $J$ corresponds to the string tension between the two partons, Eq.~\\eqref{eqErotScaling} resembles the celebrated Regge formula from particle physics, which relates the meson excitation energy to  its angular momentum and the underlying string tension \\cite{Greensite2003}. While high-energy experiments cannot tune the string tension, which is determined by the coupling constant $g$ of quantum chromodynamics, cold atom quantum simulators \\cite{Cheuk2016,Mazurenko2017,Koepsell2019,Brown2019} can be used to tune the coupling $J\/t$ in the Hubbard model and directly measure the Regge-like trajectories we predict for rotational excitations in Fig.~\\ref{figReggeTraj}.\n\n\\begin{figure}[t!]\n\\centering\n\\epsfig{file=FIG2.pdf, width=0.49\\textwidth}\n\\caption{\\textbf{Meson Regge trajectories.} We show the dependence of the excitation gap $\\Delta E$ at the nodal point $\\vec{k}=\\vec{\\pi}\/2$ on the super-exchange energy $J$. The gap was extracted from peak positions in numerically obtained spectra. The low-lying rotational resonances ($\\mathsf{1P, 1D, 1F}$) have a gap scaling linearly with $J$ (light dotted lines denote linear fits: $\\Delta E_{\\mathsf{1D}} = 1.44 J + 0.061 t$ and $\\Delta E_{\\mathsf{1P,1F}} = 2.12 J + 0.047 t$). The gap to the first vibrational peak ($\\mathsf{2S}$) scales with $t^{1\/3} J^{2\/3}$ \\cite{Bohrdt2020}. Solid lines are parameter-free calculations neglecting spinon dynamics, see methods \\ref{MethodsC}. The inset shows the same data, but with energy measured in units of $J$ instead of $t$.\n}\n\\label{figReggeTraj}\n\\end{figure}\n\nIn further analogy with the Regge formula from high-energy physics, we can study the dependence of the meson excitation energy $\\Delta E$ on its angular momentum $m_4$. While quarks in high-energy physics are described in a continuous space-time, lattice effects are strong in the Hubbard or $t-J$ models we consider. As a result, the simplistic meson model from Ref.~\\cite{Grusdt2018PRX} predicts that all rotational states with $m_4 \\neq 0$ should be degenerate when $J\/t \\ll 1$, and be located between purely vibrational states in this limit. Refined meson models predict small splittings of the rotational lines, however. We confirm in Fig.~\\ref{figReggeTraj} that all rotational excitations are close in energy, and well separated from the first vibrational peak. \n\n\n\\begin{figure}[t!]\n\\centering\n\\epsfig{file=FIG3.pdf, width=0.49\\textwidth}\n\\caption{\\textbf{Rotational spectra and meson dispersion.} We show full rotational ARPES spectra $A_{\\rm rot}^{(m_4)}(\\omega,\\vec{k})$ for momenta $\\vec{k}$ along the cuts in the Brillouin zone indicated by blue arrows in (a). For a given value of $m_4=0$ ($m_4=1,2$), the fitted peak positions of the low-lying resonances are indicated by green (gray, blue) dots in (b) and (c). The numerics were carried out on elongated $L_x=40 \\times 4$-leg cylinders, for a $t-J$ model with $t\/J=3$. At high-symmetry momenta in the Brillouin zone, see (a), selection rules explain why some meson resonances are invisible. The insets show the same spectral cuts in the spinon-chargon toy model, where discrete delta-functions were replaced by weakly broadened Gaussian peaks but without any further fitting parameters.\n}\n\\label{figSpectralLines}\n\\end{figure}\n\n\\section{Meson dispersion}\nThe dependence of the peak position on the momentum $\\vec{k}$ enables further insights into the properties of the mesonic bound states describing mobile holes in lightly doped AFM Mott insulators. In Fig.~\\ref{figSpectralLines} we show momentum cuts of the rotational ARPES spectrum, along lines indicated in part (a), for a fixed ratio of $t\/J=3$. \n\nIn order to simplify comparison between spectra at different angular momenta, we indicate the numerically extracted dispersions of all mesonic resonances by dots. Different dot colors denote the angular momentum $m_4$ where the respective peak is most pronounced. Since $m_4$ is only a conserved quantum number at C4IM, at generic momenta $\\vec{k}$ most mesonic resonances are visible for different values of $m_4$ at the same time. In addition, even at C4IM our numerics indicate further weak hybridization of rotational states which is caused by the broken $C_4$ rotational symmetry in the elongated four-leg cylinders we consider. For example, at $\\vec{k}=(\\pi,0)$ the $\\mathsf{1S}$ state has non-zero spectral weight for $m_4=2$. On the other hand, the strict vertical and horizontal reflection symmetries $\\sigma_{\\rm v\/h}$ of our finite-size cylinders prevent hybridization of states with even and odd $m_4$, respectively. \n\nOur main findings from the meson dispersions in Fig.~\\ref{figSpectralLines} are as follows. For almost all momenta, the lowest peak corresponds to angular momentum $m_4=0$ (green dots). This peak shows the strongest dispersion, with a minimum at the nodal point. In contrast, the rotational meson resonances show significantly reduced dispersion and the locations of dispersion minima depend strongly on the value of $m_4$. These features are explained by the spinon-chargon toy model presented in the methods section \\ref{MethodsC}.\n\n\n\\section{Measurement scheme}\nWe expect that the rotational ARPES spectra we predict can be experimentally measured in cuprate compounds. Specifically, we propose to perform multi-photon spectroscopy which combines the usual ARPES beam with a periodic lattice modulation. To realize the latter, we suggest to drive the buckling phonon modes in copper-oxide layers, in particular the Raman active $A_{\\rm 1g}$ and $B_{\\rm 1g}$ modes in YBCO \\cite{Devereaux1994,Rosch2004}. Overall this leads to an effective three-photon transition, consisting of a pair of Raman lasers and the usual ARPES beam. As we show in the methods \\ref{MethodsA}, the $s$- and $d$-wave symmetries of the $A_{\\rm 1g}$ and $B_{\\rm 1g}$ modes, respectively, allow to measure $m_4=0$ and $m_4=2$ rotational ARPES spectra. \n\nWe also show in the methods \\ref{MethodsA} that a similar rotational extension of scanning-tunneling microscopy (STM) can be envisioned. Even though rotational STM spectra lack momentum resolution, we demonstrate that the predicted weakly dispersive long-lived rotational meson resonances in Fig.~\\ref{figSpectralLines} can be resolved. \n\nFinally, we expect that analogous measurements will become possible in ultracold atoms in the immediate future \\cite{Brown2019,Kollath2007,Bohrdt2018}. There, the required angular momentum can be imparted into the system directly through shaking of the optical lattice and without phonons, resulting in a two-photon spectroscopy scheme; see methods \\ref{MethodsA} for details. \n\n\n\\section{Discussion and Outlook}\nWe have proposed a rotational extension of ARPES, which we used to predict the previously unknown long-lived rotational excitations of charge carriers in lightly doped 2D AFM Mott insulators. Our finding of pronounced quasiparticle peaks in the rotational spectra allow to conclude that strong interactions between spin and charge must be present. By analyzing Regge-like trajectories, describing the dependence of the excitation energy on the super-exchange $J$, we found compelling evidence that mobile holes in lightly doped AFM Mott insulators have a rich internal structure and can be understood as spinon-chargon bound states. This finding is further supported by the good agreement we report with a microscopic toy model of spinons and chargons connected by a string on the square lattice. \n\nWe will extend our numerical analysis, obtained for the one-hole doped $t-J$ model so far, to study the closely related 2D Hubbard model and analyze effects of weak non-zero doping next. We do not expect the clear numerical signatures obtained so far -- which we also confirm in exact diagonalization of $4 \\times 4$ systems in the supplements \\ref{SuppMatAddNum} -- to change significantly in this case.  \n\nOur research provides the most direct evidence yet for the decades old idea \\cite{Beran1996,Laughlin1997} that the physics of lightly doped 2D AFM Mott insulators -- and by extension high-temperature superconductors -- is captured by emergent partons. In particular our results support the picture of the pseudogap phase in cuprates as a liquid of fermionic mesons, modeling charge carriers as bound states of spinons and chargons. We emphasize the importance of a direct experimental confirmation that charge carriers have a rich internal structure: An observation of the long-lived rotational resonances we predict would provide a strong indication that this picture is correct. \n\nThe meson structure of charge carriers in lightly doped 2D AFM Mott insulators may have further theoretical implications. On the one hand it may contribute to our understanding how stripes form at low temperatures \\cite{Grusdt2020}, or shed new light on the pairing mechanism underlying high-temperature superconductivity in cuprates. On the other hand, understanding possible un-binding transitions of spinons and chargons may contribute to a deeper understanding of the rich phase diagram of cuprates. An interesting future direction would be to explore how the parton picture relates to the sudden change of carrier properties observed around a critical doping $p^*\\approx 19\\%$ \\cite{Tallon2001,Badoux2016,Chen2019}, see also \\cite{Koepsell2020a}.\n\n\n\n\n\n\n\n\n\n\\newpage\n\n\\section{Methods}\n\n\\subsection{Multi-photon lattice modulation spectroscopy}\n\\label{MethodsA}\nTo detect rotational excitations experimentally, we propose a general lattice modulation scheme. The lattice modulation causes a two-photon side-band (or, depending on the coupling scheme, a Raman side-band) in the ARPES spectrum, shifted by the rotational excitation energy. We discuss possible implementations of the rotational ARPES scheme for ultracold atoms in quantum gas microscopes \\cite{Gross2017}, and in solids where three photon beams are required to Raman-couple to suitable phonon modes. Finally we show how scanning-tunneling microscopy (STM) can be combined with the same lattice shaking scheme to probe the rotational states without momentum resolution. \n\n\n\\subsubsection{General scheme}\nTo describe the lattice modulation scheme, we denote by $\\H_{0}$ the underlying system Hamiltonian ($t-J$ or Fermi-Hubbard model; additional terms with the same discrete lattice symmetries are also allowed). $\\H_{0}$ determines the dynamics of the underlying particles $\\c_{\\vec{j},\\sigma}$ in the system. By $\\hat{V}_{\\rm A}(t)$ we denote the applied perturbation whose linear response yields the ARPES signal: \n\\begin{equation}\n \\hat{V}_{\\rm A}(t) = - \\delta t_{\\rm A} \\sin( \\omega_{\\rm A} t)  \\sum_{\\vec{j},\\sigma} \\l  \\hat{a}^\\dagger_{\\vec{j},\\sigma} \\c_{\\vec{j},\\sigma} + \\text{h.c.} \\r.\n \\label{eqDefVA}\n\\end{equation}\nHere $\\a_{\\vec{j},\\sigma}$ denotes the \"photo electron\" channel in the ARPES sequence. \n\nWe assume that the detection system is governed by a non-interacting Hamiltonian,\n\\begin{equation}\n \\H_a = \\sum_{\\vec{k},\\sigma} \\epsilon_{\\vec{k},\\sigma} \\hat{a}^\\dagger_{\\vec{k},\\sigma} \\a_{\\vec{k},\\sigma},\n\\end{equation}\nwith a known dispersion $\\epsilon_{\\vec{k},\\sigma}$. Further, we require that the energy, momentum and spin of the \"photo electrons\" created by $\\hat{a}^\\dagger_{\\vec{k},\\sigma}$ can be experimentally measured. The dispersion relation $\\epsilon_{\\vec{k},\\sigma}$ is of the form \n\\begin{equation}\n \\epsilon_{\\vec{k},\\sigma} = \\Delta + \\delta \\epsilon_{\\vec{k},\\sigma},\n\\end{equation}\nwhere $\\Delta$ is an overall energy offset and ${\\rm min}_{\\vec{k}} \\delta \\epsilon_{\\vec{k},\\sigma} = 0$. Note that $\\Delta$ corresponds to the minimum energy required to remove a fermion from the system.\n \n\\begin{figure}[t!]\n\\centering\n\\epsfig{file=FIG4.pdf, width=0.48\\textwidth}\n\\caption{\\textbf{Multi-photon rotational ARPES scheme.} To probe the rotational response of (doped) anti-ferromagnets, we propose to use a red-detuned ARPES beam $\\delta t_{\\rm A}$ and supplement the missing energy to extract a \"photo electron\" (green) by the lattice modulation $\\delta t_{\\rm L}$, i.e. $\\omega_{\\rm A} + \\omega_{\\rm L} \\approx \\Delta$. We assume that the lattice modulation frequency $\\omega_{\\rm L}$ is far-off resonant from the doublon-hole resonance at energy $\\sim U$. In solids, the lattice modulation can be realized by Raman-coupling to suitable phonon modes, see inset.\n}\n\\label{figRARPES}\n\\end{figure}\n\nThe second ingredient for our scheme is a super-lattice modulation, which periodically changes the tunnel coupling between neighboring sites $\\vec{i}$ and $\\vec{j}$:\n\\begin{equation}\n \\hat{V}_{\\rm L}(t) = - \\sum_{\\ij}  \\delta t_{\\rm L} \\sin ( \\omega_{\\rm L} t + \\phi_{\\ij} )  \\sum_\\sigma \\l \\hat{c}^\\dagger_{\\vec{j},\\sigma} \\c_{\\vec{i},\\sigma} + \\text{h.c.} \\r.\n \\label{eqDefVL}\n\\end{equation}\nWe assume that the phases $\\phi_\\ij$ on different bonds $\\ij$ can be controlled by the spatial structure of the lattice modulation. If resonant, the perturbation \\eqref{eqDefVL} creates doublon-hole pairs with energy $\\approx U \\gg t,J$. \n\nThe basic strategy to detect rotational excitations $m=0,1,2,3$ is to consider a two-photon transition involving both drives $\\hat{V}_{\\rm A}(t)$ and $\\hat{V}_{\\rm L}(t)$. The allowed transitions are shown in Fig.~\\ref{figRARPES}. We assume that both one-photon transitions are off-resonant, i.e. \n\\begin{flalign}\n |\\delta t_{\\rm A}| &\\ll \\underbrace{\\Delta -  \\omega_{\\rm A}}_{> 0}, \\\\\n  |\\delta t_{\\rm L}| &\\ll | U - \\omega_{\\rm L} |.\n\\end{flalign}\nIn this case, \"photo electrons\" $\\a_{\\vec{k},\\sigma}$ can only be created by a two-photon process. In the linear response regime, i.e. for sufficiently weak modulations $\\delta t_{\\rm A,L}$, the probability per time $\\gamma_{\\rm AL}$ for the two-photon transition is given by\n\\begin{multline}\n\\gamma_{\\rm AL}(\\vec{k}) = 2 \\pi ~ \\delta \\bigl( \\omega_{\\rm L}+\\omega_{\\rm A} - ( E_f(\\vec{k}) - E_i ) \\bigr) \\times \\\\\n\\biggl| \\sum_{r} \\biggl\\{ \\frac{\\bra{\\psi_f(\\vec{k})} \\hat{V}_{\\rm A} \\ket{r} \\bra{r} \\hat{V}_{\\rm L} \\ket{\\psi_i}}{\\omega_{\\rm L} - (E_r - E_i) + i \\Gamma_r}  + \\frac{\\bra{\\psi_f(\\vec{k})} \\hat{V}_{\\rm L} \\ket{r} \\bra{r} \\hat{V}_{\\rm A} \\ket{\\psi_i}}{\\omega_{\\rm A} - (E_r - E_i) + i \\Gamma_r}  \\biggr\\}   \\biggr|^2,\n\\label{eqGammaAL}\n\\end{multline}\nwhere we defined for $\\mu={\\rm A, L}$ the time-independent perturbations $\\hat{V}_\\mu$ by the relation\n\\begin{equation}\n \\hat{V}_\\mu(t) \\equiv e^{ - i \\omega_\\mu t} \\hat{V}_\\mu + \\text{h.c.} ~.\n\\end{equation}\nIn Eq.~\\eqref{eqGammaAL}, $E_f(\\vec{k})$ denotes the energy of the final state $\\ket{\\psi_f(\\vec{k})} = \\ket{\\Psi_f}\\ket{\\vec{k},\\sigma}$ with a \"photo electron\" $\\a_{\\vec{k},\\sigma}$ with momentum $\\vec{k}$, $E_i$ is the energy of the initial state $\\ket{\\psi_i} = \\ket{\\Psi_0} \\ket{0}$ without \"photo electrons\"; $\\ket{\\Psi_f}$ and $\\ket{\\Psi_0}$ denote the final and ground states of the many-body system. The sum $\\sum_r$ is taken over all possible intermediate states $\\ket{r}$ with energy $E_r$ and inverse lifetime $\\Gamma_r$ (where $\\Gamma_r \\to 0^+$ for infinite lifetimes); see e.g.~\\cite{Long2002}.\n\nThe rotational spectra defined in the main text correspond to the response $\\hat{V}_{\\rm L} \\hat{V}_{\\rm A}$, where the ARPES transition takes place before the lattice modulation $\\hat{V}_{\\rm L}$ is applied. This ensures that $\\hat{V}_{\\rm L}$ transfers discrete orbital ($\\hat{C}_4$) angular momentum to the many-body system but not to the \"photo electrons\" $\\a_{\\vec{k},\\sigma}$; note that $[\\hat{V}_{\\rm L}, \\hat{V}_{\\rm A}] \\neq 0$ in general. To ensure that the two-photon response in Eq.~\\eqref{eqGammaAL} is dominated by terms where $\\hat{V}_{\\rm A}$ is applied first, we require that\n\\begin{equation}\n 0 < \\Delta - \\omega_{\\rm A}  \\ll |U-\\omega_{\\rm L}|.\n\\end{equation}\nI.e. the ARPES beam is close to resonance while the lattice modulation is far off resonant, see Fig.~\\ref{figRARPES}. \n\nThe response in Eq.~\\eqref{eqGammaAL} simplifies further, if we assume that the intermediate states $\\ket{r}$ are detuned sufficiently far, \n\\begin{equation}\n|\\omega_{\\rm A} - \\Delta| \\gg t, J,\n\\end{equation}\nto neglect any dependence on the specifics of the intermediate states: $(E_r - E_i) \\approx  \\Delta$. This yields\n\\begin{multline}\n\\gamma_{\\rm AL}(\\vec{k}) = 2 \\pi ~ \\frac{| \\bra{\\psi_f(\\vec{k})} \\hat{V}_{\\rm L} \\hat{V}_{\\rm A} \\ket{\\psi_i}|^2 }{ (\\omega_{\\rm A} - \\Delta)^2 }  \\times \\\\\n   \\delta \\bigl( \\omega_{\\rm L}+\\omega_{\\rm A} - ( E_f(\\vec{k}) - E_i ) \\bigr).\n   \\label{eqALres}\n\\end{multline}\nNote however, that this approximation does not affect the energy delta-function in Eq.~\\eqref{eqGammaAL}.\n\nAlternatively, if we assume that the quasiparticle lifetimes of the intermediate states are all relatively short,\n\\begin{equation}\n \t\\Gamma_r \\approx \\Gamma \\geq |E_r - E_i| \\approx \\Delta,\n\t\\label{eqGammaApprx}\n\\end{equation}\nwe can again sum over all intermediate states and get\n\\begin{multline}\n\\gamma_{\\rm AL}(\\vec{k}) \n\\approx 2 \\pi ~ \\frac{| \\bra{\\psi_f(\\vec{k})} \\hat{V}_{\\rm L} \\hat{V}_{\\rm A} \\ket{\\psi_i}|^2 }{ | \\omega_{\\rm A} - \\Delta + i \\Gamma |^2 }  \\times \\\\\n   \\delta \\bigl( \\omega_{\\rm L}+\\omega_{\\rm A} - ( E_f(\\vec{k}) - E_i ) \\bigr).\n   \\label{eqALresGamma}\n\\end{multline}\n\n\nFinally, for $m=0,1,2,3$ we consider a specific lattice modulation $\\hat{V}_{\\rm L}$, with the following phases:\n\\begin{flalign}\n \\phi_{\\langle \\vec{j}, \\vec{j} + \\vec{e}_x \\rangle} &= \\pi m  j_x, \\label{eqPhaseMod1}\\\\\n \\phi_{\\langle \\vec{j}, \\vec{j} + \\vec{e}_y \\rangle} &= \\frac{\\pi}{2} m + \\pi m j_y.\n \\label{eqPhaseMod2}\n\\end{flalign}\nFor this choice one obtains\n\\begin{multline}\n \\hat{V}_{\\rm L} \\hat{V}_{\\rm A} \\ket{\\psi_i} = \\frac{\\delta t_{\\rm L} \\delta t_{\\rm A}}{4} \\sum_{\\vec{k}, \\sigma} \\ket{\\vec{k},\\sigma} \\frac{1}{\\sqrt{V}} \\sum_{\\vec{j}} e^{- i (\\vec{k} + \\vec{\\pi} m) \\cdot \\vec{j}}  \\\\\n \\sum_{\\vec{i}: \\ij} e^{i m \\varphi_{\\vec{i} - \\vec{j}}} \\sum_{\\sigma'} \\hat{c}^\\dagger_{\\vec{j},\\sigma'} \\c_{\\vec{i},\\sigma'} \\c_{\\vec{j},\\sigma} \\ket{\\Psi_0},\n\\end{multline}\nwhere $V=L^2$ is the area of the system, $\\vec{\\pi}=(\\pi,\\pi)^T$ and $\\varphi_{\\vec{i} - \\vec{j}}$ is the polar angle of the vector $\\vec{i} - \\vec{j}$. Plugging this result into Eq.~\\eqref{eqALres} yields the rotational ARPES spectrum discussed in the main text in spectral, or Lehmann, representation:\n\\begin{multline}\n\\gamma_{\\rm AL}^{(m)}(\\vec{k}) \\propto \\sum_{\\sigma=\\uparrow, \\downarrow} \\sum_{\\ket{f}} |\\bra{f} \\hat{R}_{m,\\sigma}(\\vec{k} + \\vec{\\pi} m) \\ket{\\Psi_0} |^2 \\\\\n\\times    \\delta \\bigl( \\omega_{\\rm L}+\\omega_{\\rm A} - ( E_f(\\vec{k}) - E_i ) \\bigr).\n\\end{multline}\nHere $\\hat{R}_{m,\\sigma}(\\vec{k})$ creates rotational states with total momentum $\\vec{k}$, as defined in Eq.~\\eqref{eqDefRmk} in the main text; we summed over all final states $\\ket{f}$ of the many-body system contributing to instances where a \"photo electron\" with momentum $\\vec{k}$ and arbitrary spin $\\sigma$ is detected. \n\nThe following corrections to the scheme presented above can be expected: (i) In a Hubbard model, the lattice modulation may lead to recombinations of virtual doublon-hole pairs. Such processes correspond to intermediate states $\\ket{r}$ with energies $E_r - E_i \\simeq J$, i.e. they are close to resonance. However, the spectral weights of such processes are strongly suppressed by the probability for virtual doublon-hole pairs which scales as $\\simeq (U\/t)^2 \\ll 1$. (ii) With the tunneling $t$, the spin-exchange $J$ is modulated accordingly at the frequency $\\omega_{\\rm L}$. This leads to another contribution to the response $\\gamma_{\\rm AL}^{(m)}$. Since $t < J$ it follows that $\\delta t < \\delta J$ and we neglect this sub-dominant contribution. As can be seen from the delta-function in Eq.~\\eqref{eqGammaAL}, all these corrections can only modify the respective spectral weights but not the positions or symmetry properties of the expected spectral lines.\n\n\n\\subsubsection{Realizations with ultracold atoms}\nTo realize the two-photon scheme introduced above for ultracold atoms in optical lattices, the \"photo electron\" channel must be replaced by non-interacting atomic states which are experimentally accessible. One option is to use a third internal atomic state, which may occupy the same lattice, but does not interact with the two spin states $\\sigma=\\uparrow, \\downarrow$ used to realize the doped Hubbard model. Such settings have been proposed \\cite{Dao2007} and realized in the continuum \\cite{Stewart2008} and in lattice systems under a quantum gas microscope \\cite{Brown2019}. A second option is to use spatially separated states to realize the ARPES channel \\cite{Kollath2007,Bohrdt2018}: Specifically, a decoupled layer in the optical lattice, adjacent to the physical system, can be utilized in quantum gas microscopes \\cite{Preiss2015,Koepsell2020PRL}. The offset $\\Delta$ corresponds to the energy offset between the physics and detection layer, and can be independently tuned in ultracold atom systems. \n\nIn all these systems, momentum resolved detection of the $\\a_{\\vec{k},\\sigma}$ particles is required. In addition to the momentum resolution itself, this yields the energy of the emitted \"photo electron\" provided its dispersion relation $\\epsilon_{\\vec{k},\\sigma}$ can be independently determined. The required momentum resolution can be obtained by time-of-flight or band-mapping techniques. In quantum gas microscopes with a restricted field of view, a $T\/4$ oscillation in a harmonic trapping potential \\cite{Murthy2014} can be utilized to map momentum to position states \\cite{Bohrdt2018,Brown2019}. A detailed discussion of the proposed ARPES scheme can be found in Ref.~\\cite{Bohrdt2018}.\n\nIf no momentum resolution is desired, the separate detection layer can be replaced by a local probe. In this case the $\\a$-particles have no dispersion: hence their final energy is determined by $\\epsilon_{\\vec{k},\\sigma} \\equiv \\Delta$ and momentum resolution is no longer required to obtain the final state energy. An experimental realization requires a localized optical potential to spatially confine the final $\\a$-state \\cite{Kollath2007}. \n\nIn the two-layer setting, the ARPES coupling itself, Eq.~\\eqref{eqDefVA}, can be realized by introducing a weak tunnel coupling $\\delta t_{\\rm A}$ between the layers and modulating it at the frequency $\\omega_{\\rm A}$. This can be easily achieved by an intensity modulation of the lattice beams. In settings where an auxiliary internal atomic state is used, the ARPES coupling corresponds to a radio-frequency, or microwave, transition with frequency $\\omega_{\\rm A}$.\n\nFinally the lattice modulation, Eq.~\\eqref{eqDefVL}, with phase choices as in Eqs.~\\eqref{eqPhaseMod1}, \\eqref{eqPhaseMod2} can be realized by modulated super-lattice potentials in $x$- and $y$-direction. The cases of $s$- and $d$-wave modulations are particularly simple and do not even require an extra super-lattice potential: Here it is sufficient to homogeneously modulate the tunnelings along $x$- and $y$-directions. For $s$-wave, modulations in phase are needed, while $d$-wave requires a $\\pi$-phase shift between the modulations along $x$ and $y$ respectively. \n\n\n\n\\subsubsection{Realizations in solids}\nIn copper-oxide layers, we propose to combine state-of-the-art ARPES with a periodically driven optical phonon mode. These phonons couple to the hopping integral of the electrons, thus realizing the desired lattice modulation $\\hat{V}_{\\rm L}(t)$. The symmetry of the involved phonon mode determines the phases $\\phi_{\\ij}$ of this lattice modulation, responsible for the transfer of angular momentum to the many-body system. \n\nIn the following we will only consider two distinct Raman-active phonon modes, with $s$- and $d$-wave symmetry, respectively. Specifically we will discuss the $A_{1g}$ and $B_{1g}$ buckling modes in the YBCO class. Because the Cu-O plane is not the symmetry plane in these materials, a crystal electric field can introduce a linear coupling of the electrons to the buckling modes of the oxygen ions \\cite{Devereaux1994,Rosch2004}. The effect of such phonons in the effective $t-J$ model of the cuprates is to modify the tunneling matrix elements as \\cite{Normand1995}:\n\\begin{equation}\n t_{\\langle \\vec{j}, \\vec{j} + \\vec{e}_\\mu \\rangle} = t  \\left[ 1 - \\lambda_t ( u_{\\vec{j}}^\\mu \/ a ) \\right] , \\qquad \\mu=x,y;\n \\label{eqTModulation}\n\\end{equation}\nsee Ref.~\\cite{Devereaux1994} for a discussion of the effect in the three-band model. Here $\\vec{e}_\\mu$ denotes the unit vector along $\\mu$-direction, $a$ is the lattice constant, $\\lambda_t$ is a dimensionless phonon coupling, and $u_{\\vec{j}}^x$ [$u_{\\vec{j}}^y$] is the displacement of the $O(2)$ [$O(3)$] oxygen ion from its equilibrium position along the $c$ axis of the crystal. Note that a similar modification as in Eq.~\\eqref{eqTModulation} is expected for the exchange integral \\cite{Normand1995,Sherman1997}, which we neglect here. \n\nThe symmetry of the phonon mode determines the relative signs of $u_{\\vec{j}}^\\mu$: For the $A_{1g}$ mode with $s$-wave symmetry, $u_{\\vec{j}}^x = u_{\\vec{j}}^y$; the $B_{1g}$ mode has $d$-wave symmetry:\n\\begin{equation}\nu_{\\vec{j}}^x = - u_{\\vec{j}}^y,\n\\end{equation}\nwhich is the central ingredient required for measuring the $d$-wave rotational ARPES spectrum. \n\nCombining the ingredients above, we can use the following electron-phonon Hamiltonian to describe the lattice modulation \\cite{Rosch2004}:\n\\begin{equation}\n \\hat{V}_{\\rm L} =   \\sum_{\\ij, \\sigma}  \\hat{c}^\\dagger_{\\vec{j},\\sigma} \\c_{\\vec{i},\\sigma} \\sum_{\\vec{q},\\nu} g_{\\ij}(\\vec{q},\\nu) \\l \\b_{\\vec{q},\\nu} + \\hat{b}^\\dagger_{-\\vec{q},\\nu}  \\r. \n \\label{eqHeffVLsolids}\n\\end{equation}\nHere $\\nu$ labels the relevant phonon modes and the coupling $g_{\\ij}(\\vec{q},\\nu) \\propto t \\lambda_t$ reflects the symmetry of the respective phonon mode. \n\nTo measure the rotational ARPES spectrum, we propose to drive the desired $\\vec{q}=0$ phonon using a pair of Raman lasers, see inset in Fig.~\\ref{figRARPES}; Thus our scheme becomes a three-photon setup. Effectively we can describe the Raman drive by replacing $\\b_{\\vec{q},\\nu} + \\hat{b}^\\dagger_{-\\vec{q},\\nu}$ in Eq.~\\eqref{eqHeffVLsolids} with $\\delta_{\\vec{q},0} \\beta_\\nu e^{-i \\omega_{\\rm L} t} \/2 + \\text{h.c.} $, where $\\omega_{\\rm L}$ is the frequency of the drive. To obtain a sizable amplitude $\\beta_\\nu$, the frequency $\\omega_{\\rm L}$ should be close to the phonon frequency $\\Omega_\\nu$. For example, the $B_{1g}$ mode in YBCO is located at $\\Omega_{B_{1g}} \\approx 42 {\\rm meV}$. As a result, we obtain $ \\hat{V}_{\\rm L}(t) $ as in Eq.~\\eqref{eqDefVL} with:\n\\begin{equation}\n \\delta t_{\\rm L} = | \\beta_\\nu ~ g_{\\ij}(\\vec{0}, \\nu)|,~~~ \\phi_{\\ij} = \\begin{cases} \n e^{i 2 \\varphi_{\\vec{i} - \\vec{j}}}, & \\nu = B_{1g} \\\\ \n1,  & \\nu = A_{1g} \\end{cases} .\n\\end{equation}\n\nSince typical super-exchange energies in YBCO materials are around $J \\approx 250 {\\rm meV}$, we obtain a situation where the ARPES beam is essentially resonant and can directly create hole excitations. I.e. the detuning between the two Raman beams should be around $\\omega_{\\rm L} \\approx 40 {\\rm meV}$. In this regime, we use Eq.~\\eqref{eqGammaApprx} and assume that the lifetime of hole excitations is short. This is justified by the overall broad structure of the conventional ARPES spectrum, see e.g. dash-dotted bottom line in Fig.~\\ref{figRotSetup} (c). To distinguish the direct and rotational ARPES signals, we propose to take a difference measurement of the photo-electron signal with ($\\delta t_{\\rm L} \\neq 0$) and without ($\\delta t_{\\rm L} = 0$) the Raman beams on.\n\n\n\\subsubsection{Extensions to STM probes}\nThe general multi-photon scheme proposed above to probe rotational excitations of mobile dopants can be straightforwardly generalized to scanning-tunneling microscopy (STM) setups. In this case the photo-electron with momentum $\\vec{k}$ created by $\\hat{V}_{\\rm A}$ is replaced by a localized electronic final state at site $\\vec{j}_0$, i.e. \n\\begin{equation}\n \\hat{V}_{\\rm A, STM}(t) = - \\delta t_{\\rm A} \\sin( \\omega_{\\rm A} t)  \\sum_{\\sigma} \\l  \\hat{a}^\\dagger_{\\vec{j}_0,\\sigma} \\c_{\\vec{j}_0,\\sigma} + \\text{h.c.} \\r.\n \\label{eqDefVAstm}\n\\end{equation}\nThe resulting rotational STM signal is obtained by integrating the rotational ARPES over all momenta:\n\\begin{equation}\nI_{\\rm rot}^{(m)}(\\omega) = \\int_{\\rm BZ} d^2\\vec{k} ~ A_{\\rm rot}^{(m)}(\\vec{k}, \\omega).\n\\end{equation}\n\nThe bandwidth of rotational excitations (with $m_4 \\neq 0$) is significantly smaller than in the vibrational ground state (with $m_4=0$). Hence the rotational STM protocol without full momentum resolution is sufficient to resolve rotational meson excitations. As an example, we show the expected STM signal for the experimentally most relevant case $t\/J=3$ in Fig.~\\ref{figSTMrot}. Indeed, in addition to the most prominent vibrational ground state at $m_4=0$, the rotational excitations at $m_4=1,2$ are clearly visible as distinct quasiparticle peaks. For $m_4=1,2$ we also observe a weak signal at the ground state energy, owing to hybridization of rotational and vibrational states at non-C4IM. \n\nIn Fig.~\\ref{figSTMrot} we also compare our numerical results to the expected integrated spectrum from the spinon-chargon toy model. In the latter we include small shifts on the energy axis and weak broadening as fit parameters to obtain better agreement. As for the ARPES spectra at the nodal point, shown in the main text, we find that the toy model reasonably predicts the overall shape even of the incoherent part of the spectrum and the strong suppression of spectral weight at high energies.\n\n\\begin{figure}[t!]\n\\centering\n\\epsfig{file=FIG5.pdf, width=0.49 \\textwidth}\n\\caption{\\textbf{Rotational STM spectra.} The integrated rotational ARPES signal $I^{(m)}_{\\rm rot}(\\omega)$ which can be measured by the proposed multi-photon rotational STM scheme is calculated for the $t-J$ model at $t\/J=3$. The rotational meson states are still clearly visible as pronounced peaks above the ground state. We consider an extended $L_x=40\\times 4$-leg cylinder, for which the rotational ARPES signal is integrated over $k_x$ and summed over all discrete values of $k_y=-\\pi\/2,0,\\pi\/2,\\pi$. We compare our numerical DMRG results (solid lines) with predictions by the spinon-chargon toy model (shaded areas); for the latter, the four lowest lying states were broadened by $J\/4$ (similar to the observed Fourier broadening in the DMRG numerics) while all higher excited states were artificially broadened by $J$. In addition the toy model curves for $m_4=1$ ($m_4=2$) were shifted by a fitted $\\Delta \\omega = 0.9 J$ ($\\Delta \\omega = 1.2 J$) towards the lowest-lying quasiparticle peak. \n}\n\\label{figSTMrot}\n\\end{figure}\n\n\n\n\\subsection{TD-DMRG simulations}\n\\label{MethodsB}\nWe use time-dependent matrix product state methods \\cite{Schollwock2011,Kjall2013,Zaletel2015,Paeckel2019a}, in particular the TeNPy package \\cite{Hauschild2018SciPost,hauschildTenpy}, to numerically calculate the rotational ARPES spectrum introduced in the main text. To this end we start from the numerically obtained ground state $\\ket{\\Psi_0}$ of the Heisenberg model on a $4 \\times L_x$ cylinder, and apply the rotational operator $\\hat{R}_{m,\\sigma}(\\vec{j})$ introduced in Eq.~\\eqref{eqDefRmj} below.\nWe assume that the resulting one-hole states are described by the $t-J$ model on the same lattice. In two dimensions and for a single hole, the Hamiltonian of the $t-J$ model becomes~\\cite{Auerbach1998},\n\\begin{equation}\n\\H_{t-J} =  -t \\sum_{\\ij, \\sigma} \\hat{\\mathcal{P}} \\left( \\hat{c}^\\dagger_{\\vec{i},\\sigma} \\c_{\\vec{j},\\sigma} + h.c. \\right) \n\\hat{\\mathcal{P}}+ J \\sum_\\ij \\hat{\\mathbf{S}}_{\\vec{i}} \\cdot \\hat{\\mathbf{S}}_{\\vec{j}},\n\\label{eq:tjmodel}\n\\end{equation}\nwhere $\\hat{\\mathcal{P}}$ projects on states with less than two fermions $\\c^{(\\dagger)}_{\\vec{j},\\sigma}$ per site. The first term describes tunneling of holes with amplitude $t$ and the second term denotes spin-exchange interactions with coupling constant $J=4 t^2\/U$, where $U$ is the Hubbard interaction. \n\nTo evaluate the rotational spectral function $ A_{\\rm rot}^{(m)}(\\vec{k},\\omega)$ from Eq.~\\eqref{eqDefArot}, we express it in real space and time,\n\\begin{equation}\n A_{\\rm rot}^{(m)}(\\vec{k},\\omega)  = \\int_0^{\\infty} dt \\sum_{\\vec{j}} e^{-i \\vec{k} \\cdot \\vec{j}} \\mathcal{G}_{\\rm rot}^{(m_4)}(\\vec{j},t),\n \\label{eqArotTime}\n\\end{equation}\nsee also Ref.~\\cite{Bohrdt2020}, where the rotational Green's function in real space is defined as \n\\begin{equation}\n\\mathcal{G}_{\\rm rot}^{(m_4)}(\\vec{j},t) = \\theta(t) \\sum_\\sigma \\bra{\\Psi_0} \\hat{R}^\\dagger_{m_4,\\sigma}(\\vec{j},t) \\hat{R}_{m_4,\\sigma}(\\vec{0},0) \\ket{\\Psi_0},\n\\end{equation}\nwith the corresponding rotational operator\n\\begin{equation}\n\\hat{R}_{m_4,\\sigma}(\\vec{j}) =   \\sum_{\\vec{i}: \\ij} e^{i m_4 \\varphi_{\\vec{i} - \\vec{j}}}  \\sum_{\\sigma'} \\hat{c}^\\dagger_{\\vec{j},\\sigma'} \\c_{\\vec{i},\\sigma'} \\c_{\\vec{j},\\sigma}.\n\\label{eqDefRmj}\n\\end{equation}\n\nWe start by numerically calculating the ground state of the Heisenberg model, using a bond dimension of $\\chi=600$. Subsequently, we apply the rotational operator as defined in Eq.~\\eqref{eqDefRmj} in the origin and time-evolve the resulting state under the $t-J$ Hamiltonian using time-dependent matrix product state methods \\cite{Kjall2013,Zaletel2015}. We thus obtain the rotational Green's function in real space and time. We use linear prediction to increase the time window and multiply our data with a Gaussian envelope in order to minimize the weight of the data generated by said linear prediction in the spectrum \\cite{Verresen2018,Bohrdt2020}. Finally, we perform a Fourier transformation in time and space to obtain the spectral function $ A_{\\rm rot}^{(m)}(\\vec{k},\\omega)$ from Eq.~\\eqref{eqDefArot}. We carefully checked our results for convergence with the bond dimension.\n\n\n\\subsection{Spinon-chargon toy model}\n\\label{MethodsC}\nThe identification of rotational and vibrational resonances in the spectrum provide compelling evidence that magnetic polarons are composite objects with an internal structure. We describe the latter by an effective theory, which models magnetic polarons as bound states of spinons and chargons connected by a string on a square lattice, see Fig.~\\ref{figRotSetup} (a), with a linear string tension calculated from spin-correlations in the undoped parent AFM \\cite{Grusdt2018PRX}. In addition, we extend earlier approaches \\cite{Bulaevskii1968,Brinkman1970,Trugman1988,Shraiman1988a,Grusdt2018PRX,Grusdt2019PRB} by including spinon dynamics explicitly in our theory. Details of our theoretical description are presented in the supplements \\ref{SuppMatToyModel}.\n\nIn Fig.~\\ref{figRotSetup} (c) we compare our DMRG spectra to predictions by the spinon-chargon toy model. To capture Fourier broadening present in our DMRG results, we broadened the lowest rotational and vibrational peaks in the toy model by $\\sigma_0=J\/4$. For a better comparison of the overall spectral weight and shape, we added small overall energy shifts $\\Delta \\omega$ separately for each $m_4$ and introduced broadening $\\sigma_1 = J$ of all higher excited states. Such broadening is expected to arise from couplings to magnon excitations in the AFM \\cite{Grusdt2018PRX,Wrzosek2020} which we neglect in our toy model calculation so far. \n\nThe resulting toy-model prediction is in good agreement with the full numerical spectra: It captures the spectral weight of the low-energy mesonic resonances $\\mathsf{1S}$, $\\mathsf{2S}$, $\\mathsf{1P}$, $\\mathsf{1D}$ and $\\mathsf{1F}$. Remarkably, even the spectral features at higher energies are correctly described. In particular this includes the strong suppression of spectral weight in the $m_4=0$ rotational spectrum (red line in Fig.~\\ref{figRotSetup}) between $- \\omega = -5 J$ to $\\omega = 0$, which is followed by a broad continuum at higher energies. This should be contrasted with the non-vanishing spectral weight found in the same frequency range for $m_4 \\neq 0$ rotational spectra and for the standard ARPES spectrum with $m_4=0$ (bottom line in Fig.~\\ref{figRotSetup}). The tails at the highest energies are also correctly described by the toy model.\n\nIn Fig.~\\ref{figReggeTraj} we compare Regge-like trajectories. Here we used the simplified toy model \\cite{Grusdt2018PRX} (solid lines in Fig.~\\ref{figReggeTraj}) which neglects spinon dynamics and explains the characteristic scaling with $J\/t$, without any free fit parameters. A comparison to the full spinon-chargon toy model including spinon dynamics is provided in Fig.~\\ref{figToyModelRegge} in the supplements (section \\ref{SecToyModelRes}): There we confirm the power-laws from Eq.~\\eqref{eqErotScaling}, \\eqref{eqEvibScaling} and find similar quantitative agreement as in Fig.~\\ref{figReggeTraj}, again without any free fit parameters. Notably, the refined spinon-chargon toy model with spinon dynamics predicts a weak splitting of $\\mathsf{1P}$\/$\\mathsf{1F}$ and $\\mathsf{1D}$ resonances as found by DMRG. Further comparison of Regge-like trajectories to the full spinon-chargon toy model can be found in the supplements \\ref{SuppMatAddNum}.\n\nIn the insets in Fig.~\\ref{figSpectralLines} (b) and (c) we show full spectral cuts at low energy predicted by the spinon-chargon toy model, without any free fit parameters. As observed in our full DMRG results, we find that the rotational mesonic states have much weaker dispersion than the $\\mathsf{1S}$ magnetic polaron ground state. The toy model predicts some entirely flat bands, similar to the flat bands predicted for mesonic bound states of two identical partons \\cite{Shraiman1988a}, in addition to weakly dispersing bands. The overall distribution of spectral weight is also qualitatively captured by the toy model.\n\n\n\n\n\\section*{Acknowledgements}\nThe authors thank M. Knap, Z.X. Shen, A. Cavalleri, E.-A. Kim, I. Morera Navarro, S. Sachdev, U. Schollw\\\"ock, I. Bloch, M. Greiner, J. Koepsell, and F. Pollmann for fruitful discussions.\nThis research was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -- EXC-2111 -- 390814868. ED was supported by the ARO grant number W911NF-20-1-0163, the NSF grant EAGER-QAC-QSA 2222-206-2014111, the NSF grant OAC-1934714, and the Harvard-MIT CUA.\n\n\n\\section*{References}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Properties of the Operators}\n\\label{app:lm:costgstar}\n\\begin{lemma}\n\\label{lm:KPrelation_0}\nConsider matrices $P,Q,R,A,B$ of appropriate dimensions with $P,Q$ being PSD matrices and $R$ being  a PD matrix. Define $\\Phi(P,K,Q,R,A,B):=Q + K^\\tp RK\n+ (A+BK)^\\tp P(A+BK)$. Then,\n\\begin{enumerate}\n\\item[(i)]\n\\begin{small}\n\\begin{align}\n\\Omega(P,Q,R,A,B) &= \\Phi(P,\\Psi(P,R,A,B),Q,R,A,B)\\notag \\\\& = \\min_K \\Phi(P,K,Q,R,A,B).\n\\label{eq:relation1}\n\\end{align}\n\\end{small}\nNote that the minimization is in the sense of partial order $\\preceq$, that is, the minimum value $\\Omega(P,Q,R,A,B) \\preceq \\Phi(P,K,Q,R,A,B)$ for all $K$.\n\n\\item[(ii)] Furthermore, for  PSD matrices $Y_1 $ and $Y_2$ such that $Y_1 \\preceq Y_2$, we have\n\\begin{align}\n\\Omega(Y_1,Q,R,A,B) \\preceq \\Omega(Y_2,Q,R,A,B).\n\\label{eq:relation2}\n\\end{align}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nThe statements in the above lemma can be found in the literature (see, for example, \\cite[Chapter 2]{whittle_Optimal_Control}). We provide a proof for completeness.\nIt can be established by straightforward algebraic manipulations that \n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n&\\Phi(P,K,Q,R,A,B) = \\Omega(P,Q,R,A,B) \\notag \\\\\n&+(K - \\Psi(P,R,A,B))^\\tp\\mathcal{R}(K - \\Psi(P,R,A,B)), \\label{eq:new_riccati_1}\n\\end{align}\n\\end{small}\nwith $\\mathcal{R}= R + B^\\tp PB$. Then \\eqref{eq:new_riccati_1} implies that  $\\Phi(P,K,Q,R,A,B)$ is minimized when $K=\\Psi(P,R,A,B)$ and the minimum value is $\\Omega(P,Q,R,A,B)$.\n\nFor PSD matrices $Y_1 $ and $Y_2$ such that $Y_1 \\preceq Y_2$, it is straightforward to see that $\\Phi(Y_1,K,Q,R,A,B) \\preceq \\Phi(Y_2,K,Q,R,A,B)$ for any $K$. Hence, \n\\begin{align}\n \\min_K \\Phi(Y_1,K,Q,R,A,B) \\preceq  \\min_K \\Phi(Y_2,K,Q,R,A,B).\n \\label{min_K_inequality}\n\\end{align}\nFrom \\eqref{min_K_inequality} and \\eqref{eq:relation1}, it follows that \\eqref{eq:relation2} is correct.\n\\end{proof}\n\n\n\\section{Proof of Lemma \\ref{lm:MJ_infinite}}\n\\label{app:lm_MJ_infinite}\nIf matrices $P^{\\diamond}_{t}(m)$, $m \\in \\mathcal{M}$, converge as $t \\to -\\infty$ to PSD matrices $P^{\\diamond}_{*}(m)$, then by continuity, the collection of PSD matrices $P^{\\diamond}_* = \\{P^{\\diamond}_*(0),\\ldots, P^{\\diamond}_*(M)\\}$ satisfy the DCARE in \\eqref{eq:CARE_infinite}. Since the DCARE \\eqref{eq:CARE_infinite} has a PSD solution  $P^{\\diamond}_* = \\{P^{\\diamond}_*(0),\\ldots, P^{\\diamond}_*(M)\\}$, then from \\cite[Proposition 7]{costa1995discrete} and the SD assumption of the MJLS, it is also a stabilizing solution of the DCARE (\\cite[Definition 3]{costa1995discrete} and \\cite[Definition 4.4]{costa2006discrete}). Then, the MJLS is SS from the definition of the stabilizing solution.\n\n\nOn the other hand, if the MJLS is SS, under the SD assumption of the MJLS, \\cite[Corollary A.16]{costa2006discrete} ensures the existence of a stabilizing solution of the DCARE in \\eqref{eq:CARE_infinite}.\nThe solution is also the unique PSD solution from \\cite[Theorem A. 15]{costa2006discrete} (by taking $X=0$ in Theorem A. 15). Then from \\cite[Proposition A. 23]{costa2006discrete}, matrices $P^{\\diamond}_{t}(m)$, $m \\in \\mathcal{M}$, converge as $t \\to -\\infty$ to PSD matrices $P^{\\diamond}_{*}(m)$.\n\n\\section{Proof of Lemma \\ref{lm:pc_2C}}\n\\label{proof_lm:pc_2C}\nBecause of Lemma \\ref{equality_recursions_2C},  $P^0_t = P^{\\diamond}_t(0)$ and  $P^1_t = P^{\\diamond}_t(1)$, where matrices $P^{\\diamond}_t(0), P^{\\diamond}_t(1)$ are defined  by \\eqref{eq:P_MJ_init} - \\eqref{P_MJ_cmp_2C_1} for the auxiliary MJLS. Thus,  we can focus on the convergence of matrices $P^{\\diamond}_t(0), P^{\\diamond}_t(1)$.\n\n\nTo investigate the convergence of $P^{\\diamond}_t(0), P^{\\diamond}_t(1)$,  we first show that \n the auxiliary MJLS described by \\eqref{eq:MJLS_A}-\\eqref{eq:MJLS_theta} is SS if and only if $p^1 < p_c^1$ where $p_c^1$ is the critical threshold given by \\eqref{eq:pc_2c}. According to Lemma \\ref{lm:ss}, the MJLS is SS if and only if there exist matrices $K^{\\diamond}(m)$, $m \\in \\{0,1\\}$, such that $\\rho(\\mathcal{A}_s) < 1$. For the MJLS described by \\eqref{eq:MJLS_A}-\\eqref{eq:MJLS_theta}, we can find $\\mathcal{A}_s$ from \\eqref{eq:bigmatrix_ss} as follows\n\\begin{align}\n\\mathcal{A}_s = \\begin{bmatrix}\nA_s(0)\\otimes A_s(0) & (1-p^1)A_s(1)\\otimes A_s(1) \\\\\n \\mathbf{0} & p^1A_s(1)\\otimes A_s(1)\n\\end{bmatrix},\n\\label{eq:bigmatrix_As}\n\\end{align}\nwhere $A_s (m) = A^{\\diamond} (m) +  B^{\\diamond}(m) K^{\\diamond}(m)$, $m \\in \\{0,1\\}$. Since the matrix $\\mathcal{A}_{s}$ is upper-triangular, it is Schur stable if and only if all its diagonal blocks are  Schur stable. \n\nSince $A^{\\diamond}(0) = A, B^{\\diamond}(0) = B$ and  $(A,B)$ is stabilizable from Assumption \\ref{assum:det_stb_2C}, there exists $K^{\\diamond}(0)$ such that  $\\rho\\big(A_s(0)\\otimes A_s(0)\\big)$, which is equal to $\\big(\\rho\\big(A_s(0)\\big)\\big)^2$, is less than $1$. Therefore,  the MJLS is SS if and only if $\\rho\\big(p^1 A_s(1)\\otimes A_s(1)\\big)<1$ for some $K^{\\diamond}(1)$. Note that $\\rho\\big(p^1 A_s(1)\\otimes A_s(1)\\big) = p^1 \\times \\big(\\rho\\big(A_s(1)\\big)\\big)^2$. Therefore, the MJLS is SS if and only if $\\frac{1}{\\sqrt{p^1}} > \\rho\\big(A_s(1)\\big)$ for some $K^{\\diamond}(1)$. Since $A^{\\diamond}(1)= A$ and $B^{\\diamond}(1) = [\\mathbf 0,B^{11}]$, it follows then that the MJLS is SS iff \n\\[\\frac{1}{\\sqrt{p^1}} > \\rho\\big(A + B^{11} \\tilde K^{\\diamond}(1)\\big),\\] for some $\\tilde K^{\\diamond}(1)$. This condition is equivalent to $p^1 < p_c^1$ where $p_c^1$ is the critical threshold given by \\eqref{eq:pc_2c}.\n\nNext, we show that the auxiliary MJLS described by \\eqref{eq:MJLS_A}-\\eqref{eq:MJLS_theta} is SD. To this end, we can follow an argument similar to the one  described above for establishing that the MJLS is SS and use part 2 of Lemma \\ref{lm:ss} to show that the MJLS is SD if and only if there exist matrices $H^{\\diamond}(0)$ and $H^{\\diamond}(1)$ such that $\\rho\\big(A_d(0)\\otimes A_d(0)\\big)<1$ and $\\rho\\big(p^1 A_d(1)\\otimes A_d(1)\\big)<1$. Since $A^{\\diamond}(0) = A^{\\diamond}(1) = A, Q^{\\diamond}(0) = Q^{\\diamond}(1) = Q$ and $(A,Q)$ is detectable from Assumption \\ref{assum:det_stb_2C},  there exist matrices $H^{\\diamond}(0)$ and $H^{\\diamond}(1)$ such that $\\rho\\big(A_d(0)\\otimes A_d(0)\\big)<1$ and $\\rho\\big(p^1 A_d(1)\\otimes A_d(1)\\big)<1$. Hence, the MJLS is SD.\n\nThus, the MJLS of \\eqref{eq:MJLS_A}-\\eqref{eq:MJLS_theta} is SD for any $p^1$ and it is SS if and only if $p^1 < p_c^1$. It then follows from Lemma \\ref{lm:MJ_infinite} that matrices $P^{\\diamond}_{t}(m)$, $m \\in \\{0,1\\}$, converge as $t \\to -\\infty$ to\n PSD matrices  $P^{\\diamond}_{*}(m), m \\in \\{0,1\\}$ that satisfy the steady state version of \\eqref{P_MJ_cmp_2C_0}-\\eqref{P_MJ_cmp_2C_1} (i.e, equations \\eqref{eq:P_finite_2C_fixed} - \\eqref{eq:tildeP_finite_2C_fixed}) if and only if $p^1 < p_c^1$.  This proves the lemma.\n\n\n\\section{Proof of Lemma \\ref{lm:Q2_2C}, parts 1 and 2}\\label{sec:lm_Q2_2C}\nLet $g^*$ denote the strategies described  by \\eqref{eq:K_finite_2C_fixed}-\\eqref{eq:estimator_inf_2C}. We want to show that for any $g \\in \\mathcal G$, $J_{\\infty}(g) \\geq J_{\\infty}(g^*)$ and that  $J_{\\infty}(g^*)$ is finite. We will make use of the following claim. \n\\begin{claim}\\label{claim:cost_optimal_2C}\nFor the strategies $g^*$ described  by \\eqref{eq:K_finite_2C_fixed}-\\eqref{eq:estimator_inf_2C}, the following equation is true:\n\\begin{align}\nJ_{T}(g^{*}) =&(T+1) \\tr (\\Lambda_*) - \\ee^{g*} [V_{T+1} ],\n\\label{eq:costforgstar_2C}\n\\end{align}\nwhere $J_T(g^*)$ is the finite horizon cost of $g^*$ over a horizon of duration $T$, $\\Lambda_* = (1-p^1) P_{*}^0+p^1 P_{*}^1$ and for any $t\\geq 0$,\n\\begin{align}\nV_t = \\hat X_{t}^\\tp P_*^0 \\hat X_t + \\tr \\big(P_*^{1}\\cov(X_{t}|H^0_{t}) \\big).\n\\label{V_t_2C}\n\\end{align}\n\\end{claim}\n\\begin{proof}\nSee Appendix \\ref{proof_claim:cost_optimal_2C} for a proof of this claim.\n\\end{proof}\nBased on Claim \\ref{claim:cost_optimal_2C}, the infinite horizon average cost  for $g^*$ is given as \n\\begin{align}\nJ_{\\infty}(g^*)\n= &\\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} J_{T}(g^*)\n\\notag\\\\\n= & \\tr (\\Lambda_*)  - \\liminf_{T\\rightarrow\\infty} \\frac{\\ee^{g^*}[V_{T+1}]}{T+1}\n\\leq \\tr (\\Lambda_*),\n\\label{bound_J_infty}\n\\end{align}\nwhere the last inequality holds because $V_{T+1} \\geq 0$. \n\nFor $n =0,1$, define $Y_{0}^n = 0$, and for $k=0,1,2,\\dots,$\n\\begin{align}\n&Y_{k+1}^0 =  \\Omega(Y_{k}^0,Q,R,A,B),\n\\label{eq:Y_k_0_2C}\n\\\\\n&Y_{k+1}^1 =  \\Omega \\big((1-p^1) Y_{k}^0+p^1 Y_{k}^1,Q,R^{11},A,B^{11}\\big).\n\\label{eq:Y_k_n_2C}\n\\end{align}\nIt's easy to check that for  $n=0,1,$  $Y_k^n = P_{T+1-k}^n$ for all  $k \\geq 0$, and that $\\lim_{k \\rightarrow \\infty} Y_k^n = \\lim_{t \\rightarrow -\\infty} P_t^n  = P^n_*$.\n\nFurther, let's define  $\\Lambda_k =  (1-p^1) Y_{k}^0+p^1 Y_{k}^1$. From \\eqref{eq:opcost_finite_2C} of Lemma \\ref{lm:opt_strategies_2C}, we know that the optimal finite horizon cost is given as \n\\begin{align}\nJ^*_{T} &=  \\sum_{t = 0}^T \\tr \\big((1-p^1)P_{t+1}^0+p^1 P_{t+1}^1 \\big) \\notag \\\\\n&=\\sum_{k = 0}^T \\tr \\big((1-p^1)P_{T+1-k}^0+p^1 P_{T+1-k}^1 \\big) \\notag \\\\\n&=\\sum_{k = 0}^T \\tr \\big((1-p^1)Y_k^0+p^1 Y_k^1 \\big) \\notag \\\\\n&= \\sum_{k = 0}^T \\tr(\\Lambda_k). \\label{eq:Jstar}\n\\end{align}\n\n\nWe can therefore write\n\\begin{align}\n \\lim_{T\\rightarrow \\infty}\\frac{1}{T+1} J^*_{T} \n  &=  \\lim_{T\\rightarrow \\infty}\\frac{1}{T+1} \\sum_{k = 0}^T \\tr (\\Lambda_k)\n= \\tr (\\Lambda^*),\n\\label{limit_optimal_finite_cost}\n\\end{align}\nwhere the last equality is correct because $\\lim_{k \\rightarrow \\infty} Y_k^n = P^n_*$ for $n=0,1$.\n\nNow, for any $g \\in \\mathcal G$,\n\\begin{align}\nJ_{\\infty}(g)\n= &\\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} J_{T}(g)\n\\notag\\\\\n\\geq &\\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} J_{T}^*\n= \\tr (\\Lambda^*)\n\\geq J_{\\infty}(g^*),\n\\label{eq:lowerbound}\n\\end{align}\nwhere the first inequality is true because by definition $J_T^* = \\inf_{g' \\in\\mathcal{G}} \nJ_T(g') \\leq J_T(g)$ for any $g \\in \\mathcal G$,  the  second equality is true because of \\eqref{limit_optimal_finite_cost} and the last inequality is true because of \\eqref{bound_J_infty}.\nHence, $g^*$ is optimal for Problem \\ref{problem_infinite_2C}, and the optimal cost is finite and equal to $\\tr (\\Lambda^*)$.\n\n\n\n\n\n\n\n\n\n\n\n\\input{new_proof_claim1}\n\n\\input{stability_proof_new}\n\n\\section{Proof of Lemma \\ref{lm:Q3}} \\label{sec:lmQ3}\n Consider the matrices $Y^n_k$, $n=0,1, k =0,1,\\ldots,$ defined by $Y^n_0 =0$ and the recursions in \\eqref{eq:Y_k_0_2C} and \\eqref{eq:Y_k_n_2C}. Since  matrices $P_t^n$, $n =0,1,$ do not converge as $t \\to -\\infty$, it follows that  matrices $Y_k^n$, $n=0,1$, do not converge as $k \\rightarrow \\infty$ (recall that $Y_k^n = P_{T+1-k}^n $ for $n=0,1$ and $k \\geq 0$). \n  \n Recall from  \\eqref{eq:Jstar} that $J^*_T = \\sum_{k = 0}^T \\tr(\\Lambda_k)$, where $\\Lambda_k =  (1-p^1) Y_{k}^0+p^1 Y_{k}^1$. Also, from the first inequality in \\eqref{eq:lowerbound}, recall that $J_{\\infty}(g) \\geq \\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} J_{T}^*$ for any strategy $g$. Therefore, to show that no strategy can achieve finite cost, it suffices to show that \n \\begin{equation}\n \\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} J_{T}^* =  \\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} \\sum_{k = 0}^T \\tr(\\Lambda_k) = \\infty.\n \\end{equation}\n\nTo do so, we first show that the sequence $\\{Y_k^n, k=0,1,\\ldots\\}$ is monotonically increasing\\footnote{ in the sense of the partial order $\\preceq$.} for $n =0,1$. To this end, note that $Y_1^n \\succeq Y_0^n=0$ for $n \\in \\{0,1\\}$. Furthermore, the monotonic property of the operator $\\Omega(\\cdot)$ (proved in part (ii) of Lemma \\ref{lm:KPrelation_0} in Appendix \\ref{app:lm:costgstar}) implies that for $n=0,1,$ and for all $k \\geq 0$, $Y_{k+1}^n \\succeq Y_{k}^n$. Now, if the sequences $\\{Y_k^n, k\\geq 0\\}$,  $n =0,1,$ are bounded, they will converge due to the monotone behavior. This contradicts the fact that these sequences do not converge as $k \\to \\infty$. Therefore, at least one of the two sequences $\\{Y_k^0, k\\geq0\\}$, $\\{Y_k^1, k\\geq0\\}$  is unbounded. Consequently, the sequence $\\{\\Lambda_k, k\\geq 0\\}$ is unbounded. Hence, $\\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} \\sum_{k = 0}^T \\tr(\\Lambda_k) = \\infty.$ This completes the proof.\n\n\n\\section{Proof of Lemma \\ref{lm:opt_strategies_new_rep}}\n\\label{proof_lm:opt_strategies_new_rep}\n\n\n\nBy comparing \\eqref{eq:P_N_init}-\\eqref{eq:P_finite} with  \\eqref{eq:barP_init}-\\eqref{eq:barP_finite_0}, it is straightforward to observe that $P_t^0 = \\bar P_t^0$ for all $t$. We will now show by induction that at any time $t$, $P_t^n = \\bar P_t^n$ for $n =1,\\ldots,N$.\nFirst note that by definition, $P_{T+1}^{n} = \\mathbf{0}$ and $\\bar P^n_{T+1} = \\mathbf{0}$ for $n =1,\\ldots,N$. Hence, \\eqref{new_rep2} is correct at time $T+1$. Now, assume that \\eqref{new_rep2} is correct at time $t+1$ (induction hypothesis). Then, from \\eqref{eq:barP_finite} and  the induction hypothesis, we have for $n =1,\\ldots,N$,\n\\begin{align}\n\\bar P_t^{n} &=  \\Omega \\big((1-p^n) P_{t+1}^0+p^n \\mathcal{L}_{zero}(P^0_{t+1}, P_{t+1}^{n},n,n), \\notag \\\\\n& \\hspace{1.0cm}  \\mathcal{L}_{zero}(Q,Q^{nn},n,n), \\mathcal{L}_{iden}(R,R^{nn},n+1), \\notag \\\\\n & \\hspace{1.0cm}  \\mathcal{L}_{zero}(A,A^{nn},n,n), \\mathcal{L}_{zero}(B,B^{nn},n,n+1) \\big) \\notag \\\\\n&=\n\\mathcal{L}_{zero}(Q,Q^{nn},n,n)  + \\mathbb{T}_1 - \\mathbb{T}_2 (\\mathbb{T}_3)^{-1} (\\mathbb{T}_2)^{\\tp},\n\\label{P_MJ_n}\n\\end{align}\nwhere \n\\begin{align}\n\\label{T_1_def}\n&\\mathbb{T}_1 = \\mathcal{L}_{zero}(A,A^{nn},n,n)^{\\tp} \\bar{\\bar{P}}_{t+1} \\mathcal{L}_{zero}(A,A^{nn},n,n) \\\\\n\\label{T_2_def}\n&\\mathbb{T}_2 = \\mathcal{L}_{zero}(A,A^{nn},n,n)^{\\tp} \\bar{\\bar{P}}_{t+1} \\mathcal{L}_{zero}(B,B^{nn},n,n+1) , \\\\\n\\label{T_3_def}\n&\\mathbb{T}_3 = \\mathcal{L}_{iden}(R,R^{nn},n+1) \n\\notag \\\\\n&+ \\mathcal{L}_{zero}(B,B^{nn},n,n+1)^{\\tp} \\bar{\\bar{P}}_{t+1} \\mathcal{L}_{zero}(B,B^{nn},n,n+1),\n\\end{align}\nand we have defined $\\bar{\\bar{P}}_{t+1} = (1-p^n) P_{t+1}^0+p^n \\mathcal{L}_{zero}(P^0_{t+1}, P_{t+1}^{n},n,n)$.\n\nNote that from the definitions of operators $\\mathcal{L}_{zero}$ and $\\mathcal{L}_{iden}$ in \\eqref{L_zero}-\\eqref{L_iden}, it is straightforward to observe that the block dimensions of $\\mathbb{T}_1, \\mathbb{T}_2, \\mathbb{T}_3$ are the same as the block dimensions of $A,B,B^{\\tp} B$, respectively (They are block matrices of sizes $N \\times N$, $N \\times (N+1)$, and $(N+1) \\times (N+1)$, respectively). Therefore, through straightforward algebraic manipulations, we can get\n\\begin{align}\n\\label{T_1}\n&\\mathbb{T}_1 = \\mathcal{L}_{zero}(A,\\mathbb{\\tilde T}_1,n,n), \\\\\n\\label{T_2}\n&\\mathbb{T}_2 =  \\mathcal{L}_{zero}(B,\\mathbb{\\tilde T}_2,n,n+1), \\\\\n\\label{T_3}\n&\\mathbb{T}_3 = \\mathcal{L}_{iden}(B^{\\tp}B,\\mathbb{\\tilde T}_3,n+1),\n\\end{align}\nwhere\n\\begin{align}\n\\label{tilde_T_1}\n&\\mathbb{\\tilde T}_1 = (A^{nn})^{\\tp}[(1-p^n)[P_{t+1}^0]_{n,n}+p^n P_{t+1}^{n}]A^{nn}, \\\\\n\\label{tilde_T_2}\n&\\mathbb{\\tilde T}_2 = (A^{nn})^{\\tp}[(1-p^n)[P_{t+1}^0]_{n,n}+p^n P_{t+1}^{n}]B^{nn}, \\\\\n\\label{tilde_T_3}\n&\\mathbb{\\tilde T}_3 = R^{nn} + (B^{nn})^{\\tp}[(1-p^n)[P_{t+1}^0]_{n,n}+p^n P_{t+1}^{n}]B^{nn}.\n\\end{align}\n\nFurther, since $\\mathbb{T}_3$ is a block diagonal matrix, we have \n\\begin{align}\n& (\\mathbb{T}_3)^{-1} =\\mathcal{L}_{iden} \\big(B^{\\tp}B,(\\mathbb{\\tilde T}_3)^{-1},n+1 \\big).\n\\label{T_3_inv}\n\\end{align}\n\nNow, using \\eqref{T_1}-\\eqref{T_3_inv} and the fact that  matrices $A, Q, BB^{\\tp}$ have the same size as matrix $P^0_t$ (They are block matrices of size $N \\times N$), \\eqref{P_MJ_n} can be simplified to\n\\begin{align}\n\\bar P_t^{n} &= \\mathcal{L}_{zero} \\big(P^0_t,  Q^{nn} + \\mathbb{\\tilde T}_1 - \\mathbb{\\tilde T}_2 (\\mathbb{\\tilde T}_3)^{-1} (\\mathbb{\\tilde T}_2)^{\\tp},n,n \\big) \\notag \\\\\n&=   \\mathcal{L}_{zero}(P^0_t, P_t^{n},n,n),\n\\end{align}\nwhere the last equality is true because of the definition of $P_t^{n}$ in \\eqref{eq:tildeP_finite}. Hence, \\eqref{new_rep2} is true at time $t$. This completes the proof.\n\n\n\\section{Proof of Lemma \\ref{lm:pc_NC}}\n\\label{proof_lm:pc}\n\n\n\\begin{figure*}[t]\n\\begin{small}\n\\begin{align}\n\\mathcal{A}_s =\n\\begin{blockarray}{ccccc}\n\\begin{block}{[ccccc]}\n                     A_s(0)\\otimes A_s(0) & (1-p^1)A_s(1)\\otimes A_s(1)& (1-p^2)A_s(2)\\otimes A_s(2) & \\ldots & (1-p^n)A_s(N)\\otimes A_s(N) \\\\ \\\\\n                     \\mathbf{0} & p^1A_s(1)\\otimes A_s(1) &\\mathbf{0} &\\ldots & \\mathbf{0} \\\\ \\\\\n                     \\vdots &\\ddots  &p^2 A_s(2)\\otimes A_s(2)  & \\ddots & \\vdots \\\\ \\\\\n                     \\vdots &   &\\ddots  &\\ddots & \\mathbf{0} \\\\ \\\\\n                     \\mathbf{0} & \\ldots &\\ldots & \\mathbf{0}  & p^n A_s(N)\\otimes A_s(N)\\\\\n\\end{block}\n\\end{blockarray}\n\\label{eq:bigmatrix3}\n\\end{align}\n\\end{small}\n\\hrule\n\\end{figure*}\n\\begin{figure*}[t]\n\\begin{small}\n\\begin{align}\nA_s(n) &= A^{\\diamond}(n) + B^{\\diamond}(n) K^{\\diamond}(n)  \\notag \\\\\n&= \\begin{blockarray}{cccl}\n\\text{$1:n-1$} &n &\\text{$n+1:N$}  &  \\\\\n\\begin{block}{[ccc]l}\n                      \\text{\\large 0}   & \\text{\\large 0} &\\text{\\large 0} & \\text{$1:n-1$} \\\\\n   B^{nn} [K^{\\diamond}(n)]_{n+1,1:n-1} & A^{nn} + B^{nn} [K^{\\diamond}(n)]_{n+1,n} & B^{nn} [K^{\\diamond}(n)]_{n+1,n+1:N} &n \\\\\n                      \\text{\\large 0}   & \\text{\\large 0} & \\text{\\large 0}&\\text{$n+1:N$}  \\\\\n\\end{block}\n\\end{blockarray}.\n\\label{A_c_l}\n\\end{align}\n\\end{small}\n\\hrule\n\\end{figure*} \n\n\nFrom Lemma \\ref{lm:opt_strategies_new_rep}, we know that the convergence of matrices $P_t^{n}$, $n \\in \\mathcal{\\overline N}$, is equivalent to the convergence of matrices $\\bar P_t^{n}$, $n \\in \\mathcal{\\overline N}$. Further, because of Lemma \\ref{equality_recursions_NC}, $\\bar P^n_t = P^{\\diamond}_t(n)$, $n \\in \\mathcal{\\overline N}$, where matrices $P^{\\diamond}_t(n)$, $n \\in \\mathcal{\\overline N}$, are defined by \\eqref{eq:P_N_MJ_init}-\\eqref{P_MJ_cmp_NC_1} for the auxiliary MJLS. Thus, in order to study the the convergence of matrices $P_t^{n}$, $n \\in \\mathcal{\\overline N}$, we can focus on the convergence of matrices $P^{\\diamond}_t(n)$, $n \\in \\mathcal{\\overline N}$.\n\nTo investigate the convergence of $P^{\\diamond}_t(n)$, $n \\in \\mathcal{\\overline N}$, we first show that the auxiliary MJLS described by \\eqref{A_mj}-\\eqref{transition_prob} is SS if and only if $p^n < p_c^n$ for all $n \\in \\mathcal{N}$. To do so, we can follow a methodology similar to the one used to prove Lemma \\ref{lm:pc_2C} with $\\mathcal{A}_s$ defined as in \\eqref{eq:bigmatrix3} and $A_s(n)$, $n =1,\\ldots,N,$ defined as in \\eqref{A_c_l}. \n\nNext, we can use part 2 of Lemma \\ref{lm:ss} to show that the auxiliary MJLS described by \\eqref{A_mj}-\\eqref{transition_prob} is SD if and only of $\\mathcal{A}_d$ defined as in \\eqref{eq:bigmatrix4} is Schur stable. Since the matrix $\\mathcal{A}_{d}$ is upper-triangular, it is Schur stable if and only if there exist matrices $H^{\\diamond}(0)$ and $H^{\\diamond}(n)$ for $n \\in \\mathcal{N}$ such that $\\rho\\big(A_d(0)\\otimes A_d(0)\\big)<1$ and $\\rho\\big(p^n A_d(n)\\otimes A_d(n)\\big)<1$, where $A_d(n)$, $n \\in \\mathcal{N}$, defined as in \\eqref{A_c_l_2}. The existence of these matrices follows from detectability of $(A,Q)$ and $\\big(A^{nn},(Q^{nn})^{1\/2} \\big)$ for  $n \\in \\mathcal{N}$ (see Assumptions \\ref{assum:det_stb} and \\ref{assum:det_stb_2}).\nHence, the MJLS is SD.\n\n\nIt then follows from Lemma \\ref{lm:MJ_infinite} that matrices $P^{\\diamond}_{t}(n)$, $n \\in \\overline{\\mathcal{N}}$, converge as $t \\to -\\infty$  if and only if $p^n < p_c^n$ for all $n \\in \\mathcal{N}$.  Consequently,  matrices $ P^0_t,\\ldots,  P^N_t$  converge as $t \\to -\\infty$ to matrices $P_*^0,\\ldots, P_*^N$ that satisfy the coupled fixed point  equations \\eqref{eq:P_finite_NC_fixed}-\\eqref{eq:tildeP_finite_NC_fixed} if and only if  $p^n < p_c^n$ for all $n \\in \\mathcal{N}$. This proves the lemma.\n\n\\begin{figure*}[t]\n\\begin{small}\n\\begin{align}\n\\mathcal{A}_d =\n\\begin{blockarray}{ccccc}\n\\begin{block}{[ccccc]}\n                     A_d(0)\\otimes A_d(0) & (1-p^1)A_d(1)\\otimes A_d(1)& (1-p^2)A_d(2)\\otimes A_d(2) & \\ldots & (1-p^n)A_d(N)\\otimes A_d(N) \\\\ \\\\\n                     \\mathbf{0} & p^1A_d(1)\\otimes A_d(1) &\\mathbf{0} &\\ldots & \\mathbf{0} \\\\ \\\\\n                     \\vdots &\\ddots  &p^2 A_d(2)\\otimes A_d(2)  & \\ddots & \\vdots \\\\ \\\\\n                     \\vdots &   &\\ddots  &\\ddots & \\mathbf{0} \\\\ \\\\\n                     \\mathbf{0} & \\ldots &\\ldots & \\mathbf{0}  & p^n A_d(N)\\otimes A_d(N)\\\\\n\\end{block}\n\\end{blockarray}\n\\label{eq:bigmatrix4}\n\\end{align}\n\\end{small}\n\\hrule\n\\end{figure*}\n\\begin{figure*}[t]\n\\begin{small}\n\\begin{align}\n&A_d(n) = A^{\\diamond} (n) +  H^{\\diamond}(n) \\big(Q^{\\diamond}(n) \\big)^{1\/2}\n= \\begin{blockarray}{cccl}\n\\text{$1:n-1$} &n &\\text{$n+1:N$}  &  \\\\\n\\begin{block}{[ccc]l}\n                      \\text{\\large 0}   & [H^{\\diamond}(n)]_{n,1:n-1} (Q^{nn})^{1\/2} &\\text{\\large 0} & \\text{$1:n-1$} \\\\\n   \\text{\\large 0} & A^{nn} + [H^{\\diamond}(n)]_{n,n} (Q^{nn})^{1\/2} & \\text{\\large 0} &n \\\\\n                      \\text{\\large 0}   & [H^{\\diamond}(n)]_{n,n+1:N} (Q^{nn})^{1\/2} & \\text{\\large 0}&\\text{$n+1:N$}  \\\\\n\\end{block}\n\\end{blockarray}.\n\\label{A_c_l_2}\n\\end{align}\n\\end{small}\n\\hrule\n\\end{figure*} \n\n\n\n\\section{Proof of Lemma \\ref{lm:Q2_NC}} \\label{sec:proof_lm_Q2_NC}\nLet $g^*$ denote the strategies defined by \\eqref{eq:K_finite_NC_fixed}-\\eqref{eq:opt_U_NC_fixed}. We want to show that for any $g \\in \\mathcal G$, $J_{\\infty}(g) \\geq J_{\\infty}(g^*)$ and that $J_{\\infty}(g^*)$ is finite. We will make use of the following claim.\n\n\\begin{claim}\\label{claim:cost_optimal_NC}\nFor the strategies $g^*$ described  by \\eqref{eq:K_finite_NC_fixed}-\\eqref{eq:opt_U_NC_fixed}, the following equation is true:\n\\begin{align}\nJ_{T}(g^{*}) =&(T+1) \\tr (\\Lambda_*) - \\ee^{g*} [V_{T+1} ]\n\\label{eq:costforgstar_NC}\n\\end{align}\nwhere $J_T(g^*)$ is the finite horizon cost of $g^*$ over a horizon of duration $T$, $\\Lambda_* =  \\sum_{n=1}^N \\big( (1-p^n) [P_{*}^0]_{n,n}+p^n P_{*}^n \\big)$ and for any $t\\geq 0$,\n\\begin{align}\nV_t = \\hat X_{t}^\\tp P_*^0 \\hat X_t + \\sum_{n=1}^N \\tr \\big(P_*^{n}\\cov(X_{t}^n|H^0_{t}) \\big).\n\\label{V_t_NC}\n\\end{align}\n\\end{claim}\n\\begin{proof}See Appendix \\ref{Cost_of_the_Strategies} for a proof of Claim \\ref{claim:cost_optimal_NC}.\n\\end{proof}\n\n\nAlong the lines of the proof of Lemma \\ref{lm:Q2_2C} in Appendix \\ref{sec:lm_Q2_2C}, we define  $Y_{0}^0 = 0$, and for $k=0,1,2,\\dots,$\n\\begin{align}\n&Y_{k+1}^0 =  \\Omega(Y_{k}^0,Q,R,A,B),\n\\end{align}\nand for $n=1,\\ldots,N,$\n\\begin{align}\n&Y^n_0 =0, \\\\\n&Y_{k+1}^n =  \\Omega \\big((1-p^n)[Y_{k}^0]_{n,n}+p^n Y_{k}^{n},Q^{nn},R^{nn},A^{nn},B^{nn}\\big).\n\\end{align}\nIt's easy to check that for  $n=0,1,\\ldots,N,$  $Y_k^n = P_{T+1-k}^n$ for all  $k \\geq 0$, and that $\\lim_{k \\rightarrow \\infty} Y_k^n = \\lim_{t \\rightarrow -\\infty} P_t^n  = P^n_*$.\n The rest of the proof for parts 1 and 2 follows the same arguments as in Appendix \\ref{sec:lm_Q2_2C} for the proof of parts 1 and 2 of Lemma \\ref{lm:Q2_2C}. \nFor the proof of part 3, define for $n=1,\\ldots,N,$ \n \\begin{align*}\n \\bar K_*^n := \n \\Psi \\big( &(1-p^n)\\bar P_*^0+p^n \\bar P_{*}^n, R^{\\diamond}(n),  A^{\\diamond}(n), B^{\\diamond}(n) \\big),\n \\end{align*}\n where $\\bar P_*^{0:N}$ are the limits of $\\bar P_t^{0:N}$ (see Lemmas \\ref{lm:opt_strategies_new_rep} and \\ref{lm:pc_NC} and the auxiliary MJLS in Section \\ref{sec:model_N_controllers}). \nThen, it can be shown that  (i) $\\bar K_*^n = \\mathcal{L}_{zero}(K_*^0,K_*^{n},n+1,n)$ and hence (ii)\\\\ $p^n\\rho( (A^{nn} +B^{nn}K^n_*) \\otimes (A^{nn} +B^{nn}K^n_*)) = p^n \\rho (( A^{\\diamond}(n) + B^{\\diamond}(n)\\bar K_*^n ) \\otimes (A^{\\diamond}(n) + B^{\\diamond}(n)\\bar K_*^n )),  $ which  is less than 1 since the auxiliary MJLS of \\eqref{A_mj}-\\eqref{transition_prob} is SD and SS (see proof of Lemma \\ref{lm:pc_NC}). The rest of the proof uses similar arguments as in Appendix \\ref{sec:stability_proof} for the proof of part 3 of Lemma 7.\n\n\n\n\n\n\\section{Proof of Lemma \\ref{lm:Q3_NC}} \\label{sec:proof_lm_Q3_NC}\nThe proof can be obtained by following the arguments  in the proof of Lemma \\ref{lm:Q3} and defining $\\Lambda_k =  \\sum_{n=1}^N \\big( (1-p^n) [Y_{k}^0]_{n,n}+p^n Y_{k}^n \\big)$, where $Y^0_k,Y^n_k$ are as defined in Appendix \\ref{sec:proof_lm_Q2_NC}.\n\n\n\n\\input{new_proof_claim2}\n\n\n\\subsection{Summary of the Approach}\n\nPrevious sections provide optimal controllers for certain infinite horizon DNCS problems.\nThe key idea is to construct an auxiliary MJLS and use MJLS properties to extend optimal controllers in finite horizons to their counterparts in the infinite horizon.\nThis approach of solving an infinite horizon DNCS problem can be summarized as the following main steps:\n\\begin{enumerate}\n\\item Solve the finite horizon version of the DNCS problem and obtain matrices $P_t^m, m \\in \\mathcal M = \\{1,2,\\dots,M\\}$ which satisfy $M$ coupled Ricatti recursions \n\\begin{align}\nP_t^m =  \\Omega(\\sum_{j} \\theta^{mj} P_{t+1}^j,Q^m,R^m,A^m,B^m)\n\\label{eq:P_finite_general}\n\\end{align}\nfor some matrices $Q^m,R^m,A^m,B^m$ and positive numbers $\\theta^{mj}$ for $m,j \\in\\mathcal M$. Note that we can scale the $\\theta$'s such that $\\sum_{j\\in\\mathcal M}\\theta^{mj} = 1$ by appropriately scaling $A^m$ and $R^m$ for all $m\\in\\mathcal M$.\n\n\n\\item Construct a $M$-mode auxiliary MJLS with transition probabilities $\\theta^{mj}$ and system matrices $Q^m,R^m,A^m,B^m$\nbased on the Ricatti recursions \\eqref{eq:P_finite_general}.\n\n\n\\item Analyze stability criteria of the auxiliary MJLS which directly translate to stability conditions for the DNCS.\n\n\\item When the auxiliary MJLS is stable, solve its associated DCARE to obtain optimal strategies for the infinite horizon DNCS problem.\n\\end{enumerate}\n\n\n\\subsection{The Information Switching}\nEven though the auxiliary MJLS is an artificial system without physical meanings (See Remark ?), the connection between a DNCS and a MJLS provides a general method to analyze switching behaviors of a DNCS. \nIn particular, a DNCS with delays and packet-drop channels can be viewed as a decentralized control problem with a randomized information structure \\cite{RPN}.\nIn other words, a DNCS can be viewed as a special class of ``switched\" systems where each controller's information is switching between different structures or patterns. \n\nThe information of the remote controller in Section \\ref{sec:model_2_controllers} switches between two structures: no observation when the packet is dropped, and perfect observation when the transmission is successful.\nFor this two-controller DNCS, each information structure of the remote controller can be treated as a ``mode\". Consequently, the closed-loop system under the optimal strategies become a two-mode switching system.\nIt is worth noting that, even though the DNCS can be thought of as a two-mode switching system, it cannot be modled by an MJLS due to its decentralized information.\nThe auxiliary MJLS is an artificial two-mode system constructed to help analyze the behavior of the DNCS with two switching information structures.\n\nIn the $(N+1)$-controller DNCS presented in Section \\ref{sec:model_N_controllers}, \nthe remote controller's information about each of the $N$ subsystem switches between two patterns: knowing $X^n_t$ or not. \nTherefore, its information is switching among $2^N$ structures.\nThe analysis of these information structures can be simplified from the decoupled dynamics.\nIn fact, the remote controller's information about each of the $N$ subsystems depends only on the $n$th link because the subsystems' states are conditional independent. Consequently, we construct an auxiliary $(N+1)$-mode MJLS where mode $0$ corresponds to the perfect information case, and each of the other $N$ modes corresponds to the failure of each of the $N$ links.\n\n\n\n\n\n\\subsection{Summary of the Approach}\nThe analysis in Sections \\ref{sec:infinite_2_controllers} and  \\ref{sec:model_N_controllers} suggests a general approach   for solving infinite horizon decentralized control\/DNCS problems. This can be summarized as follows:\n\\begin{enumerate}\n\\item Solve the finite horizon version of the DNCS\/decentralized control problem (for instance by using the common information approach \\cite{nayyar2013decentralized}). Suppose the optimal strategies  are characterized by matrices $P_t^m, m \\in \\mathcal M = \\{1,2,\\dots,M\\}$ which satisfy $M$ coupled Riccati recursions \n\\begin{align}\nP_t^m =  \\Omega(\\sum_{j} \\theta^{mj} P_{t+1}^j,Q^m,R^m,A^m,B^m),\n\\label{eq:P_finite_general}\n\\end{align}\nfor some matrices $Q^m,R^m,A^m,B^m$ and positive numbers $\\theta^{mj}$ for $m,j \\in\\mathcal M$. Note that we can scale the $\\theta$'s such that $\\sum_{j\\in\\mathcal M}\\theta^{mj} = 1$ by appropriately scaling $A^m$ and $R^m$ for all $m\\in\\mathcal M$.\n\n\n\\item Construct a $M$-mode auxiliary MJLS with transition probabilities $\\theta^{mj}$ and system matrices $Q^m,R^m,A^m,B^m$\nso that the Riccati recursions associated with optimal control of the MJLS coincide with the Riccati recursions \\eqref{eq:P_finite_general}.\n\n\n\\item Analyze stability criteria of the auxiliary MJLS to find conditions under which the Riccati recursions of the DNCS reach a steady state.\n\n\\item Verify  that the decentralized strategies characterized by the steady state DNCS Riccati equations are optimal.\n\\end{enumerate}\n\n\n\\subsection{The Information Switching}\n Even though the auxiliary MJLS we used in our analysis is an artificial system without apparent physical meaning (see Remark \\ref{rem:fictitious}), a general DNCS with random packet drops (or random packet delays) does have some aspects of a switched system. In particular, the information at a controller (e.g, the remote controller in our problem) switches between different patterns  based on the state of the underlying communication network.   The information of the remote controller in Section \\ref{sec:model_2_controllers} clearly switches between two patterns: no observation when the packet is dropped, and perfect observation when the transmission is successful. The number of such patterns seems  related to (but not always equal to) the number of modes in the MJLS used to analyze the DNCS. For the two-controller DNCS of Section  \\ref{sec:model_2_controllers}, the number of patterns between which the remote controller's information switches is the same as the number of modes in its auxiliary MJLS.  For the $N+1$ controller DNCS in Section \\ref{sec:model_N_controllers}, the remote controller's information appears to switch between $2^N$ patterns (depending on the state of the $N$ links) but its auxiliary MJLS has only $N+1$ modes. This difference between the number of information patterns and the number of modes is due to the nature of the plant dynamics which ensure that the remote controller's estimate of the $n$th plant is not affected by the state of the $m$th link if $m\\neq n$.\n\n\n\n\\subsection{Answering Q1}\n\\label{sec:Q1_2C} \nIn Q1, we want to know whether $P_t^0$ and $P_t^1$ defined by coupled recursions of \\eqref{eq:P_initial}-\\eqref{eq:tildeP_finite_2C} converge to $P_*^0$ and $P_*^1$ satisfying \\eqref{eq:P_finite_2C_fixed}-\\eqref{eq:tildeP_finite_2C_fixed}.  Our approach for answering Q1 is based on establishing a connection between the recursions for matrices $P_t^0$ and $P_t^1$ in our DNCS problem and the recursions for matrices $P_t^{\\diamond}(m)$, $m \\in \\mathcal{M},$ in the MJLS problem reviewed in Section \\ref{sec:mjls}. This approach consists of the following two steps.\n\n\n\n\n\\subsubsection*{\\textbf{Step 1: Constructing an auxiliary MJLS}}\nConsider an auxiliary MJLS where the set $\\mathcal{M}$ of modes is $\\{0,1\\}$. Then, we have the following two sequences of  matrices, $P^{\\diamond}_t(0),P^{\\diamond}_t(1),$ defined recursively using \\eqref{eq:CARE_init} and \\eqref{eq:CARE_finite} for this MJLS:\n\\begin{align}\n&P^{\\diamond}_{T+1}(0) = P^{\\diamond}_{T+1}(1) =\\mathbf{0}, \\label{eq:P_MJ_init}\\\\\n&P^{\\diamond}_t(0) =  \\notag \\\\\n&\\Omega\\big(\\theta^{00} P^{\\diamond}_{t+1}(0) + \\theta^{01} P^{\\diamond}_{t+1}(1),Q^{\\diamond}(0),R^{\\diamond}(0),A^{\\diamond}(0),B^{\\diamond}(0) \\big),\n\\label{P_MJ_cmp_2C_0}\n\\\\\n&P^{\\diamond}_t(1)=   \\notag \\\\\n& \\Omega\\big(\\theta^{10} P^{\\diamond}_{t+1}(0) + \\theta^{11} P^{\\diamond}_{t+1}(1),Q^{\\diamond}(1),R^{\\diamond}(1),A^{\\diamond}(1),B^{\\diamond}(1) \\big).\n\\label{P_MJ_cmp_2C_1}\n\\end{align}\nFurthermore, recall that from \\eqref{eq:P_initial}-\\eqref{eq:tildeP_finite_2C}, we have the following recursions for matrices $P_t^0$ and $P_t^1$ in our DNCS problem,\n\\begin{align}\n&P^0_{T+1}=P^1_{T+1}=0, \\label{eq:barP_cmp_init}\\\\\n&P_t^0 =  \\Omega(P_{t+1}^0,Q,R,A,B),\n\\label{eq:barP_cmp_0_2C}\n\\\\\n&P_t^{1} =  \\Omega((1-p^1) P_{t+1}^0+p^1 P_{t+1}^1,Q,R^{11},A,B^{11}).\n\\label{eq:barP_cmp_n_2C}\n\\end{align}\nIs it possible to find matrices $A^{\\diamond}(m),B^{\\diamond}(m), Q^{\\diamond}(m),R^{\\diamond}(m)$, $m \\in \\{0,1\\}$, and a transition probability matrix $\\Theta$ for the auxiliary MJLS such that the recursions in \\eqref{eq:P_MJ_init} - \\eqref{P_MJ_cmp_2C_1} coincide with the recursions in \\eqref{eq:barP_cmp_init} - \\eqref{eq:barP_cmp_n_2C}?\n\n\nBy comparing \\eqref{P_MJ_cmp_2C_0}-\\eqref{P_MJ_cmp_2C_1} with \\eqref{eq:barP_cmp_0_2C}-\\eqref{eq:barP_cmp_n_2C}, we find that the following definitions would make the two sets of equations identical:\n\\begin{align}\n&A^{\\diamond}(0) = A^{\\diamond}(1) = A, \\label{eq:MJLS_A} \\\\\n&B^{\\diamond}(0) = B, \\quad B^{\\diamond}(1) = [\\mathbf 0,B^{11}], \\label{eq:MJLS_B}\n\\\\\n&Q^{\\diamond}(0) = Q^{\\diamond}(1) = Q,\\label{eq:MJLS_Q} \\\\\n&R^{\\diamond}(0) = R, \\quad R^{\\diamond}(1) = \\begin{bmatrix}\n\\mathbf I & \\mathbf 0 \\\\\n\\mathbf 0 & R^{11}\n\\end{bmatrix}, \\label{eq:MJLS_R}\n\\\\\n&\\Theta = \\begin{bmatrix}\n\\theta^{00} & \\theta^{01} \\\\\n\\theta^{10} & \\theta^{11}\n\\end{bmatrix} = \\begin{bmatrix}\n1 & 0 \\\\\n1-p^1 & p^1\n\\end{bmatrix}.\n\\label{eq:MJLS_theta}\n\\end{align}\nTo complete the definition of the auxiliary MJLS, we need to define the initial state and mode probability distributions $\\pi_{X_0^{\\diamond}}$ and $\\pi_{M_0}$. These can be defined arbitrarily and for simplicity we assume that the initial state and mode of the auxiliary MJLS are fixed to be $X^{\\diamond}_0 = 0$ and $M_0 = 1$.\nThe following lemma summarizes the above discussion.\n\n\n\\begin{lemma}\n\\label{equality_recursions_2C}\nFor the auxiliary MJLS described by \\eqref{eq:MJLS_A}-\\eqref{eq:MJLS_theta}, the coupled recursions in \\eqref{eq:P_MJ_init}-\\eqref{P_MJ_cmp_2C_1} are identical to  the coupled recursions in \\eqref{eq:barP_cmp_init}-\\eqref{eq:barP_cmp_n_2C}.\n\\end{lemma}\n\\begin{proof}\nThe lemma can be proved by  straightforward algebraic manipulations.\n\\end{proof}\n\n\\begin{remark}\nNote that  we have not defined $B^{\\diamond}(1)$ to be $B^{11}$  because the MJLS model requires that the dimensions of matrices $B^{\\diamond}(0)$ and $B^{\\diamond}(1)$ be the same (see Section \\ref{sec:mjls}). Similar dimensional considerations  prevent us from defining $R^{\\diamond}(1)$ to be simply $R^{11}$.\n\\end{remark}\n\n\\begin{remark}\\label{rem:fictitious}\nIt should be noted that the auxiliary MJLS is simply a mathematical device. It cannot be seen as a reformulation or  another interpretation of our DNCS problem. In particular, the binary mode $M_t$  \\emph{is not} the same as the link state $\\Gamma^1_t$. The distinction between $M_t$ and $\\Gamma^1_t$ is immediately clear if one recalls that $M_t$ is the state of a Markov chain with transition probability matrix given in \\eqref{eq:MJLS_theta} whereas $\\Gamma^1_t, t \\geq 0,$ is an i.i.d process. \n\\end{remark}\n\\subsubsection*{\\textbf{Step 2: Using MJLS results to answer Q1}}\nNow that we have constructed an auxiliary MJLS such that $P^{\\diamond}_t(m) = P_t^{m}$ for $m \\in \\{0,1\\}$, we can use the MJLS results about convergence of  matrices $P^{\\diamond}_t(m)$ (that is, Lemmas \\ref{lm:MJ_infinite} and \\ref{lm:ss})  to answer Q1. The following lemma states this result.\n\n\n\\begin{lemma}\\label{lm:pc_2C}\nSuppose Assumption \\ref{assum:det_stb_2C} holds. Then, the matrices $P_t^0$ and $P_t^1$ defined in \\eqref{eq:P_initial}-\\eqref{eq:tildeP_finite_2C} converge as $t \\to -\\infty$ to matrices $P_*^0$ and $P_*^1$ that satisfy the coupled fixed point  equations \\eqref{eq:P_finite_2C_fixed} - \\eqref{eq:tildeP_finite_2C_fixed} if and only if $p^1 < p^1_c$, where $p^1_c$ is the critical threshold given by \n\\begin{align}\n\\frac{1}{\\sqrt{p^1_c}} =  \\min_{K  \\in \\mathbb{R}^{d^1_U \\times d_X}}\\rho(A+B^{11}K).\n\\label{eq:pc_2c}\n\\end{align} \n\\end{lemma}\n\n\\begin{proof}\nSee Appendix \\ref{proof_lm:pc_2C}.\n\\end{proof}\n\n\\subsection{Answering Q2 and Q3}\n\\label{sec:Q2_2C}\nAssuming that $P_{t}^n \\rightarrow P_*^n$ as $t \\rightarrow -\\infty$ for $n=0,1,$ we want to know whether the control strategies of \\eqref{eq:K_finite_2C_fixed}-\\eqref{eq:estimator_inf_2C} are optimal for Problem \\ref{problem_infinite_2C}. The following result shows that these control strategies are indeed optimal.\n\n\\begin{lemma}\n\\label{lm:Q2_2C}\nIf $P_{t}^n \\rightarrow P_*^n$ as $t \\rightarrow -\\infty$ for $n=0,1$, then\n\\begin{enumerate}\n\\item Problem \\ref{problem_infinite_2C} has finite optimal cost,\n\\item The strategies described by \\eqref{eq:K_finite_2C_fixed}-\\eqref{eq:estimator_inf_2C} are optimal for Problem \\ref{problem_infinite_2C},\n\\item Under the strategies described by \\eqref{eq:K_finite_2C_fixed}-\\eqref{eq:estimator_inf_2C}, $X_t$ and $(X_t-\\hat X_t)$ are mean square stable, i.e.,\n\\[ \\sup_{t \\geq 0} \\mathds{E}^{g^*}[||X_t||^2]  < \\infty \\mbox{~and~} \\sup_{t \\geq 0} \\mathds{E}^{g^*}[||(X_t-\\hat X_t)||^2]  < \\infty,\\]\n where $g^*$ denotes the strategy described by \\eqref{eq:K_finite_2C_fixed}-\\eqref{eq:estimator_inf_2C}.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{sec:lm_Q2_2C} for proof of parts 1) and 2). See Appendix \\ref{sec:stability_proof} for proof of part 3).\n\\end{proof}\n\n\nThe following lemma answers Q3.\n\\begin{lemma}\n\\label{lm:Q3}\nIf  $P_t^n$, $n=0,1,$ do not converge as $t \\to -\\infty$, then Problem \\ref{problem_infinite_2C} does not have finite optimal cost.\n\\end{lemma}\n\\begin{proof} See Appendix \\ref{sec:lmQ3}.\n\\end{proof}\n\nNow that we have answered  Q1, Q2 and Q3, we can summarize our results for the infinite horizon DNCS problem (Problem \\ref{problem_infinite_2C}).\n\n\\subsection{Summary of the Infinite Horizon Results}\nBased on the answers to Q1-Q3, the following theorem summarizes our results for Problem \\ref{problem_infinite_2C}.\n\\begin{theorem}\n\\label{thm:DC_infinite_2C}\nSuppose Assumption \\ref{assum:det_stb_2C} holds. Then, \n\\begin{enumerate}[(i)]\n\\item Problem \\ref{problem_infinite_2C} has finite optimal cost if and only if $p^1 < p^1_c$ where the critical threshold $p^1_c$ is given by \\eqref{eq:pc_2c}.\n\n\\item If $p^1 < p^1_c$, there exist symmetric positive semi-definite matrices $P_*^0, P_*^1$ that satisfy \\eqref{eq:P_finite_2C_fixed}-\\eqref{eq:tildeP_finite_2C_fixed} and the optimal strategies  for Problem \\ref{problem_infinite_2C} are given by\n\\begin{align}\n\\bmat{U^{0*}_t \\\\ U^{1*}_t \\\\} = K_*^0\\hat X_t + \\bmat{\\mathbf{0} \\\\ K_*^1 }\\left(X_t - \\hat X_t\\right),\n\\label{eq:opt_U_infinite_2C_thm}\n\\end{align}\nwhere the  estimate $\\hat X_t$ can be computed recursively using \\eqref{eq:estimator_inf_2C} with $\\hat{X}_0=0$\n and the gain matrices $K_*^0,K_*^1$ are given by\n\\begin{align}\n& K_*^0 = \\Psi(P_{*}^0,R,A,B),\n\\label{eq:K_finite_2C_fixed_final}\n\\\\\n& K_*^1 = \\Psi((1-p^1)P_{*}^0+p^1 P_{*}^1,R^{11},A,B^{11}).\n\\label{eq:tildeK_finite_2C_fixed_final}\n\\end{align} \n\\item If $p^1 < p^1_c$, then under the strategies described in part (ii) above, $X_t$ and $(X_t-\\hat X_t)$ are mean square stable.\n \n\n\n\\end{enumerate}\n\n\n\n\\end{theorem}\n\\begin{proof}\nThe result follows from Lemmas \\ref{lm:pc_2C}, \\ref{lm:Q2_2C} and \\ref{lm:Q3}.\n\\end{proof}\n\n\n\n\n\n\nIf $B^{11} = 0$, the local controller becomes just a sensor without any control ability. In this case, Theorem \\ref{thm:DC_infinite_2C} gives the \ncritical threshold as $p^1_c = \\rho(A)^{-2}$ and the closed-loop system is mean-square stable if $\\rho(A) < 1\/\\sqrt{p^1}$. This recovers the single-controller NCS result in \\cite{Imer2006optimal}. Thus, we have the following corollary of Theorem \\ref{thm:DC_infinite_2C}.\n\\begin{corollary}[Theorem 3 of \\cite{Imer2006optimal} with $\\alpha = 0$ and $\\beta = p^1$]\nSuppose the local controller is just a sensor (i.e., $B^{11} = 0$) and the remote controller is the only controller present. Then, if $\\rho(A) < 1\/\\sqrt{p^1}$, the optimal controller of this single-controller NCS is given by $U^{0*}_t$ in \\eqref{eq:opt_U_infinite_2C_thm}, and the corresponding closed-loop system is mean-square stable.\n\\end{corollary}\n\n\n\n\\begin{remark}\nThe value $\\frac{1}{\\sqrt{p^1_c}} =  \\min_{K}\\rho(A+B^{11}K)$ is actually the largest unreachable mode of $(A,B^{11})$. Therefore, it can be computed using tests for reachability such as the Popov-Belovich-Hautus (PBH) test \\cite{hespanha2009linear}. Further, if  $(A,B^{11})$ is reachable, then $\\rho(A+B^{11}K)$ can be made arbitrarily small which implies that $p^1_c = \\infty$. This is the case when the local controller can stabilize the system by itself, so the DNCS is stabilizable under any link failure probability $p^1$.\n\\end{remark}\n\n\\begin{remark}\nIf $p^1 < p^1_c$, the coupled Riccati equations in \\eqref{eq:P_finite_2C_fixed}-\\eqref{eq:tildeP_finite_2C_fixed} can be solved by iteratively carrying out the recursions in \\eqref{eq:P_initial}-\\eqref{eq:tildeP_finite_2C} until convergence. This is similar to the procedure in \\cite[Chapter 7]{costa2006discrete}.\n\\end{remark}\n\n\n\n\n\\subsection{Contributions of the Paper}\n\\begin{enumerate}\n\\item  We investigate an infinite time horizon decentralized stochastic control problem in which local controllers send their information to a\nremote controller over unreliable links. To the best of our\nknowledge, this is the first paper that solves an infinite time horizon optimal\ndecentralized control problem with unreliable communication\nbetween controllers.  The finite time horizon version of our problem was solved in \\cite{ouyang2016optimal, asghari_ouyang_nayyar_tac_2018} and our results  in this paper use the finite horizon solutions obtained there. However, unlike the finite horizon case, we have to address the possibility that no control strategy may achieve finite cost over infinite time horizon. Due to such stability related issues, our approach for the infinite horizon problem is markedly different from the  common information based approach adopted in \\cite{ouyang2016optimal, asghari_ouyang_nayyar_tac_2018}.\n\n\\item We show that there are critical thresholds for link failure probabilities above which no control strategy can achieve a finite cost in our problem. When the link failure probabilities are  below their critical thresholds, we show that the optimal control strategies of this infinite horizon decentralized control problem admit simple structures: the optimal remote controller strategy is a time-invariant linear function of the common estimates of system states and the optimal strategies for  local controllers are time-invariant linear functions of the common estimates of system states and the perfectly observed local states. The main strengths of our result are that (i) it provides simple strategies that are proven to be optimal: not only are the strategies in Theorems \\ref{thm:DC_infinite_2C} and \\ref{thm:DC_infinite_NC} linear, they use estimates that can be easily updated; (ii) it shows that the optimal strategies are completely characterized by solution of coupled Riccati equations.\n\n\n\\item If the local controllers act only as sensors and the remote controller is the only controller in the system, then our model reduces to a NCS with multiple sensors  observing different components of the system state and communicating with the remote controller over independent unreliable links. Thus, we obtain optimal strategy and critical probabilities for a multi-sensor, single-controller NCS as a corollary of our result in Theorem \\ref{thm:DC_infinite_NC}.\n \n\\item Finally, our problem can be viewed as a dynamic team problem  by viewing  each controller's actions at different time instants as the actions of distinct players \\cite{HoChu:1972}. Since we are interested in infinite time horizon, this team-theoretic viewpoint means that our dynamic team has infinitely many players. Further, due to the unreliable links, our problem does not directly fit into the partially nested LQG team problem. Thus, the standard results for partially nested LQG teams with finitely many players \\cite{HoChu:1972} do no apply to our problem.  As observed in \\cite{infinite_team}, results for teams with finitely many players cannot be directly extended to teams with infinitely many players even when the information structure is static or partially nested.\n In spite of this,  our dynamic team problem turns out to have simple optimal strategies.\n\\end{enumerate}\n\n\n\n\n\n\\subsection{Organization}\nThe rest of the paper is organized as follows. Section \\ref{sec:pre} summarizes the notations and operators used in this paper. \nIn Section \\ref{sec:model_2_controllers}, we formulate the finite horizon and infinite horizon optimal control problems for a DNCS with one remote controller and one local controller. We briefly review Markov Jump Linear Systems (MJLSs) in Section \\ref{sec:mjls}. We establish a connection between the  DNCS of Section \\ref{sec:model_2_controllers} and an auxiliary MJLS in Section \\ref{sec:infinite_2_controllers} and use this connection to provide our main results for the  DNCS of Section \\ref{sec:model_2_controllers}. In Section \\ref{sec:model_N_controllers}, we extend our DNCS model to the case with multiple local controllers and provide our main results for  this DNCS. We discuss some key aspects of our approach in Section \\ref{sec:discussion}. Section \\ref{sec:conclusion} concludes the paper.\nThe proofs of all  technical results are in the Appendices.\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\\input{Introduction.tex}\n\n\\section{Preliminaries}\n\\label{sec:pre}\n\\input{prelim.tex}\n\n\\section{System Model and Problem Formulation}\n\\label{sec:model_2_controllers}\n\\input{model_2_controllers.tex}\n\n\\section{Review of Markov Jump Linear Systems}\n\\label{sec:mjls}\n\\input{mjls.tex}\n\n\\section{Infinite Horizon Optimal Control}\n\\label{sec:infinite_2_controllers}\n\\input{infinite_horizon_2_controllers.tex}\n\n\\section{Extension to Multiple Local Controllers}\n\\label{sec:model_N_controllers}\n\\input{model_N_controllers.tex}\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\\input{discussion.tex}\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{conc.tex}\n\n\\bibliographystyle{ieeetr}\n\n\\subsubsection{Finite Horizon MJLS Optimal Control}\n\\begin{definition}\n\\label{def:ss}\nThe MJLS of \\eqref{eq:MJ_X}-\\eqref{eq:MJ_M} is stochastically stabilizable if there exist gain matrices $K^{\\diamond}(m), m\\in\\mathcal M,$ such that for any initial state and mode, $\\sum_{t=0}^\\infty \\ee[||X^{\\diamond}_t||^2]< \\infty$ where $X^{\\diamond}_{t+1} = A_s(M_t) X^{\\diamond}_t$ and  \n\\begin{align}\nA_s (M_t) = A^{\\diamond} (M_t) +  B^{\\diamond}(M_t) K^{\\diamond}(M_t).\n\\label{A_s_matrix}\n\\end{align}\nIn this case, we say the gain matrices $K^{\\diamond}(m), m \\in \\mathcal M$, stabilize the MJLS.\n\\end{definition}\n\n\\begin{definition}\n\\label{def:sd}\nThe MJLS of \\eqref{eq:MJ_X}-\\eqref{eq:MJ_c} is Stochastically Detectable (SD) if there exist gain matrices $H^{\\diamond}(m), m \\in \\mathcal{M}$, such that for any initial state and mode, $\\sum_{t=0}^\\infty \\ee[||X^{\\diamond}_t||^2]< \\infty$ where \n$X^{\\diamond}_{t+1} = A_d(M_t) X^{\\diamond}_t$ and \n\\begin{align}\nA_d(M_t) = A^{\\diamond} (M_t) +  H^{\\diamond}(M_t) \\big(Q^{\\diamond}(M_t) \\big)^{1\/2}.\n\\label{A_d_matrix}\n\\end{align}\n\\end{definition}\n\nFrom the theory of MJLS (\\cite{costa2006discrete,costa1995discrete}), we can obtain the following result for \nthe convergence of matrices $\\{P^{\\diamond}_{t}(m), t = T+1,T,T-1,\\ldots \\}$ to $P^{\\diamond}_{*}(m)$ satisfying the DCARE in \\eqref{eq:CARE_infinite}.\n\n\\begin{lemma}\n\\label{lm:MJ_infinite}\nSuppose the MJLS is stochastically detectable (SD).  Then, matrices $P^{\\diamond}_{t}(m)$, $m \\in \\mathcal{M}$, converge as $t \\to -\\infty$ to PSD matrices $P^{\\diamond}_{*}(m)$ that satisfy the DCARE in \\eqref{eq:CARE_infinite} if and only if the MJLS is stochastically stabilizable (SS).\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{app:lm_MJ_infinite}.\n\\end{proof}\n\n\nStochastically stabilizability (SS) and stochastically detectability (SD) of a MJLS can be verified from the system matrices and the transition matrix for the mode of the MJLS \\cite{costa2006discrete,fang2002stochastic}. Specifically, we have the following lemmas.\n\n\\begin{lemma}(\\cite[Theorem 3.9]{costa2006discrete} and also \\cite[Corollary 2.6]{fang2002stochastic})\n\\label{lm:ss}\n\\begin{enumerate}\n\\item A MJLS is SS if and only if there exist matrices $K^{\\diamond}(m)$, $m \\in \\mathcal M$, such the matrix\n\\begin{align}\n&\\mathcal{A}_s := \\notag \\\\\n&\\diag \\big(A_s(0)\\otimes A_s(0), \\ldots, A_s(M)\\otimes A_s(M) \\big)\n(\\Theta^{\\tp}\\otimes \\mathbf{I}),\n\\label{eq:bigmatrix_ss}\n\\end{align}\nis Schur stable, i.e. $\\rho(\\mathcal{A}_s) < 1$, where $A_s(M_t)$ is given by \\eqref{A_s_matrix}.\n\\item A MJLS is SD if and only if there exist matrices $H^{\\diamond}(m)$, $m \\in \\mathcal M$, such the matrix\n\\begin{align}\n&\\mathcal{A}_d := \\notag \\\\\n&\\diag \\big(A_d(0)\\otimes A_d(0), \\ldots, A_d(M)\\otimes A_d(M) \\big)\n(\\Theta^{\\tp}\\otimes \\mathbf{I}),\n\\label{eq:bigmatrix_sd}\n\\end{align}\nis Schur stable, i.e. $\\rho(\\mathcal{A}_d) < 1$, where $A_d(M_t)$ is given by \\eqref{A_d_matrix}.\n\\end{enumerate}\n\\end{lemma}\n\\subsection{Communication Model}\n\\label{subs:comm_model_2C}\n At each time $t$, the local controller $C^1$  observes the state $X_t$ perfectly and sends the observed state to the remote controller $C^0$ through an unreliable link with packet drop probability $p^1$. \n Let $\\Gamma_t^1$ be a Bernoulli random variable describing the state of this link, that is, $\\Gamma_t^1=0$ if the link is broken (i.e., the packet is dropped) and  $\\Gamma_t^1=1$ if the link is active. We assume that $\\Gamma_{t}^1, t \\geq 0,$ is an i.i.d. process and is independent of the noise process $W_{0:t}, t \\geq 0$.  Let $Z_t^1$ be the output of the unreliable link. Then,\n \\begin{align}\n\\Gamma_t^1 = &\\left\\{\\begin{array}{ll}\n1 & \\text{ with probability }(1-p^1),\\\\\n0 & \\text{ with probability }p^1.\n\\end{array}\\right. \\label{eq:gamma}\n\\\\\nZ_t^1 = &\n\\left\\{\\begin{array}{ll}\nX_t & \\text{ when } \\Gamma_t^1 = 1,\\\\\n\\emptyset & \\text{ when } \\Gamma_t^1 = 0.\n\\end{array}\\right.\n\\label{Model:channel_2C}\n\\end{align}\nWe assume that $Z_t^1$ is perfectly observed by $C^0$. Further,  we assume that  $C^0$ sends an acknowledgment to the local controller $C^1$ if it receives the state value. Thus, effectively,  $Z_t^1$ is perfectly observed by $C^1$ as well.  The two controllers select their control actions at time $t$ after observing $Z_t^1$. We assume that the links for sending acknowledgments as well as the links from the controllers to the plant are perfectly reliable.\n\n\\subsection{Information structure and cost}\n\\label{subs:info_cost_2C}\n\nLet $H_t^0$ and $H_t^1$ denote the information available to the controllers $C^0$ and $C^1$ to make decisions at time $t$, respectively. Then,\n\\begin{align}\nH^0_t = \\{Z_{0:t}^1, U^0_{0:t-1}\\}, \\hspace{2mm} H^1_t= \\{X_{0:t}, Z_{0:t}^1, U^1_{0:t-1}, U^0_{0:t-1}\\}. \n\\label{Model:info_2C}\n\\end{align}\n$H^0_t$ will be referred to as the \\textit{common information} among the two controllers at time $t$\\footnote{ We assumed that $U^0_{0:t-1}$  is pat of $H^1_t$. This is not a restriction because even if $U^0_{0:t-1}$ is not directly observed by $C^1$ at time $t$,  $C^1$ can still compute it using $C^0$'s strategy since it knows everything $C^0$ knows.}.\n\nLet $\\mathcal{H}^0_t$ and $\\mathcal{H}^1_t$ be the spaces of all possible \nrealizations of $H^0_t$ and $H^1_t$, respectively.\nThen, the control actions are selected according to\n\\begin{align}\nU^0_t = g^0_t(H^0_t), \\hspace{2mm} U^1_t = g^1_t(H^1_t), \n\\label{Model:strategy_2C}\n\\end{align}\nwhere the control laws $g^0_t:\\mathcal{H}^0_t \\to \\R^{d_0}$ and $g^1_t:\\mathcal{H}^1_t \\to \\R^{d_1}$ are measurable mappings.\nWe use $g:=(g^0_{0},g^0_1,\\dots,g^1_{0},g^1_1,\\dots)$ to denote the control strategies of $C^0$ and $C^1$.\n\n\nThe instantaneous cost $c(X_t,U_t)$ of the system is a  quadratic function given by\n\\begin{align}\n&c(X_t,U_t) = \nX_t^\\tp Q X_t + U_t^\\tp R U_t,\n\\label{Model:cost_2C}\n\\end{align}\nwhere $Q$ is a symmetric positive semi-definite (PSD) matrix, and $R = \\begin{bmatrix}\nR^{00} & R^{01}\\\\R^{10} & R^{11}\n\\end{bmatrix}$ is a symmetric positive definite (PD) matrix.\n\n\\subsection{Problem Formulation}\n\\label{subs:prob_formulaiton_2C}\n\nLet $\\mathcal{G}$ denote the set of all possible control strategies of $C^0$ and $C^1$. The performance of control strategies $g$ over a\nfinite horizon $T$ is measured by the total expected cost\\footnote{Because the cost function $c(X_t,U_t)$ is always non-negative, the expectation is well-defined on the the extended real line $\\R \\cup \\{\\infty\\}$.}:\n\\begin{align}\nJ_T(g):=\\ee^{g}\\left[\\sum_{t=0}^T c(X_t,U_t)\\right].\n\\label{Model:J_T_2C}\n\\end{align}\n\nWe refer to the system described by \\eqref{Model:system_2C}-\\eqref{Model:cost_2C} as the \\emph{decentralized networked control system} (DNCS).\nWe consider the problem of strategy optimization for the DNCS over finite and infinite time horizons. These two problems are formally defined below.\n\\begin{problem}[Finite Horizon DNCS Optimal Control]\n\\label{problem_finite_2C}\nFor the DNCS described by \\eqref{Model:system_2C}-\\eqref{Model:cost_2C}, determine decentralized control strategies $g$ that optimize the  total expected cost over a finite horizon of duration $T$. In other words, solve the following strategy optimization problem:\n\\begin{align}\n\\inf_{g \\in\\mathcal{G}} \nJ_T(g).\n\\label{Model:obj_finite_2C}\n\\end{align}\n\\end{problem}\n\n\n\\begin{problem}[Infinite Horizon DNCS Optimal Control]\n\\label{problem_infinite_2C}\nFor the DNCS described by \\eqref{Model:system_2C}-\\eqref{Model:cost_2C}, find decentralized strategies $g$ that minimize the  infinite horizon average cost.  In other words, solve the following strategy optimization problem:\n\\begin{align}\n&\\inf_{g \\in\\mathcal{G}} \nJ_{\\infty}(g)\n:= \\inf_{g \\in\\mathcal{G}} \n\\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} J_{T}(g).\n\\label{Model:obj_infinite_2C}\n\\end{align}\n\n\\end{problem}\n\nWe make the following  standard assumption on the system and cost matrices \\cite{Bertsekas:1995}. \n\\begin{assumption}\n\\label{assum:det_stb_2C}\n$(A,Q^{1\/2})$ is detectable and $(A,B)$ is stabilizable.\n\\end{assumption}\n\nThe finite horizon DNCS optimal control problem (Problem \\ref{problem_finite_2C}) has been solved in \\cite{ouyang2016optimal, asghari_ouyang_nayyar_tac_2018}.  We summarize the finite horizon results below. \n\n\\begin{lemma}(\\cite[Theorem 2]{ouyang2016optimal})\n\\label{lm:opt_strategies_2C}\nThe optimal control strategies of Problem \\ref{problem_finite_2C} are given by\n\\begin{align}\n\\bmat{U^{0*}_t \\\\ U^{1*}_t \\\\} = K_t^0\\hat X_t + \\bmat{\\mathbf{0} \\\\ K_t^1 }\\left(X_t - \\hat X_t\\right), \\label{eq:opt_U_2C}\n\\end{align}\nwhere $\\hat X_t = \\ee[X_t|H^0_t]$ is the  estimate (conditional expectation) of $X_t$ based on the common information $H^0_t$.  The  estimate can be computed recursively according to\n\\begin{align}\n\\hat X_0 = & 0,\n\\label{eq:estimator_0_2C}\n\\\\\n\\hat X_{t+1}\n= &\\left\\{\n\\begin{array}{ll}\n \\big(A + B K^0_t\\big)\\hat X_t& \\text{ if }Z_{t+1}= \\emptyset,\\\\\n X_{t+1} & \\text{ if }Z_{t+1} = X_{t+1}.\n\\end{array}\\right.\n\\label{eq:estimator_t_2C}\n\\end{align}\nThe gain matrices are given by\n\\begin{align}\n& K_t^0 = \\Psi(P_{t+1}^0,R,A,B),\n\\label{eq:K_finite_2C}\n\\\\\n& K_t^1 = \\Psi((1-p^1)P_{t+1}^0+p^1 P_{t+1}^1,R^{11},A,B^{11}),\n\\label{eq:tildeK_finite_2C}\n\\end{align}\nwhere $P_t^0$ and $P_t^1$ are PSD matrices obtained recursively as follows:\n\\begin{align}\n&P_{T+1}^0 =  P_{T+1}^1 = \\mathbf{0},  \\label{eq:P_initial}\\\\\n&P_t^0 =  \\Omega(P_{t+1}^0,Q,R,A,B),\n\\label{eq:P_finite_2C}\n\\\\\n&P_t^1 =  \\Omega((1-p^1)P_{t+1}^0+p^1 P_{t+1}^1,Q,R^{11},A,B^{11}).\n\\label{eq:tildeP_finite_2C}\n\\end{align}\nFurthermore, the optimal cost is given by\n\\begin{align}\nJ^*_{T} = & \\sum_{t = 0}^T \\tr \\big((1-p^1)P_{t+1}^0+p^1 P_{t+1}^1 \\big).\n\\label{eq:opcost_finite_2C}\n\\end{align}\n\\end{lemma}\n\n\\begin{remark}\nNote that remote controller's action $U^{0*}_t$ in \\eqref{eq:opt_U_2C} is a function  of $\\hat{X}_t$ only while the local controller's action $U^{1*}_t$ is a function of both $\\hat{X}_t$ and $X_t$. Further, as per \\eqref{eq:estimator_0_2C} and \\eqref{eq:estimator_t_2C}, $\\hat{X}_t$ is computed recursively based only on the knowledge of $Z_{0:t}$.\n\\end{remark}\n\nIn this paper, we will focus on solving the infinite horizon problem (Problem \\ref{problem_infinite_2C}). Our solution will employ results from Markov Jump Linear Systems (MJLS). We provide a review of the relevant results from the theory of Markov jump linear systems before describing our solution to Problem \\ref{problem_infinite_2C}.\n\\subsection{System Model and Problem Formulation}\n\nIn this section, we study an extension of the system model in Section \\ref{sec:model_2_controllers} to the case where instead of 1 local controller, we have $N$ local controllers, $C^1,C^2, \\ldots,C^N$, each associated to a co-located plant as shown in Fig. \\ref{fig:SystemModel}. We use $\\mathcal{N}$ to denote the set $\\{1,2, \\ldots, N\\}$ and $\\overline{\\mathcal{N}}$ to denote $\\{0,1,\\ldots, N\\}$.\nThe linear dynamics of plant $n \\in \\mathcal{N}$ are given by\n\\begin{align}\n&X_{t+1}^n \\!=\\! A^{nn} X_t^n + B^{nn}U^{n}_t+ B^{n0} U^0_t + W_t^n, t=0,\\dots,T,\n \\label{Model:system}\n\\end{align}\nwhere $X_t^n\\in \\R^{d_X^n}$ is the state of the plant $n$ at time $t$,\n$U^n_t \\in \\R^{d_U^n}$ is the control action of the controller $C^{n}$,  $U^0_t \\in \\R^{d_U^0}$ is the control action of the controller $C^{0}$, and $A^{nn}, B^{nn}, B^{n0}$ are matrices with appropriate dimensions.\nWe assume that $X^n_0 = 0$, and that  $W^n_t$, $n \\in \\mathcal{N}, t \\geq 0$, are i.i.d  random variables with zero mean and $\\cov(W_t^n) = \\mathbf{I}$. Note that we do not assume that random variables $W_{t}^{n}$, $n \\in \\mathcal{N}, t \\geq 0$, are Gaussian.\n\nThe overall system  dynamics can be written as\n\\begin{align}\nX_{t+1} = A X_t + BU_t + W_t,\n\\label{overall_state_dynamic}\n\\end{align}\nwhere $X_t = \\vecc(X^{1:N}_t), U_t = \\vecc(U^{0:N}_t),W_t = \\vecc(W^{0:N}_t)$ and $A,B$ are defined as\n\n\\begin{align}\nA &= \\begin{bmatrix}\n   A^{11} & & \\text{\\huge0}\\\\\n          & \\ddots & \\\\\n     \\text{\\huge0} & & A^{NN}\n\\end{bmatrix}, \nB= \n\\begin{bmatrix}\nB^{10} &  B^{11} & & \\text{\\huge0}\\\\\n\\vdots     &     & \\ddots & \\\\\nB^{N0}  &   \\text{\\huge0} & & B^{NN}\n\\end{bmatrix}.\n\\label{eq:thm_matricesABB}\n\\end{align}\n\n\\begin{singlespace}\n\n\\input{system_model}\n\\end{singlespace}\n\n\\subsubsection*{Communication Model}\n\\label{subs:comm_model}\nThe communication model is similar to the one described in  Section \\ref{subs:comm_model_2C}. In particular, for each $n \\in \\mathcal{N}$, there is an unreliable link with link failure probability $p^n$ from the local controller $C^{n}$ to the remote controller $C^0$. The local controller $C^{n}$ uses its unreliable  link to send the  state  $X_t^n$ of its co-located plant to the remote controller. The state of this link at time $t$ is described by a Bernoulli random variable $\\Gamma_t^n$ and the output of this link at time $t$ is denoted by $Z_t^n$, where $\\Gamma_t^n$ and $Z_t^n$ are described by equations  similar to \\eqref{eq:gamma} and \\eqref{Model:channel_2C}. We assume that $\\Gamma_{0:t}^{1:N}$, $t \\geq 0$, are independent random variables and that they are independent of $W_{0:t}^{1:N}$, $t \\geq 0$.\n\nUnlike the unreliable uplinks, we assume that there exist perfect links from $C^0$ to $C^{n}$, for  each $n \\in \\mathcal{N}$. Therefore, $C^0$ can share $Z_t^{1:N}$ and $U_{t-1}^0$ with all local controllers $C^{1:N}$.\nAll controllers select their control actions at time $t$ after observing $Z_t^{1:N}$  and  $U_{t-1}^0$. \nWe assume that for each $n \\in \\mathcal{N}$, the links from controllers $C^{n}$ and $C^0$ to  plant $n$ are perfect.\n\n\n\\subsubsection*{Information structure and cost}\n\\label{subs:info_cost_2C}\nLet $H^{n}_t$ denote the information available to controller $C^{n}$, $n \\in {\\color{black} \\overline{\\mathcal{N}}}$,  at time $t$.\nThen,\n\\begin{align}\nH^{n}_t&= \\{X^n_{0:t}, U^{n}_{0:t-1}, Z_{0:t}^{1:N}, U^0_{0:t-1}\\}, \\hspace{2mm} n \\in \\mathcal{N}, \\notag \\\\\nH^0_t &= \\{Z_{0:t}^{1:N}, U^0_{0:t-1}\\}. \n\\label{Model:info}\n\\end{align}\nLet $\\mathcal{H}^{n}_t$ be the space of all possible realizations of $H_t^n$.\nThen, $C^{n}$'s actions are selected according to\n\\begin{align}\nU^{n}_t &= g^{n}_t(H^{n}_t), \\hspace{2mm}  n \\in {\\color{black} \\overline{\\mathcal{N}}},\n\\label{Model:strategy}\n\\end{align}\nwhere $g^{n}_t:\\mathcal{H}^{n}_t{\\color{black} \\to } \\R^{d_U^n}$ is a Borel measurable mapping.\nWe use $g:=(g^0_{0},g^0_1,\\dots,g^1_{0},g^1_1,\\dots, g^N_{0},g^N_1,\\dots,)$ to collectively denote the control strategies of all $N+1$ controllers.\n\nThe instantaneous cost $c_t(X_t, U_t)$ of the system is a quadratic function similar to the one described in \\eqref{Model:cost_2C} where $X_t = \\vecc(X^{1:N}_t), U_t = \\vecc(U^{0:N}_t)$ and \n\\begin{align}\nQ= \\begin{bmatrix}\nQ^{11} &\\ldots &Q^{1N} \\\\\n\\vdots & \\ddots & \\vdots \\\\\nQ^{N1} & \\ldots & Q^{NN}\n\\end{bmatrix}, R= \\begin{bmatrix}\nR^{00} &R^{01} &\\ldots &R^{0N} \\\\\nR^{10} &R^{11} &\\ldots &R^{1N} \\\\\n\\vdots &\\vdots & \\ddots & \\vdots \\\\\nR^{N0}  & \\ldots & \\ldots & R^{NN} \n\\end{bmatrix}.\n\\label{matrix_structure}\n\\end{align}\n$Q$ is a symmetric positive semi-definite (PSD) matrix and $R$ is a symmetric positive definite (PD) matrix. \n\n\\subsubsection*{Problem Formulation}\n\\label{subs:prob_formulaiton}\nLet $\\mathcal{G}$ denote the set of all possible control strategies of controllers $C^{0},\\ldots,C^N$.\nThe performance of control strategies $g$ over a finite horizon $T$ is measured by $J_T(g)$ defined in \\eqref{Model:J_T_2C}.\nFor the decentralized networked control system (DNCS) described above, we consider the problem of strategy optimization  over finite and infinite time horizons. These two problems are formally defined below.\n\\begin{problem}\n\\label{problem_finite}\nFor the DNCS described above,\nsolve the following strategy optimization problem:\n\\begin{align}\n\\inf_{g \\in\\mathcal{G}} \nJ_T(g)\n\\label{Model:obj_finite}\n\\end{align}\n\\end{problem}\n\n\n\\begin{problem}\n\\label{problem_infinite}\nFor the DNCS described above,\nsolve the following strategy optimization problem:\n\\begin{align}\n&\\inf_{g \\in\\mathcal{G}} \nJ_{\\infty}(g)\n:= \\inf_{g \\in\\mathcal{G}} \n\\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} J_{T}(g).\n\\label{Model:obj_infinite}\n\\end{align}\n\\end{problem}\n\nDue to stability issues in the infinite horizon problem, we make the following  assumptions on the system and cost matrices. \n\\begin{assumption}\n\\label{assum:det_stb}\n$(A,Q^{1\/2})$ is detectable and $(A,B)$ is stabilizable.\n\\end{assumption}\n\n\\begin{assumption}\n\\label{assum:det_stb_2}\n$\\big(A^{nn},(Q^{nn})^{1\/2} \\big)$ is detectable for all $n \\in \\mathcal{N}$.\n\\end{assumption}\n\nThe finite horizon strategy optimization problem (Problem \\ref{problem_finite}) has been solved in \\cite{asghari_ouyang_nayyar_tac_2018}.  We summarize the finite horizon results below. \n\\begin{lemma}(\\cite[Theorem 2]{asghari_ouyang_nayyar_tac_2018})\n\\label{lm:opt_strategies}\nThe optimal control strategies of Problem \\ref{problem_finite} are given by\n\\begin{align}\n\\bmat{U^{0*}_t \\\\ U^{1*}_t \\\\ \\vdots \\\\ U^{N*}_t} = K_t^0 \\hat X_t + \n\\begin{bmatrix}\n\\mathbf{0} & \\ldots & \\mathbf{0} \\\\\n&&\\\\\nK_t^{1} &     &  \\text{\\huge0}  \\\\\n &\\ddots     &   \\\\\n  \\text{\\huge0} &  & K_t^{N}\n\\end{bmatrix} \\left(X_t - \\hat X_t\\right), \n\\label{eq:opt_U}\n\\end{align}\nwhere $\\hat X_t = \\vecc(\\hat X^{1:N}_t) = \\vecc(\\ee[X^1_t|H^0_t], \\ldots, \\ee[X^N_t|H^0_t]) = \\ee[X_t|H^0_t]$ is the  estimate (conditional expectation) of $X_t$ based on the common information $H^0_t$.  The estimate $\\hat X_t$ can be computed recursively as follows: for $n \\in \\mathcal{N}$,\n\\begin{align}\n\\hat X_0^n = & 0, \n\\label{eq:estimator_0}\n\\\\\n\\hat X_{t+1}^n\n= &\\left\\{\n\\begin{array}{ll}\n \\big([A]_{n,:} + [B]_{n,:} K^0_t\\big)\\hat X_t& \\text{ if }Z_{t+1}^n= \\emptyset,\\\\\n X_{t+1}^n & \\text{ if }Z_{t+1}^n = X_{t+1}^n.\n\\end{array}\\right. \n\\label{eq:estimator_t}\n\\end{align}\nThe gain matrices $K_t^0$ and $K_t^{n}$, $n \\in \\mathcal{N}$, are given by\n\\begin{align}\n& K^0_t = \\Psi(P_{t+1},R,A,B),\n\\label{eq:K_finite}\n\\\\\n&K_t^n = \\Psi \\big ((1-p^n)[P_{t+1}^0]_{n,n}+p^n P_{t+1}^{n},R^{nn},A^{nn},B^{nn} \\big),\n\\label{eq:tildeK_finite}\n\\end{align}\nwhere $P^0_t = \\begin{bmatrix}\n[P_{t}^0]_{1,1} &\\ldots &[P_{t}^0]_{1,N} \\\\\n\\vdots & \\ddots & \\vdots \\\\\n[P_{t}^0]_{N,1} & \\ldots & [P_{t}^0]_{N,N}\n\\end{bmatrix} \\in \\mathbb{R}^{(\\sum_{n=1}^N d^n_X)\\times (\\sum_{n=1}^N d^n_X)}$ and $P_t^{n} \\in \\mathbb{R}^{ d^n_X\\times d^n_X} $, for $n \\in \\mathcal{N}$, are PSD matrices obtained recursively as follows:  \nwhere $P_t^0$ and $P_t^{n}$,  $n \\in \\mathcal{N}$, are PSD matrices obtained recursively as follows:\n\\begin{align}\n&P_{T+1}^0 = \\mathbf{0}, \\quad P_{T+1}^{n} = \\mathbf{0}, \\label{eq:P_N_init}\\\\\n&P_t^0 =  \\Omega(P_{t+1}^0,Q,R,A,B),\n\\label{eq:P_finite}\n\\\\\n&P_t^{n} =  \\Omega \\big((1-p^n)[P_{t+1}^0]_{n,n}+p^n P_{t+1}^{n},Q^{nn},R^{nn},A^{nn},B^{nn}\\big).\n\\label{eq:tildeP_finite}\n\\end{align}\nFurthermore, the optimal cost is given by\n\\begin{align}\nJ^*_{T} = & \\sum_{t = 0}^T \\sum_{n = 1}^N \\tr \\Big((1-p^n)[P_{t+1}^0]_{n,n}+p^n P_{t+1}^{n} \\Big).\n\\label{eq:opcost_finite}\n\\end{align}\n\\end{lemma}\n\n\\begin{remark}\nNote that remote controller's action $U^{0*}_t$ in \\eqref{eq:opt_U} is a function  of $\\hat{X}_t$ only while the local controller $C^n$'s action $U^{n*}_t$ is a function of both $\\hat{X}_t$ and $X_t^n$. Further, as per \\eqref{eq:estimator_0} and \\eqref{eq:estimator_t}, $\\hat{X}_t$ is computed recursively based only on the knowledge of $Z_{0:t}^{1:N}$.\n\\end{remark}\n\n\\subsection{Infinite Horizon Optimal Control}\n\\label{sec:infinite_N_controllers}\nAs in Section \\ref{sec:infinite_2_controllers},  the infinite horizon  problem (Problem \\ref{problem_infinite}) can be solved by answering the following three questions:\n\n\n\n\n\n\n\n\n\\begin{enumerate}[label=(\\mylabel{Q}{\\arabic*})]\n\\item Do matrices $P_t^{0},\\ldots,P_t^N$, defined in \\eqref{eq:P_N_init}-\\eqref{eq:tildeP_finite} converge as $t \\to -\\infty$ to $P_*^0,\\ldots, P_*^N$, that satisfy the coupled fixed point  equations \\eqref{eq:P_finite_NC_fixed} - \\eqref{eq:tildeP_finite_NC_fixed} below?\n\\begin{align}\n&P_*^0 =  \\Omega \\big(P_{*}^0,Q,R,A,B\\big),\n\\label{eq:P_finite_NC_fixed}\n\\\\\n& P_*^{n} =  \\Omega \\big((1-p^n)[P_{*}^0]_{n,n}+p^n P_{*}^{n},Q^{nn},R^{nn},A^{nn},B^{nn}\\big).\n\\label{eq:tildeP_finite_NC_fixed}\n\\end{align}\n\\item If matrices $P_t^{0},\\ldots,P_t^N$ converge and we define matrices $K_*^0, \\ldots, K_*^N$, using matrices $P_*^{0},\\ldots,P_*^N$ as follows, \n\\begin{align}\n& K_*^0 = \\Psi(P_{*}^0,R,A,B),\n\\label{eq:K_finite_NC_fixed}\n\\\\\n& K_*^n = \\Psi \\big((1-p^n)[P_{*}^0]_{n,n}+p^n P_{*}^n,R^{nn},A^{nn},B^{nn}\\big),\n\\label{eq:tildeK_finite_NC_fixed}\n\\end{align}\nare the following strategies optimal for Problem \\ref{problem_infinite}?\n\\begin{align}\n\\bmat{U^{0*}_t \\\\ U^{1*}_t \\\\ \\vdots \\\\ U^{N*}_t} = K_*^0 \\hat X_t + \n\\begin{bmatrix}\n\\mathbf{0} & \\ldots & \\mathbf{0} \\\\\n&&\\\\\nK_*^{1} &     &  \\text{\\huge0}  \\\\\n &\\ddots     &   \\\\\n  \\text{\\huge0} &  & K_*^{N}\n\\end{bmatrix} \\left(X_t - \\hat X_t\\right), \n\\label{eq:opt_U_NC_fixed}\n\\end{align}\nwhere $\\hat{X}_t$ can be computed recursively using \\eqref{eq:estimator_0} - \\eqref{eq:estimator_t}  by replacing $K^0_t$ with $K^0_*$.\n\n\\item If matrices $P_t^{0},\\ldots,P_t^N$ do not converge, is it still possible to find control strategies with finite cost for Problem \\ref{problem_infinite}?\n\\end{enumerate}\n\n\n\\subsection{Answering Q1}\nAs in Section \\ref{sec:Q1_2C}, we will answer Q1 by establishing a connection between the recursions for matrices $P_t^{0},\\ldots,P^N_t$ in our DNCS problem and the recursions for matrices $P_t^{\\diamond}(m)$, $m \\in \\mathcal{M},$ in the MJLS problem.  One obstacle in making this connection is the fact that the matrices $P_t^{0},\\ldots,P^N_t$ in our DNCS problem do not have the same dimensions while the matrices $P_t^{\\diamond}(m)$, $m \\in \\mathcal{M},$ in the MJLS problem  all have the same dimensions. This obstacle was not present in Section \\ref{sec:infinite_2_controllers}. To get around this difficulty, we first provide a new representation $\\bar P_t^{0},\\ldots,\\bar P_t^{N}$ for the matrices $P_t^{0},\\ldots,P^N_t$ in our DNCS problem such that the new  matrices $\\bar P_t^{0},\\ldots,\\bar P_t^{N}$ all have the same dimensions.\n\n\n\\begin{lemma}\n\\label{lm:opt_strategies_new_rep}\nDefine matrices $\\bar P_t^{n} \\in \\mathbb{R}^{(\\sum_{k=1}^N d^k_X)\\times (\\sum_{k=1}^N d^k_X)}$, $n =0,1,\\ldots,N$, recursively as follows:\n\\begin{align}\n&\\bar P_{T+1}^n = \\mathbf{0}, \\label{eq:barP_init}\\\\\n&\\bar P_t^0 = \\Omega(\\bar P_{t+1}^0,Q,R,A,B),\n\\label{eq:barP_finite_0}\n\\\\\n&\\bar P_t^{n} = \\Omega \\big((1-p^n)\\bar P_{t+1}^0+p^n \\bar P_{t+1}^n, \n \\mathcal{L}_{zero}(Q,Q^{nn},n,n),  \\notag \\\\\n& \\hspace{1.4cm}  \\mathcal{L}_{iden}(R,R^{nn},n+1), \\mathcal{L}_{zero}(A,A^{nn},n,n), \\notag \\\\\n & \\hspace{1.4cm} \\mathcal{L}_{zero}(B,B^{nn},n,n+1) \\big),\n\\label{eq:barP_finite}\n\\end{align}\nwhere the operators $\\mathcal{L}_{zero}$ and $\\mathcal{L}_{iden}$ are as defined in Section \\ref{sec:operators}.\nThen, for $ t \\leq T+1$,\n\\begin{align}\n\\label{new_rep1}\n&\\bar P_t^0 =  P_t^0, \\\\\n& \\bar P_t^{n} = \\mathcal{L}_{zero}(P^0_t, P_t^{n},n,n), ~~n=1,\\ldots,N.\n\\label{new_rep2}\n\\end{align}\nConsequently, matrices $P_t^{0},\\ldots,P^N_t$ converge as $t \\to -\\infty$ if and only if matrices $\\bar P_t^{0},\\ldots,\\bar P_t^{N}$ converge as $t \\to -\\infty$.\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{proof_lm:opt_strategies_new_rep} for a proof.\n\\end{proof}\n\nWe can now proceed with constructing an auxiliary MJLS.\n\\subsubsection*{\\textbf{Step 1: Constructing an auxiliary MJLS}}\nConsider an auxiliary MJLS where the set $\\mathcal{M}$ of modes is $ \\{0,1,\\ldots,N\\}$.  Then, we have the following $N+1$ sequences of  matrices, $P^{\\diamond}_t(0),P^{\\diamond}_t(1),\\ldots,P^{\\diamond}_t(N)$, defined recursively using \\eqref{eq:CARE_init} and \\eqref{eq:CARE_finite} for this MJLS:\n\\begin{align}\n&P^{\\diamond}_{T+1}(m)  =\\mathbf{0}, ~\\forall m \\in \\mathcal{M}, \\label{eq:P_N_MJ_init}\\\\\n&P^{\\diamond}_t(0) =  \\notag \\\\\n&\\Omega\\big(\\sum_{k=0}^N \\theta^{0k} P^{\\diamond}_{t+1}(k),Q^{\\diamond}(0),R^{\\diamond}(0),A^{\\diamond}(0),B^{\\diamond}(0) \\big),\n\\label{P_MJ_cmp_NC_0}\n\\\\\n&P^{\\diamond}_t(n)=   \\notag \\\\\n& \\Omega\\big(\n\\sum_{k=0}^N \\theta^{nk} P^{\\diamond}_{t+1}(k),Q^{\\diamond}(n),R^{\\diamond}(n),A^{\\diamond}(n),B^{\\diamond}(n) \\big).\n\\label{P_MJ_cmp_NC_1}\n\\end{align}\nFurthermore, we have the recursions of \\eqref{eq:barP_init}-\\eqref{eq:barP_finite} for matrices $\\bar P_t^{0},\\ldots, \\bar P_t^N$ in our DNCS problem.\nBy comparing \\eqref{P_MJ_cmp_NC_0}-\\eqref{P_MJ_cmp_NC_1} with \\eqref{eq:barP_finite_0}-\\eqref{eq:barP_finite}, we find that the following definitions would make the two sets of equations identical:\n\\begin{align}\n&A^{\\diamond}(0) = A, A^{\\diamond}(n) = \\mathcal{L}_{zero}(A,A^{nn},n,n), \\hspace{1mm} n \\in \\mathcal{N},\n\\label{A_mj}\n\\end{align}\n\\begin{align}\n&B^{\\diamond}(0) = B, B^{\\diamond}(n) =  \\mathcal{L}_{zero}(B,B^{nn},n,n+1), n \\in \\mathcal{N},\n\\label{B_mj}\n\\end{align}\n\\begin{align}\n&Q^{\\diamond}(0) = Q, Q^{\\diamond}(n) =   \\mathcal{L}_{zero}(Q,Q^{nn},n,n), \\hspace{1mm} n \\in \\mathcal{N},\n\\label{Q_mj}\n\\\\\n&R^{\\diamond}(0) = R, R^{\\diamond}(n) =  \\mathcal{L}_{iden}(R,R^{nn},n+1), \\hspace{1mm} n \\in \\mathcal{N},\n\\label{R_mj} \n\\\\\n&\\Theta =\n\\begin{blockarray}{rccccc}\n&0 &1 &2  &\\ldots &N  \\\\\n\\begin{block}{r[ccccc]}\n                     0 & 1 & 0 & \\ldots &\\ldots & 0 \\\\\n                     1 & 1-p^1 & p^1 &\\ddots  & & \\vdots \\\\\n                     2 & 1-p^2 &0  &p^2  & \\ddots& \\vdots \\\\\n                     \\vdots & \\vdots & \\vdots  &\\ddots  &\\ddots & 0 \\\\\n                     N & 1-p^N & 0  &\\ldots  &0 &p^N \\\\\n\\end{block}\n\\end{blockarray}.\n\\label{transition_prob}\n\\end{align}\nTo complete the definition of the auxiliary MJLS, we need to define the initial state and mode probability distributions $\\pi_{X_0^{\\diamond}}$ and $\\pi_{M_0}$. These can be defined arbitrarily and for simplicity we assume that the initial state is fixed to be $X^{\\diamond}_0 = 0$ and the initial mode $M_0$ is uniformly distributed over the set $\\mathcal{M}$.  The following lemma summarizes the above discussion.\n\n\n\n\\begin{lemma}\n\\label{equality_recursions_NC}\nFor the auxiliary MJLS described by \\eqref{A_mj}-\\eqref{transition_prob}, \n the coupled recursions in \\eqref{eq:P_N_MJ_init}-\\eqref{P_MJ_cmp_NC_1} are identical to  the coupled recursions in \\eqref{eq:barP_init}-\\eqref{eq:barP_finite}.\n\\end{lemma}\n\\begin{proof}\nThe lemma can be proved by  straightforward algebraic manipulations.\n\\end{proof}\n\n\\subsubsection*{\\textbf{Step 2: Using MJLS results to answer Q1}}\nNow that we have constructed an auxiliary MJLS where $P^{\\diamond}_t(m) = \\bar P_t^{m}$ for $m=0,\\ldots,N$, we can use the MJLS results about convergence of  matrices $P^{\\diamond}_t(m)$ (that is, Lemmas \\ref{lm:MJ_infinite} and \\ref{lm:ss})  to answer Q1. The following lemma states this result.\n\n\\begin{lemma}\n\\label{lm:pc_NC}\nSuppose Assumptions \\ref{assum:det_stb} and \\ref{assum:det_stb_2} hold. Then, the matrices $ P^0_t,\\ldots,  P^N_t$ defined in \\eqref{eq:P_N_init}-\\eqref{eq:tildeP_finite} converge as $t \\to -\\infty$ to matrices  $P_*^0,\\ldots, P_*^N$ that satisfy the coupled fixed point  equations \\eqref{eq:P_finite_NC_fixed}-\\eqref{eq:tildeP_finite_NC_fixed} if and only if  $p^n < p_c^n$ for all $n \\in \\mathcal{N}$, where the critical thresholds $p_c^n, n \\in \\mathcal{N}$, are given by\n\\begin{align}\n\\frac{1}{\\sqrt{p_c^n}} =  \\min_{K  \\in \\mathbb{R}^{d^n_U \\times d^n_X}}\\rho(A^{nn}+B^{nn}K).\n\\label{eq:pc}\n\\end{align}\n \n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{proof_lm:pc}.\n\\end{proof}\n\n\\subsection{Answering Q2 and Q3}\nAssuming that $P_{t}^n \\rightarrow P_*^n$ as $t \\rightarrow -\\infty$ for $n=0,\\ldots,N,$ we want to know whether the control strategies of \\eqref{eq:K_finite_NC_fixed}-\\eqref{eq:opt_U_NC_fixed} are optimal for Problem \\ref{problem_infinite_2C}. The following result shows that these control strategies are indeed optimal.\n\n\n\\begin{lemma}\n\\label{lm:Q2_NC}\nIf $P_{t}^n \\rightarrow P_*^n$ as $t \\rightarrow -\\infty$ for $n=0,\\ldots,N$, then\n\\begin{enumerate}\n\\item Problem \\ref{problem_infinite} has finite optimal cost,\n\\item The strategies described by \\eqref{eq:K_finite_NC_fixed}-\\eqref{eq:opt_U_NC_fixed} are optimal for Problem \\ref{problem_infinite},\n\\item Under the strategies described by \\eqref{eq:K_finite_NC_fixed}-\\eqref{eq:opt_U_NC_fixed}, $X_t$ and $(X_t-\\hat X_t)$ are mean square stable.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{sec:proof_lm_Q2_NC}.\n\\end{proof}\nThe following lemma answers Q3.\n\\begin{lemma}\\label{lm:Q3_NC}\nIf matrices $P_t^{0},\\ldots,P_t^{N}$ do not converge as $t \\rightarrow -\\infty$, then Problem \\ref{problem_infinite} does not have finite optimal cost. \n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{sec:proof_lm_Q3_NC}.\n\\end{proof}\n\nNow that we have answered  Q1, Q2 and Q3, we can summarize our results for the infinite horizon DNCS problem (Problem \\ref{problem_infinite}).\n\n\\subsection{Summary of the Infinite Horizon Results}\nBased on the answers to Q1-Q3, the following theorem summarizes our results for Problem   \\ref{problem_infinite}.\n\\begin{theorem}\n\\label{thm:DC_infinite_NC}\nSuppose Assumptions \\ref{assum:det_stb} and \\ref{assum:det_stb_2} hold. Then, \n\\begin{enumerate}[(i)]\n\\item Problem \\ref{problem_infinite} has finite optimal cost if and only if\nfor all $n \\in \\mathcal{N}$, $p^n < p_c^n$ where the critical threshold $p_c^n$ is given by \\eqref{eq:pc}.\n\n\\item If $p^n < p_c^n$ for all $n \\in \\mathcal{N}$, there exist symmetric positive semi-definite matrices $P_*^0,\\ldots, P_*^N$ that satisfy \\eqref{eq:P_finite_NC_fixed}-\\eqref{eq:tildeP_finite_NC_fixed} and the optimal strategies for Problem \\ref{problem_infinite} are given by\n\\begin{align}\n\\bmat{U^{0*}_t \\\\ U^{1*}_t \\\\ \\vdots \\\\ U^{N*}_t} = K_*^0 \\hat X_t + \n\\begin{bmatrix}\n\\mathbf{0} & \\ldots & \\mathbf{0} \\\\\n&&\\\\\nK_*^{1} &     &  \\text{\\huge0}  \\\\\n &\\ddots     &   \\\\\n  \\text{\\huge0} &  & K_*^{N}\n\\end{bmatrix} \\left(X_t - \\hat X_t\\right), \n\\label{eq:opt_U_NC_thm}\n\\end{align}\nwhere the  estimate $\\hat X_t$ can be computed recursively using \\eqref{eq:estimator_0}-\\eqref{eq:estimator_t} by replacing $K^0_t$ with $K^0_*$ and the gain matrices $K_*^0, \\ldots,K_*^N$ are given by \n\\begin{align}\n& K_*^0 = \\Psi(P_{*}^0,R,A,B),\n\\label{eq:K_finite_NC_final}\n\\\\\n& K_*^n = \\Psi \\big((1-p^n)[P_{*}^0]_{n,n}+p^n P_{*}^n,R^{nn},A^{nn},B^{nn}\\big).\n\\label{eq:tildeK_finite_NC_final}\n\\end{align}\n\\item If $p^n < p_c^n$ for all $n \\in \\mathcal{N}$, then under the strategies described in part (ii) above, $X_t$ and $(X_t-\\hat X_t)$ are mean square stable.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{corollary}\nSuppose the local controllers are just  sensors (i.e., $B^{nn} = 0$ for $n=1,\\ldots,N$) and the remote controller is the only controller present. Then, if $\\rho(A) < 1\/\\sqrt{p^n}$ for all $n=1,\\ldots,N$, the optimal controller of this multi-sensor,  single-controller NCS is given by $U^{0*}_t$ in \\eqref{eq:opt_U_NC_thm}, and the corresponding closed-loop system is mean-square stable.\n\\end{corollary}\n\n\n\\begin{remark}\nIf Assumption \\ref{assum:det_stb_2} is not true, define, for $n=1,\\ldots,N$, $p^n_c = \\min \\{p^n_s, p^n_d\\},$  where \n\\begin{align}\n\\frac{1}{\\sqrt{p_s^n}} &=  \\min_{K  \\in \\mathbb{R}^{d^n_U \\times d^n_X}}\\rho(A^{nn}+B^{nn}K).\n\\label{eq:pc_ss}\n\\\\\n\\frac{1}{\\sqrt{p_d^n}} &=  \\min_{H \\in \\mathbb{R}^{d^n_X \\times d^n_X}}\\rho \\big(A^{nn}+H (Q^{nn})^{1\/2} \\big).\n\\label{eq:pc_sd}\n\\end{align}\nThen, using arguments similar to those used for proving Theorem \\ref{thm:DC_infinite_NC}, we can show that if $p^n < p_c^n$ for all $n \\in \\mathcal{N}$, the strategies in \\eqref{eq:opt_U_NC_thm} are optimal for Problem \\ref{problem_infinite}. Moreover, Problem \\ref{problem_infinite} has finite optimal cost and the system state is mean square stable under optimal strategies.\n\\end{remark}\n\n\n\n\n\n\\subsubsection*{Communication Model}\n\\label{subs:comm_model}\nWe assume that the communication network between the local controller $C^{n}$, $n \\in \\mathcal{N}$, and the remote controller $C^0$ is the same as one described for the local controller $C^{1}$ is Section \\ref{subs:comm_model_2C}. In particular, there exists an unreliable channel with link failure probability $p^n$ from the local controller $C^{n}$, $n \\in \\mathcal{N}$, to the remote controller $C^0$, through which the local controller $C^{n}$ sends the perfectly observed state  $X_t^n$ of its co-located plant. The state of this channel at time $t$ is described by Bernoulli random variable $\\Gamma_t^n=0$ and the output of this channel at time $t$ is denoted by $Z_t^n$ where $\\Gamma_t^n$ and $Z_t^n$ are described similar to \\eqref{Model:channel_2C}.\n\nUnlike the unreliable uplinks, we assume that there exist perfect links from $C^0$ to $C^{n}$, for $n \\in \\mathcal{N}$. Therefore, $C^0$ can share $Z_t^{1:N}$ and $U_{t-1}^0$ with all local controllers $C^{1:N}$.\nAll controllers select their control actions at time $t$ after observing $Z_t^{1:N}$. \nA schematic of the time ordering of  variables is shown in Fig. \\ref{fig:timing}. \nWe assume that for all $n \\in \\mathcal{N}$, the links from controllers $C^{n}$ and $C^0$ to the plant $n$ are perfect.\n\n\n\\subsubsection*{Information structure and cost}\n\\label{subs:info_cost_2C}\nLet $H^{n}_t$ denote the information available to controller $C^{n}$, $n \\in {\\color{black} \\overline{\\mathcal{N}}}$, to make decisions at time $t$.\nThen,\n\\begin{align}\nH^{n}_t&= \\{X^n_{0:t}, U^{n}_{0:t-1}, Z_{0:t}^{1:N}, U^0_{0:t-1}\\}, \\hspace{2mm} n \\in \\mathcal{N} \\notag \\\\\nH^0_t &= \\{Z_{0:t}^{1:N}, U^0_{0:t-1}\\}. \n\\label{Model:info}\n\\end{align}\nLet $\\mathcal{H}^{n}_t$ be the space of all possible realizations of $H_t^n$.\nThen, $C^{n}$'s actions are selected according to\n\\begin{align}\nU^{n}_t &= g^{n}_t(H^{n}_t), \\hspace{2mm}  n \\in {\\color{black} \\overline{\\mathcal{N}}},\n\\label{Model:strategy}\n\\end{align}\nwhere $g^{n}_t:\\mathcal{H}^{n}_t{\\color{black} \\to } \\R^{d_U^n}$ is a Borel measurable mapping.\nWe use $g:=(g^0_{0},g^0_1,\\dots,g^1_{0},g^1_1,\\dots, g^N_{0},g^N_1,\\dots,)$ to denote the control strategies of $C^0$ and $C^{1:N}$.\n\nThe instantaneous cost $c_t(X_t, U_t)$ of the system is a quadratic function similar to the one described in \\eqref{Model:cost_2C} where $X_t = \\vecc(X^{1:N}_t), U_t = \\vecc(U^{0:N}_t)$ and \n\\begin{align}\nQ= \\begin{bmatrix}\nQ^{11} &\\ldots &Q^{1N} \\\\\n\\vdots & \\ddots & \\vdots \\\\\nQ^{N1} & \\ldots & Q^{NN}\n\\end{bmatrix}, R= \\begin{bmatrix}\nR^{00} &R^{01} &\\ldots &R^{0N} \\\\\nR^{10} &R^{11} &\\ldots &R^{1N} \\\\\n\\vdots &\\vdots & \\ddots & \\vdots \\\\\nR^{N0}  & \\ldots & \\ldots & R^{NN} \n\\end{bmatrix}.\n\\label{matrix_structure}\n\\end{align}\n$Q$ is a symmetric positive semi-definite (PSD) matrix and $R$ is a symmetric positive definite (PD) matrix. \n\n\\subsubsection*{Problem Formulation}\n\\label{subs:prob_formulaiton}\nLet $\\mathcal{G}$ denote the set of all possible control strategies of $C^{0:N}$ that ensure that all states and control actions have finite second moments. The performance of control strategies $g$ over a finite horizon $T$ is measured by $J_T(g)$ described in \\eqref{Model:J_T_2C}.\n\nWe refer to the system described above as the \\emph{decentralized networked control system} with $N$ Local controller and 1 Remote controller (NL1R-DNCS).\nWe consider the problem of strategy optimization  for the NL1R-DNCS over finite  and infinite time horizons. These two problems are formally defined below.\n\\begin{problem}\n\\label{problem_finite}\nFor the NL1R-DNCS described above,\nsolve the following strategy optimization problem:\n\\begin{align}\n\\inf_{g \\in\\mathcal{G}} \nJ_T(g)\n\\label{Model:obj_finite}\n\\end{align}\n\\end{problem}\n\n\n\\begin{problem}\n\\label{problem_infinite}\nFor the NL1R-DNCS described above,\nsolve the following strategy optimization problem:\n\\begin{align}\n&\\inf_{g \\in\\mathcal{G}} \nJ_{\\infty}(g)\n:= \\inf_{g \\in\\mathcal{G}} \n\\limsup_{T\\rightarrow\\infty} \\frac{1}{T+1} J_{T}(g).\n\\label{Model:obj_infinite}\n\\end{align}\n\n\\end{problem}\n\\section{Proof of Claim \\ref{claim:cost_optimal_2C}}\n\\label{proof_claim:cost_optimal_2C}\nIn order to show that \\eqref{eq:costforgstar_2C} holds, it suffices to show that the following equation is true for all $t \\geq 0$:\n\\begin{align}\n\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]=  \\tr(\\Lambda_*)  + \\ee^{g^*} [V_t - V_{t+1} | H_t^0],\n\\label{main_goal_2C}\n\\end{align}\nwhere $U^*_t$ are the control actions at time $t$ under $g^*$.\nThis is because by taking the expectation of \\eqref{main_goal_2C} and summing it from $t=0$ to $T$, we obtain\n\\begin{align}\nJ_{T}(g^*) \n= & \\sum_{t=0}^T \\ee^{g^*} [ c(X_t,U_t^*) ]\n\\notag\\\\\n=&(T+1) \\tr(\\Lambda_*) \n+ \\ee^{g^*} [V_0 - V_{T+1} ]\n\\notag\\\\\n=&(T+1) \\tr(\\Lambda_*) \n- \\ee^{g^*} [V_{T+1} ],\n\\end{align}\nwhere the last equality holds because $V_0 = 0$ (recall that $X_0 = \\hat X_0 =0$). \n\n\nNow, to show that \\eqref{main_goal_2C} holds, first note that $\\ee^{g^*}[V_t | H_t^0] = V_t$ since $V_t$, given by \\eqref{V_t_2C}, is a function of $H^0_t$.  Hence, \\eqref{main_goal_2C} is equivalent to\n\\begin{align}\n\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0] + \\ee^{g^*} [ V_{t+1} | H_t^0] = \\tr(\\Lambda_*)  + V_t. \\label{eq:main_goal_2C_2}\n\\end{align}\n In the following subsections we will calculate $\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]$ and $\\ee^{g^*}[V_{t+1} | H_t^0]$ and then simplify the left hand side of \\eqref{eq:main_goal_2C_2}.  To do so, we define \n\\begin{align}\n&\\hat X_{t+1|t} := \\ee [X_{t+1}|H^0_{t}]\n\\\\\n&\\Sigma_{t+1|t} := \\cov(X_{t+1}|H^0_{t})\n \\\\\n&\\Sigma_t := \\cov(X_{t}|H^0_{t}),\n\\label{Sigma_2C}\n\\end{align}\nand recall that $\\hat{X}_t = \\ee [X_{t}|H^0_{t}]$.\n\n\n\\subsection{Calculating $\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]$}\nNote that \n\\begin{small}\n\\begin{align}\n&\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]=  \\underbrace{\\ee^{g^*} [X_t^\\tp Q X_t | H_t^0]}_{\\mathbb{T}_4} + \\underbrace{\\ee^{g^*}[U_t^{*\\tp} R U_t^* | H_t^0]}_{\\mathbb{T}_5}. \n\\label{cost_given_h_2C}\n\\end{align}\n\\end{small}\nWe can simplify the term $\\mathbb{T}_4$ as follows\n\\begin{small}\n\\begin{align}\n\\mathbb{T}_4 &=  \\hat X_t^\\tp Q\\hat X_t + \\ee^{g^*} \\Big[(X_t-\\hat X_t)^\\tp Q (X_t-\\hat X_t)|H^0_t\\Big] \\notag \\\\\n& =  \\hat X_t^\\tp Q\\hat X_t + \\tr\\big(Q\\Sigma_t \\big). \\label{eq:T4eq}\n\\end{align}\n\\end{small}\n\n From \\eqref{eq:opt_U_2C_fixed}, we have $U_t^* = K_*^0\\hat X_t + \\tilde K_*^1 (X_t - \\hat X_t)$, where $\\tilde K_*^1 =  \\bmat{\\mathbf{0} \\\\ K_*^1 }$.  Therefore, we can simplify the term $\\mathbb{T}_5$ as follows\n\\begin{small}\n\\begin{align}\n\\mathbb{T}_5 &= (K_*^0\\hat X_t)^\\tp R K_*^0\\hat X_t \\notag \\\\\n& + \\ee^{g^*} \\Big[(X_t-\\hat X_t)^\\tp (\\tilde K_*^1)^{\\tp} R \\tilde K_*^1 (X_t-\\hat X_t)|H^0_t\\Big]\n\\notag\\\\\n&= \\hat X_t^\\tp(K_*^0)^\\tp R K_*^0\\hat X_t + \\tr \\big( (\\tilde K_*^1)^\\tp R \\tilde K_*^1 \\Sigma_t \\big).\n \\label{cost_on_U_t_2C}\n\\end{align}\n\\end{small}\n\nPutting \\eqref{cost_given_h_2C}, \\eqref{eq:T4eq} and \\eqref{cost_on_U_t_2C} together, we can write \n\\begin{align}\\label{eq:calculated_c}\n&\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]= \\hat X_t^\\tp Q\\hat X_t +\\tr(Q \\Sigma_t) \\notag \\\\\n&+ \\hat X_t^\\tp(K_*^0)^\\tp R K_*^0\\hat X_t + \\tr( (\\tilde K_*^1)^\\tp R \\tilde K_*^1 \\Sigma_t).\n\\end{align}\n\n\\subsection{Calculating $\\ee^{g^*}[V_{t+1}|H^0_t]$}\nFrom the definition of $V_{t+1}$ (see \\eqref{V_t_2C})  we have\n\\begin{small}\n\\begin{align}\n \\ee^{g^*}[V_{t+1}|H^0_t] &= \\underbrace{\\ee^{g^*}\\Big[\n\\hat X_{t+1}^\\tp P_*^0 \\hat X_{t+1} \\Big| H^0_t\n\\Big]}_{\\mathbb{T}_6} \\notag \\\\\n&+ \\underbrace{\\ee^{g^*}\\Big[\n\\tr \\big(P_*^{1} \\Sigma_{t+1} \\big) \\Big| H^0_t \\Big]}_{\\mathbb{T}_7}.\\label{eq:T6T7}\n\\end{align}\n\\end{small}\nNote that if $\\Gamma_{t+1}^1 = 1$ (i.e., the link is active) $\\hat X_{t+1} = X_{t+1}$ and   $\\Sigma_{t+1} = 0$  and if $\\Gamma_{t+1}^1 = 0$ $\\hat X_{t+1} = \\hat X_{t+1|t}$ and $\\Sigma_{t+1} = \\Sigma_{t+1|t}$\\footnote{If $\\Gamma_{t+1}^1=0$, the remote controller gets no new information about $X_{t+1}$. Hence, its belief on $X_{t+1}$ given $H_{t+1}^0$ remains the same as its belief on $X_{t+1}$ given $H_{t}^0$.}. That is, \n\\begin{align}\n\\label{estimation_state_2C}\n&\\hat X_{t+1} = \\Gamma_{t+1}^1 X_{t+1} +(1-\\Gamma_{t+1}^1) \\hat X_{t+1|t},  \\\\\n &\\Sigma_{t+1}  = (1-\\Gamma_{t+1}^1) \\Sigma_{t+1|t}.\n\\label{estimation_covariance_2C}\n\\end{align}\nNow, we use \\eqref{estimation_state_2C} and \\eqref{estimation_covariance_2C} to calculate  the terms $\\mathbb{T}_6$ and $\\mathbb{T}_7$ in \\eqref{eq:T6T7}.\n\nNote that from \\eqref{estimation_state_2C}, $\\hat X_{t+1}$ is equal to $\\hat X_{t+1|t} + \\Gamma_{t+1}(X_{t+1} -\\hat X_{t+1|t})$. Therefore,  $\\mathbb{T}_6$ can be written as\n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n&\\mathbb{T}_6= \n\\ee^{g^*}\\Big[\n\\hat X_{t+1|t} ^\\tp P_*^0 \\hat X_{t+1|t} \\Big| H^0_t \\Big] \\notag \\\\\n&+ \\ee^{g^*}\\Big[\n(X_{t+1} -\\hat X_{t+1|t})^\\tp \\Gamma_{t+1}^1 P_*^0 \\Gamma_{t+1}^1 (X_{t+1} -\\hat X_{t+1|t}) \\Big| H^0_t \\Big] \\notag \\\\ & = \n\\hat X_{t+1|t} ^\\tp P_*^0 \\hat X_{t+1|t}+ (1-p^1) \\tr (P_*^0 \\Sigma_{t+1|t}).\n\\label{eq:T_4_1}\n\\end{align}\n\\end{small}\nFurthermore, using \\eqref{estimation_covariance_2C}, it is straightforward to see that \n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n&\\mathbb{T}_7\n= p^1 \\tr (P_*^1  \\Sigma_{t+1|t}).\\label{eq:T7}\n\\end{align}\n\\end{small}\n\nCombining \\eqref{eq:T6T7}, \\eqref{eq:T_4_1} and \\eqref{eq:T7}, we get\n\\begin{align}\n \\ee^{g^*}[V_{t+1}|H^0_t]= &\\hat X_{t+1|t} ^\\tp P_*^0 \\hat X_{t+1|t}+ (1-p^1) \\tr (P_*^0 \\Sigma_{t+1|t})\\notag \\\\\n& +  p^1 \\tr (P_*^1  \\Sigma_{t+1|t}). \\label{eq:vt1}\n\\end{align}\nNote that the right hand side of  \\eqref{eq:vt1} involves  $\\hat X_{t+1|t}$ and $\\Sigma_{t+1|t}$. We will now try to write these in terms of  $\\hat X_{t}$ and $\\Sigma_{t}$. For that purpose,  note that under the strategies $g^*$\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\nX_{t+1} &= AX_t + B \\Big[ \nK_*^0\\hat X_t + \\bmat{\\mathbf{0} \\\\ K_*^1 }(X_t - \\hat X_t)\n \\Big] + W_t \\notag \\\\\n& = A_s(0) \\hat X_t + A_s(1)  ( X_t - \\hat X_t) + W_t.\n\\label{X_dynamics_proof_2C}\n\\end{align}\n\\end{small}\nIn the above equation, we have defined $A_s(0) = A+ B K_*^0$ and $A_s(1) = A + B \\tilde K_*^1$ where   $\\tilde K_*^1 =  \\bmat{\\mathbf{0} \\\\ K_*^1 }$. Now using \\eqref{X_dynamics_proof_2C}, we can calculate $\\hat X_{t+1|t}$ and $\\Sigma_{t+1|t}$ as follows, \n\\begin{align}\n\\label{estimation_X_2C}\n&\\hat X_{t+1|t} = A_s(0) \\hat X_t, \\\\\n&\\Sigma_{t+1|t} =  \\mathbf I + A_s(1) \\Sigma_t A_s(1)^\\tp.\n\\label{Sigma_t_2C}\n\\end{align}\n\nUsing \\eqref{estimation_X_2C} and \\eqref{Sigma_t_2C} in \\eqref{eq:vt1}, we get\n\\begin{align}\n \\ee^{g^*}[V_{t+1}|H^0_t]&= \\hat X_t ^\\tp A_s(0)^\\tp P_*^0 A_s(0) \\hat X_t+ (1-p^1) \\tr (P_*^0) \\notag \\\\\n& +  p^1 \\tr (P_*^1) + (1-p^1) \\tr(P_*^0A_s(1) \\Sigma_t A_s(1)^\\tp) \\notag \\\\\n&+ p^1\\tr(P_*^1A_s(1) \\Sigma_t A_s(1)^\\tp). \\label{eq:vt1A}\n\\end{align}\nRecall that $\\Lambda_* = (1-p^1) P_{*}^0+p^1 P_{*}^1$. Thus, \\eqref{eq:vt1A} can be written as\n\\begin{align}\n \\ee^{g^*}[V_{t+1}|H^0_t]&= \\hat X_t ^\\tp A_s(0)^\\tp P_*^0 A_s(0) \\hat X_t+  \\tr (\\Lambda_*) \\notag \\\\\n& +   (1-p^1) \\tr(P_*^0A_s(1) \\Sigma_t A_s(1)^\\tp) \\notag \\\\\n&+ p^1\\tr(P_*^1A_s(1) \\Sigma_t A_s(1)^\\tp). \\label{eq:vt1B}\n\\end{align}\n\n\\subsection{Simplifying the left hand side of \\eqref{eq:main_goal_2C_2}}\nNow that we have calculated $\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]$ and $\\ee^{g^*}[V_{t+1} | H_t^0]$, we will try to simplify the left hand side of \\eqref{eq:main_goal_2C_2}.\n\nAdding \\eqref{eq:vt1B} and \\eqref{eq:calculated_c} and grouping together the terms involving $\\hat X_t$ and those involving $\\Sigma_t$, we can write\n\\begin{align}\n& \\ee^{g^*}[c(X_t,U_t^*)|H^0_t] + \\ee^{g^*}[V_{t+1}|H^0_t] \n= \\tr(\\Lambda_*) \\notag \\\\\n& + \\hat X_t ^\\tp \\Phi(P_*^0,K^0_*) \\hat X_t  + \\tr(\\Phi((1-p^1)P_*^0 +p^1P^1_*,\\tilde K^1_*)\\Sigma_t), \\label{eq:vt1C}\n\\end{align}\nwhere \n\\begin{small}\n\\begin{align}\n\\Phi(P_*^0,K^0_*) := Q + (K^0_*)^\\tp RK^0_*\n+ (A+BK^0_*)^\\tp P^0_*(A+BK^0_*),\n\\end{align}\n\\end{small}\nand similarly\n\\begin{small}\n\\begin{align}\n&\\Phi((1-p^1)P_*^0 +p^1P^1_*,\\tilde K^1_*):=Q + (\\tilde K_*^1)^\\tp R \\tilde K_*^1 \\notag \\\\\n&~~+(A+B\\tilde K^1_*)^\\tp\\big((1-p^1)P_*^0 +p^1P^1_* \\big)(A+B\\tilde K^1_*) \\notag \\\\\n&= Q + ( K_*^1)^\\tp R^{11}  K_*^1 \\notag \\\\\n&~~+(A+B^{11} K^1_*)^\\tp\\big((1-p^1)P_*^0 +p^1P^1_* \\big)(A+B^{11}K^1_*) \n\\end{align}\n\\end{small}\n\nUsing the fact that $K^0_* = \\Psi(P_{*}^0,R,A,B)$, it can be easily established that \n\\begin{equation}\n\\Phi(P_*^0,K^0_*) = \\Omega(P_{*}^0,Q,R,A,B).\n\\end{equation}\nFurther, since $P^0_* = \\Omega(P_{*}^0,Q,R,A,B)$, we have \n\\begin{equation}\n\\Phi(P_*^0,K^0_*) = P^0_*.\\label{eq:phi0}\n\\end{equation}\n\nSimilarly, using the fact that $K_*^1 = \\Psi((1-p^1)P_{*}^0+p^1 P_{*}^1,R^{11},A,B^{11})$, it can be established that \n\\begin{align}\n&\\Phi((1-p^1)P_*^0 +p^1P^1_*,\\tilde K^1_*) \\notag \\\\\n&= \\Omega((1-p^1)P_*^0 +p^1P^1_*,Q,R^{11},A,B^{11}).\n\\end{align}\nFurther, since $P^1_* = \\Omega((1-p^1)P_*^0 +p^1P^1_*,Q,R^{11},A,B^{11})$, we have \n\\begin{equation}\n\\Phi((1-p^1)P_*^0 +p^1P^1_*,\\tilde K^1_*) =  P^1_*.\\label{eq:phi1}\n\\end{equation}\n\nUsing \\eqref{eq:phi0} and \\eqref{eq:phi1} in \\eqref{eq:vt1C}, we get\n\\begin{align}\n& \\ee^{g^*}[c(X_t,U_t^*)|H^0_t] + \\ee^{g^*}[V_{t+1}|H^0_t]  \\notag \\\\\n&= \\tr(\\Lambda_*)\n + \\hat X_t ^\\tp P^0_* \\hat X_t  + \\tr(P^1_*\\Sigma_t), \\notag \\\\\n&= \\tr(\\Lambda_*) + V_t. \\label{eq:vt1D}\n\\end{align}\nThis establishes \\eqref{eq:main_goal_2C_2} and hence completes the proof of the claim.\n\\section{Proof of Claim \\ref{claim:cost_optimal_NC}}\n\\label{Cost_of_the_Strategies}\nIn order to show that \\eqref{eq:costforgstar_NC} holds, it suffices to show that the following equation is true for all $t \\geq 0$:\n\\begin{align}\n\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]=  \\tr(\\Lambda_*)  + \\ee^{g^*} [V_t - V_{t+1} | H_t^0],\n\\label{main_goal_NC}\n\\end{align}\nwhere $U^*_t$ are the control actions at time $t$ under $g^*$.\nThis is because by taking the expectation of \\eqref{main_goal_NC} and summing it from $t=0$ to $T$, we obtain\n\\begin{align}\nJ_{T}(g^*) \n=(T+1) \\tr(\\Lambda_*) \n- \\ee^{g^*} [V_{T+1} ].\n\\end{align}\n\nNow, to show that \\eqref{main_goal_NC} holds,  note that $\\ee^{g^*}[V_t | H_t^0] = V_t$ since $V_t$ is a function of $H^0_t$.  Hence, \\eqref{main_goal_NC} is equivalent to\n\\begin{align}\n\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0] + \\ee^{g^*} [ V_{t+1} | H_t^0] = \\tr(\\Lambda_*)  + V_t. \\label{eq:main_goal_NC_2}\n\\end{align}\n In the following subsections we will calculate $\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]$ and $\\ee^{g^*}[V_{t+1} | H_t^0]$ and then simplify the left hand side of \\eqref{eq:main_goal_NC_2}.  To do so, we define for $n=1,\\ldots,N,$\n\\begin{align}\n&\\hat X_{t+1|t}^n := \\ee [X_{t+1}^n|H^0_{t}]\n\\\\\n&\\Sigma_{t+1|t}^n := \\cov(X_{t+1}^n|H^0_{t})\n \\\\\n&\\Sigma_t^n := \\cov(X_{t}^n|H^0_{t}).\n\\label{sigma_t}\n\\end{align}\nand recall that $\\hat{X}_t^n = \\ee [X_{t}^n|H^0_{t}]$.\n \n \\subsection{Calculating $\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]$}\n Note that \n\\begin{small}\n\\begin{align}\n&\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]=  \\underbrace{\\ee^{g^*} [X_t^\\tp Q X_t | H_t^0]}_{\\mathbb{T}_4} + \\underbrace{\\ee^{g^*}[U_t^{*\\tp} R U_t^* | H_t^0]}_{\\mathbb{T}_5}. \n\\label{cost_given_h_NC}\n\\end{align}\n\\end{small}\n$\\mathbb{T}_4$ can be written as\n\\begin{small}\n\\begin{align}\n\\mathbb{T}_4 &=  \\hat X_t^\\tp Q\\hat X_t + \\ee \\Big[(X_t-\\hat X_t)^\\tp Q (X_t-\\hat X_t)|H^0_t\\Big] \\notag \\\\\n& =  \\hat X_t^\\tp Q\\hat X_t + \\sum_{n=1}^N \\tr\\big(Q^{nn}\\Sigma_t^n \\big).\n\\end{align}\n\\end{small}\nwhere the second equality is true because according to \\cite[Lemma 3]{asghari_ouyang_nayyar_tac_2018} $X_t^n$ and $X_t^m$, $m \\neq n,$ are conditionally independent given $H_t^0$.\n\nSimilarly, $\\mathbb{T}_5$ can be written as\n\\begin{small}\n\\begin{align}\n\\mathbb{T}_5 &=\\hat X_t^\\tp(K_*^0)^\\tp R K_*^0\\hat X_t + \\sum_{n=1}^N \\tr \\big((K_*^n)^{\\tp} R^{nn} K_*^n \\Sigma_{t}^n \\big ).\n\\end{align}\n\\end{small}\nThus,\n\\begin{align}\\label{eq:calculated_c_N}\n&\\ee^{g^*} [ c(X_t,U_t^*) | H_t^0]= \\hat X_t^\\tp Q\\hat X_t + \\sum_{n=1}^N \\tr\\big(Q^{nn}\\Sigma_t^n \\big) \\notag \\\\\n&~+  \\hat X_t^\\tp(K_*^0)^\\tp R K_*^0\\hat X_t + \\sum_{n=1}^N \\tr \\big((K_*^n)^{\\tp} R^{nn} K_*^n \\Sigma_{t}^n \\big ).\n\\end{align} \n \n \\subsection{Calculating $\\ee^{g^*}[V_{t+1}|H^0_t]$}\n From the definition of $V_{t+1}$ (see \\eqref{V_t_NC})  we have\n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n \\ee^{g^*}[V_{t+1}|H^0_t] &= \\underbrace{\\ee^{g^*}\\Big[\n\\hat X_{t+1}^\\tp P_*^0 \\hat X_{t+1} \\Big| H^0_t\n\\Big]}_{\\mathbb{T}_6} \\notag \\\\\n&+ \\underbrace{\\ee^{g^*}\\Big[\n\\sum_{n=1}^N \\tr \\big(P_*^n \\Sigma_{t+1}^n \\big )\n\\Big| H^0_t\n\\Big]}_{\\mathbb{T}_7}.\\label{eq:T6T7_N}\n\\end{align}\n\\end{small}\nNote that if  $\\Gamma_{t+1}^n = 1$, $\\hat X_{t+1}^n = X_{t+1}^n$ and  $\\Sigma_{t+1}^n = 0$ and if $\\Gamma_{t+1}^n = 0$, $\\hat X_{t+1}^n = \\hat X_{t+1|t}^n$ and $\\Sigma_{t+1}^n = \\Sigma_{t+1|t}^n$\\footnote{If $\\Gamma^n_{t+1}=0$, the remote controller gets no new information about $X^n_{t+1}$. Hence, its belief on $X^n_{t+1}$ given $H_{t+1}^0$ remains the same as its belief on $X^n_{t+1}$ given $H_{t}^0$.}. Let $\\Delta$ be a random block diagonal matrix defined as follows:\n\\begin{align}\n\\Delta := \\begin{bmatrix}\n   \\Gamma_{t+1}^1 \\mathbf{I}_{d_X^1} & & \\text{\\huge0}\\\\\n          & \\ddots & \\\\\n     \\text{\\huge0} & & \\Gamma_{t+1}^N \\mathbf{I}_{d_X^N}\n\\end{bmatrix}.\n\\label{Delta}\n\\end{align}\nThen, we can write \n\\begin{align}\n\\label{estimation_state}\n&\\hat X_{t+1} = \\Delta X_{t+1} +(\\mathbf{I}-\\Delta) \\hat X_{t+1|t}, \\\\\n &\\Sigma_{t+1}^n  = (1-\\Gamma_{t+1}^n) \\Sigma_{t+1|t}^n, \\quad n \\in \\mathcal{N}.\n\\label{estimation_covariance}\n\\end{align}\nNow, we use \\eqref{estimation_state} and \\eqref{estimation_covariance} to calculate  the terms $\\mathbb{T}_6$ and $\\mathbb{T}_7$ in \\eqref{eq:T6T7_N}. It can be shown through some straightforward  manipulations that \n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n&\\mathbb{T}_6= \n  \\hat X_{t+1|t} ^\\tp P_*^0 \\hat X_{t+1|t} + \\sum_{n=1}^N (1-p^n) \\tr \\big ([P_*^0]_{n,n}  \\Sigma_{t+1|t}^n  \\big).\n\\label{T_4_N}\n\\end{align}\n\\end{small}\nSimilarly, it can be shown that\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\\label{eq:T7_N}\n&\\mathbb{T}_7= \\sum_{n=1}^N p^n \\tr \\big (P_*^n \\Sigma_{t+1|t}^n  \\big).\n\\end{align}\n\\end{small}\nCombining \\eqref{eq:T6T7_N}, \\eqref{T_4_N} and \\eqref{eq:T7_N}, we get\n \\begin{align}\n &\\ee^{g^*}[V_{t+1}|H^0_t]= \\hat X_{t+1|t} ^\\tp P_*^0 \\hat X_{t+1|t}+ \\notag \\\\ &\\sum_{n=1}^N (1-p^n) \\tr \\big ([P_*^0]_{n,n}  \\Sigma_{t+1|t}^n  \\big) +\n \\sum_{n=1}^N p^n \\tr \\big (P_*^n \\Sigma_{t+1|t}^n  \\big). \\label{eq:vt1_N}\n\\end{align}\nSince the right hand side of  \\eqref{eq:vt1_N} involves  $\\hat X_{t+1|t}$ and $\\Sigma^n_{t+1|t}$, we will now try to write these in terms of  $\\hat X_{t}$ and $\\Sigma^n_{t}$. It can be easily established that \n\\begin{align}\n\\label{estimation_X_NC}\n&\\hat X_{t+1|t} = A_s(0) \\hat X_t, \\\\\n&\\Sigma_{t+1|t}^n =  \\mathbf I + A_s(n) \\Sigma_t^n A_s(n)^\\tp,\n\\label{Sigma_t_NC}\n\\end{align}\nwhere $A_s(0) = A+ B K_*^0$ and $A_s(n) = A^{nn} + B^{nn}K^n_*$ for $n=1,\\ldots,N$.\n\nUsing \\eqref{estimation_X_NC},\\eqref{Sigma_t_NC} and the definition of $\\Lambda_*$  in \\eqref{eq:vt1_N}, we get\n\\begin{align}\n &\\ee^{g^*}[V_{t+1}|H^0_t]= \\hat X_t ^\\tp A_s(0)^\\tp P_*^0 A_s(0) \\hat X_t+ \\tr(\\Lambda_*)\\notag \\\\ \n   &+\\sum_{n=1}^N \\big((1-p^n) \\tr([P_{*}^0]_{n,n}A_s(n) \\Sigma_t A_s(n)^\\tp) \\notag \\\\\n&+ p^n\\tr(P_*^nA_s(n) \\Sigma_t A_s(n)^\\tp)\\big). \\label{eq:vt1A_N}\n\\end{align}\n\n\n\n\n \\subsection{Simplifying the left hand side of \\eqref{eq:main_goal_NC_2}}\n Adding \\eqref{eq:vt1A_N} and \\eqref{eq:calculated_c_N} and grouping together the terms involving $\\hat X_t$ and those involving $\\Sigma^n_t$, we can write\n\\begin{small}\n \\begin{align}\n& \\ee^{g^*}[c(X_t,U_t^*)|H^0_t] + \\ee^{g^*}[V_{t+1}|H^0_t] \n \\notag \\\\\n& = \\tr(\\Lambda_*)+ \\hat X_t ^\\tp \\Phi(P_*^0,K^0_*) \\hat X_t  + \\notag \\\\\n&\\sum_{n=1}^N \\tr(\\Phi^n((1-p^n)[P_*^0]_{n,n} +p^nP^n_*, K^n_*)\\Sigma^n_t), \\label{eq:vt1C_N}\n\\end{align}\n\\end{small}\nwhere \\begin{small}\n\\begin{align}\n\\Phi(P_*^0,K^0_*) := Q + (K^0_*)^\\tp RK^0_*\n+ (A+BK^0_*)^\\tp P^0_*(A+BK^0_*),\n\\end{align}\n\\end{small}\nand\n\\begin{small}\n\\begin{align}\n&\\Phi^n((1-p^n)[P_*^0]_{n,n} +p^nP^n_*, K^n_*) :=Q^{nn} + (K_*^n)^{\\tp} R^{nn} K_*^n \\notag \\\\\n&+(A^{nn}+B^{nn} K^n_*)^\\tp\\big((1-p^n)[P_*^0]_{n,n} +p^nP^n_* \\big)(A^{nn}+B^{nn}K^n_*).\n\\end{align}\n\\end{small}\nUsing the fact that $K^0_* = \\Psi(P_{*}^0,R,A,B)$ and that $P^0_* = \\Omega(P_{*}^0,Q,R,A,B)$, it can be shown that \n\\begin{equation}\n\\Phi(P_*^0,K^0_*) = P^0_*.\\label{eq:phi0_N}\n\\end{equation}\nSimilarly, using the fact that $K_*^n = \\Psi((1-p^n)[P_{*}^0]_{n,n}+p^n P_{*}^n,R^{nn},A^{nn},B^{nn})$ and that $P_*^{n} =  \\Omega \\big((1-p^n)[P_{*}^0]_{n,n}+p^n P_{*}^{n},Q^{nn},R^{nn},A^{nn},B^{nn}\\big)$, it can be shown that \n\\begin{equation}\n\\Phi^n((1-p^n)[P_*^0]_{n,n} +p^nP^n_*, K^n_*) =  P^n_*.\\label{eq:phi1_N}\n\\end{equation}\n\nUsing \\eqref{eq:phi0_N} and \\eqref{eq:phi1_N} in \\eqref{eq:vt1C_N}, we get\n\\begin{small}\n\\begin{align}\n& \\ee^{g^*}[c(X_t,U_t^*)|H^0_t] + \\ee^{g^*}[V_{t+1}|H^0_t]  \\notag \\\\\n&= \\tr(\\Lambda_*)\n + \\hat X_t ^\\tp P^0_* \\hat X_t  + \\sum_{n=1}^N\\tr(P^n_*\\Sigma^n_t), \\notag \\\\\n&= \\tr(\\Lambda_*) + V_t. \\label{eq:vt1D_N}\n\\end{align}\n\\end{small}\nThis establishes \\eqref{eq:main_goal_NC_2} and hence completes the proof of the claim.\n\\section{Proof of Claim \\ref{claim:cost_optimal_2C}}\n\\label{proof_claim:cost_optimal_2C}\n\n\\blue{Partially Edited OLD proof}\n\nIn order to show that \\eqref{eq:costforgstar_2C} holds, it suffices to show that the following equation is true for all $t \\geq 0$:\n\\begin{align}\n\\ee^{g*} [ c(X_t,U_t^*) | H_t^0]=  \\tr(\\Lambda_*)  + \\ee^{g*} [V_t - V_{t+1} | H_t^0],\n\\label{main_goal_2C}\n\\end{align}\n\\blue{where $U^*_t$ are the control actions at time $t$ under $g^*$.\nThis is because by taking the expectation of \\eqref{main_goal_2C} and summing it from $t=0$ to $T$, we obtain}\n\\begin{align}\nJ_{T}(g^*) \n= & \\sum_{t=0}^T \\ee^{g*} [ c(X_t,U_t^*) ]\n\\notag\\\\\n=&(T+1) \\tr(\\Lambda_*) \n+ \\ee^{g*} [V_0 - V_{T+1} ]\n\\notag\\\\\n=&(T+1) \\tr(\\Lambda_*) \n- \\ee^{g*} [V_{T+1} ],\n\\end{align}\nwhere the last equality holds because $V_0 = 0$ (recall that $X_0 = \\hat X_0 =0$). \n\n\nNow, to show that \\eqref{main_goal_2C} holds, \\blue{first note that $\\ee^{g*}[V_t | H_t^0] = V_t$ since $V_t$, given by \\eqref{V_t_2C}, is a function of $H^0_t$}.  Hence, \\eqref{main_goal_2C} is equivalent to\n\\begin{align}\n\\ee^{g*} [ c(X_t,U_t^*) | H_t^0] + \\ee^{g*} [ V_{t+1} | H_t^0] = \\tr(\\Lambda_*)  + V_t. \\label{eq:main_goal_2C_2}\n\\end{align}\n In the following subsections we will calculate $\\ee^{g*} [ c(X_t,U_t^*) | H_t^0]$ and $\\ee^{g*}[V_{t+1} | H_t^0]$.  To do so, we define \n\\begin{align}\n&\\hat X_{t+1|t} := \\ee [X_{t+1}|H^0_{t}]\n\\\\\n&\\Sigma_{t+1|t} := \\cov(X_{t+1}|H^0_{t})\n \\\\\n&\\Sigma_t := \\cov(X_{t}|H^0_{t}),\n\\label{Sigma_2C}\n\\end{align}\nand recall that $\\hat{X}_t = \\ee [X_{t}|H^0_{t}]$.\n\n\\subsection{Calculating $\\ee^{g*} [ c(X_t,U_t^*) | H_t^0]$}\nNote that \n\\begin{small}\n\\begin{align}\n&\\ee^{g*} [ c(X_t,U_t^*) | H_t^0]=  \\underbrace{\\ee^{g*} [X_t^\\tp Q X_t | H_t^0]}_{\\mathbb{T}_4} + \\underbrace{\\ee^{g*}[U_t^{*\\tp} R U_t^* | H_t^0]}_{\\mathbb{T}_5}. \n\\label{cost_given_h_2C}\n\\end{align}\n\\end{small}\nWe can simplify the term $\\mathbb{T}_4$ as follows\n\\begin{small}\n\\begin{align}\n\\mathbb{T}_4 &=  \\hat X_t^\\tp Q\\hat X_t + \\ee^{g*} \\Big[(X_t-\\hat X_t)^\\tp Q (X_t-\\hat X_t)|H^0_t\\Big] \\notag \\\\\n& =  \\hat X_t^\\tp Q\\hat X_t + \\tr\\big(Q\\Sigma_t \\big)\n=  \\hat X_t^\\tp Q^{\\diamond}(0)\\hat X_t + \\tr\\big(Q^{\\diamond}(1)\\Sigma_t \\big), \\label{eq:T4eq}\n\\end{align}\n\\end{small}\nwhere the last equality is correct because from \\eqref{eq:MJLS_Q}, $Q^{\\diamond}(0) = Q^{\\diamond}(1) = Q$. \n\n From \\eqref{eq:opt_U_2C_fixed}, we have $U_t^* = K_*^0\\hat X_t + \\tilde K_*^1 (X_t - \\hat X_t)$, where $\\tilde K_*^1 =  \\bmat{\\mathbf{0} \\\\ K_*^1 }$.  Therefore, we can simplify the term $\\mathbb{T}_5$ as follows\n\\begin{small}\n\\begin{align}\n\\mathbb{T}_5 &= (K_*^0\\hat X_t)^\\tp R K_*^0\\hat X_t \\notag \\\\\n& + \\ee^{g*} \\Big[(X_t-\\hat X_t)^\\tp (\\tilde K_*^1)^{\\tp} R \\tilde K_*^1 (X_t-\\hat X_t)|H^0_t\\Big]\n\\notag\\\\\n&= (K_*^0\\hat X_t)^\\tp R^{\\diamond}(0) K_*^0\\hat X_t + \\tr \\big( (\\tilde K_*^1)^\\tp R^{\\diamond}(1) \\tilde K_*^1 \\Sigma_t \\big),\n \\label{cost_on_U_t_2C}\n\\end{align}\n\\end{small}\nwhere the last equality is correct from definition of $R^{\\diamond}(0), R^{\\diamond}(1)$ in \\eqref{eq:MJLS_R}.\n\nPutting \\eqref{cost_given_h_2C}, \\eqref{eq:T4eq} and \\eqref{cost_on_U_t_2C} together, we can write \n\\begin{align}\\label{eq:calculated_c}\n&\\ee^{g*} [ c(X_t,U_t^*) | H_t^0]=\\underbrace{[ \\hat X_t^\\tp Q(0)\\hat X_t +\\tr(Q(1) \\Sigma_t)]}_{=\\mathbb{T}_4} \\notag \\\\\n&+ \\underbrace{[(K_*^0\\hat X_t)^\\tp R(0) K_*^0\\hat X_t + \\tr( (\\tilde K_*^1)^\\tp R(1) \\tilde K_*^1 \\Sigma_t)]}_{=\\mathbb{T}_5}.\n\\end{align}\n\\subsection{Calculating $\\ee^{g*}[V_{t+1}|H^0_t]$}\n\n\\red{Start here}\nFrom the definition of $V_{t+1}$ (see \\eqref{V_t_2C})  we have\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n& \\ee^{g*}[V_{t+1}|H^0_t]= \\underbrace{\\ee^{g*}\\Big[\n\\hat X_{t+1}^\\tp P_*^0 \\hat X_{t+1} \\Big| H^0_t\n\\Big]}_{\\mathbb{T}_6} + \\underbrace{\\ee^{g*}\\Big[\n\\tr \\big(P_*^{1} \\Sigma_{t+1} \\big) \\Big| H^0_t \\Big]}_{\\mathbb{T}_7}.\\label{eq:T6T7}\n\\end{align}\n\\end{small}\nNote that if $\\Gamma_{t+1} = 1$ (i.e., the link is active) $\\hat X_{t+1} = X_{t+1}$ and   $\\Sigma_{t+1} = 0$  and if $\\Gamma_{t+1} = 0$ $\\hat X_{t+1} = \\hat X_{t+1|t}$ and $\\Sigma_{t+1} = \\Sigma_{t+1|t}$. That is, \n\\red{Do the above statements need some justification? FYI Something is weird about 109 in the finite horizon paper}\n\\begin{align}\n\\label{estimation_state_2C}\n&\\hat X_{t+1} = \\Gamma_{t+1}X_{t+1} +(1-\\Gamma_{t+1}) \\hat X_{t+1|t},  \\\\\n &\\Sigma_{t+1}  = (1-\\Gamma_{t+1}) \\Sigma_{t+1|t}.\n\\label{estimation_covariance_2C}\n\\end{align}\nNow, we use \\eqref{estimation_state_2C} and \\eqref{estimation_covariance_2C} to calculate  the terms $\\mathbb{T}_6$ and $\\mathbb{T}_7$ in \\eqref{eq:T6T7}.\n\nNote that from \\eqref{estimation_state_2C}, $\\hat X_{t+1}$ is equal to $\\hat X_{t+1|t} + \\Gamma_{t+1}(X_{t+1} -\\hat X_{t+1|t})$. Therefore,  $\\mathbb{T}_6$ can be written as\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n&\\mathbb{T}_6= \n\\ee^{g^*}\\Big[\n\\hat X_{t+1|t} ^\\tp P_*^0 \\hat X_{t+1|t} \\Big| H^0_t \\Big] \\notag \\\\\n&+ \\ee^{g^*}\\Big[\n(X_{t+1} -\\hat X_{t+1|t})^\\tp \\Gamma_{t+1} P_*^0 \\Gamma_{t+1}(X_{t+1} -\\hat X_{t+1|t}) \\Big| H^0_t \\Big] \\notag \\\\ & = \n\\hat X_{t+1|t} ^\\tp P_*^0 \\hat X_{t+1|t}+ (1-p^1) \\tr (P_*^0 \\Sigma_{t+1|t}).\n\\label{T_4_1}\n\\end{align}\n\\end{small}\nFurthermore, using \\eqref{estimation_covariance_2C}, it is straightforward to see that \n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n&\\mathbb{T}_7\n= p^1 \\tr (P_*^1  \\Sigma_{t+1|t}).\n\\end{align}\n\\end{small}\n\nIn the subsequent analysis, we will make use of the following definitions and lemmas.\n\\begin{definition}\\label{def:Pi_2}\nLet $P_*$ denote the set of matrices $\\{P^0_*,P^*_1\\}$.  For $n=0,1$, we define\n \\begin{align}\n\\Pi (P_*, n) := \\sum_{m=0}^1 \\theta^{nm} P_*^m,\n\\label{Pi_2_matrix}\n\\end{align}\nwhere $\\theta^{nm}$ are the mode transition probabilities  given in \\eqref{eq:MJLS_theta}. We further define, for $n=0,1$,\n\\begin{align}\n&\\Omega^{\\diamond}(P_*,n) := \\Omega\\big(\\Pi(Y, n),Q^{\\diamond}(n),R^{\\diamond}(n),A^{\\diamond}(n),B^{\\diamond}(n) \\big),\n\\label{Omega_MJ}\n\\\\\n&\\Psi^{\\diamond}(P_*,n) := \\Psi \\big(\\Pi(Y, n),R^{\\diamond}(n),A^{\\diamond}(n),B^{\\diamond}(n) \\big),\n\\label{Psi_MJ}\n\\end{align}\nwhere $\\Omega$ and $\\Psi$ are the operators defined in \\eqref{Omega} and \\eqref{Psi}.\nFinally, for a matrix $K$ and for $n=0,1$, we define\n\\begin{align}\n& \\Phi(P_*,K,n) = Q^{\\diamond}(n) + (K)^\\tp R^{\\diamond}(n)K\n\\notag\\\\\n+ &(A^{\\diamond}(n)+B^{\\diamond}(n)K)^\\tp \\Pi(Y,n)(A^{\\diamond}(n)+B^{\\diamond}(n)K).\n\\label{Phi_2_def}\n\\end{align}\n\\end{definition}\nWe have the following lemma.\n\\begin{lemma}\nFor $n=0,1,$\n\\begin{small}\n\\begin{align}\n\\Omega^{\\diamond}(P_*,n) = \\Phi(P_*,\\Psi^{\\diamond}(P_*,n),n) = \\min_K \\Phi(P_*,K,n).\n\\label{eq:relation1}\n\\end{align}\n\\end{small}\nNote that the minimization is in the sense of \\red{partial order undefined?} $\\succeq$, that is, the minimum value $\\Omega^{\\diamond}(P_*,n) \\preceq \\Phi(P_*,K,n)$ for all $K$.\n\\end{lemma}\n\\red{Proof}\n\\begin{lemma}\n\\begin{small}\n\\begin{align}\n\\Phi(P_*,K_*^0,0) = \\Phi(P_*,\\Psi^{\\diamond}(P_*,0),0) = \\Omega^{\\diamond}(P_*,0) = P_*^0, \\label{eq:claim1P0} \\\\\n\\Phi(P_*,\\tilde K_*^1,1) = \\Phi(P_*,\\Psi^{\\diamond}(P_*,1),1) = \\Omega^{\\diamond}(P_*,1) = P_*^1. \\label{eq:claim1P1}\n\\end{align}\n\\end{small}\n\\end{lemma}\n\\red{Proof}\n\nNow, using $\\mathbb{T}_6$ and $\\mathbb{T}_7$ calculated above and Definition \\ref{def:Pi_2}, $\\ee^{g^*}[V_{t+1}|H^0_t]$ can be written as,\n\\begin{align}\n\\ee^{g^*}[V_{t+1}|H^0_t]&= \n\\hat X_{t+1|t} ^\\tp \\Pi(P_*,0) \\hat X_{t+1|t}+  \\tr (\\Pi(P_*,1) \\Sigma_{t+1|t}).\n\\label{V_t_1_2C}\n\\end{align}\n\nNote that \\eqref{V_t_1_2C} involves  $\\hat X_{t+1|t}$ and $\\Sigma_{t+1|t}$. We will now try to write these in terms of  $\\hat X_{t}$ and $\\Sigma_{t}$. \n For that purpose,  note that under the strategies $g^*$\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\nX_{t+1} &= AX_t + B \\Big[ \nK_*^0\\hat X_t + \\bmat{\\mathbf{0} \\\\ K_*^1 }(X_t - \\hat X_t)\n \\Big] + W_t \\notag \\\\\n& = A_s(0) \\hat X_t + A_s(1)  ( X_t - \\hat X_t) + W_t.\n\\label{X_dynamics_proof_2C}\n\\end{align}\n\\end{small}\nIn the above equation, we have defined $A_s(0) = A^{\\diamond}(0) + B^{\\diamond}(0) K_*^0$ and $A_s(1) = A^{\\diamond}(1) + B^{\\diamond}(1) \\tilde K_*^1$ where  matrices $A^{\\diamond}(m),B^{\\diamond}(m)$, $m=0,1,$ are as defined in \\eqref{eq:MJLS_A}-\\eqref{eq:MJLS_B} and $\\tilde K_*^1 =  \\bmat{\\mathbf{0} \\\\ K_*^1 }$. Now using \\eqref{X_dynamics_proof_2C}, we can calculate $\\hat X_{t+1|t}$ and $\\Sigma_{t+1|t}$ as follows, \n\\begin{align}\n\\label{estimation_X_2C}\n&\\hat X_{t+1|t} = A_s(0) \\hat X_t, \\\\\n&\\Sigma_{t+1|t} =  \\mathbf I + A_s(1) \\Sigma_t A_s(1)^\\tp.\n\\label{Sigma_t_2C}\n\\end{align}\n\n\\red{Many matrices are missing the diamonds, I have added some, but several remain....}\n\nUsing \\eqref{estimation_X_2C}, we can write \n\\begin{small}\n\\begin{align}\n&\\hat X_{t+1|t} ^\\tp \\Pi(P_*,0) \\hat X_{t+1|t} \\notag \\\\\n& =\\hat X_t ^\\tp A_s(0)^\\tp \\Pi(P_*,0) A_s(0) \\hat X_t\n\\notag\\\\\n&= \\hat X_t ^\\tp \\Big(\\Phi(P_*,K_*^0,0) - Q(0) - (K_*^0)^\\tp R(0) K_*^0 \\Big)\\hat X_t\n\\notag\\\\\n&= \\hat X_t^\\tp P_*^0\\hat X_{t}\n - \\hat X_t^\\tp Q(0) \\hat X_{t}- (K_*^0\\hat X_t)^\\tp R(0) K_*^0\\hat X_t\n\\label{eq:Vtp1_mean_2C}\n\\end{align}\n\\end{small}\nwhere the first equality follows from the definition of $\\Phi(P_*,K,0)$ in Definition \\ref{def:Pi_2} and the second equality is correct because of \\eqref{eq:claim1P0}.\n\n\nSimilarly, using \\eqref{Sigma_t_2C}, we can write\n\\begin{small}\n\\begin{align}\n&\\tr (\\Pi(P_*,1) \\Sigma_{t+1|t})\n\\notag\\\\\n&= \\tr(\\Pi(P_*,1)) + \\tr((A_{s}(1))^\\tp \\Pi(P_*,1) A_{s}(1) \\Sigma_t)\n\\notag\\\\\n&= \\tr(\\Pi(P_*,1))\n+\\tr(\\Phi(P_*,\\tilde K_*^1,1) \\Sigma_t)\n\\notag\\\\\n&-\\tr(Q(1)\\Sigma_t)-\\tr( (\\tilde K_*^1)^\\tp R(1) \\tilde K_*^1 \\Sigma_t)\n\\notag\\\\\n&= \\tr(\\Pi(P_*,1))\n+\\tr(P_*^1 \\Sigma_t)\n\\notag\\\\\n&-\\tr(Q(1) \\Sigma_t)- \\tr( (\\tilde K_*^1)^\\tp R(1) \\tilde K_*^1 \\Sigma_t),\n\\label{eq:Vtp1_variance_2C}\n\\end{align}\n\\end{small}\nwhere the  first equality follows from the definition of $\\Phi(P_*,K,1)$ in Definition \\ref{def:Pi_2} and the second equality is correct because of \\eqref{eq:claim1P1}.\n\nPutting  \\eqref{V_t_1_2C}, \\eqref{eq:Vtp1_mean_2C} and \\eqref{eq:Vtp1_variance_2C}together, we get\n\\begin{small}\n\\begin{align}\n \\ee^{g^*}[V_{t+1}|H^0_t] &=  \\hat X_t^\\tp P_*^0\\hat X_{t}\n - \\hat X_t^\\tp Q(0) \\hat X_{t}- (K_*^0\\hat X_t)^\\tp R(0) K_*^0\\hat X_t\n \\notag \\\\\n& + \\tr(\\Pi(P_*,1)) + \\tr(P_*^1 \\Sigma_t)\n\\notag\\\\\n&-\\tr(Q(1) \\Sigma_t)- \\tr( (\\tilde K_*^1)^\\tp R(1) \\tilde K_*^1 \\Sigma_t) \\notag \\\\\n&=  \\tr(\\Pi(P_*,1)) + \\underbrace{\\hat X_t^\\tp P_*^0\\hat X_{t} + \\tr(P_*^1 \\Sigma_t)}_{=V_t} \\notag \\\\\n& - \\underbrace{[ \\hat X_t^\\tp Q(0)\\hat X_t +\\tr(Q(1) \\Sigma_t)]}_{=\\mathbb{T}_4} \\notag \\\\\n& -  \\underbrace{[(K_*^0\\hat X_t)^\\tp R(0) K_*^0\\hat X_t + \\tr( (\\tilde K_*^1)^\\tp R(1) \\tilde K_*^1 \\Sigma_t)]}_{=\\mathbb{T}_5}.\n\\label{eq:Vtp1_t_2C}\n\\end{align}\n\\end{small}\n\nAdding \\eqref{eq:Vtp1_t_2C} and \\eqref{eq:calculated_c} and noting that $ \\Pi(P_*,1) = \\Lambda_*$ , we get \n\n\\vspace{-3mm}\n\\begin{small}\n\\begin{align}\n& \\ee^{g^*}[V_{t+1}|H^0_t] + \\ee^{g^*}[c(X_t,U_t^*)|H^0_t]\n= \\tr(\\Lambda_*)\n+ V_t ,\n\\end{align}\n\\end{small}\nwhich proves \\eqref{eq:main_goal_2C_2}.\n\\section{Proof of Claim \\ref{claim:cost_optimal_NC}}\n\\label{Cost_of_the_Strategies}\n\\red{Start here}\nWe want to show that \\eqref{eq:costforgstar_NC} holds. To this end, we show that \n\\begin{align}\n\\ee [ c(X_t,U_t^*) | H_t^0]=  \\tr(\\Lambda_*)  + \\ee [V_t - V_{t+1} | H_t^0].\n\\label{main_goal_NC}\n\\end{align}\nwhere $\\Lambda_* =  \\sum_{n=1}^N \\big( (1-p^n) [P_{*}^0]_{n,n}+p^n P_{*}^n \\big)$.\nThen, taking the expectation of the above equation and summing it from $0$ to $T$, the correctness of \\eqref{eq:costforgstar_NC} is concluded as follows,\n\\begin{align}\nJ_{T}(g^*) \n= & \\sum_{t=0}^T \\ee [ c(X_t,U_t^*) ]\n\\notag\\\\\n=&(T+1) \\tr(\\Lambda_*) \n+ \\ee [V_0 - V_{T+1} ]\n\\notag\\\\\n=&(T+1) \\tr(\\Lambda_*) \n- \\ee [V_{T+1} ],\n\\end{align}\nwhere the last equality holds because $V_0 = 0$. \n\nNow, to show that \\eqref{main_goal_2C} holds, first note that $\\ee[V_t | H_t^0] = V_t$ which is given by \\eqref{V_t_2C}. Hence, we only need to calculate $\\ee [ c(X_t,U_t^*) | H_t^0]$ and $\\ee[V_{t+1} | H_t^0]$. \n\n\\subsection{Calculating $\\ee [ c(X_t,U_t^*) | H_t^0]$}\nNote that \n\\begin{small}\n\\begin{align}\n&\\ee [ c(X_t,U_t^*) | H_t^0]=  \\underbrace{\\ee [X_t^\\tp Q X_t | H_t^0]}_{\\mathbb{T}_4} + \\underbrace{\\ee[U_t^{*\\tp} R U_t^* | H_t^0]}_{\\mathbb{T}_5}. \n\\label{cost_given_h}\n\\end{align}\n\\end{small}\nWe can simplify $\\mathbb{T}_4$ as follows,\n\\begin{small}\n\\begin{align}\n\\mathbb{T}_4 &=  \\hat X_t^\\tp Q\\hat X_t + \\ee \\Big[(X_t-\\hat X_t)^\\tp Q (X_t-\\hat X_t)|H^0_t\\Big] \\notag \\\\\n& =  \\hat X_t^\\tp Q\\hat X_t + \\sum_{n=1}^N \\tr\\big([Q]_{n,n}\\Sigma_t^n \\big) \\notag \\\\\n& =  \\hat X_t^\\tp Q(0)\\hat X_t + \\sum_{n=1}^N \\tr\\big([Q(n)]_{n,n}\\Sigma_t^n \\big),\n\\end{align}\n\\end{small}\nwhere the second equality is correct because according to \\cite[Lemma 3]{asghari_optimal_2017_arXiv},\ngiven $H_t^0$, $X_t^n$ is independent of $X_t^m$ for $m \\neq n$ and\nthe last equality is correct because from \\eqref{Q_mj}, $Q(0) = Q$ and $[Q(n)]_{n,n} = [Q]_{n,n}= Q^{nn}$.\nSimilarly, since from \\eqref{eq:opt_U_NC_fixed} we have $U_t^* = K_*^0 \\hat X_t + \\sum_{n=1}^N \\tilde K_*^n (X_t - \\hat X_t)$ where we have defined $\\tilde K_*^n = \\mathcal{L}_{zero}(K_*^0, K_*^n,n+1,n)$, we can simplify $\\mathbb{T}_5$ as follows,\n\n\\vspace{-3mm}\n\\begin{small}\n\\begin{align}\n\\mathbb{T}_5 &=(K_*^0 \\hat X_t)^\\tp R K_*^0 \\hat X_t +  \\sum_{n=1}^N \\tr \\big( [(\\tilde K_*^n)^{\\tp} R \\tilde K_*^n]_{n,n} \\Sigma_{t}^n \\big )\n\\notag\\\\\n&= (K_*^0 \\hat X_t)^\\tp R(0) K_*^0 \\hat X_t  \n+ \\sum_{n=1}^N \\tr \\big( [(\\tilde K_*^n)^{\\tp} R(n) \\tilde K_*^n]_{n,n} \\Sigma_{t}^n \\big ),\n \\label{cost_on_U_t}\n\\end{align}\n\\end{small}\nwhere the last equality is correct because from \\eqref{R_mj}, $R(0) = R$ and $[(\\tilde K_*^n)^{\\tp} R \\tilde K_*^n]_{n,n} = \n[(\\tilde K_*^n)^{\\tp} R(n) \\tilde K_*^n]_{n,n} = (K_*^n)^{\\tp} R^{nn} K_*^n$.\n\n\\subsection{Calculating $\\ee[V_{t+1}|H^0_t]$}\nFor $n \\in \\mathcal{N}$, we define\n\\begin{align}\n&\\hat X_{t+1|t}^n := \\ee [X_{t+1}^n|H^0_{t}]\n\\\\\n&\\Sigma_{t+1|t}^n := \\cov(X_{t+1}^n|H^0_{t})\n \\\\\n&\\Sigma_t^n := \\cov(X_{t}^n|H^0_{t}).\n\\label{sigma_t}\n\\end{align}\nRecall that $\\hat{X}_t^n = \\ee [X_{t}^n|H^0_{t}]$.  Note that under the strategies $g^*$ defined in \\eqref{eq:costforgstar_NC} and from \\eqref{Model:system}, we have \n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\nX_{t+1} &= AX_t + B \\Big[ \nK_*^0\\hat X_t +  \\sum_{n=1}^N \\tilde K_*^n (X_t - \\hat X_t)\n \\Big] + W_t \\notag \\\\\n& = A_s(0) \\hat X_t + \\sum_{n=1}^N A_s(n)  ( X_t - \\hat X_t) + W_t.\n\\label{X_dynamics_proof_NC}\n\\end{align}\n\\end{small}\nIn the above equation, we have defined $A_s(0) = A(0) + B(0) K_*^0$ and $A_s(n) = A(n) + B(n) \\tilde K_*^n$ for $n \\in \\mathcal{N}$ where  matrices $A(n),B(n)$, $n \\in \\overline{\\mathcal{N}}$ are as defined in \\eqref{A_mj}-\\eqref{B_mj}. Note that the last equality of \\eqref{X_dynamics_proof_NC} is correct because from \\eqref{A_mj}-\\eqref{B_mj}, $A(0) = A$, $\\sum_{n=1}^N A(n) = A$, $B(0) =B$, and $B\\tilde K_*^n = B(n) \\tilde K_*^n$ for $n \\in \\mathcal{N}$. Now using \\eqref{X_dynamics_proof_2C}, we can calculate $\\hat X_{t+1|t}$ and $\\Sigma^n_{t+1|t}$ as follows, \n\\begin{align}\n\\label{estimation_X_NC}\n&\\hat X_{t+1|t} = A_s(0) \\hat X_t \\\\\n&\\Sigma_{t+1|t}^n =  \\mathbf I + [A_s(n)]_{n,n} \\Sigma_t^n [A_s(n)]_{n,n}^\\tp.\n\\label{Sigma_t_NC}\n\\end{align}\nFrom the definition of $V_{t+1}$ in \\eqref{V_t_NC} and considering \\eqref{sigma_t}, we have\n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n& \\ee[V_{t+1}|H^0_t]= \\underbrace{\\ee\\Big[\n\\hat X_{t+1}^\\tp P_*^0 \\hat X_{t+1} \\Big| H^0_t\n\\Big]}_{\\mathbb{T}_6}\n+ \\underbrace{\\ee\\Big[\n\\sum_{n=1}^N \\tr \\big(P_*^n \\Sigma_{t+1}^n \\big )\n\\Big| H^0_t\n\\Big]}_{\\mathbb{T}_7}.\n\\end{align}\n\\end{small}\n\nNote that  when $\\Gamma_{t+1}^n = 1$, $\\hat X_{t+1}^n = X_{t+1}^n$ and hence $\\Sigma_{t+1}^n = 0$. Further,  when $\\Gamma_{t+1}^n = 0$, $\\hat X_{t+1}^n = \\hat X_{t+1|t}^n$ and $\\Sigma_{t+1}^n = \\Sigma_{t+1|t}^n$. Let $\\Delta$ be a block diagonal matrix whose diagonal are block matrices $\\Gamma_{t+1}^1 \\mathbf{I}_{d_X^1}, \\ldots, \\Gamma_{t+1}^N \\mathbf{I}_{d_X^N}$, that is,\n\\begin{align}\n\\Delta := \\begin{bmatrix}\n   \\Gamma_{t+1}^1 \\mathbf{I}_{d_X^1} & & \\text{\\huge0}\\\\\n          & \\ddots & \\\\\n     \\text{\\huge0} & & \\Gamma_{t+1}^N \\mathbf{I}_{d_X^N}\n\\end{bmatrix}.\n\\label{Delta}\n\\end{align}\nNote that $\\Delta$ is a random matrix that depends on $\\Gamma_{t+1}^1,\\ldots,\\Gamma_{t+1}^N$.\nThen, we can write \n\\begin{align}\n\\label{estimation_state}\n&\\hat X_{t+1} = \\Delta X_{t+1} +(1-\\Delta) \\hat X_{t+1|t}, \\\\\n &\\Sigma_{t+1}^n  = (1-\\Gamma_{t+1}^n) \\Sigma_{t+1|t}^n, \\quad n \\in \\mathcal{N}.\n\\label{estimation_covariance}\n\\end{align}\nNow, we use \\eqref{estimation_state} and \\eqref{estimation_covariance} to calculate $\\mathbb{T}_6$ and $\\mathbb{T}_7$.\n\n\\subsubsection{Calculating $\\mathbb{T}_6$}\nNote that from \\eqref{estimation_state}, $\\hat X_{t+1}$ can be written as $ \\hat X_{t+1}= \\hat X_{t+1|t} + \\Delta (X_{t+1} -\\hat X_{t+1|t})$. Considering this, $\\mathbb{T}_6$ can be written as\n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n&\\mathbb{T}_6= \n\\ee\\Big[\n\\hat X_{t+1|t} ^\\tp P_*^0 \\hat X_{t+1|t} \\Big| H^0_t \\Big] \\notag \\\\\n&+ \\ee\\Big[\n(X_{t+1} -\\hat X_{t+1|t})^\\tp  \\Delta^{\\tp} P_*^0 \\Delta (X_{t+1} -\\hat X_{t+1|t}) \\Big| H^0_t \\Big] \\notag \\\\ & = \\hat X_{t+1|t} ^\\tp P_*^0 \\hat X_{t+1|t} + \\sum_{n=1}^N (1-p^n) \\tr \\big ([P_*^0]_{n,n}  \\Sigma_{t+1|t}^n  \\big).\n\\label{T_4_1}\n\\end{align}\n\\end{small}\n\nThe last equality of \\eqref{T_4_1} is correct because\n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n& \\ee\\Big[\n(X_{t+1} -\\hat X_{t+1|t})^\\tp  \\Delta^{\\tp} P_*^0 \\Delta (X_{t+1} -\\hat X_{t+1|t}) \\Big| H^0_t \\Big] \\notag \\\\ \n&= \n\\ee\\Big[ \\ee[\n(X_{t+1} -\\hat X_{t+1|t})^\\tp  \\Delta^{\\tp} P_*^0 \\Delta (X_{t+1} -\\hat X_{t+1|t}) \n| \\Delta,H_t^0] \\Big| H^0_t \\Big] \\notag \\\\ \n&= \n\\ee\\Big[ \\sum_{n=1}^N\n(X_{t+1}^n -\\hat X_{t+1|t}^n)^\\tp  [\\Delta^{\\tp} P_*^0 \\Delta]_{n,n} (X_{t+1}^n -\\hat X_{t+1|t}^n) \n\\Big| H^0_t \\Big] \\notag \\\\ \n&= \n\\sum_{n=1}^N \\ee\\Big[ \n(X_{t+1}^n -\\hat X_{t+1|t}^n)^\\tp (\\Gamma_{t+1}^n)^2  [P_*^0]_{n,n} (X_{t+1}^n -\\hat X_{t+1|t}^n) \n\\Big| H^0_t \\Big] \\notag \\\\ \n&=  \\sum_{n=1}^N (1-p^n) \\tr \\big ([P_*^0]_{n,n}  \\Sigma_{t+1|t}^n  \\big),\n\\label{T_4_1_1}\n\\end{align}\n\\end{small}\nwhere the first equality is correct because of tower property and the second equality is correct because according to \\cite[Lemma 3]{asghari_optimal_2017_arXiv}, given the common information $H_t^0$, $X_{t+1}^n$ is independent of $X_{t+1}^m$ for $m \\neq n$. The third equality is correct because $[\\Delta^{\\tp} P_*^0  \\Delta]_{n,n} = (\\Gamma_{t+1}^n)^2  [P_*^0]_{n,n}$ which can be verified through straightforward algebraic manipulations by using \\eqref{Delta}. Finally, the last equality is correct because each term of the summation is only non-zero when $\\Gamma_{t+1}^n =1$ which happens with probability $1-p^n$.\n\n\\subsubsection{Calculating $\\mathbb{T}_7$}\nConsidering \\eqref{estimation_covariance}, it is straightforward to see that \n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n&\\mathbb{T}_7= \\sum_{n=1}^N p^n \\tr \\big (P_*^n \\Sigma_{t+1|t}^n  \\big).\n\\end{align}\n\\end{small}\n\n\\subsubsection{Simplifying $\\ee[V_{t+1}|H^0_t]$}\nNow, using $\\mathbb{T}_6$ and $\\mathbb{T}_7$ calculated above, $\\ee[V_{t+1}|H^0_t]$ can be written as,\n\\begin{align}\n\\ee[V_{t+1}|H^0_t]&= \n\\hat X_{t+1|t} ^\\tp \\Pi(\\tilde P_*^0,0) \\hat X_{t+1|t} \\notag \\\\\n&+ \\sum_{n=1}^N \\tr \\big ([\\Pi(\\tilde P_*^n,n)]_{n,n}  \\Sigma_{t+1|t}^n  \\big),\n\\label{V_t_1}\n\\end{align}\nwhere we have defined $\\tilde P_*^0 = P_*^0$ and $\\tilde P_*^n = \\mathcal{L}_{zero}(P_*^0, P_*^n,n,n)$. Further, we have used the fact that $[\\tilde P_*^n]_{n,n} = P_*^n$ and the definition of $\\Pi(\\cdot)$ in \\eqref{Pi_matrix} where $\\Theta$ is as defined in \\eqref{eq:MJLS_theta}.\n\nUsing \\eqref{estimation_X_NC}, we can write $\\hat X_{t+1|t} ^\\tp \\Pi(\\tilde P_*,0) \\hat X_{t+1|t}$ as follows,\n\\begin{small}\n\\begin{align}\n& \\hat X_t ^\\tp A_s(0)^\\tp \\Pi(\\tilde P_*,0) A_s(0) \\hat X_t\n\\notag\\\\\n&= \\hat X_t ^\\tp \\Big(\\Phi(\\tilde P_*,K_*^0,0) - Q(0) - (K_*^0)^\\tp R(0) K_*^0 \\Big)\\hat X_t\n\\notag\\\\\n&= \\hat X_t^\\tp \\tilde P_*^0\\hat X_{t}\n - \\hat X_t^\\tp Q(0) \\hat X_{t}- (K_*^0\\hat X_t)^\\tp R(0) K_*^0\\hat X_t,\n\\label{eq:Vtp1_mean_NC}\n\\end{align}\n\\end{small}\nwhere the first equality is true from \\eqref{Phi_def} in Lemma \\ref{lm:KPrelation}. Furthermore, the last equality is correct because\n\n\\vspace{-2mm}\n\\begin{small}\n\\begin{align}\n\\Phi(\\tilde P_*,K_*^0,0) = \\Phi(\\tilde P_*,\\Psi^{\\diamond}(\\tilde P_*,0),0) = \\Omega^{\\diamond}(\\tilde P_*,0) = \\tilde P_*^0,\n\\end{align}\n\\end{small}\nwhere the first equality is true because from \\eqref{eq:K_finite_2C_fixed}, $K_*^0 = \\Psi^{\\diamond}(\\tilde P_*,0)$, the second equality is true from \\eqref{eq:relation1} in Lemma \\ref{lm:KPrelation}, and the last equality is true because of definition operator $\\Omega^{\\diamond}(\\cdot)$ in \\eqref{Omega_MJ} and \\eqref{eq:P_finite_2C_fixed}.\n\nSimilarly, using \\eqref{Sigma_t_NC} and by defining $\\hat P_*^n = \\Pi(\\tilde P_*^n,n)$, $\\tr \\big ([\\Pi(\\tilde P_*^n,n)]_{n,n}  \\Sigma_{t+1|t}^n  \\big)$ can be written as\n\\begin{small}\n\\begin{align}\n&\\tr \\big ([\\Pi(\\tilde P_*^n,n)]_{n,n}  \\Sigma_{t+1|t}^n  \\big)\n\\notag\\\\\n&=\\tr \\big ([\\hat P_*^n]_{n,n} \\big) + \\tr \\big ([A_{s}(n)]_{n,n}^{\\tp} [\\hat P_*^n]_{n,n} [A_{s}(n)]_{n,n} \\Sigma_t^n \\big)\n\\notag\\\\\n&= \\tr \\big ([\\hat P_*^n]_{n,n} \\big)  \n+\\tr(\\Phi(\\tilde P_*,\\tilde K_*^n,n) \\Sigma_t^n)\n\\notag\\\\\n&-\\tr \\big( [Q(n)]_{n,n} \\Sigma_t^n \\big)-\\tr \\big( [(\\tilde K_*^n)^\\tp R(n) \\tilde K_*^n]_{n,n} \\Sigma_t^n \\big)\n\\notag\\\\\n&= \\tr \\big ([\\hat P_*^n]_{n,n} \\big)  \n+\\tr([\\tilde P_*^n]_{n,n} \\Sigma_t^n)\n\\notag\\\\\n&-\\tr \\big( [Q(n)]_{n,n} \\Sigma_t^n \\big)- \\tr \\big( [(\\tilde K_*^n)^\\tp R(n) \\tilde K_*^n]_{n,n} \\Sigma_t^n \\big).\n\\label{eq:Vtp1_variance_NC}\n\\end{align}\n\\end{small}\nPutting \\eqref{eq:Vtp1_mean_NC} and \\eqref{eq:Vtp1_variance_NC} into \\eqref{V_t_1} and using the fact that $[\\tilde P_*^n]_{n,n} = P_*^n$, we get\n\\begin{small}\n\\begin{align}\n& \\ee[V_{t+1}|H^0_t]\n\\notag\\\\\n& =  \\hat X_t^\\tp P_*^0\\hat X_{t}\n - \\hat X_t^\\tp Q(0) \\hat X_{t}- (K_*^0\\hat X_t)^\\tp R(0) K_*^0\\hat X_t\n \\notag \\\\\n& + \\sum_{n=1}^N \\tr \\big ([\\hat P_*^n]_{n,n} \\big) +\\sum_{n=1}^N \\tr(P_*^n \\Sigma_t^n)\n\\notag\\\\\n&- \\sum_{n=1}^N \\tr \\big( [Q(n)]_{n,n} \\Sigma_t^n \\big)-  \\sum_{n=1}^N \\tr \\big( [(\\tilde K_*^n)^\\tp R(n) \\tilde K_*^n]_{n,n} \\Sigma_t^n \\big)\n \\notag \\\\\n&=\\sum_{n=1}^N \\tr \\big ([\\hat P_*^n]_{n,n} \\big) + \\underbrace{ \\hat X_t^\\tp P_*^0\\hat X_{t} + \\sum_{n=1}^N \\tr(P_*^n \\Sigma_t^n)}_{=V_t} \\notag \\\\\n& - \\underbrace{[\\hat X_t^\\tp Q(0) \\hat X_{t} + \\sum_{n=1}^N \\tr \\big( [Q(n)]_{n,n} \\Sigma_t^n \\big)]}_{=\\mathbb{T}_4} \\notag \\\\\n& - \\underbrace{ [(K_*^0\\hat X_t)^\\tp R(0) K_*^0\\hat X_t + \\sum_{n=1}^N \\tr \\big( [(\\tilde K_*^n)^\\tp R(n) \\tilde K_*^n]_{n,n} \\Sigma_t^n \\big)]\n}_{=\\mathbb{T}_5}.\n\\label{eq:Vtp1_t}\n\\end{align}\n\\end{small}\n\nNote that from \\eqref{cost_given_h}, we have $\\ee[c(X_t,U_t^*)|H^0_t] = \\mathbb{T}_4 + \\mathbb{T}_5$ and furthermore, \n\n\\vspace{-3mm}\n\\begin{small}\n\\begin{align}\n&\\sum_{n=1}^N [\\hat P_*^n]_{n,n} = \\sum_{n=1}^N [\\Pi(\\tilde P_*,n)]_{n,n} \\notag \\\\\n&= \\sum_{n=1}^N \\big( (1-p^n) [\\tilde P_*^0]_{n,n} + p^n  [\\tilde P_*^n]_{n,n} \\big) \\notag \\\\\n&=\n\\sum_{n=1}^N \\big( (1-p^n) [P_{*}^0]_{n,n}+p^n P_{*}^n \\big) = \\Lambda_*.\n\\end{align}\n\\end{small}\n\nTherefore,\n\\begin{small}\n\\begin{align}\n& \\ee[V_{t+1}|H^0_t]\n= \\Lambda_*\n+ V_t - \\ee[c(X_t,U_t^*)|H^0_t],\n\\end{align}\n\\end{small}\nwhich concludes the correctness of \\eqref{main_goal_NC}.\n\n\\subsection{Notations}\nIn general, subscripts are used as time indices while superscripts are used to index controllers.\nFor time indices $t_1\\leq t_2$, $X_{t_1:t_2}$ is a short hand notation for the collection variables $(X_{t_1},X_{t_1+1},...,X_{t_2})$.\nRandom variables\/vectors are denoted by upper case letters, their realizations by the corresponding lower case letters.\nFor a sequence of column vectors $X, Y, Z, \\ldots$, the notation $\\vecc(X,Y,Z,\\ldots)$ denotes the vector $[X^{\\tp}, Y^{\\tp}, Z^{\\tp},...]^{\\tp}$. $\\prob(\\cdot)$ denotes the probability of an event, and $\\ee[\\cdot]$ and $\\cov(\\cdot)$ denote the expectation and the covariance matrix of a random variable\/vector. The transpose, trace, and spectral radius of a matrix $A$ are denoted by $A^{\\tp}$, $\\tr(A)$, and $\\rho(A)$, respectively. \nFor two symmetric matrices $A,B$, $A\\succeq B$ (resp. $A\\succ B$) means that $(A-B)$ is positive semi-definite (PSD) (resp. positive definite (PD)).\nFor a block matrix $A$, we use $[A]_{m,:}$ to denote  the $m$-th block row and $[A]_{:,n}$ to denote  the $n$-th block column of $A$. Further, $[A]_{m,n}$ denotes the block  located at the $m$-th block row and $n$-th block column of $A$. For example, if \n\\begin{align}\nA = \\begin{bmatrix}\nA^{11} & A^{12} & A^{13} \\\\\nA^{21} & A^{22} & A^{23} \\\\\nA^{31} & A^{32} & A^{33} \n\\end{bmatrix},\\notag\n\\end{align}\nthen $[A]_{2,:} = \\begin{bmatrix}\nA^{21} & A^{22} & A^{23}\n\\end{bmatrix}$, $[A]_{:,3} = \\begin{bmatrix}\nA^{13} \\\\ A^{23} \\\\ A^{33}\n\\end{bmatrix}$, and $[A]_{2,3} = A^{23}$.\nWe use $\\R^n$ to denote the $n$-dimensional Euclidean space and $\\R^{m \\times n}$ to denote the space of all real-valued $m \\times n$ matrices. We use $\\otimes$ to denote the Kronecker product.\n\n\\subsection{Operator Definitions}\\label{sec:operators}\nWe define the following operators. \n\\begin{itemize}\n\\item Consider matrices $P,Q,R,A,B$ of appropriate dimensions with $P,Q$ being PSD matrices and $R$ being  a PD matrix. We define $\\Omega(P,Q,R,A,B)$ and $\\Psi(P,R,A,B)$ as follows: \n\\begin{align}\n\\label{Omega}\n&\\Omega(P,Q,R,A,B) \n:= Q+A^\\tp P A- \\notag\\\\\n& ~~~~~~~~ ~~~~~~~~ ~~~~~~~~A^\\tp P B(R+B^\\tp P B)^{-1}B^\\tp P A.\n\\\\\n&\\Psi(P,R,A,B) \n:=\n-(R+B^\\tp P B)^{-1}B^\\tp P A.\n\\label{Psi}\n\\end{align}\nNote that $P = \\Omega(P,Q,R,A,B) $ is the discrete time algebraic Riccati equation.\n\n\\item Let $P$ be a block matrix with $M_1$ block rows and $M_2$ block columns. Then, for numbers $m_1, m_2$ and matrix $Q$, $\\mathcal{L}_{zero}(P,Q,m_1,m_2)$ is a matrix with the same size as $P$ defined as follows:\n\\begin{align}\n&\\mathcal{L}_{zero}(P,Q,m_1,m_2) := \\notag \\\\\n&\\begin{blockarray}{cccl}\n\\text{$1:m_2-1$} &m_2 &\\text{$m_2+1:M_2$}  &  \\\\\n\\begin{block}{[ccc]l}\n                       \\mathbf{0} & \\mathbf{0} & \\mathbf{0} & \\text{$1:m_1-1$} \\\\\n   \\mathbf{0} & Q & \\mathbf{0} &m_1 \\\\\n                    \\mathbf{0} & \\mathbf{0} & \\mathbf{0} &\\text{$m_1+1:M_1$}  \\\\\n\\end{block}\n\\end{blockarray}\n\\label{L_zero}\n\\end{align}\n\n\\item Let $P$ be a block matrix with $M_1$ block rows and $M_1$ block columns. Then, for number $m_1$ and matrix $Q$, $\\mathcal{L}_{iden}(P,Q,m_1)$ is a matrix with the same size as $P$ defined as follows:\n\\begin{align}\n&\\mathcal{L}_{iden}(P,Q,m_1) := \\notag \\\\\n&\\begin{blockarray}{cccl}\n\\text{$1:m_1-1$} &m_1 &\\text{$m_1+1:M_1$}  &  \\\\\n\\begin{block}{[ccc]l}\n                       \\mathbf{I} & \\mathbf{0} & \\mathbf{0} & \\text{$1:m_1-1$} \\\\\n   \\mathbf{0} & Q & \\mathbf{0} &m_1 \\\\\n                    \\mathbf{0} & \\mathbf{0} & \\mathbf{I} &\\text{$m_1+1:M_1$}  \\\\\n\\end{block}\n\\end{blockarray}\n\\label{L_iden}\n\\end{align}\n\\end{itemize}\n\\section{Proof of Lemma \\ref{lm:pc_NC}}\n\\label{proof_lm:pc}\nThe proof can be obtained by following a methodology similar to the one used to prove Lemma \\ref{lm:pc_2C}. We can first use part 1 of Lemma \\ref{lm:ss} to show that the auxiliary MJLS described by \\eqref{A_mj}-\\eqref{transition_prob} is SS if and only if $p^n < p_c^n$ for all $n \\in \\mathcal{N}$ where $p_c^n$ is the critical threshold given by \\eqref{eq:pc}. \n\nSecond, we can use part 2 of Lemma \\ref{lm:ss} to show that the auxiliary MJLS described by \\eqref{A_mj}-\\eqref{transition_prob} is SD if and only of $\\mathcal{A}_d$ defined as in \\eqref{eq:bigmatrix3} is Schur stable. Since the matrix $\\mathcal{A}_{d}$ is upper-triangular, it is Schur stable if and only if there exist matrices $H^{\\diamond}(0)$ and $H^{\\diamond}(n)$ for $n \\in \\mathcal{N}$ such that $\\rho\\big(A_d(0)\\otimes A_d(0)\\big)<1$ and $\\rho\\big(p^n A_d(n)\\otimes A_d(n)\\big)<1$ where $A_d(n)$, $n \\in \\mathcal{N}$, defined as in \\eqref{A_c_l}. The existence of these matrices is resulted since $(A,Q)$ and $\\big(A^{nn},(Q^{nn})^{1\/2} \\big)$ for all $n \\in \\mathcal{N}$ are detectable from Assumptions \\ref{assum:det_stb} and \\ref{assum:det_stb_2}.\nHence, the MJLS is SD.\n\nIt then follows from Lemma \\ref{lm:MJ_infinite} that matrices $P^{\\diamond}_{t}(n)$, $n \\in \\overline{\\mathcal{N}}$, converge as $t \\to -\\infty$ to PSD matrices $P^{\\diamond}_{*}(n)$ that satisfy the steady state version of \\eqref{P_MJ_cmp_NC_0}-\\eqref{P_MJ_cmp_NC_1} (i.e., equations \\eqref{eq:P_finite_NC_fixed}-\\eqref{eq:tildeP_finite_NC_fixed}) if and only if $p^n < p_c^n$ for all $n \\in \\mathcal{N}$. This proves the lemma.\n\n\\begin{figure*}[t]\n\\begin{small}\n\\begin{align}\n\\mathcal{A}_d =\n\\begin{blockarray}{ccccc}\n\\begin{block}{[ccccc]}\n                     A_d(0)\\otimes A_d(0) & (1-p^1)A_d(1)\\otimes A_d(1)& (1-p^2)A_d(2)\\otimes A_d(2) & \\ldots & (1-p^n)A_d(N)\\otimes A_d(N) \\\\ \\\\\n                     \\mathbf{0} & p^1A_d(1)\\otimes A_d(1) &\\mathbf{0} &\\ldots & \\mathbf{0} \\\\ \\\\\n                     \\vdots &\\ddots  &p^2 A_d(2)\\otimes A_d(2)  & \\ddots & \\vdots \\\\ \\\\\n                     \\vdots &   &\\ddots  &\\ddots & \\mathbf{0} \\\\ \\\\\n                     \\mathbf{0} & \\ldots &\\ldots & \\mathbf{0}  & p^n A_d(N)\\otimes A_d(N)\\\\\n\\end{block}\n\\end{blockarray}\n\\label{eq:bigmatrix3}\n\\end{align}\n\\end{small}\n\\hrule\n\\end{figure*}\n\\begin{figure*}[t]\n\\begin{small}\n\\begin{align}\n&A_d(n) = A^{\\diamond} (n) +  H^{\\diamond}(n) \\big(Q^{\\diamond}(n) \\big)^{1\/2}\n= \\begin{blockarray}{cccl}\n\\text{$1:n-1$} &n &\\text{$n+1:N$}  &  \\\\\n\\begin{block}{[ccc]l}\n                      \\text{\\large 0}   & [H^{\\diamond}(n)]_{n,1:n-1} (Q^{nn})^{1\/2} &\\text{\\large 0} & \\text{$1:n-1$} \\\\\n   \\text{\\large 0} & A^{nn} + [H^{\\diamond}(n)]_{n,n} (Q^{nn})^{1\/2} & \\text{\\large 0} &n \\\\\n                      \\text{\\large 0}   & [H^{\\diamond}(n)]_{n,n+1:N} (Q^{nn})^{1\/2} & \\text{\\large 0}&\\text{$n+1:N$}  \\\\\n\\end{block}\n\\end{blockarray}.\n\\label{A_c_l}\n\\end{align}\n\\end{small}\n\\hrule\n\\end{figure*} \n\\section{Proof of Lemma \\ref{lm:opt_strategies_new_rep}}\n\\label{proof_lm:opt_strategies_new_rep}\nBy comparing \\eqref{eq:P_N_init}-\\eqref{eq:P_finite} with  \\eqref{eq:barP_init}-\\eqref{eq:barP_finite_0}, it is straightforward to observe that $P_t^0 = \\bar P_t^0$ for all $t$. We will now show by induction that at any time $t$, $P_t^n = \\bar P_t^n$ for $n =1,\\ldots,N$.\nFirst note that by definition, $P_{T+1}^{n} = \\mathbf{0}$ and $\\bar P^n_{T+1} = \\mathbf{0}$ for $n =1,\\ldots,N$. Hence, \\eqref{new_rep2} is correct at time $T+1$. Now, assume that \\eqref{new_rep2} is correct at time $t+1$ (induction hypothesis). Then, from \\eqref{eq:barP_finite} and  the induction hypothesis, we have for $n =1,\\ldots,N$,\n\\begin{align}\n\\bar P_t^{n} &=  \\Omega \\big((1-p^n) P_{t+1}^0+p^n \\mathcal{L}_{zero}(P^0_{t+1}, P_{t+1}^{n},n,n), \\notag \\\\\n& \\hspace{1.0cm}  \\mathcal{L}_{zero}(Q,Q^{nn},n,n), \\mathcal{L}_{iden}(R,R^{nn},n+1), \\notag \\\\\n & \\hspace{1.0cm}  \\mathcal{L}_{zero}(A,A^{nn},n,n), \\mathcal{L}_{zero}(B,B^{nn},n,n+1) \\big) \\notag \\\\\n&=\n\\mathcal{L}_{zero}(Q,Q^{nn},n,n)  + \\mathbb{T}_1 - \\mathbb{T}_2 (\\mathbb{T}_3)^{-1} (\\mathbb{T}_2)^{\\tp},\n\\label{P_MJ_n}\n\\end{align}\nwhere \n\\begin{align}\n\\label{T_1_def}\n&\\mathbb{T}_1 = \\mathcal{L}_{zero}(A,A^{nn},n,n)^{\\tp} \\bar{\\bar{P}}_{t+1} \\mathcal{L}_{zero}(A,A^{nn},n,n) \\\\\n\\label{T_2_def}\n&\\mathbb{T}_2 = \\mathcal{L}_{zero}(A,A^{nn},n,n)^{\\tp} \\bar{\\bar{P}}_{t+1} \\mathcal{L}_{zero}(B,B^{nn},n,n+1) , \\\\\n\\label{T_3_def}\n&\\mathbb{T}_3 = \\mathcal{L}_{iden}(R,R^{nn},n+1) \n\\notag \\\\\n&+ \\mathcal{L}_{zero}(B,B^{nn},n,n+1)^{\\tp} \\bar{\\bar{P}}_{t+1} \\mathcal{L}_{zero}(B,B^{nn},n,n+1),\n\\end{align}\nand we have defined $\\bar{\\bar{P}}_{t+1} = (1-p^n) P_{t+1}^0+p^n \\mathcal{L}_{zero}(P^0_{t+1}, P_{t+1}^{n},n,n)$.\n\nNote that from the definitions of operators $\\mathcal{L}_{zero}$ and $\\mathcal{L}_{iden}$ in \\eqref{L_zero}-\\eqref{L_iden}, it is straightforward to observe that the block dimensions of $\\mathbb{T}_1, \\mathbb{T}_2, \\mathbb{T}_3$ are the same as the block dimensions of $A,B,B^{\\tp} B$, respectively (They are block matrices of sizes $N \\times N$, $N \\times (N+1)$, and $(N+1) \\times (N+1)$, respectively). Therefore, through straightforward algebraic manipulations, we can get\n\\begin{align}\n\\label{T_1}\n&\\mathbb{T}_1 = \\mathcal{L}_{zero}(A,\\mathbb{\\tilde T}_1,n,n), \\\\\n\\label{T_2}\n&\\mathbb{T}_2 =  \\mathcal{L}_{zero}(B,\\mathbb{\\tilde T}_2,n,n+1), \\\\\n\\label{T_3}\n&\\mathbb{T}_3 = \\mathcal{L}_{iden}(B^{\\tp}B,\\mathbb{\\tilde T}_3,n+1),\n\\end{align}\nwhere\n\\begin{align}\n\\label{tilde_T_1}\n&\\mathbb{\\tilde T}_1 = (A^{nn})^{\\tp}[(1-p^n)[P_{t+1}^0]_{n,n}+p^n P_{t+1}^{n}]A^{nn}, \\\\\n\\label{tilde_T_2}\n&\\mathbb{\\tilde T}_2 = (A^{nn})^{\\tp}[(1-p^n)[P_{t+1}^0]_{n,n}+p^n P_{t+1}^{n}]B^{nn}, \\\\\n\\label{tilde_T_3}\n&\\mathbb{\\tilde T}_3 = R^{nn} + (B^{nn})^{\\tp}[(1-p^n)[P_{t+1}^0]_{n,n}+p^n P_{t+1}^{n}]B^{nn}.\n\\end{align}\n\nFurther, since $\\mathbb{T}_3$ is a block diagonal matrix, we have \n\\begin{align}\n& (\\mathbb{T}_3)^{-1} =\\mathcal{L}_{iden} \\big(B^{\\tp}B,(\\mathbb{\\tilde T}_3)^{-1},n+1 \\big).\n\\label{T_3_inv}\n\\end{align}\n\nNow, using \\eqref{T_1}-\\eqref{T_3_inv} and the fact that  matrices $A, Q, BB^{\\tp}$ have the same size as matrix $P^0_t$ (They are block matrices of size $N \\times N$), \\eqref{P_MJ_n} can be simplified to\n\\begin{align}\n\\bar P_t^{n} &= \\mathcal{L}_{zero} \\big(P^0_t,  Q^{nn} + \\mathbb{\\tilde T}_1 - \\mathbb{\\tilde T}_2 (\\mathbb{\\tilde T}_3)^{-1} (\\mathbb{\\tilde T}_2)^{\\tp},n,n \\big) \\notag \\\\\n&=   \\mathcal{L}_{zero}(P^0_t, P_t^{n},n,n),\n\\end{align}\nwhere the last equality is true because of the definition of $P_t^{n}$ in \\eqref{eq:tildeP_finite}. Hence, \\eqref{new_rep2} is true at time $t$. This completes the proof.\n}\n\n\n{\\color{blue}\n\\section{Proof of Lemma \\ref{lm:pc_NC}}\n\\label{proof_lm:pc}\nFrom Lemma \\ref{lm:opt_strategies_new_rep}, we know that the convergence of matrices $P_t^{n}$, $n \\in \\mathcal{\\overline N}$, is equivalent to the convergence of matrices $\\bar P_t^{n}$, $n \\in \\mathcal{\\overline N}$. Further, because of Lemma \\ref{equality_recursions_NC}, $\\bar P^n_t = P^{\\diamond}_t(n)$, $n \\in \\mathcal{\\overline N}$, where matrices $P^{\\diamond}_t(n)$, $n \\in \\mathcal{\\overline N}$, are defined by \\eqref{eq:P_N_MJ_init}-\\eqref{P_MJ_cmp_NC_1} for the auxiliary MJLS. Thus, in order to study the the convergence of matrices $P_t^{n}$, $n \\in \\mathcal{\\overline N}$, we can focus on the convergence of matrices $P^{\\diamond}_t(n)$, $n \\in \\mathcal{\\overline N}$.\n\nTo investigate the convergence of $P^{\\diamond}_t(n)$, $n \\in \\mathcal{\\overline N}$, we can follow a methodology similar to the one used to prove Lemma \\ref{lm:pc_2C}. We first use part 1 of Lemma \\ref{lm:ss} to show that the auxiliary MJLS described by \\eqref{A_mj}-\\eqref{transition_prob} is SS if and only if $p^n < p_c^n$ for all $n \\in \\mathcal{N}$ where $p_c^n$ is the critical threshold given by \\eqref{eq:pc}. \n\nSecond, we can use part 2 of Lemma \\ref{lm:ss} to show that the auxiliary MJLS described by \\eqref{A_mj}-\\eqref{transition_prob} is SD if and only of $\\mathcal{A}_d$ defined as in \\eqref{eq:bigmatrix3} is Schur stable. Since the matrix $\\mathcal{A}_{d}$ is upper-triangular, it is Schur stable if and only if there exist matrices $H^{\\diamond}(0)$ and $H^{\\diamond}(n)$ for $n \\in \\mathcal{N}$ such that $\\rho\\big(A_d(0)\\otimes A_d(0)\\big)<1$ and $\\rho\\big(p^n A_d(n)\\otimes A_d(n)\\big)<1$ where $A_d(n)$, $n \\in \\mathcal{N}$, defined as in \\eqref{A_c_l}. The existence of these matrices is resulted since $(A,Q)$ and $\\big(A^{nn},(Q^{nn})^{1\/2} \\big)$ for all $n \\in \\mathcal{N}$ are detectable from Assumptions \\ref{assum:det_stb} and \\ref{assum:det_stb_2}.\nHence, the MJLS is SD.\n\nIt then follows from Lemma \\ref{lm:MJ_infinite} that matrices $P^{\\diamond}_{t}(n)$, $n \\in \\overline{\\mathcal{N}}$, converge as $t \\to -\\infty$ to PSD matrices $P^{\\diamond}_{*}(n)$ that satisfy the steady state version of \\eqref{P_MJ_cmp_NC_0}-\\eqref{P_MJ_cmp_NC_1} (i.e., equations \\eqref{eq:P_finite_NC_fixed}-\\eqref{eq:tildeP_finite_NC_fixed}) if and only if $p^n < p_c^n$ for all $n \\in \\mathcal{N}$. This proves the lemma.\n\n\\begin{figure*}[t]\n\\begin{small}\n\\begin{align}\n\\mathcal{A}_d =\n\\begin{blockarray}{ccccc}\n\\begin{block}{[ccccc]}\n                     A_d(0)\\otimes A_d(0) & (1-p^1)A_d(1)\\otimes A_d(1)& (1-p^2)A_d(2)\\otimes A_d(2) & \\ldots & (1-p^n)A_d(N)\\otimes A_d(N) \\\\ \\\\\n                     \\mathbf{0} & p^1A_d(1)\\otimes A_d(1) &\\mathbf{0} &\\ldots & \\mathbf{0} \\\\ \\\\\n                     \\vdots &\\ddots  &p^2 A_d(2)\\otimes A_d(2)  & \\ddots & \\vdots \\\\ \\\\\n                     \\vdots &   &\\ddots  &\\ddots & \\mathbf{0} \\\\ \\\\\n                     \\mathbf{0} & \\ldots &\\ldots & \\mathbf{0}  & p^n A_d(N)\\otimes A_d(N)\\\\\n\\end{block}\n\\end{blockarray}\n\\label{eq:bigmatrix3}\n\\end{align}\n\\end{small}\n\\hrule\n\\end{figure*}\n\\begin{figure*}[t]\n\\begin{small}\n\\begin{align}\n&A_d(n) = A^{\\diamond} (n) +  H^{\\diamond}(n) \\big(Q^{\\diamond}(n) \\big)^{1\/2}\n= \\begin{blockarray}{cccl}\n\\text{$1:n-1$} &n &\\text{$n+1:N$}  &  \\\\\n\\begin{block}{[ccc]l}\n                      \\text{\\large 0}   & [H^{\\diamond}(n)]_{n,1:n-1} (Q^{nn})^{1\/2} &\\text{\\large 0} & \\text{$1:n-1$} \\\\\n   \\text{\\large 0} & A^{nn} + [H^{\\diamond}(n)]_{n,n} (Q^{nn})^{1\/2} & \\text{\\large 0} &n \\\\\n                      \\text{\\large 0}   & [H^{\\diamond}(n)]_{n,n+1:N} (Q^{nn})^{1\/2} & \\text{\\large 0}&\\text{$n+1:N$}  \\\\\n\\end{block}\n\\end{blockarray}.\n\\label{A_c_l}\n\\end{align}\n\\end{small}\n\\hrule\n\\end{figure*} \n}\n\\section{Proof of Lemma \\ref{lm:Q2_2C}, part 3}\\label{sec:stability_proof}\nLet $\\tilde X_t := X_t - \\hat X_t$ denote the estimation error.  It suffices to show that $\\hat X_t$ and  $\\tilde X_t$ are mean square stable.\nThe optimal strategies can be written as \n\\begin{align}\nU_t = K^0_* \\hat X_t + \n\\begin{bmatrix}\n0 \\\\ K^1_* \n\\end{bmatrix}\n\\tilde X_t.\n\\end{align}\nThen, from \\eqref{eq:estimator_inf_2C} we have\n\\begin{align}\n\\tilde X_{t+1} \n = & (1-\\Gamma_{t+1}^1)(A_s(1) \\tilde X_t + W_t),\n\\end{align}\nwhere $A_s(1) = (A + B^{11}K^1_*)$.\nIf $p^1 = 0$ or $1$, the stability result follows from standard  linear system theory arguments.\nIf $0< p^1 < 1$, the estimation error $\\tilde X_{t}$ is a MJLS with an i.i.d. switching process\\footnote{Note that this MJLS is not the same as the auxiliary MJLS constructed in Section \\ref{sec:Q1_2C}.}.\nFrom \\cite[Theorem 3.33]{costa2006discrete}, the estimation error process is mean square stable if the corresponding noiseless system (i.e., with $W_t = 0$) is mean square stable.\nBecause $\\Gamma_{t+1}^1$ is an i.i.d. process, from \\cite[Corollary 2.7]{fang2002stochastic}, the noiseless system is mean-square stable if\n$p^1\\rho( A_{s}(1) \\otimes A_{s}(1)) < 1$.\n\nNote that the gain matrices $K^0_* ,K^1_* $ are obtained from the DCARE in \\eqref{eq:CARE_infinite} for the SD and SS auxiliary MJLS described by \\eqref{eq:MJLS_A}-\\eqref{eq:MJLS_theta}, so the corresponding gains stabilize the auxiliary MJLS \\cite[Corollary A.16]{costa2006discrete}, \\cite[Theorem A.15]{costa2006discrete}. That is, the following matrix\n\\begin{align}\n\\mathcal{A}_s = \\begin{bmatrix}\nA_s(0)\\otimes A_s(0) & (1-p^1)A_s(1)\\otimes A_s(1) \\\\\n \\mathbf{0} & p^1A_s(1)\\otimes A_s(1)\n\\end{bmatrix}\n\\label{eq:A_smat}\n\\end{align}\nhas a spectral radius less than one (see the proof of Lemma \\ref{lm:pc_2C}),  where $A_s(0) = A + BK^0_*$. Thus, $\\rho(A_s(0)) <  1$ and $p^1\\rho( A_{s}(1) \\otimes A_{s}(1)) < 1$. Consequently, the estimation error $\\tilde X_{t}$ is mean-square stable.\n\n \nNow, note that the estimate evolution can be written as\n\\begin{align}\n\\hat X_{t+1} \n = & A_s(0) \\hat X_t + \\tilde W_t,\n\\end{align}\nwhere $\\tilde W_t = \\Gamma_{t+1}^1(A_s(0) \\tilde X_t + W_t)$ can be viewed as a ``noise\" process.\nThe process $\\tilde W_t$ is mean square stable because $\\Gamma_{t+1}^1 \\leq 1$, and $\\tilde X_t$ and $W_t$ are both mean square stable.\nSince $\\rho(A_s(0)) <  1$, we conclude that $\\hat X_t$ is mean square stable using standard linear system arguments \\cite[Theorem 3.4]{KumarVaraiya:1986}.\n\n\n\n\n \n\\section{Proof of Lemma \\ref{lm:Q2_2C}, part 3}\\label{sec:stability_proof}\n Let $\\tilde X_t := X_t - \\hat X_t$ denote the  estimation error.  It suffices to show that $\\hat X_t$ and  $\\tilde X_t$ are mean square stable.\n\nThe optimal strategies can be written as \n\\begin{align}\nU_t = K^0_* \\hat X_t + \n\\begin{bmatrix}\n0 \\\\ K^1_* \n\\end{bmatrix}\n\\tilde X_t.\n\\end{align}\n\nAfter some algebra, the   closed-loop dynamics of $(\\hat X_t, \\tilde X_t)$ under the optimal strategies can be written as\n\\begin{align}\n\\begin{bmatrix}\n\\hat X_{t+1}\\\\\n\\tilde X_{t+1}\n\\end{bmatrix}\n= \nA_{cl}(\\Gamma^1_{t+1})\n\\begin{bmatrix}\n\\hat X_{t} \\\\\n\\tilde X_{t} \n\\end{bmatrix}\n+ G_{cl}(\\Gamma^1_{t+1}) W_t,\n\\end{align}\nwhere\n\\begin{small}\n\\begin{align}\nA_{cl}(\\Gamma^1_{t+1}) = &\n\\begin{bmatrix}\n(A + B K^0_* ) & \\Gamma^1_{t+1}(A + B^{11}K^1_*) \\\\\n0 & (1-\\Gamma^1_{t+1})(A + B^{11}K^1_*) \n\\end{bmatrix},\n\\\\\nG_{cl}(\\Gamma^1_{t+1}) = &\n\\begin{bmatrix}\n\\Gamma^1_{t+1}\n\\\\\n(1-\\Gamma^1_{t+1})\n\\end{bmatrix},\n\\end{align}\n\\end{small}\nand $\\hat X_0 =0, \\tilde X_0=0$.\n\nIf $p^1 = 0$ or $1$, the stability result follows from standard  linear system theory arguments.\nIf $0< p^1 < 1$, the closed-loop system is a MJLS with an i.i.d. switching process\\footnote{Note that this MJLS is not the same as the auxiliary MJLS constructed in Section \\ref{sec:Q1_2C}.}.\nFrom \\cite[Theorem 3.33]{costa2006discrete}, the closed-loop system is mean square stable if the corresponding  noiseless system (i.e., with $W_t = 0$) is mean square stable.\nBecause $\\Gamma_{t+1}^1$ is an i.i.d. process, from \\cite[Corollary 2.7]{fang2002stochastic}, the noiseless system is mean-square stable if\nthe spectral radius of the matrix\n\\begin{align}\\label{eq:ms_eq1}\n& p^1 A_{cl}(0) \\otimes A_{cl}(0)+ (1 - p^1) A_{cl}(1) \\otimes A_{cl}(1) =\n\\notag\\\\\n&\n\\begin{tiny}\n\\begin{bmatrix} \nA_s(0) \\otimes \n\\begin{bmatrix}\nA_s(0) & (1-p^1)A_s(1)\\\\\n \\mathbf{0} & p^1 A_s(1)\n\\end{bmatrix}\n& \n(1-p^1) A_s(1) \\otimes \n\\begin{bmatrix}\nA_s(0) & A_s(1)\\\\\n \\mathbf{0} &  \\mathbf{0}\n\\end{bmatrix}\n\\\\\n \\mathbf{0}\n&\np^1 A_s(1) \\otimes \n\\begin{bmatrix}\nA_s(0) &  \\mathbf{0}\\\\\n \\mathbf{0} & A_s(1)\n\\end{bmatrix}\n\\end{bmatrix}\n\\end{tiny}\n\\end{align}\nis less than one, where $A_s(0) = (A + B K^0_* )$ and $A_s(1) = (A + B^{11}K^1_*)$.\n\n\nNote that the gain matrices $K^0_* ,K^1_* $ are obtained from the CARE in \\eqref{eq:CARE_infinite} for the SD and SS auxiliary MJLS described by \\eqref{eq:MJLS_A}-\\eqref{eq:MJLS_theta}, so the corresponding gains stabilize the auxiliary MJLS \\cite[Corollary A.16]{costa2006discrete}, \\cite[Theorem A.15]{costa2006discrete}. That is, the following matrix\n\\begin{align}\n\\mathcal{A}_s = \\begin{bmatrix}\nA_s(0)\\otimes A_s(0) & (1-p^1)A_s(1)\\otimes A_s(1) \\\\\n \\mathbf{0} & p^1A_s(1)\\otimes A_s(1)\n\\end{bmatrix}\n\\end{align}\nhas a spectral radius less than one (see the proof of Lemma \\ref{lm:pc_2C}). Thus, $\\rho(A_s(0)) < 1$ and $\\rho(A_s(1)) <  \\frac{1}{\\sqrt p^1}$. Consequently, the spectral radius of the matrix in \\eqref{eq:ms_eq1} is less than $1$. This establishes the mean-square stability of $(\\hat X_t, \\tilde X_t)$.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\n\n\n\nThe omnipresence of the autonomous driving and the intelligent manufacturing systems involve tasks of sampling and remotely estimating fresh status information. For example, in autonomous driving systems, status information such as the position and the instant speed of cars keep changing, and the controller has to estimate the update-to-date status based on samples collected from the surrounding sensors. To ensure efficient control and system safety, it is important to estimate the fresh status information precisely under limited communication resources and random channel conditions. \n\nTo measure the freshness of the status update information, the Age of Information (AoI) metric has been proposed in \\cite{roy_12_aoi}. By definition, AoI captures the difference between the current time and the time-stamp at which the freshest information available at the destination was generated. It is revealed that the AoI minimum sampling and transmission strategies behave differently from utility maximization and delay minimization \\cite{roy_15_isit}. Samples with fresher content should be delivered to the destination in a timely manner \\cite{sun_17_tit}. \n\nWhen the evolution of the dynamic source can be modeled by a random signal process, the mean square estimation error (MSE) based on the available information at the receiver can be used to capture freshness. Sampling to minimize the MSE of the random process in different communication networks are studied in \\cite{hajet_03_infocom,ornee_21_ton,nayyar_13_tac,sun_wiener,tsai_2021_ton}. When the random process can be observed at the sampler, the optimum sampling policy is shown to have a threshold structure, i.e., a new sample should be taken once the difference between the actual signal value and the estimate based on past samples exceed a certain threshold. The optimum threshold can be computed by iterative thresholding \\cite{chichun-19-isit} or the bi-section search \\cite{sun_wiener} if the delay distribution and the statistics of the channel are known in advance. \n\nWhen the statistics of the communication channel is unknown, the problem of sampling and transmissions for data freshness optimization can be formulated into a sequential decision making problem \\cite{aoibandit,atay2020aging, banerjee_adversarial_aoi,tripathi2021online,li2021efficient}. By using the AoI as the freshness metric,  \\cite{aoibandit,atay2020aging,banerjee_adversarial_aoi} design online link rate selection algorithms based on stochastic bandits. When the channels are time-varying and the transmitter has an average power constraint, \\cite{ceran_19_infocomwks,ceran_21_jsac,kam_rl,aba_drl_aoi,aylin_rl} employ reinforcement learning algorithms to minimize the average AoI under unknown channel statistics. Notice that in applications such as the remote estimation, a linear AoI cannot fully capture the data freshness. To solve this problem, Tripathi \\emph{et al. } model the information freshness to be a time-varying function of the AoI \\cite{tripathi2021online}, and a robust online learning algorithm is proposed. The above research tackles with unknown packet loss rate or utility functions, the problem of designing online algorithms under unknown delay statistics are not well studied. The iterative thresholding algorithm proposed in  \\cite{chichun-19-isit} can be applied in the online setting when the delay statistics is unknown, whereas the convergence rate and the optimality of the algorithm are not well understood. \n\nIn this paper, we consider an online sampling problem, where a sensor transmits status updates of the Wiener source to a destination through a channel with random delay. Our goal is to design a sampling policy that minimizes the estimation error when the delay distribution is unknown  a priori. The main contributions of this paper are as follows:\n\\begin{itemize}\n\t\\item The design of the MSE minimum sampling policy is reformulated as an optional stopping problem.\n\tBy analyzing the sufficient conditions of the optimum threshold, we propose an online sampling policy that learns the optimum threshold adaptively through stochastic approximation. Compared with \\cite{chichun-19-isit,Tang2205:Sending,tang2022age}, the operation of the proposed algorithm does not require prior knowledge of an upper bound of the optimum threshold. \n\t\\item We prove that the time averaged MSE of the proposed algorithm converges almost surely to the minimum MSE if the fourth order moment of the transmission delay is bounded (Theorem \\ref{thm:dep-converge}). In addition, it is shown that the MSE regret, i.e., the sub-optimality gap between the expected cumulative MSE of the proposed algorithm and the optimum offline policy, grows at a speed of $\\mathcal{O}(\\ln k)$, where $k$ is the number of samples (Corollary \\ref{thm:mse-rate}). The perturbed ordinary differential equation (ODE) method is a popular tool for establishing the convergence rate of stochastic approximation algorithms \\cite{Kushner2003}. However, this tool requires either the threshold being learned is in a bounded closed set, or the second moment of the updating directions are bounded. Because our algorithm does not require an upper bound on the optimum threshold, and the essential supremum of the transmission delay could be unbounded, we need to develop\n\ta new method for convergence rate analysis, which is based on the Lyapunov drift method for heavy traffic analysis. \n\t\\item Further by using the classic Le Cam's two point method, we show that for any causal algorithm that makes sampling decision based on historical information, under the worst case delay distribution, the MSE regret is lower bounded by $\\Omega(\\ln k)$  (Theorem \\ref{thm:converse}). By combining Theorem \\ref{thm:mse-rate} and Theorem \\ref{thm:converse}, we obtain that the proposed online sampling algorithm achieves the minimax order-optimal regret.\n\t\\item We validate the performance of the proposed algorithm via numerical simulations. In contrast to \\cite{chichun-19-isit}, the proposed algorithm could meet an average sampling frequency constraint. \n\\end{itemize}\n\n\n\n\n\\section{System Model and Problem Formulation}\n\n\\subsection{System Model}\nAs is depicted in Fig.~\\ref{fig:model}, we revisit the status update system in \\cite{sun_17_tit,arafa_model,sun_wiener}, where a sensor takes samples from a Wiener process and transmits the samples to a receiver through a network interface queue. The network interface serves the update packets on the First-Come-First-Serve (FCFS) basis. An ACK is sent back to the sensor once an update packet is cleared at the interface. We assume that the transmission duration after passing the network interface is negligible. \n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=.4\\textwidth]{modelicc}\n\t\\caption{System model. }\n\t\\label{fig:model}\n\\end{figure} \n\nLet $X_t\\in\\mathbb{R}$ denote the value of the Wiener process at time $t\\in\\mathbb{R}^+$. The sampling time-stamp of the $k$-th sample, denoted by $S_k$, is determined by the sensor at will. Based on the FCFS principle, the network interface will start serving the $k$-th packet after the $(k-1)$-th packet is cleared at the network interface and arrived at the receiver. We assume that the service time $D_k$ are independent and identically distributed (i.i.d) with a probability distribution $\\mathbb{P}_D$. The reception time of the $k$-th packet, denoted by $R_k$ satisfies the following recursive formula: $R_k=\\{S_k, R_{k-1}\\}+D_k$ and we define $R_0=0$ for simplicity. We assume the average transmission delay $\\overline{D}:=\\mathbb{E}_{D\\sim\\mathbb{P}_D}[D]$ is lower bounded by $\\overline{D}_{\\text{lb}}>0$. \n\\subsection{MMSE Estimation}\n\nLet $i(t):=\\max_{k\\in\\mathbb{N}}\\{k|R_k\\leq t\\}$ be the index of the latest sample received by the destination at time $t$. The information available at the receiver at time $t$ can be summarized as follows: (i). The sampling time-stamps, transmission delay and the values of previous samples $\\mathcal{M}_t:=\\{(S_j, D_j, X_{S_j})\\}_{j=1}^{i(t)}$; (ii). The fact that no packet was received during $(R_{i(t)}, t]$. Similar to \\cite{sun_17_tit,est_ifac}, we assume that the receiver estimates $X_t$ only based on $\\mathcal{M}_t$ and neglects the second part of information. The minimum mean-square error (MMSE) estimator \\cite{poor2013introduction} in this case is:\n\\begin{equation}\n\\hat{X}_t=\\mathbb{E}[X_t|\\mathcal{M}_t]=X_{S_{i(t)}}. \\label{eq:MMSEest}\n\\end{equation}\n\nWe use a sequence of sampling time instants $\\pi\\triangleq\\{S_k\\}_{k=1}^{\\infty}$ to represent a sampling policy. The expected time average mean square error (MSE) under $\\pi$ is denoted by $\\overline{\\mathcal{E}}_\\pi$, i.e., \n\\begin{equation}\n\\overline{\\mathcal{E}}_\\pi\\triangleq\\limsup_{T\\rightarrow\\infty}\\mathbb{E}\\left[\\frac{1}{T}\\int_{t=0}^T\\left(X_t-X_{S_{i(t)}}\\right)^2\\mathsf{d}t\\right].\n\\end{equation}\n\\subsection{Problem Formulation}\n\nOur goal in this work is to design one sampling policy that can minimize the MSE for the estimator when the delay distribution $\\mathbb{P}_D$ is unknown. Specifically, we focus on the set of causal policies denoted by $\\Pi$, where each policy $\\pi\\in\\Pi$ selects the sampling time $S_k$ of the $k$-th sample based on the transmission delay $\\{D_{k'}\\}_{k'< k}$ and Wiener process evolution $\\{X_t\\}_{t\\leq S_k}$ from the past. The transmission delay and the evolution of the Wiener process in the future cannot be used to decide the sampling time. Due to the energy constraint, we require that the sampling frequency should below a certain threshold. The optimal sampling problem is organized as follows:\n\\begin{pb}[MMSE minimization]\\label{pb:mse}\n\t\\begin{subequations}\n\t\t\\begin{align}\n\t\t\\mathsf{mse}_{\\mathsf{opt}}\\triangleq&\\inf\\limits_{\\pi\\in\\Pi}\\mathop{\\limsup}\\limits_{T\\rightarrow\\infty}\\mathbb{E}\\left[\\frac{1}{T}\\int_{t=0}^T\\left(\\hat{X}_t-X_t\\right)^2\\mathrm{d}t\\right],\\label{eq:primalobj}\\\\\n\t\t&\\hspace{0.5cm}\\text{s.t.}\\hspace{0.2cm}\\mathop{\\limsup}\\limits_{T\\rightarrow\\infty}\\mathbb{E}\\left[\\frac{i(T)}{T}\\right]\\leq f_{\\mathsf{max}}.\n\t\t\\end{align}\n\t\\end{subequations}\n\\end{pb}\t\n\n\\section{Problem Solution}\\label{sec:dep}\nIn this section,\nthe MSE minimization problem (i.e., Problem~\\ref{pb:mse}) is reformulated into an optional stopping problem. Let $\\pi^\\star$ be an optimum policy whose average MSE achieves $\\mathsf{mse}_{\\mathsf{opt}}$. Sufficient conditions for $\\pi^\\star$ are provided in Subsection~\\ref{sec:dep-off}. The online sampling algorithm $\\pi_{\\mathsf{online}}$ is provided in Subsection~\\ref{sec:dep-online} and Subsection~\\ref{sec:dep-analysis} characterizes the behaviors of the online sampling policy. \n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=.33\\textwidth]{estreformulate}\n\t\\caption{Illustration of the Wiener process and the estimation error. The sampling and reception time-stamp of the $k$-th sample are denoted by $S_k$ and $R_k$, respectively. For MMSE estimator, $\\hat{X}_t=X_{S_k}, \\forall t\\in [R_k, R_{k+1})$. }\n\t\\label{fig:errevolve}\n\\end{figure}\n\n\n\\subsection{Markov Decision Reformulation~\\ref{pb:mse}}\\label{sec:dep-rr}\nAccording to \\cite[Theorem 1]{sun_wiener}, policy $\\pi^\\star$ should not take a new sample before the previous sample is delivered to the destination. As is depicted in Fig.~\\ref{fig:errevolve}, the waiting time between the delivery time of the $k$-th sample and the sampling time of the $(k+1)$-th sample is denoted by $W_k$. Define frame $k$ as the time interval between the sampling time-stamp of the $k$-th and the $(k+1)$-th sample. The following corollary enables us to reformulate Problem~\\ref{pb:mse} into a Markov Decision Process. \n\\begin{lemma}\\label{coro:sig-dep-reformulate}\n\tLet $\\mathcal{I}_k:=(D_k, (X_{S_k+t}-X_{S_k})_{t\\geq 0})$ denote the recent information of the sampler in frame $k$. The set of sampling policies that determine the waiting time $W_k$ only based on the recent information $\\mathcal{I}_k$ is denoted by $\\Pi_{\\mathsf{recent}}$. Since for each frame $k$, the difference $X_{S_k+t}-X_{S_k}$ evolves as a Wiener process that is independent of the past $\\{X_{S_{k'}+t}-X_{S_{k'}}\\}_{k'<k}$, Problem~\\ref{pb:mse} can be reformulated into the following Markov decision process:\n\t\\begin{pb}[Markov Decision Process Reformulation]\\label{pb:rr}\n\t\t\\begin{subequations}\n\t\t\t\\begin{align}\n\t\t\t\\mathsf{mse}_{\\mathsf{opt}}\\!=\\!&\\mathop{\\inf}_{\\pi\\in\\Pi_{\\mathsf{recent}}}\\limsup_{K\\rightarrow\\infty}\\left(\\frac{\\sum_{k=1}^K\\mathbb{E}\\left[\\frac{1}{6}(X_{S_{k+1}}-X_{S_k})^4\\right]}{\\sum_{k=1}^K\\mathbb{E}\\left[(S_{k+1}-S_k)\\right]}+\\overline{D}\\right),\\label{eq:sig-dep-rr-goal}\\\\\n\t\t\t&\\hspace{0.55cm}\\text{s.t. }\\hspace{0.15cm}\\liminf_{K\\rightarrow\\infty}\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}\\left[(S_{k+1}-S_k)\\right]\\geq \\frac{1}{f_{\\mathsf{max}}}. \n\t\t\t\\end{align}\n\t\t\\end{subequations}\n\t\\end{pb}\n\\end{lemma}\n\nThe proof is provided in Appendix~\\ref{pf:sig-dep-reformulate}. \n\nAccording to \\cite[Theorem 1]{sun_wiener}, there exists a stationary policy $\\pi^\\star$ that selects the waiting time $W_k$ using a conditional probability distribution given the recent $\\mathcal{I}_k$. The average MSE $\\overline{\\mathcal{E}}_{\\pi^\\star}$ achieves $\\mathsf{mse}_{\\mathsf{opt}}$. Therefore, we can restrict our search of $\\pi^\\star$ within the set of stationary policies. Next, we will reveal the sufficient conditions for the optimal stationary policy.\n\n\\subsection{Designing $\\pi^\\star$ with Known $\\mathbb{P}_D$}\\label{sec:dep-off}\n\nLet $\\Pi_{\\mathsf{cons}}$ denote the set of policies that satisfy the sampling frequency constraint. Since $\\pi^\\star$ achieves the minimum expected time-average MSE among $\\Pi_{\\mathsf{cons}}$, we have: \n\\begin{equation}\n\\limsup_{K\\rightarrow\\infty}\\frac{\\sum_{k=1}^K\\mathbb{E}\\left[\\frac{1}{6}(X_{S_{k+1}}-X_{S_k})^4\\right]}{\\sum_{k=1}^K\\mathbb{E}[D_k+W_k]}\\geq \\overline{\\mathcal{E}}_{\\pi^\\star}-\\overline{D},\\pi\\in\\Pi_{\\mathsf{cons}}. \\label{eq:dep-inequal-equiv}\n\\end{equation}\n\nFor simplicity, denote $\\gamma^\\star:=\\overline{\\mathcal{E}}_{\\pi^\\star}-\\overline{D}$, which is the average cost of the MDP when the optimum policy $\\pi^\\star$ is used, i.e., $\\gamma^\\star=\\limsup_{K\\rightarrow\\infty}\\frac{\\sum_{k=1}^K\\mathbb{E}\\left[\\frac{1}{6}(X_{S_{k+1}}-X_{S_k})^4\\right]}{\\sum_{k=1}^K\\mathbb{E}[D_k+W_k]}$. Because $\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}[D_k+W_k]>0$ , for any policy $\\pi\\in\\Pi_{\\mathsf{cons}}$, inequality \\eqref{eq:dep-inequal-equiv} can be rewritten as:\n\\begin{align}\n\\theta_{\\pi}(\\gamma^\\star):=&\\liminf_{K\\rightarrow\\infty}\\left(\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}\\left[\\frac{1}{6}(X_{S_{k+1}}-X_{S_k})^4\\right]\\right.\\nonumber\\\\\n&\\left.-\\gamma^\\star\\cdot\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}[D_k+W_k]\\right)\\geq 0.  \\label{eq:dep-inequal}\n\\end{align}\n\nInequality \\eqref{eq:dep-inequal} takes the minimum value 0 if and only if policy $\\pi$ is optimum. Therefore, if the ratio $\\gamma^\\star$ is known, an optimum policy $\\pi^\\star$ can be obtained by solving the following functional optimization: \n\\begin{pb}[Functional Optimization Problem]\\label{pb:sig-frac}\n\t\\begin{subequations}\\begin{align}\n\t\t\\mathsf{mse}_{\\mathsf{opt}}=&\\inf_{\\pi\\in\\Pi}\\limsup_{K\\rightarrow\\infty}\\left(\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}\\left[\\frac{1}{6}\\left(X_{S_{k+1}}-X_{S_k}\\right)^4\\right]\\right.\\nonumber\\\\\n\t\t&\\hspace{1cm}\\left.-\\gamma^\\star\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}\\left[\\left(D_k+W_k\\right)\\right]\\right),\\label{eq:sig-frac-obj}\\\\\n\t\t&\\hspace{0.5cm}\\text{s.t.}\\hspace{0.2cm}\\liminf_{K\\rightarrow\\infty}\\mathbb{E}\\left[\\frac{1}{K}\\sum_{k=1}^K\\left(D_k+W_k\\right)\\right]\\geq\\frac{1}{f_{\\mathsf{max}}}.\\label{eq:cons} \n\t\t\\end{align}\n\t\\end{subequations}\n\\end{pb}\n\nTo solve Problem~\\ref{pb:sig-frac}, we can take the Lagrangian duality of the constraint \\eqref{eq:cons} with a dual variable $\\nu$ and obtain the Lagrange function $\\mathcal{L}(\\pi, \\gamma, \\nu)$:\n\\begin{align}\n&\\mathcal{L}(\\pi, \\gamma, \\nu)\\triangleq\\limsup_{K\\rightarrow\\infty}\\left(\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}\\left[\\frac{1}{6}(X_{S_{k+1}}-X_{S_k})^4\\right]\\right.\\nonumber\\\\\n&\\hspace{2cm}\\left.-(\\gamma+\\nu)\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}\\left[\\left(S_{k+1}-S_k\\right)\\right]\\right)+\\nu\\frac{1}{f_{\\mathsf{max}}}.\\label{eq:lagrange-dep}\n\\end{align}\n\nWe say that a stationary policy $\\pi$ has a threshold structure, if the waiting time $W_k$ is determined by:\n\\begin{equation}W_k=\\inf\\{w\\geq 0\\big||X_{S_k+D_k+w}-X_{S_k}|\\geq \\tau\\}.\\label{eq:opt-dep}\n\\end{equation}\n\nLet $Z_t$ be a Wiener process staring from $t=0$.  Let $D$ be the transmission delay following distribution $\\mathbb{P}_D$ and the value of the Wiener process at time $D$ is denoted by $Z_D$. Using the threshold policy \\eqref{eq:opt-dep}, the expected frame-length $L_k:=D_k+W_k$ and $\\frac{1}{6}(X_{S_{k+1}}-X_{S_k})^4$ has the following properties:\n\\begin{lemma}\\cite[Corollary 1 Restated]{sun_wiener}\\label{lemma:cond-l}\n\t\\begin{subequations}\n\t\t\\begin{align}\n\t\t&\\mathbb{E}[L_k]=\\mathbb{E}\\left[\\max\\{\\tau^2, Z_D^2\\}\\right],\\\\\n\t\t&\\mathbb{E}\\left[\\frac{1}{6}(X_{S_{k+1}}-X_{S_k})^4\\right]=\\frac{1}{6}\\mathbb{E}\\left[\\max\\{\\tau^2, Z_D^2\\}^2\\right]. \n\t\t\\end{align}\n\t\\end{subequations}\n\\end{lemma}\n\nAs is revealed by \\cite{sun_wiener}, the optimum policy $\\pi^\\star$ has a threshold structure as in equation \\eqref{eq:opt-dep}. To design an off-line algorithm that can learn the updating threshold $\\tau^\\star$ of $\\pi^\\star$, \nwe then reveal the necessary conditions that $\\tau^\\star$ should satisfy.\nWith slightly abuse of notations, let $\\mathcal{L}(\\tau, \\gamma, \\nu)$ denote the expected value of the Lagrange function $\\mathcal{L}(\\pi, \\gamma, \\nu)$ when a stationary policy $\\pi$ with threshold $\\tau$ is used. According to Lemma~\\ref{lemma:cond-l}, $\\mathcal{L}(\\tau, \\gamma, \\nu)$ can be computed as follows:\n\\begin{align}\n\\mathcal{L}(\\tau, \\gamma, \\nu)=&\\mathbb{E}\\left[\\frac{1}{6}\\max\\{\\tau^2, Z_D^2 \\}^2\\right]-(\\gamma+\\nu)\\mathbb{E}[\\max\\{\\tau^2, Z_D^2\\}]\\nonumber\\\\\n&+\\nu\\frac{1}{f_{\\mathsf{max}}}.\n\\end{align}\n\n\\hspace{-10pt}\\emph{Condition 1: }\\cite[Theorem 5 Restated]{sun_wiener} Let $\\tau(\\gamma, \\nu)$ be the optimum sampling threshold that minimizes function $\\mathcal{L}(\\tau, \\gamma, \\nu)$, which can be computed as follows:\n\\begin{equation}\n\\tau(\\gamma, \\nu):=\\arg\\inf_{\\tau\\geq 0}\\mathcal{L}(\\tau, \\gamma ,\\nu)=\\sqrt{3(\\gamma+\\nu)}.\\label{eq:tauopt}\n\\end{equation}\n\n\\hspace{-10pt}\\emph{Condition 2: }\\cite[Eq.~(123, 125)]{sun_wiener}\\begin{equation}\n\\nu^\\star\\left(\\mathbb{E}\\left[\\max\\{3(\\gamma^\\star+\\nu^\\star), Z_D^2\\}\\right]-\\frac{1}{f_{\\mathsf{max}}}\\right)=0, \\nu^\\star\\geq 0. \\label{eq:cs}\n\\end{equation}\n\nRecall that for any policy $\\pi\\in\\Pi_{\\text{cons}}$ with threshold $\\tau$,  inequality \\eqref{eq:dep-inequal-equiv} implies\n\\begin{equation}\n\\theta_\\pi(\\gamma^\\star)=\\frac{1}{6}\\mathbb{E}\\left[\\max\\{\\tau^2, Z_D^2\\}^2\\right]-\\gamma^\\star\\mathbb{E}\\left[\\max\\{\\tau^2, Z_D^2\\}\\right]\\geq 0. \\label{eq:theta}\n\\end{equation}\nAccording to \\eqref{eq:tauopt}, inequality \\eqref{eq:theta} holds with equality if and only if $\\pi^\\star$ with threshold $\\tau^\\star=\\sqrt{3(\\gamma^\\star+\\nu^\\star)}$ is used. Adding the CS condition \\eqref{eq:cs} on both sides of \\eqref{eq:theta}, the necessary condition for $\\gamma^\\star$ then becomes:\n\\begin{equation}\n\\overline{g}_\\nu(\\gamma^\\star)=\\theta_{\\pi^\\star}(\\gamma^\\star)=0, \\label{eq:equation-offline}\n\\end{equation}\nwhere function $\\overline{g}_\\nu(\\gamma):=\\mathbb{E}[g_\\nu(\\gamma;Z_D)]$ is the expectation of function $g_{\\nu}(\\gamma;Z_D)$ defined as follows: \n\\begin{equation}\ng_\\nu(\\gamma;Z_D):=\\frac{1}{6}\\max\\{3(\\gamma+\\nu), Z_D^2\\}^2-\\gamma\\max\\{3(\\gamma+\\nu), Z_D^2\\}.\\label{eq:gdef}\n\\end{equation}\n\nAs is shown by \\cite[Theorem 7]{sun_wiener}, the duality gap between $\\overline{\\mathcal{E}}_{\\pi^\\star}$ and $\\sup_{\\nu\\geq 0}\\inf_{\\pi}\\mathcal{L}(\\pi, \\gamma^\\star, \\nu)$ is zero, and \\eqref{eq:equation-offline} becomes a necessary and sufficient condition. \n\n\\subsection{An Online Algorithm $\\pi_{\\mathsf{online}}$}\\label{sec:dep-online}\n\nWhen $\\mathbb{P}_D$ is unknown but $\\nu^\\star$ is known, we can approximate $\\gamma^\\star$ by solving equation \\eqref{eq:equation-offline} through stochastic approximation \\cite{neely2021fast,Kushner2003,robbins_monro}. Notice that the role of $\\nu^\\star$ is to satisfy the sampling frequency constraint. To achieve this goal, we approximate $\\nu^\\star$ by maintaining a sequence $\\{U_k\\}$ that records the sampling constraint violations up to frame $k$. \n\nThe algorithm is initialized by selecting $\\gamma_1=0$ and $U_1=0\n. In each frame $k$, the sampling and updating rules are as follows:\n\n\\hspace{-10pt}\\underline{1. Sampling: }We treat $\\nu_k:=\\frac{1}{V}U_k^+$ as the dual optimizer $\\nu$, where $V>0$ is fixed as a constant. The waiting time $W_{k+1}$ is selected to minimize the Lagrange function \\eqref{eq:lagrange-dep}, and according to the statement after equation \\eqref{eq:theta}, $W_k$ is selected by:\n\\begin{equation}\nW_k=\\inf\\{w\\geq 0|\\left|X_{S_k+D_k+w}-X_{S_k}\\right|\\geq\\sqrt{3\\left(\\gamma_k+\\nu_k\\right)}\\}.\\label{eq:o-dep-wait}\n\\end{equation}\n\n\\hspace{-10pt}\\underline{2. Update $\\gamma_k$: }To search for the root $\\gamma>0$ of equation $\\overline{g}_{\\nu_k}(\\gamma)=0$, we update $\\gamma_{k}$ through the Robbins-Monro algorithm \\cite{robbins_monro}. In each frame $k$, we are given an i.i.d sample $\\delta X_k=X_{S_k+D_k}-X_{S_k}\\sim Z_D$, and the Robbins-Monro algorithm operates by:\n\\begin{align}\n&\\gamma_{k+1}=\\left(\\gamma_{k}+\\eta_kY_k\\right)^+,\\label{eq:robbins-monro-gamma}\n\\end{align}\nwhere $Y_k=g_{\\nu_k}(\\gamma_k;\\delta X_k)$ and function $g_{\\nu}(\\cdot)$ is defined in \\eqref{eq:gdef}. Recall that $\\overline{D}_{\\text{lb}}$ is a non-zero lower bound of the average delay, the step-size $\\{\\eta_k\\}$ is selected by:\n\\begin{equation}\n\\eta_k=\\frac{1}{2\\overline{D}_{\\text{lb}}}k^{-\\alpha},\\alpha\\in(0.5, 1]. \\label{eq:stepsize}\n\\end{equation}\n\n\n\\hspace{-10pt}\\underline{3. Update $U_k$: }To guarantee that the sampling frequency constraint is not violated, we update the violation $U_k$ up to the end of frame $k$ by:\n\\begin{equation}\nU_{k+1}=U_k+\\left(\\frac{1}{f_{\\mathsf{max}}}-(D_k+W_k)\\right). \\label{eq:debt-evolve}\n\\end{equation}\n\n\\subsection{Theoretical Analysis}\\label{sec:dep-analysis}\nWe analyze the convergence and optimality of algorithm $\\pi_{\\mathsf{online}}$. We assume there is no sampling frequency constraint, i.e., $f_{\\mathsf{max}}=\\infty$ and make the following assumption on distribution $\\mathbb{P}_D$:\n\\begin{assu}\n\tThe fourth order moment of the transmission delay is upper bounded by $B$, i.e., \n\t\\[\\mathbb{E}[D^4]\\leq B<\\infty.\\]\n\\end{assu}\n\nThe convergence behavior of the optimum threshold $3\\gamma^\\star$ and the MSE performance are manifested in the following theorems:\n\\begin{theorem}\\label{thm:dep-converge}\n\tThe proposed algorithm learns the optimum parameter $\\gamma^\\star$ almost surely, i.e., \n\t\\begin{equation}\n\t\\lim_{k\\rightarrow\\infty}\\gamma_k=\\gamma^\\star, \\hspace{0.3cm}\\text{w.p.1}.\\label{eq:gamma-as}\n\t\\end{equation}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:dep-converge} is obtained by the ODE method in \\cite[Chapter 5]{Kushner2003} and is provided in Appendix~\\ref{pf:dep-converge}. \n\n\\begin{theorem}\\label{thm:rate-converge}\n\tThe second moment of $(\\gamma_k-\\gamma^\\star)$ satisfies:\n\t\\begin{equation} \n\t\\sup_k\\mathbb{E}\\left[\\frac{|\\gamma_k-\\gamma^\\star|^2}{\\eta_k}\\right]<\\infty. \n\t\\end{equation}\n\tSpecifically, if $\\alpha=1$ and $\\eta_k=\\frac{1}{2\\overline{D}_{{\\rm lb}}k}$, then the mean square error decays with rate $\\mathbb{E}[(\\gamma_k-\\gamma^\\star)^2]=\\mathcal{O}(1\/k)$. \n\\end{theorem}\n\nOne challenge in the proof of Theorem \\ref{thm:rate-converge} is that $\\gamma_k$ is unbounded and the second moment of $Y_k$ is unbounded. We notice that $Y_k$ could become very large when $\\gamma_k$ is much larger than the true value $\\gamma^\\star$, but the truncation of $(\\gamma_k+\\eta_k Y_k)^+$ to non-negative part actually prevents the actual update $|(\\gamma_k+\\eta_k Y_k)^+-\\gamma_k|$ from becoming too large. Based on this observation, we adopt a method from the heavy-traffic analysis by introducing the unused rate $\\chi_k:=(-(\\gamma_k+\\eta_kY_k))^+$, then prove that the variance of the amount of the actual updating $(\\eta_kY_k+\\chi_k)$ is finite. Detailed proofs are provided in Section~\\ref{pf:rate-converge}. \n\n\n\\begin{theorem}\\label{thm:mse-as}\n\tThe average MSE under policy $\\pi_{\\mathsf{online}}$ converges to $\\overline{\\mathcal{E}}_{\\pi^\\star}$ almost surely, i.e.,\n\t\\begin{equation}\n\t\\limsup_{k\\rightarrow\\infty}\\frac{\\int_{t=0}^{S_{k+1}}(X_t-\\hat{X}_t)^2{\\rm{d}}t}{S_{k+1}}=\\overline{\\mathcal{E}}_{\\pi^\\star}, \\hspace{0.3cm}\\text{w.p.1}.\\label{eq:thm-1-}\n\t\\end{equation} \n\\end{theorem}\n\nWith the mean-square convergence of $\\gamma_k$, the proof of Theorem~\\ref{thm:mse-as} is a direct application of the perturbed ODE method \\cite{Kushner2003} and is provided in Appendix~\\ref{pf:mse-rate}.\n\nBy using Theorem~\\ref{thm:rate-converge} and Theorem~\\ref{thm:mse-as}, we can upper bound the growth rate of the cumulative MSE optimality gap in the following corollary:\n\\begin{corollary}\\label{thm:mse-rate}\n\tIf $\\alpha=1$, then the growth rate of the cumulative MSE optimality gap up to the $k$-th sample can be bounded as follows:\n\t\\begin{equation}\n\t\\left(\\mathbb{E}\\left[\\int_0^{S_{k+1}}(X_t-\\hat{X}_t)^2{\\rm d}t\\right]-\\overline{\\mathcal{E}}_{\\pi^\\star}\\mathbb{E}[S_k]\\right)=\\mathcal{O}\\left(\\ln k\\right). \n\t\\end{equation}\n\\end{corollary}\nThe proof of Corollary~\\ref{thm:mse-rate} is provided in Appendix~\\ref{pf:sig-dep-reformulate}. \n\n\n\\begin{theorem}\\label{thm:converse}\n\tFor any distribution $\\mathbb{P}$, let $\\pi^\\star(\\mathbb{P})$ denote the MSE minimum sampling policy when the delay $D\\sim\\mathbb{P}$. The threshold obtained by solving equation \\eqref{eq:equation-offline} is denoted by $\\gamma^\\star(\\mathbb{P})$. After $k$-samples are taken, the minimax estimation error $\\gamma^\\star(\\mathbb{P})$ is lower bounded by:\n\t\\begin{equation}\n\t\\inf_{\\hat{\\gamma}}\\sup_{\\mathbb{P}}\\mathbb{E}\\left[(\\hat{\\gamma}-\\gamma^\\star(\\mathbb{P}))^2\\right]=\\Omega(1\/k). \\label{eq:converse-est}\n\t\\end{equation}\n\t\n\t\n\tLet $p_w(\\mathbb{P}):={\\rm Pr}(Z_D^2\\leq 3\\gamma^\\star(\\mathbb{P})|D\\sim\\mathbb{P})$ denote the probability of waiting by using policy $\\pi^\\star(\\mathbb{P})$ and let $\\mathcal{P}_u(\\mu):=\\{\\mathbb{P}|p_w(\\mathbb{P})\\geq\\mu \\}$. Specifically, let $p_{\\rm w, \\rm{uni}}^\\star:={\\rm{Pr}}(Z_D^2\\leq 3\\gamma^\\star_{{\\rm uni}}|D\\sim{\\rm Uni}([0, 1]))$. Let $\\Pi_h$ denote the set of policies which the sampling decision $S_k$ is made based on historical information $\\mathcal{H}_{k-1}$. We have the following result for $\\mu\\leq p_{\\rm w, \\rm{uni}}^\\star\/2$:\n\t\\begin{align}\n\t&\\inf_{\\pi\\in\\Pi_h}\\sup_{\\mathbb{P}\\in\\mathcal{P}_u(\\mu)}\\left(\\mathbb{E}\\left[\\int_0^{S_{k+1}}(X_t-\\hat{X}_t)^2{\\rm d}t\\right]-\\overline{\\mathcal{E}}_{\\pi^\\star(\\mathbb{P})}\\mathbb{E}[S_{k+1}]\\right)\\nonumber\\\\\n\t&\\hspace{4.5cm}\\geq\\frac{1}{2}\\mu\\cdot\\Omega\\left(\\ln k\\right). \\label{eq:regconverse}\n\t\\end{align}\n\\end{theorem}\n\nAs the transmission delay $\\mathbb{P}_D$ considered in the paper does not belong to a specific family and could be quite general, obtaining a point-wise converse bound on $\\mathbb{E}[(\\hat{\\gamma}-\\gamma^\\star(\\mathbb{P}))^2]$ for each distribution $\\mathbb{P}$ is impossible. As an alternative, a minimax risk bound $\\mathbb{E}[(\\hat{\\gamma}-\\gamma^\\star(\\mathbb{P}))^2]$ over a general distribution set $\\mathcal{P}$ can be obtained using Le Cam's two point method for non-parametric estimation \\cite{nonpara}. The core idea is to construct two distributions $\\mathbb{P}_1, \\mathbb{P}_2$, whose $\\ell_1$ distance $|\\mathbb{P}_1^{\\otimes k}-\\mathbb{P}_2^{\\otimes k}|_1$ can be upper bounded by a constant, but $(\\gamma^\\star(\\mathbb{P}_1)-\\gamma^\\star(\\mathbb{P}_2))^2\\geq\\Omega(1\/k)$ is difficult to distinguish. Such a construction is still challenging because $\\gamma^\\star(\\mathbb{P})$ cannot be obtained in closed form even for the simpliest distribution families such as the delta distribution or exponential distribution. Notice that the estimation error of $\\gamma^\\star$ is closely related to the estimation error $\\overline{g}_\\nu(\\cdot)$ at a given point. Therefore, the construction of $\\mathbb{P}_1$ and $\\mathbb{P}_2$ for obtaining the converse bound of H\u00f6lder smooth functions \\cite[Chapter 2]{nonpara} are adopted. The proof of inequality \\eqref{eq:regconverse} is a direct application of the minimax estimation error \\eqref{eq:converse-est}. Detailed proof of Theorem~\\ref{thm:converse} is provided in Section \\ref{pf:converse}. \n\n\n\n\t\\section{Simulation Results}\n\tIn this section, we provide simulation results to verify the theoretic findings and illustrate the performance of our proposed algorithms. We notice that the MSE minimization problem is closely related to the AoI minimization problem, where the AoI at time $t$, denoted by $A(t)=t-S_{i(t)}$. For signal-ignorant sampling policies (i.e., the sensor cannot always observe the time-varying process), according to the analysis in \\cite[Section IV-B]{sun_17_tit}, policies that minimize the average AoI achieves the minimum MSE. Therefore, we choose both offline and online AoI minimization policies ($\\pi_{\\text{AoI}}^\\star$ from \\cite{sun_17_tit}, $\\pi_{\\text{itr}}$ from \\cite{chichun-19-isit}) for comparison. To show the convergence of online learning algorithm, we plotted the average MSE performance of the optimum off-line algorithm $\\pi^\\star$ from \\cite{sun_wiener}. \n\t\n\tThe transmission delay follows the log-normal distribution parameterized by $\\mu$ and $\\sigma$ such that the density function of the probability measure $\\mathbb{P}_D$ is:\n\t\\[p(x):=\\frac{\\mathbb{P}_D(\\text{d}x)}{\\text{d}x}=\\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\left(-\\frac{(\\ln x-\\mu)^2}{2\\sigma^2}\\right). \\]\n\tIn simulations, we set $\\mu=0.8$ and $\\sigma=1.2$, the expected time-averaged MSE is computed by taking the average of 20 runs. Fig.~\\ref{fig:mse-frame} depicts the time-average MSE performance up to the $k$-th frame of different sampling policies. The evolution of $\\{\\gamma_k\\}$ and the MSE regret $\\mathbb{E}\\left[\\int_0^{S_{k+1}}(X_t-\\hat{X}_t)^2\\text{d}t\\right]-\\overline{E}_{\\pi^\\star}\\mathbb{E}[S_{k+1}]$ are depicted in Fig.~\\ref{fig:gap}. From Fig.~\\ref{fig:mse-frame}, with $5\\times 10^4$ samples, the time averaged MSE is almost the same as using the optimum policy. From Fig.~\\ref{fig:gap}, the MSE regret is almost a logarithm function of frame $k$. The asymptotic MSE behaviour is consistent with the convergence results in Theorem~\\ref{thm:mse-as} and Corollary~\\ref{thm:mse-rate}. \n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=.5\\textwidth]{frameasympto}\n\t\t\\caption{The time average MSE evolution as a function of frame $k$. }\n\t\t\\label{fig:mse-frame}\n\t\\end{figure}\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=.23\\textwidth]{gamma}\n\t\t\\includegraphics[width=.23\\textwidth]{reg}\n\t\t\\caption{The evolution of the threshold estimate $\\gamma_k$ (left) and the MSE regret $\\Delta_k:=\\mathbb{E}\\left[\\int_0^{S_{k+1}}(X_t-\\hat{X}_t)^2\\text{d}t\\right]-\\overline{\\mathcal{E}}_{\\pi^\\star}\\mathbb{E}[S_{k+1}]$ (right). }\n\t\t\\label{fig:gap}\n\t\\end{figure}\n\t\n\tWhen there is a sampling frequency constraint, the average MSE and the average sampling interval achieved by policy $\\pi_{\\text{online}}$ are depicted in Fig.~\\ref{fig:cons} and Fig.~\\ref{fig:inte}, respectively. We set $f_{\\text{max}}=\\frac{1}{10\\overline{D}}$. From these figures, one can observe that the average MSE of $\\pi_{\\text{online}}$ is close to the optimum MSE $\\overline{\\mathcal{E}}_{\\pi^\\star}$ and the sampling frequency can be satisfied. In addition, by choosing  a larger $V$, a smaller MSE performance can be achieved, whereas a larger number of iterations are needed to meet the sampling frequency constraint. \n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=.5\\textwidth]{msecons}\n\t\t\\caption{The time average MSE evolution as a function of frame $k$. (Left: $V=10$, Right: $V=1$. ) }\n\t\t\\label{fig:cons}\n\t\\end{figure}\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=.5\\textwidth]{freqcons}\n\t\t\\caption{The average sampling interval under different constant $V$. (Left: $V=10$, Right: $V=1$. )}\n\t\t\\label{fig:inte}\n\t\\end{figure}\n\t\\section{Proofs of Main Results}\\label{sec:pf}\n\n\n\n\n\t\\subsection{Notations and Preliminary Lemmas}\n\tIn Table~\\ref{tab:notations}, we summarize the notations used in the following proofs. Throughout the proofs, we use $N_1, N_2, \\cdots$ to denote absolute constants and $C_1(\\cdot), C_2(\\cdot)$ to denote polynomials with finite order. For ease\n\tof exposition, the specific values and expressions of the constants and functions may vary across different context.\n\t\\begin{table}\n\t\t\\caption{Notations}\n\t\t\\label{tab:notations}\n\t\t\\begin{tabular}{cp{7cm}}\n\t\t\t\\toprule\n\t\t\tNotation&Meaning\\\\\n\t\t\t\\midrule\n\t\t\t$Z_t$& a Wiener process staring from time 0\\\\\n\t\t\t$l_\\gamma$ & length of running time using stopping rule $\\tau_\\gamma:=\\inf\\{{t\\geq D}||Z_t|\\geq\\sqrt{3\\gamma}\\}$\\\\ \n\t\t\t$\\delta X_k$ & $\\delta X_k:=X_{S_k+D_k}-X_{S_k}$\\\\\n\t\t\t$Q_k$ & $Q_k:=\\frac{1}{6}\\left(X_{S_k+D_k}-X_{S_k}\\right)^4$\\\\\n\t\t\t$L_k$ & $L_k:=S_{k+1}-S_k=D_k+W_k$, frame length $k$\\\\\n\t\t\t$E_k$ & $E_k:=\\int_{S_k}^{S_{k+1}}(X_t-\\hat{X}_t)^2\\text{d}t$, cumulative estimation error in frame $k$\\\\\n\t\t\t$q(\\gamma)$ & $q(\\gamma):=\\frac{1}{6}\\mathbb{E}\\left[\\max\\{3\\gamma_k, Z_D^2\\}^2\\right]$, the expectation of $Q_k$ when $\\gamma_k=\\gamma$\\\\\n\t\t\t$l(\\gamma)$ & $l(\\gamma):=\\mathbb{E}[\\max\\{3\\gamma, Z_D^2\\}]$, expected frame length $L_k$ when $\\gamma_k=\\gamma$\\\\\n\t\t\t$\\mathcal{I}_k$ & $(D_k, (X_{t}-X_{S_k})_{S_k\\leq t<S_{k+1}}$, information in frame $k$\\\\\n\t\t\t$\\mathcal{H}_k$ & $\\mathcal{H}_k:=\\{\\mathcal{I}_{\\kappa}\\}_{\\kappa\\leq k}$ historical information up to the end of frame $k$\\\\\n\t\t\t$\\mathbb{E}_k[\\cdot]$ & Conditional expectation  $\\mathbb{E}[\\cdot|\\mathcal{H}_{k-1}]$\\\\\n\t\t\t$t_k$ & $t_k:=\\sum_{i=1}^k\\eta_k$ or $t_k:=\\sum_{i=1}^k\\epsilon_k$, the cumulative step-sizes depending on the context\\\\\n\t\t\t$m(t)$ & $m(t)$ is the unique $k$ so that $t_k\\leq t\\leq t_{k+1}$\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\\begin{lemma}\\label{coro:gammadep-bound}\n\t\tLet $M:=\\mathbb{E}[D^2]$, the optimum ratio $\\gamma^\\star$ is upper and lower bounded by:\n\t\t\\begin{equation}\n\t\t\\frac{1}{6}\\overline{D}\\leq \\gamma^\\star\\leq\\frac{1}{2}\\frac{M+2\\overline{D}\\frac{1}{f_{\\mathsf{max}}}+\\frac{1}{f_{\\mathsf{max}}^2}}{\\overline{D}+\\frac{1}{f_{\\mathsf{max}}}}. \n\t\t\\end{equation}\n\t\\end{lemma}\n\tThe proof is provided in Appendix~\\ref{pf:gammadep-bound}.\n\t\n\t\\begin{lemma}\\label{lemma:4order}\n\t\tFor threshold $\\gamma<\\infty$, the first, second and fourth order moments of the stopping time $\\tau_\\gamma$ are bounded, i.e., \n\t\t\\begin{subequations}\n\t\t\t\\begin{align}\n\t\t\t&\\mathbb{E}[l_\\gamma]\\leq 3\\gamma+\\overline{D},\\\\\n\t\t\t&\\mathbb{E}[l_{\\gamma}^2]\\leq\\frac{10}{3}\\left((3\\gamma)^2+3\\sqrt{B}\\right), \\\\ &\\mathbb{E}\\left[l_{\\gamma}^4\\right]<4^3\\left((3\\gamma)^4+105B\\right)<\\infty. \n\t\t\t\\end{align}\n\t\t\\end{subequations}\n\t\\end{lemma}\n\tThe proof of Lemma~\\ref{lemma:4order} is provided in Appendix~\\ref{pf:4order}.\n\t\n\n\t\n\n\t\\begin{lemma}\\label{lemma:g}\n\t\tFunction $\\overline{g}_0(\\gamma)=q(\\gamma)-\\gamma l(\\gamma)$ and has the following properties:\n\t\t\\begin{itemize}\n\t\t\t\\item[(i)] $\\overline{g}_0(\\gamma)$ is concave and monotonically decreasing. The second order derivative $-3\\leq\\overline{g}_0''(\\gamma)\\leq 0$. \n\t\t\t\n\t\t\t\\item[(ii)] $\\overline{g}_0(\\gamma^\\star)=0$\n\t\t\t\n\t\t\t\\item[(iii)] For $\\gamma\\neq\\gamma^\\star$, \n\t\t\t$(\\gamma-\\gamma^\\star)\\overline{g}_0(\\gamma)\\leq-l(\\gamma^\\star)(\\gamma-\\gamma^\\star)^2\\leq 0$. \n\t\t\\end{itemize}\n\t\\end{lemma}\n\t\n\tThe proof of Lemma \\ref{lemma:g} is provided in Appendix~\\ref{pf:lemma-g}. \n\t\n\t\n\n\n\n\n\t\n\t\\begin{corollary}\\label{coro:squareub}\n\t\tFor each $\\gamma_k<\\infty$, if the fourth order moment of the delay satisfies $\\mathbb{E}[D^4]<B<\\infty$, given historical transmission $\\mathcal{H}_{k-1}$, the conditional second order moment of the cumulative error in frame $E_k=\\int_{S_k}^{S_{k+1}}(X_t-\\hat{X}_t)^2\\text{d}t$ can be bounded as follows:\n\t\t\\begin{align}\n\t\t&\\mathbb{E}_k[E_k^2]=3(X_{S_k}-X_{S_{k-1}})^2\\sqrt{B}\\nonumber\\\\\n\t\t&\\hspace{1cm}+12C_1(\\gamma_k, B)(X_{S_k}-X_{S_{k-1}})^2+3C_2(\\gamma_k, B)<\\infty,\n\t\t\\end{align}\n\t\twhere $C_1$ and $C_2$ are fourth order polynomials of $\\gamma$. \n\t\\end{corollary}\n\t\n\tThe proof of Corollary~\\ref{coro:squareub} is provided in Appendix~\\ref{pf:corosquareub}.\n\t\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\t\n\n\t\n\n\n\n\n\n\n\n\n\t\n\tWith these Lemmas we can proceed to prove main results in Section~\\ref{sec:dep}:\n\t\n\t\n\t\n\t\\subsection{Proof of Theorem~\\ref{thm:rate-converge}}\\label{pf:rate-converge}\n\tThe analysis of the convergence rate is obtained through Lyapunov analysis, where the Lyapunov function is denoted by $V(\\gamma):=\\frac{1}{2}(\\gamma-\\gamma^\\star)^2$. The proof is divided into two steps: first we will upper bound the Lyapunov drift for each $\\gamma_k$ by showing the following equation holds:\n\t\\begin{equation}\n\t\\mathbb{E}_k[V(\\gamma_{k+1})]-V(\\gamma_k)\\leq -\\eta_k\\overline{D}_{\\mathsf{lb}}V(\\gamma_k)+\\mathcal{O}(\\eta_k^2N_1). \\label{eq:step-sizedrift}\n\t\\end{equation}\n\tThen, based on \\eqref{eq:step-sizedrift}, we can then compute $\\mathbb{E}[V(\\gamma_k)]$ directly. \n\t\n\t\\hspace{-10pt}\\textbf{Step 1: Bounding the Lyapunov Drift:} The analysis is divided into two cases, for $\\gamma_k\\leq 3\\gamma^\\star$, inequality \\eqref{eq:step-sizedrift} can be verified easily; for $\\gamma_k\\geq 3\\gamma^\\star$ we will first establish the relationship between $\\mathbb{E}_k[V(\\gamma_{k+1})]-V(\\gamma_k)$ and $\\text{Var}[Y_k]$, then upper bound $\\text{Var}[Y_k]$ using the fact that $Z_D^2$ is sub-Gaussian when $D$ is fourth order bounded. \n\t\n\t\\hspace{-10pt}\\textbf{Case 1:} If $\\gamma_k\\leq 3\\gamma^\\star$, we have:\\begin{align}\n\t&\\mathbb{E}_k[V(\\gamma_{k+1})]-V(\\gamma_k)\\nonumber\\\\\n\t=&\\mathbb{E}_k\\left[\\frac{1}{2}\\left((\\gamma_k+\\eta_kY_k)^+-\\gamma^\\star\\right)^2\\right]-\\frac{1}{2}(\\gamma_k-\\gamma^\\star)^2\\nonumber\\\\\n\t\\leq&\\mathbb{E}_k\\left[\\frac{1}{2}(\\gamma_k-\\gamma^\\star+\\eta_kY_k)^2-\\frac{1}{2}(\\gamma_k-\\gamma^\\star)^2\\right]\\nonumber\\\\\n\t\\overset{(a)}{=}&(\\gamma_k-\\gamma^\\star)\\eta_k\\overline{g}_0(\\gamma_k)\\nonumber\\\\\n\t&+\\frac{1}{2}\\eta_k^2\\mathbb{E}_k\\left[\\left(\\frac{1}{6}\\max\\{3\\gamma_k, \\delta X_k^2\\}^2-\\gamma_k\\max\\{3\\gamma_k, \\delta X_k^2\\}\\right)^2\\right]\\nonumber\\\\\n\t\\overset{(b)}{\\leq}&-2\\eta_kl(\\gamma^\\star)V(\\gamma_k)\\nonumber\\\\\n\t&+\\frac{1}{2}\\eta_k^2\\left(\\frac{1}{36}((9\\gamma^\\star)^4+B)+(3\\gamma^\\star)^2((9\\gamma^\\star)^2+3\\sqrt{B})\\right),\\label{eq:stepsizez-ub1}\n\t\\end{align}\n\twhere equality $(a)$ is because $\\mathbb{E}_k[Y_k]=\\mathbb{E}_k[g_0(\\gamma_k;\\delta X_k)]=\\overline{g}_0(\\gamma_k)$; inequality $(b)$ is obtained because according to Lemma~\\ref{lemma:g}-(iii), $(\\gamma_k-\\gamma^\\star)\\overline{g}_0(\\gamma_k)\\leq -l(\\gamma^\\star)(\\gamma_k-\\gamma^\\star)^2=-2l(\\gamma^\\star)V(\\gamma_k)$ and the assumption that $\\gamma_k\\leq 3\\gamma^\\star$. \n\t\n\t\\hspace{-10pt}\\textbf{Case 2:} If $\\gamma_k\\geq 3\\gamma^\\star$,  $\\gamma_{k+1}=\\left(\\gamma_k+\\eta_kY_k\\right)^+$ is truncated into the non-negative real number. We can view the evolution of $\\gamma_k$ as a queueing system, where the queue $\\gamma_k$ is non-negative, and $\\eta_k Y_k$ is the arrival rate minus the service rate. Therefore, it is natural to introduce the ``unused rate'' from \\cite{eryilmaz2012asymptotically}, which is denoted by $\\chi_k:=\\left(-\\left(\\gamma_k+\\eta_kY_k\\right)\\right)^+$. If $\\chi_k=0$, $(\\gamma_k+\\eta_kY_k)\\chi_k=0=-\\chi_k^2$ and if $\\chi_k\\geq 0$, $\\gamma_k+\\eta_kY_k=-\\chi_k$, therefore\n\t\t\\begin{equation} (\\gamma_k+\\eta_kY_k)\\chi_k=-\\chi_k^2.\\label{eq:unused}\n\t\t\\end{equation}\n\tSince $\\gamma_k+\\eta_kY_k+\\chi_k\\geq 0$, we have:\n\t\\begin{equation}\n\t-\\mathbb{E}_k[\\gamma_k+\\eta_kY_k]\\leq\\mathbb{E}_k[\\chi_k]. \\label{eq:chiub}\n\t\\end{equation}\n\t\n\tWe can then upper bound $\\mathbb{E}_k[V(\\gamma_{k+1})-V(\\gamma_k)]$ by:\n\t\\begin{align}\n\t&\\mathbb{E}_k\\left[V(\\gamma_{k+1})-V(\\gamma_k)\\right]\\nonumber\\\\\n\t=&\\mathbb{E}_k\\left[\\frac{1}{2}\\left(\\gamma_k-\\gamma^\\star+\\eta_kY_k+\\chi_k\\right)^2-\\frac{1}{2}\\left(\\gamma_k-\\gamma^\\star\\right)^2\\right]\\nonumber\\\\\n\t=&\\mathbb{E}_k\\left[\\frac{1}{2}\\left(\\gamma_k-\\gamma^\\star+\\eta_kY_k\\right)^2-\\frac{1}{2}(\\gamma_k-\\gamma^\\star)^2\\right.\\nonumber\\\\\n\t&\\left.+\\frac{1}{2}\\chi_k^2+(\\gamma_k+\\eta_kY_k)\\chi_k-\\gamma^\\star\\chi_k\\right]\\nonumber\\\\\n\t\\overset{(c)}{=}&\\mathbb{E}_k\\left[\\frac{1}{2}(\\gamma_k-\\gamma^\\star+\\eta_kY_k)^2-\\frac{1}{2}(\\gamma_k-\\gamma^\\star)^2-\\frac{1}{2}\\chi_k^2-\\gamma^\\star\\chi_k\\right]\\nonumber\\\\\n\t\\overset{(d)}\\leq&\\frac{1}{2}\\left(\\gamma_k-\\gamma^\\star+\\eta_k\\mathbb{E}_k[Y_k]\\right)^2-\\frac{1}{2}(\\gamma_k-\\gamma^\\star)^2+\\frac{1}{2}\\eta_k^2\\text{Var}[Y_k]\\nonumber\\\\\n\t&-\\frac{1}{2}\\mathbb{E}_k[\\chi_k]^2-\\gamma^\\star\\mathbb{E}_k[\\chi_k]\\nonumber\\\\\n\t=&\\frac{1}{2}\\left(\\gamma_k-\\gamma^\\star+\\eta_k\\mathbb{E}_k[Y_k]\\right)^2-\\frac{1}{2}(\\gamma_k-\\gamma^\\star)^2+\\frac{1}{2}\\eta_k^2\\text{Var}[Y_k]\\nonumber\\\\\n\t&-\\frac{1}{2}\\left(-\\mathbb{E}_k[\\chi_k]-\\gamma^\\star\\right)^2+\\frac{1}{2}(\\gamma^\\star)^2,\\label{eq:stepsize-drift}\n\t\\end{align}\n\twhere equality $(c)$ is because equation \\eqref{eq:unused}; inequality $(d)$ is obtained because $\\mathbb{E}_k[\\chi_k^2]\\geq\\mathbb{E}_k[\\chi_k]^2\\geq 0$;\n\t\n\tWe then further divide the analysis into two cases:\n\t\n\t\\hspace{-10pt}\\underline{Case 2(a)}: If $\\mathbb{E}_k[\\gamma_k+\\eta_k Y_k]\\leq \\gamma^\\star$, we then have $\\mathbb{E}_k[\\gamma_k-\\gamma^\\star+\\eta_k Y_k]\\leq 0$. According to \\eqref{eq:chiub},  $|-\\mathbb{E}_k[\\chi_k]-\\gamma^\\star|\\geq |\\gamma_k-\\gamma^\\star+\\eta_k\\mathbb{E}_k[Y_k]|$. Therefore, inequality \\eqref{eq:stepsize-drift} can be upper bounded by:\n\t\\begin{align}\n\t&\\mathbb{E}_k\\left[V(\\gamma_{k+1})-V(\\gamma_k)\\right]\\nonumber\\\\\n\t\\leq&-\\frac{1}{2}(\\gamma_k-\\gamma^\\star)^2+\\frac{1}{2}(\\gamma^\\star)^2+\\frac{1}{2}\\eta_k^2\\text{Var}[Y_k]\\nonumber\\\\\n\t\\overset{(e)}{\\leq}&-\\frac{1}{4}(\\gamma_k-\\gamma^\\star)^2+\\frac{1}{2}\\eta_k^2\\text{Var}[Y_k]\\nonumber\\\\\n\t\\leq&-2\\eta_k\\overline{D}_{\\mathsf{lb}}V(\\gamma_k)+\\frac{1}{2}\\eta_k^2\\text{Var}[Y_k],\\label{eq:stepsizez-ub2}\n\t\\end{align} \n\twhere inequality $(e)$ is obtained because $\\frac{1}{4}(\\gamma_k-\\gamma^\\star)^2\\geq(\\gamma^\\star)^2\\geq\\frac{1}{2}(\\gamma^\\star)^2$ by assumption that $\\gamma_k\\geq 3\\gamma^\\star$ and the last inequality is obtained because $\\eta_k\\overline{D}_{\\mathsf{lb}}\\leq\\frac{1}{2}$ by the step-size selection rule in equation \\eqref{eq:stepsize}.  \n\t\n\t\\hspace{-10pt}\\underline{Case 2(b):} If $\\mathbb{E}_k[\\gamma_k+\\eta_kY_k]\\geq \\gamma^\\star$, considering that $\\mathbb{E}_k[Y_k]=\\overline{g}_0(\\gamma_k)<0$ for $\\gamma_k\\geq \\gamma^\\star$, we have $0>\\mathbb{E}_k[\\eta_kY_k]\\geq-(\\gamma_k-\\gamma^\\star)$. Inequality \\eqref{eq:stepsize-drift} can be bounded by:\n\t\\begin{align}\n\t&\\mathbb{E}_k[V(\\gamma_{k+1})-V(\\gamma_k)]\\nonumber\\\\\n\t\\overset{(f)}{\\leq}&\\frac{1}{2}(\\gamma_k-\\gamma^\\star)(\\gamma_k-\\gamma^\\star+\\eta_k\\mathbb{E}_k[Y_k])-\\frac{1}{2}(\\gamma_k-\\gamma^\\star)^2+\\frac{1}{2}\\eta_k^2\\text{Var}[Y_k]\\nonumber\\\\\n\t\\leq&\\frac{1}{2}\\eta_k(\\gamma_k-\\gamma^\\star)\\overline{g}_0(\\gamma_k)+\\frac{1}{2}\\eta_k^2\\text{Var}[Y_k]\\nonumber\\\\\n\t\\overset{(g)}{\\leq} &-\\frac{1}{2}\\eta_kl(\\gamma^\\star)(\\gamma_k-\\gamma^\\star)^2+\\frac{1}{2}\\eta_k^2\\text{Var}[Y_k]\\nonumber\\\\\n\t=&-\\eta_kl(\\gamma^\\star) V(\\gamma_k)+\\frac{1}{2}\\eta_k^2\\text{Var}[Y_k],\\label{eq:stepsizez-ub3}\n\t\\end{align}\n\twhere equality $(f)$ is because $\\left(-\\mathbb{E}_k[\\chi_k]-\\gamma^\\star\\right)^2\\geq(\\gamma^\\star)^2$ and $(\\gamma_k-\\gamma^\\star+\\eta_k\\mathbb{E}_k[Y_k])^2\\leq(\\gamma_k-\\gamma^\\star+\\eta_k\\mathbb{E}_k[Y_k])(\\gamma_k-\\gamma^\\star)$; inequality $(g)$ is due to Lemma~\\ref{lemma:g}-(iii). \n\t\n\tFor proceed to show inequality \\eqref{eq:step-sizedrift} for $\\gamma_k\\geq 3\\gamma^\\star$, we need to upper bound $\\text{Var}[Y_k]$ in inequalities \\eqref{eq:stepsizez-ub2} and \\eqref{eq:stepsizez-ub3}. First, we compute the expectation $\\mathbb{E}[Y_k]$ as follows:\n\t\\begin{align}\n\t\\mathbb{E}_k[Y_k]=&\\mathbb{E}\\left[\\frac{1}{6}\\max\\{3\\gamma_k, Z_D^2\\}^2-\\gamma_k\\max\\{3\\gamma_k, Z_D^2\\}\\right]\\nonumber\\\\\n\t=&-\\frac{3}{2}\\gamma_k^2+\\mathbb{E}\\left[(\\frac{1}{6}Z_D^4-\\gamma_kZ_D^2+\\frac{3}{2}\\gamma_k^2)\\mathbb{I}_{(Z_D^2\\geq 3\\gamma_k)}\\right]\\nonumber\\\\\n\t=&-\\frac{3}{2}\\gamma_k^2+\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma_k)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma_k)}\\right]\\nonumber\\\\\n\t\\leq&-\\frac{3}{2}\\gamma_k^2+\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2)^2\\right]\\nonumber\\\\\n\t\\leq&-\\frac{3}{2}\\gamma_k^2+\\frac{1}{2}\\mathbb{E}[D^2]\\leq-\\frac{3}{2}\\gamma_k^2+\\frac{1}{2}\\sqrt{B}. \n\t\\end{align}\n\t\n\tGiven historical information $\\mathcal{H}_{k-1}$, the variance of $Y_k$ can be computed by:\n\t\\begin{align}\n\t&\\text{Var}[Y_k|\\mathcal{H}_{k-1}]\\nonumber\\\\\n\t=&\\mathbb{E}_k\\left[(Y_k-\\mathbb{E}_k[Y_k])^2\\right]\\nonumber\\\\\n\t=&\\mathbb{E}_k\\left[\\left(-\\frac{3}{2}\\gamma_k^2-\\mathbb{E}_k[Y_k]\\right)^2\\mathbb{I}_{(Z_D^2\\leq 3\\gamma_k)}\\right]\\nonumber\\\\\n\t&+\\mathbb{E}_k\\left[\\left(\\frac{1}{6}Z_D^4-\\gamma_kZ_D^2+\\frac{3}{2}\\gamma_k^2+(-\\frac{3}{2}\\gamma_k^2-\\mathbb{E}_k[Y_k])\\right)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma_k)}\\right]\\nonumber\\\\\n\t\\overset{(h)}{\\leq} &\\frac{1}{4}B+2\\mathbb{E}_k\\left[\\left(\\frac{1}{6}Z_D^4-\\gamma_kZ_D^2+\\frac{3}{2}\\gamma_k^2\\right)^2\\mathbb{I}_{(Z_D^2> 3\\gamma_k)}\\right]\\nonumber\\\\\n\t&+2\\mathbb{E}_k\\left[\\left(-\\frac{3}{2}\\gamma_k^2-\\mathbb{E}_k[Y_k]\\right)^2\\mathbb{I}_{(Z_D^2> 3\\gamma_k)}\\right]\\nonumber\\\\\n\t\\leq&\\frac{3}{4}B+\\frac{1}{3}\\mathbb{E}_k\\left[(Z_D^2-3\\gamma_k)^4\\mathbb{I}_{(Z_D^2\\geq 3\\gamma_k)}\\right]\\nonumber\\\\\n\t\\leq&\\frac{3}{4}B+\\frac{1}{3}\\mathbb{E}[Z_D^8]\\leq(35+\\frac{3}{4})B,\n\t\\end{align}\n\twhere $(h)$ is because $\\mathbb{E}_k[Y_k]\\leq-\\frac{3}{2}\\gamma_k^2+\\frac{1}{2}\\sqrt{B}$ implies $(-\\frac{3}{2}\\gamma_k^2-\\mathbb{E}_k[Y_k])^2\\leq\\frac{1}{4}B$ and $(a+b)^2\\leq 2(a^2+b^2)$.\n\t\n\tDenote $N_1:=\\max\\{(35+\\frac{3}{4})B,\\frac{1}{36}((9\\gamma^\\star)^4+B)+(3\\gamma^\\star)^2((9\\gamma^\\star)^2+3\\sqrt{B})\\}$, inequalities \\eqref{eq:stepsizez-ub1}, \\eqref{eq:stepsizez-ub2} and \\eqref{eq:stepsizez-ub3} then lead to:\n\t\\begin{equation}\n\t\\mathbb{E}_k[V(\\gamma_{k+1})]-V(\\gamma_k)\\leq-\\eta_k\\overline{D}_{\\mathsf{lb}}V(\\gamma_k)+\\eta_k^2 N_1. \\label{eq:stepdrift-final}\n\t\\end{equation}\n\t\\textbf{Step 2: Computing $\\mathbb{E}[V(\\gamma_k)]$ through iteration}\n\tTaking the expectation on both sides of \\eqref{eq:stepdrift-final}, we have:\n\t\\begin{equation}\n\t\\mathbb{E}[V(\\gamma_{k+1})]\\leq(1-\\eta_k\\overline{D}_{\\mathsf{lb}})\\mathbb{E}[V(\\gamma_k)]+\\eta_k^2N_1. \\label{eq:perstep}\n\t\\end{equation}\n\tMultiplying inequality \\eqref{eq:perstep} from $i=1$ to $k$ yields:\n\t\\begin{equation}\n\t\\mathbb{E}[V(\\gamma_{k+1})]\\leq\\prod_{i=1}^k(1-\\eta_i\\overline{D}_{\\mathsf{lb}})V(\\gamma_0)+\\sum_{i=1}^k\\eta_i^2N_1\\cdot\\prod_{j=i+1}^k(1-\\eta_j\\overline{D}_{\\mathsf{lb}}). \\label{eq:telescope-stepsize}\n\t\\end{equation}\n\t\n\tSince the stepsize selected by \\eqref{eq:stepsize} satisfies \\[\\eta_k\\rightarrow 0, \\liminf_{k}\\min_{k\\geq i\\geq  m(t_k-T)}\\frac{\\eta_k}{\\eta_i}=1\\]\n\taccording to \\cite[p. 343, Eq. (4.8)]{Kushner2003}, term $\\prod_{i=1}^k(1-\\eta_i\\overline{D}_{\\mathsf{lb}})=\\mathcal{O}(\\eta_k)$. Therefore, \n\t\\begin{equation}\n\t\\sup_k\\mathbb{E}\\left[\\frac{(\\gamma_k-\\gamma^\\star)^2}{\\eta_k}\\right]=\\sup_k\\mathbb{E}\\left[2V(\\theta_k)\/\\eta_k\\right]=\\mathcal{O}(1). \n\t\\end{equation}\n\t\n\tThis finishes the proof of Theorem~\\ref{thm:rate-converge}. \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\t%\n\n\n\n\n\t%\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\n\n\n\n\n\n\n\n\t\n\t\n\t\n\t\n\t\n\t\\subsection{Proof of Theorem~\\ref{thm:converse}}\\label{pf:converse}\n\t\n\t\\subsubsection{Proof of Inequality \\eqref{eq:converse-est}}\n\t\n\tLet $\\mathbb{P}_1, \\mathbb{P}_2$ be two delay distributions and let $\\gamma_1^\\star, \\gamma_2^\\star$ be the solution to \\eqref{eq:equation-offline} when $D\\sim\\mathbb{P}_1$ and $D\\sim\\mathbb{P}_2$, respectively. \n\tThrough Le Cam's inequality \\cite{yu1997assouad}, we have:\n\t\\begin{equation}\n\t\\inf_{\\hat{\\gamma}}\\sup_{\\mathbb{P}}\\mathbb{E}\\left[\\left(\\hat{\\gamma}-\\gamma^\\star(\\mathbb{P})\\right)^2\\right]\\geq (\\gamma_1^\\star-\\gamma_2^\\star)^2\\cdot\\left(\\mathbb{P}_1^{\\otimes  k}\\wedge \\mathbb{P}_2^{\\otimes k}\\right),\\label{eq:lecam-gamma}\n\t\\end{equation}\n\twhere $\\mathbb{P}\\wedge \\mathbb{Q}:=\\int_{\\Omega}\\min\\{p(x), q(x)\\}\\text{d}x$ and $\\mathbb{P}^{\\otimes k}$ is the product of distribution $k$ i.i.d random variables drawn from $\\mathbb{P}$. \n\t\n\tTo use Le Cam's inequality \\eqref{eq:lecam}, we need to find two distributions $\\mathbb{P}_1$ and $\\mathbb{P}_2$, whose $\\ell_1$ distance $|\\mathbb{P}_1^{\\otimes k}-\\mathbb{P}_2^{\\otimes k}|_1$ is bounded, and the difference $(\\gamma_1^\\star-\\gamma_2^\\star)^2$ is of order $1\/k$. We consider $\\mathbb{P}_1$ to be a uniform distribution on $[0, 1]$ and let $\\gamma_1^\\star$ be the optimum ratio of distribution $\\mathbb{P}_1$.  Through Corollary~\\ref{coro:gammadep-bound}, we can obtain a loose upper bound on $\\gamma_1^\\star$ as follows:\n\t\\begin{equation}\n\t\\gamma_1^\\star<\\frac{1}{2}\\frac{\\mathbb{E}[D^2]}{\\mathbb{E}[D]}=\\frac{1}{3}. \n\t\\end{equation} \n\t\n\tLet $c\\leq\\frac{1}{2}$ be a constant and we denote\n\t\\begin{equation}\n\t\t\\delta=\\min\\{1-3\\gamma_1^\\star, 1\/3, p_{\\text{w, uni}}^\\star\/2\\}.\\label{eq:deltadef}\n\t\\end{equation} \n\tLet $\\mathbb{P}_2$ be a probability distribution with probability density function $p_2(x)$ defined as follows:\n\t\\begin{equation}\n\tp_2(x)=\\begin{cases}\n\t1-c\\sqrt{1\/k}, &x\\leq \\frac{1}{2}\\delta;\\\\\n\t1, &\\frac{1}{2}\\delta<x\\leq 1-\\frac{1}{2}\\delta;\\\\\n\t1+c\\sqrt{1\/k}, &x>1-\\frac{1}{2}\\delta;\\\\\n\t0, &\\text{otherwise}.\n\t\\end{cases}\\label{eq:p2def}\n\t\\end{equation}\n\t\n\tWe will first bound $(\\gamma_1^\\star-\\gamma_2^\\star)^2$ (in Step 1) and $\\mathbb{P}_1^{\\otimes k}\\wedge\\mathbb{P}_2^{\\otimes k}$ (in Step 2) as follows:\n\t\n\t\\hspace{-10pt}\\textbf{Step 1: Lower bounding $\\gamma_2^\\star-\\gamma_1^\\star$: }\n\tFor notational simplicity, denote function $h_1(\\gamma):=\t\\mathbb{E}_{D\\sim\\mathbb{P}_1}[\\frac{1}{6}\\max\\{3\\gamma, Z_D^2\\}^2-\\gamma\\max\\{3\\gamma, Z_D^2\\}]$ and $h_2(\\gamma):=\t\\mathbb{E}_{D\\sim\\mathbb{P}_2}[\\frac{1}{6}\\max\\{3\\gamma, Z_D^2\\}^2-\\gamma\\max\\{3\\gamma, Z_D^2\\}]$. According to the definition of $\\mathbb{P}_2$ in \\eqref{eq:p2def}, for each $\\gamma$, the difference between $h_1(\\gamma)$ and $h_2(\\gamma)$ can be computed by:\n\t\\begin{align}\n\t&h_2(\\gamma)-h_1(\\gamma)\\nonumber\\\\\n\t=&\\int_{1-\\delta\/2}^1\\frac{c}{\\sqrt{k}}\\mathbb{E}\\left[\\frac{1}{6}\\max\\{3\\gamma, Z_D^2\\}^2-\\gamma\\max\\{3\\gamma, Z_D^2\\}\\big|D=x\\right]\\text{d}x\\nonumber\\\\\n\t&-\\int_{0}^{\\delta\/2}\\frac{c}{\\sqrt{k}}\\mathbb{E}\\left[\\frac{1}{6}\\max\\{3\\gamma, Z_D^2\\}^2-\\gamma\\max\\{3\\gamma, Z_D^2\\}\\big|D=x\\right]\\text{d}x\\nonumber\\\\\n\t\\overset{(a)}{=}&\\int_{1-\\delta\/2}^1\\frac{c}{\\sqrt{k}}\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma)}|D=x\\right]\\text{d}x\\nonumber\\\\\n\t&-\\int_{0}^{\\delta\/2}\\frac{c}{\\sqrt{k}}\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma)}|D=x\\right]\\text{d}x, \\label{eq:hdiff}\n\t\\end{align}\n\twhere inequality $(a)$ is obtained because \n\t\\begin{align}\\frac{1}{6}\\max\\{3\\gamma, Z_D^2\\}^2-\\gamma\\max\\{3\\gamma, Z_D^2\\}=-\\frac{3}{2}\\gamma^2+\\frac{1}{6}(Z_D^2-3\\gamma)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma)}.\n\t\\end{align}\n\tSince $\\gamma_1^\\star$ is the optimum ratio for delay distribution $\\mathbb{P}_1$, we have $h_1(\\gamma_1^\\star)=0$. \n\n\n\n\n\n\t%\n\t%\n\n\tAccording to equation \\eqref{eq:hdiff}, function $h_2(\\gamma_1^\\star)$ can be lower bounded by:\n\t\\begin{align}\n\t&h_2(\\gamma_1^\\star)\\nonumber\\\\\n\t\\overset{(b)}{\\geq}&\\frac{c}{\\sqrt{k}}\\cdot\\int_{1-\\delta\/2}^1\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma_1^\\star)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma_1^\\star)}\\big|D=x\\right]\\text{d}x\\nonumber\\\\\n\t&-\\int_{0}^{\\delta\/2}\\frac{c}{\\sqrt{k}}\\frac{1}{2}x^2\\text{d}x\\nonumber\\\\\n\t\\geq&\\frac{c}{\\sqrt{k}}\\cdot\\int_{1-\\delta\/2}^1\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma_1^\\star)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma_1^\\star)}\\big|D=x\\right]\\text{d}x-\\frac{c}{\\sqrt{k}}\\frac{1}{6}\\left(\\frac{\\delta}{2}\\right)^3. \\label{eq:h2mid}\n\t\\end{align}\n\twhere inequality $(b)$ is because $\\mathbb{E}[\\frac{1}{6}(Z_D^2-3\\gamma)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma)}|D=x]\\leq\\mathbb{E}[\\frac{1}{6}Z_D^4|D=x]=\\frac{1}{2}x^2$. \n\t\n\tWe then proceed to lower bound $\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma_1^\\star)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma_1^\\star)}|D=x\\right]$ for each delay realization $x\\in[1-\\delta\/2, 1]$ as follows:\n\t\\begin{align}\n\t&\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma_1^\\star)^2\\mathbb{I}_{(Z_D^2\\geq 3\\gamma_1^\\star)}|D=x\\right]\\nonumber\\\\\n\t\\overset{(c)}{\\geq}&\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma_1^\\star)^2\\mathbb{I}_{(3\\gamma_1^\\star\\leq Z_D^2\\leq x)}+\\frac{1}{6}(Z_D^2-x)^2\\mathbb{I}_{(Z_D^2\\geq x)}|D=x\\right]\\nonumber\\\\\n\t\\geq&\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma_1^\\star)^2\\mathbb{I}_{(3\\gamma_1^\\star\\leq Z_D^2\\leq x)}\\right]\\nonumber\\\\\n\t&+\\frac{1}{6}\\left(\\text{Var}[Z_D^2|D=x]-x^2\\text{Pr}\\left(Z_D^2\\leq x|D=x\\right)\\right)\\nonumber\\\\\n\t\\overset{(d)}{\\geq}&\\frac{1}{6}x^2\\geq\\frac{1}{6}(1-\\delta\/2)^2,\\label{eq:h2lb}\n\t\\end{align}\n\twhere inequality $(c)$ is because $\\delta\\geq1-3\\gamma_1^\\star$ by equation \\eqref{eq:deltadef}, and for the conditional mean $\\mathbb{E}[Z_D^2|D=x]=x\\geq 1-\\delta\/2\\geq 1-\\delta \\geq3\\gamma_1^\\star$; inequality $(d)$ is because $\\text{Var}[Z_D^2|D=x]=2x^2$ and $x^2\\text{Pr}(Z_D^2\\leq x)\\leq x^2$ and $\\mathbb{E}\\left[\\frac{1}{6}(Z_D^2-3\\gamma_1^\\star)^2\\mathbb{I}_{(3\\gamma_1^\\star\\leq Z_D^2\\leq x)}\\right]\\geq 0$. \n\tPlugging inequality \\eqref{eq:h2lb} into \\eqref{eq:h2mid} and recall that $\\delta<1$ by definition, we have the lower bound of $h_2(\\gamma_1^\\star)$:\n\t\\begin{equation}\n\th_2(\\gamma_1^\\star)\\geq\\frac{c}{\\sqrt{k}}\\frac{\\delta}{2}\\frac{1}{6}\\left(\\left(1-\\frac{\\delta}{2}\\right)^2-\\left(\\frac{\\delta}{2}\\right)^2\\right)\\geq\\frac{c}{\\sqrt{k}}\\frac{\\delta}{12}(1-\\delta)>0. \\label{eq:h2lbf}\n\t\\end{equation}\n\tBy Lemma~\\ref{lemma:g}-(i), function $h_2(\\cdot)$ is monotonically decreasing. Since $h_2(\\gamma_1^\\star)> 0$ and $h_2(\\gamma_2^\\star)=0$, we can conclude that $\\gamma_2^\\star\\geq\\gamma_1^\\star$. We then proceed to bound $\\gamma_2^\\star-\\gamma_1^\\star$ through Taylor expansion at $\\gamma=\\gamma_1^\\star$.\n\t\\begin{equation}\n\th_2(\\gamma_2^\\star)=h_2(\\gamma_1^\\star)+h_2'(\\gamma)(\\gamma_2^\\star-\\gamma_1^\\star),\n\t\\end{equation}\n\twhere $\\gamma\\in[\\gamma_1^\\star, \\gamma_2^\\star]$. Therefore, $\\gamma_2^\\star$ can be computed by:\n\t\\begin{equation}\n\t\\gamma_2^\\star-\\gamma_1^\\star=-\\frac{h_2(\\gamma_1^\\star)}{h_2'(\\gamma)}. \n\t\\end{equation}\n\t%\n\n\n\n\n\t\n\tTo lower bound $\\gamma_2^\\star$, we will first find a loose upper bound of $\\gamma_2^\\star$ using Lemma~\\ref{coro:gammadep-bound}: \\begin{equation}\\gamma_2^\\star\\leq\\frac{1}{2}\\frac{\\mathbb{E}_{D\\sim\\mathbb{P}_2}[D^2]}{\\mathbb{E}_{D\\sim\\mathbb{P}_2}[D]}\\leq\\frac{1}{2}\\left(\\frac{1}{3}+\\delta \\cdot c\\sqrt{1\/k}\\right),\\label{eq:gamma2lbloose}\n\t\\end{equation}\n\tTherefore, since $\\delta<1\/3$, we have $|h_2'(\\gamma)|\\leq|h_2'(\\gamma_2^\\star)|=\\mathbb{E}[\\max\\{3\\gamma_2^\\star, Z_D^2\\}]\\leq\\overline{D}+3\\gamma_{\\text{2, ub}}\\leq 1+\\frac{1}{2}+\\frac{3}{2}c\\sqrt{\\frac{1}{k}}\\delta\\leq 2$. Then by inequality \\eqref{eq:h2lbf}, we have \n\t\\begin{equation}\n\t\\gamma_2^\\star-\\gamma_1^\\star\\geq\\frac{-h_2(\\gamma_1^\\star)}{h_2'(\\gamma_2^\\star)}\\geq\\frac{1}{24}(1-\\delta)\\delta c\\sqrt{\\frac{1}{k}}. \\label{eq:gamma2lb}\n\t\\end{equation}\n\t\n\t\\hspace{-10pt}\\textbf{Step 2: Lower bounding $\\mathbb{P}_1^{\\otimes k}\\wedge \\mathbb{P}_2^{\\otimes k}$: }Let $|\\mathbb{P}-\\mathbb{Q}|=\\int_{\\Omega}|\\text{d}\\mathbb{P}-\\text{d}\\mathbb{Q}|$ be the $\\ell_1$ distance between probability distribution $\\mathbb{P}$ and $\\mathbb{Q}$. Then\n\t\\begin{align}\n\t\\mathbb{P}_1^{\\otimes k}\\wedge \\mathbb{P}_2^{\\otimes k}=&\\int\\min\\{\\mathbb{P}_1^{\\otimes k}(\\text{d}x), \\mathbb{P}_2^{\\otimes k}(\\text{d}x)\\}\\nonumber\\\\\n\t=&\\int\\mathbb{P}_1^{\\otimes k}(\\text{d}x)\\cdot\\left(1-\\frac{\\left(\\mathbb{P}_2^{\\otimes k}(\\text{d}x)-\\mathbb{P}_1^{\\otimes k}(\\text{d}x)\\right)^+}{\\mathbb{P}_1^{\\otimes k}(\\text{d}x)}\\right)\\nonumber\\\\\n\t=&1-\\int\\left(\\mathbb{P}_2^{\\otimes k}(\\text{d}x)-\\mathbb{P}_1^{\\otimes k}(\\text{d}x)\\right)^+\\nonumber\\\\\n\t=&1-\\frac{1}{2}|\\mathbb{P}_1^{\\otimes k}-\\mathbb{P}_2^{\\otimes k}|_1.\\label{eq:wedgepq}\n\t\\end{align}\n\tEquality \\eqref{eq:wedgepq} enables us to lower bound $\\mathbb{P}_1^{\\otimes k}\\wedge \\mathbb{P}_2^{\\otimes k}$ by upper bounding the $\\ell_1$ distance $|\\mathbb{P}_1^{\\otimes k}-\\mathbb{P}_2^{\\otimes k}|_1$, which is done through the Pinsker's inequality:\n\t\\begin{align}\n\t&\\frac{1}{2}\\left|\\mathbb{P}_1^{\\otimes k}-\\mathbb{P}_2^{\\otimes k}\\right|_1\\nonumber\\\\\n\t\\leq&\\sqrt{\\frac{1}{2}D_{\\mathsf{KL}}(\\mathbb{P}_2^{\\otimes k}||\\mathbb{P}_1^{\\otimes k})}\\nonumber\\\\\n\t=&\\sqrt{\\frac{1}{2}kD_{\\mathsf{KL}}(\\mathbb{P}_2||\\mathbb{P}_1)}\\nonumber\\\\\n\t\\overset{(e)}{\\leq}&\\sqrt{\\frac{1}{2}k\\int_0^1p_2(x)\\ln p_2(x)\\text{d}x}\\nonumber\\\\\n\t\\overset{(f)}{\\leq}&\\sqrt{\\frac{1}{2}k\\int_0^1\\left(p_2(x)-1+\\frac{1}{\\min\\{p_2(x), 1\\}}(p_2(x)-1)^2\\right)\\text{d}x}\\nonumber\\\\\n\t\\overset{(g)}{\\leq}&\\sqrt{\\frac{1}{2}k\\frac{1}{\\inf_{0\\leq d\\leq 1} p_2(d)}\\int_0^1(p_2(x)-1)^2\\text{d}x}\\nonumber\\\\\n\t\\leq&\\sqrt{\\frac{1}{2}k\\frac{1}{1-c\\sqrt{1\/k}}\\delta\\frac{c^2}{k}}\\leq\\sqrt{\\delta c^2},\\label{eq:klub-last}\n\t\\end{align}\n\twhere inequality $(e)$ is because the density function $p_1(x)=1$ for uniform distribution, therefore $D_{\\mathsf{KL}}(\\mathbb{P}_2||\\mathbb{P}_1)=\\int_0^1p_2(x)\\ln p_2(x)$; inequality $(f)$ is because function $g(t):=(t\\ln t)$ is convex, its derivative $g(t)''=1\/t$, therefore, through Taylor expansion we have $g(t)\\leq g(1)+(t-1)+\\frac{1}{2}\\frac{1}{\\min\\{t, 1\\}}(t-1)^2=(t-1)+\\frac{1}{2}\\frac{1}{\\min\\{t, 1\\}}(t-1)^2$; inequality $(g)$ is because $\\int_0^1p_2(x)\\text{d}x=1$. \n\t\n\tBy choosing $c=1\/2$ and recall that $\\delta<1$, inequality \\eqref{eq:klub-last} can be upper bounded by:\n\t\\begin{equation}\n\t\\frac{1}{2}|\\mathbb{P}_1^{\\otimes k}-\\mathbb{P}_2^{\\otimes k}|_1\\leq\\frac{1}{2}. \\label{eq:klub}\n\t\\end{equation}\n\t\n\tPlugging \\eqref{eq:klub} into \\eqref{eq:wedgepq} yields:\n\t\\begin{equation}\n\t\\mathbb{P}_1^{\\otimes k}\\wedge \\mathbb{P}_2^{\\otimes k}\\geq\\frac{1}{2}. \\label{eq:infub}\n\t\\end{equation}\n\t\n\tFinally, plugging \\eqref{eq:infub} and \\eqref{eq:gamma2lb} into the Le Cam's inequality \\eqref{eq:lecam-gamma} finishes the proof of inequality \\eqref{eq:converse-est}:\n\t\\begin{equation}\n\t\\inf_{\\hat{\\gamma}}\\sup_{\\mathbb{P}}(\\hat{\\gamma}- \\gamma^{\\star}(\\mathbb{P}))^2\\geq\\frac{1}{2}\\left(\\frac{1}{24}(1-\\delta)\\delta p_{\\text{w,uni}}^\\star\\right)^2\\cdot\\frac{1}{k}.\\label{eq:minimaxf}\n\t\\end{equation} \n\t\\subsubsection{Proof of Inequality \\eqref{eq:regconverse}}\n\tThe proof is divided into three steps: consider a delay distribution $\\mathbb{P}\\in\\mathcal{P}_u(\\mu)$, first we will show for each sample policy $\\pi$ with a random sampling interval $\\tau$, let $l_\\pi:=\\mathbb{E}[\\tau]=\\mathbb{E}[Z_\\tau^2]$ denote the expected running lengt\n\t, the following inequality holds:\n\t\\begin{equation}\n\t\\mathbb{E}\\left[\\int_{t=0}^{\\tau}Z_t^2\\text{d}t\\right]-\\gamma^\\star\\mathbb{E}[\\tau]\\geq\\frac{1}{6}p_w(\\mathbb{P})\\left(l_\\pi-l^\\star(\\mathbb{P})\\right)^2,\\label{eq:regrettoerror}\n\t\\end{equation}\n\twhere $l^\\star(\\mathbb{P}):=\\mathbb{E}_{D\\sim\\mathbb{P}}[\\max\\{3\\gamma^\\star(\\mathbb{P}), Z_D^2\\}]$ is the average frame length when the optimum policy $\\pi^\\star(\\mathbb{P})$ is used. \n\tNext, We will show that given $k$ samples $\\delta X_1, \\cdots, \\delta X_k\\overset{{\\rm i.i.d}}{\\sim}Z_D$, where $D\\sim\\mathbb{P}$, the minimax estimation error satisfies:\n\t\\begin{equation}\n\t\\inf_{\\hat{l}}\\sup_{\\mathbb{P}\\in\\mathcal{P}_u(\\mu)}\\mathbb{E}\\left[\\left(\\hat{l}-l^\\star(\\mathbb{P})\\right)^2\\right]\\geq N\\cdot\\frac{1}{k}, \\label{eq:converses2}\n\t\\end{equation}\n\twhere $N$ is a constant independent of $k$ and $\\mu$ with expressions provided in equation \\eqref{eq:minimaxf}. \n\t\n\tFinally, notice that:\n\t\\begin{align}\n\t&\\mathbb{E}\\left[\\int_0^{S_k}(X_t-\\hat{X}_t)^2\\text{d}t-(\\gamma^\\star+\\overline{D})S_k\\right]\\nonumber\\\\\n\t=&\\sum_{k'=1}^k\\mathbb{E}\\left[(X_{S_{k'+1}}-X_{S_{k'}})^2D_k+\\frac{1}{6}(X_{S_{k'+1}}-X_{S_{k'}})^4\\right]\\nonumber\\\\\n\t&-(\\gamma^\\star+\\overline{D})\\sum_{k'=1}^k\\mathbb{E}[S_{k'+1}-S_{k'}]\\nonumber\\\\\n\t\\geq&\\frac{1}{6}p_w(\\mathbb{P})\\sum_{k'=1}^k\\left(\\mathbb{E}[L_k]-l^\\star\\right)^2.\\label{eq:converses3}\n\t\\end{align}\n\t\n\tTake $\\inf_{\\pi}\\sup_{\\mathbb{P}\\in\\mathcal{P}_w(\\mu)}$ on both sides of inequality \\eqref{eq:converses3}, we have:\n\t\\begin{align}\n\t&\\min_{\\pi}\\max_{\\mathbb{P}\\in\\mathcal{P}_u(\\mu)}\\mathbb{E}\\left[\\int_0^{S_k}(X_t-\\hat{X}_t)^2\\text{d}t-(\\gamma^\\star+\\overline{D})S_k\\right]\\nonumber\\\\\n\t\\geq&\\frac{1}{6}\\mu \\inf_{\\hat{l}}\\sup_{\\mathbb{P}\\in \\mathbb{P}_u(\\mu)}\\mathbb{E}[(\\hat{l}-l^\\star(\\mathbb{P}))^2]\t\\geq\\frac{1}{6}\\mu N\\sum_{k'=1}^k \\frac{1}{k'}\\geq\\Omega(\\ln k).  \n\t\\end{align}\n\t\n\tProof of inequality \\eqref{eq:regrettoerror} follows similar ideas as \\cite[Lemma 4]{Tang2205:Sending}. Details are provided in Appendix~\\ref{pf:regrettoerror} \n\tdue to space limitations. The proof of the minimax risk bound \\eqref{eq:converses2} is based on  Le Cam's two point method as follows:\n\t\n\t\\subsubsection{Proof of inequality \\eqref{eq:converses2}}\n\tLet $\\mathbb{P}_1, \\mathbb{P}_2\\in\\mathcal{P}_w(\\mu)$, \n\tthrough Le Cam's inequality \\cite{yu1997assouad}, we have:\n\t\\begin{equation}\n\t\\inf_{\\hat{l}}\\sup_{\\mathbb{P}\\in\\mathcal{P}_u(\\mu)}\\mathbb{E}\\left[\\left(\\hat{l}-l^\\star(\\mathbb{P})\\right)^2\\right]\\geq (l_1^\\star-l_2^\\star)^2\\cdot\\left(\\mathbb{P}_1^{\\otimes  k}\\wedge \\mathbb{P}_2^{\\otimes k}\\right).\\label{eq:lecam}\n\t\\end{equation}\n\t\n\tSimilarly to the proof of bounding $(\\hat{\\gamma}-\\gamma^\\star(\\mathbb{P}))$, let $\\mathbb{P}_1$ be a uniform distribution and $\\mathbb{P}_2$ through the density function in equation \\eqref{eq:p2def}. For $\\mu\\leq p_{\\text{w, uni}}^\\star\/2$, it is easy to show that $p_w(\\mathbb{P}_2)\\in\\mathcal{P}_u(\\mu)$ as follows:\n\t\\begin{align}\n\t&\\text{Pr}(Z_D^2\\leq 3\\gamma_2^\\star|D\\sim\\mathbb{P}_2)\\nonumber\\\\\n\t=&\\int_0^\\infty\\text{Pr}(Z_D^2\\leq 3\\gamma_2^\\star|D=x)p_2(x)\\text{d}x\\nonumber\\\\\n\t=&\\int_0^\\infty\\text{Pr}(Z_D^2\\leq 3\\gamma_2^\\star|D=x)p_1(x)\\text{d}x\\nonumber\\\\\n\t&-\\int_0^{\\delta\/2}\\text{Pr}(Z_D^2\\leq 3\\gamma_2^\\star|D=x)\\frac{c}{\\sqrt{k}}\\text{d}x\\nonumber\\\\\n\t&+\\int_{1-\\delta\/2}^{1}\\text{Pr}(Z_D^2\\leq 3\\gamma_2^\\star|D=x)\\frac{c}{\\sqrt{k}}\\text{d}x\\nonumber\\\\\n\t\t\\overset{(i)}{\\geq}&\\int_0^\\infty\\text{Pr}(Z_D^2\\leq 3\\gamma_1^\\star|D=x)p_1(x)\\text{d}x-c\/\\sqrt{k}\\delta\\nonumber\\\\\n\\geq&p_{\\text{w, uni}}^\\star-c\/\\sqrt{k}\\delta\\nonumber\\\\\n\t\\overset{(j)}{\\geq}&p_{\\text{w, uni}}^\\star\/2. \n\t\\end{align} \nwhere inequality $(i)$ holds because $\\gamma_1^\\star\\leq\\gamma_2^\\star$ from inequality \\eqref{eq:gamma2lb}; inequality $(j)$ holds because $\\delta<p_{\\text{w, uni}}^\\star\/2$ by definition from equation \\eqref{eq:deltadef}. The difference between $l_2^\\star-l_1^\\star$ can be computed by:\n\t\\begin{align}\n\tl_2^\\star-l_1^\\star\n\t=&\\int_{0}^1\\mathbb{E}\\left[\\max\\{3\\gamma_2^\\star, Z_D^2\\}|D=x\\right]\\text{d}x\\nonumber\\\\\n\t&+\\int_{1-\\delta\/2}^1\\frac{c}{\\sqrt{k}}\\mathbb{E}\\left[\\max\\{3\\gamma_2^\\star, Z_D^2\\}|D=x\\right]\\text{d}x\\nonumber\\\\\n\t&-\\int_{0}^{\\delta\/2}\\frac{c}{\\sqrt{k}}\\mathbb{E}\\left[\\max\\{3\\gamma_2^\\star, Z_D^2\\}|D=x\\right]\\text{d}x\\nonumber\\\\\n\t&-\\int_{0}^1\\mathbb{E}\\left[\\max\\{3\\gamma_1^\\star, Z_D^2\\}|D=x\\right]\\text{d}x.\\label{eq:ldiff}\n\t\\end{align}\n\t\n\tNotice that if $x_1\\geq x_2$, \n\t\\begin{align}\n\t\\mathbb{E}[\\max\\{3\\gamma, Z_D^2\\}|D=x_1]-\\mathbb{E}[\\max\\{3\\gamma, Z_D^2\\}|D=x_2]\\geq 0.\n\t\\end{align} Therefore, inequality \\eqref{eq:ldiff} can be bounded by:\n\t\\begin{align}\n\tl_2^\\star-l_1^\\star\\geq&\\int_{0}^1\\mathbb{E}\\left[\\max\\{3\\gamma_2^\\star, Z_D^2\\}|D=x\\right]\\text{d}x\\nonumber\\\\\n\t&-\\int_{0}^1\\mathbb{E}\\left[\\max\\{3\\gamma_1^\\star, Z_D^2\\}|D=x\\right]\\text{d}x\\nonumber\\\\\n\t\\geq&3(\\gamma_2^\\star-\\gamma_1^\\star)\\mathbb{E}_{D\\sim\\mathbb{P}_1}[\\text{Pr}(Z_D^2\\leq 3\\gamma_1^\\star)]\\nonumber\\\\\n\t\\overset{(g)}{\\geq}&\\frac{1}{24}(1-\\delta)\\delta c p_{\\text{w, uni}}^\\star\\sqrt{\\frac{1}{k}},\\label{eq:llb}\n\t\\end{align}\n\twhere inequality $(g)$ is obtained by equation \\eqref{eq:gamma2lb}. \n\t\n\n\n\n\n\t\n\t\n\tFinally, plugging \\eqref{eq:klub} and \\eqref{eq:llb} into Le Cam's inequality \\eqref{eq:lecam} finishes the proof of inequality \\eqref{eq:converses2}:\n\t\\begin{equation}\n\t\\inf_{\\hat{l}}\\sup_{\\mathbb{P}\\in\\mathcal{P}_w(\\mu)}(\\hat{l}- l^{\\star}(\\mathbb{P}))^2\\geq\\frac{1}{2}\\left(\\frac{1}{48}(1-\\delta)\\delta p_{\\text{w,uni}}^\\star\\right)^2\\cdot\\frac{1}{k}.\\label{eq:minimaxl}\n\t\\end{equation}\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\t\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\t\\section{Conclusions}\n\t\n\tIn this work, we studied the problem of sampling a Wiener process for remote estimation over a channel with unknown delay statistics. By reformulating the MSE minimization problem as a renewal-reward process, we proposed an online sampling algorithm that can adaptively learn the optimum algorithm as the number of samples grows. We showed that the average MSE obtained by the proposed algorithm converges to the minimum MSE almost surely, and the cumulative MSE has an order of $\\mathcal{O}(\\ln k)$, where $k$ is the number of samples. We then prove that the cumulative MSE regret of any algorithm is at best $\\Omega(\\ln k)$. Numerical simulation results validate the convergence behaviors of the proposed algorithm. \n\t\\begin{acks}\n\t\tThe authors are grateful to Prof. Pengkun Yang and Prof. Yuhao Wang for helpful discussions on the proof of Theorem~\\ref{thm:converse}. The work of H. Tang and L. Tassiulas was supported by the NSF CNS-2112562 AI Institute for Edge Computing Leveraging Next\n\t\tGeneration Networks (Athena) and the ONR N00014-19-1-2566. The work of Y. Sun was supported by the NSF grant CCF-1813078 and the ARO grant W911NF-21-1-0244.\n\t\\end{acks}\n\t\n\n\n\n\t\\bibliographystyle{ACM-Reference-Format}\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\subsection{Calibration}\nThe instrumental magnitudes for reference images were converted to\nthe  standard ones using linear transformations based on 817 and 570\nlocal Stetson's standards in fields W and E, respectively (Stetson\n2000; 2009 on-line version). The following transformations were\nderived:\n\\begin{eqnarray}\n{\\rm v=V -0.086(3)\\times (B-V) -1.591(3)}\\nonumber \\\\\n{\\rm b=B -0.146(4)\\times (B-V) -2.054(4)} \\\\\n{\\rm b-v=0.938(3)\\times (B-V) -0.462(3)}\\nonumber\n\\end{eqnarray}\n(field E) and\n\\begin{eqnarray}\n{\\rm v=V -0.074(3)\\times (B-V) -1.602(3)}\\nonumber \\\\\n{\\rm b=B -0.139(4)\\times (B-V) -2.038(4)} \\\\\n{\\rm b-v=0.936(3)\\times (B-V) -0.437(3)~}\\nonumber\n\\end{eqnarray}\n(field W), where capital letters denote the standard magnitudes.\n\nAs most of the monitoring was conducted in $V$ only, the standard magnitudes\nwere derived using average values of $B-V$. This introduced some systematic\nerrors in the case of color-changing objects. However, for most of the\nvariables the amplitude of color variations did nor exceed 0.1 mag,\nand the systematic error of $V$ magnitude was smaller than 0.01 mag.\nFor the main goal of our survey, which is is to detect\nnew variables, inaccuracies of this size are unimportant.\n\nThe astrometric solutions for the reference images in $V$ were found\nbased on positions of 2997 UCAC3 stars (Zacharias et al. 2010). The\naverage residuals in RA and DEC between cataloged and recovered\ncoordinates amount to 0.14 and 0.14 arcsec, respectively. For the\ndetection of variables we used methods described in Kaluzny et al.\n(2013). A total of 65 variables were found (among them 19 new ones).\nThey are listed in Table 1 along with their equatorial coordinates\nand cross-identifications for catalogs published by Weldrake et al.\n(2004), Kaluzny et al. (1998) and Samus et al. (2009). We detected\nvariability of all but one object from Weldrake et al. (2004)\npresent in the surveyed field (our photometry shows no brightness\nvariations for their star V93. \n\nFinding charts for newly detected variables are shown in Fig. 2.\n\n\\section{Variables}\n\nThe basic properties of variables detected in our survey are listed\nin Table 2. The periods in column 2 were derived with the method\nemploying periodic orthogonal polynomials to fit the observations\nand the analysis of variance statistic to evaluate the quality of\nthe fit (Schwarzenberg-Czerny 1996), as implemented in the TATRY\ncode kindly made available by the author.\n\nThe parameter $\\Delta V$ listed in column 4 gives the full range of\nthe measured $V$-magnitude, including seasonal changes of light curves. The last\ncolumn of Table 2 contains the proposed classification of variables,\nwith ``new'' indicating newly detected objects. EW, EB and EA denote\neclipsing binaries with light curves of W~UMa, $\\beta$ Lyrae and\nAlgol type, respectively; Ell stands for ellipsoidal variables, and\nstars located along or near the red giant branch on the cluster\ncolor-magnitude diagram (CMD) are labeled with RGB \/ RS CVn. The variability\nof the latter is likely caused by ellipsoidal effect and\/or\nchromospheric activity related to binarity. Among the new variables\nthere are five likely optical counterparts to X-ray sources detected\nwith the Chandra telescope by Heinke et al. (2005).\n\nFigure 3 presents the CMD of a small section of the monitored field\nwith marked locations of all but two variables listed in Tables 1\nand 2. Not included are W26 and E20 - very red Miras with $B-V>3$\nmag which were originally detected by the OGLE group (Soszynski et al.\n2011). The group of variables located to the blue of the cluster's\nmain sequence at $V\\approx19$ is composed of RR~Lyr pulsators from the\nSMC. The sequence of red long period variables starting at\n($V\\approx19$, $B-V\\approx1.1$) and extending to $B-V\\approx2.0$\nalso consists of SMC stars. Fig. 4 shows the positions in the \ncluster CMD  of a  selection of\nour variables, including all eclipsing binaries and three\nparticularly interesting objects which will be discussed below.\nPhased light curves for 14 new variables are\ndisplayed in Fig.~5 (not shown are RR-Lyrae stars, long period\nvariables, and E32 whose period is not known). Light curves for all\n65 variables detected within the present survey can be found in the\nelectronic version of this paper available from the Acta Astronomica \narchive or from CASE archive at http:\/\/case.camk.edu.pl. \n\n\\subsection{Eclipsing binaries}\nThe main result of our survey is the detection of four new detached\neclipsing binaries. All these objects are located beyond the core region of\nthe cluster and are suitable for spectroscopic follow-up studies\nwith ground-based telescopes.\n\nThe light curve of W12 shows a shallow secondary eclipse with $\\Delta V\n\\approx 0.1$~mag and a primary eclipse with $\\Delta V \\approx 0.4$~mag.\nA preliminary solution of the $V$-light curve indicates a high luminosity\nratio of the components: $L_{p}\/L_{s}=9$. The expected luminosity ratio\nfor the $I$ band equals to 6, so that the near-IR spectroscopy may allow\nfor the determination of radial velocity curves of both components.\n\nThe available light curve of E32 shows only one clear eclipse\nwith $\\Delta V=0.6$ mag. This means that the orbital period has to\nbe longer than 10 days (and may be significantly longer). We\nobtained a single spectrum of E32 with the MIKE Echelle spectrograph\non  the 6.5 m Magellan Clay telescope, and we found that the\nbinary is an SB2 system. Several additional spectra are needed to\nestablish the ephemeris with confidence. This in turn will enable\nfurther photometric observations in eclipses, and finally the\ndetermination of absolute parameters of the system. The location of\nE32 on the CMD indicates that at least one of its components is an\nevolved star at or past the turnoff. The analysis of E32 together\nwith V69 (Thompson et al. 2010) may allow interesting limits \nto be placed on the helium abundance of 47~Tuc.\n\\begin{table}\n\\caption{Equatorial coordinates of variable stars identified\nwithin the present survey \\label{tab:coords}}\n{\\scriptsize\n\\setlength{\\tabcolsep}{0.48em}\n\\begin{tabular}{|l|c|c|c|c|c|l|c|c|c|c|c|}\n\\hline\nID &  RA$_{J2000}$& Dec$_{J2000}$ & ID-W$^{a}$ & ID-K$^{b}$ & ID-S$^{c}$& ID &\n      RA$_{J2000}$& Dec$_{J2000}$ & ID-W$^{a}$ & ID-K$^{b}$ & ID-S$^{c}$\\\\\n      &  [deg]    & [deg]   &            &            &           &          & [deg]      &     [deg]  &            &         \\\\\n\\hline\n\\hline\nW1  &6.04262 &-72.06708& & &           &E8  &6.51136 &-71.97276& &231  & \\\\\nW2  &5.91532 &-72.02788& & &           &E9  &6.45689&-71.97410& V99 &226&        \\\\ \nW3  &6.05276 &-72.11105& & &V001       &E10 &6.55163&-72.03701& V27 &230&        \\\\  \nW4  &6.02083 &-72.08958& & &           &E11 &6.51274&-72.05086& V28 &229&        \\\\\nW5  &5.99704 &-72.11350& & &           &E12 &6.38434&-72.03098& V100&225  & V044 \\\\\nW6  &5.91897 &-72.09997& &217 &V009    &E13 &6.53692&-72.11706& V7  &228  & V046 \\\\\nW7  &5.95557 &-72.21893& & &           &E14 &6.42508&-72.10040& V10 &223  &      \\\\\nW8  &5.69983 &-71.94696& & &           &E15 &6.54387&-72.18549& V6  &227  & V045\\\\\nW9  &5.68282 &-71.95575& V71&234 &     &E16 &6.53915&-72.20798& V5  &255  & \\\\\nW10 &5.77913 &-72.02260&  &218&        &E17 &6.39210&-72.16616&    V9  &222  & \\\\\nW11 &5.84984 &-72.05125& & &           &E18 &6.51177&-72.24155&     &     & \\\\\nW12 &5.73147 &-72.08756& & &           &E19 &6.40987&-72.23572& & & \\\\\nW13 &5.72354 &-72.06296& V69& &        &E20 &6.40448&-72.25761& & 252   & \\\\\nW14 &5.65323 &-72.11997& & &           &E21 &6.31753&-71.93490& V30 & 238 &V047\\\\\nW15 &5.81761 &-72.18639& & &           &E22 &6.24824&-71.89658& & 241   & \\\\\nW16 &5.77493 &-72.15847& V97& &        &E23 &6.20544&-71.93885& V34 &   & \\\\\nW17 &5.67023 &-72.15625& V96  &215&    &E24 &6.25194&-72.00076& V32 &221&V043\\\\\nW18 &5.69786 &-72.22143& V95&245&V050  &E25 &6.17727&-72.10601& & & \\\\\nW19 &5.63677 &-71.99193& V70&216&      &E26 &6.17207&-72.10331& & & \\\\\nW20 &5.54779 &-71.98831& V62 & &       &E27 &6.29647&-72.20400& V14&250 &V052\\\\\nW21 &5.50268 &-72.03445& V61 &214&V042 &E28 &6.27575&-72.17319& V15&219 & \\\\\nW22 &5.46433 &-72.18055& V91 & &       &E29 &6.15076&-71.95535&  V33&239 \\\\\nW23 &5.63907 &-72.23045& V94 & 246&    &E30 &6.17717&-71.98994& V31&220 & \\\\\nW24 &5.27715 &-71.94372& V64& &        &E31 &6.05647&-71.96737& & & \\\\\nW25 &5.34974 &-72.25939& V92&243&      &E32 &6.15633&-72.06476& & & \\\\\nW26 &5.27031 &-72.25646&    &242&      &E33 &6.07805&-72.08192& & & \\\\\nE1  &6.75007 &-71.91918& V26 & &       &E34 &6.10767&-72.11765& & & \\\\\nE2  &6.69230 &-71.99538& & &           &E35 &6.09773&-72.14155& V17 & & \\\\\nE3  &6.73688 &-72.17036& V13 & 232 &   &E36 &6.09675&-72.12296& & & \\\\\nE4  &6.69681 &-72.27622& & &           &E37 &6.07745&-72.13304& & &V002\\\\\nE5  &6.67914 &-72.25578& V12 & 253&V053 &E38 &6.13241&-72.15819& V16&251 & \\\\\nE6  &6.58064 &-72.23394& V4& 254 &     &E39 &6.10552&-72.22732& & & \\\\\nE7  &6.40631 &-71.93440& V29 & &       &    &       &           &  &\\\\\n\\hline\n\\end{tabular}\n}\n{\\footnotesize$^a$ Weldrake et al. 2004;\n$^b$ Kaluzny et al. 1988; $^c$ Samus et al. 2009}\n\\end{table}\n\nFor W7 we have three MIKE\/Magellan spectra. They show that the\nvariable is an SB1 binary with  measured velocities of\n$-37.41\\pm0.32$, $-74.27\\pm0.22$ and $+16.38\\pm0.16$ km\/s at orbital\nphases 0.49 (HJD 245 5769.866), 0.353 (HJD 245 5770.839) and 0.692 \n(HJD 55836.642), respectively. \nThese measurements indicate that the binary is a\nlikely member of the cluster. The systemic velocity of 47 Tuc is\nequal to -18.0 km\/s, and the velocity dispersion at the location \nof W7 amounts to $\\sim$11 km\/s (Harris 1996). This can be\ncompared with the binary's velocity near conjunction at orbital phase 0.49.\nThe fourth detached system, E39, is placed rather far\nfrom the cluster main sequence and may be a foreground halo object.\nIts low luminosity and short orbital period make it a\ndifficult target for spectroscopy.\n\n\n\\begin{center}\n\\footnotesize{\n\\begin{longtable}{|l|l|c|c|c|c|}\n\\caption[]{Properties of variable stars identified\nwithin the present survey}\\label{tab:properties}\\\\\n\\hline\nID &  P       & $V_{max}$ & $\\Delta V$ & $<B-V>$ & Type of variability \\\\\n      &  [d]  & [mag]     & [mag]      &         &    Remarks   \\\\\n\\hline \\hline\n\\endfirsthead\n\\multicolumn{6}{l}\n{{ \\tablename\\ \\thetable{} -- continued from previous page}} \\\\\n\\hline\nID &  P       & $V_{max}$ & $\\Delta V$ & $<B-V>$ & Type of variability \\\\\n      &  [d]  & [mag]     & [mag]      &  [mag]  &  Remarks     \\\\\n\\hline\n\\hline\n\\endhead\n\\hline \\multicolumn{6}{|r|}{Continued on next page} \\\\ \\hline\n\\endfoot\n\n\\hline\n\\endlastfoot\nW1 &   0.36115544(1) & 15.607&  0.163 & 0.600 & EW, YS, X$^{a}$, {\\bf new}, {\\scriptsize Ch-0024410.2-720401}\\\\\nW2 &   0.28287568(1) & 17.565 & 0.317 & 0.543 & EW, {\\bf new} \\\\\nW3 &   -             & 11.753 & 2.872 & 1.516 & LP \\\\\nW4 &   0.35298992(1) & 15.838 & 0.163 & 0.271 & EW, BS, X, {\\bf new}, {\\scriptsize Ch-002404.9-720522}\\\\\nW5 &   0.28036018(8) & 17.143 & 0.225 & 0.420 & EW, BS, X, {\\bf new}, {\\scriptsize Ch-002359.3-720648}\\\\\nW6 &   0.73703372(1) & 13.063 & 1.024 & 0.261 & RR, {\\scriptsize OGLE-SMC-RRLYR-0051}$^{~b}$\\\\\nW7 &   3.88255(2)    & 19.091 & 0.384 & 0.815 & EA, {\\bf new} \\\\\nW8 &   12.78227(2)   & 14.150 & 0.032 & 1.173 & RS CVn, RGB, {\\bf new} \\\\\nW9 &   0.6158898(1)  & 19.084 & 0.811 & 0.439 & RR, {\\scriptsize OGLE-SMC-RRLYR-0030} \\\\\nW10&   -             & 15.976 & 0.384 & 1.652 & LP, {\\scriptsize OGLE-SMC-LPV-00128}$^{~c}$ \\\\\nW11&   0.63228616(9) & 18.948 & 0.65  & 0.18  & RR, {\\scriptsize OGLE-SMC-RRLYR-0044}  \\\\\nW12&   3.732001(2)   & 17.462 & 0.411 & 0.634 & EA, {\\bf new} \\\\\nW13&   29.53975(1)   & 16.799 & 0.616 & 0.537 & EA \\\\\nW14&   0.04595852(3) & 14.679 & 0.037 & 0.145 & SX, BS, {\\bf new}$^d$ \\\\\nW15&   71.4156(5)    & 17.138 & 0.141 & 1.557 & LP, {\\scriptsize OGLE-SMC-LPV-00143} \\\\\nW16&   0.39708061(4) & 18.859 & 0.274 & 0.834 & EB \\\\\nW17&   8.4278094(1)  & 16.638 & 0.203 & 0.931 & BY~Dra \\\\\nW18&   0.27890017(1) & 15.459 & 0.424 & 0.565 & EW, YS             \\\\\nW19&   0.36164588(6) & 19.584 & 0.624 & 0.276 & RR, {\\scriptsize OGLE-SMC-RRLYR-0026} \\\\\nW20&   66.6          & 16.991 & 0.611 & 1.792 & LP, {\\scriptsize OGLE-SMC-LPV-00066} \\\\\nW21&   0.273741797(2)& 17.919 & 0.360 & 0.610 & EW \\\\\nW22&   0.65390427(7) & 18.738 & 0.264 & 0.582 & RR (blend?), {\\scriptsize OGLE-SMC-RRLYR-0016} \\\\\nW23&   0.57218584(4) & 19.017 & 0.980 & 0.428 & RR, {\\scriptsize OGLE-SMC-RRLYR-0027} \\\\\nW24&   0.595771(2)   & 19.323 & 0.839 & 0.301 & RR, {\\scriptsize OGLE-SMC-RRLYR-0011} \\\\\nW25&   0.6256305(15) & 19.364 & 0.752 & 0.404 & RR, {\\scriptsize OGLE-SMC-RRLYR-0012} \\\\\nW26&   269.3         & 16.879 & 1.406 & 3.347 & LP, {\\scriptsize OGLE-SMC-LPV-00015} \\\\\nE1 &   0.347489131(4)& 17.025 & 0.485 & 0.493 & EW, BS  \\\\\nE2 &   0.8102273(2)  & 18.306 & 0.168 & 0.734 & RR, {\\bf new} \\\\\nE3 &   0.36349524(3) & 19.129 & 0.733 & 0.240 & RR, {\\scriptsize OGLE-SMC-RRLYR-0122} \\\\\nE4 &   79.92         & 16.850 & 0.297 & 2.053 & LP, {\\scriptsize OGLE-SMC-LPV-00446} \\\\\nE5 &   0.44625922(1) & 16.659 & 0.398 & 0.434 & EB, BS \\\\\nE6 &   -             & 16.539 & 0.254 & 1.783 & LP, {\\scriptsize OGLE-SMC-LPV-00398} \\\\\nE7 &   4.463003(6)   & 18.792 & 0.210 & 1.273 & ? \\\\\nE8 &   6.371513(8)   & 14.141 & 0.152 & 0.760 & RS CVn, RGB-clump          \\\\\nE9 &   0.64752185(2) & 18.959 & 0.852 & 0.446 & RR, {\\scriptsize OGLE-SMC-RRLYR-0098} \\\\\nE10&   4.78667(2)    & 17.474 & 0.603 & 0.917 & BY~Dra \\\\\nE11&   8.3711839(8)  & 14.916 & 0.157 & 0.911 & RS CVn, RGB \\\\\nE12&   0.234635701(3)& 19.441 & 0.787 & 0.981 & EW \\\\\nE13&   1.150686034(3)& 15.797 & 0.417 & 0.233 & EB, BS \\\\\nE14&   0.29712114(1) & 17.306 & 0.505 & 0.238 & RR, {\\scriptsize OGLE-SMC-RRLYR-0098} \\\\\nE15&   0.378854028(1)& 16.397 & 0.311 & 0.349 & EW, BS \\\\\nE16&   0.52515175(1) & 18.977 & 1.254 & 0.422 & RR, {\\scriptsize OGLE-SMC-RRLYR-0104} \\\\\nE17&   20.791(1)     & 16.576 & 0.207 & 0.733 & RS CVn, RGB  \\\\\nE18&   14.09425(2)   & 17.176 & 0.070 & 1.405 & ?, {\\bf new} \\\\\nE19&   -             & 14.344 & 0.194 & 0.989 & LP, {\\bf new} \\\\\nE20&   266.4         & 17.911 & 2.254 & 3.607 & LP, {\\scriptsize OGLE-SMC-LPV-00329} \\\\\nE21&   0.250546155(3)& 18.397 & 0.457 & 0.669 & EW \\\\\nE22&   35.3519(1)    & 16.747 & 0.111 & 1.680 & LP, {\\scriptsize OGLE-SMC-LPV-00271} \\\\\nE23&   0.24169575(1) & 18.551 & 0.337 & 0.693 & EW \\\\\nE24&   0.313434596(1)& 17.650 & 0.519 & 0.511 & EW \\\\\nE25&   3.33587(4)    & 17.849 & 0.143 & 0.589 & ?, {\\bf new} \\\\\nE26&   1.4043996(8)  & 18.528 & 0.282 & 1.276 & ?, {\\bf new} \\\\\nE27&   0.351384786(1)& 16.244 & 0.246 & 0.318 & EW, BS \\\\\nE28&   36.8914(12)   & 15.252 & 0.148 & 0.900 & RS CVn, RGB \\\\\nE29&   40.39(16)     & 16.613 & 0.111 & 1.608 & LP, {\\scriptsize OGLE-SMC-LPV-00235} \\\\\nE30&   10.77526(2)   & 16.007 & 0.320 & 0.814 & RS CVn, RGB \\\\\nE31&   20.43463(6)   & 16.564 & 0.457 & 0.901 & RS CVn, RGB, {\\bf new} \\\\\nE32&   -             & 17.109 & 0.362 & 0.569 & EA, {\\bf new}  \\\\\nE33&   0.25151879(1) & 16.391 & 0.107 & 1.044 & Ell?, X, {\\bf new}, {\\scriptsize Ch-002418.6-720455}\\\\\nE34&   9.87379(4)    & 15.495 & 0.294 & 0.878 & RGB, X, {\\bf new}, {\\scriptsize Ch-002425.8-720703 }\\\\\nE35&   0.300361921(7)& 18.021 & 0.345 & 0.648 & EW \\\\\nE36&   0.279869331(2)& 18.053 & 0.641 & 0.703 & EW, {\\bf new} \\\\\nE37&   135.43559(2)  & 13.180 & 2.480 & 1.301 & LP \\\\\nE38&   3.480885(3)   & 16.468 & 0.313 & 0.843 & RS CVn, RGB \\\\\nE39&   0.987519(2)   & 18.792 & 0.507 & 0.937 & EA, {\\bf new} \\\\\n\\end{longtable}\n}\n\\end{center}\n\\vskip -1cm\n{\\footnotesize\n$^a$ X - likely counterpart of an X-ray source \nCh-nnnnnnn.n-nnnnnn cataloged by Heinke et al. (2005)\\\\\n$^b$ OGLE-SMC-RRLYR-nnnn = stars cataloged by Soszynski et al. (2010)\\\\\n$^c$ OGLE-SMC-LPV-nnnnn = stars catalogud by Soszynski et al. (2011)\\\\\n$^d$ independently discovered by Poleski (2012)}\n\nIn addition to the five detached binaries our sample includes one\nsemi-detached system (E13; Kaluzny et al. 2007) and 15 contact or\nnearly-contact binaries. Six of them are blue stragglers, and two\n(W1 and W18) are located in the region occupied by yellow stragglers\n(see e.g. Stetson 1994). Variable yellow stragglers are rare objects\nand therefore we examined HST\/ACS images of W1 and W18 to check \nwhether or not \nthe ground based photometry is affected by blending. We found\nW18 to be an isolated object with no indication for unresolved\nvisual companions on ACS images. Still, we cannot rule out the\npossibility that it is a triple system like many (perhaps most) W\nUMa stars (Rucinski et al 2007). Spectroscopic observations can\nclarify this issue. As for W1, ACS images show three visual\ncomponents forming a blend which cannot be resolved on our ground\nbased images. Two closest components of the blend are separated by\n0.3 arcsec. Thus, based on the available evidence, W1 cannot be\nregarded as a yellow straggler.\n\n\\subsection{Notes on individual objects}\nSome of the newly detected variables deserve a short comment on\ntheir properties. The red straggler candidate E33 is located only 61 arcsec\naway from the cluster centre; however it is an isolated object whose\nground based photometry is free from blending-related problems. This\nwas confirmed by the examination of HST\/ACS\/F625W images. The star\nhas a sine-like light curve with  $\\Delta V=0.11$ mag and a period\nof 0.25~d which was coherent and stable during several observing\nseasons between 1998 and 2010. This indicates that the observed\nvariability is very likely related to the binarity of E33. We have\ntentatively classified E33 as an ellipsoidal variable. Heinke et al.\n(2005) analyzed two sets of Chandra data for 47 Tuc. The X-ray\ncounterpart of E33 was detected in 2002 with an X-ray luminosity of\n$0.7$E30 erg\/s in 0.5-2.5~keV band, but no X-rays were detected at\nE33 position in 2000. This implies a seasonal variability of the\nX-ray source connected with the star. The variable is too red to be\nan ordinary contact binary regardless of its membership status in 47\nTuc. This is evident by comparing its color with colors of contact\nsystems E21 and E23 whose orbital periods are similar to the period\nof E33 (see Fig. 4). As for the membership status of E33, the\nexamination of stacked subtracted images from seasons 1998, 1999,\n2009, and 2010 does not indicate any proper motion with respect to\ncluster stars. The qualitative method we used to estimate the proper\nmotion is based on Eyer and Wozniak (2001). Spectroscopic\ndata are needed to clarify the evolutionary and membership status of\nE33. \n\nThe variable E8 is located on the red horizontal branch on\nthe cluster CMD. The light curve shows coherent changes with $\\Delta\nV= 0.15$~mag and $P=6.4$~d. The amplitude of the light curve\nexhibits small but easily visible seasonal changes. Similar\nvariations with the same period were detected by the OGLE group in\n1993 (Kaluzny et al. 1998). Given the coherence of the variability\nit is likely that the star is a binary. If so, it would be a rare\nexample of a photometrically variable binary from the red horizontal\nbranch. With $P=6.4$~d and a radius of the red giant component of\nabout 10~$R_{\\odot}$, E8 would have to be a rather compact system.\nObviously, if the variability was induced mainly by the ellipsoidal\neffect, the actual orbital period of a binary would double to\n12.8~d.\nThe bright blue straggler W14 turned out to be an SX~Phe-type\npulsator. It is one of a few such variables detected in 47 Tuc\n(Gilliland et al. 1998; Poleski 2012). \nSX Phe stars are\ncommon among blue stragglers in metal-poor globular clusters, but\nrather rare in metal rich ones.\n\nFinally, we note the presence of several variables located on or\nslightly to the red of the red giant branch of the cluster, newly \ndetected of which are E31 and E34 (see Figs. 3 and 4). They are \ngood candidates for binaries and are easy targets for a spectroscopic \nfollow-up.\n\\section{Summary}\nWe performed a photometric study of the globular\ncluster 47 Tuc with a time baseline of more than ten years. \nBased on over 6500 $BV$ frames we have identified 65\nvariable stars, 19 of which are new detections. We provide celestial\ncoordinates of all variables, and cross-identifications of the\nvariables discovered earlier by other authors. Finding charts for\nthe new variables are also provided. Five of the new variables are\nlikely optical counterparts of X-ray sources, and another four ones\nare detached eclipsing binaries. Two detached eclipsing systems are\nlocated close to the main-sequence turnoff on the CMD of 47 Tuc, and\nwe argue that they are promising targets for detailed photometric\nand spectroscopic studies: when combined with the results of an\nearlier study of W13 (Thompson et al. 2010; their variable V69) they\nmay allow an improved constraint on the helium content of the cluster.\nThe yellow straggler W18, the red straggler E33 and the red\nhorizontal branch object E8 are another systems deserving further\nstudy which would clarify their membership and evolutionary status.\n\n\\Acknow{We are grateful to Igor Soszynski for a very detailed and helpful\nreferee report, and to Radek Poleski for pointing out the references to \nSX Phe stars in 47 Tuc. JK, MR, WP and WN were partly \nsupported by the grant NCN 2012\/05\/B\/ST9\/03931 from the Polish Ministry of Science.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nAt present, the continuum limit of antiferromagnetic Lie superalgebra spin chains is poorly understood. High degeneracies or continuous spectra~\\cite{Essler:2005ag, Saleur:2006tf, 2006cond.mat.12037I} and singular IR scattering amplitude behavior~\\cite{Essler:2005ag, Saleur:2009bf} emerge in the continuum large volume limit. We interpret this as a hint of new physical phenomena.\nStrikingly, when $q$-deformed, connections with long known puzzling theories with no obvious supergroup symmetry appear~\\cite{Saleur:2009bf}.\n\n\nPrevious, above mentioned, research has mostly dealt with  $\\osp(2|2)\\simeq \\sgl(2|1)$ spin chains.\nWe adopt a general approach, taking as a model~\\cite{Essler:2005ag}, and hoping that the greater $\\gl(M|N)$ symmetry and the relatively new cohomology techniques developed in~\\cite{Candu:2010yg} will allow us to better understand the nature of these new phenomena, thus, suggesting how to correctly interpret, treat and, eventually, apply them.\nThe goal is to ultimately formulate a meaningful factorizable scattering theory for the continuum limit of superspin chains.\nThis is an even harder task then it sounds, because there are few massive relativistic field theories with supergroup symmetry that have been understood, at least partially~\\cite{Saleur:2001cw, Saleur:2009bf, Bassi:1999ua, 2000NuPhB.583..475G}.\n\n\nSo, clarifying the continuum limit of superspin chains must ultimately result in a better understanding of formally simple field theories such as the $\\gl(M|N)$ and $\\osp(R|2S)$ Gross-Neveu models, which play an important role in disordered electronic system~\\cite{Bernard:1995as, 2000NuPhB.583..475G, LeClair:2007aj}, provide instances of continuous families of conformal field theories~\\cite{Candu:2008yw, Candu:2009ep} and appear in strong-weak coupling dualities with sigma models~\\cite{Candu:2008yw, 2000NuPhB.583..475G, Creutzig:2008an}.\nDue to strong violation of unitarity, it is not obvious how to treat these super Gross-Neveu models directly in the continuum by standard bootstrap methods~\\cite{Zamolodchikov:1978xm, ORW}.\nInsight from coordinate Bethe ansatz for chiral Gross-Neveu models~\\cite{Andrei:1979un, Andrei:1979wy, Andrei:1979sq, Destri:1987ug} has led us to consider the spin chain of $\\gl(M|N)$ alternating fundamental and dual representations as the most natural candidate for an integrable discretization of the $\\gl(M|N)$ Gross-Neveu model.\nThe spin chain of $\\gl(M|N)$ fundamental representations is discarded because it cannot lead to a relativistic field theory~\\cite{Saleur:1999cx, 1994IJMPB...8.3243E}.\nThis can be immediately seen from the fact that the dual of any $\\gl(M|N)$ representation appearing in the tensor product of $\\gl(M|N)$ fundamental representations does not belong to this tensor product if $N>0$. \n\n\n\nThe article is organized as follows.\nIn sec.~\\ref{sec:formalism} we define the spin chain and its integrable dynamics.\nIn sec.~\\ref{sec:num_diag} we explain how to perform numerical calculations for spin chains of modest size, but arbitrary $\\gl(M|N)$ symmetry, using the walled Brauer algebra. Then we present numerical results on the spectrum.\nSec.~\\ref{sec:rl} is of primary technical importance.\nWe discover a class of solutions to the $\\gl(M|N)$ Bethe ansatz equations (BAE) which can be fully characterized in terms of solutions to the  $\\gl(M-1|N-1)$ BAE. This provides an explicit embedding of the $\\gl(M-1|N-1)$ spectrum into the $\\gl(M|N)$ spectrum.\nWe give an algebraic explanation to this relationship.\nFurther, in sec.~\\ref{sec:BAE} we identify the vacuum solution of $\\gl(M|N)$ BAE and write it down explicitly in terms of the $\\gl(M-N)$ vacuum solution. We then classify $\\gl(M|N)$ excitations that lead to a $\\gl(M-N)$ like spectrum and the simplest ones that do not. For the latter, we analyze numerically the form of the solutions to the BAE.\nFinally, in sec.~\\ref{sec:cl} we consider the continuum limit of spin chains with alternating inhomogeneities.\nExcitations which can be characterized in terms of solutions to $\\gl(M-N)$ BAE lead to a spectrum of massive particles with $\\gl(M-N)$ Gross-Neveu mass ratios and $S$-matrix eigenvalues.\nWe argue that the continuum limit of the spin chain is the $\\gl(M|N)$ Gross-Neveu model.\nWe end with analyzing some low lying excitations of the $\\gl(2m|1)$ spin chain which are not of $\\gl(2m-1)$ type and lead to new massive particles.\n\n\n\n\\section{Transfer matrices, Hamiltonians and their spectra}\n\\label{sec:formalism}\n\nLet $V$ be the fundamental module of $\\gl(M|N)$, $V^*$ denote its dual and $\\rho:\\gl(M|N)\\mapsto \\gl(V)$ together with $\\bar{\\rho}:\\gl(M|N)\\mapsto \\gl(V^*)$ be the corresponding representations.\nIf $\\{e_\\alpha\\}_{\\alpha=1}^{M+N}$ is a graded basis of $V$ with grading $|\\alpha|:=|e_\\alpha|\\in \\mathbb{Z}\/2\\mathbb{Z}$ and $\\{e^\\alpha\\}_{\\alpha=1}^{M+N}$ is the dual basis $e^\\alpha(e_\\beta)=\\delta_{\\alpha\\beta}$, then $E_{\\alpha\\beta}$ are the standard generators of $\\gl(M|N)$ acting in $V$ as $E_{\\alpha\\beta}\\cdot e_\\gamma=\\delta_{\\beta\\gamma}e_\\alpha$ and in $V^*$ as $E_{\\alpha\\beta}\\cdot e^\\gamma=-(-1)^{(|\\alpha|+|\\beta|)|\\alpha|}\\delta_{\\alpha\\gamma}e^\\beta$.\nLet us label the $V$ factors of the spin chain $\\mathcal{C}(L)=(V\\otimes V^*)^{\\otimes L}$ from left to right by a subscript $1,\\dots,L$. Similarly, we label the $V^*$ factors  by a subscript $\\bar{1},\\dots,\\bar{L}$.\nFor $\\mathcal{E}\\in \\End V$ we denote by $\\mathcal{E}_k\\in\\End\\mathcal{C}(L)$  the endomorphisms acting as $\\mathcal{E}$ on $V_k$ and trivially, up to grading signs, everywhere else in the chain. Similarly, for $\\mathcal{E}\\in \\End V^*$, $\\mathcal{E}_{\\bar{k}}\\in\\End\\mathcal{C}(L)$ will act as $\\mathcal{E}$ on $V^*_{\\bar{k}}$ and trivially, up to grading signs, everywhere else.   \n\nThe $\\mathcal{C}(L)$--endomorphisms $P_{kl}=(-1)^{|\\beta|}\\rho_k(E_{\\alpha\\beta}) \\rho_l(E_{\\beta\\alpha})$ provide a representation for the symmetric group acting on the $V$ factors of $\\mathcal{C}(L)$. Similarly, $P_{\\bar{k}\\bar{l}}=(-1)^{|\\beta|}\\bar{\\rho}_{\\bar{k}}(E_{\\alpha\\beta}) \\bar{\\rho}_{\\bar{l}}(E_{\\beta\\alpha})$\ngenerate a representation for the symmetric group acting on the $V^*$ factors of $\\mathcal{C}(L)$.\nOn the other hand, $Q_{k\\bar{l}}= - (-1)^{|\\beta|}\\rho_k(E_{\\alpha\\beta}) \\bar{\\rho}_{\\bar{l}}(E_{\\beta\\alpha})$ generate a representation of the Temperley-Lieb algebra $T_{2L}(n)$ with loop weight $n=M-N$.\nTogether, the $P$'s and $Q$'s generate the $\\gl(M|N)$--centralizer algebra of $\\mathcal{C}(L)$, that is the set of all endomorphisms of the spin chain that commute with the $\\gl(M|N)$ action~\\cite{Sergeev}.\nThis centralizer algebra is a representation of the walled Brauer algebra $B_{L,L}(n)$, which can be viewed as a subalgebra of the Brauer algebra $B_{2L}(n)$. We shall discuss in detail the algebra $B_{L,L}(n)$ and its representations in sec.~\\ref{sec:wb_sub}.\n\nIn terms of walled Brauer algebra generators we introduce the $R$--matrices\n\\begin{align*}\nR_{ij}(u)&= u+P_{ij}& R_{\\bar{\\imath}j}(u)&= u-Q_{\\bar{\\imath}j}\\\\\nR_{i\\bar{\\jmath}}(u)&=u-Q_{i\\bar{\\jmath}}& R_{\\bar{\\imath}\\bar{\\jmath}}(u)&=u+P_{\\bar{\\imath}\\bar{\\jmath}}\\ ,\n\\end{align*}\nwhich satisfy the Yang-Baxter algebra\n\\begin{align}\\label{eq:YBE}\n R_{ij}(u-v)R_{ik}(u)R_{jk}(v)&=R_{jk}(v)R_{ik}(u)R_{ij}(u-v) \\\\ \\notag R_{ij}(u-v)R_{i\\bar{k}}(u)R_{j\\bar{k}}(v)& = R_{j\\bar{k}}(v)R_{i\\bar{k}}(u)R_{ij}(u-v)\\\\ \\notag\n R_{i\\bar{\\jmath}}(u-v+n)R_{ik}(u)R_{\\bar{\\jmath}k}(v)&=R_{\\bar{\\jmath}k}(v)R_{ik}(u)R_{i\\bar{\\jmath}}(u-v+n)\n\\end{align}\nand similar ``dual'' relations, that is the above relations with all barred indices replaced with unbarred ones and all unbarred indices with barred ones.\nWe define two one-parameter families of monodromies\n\\begin{align}\\label{eq:mon_mat_a}\n T_a(u)&=R_{a\\bar{L}}(u  +n\/2)R_{aL}(u)\\dots  R_{a\\bar{1}}(u+n\/2)R_{a1}(u)\\\\\n  \\bar{T}_{\\bar{a}}(u)&= R_{\\bar{a}\\bar{L}}(u)R_{\\bar{a}L}(u+n\/2) \\dots R_{\\bar{a}\\bar{1}}(u) R_{\\bar{a}1}(u+n\/2) \\label{eq:mon_mat_ab}\n\\end{align}\nacting on $V_a\\otimes\\mathcal{C}(L)$ and $V_{\\bar{a}}\\otimes \\mathcal{C}(L)$, respectively,\nand corresponding transfer matrices\n\\begin{equation}\\label{eq:tmat}\n t(u)=\\str_a T_a(u),\\qquad \\bar{t}(u)=\\str_{\\bar{a}}\\bar{T}_{\\bar{a}}(u)\\ .\n\\end{equation}\nYang-Baxter relations~\\eqref{eq:YBE} imply a Yangian structure given by the two relations\n\\begin{align}\\label{eq:comm1}\n R_{ab}(u-v)T_a(u)T_b(v)&=T_b(v)T_a(u)R_{ab}(u-v)\\\\\nR_{a\\bar{b}}(u-v+n\/2)T_a(u)\\bar{T}_{\\bar{b}}(v)&=\\bar{T}_{\\bar{b}}(v)T_a(u)R_{a\\bar{b}}(u-v+n\/2)\\label{eq:comm2}\n\\end{align}\nand their duals.\nThe following commutation relations immediately follow\n\\begin{equation*}\n [t(u),t(v)]=[t(u),\\bar{t}(v)]=[\\bar{t}(u),\\bar{t}(v)]=0\\ .\n\\end{equation*}\n\n\n\nThe nested algebraic Bethe ansatz for the most general $\\gl(M|N)$ spin chain was considered in \\cite{Belliard:2008di}. The Bethe ansatz equations and the spectrum of $t(u)$ formally depend on the nesting order, that is an ordering of a basis of $V$ and the induced ordering of the dual basis of $V^*$.\nIf  the basis $\\{e_\\alpha\\}_{\\alpha=1}^{M+N}$ of $V$  diagonalizes the Cartan subalgebra, then, without loss of generality, we label the basis vectors so that the total ordering reads\n\\begin{equation}\\label{eq:tot_ordering_bas_vecs}\n e_1>e_2>\\dots>e_{M+N}\\ ,\n\\end{equation}\nwhere, however, we keep unspecified the grading of the basis vectors.\nLet $\\wt(e_\\alpha)=\\epsilon_\\alpha$ denote the weights of basis vectors called \\emph{fundamental weights}. The ordering~\\eqref{eq:tot_ordering_bas_vecs} induces a weight space ordering $\\epsilon_1>\\dots>\\epsilon_{M+N}$\nwhich fixes the simple root system $\\Delta_0=\\{\\alpha_i:=\\epsilon_i-\\epsilon_{i+1}\\}_{i=1}^{M+N-1}$.\nThe choice of grading, which we denote by $\\Sigma=\\{\\sigma_\\alpha = |e_\\alpha|=|\\alpha|\\}_{\\alpha=1}^{M+N}$, is equivalent to the choice of a Cartan matrix, or a Dynkin diagram.\nChanging the grading can be equivalently seen as changing the total ordering~\\eqref{eq:tot_ordering_bas_vecs}. As a result, the simple root system, the positive root system and the Borel subalgebra changes with $\\Sigma$.\nSo, keep in mind that the notion of highest weight always depends on the grading choice and changes when $\\Sigma$ changes.\n\nDefine the operator matrix elements of the monodromies~(\\ref{eq:mon_mat_a}, \\ref{eq:mon_mat_ab}) as\n$T = E_{ij}\\otimes T_{ij}$, where $T_{ij}\\in \\End\\mathcal{C}(L)$ and $T=T_{a},T_{\\bar{a}}$. \nChoosing the reference state $\\Omega$ to be the highest weight state of $\\mathcal{C}(L)$, the eigenvalues of $t(u)$ can be written in terms of polynomials $(T_a)_{ii}(u)\\Omega=\\Lambda_i(u)\\Omega$\n\\begin{equation*}\n \\Lambda_i(u)=\\begin{cases}\n(u+(-1)^{|1|})^L(u+n\/2)^L, & i=1\\\\\n u^L(u+n\/2)^L, & 2\\leq i\\leq r\\\\\nu^L(u+n\/2-(-1)^{|M+N|})^L,& i=M+N\n\\end{cases}\n\\end{equation*}\nand simple root $Q$-polynomials\n\\begin{equation*}\n Q_k(u)=\\prod_{j=1}^{\\nu^{(k)}}(u-u^{(k)}_j)\n\\end{equation*}\nas follows\n\\begin{equation}\\label{eq:spec_trmat}\n \\Lambda(u)=\\sum_{k=1}^{M+N} (-1)^{|k|}\\Lambda_k(u) \\frac{Q_{k-1}(u+(-1)^{|k|})Q_k(u-(-1)^{|k|})}{Q_{k-1}(u)Q_{k}(u)}\n\\end{equation}\nwhere the $u^{(k)}_j$, $k=1,\\dots,r=M+N-1$ are the Bethe roots appearing at the $k$--th step of the nesting. There are solutions of the following system of nested Bethe ansatz equations\n\\begin{equation}\\label{eq:BAE_Qform}\n\\frac{\\Lambda_{k}(u^{(k)}_j)}{\\Lambda_{k+1}(u^{(k)}_j)}=-(-1)^{|k|+|k+1|}\\frac{Q_{k-1}(u^{(k)}_j)Q_k(u^{(k)}_j+(-1)^{|k+1|})Q_{k+1}(u^{(k)}_j-(-1)^{|k+1|})}{Q_{k-1}(u^{(k)}_j+(-1)^{|k|})Q_k(u^{(k)}_j-(-1)^{|k|})Q_{k+1}(u^{(k)}_j)}\\ ,\n\\end{equation}\nensuring the analyticity of eigenvalues~\\eqref{eq:spec_trmat}, where $k=1,\\dots,r$ and $j=1,\\dots,\\nu^{(k)}$ and it has to be understood that $Q_0(u)=Q_{M+N}(u)=1$. We stress that the BAE~\\eqref{eq:BAE_Qform} are equivalent to the analyticity requirement of the transfer matrix~\\eqref{eq:spec_trmat} if and only if none of the Bethe roots of the same type coincide, which is an essential requirement in the algebraic Bethe ansatz construction. \n\n\nThe BAE take a more familiar look~\\cite{Ogievetsky:1986hu}\n\\begin{equation}\\label{eq:BAE_short}\n \\prod_{k=1}^{L} e_{\\langle \\Lambda_{k},\\alpha \\rangle }(x_j^{(\\alpha)}-y_k)e_{\\langle \\Lambda_{\\bar{k}},\\alpha \\rangle }(x_j^{(\\alpha)}-y_{\\bar{k}})=-(-1)^{|\\alpha|}\\prod_{\\beta\\in\\Delta_0}\\prod_{i=1}^{\\nu^{\\alpha}}e_{\\langle\\alpha,\\beta\\rangle}(x_j^{(\\alpha)}-x_i^{(\\beta)})\\ ,\n\\end{equation}\nwhen written in terms of new variables\n\\begin{equation}\\label{eq:shifts}\n iu^{(k)}_{j}= x^{(k)}_j - \\frac{i}{2}\\sum_{l=1}^k(-1)^{|l|}\\ ,\n\\end{equation}\nthe Takahashi functions\n\\begin{equation*}\n e_t(x)=\\frac{x+it\/2}{x-it\/2}\\, ,\n\\end{equation*}\nthe highest weights $\\Lambda_k=\\epsilon_1$ and $\\Lambda_{\\bar{k}}=-\\epsilon_{M+N}$ of modules $V_k$ and $V^*_{\\bar{k}}$ in $\\mathcal{C}(L)$, \nand the weight space scalar product $\\langle \\epsilon_i, \\epsilon_j\\rangle = \\delta_{ij}(-1)^{|i|}$.\nThe degree of a root $\\alpha_k=\\epsilon_k-\\epsilon_{k+1}$ is defined as $|\\alpha_k|=|k|+|k+1|$.\nWe shall be mostly considering the homogeneous case~(\\ref{eq:mon_mat_a}, \\ref{eq:mon_mat_ab}) corresponding to \n$y_k = y_{\\bar{k}}= 0$, although the inhomogeneous deformation $y_k,y_{\\bar{k}}\\neq 0$ shall also be required.\nThere are $j=1,\\dots,\\nu^{\\alpha}$ equations for every $\\alpha\\in \\Delta_0$.\n\nThe weight of the reference state is $\\wt(\\Omega)=\\sum_{k=1}^L\\Lambda_k+\\sum_{\\bar{k}=1}^L\\Lambda_{\\bar{k}}=L(\\epsilon_1-\\epsilon_{M+N})$. Bethe vectors $\\omega$ described by the Bethe roots~\\eqref{eq:BAE_short} are highest weight vectors of weight\n\\begin{equation}\n \\wt(\\omega) = \\wt(\\Omega) - \\sum_{k=1}^r\\alpha_k \\nu^{(k)}\\ .\n\\label{eq:weight_BV1}\n\\end{equation}\n\nWe define the dynamics of the spin chain by the momentum and Hamiltonian operators\n\\begin{equation}\\label{eq:first_charges}\n H = \\frac{d}{du}\\bigg\\vert_{u=0}\\log \\frac{t(u)\\bar{t}(u)}{\\Lambda_1(u)\\bar{\\Lambda}_{M+N}(u)}\\ ,\\qquad \\exp{i P} = (-1)^{|1|+|M+N|}\\frac{t(0)\\bar{t}(0)}{\\Lambda_1(0)\\bar{\\Lambda}_{M+N}(0)}\\ .\n\\end{equation}\nIn order to have explicit expression for the spectrum of these operators, the eigenvalues~\\eqref{eq:spec_trmat} of $t(u)$ are not enough. One also needs to evaluate the eigenvalue of $\\bar{t}(u)$ on a given Bethe eigenvector of $t(u)$.\nA \\emph{fundamental difference} w.r.t. $\\gl(N)$ spin chains is that one cannot solve this problem by fusion. This is because tensor products of the fundamental representation $V$ of $\\sgl(M|N)$ will never generate the dual representation $V^*$ as a direct summand, nor even as a subquotient.\nTo develop a clear idea about how this should be done, let us recall that a set of Bethe vectors for $t(u)$ can be constructed in the framework of the algebraic Bethe ansatz (ABA) by using only the commutation relations~\\eqref{eq:comm1}. Then, the eigenvalue of $\\bar{t}(u)$ on such a Bethe vector can be calculated, in principle, by using the second type of commutation relations~\\eqref{eq:comm2}.\nWe shall not pursue this rather tedious route. Instead, we guess the eigenvalue of $\\bar{t}(u)$ on a given Bethe vector of $t(u)$ as follows.\n\nFirst, notice that a different set of Bethe vectors can be obtained by performing the ABA for $\\bar{t}(u)$, that is by using the commutation relations dual to eq.~\\eqref{eq:comm1}.\nWe perform the nesting by ordering the dual basis vectors $\\{e^\\alpha\\}_{\\alpha=1}^{M+N}$ of the auxiliary space $V^*$ according to their weights $\\wt(e^\\alpha)=-\\epsilon_\\alpha$\n\\begin{equation}\n e^{M+N}>\\dots >e^2>e^1\\ .\n\\label{eq:dual_ord}\n\\end{equation}\nWe denote the Bethe roots appearing at step $k$ of the nested ABA by $\\bar{u}^{(M+N-k)}_j$, because the simple root which must be associated to them is $\\wt(e^{M+N-k+1})-\\wt(e^{M+N-k})=\\alpha_{M+N-k}$.\nThen, the eigenvalues of $\\bar{t}(u)$ can be written in term of polynomials \n$(T_{\\bar{a}})_{ii}(u)\\Omega=\\bar{\\Lambda}_i(u)\\Omega$\n\\begin{equation*}\n \\bar{\\Lambda}_i(u)=\\begin{cases}\n(u-(-1)^{|1|}+n\/2)^L u^L, & i=1\\\\\n (u+n\/2)^Lu^L, & 2\\leq i\\leq r\\\\\n(u+n\/2)^L(u+(-1)^{|M+N|})^L,& i=M+N\n\\end{cases}\n\\end{equation*}\nand simple root polynomials $\\bar{Q}_k(u)=\\prod_{j=1}^{\\bar{\\nu}^{(k)}}(u-\\bar{u}^{(k)}_j)$\nin complete analogy with~\\eqref{eq:spec_trmat}\n\\begin{equation}\n\\bar{\\Lambda}(u)=\\sum_{k=1}^{M+N}(-1)^{|k|}\\bar{\\Lambda}_k(u)\\frac{\\bar{Q}_{k-1}(u-(-1)^{|k|})\\bar{Q}_k(u+(-1)^{|k|})}{\\bar{Q}_{k-1}(u)\\bar{Q}_k(u)}\\ .\n\\label{eq:eg_trmat_bar}\n\\end{equation}\nAnalyticity conditions for $\\bar{\\Lambda}(u)$ take the same form as eqs.~\\eqref{eq:BAE_short} in terms of variables\n\\begin{equation}\n \\bar{u}^{(k)}_j = \\bar{x}^{(k)}_j-\\frac{i}{2}\\sum_{l=k+1}^{M+N}(-1)^{|l|}\n\\label{eq:bshifts}\n\\end{equation}\nand parameters $\\bar{\\nu}^{(k)}$.\nA Bethe vector constructed in this way, which we denote by $\\bar{\\omega}$, has weight \n\\begin{equation*}\n\\wt(\\bar{\\omega})=\\wt(\\Omega)-\\sum_{k=1}^r \\alpha_k \\bar{\\nu^{(k)}}\\ .\n\\end{equation*}\n\nAt this point, the eigenvalues of $t(u)$ w.r.t. the second set of Bethe vectors $\\bar{\\omega}$ is not known. If we can match Bethe vectors $\\omega$ with Bethe vectors $\\bar{\\omega}$ then the problem is solved.\nDue to the subtle completeness issue of ABA for super spin chains, it is not clear at all if the matching can actually be performed.\nWe assume it can and we do it as follows: a Bethe vector $\\omega$ of $t(u)$ is identified with a Bethe vector $\\bar{\\omega}$ of $\\bar{t}(u)$ if\n\\begin{align}\\label{eq:indent_BV}\n&&\\omega =& \\bar{\\omega} & {}&\\Leftrightarrow&\n\\begin{cases}\n   \\nu^{(k)}=\\bar{\\nu}^{(k)}\\\\ \\{x^{(k)}_j\\}_{j=1}^{\\nu^{(k)}}=\\{\\bar{x}^{(k)}_j\\}_{j=1}^{\\bar{\\nu}^{(k)}} &\n \\end{cases} &  k=1,\\dots r \\ .\n\\end{align}\nThe first condition in the braces means $\\wt(\\omega)=\\wt(\\bar{\\omega})$.\nEqs.~(\\ref{eq:spec_trmat}, \\ref{eq:eg_trmat_bar}, \\ref{eq:shifts}, \\ref{eq:bshifts}) allow us to compute both the eigenvalues of $t(u)$ and $\\bar{t}(u)$ on a Bethe vector $\\omega=\\bar{\\omega}$ in~\\eqref{eq:indent_BV}.\n\nAfter this long detour, we come back to the spectrum of the operators~\\eqref{eq:first_charges}, which can now be explicitly computed\n\\begin{align}\\label{eq:spec_ham_mom}\n E &= -\\sum_{j=1}^{\\nu^{(1)}}\\frac{(-1)^{|1|}}{x^{(1)}_j {}^2+1\/4}-\\sum_{j=1}^{\\nu^{(r)}}\\frac{(-1)^{|M+N|}}{x^{(r)}_j{}^2+1\/4}\\\\\nP &\\equiv  \\sum_{i=1}^{\\nu^{(1)}}(-1)^{|1|}\\theta_1(x^{(1)}_j)+ \\sum_{i=1}^{\\nu^{(r)}}(-1)^{|M+N|}\\theta_1(x^{(r)}_j)\\mod 2\\pi \\ ,\n\\end{align}\nwhere $\\theta_t(u)=i \\log e_t(u)+\\pi=2\\tan^{-1}\\tfrac{2u}{t}$ with some fixed branch.\nFirst thing to be noticed is the explicit dependence of the definitions~\\eqref{eq:first_charges} on the grading.\nTherefore, it looks like the type of the chain -- ferromagnetic or antiferromagnetic -- also depends on it, which is also suggested by the grading signs in \\eqref{eq:spec_ham_mom}.\nHowever, the charges $H$ and $P$ can be written explicitly as a representation of an element of the periodic walled Brauer algebra\n\\begin{align}\\label{eq:ham}\n H &= \\sum_{i=1}^L -(-1)^{|1|} -(-1)^{|M+N|}+ P_{ii+1}+ P_{\\overline{i-1i}} -\\frac{2}{n}\\left(\\{ Q_{\\bar{i}i+1},Q_{i\\bar{i}}\\} + \\{Q_{\\overline{i-1}i},Q_{i\\bar{i}}\\}\\right)\n\\end{align}\nwhere all (un)barred indices are defined modulo $L$ and the affine generators are expressed in terms of non-periodic walled Brauer algebra elements $P_{L1}:=P_{1L}=P_{12}\\dots P_{L-1L}\\dots P_{12}$,  $P_{\\overline{L1}}:=P_{\\overline{1L}}=P_{\\overline{12}}\\dots P_{\\overline{L-1L}}\\dots P_{\\overline{12}}$ and $Q_{\\bar{L}1}:=Q_{1\\bar{L}} = P_{1L}Q_{L\\bar{L}}P_{1L}$.\nThe following general formulas have been used\n\\begin{align*}\n \\frac{d\\log t(0)}{du}&=\\sum_{i=1}^L R^{-1}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)\\dot{R}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)+\\sum_{i=1}^{L} R^{-1}_{i\\bar{\\imath}}\\left(\\frac{n}{2}\\right)\\dot{\\check{R}}_{ii+1}(0)R_{i\\bar{\\imath}}\\left(\\frac{n}{2}\\right)\\\\\n  \\frac{d\\log \\bar{t}(0)}{du}&=\\sum_{i=1}^L\\dot{R}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)R^{-1}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)+\\sum_{i=1}^{L}R_{i\\bar{\\imath}}\\left(\\frac{n}{2}\\right)\\dot{\\check{R}}_{\\overline{i\\imath-1}}(0)R^{-1}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)\\ .\n\\end{align*}\nThe latter can be derived using only the cyclicity of the supertrace, which holds for a graded tensor product when even endomorphisms are considered, relations of the form $P_{ij}R_{jk}(u)= R_{ik}(u)P_{ij}$, $P_{ij}R_{j\\bar{k}}(u)= R_{i\\bar{k}}(u)P_{ij}$, $\\str_a P_{aj} = \\id$ and their duals.\nWe see from eq.~\\eqref{eq:ham} that the grading enters in the definition of the Hamiltonian only as a shift, therefore having nothing to do with ferro or antiferromagnetism.\nThe equivalence of the solutions of BAE in different gradings, and therefore of $\\spec H$, is at present somewhat understood in terms of particle hole transformations~\\cite{Tsuboi:1998ne}, although the relationship between Bethe vectors in different gradings not at all.\n\nA relation between the spectrum of $H$ and $-H$ can be constructed, but one has to consider different chains. Let $H_{M|N}$ denote the integrable Hamiltonian of the $\\gl(M|N)$ spin chain. Then on has the following relation\n\\begin{equation}\\label{eq:aut}\n H_{M|N} = - H_{N|M} \\ .\n\\end{equation}\nTo prove it one uses the algebra homomorphism between the walled Brauer algebras $B_{L,L}(n)$ and $B_{L,L}(-n)$ provided by\n$P\\mapsto -P$ and $Q\\mapsto -Q$. This automorphism is realized in the spin chain representation by the shift of the grading function $|i|\\mapsto |i|+1$, which  maps $\\gl(M|N)\\mapsto \\gl(N|M)$.\nAs we shall see in the next section, the sign in front of our Hamiltonian~\\eqref{eq:ham} ensures that we are dealing with antiferromagnetic spin chains for $n=M-N>0$.\nEq.~\\eqref{eq:aut} allows to fold back the $\\gl(N|M)$ spin chains with $N<M$ to $\\gl(M|N)$ spin chains with $n>0$ by changing the sign of the Hamiltonian~\\eqref{eq:ham}, but now they will be ferromagnetic.\nAs the  $\\gl(N|N)$ chain in eq.~\\eqref{eq:ham} is poorly defined, we shall discard it from the main discussion and come back to it only in the conclusions.\nFrom now on we restrict to $\\gl(M|N)$ antiferromagnetic spin chains~\\eqref{eq:ham} for which $n=M-N>0$.\n\n\\section{Numerical diagonalization of the Hamiltonian}\n\\label{sec:num_diag}\n\n\n\n\nDiagonalizing the matrix~\\eqref{eq:ham} is not the smartest thing to do if one is solely interested in its spectrum, especially if one intends to consider on the same footing all the $\\gl(M|N)$--spin chains. This is because many eigenvalues have multiplicities corresponding to the dimension of irreducible $\\gl(M|N)$ representations appearing (generically as quotients of submodules) in the spin chain. These multiplicities quickly grow with the rank and the computing power per eigenvalue increases respectively.\nAn approach which allows to select the eigenvalues corresponding to a given $\\gl(M|N)$--irreducible representation and eliminate the corresponding degeneracy would be considerably more efficient.\nTo implement this approach one interprets the Hamiltonian $H$ in eq.~\\eqref{eq:ham} as an element $\\mathsf{H}$ of some algebra, namely the walled Brauer algebra $B_{L,L}(n=M-N)$, in a particular representation provided by the $\\gl(M|N)$--centralizer of the $\\mathcal{C}(L)$ spin chain.\nAs we shall explain shortly, the algebra $B_{L,L}(n)$ can be abstractly defined independently of the $\\gl(n+N|N)$ spin chains and, in particular, it does not depend on $N$. Centralizers of\n$\\gl(n+N|N)$ spin chains provide $N$ dependent representations for $B_{L,L}(n)$.\nThe important thing is that the representation theory of $B_{L,L}(n)$ is understood well enough, so that one can find the spectrum of the algebraic Hamiltonian $\\mathsf{H}$ by working in other representations where its spectrum is much less\ndegenerate compared to that in the $\\gl(n+N|N)$ spin chain, namely it does not depend on $N$.\n\n\\subsection{Walled Brauer algebra and its standard modules}\n\\label{sec:wb_sub}\n\n\nThe walled Brauer algebra $B_{L,L}(n)$ can be conveniently viewed as a subalgebra of the Brauer algebra $B_{2L}(n)$.\nThe defining relations of $B_{2L}(n)$ can be found in \\cite{ram}.\nThe words of $B_{2L}(n)$ admit a representation as graphs\non $4L$ labeled vertices with $2L$ edges connecting the vertices\npairwise in all $(4L-1)!!$ possible ways (crossings are allowed).\nThe identity $\\mathsf{I}$ of the Brauer algebra and the generators $\\mathsf{E}_i,\\mathsf{P}_i$\nare represented by the graphs on the left in\nfig.~\\ref{fig:gen_br_alg}.\n\n\\begin{figure}[t]%\n\\psfrag{1}{$1$}\n\\psfrag{l}{$L$}\n\\psfrag{d}{$\\cdots$}\n\\psfrag{i}{$i$}\n\\psfrag{i}{$i$}\n\\psfrag{P}{$\\mathsf{P}_i$}\n\\psfrag{I}{$\\mathsf{I}$}\n\\psfrag{E}{$\\mathsf{E}_i$}\n\\psfrag{PPP}{$\\mathsf{P}_{i-1}\\mathsf{P}_{i}\\mathsf{P}_{i+1}$}\n\\centerline{\\includegraphics[scale=0.8]{brauer_diags.eps}}\\caption{The identity $\\mathsf{I}$ and the generators $\\mathsf{E}_i,\\mathsf{P}_i$ of the Brauer algebra of\ndimension\n$(2L-1)!!$ are represented on the left; the walled Brauer algebra generator\n$\\mathsf{P}_{i-1} \\mathsf{P}_{}\\mathsf{P}_{i+1}$\nis represented on the right.}%\n\\label{fig:gen_br_alg}%\n\\end{figure}\n\nIn order to multiply the diagrams one arranges the first $2L$\nvertices horizontally with the remaining $2L$ vertices on top of\nthe first ones. The product of a diagram $d_1$ with a diagram\n$d_2$ is the diagram $d_1 d_2$ obtained by i) placing the diagram\n$d_1$ on top of the diagram $d_2$, ii) identifying the top of the\ndiagram $d_2$ with the bottom of the diagram $d_1$ and iii)\nreplacing every loop generated in this process by $n$.\nThe walled Brauer algebra $B_{L,L}(n)$ is the subalgebra of $B_{2L}(n)$ generated by the elements $\\mathsf{Q}_{i\\bar{\\imath}}:=\\mathsf{E}_{2i-1}$, $\\mathsf{Q}_{\\bar{\\imath}i+1}:=\\mathsf{E}_{2i}$, $\\mathsf{P}_{i,i+1}:= \\mathsf{P}_{2i-1} \\mathsf{P}_{2i}\\mathsf{P}_{2i-1}$, $\\mathsf{P}_{\\overline{\\imath\\imath+1}}:= \\mathsf{P}_{2i} \\mathsf{P}_{2i+1}\\mathsf{P}_{2i}$. The generators\n$\\mathsf{P}_i\\mathsf{P}_{i+1}\\mathsf{P}_i$ are represented on the right in\nfig.~\\ref{fig:gen_br_alg}.\nFor every diagram spanning $B_{2L}(n)$ with vertices labeled as in fig.~\\ref{fig:gen_br_alg}, imagine moving all odd vertices to the left of a wall, all the even ones to the right, while keeping the connectivity unchanged. Then $B_{L,L}(n)$ is spanned by the set of $B_{2L}(n)$ diagrams, such that only strictly horizontal edge cross the wall.\nIn this representation of $B_{L,L}(n)$ we label the  $L$ up and $L$ down vertices on the left of the wall by the set $1,\\dots,L$ from left to right and similarly those on the right by the set $\\bar{1},\\dots,\\bar{L}$.\nThe $\\mathsf{P}_{ii+1}$ generators act on the left of the wall, the $\\mathsf{P}_{\\overline{\\imath\\imath+1}}$ act on the right, while the  generators  $\\mathsf{Q}_{i\\bar{\\imath}}$ and $\\mathsf{Q}_{\\bar{\\imath}i+1}$ act across the wall.\n\n\nNext we give a brief description of a set of modules of $B_{L,L}(n)$ over which we actually numerically diagonalize the algebraic Hamiltonian $\\mathsf{H}$.\nThese modules shall be related in the following to $\\gl(n+N|N)$ \\emph{traceless tensors} of fixed co-- and contravariant shapes.\nFor $\\lambda$ and $\\mu$ partitions of the non-negative integers $|\\lambda|,|\\mu|=f\\leq L$, we denote by $\\Delta_{L,L}(\\lambda,\\mu)$ the \\emph{standard modules} of $B_{L,L}(n)$.\nThere are constructed in the following way.\nLet $\\Sym(f)$ denote the symmetric group on $f$ objects and $S(\\lambda)$, $S(\\mu)$ denote its simple modules labeled by the corresponding partitions.\nThen, $\\Delta_{L,L}(\\lambda,\\mu)$ has as basis the tensor products between the set of some diagrams on $2L$ points, with $L$ of them on the either side of the wall, and some basis of $S(\\lambda)$ and $S(\\mu)$.\nThe diagrams are such that every point is either free or belongs to a horizontal edge crossing the wall. The number of edges is fixed to $L-f$.\nWe give a rough idea about how the action of the walled Brauer algebra can be constructed by diagrammatic multiplication in in fig.~\\ref{fig:mult_diag}: i) in a first step, before the diagrammatic multiplication, assign labels to the free points of the diagrams on each side of the wall; ii) in a second step, after the diagrammatic multiplication, apply a surjective homomorphism from labeled diagrams to the tensor product of unlabeled diagrams with $S(\\lambda)$ and $S(\\mu)$, that is to $\\Delta_{L,L}(\\lambda,\\mu)$. All labeled diagrams with more then $L-f$ horizontal edges that can appear as a result of the diagrammatic multiplication belong to the kernel of this homomorphism.\nIn particular, $\\Delta_{f,f}(\\lambda,\\mu)\\simeq S(\\lambda)\\times S(\\mu)$.\nThe detailed definitions can be found in \\cite{martin}.\n\\begin{figure}%\n\\psfrag{x}{$\\times$}\n\\psfrag{o}{$\\otimes$}\n\\psfrag{v}{$\\mathbf{v}$}\n\\psfrag{u}{$((2,1), (1,2))\\cdot \\mathbf{v}$}\n\\psfrag{1}{$1$}\n\\psfrag{2}{$2$}\n\\psfrag{f}{$\\phantom{fffff}$}\n\\centerline{\\includegraphics[scale=0.7]{mult_diag.eps}}\n\\caption{Example of diagrammatic multiplication of a basis element of $\\Delta_{3,3}(\\lambda,\\mu)$, where $\\lambda,\\mu\\vdash 2$ and $\\mathbf{v}$ is an element of the module $S(\\lambda)\\times S(\\mu)$ of $\\Sym(2)\\times\\Sym(2)$.}\n\\label{fig:mult_diag}\n\\end{figure}\n\n\n\nWe implement the action of $B_{L,L}(n)$ in $\\Delta_{L,L}(\\lambda,\\mu)$ on a computer and investigate the spectrum of the algebraic Hamiltonian $\\mathsf{H}$. Before presenting and discussing the numerical results of sec.~\\ref{sec:spec} one should explain how to extract from these data the spectrum of the original spin chain Hamiltonian $H$.\n\n\n\\subsection{Traceless tensors}\\label{sec:tens}\n\nConsider the $\\gl(M|N)$ tensor $(V\\otimes V^*)^{\\otimes f}$.\nThere is an obvious action of $\\Sym(f)\\times\\Sym(f)$ on the $V$ and $V^*$ factors.\nLet $\\lambda,\\mu$ be partitions of $f$, which we symbolically write as $\\lambda,\\mu\\vdash f$.\nOne can apply a Young symmetrizer of shape $\\lambda$ to the $V$ factors and a Young symmetrizer of shape $\\mu$ to the $V^*$ factors to get a tensor of shape $t(\\lambda,\\mu)$ \\cite{Weyl}.\nWe say that $t(\\lambda,\\mu)$ has rank $(f,f)$, covariant shape $\\lambda$ and contravariant shape $\\mu$ or, shortly, shape $(\\lambda,\\mu)$.\nThe number of Young symmetrizers of shape $\\lambda$ is equal to the \nnumber of standard Young tableau of shape $\\lambda$, which is also equal to $\\dim S(\\lambda)$.\nTherefore, the number of tensors of shape $(\\lambda,\\mu)$ is $\\dim S(\\lambda)\\times \\dim S(\\mu)$.\nEvery tensor $t(\\lambda,\\mu)$ is a $\\gl(M|N)$ module, which appears as a direct summand in $(V\\otimes V^*)^{\\otimes f}$.\nThe symmetric group $\\Sym(f)\\times\\Sym(f)$ acts in the space of tensors of the same shape $(\\lambda,\\mu)$, transforming them into each other. The latter subspace is isomorphic to a direct sum of $S(\\lambda)\\otimes S(\\mu)$ modules of $\\Sym(f)\\times\\Sym(f)$. \nThese statements can be compactly written as follows\n\\begin{equation*}\n(V\\otimes V^*)^{\\otimes f}\\mathop{\\simeq}_{\\Sym(f)\\times \\Sym(f)}\\bigoplus_{\\lambda,\\mu \\vdash f}\\dim t(\\lambda,\\mu)S(\\lambda)\\otimes S(\\mu)\\ .\n\\end{equation*}\n\n\n\nConsider now the subspace $t_0(\\lambda,\\mu)\\subset t(\\lambda,\\mu)$ of traceless tensors. Notice that this is a $\\gl(M|N)$\nsubmodule, which will  \\emph{not} necessarily be \\emph{a direct summand} of $t(\\lambda,\\mu)$.\nMore generally, one can consider the $\\gl(M|N)$ submodules $t_{n-1}(\\lambda,\\mu)\\subset t(\\lambda,\\mu)$ composed of tensors whose all contractions of $n$ covariant indices with $n$ contravariant indices vanish.\nThese provide a filtration of the tensor $t(\\lambda,\\mu)$\n\\begin{equation}\n t_0(\\lambda,\\mu)\\subset t_1(\\lambda,\\mu)\\subset \\cdots \\subset t_f(\\lambda,\\mu)=t(\\lambda,\\mu)\\ .\n\\label{eq:filtr}\n\\end{equation}\nIt is very important to observe that the subquotients $t_{n}(\\lambda,\\mu)\/t_{n-1}(\\lambda,\\mu)$ of this filtration are isomorphic to traceless tensors of lower rank $(f-n,f-n)$.\nFor instance, taking a trace of $t_1(\\lambda,\\mu)$ one gets a traceless tensor of rank $f-1$ because, by definition, all double contractions of $t_1(\\lambda,\\mu)$ must vanish. Now, the preimage of single traces of $t_1(\\lambda,\\mu)$ modulo the kernel $t_0(\\lambda,\\mu)$ of the single trace homomorphisms is precisely to $t_1(\\lambda,\\mu)\/t_0(\\lambda,\\mu)$. Therefore, $t_1(\\lambda,\\mu)\/t_0(\\lambda,\\mu)$ is isomorphic to traceless tensors of rank $(f-1,f-1)$.\n\n\nWe see that traceless tensors $t_0(\\lambda,\\mu)$ have a fundamental role --- all direct summands of the spin chain $(V\\otimes V^*)^{\\otimes L}$ can be built out of them.\nAs a consequence, the full spectrum of the spin chain Hamiltonian can be reconstructed from the spectra in the subspaces of traceless tensors of shape $\\lambda,\\mu\\vdash f$, $f=0,1,\\dots,L$.\nIn fact, not all of these shapes are possible. We shall determine the class of shapes for which the traceless tensors do not vanish later.\n\n\nThe traceless tensors are not necessarily irreducible. One way to build submodules of a traceless tensor $t_0(\\lambda,\\mu)$ is by embedding into it quotients of traceless tensors of lower rank as follows. Let $t_0(\\lambda',\\mu')$ be a traceless tensor, $\\lambda',\\mu'\\vdash f-k$, $\\lambda'\\subset \\lambda$, $\\mu'\\subset \\mu$ and $\\mathsf{e}_{\\lambda},\\mathsf{e}_\\mu$ denote some Young symmetrizers of shape $\\lambda$, $\\mu$. The tensor  \n\\begin{equation*}\n\\mathsf{e}_{\\lambda}\\mathsf{e}_\\mu t_0(\\lambda',\\mu')\\otimes \\left((V\\otimes V^*)^{\\otimes k}\\right)^{\\gl(M|N)} \\subset t(\\lambda,\\mu)\n\\end{equation*}\nmight have an intersection with a non-trivial submodule of $t_0(\\lambda,\\mu)$. The latter will generally be isomorphic to only a quotient of $t_0(\\lambda',\\mu')$, because the Young symmetrizers $\\mathsf{e}_{\\lambda},\\mathsf{e}_\\mu$ are projectors.\nAn illustrative example is the indecomposable $\\gl(N|N)$ tensor $t(1,1)= V\\otimes V^*$. The traceless subspace $t_0(1,1)$, isomorphic to the adjoint representation, is spanned by elements of the form $G^i_j e_i\\otimes e^j$ subject to the constraint $\\str G = G^i_i (-1)^{|i|}=0$, where $\\{e_i\\}_{i=1}^{2N}$ is a basis of $V$ and $\\{e^i\\}_{i=1}^{2N}$ is the dual basis. The quotient $t_1(1,1)\/t_0(1,1)$ is one dimensional. A coset representative for this quotient is, for instance, $(-1)^{|i|}e_i\\otimes e^i$.\nThe traceless tensor $t_0(1,1)$ is also indecomposable, but reducible.\nIt has a unique proper non-trivial submodule  spanned by the $\\gl(N|N)$ traceless invariant $e_i\\otimes e^i$.\n\n\nRepresent now the covariant part of every tensor $t_0(\\lambda,\\mu)$ of fixed shape $(\\lambda,\\mu)$ and ranks $(f,f)$ by $f$ dots on the left of an imaginary wall and the contravariant part by $f$ dots on the right of that wall.\nThen the walled Brauer algebra generators $\\mathsf{P}_{ii+1}$ will act on the left of the wall as in $S(\\lambda)$ and the $\\mathsf{P}_{\\overline{\\imath\\imath+1}}$ generators will act on the right as in $S(\\mu)$ by transforming $\\dim S(\\lambda)\\times \\dim S(\\mu)$ different traceless tensors of shape $(\\lambda,\\mu)$ into each other. The  generators  $\\mathsf{Q}_{i\\bar{\\imath}}$ and $\\mathsf{Q}_{\\bar{\\imath}i+1}$ will act across the wall by contracting a covariant index with a contravariant one. In view of the tracelessness condition this action is trivial.\nThus, the space of all traceless tensors $t_0(\\lambda,\\mu)$ of the same shape $(\\lambda,\\mu)$ is isomorphic to a direct sum of $\\dim t_0(\\lambda,\\mu)$ modules $\\Delta_{f,f}(\\lambda,\\mu)\\simeq S(\\lambda)\\times S(\\mu)$ of the walled Brauer algebra.\nA very important observation is the \\emph{triviality of the centralizer} of a traceless tensor\n\\begin{equation}\n \\End_{\\gl(M|N)} t_0(\\lambda,\\mu)\\simeq \\mathbb{C}\\ .\n\\label{eq:centralizer_simple}\n\\end{equation}\nThis is a generalization of the Schur lemma for $\\gl(n)$ traceless tensors, which are irreducible. The statement~\\eqref{eq:centralizer_simple} follows immediately from the action of the walled Brauer algebra in the space of traceless tensors of shape $(\\lambda,\\mu)$ that we have just described. \nIt means that traceless tensors are a very special type of indecomposables, namely any $\\gl(M|N)$ Casimir is diagonalizable and proportional to the identity in a traceless tensor. \nIt should be noticed that this is typical of highest weight or Kac modules \\cite{Kac77a, Kac77b}.   \n\\begin{assumption}{1}\\label{ass:hw}\n Traceless tensors are highest weight modules.\n\\end{assumption}\n\\noindent This means that there is a Borel subalgebra $\\mathfrak{b}$ of $\\gl(M|N)\\simeq \\mathfrak{b}\\oplus \\mathfrak{n}^-$ and a $\\mathfrak{b}$--highest weight vector $\\mathbf{v}\\in t_0(\\lambda,\\mu)$  such that the full tensor $t_0(\\lambda,\\mu)$ can be generated from $\\mathbf{v}$ by repeated action of $\\mathfrak{n}^-$. We shall see later how to choose $\\mathfrak{b}$ for given $t_0(\\lambda,\\mu)$.\n\nConsider now the vector space $\\delta_{L,L}(\\lambda,\\mu)$ of all possible embeddings  of traceless tensors of shape $(\\lambda,\\mu)$ and ranks $(f,f)$ into the spin chain $(V\\otimes V^*)^{\\otimes L}$. It consists of tensor products of tensors $t_0(\\lambda,\\mu)$ with  $\\gl(M|N)$--invariants of $(V\\otimes V^*)^{\\otimes (L-f)}$.\nNotice that there is a unique $\\gl(M|N)$ invariant in $V\\otimes V^*$, which can be written as   $e_i\\otimes e^i$.\nRepresenting this invariant by an edge  connecting two vertices across the wall and the traceless tensors $t_0(\\lambda,\\mu)$ as we did before,\nwe can visualize $\\delta_{L,L}(\\lambda,\\mu)$ as a diagram with $L$ vertices on each side of the wall and $L-f$ edges connecting pairs of vertices across the wall.\nThus, we reconstruct the same diagrammatic representation of $\\delta_{L,L}(\\lambda,\\mu)$ as for $\\Delta_{L,L}(\\lambda,\\mu)$. \nThis proves that  all the relations between the generators of $B_{L,L}(n)$ satisfied in $\\Delta_{L,L}(\\lambda,\\mu)$\nwill be satisfied in $\\delta_{L,L}(\\lambda,\\mu)$ as well.\nThe converse is generally not true, meaning that $\\delta_{L,L}(\\lambda,\\mu)$ is generally only a quotient of ($\\dim t_0(\\lambda,\\mu)$ direct sums of) $\\Delta_{L,L}(\\lambda,\\mu)$.\nWe stress that neither $\\delta_{L,L}(\\lambda,\\mu)$ nor $\\Delta_{L,L}(\\lambda,\\mu)$ are necessarily simple $B_{L,L}(n)$ modules and, therefore, the quotient can be non-trivial.\n\n\n\nThe ABA in some grading $\\Sigma$ provides $\\gl(M|N)$  Bethe eigenvectors of highest weight with respect to the Borel subalgebra $\\mathfrak{b}_\\Sigma$ determined by $\\Sigma$ and the ordering~\\eqref{eq:tot_ordering_bas_vecs}.\nTherefore, in order to match the numerical spectrum of $\\mathsf{H}$ in $\\Delta_{L,L}(\\lambda,\\mu)$ with the exact spectrum of $H$ by the ABA we need to know the highest weight of a traceless tensor $t_0(\\lambda,\\mu)$ at least in one grading $\\Sigma$. We explain below how to evaluate it.\n\nConsider the Young diagrams corresponding to the shape $(\\lambda,\\mu)$ of a \\emph{full} tensor $t(\\lambda,\\mu)$.\nThe basis vectors in the tensor subspace of co(ntra)variant shape $\\lambda\\, (\\mu)$ can be represented by co(ntra)variant Young supertableaux of shape $\\lambda\\, (\\mu)$, that is Young diagrams of shape $\\lambda\\, (\\mu)$ with a fundamental weight $\\epsilon_i\\, (-\\epsilon_j)$ inscribed in every box.\nThe pattern of weights within the Young diagrams must satisfy the supersymmetrization rules, that is i) in the same row the bosonic (fermionic) weights are weakly (strongly) ordered w.r.t. each other, ii) in the same column the bosonic (fermionic) weights are strongly (weakly) ordered w.r.t. each other\nand iii) bosonic weights are weakly (strongly) ordered w.r.t. fermionic weight in the same row (column).\nThe weight of a supertableau is the sum of all weights it carries in its boxes.\n\nBoth $\\lambda$ and $\\mu$ must fit into a hook whose horizontal (vertical) arm is of width $M\\, (N)$, otherwise $t(\\lambda,\\mu)$ vanishes identically because it is not possible to fill in the Young diagrams and get Young supertableaux compatible with the supersymmetrization rules.\nThe highest weight of a supertableau depends on the grading.\nThe choice of grading is a splitting of the set of basis vectors into two sets $B\\, (F)=\\{\\epsilon_i\\mid |i|\\equiv 0\\, (1)\\}^{<}$ which are ordered w.r.t. the total ordering~\\eqref{eq:tot_ordering_bas_vecs}.\nEquivalently, it can be represented by paths connecting the two corners of the $(M,N)$--hooks as represented in fig.~\\ref{fig:young}.\nFix these paths and consider Young diagrams $\\lambda$, $\\mu$\nwith rows $\\lambda_i$, $\\mu_i$ and columns $\\lambda'_i$, $\\mu'_i$.\nThen, the highest weight of $t(\\lambda,\\mu)$ can be written in the following form\n\\begin{equation}\\label{eq:weight_tensor}\n \\Lambda_\\Sigma(\\lambda,\\mu) = \\sum_{i=1}^M [r_i\\epsilon_{b(i)}-\\bar{r}_i\\epsilon_{\\bar{b}(i)}]+\\sum_{i=1}^N  [c_i\\epsilon_{f(i)} -\\bar{c}_i\\epsilon_{\\bar{f}(i)}]\n\\end{equation}\nwhere $b(i)$ and $f(i)$ are the elements at position $i$ in the ordered sets $B$ and $F$,\n $\\bar{b}(i)$ and $\\bar{f}(i)$ are the elements at position $i$ in the ordered sets $-B$ and $-F$, \n $r_i=\\max(0,\\, \\lambda_i - \\sum_{j=1}^{b(i)}(1-(-1)^{|i|})\/2)$, $c_i=\\max(0,\\, \\lambda'_i- \\sum_{j=1}^i (1-(-1)^{|i|})\/2$,\n$\\bar{r}_i = \\max(0,\\, \\mu_i-\\sum_{i=\\bar{b}(i)}^M(1-(-1)^{|i|})\/2 )$ and $\\bar{c}_i = \\max(0,\\, \\mu'_i-\\sum_{i=\\bar{f}(i)}^N(1+(-1)^{|i|})\/2 )$\nare number of boxes in a row or column of $\\lambda$ or $\\mu$ overpassing the grading paths as represented in fig.~\\ref{fig:young}.\n\\begin{figure}%\n\\psfrag{mu}{$\\mu$}\n\\psfrag{lam}{$\\lambda$}\n\\psfrag{r1}{$r_1$}\n\\psfrag{r2}{$r_2$}\n\\psfrag{r3}{$r_3$}\n\\psfrag{r4}{$r_4$}\n\\psfrag{r5}{$r_5$}\n\\psfrag{x1}{$\\bar{r}_1$}\n\\psfrag{x2}{$\\bar{r}_2$}\n\\psfrag{x3}{$\\bar{r}_3$}\n\\psfrag{x4}{$\\bar{r}_4$}\n\\psfrag{x5}{$\\bar{r}_5$}\n\\psfrag{c1}{$c_1$}\n\\psfrag{c2}{$c_2$}\n\\psfrag{c3}{$c_3$}\n\\psfrag{c4}{$c_4$}\n\\psfrag{y1}{$\\bar{c}_1$}\n\\psfrag{y2}{$\\bar{c}_2$}\n\\psfrag{y3}{$\\bar{c}_3$}\n\\psfrag{y4}{$\\bar{c}_4$}\n\\psfrag{e1}{$\\epsilon_1$}\n\\psfrag{e2}{$\\epsilon_2$}\n\\psfrag{e3}{$\\epsilon_3$}\n\\psfrag{e4}{$\\epsilon_4$}\n\\psfrag{e5}{$\\epsilon_5$}\n\\psfrag{e6}{$\\epsilon_6$}\n\\psfrag{e7}{$\\epsilon_7$}\n\\psfrag{e8}{$\\epsilon_8$}\n\\psfrag{e9}{$\\epsilon_9$}\n\\psfrag{d1}{$-\\epsilon_1$}\n\\psfrag{d2}{$-\\epsilon_2$}\n\\psfrag{d3}{$-\\epsilon_3$}\n\\psfrag{d4}{$-\\epsilon_4$}\n\\psfrag{d5}{$-\\epsilon_5$}\n\\psfrag{d6}{$-\\epsilon_6$}\n\\psfrag{d7}{$-\\epsilon_7$}\n\\psfrag{d8}{$-\\epsilon_8$}\n\\psfrag{d9}{$-\\epsilon_9$}\n\\psfrag{s}{$\\Sigma$}\n\\psfrag{v}{$v_\\Sigma(\\lambda)$}\n\\psfrag{bv}{$\\bar{v}_\\Sigma(\\mu)$}\n\\psfrag{bh}{$\\bar{h}_\\Sigma(\\mu)$}\n\\psfrag{h}{$h_\\Sigma(\\lambda)$}\n\\centerline{\\includegraphics[scale=1.3]{SAVE.eps}}\n\\caption{Covariant Young tableau $\\lambda$ and contravariant Young tableau $\\mu$ of $(5,4)$--hook shape for $\\gl(5|4)$. Bosonic fundamental weights belong to $B=\\{\\epsilon_2,\\epsilon_3,\\epsilon_5,\\epsilon_6,\\epsilon_9\\}$, while fermionic weights  to $F=\\{\\epsilon_1,\\epsilon_4,\\epsilon_7,\\epsilon_8\\}$. Correspondingly $-B=\\{-\\epsilon_9,-\\epsilon_6,-\\epsilon_5,-\\epsilon_3,-\\epsilon_2\\}$ and $-F=\\{-\\epsilon_8,-\\epsilon_7,-\\epsilon_4,-\\epsilon_1\\}$. The values of $r_4$, $r_5$, $c_3$, $c_4$, $\\bar{r}_2,\\dots,\\bar{r}_5$, $\\bar{c}_3$, $\\bar{c}_4$ are zero.}\n\\label{fig:young}\n\\end{figure}\n\nThe highest weight component of a tensor $t(\\lambda,\\mu)$ w.r.t. the grading $\\Sigma$ will belong to the traceless subspace $t_0(\\lambda,\\mu)$ if the highest weight Young supertableau of shape $(\\lambda,\\mu)$ does not contain some fundamental weight $\\pm \\epsilon_i$ in both $\\lambda$ and $\\mu$.\nOtherwise, the corresponding highest weight component of $t(\\lambda,\\mu)$ w.r.t. the grading $\\Sigma$ will either i) not belong to $t_0(\\lambda,\\mu)$ or ii) generate a submodule of $t_0(\\lambda,\\mu)$ isomorphic to the embedding of a quotient of a lower rank tensor $t_0(\\lambda',\\mu')$. If the latter case holds, then the possible Young diagrams $(\\lambda',\\mu')$ are obtained from the Young diagrams $(\\lambda,\\mu)$ by removing pairs of boxes from the highest weight Young supertableau of shape $(\\lambda,\\mu)$: a box of $\\lambda$ carrying some weights $\\epsilon_i$ together with a box of $\\mu$ carrying the opposite weight $-\\epsilon_i$.\nMoreover, in the case ii) the highest weight of the top~\\footnote{The top of a module is the quotient by the intersection of all maximal ideals. The top of a Kac module is the irreducible representation of highest weight.} of $t_0(\\lambda,\\mu)$ will be smaller then the highest weight of the submodule $t_0(\\lambda',\\mu')$.\nTherefore, $t_0(\\lambda,\\mu)$ cannot be a Kac module w.r.t. $\\mathfrak{b}_\\Sigma$.\n\nWe call $\\gl(M|N)$--\\emph{admissible} the shapes $(\\lambda,\\mu)$ for which the $\\gl(M|N)$ traceless tensors $t_0(\\lambda,\\mu)$ neither vanish nor are isomorphic to lower rank traceless tensors.\nWe are now ready to answer the very important question: what are the admissible shapes of traceless tensors?\nAccording to the previous discussion on the highest weight component of a tensor $t(\\lambda,\\mu)$, a shape $(\\lambda,\\mu)$ is admissible if there \\emph{exists} a grading $\\Sigma$ such that \nthe highest weight Young supertableau of shape $(\\lambda,\\mu)$ does not contain any fundamental weight $\\pm \\epsilon_i$ both in $\\lambda$ and in $\\mu$.\nConsequently, for a traceless tensor $t_0(\\lambda,\\mu)$ of admissible shape there is a grading $\\Sigma$ and a corresponding highest weight~\\eqref{eq:weight_tensor} such that  no cancellation between the $r_i$ and $\\bar{r}_i$ or $c_i$ and $\\bar{c}_i$ terms can occur.\nIf $v_\\Sigma(\\lambda)$, $\\bar{v}_\\Sigma(\\mu)$ denote the number of nonzero ``reduced rows'' $r_i$, $\\bar{r}_i$ and $h_\\Sigma(\\lambda)$, $\\bar{h}_\\Sigma(\\mu)$ denote the number of non-zero ``reduced'' columns\n$c_i$, $\\bar{c}_i$, then one must have\n\\begin{equation}\n\\exists \\Sigma \\quad \\text{such that} \\quad v_\\Sigma(\\lambda)+\\bar{v}_\\Sigma(\\mu)\\leq M,\\qquad h_\\Sigma(\\lambda)+\\bar{h}_\\Sigma(\\mu)\\leq N\n\\label{eq:restr_form}\n\\end{equation}\nfor a $\\gl(M|N)$--admissible shape $(\\lambda,\\mu)$, as represented in fig.~\\ref{fig:young}\nThese admissible shapes nicely reduce to hook shapes for purely covariant or contravariant $\\gl(M|N)$ tensors and to staircases for $\\gl(n)$ traceless tensors \\cite{staircase}.\n\nWe say that a shape $(\\lambda,\\mu)$ of a traceless tensor $t_0(\\lambda,\\mu)$ is $\\Sigma$--admissible if the inequalities in eq.~\\eqref{eq:restr_form} are satisfied.\nLet $K_\\Sigma(\\Lambda)$ be the Kac module of highest weight $\\Lambda$ w.r.t. $\\mathfrak{b}_\\Sigma$. \nWe can  make now assumption~\\ref{ass:hw} more precise.\n\\begin{assumption}{1$'$}\\label{ass:hw2}\n The following isomorphism holds for $\\Sigma$--admissible shapes $(\\lambda,\\mu)$\n\\begin{equation*}\n t_0(\\lambda,\\mu)\\simeq K_\\Sigma(\\Lambda_\\Sigma(\\lambda,\\mu))\\ .\n\\end{equation*}\n\\end{assumption}\n\\noindent \nThis assumption in combination with the general theory of Kac modules \\cite{Kac77a, Kac77b} is very useful for counting highest weight vectors. Namely, if $(\\lambda,\\mu)$ is $\\Sigma$--admissible then the number of highest weight vectors in $t_0(\\lambda,\\mu)$ w.r.t. $\\mathfrak{b}_\\Sigma$ is equal to the number of irreducible subquotients.\nNoticing that a highest weight vector cannot generate more then a highest weight module, we prove the following claim in app.~\\ref{sec:proofs}.\n\\begin{claim}\\label{claim:cv}\nHighest weight vectors of $(V\\otimes V^*)^{\\otimes L}$ w.r.t. any Borel subalgebra belong to submodules isomorphic to traceless tensors $t_0(\\lambda,\\mu)$, $\\lambda,\\mu\\vdash f=0,1,\\dots,L$ of admissible shape.\n\\end{claim}\n\nTo sum up, we have explained the connexion between traceless tensors $t_0(\\lambda,\\mu)$ of admissible shapes $(\\lambda,\\mu)$ and standard modules $\\Delta_{L,L}(\\lambda,\\mu)$. Secondly, if $(\\lambda,\\mu)$ is $\\Sigma$-admissible, then eq.~\\eqref{eq:weight_tensor} allows to compute the highest weight of $t_0(\\lambda,\\mu)$ w.r.t. the Borel subalgebra $\\mathfrak{b}_\\Sigma$. Evaluating the highest weight of $t_0(\\lambda,\\mu)$ w.r.t. arbitrary Borel subalgebras is more delicate, mainly because of indecomposability issues.\nFinally, claim~\\ref{claim:cv} indicates a natural relationship between traceless tensors and Bethe vectors, which have the highest weight property, constructed in the framework of ABA.\n\n\n\\subsection{Spectrum}\n\\label{sec:spec}\n \n\nWe present the spectrum of $\\mathsf{H}$ in various $\\Delta_{L,L}(\\lambda,\\mu)$ standard modules of $B_{L,L}(n)$ in tab.~\\ref{tab:1}.\n\\begin{table}[t]\\label{tab:1}\n\\begin{center}\n\\caption{Lowest eigenvalue of $\\mathsf{H}$ in $\\Delta_L(\\lambda,\\mu)$ with opposite sign.}\n\\begin{tabular}{|c|c|ccccccc|}\n\\hline\n $L$& $n$ & $(1^2,2)$ & $(2,2)$ & $(1^3,21)$ & $(1^3,3)$ & $(21,21)$ & $(21,3)$ & $(3,3)$\\\\ \\hline\n \\multirow{8}{*}{5} & 1 &\n$32.130$   & $29.797$ & $29.461$ & $27.453$ & $27.156$  & $24.101$ & $21.446$  \\\\ \n&  &\n$\\pm0.591 i$&            &         &           &          &             &  \\\\ \n& 2 &\n$23.237$    & $21.904$ & $22.626$  & $20.498$ & $20.506$    & $17.999$   & $15.548$ \\\\ \n&  &\n$\\pm 0.339i$&           &           &           &          &          & \\\\\n& 3 &\n$20.763$    & $20.130$ & $20.554$ & $18.530$ & $18.762$  & $16.572$      &  $14.472$ \\\\\n&   &\n              &           &          &          &        &       &  \\\\\n& 4 &\n$19.968$ & $19.487$ & $19.588$ & $17.682$ & $18.139$   &$16.180$  &  $14.153$ \\\\\n&  &\n           &            &          &           &           &             &  \\\\ \\hline\n \\multirow{8}{*}{6} & 1 &\n41.558 & 39.639 & 38.640 &        36.835 & 36.559 & 33.762&  31.264  \\\\ \n&  &\n             &       &  $\\pm 0.355 i$ &  & $\\pm 0.513 i$&   $\\pm 0.127i$ &  \\\\ \n& 2 & 29.537  & 28.282 & 28.802  & 26.976 & 26.984 & 24.784& 22.607 \\\\ \n&  &          &           & $\\pm 0.267 i$ &       & $\\pm 0.426 i$  &  $\\pm 0.148i$  & \\\\\n& 3 & 26.123 & 25.600 & 25.884  & 24.210 & 24.365&  &  20.713\\\\\n&  &          &         & $\\pm 0.194i$ &     &  $\\pm0.258i$& 22.550  &  \\\\\n& 4 & 24.917 & 24.557 & 24.566 & 23.031 &23.416&  21.800 &  20.021\\\\\n&  &         &            &  $\\pm 0.129i$ &      &   &  &  \\\\ \\hline\n\n\\end{tabular}\n\\end{center}\n\\end{table}\nAt a first glance, it appears that the vacuum always lies in $\\Delta_{L,L}(\\emptyset,\\emptyset)$.\nTo check more thoroughly this vacuum hypothesis we need an additional assumption on the spectrum.\nConsider the spectral sets of $\\mathsf{H}$ defined as\n\\begin{equation*}\n\\spec f=\\bigcup_{\\lambda,\\mu\\vdash f } \\spec\\Delta_{L,L}(\\lambda,\\mu)\\ .\n\\end{equation*}\nNotice from tab.~\\ref{tab:1} that the lowest eigenvalue in $\\spec f$ always lies in $\\Delta_{L,L}(1^f,1^f)$, where $1^f$ denotes the Young diagram with a single column of length $f$. We have checked this observation extensively \\textbf{(more details)}.\n\\begin{assumption}{2}\\label{ass:ord}\n The lowest eigenvalues of $\\mathsf{H}$ in $\\spec f$ always lies in $\\Delta_{L,L}(1^f,1^f)$.\n\\end{assumption}\n\\noindent Comparing only the lowest eigenvalues in $\\Delta_{L,L}(1^f,1^f)$ allows us to gain several spin chain length units and check the vacuum hypothesis further, see tab.~\\ref{tab:2}.\n\n\n\n\n\n\n\n\\begin{table}\\label{tab:2}\n\\begin{center}\n\\caption{Lowest eigenvalue of $\\mathsf{H}$ in $\\Delta_L(1^k,1^k)$ with opposite sign.}\n\\begin{tabular}{|c|c|ccccccccc|}\n\\hline\n $L$& $n$ & $k=0$ & $k=1$ & $k=2$ & $k=3$ & $k=4$ & $k=5$ & $k=6$ & $k=7$ & $k=8$\\\\ \\hline\n \\multirow{4}{*}{5} & 1 & 40.000 & 38.846 & 36.564 & 32.710 & 26.385& 20.000 & & & \\\\ \n& 2 & 28.062 & 26.369 & 26.156 & 25.082 & 22.606& 20.000 & &  &\\\\ \n& 3 & 24.625 & 22.950 & 22.944 & 22.640 & 21.407& 20.000 &  &  & \\\\\n& 4 & 23.123 & 21.631 & 21.456 & 21.456 & 20.828& 20.000 &  &  & \\\\ \\hline\n\\multirow{4}{*}{6} & 1 & 48.000 & 46.723 & 45.626 & 41.538 & 36.782& 30.397 & 24.000 &  & \\\\ \n& 2 & 33.550 & 32.126 & 32.126 & 30.902 & 29.253& 26.647 & 24.000 &  & \\\\ \n& 3 & 29.388 & 27.989 & 28.024 & 27.617 & 26.885& 25.476 & 24.000 &  & \\\\ \n& 4 & 27.574 & 26.339 & 26.146 & 26.086 & 25.754& 24.921 & 24.000 &  & \\\\ \\hline\n \\multirow{4}{*}{7} & 1 & 56.000 & 55.134 & 53.631 & 50.894 & 46.015 & 40.796 & 34.399 & 28.000 &\\\\ \n& 2 & 39.054 & 37.826 & 37.699 & 36.992 & 35.260 & 33.307 & 30.660 & 28.000 & \\\\ \n& 3 & 34.172 & 32.971 & 32.939 & 32.748 & 31.954& 30.981 & 29.504 & 28.000 & \\\\\n& 4 & 32.046 & 30.991 & 30.800 & 30.800 & 30.420 & 29.885 & 28.962 & 28.000 & \\\\ \\hline\n\\multirow{4}{*}{8} & 1 & 64.000 & 63.035 & 63.296 & 59.127 & 55.589 & 50.296  & 44.799 & 38.400 & 32.000\\\\ \n& 2 & 44.569 & 43.488 & 43.488 & 42.533 & 41.457 & 39.453 & 37.325 & 34.664 & 32.000\\\\ \n& 3 & 38.970 & 37.917 & 39.480 & 37.652 & 37.146& 36.134 & 35.019 & 33.515 & 32.000\\\\ \n& 4 &  36.530    & 35.611 & 35.428 & 35.391 & 35.169& 34.603 & 33.944 & 32.982 & 32.000\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nTo extract the spectrum of the $\\gl(n+N|N)$ spin chain Hamiltonian~\\eqref{eq:ham} from the spectrum of the algebraic Hamiltonian $\\mathsf{H}$ we do the following.\nFor every pair of Young diagrams $(\\lambda,\\mu)$ with $f=0,1,\\dots, L$ boxes and of admissible shape we pick a grading $\\Sigma$ such that the inequalities in eq.~\\eqref{eq:restr_form} are satisfied.\nThen we try to reproduce the spectrum of $\\mathsf{H}$ in $\\Delta_{L,L}(\\lambda,\\mu)$ by means of eq.~\\eqref{eq:spec_ham_mom} from numerical solutions of BAE~\\eqref{eq:BAE_short} in the the form determined by the grading $\\Sigma$ and for root numbers corresponding\nto the highest weight $\\Lambda_\\Sigma(\\lambda,\\mu)$ in eqs.~(\\ref{eq:weight_BV1}, \\ref{eq:weight_tensor}).~\\footnote{One could work with a single form of BAE, say, that corresponding to the distinguished gradation $\\Sigma_0$.\nHowever, the weight of Bethe vectors  reproducing eigenvalues of $\\mathsf{H}$ in $\\Delta_{L,L}(\\lambda,\\mu)$ will no longer be given by eq.~\\eqref{eq:weight_tensor} if the shapes of $(\\lambda,\\mu)$ do not satisfy the inequalities~\\eqref{eq:restr_form} w.r.t. $\\Sigma_0$.}\nNot all eigenvalues can be reproduced in this way. This happens because, as we have explained in sec.~\\ref{sec:tens}, \nthe vector space $\\delta_{L,L}(\\lambda,\\mu)$ of all possible embeddings of $\\gl(n+N|N)$ traceless tensors  $t_0(\\lambda,\\mu)$ into $(V\\otimes V^*)^{\\otimes L}$ can be identified with only a quotient of the standard module $\\Delta_{L,L}(\\lambda,\\mu)$ of $B_{L,L}(n)$.\nWhen an eigenvalue of $\\mathsf{H}$ can be reproduced this way, we assume that a corresponding non-vanishing Bethe vector exist. Otherwise, we assume that there is no eigenstate of $H$ corresponding to that eigenvalue. So, we rely on the completeness of the ABA, at least as far as the spectrum is concerned.\n\n\nDenote the spectrum of the $\\gl(M|N)$ spin chain Hamiltonian~\\eqref{eq:ham} by $\\spec H_{M|N}$.\nThe central idea of this section was the existence of an abstract algebra $B_{L,L}(n)$ such that the centralizers of the series $N=0,1,2,\\dots$ of $\\gl(n+N|N)$ spin chains $\\mathcal{C}(L)$ provide different, $N$ dependent, representations of $B_{L,L}(n)$.\nThis suggests that the intersection $\\cap_{N\\in \\mathbb{Z}^+}\\spec H_{n+N|N}$ might be non-trivial.\nIn fact, with the cohomological techniques developed in \\cite{Candu:2010yg} one can prove the following relationship between the spectral sets $\\spec H_{N+n|N}$ with $n$ fixed\n\\begin{equation}\n\\spec H_{n|0}\\subset \\spec H_{n+1|1}\\subset \\spec H_{n+2|2}\\subset \\cdots\\subset \\mathsf{H}\\ . \n\\label{eq:embed_spec}\n\\end{equation}\nThis ``embedding of spectra'' is a very interesting and general feature of supergroup spin chains and one might wonder how does it carry on to the field theory description of the continuum limit.\nIn this respect, two scenarios are possible.\nThe first possibility is that $\\spec H_{n+N'|N'}$ becomes a very excited subset within $\\spec H_{n+N''|N''}$, where $N'<N''$, and decouples from it in the thermodynamic limit $L\\to \\infty$.\nThis means that the vacuum energy per site of $H_{n+N'|N'}$ is higher then the vacuum energy per site of $H_{n+N''|N''}$ in the thermodynamic limit.\nThe second, more interesting, possibility is that the vacuum energies per site for both Hamiltonians coincide in the thermodynamic limit.\nThe first possibility occurs, for instance, in the $V^{\\otimes L}$ chains with Hamiltonian $\\pm \\sum P_{i,i+1}$.\n\nIn next section we describe the mechanism which is behind the embedding of spectra~\\eqref{eq:embed_spec}  at the level of BAE.\nThe answer to the question of how excited $\\spec H_{n+N'|N'}$ is within $\\spec H_{n+N''|N''}$ for $N'< N''$ will have to wait until sec.~\\ref{sec:rl}.\n\n\n\n\n\\section{Restriction and lift of BAE}\n\\label{sec:rl}\n\nAt the end of the previous section we have explained the phenomenon~\\eqref{eq:embed_spec} of embedding of spectra. For it to work, it was important that there is a series  $\\{\\gl(n+N|N)\\}_{N\\in\\mathbb{Z}^+}$ of spin chains  endowed with Hamiltonians $H_{n+N|N}$ which are all different representations of the same algebraic Hamiltonian $\\mathsf{H}\\in B_{L,L}(n)$.~\\footnote{Strictly speaking, it does not make sense to talk about different representations of a single matrix $\\mathsf{H}$. What we mean here is that the $H_{n+N|N}$ are the image of $\\mathsf{H}$ in different representations of $B_{L,L}(n)$.}\nThe integrability did not matter.\nIn this section we would like to understand how this curious phenomenon arises at the level of BAE describing the spectra of very general $\\gl(M|N)$ integrable Hamiltonians and what structure is responsible for it.\n\nFix a grading $\\Sigma = \\{\\sigma_i=(-1)^{|i|}\\}_{i=1}^{M+N}$  \nof the ordered basis vectors~\\eqref{eq:tot_ordering_bas_vecs} of the $\\gl(M|N)$ fundamental representation $V$ and let $\\Delta^{M|N}_0=\\{\\alpha_j=\\epsilon_j-\\epsilon_{j+1}\\}_{j=1}^{r}$ be the corresponding simple root system, where $r=M+N-1$ is the rank.\nWe have explained how to construct the ABA in the grading $\\Sigma$ for the spin chain $(V\\otimes V^*)^{\\otimes L}$ in sec.~\\ref{sec:formalism}.\nThe Bethe vectors are highest weight vectors of weight~\\eqref{eq:weight_BV1} w.r.t. the Borel subalgebra $\\mathfrak{b}_\\Sigma$ determined by the grading $\\Sigma$ and the ordering~\\eqref{eq:tot_ordering_bas_vecs}, that is by $\\Delta^{M|N}_0$.\nWe shall restrict to Bethe vectors whose weights are given by highest weight Young supertableaux of $\\Sigma$--admissible shapes $(\\lambda,\\mu)$ according to eq.~\\eqref{eq:weight_tensor}.\nDue to fundamental r\\^ole of traceless tensors explained in sec.~\\ref{sec:tens} and claim~\\ref{claim:cv}, it is clear that with the imposed restriction, one must consider the ABA in all the gradings in order to recover the full spectrum of the spin chain Hamiltonian $H$.\nFor a fixed grading and a corresponding simple root system, fig.~\\ref{fig:young} and eq.~\\eqref{eq:weight_tensor} implies that we are restricting to Bethe vectors of weight $w=\\sum_{i=1}^{M+N}w^i \\epsilon_i$, $w^i\\in\\mathbb{Z}$ such that for every simple bosonic root $\\alpha_j=\\epsilon_j-\\epsilon_{j+1}$ one has $w^i \\geq w^{i+1}$,\nwhile for every simple fermionic  root one has the following implications\n\\begin{align}\\label{eq:cond_TL}\n w^{j\\phantom{+1}}<0 &\\longrightarrow w^{j+1}<0\\ ,& w^{j\\phantom{+1}} =0 &\\longrightarrow w^{j+1}\\leq 0\\\\ \\notag\n w^{j+1}>0 &\\longrightarrow w^{j\\phantom{+1}}>0\\ ,&  w^{j+1}=0&\\longrightarrow w^{j\\phantom{+1}} \\geq 0 \\ .\n\\end{align}\nThe main reason for introducing this restrictions and working with BAE in multiple gradings is the bounds on the number of Bethe roots resulting from eqs.~\\eqref{eq:cond_TL}.\n\n\\subsection{Restriction}\n\nWe wish to consider the BAE in the form~\\eqref{eq:BAE_Qform} corresponding to a simple root $\\alpha_k$ such that the following assumptions hold\n\\begin{description}\n\t\\item[A1] $\\alpha_k$ is odd, that is $\\sigma_k \\sigma_{k+1}=-1$\n\t\\item[A2] $\\alpha_k$ has no source terms or, equivalently, $k\\neq 1, r$\n\\end{description}\nWith these assumptions, we have $\\nu^{(k)}$ BAE for $\\alpha_{k-1}$ of the form\n\\begin{equation}\\label{eq:red1} \\frac{\\sigma_{k-1}\\Lambda_{k-1}(u^{(k-1)}_j)}{\\sigma_{k}\\Lambda_{k}(u^{(k-1)}_j)}=- \\frac{Q_{k-2}(u^{(k-1)}_j)Q_{k-1}(u^{(k-1)}_j+\\sigma_k)Q_{k}(u^{(k-1)}_j-\\sigma_k)}{Q_{k-2}(u^{(k-1)}_j+\\sigma_{k-1})Q_{k-1}(u^{(k-1)}_j-\\sigma_{k-1})Q_{k}(u^{(k-1)}_j)}\\ ,\n\\end{equation}\n$\\nu^{(k)}$ equations for $\\alpha_k$\n\\begin{equation}\\label{eq:red2} 1=\\frac{Q_{k-1}(u^{(k)}_j)Q_{k+1}(u^{(k)}_j+\\sigma_k)}{Q_{k-1}(u^{(k)}_j+\\sigma_k)Q_{k+1}(u^{(k)}_j)}\n\\end{equation}\nand $\\nu^{(k+1)}$ equations for $\\alpha_{k+1}$\n\\begin{equation}\\label{eq:red3} \\frac{\\sigma_k \\Lambda_{k+1}(u^{(k+1)}_j)}{\\sigma_{k+2}\\Lambda_{k+2}(u^{(k+1)}_j)}= \\frac{Q_{k}(u^{(k+1)}_j)Q_{k+1}(u^{(k+1)}_j+\\sigma_{k+2})Q_{k+2}(u^{(k+1)}_j-\\sigma_{k+2})}{Q_{k}(u^{(k+1)}_j-\\sigma_k)Q_{k+1}(u^{(k+1)}_j+\\sigma_k)Q_{k+2}(u^{(k+1)}_j)}\\ .\n\\end{equation}\nNotice that if $\\nu^{(k-1)}=\\nu^{(k+1)}$ then one can reduce the\nBAE~\\eqref{eq:BAE_Qform} for the  $\\gl(M|N)$ spin chain $(V\\otimes V^*)^{\\otimes L}$ to the BAE for the $\\gl(M-1|N-1)$ spin chain of the same type $(V\\otimes V^*)^{\\otimes L}$ by\n\\begin{description}\n\t\\item[R1] identifying the Bethe roots corresponding to $\\alpha_{k-1}$ and $\\alpha_{k+1}$\n\\begin{equation}  \\{u^{(k-1)}_j\\}_{j=1}^{\\nu^{(k-1)}}=\\{u^{(k+1)}_j\\}_{j=1}^{\\nu^{(k+1)}}\n\\end{equation}\n\\item[R2] multiplying the BAE for $\\alpha_{k-1}$ and $\\alpha_{k+1}$ corresponding to, say $u^{(k-1)}_j=u^{(k+1)}_j$\n\\begin{equation}\\label{eq:red}\n \\frac{\\sigma_{k-1}\\Lambda_{k-1}(u^{(k- 1)}_j)}{\\sigma_{k+2}\\Lambda_{k+2}(u^{(k- 1)}_j)}=-\n \\frac{Q_{k-2}(u^{(k- 1)}_j)Q_{k- 1}(u^{(k- 1)}_j+\\sigma_{k+2})Q_{k+2}(u^{(k- 1)}_j-\\sigma_{k+2})}{Q_{k-2}(u^{(k-1)}_j+\\sigma_{k-1})Q_{k- 1}(u^{(k- 1)}_j-\\sigma_{k-1})Q_{k+2}(u^{(k- 1)}_j)}\\ .\n\\end{equation}\n\\end{description}\nIndeed,  eq.~\\eqref{eq:red2} is trivially satisfied  because according to \\textbf{R1} one has $Q_{k-1}(u)=Q_{k+1}(u)$. \nFurthermore, multiplying BAE according to \\textbf{R2}, the Bethe roots $\\{u^{(k)}_j\\}_{j=1}^{\\nu^{(k)}}$ drop off yielding a BAE of the form~\\eqref{eq:red1} with $k$ replaced by $k+2$.\nWe call a \\emph{restriction} of BAE the procedure \\textbf{R1}--\\textbf{R2}.\n\nThe restriction procedure can be given an algebraic meaning and, therefore, partially explained as follows. \nAs described in detail in~\\cite{Candu:2010yg}, choose\n\\begin{equation}\\label{eq:q_choice}\nQ=E_{k+1k} \n\\end{equation}\nto be the odd $\\gl(M|N)$ element that squares to zero and defines the $\\gl(M-1|N-1)$ chain $(V_{M-1|N-1}\\otimes V^*_{M-1|N-1})^{\\otimes L}$ as the $Q$--cohomology of the  $\\gl(M|N)$  chain $(V_{M|N}\\otimes V^*_{M|N})^{\\otimes L}$.\nA necessary condition for a highest weight vector $\\omega\\in (V_{M|N}\\otimes V^*_{M|N})^{\\otimes L}$ to yield a non-trivial $Q$--cohomology is for it to be  in the kernel of $Q$ and, therefore, one must have\n\\begin{equation}\\label{eq:wt_ker}\n[E_{kk+1},E_{k+1k}]\\omega =0\\quad \\Rightarrow\\quad \\langle \\wt(\\omega),\\alpha_k\\rangle=0\\ .\n\\end{equation}\nApplying this constraint to a Bethe vector of weight~\\eqref{eq:weight_BV1} one recovers the condition $\\nu^{(k-1)}=\\nu^{(k+1)}$ necessary for \\textbf{R1} to hold.\nThe form of the reduced $\\gl(M-1|N-1)$ BAE can also be easily understood.\nThe subquotient $ \\big(\\ker_{\\langle\\alpha_k,-\\rangle} \\sum_{i=1}^{M+N}\\mathbb{C}\\epsilon_i\\big)\/\\mathbb{C}\\alpha_k$ of the $\\gl(M|N)$ weight space can be straightforwardly identified with the $\\gl(M-1|N-1)$ weight space $\\sum_{i\\neq k,k+1}^{M+N}\\mathbb{C}\\epsilon_i$.\nTherefore, the subquotient $ \\big(\\ker_{\\langle\\alpha_k,-\\rangle} \\Delta_0^{M|N}\\big) \/\\mathbb{C}\\alpha_k$ of the $\\gl(M|N)$ simple root system $\\Delta^{M|N}_0$ induced by cohomological reduction can be identified with a $\\gl(M-1|N-1)$ simple root system $\\Delta_0^{M-1|N-1}=  \\{\\alpha_1,\\dots,\\alpha_{k-2},\\epsilon_{k-1}-\\epsilon_{k+2},\\alpha_{k+2},\\dots,\\alpha_r\\}$.\nThe reduced BAE have the form~\\eqref{eq:BAE_short} corresponding to precisely the simple root system $\\Delta_0^{M-1|N-1}$.\n\n\n\\subsection{Lift}\n\n\nTo resume, assuming \\textbf{A1}--\\textbf{A2} holds for the BAE of the spin chain $(V_{M|N}\\otimes V^*_{M|N})^{\\otimes L}$ written w.r.t. a simple root system $\\Delta^{M|N}_0$, we showed that one can restrict them to the system of BAE for the spin chain $(V_{M-1|N-1}\\otimes V^*_{M-1|N-1})^{\\otimes L}$ written w.r.t. the simple root system $\\Delta_0^{M-1|N-1}=\\{\\alpha_1,\\dots,\\alpha_{k-2},\\epsilon_{k-1}-\\epsilon_{k+2}, \\alpha_{k+2},\\dots,\\alpha_r\\}$ induced by  cohomological reduction.\nThis restriction is represented at the level of Dynkin diagrams in fig.~\\ref{fig:col}.\n\\begin{figure}%\n\\psfrag{+}{$+$}\n\\psfrag{-}{$-$}\n\\psfrag{a1}{$\\alpha_{k-1}$}\n\\psfrag{a2}{$\\alpha_{k}$}\n\\psfrag{a3}{$\\alpha_{k+1}$}\n\\psfrag{a4}{$\\alpha_{k-1}\\simeq\\alpha_{k+1}$}\n\\psfrag{b1}{$\\gl(M|N)$}\n\\psfrag{b2}{$\\gl(M-1|N-1)$}\n\\psfrag{P}{$P$}\n\\psfrag{p}{$p$}\n\\centerline{\\includegraphics[scale=1]{col.eps}}\n\\caption{Restriction of BAE for supergroups. The dashed lines indicate possible roots on the left or on the right. For a root $\\alpha_j=\\epsilon_j-\\epsilon_{j+1}$ with $j=k,k\\pm 1$, we have indicated with $\\pm$ signs the gradings of fundamental weight $\\epsilon_j$ determining it.}\n\\label{fig:col}\n\\end{figure}\n\nNotice that we have not imposed any condition on the roots $u_j^{(k)}$.\nTherefore, it is legitimate to ask if the BAE satisfying \\textbf{A1}--\\textbf{A2} actually admit solutions of type \\textbf{R1}.\nSo, we are given a simple root system $\\Delta_0^{M|N}$ and a solution $\\{u_j^{(l)}\\}_{j=1}^{\\nu^{(l)}}$, $l=1,\\dots,k-1,k+2,\\dots,r$ of the \n$(V_{M-1|N-1}\\otimes V^*_{M-1|N-1})^{\\otimes L}$ BAE w.r.t the  simple root system $\\Delta_0^{M-1|N-1}$ induced by cohomological reduction from $\\Delta_0^{M|N}$ with $Q$ as in eq.~\\eqref{eq:q_choice}.\nWe must show that there is a solution $\\{u_j^{(l)}\\}_{j=1}^{\\nu^{(l)}}$, $l=1,\\dots,r$ of the $(V_{M|N}\\otimes V^*_{M|N})^{\\otimes L}$ BAE w.r.t. $\\Delta_0^{M|N}$ which restricts to the given one.\nFirst of all, eq.~\\eqref{eq:wt_ker} implies $\\nu^{(k+1)}=\\nu^{(k-1)}=P$. After defining $u^{(k+1)}_j=u^{(k-1)}_j$ for $j=1,\\dots,P$ the task is reduced to finding a solution $\\{u^{(k)}_j\\}_{j=1}^p$ to either eq.~\\eqref{eq:red1} or eq.~\\eqref{eq:red3}, where we have set $p=\\nu^{(k)}$.\nIf $p\\geq P$, such a solution obviously exists. More then that, for $p>P$ there is a continuum of such solutions!\nHowever, recall that for a fixed root system and corresponding BAE we have restricted to Bethe vectors such that their weights satisfy the constraints~\\eqref{eq:cond_TL}.\nIf the solution we are looking for exists then the weight  of the corresponding Bethe vector $\\omega$ can be written as $\\wt(\\omega)=\\dots+(P-p)(\\epsilon_k-\\epsilon_{k+1})+\\dots$.\nThe constraints~\\eqref{eq:cond_TL} imply $p\\leq P$.\nSo, for $p=P$ the solution \\emph{always exists}.\nIn our numerical investigations we have observed that solutions might exist even for $p<P$.\nWe call the solutions of $\\gl(M|N)$ BAE  constructed in this way from solutions of $\\gl(M-1|N-1)$ BAE \\emph{lifted solutions}.\nWe discard lifted solutions with $p< P$ for the following reason.\n\\begin{claim}\\label{claim:coh}\n If a solution of $\\gl(M-1|N-1)$ BAE admits lifts to solutions of $\\gl(M|N)$ BAE corresponding to Bethe vectors of different weights, then all of them have zero $Q$--cohomology.\n\\end{claim}\n\\noindent We prove the claim in appendix~\\ref{sec:B}.\n\nNext, we show that \\emph{the lift is unique}.\nSo, we have seen that lifted Bethe vectors correspond to lifted solutions with  $\\nu^{(k)}=\\nu^{(k\\pm 1)}=P$.\nLet us now prove that the $P$ equations, say,~\\eqref{eq:red1} regarded as a constraint on the unknowns $\\{u^{(k)}_j\\}_{j=1}^P$ admit a unique solution. We start by multiplying both sides with $Q_k(u_j^{(k-1)})$ and then expand in  $\\{u^{(k)}_j\\}_{j=1}^P$.\nWhat we get is a system of $P$ linear equations for the $P$ unknowns\n\\begin{equation*}\n s_l = \\sum_{i_1<\\cdots<i_l}u^{(k)}_{i_1}\\dots u^{(k)}_{i_l}\\ .\n\\label{eq:sym}\n\\end{equation*} \nThus, for a  given solution $\\{u_j^{(1)}\\}_{j=1}^{\\nu^{(j)}}$, $j=1,\\dots,k-1,k+2,\\dots,r$ of the $\\gl(M-1|N-1)$ BAE, this system of linear equations admits a single solution for the $\\{s_j\\}_{j=1}^P$.\nFinally, notice that the latter determine the set $\\{u^{(k)}_j\\}_{j=1}^P$ uniquely, because every Bethe root in this set is a solution of the polynomial equation of degree $P$\n\\begin{equation*}\n\\prod_{j=1}^P(u-u_j^{(k)}) = u^P +s_1 u^{P-1}+\\dots- (-1)^P s_{P-1}u + (-1)^Ps_P=0\\ .\n\\end{equation*}\n\nThere are important cases when one can compute the roots $\\{u^{(k)}_j\\}_{j=1}^P$ explicitly in terms of the other roots.\nThis happens if\n\\begin{description}\n\\item[C 1.1)] $\\alpha_{k\\pm 1}$ is odd and sourceless\n\\item[C 1.2)]  $\\nu^{(k\\pm 2)}=P$, \n\\end{description}\nwhere we meant that either we choose the plus sign or the minus sign overall. \nIn the case \\textbf{C~1} eq.~\\eqref{eq:red1} or eq.~\\eqref{eq:red3} can be rewritten in terms of variables~\\eqref{eq:shifts} as\n\\begin{equation*}\n\\prod_{i=1}^P e_1(x^{(k\\pm 1)}_j-x^{(k)}_i) =  \\prod_{i=1}^P e_1(x^{(k\\pm 1)}_j-x^{(k\\pm 2)}_i)\\ ,\\quad j =1,\\dots,P\\ .\n\\end{equation*}\nWe immediately read off the obvious solution\n\\begin{equation} \\label{eq:freezing1}\n \\{x^{(k)}_{j}\\}_{j=1}^P = \\{x^{(k\\pm 2)}_j\\}_{j=1}^P\\ ,\n\\end{equation}\nwhich we already know is unique.\nAnother important case is\n\\begin{description}\n\\item[C 2.1)] $k=2$, $\\alpha_1$ is odd and $P=L$ \n\\item[C 2.2)] $k=r-1$, $\\alpha_r$ is odd and $P=L$.\n\\end{description}\nIn the case \\textbf{C~2.1} the one has\n\\begin{equation*}\n\\prod_{i=1}^L e_1(x^{(1)}_j-x^{(2)}_i) =  \\prod_{a=1}^L e_1(x^{(1)}_j-y_a)\\ ,\\quad j =1,\\dots,L\\ ,\n\\end{equation*}\nwhile in the case \\textbf{C~2.2}\n\\begin{equation*}\n\\prod_{i=1}^L e_1(x^{(r)}_j-x^{(r-1)}_i) =  \\prod_{a=1}^L e_1(x^{(r)}_j-\\bar{y}_a)\\ ,\\quad j =1,\\dots,L\\ ,\n\\end{equation*}\nwhere we have used eqs.~\\eqref{eq:BAE_short} with non-zero arbitrary inhomogeneities.\nAgain, we  read off the unique solutions\n\\begin{equation} \\label{eq:freezing2}\n \\{x^{(2)}_j\\}_{j=1}^L=\\{y_a\\}_{a=1}^L\\ ,\\qquad\n \\{x^{(r-1)}_j\\}_{j=1}^L=\\{y_{\\bar{a}}\\}_{a=1}^L\\ .\n\\end{equation}\nIt is important to notice that inhomogeneities are essential to lift the degeneracy of solutions in the case \\textbf{C~2}.\n\n\\subsection{Generalizations}\n\n\nWe have argued that the restriction and lift of BAE have an algebraic origin. From this viewpoint, assumption \\textbf{A2} seems unnecessary. Indeed, all of the above constructions can be appropriately modified to accommodate the boundary case corresponding to \\textbf{A1} and\n\\begin{description}\n\\item[A2$'$] $k=1,r$.\n\\end{description}\nThe necessary condition~\\eqref{eq:wt_ker} for a non-vanishing $Q$--cohomology of a $\\gl(M|N)$ Bethe vector $\\omega$ implies that reducible solution of $\\gl(M|N)$ BAE must have $\\nu^{(2)}=L$ if $k=1$ and $\\nu^{(r-1)}=L$ if $k=r$.\nThe reduced BAE must have a form~\\eqref{eq:BAE_short} corresponding to simple root systems $\\Delta_0^{M-1|N-1}=\\{\\alpha_3,\\dots,\\alpha_r\\}$ and $\\Delta_0^{M-1|N-1}=\\{\\alpha_1,\\dots,\\alpha_{r-2}\\}$ induced by evaluating the subquotient $\\big(\\ker_{\\langle\\alpha_k,-\\rangle} \\Delta_0^{M|N}\\big)\/\\mathbb{C}\\alpha_k$ for $k=1$ and $k=r$ respectively.  \nTo satisfy these requirements one must replace \\textbf{R1--R2} with\n\\begin{description}\n\\item[R$'$] $\\{x_j^{(2)}\\}_{j=1}^L=\\{y_a\\}_{a=1}^L$ for  $k=1$ and \n$\\{x_j^{(r-1)}\\}_{j=1}^L=\\{\\bar{y}_a\\}_{a=1}^L$ for  $k=r$.\n\\end{description}\nInvoking again claim~\\ref{claim:coh} and the arguments that follow it, we conclude that the lift of a $\\gl(M-1|N-1)$ Bethe vector to a $\\gl(M|N)$ Bethe vector must be unique and satisfies $\\nu^{(1)}=L$ for $k=1$ and $\\nu^{(r)}=L$ for $k=r$.\nFinally, the corresponding lifted solution can be evaluated explicitly if $\\alpha_2$ is odd for $k=1$ and  $\\alpha_{r-1}$ is odd for $k=r$\n\\begin{equation*}\n \\{x^{(1)}_j\\}_{j=1}^L = \\{x^{(3)}_j\\}_{j=1}^L\\ ,\\qquad \\{x^{(r)}_j\\}_{j=1}^L = \\{x^{(r-2)}_j\\}_{j=1}^L\\ .\n\\end{equation*}\n\nSubsectors in integrable systems are not a new phenomenon. For example the homogeneous $\\gl(2n+1)$ spin chain $(V\\otimes V^*)^L$ of sec.~\\ref{sec:formalism} contains a subsector corresponding to the $2L$--th tensor power of the fundamental representation of $\\osp(1|2n)$, see \\cite{Saleur:2001cw}.\nWe shall provide more examples in sec.~\\ref{sec:BAE}.\nHowever, all these examples  lack by far the generality of the restriction and lifting procedures we have described.\nThis is because in our case there is an algebraic mechanism behind which relates the representation theories of $\\{\\gl(n+N|N)\\}_{N\\in\\mathbb{Z}^+}$ Lie superalgebras.\nThe same cohomological reduction mechanism exists for any Lie superalgebra except $\\osp(1|2n)$.\nTherefore there is no doubt that the same restriction and lift phenomena occur also for general $\\osp(R|2S)$ integrable spin chains.\nMoreover, the embedding of spectra~\\eqref{eq:embed_spec} was already observed on a (non-integrable) $\\osp(2S+2|2S)$ spin chain in \\cite{Candu:2008vw}.\n\nWe come back now to the spin chain of sec.~\\ref{sec:formalism} and\nanswer the question about how excited w.r.t. each other are the subsectors~\\eqref{eq:embed_spec}. \n\n\n\n\n\n\n\n\n\\section{Vacuum and low lying excitations}\n\\label{sec:BAE}\n\n\nIn this section we present numerical evidence showing that all integrable $\\gl(n+N|N)$ spin chains~\\eqref{eq:ham} with $n$ fixed have the same vacuum energy.\nIn view of the embedding of spectra~\\eqref{eq:embed_spec}, we introduce the notion of degree of an excitation, which is the smallest value of $N$ for which it appears in $\\spec H_{n+N|N}$.\nWe then proceed to classify the excitations of degree 0 and 1,\nand present the\nform of the\nnumerical solutions of BAE reproducing them. \n\n\n\n\\subsection{Bosonic lift}\n\\label{sec:bos_lift}\n\nFrom~\\eqref{eq:embed_spec}, the vacuum energy of the spin chain is greater or equal to the vacuum energy of the algebraic Hamiltonian.\nSupposing they coincide, from tables~(\\ref{tab:1}, \\ref{tab:2}) and the general discussion of sec.~\\ref{sec:num_diag}\nit follows that the vacuum state in the spin chain is a $\\gl(n+N|N)$ invariant tensor.\nAccording to eq.~\\eqref{eq:weight_BV1}, the number of roots in the vacuum state should then be $\\nu^{(k)}=L$.\nNumerically, we confirm\nthat there is indeed a solution to the $\\gl(n+N|N)$ BAE~\\eqref{eq:BAE_short} with $\\nu^{(k)}=L$ which reproduces the vacuum energy of the algebraic Hamiltonian $\\mathsf{H}\\in B_{L,L}(n)$.\nAccording to the terminology of sec.~\\ref{sec:rl}, this is the $\\gl(n+N|N)$ lift of the vacuum solution for the purely ``bosonic'' $\\gl(n)$ chain.\nThe latter is characterized by $n-1$ seas of real roots without holes.\n\nThe low rank cases $\\gl(2|1)$ and $\\gl(3|1)$ have to be treated independently.\nFor the $\\gl(2|1)$ case it was realized in \\cite{Essler:2005ag} that the vacuum energy of the Hamiltonian~\\eqref{eq:ham} is exactly $-4L$ and the corresponding vacuum solution in the grading $\\Sigma=\\{+,-,+\\}$ is highly degenerated $x^{(1)}_j=x^{(2)}_j=0$.\nThis solution obviously does not make sense in the framework of the ABA. \nThe problem persists in all other gradings as well.\nA regularized vacuum Bethe vector can be defined by introducing arbitrary inhomogeneities for every $V$ and $V^*$ site in the chain.\nThen, according to eqs.~\\eqref{eq:freezing2}, the vacuum solution is entirely fixed by these inhomogeneities and the vacuum Bethe vector can be constructed.\nThe vacuum state of the homogeneous chain is then defined by a limiting procedure.\nThe $\\gl(3|1)$ spin chain has the same problem, again in all the gradings, and the same cure.\n\n\nWe now realize  the embedding $\\spec H_{n|0}\\subset \\spec H_{n+N|N}$ explicitly.\nFollowing~\\cite{Candu:2010yg}, choose an odd element $Q\\in\\gl(n+N|N)$ that squares to zero, has rank $N$ and defines the $\\gl(n)$ spin chain $(V\\otimes V^*)^{\\otimes L}$ as the $Q$-cohomology of the $\\gl(n+N|N)$ spin chain $(V\\otimes V^*)^{\\otimes L}$. \nFor this choice of $Q$, one can perform the lift of all $\\gl(n)$ Bethe vectors to $\\gl(n+N|N)$ Bethe vectors as described in sec.~\\ref{sec:rl}.\nAs we shall see in a moment, the lift depends on the choice of $Q$.\nWe call this part of the spectrum the \\emph{bosonic lift}. The latter is straightforward to understand in terms of $\\gl(n)$ spin chain excitations \\cite{Andrei:1979un, Andrei:1979wy, Andrei:1979sq, Doikou:1998xi}.\n\nHow to characterize the $\\gl(n+N|N)$ symmetry of the excitations that belong to the bosonic lift?\nWe say  that the component $w_i$ of a $\\gl(n+N|N)$ weight $w=\\sum_{i=1}^{M+N}w_i\\epsilon_i$ is bosonic if $|i|\\equiv 0$ and fermionic if $|i|\\equiv 1$. \nAccording to sec.~\\ref{sec:rl}, if a $\\gl(n+N|N)$ Bethe vector $\\omega$ is the lift of a $\\gl(n)$ Bethe vector, that is $\\omega$ has degree 0, then it should be possible (in some grading) to write its weight in the form $\\wt(\\omega)=\\sum_{i=1}^{M+N}w_i\\epsilon_i$ with no fermionic components and at most $n$ bosonic components.\nEquivalently, the weight of a Bethe vector $\\omega$ of degree 0 must be representable (in some grading) by highest weight Young supertableaux of $\\gl(n)$--admissible shape $(\\lambda,\\mu)$\n\\begin{equation}\\label{eq:adm_shape_bos}\n \\lambda'_1+\\mu'_1\\leq n\\ .\n\\end{equation}\nThese shapes are $\\Sigma$--admissible and most obviously $\\gl(n)$ reducible w.r.t. the gradings\n\\begin{equation}\\label{eq:some_special_grad}\n\\Sigma=\\{\\overbrace{+,\\dots,+}^m,\\overbrace{+,-,\\dots,+,-}^{2N},\\overbrace{+,\\dots,+}^{n-m}\\}=[+^m(+-)^N+^{(n-m)}]\\ ,\n\\end{equation}\nwhere $\\lambda'_1\\leq m\\leq n-\\mu'_1$.\nThere are many choices for the nilpotent element to define the restriction and lift in the grading~\\eqref{eq:some_special_grad}.\nFor instance, the two possibilities\n\\begin{align}\\label{eq:two_q_choices}\n Q_1 \\in \\mathbb{C}E_{m+2,m+1}\\oplus \\mathbb{C} E_{m+4,m+3}\\oplus\\cdots \\oplus \\mathbb{C}E_{m+2N,m+2N-1}\\\\\nQ_2 \\in \\mathbb{C}E_{m+3,m+2}\\oplus \\mathbb{C} E_{m+5,m+4}\\oplus\\cdots \\oplus \\mathbb{C}E_{m+2N+1,m+2N}\\notag\n\\end{align}\nare represented in fig.~\\ref{fig:bos_lift}. Notice that for a fixed $Q$ the lift is unique, although it is different for different $Q$'s. \n\\begin{figure}[t]%\n\\psfrag{l}{$L$}\n\\psfrag{d}{$\\cdots$}\n\\psfrag{a}{$P$}\n\\psfrag{b}{$R$}\n\\psfrag{q1}{$Q_1$}\n\\psfrag{q2}{$Q_2$}\n\\psfrag{a1}{$\\alpha_m$}\n\\psfrag{a2}{$\\alpha_{m+2N+1}$}\n\\centerline{\\includegraphics[scale=1.0]{bos_lift.eps}\\hspace{0.5cm}\n\\includegraphics[scale=1.0]{lift.eps}}\n\\caption{The grading $[+^{m}(+-)^{N}+^{n-m}]$, $1<m<n-1$ is such that there is a chain of $2N$ odd roots. On the left: lift of the $\\gl(n)$ vacuum to the $\\gl(n+N|N)$ vacuum. On the right: Bethe roots for Bethe vectors in the bosonic lift w.r.t. the two choices~\\eqref{eq:two_q_choices} of $Q$.}%\n\\label{fig:bos_lift}%\n\\end{figure}\nMultiple gradings of the type~\\eqref{eq:some_special_grad} must be considered in order to recover the full bosonic lift.\n\nTo conclude, we write down explicitly the vacuum solution in the grading $[+^{m} (+-)^{N} +^{n-m} ]$ with $1<m<n-1$\n\\begin{align}\\label{eq:vac_sol}\n x^{(m)}_j&=x^{(m+2)}_j=\\dots=x^{(m+2N)}_j\\\\ \\notag\nx^{(m+1)}_j&=x^{(m+3)}_j=\\dots=x^{(m+2N+1)}_j\\ ,\\qquad j=1,\\dots,L\\ ,\n\\end{align}\nwhere the roots $\\{x^{(l)}_j\\}_{j=1}^L$, $l=1,\\dots,m,m+2N+1,\\dots,n+2N-1$ correspond to the $\\gl(n)$ vacuum solution. We represent the lift of the bosonic vacuum  in fig.~\\ref{fig:bos_lift}.\nWe have already mentioned, see eq.~\\eqref{eq:freezing2}, how to treat the special cases $\\gl(2|1)$ and $\\gl(3|1)$ by introducing arbitrary inhomogeneities.\n\n\n\n\n\\subsection{Degree one excitations}\n\\label{sec:deg1}\n\n\\subsubsection{Classification}\n\nFix an odd element $Q\\in\\gl(n+N|N)$ that squares to zero, has rank $N-1$ and defines the $\\gl(n+1|1)$ spin chain $(V\\otimes V^*)^{\\otimes L}$ as the $Q$-cohomology of the $\\gl(n+N|N)$ spin chain $(V\\otimes V^*)^{\\otimes L}$. \nExcitations of degree 1 correspond to $\\gl(n+N|N)$ Bethe vectors which: i) are non-reducible to $\\gl(n)$ Bethe vectors and ii) are lifted $\\gl(n+1|1)$ Bethe vectors.\nWe shall call the corresponding Bethe vectors also of degree 1.\nAccording to sec.~\\ref{sec:rl}, the weight of a Bethe vector $\\omega$ of degree 1 admits a representation (in some grading) of the form $\\wt(\\omega)=\\sum_{i=1}^{M+N}w_i \\epsilon_i$ with at most $n+1$ bosonic components and at most one fermionic component.\nNotice that due to the claim~\\ref{claim:coh} of sec.~\\ref{sec:rl}, there are also excitations of degree 1 with the same weight as excitations of degree 0, that is no fermionic components and at most $n$ bosonic components.\nTaking into account sec.~\\ref{sec:tens} as well, the weight of Bethe vectors of degree 1 must be representable (in some grading) by highest weight Young supertableaux of $\\gl(n+1|1)$--admissible shapes $(\\lambda,\\mu)$,\nthat is at least one of the two conditions holds\n\\begin{align}\\label{eq:deg_1a}\n \\lambda'_1+\\mu'_2&\\leq n+1\\\\\n\\lambda'_2+\\mu'_1&\\leq n+1\\ .\n\\label{eq:deg_1b}  \n\\end{align}\nShapes~\\eqref{eq:deg_1a} are $\\Sigma$--admissible and most obviously $\\gl(n+1|1)$ reducible w.r.t. the gradings $[(+)^{n-m}(+-)^{2N}(+)^m]$, such that $\\mu_1'\\leq m\\leq n+1-\\lambda_2'$, and the choice  $Q\\in \\mathbb{C}E_{m-n+3,m-n+2}\\oplus\\dots \\oplus \\mathbb{C}E_{m-n+2N-1,m-n+2N-2}$.\nShapes~\\eqref{eq:deg_1b} are $\\Sigma$--admissible and most obviously $\\gl(n+1|1)$ reducible w.r.t. the gradings $[(+)^m(-+)^{2N}(+)^{n-m}]$, such that $\\lambda_1'\\leq m\\leq n+1-\\mu_2'$, and the choice  $Q\\in\\mathbb{C}E_{m+3,m+2}\\oplus\\cdots \\oplus E_{m+2N-1,m+2N-2}$.\nThe corresponding root configurations are represented in fig.~\\ref{fig:deg1} on the left.\n\\begin{figure}[t]%\n\\psfrag{d}{$\\cdots$}\n\\psfrag{a}{$P$}\n\\psfrag{b}{$Q$}\n\\psfrag{c}{$R$}\n\\psfrag{e}{$S$}\n\\psfrag{a1}{$\\alpha_1$}\n\\psfrag{a2}{$\\alpha_{m-n}$}\n\\psfrag{a3}{$\\alpha_{m-n+2N}$}\n\\psfrag{a4}{$\\alpha_{n+2N-1}$}\n\\psfrag{l1}{$L-1$}\n\\psfrag{l2}{$L-m+1$}\n\\psfrag{l3}{$L-m$}\n\\psfrag{l4}{$L-m-1$}\n\\psfrag{l5}{$L-m$}\n\\psfrag{l6}{$L-m+1$}\n\\psfrag{l7}{$L-1$}\n\\psfrag{g1}{$\\gl(2m|1)$}\n\\psfrag{g2}{$\\gl(2m+1|1)$}\n\\psfrag{b1}{$\\alpha_m$}\n\\psfrag{b2}{$\\alpha_{m+2N}$}\n\\centerline{\\includegraphics[scale=1.0]{lift_1.eps}\\hspace{1.5cm}\n\\includegraphics[scale=1]{labels_exc1.eps}\n}\n\\caption{Root numbers for $\\gl(n+N|N)$ excitations of degree 1 and shape $(\\lambda,\\mu)$ satisfying, top left, eq.~\\eqref{eq:deg_1a} w.r.t. some grading $[(+)^{n-m}(+-)^{2N}(+)^m]$ with $\\mu_1'\\leq m\\leq n+1-\\lambda_2'$ or, bottom left, eq.~\\eqref{eq:deg_1b} w.r.t. some grading $[(+)^m(-+)^{2N}(+)^{n-m}]$ with  $\\lambda_1'\\leq m\\leq n+1-\\mu_2'$. \nOn the right: root numbers for the lowest lying degree 1 excitation.}%\n\\label{fig:deg1}%\n\\end{figure}\n\n\\subsubsection{Low lying excitations}\n\\label{sec:low_lying_deg1}\n\n\nAssumption~2 of sec.~\\ref{sec:spec} and tab.~\\ref{tab:2} clearly indicates that the lowest lying excitation of the $B_{L,L}(n)$ algebraic Hamiltonian which is of degree higher then 0 lies in $\\Delta_{L,L}(1^k,1^k)$ with $k=[\\tfrac{n}{2}]+1$.\nEquivalently, the lowest lying excitation of the $\\gl(n+N|N)$ spin chain Hamiltonian which is of degree higher then 0 is a traceless tensor $t_0(1^k,1^k)$. Eqs.~(\\ref{eq:deg_1a}, \\ref{eq:deg_1b}) imply\nthis excitation is of degree 1.\nTo reproduce it from $\\gl(n+1|1)$ BAE we have chosen the following gradings\n\\begin{equation}\\label{eq:grad_special}\n \\Sigma=\\begin{cases}\n[+^{m}-+^{m}]\\phantom{^{+1}}\\ ,& n=2m-1\\\\\n[+^{m} - +^{m+1}]\\ ,& n=2m\n        \\end{cases}\n\\end{equation}\nThe corresponding root numbers are represented in fig.~\\ref{fig:deg1} on the right.\nWe have performed extensive numerical calculations on the form of the solution of Bethe ansatz equations reproducing the lowest lying excitations of degree 1.~\\footnote{We will show a plot of the root distributions in the complex place for low lying excitations of degree 1 in a future version of this eprint.}\nCompared to usual Lie algebra spin chains, this excitation looks very strange, to say the least.\nStrikingly, for $n$ odd the roots $\\{x_j^{(m)}\\}_{j=1}^{L-m}$, $\\{x_j^{(m+1)}\\}_{j=1}^{L-m}$ are always complex, while for $n$ even the roots\n$\\{x^{(m+1)}_j\\}_{j=1}^{L-m-1}$ are always complex.\nWe recall that all roots are real for the  vacuum solution~\\eqref{eq:vac_sol}.\nThus, this excitation is not constructed as usual by making ``minor'' modifications to the vacuum solution.\n\nAnother strange feature is the non-selfconjugacy of the solutions, at least in general.\nThis is a consequence of the non-hermiticity of the Hamiltonian~\\eqref{eq:ham}.~\\footnote{See~\\cite{vlad} for a proof of selfconjugacy in the case of hermitian Lie algebra spin chain Hamiltonians.}\nSuch solutions have been investigated already in the study~\\cite{Essler:2005ag} of the $\\gl(2|1)$ spin chain, but also encountered in different contexts dealing with non-hermitian Hamiltonians~\\cite{Saleur:2000bq}. \nInstead, all lowest lying degree 1 solutions\nare invariant w.r.t. a modified conjugation symmetry\n\\begin{align}\\label{eq:conj_1}\n \\gl(2m|1):&& \\{x^{(k)*}_j\\}_{j=1}^{\\nu^{(k)}}&=\\{x^{(n+1-k)}_j\\}_{j=1}^{\\nu^{(n+1-k)}}\\\\\n \\gl(2m+1|1):&& \\{x^{(k)*}_j\\}_{j=1}^{\\nu^{(k)}}&=\\{x^{(n+1-k)}_j\\}_{j=1}^{\\nu^{(n+1-k)}}\\ ,\\quad k\\neq m+1\\ .\\label{eq:conj_2}\n\\end{align}\nThe symmetry~\\eqref{eq:conj_2} looks surprising, because the Dynkin diagram in fig.~\\ref{fig:deg1} is not symmetric w.r.t. the transposition $\\alpha_{k}\\mapsto\\alpha_{n+1-k}$. However, notice that the roots $\\{x_j^{(k)}\\}_{j=1}^{\\nu^{(k)}}$, $k\\neq m+1$ together with the particle hole transformed roots $\\{x_j^{(m+1)}\\}_{j=1}^{\\nu^{(m+1)}}$ also solve the BAE corresponding to the transposed Dynkin diagram of fig.~\\ref{fig:deg1}, see~\\cite{Tsuboi:1998ne} for details. \nFor $\\nu^{(m+1)}$ even, the solutions\nare (exceptionally) selfconjugate\n\\begin{equation}\\label{eq:exceptional}\n \\nu^{(m+1)} \\quad \\text{even}:\\qquad \\{x^{(k)*}_j\\}_{j=1}^{\\nu^{(k)}}=\\{x^{(k)}_j\\}_{j=1}^{\\nu^{(k)}}\\ , \\quad k=1,\\dots, n+1\\ .\n\\end{equation}\n\nThe following picture of the distribution of roots in the complex plane in the thermodynamic limit is emerging from\nour numerical analysis\n\\begin{align}\\label{eq:ideal_sol_1}\n\\gl(2m|1) &:& &\\begin{cases}\\begin{matrix}\n x^{(k)}_j &=& x^{(2m+1-k)}_j&,& x^{(k)*}_j = x^{(k)}_j&,& k \\neq m,m+1  \\\\[5pt]\nx^{(m)}_j &=& x^{(m+1)*}_j &=& x^{(\\pm)}_j \\pm \\frac{i}{4} & , &  x_j^{(\\pm)*}=x_j^{(\\pm)}\n\\end{matrix}\\end{cases}\\\\[5pt]\n\\gl(2m+1|1) &:& &\\begin{cases} \\begin{matrix}\n x^{(k)}_j &=& x^{(2m+2-k)}_j&, &  x^{(k)*}_j = x^{(k)}_j & , & k \\neq m+1\\\\[5pt]\nx^{(m+1)}_j &=&x^{(\\pm)}_j \\pm \\frac{i}{2} & ,& x_j^{(\\pm)*}=x_j^{(\\pm)} & &  \n\\end{matrix}\n\\end{cases}\\ . \\label{eq:ideal_sol_2}\n\\end{align}\nThe $\\gl(2m|1)$ complex root configurations where called $\\pm$-strange strings in~\\cite{Essler:2005ag}.\n\nWe see that the lowest degree 1 excitation looks pretty complicated.\nWhat about other degree 1 excitations? We have verified that properties~(\\ref{eq:conj_1}--\\ref{eq:conj_2}) hold for all low lying excitations with the same root numbers as in fig.~\\ref{fig:deg1} on the right and, more then that, for all low lying solutions with symmetric configurations of the root numbers $\\nu^{(k)}=\\nu^{(n+1-k)}$.\nMany of these excitations seem to tend to the form~(\\ref{eq:ideal_sol_1}, \\ref{eq:ideal_sol_2}) in the thermodynamic limit.\nThese are the best understood excitations for which we shall give a continuum description in the next section.\nWe have also noticed that among the low lying excitations with $\\nu^{(m+1)}$ even, there is always a \\emph{subset} of selfconjugate solutions of the type~\\eqref{eq:exceptional}. These have interesting properties, as we shall see in a moment.\n\nTo complete the overview of low lying degree 1 excitations, let us mention that we have also identified solutions with symmetrical configurations of the \nroot numbers, which do not seem to tend to the form~(\\ref{eq:ideal_sol_1}, \\ref{eq:ideal_sol_2}) in the thermodynamic limit.\nFor these solutions we have nothing to say, although their study might prove crucial in constructing an $S$-matrix description of the continuum theory in large volume.\nThe situation is even worse for solutions with non symmetrical configurations of the root numbers, because $\\pm$-strange string configurations cannot be clearly defined. \n\n\\subsubsection{Selfconjugate solutions}\n\nAs we have said, for $\\nu^{(m+1)}$ even and  $\\nu^{(k)}=\\nu^{(n+1-k)}$, there is always a subset of selfconjugate solutions of the type~\\eqref{eq:exceptional}.\nIt is straightforward to show that the $\\gl(2m|1)$ BAE~\\eqref{eq:BAE_short} subject to the constraints~(\\ref{eq:conj_1}, \\ref{eq:exceptional}) are equivalent to the BAE of the $\\so(2m+1)$ fundamental spin chain of length $L$ with periodic boundary conditions for the first $m-1$ types of roots and antiperiodic boundary conditions for the spinorial roots.\nThis is the generalization of the Takhtajan-Babujian subsector of the $\\gl(2|1)$ spin chain of~\\cite{Essler:2005ag}.\nThe eigenvalues of the integrable Hamiltonian for this $\\so(2m+1)$ spin chain are given by eq.~\\eqref{eq:spec_ham_mom} subject to the constraint~\\eqref{eq:exceptional}.\nWhile this correspondence between solutions of $\\gl(2m|1)$ and $\\so(2m+1)$ BAE is quite curious, the remarkable thing is that it extends to a correspondence between weakly excited solutions on both sides.\nIn particular, the vacuum solution of the $\\so(2m+1)$ fundamental chain  in the thermodynamic limit~\\cite{Martins:1990ca} (plus holes) is compatible with the form~\\eqref{eq:ideal_sol_1} when the constraint~\\eqref{eq:exceptional} is taken into account.\nWe shall see how this correspondence can be used effectively in the next section.\n\nThe situation is quite different for $\\gl(2m+1|1)$. Notice that assumptions \\textbf{A1--A2} and \\textbf{R1} of sec.~\\ref{sec:rl} hold for solutions of $\\gl(2m+1|1)$ BAE~\\eqref{eq:BAE_short} satisfying~(\\ref{eq:conj_2}, \\ref{eq:exceptional}) and, therefore, these can be restricted to solutions of $\\gl(2m)$ BAE.\nHowever, the latter are very special,  because in addition to satisfying the $\\gl(2m)$ BAE they must also admit multiple $\\gl(2m+1|1)$ lifts.\nAccording to the claim 2 of sec.~\\ref{sec:rl}, $\\gl(2m+1|1)$ lifted \\emph{Bethe vectors} correspond exclusively to $\\gl(2m)$ solutions that do not admit multiple lifts.\nPresumably, what happens is that the additional conditions satisfied by these $\\gl(2m)$ solutions obtained by restriction ensure the vanishing of the corresponding $\\gl(2m)$ Bethe vectors.\nWe have checked that\nthere are many other low lying solutions satisfying the constraints~(\\ref{eq:conj_2}, \\ref{eq:exceptional}).\nHowever, it is most confusing that many low lying solutions that \\emph{do not} satisfy the constraints~(\\ref{eq:conj_2}, \\ref{eq:exceptional}) in finite volume seem to have  a thermodynamic limit~\\eqref{eq:ideal_sol_2} that does satisfy the constraints~(\\ref{eq:conj_2}, \\ref{eq:exceptional}).\nWe are not sure how to interpret this behavior,\nalthough we are tempted to believe this indicates that the large volume limit is a subtle issue.\nTherefore, we concentrate in the next section solely on the continuum description of $\\gl(2m|1)$ degree 1 excitations of the type~(\\ref{eq:conj_1}, \\ref{eq:ideal_sol_1}).\n\n\n\n\n\n\\section{Continuous limit}\n\\label{sec:cl}\n\nIn this section we shall consider the spectrum of the spin chain of sec.~\\ref{sec:formalism} in the thermodynamic limit $L\\to \\infty$.\nThe homogeneous spin chains are gapless and the only thermodynamic quantity, as far as the spectrum is concerned, is the vacuum energy per site. The latter does not provide any insight into the the continuous limit of the chain, which is expected to be governed by a conformal field theory (CFT). \nProbing this CFT requires computing scaling corrections to the spectrum~\\cite{Cardy:1986ie}.\nHowever, these are much harder to compute, e.g. see~\\cite{pearce}, then thermodynamic quantities. Therefore, gaining insight into the CFT solely from the lattice is quite non-trivial.\nTo avoid such complications, one can introduce a smooth gap in the spin chain.\nThe continuous limit is then expected to be a massive deformation of the CFT.\nMany interesting quantities, such as the $\\beta$-function, the particle spectrum, $S$-matrices, can now be computed in the thermodynamic limit.\nThe standard way to generate a gap in the homogeneous spin chains is by introducing an alternating inhomogeneity $\\Lambda$ in the monodromies~(\\ref{eq:mon_mat_a}, \\ref{eq:mon_mat_ab}), called staggering~\\cite{Faddeev:1996iy}. The source terms of the homogeneous BAE~\\eqref{eq:BAE_short} then change to \n\\begin{equation*}\n \\left(\\frac{x^{(k)}_j+i\/2}{x^{(1)}_j-i\/2}\\right)^L\\mapsto \\left(\\frac{x^{(k)}_j-\\Lambda\/2+i\/2}{x^{(k)}_j-\\Lambda\/2 -i\/2}\\right)^{L\/2} \\left(\\frac{x^{(k)}_j+\\Lambda\/2+i\/2}{x^{(k)}_j+\\Lambda\/2 -i\/2}\\right)^{L\/2}\\ ,\\qquad k=1,r\\ ,\n\\end{equation*}\nwhile the eigenvalues of the new energy and momentum operators become\n\\begin{align}\\label{eq:spec_ham_mom_stag}\n E &= -\\sum_{j=1}^{\\nu^{(1)}}\\frac{\\frac{\\sigma_1}{2}}{\\big(x^{(1)}_j -\\frac{\\Lambda}{2}\\big)^2+\\frac{1}{4}}+\\frac{\\frac{\\sigma_1}{2}}{\\big(x^{(1)}_j -\\frac{\\Lambda}{2}\\big)^2+\\frac{1}{4}}\n-\\sum_{j=1}^{\\nu^{(r)}}\\frac{\\frac{\\sigma_{M+N}}{2}}{\\big(x^{(r)}_j-\\frac{\\Lambda}{2}\\big)^2+\\frac{1}{4}}+\\frac{\\frac{\\sigma_{M+N}}{2}}{\\big(x^{(r)}_j+\\frac{\\Lambda}{2}\\big)^2+\\frac{1}{4}}\\\\\nP &\\equiv  \\frac{\\sigma_1}{2}\\sum_{i=1}^{\\nu^{(1)}}\\theta_1\\left(x^{(1)}_j-\\frac{\\Lambda}{2}\\right)+\\theta_1\\left(x^{(1)}_j+\\frac{\\Lambda}{2}\\right)+ \\frac{\\sigma_{M+N}}{2}\\sum_{i=1}^{\\nu^{(r)}}\\theta_1\\left(x^{(r)}_j-\\frac{\\Lambda}{2}\\right)+\\theta_1\\left(x^{(r)}_j+\\frac{\\Lambda}{2}\\right)\\ , \\notag\n\\end{align}\nwhere $\\sigma_1=(-1)^{|1|}$, $\\sigma_{M+N}=(-1)^{|M+N|}$ and we have assumed $L$ to be even. These expressions reduce the the previous ones~\\ref{eq:spec_ham_mom} in the limit $\\Lambda\\to0$.\n\nIn this section we define and study the continuous limit of the staggered $\\gl(n+N|N)$ chains of sec.~\\ref{sec:formalism}. First we show that the bosonic lift of the staggered chain is described in the continuous limit by the $\\gl(n)$ Gross-Neveu (GN) model~\\cite{Andrei:1979un, Andrei:1979wy, Andrei:1979sq}.\nThis strongly suggests that the continuous limit of the staggered spin chain coincides with the $\\gl(n+N|N)$ GN model, because the latter contains the $\\gl(n)$ GN model as a cohomological subsector~\\cite{Candu:2010yg}. The identification of the continuous limit is the main result of the paper.\nIn the second subsection we explore the particle content of the $\\gl(2m|1)$ GN model which does not lie in the $\\gl(2m-1)$ GN model lift, that is the bosonic lift. \n\n\n\n\n\n\n\\subsection{Bosonic lift}\n\nAs we have explained in sec.~\\ref{sec:rl} and~\\ref{sec:bos_lift}, the solutions of $\\gl(n)$ BAE corresponding to non-vanishing Bethe vectors admit a unique lift to solutions of $\\gl(n+N|N)$ BAE and Bethe vectors. We called this subsector of the $\\gl(n+N|N)$ spin chain the bosonic lift.\nThe uniqueness of the lift implies that the fermionic roots of the lifted solutions of $\\gl(n+N|N)$ BAE, in the form schematically represented in fig.~\\ref{fig:bos_lift}, are uniquely determined by the even roots.\nBy the restriction procedure of sec.~\\ref{sec:rl}, the latter solve $\\gl(n)$ BAE.\nTherefore, all dynamical degrees of freedom (associated to holes) are determined by the even roots.\nThe odd roots behave as auxiliary quantities useful for defining lifted Bethe vectors.\nSo, we conclude that the BAE of the $\\gl(n)$ spin chain describe \\emph{entirely} the bosonic lift of the $\\gl(n+N|N)$ spin chain.\nWe recall some of the old results on the continuum limit of  $\\gl(n)$ spin chains and then reinterpret them in the $\\gl(n+N|N)$ context.\n\n\\subsubsection{Vacuum energy}\\label{sec:vac_en}\n\n\n\nAccording to sec.~\\ref{sec:bos_lift}, the vacuum energy of the $\\gl(n+N|N)$ spin chain coincides exactly with the vacuum energy of the $\\gl(n)$ spin chain.\nThe vacuum energy $e_{\\infty}=\\lim_{L\\to \\infty} \\tfrac{E_{\\text{vac}}}{2L}$ of the homogeneous $\\gl(n)$ spin chain $V^{\\otimes L}$ was computed in~\\cite{Sutherland:1975vr}. This calculation is easily generalized for the staggered $\\gl(n)$ spin chains $(V\\otimes V^*)^{\\otimes L}$\n\\begin{align}\\notag\ne_{\\infty}&= -\\frac{1}{2n}\\left[\\psi(1+\\tfrac{i\\Lambda}{n})-\\psi(\\tfrac{1}{n}+\\tfrac{i\\Lambda}{n})+\\psi(\\tfrac{1}{2}+\\tfrac{1}{n}+\\tfrac{i\\Lambda}{n})-\\psi(\\tfrac{1}{2}+\\tfrac{i\\Lambda}{n}) +c.c\\right] \\\\\n&\\phantom{=}-\\frac{1}{n}\\left[\n\\psi(1)-\\psi(\\tfrac{1}{n})+\\psi(\\tfrac{1}{2}+\\tfrac{1}{n})-\\psi(\\tfrac{1}{2})\\right] \\ , \\label{eq:vac_en}\n\\end{align}\nwhere $\\psi$ is the digamma function.\nInserting $n=1$ and $\\Lambda=0$ we get the result $e_{\\infty}=-4$ of~\\cite{Essler:2005ag}.\nFor $n=2m-1$ the vacuum energy coincides exactly with the vacuum energy of the staggered $\\so(2m+1)$ and $\\osp(1|2m-2)$ fundamental chains of length $L$.\nThese coincidences where probably first noticed in~\\cite{Saleur:2001cw, Martins:1995bb}.  The correspondence between the spectra of $\\gl(2m|1)$, $\\gl(2m-1)$, $\\so(2m+1)$ and $\\osp(1|2m-2)$ chains is easily understood at the level of BAE from fig.~\\ref{fig:emb}.\n\\begin{figure}[t]%\n\\psfrag{1}{$1$}\n\\psfrag{2}{$2m$}\n\\psfrag{3}{$2m-2$}\n\\psfrag{4}{$m$}\n\\psfrag{5}{$m-1$}\n\\psfrag{d}{$\\cdots$}\n\\psfrag{g1}{$\\gl(2m|1)\\, (V\\otimes V^*)^{\\otimes L}$}\n\\psfrag{g2}{$\\gl(2m-1)\\, (V\\otimes V^*)^{\\otimes L}$}\n\\psfrag{g3}{$\\so(2m+1)\\, V^{\\otimes L}$}\n\\psfrag{g4}{$\\osp(1|2m-2)\\, V^{\\otimes L}$}\n\\centerline{\\includegraphics[scale=1]{embedding.eps}\\hfill}\n\\caption{Dynkin diagrams represent BAE for the corresponding chains.\nDouble arrows indicate the equality of the corresponding set of roots.\n}%\n\\label{fig:emb}%\n\\end{figure}\nThe non-trivial feature of the correspondence is that it relates low lying with low lying excitations.\nIt is exactly this type of correspondences that lie at the heart of the dualities observed in~\\cite{Saleur:2009bf}.\n\n\n\\subsubsection{Particle spectrum}\n\n\nLet us first recall some of the results on the continuous limit of $\\gl(n)$ chains with real vacuum solution.\nGoing on the lines of~\\cite{Andrei:1979wy}, the $\\gl(n)$ staggered spin chain $(V\\otimes V^*)^{\\otimes L}$ can be shown to posses $n-1$ gapped excitations branches associated with the holes in the distributions of $n-1$ types of roots. For $\\Lambda$ big, the gap is of order $e^{-\\pi \\Lambda\/n}$. The continuous limit is performed by introducing a physical lattice spacing $a$ and then by taking the limit $a\\to0, \\Lambda\\to\\infty$ such that\n\\begin{equation}\\label{eq:cont_limit}\nm= a^{-1} e^{-\\pi \\Lambda\/n} \n\\end{equation}\nremains fixed, defining the mass scale of the theory.\nIf we correct for the Fermi velocity $c=4\\pi\/n$, then in \nthe continuous limit each excitation branch yields a relativistic dispersion relation with mass\n\\begin{equation}\\label{eq:bos_particles}\n m_k = 2m\\sin \\frac{\\pi k}{n},\\qquad k=1,\\dots,n-1\\ .\n\\end{equation}\nTherefore, we interpret it as a particle in some relativistic quantum field theory.\nThe lowest lying excitation of the spin chain corresponding to a traceless tensor $t_0(\\lambda,\\mu)$ of admissible shape~\\eqref{eq:adm_shape_bos} is reproduced by a real solution of BAE with root and hole numbers fixed by the highest weight of the tensor~\\cite{Doikou:1998xi}.\nThis excitation is associated to a scattering eigenstate in the field theory of energy\n\\begin{equation}\\label{eq:lowest_lambla_mu}\n E_{\\lambda,\\mu}-E_{\\text{vac}}=\\sum_{i=1}^{\\lambda_1} m_{\\lambda'_i}\\cosh \\theta_i + \\sum_{i=1}^{\\mu_1} m_{\\mu'_i}\\cosh \\bar{\\theta}_i\\ ,\n\\end{equation}\nwhere $\\theta$, $\\bar{\\theta}$ are the physical rapidities parametrizing the hole positions. \nEq.~\\eqref{eq:lowest_lambla_mu} suggests that the $k$-th particle~\\eqref{eq:bos_particles} corresponds to either a covariant $1^k$ or contravariant $\\bar{1}^k$ antisymmetric tensor.\nThis is confirmed by an $S$-matrix calculation~\\cite{Andrei:1979sq, Doikou:1998xi}.\nThe end result is the  $S$-matrix of the $\\gl(n)$ Gross-Neveu model~\\cite{PhysRevD.20.897}.\nFor instance, $1$ and $\\bar{1}$ particles scatter with themselves with\nthe $S$-matrix\n\\begin{equation}\\label{eq:smat_11}\n S_{11}(\\theta)=S_{\\bar{1}\\bar{1}}(\\theta)=Z_{11}(\\theta)\\frac{P + i n \\theta\/2\\pi}{1 + i n \\theta\/2\\pi}\\ ,\\qquad S_{1\\bar{1}}=Z_{1\\bar{1}}(\\theta)\\left(1-\\frac{Q}{n\/2+in\\theta\/2\\pi}\\right)\\ ,\n\\end{equation}\nwhere $P$ and $Q$ are permutation and Temperley-Lieb operators defined in sec.~\\ref{sec:formalism} and \n\\begin{equation*}\n Z_{11}(\\theta)=\\frac{\\Gamma(1-1\/n-i\\theta\/2\\pi)\\Gamma(1+i\\theta\/2\\pi)}{\\Gamma(1-1\/n+i\\theta\/2\\pi)\\Gamma(1-i\\theta\/2\\pi)}\\ ,\\ Z_{1\\bar{1}}(\\theta)=\\frac{\\Gamma(1\/2-i\\theta\/2\\pi)\\Gamma(1\/2-1\/n+i\\theta\/2\\pi)}{\\Gamma(1\/2+i\\theta\/2\\pi)\\Gamma(1\/2-1\/n-i\\theta\/2\\pi)} \n\\end{equation*}\nare, so called, dressing factors.\nThe GN model coupling is related to the staggering $g=\\Lambda^{-1}$ and eq.~\\eqref{eq:cont_limit} yields its $\\beta$-function. \n\nThere is one important remark. Multiparticle states have zero total $\\un(1)=\\gl(n)\/\\sgl(n)$ charge, because the  spin chain $(V\\otimes V^*)^{\\otimes L}$ has zero $\\un(1)$ charge.\nTherefore, it is not possible to fix the $\\un(1)$ charge of the individual particles in the spin chain. The allowable multiparticle configurations in the spin chain make the value of this charge irrelevant. However, the field theories with $\\un(1)$ charged and uncharged particles~\\eqref{eq:bos_particles} are clearly different: the first one has $2n-2$ irreducible multiplets of particles, while the latter, which is the GN model, has $n-1$ irreducible multiplets.\n\nWe now reinterpret the results of the above calculations in the framework of the bosonic lift of the $\\gl(n+N|N)$ spin chain.\nThe continuous limit of the $\\gl(n+N|N)$ spin chain is defined by the same eq.~\\eqref{eq:cont_limit}. It already implies that the $\\beta$-function of the continuous theory does not depend on $N$.\nThe $n-1$ excitation branches of the bosonic lift are interpreted again as relativistic particles of\nmass~\\eqref{eq:bos_particles}.\nEq.~\\eqref{eq:lowest_lambla_mu} gives now the energy of a scattering eigenstate corresponding to a $\\gl(n+N|N)$ traceless tensor $t_0(\\lambda,\\mu)$ of $\\gl(n)$ admissible shape~\\eqref{eq:adm_shape_bos}.\nHowever now, it suggests that the bosonic lift contains $2n-2$ distinct multiplets of particles corresponding to the $\\gl(n+N|N)$ covariant antisymmetric tensors $1,1^2,\\dots,1^{n-1}$ and  their antiparticles, that is the contravariant antisymmetric tensors $\\bar{1},\\bar{1}^2,\\dots,\\bar{1}^{n-1}$.\nNotice that the number of particles does not depend on whether they carry or not a $\\un(1)$ charge, which again cannot be fixed in the spin chain. This is because the $\\sgl(n+N|N)$ antisymmetric tensors  $1^k$  and $1^{n-k}$  are  not isomorphic anymore. \n\nThe suggested $\\gl(n+N|N)$ symmetry of the particles in the bosonic lift must be confirmed by an $S$-matrix calculation. While we do not know the full $S$-matrix, one can compute its \\emph{restriction} to the bosonic lift.\nThe calculation is formally the same as for the $\\gl(n)$ spin chain, because the BAE are the same.\nThus, the eigenvalues of the $\\gl(n+N|N)$ spin chain $S$-matrix restricted to the bosonic lift coincide with the eigenvalues of the $\\gl(n)$ spin chain $S$-matrix.\nMore precisely, the eigenvalue of the $\\gl(n+N|N)$ spin chain $S$-matrix on a scattering eigenstate  corresponding to a $\\gl(n+N|N)$ traceless tensor $t_0(\\lambda,\\mu)$ of $\\gl(n)$-admissible shape does not depend on $N$.\nDue to the fact that the $Q$-cohomology of a tensor product of $\\gl(n+N|N)$ representations is the tensor product of their $Q$-cohomologies~\\cite{Candu:2010yg} all these eigenvalues are compatible with our earlier assumption that the particles of the bosonic lift are co- and contravariant antisymmetric tensors of $\\gl(n)$-admissible shape.\n\nFinally, the scattering of particles in the bosonic lift can be computed exactly when no scattering eigenstates outside the bosonic lift are generated. \nThus, particles $1$ and $\\bar{1}$ scatter with the same $S$-matrix~\\eqref{eq:smat_11}, where this time $P$ and $Q$ act on tensor products of $\\gl(n+N|N)$ representations.\nNotice that we did not assume any crossing or unitarity to derive $S_{1\\bar{1}}$ from $S_{11}$.\nThe question now is what field theory can reproduce these $S$-matrices? Given its formal similarity with the $\\gl(n)$ GN $S$-matrix, the most obvious candidate is the $\\gl(n+N|N)$ GN model.\nThere are deeper reasons to believe this.\nFirst of all, the $Q$-cohomology of the $\\gl(n+N|N)$ GN model is the $\\gl(n)$ GN model. In particular, their $\\beta$-functions coincide~\\cite{Candu:2010yg}.\nSecondly, there is no doubt that the perturbative $1\/n$ calculations of the $\\gl(n)$ GN $S$-matrix can be generalized. While we did not carry out an honest perturbative $S$-matrix calculation as in~\\cite{PhysRevD.20.897}, the result should certainly be~\\eqref{eq:smat_11}.\nThis is because the Feynman rules are  $\\gl(n+N|N)$ invariant tensors and their algebra, generated by tensor multiplication and contraction, is a representation of the walled Brauer algebra $B_{L,L}(n)$ of sec.~\\ref{sec:wb_sub} and~\\ref{sec:tens}. Thus, as long as one computes an invariant tensor, such as the $S$-matrix, starting from other invariant tensors, such as the Feynman rules, the result cannot depend on $N$.\nThirdly, the detailed analysis of~\\cite{Essler:2005ag, Saleur:2006tf} suggests that the continuum limit of the homogeneous $\\gl(2|1)$ spin chain is the $\\widehat{\\sgl}(2|1)_1$ WZNW model, that is the $\\gl(2|1)$ GN model at zero coupling.\nFinally, we mention the similarity with the situation for $\\osp(R|2S)$ GN models~\\cite{Bassi:1999ua, Saleur:2001cw, Saleur:2009bf}. \nTherefore, we conjecture that the continuous limit of the $\\gl(n+N|N)$ spin chain $(V\\otimes V^*)^{\\otimes L}$ is the $\\gl(n+N|N)$ GN model. \n\n\n\n\\subsection{Degree one excitation}\n\nIn this section we investigate degree 1 excitations of the $\\gl(2m|1)$ spin chain which are of the type\n~\\eqref{eq:conj_1} and tend to the form~\\eqref{eq:ideal_sol_1} in the thermodynamic limit.\nAs explained in sec.~\\ref{sec:deg1}, the lowest lying excitation of degree 1 is precisely of this type. It corresponds to the antisymmetric tensor $t_0(1^m,1^m)$ and its root numbers are represented in fig.~\\ref{fig:deg1}.\n\nIn the thermodynamic limit the roots\n\\begin{equation}\\label{eq:identification} \\{x_j^{(k)}\\}^{\\nu^{(k)}}_{j=1}=\\{x_j^{(2m+1-k)}\\}^{\\nu^{(2m+1-k)}}_{j=1}\\ , \\quad k=1,\\dots,m-1\n\\end{equation}\nand the centers $\\{x_j^{(\\pm)}\\}_{j=1}^{N_\\pm}$, of the $\\pm$-strange strings become dense on the real line.\nWe now derive the Lieb equations satisfied by their densities.\nCollecting all BAE, in the grading $[+^m-+^m]$, we get\n\\begin{align}\\label{eq:BAE_deg_1}\n -1&=e_{1}(x-\\Lambda\/2)^{L\/2}e_{1}(x+\\Lambda\/2)^{L\/2}E_2^{(1)}(x) E_{-1}^{(2)}(x)\\ ,& x&\\in\\{x_j^{(1)}\\}_{j=1}^{\\nu^{(1)}}\\\\\\notag\n-1 &= E_{-1}^{(k-1)}(x)E_{2}^{(k)}(x) E_{-1}^{(k+1)}(x)\\ ,\\quad k=2,\\dots,m-2\\ ,& x&\\in \\{x_j^{(k)}\\}_{j=1}^{\\nu^{(k)}}\\\\\\notag\n-1&=E_{-1}^{(m-2)}(x)^2E_2^{(m-1)}(x)^2E_{-\\frac{3}{2}}^{(+)}(x)E_{-\\frac{1}{2}}^{(+)}(x)E_{-\\frac{3}{2}}^{(-)}(x)E_{-\\frac{1}{2}}^{(-)}(x)\\ ,& x&\\in \\{x_j^{(m-1)}\\}_{j=1}^{\\nu^{(m-1)}}\\\\\n+1&=E_{-\\frac{3}{2}}^{(m-1)}(x)E_{-\\frac{1}{2}}^{(m-1)}(x)E_2^{(\\pm)}(x)E_1^{(\\mp)}(x)^2\\ ,& x&\\in\\{x_j^{(\\pm)}\\}_{j=1}^{N_\\pm}\\notag\n\\end{align}\nwhere $E_t^{(k)}(x)=\\prod_{i=1}^{\\nu^{(k)}}e_t(x-x^{(k)}_j)$, for $k=1,\\dots,m-1,\\pm$.\nThe first two sets of BAE equations is just a rewriting of eqs.~\\eqref{eq:BAE_short}. To get the last three we have multiplied the equations for same root appearing symmetrically in the Dynkin diagram~\\ref{fig:deg1} at positions $m-1$ and $m+2$ or $m$ and $m+1$. \nNotice the $+$ sign appearing in the last two sets of equations.\n\nDefine the densities of roots $\\rho_k(x^{(k)}_j):=\\lim_{L\\to \\infty}1\/L(x^{(k)}_{j+1}-x^{(k)}_j)$ and holes $\\rho_k^h(x):=\\sum_{j=1}^{n^{(k)}}\\delta(x-\\xi^{(k)}_j)\/L$ for\n$k=1,\\dots,m-1,\\pm$.\nTaking the logarithm of eqs.~\\eqref{eq:BAE_deg_1} and derivating w.r.t. the spectral parameters we get the Lieb equations\n\\begin{align}\\label{eq:lieb_deg1}\n \\tfrac{1}{2}a_1*\n (\\delta_{\\frac{\\Lambda}{2}}+ \\delta_{-\\frac{\\Lambda}{2}})&=\\rho_1^h+\\rho_1+a_2*\\rho_1-a_1*\\rho_2\\\\ \\notag\n0&=\\rho_k^h +\\rho_k +a_2* \\rho_k - a_1*(\\rho_{k-1}+\\rho_{k+1})\\ ,\\quad k=2,\\dots,m-2\\\\ \\notag\n0&=\\rho_{m-1}^h + \\rho_{m-1}+a_2*\\rho_{m-1}-a_1*\\rho_{m-2}-\\tfrac{1}{2}(a_{\\frac{3}{2}}+a_{\\frac{1}{2}})*(\\rho_++\\rho_-)\\\\ \\notag\n0&=\\rho^h_{\\pm} + \\rho_{\\pm} +a_2* \\rho_\\pm +2 a_1* \\rho_\\mp - (a_{\\frac{3}{2}}+a_{\\frac{1}{2}})* \\rho_{m-1}\\ ,\n\\end{align}\nwhere $\\delta_x$ is the Dirac distribution centered at $x$, $a_t(x)=\\tfrac{1}{2\\pi}\\tfrac{d\\theta_t(x)}{dx}=\\tfrac{1}{\\pi}\\tfrac{t\/2}{x^2+t^2\/4}$ and $*$ denotes the convolution $(f*g)(x)=\\int_{-\\infty}^{+\\infty}dy\\,f(x-y)g(y)$.\nFrom eq.~\\eqref{eq:spec_ham_mom_stag}, the energy and momentum of a root configuration~\\eqref{eq:lieb_deg1} is \n\\begin{align*}\n E&=-2\\pi L\\int_{-\\infty}^{\\infty} dx\\, \\rho_1(x)[a_1(x+\\Lambda\/2)+a_1(x-\\Lambda\/2)]\\\\\nP&=L\\int_{-\\infty}^\\infty dx\\,\\rho_1(x)[\\theta_1(x+\\Lambda\/2)+\\theta_1(x-\\Lambda\/2)]\n\\end{align*}\n\nTo solve for the root densities in terms of the hole densities it is useful to define the distributions\n\\begin{align*}\n\\rho_s&=\\frac{\\rho_+ +\\rho_-}{2}\\ ,& \\rho_a&=\\frac{\\rho_+ -\\rho_-}{2}\\\\\n\\rho^h_s&=\\frac{\\rho^h_+ +\\rho^h_-}{2}\\ ,& \\rho^h_a&=\\frac{\\rho^h_+ -\\rho^h_-}{2}\\ .\n\\end{align*}\nNotice that the distributions $\\rho^h_a$, $\\rho_a$ split from the rest of eqs.~\\eqref{eq:lieb_deg1}\n\\begin{equation}\\label{eq:sing_anti}\n 0=\\rho_a^h +\\rho_a+a_2*\\rho_a -2a_1*\\rho_a\\ ,\n\\end{equation}\nwhile the Lieb equations for the remaining densities $\\rho_1,\\dots,\\rho_{m-1},\\rho_s$ look exactly like the Lieb equations for the staggered $\\so(2m+1)$ fundamental  chain of length $L$, e.g.~\\cite{Martins:1990ca}. The resolvent for the latter can be found in~\\cite{ORW}.\nComputing the solution of eqs.~\\eqref{eq:BAE_deg_1} with no holes one gets\n\\begin{align}\\label{eq:no_holes_deg1}\n\\rho^0_k(x)&=\\frac{2}{n}\\frac{\\sin\\tfrac{\\pi k}{n}\\cosh\\pi\\tfrac{2x+\\Lambda}{n}}{\\cosh\\tfrac{2\\pi(2x+\\Lambda)}{n} -\\cos \\tfrac{2\\pi k}{n}}+\\frac{2}{n}\\frac{\\sin\\tfrac{\\pi k}{n}\\cosh\\pi\\tfrac{2x-\\Lambda}{n}}{\\cosh\\tfrac{2\\pi(2x-\\Lambda)}{n} -\\cos \\tfrac{2\\pi k}{n}}\\ ,\\\\\n\\rho^0_s(x)&=\\frac{1}{2n}\\frac{1}{\\cosh \\tfrac{\\pi(2x+\\Lambda)}{n}}+\n\\frac{1}{2n}\\frac{1}{\\cosh \\tfrac{\\pi(2x-\\Lambda)}{n}}\\ , \\quad k=1,\\dots,m-1\\ ,\\notag\n\\end{align}\nwhere $n=2m-1$.\nWe want to stress that this is a \\emph{formal} solution that does not correspond to any state in the spin chain. This is because the degree 1 excitations we are considering always contain holes.\nHowever, it is  important to realize that the energy of the formal solution~\\eqref{eq:no_holes_deg1} coincides with the actual vacuum energy~\\eqref{eq:vac_en}.  We have already explained in sec.~\\ref{sec:vac_en} why this happens.\nAs usual, the energy and momenta of holes can be expressed in terms of hole less densities\n\\begin{align*}\n\\epsilon_k (\\xi) &= 4\\pi \\rho_k^0(\\xi)\\ ,& \\frac{d p_k(\\xi)}{d\\xi}&=2\\pi \\rho_k^0(\\xi)\\\\\n\\epsilon_\\pm (\\xi)&=2\\pi \\rho^0_s(\\xi)\\ ,& \\frac{d p_\\pm(\\xi)}{d\\xi}&=\\pi \\rho_k^0(\\xi)\n\\end{align*}\nRestricting to low lying states $|\\xi|\\ll \\Lambda$ one gets the expected relativistic dispersion relations\n\\begin{align}\\label{eq:old_particles}\n\\epsilon_k(\\theta) &= 2c a m_k \\cosh\\theta \\ ,& p_k(\\theta) & = 2a m_k \\sinh \\theta \\\\\n\\epsilon_\\pm(\\theta) &= c a m \\cosh\\theta \\ ,& p_\\pm(\\theta) & = a m \\sinh \\theta \\label{eq:new_particles}\n\\end{align}\nwith the $\\gl(n)$ Fermi velocity $c=4\\pi\/n$ and physical rapidity $\\theta = 2\\pi \\xi\/n$.\nThe masses are the same as in eqs.~(\\ref{eq:cont_limit}, \\ref{eq:bos_particles}).\n\nIt is certainly reassuring that degree 1 excitations have the same mass scale $m$ as degree 0 excitations. The doubling of masses~\\eqref{eq:old_particles} w.r.t. the masses of particles in the bosonic lift can be understood as follows.\nWe have observed in sec.~\\ref{sec:low_lying_deg1} that the imaginary parts of the odd roots tend to vanish for large $L$ and, therefore, we have neglected them in the thermodynamic limit.\nHowever, due to the conjugation symmetry \\eqref{eq:conj_1} of the solutions, this approximation leads to an identification~\\eqref{eq:identification} of \\emph{distinct} roots.\nTherefore, the positions of holes in the distribution of even roots $\\{x_j^{(k)}\\}_{j=1}^{\\nu^{(k)}}$ and $\\{x_j^{(2m+1-k)}\\}_{j=1}^{\\nu^{(2m+1-k)}}$ have also been identified.\nSo, the excitation~\\eqref{eq:old_particles} is in fact a state of two particles $1^k$ and $\\bar{1}^k$ each of mass $m_k$ and rapidity $\\theta$.\nThese are the old particles from the bosonic lift.\nOn the other hand, the particles~\\eqref{eq:new_particles} are new.\n\nThe artificial binding of holes~\\eqref{eq:old_particles} induced by the thermodynamic limit does not allow to derive meaningful $S$-matrices from eqs.~\\eqref{eq:lieb_deg1}\nMost probably, to correct the approach and recover the degrees of freedom lost in the thermodynamic limit one needs to distinguish between the two types of roots $\\Img x_j^{(k)}>0$ and $\\Img x_j^{(k)} < 0$, exactly as we did for the strange $\\pm$-strings~\\eqref{eq:ideal_sol_1}.\nHowever, a finite volume treatment will be required, because the imaginary parts of even roots vanish in the thermodynamic limit.\n\nThe need for a finite volume approach can also be seen from eq.~\\eqref{eq:sing_anti}.\nSolving for the Fourier transform of $\\hat{\\rho}_a(p)=\\int dp\\,\\exp(-2\\pi i p x)\\rho_a(x)$ one gets\n\\begin{equation}\\label{eq:dist_anti_four}\n\\hat{\\rho}_a(p\\mid\\xi^{(+)},\\xi^{(-)})=\\frac{i}{2L}e^{\\pi |p|-\\pi i p (\\xi^{(+)}+\\xi^{(-)})}\\frac{\\sin \\pi p (\\xi^{(+)}-\\xi^{(-)})}{\\sinh^2 \\tfrac{\\pi p }{2}}\\ ,\n\\end{equation}\nwhere we have considered only a single pair of $\\pm$-holes.\nThis is no loss of generality, because from~\\eqref{eq:lieb_deg1} the numbers of $\\pm$-holes is always equal\n\\begin{equation*}\nn^{\\pm}=2(\\nu^{(m-1)}-N_+-N_-) = 2(\\nu^{(m-1)}-\\nu^{(m)})\\ .\n\\end{equation*}\nand, therefore, they always come in pairs.\nThe Fourier transform~\\eqref{eq:dist_anti_four} is singular and clearly must be regularized, because one has to satisfy the constraint\n\\begin{equation}\\label{eq:charge_finally_found}\nb:=\\lim_{L\\to \\infty} \\frac{N_+-N_-}{2L} = L \\int_{-\\infty}^{\\infty} dx\\, \\rho_a(x)\t=\\hat{\\rho}_a(0)\\ ,\n\\end{equation}\nwhere $b$ is a fixed real number parametrizing the state.\nOn a lattice of length $L$ the the momentum can take a minimal value of $p_\\sim 1\/L$. Using this value as a regulator one gets for a pair of $\\pm$-holes from eqs.~(\\ref{eq:dist_anti_four} \\ref{eq:charge_finally_found}) \n\\begin{equation*}\n\\xi^{(+)}-\\xi^{(-)}=\\frac{\\pi b}{2i}\\ ,\n\\end{equation*}\nwhich clearly shows that for $b\\neq 0$ the deviations of strange $\\pm$-strings from the form~\\eqref{eq:ideal_sol_1} of the solution in the thermodynamic limit have to be taken into account.\nNotice that the energy does not depend on the continuous parameter~\\eqref{eq:charge_finally_found} parametrizing the state.\nTherefore, it is tempting to conclude that there is a \\emph{continuum} of new particles~\\eqref{eq:new_particles}.\n\n\n\n\\section{Conclusions and Outlook}\n\n\n\nWe have put on firm grounds the relationship between $\\gl(n+N|N)$ integrable spin chains with $n$ fixed.\nThis allowed us to prove that all $\\gl(n+N|N)$ spin chains $(V\\otimes V^*)^{\\otimes L}$ with $n,N>0$ possess in the continuum limit $2n-2$ multiplets of massive particles which scatter with $\\gl(n)$ Gross-Neveu like $S$-matrices, namely their eigenvalues do not depend on $N$.\nWe concluded that the continuum theory is the $\\gl(M|N)$ Gross-Neveu model.\nEvidence that the massive spectrum is much richer, possibly continuous, was established on the example of $\\gl(2m|1)$ chain.\nFinally, our analysis of the thermodynamic limit strongly suggests that understanding the nature of new particles requires a finite volume treatment.\n\nThe question that begs the quickest answer is how to close the fusion\nof $S$-matrices~\\eqref{eq:smat_11} of the $\\gl(M|N)$ Gross-Neveu model starting with just the vector multiplet and its antiparticles.\n\nThe $\\gl(N|N)$ spin chains require a separate treatment, which we hope to report on later.\\\\\n\n\\noindent\\textbf{Acknowledgements.} I would like to greatly thank Hubert Saleur, Volker Schomerus, Sergei Lukyanov and J\\\"org Techner for helpful discussions, important guidance and critical feedback. I am also grateful to Fabian  E{\\ss}ler, Holger Frahm and Nikolay Gromov for sharing some of their expertise in solving numerically BAE.\nThe author thanks the Rutgers NHET center for their hospitality, where an important part of this paper was written, and SFB676 for partial financial support.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $G=(V(G),E(G))$ be an undirected simple graph.\nThe \\textit{adjacency matrix}  of $G$ is the $n \\times n$ matrix   $A(G)=(a_{ij})$, where $a_{ij}=1$\nif $v_{i}$ is adjacent to $v_{j}$, and $0$ otherwise.\nThe \\textit{$Q$-matrix} (or \\textit{signless Laplacian matrix}) of $G$ is defined as $Q(G)=D(G)+A(G)$,\nwhere $D(G)$ is the diagonal matrix of vertex degrees of $G$. The largest eigenvalue of $Q(G)$,\ndenoted by $q(G)$, is called the \\textit{$Q$-index} (or \\textit{signless Laplacian spectral radius}) of $G$.\nFor two vertex disjoint graphs $G$ and $H$, we denote by $G \\cup H$ the union of $G$ and $H$, and $G \\vee H$  the join of $G$ and $H$, i.e., joining every vertex of $G$ to every vertex of $H$.\nDenote by $kG$ the union of $k$ disjoint copies of $G$.\nAs usual, denote by $K_{n}$ a \\textit{complete graph} of order $n$, and $K_{m, n}$ a \\textit{complete bipartite graph} on $m + n$ vertices.\nLet $F_{s, t}(n):\\cong K_{s-1} \\vee(p K_{t}\\cup K_{r})$, where $2 \\leq s \\leq t, n-s+1=p t+r$ and $1\\leq r\\leq t$.\nIt is easy to check that $F_{s, t}(n)$ is a $K_{s, t}$-minor free graph of order $n$. \\\nFor a graph $G$, let $\\overline{G}$ be its complement. Denote by $S^{1}(G)$ a graph obtained from a graph $G$ by subdividing once of an edge $uv$ with minimum degree sum $d_{G}(u)+d_{G}(v)$.\nDenote by $H^{\\star}$ the Petersen graph.\nLet $H_{s, t}:\\cong(\\beta-1)K_{1,s} \\cup K_{1,\\alpha}$, where  $1\\leq s\\leq t$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ and $\\alpha=t-(\\beta-1)(s+1)\\geq s$.\nGiven two graphs $G$ and $H$, $H$ is a \\textit{minor} of $G$\nif $H$ can be obtained from a subgraph of $G$ by contracting edges.\nA graph is said to be \\textit{$H$-minor free} if it does not contain $H$ as a minor.\n\nMinors play a key role in graph theory, and extremal problems on forbidding\nminors have attracted appreciable amount of interests.\nFirstly, it is very useful for\nstudying the structures and properties of graphs. For example,\nevery planar graph is $\\{K_{3,3}, K_5\\}$-minor free and every outer-planar graph is $\\{K_{2,3}, K_4\\}$-minor free.\nSecondly, one of the problems in extremal graph theory is to concern about the maximum\nnumber of edges for graphs that do not contain a given $H$ as a minor. It is known that a\nplanar graph has at most $3n-6$ edges and an outer planar graph has at most $2n-3$ edges, see \\cite{A.B}.\n\nIn 2017, Nikiforov \\cite{Nikiforov2} provided a unified extension of both the adjacency spectral\nradius and the signless Laplacian spectral radius. For a graph $G$, it was proposed by Nikiforov \\cite{Nikiforov2}\nto study the family of matrices $A_{\\alpha}(G)$ defined as\n$$A_{\\alpha}(G) = \\alpha D(G) + (1 - \\alpha)A(G),$$\nwhere $0\\leq\\alpha \\leq1$.\nThe \\textit{$A_\\alpha$-spectral radius} (or \\textbf{$\\alpha$-index} ) of $G$, denoted by $\\rho_{\\alpha}(G)$, is the largest eigenvalue of $A_{\\alpha}(G)$.\nNikiforov \\cite{Nikiforov2} posed the $A_{\\alpha}$-spectral extrema problem:\n\\begin{prob}\\label{prob-0}\nGiven a graph $F$, what is the maximum $A_\\alpha$-spectral radius of a graph $G$ of order $n$, with no\nsubgraph isomorphic to $F$?\n\\end{prob}\nAt present, for $0 \\leq \\alpha<1$, the $A_{\\alpha}$-spectral extrema problem is done when $F$ is a $K_r$ (see \\cite{Nikiforov2}), $F$ is a $K_r$-minor (see \\cite{M.T} and \\cite{C.M}), and $F$ is a star forest (see \\cite{C.M2} and \\cite{C.M}). In this paper, we pay our attention on Problem \\ref{prob-0}\nwhen $F$ is a $K_{s,t}$-minor. For special value $\\alpha\\in \\left\\lbrace 0, \\frac{1}{2}\\right\\rbrace$, there are plentiful results.\n\n\n\n\n\n\n\nFor the special value $\\alpha=0$, then $A_\\alpha(G) = A(G)$ of $G$.\nIn 2007,\nNikiforov \\cite{Nikiforov1} (resp. Zhai and Wang \\cite{M.Q1} ) obtained a sharp upper bound of adjacency  spectral radius over all $K_{2,2}$-minor free graphs of odd (resp. even) order, and determined the extremal graph.\nIn 2017, Nikiforov \\cite{Nikiforov}\nestablished a sharp upper bound of adjacency  spectral radius over all $K_{2,t}$-minor free graphs of large\norder $n$ for $t\\ge3$, and determined the extremal graph when $t \\mid n-1$. In addition, Nikiforov determined the extremal graph when $t=3$ for all $n$. In 2019,\nTait \\cite{M.T}\nobtained an upper bound of adjacency spectral radius\nover all  $K_{s,t}$-minor free graphs of large order $n$ for  $2\\leq s\\leq t$, and determined  the   extremal graph when\n$t \\mid  n-s+1$. In 2021, Wang, Chen and Fang \\cite{B.W} determined the extremal graph with maximum adjacency spectral radius\nover all  $K_{3,3}$ (resp. $K_{2,4}$)-minor free graphs of large order.\nIn 2022, Zhai and Lin \\cite{M.Q} improved  Tait's result \\cite{M.T} by removing the condition $t \\mid n-s+1$.\nMeanwhile, they\ndetermined the extremal graph with maximum adjacency spectral radius over all $K_{1,t}$-minor free graphs.\nThus the adjacency spectral extremal problem on $K_{s,t}$-minor free graphs\nis completely solved for large order.\n\n\nFor the special values $\\alpha=\\frac{1}{2}$, then $A_\\alpha(G) = Q(G)$ of $G$.\nIn 2022, Zhang and Lou \\cite{Y.T} determined the unique extremal graph with maximum $Q$-index among all $n$-vertex connected $K_{1,t}$-minor free graphs.\nFor $2 \\leq s \\leq t$, Chen and Zhang proposed the following conjecture in \\cite{C.M1}.\n\\begin{conj}(\\cite{C.M1})\\label{conj::1}\nLet $2 \\leq s \\leq t$ and $G$ be a $K_{s,t}$-minor free graph of sufficiently large order $n$. Then\n$$q(G) \\leq q(F_{s,t}(n))$$\nwith equality if and only if $G \\cong F_{s,t}(n)$.\n\\end{conj}\nThe previous results showed that the Conjecture \\ref{conj::1} is true in several cases. In 2013, Freitas, Nikiforov and Patuzzi \\cite {M.F} showed that $F_{2,2}(n)$ is the extremal graph with maximum $Q$-index among all $K_{2,2}$-minor free graphs for $n \\geq 4$.\nIn 2021, Chen and Zhang \\cite {C.M1} showed that $F_{2,3}(n)$ (resp. $F_{3,3}(n)$) is the extremal graph with maximum $Q$-index among all $K_{2,3}$ (resp. $K_{3,3}$)-minor free graphs for $n \\geq 22$ (resp. $n \\geq 1186$).\nThey also obtained an upper bound of $Q$-index\nover all $K_{2,t}$-minor free graphs of order $n\\geq t^{2} +4t+1$ with $t\\ge 3$, and proved the extremal graph is $F_{2,t}(n)$ when $t \\mid n-1$.\n\nHowever, for general $\\alpha\\in(0,1)$,\nthe related results are few. In 2022, Chen, Liu and Zhang \\cite{C.M}\nobtained an upper bound of $A_\\alpha$-spectral radius over all  $K_{s,t}$-minor free graphs of large order $n$ for  $2\\leq s\\leq t$, and proved that $F_{s,t}(n)$ is the extremal graph when\n$t \\mid  n-s+1$.\nTherefore,\nProblem \\ref{prob-0} is still open when $F$ is a $K_{s,t}$-minor for $t \\nmid  n-s+1$.\nNaturally, we want to overcome the following problem:\n\\begin{prob}\\label{1.1}\nFor $2\\leq s\\leq t$, what is the extremal graph with maximum $A_\\alpha$-spectral radius among all $K_{s, t}$-minor free graphs of sufficiently large order for $0 < \\alpha <1$?  Whether the extremal graph is consistent for $0 \\leq \\alpha <1$?\n\\end{prob}\n\n\n\nIn this paper, the \\emph{$A_\\alpha$-spectral extremal graph} is defined by a graph with maximum $A_\\alpha$-spectral radius among all $K_{s, t}$-minor free graphs of sufficiently order $n$ for $0 < \\alpha<1$ and $2\\leq s\\leq t$.\nIn order to characterize the structure of the {$A_\\alpha$-spectral extremal graph,\nour first challenge is to show that\nthe $A_\\alpha$-spectral extremal graph contains a $(s-1)$-clique dominating set, the way of which is different from the adjacency spectra. In addition, we prove that the $A_\\alpha$-spectral extremal graph has a property of\nlocal edge maximality. Meanwhile, we apply  the double eigenvectors transformation technique to the $A_{\\alpha}(G)$-matrix and get that\nthe $A_\\alpha$-spectral extremal graph has a property of local degree sequence majorization.\nFinally, we completely determined the $A_\\alpha$-spectral extremal graph as follows.\n\n\\begin{thm}\\label{thm::1.1}\nLet $0<\\alpha<1$, $2 \\leq s \\leq t$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ and $G^{*}$ be a $K_{s,t}$-minor free graph of sufficiently large order $n$ with the maximum $A_\\alpha$-spectral radius, where $n-s+1=pt+r$ and $1\\leq r\\leq t$. Then\n$$\nG^{*} \\cong \\begin{cases}K_{s-1} \\vee \\left((p-1) K_{t} \\cup \\overline{H^{\\star}}\\right) & \\text { if } r=2, t=8 \\text { and } \\beta=1 ; \\\\ K_{s-1} \\vee\\left( (p-1) K_{t} \\cup S^{1}\\left(\\overline{H_{s, t}}\\right)\\right) & \\text { if } r=\\beta=2; \\\\ K_{s-1} \\vee\\left((p-r) K_{t} \\cup r \\overline{H_{s, t}}\\right) & \\text { if } r \\leq 2(\\beta-1) \\text { except } r=\\beta=2 ; \\\\ K_{s-1} \\vee\\left(p K_{t} \\cup K_{r}\\right) & \\text { otherwise. }\\end{cases}\n$$\n\\end{thm}\n\nTaking $\\alpha=\\frac{1}{2}$ in Theorem \\ref{thm::1.1},  we  completely solve Conjecture \\ref{conj::1}.\nOn the other hand, we completely answer the front part of Problem \\ref{1.1}. Combining the result of the extremal graph with maximum adjacency spectral radius among all $K_{s,t}$-minor free graphs in \\cite{M.Q},\nwe give a positive answer to the last part of Problem \\ref{1.1}.\n\n\n\nOur proofs are based on the structural analysis of the $A_\\alpha$-spectral extremal graph $G^{*}$, which is motivated by the work of Zhai and Lin in \\cite{M.Q}. Some special notations, terminologies and lemmas will be presented in Section 2. We also prove that $G^{*}$ has a $(s-1)$-clique dominating set in Section 2. The proofs of $G^{*}$ has the structural properties of local edge maximality and local degree sequence majorization are shown in Section 3. The proof of Theorem \\ref{thm::1.1} will be shown in Section 4.\n\n\n\\section{Preliminaries}\nIn this section, we will list some symbols and  useful lemmas. Let $\\pi(G)=\\left( d_{1}, d_{2},\\dots , d_{n}\\right) $ be the non-increasing degree sequence of an $n$-vertex graph $G$.\nFor $A, B \\subseteq V(G)$, $e(A, B)$ is denoted the number of the edges of $G$ with one end vertex in $A$ and the other in $B$.\nFor a vertex  $v \\in V(G)$, we write $N_{G}(v)$ for the set of neighbors of $v$ in $G$. Let $d_{G}(v)$ be the degree of a vertex  $v$ in $G$. Let $H$ be a subgraph of $G$, we write $N_{H}(v)$ for the set of neighbors of $v$ in $V(H)$, and  $d_{H}(v)$  for the number of neighbors of  $v$  in $V(H)$, that is,  $d_{H}(v)=\\left|N_{H}(v)\\right|=|N_{G}(v) \\cap V(H)|$.  Let $\\Delta(G)$ and $\\delta(G)$ denote the maximum degree and minimum degree of $G$, respectively.\nFor $A \\subseteq V(G)$, the graph $G[A]$ is the \\textit{induced subgraph} by $A$. $G[A]$ is called a clique if it is a complete subgraph of $G$. Let $G$ be an $n$-vertex graph and $A\\subseteq V(G)$, then $A$ is called a \\textit{clique dominating set} of $G$, if $d_{G}(v)=n-1$ for any $v\\in A$.\nLet $K_{n}-e$ be a graph obtained by deleting one edge from a complete graph $K_{n}$.\nFor graph notations and concepts undefined here, readers are referred to \\cite {A.B}.\n\n\n\\begin{lem}(\\cite{Nikiforov2})\\label{lem::2.2}\nLet $0\\leq\\alpha<1$, then the $\\alpha$-index of any proper subgraph of a connected graph is smaller than the $\\alpha$-index of the original graph.\n\\end{lem}\n\nThe following result is from  the proof of Theorem 1.2 in \\cite{C.M}.\n\n\\begin{lem}(\\cite{C.M})\\label{lem::2.1}\nLet  $G$ be a $K_{s,t}$-minor free graph of sufficiently large order $n$ with maximum $\\alpha$-index, where $2 \\leq s \\leq t$ and  $0 < \\alpha < 1$. Then $G$ contains a vertex set $K=\\left\\{v_{1}, v_{2}, \\ldots, v_{s-1}\\right\\}$ such that all of $v_{i}$ have common neighborhood of size $n-s+1$ in $G$. That is, $d_{G-K}\\left(v_{i}\\right)=n-s+1$ for $i \\in\\{1,2, \\ldots, s-1\\}$.\n\\end{lem}\n\n\n\\begin{lem}(Lemma 2.1, \\cite{C.M})\\label{lem::2.1'}\nLet $0<\\alpha<1, s \\geq 2$, and $n \\geq s-1$. If $G=K_{s-1} \\vee \\overline{K}_{n-s+1}$, then $\\rho_\\alpha(G) \\geq \\alpha(n-1)+(1-\\alpha)(s-2)$.\n\\end{lem}\n\n\nRecall that $F_{s, t}(n):\\cong K_{s-1} \\vee(p K_{t}\\cup K_{r})$, where $2 \\leq s \\leq t, n-s+1=p t+r$ and $1\\leq r\\leq t$.\n\n\\begin{lem}\\label{lem::2.3}\nLet $0 < \\alpha <1$, $2 \\leq s \\leq t$ and $n \\geq  s-1+\\frac{t^{2}-1}{\\alpha}$. Then\n$\\rho_{\\alpha}(F_{s, t}(n))$ is no more than the largest root of $g(x)=0$\nand $\\rho_{\\alpha}(F_{s, t}(n))$ is larger than the largest root of $h(x)=0$, where\n\\begin{equation*}\n\\begin{aligned}\nh(x)=&x^2-(\\alpha n+s+t-3) x+\n(\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)\\\\&-(1-\\alpha)^2(s-1)(n-s),\n\\end{aligned}\n\\end{equation*}\nand\n$\ng(x)=h(x)-(1-\\alpha)^2(s-1).\n$\n\\end{lem}\n\n\\begin{proof}\nLet $\\rho_{\\alpha}=\\rho_{\\alpha}(F_{s, t}(n))$ and\n$\\mathbf{x}$ be a positive eigenvector of $A_{\\alpha}(F_{s, t}(n))$ corresponding to $\\rho_{\\alpha}$. By symmetry and the Perron-Frobenius theorem, all vertices of subgraphs $K_{s-1}$, $p \\cdot K_{t}$, or $K_{r}$ in $F_{s, t}(n):=K_{s-1} \\vee(p \\cdot K_{t} \\cup K_{r} )$ have the same eigenvector components respectively, which are denoted by $x_{1}$, $x_{2}$, $x_{3}$, respectively. We consider the following two cases.\n\t\n{\\flushleft\\bf Case 1. $r=t$.} By $A_{\\alpha}(F_{s, t}(n)) \\mathbf{x}=\\rho_{\\alpha} \\mathbf{x}$, it is easy to see that\n$$\n\\begin{aligned}\n&(\\rho_{\\alpha}-\\alpha(n-1)-(1-\\alpha)(s-2)) x_{1} =(1-\\alpha)(n-s+1) x_{2}, \\\\\n&(\\rho_{\\alpha}-\\alpha(s+t-2)-(1-\\alpha)(t-1))x_{2} =(1-\\alpha)(s-1) x_{1}.\n\\end{aligned}\n$$\nThen $\\rho_{\\alpha}$ is the largest root of $g(x)=0$, where\n$\ng(x)=h(x)-(1-\\alpha)^2(s-1).\n$\nSince $0 <\\alpha <1$ and $s \\geq2$, we find that $\\rho_{\\alpha}$ is larger than the largest root of $h(x)=0$.\n\t\n\t\n{\\flushleft\\bf Case 2. $1 \\leq r<t$.} By $A_{\\alpha}(F_{s, t}(n))\\mathbf{x}=\\rho_{\\alpha} \\mathbf{x}$, it is easy to see that\n$$\n\\begin{aligned}\n&(\\rho_{\\alpha}-\\alpha(n-1)-(1-\\alpha)(s-2)) x_{1} =(1-\\alpha)(n-s-r+1)x_{2}+(1-\\alpha)rx_{3}, \\\\\n&(\\rho_{\\alpha}-\\alpha(s+t-2)-(1-\\alpha)(t-1))x_{2} =(1-\\alpha)(s-1) x_{1},\\\\\n&(\\rho_{\\alpha}-\\alpha(s+r-2)-(1-\\alpha)(r-1))x_{3}=(1-\\alpha)(s-1) x_{1}.\n\\end{aligned}\n$$\nThen $\\rho_{\\alpha}$ is the largest root of $f(x)=0$, where\n$$\nf(x)=(x-\\alpha(s+r-2)-(1-\\alpha)(r-1)) g(x)-r(\\alpha-1)^{2}(s-1)(r-t).\n$$\nSince $\\rho_{\\alpha}>\\rho_{\\alpha}\\left(K_{s+r-1}\\right)=s+r-2$, $0 < \\alpha <1$ and $s\\geq2$, we have\n$$\\rho_{\\alpha}-\\alpha(s+r-2)-(1-\\alpha)(r-1)>(s+r-2)-\\alpha(s+r-2)-(1-\\alpha)(r-1)=(1-\\alpha)(s-1)>0.$$\nMoreover, since $1 \\leq r<t$, we see that\n$$\ng(\\rho_{\\alpha})=\\frac{r(\\alpha-1)^{2}(s-1)(r-t)}{\\rho_{\\alpha}-\\alpha(s+r-2)-(1-\\alpha)(r-1)}<0,\n$$\nwhich implies that $\\rho_{\\alpha}$ is less than the the largest root of $g(x)=0$.\n\t\nMoreover, we see that\n\\begin{equation*}\n\\begin{aligned}\nf(x)=&(x-\\alpha(s+r-2)-(1-\\alpha)(r-1)) h(x)-\n(x + 1 -\\alpha (s - 1) - r)(s - 1)(\\alpha-1)^2\\\\&-r(\\alpha-1)^{2}(s - 1)(r-t).\n\\end{aligned}\n\\end{equation*}\nThus\n$$\nh(\\rho_{\\alpha})=\\frac{[\\rho_{\\alpha}+1- \\alpha (s-1)- r(1-r+t)](\\alpha-1)^{2}(s-1)}{\\rho_{\\alpha}-\\alpha(s+r-2)-(1-\\alpha)(r-1)}.\n$$\nNotice that $G$ contains $K_{s-1} \\vee \\overline{K}_{n-s+1}$ as a proper subgraph. By Lemma \\ref{lem::2.1'}, we have\t\n\\begin{equation*}\n\\begin{aligned}\n\\rho_{\\alpha}>&\\rho_{\\alpha}\\left(K_{s-1} \\vee \\overline{K}_{n-s+1}\\right)\\geq \\alpha(n-1)+(1-\\alpha)(s-2).\n\\end{aligned}\n\\end{equation*}\nRecall that $0 < \\alpha <1$, $s\\geq2$, $1\\leq r< t$ and $n \\geq  s-1+\\frac{t^{2}-1}{\\alpha}$. We obtain\n\\begin{equation*}\n\\begin{aligned}\n\\rho_{\\alpha}+1-\\alpha (s-1)-r(1-r+t)>&\\alpha(n-1)+(1-\\alpha)(s-2)+1-\\alpha (s-1)-(t-1)(1+t)\\\\\n=&\\alpha (n-1) + s-1-\\alpha(2s-3)-t^{2}+1\\\\\n\\geq&\\alpha (n-1) -\\alpha(s-2)-t^{2}+1\\\\\n\\geq&\\alpha \\left( \\left( s-1+\\frac{t^{2}-1}{\\alpha}\\right) -1\\right)  -\\alpha(s-2)-t^{2}+1\\\\\n=&0.\n\\end{aligned}\n\\end{equation*}\nHence, $h(\\rho_{\\alpha})>0$. Furthermore, since\n\\begin{equation*}\n\\begin{aligned}\nn&\\geq s-1+\\frac{t^{2}-1}{\\alpha}=2(s-1)+\\frac{t^{2}-1-\\alpha(s-1)}{\\alpha}\\geq2(s-1)+\\frac{t^{2}-1-(s-1)}{\\alpha}\\\\\n&\\geq2(s-1)+\\frac{t-(s-1)}{\\alpha}=2(s-1)+\\frac{t-s+1}{\\alpha},\n\\end{aligned}\n\\end{equation*}\nwe have\n\\begin{equation*}\n\\begin{aligned}\n\\rho_{\\alpha}>&\\alpha(n-1)+(1-\\alpha)(s-2)\n=\\frac{\\alpha n}{2} +\\frac{\\alpha n}{2}-\\alpha (s-1)+s-2\\\\\n\\geq&\\frac{\\alpha n}{2} +\\frac{\\alpha}{2}\\left(2(s-1)+\\frac{t-s+1}{\\alpha} \\right) -\\alpha (s-1)+s-2=\\dfrac{\\alpha n+s+t-3}{2}.\n\\end{aligned}\n\\end{equation*}\nIt follows that $\\rho_{\\alpha}$ is larger than the largest root of $h(x)=0$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::2.5}\nLet $0<\\alpha<1$, $s\\geq4$, $n\\geq s$ and $G\\cong (K_{s-1}-e) \\vee \\overline{K}_{n-s+1}$. Then $\\rho_{\\alpha}(G)$ is the largest root of $f(x) = 0$, where\n\\begin{equation*}\n\\begin{aligned}\nf(x)=&x^3 -(2\\alpha n + s - 4)x^2 +(\\alpha^2n^2 +3\\alpha ns - \\alpha s^2 - 6\\alpha n +\\alpha s- ns + s^2 - 2\\alpha + n -\n4s +\\\\& 7)x -2\\alpha^2 n^2 s + \\alpha^2 ns^2 + 2 \\alpha^2 n^2 - \\alpha^2 n s + \\alpha n^2 s - \\alpha ns^2 + 2\\alpha^2 n - \\alpha n^2 + 6\\alpha ns -\n2\\alpha s^2 \\\\&-9\\alpha n + 4\\alpha s -2ns + 2s^2 - 2\\alpha + 4n - 6s + 4.\n\\end{aligned}\n\\end{equation*}\t\nMoreover, $\\rho_{\\alpha}(G) > \\alpha(n-1)+(1-\\alpha)(s-4).$\n\\end{lem}\n\n\\begin{proof}\nFirstly, denote by $\\{v_{1}, v_{2}, \\cdots, v_{s-3}, w_{1}, w_{2}\\}$ the  vertex set of $K_{s-1}-e$ in the representation $G:\\cong (K_{s-1}-e) \\vee \\overline{K}_{n-s+1}$, where $d_{G}(v_{i})=n-1$ for $i\\in \\{1,2,\\dots,s-3\\}$ and $d_{G}(w_{1})=d_{G}(w_{2})=n-2$. Set for short $\\rho_{\\alpha}=\\rho_{\\alpha}(G)$ and let $\\mathbf{x}=\\left(x_{v}\\right)_{v \\in V(G)}$ be the Perron vector of $A_{\\alpha}(G)$ with respect to $\\rho_{\\alpha}$. By symmetry, we have $x_{v_{1}}=\\cdots=x_{v_{s-3}}$ and  $x_{w_{1}}=x_{w_{2}}$. Additionally, $x_{z}=x_{z_{1}}$ for any two vertices\n$z, z_1\\in V(G)\\backslash \\{v_{1}, v_{2}, \\cdots, v_{s-3}, w_{1}, w_{2}\\}$. By eigen-equations of $A_{\\alpha}(G)$ on $v_{1}$, $w_{1}$ and $z_{1}$, we have\n$$\n\\begin{aligned}\n&(\\rho_{\\alpha}-\\alpha(n-1)-(1-\\alpha)(s-4)) x_{v_{1}} =2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}}, \\\\\n&(\\rho_{\\alpha}-\\alpha(n-2))x_{w_{1}} =(1-\\alpha)(s-3) x_{v_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\\\\\n&(\\rho_{\\alpha}-\\alpha(s-1))x_{z_{1}}=(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}.\n\\end{aligned}\n$$\nThen $\\rho_{\\alpha}$ is the largest real root of $f(x) = 0$, where\n\\begin{equation*}\n\\begin{aligned}\nf(x)=&x^3 -(2\\alpha n + s - 4)x^2 +(\\alpha^2n^2 + 3\\alpha ns - \\alpha s^2 - 6\\alpha n +\\alpha s - ns + s^2\n- 2\\alpha + n -  4s \\\\\n&+ 7)x -2\\alpha^2 n^2 s + \\alpha^2 ns^2 + 2 \\alpha^2 n^2 - \\alpha^2 n s + \\alpha n^2 s - \\alpha ns^2 + 2\\alpha^2 n - \\alpha n^2 + 6\\alpha ns -\\\\\n&2\\alpha s^2 - 9\\alpha n + 4\\alpha s -2ns + 2s^2 - 2\\alpha + 4n - 6s + 4.\n\\end{aligned}\n\\end{equation*}\nSince $0 < \\alpha < 1$, $s\\geq4$ and $n \\geq s$, we have\n\\begin{equation*}\n\\begin{aligned}\n(\\alpha(s-3)-s)(n-s+3) + 8\\leq&((s-3)-s)(n-s+3)  + 8\n=-3(n-s+3) + 8\\\\\n\\leq&-3(s-s+3) + 8\n<0,\n\\end{aligned}\n\\end{equation*}\nand so\n$f(\\alpha(n-1)+(1-\\alpha)(s-4))=((\\alpha(s-3)-s)(n-s+3) + 8)(s - 3)(\\alpha - 1)^2<0,$\nwhich implies that $\\rho_{\\alpha} >\\alpha(n-1)+(1-\\alpha)(s-4).$\n\nThis completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{lem::2.6}\nLet $0< \\alpha <1$, $t\\geq 2$, $s\\geq3$, $C\\geq\\frac{1}{\\alpha}$ and\n$$n \\geq \\max\\{ 2s-3+\\frac{t-s+4}{\\alpha}, \\frac{(1-\\alpha)(s-1)(C(s+t)+2)}{2}+s+t \\}.$$ Suppose that $H$ is a graph of order $n-s+1$ and $G\\cong (K_{s-1}-e) \\vee H$, particularly, defined by $G^{\\prime}=G$ when $H$ is $(t-1)$-regular. If $\\Delta(H) \\leq t-1$, then $\\rho_{\\alpha}(G) \\leq \\rho_{\\alpha}(G^{\\prime})$, eauality holds if and only if $G\\cong G^{\\prime}$,\nmoreover, $\\rho_{\\alpha}(G^{\\prime})$ is less than the largest root of $h(x)=0$, where\n\\begin{equation*}\n\\begin{aligned}\nh(x)=&x^2-(\\alpha n+s+t-3) x+\n(\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)\\\\&-(1-\\alpha)^2(s-1)(n-s).\n\\end{aligned}\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nDenote by $\\{v_{1}, v_{2}, \\cdots, v_{s-3}, w_{1}, w_{2}\\}$ the  vertex set of $K_{s-1}-e$ in the representation $G:\\cong (K_{s-1}-e) \\vee H$, where $d_{G}(v_{i})=n-1$ for $i\\in \\{1,2,\\dots,s-3\\}$ and $d_{G}(w_{1})=d_{G}(w_{2})=n-2$. Set for short $\\rho_{\\alpha}=\\rho_{\\alpha}(G)$ and let\n$\\mathbf{x}=\\left(x_{v}\\right)_{v \\in V(G)}$ be the Perron vector of $A_{\\alpha}(G)$ with respect to $\\rho_{\\alpha}$. Clearly, $x_{v_{1}}=\\cdots=x_{v_{s-3}}$ and $x_{w_{1}}=x_{w_{2}}$ by the symmetry. Choose a vertex $z_{1} \\in V(H)$ such that\n$$x_{z_{1}}=\\max _{v \\in V(H)} x_{v}.$$\nSince $\\Delta(H) \\leq t-1$, we have $d_{G}(z_{1})=d_{H}(z_{1})+s-1\\leq s+t-2$.\n\t\n{\\flushleft\\bf Case 1. $s=3$.}\nThen\n$\nh(x)=x^2-(\\alpha n+t) x+(\\alpha(n-2)+1)(2\\alpha+t-1)-2(1-\\alpha)^2(n-3)$.\nBy eigenequations of $A_{\\alpha}(G)$ on $w_{1}$ and $z_{1}$, we have\n$$\n\\begin{aligned}\n\\rho_{\\alpha}x_{w_{1}}&=\\alpha(n-2)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in V(H)} x_{v}\\leq\\alpha(n-2)x_{w_{1}}+(1-\\alpha)(n-2)x_{z_{1}},\\\\\n\\rho_{\\alpha}x_{z_{1}}&=\\alpha d_{G}(z_{1})x_{z_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}(z_{1})} x_{v}\n\\\\&\\leq ((t+1)\\alpha+(1-\\alpha)(t-1) )x_{z_{1}}+2(1-\\alpha)x_{w_{1}},\n\\end{aligned}\n$$\nand thus\n\\begin{equation}\\label{equ::2_}\n(\\rho_{\\alpha}-\\alpha(n-2)) x_{w_{1}} \\leq (1-\\alpha)(n-2)x_{z_{1}},\n\\end{equation}\n\\begin{equation}\\label{equ::3_}\n(\\rho_{\\alpha}-(t+1)\\alpha-(1-\\alpha)(t-1) ) x_{z_{1}} \\leq 2(1-\\alpha)x_{w_{1}}.\n\\end{equation}\nNotice that $n \\geq 2s-3+\\frac{t-s+4}{\\alpha}=3+\\frac{t+1}{\\alpha}$. Then we obtain\n\\begin{equation}\\label{equ::2-}\n\\begin{aligned}\n\\rho_{\\alpha}&\\geq\\rho_{\\alpha}(K_{2,n-2})=\\dfrac{\\alpha n+\\sqrt{(\\alpha n)^2+8(n-2)(1-2\\alpha)}}{2}\n=\\dfrac{\\alpha n+\\sqrt{(\\alpha (n-4))^2+8(\\alpha - 1)^2 (n - 2)}}{2}\\\\&\n>\\dfrac{\\alpha n+\\alpha (n-4)}{2}\n=\\alpha (n-2)\n\\geq\\alpha \\left( \\left( 3+\\frac{t+1}{\\alpha}\\right) -2\\right)\n>(t+1)\\alpha+(1-\\alpha)(t-1).\n\\end{aligned}\n\\end{equation}\nNow, we can multiply (\\ref{equ::2_})-(\\ref{equ::3_}), obtaining\n$$\n\\rho_{\\alpha}^2 -(\\alpha n + t - 1)\\rho_{\\alpha} +\\alpha nt + 3 \\alpha n - 2 \\alpha t - 6\\alpha - 2n + 4 \\leq 0.\n$$\nThis implies that $\\rho_{\\alpha}$ is no more than the largest root of $g(x)=0$, where\n$$g(x)=x^2 -(\\alpha n + t - 1)x +\\alpha nt + 3 \\alpha n - 2 \\alpha t - 6\\alpha - 2n + 4 .$$\nIf $\\rho_{\\alpha}$ is equal to the largest real root of $g(x)=0$, then equalities in (\\ref{equ::2_})-(\\ref{equ::3_}) hold. Therefore, for any vertex $z \\in V(H)$, we have $x_{z}=x_{z_{1}}$ and\n$$\n\\begin{aligned}\n\\rho_{\\alpha}-\\alpha d_{G}(z)x_{z}\n&=2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}(z)} x_{v}\\\\\n&\\leq 2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(t-1)x_{z_{1}}=\\rho_{\\alpha}-\\alpha d_{G}(z_{1})x_{z_{1}},\n\\end{aligned}\n$$\nwhich implies $d_{G}(z)=s+t-2$, that is, $d_{H}(z)=t-1$. Hence, $H$ is a $(t-1)$-regular graph, and so $G\\cong G^{\\prime}$.\nFurthermore, we see that $g(x)=h(x)+x-(t+1-2\\alpha(1-\\alpha))$. By (\\ref{equ::2-}), we have\n$\\rho_{\\alpha}>\\alpha \\left( \\left( 3+\\frac{t+1}{\\alpha}\\right) -2\\right)=t+1+\\alpha.$\nThus,\n$h(\\rho_{\\alpha})=-(\\rho_{\\alpha}-(t+1-2\\alpha(1-\\alpha)))\\leq-(\\rho_{\\alpha}-(t+1))<0$.\nTherefore, $\\rho_{\\alpha}$ is less than the largest root of $h(x)=0$.\n\t\n{\\flushleft\\bf Case 2. $s\\geq4$.}\nBy eigenequations of $A_{\\alpha}(G)$ on $v_{1}$, $w_{1}$ and $z_{1}$, we have\n\\begin{equation*}\\begin{array}{ll}\n(\\rho_{\\alpha}-\\alpha(n-1)-(1-\\alpha)(s-4))x_{v_{1}}\n&=2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in V(H)} x_{v}\\\\\n&\\leq2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\n\\end{array}\n\\end{equation*}\n\\begin{equation*}\\begin{array}{ll}\n(\\rho_{\\alpha}-\\alpha(n-2))x_{w_{1}}\n&=(1-\\alpha)(s-3) x_{v_{1}}+(1-\\alpha)\\sum_{v \\in V(H)} x_{v}\\\\\n&\\leq(1-\\alpha)(s-3) x_{v_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\n\\end{array}\n\\end{equation*}\n\\begin{equation*}\\begin{array}{ll}\n(\\rho_{\\alpha}-\\alpha(s+t-2))x_{z_{1}}\n&\\leq (\\rho_{\\alpha}-\\alpha d_{G}(z_{1})x_{z_{1}}\\\\\n&=(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}(z_{1})} x_{v}\\\\\n&\\leq(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(t-1)x_{z_{1}},\n\\end{array}\n\\end{equation*}\n\nthat is,\n\\begin{equation}\\label{equ::1}\n(\\rho_{\\alpha}-\\alpha(n-1)-(1-\\alpha)(s-4)) x_{v_{1}} \\leq2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\n\\end{equation}\n\\begin{equation}\\label{equ::2}\n(\\rho_{\\alpha}-\\alpha(n-2)) x_{w_{1}} \\leq (1-\\alpha)(s-3) x_{v_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\n\\end{equation}\n\\begin{equation}\\label{equ::3}\n(\\rho_{\\alpha}-\\alpha(s+t-2)-(1-\\alpha)(t-1)) x_{z_{1}} \\leq (1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}.\n\\end{equation}\nNotice that and $G$ contains $(K_{s-1}-e) \\vee \\overline{K}_{n-s+1}$ as a subgraph. By Lemma \\ref{lem::2.5}, we have\n$$\n\\rho_{\\alpha}\\geq \\rho_{\\alpha}\\left((K_{s-1}-e) \\vee \\overline{K}_{n-s+1} \\right)>\\alpha(n-1)+(1-\\alpha)(s-4).\n$$\nSince $0<\\alpha<1$ and $s\\geq4$, we have\n\\begin{equation}\\label{equ::4-}\n\\begin{aligned}\n\\rho_{\\alpha}\\geq \\alpha(n-1)+(1-\\alpha)(s-4)>\\alpha(n-2).\n\\end{aligned}\n\\end{equation}\nRecall that $n \\geq  2s-3+\\frac{t-s+4}{\\alpha}$. Hence, we get that\n\\begin{equation}\\label{equ::4--}\n\\begin{aligned}\n\\rho_{\\alpha}\\geq \\alpha(n-1)+(1-\\alpha)(s-4)>\\alpha(s+t-2)+(1-\\alpha)(t-1).\n\\end{aligned}\n\\end{equation}\nLet $A=1$; $B=-(2\\alpha n + s + t - 5)$; $C=a^2 n^2 + 3\\alpha ns + 2\\alpha nt - \\alpha s^2 - \\alpha st - 8\\alpha n + 2\\alpha s + \\alpha t - ns + s^2 + st - 3\\alpha + n - 5s - 4t + 11$;\n$D= 10 + 4n - 6t - 8s - 6\\alpha + 2s^2 + 7\\alpha ns - 2\\alpha^2 ns - 2\\alpha^2 n^2 s + 4\\alpha nt - 2\\alpha st -\\alpha n^2+ 4\\alpha t + 3\\alpha^2 n + 3\\alpha^2 n^2 - 2\\alpha s^2 + 6\\alpha s - 2ns - 13\\alpha n + 2st + \\alpha n^2s + \\alpha^2 ns^2 - \\alpha ns^2 - \\alpha^2 n^2 t + \\alpha^2 nst - \\alpha nst - \\alpha^2 nt$.\nNow, we can multiply (\\ref{equ::1})-(\\ref{equ::3}), obtaining\n$A\\rho_{\\alpha}^{3}+B\\rho_{\\alpha}^{2}+C\\rho_{\\alpha}+D \\leq 0$.\nThis implies that $\\rho_{\\alpha}$ is no more than the largest  root of $f(x)=0$, where\n$$\nf(x)=Ax^{3}+Bx^{2}+Cx+D.\n$$\nIf $\\rho_{\\alpha}$ is equal to the largest real root of $f(x)=0$, then equalities   in (\\ref{equ::1})-(\\ref{equ::3}) hold. Therefore, for any vertex $z \\in V(H)$, we have $x_{z}=x_{z_{1}}$ and\n$$\n\\begin{aligned}\n\\rho_{\\alpha}-\\alpha d_{G}(z)x_{z}\n&=(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}(z)} x_{v}\\\\\n&\\leq(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(t-1)x_{z_{1}}=\\rho_{\\alpha}-\\alpha d_{G}(z_{1})x_{z_{1}},\n\\end{aligned}\n$$\nwhich implies   $d_{G}(z)=s+t-2$, that is, $d_{H}(z)=t-1$. Hence, $H$ is a $(t-1)$-regular graph, and so $G\\cong G^{\\prime}$. Moreover, let\n\\begin{equation*}\n\\begin{aligned}\ng_{1}(x)=&x^2-(\\alpha n+s+t-3) x+ (\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)\\\\&-(1-\\alpha)^2(s-1)(n-s+1).\n\\end{aligned}\n\\end{equation*}\nwe find that\n$$\nf(x)=(x-(\\alpha n-2))g_{1}(x)+2(1-\\alpha)(x-((\\alpha-1) n  + s + t - 2)).\n$$\nNote that $0<\\alpha<1$. By (\\ref{equ::4-}) we have $\\rho_{\\alpha}>\\alpha(n-2)\\geq\\alpha n-2,$\nand by (\\ref{equ::4--}) we get that\n$$\n\\begin{aligned}\n\\rho_{\\alpha}-((\\alpha-1) n  + s + t - 2)\n&>\\alpha(s+t-2)+(1-\\alpha)(t-1)-((\\alpha-1) n  + s + t - 2)\\\\\n&=(1-\\alpha)(n-s+1)>0.\n\\end{aligned}\n$$\nIt follows that\n$$\nh(\\rho_{\\alpha})=-\\dfrac{2(1-\\alpha)(\\rho_{\\alpha}-((\\alpha-1) n  + s + t - 2))}{\\rho_{\\alpha}-(\\alpha n-2)}<0,\n$$\nTherefore, $\\rho_{\\alpha}$ is less than the largest root of $g_{1}(x)=0$. That is,\n$\n\\rho_{\\alpha}<\\dfrac{\\alpha n+s+t-3+\\sqrt{R}}{2},\n$\nwhere\n$$\n\\begin{aligned}\nR=&(\\alpha n+s+t-3)^{2}-4\\left( (\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)-(1-\\alpha)^2(s-1)(n-s+1)\\right)\\\\\n=&(\\alpha n +(2C-1)(s + t) + 3)^2-4(C(C-1)(s^2 + 2st+t^2)+(C\\alpha-1)ns + C\\alpha nt  +\\alpha ns +\\\\\n&3s(C-1) +(3C-2)t + \\alpha (2s + t)  +(1-\\alpha)(t+s)s + n +(3-\\alpha))\\\\\n<&(\\alpha n +(2C-1)(s + t) + 3)^2,\n\\end{aligned}\n$$\nsince $0<\\alpha<1$ and $C\\geq\\frac{1}{\\alpha}$.\nHence, we have\n$$\n\\begin{aligned}\n\\rho_{\\alpha}<\\dfrac{\\alpha n+s+t-3+\\alpha n +(2C-1)(s + t) + 3}{2}=\\alpha n+C(s+t).\n\\end{aligned}\n$$\nNow we see that\n$$\nf(x)=(x-(\\alpha n-2))h(x)+2(1-\\alpha)(x-((\\alpha-1) n  + s + t - 2))-(x-(\\alpha n-2))(1-\\alpha)^{2}(s-1).\n$$\nNote that $n\\geq\\frac{(1-\\alpha)(s-1)(C(s+t)+2)}{2}+s+t$. Then we obtain\n$$\n\\begin{aligned}\n&2(1-\\alpha)(\\rho_{\\alpha}-((\\alpha-1) n  + s + t - 2))-(\\rho_{\\alpha}-(\\alpha n-2))(1-\\alpha)^{2}(s-1)\\\\\n=&(1-\\alpha)\\left( 2(n-s-t)+(\\rho_{\\alpha}-(\\alpha n-2))\\left( -(1-\\alpha)(s-1)+2\\right) \\right) \\\\\n>&(1-\\alpha)\\left( 2(n-s-t)-(1-\\alpha)(s-1)(\\rho_{\\alpha}-(\\alpha n-2)) \\right) \\\\\n\\geq&(1-\\alpha)\\left( 2(n-s-t)-(1-\\alpha)(s-1)((\\alpha n+C(s+t))-(\\alpha n-2)) \\right) \\\\\n=&(1-\\alpha)\\left( 2(n-s-t)-(1-\\alpha)(s-1)(C(s+t)+2) \\right) \\\\\n\\geq&(1-\\alpha)\\left( 2\\left( \\frac{(1-\\alpha)(s-1)(C(s+t)+2)}{2}+s+t-s-t\\right) -(1-\\alpha)(s-1)(C(s+t)+2) \\right) \\\\\n=&0.\n\\end{aligned}\n$$\nIt follows that\n$$\nh(\\rho_{\\alpha})=-\\dfrac{2(1-\\alpha)(x-((\\alpha-1) n  + s + t - 2))-(x-(\\alpha n-2))(1-\\alpha)^{2}(s-1)}{\\rho_{\\alpha}-(\\alpha n-2)}<0,\n$$\nTherefore, $\\rho_{\\alpha}$ is less than the largest root of $h(x)=0$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::2.7}\nLet $0<\\alpha <1$, $2 \\leq s \\leq t$ and $G$ be a $K_{s,t}$-minor free graph of sufficiently large order $n$ with maximum $A_\\alpha$-spectral radius. Then $G$ contains a vertex set $K=\\left\\{v_{1}, v_{2}, \\ldots, v_{s-1}\\right\\}$ such that $d_{G}\\left(v_{i}\\right)=n-1$ for $i \\in\\{1,2, \\ldots, s-1\\}$.\n\\end{lem}\n\n\\begin{proof}\nLet\n\\begin{equation*}\n\\begin{aligned}\nh(x)=&x^2-(\\alpha n+s+t-3) x+\n(\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)\\\\&-(1-\\alpha)^2(s-1)(n-s).\n\\end{aligned}\n\\end{equation*}\nNote that $F_{s, t}(n)$ is a $K_{s, t}$-minor free graph. Then we have $\\rho_{\\alpha}(G) \\geq \\rho_{\\alpha}(F_{s, t}(n))$. Furthermore, by Lemma \\ref{lem::2.3}, we get that $\\rho_{\\alpha}(G)$ is larger than the largest root of $h(x)=0$. By Lemma \\ref{lem::2.1}, $G$ contains a vertex set $K$ of size $s-1$ such that $d_{G-K}\\left(v\\right)=n-s+1$ for any vertex $v \\in K$. Now, we need to show that $K$ induces a clique. Otherwise, we have $s\\geq3$ and $G[K] \\subseteq K_{s-1}-e$. Let $H=G-K$, i.e., $G=G[K] \\vee H$. Since $G$ is a $K_{s, t}$-minor free graph, we have $\\Delta(H)\\leq t-1$. If $G[K] \\cong K_{s-1}-e$, by Lemma \\ref{lem::2.6}, we obtain $\\rho_{\\alpha}(G)$ is less than the largest root of $h(x)=0$, a contradiction. Therefore, $G[K]$ is a proper subgraph of $K_{s-1}-e$. Let $G^{\\prime}$ be the graph obtained from $G$\nby adding edges to $G[K]$ to make it a graph $K_{s-1}-e$. Then we have $\\rho_{\\alpha}(G)<\\rho_{\\alpha}(G^{\\prime})$ by Lemma \\ref{lem::2.2}. However, from Lemma \\ref{lem::2.6}, $\\rho_{\\alpha}(G^{\\prime})$ is\nless than the largest root of $h(x)=0$, which is also a contradiction.\n\\end{proof}\n\n\\begin{lem}(\\cite{D.L})\\label{lem::2.4}\nLet $t \\geq 3$ and $n \\geq t+2$. If $G$ is an $n$-vertex connected graph with no $K_{1, t}$-minor, then $e(G) \\leq \\binom{t}{2} +n-t$, and this is best possible for all $n, t$.\n\\end{lem}\n\n\n\nFor $2\\leq s\\leq t$, we say a graph $G$ has the $(s, t)$-property, if $G$ is $K_{a, b}$-minor free for any two positive integers $a, b$ with $a+b=t+1$ and $a \\leq \\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}$.\nThe following Lemma gives a equivalent\ncondition whether a graph $G$ has  $K_{s, t}$-minor or not.\n\n\\begin{lem}(Lemma 2.3, \\cite{M.Q})\\label{lem::2.10}\nLet $2\\leq s\\leq t$ and $G$ be a graph with a clique dominating set $K$ of size $s-1$. Then $G$ is $K_{s, t}$-minor free if and only if $G-K$ has the $(s, t)$-property.\n\\end{lem}\n\nRecall that $H_{s, t}:\\cong(\\beta-1)K_{1,s} \\cup K_{1,\\alpha}$, where $1\\leq s\\leq t$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ and $\\alpha=t-(\\beta-1)(s+1)\\geq s$,\n$S^{1}\\left(\\overline{H_{s, t}}\\right)$ is a graph obtained from a graph $\\overline{H_{s, t}}$ by subdividing once of an edge $uv$ with minimum degree sum $d_{G}(u)+d_{G}(v)$ and  $H^{\\star}$ is the Petersen graph.\nThe following result is from Theorem 3.1 and the proofs of Claims $3.8$-$3.9$ in \\cite{M.Q}.\n\\begin{lem}(\\cite{M.Q})\\label{lem::2.16}\nLet $2\\leq s\\leq t$ and $t\\geq4$. Then\n\t\n(i) $\\overline{H_{s,t}}$ and $S^{1}\\left(\\overline{H_{s, t}}\\right)$ have the $(s, t)$-property. Moreover, if $\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor=2$, then $\\pi\\left(S^{1}\\left(\\overline{H_{s, t}}\\right)\\right)=(t-1, \\ldots, t-1, t-s, s+1,2)$.\n\n(ii) $\\overline{H^{\\star}}$ has the $(s, t)$-property for $t=8$.\n\\end{lem}\n\\begin{lem}(Lemma 3.1, \\cite{M.Q})\\label{lem::2.11}\nLet $2\\leq s\\leq t$, $t\\geq4$, $\\gamma = \\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}$ and $G$ be a connected graph with $|G|=t+1$. Then $G$ has the $(s, t)$-property if and only if each component of $\\overline{G}$ has at least $\\gamma+1$ vertices.\n\\end{lem}\n\nThe following result comes from the proof of Lemma 3.2 in \\cite{M.Q}.\n\\begin{lem}(\\cite{M.Q})\\label{lem::2.12}\nLet $2\\leq s\\leq t$, $t\\geq4$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ and $G$ be a connected graph with $|G|=t+1$. If $G$ is an edge-maximal graph with the $(s, t)$-property, then $e(G)=\\binom{t}{2}+\\beta-1$ and $\\overline{G}$ is a forest with $\\beta$ components.\n\\end{lem}\n\n\\begin{lem}(Lemma 3.3, \\cite{M.Q})\\label{lem::2.13}\nLet $G$ be a graph with $vw \\in E(G)$ and $uw \\notin E(G)$. If $d_{G}(u) \\geq d_{G}(v)$, then $\\pi(G) \\prec \\pi(G-\\{v w\\}+\\{u w\\})$ and $\\pi(G) \\neq \\pi(G-\\{v w\\}+\\{u w\\})$.\n\\end{lem}\n\n\\begin{lem}(Lemma 3.4, \\cite{M.Q})\\label{lem::2.15}\nLet $2\\leq s\\leq t$, $t\\geq4$, $\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor \\leq 2$ and $G$ be a connected graph with $|G|=t+2$. If $G$ has the $(s, t)$-property, then $e(G) \\leq \\binom{t}{2} +2$, and if equality holds, then $\\overline{G}$ is isomorphic to either the Petersen graph $H^{\\star}$ or some $H_{a, b, c}$ (see Fig. \\ref{fig-1}), where $a+b+c=t-1$.\n\\end{lem}\n\n\\begin{figure}[hbtp]\n\\centering\n\\begin{tikzpicture}[x=0.9mm, y=0.9mm, inner xsep=0pt, inner ysep=0pt, outer xsep=0pt, outer ysep=0pt]\n\\path[line width=0mm] (25.74,33.08) rectangle +(155.86,74.28);\n\\definecolor{L}{rgb}{0,0,0}\n\\path[line width=0.30mm, draw=L] (60.31,92.82) circle (12.55mm);\n\\path[line width=0.30mm, draw=L] (80.50,57.74) circle (12.55mm);\n\\path[line width=0.30mm, draw=L] (40.29,56.89) circle (12.55mm);\n\\definecolor{F}{rgb}{0,0,0}\n\\path[line width=0.30mm, draw=L, fill=F] (57.23,71.77) circle (1.00mm);\n\\path[line width=0.30mm, draw=L, fill=F] (63.22,71.09) circle (1.00mm);\n\\path[line width=0.30mm, draw=L, fill=F] (59.28,66.98) circle (1.00mm);\n\\path[line width=0.30mm, draw=L] (63.22,71.77) -- 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node[anchor=base west]{\\fontsize{14.23}{17.07}\\selectfont $K_b$};\n\\draw(164.24,54.28) node[anchor=base west]{\\fontsize{14.23}{17.07}\\selectfont $K_c$};\n\\draw(139.22,70.29) node[anchor=base west]{\\fontsize{14.23}{17.07}\\selectfont $u_2$};\n\\draw(141.94,61.82) node[anchor=base west]{\\fontsize{14.23}{17.07}\\selectfont $u_1$};\n\\path[line width=0.30mm, draw=L] (142.42,65.78) -- (134.89,58.08);\n\\draw(150.46,68.01) node[anchor=base west]{\\fontsize{14.23}{17.07}\\selectfont $w$};\n\\draw(141.56,36.61) node[anchor=base west]{\\fontsize{14.23}{17.07}\\selectfont $\\overline{H_{a,b,c}}$};\n\\end{tikzpicture}%\n\\caption{The graph $H_{a, b, c}$ and its complement, where $a+b+c=t-1$.}\\label{fig-1}\n\\end{figure}\n\n\\section{Two properties of the $A_\\alpha$-spectral extremal graph}\nRecall that the \\emph{$A_\\alpha$-spectral extremal graph} is a graph with maximum $A_\\alpha$-spectral radius among all $K_{s, t}$-minor free graphs of sufficiently order $n$ for $0 < \\alpha<1$ and $2\\leq s\\leq t$.\nIn the following, we always assume that $G^*$ is the $A_\\alpha$-spectral extremal graph.\nIn this section, we will prove that $G^*$ has a property of\nlocal edge maximality. Meanwhile, we will apply  the double eigenvectors transformation technique to the $A_{\\alpha}(G)$-matrix and prove that\n$G^*$  has a property of local degree sequence majorization.\n\n\nLet $\\mathbf{x}=\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)^{T}$ and $\\mathbf{y}=\\left(y_{1}, y_{2}, \\ldots, y_{n}\\right)^{T}$ be two non-increasing real vectors. We say $\\mathbf{x}$ is weakly majorized by $\\mathbf{y}$, denoted by $\\mathbf{x} \\prec_{w} \\mathbf{y}$, if and only if $\\sum_{i=1}^{k} x_{i} \\leq \\sum_{i=1}^{k} y_{i}$ for  $k=1,2, \\ldots, n$. We say  $\\mathbf{x}$ is majorized by $\\mathbf{y}$, denoted  $\\mathbf{x} \\prec \\mathbf{y}$, if and only if $\\mathbf{x} \\prec_{w} \\mathbf{y}$ and $\\sum_{i=1}^{n} x_{i}=\\sum_{i=1}^{n} y_{i}$.\n\n\\begin{lem}(\\cite{Lin})\\label{lem::2.8}\nLet $\\mathbf{x}, \\mathbf{y} \\in R^{n}$ be two non-negative and non-increasing vectors. If $\\mathbf{x} \\prec_{w} \\mathbf{y}$, then $\\|\\mathbf{x}\\|_{k} \\leq\\|\\mathbf{y}\\|_{k}$ for $k>1$, with equality if and only if $\\mathbf{x}=\\mathbf{y}$.\n\\end{lem}\n\nThe following lemma is from Exercises $5(i)$ on page $74$ of \\cite{Zhan}.\n\\begin{lem}(\\cite{Zhan})\\label{lem::2.9}\nLet $\\mathbf{x}, \\mathbf{y}, \\mathbf{z} \\in R^{n}$ be three non-increasing vectors. If $\\mathbf{x} \\prec \\mathbf{y}$, then $\\mathbf{x}^{T} \\cdot \\mathbf{z} \\leq \\mathbf{y}^{T} \\cdot \\mathbf{z}$.\n\\end{lem}\n\nSet for short $\\rho_{\\alpha} = \\rho_{\\alpha} (G^{*}) $ and let $\\mathbf{X}=(x_{v})_{v \\in V(G^{*})} \\in R^{n}$ be a Perron eigenvector of $A_{\\alpha}(G^{*})$ corresponding to $\\rho_{\\alpha}$.  By Lemma \\ref{lem::2.7},\n$G^*$ contains a clique dominating set $K$ of size $s-1$. We immediately get that $G^{*}-K$ has the $(s, t)$-property from Lemma \\ref{lem::2.10}. So, $\\Delta(G^{*}-K)<t$. For convenience, we let $x_{0}=\\sum\\limits_{v\\in K}x_{v}$. Furthermore, we have $\\rho_{\\alpha}\\geq \\rho_{\\alpha}\\left( K_{s-1} \\vee \\overline{K}_{n-s+1}\\right)\\geq \\alpha(n-1)+(1-\\alpha)(s-2)$ by Lemma \\ref{lem::2.1'}. Thus we have the following result.\n\n\\begin{lem}\\label{lem::3.1}\nLet $H$ consist of some  components of $G^{*}-K$ with $|H| \\leq N$ (a constant), $x_{1}=\\max\\limits_{v \\in V(H) }x_{v}$ and\n$x_{2}=\\min\\limits_{v \\in V(H) }x_{v}$. If $a, b$ are two constants with $a>b.$ Then $a x_{2}>b x_{1}$ and $a x_{2}^{2}>b x_{1}^{2}$.\n\\end{lem}\n\n\\begin{proof}\nSince $H\\subseteq G^{*}-K$, we have $\\Delta(H)\\leq\\Delta(G^{*}-K) < t$.\nFor each $v \\in V(H)$, we see that\n\\begin{equation}\\label{equ::a}\ns-1\\leq d_{G^{*}}(v)=d_{H}(v)+s-1 < t+s-1.\n\\end{equation}\nNote that $x_{0}=\\sum\\limits_{v\\in K}x_{v}$. Therefore, we obtain $\\rho_{\\alpha} x_{1}<(\\alpha(t+s-1)+(1-\\alpha)t)x_{1}+(1-\\alpha)x_{0}$ and $\\rho_{\\alpha} x_{2} \\geq \\alpha(s-1)x_{2}+(1-\\alpha)x_{0}$, and thus\n\\begin{equation}\\label{equ::4}\nx_{1}<\\frac{(1-\\alpha)x_{0}}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)} \\quad \\text { and } \\quad x_{2} \\geq \\frac{(1-\\alpha)x_{0}}{\\rho_{\\alpha}-\\alpha(s-1)}.\n\\end{equation}\nSince $\\rho_{\\alpha}\\geq \\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, $n$ is sufficiently large and $a, b, s, t$ are constants, we can easily get that\n$$\nax_{2}-bx_{1}>(1-\\alpha)x_{0}(\\frac{a}{\\rho_{\\alpha}-\\alpha(s-1)}-\\frac{b}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)})>0.\n$$\nSimilarly, we can obtain $a x_{2}^{2}>b x_{1}^{2}$.\n\\end{proof}\n\nThe following lemma implies that $G^{*}$ has a property of local edge maximality.\n\\begin{lem}\\label{lem::3.2}\nLet $H$ consist of some  components of $G^{*}-K$ with $|H| \\leq N$ (a constant) and $H^{\\prime}$ be a graph with $V(H^{\\prime})=V(H)$. If $H^{\\prime}$ has the $(s, t)$-property, then $e(H^{\\prime}) \\leq e(H)$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that $e(H^{\\prime})>e(H)$. Let $G^{\\prime}=G^{*}-E(H)+E\\left( H^{\\prime}\\right) $. By Lemma \\ref{lem::2.10}, $G^{\\prime}$ is $K_{s,t}$-minor free since $H^{\\prime}$ has the $(s, t)$-property.  Now, by Lemma \\ref{lem::3.1} we obtain\n$$\n\\begin{aligned}\n\\rho_{\\alpha}(G^{\\prime})-\\rho_{\\alpha} & \\geq \\mathbf{X}^{T}(A_{\\alpha}(G^{\\prime})-A_{\\alpha}(G^{*})) \\mathbf{X}\\\\\n&=\\sum_{u v \\in E(H^{\\prime})}(\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{v}+\\alpha x_{v}^{2})-\\sum_{u v \\in E(H)} (\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{v}+\\alpha x_{v}^{2}) \\\\\n& \\geq 2 e(H^{\\prime}) x_{2}^{2}-2 e(H) x_{1}^{2}>0,\n\\end{aligned}\n$$\nwhich contradicts the maximality of $\\rho_{\\alpha}$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.3}\nLet $H$ consist of some connected components of $G^{*}-K$ with $|H| \\leq N$ (a constant), $H^{\\prime}$ be a graph with $V\\left(H^{\\prime}\\right)=V(H)$ and $e\\left(H^{\\prime}\\right)=e\\left( H\\right)$. If $H^{\\prime}$ has the $(s, t)$-property, then\n\\begin{equation}\\label{equ::5}\n\\sum_{u v \\in E\\left(H^{\\prime}\\right)} (d_{H}(u)+ d_{H}(v)) \\leq \\sum_{u v \\in E(H)} (d_{H}(u)+ d_{H}(v)).\n\\end{equation}\nMoreover, if (\\ref{equ::5}) holds in equality, then\n\\begin{equation}\\label{equ::6}\n\\sum_{u v \\in E\\left(H^{\\prime}\\right)}(d_{H^{\\prime}}(u)+ d_{H^{\\prime}}(v)) \\leq \\sum_{u v \\in E(H)}(d_{H}(u)+ d_{H}(v)).\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nLet $G^{\\prime}=G^{*}-E(H)+E\\left(H^{\\prime}\\right)$ and $\\rho_{\\alpha}^{\\prime}=\\rho_{\\alpha}(G^{\\prime})$.\nClearly, $G^{\\prime}$ is also $K_{s, t}$-minor free.\nLet $x_{1}=\\max\\limits_{v \\in V(H) }x_{v}$ and $ x_{2}=\\min\\limits_{v \\in V(H) }x_{v}$. Note that $x_{0}=\\sum\\limits_{v\\in K} x_{v}$.\nThen by (\\ref{equ::a}) and the eigen-equations of $A_{\\alpha}(G^{*})$, we have\n\\begin{equation}\\label{equ::8}\n\\frac{(1-\\alpha)(d_{H}(v)x_{2} +x_{0})}{\\rho_{\\alpha}-\\alpha(s-1)}\\leq\\frac{(1-\\alpha)(d_{H}(v)x_{2} +x_{0})}{\\rho_{\\alpha}-\\alpha d_{G^{*}}(v)}\\leq x_{v},\n\\end{equation}\nand\n\\begin{equation}\\label{equ::8''}\n x_{v}\\leq \\frac{(1-\\alpha)(d_{H}(v)x_{1} +x_{0})}{\\rho_{\\alpha}-\\alpha d_{G^{*}}(v)}< \\frac{(1-\\alpha)(d_{H}(v)x_{1} +x_{0})}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)},\n\\end{equation}\nfor each vertex $v \\in V(H)$.\nLet\n$a=\\sum\\limits_{u v \\in E(H^{\\prime})} (d_{H}(u)+ d_{H}(v))$,\n$b=\\sum\\limits_{u v \\in E(H)} (d_{H}(u)+ d_{H}(v))$,\n$c_{1}=\\sum\\limits_{u v \\in E(H)} \\left( d_{H}^{2}(u)+ d_{H}^{2}(v)\\right) $,\n$c_{2}=\\sum\\limits_{u v \\in E(H)} d_{H}(u)d_{H}(v)$.\nFirstly, we will show that $a\\leq b$. Suppose to the contrary that $a\\geq b+1$.\nBy Lemma \\ref{lem::3.1}, we obtain $\\left(a-\\frac{1}{2}\\right) x_{2}>b x_{1}$. Moreover, by (\\ref{equ::4}) we have\n$$x_{0} x_{2}-c_{1} x_{1}^{2}>(1-\\alpha)x_{0}^{2}\\left(\\frac{1}{\\rho_{\\alpha}-\\alpha(s-1)}-\\frac{(1-\\alpha)c_{1}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}} \\right) >0,$$\nsince $\\rho_{\\alpha}\\geq\\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, $n$ is sufficiently large, and  $s, t, c_{1}$ are constants.\nSimilarly, we can obtain $\\frac{1}{2}x_{0} x_{2}-c_{2} x_{1}^{2}>0.$\n\nNow, we denote\n$$A_{1}=\\sum\\limits_{uv\\in E(H^{\\prime})}( (d_{H}(u)x_{2}+x_{0})^{2}+(d_{H}(v)x_{2}+x_{0})^{2}),\\\\\nA_{2}=\\sum\\limits_{uv\\in E(H)}\\left( (d_{H}(u)x_{1} +x_{0})^{2}+(d_{H}(v)x_{1} +x_{0})^{2}\\right),$$\n$$A_{3}=\\sum\\limits_{uv\\in E(H^{\\prime})}(  (d_{H}(u)x_{2}+x_{0})(d_{H}(v)x_{2}+x_{0})) \\ \\ \\mbox{ and} \\ \\ A_{4}=\\sum\\limits_{uv\\in E(H)}\\left( (d_{H}(u)x_{1} +x_{0})(d_{H}(v)x_{1} +x_{0})\\right).\n$$\nBy simple scaling, we see that\n$A_{1}\\geq 2ax_{2}x_{0}+2e(H^{\\prime})x_{0}^{2}$, $A_{2}= 2bx_{1}x_{0}+2e(H)x_{0}^{2}+c_{1}x_{1}^{2}$, $A_{3}\\geq ax_{2}x_{0}+e(H^{\\prime})x_{0}^{2}$ and $A_{4}= bx_{1}x_{0}+e(H)x_{0}^{2}+c_{2}x_{1}^{2}$.\nSince $e\\left(H^{\\prime}\\right)=e\\left( H\\right) $, we get that\n\\begin{align*}\nA_{1}-A_{2}\n&\\geq (2ax_{2}x_{0}+2e(H^{\\prime})x_{0}^{2})-(2bx_{1}x_{0}+2e(H)x_{0}^{2}+c_{1}x_{1}^{2})\\notag\\\\\n&=2ax_{2}x_{0}-2bx_{1}x_{0}-c_{1}x_{1}^{2}=2x_{0}\\left(\\left(a-\\frac{1}{2}\\right) x_{2}-b x_{1}\\right)+(x_{0} x_{2}-c_{1} x_{1}^{2})>0,\n\\end{align*}\nand\n\\begin{align*}\nA_{3}-A_{4}\n&\\geq (ax_{2}x_{0}+e(H^{\\prime})x_{0}^{2})-(bx_{1}x_{0}+e(H)x_{0}^{2}+c_{2}x_{1}^{2})\\notag\\\\\n&=ax_{2}x_{0}-bx_{1}x_{0}-c_{2}x_{1}^{2}=x_{0}\\left(\\left(a-\\frac{1}{2}\\right) x_{2}-b x_{1}\\right)+(\\frac{1}{2}x_{0} x_{2}-c_{2} x_{1}^{2})>0,\n\\end{align*}\nwhich include that $A_{1}>A_{2}$ and $A_{3}>A_{4}$.\nFurthermore, let\n$$B_{1}= \\sum_{uv\\in E(H^{\\prime})}\\left( \\frac{(d_{H}(u)x_{2}+x_{0})^{2}+(d_{H}(v)x_{2}+x_{0})^{2}}{(\\rho_{\\alpha}-\\alpha(s-1))^{2}}\\right)-\\sum_{uv\\in E(H)}\\left(\\frac{(d_{H}(u)x_{1} +x_{0})^{2}+(d_{H}(v)x_{1} +x_{0})^{2}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}}\\right),$$\nand\n$$\nB_{2}=\\sum_{u v \\in E(H^{\\prime})}\\frac{(d_{H}(u)x_{2}+x_{0})(d_{H}(v)x_{2}+x_{0})}{(\\rho_{\\alpha}-\\alpha(s-1))^{2}}-\\sum_{u v \\in E(H)} \\frac{(d_{H}(u)x_{1} +x_{0})(d_{H}(v)x_{1} +x_{0})}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}}.$$\nNotice that  $\\rho_{\\alpha}\\geq\\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, $n$ is sufficiently large. Then from (\\ref{equ::8})-(\\ref{equ::8''}), we have\n\\begin{align*}\n\\rho_{\\alpha}^{\\prime}-\\rho_{\\alpha}\\geq& \\mathbf{X}^{T}(A_{\\alpha}(G^{\\prime})-A_{\\alpha}(G^{*}))\\mathbf{X}\\\\\n=&\\sum_{u v \\in E(H^{\\prime})}(\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{v}+\\alpha x_{v}^{2})-\\sum_{u v \\in E(H)} (\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{v}+\\alpha x_{v}^{2})\\notag\\\\\n=&\\alpha\\left( \\sum_{u v \\in E(H^{\\prime})}( x_{u}^{2}+ x_{v}^{2})-\\sum_{u v \\in E(H)} (x_{u}^{2}+x_{v}^{2})\\right)+2(1-\\alpha)\\left(\\sum_{u v \\in E(H^{\\prime})}x_{u}x_{v}-\\sum_{u v \\in E(H)} x_{u}x_{v} \\right) \\\\\n\\geq&\\alpha(1-\\alpha)^{2}B_{1}+2(1-\\alpha)^{3}B_{2}\\\\\n=&\\alpha(1-\\alpha)^{2}\\left( \\frac{A_{1}}{(\\rho_{\\alpha}-\\alpha(s-1))^{2}}-\\frac{A_{2}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}}\\right) \\\\\n&+2(1-\\alpha)^{3}\\left(  \\frac{A_{3}}{(\\rho_{\\alpha}-\\alpha(s-1))^{2}}-\\frac{A_{4}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}}\\right) \\\\\n>&0,\\label{equ::6'}\n\\end{align*}\nwhich contradicts the maximality of $\\rho_{\\alpha}$.\nSo (\\ref{equ::5}) holds.\n\t\nWe next show that the inequality (\\ref{equ::6}) holds when $a=b$.\nLet $\\mathbf{Y}=(y_{v})_{v \\in V(G^{*})} \\in R^{n}$ be a Perron vector of $A_{\\alpha}(G^{\\prime})$ corresponding to $\\rho_{\\alpha}^{\\prime}$.\nAssume that $y_{0}=\\sum\\limits_{v \\in K} y_{v}$, $y_{1}=\\max\\limits_{v \\in V\\left(H^{\\prime}\\right)} y_{v}$\n and $y_{2}=\\min\\limits_{v \\in V\\left(H^{\\prime}\\right)} y_{v}$. Similar as (\\ref{equ::4}), we obtain\n\\begin{equation}\\label{equ::14}\ny_{1}<\\frac{(1-\\alpha)y_{0}}{\\rho_{\\alpha}^{\\prime}-(\\alpha(t+s-1)+(1-\\alpha)t)} \\quad \\text { and } \\quad y_{2} \\geq \\frac{(1-\\alpha)y_{0}}{\\rho^{\\prime}_{\\alpha}-\\alpha(s-1)}.\n\\end{equation}\nMoreover, for any $v\\in V(H^{\\prime})$  we have\n\\begin{equation}\\label{equ::8'}\n\\frac{(1-\\alpha)(d_{H^{\\prime}}(v)y_{2} +y_{0})}{\\rho^{\\prime}_{\\alpha}-\\alpha(s-1)}\\leq\\frac{(1-\\alpha)(d_{H^{\\prime}}(v)y_{2} +y_{0})}{\\rho^{\\prime}_{\\alpha}-\\alpha d_{G^{\\prime}}(v)}\\leq y_{v},\n\\end{equation}\nand\n\\begin{equation}\\label{equ::8'''}\ny_{v}\\leq \\frac{(1-\\alpha)(d_{H^{\\prime}}(v)y_{1} +y_{0})}{\\rho^{\\prime}_{\\alpha}-\\alpha d_{G^{\\prime}}(v)}<\\frac{(1-\\alpha)(d_{H^{\\prime}}(v)y_{1} +y_{0})}{\\rho^{\\prime}_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)}.\n\\end{equation}\nNow, we let\n\\begin{align*}\na^{\\prime}&=\\sum\\limits_{u v \\in E(H^{\\prime})} (d_{H^{\\prime}}(u)+d_{H^{\\prime}}(v)),&\nb^{\\prime}&=\\sum\\limits_{u v \\in E(H)} (d_{H^{\\prime}}(u)+d_{H^{\\prime}}(v)),\\\\\nc^{\\prime}_{1}&=\\sum\\limits_{u v \\in E(H)}(d_{H}(u) d_{H^{\\prime}}(v)+d_{H}(v) d_{H^{\\prime}}(u)),&\nc^{\\prime}_{2}&=\\sum\\limits_{u\\in V(H)} d_{H}^{2}(u)d_{H^{\\prime}}(u).\n\\end{align*}\nSince $V\\left(H^{\\prime}\\right)=V(H)$, we have\n\\begin{align*}a&=\\sum_{uv \\in E\\left(H^{\\prime}\\right)} (d_{H}(u)+d_{H}(v))=\\sum_{u \\in V\\left(H^{\\prime}\\right)} d_{H}(u) d_{H^{\\prime}}(u)=\\sum_{u \\in V(H)} d_{H}(u) d_{H^{\\prime}}(u)\\\\\n&=\\sum_{u v \\in E(H)} (d_{H^{\\prime}}(u)+d_{H^{\\prime}}(v))=b^{\\prime}.\n\\end{align*}\nFurthermore, we see that\n$\\sum\\limits_{u \\in V\\left(H^{\\prime}\\right)} d_{H^{\\prime}}^{2}(u)=\\sum\\limits_{uv \\in E\\left(H^{\\prime}\\right)} (d_{H^{\\prime}}(u)+d_{H^{\\prime}}(v))=a^{\\prime}$\nand $\\sum\\limits_{u \\in V\\left(H\\right)} d_{H}^{2}(u)= \\sum\\limits_{uv \\in E\\left(H\\right)} (d_{H}(u)+d_{H}(v))=b$.\nNow we denote\n\\begin{equation*}\n\\begin{aligned}\nA^{\\prime}_{1}=&\\sum_{u v \\in E(H^{\\prime})}((x_{0}+d_{H}(u) x_{2})(y_{0}+d_{H^{\\prime}}(v) y_{2})+(x_{0}+d_{H}(v) x_{2})(y_{0}+d_{H^{\\prime}}(u) y_{2}))\\\\\nA^{\\prime}_{2}=&\\sum_{u\\in V(H^{\\prime})} d_{H^{\\prime}}(u) \\cdot (d_{H}(u)x_{2} +x_{0}) \\cdot(d_{H^{\\prime}}(u)y_{2} +y_{0})\\\\\nA^{\\prime}_{3}=&\\sum_{u v \\in E(H)}((x_{0}+d_{H}(u) x_{1})\\cdot( y_{0}+d_{H^{\\prime}}(v) y_{1})+(x_{0}+d_{H}(v) x_{1})(y_{0}+d_{H^{\\prime}}(u) y_{1}))\\\\\nA^{\\prime}_{4}=&\\sum_{u\\in V(H)} d_{H}(u) \\cdot (d_{H}(u)x_{1} +x_{0}) \\cdot(d_{H^{\\prime}}(u)y_{1} +y_{0})\n\\end{aligned}\n\\end{equation*}\nBy simple scaling, we see that\n\\begin{align*}\nA^{\\prime}_1\\geq&2e(H^{\\prime})x_{0}y_{0} + ax_{2}y_{0}+a^{\\prime}x_{0}y_{2}, \\ \\ A^{\\prime}_3=2e(H)x_{0}y_{0} + ax_{1}y_{0}+ax_{0}y_{1}+c^{\\prime}_{1}x_{1}y_{1},\\\\\nA^{\\prime}_2\\geq &a^{\\prime}x_{0}y_{2}+ax_{2}y_{0} +2e(H^{\\prime})x_{0}y_{0},\\ \\ A^{\\prime}_4= c^{\\prime}_{2}x_{1}y_{1}+ax_{0}y_{1}+ax_{1}y_{0}+2e(H)x_{0}y_{0}.\n\\end{align*}\nSuppose that (\\ref{equ::6}) does not hold, then $a^{\\prime} \\geq b+1=a+1$.\nFrom Lemma \\ref{lem::3.1}, we find that $\\left(a+\\frac{1}{2}\\right) y_{2}>a y_{1}$.\nAdditionally,  by (\\ref{equ::4}) and (\\ref{equ::14}) we have\n\\begin{equation*}\\label{equ::22}\n\\begin{array}{ll}\n&\\frac{1}{2}x_{0} y_{2}+a y_{0}\\left(x_{2}-x_{1}\\right)-c^{\\prime}_{1} x_{1} y_{1}\\\\\n>&(1-\\alpha)x_{0} y_{0}\\left(\\frac{1}{ 2(\\rho^{\\prime}_{\\alpha}-\\alpha(s-1))}+\\frac{a}{\\rho_{\\alpha}-\\alpha(s-1)}-\\frac{a}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)}-\\frac{(1-\\alpha)c^{\\prime}_{1}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))\\left(\\rho^{\\prime}_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)\\right)}\\right)\\\\\n>&0,\n\\end{array}\n\\end{equation*}\nsince $\\rho_{\\alpha}\\geq\\alpha(n-1)+(1-\\alpha)(s-2)$, $\\rho^{\\prime}_{\\alpha}\\geq \\rho_{\\alpha}\\left( K_{s-1} \\vee \\overline{K}_{n-s+1}\\right)\\geq \\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, and $n$ is sufficiently large. Similarly, we have $\\frac{1}{2}x_{0} y_{2}+a y_{0}\\left(x_{2}-x_{1}\\right)-c^{\\prime}_{2} x_{1} y_{1}>0,$\nFurthermore, since $e\\left(H^{\\prime}\\right)=e\\left( H\\right)$, we have\n\\begin{align*}\nA^{\\prime}_{1}-A^{\\prime}_{3}\\geq&(2e(H^{\\prime})x_{0}y_{0} + ax_{2}y_{0}+a^{\\prime}x_{0}y_{2})-(2e(H)x_{0}y_{0} +ax_{1}y_{0}+ax_{0}y_{1}+c^{\\prime}_{1}x_{1}y_{1})\\\\\n=& a^{\\prime} x_{0} y_{2}+a x_{2} y_{0}-a x_{0} y_{1}-a x_{1} y_{0}-c^{\\prime}_{1} x_{1} y_{1}\\\\\n\\geq&(a+1) x_{0} y_{2}+a x_{2} y_{0}-a x_{0} y_{1}-a x_{1} y_{0}-c^{\\prime}_{1} x_{1} y_{1} \\\\\n=&x_{0}\\left(\\left(a+\\frac{1}{2}\\right) y_{2}-a y_{1}\\right)+\\left(\\frac{1}{2}x_{0} y_{2}+a\\left(x_{2}-x_{1}\\right) y_{0}-c^{\\prime}_{1} x_{1} y_{1}\\right)\\\\\n>&0,\n\\end{align*}\nand similarly, we obtain $A^{\\prime}_{2}-A^{\\prime}_{4}>0$.\nBy (\\ref{equ::8})-(\\ref{equ::8''}) and (\\ref{equ::8'})-(\\ref{equ::8'''}), we find that\n\\begin{align*}\n\\left(\\rho_{\\alpha}^{\\prime}-\\rho_{\\alpha}\\right) \\mathbf{Y}^{T}\\mathbf{X} =&\\left(A_{\\alpha}\\left(G^{\\prime}\\right) \\mathbf{Y}\\right)^{T}\\mathbf{X}-\\mathbf{Y}^{T}\\left(A_{\\alpha}\\left(G^{*}\\right) \\mathbf{X}\\right)\\\\\n=&\\left( \\sum_{u v \\in E\\left(H^{\\prime}\\right)}(1-\\alpha)\\left( x_{u} y_{v}+x_{v} y_{u}\\right)+\\sum_{u\\in V\\left(H^{\\prime}\\right)} \\alpha d_{H^{\\prime}}(u) x_{u} y_{u}\\right) \\\\\n&-( \\sum_{u v \\in E(H)}(1-\\alpha)\\left( x_{u}  y_{v}+x_{v}y_{u}\\right)+\\sum_{u\\in V\\left(H\\right)} \\alpha d_{H}(u) x_{u} y_{u})\\\\\n=&(1-\\alpha)\\left( \\sum_{u v \\in E\\left(H^{\\prime}\\right)}\\left( x_{u} y_{v}+x_{v} y_{u}\\right)-\\sum_{u v \\in E(H)}\\left( x_{u}  y_{v}+x_{v}y_{u}\\right)\\right)\\\\\n&+\\alpha\\left( \\sum_{u\\in V\\left(H^{\\prime}\\right)}  d_{H^{\\prime}}(u) x_{u} y_{u} - \\sum_{u\\in V\\left(H\\right)}d_{H}(u) x_{u} y_{u})\\right) \\\\\n\\geq&(1-\\alpha)^{3}\\left( \\frac{A^{\\prime}_{1}}{(\\rho_{\\alpha}-\\alpha(s-1))(\\rho^{\\prime}_{\\alpha}-\\alpha(s-1))}\\right.\\\\\n&\\left.-\\frac{A^{\\prime}_{3}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))(\\rho^{\\prime}_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))}\\right) \\\\\n&+\\alpha(1-\\alpha)^{2}\\left( \\frac{A^{\\prime}_{2}}{(\\rho_{\\alpha}-\\alpha(s-1))(\\rho^{\\prime}_{\\alpha}-\\alpha(s-1))}\\right.\\\\\n&\\left.-\\frac{A^{\\prime}_{4}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))(\\rho^{\\prime}_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))}\\right) \\\\\n>&0,\n\\end{align*}\nwhich  contradicts  the maximality of $\\rho_{\\alpha}$.  Therefore,  (\\ref{equ::6}) holds.\n\\end{proof}\n\nThe following lemma implies that $G^{*}$ has a property of local degree sequence majorization.\n\\begin{lem}\\label{lem::3.4}\nLet $H$ consist of some components of $G^{*}-K$ with $|H| \\leq N$ (a constant), $H^{\\prime}$ be a graph with $V\\left(H^{\\prime}\\right)=V(H)$, $e\\left(H^{\\prime}\\right)=e\\left( H\\right)$ and $H^{\\prime}$ have the $(s, t)$-property. If $\\pi(H) \\prec \\pi\\left(H^{\\prime}\\right)$, then $\\pi(H)=\\pi\\left(H^{\\prime}\\right)$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that $\\pi(H) \\neq \\pi\\left(H^{\\prime}\\right)$. Since $V(H)=V\\left(H^{\\prime}\\right)$, we have $\\pi(H)=\\left(d_{1}, d_{2}, \\ldots, d_{|H|}\\right)$ and $\\pi\\left(H^{\\prime}\\right)=\\left(d_{1}^{\\prime}, d_{2}^{\\prime}, \\ldots, d_{|H|}^{\\prime}\\right)$. Let $k=2$ in Lemma \\ref{lem::2.8}, we have\n\\begin{equation}\\label{equ::25}\n\\sum_{i=1}^{|H|} d_{i}^{2}<\\sum_{i=1}^{|H|} d_{i}^{\\prime 2}.\n\\end{equation}\nFurthermore, let $\\mathbf{x}=\\mathbf{z}=\\pi(H)$ and $\\mathbf{y}=\\pi\\left(H^{\\prime}\\right)$ in Lemma \\ref{lem::2.9}, we obtain\n\\begin{equation}\\label{equ::26}\n\\sum_{i=1}^{|H|} d_{i}^{2} \\leq \\sum_{i=1}^{|H|} d_{i} d_{i}^{\\prime}.\n\\end{equation}\nNotice that $\\sum\\limits_{i=1}^{|H|} d_{i}^{2}=\\sum\\limits_{u v \\in E(H)}\\left(d_{H}(u)+d_{H}(v)\\right)$ and\n$$\\sum_{i=1}^{|H|} d_{i} d_{i}^{\\prime}=\\sum_{v \\in V\\left(H^{\\prime}\\right)} d_{H}(v) d_{H^{\\prime}}(v)=\\sum_{u v \\in E\\left(H^{\\prime}\\right)}\\left(d_{H}(u)+d_{H}(v)\\right).$$\nThen by (\\ref{equ::5})  we have $\\sum\\limits_{i=1}^{|H|} d_{i} d_{i}^{\\prime} \\leq \\sum\\limits_{i=1}^{|H|} d_{i}^{2}$. Together with (\\ref{equ::26}), we obtain $\\sum\\limits_{i=1}^{|H|} d_{i} d_{i}^{\\prime}=\\sum\\limits_{i=1}^{|H|} d_{i}^{2}$. Therefore, (\\ref{equ::6})  holds, that is, $\\sum\\limits_{i=1}^{|H|} d_{i}^{\\prime 2}\\leq \\sum\\limits_{i=1}^{|H|} d_{i}^{2}$, which contradicts (\\ref{equ::25}).\n\\end{proof}\n\n\n\n\n\\section{The characterization of the $A_\\alpha$-spectral extremal graph}\nRecall that $G^*$ is a graph with maximum $A_\\alpha$-spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order $n$, where $0 < \\alpha<1$ and $2\\leq s\\leq t$. Moreover, $\\rho_{\\alpha} = \\rho_{\\alpha}(G^{*})$, $\\mathbf{X}=(x_{v})_{v \\in V(G^{*})} \\in R^{n}$ is a Perron eigenvector of $A_{\\alpha}(G^{*})$ corresponding to $\\rho_{\\alpha}$, $x_{0}=\\sum\\limits_{v\\in K}x_{v}$. In addition,\n$G^*$ contains a clique dominating set $K$ of size $s-1$, $G^{*}-K$ has the $(s, t)$-property and $\\Delta(G^{*}-K)<t$.\n\n\\begin{lem}\\label{lem::3.1'}\nIf $s=2$, $t=2$, $n-1=pt+r$ and $1\\leq r \\leq2$. Then $G^{*}\\cong K_{1}\\vee(pK_{2}\\cup K_{r}).$\n\\end{lem}\n\\begin{proof}\nNote that $|K|=s-1=1$. Let $v$ be the unique vertex in $K$. Thus $d_{G^{*}}(v)=n-1$. Since $G^{*}$ is a $K_{2,2}$-minor free graph, then $G^{*}$ contains no $K_{2,2}$ as a subgraph, which implies that $d_{G^{*}-v}(u)\\leq1$ for any vertex $u \\in V(G^{*}-v)$. Thus $G^{*}$ is a subgraph of $K_{1}\\vee(pK_{2}\\cup K_{r})$. By the maximality of  $\\rho_{\\alpha}$, we get that $G^{*}\\cong K_{1}\\vee(pK_{2}\\cup K_{r})$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.1''}\nIf $s=2$, $t=3$, $n-1=pt+r$ and $1\\leq r \\leq3$. Then $G^{*}\\cong K_{1}\\vee(pK_{3}\\cup K_{r}).$\n\\end{lem}\n\\begin{proof}\nNote that $|K|=s-1=1$. Let $v$ be the unique vertex in $K$. Thus $d_{G^{*}}(v)=n-1$. Since $G^{*}$ is $K_{2,3}$-minor free, we see that $G^{*}-v$ does not contain any cycle of length at least $4$ as a subgraph\nand    $d_{G^{*}-v}(u)\\leq2$ for any vertex $u \\in V(G^{*}-v)$. Therefore, each component of $G^{*}-v$ is either a triangle or a path of order at least $1$. In fact, there is at most one component of $G^{*}-v$ is a path. Otherwise adding an edge to two pendant vertices in two different components which are paths leads to a $K_{2,3}$-minor free graph with larger $A_\\alpha$-spectral radius, a contradiction. Let $P$ be the unique component of $G^{*}-v$ which is not triangle, that is,\n$P$ is a path of order $d \\geq 1$.  Now, we assert that $1 \\leq d \\leq 2$.\nLet $v_{i}\\in V(P)$ such that $P=v_1\\ldots v_d$ for $1\\leq i \\leq d$. We next consider the following three cases.\n\n\n{\\flushleft\\bf Case 1. $d=3$.} Let $G^{\\prime}=G^{*}+v_1v_d$. Clearly, $G^{\\prime}$ is a $K_{2,3}$-minor free graph and $\\rho_{\\alpha}\\left(G^{\\prime}\\right)>\\rho_{\\alpha}$ by Lemma \\ref{lem::2.2}, a contradiction.\n\n\n{\\flushleft\\bf Case 2. $d=4$.} We have $x_{v_1}=x_{v_4}$ and $x_{v_2}=x_{v_3}$ by symmetry. Let $G^{\\prime}=G^{*}-v_1 v_2+v_2 v_4$. Clearly, $G^{\\prime}$ is $K_{2,3}$-minor free and\n$$\n\\rho_{\\alpha}\\left(G^{\\prime}\\right)-\\rho_{\\alpha} \\geq\\left(\\alpha x^{2}_{v_2}+2(1-\\alpha)x_{v_2} x_{v_4}+\\alpha x^{2}_{v_4}\\right)-\\left(\\alpha x^{2}_{v_1}+2(1-\\alpha)x_{v_1} x_{v_2}+\\alpha x^{2}_{v_2}\\right)=0.\n$$\nIf $\\rho_{\\alpha}\\left(G^{\\prime}\\right)=\\rho_{\\alpha}$, then $\\mathbf{X}$ is also an unit eigenvector corresponding to $\\rho_{\\alpha}\\left(G^{\\prime}\\right)$. Since $v_2$, $v_3$, and $v_4$ are symmetric in $G^{\\prime}$, we have $x_{v_2}=x_{v_3}=x_{v_4}$, which implies that $x_{v_1}=x_{v_2}=x_{v_3}=x_{v_4}$. By eigenequations of $A_{\\alpha}(G^{*})$ on $v_1$ and $v_2$, we have\n$$\n\\rho_{\\alpha} x_{v_1}=2\\alpha x_{v_1}+(1-\\alpha)(x_{v_2}+x_v) \\quad \\text { and } \\quad \\rho_{\\alpha} x_{v_2}=3\\alpha x_{v_2}+(1-\\alpha)(x_{v_1}+x_{v_3}+x_v),\n$$\na contradiction. Thus, $\\rho_{\\alpha}\\left(G^{\\prime}\\right)>\\rho_{\\alpha}$, which is also a contradiction.\n\n\n{\\flushleft\\bf Case 3. $d\\geq5$.} We firstly assume that $d$ is odd, let $d=2j+1$ with $j \\geq 2$. Then we have $x_{v_k}=x_{v_{2j+2-k}}$ for $1 \\leq k \\leq j$ by symmetry. Let $G^{\\prime}=G^{*}-\\left\\{v_{j-1} v_j, v_{j+2} v_{j+3}\\right\\}+\\left\\{v_j v_{j+2}, v_{j-1} v_{j+3}\\right\\}$. Clearly, $G^{\\prime}$ is a $K_{2,3}$-minor free graph and\n$$\n\\begin{aligned}\n\\rho_{\\alpha}\\left(G^{\\prime}\\right)-\\rho_{\\alpha}\\geq& \\sum_{u w \\in E(G^{\\prime})}(\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{w}+\\alpha x_{w}^{2})-\\sum_{u w \\in E(G^{*})} (\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{w}+\\alpha x_{w}^{2}) \\\\\n=&2(1-\\alpha) (x_{v_j} x_{v_{j+2}}+x_{v_{j-1}} x_{v_{j+3}}-x_{v_{j-1}} x_{v_{j}}-x_{v_{j+2}} x_{v_{j+3}}) \\\\\n=& 2(1-\\alpha)( x_{v_{j}}^2+x_{v_{j-1}}^2-2x_{v_{j-1}} x_{v_{j}})  \\\\\n=& 2(1-\\alpha)(x_{v_{j-1}}-x_{v_{j}})^2 \\\\\n\\geq & 0.\n\\end{aligned}\n$$\nNow, suppose that $d$ is even, let $d=2 j$ with $j \\geq 3$. Then we have $x_{v_k}=x_{v_{2j+1-k}}$ for $1 \\leq k \\leq j$ by symmetry. Let $G^{\\prime}=G^{*}-\\left\\{v_{j-1} v_j, v_{j+2} v_{j+3}\\right\\}+\\left\\{v_j v_{j+2}, v_{j-1} v_{j+3}\\right\\}$. Clearly, $G^{\\prime}$ is $K_{2,3}$-minor free and\n$$\n\\begin{aligned}\n\\rho_{\\alpha}\\left(G^{\\prime}\\right)-\\rho_{\\alpha}\n\\geq& \\sum_{u w \\in E(G^{\\prime})}(\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{w}+\\alpha x_{w}^{2})-\\sum_{u w \\in E(G^{*})} (\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{w}+\\alpha x_{w}^{2}) \\\\\n=&2(1-\\alpha) (x_{v_{j}} x_{v_{j+2}}+ x_{v_{j-1}} x_{v_{j+3}}-x_{v_{j-1}} x_{v_{j}}-x_{v_{j+2}} x_{v_{j+3}})\\\\\n=&2(1-\\alpha) ( x_{v_{j}}(x_{v_{j+2}}-x_{v_{j-1}})- x_{v_{j+3}}(x_{v_{j+2}}-x_{v_{j-1}})) \\\\\n=&2(1-\\alpha) (x_{v_{j}}-x_{v_{j+3}})(x_{v_{j+2}}-x_{v_{j-1}})\\\\\n=&0.\n\\end{aligned}\n$$\nThus, whether $d$ is odd or even, if $\\rho_{\\alpha}\\left(G^{\\prime}\\right)=\\rho_{\\alpha}$, then $\\mathbf{X}$ is also an unit eigenvector corresponding to $\\rho_{\\alpha}\\left(G^{\\prime}\\right)$. Since $v_{v_{j}}$, $ v_{v_{j+1}}$, and $v_{v_{j+2}}$ are symmetric in $G^{\\prime}$, we have $x_{v_{j-1}}=x_{v_{j}}=x_{v_{j+1}}$. By the eigenequations of $A_{\\alpha}(G^{*})$, we have $x_{v_{1}}=\\cdots=x_{v_{d}}$. From the eigenequations of $A_{\\alpha}(G^{*})$ on $v_1$ and $v_2$, we have\n$$\n\\rho_{\\alpha} x_{v_1}=2\\alpha x_{v_1}+(1-\\alpha)(x_{v_2}+x_v) \\quad \\text { and } \\rho_{\\alpha} x_{v_2}=3 \\alpha x_{v_2}+(1-\\alpha)(x_{v_1}+x_{v_3}+x_v),\n$$\na contradiction. So, $\\rho_{\\alpha}\\left(G^{\\prime}\\right)>\\rho_{\\alpha}$, which is also a contradiction.\n\nTherefore, we get that $1 \\leq d \\leq 2$, and so $G^{*}-v$ consists of disjoint copies of triangles and at most a path of order $1$ or $2$, that is, $G^{*}\\cong K_{1}\\vee(pK_{2}\\cup K_{r}).$\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.1'''}\nIf $s=3$, $t=3$, $n-1=pt+r$ and $1\\leq r \\leq3$. Then $G^{*}\\cong K_{2}\\vee(pK_{3}\\cup K_{r}).$\n\\end{lem}\n\\begin{proof}\nNote that $|K|=s-1=2$. Let $K=\\{v, v^{\\prime}\\}$. Thus, $d_{G^{*}}(v)=d_{G^{*}}(v^{\\prime})=n-1$. Since $G^{*}$ is $K_{3,3}$-minor free, we see that $G^{*}-\\{v, v^{\\prime}\\}$ does not contain any cycle of length at least $4$ as a subgraph. Moreover, $G^{*}$ contains no $K_{3,3}$ as a subgraph, which implies that $d_{G^{*}-\\{v, v^{\\prime}\\}}(u)\\leq2$ for any vertex $u \\in V(G^{*}-\\{v, v^{\\prime}\\})$.\nTherefore, each component of $G^{*}-\\{v, v^{\\prime}\\}$ is either a triangle or a path of order at least $1$. In fact, there is at most one component of $G^{*}-\\{v, v^{\\prime}\\}$ is a path. Otherwise, adding an edge to two pendant vertices in two different components which are paths leads to a $K_{3,3}$-minor free graph with larger $A_\\alpha$-spectral radius, a contradiction. Let $P$ be the unique component of $G^{*}-\\{v, v^{\\prime}\\}$ which is not triangle, that is, $P$ is a path of order $d \\geq 1$.\nSimilar as Cases 1-3 in the proof of Lemma \\ref{lem::3.1''}, we can get that $1 \\leq d \\leq 2$.\nThus $G^{*}-\\{v, v^{\\prime}\\}$ consists of disjoint copies of triangles and at most a path of order $1$ or $2$, and so $G^{*}\\cong K_{2}\\vee(pK_{3}\\cup K_{r}).$\n\\end{proof}\n\nLemmas \\ref{lem::3.1'}-\\ref{lem::3.1'''} imply that the cases $t=2$ and $t=3$ in Theorem \\ref{thm::1.1} hold. Thus, it remains to prove the Theorem \\ref{thm::1.1} for $t\\geq4$. We now let $t\\geq4$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$, $\\gamma = \\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}$. In addition,\nwe use $H_{i}$, $H_{>i}$ and $H_{<i}$ to denote the family of components in  $G^{*}-K$ with order $i$, greater than $i$, and less than $i$, respectively.\n\n\\begin{lem}\\label{lem::3.6}\n$H_{>t+3}=\\varnothing$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that there exists a component $H \\in H_{>t+3}$. Let $|H|=pt+r$, where $p \\geq 1$ and $1 \\leq r \\leq t$. Since $H$ is $K_{1,t}$-minor free, by Lemma \\ref{lem::2.4} we have\n\\begin{equation}\\label{equ::27}\ne(H) \\leq \\binom{t}{2}+|H|-t= \\binom{t}{2} +(p-1) t+r.\n\\end{equation}\nNow, let $H^{\\prime} \\cong p K_{t} \\cup K_{r}$ with $V\\left(H^{\\prime}\\right)=V(H)$. Clearly, $H^{\\prime}$ also has the $(s, t)$-property. Since $|H|>t+3$ and $t \\geq 4$, then by (\\ref{equ::27}) we obtain\n$\ne(H)<p \\binom{t}{2} + \\binom{r}{2} =e\\left(H^{\\prime}\\right)\n$,\ncontradicting Lemma \\ref{lem::3.2}.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.10}\nThere exists at most one component in $H_{<t} \\cup H_{t+2}$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that there exists two components $D_{1}, D_{2} \\in H_{<t} \\cup H_{t+2}$ with $|D_{1}|\\leq|D_{2}|$. For $i=1,2$, we have $e\\left(D_{i}\\right) \\leq\\binom{\\left|D_{i}\\right| }{2}$ if $|D_{i}| < t$, and by\nLemma \\ref{lem::2.4} we obtain\n$e\\left(D_{i}\\right) \\leq\\binom{t }{2}+2$ if $|D_{i}|=t+2$.\nNow, let $|D_{1}|+|D_{2}|=pt+r$, where $0 \\leq p \\leq 2$ and $1 \\leq r \\leq t$.\nMoreover, let $H \\cong p K_{t} \\cup K_{r}$ with $V\\left(H\\right)=V(D_{1} \\cup D_{2})$. Clearly, $H$ has the $(s, t)$-property.\nHowever, in any case of $|D_{1}|\\leq|D_{2}|<t$, $|D_{1}|<t<|D_{2}|$ or $|D_{1}|=|D_{2}|=t+2$, we always find that $e\\left(D_{1} \\cup D_{2}\\right)<p\\binom{t}{2}+\\binom{r}{2}=e\\left(H\\right)$, contradicting Lemma \\ref{lem::3.2}.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.7}\nLet $\\eta$ be the number of the components in $H_{t}$. Then $\\eta=O\\left(\\frac{n}{t}\\right)$.\n\\end{lem}\n\n\\begin{proof}\nFrom Lemma \\ref{lem::3.6}, we have $H_{>t+3}=\\varnothing$. Furthermore, there exists at most one component in $H_{<t} \\cup H_{t+2}$ by Lemma \\ref{lem::3.10}. Note that $t\\geq4$. Hence, we just need to show that the number of the components in $H_{i}$ is less than $t$ for each $i\\in \\{t+1, t+3\\}$. Suppose to the contrary that there exists a $H_{i}$ which contains at least $t$ components for $i\\in \\{t+1, t+3\\}$. Let $H^{\\prime}_{i}$ be a disjoint union of $t$ components in $H_{i}$.\nIf $i=t+1$, then by Lemma \\ref{lem::2.12} and $\\beta=   \\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor<\\frac{t+1}{2}$, we have\n$\ne(H^{\\prime}_{i}) \\leq t\\left( \\binom{t}{2} +\\beta-1\\right)<(t+1) \\binom{t}{2} =e\\left(i K_{t}\\right).\n$\nIf $i =t+3$, then by Lemma \\ref{lem::2.4}, we obtain\n$\ne(H^{\\prime}_{i}) \\leq t\\left( \\binom{t}{2} +i-t\\right)<i\\binom{t}{2}=e\\left(i K_{t}\\right).\n$\nClearly, $iK_{t}$ has the $(s, t)$-property. In either of the above cases, we always get a contradiction to Lemma \\ref{lem::3.2}.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem::3.5}\nIf $H$ is a component in $H_{t+1}$. Then $H \\cong \\overline{H_{s, t}}$, $e\\left(\\overline{H_{s, t}}\\right)=\\binom{t}{2}+\\beta-1$ and $\\beta \\geq 2$.\n\\end{lem}\n\n\\begin{proof}\nRecall that $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ and $\\gamma = \\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}$. Clearly, $H$ has the $(s, t)$-property.\nMoreover, we find that $H$ is an edge-maximal graph by Lemma \\ref{lem::3.2}.\nHence, by Lemmas \\ref{lem::2.11} and \\ref{lem::2.12}, we get that $e(H)=\\binom{t}{2}+\\beta-1$, $\\overline{H}$ is a forest with $\\beta$ components and each of which has at least $\\gamma+1$ vertices. Suppose that $\\overline{H}$ contains a component $D$ which is not a star. Then $D$ contains two leaves $v_{1}$ and $v_{2}$ with distinct neighbors, say $N_{D}(v_{i})=\\{u_{i}\\}$ for $i=1,2$. Assume without loss of generality that $d_{D}(u_{1}) \\leq d_{D}(u_{2})$. Then  $d_{H}(u_{1}) \\geq d_{H}(u_{2})$. Let $H^{\\prime}=H-\\{u_{2} v_{1}\\}+\\{u_{1} v_{1}\\}$. By Lemma \\ref{lem::2.11}, $H^{\\prime}$ also has the $(s, t)$-property. Furthermore, by Lemma \\ref{lem::2.13} we get that $\\pi(H) \\prec \\pi\\left(H^{\\prime}\\right)$ and $\\pi(H) \\neq \\pi\\left(H^{\\prime}\\right)$, which contradicts Lemma \\ref{lem::3.4}. Therefore, $\\overline{H}$ is a star forest. Notice that $H$ is connected and each component of $\\overline{H}$ is a star of order at least $\\gamma+1$. Thus, $\\overline{H}$ has at least two components, that is, $\\beta \\geq 2$. Then by $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ we have $s\\leq\\left\\lfloor\\frac{t-1}{2}\\right\\rfloor$. Note that $\\gamma = \\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}$. So, $\\gamma=s$. That is, each component of $\\overline{H}$ has at least $s+1$ vertices.\n\nClearly, $|\\overline{H_{s,t}}|=t+1$, and we get that $\\overline{H_{s, t}}$ has the $(s, t)$-property by by Lemma \\ref{lem::2.16}(i).\nIn order to show that $H \\cong \\overline{H_{s, t}}$, we just need to show that $\\overline{H}$ has at most one component not isomorphic to $K_{1, s}$. Suppose to the contrary that there exists two components $D_{1}$ and $D_{2}$ with $|D_{2}| \\geq|D_{1}| \\geq s+2$. For $i \\in\\{1,2\\}$, let $v_{i}$ be a leaf and $u_{i}$ be the central vertex of $D_{i}$. Let $H^{\\prime}=H-\\{u_{2} v_{1}\\}+\\{u_{1} v_{1}\\}$. We get that $\\pi(H) \\prec \\pi\\left(H^{\\prime}\\right)$ and $\\pi(H) \\neq \\pi\\left(H^{\\prime}\\right)$ in a similar way as above, which contradicts Lemma \\ref{lem::3.4}. Thus, $H \\cong \\overline{H_{s, t}}$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.12}\nThere are at most $2(\\beta-1)$ components in $H_{t+1}$.\n\\end{lem}\n\n\\begin{proof}\nThe case $H_{t+1}=\\varnothing$ is trivial. Assume that $H_{t+1} \\neq \\varnothing$. Then by Lemma \\ref{lem::3.5}, we have $H \\cong \\overline{H_{s, t}}$, $e(H)=\\binom{t}{2}+\\beta-1$ and $\\beta \\geq 2$ for each component $H \\in H_{t+1}$. If there exists at least $2\\beta$ components in $H_{t+1}$. Then we select $2 \\beta$ copies of $\\overline{H_{s, t}}$ and denote it by $H^{\\prime}_{t+1}$. Now let $H^{\\prime\\prime}_{t+1} \\cong 2 \\beta K_{t} \\cup K_{2 \\beta}$. Since $2 \\beta<t$, then $H^{\\prime\\prime}_{t+1}$ also has the $(s, t)$-property. However, we find that\n$$\ne\\left(H^{\\prime\\prime}_{t+1}\\right)=2 \\beta\\binom{t}{2}+ \\binom{2 \\beta}{2} >2 \\beta\\left(\\binom{t}{2}+\\beta-1\\right)=e(H^{\\prime}_{t+1}),\n$$\nwhich contradicts Lemma \\ref{lem::3.2}. Hence, there exists at most $2\\beta-1$ components in $H_{t+1}$.\nIf there are exactly $2\\beta-1$ components in $H_{t+1}$. Then  $H_{t+1}\\cong(2 \\beta-1)\\overline{H_{s, t}}$. Let  $H^{\\prime\\prime\\prime}_{t+1} \\cong(2 \\beta-1) K_{t} \\cup K_{2 \\beta-1}$. Thus,  $e\\left(H^{\\prime\\prime\\prime}_{t+1}\\right)=e(H_{t+1})$. Note that $H_{s, t} \\cong (\\beta-1) K_{1, s} \\cup K_{1, \\alpha} $, where $\\alpha=t-(s+1)(\\beta-1) \\geq s$. Thus we obtain\n\\begin{equation}\\label{equ::30}\n\\delta\\left(\\overline{H_{s, t}}\\right)=(s+1)(\\beta-1)>2(\\beta-1),\n\\end{equation}\nwhich implies that $\\delta(H_{t+1})>2 \\beta-2$. Clearly,  $\\pi\\left(H^{\\prime\\prime\\prime}_{t+1}\\right)=(t-1, \\ldots, t-1,2 \\beta-2, \\ldots, 2 \\beta-2)$. Then $\\pi(H_{t+1}) \\prec \\pi\\left(H^{\\prime\\prime\\prime}_{t+1}\\right)$, which contradicts Lemma \\ref{lem::3.4}. Thus, there exists at most $2(\\beta-1)$ components in $H_{t+1}$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.9}\n$H_{t+3}=\\varnothing$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that there exists a component $H$ of $G^{*}-K$ with $|H|=t+3$. Then we have $e(H) \\leq\\binom{t }{2}+3$ by Lemma \\ref{lem::2.4}. Clearly, $K_{t} \\cup K_{3}$ also has the $(s, t)$-property. Then by Lemma \\ref{lem::3.2}, we have $e(H) \\geq  e\\left(K_{t} \\cup K_{3}\\right)=\\binom{t}{2}+3$. Hence, $e(H)=\\binom{t}{2}+3$.\nSuppose that $\\beta \\geq 3$. Since $|\\overline{H_{s, t}}|=t+1$, $\\overline{H_{s,t}}$ has the $(s, t)$-property by Lemma \\ref{lem::2.16}(i) and $e\\left(\\overline{H_{s, t}}\\right)=\\binom{t}{2}+\\beta-1$ by Lemma \\ref{lem::3.5}. Moreover, from Lemma \\ref{lem::3.7}, there exists two components which are isomorphic to $K_{t}$ in $G^{*}-K$. Then\n$\ne\\left(H \\cup 2K_{t}\\right)=3\\binom{t}{2}+3<3\\binom{t}{2}+3(\\beta-1)=e\\left(3\\overline{H_{s, t}}\\right),\n$\nwhich contradicts Lemma \\ref{lem::3.2}. Hence $\\beta \\leq 2$.\n\t\n\\begin{clm}\\label{clm::3.5}\n$2 \\leq d_{H}(v) \\leq t-1$ for each $v \\in V(H)$.\n\\end{clm}\n\t\n\\begin{proof}\nClearly, $\\Delta(H)\\leq\\Delta(G^{*}-K)\\leq t-1$. Thus, it remains to show that $\\delta(H) \\geq 2$. Suppose to the contrary that $\\delta(H)=1$. Take an arbitrary vertex $v \\in V(H)$ with $d_{H}(v)=1$. Then $H-\\{v\\}$ is connected and  $e(H-v)=\\binom{t}{2}+2$.\nBy Lemmas \\ref{lem::2.15} and \\ref{lem::3.2},\n$H-v$ is isomorphic to either $\\overline{H^{\\star}}$ or some $\\overline{H_{a, b, c}}$ with $a+b+c=t-1$. If $H-v$ is isomorphic to some $\\overline{H_{a, b, c}}$, by contracting the edges $u_{1} u_{2}$ and $u_{2} w$ from Fig. \\ref{fig-1}, we find that $\\overline{H_{a, b, c}}$ contains a $K_{t}$-minor, this implies that $H$ contains a $K_{1, t}$-minor, a contradiction. Hence, $H-v \\cong \\overline{H^{\\star}}$. Now, let $vuw$ be a path of length $2$ in $H$. Note that any two non-adjacent vertices in the Petersen graph $H^{\\star}$ have exactly one common neighbor. So, if we contract $uw$ in $H-\\{v\\}$, then the new vertex is of degree $t-1$ in the resulting graph. Consequently, if we contract $uw$ in $H$, then the new vertex is of degree $t$, which implies that $H$ contains a $K_{1, t}$-minor, a contradiction.\n\\end{proof}\nBy $e(H)=\\binom{t}{2}+3$ and Claim \\ref{clm::3.5}, we have $\\pi(H) \\prec (t-1, \\ldots, t-1,2,2,2)=\\pi\\left(K_{t} \\cup K_{3}\\right)$. Then by Lemma \\ref{lem::3.4}, we obtain $\\pi(H)=\\pi\\left(K_{t} \\cup K_{3}\\right)$.\nLet $M_{1}=\\{v \\in V(H) \\mid d_{H}(v)=2\\}$ and $M_{2}=\\cup_{v \\in M_{1}} N_{H}(v) \\backslash M_{1}$. Clearly, $1 \\leq d_{G[M_{1}]}(u) \\leq 3$ for any $u \\in M_{2}$.\n\t\n\\begin{clm}\\label{clm::3.6}\n$d_{G[M_{1}]}(u)=1$ for any $u \\in M_{2}$.\n\\end{clm}\n\t\n\\begin{proof}\nLet $L$ be the set of non-adjacent vertex-pairs in $M_{2}$. Since $|V(H) \\backslash M_{1}|=t$ and $d_{H}(u)=t-1$ for each $u \\in V(H) \\backslash M_{1}$, we have $|L|=\\frac{1}{2} e(M_{1}, M_{2})$, and so $1 \\leq|L| \\leq 3$. Suppose that $d_{G[M_{1}]}\\left(u_{0}\\right)=c \\in\\{2,3\\}$ for some $u_{0} \\in M_{2}$. Then $u_{0}$ has exactly $c$ non-neighbors, say $\\left\\{u_{1}, \\ldots, u_{c}\\right\\}$, in $M_{2}$. If $c=3$, then $L=\\left\\{\\left(u_{0}, u_{i}\\right) \\mid i=1,2,3\\right\\}$  and there are three paths $u_{0} v_{i} u_{i}$ in $H$, where $v_{i} \\in M_{1}$ and $i \\in\\{1,2,3\\}$. Now, if we contract two of the three paths into edges, then the resulting graph is isomorphic to $S^{1}(K_{t})$. However, $S^{1}\\left(K_{t}\\right)$ contains $K_{2, t-1}$ as a subgraph, which contradicts the fact that $H$ has the $(s,t)$-property. Therefore, $c=2$. Since $t \\geq 4$, we can find a vertex $u_{3} \\in N_{H}(u_{0}) \\backslash M_{1}$. Note that $|L| \\leq 3$. Hence, $\\{u_{1}, u_{2}\\} \\cap N_{H}(u_{3}) \\neq \\varnothing$. Moreover, if $u_{1}, u_{2} \\in N_{H}\\left(u_{3}\\right)$, then $H$ contains a double star with a non-pendant edge $u_{0} u_{3}$ and $t$ leaves, which implies that $H$ has a $K_{1, t}$-minor, a contradiction. So we may assume that  $u_{1} u_{3} \\in E(H)$ and $u_{2} u_{3} \\notin E(H)$. Since $|L| \\leq 3$, we have $u_{1} u_{2} \\in E(H)$. It follows that $P=u_{0} u_{3} u_{1} u_{2}$ is an induced path in $H$. Now, $|L|=3$ and $M_{2}=\\left\\{u_{0}, u_{1}, u_{2}, u_{3}\\right\\}$. Furthermore, $d_{G[M_{1}]}\\left(u_{i}\\right)=2$ for $i \\in\\{0,2\\}$ and  $d_{G[M_{1}]}(u_{i})=1$ for $i \\in\\{1,3\\}$. Then there exists a double star with a non-pendant edge in $E(P)$ and $t$ leaves, which implies that $H$ has a $K_{1, t}$-minor, a contradiction.\n\\end{proof}\n\t\n\\begin{clm}\\label{clm::3.7}\n$x_{u}>x_{v}$ for any two vertices $u, v$ with $u \\in M_{2}$ and $v \\in M_{1}$.\n\\end{clm}\n\\begin{proof}\nLet  $x_{1}=\\max\\limits_{v \\in V(H)}x_{v}$ and $x_{2}=\\min\\limits_{v\\in V(H)}x_{v}$.   Note that $x_{0}=\\sum\\limits_{v\\in K}x_{v}$.\nBy (\\ref{equ::8}) we have $$x_{u}\\geq\\frac{(1-\\alpha)(d_{H}(u)x_{2} +x_{0})}{\\rho_{\\alpha}-\\alpha(s-1)}\\ \\ \\mbox{and} \\ \\  x_{v}< \\frac{(1-\\alpha)(d_{H}(v)x_{1} +x_{0})}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)}.$$\nSince $d_{H}(u)>d_{H}(v)$, by Lemma \\ref{lem::3.1} we have $d_{H}(u)x_{2}>d_{H}(v)x_{1}$.\nNow we see that\n$$\nx_{u}-x_{v}> (1-\\alpha)\\left( \\frac{d_{H}(u)x_{2} +x_{0}}{\\rho_{\\alpha}-\\alpha(s-1)}-\\frac{d_{H}(v)x_{1} +x_{0}}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)}\\right)  >0,\n$$\nsince $\\rho_{\\alpha}\\geq\\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, $n$ is sufficiently large, and $s, t$ are constants.\n\\end{proof}\n\t\nBy Claim \\ref{clm::3.6}, each $u_{i}\\in M_{2}$ has a unique neighbor $v_{i} \\in M_{1}$. Thus, each $u_{i}\\in M_{2}$ has a unique non-neighbor $u_{j} \\in V(H) \\backslash M_{1}$, where $u_{j} \\in M_{2}$.\nIf $v_{i}=v_{j}$ for some $(u_{i}, u_{j}) \\in L$, then $u_{i}$ and $u_{j}$ have $t-1$ common neighbors in $H$, which implies that $H$ contains a copy of $K_{2, t-1}$, a contradiction.\nHence, $v_{i} \\neq v_{j}$ for each $\\left(u_{i}, u_{j}\\right) \\in L$. If $v_{i} v_{j} \\in E(H)$ for some $\\left(u_{i}, u_{j}\\right) \\in L$,\nthen we can also obtain a copy of $K_{2, t-1}$ by contracting the edge $v_{i} v_{j}$, a contradiction. So, $v_{i} v_{j} \\notin E(H)$ for each $\\left(u_{i}, u_{j}\\right) \\in L$. Now, Let\n$$\nH^{\\prime}=H-\\left\\{u_{i} v_{i}, u_{j} v_{j} \\mid\\left(u_{i}, u_{j}\\right) \\in L\\right\\}+\\left\\{u_{i} u_{j}, v_{i} v_{j} \\mid\\left(u_{i}, u_{j}\\right) \\in L\\right\\},\n$$\nand $G^{\\prime}=G^{*}-E(H)+E\\left(H^{\\prime}\\right)$.\nClearly, $H^{\\prime} \\cong K_{t} \\cup K_{3}$, and so $H^{\\prime}$ has the $(s,t)$-property. Furthermore, by Claim \\ref{clm::3.7} we have\n$$\n\\begin{aligned}\n\\rho_{\\alpha}(G^{\\prime})-\\rho_{\\alpha} \\geq& \\sum_{\\left(u_{i}, u_{j}\\right) \\in L} ((\\alpha x^{2}_{u_{i}} +2(1-\\alpha)x_{u_{i}}x_{u_{j}} +\\alpha x^{2}_{u_{j}})+(\\alpha x^{2}_{v_{i}}+2(1-\\alpha)x_{v_{i}}x_{v_{j}} +\\alpha x^{2}_{v_{j}})\\\\\n&-(\\alpha x^{2}_{u_{i}}+2(1-\\alpha)x_{u_{i}}x_{v_{i}} +\\alpha x^{2}_{v_{i}})-(\\alpha x^{2}_{u_{j}} +2(1-\\alpha)x_{u_{j}}x_{v_{j}}+ \\alpha x^{2}_{v_{j}})) \\\\\n=&\\sum_{\\left(u_{i}, u_{j}\\right) \\in L} 2(1-\\alpha)(x_{u_{i}}-x_{v_{j}})(x_{u_{j}}-x_{v_{i}})>0,\n\\end{aligned}\n$$\na contradiction.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem::3.8}\nIf $H$ is a component in $H_{t+2}$. Then $\\beta \\leq 2$, moreover, $H \\cong S^{1}\\left(\\overline{H_{s, t}}\\right)$ for $\\beta=2$ and $H \\cong \\overline{H^{\\star}}$ for $\\beta=1$.\n\\end{lem}\n\n\\begin{proof}\nThe proof is divided into several claims.\n\\begin{clm}\\label{clm::3.1}\n$\ne(H)=\\binom{t}{2}+2.\n$\n\\end{clm}\n\\begin{proof}\nSince $|H|=t+2$, by Lemma \\ref{lem::2.4} we have $e(H) \\leq\\binom{t }{2}+2$. Clearly, $K_{t} \\cup K_{2}$ has the $(s, t)$-property, then by Lemma \\ref{lem::3.2} we obtain $e(H) \\geq e\\left(K_{t} \\cup K_{2}\\right)=\\binom{t }{2}+1$. Suppose that  $e(H)=\\binom{t }{2}+1$. Notice that $\\Delta(H) \\leq t-1$ and $\\delta(H) \\geq 1$. Then $\\pi(H) \\prec \\pi\\left(K_{t} \\cup K_{2}\\right)$. By Lemma \\ref{lem::3.4}, we obtain\n$\\pi(H)=\\pi\\left(K_{t} \\cup K_{2}\\right)=(t-1, t-1, \\ldots, t-1,1,1)$.\nThis implies that $H$ is obtained from $K_{t}$ by deleting an edge $u_{1} u_{2}$ and adding two pendant edges $u_{1} v_{1}$ and $u_{2} v_{2}$. Since $t \\geq 4$, we have  $N_{H}\\left(u_{1}\\right) \\backslash\\left\\{v_{1}\\right\\} \\neq \\varnothing$ and\n$$\n\\left( \\rho_{\\alpha}-\\alpha(s+1)+1\\right) \\left(x_{u_{1}}-x_{v_{1}}\\right)=\\alpha(t-2)x_{u_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}\\left(u_{1}\\right) \\backslash\\left\\{v_{1}\\right\\}} x_{v}>0,\n$$\nthus $x_{u_{1}}>x_{v_{1}}$. By symmetry, we have $x_{u_{1}}=x_{u_{2}}$ and  $x_{v_{1}}=x_{v_{2}}$. Now, let $H^{\\prime}= H-\\left\\{u_{1} v_{1}, u_{2} v_{2}\\right\\}+\\left\\{u_{1} u_{2}, v_{1} v_{2}\\right\\}$ and $G^{\\prime}=G^{*}-E(H)+E\\left(H^{\\prime}\\right)$. Clearly, $H^{\\prime} \\cong K_{t} \\cup K_{2}$, thus $H^{\\prime}$ has the $(s, t)$-property. However, we see that\n$$\n\\rho_{\\alpha}\\left(G^{\\prime}\\right)-\\rho_{\\alpha} \\geq X^{T}\\left(A_{\\alpha}\\left(G^{\\prime}\\right)-A_{\\alpha}\\left(G^{*}\\right)\\right) X=2(1-\\alpha)\\left(x_{u_{1}}-x_{v_{2}}\\right)\\left(x_{u_{2}}-x_{v_{1}}\\right)>0,\n$$\na contradiction. Therefore, $e(H)=\\binom{t}{2}+2$.\n\\end{proof}\n\n\\begin{clm}\\label{clm::3.2}\n$\\beta \\leq 2$.\n\\end{clm}\n\\begin{proof}\nSuppose to the contrary that $\\beta \\geq 3$.\nThen we have $e\\left(\\overline{H_{s, t}}\\right)= \\binom{t}{2}+\\beta-1 \\geq  \\binom{t}{2}+2$ by Lemma \\ref{lem::3.5}. From Lemma \\ref{lem::3.7}, there exists a component which is isomorphic to $K_{t}$ in $G^{*}-K$.\nLet $H^{\\prime} \\cong 2\\overline{H_{s, t}}$ and $V\\left(H^{\\prime}\\right)=V\\left(H \\cup K_{t}\\right)$. By Lemma \\ref{lem::2.16}(i), $H^{\\prime}$ has the $(s, t)$-property. Then by Claim \\ref{clm::3.1}, we have $e(H^{\\prime})\\geq  2\\left( \\binom{t}{2}+2\\right) >2 \\binom{t}{2}+2=e({H} \\cup K_{t})$, which contradicts\nLemma \\ref{lem::3.2}.\n\\end{proof}\n\n\\begin{clm}\\label{clm::3.3}\nIf $\\beta=2$, then $H\\cong S^{1}\\left(\\overline{H_{s, t}}\\right)$.\n\\end{clm}\n\\begin{proof}\nClearly, $|S^{1}\\left(\\overline{H_{s, t}}\\right)|=t+2$. Moreover, Lemma \\ref{lem::2.16}(i) gives that $S^{1}\\left(\\overline{H_{s, t}}\\right)$ has the $(s, t)$-property.\nSince $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor=2$, we have $2s+1 \\leq t \\leq 3s+1$.\nThus $\\gamma=\\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}=s$.\nSince $|H|=t+2$, then together with  Claims \\ref{clm::3.1}, \\ref{clm::3.2} and Lemma \\ref{lem::2.15}, we find that $H$ is isomorphic to either $\\overline{H^{\\star}}$ or some $\\overline{H_{a, b, c}}$, where $a+b+c=t-1$.\nIf $H \\cong \\overline{H^{\\star}}$. Then $|H|=t+2=10$ and $H$ is $6$-regular. Thus, we obtain $t=8$ and $e(H)=30$. Moreover, since $S^{1}\\left(\\overline{H_{s, 8}}\\right)$ is a subdivision of $\\overline{H_{s, 8}}$, by Lemma \\ref{lem::3.5} we have $e\\left(S^{1}\\left(\\overline{H_{s, 8}}\\right)\\right)=e\\left(\\overline{H_{s, 8}}\\right)+1\n=\\binom{t}{2}+\\beta=30$. From Lemma \\ref{lem::2.16}(i), we get that\n$$\n\\pi\\left(S^{1}\\left(\\overline{H_{s, t}}\\right)\\right)=(t-1, \\ldots, t-1, t-s, s+1,2).\n$$\nSince $t=8$ and $s\\geq 2$, we have $t-s \\leq 6$. Hence, $\\pi(H) \\prec \\pi\\left(S^{1}\\left(\\overline{H_{s, 8}}\\right)\\right)$, which contradicts Lemma \\ref{lem::3.4}. Therefore, $H$ is isomorphic to some $\\overline{H_{a, b, c}}$. Now, we assert that\n\\begin{equation}\\label{equ::29}\n\\min \\{b, c\\} \\geq \\gamma.\n\\end{equation}\nIf $b \\leq \\gamma-1$, we contract the edge $u_{2} w$ in $\\overline{H_{a, b, c}}$ and call the new vertex $u$ in the resulting graph, then we get a complete bipartite subgraph with bipartite partition\n$$\\left(V(K_{b}) \\cup\\{u\\}, V(K_{a})\\cup V(K_{c}) \\cup\\{u_{1}\\}\\right).$$\nThis implies that $H$ contains a $K_{b+1, a+c+1}$-minor, contradicting the $(s, t)$-property. So, $b \\geq \\gamma$. By symmetry, we have $c \\geq \\gamma$.\n\nFurthermore, by Fig. \\ref{fig-1} we can see that\n$$\n\\pi\\left(\\overline{H_{a, b, c}}\\right)=\\left(t-1, \\ldots, t-1, a_{1}, a_{2}, a_{3}\\right),\n$$\nwhere $a_{1}, a_{2}, a_{3} \\in\\{a+2, b+1, c+1\\}$. By (\\ref{equ::29})  and $\\gamma=s$, we have $\\min \\{b, c\\} \\geq \\gamma=s$. Thus, we find that $a_{3} \\geq 2$ and $a_{2} \\geq s+1$. Hence, we see that $\\pi\\left(\\overline{H_{a, b, c}}\\right) \\prec \\pi\\left(S^{1}\\left(\\overline{H_{s, t}}\\right)\\right)$. Then we obtain $\\pi\\left(\\overline{H_{a, b, c}}\\right)=\\pi\\left(S^{1}\\left(\\overline{H_{s, t}}\\right)\\right)$ by Lemma  \\ref{lem::3.4},. This implies that $a_{3}=2$ and $a_{2}=s+1$. Note that $\\min \\{b, c\\} \\geq s \\geq 2$. It follows that $a=0$ and $\\min \\{b, c\\}=s$, that is, $\\overline{H_{a, b, c}} \\cong S^{1}\\left(\\overline{H_{s, t}}\\right)$.\n\\end{proof}\n\n\\begin{clm}\\label{clm::3.4}\nIf $\\beta=1$, then $H \\cong \\overline{H^{\\star}}$.\n\\end{clm}\n\n\\begin{proof}\nSince $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor=1$, we have $t \\leq 2s$. Thus, $\\gamma=\\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}=\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor$. Since $|H|=t+2$, then together with  Claims \\ref{clm::3.1}, \\ref{clm::3.2} and Lemma \\ref{lem::2.15}, $H$ is isomorphic to either $\\overline{H^{\\star}}$ or some $\\overline{H_{a, b, c}}$, where $a+b+c=t-1$. If $H$ is isomorphic to some $\\overline{H_{a, b, c}}$, then we get (\\ref{equ::29}) in a similar way as above. It follows that $t-1\\geq b+c \\geq 2 \\gamma=2\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor$, a contradiction.  Hence, $H$ is only possibly isomorphic to $\\overline{H^{\\star}}$. Since $|\\overline{H^{\\star}}|=t+2=10$, we have $t=8$. Then by Lemma \\ref{lem::2.16}(ii), $\\overline{H^{\\star}}$ has the $(s, t)$-property. Therefore, $H \\cong \\overline{H^{\\star}}$, as desired.\n\\end{proof}\nCombining Claims \\ref{clm::3.2}-\\ref{clm::3.4}, we obtain the result of Lemma \\ref{lem::3.8}.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.13}\nIf $H_{t+1} \\neq \\varnothing$. Then $H_{t+2} \\cup H_{<t}=\\varnothing$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that $H_{t+2} \\cup H_{<t} \\neq \\varnothing$. By Lemma \\ref{lem::3.10}, $G^{*}-K$ contains a unique component $D_{1} \\in H_{t+2} \\cup H_{<t}$. Since $H_{t+1} \\neq \\varnothing$, we take a component $D_{2} \\in H_{t+1}$. Then by Lemma \\ref{lem::3.5}, we have $\\beta \\geq 2$ and $D_{2} \\cong \\overline{H_{s, t}}$.\n\t\nIf $|D_{1}|=t+2$, then $\\beta=2$ and $D_{1} \\cong S^{1}\\left(\\overline{H_{s, t}}\\right)$ by Lemma \\ref{lem::3.8}. Note that $D_{1}$ is a subdivision of $\\overline{H_{s, t}}$. From Lemma \\ref{lem::3.5}, we obtain $e\\left(\\overline{H_{s, t}}\\right)=\\binom{t}{2}+\\beta-1$. Thus, we have\n$$\ne\\left(D_{1} \\cup D_{2}\\right)=2 e\\left(\\overline{H_{s, t}}\\right)+1=2\\left(\\binom{t}{2}+\\beta-1\\right)+1=2\\binom{t}{2}+3.\n$$\nNow let $H^{\\prime}=2 K_{t} \\cup K_{3}$. Then $|H^{\\prime}|=|D_{1} \\cup D_{2}|$ and $e\\left(H^{\\prime}\\right)=e\\left(D_{1} \\cup D_{2}\\right)$. By (\\ref{equ::30}), we have $\\delta\\left(D_{2}\\right)>2$. Since $D_{1}$ is a subdivision of $D_{2}$, $\\delta\\left(D_{1}\\right)=2$ and its vertex of degree two is unique. Which implies that $\\pi\\left(D_{1} \\cup D_{2}\\right) \\prec \\pi\\left(H^{\\prime}\\right)$ and  $\\pi\\left(D_{1} \\cup D_{2}\\right) \\neq \\pi\\left(H^{\\prime}\\right)$, this contradicts Lemma \\ref{lem::3.4}.\n\t\nIf $|D_{1}|<t$, then $D_{1} \\cong K_{|D_{1}|}$. Indeed, we have\n\\begin{equation}\\label{equ::31}\n|D_{1}| \\leq \\beta-1.\n\\end{equation}\nOtherwise, we have $|D_{1}|>\\beta-1$, then\n$$\ne\\left(D_{1} \\cup D_{2}\\right)=\\binom{\\left|D_{1}\\right| }{2}+\\binom{t}{2}+\\beta-1<\\binom{t }{2}+\\binom{\\left|D_{1}\\right| }{2}+\\left|D_{1}\\right|=e\\left(K_{t} \\cup K_{\\left|D_{1}\\right|+1}\\right),\n$$\nwhich contradicts Lemma \\ref{lem::3.2}. On the other hand,\nwe get that $G^{*}-K$ contains the disjoint union of a copy of $D_{1}$ and $|D_{1}|$ copies of $K_{t}$ by Lemma \\ref{lem::3.7}. We denote it by $H^{\\prime\\prime}$. Then $e(H^{\\prime\\prime})=  \\binom{\\left|D_{1}\\right| }{2} +\\left|D_{1}\\right|\\binom{t}{2}$. Now let $H^{\\prime\\prime\\prime}$ be the disjoint union of $|D_{1}|$ copies of $D_{2}$. Clearly, $|H^{\\prime\\prime}|=|H^{\\prime\\prime\\prime}|=|D_{1}|(t+1)$ and $e\\left(H^{\\prime\\prime\\prime}\\right)= |D_{1}|\\left(\\binom{t}{2}+\\beta-1\\right)$. By Lemma \\ref{lem::3.2}, $e\\left(H^{\\prime\\prime\\prime}\\right) \\leq e(H^{\\prime\\prime})$.\nTherefore, we get that\n\\begin{equation}\\label{equ::32}\n\\begin{array}{ll}\n|D_{1}|\\geq 2\\beta-1,\n\\end{array}\n\\end{equation}\nwhich contradicts (\\ref{equ::31}).\n\\end{proof}\n\n\n\\begin{lem}\\label{lem::3.11}\nLet $t\\geq4$, $\\beta=1$, $n-s+1=pt+r$ and $1\\leq r\\leq t$. Then\n$$\nG^{*}-K \\cong\\left\\{\\begin{array}{ll}\n(p-1) K_{t} \\cup \\overline{H^{\\star}} & \\text { for } r=2 \\text { and } t=8; \\\\\np K_{t} \\cup K_{r} & \\text { otherwise. }\n\\end{array}\\right.\n$$\n\\end{lem}\n\n\\begin{proof}\nBy Lemmas \\ref{lem::3.6} and \\ref{lem::3.9}, $H_{>t+2}=\\varnothing$. Note that $\\beta=1$. Then by Lemma \\ref{lem::3.5}, we also have $H_{t+1}=\\varnothing$. Since any component of $G^{*}- K$ of order $t$ is isomorphic to $K_{t}$, then by Lemma \\ref{lem::3.10}, there exists at most one component $H$ of $G^{*}-K$ are not isomorphic to $K_{t}$. Notice that  $|G^{*}-K|=pt+r$, where $1 \\leq r \\leq t$, we see that either $|H|=r$ or $|H|=t+r=t+2$. If $r \\neq 2$, then $H \\cong K_{r}$ and $G^{*}-K \\cong p K_{t} \\cup K_{r}$, as desired. Now assume that $r=2$. Then either $H \\cong K_{2}$ or $H \\cong \\overline{H^{\\star}}$ by Lemma \\ref{lem::3.8}. If $H \\cong \\overline{H^{\\star}}$, then $|H|=t+2=10$, and so $t=8$. Hence, if $t \\neq 8$, then $G^{*}-K \\cong p K_{t} \\cup K_{2}$. Now we consider the case $t=8$. Suppose that $H \\cong K_{2}$, then $e\\left(K_{8} \\cup K_{2}\\right)=29<30=e\\left(\\overline{H^{\\star}}\\right)$, contradicting Lemma \\ref{lem::3.2}. That is, $H\\cong \\overline{H^{\\star}}$.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem::3.14}\nLet $t\\geq4$, $\\beta \\geq 2$, $n-s+1=pt+r$ and $1\\leq r\\leq t$. Then\n$$\nG^{*}-K \\cong\\left\\{\\begin{array}{ll}\n(p-1) K_{t} \\cup S^{1}\\left(\\overline{H_{s, t}}\\right) & \\text { for } r=\\beta=2; \\\\\n(p-r) K_{t} \\cup r \\overline{H_{s, t}} & \\text { for } r \\leq 2(\\beta-1) \\text { except } r=\\beta=2; \\\\\np K_{t} \\cup K_{r} & \\text { for } r>2(\\beta-1).\n\\end{array}\\right.\n$$\n\\end{lem}\n\n\\begin{proof}\nNote that $|G^{*}-K|=pt+r$, where $1 \\leq r \\leq t$, and $H \\cong \\overline{H_{s, t}}$ for every $H \\in H_{t+1}$.\nNow, we assert that if $r \\leq 2(\\beta-1)$ then $H_{<t}=\\varnothing$. Indeed, if $G^{*}-K$ contains a component $D$ with $|D|<t$, then $H_{>t}=\\varnothing$ and there exists exactly one component in $H_{<t}$ by Lemmas \\ref{lem::3.10} and \\ref{lem::3.13}. This implies that $|D|=r$.\nFrom (\\ref{equ::32}),  $r=|D| \\geq 2\\beta-1$, a contradiction.\n\nWe firstly assume that $r\\neq 2$. Then by Lemmas \\ref{lem::3.10} and \\ref{lem::3.13}, $G^{*}-K$ is isomorphic to either $p K_{t} \\cup K_{r}$ or  $(p-r) K_{t} \\cup r \\overline{H_{s, t}}$. If $r \\leq 2(\\beta-1)$, then $r<t$, and  $H_{<t}=\\varnothing$. Thus $G^{*}-K \\cong(p-r) K_{t} \\cup r \\overline{H_{s, t}}$. If $r>2(\\beta-1)$, then by Lemma  \\ref{lem::3.12}, $G^{*}-K \\cong pK_{t} \\cup K_{r}$.\n\nNext, we assume that $r=2$. Since $\\beta \\geq 2$, we have $r\\leq 2(\\beta-1)$ and thus  $H_{<t}=\\varnothing$. If $\\beta>2$, then $H_{t+2}=\\varnothing$ by Lemma \\ref{lem::3.8}. Thus, $G^{*}-K \\cong(p-r) K_{t} \\cup r\\overline{H_{s, t}}$. It remains to prove the case $r=\\beta=2$. Now if $H_{t+1} \\neq \\varnothing$, then $H_{t+2} \\cup H_{<t}=\\varnothing$ by Lemma \\ref{lem::3.13}. Moreover, by Lemma \\ref{lem::3.12}, the number of the components in $H_{t+1}$ is at most $2(\\beta-1)=2$. This implies that  $G^{*}-K \\cong(p-2) K_{t} \\cup 2\\overline{H_{s, t}}$. Notice that $e\\left(\\overline{H_{s, t}}\\right)=\\binom{t}{2}+\\beta-1=\\binom{t}{2}+1$. Now let $H^{\\prime} \\cong K_{t} \\cup S^{1}\\left(\\overline{H_{s, t}}\\right)$. Then  $|H^{\\prime}|=2|\\overline{H_{s, t}}|=2t+2$ and\n$\ne\\left(H^{\\prime}\\right)=e\\left(K_{t}\\right)+e\\left(\\overline{H_{s, t}}\\right)+1=2 e\\left(\\overline{H_{s, t}}\\right).\n$\nSince $S^{1}\\left(\\overline{H_{s, t}}\\right)$ is a subdivision of $\\overline{H_{s, t}}$, and by (\\ref{equ::30}) we have $\\delta\\left(\\overline{H_{s, t}}\\right)>2$, we can easily see that $\\pi\\left(2\\overline{H_{s, t}}\\right) \\prec \\pi\\left(H^{\\prime}\\right)$, this contradicts Lemma \\ref{lem::3.4}. Thus, $H_{t+1}=\\varnothing$. It follows that there exists exactly one component in $H_{t+2}$. By Lemma \\ref{lem::3.8}, we have $G^{*}-K \\cong(p-1) K_{t} \\cup S^{1}\\left(\\overline{H_{s, t}}\\right)$.\n\\end{proof}\n\nCombining Lemmas \\ref{lem::3.1'}-\\ref{lem::3.1'''}, \\ref{lem::3.11} and \\ref{lem::3.14}, we obtain the result of Theorem \\ref{thm::1.1}.\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\n\n\n\n\n\n\n\nA cold, bright and compact atomic beam source is\nan important asset for any experiment featuring ultra-cold atoms, such as atom interferometers~\\cite{Cronin09}, degenerate quantum gases for quantum simulation~\\cite{Georgescu14} and, in particular, optical atomic clocks~\\cite{Ludlow2015}. In the last case, two valence electron alkaline-earth (like) metals, like Ca~\\cite{Wilpers07}, Sr~\\cite{Ushijima2015}, Mg~\\cite{Kulosa15}, and Yb\\cite{McGrew2018}, are generally used as atomic frequency discriminators, and present low vapor pressures at room temperature thus needing high temperature ovens to generate enough atomic vapour, typically followed by a space-demanding Zeeman slower (ZS). Although compact and transportable versions of the ``oven + ZS'' atomic beam system have been developed~\\cite{Poli2014,AOsense}, some concerns about the systematic effects due to collisions with the atomic beam particles~\\cite{Gibble13} and the hot black-body radiation from the oven region~\\cite{Beloy14} can arise below the $10^{-18}$ relative uncertainty level.\n\nThe two-dimensional magneto optical trap (2D-MOT) atomic source~\\cite{Dieckmann1998, Schoser2002} can be transversely loaded, hence reducing the setup dimensions, avoiding direct exposure of the atomic reference to hot metals, and at the same time obtaining an optical shutter of the atomic beam just by turning-off its cooling beams. This avoids the use of in-vacuum mechanical shutters or optical beam deflectors~\\cite{Witte92} as done for ZS or collimated oven beams. The 2D-MOT system complexity can be further reduced by its permanent magnets implementation~\\cite{Tiecke2009, Lamporesi2013}.\n\n\nIn this work, we present a novel atomic source employing a 2D-MOT source of strontium (Sr) atoms for metrological application. The mechanical implementation of the atomic source is similar to other setups built to generate lithium~\\cite{Tiecke2009}, sodium \\cite{Lamporesi2013,Colzi2018} and strontium~\\cite{Nosske2017} atomic beams. Our system is further characterized by a collimated atomic beam transmitted by a bundle of capillaries directly towards the 2D-MOT region, and a two-frequency optical molasses to enhance the atomic flux toward the trapping region. The design, engineering and characterization of the sideband-enhanced 2D-MOT strontium source is the main result of this work. This is accomplished by looking at the loading performances of a three-dimensional MOT typically used as the first cooling and trapping stage for an optical lattice clock~\\cite{Xu2003}. Monte Carlo (MC) numerical simulations are used to find the optimal optical configuration which are then compared to the experimental results. \n\nThe article is organized as follows: \nSec.~\\ref{sec:Theory} introduces the physical interpretation and significance of adding a sideband frequency to the cooling beams of the 2D-MOT; Sec.~\\ref{sec:apparatus} depicts the experimental apparatus assembled for an optical lattice clock; in Sec.~\\ref{sec:numerical simulation} we describe the numerical modeling of the atomic source and the 2D-MOT cooling and trapping processes by Monte Carlo simulations; Sec.~\\ref{sec:atomic source characterization} shows the experimental characterization of our atomic source and in Sec.~\\ref{sec:sideband enhancement} we demonstrate how the sideband-enhancement method is able to magnify the number of trapped atoms by a magneto-optical trap.\n\n\n\n\n\n\\section{\\label{sec:Theory}Principles of sideband-enhanced 2D-MOT}\n\nA 2D-MOT atomic source relies on the radiation-pressure friction force to capture and cool thermal atoms effusing from either an oven, or a background gas. In this work, we focus our attention on the 2D-MOT loaded from a collimated atomic source, so that a 1D model offers a good insight on the expected 2D-MOT flux. For the 1D model, the MOT captured atoms per second  $\\Phi_\\text{2D}$ is provided by the formula \n\\begin{equation}\\label{eq:cap_2d}\n\\Phi_\\text{2D} \\simeq n v_\\text{th} A (v_\\text{c} \/ v_\\text{th})^4, \\qquad(v_c \\ll v_\\text{th})\n\\end{equation}\nwhere $n$ is the spatial density of the thermal beam, $v_\\text{th} =  \\sqrt{2 k_B T_\\text{ov} \/ m}$ is the most probable thermal velocity (for the atomic Sr vapour at $T_\\text{ov} = \\SI{460}{\\celsius}$, $v_\\text{th} = \\SI{379}{m \\per s}$ ), $A = 2\\pi w^2$ is the MOT capture surface related to the trapping beam width $w$, and $v_c$ is the capture velocity of the trap. It is clear from (\\ref{eq:cap_2d}) that the most influential parameter is the capture velocity $v_\\text{c}$, which is related to the magnetic gradient $b$, the frequency detuning $\\Delta$ from the cooling transition, and the total saturation parameter $s=I\/I_\\text{sat}$ of the MOT optical beams, where for an atomic transition at wavelength $\\lambda$ and spontaneous emission rate $\\Gamma$ the resonant saturation intensity is $I_\\text{sat} = \\pi hc\\Gamma\/3\\lambda^3$. In the 1D model one typically computes $v_\\text{c}$ numerically by solving the semiclassical equation of motion, as shown in Fig.\\ref{fig:capt_vel}(a). Here one can observe that there are two different dynamics inside the MOT region. In the outer region, the MOT behaves like a Zeeman slower, where the friction force exerted upon any atom will be effective only if the velocity $v$ at distance $r$ from the symmetry axis will be nearly resonant with the cooling laser, i.e., if the difference of the Zeeman shift and the laser detuning equals the Doppler shift. In the inner region the motion of the atoms can be described by an overdamped harmonic oscillator model. Hence the capture velocity is strictly related with the dynamics in the outer region of the MOT and, assuming perfect compensation of the Zeeman shift and Doppler shift, it can be roughly estimated as~\\cite{Tiecke2009,Zinner98}\n\n\\begin{equation}\\label{eq:vc}\nv_\\text{c} \\lesssim v_\\text{max} = \\sqrt{a_\\text{max}\\,r_\\text{max}}    \n\\end{equation}\nwhere $a_\\text{max} = \\hbar k \\Gamma\/(2m)$ is the maximum acceleration at infinite saturation parameter, and $r_\\text{max}=\\sqrt{2}w$ is the maximum interaction distance with the MOT beams, taking into account the projection of the 2D-MOT beams at \\SI{45}{\\degree} from the atoms propagation axis. This $v_\\text{c}$  corresponds to the maximum velocity allowed in order to decelerate an atom to zero at the center of the trap. This oversimplified estimation gives us some hints on the 2D-MOT expected performance. In particular, even for infinite available power, the capture velocity would be bounded, while the capture mechanism is fundamentally limited by the natural linewidth of the cooling transition and the cooling beam radius. However, if one uses laser light which has several red-detuned sidebands, even faster atoms can be slowed down and the capture velocity increased. MOT loading enhancement was observed in alkali atomic systems by means of electro-optic modulation (EOM) of the cooling beams \\cite{Anderson1994,Lee2017}. This technique is generally not feasible at the wavelengths of alkaline-earth atoms by EOMs. Furthermore, because of the higher $\\Gamma$s excessive spectral broadening would reduce the radiation pressure force, making it no longer sufficient to keep the thermal atoms in the trap. A one-sideband 3D MOT has been previously realized to trap Ca atoms loaded directly from an effusing atomic oven~\\cite{Zinner98,Riehle:1999p6441}. In this case, with a total MOT saturation parameter $s \\sim 0.1$ and an atomic vapour temperature of 600 $^\\circ$C, an enhancement factor of 7 was observed~\\cite{Riehle:1999p6441}. However here only a very small fraction of the available atoms were trapped, hence that system would be very unfavourable in the case of a 2D-MOT source loading. \n\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{sideband_principle.pdf}\n    \\caption{1D simulation of the atomic trajectories of strontium atoms (light blue line) for different capture processes in a 2D-MOT. The color map plotted on the background depicts the acceleration value at each point of the phase-space. \\textbf{(a)} Single-frequency 2D-MOT. The total saturation and the detuning of the MOT beams are $s=7$, $\\Delta\/\\Gamma = -1.6$. The estimated capture velocity is \\SI{72(1)}{m \\per s}. \n    \\textbf{(b)} Sideband-enhanced 2D-MOT. The saturation parameter is $s=3.5$ for the 2D-MOT beam at with $\\Delta\/\\Gamma =-1.6 $, and $s_\\text{SB}=3.5$ for the sideband beam  at $\\Delta_\\text{side}\/ \\Gamma = -3.2$. The estimated capture velocity is \\SI{90(1)}{m \\per s}. The magnetic field gradient and beam width used in this calculation are $b=\\SI{0.22}{T \\per m}$ and $w=\\SI{1}{cm}$.\n    \\textbf{(c)} Acceleration profile at $r=-w\/2$ of the sideband trapping (red line) and  standard 2D-MOT  trapping (blue line).}\n   \n    \\label{fig:capt_vel}\n\\end{figure}\n\nIt is more interesting to investigate the sideband-enhanced 2D-MOT in the limit of high total saturation parameter $s\\geq 1$, where most of the low velocity class ($v\\leq v_\\text{max}$) is slowed and captured by the cooling beams. Fig.\\ref{fig:capt_vel}(a) shows a simulation of the phase-space trajectories for typical values of the experimental parameters ($\\Delta,s,b$) used in a strontium 2D-MOT~\\cite{Nosske2017}. The acceleration patterns of the sideband-enhanced 2D-MOT in the atomic phase-space are depicted in the in Fig.\\ref{fig:capt_vel}(b). As shown in the plot, the sideband beams interact with atoms from a higher velocity class, decelerating them toward the capture region of the standard MOT beam. This increment of the capture velocity is best displayed in Fig.\\ref{fig:capt_vel}(c): here we can see the MOT acceleration as function of the atomic approaching velocity. In the standard MOT (blue dashed) the force is peaked around a given velocity value, reaching $a_\\text{max}$ and the amount of power increases the spectral width of the force as $\\sqrt{s}$. On the other hand, the sideband-enhanced force (red dot-dashed) presents a second peak at higher velocity without degrading the peak acceleration. Optimal positioning of the sideband frequency thus allows an increase of the expected capture velocity $v_c$ and of the expected MOT loading rate too. \n\nAnother expected beneficial effect of the sideband-enhanced 2D-MOT with large $s$ is the reduction of the transverse temperature of the cold atomic sample compared to the standard 2D-MOT, which would yield a higher brightness (i.e. lower beam divergence). This can be explained considering that the optical power redistributed at a higher frequency weakly interacts with the atoms trapped once they reach the center of the MOT.\n\nIn order to correctly address the expected performances of a sideband-enhanced strontium 2D-MOT we performed a dedicated Monte Carlo (MC) simulation which takes into account the actual geometry of the system, the magnetic field gradient, the residual divergence of our atomic beam from the oven, and the expected loading rate for the final 3D-MOT. This is described in detail in Sec.\\ref{sec:numerical simulation}.\n\n\n\n\\section{\\label{sec:apparatus}Experimental apparatus}\n\n\\subsection{Vacuum system}\nThe schematic drawing of the vacuum system to produce and trap ultra-cold strontium atoms is depicted in Fig.~\\ref{fig:Figure1}. It has been previously described in~\\cite{Tarallo2017}, and its concept is adapted from previous works~\\cite{Tiecke2009, Lamporesi2013}. The vacuum system is conceived to host two physical regions with very different vacuum levels, the atomic source region and the science cell region and, at the same time, to be very compact. The atomic source region consists of a stainless-steel vacuum chamber with a multi-way cross  at its end, where the intersection plane of the tubes forms the 2D-MOT plane. The ultra-high vacuum region hosts a small octagonal science cell with two large vertical optical accesses (DN63CF) and seven small lateral optical windows (DN16CF) for cooling, trapping and operate a Sr optical clock. The two vacuum regions are connected by a differential pumping channel (DPC) carved in a custom bellow with \\SI{2}{mm} diameter and \\SI{22.8}{mm} length and all-metal gate valve. The DPC sets the maximum divergence of the cold atomic beam at \\SI{87}{mrad}, while a conductance of \\SI{4.3e-2}{L \\per s} allows to maintain a differential pressure of \\num{e4} between the two regions. Vacuum is maintained by two ion-getter pumps, both regions reaching a pressure below \\SI{e-10}{mbar} when the oven is not heated.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{fig_vacuum_system2.pdf}\n    \\caption{Schematic drawing of the vacuum system. It hosts a high vacuum (HV) region for the atomic beam production, and an ultra-high vacuum (UHV) region for cooling and trapping the atomic sample. A DPC connects the two regions. The size of the entire vacuum apparatus is roughly \\SI{70}{cm} $\\times$ \\SI{70}{cm} $\\times$ \\SI{45}{cm}.}\n   \n    \\label{fig:Figure1}\n\\end{figure}\n\n\n\n\n\\subsection{Collimated atomic source}\n\nThe oven consists of a simple stainless-steel cylinder with an aperture of \\SI{16}{mm} and a conflat flange DN16CF to be attached to the main body of the vacuum system on one of its circular sides.\nThe oven is attached to the multi-way cross vacuum chamber \\SI{128}{mm} away from its center.\nIn order to produce a collimated atomic beam, an array of $N_\\text{cap} \\simeq \\num{150}$ capillaries made of nickel-based alloy Monel400, with an internal radius $r_\\text{cap}=\\SI{0.2}{mm}$ and a length  $L_\\text{cap}=\\SI{20}{mm}$, is inserted at the oven aperture. The capillaries are tightened inside a holder which lays in the aperture of the oven. The heating is insured by a pair of heating cartridges. The Sr vapor is typically generated at the temperature $T_\\text{ov} =\\SI{460}{\\celsius}$. In order to avoid clogging of the capillaries with strontium, the oven hosts an extra pair of heating cartridges close to its aperture to maintain the capillaries at a temperature $T_\\text{cap}$ higher than $T_\\text{ov}$. For all experimental characterizations we maintained a differential temperature $T_\\text{cap} - T_\\text{ov}=\\SI{30}{\\celsius}$. At the typical operational oven temperature $T_\\text{ov}=\\SI{460}{\\celsius}$, the estimated vapour pressure inside is $p_\\text{ov}=\\SI{0.133}{Pa}$~\\cite{Alcock1984} from which we estimate the Sr atomic density by means of the ideal gas law $n_\\text{ov}= p_\\text{ov} \/ k_B T_\\text{ov} =\\SI{1.31e19}{atoms \\per m^3}$.\nIn the regime of negligible collisions inside the capillaries (mean free path $\\lambda_\\text{ov} = (\\sqrt{2} n_\\text{ov} \\sigma_\\text{Sr} )^{-1}\\sim \\SI{70}{mm} \\gg L_\\text{cap}$, with $\\sigma_\\text{Sr}$ = 8$\\cdot10^{-19}$ m$^2$ the elastic cross section), the atomic flux is proportional to the oven pressure $P_\\text{ov}$  and it is estimated as~\\cite{Wang1960}:\n\\begin{equation}\n    \\Phi_\\text{ov} =a \\frac{4 \\sqrt{\\pi}}{3} \\frac{n_\\text{ov} v_\\text{th} r^3_\\text{cap}}{L_\\text{cap}}N_\\text{cap}\n    \\label{eq:flux_oven}\n\\end{equation}\nwhere $a$ is the isotopic abundance.\nIn the case of $^{88}$Sr, the expected atomic flux  at $T_\\text{ov}= \\SI{460}{\\celsius}$ is $\\Phi_\\text{ov}=\\SI{5.8e14}{atoms \\per s}$. The geometrical constraint imposed by the capillaries yield a theoretical divergence angle $\\theta_\\text{cap} \\simeq r_\\text{cap} \/ L_\\text{cap}=20\\,$mrad.\n\n\n\n\\subsection{2D-MOT and cold atomic source generation}\n\nAs sketched in Fig.~\\ref{fig:Figure1}, the 2D MOT is composed of a 2D quadrupole magnetic field in combination with two orthogonal pairs of retroreflected laser beams of opposite circular polarization.\n\nThe magnetic field gradient is generated by four stacks of permanent magnets~\\cite{Tiecke2009}. Each stack is composed of \\num{9} neodymium bar magnets with size of \\SI{25}{mm} $\\times$ \\SI{10}{mm} $\\times$ \\SI{3}{mm} and magnetization \\SI{6.6(1)e5}{A \\per m}. The stacks are placed around the center of the 2D-MOT at the positions $\\mathbf{r}_\\text{m} = \\pm \\mathbf{x}_0  \\pm \\mathbf{y}_0$ where $x_0 = \\SI{110}{mm}$ and $y_0 = \\SI{90}{mm}$. \nThe magnetization of each permanent magnet has been oriented in such a way that it has the same direction with the one along the $y$-axis and opposite direction with the one faced along the $x$ axis. We estimated the generated field upon the 2D-MOT plane by finite element analysis (FEA). This shows a uniform linear gradient $\\mathbf{B}_\\text{m} (\\mathbf{r}) = b \\mathbf{x }  - b \\mathbf{z}$ close to  the center of the trap $|\\mathbf{r}|<\\SI{1}{cm}$ with $b=\\SI{0.224}{T \\per m}$. As compared to the above expression the maximum deviation of the actual magnetic field is negligible within the 2D-MOT trapping volume, as it ultimately amounts to $\\Delta_\\text{Z} = 2 \\pi\\times \\SI{5.6}{MHz} = \\SI{0.17}{\\Gamma}$ in frequency detuning.\n\n\nThe two pairs of counterpropagating beams allow magneto-optical cooling and trapping of slow atoms effusing from the oven along the $x$ and $z$ axes, while they are free to drift along the $y$ direction. Hence a nearly-resonant laser ``push'' beam is directed to the 2D-MOT center along the $y$-axis toward the UHV science cell in order to launch atoms collected in the 2D-MOT towards the MOT region. The center of the MOT in the science cell is located \\SI{370}{mm} far from the 2D-MOT center. Finally, the mandatory MOT quadrupolar field is generated by a pair of coils with the current flowing in the Anti-Helmholtz configuration, which generates a typical magnetic field gradient of \\SI{0.4}{T \\per m}. \n\n\n\n\n\\subsection{Laser system}\n\nA schematic drawing of the laser system is shown in Fig.~\\ref{fig:blue_optical_setup}.\nThe \\SI{461}{nm} laser is provided by a semiconductor-based commercial laser composed of an infrared master laser, a tapered amplifier and a second harmonic generation cavity. It is able to generate up to \\SI{600}{mW} of blue power. This blue laser is split in six main optical paths and frequency manipulated by acousto-optic modulators (AOMs). The laser frequency is stabilized to the Sr atomic transition $^1$S$_0$\\,--\\,$^1$P$_1$~by performing wavelength modulation saturation spectroscopy on an hot vapour of strontium generated in a heatpipe~\\cite{Poli2006}.\nTypically we are able to deliver about half of the available power to the atoms.\n\nA detailed scheme of the various beam paths is depicted in Fig.\\ref{fig:blue_optical_setup}. For typical experimental conditions, the 2D-MOT and sideband beams share \\SI{200}{mW} and have a $1\/e^2$ beam width $w_\\text{2D} =$~\\SI{9.5}{mm}, the MOT beams have a total power of \\SI{45}{mW} with a beam width of \\SI{6.2}{mm} and a detuning from the atomic resonance of -1.2 $\\Gamma$, the push beam has a power up to \\SI{5}{mW} and a beam width of  \\SI{0.81}{mm}, the spectroscopy beam sent inside the heatpipe has a power of \\SI{0.5}{mW} and beam width \\SI{0.37}{mm}, and finally the probe beam has \\SI{0.5}{mW} of power and \\SI{0.83}{mm} width. The detuning from the atomic resonance of the beams used in atomic source system (2D-MOT, sideband and push beams) have been scanned for optimal atomic flux, as described in Sec.\\ref{sec:atomic source characterization}.\n\n\n\n\n\n\n \\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{Figure2_art.pdf}\n    \\caption{Optical setup of the blue laser system used for cooling, trapping and probe ultracold strontium atoms. Each acousto-optical modulator (AOM) is drawn with its driving frequency (MHz) and the sign corresponds to the diffraction order. Beam shaping lenses are not shown.}\n    \\label{fig:blue_optical_setup}\n\\end{figure}\n\n \nWe generate the 2D-MOT main and sideband beams as follows: two dedicated \\SI{200}{MHz} and \\SI{350}{MHz} AOMs are employed to shift the frequencies of two beams, which are shaped with the same telescope in order to have the same beam width. They are combined in a polarizing beam splitter (PBS) cube with orthogonal linear polarizations in such a way that the 2D-MOT (sideband) beam is completely transmitted (reflected). The two beam polarization is then rotated 45$^{\\circ}$ by a half-wavelength retarding waveplate, thus the two beams are recombined into a second PBS which yields the two beams for the branches of the 2D optical molasses. The \\SI{350}{MHz} is dedicated to the sideband beam, offering a \\SI{100}{MHz} bandwidth to find the optimal frequency which maximizes the loading of the atomic source.\n\n\n\n\\section{Numerical simulation of the 2D-MOT\\label{sec:numerical simulation}}\n\n\nMonte Carlo (MC) simulation is a powerful and versatile numerical approach because it allows to study complex physical processes in a realistic environment: from the simple MOT capture process \\cite{Wohlleben2001,Kohel2003,Chaudhuri2006,Szulc2016}, to the loading process of an optical potential \\cite{Hanley2017,Mu2010}, a molecular MOT \\cite{Comparat2014} and Rydberg-dressed MOT \\cite{Bounds2018}. Knowing the atom-light interactions and the geometry of the system, we want to extract the capture efficiency of our 2D-MOT system at a given trapping configuration, defined by the 2D-MOT beams, sideband beams and push beam, as described in the previous section. The MC algorithm is implemented in Python language.\n\n We simulate $N_\\text{sim}=2 \\times 10^4$ trajectories of atoms that interact with the trap. At initial time $t=0$ the starting positions of atoms are randomly sampled in a disk region of radius $r_0 =\\SI{7.5}{mm}$ in the $(y,z)$ plane and at $x_0=\\SI{-128}{mm}$ far from 2D-MOT trap center along the direction of hot atomic flux emitted by the oven. The velocity space is sampled from the Maxwell-Boltzmann probability distribution expressed in polar coordinates. The sampling of the absolute value of the starting atomic velocity $v_0$ is limited to $v_\\text{cut} =\\SI{90}{m \\per s}$ to speed up the calculation. The polar angle $\\theta_0$ is uniformly sampled considering the geometrical constraint imposed by the capillaries $\\theta_0 \\leq \\theta_\\text{cap}$. The azimuthal angle $\\phi_0$ is randomly chosen between 0 and $2 \\pi$.\n\nThe trajectory is discretized in time with a step size $\\delta t = \\SI{50}{\\micro s}$, and computed until $t_\\text{tot}=\\SI{4}{ms}$ by using a Runge-Kutta algoritm~\\cite{Enright1989}. The time step $\\delta t$ is chosen to be greater than the internal atomic time scale $\\tau_{^1P_1} = \\Gamma^{-1}$, so that the atom-light interaction can be calculated by using the semi-classical approximation of the  Optical Bloch Equations, but shorter than the capture time for an atom moving at $v_\\text{max}$ which is about $\\Delta t_\\text{max}$ = 165 $\\mu$s. At each time step $t_i=i \\delta t$, the atom-light scattering rate with a single laser beam is computed as:\n\\begin{equation}\n    R(t_i) = \\frac{\\Gamma}{2} \\frac{s(\\mathbf{r}(t_i))}{1 + s(\\mathbf{r}(t_i)) + 4\\left(\\frac{ \\Delta_\\text{eff}(\\mathbf{r}(t_i),\\mathbf{v}(t_i))}{ \\Gamma}\\right)^2}\n    \\label{eq:scattergin_rate}\n\\end{equation}\nwhere $s(\\mathbf{r}(t_i))$ is the position-dependent saturation parameter, and $\\Delta(\\mathbf{r}(t_i),\\mathbf{v}(t_i))$ is the frequency detuning due to the Doppler and Zeeman  shift. \nThe local saturation parameter is computed as:\n\\begin{equation}\n\\qquad s(\\mathbf{r}(t_i)) = s_0 \\exp \\left( -\\frac{ 2 |\\mathbf{r}(t_i) \\times \\mathbf{\\hat{k}}|^2}{ w^2}\\right),\n\\end{equation}\nwhere $s_0$ is the saturation peak, $w$ is the width of the optical beam and where the vector product $\\mathbf{r} \\times \\mathbf{\\hat{k}}$ is the distance between the atom position and the center of the laser line propagation described by the unitary vector $\\mathbf{\\hat{k}}$. Considering the aperture of the optics elements, a spatial cut-off of $|\\mathbf{r} \\times \\mathbf{\\hat{k}}|< \\SI{1.2}{cm}$ in the local saturation parameter is also applied.\nThe frequency detuning is computed as:\n\\begin{equation}\n    \\Delta_\\text{eff}(\\mathbf{r}(t_i),\\mathbf{v}(t_i)) = \\Delta + \\mathbf{k}\\cdot \\mathbf{v}(t_i) -\\frac{\\mu_B}{\\hbar}| \\mathbf{B}(\\mathbf{r(t_i}))|\n\\end{equation}\nwhere $\\Delta$ is the laser frequency detuning from the atomic transition, $\\mathbf{k}\\cdot \\mathbf{v}(t_i)$ is the Doppler shift and the last term is the Zeeman shift induced by the atomic position in the magnetic field $\\mathbf{B}(\\mathbf{r}(t_i))$ described in Sec.\\ref{sec:apparatus}.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{mc_sim.pdf}\n    \\caption{Monte Carlo simulation of the velocity and position transverse coordinates of the 2D-MOT generated atomic beam at the end of the simulation time $t_\\text{tot}$. Density maps of the captured trajectories for (a) the single frequency 2D-MOT ($s_\\text{2D}$ = $s_{tot}$ = 6.56, $\\Delta_\\text{2D}$ = -1.6 $\\Gamma$), and (b) adding a sideband beam ($s_{SB}$ = 3.45, $s_{2D} = s_{tot} -s_{SB}$, $\\Delta_{SB}$ = -3.1 $\\Gamma$).}\n    \\label{fig:mc_sim_hist}\n\\end{figure}\n\n\nThe heating induced by the spontaneous emission process is also taken into consideration in the simulated dynamics by adding a random recoil momentum $\\hbar |k| \\sqrt{R \\delta t} \\ \\hat{\\mathbf{e}}$, where ${R\\delta t}$ is the average number of scattering events in a time interval $\\delta t$, while $\\mathbf{\\hat{e}}$ is a unitary vector randomly chosen from an isotropic distribution~\\cite{Kohel2003}. The resulting atom's acceleration induced by the 2D-MOT (and sideband) beams is described according to\n\\begin{equation}\n    \\mathbf{a}_\\text{2D,SB}= \\frac{\\hbar |k| }{m} \\sum_{n=0}^{4} \\frac{R_n}{4} \\left[   \\hat{\\mathbf{k}}_n + \\frac{\\hat{\\mathbf{e}}_n}{\\sqrt{R_n \\delta t \/ 4}}\\right], \n\\end{equation}\nwhere the saturation peak $s_\\text{2D,SB}$ is redistributed equally among the 4 beams of the 2D-MOT and sideband and  the beam directions $\\mathbf{\\hat{k}}_n$ are described by the 4 combinations of the unitary vectors $(\\pm \\mathbf{\\hat{x}} \\pm \\mathbf{\\hat{z}}) \/ \\sqrt{2}$.\nThe acceleration induced by the push beam  is computed as:\n\\begin{equation}\n    \\mathbf{a}_\\text{push}= \\frac{\\hbar |k| }{m} R_\\text{push} \\left[   \\hat{\\mathbf{y}} + \\frac{\\hat{\\mathbf{e}}}{\\sqrt{R_\\text{push} \\delta t }}\\right] .\n\\end{equation}\n\nThe total acceleration $\\mathbf{a}(t_i)$ exerted on the atom at position $\\mathbf{r}(t_i)$ with velocity $\\mathbf{v}(t_i)$  is quantified as the sum of the above processes:\n\\begin{equation}\n    \\mathbf{a} =  \\mathbf{a}_\\text{2D}  + \n     \\mathbf{a}_\\text{SB}  +\n     \\mathbf{a}_\\text{push}  \n\\end{equation}\n\n\n\nOnce $t=t_\\text{tot}$, each simulated atom is considered captured in the MOT if the divergence of the atomic trajectory computed along the push direction is lower that the geometrical constraint imposed by the MOT capture angle $\\theta_\\text{MOT} = \\SI{16}{\\milli rad}$ and if the final longitudinal velocity is below the MOT capture velocity  $v^\\text{MC}_\\text{capt} =\\SI{60}{m \\per s}$. In the selection of the captured trajectories, we also considered the losses due to collisions with hot atoms from the thermal beam, whose time scale is calculated to be $\\tau_\\text{coll}=\\SI{50}{ms}$. Hence for each trajectory the collision probability is estimated as $p_\\text{coll} = 1-e^{- \\tau_\\text{2D} \/ \\tau_\\text{coll}}$ and a unitary random number $\\varepsilon$ is generated in order to accept ($\\varepsilon>p_\\text{coll}$) or reject ($\\varepsilon<p_\\text{coll}$) each simulated atomic trajectory. Figure~\\ref{fig:mc_sim_hist} shows the results from two simulation runs, where the final transverse velocity and position coordinates are displayed versus the number of occurrences. The resulting velocity distribution is used to estimate the transverse temperature of the atomic beam.\n\n\n\nFinally, a capture efficiency ratio $r$ is defined as $r = N_\\text{tr} \/ N_\\text{sim}$, where $N_\\text{tr}$ is the number of captured trajectories for a given trapping configuration. Besides a scaling factor, the $r$ parameter is used as a comparison with the experimental data. The numerical results of such modeled atomic source are presented in Sec.\\ref{sec:atomic source characterization} and \\ref{sec:sideband enhancement} together with the experimental data.\n\n\n\n\n\n\n\\section{Atomic source characterization \\label{sec:atomic source characterization}}\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig_atomic_source_characterization.pdf}\n    \\caption{Experimental characterization of the strontium atomic source in the single-frequency 2D-MOT configuration. \\textbf{(a)} Number of atoms loaded in the MOT as a function of the 2D-MOT detuning $\\Delta_\\text{2D}$ at fixed saturation parameter $s_\\text{2D}=3.6$. The blue points are the experimental values of the number of atoms in the MOT. The turquoise regions are the MC simulations properly scaled with the experimental points. \\textbf{(b)} Number of atoms in the MOT as function of the 2D-MOT saturation $s_\\text{2D}$ at fixed detuning $\\Delta_\\text{2D}= - 1.6 \\, \\Gamma$.\n    \\textbf{(c)} Number of atoms in the MOT as a function of the push detuning $\\Delta_\\text{push}$ at fixed saturation $s_\\text{push}$= 0.34. All the data were taken at $T_\\text{ov}=\\SI{460}{\\celsius}$.}\n    \\label{fig:2DMOT_char}\n\\end{figure*}\n\n\n\n\n\nIn Fig.~\\ref{fig:2DMOT_char} we report the results of the characterization of our strontium 2D-MOT atomic source. This was obtained by looking at the loading of the MOT in the science chamber. Laser parameters of the 2D-MOT and push beams,  $\\Delta_\\text{2D}$, $s_\\text{2D}$ and $\\Delta_\\text{push}$, were scanned for optimal settings in order to find the maximum loading rate (blue points) and then compared with the expected capture ratio $r$ from MC simulation (turquoise region).\n\nRegarding the 2D-MOT beam parameters, Fig.\\ref{fig:2DMOT_char}(a) shows the number of atoms loaded in the MOT that reaches its maximum value at $\\Delta_\\text{2D} = -1.6\\, \\Gamma$, with a FWHM of the order of $1.5\\,\\Gamma$. The peak position and the spectral response are in good agreement with the simulated one. Fig.\\ref{fig:2DMOT_char}(b) shows the increase of number of atoms in the MOT as a function of the 2D-MOT optical intensity at $\\Delta_\\text{2D}=-1.6\\,\\Gamma$. Here we can observe that for $s_\\text{2D}>6$, the number of atoms in the MOT starts to saturate, however much later than the unity value. The same result is predicted by the MC simulation.\n\nWe observed an optimal push intensity around the $s_\\text{push} \\simeq  0.34$, beyond this value the MOT number of atoms decreases, as previously verified in a similar setup~\\cite{Nosske2017}. The reduced efficiency in the transfer from the 2D-MOT to the blue MOT is explained considering that atoms accelerate beyond $v_\\text{capt}$ cannot be captured in the MOT. This behaviour at higher $s_\\text{push}$ is  also observed in the MC simulation considering only the atom captured in the MOT at longitudinal $v_L$ velocity below $v^\\text{MC}_\\text{capt} \\sim \\SI{60}{ \\per s}$. Fig.\\ref{fig:2DMOT_char}(c) shows the MOT number of atoms as a function of the push beam detuning. From this plot we observe that the best transfer efficiency is obtained near the atomic resonance $\\Delta_\\text{push} = 0$, but it is not a critical parameter.\n\nAt the best trapping configuration the total atomic flux generated by the 2D-MOT source is measured by detecting the fluorescence generated by a probe beam sent along the $z$-direction,\nnearly at the center of the MOT in the science chamber. The resulting atomic flux $\\Phi_\\text{2D}$ reaches a maximum value of \\SI{6(1)e8}{atoms \\per s}, as shown in Fig.~\\ref{fig:2Dflux}. This can be compared with the MOT loading rate $L_\\text{MOT}$ and with the expected flow resulting from the capture efficiency ratio resulting from MC simulations. The MOT loading rate is simply given by $L_\\text{MOT}=N_0\/\\tau =\\SI{3.1(4)e8}{atoms \\per s}$, where $\\tau$ is the MOT relaxation time, which in our system without repumping is \\SI{17(2)}{ms} and $N_0=\\SI{5.3(2)e6}{atoms}$ is the maximum number of atoms trapped in the final MOT. It corresponds to roughly \\SI{51}{\\percent} of the total flux. The expected atomic flux generated by the 2D-MOT can be estimated as \n$$\n\\Phi^{(th)}_\\text{2D} = r f_\\text{cut} p_\\text{rad} \\Phi_\\text{ov} = \\SI{1.5e9}{atoms \\per s}.\n$$ \nIn this estimate we used $r=\\num{6.9e-2}$ from MC results, $f_\\text{cut}=\\num{1.53e-3}$ is the fraction of simulated velocities from the Maxwell-Boltzmann distribution considering a cut-off at $v_\\text{cut}=\\SI{90}{m \\per s}$, $p_\\text{rad}$ is the survival probability from optical pumping to the metastable $^3$P$_2$ state~\\cite{Xu2003} that we calculated considering a typical time spent in the 2D-MOT region $\\expval{ \\tau_\\text{2D} }= \\SI{2.9(6)}{ms}$ and a pumping rate $R=\\SI{223}{Hz}$, which give us $p_\\text{rad}\\simeq 1 - R \\expval{ \\tau_\\text{2D}} = 0.35$. The estimated theoretical flux $\\Phi^{(th)}_\\text{2D}$ provides a discrepancy from the measured $\\Phi_\\text{2D}$ of only a factor 2.4, which is remarkably close. In fact our simulations do not consider effects due to experimental imperfections, such as the misalignment of the zero magnetic field of the permanent magnets and the optimal push beam direction for optical transfer to the MOT in the science chamber\n\n \n\n\n\\begin{figure}[!b]\n \\includegraphics[width=0.45\\textwidth]{fig_atomic_flux.pdf}\n \\caption{Atomic flux generated from the atomic source as a function of the 2D-MOT saturation parameter, without (blue circles) and with (red diamonds) the use of the sideband ($P_\\text{SB}$ = 200 mW - $P_\\text{2D}$). These data are taken at $s_\\text{push} $ =  0.34,  $\\Delta_\\text{2D}\/\\Gamma$ = -1.6 and compared with MC estimates (shaded lines).}\n  \\label{fig:2Dflux}\n\\end{figure}\n\n\n\n\n\n\\section{Sideband enhancement \\label{sec:sideband enhancement}}\n\n\\subsection{Loading a MOT with sideband-enhancement}\n\nWe demonstrated sideband-enhanced loading of a 2D-MOT atomic source by overlapping a second laser beam with higher frequency detuning to the 2D-MOT cooling lasers, as described in Sec.\\ref{sec:apparatus}. Fig.\\ref{fig:sideband} shows how the power distribution between the two frequencies affects the number of atoms collected in the MOT trap at sideband detuning $\\Delta_\\text{SB}=\\SI{ -3.13}{\\Gamma}$, while the 2D-MOT beam is tuned at its previously shown maximum $\\Delta_\\text{2D}=\\SI{-1.6}{\\Gamma}$. From Fig.\\ref{fig:sideband} we see that it exists an optimal power distribution around $s_\\text{SB} \\simeq 3.5$ that maximizes the number of atoms in trapped in the MOT, reaching up to \\SI{1.2e7}{atoms} atoms, i.e., about \\num{2.3} times higher than with the total available power sent to the 2D-MOT AOM and about \\num{4} times higher than the corresponding value with the sideband beam blocked.\n\n\n\\begin{figure}[b]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{fig_enhancing.pdf}\n    \\caption{Number of atoms captured in the MOT region for different power distribution of the $s_\\text{tot}=6.56$ between the sideband beam $s_\\text{SB}$ and the 2D-MOT beam $s_\\text{2D}$. All the data are measured with a 2D-MOT detuning $\\Delta_\\text{2D} = -1.6\\,\\Gamma$ and sideband detuning $\\Delta_\\text{SB}= \\SI{-3.13}{\\Gamma}$. The (red) diamonds are the number of atoms trapped in the MOT at increasing sideband beam saturation parameter $s_\\text{SB}$. The (pink) shaded area represents the MC simulation in this experimental configuration. The (blue) circles describe corresponding number of atoms trapped in the MOT with the sideband beam AOM turned off.}\n    \\label{fig:sideband}\n\\end{figure}\n\n\nIn order to find the optimal working point of the sideband-enhanced 2D-MOT, we scanned over the sideband AOM frequency from \\SI{230}{MHz} to \\SI{335}{MHz}, which corresponds to a detuning range between $-5\\Gamma$ and $-2.2\\Gamma$, and measured the MOT trapped atoms $N$ at different sideband power. We performed this scan with a total power $P_\\text{tot}=\\SI{200}{mW}$ and \\SI{110}{mW}, i.e. a total saturation parameter $s_\\text{tot}=6.56$ and $s_\\text{tot}=3.61$, respectively. We introduce the enhancement parameter $\\eta$ as: \n\\begin{equation}\n    \\eta(s_\\text{SB} , \\Delta_\\text{SB})= \\frac{N(s_\\text{SB}= s_\\text{tot} - s_\\text{2D}  , \\Delta_\\text{SB})}{ N(s_\\text{2D} = s_\\text{tot} )  }\n    \\label{eq:enhacement_factor}\n\\end{equation}\nThe so-defined $\\eta$ parameter compares the two different trapping configurations, both sharing the same total optical power $s_\\text{tot}$. When $\\eta > 1$ sideband-enhancement is achieved. \n\nFig.\\ref{fig:SE_scan}(a) and (c) show two sets of the sideband enhancement parameter scan, where we plot the enhancement parameter ($\\eta$) with respect to $s_\\text{SB}$ and $\\Delta_\\text{SB}$ when $P_\\text{tot} = \\SI{200}{mW}$ and \\SI{110}{mW} respectively. These results are compared to their respective MC simulations (Fig.\\ref{fig:SE_scan}(b) and (d)). The data show that optimum loading efficiency of the final MOT is reached tuning the sideband frequency to $\\Delta_\\text{SB} = \\SI{-3.13}{\\Gamma}$ for both the total power regimes.  At $P_\\text{tot}=\\SI{200}{mW}$, we reached a maximum enhancement of $\\eta^\\text{exp}=2.3(1)$ when $s_\\text{SB} \\simeq 3.1$ ($P_\\text{SB}\\simeq \\SI{90}{mW}$). The MC numerical data present essentially the same main features as the experimental measurements, both having the maximum loading at the same sideband parameter point, reaching a slightly lower enhancement $\\eta^\\text{MC} = 2.13(4)$, as detailed in Fig.\\ref{fig:sideband}. At $P_\\text{tot}=\\SI{110}{mW}$, we obtained the best enhancement factor of $\\eta^\\text{exp}=1.48(7)$ when $s_\\text{SB}=1.3$ ($P_\\text{SB} \\simeq \\SI{40}{mW} $), while the MC numerical results show a slightly higher enhancement of $\\eta^\\text{MC}=1.87(3)$. Both numerical and experimental data suggest that for $s_\\text{tot}\\geq 1$, sideband-enhancement grows with the increasing available power, where the optimized power distribution can be more effective. Alternatively, the rate at which the atomic flux increases with respect to the laser power is significantly higher for the case where we add the higher-detuned frequency sideband.\n\nAt maximum $\\eta^\\text{exp}$ = 2.3(1), we measured $N=\\SI{1.25(4)e7}{atoms}$ trapped in the MOT, which corresponds to a loading rate of $L_\\text{MOT}^\\text{SB}=\\SI{7.3(9)e8}{atoms \\per s}$. A total flux measurement by fluorescence detection is also performed for the sideband-enhanced 2D-MOT and reported in Fig.~\\ref{fig:2Dflux}. In this case we measured an atomic flux of $\\Phi_\\text{SB} = \\SI{1.5(2)e9}{atoms \\per s}$. The related enhancement factor is \\num{2.4(4)} and it is in agreement with the experimental and numerical results reported for the MOT loading.\n\nCompared to other Sr atomic sources, our sideband-enhanced 2D-MOT source shows high transfer efficiency $L_\\text{MOT}^\\text{SB}\/\\Phi_\\text{SB}$ = 48(8)\\%, with a MOT loading rate slightly larger than a ZS-enhanced Sr 2D-MOT source~\\cite{Nosske2017}, and less than a factor ten lower than more complex and power-demanding high-flux source systems based, for instance, on a combination of Zeeamn slower, 2D-MOT and deflection~\\cite{Yang2015}. \n\n\n\n\n\\begin{figure*\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig_sideband_enhancing_factor_all.pdf}\n   \n   \n    \\caption{Enhancement factor $\\eta$ as function of the sideband parameter scan. (a) Experimental data compared to (b) MC numerical results at fixed total saturation parameter $s_\\text{tot}=6.56$ and $\\Delta_\\text{2D}=-1.6\\,\\Gamma$. (c) and (d) same as (a) and (b) but for $s_\\text{tot}$ = 3.6.}\n    \\label{fig:SE_scan}\n\\end{figure*}\n\n\n\\subsection{Kinetic properties of the sideband-enhanced 2D-MOT}\n\nThe kinetic properties of a Sr cold atomic beam generated by a 2D-MOT has been recently studied~\\cite{Nosske2017}, in particular as function of the push beam and 2D-MOT beam intensities. We verified these findings in our setup and extended the study to the addition of the sideband beam. \n\nThe longitudinal velocity was measured by time-of-flight technique. A push beam pulse of \\SI{5}{ms} accelerate the 2D-MOT atoms towards the science cell. The longitudinal velocity distribution is estimated recording the fluorescence time distribution $f(t)$\nas measured at the MOT center\nWe compute the longitudinal velocity distribution as $f(v) = f(d\/t)$, where $d = \\SI{36.5(5)}{cm}$ is the 2D-MOT to MOT distance. Compared to the single-frequency 2D-MOT, we did not observe any change in peak velocity or in velocity dispersion. The peak velocity for optimal push saturation parameter $s_\\text{push} = 0.34$ is $v_\\text{L} = \\SI{22.5}{m \\per s}$.\n\n\\begin{figure}[b]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{temp_art.pdf}\n    \\caption{Doppler spectrum of the transverse spectroscopy on the 2D-MOT Sr atomic beam. The inset shows the corresponding Doppler temperatures compared to the MC simulation results and the corresponding Doppler limit.}\n    \\label{fig:transvel}\n\\end{figure}\n\n\nThe atomic beam transverse velocity was measured by Doppler spectroscopy with and without powering the sideband beam. The transverse velocity was extracted from the Doppler profile by fixing the Lorentzian component due to the natural linewidth of the $^1$S$_0$\\,--\\,$^1$P$_1$~probe transition and the saturation broadening ($s_\\text{probe} = 0.1$). The measurement results are shown in Fig.\\ref{fig:transvel}, yielding a Doppler broadening $\\sigma_\\text{2D}(T) = \\SI{3.6(8)}{MHz}$ and $\\sigma_\\text{SB}(T) = 0.8(3.0)\\,\\text{MHz}$ respectively. This corresponds to a transverse temperature of \\SI{14(7)}{mK} for the 2D-MOT and $0.7(5.0)\\,\\text{mK}$ for the sideband-enhanced 2D-MOT, as shown in the inset in Fig.\\ref{fig:transvel}. Compared to the Doppler temperature at $s_\\text{2D} = 6.6$ and $\\Delta_\\text{2D}=-\\SI{1.6}{\\Gamma}$ which is equal to \\SI{2.1}{mK}, the 2D-MOT result is nearly seven times warmer, while the sideband-enhanced case shows an upper limit temperature almost three times higher. We also estimated the transverse temperature of the atomic beam resulting from MC numerical simulations, which present transverse temperatures of \\SI{2.9(1)}{mK} and \\SI{2.12(4)}{mK} respectively. While MC results confirms a colder beam for the sideband-enhanced case, they still miss the extra-heating effects which can be explained by transverse spatial intensity fluctuations of the optical molasses in the 2D-MOT~\\cite{Chaneliere05}.\n\nFrom the measured transverse and longitudinal velocities, we derive an atomic beam divergence $\\theta_\\text{2D} \\equiv v_t\/v_L = \\SI{75(17)}{ mrad}$ and $\\theta_\\text{SB}\\leq \\SI{58}{mrad}$. Finally, from the values of the beam divergence and the atomic flux, we estimated the atomic beam radiant intensity, or sometimes called beam ``brightness'', $\\mathcal{J}\\equiv \\Phi \/(\\pi\\theta^2)$. For the single-frequency 2D-MOT we obtained a brightness\n\n$$\n\\mathcal{J}_\\text{2D} = 3.5(8)\\times 10^{10} \\,\\mathrm{atoms}\\cdot\\mathrm{s}^{-1}\\cdot\\mathrm{sr}^{-1},\n$$\nwhile, for the sideband-enhanced beam, the brightness\n\n$$\n\\mathcal{J}_\\text{SB} \\geq 1.4\\times 10^{11}\\,\\mathrm{atoms}\\cdot\\mathrm{s}^{-1}\\cdot\\mathrm{sr}^{-1}.\n$$\n\nThis results represents a factor four improvement with respect to the single-frequency 2D-MOT, making the sideband-enhancement a promising technique for optimal transfer to 2D optical molasses working on narrow linewidth intercombination transition of strontium for continuous BEC production and continuous optical clock proposals~\\cite{Bennetts17}.\n\n\n\n\\subsection{Comparison with the Zeeman-slower enhancement}\n\nAn alternative method to increase the 2D-MOT capture rate is to direct another slowing beam towards the hot atomic beam generated by oven which, exploiting the decreasing tail of the 2D-MOT magnetic field, can efficiently scatter faster atoms along the beam direction similarly to a Zeeman Slower (ZS). This approach was previously demonstrated in similar setups~\\cite{Lamporesi2013,Nosske2018,Colzi2018}.\n\nWe employed the beam generated by the sideband AOM as Zeeman slowing beam, shaped to have a beam width $w_{ZS} = \\SI{6}{mm}$. We partially scanned over the ZS parameters, which resulted in a maximum number of atoms in the MOT of $N_\\text{ZS} = \\SI{3.8(1)e6}{atoms}$, obtained with $\\Delta_\\text{ZS}=\\SI{-8.1}{\\Gamma}$, $P_\\text{ZS} =\\SI{140}{mW}$, while we kept $P_\\text{2D} =\\SI{36}{mW}$ and $\\Delta_\\text{2D}=\\SI{-1.5}{\\Gamma}$ fixed. Blocking the ZS beam we observe a gain in the atomic number of the order of 4, in agreement with the experimental observation in~\\cite{Nosske2017}.\n\nBecause of the short distance (about \\SI{25}{cm}) between the oven aperture and the optical window facing it to send the ZS beam, we heated the window flange up to \\SI{350}{\\celsius} in order to prevent metalization. However we first observed a fast degradation of the atomic source flux, soon followed by the full metalization of the window. This prevented us to perform a fine optimization of the $s_\\text{ZS}$ and $\\Delta_\\text{ZS}$ as the one shown in the Fig.~\\ref{fig:SE_scan} and also to produce a stable MOT during the day.\n\nA possible way to reduce Sr metalization of the ZS window would be to increase the distance between the oven and window itself by means of a vacuum extension (at least a \\SI{1}{m} tube with shorter diameter). However this solution would compromise the compactness of the atomic source conceived by a 2D-MOT making the system more complex, power consuming and perhaps needing extra water cooling to avoid thermal stress to the vacuum system.\n\nAnother drawback of the ZS method is the fact that the quantization axis of the magnetic field along the hot atoms direction imposes that only one half of the linear polarized ZS light power has the correct circular polarization. This means that at least half of the ZS optical power is wasted in the slowing process. On the contrary, the sideband beams have a well defined polarization in the capture region so that all the employed power is effective in the cooling and trapping process.\n\n\\subsection{Application to other alkaline-earth atoms and prospects for optical clocks}\n\nIt is interesting to extend the discussion about the sideband-enhancement method to other atomic species, in particular those employed in optical clocks. We exploit the MC simulation\nin order to investigate the potential trapping performances of additional atomic species. Table~\\ref{tab:alkaline-earth} shows the main optical and atomic parameters for alkaline-earth(-like) atomic species currently in use in optical clock experiments. In particular, we consider the broad $^1$S$_0$\\,--\\,$^1$P$_1$~strong dipole transition as cooling transition, fixing the atomic vapour pressure to \\SI{0.1}{Pa} for all the species. We ran our MC simulation with these parameters, with a total available saturation parameter $s_\\text{tot}$ = 6.56, the same magnetic field gradient and the same laser beam widths as for our previously described apparatus.\n\nThe simulation workflow is the following: first we simulate the single-frequency 2D-MOT, looking for the optimal detuning $\\Delta_\\text{2D}$ at half of $s_\\text{tot}$; then we add the sideband at $\\Delta_\\text{SB} = 2\\Delta_\\text{2D}$, which is basically the result we found in Sec.\\ref{sec:sideband enhancement} for Sr, and we scan the sideband-enhanced 2D-MOT at different sideband saturation parameter $s_\\text{SB}$. Because of the extremely high saturation intensities of Cd and Hg which makes unrealistic the application of this method, they were excluded from this numerical study.\n\n\n\\begin{table*}[tb]\n\\centering\n\\begin{tabular}{l c c c c c c |c c c}\n\\hline\n\\hline\nAtom & $\\lambda$     &    $\\Gamma\/2\\pi$  & $I_\\text{sat}$ & $a_\\text{max}$ & $T(p_0)$& $v_{th}(p_0)$ & $\\Delta_\\text{2D}\/\\Gamma$ & $r^\\text{MC}f_\\text{cut}$ & $\\eta$\\\\\n & (nm) & (MHz) & (mW\/cm$^2$) & ($10^6$ m\/s$^2$) & (K) & (m\/s)&&(ppm)\\\\\n\\hline\n$^{24}$Mg  & 285.30 & 80.95 & 455 & 14.8& 641 & 679 & -2.28& 77 & 1.0\\\\\n$^{40}$Ca  & 422.79 & 34.63 & 59.9& 2.69 & 788 & 583 & -2.5 & 103& 2.1\\\\\n$^{88}$Sr  & 460.86 & 31.99 & 42.7 & 1.01 & 725 & 379 &-1.76& 260& 2.1\\\\\n$^{138}$Ba  & 553.70 & 18.33 & 14.1& 0.31 & 826 & 321&-0.89& 60&2.9 \\\\\n$^{114}$Cd  & 228 & 91& 1005 & 4.64 & 485 & 273\\\\\n$^{174}$Yb  & 398.91 & 29 & 59.8 & 0.54 & 673 & 258&-1.42& 316& 2.2\\\\\n$^{198}$Hg & 185 & 120 & 2481 & 4.24 & 286 & 157\\\\\n\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Sideband enhancement on alkaline-earth(-like) atomic species. On the left side of the table the most relevant parameters for cooling and trapping atoms on the $^1$S$_0$\\,--\\,$^1$P$_1$~ strong dipole transition are shown, whereas we estimate the thermal and kinetic properties of every atomic species at a pressure $p_0$ = 0.1 Pa. On the right, MC optimization results of the 2D-MOT detuning $\\Delta_\\text{2D}$, the fraction of trapped atoms, and the enhancement factor $\\eta$ for each alkaline-earth species.}\n\\label{tab:alkaline-earth}\n\\end{table*}\n\nThe MC simulation results are reported in Tab.~\\ref{tab:alkaline-earth}. Here we clearly see that the sideband-enhancement is more effective for those atoms having a lower value of the maximum acceleration $a_\\text{max}$. This dependence can be understood by looking at the definition of maximum capture velocity in (\\ref{eq:vc}). In fact, it can be only achieved for a light field uniformly resonant with the atomic transition and fully saturated all along the trap diameter. This means that the broader the cooling transition linewidth $\\Gamma$ is (and thus the higher $a_{max}$), the closer the MOT is to its capture limit, implying that the expected enhancement factor is lower. Furthermore, according to Eq.~\\ref{eq:cap_2d}, we would expect that the sideband-enhancement works better for light species, in particular where $v_\\text{th}$ is higher and the quartic dependence of the loading rate on the capture velocity is a more accurate approximation. Hence we can work out a sideband-enhancement factor functional dependence\n\n$$\n\\eta(X) \\propto \\frac{v_\\text{th}(X)^2}{a_\\text{max}(X)}\n$$\nwhere $X$ is the considered atomic species. By accident, either Sr, Ca and Yb have very similar ${v_\\text{th}(X)^2}\/{a_\\text{max}(X)}$ values, and the resulting $\\eta$ is the same within the numerical error.\n\nWe also report in Tab.\\ref{tab:alkaline-earth} the absolute capture efficiency for the sideband-enhanced 2D-MOT atomic source $r(X)$. MC simulations show that the highest capture rate is predicted for Yb followed by Sr, two of the strongest candidates for a possible redefinition of the second based on optical atomic clocks~\\cite{Riehle2015}.\n\n\n\n\\section{Conclusions}\\label{sec:conclusion}\n\nIn this work we demonstrated and fully characterized a robust method to enhance the atomic flux generated by a Sr 2D-MOT by adding a second frequency to the 2D-MOT beams. The experimental implementation of the sideband-enhancement method only requires a simple optical setup and a proper alignment of the sideband beam to the main 2D-MOT beam. The resulting bright atomic source can deliver more than $1.4\\times 10^{11}\\,\\mathrm{atoms}\\cdot\\mathrm{s}^{-1}\\cdot\\mathrm{sr}^{-1}$ if the total available power for the atomic source is 200 mW. This cold atomic flux can be efficiently loaded in a 3D MOT for ultracold atoms experiments, preventing direct sight to the hot atomic oven and providing an efficient optical shutter of the atomic beam. This result represents an enhancement in MOT loading by a factor 2.3 with respect to single-frequency 2D-MOT based atomic source.\n\nA dedicated Monte Carlo simulation, which well predicts the experimental data of our Sr atomic source, shows that this technique is a valid method to increase the number of atomic sources based on the other alkaline-earth species such as Yb and Ca, paving the way for compact atomic sources suitable for transportable optical clocks or optical clock transition-based gravimeters~\\cite{Hu17,Akatsuka_2017}.\n\n\n\n\\begin{acknowledgments}\nThe authors would like to thank U. Sterr for inspirational discussions about sideband-enhanced MOT, D. Racca, E. Bertacco, M. Bertinetti and A. Barbone for laboratory assistance. We acknowledge funding of the project EMPIR-USOQS, EMPIR projects are co-funded by the European Union's Horizon2020 research and innovation programme and the EMPIR Participating States.\nWe also acknowledge QuantERA project Q-Clocks, ASI, and Provincia Autonoma di Trento (PAT) for financial support. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOptical Character Recognition (OCR) has long been the fundamental task in computer vision. There are many applications such as automatic content extraction for documents \\cite{burie2015icdar2015}, translation \\cite{toyama2014mixed}, text style transfer \\cite{krishnan2021textstylebrush} and assistance for robots and visually impaired users. From the perspective of research, OCR can range from relatively easy and controlled tasks such as digit recognition~\\cite{lecun1998mnist}, to difficult scenarios such as scene text with arbitrary orientations and shapes \\cite{karatzas2015icdar,ch2017total,hassner2013computation,chng2019icdar2019,singh2021textocr}, and has become an important domain for benchmarking new machine learning techniques.\n\nNevertheless, most existing OCR approaches focus on numbers and the English alphabet due to English's status as a common \\textit{lingua franca} and its subsequent wide availability in popular datasets. Thanks to the introduction of multilingual text detection and recognition datasets and benchmarks \\cite{nayef2017icdar2017,nayef2019icdar2019}, there is now a unified platform to measure the model performance on challenging scenarios containing thousands of distinctive characters. Recent methods have shown that training different languages with different recognition heads can improve end-to-end recognition accuracy compared to combining characters from all languages in the same recognition head \\cite{huang2021multiplexed}. However, it's not clear whether an individual recognition head for each language is optimal. For example, should English and Spanish be separated into two heads? The answer is probably no since they share most of the characters. Moreover, even if separating two languages into two heads does yield the best accuracy, it might not be worth it if the accuracy gain is marginal compared to the increase in number of parameters and\/or the inference time. Therefore, one of the questions our work tries to answer is how to decide whether\/how languages should be grouped together under the constraint of a limited number of models.\n\nWithout any pre-assigned grouping, we treat each of the models at initialization as a generalist agent which looks at all tasks. Then, as the scale tips, each agent is encouraged to be increasingly specialized in one or more tasks; each agent becomes a specialist, each model can still try to learn the other tasks to some lesser extent. Due to different transferability, data variation, and similarities among the tasks, each agent can have different progress in both the specialized tasks and non-specialized tasks, and the specialties will be redistributed automatically as the agents evolve. Eventually, as confirmed by our experiments, this multi-agent system will reach an equilibrium where the specialties for each agent do not change any more, and this is when the task grouping result is finalized.\n\nTo summarize, our contributions include:\n\\begin{itemize}\n\\item To our knowledge, this is {\\em the first work} exploring the grouping of languages for multilingual OCR.\n\\item We propose an automatic grouping mechanism that allows dynamic and differentiable routing of tasks to different heads during training.\n\\item We empirically show that the automatic task grouping model outperforms both the one-task-per-head and the all-tasks-in-one-head baselines.\nWe further show that when the models have different capacities, the task assignment can potentially reflect the underlying task complexity and data distribution.\n\\end{itemize}\n\nTo promote reproduction of our work, our code is publicly available at \\url{https:\/\/github.com\/facebookresearch\/MultiplexedOCR}.\n\n\n\\section{Related work}\n\n\\subsection{Multilingual text spotting}\nText spotting systems combine text detection and text recognition modules to identify the location and content of text in images. Early works approached both modules independently: text proposals for regions containing text were generated first, followed by a recognition module to identify the text given a pre-defined character dictionary. For text detection, state-of-the-art (SotA) methods are mostly based on Region Proposal Networks (RPN)~\\cite{ren2015faster}, previously proven successful for object detection. Variants of RPNs have been proposed to account for varying text orientations~\\cite{jiang2017r2cnn}, arbitrary shapes~\\cite{liao2020mask,qin2019towards}, and character masking~\\cite{baek2019character}. For text recognition, models typically use an RNN-style design to predict sequences of characters~\\cite{su2014str,he2016reading}. Representative methods include connectionist temporal classification (CTC)~\\cite{graves2006connectionist} and attention-based decoders \\cite{bahdanau2014neural,lee2016recursive,shi2016robust}. \n\nWhile earlier systems treated detection and recognition as independent modules, most of the recent works train these modules in an end-to-end manner. Given the inter-dependability of these modules, this training methodology results in performance improvements for these systems. Some representative works include Mask TextSpotter~\\cite{liao2021mask}, FOTS~\\cite{liu2018fots}, CharNet~\\cite{xing2019charnet}, etc. For deeper insights into the text spotting systems, we refer the readers to the thorough review in \\cite{long2020scene}. Similar to these works, we also employ end-to-end training for our system. For recognition, we mainly use an attention-based decoder to make fair comparisons with previous works~\\cite{liao2020mask,huang2021multiplexed}. \n\nWith the availability of reliable multilingual datasets such as MLT19~\\cite{nayef2019icdar2019}, text spotting systems have tried to address the problem of multilingual texts. In addition to detection and recognition modules, some multilingual text spotting systems also include a script identification module~\\cite{buvsta2018e2e,huang2021multiplexed} to identify the language for text recognition. While text spotting systems such as E2E-MLT~\\cite{buvsta2018e2e} and CRAFTS~\\cite{baek2020character} present results for multilingual datasets, they do not explicitly incorporate model specific components adapted for multiple languages. Instead, they combine the characters from all languages to form a larger dictionary for the recognition head. Recently, Multiplexed Multilingual Mask TextSpotter (MMMT) \\cite{huang2021multiplexed} proposed to employ different recognition heads for different scripts, routed through a script identification module at the word level. Unlike MMMT, which employs hard assignment for routing to an appropriate recognition head, we propose to group the languages by training agents to route words to different heads in a data-driven fashion. This automatic grouping mechanism allows for dynamic and differentiable shifting of tasks to optimize the language combinations for various recognition heads.\n\n\\subsection{Multitask learning and grouping}\n\nMultitask learning methods have a long history~\\cite{caruana1997multitask,evgeniou2004regularized}. As their name implies, they jointly learn solutions for multiple tasks, sharing or transferring information between tasks to improve overall performance. Recent deep learning methods assume that the parameters for early layers, which account for low-level feature extraction, are shared among different tasks, while the parameters for later layers, which account for high-level integration of visual signals, are task-specific~\\cite{zeiler2014visualizing,inkawhich2020transferable}. Hence, information relevant to all tasks is learned by a shared trunk which later splits to multiple task-specific heads. \n\nA natural question that arises when designing multitask systems is the following: How should tasks be grouped to maximise a model's accuracy? To answer this question, Kang \\emph{et al}\\onedot \\cite{kang2011learning} proposed learning shared feature representations for related tasks by formulating task grouping as a mixed integer programming problem, where binary indicator variables are used to assign tasks to groups. Unlike this hard group assignment, Kumar \\emph{et al}\\onedot \\cite{kumar2012learning} propose to allow for parameter sharing across groups through soft, latent assignment of task features as a linear combination of a finite number of underlying basis tasks. Zhong \\emph{et al}\\onedot \\cite{zhong2016flexible} extend this work by removing constraints on the size of latent basis tasks and adding regularization terms to enforce sparsity in task weights and orthogonality to prohibit commonality among unrelated tasks. Zamir \\emph{et al}\\onedot \\cite{zamir2018taskonomy} proposed a method for modeling the relationship of different visual tasks based on the {\\em transferability} between them. Instead of learning shared representation on a trunk, Strezoski \\emph{et al}\\onedot \\cite{strezoski2019routing} introduce a {\\em Task Routing layer} that masks convolutional channels based on the task, effectively creating a sub-network per task.\n\nOur work is similar in spirit to Strezoski \\emph{et al}\\onedot \\cite{strezoski2019routing} in that we allow for dynamic routing of tasks to different heads during training. In our approach, however, the routing is done using {\\em Gumbel-Softmax} \\cite{jang2016categorical} to ensure probabilistic interpretation of each task and using a novel {\\em grouping loss} for task assignment.\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.98\\linewidth]{figures\/system_b.png}\n    \\caption{\\textbf{Our proposed task grouping framework.} Here we show a batch of $4$ inputs potentially belonging to $3$ tasks, with $2$ recognition heads. See Sec.~\\ref{sec:method} for more details.}\n    \\label{fig:task_grouper}\n\\end{figure*}\n\n\\section{Methodology}\\label{sec:method}\n\nGiven a list of tasks $T = \\{T_i\\}_{1 \\leq i \\leq t}$ and a list of models $M = \\{M_j\\}_{1 \\leq j \\leq m}$, we can define the task grouping of $T$ over $M$ as a mapping $G: T \\rightarrow M$, where $G$ is a single-valued function, which means each task will be assigned to exactly one model. On the other hand, $G$ does not need to be an injection, since multiple tasks can be assigned to the same model. $G$ need not be a surjection either, in which case some models will not be assigned with any tasks. Our goal is to find out the best assignment $G$ such that the overall performance is maximized.\n\nFigure \\ref{fig:task_grouper} shows the core architecture of the proposed task grouping framework. Given an input, which could be a batch of already cropped image patches or pre-extracted features, we first pass them through a {\\em task classification network} that predicts the probabilities of each input belonging to each of the tasks.\nUnder the context of multilingual text recognition, each task $T_i$ can be described as ``recognizing the word instance $W_k$ in language set $L_i$'', where $W_k$ is the $k$-th instance in a batch of $w$ word crops. We can thus define a probability matrix of size $w \\times t$ on the likelihood of each word belonging to each task\/language set:\n\n\\begin{equation} \\label{eq:word_task_matrix}\n    P_{WT} = \\{p(T_i | W_k)\\}_{1 \\leq k \\leq w, 1 \\leq i \\leq t}\n\\end{equation}\n\nAt inference time, $P_{WT}$ can be inferred from a task classification network such as a language prediction network \\cite{huang2021multiplexed}. This is a $t$-way classification problem, and the task classification loss can be computed using a cross entropy loss:\n\\begin{equation}\n    L_{task}(W_k) = - \\sum_{i=1}^{t} I(T_i=T_{gt}) \\log p(T_i|W_k)\n\\end{equation}\nwhere $I(T_i=T_{gt})$ is a binary indicator of the task matching the ground truth.\n\nAt training time, $P_{WT}$ can be inferred from the ground truth, if there is an annotation of which language each word belongs to: $p(T_i|W_k)$ is $1$ if $W_k$ belongs to $T_i$ and $0$ otherwise. When the ground truth annotation for the language information is not directly available but the transcription is available, we can make an educated guess of the probability by calculating the proportion of characters in $W_k$ that are supported by language set $L_i$.\n\n\\subsection{Grouping module}\n\nSince task-model mapping $G$ is a discrete function, to be able to learn it we can define the following probability matrix, of size $t \\times m$:\n\\begin{equation} \\label{eq:task_model_matrix}\n    P_{TM} = \\{p(M_j|T_i)\\}_{1 \\leq i \\leq t, 1 \\leq j \\leq m},\n\\end{equation}\nwhere $p(M_j|T_i)$ is the probability of an arbitrary word belonging to $T_i$ to be handled by model $M_j$. Then, we can compute the probability matrix of each word $W_k$ to be handled by model $M_j$ by multiplying $P_{WT}$ and $P_{TM}$:\n\n\\begin{equation} \\label{eq:word_model_matrix}\n    P_{WM} = P_{WT} \\cdot P_{TM}\n\\end{equation}\nNaive task assignment to a group based on traditional SoftMax is a discrete operation and thus non-differentiable.\nBackpropagation through only the selected task-group pairing would result in high variance gradients leading to unstable learning.\nInstead, when computing $P_{WM}$ during training, we apply a soft relaxation of the assignment operation using the Gumbel-Softmax \\cite{jang2016categorical}.\nGumbel-Softmax if fully differentiable with the reparameterization trick and results in gradient backpropagating through all possible task-group pairings, not just the one with the maximum score.\nWe instantiate learnable parameters for task-model assignment as a real-valued matrix $R_{TM} \\in \\mathbb{R}^{t \\times m }$, initialized with all ones (or any equal numbers) in the beginning, and we set the temperature $\\tau = 1.0$ throughout the training. At test time, we can just pick the model corresponding to the maximum, \\emph{i.e}\\onedot the hard mode of Gumbel-Softmax.\n\n\\subsection{Integrated loss}\n\nA key difference of our approach compared to \\cite{huang2021multiplexed} is that in our framework, we do not restrict the capability of each model, or recognition head, to support any specific task, \\emph{i.e}\\onedot, a certain recognition head can only support certain characters, from the beginning. Instead, we assume each model to be omnipotent in the beginning and has the potential to handle every task. This is necessary since otherwise there is no point in doing the grouping if each model is already designed to do certain tasks. \n\nTherefore, unlike \\cite{huang2021multiplexed}, we can directly use the negative log likelihood as the recognition loss $L_{seq(j)}$ for each model $M_j$ without worrying about the unsupported characters:\n\\begin{equation}\n    L_{seq} = - \\frac{1}{s} \\sum_{l=1}^{s} \\log p(S_l),\n\\end{equation}\nwhere $p(S_l)$ is the predicted probability of character at position $l$ of the sequence, and $s$ is the length of the sequence of character labels.\nWe can, however, perform the pruning at the output layer to remove any characters that do not belong to the task assigned to certain head, once the grouping is determined. This reduces the unnecessary weights in the final model.\n\nThe integrated loss across all probabilistic instance-model assignments can thus be calculated as the weighted sum of individual losses:\n\n\\begin{equation}\n    L_{integrated0} = \\sum_{k=1}^{w}\\sum_{j=1}^{m} p(M_j|W_k) \\cdot L_{seq(j)}(W_k, M_j),\n    \\label{eq:integrated_loss_0}\n\\end{equation}\nwhere the probability term is from $P_{WM}$ of Eq. \\eqref{eq:word_model_matrix}, which is essentially the law of total probability:\n\n\\begin{equation}\n    p(M_j|W_k) = \\sum_{i=1}^{t} p(T_i|W_k) \\cdot p(M_j|T_i)\n    \\label{eq:word_model_element}\n\\end{equation}\n\n\\subsection{Integrated loss with a base loss coefficient}\n\nWith the integrated loss (Eq. \\eqref{eq:integrated_loss_0}), we can see that in general, a task $T_{big}$ with a bigger probability $p(M_j|T_{big})$ to be assigned to a model $M_j$ will contribute a bigger loss than a task $T_{small}$ with a smaller probability $p(M_j|T_{small})$ to be assigned to the model, encouraging the model to optimize towards a better prediction for $T_{big}$, which then encourages $p(M_j|T_{big})$ to be bigger until it reaches $1$. A similar but opposite process applies to $p(M_j|T_{small})$, which would become smaller until it reaches $0$. As a result, the learned task-model assignment $P_{TM}$ will almost certainly be random and fully depending on the first few iterations due to the positive-feedback loop. We resolve this issue by adding a small positive base loss coefficient, $\\epsilon$:\n\n\\begin{equation}\n    L_{integrated} = \\sum_{k=1}^{w}\\sum_{j=1}^{m} (p(M_j|W_k)+\\epsilon) \\cdot L_{seq(j)}(W_k, M_j).\n    \\label{eq:integrated_loss}\n\\end{equation}\n\nThis ensures that the model not only tries to excel at the tasks assigned to it, but also learns the other tasks at a small but positive rate. The effect of $\\epsilon$ can be quantified from the perspective of training data ratios among different tasks. Assume the original ratio of data from any task is $1$, for any model-task pair, the maximum effective data ratio would be $1 + \\epsilon$, which is achieved when $p$ reaches $1$, and the minimum effective data ratio would be $0 + \\epsilon$, which is achieved when $p$ falls to $0$. The ratio $\\frac{1 + \\epsilon}{\\epsilon}$ can thus be used to measure how biased the model can potentially be trained towards the most vs. least important task. Based on our ablation study (Sec. \\ref{sec:ablation_study}), we set $\\epsilon=0.2$ when training from scratch, $\\epsilon=0.1$ when fine-tuning from pretrained models and $\\epsilon=0$ for the final head-wise fine-tuning.\n\n\\subsection{Grouping loss}\n\nWhile Eq. \\eqref{eq:integrated_loss} makes sure that any model has the potential to learn every task, we also would like to ensure that happens within a certain budget, i.e., given the number of different models (heads) we can support, each model is specialized in at least one task. This ensures we do not waste the modeling capacity of an idle head. Therefore, we introduce the following grouping loss\n\n\\begin{equation}\n    L_{group} = \\sum_{j=1}^{m} L_{group(j)} \\\\\n    = \\sum_{j=1}^{m}\\max(\\mu_j - \\sum_{i=1}^{t}p(M_j|T_i), 0),\n    \\label{eq:grouping_loss}\n\\end{equation}\nwhere $\\mu_j$ is the least number of tasks model $M_j$ is expected to handle. In most experiments, we set $\\mu_j=1$, meaning that if $M_j$ completely takes over at least one task, the grouping loss for $M_j$ would reach the minimum value $0$. Note that the converse does not hold - the grouping loss can reach $0$ even when certain model do not excel in any specific task. However, in practice, as long as the number of tasks is larger than or equal to the number of models, the small penalty of the grouping loss could help us achieve the minimum task assignment goal.\n\n\n\\section{Experimentals}\n\n\\subsection{Datasets}\nOur work leverages a number of public datasets. These sets are summarized in Table~\\ref{tab:datasets}. We next offer a brief description of these sets.\n\n\\minisection{ICDAR 2013 dataset (IC13)}~\\cite{karatzas2013icdar} This is the oldest set used in this work, originally released for the ICDAR 2013 Robust Reading Competition. It offers 229 training and 233 test images of English text. Text locations are given as axis aligned, rectangular bounding boxes with text annotated at a word level. \n\n\\minisection{ICDAR 2015 dataset (IC15)}~\\cite{karatzas2015icdar}\nThis dataset was introduced in ICDAR'15 and offers more images than IC13: 1000 training and 500 test. Images in this set are of scene text in English, appearing at different orientations, where words are annotated using quadrangle bounding boxes. \n\n\n\\minisection{Total Text dataset}~\\cite{ch2017total} This collection offers 1255 training and 300 test, English scene text images. The images reflect a wide range of text orientations and shapes, including curved text examples. To accommodate different shapes, text locations are provided as polygons; recognition labels are given at word level.\n\n\\minisection{ICDAR 2017 RCTW dataset (RCTW17)}~\\cite{shi2017rctw} This set was collected to promote development of OCR methods for in the wild Chinese text. It is partitioned to 8034 and 4229 subsets of training and test images, respectively. \n\n\n\\minisection{ICDAR 2019 MLT dataset (MLT19) and SynthTextMLT}~\\cite{nayef2019icdar2019} was an extension of the ICDAR 2017 MLT dataset (MLT17) \\cite{nayef2017icdar2017} for multilingual text detection, recognition and script identification, which contains 10000 training images, 2000 validation images and 10000 test images in 7 different scripts from 10 languages. The dataset contains multi-oriented scene text annotated by quadrangle boxes. A synthetic dataset (SynthTextMLT)~\\cite{buvsta2018e2e} containing over 250k synthetic data in 7 scripts was also released along with the MLT19 benchmark. Since MLT19 training and validation sets completely covers the training and validation images in MLT17, though the split is a bit different, we only use MLT19 data for training in this paper.\n\n\n\\minisection{ICDAR 2019 ArT dataset (ArT19)}~\\cite{chng2019icdar2019} Contains 5603 training and 4563 test images in both English and Chinese, sourced from Total Text~\\cite{ch2017total} and SCUT-CTW1500~\\cite{yuliang2017detecting}. Released as part of the ICDAR 2019 Robust Reading Competition, the images in this collection depict texts in challenging shapes. Similarly to Total Text, text locations are encoded as polygons. We remove all Total Text test images from this set, ensuring that any training on this set can be applied to other sets without risk of test images influencing models trained on this set.  \n\n\n\\minisection{ICDAR 2019 LSVT dataset (LSVT19)}~\\cite{sun2019lsvt}\nThis is one of the largest data sets used for developing OCR methods: 30000 training and 20000 test images. LSVT images mostly show street views with about 80\\% of them showing Chinese text and the rest examples in English. \n\n\\begin{table}[t]\n    \\scriptsize\n    \\centering\n    \\caption{\\textbf{Datasets used in our experiments.} \\#Train: number of training images. Ratio: the relative sampling ratio when the dataset is used in training. Word \/ Phrase: Annotations given at a word or phrase level. Box type: horizontal, axis aligned (H-Box), arbitrarily rotated (R-Box), quadrangle (Quad), and Polygon. \\#Lang: Number of languages provided. Note that the Total Text dataset is fully covered in ArT19, and we removed the testing set of Total Text from ArT19.}\n    \\resizebox{0.95\\columnwidth}{!}{\n    \\begin{tabular}{lccccc}\n    \\toprule\n    Name & \\#Train & Ratio & Word \/ Phrase & Box type & \\#Lang.\\\\\n    \\midrule\n    ICDAR13~\\cite{karatzas2013icdar}         & 229 & 20 & Word & H-Box & 1 \\\\  \n    ICDAR15~\\cite{karatzas2015icdar}         & 1000 & 20 & Word & Quad & 1 \\\\\n    Total Text~\\cite{ch2017total} & 1255 & 50 & Word & Polygon & 1 \\\\\nRCTW17~\\cite{shi2017rctw}          & 8034 & 20 & Phrase & R-Box & 2 \\\\    \nMLT19~\\cite{nayef2019icdar2019}           & 10000 & 100 & Word & Quad & 10 \\\\\nSynthTextMLT~\\cite{buvsta2018e2e}    & 252599 & 1 & Word & R-Box & 7 \\\\\nArT19~\\cite{chng2019icdar2019}           & 5303 & 50 & Word & Polygon & 2 \\\\\nLSVT19~\\cite{sun2019lsvt}          & 30000\t& 20 & Phrase & Polygon & 2 \\\\\n    \\bottomrule\n    \\end{tabular}\n    }\n    \\label{tab:datasets}\n\\end{table}\n\n\\subsection{Model training}\nWe base our implementation on the Multiplexer codebase\\footnote{\\url{https:\/\/github.com\/facebookresearch\/MultiplexedOCR}}.\nFor fair comparison, we adopt the same segmentation-based detection and ROI mask feature extraction modules as \\cite{huang2021multiplexed}, and freeze the pretrained weights of these layers throughout training. For language classification, \\cite{huang2021multiplexed} uses 8 classes including Arabic, Bengali, Chinese, Hindi, Japanese, Korean, Latin, and Symbol, but in our experiment we only use 7 classes by discarding the Symbol class, as it does not have any dedicated dataset and the number of the samples is too small to make a difference.\n\nTo expedite the training, we first combine every dataset to train a single recognition head with hidden size 256 and embed size of 200 covering all datasets using the ratios specified in Table \\ref{tab:datasets} for 40k iterations. We then use this weight as a universal pretrained weights for the second stage of training.\n\nNext, we perform a series of experiments that jointly train the grouping module and the recognition heads, each restricting the number of recognition heads to $m$ ($2 \\leq m \\leq 7$). For each $m$, we launch three training jobs with different random seeds. Each of the training jobs runs for 20k iterations on the MLT19 training datasets only to reduce the potential data imbalance when including the other training set. We record and summarize the final grouping results in Table \\ref{tab:task_grouping_result}, which we will discuss in \\ref{sec:results:grouping}.\n\nFinally, based on the grouping result, we fine-tune each recognition head with only the datasets within the assigned group corresponding to the head. At this stage the grouping is essentially frozen and does not change any more. We can prune the output layer of the decoder so that the characters not belonging to the group are removed, to reduce the parameter number for the final model.\n\n\n\\subsection{Task grouping results}\n\\label{sec:results:grouping}\n\nTable \\ref{tab:task_grouping_result} shows the aggregation of grouping results from $18$ task grouping experiments with $2$ to $7$ recognition heads, each repeated for $3$ times. All task assignments stabilize after about 10000 iterations.\n\nThe top 14 groups are ordered first by the number of occurrences and then by the first occurrence, \\emph{i.e}\\onedot the minimum number of recognition heads when the group first occurs. All exclusive task-model assignments (one head focusing on one task) occur at least twice, showing the effectiveness of having a dedicated model for each task. Chinese ending up as an individual task occurs in 50\\% of the cases, which is expected given its high character number and the datasets, except that it's grouped together with Japanese, which shares many characters with it, only once. On the other hand, Hindi seems to be suitable to be grouped with many different languages rather than being trained by itself. \n\nSurprisingly, the most frequent task group that has more than one task is Arabic+Korean, which occurs 5 times. This suggests that there are inherent characteristics shared either by these two scripts, or by the examples in the MLT19 dataset itself, that boost the performance for each other. Another unusual cluster is the combination of 5 tasks, Bengali+Chinese+Hindi+Japanese+Latin, which is the only grouping with more than 2 tasks that occurs more than once.\nWe note however that the scattering of the grouping results shows that there can be many local optima for this specific scenario of $7$ distinctive scripts. We shall expect higher frequencies of the same grouping results if certain tasks share greater similarity, and we will leave that as one of our future work.\nWe additionally find it interesting that despite a grouping loss to encourage each head to take on at least one task, we observe that some recognition heads might not be assigned with any task in the end when there are $6$ or $7$ tasks. This means for certain combinations of tasks, training them together could outperform training them separately, even if there is spare resource for a new head.\n\n\\begin{table}[t]\n    \\scriptsize\n    \\centering\n    \\caption{\\textbf{Task grouping result.} Task combinations that end up being grouped together. The 2nd to the 5th columns indicate task names in the final grouping, the number of tasks in the group, the number of occurrences in the 18 experiments, and the minimum number of recognition heads when the combination first occurs. }\n    \\begin{tabularx}{0.98\\linewidth}{c|X|c|c|c}\n    \\toprule\n    Rank & Group & \\#Tasks within group & \\#Occurrences & \\#Heads at first occurrence \\\\ \n    \\midrule\n    1 & Chinese (C) & 1 & 9 & 4 \\\\\n    2 & Latin (L) & 1 & 7 & 5 \\\\\n    3 & Arabic (A) & 1 & 6 & 3 \\\\\n    3 & Korean (K) & 1 & 6 & 3 \\\\\n    5 & Arabic+Korean & 2 & 5 & 2 \\\\\n    6 & Bengali (B) & 1 & 5 & 5 \\\\\n    7 & Japanese (J) & 1 & 4 & 5 \\\\\n    8 & B+C+H+J+L & 5 & 2 & 2 \\\\\n    9 & Hindi+Japanese & 2 & 3 & 3 \\\\\n    10 & Japanese+Latin & 2 & 2 & 4 \\\\\n    11 & Arabic+Hindi & 2 & 2 & 5 \\\\\n    11 & Bengali+Japanese & 2 & 2 & 5 \\\\\n    11 & Hindi+Latin & 2 & 2 & 5 \\\\\n    14 & Hindi (H) & 1 & 2 & 6 \\\\\n    \\hline\n    15 & A+K+L & 3 & 1 & 2 \\\\\n    15 & H+J+K & 3 & 1 & 2 \\\\\n    15 & A+B+C+L & 4 & 1 & 2 \\\\\n    15 & B+C+H+J & 4 & 1 & 2 \\\\\n    15 & Chinese+Hindi & 2 & 1 & 3 \\\\\n    15 & Korean+Latin & 2 & 1 & 3 \\\\\n    15 & A+B+J & 3 & 1 & 3 \\\\\n    15 & B+C+L & 3 & 1 & 3 \\\\\n    15 & Arabic+Bengali & 2 & 1 & 4 \\\\\n    15 & Bengali+Hindi & 2 & 1 & 4 \\\\\n    15 & Chinese+Latin & 2 & 1 & 4 \\\\\n    15 & A+B+H & 3 & 1 & 4 \\\\\n    15 & Japanese+Korean & 2 & 1 & 5 \\\\\n    15 & B+C+K & 3 & 1 & 6 \\\\\n    15 & Hindi+Korean & 2 & 1 & 7 \\\\\n    15 & Chinese+Japanese & 2 & 1 & 7 \\\\\n    \\bottomrule\n    \\end{tabularx}\n    \\label{tab:task_grouping_result}\n\\end{table}\n\n\n\\subsection{Ablation study}\n\\label{sec:ablation_study}\n\n\\minisection{Base integrated loss coefficient.} We train the task grouping network with different base integrated loss coefficient $\\epsilon$ defined in Eq. \\eqref{eq:integrated_loss} on MLT19 training set. The network contains 5 recognition heads that are initialized with the same pretrained weights. We record the number of task assignment changes in the first 3000 iterations. From Table \\ref{tab:base_loss_weight} we can see that, when $\\epsilon=0.0$, there's only 1 assignment change since the model does not have much chance to learn the unassigned tasks; interestingly, when $\\epsilon$ is too big ($0.3\/0.4$), there are also fewer changes happening, possibly because there is not much diversity across the models and everything moves in the same direction. The maximum number of assignment changes happen when $\\epsilon$ is $0.2$ or $0.1$. Therefore, in most of our experiments we use $0.2$ for early training and $0.1$ for fine-tuning.\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.98\\linewidth]{figures\/qual_img_grping_3.png}\n    \\caption{\\textbf{Qualitative results on MLT19 test set~\\cite{nayef2019icdar2019}.} The predicted transcription is rendered with green background, along with the detection confidence, language and the assigned group. The model has $5$ heads (groups): group 1 - Arabic (ar) and Hindi (hi), group 2 - Bengali (bn) and Japanese (ja), group 3 - Chinese (zh), group 4 - Latin (la), group 5 - Korean (ko). See Sec.~\\ref{sec:results:e2e} for more details.}\n    \\label{fig:qualitative}\n\\end{figure*}\n\n\n\\begin{table}[t]\n    \\scriptsize\n    \\centering\n    \\caption{\\textbf{Ablation study for base integrated loss coefficient $\\epsilon$.} }\n    \\begin{tabularx}{0.98\\linewidth}{X*{5}{p{0.1\\linewidth}}}\n    \\toprule\n    Base integrated loss coefficient $\\epsilon$ & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 \\\\ \n    \\midrule\n    Assignment changes within 3k iters & 1 & 3 & 5 & 2 & 2 \\\\\n    \\bottomrule\n    \\end{tabularx}\n    \\label{tab:base_loss_weight}\n\\end{table}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth, scale=0.8,clip,trim=0mm 2mm 0mm 2mm]{figures\/error_res_grping_scl_2.png}\n    \\caption{\\textbf{Error analysis on MLT19.} Detection errors are represented in red outline and recognition errors in purple. See Sec.~\\ref{sec:results:e2e}.}\n    \\label{fig:failures}\n\\end{figure}    \n\n\n\\subsection{Task assignment on models with different hyper-parameters}\n\nIn this section, we perform an interesting experiment that showcases how our design can help assign different tasks to models with different hyper-parameters based on the potential difficulty and the available data. We set the number of models (recognition heads) to be equal to the number of tasks, but set the key hyper-parameters of the models, embed size and hidden size, to be different from each other. We train the overall model on the weighted combination of all datasets listed above, and Table \\ref{tab:parameter_number_comparison} shows the assigned task corresponding to each of the models. We can clearly see the correlation between the number of parameters versus the number of characters in the corresponding character set, with the exception of Latin. This illustrated that in general, when the number of characters grow, the heavier models will outperform lighter models in the long term; however, since Latin words are dominating in all the datasets including many difficult cases like curved text, the task grouping framework learns to spend the heaviest model on it to boost the overall performance.\n\n\n\\begin{table}[t]\n    \\scriptsize\n    \\centering\n    \\caption{\\textbf{Task assignment result for models with different major hyper-parameters.} Each model supports all characters in the beginning so the total number of parameters for each head is high, but they can be pruned when the task assignment stabilizes.}\n    \\begin{tabularx}{0.98\\linewidth}{c|c|c|c|c|c}\n    \\toprule\n    Embed Size & Hidden Size & Parameter Number & Assigned Task & Charset Size & Final Parameters\\\\\n    \\hline\n    100 & 224 & 4.05M & Arabic & 80 & 1.15M \\\\\n    150 & 224 & 4.51M & Bengali & 110 & 1.18M \\\\\n    200 & 224 & 4.98M & Japanese & 2300 & 2.13M \\\\\n    100 & 256 & 4.59M & Hindi & 110 & 1.42M \\\\\n    150 & 256 & 5.06M & Korean & 1500 & 2.00M \\\\\n    200 & 256 & 5.52M & Chinese & 5200 & 3.78M \\\\\n    250 & 256 & 5.98M & Latin & 250 & 1.54M \\\\\n    \\bottomrule\n    \\end{tabularx}\n    \\label{tab:parameter_number_comparison}\n\\end{table}\n\n\n\\subsection{E2E text recognition}\\label{sec:results:e2e}\n\nTable \\ref{tab:mlt19_task4} shows the results on MLT19 \\cite{nayef2019icdar2019} end-to-end multilingual recognition benchmark. Fig.~\\ref{fig:qualitative} additionally provides qualitative examples of these multilingual. We find that using varying numbers of grouped heads can perform similarly to (and in some cases, better than) the multiplexed approach of a separate recognition head per language~\\cite{huang2021multiplexed}.\nThis is an interesting result, as it means we can significantly cut down on the computational cost and model size with little impact or even some gains to the performance.\nNotably, we also find that increasing the number of heads from a single shared head (Mask TextSpotter V3 \\cite{liao2020mask}) to even just two grouped heads leads to a significant increase in F1-score. \n\nWe provide qualitative failure cases in Fig.~\\ref{fig:failures}. While detection errors could be attributed to arbitrary text shape, blurred text, glossy surfaces and rare fonts; recognition errors could be attributed to text ordering, text resolution and challenging scripts. \n\n\\begin{table}[t]\n    \\scriptsize\n    \\centering\n    \\caption{\\textbf{End-to-end recognition results on MLT19.} Note that there are two versions results of CRAFTS, one from the official MLT19 website and one from paper \\cite{baek2020character}. Importantly, CRAFTS has a ResNet-based feature extraction which is much bigger than the one with $5$-Convs used in our experiments.}\n    \\begin{tabularx}{0.9\\linewidth}{lX*{2}X}\n    \\toprule\n    Method & F & P & R \\\\ \n    \\midrule\n    E2E-MLT \\cite{buvsta2018e2e} & 26.5 & 37.4 & 20.5 \\\\\n    RRPN+CLTDR \\cite{ma2018arbitrary} & 33.8 & 38.6 & 30.1 \\\\\n    CRAFTS \\cite{baek2020character} & 51.7 & 65.7 & 42.7 \\\\\n    CRAFTS (paper) \\cite{baek2020character} & \\textbf{58.2} & \\textbf{72.9} & \\textbf{48.5} \\\\\n    \\hline\n    Mask TextSpotter V3 (1 head) \\cite{liao2020mask} & 39.7 & \\textbf{71.8} & 27.4 \\\\ \n    Multiplexed TextSpotter (8 heads) \\cite{huang2021multiplexed} & 48.2 & 68.0 & 37.3   \\\\\n    \\hline\n    Grouped (2 heads) & 45.5 & 67.7 & 34.3 \\\\\n    Grouped (3 heads) & 47.1 & 67.0 & 36.3 \\\\\n    Grouped (4 heads) & 47.9 & 66.7 & 37.4 \\\\\n    Grouped (5 heads) & \\textbf{48.5} & 67.7 & \\textbf{37.8} \\\\\n    Grouped (6 heads) & 48.3 & 67.8 & 37.5 \\\\\n    Grouped (7 heads) & 48.2 & 68.0 & 37.3 \\\\\n    \\bottomrule\n    \\end{tabularx}\n    \\label{tab:mlt19_task4}\n\\end{table}\n\n\\section{Conclusions}\nText is one of the most ubiquitous visual object classes in real-world scenes, making understanding it practical and critically important. Processing multiple languages, however, requires substantial resources, to accurately recognize the subtleties of appearances variations of different scripts and different intra-script characters. This ability was, therefore, previously accomplished by specializing separate network heads to specific languages. We, instead, are the first to propose automatically grouping different languages together, in the same recognition heads. Our dynamic, and differentiable task shifting approach automatically routes tasks to different heads while the network trains, optimizing for the best, bottom line accuracy across all languages. Extensive tests show our method to not only achieve SotA accuracy, but to do so with fewer recognition heads and hyperparameters, consequently making is a practical design choice for real-world OCR systems. \n\n\\minisection{Future work} Our work leaves several natural follow-up directions. One interesting question relates to the scalability of our approach: How many multitask heads, for example, would be required to effectively learn hundreds of languages? Another intriguing direction is extending our multitask learning and task grouping to include neural architecture search as part of its design. Such a solution should allow growing heads with different architectures for different languages to account for, \\emph{e.g}\\onedot, harder vs. easier languages. Finally, another potential extension could be continual language learning~\\cite{Parisi2019}: adding more languages as relevant training data becomes available, without retraining or regrouping existing languages. Alternative grouping approaches based on Bayesian nonparametric approaches like the Chinese Restaurant Process~\\cite{Aldous1985} or Indian Buffet Process~\\cite{Ghahramani2006,mehta2021continual} may be natural ways to perform groupings in such settings.\n\n\n\\clearpage\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nBayesian inference often requires estimating expectations with respect to non-Gaussian distributions. To solve this problem, particularly in high dimensions, one frequently resorts to sampling methods. A commonly used class of sampling methods is based on the Langevin dynamics (LD), which uses the gradient of the log-target density to specify a stochastic differential equation (SDE) whose invariant distribution is the target (e.g., posterior) distribution of interest. Long term averages over a single trajectory of the SDE can be then used to estimate expectations of interest by appealing to the ergodicity of the stochastic process. Other LD-based approaches that reduce the mean squared error (MSE) of such estimators include the Metropolis-adjusted Langevin algorithm (MALA) \\cite{roberts1996exponential,girolami2011riemann}, the stochastic gradient Langevin dynamics (SGLD) \\cite{welling2011bayesian}, and their variants.\n\nIt is also known that certain perturbations to the LD can accelerate convergence of the dynamics to the stationary distribution. In \\cite{rey2015irreversible} the authors show that suitable reversible and irreversible perturbations to diffusion processes can decrease the spectral gap of the generator, as well as increase the large deviations rate function and decrease the asymptotic variance of the estimators. One widely celebrated choice of \\emph{reversible} perturbation is the Riemannian manifold Langevin dynamics \\cite{girolami2011riemann}, in which one defines a Riemannian metric to alter the way distances and gradients are computed. The use of \\emph{irreversible} perturbations to accelerate convergence has also been well studied in a variety of contexts and general settings \\cite{rey2015irreversible,hwang2005accelerating,rey2015variance,rey2016improving}; see also \\cite{franke2010behavior,hwang1993accelerating,Bierkens,DiaconisHolmesNeal2010} and for linear systems, \\cite{lelievre2013optimal}. Existing literature shows that augmenting the drift of the LD with a vector field that is {orthogonal} to the gradient of the log-target density will leave the invariant measure unchanged while decreasing the spectral gap. A convenient choice is simply to add the vector field induced by a skew-symmetric matrix applied to the gradient of the log posterior.\n\nAt the same time, traditional sampling methods for Bayesian inference are often intractable for extremely large datasets. While Langevin dynamics-based sampling methods only require access to the unnormalized posterior density, they need many evaluations of this unnormalized density and its gradient. When the dataset is extremely large, each evaluation of the density may be computationally intractable, as it requires the evaluation of the likelihood over the entire dataset. In the past decade the \\emph{stochastic gradient} Langevin dynamics (SGLD) has been introduced and analyzed \\cite{welling2011bayesian,teh2016consistency} to address the problem posed by large datasets. Rather than evaluating the likelihood over the entire dataset, SGLD subsamples a portion of the data (either with or without replacement) and uses the likelihood evaluated at the sampled data to estimate the true likelihood. The resulting chain can then be used to estimate ergodic averages.\n\nIn this paper we present a \\emph{state-dependent irreversible} perturbation of Riemannian manifold Langevin dynamics that is informed by the \\emph{geometry} of the manifold. This departs from existing literature, as the vector field of the resulting perturbation is \\emph{not} orthogonal to the original drift term. This geometry-informed irreversible perturbation accelerates convergence and, if desired, can be used in combination with the SGLD algorithm to exploit the computational savings of a stochastic gradient.\n\nWe demonstrate this approach on a variety of examples: a simple anisotropic Gaussian target, a posterior on the mean and variance parameters of a normal distribution, Bayesian logistic regression, and Bayesian independent component analysis (ICA). Generally, we observe that the geometry-informed irreversible perturbation improves the convergence rate of LD compared to a standard irreversible perturbation. The improvement tends to be more pronounced as the target distribution deviates from Gaussianity. Our numerical studies also show that introducing irreversibility can reduce the MSE of the resulting long-term average estimator, mainly by reducing variance. In many cases this reduction can be significant, e.g., 1--2 orders of magnitude.\n\nOne must, however, also take the effects of discretization into account. In the continuous-time setting, it is known theoretically that irreversible perturbations can at worst only leave the spectral gap fixed. In borderline cases, though---i.e., in cases where the continuous-time theoretical improvement is nearly zero---after accounting for discretization, stiffness can actually cause the resulting estimator to perform worse than if no irreversibility were applied at all. Indeed, we will describe in Appendix \\ref{sec:appendix} an illustrative Gaussian example in which the standard Langevin algorithm performs better than the algorithm with the standard irreversible perturbation. That is, an example in which additional irreversibility leads to increased bias and variance of the long term average estimator (see Remark \\ref{R:IrrevNotImprovement} for a theoretical explanation).\nAlong similar lines, the idea of applying irreversible perturbations to SGLD has recently been studied in the context of nonconvex stochastic optimization \\cite{hu2020non}. The authors also note that while irreversibility increases the rate of convergence, it increases the discretization error and amplifies the variance of the gradient, compared to a non-perturbed system with the same step size; see also \\cite{BrosseMoulinesDurmus2018} for a related discussion on the relation of SGLD to SGD and convergence properties. This reflects the increased stiffness of irreversible SGLD relative to standard SGLD.\n\nThe rest of the paper is organized as follows. In Section \\ref{sec:background} we review reversible and irreversible perturbations of the overdamped Langevin dynamics that may improve the efficiency of sampling from equilibrium. Then, in Section~\\ref{sec:giirr}, we present our new geometry-informed irreversible perturbation. In Section \\ref{sec:num}  we present simulation studies that demonstrate the good performance of this geometric perturbation, relative to a variety of other standard reversible and irreversible choices. In several of these examples, we also demonstrate the use of stochastic gradients.\nSection~\\ref{S:Conclusions} summarizes our results and outlines directions for future work. Appendix \\ref{sec:appendix} details the simple Gaussian example showing that in ``borderline'' cases---i.e., when continuous-time analysis does not predict improvements from irreversible perturbations---the stiffness created by an irreversible perturbation can, after discretization, lead to poorer performance than the unperturbed case.\n\n\n\\section{Improving the performance of Langevin samplers}\n\\label{sec:background}\nWe begin by recalling some relevant background on Langevin samplers, Riemmanian manifold Langevin dynamics, perturbations of Langevin dynamics, and the stochastic gradient Langevin dynamics algorithm. Let $f(\\theta)$ be a test function on state space $E \\subset \\mathbb{R}^d$ and let $\\pi(\\theta)$ be some unnormalized target density on $E$. In our experiments, $\\pi(\\theta)$ arises as a posterior density of the form $\\pi(\\theta) \\propto L(\\theta;X) \\pi_0(\\theta)$, where $L(\\theta; X)$ is the likelihood model, $X$ are the data, and $\\pi_0(\\theta)$ is the prior density. Define $\\{\\theta(t)\\}$ as a Langevin process that has invariant density $\\pi(\\theta)$:\n\\begin{align}\n\t\\text{d} \\theta(t) = \\beta\\nabla \\log \\pi(\\theta(t)) \\text{d} t + \\sqrt{2\\beta}\\text{d} W(t), \\label{eq:langevin}\n\\end{align}\nwhere $\\beta>0$ denotes the temperature, $W(t)$ is a standard Brownian motion in $\\mathbb{R}^d$, and the initial condition may be arbitrary. By ergodicity, we may compute expectations with respect to the posterior by the long term average of $f(\\theta)$ over a single trajectory:\n\\begin{align}\n\t\\mathbb{E}_\\pi[f(\\theta)] =  \\int _E f(\\theta) \\pi(\\theta) \\text{d} \\theta = \\lim_{t\\to\\infty} \\frac{1}{T} \\int_0^T f(\\theta(t)) \\text{d} t.  \\label{eq:ergodicavg}\n\\end{align}\nFor practical computations, we must approximate \\eqref{eq:ergodicavg} by discretizing the Langevin dynamics and choosing a large but finite $T$. Applying the Euler-Maruyama method to \\eqref{eq:langevin} with step size $h$ yields the following recurrence relation,\n\\begin{align}\n \t\\theta_{k+1} = \\theta_k + h\\beta\\nabla \\log \\pi(\\theta_t) \\text{d} t + \\sqrt{2\\beta h} \\xi_{k+1}\n \\end{align}\n where $\\xi_{k}$ are independent standard normal random variables. The total number of steps is equal to $K = T\/h$. The resulting estimator for \\eqref{eq:ergodicavg} is\n\\begin{align}\n\t\\mathbb{E}_\\pi[f(\\theta)] \\approx \\frac{1}{K} \\sum_{k = 0}^{K-1} f(\\theta_k).\n\\end{align}\nThis estimator is the \\emph{unadjusted Langevin algorithm} (ULA), which has found renewed interest in the context of high-dimensional machine learning problems \\cite{durmus2019high}. Discretization and truncation, however, introduce bias into the estimator. Moreover, there are noted examples in which the continuous-time process and the discretized version do not have the same invariant distribution  no matter the choice of the fixed, but nonzero, discretization step $h$; see \\cite{Ganguly} for a related discussion. Certain Markov chain Monte Carlo (MCMC) methods such as MALA circumvent these issues by using the dynamics to propose new points, but accepting or rejecting them according to some rule so that the resulting discrete-time Markov chain has the target distribution as its invariant distribution \\cite{roberts1996exponential,girolami2011riemann}.\n\nMany different SDEs can have the same invariant distribution. Therefore, there has been much study into how the standard Langevin dynamics of some target distribution can be altered to increase its rate of convergence. Some examples of this can be found in the work of \\cite{hwang2005accelerating,rey2016improving} and others. The standard Langevin dynamics is a reversible Markov process, meaning that the process satisfies detailed balance. The work of \\cite{rey2016improving} studies, in general terms, how reversible and irreversible perturbations to reversible processes decrease the spectral gap, increase the large deviations rate function, and decrease the asymptotic variance. Yet how to \\emph{choose} such perturbations to most efficiently accelerate convergence is yet to be thoroughly studied in settings beyond linear diffusion processes \\cite{lelievre2013optimal}. Also, with the exception of a few examples---see for instance \\cite{DuncanPavliotisZygalakis2017,lu2018analysis}---these perturbations have mainly been studied in the continuous-time setting.\n\n\\subsection{Reversible perturbations and Riemannian manifold Langevin dynamics}\nWe only review relevant aspects of reversible perturbations and RMLD in this section. For a detailed review of RMLD and its related Monte Carlo methods, we refer the reader to \\cite{girolami2011riemann,livingstone2014information,xifara2014langevin}.  Let $\\textbf{B}(\\theta)$ be a $d\\times d$ symmetric positive definite matrix. A reversible perturbation on LD \\eqref{eq:langevin} is an SDE with multiplicative noise:\n\\begin{align}\n\t\\text{d} \\theta(t) = \\beta\\left[\\mathbf{B}(\\theta)\\nabla \\log \\pi(\\theta(t))  + \\nabla \\cdot \\mathbf{B}(\\theta) \\right] \\, \\text{d} t + \\sqrt{2\\beta\\mathbf{B}(\\theta)} \\text{d} W(t).\n\t\\label{eq:revperturb}\n\\end{align}\nHere, the $i$-th component of $\\nabla \\cdot \\mathbf{B}(\\theta)$ is $\\sum_{j = 1}^d \\partial_{\\theta_{j}} \\mathbf{B}_{ij}(\\theta)$. This is equivalent to Langevin dynamics defined on a Riemannian manifold, where the metric is $\\mathbf{G}(\\theta) = \\mathbf{B}(\\theta)^{-1}$ \\cite{xifara2014langevin}. A straightforward calculation shows that \\eqref{eq:revperturb} with $\\mathbf{B}(\\theta)$ being any symmetric positive-definite matrix admits the same invariant distribution, $\\pi$. The improved rate of convergence depends on the choice of the underlying metric. The work of \\cite{girolami2011riemann} argues that choosing the expected Fisher information matrix plus the Hessian of the log-prior to be the metric improves the performance of the resulting manifold MALA method. Meanwhile, \\cite{rey2016improving} shows that under certain regularity conditions, if $\\mathbf{B}(\\theta)$ is chosen such that $\\mathbf{B}(\\theta) - \\mathbf{I}$ is positive definite, then the resulting estimator is expected to have improved performance in terms of the asymptotic variance, the spectral gap, and the large deviations rate function.\n\n\\subsection{Irreversible perturbations}\nConsider the following Langevin dynamics\n\\begin{align}\n\t\\text{d} \\theta(t) = \\left[\\beta\\nabla \\log\\pi(\\theta(t)) + \\gamma(\\theta(t)) \\right]\\text{d} t +\\sqrt{2\\beta} \\text{d} W(t).\n\\end{align}\nWhen $\\gamma(\\theta) \\equiv 0$, the process is reversible and has $\\pi(\\theta)$ as its invariant distribution. If $\\gamma\\neq 0$, then the resulting process will, in general, be time-irreversible unless $\\gamma(\\theta)$ can be written as a multiple of $\\nabla \\log\\pi(\\theta)$; see for example \\cite{pavliotis2014stochastic}. However, an irreversible perturbation can still preserve the invariant distribution of the system. By considering the Fokker-Planck equation, one can show that if $\\gamma(\\theta)$ is chosen such that $\\nabla \\cdot \\left( \\gamma\\pi \\right) = 0$, then $\\pi$ will still be the invariant distribution. A frequently used choice in the literature is $\\gamma(\\theta) = \\mathbf{J} \\nabla \\log \\pi(\\theta)$, where $\\mathbf{J}$ is a constant skew-symmetric matrix, i.e., $\\mathbf{J} = -\\mathbf{J}^T$. The computational advantage of this choice is clear since only one additional matrix-vector multiply is needed to implement this choice. The optimal choice of irreversible perturbation to linear systems that accelerated convergence fastest was completely analyzed in \\cite{lelievre2013optimal}.\n\nThe advantages of using irreversible perturbations is widely noted. The main result of \\cite{hwang2005accelerating} is that under certain conditions, the spectral gap, i.e., the difference between the leading two eigenvalues of the generator of the Markov semigroup, increases when $\\gamma \\neq 0$. In \\cite{rey2015irreversible,rey2015variance,rey2016improving}, the large deviations rate function is introduced as a measure of performance in the context of sampling from the equilibrium, and upon connecting it to the asymptotic variance of the long term average estimator, it is proven that adding an appropriate perturbation $\\gamma$ not only increases the large deviations rate function but also decreases the asymptotic variance of the estimator. The use of irreversible proposals in the MALA was studied in \\cite{ottobre2019optimal}.\n\n\n\\subsection{Irreversible perturbations for RMLD}\n\\label{sec:giirr}\n\nIn this section, we will introduce our novel geometry-informed irreversible perturbation to Langevin dynamics. Suppose that we are given a diffusion process as in \\eqref{eq:revperturb}, and we want to study how to choose an irreversible perturbation that leaves the invariant distribution fixed. Indeed, our previous choice of irreversible perturbation remains valid for this system, that is, adding $\\gamma(\\theta) = \\mathbf{J} \\nabla \\log \\pi(\\theta)$ for a constant skew-symmetric matrix $\\mathbf{J}$ to the drift term of \\eqref{eq:revperturb} will preserve the invariant density. This choice yields the following SDE:\n\\begin{align}\n\t\\text{d} \\theta(t) = \\left[(\\beta\\mathbf{B}(\\theta(t))+\\mathbf{J}) \\nabla \\log \\pi(\\theta(t)) + \\beta\\nabla \\cdot \\mathbf{B}(\\theta(t)) \\right] \\text{d} t + \\sqrt{2\\beta\\mathbf{B}(\\theta)} \\text{d} W(t)\n\t\\label{eq:RMIrr}\n\\end{align}\nWe refer to this system as Riemannian manifold Langevin with an additive irreversible perturbation (\\texttt{RMIrr}). This choice, however, does not take into account the relevant features that the reversible perturbation may provide when constructing an irreversible perturbation.\n\nThe reversible perturbation leads to a positive definite matrix (a metric, in the terminology of Riemannian geometry) that is state-dependent. In contrast, the skew-symmetric matrix $\\mathbf{J}$ is fixed in the irreversible perturbation. The skew-symmetric matrix need not be constant, however, as an irreversible perturbation $\\gamma(\\theta)$ only needs to satisfy $\\nabla \\cdot (\\gamma(\\theta) \\pi(\\theta) )= 0$. In fact, if $\\gamma(\\theta) = \\mathbf{C}(\\theta) \\nabla \\log \\pi(\\theta) + \\nabla \\cdot \\mathbf{C}(\\theta)$ for $\\mathbf{C}(\\theta) = -\\mathbf{C}(\\theta)^T$, then this irreversible perturbation will also leave the invariant density intact. Noting that $\\mathbf{C}_{ii}(\\theta) = 0$ and that $\\mathbf{C}_{ij} = -\\mathbf{C}_{ji}$, observe that\n\\begin{align*}\n\t\\nabla \\cdot ( \\gamma(\\theta) \\pi(\\theta)) &= \\nabla \\cdot ( \\mathbf{C}(\\theta) \\nabla \\pi(\\theta) + (\\nabla \\cdot \\mathbf{C}(\\theta) ) \\pi(\\theta) ) \\\\\n\t = &\\sum_{i,j = 1}^d \\frac{\\partial \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_i} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_j} + \\mathbf{C}_{ij}(\\theta) \\frac{\\partial^2 \\pi(\\theta)}{\\partial \\theta_i \\partial \\theta_j} + \\frac{\\partial^2 \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_i\\partial \\theta_j} \\pi(\\theta) + \\frac{\\partial \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_j} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_i} \\\\\n\t= &\\sum_{i>j, i = 1}^d \\frac{\\partial \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_i} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_j} + \\mathbf{C}_{ij}(\\theta) \\frac{\\partial^2 \\pi(\\theta)}{\\partial \\theta_i \\partial \\theta_j} + \\frac{\\partial^2 \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_i\\partial \\theta_j} \\pi(\\theta) + \\frac{\\partial \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_j} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_j} \\\\\n\t & + \\frac{\\partial \\mathbf{C}_{ji}(\\theta)}{\\partial \\theta_j} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_i} + \\mathbf{C}_{ji}(\\theta) \\frac{\\partial^2 \\pi(\\theta)}{\\partial \\theta_j\\partial \\theta_i} + \\frac{\\partial^2 \\mathbf{C}_{ji}(\\theta)}{\\partial \\theta_j\\partial \\theta_i} \\pi(\\theta) + \\frac{\\partial \\mathbf{C}_{ji}(\\theta)}{\\partial \\theta_i} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_i} \\\\\n\t = &\\sum_{i>j, i = 1}^d \\frac{\\partial \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_i} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_j} + \\mathbf{C}_{ij}(\\theta) \\frac{\\partial^2 \\pi(\\theta)}{\\partial \\theta_i \\partial \\theta_j} + \\frac{\\partial^2 \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_i\\partial \\theta_j} \\pi(\\theta) + \\frac{\\partial \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_j} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_i} \\\\\n\t & - \\frac{\\partial \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_j} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_i} - \\mathbf{C}_{ij}(\\theta) \\frac{\\partial^2 \\pi(\\theta)}{\\partial \\theta_j\\partial \\theta_i} - \\frac{\\partial^2 \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_j\\partial \\theta_i} \\pi(\\theta) - \\frac{\\partial \\mathbf{C}_{ij}(\\theta)}{\\partial \\theta_i} \\frac{\\partial \\pi(\\theta)}{\\partial \\theta_j}\\\\\n\t = &\\, \\,  0.\n\\end{align*}\n\nWe seek an irreversible perturbation that takes the reversible perturbation into account, with the possibility that $\\mathbf{C}(\\theta)$ is not a constant matrix, and investigate if it leads to any performance improvements of the long term average estimator. Note that in the literature, the above condition $\\nabla \\cdot (\\gamma\\pi)=0$ is typically rewritten into the following sufficient conditions: $\\nabla \\cdot \\gamma(\\theta) = 0$ and $\\gamma(\\theta) \\cdot \\nabla \\pi(\\theta) = 0$ \\cite{rey2016improving}.   One can check, however, that when $\\mathbf{C}$ is not constant, these conditions are not met, yet $\\gamma(\\theta)$ is still a valid irreversible perturbation. A simple choice of $\\mathbf{C}(\\theta)$ that incorporates $\\mathbf{B}(\\theta)$ is\n\\begin{align}\n\t\\mathbf{C}(\\theta) = \\frac{1}{2}\\mathbf{J} \\mathbf{B}(\\theta) + \\frac{1}{2}\\mathbf{B}(\\theta) \\mathbf{J} \\, ,\n\t\\label{eq:Girrp}\n\\end{align}\nwhere $\\mathbf{J}$ is a constant skew-symmetric matrix. The $\\frac{1}{2}$ factor is introduced so that if $\\mathbf{B}(\\theta) = \\mathbf{I}$, i.e., if there is no reversible perturbation, then this perturbation reverts to the standard irreversible perturbation (\\texttt{Irr}). We arrive at the following system:\n\\begin{align}\n\t\\text{d} \\theta(t) = \\left[ (\\beta\\mathbf{B}(\\theta(t)) + \\mathbf{C}(\\theta(t))) \\nabla \\log \\pi(\\theta(t)) + \\nabla \\cdot (\\beta\\mathbf{B}(\\theta(t)) + \\mathbf{C}(\\theta(t)) \\right] \\text{d} t + \\sqrt{2\\beta\\mathbf{B}(\\theta(t))} \\text{d} W(t).\n\\end{align}\n\nWe call this choice of perturbation the \\emph{geometry-informed irreversible perturbation} (\\texttt{GiIrr}).  Indeed, while there are infinitely many valid choices for $\\mathbf{C}(\\theta)$, we will investigate the choice in \\eqref{eq:Girrp} in the numerical examples. Since we will have already explicitly constructed $\\mathbf{B}(\\theta)$ and $\\mathbf{J}$ for the other systems, the additional computational cost of computing their product will be marginal. Furthermore, as mentioned earlier, this choice reduces to \\texttt{Irr} when $\\mathbf{B}(\\theta) = \\mathbf{I}$.\n\nOne may wonder when does \\texttt{GiIrr} result in improved performance over standard irreversible perturbations such as in Equation \\ref{eq:RMIrr}. Based on the numerical results and intuition, we will argue that \\texttt{GiIrr} results in better performance if the underlying reversible perturbation already improves the sampling. As we mentioned earlier, the choice of \\texttt{GiIrr} that is made in this paper is not unique, and a further investigation of its theoretical properties is left for future work; see also the discussion in the Section~\\ref{S:Conclusions}. The goal of this paper is to present this new class of irreversible perturbations and investigate it numerically in a number of representative computational studies.\n\n\n\n\\subsection{Stochastic gradient Langevin dynamics}\nIn certain Bayesian inference problems, the data are conditionally independent of each other given the parameter value. Therefore, the likelihood model can often be factorized and the posterior density can be written as follows:\n\\begin{align}\n\t\\pi(\\theta) \\propto \\pi_0(\\theta) \\prod_{i = 1}^N \\pi_i(X_i| \\theta)\n\\end{align}\nwhere $\\pi(X_i|\\theta)$ is the likelihood function for data point $X_i$. When the dataset is extremely large, i.e., when $N\\gg 1$, however, ULA becomes exceedingly expensive as it requires repeatedly evaluating the likelihood over the entire dataset for each step of the trajectory. To mitigate this challenge, the stochastic gradient Langevin dynamics was presented to reduce the computational cost of evaluating the posterior density by only evaluating the likelihood over \\emph{subsets} of the data at each step. The true likelihood is estimated based on the likelihood function evaluated at the subsampled data \\cite{welling2011bayesian}. Specifically, the gradient is estimated using a stochastic gradient\n\\begin{align}\n\t\\nabla \\log \\pi(\\theta|X)\\approx\\widehat{\\nabla \\log \\pi(\\theta|X)} = \\log \\pi_0(\\theta) + \\frac{N}{n} \\sum_{i = 1}^n \\log \\pi(X_{\\tau_i}|\\theta)\n\\end{align}\nwhere $\\tau$ is a random subset of $\\{1,\\ldots, N\\}$ of size $n$ drawn with or without replacement. Depending on the choice of $n$, this approach cuts down on the computational costs dramatically with some additional variance incurred by the random subsampling of the data. The original version of this algorithm made the step size variable, approaching zero as the number of steps taken $K$\nbecame large. SGLD applied with a variable and shrinking step size was proven to be consistent: that is, the invariant distribution of the discretized system is equivalent to that of the continuous system \\cite{teh2016consistency}. Having a decreasing step size counteracts the cost savings provided by computing the stochastic gradient, and therefore a version where the step size is fixed was presented in \\cite{vollmer2016exploration}, where theoretical characterizations of the asymptotic and finite-time bias and variance are also developed. In most of our numerical results, we use stochastic gradient version of the Langevin algorithm with fixed step size to demonstrate that SGLD can be used together with irreversible perturbations.\n\n\n\n\\begin{table}[]\n\\centering\n\\begin{tabular}{l|l|l}\n & $b(\\theta)$ & $\\sigma(\\theta)$ \\\\ \\hline\n \\textbf{\\texttt{LD}}& $\\beta\\nabla \\log \\pi(\\theta)  $ & $\\sqrt{2\\beta} \\mathbf{I}$  \\\\\n \\textbf{\\texttt{RM}}&  $\\beta\\mathbf{B}(\\theta)\\nabla \\log \\pi(\\theta)) + \\beta\\nabla \\cdot \\mathbf{B}(\\theta)$ & $\\sqrt{2\\beta\\mathbf{B}(\\theta)}$\\\\\n \\textbf{\\texttt{Irr}} & $(\\beta\\mathbf{I} + \\mathbf{J}) \\nabla \\log \\pi(\\theta) $ & $\\sqrt{2\\beta} \\mathbf{I}$ \\\\\n \\textbf{\\texttt{RMIrr}} & $(\\beta\\mathbf{B}(\\theta) + \\mathbf{J})\\nabla \\log \\pi(\\theta) +\\beta\\nabla \\cdot \\mathbf{B}(\\theta)$ & $\\sqrt{2\\beta\\mathbf{B}(\\theta)} $ \\\\\n \\textbf{\\texttt{GiIrr}} & $ (\\beta\\mathbf{B}(\\theta) + \\frac{1}{2}\\mathbf{J}\\mathbf{B}(\\theta) + \\frac{1}{2}\\mathbf{B}(\\theta) \\mathbf{J})\\nabla \\log \\pi(\\theta) + \\nabla\\cdot (\\beta\\mathbf{B}(\\theta) + \\frac{1}{2}\\mathbf{J}\\mathbf{B}(\\theta) + \\frac{1}{2}\\mathbf{B}(\\theta) \\mathbf{J})  $ & $\\sqrt{2\\beta\\mathbf{B}(\\theta)}  $\n\\end{tabular}\n\\caption{Summary of the five SDEs that share the same invariant density $\\pi(\\theta)$. Stochastic gradients can be considered instead of the deterministic gradients. All systems are of the form $\\text{d} \\theta_t = b(\\theta_t) \\text{d} t + \\sigma(\\theta_t) \\text{d} W_t$. The term $\\beta$ denotes the temperature.}\n\\label{table:dynamicssummary}\n\\end{table}\n\n\\section{Numerical examples}\n\\label{sec:num}\n\nIn the following examples, we always apply the stochastic gradient version of each Langevin system unless otherwise stated. We fix $\\beta = 1\/2$ for all examples. The efficacy of the \\texttt{GiIrr} perturbation does not change whether or not the stochastic gradient is used. We illustrate this explicitly in Section \\ref{subsec:blr}, where we report the results of all perturbations both with and without the stochastic gradient, for comparison.\n\n\\subsection{Linear Gaussian example}\n\\label{sec:linear}\n\n\nSuppose we have data $\\{X_i\\}_{i = 1}^N \\subset \\mathbb{R}^d$ generated from a bivariate normal distribution with mean $\\theta \\in \\mathbb{R}^d$ and known precision matrix $\\bm{\\Gamma}_X \\in \\mathbb{R}^{d\\times d}$. From the data, we infer the value of $\\theta$. Endow $\\theta$ with a normal prior with mean zero and precision $\\bm{\\Gamma}_\\theta \\in \\mathbb{R}^{d\\times d}$. Then the posterior distribution is Gaussian with mean and precision\n\\begin{align}\n\t\\mu_p = (\\bm{\\Gamma}_\\theta+ N \\bm{\\Gamma}_X)^{-1}\\bm{\\Gamma}_X \\sum_{i = 1}^N X_i\\,\\,\\,\\, \\text{and} \\,\\,\\,\\, \\bm{\\Gamma}_p = (\\bm{\\Gamma}_\\theta+N  \\bm{\\Gamma}_X),\n\\end{align}\nrespectively. The Euler-Maruyama discretization with constant step size $h$ applied to the corresponding Langevin dynamics is\n\\begin{align}\n\t\\theta_{k+1} = \\left(\\mathbf{I} - \\bar{\\mathbf{A}}h \\right)\\theta_k + \\bar{\\mathbf{D}}_k h + \\sqrt{h}\\xi_k\n\\end{align}\nwhere\n\\begin{align*}\n\t\\bar{\\mathbf{A}} = \\frac{1}{2} (\\bm{\\Gamma}_\\theta + N \\bm{\\Gamma}_X), \\,\\,\\,\\, \\bar{\\mathbf{D}}_k = \\frac{1}{2}\\bm{\\Gamma}_X \\sum_{i = 1}^N X_i, \\,\\,\\,\\, \\xi_k \\sim \\mathcal{N}(0,\\mathbf{I}).\n\\end{align*}\nUsing stochastic gradients yields the same recurrence above except with\n\\begin{align}\n\t\\bar{\\mathbf{D}}_k = \\frac{1}{2}\\bm{\\Gamma}_X \\frac{N}{n} \\sum_{i = 1}^n X_{\\tau^k_i}\n\\end{align}\nwhere $n \\le N$ and $\\tau^k_i\\in\\{1,\\ldots, N\\}$ is randomly sampled (with or without replacement) \\cite{welling2011bayesian}. Expectations with respect to the posterior are approximated by an long term average of the observable over the course of a trajectory. It has been shown that despite subsampling the data at each step in the dynamics, this estimator has comparable performance as the estimator produced by the regular Langevin dynamics with the full likelihood or MALA \\cite{welling2011bayesian,vollmer2016exploration}.\n\nNow, we consider the case where the dynamics are perturbed by an irreversible term that preserves the invariant distribution of the dynamics.  We demonstrate that this leads to a lower MSE than standard SGLD or Langevin dynamics. In this case, we replace $\\bar{\\mathbf{A}}$ and $\\bar{\\mathbf{D}}_k$ with ${\\mathbf{A}}$ and ${\\mathbf{D}}_k$, which are\n\\begin{align}\n\t{\\mathbf{A}} = \\frac{1}{2}(\\mathbf{I}+\\mathbf{J})(\\bm{\\Gamma}_\\theta + N\\bm{\\Gamma}_X), \\,\\,\\,\\, {\\mathbf{D}}_k  = \\frac{1}{2}(\\mathbf{I} + \\mathbf{J})\\bm{\\Gamma}_X \\frac{N}{n} \\sum_{i = 1}^n X_{\\tau_i^k}.\n\\end{align}\nand $\\mathbf{J}$ is a skew-symmetric matrix.\n\n\nFor the numerical experiments, we choose $d = 3$, $N = 10$, where the mini-batches are of size $n = 2$. We have $\\bm{\\Gamma}_X = 0.25\\mathbf{I}$, $\\bm{\\Gamma}_\\theta$ is a precision matrix with eigenvalues $0.2,0.01,0.05$ and eigenvectors that are randomly generated, and $h = 0.005$. Note that these matrices were chosen so that the resulting reversible perturbation has eigenvalues greater than one. To construct the perturbations, we choose $\\mathbf{B} = \\bm{\\Gamma}_p^{-1}$ and $\\mathbf{J}$ to be\n\\begin{align}\n\t\\mathbf{J} = \\delta \\begin{bmatrix}\n\t\t0 & 1 & 1 \\\\ -1 & 0 & 1 \\\\ -1 & -1 & 0\n\t\\end{bmatrix}\n\t\\label{eq:signpattern}\n\\end{align}\nfor $\\delta \\in \\mathbb{R}$. We consider the five different SDE systems presented in Table \\ref{table:dynamicssummary} and investigate how the MSE, bias, and variance differs for each case. For this example, since a constant metric is used, the geometry-informed irreversible perturbation simply produces a different constant skew-symmetric matrix than the other irreversible perturbations. Each system is simulated for $K = 10^5$ steps with step size $h = 5\\times10^{-3}$. In Figure \\ref{fig:3Dgauss}, we plot the MSE of the running average for each case when the observables are the sums of the first and second moments. To compute the asymptotic variances of the observables we use the batch means method in \\cite{asmussen2007stochastic}. After the burn-in period, we evaluate the observable over each chains. Each observable chain is then batched into twenty separate chains, and their means are evaluated. The asymptotic variance is estimated by computing the empirical variance of those means and then multiplying by the length of each of the subsampled trajectories. In Table \\ref{table:lineargaussian}, we report the asymptotic variance of the estimator for each system.\\footnote{The asymptotic variance reported here is $\\sigma^{2}$ where $\\text{Var}(Y(t)) \\sim \\sigma^2\/t$,  $Y(t)=\\frac{1}{t}\\int_{0}^{t}\\phi(\\theta_{t})dt$, and $\\phi$ is an observable.}\n We see that irreversible perturbations definitely improve the performance of the estimators, although the improvement provided by the geometry-informed irreversible perturbation seems marginal over \\texttt{RMIrr} when estimating the second moments.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = \\textwidth]{gausiannew_mean-eps-converted-to}\n\t\\includegraphics[width = \\textwidth]{gaussiannew_sec-eps-converted-to}\n\t\\caption{MSE of the running average for the first and second moments. Stochastic gradients are used in this example.}\n\t\\label{fig:3Dgauss}\n\\end{figure}\n\nWhen the reversible perturbation is chosen such that the drift matrix is exactly the identity (for example, when the matrix is chosen to be the covariance matrix of the posterior), additional irreversibility cannot widen the spectral gap of the system. This fact can be deduced from the results of \\cite{lelievre2013optimal}. The improved performance of the geometry-informed irreversible perturbation is mostly due to the fact that the norm of the corresponding skew-symmetric matrix is greater than that of simple irreversibility. Even though one can scale the skew-symmetric matrix for the other two cases to observe similar performance as geometry-informed irreversibility, \\texttt{GiIrr} accomplishes that in a more systematic way.\n\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n                       & $\\mathbb{E}[\\text{AVar}_{\\phi_1}]$    & $\\text{Std}[\\text{AVar}_{\\phi_1}]$ & $\\mathbb{E}[\\text{AVar}_{\\phi_2}]$    & $\\text{Std}[\\text{AVar}_{\\phi_2}]$  \\\\ \\hline\n\\texttt{LD}                     & $37.75$ & $11.94$ &$209.4$ & $84.38$  \\\\ \\hline\n\\texttt{RM}                     & $20.09$ & $6.420$ & $132.8$& $49.21$       \\\\ \\hline\n\\texttt{Irr}       & $15.72$ & $5.008$ &$135.4$ & $47.91$              \\\\ \\hline\n\\texttt{RMIrr}  & $12.36$ & $3.937$ & $115.9$ & $ 40.10$             \\\\ \\hline\n\\texttt{GiIrr}   & $\\mathbf{7.444}$ & $2.336$ & $\\mathbf{103.7}$ & ${36.78}$              \\\\ \\hline\n\\end{tabular}\n\\caption{Asymptotic variance estimates for the linear Gaussian example. }\n\\label{table:lineargaussian}\n\\end{table}\n\n\nWhile it is known that irreversible perturbations can, at worst, maintain the same performance as standard Langevin in the continuous-time setting \\cite{rey2016improving}, when considering discretization and in borderline cases (i.e., when one does not expect much or any improvement in continuous time), irreversibility may actually harm the performance of the estimator as it introduces additional stiffness into the system without resulting in faster convergence to the invariant density. A detailed exploration of this effect is presented in Appendix \\ref{sec:appendix}, in which we compute the bias and variance of the long term average estimator for a simple linear Gaussian problem where the posterior precision is a scalar multiple of the identity matrix. As further discussed in Remark \\ref{R:IrrevNotImprovement}, in this case, the irreversible perturbation is not expected to lead to improvement in the sampling properties from the equilibrium. Hence, the stiffness induced upon discretization has a more profound impact on the practical performance of the irreversible perturbation.\n\nIn the current numerical study, the posterior precision is diagonal, but not a scalar multiple of the identity matrix. The eigenvalues of the resulting drift matrix are therefore distinct, and by the theory in \\cite{lelievre2013optimal}, irreversible perturbations are able to reduce the spectral gap and result in improved performance. This is in contrast with the example studied in the appendix.\n\n\\subsection{Parameters of a normal distribution}\nThis example is identical to that used in \\cite[Section 5]{girolami2011riemann} to demonstrate the performance of RMLD. Given a dataset of $\\mathbb{R}$-valued data $\\mathbf{X} = \\{X_i\\}_{i = 1}^N \\sim \\mathcal{N}(\\mu,\\sigma^2)$, we infer the parameters $\\mu, \\sigma$. To be clear, in this example the state is $\\theta = [\\mu,\\sigma]^T$. The prior on $\\mu,\\sigma$ is chosen to be flat (and, therefore, improper). The log-posterior is\n\\begin{align}\n\t\\log p(\\mu,\\sigma|\\mathbf{X}) &= \\frac{N}{2} \\log 2\\pi - N \\log \\sigma - \\sum_{i = 1}^N \\frac{(X_i-\\mu)^2}{2\\sigma^2}.\n\\end{align}\nThe gradient is\n\\begin{align}\n\t\\nabla \\log p(\\mu,\\sigma|\\mathbf{X}) &= \\begin{bmatrix}\tm_1(\\mu)\/\\sigma^2 \\\\ -N\/\\sigma + m_2(\\mu)\/\\sigma^3\n\t\\end{bmatrix}\n\\end{align}\nwhere $m_1(\\mu) = \\sum_{i = 1}^N (X_i - \\mu)$, and $m_2(\\mu) = \\sum_{i = 1}^N (X_i-\\mu)^2$.\nIn \\cite{girolami2011riemann}, the authors propose using the geometry of the manifold defined by the parameter space of the posterior distribution to accelerate the resulting Metropolis-adjusted Langevin algorithm.   The authors in \\cite{girolami2011riemann} suggest using the expected Fisher information matrix to define the Riemannian metric, which in the context of reversible diffusions \\cite{rey2016improving}, is equivalent to choosing $\\mathbf{B}(\\mu,\\sigma)$ to be the inverse of the sum of the expected Fisher information matrix and the negative Hessian of the log-prior. Straightforward computations yield\n\\begin{align}\n\t\\mathbf{B} = \\frac{\\sigma^2}{N} \\begin{bmatrix}\n\t\t1 & 0 \\\\ 0 & 1\/2\n\t\\end{bmatrix}, \\;\\; \\sqrt{\\mathbf{B}} = \\frac{\\sigma}{\\sqrt{N}} \\begin{bmatrix}\n\t\t1 & 0 \\\\ 0 & 1\/\\sqrt{2}\n\t\\end{bmatrix},\\;\\; \\nabla \\cdot \\mathbf{B} = \\begin{bmatrix}\n\t\t0 \\\\ \\sigma\/N\n\t\\end{bmatrix}.\n\\end{align}\nAs for the geometry-informed irreversible perturbation, let $\\mathbf{J} = \\delta \\begin{bmatrix}0 & 1 \\\\ -1 & 0 \\end{bmatrix}$, for $\\delta \\in \\mathbb{R}$. Then the relevant quantities are\n\\begin{align}\n\t\\frac{1}{2}\\mathbf{J} \\mathbf{B} + \\frac{1}{2}\\mathbf{B}\\mathbf{J} = \\frac{3 \\sigma^2}{4N}\\mathbf{J}, \\;\\; \\frac{1}{2}\\nabla\\cdot(\\mathbf{J}\\mathbf{B} + \\mathbf{B}\\mathbf{J}) = \\frac{3\\delta\\sigma}{2N}\\begin{bmatrix}\n\t\t1 \\\\0\n\t\\end{bmatrix}.\n\\end{align}\n\nIn the experiments, we have $N = 30$, $h = 10^{-3}$, $\\delta = 2$. and simulate $M = 1000$ independent trajectories of each system up to $T = 1000$ for a total of $K = 10^6$ steps. The data is subsampled at a rate of $n = 6$ per stochastic gradient computation. Each trajectory is allotted a burn-in time of $T_b = 10$. The dataset is generated by drawing samples from a normal distribution with $\\mu_{true} = 0$ and $\\sigma_{true} = 10$. The observables we study are $\\phi_1(\\mu,\\sigma) = \\mu + \\sigma$, and $\\phi_2(\\mu,\\sigma) = \\mu^2 + \\sigma^2$. We plot the MSE, squared bias, and variance of resulting estimators for each observable in Figures \\ref{fig:1dn2sg} and \\ref{fig:2dn2sg}. Moreover, in Table \\ref{table:asympvarsg} we report the asymptotic variance of the estimators of each of the five systems. The main takeaway is that an irreversible perturbation that is adapted to the existing reversible perturbation performs much better than if the irreversible perturbation were applied without regard to the underlying geometry. Notice that the reversible perturbation considered here still improves the performance of the long term average estimator despite the fact that $\\mathbf{B} -\\mathbf{I}$ is not positive definite on the state space. Indeed, while $\\mathbf{B}-\\mathbf{I}$ being positive definite is a sufficient condition to obtain improved performance, it is not a necessary one \\cite{rey2016improving}. The reason for the reduced asymptotic variance we observed here is because the reversible perturbation $\\mathbf{B}$ has eigenvalues larger than one where the bulk of the posterior distribution lies.\n\nFigure \\ref{fig:trajdn2} show single and mean trajectories of the burn-in period of trajectories from each of the five systems. The plot shows that the geometry-informed irreversible perturbation is able to find the bulk of the distribution sooner than the other systems without incurring additional errors due to stiffness.\n\nTo show that the \\texttt{GiIrr} perturbation is not intimately tied to the stochastic gradient, we also report the results for each system when the gradients are computed exactly in Table \\ref{table:asympvar}. We see that there is little meaningful difference in the results compared to when stochastic gradients are used.\n\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width = 0.8\\linewidth]{burninrace-eps-converted-to}\n\t\\caption{Trajectory burn-in: each trajectory is run for $T = 2.5$. Left: single trajectories, right: mean paths. The gradients are computed exactly here. }\n\t\\label{fig:trajdn2}\n\\end{figure}\n\n\n\n\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n                       & $\\mathbb{E}[\\text{AVar}_{\\phi_1}]$    & $\\text{Std}[\\text{AVar}_{\\phi_1}]$ & $\\mathbb{E}[\\text{AVar}_{\\phi_2}]$    & $\\text{Std}[\\text{AVar}_{\\phi_2}]$  \\\\ \\hline\n\\texttt{\\texttt{LD}}                     & $55.29$ & $21.52$ &$8332$ & $4359$  \\\\ \\hline\n\\texttt{\\texttt{RM}}                     & $20.63$ & $6.019$ & $4034$& $1378$       \\\\ \\hline\n\\texttt{\\texttt{Irr}}       & $5.791$ & $2.638$ &$2169$ & $1072$              \\\\ \\hline\n\\texttt{\\texttt{RMIrr}}  & $6.512$ & $2.226$ & $1729$ & $ 631.2$             \\\\ \\hline\n\\texttt{\\texttt{GiIrr}}   & $\\mathbf{1.400}$ & $0.4697$ & $\\mathbf{479.4}$ & $170.8$              \\\\ \\hline\n\\end{tabular}\n\\caption{Asymptotic variance estimates for the parameters of a normal distribution example. Stochastic gradients are employed. }\n\\label{table:asympvarsg}\n\\end{table}\n\n\\begin{figure}[H]\n\\centering\n\t\\includegraphics[width = \\linewidth]{obs1-eps-converted-to}\n\t\\caption{Observable: $\\phi_1(\\mu,\\sigma) = \\mu+\\sigma$, $\\delta = 2$. Stochastic gradients are computed. }\n\t\\label{fig:1dn2sg}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\t\\includegraphics[width = \\linewidth]{obs2-eps-converted-to}\n\t\\caption{Observable: $\\phi_2(\\mu,\\sigma) = \\mu^2 + \\sigma^2$, $\\delta = 2$. Stochastic gradients are computed.}\n\t\\label{fig:2dn2sg}\n\\end{figure}\n\n\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n                       & $\\mathbb{E}[\\text{AVar}_{\\phi_1}]$    & $\\text{Std}[\\text{AVar}_{\\phi_1}]$ & $\\mathbb{E}[\\text{AVar}_{\\phi_2}]$    & $\\text{Std}[\\text{AVar}_{\\phi_2}]$  \\\\ \\hline\n\\texttt{\\texttt{LD}} (no SG)                     & $48.51$ & $17.53$ &$7339$ & $3707$  \\\\ \\hline\n\\texttt{\\texttt{RM}}  (no SG)                   & $20.91$ & $6.445$ & $3855$& $1406$       \\\\ \\hline\n\\texttt{\\texttt{Irr}} (no SG)      & $5.658$ & $2.108$ &$2265$ & $1191$              \\\\ \\hline\n\\texttt{\\texttt{RMIrr}} (no SG) & $6.276$ & $2.075$ & $1648$ & $ 565.1$             \\\\ \\hline\n\\texttt{\\texttt{GiIrr}} (no SG)  & $\\mathbf{1.363}$ & $0.4223$ & $\\mathbf{492.9}$ & $183.8$              \\\\ \\hline\n\\end{tabular}\n\\caption{Asymptotic variance estimates for the parameters of a normal distribution example. The gradients are computed exactly. }\n\\label{table:asympvar}\n\\end{table}\n\n\\subsection{Bayesian logistic regression}\n\\label{subsec:blr}\nNext we consider Bayesian logistic regression. Given data $\\{(\\mathbf{x}_i,t_i)\\}_{i = 1}^N$, where $\\mathbf{x}_i \\in \\mathbb{R}^d$, and $t_i \\in \\{0,1\\}$, we seek a logistic function, parameterized by weights $\\mathbf{w} \\in \\mathbb{R}^d$, that best fits the data. The weights are obtained in a Bayesian fashion, in which we endow the weights with a prior and seek to characterize its posterior distribution via sampling. Define $\\varphi(y)$ to be the logistic function $\\varphi(y)= ({1+\\exp(-y) })^{-1}$.\nThe log-likelihood function is\n\\begin{align}\n\tl(\\mathbf{w}) &= \\sum_{i = 1}^N t_i \\mathbf{x}_i^T \\mathbf{w} - \\sum_{i = 1}^N \\log (1 + \\exp(\\mathbf{x}_i^T \\mathbf{w}) ).\n\\end{align}\nThe prior for the weights is normally distributed with mean zero and covariance $\\alpha^{-1} \\mathbf{I}$. The gradient of the log-posterior is\n\\begin{align}\n\t\\nabla_{\\mathbf{w}} \\log \\pi(\\mathbf{w} | \\mathbf{X}) = -\\alpha\\mathbf{w} + \\sum_{i = 1}^N t_i \\mathbf{x}_i - \\sum_{i = 1}^N \\varphi(\\mathbf{x}_i^T \\mathbf{w}) \\mathbf{x}_i.\n\\end{align}\nThis term is used in the drift part of the Langevin dynamics that fully computes the gradient of the log-likelihood at every step. If the data are subsampled as in SGLD, we instead compute\n\\begin{align}\n\t\\nabla_{\\mathbf{w}}\\log \\tilde{\\pi}(\\mathbf{w}|\\mathbf{X}) = -\\alpha\\mathbf{w} + \\frac{N}{n}\\sum_{i = 1}^n t_{\\tau_i} \\mathbf{x}_{\\tau_i} - \\frac{N}{n}\\sum_{i = 1} ^n \\phi(\\mathbf{x}_{\\tau_i}^T\\mathbf{w}) \\mathbf{x}_{\\tau_i}.\n\\end{align}\n\nWe use the \\texttt{german} data set described in \\cite{gershman2012nonparametric} for the numerical experiments. In this problem, there are 20 weight parameters to be learned. The training dataset is of size $N = 400$ and we choose to subsample at a rate of $n = 10$ per likelihood computation. The time step we choose is $h = 10^{-4}$ and $ K = 4\\times10^{5}$ steps. We generate the skew-symmetric matrix by constructing a lower triangular matrix with entries randomly drawn from $\\{1,-1\\}$ and then subtracting its transpose. The diagonal is then set to zero and the matrix is scaled to have norm one.\n\nAs for the Riemannian manifold Langevin dynamics, in \\cite{girolami2011riemann} the authors use the expected Fisher information matrix plus the negative Hessian of the log-prior as the underlying metric, which in this case is equal to\n\\begin{align}\n\t\\mathbf{G}(w) = \\alpha^{-1}\\mathbf{I} + \\mathbf{X}\\bm{\\Lambda}(w) \\mathbf{X}^T\n\\end{align}\nwhere $\\bm{\\Lambda}$ is a diagonal matrix with entries $\\bm{\\Lambda}_{ii}(w) = (1-\\varphi(\\mathbf{x}_i^T w) ) \\varphi(\\mathbf{x}_i^Tw)$ and $\\mathbf{x}_i$ is the $i$-th column of $\\mathbf{X}$. The resulting reversible perturbation uses the inverse of $\\mathbf{G}(w)$. This perturbation, however, does not lead to accelerated convergence to the invariant measure since the eigenvalues of $\\mathbf{G}$ are large. This implies that the eigenvalues of $\\mathbf{G}^{-1}$ are less than one and so $\\mathbf{G}^{-1}(w) -\\mathbf{I}$ is not positive definite, a condition that needs to be satisfied to guarantee accelerated convergence \\cite{rey2015irreversible}. To alleviate this issue, we consider the reversible perturbation $\\mathbf{B}(w) = \\mathbf{I} + \\mathbf{G}^{-1}(w)$. This guarantees $\\mathbf{B}(w)$ to be positive definite for all $w$, but the drawback is that computing the square root of $\\mathbf{B}(w)$ requires explicitly computing or at least approximating the inverse of $\\mathbf{G}(w)$ repeatedly in the simulation (and not just computing the action of the inverse). This additional computational cost is incurred for all examples that consider a geometry-informed perturbation, both reversible and irreversible. We show the result of this state-dependent perturbation in Figure \\ref{fig:blrvariable} and report the asymptotic variance in Table \\ref{table:blr}. The geometry-informed irreversible perturbation does provide improvement over all other perturbations. We observe that the asymptotic variance is reduced by half over \\texttt{RM}, with only little additional computational effort. Most of the computational cost of applying \\texttt{GiIrr} is due to the evaluation of the reversible perturbation. Therefore we emphasize that if one is already applying the reversible perturbation to the Langevin dynamics, the marginal cost of applying the \\texttt{GiIrr} perturbation is negligible.\n\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = \\textwidth]{variablefirst-eps-converted-to}\n\t\\includegraphics[width = \\textwidth]{variablesec-eps-converted-to}\n\t\\caption{Bayesian logistic regression with a variable metric. Here, $d = 20$. }\n\t\\label{fig:blrvariable}\n\\end{figure}\n\n\n\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n                      & $\\mathbb{E}[\\text{AVar}_{\\phi_1}]$    & $\\text{Std}[\\text{AVar}_{\\phi_1}]$ & $\\mathbb{E}[\\text{AVar}_{\\phi_2}]$    & $\\text{Std}[\\text{AVar}_{\\phi_2}]$  \\\\ \\hline\n\\texttt{\\texttt{LD}}                     & $1.967$ & $0.9995$ &$23.77$ & $12.52$  \\\\ \\hline\n\\texttt{RM}                     & $1.328$ & $0.6538$ & $15.35$& $7.348$       \\\\ \\hline\n\\texttt{Irr}       & $1.163$ & $0.5698$ &$14.84$ & $7.738$              \\\\ \\hline\n\\texttt{RMIrr}  & $0.8775$ & $0.4228$ & $10.68$ & $ 5.306$             \\\\ \\hline\n\\texttt{GiIrr}   & $\\mathbf{0.7148}$ & $0.3450$ & $\\mathbf{8.798}$ & ${4.490}$              \\\\ \\hline\n\\end{tabular}\n\\caption{Asymptotic variance estimates for the Bayesian logistic regression example with a state-dependent metric. }\n\\label{table:blr}\n\\end{table}\n\n\n\\subsection{Independent component analysis}\n\n\n\\label{subsec:ica}\nOur last example considers the problem of blind signal separation addressed in \\cite{welling2011bayesian} and \\cite{amari1996new}. This problem yields a posterior that is strongly non-Gaussian and multi-modal, and we show that \\texttt{GiIrr} has substantially better sampling performance over standard reversible and irreversible perturbations. Suppose there are $m$\n separate unknown independent signals $s^i(t)$ for $i = 1,\\ldots, m$ that are mixed by mixing matrix $\\mathbf{M}\\in \\mathbb{R}^{d\\times d}$. Suppose we can observe the mixed signals $X(t) = \\mathbf{M} s(t)$ for $N$ instances in time. The goal of independent component analysis is to infer a de-mixing matrix $\\mathbf{W}$ such that the $m$ signals are recovered up to a nonzero constant and permutation. As such, this problem is generally ill-posed, but is suitable to be considered in a Bayesian context. The ICA literature states that, based on real-world data, it is best to assume a likelihood model with large kurtosis. Following \\cite{welling2011bayesian,amari1996new}, let\n$ \tp(y_i) = \\frac{1}{4} \\text{sech}^2\\left( \\frac{1}{2} y_i\\right). $\n The prior on the weights $\\mathbf{W}_{ij}$ is Gaussian with zero mean and precision $\\lambda$. The posterior is equal to\n \\begin{align}\n \tp(\\mathbf{W}|X) \\propto |\\det \\mathbf{W}| \\prod_{i = 1}^m p(\\mathbf{w}_i^T \\mathbf{x}) \\prod_{ij} \\mathcal{N}(\\mathbf{W}_{ij}; 0,\\lambda^{-1}).\n \\end{align}\n The gradient of the log posterior with respect to the matrix $\\mathbf{W}$ is then\n \\begin{align}\n \tf(\\mathbf{W}) = \\nabla_\\mathbf{W} \\log p(\\mathbf{W}|X) = \\left( N(\\mathbf{W}^T)^{-1} - \\sum_{n = 1}^N \\tanh\\left( \\frac{1}{2} \\mathbf{y}_n \\right) \\mathbf{x}_n^T \\right) - \\lambda \\mathbf{W}.\n \\end{align}\n It is suggested in \\cite{amari1996new} that the natural gradient should be used instead of the gradient we see here above to account for the information geometry of the problem. Specifically, \\cite{teh2016consistency,amari1996new} post-multiply the gradient by $\\mathbf{W}^T\\mathbf{W}$ and arrive at the so-called natural gradient of the system\n \\begin{align}\n \t\\mathcal{D}_{\\mathbf{W}} := \\left( N\\mathbf{I} - \\sum_{n = 1}^N \\tanh\\left( \\frac{1}{2} \\mathbf{y}_n \\right) \\mathbf{y}_n^T \\right)\\mathbf{W} - \\lambda \\mathbf{W}\\W^T\\mathbf{W}.\n \\end{align}\n In the context of RMLD, this is equivalent to perturbing the system with a reversible perturbation with $\\mathbf{B}(\\mathbf{W}) = \\mathbf{W}^T\\mathbf{W}\\otimes \\mathbf{I}$ pre-multipled in front of the vectorized gradient. That is, we have\n \\begin{align*}\n \t\\text{vec}(f(\\mathbf{W})\\mathbf{W}^T\\mathbf{W}) = (\\mathbf{W}^T\\mathbf{W} \\otimes \\mathbf{I}) \\text{vec}f(\\mathbf{W}).\n \\end{align*}\n\nWe construct the \\texttt{GiIrr} term as follows. To take advantage of the matrix structure of the reversible perturbation, we choose the skew-symmetric matrix such that it acts within the computation of the natural gradient. We choose $\\mathbf{J} = (\\mathbf{I} \\otimes \\mathbf{C}_0) + (\\mathbf{C}_0 \\otimes \\mathbf{I})$ where $\\mathbf{C}_0$ has the same sign pattern as \\eqref{eq:signpattern} but such that $\\mathbf{J}$ has matrix norm equal to 2. Then the geometry-informed irreversible perturbation is\n \\begin{align*}\n \t\\frac{1}{2}\\mathbf{B}(\\mathbf{W})\\mathbf{J} + \\frac{1}{2}\\mathbf{J} \\mathbf{B}(\\mathbf{W}) = (\\mathbf{W}^T\\mathbf{W} \\otimes \\mathbf{C}_0) + \\frac{1}{2}(\\mathbf{W}^T\\mathbf{W} \\mathbf{C}_0 \\otimes \\mathbf{I}) + \\frac{1}{2}(\\mathbf{C}_0 \\mathbf{W}^T\\mathbf{W} \\otimes \\mathbf{I}).\n \\end{align*}\nTo simulate the \\texttt{RM} and \\texttt{GiIrr} systems, correction terms (such as $\\nabla \\cdot \\mathbf{B}(\\theta)$) need to be computed. The correction terms are derived using the symbolic algebra toolbox in MATLAB. Since the perturbations are vectors of polynomials, the symbolic algebra toolbox can easily derive and efficiently evaluate the correction terms.\n\n\nFor the numerical experiments, we synthetically generate $m = 3$ signals,\none of which is Laplace distributed, and two are distributed according to the squared hyperbolic secant distribution. The posterior distribution is $d = 9$ dimensional, there are a total of $N = 400$ data points, and the gradient is approximated by subsampling $n = 40$ data points per estimate. Since the posterior is nine-dimensional and highly multimodal, it is difficult to evaluate its marginal densities directly, i.e., without sampling. Instead, we establish a baseline reference density by simulating the standard Langevin dynamics with exact computation of the likelihood over all the data over $T = 10000$ with $h = 10^{-4}$. One- and two-dimensional marginals of this baseline posterior distribution are plotted in Figure \\ref{fig:baselinepos}. The two-dimensional marginals highlight the challenges of sampling from this posterior. In Figure \\ref{fig:traceplots}, we plot trace plots of the $\\mathbf{W}_{21}$ variable for each system. By visual inspection, we see that that mixing is best for the geometry-informed irreversibly perturbed system. One can intuitively expect that with better mixing, the geometry-informed irreversibility should yield better estimation performance than the other systems. We assess this quantitatively below.\n\n\\begin{figure}\n\\centering\n\t\\includegraphics[width = 0.8\\textwidth]{baserefT10k.png}\n\t\\caption{Posterior distribution sampled with standard Langevin with a deterministic gradient with $T = 10000$ and $h = 10^{-4}$. Notice that the system is very multimodal and non-Gaussian.}\n\t\\label{fig:baselinepos}\n\\end{figure}\n\nAs in the previous example, we simulate the five systems and compute the asymptotic variances of two observables for each system. Each system is simulated independently 100 times up to time $T = 2000$ with $h = 2\\times 10^{-5}$. The smaller step size is to account for the additional stiffness irreversible perturbations introduce. Since the true mean of the posterior distribution is unknown, and because standard sampling methods fail to adequately sample from the posterior distribution to get a reasonable estimate for the mean, we only plot the variance of the two observables with respect to $K$ in Figure \\ref{fig:varica}. To compute the asymptotic variance, we allot a burn-in time of $T_b = 20$. The observables we estimate are $\\phi_1(\\mathbf{W}) = \\sum_{i,j} \\mathbf{W}_{ij}$, and $\\phi_2(\\mathbf{W}) = \\left( \\sum_{i,j} \\mathbf{W}_{ij}\\right)^2$. The asymptotic variance numbers confirm that the faster mixing observed in the geometry-adapted irreversible perturbation does lead to a better sampling method. The values of the asymptotic variance are reported in Table \\ref{table:ica}. Notice that the geometry-informed irreversible perturbation far outperforms standard irreversibility applied to the reversible perturbation. When estimating the posterior mean, \\texttt{GiIrr} yields an asymptotic variance that is more than 60 times smaller than that of \\texttt{RM}.\n\n\\begin{figure}\n \t\\centering\n \t\\includegraphics[width = \\textwidth]{w21trace-eps-converted-to}\n \t\\caption{Trace plots of the $W_{21}$ marginal.}\n \t\\label{fig:traceplots}\n \\end{figure}\n\n\n \\begin{figure}\n \t\\centering\t\n \t\\includegraphics[width = 0.8\\textwidth]{vardecay-eps-converted-to}\n \t\\caption{Variance of running average estimators}\n \t\\label{fig:varica}\n \\end{figure}\n\n\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n                      & $\\mathbb{E}[\\text{AVar}_{\\phi_1}]$    & $\\text{Std}[\\text{AVar}_{\\phi_1}]$ & $\\mathbb{E}[\\text{AVar}_{\\phi_2}]$    & $\\text{Std}[\\text{AVar}_{\\phi_2}]$  \\\\ \\hline\n\\texttt{LD}                     & $81.76$ & $23.88$ &$50.21$ & $17.92$  \\\\ \\hline\n\\texttt{RM}                     & $71.96$ & $17.00$ & $41.48$& $14.24 $       \\\\ \\hline\n\\texttt{Irr}       & $37.90$ & $13.46$ &$20.63$ & $9.147$              \\\\ \\hline\n\\texttt{RMIrr}  & $32.84$ & $10.76$ & $10.89$ & $ 3.140$             \\\\ \\hline\n\\texttt{GiIrr}   & $\\mathbf{1.182}$ & $0.3881$ & $\\mathbf{0.7419}$ & $0.2391$              \\\\ \\hline\n\\end{tabular}\n\\caption{Asymptotic variance estimates for the ICA example. }\n\\label{table:ica}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{S:Conclusions}\nWe presented a novel irreversible perturbation, \\texttt{GiIrr}, that accelerates the convergence of Langevin dynamics.\n By introducing an irreversible perturbation that incorporates any given underlying reversible perturbation, which can also be interpreted as defining a Riemannian metric, we have shown through numerical examples that geometry-informed irreversible perturbations outperform those that are not informed as such. In the examples, we found that \\texttt{GiIrr} seems to perform best when the target distribution is highly non-Gaussian.\n\nMost of our numerical examples used stochastic gradients to cut down on computational effort in sampling each trajectory. This demonstrates that SGLD can be used in conjunction with irreversibility for practical computations.\n\nWe also provided some analysis on how irreversibility interacts with discretization of the SDE systems. Irreversibility introduces additional stiffness into the system, which may lead to additional bias or variance in the estimator. For practical purposes, one can simply choose a small enough step size so that the asymptotic bias and variance are sufficiently small. At the same time, we note an example (see Appendix \\ref{sec:appendix}) where the introduction of the irreversible term, once discretized, leads to no improvement in the long term average estimator.\n\nFuture work could study the use of novel integrators which circumvent stiffness. For example, \\cite{lu2018analysis} uses a multiscale integrator, but it is not readily adapted to the data-driven setting of Bayesian inference. Another direction for future work is to theoretically characterize the performance of the geometry-informed irreversible perturbation and to compare it with that of other perturbations. A starting point for such an analysis could be the general results of \\cite{rey2016improving}, in particular the large deviations Theorem 1 together with Propositions 2--4 therein. Preliminary investigation of this direction showed that it is a promising avenue for a theoretical investigation, but non-trivial work and a finer analysis are needed to demonstrate the effects of this class of irreversible perturbations. We leave this for future work, as such an analysis is outside the scope of this paper. Our goal in this paper has been to introduce the perturbation and showcase its potential through simulation examples.\n\n\n\n\\begin{appendices}\n\\section{The effects of discretization}\n\\label{sec:appendix}\n\n\nIn this section, we study the effects of discretization in the setting of an irreversibly perturbed Langevin system. Results in full generality are, as yet, elusive; therefore we only consider a Gaussian example, as it still provides insight into how irreversibility interacts with discretization in impacting the asymptotic and finite sample bias and variance of the long term average estimator. While we do not present the results when a stochastic gradient is used, we note that the results are similar and can be easily extended based on what we present here. Recall that,\n\\begin{align*}\n\t\\mathbf{A} = \\frac{1}{2}(\\mathbf{I}+\\mathbf{J})(\\bm{\\Gamma}_\\theta + N \\bm{\\Gamma}_X),\\,\\,\\,\\, \\mathbf{D} = \\frac{1}{2}(\\mathbf{I}+\\mathbf{J}) \\left(\\bm{\\Gamma}_X \\sum_{i = 1}^N X_{i} \\right), \\text{ where } \\mathbf{J} = \\delta \\begin{bmatrix} 0 & 1 \\\\ -1 & 0\\end{bmatrix}\n\\end{align*}\nFor this analysis, all precision matrices are $2\\times 2$ scalar matrices. That is, we assume $\\bm{\\Gamma}_\\theta = \\sigma_\\theta^{-2}\\mathbf{I}$, $\\bm{\\Gamma}_X = \\sigma_X^{-2}\\mathbf{I}$. This is distinct from the example in Section \\ref{sec:linear}, since the precision matrices there are diagonal but not scalar. Let $b = \\frac{1}{2\\sigma_X^2}$ and $S_X = \\sum_{k = 1}^N X_i$, so that $\\mathbf{D} = b(\\mathbf{I}+\\mathbf{J}) S_X$.\n\n\nWe summarize our findings here. For fixed discretization size $h$ and scalar precision matrices as defined above, and introducing the irreversible perturbation scaled by $\\delta$, we find the following:\n\\begin{itemize}\n\t\\item The asymptotic bias for linear observables is zero, that is, $\\mathbb{E}[\\theta_\\infty] = \\mu_p$;\n\t\\item The asymptotic variance for linear observables increases. We found that\n\t\\begin{align}\n\t\n\t\t\t\\text{Tr}\\, \\text{Var}[\\theta_\\infty] = \\frac{2}{2a-ha^2(1+\\delta^2)}\n\t\\end{align}\n\twhere $a = 0.5 (1\/\\sigma_\\theta^2 + N\/\\sigma_X^2)$;\n\t\\item The finite time estimator for the observable $\\phi(\\theta) = \\theta_1 + \\theta_2$ has lower bias and variance;\n\t\\item The finite time estimator for the observable $\\phi(\\theta) = \\|\\theta\\|^2$ has higher bias and variance.\n\\end{itemize}\nWe focus on the finite time results and omit the asymptotic results, since the the former case is of more practical interest. The computations related to both are similar.\n\n\n\n\\paragraph{Finite time analysis: bias for linear observables.}\nWe study how the magnitude of the irreversibility, characterized by $\\delta$, impacts the mean-squared error $\\text{MSE} = \\mathbb{E}\\left[ \\|\\bar{\\theta}_K-\\mu_p\\|^2\\right]$ where $\\bar{\\theta}_K = \\frac{1}{K}\\sum_{k = 0}^{K-1} \\theta_k$. We approach this quantity via its bias-variance decomposition:\n\\begin{align}\n\t\\text{MSE} = \\left\\|\\mathbb{E}[\\bar{\\theta}_{K}] - \\mu _p \\right\\|^2 + \\text{Tr}\\,\\text{Var}\\left( \\bar{\\theta}_K\\right) .\n\\end{align}\nFirst, we compute the expected value of the sample average $\\mathbb{E}\\left[\\bar{\\theta}_K\\right] = \\frac{1}{K} \\sum_{k = 0}^{K-1} \\mathbb{E}[\\theta_k].$ For simplicity, we assume that the initial condition is always $\\theta_0 = \\mathbf{0}$. For any $k$, we have\n\\begin{align*}\n\t\\mathbb{E}[\\theta_k] &= (\\mathbf{I}-h\\mathbf{A})\\mathbb{E}[\\theta_{k-1}] + h\\mathbf{D} \\\\\n\t& = (\\mathbf{I}-h\\mathbf{A})^k \\theta_0 + h\\sum_{n = 0}^{k-1} (\\mathbf{I}-h\\mathbf{A})^n  \\mathbf{D} \\\\\n\t& =  h(\\mathbf{A} h)^{-1}(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^{k})\\mathbf{D} \\\\\n\t& =  \\mathbf{A}^{-1}\\mathbf{D} - \\mathbf{A}^{-1}(I-h\\mathbf{A})^k \\mathbf{D}.\n\\end{align*}\nThis yields\n\\begin{align*}\n\t\\mathbb{E}[\\bar{\\theta}_K] &= \\frac{1}{K}\\sum_{k = 0}^{K-1} \\left(\\mathbf{A}^{-1}\\mathbf{D} - \\mathbf{A}^{-1}(\\mathbf{I}-h\\mathbf{A})^k \\mathbf{D} \\right) \\\\\n\t&= \\mathbf{A}^{-1}\\mathbf{D} - \\frac{1}{K}\\mathbf{A}^{-1}(\\mathbf{A} h)^{-1}(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K)\\mathbf{D}.\n\\end{align*}\nSince $\\mu_p = \\mathbf{A}^{-1}\\mathbf{D}$, the bias is\n\\begin{align*}\n\t\\text{bias} = -\\frac{1}{Kh}\\mathbf{A}^{-2}(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K)\\mathbf{D}.\n\\end{align*}\nThe norm of the bias can in fact be computed. Note that $\\mathbf{A}^2 = (1+\\delta^2)a^2\\mathbf{I}$ and we have\n\\begin{align*}\n\t\\|\\text{bias}\\|^2 &= \\frac{1}{K^2h^2} \\mathbf{D}^T (\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A}^T)^K)\\mathbf{A}^{-2T}\\mathbf{A}^{-2}(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K) \\mathbf{D} \\\\\n\t& = \\frac{1}{K^2h^2a^4(1+\\delta^2)^2}\\mathbf{D}^T(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A}^T)^K)(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K) \\mathbf{D} \\\\\n\t& = \\frac{b^2}{K^2h^2a^4(1+\\delta^2)^2}S_X^T(\\mathbf{I}+\\mathbf{J})^T(\\mathbf{I}-(\\mathbf{I}-\\mathbf{A}^Th)^K)(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K) (\\mathbf{I}+\\mathbf{J}) S_X.\n\\end{align*}\nThe inner matrix can be computed. Since each matrix above is simultaneously diagonalizable, we only need to consider the eigenvalues of each of the above matrices. Note that $\\mathbf{I}+\\mathbf{J}$ is a normal matrix, so we may write the eigenvalue decomposition $\\mathbf{I}+\\mathbf{J} = \\mathbf{PDP}^*$, where $^*$ denotes conjugate transpose, $\\mathbf{Q} = \\text{diag}(1+i\\delta,1-i\\delta)$, and\n\\begin{align*}\n\t\\mathbf{P} = \\frac{1}{\\sqrt{2}}\\begin{bmatrix}\n\t\t1 & 1 \\\\ i & -i\n\t\\end{bmatrix}\n\\end{align*}\nis orthogonal. Now note that\n\\begin{align*}\n\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K = \\mathbf{P} \\begin{bmatrix}\n\t\t1-(1-ah(1+i\\delta))^K & 0 \\\\0 & 1-(1-ah(1-i\\delta))^K\n\t\\end{bmatrix} \\mathbf{P}^*,\n\\end{align*}\nwhich implies\n\\begin{align*}\n\t(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A}^T)^K)(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K)= |1-(1-ah(1+i\\delta)^K)|^2\\mathbf{I}.\n\\end{align*}\nUsing the fact that $(\\mathbf{I}+\\mathbf{J})^T(\\mathbf{I}+\\mathbf{J}) = (1+\\delta^2)I$, and we have the following\n\\begin{align}\n\t\\|\\text{bias}\\|^2 &= \\frac{b^2}{K^2h^2a^4(1+\\delta^2)} |1-(1-a(1+i\\delta)h)^K|^2 \\|S_X\\|.\n\\end{align}\nTo simplify further, we write $1-a(1+i\\delta)h = re^{i\\theta}$ where $r^2 = (1-ah)^2 + \\delta^2a^2h^2$, and $\\tan \\theta = \\delta a h\/ (1-ah)$. Then we obtain\n\\begin{align*}\n\t\\|\\text{bias}\\|^2 &= \\frac{b^2}{K^2h^2a^4(1+\\delta^2)} |1-r^Ke^{i\\theta K}|^2 \\| S_X \\|^2 \\\\\n\t& = \\frac{b^2}{K^2h^2a^4(1+\\delta^2)} (1+r^{2K}-2r^K \\cos K\\theta )\\| S_X \\|^2.\n\\end{align*}\nWe know that $r<1$, since otherwise, the numerical scheme would be unstable. It is easy to see that for large, but not infinite, $K$, the bias decays as $\\mathcal{O}(1\/(Kh\\sqrt{1+\\delta^2}))$, so the introduction of irreversibility decreases the constant in front of the expression and therefore slightly improves the convergence of the bias.\n\n\\paragraph{Finite time analysis: variance for linear observables.}\nFor simplicity, we assume $\\theta_0 = 0$. We compute $\\text{TrVar}(\\bar{\\theta}_K)$. We begin with\n\\begin{align*}\n\t\\text{Tr}\\,\\text{Var}(\\bar{\\theta}_K) = \\text{Tr}\\,\\mathbb{E}[\\bar{\\theta}_K\\bar{\\theta}_K^T] - \\text{Tr}\\,\\mathbb{E}[\\bar{\\theta}_K]\\mathbb{E}[\\bar{\\theta}_K]^T\n\\end{align*}\nand compute these terms separately. It is difficult to surmise a relationship between $\\delta$ and $\\text{Tr}\\,\\text{Var}(\\bar{\\theta}_K)$ even with exact formulas, so we appeal to plots of the expressions to see that the variance decreases with irreversibility. We computed $\\mathbb{E}[\\bar{\\theta}_K]$ in the previous section. With the observation that\n\\begin{align*}\n\t\\mathbf{A}^{-2}(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K) \\mathbb{E}[\\mathbf{D}] = \\frac{b}{a^2}\\mathbf{PQ'P}^*\\mathbb{E}[S_X]\n\\end{align*}\nwhere\n\\begin{align*}\n\t\\mathbf{Q}' = \\begin{bmatrix}\n\t\t\\frac{1-(1-ah(1+i\\delta))^K}{1+i\\delta} & 0 \\\\ 0 & \\frac{1-(1-ah(1-i\\delta))^K}{1-i\\delta},\n\t\\end{bmatrix}\n\\end{align*}\nand $\\mathbf{P}$ is defined in the previous section. We compute that\n\\begin{align}\n\t\\text{Tr}\\,\\mathbb{E}[\\bar{\\theta}_K] \\mathbb{E}[\\bar{\\theta}_K]^T = \\|\\mu_p\\|^2 + \\|\\text{bias}\\|^2 - \\frac{2b^2}{Kha^3(1+\\delta^2)}\\text{Re}\\{(1-i\\delta)(1-ah(1+i\\delta))^K \\} \\|\\mathbb{E}[S_X]\\|^2.\n\\end{align}\n\n\n\nThe other term is more complicated and needs to be approached more gingerly. Observe that\n\\begin{align*}\n\t\\text{Tr}\\,\\mathbb{E}[\\bar{\\theta}_K\\bar{\\theta}_K^T ] = \\frac{1}{K^2} \\sum_{i,j = 1}^K \\text{Tr}\\,\\mathbb{E}[\\theta_i\\theta_j^T] = \\frac{1}{K^2} \\left(\\sum_{i = 0}^{K-1} \\text{Tr}\\,\\mathbb{E}[\\theta_i\\theta_i^T] + 2\\sum_{i<j = 0}^{K-1} \\text{Tr}\\,\\mathbb{E}[\\theta_i\\theta_j^T] \\right).\n\\end{align*}\nWe take each term individually. To compute $\\mathbb{E}[\\theta_k\\theta_k^T]$, it is actually better to consider the covariance matrix of $\\theta_k$, $\\Sigma_{k} = \\mathbb{E}[\\theta_k\\theta_k^T] - \\mathbb{E}[\\theta_k]\\mathbb{E}[\\theta_k]^T$.\n\nWe first compute\n \\begin{align*}\n \t\\mathbb{E}[ \\theta_{k} \\theta_{k}^T ] = &(\\mathbf{I}-h\\mathbf{A}) \\mathbb{E}[\\theta_{k-1}\\theta_{k-1}^T] (\\mathbf{I}-h\\mathbf{A})^T + h^2 \\mathbf{D} \\mathbf{D}^T + h \\mathbf{I} \\\\&+ (\\mathbf{I}-h\\mathbf{A}) \\mathbb{E}[\\theta_{k-1}] \\mathbf{D}^T h + h\\mathbf{D} \\mathbb{E}[ \\theta_{k-1}^T](\\mathbf{I}-h\\mathbf{A})^T \\\\\n \n \t\\mathbb{E}[\\theta_{k}]\\mathbb{E}[\\theta_{k}]^T =& (\\mathbf{I}-h\\mathbf{A}) \\mathbb{E}[\\theta_{k-1}]\\mathbb{E}[\\theta_{k-1}]^T (\\mathbf{I}-h\\mathbf{A})^T + \\mathbf{D}\\mathbb{E}[\\theta_{k-1}]^T (\\mathbf{I}-h\\mathbf{A})^T \\\\& + (\\mathbf{I}-h\\mathbf{A}) \\mathbb{E}[\\theta_{k-1}]\\mathbf{D}+ h^2\\mathbf{D}\\D^T\n \\end{align*}\n which imply the following recurrence relation. Assuming $\\Sigma_0 = \\mathbf{0}$, we have\n \\begin{align*}\n \t\\Sigma_{k} = (\\mathbf{I}-h\\mathbf{A}) \\Sigma_{k-1}(\\mathbf{I}-h\\mathbf{A})^T + h\\mathbf{I} &= h \\sum_{n = 0}^{k-1}\\left( (\\mathbf{I}-h\\mathbf{A})(\\mathbf{I}-h\\mathbf{A})^{T}\\right)^n \\\\\n\t&= h\\sum_{n = 0}^{k-1}(\\mathbf{I}-(\\mathbf{A}+\\mathbf{A}^T)h + h^2\\mathbf{A}\\A^T)^n  \\\\ &= ((\\mathbf{A}+\\mathbf{A}^T)-h^2\\mathbf{A}\\A^T)^{-1}(\\mathbf{I}-(\\mathbf{I}-(\\mathbf{A}+\\mathbf{A}^T)h+\\mathbf{A}\\A^T h^2)^k).\n\\end{align*}\nLet $s = 1-2ah + h^2a^2(1+\\delta^2)$, then, by recalling that $\\mathbf{A}+\\mathbf{A}^T = 2a\\mathbf{I}$ and $ \\mathbf{A}\\A^T = a^2(1+\\delta^2)\\mathbf{I}$, the above sum is equal to\t$\\frac{1-s^k}{1-s} h\\mathbf{I}$.\nTherefore,\n\\begin{align}\n\t\\text{Tr}\\, \\Sigma_k = \\frac{2h(1-s^k)}{1-s}.\n\\end{align}\nMeanwhile note that\n\\begin{align*}\n\t\\mathbb{E}[\\theta_k] &= \\mu_p -\\mathbf{A}^{-1} (\\mathbf{I}-h\\mathbf{A})^k \\mathbf{D}.\n\\end{align*}\nTherefore,\n\\begin{align*}\n\t\\text{Tr}\\,\\mathbb{E}[\\theta_k]\\mathbb{E}[\\theta_k]^T = \\mathbb{E}[\\theta_k]^T\\mathbb{E}[\\theta_k] &= \\|\\mu_p\\|^2 + \\mathbf{D}^T(\\mathbf{I}-h\\mathbf{A}^T)^k \\mathbf{A}^{-T}\\mathbf{A}^{-1}(\\mathbf{I}-h\\mathbf{A})^k\\mathbf{D}-2\\mu_p^T\\mathbf{A}^{-1}(\\mathbf{I}-h\\mathbf{A})^k \\mathbf{D} \\\\\n\t& = \\|\\mu_p\\|^2 + \\frac{s^k b^2}{a^2(1+\\delta^2)}\\|S_X\\|^2-2\\mu_p^T\\mathbf{A}^{-1}(\\mathbf{I}-h\\mathbf{A})^k \\mathbf{D}.\n\\end{align*}\nWe now take the sum for each expression from $k = 0$ to $K-1$. We have\n\\begin{align*}\n\t\\sum_{i =0}^{K-1} \\text{Tr}\\,\\Sigma_i &= 2h\\left(\\frac{K}{1-s}- \\sum_{i = 0}^{K-1}\\frac{s^i}{1-s}\\right)  \\\\\n\t& = 2h\\left(\\frac{K}{1-s}- \\frac{1-s^K}{(1-s)^2}\\right)\n\n\\end{align*}\nand\n\\begin{align*}\n\t\\sum_{i = 0}^{K-1}\\mathbb{E}[\\theta_i]^T\\mathbb{E}[\\theta_i]\n\t& = K\\|\\mu_p\\|^2 + \\frac{(1-s^K) b^2}{a^2(1+\\delta^2)(1-s)}\\|S_X\\|^2-2\\mu_p^T\\mathbf{A}^{-1}(h\\mathbf{A})^{-1}(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K) \\mathbf{D}\\\\\n\t& = K\\|\\mu_p\\|^2 + \\frac{(1-s^K) b^2}{a^2(1+\\delta^2)(1-s)}\\|S_X\\|^2-2\\mu_p^Th^{-1}\\mathbf{A}^{-2}(\\mathbf{I}-(\\mathbf{I}-h\\mathbf{A})^K) \\mathbf{D}.\n\\end{align*}\n\nFor the cross-terms, observe that we may write\n\\begin{align}\n\t\\sum_{i<j = 0}^{K-1} \\theta_i\\theta_j^T = \\sum_{i = 0}^{K-1} \\theta_i \\sum_{j = i+1}^{K-1} \\theta_j^T\n\\end{align}\nwhich can be simplified further. First note that\n\\begin{align*}\n\t\\theta_j &= (\\mathbf{I}-h\\mathbf{A}) \\theta_{j-1} + \\mathbf{D} h + \\sqrt{h}\\xi_{j-1} \\\\\n\t& = (\\mathbf{I}-h\\mathbf{A})^{j-i} \\theta_i + h\\sum_{n = 0}^{j-1-i} (\\mathbf{I}-h\\mathbf{A})^{n} \\mathbf{D} + \\sqrt{h} \\sum_{n = 0}^{j-i-1} (\\mathbf{I}-h\\mathbf{A})^{n} \\xi_{j-1-n}.\n\\end{align*}\nPlugging this expression into the double sum above, we have\n\\begin{align}\n\t\\sum_{i =0}^{K-1} \\theta_i \\sum_{j = i+1}^{K-1}\\left[\\theta_i^T(\\mathbf{I}-\\mathbf{A}^Th)^{j-i} + h \\sum_{n = 0}^{j-1-i} \\mathbf{D}^T(\\mathbf{I}-\\mathbf{A}^Th)^n + \\sqrt{h}\\sum_{n = 0}^{j-i-1}\\xi_{j-1-n}^T(\\mathbf{I}-\\mathbf{A}^Th)^n \\right].\n\\end{align}\nTaking expectations, we have\n\\begin{align*}\n\t& \\sum_{i = 0}^{K-1}  \\sum_{j = i+1}^{K-1} \\left[\\mathbb{E}[\\theta_i\\theta_i^T](\\mathbf{I}-h\\mathbf{A}^T)^{j-i} + h\\mathbb{E}[\\theta_i]\\mathbf{D}^T \\sum_{n = 0}^{j-1-i} (\\mathbf{I}-h\\mathbf{A}^T)^n  \\right] \\\\\n\t= & \\sum_{i = 0}^{K-1}  \\sum_{j = i+1}^{K-1} \\left[\\mathbb{E}[\\theta_i\\theta_i^T](\\mathbf{I}-h\\mathbf{A}^T)^{j-i} + h\\mathbb{E}[\\theta_i]\\mathbf{D}^T (\\mathbf{A}^Th)^{-1}(\\mathbf{I}-(\\mathbf{I}-\\mathbf{A}^Th)^{j-i})  \\right].\n\\end{align*}\nCarrying out the computation for the first term, we have\n\\begin{align*}\n\tF = \\sum_{i = 0}^{K-1} \\mathbb{E}[\\theta_i\\theta_i^T] \\sum_{j = i+1}^{K-1} (\\mathbf{I}-\\mathbf{A}^Th)^{j-i} = \\sum_{i = 0}^{K-1} \\mathbb{E}[\\theta_i\\theta_i^T] (\\mathbf{I}-\\mathbf{A}^Th)(\\mathbf{A}^Th)^{-1}(\\mathbf{I}-(\\mathbf{I}-\\mathbf{A}^Th)^{K-1-i}).\n\\end{align*}\nFor the second term we have,\n\\begin{align*}\n\t&\\sum_{i = 0}^{K-1} \\mathbb{E}[\\theta_i]\\mu_p^T \\sum_{j = i+1}^{K-1} (\\mathbf{I}-(\\mathbf{I}-\\mathbf{A}^Th)^{j-i})  \\nonumber\\\\\n&\\qquad= \\sum_{i = 0}^{K-1} \\mathbb{E}[\\theta_i] \\mu_p^T \\left[(K-1-i)\\mathbf{I} - (\\mathbf{I}-\\mathbf{A}^Th)(\\mathbf{A}^Th)^{-1} (\\mathbf{I}-(\\mathbf{I}-\\mathbf{A}^Th)^{K-1-i})  \\right].\n\\end{align*}\nThe summations are difficult to compute precisely, so we compute them by direct evaluation instead.  For simplicity, we assume that $\\mu_p = \\mathbb{E}[S_X] = [0,0]^T$, $\\sigma_X$ and $\\sigma_\\theta$ are chosen such that $a=1$. For this scenario, the bias is zero and only the variance contributes to the MSE. The variance is\n\\begin{align*}\n\t\\text{Tr}\\,[\\text{Var}\\overline{\\theta}_K] = \\frac{1}{K^2}\\left( 2h\\left(\\frac{K}{1-s} - \\frac{1-s^K}{(1-s)^2} \\right) + 2\\text{Tr}\\, F\\right)\n\\end{align*}\nwhere $\\mathbb{E}[\\theta_i\\theta_i^T] = \\Sigma_i = \\frac{1-s^i}{1-s}h\\mathbf{I}.$\n\n\n\n\nIn Figure \\ref{fig:biasvardirecteval} we plot the variance for varying choices of $\\delta$. In this plots, $h = 0.001$, $K = 2\\times 10^5$, and $\\delta$ varies between zero and ten. We can clearly see that strengthening the irreversible perturbation leads to improvement of the squared bias and variance of the long term average estimator.\n\n\\begin{figure}\n\\centering\n\t\\includegraphics[width = 0.4\\textwidth]{firstmomfig-eps-converted-to}\n\t\\caption{Variance for different $\\delta$, fixed $h$.}\n\t\\label{fig:biasvardirecteval}\n\\end{figure}\n\n\n\\paragraph{Finite sample analysis for the quadratic  observable $\\phi(\\theta) = \\|\\theta\\|^2$.}\n\nThe previous finite sample results for the observable $\\phi(\\theta) = \\theta_1 + \\theta_2$ suggests that both the bias and variance of the long term average estimator goes down with a larger irreversible term. In this section, we show that this is actually a special case, and that when the observable is not linear, then the bias and variance may increase. We analyze the bias and variance of the long term average estimator of the observable $\\phi(\\theta) = \\|\\theta\\|^2$. Define\n\\begin{align}\n\t\\overline{\\phi} = \\int \\phi(\\theta) \\pi(\\theta) \\text{d}\\theta, \\;\\;\\;\\; \\overline{\\phi}_K = \\frac{1}{K}\\sum_{k = 0}^{K-1} \\phi(\\theta_k).\n\\end{align}\n\n  As before, we assume that $\\mu_p = [0,0]^T$, $\\mathbb{E}[S_X] = \\mathbf{0}$, and $\\sigma_x$ and $\\sigma_\\theta$ are chosen such that $a=1$. We compute $|\\mathbb{E}\\overline{\\phi}_K - \\overline{\\phi}|^2$ and $\\text{Var} \\overline{\\phi}_k$ and see how they vary with $\\delta$. From previous computations, we can show that\n\t\\begin{align}\n\t\\mathbb{E}\\overline{\\phi}_K = 2h \\left(\\frac{1}{1-s} - \\frac{1-s^K}{K(1-s)^2} \\right),\n\t\\end{align}\n\twhere $s = 1-2ah + a^2h^2(1+\\delta^2)$. Given this, the only term left to compute is the variance of the second moment of this observable:\n\\begin{align}\n\t\\mathbb{E}\\left[\\left( \\overline{\\phi}_K\\right)^2 \\right] &= \\frac{1}{K^2} \\mathbb{E}\\left[\\left(\\sum_{k = 0}^{K-1} \\theta_k^T\\theta_k \\right)^2 \\right] \\nonumber\\\\\n\t& = \\frac{1}{K^2}\\sum_{k = 0}^{K-1} \\mathbb{E}\\left[(\\theta_k^T\\theta_k)^2 \\right] + \\frac{2}{K^2}\\mathbb{E} \\sum_{k = 0}^{K-1} \\sum_{l = k+1}^{K-1} (\\theta_k^T\\theta_k) (\\theta_l^T \\theta_l).\n\\end{align}\nTo compute the first sum, consider the following:\n\\begin{align*}\n\t\\theta_k^T\\theta_k &= \\theta_{k-1}^T(\\mathbf{I}-h\\mathbf{A})^T(\\mathbf{I}-h\\mathbf{A})\\theta_{k-1} + h\\xi_{k-1}^T\\xi_{k-1} + 2\\sqrt{h} \\theta_{k-1}^T(\\mathbf{I}-h\\mathbf{A})^T \\xi_{k-1}\n\\end{align*}\nand so we have\n\\begin{align*}\n\t(\\theta_k^T \\theta_k)^2 =& s^2(\\theta_{k-1}^T\\theta_{k-1})^2 + h^2 (\\xi_{k-1}^T \\xi_{k-1})^2 + 4h (\\xi_{k-1}^T(\\mathbf{I}-h\\mathbf{A})\\theta_{k-1})^2+2s\\theta_{k-1}^T\\theta_{k-1} h \\xi_{k-1}^T \\xi_{k-1} \\\\\n\t&+ 4 s\\sqrt{h}(\\theta_{k-1}^T\\theta_{k-1}) \\theta_{k-1}^T(\\mathbf{I}-h\\mathbf{A})^T\\xi_{k-1}+4h^{3\/2}(\\xi_{k-1}^T\\xi_{k-1})\\theta_{k-1}^T(\\mathbf{I}-h\\mathbf{A})^T\\xi_{k-1}.\n\\end{align*}\nTaking the expectation, we have\n\\begin{align*}\n\t\\mathbb{E}[(\\theta_k^T\\theta_k)^2] = &s^2 \\mathbb{E}[(\\theta_{k-1}^T\\theta_{k-1})^2] + h^2 \\mathbb{E}[(\\xi_{k-1}^T\\xi_{k-1})^2] + 4h\\mathbb{E}\\left[(\\xi_{k-1}^T(\\mathbf{I}-h\\mathbf{A})\\theta_{k-1})^2 \\right] \\\\&+2sh\\mathbb{E}\\left[(\\theta_{k-1}^T\\theta_{k-1})(\\xi_{k-1}^T\\xi_{k-1}) \\right].\n\\end{align*}\nAfter simplifying, we arrive at the following recurrence relation:\n\\begin{align}\n\t\\mathbb{E}\\left[(\\theta_k^T\\theta_k)^2 \\right] = s^2 \\mathbb{E}\\left[ (\\theta_{k-1}^T\\theta_{k-1})^2\\right] + 8h^2 + 8sh \\mathbb{E}\\left[\\theta_{k-1}^T\\theta_{k-1}\\right].\n\\end{align}\nLet $\\beta_k = \\mathbb{E}[(\\theta_k^T\\theta_k)^2]$, $\\zeta_k = 8sh\\mathbb{E}[\\theta_k^T\\theta_k]$, and $\\kappa = 8h^2$. We have the following recurrence, which we solve\n\\begin{align*}\n\t\\beta_k &= s^2 \\beta_{k-1} + \\zeta_{k-1} + \\kappa \\\\\n\t& = s^{2k} \\beta_0 + \\sum_{n = 0}^{k-1} s^{2n} \\zeta_{k-n-1} + \\sum_{n = 0}^{k-1} s^{2n} \\kappa.\n\\end{align*}\nFrom previous for the term $\\zeta_k$, we have\n\\begin{align*}\n\t\\beta_k &= \\sum_{n = 0}^{k-1} s^{2n} 8sh \\cdot 2h \\frac{1-s^{k-n-1}}{1-s} + \\kappa \\frac{1-s^{2k}}{1-s^2} \\\\\n\t&= \\frac{16sh^2}{1-s} \\sum_{n = 0}^{k-1} (s^{2n} - s^{k+n-1}) + \\frac{8h^2(1-s^{2k})}{1-s^2} \\\\\n\t& = \\frac{16sh^2}{1-s} \\left(\\frac{1-s^{2k}}{1-s^2} - s^{k-1} \\frac{1-s^k}{1-s} \\right) + \\frac{8h^2(1-s^{2k})}{1-s^2}.\n\\end{align*}\nNext we compute the summation of the cross terms. Define $R_k$ such that\n\\begin{align}\n\t\\sum_{k = 0}^{K-1} R_k = \\sum_{k = 0}^{K-1} \\sum_{l = k+1}^{K-1} \\mathbb{E}\\left[(\\theta_k^T\\theta_k)(\\theta_l^T\\theta_l)\\right].\n\\end{align}\nWe write\n\\begin{align*}\n\t\\theta_l^T \\theta_l& = s\\theta_{l-1}^T\\theta_{l-1} + h \\xi_{l-1}^T\\xi_{l-1} + 2\\sqrt{h} \\xi_{l-1}^T (\\mathbf{I}-h\\mathbf{A}) \\theta_{l-1} \\\\\n\t& = s^{l-k} \\theta_k^T\\theta_k + \\sum_{n = 0}^{l-k-1} hs^n \\xi_{l-n-1}^T\\xi_{l-n-1} + 2\\sqrt{h} s^n \\xi_{l-n-1}^T(\\mathbf{I}-h\\mathbf{A}) \\theta_{l-n-1}.\n\\end{align*}\nThis implies that\n\\begin{align*}\n\tR_k &= \\sum_{l = k+1}^{K-1} \\left(s^{l-k}\\mathbb{E}[ (\\theta_k^T\\theta_k)^2]\t+ \\sum_{n = 0}^{l-k-1} 2h s^n \\mathbb{E}[\\theta_k^T\\theta_k] \\right)\\\\\n\t& = \\sum_{l = k+1}^{K-1} \\beta_ks^{l-k} + 2h\\mathbb{E}[\\theta_k^T\\theta_k]\\sum_{n = 0}^{l-k-1} s^n  \\\\\n\t& = \\sum_{l = k+1}^{K-1} \\beta_k s^{l-k} + 2h\\mathbb{E}[\\theta_k^T\\theta_k] \\frac{1-s^{l-k}}{1-s} \\\\\n\t& = \\beta_k \\sum_{l = k+1}^{K-1} s^{l-k} + \\frac{2h\\mathbb{E}[\\theta_k^T\\theta_k]}{1-s} \\sum_{l = k+1}^{K-1} 1-s^{l-k} \\\\\n\t& = \\beta_k \\frac{s-s^{K-k}}{1-s} + \\frac{2h\\mathbb{E}[\\theta_k^T\\theta_k]}{1-s}\\left(K-1-k - \\frac{s-s^{K-k}}{1-s} \\right).\n\\end{align*}\nTo summarize, we have\n\\begin{align}\n\t\\mathbb{E}\\left[\\left(\\overline{\\phi}_K \\right)^2 \\right] = \\frac{1}{K^2} \\sum_{k = 0}^{K-1}\\left( \\beta_k +2 R_k \\right),\n\\end{align}\nthe squared bias is $(\\mathbb{E}\\overline{\\phi}_K - 1)^2$ and the variance is $\\mathbb{E}[(\\overline{\\phi}_K)^2] - \\mathbb{E}[\\overline{\\phi}_K]^2$. These expressions are not simplifiable easily, so we plot these expressions and study their trends. In Figure \\ref{fig:secobs}, we plot the squared bias and variance for fixed $h$ and $K$ and varying $\\delta$. In these plots, $h = 0.001$, $K = 2\\times 10^5$, and $\\delta$ varies between zero and ten. Notice that for these choices, both the squared bias and variance increases as $\\delta$ grows, showing that for irreversibility provides no benefit, and in fact, harms the performance of the standard estimator.\n\n\\begin{figure}\n\\centering\n\t\\includegraphics[width = \\textwidth]{secmomanalysisfig-eps-converted-to}\n\t\\caption{Bias and variance of $\\phi(\\theta) = \\|\\theta\\|^2$ for varying levels of irreversibility.}\\label{fig:secobs}\n\\end{figure}\n\n\\begin{remark}\\label{R:IrrevNotImprovement}\nWhen discretization is considered, sampling properties in this simple example in which the drift is proportional to the state cannot be improved upon using irreversibility. Let us now explain this phenomenon from a theoretical point of view. In \\cite{rey2016improving}, the authors show that adding an irreversible perturbation to the generator of the diffusion process may decrease the spectral gap, and will never increase it. They further prove that in the continuous case, decreasing the spectral gap then decreases the asymptotic variance. However this improvement is not strict, that is, irreversibility is only guaranteed to not increase the spectral gap.\n\nMeanwhile, in \\cite{lelievre2013optimal}, the authors consider irreversibility only in the context of linear systems, and rigorously study optimal irreversible perturbations that accelerate convergence to the invariant distribution. Their results show that when the drift matrix is proportional to the identity matrix, the spectral gap cannot be widened. Proposition 4 in \\cite{lelievre2013optimal} shows that the leading nonzero eigenvalue of the irreversibly perturbed drift matrix is bounded above by the leading nonzero eigenvalue of the original drift matrix and below by the trace of the original drift matrix over the dimension of the state space. The lower bound is then the optimal spectral gap. For a drift matrix that is a multiple of the identity, the upper and lower bounds are the same, which implies that the spectral gap can never decrease from its original value in the continuous case. After factoring in discretization, the irreversible perturbation increases stiffness of the system, which contributes to increased bias and variance in the resulting estimator.\n\\end{remark}\n\n\\end{appendices}\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nExtra-solar planets have continued to be discovered, since the first detection of a giant planet orbiting a solar-like star \\citep[][]{Mayor-1995}, including several planets in the Neptune-mass regime \\citep[e.g.][]{McArthur-2004, Santos-2004b, butler-2004, Vogt-2005, Rivera-2005, Bonfils-2005, Bonfils-2007, Udry-2006, Udry-2007, Lovis-2006, Melo-2007}. This has been possible because of to the development of a new generation of instruments capable of radial-velocity measurements of unprecedented quality. One such instrument is undoubtedly the ESO high-resolution HARPS fiber-fed echelle spectrograph, especially designed for planet-search programmes and asteroseismology. HARPS has proved to be the most precise spectro-velocimeter to date, which has achieved an instrumental radial-velocity accuracy superior to 1 $m s^{-1}$ for many years \\citep[][]{Mayor-2003, Lovis-2005}.\n\n\nA by-product of these planet searches are the high-quality spectra, of high S\/N and high-resolution, that are collected in every run in a consistent and systematic way for selected candidate stars. The result is a vast quantity of spectra, mainly of solar-type stars, which can be analyzed in a consistent way for the determination of accurate spectroscopic stellar parameters.\n\nWe present a catalogue of spectroscopic stellar parameters for the HARPS Guaranteed Time Observations (GTO) ``high precision'' sample. In Sect. 2 we describe the observations with HARPS. Section 3 describes the procedure used to derive accurate spectroscopic stellar parameters. In Sect. 4, we present the results for our sample of 451 solar-type stars. We then compare, in Sect. 5, our results with those of several other authors obtained using different methods. In Sect. 6, we present a useful calibration of the effective temperature as a function of $B-V$ and [Fe\/H]. Finally, we perform a simple statistical analysis of our results in the context of the metallicity-planet correlation.\n\n\\section{Stellar Sample \\& Observations}\n\nThe objective of the HARPS ``high-precision'' GTO project is to detect very low-mass extra-solar planets by increasing the radial velocity measurement accuracy below 1 $m s^{-1}$. The stars in this project were selected from a volume-limited stellar-sample observed by the CORALIE spectrograph at La Silla observatory \\citep[][]{Udry-2000}. These stars are slowly-rotating, non-evolved, and low-activity stars that presented no obvious radial-velocity variations at the level of the CORALIE measurement precision (typically 5-10 $m s^{-1}$ and dominated by photon-noise). Southern planet hosts were also added to this program to search for further low-mass planets. Therefore, a significant number of stars (66) from this catalogue are already confirmed planet hosts. For more details we point the reader to a description of this sample by \\citet[][]{Mayor-2003}. In summary, the sample is composed of 451 FGK stars with apparent magnitudes that range from 3.5 to 10.2 and have distances less than 56 parsec.\n\nAsteroseismology work using HARPS has proved that the precision of the instrument is no longer set by instrumental characteristics but rather by the stars themselves \\citep[][]{Mayor-2003, Bouchy-2005}. Even a ``quiet'' G dwarf shows oscillation modes of several tens of $cm s^{-1}$ each, which might add up to radial-velocity amplitudes as large as several $m s^{-1}$ \\citep[][]{Bouchy-2005}. As a consequence, any exposure with a shorter integration time than the oscillation period of the star, might fall arbitrarily on a peak or on a valley of these mode interferences and thus introduce additional radial-velocity ``noise''. This phenomenon could seriously compromise the ability to detect very low-mass planets around solar-type stars by means of the radial-velocity technique.\n\nTo average out stellar oscillations, the observations are designed to last at least 15 minutes on the target, splitting them into several exposures as required, to avoid CCD saturation. The final result is a high-quality spectrum for each star in the sample. \n\nThe individual spectra of each star were reduced using the HARPS pipeline and combined using IRAF \\footnote{IRAF is distributed by National Optical Astronomy Observatories, operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation, U.S.A.} after correcting for its radial velocity. The final spectra have a resolution of R $\\sim$ 110\\,000 and signal-to-noise ratios that vary from $\\sim 70$ to $\\sim 2000$, depending on the amount and quality of the original spectra (90\\% of the spectra have S\/N higher than 200).\n\n\n\n\\section{Stellar parameters - Spectroscopic Analysis}\n\n\nWe followed the procedure presented in \\citet[][]{Santos-2004b} for deriving stellar parameters and metallicities, where [Fe\/H] is used as a proxy for overall stellar metallicity. As reported in \\citet[][]{Sousa-2007} however, we attempted to increase the number of spectral lines and decrease the errors involved in the determination of stellar parameters. Therefore, we used the large, extended line-list presented in \\citet[][]{Sousa-2007} and applied this method to the sample.\n\nThe spectroscopic analysis was completed assuming LTE, and using the 2002 version of the code MOOG\\footnote{The source code of MOOG2002 can be\ndownloaded at http:\/\/verdi.as.utexas.edu\/moog.html} \\citep[][]{Sneden-1973} and a grid of Kurucz Atlas\\,9 plane-parallel model atmospheres \\citep[][]{Kurucz-1993}. Stellar parameters and metallicities were derived as in previous works \\citep[][]{Santos-2004b,Santos-2005,Sousa-2006}, based on the equivalent widths of \\ion{Fe}{i} and \\ion{Fe}{ii} weak lines, by imposing excitation and ionization equilibrium.\n\n\\begin{table}[t]\n\\caption[]{Examples of atomic parameters and measured solar equivalent widths for\n\\ion{Fe}{ii} and \\ion{Fe}{i} lines.}\n\\begin{tabular}{lcccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\n$\\lambda$ (\\AA) & $\\chi_{l}$ & $\\log{gf}$ & Ele. & EW$_{\\sun}$ (m\\AA)\\\\\n\\hline\n4508.28  &   2.86  &   -2.403 & FeII  &   87.3 \\\\\n4520.22  &   2.81  &   -2.563 & FeII  &   81.9 \\\\\n4523.40  &   3.65  &   -1.871 & FeI   &   44.2 \\\\\n4531.62  &   3.21  &   -1.801 & FeI   &   66.4 \\\\\n4534.17  &   2.86  &   -3.203 & FeII  &   53.7 \\\\\n4537.67  &   3.27  &   -2.870 & FeI   &   17.4 \\\\\n4541.52  &   2.86  &   -2.762 & FeII  &   71.5 \\\\\n4551.65  &   3.94  &   -1.928 & FeI   &   29.1 \\\\\n4554.46  &   2.87  &   -2.752 & FeI   &   37.4 \\\\\n4556.93  &   3.25  &   -2.644 & FeI   &   26.3 \\\\\n4576.34  &   2.84  &   -2.947 & FeII  &   64.9 \\\\\n  ...    &    ...  &     ...  & ...   &   ...  \\\\\n\\hline\n\\end{tabular}\n\\label{tab1}\n\\end{table}\n\n\\subsection{A large list of stable lines}\n\nWe considered the line list used for the spectroscopic analysis given in \\citet[][]{Sousa-2007}. As a first step, to check the quality of the line list, we derived preliminary stellar parameters for our sample. The stability of the lines were then checked by comparing the abundances of each individual line with the mean abundance obtained for each star.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=8cm]{9698fig1.ps}\n\\caption[]{Selection of the stable lines from the list provided by \\citet[][]{Sousa-2007}. We present the mean difference between each line abundance and the mean abundance of the respective star as a function of the wavelength. We divided the analysis into three different ranges of effective temperatures. The dashed lines represent the 1.5 sigma threshold. Lines in the shaded area ($< 4500$\\AA) were also removed since this region was strongly affected by blending effects, in particular for cooler stars.}\n\\label{fig_linelist}\n\\end{figure}\n\nSince the stars in our sample have a significant range of spectral type with effective temperatures ranging from $\\sim$4500 K to $\\sim$6400 K, we expect a significant different local environment around each line, due to variations in blending effects. To take this problem into account, we divided our sample into three subsamples, taking into consideration the preliminary determination of the effective temperature ($T_{\\mathrm{eff}} < 5300K$, $5300K < T_{\\mathrm{eff}} < 6000 K$, $T_{\\mathrm{eff}} > 6000K$).\n\n\nThis allows the search for systematic non-stable lines in the large list for the three ranges of stellar spectral class. The lines that showed a systematic offset from the mean value, that was larger than 1.5 sigma were considered ``bad'' lines, which provide false abundance measurements either due to line-blending events or because the continuum could not be fitted readily in a particular spectral region. Lines at wavelengths lower than 4500\\AA\\ were removed because, for cooler stars, this region is highly line crowded and therefore more likely to produce blending effects. These lines were removed from the original large line list.\n\nTable \\ref{tab1} shows the final list composed of 263 \\ion{Fe}{i} and 36 \\ion{Fe}{ii} weak lines to be used on HARPS spectra for the determination of the stellar parameters and iron abundance.\n\nWe note that this procedure is difficult to apply to cooler stars, because the spectra of these stars are crowded with lines or the atomic parameters that we adopt become inaccurate as we study spectral classes increasingly different from solar. These two effects can be seen clearly in the dispersion of measurements shown in Fig. \\ref{fig_linelist} (upper panel) that is significantly higher than for the two other cases (middle and lower panel).\n\nDue to these problems, the solutions for cooler stars are more difficult to converge into the ``true'' stellar parameters. In a few cases, we had to discard some lines from the final stellar-line list with equivalent widths greater that 200 m\\AA\\ and smaller than 10m\\AA\\ to overcome this issue.\n\n\n\\subsection{Using ARES}\n\nThe equivalent widths of the lines were measured automatically in all spectra using the ARES\\footnote{The ARES code can be downloaded at http:\/\/www.astro.up.pt\/$\\sim$sousasag\/ares} code \\citep[Automatic Routine for line Equivalent widths in stellar Spectra - ][]{Sousa-2007}. \n\nThe ARES code considers a one-dimensional spectrum, a list of absorption lines to be measured, and some calibration parameters. First, ARES estimates the continuum about each individual line and calculates a local normalization. The location and number of lines for which a Gaussian fit can be attempted, are then determined. After fitting the local, normalized spectrum, ARES uses the fit parameters to compute the equivalent width. The result is then outputted to a file.\n\nA number of ARES parameters were fixed during the analysis of all spectra in our sample: $smoothder = 4$, $space = 3$, $lineresol = 0.07$, $miniline = 2$. The \\textit{smoothder} value was used to reduce the noise in the computed derivatives of the local spectrum to find the number and location of the lines to be fitted. The \\textit{space} parameter was set to be the total length of spectrum about each line required to fit the local continuum; therefore, for a value of 3 \\AA, ARES fitted the continuum over a total interval of 6 \\AA. The \\textit{lineresol} parameter defines the minimum separation allowed for consecutive lines. This parameter is used to limit the effect of noise on the determination of the number and location of lines, i.e. if ARES identify lines that are close together according to the value indicated in \\textit{lineresol}, then it ignores one line and then measures a mean position. The \\textit{miniline} parameter is the lower value of EW that will be returned by ARES in the output file.\n\n\\begin{center}\n\\begin{table}[th]\n\\caption[]{Dependence of the \\textit{rejt} parameter on the S\/N of HARPS spectra. ARES users should use this table as a reference for the rejt parameter.}\n\\begin{tabular}{cc|cc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nS\/N condition   & \\textit{rejt}&            S\/N condition        & \\textit{rejt}\\\\\n\\hline\n~~~~~~~~~~   S\/N $<$ 100 \t&\t0.985 & \t\t\t\t&\t       \\\\\n100~  $\\le$ S\/N $<$ 125 \t&\t0.990 & 250~  $\\le$ S\/N $<$ 300~\t&\t0.995  \\\\\n125~  $\\le$ S\/N $<$ 150 \t&\t0.991 & 300~  $\\le$ S\/N $<$ 500~\t&\t0.996  \\\\\n150~  $\\le$ S\/N $<$ 200 \t&\t0.992 & 500~  $\\le$ S\/N $<$ 800~\t&\t0.997  \\\\\n200~  $\\le$ S\/N $<$ 225 \t&\t0.993 & 800~  $\\le$ S\/N $<$ 1500\t&\t0.998  \\\\\n225~  $\\le$ S\/N $<$ 250 \t&\t0.994 & 1500  $\\le$ S\/N ~~~~~~~~~~~~    &\t0.999  \\\\\n\\hline\n\\end{tabular}\n\\label{tab2}\n\\end{table}\n\\end{center}\n\n\n\n\n\n\\begin{table*}[t]\n\\caption[]{Sample table of the HARPS GTO ``high precision'' spectroscopic catalogue. We provide the star name, effective temperature, surface gravity, micro turbulence, [Fe\/H], the number of iron lines used in the spectroscopic analysis, a mass estimate, the surface gravity computed from the parallaxes, the luminosity of the star, and an indication of whether it hosts a planet. In the electronic table, we also indicate the error in the luminosity and the computed absolute magnitude of the stars.}\n\\begin{tabular}{lcccrccccc}\n\\hline\n\\hline\nStar ID     & T$_{\\mathrm{eff}}$ & $\\log{g}_{spec}$ & $\\xi_{\\mathrm{t}}$ & \\multicolumn{1}{c}{[Fe\/H]} & N(\\ion{Fe}{i},\\ion{Fe}{ii}) & Mass       & $\\log{g}_{hipp}$ & Luminosity & planet host?\\\\\n            & [K]                & [cm\\,s$^{-2}$]   &  [km\\,s$^{-1}$]    &                            &                             & [M$_{\\sun}$]     & [cm\\,s$^{-2}$] &   [$L_{\\sun}$]   &  \\\\\n\\hline\n\n\n...                    &        ...  &  ...  &  ...  &   ...  &  ...  &  ...  &   ...    & ...     &...\\\\\n\\object{HD\\,63454} &            4840   \\ $\\pm$\\      66  & 4.30   \\ $\\pm$\\     0.16 &  0.81  \\ $\\pm$\\     0.14 &   0.06\\ $\\pm$\\ 0.03  &  227, 27  &   0.66     &        4.55      &     0.25    &      yes   \\\\\n\\object{HD\\,63765} &             5432   \\ $\\pm$\\      19  & 4.42   \\ $\\pm$\\     0.03 &  0.82  \\ $\\pm$\\     0.03 &  -0.16\\ $\\pm$\\ 0.01  &  249, 32  &   0.83     &        4.51     &      0.55    &       no    \\\\\n\\object{HD\\,65216 } &            5612  \\ $\\pm$\\       16  & 4.44    \\ $\\pm$\\    0.02 &  0.78  \\ $\\pm$\\     0.03 &  -0.17\\ $\\pm$\\ 0.01  &  256, 34  &   0.87     &        4.48     &      0.71    &      yes   \\\\\n\\object{HD\\,65277} &             4802   \\ $\\pm$\\      88  & 4.43    \\ $\\pm$\\    0.18 &  0.55  \\ $\\pm$\\     0.34 &  -0.31\\ $\\pm$\\ 0.04  &  257, 32   &  0.49     &        4.49     &      0.21     &      no    \\\\\n\\object{HD\\,65562} &             5076   \\ $\\pm$\\      47  & 4.39    \\ $\\pm$\\    0.09 &  0.45  \\ $\\pm$\\     0.18 &  -0.10\\ $\\pm$\\ 0.03  &  252, 34   &  0.72     &        4.51      &     0.37     &      no    \\\\\n\\object{HD\\,65907A} &            5945   \\ $\\pm$\\      16  & 4.52   \\ $\\pm$\\     0.02 &  1.05  \\ $\\pm$\\     0.02 &  -0.31\\ $\\pm$\\ 0.01 &   256, 34  &   0.93     &        4.37      &     1.24    &       no    \\\\\n\\object{HD\\,66221} &             5635   \\ $\\pm$\\      25  & 4.40    \\ $\\pm$\\    0.04 &  0.92  \\ $\\pm$\\     0.03  &  0.17\\ $\\pm$\\ 0.02  &  260, 35  &   0.96     &        4.40      &     0.95     &      no   \\\\ \n\\object{HD\\,66428} &             5705   \\ $\\pm$\\      27  & 4.31    \\ $\\pm$\\    0.06 &  0.96  \\ $\\pm$\\     0.03 &   0.25\\ $\\pm$\\ 0.02  &  258, 35  &   1.01      &       4.31      &     1.28     &     yes   \\\\\n\\object{HD\\,67458} &             5891   \\ $\\pm$\\      12  & 4.53    \\ $\\pm$\\    0.02 &  1.04  \\ $\\pm$\\     0.01 &  -0.16\\ $\\pm$\\ 0.01  &  256, 35   &  0.98      &       4.45     &      1.02    &       no    \\\\\n\\object{HD\\,68607 } &            5215    \\ $\\pm$\\     45  & 4.41    \\ $\\pm$\\    0.08 &  0.82  \\ $\\pm$\\     0.08  &  0.07\\ $\\pm$\\ 0.03  &  251, 35  &   0.81      &       4.52     &      0.45    &       no    \\\\\n\\object{HD\\,68978A} &            5965   \\ $\\pm$\\      22  & 4.48    \\ $\\pm$\\    0.02  & 1.09  \\ $\\pm$\\     0.02  &  0.05\\ $\\pm$\\ 0.02  &  260, 34   &  1.08      &       4.43     &      1.24    &       no    \\\\\n\\object{HD\\,69655 } &            5961   \\ $\\pm$\\      12  & 4.44     \\ $\\pm$\\   0.03  & 1.15  \\ $\\pm$\\     0.01 &  -0.19\\ $\\pm$\\ 0.01  &  249, 34  &   0.98      &       4.36     &      1.33    &       no    \\\\\n\\object{HD\\,69830} &             5402   \\ $\\pm$\\      28  & 4.40     \\ $\\pm$\\   0.04  & 0.80  \\ $\\pm$\\     0.04 &  -0.06\\ $\\pm$\\ 0.02  &  255, 36  &   0.82       &      4.46     &      0.60    &      yes   \\\\\n\\object{HD\\,70642 } &            5668   \\ $\\pm$\\      22  & 4.40     \\ $\\pm$\\   0.04  & 0.82  \\ $\\pm$\\     0.03  &  0.18\\ $\\pm$\\ 0.02  &  253, 36  &   0.98       &      4.42     &      0.95    &      yes   \\\\\n...                    &        ...  &  ...  &  ...  &   ...  &  ...  &  ...  &   ...    & ...     &...\\\\\n\n\n\n\n\\hline\n\\end{tabular}\n\n\\label{tab3}\n\\end{table*}\n\nAs described in \\citet[][]{Sousa-2007}, the most important parameter for the correct automatic determination of equivalent widths using ARES is the \\textit{rejt} parameter. This parameter strongly depends on the signal-to-noise ratio (S\/N) of the spectra. Values close to 1 are indicative of lower spectrum noise levels. Table \\ref{tab2} shows how the parameter was defined by considering the spectrum S\/N (measured at 6050 \\AA). Future ARES users may use this table as a reference for the most appropriate choice of the \\textit{rejt} parameter. This table was compiled for HARPS spectra, but the data should be applicable to other high-resolution spectra.\n\n\n\\subsection{Stellar mass and luminosity}\n\nStellar masses were estimated, as in previous works e.g. \\citet[][]{Santos-2004b}, by interpolating the theoretical isochrones of \\citet[][]{Schaller-1992} and \\citet[][]{Schaerer-1993b,Schaerer-1993a}, using $M_V$ computed for the Hipparcos parallaxes and V magnitudes \\citep[][]{ESA-1997}, a bolometric correction from \\citet[][]{Flower-1996}, and the effective temperature derived from the spectroscopic analysis. We adopt a typical relative error of 0.10 $M_{\\sun}$ for the masses. In some cases, the value of mass was not determined because the calculation would have required a large extrapolation of the existing isochrones.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=6.9cm]{9698fig2.ps}\n\\caption[]{In the top panel, we show the distribution of the sample stars in the H-R diagram. The filled circles represent the planet hosts in our sample. We also plot some evolutionary tracks computed with CESAM for a 1.0, 1.1 and 1.2 M$_{\\sun}$. In the bottom panel, we present the metallicity distribution of the sample.}\n\\label{fig_analise}\n\\end{figure}\n\n\nThe luminosity was computed by considering the Hipparcos parallaxes, V magnitude and the bolometric correction. Its error was derived based on the parallax errors from Hipparcos, which was the main source of uncertainty in the calculation of luminosity. The error in the luminosity is available in the electronic table only.\n\n\n\\section{The spectroscopic catalogue of the HARPS GTO ``high precision'' sample}\n\\begin{figure*}[hth]\n\\centering\n\\includegraphics[width=17cm]{9698fig3.ps}\n\\caption[]{Comparison of our spectroscopic results for the effective temperature with other values found in the literature. We also show the comparison with IRFM using either the Kurucz or Phoenix models.}\n\\label{fig_spec_teff}\n\\end{figure*}\n\nFigure \\ref{fig_analise} presents some characteristics of the sample. The top plot shows the distribution of the sample on the Hertzsprung-Russell diagram, where we represent evolutionary tracks for 1.0, 1.1, and 1.2 $M_{\\sun}$, computed using the CESAM code \\citep[][]{Morel-1997, Marques-2008} \\footnote{http:\/\/www.astro.up.pt\/corot\/models\/}. Moreover, we present the typical error boxes on this specific diagram, where the error in the luminosity is dependet mostly on the error in the parallax of each star, and the error in the temperature is derived from our spectroscopic method. This plot shows that the sample is composed mainly of main-sequence solar-type stars. In the bottom plot, we present the metallicity distribution that has a mean value of about $-0.09$ dex. This is compatible with the comparison samples presented in the work of \\citet[][]{Santos-2004b}, when taken into consideration the amount of planet hosts (more metal-rich on average) that were added to the original HARPS GTO ``high precision'' sample.\n\nTable \\ref{tab3} shows a sample of the catalogue with some of the stellar parameters that have been determined. This catalog is electronically available on CDS and also at the author webpage\\footnote{http:\/\/www.astro.up.pt\/$\\sim$sousasag\/harps\\_gto\\_catalogue.html}.\n\n\n\n\\section{Comparison with previous works}\n\nWe compared our results with those of other authors to acess the consistency between different methods, using spectroscopy or photometry to derive stellar parameters. We used the spectroscopic results of \\citet[][]{Edvardsson-1993}, \\citet[][]{Bensby-2003}, \\citet[][]{Valenti-2005}, \\citet[][]{Santos-2004b}, \\citet[][]{ Fuhrmann-2004}, and the photometric results of \\citet[][]{Nordstrom-2004}. We also checked the consistency with results derived using the Infra-Red Flux Method (IRFM), following the work of \\citet[][]{Casagrande-2006}.\n\nFrom this point on, all differences presented are relative to our own work, i.e. ``other works''$-$``this work''.\n\n\n\\subsection{Effective Temperature}\n\nFor the effective temperature the results of the comparison, presented in Fig. \\ref{fig_spec_teff}, are compatible with other spectroscopic studies. We found a mean difference of only $-18 \\pm 47$ K for the 20 stars in common with the work of \\citet[][]{Bensby-2003}. These authors used the same procedure as this paper for determining the effective temperature, but in their case for different stellar-atmosphere models (Uppsala MARCS code). In their work, they measured equivalent widths from spectra taken with the FEROS spectrograph on the 1.52\\,m ESO telescope.\n\n\\citet[][]{Fuhrmann-2004} used a different procedure for the determination of the effective temperature, based on hydrogen line profiles. However, our and their results are in excellent agreement for the 15 stars that we have in common, for which we find a mean difference of $+1 \\pm 44$ K.\n\n\\citet[][]{Valenti-2005} presented stellar parameters for 1040 F, G, and K dwarfs. We were able to complete a comparison using data for a larger number of stars and found good agreement with a mean difference $-16 \\pm 44$ K in the effective temperature. However, a small offset can be seen in the plot for higher temperatures ($T_{\\mathrm{eff}} > 6000$~K). These authors used spectral synthesis for data taken with HIRES, UCLES, and the Lick Observatory Hamilton Echelle Spectrometer.\n\n\\begin{figure*}[htp]\n\\centering\n\\includegraphics[width=17cm]{9698fig4.ps}\n\\caption[]{Comparison of our spectroscopic results for surface gravity with other measurements in the literature.}\n\\label{fig_logg}\n\\end{figure*}\n\nWe compared our measurements derived here with our own previous results \\citep[][]{Santos-2004b}, which were obtained using the same procedure but a different line list. The comparison indicated good agreement for effective temperatures with a mean difference of $-18 \\pm 57$ K. For this specific comparison, most stars in common with the sample of \\citet[][]{Santos-2004b} were observed with the CORALIE spectrograph and only a few with the FEROS spectrograph.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn their careful spectroscopic analysis of 189 nearby field F and G disk dwarfs, \\citet[][]{Edvardsson-1993} determined effective temperatures using  photometry. For the stars in common, we found very similar effective temperatures, with a mean difference of only $-27 \\pm 71$~K. \n\nWe also compared our effective temperatures with those in \\citet[][]{Nordstrom-2004} that used the IRFM--based calibration of \\citet[][]{Alonso-1996}. There is a systematic deviation between our results and those of \\citet[][]{Nordstrom-2004}, with a mean difference of $-108 \\pm 68$~K. In particular, a clear offset appears to be present for effective temperatures above $5500$~K. We do not know the reason for this, but it is interesting to note that the \\citet[][]{Alonso-1996} temperature scale is based on the Vega absolute calibration derived by \\citet[][]{Alonso-1994}. In that work, it was found that the Vega absolute calibration derived from hot ($T_{\\mathrm{eff}} > 6000$~K) and cool ($T_{\\mathrm{eff}} < 5000$~K) stars differed by a few percent, which implied that effective temperatures for cool and hot stars were not on exactly the same scale. The Vega absolute calibration used by \\citet[][]{Alonso-1996} was a weighted average of that for both hot and cool stars: it is in fact interesting to note that the offset in Fig. \\ref{fig_spec_teff} appears approximately where the method showed some problems.\n\n\n\\subsection{Effective Temperature - The IRFM}\n\nCasagrande et al. (2006) implemented a revised form of the IRFM, as described in their paper. They found good agreement with various spectroscopic temperature scales; they explained a $\\sim100$~K systematic difference with respect to previous implementation of the IRFM, as being most likely due the adoption of the new Vega zero points and absolute calibration of 2MASS \\citep[][]{Cohen-2003}. The comparison with a large set of solar analogs confirm that the temperature scale is calibrated well for solar-type stars \\citep[][]{Casagrande-2006}. \n\n\n\nThe main concept behind the IRFM is to assume that the ratio between the bolometric and infrared monochromatic fluxes is a sensitive indicator of effective temperature. The bolometric flux is recovered using multi--band photometry, right across the optical and near--infrared. The flux outside the photometric filters is estimated using model atmospheres, and the infrared monochromatic fluxes are computed from 2MASS $JHK_K$ photometry. Accurate infrared photometry is crucial for recovering effective temperatures precisely and only stars with total photometric errors (i.e. ``j\\_''$+$``h\\_''$+$``k\\_msigcom'' as given from 2MASS) smaller than 0.10~mag are used. \n\n\nFor 66 stars in the HARPS GTO sample we obtained accurate Johnson--Cousins $BV(RI)_C$ colors from the General Catalogue of Photometric Data \\citep[][]{Mermilliod-1997} and $JHK_S$ from 2MASS, and executed the IRFM as described in \\citet[][]{Casagrande-2006}. To increase the number of stars for which it was possible to apply the IRFM, we tested the results only if Johnson $BV$ and 2MASS $JHK_S$ magnitudes were used. We found the average difference to be $\\Delta T_{eff} = 5 \\pm 15$~K, implying that the IRFM could be applied to all HARPS GTO stars which with {\\it Hipparcos} Johnson photometry \\citep[][]{ESA-1997}. We also checked that if {\\it Hipparcos} Tycho $B_TV_T$ magnitudes, transformed onto the Johnson system \\citep[][]{Bessell-2000}, were used instead that the differences in our results would be negligible.\n\n\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=6.5cm]{9698fig5.ps}\n\\caption[]{Comparison of our spectroscopic measurements of [Fe\/H] to others found in the literature.}\n\\label{fig_feh}\n\\end{figure}\n\nFigure \\ref{fig_spec_teff} (three right plots) shows the comparison with effective temperatures derived by the spectroscopic analysis described in Sect. 3. The agreement is excellent, corresponding to a mean difference of $-1 \\pm 63$~K. \nWe measure a slight disagreement when we consider cooler stars alone, which may due to inconsistencies in stellar model atmospheres at the lowest temperatures covered by the present study. In the case of the IRFM, we use the ATLAS9--ODFNEW models \\citep[][]{Castelli-2003} to recover the bolometric flux that is missing from our multi--band photometry. We also tested the use of the latest Phoenix models \\citep[][]{Brott-2005}, which include an extensive molecular-line list, which is particularly important for cooler stars. Using the IRFM, we found that results derived using the ATLAS9--ODFNEW models agree incredibly well with those for the Phoenix models, corresponding to a mean difference in temperature of only $-5 \\pm 11$~K.\nSuch agreement is in part due to our implementation of the IRFM, which uses more observational data and highlights the high degree of consistency between physical descriptions in independent spectral synthesis codes.\n\n\nThe comparison with the IRFM in Fig. \\ref{fig_spec_teff} implies that our spectroscopic method decreases its accuracy when applied to stars cooler than $\\sim 4800$~K. This conclusion is expected since the method described in Sect. 3 is based on a differential analysis with respect to the Sun; it is also supported by considering the error determination for the stars: since errors are based on the distribution of the abundance determination, the dispersion is larger for cooler stars. Above $4800$~K, we find, however, very good agreement.\n\n\n\\subsection{Surface Gravity}\n\nFigure \\ref{fig_logg} shows the comparison between surface gravity measurements derived in our work and those from other authors. We also computed spectroscopy surface gravities using Hipparcos parallaxes and the estimated masses of stars as it is described in \\citet[][]{Santos-2004b}. For the comparison between these computed gravities and the spectroscopic gravities it is observed a dispersion and offset for lower gravities. This is probably due to uncertainties in the estimated value of stellar mass, which is required in this approach to compute the gravity. The large mass uncertainty can be attributed to the considerable number of degeneracies in stellar interior models: it is possible, for example, to describe the data for a star in the H-R diagram using several models with different initial conditions \\citep[see e.g.][]{Fernandes-2003}. It is interesting that some high gravity values (close to 5.0 dex) are computed using the parallaxes. These high values are not expected, since we are dealing with dwarf main sequence stars.\n\nThe surface gravities presented by \\citet[][]{Bensby-2003} were determined using the stellar mass and parallax of stars. In Fig. \\ref{fig_logg}, we compare the values of $\\log g$ from \\citet[][]{Bensby-2003} with our surface gravities computed from parallaxes. We observe perfect agreement with a mean difference of $-0.02 \\pm 0.05$ dex.\n\nWe also find an excellent agreement with the surface gravities determined by \\citet[][]{Fuhrmann-2004} in addition to our old results using the same method but a smaller line list \\citep[][]{Santos-2004b}.\n\n\\citet[][]{Edvardsson-1993} measured the surface gravity using a different method, which used the Balmer discontinuity index. When comparing their values with our spectroscopic surface gravities, we find a rather large deviation with a mean difference of $-0.14 \\pm 0.11$ dex.\n\nThe mean difference between values obtained by \\citet[][]{Valenti-2005}, using spectral synthesis, and ourselves is only $0.04 \\pm 0.13$ dex, which indicates good agreement between the measurements. There is, however, significant dispersion in this comparison. For a few stars, \\citet[][]{Valenti-2005} derive unusually high values of gravity; in the lower regime of gravity, they also measure higher values than the ones we present in this work. A comparison between \\citet[][]{Valenti-2005} gravities and independent computation of gravity using Hipparcos parallaxes reveals a larger offset for smaller gravities. We note that this does not imply that gravities given by these authors are incorrect, but that the offset is greater than for our values when comparion to the computed gravities using parallaxes. We also recall that the mass estimates have a large uncertainty, which may contribute to the observed offsets.\n\n\n\n\n\\subsection{Metallicity}\n\n\nFigure \\ref{fig_feh} shows the comparison of the metallicity estimates in this work with those derived in other works.\nAs for the other parameters, all metallicities agree well within the errors. The results from \\citet[][]{Edvardsson-1993} differ the most from our values (with only 0.05 dex), in particular for metallicities above $\\sim$$0.1$\\,dex.\n\n\n\n\n\n\n\\section{A calibration of the effective temperature as a function of \\textit{B-V} and [Fe\/H]}\n\nUsing the parameters presented in Table \\ref{tab3} and the $B-V$ value taken from the Hipparcos catalogue \\citep[][]{ESA-1997}, we derived a new calibration of the effective temperature as a function of \\textit{B-V} and [Fe\/H]. The result is illustrated in Fig. \\ref{fig_teff_bv_feh} and the calibration is expressed by:\n\n\n\\begin{equation}\n T_{\\mathrm{eff}} = 9114 - 6827(B{-}V) +2638(B{-}V)^2 + \\text{368[Fe\/H].}\n\\end{equation}\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=8cm]{9698fig6.ps}\n\\caption[]{Calibration of the effective temperature as a function of the color index \\textit{B-V} and [Fe\/H]. The 3 fitted lines correspond to lines with constant values of [Fe\/H] (-0.5, 0.0, 0.5). }\n\\label{fig_teff_bv_feh}\n\\end{figure}\n\n\nThe standard deviation of the fit is only 47 K, illustrating the quality of the relation. Such a result was already observed in a calibration completed by \\citet[][]{Santos-2004b}, using a much smaller number of stars. This calibration can be useful and applied to stars without the need for a detailed spectroscopic analysis, with the guarantee that the result will be in the same effective temperature scale.\nThis calibration is valid in the following intervals: $4500 K < T_{eff} <6400 K$, $-0.85 < [Fe\/H] < 0.40$, and $ 0.51 < B{-}V < 1.20$. The small dispersion also attest to the quality of the metallicity values derived in this work.\n\n\n\n\\section{Planet-hosts in the sample}\n\n\n\nThe well-established correlation between the presence of a giant planet and the metallicity of its host star \\citep[][]{Santos-2004b,Fischer_Valenti-2005} is illustrated by our results. This trend is observed in Fig. \\ref{fig_planets2}, where the metallicity\ndistribution of jovian-like planets is clearly located towards the metal-rich regime. The subsample of 66 planet hosts has \na mean metallicity of $+0.09$ dex. For comparison, the sample presented in this work has a mean \nvalue of $-0.12$ dex, if we only include stars without planets, and $-$0.09, if we include all the stars (see also Fig.\\,\\ref{fig_analise}). \n\nFrom the 66 planet hosts belonging to our sample, three host neptunian planets (HD4308, HD69830 and HD160691). The first two of these only harbor\nneptunian planets \\citep[][]{Udry-2006,Lovis-2006}, while the third also has Jupiter-like planets \\citep[][]{Pepe-2007}. Others Neptune-like planets were discovered with HARPS, but do not belong to this sample. In Fig.\\ref{fig_planets2}, we present the metallicity distribution of these two types of host stars. Althought the numbers are small, we find a wider spread of metallicities for stars hosting Neptunian-like planets.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=8.5cm]{9698fig7.ps}\n\\caption[]{Metallicity distribution of jovian planet hosts and neptunian planet hosts within the HARPS GTO ``high precision'' spectroscopic catalogue.}\n\\label{fig_planets2}\n\\end{figure}\n\n\nIn Table \\ref{tab4}, we compile a list of [Fe\/H] values for all known planet hosts orbited by neptunes or super-earths. This list was compiled using the extrasolar-planets encyclopaedia (http:\/\/www.exoplanet.eu), for which we select the discovered planets with masses ($M_p\\sin i$) less than 25\\,M$_\\oplus$. In the table, we indicate whether each star also hosts jovian planet(s) (``yes''), or if it only hosts neptunian planets or super-earths (``no''). The [Fe\/H] values are listed in the table for each star.\n\n\nTo increase the number of neptunian-planet hosts, we must include M stars presented in the table. We note that the metallicity determination of M stars can be difficult; we therefore use the mean of the [Fe\/H] values quoted in the literature.\n\nWe should mention that we do not expect the inclusion of M-dwarfs in our comparison to strongly compromise or bias our conclusions. First, the metallicity distribution of M-dwarfs in the solar neighborhood differs only slightly from that of field FGK-dwarfs. The offset on the metal distribution is small ($\\sim$0.07\\,dex), and possibly within the margins of errors \\citep[][]{Bonfils-2005}. Secondly, the fact that these metallicities are more uncertain will produce a scatter, but likely not a systematic offset, in the [Fe\/H] distribution of stars with Neptune-like planets. Finally, it is reasonable to assume that the influence of metallicity on planet-formation efficiency is independent of either host mass or spectral type\n\n\n\nUsing the values of [Fe\/H] listed in Table \\ref{tab4}, we compared the metallicity distribution of the spectroscopic catalogue presented in this paper, which corresponds to the [Fe\/H] distribution of the solar neighborhood, with two different distributions. The first, composed of all stars presented in Table\\,\\ref{tab4}, and the second, composed only of stars that do not host jovian planets (``no'' in the table). The result can be observed in Fig.\\,\\ref{fig_planets3}. The metallicity distribution of the catalogue presented in this work has a mean metallicity of $<[Fe\/H]>=-0.09$, while the metallicity distribution of stars known to harbor at least one neptunian planet corresponds to a mean metallicity of $<[Fe\/H]>=-0.03$. Finally, stars known to host only neptunian planets have a mean metallicity of $<[Fe\/H]>=-0.21$.\n\nWe performed a Kolmogorov-Smirnov  test to check the probability that samples are part of the same population. We compare the sample composed of all stars in Table\\,\\ref{tab4}, to that composed only of giant-planet hosts in the catalogue. According to a Kolmogorov-Smirnov test, there is a probability of ~3\\% that the two samples belong to the same population. We applied the same test to stars that host only neptunian planets (``no'' in the table) and found a probability of only 0.5\\%. This result, although preliminary, implies that these samples are unlikely to belong to the same population.\n\nAlthough the sizes of the datasets are small, we attempt to estimate a lower limit value of the frequency of Neptune-like planets (or super-earths) found, as a function of stellar metallicity. For each of the three metallicity bins $[Fe\/H]<-0.15$, $-0.15<[Fe\/H]\\le0.15$, and $[Fe\/H]\\ge0.15$, we computed the ratio of the number of stars with Neptune-like planets in Table \\ref{tab4} to the number of stars in our entire sample. In this estimate, we assume that the metallicity distribution of the 451 stars represents the metallicity distribution of the planet search samples used to find the planets listed in Table \\ref{tab4}.\n\nThe results show that the frequency of stars hosting neptunians is 1.8\\%, 2.0\\%, and 5.1\\%, respectively for each of the metallicity bins mentioned above. These numbers may, however, be biased towards the high-metallicity regime, since several stars in Table \\ref{tab4} also have jovian planets. \nWe further note that we attempted to find Neptune-like planets orbiting some of the FGK stars in Table\\,\\ref{tab4} because of their high metallicity \\citep[e.g.][]{Melo-2007}. In other cases, the approach taken to detect a Neptune-like planet assumed that the star hosted already at least one jupiter-like companion \\citep[e.g.][]{Santos-2004a,McArthur-2004}. If we consider stars hosting $only$ Neptune-like planets, these frequencies change to 1.8, 1.5, and 0.0\\% (there are no stars that host Neptune-like planets, that do not host Jupiter-like planets with [Fe\/H]$>-$0.06).\n\n\n\\begin{table}[t]\n\\caption[]{Neptunian hosts and their metal content. We also list the mass (in earth masses) of the least massive planet \norbiting the star, as well as an indication of whether there is one or more jovian companions.}\n\\begin{tabular}{lrccl}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nstar & [Fe\/H] & jov.? & $M_p\\sin i$ & [Fe\/H] reference\\\\\n\\hline\n\\object{Gl 581 }   & -0.33 &  no &   5.09 & \\citet[][]{Bean-2006}\\\\\n                   & -0.25  &     &        & \\citet[][]{Bonfils-2005b}\\\\\n                   & -0.29  &     &        & Mean value\\\\\n\\object{Gl 876 }   & -0.12 & yes &   5.72 & \\citet[][]{Bean-2006}\\\\\n                   & -0.03  &     &        & \\citet[][]{Bonfils-2005b}\\\\\n                   & -0.08  &     &        & Mean value\\\\\n\\object{HD69830}   & -0.06 &  no & 10.49  & This work\\\\\n\\object{HD160691}  &  0.30 & yes & 13.99  & This work\\\\\n\\object{Gl 674 }   & -0.28 &  no & 11.76  & \\citet[][]{Bonfils-2007}\\\\\n\\object{55 Cnc e}  &  0.33 & yes & 10.81  & \\citet[][]{Santos-2004b}\\\\\n\\object{HD4308}    & -0.34 &  no & 14.94  & This work\\\\\n\\object{HD190360}  &  0.24 & yes & 18.12  & \\citet[][]{Santos-2004b}\\\\\n\\object{HD219828}  &  0.19 & yes & 20.98  & \\citet[][]{Melo-2007}\\\\\n\\object{GJ 436}    & -0.32 &  no & 22.89  & \\citet[][]{Bean-2006}\\\\\n                   &  0.02 &     &        & \\citet[][]{Bonfils-2005b}\\\\\n                   & -0.15 &     &        & Mean value\\\\\n\\object{Gl 176}    & -0.10 &  no & 24.16  & \\citet[][]{Endl-2007}\\\\\n\n\n\\hline\n\\end{tabular}\n\\label{tab4}\n\\end{table}\n\n\n\n\n\n\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=8.5cm]{9698fig8.ps}\n\\caption[]{Metallicity distribution for the sample presented in this work (dotted line), for stars hosting neptunian planets (dashed line), and for stars exclusively hosting neptunian planets (full line). The mean metallicity of each distribution is indicated on the plot.}\n\\label{fig_planets3}\n\\end{figure}\n\nThese numbers imply that the few low-mass planets found appear to follow a trend in metallicity that differ from well-established relations for giant planets first pointed out by \\citet[][]{Udry-2006}, this result, is supported by planet-formation models based on the core-accretion paradigm \\citep[][]{IdaLin-2004, Benz-2006}. These works suggest that neptunian planets should be found in a wider range of stellar metallicities. Lower-mass planets could even be preferentially found orbiting metal-poorer stars.\n\n\nUsing the numbers presented in Table\\,\\ref{tab4}, we measure a preliminary value for the \nJupiter-to-Neptune ratio as a function of stellar metallicity. We estimate the value using the metallicity distribution\nof stars with only jovian planets in our HARPS sample, and compared it with the same distribution\nfor stars orbited only by Neptune-like planets. For this, we used the sample of three stars \nmentioned above (HD4308, HD69830 and HD160691, belonging to the sample presented in this paper -- case\\,1), \nall stars $only$ with neptunian planets listed in Table\\,\\ref{tab4} (case\\,2), and all stars in the table (case\\,3). \n\nThe results show that in the same three metallicity intervals mentioned above, the Jupiter-to-Neptune ratios \nare, by order of increasing metallicity, 5:1, 28:1, and 30:1 (case\\,1), 5:3, 28:3, 30:0 (case\\,2), and 5:3 28:4 30:4 (case\\,3). \nIn other words, this result tentatively suggests that the Jupiter-to-Neptune ratio may be higher at higher metallicities.\nThis is expected by models of planet formation based on the core-accretion process \\citep[][]{Benz-2006}.\n\nThese final results should be interpreted with caution, however, since most of the stars that have $only$ with Neptune-like planets listed\nin Table\\,\\ref{tab4} are M-dwarfs. It has been proposed that M-dwarf stars may have a lower rate of Jupiter-like planets, but (at least in the short planet period range) a high incidence of Neptune-like planet detections \\citep[][]{Endl-2006,Bonfils-2007}.\nThe above mentioned bias in the case of a search for neptunian planets around metal-rich (or planet-host) stars may also\ninduce important biases in this analysis.\n\n\\section{Summary}\n\n\nWe have measured accurate stellar parameters for a well-defined sample of solar-type stars, which were originally observed as a part of a search for very low-mass planets. We have presented a catalog of stellar parameters for 451 stars which is available online. \nWe compared our results with other works in the literature. The effective temperatures obtained were also compared \nwith values derived using IRFM. All comparisons indicated a very good agreement in despite of the different \nmethods used by the different authors.\n\nA new calibration was developed for the determination of effective temperature using the index color $B-V$ and metallicity. \n\nThe results presented here were explored to study the metallicity-planet correlation, in particular\nfor stars hosting planets in the Neptune-mass range. We showed that, when compared with their higher-mass counterparts, \nneptunian planets did not follow the same trend. Low-mass planets appeared to be formed in a lower metal content \nenvironment. Interestingly, the Jupiter-to-Neptune ratio was found to be an increasing function of stellar metallicity.\nThe results presented in this paper have shown that the study of the chemical abundances of Neptune-host stars\nprovides new important constraints on models of planet formation and evolution.\n\n\n\n\n\n\n\n\\begin{acknowledgements}\nS.G.S and N.C.S. would like to acknowledge the support from the Funda\\c{c}\\~ao para a Ci\\^encia e Tecnologia (Portugal) in the form of fellowships and grants SFRH\/BD\/17952\/2004, POCI\/CTE-AST\/56453\/2004, PPCDT\/CTE-AST\/56453\/2004 and PTDC\/CTE-AST\/66181\/2006, with funds from the European program FEDER. We thank the anonymous referee for the useful comments and suggestions.\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPresentations of monoids are known to be essential in the study of actions on sets.\n In fact, given a monoid $M$ with\npresentation $\\left\\langle X,\\mathbf{r}  \\right\\rangle$, to describe an action of $M$ on a set\n$S$ \n  it is enough to give a set of endomorphisms $A_x\\colon S\\to S$, $x\\in X$,\nsatisfying the\nrelations in $\\mathbf{r} $.\n\nIn recent years the interest for actions of monoids on categories raised. \nIn this case presentations have to be replaced by the so called \\emph{coherent\npresentations}. \nSuppose we are given \n an action of $M$ on  $\\mathcal{C}$. Then we have a collection\nof endofunctors $F_m\\colon \\mathcal{C} \\to \\mathcal{C}$, $m\\in M$, for which  $F_m \\circ F_n = F_{mn}$ does not hold in general. Instead, we have a collection of natural\nisomorphisms $\\lambda_{m,n}\\colon F_m \\circ F_n \\to F_{mn}$ such that the\nnatural transformations\n\\begin{equation}\n\\label{equal}\n\\begin{gathered}\nF_{l}\\circ F_{m} \\circ F_n \\xrightarrow{\\lambda_{l,m}\\circ F_n} F_{lm} \\circ F_n\n\\xrightarrow{\\lambda_{lk,n}}F_{lmn}\\\\\nF_{l}\\circ F_{m} \\circ F_n \\xrightarrow{F_l\\circ \\lambda_{m,n}} F_{l} \\circ\nF_{mn} \\xrightarrow{\\lambda_{l,kn}}F_{lmn}\n\\end{gathered}\n\\end{equation}\nare equal for all $l$, $m$, $n\\in M$. Further, using\n$\\left\\{\\,\\lambda_{m,n}\\,\\middle|\\, m,n\\in M\\,\\right\\}$, we can construct\nnatural isomorphisms\n\\begin{equation*}\n\\lambda_{m_1,\\dots, m_k} \\colon F_{m_1}\\circ \\dots \\circ F_{m_k} \\to\nF_{m_1\\dots m_k}\n\\end{equation*}\nfor all $m_1$,\\dots, $m_k\\in M$. \n Now for  every relation\n\\begin{equation*}\nr = (x_{1} \\dots x_{s} , y_{1}\\dots y_{t})\n\\end{equation*}\nin $\\mathbf{r} $ we define the natural isomorphism $\\tau_r = \\lambda_{y_1,\\dots,\ny_t}^{-1} \\lambda_{x_1,\\dots,x_s}$. It can be shown that one can reconstruct\n(up\nto an isomorphism) the action \n\\begin{equation*}\n\\left\\{\\, F_m,\\ \\lambda_{m,n} \\,\\middle|\\, m,n\\in\nM \\right\\} \n\\end{equation*}\nof $M$ on $\\mathcal{C}$ from the collection $\\left\\{\\, F_x,\\ \\tau_r \\,\\middle|\\,\nx\\in x,\\ \\tau_r \\right\\}$. \n\nNot every collection $\\left\\{\\, F_x,\\ \\tau_r \\,\\middle|\\, x\\in X,\\ r\\in \\mathbf{r} \n\\right\\}$ of endofunctors of $\\mathcal{C}$ and natural transformations between them can\nbe obtained from an action of $M$ on $\\mathcal{C}$. The obstruction comes from the\naxiom~\\eqref{equal}. This obstruction can be translated into a set $\\mathcal{E}$  of equations\nof the form\n\\begin{equation*}\n\\tau_{r_1} \\dots \\tau_{r_k} = \\tau_{s_1}\\dots \\tau_{s_l}\n\\end{equation*}\nwith $r_1$, \\dots $r_k$, $s_1$,\\dots, $s_l\\in \\mathbf{r} $. Such a  set $\\mathcal{E}$ is called\na complete\nset of relations between relations from $\\mathbf{r} $, and $\\left\\langle X,\\mathbf{r} ,\\mathcal{E}\n\\right\\rangle$ is called a\n\\emph{coherent presentation} of $M$. \n\n\n\nTo find coherent presentations of some monoids,\nGuiraud and Malbos~\\cite{plactic} used higher dimensional rewriting theory.\nApplying the same technique, Gausent, Guiraud, and Malbos obtained a coherent\npresentation of Artin braid groups associated with Coxeter\nsystems~\\cite{braid}.\n\nIn this article, we use  a new approach to determine coherent presentations of\nmonoids. Namely, we use the notion of decreasing\ndiagrams, that was introduced by van Oostrom in~\\cite{oostrom} to obtain a\nsufficient condition for a  locally confluent abstract reduction system to \nbe confluent. In~\\cite{klop} Klop, van Oostrom, and de Vrijer gave a\ngeometrical proof of the result of van Oostrom. In their proof they discovered\nthat if an abstract reduction system admits enough \ndecreasing elementary diagrams, then every reduction diagram can be patched by \nthese elementary diagrams. Their result allows us to give a sufficient\ncondition for $\\left\\langle X,\\mathbf{r} , \\mathcal{E} \\right\\rangle$ to be a coherent\npresentation of a monoid $\\left\\langle X,\\mathbf{r}  \\right\\rangle$ (see\nTheorem~\\ref{main:presentation}). \nUsing \nthis sufficient condition we establish a coherent presentation of the $0$-Hecke\nmonoid. This presentation was used in the joint work  of the\nauthor with A. P.\nSantana~\\cite{0hecke} to exhibit an action of the  \\mbox{$0$-Hecke} monoid on the category of rational\nmodules for the quantum Borel group. \nNote that our coherent presentation contains the  coherent presentation of the\nbraid group obtained by Guiraud \n\\emph{et al.} in~\\cite{braid}. \n\nThe paper is organised as follows. In Section~\\ref{ARS} we collect results on\nabstract reduction systems used throughout  the paper. In\nSection~\\ref{cat} we establish  a relationship between abstract reduction systems\nand categories, and show how decreasing diagrams can be used to deduce\nthe commutativity of an infinite set of diagrams, from the commutativity of a given,\noften finite, set of diagrams (Theorem~\\ref{main:theory}). Section~\\ref{action}\ncontains the  definition of\nan action of a monoid on a category (following~\\cite{deligne}). \nIn Section~\\ref{monoids}, we apply Theorem~\\ref{main:theory} to an abstract\nreduction system associated to a presentation of a monoid. The main result of\nSection~\\ref{coherent} is Theorem~\\ref{main:presentation}, that gives a\nsufficient condition for $\\left\\langle X,\\mathbf{r} ,\\mathcal{E} \\right\\rangle$ to be a coherent presentation. In Section~\\ref{hecke} we\ndescribe a coherent presentation of the $0$-Hecke monoid $\\rs{n+1}$ of the\nsymmetric group $\\Sigma_{n+1}$.\n\n\n\n\\section{Abstract reduction system}\n\\label{ARS}\n\nAn \\emph{abstract reduction system} is a set $A$ with a relation $R\\subset\nA\\times A$ on $A$ which is called a set of \\emph{rewriting rules}. The elements\nof $R$ will be sometimes depicted by $a\\to b$ for $\\left( a,b \\right)\\in R$. \nThe sequence of elements $a_0$, \\dots, $a_k$ is called  a \\emph{reduction path}\nfrom $a_0$ to $a_k$ if \n$(a_{i-1},a_{i})\\in R$ for all $1\\le i\\le k $. \nIf there is a reduction path from $a\\in A$ to $b\\in A$, we write $a\\twoheadrightarrow b$. \n\n\nA \\emph{reduction diagram} for $\\left\\langle A,R \\right\\rangle$ is an oriented planar graph $\\Gamma$, such that:\n\\begin{enumerate}[(1)]\n\t\t\\setlength{\\itemsep}{-2pt}\n\t\\item  All the arrows of $\\Gamma$ go either from left to right\n\t\tor from top to bottom.  \n\t\t\\item Some arrows of $\\Gamma$ are solid and some arrows of\n\t\t\t$\\Gamma$ are dashed. \n\t\\item  The nodes of $\\Gamma$ are labeled by elements of $A$. \n\t\tThe label of a node $x$ will be denoted by $l\\left( x\n\t\t\\right)\\in A$.\n\\item If the nodes with labels $a$ and $b$ are connected by a solid arrow then\n\t\t${(a,b)\\in R}$.\n\t\\item If two nodes are connected by a dashed arrow then they have \n\t\tequal labels. \n\t\\item  \\label{tricky}If from a node $x\\in \\Gamma$\n\t\tthere is a horizontal arrow to $y\\in \\Gamma$ and a vertical arrow to\n\t\t$z\\in \\Gamma$ then one of the two mutually exclusive\npossibilities holds\n\t\t\\begin{enumerate}[(a)]\n\t\t\t\\item There is no vertex which is simultaneously  strictly bellow and\nstrictly to the right of $x$.  \t\t In this case\n\t\twe say that $x$ is an \\emph{open corner}. \n\t\\item There is a node $w\\in\n\t\t\\Gamma$, a vertical path from $y$ to $w$ in $\\Gamma$ and a\n\t\thorizontal path from $z$ to $w$ in $\\Gamma$. These paths are\n\t\tcalled \\emph{convergence} paths. \n\t\tIf $x$ and $y$ are connected by a dashed arrow, then $z$ and\n\t\t$w$ are connected by a dashed arrow as well, and the path from\n\t\t$y$ to $w$ contains just one arrow. Similarly, if $x$ and\n\t\t$z$ are connected by a dashed arrow then $y$ and $w$ are\n\t\tconnected by a dashed arrow and the path from $z$ to $w$\n\t\tcontains just one arrow. \n\t\t\\end{enumerate}\n\t\t  \\end{enumerate}\nBellow is an example of a reduction diagram\n\\begin{equation*}\n\\xymatrix{a_{11} \\ar[r] \\ar[dd] & a_{12} \\ar[r] \\ar[d] & a_{13} \\ar[rr] \\ar[d] &\n& a_{15} \\ar[d] \\\\ & a_{22} \\ar@{-->}[r] \\ar[d]  & a_{22} \\ar[r] \\ar[d] &\n a_{24}\\ar[r] & a_{25}\\\\\na_{31} \\ar[r] \\ar@{-->}[d] & a_{32} \\ar@{-->}[r] \\ar@{-->}[d] & a_{32} \\\\ a_{31}\n\\ar[r] &\na_{32} &&&& .}\n\\end{equation*}\nIn the above reduction diagram there are two open corners with labels\n$a_{22}$ and $a_{32}$, whose (matrix) coordinates are $(2,3)$ and $(3,2)$,\nrespectively. \nThe following graph does not satisfy the  axioms of a reduction diagram\n\\begin{equation*}\n\\xymatrix{\na_{11} \\ar[r] \\ar[d] & a_{12} \\ar[dd] \\\\\na_{21} \\ar[d] \\\\\na_{32} \\ar[r] & a_{33}\\, .\n} \n\\end{equation*}\nIn fact, the top left corner of the above graph is neither an open corner nor\nthere are convergence paths for the horizontal arrow $a_{11}\\to a_{12}$ and the\nvertical arrow $a_{11}\\to a_{21} $.\n\n  \\begin{definition}\n\t\t  A reduction diagram is called \\emph{complete} if it does not\n\t  \t\t  contain any open corners. \\end{definition}\nAn example of a complete diagram is given by \n$$\n\\xymatrix{a_{11} \\ar[r] \\ar[dd] & a_{12} \\ar[r] \\ar[d] & a_{13} \\ar[rr] \\ar[d] &\n& a_{15} \\ar[d] \\\\ & a_{22} \\ar@{-->}[r] \\ar[d]  & a_{22} \\ar[r] \\ar[d] &\na_{24}\\ar[r] \\ar@{-->}[d]& a_{25}\\ar@{-->}[d]\\\\\na_{31} \\ar[r] \\ar[d] & a_{32} \\ar@{-->}[r] \\ar@{-->}[d] &\na_{32}\\ar[r]\\ar@{-->}[d] &\na_{24} \\ar[r] \\ar@{-->}[d] & a_{25} \\ar@{-->}[d]  \\\\ a_{41} \\ar[r] &\na_{32}\\ar@{-->}[r] &a_{32}\\ar[r] & a_{24}\\ar[r] & a_{25} }\n$$\n\t\t  \\begin{definition}\n\t\t\t  An \\emph{elementary diagram} (e.d.) for $\\left\\langle A,R\n\t\t\t  \\right\\rangle$ is a reduction diagram $\\Gamma$ such\n\t\t\t  that the edges of $\\Gamma$ constitute the boundary of a\n\t\t\t  rectangular. \n\t\t  \\end{definition}\n\t\t  Suppose $\\Gamma$ is an e.d. Then the top side and the left \n\t\t  side of $\\Gamma$ contain just one arrow each, as otherwise the top-left\n\t\t  corner of $\\Gamma$ would not satisfy the axiom \\eqref{tricky}\nfor a reduction diagram.\nTherefore there\n\t\t  are four different types of e.d.s:\n\t\t  $$\n\t\t  \\xymatrix{a \\ar[r] \\ar[d] & b \\ar@{->>}[d] \\\\ c \\ar@{->>}[r] &\n\t\t  d}\\ \\ \\ \\  \n\t\t  \\xymatrix{a \\ar@{-->}[r] \\ar[d] & a \\ar[d] \\\\ c \\ar@{-->}[r] &\n\t\t  c}\\ \\ \\ \\ \n\t\t  \\xymatrix{a \\ar[r] \\ar@{-->}[d] & b \\ar@{-->}[d] \\\\ a \\ar[r] &\n\t\t  b}\\ \\ \\ \\  \n\t\t  \\xymatrix{a \\ar@{-->}[r] \\ar@{-->}[d] & a \\ar@{-->}[d] \\\\ a \\ar@{-->}[r] &\n\t\t  a} \n\t\t  $$\n\t\t  where two headed arrows are used as an abbreviation of\n\t\t  a path. \n\t\t  The e.d.s of first type are called \\emph{proper} and the\n\t\t  rest of e.d.s are called \\emph{improper}. \n\n\t\t  Let $\\Gamma$ be a non-complete reduction diagram for\n\t\t $\\left\\langle A,R \\right\\rangle$ and $x$ is an open corner in\n\t\t $\\Gamma$ with the horizontal arrow to $y$ and  the vertical arrow to\n\t\t $z$.  Suppose that we have an e.d. $\\Gamma'$  with the labels $l\\left( x\n\t\t \\right)$, $l\\left( y \\right)$, $l\\left( z \\right)$ at the top\n\t\t left, top right, and bottom left corners, respectively. Then we\n\t\t can glue (suitably stretched) $\\Gamma'$ into $\\Gamma$ identifying the top left \n\t\t corner of $\\Gamma'$ with $x$, the top right  corner of $\\Gamma'$\n\t\t with $y$, and the bottom left  corner of $\\Gamma'$ with $z$.  \n\t\t This process is called \\emph{adjoining} of the  e.d. $\\Gamma'$ to\n\t\t $\\Gamma$ at $x$.\nIt is obvious, that the resulting diagram is again a reduction diagram. \n\n\t\t \\begin{definition}\n\t\t\t We say that $\\left\\langle A,R \\right\\rangle$ is \n\t\t\t \\emph{locally confluent} if for any $a\\to b$,\n\t\t\t $a\\to c$ there is an e.d. with labels $a$, $b$, $c$ at the\n\t\t\t top left, top right, and bottom left corners,\n\t\t\t respectively.\n\t\t \\end{definition}\n\t\t \n\\begin{definition}\n\tA reduction diagram $\\Gamma$ for $\\left\\langle A,R \\right\\rangle$ is\n\tcalled \\emph{initial} if its edges constitute the top and the left sides of a\n\trectangular. \n\\end{definition}\n\n\nSuppose that $\\left\\langle A,R \\right\\rangle$ is a locally confluent ARS. Let\n$\\mathcal{E}$ be a family of e.d.s such that  for every ordered pair $a\\to b$, $a\\to c$ in $R$\nthere is an e.d. $E\\in \\mathcal{E}$ whose top arrow is $a\\to b$ and left arrow\nis $a\\to c$.  In this case we say that $\\mathcal{E}$ is a complete set of e.d.s. \n It is\nproved in Section~4 of \\cite{klop} that the recursive process of adjoining of \ne.d.s from a complete set of e.d.s   to any initial finite diagram $\\Gamma$ results in a  complete\ndiagram $\\Gamma'$ in at most  a countable number of   steps. \n\n\nRecall that a \\emph{preorder}  is a reflexive and transitive binary\nrelation.\nSuppose now that the set $R$ is equipped with a preorder $\\succeq$. \nWe write $r_1 \\succ r_2$ if $r_1 \\succeq r_2$ but not $r_2 \\succeq r_1$. If\n$r_1 \\succeq r_2$  and $r_2\\succeq r_1$ simultaneously, then we write $r_1 \\sim r_2$. It is\nimmediate that  $\\sim$ is an equivalence relation on $R$. \n\\begin{definition}\nWe say that the e.d.  \n\\begin{equation*}\n\\xymatrix{\nx \\ar[rrr]^{u} \\ar[ddd]_{l} &&& y_1 \\ar[d]^{r_1} \\\\\n&  && y_1 \\ar@{.}[d] \\\\\n&&& y_m \\ar[d]^{r_m} \\\\\nz_1 \\ar[r]^{d_1} & z_2 \\ar@{.}[r] & z_n \\ar[r]^{d_n} & w\n}\n\\end{equation*}\nis \\emph{decreasing} if the following two condition hold\n\\begin{multicols}{2}\n\\begin{enumerate}[1)]\n\\item there is $0\\le j \\le n$ such that \n\\begin{enumerate}[i)]\n\\item $u \\sim d_j$ in the case $j\\not= 0$;\n\\item $l \\succ d_k$ for all $k<j$;\n\\item $l \\succ d_k$ or $u \\succ d_k$ for all $k >j$;\n\\end{enumerate}\n\\item there is $0\\le s \\le m$ such that \n\\begin{enumerate}[i)]\n\\item $l \\sim r_s$ in the case  $s\\not=0$;\n\\item $ u \\succ r_t$ for all $t<s$;\n\\item $u \\succ r_t$ or $l \\succ r_t$ for all $t>s$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{multicols}\nMore informally we require that either  the reduction path $r_1$, \\dots, $r_m$ consists\nof the steps that are strictly less than $l$ or $u$, or if it starts with the rules\nthat are strictly less than $u$, then there is a step $r_t$ which is equivalent\nto $l$ and all other steps are strictly less than $u$ or $l$. \n\nSimilarly the reduction path $d_1$, \\dots, $d_n$ either consists\nof the steps that are strictly less than $l$ or $u$, or if it starts with the rules\nthat are strictly less than $l$, then there is a step $d_t$ which is equivalent\nto $u$ and all other steps are strictly less than $u$ or $l$. \n\\end{definition}\nWe say that the preorder $\\succeq$ is \\emph{well-founded} if for every  sequence\n\\begin{equation*}\nr_1 \\succeq r_2 \\succeq \\dots \\succeq r_m \\succeq \\dots\n\\end{equation*}\nof elements in $R$ there is an integer  $n$ such that for all  $N\\ge n$ we have $r_N \\sim\nr_n$; in other words, any decreasing sequence in $(R,\\succeq)$ stabilizes. \n\nThe following theorem is a reformulation of \\cite[Proposition~15]{klop}. \n\\begin{theorem}\\label{geometric}\nSuppose $\\left\\langle A,R \\right\\rangle$ is a locally confluent ARS and there is\na well-founded preorder $\\succeq$ on $R$ and a complete set $\\mathcal{E}$ of decreasing\ne.d.s.   Then every process of adjoining chosen\ne.d.s to any initial finite diagram $\\Gamma$ results in a complete\ndiagram $\\Gamma'$ in a finite number of steps. \n\\end{theorem}\nWe will say that the sequence of elements\n\\begin{equation*}\na_0, a_1, \\dots, a_k\n\\end{equation*}\nis a zigzag in $\\left\\langle A,R \\right\\rangle$ from $a_0$ to $a_k$, if for ever $1\\le j\\le k$, we have\n$(a_{j-1},a_j)\\in R$ or $(a_j, a_{j-1}) \\in R$. We will denote zigzags by $a_0 \\rightsquigarrow\na_k$. \nLet $\\left\\langle A,R \\right\\rangle$ be an ARS, $\\succ$ a well-founded preorder\non $R$,  and $\\mathcal{E}$ a complete set of decreasing e.d.s. \nSuppose we are given a reduction diagram of the form\n\\begin{equation}\n\\label{zigzag}\n\\xymatrix{\n&& a_1 \\ar@{->>}[r] \\ar@{->>}[d] & b_0 \\\\\n&& b_1 \\ar@{.}[ld]\\\\\na_k \\ar@{->>}[r] \\ar@{->>}[d] & b_{k-1}\\\\\nb_k\n}\n\\end{equation}\nThen applying Theorem~\\ref{geometric} to the diagrams\n\\begin{equation*}\n\\xymatrix{\na_j \\ar@{->>}[r] \\ar@{->>}[d] & b_{j-1} \\\\\nb_j  , \n}\n\\end{equation*}\nwe can get in a finite number of steps the diagram\n\\begin{equation}\n\\label{zigzag2}\n\\xymatrix{\n&&& a_1 \\ar@{->>}[r] \\ar@{->>}[d] & b_0 \\ar@{->>}[d]  \\\\\n&&& b_1 \\ar@{.}[ld] \\ar@{->>}[r] & c_0\\\\\n& a_{k-2} \\ar@{->>}[r] \\ar@{->>}[d] & b_{k-2} \\ar@{->>}[d] \\\\\na_k \\ar@{->>}[r] \\ar@{->>}[d] & b_{k-1} \\ar@{->>}[d] \\ar@{->>}[r] & c_{k-2} \\\\\nb_k \\ar@{->>}[r]  & c_{k-1} ,\n}\n\\end{equation}\nwhere every square is in fact tiled into an elementary diagrams from $\\mathcal{E}$. Note\nthat the new diagram has $k-1$ open corners, that is one open corner less than~\\eqref{zigzag}.  Now\nwe can apply Theorem~\\ref{geometric} to the diagrams\n\\begin{equation*}\n\\xymatrix{\nb_j \\ar@{->>}[r] \\ar@{->>}[d] & c_{j-1} \\\\\nc_j&  . \n}\n\\end{equation*}\nAs a result we get a new reduction diagram with $k-2$ open corners. Continuing,\nwe get a reduction diagram\n\\begin{equation}\n\\label{completezz}\n\\xymatrix{\n&& a_1 \\ar@{->>}[r] \\ar@{->>}[d] & b_0  \\ar@{->>}[ddd]\\\\\n&& b_1 \\ar@{.}[ld]\\\\\na_k \\ar@{->>}[r] \\ar@{->>}[d] & b_{k-1}\\\\\nb_k \\ar@{->>}[rrr] &&& z, \n}\n\\end{equation}\nwhose interior is tiled by e.d.s from $\\mathcal{E}$. \nThus we get\n\\begin{corollary}\n\\label{cor:zigzag}\nSuppose $\\left\\langle A,R \\right\\rangle$\nis an ARS, $\\succeq$ a well-founded preorder on $R$, and $\\mathcal{E}$ a complete set of\ndecreasing e.d.s. Then any reduction diagram of the form \\eqref{zigzag} can be\ncompleted to a diagram of the form \\eqref{completezz} in a finite number of steps. \n\\end{corollary}\n\nOne of the applications of Proposition~\\ref{geometric} is to prove\n\\emph{confluency} of\nan ARS. Similarly, Corollary~\\ref{cor:zigzag} can be used to prove the Church-Rosser\nproperty of $\\left\\langle A,R \\right\\rangle$. In this paper we will apply\nProposition~\\ref{geometric} and Corollary~\\ref{cor:zigzag} to prove\ncommutativity of certain diagrams. It should be noted that\nProposition~\\ref{geometric} enlightens the long time observed connection between\nconfluent ARSs and coherence results in category theory.\n\n\n\\section{ARSs and categories}\n\\label{cat}\nWe will start by introducing notion that generalises normal forms for\nterminating ARS to the case when $\\twoheadrightarrow$ is a well-founded but not necessarily\nterminating relation on $A$.\n\\begin{definition}\nWe say that $a\\in A$ is  \\emph{semi-normal} if for every path $a\\twoheadrightarrow b$ in\n$\\left\\langle A,R \\right\\rangle$ there is a path $b\\twoheadrightarrow a$ in $\\left\\langle\nA,R \\right\\rangle$. If $c\\in A$ and there is a path $c\\twoheadrightarrow a$ to a semi-normal\nelement $a$, we say that $a$ is a \\emph{semi-normal form} of $c$.  \n\\end{definition}\n\nWe will denote\nby $\\bidir$ the equivalence relation on $A$ defined by\n\\begin{equation*}\na \\bidir b \\Leftrightarrow a \\twoheadrightarrow b\\,\\, \\&\\,\\, b \\twoheadrightarrow a.\n\\end{equation*}\n\nSuppose the preorder $\\twoheadrightarrow$ on $A$ is well-founded. Then for every element\nof $A$ there is a semi-normal form. \n\nSuppose $\\left\\langle A,R \\right\\rangle$ is a confluent system and $b_1$, $b_2$ are two semi-normal forms of $a\\in A$. Since\n$\\left\\langle A,R \\right\\rangle$ is confluent, there is $c\\in A$ and two paths\n$b_1 \\twoheadrightarrow c$, $b_2 \\twoheadrightarrow c$. As $b_2$ is semi-normal there is  a path\n$c\\twoheadrightarrow b_2$. Therefore we get the path $b_1\\twoheadrightarrow c \\twoheadrightarrow b_2$. Since also $b_1$ is\nsemi-normal,   there is a path $b_2\\twoheadrightarrow b_1$. Hence $b_1 \\bidir\nb_2$. \nThus we get\n\\begin{proposition}\n\\label{seminormal}\nSuppose $\\left\\langle A,R \\right\\rangle$ is a confluent ARS, and $\\twoheadrightarrow$ is a\nwell-founded preorder on $A$. Then for every element $a\\in A$ there is a\nsemi-normal form $b$ which is unique up to equivalence with respect to $\\bidir$. \n\\end{proposition}\n\\begin{remark}\nProposition~\\ref{seminormal} is a generalization of the well-known fact that\nevery element in a  confluent terminating ARS has a unique normal form. \n\\end{remark}\nWe will call the set $\\attr(a)$ of all semi-normal forms of $a\\in A$ an\n\\emph{attractor} of $a$. \nWe also will denote by $\\attr(A)$ the set of all semi-normal elements in\n$A$. \nSince $a\\in \\attr(A)$ and $(a,b)\\in R$ imply that $b\\in \\attr(A)$, we see that\nthe relation $R$ can be restricted to $\\attr(A)$. We will denote the resulting\nrelation on $\\attr(A)$ by $\\attr(R)$. We will sometimes denote the ARS\n$\\left\\langle \\attr(A), \\attr(R) \\right\\rangle$ by $\\attr(\\left\\langle A,R\n\\right\\rangle)$. \n\n\nNow, let $\\mathcal{C}$ be a category and $\\left\\langle A,R \\right\\rangle$ an ARS. Both $\\left\\langle A,R \\right\\rangle$ and $\\mathcal{C}$ can\nbe considered as graphs. Suppose $f\\colon \\left\\langle A,R \\right\\rangle\\to\n\\mathcal{C}$ is  a map of graphs. \n Then using the composition of morphisms in $\\mathcal{C}$, we can extend $f$ to the paths \n$a\\twoheadrightarrow b$\n  in $\\left\\langle A,R \\right\\rangle$. In particular, given an empty path\n$a\\dashrightarrow a$, we set $f(a\\dashrightarrow a) = 1_{f(a)}$. \nOur aim is to find sufficient conditions on $f$ that guarantee that for any two\npaths $p$, $q\\colon a\\twoheadrightarrow b$ one gets $f(p) = f(q)$. \n\\begin{theorem}\\label{main:theory}\nLet $\\left\\langle A,R \\right\\rangle$ be an ARS, $\\succeq$ be a well-founded preorder on $R$ and $\\mathcal{E}$ a complete set of\ndecreasing e.d.s.  Suppose that\n\\begin{enumerate}[i)]\n\\item for every $E\\in \\mathcal{E}$ the diagram $f(E)$ is commutative;\n\\item for every $b\\in \\attr(A)$ and every path $p\\colon b\\twoheadrightarrow b$ the map $f(p)$\nis equal to~$1_{f(b)}$. \n\\end{enumerate}\nThen for any two paths $p$, $q\\colon a\\twoheadrightarrow b$ with $b\\in \\attr(a)$, we get $f(p) = f(q)$. \n\\end{theorem}\n\\begin{proof}\nBy Theorem~\\ref{geometric} we can construct a finite complete reduction\n\tdiagram $\\Gamma$ in\n\t$\\left\\langle A,R \\right\\rangle$ whose upper side is $p$,  left side\n\tis $q$ and which is tiled by e.d.s in $\\mathcal{E}$. It follows that $f\\left( \\Gamma \\right)$ is a\n\tcommutative diagram in $\\mathcal{C}$. \nSuppose that the label of the bottom right corner of $\\Gamma$ is $c$. Then\nthe right  side of $\\Gamma$ gives a path\n$p'\\colon b\\twoheadrightarrow c$ and the bottom side of $\\Gamma$ gives a path $q'\\colon b\\twoheadrightarrow\nc$. From the commutativity of $\\Gamma$ we get\n\\begin{equation}\n\\label{eq:ilike}\nf\\left( p' \\right) f\\left(\np \\right) = f\\left( q' \\right)f\\left( q \\right).\n\\end{equation}\nNow, since $b$ is semi-normal, there is a path $t\\colon c\\twoheadrightarrow b$. By the theorem\nassumptions we have $f(tp') = 1_{f(b)} = f(tq')$. \nThus from \\eqref{eq:ilike}, we get\n\\begin{equation*}\nf(p) = f(tp') f(p)=  f(t)f(p')f(p) = f(t)f(q')f(q) = f(tq')f(q) =f(q). \n\\end{equation*}\n\\end{proof}\nWe will get several corollaries of Theorem~\\ref{main:theory}.\n\\begin{corollary}\\label{cor:mono}\nLet $\\left\\langle A,R \\right\\rangle$ be an ARS, $\\succeq$ be a well-founded preorder on $R$ and $\\mathcal{E}$ a complete set of\ndecreasing e.d.s.  Suppose that\n\\begin{enumerate}[i)]\n\\item $\\twoheadrightarrow$ is a well-founded preorder on $A$;\n\\item for every $E\\in \\mathcal{E}$ the diagram $f(E)$ is commutative;\n\\item for every $r\\in R$ the map $f(r)$ is a monomorphism; \n\\item for every $b\\in \\attr(A)$ and every path $p\\colon b\\twoheadrightarrow b$ the map $f(p)$\nis equal to~$1_{f(b)}$.\n\\end{enumerate}\nThen for any two paths $p$, $q\\colon a\\twoheadrightarrow b$ in $\\left\\langle A,R\n\\right\\rangle$, we get $f(p) = f(q)$. \n\\end{corollary}\n\\begin{proof}\nNote that $\\attr(b)$ is non-empty since $\\twoheadrightarrow$ is a well-founded preorder. \nLet $b'\\in \\attr(b) = \\attr(a)$.  Then there is a path $s\\colon b \\twoheadrightarrow b'$. \nWe get two composed paths $sp$, $sq\\colon a \\twoheadrightarrow b'$.  We\nhave $f(sp) = f(s)f(p)$ and  $f(sq) = f(s)f(q)$. By Theorem~\\ref{main:theory} we\nobtain $f(sp) = f(sq)$. Since $f(s)$ is a monomorphism\n$f(s)f(p) = f(s)f(q)$ implies that $f(p)=f(q)$. \n\\end{proof}\nIf $f\\colon \\left\\langle A,R \\right\\rangle \\to \\mathcal{C}$\nis such that the map $f(r)$ is invertible for every\n$r\\in R$, then we can\ndefine, in the obvious way, \na morphism $f(z)\\colon f(a) \\to f(b)$ for every zigzag $z\\colon a\\rightsquigarrow b$. \n\n\\begin{corollary}\n\\label{cor:iso}\nLet $\\left\\langle A,R \\right\\rangle$ be an ARS, $\\succeq$ be a well-founded preorder on $R$ and $\\mathcal{E}$ a complete set of\ndecreasing e.d.s.  Suppose that\n\\begin{enumerate}[i)]\n\\item $\\twoheadrightarrow$ is a well-founded preorder on $A$;\n\\item for every $E\\in \\mathcal{E}$ the diagram $f(E)$ is commutative;\n\\item for every $r\\in R$ the map $f(r)$ is an isomorphism; \n\\item for every $b\\in \\attr(A)$ and every path $p\\colon b\\twoheadrightarrow b$ the map $f(p)$\nis equal to $1_{f(b)}$.\n\\end{enumerate}\nThen for any two zigzags $z$, $z'\\colon a\\rightsquigarrow b$, we get $f(z) = f(z')$. \n\\end{corollary}\n\\begin{proof}\nThe zigzags $z$ and $z'$ fit in the following reduction diagram\n\\begin{equation*}\n\\xymatrix{\n&& b\\\\\n& a \\ar@{~>}[ur]^{z} \\ar@{~>}[dl]_{z'} \\\\ b\n}\n\\end{equation*}\nof type~\\eqref{zigzag}. Now, by Corollary~\\ref{cor:zigzag}, there is a diagram\nof type \n\\begin{equation*}\n\\xymatrix{\n&& b \\ar[dd]^{p} \\\\\n& a \\ar@{~>}[ur]^{z} \\ar@{~>}[dl]_{z'} \\\\ b \\ar[rr]^{q} && c,\n}\n\\end{equation*}\nwhich is tiled by diagrams in $\\mathcal{E}$. \nNote, that by Corollary~\\ref{cor:mono}, we have $f(p) = f(q)$. \nSince every diagram $f(E)$, $E\\in \\mathcal{E}$, is commutative and every map $f(r)$,\n$r\\in R$, is an isomorphism, we get that \n\\begin{equation*}\nf(z) f(z')^{-1} = f(p)^{-1} f(q) = 1_{f(b)}. \n\\end{equation*}\nTherefore $f(z) = f(z')$. \n\\end{proof}\n\\begin{example}\n\\label{terminating}\nSuppose $\\left\\langle A,R \\right\\rangle$ is a terminating locally confluent ARS. \nBy \\cite[Corollary~4.4]{oostrom}\nall elementary diagrams for a terminating ARS can be made decreasing.\nFurther, for every $b\\in \\attr(A)$ the only path $b\\twoheadrightarrow b$ is the empty one.\nThus we have $f(b\\twoheadrightarrow b) = 1_{f(b)}$ for any map of graphs $f\\colon\n\\left\\langle A,R \\right\\rangle \\to \\mathcal{C}$.  \nHence if $f $ is such that\n$f(r)$ is an isomorphism for all $r\\in R$ and $f(E)$ is commutative for all\n$E$ in a complete set $\\mathcal{E}$ of e.d.s, then $f(z)=f(z')$ for any two zigzags\n$z$, $z'\\colon a\\rightsquigarrow b$ in $\\left\\langle A,R \\right\\rangle$. \n\\end{example}\n\n\n\n\\section{Actions of monoids on categories}\n\\label{action}\n\nLet $\\mathcal{C}$ be a category and $M$ a monoid with  neutral element $e$. Following\n\\cite{deligne} we define a \\emph{(pseudo)action}\n$(\\mathcal{F},\\lambda)$ of $M$ on $\\mathcal{C}$ as a collection of\n\\begin{enumerate}[i)]\n\\item   endofunctors $F_a\\colon \\mathcal{C} \\to \\mathcal{C}$, $a\\in M$,\nsuch that  $F_e \\cong Id $ via the natural isomorphism $\\eta$;\n\\item natural isomorphisms $\\lambda_{a,b}\\colon F_{a}F_b\\to F_{ab}$,\nsuch that for all $a$, $b$, $c\\in M$ the diagram\n\\begin{equation*}\n\\xymatrix{ F_aF_bF_c \\ar[r]^-{\\lambda_{a,b}F_c} \\ar[d]_-{F_a \\lambda_{b,c}} &\nF_{ab}F_c\\ar[d]^-{\\lambda_{ab,c}} \\\\\nF_a F_{bc} \\ar[r]^-{\\lambda_{a,bc}} & F_{abc}}\n\\end{equation*}\ncommutes, and $\\lambda_{e,a}$, $ \\lambda_{a,e}$ \nare induced by $\\eta$. \n\\end{enumerate}\nGiven an action $(F,\\lambda)$ on $\\mathcal{C}$ and a sequence of elements $a_1$, \\dots,\n$a_k$, with $k\\ge 3$,  we will define recursively the natural isomorphism $\\lambda_{a_1,\\dots,a_k}$  from \n$F_{a_1}\\dots F_{a_k}$ to $F_{a_1\\dots a_k}$ by \n\\begin{equation*}\n\\lambda_{a_1,\\dots,a_k} = \\lambda_{a_1,a_2 \\cdots a_k} \\circ\nF_{a_1} (\\lambda_{a_2,\\dots,a_k}).\\end{equation*}\n\n\n\n\nThe actions of $M$ on $\\mathcal{C}$ form a category $[M;\\mathcal{C}]$, where\na morphism from $(F,\\lambda)$ to $(F',\\lambda')$ is given by a collection of\nnatural transformations $\\rho_a\\colon F_a \\to F'_a$, $a\\in M$, such that the\ndiagrams\n\\begin{equation*}\n\\xymatrix{\nF_a F_b \\ar[r]^{\\lambda_{a,b}} \\ar[d]_{\\rho_a \\rho_b} &  F_{ab}\n\\ar[d]^{\\rho_{ab}} \\\\\nF'_a F'_b \\ar[r]^{\\lambda'_{a,b}} & F'_{ab}\n}\n\\quad\n\\xymatrix{\nF_e \\ar[r]^{\\eta} \\ar[d]_{\\rho_e} & \\mathrm{Id}\\\\ F'_e \\ar[ru]_{\\eta'} & \n}\n\\end{equation*}\ncommute.\nFrom a more abstract point of view, the category of actions of $M$ on $\\mathcal{C}$ is\nthe category of pseudofunctors from the category $(*,M)$ \nto the $2$-category $\\mathrm{Cat}$ of categories.\n\n\nSuppose $\\left\\langle X,\\mathbf{r}  \\right\\rangle$ is a presentation of $M$. \nGiven $(F,\\lambda) \\in [M;\\mathcal{C}]$, we get a collection of functors $F_x\\colon\n\\mathcal{C}\\to \\mathcal{C}$ and natural isomorphisms \n\\begin{equation*}\n\\tau_r \\colon F_{a_1} \\dots F_{a_k} \\xrightarrow{\\lambda_{a_1,\\dots, a_k}}\nF_{a_1\\dots a_k} = F_{b_1\\dots b_l}\n\\xrightarrow{\\lambda_{b_1,\\dots,b_l}^{-1}} F_{b_1}\\dots F_{b_l}\n\\end{equation*}\nfor every relation $r=(a_1\\dots a_k,b_1\\dots b_l)$ in $\\mathbf{r} $. \nLet us denote by $[X,r; \\mathcal{C}]$ the category whose objects are pairs \n$( (F_x)_{x\\in X},(\\tau_r)_{r\\in \\mathbf{r} })$ \nwhere $F_x$ are endofunctors of $\\mathcal{C}$ and, for every $r=(a_1\\dots a_k, b_1\\dots,\nb_l)$, $\\tau_r$ is a natural isomorphism from $F_{a_1}\\dots F_{a_k}$ to\n$F_{b_1}\\dots F_{b_l}$.  \nThe  morphisms from $(F,\\tau)$ to\n$(F',\\tau')$ in $[X,r;\\mathcal{C}]$  are  families of  natural transformations $\\rho_x\\colon F_x\\to\nF'_x$, $x\\in X$, such that for all $r=(a_1\\dots a_k,b_1\\dots b_l)\\in \\mathbf{r} $ the diagrams\n\\begin{equation*}\n\\xymatrix{\n F_{a_1} \\dots F_{a_k} \\ar[r]^{\\tau_r} \\ar[d]_{\\rho_{a_1}\\dots \\rho_{a_k}} &\nF_{b_1}\\dots F_{b_l} \\ar[d]^{\\rho_{b_1}\\dots \\rho_{b_l}}\n\\\\\n  F'_{a_1} \\dots F'_{a_k}  \\ar[r]^{\\tau'_r} & F'_{b_1}\\dots F'_{b_l}\n}\n\\end{equation*}\nare commutative.\nThen we get from the construction described above the restriction functor\n$\\mathrm{Res}\\colon [M;\\mathcal{C}] \\to [X,\\mathbf{r} ;\\mathcal{C}]$. \n\\begin{theorem}\\label{trm:subcat}\nThe functor $\\mathrm{Res}\\colon [M;\\mathcal{C}] \\to [X,\\mathbf{r} ;\\mathcal{C}]$ is full and faithful.\n\\end{theorem}\nTheorem~\\ref{trm:subcat} should be well-known. \nIn Section~6, we reobtain it as a consequence of Corollary~\\ref{cor:iso}. \n\n\nIt is clear that it is easier to specify objects in $[X,\\mathbf{r} ;\\mathcal{C}]$ than\n objects in $[M;\\mathcal{C}]$. Therefore it is important to have a description of the\nessential image of the functor $\\mathrm{Res}$. This can be done using the coherent\npresentations of $M$ described in the next section.  \n\n\\section{Monoids and ARS}\n\\label{monoids}\nLet $M$ be a monoid with neutral element $e$ and $\\left\\langle X,\\mathbf{r} \n\\right\\rangle$ a presentation of $M$. \nWe will denote by   $X^*$  the set of all finite words over the alphabet $X$. \nThe set $X^*$ will be considered  as a free monoid with multiplication given by\n concatenation of words and  neutral element given by the empty word\n$\\varnothing$. \nDenote by $\\phi$ the canonical epimorphism from $X^*$ to $M$. \n Let\n\\begin{equation*}\nX^* \\mathbf{r}  X^*:= \\left\\{\\, (w_1w w_2, w_1 w'w_2) \\,\\middle|\\, w_1,w_2\\in X^*,\\, (w,w')\\in \\mathbf{r} \n\\right\\}\\subset X^* \\times X^*. \n\\end{equation*} \nWe will sometimes write the elements of $X^*\\mathbf{r}  X^*$ in the form $w_1 r w_2$\nwith $r\\in \\mathbf{r} $. \nLet us consider the ARS $\\left\\langle X^*,X^* \\mathbf{r}  X^* \n\\right\\rangle$. \nIt is clear that if $w_1 \\twoheadrightarrow w_2$ in $\\left\\langle X^*,X^*\\mathbf{r}  X^*\n\\right\\rangle$, then $\\phi(w_1) = \\phi(w_2)$. \n\\begin{proposition}\nSuppose $\\left\\langle X^*,X^*\\mathbf{r}  X^* \\right\\rangle$ is confluent and $\\twoheadrightarrow$\nis a well-founded preorder. Then $\\phi(u) = \\phi(v)$ if and only if\n$\\attr(u) = \\attr(v)$.\n\\end{proposition}\n\\begin{proof}\nThe ``if'' part is obvious. Suppose $\\phi(u) = \\phi(v)$. Then there is a\nsequence of words\n\\begin{equation*}\nu=w_0, w_1, \\dots, w_k = v\n\\end{equation*}\nsuch  that $(w_j, w_{j-1}) \\in X^* \\mathbf{r}  X^*$ or $(w_{j-1},w_j) \\in X^*\\mathbf{r}  X^*$\nfor all $1\\le j\\le k$. In other words we have a zigzag $u\\rightsquigarrow v$ in\n$\\left\\langle X^*, X^* \\mathbf{r}  X^* \\right\\rangle$. Using the fact that $\\left\\langle X^*,\nX^*\\mathbf{r}  X^*\n\\right\\rangle$ is confluent and following the same reasoning as in the\nproof of Corollary~\\ref{cor:zigzag}, we conclude that there are $w \\in X^*$ and\ntwo\npaths $u\\twoheadrightarrow w$, $v\\twoheadrightarrow w$ in $\\left\\langle X^*, X^*\\mathbf{r}  \\right\\rangle$. Thus\n\\begin{equation*}\n\\attr(u) = \\attr(w) = \\attr(v). \n\\end{equation*} \n\\end{proof}\nWe will denote by $l(w)$ the length of $w\\in X^*$. \nIf $r = (u,v) \\in \\mathbf{r} $, then we define $s(r) = u$ and $t(r)=v$. \n\\begin{definition}\n\\label{criticalpair}\nA \\emph{critical pair} is a \npair of elements in $X^* \\mathbf{r}  X^*$ of one of the forms\n\\begin{enumerate}[i)]\n\\item  $(ur,r'v)$ with $us(r) = s(r')v$ in $X^*$;\n\\item $(r, ur'v)$ with $s(r) = u s(r') v$. \n\\end{enumerate}\nWe say that a critical pair of the first type is \\emph{convergent} if there is\n$w\\in X^*$ and there are paths $ut(r) \\twoheadrightarrow w$, $t(r')v \\twoheadrightarrow w$. A critical\npair of the second type is called \\emph{convergent} if there is $w\\in X^*$ and\nthere are paths $t(r) \\twoheadrightarrow w$, $ut(r')v\\twoheadrightarrow w$ in $\\left\\langle X^*, X^* \\mathbf{r} \nX^*\n\\right\\rangle$. \n\\end{definition}\nGiven convergent critical pairs $(ur,r'v)$, $(r,ur'v)$ and convergence paths\n$ut(r) \\twoheadrightarrow w$, $t(r')v\\twoheadrightarrow w$, $t(r) \\twoheadrightarrow w'$, $ut(r')v\\twoheadrightarrow w'$, we\ndefine the following  e.d.s in $\\left\\langle X^*,X^*\\mathbf{r}  X^* \\right\\rangle$\n\\begin{equation}\n\\label{criticaled}\n\\begin{aligned}\n\\xymatrix{\n{\\phantom{s(r)u=}  s(r')v = us(r)} \\ar[r]^-{ur} \\ar[d]_{r'v} & u t(r) \\ar@{->>}[d] \\\\\n t(r')v \\ar@{->>}[r] & w\n}& \n\\xymatrix{\n{\\phantom{s(r)=} us(r')v = s(r)} \\ar[d]_{ur'v} \\ar[r]^-{r} & t(r)\n\\ar@{->>}[d]\\\\\nut(r')v \\ar@{->>}[r] & w'.\n}\n\\end{aligned}\n\\end{equation}\nWe will call the  e.d.s \\eqref{criticaled} a  \\emph{critical e.d.s}.\nLet $\\mathcal{Y}$ be a set of critical e.d.s. We say that $\\mathcal{Y}$ is complete\nif for every critical pair there is at least one corresponding  critical e.d. in\n$\\mathcal{Y}$.\n\n Let $r$, $r'\\in \\mathbf{r} $ and $w \\in W$. We will denote by $\\mathcal{N}(r,w,r')$ the e.d. \\begin{equation*}\n\\xymatrix{\ns(r) w s(r') \\ar[r]^{rws(r')} \\ar[d]_{s(r)w r'} & t(r)ws(r')\\ar[d]^{t(r)w r'}\\\\\ns(r) w t(r') \\ar[r]^{rwt(r')}  & t(r) wt(r').\n}\n\\end{equation*}\nThe e.d. $\\mathcal{N}(r,w,r')$ is called a \\emph{natural e.d.}\n We write $\\mathcal{N}$ for the set of all natural e.d.s.\nGiven an e.d. \n\\begin{equation*}\nE :=\n\\begin{gathered}\n \\xymatrix{\nu \\ar[r] \\ar[d] & v \\ar@{->>}[d]  \\\\\nw \\ar@{->>}[r] & z\n}\n\\end{gathered}\n\\end{equation*}\n in $\\left\\langle X^*, X^*\\mathbf{r}  X^* \\right\\rangle$ and words\n$w_1$, $w_2\\in X^*$, we define \\begin{equation*}\nw_1 E w_2 =\n\\begin{gathered}\n \\xymatrix{\nw_1 uw_2 \\ar[r] \\ar[d] & w_1 v w_2\\ar@{->>}[d]  \\\\\nw_1 w w_2\\ar@{->>}[r] & w_1 z w_2\n}\n\\end{gathered},\n\\quad  \\quad \nE^t =\n\\begin{gathered}\n\\xymatrix{\nu \\ar[r] \\ar[d] & w \\ar@{->>}[d]  \\\\\nv \\ar@{->>}[r] & z\n}\n\\end{gathered}.\n\\end{equation*} \nLet $\\mathcal{Y}$ be a complete set of critical e.d.s. Then the set \n\\begin{equation}\n\\label{eds}\n\\mathcal{E} = X^* \\mathcal{N}\nX^* \\sqcup X^* \\mathcal{N}^t X^* \\sqcup X^* \\mathcal{Y} X^* \\sqcup X^* \\mathcal{Y}^t X^*\n\\end{equation}\n is a complete set of e.d.s for the ARS $\\left\\langle\nX^*, X^* \\mathbf{r}  X^* \\right\\rangle$.\n\nWe say that a preorder $\\succeq$ on $X^* \\mathbf{r}  X^*$ is \\emph{monomial} if \n\\begin{enumerate}[1)]\n\\item for every\n$\\rho_1$, $\\rho_2\\in X^*\\mathbf{r}  X^*$ such that \n$\\rho_1 \\succ \\rho_2$ and for every\n$w\\in X^*$, we have\n\\begin{equation*}\nw \\rho_1 \\succ w \\rho_2, \\quad \\rho_1 w \\succ \\rho_2 w;\n\\end{equation*}\n\\item for every\n$\\rho_1$, $\\rho_2\\in X^*\\mathbf{r}  X^*$ such that \n$\\rho_1 \\sim \\rho_2$ and for every\n$w\\in X^*$, we have\n\\begin{equation*}\nw \\rho_1 \\sim w \\rho_2, \\quad \\rho_1 w \\sim \\rho_2 w.\n\\end{equation*}\n\\end{enumerate}\nLet us state for the late use \n\\begin{proposition}\n\\label{decreasinged}\nLet $\\left\\langle X,\\mathbf{r}  \\right\\rangle$ be a presentation of a monoid $M$,\n$\\mathcal{Y}$ a complete set of critical e.d.s and $\\mathcal{N}$ the set of all natural\ne.d.s. Define $\\mathcal{E}$  as in~\\eqref{eds}.\nSuppose $\\succeq$ is a monomial preorder on $X^* \\mathbf{r}  X^*$ such that all the\ne.d.s in $\\mathcal{Y}$ and $\\mathcal{N}$ are decreasing. Then all e.d.s in $\\mathcal{E}$ are \ndecreasing as well. \n\\end{proposition}\n\n\n\\section{Coherent presentation}\n\\label{coherent}\nLet $\\left\\langle X,\\mathbf{r}  \\right\\rangle$ be a presentation of a monoid $M$. \nSuppose $(F,\\tau)\\in\\left[ X,\\mathbf{r} ;\\mathcal{C} \\right]$.\nDenote by $\\End(\\mathcal{C})$ the category of endofunctors of $\\mathcal{C}$. \n Define the map of graphs\n$f_{F,\\tau} \\colon \\left\\langle X^*,X^*\\mathbf{r}  X^* \\right\\rangle \\to \\End(\\mathcal{C})$ by\n\\begin{equation*}\n\\begin{aligned}\nf_{F,\\tau} (x_1\\dots x_k) &= F_{x_1} \\dots F_{x_k}, \\quad x_i\\in X\\\\\nf_{F,\\tau} (x_1 \\dots x_k r y_1 \\dots y_l)& =  F_{x_1} \\dots F_{x_k} \\tau_r\n F_{y_1} \\dots F_{y_l}, \\quad x_i,y_j \\in X,\\ r\\in\\mathbf{r} .\n\\end{aligned}\n\\end{equation*}\nWe will also use the following  natural abbreviations\n\\begin{equation*}\n\\begin{aligned}\nF_w &:= f_{F,\\tau}(w),\\ w\\in X^*; &  \\tau_{\\rho} := f_{F,\\tau} (\\rho), \\rho\\in\nX^* \\mathbf{r}  X^*\n\\end{aligned}\n\\end{equation*}\n For every path $p$\n\\begin{equation*}\nw_0 \\xrightarrow{r_1} w_1 \\rightarrow \\dots \\xrightarrow{r_k} w_k\n\\end{equation*}\nin $\\left\\langle X^*, X^*\\mathbf{r}  X^* \\right\\rangle$ we denote by $\\tau_p$ the\nnatural isomorphism $\\tau_{r_k} \\dots \\tau_{r_1}$. \nIf $p\\colon w \\twoheadrightarrow w$ is of length zero then $\\tau_p$ is the identity\ntransformation of $F_w$. Now, given a zigzag $z$\n\\begin{equation*}\nw_0 \\stackrel{p_1}{\\twoheadrightarrow} w_1 \\stackrel{p_2}{\\twoheadleftarrow} w_2 \\twoheadrightarrow \\dots\nw_{k-1} \\stackrel{p_{2k}}{\\twoheadleftarrow} w_{2k},\n\\end{equation*}\nwhere some paths could be empty, \nwe define $\\tau_z$ to be the product\n\\begin{equation*}\n\\tau_{p_1} \\tau_{p_2}^{-1} \\tau_{p_3} \\dots \\tau_{p_{2k-1}}^{-1}. \n\\end{equation*}\n\nWe say that two zigzags $z_1$ and $z_2$ are parallel if they have the same\nsource and target. \nGiven a set $\\mathcal{Z}$ of pairs of parallel zigzags,  we will define $\\left[ X,\\mathbf{r} ,\\mathcal{Z};\\mathcal{C} \\right]$\nto be the full subcategory of $[X,\\mathbf{r}  ; \\mathcal{C}]$ with objects  $(F,\\tau)$ such\nthat $\\tau_p=\\tau_q$ for all $(p,q)\\in \\mathcal{Z}$. \n\nIn what follows, if $E$ is an e.d. then $E\\in \\mathcal{Z}$ means that the two paths,\nthat one can obtain from $E$, constitute a pair in $\\mathcal{Z}$. Thus if $E\\in\n\\mathcal{Z}$ and $(F,\\tau)\\in [X,\\mathbf{r} ,\\mathcal{Z};\\mathcal{C}]$, then $f_{F,\\tau}(E)$ is a\ncommutative diagram.  \n\nNote that if $E$ is a natural e.d. then $f_{F,\\tau}(E)$ is commutative since\n$\\tau_r$ are natural transformations. Similarly, if $E$ is an e.d. such that\n$f_{F,\\tau}(E)$ is commutative, then also $f_{F,\\tau}(u E v)$ and\n$f_{F,\\tau}(u E^t v)$  are commutative for all $u$, $v\\in X^* $. \nThus if $\\mathcal{Y}$ is a set of critical e.d.s and $\\mathcal{E}$ is defined as\nin~\\eqref{eds}\nthen\n\\begin{equation}\\label{ced=ed}\n[X,\\mathbf{r} ,\\mathcal{Y}\\sqcup \\mathcal{Z};\\mathcal{C}] = [X,\\mathbf{r} ,\\mathcal{E} \\sqcup \\mathcal{Z};\\mathcal{C}] \n\\end{equation}\nfor any collection $\\mathcal{Z}$ of pairs of parallel zigzags. \n\\begin{example}\n\\label{full}\nTo avoid ambiguity we write elements in $M^k$ as $m_1|m_2|\\cdots|m_k$. Then\nwe have a\npresentation $\\left\\langle M,\\widetilde{\\mathbf{r} } \\right\\rangle$ of $M$, with\n\\begin{equation*}\n\\widetilde{\\mathbf{r} } = \\left\\{\\, \\left(m_1| m_2, m_1 m_2\\right) \\,:\\, m_1,m_2\\in M \\right\\}\n\\sqcup\n\\{(e,\\varnothing)\\}.\n\\end{equation*}\nEvery path in the resulting ARS $\\left\\langle M^*,M^*\\widetilde{\\mathbf{r} } M^* \\right\\rangle$\nends either at $m\\in M^1$, $m\\not=e$, or at $\\varnothing$. \nThus $\\left\\langle M^*,M^*\\widetilde{\\mathbf{r} } M^* \\right\\rangle$ is terminating. \nEvery critical e.d. for $\\left\\langle M,\\widetilde{\\mathbf{r} } \\right\\rangle$  has one of the following forms\n\\begin{equation*}\n\\xymatrix{\nm_1|m_2|m_3 \\ar[r] \\ar[d] & m_1|m_2m_3\\ar[d] \\\\\nm_1m_2 | m_3 \\ar[r] & m_1m_2m_3\n}\\\n\\xymatrix{\nm|e \\ar[r]^{(m|e,m)} \\ar[d]_{m|(e,\\varnothing)} & m \\ar@{-->}[d]\\\\\nm \\ar@{-->}[r]& m\n}\\\n\\xymatrix{\ne|m \\ar[r]^{(e|m,m)} \\ar[d]_{(e,\\varnothing)|m} & m\\ar@{-->}[d] \\\\\nm \\ar@{-->}[r] & m.\n}\n\\end{equation*}\nLet us denote by $\\mathcal{Y}$ the set of all critical e.d.s. Then, by comparing\ndefinitions, we get $[M,\\widetilde{\\mathbf{r} },\\mathcal{Y};\\mathcal{C}] = [M;\\mathcal{C}]$. \nLet $\\mathcal{E}$ be defined by~\\eqref{eds}. Then, using~\\eqref{ced=ed}, we get\n\\begin{equation*}\n[M,\\widetilde{\\mathbf{r} },\\mathcal{E};\\mathcal{C}] = [M;\\mathcal{C}].\n\\end{equation*}\nLet $(F,\\lambda) \\in [M,\\widetilde{\\mathbf{r} },\\mathcal{E};\\mathcal{C}]$.  By\nExample~\\ref{terminating}, it follows that for any two zigzags $z$,$z'\\colon m_1|\\cdots|m_k \\to m'_1|\\cdots|m'_l$\nin $\\left\\langle M^*,M^*\\widetilde{\\mathbf{r} } M^* \\right\\rangle$ one has  $f_{F,\\lambda}(z) =\nf_{F,\\lambda}(z')$. \n\\end{example}\n\n\\begin{proposition}\n\\label{factorize}\nLet $\\left\\langle X,\\mathbf{r}  \\right\\rangle$ be a presentation of a monoid $M$. \n For any set $\\mathcal{Z}$ of pairs of parallel zigzags in $\\left\\langle X^*,X^*\\mathbf{r}  X^* \\right\\rangle$ the restriction functor $\\mathrm{Res}$ defined in\nSection~\\ref{action} factors via the embedding $[ X,\\mathbf{r} ,\\mathcal{Z};\\mathcal{C}] \\to\n[X,\\mathbf{r} ;\\mathcal{C}]$. We will denote the resulting functor $[M;\\mathcal{C}]\\to\n[X,\\mathbf{r} ,\\mathcal{Z};\\mathcal{C}]$ by $\\mathrm{Res}_\\mathcal{P}$. \n\\end{proposition}\n\\begin{proof}\nLet $(F,\\lambda) \\in [M;\\mathcal{C}]$ and $\\mathrm{Res}(F,\\lambda) = (F,\\tau)\\in [X,\\mathbf{r} ,\\mathcal{E};\\mathcal{C}]$. \nBy the definition of $\\tau$, every natural isomorphism $\\tau_r$, $r\\in \\mathbf{r} $, is a\nvalue of $f_{F,\\lambda}$ on a suitable zigzag in the defined above ARS\n$\\left\\langle M^*, M^*\\widetilde{\\mathbf{r} }M^*  \\right\\rangle$. \nNow let \n$(p,q) \\in \\mathcal{Z}$, with $p$, $q\\colon u\\rightsquigarrow w$. We have to prove that $\\tau_p = \\tau_q$.\nSince every $\\tau_r$ can be replaced by $\\lambda_z$ for a suitable zigzag\n$z$ in $\\left\\langle M^*, M^* \\widetilde{\\mathbf{r} } M^* \\right\\rangle$, we see that\nthere are zigzags $z'$, $z'' \\colon u \\rightsquigarrow w$ in  $\\left\\langle M^*, M^* \\widetilde{\\mathbf{r} } M^* \\right\\rangle$\nsuch that $\\tau_p = \\lambda_{z'}$ and $\\tau_q = \\lambda_{z''}$. But by\nExample~\\ref{full}, we have $\\lambda_{z'} = \\lambda_{z''}$. Thus also $\\tau_p =\n\\tau_q$. \n\\end{proof}\n\\begin{proof}[Proof of Theorem~\\ref{trm:subcat}]\nThe functor $\\mathrm{Res}$ is faithful as every morphism $\\rho \\colon (F,\\lambda) \\to\n(G,\\mu)$ in $[M;\\mathcal{C}]$ can be uniquely reconstructed from  its image $\\nu = \\mathrm{Res}(\\rho)$ by\nuse of  the diagrams\n\\begin{equation}\n\\label{diag}\n\\begin{gathered}\n\\xymatrix@C5em@R4em{\nF_{x_1} \\dots F_{x_k} \\ar[r]^{\\lambda_{x_1,\\dots,x_k}} \\ar[d]_{\\nu_{x_1}\\dots\n\\nu_{x_k}} & F_{x_1\\dots x_k} \\ar[d]^{\\rho_{x_1\\dots x_k}} \\\\\nG_{x_1} \\dots G_{x_k} \\ar[r]^{\\mu_{x_1,\\dots,x_k}} \n & G_{x_1\\dots x_k}.\n}\n\\end{gathered}\n\\end{equation}\nNow we show that the functor $\\mathrm{Res}$ is full. \nDenote the images of $(F,\\lambda)$ and of $(G,\\mu)$ under $\\mathrm{Res}$ by $(F,\\tau)$\nand $(G,\\sigma)$, respectively. Take a morphism $\\nu\\colon (F,\\tau)\\to\n(G,\\sigma)$ in $[X,\\mathbf{r} ;\\mathcal{C}]$. Let $m\\in M$. Then we can write $m$ as a product\nof elements in $X$, say $m = x_1\\dots x_k$. We define $\\rho_m$ by using the\ndiagram~\\eqref{diag}\n\\begin{equation*}\n\\rho_m = \\mu_{x_1\\dots x_k} \\circ \\nu_{x_1} \\dots \\nu_{x_k} \\circ \\lambda_{x_1\\dots\nx_k}^{-1}.\n\\end{equation*}\n We have to check that the natural transformation  $\\rho_m$ is well-defined. \nSuppose $y_1\\dots y_l = m $  with $y_j\\in X$. As $\\left\\langle X,\\mathbf{r} \n\\right\\rangle$ is a presentation of $M$, there is a zigzag $z\\colon x_1\\dots x_k\n\\to y_1\\dots y_l$ in $\\left\\langle X^*, X^* \\mathbf{r}  X^* \\right\\rangle$.  \nSince $\\nu$ is a morphism in $[X,\\mathbf{r} ;\\mathcal{C}]$ we get that the diagram\n\\begin{equation*}\n\\xymatrix{\nF_{x_1} \\dots F_{x_k} \\ar[r]^{\\tau_{z}} \\ar[d]_{\\nu_{x_1}\\dots\n\\nu_{x_k}} &\n F_{y_1} \\dots F_{y_l}\\ar[d]^{\\nu_{y_1}\\dots\n\\nu_{y_l}} \\\\\nG_{x_1} \\dots G_{x_k} \\ar[r]^{\\sigma_{z}} \n & G_{y_1} \\dots G_{y_l}\n}\n\\end{equation*}\ncommutes.\nAs in the proof of Proposition~\\ref{factorize}, we can find a zigzag $z'$ in\n$\\left\\langle M^*, M^* \\widetilde{\\mathbf{r} }M^* \\right\\rangle$ such that\n$\\lambda_{z'} = \\tau_z$ and $\\mu_{z'} = \\sigma_z$. \nNow, consider the zigzag\n\\begin{equation*}\nz'' \\colon x_1|\\dots | x_k \\twoheadrightarrow m \\twoheadleftarrow y_1|\\dots|y_l\n\\end{equation*}\nin $\\left\\langle M^*,M^*\\widetilde{\\mathbf{r} }M^* \\right\\rangle$. Since $z'$ and\n$z''$ have the same source and target, we get as in the proof of\nProposition~\\ref{factorize}, that $\\lambda_{z'} = \\lambda_{z''}$ and\n$\\mu_{z'}= \\mu_{z''}$. \nThus we have the commutative diagram\n\\begin{equation*}\n\\xymatrix{\nF_{y_1} \\dots F_{y_l} \\ar[r]^{\\lambda_{z''} }\\ar@\/^6ex\/[rr]^{\\lambda_{y_1,\\dots\n,y_l}}\n\\ar[d]_{\\nu_{y_1}\\dots\n\\nu_{y_k}}\n & F_{x_1} \\dots F_{x_k} \\ar[r]^{\\lambda_{x_1,\\dots,x_k}} \\ar[d]_{\\nu_{x_1}\\dots\n\\nu_{x_k}} & F_{x_1\\dots x_k} \\ar[d]^{\\rho_{m}} \\\\\n G_{y_1} \\dots G_{y_l}\\ar[r]^{\\mu_{z''} }\\ar@\/_6ex\/[rr]_{\\mu_{y_1, \\dots\n,y_l}} &\n G_{x_1} \\dots G_{x_k} \\ar[r]^{\\mu_{x_1,\\dots,x_k}} \n & G_{x_1\\dots x_k},\n}\n\\end{equation*}\nwhich shows that $\\rho_m$ does not depend on the choice of the presentation of\n$m$ in~$\\left\\langle X,\\mathbf{r}  \\right\\rangle$.  \n\nNow we have to check that $(\\rho_m)_{m\\in M}$ is a well-defined morphism in\n$[M;\\mathcal{C}]$,i.e. that  for every $m_1$, $m_2\\in M$ the diagram\n\\begin{equation*}\n\\xymatrix{\nF_{m_1} F_{m_2} \\ar[r]^{\\lambda_{m_1,m_2}} \\ar[d]_{\\rho_{m_1} \\rho_{m_2}} &\nF_{m_1m_2} \\ar[d]^{\\rho_{m_1m_2}} \\\\\nG_{m_1} G_{m_2} \\ar[r]^{\\mu_{m_1,m_2}}  &\nG_{m_1m_2}.\n}\n\\end{equation*}\ncommutes. Suppose $m_1 = x_1\\dots x_k$ and $m_2 =\ny_1\\dots y_l$ with $x_i$, $y_j\\in X$. Then since we can use the presentation\n$x_1\\dots x_k y_1\\dots y_l$ of $m_1m_2$ to define $\\rho_{m_1m_2}$, we get that\nin the diagram \n\\begin{equation*}\n\\xymatrix@C7em{\n F_{x_1} \\dots F_{x_k}F_{y_1} \\dots F_{y_l} \\ar@\/^6ex\/[rr]^-{\n\\lambda_{x_1,\\dots,x_k,y_1,\\dots\n,y_l}}\\ar[r]^-{\\lambda_{x_1,\\dots,x_k}\\lambda_{y_1,\\dots,\n,y_l}}\n\\ar[d]_{\\nu_{x_1}\\dots \\nu_{x_k} \\nu_{y_1} \\dots \\nu_{y_l}}\n& F_{m_1} F_{m_2} \\ar[r]^{\\lambda_{m_1,m_2}} \\ar[d]_{\\rho_{m_1} \\rho_{m_2}} &\nF_{m_1m_2} \\ar[d]^{\\rho_{m_1m_2}} \\\\\n G_{x_1} \\dots G_{x_k}G_{y_1} \\dots G_{y_l} \\ar@\/_6ex\/[rr]_-{\n\\mu_{x_1,\\dots,x_k,y_1,\\dots\n,y_l}}\\ar[r]^-{\\mu_{x_1,\\dots,x_k}\\mu_{y_1,\\dots,\n,y_l}}\n& G_{m_1} G_{m_2} \\ar[r]^{\\mu_{m_1,m_2}}  &\nG_{m_1m_2},\n}\n\\end{equation*}\nthe triangles commute by the definition of the $\\lambda$'s, the left rectangle\ncommutes by the definition of $\\rho_{m_1}$ and $\\rho_{m_2}$, and the external\nrectangle commutes by the definition of $\\rho_{m_1m_2}$. Since all $\\lambda$'s are\nisomorphisms, we get that also the right rectangle commutes. \nThis shows that $\\rho$ is a well-defined morphism from $(F,\\lambda)$ to\n$(G,\\mu)$ in $[M;\\mathcal{C}]$. \n\\end{proof}\nWe say that a set $\\mathcal{Z}$ of pairs of parallel zigzags in $\\left\\langle X^*,X^*\\mathbf{r}  X^*\n\\right\\rangle$ defines a \\emph{coherent presentation} of $M$, if the functor\n$\\mathrm{Res}_{\\mathcal{Z}}\\colon [M;\\mathcal{C}] \\to [X,\\mathbf{r} ,\\mathcal{Z};\\mathcal{C}]$ is an equivalence of\ncategories. \nThe main result of this section is the following theorem.\n\\begin{theorem}\\label{main:presentation}\nLet $\\left\\langle X,\\mathbf{r}  \\right\\rangle$ be a presentation of a monoid $M$ with\nneutral element~$e$. Suppose that \n\\begin{enumerate}[1)]\n\\item the transitive closure $\\twoheadrightarrow$ of $X^* \\mathbf{r}  X^*$ is\na well-founded preorder on $X^*$;\n\\item  there exists a well-founded monomial preorder $\\succeq$ on $X^* \\mathbf{r} \nX^*$ such that all natural e.d.s are decreasing;\n\\item there is a complete set $\\mathcal{Y}$ of critical e.d.s that are decreasing with\nrespect to~$\\succeq$.\n\\end{enumerate}\n Denote by\n$\\mathcal{L}$ the collection of all pairs $(p,\\varnothing_b)$, where\n$b\\in\\attr(A)$, $\\varnothing_b$ is the empty path at $b$, and $p\\colon b\\twoheadrightarrow\nb$. Then $\\mathrm{Res}_{\\mathcal{Y}\\sqcup \\mathcal{L}}$ is an equivalence of categories. \n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal{E}$ be the collection of e.d.s defined by \\eqref{eds}.  In view of \\eqref{ced=ed}, it is enough to\nshow that\n\\begin{equation*}\n\\mathrm{Res}_{\\mathcal{E}\\sqcup \\mathcal{L}}\\colon [M;\\mathcal{C}] \\to [X,\\mathbf{r} , \\mathcal{E}\\sqcup \\mathcal{L}; \\mathcal{C}]\n\\end{equation*}\nis an equivalence of categories. \nSince $\\mathrm{Res}_{\\mathcal{E}\\sqcup \\mathcal{L}}$ is fully faithful, it is enough to check that it\nis a dense functor. \nLet $(F,\\tau)\\in [X,\\mathbf{r} ,\\mathcal{E}\\sqcup \\mathcal{L} ;\\mathcal{C}]$. Then\n\\begin{equation*}\nf_{F,\\tau} \\colon \\left\\langle X^*,X^* \\mathbf{r}  X^* \\right\\rangle \\to \\End(\\mathcal{C})\n\\end{equation*}\nsatisfies conditions of Corollary~\\ref{cor:iso}. \nIn particular, for any two zigzags $z$, $z'\\colon u\\rightsquigarrow v$ in $\\left\\langle X^*, X^* \\mathbf{r}  X^*\n\\right\\rangle$ we have $\\tau_z = \\tau_{z'}$. \n\nNow, for every $m$ in $M$ we choose a presentation $x_1\\dots x_k$ of $m$ in\n$\\left\\langle X,\\mathbf{r}  \\right\\rangle$. We will assume that if $m\\in X$, then its\nchosen presentation is $m$ itself. \nWe define $G_m = F_{x_1}\\dots F_{x_k}$ using the above chosen presentation.\nNow, let $m'\\in M$ and $y_1\\dots y_l$ be its chosen presentation. Then\n$x_1\\dots x_k y_1\\dots y_l$ is a presentation of $m m'$, which, in general, is \ndifferent of the chosen presentation, say, $z_1\\dots z_n$ of $mm'$. \nSince $\\left\\langle X,\\mathbf{r}  \\right\\rangle$ is a presentation of $M$, we get that\nthere is zigzag $\\zeta\\colon x_1\\dots x_k y_1 \\dots y_l \\rightsquigarrow z_1 \\dots z_n$\nin $\\left\\langle X^*, X^* \\mathbf{r}  X^* \\right\\rangle$. We define $\\lambda_{m,m'} =\n\\tau_{\\zeta}$. As we already mentioned the resulting natural isomorphism is independent\nof the choice of $\\zeta$.\nNow, if $m''\\in M$, the compositions\n\\begin{equation}\n\\begin{aligned}\n\\label{comp}\n& G_m G_{m'} G_{m''} \\xrightarrow{G_m \\lambda_{m',m''}} G_m G_{m'm''}\n\\xrightarrow{\\lambda_{m,m'm''}} G_{mm'm''}\\\\\n& G_m G_{m'} G_{m''} \\xrightarrow{ \\lambda_{m,m'}G_{m''}}  G_{mm'}G_{m''}\n\\xrightarrow{\\lambda_{mm',m''}} G_{mm'm''}\n\\end{aligned}\n\\end{equation}\nare equal to $\\tau_{\\zeta'}$ and $\\tau_{\\zeta''}$ for suitable parallel zigzags\n$\\zeta'$, $\\zeta''$ in\n$\\left\\langle X*,X^*\\mathbf{r}  X^* \\right\\rangle$.\nThus the natural transformations \\eqref{comp} are equal.\nHence we get that  $(G,\\lambda)$ is an object of $[M;\\mathcal{C}]$. \n\nWe have to check that $\\mathrm{Res}(G,\\lambda) = (F,\\tau)$. Since for every $x\\in X$, we\nhave $G_x = F_x$, we get $\\mathrm{Res}(G,\\lambda) = (F,\\sigma)$. Thus, we have only to\ncheck that $\\sigma = \\tau$. For every $r = (u,v)\\in \\mathbf{r} $, we defined $\\sigma_r$ as\n$\\lambda_z$ for  a suitable zigzag $z$ in $\\left\\langle M^*, M^*\n\\widetilde{\\mathbf{r} } M^* \\right\\rangle$. Since every $\\lambda_{m,m'}$ is of the form\n$\\tau_{z'}$ for a zigzag $z'$ in $\\left\\langle X^*, X^* \\mathbf{r}  X^* \\right\\rangle$,\nwe get that there is a zigzag $\\zeta\\colon u \\rightsquigarrow v$ in $\\left\\langle X^*,\nX^*\\mathbf{r}  X^*\n\\right\\rangle$ such that $\\lambda_z = \\tau_\\zeta$. But now $\\tau_\\zeta =\n\\tau_r$. This shows that $\\sigma_r = \\tau_r$.  \n\\end{proof} \n\\begin{example}\n\\label{knuth}\nSuppose $\\left\\langle X^*, X^* \\mathbf{r}  X^* \\right\\rangle$ is terminating and\n$\\mathbf{r} $ satisfies the Knuth-Bendix condition. Let $\\mathcal{Y}$ be a complete set of\ncritical e.d.s. Then, in view of Example~\\ref{terminating}, we get that\n$\\mathrm{Res}_\\mathcal{Y} $  is an equivalence of categories. Thus one of the ways to find a\ncoherent presentation of a monoid $M$ is to perform the Knuth-Bendix completion\nprocedure on a given presentation. Unfortunately, this process can either not\nto finish or to lead to an enormous coherent presentation which is not useful for practical\npurposes.  \n\\end{example}\n\n\n\\section{Coherent presentation for the  $0$-Hecke monoid}\n\\label{hecke}\n\nThe $0$-Hecke monoid $\\rs{n+1}$ is defined by the following presentation\n\\begin{equation*}\n\\begin{aligned}\n& X = \\left\\{ T_1,\\dots, T_n \\right\\}\\\\[3ex]\n&{\\mathbf{r} '}=\\left\\{ a_{i}\\,\\middle|\\, 1\\le i\\le n \\right\\}\\sqcup \\left\\{\\, b_{i+1}\n\\,\\middle|\\, 1\\le i\\le n-1 \\right\\} \\sqcup \\left\\{\\, c_{ji} \\,\\middle|\\,\n 1 \\le i \\le j-2 \\le n-2\n\\right\\}, \n\\end{aligned}\n\\end{equation*}\nwhere\n\\begin{equation*}\na_{i} = (T_iT_i, T_i) ,\\quad b_{i+1} = (T_{i+1}T_i T_{i+1}, T_i\nT_{i+1}T_i),\\quad c_{ji} = (T_{j}T_i , T_i T_j).\n\\end{equation*}\nIn this section we find a coherent presentation of $\\rs{n+1}$ that extends the\nabove presentation of $\\rs{n+1}$. \nIt is easy to see that $\\left\\langle X^*,X^* {\\mathbf{r} '} X^* \\right\\rangle$\nis not (locally) confluent.\nWe  denote by $\\mathbf{r} ''$ the set $\\mathbf{r} '\\sqcup \\left\\{\\, c_{ij} \\,\\middle|\\,\ni\\le j-2\n\\right\\}$, where $c_{ij} = (T_iT_j, T_j T_i)$. \nIt is well-know that $\\left\\langle X^*, X^* \\mathbf{r} '' X^* \\right\\rangle$ is\nconfluent.  \n\nIn the diagrams below we will write $i$ in place of $T_i$. \nWe will also use ${}'$ for decrement and $\\,\\widehat{\\ }\\,$ for increment, thus for an integer\n$k$\n\\begin{equation*}\nk' = k-1, \\quad \\widehat{k} = k+1.\n\\end{equation*}\nWe will show that the set $\\mathcal{P}$ of  pairs of paths  in\n$\\left\\langle X^*, X^*\\mathbf{r} '' X^* \\right\\rangle$ defined below gives a coherent\npresentation of $\\rs{n+1}$, i.e. that \n\\begin{equation*}\n\\mathrm{Res}_{\\mathcal{P}}\\colon [\\rs{n+1}; \\mathcal{C}]\n\\to [ X,\\mathbf{r} '', \\mathcal{P} ; \\mathcal{C}] \n\\end{equation*}\nis an\nequivalence of categories. \nThe set $\\mathcal{P}$ consists of the pairs of paths that one obtains from the\nfollowing diagrams\n\\begin{equation}\n\\label{loops}\n\\begin{gathered}\n\\xymatrix{\nst \\ar[r]^{c_{st}} \\ar@\/_3ex\/@{-->}[rr] & ts \\ar[r]^{c_{ts}} & st\n}\n\\end{gathered}\n\\end{equation}\n\n\\begin{equation}\n\\label{aa}\n\\begin{gathered}\n\\xymatrix{\nkkk \\ar[r]^{ka_k} \\ar[d]_{a_k k} & kk\\ar[d]^{a_k}\\\\\nkk \\ar[r]^{a_k} & k\n}\n\\end{gathered}\n\\end{equation}\n\\begin{equation}\n\\label{ba}\n\\begin{gathered}\n\\xymatrix@C4em{\nkk'kk \\ar[rr]^-{kk'a_k} \\ar[d]_{b_kk} && kk'k \\ar[d]^{b_k} \\\\\nk'kk'k \\ar[r]^-{k'b_k} & k'k'kk' \\ar[r]^{a_{k'}k k'}  & k'kk'\n}\n\\end{gathered}\n\\end{equation}\n\\begin{equation}\n\\label{ab}\n\\begin{gathered}\n\\xymatrix{\nkkk'k \\ar[r]^{kb_k} \\ar[dd]_{a_kk'k} & kk'kk'\\ar[d]^{b_kk'} \\\\\n& k'kk'k' \\ar[d]^{k'ka_{k'}} \\\\\nkk'k \\ar[r]^{b_k} & k'kk'\n}\n\\end{gathered}\n\\end{equation}\n\n\\begin{equation}\n\\label{ac}\n\\begin{gathered}\n\\xymatrix{\nstt \\ar[rr]^-{sa_t}  \\ar[d]_{c_{st}t} && st \\ar[d]^{c_{st}}\\\\\ntst \\ar[r]^-{tc_{st}} & tt s \\ar[r]^{a_t} & ts\n}\n\\end{gathered}\n\\end{equation}\n\n\\begin{equation}\n\\label{bb}\n\\begin{gathered}\n\\xymatrix{\nkk'kk'k  \\ar[d]_{b_k k'k} \\ar[r]^-{kk'b_k} & kk'k'kk' \\ar[r]^{ka_{k'} kk'} & kk'kk' \\ar[d]^{b_k k'} \\\\\nk'kk'k'k \\ar[d]_{k'ka_{k'} k} && k'kk'k' \\ar[d]^{k'k a_{k'}}  \\\\\nk'kk'k \\ar[r]^-{k'b_k} & k'k'kk' \\ar[r]^{a_{k'}k k'} & k'kk'\n}\n\\end{gathered}\n\\end{equation}\n\n\\begin{equation}\n\\label{bc}\n\\begin{gathered}\n\\xymatrix{\ntss's \\ar[r]^-{tb_s} \\ar[d]_{c_{ts} s's} & t s'ss' \\ar[r]^-{c_{ts'}ss'} & s'tss'\n\\ar[d]^{s'c_{ts} s'} \\\\\nsts's \\ar[d]_{sc_{ts'} s} && s'sts' \\ar[d]^{s's c_{ts'}} \\\\\nss'ts \\ar[r]^{ss'c_{ts} } & ss'st \\ar[r]^{b_s t} & s'ss't\n}\n\\end{gathered}\n\\end{equation}\n\n\\begin{equation}\n\\label{cc}\n\\begin{gathered}\n\\xymatrix{\nkji \\ar[r]^-{kc_{ji}} \\ar[d]_{c_{kj}i} &  kij \\ar[r]^{c_{ki}j} & ikj\n\\ar[d]^{ic_{kj}} \\\\\njki \\ar[r]^-{jc_{ki}} & jik \\ar[r]^{c_{ji}k} & ijk\n}\n\\end{gathered}\n\\end{equation}\n\n\\begin{equation}\n\\label{TZ}\n\\begin{gathered}\n\\xymatrix@C3.4em{\n\\widehat{k}kk'\\widehat{k}k\\widehat{k} \\ar[r]^-{\\widehat{k}kk'b_{\\widehat{k}}} \\ar[d]_{\\widehat{k}kc_{k'\\widehat{k}} k\\widehat{k}} & \n\\widehat{k}kk'k\\widehat{k}k \\ar[r]^-{ \\widehat{k}b_k \\widehat{k}k} & \\widehat{k}k'kk' \\widehat{k}k \\ar[r]^-{c_{\\widehat{k}k'} kk'\\widehat{k}k} & k'\\widehat{k}kk'\\widehat{k}k\n\\ar[d]^{k'\\widehat{k}kc_{k'\\widehat{k}}k} \\\\\n\\widehat{k}k\\widehat{k}k'k\\widehat{k}  \\ar[d]_{b_{\\widehat{k}} k'k\\widehat{k}} \n&&& k'\\widehat{k}k\\widehat{k}k'k \\ar[d]^{k'b_{\\widehat{k}} k'k} \\\\\nk\\widehat{k}kk'k\\widehat{k} \\ar[d]_{k\\widehat{k} b_k \\widehat{k}} \n&&& k'k\\widehat{k}kk'k \\ar[d]^{k'k\\widehat{k} b_k} \\\\\nk\\widehat{k}k'kk'\\widehat{k} \\ar[d]_{kc_{\\widehat{k}k'} kk'\\widehat{k}} \n&&& k'k\\widehat{k}k'kk' \\ar[d]^{k'kc_{\\widehat{k}k'} kk'} \\\\\nkk'\\widehat{k}kk'\\widehat{k} \\ar[r]^-{kk'\\widehat{k}kc_{k'\\widehat{k}}} & kk'\\widehat{k}k\\widehat{k}k' \\ar[r]^-{kk'b_{\\widehat{k}}k'} &\nkk'k\\widehat{k}kk' \\ar[r]^-{ b_k \\widehat{k}kk'} & k'kk'\\widehat{k}kk'.\n}\n\\end{gathered}\\,\\,\\,\\,\\,\\,\n\\end{equation}\nThe diagrams \\eqref{cc} and \\eqref{TZ} were called\n\\emph{Tits-Zamolodchikov $3$-cells} in~\\cite{braid}. \n\nWe will say that a path $p$ in $\\left\\langle X^*, X^* \\mathbf{r} '' X^* \\right\\rangle$\nis a \\emph{$c$-path} if all the steps in $p$ are of the form $X^* c_{st}X^*$ for\nsome $s$, $t$ such that $|s-t| \\ge 2$. \n\n\\begin{proposition}\n\\label{cpath}\nSuppose $(F,\\tau) \\in \\left[ X,\\mathbf{r} '', \\mathcal{P}; \\mathcal{C} \\right]$. \nIf $p$, $p'\\colon u\\twoheadrightarrow v$ are two $c$-paths in $\\left\\langle X^*, X^*\\mathbf{r} '' X^*\n\\right\\rangle$, then $\\tau_{p} = \\tau_{p'}$. \n\\end{proposition}\n\\begin{proof}\nLet $\\mathbf{c}$ be the subset $\\mathbf{r} ''$ consisting of all $c_{ji}$ with $j\\ge i+2$.\nUsing~\\eqref{loops}, we see that $\\tau_{c_{ij}} = \\tau_{c_{ji}}^{-1}$ for all\n$j\\ge i+2$. Therefore, there are zigzags $z$, $z'$ in $\\left\\langle X^*, X^*\n\\mathbf{c} X^*\n\\right\\rangle$ such that $\\tau_{p} = \\tau_z$ and $\\tau_{p'} = \\tau_{z'}$.\n\nThe ARS $\\left\\langle X^*, X^* \\mathbf{c} X^* \\right\\rangle$ is terminating and\nlocally confluent, with the only critical e.d.s given by~\\eqref{cc} with\n$k\\ge j+2$, $j\\ge i+2$. Therefore, by Example~\\ref{terminating}, we get that\n$\\tau_{z} = \\tau_{z'}$. \n\\end{proof}\nThe easiest way to prove our result would be to have a complete set of\ndecreasing critical e.d.s for $\\left\\langle X^*, X^*\\mathbf{r} '' X^* \\right\\rangle$.\nThe author does not know at the moment if this is possible. Instead, we will\nproceed as follows\n\\begin{enumerate}[1)]\n\\item first we replace $\\mathbf{r} ''$ with a bigger set $\\mathbf{r} $ of generating rules;\n\\item then we define a preorder on  $X^* \\mathbf{r}  X^*$ so that there is a complete\nset of decreasing critical e.d.s for $\\left\\langle X, \\mathbf{r}  \\right\\rangle$ and\nall natural e.d.s are decreasing;\n\\item further we show that all the chosen critical e.d.s can be subdivided into\ndiagrams in $\\mathcal{P}$;\n\\item finally we show that for all $(F,\\tau)\\in [X,\\mathbf{r} , \\mathcal{P};\\mathcal{C}]$ and any\nattractor loop $p\\colon b\\twoheadrightarrow b$ one has $\\tau_p = 1_{F_b}$. \n\\end{enumerate}\nLet\n\\begin{equation*}\n\\mathbf{r}  = \\left\\{\\, a_{i} \\,\\middle|\\,  1\\le i \\le n \\right\\} \\sqcup \n\\left\\{\\, b_{ji} \\,\\middle|\\,  1\\le i <j \\le n \\right\\} \\sqcup \\left\\{\\,\nc_{st}\n\\,\\middle|\\, |s-t| \\ge 2 \\right\\}.\n\\end{equation*}\nHere $a_i$, $c_{st}$ are defined as before and \n\\begin{equation*}\nb_{ji} = (j\\dots ij,\\, j'jj'\\dots i), \n\\end{equation*}\nwhere $j\\dots i$ denotes the interval of the arithmetic progression with the step\n$-1$. \nNote that in particular $b_{jj'} = b_j$, and therefore $\\mathbf{r} ''\\subset \\mathbf{r} $.\nWe define the preorder $\\succeq$ on $X^* \\mathbf{r}  X^*$ as follows. Given \n$r$, $r'$, we write $r \\gg r'$ to indicate that $urv \\succ u'r'v'$ for all\n$u$, $v$, $u'$, $v'\\in X^*$. Moreover, for example, $c \\gg a$ will indicate that\n$c_{st} \\gg a_i$ for all $s$, $t$, and $i$.  We will write\n$\\#_j u$ for the number of occurrences of $j$ in $u\\in X^*$. \nThe preorder $\\succeq$ is defined by the rules\n\\begin{enumerate}[$1)$]\n\\item  for $i\\le j''$, $b_{ji} \\sim b_{jj'} j''\\dots i$, i.e. to compare two elements in $X^*\\mathbf{r}  X^*$,\nwe first replace, if necessary, all $b_{ji}$ by $b_{jj'}j''\\dots i$, and then proceed with\nthe rules below;\n\\item we order the relations in $\\mathbf{r} $ by \n\\begin{enumerate}[i)]\n\\item $b \\gg c\\gg a$;\n\\item $b_{i+1,i} \\gg b_{i,i-1} $;\n\\item $c_{ij} \\gg c_{ts}$ for all $j\\ge i+2$ and $t\\ge s+2$;\n\\item for $k\\ge j+2$ and $t\\ge s+2$, $c_{kj} \\gg c_{ts}$, if $k> t$ or\n$k = t$ and $j > s $; \n\\item  $\\left\\{\\, c_{ij} \\,\\middle|\\,  j\\ge i+2 \\right\\}$ are ordered arbitrary;\n\\end{enumerate}\n\\item if words contain  the same generating rule $r\\in \\mathbf{r} $ we proceed as\nfollows:\n\\begin{enumerate}[i)]\n\\item if $r=a_i$, then we just compare lengths of the words: the\nlonger word is greater; \n\\item if $r=c_{ji}$ with $j\\ge i+2$, then $u c_{ji} v \\succ u' c_{ji} v'$ if\n\\begin{enumerate}[a)]\n\\item $\\sum\\limits_{k\\ge j} \\#_k u > \\sum\\limits_{k\\ge j} \\#_k u'$; \\vspace{2ex}\n\\item $\\sum\\limits_{k\\ge j} \\#_k u = \\sum\\limits_{k\\ge j} \\#_k u'$\nand $\\sum\\limits_{k\\le i} \\#_k v > \\sum\\limits_{k\\le i} \\#_k v'$. \\vspace{3ex}\n\\end{enumerate}\n\\item if  $r=b_{kk'}$, then  $ub_{kk'} v \\succ u'b_{kk'} v'$ \n if \n\\begin{equation*}\n(\\#_k uv , \\#_{k'} u v, \\dots, \\#_1 u  v) >\n(\\#_k u'  v' , \\#_{k'} u'  v', \\dots, \\#_1 u' v') \n\\end{equation*}\nwith respect to the lexicographical order on $\\mathbb{N}^k$. \n\\end{enumerate}\n\\end{enumerate}\nIf for two words $w$, $w'\\in X^*\\mathbf{r}  X^*$ we cannot conclude either $w\\succ w'$\nor $w'\\succ w$, according to the above rules, then $w\\sim w'$. \nIt is obvious that the preorder $\\succeq$ is monomial. \n\\begin{proposition}\nAll the natural e.d.s of the ARS $\\left\\langle X^*, X^*\\mathbf{r}  X^* \\right\\rangle$\nare decreasing with respect to the preorder $\\succeq$. \n\\end{proposition}\n\\begin{proof}\nLet $r,r'\\in \\mathbf{r} $ and $w\\in X^*$. Then $\\mathcal{N}(r,w,r')$ has the top arrow\n$rws(r')$,  the bottom arrow $rwt(r')$, the left arrow $s(r)wr'$, and the right\narrow $t(r)wr'$. \nIf $rws(r')\\succeq rwt(r')$ and $s(r)wr'\\succeq t(r)wr'$ then the e.d.\n$\\mathcal{N}(r,w,r')$ is obviously decreasing. \n\nThus, we have to find triples $(r,w,r')$ for which $rwt(r')\\succ rws(r')$ or\n$t(r)wr' \\succ s(r)wr'$. \n\nFirst we identify triples $(r,w,r')$ such that $rwt(r') \\succ rws(r')$. \nIf $r$ is of type $a$, then $rwt(r')\\succ rws(r')$ if and only if the length of\n$t(r')$ is greater than the length of $s(r')$. But there is no rule $r'\\in \\mathbf{r} $\nwith such property. Thus $r$ cannot be of type $a$. \n\nNow, suppose $r$ if of type $c$. We have two cases: either $r=c_{ij}$ or\n$r=c_{ji}$, with $j\\ge i+2$. If $r=c_{ij}$ then $rws(r') \\sim rwt(r')$. If\n$r=c_{ji}$ then $rwt(r') \\succ rws(r')$ if and only if \n\\begin{equation}\n\\label{kir}\n\\sum_{k\\le i} \\#_k t(r') > \\sum_{k\\le i} \\#_k s(r'). \n\\end{equation}\nIt is clear that $r'$ can be of type $a$ or $c$ as for such rules we have\n$\\#_k t(r') = \\#_k s(r')$ for arbitrary $k$. If $r'=b_{ml}$ with $m>l$, then \n\\begin{equation}\n\\label{jtrp}\n\\#_k t(r') = \n\\begin{cases}\n\\#_k s(r') -1, & k= m\\\\\n\\#_k s(r') + 1, & j=m-1\\\\\n\\#_k s(r'), & \\mbox{otherwise.}\n\\end{cases}\n\\end{equation}\nThus \\eqref{kir} holds only if $i=m-1$, i.e. $r'=b_{i+1,l}$ with $i+1>l $. \nAs $b\\gg c$ we get that $s(r)wr' \\succ rwt(r')$. Moreover, \n$s(r)wr'\\sim t(r)wr'$ as $\\#_k s(c_{ji}) = \\#_k t(c_{ji})$ for all $k$. \nThus $\\mathcal{N}(r,w,r')$ is decreasing in this case. \n\nLet us consider the case $r=b_{ji}$ for some $j>i$. Then we have $rwt(r')> rws(r')$ if\nand only if \n\\begin{equation*}\n\\left( \\#_j t(r'),\\dots \\#_1 t(r') \\right) > (\\#_j s(r'),\\dots, \\#_1 s(r')).\n\\end{equation*}\nThis is possible only in the case $r'=b_{j+1,k}$ for some $k<j+1$. \nAs $b_{j+1,j} \\gg b_{jj'}$, we get that $s(r)wr'\\succ rwt(r')$. Moreover\n\\begin{equation*}\n\\begin{aligned}\n\\#_{j+1}s(r) = \\#_{j+1}(j\\dots ij)=0=\\#_{j+1}(j'j\\dots i) =\n\\#_{j+1}t(r)\\\\\n\\#_{j}s(r) = \\#_j (j\\dots i j)  = 2 > 1 = \\#_j (j'j\\dots i) = \\#_j t(r).\n\\end{aligned}\n\\end{equation*} \nThus $s(r)wr'\\succ t(r)wr'$ and we get that $\\mathcal{N}(r,w,r')$ is decreasing. \n\n\nNow we will analyze the triples $(r,w,r')$ such that $t(r)wr' \\succ s(r)wr'$. \nIt is impossible to have $r'$ of type $a$ as the length of $t(r)w$ never exceeds\nthe length of $s(r)w$. \nSuppose $r'$ is of type $c$. Then $r'=c_{ij}$ or $r'=c_{ji}$ for some $j$,\n$i$ such that $j>i+1$. If $r'=c_{ij}$ then $t(r)wr' \\sim s(r) wr'$. Thus we have\nonly to consider the case $r'=c_{ji}$. In this situation $t(r)wr'\\succ\ns(r)wr'$ if and only if \n\\begin{equation}\n\\label{kjr}\n\\sum_{k\\ge j }\\#_k t(r) > \\sum_{k\\ge j} \\#_k s(r). \n\\end{equation}\nAs for rules $r''$ of type $a$ or $c$, the inequality $\\#_k t(r'') \\le \\#_k\ns(r'')$ holds for\nevery $k$, we get that $r$ cannot  be of type $a$ or $c$. If $r=b_{ml}$ for some\n$m>l$, then the sums in \\eqref{kjr} are equal unless $j=m$, in which case the\nleft hand side of \\eqref{kjr} is less than the right hand side of\n\\eqref{kjr}. This shows that $r'$ cannot have type $c$.\n\nSuppose $r'=b_{ji}$ for some $j>i$. Then $t(r)wr' \\succ s(r)wr'$ if and only if \n\\begin{equation*}\n\\left( \\#_j t(r),\\dots, \\#_1 t(r) \\right) > \\left( \\#_j s(r),\\dots, \\#_1\ns(r) \\right)\n\\end{equation*}\nwith respect to the lexicographical order on $\\mathbb{N}^j$. This is possible only in\nthe case $r=b_{j+1,l}$ for some $l<j+1$.  \nAs $b_{j+1,j} \\gg b_{jj'}$, we get that $rws(r') \\succ t(r)wr'$. Moreover, as \n$\\#_{j+1} s(r') = \\#_{j+1} t(r')$ and $\\#_j s(r') > \\#_j t(r')$, we have\n$rws(r') \\succ rwt(r')$.  This shows that $\\mathcal{N}(r,w,r')$ is decreasing in this\ncase. \n\\end{proof}\n\nNow we construct a complete set of decreasing critical e.d.s\nfor the ARS $\\left\\langle X^*, X^*\\mathbf{r}  X^* \\right\\rangle$. \n\nNote that there are not any rules $r$, $r'\\in \\mathbf{r} $\nsuch that $s(r')$ is a proper subword of $s(r)$. Therefore all the cricital\npairs for $\\mathbf{r} $ are of the form $(ru,vr')$ for some $r$, $r'\\in \\mathbf{r} $ and\nnon-empty $u$, $v\\in X^*$. \n\n If $r=c_{ij}$ with\n$j\\ge i+2$, then the following commutative diagram is decreasing and  resolving\nfor this critical\npair  \n\\begin{equation*}\n\\begin{gathered}\n\\xymatrix{\niju \\ar[r]^{c_{ij}u} \\ar[dd]_{vr'} & jiu\\ar[d]^{c_{ji}u} \\\\\n& iju \\ar[d]^{vr'} \\\\\nt(r) \\ar@{-->}[r] & t(r).\n}\n\\end{gathered}\n\\end{equation*} \nIn fact, $c_{ij} \\gg c_{ji}$ implies that $c_{ij}u \\succ c_{ji} u $.  As\n$\\succeq$ is monomial, we get \nthat $vr'\\sim vr'$. \nSimilarly, if $r'=c_{ij}$ with $j\\ge i+2$, the diagram\n\\begin{equation}\n\\label{conc}\n\\begin{gathered}\n\\xymatrix{\nuij \\ar[r]^{uc_{ij}} \\ar[dd]_{ru} & uji\\ar[d]^{uc_{ji}} \\\\\n& uij \\ar[d]^{ru} \\\\\nt(r) \\ar@{-->}[r] & t(r).\n}\\end{gathered}\n\\end{equation}\nis decreasing and resolves the critical pair $(ru,vc_{ij})$. \n\nIt is left to consider the critical pairs with $r$ and $r'$ equal to one of \n$a_k$, $b_{ji}$, $j\\ge i+1$, $c_{ts}$, $t\\ge s+2$. \nIn the diagrams below we will abbreviate $c$-paths by~$\\stackrel{c^*}{\\twoheadrightarrow}$. This does not create any ambiguity in view of\nProposition~\\ref{cpath}. \n\nWe first consider all the critical pairs involving at least one $a_k$. \nThe diagram~\\eqref{aa} is  a decreasing critical e.d. for the critical pair\n$(ka_k, a_kk)$. \nFor the critical pair $(a_k k'\\dots jk, kb_{kj})$, we consider the diagram\n\n\\begin{equation}\n\\label{aB}\nE(a_k,b_{kj})=\\begin{gathered}\n\\xymatrix{\nkk\\dots jk \\ar[r]^-{kb_{kj}} \\ar[dd]_{a_k k'\\dots jk} & kk'k\\dots j \\ar[d]^{b_ k'\\dots  j} \\\\\n& k'k k'k'\\dots j \\ar[d]^{k'k a_{k'} k''\\dots j}\\\\\nk\\dots jk \\ar[r]^-{b_{kj}} & k'k\\dots j. \n}\n\\end{gathered}\n\\end{equation}\nThe diagram~\\eqref{aB} is decreasing since $kb_{kj}$ is greater than all the other\narrows. Note, that for $j=k'$ the word $k''\\dots j$ is empty and we\nrecover~\\eqref{ab}. \n\nFor the critical pair $(b_{kj}k, k\\dots j a_k)$, we consider the diagram\n\\begin{equation}\n\\label{Ba}\nE(b_{kj}, a_k) = \\begin{gathered}\n\\xymatrix@C4em{\nk\\dots jkk \\ar[rr]^-{k\\dots ja_k} \\ar[d]_{b_{kj} k} && k\\dots jk\\ar[d]^{b_{kj}}\\\\\nk'k\\dots j k \\ar[r]^-{k'b_{kj}} &  k'k'k\\dots j\\ar[r]^-{a_{k'} k\\dots j} & k'k\\dots\nj.\n}\n\\end{gathered}\n\\end{equation}\nThe diagram~\\eqref{Ba} is decreasing since $b_{kj}k$ is greater then any other\narrow. \nIn the case $j=k'$, we recover the diagram~\\eqref{ba}. \n\nAll the critical pairs involving two $b$-rules are of the form $(b_{kj}k'\\dots\ni, k\\dots jb_{ki})$. For $j=k'$, we get\n\\begin{equation}\n\\label{bB}\n{}_{E(b_{kk'}, b_{ki}) =} \\begin{gathered}\n\\xymatrix{\nkk'k\\dots ik \\ar[rrr]^-{kk'b_{ki}} \\ar[ddd]^{b_{kk'}k'\\dots ik} &&& kk'k'k\\dots\ni \\ar[d]^{ka_{k'} k\\dots i}\\\\\n&&& kk'k\\dots i \\ar[d]^{b_{kk'}k'\\dots i} \\\\\n&&& k'kk'k'\\dots i \\ar[d]^{k'ka_{k'} k''\\dots i}\\\\\nk'kk'k'\\dots ik \\ar@\/^4ex\/[r]^-{k'ka_{k'}k''\\dots ik } & k'k\\dots ik\n\\ar[r]^-{k'b_{ki}} & k'k'k\\dots i\\ar[r]^-{a_{k'}k\\dots i} & k'k\\dots i.\n}\n\\end{gathered}\n\\end{equation}\nThe diagram~\\eqref{bB} is decreasing since $b_{kk'}k'\\dots ik\\sim\nkk'b_{ki}$ dominates all\nthe other arrows. Note that for $i=k'$ the subword $k''\\dots i$ is empty and we\nrecover the diagram~\\eqref{bb}. \n\nThe diagram corresponding to  the case $j\\le k''$ is\nnot drawn in the rectangular form for the\ntypographical\nreasons. We also write $S$ for the word $k'''\\dots j k''\\dots i$. \n\n\\begin{equation}\n\\label{BB}\n{}_{E(b_{kj}, b_{ki})=} \\begin{gathered}\n \\xymatrix@R8ex@C0.2em{\n& k\\dots jk\\dots ik \\ar[ld]_{b_{kj} k'\\dots ik} \\ar[rd]^{k\\dots\njb_{ki}} \\\\ \nk'k\\dots jk'\\dots ik \\ar@{->>}[d]_{c^*}  & &  k\\dots jk'k\\dots i\n\\ar@{->>}[d]^{c^*} \\\\\nk'k k' k'' k' k S \\ar[d]_{k'k b_{k'k''} k S} && \nkk' k'' k' k k' S \\ar[d]^{kb_{k'k''} kk'S } \\\\\nk'kk'' k'k'' k S  \\ar@{->>}[d]_{c^*}&&\nkk''k'k'' kk' S \\ar@{->>}[d]^{c^*} \\\\\nk'k''kk'kk'' S \\ar[d]_{k'k'' b_{kk'} k'' S}  && k''kk'kk'' k' S\n\\ar[d]^{k''b_{kk'} k'' k' S} \n\\\\\nk'k''k'kk'k'' S  \\ar[d]_{b_{k'k''} kk'k''S} && k''k'k k' k'' k' S\n\\ar[d]^{k'' k'k b_{k'k''}S} \\\\ \nk''k'k'' kk'k''  S\\ar@{->>}[rr]^{c^*} && k'' k' k k'' k' k'' S\n}\n\\end{gathered}\n\\end{equation}\nIn~\\eqref{BB} all the arrows are dominated by the two top arrows \n$b_{kj} k'\\dots ik \\sim k\\dots jb_{ki}$. For $j=k''$ and $i=k'$,\n\\eqref{BB} becomes \\eqref{TZ}. \n\nNow we will consider critical pairs involving one $b$-rule and one $c$-rule of\nthe form $c_{kj}$ with $k\\ge j+2$. \n\nFor $(c_{kj}j'\\dots ij, k b_{ji})$, $k\\ge j+2$, $j\\ge i+1$, \nwe consider the diagram\n\\begin{equation}\n\\label{cB}\n{}_{E(c_{kj}, b_{ji})} = \\begin{gathered}\n\\xymatrix{\nkj\\dots ij \\ar[rrr]^-{kb_{ji}} \\ar[d]_{c_{kj} j'\\dots ij} &&& kj'j\\dots i\n\\ar@{->>}[d]^{c^*} \\\\\njkj'\\dots ij \\ar@{->>}[r]^{c^*} & j\\dots i k j \\ar[r]^{j\\dots i c_{kj}} &  j\\dots ij k \\ar[r]^{b_{ji} k} & j'j\\dots ik.\n}\n\\end{gathered}\n\\end{equation}\nIn~\\eqref{cB}, we have $kb_{ji} \\sim b_{ji} k$ and $kb_{ji}$  dominates every\narrow in the vertical $c$-path. Thus we have to show that $c_{kj} j'\\dots ij$\ndominates every arrow in the horizontal $c$-path. The horizontal $c$-path\ninvolves the rules generated by $c_{ks}$ with $s<j$ and at the last step the\nrule $j\\dots i c_{kj}$. Since, by definition, $c_{kj} \\gg c_{ks}$ if $j>s$, we\nsee that $c_{kj} j'\\dots ij$ dominates all the horizontal $c$-arrows except\nprobably the last one. But it is easy to see that also\n\\begin{equation*}\nc_{kj} j'\\dots ij \\succ j\\dots i c_{kj}. \n\\end{equation*}\n\nNow suppose $k\\ge s+2$ and $k\\ge j+1$. For the critical pair $(b_{kj} s, k\\dots\nj c_{ks})$ we will consider several cases. If $j\\ge s+2$, then we can take a\ndiagram similar to~\\eqref{cB}\n\\begin{equation}\n\\label{Bc1}\nE(b_{kj},c_{ks})=\\begin{gathered}\n\\xymatrix{\nk\\dots jk s \\ar[r]^-{k\\dots jc_{ks}} \\ar[ddd]_{b_{kj} s}  & k\\dots jsk\n\\ar@{->>}[d]^{c^*}\\\\ \n& ks k'\\dots jk \\ar[d]^{c_{ks} k'\\dots jk} \\\\\n& s k\\dots j k \\ar[d]^{s b_{kj}}\\\\\nk'k\\dots j s \\ar@{->>}[r]^-{c^*} & s k'k\\dots j. \n}\n\\end{gathered}\n\\end{equation}\nIn~\\eqref{Bc1}, $b_{kj}s \\sim sb_{kj}$ and dominates every arrow in the\nhorizontal $c$-path. We have to check that $k\\dots j c_{ks}$ dominates all\narrows in the vertical $c$-path. This is done using that $c_{ks} \\gg\nc_{ts}$ for all $k'\\ge t \\ge j$, and that $k\\dots j c_{ks} \\succ c_{ks} k'\\dots\njk$. \n\nFor $j=s+1$ we consider the diagram\n\\begin{equation}\n\\label{Bc2}\nE(b_{kj}, c_{kj'}) = \\begin{gathered}\n\\xymatrix@C6em{\nk\\dots jkj' \\ar[r]^{k\\dots j c_{kj'}} \\ar[d]_{b_{kj} j'} & k\\dots j'k\n\\ar[d]^{b_{kj'}} \\\\\nk'k\\dots j' \\ar@{-->}[r] & k'k\\dots j'.\n}\n\\end{gathered}\n\\end{equation}\nThe diagram~\\eqref{Bc2} is decreasing since $b_{kj} j' \\sim b_{kk'} k''\\dots j'\n\\sim b_{kj'}$. \n\nFor $j=s$, we consider the diagram\n\\begin{equation}\n\\label{Bc3}\nE(b_{kj}, c_{kj}) = \\begin{gathered}\n\\xymatrix@C6em{\nk\\dots jkj \\ar[r]^{k\\dots j c_{kj}} \\ar[dd]_{b_{kj} j} & k\\dots jjk\n\\ar[d]^{k\\dots \\widehat{\\jmath} a_j k} \\\\\n& k\\dots jk \\ar[d]^{b_{kj}} \\\\\nk'k\\dots jj \\ar[r]^{k\\dots \\widehat{\\jmath} a_j} & k'k\\dots j\n}\n\\end{gathered}\n\\end{equation}\nIn~\\eqref{Bc3}, $b_{kj}j$ dominates all the arrows. \n\nFor $k'' \\ge s \\ge j+1$, we take the diagram\n\\begin{equation}\n\\label{Bc4}\nE(b_{kj}, c_{ks}) = \\begin{gathered}\n\\xymatrix@C6em{\nk\\dots jks \\ar[r]^{k\\dots j c_{ks}} \\ar[ddd]_{b_{kj} s} & k\\dots js k\n\\ar@{->>}[d]^{c^*} \\\\\n& k\\dots \\widehat{s} k s\\dots js \\ar[d]^{k \\dots \\widehat{s}k b_{sj}} \\\\\n& k \\dots \\widehat{s}k s's\\dots j \\ar[d]^{b_{k\\widehat{s}} s's \\dots j} \\\\\nk'k\\dots \\widehat{s} s \\dots js \\ar[r]^-{k'k \\dots \\widehat{s} b_{sj}} & k'k\\dots\n\\widehat{s} s's \\dots j.\n}\n\\end{gathered}\n\\end{equation}\nNote that for $s=k''$ the subword $k''\\dots \\widehat{s}$ is empty. We claim that\n$b_{kj}s$ dominates all the arrows. This is obvious for all arrows except\n$b_{k\\widehat{s} } s'\\dots j$. \nWe have $b_{kj} s\\sim b_{kk'} k''\\dots j s$ and $b_{k\\widehat{s}} s's\\dots j\n\\sim b_{kk'} k''\\dots \\widehat{s} s' s \\dots j$. Now for all $s+1 \\le t \\le k$, we\nhave\n\\begin{equation*}\n\\begin{aligned}\n\\#_t (k''\\dots j s)  & =\n\\begin{cases}\n0 ,& t=k,k'\\\\\n1, & s+1\\le t\\le k''\n\\end{cases}\n\\\\[3ex] & = \\#_t (k''\\dots \\widehat{s} s's \\dots j).\n\\end{aligned}\n\\end{equation*}\nFurther,\n\\begin{equation*}\n\\#_s (k''\\dots js ) = 2 > 1 = \\#_s (k''\\dots \\widehat{s} s's \\dots j).\n\\end{equation*}\nThus\n\\begin{equation*}\nb_{kj} s \\succ b_{k\\widehat{s}}  s's \\dots j. \n\\end{equation*}\nIt is left to consider the critical pairs involving two $c$-rules of the form\n$c_{ts}$ with $t\\ge s+2$. They are all of\nthe type $(c_{kj}i, kc_{ji})$. The diagram~\\eqref{cc} gives a decreasing\nconvergence diagram for this pair as $c_{kj}i$ dominates all the arrows\nin~\\eqref{cc}. This is true as $c_{kj} \\gg c_{ki} \\gg c_{ji}$ and $c_{kj}i >\nic_{kj}$ for $i<j$. \n\nWe will denote the set of pairs of paths that one obtains from\nthe chosen critical e.d.s\nby $\\mathcal{Y}$. \nWe also write $\\mathcal{L}$ for the set $(p,\\varnothing_w)$, were  $p\\colon\nw\\twoheadrightarrow w$  are the loops at semi-normal elements of $\\left\\langle X^*, X^* \\mathbf{r} \nX^*\n\\right\\rangle$. By Theorem~\\ref{main:presentation}, we get that\n$\\mathrm{Res}_{\\mathcal{Y}\\sqcup \\mathcal{L}}$ is an equivalence of categories. \n\n\nNow we identify pairs of paths coming from loops in the attractor of\n$\\left\\langle X^*, X^* \\mathbf{r}  X^* \\right\\rangle$. \n\\begin{proposition}\nThe preorder $\\twoheadrightarrow$ on $\\left\\langle X^*, X^*\\mathbf{r} \nX^*\n\\right\\rangle$ is well-founded. Equivalently $\\attr(w) \\not=\\varnothing$\nfor every $w\\in X^*$. \n\\end{proposition}\n\\begin{proof}\nSuppose \n\\begin{equation}\n\\label{seq}\nw_0 \\to w_1 \\to \\dots \\to w_k \\to \\dots\n\\end{equation}\nis an infinite sequence in $\\left\\langle X^*, X^*\\mathbf{r} \nX^*\n\\right\\rangle$. Then we have \n\\begin{equation*}\nl(w_0) \\ge l(w_1) \\ge \\dots \\ge l(w_k) \\ge \\dots\n\\end{equation*}\nSince the set of words of length not greater than $l=l(w_0)$ is finite, we see\nthat there is a word $w$ that appears in~\\eqref{seq} infinitely many times. Let\n$w_k = w$ be the first appearance of $w$ in~\\eqref{seq}. Then for all $m>k$,\nthere is $n > m$ such that $w_n = w$. Thus we get that for all $m>k$ there are\npaths $w = w_k \\twoheadrightarrow w_m$ and $w_m \\twoheadrightarrow w_n = w$. This shows that for every\n$m>k$, we have $w \\bidir w_m$, that is $\\twoheadrightarrow$ is a  well-founded preorder.\n\\end{proof}\n\\begin{proposition}\n\\label{attractor}\nLet $w$ be a semi-normal element in $\\left\\langle X^*, X^*\\mathbf{r} \nX^*\n\\right\\rangle$ and $p\\colon w\\twoheadrightarrow w$ a path in $\\left\\langle X^*, X^*\\mathbf{r} \nX^*\n\\right\\rangle$. Then $p$ is a $c$-path.\n\\end{proposition}\n\\begin{proof}\n Define the map\n\\begin{equation*}\n\\begin{aligned}\n\\underline{l} \\colon X^* & \\to \\mathbb{N}^n\\\\\nu & \\mapsto (l(u), \\#_n(u), \\#_{n-1}(u), \\dots, \\#_2(u)). \n\\end{aligned}\n\\end{equation*}\nWe will write $u \\ge v$ if $\\underline{l}(u) \\ge \\underline{l}(v)$ with respect to\nthe lexicographical order on $\\mathbb{N}^n$. If $u\\ge v$ and $v\\ge u$, then we write\n$u \\equiv v$. \nIt is obvious that $(u,v) \\in X^* \\mathbf{r}  X^*$ implies that $u\\ge v$ and $u\\equiv\nv$ if and only if $(u,v) = w'c_{ij}w''$ for some $w'$, $w''\\in X^*$. Thus\n$p$ is a path in $\\left\\langle X^*, X^*c\nX^*\n\\right\\rangle$. \n\\end{proof}\nLet $\\mathcal{L}'$ be a subset of $\\mathcal{L}$ consisting of \\eqref{loops}.\nCombining Proposition~\\ref{cpath} and Proposition~\\ref{attractor}, we get\n\\begin{corollary}\nThe functor $\\mathrm{Res}_{\\mathcal{Y} \\sqcup \\mathcal{L}'}$ is an equivalence of categories. \n\\end{corollary}\nNext we are going to relate the categories $[X,\\mathbf{r} , \\mathcal{Y} \\cup \\mathcal{L}'; \\mathcal{C}]$\nand $[X,\\mathbf{r} '',\\mathcal{P}; \\mathcal{C}]$, where $\\mathcal{P}$ was defined on\npage~\\pageref{loops}. \n\nWe have two functors $\\mathrm{Res} \\colon [X,\\mathbf{r} ; \\mathcal{C}] \\to [X,\\mathbf{r} '';\\mathcal{C}]$ and\n$\\mathrm{P} \\colon [X,\\mathbf{r} '';\\mathcal{C}]\\to [X,\\mathbf{r} ; \\mathcal{C}]$. The first functor is defined by $(F,\\tau)\\to\n(F, \\mathrm{res}(\\tau))$, where $\\mathrm{res}(\\tau)$ is the restriction of\n$\\tau$ to $\\mathbf{r} ''$.\nThe functor $\\mathrm{P} \\colon [X,\\mathbf{r} '';\\mathcal{C}]\\to [X,\\mathbf{r} ;\\mathcal{C}]$\nis defined by $\\mathrm{P}(F,\\tau) = (F,\\tilde{\\tau})$\nwhere\n\\begin{equation*}\n\\begin{aligned}\n\\tilde{\\tau}(a_i) & = \\tau(a_i) ,&  \\tilde{\\tau}(c_{st}) & = \\tau(c_{st}), &\n\\tilde{\\tau}(b_{kk'}) = \\tau(b_k)\n\\end{aligned}\n\\end{equation*}\nand $\\tilde{\\tau}(b_{kj})$ for $j\\le k''$ is computed  recursively\nusing the relation\n\\begin{equation*}\n\\tilde{\\tau}(b_{kj}) = (\\tilde{\\tau}(b_{k,j+1})F_j) \\circ (F_k\\dots F_{j+1}\n\\tau(c_{jk})).\n\\end{equation*}\n\nLet $\\mathcal{Z}$ be the set of pairs of paths in $\\left\\langle X^*, X^*\\mathbf{r}  X^*\n\\right\\rangle$obtained from~\\eqref{Bc2}. Then \nit is easy to see that $\\mathrm{Res}$ and $\\mathrm{P}$ induce mutually inverse equivalences of\ncategories\n\\begin{equation*}\n\\mathrm{Res} \\colon [X,\\mathbf{r} ,\\mathcal{Z}; \\mathcal{C}] \\rightleftarrows [X,\\mathbf{r} '';\\mathcal{C}]\\colon \\mathrm{P}\n\\end{equation*}\nIt is also clear that\nif $(F,\\tau) \\in [X,r,\\mathcal{Y} \\sqcup \\mathcal{L}'; \\mathcal{C}]$ then $\\mathrm{Res}(F,\\tau) \\in\n[X,\\mathbf{r} '', \\mathcal{P};\\mathcal{C}]$ since the images of the pairs in $\\mathcal{P}$ under\n$\\mathrm{P}$ appear among the pairs\n$\\mathcal{Y} \\sqcup \\mathcal{L}' $. \n\nNow we are going to show that for every $(F,\\tau) \\in [X,\\mathbf{r} '', \\mathcal{P}; \\mathcal{C}]$,\nwe get $\\mathrm{P} (F,\\tau) \\in [X,\\mathbf{r} , \\mathcal{Y}\\sqcup \\mathcal{L}';\\mathcal{C}]$. Let us write\n$(F,\\tilde{\\tau})$ \nfor $\\mathrm{P}(F,\\tau)$. We have to show that every diagram in $\\mathcal{Y} \\sqcup \\mathcal{L}'$ is mapped into a\ncommutative diagram under $f_{F,\\tilde{\\tau}}$.  For $\\mathcal{L}'$ this is obvious,\nsince $\\mathcal{L}' \\subset \\mathcal{P}$ and $(F,\\tau) \\in [X,\\mathbf{r} '',\\mathcal{P};\\mathcal{C}]$. \nNow, we have to check that all the diagrams (\\ref{conc}-\\ref{Bc4}) become\ncommutative under $f_{F,\\tilde{\\tau}}$. \n\nFor~\\eqref{conc} this is obvious, since $\\tau(c_{st}) \\tau(c_{ts}) = \\mathrm{id}$ for\nall $|s-t| \\ge 2$.  \nThe commutativity of the other diagrams follows by an induction argument and the  \npatching diagrams listed bellow. \nWe label natural e.d.s by $\\mathcal{N}$. \n\nAs we already noted before~\\eqref{aB} for $j=k'$ becomes~\\eqref{ab}.\nNow, for $j\\le k''$, we have\n\\begin{equation}\n\\label{aBp}\n\\begin{gathered}\n\\xymatrix{ \nkk\\dots j k \\ar[rr]^-{ kk\\dots \\widehat{\\jmath} c_{jk} }  \\ar[dd]_{a_k k'\\dots jk } && kk\\dots \\widehat{\\jmath}kj \n\\ar[dd]^{a_k k'\\dots \\widehat{\\jmath}kj} \\ar[rr]^-{k b_{k\\widehat{\\jmath}}j} && kk'k\\dots \\widehat{\\jmath}j\\ar@{->>}[dd] \\\\\n& \\mathcal{N} && E(a_k,b_{k\\widehat{\\jmath}})j \\\\\nk\\dots jk \\ar[rr]^{k\\dots \\widehat{\\jmath}c_{jk}} && k \\dots \\widehat{\\jmath}kj \\ar@{->>}[rr] && k'k\\dots j.\n}\n\\end{gathered}\n\\end{equation}\n\nFurther~\\eqref{Ba} for $j=k'$ is~\\eqref{ba}, and for $j\\le k''$, we have the\ndiagram\n\\begin{equation}\n\\label{Bap}\n\\begin{gathered}\n\\xymatrix{\nk\\dots jkk\\ar[dd]_{k\\dots \\widehat{\\jmath} c_{jk} k} \\ar[rrrr]^-{k\\dots j a_k} &&&& k\\dots jk  \\ar[dd]^{k\\dots \\widehat{\\jmath} c_{jk}}\\\\\n&& \\eqref{ac}  \\\\\nk\\dots \\widehat{\\jmath} kjk \\ar[rr]^-{k\\dots \\widehat{\\jmath} kc_{jk}}\n\\ar[dd]_{b_{k\\widehat{\\jmath}}jk}\n && k\\dots \\widehat{\\jmath} kkj \\ar[rr]^{k\\dots \\widehat{\\jmath}  a_kj} \\ar[dd]_{b_{k\\widehat{\\jmath}}kj}&& k\\dots \\widehat{\\jmath} kj  \\ar@{->>}[dd]\\\\ \n& \\mathcal{N} && E(b_{k\\widehat{\\jmath}},a_k) j\n\\\\\nk'k \\dots j k \\ar[rr]^{k'k\\dots \\widehat{\\jmath} c_{jk}} && k'k \\dots \\widehat{\\jmath} kj \\ar@{->>}[rr] &&\nk'k\\dots \\widehat{\\jmath}j.\n}\n\\end{gathered}\n\\end{equation}\nThe diagram~\\eqref{bB} for $i=k'$ is \\eqref{bb}. For $i\\le k''$, we use the\ndiagram\n\\begin{equation}\n\\label{bbp}\n\\begin{gathered}\n\\xymatrix@C2em{\nkk'k\\dots ik\\ar[dd]_{b_{kk'}k'\\dots ik}  \\ar[rr]^{kk'k\\dots \\widehat{\\imath} c_{ik}} && kk'k \\dots \\widehat{\\imath}ki\n\\ar[rr]^-{kk'b_{k\\widehat{\\imath}}}\\ar[dd]_{b_{kk'} k'\\dots \\widehat{\\imath}ki}  && kk'k' k \\dots \\widehat{\\imath}i\n\\ar@{->>}[dd] \\\\\n& \\mathcal{N} && E(b_{kk'}, b_{k\\widehat{\\imath}}) i \\\\\nk'kk'k' \\dots ik \\ar[rr]^-{k'kk'k' \\dots \\widehat{\\imath} c_{ik}} && k'kk'k' \\dots \\widehat{\\imath}k i\n\\ar@{->>}[rr] && k'k\\dots \\widehat{\\imath}i.\n}\n\\end{gathered}\n\\end{equation}\n\nThe diagram~\\eqref{BB} for $j=k''$ and $i=k'$ is~\\eqref{TZ}. Let us first\nconsider the case $i=k'$. Then we use the patching diagram\n\\begin{equation}\n\\label{BBp1}\n\\begin{gathered}\n\\xymatrix@C0em{\nk\\dots j kk'k\\ar[dd]_{k\\dots \\widehat{\\jmath} c_{jk} k'k}  \\ar[rrrr]^-{k\\dots j b_{kk'}} &&&&\nk\\dots jk'kk' \\ar@{->>}[dd]^{c^*} \\\\\n&& \\eqref{bc} \\\\\nk \\dots \\widehat{\\jmath} kjk'k \\ar[rr]^{c^*} \\ar[dd]_{b_{k\\widehat{\\jmath}} jk'k } && k\\dots \\widehat{\\jmath} kk' kj\n\\ar[dd]_{b_{k\\widehat{\\jmath}}k'kj} \n\\ar[rr]^{k\\dots \\widehat{\\jmath} b_k j} && k \\dots \\widehat{\\jmath} k'kk' j \\ar@{->>}[dd] \\\\\n& \\mathcal{N} && E(b_{k\\widehat{\\jmath}},b_{kk'}) j\\\\\nk'k\\dots \\widehat{\\jmath}j k' k  \\ar[rr]^{c^*} && k'k \\dots \\widehat{\\jmath} k'k j \\ar@{->>}[rr] && k'' k' k\nk'' k' k'' \\dots \\widehat{\\jmath} j.\n}\n\\end{gathered}\n\\end{equation}\nFor $i\\le k''$, we use \n\\begin{equation}\n\\label{BBp2}\n\\begin{gathered}\n\\xymatrix@C0em{\nk\\dots j k\\dots i k \\ar[dd]_{b_{kj} k'\\dots ik}  \\ar@\/^3ex\/[rr]^-{k\\dots j\nk\\dots \\widehat{\\imath} c_{ik}} && k\\dots j k\\dots \\widehat{\\imath} ki \\ar[dd]_(0.7){b_{kj} k' \\dots \\widehat{\\imath}ki}\n\\ar[rr]^{ k\\dots j b_{k\\widehat{\\imath}}i} && k\\dots j k' k \\dots \\widehat{\\imath} i \\ar@{->>}[dd] \\\\\n& \\mathcal{N} && E(b_{kj}, b_{k\\widehat{\\imath}}) i \\\\\nk'k\\dots jk'\\dots ik \\ar@\/_4ex\/[rr]_-{k'k\\dots j k' \\dots \\widehat{\\imath}c_{ik}} && \nk'k \\dots jk' \\dots \\widehat{\\imath}ki \\ar@{->>} [rr] && k''k'kk''k'k''\\dots jk''\\dots \\widehat{\\imath}i.\n}\n\\end{gathered}\n\\end{equation}\n\nThe diagram~\\eqref{cB} for $i=j'$ is \\eqref{bc}. For $i\\le j''$, we consider the\ndiagram\n\\begin{equation}\n\\label{cBp}\n\\begin{gathered}\n\\xymatrix@C0em{\nkj\\dots ij \\ar@\/^3ex\/[rr]^{kj\\dots vc_{ij}}\\ar[dd]_{c_{kj} j'\\dots ij} &&\nkj\\dots v ji \\ar[rr]^{kb_{jv}i} \\ar[dd]^(0.7){c_{kj} j' \\dots vji} &&\nkj'j\\dots vi \\ar@{->>}[dd]\\\\\n& \\mathcal{N} && E(c_{kj}, b_{jv}) i \\\\\njkj' \\dots ij \\ar@\/_4ex\/[rr]_-{jkj' \\dots v c_{ij}} && jkj' \\dots v ji \\ar@{->>}[rr] &&\nj'j \\dots vk i.\n}\n\\end{gathered}\n\\end{equation}\nThe commutativity of $f_{F,\\tilde\\tau}(E(b_{kk'}, c_{ks}))$ for $s\\le k'''$ follows\nfrom the commutativity of the diagram obtained by application  $f_{F,\\tau}$ to~\\eqref{bc}.\nNow, we show that $f_{F,\\tilde\\tau}(E(b_{kj}, c_{ks}))$ is commutative for\n$s\\le k'''$ and $j\\le k''$ by induction on $j$, using the diagram\n\\begin{equation}\n\\label{Bc1p}\n\\begin{gathered}\n\\xymatrix@C0em{\n\\ar[dd]_{k\\dots \\widehat{\\jmath} c_{jk} s} k\\dots j ks \\ar[rrrr]^-{k\\dots j c_{ks}} &&&& k\\dots j sk\\ar[dd]^{c^*} \\\\\n && \\eqref{cc}\\\\\n\\ar[dd]_{b_{k\\widehat{\\jmath}}js} k\\dots \\widehat{\\jmath} kjs \\ar@\/^3ex\/[rr]^{k\\dots \\widehat{\\jmath} kc_{js}}&&  k\\dots \\widehat{\\jmath} ks j\n\\ar[rr]^{k\\dots \\widehat{\\jmath} c_{ks} j} \\ar[dd]_(0.7){b_{k\\widehat{\\jmath}}sj}  && \\ar@{->>}[dd]  k \\dots \\widehat{\\jmath}sk j \\\\\n& \\mathcal{N} && E(b_{k\\widehat{\\jmath}},c_{ks})j \\\\\nk'k\\dots js \\ar@\/_4ex\/[rr]_{k'k\\dots \\widehat{\\jmath} c_{js}} && k'k\\dots \\widehat{\\jmath} sj \\ar@{->>}[rr] && sk'k\n\\dots \\widehat{\\jmath}j.\n}\n\\end{gathered}\n\\end{equation}\nThe commutativity of the diagrams $f_{F,\\tilde\\tau}(E(b_{kj}, c_{kj'})) $ follows from the definition\nof $\\tilde\\tau$. \nNext we check that $f_{F,\\tilde\\tau}(E(b_{kj}, c_{kj}))$ is commutative. \n\\begin{equation}\n\\label{Bc3p2}\n\\begin{gathered}\n\\xymatrix@C1.2em{\n\\ar@\/_3ex\/[dd]_{k\\dots v c_{j k} j}^{{\\eqref{loops}}} k \\dots j k j\n\\ar[rrrr]^{k\\dots j c_{kj}} &&&& k\\dots jjk  \\ar[dd]^{ k\\dots v a_j k}\\\\ && \\eqref{ac}\n\\\\\n \\ar@\/_3ex\/[uu]_(0.3){k\\dots v c_{kj}\nj}\nk \\dots v kjj \\ar[rr]_{k\\dots v ka_j} \\ar[dd]_{b_{kv}jj}  && k \\dots v kj \\ar[rr]^{k\\dots v\nc_{kj}}\\ar[dd]^{b_{kv} j} && k\\dots v jk \\ar[dd]^{b_{kj}} \\\\ & \\mathcal{N} && E(b_{kv}, c_{kj}) \n\\\\\nk'k\\dots v kjj \\ar[rr]^{k'k\\dots v k a_j}  && k'k\\dots v j  \\ar@{-->}[rr] &&\nk'k\\dots v_j\n}\n\\end{gathered}\n\\end{equation}\n\nTo prove the commutativity of $f_{F,\\tilde{\\tau}}(E(b_{kj},c_{ks}))$, one notices\nthat commutativity of  $f_{F,\\tilde\\tau} (E(b_{kj}, c_{kj'})$ for all $j\\le k'$, implies that \nfor any $k''\\ge s \\ge j+1$, the following diagram commutes uppon\napplication of $f_{F,\\tilde\\tau}$ to it:\n\\begin{equation}\n\\label{asdf}\n\\begin{gathered}\n\\xymatrix{\nk\\dots j k s \\ar[r] \\ar[r]^-{c^*} \\ar[d]_{b_{k j} s} & k\\dots \\widehat{s}k s\n\\dots j s \\ar[d]^{b_{k\\widehat{s}} s \\dots js} \\\\\nk'k \\dots j s \\ar@{-->}[r] & k' k \\dots j s .\n}\n\\end{gathered}\n\\end{equation}\nFurther one uses that $\\tilde\\tau (c_{sk})\\tilde\\tau ( c_{ks}) = \\mathrm{id}$ and the\nnatural e.d. with $r=b_{k\\widehat{s}}$, $r' = b_{sj}$, and $w = \\varnothing$. \n\nThis finishes the proof that the diagrams (\\ref{aa}-\\ref{TZ}) give a coherent\npresentation for the $0$-Hecke monoid $\\rs{n+1}$. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\label{sectionIntroduction}\n\nDuring the past two decades there has been a huge expansion in the development and use of agent-based models for a variety of societal systems and economic phenomena, ranging from markets of various types and societal activities such as energy systems and land use, see e.g., \\cite{ComputationalEconomicsHandbook2,ArtificialStockMarket1997,ABMFinancial2007,WITCH2006,SimLUC2003} for a few illustrative examples. A key issue in the construction of such models is the design of the agents. To what extent are the agents rational? And what does it mean that an agent is rational? This has divided scholars into different camps. Herbert Simon \\cite{Simon1957Bounded} introduced the concept of \"bounded rationality\", which can be implemented in a variety of ways in agent-based models, assuming that there is some limitation on the reasoning capability of the agent. Game theory provides useful methods for the analysis of interaction between agents and their behaviour. But it is also well known, from experiments of human behaviour in game-theoretic situations \\cite{CamererBehavioral}, that human subjects do not always follow the behaviour predicted by game theory -- and for good reasons. People can in many cases establish cooperation for mutual benefit where game theory would predict the opposite. This discrepancy between \"rational\" game-theoretic agents and human ones is often attributed to either limited rationality of human reasoning or to social preferences. The latter can in some cases be referred to as rule-based rationality \\cite{Aumann2008RuleRationality} under which rules-of-thumb may have developed over time in cultural evolution under positive selective feedback from the benefits of cooperation. \n\nIn the modeling and construction of agents it is therefore of high importance that the assumptions made on rationality and the reasoning process are made explicit. Binmore discusses this in his classic papers \"Modeling rational players\" \\cite{Binmore1987Modeling,Binmore1987Modeling2}. He makes the distinction between {\\em eductive} and {\\em evolutive} processes leading to an equilibrium in a game. The former refers to a process internal to the agent representing a reasoning process, while the latter may work with much simpler characterisation of the agents where evolutionary processes lead to an equilibrium by mutation and selection on the population level. In evolutionary game theory and agent-based modeling it is common to use a combination of these, but one seldom designs agents who carefully reason about possible actions and their consequences. And the fundamental question still remains: What does it mean for an agent to be rational? \n\nOne of the major achievements in game theory is the establishment of the Nash equilibrium concept and the existence proof that any finite game has at least one such equilibrium \\cite{Nash1950}. The Nash equilibrium is a situation where no player can gain by unilateral change of strategy, and in that sense this can be seen as a rational equilibrium in providing the player a best response to other players' actions. The Nash equilibrium is thus often regarded as a result of rational reasoning, reflecting the behaviour of rational players. Importantly, for our discussion, this view of the Nash equilibrium has also been carried over from single-shot to repeated games.\n\nIn several finitely repeated games, in which the number of rounds is known, the solution of how players choose actions can be guided by the backward induction procedure. This is often exemplified by the Prisoners' Dilemma, for which the single round game has a unique Nash equilibrium with both players defecting, while the indefinitely repeated game has an uncountable infinity of equilibria allowing for cooperation. However, when the exact number of rounds, $n$, is known, a player can start with considering the last round, in which the score is maximized by defecting. So, with both players being rational in the sense that they want to maximize their own score, the outcome of the last round is clear---mutual defection. But then the next-to-last round turns into the last unresolved round, and the same reasoning applies again resulting in mutual defection also for round $n-1$. The assumption needed is that each player knows that the other one is rational. The procedure then repeats all the way to the first round, showing that the Nash equilibrium is mutual defection from the start of the game. Since the result of the backward induction procedure does not seem to lead to an intuitively rational result it is often referred to as the \"backward induction paradox\" \\cite{PetitSugden1989Backward}. Note that backward induction also applies to certain one-shot games, and the discrepancy between theory and observation has been discussed also for several such examples, e.g., the \"Beauty contest\" \\cite{BoschDomenech2002} and the \"Traveler's dilemma\" \\cite{Kaushik1994}.\n\nThere are at least two important objections against the generality of the reasoning based on backward induction. The first objection is empirical, since studies on how human players behave in the game show a substantial level of cooperation, but with a transition to lower levels of cooperation towards the end. Explanations are several, and this implies that several mechanisms are in play. For example, it has been observed in the laboratory that subjects cooperate initially but attempt to cheat each other by deviation in the end \\cite{SeltenStocker1986,CamererBehavioral}.\n\nThe second objection is conceptual and strongly connected to the notion of rationality and what can be considered as a rational way of reasoning. The only equilibrium that can exist in a given finite repetition is the Nash equilibrium, but whether that is to be considered as rational is the question. A critical point concerns what conclusion a player should draw if the opponent deviates from what backward induction implies and instead cooperates \\emph{in the first round}. It is then clear that the opponent is not playing according to the Nash equilibrium, and there is a chance to get a higher score for some time if cooperation can be established.\n\nIn this situation, the choice between (i) following backward induction and defecting from start and (ii) deviating from backward induction by starting with cooperation becomes a strategic decision. One can imagine \"rational\" players in both categories. In the first category, there are then two options, either one just plays defect throughout the game whatever the opponent does, as backward induction suggests, or one switches to cooperation if the opponent cooperates. In the second category, both players are again faced with the question of, provided the opponent is cooperating, when to switch to defection. Obviously, there cannot exist a fixed procedure for deciding on when it is optimal to switch from cooperation to defection, since such a strategy would be dominated by the one that switches one round before. However, the interaction and survival of different ways to handle first round cooperation can be studied using evolutionary methods.\n\nThe purpose of this paper is to investigate in an evolutionary context the performance of strategies representing the strategic choices discussed above in the finitely repeated Prisoners' Dilemma. In Binmore's terms, we focus on an evolutive process, in which each agent has a certain, relatively simple strategy for the game, and the mix of strategies and their evolution is investigated on the population level. Importantly, the chosen strategies can all be seen as components in the reasoning processes discussed above: both (i) the steps involved in the backward induction process, and (ii) the steps initiating and responding to cooperation in the first round which then reflects the possibility for strategies to deviate from equilibrium play. It is well known that evolutionary drift or mutations, at least if sufficiently strong, can drive the population away from a fixed point corresponding to the Nash equilibrium. Under what circumstances does the evolutionary dynamics lead to the same result as the backward induction process with a Nash equilibrium as its fixed point, and when can deviation from Nash equilibrium play alter that process? The answer, which is elaborated in this paper, depends on choices of a number of critical model characteristics and parameters: selected strategy space, mutation rate, payoff matrix, and the length of the game.\n\nWe prove that, for a simple set of strategies, {\\em i.e.}, conditional cooperators, the replicator-mutation dynamics is always characterised by a stable fixed point corresponding to the Nash equilibrium in the limit of zero (but positive) mutation rate. Numerically we show that such a result does not hold in general, even if the only Nash equilibrium is characterised by defection from the start of the game. The strategy set that allows for different reactions on the first round may lead to a path of actions different from what is considered in the backward induction process. On the population level, this turns out to destabilise the dynamics, and for a large part of parameter space, the evolution does not bring the system to a stable fixed point, even in the limit of zero (but positive) mutation rate. The dynamics is instead characterised by oscillatory behaviour.\n\nMost of the work related to evolutionary dynamics, backward induction, and the finitely repeated Prisoners' Dilemma concerns the replicator dynamics \\cite{SchusterSigmund1983RepliatorDynamics} with no mutation, as this class has been examined analytically more thoroughly than the replicator-mutation dynamics \\cite{HofbauerSelectionmutation1985}. Nachbar \\cite{Nachbar1992Evolution} studies convergence in the dynamics and shows that for 2-stage games all cooperation goes extinct when starting from a mixed population. This result is extended by Cressman \\cite{Cressman1996Evolutionary, cressman2003evolutionary} to the finitely repeated Prisoners' Dilemma of arbitrary length.\n\nSeveral authors have investigated various types of evolutionary dynamics under the effect of perturbations or mutations (\\cite{Noldeke1993, CressmanSchlag1998Dynamic, BinmoreSamuelsson1999Drift, Sergiu2002, HofbauerSandholm2011}). Here the focus has primarily been on other games than the finitely repeated Prisoners' Dilemma. Gintis et al. \\cite{GintisCressmanRuijgrok2009} consider the replicator dynamics subject to recurrent mutation when mutation rate goes to zero. For finite noncooperative games, they show that the dynamics need not converge to the subgame perfect equilibrium, and limiting equilibria can be far away from the this equilibrium. They also show that in the $n$-player Centipede game, there exists a unique limiting equilibrium as mutation rate goes to zero, which is far from the subgame perfect equilibrium but equal in payoffs. \n\nPonti \\cite{Ponti2000Cycles} studies replicator-mutation dynamics in the Centipede game \\cite{Rosenthal1981Centipede}. Using simulations, he finds recurrent phases of cooperation in the evolutionary dynamics, and for particular payoffs of the game, this is shown to depend both on mutation rate as well as the length of the game. It is left as an open question whether such behaviour would disappear in the long run, or persist for negligible mutation rate. This result is most relevant to the present paper in the context of the finitely repeated Prisoners' Dilemma.\n\nOur work is also related to the literature on the \"backward induction paradox\" \\cite{PetitSugden1989Backward}, which has focused on deviation from equilibrium play in the first round of extensive games. For the backward induction reasoning to be rational, it has been assumed that each player has full knowledge about the rationality of the other player, and that both players know this, et cetera, known as \"common knowledge of rationality\". It has been proved by Aumann \\cite{Aumann1995CKR} that such common knowledge implies backward induction in games with a unique subgame perfect equilibrium. However, backward induction provides no firm basis to act when a player deviates from equilibrium play \\cite{Aumann1996Reply,Binmore1997Backward,Gintis2010Towards}. Our study can be viewed as an evolutionary study of a population in a setting where reacting to out-of-equilibrium play in the first round is possible.\n\n\\section{Evolutionary Dynamics}\n\\label{sect_evoldynamics}\n\nThe evolution of strategies in the finitely repeated Prisoners' Dilemma is studied using replicator dynamics with a uniform mutation rate. This is a model of an infinite population where all interact with all, and in which each strategy $i$ occupies a certain fraction $x_i$ of the population. The selection process gradually changes the population structure, based on a comparison of the average score $s_i$ of strategy $i$ with the average score $s$ in the population, \n\\begin{align}\ns_i &= \\sum_{j=1}^n u(i, j) x_{j} \\\\\ns &= \\sum_{i=1}^n s_i x_i \\;\n\\end{align}\nwhere $u(i,j)$ is the score for strategy $i$ meeting strategy $j$ in the $N$-round Prisoners' Dilemma. Assuming a uniform mutation rate of $\\epsilon$, the replicator-mutation dynamics can be written,\n\\begin{eqnarray}\n\\frac{dx_i}{dt} = x_i \\left( s_i - s - \\epsilon \\right) + \\epsilon\/n \\;\n\\label{eqnMutatorReplicator}\n\\end{eqnarray}\nwhere $n$ is the number of different strategies. The dynamics conserve the normalisation of $x$.\n\nThe score $u(i,j)$ between two strategies, $i$ and $j$, is calculated from $N$ rounds of the Prisoners' Dilemma with a payoff matrix for the row player given by\n\\begin{center}\n\\begin{game}{2}{2}\n& $C$ & $D$\\\\\n$C$ &$1$ &$0$\\\\\n$D$ &$T$ &$P$\n\\end{game} \\\\\n\\end{center}\nThis means that the reward $R$ for mutual cooperation (C, C) has been set to 1, and the \"suckers payoff\" $S$ for cooperation against defection (C, D) is 0. The temptation to defect against a cooperator (D, C) is associated with a score $T$, and for mutual defection (D, D) the score is $P$. With the usual constraints of $T > R > P > S$ and $2R > T + S$ it remains to study the parameter region given by $T \\in [1,2]$ and $P \\in [0,1]$. There are in principle three independent parameters in the game, but in combination with the replicator-mutation dynamics, the third one is incorporated in the mutation rate $\\epsilon$ (which can be derived by subtraction and division by $S$ and $R-S$, respectively, in the replicator-mutation dynamics equation, Equation~(\\ref{eqnMutatorReplicator})).\n\n\\newpage\n\\subsection{Selecting the Strategy Set}\n\\label{subsect_selectstrat}\n\nWe investigate the evolutionary behaviour considering two sets of strategies. The first one is a strategy set $\\Gamma_1$ that represents various levels of depth in applying the backward induction procedure to conditional cooperation. A strategy in this set is denoted $S_k$ (with $k \\in \\{0, 1, ..., N\\}$), which means that the strategy is playing conditional cooperation up to a round $k$ and then defects throughout the game, see Figure~\\ref{fig_statespace0}. Conditional cooperation means that if the opponent defects, one switches to unconditional defection for the rest of the game. For example, $S_N$ is prepared to cooperate through all rounds (or as long the opponent does), while $S_0$ defects from the first round. It is then clear that strategy $S_k$ dominates $S_{k+1}$ (for $k \\in \\{0, 1, ..., N-1\\}$), and backward induction leads us to the single Nash Equilibrium in which both players choose strategy $S_0$. \n\n\n\\begin{figure}\n\\center\n\\framebox{\n\\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,semithick]\n\\tikzstyle{every state}=[circle,draw]\n\n\\node[state, label=220:$S_{i \\in \\{1..N\\}}$] (C) {$C$};\n\\node[state, label=300:$S_0$] (D) [right of=C] {$D$};\n\n\\path (C) edge node {$D \\lor (t \\geq i)$} (D) (C) edge [loop above, distance=2.5cm, out=130, in=50, looseness=0.8] node {$C \\land (t < i)$} (C) (D) edge [loop above, distance=2.5cm, out=130, in=50, looseness=0.8] node {} (D);\n\\end{tikzpicture}\n}\n\\caption{Finite state machine illustrating the first strategy set, $\\Gamma_1$, of the $S_i$ strategies. The action to perform is in the node, and transitions are taken on the basis of the opponent's action in the previous round (or based on the previous round number $t$). All strategies start in the left node (C), except $S_0$ that starts with defection in the right node (D). $S_i$ cooperates conditionally for $i$ rounds after which it starts to defect.}\n\\label{fig_statespace0}\n\\end{figure}\n\n\\begin{figure}\n\\center\n\\framebox{\n\\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,semithick]\n\\tikzstyle{every state}=[circle,draw]\n\n\\node[state, label=220:$S_{i \\in \\{1..N\\}}$] (C) {$C$};\n\\node[state, label=left:$CC_{i \\in \\{2..N\\}}$] (A) [left of=C] {$C$};\n\\node[state, label=left:$DC_{i \\in \\{2..N\\}}$] (B) [below of=C] {$D$};\n\\node[state, label=300:$S_0$] (D) [right of=C] {$D$};\n\n\\path (A) edge node {} (C) (B) edge node {$C$} (C) (B) edge node {$D$} (D) (C) edge node {$D \\lor (t \\geq i)$} (D) (C) edge [loop above, distance=2.5cm, out=130, in=50, looseness=0.8] node {$C \\land (t < i)$} (C) (D) edge [loop above, distance=2.5cm, out=130, in=50, looseness=0.8] node {} (D);\n\\end{tikzpicture}\n}\n\\caption{\\label{fig_statespace1} Finite state machine illustrating the extended strategy set $\\Gamma_2$ consisting of the strategies $S_i,CC_i$, and $DC_i$. $S_i$ are the conditional cooperators as described in Figure~\\ref{fig_statespace0}. The Convincers are denoted $CC_i$ and the Followers $DC_i$. Strategies start in the state at which the name is placed. The strategies $CC_i$ start with at least two rounds of cooperation, which may trigger $DC_i$ to switch from defection to cooperation. After that the latter two strategies act as the conditional cooperators $S_i$.}\n\\end{figure}\n\nNote that the entire strategy set $\\Omega$ for the $N$-round Prisoners' Dilemma is very large, as a strategy requires a specification how to react on each possible history (involving the opponent's actions) for every round of the game. This results in $2^{2^N-1}$ possible strategies, e.g. for $N=10$ rounds this is in the order of $10^{308}$. The selection of strategies to consider is critical. One can certainly introduce strategies so that other Nash equilibria are introduced along with those characterized by always defection. For example, selecting only three strategies, e.g., $\\{S_0,S_5,S_{10}\\}$ will lead to a game with three equilibria, one for each strategy. But this is an artefact of the specific selection made. Here, we do not want to create new Nash equilibria but we want to investigate how and if the evolutionary dynamics brings the population to a fixed point dominated by defect actions corresponding to the original Nash equilibrium. One way to achieve this is to make sure that iterated elimination of weakly dominated strategies can be applied to the constructed strategy set, in a way so that only strategies that defect throughout the game remain, keeping the non-cooperative characteristic of the Nash equilibrium. \n\n\nThe second set of strategies $\\Gamma_2$ (in Figure~\\ref{fig_statespace1}) that we consider is given by extending $\\Gamma_1$ to include also strategies corresponding to steps of reasoning in which one (i) tries to establish cooperation even if the opponent defects in the first round and (ii) responds to such attempts by switching to cooperation for a certain number of rounds. We refer to such strategies as \"Convincers\" and \"Followers\", respectively. \n\nA Convincer strategy $CC_k$ starts with cooperation twice in a row, regardless of the opponent's first round action, and in this way it can be seen as an attempt to establish cooperation even with first round defectors. From round 3, the strategy plays as $S_k$, {\\em i.e.}, conditional cooperation up to a certain round $k \\in \\{2, 3, ..., N\\}$.\n\nA Follower strategy $DC_k$ starts with defection, but can be triggered to cooperation by a Convincer (or $S_k$ where $k>0$). So the Follower switches to cooperation in the second round, after which it, like the Convincer, plays as $S_k$, {\\em i.e.}, conditional cooperation up to round $k \\in \\{2, 3, ..., N\\}$. (A Follower strategy is also triggered to cooperation by an $S_k$ strategy, but $S_k$ does not forgive the first round defect action and cooperation cannot be established.) \n\nFor the extended strategy set, $\\Gamma_2$, it is straightforward to see that iterated elimination of weakly dominated strategies, starting with those cooperating throughout the game, leads to a Nash equilibrium with only defectors.\n\nFor the first strategy set, $\\Gamma_1$, the Nash equilibrium of $(S_0,S_0)$ is strict since any player deviating would score less. For the second strategy set, $\\Gamma_2$, this Nash equilibrium is no longer strict as one of the players could switch to a Follower strategy, still defecting and scoring the same. For the first strategy set, the NE is unique, but for the second one that is not necessarily the case. Since backward induction still applies in the second set, we know that any NE is characterized by defection only, which can be represented by a pair $(S_0,DC_k)$. This is a NE only if the $S_0$ player cannot gain by switching to $CC_{k-1}$ (or to $CC_2$ if $k=2$). This translates into $k P > S + (k-2)R + T$ (for $k>2$) or $T < k P - (k-2)$ for our parameter space, while for $k=2$, we have $2P>S+R$ or $P>1\/2$. Furthermore, if $(S_0, DC_k)$ is a NE, then also $(DC_j, DC_k)$, with $j\\le k$, is a NE. \n\nThis means that in a part of the payoff parameter region, for $P<1\/2$, there is only one NE, while for larger $P$ values there are several NEs, with increasing number the closer to $1$ the parameter $P$ is. Note, though, that any additional NE here is characterized by defection from the first round, corresponding to the unique subgame perfect Nash equilibrium $(S_0, S_0)$. This illustrates the fact that in the NE one can switch from a pure defector to a Follower without reducing the payoff. If this happens under genetic drift in evolutionary dynamics, the situation may change so that Convincers may benefit and cooperation can emerge.\n\n\\section{Dynamic Behaviour and Stability Analysis}\n\\label{numerical_study}\n\nThe dynamic behaviour and the stability properties of the fixed points are investigated both analytically and numerically, for the two strategy sets presented in Section \\ref{subsect_selectstrat}. In each case, we investigate the dependence of the payoff parameters, $T$ and $P$, as well as the mutation rate and game length, $\\epsilon$ and $N$, respectively. We are mainly interested in which influence the presence of Convincers and Followers has on the stability of fixed points and its impact on the dynamics. This will be illustrated in three different ways as follows. First, we present and examine a few specific examples of the evolutionary dynamics and discuss the qualitative difference between the strategy sets. For the simple strategy set, we show analytically that the fixed point associated with the Nash equilibrium remains stable for a sufficiently small mutation rate. A numerical investigation is then performed for the extended strategy set, characterising the fixed point stability. Finally, for different lengths of the game, we study realisations of the evolutionary dynamics from an initial condition of full cooperation ($x_{N} = 1$ and $x_i=0$ for $i \\neq N$, with $x_k$ denoting the fraction of strategy $S_k$ in the population) to examine to which degree cooperation survives and whether the dynamics is attracted to a fixed point or characterized by oscillations. By studying variation of the dynamics over the different regions in the parameter space and changing the payoff parameters $T$ and $P$ (by adhering to the inequalities between $T,P,R,S$ as described in Section \\ref{sect_evoldynamics}), it is studied how the population evolves in various games over time. \n\n\\subsection{Dynamics with the Simple and the Extended Strategy Set}\n\\label{exampledynamics}\n\n\\begin{figure}\n\\centerline{\n$\\begin{array}{cc}\n\\noindent \\includegraphics[width=8cm]{StandardRPDScanPAPERPLOTS__N10__E2E_8.pdf} &\n\\includegraphics[width=8cm]{StandardRPDScanPAPERPLOTS__N10__E2E_12.pdf} \\\\ \n\\includegraphics[width=8cm]{ExtendedScanPAPERPLOTS__N10__E2E_8.pdf} &\n\\includegraphics[width=8cm]{ExtendedScanPAPERPLOTS__N10__E2E_12.pdf}\n\\end{array}$\n}\n\\caption{\\label{fig_example1} Illustration of the dynamics for a particular game (P=0.2, T=1.33) for 10-round Repeated Prisoners' Dilemma ($N=10$). Each main plot displays how the frequency of different strategies changes in the population as a function of time. Smaller subplots show how the population average payoff changes with time, with $NR$ and $NP$ denoting full cooperation and defection, respectively. Top: simple strategy set ($S_0,...,S_{10}$). Below: the extended strategy set, which includes also Convincers ($CC_2,...,CC_{10}$) and Followers ($DC_2,...,DC_{10}$). When lowering $\\epsilon$, the extended strategy set exhibits meta-stability with recurring cooperation, while for the simple strategy set cooperation disappears.}\n\\end{figure}\n\n\nRealisations of the evolutionary dynamics, Equation~(\\ref{eqnMutatorReplicator}), for both the simple and the extended strategy sets are shown in Figure~\\ref{fig_example1}. For particular payoff values and a 10-round game, the mutation rate $\\epsilon$ is varied to illustrate how it affects the dynamics.\n\n\\newpage First, we consider the case with the simple strategies ($S_0,...,S_{10}$) in Figure~\\ref{fig_example1}. From an initial state of full cooperation, with a population consisting only of fully cooperative $S_{10}$ players, the dynamics will, for both levels of mutation rate, lead to a gradual unraveling of cooperation to a point where $S_0$, full defection, dominates the population. The first step of this unraveling occurs because $S_9$ defecting in the final round will have higher payoff than $S_{10}$. At this stage $S_0$ is much worse off, but the population goes through a series of transitions which reminds of a backward induction process. This can also be seen in terms of average payoff, as illustrated in Figure~\\ref{fig_example1}. When the population is in this non-cooperative mode a positive mutation-rate $\\epsilon$ may offset the situation. For the higher mutation rate, cooperative strategies re-emerge after a period of influx from other strategies. The mutations gradually introduce cooperative behaviour to a critical point where some degree of cooperation has a selective advantage over full defection, and we see a shift in the level of cooperation. After a while, cooperative behaviour is again overtaken by full defection and a cyclic behaviour becomes apparent. Comparing with the next realisation of the simple strategies we see that this can happen only when the mutation rate is high enough. As mutation rate becomes smaller, here illustrated with $\\epsilon=2^{-12}$, there is no re-appearance of cooperation. When the mutation rate gets too low, strategies other than defection are kept on a level that is too low to promote further cooperation. This demonstrates that the mutation rate can affect whether cooperation re-appears or not.\n\nSecond, we consider the case with extended strategies (the $3N-1$ strategies $S_0,...S_{10}$, $CC_2,...,CC_{10}$, $DC_2,...,DC_{10}$) shown at the bottom of Figure~\\ref{fig_example1}. We observe that adding Convincer and Follower strategies changes the dynamics, but with a similar unraveling of cooperation. For both mutation rates, the trajectories have periods with defection dominating in the population, indicated by average payoff, but the population does not seem to stabilize. Lowering the mutation rate increases the time between the outbreaks of cooperation. Contrary to the simple strategy set, the system does not seem to settle close to full defection. The explanation is due to the Follower strategies being able to gradually enter the population by getting the same payoff as the strategy of full defection. At a critical point, there are sufficiently many Followers, which make mutations to Convincers successful and cooperation re-enters the population for a period.\n\nIn the next section, we will investigate the dynamics and the stability characteristics of both the simple and the extended strategy sets in detail, varying the payoff parameters over the full ranges, and investigating the behaviour in the limit of diminishing mutation rate.\n\n\\subsection{Existence of Stable Fixed Points}\n\\label{fixedPointExistence}\n\nWe now turn to examine the existence of stable fixed points in the dynamics for low mutation rates. For the simple strategy set the following proposition holds.\n\n\\paragraph{Proposition 1:} \n\n\nFor the simple set of strategies, $\\Gamma_1$, the fixed point associated with the Nash equilibrium, dominated by strategy $S_0$, is stable under the replicator-mutation dynamics, if the mutation rate is sufficiently small (but positive). \n\n\\paragraph{Proof 1:} See Appendix A.\\\\\n\nOur results for the extended strategy set, $\\Gamma_2$, are based on numerical investigations: by using an eigenvalue analysis of the Jacobian of the replicator-mutation dynamics, Equation~(\\ref{eqnMutatorReplicator}), we determine for which parameters $T, P$ and $\\epsilon$ the dynamics is characterised by stable fixed points. We are especially interested in the case of lowering $\\epsilon$ towards 0 to examine whether fixed points become stable when mutation rate is sufficiently small, like in the case of the simple strategy set.\n\nThis stability analysis over the parameter space and the results are presented for different lengths of the game in Figure~\\ref{fig_fixpoints}. Stable fixed points exist to the \\emph{right} of the line corresponding to a specific $\\epsilon$, while to the \\emph{left} of the line no stable fixed points were found. When decreasing the mutation rate $\\epsilon$ towards 0, we see that an increasing fraction of parameter space is characterised by a stable fixed point. Unlike the case for the simple strategy set, the numerical investigation shows a convergence of the delimiting line between stable and unstable fixed points, indicating that there is a remaining region in parameter space (with low $P$ and low $T$) for which fixed points are unstable in the limit of zero mutation rate.\n\nNote from the discussion in Section \\ref{sectionIntroduction} that while these results show the existence of stable fixed points, it is left to consider whether the population dynamics would actually converge to such states. Next, we turn to consider population dynamics over the parameter space and its outcome.\n\n\\begin{figure}\n\\centerline{\n$\\begin{array}{cc}\n\\includegraphics[width=8cm]{FixpointExtendedN5.pdf} &\n\\includegraphics[width=8cm]{FixpointExtendedN6.pdf} \\\\\n\\includegraphics[width=8cm]{FixpointExtendedN7.pdf} &\n\\includegraphics[width=8cm]{PlotStableUnstableBoundary_N5_VaryEPS_SimpleAndExtended_Ver2.pdf} \n\\end{array}$\n}\n\\caption{\\small \\label{fig_fixpoints} Numerical analysis showing which games have stable fixed points. Stable fixed points have been found to the right of a given line, at which they disappear if $P$ is decreased further. Lowering the mutation rate $\\epsilon$ turns more evolutionary dynamics into having stable fixed points. Runs for $N=5,6,7$ indicate a convergence, as the mutation rate is decreased, towards a fraction of games being without stable fixed points, as seen in the lower left of the parameter space for the different game lengths. To the bottom right the boundary is shown for a particular $T=1.1$.\n}\n\\end{figure}\n\n\\subsection{Recurring Phases of Cooperation}\n\\label{repeated}\n\nMotivated by the findings above in Section \\ref{exampledynamics}, we now turn to study recurrent cooperative behaviour in the evolutionary dynamics. We investigate if and for which games a population, starting from initial cooperation as before, settles into a mode of oscillations that at some point in the cycle brings the system back to a higher level of cooperation \\footnote{An instance of the dynamics was counted as oscillating when the average payoff $A$ repeatedly returns to at least $5\\%$ above full defection, {\\em i.e.} $A>1.05NP$. Frequently, it was the case that the oscillations had phases of cooperation well above the $5\\%$ threshold.}. \n\nThe interesting case is when mutation rates are small: higher mutation rates introduce a background of all different strategies, which can be seen as artificially keeping up cooperative behaviour in the population. To avoid this effect, we investigate the dynamics with $0<\\epsilon<<1$, and, in particular, numerical analyses were made with a series of decreasing mutation rates. The population is initialized as before, described in Section \\ref{exampledynamics}.\n\nFirst, we characterise the simple and the extended strategy set for different game lengths, varying mutation rates, and varying the parameters of $T$ and $P$. Figure~\\ref{phaseplots1} displays the results for games of 5 and 10 rounds in the top and the middle row. For a given $T$ a line denotes the border for which games to the \\emph{left} have recurring phases of cooperation, while games to the \\emph{right} are characterised by a fixed point dominated by defection. For the simple strategy set, we observe that by decreasing mutation rate, the fraction of games where cooperation recurs becomes smaller and disappear. This suggests that as we make the mutation rate sufficiently small, cooperation will die out in the case of simple strategies, as is expected from Proposition 1 stating that the fixed point dominated by the $S_0$ strategy becomes stable. On the other hand, for the extended strategy set, a considerable fraction of games seem to offer recurring phases of cooperation despite lowering the mutation. For the 10-round game, the line describing the critical parameters is seen to converge as the mutation rate decreases, {\\em i.e.}, there is a large part of the parameter region for which the dynamics is not attracted to a fixed point. For the $5$-round game, this occurs at least in part of the parameter space. The bottom graphs in Figure~\\ref{phaseplots1} show that the longer the game, the larger is the parameter region for $T$ and $P$ in which the fixed point is avoided and recurring periods of cooperation are sustained. It should be noted, though, that as the mutation rate decreases, the part of the cycle in which there is a significant level of cooperation decreases towards zero. This is due to the slow genetic drift that brings $DC_k$ strategies back into the population, which eventually make it possible for the $CC_k$ strategies to re-establish a significant level of cooperation.\n\n\\begin{figure}[htbp]\n\\centerline{\n$\\begin{array}{cc}\n\\includegraphics[width=8cm]{ParseOscillationsPAPERPLOTS__N5Simple.pdf} &\n\\includegraphics[width=8cm]{ParseOscillationsPAPERPLOTS__N10Simple.pdf} \\\\\n\\includegraphics[width=8cm]{ParseOscillationsPAPERPLOTS__N5Extended.pdf} & \n\\includegraphics[width=8cm]{ParseOscillationsPAPERPLOTS__N10ExtendedAdditionalEps.pdf} \\\\\n\\includegraphics[width=8cm]{ParseOscillationsPAPERPLOTS__epsE2_8.pdf} &\n\\includegraphics[width=8cm]{ParseOscillationsPAPERPLOTS__epsE2_12.pdf} \\\\\n\\end{array}$\n}\n\\caption{\\label{phaseplots1} Parameter diagram showing which games have recurrent cooperation in the evolutionary dynamics from a starting point of initial cooperation. Top row: simple strategy set. Middle row: extended strategy set with Convincers and Followers. In the graphs, recurrent cooperation exists to the left of the line and a fixed point with defectors characterizes the behaviour to the right. For simple strategies, lowering $\\epsilon$ steadily reduces the part of parameter space dominated by recurrent cooperation. In the bottom row, the delimiting line is shown for a variety of game lengths and for two levels of mutation (left and right), illustrating the fact that the longer the game the larger the parameter region supporting cooperative phases in the evolutionary dynamics.\n}\n\\end{figure}\n\n\\subsection{Co-existence between Fixed Point Existence and Recurring Cooperation}\n\nThe discussion in Section \\ref{fixedPointExistence} left the question of whether stable fixed points are reached by population dynamics, and we found that it is not necessarily so in Section \\ref{repeated}. This was illustrated for the extended strategy set for low $\\epsilon$. Combining the different findings by considering their boundaries in the parameter space, this suggests an additional property of the evolutionary dynamics. By considering the joint results of our findings (in Figures~\\ref{fig_fixpoints} and \\ref{phaseplots1}), one can note that there is a part of parameter space for which there is co-existence between a stable fixed point of defect strategies and a stable cycle with recurring cooperation. \n\n\\newpage\n\\section{Discussion and Conclusion}\n\\label{discussion_conclusion}\n\nA key point in game theory is that a player's strategic choice must consider the strategic choice the opponent is making. For finitely repeated games, backward induction as a solution concept has become established by assuming player beliefs being based on common knowledge of rationality. However, this assumption says nothing about how players would react to a deviation from full defection---a deviation from the Nash equilibrium---since it a priori rules out actions and reactions that exemplify other ways of reasoning. \n\nMotivated by the general importance of backward induction and what has been called its \"paradox\", we have introduced an evolutionary analysis of the interaction in a population of strategies that react differently to out-of-equilibrium play in the first round of the game. We have shown how extending a strategy set for this possibility, in the special case of the repeated Prisoners' Dilemma, allows for stable limit cycles in which cooperative players return after a period of defection. The introduction of Convincers and Followers, representing both strategies that try to establish cooperation and strategies that are capable of responding to that, are made in a way to preserve the structure of the selected strategy set so that elimination of weakly dominated strategies leads to full defection.\n\nFor the simple strategy set, as the mutation rate becomes sufficiently small, the cyclic behaviour disappears and the system is attracted to a stable fixed point. The stability of this fixed point was shown analytically for a sufficiently small mutation rate $\\epsilon$.\n\nFor the extended strategy set, for low levels of mutation, the numerical investigation of fixed point stability and oscillatory modes indicates that, for a certain part of payoff parameter space, the evolutionary dynamics does not reach a stable fixed point but stays in an oscillatory mode, unlike the case of simple strategy set. We characterise our results by a detailed quantitative analysis of where this occurs: showing how the length of the repeated game and the mutation rate affects the boundaries of this region.\n\nOne of the main results of the study is an affirmative answer to the question whether different responses to out-of-equilibrium play in the first round can make the dynamics avoid fixed points, and the corresponding Nash equilibrium. Additionally, the fixed point analysis showed the co-existence of a stable fixed point and stable oscillations with recurring phases of cooperation. This means that a system with different responses to out-of-equilibrium play may be found far from its possible stable fixed point. Taken together, this illustrates that the Nash equilibrium play can be unstable at the population level when mutations make explorations off the equilibrium path possible.\n\nThis paper contributes to the backward induction discussion in game theory, but more broadly to the study of repeated social and economic interaction. Many models, typically much larger and less transparent ones, of social and economic systems involve agents. If solving these systems means finding the Nash equilibria, then one may doubt whether that is a good representation of rational behaviour except under certain conditions as we have discussed in the paper. We have shown that strategies corresponding to the Nash equilibrium cannot be taken for granted when they interact and compete with strategies that act and respond differently to out-of-equilibrium play.\n\n\\newpage\n\\section*{Acknowledgements}\n\nFinancial support from the Swedish Energy Agency is gratefully acknowledged. We would also like to thank two anonymous reviewers for constructive criticism and for inspiring us to prove Proposition 1. We also thank David Bryngelsson for valuable comments on the introduction.\n\n\\begin{appendix}\n\\section*{Appendix A. Stability of fixed point in the simple strategy set for small mutation rates}\n\nIn order to show that the Nash equilibrium fixed point at zero mutation rate, $\\epsilon = 0$, continuously translates into a stable fixed point as the mutation rate becomes positive, $\\epsilon > 0$, we investigate the fixed point more thoroughly. First, we note that the stability of the fixed point without mutations ($\\epsilon=0$), characterised by $x_1=1$ (strategy $S_0$ dominating), can be determined by the largest eigenvalue of the Jacobian matrix ($\\partial \\dot{x}_i \/ \\partial x_j$) derived from the dynamics, Equation~(\\ref{eqnMutatorReplicator}), where we use the notation $\\dot{x}_i = dx_i\/dt$. One finds that the largest eigenvalue is given by $\\lambda_\\text{max}= - P < 0$, which shows that the fixed point is stable. This is also known from the stability analysis of the finitely iterated game by Cressman \\cite{Cressman1996Evolutionary}. \n\nWe will now proceed with reformulating the fixed point condition, for positive mutation rate $\\epsilon>0$, which is a set of $n$ polynomial equations given by $\\dot{x}_i = 0$ (with $i=1,...,n$), into an equation of only one variable $x_1$. Based on this we show that the Nash equilibrium fixed point for zero mutation rate $\\epsilon=0$, characterised by $x_1=1$, continuously moves into the interval $x_1 \\in [0,1[$, with retained stability.\n\nWe let index $i$ denote strategy $S_{i-1}$ in the simple strategy set ($i=1,...,n$), where the number of strategies is $n=N+1$. Due to the structure of the repeated game for the simple strategy set, determining $u(i,j)$, we can establish pair-wise relations between $x_k$ and $x_{k+1}$ for $k=1,2,...,n-1$, at any fixed point. We use the slightly more compact notation $s_{i,j}=u(i,j)$, for the score for a strategy $i$ against $j$.\n\nFor $x_1$ and $x_2$ (strategies $S_0$ and $S_1$), using Equation~(\\ref{eqnMutatorReplicator}), we have\n\\begin{align}\n\\frac{d x_1}{d t} = x_1 \\left( s_{1,1}x_1 + s_{1,2}(1-x_1) - \\bar{s} - \\epsilon \\right) + \\frac{\\epsilon}{n} = 0\n\\end{align}\n\\begin{align}\n\\frac{d x_2}{d t} = x_2 \\left( s_{2,1}x_1 + s_{2,2}x_2 + s_{2,3}(1-x_1-x_2) - \\bar{s} - \\epsilon \\right) + \\frac{\\epsilon}{n} = 0 \\;\n\\end{align}\nwhere we have used the fact that $s_{1,2}=s_{1,j}$ (for $j>2$), and $s_{2,3}=s_{2,j}$ (for $j>3$). At a fixed point, for $\\epsilon>0$, all strategies are present, $x_i>0$, and we can divide these equations by $x_1$ and $x_2$, respectively. Then, taking the difference between the equations gives us an equation for the relation between $x_1$ and $x_2$,\n\\begin{align}\n\\frac{\\epsilon}{n} \\left( \\frac{1}{x_1}-\\frac{1}{x_2} \\right) + x_1 + (T-P) x_2 - (1-P) = 0 \\;\n\\end{align}\nwhere we have used that $s_{2,3}-s_{1,2}=1-P$, $s_{1,1}-s_{2,1}-s_{1,2}+s_{2,3}=1$, and $s_{2,3}-s_{2,2}=T-P$. Solving this quadratic equation gives $x_2$ as a function of $x_1$,\n\\begin{align}\nx_2=f_A(x_1)\n\\end{align}\nwith the function $f_A(x)$ defined by\n\\begin{align}\n\\label{eqn_fA}\nf_A(x) = \\frac {1} {2 (T - P)} \\left (x - (1 - P) + \\frac {\\epsilon'} {x} \\right) \\left (\n-1 + \n\\sqrt {1 + \\frac{4 (T-P) \\epsilon'}{\\left (x - (1 - P) + \\frac {\\epsilon'} {x} \\right)^2}} \\; \\right) \\,\n\\end{align}\nHere we have introduced $\\epsilon'=\\epsilon \/ n$. In the same way, for $x_{k-1}$ and $x_k$ (for $k=3, ..., N-1$), the fixed point implies that \n\\begin{align}\n\\frac{d x_{k-1}}{d t} = x_{k-1} \\left( \\sum_{j=1}^{k-2} s_{k-1,j}x_j + s_{k-1,k-1}x_{k-1}\n+ s_{k-1,k}\\left(1-\\sum_{j=1}^{k-1} x_j \\right) - \\bar{s} - \\epsilon \\right) + \\frac{\\epsilon}{n} = 0\n\\end{align}\n\\begin{align}\n\\frac{d x_k}{d t} = x_k \\left( \\sum_{j=1}^{k-2} s_{k,j} x_j + s_{k,k-1}x_{k-1}+ s_{k,k}x_{k}\n+ s_{k,k+1}\\left(1-\\sum_{j=1}^{k} x_j \\right) - \\bar{s} - \\epsilon \\right) + \\frac{\\epsilon}{n} = 0 \\;\n\\end{align}\nwhere we have used the fact that $s_{k,j}=s_{k,k+1}$ for $j>k$. Again, the difference between these equations results in a relation between $x_k$ and all $x_j$ with $j<k$ (for $k=3, ..., N-1$). Since $s_{k-1,j}=s_{k,j}$ for $j<k-1$, the difference can be written\n\\begin{align}\n\\frac{\\epsilon}{n} \\left( \\frac{1}{x_{k-1}}-\\frac{1}{x_k} \\right) + P x_{k-1} + (T-P) x_k - (1-P)\\left( 1-\\sum_{j=1}^{k-1} x_j \\right) = 0 \\;\n\\end{align}\nwhere the score differences, $s_{k-1,k-1}-s_{k,k-1}= P-S=P$, $s_{k,k+1}-s_{k,k}= T-P$ and, $s_{k-1,k}-s_{k,k+1}= P-R=P-1$, have been used. This results in an expression for $x_k$ in terms of all $x_j$ (with $j<k$), \n\\begin{align}\nx_{k} = f_B\\left(x_{k-1},1-\\sum_{j=1}^{k-1} x_j\\right) \n\\end{align}\nwith the function $f_B(x,w)$ defined by\n\\begin{align}\n\\label{eqn_fB}\nf_B(x,w) &=\n\\frac {1} {2 (T - P)} \\left (P x - (1 - P) w + \\frac {\\epsilon'} {x} \\right) \\left (\n- 1 + \n\\sqrt {1 + \\frac{4 (T-P) \\epsilon'}{\\left (P x - (1 - P) w + \\frac {\\epsilon'} {x} \\right)^2}} \\; \\right) \\,\n\\end{align}\nFinally, using the same approach for the last two variables, $x_{n-1}$ and $x_{n}$, the equation for the relation between these can be written\n\\begin{align}\n\\frac{\\epsilon}{n} \\left( \\frac{1}{x_{n-1}}-\\frac{1}{x_n} \\right) + P x_{n-1} + (T - 1) x_n = 0 \\;\n\\end{align}\nThe last variable can then be expressed as\n\\begin{align}\nx_{n} = f_C(x_{n-1}) \n\\end{align}\nwith the function $f_C(x)$ defined by\n\\begin{align}\n\\label{eqn_fC}\nf_C(x) = \n\\frac{1}{2(T-1)}\\left(P x+\\frac{\\epsilon' }{x}\\right) \\left(-1+\n\\sqrt{1 + \\frac{4 (T-1) \\epsilon'}{\\left (Px + \\frac {\\epsilon'} {x} \\right)^2}} \\; \\right) \\,\n\\end{align}\nThis means that we can recursively express the fixed point abundancies $x_k$ of strategies $k=2,...,n$ in terms of $x_1$, using Eqs.~(\\ref{eqn_fA}), (\\ref{eqn_fB}), and (\\ref{eqn_fC}). Together with the normalisation constraint, this results in an equation with only one variable, $x_1$, that determines the fixed points. In other words: Summation over all $x_k$ and subtraction of 1 gives us a function of $x_1$, $F(x_1)$, with zeroes at the fixed points, \n\\begin{align}\nF(x_1) = \\sum_{k=1}^{n} x_k - 1 \\;\n\\end{align}\nNote that we can extend the function $F$ also to the case $\\epsilon=0$, since all functions $f_A, f_B$ and $f_C$ are identical to zero in that case and we simply get $F(x_1)=x_1-1$, capturing the Nash equilibrium fixed point $x_1=1$. (Note, though, that in this extension none of the other fixed points at $\\epsilon=0$ are captured; any $x_k=1$ defines a fixed point for zero mutation rate, but all except the first one are of no interest for us.)\n\nAll functions $f_A, f_B$ and $f_C$ are continuous and bounded, since $g(y)=y(1-\\sqrt{1+1\/y^2})$ is. We also see that $g(y)\\rightarrow 0$ as $|y| \\rightarrow \\infty$. Thus, $F(x)$ being composed of these functions is continuous in the interval $[0,1]$. \n\nAs an example, Figure~\\ref{fig_F} illustrates $F(x)$ for $n=N+1=6, T=1.05, P=0.15$ with four different choices of the mutation rate $\\epsilon$. The graph shows that as the mutation rate increases the fixed point at $x_1=1$ moves to the left and also that new fixed points may emerge. Most importantly, though, the graph indicates a discontinuous change of the function for $x$ below about $0.95$, when going from $\\epsilon=0$ to $\\epsilon>0$. In order to show that the fixed point always moves continuously from $x_1=1$, we need to be sure that any such a discontinuity is bounded away from $x_1=1$.\n\n\n\\begin{figure}\n\\center\n\\includegraphics[width=10cm]{Fgraph2.pdf}\n\\caption{\\small \\label{fig_F} The function $F(x)$ for the 5-round PD game with $T=1.05, P=0.15$, and four different mutation rates $\\epsilon$.}\n\\end{figure}\n\nWe already know that for $\\epsilon=0$ there is a zero in $x=(1,0,...,0)$. We want to show that as $\\epsilon$ increases, this fixed point, characterised by $F(x_1)=0$ for $x_1=1$, continuously moves into the unit interval, $0<x_1<1$. We accomplish this by performing a series expansion of $F(x)$ around $x=1$ and $\\epsilon=0$, by showing that $F'(x_1)=1$ for $x_1=1$, that $F$ is increasing with $\\epsilon$ in the neighbourhood of $x_1=1$, and that the coordinates $x_k$ ($k>1$) of the fixed point move continuously with $x_1$ when sufficiently close to 1.\n\nThe first part is straightforward: At zero mutation rate, $\\epsilon=0$, we have already noted that $F(x_1)=x_1-1$, and thus the derivative $F'(x_1)=1$.\n\nNext, we need to show that, at least sufficiently close to the fixed point and for sufficiently small $\\epsilon$, $F$ increases with $\\epsilon$. We do this term by term in the sum of $F$. First assume that the point $x_1$ is close to fixed point value $1$ at zero mutation rate, {\\em i.e.}, $x_1=1-\\delta$, and that $\\epsilon$ is small, so that $\\epsilon << P$ and $\\delta << P$. The first term in $F$ is $x_1$ and does not depend on $\\epsilon$. The second term, $x_1$, is given by $f_A$, \n\\begin{align}\nx_2 &= f_A(1-\\delta) = \n\\frac {1} {2 (T - P)} \\left (P-\\delta + \\frac {\\epsilon'} {1-\\delta} \\right) \\left (\n-1 + \\sqrt {1 + \\frac{4 (T-P) \\epsilon'}{\\left (P-\\delta + \\frac {\\epsilon'} {1-\\delta} \\right)^2}} \\; \\right) = \\nonumber \\\\\n& = \\frac{\\epsilon'}{P} \\left( 1 + \\frac{\\delta}{P} + O(\\epsilon') \\right) + O(\\epsilon')^2+ O(\\delta)^2 \\;\n\\end{align}\nThis term thus increases with $\\epsilon'$. For the next term $x_3$, using Equation~(\\ref{eqn_fB}), we find that to first order in $\\epsilon'$ and $\\delta$,\n\\begin{align}\nx_3 &= f_B(x_2, 1-x_1-x_2) = f_B\\left(\\frac{\\epsilon'}{P}, \\delta-\\frac{\\epsilon'}{P} \\left( 1 + \\frac{\\delta}{P} \\right)\\right) = \\nonumber \\\\ \n& = \\frac{\\epsilon'}{P} \\left(1 + \\frac{2-P}{P}\\delta+ O(\\delta)^2 \\right) + O(\\epsilon')^2 + O(\\delta)^2 \\;\n\\end{align}\nMore generally, for $2<k<n$, repeatedly using Equation~(\\ref{eqn_fB}), we find \n\\begin{align}\nx_k &= \\frac{\\epsilon'}{P} \\left(1 + \\frac{k-P(k-1)}{P}\\delta+ O(\\delta)^2 \\right) + O(\\epsilon')^2 + O(\\delta)^2 \\;\n\\end{align}\nand finally, using Equation~(\\ref{eqn_fC}), that \n\\begin{align}\nx_{n} &= \\frac{\\epsilon'}{P} \\left(1 + \\frac{(N-1)-P(N-2)}{P}\\delta+ O(\\delta)^2 \\right) + O(\\epsilon')^2 + O(\\delta)^2 \\;\n\\end{align}\nFrom the linearisation we can conclude that when $x_1$ is sufficiently close to 1, {\\em i.e.}, $\\delta << P$, the change in $F(x_1)$ from an increase of $\\epsilon'$ is given by the $n-1$ terms $x_2, ..., x_n$, or $dF(x_1)\/d\\epsilon' \\sim (n-1)\/P > 0$. Adding the fixed point constraint, $F(x_1)=0$, to the linearisation determines $x_1$ and thus $\\delta$ in terms of $\\epsilon'$. To first order in $\\epsilon$ the fixed point is given by\n\\begin{align}\nx_1 &\\sim 1 - (n-1) \\frac{\\epsilon'}{P} + O(\\epsilon)^2 = 1-(n-1) \\frac{\\epsilon}{n P} + O(\\epsilon)^2 \\;, \\text{and} \\\\\nx_k &\\sim \\frac{\\epsilon}{n P} + O(\\epsilon)^2 \\; \\text{ for } k=2,..., n\\;\n\\end{align}\nwhere we have switched back to the original $\\epsilon$. This analysis shows that as $\\epsilon$ increases from 0, the fixed point gradually moves into the unit interval. \n\nSince the fixed point is changed continuously as the mutation rate increases from $\\epsilon=0$, the eigenvalues of the Jacobian also change continuously. The fixed point at $\\epsilon=0$ is stable, with the largest eigenvalue being $\\lambda_\\text{max}=-P<0$, and we can conclude that for sufficiently small $\\epsilon>0$, the real part of the largest eigenvalue remains negative.\nFrom this we can conclude that \\emph{the fixed point associated with the Nash equilibrium in the finitely repeated Prisoner's Dilemma remains stable in the simple strategy set if the mutation rate is sufficiently small}. This concludes the proof of Proposition 1.\n\n\\end{appendix}\n\n\\bibliographystyle{ieeetr}\n\\makeatletter\n\\renewcommand\\@biblabel[1]{#1. }\n\\makeatother\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\newcommand{\\TA}[1]{{\\color{red1} \\textbf{TA:}\\ \\texttt{#1}}}\n\\newcommand{\\CO}[2]{{\\color{red1} \\textbf{#1:}\\ \\texttt{#2}}}\n\\newcommand{\\SO}[1]{{\\color{red1} \\sout{#1}}}\n\nA large fraction of the sources that have so far been detected in the very high energy (VHE, E$_{\\gamma}>100$~GeV) regime are blazars \\citep{tevcat}. These active galactic nuclei (AGN) have relativistic jets orientated in such a way that the emission is beamed towards Earth, providing a large boost to the observed flux.  Despite this, our ability to detect blazars at energies greater than some 10's to 100's of GeV is hindered by the existence of the Extragalactic Background Light (EBL), which attenuates gamma-rays by way of pair production with lower energy photons (\\cite{Nikishov1962}, \\cite{1967PhRv..155.1404G}, \\cite{1967PhRv..155.1408G} and \\cite{2013APh....43..112D} for a recent review). \n\nThe EBL is thought to consist of UV-FIR (far infrared) radiation from stars (with UV-optical representing the main attenuation band for gamma-rays at energies below 100~GeV), either produced directly from their surfaces or via reprocessing by dust within their host galaxies, with an additional component coming from optically bright AGN \\citep{fermiEBL}. The evolution of the EBL density is of considerable interest as it probes models of galaxy and star formation\/evolution. However, direct measurements are difficult due to the presence of foreground zodiacal and Galactic light \\citep{2001ARA&A..39..249H}. Additionally, measurements of the EBL in the local universe ($z=0$) provide little information about the evolution of the EBL through past epochs. This evolution is especially important when considering the attenuation of gamma-rays from distant sources and its understanding therefore represents one of the major science goals of the \\textit{Fermi}-LAT space based instrument, which is able to observe gamma-rays from 100~MeV to above 300~GeV \\citep{fermi}.\n\nFor AGN at a redshift of \\textit{z}=1, the attenuation from the EBL quickly becomes significant above 10~GeV, therefore the \\textit{Fermi}-LAT instrument is well suited to observe, and place constraints on, the effect of the EBL. By modelling the un-absorbed part of a given AGN spectrum, it is possible to obtain an indication of the intrinsic source spectrum. Combining this with different EBL models, the amount of EBL absorption, and therefore density, can be inferred by evaluating the magnitude of the softening of the spectrum above $\\sim$10~GeV. This method has been adopted to derive EBL limits by several authors (\\cite{fermiEBL}, \\cite{2012Sci...338.1190A} \\cite{2013A&A...550A...4H}, \\cite{2015ApJ...812...60B}, \\cite{2016arXiv160901095M}, \\cite{2016A&A...590A..24A} and \\cite{2016arXiv161009633M}). \n\n\n\n\n \n\n \nAn alternative method which can be used to set limits on the EBL is to measure the maximum observed photon energy from distant gamma-ray emitters, such as AGN or gamma-ray bursts (GRBs). The observation of high energy photons from large redshifts, where the universe is thought to be optically thick to gamma-rays, can challenge current EBL models.  Currently, the most distant AGN detected in the VHE band are the gravitationally-lensed blazar B0 0218+357 ($z=0.94$) \\citep{2016arXiv160901095M} and the FSRQ PKS 1441+25 ($z=0.939$)  \\citep{Ahnen2015}, both detected with the MAGIC telescope, with the latter also detected by VERITAS \\citep{2015arXiv150807251B}. Additionally, the second \\textit{Fermi}-LAT catalogue of hard sources (50~GeV - 2~TeV) contains a large sample of 271 AGN with a redshift range out to 2.1 \\citep{2fhl}. The two GRBs, GRB 090902B ($z=1.822$) \\citep{GRB090902B} and GRB 08916C ($z=4.35$) \\citep{GRB080916C}, have also aided in providing limits to the EBL density.\n\nIn an attempt to increase the sample of VHE AGN, previous work (\\cite{Arm2015}, \\cite{Arm2016}) has focused on developing clustering methods to identify extragalactic sources with $E_{\\gamma} > 100$~GeV in the \\textit{Fermi}-LAT data set.  The chosen algorithm, \\textsc{dbscan}, was first proposed in \\cite{dbscan} and was designed to efficiently detect clusters of arbitrary shape in noisy data sets. The algorithm works by evaluating the number of events around a detected photon within the data set; if these exceed a given number then they are added to a cluster and in turn evaluated, accumulating further events to the cluster that satisfy the same condition\\footnote{For the \\textsc{dbscan} sources presented in this work that were found in \\cite{Arm2016}, the search radius around a given point was 0.4 degrees, the PSF at 100~GeV, and the number of required events started at 2 and scaled upward with the integrated Galactic background emission}. The \\textsc{dbscan} algorithm makes no assumption regarding the underlying source spectrum. The ability to identify clusters within sparse noisy data sets made \\textsc{dbscan} an ideal choice for analysing \\textit{Fermi}-LAT VHE data. So far this method has proven to be successful, discovering emission with $E_{\\gamma} > 100$~GeV from 45 sources. The sources detected are all either AGN or unassociated sources. One of the aims of this work is to draw attention to six high redshift AGN, PKS 0426-380 ($z=1.11$), 4C +55.17 ($z=0.899$), Ton 116 ($z=1.088$), PG 1246+586 ($z=0.857$), RBS 1432 ($z=1.508$) and TXS 1452+516\\footnote{The Blazar TXS 1452+516 ($z=1.522$) was found in a separate work [in prep], using the same method but for $50~\\textrm{GeV} > E_{\\gamma}>2$~TeV.}, that were found using this method and could have implications for EBL limits. \n\nThis paper is organised as follows, in Section \\ref{S:EBL} a brief overview of the EBL and EBL models used in this work will be given. In Section \\ref{S:srcInfo} a description of each source of interest, with a focus on the reliability of their redshift determination, will be presented. Section \\ref{S:data} will describe the data selection process and Sections \\ref{S:specAn} and \\ref{S:VHEphoton} will discuss the spectral and `highest energy photon' analyses respectively. Finally Section \\ref{S:conc} will conclude and summarise the results presented. \n\n\n\n\n\\section{EBL Models}\n\\label{S:EBL}\n\n\nSince direct measurements of the EBL are hindered due to the presence of foreground emission, and because this reveals little about the evolution on cosmological scales, it is necessary to use complex models to estimate the density of EBL photons and therefore the absorption of gamma-rays. There is currently a wide range of models available which adopt different methods to determine the evolution of the EBL density as a function of redshift. \n\nIn this work, five models have been considered: \\cite{Gilmore2012} (GIL12) which uses a semi-analytic model of the star formation rate, initial mass function and dust extinction, \\cite{Fink2010} (FIN10) and \\cite{Kneiske2010} (K\\&D10) which use forward evolution models based on observations of \\textit{Spitzer, ISO, Hubble, COBE, BLAST} and \\textit{GALEX} from which the cosmic star formation rate is inferred, and finally \\cite{FRAN08} (FRA08) and \\cite{Dominguez11} (DOM11) which use backward evolution models to model the redshift evolution of the luminosity function of galaxies based on number counts. For an overview of the different model types, see \\cite{2013APh....43..112D}.\n\nThe absorption of distant gamma-rays depends on the optical depth of the EBL, expressed as  $\\tau(E,z,n)$, which is dependent on the gamma-ray energy ($E$), the distance ($z$) and the density ($n$) of EBL photons, the last being defined by the choice of model. The total is based on an integral along the line of sight to the target source. For an individual source, the spectrum is therefore attenuated as a function of its energy such that \n\n\\begin{equation}\n\\frac{dN}{dE}_{obs}=e^{-\\tau (E,z,n)} \\frac{dN}{dE}_{int}\n\\label{E:eblspec}\n\\end{equation}\n\nwhere $dN\/dE_{obs}$ is the observed spectrum and $dN\/dE_{int}$ is the intrinsic\/unabsorbed spectrum. For the models used in this work, tabulated data of $\\tau$ as a function of energy, available for a range of redshift values, were obtained from online resources\\footnote{\\url{http:\/\/www.phy.ohiou.edu\/~finke\/EBL\/} for \\cite{Fink2010}, \\url{http:\/\/www.ias.u-psud.fr\/irgalaxies\/cib.php} for \\cite{Kneiske2010}, \\url{http:\/\/physics.ucsc.edu\/~joel\/EBLdata-Gilmore2012} for \\cite{Gilmore2012} and \\url{http:\/\/www.astro.unipd.it\/background\/} for \\cite{FRAN08}. Data from Dominguez were obtained from \\url{https:\/\/github.com\/me-manu\/eblstud\/blob\/master\/ebl\/ebl_model_files\/tau_dominguez10.dat} due to the inactivity of the original link in \\cite{Dominguez11}.}. By interpolating these data sets, a function for determining $\\tau$ based on the energy and redshift is obtained. \n\n\n\\section{Source Information}\n\\label{S:srcInfo}\nAn overview of each source analysed in this work is given here, with a focus on determining the most reliable redshift from literature. The sources are divided into two categories: 1) DBSCAN VHE sources, which were identified in \\cite{Arm2016} to have significant emission in the 100~GeV - 3~TeV energy range and 2) a sample of 10 2FHL objects with the highest quoted redshifts within the catalogue which also coincided with the redshift range of the first sample. Two sources where however excluded from the selection, MG4 J000800+4712 which is quoted to have a redshift of 2.1 but is found with \\textit{z}=0.28 in all other literature, and PKS 0823-223 due to its proximity to the Galactic plane. The second category of sources was included in order to obtain a more reliable limit for the EBL when considering the fit to the intrinsic and absorbed spectra (see Section \\ref{S:specAn}).\n\n\n\n\\begin{table*}\n\\centering\n\\begin{tabular}{lcccccr}\nSource & Type & RA & Dec & $z$ & $z_{err}$ & $z_{ref}$ \\\\ \\hline\nPKS 0426-380 & BLL & 67.18 & 37.93 & 1.111 & ... & \\cite{heidt} \\\\\n4C +55.17 & FSRQ & 149.41 & 55.38 & 0.901 & 0.00019 & \\cite{sdssdr13} \\\\\nTon 116 & BLL & 190.80 & 36.48 & 1.066 & 0.00150 & \\cite{sdssdr7} \\\\\nPG 1246+586 &  BLL & 192.08 & 58.34 & 0.847 & 0.00169 & \\cite{sdssdr7} \\\\\nRBS 1432 & BLL & 221.50  & 36.35 & 1.565 & +0.275-0.125 & \\cite{Richards} \\\\\nTXS 1452+516 & BLL & 223.61 & 51.41 & 1.522 & 0.00183 & \\cite{sdssdr12} \\\\\nGB6 J0043+3426 & FSRQ & 10.95 & 34.44 & 0.966 & ... & \\cite{shaw12} \\\\\nB0218+357 & FSRQ & 35.27 & 35.94 & 0.960 & ... & \\cite{Lawrence1996} \\\\\nPKS 0235+164 &  BLL & 39.66 & 16.62 & 0.940 & ... & \\cite{2massbll} \\\\\nMG2 J043337+2905 & BLL & 68.41 & 29.10 & 0.970 & ... & \\cite{bzcat} \\\\\nPKS 0454-234 & FSRQ & 74.26 & -23.41 & 1.003 & ... & \\cite{Stickel1989} \\\\\nPKS 0537-441 & BLL & 84.70 & -44.09 & 0.892 & ... & \\cite{peterson} \\\\\nTXS 0628-240 & BLL & 97.75 & -24.11 & 1.6 & +0.10-0.05 & \\cite{photometryz} \\\\\nOJ 014 & BLL & 122.86 & 1.78 & 1.148 & ... & \\cite{esonewz} \\\\\nPKS B1424-418 & FSRQ & 216.98 & -42.11 & 1.522 & 0.002 & \\cite{southernZ} \\\\ \nB2 2114+33 & BLL & 319.06 & 33.66 & 1.596 & ... & \\cite{shaw} \\\\\\hline\n\\end{tabular}\n\\caption{Summary table of source data used in this study.}\n\\label{T:srcsum}\n\\end{table*}\n\n\n\n\\subsection{DBSCAN VHE Sources}\n\\label{SS:DBSCANsrc}\n\n\\begin{enumerate}\n\\item[\\textbf{PKS 0426-380:}]\nClassed as a BL Lac in the 3FGL and situated in the southern hemisphere at RA=67.18$^{\\circ}$, Dec=-37.93$^{\\circ}$. The 2FHL redshift is quoted as $z=1.111$ as found in \\cite{heidt}, where the existence of a closer source at $z=0.559$ implies micro-lensing may be present. This source has previously been noted for producing VHE photons in \\cite{Tanaka2013} and \\cite{Neronov2015}, which both use Pass 7 data. The new Pass 8 data have reclassified the energy of these photons. \n\\item[\\textbf{4C +55.17:}] \nDefined as a FSRQ, and located in the northern hemisphere at RA=149.41$^{\\circ}$, Dec=55.38$^{\\circ}$, in the 2FHL the redshift is quoted as 0.899, most likely obtained from \\cite{improvedsdss}. However, more recent measurements from the 13th SDSS data release \\citep{sdssdr13} finds a redshift of 0.901$\\pm$0.00019 which will be adopted in this work. This source was considered in \\cite{Neronov2015} where a redshift of 0.8955 was adopted.\n\\item[\\textbf{Ton 116:}]\nA BL Lac located in the northern hemisphere at RA=190.80$^{\\circ}$, Dec=36.46$^{\\circ}$ with a redshift of 0.0 in the 2FHL indicating that no redshift was available at the time or that it was determined to be unreliable. However, measurements from SDSS indicate a redshift of 1.066$\\pm$0.00150 (data release 7, \\citep{sdssdr7}) or 1.182$\\pm$0.00132 (data release 13 \\citep{sdssdr13}); both have data flags associated with the spectrum. This source was also considered in \\cite{Neronov2015} in which a redshift of 1.065 was used. Our work adopts the SDSS data release 7 value, which is the same within errors. \n\\item[\\textbf{PG 1246+586}]\nA BL Lac located in the northern hemisphere at RA=192.08$^{\\circ}$, Dec=58.34$^{\\circ}$ and with an unknown or uncertain redshift in the 2FHL. Measurements from SDSS (data release 7 \\citep{sdssdr7}) show a redshift of 0.847$\\pm$0.00169. This source was considered in \\cite{Neronov2015} who applied the same redshift as used here.\n\\item[\\textbf{RBS 1432:}]\nA northern hemisphere BL Lac at RA=221.50$^{\\circ}$, Dec=36.35$^{\\circ}$ for which the 2FHL again defines the redshift as unknown or uncertain. The redshift in the literature is generally accepted as 1.565 which originates from \\cite{Richards}, although the redshift is quoted as being between 1.440 and 1.840.\n\\item[\\textbf{TXS 1452+516:}]\nThis source was found in a comparative study of DBSCAN and the 2FHL [paper in prep], using the same energy range and observational period. As it was not found to be significant in VHE range (E$_{\\gamma}>$100~GeV) range, as with the rest of this sample, we do not claim in to be a VHE source. However as it is not in the 2FHL (but is in the 3FGL) it is included in this list. It is a BL Lac located in the northern hemisphere at RA=223.61$^{\\circ}$, Dec=51.41$^{\\circ}$ and has a redshift of 1.522$\\pm$0.00183, which originates from the 12th SDSS data release \\citep{sdssdr12}.\n\n\n\n\\end{enumerate}\n\n\n\\subsection{High-$z$ 2FHL Sample}\n\\label{SS:highz2FHL}\n\n\\begin{enumerate}\n\\item[\\textbf{GB6 J0043+3426:}]\nA northern hemisphere FSRQ at RA=10.95$^{\\circ}$, Dec=34.44$^{\\circ}$ with a redshift of 0.966 based on observations at the W. M. Keck Observatory and presented in \\cite{shaw12}.\n\\item[\\textbf{B0218+357:}]\nA northern hemisphere FSRQ at RA=35.27$^{\\circ}$, Dec=35.94$^{\\circ}$ which is thought to be a gravitationally lensed object with two images of the FSRQ at a redshift of 0.960 \\citep{Lawrence1996} and a lensing source at 0.685 \\citep{glensz1}. This FSRQ has been observed with MAGIC \\citep{2016arXiv160901095M} and was used to determine an EBL correction factor\n\\item[\\textbf{PKS 0235+164:}]\nA northern hemisphere BL Lac at RA=39.66$^{\\circ}$, Dec=16.62$^{\\circ}$ which has an associated redshift of 0.940 as stated in \\cite{2massbll}. \n\\item[\\textbf{MG2 J043337+2905:}]\nA northern hemisphere BL Lac at RA=68.41$^{\\circ}$, Dec=29.10$^{\\circ}$ with a redshift of 0.970 found in \\cite{bzcat}.\n\\item[\\textbf{PKS 0454-234:}]\nA southern hemisphere FSRQ at RA=74.26$^{\\circ}$, Dec=-23.41$^{\\circ}$. The redshift is given as 1.003 based on absorption lines in \\cite{Stickel1989}. \n\\item[\\textbf{PKS 0537-441:}]\nA southern hemisphere BL Lac at RA=84.71$^{\\circ}$, Dec=-44.09$^{\\circ}$ with a redshift of 0.892 \\citep{peterson}.\n\\item[\\textbf{TXS 0628-240:}]\nA southern hemisphere BL Lac at RA=97.75$^{\\circ}$, Dec=-24.11$^{\\circ}$ which has an associated photometric redshift of 1.6$^{+0.10}_{-0.05}$ \\citep{photometryz} or limits of 1.239 to 1.91 based on spectroscopic measurements \\citep{shaw}. In this work we use the photometric redshift. \n\\item[\\textbf{OJ 014:}]\nA southern hemisphere BL Lac at RA=122.86$^{\\circ}$, Dec=1.78$^{\\circ}$ with a redshift of 1.148 from \\cite{esonewz}.\n\\item[\\textbf{PKS B1424-418:}]\nA southern hemisphere FSRQ at RA=216.98$^{\\circ}$, Dec=-42.11$^{\\circ}$ with redshift 1.522$\\pm$0.002 from \\cite{southernZ}.\n\\item[\\textbf{B2 2114+33:}]\nA southern hemisphere BL Lac located at RA=319.06$^{\\circ}$, Dec=33.66$^{\\circ}$. A redshift is given as 1.596 based on significant broad emission feature identified with C$_{\\textrm{IV}}$ consistent with a weak bump in the far blue at Ly$\\alpha$.  It is suggested that if the emission feature is false that this would then at least represent a lower limit \\citep{shaw}. \n\\end{enumerate}\n\nA summary of all these sources and their basic data can be found in Table \\ref{T:srcsum}.\n\n\n\\section{Data Selection}\n\\label{S:data}\n\n\nEach source presented in this work was evaluated using the Pass 8 processed data \\citep{pass8},  which provides several improvements over the previous Pass 7 reprocessed data set \\citep{pass7rep}\\footnote{Pass 7 was previously used to evaluate some of the sources presented in this work (\\cite{Tanaka2013} and \\cite{Neronov2015}).}. These include better energy and angular resolution, increased effective area, an extended observable energy range (around 10~MeV \\textless~$E_{\\gamma}$ \\textless 3~TeV) and better background characterisation, resulting in improved point source sensitivity. As a result of this re-evaluation of the data, the energies of many events have been updated. \n\nTo investigate each source, 8 years of data ranging from August 2008 to September 2016 (MET  239557417 - 494930839) and with energies from 100~MeV to 3~TeV were downloaded from the \\textit{Fermi} data servers\\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/}. All source class events were retained for both front and back converting photons. Additionally, the recommended filter expression \\textit{`DATA\\_QUAL$>$0 \\&\\& LAT\\_CONFIG==1'} was applied in order to remove any sub-optimal data affected by spacecraft events. Finally a zenith cut of 90$^{\\circ}$ was implemented in order to remove any $\\gamma$-rays induced by cosmic-ray interactions in the Earth's atmosphere. \n\n\n\\section{Spectral Analysis}\n\\label{S:specAn}\nTo investigate the effect of the EBL absorption on the spectrum of each source presented in Section \\ref{S:srcInfo}, a binned likelihood analysis was performed. An initial fit was obtained for each source below a threshold energy $E_{max}$, here defined to be the point where there is less than 0.1\\% of emission attenuated by the EBL, in order to represent the intrinsic spectrum. For a source at redshift $\\sim$1 this corresponds to an energy of $\\sim$10~GeV. The values for each source can be found in Table \\ref{T:SPEC}. \n\nFrom each source data set comprising a 10$^{\\circ}$ region of interest (ROI) around the source, an initial model is constructed from the 3FGL which includes sources out to a further 10$^{\\circ}$ fixed to the 3FGL values. The normalisations and spectral indices of the sources within the ROI were allowed to vary. Also included were the extragalactic diffuse emission model (iso\\_P8R2\\_SOURCE\\_V6\\_v06.txt) with a free normalisation, and the Galactic diffuse template (gll\\_iem\\_v06.fits), which was multiplied by a power law in energy and the normalisation of which was free to vary. The sources of interest were modelled with both a power law and a log parabola in the form\n\n\\begin{equation}\n\\frac{dN}{dE}=N_{0} \\left(\\frac{E}{E_0}\\right)^{\\Gamma}, ~~~~~~~ \\frac{dN}{dE}=N_{0} \\left( \\frac{E}{E_0}\\right) ^{-(a+b~\\textrm{log}(E\/E_0))}\n\\label{E:pwllgp}\n\\end{equation}\n\nwhere $N_0$ is the normalisation, $\\Gamma$ is the spectral index of the power law, $a$ is the log parabola index, $b$ is the curvature and $E_0$ is a scaling factor. In the analysis of each source, all parameters excluding $E_{0}$ were left free to vary and a full binned likelihood analysis was performed for each ROI, returning the best fit model. The best fit intrinsic model parameters can be found in Table \\ref{T:SPEC}. \n\n\\begin{table}\n\\centering\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{lcccc}\n \t\t\t&\t$N_{0}$ $\\times 10^{-12}$\t\t\t & $\\Gamma$  & ($a,b$) &$E_{max}$   \\\\\nSource &  ph~cm$^{-2}~$s$^{-1}~$MeV$^{-1}$ &  (PL) & \t\t\t (LGP) & (GeV) \\\\ \\hline\nPKS 0426-380 \t\t\t& \t54.09 \t& -1.99 \t&  \t1.99  \t&  9.00 \\\\%LGP\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.06\t\t&  \\\\ \n4C +55.17 \t\t\t\t&  10.64\t& -1.90\t&\t\t1.92\t\t&\t10.58 \\\\% LGP\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.06\t\t&  \\\\ \nTon 116 \t\t\t\t\t&  1.57\t\t&\t-1.62\t&\t\t1.64\t\t&\t 9.42 \\\\ \n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t-0.02\t\t&  \\\\ \nPG 1246+586 \t\t\t&  4.27\t\t&\t-1.81\t&\t\t1.82\t\t&\t 11.20 \\\\\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t-0.02\t\t&  \\\\ \nRBS 1432 \t\t\t\t\t& 1.21\t\t&\t-1.69\t&\t\t1.64\t\t&\t 6.96\\\\\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.07\t\t&  \\\\ \nTXS 1452+516 \t\t\t& 3.3 \t\t&\t-1.98\t&\t\t1.99\t\t&\t 6.89\\\\ \n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.02\t\t&  \\\\ \nGB6 J0043+3426\t\t& 2.62\t\t&\t-1.90\t&\t\t1.91\t\t&\t 9.94\\\\ \n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t-0.03\t\t&  \\\\ \nB0218+357 \t\t\t\t& 10.25\t\t&\t-2.25\t&\t\t2.29\t\t&\t 10.13\\\\ \n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.04\t\t&  \\\\ \nPKS 0235+164 \t\t\t&  14.80\t&\t-2.08\t&\t\t2.12\t\t&\t 10.17\\\\\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.07\t\t& \\\\ \nMG2 J043337+2905 \t&  3.26\t\t&\t-2.00\t&\t\t1.99\t\t&\t 9.91\\\\\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.03\t\t&  \\\\ \nPKS 0454-234 \t\t\t& \t27.09\t&\t-2.12\t&\t\t2.19\t\t& 9.65\t \\\\\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.08\t\t& \\\\ \nPKS 0537-441 \t\t\t&  23.95\t&\t-2.03\t&\t\t2.07\t\t&\t 10.66\\\\% LGP\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.05\t\t&\\\\ \nTXS 0628-240 \t\t\t& \t3.23\t\t&\t-1.73\t&\t\t1.72\t\t&\t 8.12\\\\ \n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.02\t\t&  \\\\ \nOJ 014 \t\t\t \t\t\t&  3.84\t\t&\t-2.00\t&\t\t2.01\t\t&\t 8.75\\\\\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.04\t\t&  \\\\ \nPKS B1424-418 \t\t&  47.58\t&\t-2.09\t&\t\t2.15\t\t&\t 6.75\\\\\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.05\t&\\\\\nB2 2114+33 \t\t\t\t&  2.42\t\t&\t-1.64 &\t\t1.57\t\t&\t 6.56\\\\\n\t\t\t\t\t\t\t\t&\t\t\t\t&\t\t\t&\t\t0.10\t\t&  \\\\  \\hline\n\\end{tabular}\n}\n\\caption{Intrinsic model parameters for each source derived from the binned likelihood analysis below $E_{max}$. The parameters correspond to those found in equation \\ref{E:pwllgp}. PL = power law model and LGP = log-parabola model.}\n\\label{T:SPEC}\n\\end{table}\n\n\nIn order to investigate the level of EBL absorption present, the initial fit was scaled with $e^{-\\alpha \\cdot \\tau (E,z,n)}$, as in Equation \\ref{E:eblspec}, where the scaling factor $\\alpha$ has now been introduced (here a value of $\\alpha=1$ would return the original EBL model absorption level). By taking the scaled spectral model, and fitting this to to data between 100~MeV and 300~GeV, which includes both the absorbed and unabsorbed sections of the spectrum, it is possible to scan through values of $\\alpha$ and identify the optimal level of EBL absorption (as was performed in \\cite{2012Sci...338.1190A} and \\cite{2013A&A...550A...4H}; a more detailed description of the process is given in Appendix \\ref{A:1}).\n\nFor each EBL scaling factor, a likelihood was generated for the attenuated power law and log parabola models. The latter was chosen if the Test Statistic (TS) between the two models, with the maximum likelihood value of $\\alpha$ applied, was greater than 16 (the power law is the null hypothesis, i.e. TS=2log$[\\mathcal{L}_{\\textrm{max}}(Log Parabola)\/\\mathcal{L}_{\\textrm{max}}(Power Law)]$)\\footnote{This is the value used by the \\textit{Fermi}-LAT collaboration in the 3FGL to chose a given spectral model over a power law \\cite{2015ApJS..218...23A}.}. In Figures \\ref{F:sed1} and \\ref{F:sed2} the delta logLikelihood can be seen for each energy bin of the full spectral energy distribution along with the intrinsic spectrum (black dotted line) which is calculated based on events with energies below that indicated by the vertical dotted line. The black dashed line shows the absorbed spectrum calculated using the chosen EBL model (in this case GIL12) and the solid black line the resulting the best maximum likelihood fit when GIL12 is modified by a scaling factor $\\alpha$. The individual EBL scaling factors for each source are given in the Figures.\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.89\\textwidth]{images\/sedfit_p1.pdf}\n\\caption{Spectral  energy distribution for each source between 100~MeV and 300~GeV. For each energy bin the delta log likelihood determined from the binned likelihood analysis is plotted, where all but the normalisation is fixed to the best fit intrinsic model. The vertical dotted line shows the boundary between the intrinsic and absorbed spectrum. Also shown are the intrinsic spectrum fit (black dotted line), the intrinsic model including EBL absorption, pure model (GIL12, black dashed line) and best fit modification to GIL12 (black solid line). Lastly, as an indication of whether these sources would be observable by future ground-based gamma-ray observatories, the predicted sensitivity of the future CTA observatory is shown (\\url{https:\/\/www.cta-observatory.org\/science\/cta-performance\/}).}\n\\label{F:sed1}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.89\\textwidth]{images\/sedfit_p2.pdf}\n\\caption{See Figure \\ref{F:sed1} caption.}\n\\label{F:sed2}\n\\end{figure*}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{images\/EBLcorrGIL.pdf}\n\\caption{Combined TS (solid black) from each individual source (grey lines) as a function of EBL scaling factor. This is based on the GIL12 model for which the best fit EBL scaling factor corresponds to $\\alpha=0.96\\pm0.05$ where the error is the $1\\sigma$ standard deviation. }\n\\label{F:alphascan}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lcc}\n\t\t & \\multicolumn{2}{c}{EBL correction Factor} \\\\ \nModel \t\t\t\t\t\t& Best Fit \t& LogParabola \\\\ \\hline\n\\citep{Fink2010} \t\t& 1.31$\\pm$0.08 & 1.24$\\pm$0.08\\\\\n \\citep{Kneiske2010} & 1.31$\\pm$0.07 & 1.24$\\pm$0.07\\\\\n \\citep{Gilmore2012} \t& 0.95$\\pm$0.05 & 0.90$\\pm$0.05 \\\\\n \\citep{Dominguez11} & 1.85$\\pm$0.11 & 1.75$\\pm$0.11 \\\\\n \\citep{FRAN08} \t\t& 1.85$\\pm$0.11 & 1.71$\\pm$0.11\\\\ \\hline\n\\end{tabular}\n\\caption{Derived EBL correction factor for each of the 5 models used in this work. The `Best Fit' column is based on whichever of the power law or log parabola models is a better fit for each source, while the last column uses only log parabola spectral models for every source, regardless of whether this is the best fit model.}\n\\label{T:eblcorr}\n\\end{table}\n\n\nFrom the likelihood distribution for each source as a function of the EBL correction factor, the TS was calculated with respect to the null hypothesis that there is no absorption from the EBL (i.e. TS=2log$[\\mathcal{L}(\\alpha)\/\\mathcal{L}(\\alpha=0)]$). These distributions were then summed in order to obtain a combined TS distribution, allowing an overall EBL correction to be calculated. The result of this process applied to the GIL12 model can be seen in Figure \\ref{F:alphascan} from which an optimal correction factor of $\\alpha_{\\textrm{GIL12}}=0.95\\pm0.05$ was derived (1 standard deviation uncertainty).  The EBL normalisation for the other models gives $\\alpha_{\\textrm{K\\&D10}}=1.31\\pm0.07$, $\\alpha_{\\textrm{FIN10}}=1.31\\pm0.08$, $\\alpha_{\\textrm{DOM11}}=1.85\\pm0.11$ and $\\alpha_{\\textrm{FRA08}}=1.85\\pm0.11$ for the redshift range $0.897<z<1.596$. It is clear that the GIL12 model is consistent with the results, while it is suggested that the other models underestimate the EBL density. \n\nAs noted above, the quoted errors are statistical only. One of the main sources of systematic error in this analysis comes from the choice of spectral model, where it is clear that an incorrect choice will strongly affect the derived EBL scaling factor. This was addressed by \\cite{2012Sci...338.1190A} who concluded that the assumption that spectral cutoffs existed in the EBL absorption energy range was acceptable. However, \\cite{2012Sci...338.1190A} assumed each source to be best represented by a log parabola. We have therefore repeated our analysis using only log parabola models, which provides a lower limit for the EBL correction factor. This leads to the following derived scaling factors: $\\alpha_{\\textrm{GIL12}}=0.90\\pm0.05$, $\\alpha_{\\textrm{K\\&D10}}=1.24\\pm0.07$, $\\alpha_{\\textrm{FIN10}}=1.24\\pm0.08$, $\\alpha_{\\textrm{FRA08}}=1.71\\pm0.11$ and $\\alpha_{\\textrm{DOM11}}=1.75\\pm0.11$. The derived EBL scaling factors are summarised in Table \\ref{T:eblcorr}.\n\n\n\nIn each panel of Figures \\ref{F:sed1} and \\ref{F:sed2}, the expected sensitivity for 50 hours of observation of the two arrays of the future ground based gamma-ray observatory, CTA, are shown (publicly available at \\url{https:\/\/www.cta-observatory.org\/science\/cta-performance\/}, see \\cite{2013APh....43..171B} for description of method). Since CTA will possess a much improved sensitivity compared to current telescopes, it will be well placed to develop further our understanding of the EBL by observing AGN in the range $0<z<1$ and at higher energies than is possible with \\textit{Fermi}. This work shows that several of our sources should be detectable with 50 hours of observation, while others may also be detectable with deeper observations. Further observations with CTA would aid in reducing systematics in the choice of best fit spectral model.  \n\n\n\\section{VHE Photons}\n\\label{S:VHEphoton}\n\nAn alternative method for probing the EBL is to identify the highest energy photons associated with distant AGN. Here, detection of a photon from a high-$z$ source at energies where the density of the EBL is thought to attenuate the signal almost entirely can bring EBL models into question. In this section, we investigate whether this is the case for the sources presented in the DBSCAN VHE sample which have already exhibited significant emission above 100~GeV (which therefore excludes TXS 1452+516), despite their large redshifts.   \n\nFor this analysis, it is particularly important to ensure that the VHE photons are genuinely associated with the AGN. To verify this, a subset of data was selected for each source covering energies above 100~GeV only for a 1$^{\\circ}$ region around the source position (much larger than the PSF at these energies). The \\textit{Fermi} science tool \\textsc{gtsrcprob} was applied to this dataset in order to calculate the probability that each photon is associated with a source in or around the ROI. This is based on the likelihood of each photon and convolves a source model with the instrument response functions. For the source model, the best fit derived in the previous section without EBL absorption was used. Additionally, in order to account for diffuse components such as the Galactic diffuse model, the \\textit{Fermi} science tool \\textsc{gtdiffrsp} was applied in order to add a diffuse response to the input data. The energies of the highest energy photons and the probabilities of their being associated with the source in question can be seen in Table \\ref{T:HEPprob}. It is worth noting that, if the attenuation from the EBL is included, these probabilities will reduce. For example, the highest energy photon probability associated with PKS 0426-380 drops from 0.99983 to 0.99899 when the GIL12 model is included. For out purpouse we assume this to be a negligible difference.\n\n\\begin{table}\n\\centering\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{lcccccc}\nsource & E1 & P(E1) & $\\sigma$(E1) & E2 & P(E2) & $\\sigma$(E2) \\\\ \\hline\nPKS 0426-380 & 122.0 & 0.99983 & 3.76 & 115.8 & 0.99991 & 3.92 \\\\\n4C +55.17 & 150.6 & 0.99990 & 3.89 & 94.1 & 0.99840 & 3.16 \\\\\nTon 116 & 148.8 & 0.96454 & 2.10 & 132.2 & 0.99941 & 3.44 \\\\\nPG 1246+586 & 251.8 & 0.99983 & 3.76 & 198.1 & 0.99826 & 3.13 \\\\\nRBS 1432 & 140.4 & 0.99798 & 3.09 & 111.0 & 0.99932 & 3.40 \\\\ \\hline\n\\end{tabular}\n}\n\\caption{The highest energy photons and their associated probability of originating from the proposed source.}\n\\label{T:HEPprob}\n\\end{table}\n\nAs an initial consideration, the maximum energies found in Table \\ref{T:HEPprob} and the redshift of the proposed sources were compared to objects in the rest of the 2FHL (where the 2FHL catalogue contains the maximum energy detected). This can be seen in Figure \\ref{F:fazsteck} in which different levels of optical depth, starting at  $\\tau=1$ (where the universe becomes optically thick to gamma-rays) have also been shown. What is immediately apparent is that the majority of 2FHL sources lie within, or close to, the optically thin ($\\tau<1$) region, while the sources found in the DBSCAN VHE source sample are pushing out to larger values of $\\tau$.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{images\/fazio_lin.pdf}\n\\caption{Distribution of the detected energy compared to the redshift for all 2FHL sources with an available redshift, and the DBSCAN sources presented in this work. This is often referred to as the Fazio Stecker relation \\citep{faziostecker}. Also shown are the different levels of EBL absorption, where the area covers the four different models used in this work.}\n\\label{F:fazsteck}\n\\end{figure}\n\nOf the VHE sources, the largest derived optical depth corresponds to $\\tau_{GIL12}=4.39$ for RBS 1432, indicating that a photon was detected despite 98.8\\%  of the flux being attenuated. In order to evaluate the probability of observing the highest energy photons from each source, Monte Carlo simulations were performed as in \\cite{fermiEBL}. Here, spectral parameters were drawn from a distribution based on the best fit intrinsic model and corresponding errors from section \\ref{S:specAn}, including the spectral EBL absorption from the GIL12 model. Then using the \\textit{Fermi} science tool \\textsc{gtobsim},  eight year observations between 10~GeV and 1~TeV were simulated, taking into account the instrument response functions and spacecraft pointing history. Using 1000 simulations for each source, the number of observations of photons of a given energy was determined. An example of this for PG 1246+586 can be seen in Figure \\ref{F:maxEdist}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{images\/maximumEeg2.pdf}\n\\caption{Distribution of simulated maximum photon energy for source PG 1246+586.}\n\\label{F:maxEdist}\n\\end{figure}\n\nFrom these simulations, it is evident that, in the 8 years of \\textit{Fermi}-LAT observations, the highest energy photons detected are within expectation, with all but the 251~GeV photon from PG 1246+586 at 2.2$\\sigma$ found at an energy less than one standard deviation from the simulated mean value. This result confirms that found in Section \\ref{S:specAn}, that the observations presented in this work strengthen our confidence in the EBL models. \n\nGiven that it has been shown that photons with energies greater than 100~GeV are expected from these sources, this also provides further evidence that they should be observable with ground based gamma-ray observatories which, with their much greater sensitivity, could provide stricter limits on EBL models. An important caveat here is that the analysis carried out in this work makes use of the near continuous \\textit{Fermi}-LAT observations over 8 years. The assumption is therefore that these sources are non-variable over that time scale, or more importantly that the spectrum remains unchanged. The published variability index from the 3FGL suggests that this assumption is reasonable for all but PKS 0426-380 and RBS 1432, with the former exhibiting strong evidence for variability. Additionally, many of the sources from Section\\ref{SS:highz2FHL} also show evidence of variability. It is expected that this may bias the results presented in this and the previous section. In a following paper, the effect of variability on the estimate of the EBL will be studied in detail.\n\n\n\n\\section{Discussion and Conclusion}\n\\label{S:conc}\n\n\n\n\\textit{Fermi}-LAT's excellent energy coverage enables it to be used to observe attenuation effects from the EBL. In this work, analysis of 16 high redshift sources has been presented, in which the EBL absorption is apparent. By modelling the energy spectrum from 100~MeV to around $\\sim$10~GeV (redshift dependent), over which the effect of the EBL is negligible, it is possible to obtain an idea of the intrinsic spectrum of the source of interest.  Using this, the full data set from 100~MeV to 300~GeV and the absorption from a selection of EBL models, scaled by a correction factor $\\alpha$, a measurement of the EBL scale was derived. By using a combination of power law and log parabola models, a best fit to the combined TS of each individual source revealed scaling factors of $\\alpha_{\\textrm{GIL12}}=0.95\\pm0.05$, $\\alpha_{\\textrm{K\\&D10}}=1.31\\pm0.07$, $\\alpha_{\\textrm{FIN10}}=1.31\\pm0.08$, $\\alpha_{\\textrm{FRA08}}=1.85\\pm0.11$ and $\\alpha_{\\textrm{DOM11}}=1.85\\pm0.11$ for the redshift range $0.897<z<1.596$. All models apart from GIL12 predict an EBL density less than suggested by this study. As the choice of spectral model introduces an unknown uncertainty, we also derived a conservative EBL scaling factor based solely on the log parabola model, From this it was found that $\\alpha_{\\textrm{GIL12}}=0.90\\pm0.05$ $\\alpha_{\\textrm{K\\&D10}}=1.24\\pm0.07$, $\\alpha_{\\textrm{FIN10}}=1.24\\pm0.08$, $\\alpha_{\\textrm{FRA08}}=1.71\\pm0.11$ and $\\alpha_{\\textrm{DOM11}}=1.75\\pm0.11$.\n\n\nAs mentioned previously, the source B0 218+357 has been observed using the ground-based MAGIC telescopes \\citep{2016arXiv160901095M}. They used a combination of MAGIC and \\textit{Fermi}-LAT data for a period of activity between the 11th of July 2014 and the 6th of August 2014, and modelling B0 218+357 with a power law, to obtain EBL correction factors for the following models\\footnote{In the analysis of B0 218+357, a larger range of models were investigated. We present only their results for the models also considered in this work.}: $\\alpha_{\\textrm{FIN10}}=0.91 \\pm 0.32_{stat} \\pm 0.19_{sys}$, $\\alpha_{\\textrm{FRA08}}=1.19 \\pm 0.42_{stat} \\pm 0.25_{sys}$, $\\alpha_{\\textrm{DOM11}}=1.19 \\pm 0.42_{stat} \\pm 0.25_{sys}$ and $\\alpha_\\textrm{{GIL12}}=0.99 \\pm 0.34_{stat}~^{+0.15 sys}_{-0.18 sys}$.\n\nIn addition to B0 218+357, the MAGIC collaboration used 11.8 hours of observations from a 2014 flare of 1ES 1011+496 ($z=0.212$) \\citep{2016A&A...590A..24A} to obtain a scaling factor for the EBL models  $\\alpha_{\\textrm{DOM11}}=1.07~ (-0.20,+0.24)_{stat+sys}$ and $\\alpha_{\\textrm{FRA08}}=1.14~ (-0.14,+0.09)_{stat}$. This was followed by a study using 8 high frequency peaked BL Lac and 4 FSRQs within the redshift range  of $0.031<z<0.944$ (which also includes the observations of the two aforementioned sources) where a scaling factor of $\\alpha_{\\textrm{DOM11}}=0.99~(-0.56,+0.15)_{stat+sys}$ was derived \\citep{2016arXiv161009633M}\n\nCorrection factors for the EBL have also been derived for less distant AGN by the HESS collaboration \\citep{2013A&A...550A...4H}. Here data for 7 bright Blazars (Mrk 421 [\\textit{z}=0.031], PKS 2005-489 [\\textit{z}=0.071], PKS 2155-304 [\\textit{z}=0.116], 1ES 0229+200 [\\textit{z}=0.14], H 2356-309 [\\textit{z}=0.165], 1ES 1101-232 [\\textit{z}=0.186], and 1ES 0347-121 [\\textit{z}=0.188]), over different periods of activity, were modelled in a similar way to that presented here. By fitting a range of different spectral models to the observed data, scanning the EBL correction parameter space, a value was obtained that maximises the likelihood. In the paper, only the \\citet{FRAN08} model was considered, for which a correction factor of $\\alpha_{\\textrm{FRA08}}=1.27_{-0.15_{stat}}^{+0.18_{stat}} \\pm 0.24_{sys}$ was obtained.\n\n\nThe most encompassing study using ground based observations was performed in \\citet{2015ApJ...812...60B}. Here spectra from 38 sources in the redshift range $0.019<z<0.64$ were evaluated, using data from HESS, MAGIC, VERITAS, Whipple, ARGO-YBJ, TACTIC, HEGRA, Tibet and CAT. Using these data sets, along with local constraints taken from \\citet{2013APh....43..112D}, the spectra were fitted with power law, log parabola and exponential cutoff powerlaw models, resulting in a derived EBL spectrum. Also presented were correction factors for current EBL models: \n$\\alpha_{\\textrm{GIL12}}=1.13\\pm0.07$ \n$\\alpha_{\\textrm{K\\&D10}}=1.52\\pm0.14$, \n$\\alpha_{\\textrm{FIN10}}=1.48\\pm0.07$, \n$\\alpha_{\\textrm{FRA08}}=1.05\\pm0.07$ and \n$\\alpha_{\\textrm{DOM11}}=1.16\\pm0.05$.\n\n\nOne of the main scientific objectives of the \\textit{Fermi}-LAT instrument was to measure the effect of the EBL attenuation of high energy gamma-rays. In \\citet{2012Sci...338.1190A}, data from the first 46 months of observation were used to model the EBL attenuation of 150 BL Lac type blazars. The method to determine the scaling of the EBL was the same as presented in this work, first modelling the intrinsic blazar spectrum and then fitting the higher energies, up to 500~GeV. In doing this, scaling factors for a large range of EBL models were determined. For the models used in this work, the corresponding weighted averages were found to be $\\alpha_{\\textrm{GIL12}}=0.67\\pm0.14$ $\\alpha_{\\textrm{K\\&D10}}=0.90\\pm0.19$, $\\alpha_{\\textrm{FIN10}}=0.86\\pm0.23$, $\\alpha_{\\textrm{FRA08}}=1.02\\pm0.23$ and $\\alpha_{\\textrm{DOM11}}=1.02\\pm0.23$. Also presented was the binned results for the \\citet{FRAN08} model, giving $\\alpha_{\\textrm{FRA08}}=1.18^{+0.94}_{-0.81}$ ($z<0.2$), $\\alpha_{\\textrm{FRA08}}=0.82^{+0.41}_{-0.30}$ ($0.2<z<0.5$) and $\\alpha_{\\textrm{FRA08}}=1.29^{+0.43}_{-0.36}$ ($0.5<z<1.6$). \n\nThe results from \\citet{2016arXiv160901095M}, \\citet{2016A&A...590A..24A}, \\citet{2016arXiv161009633M}, \\citet{2013A&A...550A...4H}, \\citet{2012Sci...338.1190A} and \\citet{2015ApJ...812...60B} are shown in comparison to the results derived in this paper in Figure \\ref{F:resfra}. It can be seen that, for all but the GIL12, our results imply a larger density of the EBL at high redshifts. For all models we find a larger estimate than found by the \\textit{Fermi}-LAT in \\citet{2012Sci...338.1190A}; however they are consistent with the results found by the MAGIC collaboration \\citep{2016arXiv160901095M}. When considering the binned \\textit{Fermi}-LAT results for the FRA08 model, the last bin which encompasses the redshift range under investigation in this paper, is consistent with our results for log parabola models. When considering the EBL model types, it is suggested that the backward evolution models DOM11 and FRA08, which are based on the luminosity function of galaxies based on number counts, underestimate the EBL at large redshifts. This is not an unexpected result as these models match observed lower limits which are based on incomplete surveys. In a following paper we will attempt to investigate these trends in more detail, taking into account variability and source class.\n\n\n\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{images\/results.pdf}\n\\caption{Derived scaling factors for the EBL models considered in this work as a function of redshift. The HESS measurements are taken from  \\citep{2013A&A...550A...4H} (red diamond), the MAGIC from  \\citep{2016arXiv160901095M}, \\citep{2016A&A...590A..24A} and \\citep{2016arXiv161009633M} (purple square), the \\textit{Fermi}-LAT points from \\citep{2012Sci...338.1190A} and the combined ground based points from \\citep{2015ApJ...812...60B} (orange circles). The black upward triangle is the value derived in this work using a combination of power law and log parabola models, whereas the green downward triangle shows the value derived when only considering log parabola models.}\n\\label{F:resfra}\n\\end{figure*}\n\n\nThe second part of this work focused on  investigating the highest energy photons from the VHE sources found with DBSCAN. By considering the often used Fazio-Stecker relation, it seemed that these sources were pushing out to large optical depths. Upon further investigation using Monte Carlo simulations, it was shown that the observed photon energies were not unexpected (less than 1 standard deviation from the mean energy or $2.2\\sigma$ for PG 1246+586), illustrating the pitfalls of relying solely on the Fazio-Stecker relation.\n\nGiven the expected performance of the future ground based observatory, CTA, which will be able to place even stronger constraints on the EBL, it was shown that several of the sources presented in the work should be observable by CTA (See Figure \\ref{F:sed1} and \\ref{F:sed2}). Indeed, the observation of VHE emission from PKS 0426-380, 4C +55.17, Ton 116, PG 1246+586 and RBS 1432 is expected, providing further evidence that these sources should be observable with CTA.\n\n\n\\section*{Acknowledgments}\n\n TPA would like to acknowledge the support from the UK Science and Technology Facilities Council grant ST\/K501979\/1. AMB would like to acknowledge the financial support of Durham University. This work has made use of publicly available \\textit{Fermi}-LAT data from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASAs Goddard Space Flight Center. Finally, we would like to thank the reviewer for their insightful comments which helped improve the quality of this work.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n Slipping contact between solids is responsible for various important dynamic phenomena, such as dynamic jamming, friction-induced vibrations, brake squeal, stick-slip and sprag-slip oscillations. For many engineering applications, it is crucial to understand and predict the onset of these phenomena, as they are associated with impulsive contact forces, mechanical damage, unwanted noise and intensive wear of the interacting surfaces \\citep{ibrahim1994friction,sinou2007mode,butlin2013friction,mo2013effect,le2013friction,kruse2015influence}. \n \nMoving solids with contact interactions are often modelled as rigid multi-body systems with unilateral point contacts subject to Coulomb friction. Even within this simple framework, it is an open question what kind of dynamic phenomena may occur during slip motion and how they can be  predicted. The only exception is the dynamics of one single point contact in two dimensions, which has been analyzed in detail by several authors. Such systems may undergo a singularity called dynamic jam during slip \\citep{Genot1999}, in addition to the classical slip-stick, slip-reversal and slip-liftoff transitions. The same systems may also exhibit other forms of contact dynamics such as self-excited bouncing motion (a.k.a inverse chattering) \\citep{paper2} and impulsive contact forces or \"impact without collision\" \\citep{LeSuanAn_book,Zhao15} albeit transition into such modes does not occur during slip motion.\n\nThe goal of the present paper is to review these results and to investigate systematically the generic transitions and singularities of  multicontact systems. We perform a detailed analysis of systems with two slipping point contacts in two dimensional physical space, and many new phenomena are discovered. Our new findings include\n\\begin{itemize}\n\\item the sudden onset of impulsive contact forces (i.e. impact without collision or IWC) blocking slip motion\n\\item the onset of self-excited oscillations during slip, which may either remain microscopic, or they may grow exponentially to a macroscopic scale\n\\item convergence to two different types of codimension 2 singular manifolds where the dynamics becomes ill-defined within rigid body theory. One of these singularities is analogous to dynamic jamming, whereas the other one does not have an analogue in the single-contact case.\n\\end{itemize}\nWe then turn to systems with arbitrary number of point contacts and highlight some general properties of the dynamic jamming singularity.\n\nThere are two classical ways to deal with the piecewise smooth character of rigid unilateral contacts and Coulomb friction. The appealing mathematical framework of complementarity problems and measure differential inclusions offers a unified approach \\citep{Leine_book,Brogliato1999} or one can  distinguish between various contact modes (slip, stick and separation) each of which implies different smooth behavior. Since our goal includes predicting and understanding transitions between various contact modes, it is a natural choice here to follow the contact mode-based approach. \n\nIt is well-known that rigid body theory cannot be developed into a complete and consistent modeling framework in the presence of contacts. Solution inconsistency and indeterminacy have been known for a long time \\citep{Jellet1872,Painleve1895}, and many examples of this phenomenon have been found and studied more recently \\citep{champneys2016painleve}. In addition,  the solution may become ill-defined at attractive singular points \\citep{Genot1999,szalai2014nondeterministic}. The limitations of rigid models can often be resolved by contact regularization: the rigid contacts are replaced by compliant ones, and the behavior of the regularized system is studied in the quasi-rigid limit $\\epsilon\\rightarrow 0$ of a compliance parameter $\\epsilon$. We will make extensive use of contact regularization in the present paper.\n\nIn Sec. 2, we introduce our notation (Sec. 2.1) and review relevant results of the contact mode-based approach and of contact regularization wherein we follow the previous works \\citet{champneys2016painleve} and \\citet{varkonyi2017dynamics}. We also introduce a conceptual model system in Sec. 2.4 (inspired by a similar model system from \\citet{nordmark2017dynamics}), which is used to illustrate certain phenomena by numerical simulation as well as to construct examples of other phenomena analytically. \n\nSec. 3 focuses on the transitions of systems with slipping contacts. Existing results regarding the case of $n=1$ point contact are reviewed first, and we also formulate a theorem, which shows how and to what extent these systems stay away from those states where rigid models become ill-defined. We then turn towards systems with $n=2$ point contacts where a detailed analysis of transitions is given. Finally, we  discuss the case of general $n$, and a theorem related to dynamic jamming is generalized to this case. The paper is closed by a Discussion section, and by an appendix where some technical proofs are presented. \n\n\n\n\n\n\n\n\\section{Instantaneous behaviour of systems with sliding contacts}\n\n\n\\subsection{General formulation}\nWe consider the motion of an autonomous  mechanical system consisting of one or more rigid elements in two dimensions, subject to any number of ideal constraints (i.e. bilateral, frictionless connections) as well as $n\\geq 1$ unilateral point contacts. We assume Coulomb friction with given coefficients of friction at all contact points. We use lowercase letters for scalars and vectors, whereas capital letters denote matrix quantities. The state of the system is characterized by a vector of generalized coordinates $q$ as well as the generalized velocities $\\dot q$.\nThe continuous dynamics of such a system is given by a differential equation of the form\n\\begin{align}\n\\ddot{q}=f(q,\\dot q)+G_T(q)\\lambda_T+G_N(q)\\lambda\n\\label{eq:basiceq}\n\\end{align}\nwhere $\\lambda\\in\\mathbb{R}^n$ is a column vector of $n$ non-negative normal contact forces $\\lambda_i$ and $\\lambda_T$ is a similar vector containing tangential contact forces. The contact forces are determined by unilateral contact constraints and by the friction model. (Note that the equation above is not valid during impacts, since then the contact forces are impulsive.)\n\nLet $x_i(q)$ and $z_i(q)$ denote tangential and normal coordinates of the point contacts such that $z_i=0$ corresponds to a closed contact and $z_i>0$ to an open one, and $z_i=\\dot z_i=\\dot x_i=0$ means stick. \n\nWe will examine slip motion, which means that contacts are initially closed with nonzero tangential velocity. We assume without loss of generality that $\\dot x_i>0$ for all $i$. Then, the tangential contact forces $\\lambda_{t,i}$ become dependent on the normal forces $\\lambda_{i}$ by Coulomb's law\n$$\n\\lambda_{t,i}=-\\mu_i(q)\\lambda_i\n$$\n(where $\\mu_i$  is known and velocity-independent). In this case, \\eqref{eq:basiceq} can be written as\n\\begin{align}\n\\ddot{q}=f(q,\\dot q)+G(q)\\lambda\n\\label{eq:basiceq2}\n\\end{align}\n\nIf the functions $x_i(q)$, $z_i(q)$, $f(q,\\dot q)$, and $G(q)$ are known, then one determine two sets of equations of the form\n\\begin{align}\n\\ddot x&=a(q,\\dot q)+K(q)\\lambda\n\\label{eq:xdyn}\\\\\n\\ddot z&=b(q,\\dot q)+P(q)\\lambda\n\\label{eq:zdyn}\n\\end{align}\ndescribing the dynamics of the contact points.\nHere, $z=[z_1,z_2,...,z_n]^T$, $x=[x_1,x_2,...,x_n]^T$. The vectors $a,b\\in\\mathbb{R}^n$ and the $n$-by-$n$ matrices $K,P$ depend on the system in question. $P$ is sometimes referred to as the Delassus matrix of the system. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=4cm] {rod.pdf}\n\\caption{Rod slipping at one endpoint}\\label{fig:rod}\n\\end{center}\n\\end{figure}\n\n\\textbf{Example:} in the case of a  rigid rod (Fig. \\ref{fig:rod}) of length $2l$, mass $m$, radius of inertia $\\rho$ under gravitational force $mg$, with one endpoint in unilateral contact with a flat, horizontal surface, we can use the generalized coordinates $q=[x_c,z_c,\\phi]^T$. The equations of motion are\n\\begin{align}\n\\ddot x_c&=-m^{-1}\\mu\\lambda\\\\\n\\ddot z_c&=-g+m^{-1}\\lambda\\\\\n\\ddot\\phi&=m^{-1}\\rho^{-2}l\\lambda(-\\cos\\phi+\\mu\\sin\\phi)\n\\end{align}\nand the contact coordinates can be expressed as\n\\begin{align}\nx&=x_c+l\\cos\\phi\\\\\nz&=z_c-l\\sin\\phi\n\\end{align}\n The Lagrange equations and the kinematics of the rod then imply \\citep{Genot1999}\n\\begin{align}\na&=-l\\dot\\phi^2\\cos\\phi\\\\\nb&=l(\\dot\\phi^2\\sin\\phi-g)\\label{eq:rodb}\\\\\nK&=m^{-1}\\left[-1+l^2\\rho^{-2}(-\\mu\\sin^2\\phi+\\cos\\phi\\sin\\phi)\\right]\\\\\\\\\nP&=m^{-1}\\left[1+l^2\\rho^{-2}\\cos^2\\phi-\\mu\\cos\\phi\\sin\\phi\\right]\\label{eq:rodP}\\\\\n\\end{align}\nwhere $\\mu$ is the coefficient of friction . Note that $a,b,K,P$ are all scalars, for now $n=1$.\n\nIn what follows, we denote the $j^{th}$ column vector of $K,P$ by $k_j$ and $p_j$ and the $i^{th}$ element of vectors $a,b,k_j,p_j$ by $a_i,b_i,k_{ij},p_{ij}$.\nWe drop arguments like $t,q(t),\\dot q(t)$ for brevity. We will use a lower index $_{init}$ to denote initial conditions at some initial time $t_{init}<0$ and index $_{0}$ for quantities evaluated at $t=0$. Typically, we will choose initial conditions in such a way that transitions from slip occur at $t=0$. \n\nSlip motion requires an initial state $q_{init},\\dot q_{init}$ inducing $z_{init}=\\dot z_{init}=0$, $\\dot x_{init}>0$ (where the inequality should be satisfied by all elements of vector $x$). In this situation, each contact point may undergo sustained slip in the positive $x_i$ direction (contact mode S) or lift-off (contact mode F). Stick and negative slip  are not possible since $\\dot x_{init}>0$. Each one of these contact modes has a corresponding equality and an inequality constraint:\n\\begin{align}\n\\ddot z_i=0\n\\label{eq:slip-equality}\\\\\n\\lambda_i\\geq 0\n\\label{eq:slip-inequality}\n\\end{align}\n for S mode and \n \\begin{align}\n\\lambda_i=0\n\\label{eq:liftoff-equality}\\\\\n\\ddot z_i\\geq 0\n\\label{eq:liftoff-inequality}\n\\end{align}\nfor F mode. The contact mode of a full system can be represented by an $n$-letter word from the two-letter alphabet $\\{S,F\\}$. We can check the feasibility of all $2^n$  contact modes by solving \\eqref{eq:zdyn} together with the $n$ equality constraints, and by testing the inequality constraints of the contact modes. The fact that we may find multiple or no feasible contact mode is known as Painlev\\'e paradox \\citep{champneys2016painleve}. This kind of limitation of rigid models is the primary motivation for contact regularization techniques introduced below.\n\n\\subsection{Contact regularization}\n\nThe non-uniqueness and non-existence of solution associated with rigid models is often resolvable by contact regularization techniques. A regularized contact model allows small penetrations $z_i<0$ at the contact points and assumes a certain relation between the contact force and the penetration. Here, we will use a standard, linear, unilateral viscoelastic contact law \n\\begin{align}\n\\lambda_i=\n\\begin{cases}\n0 & if\\;\\;  z_i>0\\\\\n\\max(0,-\\epsilon^{-2}k_iz_i-\\epsilon^{-1}\\nu_i\\dot z_i) & if\\;\\;  z_i \\leq 0\n\\end{cases}\n\\label{eq:contactlaw}\n\\end{align}\nwhere, $k_i$ and $\\nu_i$ are scaled stiffness and damping coefficients, whereas $\\epsilon$ is a scaling factor. Now we will say that a contact is closed if $\\lambda_i>0$ (as opposed to $z_i=0$ in the rigid case). Note that the contact model is smooth as long as the contact is closed. \n\n\nThen, the analysis of a rigid model is replaced by the quasi-rigid limit $\\epsilon\\rightarrow 0$ of the regularized model, which often yields deeper insight into the behavior of the system. The regularized model is a slow-fast system in which $a,b,P,K$ evolve on a slow time scale, and the internal dynamics of contacts ($z,\\dot z$) represents the fast subsystem. In Section 2 of the paper, we will focus exclusively on the fast dynamics and we do not even specify  the full slow-fast dynamics or the reduced problem. Then in Section 3, we discuss sudden transitions of the fast dynamics induced by the slow dynamics.\n\nWhen all $n$ contact points are closed ($\\lambda_i>0$ for $i=1,2,...,n$), then  the equations \\eqref{eq:zdyn} and \\eqref{eq:contactlaw} induce the fast dynamics:\n\\begin{align}\ng' = \n\\begin{bmatrix}\nO_n & I_n \\\\\n -PK & -PN  \n\\end{bmatrix}\ng+\n\\begin{bmatrix}\no_n,\\\\b  \n\\end{bmatrix}\n\\label{eq:contactdynamics_generaln}\n\\end{align}\nwhere $'$ represents differentiation with respect to fast time $\\tau=\\epsilon^{-1}t$; $O_n$ and $I_n$ are zero and identity matrices of size $n\\times n$, $o_n$ is a column vector of $n$ zeros, and\n\\begin{align}\ng=\n\\epsilon^{-2}\\begin{bmatrix}\nz\\\\\nz'  \n\\end{bmatrix},\nK=\n\\begin{bmatrix}\nk_1&0&...&&0\\\\\n0&k_2&0&...&0\\\\\n\\vdots&&&&\\\\\n0&...&&0&k_n \n\\end{bmatrix},\nN=\n\\begin{bmatrix}\n\\nu_1&0&...&&0\\\\\n0&\\nu_2&0&...&0\\\\\n\\vdots&&&&\\\\\n0&...&&0&\\nu_n \n\\end{bmatrix}\n\\end{align}\nNote that $g$ is a rescaled vector depending on those variables, which represent motion in the directions of the contact normals. Such a reduction is possible because the fast dynamics of $g$ decouples from all other components of the dynamics including the dynamics of $x$ and its time derivative. \n\nWhen the regularized model is used, we identify slip motion with motion during which all contacts are closed and the fast dynamics remains stationary at the invariant point \n$$\ng=-\\begin{bmatrix}\nO_n & I_n \\\\\n -PK & -PN  \n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\no_n,\\\\b  \n\\end{bmatrix}\n$$\nof the linear system \\eqref{eq:contactdynamics_generaln}. Such an invariant point may be asymptotically stable or unstable depending on the eigenvalues of the coefficient matrix. Hence, regularization shows that certain modes involving slipping contacts are unstable and do not occur in practice, which can eliminate some forms of solution non-uniqueness. \n\n\n\nAs we point out later, contact regularization has other benefits as well. Most importantly, we will encounter situations, when the invariant point is asymptotically unstable and  $z_i$ diverges towards $-\\infty$. Such an event induces rapidly increasing contact force $\\lambda_i$. We will identify this kind of behaviour with the impact without collision (IWC) phenomenon of rigid models. Detailed analysis of dynamics induced by diverging contact forces (similar to \\citet{Zhao15} in the case of $n=1$) is however beyond the scope of this work.   \n\n\n\\subsection{Systems with a single point contact}\\label{sec:1contact-instantaneous}\nIn the case of $n=1$\\citep{champneys2016painleve}, $a,b,P,K,\\lambda$ are scalars. Slip motion means that \n\\begin{align}\n\\lambda=-b\/P\n\\label{eq:lambda-1contact}\n\\end{align}\naccording to \\eqref{eq:zdyn} and \\eqref{eq:slip-equality}. Then the feasibility condition \\eqref{eq:slip-inequality} is satisfied if $b$ and $P$ have opposite signs. Similarly, liftoff is feasible if $b>0$. Since both $b$ and $P$ may have any sign, either one, both or none of the two modes may be feasible (Table \\ref{table:1contact}). The issue of non-existence is resolved by considering a third type of solution involving impulsive contact forces. Whenever $P<0$, an impact without collision (IWC) may occur: an impact occurs, which causes instantaneous jump in tangential velocity into stick or reverse slip \\citep{LeSuanAn_book,Zhao15,hogan2017regularization}. The regularized contact model \\eqref{eq:contactlaw} predicts rapid divergence of $\\lambda$ to $\\infty$ in such situations, which is consistent with the assumption of an IWC in the $\\epsilon\\rightarrow 0$ limit.\n\nStability analysis using regularized contact model reveals that slip is stable if and only if $P>0$, yet stability analysis is not able to eliminate non-uniqueness if $P<0<b$.\\\\\n\n\\textbf{Example:} the parameters $b,P$ of the sliding rod (Fig.\\ref{fig:rod}) given by \\eqref{eq:rodb},\\eqref{eq:rodP} may have either sign. In particular, $b$ is always negative if $\\dot\\phi$ is small, but may become positive for large $\\dot\\phi$, whereas $P$ is always positive for small $\\mu$ and may have either sign if $\\mu$ is large. \n\n\\begin{table} \n\\centering\n\\caption{Feasibility, and stability of contact modes and IWC in the case of one point contact}\n\\label{table:1contact}\n\\begin{tabulary}{0.6\\textwidth}{ccc}\n\\hline\n& b\\textless 0 & b\\textgreater 0                                                            \\\\ \n\\hline\nP\\textless 0    & IWC         & \\begin{tabular}{c}IWC\\\\ liftoff\\\\ slip (unstable)\\end{tabular}   \n\\\\ \\hline\nP\\textgreater0 & liftoff     & slip (stable)                                                             \\\\ \\hline\n\\end{tabulary}\n\\end{table}\n\n\\subsection{Systems with two point contacts}\\label{sec:2contact-instantaneous}\n\nIn the case of 2 contacts, there are 4 possible contact modes: SS, FF, SF, and FS. The feasibility and stability analysis of these systems was recently investigated by  \\citet{varkonyi2017dynamics}. The analysis reveals that the most important model parameters are the angle $\\beta$ between vector $b$ and the unit vector $[1,0]$ (measured in the direction of $[0,1]$), as well as the analogous angles $\\gamma_1$ and $\\gamma_2$ associated with the vectors $p_1$, $p_2$ (Figure \\ref{fig:gammabeta}). The feasibility of the four contact modes depends exclusively on these three parameters. For example, FF motion (liftoff at both points) is feasible if $b_1,b_2>0$, i.e. if both points accelerate in the absence of contact forces away from the contact surfaces. This can also be expressed as $0\\leq\\beta\\leq\\pi\/2$. Contact mode FS is feasible if we have positive contact force at point 2, and positive normal acceleration at point 1 i.e. if\n\\begin{align}\n-b_2\/p_{22}>0 \\label{eq:FS1}\\\\\nb_1-p_{21}b_2\/p_{22}>0 \\label{eq:FS2}\n\\end{align}\nAs we know from Sec. \\ref{sec:1contact-instantaneous}, the stability of FS additionally requires $p_{22}>0$. These conditions depend exclusively on the angles $\\gamma_1,\\beta$ (Fig. \\ref{fig: 2contact feasibility}(a)). The feasibility and stability conditions of SF are analogous.\n\nFor simultaneous slip at both contact points, the contact force is determined by \\eqref{eq:zdyn}:\n\\begin{align}\n\\lambda=-P^{-1}b\n\\label{eq:lambda-2contact}\n\\end{align}\nThe feasibility condition \\eqref{eq:slip-inequality} is satisfied exactly when $-b$ is in the cone spanned by $p_1$ and $p_2$, which can be expressed in terms of the angles $\\beta,\\gamma_1,\\gamma_2$ as illustrated by Fig. \\ref{fig: 2contact feasibility}(b). The figure also shows the region of stability obtained by eigenvalue analysis. This region will soon be analyzed in more detail (see Fig. \\ref{fig:SSrange} below) \n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=6cm] {gammabeta.pdf}\n\\caption{Definition of the angles $\\gamma_i$ ($i=1,2$) and $\\beta$.}\\label{fig:gammabeta}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=13cm] {feasibility.pdf}\n\\caption{Feasibility and stability of the contact modes SF (a) and SS (b). Grey areas or volumes show feasibility and hatching marks regions where motion is also stable. } \\label{fig: 2contact feasibility}\n\\end{center}\n\\end{figure}\n\nThe feasibility of impulsive contact forces (IWC) has also been examined in \\citet{varkonyi2017dynamics}. It was found that there are three different types of IWC since point 1, point 2 or both points may exhibit impulsive contact forces. Unlike in the case of $n=1$, contact regularization does not yield uniform answer with respect to the feasibility of IWCs: in certain cases, the feasibility of the IWC may depend on the stiffness and dissipation coefficients $k_1,k_2,\\nu_1,\\nu_2$ of the regularized contact model.\n\nThe stability analysis of the contact modes  via contact regularization yields the following results:\n\\begin{enumerate}\n\\item the stability of the SF and FS modes is  determined by $\\gamma_1,\\gamma_2$ and it is independent of any other parameter. The same is true for SS in a large range of $\\gamma_1,\\gamma_2$\n\\item in certain ranges of $\\gamma_1,\\gamma_2$, the stability of the SS mode may also depend on the model parameters $k_1,k_2,\\nu_1,\\nu_2$\n\\item stability of any mode is independent of $\\beta$\n\n\\end{enumerate}\n\nFinally, \\citep{varkonyi2017dynamics} shows that the regularized model predicts unusual forms of contact dynamics in some ranges of $\\gamma_1,\\gamma_2,\\beta$, including limit cycles of the fast dynamics, and self-excited exponentially growing oscillations. A comprehensive list of such dynamic phenomena is not known.\n\n\\subsection{An illustrating example}\\label{sec:numericexample}\nWe will illustrate several types of dynamic behavior by analytical investigation and numerical simulation of a conceptual model system with $n=2$ contacts. Our model captures the algebraic structure of the equations of motion, but does not correspond to a specific mechanical system. \n\nWe have seen that $P$ is a function of $q$ whereas $b$ may additionally depend on $\\dot q$. Similarly to the equations \\eqref{eq:xdyn}-\\eqref{eq:zdyn} developed for the dependent variables $x$ and $z$, we can also develop a set of expressions for $\\dot P$ and $\\dot b$ by using \\eqref{eq:basiceq2} and the chain rule. In the present paper, we do not explicitly construct such equations for any system, but we note that they can always be written as\n\\begin{align}\n\\dot P = A_1(q,\\dot q)\\\\\n\\dot b =  \\alpha_2(q,\\dot q)+ A_3(q,\\dot q)\\lambda \n\\end{align}\nwhere the functions $A_1,A_3\\in\\mathbb {R}^{2\\times 2}$, $\\alpha_2\\in\\mathbb {R}^{2}$ are system-specific. For example, the  values of $A_1,\\alpha_2,A_3$ corresponding to a frictional impact oscillator originally introduced by \\citet{leine2002} are shown in Appendix B of \\citet{nordmark2017dynamics}.\n\nFor the sake of illustration, we will now consider the case when $A_1,\\alpha_2,A_3$ are constants:\n\\begin{align}\n\\dot P = \\alpha_1\\label{eq:dotP}\\\\\n\\dot b =  \\alpha_2+ A_3\\lambda \\label{eq:dotb}\n\\end{align}\nThis model system can be combined with the rigid contact model, where we have $\\lambda=0$ in F mode and $\\lambda$ is given by \\eqref{eq:lambda-2contact} in S mode. Then the equations \\eqref{eq:dotP}-\\eqref{eq:dotb} decouple from all other equations of motion, thus we are able to investigate the dynamics of $P$ and $b$ in isolation. This system will be used to demonstrate the onset of some phenomena analytically. \nAlternatively, we can also combine \\eqref{eq:dotP}-\\eqref{eq:dotb} with the regularized contact model. Then, the equations \\eqref{eq:zdyn}, \\eqref{eq:contactlaw}, \\eqref{eq:dotP} and\\eqref{eq:dotb} decouple from all other equations of motion. We will use this system to illustrate our findings by a series of numerical simulations (Fig. \\ref{fig:sim1}-Fig. \\ref{fig:sim4}). In each figure, we show the time history of $z_1$ and $z_2$ with circles denoting liftoff and landing at one of the contact points. In some of the figures, the time history of the angles $\\beta$, $\\gamma_i$ is also shown. The model parameters and initial conditions used in these simulations are given in the appendix.   \n\n\\section{Generic transitions from slip}\nThe main goal of the paper is to identify and to give a qualitative description of generic transitions and singularities of a system, which is initially in slip mode. Clearly, such events occur when the actual contact mode of the system becomes either unfeasible or unstable. In what follows, we begin with the $n=1$ case, where existing results are reviewed and some new results are also presented. This is followed by detailed analysis for $n=2$ and some results for general $n$.\n\n\n\\subsection{Transitions of systems with a single point contact}\n\nAssume now that a system with a single point contact undergoes slip motion with nonzero velocity\n and this mode of motion is feasible and stable. As we know from Sec. \\ref{sec:1contact-instantaneous}, this means \n\\begin{align}\n\\dot x_1&>0\n\\label{eq:1pointslidingcondition1}\\\\\nb&<0<P\n\\label{eq:1pointslidingcondition2}\n\\end{align} \nSince $\\dot x_1,b$ are functions of all state variables ($q$ and $\\dot q$) and $P$ depends on $q$, all three variables vary over time.\n\nA sudden transition occurs, if one or several of the conditions \\eqref{eq:1pointslidingcondition1}-\\eqref{eq:1pointslidingcondition2} break down. There are two trivial transitions. First, reaching the border $\\dot x_1=0$ means transition to stick or slip reversal. Moreover,\\citet{nordmark2017dynamics} show using regularized contacts that this transition is in some cases immediately followed by exponentially growing bouncing motion of $z$, which is termed by those authors \\textit{reverse chatter}. \nThe second simple option is reaching the \\textit{liftoff manifold} $b=0$, where slip becomes infeasible and the contact lifts of. It is tempting to declare that crossing $P=0$ (where slip again becomes infeasible) is also a generic event. Nevertheless $P=0$, which we will refer to the  \\textit{\\Pain manifold}, corresponds to a singularity of the contact force according to \\eqref{eq:lambda-1contact}. What happens instead was uncovered first by \\citet{Genot1999}. They studied the problem of a sliding rod, and found that as $P$ approaches 0, the large contact force induces large acceleration of the rod. The value of $b$ is a function of the velocity $\\dot q$ whereas $P$ is independent of it. In the case of the rod, $b$ rapidly increases in response to large contact forces and thus it may approach $b=0$. Thereby the system may converge to the codimension-2 manifold given by $b=P=0$. We will refer to this as the \\textit{G\\'enot-Brogliato manifold}. Here, the contact force is left ill-defined by\\eqref{eq:lambda-1contact}. Local analysis of the system near the \\GB manifold reveals that it attracts an open set of initial conditions, and the contact force diverges to infinity, which inspired the name \\textit{dynamic jamming} of this phenomenon. What happens beyond that point is a subtle question, which was only partially uncovered by using asymptotic analysis and geometric singular perturbation techniques \\citep{nordmark2017dynamics,kristiansen2017canard}.\n\nThe local analysis of the rod was extended in unpublished work by Nordmark, Dankowicz, and Champneys (see also Fig. 13 in \\citet{champneys2016painleve}) to general systems with a single point contact. They found that the G\\'enot-Brogliato manifold is often attractive and the contact force may or may not diverge to infinity when approaching it. \n\nThese local results suggest that the $P=0$ boundary is never crossed by  systems with $n=1$ away from the \\GB manifold. Nevertheless a global extension of this result has never been published. Here we give a formal proof of this property:\n\n\\begin{theorem}\nAssume that a mechanical system  with a single point contact is initially in slip mode, in a state with $b<0<P$, furthermore $P$ is Lipschitz with respect to $q$ and $K$, $a$, $b$ are continuous functions of their arguments. Then slip motion of the system may not reach a state where $ P = 0 <  b$\n\\label{thm: no_painleve_1_contact}\n\\end{theorem}\n\nThe proof of this statement is based on the observation that crossing the boundary would induce a $1\/x$ type singularity in the contact force, which is non-integrable. This implies that the contact force would eliminate slip motion for any finite initial tangential velocity. The detailed proof is given below.\n\\begin{proof} [Proof by contradiction:] \n\nAssume that the system is launched at $t_{init}<0$ and a state with $P=0> b$ is reached at time $t=0$.\n \n\nThe velocity $\\dot q$ is a continuous function of time during impact-free motion, hence $|\\dot q|$ has a global maximum $\\dot q_{max}$ over the closed interval $t\\in[t_{init},0]$. This means that the following bound applies to $P$:\n\\begin{align}\n0<P(q(t))<\\mathcal{L}|q(t)-q_{0}| \\leq \\mathcal{L}\\dot q_{max}(-t)\n\\label{eq:Pbound}\n\\end{align}\nwhere $\\mathcal L$ is a Lipschitz constant of $P$. This relation shows that the contact force \\eqref{eq:lambda-1contact} undergoes an $f(x)=1\/x$ type singularity.\n\nNext we investigate how the tangential velocity of the contact point varies in response to the singular contact force. The tangential acceleration of the contact point can be expressed according to \\eqref{eq:xdyn}, and \\eqref{eq:lambda-1contact} as\n\\begin{align}\n\\ddot x&=a+K\\lambda\\\\\n&=a-KP^{-1}b\n\\end{align}\nThis can be rearranged as\n\\begin{align}\nP\\ddot x=Pa-Kb\n\\label{eq:tanaccformula}\n\\end{align}\nimplying\n\\begin{align}\n(P\\ddot x)|_{t=0}&=\n\\underbrace{P_0}_0a_0-K_{0}b_{0} <0\n\\end{align}\nwhere the index $_0$ refers to values at $t=0$. The last inequality holds because $K_{0}<0$ follows from $P_{0}=0$, since any external force must cause an acceleration, whose scalar product with the force vector is positive (unless the force is directly opposed by a hard constraint). \n\nThe right-hand side of \\eqref{eq:tanaccformula} is continuous, thus there exists a time\n$t_1$ sufficiently close to $0$ such that for all $t_1\\leq t\\leq 0$, $P\\ddot x$ is strictly negative with some finite bound $E$:\n\\begin{align}\nP\\ddot x<-E<0\n\\end{align}\n\nThen, we can combine this with \\eqref{eq:Pbound} to obtain \n\\begin{align}\n\\ddot x&<-P^{-1}E\\\\\n     & < -\\frac{E} {\\mathcal{L}\\dot q_{max}}(-t)^{-1}\n\\end{align}\nThis bound involves a non-integrable singularity, which means that the variation of $\\dot x$ in the time interval $[t_1,0]$ is unbounded. In other words, for any positive sliding velocity at $t_1$, $\\dot x = 0$ is reached for some $t<0$. Then a slip-stick transition occurs, contradicting the initial assumption that the system slips until $t=0$.\n\\end{proof}\n\nWe can summarize the results of this subsection as follows. There are four generic transitions from slip: transition to stick or slip reversal (possibly followed by inverse chattering); lift-off; and the dynamic jamming singularity. Crossing the \\Pain manifold $P=0$ is impossible, i.e. such systems never evolve into states suffering from Painlev\\'e paradox except for the codimension 2 \\GB manifold. Notice that the last result required the Lipschitz property of $P$. This is violated if the friction coefficient jumps abruptly, in which case a system may slip into a state subject to \\Pain paradox away from the \\GB manifold.\n\n\\subsection{Transitions of  systems with two point contacts}\nAs before, we assume that the system under consideration is in slip state initially with nonzero tangential velocity at both contact points, furthermore slip motion is feasible and stable. In the next subsection, we find all generic boundaries at which feasibility and stability can be lost. Then the remainder of the section focuses on the question: what happens after reaching each of these boundaries?\n\n\\subsubsection{Feasibility and stability boundaries}\nReaching zero tangential velocity is an obvious violation of feasibility. In addition to that, detailed analysis of \\citet{varkonyi2017dynamics} shows that we have feasibility and stability if the system parameters $\\beta$, $\\gamma_1$, $\\gamma_2$ are in  a certain range (Fig. \\ref{fig:SSrange}). This range is bounded by 5 smooth surfaces as follows:\n\\begin{enumerate}\n\\item two planar \\textit{liftoff boundaries} L1, L2 given by $\\beta=\\gamma_2$ (L1) and $\\beta=\\gamma_1$ (L2). One of the contact forces $\\lambda_1$ and $\\lambda_2$ becomes zero at each of these surfaces.\n\\item the \\textit{Painlev\\'e boundary} P given by $|\\gamma_1-\\gamma_2|=\\pi$ where matrix $P$ is singular, and the (degenerate) cone spanned by $p_1$ and $p_2$ is a full line. Here, the contact forces \\eqref{eq:lambda-1contact} become infinitely large.\n\\item the stability boundaries S1, S2, which are curved surfaces composed of straight lines parallel to the $\\beta$ axis. The coefficient matrix of \\eqref{eq:contactdynamics_generaln} has a pair of purely imaginary eigenvalues along these surfaces, and thus slip motion becomes unstable. The exact locations of these surfaces depend on parameters $|p_i|,k_i,\\nu_i$ but they always lie within the boundaries shown in Fig. \\ref{fig:L1parts}. \n\\end{enumerate}\nThere are three additional curves along the boundary, which deserve special attention as they are associated with singular behaviour:\n\\begin{enumerate}\n\\item the line sections GB1,GB2 are at the intersections of L1 and L2 with P. As we will see, these are analogous to the G\\'enot-Brogliato manifolds of the single-contact case.\n\\item the line L12 is at the intersection of the liftoff manifolds L1 with L2 and it is given by $\\beta=\\gamma_1=\\gamma_2$. Along L12, $P$ is singular and the cone spanned by $p_1$ and $p_2$ is a half-line. \n\\end{enumerate}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=8cm] {SSregion.pdf}\n\\caption{The range of angles $\\beta$,$\\gamma_1$, $\\gamma_2$ where the SS contact mode is feasible and stable. Note that the intersection of the coordinate axes is shifted from 0 for better visibility. The exact location of the curved surfaces S1, S2 depends on $|p_1|, |p_2|$, $k_1$, $k_2$, $\\nu_1$, and $\\nu_2$ The figure shows the case $|p_1|=|p_2|=k_1=1$, $k_2=10$, $\\nu_1=\\nu_2=0$. For more information about possible variations of the stability boundaries, see Fig. \\ref{fig:L1parts}.}\\label{fig:SSrange}\n\\end{center}\n\\end{figure}\n\nSlip motion persists until one of the feasibilility or stability conditions is violated. In what follows, we analyze the consequences of crossing each of the boundaries listed above.\n\n\n\\subsubsection{Reaching zero tangential velocity}\nThe system may undergo slip-stick transition or slip reversal, if the tangential velocity at one or both contact points drops to zero. Note that simultaneous zero crossing is generic in the presence of certain kinematic constraints, as for example in the case of a single rigid object with two point contacts. Similarly to the case of a single point contact \\citep{paper2}, it is likely that the regularized contact model sometimes predicts self-excited bouncing motion resembling inverse chattering in this situation, however we do not examine this possibility in the present paper.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=9cm] {L1parts.pdf}\n\\caption{Projection of Fig. \\ref{fig:SSrange} to the $\\gamma_1-\\gamma_2$ plane (dark grey area), possible locations of the projection if parameters $|p_i|,k_i,\\nu_i$ are varied (light grey area), and subdivision of the L1 surface (solid lines and two-digit numbers).}\\label{fig:L1parts}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Crossing L1} \nAs we have seen, the contact force $\\lambda_1$ becomes zero at L1, i.e. contact point 1 lifts off while point 2 continues to slip. We can now test the feasibility and stability of the FS mode. The result depends on where the L1 surface was crossed. Fig. \\ref{fig:L1parts} shows a subdivision of L1 into regions marked by 2-digit numbers. The numbers refer to a classification scheme introduced by \\citet{varkonyi2017dynamics}. The meaning of the labels is not explained here. Within ranges 23,12,13,41,42,31, the feasibility conditions \\eqref{eq:FS1}-\\eqref{eq:FS2} are satisfied, and we also have $p_{22}>0$, i.e. slip motion on contact 2 is feasible and stable (see Sec. \\ref{sec:1contact-instantaneous}). Thus, we conclude that the system undergoes an SS$\\rightarrow$FS contact mode transition. This scenario is illustrated by numerical simulation in Fig. \\ref{fig:sim1}(a).\n\nIn contrast, if L1 is crossed in region 14 or 46, then we have $p_{22}<0<b_{22}$, implying that slip at contact point 2 is feasible but unstable. According to Table \\ref{table:1contact}, the fast dynamics of point 2 may evolve either towards liftoff or towards an IWC (as long as contact 1 is separated). The unstable slip mode of point 2 corresponds to a saddle point of the fast contact dynamics, and its stable manifold separates the basins of attractions of liftoff and IWC. In order to determine, what happens to our system at this point, one should find the dynamic state of point 2 when point 1 lifts off to see within which basin of attraction it lies. Unfortunately, this simple approach is inconclusive here since it is straightforward to show that contact point 2 is exactly at the saddle point (which is part of the separatrix between liftoff and IWC). Hence, the analysis of the fast dynamics itself does not enable us to decide, in which direction the system evolves from here. The indeterminacy can perhaps be removed by considering the full slow-fast system. The full system evolves initially along the critical manifold corresponding to motion in SS mode. This manifold is probably off the separatrix between liftoff and IWC in the full slow-fast system. Nevertheless such a detailed analysis is beyond the scope of the present paper. Instead, we perform  numerical simulations using the system introduced in Sec. \\ref{sec:numericexample}. Our results suggest evolution towards liftoff whenever the L1 surface is crossed in region 14, and towards IWC in region 46 (Fig. \\ref{fig:sim1}(b)-(c)). Proving or refuting that the opposite scenarios (liftoff in region 46 and IWC in 14) are impossible remains an open question.\n\n In those two cases, which did occur in the simulations, the fate of the system can be deduced analytically:\n\\begin{itemize}\n\\item If point 2 evolves towards liftoff and we are in region 14, then the gradually decreasing contact force $\\lambda_2$ and the negative signs of $p_{21},p_{22}$ imply that both contact points have increasing normal accelerations and eventually contact point 2 also lifts off. Then,  $b_{1},b_2>0$ imply that the FF mode persists, i.e. the system undergoes SS$\\rightarrow$FF transition.\n\\item If point 2 evolves towards IWC, and we are in region 46, then $p_{21}>0$ implies that the large contact force  lifts up contact 1, i.e. it remains in separation. In this case an IWC develops at point 2, while point 1 is in F mode. IWC means that slip motion at point 2 stops rapidly, after which contact 2 may jump into separation with nonzero velocity\n\\end{itemize}  .\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\subfigure[]{\\includegraphics [width=0.32\\textwidth] {caseL1egyeb.pdf}}\n\\subfigure[]{\\includegraphics [width=0.32\\textwidth] {case14.pdf}}\n\\subfigure[]{\\includegraphics [width=0.32\\textwidth] {case46.pdf}}\n\\caption{Crossing the L1 surface in numerical simulation of the model system introduced in Sec. \\ref{sec:numericexample}.  L1 is crossed at $t=0$ in region 13 (a), 14 (b), or 46 (c). The diagrams show $z_1$(solid curve) and $z_2$ (dashed curve) versus time $t$. Circles mark the liftoff ($\\lambda=0$) of the contact points. The corresponding model parameters and initial conditions are given in the appendix.\n}\n\\label{fig:sim1}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nThe results related to region 46 are particularly interesting, because Theorem \\ref{thm: no_painleve_1_contact} and other results from \\citet{champneys2016painleve} imply that a system with a single point contact never exhibits an IWC inside the \\Pain regime ($P<0$). Nevertheless we see here that the analogous statement does not hold for systems with 2 point contacts. That is, we uncovered a generic mechanism, which does not occur in previously studied simple model systems.  \n\n\\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_01.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_01_detail.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_02_detail.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_03_detail.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_04.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_04_detail.pdf}}\n\\caption{Crossing the S1 surface in numerical simulation of the model system.  Panel (a) shows inverse chattering  and (b) is a magnified detail of the same diagram. (c) is an example of simulatneous liftoff at both contact points, whereas (d) shows liftoff at point 1 accompanied by the onset of an IWC at point 2. (e) shows converges to a microscopic limit cycle of the fast subsystem and (f) is a magnified detail of the same diagram. \n}\n\\label{fig:sim2}\n\\end{figure}\n\n\n\\subsubsection{Crossing S1}\nWe have seen that the system matrix of \\eqref{eq:contactdynamics_generaln} has a pair of purely imaginary eigenvalues at S1. As the system crosses the stability boundary, the invariant point of the regularized fast dynamics corresponding to the SS mode becomes unstable and $z_1$ and $z_2$ exhibit gradually growing harmonic oscillations. Soon, one of the two points will lift off. Beyond liftoff, the regularized contact law switches between the two cases of \\eqref{eq:contactlaw} and it is not immediately clear what is going to happen. It can be shown that the FS and SF contact modes are either inconsistent or unstable, and FF is inconsistent except for a relatively small region within the S1 surface where $0<\\beta<\\pi\/2$. During systematic numerical simulations, we found 4 different behaviors (Figure \\ref{fig:sim2}): \n\\begin{enumerate}\n\\item rapidly growing oscillations with repeated impacts and lift-off at both contact points ((Figure \\ref{fig:sim2}(a-b))). The figure shows a regular oscillating pattern where the amplitude of the oscillation grows roughly by a factor of 10 in each cycle of the oscillation and thus only the last two cycles are visible. \n\\item transition to FF mode ((Figure \\ref{fig:sim2}(c))). The figure shows that $z_1$ and $z_2$ become positive and continue to grow rapidly.\n\\item IWC at point 1, while point 2 lifts off ((Figure \\ref{fig:sim2}(d))) The figure shows that $z_1$ becomes positive and continues to grow rapidly, while $z_2$ simultaneously diverges towards minus infinity.\n\\item limit cycle of the fast dynamics involving repeated liftoff and reestablishment of the contacts. This behavior  tends to occur when $0<\\gamma_2<\\pi\/2$ and $\\beta+\\pi$ is close to $\\gamma_2$. The results depicted in Figure \\ref{fig:sim2}(e-f) show oscillations of slowly growing amplitude (rather than exact limit cycles), since a finite value of $\\epsilon$ was used in the simulation. \n\\end{enumerate}\nIt is possible that other qualitatively different phenomena may also occur.\nThe first scenario listed above  appears to an external macroscopic observer as self-excited bouncing motion of increasing amplitude, i.e. inverse chattering \\citep{paper2}. The last one appears as sliding combined with sustained microscopic, high-frequency vibration. Similar phenomena are known to lie behind brake squeal \\citep{kinkaid2003automotive}.\n\n\n\n\n\\subsubsection{Crossing L2 or S2}\nThese scenarios are equivalent of the cases of crossing L1 and S1 with the only difference being that the roles of contact point 1 and 2 are reversed, furthermore the roles of regions 14 and 31, of 13 and 41, and of 23 and 46 are also reversed.\n\n\n\\subsubsection{Crossing P}\nMatrix $P$ is singular along the surface P, hence \\eqref{eq:lambda-2contact} shows that the contact forces diverge to infinity unless vector $b$ is a linear combination of the column vectors $p_i$ (which is satisfied at the GB1,GB2 lines). This property of surface P, and lines GB1, GB2 makes them similar to the \\Pain manifold and the \\GB manifold in the case of 1 point contact. This similarity suggests that a generalization of Theorem \\ref{thm: no_painleve_1_contact} is true for 2 contacts as well. Indeed we will present a general theorem for system with arbitrary $n$ in Sec. \\ref{sec:general n}, which implies that P is never reached away from the GB1, GB2 curves. The codimension 2 curves are investigated below.\n\n\\subsubsection{Attractive codimension 2 manifolds}\n\nWe have seen that the contact force has singularities at the codimension 2 manifolds L12, GB1, and GB2. In other words, the right-hand sides of the governing equations are discontinuous at these points. In such situations, the solution may be non-unique forward and\/or backward in time. As a consequence, it is theoretically possible that any of these co-dimension 2 manifolds can be reached in finite time from an open set of initial conditions. In what follows, we construct conceptual examples to demonstrate that all of these manifolds may be reached from generic initial conditions, and we also investigate the consequences of such a transition.\n\n\\textbf{Crossing L12 (double liftoff singularity):} L12 is at the intersection of 2 liftoff boundaries, which inspires the name proposed for this phenomenon. We consider the system \\eqref{eq:dotP}-\\eqref{eq:dotb} with ideally rigid contacts. Then, by \\eqref{eq:lambda-2contact} the system becomes\n\\begin{align}\n\\dot P = A_1\\label{eq:dotP2}\\\\\n\\dot b =  \\alpha_2- A_3P^{-1}b \\label{eq:dotb2}\n\\end{align}\nThe dynamics of $P$ is determined uniquely by the choice of\n\\begin{align}\nA_1=\\left[\n\\begin{matrix}\n-1&1\\\\\n0&0\n\\end{matrix}\n\\right]\n,P_0=\\left[\n\\begin{matrix}\n0&0\\\\\n1&1\n\\end{matrix}\n\\right]\n\\label{eq:alpha1P0}\n\\end{align}\nThis choice implies that the angle $\\gamma_1$ grows while $\\gamma_2$ decreases and both angles become equal to $\\pi\/2$ at $t=0$. \nIn addition, we choose the values\n\\begin{align}\n\\alpha_2=\n\\begin{bmatrix}\n0\\\\\n0\n\\end{bmatrix}\n,A_3=\\left[\n\\begin{matrix}\n2&-2\\\\\n0&0\n\\end{matrix}\n\\right]\n\\label{eq:alpha23}\n\\end{align}\n\nAssume now that the system has an initial state at some $t_{init}<0$ where  the system is initially in SS mode and the system parameters are within the region of feasibility and stability of SS, i.e.:\n\\begin{align}\n\\gamma_1<\\beta+\\pi<\\gamma_2\n\\label{eq:SSfeasible-stable}\n\\end{align}\nwhere $\\beta$ and $\\gamma_i$ are the angles illustrated by Fig. \\ref{fig:gammabeta}. As we approach $t=0$, the gap between the $\\gamma_1$ and $\\gamma_2$ angles shrinks to 0. Nevertheless,\n\\begin{lemma}\n For any initial condition satisfying \\eqref{eq:SSfeasible-stable} at $t_{init}<0$, the same condition is not violated as long as $t<0$ and thus the system remains in SS mode until reaching the L12 manifold at $t=0$. \n \\label{lem:L12}\n\\end{lemma}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=10cm] {lemma.pdf}\n\\caption{Illustrations of the statements of Lemma \\ref{lem:L12} (a) and Lemma \\ref{lem:GB1} (b). Solid curves indicate the evolution of the angles $\\gamma_i$. The dashed curves show the evolution of the angle $\\beta+\\pi$ for various initial conditions. The SS mode is feasible and stable in the grey areas. Filled circles mark the attractive singular points.}\n\\label{fig:lemmas}\n\\end{center}\n\\end{figure}\nIn the proof we show that whenever $\\beta=\\gamma_1$, the resulting contact force causes $\\beta$ to increase faster then $\\gamma_1$, whereas in the case of $\\beta=\\gamma_2$, the resulting contact force makes $\\beta$ decrease faster than $\\gamma_2$. This mechanism keeps $\\beta$ in the shrinking interval $(\\gamma_1,\\gamma_2)$ until the L12 manifold is reached at $t=0$ (Fig. \\ref{fig:lemmas}(a)). The detailed proof, given in the appendix, borrows ideas from  \\citet{Genot1999}. We apply a singular rescaling of time and show that the L12 manifold turns into an attractive invariant point of the rescaled dynamics. We note that the proof of Lemma \\ref{lem:L12} does not rely on the specific values of $A_1,A_3,\\alpha_2$ chosen above, hence the mechanism illustrated by this example may occur in many systems.\n\nAt the L12 manifold, the contact force $\\lambda$ becomes undefined due to the singularity of $P$, and the dynamics of $b$ given by \\eqref{eq:dotb2} becomes ill-defined. The indeterminacy may be resolvable by contact regularization, nevertheless such an analysis is beyond the scope of the present work. For the sake of illustration, we show examples of numerical simulation using regularized contact in Fig. \\ref{fig:sim3}. The results confirm that angle $\\beta$ remains in the shrinking interval $(\\gamma_1,\\gamma_2)$ until the singularity is reached. After crossing the singularity contact 1 lifts off while 2 remains in slip state for some initial conditions. Shortly thereafter, contact 1 becomes active again, and it exhibits rapidly increasing contact force (indicating the onset of an IWC), while contact 2 lifts off. For other initial conditions, contact 2 lifts off and contact one enters an IWC directly. This behavior appears to be qualitatively similar to one of the possible scenarios during the dynamic jamming singularity of the slipping rod reported by \\citet{nordmark2017dynamics}.\n\n\n\\textbf{Crossing GB1 (dynamic jam):} we again consider the system introduced in Sec. \\ref{sec:numericexample} with rigid contacts. This time, we choose \n\\begin{align}\nA_1=\\left[\n\\begin{matrix}\n-1&-1\\\\\n0&0\n\\end{matrix}\n\\right]\n,P_0=\\left[\n\\begin{matrix}\n0&0\\\\\n-1&1\n\\end{matrix}\n\\right]\n,\n\\alpha_2=\n\\begin{bmatrix}\n0\\\\\n0\n\\end{bmatrix}\n,\nA_3=\\left[\n\\begin{matrix}\n0.5&0.5\\\\\n0&0\n\\end{matrix}\n\\right]\n\\label{eq:alpha1P0masodik}\n\\end{align}\nThen, the angle $\\gamma_1$ decreases while $\\gamma_2$ increases, and we have $\\gamma_1=\\gamma_2+\\pi=3\\pi\/2$ at $t=0$. Furthermore,\n\\begin{lemma}\n For any initial condition such that the system is in SS mode at time $t_{init}<0$ and $0<\\beta+\\pi<\\gamma_2$, the SS mode persists until $t=0$, where the angle $\\beta$ converges to $3\\pi\/2$, i.e. the system reaches the GB1 manifold.\n \\label{lem:GB1}\n \\end{lemma}\nThe proof is based on the observation that large $\\lambda$ causes $b_1$ to converge towards 0 rapidly (Fig. \\ref{fig:lemmas}(b)). Technically, the proof is very similar to that of Lemma \\ref{lem:L12} and it is again given in the appendix.\n\nUpon reaching the GB1 or GB2 manifold, the contact force again becomes ill-defined due to the singularity of $P$. Systematic investigations of what happens after this point is beyond the scope of the paper. Based on lessons learned from the single-contact case \\citep{nordmark2017dynamics,kristiansen2017canard}, we expect that the regularized system exhibits liftoff, or impulsive contact forces. Numerical simulations (Fig. \\ref{fig:sim4}) are consistent with our findings and expectations. Depending on the initial conditions, solution trajectories converge to $\\beta=\\pi\/2$ or $3\\pi\/2$ at $t=0$, i.e. the system is attracted by the GB1 or by the GB2 manifold.In the first case, contact 1 lifts off, whereas both ones lift off in the second. Further simulations (not shown) indicated that impulsive contact forces are possible, furthermore the initial liftoff is often followed by additional mode transitions, similarly to the results of Fig. \\ref{fig:sim3} and to numerical results of \\citet{nordmark2017dynamics} in the case of $n=1$.  \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=6.5cm] {L12szog.pdf}\n\\includegraphics [width=6.5cm] {L12y.pdf}\n\\caption{Numerical simulation of the double liftoff singularity. Left: $\\beta+\\pi$ (dashed curves) and $\\gamma_i$ (solid curves) versus time for four different initial conditions. Right: $z_1$ (solid curve) and $z_2$ (dashed curves) in the same simulations. \t\n}\n\\label{fig:sim3}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=6.5cm] {GB1szog.pdf}\n\\includegraphics [width=6.5cm] {GB1y.pdf}\n\\caption{Numerical simulation of dynamic jamming. Left: $\\beta+\\pi$ (dashed curves) and $\\gamma_i$ (solid curves) versus time for 8 different initial conditions. Right: $z_1$ (solid curves) and $z_2$ (dashed curves) in the same simulations.\n}\n\\label{fig:sim4}\n\\end{center}\n\\end{figure}\n\n\\subsection{Dynamic jamming of general systems} \\label{sec:general n}\n\nWe have seen previously that the boundary of the region of feasibility and stability of slip motion includes the codimension-1  \\Pain surfaces where $P=0$ ($n=1$) or $detP=0$ ($n=2$). Nevertheless general points of these surfaces are inpenetrable to the systems. In addition, we have also seen that some codimension-2 surfaces with $detP=0$ may become attractive in turn. In what follows, we generalize Theorem \\ref{thm: no_painleve_1_contact} to systems with arbitrary number of point contacts in two dimensions and thus show that the inpenetrability of the \\Pain surfaces is a general property in contact mechanics of planar systems. \n\nAssume that a mechanical system  with $n$  point contacts initially undergoes slip motion at all contact points with $\\dot x_i>0$. Similarly to the case $n=2$, slip motion must be feasible, which means that $-b$ must be in the cone spanned by the column vectors of $P$.\n\nFirst we define the \\Pain manifold as follows:\n\\begin{definition}\n The \\Pain manifold in the state space consists of those points where $P$ is singular and the cone spanned by column vectors of $P$ is a full $(n-1)$-dimensional space.\n\\end{definition}\n\nIf we approach a generic point of the \\Pain manifold in state space transversally, the cone spanned by the column vectors of $P$ gradually grows and converges to an $n$-dimensional half-space. When crossing the manifold, the cone collapses to an $n-1$ dimensional space and then it turns inside out, i.e. it becomes a half-space again. Because of this property, slip motion is feasible on exactly one side of the \\Pain manifold, hence the \\Pain manifold may form a boundary of the range of feasible and stable slip motion in state space.\n\nNext, we define the \\GB manifold as follows:\n\\begin{definition}\nThe \\GB manifold is a submanifold of the Painlev\\'e manifold including those points where $b$ is contained in the $n-1$ dimensional space spanned by the column vectors of $P$.\n \\end{definition}\nIt is easy to show that by approaching the \\GB manifold in a direction transversal to the \\Pain manifold, one  of the contact forces in slip mode dictated by \\eqref{eq:lambda-1contact}  converges to 0. Hence, the \\GB manifold contains those points within the \\Pain manifold, which are at the verge of liftoff (in full analogy with the $n=1$ case).\n\nNow we can state the following theorem: \n\\begin{theorem}\nIf $P$ is Lipschitz with respect to $q$ and $a,b,K$ are continuous function of their arguments, then the system never reaches the Painlev\\'e manifold except for points of the \\GB manifold. \n\\label{thm: no_painleve_n_contact}\n\\end{theorem}\n\nThe proof is highly similar to that of Theorem \\ref{thm: no_painleve_1_contact} but involves some additional technical steps. We present the proof in the Appendix.\n\n\n\n\\section{Conclusions}\n\nIn this paper, we investigated singularities and transitions of mechanical systems with frictional point contacts during slip motion. We found that the presence of 2 (or more) point contacts induces several dynamic phenomena, which are not possible in previously studied single-contact systems. Slip motion may terminate due to destabilization or loss of feasibility. In both cases, various transitions may occur, including lift-off at one or both contact points, self-excited microscopic limit cycle oscillations or exponentially growing macroscopic oscillations, as well as impact without collisions. The last three transitions are not possible in the single-contact case. The list of phenomena is not comprehensive, as other types of behavior may be possible. On the other hand, the author believes that the patterns found by this study occur frequently in systems with any number of contacts and thus they are of practical interest.\n\nWe also demonstrated that two different types of singularity are generic in the case of $n=2$. The first one occurs when a system converges to a codimension-2 manifold, where the associated matrix $P$ is singular and simultaneously the system is at the boundary of lift-off at one contact point.  This singularity is essentially identical to the previously uncovered \\emph{dynamic jamming} singularity of single-contact systems first described by \\citet{Genot1999}. Based on earlier results, we expect that the contact force may or may not diverge to infinity, and passing the singularity may be followed by impulsive contact forces or lift-off at one point. A second type of singularity occurs when the system converges to another codimension-2 manifold, which is at the boundary of liftoff at both contact points. The double liftoff singularity is a novel phenomenon. Here, the contact force does not diverge to infinity, nevertheless the system shows indeterminacy at the point of passing the singularity. Either one of the contact points may lift off, while the other contact may continue to slip or exhibit impulsive contact forces. The underlying mechanism and the induced indeterminacy are both similar to the well-known two-fold singularity of Filippov system \\citep{Springerbook}. \n\n\nThe analysis was based on the assumption of a picewise linear, regularized contact model with two parameters ($k_i,\\nu_i$). The phenomena uncovered in the paper did not rely on choosing special values of the parameters $k_i,\\nu_i$ of the model, hence they appear to be generic. The assumption of linearity was motivated by its simplicity, but it is not crucial either. In the case of a nonlinear contact model (such as Hertz law), the stability of contact modes can be investigated after linearization, and the result of the analysis is often independent of the presence or the exact form of nonlinearity \\citep{champneys2016painleve}. Hence we believe that the uncovered phenomena also occur in systems with nonlinear contacts.  \n\nThis paper leaves many questions open for future work. Our aim was to provide a general overview of generic transitions, and we skipped the detailed analysis of these transitions. Open questions include\n\\begin{itemize}\n\\item detailed description of the dynamics after crossing the liftoff manifold within region 14 and 46\n\\item comprehensive list of transitions for $n=2$ (or more) contacts after crossing the stability boundaries and conditions under which they occur\n\\item general characterization of the double lift-off and dynamic jamming singularities for nonlinear systems with $n=2$ (or more) contacts\n\\end{itemize}\nAlso, we did not present real examples of the most interesting transitions and singularities. It was demonstrated that these singularities occur in a conceptual model, nevertheless finding real-world examples will be subject of future work.\n\n\\begin{acknowledgement}\nThe author thanks two anonymous reviewers for their useful suggestions. This work was supported by Grant 124002 of the National Research, Development, and Innovation Office of Hungary.\n\\end{acknowledgement}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{s1}\n\n\n\nAn important open problem in Loop Quantum Gravity (LQG)~\\cite{alrev,crbook,crlrr,ttbook} is to obtain a well \ndefined method of perturbatively computing its dynamics. The covariant approach given by Spin Foam Models\n(SFM)~\\cite{perezrev,jb-BF,crbook} provides an avenue to obtain such a method, however there are still several open issues regarding its precise\n relation to the Hamiltonian theory.  \n\nIf SFM and canonical LQG  are to be the covariant and canonical descriptions of a single quantum theory of gravity, one\nshould be able to derive one from the other. We are still far from rigorously establishing such connection,\nhowever important progress has arisen in recent years. These include, in the {\\small covariant $\\to$ canonical}  direction,\nthe derivation of the LQG Hilbert space as well as the spectra of geometrical\noperators~\\cite{newlook,engleper,dingrov}, and, in the {\\small canonical $\\to$ covariant}  direction, the extension of the EPRL\namplitude to arbitrary valent spin foams vertices thereby allowing general histories of graphs~\\cite{kkl}. \nThe latter direction leads also to the picture of regarding spin foams as spin networks histories~\\cite{rr}. \nThis interpretation however, has not gone beyond the heuristic level (as far the full four dimensional theory \nis concerned; in the three dimensional case the connection between the canonical and covariant descriptions is well established~\\cite{np}). \n\nRecently, the {\\small canonical $\\to$ covariant}  direction was analyzed at the symmetry reduced level of homogeneous and isotropic\ncosmology~\\cite{ach1,ach2}. Here we extend that analysis and consider a non-isotropic cosmological model. The\nadditional degrees of freedom allow for a richer discussion than in the isotropic case. In particular, besides\nthe `vertex expansion' present in the Friedman-Robertson-Walker (FRW) case, there are additional sums over the\nextra parameters, which are interpreted as giving a `colouring' of the graph, thus strengthening the analogy with spin foams.\n\nThe aim here is to obtain a `sum over histories' description within \nLoop Quantum Cosmology (LQC)~\\cite{mblrr}. Our construction is then different from that of `spinfoam cosmology' \\cite{rv1,brv},\n where cosmological amplitudes are obtained using SFM as the starting point. An interesting question which we do not address here\n is whether there is any precise relation among the two constructions. Let us also mention that the idea of looking for a \\mbox{{\\small  canonical}-{\\small  covariant} }\n connection at the homogeneous level has appeared before in the context of Plebanksi theory~\\cite{npv}; such approach could shed light\n into the previous question of relating  `cosmological spinfoams' with `spinfoam cosmologies'.\n\nThe paper is organized as follows. In Section~\\ref{s2} we introduce the model we will work with, namely the quantum Bianchi~I\ncosmology with a massless scalar field, as obtained by Ashtekar and Wilson-Ewing in~\\cite{awe}. In Section~\\ref{s3} we construct\nthe sum over histories description of the model. We outline  how individual amplitudes are to be calculated and illustrate the\nprocedure for a simple history. In Section \\ref{s4} the vertex expansion of the FRW~\\cite{ach1} model is recovered by integrating out\n the anisotropies of our vertex expansion. We finish the paper with a discussion in Section~\\ref{s5}.\n\n\\section{Loop Quantum Cosmology of the Bianchi~I model} \\label{s2}\n\nWe are interested in the Bianchi~I cosmological model, which is the \nsimplest non-isotropic homogeneous cosmology, coupled with a massless scalar field $\\phi$.\nAs in the isotropic case, one can fix a fiducial 3-metric $ds_0^2$ and choose Cartesian coordinates on the spacial slice\n such that ${\\rm d}s_0^2= {\\rm d}z_1^2 + \n{\\rm d}z_2^2 + {\\rm d}z_3^2$. The physical 3-metric is then determined by three  directional scales factors, $a_1,a_2$ and $a_3$,\n\\begin{equation}\n{\\rm d}s^2 =  a_1^2 ~ {\\rm d}z_1^2 + a_2^2 ~ {\\rm d}z_2^2 + a_3^2 ~ {\\rm d}z_3^2~.\n\\end{equation}\nThe Hamiltonian analysis requires one to choose a fiducial cell $\\mathcal{V}$, for which we take the rectangular prism  $0 \\leq z_i\n\\leq L_i$, for each direction $i=1,2,3$. The physical volume of the cell is then given by $V= |a_1 a_2 a_3| L_1 L_2 L_3$.\nNote that the choice of the fiducial cell $\\mathcal{V}$ (i.e.\\ the choice of $L_1$, $L_2$ and $L_3$)\n is arbitrary, and one has to ensure that physical results are insensitive to that choice (see \\cite{awe} for further discussion).\n\nWhen one goes to the quantum theory \\cite{awe}, it is convenient to work with a new set of variables, $(\\lambda_1,\\lambda_2,v)$,\ndefined by \n\\begin{eqnarray} \\label{lambda}\n\\lambda_1 & := & \\frac{\\sgn(a_1) \\sqrt{|a_2 a_3| L_2 L_3}}{(4 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o )^{1\/3}}~, \\\\\n\\lambda_2 & := & \\frac{\\sgn(a_2) \\sqrt{|a_3 a_1| L_3 L_1}}{(4 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o )^{1\/3}}~, \\\\\nv & := & \\frac{\\sgn(a_1 a_2 a_3) |a_1 a_2 a_3| L_1 L_2 L_3}{2 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o } = 2 \\lambda_1 \\lambda_2 \\lambda_3.\n\\end{eqnarray}\nHere $\\ell_o$ is the square root of the `area gap' $\\Delta =4\\sqrt{3}\\pi\n\\gamma\\, {\\ell}_{\\rm Pl}^2$, $\\gamma$ is the Barbero-Immirizi parameter, and $\\lambda_3$ is defined in a similar way as $\\lambda_1$ and $\\lambda_2$. In\nthis representation, the gravitational Hilbert space $\\H_{\\rm kin}^{\\rm grav}$ consists of functions\n$\\Psi(\\lambda_1,\\lambda_2,v)$ with support on a\ncountable number of points and with finite norm $||\\Psi||^2 :=\n\\sum_{\\lambda_1,\\lambda_2,v}\\,|\\Psi(\\lambda_1,\\lambda_2,v)|^2 <\\infty$. The matter Hilbert space is the standard\none: $\\H_{\\rm kin}^{\\rm matt} = L^2(\\mathbb{R}, \\textrm{d}\\phi)$. The total kinematical Hilbert space is thus a tensor product $\\H_{\\rm kin}=\\H_{\\rm kin}^{\\rm grav} \\otimes\n\\H_{\\rm kin}^{\\rm matt}$ and, as usual in LQC, the dynamics of the system are encoded in the constraint equation\n\\begin{equation}  -\\,C \\Psi \\equiv \\partial^2_\\phi \\Psi + \\Theta \\Psi = 0~,\n\\end{equation}\n\nwhere $\\Theta$ is a symmetric operator that acts on the gravitational part. As in \\cite{awe}, we restrict attention to the\n`positive octant' ($v,\\lambda_1,\\lambda_2 \\geq 0$). The action of $\\Theta$ takes the form,  \n\\begin{align} \\label{qham6} \\left(\\Theta \\Psi \\right)(\\lambda_1,\\lambda_2,v) =& \\frac{\\pi G}\n{4}\\sqrt{v}\\Big[(v+2)\\sqrt{v+4}\\,\\Psi^+_4(\\lambda_1,\\lambda_2,v) - (v+2)\\sqrt\nv\\, \\Psi^+_0( \\lambda_1,\\lambda_2,v)\\nonumber \\\\& -(v-2)\\sqrt v\\,\n\\Psi^-_0(\\lambda_1,\\lambda_2,v) + (v-2)\n\\sqrt{|v-4|}\\,\\Psi^-_4(\\lambda_1,\\lambda_2,v)\\Big], \\end{align}\nwhere $\\Psi^\\pm_{0,4}$ are defined as:\n\\begin{align} \\label{qham7}\\Psi^\\pm_4(\\lambda_1,\\lambda_2,v)=& \\:\\Psi\n\\left(\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_1,\\frac{v\\pm2}{v}\\cdot\\lambda_2,\nv\\pm4\\right)+\\Psi\\left(\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_1,\\lambda_2,\nv\\pm4\\right)\\nonumber\\\\& +\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_1,\n\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_2,v\\pm4\\right)+\\Psi\n\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_1, \\lambda_2,v\\pm4\\right)\\nonumber\n\\\\&+\\Psi\\left(\\lambda_1,\\frac{v\\pm2}{v}\\cdot\\lambda_2,\nv\\pm4\\right)+\\Psi\\left(\\lambda_1,\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_2,v\\pm4\\right),\n\\end{align}\nand\n\\begin{align} \\label{qham8} \\Psi^\\pm_0(\\lambda_1,\\lambda_2,v)=\n& \\:\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_1, \\frac{v}{v\\pm2}\\cdot\\lambda_2,v\\right)\n+\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_1,\\lambda_2,v \\right)\\nonumber \\\\&\n+\\Psi\\left(\\frac{v}{v\\pm2}\\cdot\\lambda_1,\\frac{v\\pm2}{v}\\cdot\\lambda_2,v\\right)+\\Psi\\left(\\frac{v}{v\\pm2}\\cdot\\lambda_1,\\lambda_2,v\\right)\\nonumber\n\\\\& +\n\\Psi\\left(\\lambda_1,\\frac{v}{v\\pm2}\\cdot\\lambda_2,v\\right)+\\Psi\\left(\\lambda_1,\\frac{v\\pm2}{v}\n\\cdot\\lambda_2,v\\right)\\, .\\end{align}\nAs noted in \\cite{awe}, the operator preserves the subspaces ${\\cal H}_\\epsilon \\subset \\H_{\\rm kin}^{\\rm grav}$ of states whose support lies on the lattice\n\\begin{equation}\nv = \\epsilon + 4 \\mathbb{Z}~.\n\\end{equation}\nThese superselection sectors have the same form as in the isotropic case \\cite{acs}, and, as done there, we will restrict\nto the physically interesting $\\epsilon = 0$ sector (the one that contains the classically singular value  $v=0$).  The space we\nfinally work with is  the space of vectors with support on the positive octant and the $\\epsilon=0$ lattice, which we denote\nby ${\\cal H}^{+++}_{0}$.\n\nWe now introduce a different representation of the space ${\\cal H}^{+++}_{0} \\subset \\H_{\\rm kin}^{\\rm grav}$, by changing coordinates\n$(\\lambda_1,\\lambda_2,v) \\to (n,{\\bf x})$ where\n\\begin{eqnarray} \\label{eq:intertwiner}\nn  & := & \\frac{1}{4} v \\in \\mathbb{N}~, \\\\\n{\\bf x}   =  (x_1,x_2) & := & (\\log \\lambda_1,\\log \\lambda_2) \\in \\mathbb{R}^2~.\n\\end{eqnarray}\nIn this representation, states are described by functions $\\Psi(n,{\\bf x})$, which again have support on a countable\nnumber of points and have a finite norm $||\\Psi||^2 = \\sum_{{\\bf x},v}\\,|\\Psi({\\bf x},v)|^2<\\infty$.\nThey represent the components of the state in a\n$\\{ |n,{\\bf x} \\rangle \\}$ basis, which is characterized by eigenvalues of  $\\widehat{\\lambda}_1,\\widehat{\\lambda}_2$ and\n$\\widehat{V}=4 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o \\widehat{\\lambda}_1\\widehat{\\lambda}_2\\widehat{\\lambda}_3$ as follows \n\\begin{eqnarray}\n\\widehat{V} |n,{\\bf x} \\rangle & = & 8 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o n |n,{\\bf x} \\rangle~, \\\\\n\\widehat{\\lambda}_i |n, {\\bf x} \\rangle & = & \\log x_i |n,{\\bf x} \\rangle , \\quad i=1,2~,\n\\end{eqnarray}\nwith their normalization given by the product of Kronecker deltas:\n\\begin{equation}\n\\langle n', {\\bf x}'| n,  {\\bf x} \\rangle = \\delta_{n,n'} \\delta_{{\\bf x},{\\bf x}'}.\n\\end{equation}\n\nIn the subsequent sections, we will regard ${\\cal H}^{+++}_{0}$ as a tensor product of the `volume' factor and the `anisotropy' factor:\n\\begin{eqnarray}\n{\\cal H}^{+++}_{0} & = &{\\cal H}_V \\otimes {\\cal H}_\\lambda~, \\\\\n|n, {\\bf x} \\rangle & = & |n \\rangle \\otimes | {\\bf x} \\rangle~.\n\\end{eqnarray}\nWithin this splitting, $\\Theta$ is expressed as a sum of the tensor product of operators acting on ${\\cal H}_V$ and ${\\cal H}_\\lambda$,\n\\begin{equation} \\label{theta}\n\\Theta = \\sum_n | n+1 \\rangle \\langle n| \\otimes \\Theta_{\\left(n+1\\right) n} + | n \\rangle \\langle n| \\otimes \\Theta_{nn} +\n | n-1 \\rangle \\langle n| \\otimes \\Theta_{\\left(n-1\\right) n}~.\n\\end{equation}\nThe form of the operators acting on the anisotropy factor is quite simple: they are composed of translations in the\n ${\\bf x}$ plane of lengths\n\\begin{equation} \\label{stepsize}\na^\\pm_n  :=\\log \\frac{2n\\pm 1}{2n}~,\n\\end{equation}\nwhich depend on the volume $n$. If we write the operator generating the translation by $a_n$ in the $x_1$ direction as\n\\begin{equation}\n\\left(e^{i a_n p_1} \\Psi \\right) (n, x_1 ,x_2):= \\Psi(n, x_1 +a_n ,x_2)~,\n\\end{equation}\nand similarly for translations in the $x_2$ direction, the operators acting on ${\\cal H}_\\lambda$ take the form\n\\begin{equation} \\label{diag}\n \\Theta_{n n}   =   2 \\pi G \\left( n (2n + 1)\\left[ \\cos a^+_n p_1 + \\cos a^+_n p_2 + \\cos (a^+_n p_2-a^+_n p_1)\n\\right]+n(2n-1) [ a^+_n \\to a^-_n ] \\right)~,\n\\end{equation} \n\\begin{equation} \\label{offdiag}\n\\Theta_{n\\pm1\\; n}  =   - \\pi G \\sqrt{n(n \\pm 1)}(2n \\pm 1)  \\left( e^{-i a^\\pm_n p_1} + e^{i a^\\pm_{n\\pm 1\/2} p_1}\n+ e^{-i a^\\pm_n p_1} e^{i a^\\pm_{n\\pm 1\/2} p_2} + p_1 \\leftrightarrow p_2 \\right)~.\n\\end{equation}\nOne can easily verify that the action of Eq.~(\\ref{theta}), when written in terms of the original representation, reproduces \nEq.~(\\ref{qham6}).\n\nLet us conclude this section by mentioning a remarkable property of the $\\Theta$ operator. As found in \\cite{awe}, one can recover the\nFRW cosmology by `integrating out' the anisotropies of the Bianchi I model. Specifically, it was shown that there is a\nprojection map from the Bianchi~I space to the FRW space defined by\n\\begin{equation}\n\\sum_{{\\bf x}} \\Psi(n,{\\bf x};\\phi) = \\Psi^\\text{FRW} (n ; \\phi)~,\n\\end{equation}\nin which the $\\Theta$ operator is mapped to the $\\Theta^\\text{FRW}$ operator of the FRW model\\footnote{See~\\cite{Nelson:2009yn}\nfor an alternative projection that produces isotropic states, but not the $\\Theta^\\text{FRW}$ associated\nwith the $\\nu$-quantization procedure.} namely, \n\\begin{equation}\n\\sum_{{\\bf x}} \\Theta \\Psi(n,{\\bf x} ; \\phi) = \\Theta^\\text{FRW} \\Psi^\\text{FRW} (n ; \\phi).\n\\end{equation}\nWe will later see how this projections holds order by order in the vertex expansion.\n\n\n\n\n\\section{Sum over histories} \\label{s3}\n\nAs in \\cite{ach1,ach2}, the natural object on which to construct a sum over histories expansion is the physical inner product between\n `initial' $| [n_i, {\\bf x}_i, \\phi_i] \\rangle$ and `final' $| [n_f,{\\bf x}_f, \\phi_f] \\rangle$ physical states. This inner product is constructed \nfrom a group averaging formula involving the kinematical states $|n_i, {\\bf x}_i, \\phi_i \\rangle$ and $|n_f, {\\bf x}_f, \\phi_f \\rangle$,\n\\begin{equation} \\label{physip-0}\n ([n_f,{\\bf x}_f, \\phi_f] , [n_i, {\\bf x}_i, \\phi_i] ) =\n 2 \\langle n_f, {\\bf x}_f, \\phi_f | \\textstyle{\\int}_{-\\infty}^{\\infty}\n \\textrm{d}{\\alpha} \\,e^{i \\alpha C} \\ |{p}_{\\phi}|\\, | n_i, {\\bf x}_i,\\phi_i \\rangle,\n\\end{equation}\n(the $|{p}_{\\phi}|$ term is there so that the normalization agrees with the one used in \\cite{ach2}).\nAs in the FRW case \\cite{ach1,ach2}, a  key\nsimplification comes from the fact that the constraint $C$ is a\nsum of two commuting pieces that act separately on $\\H_{\\rm kin}^{\\rm matt}$ and\n$\\H_{\\rm kin}^{\\rm grav}$. Consequently, the integrand of Eq.~(\\ref{physip-0}) splits into a matter and gravitational factors:\n\\begin{equation} \\label{integrandsplit}\n2 \\langle n_f, {\\bf x}_f, \\phi_f |  \\,e^{i \\alpha C} \\ |{p}_{\\phi}|\\, | n_i, {\\bf x}_i,\\phi_i \\rangle  = 2\\,\\langle \\phi_f |\ne^{i \\alpha p_{\\phi}^2 }\\, |p_{\\phi}|  | \\phi_i \\rangle \\langle n_f, {\\bf x}_f | e^{-i \\alpha \\Theta} | n_i, {\\bf x}_i \\rangle.\n\\end{equation}\nThe matter part can be easily evaluated as,\n\\begin{equation}\n\\label{aphi} 2\\,\\langle \\phi_f |  e^{i \\alpha\np_{\\phi}^2 }\\, |p_{\\phi}|  | \\phi_i \\rangle\\,=\\, 2\\,\\textstyle{\\int} \\frac{\\textrm{d} p_\\phi}{2 \\pi}\ne^{i \\alpha p_\\phi^2}\\, e^{i p_\\phi (\\phi_f-\\phi_i)}\\, |p_\\phi|~.\n\\end{equation}\nThe non-triviality of Eq.~(\\ref{physip-0}) lies in the gravitational part, $\\langle n_f, {\\bf x}_f | e^{-i \\alpha \\Theta} | n_i, {\\bf x}_i \\rangle$. \n Following the strategy depicted in \\cite{ach1}, we will express such term as a sum over histories. This can be achieved by observing that\n the term has the form of a matrix element of a fictitious evolution operator $e^{-i \\alpha \\Theta}$, with $\\Theta$ playing the role of\n Hamiltonian and $\\alpha$ that of time. \n\nOnce the gravitational factor is written as a sum over histories, the idea is to perform the integral over $\\alpha$ for each history\n separately, obtaining at the end a sum over histories expansion of the physical inner product. These steps will be discusses in the\n following subsections.\n\n\n\n\\subsection{Sum over histories for the gravitational amplitude}\n\n\nTo construct a `sum over histories' expansion of the gravitational amplitude $\\langle n_f, {\\bf x}_f | e^{-i \\alpha \\Theta} |\nn_i, {\\bf x}_i \\rangle$, one would proceed with a Feynman-like procedure of dividing the `time' $\\alpha$ into $N$ steps of length\n$\\epsilon = \\alpha\/N$, inserting a complete basis in between each factor, and finally taking the $N \\to \\infty$ limit. In~\\cite{ach2}\nit was shown (in the FRW context, but the result is generic for any discrete labeled basis) that the resulting limit is equivalent to a specific perturbative expansion of the `evolution' operator under study. We will use this result here to construct the sum over histories directly from the perturbation series.\n\nThe starting point in such a derivation is to write the fictitious Hamiltonian $\\Theta$ as an `unperturbed part'\n$\\Theta_0 $ plus a `perturbation' $\\Theta_1$,\n\\begin{equation}\\label{eq:diag_offdiag}\n\\Theta=\\Theta_0 + \\Theta_1~.\n\\end{equation}\n\nIn a spin network\/spin foam picture, the above splitting would correspond to a graph preserving piece, $\\Theta_0$, plus the\n remaining graph changing part, $\\Theta_1$. In our case, we choose to interpret the label $n$ as containing the information \nof the `graph' , and the remaining ${\\bf x}$ label as the colouring of the graph. Thus, $\\Theta_0$ and $\\Theta_1$ are respectively\n diagonal and off-diagonal in $n$. In the tensorial notation used in Eq. (\\ref{theta}), these operators are given by,\n\\begin{eqnarray} \n\\Theta_0 & = & \\sum_n   | n \\rangle \\langle n| \\otimes \\Theta_{n n}~,  \\\\\n\\Theta_1 & = & \\sum_n | n+1 \\rangle \\langle n| \\otimes \\Theta_{\\left(n+1\\right) n} + | n-1 \\rangle \\langle n| \\otimes \\Theta_{\\left(n-1\\right) n}~.\n\\end{eqnarray}\n\nThe construction now follows as in the FRW case \\cite{ach2}, where the same label was used to trigger the transitions.\nUsing standard perturbation theory in the interaction picture,  the transition amplitude is be written as\n\\begin{align} \\label{exp1-1}\n\\langle n_f,{\\bf x}_f | \\ e^{-i \\alpha \\Theta}\\ | n_i, {\\bf x}_i \\rangle & =  \\langle n_f,{\\bf x}_f| \\bigg[ \\sum_{M=0}^{\\infty} (-i)^M\n \\int_{0}^{\\alpha} \\textrm{d} \\tau_{M}\n\\ldots \\int_{0}^{\\tau_2}\\! \\textrm{d} \\tau_1  \\nonumber \\\\\n   & e^{-i (\\alpha-\\tau_M)\\Theta_0} \\Theta_1 e^{-i (\\tau_M-\\tau_{M-1})\\Theta_0} \\Theta_1 \\ldots \ne^{-i (\\tau_2-\\tau_{1})\\Theta_0} \\Theta_1  e^{-i \\tau_{1}\\Theta_0} \\bigg]  | n_i, {\\bf x}_i\\rangle~.\n\\end{align}\nThe $M$-th term of the sum generates all histories with $M$ transitions. These histories are obtained by inserting $M-1$\nidentities in the form $\\mathbf{1}=\\sum_{n_m} \\left(| n_m \\rangle \\langle n_m | \\right) \\otimes \\mathbf{1}_{\\lambda}$, $m=1,\\ldots,\nM-1$ next to each $\\Theta_1$ factor. This results in a sum over a sequence of volumes $(n_0,n_1,\\ldots,n_M)$ (with $n_0 \\equiv n_i$\nand $n_M \\equiv n_f$ held fixed), given by\n\\begin{equation} \\label{sohg}\n\\langle n_f,{\\bf x}_f| \\ e^{-i \\alpha \\Theta}\\ | n_i, {\\bf x}_i\\rangle = \\sum_{M=0}^\\infty \\Big[\\sum_{n_{M-1},\\ldots,n_{1}}\n\\langle {\\bf x}_f | A(n_M,\\ldots,n_0;\\alpha) |{\\bf x}_i \\rangle \\Big]~,\n\\end{equation}\nwhere\n\\begin{align} \\label{amphg}\n& A(n_M,\\ldots,n_0;\\alpha)  :=  (-i)^M \\int_{0}^{\\alpha} \\textrm{d} \\tau_{M} \\ldots  \\int_{0}^{\\tau_2}\\! \\textrm{d} \\tau_1 \\;  A(n_M,\\ldots,n_0;\n\\tau_M,\\ldots,\\tau_1;\\alpha)~, \\\\ \n& A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)  :=  A_{n_M}(\\alpha-\\tau_M) V_{n_M n_{M-1}}A_{n_{M-1}}(\\tau_M-\\tau_{M-1}) \\ldots\nV_{n_1 n_0}A_{n_0}(\\tau_1)~,\n\\label{amphgt} \n\\end{align} \nwith  $A_n\\left(\\tau\\right)$ and $V_{n'n}$  defined as,\n\\begin{eqnarray} \n&A_n(\\tau) &:= e^{-i \\tau \\Theta_{nn}}  \\label{A-1} \\\\ \n&V_{n' n}& : =  \\left\\{ \\begin{array}{ll} \\Theta_{n' n} \\label{V-1}\n& \\quad n' \\neq n \\\\\n0 & \\quad n' = n.\n\\end{array}\\right.\n\\end{eqnarray}\n\nNote that all the factors in Eqs.~(\\ref{amphg} - \\ref{V-1}) are operators on ${\\cal H}_\\lambda$ , whilst the actual amplitude,\nEq.~(\\ref{sohg}), involves matrix elements of the operator defined by Eq.~(\\ref{amphgt}). Note also that the only\nsequences entering in the sum are such that $n_{m}=n_{m-1} \\pm 1, m=1,\\ldots,M$. In particular, for $M$ fixed, there\nis a finite number of terms.\n\nThe construction above has the same form as in the FRW case. The distinction however lies on the fact that there are\nadditional degrees of freedom, given by the anisotropies ${\\bf x}$. However, in the description given so far, intermediate\nanisotropies do not appear since they are implicitly `summed over'. To make these additional sums explicit, we insert\nidentities in the form $\\mathbf{1}_\\lambda= \\sum_{{\\bf x}_m}| {\\bf x}_m \\rangle \\langle {\\bf x}_m |$  and \n$\\mathbf{1}_\\lambda= \\sum_{{\\bf y}_m}| {\\bf y}_m \\rangle \\langle {\\bf y}_m |$ to the right and left of the $A_{n_m}(\\tau_{m+1}-\\tau_m)$ operators in\nEq.~(\\ref{amphgt}). The gravitational amplitude (\\ref{exp1-1}) then takes the form,\n\n\\begin{align}\\label{diagram}\n  & \\langle n_f,{\\bf x}_f| \\ e^{-i \\alpha \\Theta}\\ | n_i, {\\bf x}_i\\rangle = \\nonumber \\\\ \n& \\sum_{M=0}^\\infty  \\sum_{n_{M-1},\\ldots,n_{1}} \\sum_{\\substack{{\\bf x}_1,\\ldots,{\\bf x}_M\\\\{\\bf y}_0,\\ldots,{\\bf y}_{M-1}}}\n A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0 ; \\alpha)\n\\end{align}\nwhere now we have, on top of the `graph history' sum (given by the volume sequence), a sum over all possible `colourings' for\n each `graph history'. The amplitude for such history is given by\n\\begin{eqnarray} \\label{amphg2}\n&&A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0 ; \\alpha) :=\\\\\n&&\\nonumber  (-i)^M \\int_{0}^{\\alpha} \\textrm{d} \\tau_{M} \\ldots  \\int_{0}^{\\tau_2}\\! \\textrm{d} \\tau_1 \\;  A({\\bf y}_M, n_M, {\\bf x}_M,\n {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0  ;\n\\tau_M,\\ldots,\\tau_1;\\alpha)~,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \\label{ampanis}\n&&A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0  ;\n\\tau_M,\\ldots,\\tau_1;\\alpha) :=\\\\\n&&\\nonumber  A_{n_M {\\bf y}_M {\\bf x}_M }(\\alpha-\\tau_M)V_{n_M {\\bf x_M} n_{M-1} {\\bf y}_{M-1}} A_{n_{M-1} {\\bf y}_{M-1}\n {\\bf x}_{M-1}}(\\tau_M-\\tau_{M-1}) \\ldots\nV_{n_1  {\\bf x_1} n_{0} {\\bf y}_{0}}A_{n_0}(\\tau_1)~,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n&A_{n {\\bf y} {\\bf x}}(\\tau) & := \\langle {\\bf y} | A_n(\\tau) | {\\bf x} \\rangle \\label{A-2} \\\\\n&V_{n' {\\bf x} n {\\bf y}} &:= \\langle {\\bf x}| V_{n'n} | {\\bf y} \\rangle \\quad \\label{V-2}.\n\\end{eqnarray}\nNote that, in spite of their appearance, the sums over intermediate anisotropies in (\\ref{diagram}) are well defined.\nAlthough in principle the anisotropies can take any real value, in practice only a countable subset of the real numbers is\ninvolved in the sum (the amplitude vanishes elsewhere).  More details on this point are given in Section~\\ref{histamp}.\n\n\n\n\\subsection{Vertex expansion of the physical inner product}\n\nWe now use the above construction to obtain an expansion for the physical inner product, Eq.~(\\ref{physip-0}). First, one\nrewrites the integrand of Eq.~(\\ref{physip-0}) as in Eq.~(\\ref{integrandsplit}) and then the gravitational factor\nis written using the expansion given in Eq.~(\\ref{diagram}).\nOne then interchanges the integral over $\\alpha$ with the sums over $M$ and the intermediate labels, to arrive at a `sum over histories' expansion of the\nphysical inner product,\n\\begin{eqnarray} \n ([n_f,{\\bf x}_f, \\phi_f] , [n_i, {\\bf x}_i, \\phi_i] ) & = & \n \\sum_{M=0}^\\infty  \\sum_{n_{M-1},\\ldots,n_{1}} \\sum_{\\substack{{\\bf x}_1,\\ldots,{\\bf x}_M\\\\{\\bf y}_0,\\ldots,{\\bf y}_{M-1}}}A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0) \\nonumber \\\\\n&=& \\sum_{M=0}^\\infty  \\sum_{n_{M-1},\\ldots,n_{1}} A(n_M,\\ldots,n_0;{\\bf x}_i,{\\bf x}_f;\\phi_i,\\phi_f) \\label{physip-1}\n\\end{eqnarray}\nwhere\n\n\\begin{align} \\label{amppath}\n A({\\bf y}_M, n_M, {\\bf x}_M,  {\\bf y}_{M-1}, n_{M-1}, & \\ldots,  {\\bf y}_0, n_0, {\\bf x}_0) :=2\\,\\textstyle{\\int} \\frac{\\textrm{d} p_\\phi}{2\\pi} \\;\n e^{i p_\\phi (\\phi_f-\\phi_i)}\\, |p_\\phi| \\textstyle{\\int}_{-\\infty}^{\\infty}\n \\textrm{d}{\\alpha} \\, \\\\\n & e^{i \\alpha p_\\phi^2} A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0 ;\n\\alpha)~.\\nonumber \n\\end{align}\n\\begin{figure}\n \\includegraphics{spin-network_final.eps}\n \\caption{\\label{fig:1} The Bianchi~I model can be written in terms of three parameters $(n, x_1, x_2)$. $n$ dictates the\nvolume, whilst $x_1$ and $x_2$ are analogous to the spin labels\nof LQG and are represented here as two edges. Note that this is not intended to be a true LQG graph,\nrather it is a pictorial description of the degrees of freedom that our states have.} \n\\end{figure}\nPictorially, we can represent the expansion as follows. First, we represent the gravitational ket $| n, {\\bf x } \\rangle$\nas depicted in Fig.~\\ref{fig:1}. A `history' with one transition in $n\\rightarrow \\bar{n}$ is then represented in Fig.~\\ref{fig:2}.\n\n\n\\begin{figure}\n \\includegraphics{spin-foam_final.eps}\n \\caption{\\label{fig:2} The history of the spin network state i.e.\\ the spin foam analogue of our vertex expansion,\nwith $M=1$ is represented pictorially. The initial state has volume $n$ and anisotropies $x_i$. The state then `evolves', keeping $n$ constant but allowing the anisotropies to vary. Eventually there is a transition to a new volume, $\\bar{n}$. The anisotropy before and after the transition are $y_i$ and $\\bar{x}_i$ respectively. The final state has labels $\\bar{n}$ and $\\bar{y}_i$. \n The dashed lines indicate that the `spin' labels ($x_i$) evolve along each of the constant volume pieces. The amplitude for such process, in the notation of Eq. (\\ref{amppath}), is given by $A(\\bar{{\\bf y}}, \\bar{n}, \\bar{{\\bf x}},  {\\bf y}, n, {\\bf x})$.}\n\\end{figure}\n\nNote that here the analogue\nto spin foams is not exact, since in a spin foam, the spin labels on each face of the triangulation are constant. In the expansion\nderived here there is non-trivial dynamics for the `spins', ${\\bf x}_i$, even with a fixed `graph' $n$. In general there are\ntwo distinct labels for a face, those at the beginning, ${\\bf x}_i$ and those at the end, ${\\bf y}_i$, and  ${\\bf y}_i \\neq\n{\\bf x}_i$. Also, in a \nspin foam there is no restriction on the spins across a vertex, whereas in our case ${\\bf y}_i$ and ${\\bf x}_{i+1}$\n(the `spin' labels on either side of the $(i+1)^{\\rm th}$ vertex ) are very closely related, by the form of\nEq.~(\\ref{offdiag}). Thus, although the analogue is not complete, the form of the expansions are qualitatively the\nsame. \n\n\n\\subsection{Histories amplitudes} \\label{histamp}\n\nWe now discuss how path amplitudes appearing in Eq.~(\\ref{physip-1}) can be calculated and illustrate the procedure\nin the simplest case.  The first step is to evaluate the matrix element $\\langle {\\bf x}_f |A(n_M,\\ldots,n_0; \\tau_M,\n\\ldots, \\tau_1;\\alpha)|{\\bf x}_i \\rangle$ of the operator defined in Eq.~(\\ref{amphgt}).  This operator consist of compositions\n of operators $A_n(\\tau)$ and $V_{n' n}$, given respectively in Eq.~(\\ref{A-1}) and Eq.~(\\ref{V-1}), which themselves are\nconstructed from translations in the  ${\\bf x}$ plane.  Because of the translation invariance, we will consider the matrix\nelements between  $\\langle {\\bf x}|$ and $|0 \\rangle$; the original matrix element is then recovered by the substitution\n${\\bf x} \\to {\\bf x}_f-{\\bf x}_i$.\n\nLet us discuss the structure of the operators in more detail.  As already noted, $V_{ n' n}$ vanishes unless $n'=n\\pm 1$, in which\ncase it is given by Eq.~(\\ref{offdiag}).  It consist of an overall factor times six simple shifts involving lengths of $a^\\pm_n$ and\n $a^\\pm_{n \\pm 1\/2}$.  The $A_n(\\tau)$ operator is the exponentiation of ($-i \\tau$ times) the operator given in\nEq.~(\\ref{diag}).  It can be factored into two terms,\n\\begin{equation}\nA_n(\\tau)=A^+_n(\\tau)A^-_n(\\tau)~,\n\\end{equation}\nwhere\n\\begin{equation}\nA^\\pm_n(\\tau)=e^{-i \\tau 2 \\pi G \\, n (2n \\pm 1)\\left[ \\cos a^\\pm_n p_1 + \\cos a^\\pm_n p_2 + \\cos (a^\\pm_n p_2-a^\\pm_n p_1) \\right]}~,\n\\end{equation}\nwhich only involves shifts with step-size $a^\\pm_n$.  The total operator $A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)$ is then a product\n of operators involving shifts of lengths $a^\\pm_j$ with $j=n_m$ or $j=n_m \\pm 1\/2$ ($m=0,\\ldots,M$).\n As a result the matrix element between $\\langle {\\bf x} |$ and $|0 \\rangle $ vanishes unless\n${\\bf x}$ lies in the `lattice' generated by the $a^\\pm_j$ steps. The computation simplifies by selecting among the\n$a^\\pm_j$'s, a set of independent (i.e. incommensurate) lengths that generate the `lattice'. \n\nLet us illustrate the situation by considering the simplest path, namely the $M=0$ case. In this case the amplitude is given by\n the matrix element $\\langle {\\bf x} |A_n(\\tau) |0 \\rangle$.   As already noticed, this operator contains two step-sizes, $a^+_n$ and\n $a^-_n$, and so the nontrivial matrix elements occur whenever ${\\bf x}$\nlies in the lattice~\\footnote{Notice that the lattices we will referring to are not the usual ones (as for instance a square\n lattice $a \\mathbb{Z}^2 \\subset \\mathbb{R}^2$) where points form a grid. Rather they fill in the entire plane. For instance one can show that\n the points of the lattice defined by   Eq.~(\\ref{lattice}) form a dense subset of $\\mathbb{R}^2$ (because $a^+_n$ and $a^-_n$ are\n incommensurate numbers).},\n\\begin{equation} \\label{lattice}\n{\\bf x} = {\\bf k}^+ a^+_n+{\\bf k}^- a^-_n \\ ; \\qquad   {\\bf k}^+ = (k^+_1,k^+_2) \\in \\mathbb{Z}^2, \\; {\\bf k}^- = (k^-_1,k^-_2)\\in \\mathbb{Z}^2~.\n\\end{equation}\nFrom the definition of $a^+_n$ and $a^-_n$, Eq.~(\\ref{stepsize}), one can check that these two numbers are always incommensurate,\nand so they form an independent set of generators of the lattice.  That is to say, a point in the lattice defined by Eq.~(\\ref{lattice}) is\nuniquely decomposed into its $a^+_n$ and $a^-_n$ components, i.e., if  ${\\bf k}^+ a^+_n+{\\bf k}^- a^-_n = {\\bf l}^+ a^+_n+\n{\\bf l}^- a^-_n$ then ${\\bf k}^\\pm={\\bf l}^\\pm$. \n\nThus, the kets $|{\\bf x}= {\\bf k}^+ a^+_n+{\\bf k}^- a^-_n \\rangle$, form a basis of the subspace of vectors which give a\nnon-vanishing matrix element.  This space has the structure of a tensor product of two copies of $L^2(\\mathbb{Z}^2)$,\n\\begin{equation}\n|{\\bf x}= {\\bf k}^+ a^+_n+{\\bf k}^- a^-_n \\rangle~= |{\\bf k}^+ \\rangle \\otimes |{\\bf k}^- \\rangle \n\\end{equation}\nwhere $|{\\bf k}^+ \\rangle$ and $|{\\bf k}^- \\rangle$ are viewed as basis of two abstract $L^2(\\mathbb{Z}^2)$ spaces.\n\nViewed in this way, the $A^\\pm_n(\\tau)$ operators act separately on each $L^2(\\mathbb{Z}^2)$ factor:\n\\begin{equation} \\label{mea}\n\\langle {\\bf x}= {\\bf k}^+ a^+_n+{\\bf k}^- a^-_n |A_n(\\tau) | 0 \\rangle = \\langle {\\bf k}^+|A^+_n(\\tau)|0 \\rangle \\langle {\\bf k}^-|A^-_n(\\tau)|0 \\rangle~.\n\\end{equation}\nEach factor is the matrix element of a (translation invariant) operator in a single $L^2(\\mathbb{Z}^2)$ space, and so can\nbe evaluated by taking the Fourier transform.  Using Eq.~(\\ref{diag}) one finds \n\\begin{equation} \\label{apm}\n\\langle {\\bf k}^\\pm|A^\\pm_n(\\tau)|0 \\rangle=\\int_0^{2 \\pi}\\frac{\\textrm{d}{\\theta_1}}{2 \\pi} \\int_0^{2 \\pi}\\frac{\\textrm{d}{\\theta_2}}{2 \\pi}\ne^{i k^\\pm_1 \\theta_1+i k^\\pm_2 \\theta_2}e^{-i 2\\pi G n(2n \\pm 1) \\tau \\left(\\cos \\theta_1 + \\cos \\theta_2 + \\cos\n(\\theta_2-\\theta_1) \\right)}~.\n\\end{equation}\n\nIn the general case of a path with $M$ transitions and $g$ independent generators, the space of vectors giving a non-vanishing matrix elements will have now the structure of a tensor\nproduct of $g$ copies of $L^2(\\mathbb{Z}^2)$. Because the vertex $V_{n' n}$ is the sum of six terms, one will generically have a\ntotal of $\\sim6^M$ terms, each of them involving $g$ Fourier integrals of the type described above.  The expressions for\nthese integrals can be directly read off from Eqs.~(\\ref{diag}) and (\\ref{offdiag}). \n\n\nAfter the evaluation of the matrix element $\\langle {\\bf x}_f |A(n_M,\\ldots,n_0; \\tau_M,\\ldots, \\tau_1;\\alpha)|{\\bf x}_i \\rangle$,\n one has to perform the $\\tau$ integrals in Eq.~(\\ref{amphgt}), and the $p_\\phi$ and $\\alpha$ integrals in Eq.~(\\ref{amppath}).\nIn our $M=0$ case, there are no $\\tau$ integrals to perform, and the\n $\\alpha$ integral can be done if interchanged with the Fourier integrals of Eq.~(\\ref{apm}). This gives a Dirac delta, which\n in turn allows one to evaluate the integral over $p_\\phi$. The result is,\n\\begin{equation}\nA(n;{\\bf x}_i,{\\bf x}_f;\\phi_i,\\phi_f)=\\int_D \\frac{\\textrm{d}^4\\theta}{(2\\pi)^4} e^{i \\theta_1  k^+_1+\\theta_2 k^+_2+\\theta_3\nk^-_1+\\theta_4  k^-_2} e^{i (\\phi_f-\\phi_i) \\sqrt{2 \\pi G n}\\sqrt{(2n + 1)h(\\theta_1,\\theta_2)+ (2n - 1)h(\\theta_3,\\theta_4)}}~,\n\\end{equation}\nwhere ${\\bf x}_f-{\\bf x}_i={\\bf k}^+ a^+_n+{\\bf k}^- a^-_n$, \\,  $h(\\theta_1,\\theta_2)=\\cos \\theta_1 + \\cos \\theta_2 + \\cos \n(\\theta_2-\\theta_1)$\nand the domain of integration is \n\\begin{equation}\nD= \\{(\\theta_1,\\theta_2,\\theta_3,\\theta_4) \\in [0,2\\pi)^4 \\; \/ \\, (2n + 1)h(\\theta_1,\\theta_2)+ (2n - 1)h(\\theta_3,\\theta_4) > 0 \\}.\n\\end{equation}\n\n\nSo far we have discussed histories amplitudes of the form $A(n_M,\\ldots,n_0;{\\bf x}_i,{\\bf x}_f;\\phi_i,\\phi_f)$, where intermediate\n anisotropies are already summed over. Let us now discuss briefly how one would compute the amplitude for an individual\n`colouring' of such a history, $A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0)$.\nThe building blocks  in this case are the amplitudes $A_{n {\\bf y} {\\bf x}}(\\tau)$ and $V_{n' {\\bf x} n {\\bf y}}$,\nEq.~(\\ref{A-2}) and Eq.~(\\ref{V-2}). The first  amplitude coincides with the $M=0$ case discussed above. We thus have that\n $A_{n {\\bf y} {\\bf x}}(\\tau)$ vanishes unless ${\\bf y}- \n{\\bf x}={\\bf k}^+ a^+_n+{\\bf k}^- a^-_n$ in which case it is given by Eqs.~(\\ref{mea}) and (\\ref{apm}). On the other hand, the value\nfor $V_{n' {\\bf x} n {\\bf y}}$  can be easily read off from its definition:\nit vanishes unless $n'=n\\pm 1$ and ${\\bf y}- {\\bf x}$ lies in one of the following six points, $\\{(a^{\\pm}_n,0),(a^{\\pm}_{n\\pm 1\/2},0),\n(a^{\\pm}_n,a^{\\pm}_{n\\pm 1\/2}),(0,a^{\\pm}_n),(0,a^{\\pm}_{n\\pm 1\/2}),(a^{\\pm}_{n\\pm 1\/2},a^{\\pm}_n) \\}$, in which case it takes the value\n$- \\pi G \\sqrt{n(n \\pm 1)}(2n \\pm 1)$. After multiplying by the remaining factors in Eq.~(\\ref{ampanis}), one has to perform the\nsame integrals as previously, in this case given in Eq.~(\\ref{amphg2}) and Eq.~(\\ref{amppath}).\n\n\\subsection{Vacuum case}\n\nThe presence of matter entered only at the very end of the construction. If we did not have matter at all, we could still follow the\nsame  procedure and arrive at the vacuum equivalent of Eq.~(\\ref{physip-1}),\n\\begin{equation} \n ([n_f,{\\bf x}_f] , [n_i, {\\bf x}_i] ) = \\sum_{M=0}^\\infty \\Big[ \\sum_{n_{M-1},\\ldots,n_{1}} A^{\\text{vacuum}}(n_M,\\ldots,\nn_0;{\\bf x}_i,{\\bf x}_f) \\Big]~.\n\\end{equation}  \nThe difference between the matter and vacuum cases lies only in the form of the amplitudes, which in the vacuum case are formally\ngiven by,\n\\begin{equation} \\label{ampvac}\nA^{\\text{vacuum}}(n_0,\\ldots,n_M;{\\bf x}_i,{\\bf x}_f)= \\textstyle{\\int}_{-\\infty}^{\\infty}\n \\textrm{d}{\\alpha} \\,  \\langle {\\bf x}_f | A(n_0,\\ldots,n_M;\\alpha) |{\\bf x}_i \\rangle~.\n\\end{equation}\nThese amplitudes can then be evaluated following the strategy given in the previous section, the only difference being the\nabsence of the final integral over $p_\\phi$. It is not obvious whether the integral in Eq.~(\\ref{ampvac}) converges for all\n paths thus giving a meaningful expansion. Nevertheless, by looking at the generic behaviour of these integral, one finds\nsome evidence that it may converge. For instance, in the constant volume ($M=0$) path, Eq.~(\\ref{ampvac}) gives\n\\begin{equation}\nA^{\\text{vacuum}}(n;{\\bf x}_i,{\\bf x}_f)=\\frac{1}{2 \\pi G n}\\int \\frac{\\textrm{d}^4\\theta}{(2\\pi)^4} e^{i \\theta_1  k^+_1+\n\\theta_2  k^+_2+ \\theta_3  k^-_1+\\theta_4  k^-_2} \\delta\\left( (2n + 1)h(\\theta_1,\\theta_2)+ (2n - 1)\nh(\\theta_3,\\theta_4) \\right)~\n\\end{equation}\nwhich is clearly finite (at least for $n>0$). This is to be contrasted with the vacuum FRW case~\\cite{ach2}, and the\nexample in~\\cite{rv2}, where a regulator is required in order to render finite the otherwise divergent amplitudes,\neven in this $M=0$ case. Further study of the vacuum case is in progress~\\cite{Ed_Adam}. \n\n\n\\section{Projection to FRW}\\label{s4}\nAt the end of section~\\ref{s2}, we commented on the projection from  Bianchi~I to FRW. We show here that\nwhen such projection is done at the level of the vertex expansion, Eq~(\\ref{physip-1}), one recovers the FRW vertex\nexpansion of \\cite{ach2}.\n\nThe structure in both cases is almost identical. One has a  sum over volume sequences $(n_M,\\ldots,n_0)$, and each amplitude\nis constructed by first obtaining a gravitational amplitude, and then performing the group averaging and scalar field\nintegrations. Thus, all that remains is to show that the amplitude $\\langle {\\bf x}_f |  A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)\n | {\\bf x}_i \\rangle$ given in Eq.~(\\ref{amphgt}) projects to the corresponding FRW one,\n\\begin{align} \\label{ampfrw}\nA^\\text{FRW}(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)= & \\; e^{-i(\\alpha - \\tau_M) \\Theta^\\text{FRW}_{n_M n_M}}\\,\\,\n \\Theta^\\text{FRW}_{n_M n_{M-1}}\\,\\,\n\\times \\nonumber\\\\\n&\\ldots\\,\\, e^{-i(\\tau_2-\\tau_1) \\Theta^\\text{FRW}_{n_1 n_1}}\\,\\,\n \\Theta^\\text{FRW}_{n_1 n_{0}}\\,\\, e^{-i\\tau_1 \n\\Theta^\\text{FRW}_{n_0 n_0}}~,\n\\end{align}\nwhen summing over all possible values of ${\\bf x}_f$. \n\nTo show this, it is convenient to explicitly write the intermediate anisotropies in the amplitude $\\langle {\\bf x}_f |  A(n_M,\\ldots,n_0;\n\\tau_M,\\ldots,\\tau_1;\\alpha) | {\\bf x}_i \\rangle$. As before, this is done by introducing complete basis $\\mathbf{1}_\\lambda= \\sum_{{\\bf x}_m}\n| {\\bf x}_m \\rangle \\langle {\\bf x}_m |$  and  $\\mathbf{1}_\\lambda= \\sum_{{\\bf y}_m}| {\\bf y}_m \\rangle \\langle {\\bf y}_m |$ to the right and left\nof the $A_{n_m}(\\tau_{m+1}-\\tau_m)$ operator in Eq.~(\\ref{amphgt}). Calling ${\\bf x}_f \\equiv {\\bf y}_M$ and ${\\bf x}_i \\equiv {\\bf x}_0$\nwe have,\n\\begin{align}  \\label{amphgtl}\n   &\\sum_{{\\bf y}_M} \\langle {\\bf y}_M |  A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha) | {\\bf x}_0 \\rangle  = \n \\sum_{\\substack{{\\bf x}_1,\\ldots,{\\bf x}_M\\\\{\\bf y}_0, \\ldots,{\\bf y}_{M}}}\n  \\langle {\\bf y}_M | A_{n_M}(\\alpha-\\tau_M) |{\\bf x}_M \\rangle  \\nonumber \\\\ \n   & \\langle {\\bf x}_M | V_{n_M n_{M-1}} |{\\bf y}_{M-1}  \\rangle \\langle {\\bf y}_{M-1}|A_{n_{M-1}}(\\tau_M-\\tau_{M-1})|{\\bf x}_{M-1}\n  \\rangle \\ldots \\langle {\\bf x}_1 |  V_{n_1 n_0}|{\\bf y}_{0} \\rangle \\langle {\\bf y}_{0}|A_{n_0}(\\tau_1)|{\\bf x}_0 \\rangle~.\n\\end{align}\nWe now use the translation invariance of the operators to write each matrix elements in Eq.~(\\ref{amphgtl}) as $\\langle {\\bf x}'| f |\n{\\bf x} \\rangle = \\langle {\\bf x}'-{\\bf x}| f | 0 \\rangle$ where $f$ is either the $A$ or $V$ operators. We then change the\nsummation variables to ${\\bf y}'_m={\\bf y}_m-{\\bf x}_m$ and ${\\bf x}'_m={\\bf x}_m-{\\bf y}_{m-1}$.\nThe different summations in Eq.~(\\ref{amphgtl}) then decouple giving,\n\\begin{align} \\label{amphgtl2}\n   &\\sum_{{\\bf y}_M} \\langle {\\bf y}_M |  A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha) | {\\bf x}_0 \\rangle  =  \\left(\\sum_{{\\bf y}'_M}\n  \\langle {\\bf y}'_M | A_{n_M}(\\alpha-\\tau_M) |0 \\rangle\\right) \\left(\\sum_{{\\bf x}'_M} \\langle {\\bf x}'_M | V_{n_M n_{M-1}} |0  \\rangle \\right) \n  \\nonumber \\\\   &  \\left(\\sum_{{\\bf y}'_{M-1}} \\langle {\\bf y}'_{M-1}|A_{n_{M-1}}(\\tau_M-\\tau_{M-1})|{\\bf x}_{M-1} \\rangle \\right) \\ldots\n  \\left(\\sum_{{\\bf x}'_1} \\langle {\\bf x}'_1 | V_{n_1 n_0}|0 \\rangle \\right) \\left(\\sum_{{\\bf y}'_0} \\langle {\\bf y}'_{0}|A_{n_0}(\\tau_1)|0\n \\rangle\\right)~.\n\\end{align}\nComparing Eq.~(\\ref{amphgtl2}) and Eq.~(\\ref{ampfrw}) we see that our task reduces to showing that\n $\\sum_{\\bf x} \\langle {\\bf x} | A_{n}(\\tau) |0 \\rangle = e^{-i\\tau \\Theta^{FRW}_{n n}}$ and $\\sum_{\\bf x}\n\\langle {\\bf x} | V_{n' n} |0 \\rangle = \\Theta^{FRW}_{n' n}$. That this\n is so can be seen as a direct consequence of the result in~\\cite{awe}. Let us nevertheless show it explicitly.\n\nFor the $V$ term, it suffices to look at $V_{n\\pm1 n}$. The operator, given\nin Eq.~(\\ref{offdiag}) consist in an overall constant times six different translations in the ${\\bf x}$ plane. Each term thus\nwill pick up a single ${\\bf x}$ from the sum. For instance, the first term gives a nonzero value only for ${\\bf x}=(a^\\pm_n,0)$,\nin which case it gives a contribution of $- \\pi G \\sqrt{n(n \\pm 1)}(2n \\pm 1)$. We then conclude that\n\\begin{equation} \\label{vfrw}\n\\sum_{{\\bf x}} \\langle {\\bf x}| V_{n\\pm1 n} |0 \\rangle = - 6 \\pi G \\sqrt{n(n \\pm 1)}(2n \\pm 1) = \\Theta^{FRW}_{n\\pm 1 n}~,\n\\end{equation}\nas required.\n\nFor the $A_n(\\tau)$ term we have\n\\begin{align} \\label{afrw}\n& \\sum_{\\bf x} \\langle {\\bf x} | A_{n}(\\tau) |0 \\rangle = \\nonumber \\\\\n & \\sum_{{\\bf k}^+,{\\bf k}^- \\in \\mathbb{Z}^2} \\langle {\\bf k}^+|A^+_n(\\tau)|0 \\rangle \\langle {\\bf k}^-|A^-_n(\\tau)|0 \\rangle = \\nonumber \\\\\n& e^{-i \\tau 24 \\pi G n^2} = e^{-i \\tau \\Theta^{FRW}_{n n}}~,\n\\end{align}\nwhere in going from the first to second line, we used Eq.~(\\ref{mea}). In going from the second\nto third line, we used Eq.~(\\ref{apm}) and the identity $\\sum_{k \\in \\mathbb{Z}} e^{i k \\theta}\/2\\pi = \\delta(\\theta)$\nto directly evaluate the Fourier integrals. Using Eqs.~(\\ref{vfrw}) and (\\ref{afrw}) we then have\n\\begin{equation}\n\\sum_{{\\bf x}_f} \\langle {\\bf x}_f |  A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha) | {\\bf x}_i \\rangle = A^\\text{FRW}(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)~,\n\\end{equation}\nwhich implies\n\\begin{equation}\n\\sum_{{\\bf x}_f} A(n_M,\\ldots,n_0;{\\bf x}_i,{\\bf x}_f;\\phi_i,\\phi_f) =  A^\\text{FRW}(n_M,\\ldots,n_0;\\phi_i,\\phi_f)~.\n\\end{equation}\nThus we see that the vertex expansion for our Bianchi model, Eq.~(\\ref{physip-1}) projects down to the vertex expansion\nof the FRW model, order by order.\n\n\n\\section{Discussion}\\label{s5}\n\nRecently it has been shown~\\cite{ach1} that one can take the Loop Quantum version of FRW cosmology\nand expand it as a sum over volume transition of amplitudes compatible with given initial and final states i.e.\\\nthat the cosmological model of Loop Quantum Gravity can be re-written in terms of a sum over amplitudes, analogous to\nthe spin foam approach. This sum over transition amplitudes is produced as a perturbation expansion of the\nconstraint operator of LQC, thus linking perturbative dynamics of LQC to (the analogue of) spin foams.\nThis analogue provided a useful link between the two theories, however because there is only\none dynamic parameter in FRW cosmologies -- the volume -- the system has no analogue of the spin labels. In this\npaper we have extended the approach of~\\cite{ach1} to the Bianchi~I cosmological model, which, in addition to volume,\nhas anisotropic degrees of freedom. We have shown that it is again possible to expand the dynamics of the\nmodel in terms of sums of amplitudes over volume transitions compatible with initial and final states. The additional\nanisotropic degrees of freedom of this model are analogous to the spin labels of spin networks, thus significantly\nimproving the analogue to spin foams.\n\nThe analogue remains at a formal level however, because one cannot directly associate the amplitudes with the \nchanging of an underlying spin network. Despite this the association of the anisotropic degrees of freedom with\nthe spin labels is well motivated by the fact that in LQC they give the area (of the fiducial cell), which is\nprecisely the role played by the spin labels (and edges) in a spin network. In addition to showing that the resulting\nsummation over `spin' labels is finite, we show that the projection to the FRW system occurs order by order\nin the expansion, thus recovering the results of~\\cite{ach1}.\n\nFinally, although spin foams are typically taken to have spin changes only at vertices, it is generally\nexpected that spin dynamics in the absence of graph changing vertices will play an important role in the final\ntheory~\\cite{freidel}. More precisely, that the action of the full constraint is non-trivial, even in the absence of vertices\nand hence that the amplitude for each vertex-free segment of the spin foam will be non-diagonal in the spin labels. In\nthe analogue produced here we show that indeed the `spin' changing amplitude is non-trivial, even in the absence of volume\nchanging `vertices'. Thus our full expansion is the analogue of a generalization of spin foams, allowing for\n`spin' dynamics. \n\n\n\n\\section*{Acknowledgments}\n\nWe would like to thank Abhay Ashtekar and Edward Wilson-Ewing for illuminating discussions.\nThis work was supported in part by NSF grants PHY0748336,\nPHY0854743, The George A.\\ and Margaret M.~Downsbrough Endowment and\nthe Eberly research funds of Penn State.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Cholesky decomposition - Introduction}\nFor a positive-definite symmetric matrix Cholesky decomposition provides a unique representation in the form of $\\Lb\\Lb^T$, \nwith a lower triangular matrix $\\Lb$ and the upper triangular $\\Lb^T$. \nOffered by a convenient $O(n^3)$ algorithm,\nCholesky decomposition is favored by many for expressing the covariance matrix~\\cite{Pourahmadi2011}. \n The matrix $\\Lb$ itself can be used to transform independent normal variables into dependent multinormal~\\cite{Moonan57} which is particularly useful for Monte Carlo simulations. \n\nExplicit forms of $\\Lb$ are known for limited correlation structures such as the equicorrelated~\\citep[pp. 104]{Tong1990}, tridiagonal~\\cite{MiwaHayterKuriki2003}, and the multinomial~\\cite{TanabeSagae1992}. The general correlated case is typically computed by using spherical parametrizations~\\cite{PinheiroBates1996,RapisardaBrigoMercurio2007,RebonatoJackel2007,MittelbachMatthiesenJorswieck2012}, a multiplicative ensemble of trigonometric functions of the angles between pairs of vectors. Others may use Cholesky matrix~\\cite[pp. 49]{CookeEtAl2011} that utilizes the multiplication of partial correlations. \n \nIn this paper, we will present two explicit parametrizations of Cholesky factor for a positive-definite correlation matrix. \nBoth parametrizations offer a preferable, simpler alternatives to the multiplicative forms of spherical parametrization and partial correlations. \nIn Section~\\ref{sec:1stL} we show that the nonzero elements of Cholesky factor are the \\textit{semi-partial correlation coefficients}\n\\[\n  \\rhob_{ij(1,\\ldots,i-1)} \n= \\frac{\\rho_{ij} - \\rhob_i^{*j} \\Rb_{i-1}^{-1} \\rhob_i}\n          {\\sqrt{1 - \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T}} ,\n\\]\nwhere $\\Rb_{i-1}^{-1}$ is the inverse of the correlation matrix $\\Rb_i = (\\rho_{kj})_{k,j=1}^{i-1}$, $\\rhob_i^{*j} = (\\rho_{1j}, \\rho_{2j}, \\ldots, \\rho_{i-1,j})$ and $\\rhob_i = \\rhob_i^{*i}$. The order of the $\\rhob_{ij(1,\\ldots,i-1)}$s is determined by Cholesky factorization, and the notations are borrowed from Huber's trivariate discussion of semi-partial correlation in regression~\\cite{Huber1981}.\n In Section~\\ref{sec:2ndL} we uncover that the squares, $\\rhob_{ij(1,\\ldots,i-1)}^2$, are equivalent to the differences between two successive ratios of determinants, \nand we use this equivalence to construct the second parametrization for $\\Lb$. In Section~\\ref{sec:gencov} we extend the representation of $\\Lb$ to the structure of a covariance matrix, \nand in Section~\\ref{sec:welldefL} we study two inequality conditions that are essential for the positive-definiteness of $\\Lb\\Lb^T$.\nWe conclude this paper by offering two possible applications, one for each of the suggested forms. \nIn Section~\\ref{sec:ttest} we present a simple $t$-test that employs the semi-partial correlation structure for testing the dependence of a single variable upon a set of multivariate normals. \nIn Section~\\ref{sec:randomcorr} we utilize the second parametrization to design a simple algorithm for the generation of random positive-definite correlation matrices. \nWe end the paper with the simple case of generalization of random AR(1) correlation in Section~\\ref{sec:AR1}.\n\n\\section{The first parametrization for Cholesky factor}\\label{sec:1stL}\nLet $\\Rb_n = (\\rho_{ij})_{ij=1}^n$ be a positive-definite correlation matrix, for which each sub-matrix $\\Rb_k = (\\rho_{ij})_{ij=1}^k$ is positive-definite. \nLet also $\\Lb = (l_{ij})_{ij = 1}^n$ be Cholesky factor of $\\Rb$, \n$|\\Rb|$ be the determinant of $\\Rb$, $\\Rb^{-1}$ its inverse, and $\\rhob_i^{*j} = (\\rho_{1j}, \\rho_{2j}, \\ldots, \\rho_{i-1,j})$ for $j\\ge i$, so $\\rhob_i \\equiv \\rhob_i^{*i}$. \n To simplify writing also set $\\Rb_0^{-1} \\equiv 1$. \n The first representation of $\\Lb$ will use \n the semi-partial correlations $l_{ji} = \\rhob_{ij(1,\\ldots,i-1)} = \\frac{\\rho_{ij}-\\rhob_i^{*j}\\Rb_{i-1}^{-1}\\rhob_i^T}{\\sqrt{1 - \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T}}$, $i\\le j$,\n\\begin{equation}\\label{chol:L1}\n     \\Lb = \\left( \\begin{array}{ccccc}\n       1 & 0 & 0 & \\cdots & 0 \\cr\n       \\rho_{12} & \\sqrt{1 - \\rho_{12}^2} & 0 \n& \\cdots & 0\\cr \n       \\rho_{13} & \\frac{\\rho_{23} - \\rho_{12}\\rho_{13}}{\\sqrt{ 1 - \\rho_{12}^2}} & \n         \\sqrt{1 -  \\rhob_3 \\Rb_2^{-1} \\rhob_3^T} & \\cdots & 0\\cr  \n\\vdots & \\vdots & \\vdots \n& \\ddots & 0 \\cr\n\\rho_{1n} &  \n     \\frac{\\rho_{2n} - \\rho_{12}\\rho_{1n}}{\\sqrt{1 - \\rho_{12}^2}} &  \n           \\frac{\\rho_{3n} - \\rhob_3^{*n} \\Rb_2^{-1} \\rhob_3^T}{\\sqrt{1 -  \\rhob_3 \\Rb_2^{-1} \\rhob_3^T}} & \n      \\cdots & \\sqrt{1 -  \\rhob_n \\Rb_{n-1}^{-1} \\rhob_n^T} \n\\end{array}\\right)\n\\end{equation}\nFor $i=j$, we have $\\rho_{ii(1,\\ldots,i-1)} = \\sqrt{1-\\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T}$, and $1 - \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T > 0$ for positive-definite $\\Rb_i$. \nSome may recognize $1 - \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T$ as the \\textit{schur-complement} \nof the matrix $\\Rb_{i-1}$ inside $\\Rb_i$ from the formula for computing the determinant of $\\Rb_i$, using the block matrix $\\Rb_{i-1}$~\\citep[pp. 188]{Harville1997},\n\\begin{equation}\\label{eq:schur_det}\n   |\\Rb_i|  =  \\left|\n \\begin{array}{cc} \\Rb_{i-1} & \n \\rhob_i^T \\\\ \\rhob_i & 1\\end{array}\n  \\right| \n   = |\\Rb_{i-1}| (1-\\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T).\n\\end{equation} \n \nTo show that $\\Rb_n = \\Lb\\Lb^T$ we introduce Theorem~\\ref{thm:CholSums}.  \n\\begin{thm}\\label{thm:CholSums}\nFor $i \\ge 1$ and $n \\ge j \\ge i+1$,\n\\begin{equation}\\label{Q_recursive}\n    \\rhob_{i+1}^{*j}\\Rb_i^{-1} \\rhob_{i+1}^T  \n           = \\sum_{k=1}^i \\rho_{ki(1,\\ldots ,k-1)} \\cdot \\rho_{kj(1,\\ldots ,k-1)} . \n\\end{equation}\n\\end{thm} \nBy the virtue that Cholesky factor of a positive-definite matrix has a unique representation, Theorem~\\ref{thm:CholSums} will serve as a general proof for the form (\\ref{chol:L1}). \nSome may recognize the Eq.~(\\ref{Q_recursive}) in Theorem~\\ref{thm:CholSums} as the inner-product used for the familiar algorithm of Cholesky Decomposition~\\citep[pp. 235]{Harville1997}:\n\\[\n   l_{ii} = \\left(1 - \\sum_{k=1}^{i-1} l_{ik}^2\\right)^{1\/2} \n   \\mbox{ and } \n   l_{ji} = \\left(\\rho_{ij} - \\sum_{k=1}^{i-1} l_{jk} l_{ik} \\right)\/l_{ii} .\n\\]\nSurprisingly, the equality in Theorem~\\ref{thm:CholSums} seems to be unknown or neglected. \nThe proof for Theorem~\\ref{thm:CholSums} will be given in~\\ref{sec:thm1proof}, and will be heavily based on the recursive arguments of the Lemma~\\ref{lem:recursive}: \n\\begin{lem} \\label{lem:recursive}\nFor $i \\ge 1$ and $n \\ge j \\ge i+1$,\n\\begin{equation}\\label{lem:recursive_eq}\n       \\rhob_{i+1}^{*j}\\Rb_i^{-1} \\rhob_{i+1}^T  \n     =\n       \\rhob_i^{*j}\\Rb_{i-1}^{-1} (\\rhob_i^{*i+1})^T  \n     + \\frac{ (\\rho_{i,i+1} - \\rhob_i^{*i+1} \\Rb_{i-1}^{-1} \\rhob_i^T)\n              (\\rho_{ij} - \\rhob_i^{*j} \\Rb_{i-1}^{-1} \\rhob_i^T)\n      }\n       {1 -  \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T} . \n\\end{equation}\n\\end{lem}\nThe proof for Lemma~\\ref{lem:recursive} will be given in~\\ref{sec:lemproof}.\n\n\\textbf{Remark.} Lemma~\\ref{lem:recursive} is useful to illustrate the recursive nature of the computation of the part $\\rhob_{i+1}^{*j}\\Rb_i^{-1} \\rhob_{i+1}^T$. \nAs an exercise, we suggest to verify that $\\rhob_2^{*j}\\Rb_1^{-1} \\rhob_2^T = \\rho_{1j}\\rho_{12}$, and then to compute $\\rhob_3^{*j}\\Rb_2^{-1} \\rhob_3^T$ by using Eq.~(\\ref{lem:recursive_eq}).\nThe result should be equal to $\\rhob_3^{*j}\\Rb_2^{-1} \\rhob_3^T  = \\rho_{13} \\rho_{1j}  + \\rho_{23(1)}\\rho_{2j(1)}$ as claimed by Theorem~\\ref{thm:CholSums}.\n \n \n\n\n \\section{The second parametrization for Cholesky factor}\\label{sec:2ndL}\nThe second parametrization for $\\Lb$ will follow directly from~(\\ref{chol:L1}) by applying Lemma~\\ref{lem:ab} \nthat claims an equivalence between semi-partial correlation coefficients to the difference between two successive schur-complements.\n\\begin{lem}\\label{lem:ab}\nLet us use the above notations and the positive-definiteness assumptions as before, and define $\\Rb_i^{*j} \\equiv  \\left( \\begin{array}{cc} \\Rb_{i-1} & (\\rhob_i^{*j})^T \\\\ \\rhob_i^{*j} & 1 \\end{array}\\right)$.\nThen for $j \\ge i+1 \\ge 3$ the difference between two successive ratios of determinants(or schur-complements) is\n\\begin{equation}\\label{eq:lem2}\n  |\\Rb_i^{*j}|\/|\\Rb_{i-1}| - |\\Rb_{i+1}^{*j}|\/|\\Rb_i|  \n     = \\left(\\rho_{ij}- \\rhob_i^{*j} \\Rb_{i-1}^{-1} \\rhob_i^T \\right)^2\n  |\\Rb_{i-1}|\/|\\Rb_i|.\n\\end{equation}\n\\end{lem}\nThe proof for Lemma~\\ref{lem:ab} will be found in~\\ref{sec:lemproof}.  \nSetting $s_{ij} \\equiv sign(\\rho_{ij(1,\\ldots,i-1)})$, it is possible to write Cholesky factor~(\\ref{chol:L1}) by the equivalent form\n\\begin{equation}\\label{chol:L2}\n    \\left( \\begin{array}{ccccc} \n             1 & 0 & 0 & \\cdots & 0 \\cr\n            s_{12}\\sqrt{1-\\frac{|\\Rb_2|}{1}} & \\sqrt{|\\Rb_2|} & 0 & \\cdots & 0\\cr\n            s_{13}\\sqrt{1-\\frac{|\\Rb_2^{*3}|}{1}} & s_{23}\\sqrt{\\frac{|\\Rb_2^{*3}|}{1} - \\frac{|\\Rb_3|}{|\\Rb_2|}} &  \\sqrt{\\frac{|\\Rb_3|}{|\\Rb_2|}} \n& \\cdots & 0\\cr \n\\vdots & \\vdots & \\vdots & \\ddots & 0 \\cr\ns_{1n}\\sqrt{1-\\frac{|\\Rb_2^{*n}|}{1}} &  \n       s_{2n}\\sqrt{\\frac{|\\Rb_2^{*n}|}{1} - \\frac{|\\Rb_3^{*n}|}{|\\Rb_2|}} &  \n       s_{3n}\\sqrt{ \\frac{|\\Rb_3^{*n}|}{|\\Rb_2|} -  \\frac{|\\Rb_4^{*n}|}{|\\Rb_3|}} & \n      \\cdots & \\sqrt{\\frac{|\\Rb_n|}{|\\Rb_{n-1}|}  }\n\\end{array}\\right)\n\\end{equation}\n\n\\subsection{The extension to nonsingular covariance matrix}\\label{sec:gencov}\nThe Cholesky factor~(\\ref{chol:L2}) can be easily extended to the general structure of nonsingular covariance when we replace each ratio $\\sigma_j^2|R_i^{*j}|\/|R_{i-1}|$ by its equivalent $|\\Sigmab_i^{*j}|\/|\\Sigmab_{i-1}|$.\nWe summarize this result into Theorem~\\ref{thm:Chlsky}.   \n\\begin{thm} \\label{thm:Chlsky}\nLet $\\Sigmab_n$ be a nonsingular covariance matrix with entries $\\sigma_{ij} = \\sigma_i\\sigma_j \\rho_{ij}$. \nLet $\\sigmab_i^{*j} \\equiv \\left(\\sigma_{1j},\\sigma_{2j}, \\cdots, \\sigma_{i-1,j}\\right)$, \nset $|\\Sigmab_1^{*j}|\\equiv \\sigma_j^2$, $|\\Sigmab_0| \\equiv 1$,\nand $\\Sigmab_i^{*j} \\equiv \\left( \\begin{array}{cc} \\Sigmab_{i-1} & \\left(\\sigmab_i^{*j}\\right)^T\n \\cr \\sigmab_i^{*j} \n & \\sigma_i^2 \\end{array}\\right)$.\nThen, Cholesky factor $\\Lb = (l_{ij})_{i,j=1}^n$ for $\\Sigmab_n$ is given by $l_{ji} = 0$ (for $j < i$), and by  \n\\[\n      l_{ji} = \n      \\left\\{\\begin{array}{ll}\n           sign(\\rho_{ij(1,\\ldots,i-1)}) \\sqrt{|\\Sigmab_i^{*j}|\/|\\Sigmab_{i-1}| \n                 - |\\Sigmab_{i+1}^{*j}|\/|\\Sigmab_i|} & \\mbox{ if } j >  i \\\\\n           \\sqrt{|\\Sigmab_i|\/|\\Sigmab_{i-1}|}          &  \\mbox{ if } i = j            \\end{array}\\right. .\n    \n\\]\n\\end{thm}  \n\\textbf{Remark.} Mathematically, Theorem~\\ref{thm:Chlsky} might as well describe Cholesky factor for arbitrary nonsingular symmetric matrix, when allowing the entries of $\\Lb$ to be complex numbers. \n\n\n\\subsection{Two order conditions on the magnitudes of sub-determinants essential to a well defined $\\Lb$}\\label{sec:welldefL}\nWe start with the well known order relation between the magnitudes of successive determinants~\\cite[pp. 525]{Yule1907,StuartOrdArnold2010}\n\\begin{equation}\\label{det:order}\n    1 \\ge |\\Rb_2| \\ge |\\Rb_3| \\ge \\cdots \\ge |\\Rb_{n-1}| \\ge |\\Rb_n| > 0.\n\\end{equation}\n One may alternatively see the order (\\ref{det:order}) as a direct result of Eq.~(\\ref{eq:schur_det}) when adding equal signs to account for $(\\rhob_i = 0)$'s.\nTo the order~(\\ref{det:order}) we will add a second order that arises from the positivity of the right-hand side of the Eq.~(\\ref{eq:lem2}) and seems to be rather new. \nFor $j=2,3, \\ldots,n$,\n\\begin{equation}\\label{detratio:order}\n    1 \\ge |\\Rb_2^{*j}| \\ge |\\Rb_3^{*j}|\/|\\Rb_2|  \n                   \\ge \\cdots \\ge  \n    |\\Rb_{j-1}^{*j}|\/|\\Rb_{j-2}|\\ge  \n    |\\Rb_j|\/|\\Rb_{j-1}| > 0 .\n \\end{equation}\n It is possible to view the order relations (\\ref{det:order}) and (\\ref{detratio:order}) as posing necessary and sufficient conditions for the correlation matrix $\\Rb = \\Lb \\Lb^T$ to be positive-definite. \nSince both order relations follow from the positive-definiteness property of $\\Rb$, and on the other hand, any failure to satisfy any of the determinant ordering in (\\ref{det:order}) or (\\ref{detratio:order}) will lead to ill defined $\\Lb$. \nIn Section~\\ref{sec:randomcorr} we shall show how to use the conditions (\\ref{det:order}) and (\\ref{detratio:order}) to generate positive-definite random correlation structures. \n\n\n\n\n\\section{Application I  - A simple $t$-test for linear dependence}\\label{sec:ttest}  \nAs a first application we will establish a procedure for testing the linear dependence of a single variable upon other variables by employing the first parametrization of Cholesky factor. \nLet $\\xb = (\\xb_1,\\cdots,\\xb_p)$ be a matrix of $N$ samples from a $p$-variate random variable that is multinormally distributed. Assume that $N>p$ and let $\\hat\\Rb = (r_{ij})_{i,j=1}^p$ be the estimated correlation sample matrix and $\\{r_{ij(1,\\ldots,i-1)}\\}_{i \\le j}$ be the nonzero elements of Cholesky factor for $\\hat\\Rb$. \nSuppose that, for some $k<p$, we wish to test the linear dependence of $\\xb_p$ upon $\\xb_1,\\xb_2,\\cdots,\\xb_k$. \nConsider the test \n\\[ \n  H_{0k} : \\rho_{1p} = \\rho_{2p(1)} = \\cdots = \\rho_{kp(1,2,\\ldots,k-1)} = 0\n\\] \nagainst the alternative \n\\[\nH_{1k} : \\rho_{ip(1,2,\\ldots,i-1)} \\ne 0 \\mbox{ for some } i<k.\n\\]\nUnder $H_{0k}$, the estimator $r_{kp(1,2 , \\ldots ,k-1)}$ forms a simple $t$ statistic~\\cite{Morrison2004}\n\\[\n  T_{kp:N} = \\frac{\\sqrt{N-k} \\cdot r_{kp(1,2 , \\ldots ,k-1)}}\n                 {\\sqrt{1 - r_{kp(1,2 , \\ldots ,k-1)}^2}} \n                 {H_{0k} \\atop \\sim } \\; t_{N-k}.\n\\] \nThe hypothesis $H_{0k}$ can be rejected, at level $\\alpha$ ($0<\\alpha <1$), if $|T_{kp:N}|>t_{\\alpha\/2,N-k}$. \nOne may further establish a sequential testing procedure that searches for the largest $k$ for which $H_{0k}$ can be rejected. \n\n\\textbf{Remark.} We leave it to the reader to verify that the null hypotheses $H_{0k}$ is equivalent to $\\rho_{1p} = \\rho_{2p} = \\cdots = \\rho_{kp} = 0$.   \n \n\\section{Application II - Generating realistic random correlations}\\label{sec:randomcorr}\nThe problem of generating random correlation structures is well discussed at the literature~\\cite{MarsagaliaOlkin84,Joe2006,MittelbachMatthiesenJorswieck2012}.\nHowever, in practice, many of the suggested procedures are not so easy to apply~\\cite{Holmes1991}, and when applied, some typically fail to provide a sufficient number of realistic correlation matrices~\\cite{BohmHornik2014}. \nMore recent algorithms for the generation of random correlations either utilize a beta distributio\n~\\cite{Joe2006,LewandowskiEtAl2009}, \nor employ uniform angular values~\\cite{RapisardaBrigoMercurio2007,RebonatoJackel2007,MittelbachMatthiesenJorswieck2012}.\nThe algorithm we will suggest in this section will be considerably simple. \nIt will be based on uniform values that are assigned to reflect the ratios $\\{|\\Rb_i^{*j}|\/|\\Rb_{i-1}|\\}_{i\\le j}$ which constitute the parametrization~(\\ref{chol:L2}). \nThe order of the values of $\\{|\\Rb_i^{*j}|\/|\\Rb_{i-1}|\\}_{i \\le j}$ will be chosen to preserve the ordering in (\\ref{det:order}) and (\\ref{detratio:order}), to ensure the positive-definiteness of $\\Lb\\Lb^T$. \nWe will start by choosing $n-1$ random uniform values in $(0,1]$, that will be further assigned by their size to reflect the determinants $\\{|\\Rb_j|\\}_{j=2}^n$, as directed by (\\ref{det:order}). \nThe diagonal of $\\Lb$ will be constructed from the ratios $\\{|\\Rb_j|\/|\\Rb_{j-1}|\\}_{j=1}^n$.  \nThen, for each row of $\\Lb$, $j$, an additional set of $j-2$ random uniform values will be chosen to serve as the ratios $\\{|\\Rb_i^{*j}|\/|\\Rb_{i-1}|\\}_{i=2}^{j-1}$, organized to keep the order as in (\\ref{detratio:order}).\nThe signs $s_{ij}$ will be chosen to be $(-1)^{Bernoulli(0.5)}$ and the matrix $\\Lb$ will be computed according to (\\ref{chol:L2}).  \n\n\\begin{enumerate}[Step 1.]\n\\item[\\textbf{Algorithm}] \\textbf{for generating realistic random correlation matrix $\\Lb\\Lb^T$}.\n\\item{\\textbf{Diagonals:}} Choose $n-1$ random uniform values from the interval $(0,1]$. \nOrder them in decreasing order, $U_{(2)} \\ge U_{(3)} \\ge \\cdots \\ge U_{(n)}$, \n and set $l_{11} = 1$, $l_{22}=\\sqrt{U_{(2)}}$, and $l_{jj} = \\sqrt{U_{(j)}\/U_{(j-1)}}$ for $j=3,\\ldots,n$;\n\\item{\\textbf{Rows:}} Repeat this step for each $j = 2,3, \\ldots, n-1$. Set $U_{(1)}^{(j)}\\equiv 1$ and $U_{(j)}^{(j)}\\equiv l_{jj}^2$.\n For $j\\ge 3$, choose $j-2$ more (additional) random uniform values $\\{U_i^{(j)}\\}_{i=2}^{j-1}$ inside $[l_{jj}^2,1]$ and sort them in decreasing order $U_{(2)}^{(j)} \\ge \\cdots \\ge U_{(j-1)}^{(j)}$.\nCompute $l_{ji} = \\sqrt{U_{(i)}^{(j)} - U_{(i+1)}^{(j)}}$ for $i=1,2,\\ldots,j-1$;\n\\item{\\textbf{Signs:}} Choose $n(n-1)\/2$ Bernoulli$(0.5)$ values $B_{ij}$'s for $j>i$, and multiply each $l_{ji}$ by $s_{ij} = (-1)^{B_{ij}}$;\n\\item Compute $\\Rb$ by $\\Lb \\Lb^T$ to obtain the actual correlation structure.\n\\end{enumerate}\n\n\n\\subsection{Generating random AR(1) structures}\\label{sec:AR1}\nWe end this paper by revealing the simple form of Cholesky factor for the $AR(1)$ structure. The AR(1) correlation matrix is defined by $\\rho_{ij} = \\rho^{|i-j|}$, \nand it is possible to verify that $|\\Rb_n| = (1-\\rho^2)^{n-1}$, $\\rhob_i^{*j} = \\rho^{j-i}\\rhob_i$ and     \n\\[ \n  \\rho_{ij} - \\rhob_i^{*j}\\Rb_{i-1}^{-1}\\rhob_i^T \n  = \\rho^{j-i}\\left(1-\\rhob_i\\Rb_{i-1}^{-1}\\rhob_i^T\\right) \n  = \\rho^{j-i} |\\Rb_i|\/|\\Rb_{i-1}|.\n\\]\nCholesky factor for the AR(1) structure enjoys the simple form: \n\\[\nl_{ji} = \\left\\{\\begin{array}{cc}\n\\rho^{j-1} & j \\ge i=1 \\cr\n\\rho^{j-i}\\sqrt{1-\\rho^2} & j \\ge i \\ge 2\n\\end{array}\\right. \n\\] \nHence, for any choice of $\\rho$, $|\\rho| < 1$, it is possible to transform the standard normals $(X_i)_{i=1}^n$ into the autocorrelated normals \n\\[\n  Y_i = \\rho^{i-1} X_1 + \\sqrt{1-\\rho}\\sum_{k=2}^i \\rho^{i-k} X_k, \n      \\qquad i = 1, \\ldots,n.\n\\] \n\n\\section*{Acknowledgments} I am thankful to the Editor and the Reviewers for their helpful discussion and their literature suggestion. Special thanks to Professor Sen for encouraging me to submit this paper and for his resourceful suggestion to add a test of hypotheses for the application part. Thanks to Professor Speed for introducing me to Yule's paper, and to Dr. Batista for her proofreading.  \n\n\\section*{References}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\section*{Appendix \\theappndx: #1} }\n\n\\def\\ba#1{\\begin{array}{#1}}\n\\def\\end{array}{\\end{array}}\n\\def\\beq#1{\\begin{equation}\\label{#1}}\n\\def\\end{equation}{\\end{equation}}\n\\newcommand{\\ab}[1]{\\left\\langle{#1}\\right\\rangle}\n\\newcommand{\\mathrm{const}\\,}{\\mathrm{const}\\,}\n\\newcommand{\\mathrm{dist}\\,}{\\mathrm{dist}\\,}\n\\newcommand{\\displaystyle}{\\displaystyle}\n\\newcommand{\\mathbb{R}}{\\mathbb{R}}\n\\newcommand{\\mathbb{N}}{\\mathbb{N}}\n\\newcommand{\\mathbb{Q}}{\\mathbb{Q}}\n\\newcommand{\\mathbb{Z}}{\\mathbb{Z}}\n\\newcommand{\\mathbf{e}}{\\mathbf{e}}\n\\newcommand{\\mathbf{x}}{\\mathbf{x}}\n\\newcommand{\\mathbf{r}}{\\mathbf{r}}\n\\newcommand{\\mathbf{y}}{\\mathbf{y}}\n\\newcommand{\\mathbf{z}}{\\mathbf{z}}\n\\newcommand{\\varepsilon}{\\varepsilon}\n\\newcommand{\\tilde\\alpha}{\\tilde\\alpha}\n\\newcommand{\\tilde\\gamma}{\\tilde\\gamma}\n\\newcommand{\\tilde\\xi}{\\tilde\\xi}\n\\newcommand{\\dlnxi}{\\lambda}\n\\newcommand{\\alpha_{\\infty}}{\\alpha_{\\infty}}\n\\newcommand{X^{0}}{X^{0}}\n\\newcommand{X^{\\!\\times}}{X^{\\!\\times}}\n\\newcommand{t^{0}}{t^{0}}\n\\newcommand{t^{\\ell}}{t^{\\ell}}\n\\newcommand{t^{r}}{t^{r}}\n\\newcommand{\\stackrel{\\text{\\scriptsize\\rm def}}{=}}{\\stackrel{\\text{\\scriptsize\\rm def}}{=}}\n\n\\newcommand{X^*}{X^*}\n\\newcommand{X^{**}}{X^{**}}\n\n\n\\newcommand{g_{\\mathrm{asym}}}{g_{\\mathrm{asym}}}\n\n\\newcommand{\\color{black}\\scriptsize}{\\color{black}\\scriptsize}\n\\newcommand{\\color{blue}\\scriptsize}{\\color{blue}\\scriptsize}\n\n\\newcommand{\\eop}{\\hfill$\\Box$\n  \n\\usepackage{pgf,tikz,pgfplots}\n\\usepackage{mathrsfs}\n\\usetikzlibrary{arrows}\n\n\\title{Precise asymptotics with log-periodic term in an elementary optimization problem}\n\n\\author{Sergey Sadov\\footnotemark[1]\\footnote{E-mail: serge.sadov@gmail.com}}\n\\date{}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nThe function $\\inf_n nx^{1\/n}$ has the asymptotics $eu+e d^2(u)\/(2u)+O(1\/u^2)$ as $x\\to\\infty$, where\n$u=\\log x$ and $d(u)$ is the distance from $u$ to the nearest integer. We generalize this observation. \n\nFirst,\nthe curves $y=nx^{1\/n}$ can be written parametrically\nas $\\log x=nt$, $y=nt$. In general, let \n$(u_n(t),v_n(t))$ be a family of parametric\ncurves  with asymptotics \n$u_n=n p_1(t)+q_1(t)+r_1(t)\/n+O(1\/n^2)$ and \n$v_n=n p_2(t)+q_2(t)+r_2(t)\/n+O(1\/n^2)$.\nSuppose the function $p_1(t)\/p_0(t)$ has a unique \nnondegenerate minimum in the parameter domain. \nIt is shown that the asymptotics of their lower envelope\n$v(u)=\\inf_{n,t} v_n(t)$ while $u=u_n(t)$, has the asymptotics of the form $v(u)=a_0 u+a_1+\\Phi(u)\/u+O(1\/u^2)$, where $\\Phi(\\cdot)$ is an affinely transformed function $d^2(\\cdot)$.\n\nSecond, note that $nx^{1\/n}$ is the minimum\nof the sum $t_1+t_2\/t_1+\\dots+t_{n}\/t_{n-1}$ subject to the constraint $t_n=x$. We consider a similar\nasymptotic problem for the sums $t_1+t_2\/(t_1+1)+\\dots+t_n\/(t_{n-1}+1)$. \nLet $F_n(x)$ is the minimum value of the $n$-term sum under the constraint $t_n=x$. Define $F(x)=\\inf_n F_n(x)$. We show that $F(x)=eu-A+e d^2(u+b)\/(2u)+O(1\/u^2)$, $u=\\log x$, with certain numerical constants $A$ and $b$. We present alternative forms of this optimization problem, in particular, \na ``least action'' formulation. Also we find the asymptotics $F_n^{(p)}(x)=e\\log n-A(p)+O(1\/\\log n)$\nfor the function arising from the sums with denominators of the form $t_j+p$ with arbitrary $p>0$ and establish\nsome facts about the function $A(p)$. \n\n\\medskip\n{\\em Keywords}: AM-GM inequality, asymptotics, dynamic programming, enveloping curve, recurrence relations.\n\n\\medskip\nMSC: \n26D15,\n26D20 \n\\end{abstract}\n\n\n\\section{Statement of results}\n\\label{sec:intro}\n\nThe main result of this work concerns the asymptotic behaviour as $x\\to+\\infty$ of the function\n$$\n F(x)=\\inf_{n\\in\\mathbb{N}} F_n(x),\n$$\nwhere\\footnote{\nThe function $F(x)$ in another guise (see Proposition~\\ref{prop:altoptfn}) appeared in the study of a certain cyclic inequality\n\\cite{Sadov_2022_maxcyc}, which motivated this paper.} \n\\begin{align}\n&F_n(x)=\\inf_{t_1,\\dots,t_{n-1}\\geq 0} S(t_1,\\dots,t_{n-1},x),\n\\label{op_main}\n\\\\[1ex]\n& S(t_1,\\dots,t_n)= t_1+\\frac{t_2}{t_1+1}+\\dots+\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{t_n}{t_{n-1}+1}\n\\label{S1}\n\\end{align}\nIn other words, $C=F(x)$ is the best constant, independent of $n$ and $\\{t_j\\}$, in the inequality\n$$\n t_1+\\frac{t_2}{t_1+1}+\\dots+\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{x}{t_{n-1}+1}\\geq C.\n$$ \n\nWe will use the notation\n$$\n \\ab{x}=\\mathrm{dist}\\,(x,\\mathbb{Z}).\n$$\nThe function $x\\mapsto\\ab{x}$ is a $1$-periodic piecewise-linear, continuous function oscillating between $0$ and $1\/2$.\n\n\\begin{theorem}\n\\label{thm:main}\nLet\n$\n u=\\log x.\n$\nThere exist numerical constants\n$$\n\\ba{l}\n A\\approx 1.7046560372,\n\\\\\n b\\approx 0.6973885601, \n\\end{array}\n$$\nsuch that \nthe function $F(x)$\nhas the asymptotics \n\\begin{equation}\n\\label{asf}\nF(x)=eu-A+\\frac{e}{2}\\,\\frac{{\\ab{u+b}}^2}{u}\n+O\\left(\\frac{1}{u^2}\\right)\n\\end{equation}\nas $x\\to\\infty$.\n\\end{theorem}\n\nTheorem~\\ref{thm:main} will be better understood \nin the context of two \ntheorems stated below: \nTheorem~\\ref{thm:AMGM}, which is a simpler result of the same kind, and Theorem~\\ref{thm:paramcurves} of a technical nature, which\nshows that a periodic (up to a small error)\nremainder term appears in a class of optimization problems\nconcerning the lower envelope of parametric curves. \nThe asymptotic formula \\eqref{asf} will be eventually obtained through Theorem~\\ref{thm:paramcurves}.\n\n\\medskip\nLet us consider a simpler analogue of the functions\n$F_n(x)$: \n\\begin{equation}\n\\label{ep_AM-GM}\nF^{(0)}_n(x)=\\inf_{t_1,\\dots,t_{n-1}>0} \\left(\n t_1+\\frac{t_2}{t_1}+\\dots+\\frac{t_{n-1}}{t_{n-2}}+\\frac{x}{t_{n-1}}\n \\right).\n\\end{equation}\n\nThe definition of $F^{(0)}_n(x)$ can be written as\n$$\n F^{(0)}_n(x)=\\inf_{a_1,\\dots,a_{n}>0}\n(a_1+\\dots+a_n)\n\\quad\\text{subject to $\\;a_1\\cdot\\dots\\cdot a_n=x$}.\n$$\n\nOne recognizes the constrained optimization problem associated with the inequality between the arithmetic and geometric means, hence\n$$\n F^{(0)}_n(x)=n x^{1\/n}.\n$$ \n\n\\begin{theorem}\n\\label{thm:AMGM}\nLet $u=\\log x$. The function\n$$\n F^{(0)}(x)=\\inf_{n\\in\\mathbb{N}} F^{(0)}_n(x)\n$$\nhas the asymptotics\n\\begin{equation}\n\\label{asf0}\n F^{(0)}(x)=eu+\\frac{e}{2}\\,\\frac{\\ab{u}^2}{u}+O\\left(\\frac{1}{u^2}\\right)\n\\end{equation}\nas $x\\to\\infty$.\n\\end{theorem}\n\n\\begin{figure}\n\\begin{picture}(260,175)\n\\put(0,0){\\includegraphics[scale=0.5]{f0_approximation}}\n\\put(265,16){$u$}\n\\end{picture}\n\\caption{Illustration of Theorem~\\ref{thm:AMGM}. Blue line: the function $u\\mapsto F^{(0)}(x)-e u$, $u=\\log x$. Gray line:\nthe correction term $(e\/2){\\ab{u}}^2\/u$.}\n\\label{fig:f0corr}\n\\end{figure}\n\nThe asymptotic approximation \\eqref{asf0} is illustrated in Fig.~\\ref{fig:f0corr}.\n\n\n\\medskip\nTheorems~\\ref{thm:main} and \\ref{thm:paramcurves} both\ndeal with a function defined as infimum over a family.\nFor instance, the graph of the function $F^{(0)}(x)$ is the lower envelope of the curves $y=nx^{1\/n}$, which can be written parametrically as $\\log x=nt$, $y=ne^t$. From this point of view Theorem~\\ref{thm:AMGM} is an easy consequence of our next theorem. To keep things simple where possible, we will prove Theorem~\\ref{thm:AMGM} directly; however, Theorem~\\ref{thm:paramcurves} will be fully relevant for the proof of Theorem~\\ref{thm:main}.\nThe relevance of the general asymptotic pattern\n\\eqref{asfabs} is apparent already.  \n\n\n\\begin{theorem}\n\\label{thm:paramcurves}\nConsider a family of parametric curves\n$u=\\xi_n(t)$, $v=\\eta_n(t)$, $t\\in I$, where $I$ is a segment of the real line. \nSuppose the functions $\\xi_n(t)$, $\\eta_n(t)$ have the asymptotic behaviour%\n$$\n\\ba{l}\n\\displaystyle\n \\xi_n(t)=p_0(t) n+ q_0(t)+ r_0(t)n^{-1}+ O(n^{-2}),\n\\displaystyle\n\\\\\n \\eta_n(t)=p_1(t) n+ q_1(t)+ r_1(t)n^{-1}+ O(n^{-2})\n\\end{array}\n$$\nas $n\\to\\infty$, uniformly in $t\\in I$.\n\nLet $v=f(u)$ be the lower envelope of the curves so defined. That is, given $u$, we determine the set%\n\\footnote{This set is nonempty if $u$ is sufficiently large.}\nof those $n$ for which the equation $\\xi_n(t)=u$ has a solution and define\n$$\n f(u)=\\inf_{(n,t):\\, \\xi_n(t)=u} \\eta_n(t).\n$$\n\nDenote\n$$\n \\beta(t)=\\frac{p_1(t)}{p_0(t)}, \n\\qquad \n\\delta(t)=p_0(t)r_1(t)-p_1(t)r_0(t).\n$$\nWe make the following assumptions.\n\n\\smallskip\n{\\rm(i)} The \ncoefficients $p_i(\\cdot)$, $q_i(\\cdot)$, $r_i(\\cdot)$ are of class $C^2(I)$. \n\n\\smallskip\n{\\rm(ii)}  $p_0(t)>0$, $p_1(t)>0$, $p_0(t)'>0$ on $I$.\n\n\\smallskip\n{\\rm(iii)} The function $\\beta(t)$ has the unique point of minimum $t=t_0$ on $I$ and\n$$\n\\beta(t)=b_0+\\frac{b_2}{2}(t-t_0)^2+o(t-t_0)^2\n\\quad \\text{near $t_0$}.  \n$$\n\nThen\n\\beq{asfabs}\n f(u)=a_0 u+a_1+\\frac{\\Phi(u)}{u}+O\\left(\\frac{1}{u^2}\\right)\n\\quad\\text{as $u\\to\\infty$},\n\\end{equation}\nwhere\n\\beq{coefa0a1}\n a_0=b_0,\n\\qquad\n a_1=q_1(t_0)-b_0 q_0(t_0),\n\\end{equation}\nand\n\\begin{align}\n& \\Phi(u)=a_2+a_3\\ab{\\frac{u-q_0(t_0)}{p_0(t_0)}}^2,\n\\label{asgenPhi}\n\\\\[2ex]\\displaystyle\n& a_2\n=-\\frac{(q_1'(t_0)-b_0 q_0'(t_0))^2}{2b_2}+\\delta(t_0),\n\\qquad\na_3=\\frac{b_2}{2}\\,\\left(\\frac{(p_0(t_0))^2}{p_0'(t_0)}\\right)^2.\n\\label{coefa2a3}\n\\end{align}\n\\end{theorem}\n\nProof of Theorem~\\ref{thm:paramcurves} is given in Appendix~\\ref{app:paramcurves}.\n\n\\medskip\nOur final theorem concerns an interpolation between\nTheorems~\\ref{thm:main} and~\\ref{thm:AMGM}.\n\nIndeed, it is natural to treat the functions $F^{(0)}(x)$\nand $F(x)=F^{(1)}(x)$ as members of a one-parameter family\nof functions\n$$\n F^{(p)}(x)=\\inf_{n\\in\\mathbb{N}} F^{(p)}_n(x)\n$$\nwith any $p\\geq 0$,\nwhere\n$$\n F^{(p)}_n(x)=\\inf_{t_1,\\dots,t_{n-1}\\geq 0,\\;t_n=x} \nS^{(p)}_n\n(t_1,\\dots,t_n),\n$$\n\\begin{equation}\n\\label{Snpt}\n S^{(p)}_n(t_1,\\dots,t_n)=t_1+\\sum_{j=2}^{n}\\frac{t_j}{t_{j-1}+p}.\n\\end{equation}\nIn particular,\n$$\n F^{(p)}_1(x)=x,\\qquad F^{(p)}_2(x,p)=\\inf_{t\\geq 0}\\left(t+\\frac{x}{t+p}\\right)=\\begin{cases}\n2\\sqrt{x}-p, \\quad x\\geq p^2,\\\\\nx\/p, \\qquad x\\leq p^2.\n\\end{cases}\n$$\n\nWe will not chase the asymptotics of $F^{(p)}(x)$ as precisely as we did in Theorems~\\ref{thm:main}--\\ref{thm:AMGM} but rather\nfocus on the constant term in the asymptotics. In particular, we determine its behaviour as $p\\to +0$. We also reveal that the parameter value $p=1$ separates the regions with different analytic form of $F^{(p)}$: it is much simpler for $p>1$. This latter fact gives an additional support to our attention to the function $F(x)=F^{(1)}(x)$ as the main subject of this paper. \n\n\\begin{theorem}\n\\label{thm:genpar}\nLet $u=\\log x$. \n\n\\smallskip\n{\\rm(a)} For any $p>0$ \n\\begin{equation}\n\\label{Fpasym}\n F^{(p)}(x)=eu-A(p)+O(u^{-1}) \\quad\\text{as $x\\to\\infty$}.\n\\end{equation}\nThe function $A(\\cdot)$ is increasing.\n\n\\smallskip\n{\\rm(b)} If $p>1$, then\n\\begin{equation}\n\\label{Fpgt1}\n F^{(p)}(x)=F\\left(\\frac{x}{p}\\right),\n\\end{equation}\nso $A(p)=A+e\\log p$.\n\n\\smallskip\n{\\rm(c)} If $0<p<1$, then\nthe function $A(p)$ satisfies the functional equation\n\\begin{equation}\n\\label{feq-Ap}\n A(p)=\\max_{1\\leq u\\leq e} \\left(A\\left(\\frac{p}{u}\\right)-u+e\\log u\\right)+p.\n\\end{equation}\nThe asymptotics of $A(p)$ as $p\\to +0$ is\n$$\nA(p)=\\frac{p}{1-e^{-1}}+o(p).\n$$\n\\end{theorem}\n\n\n\\medskip\nThe plan of the paper is as follows.\n\n\\smallskip\nIn Section~\\ref{sec:variants} we describe a few guises in which the optimization problem leading to the functions $F_n(x)$ or $F^{(p)}_n(x)$ may arise. In particular,\nwe discuss a ``least action formulation'' and introduce a recurrence that determines the ``extremal trajectory'' for each $n$.\n\n\\smallskip\nIn Section~\\ref{sec:simpleproofs} we prove what is easy to prove: Theorem~\\ref{thm:AMGM} and the crude asymptotics\n$F(x)=e\\log n+O(1)$. \n\n\\smallskip\nIn Section~\\ref{sec:minimizers} we begin the analysis\nof candidate minimizers (extremal trajectories) for\nthe main problem. We identify some special points\nfor every $n$ that play specific roles in the analysis.\nWe illustrate the introduced concept with graphs\nand point out some pitfalls.\n\n\\smallskip\nIn Section~\\ref{sec:proofmain} the facts observed in\nSection~\\ref{sec:minimizers} are fully formalized and proved and the proof of Theorem~\\ref{thm:main} is completed by reducing it to a form suitable for application of Theorem~\\ref{thm:paramcurves}.\n \n\\smallskip\nIn Section~\\ref{sec:Ap} we prove Theorem~\\ref{thm:genpar}\nand describe an efficient algorithm, based on a recurrence,\nto tabulate the function $A(p)$ for $0<p<1$.\n\n\\smallskip\nThere are two appendices containing proofs technically unrelated to the main part of the paper.\n\n\\smallskip\nIn Appendix~\\ref{app:paramcurves} Theorem~\\ref{thm:paramcurves} is proved. \n\n\\smallskip\nIn Appendix~\\ref{app:alpha} we rigorously prove a visually obvious property (Proposition~\\ref{prop:alpha})\nof a function $\\alpha_\\infty(x)$, which is important in the main proof.\n\nThe proof uses contour integration for estimating derivatives of a real function.\nSome elements of the proof refer to numerical evaluations but always in a ``provable'' way. \n\n\n\\iffalse\nSolving an optimization problem $S(\\mathbf{t})\\mathop{\\to}\\min$ is equivalent to establishing the inequality $S(\\mathbf{t})\\geq f$\nwhere the value $f=S_{min}$ is best possible.\nThroughout the paper the letters $S$ and $f$, possibly with sub\/superscripts, will designate, respectively,\nthe objective functions and their optimal values.\nThe optimization variables --- the components of \nthe vector $\\mathbf{t}$ --- will be assumed nonnegative (in denominators --- positive).\n\nIn the said sense, the classical inequality between the arithmetic and geometric means (AM-GM) can be considered as a closed form solution of the constrained optimization problem\n\\beq{cons0}\nS(\\mathbf{t}) \\stackrel{\\text{\\scriptsize\\rm def}}{=} \nt_1+\\dots+t_n\\to \\min \\quad\\text{\\rm subject to}\\;\\; t_1\\cdot\\dots\\cdot t_n=x;\n\\end{equation}\nthe minimum value is\n$$\nf_n^{(0)}(x)=nx^{1\/n}.\n$$\n\n\nIn this paper we study a recurrence that has a similar look to \\eqref{recurf0}:\n\\beq{recurf}\n f_n(x)=\\min_{y\\geq 0}\\left(f_{n-1}(y)+\\frac{x}{y+1}\\right),\n\\quad n\\geq 2,\n\\end{equation}\nwith initial condition identical to that for $f_1^{(0)}$:\n\\beq{recurf1}\n f_1(x)=x.\n\\end{equation}\n\nFor example, \n\\beq{f2}\n f_2(x)=\\min_{y\\geq 0}\\left(y+\\frac{x}{y+1}\\right)\n=\n\\begin{cases}\nx,\\; \\text{if $x\\leq 1$},\\\\\n2\\sqrt{x}-1,\\; \\text{if $x> 1$}.\n\\end{cases}\n\\end{equation}\nA longer calculation shows that\n$$\n f_3(x)=\n\\begin{cases}\nf_2(x),\\; \\text{if $x\\leq 4$},\\\\\n-1+2u+\\sqrt{\\frac{x}{u}}=3x^{1\/3}+O(1),\\; \\text{if $x> 4$},\n\\end{cases}\n$$ \nwhere $u=u(x)$ is the larger (of the two) positive root of the equation $(u^2+1)^2=xu$.\n\nA legitimate quibble concerning the use of the symbol `$\\min$' as opposed to `$\\inf$' in \\eqref{recurf} will be addressed in Sec.~\\ref{sec:prelim}.\n\nOne simple fact about the sequence $(f_n(x))$ is that it is nonincreasing (Proposition~\\ref{prop:monotfn}). It allows one to define the function \n\\beq{deflimf}\n f(x)=\\lim_{n\\to\\infty} f_n(x)=\\inf_{n\\geq 1} f_n(x).\n\\end{equation}\nMoreover, we show (Proposition~\\ref{prop:fnstab}) that for any fixed $x>0$ the sequence $(f_n(x))$ stabilizes: \nthere exists a non-decreasing function $x\\mapsto \\nu(x)$ taking values in positive integers such that\n$$\n f(x)= f_n(x)\n, \\quad \\forall n\\geq \\nu(x).\n$$\n\nOur main result \nconcerns the asymptotics of $f(x)$ as $x\\to\\infty$.\n\nTo compare, the sequence $f_n^{(0)}(x)=nx^{1\/n}$\nis not monotone. It has a minimum for every fixed $x>0$; the function\n\\beq{defminf0}\n f^{(0)}(x)=\\min_{n\\geq 1} f_n^{(0)}(x),\n\\end{equation}\nhas the asymptotics $f^{(0)}(x)=e\\log x+O(1\/\\log x)$,\nsame as in \\eqref{asf1} with constant term $A=0$; also likewise in \\eqref{asf1}\nthe remaider is not $o(1\/\\log x)$. This is a simple fact, see Proposition~\\ref{prop:asf0} in Sec.~\\ref{sec:prelim}.\nA somewhat more elaborate yet also rather quick result\nis\nthe crude\nasymptotic formula \n$f(x)=e\\log x+O(1)$ (Proposition~\\ref{prop:crudeas} in Sec.~\\ref{ssec:crudeas}).\nA proof of the two-term asymptotics \\eqref{asf1} requires much more effort.\n\nThe described analogy is meant to justify the attention to the function $f(x)$ and the sequence $(f_n(x))$ in the eyes of a reader unconcerned about author's contemplation.\n\nIn truth, my motives to\n(a) study the function $f(x)$, and (b) work out its asymptotics in the precise form, were different. \n\nFor (a), the function $f(x)$ --- and the need to analyse its asymptotic behavious --- appeared in my study \\cite{Sadov_2022maxcyc} of a certain cyclic inequality (or, better, an extremal problem for a cyclic sum) that, unlike a number of previously published variations, allows for ``uncycling''. \nAs a matter of fact, the definition of the function $f(x)$ in \\cite{Sadov_2022maxcyc} is not quite the same as above but equivalent, see \\eqref{fe1} in Sec.~\\ref{sec:prelim}. The equivalence is proved in\nProposition~\\ref{prop:funeqf}.\n\nAs concerns (b), it is a purely aesthetical or sporting\ndrive. A result of a similar nature is proved in my recent paper \\cite{Sadov_2021}.\nIn spite of some semblance of the optimization problems --- see Remark in Sec.~\\ref{ssec:leastaction} --- technical parallels between that case and the present one don't go far\nand similarity is more in spirit. The eventual proof of the asymptotic formula in both cases depends on the analysis of trajectories of discrete dynamical systems determining the critical points of the objective functions. That analysis, in essense, consists of numerous steps, some plain and some delicate,\njustifying visually observed properties or relevant functions and sequences, such as monotonicity and convexity.\n\\fi\n\n\\iffalse \n--------------\n\n\nPrevious version \n\n\nLet $f(x)=x$ for $0<x\\leq 1$ and define $f(x)$\nin the intervals $(1,2], (1,3], \\dots$ recursively by means of  \nthe functional equation\n\\beq{fe1}\n f(x)=\\min_{0< y<x-1}\\left(f(y)+\\frac{x}{y+1}\\right).\n\\end{equation}\n\nFor example, if $1<x\\leq 2$, then\n$$\n f(x)=\\min_{0< y<x-1}\\left(y+\\frac{x}{y+1}\\right)\n=-1+\\min_{1<\\tilde y<x}\\left(\\tilde y+\\frac{x}{\\tilde y}\\right)\n=2\\sqrt{x}-1.\n$$\n\nThe main result \nconcerns the asymptotics of $f(x)$ as $x\\to\\infty$.\n\n\\begin{theorem}\n\\label{thm:mainasym}\nThere exists a constant \n$$\n A\\approx 0.70465603\n$$\nsuch that\n\\beq{asf1}\n f(x)=e\\log x-A+ O\\left(\\frac{1}{\\log x}\\right)\n\\end{equation}\nas $x\\to\\infty$.\nThe symbol $O$ in the remainder term\ncannot be replaced by $o$.\n\\end{theorem}\n\nAlong with function $f(x)$ we consider the sequence of functions $f_n(x)$, $n=1,2,\\dots$, defined recurrently for $x>0$\nby \n\\beq{recurf}\n f_n(x)=\\min_{y\\geq 0}\\left(f_{n-1}(y)+\\frac{x}{y+1}\\right),\n\\quad n\\geq 2,\n\\end{equation}\nwith initial condition\n$$\n f_1(x)=x.\n$$\nWe will show that there exists a non-decreasing function $x\\mapsto \\nu(x)$ taking values in positive integers such that\n$$\n f(x)=\\lim_{n\\to\\infty} f_n(x)=\\min_{n\\geq 1} f_n(x)\n=f_{k}(x), \\quad \\forall k\\geq \\nu(x).\n$$\n\nThe recurrence \\eqref{recurf} has a similar look to the recurrence\n\\beq{recurf0}\n f^{(0)}_n(x)=\\min_{y\\geq 0}\\left(f_{n-1}(y)+\\frac{x}{y}\\right),\n\\quad n\\geq 2,\n\\end{equation}\nwith the initial condition \n$f^{(0)}_1(x)=x$.\nOne recognizes the latter as a \ndynamical programming formulation of the constrained optimization problem\n\\beq{cons0}\nf^{(0)}_n(x)=\\min_{t_1\\cdot\\dots\\cdot t_n=x} (t_1+\\dots+t_n),\n\\end{equation}\nThe classical inequality between the arithmetic and geometric means (AM-GM) amounts to the fact that \n$$\nf_n^{(0)}(x)=nx^{1\/n}.\n$$\nIn \\cite[\\S~7]{BeckBel_1961} a ``dual'' dynamical programming formulation is considered, in which one maximizes the product of $n$ undeterminates subject to the prescribed value of their sum.\n\nThe function \n$$\n f^{(0)}(x)=\\min_{n\\geq 1} f_n^{(0)}(x)\n$$\nhas the asymptotics $f^{(0)}(x)\\sim e\\ln x$\nsimilar to \\eqref{asf1}. However, $f^{(0)}(x)\\neq \\lim_{n\\to\\infty}f_n^{(0)}(x)$. \nSome mark points in our analysis of the\nfunctions $f_n(\\cdot)$ are analogous to their much simpler counterparts pertaining to the functions $f_n^{(0)}(x)$. \n\nThe described analogy is meant to justify the attention to the function $f(x)$ and the sequence $(f_n(x))$ in the eyes of a general reader,\nand, from the author's perspective, to justify a\nseparate title for an item that can be of interest in its own right notwithstanding motives that brought it\ninto being.\n\nIn truth, I needed Theorem~\\ref{thm:mainasym} as part of  the study of a certain cyclic inequality (or, better, an extremal problem for a cyclic sum) that allows for ``uncycling'' \\cite{Sadov_2022maxcyc}. \n\nLastly, and to mention the author's other internal motive, let me point out the possibility to treat the function $f_n(x)$ in terms of\nthe ``least action principle'' with discrete time,\n\\beq{lapfn}\n f_n(x)=\\min_{\\{(t_0,\\dots,t_n)\\mid t_0=0,t_n=x\\}}\nS_n(t_0,\\dots,t_n).\n\\end{equation}\nThe ``action function'' $S_n(\\dots)$ is analogous to the action in classical mechanics (the integral of the Lagrangian along a virtual trajectory)\n\\beq{actionn}\n S_n(t_0,t_1,\\dots,t_n)=\\sum_{j=1}^n L(t_{j-1},t_j).\n\\end{equation}\nThe Lagrangian in the present case is\n\\beq{lagr1}\n L(p,q)=\\frac{p}{q+1}.\n\\end{equation}\nAn asymptotic problem of the similar nature, with the Lagrangian\n$$\n L(p,q)=q+\\frac{1+p}{q}\n$$\nhas been studied in the author's recent paper \\cite{Sadov_2021}. The corresponding asymptotics of the constrained minimum defined as in \\eqref{lapfn} is\n$\\min S_n=3n-C+O(\\rho^{-n})$ with specific constants $C$ and (best possible) $\\rho<1$.\nThere, as well as here, we do not attempt to generailze (introducing more parameters, say), but rather focus on the precise estimate of the remainder term in the asymptotics.\nIn both cases the analysis of optimal trajectories plays the crusial role and the proof, in essence, consists of steps justifying experimentally observed properties or relevant functions and sequences (monotonicity, convexity). However, in spite of some visual resemblance of the two Lagrangians, the details are very different.\n\n\n\n\nThe plan is as follows.\nWe begin with some simple observations concerning the functions $f(x)$, $f_n(x)$, and their relations with\n$f^{(0)}(x)$, $f_n^{(0)}(x)$. Also in Sec.~\\ref{sec:prelim} we prove the crude asymptotics\n$f(x)=e\\log x+O(1)$, which is much simpler result than\nTheorem~\\ref{thm:mainasym}.\nThen in Sec.~\\ref{sec:proofmain} we prove Theorem~\\ref{thm:mainasym} modulo an auxiliary theorem concerning the properties of a function that determines the ``asymptotic shape'' of the optimal trajectories for the problem \\eqref{lapfn}.\nThat auxiliary theorem is proved in Sec.~\\ref{sec:proofaux}. Some elements of the (admittedly boring) proof rely on numerical evaluations; however, we never recourse to an ``experimental evidence'' as the last argument. \n\n\\fi\n\n\\section{Alternative forms of the extremal problem}\n\\label{sec:variants}\n\n\\subsection{Prototype: formulations for the AM-GM}\n\nThe AM-GM optimization problem as stated in \\eqref{ep_AM-GM} involves\nonly ``soft'' constraints $t_j>0$.\nThe objective function can be written more symmetrically by introducing two extra indeterminates $t_0$ and $t_n$ and subjecting them to the artificial, boundary-value constraints:  \n\\begin{align}\n& F^{(0)}_n(x)=\\min_{\\mathbf{t}>0,\\; t_0=1,t_n=x} \\hat S_n^{(0)}(t_0,t_1,\\dots,t_n),\n\\label{optfn0}\n\\\\[3ex]\n& \\hat S_n^{(0)}(t_0,t_1,\\dots,t_n)\\stackrel{\\text{\\scriptsize\\rm def}}{=}\n\\frac{t_1}{t_0}+\\frac{t_2}{t_1}+\\dots+\\frac{t_{n-1}}{t_{n-2}}+\\frac{t_n}{t_{n-1}}\n\\label{L0n}\n.\n\\end{align}\n\nA proof of the AM-GM inequality by R.~Bellmann's dynamic programming approach%\n\\footnote{\\cite[\\S~7]{BeckBel_1961} presents a dual dynamic programming formulation of the AM-GM inequality, in which one maximizes the product of $n$ indeterminates subject to the prescribed value of their sum.}\n amounts to replacing the multivariate optimization problem by\na sequence of univariate optimization problems, \nwhere the functions $F_n^{(0)}$ are defined recurrently by\nputting $F^{(0)}_1(x)\\stackrel{\\text{\\scriptsize\\rm def}}{=} x$ and \n\\begin{equation}\n\\label{recurf0}\n F^{(0)}_n(x)=\\min_{y> 0}\\left(F_{n-1}^{(0)}(y)+\\frac{x}{y}\\right),\n\\quad n\\geq 2.\n\\end{equation}\n\n\\smallskip\n\n\\subsection{Formulation with boundary constraints}\n\\label{ssec:bconst}\n\nThe constrained optimization problem that parallels\n\\eqref{optfn0} is\n\\begin{align}\n& F_n(x)=\\inf_{\\mathbf{t}\\geq 0,\\; t_0=0,t_n=x}\n\\hat S_n(t_0,t_1,\\dots,t_n), \n\\label{optfn}\n\\\\[3ex]\n& \\hat S_n(t_0,t_1,\\dots,t_n)\\stackrel{\\text{\\scriptsize\\rm def}}{=}\n\\frac{t_1}{t_0+1}+\\frac{t_2}{t_1+1}+\\dots+\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{t_n}{t_{n-1}+1}\n\\label{Ln}\n.\n\\end{align}\n\nIn Sec.~\\ref{ssec:leastaction} we will elaborate on this\npresentation of the problem.\n\n\\subsection{Formulation with additive constraint}\n\n\\begin{prop}\n\\label{prop:altoptfn}\nFor every $n\\in\\mathbb{N}$, the function $\\bar F_n(x)$ defined by\n\\begin{equation}\n\\label{op_additive}\n \\bar F_n(x)=\\inf_{\\{u_1,\\dots,u_n\\mid \\sum u_j=1\\}} \\left(\\frac{u_1}{u_2}+\\dots+\\frac{u_{n-1}}{u_n}+xu_n\\right)\n\\end{equation}\nis identical to $F_n(x)$. \n\\end{prop}\n\nThe objective function in \\eqref{op_additive}\ncan be written as $\\hat S^{(0)}_{n}(1\/u_1,\\dots,1\/u_n, x)$,\nusing the notation \\eqref{L0n}. Thus \\eqref{op_additive}\ncan also be seen as a result of the replacement of\nthe boundary condition $t_0=1$ in \\eqref{optfn0} by the nonlocal constraint $1\/t_0+\\dots+1\/t_{n-1}=1$.\n\n\nThe functions $F_n(x)$ in the guise \\eqref{op_additive} appeared in my work \\cite{Sadov_2022_maxcyc}.\n\n\\begin{proof}\n\\iffalse\nClearly, $\\tilde f_1(x)=x$. We make the inductive assumption that\n$$\n \\tilde f_{n-1}(y)=\\min_{\\{p'_1,\\dots,p'_{n-1}\\mid \\sum p'_j=1\\}} \\left(\\frac{p'_1}{p'_2}+\\dots+\\frac{p'_{n-2}}{p'_{n-1}}+yp'_{n-1}\\right).\n$$\nNow we put $p_n=(1+y)^{-1}$ and \n$$\n p_j=\\frac{p'_j}{y+1},\\quad j=1,\\dots,n-1.\n$$\n\\fi\n\\iffalse\nConsider the case $n=2$. Here\n$$\n f_2(x)=\\min_{t_1>0}\\left(t_1+\\frac{x}{t_1+1}\\right),\n$$\nwhile\n$$\n \\tilde f_2(x)=\\min_{p_1+p_2=1}\\left(\\frac{p_1}{p_2}+xp_2\\right).\n$$\nPut\n$$\n p_2=\\frac{1}{t_1+1},\\qquad p_1=t_1 p_2=\\frac{t_1}{t_1+1}.\n$$\nClearly, $p_1+p_2=1$. Conversely, given $p_1\\in(0,1)$\nand $p_2=1-p_1$, put\n$t_1=p_1\/p_2$.\n\\fi\nDenote the objective function in \\eqref{op_additive}\nby $\\bar S_n(\\dots)$.\nWe will exhibit a one-to-one correspondence \nbetween the points $(u_1,\\dots,u_n)$ of the simplex $\\sum u_j=1$ and the points\n$(t_1,\\dots,t_{n-1})$ of  $\\mathbb{R}_+^{n-1}$ preserving the\nobjective function: $S_n(t_1,\\dots,t_{n-1},x)=\\bar S_n(u_1,\\dots,u_n,x)$. \n\nDenote\n$$\n s_j=u_1+\\dots+u_j \\quad (j=1,\\dots,n).\n$$\n\nTo a vector $\\mathbf{u}$ with $\\sum u_j=1$ (that is,\n$s_n=1$) we\nput in correspondence the vector $\\mathbf{t}=(t_{1},\\dots,t_{n-1})$ with\ncoordinates\n$$\n t_{j}=\\frac{s_j}{u_{j+1}}\\quad (j=1,\\dots,n-1).\n$$\nThen \n$$\n t_j+1=\\frac{s_j+u_{j+1}}{u_{j+1}}=\\frac{s_{j+1}}{u_{j+1}},\n$$\nso\n$$\nS_n(t_1,\\dots,t_{n-1},x)=\\frac{s_1}{u_2}+\\sum_{j=2}^{n-1}\n\\frac{u_j}{u_{j-1}}+\\frac{xu_n}{s_n}=\n\\bar S_n(u_1,\\dots,u_n,x),\n$$\nsince $s_1=u_1$ and $s_n=1$.\n\n\\iffalse\n$$\n\\frac{t_j}{t_{j-1}+1}=\\frac{p_j}{p_{j+1}}\n\\quad (2\\leq j\\leq n-1)\n$$\nand\n$$\n t_1=\\frac{s_1}{p_2}=\\frac{p_1}{p_2}.\n$$\nFinally,\n$$\n \\frac{x}{t_{n-1}+1}=\\frac{xp_n}{s_n}=xp_n.\n$$\nWe see that the objective function from\nthe definition of $\\tilde f_n(x)$  is transformed term-wise to the objective function from \\eqref{op_main}.\n\\fi\n\n\nThe inverse transformation $\\mathbf{t}\\mapsto \\mathbf{u}$ is defined by\nthe formulas\n$$\n u_n=\\frac{1}{t_{n-1}+1},\n$$\nthen recurrently for $j=n-1,\\dots,2$\n$$\nu_j=\\frac{t_j u_{j+1}}{t_{j-1}+1}\n=\\frac{1}{t_{j-1}+1}\\prod_{i=j}^{n-1}\\frac{t_i}{t_i+1},\n$$\nand lastly,\n$$\n u_1=t_1 u_2.\n$$\nWe prove by induction (on ascending $j$) that \n$s_j=t_j u_{j+1}$, $1\\leq j\\leq n-1$.\nIt is true for $j=1$, since $s_1=u_1$. The induction step ($2\\leq j\\leq n-1)$ goes: $s_j=s_{j-1}+u_j=(t_{j-1}+1)u_j=t_j u_{j+1}$.\nFinally,\n$s_n=(t_{n-1}+1)u_n=1$, so that \nthe $\\mathbf{u}$ satisfies the required constraint.\n\\end{proof}\n\n\\subsection{Existence of a minimizer}\n\\label{ssec:existence}\nWe have written `$\\min$' in Eq.~\\eqref{optfn0}, since the\nminimum of the objective function in is attained at critical point where $t_j=x^{j\/n}$ ($j=0,1,\\dots,n)$. It is also true, thouhg not immediately obvious, that the greatest lower bound of the objective function in \\eqref{optfn} is attained. We prove this now along with first elementary properties of the functions $F_n(\\cdot)$.\n\n\\iffalse\nWe can, of course, introduce an interpolatory family of optimization problems:\n$$\n\\ba{l}\\displaystyle\n f_n(x|s)=\n\\min_{t_1,\\dots,t_{n-1}\\geq 0} \\left(t_1+\\frac{t_2}{t_1+s}+\\dots+\\frac{t_{n-1}}{t_{n-2}+s}+\\frac{x}{t_{n-1}+s}\n\\right),\n\\\\[2ex]\nf_0(x|s)=x.\n\\end{array}\n$$\nEquivalently (upon substitution $t_j\\mapsto st_j$): $f_n(x|s)=g_n(x\/s\\mid s)$, where\n$$\n\\ba{l}\\displaystyle\n g_n(x|s)=\n\\min_{t_1,\\dots,t_{n-1}\\geq 0} \\left(st_1+\\frac{t_2}{t_1+1}+\\dots+\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{x}{t_{n-1}+1}\n\\right),\n\\\\[2ex]\ng_1(x|s)=sx.\n\\end{array}\n$$\nThere is the recursive definition, the same as one for $f_n$:\n$$\n g_n(x|s)=\\min_{y>0}\\left(g_{n-1}(y)+\\frac{x}{y+1}\\right),\\quad n\\geq 2.\n$$\nOnly the initial condition (case $n=1$) depends on $s$.\n\\fi\n\n\\begin{prop}\n\\label{prop:monotfn}\n{\\rm(a)} For every $n$, the function $F_n(\\cdot)$ is nondecreasing.\n\n{\\rm(b)} For every fixed $x$, the sequence $(F_n(x))$ is nonincreasing, hence the function $F(x)$ can be defined as\na monotone limit\n$$\nF(x)=\\downarrow\\lim_{n\\to\\infty} F_n(x).\n$$\n\n{\\rm(c)} \n\\iffalse\nFor every $n$ the inequality \n\\begin{equation}\n\\label{compf0fn}\n\\frac{1}{2}F^{(0)}(x)\\leq F_n(x)\\leq F_n^{(0)}(x)\n\\end{equation}\nholds. Consequently,\n\\begin{equation}\n\\label{compf0f}\n\\frac{1}{2}F^{(0)}(x)\\leq F(x)\\leq F^{(0)}(x).\n\\end{equation}\n\n\n{\\rm(d)} \n\\fi\nThe objective function $(t_1,\\dots,t_{n-1})\\mapsto L_n(0,t_1,\\dots,t_{n-1},x)$ attains its minimum value at some nonnegative\n$(n-1)$-tuple.\nHence the symbol `$\\inf$' in  \\eqref{op_main}, \\eqref{optfn} and \\eqref{op_additive} can be replaced by `$\\min$'. \n\\end{prop}\n\n\\begin{proof}\n(a) Trivial: the function $x\\mapsto S_n(t_0,\\dots,t_{n-1},x)$ is increasing for every\n$n$-tuple \n$(t_0,\\dots,t_{n-1})$.\n\n\\smallskip\n(b) The inequality $F_{n}(x)\\leq F_{n-1}(x)$ \nis due to the fact that imposing the additional constraint $t_1=0$ in \\eqref{optfn} yields $F_{n-1}(x)$. \nThat is,\n$$\nL_{n-1}(0,t_2,\\dots,t_n)=L_n(0,0,t_2,\\dots,t_n).\n$$\n\n\\iffalse\n(c) \nThe right inequality in \\eqref{compf0fn} is obvious: $L_n^{(0)}(\\mathbf{t})>L_n(\\mathbf{t})$ for any\n$\\mathbf{t}>0$.\n\nThe left inequality is equivalent to the claim:\nfor any $n$, $x$ and $t_1,\\dots,t_{n-1}$ there exists $k$ such that\n$$\n L_n(0,t_1,\\dots,t_{n-1},x)\\geq \\frac{1}{2} F_k^{(0)}(x).\n$$\nWe prove this by induction on $n$.\nThe base case $n=1$ is obvious, since \n$L_1(0,x)=x$ and one can take $k=1$.\n\nAssuming that the claim is true with $n-1$ in place of $n$ and some $k'$ in place of $k$, we write\n$$\n L_n(0,t_1,\\dots,t_{n-1},x)=\nL_{n-1}(0,t_1,\\dots,t_{n-1})+\\frac{x}{t_{n-1}+1}.\n$$\nIf $t_{n-1}\\leq 1$, then $x\/(t_{n-1}+1)\\geq x\/2$,\nso the claim is true with $k=1$.\nIf $t_{n-1}> 1$, then $x\/(t_{n-1}+1)> x\/(2t_{n-1})$.\nUsing the recursive definition \\eqref{recurf0}, we infer\n$$\n S_n(0,t_1,\\dots,t_{n-1},x)\n\\geq\n\\frac{1}{2}\\left(f_{k'}^{(0)}(x)+\\frac{x}{t_{n-1}}\\right)\n\\geq \\frac{1}{2}f_{k}^{(0)}(x),\n$$\nwhere $k=k'+1$. \n\nThe left-hand side of \\eqref{compf0fn} does not depend on $n$, hence  \\eqref{compf0fn} implies \\eqref{compf0f}.\n\n\\smallskip\n(d) \n\\fi\n\n(c) \nIt suffices to show that the optimization region\nin \\eqref{op_main} can be reduced to a compact.\nLet us fix $x$ and put $C=F_n(x)$. \nIf $t_1>C+1$, then $S(t_1,\\dots,t_{n-1},x)>C+1$,\nso the optimization region can be reduced to\n$\\{0\\leq t_1\\leq C+1,\\;t_2,\\dots,t_{n-1}\\geq 0\\}$.\n\nSuppose that $t_1\\leq C_1=C+1$ and define the constants\n$C_k$ recurrently: $C_k=(C_{k-1}+1)(C+1)$.\nIf $k\\in\\{2,\\dots,n-1\\}$ is the least index such that $t_k>C_k$ (assuming such an index exists),\nwe have $t_k\/(t_{k-1}+1)>C+1$. Hence $S_n(t_1,\\dots,t_{n-1},x)>C+1$ in the region \n$\\{t_1>C_1\\}\\cup\\{t_2>C_2\\}\\cup\\dots\\{t_{n-1}>C_{n-1}\\}$.\nTherefore the optimization region is reduced\nto the parallelotop $\\{0\\leq t_k\\leq C_k,\\;\\;k=1,\\dots,n-1\\}$.\n\\end{proof}\n\n\\begin{definition}\nAn $(n-1)$-tuple $(t_1,\\dots,t_{n-1})$ is called a {\\em minimizer}\\ for the problem \\eqref{op_main} if\n$S_{n-1}(t_1,\\dots,t_{n-1},x)=F_{n-1}(x)$.\nWe will also say ``a minimizer for $F_n(x)$''. \n\nWith reference to the problem \\eqref{optfn} we will call\nthe minimizing $(n+1)$-tuple $(0,t_1,\\dots,t_{n-1},x)$\na minimizer. \n\\end{definition}\n\nIn preparation to the proof of Proposition~\\ref{prop:funeqf} let us prove the first result\non minimizers. \n\n\\begin{lemma}\n\\label{lem:minimizer0}\nIf $x\\leq 1$, then the unique minimizer for $F_n(x)$\nis the zero tuple.\n\\end{lemma}\n\n\\begin{proof}\nWe need to prove that if not all $t_j$ equal zero, then\n$S_{n}(t_1,\\dots,t_{n-1},x)>S_{n}(0,\\dots,0,x)=x$.\nEquivalently, the inequality to prove is: if $\\mathbf{t}\\neq\\mathbf{0}$, then (in the worst case, when $x=1$) \n$$\n S_{n-1}(t_1,\\dots,t_{n-1})>\\frac{t_{n-1}}{t_{n-1}+1}.\n$$\n\nIt is obvious, if $n-1=1$. By induction we proceed from the estimate for $S_{n-1}$ to the one for $S_n$.\n\nIf $t_1=\\dots=t_{n-1}=0$ and $t_{n}>0$, then\nwe are in the same situation as in the case $n-1=1$.\nIf not all $t_j$ with $j\\leq n-1$ are equal to 0, then\nby the induction hypothesis we have\n$$\n S_{n}(t_1,\\dots,t_{n})=S_{n-1}(t_1,\\dots,t_{n-1})+\\frac{t_n}{t_{n-1}+1}\n>\\frac{t_{n-1}+t_n}{t_{n-1}+1}.\n$$\nSince\n$$\n \\frac{t_{n-1}+t_n}{t_{n-1}+1}-\\frac{t_{n}}{t_{n}+1}\n=\\frac{t_n^2+t_{n-1}}{(t_{n-1}+1)(t_{n}+1)}\\geq 0,\n$$\nthe induction step is complete.\n\\end{proof}\n\n\\subsection{Dynamic programming formulation}\n\\label{ssec:dynprog}\n\\smallskip\nWe give recurrent equations for\n$F_n(\\cdot)$ analogous to \\eqref{recurf0}.\n\nAlso it appears possible to define the function $F(x)$ is expressed without explicit reference to the $F_n$'s\n--- by means of\nthe functional equation \\eqref{fe1}.\n\n\\begin{prop}\n\\label{prop:funeqf}\nLet $\\tilde F_n(x)$, $n=1,2,\\dots$, be a sequence of functions\ndefined for $x>0$ as follows.\n\n\\smallskip\n{\\rm (i)} For $0<x\\leq 1$ and all $n\\;\\;$ $\\tilde F_n(x)= x$.\n\n\\smallskip\n{\\rm (ii)}  \nFor $x>1$ and $n\\geq 2$\n\\begin{equation} \n\\label{recurf}\n \\tilde F_n(x)= \\min_{0< y<x-1}\\left(\\tilde F_{n-1}(y)+\\frac{x}{y+1}\\right).\n\\end{equation} \n\nDefine also the function $\\tilde F(x)$, $x>0$, as follows.\n\n\\smallskip\n{\\rm (i$'$)} If $0<x\\leq 1$, then $\\tilde F(x)= x$.\n\n\\smallskip\n{\\rm (ii$'$)} In the intervals $(1,2], (1,3], \\dots$ the values $\\tilde F(x)$ are defined recursively by means of the functional equation\n\\begin{equation} \n\\label{fe1}\n \\tilde F(x)=\\min_{0< y<x-1}\\left(\\tilde F(y)+\\frac{x}{y+1}\\right).\n\\end{equation} \n(The right-hand side refers to values of $\\tilde F(\\cdot)$ defined earlier.)\n\nThen \\\\\n{\\rm (a)} \n$\\tilde F(x)=\\tilde F_n(x)$ for $0<x\\leq n$;\n\\\\\n{\\rm (b)} \n$\\tilde F_n(x)=F_n(x)$ for all $n$ and $x$, and\n\\\\\n{\\rm (c)} \n$\\tilde F(x)=F(x)$ for all $x$. \n\\end{prop}\n\n\\begin{proof}\n(a) $\\tilde F_1(x)=\\tilde F(x)=1$ for $0<x\\leq 1$ by definition.\nThe claimed identity for any $n\\geq 2$ follows by induction, comparing the recursive definitions of $\\tilde F_n(x)$ and $\\tilde F(x)$.\n\n\\smallskip\n(b)\nWe will prove by induction, simultaneously, that (I) $\\tilde F_n(x)=F_n(x)$,\n$0<x\\leq n$; and (II) $0<t_{n-1}^*<x-1$, so that $t_{n-1}^*$ can be identified with the point of minimum\n$y_*$ in the right-hand side of \\eqref{recurf}.\n\nFor $n=1$, the claim (I) is trivial and the claim (II) is vacuous.\n\nSuppose that the objective function in \\eqref{op_main} attains its minimum at $\\mathbf{t}^*=(t_1^*,\\dots,t_n^*)$,\nso that\n$$\n F_n(x)=S_{n-1}(t^*_1,\\dots,t^*_{n-1})+\\frac{x}{t^*_{n-1}+1}.\n$$\nThe existence of such $\\mathbf{t}^*$ is ensured by Proposition~\\ref{prop:monotfn}.\n\nWe have\n$$\n\\frac{\\partial S_n(\\mathbf{t},x)}{\\partial t_{n-1}}\n=\\frac{\\partial}{\\partial t_{n-1}}\n\\left(\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{x}{t_{n-1}+1}\\right)=\\frac{1}{t_{n-2}+1}-\\frac{x}{(t_{n-1}+1)^2}.\n$$\n(For $n=2$, we formaly take $t^*_{0}=0$.) \n\nSince $x>1$, there can be no minimum at $t_{n-1}=0$. \nHence $t^*_{n-1}>0$. \n\nBy the choice of $\\mathbf{t}^*$ we have $S_{n-1}(t^*_1,\\dots,t^*_{n-2},t^*_{n-1})\n=F_{n-1}(t_{n-1}^*)$. In order to complete the induction step\nit remains to prove that $t^*_{n-1}<x-1$.\n\nThe necessary condition of extremum yields\n$$\n (t^*_{n-1}+1)^2=x(t^*_{n-2}+1).\n$$\nSuppose that $t^*_{n-1}+1\\geq x$. Then $t^*_{n-1}\\leq t^*_{n-2}$. If $n=2$, we get $1<x\\leq t^*_1+1\\leq t^*_0+1=1$, a contradiction. \n\nIf $n>2$ and $t^*_{n-1}>1$, then by part (II) of the induction hypothesis the last component $t^*_{n-2}$ in the minimizer for $F_{n-1}(t^*_{n-1})$ satisfies the inequality $t^*_{n-2}<t^*_{n-1}-1$.\n\nFinally, if $n>2$ and $t^*_{n-1}\\leq 1$, then by Lemma~\\ref{lem:minimizer0} $t^*_{n-2}=0$.\n\nIn both cases, we arrive to a contradiction with assumption\n$t^*_{n-1}\\leq t^*_{n-2}$. \n\nThe induction step is complete.\n\n\\smallskip\n(c) By parts (a) and (b) we get $F_n(x)=\\tilde F(x)$\nfor any integer $n\\geq x$. Since the sequence $F_n(x)$\nis monotone (Proposition~\\ref{prop:monotfn}(b)), we conclude that $F(x)=\\tilde F(x)$.\n\\end{proof}\n\n\\subsection%\n{General look: least action formulation}\n\\label{ssec:leastaction}\n\n\n\\iffalse\nConsider the optimization problem in the form~\\eqref{op_main}. A {\\em minimizer} is .\nThe existence of a minimizer \nhas been proved in Proposition~\\ref{prop:altoptfn}.\nSince there are no constraints in the form of equalities here, there are two possibilities for a minimizer:\n(i) a minimizer\nlies in the interior of the admissible domain;\nthen it is a critical point of the \nobjective function;\n(ii) a minimizer is a boundary point, that is, at least one of $t_j$ ($1\\leq j\\leq n-1$) is zero. \n\nCase (ii) yields a minimization problem with fewer variables and its critical points are to be analysed.\n\nFor a particular value of $n$ there may be more than one candidate point for a minimizer, and subsequent minimization over $n$ (to find a minimizer for $f(x)$)\npotentially involves another act of selection. \n\\fi\n\nThe boundary value formulation in Sec.~\\ref{ssec:bconst} and the dynamic programming formulation in Sec.~\\ref{ssec:dynprog} do not by themselved offer\nmuch of a progress in finding the asymptotics of $F(x)$.\nWe will need to study minimizers. For that purpose it\nis useful to look at the optimization problem from a more general point of view.\n\nConsider a minimization problem with objective function of the form\n\\beq{Sn-general}\n\\mathcal{S}_n(\\mathbf{t})=\\sum_{j=1}^n L(t_{j-1},t_j).\n\\end{equation} \nBy analogy with classical mechanics, we call the function $L(\\cdot,\\cdot)$ the {\\em Lagrangian}.  \nA {\\em virtual trajectory}\\ $\\mathbf{t}=(t_0,\\dots,t_n)$\nis subject to the constraints $t_0=x_0$, $t_n=x$;\nthe boundary values $x_0$ and $x$ are assumed given.\nThe function $\\mathcal{S}_n(\\mathbf{t})$ (``integral of the Lagrangian along a virtual trajectory'') is the analog of {\\em action} in mechanics. The object of our attention is the extremal value\n\\beq{fn-general}\n g_n(x_0,x)=\\min_{(t_0,\\dots,t_n)\\mid\\, t_0=x_0,\\,t_n=x}\n\\mathcal{S}_n(t_0,\\dots,t_n).\n\\end{equation}\nWe recognize in this setting ``the least action principle'' with discrete time.\n\nIn mechanics, one usually pays little attention to the minimum value of the action as such; the goal is to determine the extremal trajectory, which describes the actual evolution of a mechanical system. \nHere, we are originally interested in extremal values $g_n$, but the focus will eventually shift to extremal trajectories.\n\nWe assume that the Lagrangian is differentiable in its domain and denote $L_1(u,v)=\\partial L(u,v)\/\\partial u$\nand $L_2(u,v)=\\partial L(u,v)\/\\partial v$.\n\nLet $\\mathbf{t}^*$ be an extremal (more precisely, minimizing) trajectory\nfor the problem \\eqref{fn-general}. Suppose that $(t^*_1,\\dots,t^*_{n-1})$ is an interior point of the domain of the function $\\mathcal{S}_n|_{t_0=x_0,\\;t_n=x}$.\nThe necessary conditions of extremum $\\partial \\mathcal{S}_n\/\\partial t_j|_{\\mathbf t=\\mathbf t^*}=0$ ($1\\leq j\\leq n-1$), known in mechanics as the Euler-Lagrange equations, in the expanded form read\n\\beq{ELeq-general}\n L_2(t_{j-1},t_j)+\n L_1(t_{j},t_{j+1})=0,\\quad j=1,\\dots,n-1.\n\\end{equation}\n\nSuppose that the equation $L_1(u,v)+p=0$ with given $u$ and $p$ is uniquely solvable for $v$ in all occurrences and denote the solution as $v=V(u,p)$.\nThen the system \\eqref{ELeq-general} can be written in the form of the second order recurrence equations\n\\beq{ELeq-recur}\n t_{j+1}=V(t_j,L_2(t_{j-1},t_j)),\\quad j=1,\\dots,n-1.\n\\end{equation}\nMore precisely, we have the boundary value problem comprising the equations \\eqref{ELeq-recur} and the\nboundary conditions\n$$\n t_0=x_0,\\qquad t_n=x.\n$$\nIntroduce a free parameter $\\tau$,\nset $T_0(\\tau)=x_0$, $T_1(\\tau)=\\tau$, and define  further functions $T_j(\\tau)$ by \nmaking the substitutions \n$t_i\\mapsto T_i(\\tau)$ in the recurrence \\eqref{ELeq-recur}:\n\\beq{defTj}\nT_{j+1}(\\tau)=V(T_j(\\tau),L_2(T_{j-1}(\\tau),T_j(\\tau))), \\quad j=1,\\dots,n-1.\n\\end{equation}\nThe boundary value problem can be in principle solved by the shooting method: \nthe equation \n$\nT_n(\\tau)=x,\n$\ndetermines the value of $\\tau$ and hence the whole extremal trajectory. The existence and uniqueness of solution are two immediate concerns. We will attend to them in our concrete case --- first, in Section~\\ref{sec:minimizers}, by presenting numerical results and revealing fine points, and then, analytically.\n\nThe dynamic programming approach to the minimization\nproblem \\eqref{fn-general} leads to the recurrence\n\\beq{recg}\n g_{j+1}(x)=\\min_y (g_{j}(y)+L(y,x)),\\quad j\\geq 1,\n\\end{equation}\nwith initial condition  \n$$\n g_1(x)=L(x_0,x).\n$$\nAn extremal trajectory defined by the recurrence \\eqref{defTj} induces the recurrence\n\\beq{Gn-rec}\n G_{j+1}(\\tau)=G_j(\\tau)+L(T_j(\\tau),T_{j+1}(\\tau)),\n\\quad j\\geq 0,\n\\end{equation}\nwith initial condition  \n$$\n G_0(\\tau)=0.\n$$\nIn a simple scenario, we expect that $G_n(\\tau)$ is the minimum value $g_n(x_0,x)$ sought in \\eqref{fn-general}. However, if there are several \nsolutions of the equation $T_n(\\tau)=x$ and\nseveral extremal trajectories, one needs to pick the minimum among the several corresponding\nvalues $G_n(\\tau)$.  Also one should not ignore the possibility that the minimum value may be attained at the boundary of the parameter domain.\n\n\\begin{remark}\nUpon the change of variables $(u,v)\\mapsto (u,p)$, $p=L_2(u,v)$ applied to all pairs $(u,v)=(t_{j},t_{j+1})$, $j=0,\\dots,n-1$,\n the recurrence \\eqref{ELeq-recur} can be recast in the form \n\\beq{EL-H}\n p_{j+1}=L_2(t_j,V(t_j,p_j)),\n\\qquad t_{j+1}=V(t_j,p_j),\n\\qquad j=1,\\dots,n-1.\n\\end{equation}\nThis can be viewed as a ``Hamiltonian system with discrete time'' referring to the fact that\nthe map $(t_j,p_j)\\mapsto(t_{j+1},p_{j+1})$ is symplectic, that is, its Jacobian is equal to $1$. \nCf.~\\cite[end of Sec.~9.1]{McDuff-Salomon_1998}.\n\nIn the present paper, the Lagrangian is\n$$\n L(u,v)=\\frac{v}{u+1}.\n$$\n\nIncidentally, another asymptotic problem solved recently by the author \\cite{Sadov_2021} involves the Lagrangian \n$$\n L(u,v)=u+\\frac{1+v}{u}\n$$\nand the boundary conditions $p_0=t_n=0$ for the system in the Hamiltonian form \\eqref{EL-H}.%\n\\footnote{The comparison applies \nto the trajectory $(t_0,\\dots,t_n)$ \nin \\cite{Sadov_2021}\nre-indexed backwards.}\n(Unlike in the present case, there is no variable $x$; the integer $n$ is the only parameter.)\nCrusial to the asymptotic analysis in \\cite{Sadov_2021} is the presence of a fixed point  \nof the map $(u_j,p_j)\\mapsto(u_{j+1},p_{j+1})$, which is not the case here. \n\\end{remark}\n\n\\section{Asymptotics that allow for simple proofs}\n\\label{sec:simpleproofs}\n\nHere we prove two results that do not rely on the analysis of extremal trajectories. The material of this section is not used in the sequel.\n\n\\subsection{Proof of Theorem~\\ref{thm:AMGM}}\n\\label{ssec:asf0}\n\nWe write $u=\\log x$, as in the formulation of the Theorem.\n\nLet $u_n=n(n+1)\\log(1+n^{-1})=n+1\/2+O(1\/n)$. We have:\n$F^{(0)}_{n+1}(x)>F^{(0)}_n(x)$ if and only if $u>u_n$. \n\nSuppose $n>1$, $u\\in I_n=[u_{n-1},u_n]$.\nThen $F^{(0)}(x)=F^{(0)}_{n}(x)=u\\varphi(u\/n)$,\nwhere $\\varphi(t)=t^{-1}e^t$. The function $\\varphi(t)$ has the minimum at $t=1$\nand $\\varphi(1+\\varepsilon)=e(1+\\varepsilon^2\/2)+O(\\varepsilon^3)$ as $\\varepsilon\\to 0$. \n\nPut $u=n+s$, $|s|\\leq 1\/2+O(1\/n)$. We have $\\varphi(u\/n)=e+(e\/2)(s\/n)^2+O(n^{-3})$. \n\nNote that $|s|=|\\left\\langle  u\\right\\rangle|+O(1\/n)$;\nindeed, $s\\neq \\left\\langle  u\\right\\rangle$ can happen\nonly near half-integer values of $n$, where $s$ is close to $\\pm 1\/2$.\n\nSince $n^{-1}=u^{-1}+O(u^{-2})$, we \nobtain the asymptotics \\eqref{asf0}. \n\\qed\n\n\\begin{remark}\nThe left and right derivative numbers of $F^{(0)}(x)$\nat the endpoints of the intervals $I_n$ are different.\nIn Section~\\ref{ssec:experiment} we will see that the situation with function $F(x)$ is more interesting in this respect.\n\\end{remark}\n\n\\iffalse\nWe have $f^{(0)}(x)=\\log x\\cdot \\min_{\\mathbb{Z}\\ni n\\geq 1}\\varphi(\\log x\/n)$,\nwhere $\\varphi(t)=t^{-1}e^t$. \nThe function $\\varphi(t)$ has the minimum at $t=1$\nand $\\varphi(1)=e$. \n\nPut $u=\\log x$.\nThe value $n_*=n_*(x)$ for which\n$f^{(0)}(x)=u\\varphi(u\/n_*)$ is always one of the two integers nearest to $u$: either $n_*=\\lfloor u\\rfloor$ or $n_*=\\lceil u\\rceil$. Hence\n$u\/n_*=1+\\epsilon$, where $\\epsilon=O(1\/u)$ but $\\epsilon\\neq o(1\/u)$.\n\nSince $\\varphi(1+\\epsilon)=e+c\\epsilon^2+ o(\\epsilon^2)$ with $c>0$, we conclude that $\\varphi(u\/n_*)=e+O(1\/u^2)$ and the \nremainder estimate is optimal. The claimed asymptotics \nof $f^{(0)}(x)$ follows.\n\\fi\n\n\\subsection{Crude asymptotics of the function \\texorpdfstring{$F(x)$}{F(x)}}\n\\label{ssec:crudeas}\n\n\\begin{prop}\n\\label{prop:crudeas}\n\\iffalse\nLet $f(x)$ be a real-valued function defined for $x> 0$ recursively as follows:\n$$\n f(x)=x, \\qquad 0< x\\leq 1,\n$$\nand\n\\begin{equation}\n\\label{recg}\n f(x)=\\inf_{0<y\\leq x-1}\\left(f(y)+\\frac{x}{y+1}\\right),\n \\qquad x>1.\n\\end{equation}\n\\fi\nThe function $F(x)$ has the asymptotic behavior\n$F(x)=e\\log x+O(1)$ as $x\\to\\infty$. Specifically,\nif $x\\geq 1$, then\n$F(x)$ satisfies the inequalities\n\\begin{equation}\n\\label{rbndf}\n-a_1+\\frac{b_1}{x+1}\\leq\nF(x)-e\\log(x+1)\\leq -a_2+\\frac{b_2}{x+1},\n\\end{equation}\nwhere \n$b_1\\approx 1.77$\nis the (smaller of the two) root of the equation \n\\begin{equation}\n\\label{eqb1}\n2\\log\\frac{b+1}{2}=\\frac{b}{e};\n\\end{equation}\nthe constant $a_1$\nis defined by\n$$\na_1=\\max_{0\\leq x\\leq 1}\\left(e\\log(x+1)+\\frac{b_1}{x+1}-x\\right)\n\\approx 1.78,\n$$\nand $a_2=b_2=e\/(e-1)\\approx 1.58$.\n\\end{prop}\n\n\\begin{proof}\nWe prove that \\eqref{rbndf} is true for $0<x\\leq n$ by induction on $n$.  For $n=1$ the left inequality is fullfilled by the definition of $a_1$, while the left inequality is due to the fact that \nthe function $y\\mapsto  e\\log(x+1)+b_2\/(x+1)-x$ attains it minimum on $[0,1]$ at $x=0$.\n\nFor the induction step consider the function\n$t\\mapsto e\\log t+u\/t$ with given $u>0$. The critical\npoint is $t^*=u\/e$, hence\n\\beq{minf_tmp}\n \\min_{t>0}\\left(e\\log t+\\frac{u}{t}\\right)=\ne\\log t^*+\\frac{u}{t^*}=\ne\\log u.\n\\end{equation}\n\nLet us first prove the left inequality\nin \\eqref{rbndf} for $x\\leq n+1$.\nUsing the inductive assumption and the formula \\eqref{fe1} from Proposition~\\ref{prop:funeqf}, we get \n$$\n F(x)\\geq e\\log(y+1)-a_1+\\frac{b_1}{y+1}+\\frac{x}{y+1}\n$$\nfor any $y\\in(0,x-1]$. By \\eqref{minf_tmp} with $u=b_1+x$, we have\n$$\n F(x)\\geq e \\log(b_1+x)-a_1.\n$$\nIt remains to check the inequality\n$$\ne\\log(b_1+x)\\geq e\\log(x+1)+\\frac{b_1}{x+1};\n$$\nequivalently,\n$$\n\\frac{\\log(1+(b_1-1)s)}{s}\\geq \\frac{b_1}{ e},\n$$\nwhere\n$s=(x+1)^{-1}$. \n\nThe left-hand side is a decreasing function of $s$.  We may assume that $x>1$, hence $s<1\/2$.\nTherefore\n$$\n\\frac{\\log(1+(b_1-1)s)}{s}> 2\\log(1+(b_1-1)\/2).\n$$\nBy  definition of $b_1$, the right-hand side equals $b_1\/e$. The proof of the lower estimate for $F(x)$ is complete.\n\nWe prove the right inequality\nin \\eqref{rbndf} for $x\\leq n+1$ similarly.\nAgain, using the inductive assumption and the formula \\eqref{fe1}, we get \n$$\n F(x)\\leq \\min_{y<x-1} \\left(e\\log(y+1)-a_2+\\frac{b_2}{y+1}+\\frac{x}{y+1}\\right).\n$$\nPut $u=b_2+x$ in \\eqref{minf_tmp}. The critical point\n$t^*=(b_2+x)\/e$ corresponds to $y^*=t^*+1$. \nNow, $y^*$ lies in the admissible interval \n$0<y<x-1$ provided that\n$t_*< x$, that is, $x> b_2\/(e-1)\\approx 0.92$.\nThis condition is fulfilled, since we assume $x\\geq 1$.\nSo\n$$\n F(x)\\leq e\\log(b_2+x)-a_2.\n$$\nIt remains to prove that\n$$\ne\\log(b_2+x)\\leq e\\log(x+1)+\\frac{b_2}{x+1}.\n$$\nSubtracting $e\\log(x+1)$ from both sides, we estimate:\n$$\n e\\log\\frac{x+b_2}{x+1}=e\\log\\left(1+\\frac{b_2-1}{x+1}\\right)< e\\frac{b_2-1}{x+1}.\n$$\nSince $e(b_2-1)=b_2$, the proof is complete.\n\\end{proof}\n\n\\begin{remark}\n\\label{rem:ill-mainasym}\n The double-sided estimate\n\\eqref{rbndf} agrees with\nasymptotic formula \\eqref{asf}, since $a_2<A<a_1$, while the rational\nterms $\\mathrm{const}\\cdot(x+1)^{-1}$, as well as the difference between $\\log x$ and $\\log(x+1)$, are absorbed by the remainder.\nA freedom to choose a particular form of asymptotically negligible terms can be used to \ndevise an approximation to $F(x)$ that would be practical not only for large values of $x$ but also for small ones.\n\nThe approximation with an empirically chosen \ncorrection $b_2\/(x+1)$\n\\begin{equation}\n\\label{empir_approx}\nF(x)\\approx\nF_{\\rm approx}(x)\\stackrel{\\text{\\scriptsize\\rm def}}{=} e\\ln(x+1)-A+\\frac{b_2}{x+1},\n\\end{equation}\nis illustrated in Figure~\\ref{fig:f1corr}(a).\nNote that if we had plotted only the initial part of the graph, up to approximately $\\log(x+1)=8$ (shown in red), then it could\nlead to a misleading assumption that already the second term of the asymptotics, that of order $O(1)$, is periodic.\nThe right part of the graph shows that the error of approximation in fact decays at a rate $O((\\log x)^{-1})$,\nas the theory predicts. The equation of the dashed majorizing curve is $y=(e\/8)(\\log(x+1)+7)^{-1}$. (Here $e\/8=e\/2\\cdot\\max\\ab{\\cdot}^2$. The ``magic'' number $7$ makes the enveloping curve closer to the graph without affecting the asymptotics up to terms of order $O((\\log x)^{-1})$.)\n\nThe right part of the same graph is shown in larger scale on Fig.~\\ref{fig:f1corr}(b) in comparison with third term of asymptotics from Theorem~\\ref{thm:main}.\n\\end{remark}\n\n\\iffalse\n\\begin{figure}\n\\begin{picture}(260,190)\n\\put(0,0){\\includegraphics[scale=0.5]\nempirical_approximation_error-logx}\n}\n\\put(315,25){\\small $\\log (x+1)$}\n\\end{picture}\n\\caption{Difference $f(x)-f_{\\rm approx}(x)$ vs $\\log(x+1)$} \n\\label{fig:emperror}\n\\end{figure}\n\\fi\n\n\\begin{figure}\n\\begin{picture}(311,520)\n\\put(50,260)%\n{\\includegraphics[scale=0.5]{f1_approximation_error_logx}}\n\\put(315,274)%\n{\\small $\\log (x+1)$}\n\n\\put(50,0){\\includegraphics[trim={0 0 0 50pt},clip,scale=0.5]{f1_approximation_34-40}}\n\\put(315,16){$\\log(x+1)$}\n\\put(5,380){(a)}\n\\put(5,120){(b)}\n\\end{picture}\n\\caption\n(a) The error of empirical approximation \\eqref{empir_approx}. (b) Blue line: the same error; gray line: the correction term $\\Phi(u)\/u$ from Theorem~\\ref{thm:main}.}\n\\label{fig:f1corr}\n\\end{figure}\n\n\n\n\\section{Extremal trajectories and minimizers}\n\\label{sec:minimizers}\n\nHere we make initial steps towards the reduction of Theorem~\\ref{thm:main} to Theorem~\\ref{thm:paramcurves}.\n\nTaking Eq.~\\eqref{optfn} as the definition of $F_n(x)$, the minimizer candidates are critical points of $\\hat S_n(\\mathbf{t})|_{t_0=0,t_n=x}$ inside the admissible domain $\\{t_i>0,\\,i=1,\\dots,n-1\\}$ and on its boundary, where $t_i=0$ for at least one $i\\geq 1$.\n\n\n\nOn the other hand, from Proposition~\\ref{prop:funeqf}(c) we know that $F(x)=F_n(x)$ for all sufficiently large $n$\n(e.g. $n\\geq x$), and we will observe the stabilization of\nminimizers in a precise componentwise sense. \n\nIn this section we will use the termiology of the general least action problem with discrete time as described in Section~\\ref{ssec:leastaction}.\n\n\n\\iffalse\nConsider the optimization problem in the form~\\eqref{op_main}. A {\\em minimizer} is .\nThe existence of a minimizer \nhas been proved in Proposition~\\ref{prop:altoptfn}.\nSince there are no constraints in the form of equalities here, there are two possibilities for a minimizer:\n(i) a minimizer\nlies in the interior of the admissible domain;\nthen it is a critical point of the \nobjective function;\n(ii) a minimizer is a boundary point, that is, at least one of $t_j$ ($1\\leq j\\leq n-1$) is zero. \n\nCase (ii) yields a minimization problem with fewer variables and its critical points are to be analysed.\n\nFor a particular value of $n$ there may be more than one candidate point for a minimizer, and subsequent minimization over $n$ (to find a minimizer for $f(x)$)\npotentially involves another act of selection. \n\\fi\n\n\\iffalse\n\\subsection{Critical points, extremal trajectories}\n\\label{ssec:leastaction}\n\nFrom the formulation \\eqref{op_main} of the optimization problem defining $f_n(x)$ \nit is clear that the point of minimum (which exists, as we have proved in Proposition~\\ref{prop:altoptfn})\nis either a critical point, where all partial derivatives of the objective function turn to $0$, or a boundary point, where at least one of $t_j$ ($1\\leq j\\leq n-1$) is zero. \n\nUsing analogy with classical mechanics, we call the objective function $S_n(\\mathbf{t})$ in the optimization problem \\eqref{optfn}\n {\\em the action}\\ and\nthe vector $\\mathbf{t}=(t_0,\\dots,t_n)$ a {\\em virtual trajectory}. It must satisfy the boundary conditions $t_0=0$, $t_n=x$.\n\nThe action is ``the integral of the Lagrangian along\na virtual trajectory''; here this description corresponds to the expression\n\\begin{equation}\n\\label{action}\nS_n(\\mathbf{t})=\\sum_{j=1}^n L(t_{j-1},t_j),\n\\end{equation}\nwhere the {\\em Lagrangian} is\n$$\n L(u,v)=\\frac{u}{v+1}.\n$$\n\nThe problem of minimization of the function $S_n(\\mathbf{t})$ under the given constraints\ncan be thought of as \nthe ``least action principle'' with discrete time.\nIn mechanics, one usually pays little attention to the minimum value of the action:\nIn mechanics, \nthe focus is on describing the actual (extremal) trajectory and so it will be in this section.\n\n\nSuppose $\\mathbf{t}$  is the minimizing vector \nfor $S_n(\\cdot)$.\nIf \n$t_j\\neq 0$ for some $j$ (between $1$ and $n-1$), then the necessary condition of extremum, $\\partial S_n\/\\partial t_j=0$,\nmust be fulfilled. This equation can be written as  \n\\beq{ELj}\n\\partial_v L(t_{j-1},t_j)+\\partial_u L(t_{j},t_{j+1})=0.\n\\end{equation}\nExtending analogy with mechanics, we call it the Euler-Lagrange equation.\nIt determines $t_{j+1}$ in terms of $t_{j-1}$ and $t_j$ (provided the solution\nis unique).\n\nThe equation \\eqref{ELj} does not need to hold if $t_j=0$. \n\nIn the asymptotic problem from \\cite{Sadov_2021} mentioned in the Introduction the Lagrangian is\n$$\n L(u,v)=v+\\frac{1+u}{v}.\n$$\nNo parameter $x$ is present there and the boundary condition is one-sided, $t_0=0$.\nIn both cases the Lagrangian $L(u,v)$ is linear in $u$, Eq.~\\eqref{ELj} is uniquely solvable for $t_{j+1}$; however, technical parallels do not go much further. \n\nFrom now on we engage with explicit details of \nproblem \\eqref{optfn}.\n\\fi\n\n\\subsection{Extremal trajectories: basic properties}\n\\label{ssec:basicrelations}\n\nThe Euler-Lagrange equations \\eqref{ELeq-recur}\nfor the concrete problem \\eqref{optfn}\nbecome\n$$\n\\frac{1}{t_{j-1}+1}-\\frac{t_{j+1}}{(t_{j}+1)^2}=0.\n$$ \nThe recurrence relations\n\\eqref{ELeq-recur} and \\eqref{Gn-rec} take the form\n$$\n T_{j+1}(\\tau)=\\frac{(T_{j}(\\tau)+1)^2}{T_{j-1}(\\tau)+1},\n\\qquad\nG_{j+1}(\\tau)=G_j(\\tau)+\\frac{T_{j+1}(\\tau)}{T_j(\\tau)+1}.\n$$\n\nTo make the recurrence relations more compact, we introduce the functions where the parameter and the values are shifted by $1$: \n$$\n \\xi_j(t)=T_j(t-1)+1,\n\\qquad \n \\eta_j(t)=G_j(t-1)+1, \\quad t\\geq 1.\n$$ \nThen\n\\begin{equation}\n\\label{recxi}\n\\xi_{j+1}(t)=\\frac{\\xi_{j}^2(t)}{\\xi_{j-1}(t)}+1.\n\\end{equation}\nIntroduce also the auxiliary functions\n\\begin{equation}\n\\label{defalpha}\n \\alpha_j(t)=\\frac{\\xi_j(t)}{\\xi_{j-1}(t)}.\n\\end{equation}\nIn places where $t$ does not change, we will often simply write $\\xi_j$, $\\eta_j$, $\\alpha_j$. \n\nThe governing system of recurrence relations becomes\n\\begin{equation}\n\\label{recrelalxi}\n\\begin{array}{l}\n\\displaystyle\n\\alpha_{j+1}=\\alpha_j+\\frac{1}{\\xi_j},\n\\\\[2ex]\n\\xi_{j+1}=\\alpha_{j+1} \\xi_{j}=\\alpha_{j}\\xi_j+1.\n\\end{array}\n\\end{equation}\nIt is complemented by the subordinate recurrence\n\\begin{equation}\n\\label{recreleta}\n\\eta_{j+1}\n=\\eta_j+\\alpha_{j}.\n\\end{equation}\n\nClearly, all the introduced functions are rational functions of $t$.\nHere are the first few, including the initial values ($n=0,1$) set by definition:\n\n\\bigskip\\noindent\n\\begin{tabular}{c|c|c|c}\n$n$ & $\\alpha_n$ & $\\xi_n$ & $\\eta_n$ \\\\\n\\hline\n$0$ & $t-1$    & $1$      & $1$        \\\\[1ex]\n$1$ & $t$ & $t$ & $t$ \\\\[2ex]\n$2$ & $t+t^{-1}$ & $t^2+1$ & $2t$\\\\[2ex]\n$3$ & $\\displaystyle\\frac{t^2+1}{t}+\\frac{1}{t^2+1}$ & $t^3+2t+1+t^{-1}$ & $3t+t^{-1}$\\\\[2ex]\n$4$ & $\\cdots$ & $\\displaystyle\\frac{(t^2+1)^{3}}{t^2}+\\frac{2(t^2+1)}{t}+\\frac{1}{t^2+1}+1$ & $\\displaystyle\\frac{4t^4+6t^2+t+2}{t(t^2+1)}$\\\\\n\\end{tabular}\n\n\\bigskip \n\nSome basice consequences of the recurrence relations\n\\eqref{recxi}--\\eqref{recreleta} are collected in the following proposition.\n\n\\begin{prop}\n\\label{prop:xieta-basic}\n{\\rm(a)} The introduced rational functions behave at inifinity as follows (for any fixed $n$): $\\alpha_n(t)\\sim t$, $\\xi_n(t)\\sim t^n$,\n$\\eta_n(t)\\sim nt$.\n\n\\smallskip\n{\\rm(b)} For any $t$, the sequences $(\\alpha_n(t))$, \n$(\\xi_n(t))$ and $(\\eta_n(t))$ are increasing.\nThe inequalities\n$\n \\alpha_n(t)\\geq 2 \n$ $(n\\geq 1)$\nand \n$\n \\xi_n(t)\\geq 2^{n-1}\n$\n$(n\\geq 2)$\nhold true. Also $\\alpha_n(t)<3$ for all $n$.\nConsequently, there exists the finite limit\n$$\n \\alpha_{\\infty}(t)=\\lim_{n\\to\\infty}\\alpha_n(t).\n$$\n\n\\smallskip\n{\\rm(c)} For $n\\geq 1$, define  $X^{0}_n=\\min_{t\\geq 1}\\xi_n(t)$.\nThen $\\xi_n(\\cdot)$ maps $[1,\\infty)$ to $[X^{0}_n,\\infty)$. The sequence $(X^{0}_n)$ is increasing and $X^{0}_n\\geq 2^{n-1}$. \n\n\\smallskip\n{\\rm(d)} For any $n\\geq 1$ the following relations hold true:\n\\begin{equation}\n\\label{ic12}\n\\xi_n(1)=\\xi_{n-1}(2), \n\\qquad\n\\eta_n(1)=\\eta_{n-1}(2),\n\\qquad\n\\alpha_n(1)=\\alpha_{n-1}(2),\n\\end{equation}\nand\n\\begin{equation}\n\\label{dxieta}\n\\eta'_n(t)=\\frac{\\xi'_n(t)}{\\xi_{n-1}(t)}.\n\\end{equation}\n\\end{prop}\n\n\\begin{remark}\n1. In (c), it is not true generally that $\\xi_n(1)=X_n^0$. See Sec.~\\ref{ssec:experiment}. \n\n\\smallskip\n2. To elucidate the formula \\eqref{dxieta}, let us look at the dynamic programming formulation \\eqref{recg}.\nWe have the \nobjective function $\\Lambda_j(x,y)=g_j(y)+L(y,x)$. Let \n$y^*=y^*(x)$ be some critical point. \nThen $\\frac{\\partial}{\\partial y}\\Lambda_j(x,y^*)=0$.\nHence\n$g_{j+1}'(x)=\n\\frac{\\partial}{\\partial x}\\Lambda_j(x,y^*)=L_2(y^*,x)\n$. With our Lagrangian, and putting $j=n-1$, we get  $L_2(y^*,x)=L_2(T_{n-1}(\\tau),T_{n}(\\tau))=(T_{n-1}(\\tau)+1)^{-1}$.\nThe result is the relation $d\\eta_{n}\/d\\xi_{n}=\\xi_{n-1}^{-1}$, equivalent to \\eqref{dxieta}.\n\\end{remark}\n\n\\begin{proof}\n(a) Immediate by induction.\n\n(b) Monotonicity of $(\\alpha_n)$ and $(\\eta_n)$\nis obvious from \\eqref{recrelalxi} and \\eqref{recreleta}. Since $\\min\\alpha_2=2$, the inequalities $\\alpha_n> 2$ and $\\xi_{n+1}>2\\xi_n$ for $n>2$ follow. \n\nTherefore, for $n>2$ we have $\\xi_n>2^{n-2}\\xi_2\\geq 2^{n-1}$. By \\eqref{recrelalxi}, \n$\\alpha_n-\\alpha_2<\\sum_{j=3}^{n-1} 2^{1-j}<1$.\nThus the sequence $(\\alpha_n(t))$ is bounded and the\n$\\alpha_{\\infty}(t)<\\infty$. \n\n\\smallskip\n(c) We know from (b) that $\\alpha_n\\geq 2$ for $n\\geq 2$. Now \\eqref{recrelalxi} implies $X^{0}_{n+1}\\geq 2X^{0}_n+1$, hence the strict monotonicity of $(X^{0}_n)$ and the estimate $X^{0}_n\\geq 2^{n-1}$. \n\n\\smallskip\n(d)\nDue to the identical initial conditions $(\\alpha_0(2),\\xi_0(2),\\eta_0(2))=(1,1,1)=\n(\\alpha_1(1),\\xi_1(1),\\eta_1(1))$, \nthe sequences $(\\alpha_n(2))$ etc.\\ are identical to \n$(\\alpha_{n+1}(1))$ etc.\n\nThe relations \\eqref{dxieta} follow by induction:\nfor $n=1$ we have $\\eta'_1(t)=1=\\xi'_1(t)\/\\xi_0(t)$; \nthe induction step goes:\n$$\n \\eta_{n+1}'=\\eta_n'+\\left(\\frac{\\xi_{n+1}-1}{\\xi_n}\\right)'=\\frac{\\xi'_n}{\\xi_{n-1}}+\n\\frac{\\xi'_{n+1}}{\\xi_n}-\\frac{(\\xi_{n+1}-1)\\xi'_n}{\\xi_n^2}=\\frac{\\xi'_{n+1}}{\\xi_n},\n$$\ndue to \\eqref{recxi}.\n\\iffalse\n\\smallskip\n(e) Put $\\lambda_n(t)=\\xi'_n(t)\/\\xi_n(t)$. From \\eqref{recxi} we have the recurrence relation \n\\begin{equation}\n\\label{logdifxirec}\n \\lambda_{n+1}=(1-\\xi_{n+1}^{-1})(2\\lambda_n-\\lambda_{n-1}).\n\\end{equation}\nHence%\n\\footnote{This is only true if $\\lambda_j\\geq 0$ up to $j=n+1$.}\n the sequence $(\\lambda_n)$ is convex down for any $t\\geq 1$. The same is true for the sequence\n$\\tilde\\lambda_{n}=\\lambda_{n}-n\\lambda_1$. \nWe have $\\tilde\\lambda_1=0$ and $\\tilde\\lambda_2(t)=2t\/(1+t^2)-2\/t<0$, hence by induction\nfor any $n\\geq 2$\n$$\n \\frac{n-1}{n}\\tilde\\lambda_{n+1}\\leq\\tilde\\lambda_n-\n\\frac{1}{n}\\tilde\\lambda_1<0.\n\\eqno\\qedhere\n$$\n\\fi\n\\end{proof}\n\n\\subsection{Experimental observations}\n\\label{ssec:experiment}\n\nFor $n=1,2,\\dots$, denote by  $\\gamma_n$, resp., $\\gamma^T_n$, the parametric curve defined in the coordinate $(x,y)$-plane by the equations\n$$\n x=\\xi_n(t), \\quad y=\\eta_n(t), \\quad t\\geq 1,\n$$\nresp.,\n$$\n x=T_n(\\tau), \\quad y=G_n(\\tau), \\quad \\tau\\geq 0.\n$$\nThe curve $\\gamma_n$ is obtained from $\\gamma^T_n$\nby the shift $(x,y)\\mapsto(x+1,y+1)$.\n\nLet $\\gamma_n[1,2]$ be the part of the curve $\\gamma_n$\ncorresponding to the parameter values $1\\leq t\\leq 2$.\nProposition~\\ref{prop:xieta-basic}(d) asserts that \n$\\gamma_{n-1}[1,2]$ and $\\gamma_n[1,2]$ are geometrically adjacent to each other and have a common tangent at the adjacency point. \n\nSimilarly defined partial curves $\\gamma^T_n[0,1]$\nwill be used with reference to the curves $\\gamma^T_n$.\n\n\nThe lower envelope of the the graphs of the functions $F_n(x)$\nmakes, by definition, the graph of the function $F(x)$.\n\nOne anticipates a relation between \nthe graph of $F_n(x)$ \nand the curve\n$\\gamma^T_n$ \nbased on the premise that the abscissa $x=T_n(\\tau)$ corresponds to the value of $\\tau$ that determines the extremal trajectory yielding the extremal value $y=G_n(\\tau)$ of the action. We take this view as a ``first approximation''\nand discuss necessary corrections below.\n\nThe curve $\\gamma^T_n$ lies in the half-plane\n$x\\geqX^{0}_n-1$ (so that the equation $T_n(\\tau)=x$ has a solution).\nConsequently, $\\gamma^T_n$ cannot represent the whole\nof the graph of $F_n(x)$\n(which is defined for all $x>0$). \nParts of the graph will be represented by segments\nof curves $\\gamma^T_k$, $k<n$; we elaborate on this \nin \\S~\\ref{ssec:bndmin}.\n\n\\begin{figure}[htb]\n\\begin{picture}(260,266)\n\\put(0,0){\\includegraphics[scale=0.5]{xieta1-7-logxy}\n}\n\\put(201,116){\\small $x$}\n\\put(44,65){\\small $y$}\n\\put(42,171){\\small $\\tilde\\gamma_1$}\n\\put(66,76){\\small $\\tilde\\gamma_2$}\n\\put(96,31){\\small $\\tilde\\gamma_3$}\n\\put(126,16){\\small $\\tilde\\gamma_4$}\n\\put(161,10){\\small $\\tilde\\gamma_5$}\n\\put(196,8){\\small $\\tilde\\gamma_6$}\n\\put(232,7.4){\\small $\\tilde\\gamma_7$}\n\\put(141,246){\\framebox{\\parbox{3.6cm}{\\small {Solid lines: $1\\leq t\\leq 2$}\\\\\n{Dotted lines: $t> 2$}}}}\n\\put(211,3){\\circle{5}}\n\\end{picture}\n\\caption{Curves $\\tilde\\gamma_n$, $n=1,\\dots,7$,\nwith parametric equations $x=\\log\\xi_n(t)$, $y=\\eta_n(t)-e x$. Inside the little circle is a region displayed in Fig.~\\ref{fig:xieta6-7}.} \n\\label{fig:xieta1-6}\n\\end{figure}\n\nThe mutual position of the curves $\\gamma_n$ (or $\\gamma^T_n$) with $1\\leq n\\leq 7$ is illustrated in Figure \\ref{fig:xieta1-6}.  \nAs a matter of fact, shown are the curves $\\tilde\\gamma_n$ obtained from $\\gamma_n$ by the transformation $(x,y)\\mapsto (\\log x,y-e\\log x)$ (scaling conveniently and removing the asymptotic \ndrift).\nThe\nunion of the solid parts forms a part of the graph of the function $F(e^{x}-1)+1-ex$.\n\nFigure~\\ref{fig:xieta1-6} seems to support the view that: (a) the parametric curves $\\gamma^T_n$ are the graphs of the functions\n$F_n(x)$\nrestricted to $x\\geq X^{0}_n-1$; \n(b) moreover, $X^{0}_n=\\xi_n(1)$; and (c) the lower envelope of the curves \n$\\gamma_n$\ncoincides with union of the segments $\\gamma_n[1,2]$. \n\n\\begin{figure}[ht]\n\\begin{picture}(260,255)\n\\put(0,0){\\includegraphics[scale=0.5]{xieta30-32-logxy}\n}\n\\put(250,235){\\small $x$}\n\\put(30,-4){\\small $y$}\n\\put(53,120){\\small $\\tilde\\gamma_{30}$}\n\\put(123,10){\\small $\\tilde\\gamma_{31}$}\n\\put(249,120){\\small $\\tilde\\gamma_{32}$}\n\\put(111,180){\\small $t=1$}\n\\put(207,180){\\small $t=2$}\n\\put(83.2,68.7){\\circle*{4}}\n\\put(78,74){\\small $t=t^{0}_{31}$}\n\\put(101,28){\\circle*{4}}\n\\put(64,27){\\small $t=t^{\\ell}_{31}$}\n\\put(156.9,27.41){\\circle*{4}}\n\\put(166,27){\\small $t=t^{r}_{31}$}\n\\end{picture}\n\\caption{Parametric curves $\\tilde\\gamma_n$, $n=30,31,32$.\nThe coordinates and legend are the same as in Fig.~\\ref{fig:xieta1-6}. The $t$-value marks pertain to $\\tilde\\gamma_{31}$.} \n\\label{fig:xieta30-32}\n\\end{figure}\n\nThese impressions \nare refuted by observing the curves with greater values of $n$.\nFigure~\\ref{fig:xieta30-32} shows that:\n(a) the curve $\\tilde\\gamma_n$, and hence $\\gamma_n$ or $\\gamma^T_n$, in general is not a graph of a single-valued function;\n(b) the value $X^{0}_n$ is the abscissa of the cusp\nand corresponds to some $t^{0}_n\\in (1,2)$;\n(c) the lower envelope of the\ncurves $\\gamma_n$ \nis a proper subset of the\nunion of the segments $\\gamma_n[1,2]$. \n\n\n\\begin{figure}[thb]\n\\begin{picture}(257,255)\n\\put(0,0){\\includegraphics[scale=0.5]{xieta6-7b-logxy}\n}\n\\put(251,16){\\small $x$}\n\\put(40,244){\\small $y$}\n\\put(64,39){\\small $\\tilde\\gamma_6$}\n\\put(156,129){\\small $\\tilde\\gamma_7$}\n\\put(96,18){\\small $\\logX^{\\!\\times}_7$}\n\\put(191,18){\\small $\\logX^{0}_7$}\n\\end{picture}\n\\caption\nA magnified\nview of the curves\n$\\tilde\\gamma_6$ in parameter region $t\\to 2^-$ and $\\tilde\\gamma_7$\nin parameter region $t\\to 1^+$. Solid lines form a part of the graph of\n$F(x)$.\n} \n\\label{fig:xieta6-7}\n\\end{figure}\n\nLet us describe what {\\em is}\\ true and will be proved in the sequel (Sec.~\\ref{ssec:special-points}).\n\n\\smallskip\n(i) Case $n\\leq 6$. The observations based on Fig.~\\ref{fig:xieta1-6} are mostly adequate (with subtle exception described in (iii) below), the crucial fact being that the functions $\\xi_n(t)$ and $\\eta_n(t)$ are monotone increasing in $t\\in[1,\\infty)$.\nAt $x=T_n(0)=T_{n-1}(1)=X^{0}_n-1$, we have $F(x)=G_n(0)=G_{n-1}(1)=\\eta_n(1)-1$; \nthe function $F(\\cdot)$ has a continuous derivative at that point. \n\n\\smallskip\n(ii) Case $n\\geq 7$. There exist three special values of the parameter: $1<t^{0}_n<t^{\\ell}_n<t^{r}_n<2$.\nThe functions $\\xi_n(t)$ and $\\eta_n(t)$ are convex in $t\\in[1,\\infty)$;\nthey decrease in $[1,t^{0}_n]$ and increase in $[t^{0}_n,\\infty)$.\nThe value $X^{0}_n=\\min_t \\xi_n(t)$ equals $\\xi_n(t^{0}_n)$.\nThe curve $\\gamma_n$ has a cusp at the point\n$(X^{0}_n,\\eta_n(t^{0}))$.\nThe lower branches of the curves $\\gamma_n$ and\n$\\gamma_{n-1}$ meet at the point with coordinates\n$(\\xi_n(t^{\\ell}_n),\\eta_n(t^{\\ell}_n))=\n(\\xi_{n-1}(t^{r}_{n-1}),\\eta_{n-1}(t^{r}_{n-1}))$.\nSet by definition $X^{\\!\\times}_n=\\xi_n(t^{\\ell}_n)$.\nSo $X^{\\!\\times}_{n+1}=\\xi_n(t^{r}_n)$.\n\nThe branch of $\\gamma_n$ corresponding to parameter values $t\\geqt^{\\ell}_n$ is the graph of the function $F_n(x-1)+1$ restricted to $x\\geq X^{\\!\\times}_n$.\n(As to why $X^{\\!\\times}_n$ and not $X^{0}_n$ here -- see the end of \\S~\\ref{ssec:bndmin}.)\nThe point $(T_n(0),G_n(0))$ does not belong to the lower branch of the curve $\\gamma_n$.\nThe equality\n$F(x)=F_n(x)$ is valid for $x\\in[X^{\\!\\times}_n,X^{\\!\\times}_{n+1}]$.\n The left and right derivative numbers of $f(x)$\nat $x=X^{\\!\\times}_n$ are not equal to each other. \n\n\\smallskip\n(iii) The exception mentioned in (a) applies to \nthe case $n=6$.\nLooking at Fig.~\\ref{fig:xieta1-6},\none is led to mistakenly believe that the equality\n$F(x)=F_6(x)$ holds true for $\\xi_6(1)\\leq x\\leq \\xi_6(2)$,\nwhile in fact it is valid in the shorter interval\n$\\xi_6(1)\\leq x\\leqX^{\\!\\times}_7$, see Figure~\\ref{fig:xieta6-7}. Let us clarify that\nthe value $n=7$ is the first for which $X^{\\!\\times}_n$ is defined,\nsince $\\gamma_7$ is the first curve having a cusp.\nThe value $t^{\\ell}_6$ is not defined, but $t^{r}_6\\approx 1.9975$\nis defined and \n$X^{\\!\\times}_7=\\xi_7(t^{\\ell}_7)=\\xi_6(t^{r}_6)$. \n\n\n\\subsection{Case of a minimizer at the boundary}\n\\label{ssec:bndmin}\n\nWe will show that if the minimum of the objective function\n$\\hat S_n(\\mathbf{t})$ is attained at the boundary of the admissible region, then the minimizer corresponds to \nan extremal trajectory of a shorter length.\n\n\n\\begin{lemma}\n\\label{lem:min-at-bnd}\n(a) Suppose $\\mathbf{t^*}=(0,t^*_1,\\dots,t^*_{n-1},x)$ is a minimizer for the problem \\eqref{optfn}.\nIf $t^*_j=0$ for some $j\\in\\{1,\\dots,n-1\\}$, then\n$t^*_i=0$ for any $i\\in\\{1,\\dots,j-1\\}$.\nIn this case $F_n(x)=F_{n-j}(x)$.\n\n(b) If,\nin addition, \n$t^*_{j+1}\\neq 0$,\nthen the truncated vector\n$$\\hat{\\mathbf{t}}^*=(0=t_{j}^*,t_{j+1}^*,\\dots,t_{n-1}^*,x)\n$$\nis an extremal trajectory for the problem \\eqref{optfn}\nwith $n$ replaced by $n-j$. In this case $T_{n-j}(t^*_{j+1})=x$ and $G_{n-j}(t^*_{j+1})=F_n(x)$.\n\\end{lemma}\n\n\\iffalse\n\\begin{prop}\n\\label{prop:fnstab}\nFor every $n\\geq 1$ there exists $X_n$ such that\n$f_{n+1}(x)=f_{n}(x)$ for $0\\leq x\\leq X_n$.\nFor instance, $X_1=1$.\n\\end{prop}\n\n\\begin{proof}\nSuppose the $n$-tuple $(t_1,\\dots,t_{n}$ \nminimizes $S_{n-1}(t_1,\\dots,t_n)$ under constraint\n$t_n=x$. If \n$f_{n+1}(x)<f_{n}(x)$, then \nthere exists an $(n+1)$-tuple $t'_0,\\dots,t'_{n-1},t_n$ with $t'_n=x$ such that\n$$\nS_n(t'_0,\\dots,t'_{n-1},x)<f_{n-1}(x)=S_{n-1}(t_1,\\dots,t_n).\n$$\n\n\nWe need to show that for some $X_n$\nand any $t_1,\\dots,t_n$ with $t_n\\leq X_n$ the inequality\n$$\n S_n(t_0,\\dots,t_n)\\leq S_{n-1}(t_1,\\dots,t_n)\n$$\nimplies $t_0=0$.\n\\end{proof}\n\\fi\n\n\\begin{proof}\n(a) Put $i=j-1$. If $i>0$ and $t^*_i>0$, then the necessary condition of\nextremum $\\partial \\hat S_n\/\\partial t_i|_{\\mathbf{t}=\\mathbf{t}^*}=0$\nyields\n$$\n\\frac{1}{t_{i-1}^*+1}-\\frac{t_{j}^*}{(t_{i}^*+1)^2}=0,\n$$ \nwhich contradicts the assumption $t_j^*=0$. Hence $t_{j-1}^*=0$. By downward induction on $i$ we get $t_i=0$ for any $i<j$. \n\nSince\n$$\n \\hat S_n(0,\\dots,0,t_{j+1},\\dots,x)=S_{n-j}(0,t_{j+1},\\dots,x),\n$$\nwe have $F_n(x)=\\hat S_{n-j}(\\hat{\\mathbf{t}}^*)\\geq F_{n-j}(x)$. It follows that $F_n(x)=F_{n-j}(x)$.\n\n\\smallskip\n(b) For the same reason, $\\hat{\\mathbf{t}}^*$ is the minimizer for\n\\eqref{optfn}\nwith $n$ replaced by $n-j$. Moreover,\n$t^*_i>0$ for $i=j+1,\\dots,n$. Hence $\\hat{\\mathbf{t}}^*$ is an extremal trajectory of length $n+1-j$ and $T_{n-j}(t_{j+1}^*)=x$.\nFinally, $\\hat S_{n-j}(\\hat{\\mathbf{t}}^*)=G_{n-j}(t^*_{j+1})$,\nhence $F_n(x)=G_{n-j}(t^*_{j+1})$.\n\\end{proof}\n\nWe see that a minimizer that lies on the boundary of\nthe admissible domain is represented by an extremal trajectory of length $k\\in\\{2,\\dots,n\\}$. \nConsequently, the graph of $F_n(x)$ \nis the lower envelope of the parametric curves \n$\\gamma^T_k$ with $1\\leq k\\leq n-1$.\n\nFor $n\\leq 6$ and $1\\leq k\\leq n-1$, the graph of the restriction $F_n(x)|_{[T_k(0),T_k(1)]}$ coinsides with\n$\\gamma^T_k[1,2]$.\n\nFor $n\\geq 7$, the part of the curve $\\gamma^T_n$ corresponding to parameter values $t^{0}_n<t<t^{\\ell}_n$\nlies above the curve \n$\\gamma^T_{n+1}$ and hence does not belong to the\ngraph of $F_n(x)$. \n\n\n\\subsection{Monotonicity of the critical index}\n\\label{ssec:crind}\n\n\\begin{definition}\n\\label{def:crind}\nFor the given $x>0$, the {\\em critical index} $\\nu(x)$ is the integer equal to minimum value of $n$ such that $F(x)=F_{n}(x)$.\n\\end{definition}\n\n\\begin{prop}\n\\label{prop:crind}\nThe function $x\\mapsto \\nu(x)$ is nondecreasing.\n\\end{prop}\n\n\\begin{proof}\nBy the definition of extremal trajectories and in view ofLemma~\\ref{lem:min-at-bnd}, the equality $n=\\nu(x)$ holds if and only if both of the following\nare true:\n\n\\smallskip\n(i) there exists $\\tau>0$ \n such that $T_n(\\tau)=x$ and $G_n(\\tau)=F(x)$;\n equivalently, the vector $\\mathbf{t}\\in\\mathbb{R}^{n+1}$ with \n $t_0=0$ and $t_{j}=T_j(\\tau)$\n($j=1,\\dots,n)$ is a minimizer for the problem \\eqref{optfn};\n\n\\smallskip\n(ii) for any $k<n$ either the equation $T_{k}(\\tau)=x$ does not have a solution or, if \n$\\tau$ is a solution, then $G_{k}(\\tau)>F(x)$.  \n\n\\smallskip\nConsequently, if $\\mathbf{t}$ is an extremal trajectory specified in (i), then $\\nu(t_j)=j$ ($1\\leq j\\leq n$).  \n\nUsing this observation, we will show  by induction on $n$ that the inequality $n=\\nu(x)>\\nu(x')$ implies $x>x'$.\n\nLet us assume that the said implication is true with $n-1$ instead of $n$. (The special case $n-1=1$ is included.) \n\nSuppose that the induction step fails.\nIt means that there exist $x$ and $x'$ such that $x<x'$ and $n=\\nu(x)>\\nu(x')=k$.\n\nLet $\\mathbf{t}=(0,t_1,\\dots,t_{n-1},x)$ and\n$\\mathbf{t'}=(0,t'_1,\\dots,t'_{k-1},x')$\nbe the extremal trajectories with $G_n(t_1)=F(x)$\nand $G_k(t'_1)=F(x')$.\nThen $\\nu(t_{n-1})=n-1$ and $\\nu(t'_{k-1})=k-1$. By the inductive assumption, $t'_{k-1}<t_{n-1}$.\n(If $n=2$, then this is a tautology: $0=t'_0<t_1$.)\nWe have\n$$\n F(x')=S_k(\\mathbf{t'})=\\hat S_k(0,t'_1,\\dots,t'_{k-1},x)\n+\\frac{x'-x}{1+t'_{k-1}}.\n$$\nBy the assumption of {\\em ad absurdum}\\ argument, $k<n$, so\n$$\n\\hat S_k(0,t'_1,\\dots,t'_{k-1},x)\\geq F_k(x)>F(x)\n=\\hat S_n(0,t_1,\\dots,t_{n-1},x).\n$$\nDue to the inequality $t'_{k-1}<t_{n-1}$, \n$$\n \\frac{x'-x}{1+t'_{k-1}}>\\frac{x'-x}{1+t_{n-1}}.\n$$\nTherefore\n$$\n F(x')> \\hat S_n(0,t_1,\\dots,t_{n-1},x)\n+\\frac{x'-x}{1+t_{n-1}}=\\hat S_n(0,t_1,\\dots,t_{n-1},x'),\n$$\nwhich contradicts the assumption that $\\mathbf{t}$\nis a minimizer for $F(x')$.\n\\end{proof}\n\n\\section{Reduction of Theorem~\\ref{thm:main} to  Theorem~\\ref{thm:paramcurves}}\n\\label{sec:proofmain}\n\nIn the course of the proof, which involves many small technical steps, we will prove that Fig.~\\ref{fig:xieta30-32} adequately illustrates the relevant features of the curves $\\gamma_n$ \n(defined in Sec.~\\ref{sec:minimizers})\nwith large enough $n$.\n\nWe will explore in great detail a parametrization of the curves $\\gamma_n$. As a result, we will be able to describe\na piece-wise parametrization of their lower envelope\nand to derive the asymptotics of the function $f(x)$.\n\n\n\\subsection{Only the partial curves \\texorpdfstring{$\\gamma_n[1,2]$}{gamma[1,2]} are relevant}\n\\label{ssec:gamma12}\n\nObserving the dotted lines in Figs.~\\ref{fig:xieta1-6} and \\ref{fig:xieta30-32},\none is led to conjecture that the parts of the curves $\\gamma_n$\ncorresponding to the parameter values $t>2$\ndo not contribute to the lower envelope of the\ncurves $\\gamma_n$. Equivalently, \nthe parts of the curves $\\gamma^T_n$\ncorresponding to the parameter values $\\tau>1$\ndo not contribute to the graph of the function $F(x)$.\nWe will prove this conjecture now.\n\nAs shown in \\S~\\ref{ssec:bndmin}, the coordinates of any point of the graph of $F(x)$ can be written as\n$x=T_n(\\tau)$, $F(x)=G_n(\\tau)$ with some $n\\geq 1$\nand $\\tau>0$. \n\n\\begin{prop}\n\\label{prop:tau01}\nGiven $x>0$, suppose that $n$ and $\\tau$ are such that $x=T_n(\\tau)$ and $F(x)=F_n(x)=G_n(\\tau)$. Then $\\tau\\leq 1$.\n\\end{prop}\n\n\\begin{proof}\nWe have\n$$\n F_n(x)=\\hat S_n(0,t^*_1,\\dots,t^*_n),\n$$\nwhere $t^*_1=T_1(\\tau)=\\tau$ and $t^*_n=T_n(\\tau)=x$.\n\nSuppose, contrary to what is claimed, that $\\tau>1$.\nDefine a vector $\\mathbf{t}=(0,t_1,\\dots,t_{n+1})\n\\in\\mathbb{R}_+^{n+2}$ as follows:\n$$\n t_1=s, \\quad\\text{and}\\quad\n t_{j+1}=t^*_j \\;\\; (j=1,\\dots,n),\n$$\nwhere $s$ is an arbitrary number such that $0<s<\\tau-1$.\nThen\n$$\n \\hat S_n(\\mathbf{t^*})-\\hat S_{n+1}(\\mathbf{t})\n=t^*_1-\\left(t_1+\\frac{t_2}{t_1+1}\\right)=\n\\tau-\\left(s+\\frac{\\tau}{s+1}\\right)=\\frac{s(\\tau-1-s)}{s+1}>0,\n$$\nso $F(x)\\leq F_{n+1}(x)\\leq \\hat S_{n+1}(\\mathbf{t})<\\hat S_n(\\mathbf{t^*})=F_n(x)$, a contradiction. \n\\end{proof}\n\n\n\\subsection{Convexity of the functions \\texorpdfstring{$\\alpha_n(t)$ and $\\alpha_\\infty(t)$,\n$t\\in[1,2]$}{alpha[n] on [1,2]}}\n\\label{ssec:alpha}\n\nRecall that the functions $\\alpha_j(t)$ is defined as\n$\\alpha_j(t)=\\xi_j(t)\/\\xi_{j-1}(t)$.\nThey are a part of the recurrent scheme \\eqref{recrelalxi} defining extremal trajectories.\nThe function $\\alpha_\\infty(t)=\\uparrow\\lim_{j\\to\\infty}\\alpha_j(t)$ determines the eventual rate of the exponential \ngrowth of the components of an extremal trajectory\nstarting at $\\tau=t-1$. The graph of the function\n$\\alpha_\\infty(t)$ on the interval $[1,2]$ is shown in\nFig.~\\ref{fig:alphaplot}. Crusial for the derivation of the asymptotic formulas is the existence of the solution\nof the equation $\\alpha_\\infty(t)=e$ in $[1,2]$.\nThe roots are marked $t_a$ and $t_b$ on the figure.\n\n\\begin{prop}\n\\label{prop:alpha}\nThe function $\\alpha_\\infty(t)$ is real-analytic in $[1,2]$, convex, and has one point of minimum at $t_o\\in(1,2)$.\nThe inequality\n$$\n\\alpha_\\infty(t_o)<e<\\alpha_\\infty(1)=\\alpha_\\infty(2)\n$$\nholds. \nConsequently, there are uniquely determined $t_a$, $t_b$ with $1<t_a<t_o<t_b<2$ and $\\alpha_\\infty(t_a)=\\alpha_\\infty(t_b)=e$.\n\nAlso the functions $\\alpha_n(t)$ are convex.\n\\end{prop}\n\n\\begin{figure}[htb]\n\\begin{picture}(260,255)\n\\put(0,0){\\includegraphics[scale=0.5]{alphaplot}\n}\n\\put(260,14){\\small $t$}\n\\put(60,16){\\small $t_a$}\n\\put(132,16){\\small $t_o$}\n\\put(199,16){\\small $t_b$}\n\\end{picture}\n\\caption{Function $\\alpha_\\infty(t)$ on the interval $[1,2]$} \n\\label{fig:alphaplot}\n\\end{figure}\n\nIn \\eqref{num-alpha} the numerical values are given for reference. As a matter of fact, only\n$t_b$ will be relevant in the proof of Theorem~\\ref{thm:main}.\n\n\\begin{equation}\n\\label{num-alpha}\n\\ba{l}\n\\alpha_\\infty(1)=\\alpha_\\infty(2)\\approx 2.815572650,\n\\\\[0.7ex]\nt_o\\approx 1.447847, \n\\quad \n\\alpha_\\infty(t_o)\\approx 2.673953412\n\\\\[0.7ex] \nt_a\\approx 1.185591828,\n \\quad\nt_b\\approx 1.742084284.\n\\end{array}\n\\end{equation}\n\nThe behaviour of $\\alpha_\\infty$ is simple, but to prove it analytically is not an easy task.\nOur proof is rather long and technical. It is given in Appendix~\\ref{app:alpha}.\n\n\\smallskip\nThe next lemma, which is a simple corollary\nof Proposition~\\ref{prop:alpha}, will be important in\nthe proof of Proposition~\\ref{prop:Xmin}(b) below.\n\n\\begin{lemma}\n\\label{lem:conv-xi}\nThe functions $\\xi_n(t)$, $n=1,2,\\dots$, are convex\nin $t\\in[1,2]$.\n\\end{lemma}\n\n\\begin{proof}\nThe claim is true for $n=1$, since $\\xi_1(t)=t$.\nThe general case follows by induction due to\nthe identity $\\xi_{n+1}(t)=\\alpha_n(t)\\xi_n(t)+1$\nand the convexity of $\\alpha_n(t)$. \n\\end{proof}\n\n\\subsection{Special points on the curves \\texorpdfstring{$\\gamma_n[1,2]$}{gamma[n]}}\n\\label{ssec:special-points}\n\nRecall that the special values $X^{0}_n=\\min_{t\\geq 1}\\xi_n(t)$ \nhave been defined in Proposition~\\ref{prop:xieta-basic}(c).\nThus $X^{0}_n$ is such a value that the equation\n\\beq{Tnx}\nT_n(\\tau)=x\n\\end{equation}\nhas a solution if and only if $x\\geqX^{0}_n-1$.\n\nIn Sec.~\\ref{ssec:experiment} we have identified and illustrated (Fig.~\\ref{fig:xieta30-32}) special values of parameter $t$: $t^{0}_n$, $t^{\\ell}_n$, and $t^{r}_n$.\nHere we define them analytically. \n($t^{\\ell}_n$ is defined for $n\\geq 7$.) For the curves $\\gamma_n^T$, the corresponding values of $\\tau=t-1$  will be denoted similarly: $\\tau^{0}_n$, $\\tau^{\\ell}_n$, $\\tau^{r}_n$.\n\n\\begin{prop}\n\\label{prop:Xmin}\n{\\rm(a)} $\\xi_n'(t)>0$ for all $t\\geq 2$ and $n\\geq 1$.\nConsequently, the solution $t=t^{0}_n$ of the equation  $\\xi_n(t)=X^{0}_n$, belongs to $[1,2)$.\nEquivalently, the solution $\\tau_n^0$ of the equation \n\\eqref{Tnx} with $x=X^{0}_n-1$ lies in $[0,1)$.\n\n\\smallskip\n{\\rm(b)} For every $n\\geq 1$ and every $x\\geq X^{0}_n-1$\nthe equation~\\eqref{Tnx} has at most two solutions. \n\n\\smallskip\n{\\rm(c)} The sequences $(t_n^0)$ (hence $(\\tau_n^0)$) and $(X^{0}_n)$ are nondecreasing. More precisely, \n$t_n^0=1$ for $n\\leq 6$ and $t_n^0>t_{n-1}^0$ for $n\\geq 7$. \n\\end{prop}\n\n\\begin{definition}\n\\label{def:t_pm}\nIf the equation~\\eqref{Tnx} has two distinct solutions,\nthe smaller will be denoted $\\tau_n^-(x)$ and the larger $\\tau_n^+(x)$. \n\nIf there is a unique solution, it will be denoted $\\tau_n^+(x)$. In this case we leave $\\tau_n^-(x)$ undefined, except when $x=X^{0}_n-1$. We put $\\tau_n^-(X^{0}_n-1)=\\tau_n^+(X^{0}_n-1)=\\tau_n^0$.  \n\nSimilar notation $t_n^{\\pm}(x)$ will be used in reference to the equation $\\xi_n(t)=x$.\n\\end{definition}\n\n\\begin{proof}\n(a) Since $\\xi_n=\\alpha_n\\alpha_{n-1}\\dots\\alpha_1$,\nit suffices to prove that $\\alpha'_n>0$ for\n$t\\geq 2$ and all $n\\geq 1$. We take this inequality\nas the induction hypothesis. It is true for $n=1$.\n\nWe have $\\alpha_n=\\alpha_1+\\sum_{j=1}^{n-1}\\xi_j^{-1}$\nand $\\alpha_1=t$.\nWe need to prove that\n$$\n \\sum_{j=1}^{n-1}\\frac{\\xi'_j}{\\xi_j^2}< 1=\\alpha_1'.\n$$\nAs a consequence of the induction hypothesis, \n$\\xi'_k>0$ for $1\\leq k\\leq n-1$ and $t\\geq 2$.\nHence \n$\\alpha'_j<\\alpha'_1$, so $0<\\xi'_j\/\\xi_j=\\sum_{k=1}^j\n\\alpha'_k\/\\alpha_k<j\/t\\leq j\/2$ for $j=1,\\dots,n-1$.\nAlso, $\\xi_j\\geq\\alpha_1^j\\geq 2^j$. The required estimate follows:\n$$\n \\sum_{j=1}^{n-1}\\frac{\\xi'_j}{\\xi_j^2}\\leq\n\\sum_{j=1}^{n-1}\\frac{j}{2^{j+1}}<\\sum_1^\\infty\\frac{j}{2^{j+1}}=1.\n$$\n\n\\smallskip\n(b) The functions $\\xi_n(t)$, are convex in $1\\leq t\\leq 2$ by Lemma~\\ref{lem:conv-xi}; more precisely, for $n\\geq 2$ they are strictly convex. They are increasing in $[2,\\infty)$ by part (a).\nTherefore the equation $\\xi_n(t)=x$ has at most two solutions.\n\n\\smallskip\n(c) Due to the convexity of $\\xi_n(\\cdot)$, there are two possibilities: either (i) $t^0_n=1$ and $X^{0}_n=\\xi_n(1)$ or (ii) $t^0_n>1$, then $\\xi'(t^0_n)=0$ and the function $\\xi(\\cdot)$ decreases from $\\xi_n(1)$ to $X^{0}_n$ as $t$ changes from $1$ to $t^0_n$. (Cf.\\ the backtracking segment of the curve $\\tilde\\gamma_7$ in Fig.~\\ref{fig:xieta6-7}.)\n\nIn both cases, $\\xi_n'(t)>0$ for $t>t_n^0$.\n\nSuppose that the sequence $(t_n^0)$ is not monotone as claimed. Let $n$ be the least index for which $t^0_n>t^0_{n+1}$. Then $\\xi'_{n+1}(t^0_n)<0$ and $\\xi'_n(t^0_n)=0$. Also\n $t^0_{n-1}\\leq t^0_n$, so $\\xi'_{n-1}(t^0_n)\\leq 0$. \nThe combination of signs $\\xi'_{n+1}<0$, $\\xi'_n=0$,\n$\\xi'_{n-1}\\leq 0$ contradicts the recurrence relation\n$$\n \\frac{\\xi'_{n+1}}{\\xi_{n+1}-1}=\\frac{2\\xi'_n}{\\xi_n}-\\frac{\\xi'_{n-1}}{\\xi_{n-1}},\n$$ \nwhich follows from \\eqref{recxi} by logarithmic differentiation.\n\nBy the recurrence relations \\eqref{recxi} and the one above we find that $\\xi'_n(1)>0$ for $1\\leq n\\leq 6$,\nwhile $\\xi'_7(1)=-\\frac{19661554943536}{328636389375}<0$. \nTherefore $t^0_n>1$ for $n\\geq 7$.\n\n\\smallskip\nFor any $t$, by definition of $X^{0}_n$, we have\n$X^{0}_n\\leq \\xi_n(t)$. Since $\\xi_n(t)<\\xi_{n+1}(t)$,\nit follows that\n$X^{0}_n\\leq\\inf_{t\\geq 1}\\xi_{n+1}(t)=X^{0}_{n+1}$.\nIt is easy to see that the inequality is strict when \n$t^0_n<t^0_{n+1}$, i.e. for $n\\geq 6$.\n\\end{proof}\n\nNext we will analytically prove that the branches of the curve $\\gamma_n$ are always situated as shown in  Fig.~\\ref{fig:xieta30-32} or Fig.~\\ref{fig:xieta6-7}:\nthe ``backward'' branch (where $t<t^0_n$)\nlies above the ``forward'' branch. \n\n\\begin{prop}\n\\label{prop:pmbranches}\nIf the equation $\\xi_n(t)=x$ has two distinct solutions, then $\\eta_n(t_n^+(x))<\\eta_n(t_n^-(x))$.\nEquivalently,\n$G_n(\\tau_n^+(x))<G_n(\\tau_n^-(x))$.\n\\end{prop}\n\n\\begin{proof}\nThe equation $\\xi_n(t)=y$ has two distinct solutions for\nall $y\\in(X^{0}_n,x]$. For $y=X^{0}_n$ the value\n$t_n^-(y)=t^0_n=t_n^+(y)$ is also defined. Put\n$$\n \\kappa(y)=\\xi_{n-1}(t_n^{+}(y))-\\xi_{n-1}(t_n^{-}(y)).\n$$\n\nClearly, $\\kappa(X^{0}_n)=0$.\nWe will prove that: \n(i) $\\kappa(X^{0}_n+\\varepsilon)>0$\nfor sufficiently small $\\varepsilon>0$, \nand \n(ii) $\n \\kappa(y)\\neq 0 \n$\nfor any $y\\in(X^{0}_n,x]$.\nBy continuity of $\\kappa(\\cdot)$ it follows then that\n$\\kappa(x)>0$. \n\n\\smallskip\nProof of (i): Due to the inequality $t_n^0>t_{n-1}^0$ (Proposition~\\ref{prop:Xmin}(c)), the function \n$\\xi_{n-1}(t)$ increases in the neighborhood of $t^0_n$.\n\n\n\\smallskip\nProof of (ii): Suppose, by way of contradiction, that \n$\\kappa(y_*)=0$ for some $y_*>X^{0}_n$. \nLet $t_*^\\pm=t_n^\\pm(y_*)$.\nThen \n$y=\\xi_{n}(t_*^-)=\\xi_{n}(t_*^+)$\nand $\\xi_{n-1}(t_*^-)=\\xi_{n-1}(t_*^+)$.\nBy the inverse recurrence relation we conclude that\n$\\xi_{j}(t_*^-)=\\xi_{j}(t_*^+)$ for $j$ from $n$ down to $1$. For $j=1$ it results in $t_*^-=t_*^+$,\na contradiction.\n\nNow, by \\eqref{dxieta}\n$$\n \\frac{d\\eta_n(t^{\\pm}(y))}{d y}=\\frac{1}{\\xi_{n-1}(t^{\\pm}(y))}.\n$$\nThe inequality $\\kappa(y)>0$ implies\n$$\n \\frac{d\\eta_n(t^{+}(y))}{d y}<\n\\frac{d\\eta_n(t^{-}(y))}{d y}.\n$$\nIntegrating from $y=X^{0}_n$ to $x$ we get $\\eta_n(t^{+}(x))<\\eta_n(t^{-}(x))$.\n\\end{proof}\n\n\\begin{definition}\n\\label{def:xcross}\nFor $n\\in\\mathbb{N}$, let $X^{\\!\\times}_n$ be defined by%\n\\footnote{The subtraction of $1$ \n makes this definition consistent with notation $X^{\\!\\times}_n$ in Sec.~\\ref{ssec:experiment}.}\n$X^{\\!\\times}_n-1=\\inf\\{x\\mid\\nu(x)=n\\}\n=\\max\\{x\\mid\\nu(x)=n-1\\}$, where $\\nu(\\cdot)$\nis the critical index (Definition~\\ref{def:crind}).\n\\end{definition}\n\n\\begin{prop}\n\\label{prop:argmin_intervals}\n{\\rm (a)} $F(x)=F(x_{n-1})$ for $0<x\\leq X^{\\!\\times}_n$\nand  $F(x)<F(x_{n-1})$ for $x>X^{\\!\\times}_n-1$.\nThe point $(X^{\\!\\times}_n-1,\\,F(X^{\\!\\times}_n-1))$ is common to the\ncurves $\\gamma_{n}^T$ and $\\gamma_{n+1}^T$.\nEquivalently, the point $(X^{\\!\\times}_n,\\,1+F(X^{\\!\\times}_n-1))$ is common to the\ncurves $\\gamma_{n}$ and $\\gamma_{n+1}$.\n\n\\smallskip\n{\\rm (b)} $X^{\\!\\times}_n=X^{0}_{n}=\\xi_n(1)$ for $1\\leq n\\leq 6$\nand $X^{\\!\\times}_n<\\xi_n(1)$ for $n\\geq 7$.\n\\end{prop}\n\n\\begin{proof}\n(a) If $x\\leqX^{\\!\\times}_n-1$, then by the monotonicity of\n$\\nu(\\cdot)$ (Proposition~\\ref{prop:crind}) $\\nu(x)\\leq n-1$, hence $F(x)=F_{\\nu(x)}=F_{n-1}(x)$. \n\nIf $x>X^{\\!\\times}_n-1$, then $\\nu(x)>n-1$, hence by Definition~\\ref{def:crind} (of $\\nu(x)$) $F_{n-1}(x)>F(x)$.\n\nThe point $(X^{\\!\\times}_n-1,F(X^{\\!\\times}_n-1))$ lies on the graph of the function $F(\\cdot)$. In the left neighbourhood of\nthis point the graph of $F$ coincides with graph of $F_{n-1}$ and in the right neighbourhood -- with graph of $F_n$. Hence the curves $\\gamma_{n-1}$ and $\\gamma_n$ \nmeet at this point.\n\n\\smallskip\n(b) For $1\\leq n\\leq 6$ the intervals $[\\xi_n(1),\\xi_n(2))$\ndo not overlap, hence $\\nu(x-1)=n-1$ for $x\\leq \\xi_n(1)$. \n\nIn general, there are no points on the curve $\\gamma_{n-1}[1,2]$ with abscissas greater than $\\xi_{n-1}(2)$, since $\\xi_{n-1}'(t)>0$ for $t\\geq 2$ (Proposition~\\ref{prop:Xmin}(a)). \n\nFor $n\\geq 7$, $X^{0}_n<\\xi_n(1)=\\xi_{n-1}(2)$.\nThe equality $X^{\\!\\times}_n=\\xi_{n}(1)$ is impossible, since\nthe backward branch of the curve $\\gamma_n$ starting\nat $(\\xi_n(1),\\eta_n(1))$ lies above the branch with\nparameter values $t>X^{0}_n$ by Proposition~\\ref{prop:pmbranches}.\n\\end{proof}\n\nThe graph of the function $F(x)$ is the union of the segments of the curves $\\gamma_n^T$ corresponding to the parameter values $[\\tau^\\ell_n, \\tau^r_n]\\subset(\\tau^0_n,1)$. We have\n$$\n T_n(\\tau^\\ell_n)=T_{n-1}(\\tau^r_{n-1})=X^{\\!\\times}_n\n$$ \nand \n$$\n G_n(\\tau^\\ell_n)=G_{n-1}(\\tau^r_{n-1}).\n$$\nWith reference to the lower envelope of the curves\n$\\gamma_n$, the parameter values $t_n^\\ell=\\tau_n^\\ell+1$\nand $t_n^r=\\tau_n^r+1$ play the same role. \n\n\\iffalse\n\\section{More general problem}\n\n* AM-GM:\n$$\n \\frac{x_n}{x_{n-1}}+ \\frac{x_{n-1}}{x_{n-2}}+\\dots+\n \\frac{x_1}{x_0}+ \\frac{x_0}{p}\\to\\min \n\\quad\\text{\\rm over $x$ and $n$}.\n$$\n\n$$\n RHS\\geq n\\left(\\frac{x_n}{p}\\right)^{1\/n}.\n$$\nThere must be $x_n\/p\\geq 1$. Since $1\/p=N$ (by interpretation in Part I), we get $n\\sim\\ln N$.\n\n* Bellmann, Lee \\cite{BelLee1978} consider\n$$\n f(p)=\\max_q H(p,q,f(T(p,q))\n$$\nIn our case, $p$ becomes $\\xi$, $q$ becomes $t$ ($0<q< 1$); $T(p,q)=q(=t)$; $H(p,q,z)=ph(q)+z$.\n\nOr (?) $q=1\/(1+t)$; $t=q^{-1}-1$; $T(p,q)=T(q)=q^{-1}-1$.\n\n* Our case (one-parameter extension of the dynamic programming method as applied to the AM-GM problem)\n$$\n\\ba{l}\n g_{n+1}(\\xi)=\\inf\\left(g_n(t)+\\xi h(t)\\right)=\\inf_t G_n(t,\\xi)=g_n(t_n)+\\xi h(t_n),\n\\\\[2ex]\n g_1(\\xi)=s\\xi\n\\end{array}\n$$\nThe critical point $t_n$ is a root of the equation $\\partial_t G_n(t,\\xi)=0$, i.e.\n$$\n \\xi h'(t_n)=-g_n'(t_n).\n$$\nOn the other hand,\n$$\n g'_{n+1}(\\xi)=\\partial_t G_n(t,\\xi)|_{t=t_n}\\cdot t_n'(\\xi)+\n\\partial_\\xi G_n(t,\\xi)=0+\\partial_\\xi G_n(t,\\xi)=\nh(t_n).\n$$\nShifting the index, \nwe get $g_{n}(t_{n-1})=-\\xi h'_{t_{n-1}}$.\nHence\n$$\n \\xi h'(t_{n})=-h(t_{n-1}),\n$$\nthe implicit recurrence equation for determining $t_n(\\xi)$.\n\nIt begins with $s+\\xi h'(t_1)=0$, that is,\n$$\n h'(t_1)=-\\frac{s}{\\xi}.\n$$\n\nLet us assume that for any initial value $\\tau\\geq 0$ the sequence \n$(t_1=\\tau,t_2,\\dots)$ is determined uniquely.\nWe will write\n$$\n t_j=T_j(\\tau) \\quad (j=1,2,\\dots).\n$$\n\nOur minimization problem can be stated in the form\n$$\n g_n(\\xi)=\\min S_n(t_0,\\dots,t_n),\n$$\nwhere \n$$\n S_n(t_0,t_1,\\dots,t_n)=\\sum_{k=1}^n L(t_{k-1},t_k),\n$$\n$$\n L(p,q)=q h(p),\n$$\nwith boundary conditions $t_0=t_*$ (given) and $t_n=\\xi$.\n\n\n* Put\n$$\n G_n(t_0,\\dots,t_n\\mid c)=t_0+\\frac{t_1}{t_0+c}+\n\\frac{t_2}{t_1+c}+\\dots+\\frac{t_n}{t_{n-1}+c}.\n$$\nIf $t_j=ct'_j$, then\n$$\nG_n(0,t_1,\\dots,t_n\\mid c)=G_{n-1}(t'_1,\\dots,t'_n\\mid 1).\n$$\n\nDefine\n$$\n g_n(\\xi)=\\min_{(t_0,\\dots,t_{n-1})} G_n(t_0,\\dots,t_n\\mid c)|_{t_n=\\xi}\n=\\min_{t(=t_{n-1})}\\left(g_{n-1}(t)+\\frac{\\xi}{t+c}\n\\right).\n$$\n\n\n* Given a function $h$, define\n\\beq{recurgh}\n g(\\xi)=\\inf_{0<\\rho\\leq\\xi-1}\\left(g(\\rho)+ h(\\xi,\\rho)\\right)\n\\end{equation}\nwith initial condition\n$$\n g(\\xi)=\\xi, \\qquad 0<\\xi\\leq 1.\n$$\n\n* Given a function $h$, define\n\\beq{recurgh}\n g(\\xi)=\\inf_{0<\\rho\\leq\\xi-1}\\left(g(\\rho)+\\xi h(\\rho)\\right)\n\\end{equation}\nwith initial condition\n$$\n g(\\xi)=\\xi, \\qquad 0<\\xi\\leq 1.\n$$\n\n\n* For $s>0$:\n\\beq{recurgs}\n g_s(\\xi)=\\inf_{0<\\rho\\leq\\xi-1}\\left(g(\\rho)+\\frac{\\xi}{\\rho+s}\\right)\n\\end{equation}\nwith initial condition\n$$\n g(\\xi)=\\xi, \\qquad 0<\\xi\\leq 1.\n$$\n\n\\begin{prop}\n$g_0(\\xi)=g_1(\\xi-1).$\n\\end{prop}\n\n\n\n\\section{Analysis of the sequence $F^*(p)$}\n\nThe proofs of qualitative results rely on computations only insofar as the required accuracy is rather low and there is no doubt about stability etc.\nWe also provide graphs and more precise values of various numerical constants but not as part of rigorous arguments.  \n\nWhen referring to decimal numerals, we indicate rounding down or up by superscripts $+$ and $-$; precise decimal results are indicated\nby the $\\doteq$ sign as in the following example: \n$$\n\\frac{45}{32}=1.4^+,\\qquad\n\\frac{45}{32}=1.407^-,\n\\qquad \n\\frac{45}{32}\\doteq 1.40625.\n$$\n\nWe won't be excessively pedantic. The choice of parameters is not critical and no attempt is made to optimize them in numerical evaluations. The deviation from \"absolute rigor\" is no more than the customary allowances of using the same name for technically non-identical objects such as, say, group as a 4-tuple $(G,e\\in G, \\mathrm{mult}:G\\times G\\to G, \\mathrm{inv}:G\\to G)$ and the underlying set $G$.\n\nIn this section we prove that there exists constant \n$$\nB\\approx-0.70465603\n$$\nsuch that the solution of the recursion\n$$\n g(\\xi)=\\inf_{0<\\rho\\leq \\xi-1} \\left( g(\\rho)+\\frac{\\xi}{\\rho+1}\\right)\n$$\nwith initial condition\n$$\n g(\\xi)=\\xi,\\quad 0<\\xi\\leq 1,\n$$\nhas asymptotics\n$$\n g(\\xi)=e\\log\\xi+B+O(1\/\\log\\xi)\n$$\nas $\\xi\\to\\infty$.\n\nThis fact should not be an isolated peculiarity of the particular equation, but we do not pursue its investigation as a structural property of a class of similar equation, but rather take advantage of the specifics of our system explicitly --- notably the Eq.~\\eqref{gprime_rec}. \n\\fi\n\n\\iffalse\n---------------\n\n... We supply some additional facts that are not strictly necessary for our main line, but add to understanding of the fine features of the \nrecurrent sequences...\n\n-----------------------\n\nPut\n$$\n \\chi_n(t)=\\eta_n(t)-e\\log\\xi_n(t).\n$$\n\nProperties of the two roots of $\\alpha_\\infty(t)=e$:\n\n\\begin{lemma}\n(a) There exists $n_0$ such that for any $n\\geq n_0$\nthere are $t_1^{(n)}$ and $t_2^{(n)}$ in $(1,2)$ uniquely determined by the conditions:\n\\beq{deft1n}\n\\chi_n(t_1^{(n)})=0 \n\\end{equation}\nand\n\\beq{deft2n}\nt_2^{(n)} \\;\\text{\\rm is the point of minimum of $\\chi_n(t)$ in $[1,2]$}.\n\\end{equation}\n\n(b) $t_1=\\lim_{n\\to\\infty} t_1^{(n)}$ and\n$t_2=\\lim_{n\\to\\infty} t_2^{(n)}$.\n\\end{lemma}\n\\fi\n\n\\subsection{Asymptotics of\n\\texorpdfstring{$\\xi_n$}{xi[n]} and \n\\texorpdfstring{$\\eta_n$ as $n\\to\\infty$}{eta[n]}}\n\\label{ssec:asxieta}\n\nPut \n\\begin{equation}\n\\label{deltan}\n\\delta_{n}(t)=\\sum_{j=n}^\\infty \\frac{1}{\\xi_j(t)}=\\alpha_\\infty(t)-\\alpha_{n}(t).\n\\end{equation}\nand define the functions $\\phi(t)$ and $\\psi(t)$ for $t\\in[1,2]$ by\n\\begin{equation}\n\\label{phi}\n\\phi(t)=\\sum_{j=1}^\\infty\\log\\left(1-\\frac{\\delta_j(t)}{\\alpha_\\infty(t)}\\right),\n\\end{equation}\n\\begin{equation}\n\\label{psi}\n\\psi(t)=t-\\alpha_\\infty(t)-\\sum_{j=1}^\\infty \\delta_j(t).\n\\end{equation}\n\n\n\\begin{prop}\n\\label{prop:asphipsi}\nThe functions $\\phi(t)$ and $\\psi(t)$ are real-analytic and\nthe following asymptotic formulas hold:\n\\beq{asxin}\n\\log\\xi_n(t)=n\\log\\alpha_\\infty(t)+ \\phi(t)+\nO\\left(2^{-n}\\right),\n\\end{equation}\n\\beq{asetan}\n\\eta_n(t)=n\\alpha_\\infty(t)+ \\psi(t)+\nO\\left(2^{-n}\\right).\n\\end{equation}\n\\end{prop}\n\n\\begin{proof} \nThe recurrence relations \\eqref{recrelalxi} imply\n$$\n \\log\\xi_n(t)=\\log\\xi_0(t)+\\sum_{j=1}^n\\log\\alpha_j(t)\n=\\log t+\\sum_{j=1}^n\\log\\frac{\\alpha_j(t)}{\\alpha_\\infty(t)}\n+n\\log\\alpha_\\infty(t).\n.\n$$\nBy definition, \n$\\alpha_j(t)=\\alpha_\\infty(t)-\\delta_j(t)$, so \n$$\n\\log\\xi_n(t)-\\phi(t)-n\\log\\alpha_\\infty(t)=-\n\\sum_{j=n+1}^\\infty\\log\\left(1-\\frac{\\delta_j(t)}{\\alpha_\\infty(t)}\\right).\n$$\nSince the sequence $\\xi_j(t)$ grows exponentially, taking into account the uniform estimate $\\alpha_n(t)\\geq 2$ from Proposition~\\ref{prop:xieta-basic}(b), \nfor sufficiently large $n_0$ the conditions of Lemma~\n\\ref{lem:est-al-xi} are met with $a=2$ and it follows by \\eqref{estdeltan} that $\\delta_j(t)=O(2^{-j})$, $j>n_0$. This estimate implies \\eqref{asxin}.\n\nSimilarly, by the recurrence relations \\eqref{recreleta}\n$$\n\\ba{rcl}\n\\displaystyle\n \\eta_n(t)=\\eta_0(t)+\\sum_{j=0}^{n-1}\\alpha_j(t)\n&=&\n\\displaystyle\n1+n\\alpha_\\infty(t)-\\sum_{j=0}^{n-1}\\delta_j(t)\n\\\\[2ex]\n&=&\n\\displaystyle\n\\psi(t)+n\\alpha_\\infty(t)+\\sum_{j=n}^{\\infty}\\delta_j(t).\n\\end{array}\n$$\nThe remainder term is estimated as above and we come to \\eqref{asetan}.\n\\end{proof}\n\n\\subsection{Narrowing down the parameter domain}\n\\label{ssec:narrowing}\n\nFrom the crude asymptotic result --- Proposition~\\ref{prop:crudeas} --- we know that \n$$\\frac{\\eta_n(t)}{\\log\\xi_n(t)}=e+o(1)$$ \non the segment of the\ncurve $\\gamma_n$ that belongs to the lower envelope.\nHence $\\log\\alpha_\\infty(t)\/\\alpha_\\infty(t)=e+o(1)$\nfor the corresponding values of the parameter and\nwe conclude that $\\alpha_\\infty(t)$ must be close to $e$.\nTherefore $t$ must be close to one of the two roots,\n$t_a$ or $t_b$, of the equation $\\alpha_\\infty(t)=e$.\n\nBy Proposition~\\ref{prop:pmbranches}, \namong the two values $t_1<t_2$ corresponding\nto the same value of $\\xi_n$, the inequality\n$\\eta_n(t_2)<\\eta_n(t_1)$ takes place. Therefore\nonly the neighbourhood of $t_b$ is of interest for\ndetermining the lower envelope of the curves $\\gamma_n$.\n\nSummarizing, we state\n\n\\begin{prop}\n\\label{prop:minint}\nThe values $t_n^\\ell$ and $t_n^r$ defined at the end of\nSec.~\\ref{ssec:special-points} tend to $t_b$ as $n\\to\\infty$. \n\\end{prop} \n\n\\subsection{End of proof of Theorem~\\ref{thm:main}}\n\\label{ssec:eop-main}\nDenote\n$$\n \\beta(t)=\\frac{\\alpha_\\infty(t)}{\\log\\alpha_\\infty(t)}\n$$\nand\n$$\n\\zeta(t)=\\psi(t)-\\beta(t)\\,\\phi(t).\n$$\nDue to Proposition~\\ref{prop:minint}, in the asymptotic calculation we may choose some compact neighborhood $I\\subset(t_o,2)$ \nof the point $t_b$\nas the parameter interval. \n\nSince $\\beta'(t)=(\\log\\alpha_\\infty(t)-1)\\alpha_\\infty'(t)\/(\\log\\alpha_\\infty(t))^2$, the \nfunction $\\beta(t)$ has a unique critical point (the minimum) on $I$, namely, $t=t_b$. By Proposition~\\ref{prop:alpha}, $\\alpha'(t_b)>0$.\n\nThe family of parametric curves $x=\\log\\xi_n(t)$, $y=\\eta_n(t)$ is shown to satisfy all conditions of\nTheorem~\\ref{thm:paramcurves}. \n\nUsing the asymptotic formulas \\eqref{asxin} and \\eqref{asetan}\nLet us find the coefficients in the final asymptotic formula \\eqref{asfabs} in our case.\n\n\nThe point $t_0$ in Theorem~\\ref{thm:paramcurves}\ncorresponds to $t_b$ in this context.\nThe terms $r_0(t)$ and $r_1(t)$ of the general formulation\nare not present in Eqs.~\\eqref{asxin}--\\eqref{asetan},\nso $\\delta(t)\\equiv 0$. We have \n$a_0=\\beta(t_b)=e$,\n$a_1=\\zeta(t_b)\\approx -0.704656$. The constant $A$ in Theorem~\\ref{thm:main} equals $1-\\zeta(t_b)$ (remember that $G_n(t-1)=\\eta_n(t)-1$ on the original extremal trajectory).\n\nIn the formula \\eqref{asgenPhi}, $p_0(t_0)=\\log\\alpha_\\infty(t_b)=1$\nand $q_0(t_0)=\\phi(t_b)\\approx 0.6974$;\nthis is the numeric constant $b$ in Theorem~\\ref{thm:main}.\n\nWe will derive the coefficients \\eqref{coefa2a3} analytically in the concluding two lemmas, showing that $a_3=e\/2$ (Lemma~\\ref{lem:a3})\nand $\\zeta'(t_b)=0$ (Lemma~\\ref{lem:Dzeta}) hence $a_2=0$.\n\nThus the proof of Theorem~\\ref{thm:main} is complete.\n\\qed\n\n\\begin{lemma}\n\\label{lem:a3}\nThere holds the identity\n$$\n \\beta''(t_b)\\left(\\frac{(p_0(t_b))^2}{p_0'(t_b)}\n\\right)^2=e,\n$$\nwhere $p_0(t)=\\log\\alpha_\\infty(t)$.\n\\end{lemma}\n\n\\begin{proof} Since $\\beta=e^{p_0}\/p_0$,\nwe have $\\beta'=p_0'\\cdot e^{p_0}\/p_0^2\\cdot (p_0-1)$.\nTo evaluate $\\beta''(t_b)$, it suffices to differentiate\nthe last factor (which vanishes at $t_b$):\n$$\n\\beta''(t_b)=\\left.\\frac{p_0'(t)\\cdot e^{p_0(t)}}{p_0^2(t)}\\cdot p_0'(t)\\right|_{t=t_b}.\n$$\nTherefore \n$$\n\\beta''(t_b)\\left(\\frac{(p_0(t_b))^2}{p_0'(t_b)}\n\\right)^2=e^{p_0(t_b)}p_0^2(t_b)=e\\cdot 1^2=e.\n\\eqno\\qedhere\n$$\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:Dzeta}\nThere holds the identity\n$\\zeta'(t_b)=0$, that is, $\\psi'(t_b)-e\\phi'(t_b)=0$.\n\\end{lemma}\n\n\\begin{proof}\nWriting $\\psi'=-\\sum_{j=0}^\\infty (\\alpha'_\\infty-\\alpha'_j)$ and $\\phi'=\\sum_{j=1}^\\infty\n(\\alpha'_j\/\\alpha_j-\\alpha'_\\infty\/\\alpha_\\infty)$,\nwe see that\n$$\n\\left[\\psi'-e\\phi'\\right]_{t_b}\n=\n1-\\alpha'_\\infty(t_b)-\\sum_{j=1}^\\infty\\left[\n\\alpha'_\\infty\\left(1-\\frac{e}{\\alpha_\\infty}\n\\right)-\\alpha'_j\\left(1-\\frac{e}{\\alpha_j}\n\\right)\\right]_{t_b}\n$$\nIn view of the identities $\\alpha_\\infty(t_b)-e=0$\nand $\\alpha_j(t_b)-e=-\\delta_j(t_b)$, the right-hand side\nsimplifies to \n$$\n 1-\\alpha'_\\infty(t_b)-\\sum_{j=1}^\\infty\\delta_j(t_b)\\,\\frac{\\alpha'_j(t_b)}{\\alpha_j(t_b)}.\n$$\nWe will prove that in general\n\\begin{equation}\n\\label{alpha_identity}\n \\sum_{j=1}^\\infty \\delta_j(t) \\frac{\\alpha_j'(t)}{\\alpha_j(t)}=1-\\alpha_\\infty'(t).\n\\end{equation}\n\nBy the definition \\eqref{defalpha} of $\\alpha_j$,\n$$\n \\frac{\\alpha_j'}{\\alpha_j}=\n\\frac{\\xi_j'}{\\xi_{j}}-\\frac{\\xi_{j-1}}{\\xi_{j-1}}.\n$$\nApplying Abel's summation-by-parts formula to the partial sum in the l.h.s. of \\eqref{alpha_identity}, we get\n$$\n \\sum_{j=1}^N \\delta_j \\frac{\\alpha_j'}{\\alpha_j}=\n\\delta_N\\frac{\\xi_N'}{\\xi_N}-\\delta_1\\frac{\\xi_0'}{\\xi_0}\n+\\sum_{j=1}^{N-1} (\\delta_j-\\delta_{j+1}) \\frac{\\xi_j'}{\\xi_{j}}.\n$$\n\nWe have $\\xi_0=1$ and $\\xi'_0=0$; $\\xi'_N\/\\xi_N=O(1)$\nand $\\delta_N=o(1)$ as $N\\to\\infty$, so the boundary terms are $o(1)$.\nNow, \n$\\delta_{j+1}-\\delta_j=\\alpha_j-\\alpha_{j+1}=-\\xi_j^{-1}$. Therefore\n$$\n\\sum_{j=1}^N \\delta_j \\frac{\\alpha_j'}{\\alpha_j}\n=\\sum_{j=1}^{N-1}(\\xi_j^{-1})'+o(1)\n=\\alpha'_1-\\alpha'_N+o(1),\n$$\nand the limit is $\\alpha'_1-\\alpha'_\\infty=1-\\alpha'_\\infty$.\n\\end{proof}\n\n\\iffalse\n\\begin{corollary}\nThe critical index $\\nu(x)$ satisfies the asymptotic estimate\n$$\n e=\\min_{1\\leq t\\leq 2}\\log\\alpha_\\infty(t)\\leq\\liminf_{x\\to\\infty}\\frac{\\log x}{\\nu(x)}\n\\leq\\limsup_{x\\to\\infty}\\frac{\\log x}{\\nu(x)} \n\\leq \\max_{1\\leq t\\leq 2}\\log\\alpha_\\infty(t).\n$$\n\\end{corollary}\n\n\\begin{proof}\nWe combine Proposition~\\ref{prop:tau01} with the asymptotics of $\\log\\xi_n(t)$ from Lemma~\\ref{lem:asphipsi}.\n\n[Need details]\n\n(f) [Not quite; it's too early here. Put where discuss the asymptotics of the minimizing critical point.] \n\nA refinement of the functional equation \n\\eqref{fe1} reads\n\\beq{fe2}\n f(x)=\\min_{0<y\\leqX^{**}_{\\nu(x)-1}}\\left(f(y)+\\frac{x}{y+1}\\right).\n\\end{equation}\n\\end{proof}\n\\fi\n\n\n\n\\input{sec_Ap}\n\n\\input{app_proof-paramcurves}\n\n\\input{app_alpha}\n\n\\iffalse\n\\appndx{Stability analysis of the recurrence \\eqref{recrelalxi}}{app:stability}\n\n\\begin{prop}\nLet $(\\tilde\\alpha_n)_{n\\geq 0}$, $(\\tilde\\xi_n)_{n\\geq 0}$,\n$(\\varepsilon_n)_{n\\geq 0}$ and $(\\mu_n)_{n\\geq 0}$ be four sequences of positive numbers. \nSuppose that\n$$\n a=\\sum_{n=1}^\\infty\\frac{1}{\\tilde\\alpha_n}<\\infty\n$$\nand \n$$\n b=\\sum_{n=0}^\\infty\\frac{1}{\\tilde\\xi_n}<\\infty.\n$$\nGiven some parameter $c>1$, \nsuppose further that\n$$\n k=1-c^2 ab>0.\n$$\nFinally, suppose that $(\\varepsilon_n)$ and $(\\mu_n)$ \nsatisfy the relations\n$$\n \\ba{l}\n\\displaystyle\n \\varepsilon_{n+1}<\\varepsilon_n+c\\frac{\\mu_n}{\\tilde\\xi_n},\n\\\\[2ex]\n \\displaystyle\n \\mu_{n+1}<\\mu_n+c\\frac{\\varepsilon_{n+1}}{\\tilde\\alpha_{n+1}}.\n\\end{array}\n$$\nDenote\n$$\n\\ba{l}\n\\displaystyle\n \\bar{\\varepsilon}=\\frac{\\varepsilon_0+cb\\mu_0}{k},\n\\\\[2ex]\\displaystyle\n \\bar{\\mu}=\\frac{\\mu_0+ca\\varepsilon_0}{k}.\n\\end{array}\n$$\nThen for any $n\\geq 0$ the inequalities\n$\n \\varepsilon_{n}<\\bar\\varepsilon\n$\nand\n$\n \\mu_{n}<\\bar\\mu\n$\nhold true.\n\\end{prop}\n\n\\begin{proof}\nThe claim is obviously true for $n=0$.\nAssuming it is true up to a certain $n$, we have\n$$\n \\varepsilon_{n+1}<\\varepsilon_0+c\\bar\\mu\\sum_{j=0}^n\\frac{1}{\\tilde\\xi_j}\n<\\varepsilon_0+\\frac{cb}{k}(\\mu_0+ca\\varepsilon_0)=\\bar\\varepsilon\n$$ \nand\n$$\n \\mu_{n+1}<\\mu_0+c\\bar\\varepsilon\\sum_{j=1}^{n+1}\\frac{1}{\\tilde\\alpha_j}\n<\\mu_0+\\frac{ca}{k}(\\varepsilon_0+cb\\mu_0)=\\bar\\mu.\n$$\nThe validity of our claim for all $n$ follows by induction.\n\\end{proof}\n\n\n\\appndx{Monotonicity of critical points\nand the mutual position of branches of the curve\n$\\eta_n(\\xi_n)$}{app:critpts}\n\n\\begin{prop}\nIf $\\tau_{on}$ is the critical point of the function\n$\\alpha_n(\\tau)$ (which exists for $n\\geq 5$), then\n$t_{on}<t_{o{n+1}}$ ($n\\geq 5$).\n\\end{prop}\n\n\\begin{proof}\nSince $\\alpha_n''(\\tau)>0$ (item 3 in proof of Proposition~\\ref{prop:alpha}) and $\\xi_1(\\tau)''\\equiv 0$, we conclude by induction that the functions\n$\\xi_n(\\tau)$ are convex for $1\\leq \\tau\\leq 2$\nand any $n$.\n\nTherefore the equation $\\xi_n'(\\tau)=0$ can have at most one root in $[1,2]$ and hence in $[1,\\infty)$. (Recall that for $\\tau>2$ the inequality $\\xi_n'(\\tau)>0$ is asserted by Corollary of Lemma~\\ref{lem:estUn}.)\n\nOur inductive assumption is: $\\xi'_{n-1}(\\tau)>0$ for \nall $\\tau\\geq \\tau_{on}$. \n\nDifferentiating the recurrence relation, we get\n$$\n \\xi'_{n+1}(\\tau)=\\frac{2(\\xi_n(\\tau)+1)}{\\xi_{n-1}(\\tau)+1}\\xi'_{n}(\\tau)-\\left(\\frac{\\xi_n(\\tau)+1}{\\xi_{n-1}(\\tau)+1}\\right)^2\\xi'_{n-1}(\\tau).\n$$\nAt $\\tau=\\tau_{on}$, $\\xi'_n(\\tau)=0$. Hence\n$\\xi'_{n+1}(\\tau_{on})$ and $\\xi'_{n-1}(\\tau_{on})$ have different signs. \n\nThe inequality $\\xi'_{n-1}(\\tau_{on})<0$ contradicts the inductive assumption. Hence $\\xi'_{n+1}(\\tau_{on})<0$. Therefore\n$\\xi'_{n+1}(\\cdot)$ has a root in $(\\tau_{on},1]$.\nThat root is unique and equals $\\tau_{o,n+1}$.\nWe conclude that $\\tau_{o,n+1}>\\tau_{on}$ and\n$\\xi_n'(\\tau)>0$ for $\\tau\\geq\\tau_{o,n+1}$,\ncompleting the induction step.\n\\end{proof}\n\n\n\\begin{prop}\nIf $\\tau_{1n}<\\tau_{2n}$ are the roots of the function $\\alpha_n(\\tau)=e$, then\n$\\eta_{1n}>\\eta_{2n}$. In general, \nif $\\xi_n(\\tau')=\\xi_n(\\tau'')$ with $\\tau'<\\tau''$,\nthen $\\eta_n(\\tau')>\\eta_n(\\tau'')$.\n\\end{prop}\n\n\\begin{proof}\n\\end{proof}\n\\fi\n\n\n\\section{Constants \\texorpdfstring{$A(p)$}{A(p)}: existence and computation}\n\\label{sec:Ap}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:genpar}} \n\n(a) For any $n$-tuple $\\mathbf{t}$, the function\n$p\\mapsto F^{(p)}(\\mathbf{t})$ is decreasing. Hence,\nif the asymptotics \\eqref{Fpasym} takes place, the function\n$A(p)$ is at least nondecreasing.\n\nWe will consider the cases $p>1$ and $p<1$ separately \nto justify the formula \\eqref{Fpasym} and to deduce that $A(p)$ is strictly increasing. That way, a proof of (a) will be finished.\n\nLet us note an identity that will be useful in both cases\n$p>1$ and $p<1$.\n\nFor any $q>0$, putting $\\tilde t_j=t_j\/q$, $j=1,\\dots,n$, we get\n$t_j\/(t_{j-1}+p)=\\tilde t_j\/(\\tilde t_{j-1}+p\/q)$.\nHence\n\\begin{align}\nS^{(p)}_n(t_1,\\dots,t_n)\n&=t_1+\\frac{\\tilde{t}_2}{\\tilde t_1+p\/q}+\\dots+\\frac{\\tilde t_n}{\\tilde{t}_{n-1}+p\/q}\n\\label{Sp-transform1}\n\\\\[1ex]\n&=t_1-\\tilde{t}_1+S^{(p\/q)}_n(\\tilde{t}_1,\\dots,\\tilde {t}_n).\n\\label{Sp-transform}\n\\end{align}\n\n\\medskip\n(b) Case $p>1$. Let us take $q=p$ in \\eqref{Sp-transform}.\nWe get\n$$\nS^{(p)}_n(t_1,\\dots,t_n)\n=t_1\\left(1-\\frac{1}{p}\\right)\n+S_n(\\tilde t_1,\\dots,\\tilde t_n)\\geq \nS_n(\\tilde t_1,\\dots,\\tilde t_n).\n$$\nHence $F^{(p)}_n(x)\\geq F_n(x\/p)$ and $F^{(p)}(x)\\geq F(x\/p)$.\n\nOn the other hand, taking again $\\tilde t_j=t_j\/p$\nand setting $t_1=0$, we obtain  \n$$\n\\left. S^{(p)}_n(t_1,\\dots,t_n)\n\\right|_{t_1=0}=S_{n-1}(\\tilde t_1,\\dots,\\tilde t_{n-1}).\n$$\nHence $F^{(p)}_n(x)\\leq F_{n-1}(x\/p)$, so $F^{(p)}(x)\\leq F(x\/p)$.\n\nThe identity \\eqref{Fpgt1} is thus proved. The asymptotics and the expression for $A(p)$ follow\nfrom Theorem~\\ref{thm:main}.\n\n\\medskip\n(c) Case $0<p<1$. Here we give only a sketch of proof, using heuristics at some steps.\n\nThe asymptotics \\eqref{Fpasym} can be derived in the same way as we did it for $p=1$, by obtaining parametric\ndescription of the curves $y=F^{(p)}_n(x)$.\n\nTo derive the functional equation, take $q=t_1+p$ in Eq.~\\eqref{Sp-transform1}. Then $\\tilde t_1+p\/q=1$,\nhence\n$$\n S^{(p)}_n(t_1,\\dots,t_n)=t_1+S^{(p\/q)}_{n-1}(\\tilde t_2,\\dots,\\tilde t_n).\n$$\nIt follows that\n$$\n F^{(p)}_n(x)=\\inf_{q\\geq p} \\left(q-p+F^{(p\/q)}_{n-1}(x\/q)\\right).\n$$\nConsequently,\n$$\n F^{(p)}(x)=\\inf_{q\\geq p} \\left(q-p+F^{(p\/q)}(x\/q)\\right).\n$$\nFrom \\eqref{Sp-transform} we see that \n$F^{(p)}(x)\\leq F^{(p)}(x\/q)$ for $q\\leq 1$. As $q-p\\geq 0$, the minimization range can be reduced to $q\\geq 1$.\n\nThe next step needs a justification that we omit. Instead of the functions $F^{(\\dots)}(\\dots)$ we substitute their asymptotics\n(and change the letter $q$ into $u$).\nThe result is \n$$\n e\\log x-A(p)+o(1)=\\inf_{u\\geq 1} \\left(u-p+e\\log\\frac{x}{u}-A\\left(\\frac{p}{u}\\right)+o(1)\\right).\n$$\nAfter cancellation of the $e\\log x$ terms the remaining leading terms are constants (with respect to $x$). We obtain the equation \\eqref{feq-Ap} with a weaker constraint: $u\\geq 1$ rather that $1\\leq u\\leq e$.\nHowever, the values $u>e$ can be excluded from the maximization range since the function $A(p\/u)$ is nonincreasing w.r.t. $u$ and the function $e\\log u-u$ strictly decreases for $u>e$.\n\n\\smallskip\nFinally, let us explain the asymptotics of $A(p)$\nas $p\\to +0$. \nChoosing $u=e$ in the right-hand side of \\eqref{feq-Ap} we\nget $A(p)\\geq A(p\/e)+p$.\nTherefore\n$A(p)\\geq A(pe^{-k})+p\\sum_{j=0}^{k-1} e^{-j} $ for any $k\\in\\mathbb{N}$. By monotonicity of $A(\\cdot)$ we get\n$A(p)\\geq p(1-e^{-1})^{-1}$. Putting $k=\\liminf_{p\\to +0} A(p)\/p$, we see that $k\\geq k_0=(1-e^{-1})^{-1}$. \n\nSkipping a necessary justification, we assume that the $\\lim_{p\\to +0} A(p)\/p$ exists; its value is thus $k$.\nLet us show that $k=k_0$. \n\nTake small $\\varepsilon>0$. For sufficiently small $p$ \nand any $u\\in[1,e]$ we have\n$A(p\/u)<(k+\\varepsilon)p\/u$, so \n$$\n A(p)\\leq \\max_{1\\leq u\\leq e}\\left((k+\\varepsilon)\\frac{p}{u}+e\\log u-u+p\\right).\n$$\nDifferentiating, we see that the maximum is attained at the point $u=u^+$ which is the larger root (close to $e$) of the quadratic equation\n$$\n u^2-eu+(k+\\varepsilon)p=0.\n$$\nThe smaller root is $u_-=e-u_+=(k+\\varepsilon)p\/u_+\\sim (k+\\varepsilon)p\/e$. \nSince $|e\\log u-u|=O(|e-u|^2)$ as $u\\to e$, we have\n$$ \nA(p) \\leq (k+\\varepsilon)\\frac{p}{u_+}+O(p^2)+p=\\left(\\frac{k+\\varepsilon}{e}+1\\right)p+O(p^2).\n$$\nTherefore \n$$\n k=\\lim_{p\\to +0}\\frac{A(p)}{p}\\leq \\frac{k+\\varepsilon}{e}+1,\n$$\nso $k\\leq e\/(e-1-\\varepsilon)$. Making $\\varepsilon\\to 0$, we obtain\n$k\\leq k_0$, hence $k=k_0$.\n\n\\subsection{Tabulation of the function \\texorpdfstring{$A(p)$}{A(p)}}\nLet us rewrite the functional equation\n\\eqref{feq-Ap} in the form\n$$\n A(p)=\\max_{p\/e\\leq s\\leq p} \\left(A(s)+\\theta\\left(\\frac{p}{s}\\right)\\right)+p ,\n$$\nwhere\n$$\n\\theta(t)=e\\log t-t.\n$$\nAssuming that $A(\\cdot)$ is differentiable, we have the\ncondition of extremum:\n$$\n A'(s_*)-\\frac{p}{s_*^2}\\cdot\\theta'\\left(\\frac{p}{s_*}\\right)=0.\n$$\nEquivalently,\n$$\n  s_*^2\\,A'(s_*)-es_*+p=0.\n$$\n\nWe have\n$$\nA'(p)=1+\\frac{1}{s_*}\\theta' \\left(\\frac{p}{s_*}\\right)\n=1+\\frac{e}{p}-\\frac{1}{s_*}.\n$$\n\nConsider the triples $(s_*,A'(s_*),A(s_*))$\nand $(p,A'(p),A(p))$ as consequtive points of the iteration process:\n\\begin{align}\n(x_{n-1},y_{n-1},z_{n-1})&= (s_*,A'(s_*),A(s_*)),\n\\nonumber\n\\\\\n(x_n,y_n,z_n)&=(p,A'(p),A(p)).\n\\nonumber\n\\end{align}  \nThe equations defining the recurrence are\n$$\n \\ba{l}\n\\displaystyle\n x_n=ex_{n-1}-x_{n-1}^2 y_{n-1},\n\\\\[1ex]\n\\displaystyle \ny_n=1-\\frac{1}{x_{n-1}}+\\frac{e}{x_n},\n\\\\[2ex]\n\\displaystyle\nz_n=z_{n-1}+x_n+\\theta\\left(e-x_{n-1}y_{n-1}\\right).\n\\end{array}\n$$\nStarting with some arbitrary small $x_1$ and setting\n$y_1=k_0$, $z_1=k_0 x_1$, we can continue iterations\nuntil $x_n$ exceeds $1$ for the first time. This way we can tabulate the function $A(p)$, $0<p<1$.\n\nThe actual calculation of $A(1)$ performed by this method\ndemonstrated an\nagreement with the\nvalue stated in Theorem~\\ref{thm:main}, which was originally found using Theorem~\\ref{thm:paramcurves}\nand the functions \\eqref{phi}, \\eqref{psi}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany countries are ethnically diverse, regardless of their level of development.\\footnote{See \\cite{Alesina2003DevleeschauwerEasterlyKurlatWacziarg2003} and \\cite{Fearon2003}, who propose time-invariant  measures of ethnic diversity that are widely used in the literature, and report relatively high levels of ethnic fractionalization by country (as well as significant cross-country variation). For alternative --- time-variant --- measures, see \\cite{PatsiurkoCampbellHall2012}, who show that even the most historically homogenous countries (i.e. the OECD countries) are rapidly becoming more heterogeneous. In addition, see \\cite{VogtBormannRueggerCedermanHunzikerGirardin2015}, who propose the Ethnic Power Relations (EPR) Core Dataset, a very rich dataset that is extensively used in this paper.}  Despite the benefits of ethnic heterogeneity,\\footnote{One example is increased productivity, as ethnic diversity may be associated with more trade and innovation. However, ethnic diversity is also associated with negative outcomes, particularly lower levels of generalized trust \\citep{AlesinaLaFerrara2005} and an underprovision of public goods \\citep{AlesinaBaqirEasterly1999}. See \\cite{AlesinaLaFerrara2005} for a review of the possible costs and benefits of ethnic diversity, and \\cite{LaitinJeon2015} for a longer list of the potential benefits of ethnic diversity.} ethnic-based political inequality and discrimination are pervasive.\\footnote{From 1946 to 2017, at least 32\\% of ethnic groups were explicitly politically discriminated against at some point (see EPR Core Dataset). Group discrimination issues have also been on the agenda of the United Nations for many years (for example, see Resolution 47\/135 of Dec. 18, 1992) and, specifically, it has been argued that it is not sufficient for states to siimply ensure formal political participation for minority groups, but that this political participation must have a substantial influence on decisions that are taken (see OHCHR Forum on Minority Issues, 2009).}  Why is this?  This study suggests that part of the variation in ethnic-based political inequality depends of the relative size of ethnic groups within a country. \n\nUsing group-level data for 569 ethnic groups in 175 countries from 1946 to 2017, I find evidence of an inverted-U-shaped relationship between the relative size of an ethnic group and its access to power. This single-peaked relationship is robust to many alternative specifications, which include group fixed effects, time-varying group-specific controls, country-time fixed effects and group-specific linear trends. This relationship is also robust to using lagged relative group size as an instrument for its current value.  \n\nThough caution is necessary in interpreting the results as causal, they are consistent with relative group size affecting access to power. Taking this into consideration, I propose an explanation based on an initial high level of political inequality, and on the incentives that more powerful groups have,  \\emph{ex ante}, to continue limiting other groups' access to power. Through a very simple model, I argue that these incentives crucially depend on the relative size of politically marginalized groups, and follow an inverted-U-shaped pattern. First, when a politically marginalized group is very small, the group tends to be irrelevant, so more powerful groups have little incentive to cede any control to this small group. Second, when a politically marginalized group is relatively large, individuals in very powerful --- but also relatively small --- groups greatly value a government controlled by their own group, as it allows them to extract very high per capita rents. Therefore, individuals in small powerful groups do not have incentives to share power because it would limit their ability to keep rents for themselves. Only when a politically excluded group is neither very big nor very small can it expect to receive greater access to power; the excluded group is big enough to potentially overthrow the government if it isn't voluntarily given access to power, but small enough that it wouldn't detract too much from the incumbent group's ability to extract rents in a power-sharing arrangement. \n\nThe model I propose to explain the inverted-U relationship between group size and access to power leads to a new empirical prediction that identifies a crucial role of the persistence of political institutions: the single-peaked pattern should be weaker in countries where political institutions have historically been less open. Insofar as undemocratic institutions tend to persist, in countries with political institutions that have traditionally been less open, any increase in the \\emph{de jure} access to power of politically marginalized groups might be seen as less threatening by more powerful groups. Thus, insofar as the costs of maintaining the status quo increase with the size of marginalized groups, in countries where political institutions have historically been less open, the model predicts a monotonically increasing relation between group size and access to power. \n\nUsing PolityIV's measures of openness and competitiveness of executive recruitment as a proxy for the openness of political institutions, I find evidence consistent with this additional prediction: the inverted-U relationship between group size and access to power is specific to countries where political institutions have been more open in the past; in countries with less open political institutions, this relationship tends to be  monotonically increasing. \n\nThe rest of the paper is organized as follows. Section \\ref{background} discusses the existing literature on the relationship between political exclusion and group size. The data and empirical strategy are discussed in Section \\ref{dataandidentification}. The main results are presented in Section \\ref{baselineresults}.  Section \\ref{model} presents the model, and Section \\ref{additionalevidence} provides additional evidence related to the role of the persistence of political institutions. Section \\ref{conclusion} concludes.\n\n\n\\section{Related literature}\\label{background}\n\nThe sociology and economics literatures have developed distinct approaches to examining the relationship between group discrimination and group size. The classical sociological approach identifies the size of the groups as a crucial factor in explaining how individuals in dominant groups perceive potential threats posed by individuals in other groups. This literature has proposed two hypotheses about the specific relationship between this perception and the size of the subordinate groups. First, it may be that the larger the size of a subordinate group, the more individuals in a dominant group perceive the subordinate group as a threat to their economic and social privileges and, therefore, the higher their motivation to discriminate against the subordinate group (see for instance \\citealt{Blumer1958} or  \\citealt{Blalock1967}).\\footnote{This idea constitutes the core of what is known as the ``power-threat'' or ``group threat'' theory.} Second, it may be that the larger the relative size of a subordinate group, the greater the opportunity for positive intergroup contact, which reduces the motivation for individuals in the dominant group to discriminate against individuals in the subordinate group (see for instance \\citealt{Allport1954} or  \\citealt{Pettigrew1998}).\\footnote{This idea constitutes the core of what in known as the ``intergroup contact'' theory.} The empirical evidence for these two theories is inconclusive (see \\cite{SchlueterScheepers2010} for a critical review), with problems related to identification and external validity.\n\nThe economics literature relating discrimination and group size is less abundant, and focuses on discrimination in labor markets and  segregation. Departing from a standard model of statistical discrimination in labor markets, \\cite{MoroNorman2004}  propose a theoretical extension that examines how group size affects conflicting interests between groups. Specifically, they find that as the marginalized group grows, the per capita gains from discrimination for individuals in the dominant group also grow, as do the incentives of individuals in the subordinate group to invest in human capital. This makes it harder to sustain a discriminatory equilibrium, and suggests one rationale for apartheid or other discriminatory measures; however, \\cite{MoroNorman2004} do not model this explicitly. \n\n\\cite{Eeckhout2006} proposes a model of segregation in which segregation can make minority groups worse off while making the majority better off, so that any change in equilibrium that eliminates segregation will make the majority worse off. Even though it is not explicitly modeled, \\citeauthor{Eeckhout2006}'s main result is consistent with \\citeauthor{MoroNorman2004}'s rationale for the existence of institutional devices to ensure that the preferred equilibrium for the group with political control is realized.\n\nThis paper seeks to contribute to the sociological literature by providing new and relatively well-identified evidence about the relationship between group discrimination and group size, with a specific focus on the access to power of individuals in subordinate groups (rather than on out-group attitudes). In addition, this paper proposes a new and very stylized micro-funded theory that allows for the possibility of a non-monotonic relationship.  This paper seeks to contribute to the economics literature by providing evidence about the relationship between group discrimination and group size, and on the role of institutional mechanisms that affect this relationship.\n \n\n \n\\section{Empirical specification and identification}\\label{dataandidentification}\n\n\\subsection{Data}\n\nI study 569 ethnic groups in 175 countries from 1946 to 2017. The baseline specification is divided into 10-year sub-periods (starting in 1945) to account for potential lags in the relationship between group size and group access to state power.\\footnote{I also construct a five-year panel, which achieves similar results.}  To construct observations for each sub-period, I take an average of the annual data from that sub-period.\\footnote{I explore robustness to an alternative specification in which, instead of averaging the 10-year data, I take one year's observation from within each sub-period (i.e. one observation every tenth year). The motivation behind this robustness check is the additional serial correlation that averaging may introduce. As will be shown below, the results are robust to the use of this alternative specification.} \n\nThe analysis is based on the Ethnic Power Relations (EPR) Core Dataset 2018, which identifies all politically relevant ethnic groups and their access to state power in every country of the world from 1946 to 2017 \\citep{VogtBormannRueggerCedermanHunzikerGirardin2015}.\\footnote{This data can be found at \\url{https:\/\/icr.ethz.ch\/data\/epr\/core\/}.} It provides annual data on politically relevant ethnic groups,\\footnote{The EPR dataset defines ``ethnicity'' as a subjectively experienced sense of commonality based on a belief of common ancestry and a shared culture. It includes ethnolinguistic, ethnoreligious, and ethnosomatic (or ``racial'') groups.} their relative sizes as a share of the total\npopulation, and their access to executive  state power in all countries with a population of at least 250,000.\\footnote{For a more detailed description of this dataset, see \\cite{BesleyMueller2018}, who also use this dataset.} \n\n\nThe EPR dataset characterizes an ethnic group as politically relevant if at least one significant political actor claims to represent the interests of that group in the national political arena, or if group members are systematically and intentionally discriminated against in public politics. The EPR measures access to power with a roughly ordinal scale composed of three main categories (and several subcategories), depending on whether a group controls power alone, shares power with other ethnic groups, or is excluded from executive state power. Specifically,  the EPR measures power access according to the following categories and subcategories: \n\\begin{itemize}[noitemsep]\n  \\item[1] The group rules alone:\n  \\begin{itemize}\n  \\item[1.1] Monopoly -- Members hold monopoly power in the executive. \n  \\item[1.2]   Dominance -- Members hold dominant power in the executive.\n\\end{itemize}  \n  \\item[2]  The group shares central power:\n  \\begin{itemize}\n  \\item[2.1]   Senior partner -- Members participate as senior partners in a\nformal or informal power-sharing arrangement. \n  \\item[2.2]   Junior partner -- Members participate as junior partners in government.\n\\end{itemize}\n   \\item[3]  The group is excluded from central power:\n   \\begin{itemize}\n  \\item[3.1]   Powerless -- Members  do not have influence on decision-making at the national level of executive power, but are not explicitly discriminated against.\n  \\item[3.2]  Discrimination -- Members are subjected to active, intentional and targeted discrimination by the state, with the intent of excluding them from political power.\n  \\item[3.3]   Self-exclusion (or separatist autonomy) -- Members have excluded themselves from central state power.\n   \\item[3.4]   Irrelevance -- Members' ethnicity is not politicized (but the members comprise a sufficiently large group).\n\\end{itemize} \n\\end{itemize}\nAlthough these categories are qualitative, they have a clear hierarchy: ``monopoly'' means more access to power than ``dominance,'' which is better than sharing power. For groups that are excluded, being powerless plausibly implies having more access to power than being irrelevant or self-excluded, and the worst scenario is likely discrimination.\\footnote{The ERC dataset does not provide reasons why a group is irrelevant, so a group may become irrelevant after being heavily discriminated against. Thus, there is a case to be made that irrelevance is worse than discrimination. In addition, it is not clear whether self-exclusion is better (or worse) than irrelevance. I discuss these issues in more detail below.} I exploit this implicit order to construct a scale of access to power from 0 to 5.\\footnote{In this sample, the average index of access to central power is 1.545 (and the standard deviation is 1.216).}\n\n\\begin{table}[H]\n  \\centering \n  \\caption{ }\\label{indexaccesspower_tab}\n  \\begin{tabular}{|c|c|c|}\n\\hline\nEPR classification  &  Access to power score \\\\\n\\hline\n Monopoly  &  5  \\\\\n Dominance  &  4  \\\\\n Senior partner &  3  \\\\\n Junior partner &  2  \\\\\n Powerless\/Irrelevance\/Self-exclusion &  1  \\\\\nDiscrimination &  0  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe index in Table \\ref{indexaccesspower_tab} relies on two main assumptions. First, groups that are discriminated against have less access to central power than groups that are irrelevant, powerless or self-excluded. The justification for this assumption is that being explicitly discriminated against implies greater costs --- for accessing power --- than being  powerless, irrelevant or self-excluded.  Second, groups that are powerless, irrelevant or self-excluded have the same (low) level of access to central power. The justification for this assumption is based on the notion of irrelevance: if a group is self-excluded or powerless, dominant groups will likely expect that the other group's members will be unable to gain access to central power.\\footnote{However, since these two assumptions may be still seen as ad hoc, I explore robustness to the use of an alternative measure that relies on the possibility of being part of a government (e.g. as a junior partner). The main results are robust to the use of a dichotomous measure of access to central power, which is equal to 1 if the group is not excluded and 0 if it is excluded. In addition, the results are robust to dropping all the groups with an irrelevant level of access to power.} \n\nThe other main dataset used in the paper is the Polity IV dataset, which evaluates the openness and competitiveness of countries' political institutions. Specifically, I use  Polity IV's ``Openness of Executive Recruitment'' and ``Competitiveness of Executive Recruitment'' to measure how politically open and competitive a country's institutions are in a given year (or decade).\\footnote{See Section \\ref{additionalevidence} for a description of how I code these measures.} \n\n\n\n\\subsection{Empirical Strategy and Identification}\n\nAs previously mentioned, there is no consensus among scholars about the relationship between group size and access to power. My empirical analysis allows for the possibility of either a negative or positive relationship. \n\nIn the main specification, I model the outcome $y_{gct}$ (my index of access to central power based on the EPR power access characterization)\\footnote{In the baseline specification, the analysis is limited to groups that are not dominant, i.e. that don't have monopoly power or are not a senior partner (i.e. groups with an access to power score of 2 or less in Table \\ref{indexaccesspower_tab}).} of group $g$ in country $i$ and 10-year period $t$ as\n\\begin{equation}\n\\label{eqbaseline}\naccess\\_to\\_power_{gct}=\\alpha+\\beta_1 \\times size_{gct} +\\beta_2 \\times size^2_{gct}+ \\gamma_{ct}+\\eta_g+\\epsilon_{gct}\n\\end{equation}\nwhere $size_{gct}$ is the relative size of $g$  in country $i$ and period $t$, $size^2_{gct}$ is the square of this size, $\\gamma_{ct}$ are country\/period fixed effects,  $\\eta_g$ are group fixed effects and $\\epsilon_{gct}$ is the error term.   In some specifications, I also add group-specific linear trends, and control for the presence of each group in other countries (which varies at the group  $\\times$ 10-year level). The two coefficients of interest are $\\beta_1$ and $\\beta_2$, which are associated with the relative size of the groups; $\\beta_2$ is introduced to allow for a nonlinear relationship.\n\n\\medskip\n\nWith respect to identification, since the baseline specification (\\ref{eqbaseline}) includes country\/period fixed effects, it controls for any omitted variable varying at the country $\\times$ 10-year level.\\footnote{As previously mentioned, all the results are robust to using five-year panel, which includes country\/5-year period fixed effects.} In addition, the inclusion of group-specific linear trends helps control for any omitted linear trend that varies at the group level. Finally, note that the control for the presence  of each group in other countries accounts for a potentially omitted (and crucial) channel: a group may have less (or more) access to central power because its members were (or haven't been) forced to migrate to other countries, and this migration (or lack of it) may also explain the small (or large) group size. \n\nSimultaneity is also a concern in (\\ref{eqbaseline}) if, for instance, groups with less access to power have lower fertility rates. To address this, I first replace the explanatory variables (group size and the square of group size) with their lagged values. Second, I instrument group size (and the square of group size) with a lagged group size (and a lag of the squared group size). As long as the lagged values for group size do not themselves exert a direct effect on power access, this second alternative specification provides an effective estimation strategy.\n\nEven though I recognize that fixed effects regressions (such as (\\ref{eqbaseline})) cannot necessarily estimate the causal effect of group size on power access (e.g. there may be still omitted variables varying non-linearly at the group $\\times$ period level that are weakly correlated with a group's presence in other countries),\\footnote{One example may be the level of cohesiveness of the groups, or, more specifically, the level of trust of individuals within these groups. However, it can be argued that although this level changes over time, it does so only over long periods \\cite[in this respect see for instance][who show that inherited trust is strongly persistent]{AlganCahuc2010}.} the results are robust to many alternative specifications, which significantly alleviates these concerns.\n\n\n\n\\section{Baseline empirical results}\\label{baselineresults}\n\nFigure \\ref{integrVSsizeOLSav10yr_fig} shows, for each non-dominant minority group in each 10-year period, the correlation between the group's access to power score and the relative size of the group (i.e. the group's membership as a proportion of the country's population). It shows a clear inverted-U shape.\n\nTable \\ref{integrVSsizeOLSav10yr_tab} reports the estimated coefficients for variations of the model in Eq. (\\ref{eqbaseline}). The specification in column (1) does not include any controls or fixed effects, and corresponds to the specification used in Figure \\ref{integrVSsizeOLSav10yr_fig}. Column (2) includes country and 10-year-period fixed effects, and column (3) adds  country\/10-year period fixed effects. Columns (4) and (5) include group fixed effects and group-specific linear trends.  In all the specifications in Table \\ref{integrVSsizeOLSav10yr_tab}, the estimated coefficients for $\\beta_1$ are positive, the coefficients for $\\beta_2$ are negative, and all are individually and jointly significant. In addition, the peak is located at 0.47, which lies comfortably within the range of values for relative group size (i.e, between 0 and 1).\\footnote{More specifically, the value 0.47 falls between the 75th and 85th percentiles of relative size.} Thus, these results confirms the inverted-U shape in Figure \\ref{integrVSsizeOLSav10yr_fig}.\n\nI also provide results for several additional specifications to verify the robustness of the results in Table \\ref{integrVSsizeOLSav10yr_tab}. First, instead of averaging annual results over a 10-year period, I use one observation from within each 10-year period (e.g one observation every tenth year). These estimates are shown in Panel A of Table \\ref{integrVSsizeOLSav10yrrobustness1_tab}. Second, instead of using 10-year periods, I use a five-year panel. These estimates are shown in Panel B of Table \\ref{integrVSsizeOLSav10yrrobustness1_tab}. Third, instead of the continuous measure for access to state power, I use a dichotomous measure of access to power that is equal to 1 if a group is not excluded from central power and 0 if it is excluded. These estimates are shown in Panel C of Table \\ref{integrVSsizeOLSav10yrrobustness1_tab}. In all three additional specifications, the results from Table \\ref{integrVSsizeOLSav10yr_tab} hold, i.e. all specifications show a clear inverted-U-shaped relationship between the relative size of a group and its level of access to central power.\n\nIn interpreting causality, we must be cautious of reverse causality bias: a group's access to power may affect its relative size (e.g. groups that are discriminated against may migrate to other countries and\/or have lower fertility rates). In addition, groups that differ in relative size might also differ in other unobserved and non-linearly time-varying characteristics that affect their access to central power (e.g. members of large groups may receive support from exiles in other countries, and this may improve their chances to gaining access to central power). If this is the case, the previous results may be affected by omitted-variable bias. \n\nTable \\ref{integrVSsizeOLSav10yrlagIVcontrol_tab} addresses these potential issues.  Columns (1) to (3) of panel A consider specifications that use lagged values of group size as an independent variable instead of current group size. Columns (4) and (5) of panel A consider specifications that use the lagged values of relative group size as instruments for the contemporaneous values. Under certain conditions, these two strategies help to avoid the inconsistency problems associated with simultaneity \\cite[see][]{Reed2015}. All the estimates are virtually the same as those in Table \\ref{integrVSsizeOLSav10yr_tab}, which indicates that simultaneity is likely not a problem. \n\nOmitted-variable bias may be less worrisome given the different fixed effects included in the baseline specifications. In addition, the specifications in panel B of Table \\ref{integrVSsizeOLSav10yrlagIVcontrol_tab} add one crucial covariate: the presence of members of the same group in other countries. All the estimates remain the same whether we consider contemporaneous values for the relative size (columns (1) to (3)), lagged values (column (4)), or lagged values as instruments for the contemporaneous values (column (5)). \n \nWhile one must be cautious in claiming that the estimates in Tables  \\ref{integrVSsizeOLSav10yr_tab} to \\ref{integrVSsizeOLSav10yrlagIVcontrol_tab} are caused by purely exogenous shocks, the fact that the results in Table \\ref{integrVSsizeOLSav10yr_tab} are robust to all the specifications in Table \\ref{integrVSsizeOLSav10yrlagIVcontrol_tab} guards against the most obvious forms of endogeneity. \n\n\n\\section{Explaining the inverted U: A simple model}\\label{model}\n\nIn this section, I propose an explanation of the single-peaked relationship between access to central power and relative group size. I use a very simple stylized model in which one group, with an \\emph{ex ante} monopoly of political power, decides whether to continue limiting other groups' access to central power. \n\n\\subsection{Set-up}\n\nConsider a society composed of individuals who belong to \\emph{n} internally homogenous groups. Let $N$ be the total number of individuals in the society, with $N_i$ representing the number of individuals in group $i$, so that $\\sum_{i=1}^nN_i=N$. There are two time periods, each with a government that may be controlled by a single group or by several groups. Assume that in the first period, one group (or a cluster of groups) fully controls the government and decides whether or not to continue limiting other groups' access to power in the second period. Without loss of generality, we set $n=2$, and assume that group 1 has an \\emph{ex ante} monopoly over political power in period 1. Group 2 can be interpreted as a cluster of groups with limited access to power in period 1; \\emph{de jure}, this group is totally excluded from government decisions in period 1 (and potentially in period 2). If  group 2's access to power in period 2 is not restricted, both group 1 and group 2 will participate in the period-2 government. Crucially, I assume that if group 2's access to power in period 2 is not \\emph{de jure} limited, its level of control over the period-2 government will be proportional to its share of the population. Specifically, let $s_i$ denote group $i$'s level of control over the  period-2 government, and assume that\\footnote{\\label{footnotepi}This would correspond to an electoral system with perfectly proportional representation; in this scenario, $s_i$ could represent the number of seats in the national parliament for a political party that represents group $i$. Note that (\\ref{si}) assumes that when group 2 has access to power, its level of control over the government depends only on its relative size. This  is a strong assumption, as one might expect that group 1 could use its control of the government in period 1 to \\emph{de facto} limit group 2's level of control in period 2 (e.g. by passing constitutional measures that restrict the extent of political reforms that the government can pass in period 2). More generally, this assumption implies that a country can easily transition from having a very closed political system to having a very open system, which may not be realistic. In the next subsection, I relax this assumption and obtain new predictions.} \n\\begin{equation}\n\\label{si}\ns_i=\\frac{N_i}{\\sum_{j=1}^2N_j}.\n\\end{equation}\nAs previously mentioned, the analysis focuses on group 1's decision about whether or not to,  \\emph{de jure}, limit  group's 2 access to power in period 2. This decision depends on the benefits that individuals in group 1 obtain from a government fully or partially controlled by their own group. Specifically, I assume that the government has an exogenously determined budget that it divides and transfers to individuals using the following approach. First, each group that participates in the government receives an allocation of the budget proportional to that group's level of control of the government. Second, the budgetary allocation for that group is divided equally among the group's members. In other words, if group $i$ has monopoly control of the government, and if the \\emph{per capita} value of the government budget is normalized to unity, then each member of this group will receive $N\/N_i$.\\footnote{The model can be generalized to allow part of the budget to be allocated toward public goods that benefit all groups. In this alternative scenario, we could weight the two different categories of expenditures, and the share that is allocated to public goods could be interpreted as an indicator of the importance of the public good payoff relative to the ``monetary'' payoff.  All results are robust to this extension.}  \n\nIndividuals use their government transfer to purchase a private good in period 2, from which they derive utility that increases linearly in their consumption of the private good.\\footnote{Alternatively, if we include the possibility of public goods, and if group's 2 access to power is not limited, we could define $\\alpha_i$  as the utility that a member of group 1 derives from the public good if a single unit per capita of the optimal mix for group $i$ is produced (using the private good as numeraire). Then, the  \\emph{per capita} payoff to group 1 could be written as $\\alpha_1+(N\/N_1)$ if group 1 fully controls the government in period 2, and $\\alpha_2$ if group 2 fully controls the government in period 2. Note that it must be the case that $\\alpha_1\\geq \\alpha_2$. As previously mentioned, the results are robust to this extension.}  More precisely, we define the overall payoff to an individual in group 1 when group's 2 access to power is not restricted as\n\\begin{equation}\n\\label{payofflimited}\ns_1(N\/N_1).\n\\end{equation}\nWhen group 1 decides to restrict group's 2 access to power, I assume that group 2 attempts to overthrow group 1's government, which results in a costly contest between the two groups.  The group that wins this contest fully controls the government in period 2. The success of each group depends on its relative size and on its capacity to coordinate its actions. Specifically, I assume the following contest function:\n\\begin{equation}\n\\label{pi}\np_i=\\frac{N_i+a_i}{\\sum_{j=1}^2(N_j+a_j)}\n\\end{equation}\nwhere  $a_i\\geq 0$ for $i=1,2$ is a parameter that captures how sensitive a group's probability of success is to the group's relative population. This can be interpreted as the ability of each group to solve the collective-action problem.\\footnote{\\label{noises}Specifically, $a_i$ can be understood as the \\emph{de facto} capacity of individuals in group 2 to coordinate their actions to overthrow the existing regime, or the capacity of individuals in group 1 to repress group 2. This assumption helps to simply correct an undesirable feature of Tullock's success function \\cite[see][]{Hirshleifer1989, Blavatskyy2010, CorchonDahm2010, Jia2012}.}  \n\nI model the costs of the conflict as follows. I assume that a fraction of the government budget is destroyed due to violence, and only the reminder can be divided among the members of the winning group. The  fraction of the budget that is destroyed is assumed to be proportional to the relative size of group 2, since the bigger the group without political rights, the greater its capacity to cause turbulence and destruction. I therefore summarize the overall  payoff to an individual in group 1 when group's 2 access to power is limited as\n\\begin{equation}\n\\label{payoffnotlimited}\np_i(N\/N_1)(1-\\lambda(N_2\/N))\n\\end{equation}\nwhere $\\lambda(N_2\/N)$ is the fraction of  the government budget that is destroyed, and $\\lambda\\in(0,1)$ is a parameter that captures the marginal increase in the level of destruction as the relative size of group 2 increases.\\footnote{The main results hold if the proportion of the government budget that is destroyed also depends on $a_2$. However, to simplify the exposition, I assume that $\\lambda$ only depends on group size.}  \n\nFinally, I make two assumptions about the parameters $\\lambda$, $a_1$ and $a_2$. First, I assume that $a_1>a_2$, and normalize $a_2=0$, with which $a_1>0$. As discussed in footnote \\ref{noises}, this implies that the \\emph{de facto} power of group 1 in period 2 is greater than that of group 2. This assumption can be justified by the fact that group 1 had the monopoly of power in period 1, so it is plausible that this group,  in period 2, has more resources to invest in increasing its  \\emph{de facto} political power.\\footnote{Even though the model assumes \\emph{de facto} power is exogenous, this assumption is consistent with some literature that endogenizes \\emph{de facto} power \\cite[e.g. see][]{AcemogluRobinson2008AER}.}\n\nSecond, I assume that  \n \\begin{equation}\n\\label{assumption1}\n\\vfrac{\\lambda}{(1-\\lambda)}>a_1.\n\\end{equation}\nBy implying that the costs of conflict are sufficiently high and that the \\emph{de facto} power of group 1 is sufficiently low, this second assumption basically guarantees that limiting group 2's access to power is not a dominant strategy for group 1. \n\n\n\\subsection{Equilibrium}\n\nThe main objective of this section is to propose an explanation for the empirical relationship between access to power and relative size found in Section \\ref{baselineresults}. This section examines the conditions under which individuals in group 1 decide whether to limit group 2's access to power in period 2 and, specifically, how those conditions depend on the relative size of the two groups.\n\nFrom the previous analysis, it is clear that group 1 will not give group 2 access to power in period 2 when the expression in (\\ref{payoffnotlimited}) is greater than the expression in (\\ref{payofflimited}), i.e. when \n\\begin{equation}\n\\label{condnotlimiting}\np_i(N\/N_1)(1-\\lambda(N_2\/N))\\geq s_1(N\/N_1).\n\\end{equation}\nTo facilitate the exposition and interpretation of the results, I normalize the population to 1 (i.e. $N=1$), and define $\\delta$ as the relative size of group 2 (so $N_1=1-\\delta$ and  $N_2=\\delta$). Thus, replacing the expressions for $s_i$ and $p_i$ in (\\ref{condnotlimiting}) and rearranging, it is easy to see that (\\ref{condnotlimiting}) is equivalent to \n\\begin{equation}\n\\label{maincond}\n \\frac{(1-\\delta+a_1)}{(1+a_1)}\\frac{(1-\\lambda \\delta)}{(1-\\delta)}-1\\geq 0.\n\\end{equation}\nThe expression in (\\ref{maincond}) constitutes the main finding of this section. Importantly, it entails an  inverted-U relationship between group 2's relative size (i.e. $\\delta$) and its access to central power. This result is summarized in the following proposition: \n\n  \\begin{prop1} Consider the above-described game. In equilibrium, the relationship between access to central power for a group with an ex ante limited access and its relative size ($\\delta$) follows an inverted-U pattern:  this group's access to central power is expected to be lower for smaller and larger values of  $\\delta$, and higher for intermediate values of $\\delta$. \n \\end{prop1}\n \\begin{proof}\nSee the Appendix.\n\\end{proof} \nThe inverted-U shape can be explained as follows. First, when the relative size of group 2 is very small, individuals in group 1 have few incentives to increase group 2's access to power because the cost of maintaining the status quo, which benefits group 1, is small: the ensuing contest with group 2 is expected to be minimally destructive and group 1's chance of success is very high. As the relative size of group 2 increases, a contest becomes more destructive, so the period-2 budget associated with this scenario is reduced; this gives individuals in group 1 an incentive to increase group 2's access to power.\\footnote{Provided that the size of group 1 is not very small so that the budget in the contest scenario would be divided among a still large number of individuals, and that the effect of the bias in favor of group 1 in the contest success function is still small.} Finally, when the relative size of group 2 is very big, this means group 1 is relatively very small, which implies that individuals in group 1 greatly value a government fully controlled by their group (because the \\emph{per capita} transfers they receive are very large). In these circumstances, individuals in group 1 have few incentives to increase group 2's access to power, even though this implies that the government will have a significantly reduced budget in period 2 (which is mitigated somewhat by a contest function that is biased in favor of group 1). \n \nImportantly, note that the  U-shaped relationship described in Prop. 1 is consistent with the empirical evidence in Section 3. In particular, it rationalizes the results in Figure \\ref{integrVSsizeOLSav10yr_fig} and Columns (1) to (6) in Table \\ref{integrVSsizeOLSav10yr_tab}.  It is also important to note that even though the explanation I propose is new, it is consistent with the main predictions of the existing theoretical literature on group discrimination. Specifically, it is consistent with the idea that the smaller a minority, the more it suffers \\citep{Eeckhout2006}, and the larger a minority, the more likely the coercive measures against it \\citep{MoroNorman2004}, and the greater the antipathy felt towards it  \\citep{Blumer1958, Blalock1967, OliverWong2003}. One of the contributions of this paper is to propose a very simple mechanism that explains (and tests) both predictions. \n \n\n\n\\subsection{Persistence of political institutions}\\label{modelpesistence}\n\nIn the last subsection it was assumed that if group 2 is given access to power in period 2, its level of power will be proportional to its size. As mentioned in footnote \\ref{footnotepi}, this means that the institutions that affect group 2's access to political power can transition from being completely closed to being completely open. In this subsection, I remove this assumption to allow for the possibility that institutions in period 2 are biased in favor of group 1 even if group 2's access to power is not limited. The motivation behind this extension is based on the idea that institutions tend to persist for long periods \\citep{AcemogluJohnsonRobinson2001, AcemogluJohnsonRobinson2002, BanerjeeIyer2005}. \n  \nSpecifically, I assume that when group 2's access to power is not \\emph{de jure} limited,  the participation of group 2 in the period-2 government is  given by\n  \\begin{equation}\n\\label{q2}\nq_2=\\gamma s_2\n\\end{equation}\nwhere $\\gamma\\in[0,1]$ is a parameter that measures how unbiased political institutions are expected to be in period 2.  Since period-1 political institutions are assumed to be very biased in favor group 1, $\\gamma$ can be understood as the  inverse of how persistent institutions are.\n\nFrom (\\ref{q2}), note that since $q_1=1-q_2$, we have that $q_1=\\gamma s_1+1-\\gamma$. From these expressions, note that when $\\gamma=1$, $q_i=s_i$ for $i=1,2$, which corresponds to the scenario examined in the last subsection. Importantly, note that $q_1\\geq s_1$, $q_2\\leq s_2$, and that as $\\gamma$ decreases $q_1$ increases, meaning that group 1's level of control in the period-2 government can be disproportionally larger despite the fact that group 2's access to power is not \\emph{de jure} limited. \n\nReplacing $q_2$ in (\\ref{maincond}) and rearranging, it is easy to see that we have\n\\begin{equation}\n\\label{democcond}\n \\frac{(1-\\delta+a_1)}{(1+a_1)}\\frac{(1-\\lambda \\delta)}{(1-\\delta)}-\\frac{(1-\\gamma\\delta)}{(1-\\delta)} \\geq 0.\n\\end{equation}\nBy analyzing (\\ref{democcond}), the following proposition summarizes how the results in Prop. 1 change when institutions in period 2 are expected to be biased in favor of group 1, even though both groups have access to power. \n\n  \\begin{prop2} Consider the above-described game. In equilibrium, if political institutions are sufficiently persistent, then the relationship between a group's access to central power and its relative size does not follow an inverted-U pattern and, more specifically, the larger a group's relative size, the more likely it will have access to power. \n \\end{prop2}\n \\begin{proof}\nSee the Appendix\n\\end{proof} \n\nThe intuition behind Prop. 2 is straightforward: the more biased institutions are expected to be in favor of group 1, the less costly it is for group 1 to give group 2 access to power, particularly when group 2 is relatively large. In addition, and importantly, Prop. 2 establishes a new empirical prediction: the inverted-U relationship between relative size and access to power crucially depends on the historical quality of institutions; for historically closed political institutions, it is more likely that this relationship, if existent, is positive (rather than non-monotonic). In the next section I examine the empirical plausibility of this new prediction. \n\n\n \n \\section{Additional evidence}\\label{additionalevidence}\n \nThis section empirically examines whether the results in Section \\ref{baselineresults} depend on the historical level of openness of the political institutions, as predicted in Section \\ref{model}. To measure the openness of political institutions, I use two proxies from PolityIV. First, I use a variable called ``Openness of Executive Recruitment (\\emph{xropen}).\" This variable measures  ``the extent that the politically active population has an opportunity, in principle, to attain the position through a regularized process\" \\cite[see][]{PolityIV2018}. It ranges from 0 to 4, with 4 representing the most open institutions. Second, I use a variable called ``Competitiveness of Executive Recruitment (\\emph{xrcomp}),'' which measures  ``the extent that prevailing modes of advancement give subordinates equal opportunities to become superordinates.'' This variable ranges from 0 to 3, with 3 representing the most competitive institutions.  Since the hypothesis regarding how the level of political openness should affect each group's access to power is based on the persistence of institutions, I compute, for each country-year, the average for each variable since 1800. Then, I define a dummy variable equal to 1 if, for the case of the variable \\emph{xropen}, its historical average is equal to 4 (which is very close to the median, and represents a highly open system), and for the case of the variable (\\emph{xrcomp}), its historical average is above the median (which represents a highly competitive system). \n\nTable \\ref{integrVSsizeOLSav10yropenness_tab} shows estimates of the same specifications used in columns (4) and (5) of Table \\ref{integrVSsizeOLSav10yr_tab}, but distinguishes between countries with high and low historical levels of openness (Panel A), and between countries with high and low historical levels of competitiveness (Panel B).  These estimates show that the non-monotonic (inverted-U-shaped) effect of group size on access to power is stronger in countries with historically high levels of openness and competitiveness (columns (1) and (2)), and weaker (and almost nonexistent) in countries that are less open and competitive (columns (3) and (4)). In addition, for countries with historically low levels of openness and competitiveness, the relationship between group size on access to power appears to follow a linear and positive pattern (see column (5)).   Table \\ref{integrVSsizeOLSav10yropennesslagIV_tab} explores the robustness of the previous results to the use of lagged values for group size as i) independent variables (columns (1) to (3)) and as ii) instruments for the contemporaneous values (columns (4) to (6)).  The robustness results are all consistent with those in Table \\ref{integrVSsizeOLSav10yropenness_tab}. \n\nImportantly, the results in Tables \\ref{integrVSsizeOLSav10yropenness_tab}  and \\ref{integrVSsizeOLSav10yropennesslagIV_tab} are consistent with the prediction of the model proposed in the Section \\ref{model}. In particular, they show that as we move from high to low levels of openness, the relationship between group size on access to power passes from having a marked inverted-U shape to being better described as monotonically increasing. As mentioned in the last section, this can be explained by a decrease in the costs (to the group with access to power) associated with not limiting other groups' access to power. \n \n \n\n\n \\section{Conclusion}\\label{conclusion}\n \nThis study examines whether the relative size of ethnic groups within a country affects the extent to which governments limit these groups' access to central power. Using data at the group level in 175 countries from 1946 to 2017, I find evidence of an inverted-U-shaped relationship between the relative size of groups and their access to central power. This single-peaked relationship is robust to many alternative specifications, and several robustness checks suggest that relative size causes access to power. Through a very simple model, I propose an explanation based on an initial high level of political inequality, and on the incentives that more powerful groups have to continue limiting other groups' access to power. This explanation incorporates essential elements of existing theories about the relationship between group size and political and social exclusion, and leads to a new empirical prediction: the single-peaked pattern should be weaker in countries where political institutions have been less open in the past. This additional prediction is supported by the data.\n \nSeveral opportunities exist for future research. One could examine the effect of ethnic groups' relative size on other --- more de facto --- forms of political participation, such as the formation of political organizations and participation in protests. It would also be interesting to examine whether other kind of institutions matter (e.g. less formal political institutions and cultural institutions). Finally, there is a question of whether individuals in dominant groups actually agree that the institutions that their groups control limit access to the power of minority groups.\n \n \n\n\n\\hbox {}\n\\hbox {} \\newpage\n\\section*{Graphs and Tables}\n\n\n\\begin{figure}[H]\n\\begin{center}\n\\caption{Scatter plot of relative size and access to central power}\\label{integrVSsizeOLSav10yr_fig}\n\\resizebox{10cm}{8cm}{\\includegraphics[width=4in]{figure1.png}}\n\\caption*{\\footnotesize The figure plots a measure of the size of each ethnic group on the x-axis against each group's access to  power score --- as defined in Table \\ref{indexaccesspower_tab} --- on the y-axis. Each point represents a group over a 10-year period. The quadratic curve that is overlaid is reported in column (1) of Table \\ref{integrVSsizeOLSav10yr_tab}.}\n\\end{center}\n\\end{figure}\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{4pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power}  \\label{integrVSsizeOLSav10yr_tab}\n\\begin{center}  \n\\vspace{-0.5cm}\\begin{tabular}{lccccc}\n\\hline\\hline  \\addlinespace[0.15cm]\n    & \\multicolumn{5}{c}{Dep. variable: Level of access to central power} \\\\\\cmidrule[0.2pt](l){2-6}\n& (1)& (2) & (3)& (4)& (5) \\\\  \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm] \n\\primitiveinput{tableII.tex}\n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]           \nCountry fixed effects & N & Y & Y & Y & Y   \\\\         \n10-year period fixed effects & N & Y & Y & Y & Y   \\\\   \n  Country\/period fixed effects & N & N & Y & Y & Y \\\\       \n Group fixed effects & N & N & N & Y & Y \\\\    \n  Group-specific linear trends & N & N & N & N & Y \\\\    \n      \\addlinespace[0.15cm] \\hline\\hline         \n\\multicolumn{6}{p{14cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}). The dependent variable in all columns is each ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less, and the index is averaged over a 10-year period.  In this subsample, the average relative size is 0.117 (with s.d. 0.224) and the average level of access to central power is 1.036 (with s.d. 0.575).  Robust standard errors  clustered by country are in parentheses. * denotes results are statistically significant at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{4pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power (robustness to alternatives measures of access to central power)}    \\label{integrVSsizeOLSav10yrrobustness1_tab}\n\\begin{center}\n\\vspace{-0.6cm} \\begin{tabular}{lccccc}\n\\hline\\hline  \\addlinespace[0.15cm]\n    & \\multicolumn{5}{c}{Dep. variable: Access to central power} \\\\\\cmidrule[0.2pt](l){2-6}\n& (1)& (2) & (3)& (4)& (5) \\\\    \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm] \n &\\multicolumn{5}{c}{Panel A: Using first ob. within each 10-year period}     \\\\\\cmidrule[0.2pt](l){2-6}   \\addlinespace[0.15cm]\n\\primitiveinput{tableIIIa.tex} \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]       \n &\\multicolumn{5}{c}{Panel B: Using averages over 5-year periods}     \\\\\\cmidrule[0.2pt](l){2-6}   \\addlinespace[0.15cm]\n\\primitiveinput{tableIIIb.tex} \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]       \n &\\multicolumn{5}{c}{Panel C: Dep. var. is prob. of access to power}     \\\\\\cmidrule[0.2pt](l){2-6}   \\addlinespace[0.15cm]\n\\primitiveinput{tableIIIc.tex} \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]             \nCountry fixed effects & N & Y & Y & Y & Y   \\\\         \n Period fixed effects & N & Y & Y & Y & Y   \\\\   \n  Country\/period fixed effects & N & N & Y & Y & Y \\\\       \n Group fixed effects & N & N & N & Y & Y \\\\    \n  Group-specific linear trends & N & N & N & N & Y \\\\    \n       \\addlinespace[0.15cm] \\hline\\hline                       \n\\multicolumn{6}{p{14.8cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}). The dependent variable in all columns is based on ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less. Panel A uses 10-year panel, but rather than averaging the 10-year data, it takes one observation within each sub-period (e.g one every tenth year). In this sample, the average relative size is 0.121 (with s.d. 0.228) and the average level of access to power is 1.023 (with s.d. 0.609). Panel B uses a 5-year panel. In this sample, the average relative size is 0.118 (with s.d. 0.226) and the average level of access to power is 1.030 (with s.d. 0.586). Panel C uses 10-year panel and a dichotomous measure of access to power which is equal to one if the group is not excluded from central power and zero if it is excluded. In this sample, the average probability of not being excluded from central power is 0.216 (with s.d. 0.411). Robust standard errors  clustered by country are in parentheses. * denotes results are statistically significant at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{4pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power (robustness to the use lagged size, IV and control for presence of each group in other countries)}  \\label{integrVSsizeOLSav10yrlagIVcontrol_tab}\n\\begin{center}  \n\\vspace{-0.5cm}\\begin{tabular}{lccccc}\n\\hline\\hline  \\addlinespace[0.15cm]\n    & \\multicolumn{5}{c}{Dep. variable: Level of access to central power} \\\\\\cmidrule[0.2pt](l){2-6}\n    & (1)& (2) & (3)& (4)& (5) \\\\  \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm] \n &\\multicolumn{5}{c}{Panel A:}     \\\\\\cmidrule[0.2pt](l){2-6} \n        & \\multicolumn{3}{c}{Lagged effect} &\\multicolumn{2}{c}{IV} \\\\\\cmidrule[0.2pt](l){2-4}\\cmidrule[0.2pt](l){5-6}\n\\primitiveinput{tableIVa.tex} \n\\cmidrule[0.2pt](l){2-6} \n  Country and period fixed effects & Y & Y & Y & Y & Y \\\\       \n  Country\/period fixed effects & Y & Y & Y & N & Y \\\\       \n Group fixed effects & N & Y & Y & N & N \\\\    \n  Group-specific linear trends & N & N & Y & N & N \\\\    \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]       \n &\\multicolumn{5}{c}{Panel B: }     \\\\\\cmidrule[0.2pt](l){2-6} \n         & \\multicolumn{3}{c}{Baseline specification} & \\multicolumn{1}{c}{Lagged} & \\multicolumn{1}{c}{IV} \\\\\\cmidrule[0.2pt](l){2-4}\\cmidrule[0.2pt](l){5-5}\\cmidrule[0.2pt](l){6-6}\n \\primitiveinput{tableIVb.tex} \n \\cmidrule[0.2pt](l){2-6}        \n  Country\/period fixed effects & Y & Y & Y & Y & Y  \\\\       \n Group fixed effects & N & Y & Y& N  & N  \\\\        \n   Group-specific linear trends & N & N & Y & N & N  \\\\    \n      \\addlinespace[0.15cm] \\hline\\hline         \n\\multicolumn{6}{p{15.2cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}).  The dependent variable in all columns is each ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less, and the index is averaged over a 10-year period. In this subsample, the average relative size is 0.117 (with s.d. 0.224) and the average level of access to central power is 1.036 (with s.d. 0.575).  Robust standard errors  clustered by country are in parentheses. * denotes results are statistically significant at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\\begin{figure}[H]\n\\caption{Relative size and access to central power by level of political openness}\\label{intVSsizeCORRav10yrsindexropen_fig}\n\\begin{center}\n\\begin{subfigure}{0.45\\textwidth}\n\\caption{High degree of openness}\n\\includegraphics[width=3in,height=2.6in]{figure2a.png}\n\\end{subfigure}\n\\begin{subfigure}{0.45\\textwidth}\n\\caption{Low degree of openness}\n\\includegraphics[width=3in,height=2.6in]{figure2b.png}\n\\end{subfigure}\n\\begin{minipage}{15cm} \\footnotesize The figures plot a measure of the size of each ethnic group on the x-axis against each group's access to power score on the y-axis, by historical level of political openness (i.e. PolityIV's measure of Openness of Executive Recruitment, computed since 1800).  Each point represents a group over a 10-year period. Figure (a) includes countries with an above-median level of historical political openness, and Figure (b) includes countries with a below-median level of historical political openness. \n\\end{minipage}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\caption{Relative size and access to central power  by level of political competitiveness}\\label{intVSsizeCORRav10yrsindexrcomp_fig}\n\\begin{center}\n\\begin{subfigure}{0.45\\textwidth}\n\\caption{High degree of competitiveness}\n\\includegraphics[width=3in,height=2.6in]{figure3a.png}\n\\end{subfigure}\n\\begin{subfigure}{0.45\\textwidth}\n\\caption{Low degree of competitiveness}\n\\includegraphics[width=3in,height=2.6in]{figure3b.png}\n\\end{subfigure}\n\\begin{minipage}{15cm}  \\footnotesize The figures plot a measure of the size of each ethnic group on the x-axis against each group's access to power score on the y-axis, by historical level of political competitiveness (i.e. PolityIV's measure of Competitiveness of Executive Recruitment, computed since 1800).  Each point represents a group over a 10-year period. Figure (a) includes countries with an above-median level of historical political competitiveness, and Figure (b) includes countries with a below-median level of historical political competitiveness. \n\\end{minipage}\n\\end{center}\n\\end{figure}\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{5pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power by level of political openness and political competitiveness}    \\label{integrVSsizeOLSav10yropenness_tab}\n\\begin{center}\n \\begin{tabular}{lcccccc}\n\\hline\\hline \\addlinespace[0.15cm]\n    & \\multicolumn{5}{c}{Dep. variable: Level of access to central power} \\\\\\cmidrule[0.2pt](l){2-6}\n& (1)& (2) & (3)& (4)& (5) \\\\    \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm]     \n     &\\multicolumn{5}{c}{Panel A: By openness of executive recruitment}      \\\\\\cmidrule[0.2pt](l){2-6}        \n            & \\multicolumn{2}{c}{High degree of openness} & \\multicolumn{3}{c}{Low degree of openness} \\\\\\\n        & \\multicolumn{2}{c}{(avg. xropen$=$4)} & \\multicolumn{3}{c}{(avg. xropen$<$4)} \\\\\\cmidrule[0.2pt](l){2-3}\\cmidrule[0.2pt](l){4-6}\n\\primitiveinput{tableVa.tex}\n\\addlinespace[0.15cm]\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]       \n &\\multicolumn{5}{c}{Panel B: By competitiveness of executive recruitment}     \\\\\\cmidrule[0.2pt](l){2-6}     \n             & \\multicolumn{2}{c}{High degree of competitiveness} & \\multicolumn{3}{c}{Low degree of competitiveness} \\\\\\\n        & \\multicolumn{2}{c}{(avg. xrcomp above median)} & \\multicolumn{3}{c}{(avg. xrcomp below median)} \\\\\\cmidrule[0.2pt](l){2-3}\\cmidrule[0.2pt](l){4-6}\n \\primitiveinput{tableVb.tex}\n \\addlinespace[0.15cm]\n \\hline \\addlinespace[0.15cm]      \nCountry fixed effects & Y & Y & Y & Y  & Y \\\\         \n Period fixed effects & Y & Y & Y & Y &  Y \\\\       \n  Country\/period fixed effects & Y & Y & Y & Y  & Y \\\\         \n Group fixed effects & N & Y & N & Y &  Y \\\\     \n       \\addlinespace[0.15cm] \\hline\\hline                       \n\\multicolumn{6}{p{17cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}).  The dependent variable in all columns is each ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less, and the index is averaged over a 10-year period. In this subsample, the average relative size is 0.117 (with s.d. 0.224) and the average level of access to central power is 1.036 (with s.d. 0.575).  Columns (1) and (2) in Panel A include countries with an above-median level of historical political openness, and Columns (3) to (5) in Panel A include countries with a below-median level of historical political openness (i.e. PolityIV's measure of  Openness of Executive Recruitment, computed since 1800).   Columns (1) and (2) in Panel B include countries with an above-median level of historical political competitiveness, and Columns (3) to (5) in Panel B include countries with a below-median level of historical political competitiveness (i.e. PolityIV's measure of Competitiveness of Executive Recruitment, computed since 1800). Robust standard errors  clustered by country are in parentheses. $\\dagger$ denotes results are statistically significant at the 15\\% level, * at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{2pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power by level of political openness and political competitiveness (lagged size and IV)}    \\label{integrVSsizeOLSav10yropennesslagIV_tab}\n\\begin{center}\n \\begin{tabular}{lcccccc}\n\\hline\\hline  \\addlinespace[0.15cm]\n    & \\multicolumn{6}{c}{Dep. variable: Level of access to state power} \\\\\\cmidrule[0.2pt](l){2-6}\n            & \\multicolumn{3}{c}{Lagged effect} & \\multicolumn{3}{c}{IV} \\\\\\cmidrule[0.2pt](l){2-4}\\cmidrule[0.2pt](l){5-7}\n            & (1)& (2) & (3)& (4)& (5)& (6) \\\\    \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm] \n                 &\\multicolumn{6}{c}{Panel A: By openness of executive recruitment}      \\\\\\cmidrule[0.2pt](l){2-7}       \n        & \\multicolumn{1}{c}{High openness} & \\multicolumn{2}{c}{Low openness}& \\multicolumn{1}{c}{High openness} & \\multicolumn{2}{c}{Low openness} \\\\\\cmidrule[0.2pt](l){2-2}\\cmidrule[0.2pt](l){3-4}\\cmidrule[0.2pt](l){5-5}\\cmidrule[0.2pt](l){6-7}\n\\primitiveinput{tableVIa.tex} \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]     \n                 &\\multicolumn{6}{c}{Panel B: By competitiveness of executive recruitment}      \\\\\\cmidrule[0.2pt](l){2-7}       \n        & \\multicolumn{1}{c}{High compet.} & \\multicolumn{2}{c}{Low compet.}& \\multicolumn{1}{c}{High compet.} & \\multicolumn{2}{c}{Low compet.} \\\\\\cmidrule[0.2pt](l){2-2}\\cmidrule[0.2pt](l){3-4}\\cmidrule[0.2pt](l){5-5}\\cmidrule[0.2pt](l){6-7}\n         \\primitiveinput{tableVIb.tex}\n         \\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]                \n  Country\/period fixed effects & Y & Y & Y & Y & N & Y \\\\       \n Group fixed effects & N & N & Y & Y & N & N \\\\    \n  Group-specific linear trends & N & N & N & Y & N & N \\\\    \n       \\addlinespace[0.15cm] \\hline\\hline                       \n\\multicolumn{7}{p{16.6cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}). The dependent variable in all columns is each ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less, and the index is averaged over a 10-year period.  In this subsample, the average relative size is 0.117 (with s.d. 0.224) and the average level of access to central power is 1.036 (with s.d. 0.575).   Columns (1) and (2) in Panel A include countries with an above-median level of historical political openness, and Columns (3) to (5) in Panel A include countries with a below-median level of historical political openness (i.e. PolityIV's measure of  Openness of Executive Recruitment, computed since 1800).   Columns (1) and (2) in Panel B include countries with an above-median level of historical political competitiveness, and Columns (3) to (5) in Panel B include countries with a below-median level of historical political competitiveness (i.e. PolityIV's measure of Competitiveness of Executive Recruitment, computed since 1800).   Robust standard errors  clustered by country are in parentheses. $\\dagger$ denotes results are statistically significant at the 15\\% level, * at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\n\n\\hbox {}\n\\hbox {} \\newpage\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nPerhaps the most remarkable demonstrations of coherent interaction between atoms and photons are electromagnetically induced transparency (EIT), electromagnetically induced absorption (EIA) and coherent population trapping  \\cite{boller1991observation,fulton1995effects,alezama1999eia,renzoni1997coherent}. These processes can be interpreted as a consequence of quantum interference -- they are based on the fact that an optical field can transform atomic states such that an atomic transition can be entirely suppressed and subsequent absorption eliminated. Quantum interference shows an exceptional sensitivity to frequency shifts, including those induced by magnetic fields. This makes atomic ensembles excellent tool for magnetometry, with potential application across research fields as diverse as biomedicine, seismology, defense, and general metrology \\cite{kitching2011atomic,wiesendanger2011single}. \n\nAtomic magnetometers can now reach excellent sensitivites, comparable to, and even surpassing those of superconducting quantum interference devices (SQUIDs) \\cite{drung1990low,pannetier2004femtotesla,kominis2003subfemtotesla}.  Since the first demonstration of EIT-based scalar magnetometers \\cite{fleischhauer1994quantum}, various schemes have been reported, relying on the zero field resonance observed in Hanle-type experiments \\cite{alipieva2003narrow,gateva2007shape}, on optical pumping \\cite{acosta2006nonlinear,afach2015highly,bison2018sensitive,zhang2019multi}, or the nonlinear Faraday effect in a manifold of a single ground state \\cite{novikova2001compensation,pustelny2006pump,budker1998nonlinear,budker2000sensitive,novikova2000ac,pustelny2008magnetometry}. Miniaturisation presents a challenge for atomic magnetometers, but over the last two decades devices have been developed that combine extreme sensitivity with minute detection volumes \\cite{kominis2003subfemtotesla,shah2007subpicotesla}.\n\nMost atomic magnetometers perform scalar magnetic field metrology, i.e.~determined the magnetic field along a pre-defined axis. Simultaneously measuring the strength and direction of a magnetic field would be of great importance in specific areas such as satellite navigation and biological magnetic field measurement \\cite{budker2007optical,le2013optical}.\nThe direction of a magnetic field can be addressed by vector magnetometers, first demonstrated in \\cite{lee1998sensitive}. Since then, various schemes characterized by EIT or its counterpart, EIA, have been extensively studied \\cite{dimitrijevic2008role,yudin2010vector,cox2011measurements,margalit2013degenerate}. The full vector nature of a magnetic field may be accessed by simultaneously probing the magnetic field in orthogonal directions by separate probe beams.  Alternatively, adding an external transverse magnetic field (TMF) can make EIT-based methods sensitive to different magnetic field components by considering polarization rotation or resonance amplitudes \\cite{yudin2010vector,cox2011measurements}. \n\nIn this work we explore the possibility to detect both the strength and alignment of a magnetic vector field from the interaction of a warm atomic vapor with a vector beam (VB), i.e.~a light field that has a polarisation pattern that is varying across the beam profile.    \nThe interaction of vector beams with atoms is a relatively new concept \\cite{zhan2009cylindrical,wang2020vectorial}, which has been used to explore spatial anisotropy \\cite{fatemi2011cylindrical,wang2018optically,yang2019manipulating,wang2019directly,wang2020optically}, nonlinear effects \\cite{shi2015magnetic,stern2016controlling,bouchard2016polarization,hu2019nonlinear} and quantum storage \\cite{parigi2015storage,ye2019experimental}. \n\nOf particular interest to this work is the extension of EIT, conventionally observed as spectral features with homogeneously polarised probe beams, to spatially resolved EIT resulting from inhomogeneously polarised VBs.  This effect has been observed both in cold \\cite{radwell2015spatially} and warm \\cite{yang2019observing} atomic systems. In the former case, a weak TMF closes the EIT transitions, thereby generating phase-dependent dark states and, in turn, spatially dependent transparency. As the spatially observed transparency patterns and applied magnetic fields are directly coupled, this offers the possibility of detecting magnetic fields from absorption profiles  \\cite{clark2016sculpting,castellucci2021atomic}.\n\n\nIn this paper, an experimental setup is presented to visually observe the magnetic field based on Hanle resonances in a warm atomic vapor. Importantly, we analyze spatially resolved absorption patterns instead of the time-resolved spectrum, which is fundamentally different from other aforementioned methods.\nBy employing VBs, we show that the absorption pattern is sensitive to the TMF strength, visible particularly in the degree of absorption, whereas maximal transmission remains unchanged. It's worth noting that the above results will change depending on experimental parameters, \\textit{e.g.} increasing the temperature of the gas should lead to a reduction of transparency throughout the whole beam profile. Furthermore, the transmitted pattern of VBs can be rotated arbitrarily according to the alignment of TMF. For the general case in the current work, the spatial magnetic field can be decomposed into a TMF and a longitudinal magnetic field (LMF) according to the quantization axis. The absorption patterns and corresponding polar plots can then be analyzed to recover the full magnetic field information. Such a procedure could prove to be a powerful tool to measure the three-dimensional (3D) magnetic distribution, and can even be applied in room temperature atomic vapors, simplifying future atomic magnetometer design.\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figure1.eps}\n\\caption{The schematic of the experimental setup and atomic energy levels. M: mirror; HWP: half-wave plate; QWP: quarter-wave plate; L: lens; PBS: polarization beam splitter; PD: photodetector; SMF: single mode fiber; VRP: vortex retarder plate; CCD: charge-coupled device camera; MFS: Magnetic field shielding; PM: Projection measurement. SAS: Saturated absorption spectroscopy; VBG: Vector beam generation.}\n\\label{fig1}\n\\end{figure} \n\n\\section{Experimental setup}\nThe experimental setup is shown in Fig. \\ref{fig1}. The output of a frequency locked 795 nm external cavity diode laser is sent through a single-mode fiber (SMF) to improve the mode quality of the Gaussian beam. \nAfter the fiber, the beam passes through a half-wave plate and a polarizing beam splitter (PBS) to adjust the beam intensity and fix the polarized state of the beam as horizontal polarization. A telescope is applied to expand the beam size and the achieved high-quality Gaussian beam waist is 4 mm. The VBs are generated by sending the linearly polarized beam through a vortex retarder plate (VRP), a liquid-crystal-based retardation wave plate with an inhomogeneous optical axis which displays an azimuthal topological charge \\cite{marrucci2006optical,marrucci2006pancharatnam}. \nThe laser frequency is locked to the $5{S_{{1 \\mathord{\\left\/\n {\\vphantom {1 2}} \\right.\n \\kern-\\nulldelimiterspace} 2}}},F = 2 \\to 5{P_{{1 \\mathord{\\left\/\n {\\vphantom {1 2}} \\right.\n \\kern-\\nulldelimiterspace} 2}}},F' = 1$ transition of the $^{87}$Rb D1 line.\nThe Rb cell has a length of 50 mm. A three-layer $\\mu$-metal magnetic shield is used to isolate the atoms from the environmental magnetic fields. The temperature of the cell is set at 60\u00b0C with a temperature controller. A solenoid coil (not shown in figure) inside the inner layer generates a uniform LMF, oriented along the light propagation direction, $\\mathbf{k}$. A TMF, in the plane perpendicular to $\\mathbf{k}$ and coverin the whole cell, is generated by two pairs of orthogonal Helmholtz coils, each pair independently controlled by a high precision current supply driver.  By adjusting the current ratio, and hence the horizontal and vertical TMF component, it is possible to produce an arbitrary TMF. The power of the incident laser beam is 3.4 $\\rm mW\/cm^{2}$ ( $\\approx$ 0.75 $I_{\\rm sat}$). After passing through the cell, the spatial intensity distribution of the beam is recorded by a charge-coupled device camera (CCD). \n\nThe polarization distribution of the probe VBs can be reconstructed by measuring the Stokes parameters, which represent the full polarization information of  the light \\cite{milione2011higher}. Experimentally, the Stokes parameters can be obtained by using the projection measurement system consisting of a QWP, a polarizer and a CCD. Fig. \\ref{fig2} (a) shows the polarization distribution of the generated VB with $m=$1, which is also known as a radially polarized beam. Here, $m$ is the polarization topological charge of the VB. It can be seen that this distribution varies periodically with the azimuthal angle in the plane of the beam. The electric field vector of the VBs considered here can be expressed as:\n\\begin{equation}\n\\mathbf{E}(r, \\phi,z)={E_0}(r, \\phi,z)\n\\begin{pmatrix}\n \\cos(m\\phi)\\\\\n \\sin(m\\phi)\\\\\n 0\n\\end{pmatrix}.\n\\label{1}\n\\end{equation}\nHere $r$ is the radial distance, $\\phi$ denotes the azimuthal angle and $z$ is the propagation distance. The position-dependent complex amplitude of the light is ${E_0}(r, \\phi,z)$, where $m$ is an integer.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure2.eps}\n\\caption{The experimental results of the radially polarized beam in presence of TMF. (a) Intensity and polarization distribution without atoms. (b) - (h) Intensity distributions after passing through atoms under vertical TMF of varying strength: ${\\rm B}_{\\rm TMF}$ = 0 mG, 23 mG, 61 mG, 123 mG, 146 mG, 206 mG and 230 mG, respectively. (i) The dependence of transmitted intensity for two selected regions against ${\\rm B}_{\\rm TMF}$.}\n\\label{fig2}\n\\end{figure}\n\n\\section{Experimental results}\n\nNow we turn to how the magnetic field influences the interaction between the vector beam and the atoms: the transmission in particular. Firstly, when the magnetic field is not applied, there is very little absorption of the vector beam, as compared to the profile without atoms (Fig. \\ref{fig2} (a) and (b)). As a vertical TMF is applied to the atoms however, a petal-like transmission pattern gradually appears, as shown in Fig. \\ref{fig2} (c)-(h). In general, one predicts $2\\times m$ petals, with the exception of \n$m=0$. In our case, we consider $m=1$, and so we observe a two-fold symmetry, as considered in more detail below.\n\nIncreasing the magnitude of the TMF, we observe that maximum transmission always occurs in the region where the linear polarization is parallel or antiparallel to the TMF. The strongest absorption occurs in regions where the local linear polarization is perpendicular to the TMF axis, and it increases with increasing magnitude of the TMF. We note that positive and negative TMFs both lead to the same pattern, as atomic transitions respond to the alignment but not orientation of linear polarization. The variation of transparency  and absorption  are plotted against the TMF strength (from $-230$ mG to $+230$ mG) in Fig. \\ref{fig2} (i). To make a systematic comparison, a point of maximum transmission and absorption respectively is chosen, and the corresponding probe intensity ($\\rm{I_{\\parallel}}$ and $\\rm{I_{\\perp}}$) is determined, averaged over a square area of 25 pixels, to reduce experimental error. The red curve then shows the local sensitivity of the VBs to the TMF strength, in agreement with the Hanle-EIT profile. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure3.eps}\n\\caption{The experimental results of the radially polarized beam in presence of an LMF. (a) - (f): the intensity distributions after passing through the atom vapor under the varied LMF: ${\\rm B}_{\\rm LMF}$ = 0 mG, 50 mG, 100 mG, 120 mG, 160 mG and 200 mG, respectively. (g) The dependence of transmitted intensity for whole beam against the ${\\rm B}_{\\rm LMF}$.}\n\\label{fig3}\n\\end{figure}\n\nAs expected, the absorption of the optical ${\\mathbf E}$-field components aligned with the $\\mathbf B$-field orientation is independent of the magnitude of the applied magnetic field strength. We can therefore monitor these spatial positions to consider the effects of an LMF. Accordingly, applying an LMF of varying magnitude, between $0~mG$ and \n$200~mG$, we observe uniform absorption across the whole beam, that increases with the magnitude of the TMF (Fig.~\\ref{fig3}). The variation of transparency for the whole beam is shown in Fig. \\ref{fig3} (g), displaying the same Hanle-EIT profile as Fig. \\ref{fig2} (i). Such results are adequately described by the Zeeman effect and are no different to prior experiments that rely on linearly polarized light.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure4.eps}\n\\caption{Transmission profiles as function of TMFs alignment. (a) and (b): intensity and polarization distributions for vertical and diagonal TMF alignment. (c) Image axis of the transmission profile as a function of TMF alignment. Insets: examples of observed transmission profiles.}\n\\label{fig4}\n\\end{figure}\n\nThe transmitted vector beam, as a whole, is highly sensitive to the TMF however, particularly in regard to the $B$-field's alignment. To characterize the axis of the TMF, we select a radially polarized VB which generates the two-petal pattern after passing through the vapor cell. We then define the {\\it image axis} as the line that passes through the maximal transmission regions of the two petals and the beam center. As mentioned previously, the visibility of the transmission profile can be controlled by the magnitude of the TMF, with a stronger magnetic field corresponding to stronger maximal absorption. We set the TMF to $230~mG$ to ensure maximal contrast, allowing us to identify the image axis as precisely as possible. We further note that the linear polarization of the transmitted region is also parallel to the image axis, providing an alternative way to identify the TMF axis. \n\nFig. \\ref{fig4} (a) and (b) show the relationship between the TMF alignment and the transmitted polarization, as reconstructed from CCD measurements of the Stokes parameters. Fig. \\ref{fig4} (c) shows the angle of the image axis (green arrow) when rotating the TMF axis from $0$ to $\\pi~$rad. Moreover, the VBs' polarization can also be manipulated by rotating the half wave plate and the VRP, producing the same rotational results as when the angle of the TMF axis is fixed. The axis of TMF can thus be easily observed and the TMF's strength can be measured similarly to the procedure outlined in Fig. \\ref{fig2}.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure5.eps}\n\\caption{The experimental results of the radially polarized beam in presence of the spatial magnetic field with fixed strength ($\\left | \\mathbf{B} \\right |=230mG$). (a): $\\left | \\mathbf{B} \\right |=0mG$. (b) - (f): transmitted patterns with $\\theta = \\pi\/6, \\pi\/4, \\pi\/3, 5\\pi\/12, \\pi\/2$, respectively. (g) Polar plots for patterns at different angle $\\theta$ at the radius indicated in (f).}\n\\label{fig5}\n\\end{figure}\n\nTo visualise arbitrary magnetic field alignments, we can combine our observations for a TMF and an LMF with further experiments that consider an arbitrary inclination angle, $\\theta$, between the $\\mathbf{B}$-field and the propagation axis, $\\mathbf{k}$. For a TMF along the vertical axis, $\\theta$ then denotes the angle in the $y-z$ plane and the magnetic field can be written as $\\mathbf{B}=\\mathbf{|B|} (0, \\sin{\\theta}, \\cos{\\theta})^T$, as reported in Fig. \\ref{fig5}. \nIn the absence of a magnetic field, there is no petal-like pattern and the transmission of the radially polarized VB is uniformly distributed along the azimuthal angle, similarly to Fig. \\ref{fig2} (b) and Fig. \\ref{fig3} (a). \n\n\nWe then set the strength of the magnetic field to $\\mathbf{|B|}$ = 230 mG. When $\\theta = 0$, the magnetic field is aligned to $\\mathbf{k}$ , corresponding to a pure LMF, destroying Hanle resonances and resulting in strong homogeneous absorption as discussed in the context of Fig. \\ref{fig3}. By increasing $\\theta$, the petal-like pattern gradually appears according to the strength of the TMF component. In the case of $\\theta = \\pi\/2$, the magnetic field is purely transverse, and the transmission profile is the same as for Fig. \\ref{fig4}. As expected, regions where the polarization is perpendicular to the $\\mathbf{B}$-$\\mathbf{k}$ plane experience maximal absorption. However, with larger angle $\\theta$, the TMF component of $\\mathbf{B}$ increases and induces transparency in regions with parallel polarization. The relationship and sensitivity of transmitted patterns to the angle $\\theta$ are captured in Fig. \\ref{fig5} (g). The visibility of the pattern also depends on the magnitude of the field. A rotation of the magnetic field around the azimuthal angle $\\phi$ would result in a corresponding shift of the absorption pattern. Visual inspection of the absorption pattern therefore gives maximal information on the magnetic field alignment, subject to the symmetry of the probe pattern.\n\nSo far, we have considered only radially polarized VBs, but similar observations hold for general VBs. We demonstrate this in Fig. \\ref{fig6} by comparing VBs with differing topological charge: $m$ = 1 and $m$ = 2.  \nFig. \\ref{fig6} (a) and (d) show the polarization and intensity profile of the generated VBs with without atoms, respectively. These donut-like profiles change very little without appropriate shielding from magnetic fields, as shown in Fig. \\ref{fig6} (b) and (e).  However, in the presence of a TMF, the transmitted beams show a two-fold and four-fold petal pattern: as shown in Fig. \\ref{fig6} (c) and (f). Polar plots of the absorption profile with atoms, for $m$ = 1 and $m$ = 2, are shown in Fig. \\ref{fig6} (g) and are in agreement with the original observation in cold atoms \\cite{radwell2015spatially}.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure6.eps}\n\\caption{Polarization and intensity profiles for VBs with different polarization topological charges with  $m$ = 1 (top row) and $m$ = 2 (bottom row). (a) and (d) profiles of VBs without atoms, (b) and (e) after passing through atoms in the absence of a magnetic field, (c) and (f) petal-like patterns under $\\mathbf{B}_{\\rm TMF}$ = 230 mG. (g) Polar plots of the absorption profile in (c) along the radius of largest contrast.}\n\\label{fig6}\n\\end{figure}\n\n\\section{Theoretical interpretation and discussion}\n\nAs is now well known, atom-light interaction is strongly polarization dependent. There are however, infinitely many ways to decompose a field's polarization and to choose an atom's quantization axis. For the former, the spherical-basis has many useful properties \\cite{yudin2010vector,lee1998sensitive,auzinsh2010optically}.  Here, light polarized perpendicularly to the atom's quantization axis drives a superposition of $\\sigma^{+}$ and $\\sigma^{-}$ transitions ($\\Delta m_{\\rm F} = \\pm1$), while light polarized parallel to the quantization axis drives the associated $\\pi$ transition ($\\Delta m_{\\rm F} = 0$). In the absence of a magnetic field, it is convenient to choose the quantization axis, $z$, along the propagation axis, $\\mathbf{k}$, so that any light, $\\mathbf{E}$,  polarized in the $x-y$ plane is simply formed from the superposition of two orthogonal circular components with equal amplitude and a varying phase difference. In the presence of a magnetic field however, it is helpful to choose the quantization axis along the axis of the magnetic field, $\\mathbf{B}$,  so that the interaction is only dependent on the angle between $\\mathbf{B}$ and $\\mathbf{E}$. \n\n\nThus, we define the optical field in the spherical basis$\\left \\{ \\mathbf{e_{0}=e_{z}, e_{\\pm1} = \\mp (e_{x}\\pm \\textrm{i} e_{y})\/\\sqrt{2}} \\right \\}$ \\cite{yudin2010vector}:\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E} \n&=E_{0}\\left((\\cos{\\alpha})\\mathbf{e_{0}}+\\frac{\\sin{\\alpha}}{\\sqrt{2}}(-{\\rm e}^{-i \\beta_{1}}\\mathbf{e_{+1}}+{\\rm e}^{+i \\beta_{2}}\\mathbf{e_{-1}})\\right),\n\\label{2}\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{e}_{i}~\\forall i \\in \\{x,y,z\\}$, are the basis unit vectors in Cartesian coordinates and $\\mathbf{e}_{j}~\\forall j \\in \\{0,+1,-1\\}$ represent the spherical basis with the quantization axis set by the magnetic field. Here, $\\pi$-polarized, left and right circularly polarized light corresponds to $\\mathbf{e}_{0,+1,-1}$ respectively, $E_{0}$ is the amplitude of light, $\\alpha$ is the angle between $\\mathbf{B}$ and $\\mathbf{E}$ and $\\beta_{1}$ and $\\beta_{2}$ are the phase of each circular polarization. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure7.eps}\n\\caption{Excitation scheme for the LMF and the TMF. Coherent dark state (b) and decoherent state (c) induced by Zeeman splitting. Bare dark state without (e) and with (f) Zeeman splitting. In presence of the magnetic field,\nmagnetic sublevels are shifted by an amount $\\mu_{\\rm B}g_{\\rm F}m_{\\rm F}B$, where $\\mu_{\\rm B}$ is the Bohr magneton, $g_{\\rm F}$ is the Land$\\acute{e}$-factor, and $B$ is the magnetic field strength.}\n\\label{fig7}\n\\end{figure}\n\n\\subsection{Interaction under an LMF}\n\nAs shown in Fig. \\ref{fig7} (a), when an LMF is applied, we choose the quantization axis  along the LMF, coinciding with the propagation direction of the light. In this case, all the linearly polarized components of the probe VBs are perpendicular to the quantization axis, and the light will connect all Zeeman sublevels of the $F = 2 \\to F^{'} = 1$ transition via multiple $\\Lambda$ schemes with simultaneous $\\sigma_{\\pm1}$ excitations \\cite{cox2011measurements,dancheva2000coherent,meshulam2007transfer}. According to the light polarization in Eq. (\\ref{2}), the VBs shown in Eq. (\\ref{1}) can be rewritten as:\n\\begin{equation}\n\\mathbf{E} = \\frac{E_{0}}{\\sqrt{2}} (\\cos{(m\\phi)} + \\sin{(m\\phi)})(-{\\rm e}^{-i \\beta_{1}(\\phi)}\\mathbf{e_{+1}} + {\\rm e}^{+i \\beta_{2}(\\phi)}\\mathbf{e_{-1}}),\n\\label{3}\n\\end{equation}\nwhere, $\\beta_{1}$ and $\\beta_{2}$ are dependent on the azimuth.\n\nWhen the LMF is zero, the Zeeman sublevels are degenerate, and atoms are pumped into a non-absorbing state induced by coherent population trapping. This is similar to the case of standard EIT, where the left and right circularly polarized components of an optical field resonate with the atomic levels to form the $\\varLambda$ structure shown in Fig. \\ref{fig7} (b). Here, a dark state due to coherent superposition of atomic energy levels is formed, which causes transparency of the whole VBs' profile. Increasing the strength of the LMF results in splitting of the Zeeman sublevels and an effective detuning as shown in Fig. \\ref{fig7} (c). Thus, the coherent dark state is destroyed, the atoms can now interact and this leads to absorption of the probe VBs. For any linear polarization exciting the $\\sigma_{\\pm}$ transitions, the transparency is sensitive to the magnitude of the magnetic field and shows the Hanle-EIT profile \\cite{anupriya2010hanle}.\n\n\\subsection{Interaction under a TMF}\n\nWhen a pure TMF is applied, as shown in Fig. \\ref{fig7} (d), the quantization axis is chosen to be aligned with the axis of the TMF. In contrast to the former case, the interaction of the linearly polarized components of the probe VBs is strongly dependent on the azimuthal angle. The components whose linear polarization is parallel to the TMF axis operate on the $\\pi$ transition \\cite{happer1972optical,huss2006polarization,yin2016tunable}, while the orthogonal components activate the $\\sigma_{\\pm}$ transitions. Other linearly polarized components can be considered as superpositions of these special cases and the inclined angle between the $\\mathbf{B}_{\\rm TMF}$ and the $\\mathbf{E}$ determines which transition is dominant. Here, the situation of perpendicular components is similar to interaction under the $\\mathbf{B}_{\\rm TMF}$ and shows the same sensitivity to the magnetic field strength. Assuming the $\\mathbf{B}_{\\rm TMF}$ is along the $y$ axis and combining Eq. (\\ref{1}) and Eq. (\\ref{2}), then the $\\mathbf{E}$-field of the VBs can be rewritten as:\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E}&={E_{0}}\\sin{(m\\phi)}\\mathbf{e_{0}}+\\frac{E_{0}}{\\sqrt{2}}\\cos{(m\\phi)}(-{\\rm e}^{-i \\beta_{1}}\\mathbf{e_{+1}}+{\\rm e}^{+i \\beta_{2}}\\mathbf{e_{-1}})\\\\\n&=E_{0}\\sin{(m\\phi)}\\mathbf{e_{\\parallel}}+E_{0}\\cos{(m\\phi)}\\mathbf{e_{\\perp}},\n\\label{4}\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{e_{\\parallel}}=\\mathbf{e_{0}}$ and $\\mathbf{e_{\\perp}}=(-{\\rm e}^{-i \\beta_{1}}\\mathbf{e_{+1}}+{\\rm e}^{+i \\beta_{2}}\\mathbf{e_{-1}})\/\\sqrt{2}$.\nThe first term in Eq. (\\ref{4}) is the linear component with polarization direction along the TMF axis driving $\\pi$ transitions. In this situation, atoms are optically pumped into the stretch states by means of spontaneous emission and removed from the optical transition. Here, another type of dark state, due to strong optical pumping, is formed which will cause transparency of parallel components and shows insensitivity to the magnetic field (Fig. \\ref{fig7} (e) and (f)). The second term in Eq. (\\ref{4}) is the linear component with polarization perpendicular to the TMF axis. In this position, the analysis is similar to the interaction under the $\\mathbf{B}_{\\rm TMF}$ as discussed before. Increasing $\\mathbf{B}_{\\rm TMF}$ enhances the absorption (destroying the coherence) of perpendicular components and the splitting of the beam profile, since parallel components are always transmitted. Thus, the transmission profile of the probe VBs under the non-zero TMF fellows the equation as:\n\\begin{equation}\nI\\propto\\left |\\sin{(m\\phi)}\\right |^{2},\n\\label{5}\n\\end{equation}\nsatisfying the $2m$ sinusoidal transmission profile observed in Fig. \\ref{fig6} (g).\n\n\\subsection{Interaction with arbitrarily oriented B-fields and associated applications}\n\nGenerally, when $\\mathbf{B}$ is not applied along the axis of light propagation, it can be decomposed into contributions of an LMF and a TMF. Thus, for the general case, the interaction of the light with atoms can be viewed as a combination of the cases discussed in sections $\\mathbf{A}$ and  $\\mathbf{B}$, including all three transitions. A coherent dark state only appears when $\\mathbf{B}=0$ and forms from pure $\\sigma_{+}$ and $\\sigma_{-}$ transitions. Frequency detunings (Zeeman splitting) and $\\pi$ transitions induced by the magnetic field break the coherence of these dark states. Although the $\\pi$ transition associated with the TMF induces bare dark states which are insensitive to the magnetic field.\n\nBased on the above analysis, the transmitted pattern of VBs after passing through the atoms is strongly dependent on the $\\mathbf{B}$, making this configuration a potentially useful tool for exploring spatially varying magnetic fields. The radially polarized beam whose polarization distribution resembles a compass would be used as the probe and the transmitted pattern has two petals whose orientation clearly show the axis of the TMF. Besides, by comparing maximum intensity of the transmitted pattern with initial intensity of the beam in the same position, the angle between the magnetic field and the plane of the beam profile can be easily obtained as shown in Fig. \\ref{fig5} (g). By combining the included angle with the axis of TMF, the axis of the spatial $\\mathbf{B}$ could be defined. Not only the alignment of the magnetic field, but also its strength influence the observed transmission pattern. The strength of the magnetic field  is associated with the intensity of the region perpendicular to two petals, since the polarization of this part is always perpendicular to the quantization axis set by the $\\mathbf{B}$. The transparency of this region depends on the Zeeman splitting of atoms influenced by the magnetic field, which means the magnitude of the magnetic field can be deduced by measuring the intensity of this region.\n\n\nUltimately, the alignmnt and strength of a spatial $\\mathbf{B}$-field can be seen and quantified from the transmitted vector beam. There are experimental limitations however. First of all, the measurement range of the magnetic field is limited by the atomic coherence. In this experiment, the Zeeman energy levels were used to build the atomic coherence, but hyperfine levels could expand the range of measurement of this configuration. Secondly, the direction of the spatial $\\mathbf{B}$-field can not be obtained here. Since only resonant light is used in this experiment, the frequency detuning induced by spatial $\\mathbf{B}$ has the same influence on both $\\sigma_{+}$ and $\\sigma_{-}$ transitions. After passing through atoms, the linear polarization can always be obtained which indicates two circular components have equal amplitudes and experience same absorption. Further study will be carried out to solve this problem by utilizing detuned light and measuring the polarization ellipse of transmitted parts \\cite{selyem2019three, clark2016sculpting}.\n\n\n\\section{Conclusion}\n\nIn conclusion, we have investigated the transmission properties and pattern formation of vector beams in an atomic vapor, as influenced by a magnetic field. In particular, there are two limiting cases: corresponding to two kinds of dark state. When an LMF is applied, an incoming probe beam undergoes uniform absorption, and the coherence between Zeeman sublevels (coherent dark states) can be destroyed by increasing the strength of the LMF. Applying a TMF however, will generate bare dark states and produce a petal-like pattern which is dependent on the azimuthal angle and topological charge of the polarization. The general case, where the magnetic field is applied along an arbitrary axis, is also studied: revealing the general influence of a spatial $\\mathbf{B}$-field on transmitted patterns. Thus, the information about both the alignment and the strength of a spatial $\\mathbf{B}$-field can be seen in the transmitted pattern of a vector beam, providing a powerful tool in the investigation of 3D magnetic fields. Recent works on chip-scale VBs generation \\cite{chen2020vector} and atomic components \\cite{garrido2019compact,stern2019chip,mcgilligan2020laser} would be an exciting next step for realizing miniaturization. We also note that similar effects are seen in related work in the diamond (nitrogen-vacancy center) \\cite{chen2020calibration} and cold atoms \\cite{castellucci2021atomic} carried out, confirming the suitability of VBs for visual observation of the alignment of magnetic fields in 3D space.\n\n\\section*{Funding} This work was supported by the National Natural Science Foundation of China (92050103, 11774286, 11534008, 11604257 and 11574247) and the Fundamental Research Funds for the Central Universities of China. FC and SF-A acknowledge financial support\nfrom the European Training Network ColOpt, which is funded by the European Union (EU) Horizon 2020 program\nunder the Marie Sklodowska-Curie Action, Grant Agreement No. 721465.\nTWC acknowledges support by the National Research, Development and Innovation Office of Hungary (NKFIH) within the Quantum Technology National Excellence\nProgram (Project No. 2017-1.2.1-NKP-2017-00001). \n\n\\section*{Disclosures} The authors declare no conflicts of interest.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace*{-0.5pt}\n\\noindent\n\tIn spite of its great successes, the Glashow-Weinberg-Salam (GWS) model, based on the \ngroup $SU(2) \\otimes U(1)$, seems somewhat unsatisfactory, in that its lagrangian contains a number of \narbitrary parameters and several terms that seem, at least at first sight, rather \\letra{ad hoc}. There have \nbeen different types of ideas to improve this situation. The most popular one by far has been to \nuse a large group of which $SU(2) \\otimes U(1)$\\ is just a small subgroup. Less popular has been the \nintroduction of symmetries that do not enlarge the group too much, a step-at-a-time policy, so to \nspeak. This has been done along three different lines, depending on what additional symmetry is \nimposed, be it the group $SU(2)_L \\otimes SU(2)_R$,\\citar{1} a discrete symmetry that restricts the model's \nparameters and allows for the calculation of mass corrections to quarks,\\citar{2} or \\su3. Here we \nshall concentrate our efforts on this last idea, not following the usual Yang-Mills scheme,\\citar{3} but \ninstead considering a modification of it.\n\n\tIt is interesting that \\su3\\ contains $SU(2) \\otimes U(1)$\\ as a subgroup and that it naturally \nassigns the correct hypercharge $Y$ and isospin $T_3$ quantum numbers to the electron and the \nneutrino if we place them in its fundamental representation so that they form the chiral triplet\n\\ecuacion{\n\\psi_L = \\pmatrix{ \\nu_L \\cr e_L \\cr e_L^c }\n\\label{chiral}\n}.\n\n\tIt is necessary to express the $e_R$\\ degree of freedom using the antiparticle formulation $e_L^c$\\ in order \nto obtain the hypercharge with the correct sign, as we shall soon explain. The main problem that \npresents itself immediately when one tries to represent the GWS model using \\su3\\ stems from \nthe boson sector, because 8 vector fields are required to gauge the theory, one for every generator \nof the adjoint representation. This means that there would be four more boson vector fields than \nare experimentally observed.\n\n\tA different approach is based not on Lie but rather on graded groups\\citar{4} and uses $SU(2\/1)$. \nThe excessive number of vector bosons introduced by the gauging of \\su3\\ is reduced by \nassuming that some of them are scalar instead of vectorial. It is possible to use the Higgs bosons \nas gauge fields, thus adding to the logical simplicity of the model. Fermions are placed in the \nfundamental representation of $SU(2\/1)$\\ using the nonchiral triplet\n\\ecuacion{\n\\psi = \\pmatrix{ \\nu_L \\cr e_L \\cr e_R }\n\\label{nonchiral}\n}, \nthat differs from the chiral one in the third component. The generators of the fundamental \nrepresentations of \\su3\\ and $SU(2\/1)$\\ differ in only one component of one generator. We are \ngoing to use the generators $T^a=\\frac{1}{2} \\ld{a}$, where the $\\ld{a}$ are the usual Gell-Mann matrices, so that \nthey are normalized according to\n\\ecuacion{\n\\widetilde{\\mathop {\\rm Tr}\\nolimits}\\, T^a T^b = \\frac{1}{2} \\delta_{ab}\n\\label{norma}\n},\nwhere we have put a tilde over the trace symbol to distinguish it from the traces over Dirac \nmatrices that we shall begin to use next section. The two generators that differ are both diagonal: \nthe last \\su3\\ Gell-Mann generator\n\\ecuacion{\n\\ld{8} = \\frac{1}{\\sqrt{3}} \\mathop {\\rm diag} (1,1,-2)\n\\label{gene8}\n},\nand its partner in $SU(2\/1)$\n\\ecuacion{\n\\ld{0} = \\frac{1}{\\sqrt{3}} \\mathop {\\rm diag} (1,1,2)\n\\label{gene0}\n}.\n\n\tThe isospins of $\\nu_L$, $e_L$\\ and $e_R$\\ are $\\frac{1}{2}$, $-\\frac{1}{2}$ and $0$, and the hypercharges are $1$, $1$ and $2$, respectively, \nprecisely as given by $\\ld{0}$. The electric charges of these particles can be calculated from the Gell-Mann-Nishijima \nrelation $Q=T_3-\\frac{1}{2} Y$, that relates the charge $Q$ to the isospin $T_3$ and the \nhypercharge $Y$. From these considerations it is clear that $\\ld{8}$ would assign a hypercharge $-2$ for \n$e_R$, that has the wrong sign, which is the reason why we had to use the antiparticle formulation \n$e_L^c$\\ in \\refeq{chiral}. The $1\/\\sqrt{3}$ normalization factor neatly goes into converting the single coupling constant \n$g$ of the model into the GWS model's two coupling constants $g$ and $g'\\approx g\/\\sqrt{3}$.\n\t\n\tFrom the beginning it was noticed the graded group approach presented two serious \ndifficulties: The first one was the so-called ``sign problem'', that arises from the fact that in non-abelian \ngauge (graded) theories the vector boson kinetic energy is constructed from the trace \n(supertrace) of the product of generators, that are (are not) positive-definite. A matrix \nnormalization that is not positive-definite will give the wrong sign to the kinetic energy of some of \nthe vector bosons.\\citar{5} The second difficulty, that we are going to call the ``statistics problem'', had \nits origin in the fact that, if one is trying to reproduce the GWS model, Higgs fields have to be \nplaced in the odd (or a-type) sector\\citar{6} of the adjoint of $SU(2\/1)$, and therefore necessarily have to \nanticommute amongst themselves. On the other hand, they have spin zero and must therefore \nobey Bose-Einstein statistics, or problems with unitarity of the scattering matrix will ensue; thus, \nin contradiction with the graded group requirement, they have to commute among themselves. To \novercome these difficulties was the motivation for much effort at the time.\\citar{7} Many different ideas \nwere tried such as adding extra odd dimensions to the spacetime manifold, and taking the \nGrassmann fields to be ghosts and not bosons, but, since neither difficulty could be resolved \nwithout causing worse difficulties, eventually interest on the subject waned.\n \n\tA few years ago a new branch of mathematics, noncommutative geometry,\\citar{8} was \ndeveloped. Its methods have been applied to the direct product of spacetime and spaces with a \ndiscrete number of points, with the result that some sectors of the standard model\\citar{9} have been \nobtained. It has also been applied to other areas of theoretical physics.\\citar{10} In the former case \nthere appears a graded algebra of forms invariant under the algebra $su(2\/1)$. The GWS model \nwith its $SU(2) \\otimes U(1)$\\ local gauge invariance appears if we define the lagrangian to be the trace \n(\\letra{not} the supertrace) of group invariants. This way the sign problem is solved \\letra{ab initio}. It is not \nclear to us just how successful this approach is in simplifying or unifying the Standard Model. \nAlgebraic structures and ideas vary from paper to paper, sometimes even by the same author, so \nthat, at this stage, we hesitate upon taking a position. \n\n\tAn attempt at generalizing a Yang-Mills theory using a graded gauge group will \ninevitably lead to the sign and the Higgs' statistics problems. Thus the question of attempting a \ngeneralization of the Yang-Mills covariant derivative using a non-graded Lie group arises \nnaturally.\\citar{11} A Higgs field assigned to the adjoint of the Lie group will now have the correct \nstatistics since it would be even (that is, c-type). Furthermore, since Lie groups are invariant \nunder traces, not supertraces, the kinetic energies of all the bosons will come out with the right \nsign. In this paper we show how to construct generalized Yang-Mills theories that are invariant \nunder local gauge transformations of a Lie (not graded) group and use a covariant derivative with \nboth scalar and vector fields. In the construction of these theories we also honor the condition, \nwhich we consider to be of an essential nature, that the lagrangians do not contain any \ndifferentiation operators acting indefinitely to the right. Terms of this type arise from powers of \nthe covariant derivative, but we require that somehow they all cancel out. It goes without saying \nthat such theories are also required to be Lorentz-invariant. It turns out that the only way to have \na covariant derivative contain both vector and boson gauge fields is for it to transform as a \n4-vector contracted with the Dirac matrices, that is, as a slashed 4-vector.\n\n\tWe apply these ideas to try to unify and reduce the number of different kinds of \nterms of the GWS model. If now one uses \\su3\\ as a gauge group, one \\letra{almost} obtains the GWS \nmodel, the difference being that the hypercharge of the Higgs boson comes out wrong. An even \nmore interesting result is that, if one uses the group \\uu3\\ and chooses a certain representation of \nthe generators, the GWS comes out exactly, plus an additional scalar boson that is totally \nuncoupled both from all the other particles of the model. This unified theory has only two terms: \nthe fermion kinetic energy, that is, the covariant derivative between two fermion fields, and the \nboson kinetic energy, that is, traces of powers of the covariant derivative, what is often called the \ncurvature. We have not added a new level of symmetry breaking and in this sense we\nhave not performed a ``unification'' in the traditional sense: this theory has the\nsame number of Higgs bosons as the original GWS and they break the symmetry in\nbasically the same way. However, with the new covariant derivative the GWS can be\nwritten in a simpler way in a larger group.\n\nIn this paper we do not go into the quark sector of the standard model. The presence\nof two right quarks (as opposed to only one right lepton) per family results in a\ndifferent, more complicated situation, that we plan to address in a future. \n\n\tIn section 2 we show how to write a traditional abelian gauge theory with the \ncovariant derivative transforming as the tensor product of two spinorial transformations. In \nsection 3 we introduce the concept of scalar fields as gauge fields. In section 4 we present non-abelian \nYang-Mills theories with mixed gauge fields. In section 5 we present two attempts at \nunification of the GWS model in the context of these ideas, using the Lie groups \\su3 and \\uu3. \nFinally, in section 6 we make a summary and conclude with a couple of remarks of what is wanting \nwith the model.\n\n\n\\section{A vector field transforming under the tensor product of two spinorial representations}\n\\noindent\n\tIn this section we are going to rewrite the quantum electrodynamics lagrangian\n\\ecuacion{\n{\\cal L}_{QED} = \\overline{\\psi} i \\hbox{$\\not \\! \\! D$} \\psi - \\frac{1}{4} F^{\\mu \\nu} F_{\\mu \\nu}\n\\label{lagqed}\n},\nwhere $D_\\mu \\equiv \\deriv{\\mu} + ie A_\\mu$ and $F_{\\mu \\nu} \\equiv \\deriv{\\mu} A_\\nu - \\deriv{\\nu} A_\\mu = -ie^{-1} [D_\\mu, D_\\nu]$ using only contractions of 4-vectors \nwith Dirac matrices, that is, avoiding dot products between the 4-vectors themselves. The reason \nfor this is that there does not seem to be a way of generalizing gauge theories keeping the \ncovariant derivative in the vector representation of the Lorentz group. One has to learn how to do \neverything with spinorial representations.\n\n\tThe fermion field $\\psi$ transforms under a local \\uu1\\ Lie group, so that, if $U=e^{-i\\alpha(x)}$ \nis an element of this group, then $\\psi \\to U \\psi$. To maintain gauge invariance it is necessary that the \nvector potential undergoes a gauge transformation, too, of the sort\n\\ecuacion{\nA_\\mu \\to A_\\mu + e^{-1} \\deriv{\\mu} \\alpha\n\\label{transfgauge}\n},\n\n\tFrom this transformation law and the definition of $D_\\mu$, it is evident that the covariant derivative \nmust transform as\n\\ecuacion{\nD_\\mu \\to U D_\\mu U^{-1}\n\\label{Dgauge}\n}.\n\n\tWe call a differentiation operator that acts only on immediately succeeding \nfunctions, but whose action then stops and does not differentiate any further functions to the \nright, \\letra{a restrained operator}. Likewise, we call a differentiation operator \\letra{unrestrained} if it keeps \nacting indefinitely to the right. As an example of the latter take the $\\deriv{\\mu}$'s in the covariant \nderivatives in \\refeq{Dgauge}, that are acting to the right for we do not know how far. It is not admissible to \nleave unrestrained operators in a lagrangian because, first, what they can mean physically or \nmathematically is not clear, and, second, they are not gauge invariant. The boson kinetic energy in \n\\refeq{lagqed} is proportional to $[D_\\mu,D_\\nu][D^\\mu,D^\\nu]$, each of whose factors is gauge invariant. On the other \nhand $D_\\mu D_\\nu$ is not gauge invariant as can be seen by direct calculation. The cause for this different \nbehavior is evident from the equation\n\\ecuacion{\n[D_\\mu,D_\\nu] f = \\deriv{\\mu} A_\\nu f - A_\\nu \\deriv{\\mu} f - \\deriv{\\nu} A_\\mu f + A_\\mu \\deriv{\\nu} f = (\\deriv{\\mu} A_\\nu) f - (\\deriv{\\nu} A_\\mu) f\n\\label{DmuDnu}\n},\nwhere it is seen how four unrestrained operators result in two restrained ones, thanks to Leibnitz' \nrule.\n\n\tIf we represent by $S$ an element of the Lorentz spinor transformation group, so \nthat $\\psi \\to S \\psi$ under a Lorentz transformation, then, due to the homomorphism that exists between \nthe vector and spinor representations of the Lorentz group, we have that $\\hbox{$\\not \\! \\! A$} \\to S \\hbox{$\\not \\! \\! A$} S^{-1}$. This \nhomomorphism allowed Dirac to write the equation of motion of electrons. But, is it possible to \nwrite the \\letra{boson} kinetic energy with the vector potential transforming this same way? The \nfollowing theorem answers this question in the affirmative:\n\n\\vspace*{12pt}\n\\noindent\n{\\bf Theorem:} Let $D_\\mu = \\deriv{\\mu}+B_\\mu$, where $B_\\mu$ is a vector field. Then:\n\\ecuacion{\n(\\deriv{\\mu} B_\\nu - \\deriv{\\nu} B_\\mu) (\\partial^\\mu B^\\nu - \\partial^\\nu B^\\mu) = \\frac{1}{8} \\mathop {\\rm Tr}\\nolimits^2 \\hbox{$\\not \\! \\! D$}^2 - \\frac{1}{2} \\mathop {\\rm Tr}\\nolimits \\hbox{$\\not \\! \\! D$}^4\n\\label{Teo}\n},\nwhere the trace is to be taken over the Dirac matrices. Notice the partials on the left of the \nequation are restrained, the ones on the right are not. To prove the Theorem we expand the \ncovariant derivatives on the right side of \\refeq{DmuDnu}, and then use the following trick, which makes the \nalgebra manageable, in this and in more difficult examples in other sections. First, we define the \ndifferential operator $O \\equiv \\partial^2+2B \\cdot \\partial + B^2$. Notice that it does not contain any contractions with \nDirac matrices, so that $\\mathop {\\rm Tr}\\nolimits O = 4O$, $\\mathop {\\rm Tr}\\nolimits O (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$}) = 4O(\\partial \\cdot B)$, etc. It is not difficult to see then that \n$\\hbox{$\\not \\! \\! D$}^2 = O + (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})$, where the slashed partial is acting \\letra{only} on the succeeding slashed field. Now:\n\\ecuarreglo{\n\\frac{1}{8} \\mathop {\\rm Tr}\\nolimits^2 [O + (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})] - \\frac{1}{2} \\mathop {\\rm Tr}\\nolimits [O + (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})]^2 &=& 2(\\partial \\cdot B)^2 - \\frac{1}{2} \\mathop {\\rm Tr}\\nolimits [(\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})(\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})] \\nonumber \\\\\n&=& (\\deriv{\\mu} B_\\nu - \\deriv{\\nu} B_\\mu) (\\partial^\\mu B^\\nu - \\partial^\\nu B^\\mu)\n\\label{prueba}\n}{}\nand we have finished proving the Theorem. \n\n\tWith the aid of the Theorem we can rewrite the QED lagrangian in the form\n\\ecuacion{\n{\\cal L}_{QED} = \\overline{\\psi} i \\hbox{$\\not \\! \\! D$} \\psi + e^{-2} \\left( \\frac{1}{32} \\mathop {\\rm Tr}\\nolimits^2 \\hbox{$\\not \\! \\! D$}^2 - \\frac{1}{8} \\mathop {\\rm Tr}\\nolimits \\hbox{$\\not \\! \\! D$}^4 \\right)\n\\label{lagQED}\n},\nwhose Lorentz invariance can be proven using $\\g{\\mu} \\to S \\g{\\mu} S^{-1}$ and the cyclic properties \nof the trace. We prove, as an example, the Lorentz invariance of $\\hbox{$\\not \\! \\! D$}^2$: $\\mathop {\\rm Tr}\\nolimits \\hbox{$\\not \\! \\! D$}^2 \\to \\mathop {\\rm Tr}\\nolimits S \\hbox{$\\not \\! \\! D$} S^{-1} S \\hbox{$\\not \\! \\! D$} S^{-1} = \\mathop {\\rm Tr}\\nolimits \\hbox{$\\not \\! \\! D$}^2$. \nGenerally speaking, one could say that the cause for the term with \nthe squared trace in \\refeq{Teo} is the requirement that all differentiation operators be restrained.\n\n\n\\section{Abelian gauge theory using scalar boson fields}\n\\noindent\n\tWe want to construct an analogue to quantum electrodynamics, but using scalar \nbosons as gauge fields. This is, of course, not the so-called scalar electrodynamics, where the \nfermion matter fields are substituted by scalar bosons. Here it is the gauge field itself that has \nbecome a real scalar boson $\\varphi$. The transformation group is the same as in last section, and again \nto maintain gauge invariance \\refeq{Dgauge} must hold. We now define the covariant derivative to be\n\\ecuacion{\nD_\\varphi = \\hbox{$\\not \\! \\partial$} - e \\g5 \\varphi\n},\nand we require the gauge field to transform as $\\g5 \\varphi \\to \\g5 \\varphi - ie^{-1} \\hbox{$\\not \\! \\partial$} \\alpha$. These conditions immediately \nassure us that transformation \\refeq{Dgauge} holds in this case, too. There is an interesting point to be made \nhere: if we now substitute this covariant derivative in the QED lagrangian \\refeq{lagQED}, that was designed \nfor \\letra{vector} fields, we obtain the usual lagrangian for the interaction between a fermion and a real \n\\letra{scalar} boson field. That is, if we \\letra{assume} the lagrangian of the interaction to be\n\\ecuacion{\n{\\cal L}_{S} = \\overline{\\psi} i D_\\varphi \\psi + e^{-2} \\left( \\frac{1}{32} \\mathop {\\rm Tr}\\nolimits^2 D_\\varphi^2 - \\frac{1}{8} \\mathop {\\rm Tr}\\nolimits D_\\varphi^4 \\right)\n\\label{lagS}\n},\nthen the expansion of the covariant derivative results in \n\\ecuacion{\n{\\cal L}_{S} = \\overline{\\psi} i \\hbox{$\\not \\! \\partial$} \\psi - e \\overline{\\psi} i \\g5 \\varphi \\psi + \\frac{1}{2} (\\deriv{\\mu} \\varphi)(\\partial^\\mu \\varphi)\n},\nafter a bit of algebra. This calculation is similar to the one done last section, but with the $\\g5$ taking \nthe place of the $\\g{\\mu}$'s of that previous calculation. The function of the $\\g5$ is to ensure that the \npartial derivatives become restricted. If the gauge group had been non-abelian, then it would also \nensure that we obtain commutators, not anticommutators, of all the boson fields.\n\n\n\\section{Non-abelian Yang-Mills theory with mixed gauge fields}\n\\noindent\n\tConsider a lagrangian that transforms under a non-abelian local Lie group that has \n$N$ generators. The fermion or matter sector of the non-abelian lagrangian has the form $\\overline{\\psi} i \\hbox{$\\not \\! \\! D$} \\psi$, \nwhere $D_\\mu$ is a covariant derivative chosen to maintain gauge invariance. This term is invariant \nunder the transformation $\\psi \\to U \\psi$, where $U=U(x)$ is an element of the fundamental \nrepresentation of the group. The covariant derivative is $D_\\mu = \\deriv{\\mu} + A_\\mu$, where $A_\\mu = ig A_\\mu^{\\ a} (x) T^a$ is \nan element of the Lie algebra and $g$ is a coupling constant. We are assuming here that the set of \nmatrices $\\{ T^a \\}$ is a representation of the group's generators. Gauge invariance of the matter term \nis assured if\n\\ecuacion{\nA_\\mu \\to U A_\\mu U^{-1} - (\\deriv{\\mu} U) U^{-1}\n},\nor, what is the same,\n\\ecuacion{\n\\hbox{$\\not \\! \\! A$} \\to U \\hbox{$\\not \\! \\! A$} U^{-1} - (\\hbox{$\\not \\! \\partial$} U) U^{-1}\n}.\n\n\tWe have already seen how scalar fields can function as gauge fields. Our aim in \nthis section is to construct a non-abelian theory that uses both scalar and vector gauge fields. We \nproceed as follows. For every generator in the Lie group we choose one gauge field, it does not \nmatter whether vector or scalar. As an example, suppose there are $N$ generators in the Lie group; \nwe choose the first $N_V$ to be associated with an equal number of vector gauge fields and the last \n$N_S$ to be associated with an equal number of scalar gauge fields. Naturally $N_V + N_S = N$. Now we \nconstruct a covariant derivative $D$ by taking each one of the generators and multiplying it by one \nof its associated gauge fields and summing them together. The result is\n\\ecuacion{\nD \\equiv \\hbox{$\\not \\! \\partial$} + \\hbox{$\\not \\! \\! A$} + \\Phi\n\\label{general}\n},\n\\ecuarreglo{\n\\hbox{$\\not \\! \\! A$} & \\equiv & \\g{\\mu} A_\\mu \\equiv ig \\g{\\mu} A_\\mu^{\\ a} T^a \\coma, \\qquad \\qquad a = 1, \\ldots, N_V \\ptc,\n\\Phi    & \\equiv & \\g5 \\varphi \\equiv -g \\g5 \\varphi^b T^b \\coma, \\qquad \\qquad b = N_V+1, \\ldots, N  \\nonumber\n}.\n\nNotice the difference between $A_\\mu$ and $A_\\mu^{\\ a}$, and between $\\varphi$ and $\\varphi^b$. We take the gauge \ntransformation for these fields to be\n\\ecuacion{\n\\hbox{$\\not \\! \\! A$} + \\Phi \\to U (\\hbox{$\\not \\! \\! A$} + \\Phi) U^{-1} - (\\hbox{$\\not \\! \\partial$} U) U^{-1}\n},\nfrom which one can conclude that\n\\ecuacion{\nD \\to U D U^{-1}\n}.\n\nThe following lagrangian is constructed based on the requirements that it contains only matter \nfields and covariant derivatives, and that it possesses both Lorentz and gauge invariance:\n\\ecuacion{\n{\\cal L}_{NA} = \\overline{\\psi} i D \\psi + \\frac{1}{2g^2} \\widetilde{\\mathop {\\rm Tr}\\nolimits} \\left( \\frac{1}{8} \\mathop {\\rm Tr}\\nolimits^2 D^2 - \\frac{1}{2} \\mathop {\\rm Tr}\\nolimits D^4 \\right)\n\\label{lagNA}\n},\nwhere the trace with the tilde is over the Lie group matrices and the one without it is over \nmatrices of the spinorial representation of the Lorentz group. The additional factor of $1\/2$ that the \ntraces of \\refeq{lagNA} have with respect to \\refeq{Teo} comes from normalization \\refeq{norma}, that is the usual \none in the non-abelian case.\n\n\tAlthough lagrangian ${\\cal L}_{NA}$ was constructed based only on the requirements just \nmentioned, it is an interesting fact that the expansion of the covariant derivative into its \ncomponent fields results in expressions that are traditional in Yang-Mills theories. The reader who \nwishes to make the expansion herself can substitute \\refeq{general} in \\refeq{lagNA}, keeping in mind the derivatives \nare unrestrained, and aim first for the intermediate result\n\\ecuarreglo{\n\\frac{1}{16} \\mathop {\\rm Tr}\\nolimits^2 D^2 - \\frac{1}{4} \\mathop {\\rm Tr}\\nolimits D^4 &=& \\left( (\\partial \\cdot A)+A^2 \\right)^2 - \\frac{1}{4} \\mathop {\\rm Tr}\\nolimits \\left( (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! A$})+\\hbox{$\\not \\! \\! A$} \\hbox{$\\not \\! \\! A$} \\right)^2 \\nonumber \\\\\n   & & - \\frac{1}{4} \\mathop {\\rm Tr}\\nolimits \\left( (\\hbox{$\\not \\! \\partial$} \\Phi)+\\{ \\hbox{$\\not \\! \\! A$} \\Phi \\} \\right)^2\n\\label{expandido}\n},\nwhere the curly brackets denote an anticommutator. (We recommend to use here the same trick \nexplained in section 2.) Notice in this expression that the differentiation operators are restrained, \nand that the two different types of gauge fields appear in an anticommutator. One of the effects of \nthe $\\g5$ in \\refeq{general} is to turn this anticommutator into a commutator through the use of the properties \nof Clifford algebras. Substituting \\refeq{general} in \\refeq{expandido} and in the matter term of \\refeq{lagNA} we obtain the \nnon-abelian lagrangian in expanded form:\n\\ecuarreglo{\n{\\cal L}_{NA} &=& \\overline{\\psi} i(\\hbox{$\\not \\! \\partial$} + \\hbox{$\\not \\! \\! A$}) \\psi - g \\overline{\\psi} i \\g5 \\varphi^b T^b \\psi + \\frac{1}{2 g^2} \\widetilde{\\mathop {\\rm Tr}\\nolimits} \\left( \\deriv{\\mu} A_\\nu - \\deriv{\\nu} A_\\mu + [A_\\mu, A_\\nu] \\right)^2 \\nonumber \\\\\n   & & + \\frac{1}{g^2} \\widetilde{\\mathop {\\rm Tr}\\nolimits} \\left( \\deriv{\\mu} \\varphi + [A_{\\mu}, \\varphi] \\right)^2\n\\label{lagexpandido}\n}.\n\nThe reader will recognize familiar structures: the first term on the right looks like the usual matter \nterm of a gauge theory, the second like a Yukawa term, the third like the kinetic energy of vector \nbosons in a Yang-Mills theory and the fourth like the gauge-invariant kinetic energy of scalar \nbosons in the non-abelian adjoint representation. It is also interesting to observe that, if in the last \nterm we set the vector bosons equal to zero, then the term simply becomes $\\sum_{b,\\mu} \\frac{1}{2} \\deriv{\\mu}\\varphi^b \\partial^\\mu \\varphi^b$, the \nkinetic energy of the scalar bosons. We have constructed a generic non-abelian gauge theory with \ngauge fields that can be either scalar or vector.\n\n\n\\section{The GWS model using \\bfsu3\\ and \\bfuu3}\n\\noindent\n\tThe obvious choice for a small group to simplify and unify the GWS model using a \ngeneralized gauge theory is \\su3, because it has the correct hypercharge numbers for the leptons \nof a chiral multiplet. Furthermore, it has 8 generators, while the GWS model has precisely 8 \nbosons: 4 vector ones and the 4 Higgs real scalar fields. Unfortunately, this choice does not work \nas we shall soon see. Let $A_\\mu^{\\ a}$, $a=1,2,3$, and $B_\\mu$ be four vector fields to which we associate the \nfour Gell-Mann generators $\\ld{a}$, $a=1,2,3$, and $\\ld{8}$, respectively. Let $\\varphi^b$, $b=4,5,6,7$, be four real \nscalar fields, to which we associate the remaining Gell-Mann generators $\\ld{b}$, $b=4,5,6,7$. The \ncovariant derivative can be found following the prescription given in \\refeq{general}. Using the usual \nrepresentation of the Gell-Mann matrices and $g$ as coupling constant it can be explicitly written \nas follows:\n\\ecuacion{\nD = \\hbox{$\\not \\! \\partial$} + \\frac{g}{2} \\pmatrix{i \\hbox{$\\bf \\not \\! \\! A$} \\cdot \\hbox{{\\boldmath $\\sigma$}} + i \\hbox{$\\not \\! \\! B$} {\\bf 1} \/ \\sqrt{3} & -\\sqrt{2} \\g5 \\widehat{\\varphi} \\cr\n-\\sqrt{2} {\\widehat{\\varphi}}^{\\dag} \\g5 & -2i \\hbox{$\\not \\! \\! B$} \/ \\sqrt{3} }\n\\label{DSU3}\n},\nwhere the $\\sigma^a$, $a=1,2,3$, are the Pauli matrices, ${\\bf 1}$ is a $2 \\times 2$ matrix, and\n\\ecuacion{\n\\widehat{\\varphi} = \\frac{1}{\\sqrt{2}} \\pmatrix{ \\varphi^4 - i \\varphi^5 \\cr \\varphi^6 - i \\varphi^7 }\n}.\n\nThe lagrangian of the model is then precisely ${\\cal L}_{NA}$, as expressed by \\refeq{lagNA}, with the covariant \nderivative given by \\refeq{DSU3} and the fermion triplet by $\\psi_L$, the chiral triplet of equation \\refeq{chiral}. To \nobtain the expanded form of the boson kinetic energy sector through straightforward calculation \nis very messy, but through the use of formula \\refeq{lagexpandido} it is possible to arrive at the following \nexpression without much trouble:\n\\ecuarreglo{\n{\\cal L}_{NA} &=&\\overline{\\theta_L} (i\\hbox{$\\not \\! \\partial$} - \\frac{1}{2} g \\hbox{$\\not \\! \\! A$}^a \\sigma^a - \\frac{1}{2} g' \\hbox{$\\not \\! \\! B$}) \\theta_L + \\overline{e_R} (i \\hbox{$\\not \\! \\partial$} - g' \\hbox{$\\not \\! \\! B$}) e_R \\nonumber \\\\\n          & &+i \\frac{g}{\\sqrt{2}} \\overline{e_L^c} \\widehat{\\varphi}^{\\dag} \\theta_L +i \\frac{g}{\\sqrt{2}} \\overline{\\theta_L} \\widehat{\\varphi} e_L^c \\nonumber \\\\\n          & &-\\frac{1}{4} {\\bf A_{\\mu \\nu} \\cdot A^{\\mu \\nu}} -\\frac{1}{4} B_{\\mu \\nu} B^{\\mu \\nu} \\nonumber \\\\\n          & &+\\left| (\\deriv{\\mu} +\\frac{1}{2} ig A_\\mu^{\\ a} \\sigma^a + \\frac{3}{2} ig' B_\\mu) \\widehat{\\varphi} \\right|^2\n\\label{lagSU3}\n},\nwhere $\\theta_L = (\\nu_L, e_L)^T$ and we have introduced the symbol $g'\\equiv g\/\\sqrt{3}$. This is \\letra{almost} the lagrangian \nof the GWS model for a Weinberg angle $\\theta_W = 30^\\circ$ (a value very close to the experimental one), \nexcept that the hypercharge of the Higgs comes out as $3$ and not $-1$ as it should be. This detail \ndooms the model. Notice too the Yukawa terms are actually null, as one can see from chirality \nconsiderations.\n\n\tLet us look at the problem of the hypercharge in more detail. Let\n\\ecuacion{\n\\ld{Y} = \\frac{1}{\\sqrt{3}} \\mathop {\\rm diag} (x,x,y)\n}{}\nbe some generator associated with the vector field $B_\\mu$, the one that eventually becomes the \nisospin singlet in the GWS model; then the hypercharge of the Higgs is $Y=x-y$. In the example \nabove we were using $\\ld{8}$, shown in \\refeq{gene8}, so that $Y=1-(-2)=3$. If instead of \\su3\\ we had been \nusing $SU(2\/1)$\\ then the choice for $\\ld{Y}$ would have been $\\ld{0}$, shown in \\refeq{gene0}, and $Y=1-(+2)=-1$, the \ncorrect value. So it seems we have reached an \\letra{impasse}: the correct hypercharge is given precisely \nby the group we do not want to use. The way out of it is to realize that a very similar group, \n\\uu3, has a representation where $\\ld{0}$ appears. In other words, it is not necessary to go to graded \ngroups to obtain the correct hypercharge. We will see now how this comes about.\n\n\tA representation of \\uu3\\ is, for instance, the 8 Gell-Mann generators $\\ld{a}$, $a=1,\\ldots ,8$ plus \nanother (not traceless) $3 \\times 3$ matrix which we take to be the normalized unit matrix:\n\\ecuacion{\n\\ld{9} = \\sqrt{\\frac{2}{3}} \\mathop {\\rm diag} (1,1,1)\n}.\n\nWe now define a new matrix\n\\ecuacion{\n\\ld{10} = \\sqrt{\\frac{2}{3}} \\mathop {\\rm diag} (1,1,-1)\n},\nand make the observation that it and $\\ld{0}$ can be expressed as linear combinations of $\\ld{8}$ and $\\ld{9}$:\n\\ecuarreglo{\n\\ld{10} &=& \\frac{2 \\sqrt{2}}{3} \\ld{8} + \\frac{1}{3} \\ld{9} \\ptc,\n\\ld{0} &=& -\\frac{1}{3} \\ld{8} + \\frac{2 \\sqrt{2}}{3} \\ld{9}\n}.\nNotice that $\\ld{10}$ and $\\ld{0}$ are orthonormal under \\refeq{norma}. The set of 9 matrices $\\ld{10}$, $\\ld{0}$ and $\\ld{a}$, \n$a=1,\\ldots ,7$ forms a representation of \\uu3, since each matrix is a linear combination of the \ngenerators of another representation. So we have found $\\ld{0}$ in \\uu3, and associating it with $B_\\mu$ we \nmake sure that the hypercharge for the Higgs comes out correctly.\n\n\tThis new group has 9 generators, and, therefore, 9 bosons. This is cause for concern, since \nthe extra boson could upset the precise clockwork phenomenology of the GWS model. We have \nin principle the option of taking it to be either scalar or vector, but the second option would \npresent us with an unphysical extra vector boson. On the other hand, if we take it to be a scalar \nboson, it is an interesting fact that the boson completely decouples from the rest of the particles \nand we obtain again lagrangian \\refeq{lagSU3}, but with the correct hypercharge for the Higgs bosons. Now \n$\\psi$ is given by \\refeq{nonchiral} and not by \\refeq{chiral}, as in the \\su3\\ case.\n\n\tTo understand why the new scalar boson, that we shall call $\\Upsilon$, decouples, go back to \\refeq{lagexpandido}, \nthe non-abelian lagrangian, and notice that the scalar bosons only appear in the second and fourth \nterms of the right of the equation. Since the scalar boson is associated with $\\ld{10}$, which is a \ndiagonal generator, in the second term the spinorial degrees of freedom are all multiplying their \nrespective conjugates, that have the opposite chirality, and therefore all products are null. In the \nfourth term within the parenthesis there appears, for the case of the new scalar, the expression $[A_\\mu, \\ld{10} \\Upsilon]$,\nthat is, the commutator of a block diagonal matrix and $\\ld{10}$, and, therefore, zero. The \nonly term that remains in the lagrangian that contains $\\Upsilon$ is its kinetic energy. It would seem this \nfield is massless, as it does not couple with the Higgs.\n\n\tWe again expand ${\\cal L}_{NA}$, this time using the new set of generators and the nonchiral \nfermion triplet $\\psi$ of equation \\refeq{nonchiral}, with the result:\n\\ecuarreglo{\n{\\cal L}_{NA} &=&\\overline{\\theta_L} (i\\hbox{$\\not \\! \\partial$} - \\frac{1}{2} g \\hbox{$\\not \\! \\! A$}^a \\sigma^a - \\frac{1}{2} g' \\hbox{$\\not \\! \\! B$}) \\theta_L + \\overline{e_R} (i\\hbox{$\\not \\! \\partial$} - g' \\hbox{$\\not \\! \\! B$}) e_R \\cr\n          & &+i \\frac{g}{\\sqrt{2}} \\overline{e_R} \\widehat{\\varphi}^{\\dag} \\theta_L -i \\frac{g}{\\sqrt{2}} \\overline{\\theta_L} \\widehat{\\varphi} e_R \\cr\n          & &-\\frac{1}{4} {\\bf A_{\\mu \\nu} \\cdot A^{\\mu \\nu}} -\\frac{1}{4} B_{\\mu \\nu} B^{\\mu \\nu} \\cr\n          & &+\\left| (\\deriv{\\mu} +\\frac{1}{2} ig A_\\mu^{\\ a} \\sigma^a - \\frac{1}{2} ig' B_\\mu) \\widehat{\\varphi} \\right|^2 + \\frac{1}{2} (\\deriv{\\mu} \\Upsilon)(\\partial^\\mu \\Upsilon)\n\\label{lagU3}\n},\nwhich is the lagrangian of the GWS model. Notice the two Yukawa terms do not vanish in this \nlagrangian, as they did in \\refeq{lagSU3}. The difference of sign between them does not matter, as it simply \ndepends on what phase of the $e_R$\\ we decide to use.\n \n\tIn Yang-Mills theories when the covariant derivative acts on a field $X$, its gauge fields \nacquire certain coefficients called the ``charges'' of each gauge field with respect to $X$. In this \ngeneralized gauge theory the same thing has to be done. Thus, when $D$ acts on the leptonic \ntriplet, its gauge fields are going to be multiplied by constants we shall call ``leptonic charges'' $Q_V$ \nand $Q_S$, with the result $D \\psi = (\\hbox{$\\not \\! \\partial$} + Q_V \\hbox{$\\not \\! \\! A$} + Q_S \\Phi) \\psi$. From our knowledge of the standard model we \nconclude that $Q_V=1$, and that there are three $Q_S$, one for each generation, and have rather small \nvalues. Lagrangian \\refeq{lagU3} is actually for one generation only, and it should include its respective $Q_S$ \nas a coefficient in the two Yukawa terms. We have no \\letra{a priori} knowledge of the values of the \nleptonic charges.\n\n\n\\section{Summary and remarks}\n\\noindent\n\tWe found a particular way of expressing the kinetic energy of vector bosons (what in \nprincipal vector bundles is called the curvature) so that those fields appear contracted only with \nDirac matrices. From here we were able to generalize the concept of covariant derivative, so that \nit included both scalar and vector bosons. Using this generalized derivative one can write the \nGWS model using only two terms (a curvature and a matter term), and unify the two groups (and \ntheir coupling constants) into one. This derivative has to transform as a slashed 4-vector.\n\n\tThe unification group is not \\su3, that gives the wrong value for the hypercharge of the \nHiggs field, but \\uu3\\ instead, that predicts correctly all the quantum numbers of the GWS model. \nThis is possible because there is a representation of this group that contains the same generator of \n$SU(2\/1)$\\ that gives the right hypercharge to the Higgs bosons. Since we did not use a graded \ngroup, the Higgs fields obey, correctly, Bose-Einstein statistics, and the kinetic energy terms of \n\\letra{all} vector bosons have the right sign.\tAn extra scalar boson has to included (since \\uu3\\ has one \nmore generator than \\su3) but it automatically decouples from the rest of the model becoming \nan unobservable particle. It is probably massless, since it does not couple with the Higgs field, \neither.\n\nIn the GWS model symmetry breaking is achieved spontaneously through the\nintroduction of a potential of the form $V(\\varphi)$. In the model presented here \nthis potential also has to be introduced explicitly, as it does not appear in its\nsole\ntwo terms, kinetic energy of bosons and of fermions. In this it differs from the\nnoncommutative geometry results for the standard model, that implicitly include\nthis potential. The terms for the Higgs potential appear in our model, but they \ncancel since we introduce counterterms for the purpose of avoiding the presence of\ndifferentiation operators acting indefinitely to the right. (See equation\n\\refeq{Teo}.) These counterterms also avoid other unwanted self-interacting terms.\n\nThe Yukawa coupling constants appear in a natural way as generalized gauge charges\nof the lepton triplet $\\psi$, but the model sheds no light about what those values\nmay be, or why there is a different value for each generation.\n\n\\nonumsection{References}\n\\noindent\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Keywords:} Social Robotics, Autism, Music Therapy, Turn-Taking, Motor Control, Emotion Classification}\n\t\\end{abstract}\n\t\n\t\\section{Introduction}\n\t\\ Throughout history, music has been used medicinally due to its notable impact on the mental and physical health of its listeners. This practice became so popular over time that it ultimately transitioned into its own type of therapy called music therapy. The interactive nature of music therapy has also made it especially well-suited for children. Children's music therapy is performed either in a one-on-one session or a group session and by listening, singing, playing instruments, and moving, patients can acquire new knowledge and skills in a meaningful way. It has previously been shown to help children with communication, attention, and motivation problems as well as behavioral issues \\cite{cibrian2020supporting,gifford2011using}.\n\tWhile music therapy is extremely versatile and has a broad range of applications, previous research studies have found  that using music as an assistive method is especially useful for children with autism \\cite{cibrian2020supporting,mossler2019therapeutic, dellatan2003use,brownell2002musically,starr1998understanding}. This interest in music therapy as a treatment for children with autism has even dated as far back as the early $20^{th}$ century. In fact, in the 1940s, music therapy was used at psychiatric hospitals, institutions, and even schools for children with autism. Due to the fact both the diagnostic criteria for autism and music therapy as a profession were only just emerging at this time, no official documentation or research about this early work exists. In the next decade, the apparent unusual musical abilities of children with autism intrigued many music therapists \\cite{warwick1991music}. By the end of the 1960s, music therapists started delineating goals and objectives for autism therapy \\cite{reschke2011history}. Since the beginning of the 1970s and onward,  theoretically grounded music therapists have been working toward a more clearly defined approach to improve the lives of individuals with autism spectrum disorders (ASD). However, for decades, music therapists have not used a consistent assessment method when working with individuals with ASD. A lack of a quality universal assessment tool has caused therapy to drift towards non-goal driven treatment and has reduced the capacity to study the efficacy of the treatment in a scientific setting \\cite{thaut2000scientific}.\n\t\n\tHumanoid socially assistive robots are ideal to address some of these concerns \\cite{scassellati2012robots, diehl2012clinical}. Precise programming can be implemented to ensure the robots deliver therapy in a consistent assessment method each session. Social robots are also unique in their prior success in working with children with ASD \\cite{boucenna2014interactive, pennisi2016autism}. Previous studies have found children with autism have less interest in communicating with humans due to how easily these interactions can become overwhelming \\cite{marinoiu20183d,di2018deep,richardson2018robot,feng2013can} and instead are more willing to interact with humanoid social robots in daily life due to their relatively still faces and less intimidating characteristics. Some research has found that children with ASD speak more while interacting with a non-humanoid robot compared to regular human-human interactions \\cite{kim2013social}. A ball-like robot called Sphero has been used to examine different play patterns and emotional responses of children with ASD \\cite{boccanfuso2016emotional, boucenna2014learning}. Joint attention with body movement has also been tested and evaluated using a robot called NAO from \\cite{anzalone2014children}. Music interaction has been recently introduced in Human-Robot-Interaction (HRI) as well \\cite{taheri2019teaching}. The NAO robot has been widely used in this field such as music and dance imitation \\cite{beer2016robot} and body gesture imitation \\cite{guedjou2017influence,zheng2015robot,boucenna2016robots}.\n\tTo expand upon the aforementioned implications of socially assistive robotics and music, \\textbf{the scientific question of this research is whether music-therapy delivered by a socially assistive robot is engaging and effective enough to serve as a viable treatment option for children with ASD}.\n\tThe main contribution of this paper is as follows. First, we propose a fully autonomous assistive robot-based music intervention platform for children with autism. Second, we designed HRI sessions and conducted studies based on the current platform, where motor control and turn-taking skills were practiced by a group of children with ASD. Third, we utilized and integrated machine learning techniques into our music-therapy HRI sessions to recognize and classify event-based emotion expressed by the study participants.\n\t\n\tThe remainder of this paper is organized as follows. Section 2 presents related works concerning human-robot interaction in multiple intervention methods. Section 3 elaborates on the experiment design process including the details of the hardware and HRI sessions. We present our music therapy platform in Section 4 and the experimental results in Section 5. Finally, Section 6 concludes the paper with remarks for future work.\n\t\n\t\\section{Related Works}\n\t\n\t\\ Music is an effective method to involve children with autism in rhythmic and non-verbal communication and has often been used in therapeutic sessions with children who suffer from mental and behavioral disabilities \\cite{lagasse2019assessing,boso2007effect,roper2003melodic}. Nowadays, at least 12\\% of all treatments for individuals with autism consist of music-based therapies \\cite{bhat2013review}. Specifically, playing music to children with ASD in therapy sessions has shown a positive impact on improving social communication skills \\cite{lagasse2019assessing,lim2011effects}. Many studies have utilized both recorded and live music in interventional sessions for single and multiple participants \\cite{dvir2020body,bhat2013review, corbett2008brief}. Different social skills have been targeted and reported (i.e., eye-gaze attention, joint attention and turn-taking activities) in music-based therapy sessions \\cite{stephens2008spontaneous, kim2008effects}. Improving gross and fine motor skills for children with ASD through music interventions is  noticeably absent in this field of studies \\cite{bhat2013review} and thus is one of the core features in the proposed study.\n\tSocially assistive robots are becoming increasingly popular as interventions for youth with autism. Previous studies have focused on eye contact and joint attention \\cite{mihalache2020perceiving,  mavadati2014comparing,feng2013can}, showing that the pattern of gaze perception in the ASD group is similar to Typically Developing (TD) children as well as the fact eye contact skills can be significantly improved after interventional sessions. These findings also provide strong evidence that ASD children are more inclined to engage with humanoid robots in various types of social activities, especially if the robots are socially intelligent \\cite{anzalone2015evaluating}. Other researchers have begun to use robots to conduct music-based therapy sessions. In such studies, children with autism are asked to imitate music based on the \\textit{Wizard of Oz} and Applied Behavior Analysis (ABA) models using humanoid robots in interventional sessions to practice eye-gaze and joint attention skills \\cite{askari2018pilot,taheri2015impact, taheri2016social}.\n\t\n\tHowever, past research has often had certain disadvantages, such as a lack of an automated system in human-robot interaction. This research attempts to address these shortcomings with our proposed platform. In addition, music can be used as a unique window into the world of autism as a growing body of evidence suggests that individuals with ASD are able to understand simple and complex emotions in childhood using music-based therapy sessions \\cite{molnar2012music}. Despite the obvious advantages of using music-therapy to study emotion comprehension in children with ASD, limited research has been found, especially studies utilizing physiological signals for emotion recognition in ASD and TD children \\cite{feng2018wavelet}. The current study attempted to address this gap by using electrodermal activity (EDA) as a measure of emotional arousal to better understand and explore the relationship between activities and emotion changes in children with ASD. To this end, the current research presents an automated music-based social robot platform with an activity-based emotion recognition system. The purpose of this platform is to provide a possible solution for assisting children with autism and help improve their motor and turn-taking skills. Furthermore, by using bio-signals with Complex-Morlet (C-Morlet) wavelet feature extraction \\cite{feng2018wavelet}, emotion classification and emotion fluctuation can be analyzed based on different activities. TD children are included as a control group to compare the intervention results with the ASD group.\n\t\n\t\\section{Experiment Design}\n\t\\subsection{Participants Selection}\n\tNine high functioning ASD participants (average age: 11.73, std: 3.11) and seven TD participants (average age: 10.22, std: 2.06) were recruited for this study. Only one girl was included in the ASD group, which according to \\cite{loomes2017male}, is under the estimated ratio of 3:1 for male and females with ASD. ASD participants who had participated in previous unrelated research studies in the past were invited to return for this study. Participants were selected from this pool and the community with help from the University of Denver Psychology department. Six out of nine ASD participants had previous human-robot-interaction with another social robot named ZENO \\cite{mihalache2020perceiving}, and three out of the nine kids from the ASD group had previous music experience on instruments other than the Xylophone (saxophone and violin). All children in the ASD group had a previous diagnosis of ASD in accordance with diagnostic criteria outlined in \\cite{DSMIV2000}, including an ADOS report on record. Additionally, their parents completed the Social Responsiveness Scale (SRS) \\cite{constantino2012social} for their child. The SRS provides a quantitative measure of traits associated with autism among children and adolescents between four and eighteen years-of-age. All ARS T-scores in our sample were above 65. Unlike the ASD group, TD kids had no experience with NAO or other robots prior to this study. Most of them, on the other hand, did participate in music lessons previously. Experimental protocols were approved by the University of Denver Institutional Review Board.\n\t\n\t\\subsection{NAO: A Humanoid Robot}\n\tNAO, a humanoid robot merchandised by SoftBank Group Corporation, was selected for the current research. NAO is 58 cm (23 inches) tall and has 25 degrees of freedom, allowing him to perform most human body movements. According to the official Aldebaran manufacturer documentation, NAO's microphones have a sensitivity of 20mV\/Pa +\/-3dB at 1kHz and an input frequency range of 150Hz - 12kHz. Data is recorded as a 16 bit, 48000Hz, 4 channel wave file which meets the requirements for designing the online feedback audio score system. NAO's computer vision module includes face and shape recognition units. By using the vision feature of the robot, NAO can see an instrument with its lower camera and implement an eye-hand self-initiative system that allows the robot to micro-adjust the coordination of its arm joints in real-time, which is very useful to correct cases of inproper positioning prior to engaging with the Xylophone.\n\t\n\tNOA's arms have a length of approximately 31 cm. Position feedback sensors are equipped in each of the robot's joints to obtain real-time localization information. Each robot arm has five degrees of freedom and is equipped with sensors to measure the position of joint movement. To determine the position of the Xylophone and the mallets' heads, the robot analyzed images from the lower monocular camera located in its head, which has a diagonal field of view of 73 degrees. By using these dimensions, properly sized instruments can be selected and more accessories can be built.\n\t\n\t\\subsection{Hardware Accessories}\n\tIn order to have a well-functioning toy-size humanoid robot play music for children with autism, some necessary accessories needed to be made before the robot was capable of completing this task. All accessories will be discussed in the following paragraphs.\n\t\n\tIn this system, due to the length of NAO's outstretched arms, a Sonor Toy Sound SM Soprano Glockenspiel with 11 sound bars 2cm in width was selected and utilized in this research. The instrument is $31cm \\times 9.5cm \\times 4cm$, including the resonating body.\n\tThe smallest sound bar is playable in an area of $2.8cm \\times 2cm$, the largest in an area of $4.8cm \\times 2cm$. The instrument is diatonically tuned in C major\/A minor. The 11 bars of the Xylophone represent 11 different notes, or frequencies, that cover one and a half octave scales, from C6 to F7.\n\tThe Xylophone, also known as the marimba or the glockenspiel, is categorized as a percussion instrument that consists of a set of metal\/wooden bars that are struck with mallets to produce delicate musical tones. Much like the keyboard or drums, to play the Xylophone properly, a unique and specific technique needs to be applied. A precise striking movement is required to produce a beautiful note, an action perfect for practicing motor control, and the melody played by the user can support learning emotions through music.\n\t\n\tFor the mallets, we used the pair that came with the Xylophone, but added a modified 3D-printed gripper that allowed the robot hands to hold them properly (see Figure \\ref{griper}). The mallets are approximately 21 cm in length and include a head with a 0.8 cm radius. Compared to other designs, the mallet gripper we added encourages a natural holding position and allows the robot to properly model how participants should hold the mallet stick.\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\t\\epsfig{figure=.\/fig\/grip.eps, scale = 0.4}\\label{griper} \\\\\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Mallet Griper} \\label{griper}\n\t\t\\end{center}\n\t\\end{figure}\n\tUsing carefully measured dimensions, a wooden base was designed and laser cut to hold the Xylophone at the proper height for the robot to play in a crouching position. In this position, the robot could easily be fixed in a location and have the same height as the participants, making it more natural for the robot to teach activities (see Figure \\ref{stand}).\\\\\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\t\\epsfig{figure=.\/fig\/front_view.eps, scale = 0.4} \\label{front}\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Instrument Stand Front View.} \\label{stand}\n\t\t\\end{center}\n\t\\end{figure}\n\t\\subsection{Q-Sensor}\n\tOne Q-sensor \\cite{kappas2013validation} was used in this study. Participants were required to wear this device during each session.  EDA signal\n\t(frequency rate 32Hz) was collected from the Q-sensor attached to the wrists (wrist side varied and was determined by the participants). Due to the fact participants often required breaks during sessions, 2 to 3 seperated EDA files were recorded. These files were annotated by comparing the time stamps with the videos manually.\n\t\n\t\\subsection{Experiment Room Setup}\n\tAll experiment sessions were held in an $11ft \\times 9.5ft \\times 10ft$ room with six HD surveillance cameras installed in the corners, on the sidewalls, and on the ceiling of the room (see Figure \\ref{room}).The observation room was located behind a one-way mirror and participants were positioned with their backs toward this portion of the room to avoid distractions. During experiment sessions, an external, hand-held audio recorder was set in front of participants to collect high-quality audio to use for future research.\n\t\\begin{figure*}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\t\\epsfig{figure=.\/fig\/room_pana.eps, scale = .25}\\label{room} \n\t\t\t\\end{tabular}\n\t\t\t\\caption{Experiment room.} \\label{room}\n\t\t\\end{center}\n\t\\end{figure*}\n\t Utilizing a miniature microphone attatched to the ceiling camera, real-time video and audio were broadcasted to the observation room during sessions, allowing researchers to observe and record throughout the sessions. Parents or caregivers in the observation room could also watch and call off sessions in the case of an emergency.\n\t \n\t\\subsection{Experiment Sessions}\n\tFor each participant in the ASD group, six total sessions were delivered, including one baseline session, four interventional sessions, and one exit session. Only the baseline and exit sessions were required for the TD group. Baseline and exit sessions contained two activities: 1) music practice and 2) music gameplay. Intervention sessions contained three parts: S1) warm-up; S2) single activity practice (with a color hint); and S3) music gameplay. Every session lasted for a total of 30-60 minutes depending on the difficulty of each session and the performance of individuals. Typically, each participant had baseline and exit session lengths that were comparable. For intervention sessions, the duration gradually increased in accordance with the increasing difficulty of subsequent sessions. Additionally, a user-customized song was used in each interventional session to have participants involved in multiple repetitive activities. The single activity practice was based on music practice from the baseline\/exit session. In each interventional session, the single activity practice only had one type of music practice. For instance, single-note play was delivered in the first intervention session. For the next intervention session, the single activity practice increased in difficulty to multiple notes. The level of difficulty for music play was gradually increased across sessions and all music activities were designed to elicit an emotional reaction.\n\t\n\tTo aid in desired music-based social interaction, NAO delivered verbal and visual cues indicating when the participant should take action. A verbal cue consisted of the robot prompting the participant with \"Now, you shall play right after my eye flashes.\" Participants could also reference NAO's eyes changing color as a cue to start playing. Aftter the cues, participants were then given five to 10 seconds to replicate NAO's strikes on the Xylophone. The start of each session was considered a warmup and participants were allowed to imitate NAO freely without accuracy feedback. The purpose of having a warm-up section was to have participants focus on practicing motor control skills while also refreshing their memory on previous activities. Following the warmup, structured activities would begin with the goal of the participant ultimately completing a full song. Different from the warmup section, notes played during this section in the correct sequence were considered to be a good-count strike and notes incorrectly played were tracked for feedback and additional training purposes. To learn a song, NAO gradually introduced elements of the song, starting with a single note and color hint. Subsequently, further notes were introduced, then the participants were asked to play half the song, and then finally play the full song after they perfected all previous tasks. Once the music practice was completed, a freshly designed music game containing three novel entertaining game modes was presented to the participants. Participants could then communicate with the robot regarding which mode to play with. Figure \\ref{interact} shows the complete interaction process in our HRI study.\n\t\n\t\\begin{figure*}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\t\\epsfig{figure=.\/fig\/interact.eps, scale = .45}\\label{r} \n\t\t\t\\end{center}\n\t\t\t\\caption{Experiment session illustration.} \\label{interact}\n\t\t\\end{center}\n\t\\end{figure*}\n\t\n\tThe laboratory where these experiments took place is 300-square feet and has environmental controls that allowed the temperature and humidity of the testing area to be kept consistent. For experimental purposes in the laboratory, the ambient temperature and humidity were kept constant to guarantee the proper functionality of the Q-sensor as well as the comfort of subjects in light clothing. To obtain appropriate results and ensure the best possible function of the device, the sensor was cleaned before each usage. Annotators were designated to determine the temporal relation between the video frames and the recorded EDA sequences of every participant. To do this, annotators went through video files of every session, frame by frame, and designated the initiation and conclusion of an emotion. Corresponding sequences of EDA signals were then identified and utilized to create a dataset for every perceived emotion. \n\t\n\tFigure \\ref{eda_anno} shows the above-described procedure diagrammatically. Due the fact that it is hard to conclude these emotions with specific facial expressions, event numbers will be used in the following analysis section representing emotion comparison.  The rest of the emotion labels respresent the dominant feeling of each music activity according to the observations of both annotators and the research assistant who ran the experiment sessions. \n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/fig5.eps}\\label{eda_anno}\n\t\t\t\\end{center}\n\t\t\t\\caption{The distribution of the targeted emotions across all subjects and events. \n\t\t\t} \\label{eda_anno}\n\t\t\\end{center}\n\t\\end{figure}\n\n\t\\;\\noindent\\textbf{Mode 1):} The robot randomly picks a song from its song bank and plays it for participants. After each song, participants were asked to identify a feeling they ascribed to the music to find out whether music emotion can be recognized. \n\t\n\t\\textbf{Mode 2):} A sequence of melodies were randomly generated by the robot with a consonance (happy or comfortable feeling) or dissonance (sad or uncomfortable feeling) style. Participants were asked to express their emotions verbally and a playback was required afterwards.\n\t\n\t\\textbf{Mode 3):} Participants have five seconds of free play then challenge the robot to imitate what was just played. After the robot was done playing, the participant rated the performance, providing a reversed therapy-like experience for all human volunteers. \n\t\n\tThere was no limit on how many trials or modes each individual could play for each session, but each mode had to be played at least once in a single session. The only difference between the baseline and exit sessions was the song that was used in them. In the baseline session, \"Twinkle, Twinkle, Little Star\" was used as an entry-level song for every participant. For the exit session, participants were allowed to select a song of their choosing so that they would be more motivated to learn the music. In turn, this made the exit session more difficult than the baseline session. By using the Module-Based Acoustic Music Interactive System, inputting multiple songs became possible and less time-consuming. More than 10 songs were collected in the song bank including \"Can Can\" by Offenbach, \"Shake It Off\" by Taylor Swift, the \"SpongeBob SquarePants\" theme from the animated show SpongeBob SquarePants, and \"You Are My Sunshine\" by Johnny Cash. Music styles covered kid's songs, classic, pop, ACG (Anime, Comic and Games), and folk, highlighting the versatile nature of this platform. In addition to increasing the flexibility of the platform, having the ability to utilize varying music styles allows the robot to accommodate a diverse range of personal preferences, further motivating users to learn and improve their performance.\n\t\n\t\\section{Module-Based Acoustic Music Interactive System Design}\n\tIn this section, a novel, module-based robot-music therapy system will be presented. For this system to be successful, several tasks had to be accomplished: a) allow the robot to play a sequence of notes or melody fluently; b) allow the robot to play notes accurately; c) allow the robot to adapt to multiple songs easily; d) allow the robot to be able to have social communication with participants; e) allow the robot to be able to deliver learning and therapy experiences to participants; and f) allow the robot to have fast responses and accurate decision-making. To accomplish these tasks, a module-based acoustic music interactive system was designed. Three modules were built in this intelligent system: Module 1: eye-hand self-calibration micro-adjustment; Module 2: joint trajectory generator; and Module 3: real-time performance scoring feedback (see Figure \\ref{module}).\n\t\n\t\\begin{figure}\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\n\t\t\t\t\\includegraphics[width=0.9\\linewidth]{.\/fig\/module_blocks.eps}\\label{module}\\\\\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Block diagram of Module-based acoustic music interactive system.} \\label{module}\n\t\t\\end{center}\n\t\\end{figure}\n\t\\subsection{Module 1: Color-Based Position Initiative}\n\tKnowledge about the parameters of the robot's kinematic model was essential for programming tasks requiring high precision, such as playing the Xylophone. While the kinematic structure was known because of the construction plan, errors could occur because of factors such as imperfect manufacturing. After multiple rounds of testing, it was determined that the targeted angle chain of arms never equals the returned chain. Therefore, we used a calibration method to eliminate this error.\n\t\n\tTo play the Xylophone, the robot had to be able to adjust its motions according to the estimated relative position of the Xylophone and the heads of the mallets it was holding. To estimate the poses adopted in this paper, we used a color-based technique.\n\t\n\tThe main idea in object tracking is that, based on the RGB color of the center blue bar, given a hypothesis about the Xylophone's position, one can project the contour of the Xylophone's model into the camera image and compare them to an observed contour. In this way, it is possible to estimate the likelihood of the position hypothesis. Using this method, the robot can track the Xylophone with extremely low cost in real-time (see Figure \\ref{color_detection}).\n\t\n\t\\begin{figure}\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\n\t\t\t\t\\includegraphics[width=0.4\\linewidth]{.\/fig\/blue.eps}\\label{single_color_a}\\\\\n\t\t\t\t(a)\\\\\n\t\t\t\n\t\t\t\t\\includegraphics[width=0.4\\linewidth]{.\/fig\/all_color.eps}\\label{single_color_b}\\\\\n\t\t\t\t(b)\\\\\n\t\t\t\n\t\t\t\t\\includegraphics[width=0.75\\linewidth]{.\/fig\/color_detection.eps}\\label{color_detection_c}\\\\\n\t\t\t\t(c)\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Color detection from NAO's bottom camera: (a) single blue color detection (b) full instrument color detection (c) color based edge detection.} \\label{color_detection}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\\subsection{Module 2: Joint Trajectory Generator}\n\tOur system parsed a list of hexadecimal numbers (from 1 to b) to obtain the sequence of notes to play. The system converted the notes into a joint trajectory using the striking configurations obtained from inverse kinematics as control points. The timestamps for the control points are defined by the user to meet the experiment requirement. The trajectory was then computed by the manufacturer-provided API, using Bezier curve \\cite{han2008novel} interpolation in the joint space, and then sent to the robot controller for execution. This process allowed the robot to play in-time with songs. \n\t\n\t\\subsection{Module 3: Real-Time Performance Scoring Feedback}\n\tTwo core features were designed to complete the task in the proposed scoring system: 1) music detection and 2) intelligent scoring feedback. These two functions could provide a real-life interaction experience using a music therapy scenario to teach participants social skills.\n\t\n\t\\subsubsection{Music Detection}\n\tMusic, from a science and technology perspective, is a combination of time and frequency. To make the robot detect a sequence of frequencies, we adopted the Short-time Fourier transform (STFT) for its audio feedback system. Doing so allowed the robot to be able to understand the music played by users and provide proper feedback as a music instructor.\n\t\n\tThe STFT is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then separately compute the Fourier transform for each shorter segment. Doing so reveals the Fourier spectrum on each shorter segment. The changing spectra can then be plotted as a function of time. In the case of discrete-time, data to be transformed can be broken up into chunks of frames that usually overlap each other to reduce artifacts at boundaries. Each chunk is Fourier transformed. The complex results are then added to a matrix that records magnitude and phase for each point in time and frequency (see Figure \\ref{stft}). This can be expressed as:\n\t\\begin{equation}\n\t\t\\resizebox{.6\\hsize}{!}{${\\displaystyle \\mathbf {STFT} \\{x[n]\\}(m,\\omega )\\equiv X(m,\\omega )=\\sum _{n=-\\infty }^{\\infty }x[n]w[n-m]e^{-j\\omega n}}$}\n\t\\end{equation}\n\t\\noindent likewise, with signal x[n] and window w[n]. In this case, m is discrete and $\\omega$ is continuous, but in most applications, the STFT is performed on a computer using the Fast Fourier Transform, so both variables are discrete and quantized. The magnitude squared of the STFT yields the spectrogram representation of the Power Spectral Density of the function:\n\t\\begin{equation}\n\t\t\\resizebox{.4\\hsize}{!}{${\\displaystyle \\operatorname {spectrogram} \\{x(t)\\}(\\tau ,\\omega )\\equiv |X(\\tau ,\\omega )|^{2}}$}\n\t\\end{equation}\n\t\\noindent After the robot detects the notes from user input, a list of hexadecimal numbers are returned. This list is used for two purposes: 1) to compare with the target list for scoring and sending feedback to the user and 2) to create a new input for having robot playback in the game session.\n\t\n\t\\begin{figure}\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/stft.eps}\\label{stft}\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Melody detection with Short Time Fourier Transform} \\label{stft}\n\t\t\\end{center}\n\t\\end{figure}\n\t\\subsubsection{Intelligent Scoring-Feedback System}\n\tTo compare the detected and target notes, we used Levenshtein distance, an algorithm that is typically used in information theory linguistics. This algorithm is a string metric for measuring the difference between two sequences.\n\t\n\tIn our case, the Levenshtein distance between two string-like hex-decimal numbers \n\t${\\displaystyle a,b}$ (of length ${\\displaystyle |a|}$ and ${\\displaystyle |b|}$ respectively) \n\tis given by ${\\displaystyle \\operatorname {lev} _{a,b}(|a|,|b|)}$,\n\t\\begin{equation}\n\t\t\\resizebox{.8\\hsize}{!}{${\\displaystyle \\qquad \\operatorname {lev} _{a,b}(i,j)={\\begin{cases}\\max(i,j)&{\\text{ if }}\\min(i,j)=0,\\\\\\min {\\begin{cases}\\operatorname {lev} _{a,b}(i-1,j)+1\\\\\\operatorname {lev} _{a,b}(i,j-1)+1\\\\\\operatorname {lev} _{a,b}(i-1,j-1)+1_{(a_{i}\\neq b_{j})}\\end{cases}}&{\\text{ otherwise.}}\\end{cases}}}$}\\\\\n\t\\end{equation}\n\t\\noindent where ${\\displaystyle 1_{(a_{i}\\neq b_{j})}}$ is the indicator function equal to 0 when \n\t${\\displaystyle a_{i}=b_{j}}$ and equal to 1 otherwise, and ${\\displaystyle \\operatorname {lev} _{a,b}(i,j)}$ \n\tis the distance between the first ${\\displaystyle i}$ characters of ${\\displaystyle a}$ and the\n\tfirst ${\\displaystyle j}$ characters of ${\\displaystyle b}$.\n\tNote that the first element in the minimum corresponds to deletion (from ${\\displaystyle a}$ to \n\t${\\displaystyle b}$), the second to insertion and the third to match or mismatch, depending on \n\twhether the respective symbols are the same. \n\t\n\tBased on the real-life situation, we defined a likelihood margin for determining whether the result\n\tis good or bad:\n\t\\begin{equation}\n\t\t\\resizebox{.45\\hsize}{!}{${likelihood = \\dfrac{len(target) - lev_{target,source}}{len(target)}}$}\n\t\\end{equation}\n\t\\noindent where, if the likelihood is more than 66\\% (not including single note practice since, in that case, it is only correct or incorrect), the system will consider it to be a good result. This result is then passed to the accuracy calculation system to have the robot decide whether it needs to add additional practice trials (e.g., 6 correct out of a total of 10 trials).\n\t\n\t\\section{Social Behavior Results}\n\t\\subsection{Motor Control}\n\tNine ASD and seven TD participants completed the study over the course of eight months. All ASD participants completed all six sessions (baseline, intervention, and exit) while all the TD subjects completed the required baseline and exit sessions. Conducting a Wizard of Oz experiment, a well-trained researcher was involved in the baseline and exit sessions in order for there to be high-quality observations and performance evaluations. With well-designed, fully automated intervention sessions, NAO was able to initiate music-based therapy activities with participants.\n\t\n\tSince the music detection method was sensitive to the audio input, a clear and long-lasting sound from the Xylophone was required. As seen in Figure \\ref{fig9}, it is evident that most of the children were able to strike the Xylophone properly after one or two sessions (the average accuracy is improved over sessions). Notice that participants 101 and 102 had significant improvement during intervention sessions. Some of the participants started at a higher accuracy rate and sustained a rate above 80\\% for the duration of the study. Even with some oscillation, participants with this type of accuracy rate were considered to have consistent motor control performance. \n\t\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/figure9.eps}\\label{fig9}\\\\\n\t\t\t\\end{center}\n\t\t\t\\caption{Motor control accuracy result.} \\label{fig9}\n\t\t\\end{center}\n\t\\end{figure}\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/figure10.eps}\\label{fig10}\\\\\n\t\t\t\\end{center}\n\t\t\t\\caption{Main music therapy performance accuracy.} \\label{fig10}\n\t\t\\end{center}\n\t\\end{figure}\n\tFigure \\ref{fig10} shows the accuracy of the first music therapy activity that was part of the intervention sessions across all participants. As described in the previous section, the difficulty level of this activity was designed to increase across sessions. With this reasoning, the accuracy of participant performance was expected to decrease or remain consistent. This activity required participants to be able to concentrate, use joint attention skills during the robot therapy stage, and respond properly afterwards.  As seen in Figure \\ref{fig10}, most of the participants were able to complete single\/multiple notes practice with an average of a 77.36\\%\/69.38\\% accuracy rate although, even with color hints, the pitch difference of notes was still a primary challenge. Due to the difficulty of sessions 4 and 5, a worse performance, when compared to the previous two sessions, was acceptable. However, more than half of the participants consistently showed a high accuracy performance or an improved performance than previous sessions. \n\t\n\t\\subsection{Turn-Taking Behavior}\n    The American Psychological Association \\cite{vandenbos2007american} defines turn-taking as, \"in social interactions, alternating behavior between two or more individuals, such as the exchange of speaking turns between people in conversation or the back-and-forth grooming behavior that occurs among some nonhuman animals. Basic turn-taking skills are essential for effective communication and good interpersonal relations, and their development may be a focus of clinical intervention for children with certain disorders (e.g., Autism).\" Learning how to play one's favorite song can be a motivation that helps ASD participants understand and learn turn-taking skills. In our evaluation system, the annotators scored turn-taking based on how well kids were able to follow directions and display appropriate behavior during music-play interaction. In other words, participants needed to listen to the robot's instructions and then play, and vice versa. Any actions not following the interaction routine were considered as less or incomplete turn-taking. Measurements are described as follows: measuring turn-taking behavior can be subjective, so to quantify this, a grading system was designed. Four different behaviors were defined in the grading system: (a) \"well-done\", this level is considered good behavior where the participant should be able to finish listening to the instructions from NAO, start playback after receiving the command, and wait for the result without interrupting (3 points for each \"well-done\"); (b) \"light-interrupt\", in this level, the participant may exhibit slight impatience, such as not waiting for the proper moment to play or not paying attention to the result (2 points given in this level); (c) \"heavy-interrupt\", more interruptions may accrue in this level and the participant may interrupt the conversation, but is still willing to play back to the robot (1 point for this level); and (d) \"indifferent\", participant shows little interest in music activities including, but not limited to, not following behaviors, not being willing to play, not listening to the robot or playing irrelevant music or notes (score of 0 points). The higher the score, the better turn-taking behavior the participant demonstrated. All scores were normalized into percentages due to the difference of total \"conversation\" numbers. Figure \\ref{fig11} shows the total results among all participants. Combining the report from video annotators, 6 out of 9 participants exhibited stable, positive behavior when playing music, especially after the first few sessions. Improved learn-and-play turn-taking rotation was demonstrated over time. Three participants demonstrated a significant increase in performance, suggesting turn-taking skills were taught in this activity. Note that two participants (103 \\& 107) had a difficult time playing the Xylophone and following turn-taking cues given by the robot.\n\t\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/figure11.eps}\\label{fig11}\\\\\n\t\t\t\\end{center}\n\t\t\t\\caption{Normalized turn-taking behavior result for all subjects during intervention sessions.\n\t\t\t} \\label{fig11}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\\section{Music Emotion Classification Results}\n\t\\ Since we developed our emotion classification method based on the time-frequency analysis of EDA signals, the main properties of The Continuous Wavelet Transforms (CWT), assuming that it is a C-Morlet wavelet, is presented here. Then, the pre-processing steps, as well as the wavelet-based feature extraction scheme, are discussed. Finally, we briefly review the characteristics of the SVM, the classifier used in our approach.\n\t\n\tEDA signals were collected in this study. By using the annotation and analysis method from our previous study \\cite{feng2018wavelet}, we were able to produce a music-event-based emotion classification result that is presented below. To determine the emotions of the ASD group, multiple comparisons were made after annotating the videos. Note that on average, 21\\% of emotions were not clear during the annotation stage. It is necessary to have them as unclear rather than label them with specific categories.\n\t\n\tDuring the first part of the annotation, it was not obvious to conclude the facial expression changes from different activities, however, in S1 participants showed more \"calm\" for most of the time as the dominant emotion due to the simplicity of completing the task. In S2, the annotator could not have a precise conclusion by reading the facial expression from participants as well. Most of the participants intended to play music and complete the task, however, feelings of \"frustration\" were often invoked in this group. A very similar feeling can be found in S3 when compared to S1. In S3, all the activities were designed to create a role changing environment for users in order to stimulate a different emotion. Most of the children showed \"happy\" emotion during the music game section.\n\t We noticed that different activities can result in a change in emotional arousal. As mentioned above, the warm-up section and single activity practice section use the same activity with different intensities levels. The gameplay section has the lowest difficulty and is purposely designed to be more relaxing.\n\t\\begin{table*}\n\t\t\\label{tab1}\n\t\t\\begin{center}\n\t\t\t\\caption{Emotion change in different events using wavelet-based feature extraction under SVM classifier. }\n\t\t\t\\vspace{3mm}\n\t\t\t\\begin{tabular}{llllll}\n\t\t\t\t& Kernels                     & Accuracy(\\%) & AUC & Precision(\\%) & Recall(\\%) \\\\\n\t\t\t\t\\hline\n\t\t\t\tS1 vs S2       & \\multirow{4}{*}{\\textbf{Linear}}     & 75       & .78  & 76        & 72     \\\\\n\t\t\t\tS1 vs S3       &                             & 57       & .59  & 56        & 69     \\\\\n\t\t\t\tS2 vs S3       &                             & 69       & .72  & 64        & 86     \\\\\n\t\t\t\tS1 vs S2 vs S3 &                             & \\multicolumn{4}{l}{52}              \\\\\n\t\t\t\t\\hline\n\t\t\t\tS1 vs S2       & \\multirow{4}{*}{\\textbf{Polynomial}} & 66       & .70  & 70        & 54     \\\\\n\t\t\t\tS1 vs S3       &                             & 64       & .66  & 62        & 68     \\\\\n\t\t\t\tS2 vs S3       &                             & 65       & .68  & 62        & 79     \\\\\n\t\t\t\tS1 vs S2 vs S3 &                             & \\multicolumn{4}{l}{50}              \\\\\n\t\t\t\t\\hline\n\t\t\t\tS1 vs S2       & \\multirow{4}{*}{\\textbf{RBF}}        & 76       & .81  & 76        & 75     \\\\\n\t\t\t\tS1 vs S3       &                             & 57       & .62  & 57        & 69     \\\\\n\t\t\t\tS2 vs S3       &                             & 70       & .76  & 66        & 83     \\\\\n\t\t\t\tS1 vs S2 vs S3 &                             & \\multicolumn{4}{l}{53}              \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\t\\label{tab1}\n\t\t\\end{center}\n\t\\end{table*}\n\t\n\n\t\\begin{table*}[tbp]\n\t\t\\label{tab2}\n\t\t\\begin{center}\n\t\t\t\\caption{Emotion change classification performance in single event with segmentation using both SVM and KNN classifier. }\n\t\t\t\\vspace{3mm}\n\t\t\t\\resizebox{\\columnwidth}{!}{\n\t\t\t\t\\begin{tabular}{ccccccccc}\n\t\t\t\t\t\\multicolumn{1}{l}{\\multirow{3}{*}{}} & \\multicolumn{5}{c}{Segmentation Comparison in Single Task}                                                                                                       \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multicolumn{1}{l}{}                  & \\multicolumn{4}{c}{Warm up Section}                                                   & \\multicolumn{4}{c}{Song Practice Section}                                                  \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multicolumn{1}{l}{}                  & Kernels                     & Accuracy (\\%) & K value                & Accuracy(\\%) & Kernels                     & Accuracy(\\%) & K value                & Accuracy(\\%) \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tlearn vs play                                   & \\multirow{4}{*}{Linar}      & 52.62    & \\multirow{4}{*}{K = 1} & 54       & \\multirow{4}{*}{Linar}      & 53.79    & \\multirow{4}{*}{K = 1} & 52.41    \\\\\n\t\t\t\t\tlearn vs feedback                                   &                             & 53.38    &                        & 50.13    &                             & 53.1     &                        & 51.72    \\\\\n\t\t\t\t\tplay vs feedback                                   &                             & 47.5     &                        & 50.38    &                             & 54.31    &                        & 50.86    \\\\\n\t\t\t\t\tlearn vs play vs feedback                                 &                             & 35.08    &                        & 36.25    &                             & 35.52    &                        & 36.55    \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tlearn vs play                                   & \\multirow{4}{*}{Polynomial} & 49       & \\multirow{4}{*}{K = 3} & 50.25    & \\multirow{4}{*}{Polynomial} & 53.79    & \\multirow{4}{*}{K = 3} & 50.69    \\\\\n\t\t\t\t\tlearn vs feedback                                 &                             & 50.75    &                        & 50.13    &                             & 50.86    &                        & 50.34    \\\\\n\t\t\t\t\tplay vs feedback                                   &                             & 49.87    &                        & 49.5     &                             & 49.14    &                        & 52.07    \\\\\n\t\t\t\t\tlearn vs play vs feedback                                 &                             & 33.92    &                        & 35.83    &                             & 34.71    &                        & 35.29    \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tlearn vs play                                   & \\multirow{4}{*}{RBF}        & 54.38    & \\multirow{4}{*}{K = 5} & 48.37    & \\multirow{4}{*}{RBF}        & 50.86    & \\multirow{4}{*}{K = 5} & 50.17    \\\\\n\t\t\t\t\tlearn vs feedback                                   &                             & 55.75    &                        & 52.75    &                             & 53.97    &                        & 50.17    \\\\\n\t\t\t\t\tplay vs feedback                                   &                             & 51.12    &                        & 50       &                             & 53.79    &                        & 52.93    \\\\\n\t\t\t\t\tlearn vs play vs feedback                                 &                             & 36.83    &                        & 34.17    &                             & 34.83    &                        & 33.1    \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\t\n\t\t\t}\n\t\t\t\\label{tab2}\n\t\t\\end{center}\n\t\n\t\\end{table*}\n\t\\begin{table*}[]\n\t\t\n\t\t\\begin{center}\n\t\t\t\\caption{Classification rate in children learn, children play and robot feedback across\n\t\t\t\twarm up (S1) and music practice (S2) sessions.}\n\t\t\t\\vspace{3mm}\n\t\t\t\\label{tab3}\n\t\t\t\\begin{tabular}{lcccccc}\n\t\t\t\t\\multicolumn{1}{c}{\\multirow{2}{*}{}} & \\multicolumn{3}{c}{Accuracy of SVM} & \\multicolumn{3}{c}{Accuracy of KNN} \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{1}{c}{}                  & Linear   & Polynomial    & RBF    & K = 1   & K = 3     & K = 5     \\\\\n\t\t\t\t\\hline\n\t\t\t\tlearn 1 vs learn 2                                 & 73.45    & 69.31   & 80.86  & 73.28   & 71.03   & 65      \\\\\n\t\t\t\t\\hline\n\t\t\t\tplay 1 vs play 2                                 & 75.34    & 68.79   & 80     & 74.48   & 69.14   & 64.31   \\\\\n\t\t\t\t\\hline\n\t\t\t\tfeedback 1 vs feedback 2                                 & 76.38    & 69.48   & 80.34  & 74.14   & 69.14   & 66.9   \n\t\t\t\\end{tabular}\n\t\t\t\n\t\t\t\\label{tab3}\n\t\t\\end{center}\n\t\\end{table*}\n\t\n\t\\begin{table}[]\n\t\t\\label{tab4}\n\t\t\n\t\t\\begin{center}\n\t\t\t\\caption{TD vs ASD Emotion Changes from Baseline and Exit Sessions.}\n\t\t\t\\vspace{3mm}\n\t\t\t\\begin{tabular}{llllll}\n\t\t\t\t& \\textbf{Linear} & \\textbf{Polynomial} & \\textbf{RBF} \\\\\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\t\\textbf{Accuracy}                          & 75              & 62.5                & 80           \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multirow{2}{*}{\\textbf{Confusion Matrix}} & 63  37          & 50  50              & 81  19       \\\\\n\t\t\t\t& 12  88          & 25  75              & 25  75       \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\t\\label{tab4}\n\t\t\\end{center}\n\t\\end{table}\n\t\n\tIn the first part of the analysis, EDA signals were segmented into small event-based pieces according to the number of \"conversations\" in each section. One \"conversation\" was defined by three movements: 1) robot\/participant demonstrates the note(s) to play; 2) robot\/participant repeats the note(s); and 3) robot\/participant presents the result. Each segmentation lasts about 45 seconds. The CWT of the data, assuming the use of the C-Morlet wavelet function, was used inside a frequency range of (0.5, 50)Hz. A Support Vector Machines (SVM) classifier was then employed to classify \"conversation\" segmentation among three sections using the wavelet-based features. Table \\ref{tab1}shows the classification accuracy for the SVM classifier with different kernel functions. As seen in this table, emotion arousal change between warmup (S1) and music practice (S2) and S2 and music game (S3) sections can be classified using a wavelet-based feature extraction SVM classifier with an average accuracy of 76\\% and 70\\%, respectively. With the highest percentage of accuracy for S1 and S3 being 64\\%, fewer emotion changes between the S1 and S3 sections may be indicated. \n\t\n\tIn the second part of the analysis, EDA signals were segmented into small event-based pieces according to the number of \"conversations\" in each section as mentioned previously. In order to discover the emotion fluctuation inside one task, each \"conversation\" section was carefully divided into 3 segments, as described before. Again, each segmentation lasts about 45 seconds.\n\t Each segment lasts about 10 - 20 seconds. Table \\ref{tab2} shows the full result of emotion fluctuation in the warm-up (S1) and music practice (S2) sections from the intervention session. Notice that all of the segments cannot be appropriately classified using the existing method. Both SVM and KNN show stable results. This may suggest that the ASD group has less emotion fluctuation or arousal change once the task starts, despite varying activities in it. Stable emotion arousal in a single task could also benefit from the proper activity content, including robot agents playing music and language used during the conversation. Friendly voice feedback was based on the performance delivered to participants who were well prepared and stored in memory. Favorable feedback occurred while receiving correct input and in the case of incorrect playing, participants were given encouragement. Since emotion fluctuation can affect learning progress, less arousal change indicates the design of intervention sessions that are robust. \n\t\n\tCross-section comparison is also presented below. Since each \"conversation\" contains 3 segments, it is necessary to have specific segments from one task to compare with the other task it corresponded to. Table \\ref{tab3} shows the classification rate in children learn, children play, and robot feedback across warm-up (S1) and music practice (S2) sessions. By using RBF kernel, wavelet-based SVM classification rate had an accuracy of ~80\\% for all 3 comparisons. This result also matches the result from Table \\ref{tab1}. \n\t\n\tThe types of activities and procedures between the baseline and exit session for both groups were the same. Using the \"conversation\" concept above, each were segmented. Comparing with target and control groups using the same classifier, an accuracy rate of 80\\% for detecting different groups was found (see Table \\ref{tab3}). Video annotators also reported \"unclear\" in reading facial expressions from the ASD group. Taken together, these results suggest that even with the same activities, TD and ASD groups display different bio-reactions. It has also been reported that a significant improvement of music performance was shown in the ASD group (see Table \\ref{tab4}), although both groups had similar performance at their baseline sessions. Furthermore, the TD group was found to be more motivated to improve and ultimately perfect their performance, even when they made mistakes.\n\t\n\tWhen comparing emotion patterns from baseline and exit sessions between TD and ASD groups in Table \\ref{tab3}, differences can be found. This may suggest that we have discovered a potential way of using biosignals to help diagnose autism at an early age. According to annotators and observers, TD participants showed a strong passion for this research. Excitement, stress, and disappointment were easy to recognize and label when watching the recorded videos. On the other hand, limited facial expression changes were detected in the ASD group. This makes it challenging to determine whether the ASD participants had different feelings or had the same feelings but different biosignal activities compared to the TD group.\n\t\n\t\\section{Discussion, Conclusion and Future Work}\n\t\\ As shown by others \\cite{lagasse2019assessing,lim2011effects} as well as this study, playing music to children with ASD in therapy sessions has a positive impact on improving their social communication skills. Compared to the state of the art \\cite{dvir2020body,bhat2013review, corbett2008brief}, where they utilized both recorded and live music in interventional sessions for single and multiple participants, our proposed robot-based music-therapy platform is a promising intervention tool in improving social behaviors, such as motor control and turn-taking skills. As our human robot interaction studies (over 200 sessions were conducted with children), most of the participants were able to complete motor control tasks with ~70\\% accuracy and 6 out of 9 participants demonstrated stable turn-taking behavior when playing music. The emotion classification SVM classifier presented illustrated that emotion arousal from the ASD group can be detected and well recognized via EDA signals. \n\t\n\tThe automated music detection system created a self-adjusting environment for participants in early sessions. Most of the ASD participants began to develop the strike movement in the initial two intervention sessions; some even mastered the motor ability throughout the very first warm-up event. The robot was able to provide verbal directions and demonstrations to participants by providing voice command input as applicable, however, the majority of the participants did not request this feedback, and instead just focused on playing with NAO. This finding suggests that the young ASD population can learn fine motor control ability from specific, well-designed activities.\n\t\n\tThe purpose of using a music therapy scenario as the main activity in the current research was to create an opportunity to practice a natural turn-taking behavior during social interaction. Observing all experimental sessions, six out of nine participants exhibited proper turn-taking after one or two sessions, suggesting the practice helped improve this behavior. Specifically, participant 107 significantly improved in the last few sessions when comparing results of the baseline and final session. Participant 109 had trouble focusing on listening to the robot most of the time, however, with prompting from the researcher, they performed better at the music turn-taking activity for a short period of time. For practicing turn-taking skills, fun, motivating activities, such as the described music therapy sessions incorporating individual song preference, should be designed for children with autism.\n\t\n\tDuring the latter half of the sessions, participants started to recognize their favorite songs. Even though the difficulty for playing proper notes was much higher, over half of the participants became more focused in the activities. Upon observations, it became clear that older participants spent more time interacting with the activities during the song practice session compared to younger participants, especially during the half\/whole song play sessions. This could be for several reasons. First, the more complex the music, the more challenging it is, and the more concentration participants need to be successful. Older individuals may be more willing to accept the challenge and are better able to enjoy the sense of accomplishment they receive from their verbal feedback at the end of each session. Prior music knowledge could also be another reason for this result as older participants may have had more opportunities to learn music at school. \n\t\n\tThe game section of each session reflected the highest interest level, not only because it was relaxing and fun to play, but also because it was an opportunity for the participants to challenge the robot to mirror their free play. This exciting phenomenon could be seen as a game of \"revenge.\" Participant 106 exhibited this behavior by spending a significant amount of time in free play game mode. According to the session executioner and video annotators, this participant, the only girl, showed very high level of involvement for all the activities, including free play. Based on the conversation and music performance with the robot, participant 106 showed a strong interest in challenging the robot in a friendly way. High levels of engagement further supports the idea that the proposed robot-based music-therapy platform is a viable interventional tool. Additionally, these findings also highlight the need for fun and motivating activities, such as music games, to be incorporated into interventions designed for children with autism.\n\t\n\tConducting emotion studies with children with autism can be difficult and bio-signals provide a possible way to work around the unique challenges of this population. The event-based emotion classification method adopted in this research suggests that not only are physiological signals a good tool to detect emotion in ASD populations, but it is also possible to recognize and categorize emotions using this technique. The same activity with different intensities can cause emotion change in the arousal dimension, although, for the ASD group, it is difficult to label emotions based on facial expression changes in the video annotation phase. Fewer emotion fluctuations in a particular activity, as seen in Table \\ref{tab2}, suggests that a mild, friendly, game-like therapy system may encourage better social content learning for children with autism, even when there are repetitive movements. These well-designed activities could provide a relaxed learning environment that helps participants focus on learning music content with proper communication behaviors. This may explain the improvement in music play performance during the song practice (S2) section of intervention sessions, as seen in Figure \\ref{fig10}.\n\t\n\tThere were several limitations of this study that should be considered. As is the case with most research studies, a larger sample size is needed to better understand the impact of this therapy on children with ASD. The ASD group in this paper also only included one female. Future work should include more females to better portray and understand an already underrepresented portion of this population of interest. In addition, improving skills as complex as turn-taking for example, likely requires more sessions over a longer period of time to better enhance the treatment and resulting behavioral changes. Music practice can also be tedious if it is the only activity in an existing interaction system and may have influenced the degree of interaction in which participants engaged with the robot. Future research could include more activities or multiple instruments, as opposed to just one, to further diversify sessions. The choice of instrument, and its resulting limitations, may also have an impact. The Xylophone, for example, is somewhat static and future research could modify or incorporate a different instrument that can produce more melodies, accommodate more complex songs, and portray a wider range of musical emotion and expression. Finally, another limitation of this study was the type of communication the robot and participant engaged in. Verbal communication was not rich between participants and NAO, and instead, most of the time, participants could follow the instructions from the robot without asking it for help. While this was likely not an issue for the specific non-verbal behavioral goals of this intervention, future work should expand upon this study by incorporating other elements of behavior that often have unique deficits in this population, including speech.\n\t\n\tIn summary, this paper presented a novel robot-based music-therapy platform to model and improve social behaviors in children with ASD. In addition to the novel platform introduced, this study also incorporated emotion recognition and classification utilizing EDA physiological signals of arousal. Results of this study are consistent with findings in the literature for TD and ASD children and suggest that the proposed platform is a viable tool to facilitate the improvement of fine motor and turn-taking skills in children with ASD.\n\t\n\t\\section*{Acknowledgment}\n\t\\ This research was partially supported by gifts for research on autism to the University of Denver from several family members. The researchers would like to thank the children and their family members for their dedication and willingness to volunteer their time and make this research possible.\n\t\n\t\\bibliographystyle{frontiersinSCNS_ENG_HUMS}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nLet $n \\geq 1$ and let $P_n$ be a directed path on $[n] = \\{1,2,\\ldots,n\\}$ with directed edges from $i$ to $i-1$ for $i=2,3,\\ldots,n.$ Let $1 \\leq m \\leq n$ and assume that $m$ drivers arrive at vertex $n$ one by one, with the $i$th driver willing to park in vertex $X_i \\in [n].$ If the $i$th driver finds $X_i$ empty, they park there. If not, they continue their drive towards $1,$ parking in the first available parking space. If no such spot can be found, the driver leaves the path without parking. We say that $(x_1, \\ldots, x_m),$ with $x_1, \\ldots, x_m \\in [n],$ is a \\em parking function \\em for $P_n$ if for $X_i = x_i$ for $1 \\leq i \\leq m,$ all $m$ drivers park on the path.\n\nParking functions were first studied in the 1960s by Konheim and Weiss \\cite{KonheimWeiss-parkingPath}. They evaluated the number of parking functions, which is equivalent to evaluating the probability that an $m$-tuple of independent random variables uniformly distributed on $[n]$ gives a parking function. A similar model, with $P_n$ replaced by a uniform random rooted Cayley tree on $[n]$ was studied by Lackner and Panholzer \\cite{LacknerPanholzer}. Motivated by finding a probabilistic explanation for some phenomenons observed in \\cite{LacknerPanholzer}, Goldschmidt and Przykucki \\cite{GoldschmidtPrzykucki} analyzed the parking processes on critical Galton-Watson trees, as well as on trees with Poisson(1) offspring distribution conditioned on non-extinction, in both cases with the edges directed towards the root. Note that in all the setups above, drivers have only one choice of route at any time of the process.\n\nIn this paper, we are concerned with a related model, introduced by Damron, Gravner, Junge, Lyu, and Sivakoff \\cite{DGJLS}, in which the cars move at random. Let $L=(V,E)$ be a Cayley graph on a group $V$ with generating set $R$, and let  $\\mu$ be a probability distribution on $R$: we will refer to such a triple $(L,R,\\mu)$ as a {\\em car park triple}. At time $0$, each position $v \\in V$ is independently assigned a car with probability $p$ \nor a parking space with probability $1-p$. The cars follow independent random walks with increments  $\\mu$ \nand each car continues to follow the random walk until it finds a free space where it parks (if more than one car arrives at a free space at the same time, then one is chosen to park according to some rule).\\footnote{We note that Damron, Gravner, Junge, Lyu, and Sivakoff \\cite{DGJLS} work in a slightly more general setting, see Section 2 in \\cite{DGJLS}. While our results in Section \\ref{sec:definitions} also hold in the setting in \\cite{DGJLS}, we believe that the class of Cayley graphs is a fairly general setting and the link with lattices is a little clearer.}\n\nWe are interested in the distribution of journey lengths of cars. We introduce the stopping time $\\tau^v$ where $\\tau^v = 0$ if position $v$ is a parking space, and otherwise $\\tau^v$ is the time the car starting at $v$ takes to park ($\\tau^v= \\infty$ if the car never parks). We also write $\\tau = \\tau^0$ (by symmetry we only need to consider $v=0$). Given $t \\ge 0$ and a vertex $v,$ let\n\\[\nV^v(t)= \\left|\\left\\{(u,s) \\in V \\times [t] : \\mbox{car $u$ visits $v$ at time $s$}\\right\\}\\right| + \\mathds{1}_{\\{v \\mbox{ \\small is a car}\\}}\n\\]\nbe the number of cars that visit $v$ up to time $t.$\n\nIn the particular case of the lattice ${\\mathbb Z}^d$ (with edges joining lattice points at Euclidean distance 1), Damron, Gravner, Junge, Lyu, and Sivakoff \\cite{DGJLS} prove the following theorem.\n\\begin{thm}\\label{thm:DGJLS}\nConsider the parking process on $\\mathbb{Z}^d$ with simple symmetric random walks.\n\\begin{enumerate}\n \\item If $p \\geq 1\/2$ then  $\\mathbb{E}[\\tau] = \\infty$ with $\\mathbb{E}[\\min\\{\\tau,t\\}] = (2p-1)t + o(t).$\n \\item If $p < 1\/2$ then $\\tau$ is almost surely finite. Moreover, if $p < (256d^6e^2)^{-1}$ then $\\mathbb{E}[\\tau] < \\infty.$\n\\end{enumerate}\n\\end{thm}\n\nFor $p > 1\/2,$ Theorem \\ref{thm:DGJLS} gives good asymptotics for $\\mathbb{E}[\\min\\{\\tau,t\\}].$ However, \nfor $p = 1\/2$ Theorem \\ref{thm:DGJLS} only tells us that $\\mathbb{E}[\\min\\{\\tau,t\\}]$ is $o(t),$ while the authors of \\cite{DGJLS} conjecture that for $d=1$ and $p=1\/2$ we have $\\mathbb{E}[\\min\\{\\tau,t\\}] = \\Theta(t^{3\/4})$ \\cite{Junge-seminar}. \nMoreover, \nfor $d=1$,  Theorem \\ref{thm:DGJLS} only gives $\\mathbb{E}[\\tau] < \\infty$ for $p < 0.000528,$ while it is conjectured that this holds for all $p < 1\/2.$\n\nHere, we address both conjectures when $d=1,$ and prove the following two theorems.\n\n\\begin{thm}\\label{thm:p=1\/2}\nFor the parking problem on $\\mathbb{Z},$ when $p=1\/2,$ there exist constants $C, c > 0$ such that\n\\[\n c t^{3\/4} (\\log t)^{-1\/4} \\leq \\mathbb{E}[\\min\\{\\tau,t\\}] \\leq C t^{3\/4}.\n\\]\n\\end{thm}\n\n\\begin{thm}\\label{thm:p<1\/2}\nFor the parking problem on $\\mathbb{Z},$ when $p<1\/2$ we have $\\mathbb{E}[\\tau] < \\infty.$\n\\end{thm}\n\nFor the parking process on $\\mathbb{Z}$ with $p=1\/2,$ Theorem \\ref{thm:p=1\/2} gives good bounds on the asymptotic growth of $\\mathbb{E}[\\min\\{\\tau,t\\}]$ by showing that it indeed equals $t^{3\/4}$ up to a fractional power of $\\log t.$ For $p < 1\/2,$ Theorem \\ref{thm:p<1\/2} confirms that the expected journey length of a car is finite as predicted. Damron, Gravner, Junge, Lyu and Sivakoff \\cite{DGJLS} also ask whether (for a large family of parking processes)  there is a critical exponent $\\gamma >0$ such that, for some constant $C > 0$, $\\mathbb{E}[\\tau] \\sim C\\left(1\/2 -p\\right)^{-\\gamma}$ as $p$ increases to $1\/2.$ For the parking problem on $\\mathbb{Z}$, we have a partial result in this direction (though we have no conjecture as to the value, or even the existence of a critical exponent).\n\n\\begin{thm}\\label{thm:critexp}\nFor the parking problem on $\\mathbb{Z},$ $\\mathbb{E}[\\tau] = O\\left((1\/2 - p)^{-6}\\right)$ as $p\\nearrow1\/2.$\n\\end{thm}\n\nIn this paper, we will consider strategies that modify the car-parking process. \nWe will introduce two types of strategy: parking strategies where we allow cars to choose whether or not to park in an available space, and car removal strategies where we remove cars from the parking process (we defer formal definitions to Section \\ref{sec:definitions}).\n For a  car park triple $(L,R,\\mu)$ and a strategy $S$, we will write $V^v_S(t)$ for the value of $V^v(t)$ when strategy $S$ is followed, and similarly $\\tau^v_S$; we write $G$ for the greedy strategy (i.e.~the original process).\n\n\nThe key properties of parking and car removal strategies that we shall use are given in the following theorems, which show that no parking strategy is quicker than the greedy one, and that adding car removal makes parking easier. \nWe note that these results hold in the more general setting of Cayley graphs.\n\n\\begin{thm}\\label{thm:supermarket}\nLet $S$ be a parking strategy on the car park triple $(L,R,\\mu)$. Then for all $t \\ge 0$ and vertices $v$, the distribution of $V^v_G(t)$ majorises $V^v_S(t)$. That is for all $t, k \\ge 0$,\n\\[\n\\mathbb{P}\\left[V^v_G(t) \\le k\\right] \\ge \\mathbb{P}\\left[V^v_S(t) \\le k\\right].\n\\]\n\\end{thm}\n\n\\begin{thm}\n\\label{thm:pinkfloyd}\nLet $Q$ be a car removal strategy on the car park triple $(L,R,\\mu)$. Then $\\tau^v_Q \\le \\tau^v_G$, and for all $t \\ge 0$ we have $V^v_Q(t) \\le V^v_G(t)$.\n\\end{thm}\n\nIn order to prove Theorem \\ref{thm:supermarket} we introduce a new model, in which the cars follow directions stored at the vertices they visit, rather than their own individual random walks.  We will refer to this as the {\\em space-based model}, in contrast to the {\\em car-based model} described above.  Even though the stochastic properties of the two models are equivalent, the new model allows us to control the quantity $V^v(t)$ better, and we are then able to easily deduce the desired result for the original parking problem.\n\nThe paper is organised as follows. In Section \\ref{sec:definitions} we define the parking processes, introduce the notions of parking strategies and car removal, and prove Theorems \\ref{thm:supermarket} and \\ref{thm:pinkfloyd}. This allows us to consider both more and less restrictive parking problems, which we use in our arguments. In Section \\ref{sec:probabilities} we recall some known probability bounds that are used in this paper. In Section \\ref{sec:p=1\/2upper} we prove the upper bound on $\\mathbb{E}[\\min\\{\\tau,t\\}]$ in Theorem \\ref{thm:p=1\/2}, and in Section \\ref{sec:p=1\/2lower} we prove the lower bound. In Section \\ref{sec:p<1\/2} we prove Theorems \\ref{thm:p<1\/2}. and \\ref{thm:critexp}. Finally in Section \\ref{yaymorewaffle} we conclude the paper with some related problems and open questions.\n\nThroughout this paper, we use the notation $a \\wedge b = \\min\\{a,b\\}.$ For a normally distributed random variable $Z$ with mean $0$ and variance $1,$ we write $\\Phi(x) = \\mathbb{P}\\left[Z \\le x\\right].$\n\n\\section{Model specifics, parking strategies, and car removal}\n\\label{sec:definitions}\n\nWe will want to consider slight modifications of the original parking problem on $\\mathbb{Z}.$ In this section, we introduce new notation for these modifications and also compare these modifications to the original problem. The first modification is the addition of \\emph{parking strategies}. The second is the addition of \\emph{car removal} to the process. We compare the expected journey length of a car by time $t,$ showing that non-trivial parking strategies increase expected journey times while car removal decreases them. In fact, we are able to show that these bounds hold for any car park triple.\n\n\\subsection{The car-based parking model}\n\nLet us recall some definitions.\nLet $H$ be a group and $R$ be a generating set for $H$. The \\em Cayley graph \\em of $H$ with respect to $R$ is the edge-coloured directed graph $L = (H,E)$ where $$E := \\{(h,hr) : h \\in H, r \\in R\\},$$ and the edge $(h,hr)$ is coloured $r$.\nNote that if $R$ is closed under taking inverses then $(x,y) \\in E$ if and only if $(y,x) \\in E,$ and so we can just consider the underlying graph. For example, the $d$-dimensional integer-lattice $\\mathbb{Z}^d$ can be thought of as the abelian group with generating set $\\{e_1,-e_1,\\ldots,e_d,-e_d\\} \\subset \\mathbb{Z}^d$ where the $i$-th co-ordinate of $e_i$ is $1$ and all others are $0$.\n\n\n\nA {\\em car park triple} is an ordered triple $(L,R,\\mu)$ , where\n$L=(V,E)$ is a Cayley graph on a group $V$ with generating set $R$ and $\\mu$ is a probability distribution on $R$ \n (In later sections we will be interested in the parking problem on $\\mathbb{Z}$, namely the triple $(\\mathbb{Z},\\{-1,+1\\},\\mu^{\\mathbb{Z}})$ where $\\mu^{\\mathbb{Z}}(-1) = \\mu^{\\mathbb{Z}}(+1) = 1\/2$.  However, the results in this section hold in the more geenral model.)\n\nWe define the parking problem on the car park triple $(L,R,\\mu)$ as follows.\n\n\\begin{defn}\\label{defn:free parking}\nIndependently for each vertex $v \\in V$, let: \n\\begin{itemize}\n\\item $X^v = (X^v_0,X^v_1,\\ldots)$ be a Markov chain on $L$ with $X^v_0=v$, and transition matrix $(p_{u,w})$ where $p_{u,ur} = \\mu(r)$ for each $u\\in V$ and $r\\in R$, and $p_{u,w} = 0$ otherwise.\n\n\\item $(U^v_s)_{s \\in \\mathbb{N}}$ be a sequence of independent $\\mathrm{Unif}([0,1])$ random variables. \n\n\\item $B^v$ be a $\\mbox{Bernoulli}(p)$ random variable. We initially place a car at $v$ when $B^v = 1$ and otherwise a parking space with the capacity for one car.\n\\end{itemize}\n\nA car starting at vertex $v$ moves according to the Markov chain $X^v$ until it finds a free parking space and parks there. (We do not use the random walks $X^v$ for those $v$ where we initially place a parking space; we define them just for the simplicity of the model.) If cars $v_1,\\ldots,v_k$ all arrive at the same free parking space at time $s,$ we park car $v_j$ with smallest $U^{v_j}_s.$\n\\end{defn}\n\nWe shall sometimes refer to the model in Definition \\ref{defn:free parking} as the \\emph{car-based parking model}.\n\nLet $(\\Omega,\\mathcal{F},\\mathbb{P})$ be a probability space. A \\emph{filtration} is a sequence $\\mathcal{F}_0 \\subseteq \\mathcal{F}_{1} \\subseteq  \\ldots$ of $\\sigma$-algebras. A random variable $\\tau : \\omega \\rightarrow \\mathbb{N}$ is a \\emph{stopping time} with respect to a filtration $(\\mathcal{F}_{t})^{\\infty}_{t=0}$ if $\\tau^{-1}(\\{t\\}) \\in \\mathcal{F}_{t}$ for each $t \\in \\mathbb{N}.$  In the car-based model, for the parking problem on the car park triple $(L,R,\\mu)$, the probability space $(\\Omega,\\mathcal{F},(\\mathcal{F}_t)_{t\\ge0},\\mathbb{P})$ is as expected, with a filtration $\\left(\\mathcal{F}_t\\right)_{t \\ge 0}$ defined by\n\\[\n\\mathcal{F}_t = \\sigma((B^v)_{v \\in V}, (X^v_s)_{v \\in V,0 \\leq s \\leq t}, (U^v_s)_{v \\in V,1 \\leq s \\leq t})\n\\]\nfor all $t \\geq 0.$\n\n\\subsection{Parking strategies and the space-based model}\n\nIn the model we have defined, all cars try to park as soon as they reach a free parking space. This can be thought of as a \\em parking strategy\\em. Let $G$ denote this ``greedy\" parking strategy: a car parks as soon as it can. It will be useful to consider different (possibly random) parking strategies as a way of controlling where cars park. In the definition below we introduce parking strategies more formally; $S_t(v,w)=1$ should be thought of as the event that the car starting from $v$ parks in $w$ at time $t.$\n\n\\begin{defn}\\label{parkdefn}\nLet $(L,R,\\mu)$ be a car park triple. A \\emph{parking strategy} $S = (S_t(v,w))_{t\\ge 1, v,w \\in V}$ for the car-based model on $(L,R,\\mu)$ is a sequence of random variables taking values in $\\{0,1\\}$ with the following properties:\n\\begin{itemize}\n\\item $S_t(v,w)$ is $\\mathcal{F}_t$-measurable for each $v,w \\in V$ and $t \\ge 1.$\n\\item $\\sum_{t\\ge 1, w \\in \\mathbb{Z}} S_t(v,w) \\le 1$ (a car parks at most once).\n\\item $\\sum_{t \\ge 1, v \\in \\mathbb{Z}} S_t(v,w) \\le 1$ (a parking space can hold only one car).\n\\item $S_t(v,w) = 0$ whenever $B^v = 0$ (a parking space cannot be filled by a non-existent car).\n\\item $S_t(v,w) = 0$ whenever $B^w = 1$ (a car cannot park where there is no parking space).\n\\item $S_t(v,w) = 0$ whenever $X^v_t \\neq w$ (a car cannot park in a space which is not its current position).\n\\end{itemize}\nA car starting at $v$ \\em parks in space $w$ at time $t$ \\em if and only if $S_t(v,w) = 1.$\n\\end{defn}\nNote that $S_t(v,w)$ being $\\mathcal{F}_t$-measureable means that our parking strategy is previsible, and that the parking time of a car is a stopping time.\n\nFor a parking strategy $S$ and an event $E$ we let $\\mathbb{P}^S\\left[E\\right]$ denote the probability of $E$ when all cars follow strategy $S$ (note that $\\mathbb{P} = \\mathbb{P}^G$). We will also allow random parking strategies, which require suitable adjustments to the $\\sigma$-algebra and the filtration (for example, we may independently flip a coin at the start and choose different parking strategies depending on whether the coin is heads or tails).\n\nEquipped with these new definitions, we are nearly ready to prove Theorem \\ref{thm:supermarket}. The final element we shall need is a stochastically equivalent parking process, where the moves of cars are attached to spaces rather than the cars; we shall refer to this model as the \\emph{space-based parking model}.\n\n\\begin{defn}\\label{defn:space parking}\nLet $(L,R,\\mu)$ be a car park triple. Independently for each vertex $v \\in V$, let: \n\\begin{itemize}\n\\item $(E^v_n)_{n \\in \\mathbb{N}}$ be a sequence of independent $\\mu$-random variables,\n\n\\item $(\\tilde{U}^v_s)_{s \\in \\mathbb{N}}$ be a sequence of independent $\\mathrm{Unif}([0,1])$ random variables. \n\n\\item $\\tilde{B}^v$ be a $\\mbox{Bernoulli}(p)$ random variable. We initially place a car at $v$ when $\\tilde{B}^v = 1$ and otherwise a parking space with the capacity for one car.\n\\end{itemize}\n\nWhen a single car arrives (but does not park) at position $v$, it leaves in the next time step according to the first unused $E^v_n$. If the set of cars $\\{w_1,\\ldots,w_r\\}$ arrives at $v$ at time $s$ and do not park, they collect the next $r$ unused directions $E^v_n, E^v_{n+1}, \\ldots, E^v_{n+r-1}$, in the order determined by their increasing values of $U^{w_{\\ell}}_s$.\n\\end{defn}\n\nFor the space-based parking model on the car park triple $(L,R,\\mu)$, the probability space $(\\tilde{\\Omega},\\tilde{\\mathcal{F}},(\\tilde{\\mathcal{F}}_t)_{t\\ge0},\\tilde{\\mathbb{P}})$ is slightly less obvious than in the car-based parking model. This is because the number of directions collected from $E^v$ by cars that visit $v$ by time $t$ but do not park there depends on the behaviour of cars starting at distance at most $t$ from $v$ in the first $t$ steps of the process. Hence, we can define the filtration $\\left(\\tilde{\\mathcal{F}}_t\\right)_{t \\ge 0}$ to be\n\\[\n\\tilde{\\mathcal{F}}_t = \\sigma((\\tilde{B}^v)_{v \\in V}, (E^v_n)_{v \\in V, n < \\infty}, (\\tilde{U}^v_s)_{v \\in V,1 \\leq s \\leq t})\n\\]\nfor all $t \\geq 0$.\n\n\\begin{defn}\\label{spaceparkdefn}\nLet $(L,R,\\mu)$ be a car park triple. A \\emph{parking strategy} $\\tilde{S} = (\\tilde{S}_t(v,w))_{t\\ge 1, v,w \\in V}$ for the space-based model on $(L,R,\\mu)$ is a sequence of random variables taking values in $\\{0,1\\}$ with the following properties:\n\\begin{itemize}\n\\item $\\tilde{S}_t(v,w)$ is $\\tilde{\\mathcal{F}}_t$-measurable for each $v,w \\in V$ and $t \\ge 1$.\n\\item $\\sum_{t\\ge 1, w \\in \\mathbb{Z}} \\tilde{S}_t(v,w) \\le 1$ (a car parks at most once).\n\\item $\\sum_{t \\ge 1, v \\in \\mathbb{Z}} \\tilde{S}_t(v,w) \\le 1$ (a parking space can hold only one car).\n\\item $\\tilde{S}_t(v,w) = 0$ whenever $\\tilde{B}^w = 1$ (a car cannot park where there is no parking space).\n\\item $\\tilde{S}_t(v,w) = 0$ whenever $\\tilde{B}^v = 0$ (a parking space cannot be filled by a non-existent car).\n\\item For all $v \\in L$ such that:\n\\begin{itemize}\n\\item $\\tilde{B}^v = 1$, and\n\\item for all $u \\in L$ and $s \\leq t-1$ we have $\\tilde{S}_s(v,u) = 0$,\n\\end{itemize}\nlet $E^{v_1}_{n_1}, E^{v_2}_{n_2}, \\ldots, E^{v_t}_{n_t}$ be the directions selected by $v$ in the first $t$ steps of its walk (note that we have $v_1 = v$). Then $\\tilde{S}_t(v,w) = 0$ if the walk obtained by starting at $v$ and following these directions does not end at $w$\n(a car cannot park in a space which is not its current position).\n\\end{itemize}\nA car starting at $v$ \\em parks in space $w$ at time $t$ \\em if and only if $\\tilde{S}_t(v,w) = 1.$\n\\end{defn}\n\nWe let $\\tilde{G}$ denote the greedy parking strategy in the space-based model.In the following proposition we show that parking strategies in the car-based parking process are stochastically equivalent to corresponding parking strategies in the space-based parking proces.\n\n\\begin{prop}\n\\label{prop:sameOld}\nLet $(L,R,\\mu)$ be a car park triple. Let $S$ and $\\tilde{S}$ be parking strategies for the car-based model and the space-based model on $(L,R,\\mu)$ respectively, and assume that for all $t \\geq 1$ and $v, w \\in L$ we have $S_t(v,w) = \\tilde{S}_t(v,w)$ whenever the following conditions hold:\n\\begin{enumerate}\n\\item $B^v = \\tilde{B}^v$ for all $v \\in L$ (the same cars appear in both models),\n\\item for all $1 \\leq s < t$ and $v, w \\in L$ we have $S_s(v,w) = \\tilde{S}_s(v,w)$ (at every time $1 \\leq s < t$, the same cars park in the same parking places in both models), and \n\\item for all $1 \\leq s \\leq t$, every car that does not park before time $s$, occupies the same position at time $s$ in both models\n\\end{enumerate}\n(i.e. the strategies $S$ and $\\tilde{S}$ behave identically whenever the cars behave identically up to time $t$ in the two processes). Then for any two sets $X \\subset L \\times L \\times \\mathbb{N}, Y \\subset L$, and the event\n\\[\n A_{X, Y} = [ \\mbox{ for all } (v_i, w_i, t_i) \\in X, \\mbox{car } v_i \\mbox{ is in } w_i \\mbox{ at time } t_i; \\mbox{ for all } w_j \\in Y, w_j \\mbox{ is a parking space } ]\n\\]\nwe have $\\mathbb{P}^S[A_{X,Y}] = \\tilde{\\mathbb{P}}^{\\tilde{S}}[A_{X,Y}]$.\n\\end{prop}\n\\begin{proof}\nWe have $\\mathbb{P}^S[A_{X,Y}], \\mathbb{P}^{\\tilde{S}}[A_{X,Y}] \\leq (1-p)^{|Y|}$, so if $|Y| = \\infty$ then $\\mathbb{P}^S[A_{X,Y}], \\mathbb{P}^{\\tilde{S}}[A_{X,Y}] = 0$ and the proposition holds.\n\nIf $|X| = \\infty$ then $A_{X,Y}$ must either describe the moves of infinitely many cars, or there must be a car $v$ such that $A_{X,Y}$ gives the position of $v$ at infinitely many times, or there are some $w_1 \\neq w_2$ and some $v \\in L, t \\in \\mathbb{N}$, such that $(v, w_1, t), (v, w_2, t) \\in X$. In all of these cases we have $\\mathbb{P}^S[A_{X,Y}], \\mathbb{P}^{\\tilde{S}}[A_{X,Y}] = 0$.\n\nHence we can assume that $|X|, |Y| < \\infty$. Then, let\n\\[\nU = \\{v : (v,w,t) \\in X \\} \\cup \\{w : (v,w,t) \\in X \\} \\cup Y,\n\\]\nand let $T = \\max \\{t : (v,w,t) \\in X\\}$. Then, in the car-based model, we can express $A_{X,Y}$ as a finite union of finite events concerning the variables $B^v, X^v_t, U^v_t$, for $t \\leq T$ and $v$ at distance at most $T$ from some element in $U$, describing the car\/parking space status and the step-by-step moves of cars in the $T$-neighbourhood of the elements if $U$. Analogously, in the space-based model, we can express $A_{X,Y}$ as a finite union of finite events concerning the variables $\\tilde{B}^v, E^v_n, \\tilde{U}^v_t$, for $t \\leq T$, $n \\leq T^2$, and $v$ at distance at most $T$ from some element in $U$. The proposition now follows from the properties of $S$ and $\\tilde{S}$, from the identical distributions and independence of $(B^v)_{v \\in V}$ and $(\\tilde{B}^v)_{v \\in V}$, of the $(U^v_t)_{v \\in V,t \\geq 0}$ and $(\\tilde{U}^v_t)_{v \\in V,t \\geq 0}$, as well as of $(X^v)_{v \\in \\mathbb{Z}}$ and $((E^v_n)_{n \\in \\mathbb{N}})_{v \\in \\mathbb{Z}}$ (observe that each of $E^v_n$ is used at most once in the process).\n\\end{proof}\n\nProposition \\ref{prop:sameOld} will allow us to deduce Theorem \\ref{thm:supermarket}  from the following theorem.\n\n\\begin{thm}\\label{thm:supermarketSpaces}\nLet $\\tilde{S}$ be a parking strategy for the space-based parking process on the car park triple $(L,R,\\mu)$. For a vertex $v$, we write $V^v_{\\tilde{S}}(t)$ for the value of $V^v(t)$ when strategy $\\tilde{S}$ is followed, and $V^v_{\\tilde{G}}(t)$ for the value of $V^v(t)$ when the greedy strategy is followed. Then for all $t \\ge 0,$ we have $V^v_{\\tilde{S}}(s) \\geq V^v_{\\tilde{G}}(s)$.\n\\end{thm}\n\n\\begin{proof}\nConsider the space-based parking process on a parking triple $(L,R,\\mu)$. Let $T^{v,r^{-1}}(t-1)$ be the number of cars that arrived at $vr^{-1}$ in the first $t-1$ time steps and then picked up $E^{vr^{-1}}_n = r$. Observe that $V^v(t)$ is equal to the sum over $r\\in R$ of $T^{v,r^{-1}}(t-1)$, plus $1$ if a car started at $v$ initially. By induction on $t$ we prove the following claim: for all $t \\geq 0$ we simultaneously have $T^{v,r^{-1}}_{\\tilde{S}}(t-1) \\geq T^{v,r^{-1}}_{\\tilde{G}}(t-1)$ and $V^v_{\\tilde{S}}(t) \\ge V^v_{\\tilde{G}}(t),$ for all $r \\in R$ (where again $T_{\\tilde{S}}$ and $T_{\\tilde{G}}$ denote the quantities when all cars follow strategy ${\\tilde{S}}$ or ${\\tilde{G}}$ respectively).\n\nIf a car parks at $v$ in the first $t$ time steps under ${\\tilde{S}}$ then $v$ must have initially been a parking space; then, if at least one car drove to $v$ under ${\\tilde{G}}$, it follows that some car parked in $v$ under ${\\tilde{G}}$ as well. Hence if the number of cars arriving at any vertex in the first $t$ time steps is at least as large under ${\\tilde{S}}$ as under ${\\tilde{G}}$, the same applies to the number of cars leaving $v$ in the first $t+1$ time steps. Moreover, for each $r \\in R,$ since the directions $E^{vr^{-1}}_n$ are selected one-by-one in a fixed order, $V^{vr^{-1}}_{\\tilde{S}} (t) \\ge V^{vr^{-1}}_{\\tilde{G}} (t)$ implies $T^{v,r^{-1}}_{\\tilde{S}} (t) \\geq T^{v,r^{-1}}_{\\tilde{G}} (t).$\n\nThe base case $t = 0$ of the induction is trivial. Hence suppose that our claim is true for $t = s -1 \\ge 0.$  By induction, for each $r \\in R,$ we have $V^{vr^{-1}}_{\\tilde{S}}(s-1) \\ge V^{vr^{-1}}_{\\tilde{G}}(s-1)$; hence we have $T^{v,r^{-1}}_{\\tilde{S}}(s-1) \\geq T^{v,r^{-1}}_{\\tilde{G}}(s-1).$ We then obtain\n\\begin{align*}\nV^v_{\\tilde{S}}(s) & = \\sum_{r\\in R} T^{v,r^{-1}}_{\\tilde{S}}(s-1) + \\mathds{1}_{\\{v \\mbox{ \\small is a car}\\}} \\\\\n & \\geq \\sum_{r\\in R} T^{v,r^{-1}}_{\\tilde{G}}(s-1) + \\mathds{1}_{\\{v \\mbox{ \\small is a car}\\}} \\\\\n & = V^v_{\\tilde{G}}(s).\n\\end{align*}\nThis completes the proof of Theorem \\ref{thm:supermarketSpaces}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem  \\ref{thm:supermarket}] \nLet $S$ be a parking strategy for the car-based model on the car park triple $(L,R,\\mu)$, let $v \\in L$, and let $t, k \\ge 0$. Observe that for parking strategies in the space-based model, the filtration $\\tilde{\\mathcal{F}}_t$ carries all the information about the moves of all cars up to time $t$ (and, in fact, through $(E^v_n)_{v \\in V, n < \\infty}$ also about the movements to come). Therefore we can design a parking strategy $\\tilde{S}$ for the space-based model, such that the assumptions of Proposition \\ref{prop:sameOld} are satisfied for $S$ and $\\tilde{S}$.\n\nNext, we can express the event $[V^v_S(t) \\le k]$ as a finite union of events $A_{X,Y}$, defined as in Proposition \\ref{prop:sameOld}, describing the car\/parking space status and movements of cars starting at distance at most $t$ from $v$, such that at most $k$ cars arrive at $v$ by time $t$ under $S$. By Proposition \\ref{prop:sameOld}, we have $\\mathbb{P}^S[A_{X,Y}] = \\tilde{\\mathbb{P}}^{\\tilde{S}}[A_{X,Y}]$. By Theorem \\ref{thm:supermarketSpaces} we have $V^v_{\\tilde{S}}(s) \\geq V^v_{\\tilde{G}}(s)$ deterministically, hence if $A_{X,Y} \\subseteq [V^v_{\\tilde{S}}(t) \\le k]$, then also $A_{X,Y} \\subseteq [V^v_{\\tilde{G}}(t) \\le k]$. Thus we have $\\mathbb{P}[V^v_{\\tilde{G}}(t) \\le k] \\geq \\tilde{\\mathbb{P}}[V^v_{\\tilde{S}}(t) \\le k]$, and since by applying Proposition \\ref{prop:sameOld} again we find that $\\mathbb{P}^G[A_{X,Y}] = \\tilde{\\mathbb{P}}^{\\tilde{G}}[A_{X,Y}]$, we finally obtain $\\mathbb{P}[V^v_G(t) \\le k] \\geq \\mathbb{P}[V^v_S(t) \\le k]$ as claimed.\n\\end{proof}\n\nIn the rest of this paper, we shall consider the car-based parking model only. We remark that for parking times we may not make a conclusion similar to Theorem \\ref{thm:supermarket}. For example, consider the parking strategy where all but one car is instructed to never park. The chosen car will have a much easier job of finding a parking space. To combat this, we need some symmetry that will allow us compare visits to a space and parking times of cars, and therefore make use of Theorem \\ref{thm:supermarket}\n\nWe say that a parking strategy $S$ on the car park triple $(L,R,\\mu)$ is \\emph{weakly translation invariant} if for all $v,w \\in V$, $r\\in R$  and $t \\ge 0$, \n\\[\n \\mathbb{P}^S\\left[S_t(v,w)=1\\right] = \\mathbb{P}^S\\left[S_t(vr,wr)=1\\right].\n\\]\nAn equivalent property is that for all $v,w \\in V$, $r\\in R$ and $t \\geq 1$,\n\\[\n\\mathbb{P}^S\\left[\\mbox{car $v$ arrives at spot $w$ at time $t$}\\right] = \\mathbb{P}^S\\left[\\mbox{car $vr$ arrives at spot $wr$ at time $t$}\\right].\n\\] \n\n\\begin{rem}\nThis is a rather weak form of translation invariance -- it does not control joint events in any sense. Since in this paper we are predominantly working with expectations, we do not need to worry about this. A more natural form of translation invariance is the following form: a parking strategy $S$ on the car park triple $(L,R,\\mu)$ is \\emph{strongly translation invariant} if for any $r\\in R,$ the probability measure $\\mathbb{P}$ is invariant with respect to a translation by $r$. (The same is true for car removal strategies which we introduce later.) We note that the parking strategy (respectively, car removal strategy) we use in Section \\ref{sec:p=1\/2upper} (respectively, Section \\ref{sec:p=1\/2lower}) are in fact strongly translation invariant.\n\\end{rem}\n\nWeak translation invariance allows us to equate car journey lengths with total number of visits to a position in $V.$\n\n\\begin{lem}\\label{lem:dual}\nLet $S$ be a weakly translation invariant strategy on the car park triple $(L,R,\\mu)$. Then for all $t \\ge 0$ and $v \\in V,$\n\\[\n \\mathbb{E}^S[\\tau \\wedge t] = \\mathbb{E}^S[V^v(t)].\n\\]\n\\end{lem}\n\nWe remark that this result is a special case of the well known and more general Mass-Transport principle \\cite[Theorem 8.7]{L+P} and a similar result was noted at \\cite[Lemma 4.1]{DGJLS}. Since the proof is very short in our setting, we include it for self-containment.\n\n\\begin{proof}\nLet $t \\ge 0$ and fix an arbitrary $v \\in V.$ Write $B_t(v)$ for the vertices of $L$ connected to $v$ by a path of length at most $t$. By translation invariance\n\\begin{align*}\n \\mathbb{E}^S[\\tau \\wedge t] = \\mathbb{E}^S[\\tau^v \\wedge t] & = \\sum_{s \\in [t]}\\sum_{w \\in B_{t}(0)} \\mathbb{P}^S\\left[\\mbox{car $v$ arrives at spot $vw$ at time $s$}\\right] \\\\\n & = \\sum_{s \\in [t]}\\sum_{w \\in B_t(0)} \\mathbb{P}^S\\left[\\mbox{car $vw^{-1}$ arrives at spot $v$ at time $s$}\\right] \\\\\n & = \\mathbb{E}^S[V^v(t)].\n\\end{align*}\n\\end{proof}\n\nThe following easy corollary of Theorem \\ref{thm:supermarket} and Lemma \\ref{lem:dual} is crucial for our arguments, and considers the expected journey of a car up to time $t$ under different parking strategies. It will allow us to derive upper bounds on $\\mathbb{E}^G[\\tau \\wedge t]$ by considering a different parking strategy which is easier to control.\n\n\\begin{cor}\n\\label{lem:supermarket}\nLet $S$ be a weakly translation invariant parking strategy on the car park triple $(L,R,\\mu)$. Then for all $t \\ge 0,$\n\\[\n \\mathbb{E}^S[\\tau \\wedge t] \\ge \\mathbb{E}^G[\\tau \\wedge t].\n\\]\n\\end{cor}\n\n\\subsection{Car removal strategies}\n\nAnother way to modify the car parking problem is through \\emph{car removal strategies}. Under certain circumstances it will be helpful to pretend that a car has been removed from the process. A car is removed during a step, and it is parked off $V$. So if car $v$ is at position $w$ at time $t,$ and is removed during step $t+1$, we remove the car from the process without it taking up a parking space and set $\\tau^v = t+1.$ We remark that we will always assume a greedy parking strategy when we have a non-trivial car removal strategy.\n\n\\begin{defn}\\label{teledefn}\nLet $(L,R,\\mu)$ be a car park triple. A \\emph{car removal strategy} $Q = (Q_t(v))_{t\\ge 1, v\\in V}$ on $(L,R,\\mu)$ is a sequence of random variables taking values in $\\{0,1\\}$ with the following properties:\n\\begin{itemize}\n\\item $Q_t(v)$ is $\\mathcal{F}_t$-measurable for each $v \\in V$ and $t \\ge 1.$\n\\item $Q_t(v) = 0$ whenever $B^v = 0$ (a non-existent car cannot be removed).\n\\item $\\sum_{t\\ge 1} Q_t(v) \\le 1$ (a car can only be removed once).\n\\end{itemize}\nA car starting at $v$ is removed in the $t$-th time step if and only if $Q_t(v) = 1.$\n\\end{defn}\nAs we did for parking strategies, we define $\\mathbb{P}^Q$ for a car removal strategy $Q$. Whenever we explicitly consider a process involving car removal strategies, we assume that all vehicles follow the greedy parking strategy.\n\nWe are now ready to prove Theorem \\ref{thm:pinkfloyd}.  In the one-dimensional setting this will allow us to derive lower bounds on $\\mathbb{E}[\\tau \\wedge t]$ by considering an interval and removing cars that enter or leave the interval.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:pinkfloyd}.]\nFor each $w \\in V$ and $t \\ge 0,$ let $W^w_Q(t)$ be the set of unparked cars at position $w$ at time $t$ under $Q$, and let $W^w_G(t)$ denote the same quantity under $G$ (recall that under $G$, which is the greedy parking strategy, there is no car removal). We start by showing that at every position $w \\in V$ and for every time $t \\ge 0$ we have $W^w_Q(t) \\subseteq W^w_G(t)$. We prove this by induction on $t \\ge 0$. The base case $t=0$ is trivial, hence suppose that the claim is true up to and including time $t-1$.\n\nFix a position $w$ and observe first that if a parking space $w$ is filled at time $t$ under $Q$, then a car $v$ from $W^{wr^{-1}}_Q(t-1)$ must arrive at $w$ at time $t$ for some $r \\in R$. By the inductive hypothesis, $v$ must be in the appropriate set in $W^{wr^{-1}}_G(t-1)$, and so it must arrive at $w$ at time $t$ under $G$ (note that we are in the original parking process, where cars have random walks attached to them, rather than the space-based parking process considered in the proof of Theorem \\ref{thm:supermarketSpaces}). Therefore under $G$ either spot $w$ must already be filled before $t$, or a car must park in spot $w$ at time $t$. Therefore any parking space filled under $Q$ at time $t$ must be filled under $G$ at time not later than $t$.\n\nNow, by the inductive hypothesis, any car arriving at position $w$ under $Q$ at time $t$ must arrive at position $w$ under $G$ at time $t$. If $w$ is not a free parking space under $G$ at time $t-1$, then $W^w_Q(t) \\subseteq W^w_G(t)$ and the claim holds. Thus suppose that $w$ is a free parking space at time $t-1$ under $G$. Then by the argument above, $w$ must be a free parking space at time $t-1$ under $Q$. Further, if under $G$ a car not from $W^w_Q(t)$ parks at $w$ at time $t$, then again $W^w_Q(t) \\subseteq W^w_G(t)$ and again we are done. So suppose that under $G$ a car $v \\in W^w_Q(t)$ parks at position $w$ at time $t$. By the tie-breaking procedure, $v$ must have the smallest $U^x_t$ value over the cars $x$ that arrive at $w$ under $G$, and so must have the smallest $U^x_t$ value over cars $x$ that arrive at $w$ under $Q$. Therefore under $Q$ the car $v$ must also park at $w$ at time $t$, and so once again we have $W^w_Q(t) \\subseteq W^w_G(t)$.\n\nNow, consider that the set of unparked cars at time $t$ is the union $\\bigcup_{w \\in V}W^w(t)$, hence if a car $v$ is still unparked under $Q$ at time $t$, then there is some $w \\in L$ such that $v \\in W^w_Q(t)$. But we know that $W^w_Q(t) \\subseteq W^w_G(t)$, therefore we have $v \\in W^w_G(t)$, implying that $\\tau^v_Q \\le \\tau^v_G$ as desired.\n\nAlso, the number of visits to $w$ at time $t$ is\n\\[\n|W^w(t)|+ \\mathds{1}_{\\{\\mbox{\\small a car parks at } w \\mbox{ \\small at time } t\\}}.\n\\]\nSince $W^w_Q(t) \\subseteq W^w_G(t)$, and additionally we know that\n\\[\n\\sum_{s=1}^t \\mathds{1}_{\\{\\mbox{\\small a car parks under } Q \\mbox{ at } w \\mbox{ \\small at time } s\\}} \\leq \\sum_{s=1}^t \\mathds{1}_{\\{\\mbox{\\small a car parks under } G \\mbox{ at } w \\mbox{ \\small at time } s\\}},\n\\]\nthe inequality $V^v_Q(t) \\le V^v_G(t)$ follows.\n\\end{proof}\n\n\\section{Probabilistic bounds}\n\\label{sec:probabilities}\nIn this section, we state some probabilistic bounds that are needed for the proofs in Sections \\ref{sec:p=1\/2upper}, \\ref{sec:p=1\/2lower}, and \\ref{sec:p<1\/2}. \n\nWe make use of the following variant of the Chernoff bound (see \\cite[Chapter~4]{Chernoffcite}).\n\n\\begin{lem}\\label{feelthechern}\nLet $p \\in (0,1),$ $N \\in \\mathbb{N},$ and $\\varepsilon > 0.$ Then $$\\mathbb{P}\\left[\\Bin(N,p) \\ge N(p+\\varepsilon)\\right] \\le e^{-2\\varepsilon^2N}.$$\n\\end{lem}\n\nWe need some facts about hitting times of the simple symmetric random walk.\n\\begin{lem}\\label{lbeforer}\nLet $a, b > 0$ be positive integers. Let $\\{X_n\\}_{n \\geq 0}$ be a simple symmetric random walk on $\\mathbb{Z}$ with $X_0 = 0.$ For $i \\in \\mathbb{Z},$ let $H_i = \\min\\{s : X_s = i\\}.$ Then\n\t\\begin{itemize}\n\t\t\\item[(i)] $\\mathbb{P}[H_b < H_{-a}] = \\frac{a}{a+b}.$ \n\t\t\\item[(ii)] $\\mathbb{E}[H_b | H_b < H_{-a}] = \\frac{b(b+2a)}{3}.$\n\t\t\\item[(iii)] $\\mathbb{E}[H_{-a} \\wedge H_{a}] = a^2.$\n\t\\end{itemize}\n\\end{lem}\n\n\\begin{proof}\nAll of this is standard. Part (i) is Gambler's ruin (see \\cite[XIV.2]{Feller-Probability}). Part (iii) follows from (ii) by symmetry and a simple calculation.\n\nFor part (ii), we first prove the statement in a slightly different setup. Let $c,d$ be positive integers with $0 < c < d$ and assume that $X_0 = c.$ We show that\n\\[\n\\mathbb{E}[H_d | H_d < H_0] = \\frac{(d-c)(d+c)}{3}.\n\\]\nPart (ii) of the lemma then follows immediately by taking $d = a+b$ and $c=a$. Let $Z_n = X_n^3 - 3nX_n$ and let $S =  X_{n+1}-X_n \\in \\{-1,1\\}$. Then, since $S^2 = 1$ and $S^3 = S$, we have\n\\begin{align*}\nZ_{n+1} & = \\left(X_n+S\\right)^3 - 3(n+1)\\left(X_n+S\\right) \\\\\n & = Z_n + 3X_n^2S+3X_n S^2+S^3 - 3nS - 3X_n -3 S \\\\\n & = Z_n + S\\left(3X_n^2-2-3n\\right).\n\\end{align*}\n\nSince $X_{n+1}-X_n$ takes values in $\\{-1,+1\\}$ with mean $0$ independently of $\\mathcal{F}_n,$ and since $X_n$ is $\\mathcal{F}_n$-measurable, we have\n\\begin{align*}\n\t\t\\mathbb{E}[Z_{n+1} | \\mathcal{F}_n] & = \\mathbb{E}[Z_n + \\left(X_{n+1}-X_n\\right)\\left(3X_n^2-2-3n\\right) | \\mathcal{F}_n] \\\\\n\t\t& = Z_n + \\left(3X_n^2-2-3n\\right)\\mathbb{E}[X_{n+1}-X_n] = Z_n,\n\\end{align*}\nand so $Z$ is a martingale.\n\nFor $n \\in \\mathbb{N},$ Doob's Optional Stopping Theorem gives $\\mathbb{E}[Z_{n\\wedge H_0 \\wedge H_d}] = \\mathbb{E}[Z_0] = c^3.$ At the same time, $|Z_{n\\wedge H_0 \\wedge H_d}|$ is bounded by $3d^3 + 3(H_0 \\wedge H_d)d$ for all $n.$\nAdditionally, $H_0 \\wedge H_d$ is integrable and so by the Dominated Convergence Theorem we have\n\\[\n  \\mathbb{E}[Z_{H_0 \\wedge H_d}] = \\lim_{n \\to \\infty} \\mathbb{E}[Z_{n\\wedge H_0 \\wedge H_d}] = c^3.\n\\]\nBut $Z_{H_0 \\wedge H_d} = \\mathds{1}_{H_d < H_0}(d^3 -3dH_d).$ Therefore\n\t\\begin{align*}\n\t\tc^3 = \\mathbb{E}[Z_{H_0 \\wedge H_d}] & = \\mathbb{E}[\\mathds{1}_{H_d < H_0}(d^3 -3dH_d)]  \\\\\n\t\t& = \\mathbb{P}\\left[H_d < H_0\\right](d^3 - 3d\\mathbb{E}[H_d | H_d < H_0]).\n\t\\end{align*}\nBy (i), $\\mathbb{P}\\left[H_d < H_0\\right] = c\/d,$ and so $\\mathbb{E}[H_d | H_d < H_0] = \\frac{d^3 - c^2d}{3d} = \\frac{d^2-c^2}{3}.$\n\\end{proof}\n\n\nLet $M_n$ denote the maximum value in the first $n$ time steps of the simple symmetric random walk starting at $0,$ and let $m_n$ denote its corresponding minimum value. Define $p_{n,r} = \\binom{n}{\\frac{n+r}{2}} 2^{-n}.$ It can be shown (see, e.g., Feller \\cite[Theorem III.7.1]{Feller-Probability}) that for $r \\geq 0$ we have\n\\[\n \\mathbb{P}\\left[M_n = r\\right] = \\mathbb{P}\\left[m_n = -r\\right] = \\begin{cases}\np_{n,r} \\quad & \\mbox{if $n - r$ is even}, \\\\\np_{n,r+1} \\quad & \\mbox {otherwise.}\n\\end{cases}\n\\]\n\nLet $Y \\sim \\Bin(n,1\/2)$, where we will assume that $n$ is even, so that for $k \\le n\/2,$ $\\mathbb{P}\\left[Y = \\frac{n+2k}{2}\\right] = p_{n,2k}.$ We now conclude this section with some tail bounds for the maximum of the random walk.\n\n\\begin{lem}\\label{minssrw}\n\\begin{itemize}\n\\item[(i)] $\\mathbb{P}\\left[M_n \\ge 2 \\alpha  \\sqrt{n \\log n}\\right] \\le 2 n^{-2 \\alpha^2}.$\n\\item[(ii)] $\\mathbb{P}\\left[M_n \\ge c\\sqrt{n}\\right]$ and $\\mathbb{P}\\left[M_n \\le c\\sqrt{n}\\right]$ are bounded away from zero for each $c>0.$\n\\end{itemize}\n\\end{lem}\n\nWe remark that the analogous results hold for $m_n$ by symmetry.\n\n\\begin{proof}\n\t\\begin{align}\n\t\t\\mathbb{P}\\left[M_n \\ge 2k\\right] & = \\mathbb{P}\\left[M_n = 2k\\right] + \\sum_{\\ell = k+1}^{n\/2} \\left ( \\mathbb{P}\\left[M_n = 2\\ell-1\\right] + \\mathbb{P}\\left[M_n=2\\ell \\right] \\right ) \\nonumber \\\\\n\t\t& = p_{n,2k} + \\sum_{\\ell = k+1}^{n\/2} \\left ( p_{n,2\\ell-1} + p_{n,2\\ell} \\right ) \\nonumber \\\\\n\t\t& = p_{n,2k} + 2 \\sum_{\\ell = k+1}^{n\/2} p_{n,2\\ell} \\nonumber \\\\\n\t\t& = \\mathbb{P}\\left[Y = \\frac{n+2k}{2}\\right] + 2 \\sum_{\\ell=k+1}^{n\/2} \\mathbb{P}\\left[Y = \\frac{n+2\\ell}{2}\\right] \\label{double-ish} \\\\\n\t\t& \\leq 2\\mathbb{P}\\left[Y \\ge \\frac{n+2k}{2}\\right]. \\nonumber\n\t\\end{align}\n\nThe same holds for odd $n,$ and so we see that $\\mathbb{P}\\left[M_n \\ge 2k\\right] \\le 2\\mathbb{P}\\left[Y \\ge n(1\/2 + k\/n)\\right].$ Setting $k = \\alpha \\sqrt{n\\log n}$ and applying Lemma \\ref{feelthechern} gives\n\t\\[\n\t\t\\mathbb{P}\\left[M_n \\ge 2 \\sqrt{n \\log n}\\right] \\le 2n^{-2 \\alpha^2}.\n\t\\]\n\nSetting $k = c\\sqrt{n}$ we get\n\t\\begin{align*}\n\t\t\\mathbb{P}\\left[M_n \\le 2k\\right] & \\ge 1-\\mathbb{P}\\left[M_n \\ge 2k\\right]  \\\\\n\t\t& \\ge 1 - 2 \\mathbb{P}\\left[Y \\ge n(1\/2 + k\/n)\\right]  \\\\\n\t\t& = 1 - 2\\mathbb{P}\\left[\\frac{Y-n\/2}{\\sqrt{n\/4}} \\ge 2c\\right]  \\\\\n\t\t& \\rightarrow 1 - 2(1-\\Phi(2c)),   \\\\\n\t\t& = 2\\Phi(2c)-1, \n\t\\end{align*}\nas $n \\rightarrow \\infty$ by the Central Limit Theorem. Since $c > 0$ we have $\\Phi(c) > 1\/2,$ and so $\\mathbb{P}[M_n \\le c\\sqrt{n}]$ is bounded away from zero for each $c>0.$\n\nFrom \\eqref{double-ish}, we may read off $\\mathbb{P}\\left[M_n \\ge 2k\\right] \\ge 2 \\mathbb{P}\\left[Y \\ge \\frac{n+2k+2}{2}\\right] = 2 \\mathbb{P}\\left[Y \\ge \\frac{n+2k}{2}\\right]-O(n^{-1\/2}),$ and so again for $k = c\\sqrt{n},$ \n\t\\begin{align*}\n\t\t\\mathbb{P}\\left[M_n \\ge 2k\\right] & \\geq 2 \\mathbb{P}\\left[Y \\ge \\frac{n+2k}{2}\\right] -o(1) \\\\\n\t\t& = 2 \\mathbb{P}\\left[\\frac{Y-n\/2}{\\sqrt{n\/4}} \\ge 2c\\right] -o(1) \\\\\n\t\t& \\rightarrow 2-2\\Phi(2c) > 0 \n\t\\end{align*}\nas $n \\rightarrow \\infty$ by the Central Limit Theorem. Therefore $\\mathbb{P}\\left[M_n \\ge c\\sqrt{n}\\right]$ is bounded away from zero for each $c>0.$\n\\end{proof}\n\\section{Upper bound on $\\mathbb{E} [\\tau \\wedge t]$}\n\\label{sec:p=1\/2upper}\n\nIn this section, we prove the upper bound in Theorem \\ref{thm:p=1\/2}. We fix a target time $t$ and consider a particular weakly translation invariant parking strategy with additional properties. The parking strategy assigns (at time $0$) a parking space to most of the cars and tells the other cars they can never park. Each car then drives until it reaches its assigned parking place (or just keeps driving if it has no assigned space). The work left to do is to show that many cars are assigned parking spaces that they will reach in a short expected amount of time. We split this section into two parts; the first one detailing the parking strategy and showing some of its properties, and the second one bringing everything together to prove the desired upper bound.\n\n\\subsection{The parking strategy}\nFix $t \\ge 1.$ We define the parking strategy $T = T_t$ as follows. We first divide $\\mathbb{Z}$ into intervals of length $\\lceil \\sqrt{t} \\rceil.$ On each interval $I,$ we run through the locations from right to left, attempting to assign to each car $i$ a parking space $P(i)$ somewhere in $I$ and to the left of $i.$ If there is no unassigned parking space available within distance $O(t^{1\/4})$ then car $i$ will not try to park, and we set $P(i) = \\star;$ and if $i$ is a parking space we set $P(i) = i.$ This defines a strategy that is periodic, but not weakly translation invariant (because the intervals have specified endpoints). So we begin by applying a random shift to our intervals to make the strategy weakly translation invariant.\n\nMore formally, let $\\zeta = \\lceil \\sqrt{t} \\rceil$ and $\\nu = \\lceil t^{1\/4} \\rceil.$ First let $Z$ be uniformly distributed on $[\\zeta]$ independently from the original model. Then, given $Z=z,$ for each interval $[z+k\\zeta,z+(k+1)\\zeta-1]$ we assign specific parking spaces to cars as follows:\n\n\\begin{algorithm}[H]\n\\SetKw{KwFn}{Initialization}\n    \\KwFn{Set $m=z+(k+1)\\zeta-1,$ $W=\\emptyset$}\\;\n    \\While{$m \\ge z+k\\zeta$}{\n    \\If{\\rm{There is initially a parking space at $m$}}{\n    \t    Set $P(m) = m$\\;\n    \t    \\If{$W \\neq \\emptyset$}{\n            \tLet $v$ be the largest element of $W.$\\:\n            \tRemove $v$ from $W$ and set $P(v) = m$\\;\n            }\n        }\n      \\If{\\rm{There is initially a car at $m$}}{\n            Add $m$ to $W$\\;\n        }\n     \\If{$|W| = \\nu$}{\n            Let $v$ be the largest element of $W.$\\:\n            Remove $v$ from $W$ and set $P(v) = \\star$\\;\n        }\n    Set $m := m-1$\\;\n    }\n\\SetKw{KwFm}{Finalization}\n  \\KwFm{For all $v \\in W,$ set $P(v) = \\star.$}\n\\label{discardalgorithmpark}\n\\end{algorithm}\n\\bigskip\n\nThe strategy $T$ is defined as follows: for each car $i,$\n\\begin{itemize}\n\\item if $P(i) = \\star,$ then $S_t(i,j)=0$ for all $t \\ge 1,$ $j \\in \\mathbb{Z}$ (car $i$ never parks).\n\\item if $P(i) \\neq \\star,$ then $S_t(i,P(i)) = 1$ for the first time $t$ when car $i$ visits $P(i),$ and $S_t(i,j) = 0$ otherwise.\n\\end{itemize}\n\nNote that the random variable $Z$ causes this parking strategy to be weakly translation invariant, and so it is sufficient to show that $\\mathbb{E}^T[\\tau \\wedge t] = O(t^{3\/4})$ to prove the upper bound in Theorem \\ref{thm:p=1\/2}.\n\nThe benefit of this parking strategy is that it is much easier to give bounds on the expected hitting time of a fixed vertex rather than an arbitrary empty parking space. However, there are a couple of potential problems: the parking strategy might assign cars to distant parking spaces; and the parking strategy might dictate that many cars never park ($P(v) = \\star$ for too many $v$). The next two lemmas resolve these problems.\n\n\\begin{lem}\\label{sorting}\nFor all $i$ we have $P(i) = \\star$ or $i-P(i) \\le 3\\nu.$\n\\end{lem}\n\n\\begin{lem}\\label{markov}\nFor all $i \\in \\mathbb{Z},$ $\\mathbb{P}\\left[P(i) = \\star\\right]  = O(t^{-1\/4}).$\n\\end{lem}\n\nLemma \\ref{sorting} follows from our choice to abandon the oldest car when the queue is too long.\n\\begin{proof}[Proof of Lemma \\ref{sorting}]\nSince we define $P(i) = i$ whenever $i$ is a parking space, the lemma is equivalent to saying that a vertex does not stay in $W$ for too long. Indeed suppose that $i$ joins $W.$ Since $W$ contains at most $\\nu$ elements at any time and elements of $W$ get assigned a parking space (or $\\star$) in the order of when they join $W,$ if $P(i) \\neq \\star,$ then $i$ will be assigned a parking space $P(i)$ with $i-P(i) \\le 3\\nu$ if there are at least $\\nu$ parking spaces in the next $3\\nu$ iterations of the while loop of the algorithm above.\n\nSuppose that there are fewer than $\\nu$ parking spaces in the next $3\\nu$ iterations of the while loop. Then the number of elements joining $W$ is at least $2\\nu.$ Since $W$ contains at most $\\nu$ elements at any time, at least $\\nu$ elements must leave $W$ during these $3\\nu$ iterations of the while loop. Therefore $i$ must leave $W$ during these $3\\nu$ iterations of the loop, either with $i-P(i) \\leq 3\\nu$ or with $P(i) = \\star.$\n\\end{proof}\n\nThe proof of Lemma \\ref{markov} is a little more involved. We use some elementary properties of irreducible, aperiodic Markov chains.\n\n\\begin{proof}[Proof of Lemma \\ref{markov}]\nWe may assume without loss of generality that $Z$ is $0$, and we consider the interval obtained by taking $k=0$. By symmetry and translation invariance, we see that for any $i \\in \\mathbb{Z}$\n\t\\begin{equation}\n\t\t\\mathbb{P}^T\\left[P(i) = \\star\\right] = \\zeta^{-1}\\mathbb{E}^T[\\left|\\{j \\in [0,\\zeta-1] : P(j) = \\star\\}\\right|]. \\label{symsneak}\n\t\\end{equation}\n\nLet $C_n$ be the size of $W$ just before the last \\textbf{if} clause of the loop when $m = \\zeta-n,$ and set $C_0 = 0.$ In most situations we can only have $C_{n+1}-C_n$ equal to either 1 (if $\\zeta-n-1$ is a car) or $-1$ (if $\\zeta-n-1$ is a parking space). However, there are two exceptions to that rule. If $C_n = 0,$ i.e., if $W = \\emptyset$ after we observe $\\zeta-n,$ and if $\\zeta-n-1$ is a parking space, then $C_{n+1} = 0$ as well. Moreover, if $C_n = \\nu$ then in the last \\textbf{if} clause of the loop we deterministically remove one element from $W.$ Thus depending on the value of whether $\\zeta-n-1$ is a car o a parking space, we might have either $C_{n+1} = \\nu$ or $C_{n+1} = \\nu-2.$ Hence $C = (C_0, C_1, \\ldots)$ is a Markov chain with transition probabilities $(p_{k,l})_{k,l \\in \\{0,\\ldots,\\nu\\}}$ satisfying:\n\t\\begin{itemize}\n\t\t\\item $p_{0,0} = 1\/2$ (there is a parking space but no queue),\n\t\t\\item $p_{0,1} = 1\/2$ (a car joins an empty queue),\n\t\t\\item $p_{k,k-1} = 1\/2$ when $i \\in \\{1,\\ldots,\\nu-1\\}$ (a car in the queue is assigned a parking space),\n\t\t\\item $p_{k,k+1} = 1\/2$ when $i \\in \\{1,\\ldots,\\nu-1\\}$ (a new car joins the queue),\n\t\t\\item $p_{\\nu,\\nu-2} = 1\/2$ (we tell an old car to leave the queue, and assign another queueing car to a parking space),\n\t\t\\item $p_{\\nu,\\nu} = 1\/2$ (we tell an old car to leave the queue, and a new car joins the queue),\n\t\t\\item $p_{k,l} = 0$ otherwise.\n\t\\end{itemize}\n\t\nWe see that some vertex gets assigned $\\star$ each time $C$ hits $\\nu.$ Additionally, the $C_{\\zeta}$ vertices remaining in $W$ at the end of the execution of the algorithm also get assigned $P(v) = \\star.$ Therefore\n\t\\begin{eqnarray}\n\t\t\\left|\\{j \\in [0,\\zeta-1] : P(j) = \\star\\}\\right| = C_{\\zeta} + \\sum_{n=0,\\ldots,\\zeta-1} \\mathds{1}_{C_n = \\nu} \\label{compare}\n\t\\end{eqnarray}\nIn our algorithm, we initially impose that $W = \\emptyset.$ If, however, we started the algorithm with $W'$ containing some cars, then at every step in the algorithm, we would have $W \\subseteq W'.$ Let $C'_n$ be the size of $W'$ just before the last \\textbf{if} clause of the loop when $m = \\zeta - n.$ Then we see that $\\{C'_n\\}$ is a Markov chain with transition probabilities $(p_{k,l})_{k,l \\in \\{0,\\ldots,\\zeta\\}}$ which majorises $C.$ Thus, if $|W'|$ initially has distribution $\\mu,$ we see\n\t\\[\n\t\t\\mathbb{P}[C_n = \\nu] \\le \\mathbb{P}\\left[C'_n = \\nu\\right] = \\mathbb{P}_{C_0 \\sim \\mu}\\left[C_n = \\nu\\right].\n\t\\]\nIn particular, if we let $\\pi$ be a stationary distribution of $C,$ then for all $n$\n\t\\[\n\t\t\\mathbb{P}_{C_0=0}\\left[C_n = \\nu\\right] \\le \\mathbb{P}_{C_0 \\sim \\pi}\\left[C_n= \\nu\\right] = \\pi(\\nu).\n\t\\]\nHence if we take the expectation of \\eqref{compare} we obtain\n\t\\[\n\t\t\\mathbb{E}^T[\\left|\\{j \\in [0,\\zeta-1] : P(j) = \\star\\}\\right|] \\le \\mathbb{E}^T[C_{\\zeta}] + \\zeta\\pi(\\nu).\n\t\\]\n\nSince $C$ is irreducible and aperiodic, and has a finite state space, it has a unique stationary distribution $\\pi.$ One can then verify that $\\pi(k) = \\frac{1}{\\nu}$ for $k=0,\\ldots \\nu-2,$ and $\\pi(k) = \\frac{1}{2\\nu}$ for $k = \\nu-1,\\nu.$ Since $C$ takes values in $0,\\ldots,\\nu,$ we may bound $\\mathbb{E}[X_{\\zeta}]$ by $\\nu$ to find\n\t\\[\n\t\t\\mathbb{E}^T[\\left|\\{j \\in [0,\\zeta-1] : P(j) = \\star\\}\\right|] \\le \\nu + \\frac{\\zeta}{2\\nu}.\n\t\\]\nTogether with \\eqref{symsneak} we obtain $\\mathbb{P}^T \\left[P(i) = \\star\\right] \\le \\frac{\\nu}{\\zeta} + \\frac{1}{2\\nu}= O(t^{-1\/4}).$\n\\end{proof}\n\n\\subsection{Proof of the upper bound}\nWe now have all the ingredients necessary to prove the upper bound in Theorem \\ref{thm:p=1\/2}. We will do this by bounding $\\mathbb{E}^T[\\tau \\wedge t]$ and then appealing to Corollary \\ref{lem:supermarket}.\n\\begin{proof}[Proof of the upper bound in Theorem \\ref{thm:p=1\/2}]\nLet $t \\ge 0.$ Without loss of generality we can consider $\\tau = \\tau^0.$ Then\n\t\\begin{align*}\n\t\t\\mathbb{E}^T[\\tau \\wedge t] &= \\mathbb{E}^T[\\tau^0 \\wedge t] \\\\\n\t\t&= \\mathbb{E}^T[\\tau^0 \\wedge t | P(0) = \\star]\\mathbb{P}^T\\left[P(0) = \\star\\right] + \\mathbb{E}^T[\\tau^0 \\wedge t | P(0) \\neq \\star]\\mathbb{P}^T\\left[P(0) \\neq \\star\\right] \\\\\n\t\t&\\le t\\mathbb{P}^T\\left[P(0) = \\star\\right] + \\mathbb{E}^T[\\tau^0 \\wedge t | P(0) \\neq \\star].\n\t\\end{align*}\n\t\nLemma \\ref{markov} gives $\\mathbb{P}^T[P(v) = \\star] = O(t^{-1\/4})$ and so\n\t\\begin{equation}\n      \\label{uyp1}\n\t\t\\mathbb{E}^T[\\tau^0 \\wedge t] \\le \\mathbb{E}^T[\\tau^0 \\wedge t | P(0) \\neq \\star] + O(t^{3\/4}).\n\t\\end{equation}\n\nLet $a = 3\\nu = 3 \\lceil t^{1\/4} \\rceil, b = \\zeta = \\lceil \\sqrt{t} \\rceil$. For an integer $m,$ let $H_m$ be the first hitting time of the random walk $X^0$ to $m.$ Lemma \\ref{sorting} tells us that if $P(0) \\neq \\star,$ then $P(0) \\ge -a.$ We therefore see $\\tau^0 \\wedge t = H_{P(0)} \\wedge t \\le H_{-a}.$ When $H_{-a} > H_b,$ we may trivially bound $\\tau^0 \\wedge t$ by $t.$ Putting this into \\eqref{uyp1} gives\n\t\\begin{align*}\n\t\t\\mathbb{E}^T[\\tau^0 \\wedge t]  \\le \\mathbb{E}^T[H_{-a} &| H_{-a} < H_b, P(0) \\neq \\star]\\mathbb{P}^T[H_{-a} < H_b | P(0) \\neq \\star] \\\\\n\t\t& + t\\mathbb{P}^T\\left[H_{-a} > H_b | P(0) \\neq \\star\\right] + O(t^{3\/4}).\n\t\\end{align*}\n\t\nClearly $X^0$ is independent from $P(0),$ which only depends on the initial configuration, and so\n\t\\begin{eqnarray*}\n\t\t\\mathbb{E}^T[\\tau^0 \\wedge t] \\le \\mathbb{E}[H_{-a} | H_{-a} < H_b] + t\\mathbb{P}\\left[H_{-a} > H_b\\right] + O(t^{3\/4}).\n\t\\end{eqnarray*}\n\t\nLemma \\ref{lbeforer} (i) and (ii) tells us that $\\mathbb{P}\\left[H_b<H_{-a}\\right] = O(t^{-1\/4})$ and $\\mathbb{E}[H_{-a} | H_{-a}<H_b] = O(t^{3\/4}).$ We therefore see that\n\t\\begin{eqnarray*}\n\t\t\\mathbb{E}^T[\\tau^0 \\wedge t] \\le O(t^{3\/4}) + tO(t^{-1\/4}) + O(t^{3\/4}) = O(t^{3\/4}).\n\t\\end{eqnarray*}\n\t\nFinally, we appeal to Corollary \\ref{lem:supermarket} to obtain\n\\[\n\\mathbb{E}[\\tau \\wedge t] = \\mathbb{E}^G[\\tau \\wedge t] \\leq \\mathbb{E}^T[\\tau \\wedge t] = O(t^{3\/4}).\n\\]\n\\end{proof}\n\n\n\\section{Lower bound on $\\mathbb{E}^T[\\tau \\wedge t]$}\n\\label{sec:p=1\/2lower}\n\nIn this section, we prove the lower bound in Theorem \\ref{thm:p=1\/2}. We do this by considering a parking process on an interval, and appealing to various properties of the simple symmetric random walk. While the underlying ideas are relatively simple, proving them rigorously requires a number of steps and some new ideas. We start with an outline of the proof.\n\n\\subsection{Outline of proof}\nInstead of considering the expected journey length up to time $t,$ we consider the expected proportion of cars that have parked by time $t.$ It is helpful to restrict ourselves to a finite interval, and this is where car removal strategies become useful. We know by Theorem~\\ref{thm:pinkfloyd} that by removing cars from the process we make it easier for the remaining cars to park. Therefore, any lower bound over an interval for the proportion of unparked cars gives a lower bound for $\\mathbb{E}[\\tau \\wedge t].$\n\nFrom here, we consider a long interval $L \\cup M \\cup R,$ where $L,M,R$ are the left, middle, and right subintervals respectively. We will choose the sizes of $L$ and $R$ so that with high probability no car from $M$ leaves $L \\cup M \\cup R$ by time $t.$ The idea is that with positive probability the number of cars starting in $M$ is a few standard deviations above the mean, creating an excess of cars, and that this excess is not relieved by what happens in $L$ and $R.$ To be able to quantify this, we introduce \\em swapping: \\em this is a way of switching positions of cars so that at any time, from left to right, we see the cars that started in $L,$ then $M,$ and then $R.$ This modification does not change the stochastic properties of the process, but does allow us to say how much relief $L$ and $R$ provide by way of parking spaces available to cars starting in $M.$\n\nFinally, we will bring everything together and appeal to Theorem \\ref{thm:pinkfloyd} to obtain the desired lower bound on $\\mathbb{E}[\\tau \\wedge t].$\n\n\\subsection{The car removal strategy and the swap-modification}\nWe define the car removal strategy $Q$ as follows. Fix integers $k >8$ and $\\ell >4.$ Let $\\zeta = \\lceil \\sqrt{t\\log t} \\rceil$. Then for each integer $r \\in \\mathbb{Z}$ we remove any car which attempts to make a step (in either direction) between $r(2(k+\\ell)\\zeta+1)$ and $r(2(k+\\ell)\\zeta+1)+1$.\n\nWe show that a proportion $(t\\log t)^{-1\/4}$ of cars remains active (i.e., unparked and not removed) at time $t$ under the car removal strategy $Q.$ To establish this, it is sufficient to consider the parking process on an interval of length $2(k+\\ell)\\zeta+1$ where we assume that cars leaving the interval at either end are removed. Let $L = \\mathbb{Z} \\cap [-(k+\\ell)\\zeta,-k\\zeta),$ let $M = \\mathbb{Z} \\cap [-k\\zeta,k\\zeta]$ and $R = \\mathbb{Z} \\cap (k\\zeta,(k+\\ell)\\zeta].$\n\nWe want to show that with positive probability we start with an excess of cars in $M$ which do not escape $L \\cup M \\cup R$ and that $L$ and $R$ do not offer up enough spare parking capacity. It turns out that quantifying what capacity $R$ and $L$ provide is not straightforward since one cannot easily separate what happens to the cars with respect to their starting positions. Particularly problematic is that cars starting in different sections ($L, M$, or $R$) may swap positions. The following modification of the process ensures that at any given time the active cars, as seen from left to right, started their journeys in $L$, then in $M$, and finally in $R$, and will prove very useful.\n\n\\begin{defn}[The modified parking process]\\label{con:modify}\nGiven the parking process $X$, we define a modified process $Y$ as follows. At time $0,$ label cars according to their starting intervals $L,M$ or $R.$ For $s\\geq 0$ we write $C(s)$ for the set of starting positions (in $L \\cup M \\cup R$) of the cars that are still active at time $s$ (hence $C(0)$ is the set of $i$ such that we initially place an active car at $i$). Further we write $C_L(s)$ for the set of starting positions of the cars that started in $L$ and are still active at time $s;$ we similarly define $C_M(s)$ and $C_R(s)$. For a car starting at $i$ which is still active at time $s$ we write $Y^i(s)$ to denote its position at time $s.$\n\nGiven the set $C(s)$ of cars active at time $s,$ and their positions $(Y^i(s): i \\in C(s)),$ we want to define $C(s+1)$ and the positions $(Y^i(s+1) : i \\in C(s+1)).$ We do this in several steps: at each step, we move the cars around in a way that preserves the number of cars at each location. We use $Z^i_1,Z_2^i,$ and $Z_3^i$ to denote intermediate rearrangements, preserving $Y^i$ for the final position. \n\nRoughly speaking: $Z_1$ is where the cars move according to their respective random walks. From $Z_1$ to $Z_2$ we swap cars so that no $L$-car is to the right of an $R$-car. From $Z_2$ to $Z_3$ we swap cars so that no $L$-car is to the right of an $M$-car. Finally, from $Z_3$ to $Y$ we swap cars so that no $M$-car is to the right of an $R$-car. The end result is a swapping of cars which preserves the number of cars at each vertex, is such that cars move by at most one in a single time step, and is such that from left to right the cars have labels $L$, then $M$, and then $R$.\n\n\t\\begin{itemize}\n\t\t\\item For any car active at time $s,$ define $Z^i_1(s+1) = Y^i(s) + (X^i(s+1)-X^i(s)).$\n\t\t\\item Let $i_1, \\ldots, i_{x} \\in L$ (with $Z_1^{i_k}(s+1)$ increasing in $k$) be the starting positions of cars labelled $L$ that are active at time $s$ and such that the move at time $s+1$ places them to the right of some active car labelled $R.$ Similarly, let $j_1,\\ldots,j_{y} \\in R$ (with $Z_1^{j_k}(s+1)$ increasing in $k$) be the starting positions of cars labelled $R$ that are active at time $s$ and such that the move at time $s+1$ places them to the left of some active car labelled $L.$ \n\n\t\tWe rearrange the cars as follows: for all $i \\notin \\{i_1, \\ldots, i_{x}, j_1,\\ldots,j_{y} \\}$ let $Z^i_2(s+1) = Z^i_1(s+1).$ Let $(m_1, \\ldots, m_{x+y})$ be a permutation of $\\{i_1, \\ldots, i_{x}, j_1, \\ldots, j_{y}\\}$ with $Z^{m_k}_1(s+1)$ increasing in $k$. Then, for $1 \\leq \\ell \\leq x,$ let $Z^{i_\\ell}_2(s+1) = Z^{m_\\ell}_1(s+1),$ and for $1 \\leq \\ell \\leq y,$ let $Z^{j_\\ell}_2(s+1) = Z^{m_{x+\\ell}}_1(s+1).$ After this procedure, no car labelled $L$ is to the right of a car labelled $R.$\t\t\n\t\t\\item Given $Z^i_2(s+1)$ for all $i \\in C(s),$ we define $Z^i_3(s+1)$ by reordering in a similar way the positions $Z^i_2(s+1)$ of the cars that started in $L$ or in $M$ in such a way that no car that started in $L$ has a car that started in $M$ to its left:\n\nLet $i_1, \\ldots, i_{x} \\in L$ (with $Z_2^{i_k}(s+1)$ increasing in $k$) be the starting positions of cars labelled $L$ that are active at time $s$ and such that the move at time $s+1$ and the previous rerrangement places them to the right of some active car labelled $M.$ Similarly, let $j_1,\\ldots,j_{y} \\in M$ (with $Z_2^{j_k}(s+1)$ increasing in $k$) be the starting positions of cars labelled $M$ that are active at time $s$ and such that the move at time $s+1$ and previous rearrangement places them to the left of some active car labelled $L.$ \n\n\t\tWe rearrange the cars as follows: for all $i \\notin \\{i_1, \\ldots, i_{x}, j_1,\\ldots,j_{y} \\}$ let $Z^i_3(s+1) = Z^i_2(s+1).$ Let $(m_1, \\ldots, m_{x+y})$ be a permutation of $\\{i_1, \\ldots, i_{x}, j_1, \\ldots, j_{y}\\}$ with $Z^{m_k}_2(s+1)$ increasing in $k$. Then, for $1 \\leq \\ell \\leq x,$ let $Z^{i_\\ell}_3(s+1) = Z^{m_\\ell}_2(s+1),$ and for $1 \\leq \\ell \\leq y,$ let $Z^{j_\\ell}_3(s+1) = Z^{m_{x+\\ell}}_2(s+1).$\n\nNote that this operation can only move cars labelled $L$ to the left; hence we still have no car labelled $L$ to the right of a car labelled $R.$\n\t\t\\item Finally, given $Z^i_3(s+1)$ for all $i \\in C(s),$ we define $Y^i(s+1)$ by reordering in a similar way the positions $Z^i_3(s+1)$ of the cars that started in $M$ or in $R$ in such a way that no car that started in $R$ has a car that started in $M$ to its right. Again, note that this operation only moves  cars labelled $R$ to the right, hence we still have no car labelled $L$ to the right of a car labelled $R.$ Moreover, a car labelled $M$ can only be moved to a position $Z^i_3$ previously occupied by a car labelled $M$ or $R,$ which we know has no car labelled $L$ to its right; hence the same holds about cars labelled $M$ after the rearrangement.\n\t\\end{itemize}\n\nIf a single car starting at $i$ reaches an empty parking space at $Y^i(t),$ then it parks there. When at least two cars simultaneously arrive at a parking space $v$ at time $t,$ we choose the car $i$ labelled $L$ with smallest $U^i_t$ to park there; in the absence of a car labelled $L,$ the car $i$ labelled $R$ with smallest $U^i_t$ parks there; finally, if only cars labelled $M$ meet at $v,$ the car $i$ with smallest $U^i_t$ parks there. When a car leaves $L \\cup M \\cup R,$ we say it is \\em inactive \\em and remove it from the process. We say that a car becomes \\emph{left-inactive} if it reaches $\\min L -1,$ and it becomes \\emph{right-inactive} if it reaches $\\max R +1.$ Finally, let $C(s+1) \\subseteq C(s)$ be the set of cars active at time $s$ that have neither parked nor become inactive at time $s+1.$\n\\end{defn}\n\n\\begin{rem}\n \\label{rem:limitedDrive}\nIn the process described in Definition \\ref{con:modify}, for any $i \\in \\mathbb{Z},$ $Y^i(s+1) - Y^i(s) \\in \\{-1,0,+1\\}$ -- the total move of a car in a step is at most one. Indeed, consider an arbitrary car $i$ labelled $M$ with $Y^i(s) = j$. At time $s$ it has no cars labelled $L$ strictly to its right and no cars labelled $R$ strictly to its left. At time $s+1,$ all cars labelled $L$ can only drive to positions at most $j+1$ (so $Z^k_1(s+1) \\le j+1$ for each $k \\in C_L(s)$), and cars labelled $R$ drive to positions at least $j-1$ (so $Z^k_1(s+1) \\ge j-1$ for each $k \\in C_R(s)$). It is not possible to move $i$ to a position strictly to the left of the left-most (according to $Z_1$) car labelled $R$ so that $Y^i(s) \\ge j-1$. Similarly it is not possible to move $i$ to a position strictly to the right of the right-most (according to $Z_1$) car labelled $L$ so that $Y^i(s) \\leq j+1$. Similar arguments apply to cars labelled $L$ or $R.$\n\\end{rem}\n\nLet $\\widetilde\\mathbb{P}$ be the probability measure with respect to the modified parking process. If we ignore the labels of the cars, then the difference from the original parking process under $Q$ is that we swap some future trajectories of cars. Since the swapping is determined by past trajectories, the unlabelled modified process has the same distribution as the original parking process with car removal strategy $Q$. Thus \n\t\\begin{align}\n\t\t\\widetilde\\mathbb{E}[\\#\\mbox{active cars in $L \\cup M \\cup R$ at time $t$}] = \\mathbb{E}^Q[\\#\\mbox{active cars in $L \\cup M \\cup R$ at time $t$}]. \\label{telematch}\n\t\\end{align}\n\n\\subsection{Proof of the lower bound}\nBefore completing the proof of Theorem \\ref{thm:p=1\/2}, we prove some preliminary lemmas concerning the modified parking process. Unless stated otherwise, we assume that we are dealing with the modified parking process (Definition \\ref{con:modify}) throughout this section.\n\nFirst we consider how many cars from $L$ and $R$ become inactive. Intuitively this should be maximised if the cars drive monotonically towards the ends of the interval. Given the initial arrangement of cars and parking spaces on $L,$ let $D_L = D_L(t)$ be the number of cars starting in $L$ which would become left-inactive by time $t$ should all cars with label $L$ move left deterministically. Similarly let $D_R = D_R(t)$ denote the number of cars with label $R$ that become right-inactive by time $t$ in the process where all cars with label $R$ move right deterministically. The next lemma shows that this intuition is correct.\n\n\\begin{lem}\\label{claim:lemming}\nThe number of cars with label $L$ which become left-inactive by time $t$ is at most $D_L.$\n\\end{lem}\n\n\\begin{proof}\nUnder $\\widetilde\\mathbb{P},$ suppose that $j$ is the smallest integer which has a parking space either unfilled or filled by a car labelled $M$ or $R$ by time $t.$ Let $J = L \\cap [-(k+\\ell)\\zeta,j-1].$ We claim that the only cars labelled $L$ that can become left-inactive are the cars from $J,$ and only cars originating in $J$ park in $J.$ First suppose that $j$ is unfilled. No car from the right of $j$ passes through $j$ (else it would park there) and so no car from the right of $j$ can become left-inactive.\n\nSo suppose that car $w$ (labelled $M$ or $R$) parks in $j$. Under $\\widetilde\\mathbb{P}$ at any time, from left to right, the unparked cars have labels $L,$ then $M,$ and then $R.$ Therefore, any car $v$ labelled $L$ originating from an integer greater than $j,$ before it parks, must stay to the left of the car $w$ which parks in $j.$ Since cars in the modified process move at most one step at each time, the car $v$ cannot be unparked at time $t$ since it would have visited $j$ before $w$ parks there. Similarly, $v$ cannot park to the left of $j$ since it would first pass through $j$ (before $w$ parks there). Therefore, any car labelled $L$ originating from an integer greater than $j$ must have parked in a spot greater than $j.$\n\nSuppose that cars starting at positions $i_1<\\dots<i_N < j$ become left-inactive starting from $J.$ Observe that every parking space to the left of $i_N$ must be filled by a car originating from $J$ (otherwise, the car starting in $i_N$ must reach a free parking space on its route to $\\min L -1$). Let $p=j-1$ if all parking places in $J$ are filled in the process, and otherwise let $p+1$ be the leftmost empty parking space in $J$ at time $t.$ We see that all parking spaces to the left of $p+1$ must be filled by cars originating from the left of $p+1$ (a car starting to the right of $p$ would fill $p+1$ first). But then there must be a surplus of $N$ cars to the left of $p+1.$\n\nIf all the cars drove left deterministically, this surplus would result in at least $N$ cars, starting to the left of $p+1,$ becoming left-inactive. Thus we have $D_L \\geq N,$ proving the claim.\n\\end{proof}\n\n\\begin{rem}\nClearly, the analogous claim that the number of cars from $R$ becoming right-inactive is bounded by $D_R$, also holds.\n\\end{rem}\n\nNote that $D_L$ and $D_R$ are dependent only on the initial car configuration $(B^i)_{i \\in \\mathbb{Z}}.$ Let $S_L$ be the number of cars which start in $L$ and let $P_L$ be the number of parking spaces in $L$ (hence clearly $S_L+P_L = |L|$). Similarly define $S_R$ and $P_R.$\n\n\\begin{lem}\\label{claim:easing}\nThere exists $\\varepsilon > 0$ (independent of $t$) such that\n\\[\n \\mathbb{P}\\left[S_L-P_L-D_L \\ge -(t\\log t)^{1\/4}\\right] > \\varepsilon.\n\\]\n\\end{lem}\n\n\\begin{proof}\nConsider the simple symmetric random walk starting at $0$ which increases at time $i \\geq 1$ if the $i$th rightmost point in $L$ initially contains a car, and decreases if the $i$th rightmost point in $L$ contains a parking space. Suppose that while traversing $L,$ the walk last attains its minimum value $-m \\leq 0$ at time $j,$ and let $x$ be the $j$th rightmost point in $L.$ Then, in the process where all cars in $L$ deterministically drive left, every car starting to the right of $x$ finds a parking place, the process ends with $m$ empty spots to the right of $x-1,$ every spot to the left of $x$ is filled by a car, and all the cars that do not park reach the left end of the interval and become left-inactive.\n\nThe number of parked cars in this process is $S_L-D_L,$ and so the number of unfilled parking spaces is $P_L-S_L+D_L.$ Therefore $S_L - P_L-D_L = -m.$ From the previous paragraph, we see that $S_L - P_L-D_L$ is distributed like the minimum of a simple symmetric random walk of length $\\ell \\zeta.$ So by Lemma \\ref{minssrw}(ii) it is at least $-(t\\log t)^{1\/4}$ with probability bounded away from zero.\n\\end{proof}\n\\begin{rem}\n An analogous claim holds if we replace $S_L, P_L, D_L$ with $S_R, P_R, D_R$ respectively.\n\\end{rem}\n\nWe would like to say that no car from $L$ becomes right-inactive. Indeed, we could then say that at time $t,$ the number of cars from $L$ (possibly parked) still in $L\\cup M \\cup R$ minus the number of parking spaces (filled or unfilled) in $L$ is at least $S_L-P_L-D_L \\ge -(t\\log t)^{1\/4}$ with probability at least $\\varepsilon.$ The next result shows that this occurs, and also that no car from $M$ becomes inactive.\n\n\\begin{lem}\\label{claim:restrict}\nWith probability $1-o(1\/t),$ the random walks $\\left(X^i\\right)_{i\\in L \\cup M \\cup R}$ are such that for all possible starting configurations of active cars and parking places in $L \\cup M \\cup R,$ in the first $t$ time steps: no car starting in $M$ becomes inactive, no car starting in $L$ reaches $R,$ and no car starting in $R$ reaches $L.$\n\\end{lem}\n\\begin{proof}\nFor each $i \\in L \\cup M \\cup R,$ let $M^i$ be the maximum of $\\{X^i_s - i : s \\le t\\},$ and $m^i$ the minimum of $\\{X^i_s -i : s \\le t\\}.$ By Lemma \\ref{minssrw} (i), $\\mathbb{P}\\left[m^i \\le - 4\\sqrt{t \\log t}\\right]=\\mathbb{P}\\left[M^i \\ge 4\\sqrt{t \\log t}\\right] \\le 2t^{-8}.$ Hence by the union bound, with failure probability $o(t^{-1}),$ for all $i \\in L \\cup M \\cup R$ the random walks $X^i$ are at distance at most $4\\zeta$ from their corresponding starting point $i$ until time $t.$\n\nAssume that for all $i \\in L \\cup M \\cup R,$ $X^i$ is at distance at most $4\\zeta$ from $i$ until time $t.$ We now show that for all starting configurations of active cars and parking places in $L \\cup M \\cup R,$ in the first $t$ time steps, no car starting in $M$ becomes inactive, no car starting in $L$ reaches $R,$ and no car starting in $R$ reaches $L.$\n\nConsider a car starting at $i \\in L.$ If the car is still active at time $s$ in the modified parking process, then $Y^i(s) \\leq X^i(s),$ as if the position of the car is ever changed as a result of landing to the right of a car labelled $M$ or $R,$ then it can only be pushed further left. Therefore it stays to the left of $(4-k)\\zeta.$ Similarly all cars labelled $R$ stay to the right of $(k-4)\\zeta.$ Since $k > 8,$ no car from $L$ reaches $R,$ and vice versa.\n\nNow consider a car starting at $i \\in M.$ If the position of the car is never changed due to moving past a car labelled $L$ or $R,$ then it never reaches a point more than $4\\zeta$ from $i$ and so cannot become inactive (recall that $\\ell > 4$).\n\nHence suppose the car at some point has its position changed due to finding itself to the left of a car labelled $L.$ This implies that the car must at some point be to the left of $(4-k)\\zeta$ (or else it cannot pass a car labelled $L$). If the car reaches $(k-4)\\zeta + 1$ at some point, then there must be a passage of the car between $(4-k)\\zeta$ and $(k-4)\\zeta$ contained within $[(4-k)\\zeta,(k-4)\\zeta].$ In this segment, the position of the car cannot be changed as it keeps all cars labelled $L$ to its left, and all cars labelled $R$ to its right. Therefore it moves according to $X^i,$ and so $X^i$ reaches points $2(k-4)\\zeta > 8 \\zeta$ apart (recall that $k > 8$). This cannot happen since the maximum modulus of $X^i-i$ is at most $4\\zeta.$\n\nTherefore the car does not have its position changed due to being to the right of a car labelled $R.$ So while the car remains active, its position is bounded below by $X^i$ (having its position changed can only push its the car to the right). Since the car does not reach $(k-4)\\zeta,$ we see that the position of the car is contained in $[(-k-4)\\zeta,(k-4)\\zeta]$ and so the car cannot become inactive (as $\\ell > 4$).\n\nThe argument for a car which at some point finds itself to the right of a car labelled $R$ is identical. We conclude that no car originating from $M$ becomes inactive.\n\\end{proof}\n\nWe are now in a position to prove Theorem \\ref{thm:p=1\/2}.\n\n\\begin{proof}[Proof of the lower bound in Theorem \\ref{thm:p=1\/2}]\nIt is enough to show that with probability bounded away from zero (say at least $\\delta > 0$), at time $t$ there are at least $(t \\log t)^{1\/4}$ active cars in $L \\cup M \\cup R$ in the modified process. If this holds, then the result easily follows by symmetry, Theorem \\ref{thm:pinkfloyd} and \\eqref{telematch}:\n\t\\begin{align*}\n\t\t\\mathbb{E}[\\tau \\wedge t] & = \\frac{\\mathbb{E}^N\\left[\\sum_{v \\in L\\cup M \\cup R} \\tau^v \\wedge t\\right]}{|L \\cup M \\cup R|} \\\\\n\t\t& \\ge \\frac{\\mathbb{E}^Q\\left[\\sum_{v \\in L\\cup M \\cup R} \\tau^v \\wedge t\\right]}{|L \\cup M \\cup R|} \\\\\n\t\t& \\ge \\frac{\\widetilde\\mathbb{E}[\\# \\mbox{active cars at time $t$ in $L \\cup M \\cup R$}]}{2(k+\\ell)\\zeta+1} \\cdot t  \\\\\n\t\t& = \\frac{\\widetilde\\mathbb{E}[\\# \\mbox{active cars at time $t$ in $L \\cup M \\cup R$}]}{2(k+\\ell)\\zeta+1} \\cdot t  \\\\\n\t\t& \\ge \\frac{\\delta (t \\log t)^{1\/4}}{2(k+\\ell)\\zeta+1} \\cdot t  \\\\\n\t\t& = \\Omega(t^{3\/4} \\log^{-1\/4} t).\n\t\\end{align*}\n\nLet $I_L$ be the number of cars starting in $L$ that become left-inactive and let $I_R$ be the number of cars starting in $R$ that become right-inactive. Analogously to $L$ and $R$, let $S_M$ be the number of cars which start in $M$ and let $P_M$ be the number of initial parking places in $M.$ Hence, in total there are $P_L+P_M+P_R$ parking places in $L \\cup M \\cup R.$\n\nSuppose that in the first $t$ steps of the process, no car starting in $M$ becomes inactive, no car starting in $L$ reaches $R,$ and no car starting in $R$ reaches $L.$ Then at time $t,$ the number of cars (active or parked) in $L \\cup M \\cup R$ is $S_M  + (S_L - I_L) + (S_R  - I_R).$ By Lemma \\ref{claim:lemming} this is at least $S_M  + (S_L - D_L) + (S_R  - D_R).$ Since only one car can park in a parking space, the number of active cars in $L \\cup M \\cup R$ at time $n$ must be at least\n\t\\begin{eqnarray}\n\t\t(S_M-P_M) + (S_L-P_L)-D_L + (S_R-P_R)-D_R. \\label{eq:nexcess}\n\t\\end{eqnarray}\n\nObserve that $S_M-P_M$ is determined by the starting configuration in $M,$ $S_L-P_L-D_L$ is determined by the starting configuration in $L,$ and $S_R-P_R-D_R$ is determined by the starting configuration in $R.$ Therefore these random variables are mutually independent. Let $C^M$ be the event that $S_M - P_M$ is at least $3(t\\log t)^{1\/4},$ let $C^L$ be the event that $S_L-P_L-D_L \\geq -(t\\log t)^{1\/4},$ and let $C^R$ be the event that $S_R-P_R-D_R \\geq - (t\\log t)^{1\/4}.$\n\nLet $A$ be the random event, depending on the random walks $X^i$ only, that for all initial configurations of cars and parking places in $L \\cup M \\cup R,$ no car from $M$ becomes inactive, no car from $L$ reaches $R,$ and no car from $R$ reaches $L.$ Observe that $A,C^L, C^M,C^R$ are mutually independent events. By Lemma \\ref{claim:restrict}, $A$ occurs with high probability. By Lemma \\ref{claim:easing} both $C^L$ and $C^R$ occur with probability bounded away from zero. Let $K = 2k\\zeta + 1 \\approx 2k\\sqrt{t\\log t}.$ Since $$S_M - P_M = 2k\\zeta +1 - 2P_M \\sim K - 2\\mathrm{Bin}(K,1\/2),$$ we have\n\t\\begin{align*}\n\t\t\\mathbb{P}\\left[C^M\\right] & = \\mathbb{P}\\left[\\mathrm{Bin}(K,1\/2) \\leq \\frac{K - 3(t\\log t)^{1\/4}}{2}\\right]  \\\\\n\t\t& = \\mathbb{P}\\left[\\frac{\\mathrm{Bin}(K,1\/2) - \\frac{K}{2}}{\\sqrt{\\frac{K}{4}}} \\leq \\frac{-6(t\\log t)^{1\/4}}{\\sqrt{K}}\\right].\n\t\\end{align*}\nBy the Central Limit Theorem, this probability tends to $\\Phi(-\\frac{6}{\\sqrt{2k}})$ as $t$ tends to infinity. Therefore $C^M$ occurs with probability bounded away from zero. So all four events $A,C^L, C^M,C^R$ occur simultaneously with probability bounded away from zero.\n\nSuppose that the events $A,C^L,C^M,C^R$ all occur. Then recalling equation \\eqref{eq:nexcess} we see that the number of active cars in $L \\cup M \\cup R$ at time $t$ is at least\n\\begin{align*}\n (S_M-P_M) + (S_R-P_R)-D_R + (S_L-P_L)-D_L & \\ge 3 (t\\log t)^{1\/4} - (t\\log t)^{1\/4} - (t\\log t)^{1\/4} \\\\\n  & = (t\\log t)^{1\/4}.\n\\end{align*}\n\nWe conclude that with probability bounded away from zero, there are at least $(t\\log t)^{1\/4}$ active cars in $L \\cup M \\cup R$ at time $t.$ The lower bound $\\Omega(t^{3\/4} \\log^{-1\/4} t)$ on $\\mathbb{E}[\\tau \\wedge t]$ follows.\n\\end{proof}\n\n\\section{Subcritical parking on $\\mathbb{Z}$}\n\\label{sec:p<1\/2}\n\nIn this section we prove Theorems \\ref{thm:p<1\/2} and \\ref{thm:critexp}. This is done in two parts. First, for a car starting at $0$, we consider the smallest $J$ (depending only on the initial configuration of cars) such that no matter what the other cars do, there is always a free parking space in both $[1,J]$ and $[-J,-1]$. Given $J$, we know that the car starting at $0$ cannot reach either $-J$ or $J$ before it parks. Calculating the expected journey length of $0$ is then carried out by proving tail bounds on the random variable $J$.\n\nWe start with the following simple lemma, which we state here without proof.\n\n\\begin{lem}\\label{lem:markstat}\nLet $p \\in (0,1\/2),$ and let $Y=Y(p)$ be a Markov chain on $\\mathbb{N} \\cup \\{0\\}$ with $Y_0=0$ and transition probabilities $(p_{i,j})_{i,j \\in \\mathbb{N} \\cup \\{0\\}}$ where\n\\[\np_{i,j} = \\begin{cases}\np, \\quad & j = i+1, \\\\\n1-p, \\quad & j = i - 1 \\ge 0 \\mbox{ or } i=j=0, \\\\\n0, \\quad & \\mbox {otherwise.}\n\\end{cases}\n\\]\nThen $Y$ has stationary distribution $\\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$. Furthermore, since $Y$ is an aperiodic and irreducible Markov chain, $Y_t \\rightarrow \\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$ in distribution as $t \\rightarrow \\infty$.\n\\end{lem}\n\nLet $E^L(t)$ be the number of cars in $[-t,-1]$ that would reach $0$ if all cars deterministically drove right. We also define $E^R(t)$ to be the number of cars in $[1,t]$ which would reach $0$ if all cars deterministically drove left. Note that $(E^L(t))_{t \\in \\mathbb{N}}$ is an increasing sequence of random variables. Finally, let $E^L$ be the number of cars in $(-\\infty,-1]$ that would reach $0$ if all cars deterministically drove right (and analogously define $E^R$). Note that $E^L(t)$ increases almost surely to $E^L$ as $t \\rightarrow \\infty$.\n\n\\begin{lem}\n\\label{lem:thedescent}\nFor all $p < 1\/2,$ $E^L \\sim \\Geom_{\\ge 0}(\\tfrac{1-2p}{1-p})$.\n\\end{lem}\n\\begin{proof}\nSince $(E^L(t))_{t \\ge 1}$ increases almost surely to $E^L$, it is sufficient to show that $E^L(t) \\to \\Geom_{\\ge 0}(\\tfrac{1-2p}{1-p})$ in distribution as $t \\to \\infty$. To compute $E^L(t)$, consider forming a queue of cars from left to right in $[-t,-1]$: Let $Q_0 = 0$ (there is initially no queue), then given $Q_i$, we set $Q_{i+1} = Q_i+1$ if there is initially a car at $i-t$ (a car is added to the queue), $Q_{i+1} = Q_i-1$ if $Q_i > 0$ and there is initially a parking space at position $i-t$ (a car from the queue is parked), and $Q_{i+1}=0$ otherwise. Then $Q_t = E^L(t)$. On the other hand, $(Q_s : s \\le t)$ is distributed like $(Y_s : s \\le t)$ in Lemma \\ref{lem:markstat}, and so $E^L(t)$ has the same distribution as $Y_t$ (with $Y_0 = 0$). By Lemma~\\ref{lem:markstat}, $E^L(t) \\rightarrow \\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$ in distribution as $t \\rightarrow \\infty$.\n\\end{proof}\n\nClearly, $E^R(t)$ also increases almost surely to the random variable $E^R$ which is distributed like a $\\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$ random variable (and is independent of $E^L$).\n\nFor all $r \\ge 0$, let ${E}_r^R(t)$ be the number of cars in $[r+1,r+t]$ that would reach $r$ if all cars deterministically drove left, and similarly let ${E}_r^L(t)$ be the number of cars in $[-r-t,-r-1]$ that would reach $-r$ if all cars deterministically drove right. Let ${E}_r^R$ and ${E}_r^L$ be the limits as $t \\to \\infty$ respectively of ${E}_r^R(t)$ and ${E}_r^L(t)$. Note that Lemma \\ref{lem:thedescent} holds with $E^L$ replaced by $E^R_r$, as well as by $E^L_r$. For all $r \\ge 1$, let $S_r^R$ and $S_r^L$ be the number of cars that start in $[1,r]$ and $[-r,-1]$ respectively.\n\nIn the proof of Theorem \\ref{thm:p<1\/2}, we show that at most ${E}_K^R+E^L + S_K^R$ cars from $\\mathbb{Z}\\setminus \\{0\\}$ can be present in $[1,K]$ at any time. This means that at most ${E}_K^R+E^L + S_K^R$ parking spaces in $[1,K]$ can be filled by cars from $\\mathbb{Z} \\setminus \\{0\\}$. Therefore, if ${E}_K^R+E^L + S_K^R < K\/2,$ then there must be a parking space in $[1,K]$ not filled by a car from $\\mathbb{Z} \\setminus \\{0\\}$ (consider that there are initially $K-S_K^R$ parking spaces in $[1,K]$). It follows that a car starting at $0$ parks before reaching $K$.\n\nIn the proof of Theorem \\ref{thm:p<1\/2}, we first condition on the smallest $K$ such that both ${E}_K^R+E^L + S_K^R < K\/2$ and ${E}_{K}^L+E^R + S_K^L < K\/2.$ These conditions mean that a car starting at $0$ will have parked by the time its associated random walk $X^i$ hits either $-K$ or $K.$\n\n\\begin{proof}[Proof of Theorem \\ref{thm:p<1\/2}]\nLet $p < 1\/2$ and let $J$ be the smallest $K$ such that ${E}_K^R+E^L + S_K^R < K\/2$ and ${E}_{K}^L+E^R + S_K^L < K\/2$ if such a $K$ exists, and let $J = \\infty$ otherwise. For a given $a \\in \\mathbb{N},$ let $H_a$ be the first hitting time of $a$ by the random walk $X^0.$ We claim that if $J=N,$ then $\\tau^0 \\le H_{-N} \\wedge H_N.$ We justify this by showing that at any time $t \\ge 0,$ there are at most ${E}_N^R+E^L + S_N^R$ cars excluding car $0$ (parked or not) present in $[1,N]$ at time $t.$ A similar statement can be shown for cars present in $[-N,-1].$\n\nLet us temporarily exclude the car starting at $0$ from the parking process (e.g., assume that this car never decides to park) and suppose that at time $t,$ there are $B$ cars that started in $[N-t+1,N]\\setminus \\{0\\}$ parked in $[N+1,N+t].$ Let $R$ be the number of cars that start in $[N+1,N+t]$ that are in $[N-t+1,N]$ at time $t.$ By an argument identical to that of Lemma \\ref{claim:lemming}, we have the bound $R \\le B + E_N^R(t)$ since each parked car from $[N-t+1,N]$ that parks inside $[N+1,N+t]$ can only increase the number of cars that reach $N$ from $[N+1,N+t]$ by $1.$ Similarly, if $C$ is the number of cars that started in $[1,t]$ and parked in $[-t,-1],$ and $L$ is the number of cars that start in $[-t,-1]$ present in $[1,N]$ at time $t,$ we have $L \\le C + E^L(t).$ So the number of cars present in $[1,N]$ at time $t$ is\n\t\\begin{align*}\n\t\tS_N^R + R + L - B - C & \\leq S_N^R + E^L(t) + E_N^R(t) \\\\\n\t\t& \\leq S_N^R + E^L + E^R_N.\n\t\\end{align*}\nSince $J = N,$ this quantity is strictly less than $N\/2.$ On the other hand, there are initially $N-S_N^R > N\/2$\nparking spaces in $[1,N]$ and so car $0$ must go through an empty parking space before reaching $N.$ A similar argument applies to $[-N,-1].$ In the real process, where car $0$ tries to park, this implies car $0$ parks before reaching $N$ or $-N.$\n\nIf $J < \\infty$ almost surely, we therefore have\n\t\\[\n\t\t\\mathbb{E}[\\tau^0] \\le \\sum_{N \\ge 1}\\mathbb{P}\\left[J=N\\right]\\mathbb{E}[H_{-N} \\wedge H_N | J = N].\n\t\\]\nBy independence and Lemma \\ref{lbeforer} (iii) we have\n\\[\n\t\\mathbb{E}[H_{-N} \\wedge H_N | J = N] = \\mathbb{E}[H_{-N} \\wedge H_N] = N^2,\n\\]\nand so, assuming again that $J < \\infty$ with probability $1$,\n\t\\begin{align}\n\t\t\\mathbb{E}[\\tau^0] \\le \\sum_{N \\ge 1}N^2 \\mathbb{P}\\left[J=N\\right]. \\label{Qbound}\n\t\\end{align}\n\nWe now consider the distribution of $J$. If $J$ is at least $N,$ then by averaging one of the following must happen:\n\\begin{itemize}\n\\item[(i)] One of $S_N^R$ and $S_N^L$ is at least $N(p + (1\/4 - p\/2)).$\n\\item[(ii)] One of $E^L, E^R, {E}_{N}^L$, and ${E}_N^R$ is at least $N(1\/8-p\/4).$\n\\end{itemize}\nClearly $S_N^R$ and $S_N^L$ are both distributed like $\\Bin(N,p)$ random variables and so, by Lemma \\ref{feelthechern}, the probability that (i) occurs is at most $2e^{-(1\/2-p)^2N\/2}.$ On the other hand, by Lemma \\ref{lem:thedescent}, we know that $E^L, E^R, {E}_N^L$ and ${E}_{N}^R$ are all distributed like $\\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$ random variables. If $X~\\sim~\\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$, then $\\mathbb{P}\\left[X \\geq N(1\/8-p\/4)\\right] \\leq \\bigl(1-\\frac{1-2p}{1-p}\\bigr)^{N(1\/8-p\/4)}$, and so the probability that (ii) occurs is at most\n\\[\n4\\mathbb{P}\\left[\\Geom_{\\ge 0}\\left(\\frac{1-2p}{1-p} \\right) \\ge N(1\/8-p\/4)\\right] \\leq 4\\left(1-\\frac{1-2p}{1-p}\\right)^{N(1\/8-p\/4)}.\n\\]\nPutting these together we see that for all $N \\ge 1$ we have\n\\begin{align*} \n\t\t\\mathbb{P}\\left[J = N\\right] \\le \\mathbb{P}\\left[J \\ge N\\right] & \\le 2e^{-(1\/2-p)^2N\/2} + 4\\biggl(1-\\frac{1-2p}{1-p}\\biggr)^{N(1\/8-p\/4)} \\\\\n\t\t& \\le 2e^{-(1\/2-p)^2N\/2} + 4e^{-\\frac{1-2p}{1-p}N(1\/8-p\/4)}.\n\\end{align*}\nAs the above bound on $\\mathbb{P}\\left[J \\ge N\\right]$ tends to $0$ as $N \\to \\infty$, we see that $J < \\infty$ almost surely. Hence, putting the obtained bound into \\eqref{Qbound} gives\n\t\\begin{align}\n\t\t\\mathbb{E}[\\tau^0] \\le \\sum_{N \\ge 1}N^2\\left[2e^{-(1\/2-p)^2N\/2} + 4e^{-\\frac{1-2p}{1-p}N(1\/8-p\/4)}\\right]. \\label{critequat}\n\t\\end{align}\nThis series converges for any $p < 1\/2$ and so $\\mathbb{E}[\\tau^0]$ is finite.\n\\end{proof}\n\nFinally, we come to proving Theorem \\ref{thm:critexp}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:critexp}]\nLet $p < 1\/2$ and write $\\varepsilon = 1\/2 - p$. By \\eqref{critequat} \n\n\t\\begin{align*}\n\t\t\\mathbb{E}[\\tau^0] &\\le \\sum_{N \\ge 1}N^2\\left[2e^{-\\frac{\\varepsilon^2}{2}N} + 4e^{-\\frac{\\varepsilon^2}{1+2\\varepsilon}N}\\right]  \\le 6\\sum_{N \\ge 1} N^2e^{-\\varepsilon^2N\/2}.\n\t\\end{align*}\n\nThis sum can be approximated by the integral $\\int_{0}^{\\infty}x^2e^{-\\varepsilon^2x\/2}dx$. By considering the pdf of a $\\Gamma\\left(3,\\varepsilon^2\/2\\right)$ random variable we get a bound of the form $O\\left(\\varepsilon^{-6}\\right).$\n\n\\end{proof}\n\n\\section{Further questions}\\label{yaymorewaffle}\nThere is still a gap between the upper and lower bounds in Theorem \\ref{thm:p=1\/2}. Following the conjecture presented in \\cite{Junge-seminar}, we also believe that the upper bound gives the right order $t^{3\/4}.$\n\nIt would be interesting to know what happens in higher dimensions, where the problems seem to become more difficult and are likely to require additional ideas. It is also natural to ask what happens in other lattices: for example, are there analogous results to Theorems \\ref{thm:p=1\/2} and \\ref{thm:p<1\/2} that hold for the hexagonal lattice? We remark that Damron, Gravner, Junge, Lyu and Sivakoff \\cite[Open Questions 1 and 2]{DGJLS} have conjectures here (which we believe to be true).\n\nFinally, what can we say for more general jump distributions? We conjecture that if the increments of the random walks $X^i$ on $\\mathbb{Z}$ are bounded, then Theorems \\ref{thm:p=1\/2} and \\ref{thm:p<1\/2} should still hold. Although similar methods could work, one would have to be careful about specifying parking places for cars (as in the parking strategy $T$ in Section~\\ref{sec:p=1\/2upper}) as cars might jump over them.\n\n\\bibliographystyle{amsplain}\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe Hawking radiance of a black hole is usually characterized by the mean number of quanta emitted in each mode~\\cite{Hawking0}.  A more precise characterization entails specification of the probability distribution of the number of quanta in each mode.  Such distributions were first calculated early~\\cite{Parker,Wald,Hawking}.  In later years the issue of unitary evolution has convinced many workers that the radiance must be in a pure quantum state.  Full characterization of such state would entail, at the very least, giving the \\emph{joint} probability distribution of the number of quanta emitted in the various modes.  Since the purity of the radiance raises still unsolved problems, we focus here on the one-mode marginal probability distributions; by marginalizing the joint distribution we remain neutral with regard to the ultimate outcome of the said controversy.  \n\nMore specifically, we here investigate the relation between the one-mode response of a Kerr black hole to incident bosonic radiation and the energy spectrum for Kerr black holes.  The question we focus on is whether any insight into quantum gravity can be had from such investigation. The ready answer would be that the thermality of the Hawking radiance, which seems incontestable if the purity status is rejected, should make it blind to any specifics of the black hole structure and dynamics, so that nothing can be learned from it about quantum gravity.  In our view this is a simplistic response.  To draw an analogy, the radiation emitted by an atom says a lot about its quantum structure. Only when numerous atoms radiate together incoherently does the radiation assume a thermal guise. Now to some extent the black hole, in its simplicity, is the hydrogen atom of quantum gravity~\\cite{BekMG,Corda}.  By analogy we could expect to glean insights into this subject from detailed analysis of the black hole radiance which, after all, emanates from a coherent gravitational structure, not from an assembly of quantum systems.  Indeed, as we shall show, a definite assumption about the horizon area spectrum of a black hole recovers, by simple argumentation, the correct one-mode response probability distribution.\n\nWe shall focus here on the model of a quantum black hole espoused in Refs.~\\cite{spectrum,Mukh,BekMukh}.   Its central claim is that the horizon area $A$ (or entropy $S$) is quantized in equally spaced steps.  For the Kerr black hole (mass $M$ and angular momentum $J$) this means (we take $c=1$)\n\\begin{equation}\nA=4\\pi G^2 \\big(M+\\sqrt{M^2-J^2\/G}\\big)^2 =\\alpha L_P{}^2\\, n;\\quad  n=1,2,\\cdots,\n\\label{Aq}\n\\end{equation}\nwhere $\\alpha$ is a dimensionless positive constant.  Two possible values of $\\alpha$ are often mentioned: $\\alpha=8\\pi$~\\cite{spectrum,Maggiore} and $\\alpha=4 \\log k$ with $k=2,3,\\cdots$~\\cite{Mukh,BekMukh,Hod}.  We shall here focus on macroscopic holes, those with very large $n$. We assume that the fundamental transition of a black hole with consequent emission or absorption of one radiation quantum is one with $n\\Rightarrow n\\mp1$.  Conservation laws then fix the frequency and azimuthal quantum number of the quanta; for emission $\\hbar\\omega=-\\Delta M$ and $\\hbar\\tilde m=-\\Delta J$ with $\\tilde m\\in (\\cdots,-2,-1,0,1,2,\\cdots)$.  Thus differencing (more practically differentiating) \\Eq{Aq} gives as the characteristic parameter for the fundamental transition\n\\begin{equation}\nx\\equiv \\hbar(\\omega-\\tilde m\\Omega)\/T_{\\rm H} =\\alpha,\n\\label{x}\n\\end{equation}\nwhere $T_{\\rm H}$ and $\\Omega$ are the Hawking temperature and rotational angular frequency for the Kerr black hole.  [Use has been made of the area formula, $S_{\\rm BH}=A(4L_P{}^2)^{-1}$, as well as the thermodynamic relations $(\\partial S_{\\rm BH}\/\\partial M)_J=1\/T_{\\rm H}$ and $(\\partial S_{\\rm BH}\/\\partial J)_M=-\\Omega\/T_{\\rm H}$].  It may be seen that the fundamental transition's $x$ parameter, or $\\omega$ for fixed $\\tilde m$, is constant over a moderate range of $M$ or $A$.  This means that a moderately long series of quanta with like values of $\\omega$ and $\\tilde m$ can be emitted by a macroscopic black hole making successive $n\\Rightarrow n-1$ transitions before the frequency begins to shift and a switch to another mode can occur.  Likewise, transitions with $n\\Rightarrow n-2, n\\Rightarrow n-3, \\cdots$ can also take place, with values of the parameter $x$ which are integral multiples of that in \\Eq{x}.  These too can occur in series.\nEvery such series gives rise to a spectral line; there are many spectral lines for $n\\Rightarrow n-1$ (likewise for $n\\Rightarrow n-2$, etc.), each for a different value of $\\tilde m$.\n\nWe remark that any other quantization rule according to which $A$ is a smooth function of a quantum number analogous to $n$ will lead to similar conclusions regarding the allowed values of $\\omega$.  Not so when a black hole parameter different from $A$ is subject to quantization. Likewise, completely different results would be expected for a continuum horizon area spectrum.\n\n\n\\section{Jump probabilities}\n\\label{sec:jump}\n\nFrom now on we refer to transitions between $A$ levels as jumps.  Let us focus on two adjacent area levels of an isolated black hole, $u$ and $l$.  Each quantum state belonging to the upper level $u$ has a definite probability to jump-down to a state belonging to the lower level $l$ whereby a quantum with a definite $\\tilde m$ is emitted; we denote this \\emph{elementary probability} by $e^{-\\beta}$ with $\\beta$ real and positive~\\cite{Fulfill}; to reduce clutter we do not the $x$ or $\\tilde m$ of the mode.  Regarding the time span to which this probability refers, we shall take it as the time associated with emission of the mode of black hole radiation which emerges from the said transition.  Of course the time depends on how monochromatic we take the mode to be; very monochromatic modes are emitted over a long time.    Once the black hole has jumped down one area level, it is in a situation similar to the initial one, so another jump to one level lower should also take place with the same probability $e^{-\\beta}$.  This jumping down can repeat with the quanta emitted from each belonging to the same mode.  However, once $A$ has changed significantly, the emission can be considered to have shifted to a different mode since $x$ will also have changed somewhat.  Similar remarks apply to jumps with $n\\Rightarrow n-2, n\\Rightarrow n-3,\\cdots$.\n\nIt is an assumption here that the chance of a jump is uninfluenced by whether the present level was the initial one or not, the chain of jumps being regarded as a quantum Markov process.   Obviously, with probability $1-e^{-\\beta}$ no jump takes place during the mode's time span, thus interrupting the chain.    Accordingly, the probability for $j$ successive jump-downs with given $x$ and $\\tilde m$ is\n\\begin{equation}\np_{\\downarrow j}= (1-e^{-\\beta})\\,e^{-\\beta j}.\n\\label{spontaneous} \n\\end{equation}\nThis probability distribution is already normalized: $\\sum_{j=0}^\\infty p_{\\downarrow j}=1$.\n\nThe probability $e^{-\\beta}$ refers to a spontaneous jump-down in the area spectrum with emission of one quantum.  There should also be an \\emph{elementary probability} for the hole in some state belonging to $l$  to jump-up to any state belonging to $u$~\\cite{BekSchiff} when a quantum of the appropriate $x$ and $\\tilde m$ is incident.  We denote this probability by $e^{-\\mu}$ with $\\mu$ also real and positive.  \n\nNow $e^{-\\mu}$ and $e^{-\\beta}$ are connected via the principle of detailed balance.  For example, we can consider the hole in equilibrium with a radiation bath at temperature $T_{\\rm H}$.   Then according to Planck the mean number of quanta incident on the black hole in a radiation mode labeled by $x$ is $(e^x-1)^{-1}$; we, of course, exclude the customary phase-space factor-it relates to the number of modes. The mean rate at which the hole jumps from the state in $l$ to that in $u$ must be proportional to this mean number.  It will be shown in Sec.~\\ref{sec:stimulated} that spontaneous emission must be accompanied by stimulated emission.  We assume that the hole's jump-down probability is uninfluenced by whether quanta in the relevant mode are already incident. Thus we can write\n\\begin{equation}\ne^{-\\mu}(e^x-1)^{-1}=e^{-\\beta}+e^{-\\beta}(e^x-1)^{-1},\n\\end{equation}\nwhich tells us that\n\\begin{equation}\ne^{-\\mu}=e^{-\\beta+x}.\n\\label{detailed}\n\\end{equation}\n\n\nWith $e^{-\\mu}$ in hand we can now calculate the probability of total absorption of $k$ quanta incident on the black hole in a single mode whose $x$ matches the spacing between levels $u$ and $l$.  Assuming that, as long as there are quanta that can be absorbed, the probability of a jump-up of the hole is independent of what happened before, we get for the probability of $k$ jump-ups in sequence with consequent absorption of $k$ quanta\n\\begin{equation}\np_{\\uparrow k}=(1-e^{-\\beta})e^{-\\mu k};\\qquad k=1,2,\\cdots .\n\\label{pup}\n\\end{equation}\nThe factor $(1-e^{-\\beta})$ again represents the probability that the hole does not experience a jump-down which would interrupt the sequence of $k$ jumps-up.   Note that in the present case the sum over $k$ in \\Eq{pup} need not be unity since different $k$s do not label different outcomes of one initial set up, but rather different initial set ups.\n\nOf course an incident quanta may fail to be absorbed, i.e. the black hole may not jump-up in its presence.  Now, the probability complementary to $e^{-\\mu}$ is $1-e^{-\\mu}$.  This is not the full contribution; the black hole may also spontaneously jump-down while failing to jump-up.  Thus the probability that the black hole avoids either and remains in its initial state despite the presence of an incident quantum is\n\\begin{equation}\np_0=(1-e^{-\\mu})(1-e^{-\\beta}).\n\\label{no}\n\\end{equation}\n\n\\section{Statistics of absorption and emission}\n\\label{sec:statistics}\n\nAs already hinted, \\Eq{spontaneous} can be regarded as the probability of \\emph{spontaneous} emission of $j$ quanta in the one mode considered.  We could write this as $p(j|0)$, where $p(m|n)$ is  generally defined as the \\emph{conditional probability} that $m$ quanta are emitted when $n$ are incident in the mode in question~\\cite{BekMeis}.  Of course we must require $\\sum_{m=0}^\\infty p(m|n)=1$ for all $n$.  Likewise we may interpret \\Eq{pup} to give $p(0|k)$, the probability that if $k$ quanta impinge on the black hole in a given mode, all are absorbed.  We now compute $p(m|n)$ for all other cases.\n\nSuppose $n$ indistinguishable quanta are incident on the black hole in one mode.  And suppose $k\\leq n$ of these fail to cause any jump-up of the black hole, and thus survive to occupy the outgoing counterpart of the incident mode. Assuming that each quantum acts independently and taking \\Eq{no} into account, the probability for this partial outcome must be $(1-e^{-\\mu})^k(1-e^{-\\beta})^k$.    The probability that the hole makes $n-k$ consecutive jump-ups as it absorbs the other $n-k$ incident quanta is evidently $e^{-\\mu (n-k)}$.   And the number of independent ways in which the $n$ incident quanta can be partitioned into absorbed and surviving quanta is $n!\/[(n-k)!k!]$.  \nWe thus get the \\emph{partial} probability (as always $0!=1$)\n\\begin{equation}\n\\frac{n!}{(n-k)!k!}(1-e^{-\\mu})^k(1-e^{-\\beta})^k e^{-\\mu (n-k)}.\n\\end{equation}\n\nThe number of quanta in the  outgoing counterpart of the said mode is $m$.  Of these $k\\leq m$ are surviving incident ones.  Thus the hole must emit $m-k$ quanta as it jumps down $m-k$ levels.  According to \\Eq{spontaneous} the  probability for this is $(1-e^{-\\beta})e^{-\\beta (m-k)}$.  The number of independent ways in which the $m!$ outgoers can be selected from surviving and freshly emitted quanta is $m!\/[(m-k)!k!]$.  We thus have the second partial probability\n\\begin{equation}\n\\frac{m!}{(m-k)!k!}(1-e^{-\\beta}) e^{-\\beta (m-k)}.\n\\end{equation}\nThe conditional emission probability $p(m|n)$ must obviously contain the product of the above partial probabilities.  In addition, the number $k$, being not observable, must be summed over from zero to the smaller of $n$ or $m$.  Thus\n\\begin{eqnarray}\np(m|n)&=&(1-e^{-\\beta})e^{-(\\mu n+\\beta m)}\\sum_{k=0}^{{\\rm min}(m,n)}\\frac{n!m!X^k}{(n-k)!(m-k)(k!)^2};\n\\label{pmn}\n\\\\\nX&\\equiv &(e^\\beta-1)(e^\\mu-1).\n\\label{X}\n\\end{eqnarray}\n\nTo check this result we first set $m=0$; we get $p(0,n)=(1-e^{-\\beta})e^{-\\mu n}$ which is what we would expect from \\Eq{pup}.  Setting rather $n=0$ we get $p(m,0)=(1-e^{-\\beta})e^{-\\beta m}$ which is the expected result for spontaneous emission, i.e., \\Eq{spontaneous}.  We next check that $p(m|n)$ is normalized.  Summing $p(m|n)$ over all $m$ and interchanging the two sums while respecting the constraints $k\\leq m,n$ gives\n\\begin{equation}\n\\sum_{m=0}^\\infty p(m|n)=(1-e^{-\\beta})e^{-\\mu n}\\sum_{k=0}^n\\frac{n!X^k}{(n-k)!(k!)^2}\\sum_{m=k}^\\infty \\frac{m!}{(m-k)!}e^{-\\beta m}.\n\\label{inter}\n\\end{equation}\nThe inner sum is done in Appendix A (\\Eq{first}); substituting it here gives\n\\begin{equation}\n\\sum_{m=0}^\\infty p(m|n)=\\sum_{k=0}^n\\frac{n!}{(n-k)!k!}(1-e^{-\\mu})^k e^{-\\mu (n-k)},\n\\end{equation}\nwhich equals unity by Newton's binomial theorem. We remark that the sum in \\Eq{pmn} is symmetric in $m$ and $n$ and can, in fact, be identified with the hypergeometric function ${}_2 F_1(-m,-n,1,X)$~\\cite{AbraSteg}.  The nonpositive integer nature of the first two arguments is responsible for cutting off the hypergeometric series and rendering it a polynomial in $X$; this turns out to be the Jacobi polynomial $P^{(0,-m-n-1)}(1-2X)$ (Eq.~15.4.6 of ref.~\\cite{AbraSteg}).  \n\nWe now express $e^{-\\beta}$ and $e^{-\\mu}$, the microscopic parameters of the black hole transition,  in terms of observable black hole parameters: $x$, as defined above, and $\\Gamma$, the so called absorptivity of the black hole in the said mode (also known as the grey-body factor or barrier tunneling coefficient).  First we calculate the mean number of quanta spontaneously emitted as a result of the jumps associated with the said mode (see \\Eq{spontaneous}):\n\\begin{equation}\n\\langle m\\rangle_{\\rm sp}=\\sum_{m=0}^\\infty (1-e^{-\\beta})\\,m\\,e^{-\\beta m}=(e^\\beta-1)^{-1}.\n\\label{mean0}\n\\end{equation} \nIn Hawking radiance this mean is $\\Gamma$ times what could be attributed to a \\emph{blackbody} with temperature $T=T_{\\rm H}$.  Although in the present paper the black hole spectrum is regarded as being made up of (broadened) lines, we can still use \\Eq{mean0} for the modes making up each line. \n\nNow an ideal black body would emit into the mode a Planck's mean number,\n\\begin{equation}\n\\langle m\\rangle_{\\rm bb}=(e^x-1)^{-1},\n\\end{equation}\nof quanta.  Thus the black hole spontaneously emits a mean number\n\\begin{equation}\n\\langle m\\rangle_{\\rm sp}=\\Gamma(e^x-1)^{-1}.\n\\label{spon}\n\\end{equation}\nComparing this with \\Eq{mean0} gives\n\\begin{equation}\ne^{-\\beta}=[1+(e^x-1)\\Gamma^{-1}]^{-1}.\n\\label{embeta1}\n\\end{equation}\n\nObviously for emission from a Schwarzschild black hole $x>0$ and $\\Gamma<1$; hence $e^{-\\beta}$ turns out to be smaller than unity, as it should.  For emission from the Kerr black hole $x=\\hbar(\\omega-\\tilde m\\Omega)\/T_{\\rm H}$.  For modes with $x>0$ the story is the same as for the Schwarzschild case.  Modes with $x<0$ are known as superradiant modes because incident radiation in such modes is amplified by the hole, so that $\\Gamma<0$ for every such mode.  For these modes it again follows from \\Eq{embeta1} that $e^{-\\beta}<1$.\n\nSubstitution of \\Eq{embeta1} and (\\ref{detailed}) into \\Eq{pmn} now gives us\n\\begin{equation}\np(m|n)=\\frac{(e^x-1)e^{xn}\\Gamma^{m+n}}{(e^x-1+\\Gamma)^{m+n+1}}\\ {}_2 F_1(-m,-n,1,X).\n\\label{pmn1}\n\\end{equation}\nThis expression for the conditional probability $p(m|n)$ is our central result here.  To recapitulate, it describes the statistics of black hole radiance in the particular mode characterized by the values of $x$ and $\\Gamma$.  It results from a set of simple assumptions as to the quantum jumps that the black hole can undergo, either spontaneous jumps, or those made possible by incident quanta in the same mode. \n\n\\section{Comparison with the early results}\n\\label{sec:comparison}\n\nThe consistency of that set of assumptions will now be verified by comparison of \\Eq{pmn1} with the expression for the conditional probability obtained by two independent methods that do not focus on the black hole's quantum structure. The $p(m|n)$ was first calculated in Ref.~\\cite{BekMeis} by applying the maximum entropy method of information theory to the statistics of radiation outgoing from a black hole which is bathed by external thermal radiation.  In Ref.~\\cite{PananWald} the same result was obtained by applying methods of quantum field theory in curved spacetime to the radiance.  Both of these approaches focus on the radiation, and treat the black hole as a fixed background.  It may be noted that an ideal material grey body also exhibits the same conditional probabilities~\\cite{BekSchiff}.\n\nThe results of Refs.~\\cite{BekMeis} and~\\cite{PananWald} can both be put in the form\n\\begin{eqnarray}\np(m|n)&=&\\frac{(e^x-1)e^{xn}\\Gamma^{m+n}}{(e^x-1+\\Gamma)^{m+n+1}}\\sum_{k=0}^{{\\rm min}(m,n)}\\frac{(m+n-k)!(X-1)^k}{(n-k)!(m-k)!k!};\n\\label{pmn2}\n\\\\\nX&=&\\frac{(e^x-1)^2e^{-x}(1-\\Gamma)}{\\Gamma^2},\n\\label{X2}\n\\end{eqnarray}\nwhere we have denoted the parameter also by $X$ since by \\Eqs{detailed} and (\\ref{embeta1}) it is equivalent to that in \\Eq{X}.\nNow the sum in \\Eq{pmn2} may be identified with \n\\begin{equation}\n\\frac{(m+n)!}{m!n!}{}_2F_1(-m,-n,-m-n,X-1);\n\\label{F}\n\\end{equation}\nthis is again a polynomial because the first two arguments are nonpositive integers.  The expression in \\Eq{F} is identical to the hypergeometric function appearing in \\Eq{pmn1}.\nHence, the expressions for $p(m|n)$ in \\Eqs{pmn1} and (\\ref{pmn2}) are identical.\n\nThis agreement shows that the assumptions made in Secs.~\\ref{sec:jump} and \\ref{sec:statistics} regarding the black hole area spectrum and transition dynamics are consistent with what is already known about black hole radiance, regardless of whether the hole is in vacuum or immersed in radiation.  In the previous section we composed $p(m|n)$ from the probability distributions for quanta which miss being absorbed in a black hole jump-up and for quanta which are emitted as a result of a hole jump-down.  In the next sections we shall rather compose $p(m|n)$ from the probability distributions for scattering, spontaneous and stimulated emission of quanta; this last approach was introduced in Ref.~\\cite{Fulfill}.\n\n\\section{Scattering}\n\\label{sec:scattering}\n\nImagine an incident quantum in a definite mode.  With probability $e^{-\\mu}$ it gets absorbed concurrently with a jump-up of the black hole.  This may be followed by a jump-down of the black hole or several such in sequence.  The total probability of these independent events is\n\\begin{equation}\n\\Gamma_0=e^{-\\mu}(1+e^{-\\beta}+e^{-2\\beta}+\\cdots)=e^{-\\mu}(1-e^{-\\beta})^{-1}.\n\\label{Gamma0}\n\\end{equation}\nEvidently, this is the probability that the incident quantum disappears into the hole with all possible consequences for the hole.  We denote it by $\\Gamma_0$ because in a real sense it is the true absorption probability for a quantum of the given mode whereas, as we shall show presently, $\\Gamma$ stands for a somewhat different thing.  \n\nAn external observer unaware of the inner workings of the black hole is surely aware of scattering off the hole's geometry.  What is the scattering probability distribution?  Following the above tack, we argue that since $\\Gamma_0$ is the absorption probability, $1-\\Gamma_0$ must be the probability that a quantum incident on the black hole in the given mode is scattered (reflected) into the outgoing form of the mode  regardless of what happened to the hole. Conservation of energy and angular momentum insure that the quantum remains associated with the same mode. Now if $n$ indistinguishable quanta are incident in one mode, the probability that a number $l\\leq n$ of these are scattered  is evidently\n\\begin{equation}\np_{\\rm sc}(l|n)=\\frac{n!}{l! (n-l)!}(1-\\Gamma_0)^l\\, \\Gamma_0{}^{n-l}.\n\\label{scatt}\n\\end{equation}\nUse of \\Eq{Gamma0} allows us to put the desired distribution in terms of observable parameters:\n\\begin{equation}\np_{\\rm sc}(l|n)=\\frac{n!}{l! (n-l)!} \\frac{e^{-\\mu n}e^{-\\beta l}(X-1)^{l} }{(1-e^{-\\beta})^n}.\n\\end{equation}\n\n\\section{Stimulated emission}\n\\label{sec:stimulated}\n\nIf we replace $e^{-\\mu}$ in \\Eq{Gamma0} by means of \\Eq{detailed} and then replace $e^{-\\beta}$ in the result with help of \\Eq{embeta1} we get\n\\begin{equation}\n\\Gamma=\\Gamma_0(1-e^{-x})\n\\label{Gamma1}.\n\\end{equation}\nThis result already shows that a black hole is capable of stimulated emission~\\cite{BekMeis,Fulfill}.  To see this consider how $\\Gamma$ is calculated in practice.  One imagines a classical wave in the form of a mode function of the appropriate field directed onto the hole; $1-\\Gamma$ is identified with the fraction of the energy of the initial wave which is returned outward as a result of the scattering by the hole.   Recall that $\\Gamma_0$, being a probability, is always positive.  Now for a mode with $x<0$, \\Eq{Gamma1} predicts that $\\Gamma<0$ so that $1-\\Gamma>1$: the incident wave is thus predicted to be amplified, a sure sign of stimulated emission.  The corresponding modes are called super radiant, and are well studied numerically.\n\nTurn now to a mode with $x>0$. \\Eq{Gamma1} shows that $0<\\Gamma<1$ so that $1-\\Gamma$ \\emph{exceeds} the scattering probability $1-\\Gamma_0$ (the ratio of the two last two being independent of the wave's initial amplitude).  How is this possible?  Only if the black hole is stimulated by the incident wave to emit the same kind of wave and thus strengthen the outgoing wave in direct proportion to the incident wave's strength.  In this non-superradiant mode  the outgoing wave is weaker than the incident one, but not as weak as would be expected from the size of $\\Gamma_0$.  We must conclude~\\cite{BekMeis,Fulfill} that stimulated emission occurs in these modes also, but with strength insufficient to actually amplify the incident wave.  \n\nTurning to our main concern, we ask what is $p_{\\rm st}(m|n)$, the conditional probability for emission of $m$ quanta given that $n$ are incident in the same mode?  In ref.~\\cite{Fulfill} it was obtained by judiciously decomposing $p(m|n)$ into a the scattering contribution from \\Eq{scatt} and a part which could be interpreted in terms of emission.  Here we shall derive  $p_{\\rm st}(m|n)$ using an approach like the one we followed to obtain $p(m|n)$ in Sec.~\\ref{sec:statistics} of the present paper.\n\nStimulated emission means that each of the $n$ incident quanta may be responsible for emission of some of the outgoing $m$ quanta.  In how many ways can one associate the outgoing quanta, each to some incident one, all this in harmony with the bosonic character of the quanta?  Recall the standard textbook question, in how many ways can one arrange $m$ bosons in $g$ cells given that the bosons are identical?~\\cite{LLSP}.  The answer is \n\\begin{equation}\n\\frac{(m+g-1)!}{m!(g-1)!}.\n\\label{comb}\n\\end{equation}  \nThis last problem maps directly into ours; we get the desired number by replacing $g\\Rightarrow n$ above.\n\nWe shall assume that the dynamics of black-hole jump-down and quantum emission is the same whether occurring spontaneously or whether it is induced by an incident quantum.  This is in keeping with what we know from atomic physics (equality of the properly defined Einstein coefficients for spontaneous and stimulated emission).  The one difference here is that it is possible for the black hole to be induced to make several jump-downs in sequence and place the consequently emitted quanta in the same mode.  (This last is very rare, if at all possible, in atomic physics.)  To the $m_j$ emitted quanta associated with incident quantum $j$ we must associate a probability $e^{-\\beta m_j}$ corresponding to $m_j$ successive black hole jump-downs.  After those the jumping-down is arrested with probability $1-e^{-\\beta}$.  Thus the overall probability $p_{\\rm st}(m|n)$ is given by\n\\begin{equation}\n(1-e^{-\\beta})e^{-\\beta m_1}\\cdot (1-e^{-\\beta})e^{-\\beta m_2} \\cdots (1-e^{-\\beta})e^{-\\beta m_{n}}.\n\\end{equation}\nIn light of \\Eq{comb} and the fact that $\\sum_j m_j=m$, we have our final formula\n\\begin{equation}\np_{\\rm st}(m|n)=\\frac{(m+n-1)!}{m!(n-1)!}(1-e^{-\\beta})^n e^{-\\beta m}.\n\\label{pst}\n\\end{equation}\nOf course this formula can be used only for $n\\geq 1$.\n\nTo analyze this result we first compute the \\emph{conditional mean} number of quanta emitted given that $n$ are incident:\n\\begin{equation}\n\\langle m|n\\rangle_{\\rm st}=\\frac{(1-e^{-\\beta})^{n}}{(n-1)!} \\sum_{m=0}^\\infty \\frac{(m+n)!}{m!}\\, e^{-\\beta (m+1)}\n\\label{sum3}\n\\end{equation}\nUsing \\Eq{sum3'} of the Appendix we have\n\\begin{equation}\n\\langle m|n\\rangle_{\\rm st}=\\frac{n}{e^\\beta-1}=n\\langle m\\rangle_{\\rm sp}.\n\\end{equation}\nThis is entirely analogous to the result in atomic physics that the mean number of quanta from stimulated emission is proportional to the number of incident quanta, with the proportionality coefficient equal to the mean number from spontaneous emission.  Our result here thus strengthens the claim that a black hole is capable of stimulated emission even for non-superradiant modes.\n\nActually in atomic physics one computes the \\emph{probability} of stimulated emission of \\emph{one} photon given that $n$ are incident, and finds it to be $n$ times the probability of one-photon spontaneous emission.   Were we to do likewise here we would get\n\\begin{equation}\n\\frac{p_{\\rm st}(1|n)}{p(1|0)}=n(1-e^{-\\beta})^{n-1}.\n\\end{equation}\nThis corresponds to the atomic physics result only for $n=1$ or when $\\beta\\to\\infty$ for any $n\\geq 2$.  Now by \\Eq{spon} $\\beta\\to\\infty$ is equivalent to $x\\to\\infty$, i.e., to the case $\\hbar\\omega\\gg T$ (Wien regime).  In the intermediate and the Rayleigh-Jeans regimes the coefficient is smaller than that in the atomic physics calculation.  We interpret this discrepancy to reflect the fact that the mentioned atomic physics result is a first-order perturbation one, whereas \\Eq{pst} here is nonperturbative.  This last is obvious since the formula in \\Eq{spon} works for multi-quanta emission which would require application of arbitrarily high order perturbation theory.\n\n\\section{Composition of emission with scattering}\n\\label{sec:composition}\n\nLet us now compose the distribution $p_{\\rm st}(m|n)$ with that for spontaneous emission, $p(m|0)$, to get the conditional probability distribution for generic emission, $p_{\\rm em}(m|n)$:\n\\begin{equation}\np_{\\rm em}(m|n)=\\sum_{k=0}^m p_{\\rm st}(k|n)\\, p(m-k|0),\n\\end{equation}\nwhich equals\n\\begin{equation}\n(1-e^{-\\beta})^{n+1}e^{-\\beta m}\\sum_{k=0}^m \\frac{(n+k-1)!}{k!(n-1)!}.\n\\label{sum4}\n\\end{equation}\nThe sum is worked out in the Appendix (\\Eq{second}); we find\n\\begin{equation}\np_{\\rm em}(m|n)=\\frac{(m+n)!}{m!n!}(1-e^{-\\beta})^{n+1} e^{-\\beta m}.\n\\end{equation}\n\nWe note that $p_{\\rm em}(m|n)=p_{\\rm st}(m|n+1)$.  This accords with what happens in atomic physics: the total emission in the presence of $n$ quanta is as if $n+1$ quanta were responsible for purely stimulated emission, the extra unity being ascribed to the effect of vacuum fluctuations. \n\nWe now compose the distributions $p_{\\rm sc}$ and $p_{\\rm em}$; this should give us $p(m|n)$:\n\\begin{equation}\np(m|n)=\\sum_{k=0}^{{\\rm min}(m,n)}  p_{\\rm em}(m-k|n)\\,p_{\\rm sc}(k|n),\n\\end{equation}\nwhere the upper limit of the summation reflects the fact that no more quanta can be scattered than the number incident but also cannot exceed the total number of outgoing quanta.  Thus\n\\begin{equation}\np(m|n)=(1-e^{-\\beta})e^{-(\\mu n+\\beta m)}\\sum_{k=0}^{{\\rm min}(m,n)}  \\frac{(m+n-k)!(X-1)^{k}}{k!(m-k)! (n-k)!}  \n\\label{pmn3}\n\\end{equation}\nThe coefficient of the sum here is exactly equivalent to that in \\Eq{pmn2}.  Thus by focusing on the scattering contribution we have here derived an alternative form of $p(m|n)$, identical to that found in refs.~\\cite{BekMeis,PananWald}, and which is also known to be equivalent to the $p(m|n)$ deduced in Sec.~\\ref{sec:statistics} of this paper (see Sec.~\\ref{sec:comparison}). \n\n\\section{Summary}\n\nIt is widely accepted that a thermally radiating system cannot disclose details about its internal dynamics due to the loss of coherence implicit in the thermalization.  A black hole may be different in that, unlike ordinary matter, it is not demonstrably a collection of independent systems.   In this paper we have leaned on the hypothesis that a Kerr black hole's horizon area has a discrete spectrum to derive the black hole's response to incident radiation, specifically the conditional probability distribution for the number of emitted quanta.  We reproduce in two different ways the known probability distribution, obtaining to boot a closed form of it in terms of an hypergeometric function.  This accomplishment of the ``atomic'' model of the black hole shows that the widely hypothesized discreteness of the area spectrum is consistent with the expected statistics of the black hole radiance.   \n\nOur approach here is related to the approach to quantum gravity of a black hole based on its quasinormal modes~\\cite{BekMG,Hod,Maggiore}.   Quasinormal modes, however, are purely classical responses of the black hole to perturbations, and one cannot immediately discount the possibility that they are irrelevant to questions of quantum structure.  Nevertheless, the quasinormal mode approach does point to a discrete horizon area spectrum, often a uniformly spaced one~\\cite{Skakala},~ and so agrees with our conclusion here that the black hole response to external radiation is consistent with a discrete area spectrum.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nIn this work we investigate the regularity of $p$-orthotropic functions in the plane for $1<p<2$.\nLet $\\Omega\\subset\\mathbb{R}^2$ be an open set. A weak solution of the orthotropic $p$-Laplace equation (also known as pseudo $p$-Laplace equation) is a function $u\\in W^{1,p}(\\Omega)$ such that\n\\begin{equation}\\label{orthodeg}\n\\sum_{i=1}^2 \\int_\\Omega |\\partial_i u|^{p-2} \\partial_i u \\, \\partial_i \\phi \\dx=0 \\quad \\text{for all} \\quad \\phi\\in W_0^{1,p}(\\Omega).\n\\end{equation}\nEquation \\eqref{orthodeg} arises as the Euler-Lagrange equation for the functional\n\\begin{equation}\nI_{\\Omega}(v) = \\sum_{i=1}^2\\int_{\\Omega} \\frac{|\\partial_i v|^p}{p}\\dx.\n\\end{equation}\nThe equation is singular when either one of the derivatives vanishes, and does not fall into the category of equations with $p$-Laplacian structure. \nIt was proved by Bousquet and Brasco in \\cite{BB} that weak solutions of \\eqref{orthodeg} for $1<p<\\infty$ are $C^1(\\Omega)$. A simple proof which gives a logarithmic modulus of continuity for the derivatives is contained in \\cite{LR} for the case $p\\geq 2$. The latter relies on a lemma on the oscillation of monotone functions due to Lebesgue \\cite{L} and the fact that derivatives of solutions are monotone (in the sense of Lebesgue). The purpose of this work is to extend this result to the case $1<p<2$ employing methods developed in \\cite{LR}. We obtain the following:\n\\begin{theorem}\\label{C1}\nLet $\\Omega\\subset\\mathbb{R}^2$ and $u\\in W^{1,p}(\\Omega)$ be a solution of the  equation \\eqref{orthodeg} for $1<p<2$. Fix a ball  $B_{R}\\subset\\subset \\Omega$. Then,    \n for all $j\\in\\{1,2\\}$ and $B_r\\subset\\subset B_{R\/2}$, we have\n\\begin{equation}\\label{oscEst2}\n\\osc{B_r} (\\partial_ju)\\leq C_p \n\\left(\\log\\left(\\frac{R}{r}\\right)\\right)^{-\\frac{1}{2}}\\left( \\fint_{B_{R}}  |\\nabla u|^p \\dx\\right)^\\frac{1}{p},\n\\end{equation}\nwhere $C_p$ is a constant depending only on $p$.\n\\end{theorem}\n\n\n\n\n\\paragraph{\\bf Notation.}\nWe indicate balls by $B_r=B_r(a)=\\{x\\in\\mathbb{R}^2\\;:\\; |x-a|<r\\}$ and we omit the center when not relevant. Whenever two balls $B_r\\subset B_R$ appear in a statement they are implicitly assumed to be concentric. The variable $x$ denotes the vector $(x_1, x_2)$ and we denote the partial derivatives of a function $f$ with respect to $x_j$ as $\\partial_j f$.\n\n\n\n\\section{Regularization}\nWe will consider a regularized problem by introducing a non degeneracy parameter $\\epsilon>0$. \n\\\\\n\nFix $B_R\\subset\\subset \\Omega\\subset \\mathbb{R}^2$ and  consider  the regularized Dirichlet problem\n\\begin{equation}\\label{orthonondeg2}\n\\begin{split}\n\\begin{cases}\n\\sum_{i=1}^2 \\int_{B_R} (|\\partial_i u^\\epsilon|^2 +\\epsilon)^\\frac{p-2}{2} \\partial_i u^\\epsilon\\, \\partial_i \\phi \\dx=0\\\\\nu^\\epsilon-u \\in W_0^{1,p}(B_R).\n\\end{cases}\n\\end{split}\n\\end{equation}\nNote that $u^\\epsilon$ is the unique minimizer of the regularized functional\n\\begin{equation}\nI^\\epsilon_{B_R}(v)=\\sum_{i=1}^2\\int_{B_R}\\frac{1}{p}(|\\partial_i v|^2+\\epsilon)^\\frac{p}{2}\\dx\n\\end{equation}\namong $W^{1,p}(B_R)$ functions $v$ such that $v-u\\in W^{l,p}_0(B_R)$.\nBy elliptic regularity theory, the unique solution $u^\\epsilon$ of \\eqref{orthonondeg2} is smooth in $B_R$.\\\\\nFix an index $j\\in\\{1,2\\}$. Then, replacing $\\phi$ by $\\partial_j\\phi$ in equation \\eqref{orthonondeg2} and integrating by parts, we find that the derivative $\\partial_j u^\\epsilon$ satisfies the following equation\n\\begin{equation}\\label{orthoder2}\n\\sum_{i=1}^2 \\int_{B_R} (\\epsilon+|\\partial_i u^\\epsilon|^2)^\\frac{p-4}{2} (\\epsilon+(p-1)|\\partial_i u^\\epsilon|^2)\\,  \\partial_i \\partial_j u^\\epsilon \\,\\partial_i\\phi \\dx =0\n\\end{equation}\nfor all $\\phi\\in C_0^\\infty(B_R)$.\\\\\n\n\nWe now collect some uniform estimates and convergences (see also \\cite{BB}).\n\n\\begin{Lemma}\nLet $u\\in W^{1,p}(\\Omega)$ be a solution of \\eqref{orthodeg} and $u^\\epsilon$ be a solution of \\eqref{orthonondeg2} for $1<p<2$. Then we have\n\\begin{equation}\\label{energy2}\n\\int_{B_R} |\\nabla u^\\epsilon|^p\\dx \\leq \nC_p \\left(  \\int_{B_{R}} |\\nabla u|^p\\dx +\\epsilon^\\frac{p}{2}R^2 \\right)\n\\end{equation}\nwhere $C_p$ is a constant depending only on $p$.\n\\end{Lemma}\n\n\\begin{proof}\nThe estimate follows from $$I^\\epsilon_{B_R}(u^\\epsilon)\\leq I^\\epsilon_{B_R}(u).$$\n\\end{proof}\n\n\n\\begin{Prop}\nLet $u\\in W^{1,p}(\\Omega)$ be a solution of \\eqref{orthodeg} and $u^\\epsilon$ be a solution of \\eqref{orthonondeg2} for $1<p<2$. Then, for all $j\\in\\{1,2\\}$, we have\n\\begin{align}\n\\sup_{B_{R\/2}} (\\epsilon+|\\nabla u^\\epsilon|^2)&\\leq C_p\\left( \\fint_{B_R}(\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{p}{2}\\dx \\right)^\\frac{2}{p}, \\label{lip2} \\\\ \n\\int_{B_{R\/2}}|\\nabla \\partial_j u^\\epsilon|^2\\dx &\\leq C_p\\left(\\fint_{B_R}(|\\nabla u|^p+\\epsilon^\\frac{p}{2})\\dx\\right)^\\frac{2}{p} \\label{grad2}\n\\end{align}\nwhere $C_p$ is a constant depending only on $p$.\n\\end{Prop}\n\n\\begin{proof}\nThe proof of the Lipschitz bound can be found in \\cite{FF}  while \\eqref{grad2} appears in \\cite{BB}. We provide details for completeness.\nNote that by a change of variables, the function $u^\\epsilon_R(x)=u^\\epsilon(x_0+Rx)$ satisfies the equation \n\\begin{equation}\\label{orthonondeg2scaled}\n\\sum_{i=1}^2\\int_{B_1}(|\\partial_iu^\\epsilon_R|^2+R^2\\epsilon)^\\frac{p-2}{2}\\partial_iu^\\epsilon_R\\partial_i\\phi\\dx=0 \\quad \\text{for all} \\quad \\phi\\in W_0^{1,p}(B_1).\n\\end{equation}\nIntroduce the notation $w=\\epsilon R^2+|\\nabla u^\\epsilon_R|^2$ and $a_i(z)=a_i(z_i)=(\\epsilon R^2+|z_i|^2)^\\frac{p-2}{2}z_i$ so that equation \\eqref{orthonondeg2scaled} rewrites as $$\\sum_{i=1}^2\\int_{B_1}a_i(\\partial_iu^\\epsilon_R)\\partial_i\\phi\\dx=0 \\quad \\text{for all}\\quad \\phi\\in W_0^{1,p}(B_1).$$ \nFor $j\\in\\{1,2\\}$ and $\\alpha\\geq 0$ take $\\phi=\\partial_j(\\partial_ju^\\epsilon_R\\,w^\\frac{\\alpha}{2}\\xi^2)$ so that\n$\\partial_i\\phi=\\partial_j(\\partial_i\\partial_ju^\\epsilon_R w^\\frac{\\alpha}{2} \\xi^2+\\frac{\\alpha}{2}\\partial_iw \\, w^\\frac{\\alpha-2}{2}\\,\\partial_ju^\\epsilon_R\\, \\xi^2) + 2\\partial_j(\\xi\\partial_i\\xi\\, w^\\frac{\\alpha}{2}\\,\\partial_ju^\\epsilon_R)$. Sum in $j$ to get\n\\begin{equation*}\n\\begin{split}\nA+B:&= \\sum_{i,j=1}^2\\int_{B_1} a_i(\\partial_iu^\\epsilon_R)\\partial_j(\\partial_i\\partial_ju^\\epsilon_R w^\\frac{\\alpha}{2} \\xi^2+\\frac{\\alpha}{2}\\partial_iw \\, w^\\frac{\\alpha-2}{2}\\,\\partial_ju^\\epsilon_R\\, \\xi^2)\\dx \\\\\n&+2\\sum_{i,j=1}^2\\int_{B_1}  a_i(\\partial_iu^\\epsilon_R) \\partial_j(\\xi\\partial_i\\xi\\, w^\\frac{\\alpha}{2}\\,\\partial_ju^\\epsilon_R)\\dx=0.\n\\end{split}\n\\end{equation*}\nNote that $\\partial_i w=2\\sum_{j=1}^2\\partial_i\\partial_ju^\\epsilon_R\\,\\partial_ju^\\epsilon_R$ and $\\partial_ia_i(\\partial_iu^\\epsilon_R)\\geq c_pw^\\frac{p-2}{2}$ since $1<p<2$. Integrate by parts in $A$. We get $A=A_1+A_2$ where\n\\begin{equation*}\n\\begin{split}\nA_1&:=\\sum_{i,j=1}^2\\int_{B_1}\\partial_ia_i(\\partial_iu^\\epsilon_R)(\\partial_i\\partial_ju^\\epsilon_R)^2\\,w^\\frac{\\alpha}{2}\\,\\xi^2\\dx\n\\geq c_p\\sum_{j=1}^2 \\int_{B_1} w^\\frac{p-2+\\alpha}{2}|\\nabla\\partial_ju^\\epsilon_R|^2\\xi^2\\dx,\\\\\nA_2&:=c\\alpha\\sum_{i,j=1}^2\\int_{B_1}\\partial_ia_i(\\partial_iu^\\epsilon_R)\\partial_i\\partial_ju^\\epsilon_R\\,\\partial_ju^\\epsilon _R\\,\\partial_iw\\, w^\\frac{\\alpha-2}{2}\\,\\xi^2\\dx\n=c\\alpha\\sum_{i=1}^2 \\int_{B_1}\\partial_ia_i(\\partial_iu^\\epsilon_R) (\\partial_i w)^2 w^\\frac{\\alpha-2}{2}\\,\\xi^2\\dx\\\\\n&\\geq c_p\\alpha \\int_{B_1}w^\\frac{p-4+\\alpha}{2}|\\nabla w|^2\\xi^2\\dx.\n\\end{split}\n\\end{equation*}\nNow we estimate $B=B_1+B_2+B_3$. \n\\begin{equation*}\n\\begin{split}\nB_1:&=\\sum_{i,j=1}^2\\int_{B_1}a_i(\\partial_iu^\\epsilon_R)w^\\frac{\\alpha}{2}|\\partial_ju^\\epsilon_R|\\,|\\partial_j(\\xi\\partial_i\\xi)|\\dx\n\\leq C_p\\int_{B_1}w^\\frac{p+\\alpha}{2} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\dx,\\\\\nB_2:&=\\frac{\\alpha}{2}\\sum_{i,j=1}^2\\int_{B_1} a_i(\\partial_iu^\\epsilon_R)w^\\frac{\\alpha-2}{2}|\\partial_jw|\\,|\\partial_ju^\\epsilon_R| \\,\\xi\\,|\\partial_i\\xi|\\dx \n\\leq C\\alpha\\int_{B_1} w^\\frac{p+\\alpha-2}{2}|\\nabla w|\\,\\xi\\,|\\nabla\\xi|\\dx \\\\\n&\\leq \\eta  \\alpha \\int_{B_1}w^\\frac{p-4+\\alpha}{2}|\\nabla w|^2\\xi^2\\dx+\\frac{C \\alpha}{\\eta}\\int_{B_1}|\\nabla\\xi|^2\\,w^\\frac{p+\\alpha}{2}\\dx,\\\\\nB_3:&=\\sum_{i,j=1}^2\\int_{B_1}a_i(\\partial_iu^\\epsilon_R)w^\\frac{\\alpha}{2}|\\partial_j\\partial_ju^\\epsilon_R|\\,|\\partial_ju^\\epsilon_R| \\,\\xi\\,|\\partial_i\\xi|\\dx \n\\leq \\sum_{j=1}^2\\int_{B_1}w^\\frac{p-1+\\alpha}{2}|\\nabla\\partial_ju^\\epsilon_R|\\,\\xi\\,|\\nabla\\xi|\\dx\\\\\n&\\leq \\eta\\sum_{j=1}^2 \\int_{B_1} w^\\frac{p-2+\\alpha}{2}|\\nabla\\partial_ju^\\epsilon_R|^2\\xi^2\\dx+\\frac{C}{\\eta}\\int_{B_1}|\\nabla\\xi|^2\\, w^\\frac{p+\\alpha}{2}\\dx\n\\end{split}\n\\end{equation*}\nwhere we used $a_i(\\partial_iu^\\epsilon_R)\\leq w^\\frac{p-1}{2}$ and Young's inequality with a parameter $\\eta$ to be chosen suitably small. We get\n\\begin{equation}\\label{mixEst2}\nc_p\\sum_{j=1}^2 \\int_{B_1} w^\\frac{p-2+\\alpha}{2}|\\nabla\\partial_ju^\\epsilon_R|^2\\xi^2\\dx\n+ c_p\\alpha \\int_{B_1}w^\\frac{p-4+\\alpha}{2}|\\nabla w|^2\\xi^2\\dx\n\\leq C_p(\\alpha+1)\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p+\\alpha}{2}\\dx.\n\\end{equation}\nNote that for $\\alpha=0$ we get for all $j\\in\\{1,2\\}$\n\\begin{equation}\\label{gradientEst2}\n\\int_{B_1} w^\\frac{p-2}{2}|\\nabla\\partial_ju^\\epsilon_R|^2\\xi^2\\dx\n\\leq C_p\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p}{2}\\dx,\n\\end{equation}\nand since $|\\nabla w|^2\\leq c\\sum_j |\\nabla \\partial_j u^\\epsilon_R|^2|\\nabla u^\\epsilon_R|^2$ we have\n\\begin{equation}\\label{alpha0}\n\\begin{split}\n\\int_{B_1}w^\\frac{p-4}{2}|\\nabla w|^2\\xi^2\\dx\n&\\leq c \\sum_{j=1}^2\\int_{B_1}w^\\frac{p-4}{2}|\\nabla u^\\epsilon_R|^2|\\nabla \\partial_j u^\\epsilon|^2\\xi^2\\dx\n\\leq c \\sum_{j=1}^2\\int_{B_1}w^\\frac{p-2}{2}|\\nabla \\partial_j u^\\epsilon_R|^2\\xi^2\\\\\n&\\leq C_p\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p}{2}\\dx.\n\\end{split}\n\\end{equation}\nNow for $\\alpha\\geq 1$, \\eqref{mixEst2} implies\n\\begin{equation}\n \\int_{B_1}w^\\frac{p-4+\\alpha}{2}|\\nabla w|^2\\xi^2\\dx\n\\leq C_p \\frac{\\alpha+1}{\\alpha}\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p+\\alpha}{2}\\dx\n\\end{equation}\nand combining with \\eqref{alpha0} we get\n\\begin{equation*}\n \\int_{B_1}\\lvert\\nabla (w^\\frac{p+\\alpha}{4}\\xi)\\rvert^2\\dx\n\\leq C (p+\\alpha)^2\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p+\\alpha}{2}\\dx\n\\end{equation*}\nfor all $\\alpha\\geq 0$.\nUsing Sobolev's embedding $W_0^{1,2}(B_1)\\hookrightarrow L^{2q}(B_1)$ for a fixed $q>1$ we get\n\\begin{equation}\\label{moser2}\n \\left(\\int_{B_1} w^{q\\frac{p+\\alpha}{2}}\\xi^{2q}\\dx\\right)^\\frac{1}{q}\n\\leq C_p (p+\\alpha)^2\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p+\\alpha}{2}\\dx.\n\\end{equation} \nNow choose a sequence of radii $r_i=1\/2^i+(1-1\/2^{i})\\frac{1}{2}$, cut-off functions $\\xi$ between $r_i$ and $r_{i+1}$ and $\\alpha_i=q^ip-p$ so that $\\frac{p+\\alpha_i}{2}=\\frac{p}{2}q^i$. Using these in \\eqref{moser2}, raising to the power $1\/q^{i}$ and iterating  we get for all $i\\in\\mathbb{N}$\n\\begin{equation*}\n\\left(\\int_{B_{r_{i+1}}} w^{\\frac{p}{2}q^{i+1}}\\dx \\right)^\\frac{1}{q^{i+1}}\n\\leq (C_p q^{2i} 2^i)^\\frac{1}{q^i}\\left( \\int_{B_{r_i}} w^{\\frac{p}{2}q^i}\\dx\\right)^\\frac{1}{q^i}\n\\leq \\prod_{j=0}^i(C_p q^{2j} 2^j)^\\frac{1}{q^j} \\int_{B_{1}} w^{\\frac{p}{2}}\\dx.\n\\end{equation*} \nObserve that $\\prod_{i=0}^\\infty(C_p q^{2i} 2^i)^\\frac{1}{q^i}=C(p,q)<\\infty$ so passing to the limit as $i\\to\\infty$ we get\n$$\\sup_{B_{1\/2}}w^\\frac{p}{2}\\leq C(p,q)\\int_{B_{1}} w^{\\frac{p}{2}}\\dx$$\nwhich, after rescaling, proves \\eqref{lip2}. Now going back to \\eqref{gradientEst2}, choosing a cut-off function between $B_{R\/2}$ and $B_R$ and using $1<p<2$ we get\n\\begin{equation*}\n\\int_{B_{R\/2}} |\\nabla \\partial_j u^\\epsilon|^2\\dx \\leq \nC_p\\sup_{B_{R\/2}} (\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{2-p}{p} \\fint_{B_R}(\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{p}{2}\\dx.\n\\end{equation*}\nUsing \\eqref{lip2} and \\eqref{energy2} we obtain \\eqref{grad2}.\n\\end{proof}\n\nNext we collect some facts about the convergence of $u^\\epsilon$ to the solution of the degenerate equation. These are sufficient for our purposes.\n\\begin{Prop}\\label{uni2}\nLet $u^\\epsilon$ be the solution of \\eqref{orthonondeg2} for $1<p<2$ and $u\\in W^{1,p}(\\Omega)$ the solution of \\eqref{orthodeg}. We have\n\\begin{itemize}\n\\item $u^\\epsilon$ converges to $u$ locally uniformly in $B_R$,\\\\\n\\item $\\nabla u^\\epsilon$ converges to $\\nabla u$ in $L^p(B_R)$.\n\\end{itemize}\n\\end{Prop}\n\n\\begin{proof}\nFrom the energy estimate \\eqref{energy2} we obtain a uniform bound for the $L^p$ norm of $\\nabla u^\\epsilon$. Therefore (up to a subsequence) $u^\\epsilon$ converges to some $v\\in W^{1,p}(B_R)$ weakly in $W^{1,p}(B_R)$ and strongly in $L^p(B_R)$. Note that we have $v-u\\in W_0^{1,p}(B_R)$. By weakly lower semicontinuity we get\n\\begin{equation*}\n\\begin{split}\nI_{B_R}(v) = \\sum_{i=1}^2\\int_{B_R}\\frac{|\\partial_i v|^p}{p}\\dx\n&\\leq \\liminf_{\\epsilon\\to 0} \\sum_{i=1}^2\\int_{B_R}\\frac{|\\partial_i u^\\epsilon|^p}{p}\\dx\\\\\n&\\leq \\liminf_{\\epsilon\\to 0} \\sum_{i=1}^2\\int_{B_R}\\frac{1}{p}(|\\partial_i u^\\epsilon|^2+\\epsilon)^\\frac{p}{2}\\dx\\\\\n&\\leq \\liminf_{\\epsilon\\to 0} \\sum_{i=1}^2\\int_{B_R}\\frac{1}{p}(|\\partial_i u|^2+\\epsilon)^\\frac{p}{2}\\dx\\\\\n&=\\sum_{i=1}^2\\int_{B_R}\\frac{1}{p}|\\partial_i u|^p\\dx\n=I_{B_R}(u).\n\\end{split}\n\\end{equation*}\nNote that in the third inequality we used the minimality of $u^\\epsilon$ subject to the boundary condition $u^\\epsilon-u\\in W_0^{1,p}(B_R)$. By uniqueness of the minimizer of $I_{B_R}$ among functions with boundary values $u$ in $B_R$, we get $v=u$. \nBy the uniform Lipschitz estimate \\eqref{lip2} and Ascoli-Arzela' theorem we obtain that the convergence is uniform.\\\\\n\nNow we show $L^p(B_R)$ convergence of the gradient. Use $\\phi = u^\\epsilon-u$ as a test function in \\eqref{orthonondeg2}, add and subtract the term $(|\\partial_i u|^2+\\epsilon)^\\frac{p-2}{2}\\partial_i u$ to get\n\\begin{equation*}\n\\begin{split}\n\\sum_{i=1}^2\\int_{B_R} &\\left((|\\partial_i u^\\epsilon|^2+\\epsilon)^\\frac{p-2}{2}\\partial_i u^\\epsilon-(|\\partial_i u|^2+\\epsilon)^\\frac{p-2}{2}\\partial_i u\\right)\\left(\\partial_i u^\\epsilon-\\partial_i u\\right)\\dx\\\\\n&=\\sum_{i=1}^2\\int_{B_R} (|\\partial_i u|^2+\\epsilon)^\\frac{p-2}{2}\\partial_i u (\\partial_i u-\\partial_i u^\\epsilon)\\dx.\n\\end{split}\n\\end{equation*}\nSince  $\\partial_i u-\\partial_i u^\\epsilon$  converges to $0$ weakly in $L^p(B_R)$, the integral in the right hand side converges to $0$. We can minorize the integral in the left hand side using the inequality \n$$ |a-b|^2(\\epsilon+|a|^2+|b^2|)^\\frac{p-2}{2}\\leq C_p ((\\epsilon+|a|^2)^\\frac{p-2}{2}a-(\\epsilon+|b|^2)^\\frac{p-2}{2}b)(a-b) $$\nvalid for $1<p<2$, and obtain that\n\\begin{equation}\\label{conv0}\n\\int_{B_R}\\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p-2}{2}|\\partial_i u^\\epsilon-\\partial_i u|^2\\dx \\longrightarrow 0\n\\end{equation} \nas $\\epsilon\\to 0$, for $i=1$, $2$.\nFinally by H\\\"older's inequality\n\\begin{equation*}\n\\begin{split}\n\\int_{B_R} |\\partial_iu^\\epsilon-\\partial_iu|^p\\dx\n&= \\int_{B_R}  |\\partial_iu^\\epsilon-\\partial_iu|^p \\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p(p-2)}{2}\\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p(2-p)}{2}\\dx\\\\\n&\\leq \\left( \\int_{B_R} |\\partial_iu^\\epsilon-\\partial_iu|^2 \\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p-2}{2}\\dx\\right)^\\frac{p}{2}\\cdot\\\\\n&\\qquad\\qquad\\qquad\\quad\\cdot\\left( \\int_{B_R} \\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p}{2}\\dx\\right)^\\frac{2-p}{2}.\n\\end{split}\n\\end{equation*}\nSince the last integral is uniformly bounded in $\\epsilon$, using \\eqref{conv0} we get  that $\\partial_i u^\\epsilon$ converges to $\\partial_i u$ in $L^p(B_R)$.\n\\end{proof}\n\n\\section{Monotone functions and Lebesgue's lemma}\nA continuous function $v:\\Omega\\longrightarrow\\mathbb{R}$ is monotone (in the sense of Lebesgue) if\n$$\\max_{\\overline{D}}v=\\max_{\\partial D}v\\quad \\text{and}\\quad  \\min_{\\overline{D}}v=\\min_{\\partial D}v$$\nfor all subdomains $D\\subset\\subset \\Omega$. Monotone functions are further discussed in \\cite{Manf}.\n\nThe next Lemma is due to Lebesgue \\cite{L}.\n\\begin{Lemma}\\label{oscLeb}\nLet $B_R\\subset \\mathbb{R}^2$ and  $v\\in C(B_R )\\cap W^{1,2}(B_R)$ be monotone in the sense of Lebesgue. Then\n\\begin{equation*}\n(\\osc{ B_r} v)^2\\log\\left(\\frac{R}{r}\\right)\n\\leq  \\pi \\int_{B_R\\setminus B_r} |\\nabla v(x)|^2 \\dx\n\\end{equation*}\nfor every $r<R$.\n\\end{Lemma}\n\n\\begin{proof}\nAssume $v$ is smooth. Let $(\\eta, \\zeta)$ be the center of $B_R$. Let $x_1$ and $x_2$ be two points on the circle of radius $t$, and let $\\gamma:[0,2\\pi]\\longrightarrow \\mathbb{R}^2$, $\\gamma(s)=(\\eta+t\\cos(s),\\zeta+ t\\sin(s))$ be a parametrization of the circle such that $\\gamma(a)=x_1$ and $\\gamma(b)=x_2$. Then we have\n\\begin{equation*}\n\\begin{split}\nv(x_1)-v(x_2)=\\int_a^b \\frac{d}{ds} v(\\gamma(s)) \\ds \n= \\int_a^b \\langle  \\nabla v(\\gamma(s)), \\gamma '(s) \\rangle \\ds\n\\leq \\int_a^{b} t\\, |\\nabla v(\\gamma(s))| \\ds.\n\\end{split}\n\\end{equation*}\nTaking the supremum on angles $a$ and $b$ such that $|a-b|\\leq \\pi$ and using H\\\"older's inequality, we get\n\\begin{equation*}\n(\\osc{\\partial B_t}v )^2 \n\\leq \\pi t^2  \\int_0^{2\\pi}  |\\nabla v (\\gamma(s))|^2 \\ds.\n\\end{equation*}\nNow diving by $t$, integrating from $r$ to $R$, and using polar coordinates we get\n\\begin{equation*}\n\\int_r^R\\frac{(\\text{osc}_{\\partial B_t}v )^2}{t} \\dt\n\\leq \\pi \\int_r^R \\int_0^{2\\pi} t\\,|\\nabla v(\\gamma(s))|^2 \\ds \\dt\n= \\pi \\int_{B_R\\setminus B_r} |\\nabla v(x)|^2 \\dx.\n\\end{equation*}\nThanks to the monotonicity of $v$, for $t\\geq r$ we have\n\\begin{equation*}\n\\osc{\\partial B_t} \\partial_ju^\\epsilon \\geq\n   \\osc{B_t} \\partial_ju^\\epsilon\n\\geq \\osc{B_r} \\partial_ju^\\epsilon\n\\end{equation*}\nand we get the result for a smooth function. The general statement follows by approximation.\n\\end{proof}\n\n\nThe following is credited to \\cite{BB} (see Lemma 2.14 for the minimum principle).\n\\begin{Lemma}\\label{maxmin2}[Minimum and Maximum principles for the derivatives]\\\\\nLet $u^\\epsilon$ be the solution of \\eqref{orthonondeg2}. Then \n$$\\min_{\\partial B_r}\\partial_j u^\\epsilon\\leq \\partial_j u^\\epsilon (x)\\leq\\max_{\\partial B_r}\\partial_j u^\\epsilon $$\nfor all $x\\in B_r$, $B_r\\subset\\subset B_R$ and $j=1$, $2$. In particular, $\\partial_j u^\\epsilon$ is monotone in the sense of Lebesgue.\n\n\\end{Lemma}\n\\begin{proof}\nWe are going to show that given a constant $C$, if $\\partial_j u^\\epsilon \\leq C$ (resp. $\\partial_j u^\\epsilon \\geq C$) in $\\partial B_r$ then $\\partial_j u^\\epsilon \\leq C$ (resp. $\\partial_j u^\\epsilon \\geq C$) in $B_r$.\nLet $\\phi^{\\pm}= 1_{B_r}(\\partial_j u^\\epsilon-C)^\\pm =1_{B_r}\\max\\{\\pm(\\partial_j u^\\epsilon-C),0\\} $ in the equation satisfied by the derivative \\eqref{orthoder2}. Since $u^\\epsilon$ is smooth and $\\partial_j u^\\epsilon \\geq C$  (resp. $\\partial_j u^\\epsilon \\leq C$) on $\\partial B_r$ we have $\\phi^\\pm\\in W_0^{1,2}(\\Omega)$, so they are admissible functions. We get\n\\begin{equation*}\n\\begin{split}\n0&=\\sum_{i=1}^2 \\int_{B_r} (\\epsilon+|\\partial_i u^\\epsilon|^2)^\\frac{p-4}{2} (\\epsilon+(p-1)|\\partial_i u^\\epsilon|^2)\\, |\\partial_i (\\partial_j u^\\epsilon -C)^\\pm |^2 \\dx \\\\\n&\\geq \\epsilon \\sum_{i=1}^2\\int_{B_r}(\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{p-4}{2}  |\\partial_i (\\partial_j u^\\epsilon -C)^\\pm |^2 \\dx \\\\\n&= \\epsilon \\int_{B_r}(\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{p-4}{2}  |\\nabla (\\partial_j u^\\epsilon -C)^\\pm |^2 \\dx .\n\\end{split}\n\\end{equation*}\nThis implies $(\\partial_j u^\\epsilon -C)^\\pm$ is constant in $B_r$, and since it is $0$ in $\\partial B_r$ then $(\\partial_j u^\\epsilon -C)^\\pm=0$ in $B_r$.\n\\end{proof}\n\\\n\n\\section{Proof of the Main Theorem}\n\\begin{proof}\n[Proof of Theorem \\eqref{C1}]\nApplying Lemma \\eqref{oscLeb} and estimate \\eqref{grad2} we get for all $r<R\/2$\n\\begin{equation}\n(\\osc{B_r} \\partial_ju^\\epsilon)^2 \\log(\\frac{R}{r}) \\leq C \\norm{\\nabla \\partial_ju^\\epsilon}^2_{L^2(B_{R\/2})} \\leq C \\left(\\fint_{B_{R}} |\\nabla u|^p\\dx+\\epsilon^\\frac{p}{2}\\right)^\\frac{2}{p}\n\\end{equation}\nand hence for all $r<R\/2$\n\\begin{equation}\\label{osc2}\n\\osc{B_r} \\partial_ju^\\epsilon  \\leq C \\left(\\log\\left(\\frac{R}{r}\\right)\\right)^{-\\frac{1}{2}}\n\\left(\\fint_{B_{R}} |\\nabla u|^p\\dx+\\epsilon^\\frac{p}{2}\\right)^\\frac{1}{p}\n\\end{equation}\nwhere $C$ is a constant independent of $\\epsilon$.\n\nThanks to Proposition \\eqref{uni2} we can pass to the limit and get \\eqref{oscEst2}.\n\\end{proof}\n\n\n\\subsection{Acknowledgements}\nI thank Peter Lindqvist for useful comments and suggestions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe success of supervised deep learning approaches can be attributed to the availability of large quantities of labeled data that is essential for network training \\cite{basak2020single}. However, in the biomedical domain \\cite{shurrab2022self, ronneberger2015unet, pavlopoulos2019survey}, it is difficult to acquire such large quantities of annotated data as the annotations are to be done by trained medical professionals. Although supervised learning has been used extensively in the past decade in biomedical imaging \\cite{pavlopoulos2019survey, basak2022mfsnet}, Self-supervised Learning (SSL) \\cite{doersch2015unsupervised, pathak2016context, chen2020simple, he2020momentum, manna2022swis} provides more traction for sustaining deep learning methods in the medical vision domain \\cite{basak2022embarrassingly, chen2019self, taleb20203d}. SSL-based pre-training alleviates the data annotation problem by utilizing only unstructured data to learn distinctive information which can further be utilized in downstream applications, typically in a semi-supervised fashion \\cite{basak2022addressing, chaitanya2020contrastive, basak2022embarrassingly}. Typically, SSL have shown to be exceedingly promising in domains with enormous amounts of data such as natural images \\cite{chen2020simple, he2020momentum} with a focus on both contrastive \\cite{oord2018representation, he2020momentum, chen2020simple, wang2021dense} and non-contrastive variants \\cite{zbontar2021barlow, bardes2021vicreg}. Recently, SSL has begun to be extensively employed for medical images as well, as evidenced by the survey paper by Shurrab \\textit{et. al. } \\cite{shurrab2022self}.\n\n\\begin{figure*}[!t]\n\\begin{center}\n  \\includegraphics[width=\\linewidth]{teaser_landscape.pdf}\n\\end{center}\n\\vspace{-0.4cm}\n \\caption{Schematic representation of contrastive learning-based pre-training of (a) baseline model \\cite{chaitanya2020contrastive} (b) our IDEAL model. $\\mathcal{E}$ represents the encoder backbone; $GAP =$ Global Average Pooling; $x_q$ and $x_k$ are the query and key images respectively. The convolution projection head in the IDEAL model generates dense projections that give better local clusters than using a traditional global pooling followed by a Multi-Layer Perceptron (MLP).}\n\\vspace{-0.4cm}\n\\label{fig:teaser}\n\\end{figure*}\n\nOne of the most popular ways of employing SSL is contrastive learning \\cite{oord2018representation, he2020momentum, chen2020simple}, which owing to its prolific performance across various vision tasks has almost become a de facto standard for self-supervision. The intuition of contrastive learning is to \\textit{pull} embeddings of semantically similar objects closer and simultaneously \\textit{push} away the representations of dissimilar objects. In medical image segmentation, contrastive learning has been leveraged in a few prior works \\cite{chaitanya2020contrastive, zeng2021positional}.  However, naive contrastive learning frameworks such as SimCLR \\cite{chen2020simple} and MoCo \\cite{he2020momentum} learn a global representation and thus, are not suitable to be applied directly to segmentation, a pixel-level task. The seminal work by Chaitanya \\textit{et. al. } \\cite{chaitanya2020contrastive} utilized a contrastive pre-training strategy to learn useful representations for a segmentation task in a low-data regime. Despite having success, their method has a few limitations, which we attempt to address and improve upon in this paper. In particular, we take a cue from the aforementioned work \\cite{chaitanya2020contrastive} to propose a novel contrastive learning-based medical image segmentation framework.\n\nAlthough our work is built upon \\cite{chaitanya2020contrastive}, there are several salient points of difference between the two models that render our contributions non-trivial and unique. First of all, our proposed method preserves spatial information and constructs a dense output projection that preserves locality. This is in contrast to the projection used in \\cite{chaitanya2020contrastive} where a global pooling is applied to the backbone encoder, thereby obtaining a single global feature representation vector for every input image. In other words, \\cite{chaitanya2020contrastive}  employed global contrastive learning for encoder pre-training and the local contrastive learning is only restricted to the decoder fine-tuning. in contrast, our method involves pre-training of the encoder on dense local features. The intuitive difference between our proposed framework and that of \\cite{chaitanya2020contrastive} has been depicted in Fig. \\ref{fig:teaser}. We argue that this will benefit pixel-level informed downstream tasks such as segmentation. This statement is supported by our findings as well (refer to Section \\ref{sec:expt} for further details). \n\nSecondly, our definition of finding positive and negative pairs for contrastive learning is different from that of traditional contrastive \\cite{chen2020simple, he2020momentum} methods. This is because, in our case, we find dense correspondences across views and define positives and negatives accordingly, i.e., we perform feature-level Contrastive Learning (refer to Section \\ref{sec:method}). Further, we adopt a novel bidirectional consistency regularization mechanism that perturbs the network instead of image perturbation used by most frameworks \\cite{he2020momentum, zou2020pseudoseg, basak2022embarrassingly}. By this approach, we enforce both streams of data flow to learn better features competitively, unlike the traditional image augmentation-based consistency regularization approaches where only one model stream learns through a single backpropagation. The empirical evaluation suggests that our proposed consistency regularization yields a superior segmentation performance, outperforming existing state-of-the-art semi-supervised approaches in literature $^\\dagger$ \\cite{bai2017semi, chaitanya2019semi}. Our contributions are as follows:\n\\begin{enumerate}\n    \\item We propose an SSL strategy that leverages dense projection head representations for contrastive learning for learning robust local features. \n    \n    \\item We redefine 'positive' and 'negative' samples in contrastive learning and extend the InfoNCE loss objective to adapt to the dense representations during the pre-training phase.\n    \n    \\item Our work also employs a uniquely devised cross-consistency regularization for fine-tuning the network to the downstream segmentation task.\n\\end{enumerate}\n\n\\begin{figure*}[tbp]\n    \\centering\n    \\includegraphics[width=\\linewidth]{mainfig_landscape.pdf}\n    \\caption{An overview of the proposed IDEAL framework: (a) \\textit{Pre-training}- $x_q$ and $x_k$ are the query and key images, $\\mathcal{E}$ and $\\mathcal{G}$ represent the encoder and projection head, respectively. The projection head employs a 1x1 convolution layer instead of a traditional MLP for dense feature extraction, resulting in better local clustering of features. (b) \\textit{Fine-tuning}- Two perturbed branches with the same input are employed. $\\mathcal{E}(\\theta)$ is the shared encoder initialized similarly for both streams, $\\mathcal{D}(\\theta_1)$ and $\\mathcal{D}(\\theta_2)$ represent two different decoder architectures; $p_1$ and $p_2$ are the predicted output segmentation maps which are thresholded to obtain $y_1$ and $y_2$ respectively. $y_1$ backpropagates through the second stream and $y_2$ backpropagates through the first stream enforcing cross-consistency in segmentation.}\n    \\label{fig:mainfig}\n\\end{figure*}\n\n    \n    \n\n\n\n\n\\def$\\dagger$}\\footnotetext{\\hspace{-1.5mm}Code will be released upon acceptance}\\def\\thefootnote{\\arabic{footnote}{$\\dagger$}\\footnotetext{\\hspace{-1.5mm}Code will be released upon acceptance}\\def$\\dagger$}\\footnotetext{\\hspace{-1.5mm}Code will be released upon acceptance}\\def\\thefootnote{\\arabic{footnote}{\\arabic{footnote}}\n\\section{Methodology}\\label{sec:method}\nOur proposed method consists of two parts: (1) Self-Supervised pre-training of the encoder, and (2) Semi-Supervised fine-tuning of the network using consistency regularization. Taking reference from \\cite{chaitanya2020contrastive}, we propose a pairwise contrastive (dis)similarity learning, employing global and local feature exploration for pre-training the encoder. The pre-trained encoder is thereafter transferred for the downstream task of cross-supervision-based consistency regularization, which aims to fine-tune the network. \n\n\\subsection{Background: Global Contrastive Learning}\\label{sec:GCL}\nThe widely employed variant of self-supervised representation learning is contrastive learning \\cite{he2020momentum, chen2020simple, oord2018representation}, where the images and their perturbations are fed into a cascade of the shared encoder and projection head. The obtained features (global representations) of each of these images in the mini-batches are then utilized for a contrastive loss function, which is designed considering contrastive learning as a dictionary lookup task \\cite{wang2021dense}. Out of all the encoded keys in the set $k$ for each encoded query $q$, a single positive key $k^+$ is matched, whilst the remainder of the keys (negative keys) represent different images of the mini-batch. A contrastive loss function is represented as follows:\n\\begin{equation}\n    \\mathcal{L}_g = -\\log \\frac{\\exp(q\\cdot k^+\/\\tau)}{\\exp(q\\cdot k^+\/\\tau) + \\sum\\limits_{k^-}\\exp(q\\cdot k^-\/\\tau)}\n\\end{equation}\n\n\\subsection{Improved Local Contrastive Learning}\\label{sec:LCL}\nWe employ local contrastive Learning that extends the existing global contrastive framework into a dense paradigm. The key difference between the traditional contrastive framework and the current work lies in the formulation of the loss function and the projection head. The traditional contrastive learning follows the paradigm where features are extracted by a backbone encoder (e.g. ResNet \\cite{he2016deep}), followed by a projection head. The projection head typically consists of a global pooling operation and an MLP module comprising a few FC layers with ReLU layers in between them. However, in our case, we remove the entire global pooling and MLP part and replace it with a $1\\times1$ convolution operation ($\\mathcal{G}$). This projection head generates dense feature representations, with an equal number of parameters as the traditional ones. \n\nWe define a set of keys $\\{k_0, k_1, ....\\}$ extracted from the encoder for every query $q$, where the encoded keys represent the local part of the image, which is very different from traditional contrastive learning where every key corresponds to a single image view. These keys represent individual features of the $D_h\\times D_w$ vectors extracted from the encoder followed by $1\\times1$ convolution, where $D_h$ and $D_w$ represent the spatial dimension of the extracted feature maps. The positive keys $k^+$ are pooled from the extracted correspondence across views, i.e. one of the $D_h\\times D_w$ features extracted from different views of the same image. The negative key $k^-$ is assigned from features from a view of different images. Having this assignment strategy and taking reference from InfoNCE \\cite{oord2018representation}, we define the local contrastive loss as:\n\n\\begin{equation}\n    \\mathcal{L}_{loc} = \\frac{-1}{D_h D_w} \\sum\\limits_{i=1}^{D_h D_w} \\log \\frac{\\exp(q_i\\cdot k_i^+\/\\tau)}{\\exp(q_i\\cdot k_i^+\/\\tau) + \\sum\\limits_{k_i^-}\\exp(q_i\\cdot k_i^-\/\\tau)}\n\\end{equation}\n\n\\subsection{Semi-Supervised Fine-tuning}\nThe SSL pre-training phase is followed by a semi-supervised fine-tuning of the trained encoder network for our downstream segmentation task. Consistency regularization \\cite{basak2022embarrassingly, chaitanya2020contrastive}, a very popular semi-supervised approach for segmentation, has been employed here, however, with two very unique and non-trivial customization, which are discussed below. \n\nIn typical consistency regularization schemes, such as in MoCo \\cite{he2020momentum}, dampened sharing of weights takes place i.e. one branch is learned directly following the gradient updates whereas the other branch is learned by employing an Exponential Moving Average (EMA) weight transfer strategy. Such a learning paradigm limits the feature learning ability of the second branch. To alleviate this, we consider two decoders of different architecture initialization. Furthermore, the up-scaling convolutions corresponding to each of them are made to work differently, i.e., one of them has ConvTranspose (convolution with trainable kernels) whilst the other one has an Up-sampling layer (simple bilinear interpolation). Outputs of each of these decoders are passed through sigmoid activation functions resulting in $p_i$ which are then threshold-ed to generate one-hot $y_i$ (see Fig. \\ref{fig:mainfig}). For labeled samples, we compute a simple cross-entropy loss between target mask $m$ and the predicted outputs $p_i$. Otherwise, a cross-consistency loss is enforced between the output probabilities $p_i$ from the two decoder branches, guided by the one-hot pseudo-label $y_j$ coming from the other branch. In other words, $y_1$ acts as a pseudo-supervisory signal for $p_2$, and $y_2$ for $p_1$. Thus, one branch learns based on the other branch's output, leading to a \\emph{competitive learning} between two branches, resulting more confident predictions. The loss function is defined as:\n\n\\begin{equation}\n\\mathcal{L}_{SSF} =\n\\begin{cases}\nCE(p_1, m) + CE(p_2, m),     \\text{ for labeled set}\\\\\nCE(p_1, y_2) + CE(p_2, y_1), \\text{ for unlabeled set}\n\\end{cases}\n\\end{equation}\n\n\n\n\n\n\\section{Experiments and Results}\\label{sec:expt}\n\\subsection{Experimental Setup}\n\\keypoint{Datasets:}\\label{sec:data} The proposed method has been evaluated on two publicly available MRI datasets - (1) the ACDC dataset \\cite{bernard2018deep}, consisting of 100 cardiac 3D shot-axis MRI volumes with expert manual annotations for three structures, namely, the left and right ventricles and the myocardium; and (2) the MMWHS dataset \\cite{zhuang2013challenges, zhuang2016multi}, consisting of 20 cardiac 3D MRI volumes with expert manual annotations for seven structures: the left and right ventricles, the left and right atria, the pulmonary artery, the myocardium, and the ascending aorta. The datasets are split into train, validation, and test sets, where the validation set for each dataset consists of 2 volumes; the test set consists of 20 volumes for the ACDC dataset and 10 volumes for the MMWHS dataset. The rest of the volumes (78 for ACDC and 8 for MMWHS) are used for training. The training and validation set together form the pre-training set, while the testing set was used \\emph{solely} for model evaluation.\n\n\n\\keypoint{Implementation Details:}\\label{sec:impl} \nIDEAL uses a ResNet-50 \\cite{he2016deep} encoder backbone with ADAM optimizer and an initial learning rate of $1e-5$, and a U-Net \\cite{ronneberger2015unet} decoder with 4 upscaling layers. The IDEAL model was implemented in PyTorch \\cite{paszke2019pytorch} and accelerated by an NVIDIA Tesla K80 GPU.\n\n\\keypoint{Evaluation Metrics:}\\label{sec:eval} Three different metrics have been used to evaluate the IDEAL model on the segmentation tasks: Dice Similarity Coefficient (DSC), Average Symmetric Distance (ASD), and Hausdorff Distance (HD). The average scores over 5 runs for all the metrics have been reported for all the fine-tuning experiments.\n\n\n\n\n\\subsection{Performance Analysis}\\label{sec:performance}\n\n\\subsubsection{Fine-tuning performances on limited annotations}\n\n\\begin{table}[!ht]\n    \\centering\n    \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{l cccc cccc}\n\n    \\toprule\n    \\textbf{Metric} & \\multicolumn{4}{c}{\\textbf{ACDC}} & \\multicolumn{4}{c}{\\textbf{MMWHS}}\\\\[0.5ex]\n    \\cmidrule(lr){2-5}\n    \\cmidrule(lr){6-9}\n    & \\textbf{L=1.25\\%} & \\textbf{L=2.5\\%} & \\textbf{L=10\\%} & \\textbf{L=100\\%} & \\textbf{L=10\\%} & \\textbf{L=20\\%} & \\textbf{L=40\\%} & \\textbf{L=100\\%} \\\\[0.5ex]\n    \\midrule\n    \n    \\textbf{ASD} $\\downarrow$  & 0.677\t& 0.614\t& 0.556\t& 0.489\t& 2.397\t& 1.798\t& 1.355\t& 1.317 \\\\[.5ex]\n    \n    \\textbf{HD} $\\downarrow$ & 2.409 & 2.209 & 2.094 & 1.999 & 5.001 & 3.499 & 2.221 & 2.183 \\\\[.5ex]\n\n    \\textbf{DSC} $\\uparrow$ & 0.738 & 0.846 & 0.879 & 0.882 & 0.626 & 0.791 & 0.815 & 0.826 \\\\[.5ex]\n    \n    \\bottomrule\n    \n    \\end{tabular} %\n    }\n    \\caption{Results obtained by the IDEAL framework with varying amounts of labeled data on the ACDC and MMWHS datasets. `$L$' represents the amount of labeled data used.}\n    \\label{tab:metrics}\n\\end{table}\n\nTo study the segmentation performance under limited labels during the fine-tuning phase, we have experimented with 1.25\\%, 2.5\\%, and 10\\% labeled volumes (denoted by `$L$' henceforth) for ACDC and 10\\%, 20\\%, and 40\\% labeled volumes for the MMWHS dataset, following the existing works in literature that have used the same datasets \\cite{basak2022embarrassingly, chaitanya2020contrastive}. The amount of labeled data is expressed as a percentage of the training set. The results are tabulated in Table \\ref{tab:metrics}. From the results, it can be inferred that on both datasets, the expected trend of improving metrics with an increase in labeled data for the fine-tuning phase. Further, it is promising to observe that even with a fraction of labels available (10\\% for ACDC and 40\\% for MMWHS), the respective semi-supervised setups asymptotically approach the fully supervised setup (differs by 0.3\\% in ACDC and 1.1\\% in MMWHS), thus depicting the efficacy of our framework under limited annotations. \n\n\n\n\n\n\\subsubsection{Comparison to state-of-the-art}\n\n\nWe have also compared our proposed framework with several existing state-of-the-art semi-supervised cardiac MRI segmentation methods in literature \\cite{chaitanya2020contrastive, chen2020simple, zeng2021positional, chen2019self, hu2021semi, chaitanya2019semi, bai2017semi, wu2022mutual}, the results of which are tabulated in Table \\ref{tab:sotas}. Additionally, we also provide a visual comparison of our segmentation performances with the SoTA methods in Fig. \\ref{fig:segoutputs}. \n\n\n\\begin{table}[t]\n    \\centering\n   \n    \\caption{Performance Comparison (DSC scores) of the proposed IDEAL framework with SoTA methods in the literature on the ACDC and MMWHS datasets.}\n    \\label{tab:sotas}\n    \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{l ccc ccc}\n\n    \\toprule\n    \\textbf{Method} & \\multicolumn{3}{c}{\\textbf{Average DSC (ACDC)}} & \\multicolumn{3}{c}{\\textbf{Average DSC (MMWHS)}}\\\\[0.5ex]\n    \\cmidrule(lr){2-4}\n    \\cmidrule(lr){5-7}\n    & \\textbf{L=1.25\\%} & \\textbf{L=2.5\\%} & \\textbf{L=10\\%} & \\textbf{L=10\\%} & \\textbf{L=20\\%} & \\textbf{L=40\\%}\\\\[0.5ex]\n    \\midrule\n    \n    Chaitanya \\textit{et. al. } \\cite{chaitanya2020contrastive} & 0.725 & 0.789 & 0.872 & 0.569 & 0.694 & 0.794 \\\\[.5ex]\n\n\n    Global CL \\cite{chen2020simple} & 0.729 & \\textbf{--} & 0.847 & 0.500 & 0.659 & 0.785 \\\\[.5ex]\n    \n    \n    PCL \\cite{zeng2021positional} & 0.671 & \\textbf{0.850} & \\textbf{0.885} & \\textbf{--} & \\textbf{--} & \\textbf{--} \\\\[.5ex]\n    \n    \n    Context Restoration \\cite{chen2019self} & 0.625 & 0.714 & 0.851 & 0.482 & 0.654 & 0.783 \\\\[.5ex]\n    \n    MC-Net \\cite{wu2022mutual} & 0.677 & 0.724 & 0.855 & 0.551 & 0.654  & 0.798 \\\\[.5ex]\n    \n    Label Efficient \\cite{hu2021semi} & \\textbf{--} & \\textbf{--} & \\textbf{--} & 0.382 & 0.553 & 0.764 \\\\[.5ex]\n    \n    \n    Data Augmentation \\cite{chaitanya2019semi} & 0.731 & 0.786 & 0.865 & 0.529 & 0.661 & 0.785 \\\\[.5ex]\n     \n     \n    Self Train \\cite{bai2017semi} & 0.690 & 0.749 & 0.860 & 0.563 & 0.691 & 0.801 \\\\[.5ex]\n    \n    \n    \\rowcolor{LightCyan}\n    Ours & \\textbf{0.738} & 0.846 & 0.879 & \\textbf{0.626} & \\textbf{0.791} & \\textbf{0.815} \\\\[.5ex]\n    \n    \\bottomrule\n    \n    \\end{tabular} %\n    }\n    \n\\end{table}\n\n\n\\keypoint{ACDC Dataset} From Table \\ref{tab:sotas}, it can be observed that our IDEAL model performs the best in $ L=1.25\\%$ setting, while being highly competitive to  PCL \\cite{zeng2021positional} in $L=2.5\\% (\\sim0.004)$ and $L=10\\% (\\sim0.006)$. This shows that even in severely constrained conditions, IDEAL can perform optimally.\n\n\\keypoint{MMWHS Dataset} Compared to existing state-of-the-art methods, our IDEAL model outperforms all previous frameworks by large margins, as inferred from Table \\ref{tab:sotas}. Chaitanya \\textit{et. al. } \\cite{chaitanya2020contrastive} achieved the next best performance with DSC values of 56.9\\% ($L=10\\%$), 69.4\\% ($\\sim$10\\% less than ours) with 20\\% labeled data and 79.4\\% with 40\\% labeled data.\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale = 0.5]{segoutputs.pdf}\n    \\caption{Visual comparison of our results with SoTA methods and ground truth, thus qualitatively validating the superiority of IDEAL in terms of segmentation performance.}\n    \\label{fig:segoutputs}\n\\end{figure}\n\n\n\n\\section{Conclusion}\nIn this paper, we propose a semi-supervised approach for cardiac MRI segmentation to tackle the labeling bottleneck in medical imaging. Our framework leverages self-supervised contrastive learning with a novel projection head to capture dense local feature representation during pre-training, followed by employing a unique cross-consistency regularization scheme during model fine-tuning for the downstream segmentation task. Results show that our IDEAL framework outperforms several state-of-the-art methods on two widely used cardiac MRI datasets. We believe that such a paradigm can also be employed in cross-modal and cross-domain scenarios, as well as extended to natural images, something we intend to explore in near future.\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\n\nThe extended \\hi\\ disks of spiral galaxies exhibit characteristic 21 cm line\nwidths of 5--15\\ \\kms\\ on small scales.  Line widths are observed to be\n$\\sim6$--10~\\kms\\ even beyond $r_{25}$, where the star formation rate (SFR)\nand the associated feedback are presumably inefficient\n\\citep*{vanderkruit1982, Dickey1990, Kamphuis1993, Rownd1994, vanzee1999}.  In\nsome observed cases, the \\hi\\ disks exhibit an overall outward decrease of the\nvelocity dispersion to 6--8\\ \\kms\\ \\citep{Boulanger1992,Petric2007}.\n\nThe observed line widths are attributable in part to thermal broadening and in\npart to turbulence.  At pressures typical of disk galaxies, interstellar\natomic gas has two temperatures at which it can remain in stable thermal\nequilibrium \\citep{Wolfire1995}. The gas is observed to be distributed with\nroughly 60\\% of the mass evenly distributed between the equilibrium\ntemperatures \\citep{Kulkarni1987, DickeyLockman1990, Dickey1993} and 40\\%\ntransiting between them \\citep{Heiles2001}.  The cold ($\\sim100$~K) neutral\ngas emits lines with a characteristic line width of $\\sim1$~\\kms, while the\nwarm ($\\sim8000$~K) neutral gas has a line width of $\\sim8$~\\kms.  Spectral\nlines observed to be broader than $\\sim8$~\\kms\\ can be attributed to\nturbulence -- stochastic fluctuations of velocity and pressure on scales much\nlarger than the mean free path of the gas.  Heating by a galactic interior or\nextragalactic background of ultraviolet (UV) radiation \\citep{Schaye2004} may\nbe significant in the outermost regions of \\hi\\ disks, where velocity\ndispersions approach thermal values for the warm phase.\n \nThere are several energetic sources that can induce turbulent motions in the\ninterstellar medium (ISM), among which feedback from recently formed stellar\npopulations and magnetorotational instabilities\n\\citep[MRI;][]{Balbus1991,Balbus1998,Sellwood1999} are some of the most\nimportant.  The stirred turbulence drives motions at smaller and smaller\nscales until viscosity dissipates the motion into thermal energy.  This\nhierarchical description was formulated by \\citet{Kolmogorov1941} for an\nincompressible turbulent fluid, though it applies well to the small scale\ndynamics of the gas in galaxy disks \\citep*{Elmegreen2003}.  Ultimately, the\nISM radiates away the thermal energy, e.g, by line cooling.  Since the typical\nradiative cooling time \\citep[$\\sim10^3$~years;][]{Wolfire2003} is shorter\nthan the time scale for turbulent energy to decay\n\\citep[$\\sim10$~Myr;][]{maclow1999}, the turbulent energy must be continuously\nreplenished on the decay time scale.  The direct identification of the energy\nsources that sustain the turbulence at this rate has drawn considerable\ninterest for well over three decades, but has not yet been concluded\n\\citep[e.g.][]{maclow2004, elmegreen2004}.\n\nNumerical simulations have demonstrated that both hydrodynamical and\nmagnetohydrodynamic turbulence decay over a relatively rapid time scale,\ncomparable to the local dynamical time, i.e. $\\sim 10$~Myr \\citep*{maclow1998,\n  Stone1998, maclow1999, Padoan1999, Avila2001, Ostriker2001}, thus requiring\na driving mechanism that continuously provides energy to the medium to\nmaintain a steady state \\citep{Gammie1996, maclow1999}.\n\nWhich mechanisms drive ISM turbulence remains a matter of debate.  As first\nenvisioned by \\citet{Spitzer1978}, star formation itself injects energy back\nto the ISM and drives turbulence through stellar winds of O\/B and Wolf-Rayet\nstars, UV ionizing radiation, and supernova (SN) explosions\n\\citep*[e.g.][]{Chiang1985, Rosen1993, Kim2001, Dib2006, Joung2006,\n  Avillez2007}. The energy input from SNe likely dominates stellar feedback\n\\citep{Kornreich2000, maclow2004}.  The overall galaxy kinematics can also\ngenerate some turbulence through spiral wave shocks \\citep{roberts69},\nrotational shear \\citep{Schaye2004}, swing amplified shear instabilities\n\\citep*{Huber2001,Wada2002}, gravitational potential energy of the arms\n\\citep{Elmegreen2003}, and non-linear development of gravitational instability\n\\citep*{li05,li06}.  Magnetized disk instabilities including the\nmagnetorotational instability \\citep*{Balbus1991, Balbus1998,\n  Sellwood1999,Dziourkevitch2004, Piontek2005}, and the magneto-Jeans\ninstability \\citep{Wada2004, Kim2006, Kimetal2006}.  Finally, there could also\nbe a contribution from thermal instability \\citep{\n  Koyama2002,Kritsuk2002,Dib2005,Hennebelle2007}.\n\n\nIn this paper we attempt a direct observational identification of the energy\nsources that drive the turbulence. Our results suggest that the energy input\nfrom young stars determines the \\hi\\ line width within the star forming region\nof galactic disks, while at large radii, where star formation is absent or\nfeeble, the line width is either sustained by the MRI \\citep{Sellwood1999}, or\nby thermal broadening from UV heating.  We examine whether SN feedback, gas\ntemperature, and MRI can maintain effectively the observed ISM line width,\nassuming that the energy loss occurs on the self-dissipation timescales\npredicted for the turbulence \\citep[][]{maclow1999}.  \\citet{maclow2004}\\ \nargue that SNe, among other mechanisms of energy injection, are the most\neffective in actively star forming regions, while in the outermost regions of\ngalaxy disks, where the SFR is low or zero, the MRI becomes important in\nsustaining both the magnetic field, even in low density regions, and the gas\nvelocity dispersion \\citep{Sellwood1999}.\n\n\nWe use multi-wavelength data in the radio, infrared (IR), and far UV\n(\\S~\\ref{sec:data}), to compare detailed maps of the ISM kinetic energy\ndensity implied by 21~cm line widths to maps of the SFR density for a sample\nof nearby spiral and dwarf galaxies. In this comparison, we take the SFR\ndensity as a proxy for the energy input rate to the ISM.  Relying on the\nmeasurement of the velocity dispersion as a tracer of turbulence in the\nneutral gas, we look at both the correlation between SFR and turbulent motions\nin the \\hi\\ on a pixel-by-pixel basis and at their mean values as function of\ngalactocentric distance (\\S~\\ref{sec:analysis}).  We find that the azimuthal\naverage of the \\hi\\ velocity dispersion decreases radially much more slowly\nthan the SFR (\\S~\\ref{sec:results}); specifically, the radial profile of \\hi\\ \nvelocity dispersion approaches the \\hi\\ thermal value for the warm neutral\nmedium where the radial profile of SFR truncates.  Therefore, we investigate\n(1) pixel-by-pixel how the kinetic energy of the gas turbulence correlates\nwith the SFR, especially in terms of SN rate, and (2) what role thermal\nbroadening and MRI play in regions of low SFR and considerable velocity\ndispersion (\\S~\\ref{sec:discussion}). In particular, we compare the current\nturbulent kinetic energy of the gas turbulence to the energy input from both\nSNe and MRI (and the efficiencies of this input), on the basis that the\nturbulence dissipation timescales, provided by extensive theoretical modeling,\napproximate the energy loss timescales.\n\n\n\n\\section{DATA}\\label{sec:data}\n\nThe main observational data for our analysis are maps of the ISM kinetic\nenergy surface density derived from high resolution \\hi\\ data.  For a\nsub-sample we include carbon monoxide (CO) emission maps to explore the\ncontribution of the molecular gas to the kinetic energy budget. These maps are\nthen compared, pixel by pixel, to analogous maps of the SFR surface density,\nderived from UV and thermal IR maps for the same objects.\n\nWe use maps produced by The \\hi\\ Nearby Galaxy Survey\n\\citep[THINGS;][]{walter2008}, a survey of the 21 cm emission line carried out\nwith the NRAO\\footnote{The National Radio Astronomy Observatory is a facility\n  of the National Science Foundation operated under cooperative agreement by\n  Associated Universities, Inc.}\\ Very Large Array, to obtain the column\ndensity \\shi\\ and velocity dispersion \\sighi\\ of neutral hydrogen.  The THINGS\ndata have an angular resolution of $\\sim7''$, and a velocity resolution of\neither 2.6 or 5.2~\\kms\\ (see Table~\\ref{tab:objs}), and a $3\\,\\sigma$\nsensitivity corresponding to a column density of $N({\\rm HI})\\ge 3.2\n\\times10^{20}$~cm$^{-2}$. To avoid projection effects in the velocity\ndispersion maps, we consider only those galaxies that are more face-on than\n$\\sim 50$\\dg\\ \\citep[see, e.g.,][]{Leroy2008}, leaving 11 disk galaxies in\ntotal, including three dwarf galaxies (Holmberg~II, IC~2574, and NGC~4214),\nlisted in Table~\\ref{tab:objs}.\n\nWe complement the analysis of the \\hi\\ gas with the mass surface density and\nvelocity dispersion of molecular hydrogen (H$_2$) gas for about half of our\ngalaxies: NGC~628, NGC~3351, NGC~3184, NGC~4214, NGC~4736, and NGC~6946.  The\nH$_2$ maps are derived from CO $J=2\\rightarrow1$ emission maps, obtained at\nthe IRAM 30-m telescope by \\citet{Leroy2008b}\\ using the HERA focal plane\narray. These maps have resolution of $\\sim11''$ and 2.6~\\kms, and sensitivity\nof $\\Sigma_{\\rm H2}\\ge 4$~\\msun~pc$^{-2}$. To derive $\\Sigma_{\\rm H2}$ we\nassume a conversion factor $X_{\\rm CO}=2\\times10^{20}$~cm$^{-2}$~K~\\kms\\ and a\nline ratio $N(J=2\\rightarrow1)\/N(J=1\\rightarrow0)=0.8$.\n\nThe SFR density maps were derived by \\citet{Leroy2008} from 24~\\mum\\ and\nfar-UV (FUV) maps, combining dust-enshrouded star formation activity and\nunobscured star formation, respectively.  The MIPS 24~\\mum\\ emission maps\n\\citep{Rieke2004}\\ are taken from the Spitzer Infrared Nearby Galaxies Survey\n\\citep[SINGS;][]{kennicutt03}, and the FUV emission maps are taken from the\nGalaxy Evolution Explorer Nearby Galaxy Survey\n\\citep[GALEX-NGS;][]{gildepaz2007}, for a set of disk galaxies in common with\nthe THINGS sample.  The MIPS instrument has a resolution of $\\simeq6''$ at\n24~\\mum, and a wide areal coverage, which typically covers $\\sim2\\,r_{25}$.\nThe GALEX FUV images have a resolution (FWHM) of $5.6''$ within the\n$\\lambda=1350-1750$~\\AA\\ band and have a field of view of $\\sim1.25$\\dg.\n\n\\section{DATA ANALYSIS}\\label{sec:analysis}\n\nAs a first step in comparing the kinetic energy of the gas to possible drivers\nof turbulence, we convert the previously described maps (\\S~\\ref{sec:data}) of\n\\hi\\ and H$_2$ gas column density and kinematics, and the 24~\\mum~+~FUV\nsurface brightness into physical quantities. Specifically, we obtain mass\nsurface density, \\shi, $\\Sigma_{\\rm H2}$, velocity dispersion, \\sighi,\n$\\sigma_{\\rm H2}$, and SFR surface density, \\sfr.  Afterwards, we proceed with\na comparison between the turbulence kinetic energy, obtained from the gas mass\nsurface density and velocity dispersion, and the SFR on a pixel-by-pixel\nbasis.\n\n\\subsection{Mass Surface Density and Velocity Dispersion}\\label{sec:moments}\n\nWe determine the \\hi\\ mass surface density, \\shi, and the velocity dispersion,\n\\sighi, by taking the moments of fully reduced data cubes for our 11 sample\ngalaxies \\citep[the prior reduction steps of are described in][]{walter2008}.\nWe obtain the pixel-by-pixel maps of $\\Sigma_{\\rm HI}(x,y)$, $\\bar{v}(x,y)$,\nand $\\sigma(x,y)$, by calculating integrals over the data cubes for all the\npositions on the sky $S_i(x,y,v_i)$.  All the sums are calculated over those\nchannels with signal identified by the THINGS masking \\citep{walter2008}; we\nonly calculated moments for those ($x,y$) line-of-sight positions with signal\nin at least 3 channels.\n\nThe zero-th moment, $\\Sigma_{\\rm HI}$, is calculated by integrating the \\hi\\ \ndata cube along the velocity dimension for a total number $N$ of velocity\nchannels using:\n\\begin{equation}  \\label{eq:m_zero}\n  \\Sigma_{\\rm HI} =  \\sum_{i=1}^N S_i,\n\\end{equation}\nwhere $S_i$ denotes the signal within the $i$-th velocity channel. Afterwards,\n\\shi\\ is corrected for inclination; the adopted inclinations are listed in\nTable~\\ref{tab:objs}.  For each sample galaxy, the spectral dimension of the\n\\hi\\ data cubes is sufficiently broad to encompass all relevant velocity\nvalues.\n\nThe first moment map, the intensity-weighted mean velocity along the line of\nsight, is given by:\n \\begin{equation}\\label{eq:m_one}\n\\bar{v} =\\frac{1 }{\\Sigma_{\\rm HI}}\\;\\sum_{i=1}^N v_i\\,S_i,\n\\end{equation}\nwhere $v_i$ is the velocity value of the $i$-th channel. We only use $\\bar{v}$\nin this context to calculate the central 2nd-moment.\n\nThe variance, corresponding to the square of the line-of-sight velocity\ndispersion, is given by the intensity-weighted mean deviation:\n \\begin{equation}\\label{eq:mom_two}\n\\sigma^2\n=\\frac{1 }{\\Sigma_{\\rm HI}}\\;\\sum_{i=1}^N  (v_i-\\bar{v})^2\\,S_i.\n\\end{equation}\n\nThroughout our analysis we restrict ourselves to the moment definition of\nEq.~\\ref{eq:mom_two}\\ to define the variance.  We presume that $\\sigma$\n(Eq.~\\ref{eq:mom_two}) is the most sensible tracer of turbulent or more\nprecisely, disordered motions, since it is applicable to any line profile.  We\naddress the discussion of this assumption in \\S~\\ref{sec:realities}.\n\n\n\nWe complement the \\hi\\ maps with H$_2$ column density and kinematics data\nobtained from CO emission (\\S~\\ref{sec:data}). Following the same procedure\npreviously adopted to calculate the relevant \\hi\\ maps, we obtain the mass\nsurface density, $\\Sigma_{\\rm H2}$, and the velocity dispersion, $\\sigma_{\\rm\n  H2}$, for the H$_2$ gas.  As described in the following sections, while\nH$_2$ is an important contributor to the total mass surface density, its\ncontribution to the total turbulent kinetic energy of gas turns out to be\nrather small in our sample.\n\n\\subsection{Realities of Measuring the Velocity \n  Dispersion in THINGS}\\label{sec:realities}\n\nIn the case of perfect signal, the second moment is exactly the kinetic\nenergy of \\hi\\ gas. Even at limited signal-to-noise ratio ($\\rm S\/N$) but with\nnormally-distributed noise, we expect the measured variance to scatter about\nthe true value in a well-behaved way, so that by averaging measurements over\nenough area, we can approximate the true value.\n\n\nAs an alternative approach, assuming and then fitting a fixed line profile\nshape (e.g., a Gaussian) fails in two ways. First, towards the centers of\ngalaxies, line profiles tend to be broad and often contain multiple emission\ncomponents. Second, in the outskirts of galaxies, where individual lines of\nsight tend to have relatively low $\\rm S\/N$, a simple Gaussian fit tends to\nunderestimate the line width by latching onto only the peak of the emission.\nIn the low $\\rm S\/N$ regime, the line profiles are approximately Gaussian.\nFor a few sample galaxies (NGC~3184, NGC~5194, NGC~6946, and NGC~7793) we find\nthat the Gaussian $\\sigma$ value is systematically lower than the second\nmoment. For these galaxies, however, the difference between Gaussian $\\sigma$\nand 2nd-moment is lower than $\\sim 2-3$~\\kms\\ in average. Since we do not know\na priori the \\hi\\ gas line-of-sight velocity distribution, and the profiles\nare not necessarily Gaussian, the best tracer of kinetic energy, as a model\nindependent estimate, is the variance.\n\nTherefore, we avoid fitting and attempt to minimize systematics by measuring\nthe second moment only on regions that have been identified to contain signal.\nThe production of these ``blanked cubes'' is described in \\citet{walter2008},\nand is briefly summarized here. First, the cubes were smoothed to 30'' angular\nresolution. Then, all areas without $\\rm S\/N> 2$ in 2 consecutive velocity\nchannels are blanked (set to 0). This does an excellent job of identifying all\nareas with significant emission but still left some false positives. These\nwere removed by a by-eye inspection that focused on agreement with the overall\nvelocity structure and morphology of the galaxy. We test the systematics\ninduced by such blanking on a face-on galaxy by measuring $\\sigma$ on the\nblanked cube and then expanding the blanking mask and remeasuring $\\sigma$. We\nfind that small expansions of the mask on both directions along the velocity\ndimension, e.g. by $\\pm 15$~km~s$^{-1}$, have almost no impact on the measured\ndispersion. Larger expansion begin to show systematic effects that we want to\navoid (particularly at low $\\rm S\/N$).\n\nThese blanked cubes minimize the effect of artifacts in the THINGS cubes and\nyield a more robust estimate of kinetic energy than fitting an assumed line\nprofile. Therefore, the profiles of $\\sigma$ and $E_k$ represent the best\npossible estimates using the available data. However, we emphasize that a\nrigorous attempt to match other observations or simulations to our\nmeasurements should bear in mind (and ideally duplicate or simulate) this\nblanking procedure.\n\n\\subsection{Turbulent Kinetic Energy}\n\nWith the moment definition of \\shi\\ and \\sighi, the \\hi\\ kinetic energy per\nunit area is given by $E_k=3\/2\\;\\Sigma_{\\rm HI}\\;\\sigma_{\\rm HI}^2$, where the\nfactor 3\/2 takes into account all three velocity components assuming that the\nvelocity dispersion is isotropic.  For the galaxies with molecular gas maps,\nwe define the total kinetic energy as $E_k=3\/2\\times(\\Sigma_{\\rm\n  HI}\\;\\sigma_{\\rm HI}^2+\\Sigma_{\\rm H2}\\;\\sigma_{\\rm H2}^2)$.  In\n\\S~\\ref{sec:results}\\ we describe the relationship between the \\hi\\ (or\n\\hi~+~H$_2$) gas kinetic energy $E_k$ and the \\sfr\\ maps in a pixel-by-pixel\nscatter plot.\n\n\\subsection{Star Formation Rate}\\label{sec:limitations}\n\nWe use maps of SFR surface density, \\sfr, derived by \\citet{Leroy2008}, from\ncombining the 24~\\mum\\ and the FUV emission maps taken from the SINGS and the\nGALEX-NGS surveys (\\S~\\ref{sec:data}).  \\citet{Leroy2008}\\ have calibrated the\nSFR represented by the UV and IR emission by comparing these maps to \\halpha,\n24~\\mum, and Pa$\\alpha$ emission from \\hii\\ regions\\ and young compact stellar\nclusters \\citep{calzetti07}.  The 24~\\mum\\ band emission is mostly radiation\nfrom hot dust heated by the UV light from young massive stars and therefore\ntraces dust-enshrouded, ongoing star formation over a time scale $3-10$~Myr\n\\citep{Calzetti2005,perez2006,tamburro2008}, although part of the emission\nproceeds from outside the \\hii\\ regions, thus tracing older ($>10$~Myr)\nstellar populations.  FUV emission is mostly photospheric emission from O and\nB stars. It thus complements the 24~\\mum\\ emission in regions poor in dust\ncontent, probing therefore low-metallicity and older regions of star formation\nover timescales of $\\tau\\sim10-100$~Myr \\citep{Calzetti2005,Salim2007}.\n\n\n\nThe calibration provides SFR estimates with an uncertainty of 40\\% at most at\nhigh SFR, depending on variations in geometry, dust temperature, and age of\nstellar populations \\citep{Leroy2008}.  At low SFR, a substantial part of the\n24~\\mum\\ emission proceeds from diffuse dust in the ISM, which may not be\nassociated directly to recent star formation, with the dust being heated by\nnearby (young and old) star clusters.  \\citet{Leroy2008} argue that below a\nfiducial threshold of $\\Sigma_{\\rm SFR}=10^{-10}\\; \\rm\nM_\\odot\\;yr^{-1}\\;pc^{-2}$ the SFR maps represent upper limits to the true SFR\nbecause of the contribution of this diffuse dust component.\n\n\n\n\n\\section{RESULTS}\\label{sec:results}\n\nIn this section we describe our two main results: (1) the observation that a\nradial decline of the \\hi\\ velocity dispersion is pervasive throughout the\nentire sample with little dependence on galaxy type; and (2) that we find\ncorrelation between the kinetic energy of gas and SFR with a slope close to\nunity and a similar proportionality constant in all objects.  Both results\ndepend on the exceptional quality of the data in terms of spatial and velocity\nresolution and the wide field of view. We reserve the broader interpretation\nof our results for \\S~\\ref{sec:discussion}.\n\nIn the following, we use units of $r_{25}$, which provides a convenient\nnormalization. \\citet{Leroy2008, tamburro2008}\\ measured the exponential scale\nlengths of near-IR emission from our sample, a good proxy for stellar mass.\n$r_{25}$ is typically $4.6\\pm0.8$ times the near-IR scale length, and the SFR\nmaps yield comparable scale lengths to the near-IR. When we normalize by\n$r_{25}$ then, it is roughly equivalent to normalizing by the scale length of\nthe disk.\n\n\\subsection{Radial Profiles of $\\sigma_{\\rm HI}$}\\label{sec:radial}\n\nTo start, we determine the azimuthally averaged radial profiles of \\hi\\ \nvelocity dispersion, SFR, and gas kinetic energy, by calculating the average\nvalues of \\sighi\\ and SFR within annuli of $\\sim15''$ width.  The resulting\n\\sighi$(r)$ and \\sfr$(r)$ profiles are shown in Fig.~\\ref{fig:m2_vs_r_fit}.\nThey exhibit a radial decline of \\hi\\ velocity dispersion as a common\ncharacteristic for all sample galaxies independent of their dynamical mass.\nWhile previous studies have reported this individually for a few disk galaxies\n\\citep[i.e.,][]{Boulanger1992, Petric2007}, we show for the first time that\nthis is true for a significant sample of dwarf and normal spiral galaxies.\n\nTo characterize the \\sighi\\ radial gradients, we fit a linear relation to the\nradial profiles of \\sighi\\ for all $r\\ge r_{25}$, a regime where $\\sigma(r)$\nis approximately linear with $r$ for all the galaxies of the sample. With\n$\\sigma(r)$ going from $\\gtrsim20$~\\kms\\ down to $\\sim5\\pm2$~\\kms\\ near the\noutermost observed radius (Fig.~\\ref{fig:m2_vs_r_fit}), the velocity\ndispersion decreases with radius by $\\simeq3$--5~\\kms\\ per $\\Delta r_{25}$;\nfor comparison, azimuthal variations at a fixed radius are $\\simeq 5$~\\kms.\nIn Fig.~\\ref{fig:s25_avg}, we display and summarize the intercept value of\n\\sighi\\ at $r_{25}$ and the \\hi\\ mass weighted median of \\sighi\\ for all\nsample galaxies individually. Our observations resolve the outward radial\n\\sighi\\ decline well, since a typical $r_{25}$ for our sample galaxies\ncorresponds to $\\sim30$--40 times the 7'' \\hi\\ resolution limit and the\ntypical radial extent of the \\hi\\ emission is $2-4\\times r_{25}$.\n\nRemarkably, we find that all galaxies have the same \\hi\\ velocity dispersion\nat their respective $r_{25}$: a general value of\n$\\sigma(r_{25})\\simeq10\\pm2$~\\kms, which displays no apparent trend with the\ndynamical mass and morphological type, and is consistent with the \\hi\\ \nmass-weighted median value $\\langle\\sigma\\rangle$ (see Fig.~\\ref{fig:s25_avg}\\ \nand Table~\\ref{tab:objs}).  The radial \\sighi\\ gradient has in all cases the\nsame sign as the much steeper decline of the mean SFR as function of radius as\nshown for comparison in logarithmic scale in Fig.~\\ref{fig:m2_vs_r_fit}.\n\nSince the sample galaxies are more face-on than 50\\dg, the systematic increase\nof \\sighi\\ towards the center cannot be due to increasing beam smearing\neffects.  We analyze the combined effect of beam smearing, inclination and\nrotation curve on the velocity dispersion by constructing a sample data cube\ncontaining signal in only one velocity channel per each ($x,y$) spatial\nposition (i.e. intrinsic \\sighi~=~0) and characterized by the same inclination\nand rotation curve of NGC~5055 -- the most inclined disk of the sample with a\nfast rotation speed.  After convolving this sample data cube with a kernel of\n7'' FWHM (the resolution of our \\hi\\ maps), we calculate the velocity\ndispersion using second moments (Eq.~\\ref{eq:mom_two}) as done throughout our\nanalysis (\\S~\\ref{sec:moments}).  The resulting line broadening from beam\nsmearing for a disk with the rotation curve and orientation of NGC~5055 is\nonly $\\lesssim 5 $~\\kms, lower than the observed velocity dispersion in\nNGC~5055 by $\\sim10$--20~\\kms.  The fractional contribution of the velocity\ndispersion from beam smearing to the total observed velocity dispersion is\nonly 20\\% in the central region ($r<1\/4\\; r_{25}$) and at most 10\\% at larger\ngalactocentric radii.\n\n\n\\subsection{The H{\\sc i} Kinetic Energy Density as a Function of Radius}\\label{sec:radialek}\n\nThe \\hi\\ kinetic energy density, $E_k$, in each pixel exhibits a clear radial\ndecline, as shown in the full pixel-by-pixel distribution\n(Fig.~\\ref{fig:Ek_vs_r}), where the black contours and color scale indicate\nthe density of pixels at each \\hi\\ kinetic energy and radius, while the red\ncontours show the sum of atomic and molecular kinetic energy.  The latter\ndiverges from the \\hi\\ kinetic energy in the inner parts of some galaxies,\nindicating that the cold molecular gas contributes to the kinetic energy\nbudget. Since every galaxy of the sample includes $\\sim2\\times10^5$ pixels\nwith significant signal, we display in Fig.~\\ref{fig:Ek_vs_r} the density\ncontours of the data points.\n\nWe include the analysis of the H$_2$ mass surface density and velocity\ndispersion derived from the CO emission for a few sample galaxies\n(\\S~\\ref{sec:analysis}), to quantify the contribution of the molecular gas to\nthe total kinetic energy at high H$_2$-to-\\hi\\ mass ratio.  The total kinetic\nenergy $E_k=E_{\\rm HI}+E_{\\rm H2}$ is plotted in Fig.~\\ref{fig:Ek_vs_r}\\ (with\nred contours) for the galaxies with CO data. For the galaxies NGC~4736,\nNGC~5055, and NGC~6946, $E_{\\rm H2}$ is comparable to $E_{\\rm HI}$ or even\ndominant in the central regions of galaxy disks; for these galaxies the H$_2$\nmass and spatial extent is considerable. For those galaxies where the \\hi\\ gas\ndominates the total gas mass, i.e., NGC~628, NGC~3184, NGC~3351, and the dwarf\ngalaxies Holmberg~II, IC~2474, and NGC~4214, characterized by little or no\ndetected molecular gas, the molecular gas does not contribute much to the\ntotal kinetic energy.\n\n\n\n\\subsection{Correlation between $E_k$ and \\sfr}\\label{sec:pixel}\n\nIn Fig.~\\ref{fig:Ek_vs_sfr} we compare pixel-by-pixel the relation between the\nkinetic energy density of the \\hi\\ gas, $E_k=3\/2\\,\\Sigma_{\\rm HI}\\,\\sigma_{\\rm\n  HI}^2$, and the SFR surface density -- a proxy for the energy input rate by\nSN.  We find that in all galaxies these quantities are well correlated with a\nslope close to unity and with no evident dependence on dynamical mass\n(Fig.~\\ref{fig:Ek_vs_sfr}).  Note that a considerable fraction of all data\npoints in Fig.~\\ref{fig:Ek_vs_sfr}\\ lie below the noise estimated for the\n\\sfr\\ maps (\\S~\\ref{sec:limitations}). These data points lie in the outermost\nparts of galaxy disks ($r>2\\times r_{25}$), where there is little or no\nongoing star formation (cf. Fig.~\\ref{fig:Ek_vs_r}).\n\nWe note that the slope of all correlations in Fig.~\\ref{fig:Ek_vs_sfr}\\ \nflattens at high $E_k$ and \\sfr, although we argue in \\S~\\ref{sec:radialek}\nthat this is not caused by a higher abundance of molecular gas at higher \\sfr.\nThe total kinetic energy $E_k=E_{\\rm HI}+E_{\\rm H2}$ is plotted in\nFig.~\\ref{fig:Ek_vs_sfr} (with red contours) for the six sample galaxies with\nH$_2$ data. Only for NGC~5055 and NGC~6946 is the kinetic energy of the\nmolecular gas, $E_{\\rm H2}$ important; in those cases the $E_{\\rm HI}+E_{\\rm\n  H2}$ vs.\\ \\sfr\\ relation is linear and the slope is close to unity.\n\n\\section{DISCUSSION}\\label{sec:discussion}\n\nWhat scenario does our data support for producing the observed line widths?\nSN-driven turbulence seems likely to be the dominant factor broadening\nline widths within the radius of active star formation, since we find that the\nlevel of predicted SN energy is sufficient to account for the turbulent\nkinetic energy implied by the line width as a function of radius.\n\nThe radial slopes of SN energy and kinetic energy of the neutral (\\hi\\ and\nH$_2$) gas agree qualitatively so that the kinetic energy of the gas is\nproportional to the local star formation rate.  Yet, the fact that the \\hi\\ \nvelocity dispersion approaches its thermal value of roughly 6~\\kms\\ well\nbeyond the radius of detectable star formation indicates that either (1) the\nline broadening is due to UV heating, with a warm neutral medium temperature\nof $\\sim 5000$~K resulting in \\sighi~$\\sim 6$~\\kms, or (2) the gas is actually\nturbulent and another mechanism such as the MRI is driving the turbulence.  We\nnow explore whether our new data support this scenario.\n\nIn the following, we compare the observed line widths with the most plausible\nmechanisms for generating them: SNe (\\S~\\ref{sec:supernova}), UV heating\n(\\S~\\ref{sec:thermal}), and MRI (\\S~\\ref{sec:mri}). SNe and MRI both produce\nbroad line widths by driving turbulence, while UV heating can produce thermal\nbroadening.  The required energy injection rate depends both on the kinetic or\nthermal energy of the gas, which can be derived from the observed line widths\nand the turbulence decay or cooling timescales, which must be derived from\nmodels.  More precisely, the gas kinetic energy implied by the linewidth\nconsists of a combination of turbulence and the thermal energy associated with\nthe (warm) gas temperature, i.e. $E_k=E_{\\rm turb}+E_{\\rm therm}$.  If the\nthermal broadening is much less effective than the turbulence, then $E_k\\simeq\nE_{\\rm turb}$.\n\nIf the gas turbulence is mainly driven by SNe and MRI, then we expect the\ndissipation rate of turbulence to equal the sum of the energy input rates of\nSNe and MRI,\n\\begin{equation} \n\\dot{E}_k \\simeq \\epsilon_{\\rm SN}\\,E_{\\rm\n  SN} \/\\tau_{\\rm SN}+ \\epsilon_{\\rm MRI} E_{\\rm MRI} \/ \\tau_{\\rm MRI},\n\\end{equation}\nwhere $\\epsilon_{\\rm SN}$ and $\\epsilon_{\\rm MRI}$ are the efficiencies, and\n$\\tau_{\\rm SN}$ and $\\tau_{\\rm MRI}$ are the decay times of turbulence driven\nby the two mechanisms.  Different mechanisms can result in different decay\nrates because they have different driving scales and magnitudes\n\\citep{Stone1998,maclow1999}.\n\n\n\\subsection{Supernova Energy}\\label{sec:supernova}\n\nAssuming steady state equilibrium between the energy input rate from SNe to\nturbulent gas motions and the energy loss rate from dissipation of this\nturbulence, then the resulting kinetic energy $E_k = \\epsilon_{\\rm SN}\n\\dot{E}_{\\rm SN} \\tau_{SN}$, where $\\dot{E}_{\\rm SN}$ is the rate of released\nSN energy, which we estimate from the SFR, and the SN feedback efficiency\n$\\epsilon_{\\rm SN}$ is the fraction of SN energy converted to turbulent\nmotions in the cold gas.  \\citet{maclow1999} finds that the dissipation rate\nof turbulence depends on the driving scale $\\lambda$ and the velocity\ndispersion $\\sigma$ as\n\\begin{equation}\n\\tau_D\\simeq9.8\\;(\\lambda_{100}\/\\sigma_{10})\\;\\rm Myr,\n\\end{equation}\nwhere $\\lambda_{100} = \\lambda \/ 100$~pc and $\\sigma_{10} = \\sigma \/\n10$~km~s$^{-1}$.  Numerical simulations of SN-driven turbulence yield\n$\\lambda=100\\pm30$~pc \\citep{Joung2006, Avillez2007}, and our own analysis\ngives an average velocity dispersion $\\sigma=10$~\\kms (\\S~\\ref{sec:radial}).\nThe SN energy input rate, $\\dot{E}_{\\rm SN}$, can be estimated from the SN\nrate implied by our SFR maps.  The SN rate per unit area, $\\eta$, depends on\nthe fraction $f_{*\\rightarrow \\rm SN}$ of all recently formed stars that\nterminate in core-collapse SNe:\n\\begin{equation}\n  \\label{eq:sn_rate}\n\\eta= \\frac {\\rm SFR}{\\avg{m}}  \\times  f_{*\\rightarrow \\rm SN},\n\\end{equation}\nwhere $\\avg{m}$ is the average mass of stars of the population.  We assume\nthat only those stars in the mass range ($8-120$)\\,\\msun\\ can form\ncore-collapse SNe. The SFR maps used in our analysis \n  assume an initial mass function (IMF)\n$\\phi(m)=m^{-\\alpha}$, where $\\alpha=1.3$ for the mass range\n($0.1-0.5$)\\,\\msun\\ and $\\alpha=2.3$ for the mass range ($0.5-120$)\\,\\msun\\ \n\\citep[cf.][]{Leitherer1999, calzetti07}.  Then, the SN fraction\n\\begin{equation}\n  \\label{eq:frac_sn}\n  f_{*\\rightarrow \\rm SN}\/\\avg{m}=\n\\frac{\\int_{8\\,\\rm M_\\odot}^{120\\,\\rm M_\\odot}\n  \\phi(m)\\, dm} {\\int_{0.1\\,\\rm M_\\odot}^{120\\,\\rm M_\\odot}\n  \\phi(m)\\, dm} \/ \\frac{\\int_{0.1\\,\\rm M_\\odot}^{120\\,\\rm M_\\odot}\n  m \\phi(m)\\, dm} {\\int_{0.1\\,\\rm M_\\odot}^{120\\,\\rm M_\\odot}\n  \\phi(m)\\, dm},\n\\end{equation}\nyielding $f_{*\\rightarrow \\rm SN}\/\\avg{m}\\simeq1.3\\times10^{-2}\\;\\rm\nM_\\odot^{-1}$.  If we were to reduce the upper mass limit to 50~\\msun\\ yields\n$f_{*\\rightarrow \\rm SN}\/\\avg{m}\\simeq1.2\\times10^{-2}\\;\\rm M_\\odot^{-1}$, and\nan upper mass limit of 20~\\msun\\ yields $0.9\\times10^{-2}\\;\\rm M_\\odot^{-1}$.\nThe effect of these variations on $\\eta$ is unclear, because the upper mass\nlimit of the IMF also affects our translation of UV and IR light into SFR.\nThe UV and IR maps are primarily sensitive to high mass stars. Therefore for a\nlower upper mass limit, therefore, less UV and IR light is emitted per unit\nstar formed and if the upper mass limit is actually lower than we have\nassumed, then we have underestimated the true SFR. In calculating $\\eta$,\nthese higher SFR and lower $f_{*\\rightarrow \\rm SN}\/\\avg{m}$ have opposite\neffects, leaving the impact of changing the upper mass limit on $\\eta$\nunclear. As the upper mass limit decreases, we estimate less intrinsic UV and\nIR emission per high-mass star, which we have used to construct the maps. We\nneglect the contribution of type Ia SNe whose rate is $\\sim1\/3$ of the\ncore-collapse rate for the morphological types of our sample galaxies\n\\citep{Mannucci2005}.  In these circumstances, the SN rate can be\nstraightforwardly calculated as a function of SFR, $\\eta=\\eta(\\Sigma_{\\rm\n  SFR})$, and, depending on the adopted assumptions, the SN rate is\ncharacterized by an uncertainty of a factor $\\sim1\/3$.  We assume that for\neach SN explosion only a fraction $\\epsilon_{\\rm SN}\\le1$ of $10^{51}$~erg --\nroughly the energy released by a single SN event \\citep{Heiles1987} -- is\nconverted into turbulence. Then in steady state the kinetic energy of the gas\nturbulence $E_k=\\eta\\times(\\epsilon_{\\rm SN}\\,10^{51}\\;{\\rm erg})\\tau_D$.\n\nIn Fig.s~\\ref{fig:Ek_vs_r} and~\\ref{fig:Ek_vs_sfr}, we compare our estimate\nfor the total energy from SNe produced within a single turbulent decay time\n$\\tau_D=9.8$~Myr with $E_k$, the kinetic energy derived from the line width,\nas a function of radius and of SFR, respectively, for different values of SN\nefficiency.  Fig.~\\ref{fig:Ek_vs_r} shows the azimuthally averaged SN energy\ndecreasing as a function of radius.  A universal $\\epsilon_{\\rm SN}$ would\nproduce a linear correlation between $E_k$ and \\sfr\\ as shown in\nFig.~\\ref{fig:Ek_vs_sfr}, where lines of unity slope and constant values of\n$\\epsilon_{\\rm SN}$ represent SN energy input as a function of SFR.\nFig.~\\ref{fig:Ek_vs_r} and Fig.~\\ref{fig:Ek_vs_sfr}\\ show that, at least in\nthe high star formation regime ($\\Sigma_{\\rm SFR} >10^{-9}\\; {\\rm\n  M}_{\\odot}$~yr$^{-1}$~pc$^{-2}$), the data points typically agree with an\nefficiency $0.1\\le\\epsilon_{\\rm SN}\\times(10^7\\;{\\rm yr}\/\\tau_D)\\le1$.\n\nThis estimate of the efficiency is consistent with numerical simulations that\nestimate $\\avg{\\epsilon_{\\rm SN}}\\simeq0.1$ \\citep{Thornton1998}.  In the low\nSFR regime ($\\Sigma_{\\rm SFR} < 10^{-9}\\; {\\rm\n  M}_{\\odot}$~yr$^{-1}$~pc$^{-2}$), many data points in\nFig.s~\\ref{fig:Ek_vs_r} and~\\ref{fig:Ek_vs_sfr} lie close to the line of\nmaximum efficiency $\\epsilon_{\\rm SN}=1$.  At $\\Sigma_{\\rm SFR} <\n10^{-10}\\;{\\rm M}_{\\odot}$~yr$^{-1}$~pc$^{-2}$, in particular at large\ngalactocentric radii, the estimated \\sfr\\ rates fall to within the noise (see\n\\S~\\ref{sec:limitations}), whilst the measured $E_k$ is typically well\nconstrained.  Nevertheless, $\\epsilon_{\\rm SN} = 1$ remains problematic, since\nit would imply that all the kinetic energy of SN remnants would be deposited\nas kinetic energy of the gas.  In fact \\citet{TenorioTagle1991}\\ argue that\nthe expected SN efficiency $\\epsilon_{\\rm SN}$ should be at most $\\sim0.5$.\nValues graeter than 0.5, therefore, either imply that other sources inject\nenergy into the ISM (see following sections), or that the dissipation\ntimescales are shorter than we assume. In the next sections, we examine\nalternative mechanisms to explain the line width in these regions.\n\n\n\\subsection{Thermal Broadening}\\label{sec:thermal}\n\nThe temperature of the warm phase of the \\hi\\ is maintained by UV heating.\nHowever, neutral gas never reaches temperatures high enough to explain line\nwidths as high as 10~\\kms, which are instead attributed to supersonic\nturbulent motions \\citep{Wolfire2003, Lacour2005}.  As\n\\sighi~$\\gtrsim10$~\\kms\\ for $r\\lesssim r_{25}$ for all our sample galaxies,\nwe argue that thermal effects can be neglected within the star forming radius.\nIn regions of active star formation, stellar winds and ionizing radiation from\nstars appear less effective together than the SNe from the same stellar\npopulation at driving turbulence \\citep{maclow2004}. Thus, SN driven\nturbulence looks likely to dominate there.\n\nIn the outer regions of \\hi\\ disks, on the other hand, the velocity dispersion\napproaches its thermal value.  There, the warm neutral atomic phase\ntemperature is typically $\\sim5000$~K, as measured, e.g., in the solar\nneighborhood \\citep{Heiles2003, Redfield2004}, where the SFR is a few\n$\\times10^{-9}\\; \\rm M_\\odot\\;yr^{-1}\\;pc^{-2}$.  This temperature gives a\nthermal width of \\sighi~$\\sim6$~\\kms. Typical temperatures of the warm medium\ncan even be as high as $\\sim8500$~K \\citep[corresponding to\n$\\sim8$~\\kms;][]{Wolfire1995}.  If thermal broadening is effective, the\nobserved \\sighi~$\\sim6$~\\kms\\ outside the star forming radius does not\nnecessarily involve turbulence.  However, such temperature levels, especially\nat $r\\gtrsim2\\times r_{25}$, where the SFR is below $10^{-10}\\; \\rm\nM_\\odot\\;yr^{-1}\\;pc^{-2}$ (cf.  Fig.~\\ref{fig:m2_vs_r_fit}), still require a\ncontinuous UV background source warming up the \\hi\\ disks of galaxies. At\n$r>r_{25}$, the source of such UV radiation is presumably not local; it could\nbe extragalactic.\n\n\nThe local thermal pressure estimate and the actual velocity dispersion in the\nouter parts of \\hi\\ disks are consistent with the existence of a warm phase\nfor the gas.  For example, \\citep{Leroy2008} estimate a local pressure\n$P\/k\\sim300$~K~cm$^{-3}$ at $r\\simeq2\\times r_{25}$ for the galaxy NGC~4214,\nassuming hydrostatic equilibrium \\citep[cf.][]{Elmegreen1989}.  At a radius\n$2\\times r_{25}$, the \\hi\\ velocity dispersion in NGC~4214 is $\\sim7$~\\kms,\ncorresponding to a temperature of $\\sim5900$~K, which, solving $P\\propto\nn\\,\\sigma^2$, yields a density $n\\simeq0.05$~cm$^{-3}$. This is consistent\nwith the gas in a warm phase according to Figure~7 in \\citet{Wolfire2003},\nwhere a pressure of $\\sim300$~K~cm$^{-3}$ corresponds to\n$n\\sim0.04$~cm$^{-3}$, at least for the outer parts ($r\\sim 15-18$~kpc) of the\nMilky Way, which likely have a similar UV background.  For pressure values\nlower than $300$~K~cm$^{-3}$, all the gas is warm.\n\n\n\\subsection{MRI Energy}\\label{sec:mri}\n\nIf external UV heating is insufficient to maintain a warm phase, then a\nnon-stellar energy source is needed to explain the observed turbulence\n(\\S~\\ref{sec:radialek}). MRI is a plausible candidate. It develops in\ndifferentially rotating disks with angular velocity decreasing outwards, as in\nall but the smallest galactic disks, as long as some weak magnetic field\nthreads the disk.  It has been argued to sustain both the interstellar gas\nturbulence and the galactic magnetic field to a few $\\mu$G \\citep{Piontek2005,\n  Piontek2007}. MRI requires a minimum magnetic field as low as $10^{-25}$~G\nto originate, which is much lower than the seed galactic magnetic fields\n\\citep{Kitchatinov2004}.\n\nFollowing the same line of reasoning as \\S~\\ref{sec:supernova}, if we assume\nthat a fraction $\\epsilon_{\\rm MRI}\\le1$ of the MRI energy is transformed into\nkinetic energy, then in steady state the observed kinetic energy must equal\nthe MRI energy input within the turbulence decay time: $E_k=\\epsilon_{\\rm\n  MRI}\\,\\dot{E}_{\\rm MRI}\\;\n\\tau_{\\rm MRI}$. Theoretical calculations \\citep{Sellwood1999, maclow2004,\n  maclow2008} estimate the production energy rate of MRI to be\n\\begin{equation}\n\\begin{array}{cl}\n\\dot{E}_{\\rm MRI}= &\n3.7\\times10^{-8}\\;{\\rm erg\\;cm^{-2}\\;s^{-1}}\\times \\\\\n & \\;\\left(\\frac{h_z}{100\\;{\\rm pc}}\\right)\\;\n \\left(\\frac{B}{6\\;\\mu{\\rm G}}\\right)^2\\;\\frac{\\Omega}{(220\\;{\\rm Myr})^{-1}},\\\\\n\\end{array}\n\\end{equation}\nwhere $\\Omega\\equiv v\/r$ is the angular velocity, $h_z$ is the vertical\nthickness of the \\hi\\ disk, and $B$ is the magnetic field.  In our analysis,\nwe assume a constant thickness $h_z=100$~pc for all galaxies. This is\nappropriate for the inner Milky Way out to the solar circle\n\\citep{Wolfire2003}, although in the outer regions of disks and in dwarf\ngalaxies $h_z$ may be higher \\citep[up to $\\sim300$~pc;][]{Walterbrinks1999}.\nWe also assume a constant magnetic field $B=6\\;\\mu{\\rm G}$ as a typical\ngalactic magnetic field \\citep{Beck1996, Heiles2005}. Taking the turbulent\ndecay timescale again to be $\\tau_{\\rm MRI}\n\\simeq9.8\\;(\\lambda_{100}\/\\sigma_{10})\\;\\rm Myr$ we can estimate the driving\nscale $\\lambda=\\lambda_c$.  The critical wavelength for the fastest MRI growth\n\\citep{Balbus1998}\n\\begin{equation}\n\\lambda_c=2\\pi\\:v_A\\; \n\\left[ -\\frac{3+\\alpha}{4}\\;\\frac{d\\Omega^2}{d\\ln r}\\right]^{-1\/2},\n\\end{equation}\nwhere $\\alpha\\equiv d\\ln v\/d\\ln r$, and the Alfv\\'en velocity $v_A^2 = B^2 \/ 4\n\\pi \\rho$.  For a typical galactic magnetic field of $B=6\\;\\mu{\\rm G}$ and a\ndensity of $2\\times10^{-24}$~g~cm$^{-3}$, $\\lambda_c\\sim10^2$~pc.\n\n\nIn Fig.~\\ref{fig:Ek_vs_r}, we compare the energy produced by MRI within a\ndecay time of turbulence, $\\epsilon_{\\rm MRI}\\,\\dot{E}_{\\rm MRI}\\,\\tau_{\\rm\n  MRI}$, with the observed kinetic energy $E_k$ as a function of radius for\ndifferent values of the MRI efficiency. Note that although both $\\dot{E}_{\\rm\n  MRI}$ and $\\tau_{\\rm MRI}$ are functions of radius, the MRI energy input\nonly decreases with radius slowly, not exponentially as $E_k$. In\nFig.~\\ref{fig:Ek_vs_r}, we show a comparison between the SN energy\n(\\S~\\ref{sec:supernova}) and the MRI energy plotted as a function of radius,\nand indicated with green and blue solid lines, respectively. We also plot in\nthe pixel-by-pixel $E_k$ vs.\\ \\sfr\\ plot of Figure~\\ref{fig:Ek_vs_sfr}\\ the\naverage value of the MRI for pixels with each \\sfr, for $\\epsilon_{\\rm\n  MRI}=1$.\n    \nWhile SNe can be identified as the dominant source of energy at $\\Sigma_{\\rm\n  SFR}>-9\\;{\\rm M_\\odot\\;yr^{-1}\\;pc^{-2}}$, the MRI contribution becomes\nimportant at $\\Sigma_{\\rm SFR}<-9\\;{\\rm M_\\odot\\;yr^{-1}\\;pc^{-2}}$ and\ndominant at large galactocentric radii. On the other hand, the kinetic energy\nof turbulence within $r_{25}$ is much higher than the predicted MRI energy,\nindicating that MRI can not account for the observed gas turbulence in regions\nof active star formation.\n\nThe observed $E_k$ is higher than the SN energy for the dwarf galaxies\nHolmberg~II and, more significantly, IC~2574. In these two cases, the gas\nturbulence cannot be explained by SNe \\citep[cf.][]{Stanimirovic2001, Dib2005,\n  Pasquali2008}; still, the MRI could account for the observed regime of\nturbulence.  Our analysis suggests that MRI could dominate regions of low SFR,\nand rules out MRI as an effective turbulence driving mechanism at high SFR.\n\n\n\\subsection{Robustness  of the Approach}\\label{sec:errors}\n\nThree major sources of uncertainty enter our analysis.  The first and largest\nof these are the empirical conversions from UV and IR emission to \\sfr, which\nmay introduce up to a 40\\% uncertainty (\\S~\\ref{sec:limitations}). The\nconversion from \\sfr\\ to SN rate relying on a universal IMF introduces an\nadditional 30\\% uncertainty (\\S~\\ref{sec:supernova}).  A second uncertainty\nenters from the reliance on numerical simulation to evaluate the driving scale\nof SN-driven turbulence and thus the dissipation timescale $\\tau_{\\rm SN}$.\nThe simulations give a 30\\% error for their estimate of the driving scale\n\\citep{Joung2006, Avillez2007}.  The dissipation timescale also depends on the\nmean velocity dispersion, which is, of course, not constant as assumed in our\nestimate. A third uncertainty enters in our estimates of MRI energy, where we\nassume a constant magnetic field and vertical thickness of the disk for all\nthe galaxies of the sample, which may not be the case. The vertical thickness\nof the gaseous component in galaxies increases outwards and may double the\nvalue assumed in our analysis of $h_z=100$~pc. The typical strength of\nmagnetic fields observed in the Milky Way and other galaxies declines slowly\nas a function of galactocentric distance, although the variations of the\nmagnetic field can be much larger azimuthally than radially\n\\citep{Fletcher2004, Han2006, Beck2007}.  These three sources of errors could\nin principle explain those data points lying near $\\epsilon_{\\rm SN}\\gtrsim1$\nand $\\epsilon_{\\rm MRI}\\gtrsim1$.\n\nAside from the potential sources of uncertainties, if we were to interpret our\nempirical finding by taking into account only SNe explosions and MRI, still a\nminor part of the observed turbulence would require $\\epsilon_{\\rm SN}$ and\n$\\epsilon_{\\rm MRI}$ efficiencies uncomfortably high, as high as 100\\%.\nTherefore, we do not exclude that other mechanisms could be efficiently\ndriving some turbulence in the gas. Potentially, within the star forming\nradius ($\\lesssim r_{25}$) stellar winds could be the most effective, while in\nthe outermost regions of \\hi\\ disks, i.e. $r\\gtrsim2\\times r_{25}$, where the\nSNe and the star formation effects are not likely to produce feedback, a floor\nlevel of \\hi\\ velocity dispersion of $\\sim6$~\\kms\\ could be attributed to\nthermal broadening.  Observation of the \\hi\\ in absorption at large\ngalactocentric radii might ultimately help in determining whether the gas is\ncold and turbulent or warm \\citep[see][]{Dickey1993}.\n\n\n\n\\subsection{Is \\shi\\  Controlling the $E_k$ vs \\sfr\\ Relation?}\\label{sec:controlling}\n\nThe gas kinetic energy, $E_k$, is the correct physical quantity to study the\nenergy balance in the ISM. However, gas surface density and SFR are well known\nto correlate \\citep[e.g.][]{Kennicutt1998, Bigiel2008}, so one may wonder\nwhether this drives our observed correlation between $E_k$ and \\sfr.  In other\nwords, the observed covariation between $E_k$ and \\sfr\\ might result from the\ntwo facts: at higher gas mass density, galaxies form stars at higher rate, and\nhigher gas mass bears higher kinetic energy. In order to verify that the\ncorrelation between $E_k$ and \\sfr\\ is not controlled by \\shi, we remove the\neffect of \\shi\\ on the $E_k$ vs \\sfr\\ relation by calculating the partial\ncorrelation coefficient:\n\\begin{equation}\\label{eq:partcorr}\n\\rho_{12.3} = \\frac { \\rho_{12} - \\rho_{13} \\rho_{23} }\n {\\sqrt{ (1-\\rho^2_{13}) (1-\\rho^2_{23}) }},\n\\end{equation}\nwhere $\\rho_{12.3}$ is the partial correlation between $x_1\\equiv\\log\n\\Sigma_{\\rm SFR}$ and $x_2\\equiv\\log E_k$ while controlling for $x_3\\equiv\\log\n\\Sigma_{\\rm HI}$, and $\\rho_{ij}$ is the Pearson's correlation coefficient\nbetween two data sets $x_i$ and $x_j$. Here, considering the quantities in\nlogarithmic scale is convenient as they are correlated as power laws.\nRetaining only data points above a fiducial value for the \\hi\\ mass density,\ni.e.  $\\Sigma_{\\rm HI}\\ge 3$~\\msun\\ \\citep[cf.][]{Bigiel2008}, we obtain the\nvalues listed in Table~\\ref{tab:corrs} for our sample galaxies.  If the\ncorrelation between $E_k$ and \\sfr\\ were completely controlled by \\shi\\ we\nwould expect $\\rho_{12.3} = 0$.  Yet, we find that $0.2\\le\\rho_{12.3}\\le0.6$\nfor our sample galaxies, indicating that the correlation between $E_k$ and\n\\sfr\\ is real.  Equivalently, we show that the correlation $E_k$ vs \\sfr\\ at\nconstant \\shi\\ holds a positive slope in Fig.\\ref{fig:Ek_res_vs_sfr}, in which\nwe remove the contribution from \\shi\\ to $E_k$ by subtracting the average\n$\\avg{E_k}_{\\Sigma \\rm HI}$ within bins of \\shi. The residuals $E_k\n-\\avg{E_k}_{\\Sigma \\rm HI}$ vs \\sfr\\ exhibit a positive correlation,\nindicating that it is not \\shi\\ alone that determines the observed $E_k$ vs\n\\sfr\\ correlation.  Positive slopes in Fig.~\\ref{fig:Ek_res_vs_sfr} and\npositive partial correlation coefficients imply that there is a real, physical\nrelationship between $E_k$ and SFR even at fixed \\shi. The relatively weak\nslopes in Fig.~\\ref{fig:Ek_res_vs_sfr} indicate that higher gas mass density\ncorrelates indeed with higher kinetic energy, simply because it generates more\nstar formation.  Fig.~\\ref{fig:Ek_res_vs_sfr}\\ and the partial correlation\ncoefficients allow us to detect a relationship between $E_k$ and SFR that is\nindependent of \\shi. Comparing Fig.~\\ref{fig:Ek_vs_sfr}\\ and\nFig.~\\ref{fig:Ek_res_vs_sfr}, however, it is clear that most of the\ncorrelation between $E_k$ and SFR in Fig.~\\ref{fig:Ek_res_vs_sfr} closely\ninvolves \\shi\\ (the distributions in Fig.~\\ref{fig:Ek_res_vs_sfr}\\ are very\nflat compared to those in Fig.~\\ref{fig:Ek_vs_sfr}). The basic effect seems to\nbe that higher \\shi\\ results in higher SFR, which creates more $E_k$.  Also,\nnote that if the turbulence, as traced by $E_k$, were effectively suppressing\nstar formation, we would have observed a negative correlation here.\n\n\n\\subsection{Does Turbulence Drive Stochastic Star Formation?}\n\nThe data analysis suggests that the SFR drives the \\hi\\ turbulence through SN\nfeedback. However, the observed correlation could also be interpreted in the\nopposite logical direction, i.e., that the turbulence is driving star\nformation. In fact, as it has been argued \\citep{maclow2004}, turbulence in\nthe ISM has a dual role: (1) to quench star formation by providing pressure\nsupport to the ISM and preventing collapse, and (2) to promote star formation\nby generating stochastic super-critical density enhancements. If the\nturbulence were to drive substantial stochastic star formation, we would\nindeed expect a positive correlation between \\sighi\\ and \\sfr. However,\nFig.~\\ref{fig:m2_vs_r_fit}\\ shows that \\sighi\\ and \\sfr\\ occupy quite\ndifferent dynamic ranges.  While the \\sfr\\ ranges over several orders of\nmagnitude, \\sighi\\ ranges from $\\sim20$ to $\\sim5$~\\kms\\ and is characterized\nby large azimuthal variations.  Although \\sfr\\ positively correlates with\n\\sighi\\ on galactic scales, the large azimuthal variations imply that \\sfr\\ is\nnot well defined for any given value of \\sighi.  This does not preclude\nturbulent induction of star formation in individual regions, but does suggest\nthat this process does not dominate over large scales.  The physical\nexplanation might be that supercritical density fluctuations are often\ndispersed on timescales shorter than the free-fall time, arresting the\ncollapse \\citep{Klessen2000, Elmegreen2002, Joung2006}.\n\n\\section{Effects of Spiral Arm Kinematics and Tidal Interactions}\n\n\nIn the following we discuss other possible mechanisms to produce ISM\nturbulence, such as  spiral arm\nkinematics, tidal interactions, and streamers.\n\nTable~\\ref{tab:objs}, which lists the morphological types of the galaxies of\nour sample, shows that there is no evident trend of the mean velocity\ndispersion, $\\avg{\\sigma}$, among individual galaxies or morphological type.\nSpiral galaxies with strong spiral pattern, e.g. NGC~628 and NGC~3184, have\nsimilar values of typical \\hi\\ velocity dispersion as galaxies with no clear\nspiral structure, e.g. Holmberg~II and IC~2574. Since the spiral arm strength\nshould vary within the sample, we argue that the spiral arm kinematics in our\nsample galaxies are not an important effect in driving turbulence into the\nISM.\n\nAlthough the galaxies Holmberg~II and IC~2574 belong to the M81 \ngroup, they do not show signatures of tidal distortion. In our sample, only NGC~5194 is an interacting galaxy. The \\hi\\ velocity dispersion\nin NGC~5194 is significantly higher than the average for the galaxies in the\nsample (see Fig.~\\ref{fig:s25_avg}).  On the basis of our results, we\nspeculate that  the tidal interaction with the companion NGC~5195\n enhanced the SFR in the disk of NGC~5194, which has\nconsequently driven the velocity dispersion in the \\hi\\ gas to higher values.\n\n\nExtended streamers characterize the galaxies NCG~4736 and NGC~5055.  Their\nradial profiles of the \\hi\\ velocity dispersion exhibit a local increase\noutside the radius of active star formation, i.e., at $r\\sim4'$ ($\\sim\nr_{25}$) for NCG~4736 and at $r\\sim9'$ ($\\sim1.5\\, r_{25}$) for NGC~5055.\nHowever, these local peaks in \\sighi($r$) and the streamers have different\ngalactocentric locations, corresponding to $r\\sim8'$ and $r>11'$ for NCG~4736\nand NGC~5055, respectively. Therefore, we argue that the presence of extended\nstreamers is not likely to be connected to higher \\hi\\ velocity dispersion.\n\n\n\\section{CONCLUSIONS}\n\nCombining high quality maps of \\hi\\ column density and line width provided by\nTHINGS for a sample of dwarf and spiral galaxies, we obtain the following\nresults.\n\\begin{enumerate}\n\\item The \\hi\\ velocity dispersion, \\sighi, declines uniformly as a function\n  of galactocentric distance in all analyzed galaxies.\n  \n\\item At $r_{25}$, the edge of the star-forming region, the \\hi\\ velocity\n  dispersion $\\sigma(r_{25})\\simeq10\\pm2$~\\kms, which is consistent with the\n  mass-weighted median \\hi\\ velocity dispersion $\\avg{\\sigma}$. These findings\n  are independent of the dynamical mass of the galaxy and of their\n  morphological type.\n  \n\\item Within the radius of active star formation ($r\\le r_{25}$), the\n  estimated SN rate and the corresponding energy input rate are sufficient to\n  account for the bulk of observed kinetic energy of turbulence.  For those\n  galaxies of the sample with considerable H$_2$ gas, the SNe can well account\n  for the combined \\hi\\ and H$_2$ gas turbulence.  In this region, the\n  observed instantaneous kinetic energy of the \\hi\\ gas is consistent with the\n  balance between the energy input from the total number of SNe calculated\n  from the observed SFR and the turbulent dissipation predicted by numerical\n  models.  The proportionality between gas $E_k$ and SN energy input rate\n  derived from the SFR provides direct evidence that \\hi\\ turbulence comes\n  from SNe in regions of active SFR. The resulting SN feedback efficiencies\n  are typically $\\epsilon_{\\rm SN}\\times(10^7\\;{\\rm yr}\/\\tau_D)\\simeq0.1$ at\n  SFR levels $\\Sigma_{\\rm SFR} >10^{-9}$~M$_{\\odot}$~yr$^{-1}$~pc$^{-2}$, with\n  the dissipation timescale of turbulence $\\tau_D\\simeq10^7$~yr.\n  \n\\item Within the star forming disk ($r\\le r_{25}$), neither thermal broadening\n  nor MRI can produce the observed \\hi\\ velocity dispersion.  At low SFR,\n  $\\Sigma_{\\rm SFR} <10^{-9}$~M$_{\\odot}$~yr$^{-1}$~pc$^{-2}$, corresponding\n  to large radial distances ($r>r_{25}$), an additional mechanism driving the\n  \\hi\\ velocity dispersion is required to avoid SN efficiencies $\\epsilon_{\\rm\n    SN} > 1$.\n  \n\\item The thermal broadening of the spectral lines, associated to a\n  temperature of $\\sim5000$~K, may be able to explain the observed\n  $\\sigma_{\\rm HI}\\sim 6$~\\kms\\ in the outermost regions of \\hi\\ disks in our\n  sample galaxies, if the required UV radiation to maintain these temperatures\n  is present.  The energy input from MRI can account for the kinetic energy\n  observed in regions of low SFR, $\\Sigma_{\\rm SFR}\n  <10^{-9}$~M$_{\\odot}$~yr$^{-1}$~pc$^{-2}$, at large galactocentric\n  distances.\n  \n\\item We can not unambiguously separate the temperature of the warm and\n  non-turbulent neutral medium from the effect of MRI stirring the ISM.  Both\n  mechanisms are equivalently plausible drivers of the \\hi\\ velocity\n  dispersion observed in the outer parts ($r>r_{25}$) of galaxy disks. We\n  suggest that testing the \\hi\\ line profiles of the gas against a bright\n  background source could ultimately clarify whether the gas in regions of\n  weak star formation is uniformly warm, or contains a cold, turbulent phase,\n  presumably stirred by MRI.  If the gas is actually turbulent, the gas\n  kinetic energy for both high and low star forming regions is consistent in\n  all cases with realistic values of $\\epsilon_{\\rm SN}$ and $\\epsilon_{\\rm\n    MRI}$ efficiencies, suggesting that the feedback provided by both SN\n  explosions and MRI is sufficient to drive the bulk of the observed \\hi\\ \n  turbulence.\n\n\\end{enumerate}\n\n\n\\subsubsection*{Acknowledgments}\n\nWe are grateful to the anonymous referee for the valuable and\ninteresting comments that improved the quality of the paper.\nWe acknowledge helpful discussions with E. Bell, S. Dib, N.\nDziourkevitch, R. Klessen, and A. Pasquali.  M-MML thanks the\nMax-Planck-Gesellschaft for support during his visit to the MPIA.\n\n\\renewcommand{\\baselinestretch}{1}\\normalsize\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStars form via the gravitational collapse of $\\lesssim 0.1$\\,pc dense cores, which are embedded within molecular clouds \\citep{Andre00,Ward-Thompson07,DiFrancesco07}. Prestellar cores become unstable and collapse due to their own gravitational potential. One or several stellar embryos form in their center. This is the beginning of the main accretion phase called the protostellar phase.\nObservations of the molecular line emission from large samples of cores in close star-forming regions revealed that velocity gradients are ubiquitous to prestellar structures at scales of 0.1$-$0.5~pc  \\citep{Goodman93,Caselli02}. These were interpreted as slow rotation inherited from their formation process \\citep{Goodman93,Caselli02, Ohashi99, Redman04, Williams06}. Assuming these gradients trace organized rotational motions, the observed velocities lead to a typical angular rotation velocity of $\\Omega \\sim$ 2~km~s$^{-1}$~pc$^{-1}$ and specific angular momentum values of $j=\\rm{v} \\times r \\sim 10^{-3} - 10^{-1}$~km~s$^{-1}$~pc. \n\n\nDuring the collapse, if the angular momentum of the parent prestellar cores is totally transferred to the stellar embryo, the gravitational force can not counteract the centrifugal force and the stellar embryo fragments prematurely before reaching the main sequence. This is the angular momentum problem for star formation \\citep{Bodenheimer95}.\nAlthough observational studies suggest a trend of decreasing specific angular momentum toward smaller core sizes of $j \\propto r^{1.6}$, the $j$ measured in prestellar structures at scales of 10000~au ($\\sim 10^{-3}$ km~s$^{-1}$~pc, \\citealt{Caselli02}) is still typically three orders of magnitude higher than the one associated with the maximum rotational energy that a solar-type star can sustain ($j_\\mathrm{break} \\sim$ 10$^{18}$ cm$^{2}$~s$^{-1}$ $\\sim$ 3 $\\times$10$^{-6}$~km~s$^{-1}$~pc). The physical mechanisms responsible for the angular momentum redistribution before the matter is accreted by the central stellar object have still to be identified.\n\n\nDuring the star formation process, disk formation is expected to be a consequence of angular momentum conservation during the collapse of rotating cores \\citep{Cassen81, Terebey84}. From observational studies, disks are common in Class~II objects \\citep{Andrews09, Isella09, Ricci10, Spezzi13, Pietu14, Cieza19}.\nThus, disk formation has been naturally considered as a possible solution to the angular momentum problem by redistributing the four orders of magnitude of $j$ measured from prestellar cores to the T-Tauri stars ($j \\sim$2 $\\times$10$^{-7}$~km~s$^{-1}$~pc, \\citealt{Bouvier93}): the disk would store and evacuate the angular momentum of the matter by viscous friction \\citep{Lynden-Bell74, Hartmann98, Najita18} or thanks to disk winds \\citep{Blandford82,Pelletier92,Pudritz07} before the matter is accreted by the central stellar object. However, the spatial distribution of angular momentum during disk formation within star-forming structures at scales between the outer core radius and the stellar surface are still largely unconstrained.\n\n\nClass~0 protostars are the first (proto)stellar objects observed after the collapse in prestellar cores \\citep{Andre93, Andre00}. Due to their youth, most of their mass is still in the form of a dense, collapsing, reservoir envelope surrounding the central stellar embryo (M$_\\mathrm{env} \\gg$ M$_{\\star}$). Thus, they are likely to retain the initial conditions inherited from prestellar cores, in particular regarding angular momentum. \nThe young stellar embryo mass increases via the accretion of the gaseous and dusty envelope in a short timescale ($t<$10$^5$ yr, \\citealt{Evans09,Maury11}). During this main accretion phase, most of the final stellar mass is accreted and, at the same time, the infalling gas must redistribute most of its initial angular momentum before reaching the central stellar embryo. Class~0 protostars are therefore key objects to understand the distribution of angular momentum of the material directly involved in the star formation process and constrain physical mechanisms responsible for the redistribution as disk formation.\n\n\n\n\nClear signatures of rotation \\citep{Belloche02, Belloche04, Chen07} and infalling gas are generally detected in the envelopes of Class~0 protostars (see the review by \\citealt{Ward-Thompson07}). \nThanks to observations of the dense molecular gas emission, rotational motions where characterized in seven Class~0 or I protostellar envelopes at scales between 3500 and 10000~au \\citep{Ohashi97,Belloche02,Chen07}. These envelopes exhibit an average angular momentum of $\\sim$10$^{-3}$\\,km~s$^{-1}$~pc at scales of $r<$5000~au, consistent with the $j$ measured in prestellar cores by \\cite{Caselli02} ($\\sim 10^{-3}$ km~s$^{-1}$~pc). These studies suggest an angular momentum that is constant with radius in Class~0 protostellar envelopes. These flat profiles are generally interpreted as the conservation of the angular momentum. \nFrom hydrodynamical simulations, the conservation of these typical values of angular momentum results in the formation of large rotationally supported disks with radii $>$100~au in a few thousand years \\citep{Yorke99}. However, from observational studies, large disks are rare around Class~0 protostars (r$<$100~au, \\citealt{Maury10, Kurono13, Yen13, Segura-Cox16, Maury18}).\nOnly recent numerical simulations including magnetohydrodynamics and non-ideal effects, such as ambipolar diffusion, Ohmic dissipation, or the Hall effect, allow small rotationally supported disks to be formed ($<$100~au; \\citealt{Machida14, Tsukamoto15, Masson16}).\n\n\nVery few studies have been able to produce resolved profiles of angular momentum to characterize the actual amount of angular momentum present at the smallest scales ($r \\lesssim$1000~au) within protostellar envelopes. From interferometric observations, \\cite{Yen15b} derive specific angular momentum values of $\\sim$2 $\\times$10$^{-4}$~km~s$^{-1}$~pc in seven Class~0 envelopes at scales of $r \\sim$1000~au, values which are below the trend observed by \\cite{Ohashi97}. These studies have put constraints on the angular momentum properties of Class~0 protostellar envelopes and suggest the material at $r \\sim$1000~au must reduce its angular momentum by at least one order of magnitude from outer envelope to disk scales. \\citet{Yen11,Yen17} show specific angular momentum profiles down to $\\sim$350~au in two Class~0 protostellar envelopes: $j \\sim$ 6 $\\times$10$^{-4}$~km~s$^{-1}$~pc at $r \\sim$1000~au and $j \\lesssim$10$^{-4}$~km~s$^{-1}$~pc at $r \\sim$350~au. In this case, conservation of angular momentum during rotating protostellar collapse might not be the dominant process leading to the formation of disks and stellar multiple systems. It is therefore crucial to obtain robust estimates of the angular momentum of the infalling material in protostellar envelopes during the main accretion phase by analyzing the kinematics from the outer regions of the envelope (10000~au, \\citealt{Motte01}) to the protostellar disk ($<$50~au, \\citealt{Maury18}).\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The CALYPSO survey}\nThe Continuum And Lines in Young ProtoStellar Objects (CALYPSO\\footnote{See \\url{http:\/\/irfu.cea.fr\/Projets\/Calypso\/} and \\url{http:\/\/www.iram-institute.org\/EN\/content-page-317-7-158-240-317-0.html}}) IRAM Large Program is a survey of 16 nearby Class 0 protostars (d$<$450~pc), carried out with the IRAM Plateau de Bure interferometer (PdBI) and IRAM 30-meter telescope (30m) at wavelengths of 1.29, 1.37, and 3.18~mm.\nThe CALYPSO sources are among the youngest known solar-type Class~0 objects \\citep{Andre00} with envelope masses of $M_\\mathrm{env} \\sim $1.5~$M_{\\odot}$ and internal luminosities of $L_\\mathrm{bol} \\sim$0.1$-$30~$L_{\\odot}$ \\citep{Maury18}.\n\nThe CALYPSO program allows us to study in detail the Class~0 envelope chemistry \\citep{Maury14,  Anderl16, Simone17,Belloche20}, disk properties \\citep{Maret14,  Maury18, Maret20} and protostellar jets (\\citealt{Codella14, Santangelo15, Podio16, Lefevre17}; Anderl et al. subm.; Podio et al. in prep.).\nOne of the main goals of this large observing program is to understand how the circumstellar envelope is accreted onto the central protostellar object during the Class~0 phase, and ultimately tackle the angular momentum problem of star formation. This paper presents an analysis of envelope kinematics, for the 12 sources from the CALYPSO sample located at $d \\leq$350~pc (see Table \\ref{table:sample}) and discuss our results on the properties of the angular momentum in Class~0 protostellar envelopes. \n\n\n\nWe adopt the dust continuum peak at 1.3~mm (225~GHz) determined from the PdBI datasets by \\cite{Maury18} as origin of the coordinate offsets of the protostellar envelopes (see Table \\ref{table:sample}). We report for each source in Table \\ref{table:sample} the outflow axis considered as the rotation axis and estimated by Podio \\& CALYPSO (in prep.) from high-velocity emission of $^{12}$CO, SiO, and SO at scales $<$10\\hbox{$^{\\prime\\prime}$}. We assume the equatorial axis of the protostellar envelopes, namely the intersection of the equatorial plane with the plane of the sky at the distance of the source, to be perpendicular to the rotation axis.\nThe SiO emission in the CALYPSO maps is very collimated, so the uncertainties on the direction of the rotation axis, and thus on the direction of the equatorial axis, are smaller than $\\pm$10$^{\\circ}$ (Podio \\& CALYPSO, in prep.). For L1521-F, IRAM04191, and GF9-2, no collimated SiO jet is detected, thus, the uncertainties are a bit larger ($\\pm$20$^{\\circ}$). We also use estimates of the inclination of the equatorial plane with respect to the line of sight from the literature. These estimates, which come from geometric models that best reproduce the outflow kinematics observed in molecular emission, are highly uncertain since we do not have access to the 3D-structure of each source.\n\n\n\n\\begin{table*}[!ht]\n\\caption{Sample of CALYPSO Class~0 protostars considered for this analysis.}\n\\begin{center}\n\\resizebox{\\hsize}{!}{\\input{tables\/Table-sample-CALYPSO}}\n\\end{center}\n\\tablefoot{\n\\tablefoottext{a}{Name of the protostars with the multiple components resolved by the 1.3~mm continuum emission from PdBI observations \\citep{Maury18}.}\n\\tablefoottext{b}{Coordinates of the continuum emission peak at 1.3~mm from \\cite{Maury18}.}\n\\tablefoottext{c}{Distance assumed for the individual sources. We adopt a value of 140~pc for the Taurus distance estimated from a VLBA measurement \\citep{Torres09}. The distances of Perseus and Cepheus are taken following recent Gaia parallax measurements that have determined a distance of (293 $\\pm$ 20)~pc \\citep{OrtizLeon18} and (352 $\\pm$ 18)~pc \\citep{Zucker19}, respectively. We adopt a value of 200~pc for the GF9-2 cloud distance \\citep{Wiesemeyer97, Wiesemeyer98} but this distance is very uncertain and some studies estimated a higher distance between 440-470~pc (\\citealt{Viotti69}, C. Zucker, priv. comm.) and 900~pc \\citep{Reid16}.\n}\n\\tablefoottext{d}{Internal luminosities which come from the analysis of \\textit{Herschel} maps from the Gould Belt survey (HGBS, \\citealt{Andre10} and Ladjelate et al. in prep.) and corrected by the assumed distance.}\n\\tablefoottext{e}{Envelope mass corrected by the assumed distance.}\n\\tablefoottext{f}{Outer radius of the individual protostellar envelope determined from dust continuum emission, corrected by the assumed distance. We adopt the radius from PdBI dust continuum emission \\citep{Maury18} when we do not have any information on the 30m continuum from \\cite{Motte01} and for IRAS4A which is known to be embedded into a compressing cloud \\citep{Belloche06}.}\n\\tablefoottext{g}{Position angle of the blue lobe of the outflows estimated from CALYPSO PdBI $^{12}$CO and SiO emission maps (Podio \\& CALYPSO, in prep.). PA is defined east from north. Sources indicated with} \\tablefoottext{$\\star$}{have an asymmetric outflow and the position angles of both lobes are reported. For IRAS2A, IRAS4A, and L1157, previous works done by \\cite{Codella14-bis}, \\cite{Santangelo15}, and \\cite{Podio16}, respectively, show a detailed CALYPSO view of the jets. For L1521F, we use the PA estimated by \\cite{Tokuda14, Tokuda16}.}\n\\tablefoottext{h}{Inclination angle of the equatorial plane with respect to the line of sight. Sources indicated with} \\tablefoottext{$\\star \\star$}{have an inclination angle not well constrained, so we assumed a default value of (30 $\\pm$ 20)$^{\\circ}$.} \n\\tablefoottext{i}{References for the protostar discovery paper, the envelope mass, the envelope radius and then the inclination are reported here.}\n}\n\\tablebib{(1) \\cite{Olinger99}; (2) \\cite{Enoch09}; (3) \\cite{Maury18}; (4) \\cite{Tobin07}; (5) \\cite{Curiel90}; (6) \\cite{Sadavoy14}; (7) \\cite{Motte01}; (8) \\cite{Kwon06}; (9) \\cite{Anglada89}; (10) \\cite{Bachiller95}; (11) \\cite{Girart01}; (12) \\cite{Jennings87}; (13) \\cite{Karska13}; (14) \\cite{Codella04}; (15) \\cite{Maret14}; (16) \\cite{Grossman87}; (17) \\cite{Chini97}; (18) \\cite{Ching16}; (19) \\cite{Desmurs09}; (20) \\cite{Andre99}; (21) \\cite{Andre00}; (22) \\cite{Belloche02}; (23) \\cite{Mizuno94}; (24) \\cite{Tokuda16}; (25) \\cite{Terebey09}; (26) \\cite{Ladd91}; (27) \\cite{Umemoto92}; (28) \\cite{Gueth96}; (29) \\cite{Bachiller01}; (30) \\cite{Schneider79}; (31) \\cite{Wiesemeyer97}.\n}\n\\label{table:sample}\n\\end{table*}\n\n\\section{Observations and dataset reduction}\n\nTo probe the dense gas in our sample of protostellar envelopes, we use high spectral resolution observations of the emission of two molecular lines, C$^{18}$O (2$-$1) at 219.560~GHz and N$_{2}$H$^{+}$ (1$-$0) at 93.171~GHz. In this section, we describe the dataset\\footnote{The datasets used in this paper are available at \\url{http:\/\/www.iram.fr\/ILPA\/LP010\/}.} properties exploited to characterize the kinematics of the envelopes at radii between $r\\sim$50 and 5000~au from the central object. \n\n\\subsection{Observations with the IRAM Plateau de Bure Interferometer}\nObservations of the 12 protostellar envelopes considered here were carried out with the IRAM Plateau de Bure Interferometer (PdBI) between September 2010 and March 2013.\nWe used the 6-antenna array in two configurations (A and C), providing baselines ranging from 16 to 760~m, to carry out observations of the dust continuum emission and a dozen molecular lines, using three spectral setups (around 94~GHz, 219~GHz, and 231~GHz).\nGain and flux were calibrated using CLIC which is part of the GILDAS\\footnote{See \\url{http:\/\/www.iram.fr\/IRAMFR\/GILDAS\/} for more information about the GILDAS software \\citep{Pety05}.} software. \nThe details of CALYPSO observations and the calibration carried out are presented in \\citet{Maury18}. The phase self-calibration corrections derived from the continuum emission gain curves, described in \\citet{Maury18}, were also applied to the line visibility dataset (for all sources in the restricted sample studied here except the faintest sources IRAM04191, L1521F, GF9-2, and L1448-2A).\nHere, we focus on the C$^{18}$O (2$-$1) emission line at 219560.3190~MHz and the N$_{2}$H$^{+}$ (1$-$0) emission line at 93176.2595~MHz, observed with high spectral resolution (39~kHz channels, i.e., a spectral resolution of 0.05~km~s$^{-1}$ at 1.3~mm and 0.13~km~s$^{-1}$ at 3~mm).\nThe C$^{18}$O (2$-$1) maps were produced from the continuum-subtracted visibility tables using either (i) a robust weighting of 1 for the brightest sources to minimize the side-lobes, or (ii) a natural weighting for the faintest sources (IRAM04191, L1521F, GF9-2, and L1448-2A) to minimize the rms noise values. We resampled the spectral resolution to 0.2~km~s$^{-1}$ to improve the signal-to-noise ratio of compact emission. \nThe N$_{2}$H$^{+}$ (1$-$0) maps were produced from the continuum-subtracted visibility tables using a natural weighting for all sources. In all cases, deconvolution was carried out using the Hogbom algorithm in the MAPPING program of the GILDAS software.\n\n\n\n\\subsection{Short-spacing observations from the IRAM 30-meter telescope}\nThe short-spacing observations were obtained at the IRAM 30-meter telescope (30m) between November 2011 and November 2014. Details of the observations for each source are reported in Table \\ref{table:temps-observations-30m}. We observed the C$^{18}$O (2$-$1) and N$_{2}$H$^{+}$ (1$-$0) lines using the heterodyne Eight MIxer Receiver (EMIR) in two atmospheric windows: E230 band at 1.3~mm and E090 band at 3~mm \\citep{Carter12}. The Fast Fourier Transform Spectrometer (FTS) and the VErsatile SPectrometer Array (VESPA) were connected to the EMIR receiver in both cases. \nThe FTS200 backend provided a large bandwidth (4~GHz) with a spectral resolution of 200~kHz (0.27~km~s$^{-1}$) for the C$^{18}$O line, while VESPA provided  high spectral resolution observations (20~kHz channel or 0.063~km~s$^{-1}$) of the N$_{2}$H$^{+}$ line.\nWe used the on-the-fly spectral line mapping, with the telescope beam moving at a constant angular velocity to sample regularly the region of interest (1$\\arcmin$ $\\times$ 1$\\arcmin$ coverage for the C$^{18}$O emission and 2$\\arcmin$ $\\times$ 2$\\arcmin$ coverage for the N$_{2}$H$^{+}$ emission).\nThe mean atmospheric opacity at 225 GHz was $\\tau_{225} \\sim 0.2$ during the observations at 1.3~mm and $\\tau_{225} \\sim 0.5$ during the observations at 3~mm.\nThe mean values of atmospheric opacity are reported for each source in Table \\ref{table:temps-observations-30m}.\nThe telescope pointing was checked every 2$-$3~h on quasars close to the CALYPSO sources, and the telescope focus was corrected every 4$-$5~h using the planets available in the sky.\nThe single-dish dataset were reduced using the MIRA and CLASS programs of the GILDAS software following the standard steps: flagging of incorrect channels, temperature calibration, baseline subtraction, and gridding of individual spectra to produce regularly-sampled maps.\n\n\n\\subsection{Combination of the PdBI and 30m data}\nThe IRAM PdBI observations are mostly sensitive to compact emission from the inner envelope.\nInversely, single-dish dataset contains information at envelope scales ($r \\sim 5-40\\hbox{$^{\\prime\\prime}$}$) but its angular resolution does not allow us to characterize the inner envelope emission at scales smaller than the beamwidth. \nTo constrain the kinematics at all relevant scales of the envelope, one has to build high angular resolution dataset which recovers all emission of protostellar envelopes.\nWe merged the PdBI and the 30m datasets (hereafter PdBI+30m) for each tracer using the pseudo-visibility method\\footnote{For details, see \\url{http:\/\/www.iram.fr\/IRAMFR\/GILDAS\/doc\/pdf\/map.pdf}.}: we generated pseudo-visibilities from the Fourier transformed 30m image data, which are then merged to the PdBI dataset in the MAPPING program of the GILDAS software. This process degrades the angular resolution of the PdBI dataset but recovers a large fraction of the extended emission. The spectral resolution of the combined PdBI+30m dataset is limited by the 30m dataset at 1.3~mm (0.27~km~s$^{-1}$) and the PdBI one at 3~mm (0.13~km~s$^{-1}$). \n\n\nWe produced the N$_{2}$H$^{+}$ (1$-$0) combined datacubes in such a way to have a synthesized beam size $<2\\hbox{$^{\\prime\\prime}$}$ and a noise level $<$10~mJy~beam$^{-1}$. As the N$_{2}$H$^{+}$ emission traces preferably the outer protostellar envelope, we used a natural weighting to build the combined maps to minimize the noise rather than to maximize angular resolution.\nThe C$^{18}$O (2$-$1) combined maps were produced using a robust weighting scheme, in order to obtain synthesized beam sizes close to the PdBI ones, and to minimize the side-lobes. In all cases, the deconvolution was carried out using the Hogbom algorithm in MAPPING.\n\n\n\n\n\\subsection{Properties of the analyzed maps}\nFollowing the procedure described above, we have obtained, for each source of the sample, a set of three cubes for each of the two molecular tracers C$^{18}$O (2$-$1) and N$_{2}$H$^{+}$ (1$-$0), probing the emission at different spatial scales (PdBI map, combined PdBI+30m map, and 30m map). \nIn order to build maps with pixels that contain independent dataset and avoid oversampling, we inversed visibilities from the PdBI and the combined PdBI+30m datasets using only 4 pixels per synthesized beam, and we smoothed the resulting maps afterwards to obtain 2 pixels per element of resolution. \nThe properties of the resulting maps are reported in Appendix \\ref{properties-maps}. The spatial resolution of the molecular line emission maps is reported in Tables \\ref{table:beams-c18o} and \\ref{table:beams-n2hp}. \nThe spatial extent of the molecular emission, the rms noise levels, and the integrated fluxes are reported in Tables \\ref{table:details-obs-c18o} and \\ref{table:details-obs-n2hp}.\n\n\n\n\n\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L14482A\/intensity-maps-L1448-2A-mean.pdf}\n\\caption{Integrated intensity maps of N$_{2}$H$^{+}$ (1$-$0) (top) and C$^{18}$O (2$-$1) (bottom) emission from the PdBI (left), combined (middle), and 30m (right) datasets for L1448-2A.\nThe white crosses represent the positions of the binary system determined from the 1.3~mm dust continuum emission.\nThe black cross represents the middle position between the binary system. The clean beam is shown by an ellipse on the bottom left of each map.\nThe black lines represent the integrated intensity contours of each tracer starting at 5$\\sigma$ and increasing in steps of 25$\\sigma$ for N$_{2}$H$^{+}$ and 10$\\sigma$ for C$^{18}$O (see Tables \\ref{table:details-obs-c18o} and \\ref{table:details-obs-n2hp}). Be careful, the spatial scales of the maps are not uniform in all panels.}\n\\label{fig:intensity-maps-L1448-2A}\n\\end{figure*}\n\n\n\n\n\n\\section{Envelope kinematics from high dynamic range datasets}\n\n\n\\subsection{Integrated intensity maps}\n\nTo identify at which scales of the protostellar envelopes the different datasets are sensitive to, we produced integrated intensity maps by integrating spectra of each pixel for the molecular lines C$^{18}$O (2$-$1) and N$_{2}$H$^{+}$ (1$-$0) from the PdBI, combined, and 30m datasets for each source. \nFor C$^{18}$O (2$-$1), we integrated each spectrum on a velocity range of $\\pm$ 2.5~km~s$^{-1}$ around the velocity of the peak of the mean spectrum of each source. The 1$-$0 line of N$_{2}$H$^{+}$ has a hyperfine structure with seven components (see Fig. \\ref{fig:spectra-30m-L1448C}). We integrated the N$_{2}$H$^{+}$ spectra over a range of 20~km~s$^{-1}$ encompassing the seven \ncomponents. Figure~\\ref{fig:intensity-maps-L1448-2A} shows as an example the integrated intensity maps obtained for L1448-2A. The integrated intensity maps of the other sources are provided in Appendix~\\ref{sec:comments-indiv-sources}.\n\nWe used the integrated intensity maps to measure the average emission size of each tracer in each dataset above a 5$\\sigma$ threshold.\nThe values reported in Tables \\ref{table:details-obs-c18o} and \\ref{table:details-obs-n2hp} are the average of two measurements: an intensity cut along the equatorial axis and circular averages at different radii around the intensity peak position of the source. Only pixels whose intensity is at least 5 times higher than the noise in the map are considered to build these intensity profiles. The FWHM of the adjustment by a Gaussian function allows us to determine the average emission size of the sources. For both tracers and for all sources in our sample, the emission is detected above 5$\\sigma$ in an area larger in the combined datasets than in the PdBI datasets, and smaller than in the 30m ones (see Tables \\ref{table:details-obs-c18o} and \\ref{table:details-obs-n2hp}). Our three datasets are thus not sensitive to the same scales and allow us to probe different scales within the 12 sampled protostellar envelopes: the 30m datasets trace the outer envelope, the PdBI datasets the inner part and the combined ones the intermediate scales.\n\n\nThe C$^{18}$O and N$_{2}$H$^{+}$ molecules do not trace the same regions of the protostellar envelope either: \\cite{Anderl16} report from an analysis of the CALYPSO survey that the N$_{2}$H$^{+}$ emission forms a ring around the central C18O emission in four sources.\nPrevious studies \\citep{Bergin02,Maret02,Maret07,Anderl16} show that N$_{2}$H$^{+}$, which is abundant \nin the outer envelope, is chemically destroyed when the temperature in the envelope reaches the critical temperature ($T \\gtrsim$20K) at which CO desorbs from dust ice mantles. Thus, while N$_{2}$H$^{+}$ can be used to probe the envelope kinematics at outer envelope scales, C$^{18}$O can be used as a complementary tracer of the gas kinematics at smaller radii where the embedded protostellar embryo heats the gas to higher temperatures. \n\nThe C$^{18}$O emission is robustly detected ($>$5$\\sigma$) in our PdBI observations for most sources, except for L1521F and IRAM04191 which are the lowest luminosity sources of our sample (see Table \\ref{table:sample}), and for SVS13-B where the emission is dominated by its companion, the Class~I protostar SVS13-A. For most sources, the interferometric map obtained with the PdBI shows mostly compact emission ($r < 3 \\hbox{$^{\\prime\\prime}$}$, see Table \\ref{table:details-obs-c18o}). However, the C$^{18}$O emission from the 30m datasets shows more complex structures (see Appendix~\\ref{sec:comments-indiv-sources}). Assuming that, under the hypothesis of spherical geometry, the emission from a protostellar envelope is compact ($r \\lesssim$40$\\hbox{$^{\\prime\\prime}$}$, i.e., $\\lesssim$10000~au, see Table \\ref{table:sample}) and stands out from the environment in which it is embedded, the 30m emission of L1448-2A, L1448-C, and IRAS4A comes mainly from the envelope.\n\nThe N$_{2}$H$^{+}$ emission is detected in our combined observations for all sources. In the four sources studied by \\cite{Anderl16}, they do not detect the emission at the 1.3~mm continuum peak, but emission rings around the C$^{18}$O central emission. From Table \\ref{table:details-obs-n2hp}, we noticed two types of emission morphologies based on the PdBI dataset: compact ($r <$7$\\hbox{$^{\\prime\\prime}$}$, see Table \\ref{table:details-obs-n2hp}) or filamentary ($r \\geq$9$\\hbox{$^{\\prime\\prime}$}$). In the same way as the C$^{18}$O emission, the N$_{2}$H$^{+}$ emission from the 30m datasets shows complex structures with radius $r \\gtrsim$40$\\hbox{$^{\\prime\\prime}$}$ for most sources, except for five sources (IRAM04191, L1521F, L1448-NB, L1448-C, and L1157) where the emission is consistent with the compact emission of the protostellar envelope.\n\nThe C$^{18}$O emission from the PdBI is not centered on the continuum peak for three sources in our sample: IRAS4A, L1448-NB, and L1448-2A (see Appendix~\\ref{sec:comments-indiv-sources}). \nFor each of these sources, the PdBI 1.3~mm dust continuum emission map resolves a close binary system ($<$600~au) with both components embedded in the same protostellar envelope (\\citealt{Maury18}, see Table \\ref{table:sample}). The origin of the coordinate offsets is chosen to be the main protostar, secondary protostar, and the middle of the binary system for IRAS4A, L1448-NB, and L1448-2A, respectively, to study the kinematics in a symmetrical way.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.3,angle=0,trim=0.3cm 0cm 0.4cm 4cm,clip=true]{fig-L14482A\/mean-spectrum-N2H+-L1448-2A.pdf}\n\\includegraphics[scale=0.3,angle=0,trim=0.3cm 0cm 0.4cm 4cm,clip=true]{fig-L14482A\/mean-spectrum-C18O-L1448-2A.pdf}\n\\caption{Mean spectra of the N$_2$H$^{+}$ (top) and C$^{18}$O (bottom) molecular lines from the 30m datasets for L1448-2A. The best fits of the spectra, by a hyperfine structure and a Gaussian line profile models respectively, are represented in green solid lines. In the top panel, the velocity axis corresponds to the isolated HFS component $1_{01}-0_{12}$. The systemic velocity is estimated to be 4.10~km~s$^{-1}$ for this source (see Table \\ref{table:vitesse-systemique-n2hp-30m}). }\n\\label{fig:spectra-30m-L1448C}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\subsection{Velocity gradients in protostellar envelopes} \\label{sec:velocity-maps} \nTo quantify centroid velocity variations at all scales of the protostellar envelopes, we produced centroid velocity maps of each Class~0 protostellar envelope by fitting all individual spectra (pixel by pixel) by line profile models in the CLASS program of the GILDAS software. We only considered the line intensity detected with a signal-to-noise ratio higher than 5.\nWe fit the spectra to be able to deal with multiple velocity components. Indeed, because protostellar envelopes are embedded in large-scale clouds, multiple velocity components can be expected on some lines of sight where both the protostellar envelope and the cloud emit. For example, \\cite{Belloche06} find several velocity components in their 30m of the N$_2$H$^+$ emission of IRAS4A (see Appendix \\ref{sec:comments-IRAS4A}). \nExcept for IRAS4A and IRAS4B for which we fit two velocity components (see details in Appendix \\ref{sec:comments-indiv-sources}), for most sources we used a Gaussian line profile to model the C$^{18}$O (2$-$1) emission, with the line intensity, full width at half maximum (FWHM), and centroid velocity let as free parameters (see Fig. \\ref{fig:spectra-30m-L1448C}). In the case of N$_{2}$H$^{+}$ (1$-$0), we used a hyperfine structure (HFS) line profile to determine the FWHM and centroid velocity of the molecular line emission (see Fig. \\ref{fig:spectra-30m-L1448C}). Figures~\\ref{fig:velocity-maps-L1448-2A} to \\ref{fig:velocity-maps-GF92} show the centroid velocity maps obtained for each source of the sample using the PdBI, combined, and 30m datasets for both the C$^{18}$O and N$_{2}$H$^{+}$ emission.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L14482A\/velocity-maps-L1448-2A-mean.pdf}\n\\caption{Centroid velocity maps of N$_{2}$H$^{+}$ (1$-$0) (top) and C$^{18}$O (2$-$1) (bottom) emission from the PdBI (left), combined (middle), and 30m (right) datasets for L1448-2A. The blue and red solid arrows represent the directions of the blue- and red-shifted outflow lobes, respectively. The white crosses represent the positions of the binary system determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}).\nThe black cross represents the middle position between the binary system. The clean beam is shown by an ellipse on the bottom left. The integrated intensity contours in black are the same as in Fig. \\ref{fig:intensity-maps-L1448-2A}.\n}\n\\label{fig:velocity-maps-L1448-2A}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1448N\/velocity-maps-L1448-NB2.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for L1448-NB. The white cross represents the position of main protostar L1448-NB1 determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}). The black cross represents the position of the secondary protostar L1448-NB2 of the multiple system.\n}\n\\label{fig:velocity-maps-L1448NB}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1448C\/velocity-maps-L1448C.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for L1448-C. \n}\n\\label{fig:velocity-maps-L1448C}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-IRAS2A\/velocity-maps-IRAS2A.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for IRAS2A. \n}\n\\label{fig:velocity-maps-IRAS2A}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-SVS13B\/velocity-maps-SVS13B.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for SVS13-B. The white cross represents the position of the Class~I protostar SVS13-A determined from the 1.3~mm dust continuum emission \\citep{Maury18}.\n}\n\\label{fig:velocity-maps-SVS13B}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-IRAS4A\/velocity-maps-IRAS4A1.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for IRAS4A. The white cross represents the position of secondary protostar IRAS4A2 determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}). The black cross represents the position of the main protostar IRAS4A1 of the multiple system. \n}\n\\label{fig:velocity-maps-IRAS4A}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-IRAS4B\/velocity-maps-IRAS4B.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for IRAS4B. The white crosses represent the position of secondary protostar IRAS4B2 and the position of IRAS4A, respectively, determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}). The black cross represents the position of the secondary protostar L1448-NB2 of the multiple system.\n}\n\\label{fig:velocity-maps-IRAS4B}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-IRAM04191\/velocity-maps-IRAM04191.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for IRAM04191. The white cross represents the position of the Class~I protostar IRAS04191. The black cross represents the position of IRAM04191 determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}).\n}\n\\label{fig:velocity-maps-IRAM04191}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1521F\/velocity-maps-L1521F.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A} for L1521F. The white cross represents the position of the starless dense core MMS-2.\n}\n\\label{fig:velocity-maps-L1521F}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1527\/velocity-maps-L1527.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for L1527. \n}\n\\label{fig:velocity-maps-L1527}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1157\/velocity-maps-L1157.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for L1157. \n}\n\\label{fig:velocity-maps-L1157}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-GF92\/velocity-maps-GF92.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for GF9-2. \n}\n\\label{fig:velocity-maps-GF92}\n\\end{figure*}\n\nFor most sources in our sample, these centroid velocity maps reveal organized velocity patterns with blue-shifted and red-shifted velocity components on both sides of the central stellar embryo, along the equatorial axis where such velocity gradients could be due to rotation of the envelopes.\nThe global kinematics in Class~0 envelopes is a complex combination of rotation, infall, and outflow motions. The observed velocities are projected on the line of sight and thus, are a mix of the various gas motions. Therefore, it is not straightforward to interpret a velocity gradient in terms of the underlying physical process producing it.\nIn order to have an indication of the origin of these gradients, we performed a least-square minimization of a linear velocity gradient model on the velocity maps following:\n\\begin{equation}\n\\rm{v}_\\mathrm{grad}=\\rm{v}_0 + a \\Delta \\alpha + b \\Delta \\beta, \n\\label{eq:vgrad}\n\\end{equation}\nwith $\\Delta \\alpha$ and $\\Delta \\beta$ the offsets with respect to the central source \\citep{Goodman93}.\n\n\nThis simple model provides an estimate of the reference velocity $\\mathrm{v}_0$ called systemic velocity, the direction $\\Theta$, and the amplitude $G$ of the mean velocity gradient. One would expect a mean gradient perpendicular to the outflow axis if the velocity gradient was due to rotational motions in an axisymmetric envelope. A mean gradient oriented along the outflow axis could be due to jets and outflows or infall in a flattened geometry. \nThe gradients were fit on the region of the velocity maps shown in Fig. \\ref{fig:velocity-maps-L1448-2A}, namely 10$\\hbox{$^{\\prime\\prime}$} \\times$ 10$\\hbox{$^{\\prime\\prime}$}$ in the PdBI and combined datasets for the C$^{18}$O emission (lower left and central panels), 40$\\hbox{$^{\\prime\\prime}$} \\times$ 40$\\hbox{$^{\\prime\\prime}$}$ for the N$_{2}$H$^{+}$ emission from the PdBI and combined datasets (upper left and central panels), and 80$\\hbox{$^{\\prime\\prime}$} \\times$ 80$\\hbox{$^{\\prime\\prime}$}$ and 160$\\hbox{$^{\\prime\\prime}$} \\times$ 160$\\hbox{$^{\\prime\\prime}$}$, respectively for the C$^{18}$O and N$_{2}$H$^{+}$ emission from the 30m datasets (right panels).\nTable \\ref{table:gradient-velocity-fit} reports for each source the significant mean velocity gradients detected with an amplitude higher than 2$\\sigma$. No significant velocity gradient is observed for IRAM04191, L1521F, and SVS13-B in C$^{18}$O emission at scales of $r<$5\\hbox{$^{\\prime\\prime}$} or for L1448-C and IRAS4B at $r>$30\\hbox{$^{\\prime\\prime}$} (see Table \\ref{table:gradient-velocity-fit}).\n\nSeven of the 12 sources in our sample show a mean gradient in C$^{18}$O emission aligned with the equatorial axis ($\\Delta \\Theta <$30$^{\\circ}$) which could trace rotational motions of the envelope at scales of $r<$5\\hbox{$^{\\prime\\prime}$}. \nAt similar scales, four sources (L1448-NB, L1521F, L1157, GF9-2) show gradients with intermediate orientation (30$^{\\circ}< \\Delta \\Theta <$60$^{\\circ}$). \nFinally, L1448-2A shows a mean gradient aligned to the outflow axis rather than the equatorial axis ($\\Delta \\Theta >$60$^{\\circ}$). For these last five sources, the gradients observed could be due to a combination of rotation, ejection, and infall motions. \nFor all sources, we noticed a systematic dispersion of the direction of the velocity gradient from inner to outer scales in the envelope (see Fig. \\ref{fig:evolution-theta-plot}). We discuss in Sect. \\ref{sec:velocity-gradient-1000au} whether this shift in direction of the velocity gradient is due to a transition from rotation-dominated inner envelope to collapse-dominated outer envelope at $r>$1500~au, or is due to the different molecular tracers used for this analysis.\nIn most sources, the gradient moves away from the equatorial axis as the scale increases. Only three sources (IRAM04191, L1521F, and L1527) show a gradient close to the equatorial axis with $\\Delta \\Theta <$30$^{\\circ}$ at 2000~au in N$_{2}$H$^{+}$ emission from the combined dataset while four sources show a complex gradient and five sources have a $\\Delta \\Theta >$60$^{\\circ}$.\n\n\n\\begin{sidewaystable*}\n\\centering\n\\caption{Estimation of the systemic velocity, the mean velocity gradient amplitude and its orientation from linear gradient fit of centroid velocity maps in C$^{18}$O and N$_{2}$H$^{+}$ emission from the PdBI, the combined, and the 30m datasets for the CALYPSO sample sources.}\n\\label{table:gradient-velocity-fit}\n\\resizebox{1\\textwidth}{!}{\\input{tables\/Table-velocity-gradient-fit}}\n\\tablefoot{ \\tablefoottext{a}{Position angle of the redshifted lobe of the velocity gradient defined from north to east.}\n\\tablefoottext{b}{Absolute value, between 0$^{\\circ}$ and 90$^{\\circ}$, of the difference between the angle of the mean gradient and the angle of the equatorial axis. The equatorial axis is defined perpendicularly to the direction of the outflows (see Table \\ref{table:sample}).}\n\\tablefoottext{$\\star$}{For IRAM04191, the N$_{2}$H$^{+}$ emission in the 30m dataset was fit ignoring the pixels close to the Class~I protostar IRAS04191 in the field of view (see Appendix \\ref{sec:comments-indiv-sources}).}}\n\\end{sidewaystable*}\n\n\n\n\n\n\\subsection{High dynamic range position-velocity diagrams to probe rotational motions} \\label{subsec:diagram-PV-construction}\nTo investigate rotational motions and characterize the angular momentum properties in our sample of Class~0 protostellar envelopes, we build the position-velocity (PV$_\\mathrm{rot}$) diagrams along the equatorial axis. We assumed the position angle of the equatorial axis as orthogonal to the jet axis reported in Table \\ref{table:sample}. \nThe choice of this equatorial axis allows us to maximize sensitivity to rotational motions and minimize potential contamination on the line of sight due to collapsing or outflowing gas \\citep{Yen13}. The velocities reported in the PV$_\\mathrm{rot}$ diagram are corrected for the inclination $i$ of the equatorial plane with respect to the line of sight (see Table \\ref{table:sample}). \nWe note that the correction for inclination is a multiplicative factor, thus if this inclination angle is not correctly estimated, the global observed shape is not distorted.\n\n\nThe analysis described in detail in Appendix \\ref{details-diagram-PV-construction} allows us to build a PV$_\\mathrm{rot}$ diagram with a high dynamic range from 50~au up to 5000~au for each source as follows (see the example of L1527 in Fig. \\ref{fig:PV-diagram-construction}):\\\\\n\\indent $\\bullet$ To constrain the PV$_\\mathrm{rot}$ diagram at the smallest scales resolved by our dataset ($\\sim$0.5$\\hbox{$^{\\prime\\prime}$}$), we use the PdBI C$^{18}$O datasets that we analyze in the (u,v) plane to avoid imaging and deconvolution processes (see label \"C$^{18}$O PdBI\" in Fig. \\ref{fig:PV-diagram-construction}). We only kept central emission positions in the channel maps at a position angle $<$|45$^{\\circ}|$ with respect to the equatorial axis (see Appendix \\ref{details-diagram-PV-construction}). \\\\\n\\indent $\\bullet$ Since the C$^{18}$O extended emission is filtered out by the interferometer, we used the combined C$^{18}$O emission to populate the PV$_\\mathrm{rot}$ diagram at the intermediate scales of the protostellar envelopes (see label \"C$^{18}$O combined\" in Fig. \\ref{fig:PV-diagram-construction}). The C$^{18}$O molecule remains the most precise tracer when the temperature is higher than $\\sim$20K because below, the C$^{18}$O molecule freezes onto dust ice mantles. To determine the transition radius $R_\\mathrm{trans}$ between the two tracers, we calculate the C$^{18}$O and N$_{2}$H$^{+}$ column densities along the equatorial axis from the combined integrated intensity maps (see Appendix \\ref{sec:column-density} and green points in Fig. \\ref{fig:PV-diagram-construction}).\\\\\n\\indent $\\bullet$ At radii $r>R_\\mathrm{trans}$, the N$_{2}$H$^{+}$ emission traces better the envelope dense gas. We use the combined N$_{2}$H$^{+}$ emission maps to analyze the envelope kinematics at larger intermediate scales. When the N$_{2}$H$^{+}$ column density profile reaches a minimum value due to the sensitivity of the combined dataset, this dataset is no longer the better dataset to provide a robust information on the velocity (see label \"N$_{2}$H$^{+}$ combined\" in Fig. \\ref{fig:PV-diagram-construction}). \\\\\n\\indent $\\bullet$ Finally, we use the 30m N$_{2}$H$^{+}$ emission map to populate the PV$_\\mathrm{rot}$ diagram at the largest scales of the envelope (see label \"N$_{2}$H$^{+}$ 30m\" in Fig. \\ref{fig:PV-diagram-construction}).\\\\\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=9cm]{figures-discussion\/schema-construction-PVdiagram.pdf}\n\\caption{Plot summarizing the combination of tracers and datasets used to build high dynamic range PV$_\\mathrm{rot}$ diagrams in the L1527 envelope.\nThe transition radii between the different datasets (PdBI, combined, and 30m) and the two C$^{18}$O and N$_{2}$H$^{+}$ tracers represented by the dashed lines are given in Table \\ref{table:radius-lines}. The green points show the column density profiles along the equatorial axis of C$^{18}$O and N$_{2}$H$^{+}$ estimated from the combined datasets (see Appendix \\ref{sec:column-density}).\n}\n\\label{fig:PV-diagram-construction}\n\\end{figure}\n\nThe CALYPSO datasets allow us to continuously estimate the velocity variations along the equatorial axis in the envelope over scales from 50~au up to 5000~au homogeneously for each protostar. \nFigure~\\ref{fig:PV-diagrams-1} shows the PV$_\\mathrm{rot}$ diagrams built for all sources of the sample. The systemic velocity used in the PV$_\\mathrm{rot}$ diagrams are determined in Appendix \\ref{sec:systemic-velocity}.\n\n\nThe method of building PV$_\\mathrm{rot}$ diagrams described above and in Appendix \\ref{details-diagram-PV-construction} corresponds to an ideal case with a detection of a continuous blue-red velocity gradient along the direction perpendicular to the outflow axis in the velocity maps. In practice, the direction of velocity gradients is not always continuous at all scales probed by our observations (see Table \\ref{table:gradient-velocity-fit} and Figure \\ref{fig:evolution-theta-plot}).\nFor some sources, to constrain the PV$_\\mathrm{rot}$ diagrams, we did not take the kinematic information at all scales of the envelope into account. Velocity gradients can be considered as probing rotational motions if the following criteria are met:\\\\\n$\\indent \\bullet$ We only consider the significant velocity gradients reported in Table \\ref{table:gradient-velocity-fit} with a blue- and a red-shifted velocity components observed on each side of the protostellar embryo, itself at the systemic velocity $\\mathrm{v}_0$. For example, we only take the C$^{18}$O emission from the 30m map into account for L1521F (see Fig. \\ref{fig:velocity-maps-L1521F}). \\\\\n$\\indent \\bullet$ We only take the velocity gradients aligned with the equatorial axis ($\\Delta \\Theta <$60$^{\\circ}$) into account in order to minimize contamination by infall and ejection motions. For example, we do not report in the PV$_\\mathrm{rot}$ diagrams the N$_{2}$H$^{+}$ velocity gradients from the combined maps for L1448-NB, IRAS2A, SVS13-B, and GF9-2 (see Table \\ref{table:gradient-velocity-fit}).\\\\\n$\\indent \\bullet$ We do not consider the discontinuous velocity gradients which show an inversion of the blue- and red-shifted velocity components along the equatorial axis from inner to outer envelope scales. For example, we do not report in the PV$_\\mathrm{rot}$ diagrams the N$_{2}$H$^{+}$ velocity gradients from the 30m maps for IRAM04191 and L1157 (see Figs. \\ref{fig:velocity-maps-IRAM04191} and \\ref{fig:velocity-maps-L1157}). \\\\\n\nWhen velocity gradients with a blue- and a red-shifted velocity components observed on each side of the protostellar embryo are continuous from inner to outer envelope scales but shifted from the equatorial axis ($\\Delta \\Theta \\geq$60$^{\\circ}$), we only report upper limits on rotational velocities in the PV$_\\mathrm{rot}$ diagrams.\nThe sources in our sample show specific individual behaviors, therefore we followed as closely as possible the method of building the PV$_\\mathrm{rot}$ diagram adapting it on a case-by-case basis. \n\n\n \n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=9cm]{fig-L14482A\/overplot-rotation-column-density-log-L1448-2A.pdf}\n\\includegraphics[width=9cm]{fig-L1448N\/overplot-rotation-column-density-log-L1448-NB2.pdf}\n\\includegraphics[width=9cm]{fig-L1448C\/overplot-rotation-column-density-log-L1448C.pdf}\n\\includegraphics[width=9cm]{fig-IRAS2A\/overplot-rotation-column-density-log-IRAS2A.pdf}\n\\includegraphics[width=9cm]{fig-SVS13B\/overplot-rotation-column-density-log-SVS13B.pdf}\n\\includegraphics[width=9cm]{fig-IRAS4A\/overplot-rotation-column-density-log-IRAS4A.pdf}\n\\caption{Position-velocity diagram along the equatorial axis of the CALYPSO protostellar envelopes. Blue and red dots show the blue- and red-shifted velocities, respectively. The arrows display the upper limits of $\\mathrm{v}_\\mathrm{rot}$ determined from velocity maps that do not exhibit a spatial distribution of velocities as organized as one would expect from rotation motions (see Sect. \\ref{subsec:diagram-PV-construction} and Appendix \\ref{details-diagram-PV-construction}). Green dots show the column density profiles along the equatorial axis. Dots and large dots show the C$^{18}$O and N$_{2}$H$^{+}$ data, respectively. The dashed curve shows the best fit with a power-law model leaving the index $\\alpha$ as a free parameter ($\\mathrm{v}_\\mathrm{rot} \\propto r^{\\alpha}$) whereas the dotted curve shows the best fit with a power-law model with a fixed index $\\alpha$=-1. The vertical dashed lines show the transition radii between the different datasets (PdBI, combined, and 30m) and the two tracers as illustrated in Fig. \\ref{fig:PV-diagram-construction} and given in Table \\ref{table:radius-lines}.\n}\n\\label{fig:PV-diagrams-1}\n\\end{figure*}\n\n\n\n\\begin{figure*}[!ht]\n\\addtocounter{figure}{-1}\n\\centering\n\\includegraphics[width=9cm]{fig-IRAS4B\/overplot-rotation-column-density-log-IRAS4B.pdf}\n\\includegraphics[width=9cm]{fig-IRAM04191\/overplot-rotation-column-density-log-IRAM04191.pdf}\n\\includegraphics[width=9cm]{fig-L1521F\/overplot-rotation-column-density-log-L1521F.pdf}\n\\includegraphics[width=9cm]{fig-L1527\/overplot-rotation-column-density-log-L1527.pdf}\n\\includegraphics[width=9cm]{fig-L1157\/overplot-rotation-column-density-log-L1157.pdf}\n\\includegraphics[width=9cm]{fig-GF92\/overplot-rotation-column-density-log-GF92.pdf}\n\\caption{Continued.\n}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\nIn this section, we discuss the presence of rotation in the protostellar envelopes from the PV$_\\mathrm{rot}$ diagrams (see Sect. \\ref{sec:results-PV-diagram} and Fig. \\ref{fig:PV-diagrams-1}). We build the distribution of specific angular momentum associated with the PV$_\\mathrm{rot}$ diagrams (see Sect. \\ref{subsec:distribution-j}) and explore the possible solutions to explain the $j(r)$ profiles observed in the inner ($r<$1600~au, see Sect. \\ref{sec:j-inner-envelopes}) and outer ($r>$1600~au, see Sect. \\ref{sec:velocity-gradient-1000au}) parts of the envelopes.\n\n\\subsection{Characterization of rotational motions} \\label{sec:results-PV-diagram}\nWe assume that the protostellar envelopes are axisymmetric around their rotation axes, and thus, the velocity gradients observed along the equatorial axis, and reported in the PV$_\\mathrm{rot}$ diagrams, are mostly due to the rotational motions of the envelopes. We model the rotational velocity variations by a simple power-law model $\\mathrm{v} \\propto r^{\\alpha}$ without taking the upper limits into account. This method has been tested with an axisymmetric model of collapsing-rotating envelopes by \\cite{Yen13}. As long as rotation dominates the velocity field on the line of sight, which depends on the inclination and flattening of the envelope, \\cite{Yen13} obtain robust estimates of the rotation motions at work in the envelopes.\nFirst, we fix the power-law index at $\\alpha$=-1 to compare to what is theoretically expected for an infalling and rotating envelope from a progenitor core in solid-body rotation \\citep{Ulrich76, Cassen81, Terebey84, Basu98}. The reduced $\\chi^{2}$ values of fits by an orthogonal least-square model are reported in the second column of Table \\ref{table:chi2-fit-profil-rotation}.\nThen, we let the power-law index vary as a free parameter: the best power-law index and the reduced $\\chi^{2}$ found for each protostellar envelope in our sample are reported in the third column of Table \\ref{table:chi2-fit-profil-rotation}.\nFigure \\ref{fig:PV-diagrams-1} shows the PV$_\\mathrm{rot}$ diagrams adjusted by a power-law for the sources of the CALYPSO sample. \n\n\n\\begin{table}[h]\n\\centering\n\\caption{Parameters of best power-law fits to the PV$_\\mathrm{rot}$ diagrams.}\n\\label{table:chi2-fit-profil-rotation}\n\\resizebox{\\hsize}{!}{\\input{tables\/Table-fits-PVdiagram}}\n\\tablefoot{\n\\tablefoottext{a}{Range of radii over which the PV$_\\mathrm{rot}$ diagrams were built and the fits were performed.}\n\\tablefoottext{b}{Number of degrees of freedom we used for the modeling and reduced $\\chi^{2}$ value associated with the best fit with a power-law function $\\mathrm{v} \\propto r^{-1}$.}\n\\tablefoottext{c}{Number of degrees of freedom we used for the modeling, index of fit with a power-law function ($\\mathrm{v} \\propto r^{\\alpha}$) and the reduced $\\chi^{2}$ value associated with this best fit model.} \n}\n\\end{table}\n\n\n\nThe power-law indices of the PV$_\\mathrm{rot}$ diagrams from our sample are between -1.1 and 0.8.\nFive sources (L1448-2A, IRAS2A, SVS13-B, L1527, and GF9-2) show rotational velocity variations in the envelope scaling as a power law with an index close to -1. This is consistent with the expected index for collapsing and rotating protostellar envelopes. The reduced $\\chi^2$ are $\\sim$1.5 for these sources except for IRAS2A and SVS13-B for which it is better ($\\sim$0.2).\nL1521F and L1157 show a power-law index close to 0 with a very low reduced $\\chi^2$ ($\\leq$0.3, see Table \\ref{table:chi2-fit-profil-rotation}). These flat PV$_\\mathrm{rot}$ diagrams ($\\mathrm{v}_\\mathrm{rot} \\sim$ constant) suggest differential rotation of the envelope with an angular velocity of $\\Omega=\\frac{v_\\mathrm{rot}}{r} \\propto r^{-1}$.\nFor two other sources (IRAS4B and IRAM04191), the best indices are compatible with -0.5, which could suggest Keplerian rotation at scales of $r<$1000~au. However, the reduced $\\chi^2$ are also satisfactory ($\\sim$1) when we fix the power-law index at $\\alpha$=-1 (see Table \\ref{table:chi2-fit-profil-rotation}). Thus, for these two sources, our CALYPSO datasets only allow us to estimate a range of the power-law indices between -1 and -0.5 (see Table \\ref{table:chi2-fit-profil-rotation}).\n\nRotational velocity variations along the equatorial axis between 50 and 5000~au in L1448-NB cannot be reproduced satisfactorily by any single power-law model ($\\chi^2 >$2, see Table~\\ref{table:chi2-fit-profil-rotation}).\nHowever, considering only the points at $r<$400~au for L1448-NB, we obtain a power-law index of -0.9$\\pm$0.2 with a good reduced $\\chi^2$ of 0.4, as expected for a collapsing and rotating envelope (see Table \\ref{table:chi2-fit-profil-rotation}).\n\nWe found a positive index $\\alpha$ for IRAS4A of 0.8 (see Table \\ref{table:chi2-fit-profil-rotation}). It could be an indication of solid-body rotation of $\\Omega=\\frac{v_\\mathrm{rot}}{r} \\sim$constant. However, we observe that the velocity in the PV$_\\mathrm{rot}$ diagram decreases from 2000 to $\\sim$600~au and re-increases at small scales (see panel (f) of Fig. \\ref{fig:PV-diagrams-1}). Thus, the velocity gradient is not uniform on the scales traced by the PV$_\\mathrm{rot}$ diagram as would be expected for a solid-body rotation ($\\mathrm{v}_\\mathrm{rot} \\propto r$). \nMoreover, points at radii $<$600~au are consistent with an infalling and rotating envelope (see panel (f) of Fig. \\ref{fig:PV-diagrams-1}): considering only these points, we obtain a power-law index of -1.3$\\pm$0.6 with a good reduced $\\chi^2$ value of 0.6 (see Table \\ref{table:chi2-fit-profil-rotation}). There is a dip in the C$^{18}$O emission at $r<$350~au that could be due to the opacity (see Figures \\ref{fig:intensity-maps-IRAS4A} and \\ref{fig:column-density-maps-IRAS4A}), thus, below this radius the information on velocities could be altered. \nTo date, no observations have identified any solid-body rotating protostellar envelope. Numerical models also favor differential rotation of the envelope \\citep{Basu98}. The interpretation of the velocity field as tracing solid-body rotation in the envelope of IRAS4A is therefore unlikely to be correct.\n\nFor the sources IRAS2A, IRAM04191, and L1157, the reduced $\\chi^2$ is also good ($\\sim$1) when the PV$_\\mathrm{rot}$ diagrams of these sources are ajusted by a model with a fixed index of $\\alpha$=-1 (see Table \\ref{table:chi2-fit-profil-rotation}). We determine position and velocity from four different and independent methods and we did not consider the uncertainties of the connection between the different tracers and datasets. The uncertainty on the indices reported in Table~\\ref{table:chi2-fit-profil-rotation} may thus be underestimated.\nOn the other hand, although we determined the systemic velocity by maximizing the overlap of the blue and red points, this method does not allow a more accurate determination than 0.05~km~s$^{-1}$. The systematic error of 0.05~km~s$^{-1}$, added to previous velocity errors of the points in the PV$_\\mathrm{rot}$ diagrams to take this uncertainty on the systemic velocity into account (see Appendix \\ref{sec:systemic-velocity}), can be overestimated and thus lead us to underestimate the $\\chi^2$. For these three sources, the CALYPSO dataset only allow us to estimate a range of power-law indices between -1 and the $\\alpha$ value reported in the fifth column of Table~\\ref{table:chi2-fit-profil-rotation}.\nThe uncertainties on the indices reported in Table \\ref{sec:systemic-velocity} are statistical errors and a systematic uncertainty of $\\pm$0.1 has to be added to account for the uncertainties in the outflow directions and thus the equatorial axis directions (see Table \\ref{table:sample}).\nMoreover, despite the choice of the equatorial axis, the rotational velocities could be contaminated by infall at the small scales along this axis due to the envelope geometry.\n\nTo conclude, the organized motions reported in the PV$_\\mathrm{rot}$ diagrams and modeled by a power-law function with an index $\\alpha$ ranging from -2 to 0 are consistent with differential rotational motions ($\\Omega \\propto r^{\\epsilon}$, with -3$< \\epsilon <$-1 here). We identified rotational motions in all protostellar envelopes in our sample except in IRAS4A.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Distribution of specific angular momentum in the CALYPSO Class~0 envelopes} \\label{subsec:distribution-j}\n\n\\subsubsection{Specific angular momentum due to rotation motions}\n\nAssuming the motions detected along the equatorial axis are dominated by differential rotation for 11 of the 12 sources in our sample, we use the measurements reported in the PV$_\\mathrm{rot}$ diagrams to derive the radial distribution of specific angular momentum in the protostellar envelopes due to rotation. In this part of the study, IRAS4A is excluded. The specific angular momentum is $j=\\frac{J}{M}=\\frac{I \\Omega}{M}$ with the moment of inertia $I$ defined as $I \\propto M r^2$ \\citep{Belloche13}. Thus, the specific angular momentum is calculated from the rotational velocity: $j= \\mathrm{v}_\\mathrm{rot}(r) \\times r$. \nWe plot all the specific angular momentum profiles obtained for the CALYPSO subsample in panel (b) of Fig. \\ref{fig:diagramme-j-Belloche13+CALYPSO}.\nThe individual distribution of specific angular momentum $j(r)$ for each source is given in Appendix \\ref{sec:comments-indiv-sources}. This is the first time that the specific angular momentum distribution as a function of radius within a protostellar envelope is determined homogeneously for a large sample of 11 Class~0 protostars.\nWe performed a least-square fit of the $j(r)$ profiles for each source individually, using a model of a simple power-law and a broken power-law model to identify the change of regimes. \nThe broken power-law model function is defined with a break radius $r_\\mathrm{break}$ as follows:\n$$j(r) \\propto \\left( \\frac{r}{r_\\mathrm{break}} \\right)^{\\beta_{1}}~~\\mathrm{when}~~r<r_\\mathrm{break},$$\n$$j(r) \\propto \\left( \\frac{r}{r_\\mathrm{break}} \\right)^{\\beta_2}~~\\mathrm{when}~~r>r_\\mathrm{break}.$$\nWe report in Table \\ref{table:chi2-fit-profil-moment-ang} the power-law indices fitting the best individual profiles and the associated reduced $\\chi^2$. For the broken power-law fits, only results with a reduced $\\chi^2$ better than the one obtained with a simple power-law model and with a break radius value $r_\\mathrm{break}$ to which the $j(r)$ profile is really sensitive, have been retained.\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.7,angle=0,trim=0cm 4cm 0cm 5cm,clip=true]{figures-discussion\/plot-diagramme-j-Belloche13+CALYPSO.pdf}\n\\caption{Radial distribution of specific angular momentum. Panel (a): Figure adapted from Figure 8 of \\cite{Belloche13} and from \\cite{Ohashi97}. Panel (b): zoom on the region where the angular momentum profiles due to rotation of the CALYPSO sources lie. The gray curve shows the median profile $j(r)$. In the two panels, the solid black line shows the best fit with a broken power-law model.\n}\n\\label{fig:diagramme-j-Belloche13+CALYPSO}\n\\end{figure*}\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=10cm]{figures-discussion\/comparaison-profil-specific-angular-momentum-log-tous-gradients.pdf}\n\\caption{Radial distribution of apparent specific angular momentum $|j_\\mathrm{app}|= |v| \\times r$ along the equatorial axis of the CALYPSO sources, considering all the velocity gradients observed at all envelope scales, including the reversed gradients and the shifted ones at scales of $r \\gtrsim$1600~au (see Fig. \\ref{fig:evolution-theta-plot} and Sect. \\ref{sec:velocity-gradient-1000au}) which were excluded in the construction of the PV$_\\mathrm{rot}$ diagrams in Fig. \\ref{fig:PV-diagrams-1}, and in panel (b) of Fig. \\ref{fig:diagramme-j-Belloche13+CALYPSO} for our analysis of rotational motions. The empty circles show the negative apparent specific angular momentum from the reversed gradients along the equatorial axis in the outer scales ($r >$1600~au) of the L1448-2A, IRAS2A, IRAM04191, L1527, L1157 and GF9-2 envelopes (see Sect. \\ref{subsec:Counter-rotation}). The gray curve shows the median profile |$j_\\mathrm{app}$| and the solid black line shows the best fit with a broken power-law model. }\n\\label{fig:profil-j-CALYPSO-tous-gradients}\n\\end{figure*}\n\nTwo sources (L1448-NB and L1448-C) are better reproduced by a broken power-law model than a simple power-law model where the $\\chi^2$ are $\\sim$2: this allows us to identify a change of slope from a relatively flat profile to an increasing profile at larger radius in the envelope ($\\beta \\sim$1), with break radii between 500 and 700~au. \nFor the other sources, we also identified at scales of $r<$1300~au a flat profile of specific angular momentum with $\\beta<0.5$ (L1448-2A, IRAS2A, SVS13-B, IRAS4B, L1527, and GF9-2) while the specific angular momentum profile at scales of $r>$1000~au shows a steeper slope with $\\beta \\sim$1 (L1521-F). \nHowever, two sources of the sample (IRAM04191 and L1157) stand out as the sources showing a steep increase in their specific angular momentum profile at scales of $r<$1000~au ($\\beta \\geq$0.7), similar to the indices found at large radii in the sources showing a break in their $j(r)$ profiles.\nWe note that for the flat profiles ($\\beta <$0.5; L1448-2A, SVS13-B, IRAS4B, and GF9-2), and IRAM04191 and L1157, the specific angular momentum distribution is only constrained at scales $<$1300~au.\nMost of the sources in our sample are better reproduced by a broken power-law model with a break radius (1000 $\\pm$ 500)~au and an increasing profile at larger radius in the envelope ($\\beta \\sim$1.4) than a simple power-law model.\n\n\nIn his review, \\cite{Belloche13} plotted the observed specific angular momentum as a function of rotation radius for several objects along the star-forming sequence. In this plot (panel (a) of Figure \\ref{fig:diagramme-j-Belloche13+CALYPSO}), he identifies three regimes in the distribution of specific angular momentum, that can be broadly associated with different evolutionary stages: \\\\\n\\indent $\\bullet$ prestellar regime: on large scales, the apparent angular momentum of molecular clouds \\citep{Goldsmith85} and dense cores \\citep{Goodman93, Caselli02} appears to follow the power-law relation $j \\propto r^{1.6}$,\\\\\n\\indent $\\bullet$ protostellar regime: between 100~au and $\\sim$6000~au (0.03~pc), a few points in different protostellar envelopes suggest the specific angular momentum is relatively constant ($j \\sim 10^{-3}$~km~s$^{-1}$~pc, \\citealt{Ohashi97, Belloche02, Chen07}),\\\\\n\\indent $\\bullet$ disk and binary regime: below 100~au, measurements in disks and Class~II binaries \\citep{Chen07} show a decrease of $j$ following a trend characteristic of Keplerian rotation ($j \\propto r^{0.5}$).\\\\\n\nThus, from previous observational studies on rotational motions, finding a break at $r \\sim$1000~au between two trends of specific angular momentum within Class~0 protostars was unexpected. Although the velocity gradients observed in the outer part of the protostellar envelopes ($r>$1000~au) are consistent with rotational motions, the observed $j$ regime at these scales is not expected from pure rotational motions.\n\n\n\n\\subsubsection{Apparent specific angular momentum}\n\nThe radius range of $j(r)$ distribution due to rotation motions is not homogeneous between sources (see Table \\ref{table:chi2-fit-profil-moment-ang}). To identify whether the radius of $\\sim$1000~au is a critical radius between two trends of specific angular momentum in each source, we derive the radial distribution of the apparent specific angular momentum |$j_\\mathrm{app}$| at all scales in the envelopes. To build |$j_\\mathrm{app}$|$(r)$ distribution, we consider the gradients observed at all envelope scales, including also the reversed gradients and the shifted ones at scales of $r \\gtrsim$1000~au (see Fig. \\ref{fig:evolution-theta-plot} and Sect. \\ref{sec:velocity-gradient-1000au}) which were excluded in the construction of the PV$_\\mathrm{rot}$ diagrams in Fig. \\ref{fig:PV-diagrams-1} because they are not consistent with rotational motions. By considering these velocity gradients, we add points in the outer envelopes but the trend observed in the inner envelopes do not change (see Tables \\ref{table:chi2-fit-profil-moment-ang} and \\ref{table:chi2-fit-profil-japp}). Thus, the |$j_\\mathrm{app}$|$(r)$ distribution helps us to understand the origin of the trend and the velocity gradients observed at $r >$1000~au.\nWe plot all the apparent specific angular momentum profiles obtained for the CALYPSO subsample in Fig. \\ref{fig:profil-j-CALYPSO-tous-gradients}. We also report the apparent specific angular momentum of IRAS4A which was identified as the only source that did not show any rotational motions in our sample (see Sect. \\ref{sec:results-PV-diagram}). As for $j$ profiles, we performed a least-square fit of the |$j_\\mathrm{app}$|$(r)$ profiles for each source individually and we report the indices of the power-law models in Table \\ref{table:chi2-fit-profil-japp}.\n\n\nWe create the median |$j_\\mathrm{app}$|$(r)$ profile of the CALYPSO subsample. We first resampled the individual profile of each source in steps of 100~au and normalized it by the value at 600~au, then we took the median value of individual profiles at each radius step. The median profile is shown in gray on Figure \\ref{fig:profil-j-CALYPSO-tous-gradients}.\nFrom a broken power-law fit, we obtain a relatively flat profile ($j_\\mathrm{app} \\propto r^{0.3 \\pm 0.3}$) at radii smaller than 1570$\\pm$300~au and an increasing profile ($j_\\mathrm{app} \\propto r^{1.6 \\pm 0.2}$) in the outer envelope. The radius of $\\sim$1600~au therefore appears to be a critical radius which delimits two regimes of angular momentum in protostellar envelopes: the specific angular momentum decreases down to $\\sim$1600~au and then tends to become constant.\n\n\nThe change of behavior of $j_\\mathrm{app}$ above the break radius could be due to a change of tracer to study the kinematics in the outer envelope. However, we do not find any systematic consistency between $r_\\mathrm{app,break}$ and the transition radius $R_\\mathrm{trans}$ between the two tracers C$^{18}$O and N$_{2}$H$^{+}$. Even if for SVS13-B, $R_\\mathrm{trans}$ is in the error bars of $r_\\mathrm{app,break}$, for three sources (L1448-NB, L1448-C, and IRAS4A) it is not consistent, and for IRAS4B, we do not observe a change of regime for $j_\\mathrm{app}$ at $r \\sim$1600~au (see Tables \\ref{table:chi2-fit-profil-moment-ang} and \\ref{table:chi2-fit-profil-japp}). Moreover, for L1521F, only the C$^{18}$O emission shows a velocity gradient allowing us to constrain the kinematics at scales of $r>$1600~au (see Fig. \\ref{fig:velocity-maps-L1521F}) and we find the same trend of $j_{app}$ ($\\beta_\\mathrm{app} \\sim$1.2) than in all other sources where we used N$_{2}$H$^{+}$ to constrain the outer part of the envelopes. \nThe other sources (L1448-2A, IRAS2A, IRAM04191, L1527, L1157, and GF9-2) show a negative value of the apparent angular momentum at outer envelope scales due to a renversal of the velocity gradients (see Fig. \\ref{fig:profil-j-CALYPSO-tous-gradients}, Table \\ref{table:chi2-fit-profil-japp}, and Sect. \\ref{subsec:Counter-rotation}). For two of these sources (L1448-2A and GF9-2) the radius where the gradient reverses along the equatorial axis, resulting in a negative $j_\\mathrm{app}$ with respect to the inner envelope scales, is consistent with $R_\\mathrm{trans}$ and $r_\\mathrm{app,break}$. For two sources (IRAS2A and IRAM04191), $R_\\mathrm{trans}$ is consistent with the radius where the gradient reverses along the equatorial axis but not with $r_\\mathrm{app,break}$. For the last two sources (L1527 and L1157), the three radii are all different from each other.\nThe different individual behaviors in the CALYPSO sample allow us to conclude that our finding that protostellar envelopes are characterized by two regimes of angular momentum does not result from our use of two different tracers.\n\n\nFrom the median |$j_\\mathrm{app}$|$(r)$ profile without normalization of the individual profiles at 600~au, we find a mean value of specific angular momentum in the inner parts of the envelopes ($r<$1600~au) of $\\sim$6 $\\times$10$^{-4}$~km~s$^{-1}$~pc. This value is slightly lower but compatible with the estimates made by \\cite{Ohashi97} and \\cite{Chen07} in four Class~0 or I sources ($j \\sim$10$^{-3}$~km~s$^{-1}$~pc at $r<$5000~au). It is also consistent with the studies by \\cite{Yen15b} and \\cite{Yen15} which find values between 5 $\\times$ 10$^{-3}$~km~s$^{-1}$~pc and 5 $\\times$ 10$^{-5}$~km~s$^{-1}$~pc in the inner envelope (r$<$1500~au). \n\\cite{Yen15b} estimate a specific angular momentum of $\\sim 5 \\times$ 10$^{-4}$~km~s$^{-1}$~pc at $r \\sim$100~au for L1448-C and L1527. Moreover, our values for L1157 are consistent with their upper limit estimate of $5 \\times$ 10$^{-5}$~km~s$^{-1}$~pc in the inner envelope ($r<$100~au) of L1157. \n\n\nThe high angular resolution and the high dynamic range of the CALYPSO observations allow us to identify the first two regimes within individual protostellar envelopes: values at radii $\\gtrsim$1600~au ($j_\\mathrm{app} \\propto r^{1.6}$ on average, see Table \\ref{table:chi2-fit-profil-japp}) seem to correspond to the trend found in dense cores at scales $>$6000~au while the values stabilize around $\\sim$6 $\\times$10$^{-4}$~km~s$^{-1}$~pc on average at radii $<$1600~au. This study resolves for the first time the break radius between these two regimes deeper within the protostellar envelopes at around $\\sim$1600~au instead of $\\sim$6000~au. \nIn a study of ammonia emission in the outer envelopes of two Class~0 objects, \\cite{Pineda19} find an increasing angular momentum profile scaling as $r^{1.8}$ from 1000~au to 10000~au, with values $\\sim$3 $\\times$10$^{-4}$~km~s$^{-1}$~pc at radii $\\sim$1000~au. They do not detect the break around $\\sim$1600~au found in the CALYPSO sample.\nThis break radius from which the profiles are found to be flat in the inner envelope may depend on the evolutionary stage of the accretion process during the Class~0 phase as suggested by \\cite{Yen15}. It could be due to the propagation of the inside-out expansion wave during the collapse \\citep{Shu77}: assuming a median lifetime or half life of $\\sim$5 $\\times$10$^4$~yr for Class~0 protostar envelopes (\\citealt{Maury11}; see also \\citealt{Evans09}) at sound velocity ($\\sim$ 0.2~km~s$^{-1}$), one obtains a radius $\\sim$2000~au.\nThis radius is on the same order of magnitude as the observed break radius between the two regimes observed in the distribution of specific angular momentum of sources in our sample. In this case, the break radius could be an indication of the age of the protostars, except for four sources (L1448-NB, IRAM04191, L1521F, and L1157) in our sample where we do not observe this break radius. \nBeyond this radius, the outer envelope may not have collapsed yet, and could therefore retain the initial conditions in angular momentum of the progenitor prestellar core. \n\nThis could be an explanation for the increase in angular momentum observed at the scales of $r>$1600~au ($j_\\mathrm{app} \\propto r^{1.6}$ on average, see Table \\ref{table:chi2-fit-profil-japp}), consistent with the prestellar stage ($j \\propto r^{1.6}$). \nWe discuss the properties and physical origin of these two regimes in more details in the next sections.\n\n\n\n\n\n\\subsection{Conservation of angular momentum in Class~0 inner envelopes} \\label{sec:j-inner-envelopes}\n\n\nIn this section, we focus on the relatively constant values of specific angular momentum observed in the inner envelopes at scales of $r \\le$1600~au in the $j(r)$ profiles due to rotation motions (see Fig. \\ref{fig:diagramme-j-Belloche13+CALYPSO}). From these flat profiles, we find that the matter directly involved in the formation of the stellar embryo has a specific angular momentum $\\sim$3 orders of magnitude higher than the one in T-Tauri stars ($j \\sim$2 $\\times$10$^{-7}$~km~s$^{-1}$~pc, \\citealt{Bouvier93}). We discuss constant values of specific angular momentum as conservation of angular momentum to test disk formation as a possible solution to the angular momentum problem.\n\n\nIt is difficult to constrain the time evolution of specific angular momentum for a given particle from angular momentum distributions which are snapshots of the angular momentum distribution of all particles at a given time during the collapse phase. During the collapse of a core initially in either solid-body rotation or differential rotation, particles conserve their specific angular momentum during the accretion on the stellar embryo \\citep{Cassen81,Terebey84, Goodwin04}. \nIn the case of a protostellar envelope with a density profile $\\rho \\propto r^{-2}$, an observed flat profile $j(r)=$constant requires, since each particle at different radii has the same specific angular momentum, an initially uniform distribution of angular momentum. This does not agree with the steep increase in specific angular momentum we observe at scales of $r>$1600~au in the $j(r)$ profiles. The break in the specific angular momentum profile could be due either to a faster collapse of the inner envelope caused by an initial inner density plateau \\citep{Takahashi16} or to a change of dominant mechanisms responsible for the observed velocity gradients from inner to outer scales of the envelope.\n\n\nIn our sample, we distinguish eight sources with a relatively flat $j(r)$ profile in the inner envelope ($\\beta<$0.5, see Table \\ref{table:chi2-fit-profil-moment-ang}): L1448-2A, L1448-NB, L1448-C, IRAS2A, SVS13-B, IRAS4B, L1527, and GF9-2. \nWe estimate a centrifugal radius that would be obtained when the mass currently observed at $\\sim$100~au collapses and based on the mean value of specific angular momentum observed today $<j_\\mathrm{100~au}>$ as follows:\n\\begin{equation}\nR_\\mathrm{cent}=\\frac{<j_\\mathrm{100~au}>^2}{G~M_\\mathrm{100~au}}.\n\\label{eq:Rcent}\n\\end{equation}\n\n\nThe lower limit of the mass enclosed within 100~au, $M_\\mathrm{100~au}$, is the mass of the envelope $M_\\mathrm{100~au}^\\mathrm{dust}$ estimated from the PdBI 1.3~mm dust continuum flux \\citep{Maury18}, assuming optically thin emission, a dust temperature at 100~au computed with Eq. \\eqref{Tdust} and corrected by the assumed distance (see Table \\ref{table:sample}). This mass estimate does not include the mass of the central stellar object, $M_{\\star}$: since the embryo mass is unknown for most sources in our sample, we consider an upper limit of $M_\\mathrm{100~au}=M_{\\star}+M_\\mathrm{100~au}^\\mathrm{dust}$ assuming $M_{\\star}=$ 0.2~M$_{\\odot}$ for each source in our sample. This value of 0.2~M$_{\\odot}$ corresponds to the stellar mass in the Class~0\/I protostar L1527 from kinematic models of the Keplerian pattern in the disk \\citep{Tobin12, Ohashi14, Aso17}. The range of values for $M_\\mathrm{100~au}$ are reported in the third column in Table \\ref{table:rayon-disque-brot}. The calculated range of centrifugal radii associated with $M_\\mathrm{100~au}$ is listed for each source in the fourth column in Table \\ref{table:rayon-disque-brot}. We note that if $M_{\\star}$ of a source is smaller than that of L1527, then the centrifugal radius value we calculated is underestimated.\n\nSince the embryo mass is uncertain and $M_\\mathrm{100~au}$ may be underestimated if the dust emission is not optically thin, we compute the mass enclosed within $r<$100~au, including the stellar embryo mass, needed to form a disk the size of $R_\\mathrm{disk}^\\mathrm{dust}$ with the $<j_\\mathrm{100~au}>$ observed. The values are reported in the last column of Table \\ref{table:rayon-disque-brot}.\n\n\n\n\\begin{table*}[!ht]\n\\centering\n\\caption{Centrifugal radius $R_\\mathrm{cent}$ assuming angular momentum conservation.}\n\\label{table:rayon-disque-brot}\n\\input{tables\/Table-Rcent}\n\\tablefoot{\n\\tablefoottext{a}{Weighted mean of specific angular momentum in the inner envelopes (50~au$< r \\le$1600~au).}\n\\tablefoottext{b}{Range of the object mass at 100~au, the minimum and maximum values are defined in Sect. \\ref{sec:j-inner-envelopes}.}\n\\tablefoottext{c}{Centrifugal radii estimated from $<j_\\mathrm{100~au}>$ and $M_\\mathrm{100~au}$ using Eq. \\eqref{eq:Rcent}, assuming conservation of angular momentum.}\n\\tablefoottext{d}{Candidate disk radius determined from the CALYPSO study of PdBI dust continuum emission at 1.3 and 3~mm \\citep{Maury18}, corrected by the assumed distance (see Table \\ref{table:sample}).}\n\\tablefoottext{e}{Total minimum mass that needs to be enclosed at $r<$100~au to form a disk equal to $R_\\mathrm{disk}^\\mathrm{dust}$ if the angular momentum $<j_\\mathrm{100~au}>$ was conserved. This minimum mass considers the mass of the stellar embryo and the mass of the optically thick inner envelope enclosed within 100~au.}\n}\n\\end{table*}\n\n\nFor all the sources in our sample, the upper limits of the $R_\\mathrm{cent}$ range are larger than 150~au and systematically larger than the continuum disk candidate radii $R_\\mathrm{disk}^\\mathrm{dust}$ from \\cite{Maury18} reported in the fifth column of Table \\ref{table:rayon-disque-brot}. \nMoreover, \\cite{Maret20} only detect possible Keplerian rotation in two protostars in our sample (L1527 at radii $\\sim$90~au and L1448-C at $r \\sim$200~au) from the CALYPSO data. Thus, most $R_\\mathrm{cent}$ values is expected to be less than 100~au. The upper $R_\\mathrm{cent}$ values are probably overestimated because the contribution of the embryo mass to $M_\\mathrm{100~au}$ is excluded.\n\n\n\n\nComparing the lower limits of the $R_\\mathrm{cent}$ range with the candidate disk radius, we find a good agreement for most sources in our sample except for L1448-NB.\nWe find a larger centrifugal radius ($\\sim$500~au) than the observed disk size ($<$50~au) calculated considering only the main protostar L1448-NB1 of the binary system. Since in this study, we are interested in the kinematics of the whole system, we must consider all the continuum structure and not only that of the main protostar. Considering NB1 and NB2, \\cite{Maury18} resolve a circumbinary structure with a radius of (320 $\\pm$ 90)~au centered on the middle of the two components. Given the uncertainties, the latter value is consistent with the lower centrifugal radius estimated here. At these scales, \\cite{Tobin16} observe a spiral structure surrounding the multiple system and interpreted it as a gravitationally unstable circumbinary disk. On the other hand, \\cite{Maury18} suggest that this component is due to orbital motions and tidal arms between the companions and \\cite{Maret20} do not detect any Keplerian rotation at radii $<$170~au. Thus, the nature of this additional structure surrounding the multiple system is still unclear. As a consequence, the increase in specific angular momentum we measured at small scales could not only trace the rotation of {the disk or} the envelope but may be contaminated by gravitational instabilities due to orbital motions or a fragmented disk surrounding the system.\nGiven the large uncertainties on the dust disk radii, we found a good agreement between centrifugal radii and $R_\\mathrm{disk}^\\mathrm{dust}$ for L1448-C and L1527. Moreover, the dust radius (50~au in L1527, \\citealt{Maury18}) does not necessarily exactly correspond to the centrifugal radius which was first detected in L1527 from observations of SO emission at 100$\\pm$20~au \\citep{Sakai14Nat}. For this source, our estimate of $R_\\mathrm{cent}$ ($\\sim$70~au) is consistent with previous kinematic studies which detect a proto-planetary disk candidate with a radius of 50$-$90~au \\citep{Ohashi14, Aso17, Maret20}. Moreover, we observe a slight increase in the specific angular momentum we measured at $r<$80~au. It could be due to the transition from the envelope to the disk.\n\n\n\nThe hypothesis of collapsing material with conservation of angular momentum, resulting in disk formation, at $r<$100~au is therefore plausible for most sources in our sample. \nWe notice that L1448-NB, in which \\cite{Tobin16} claim the detection of a large candidate disk, shows the highest value of specific angular momentum at $r<$1600~au of the CALYPSO sample, consistent with the angular momentum observed in the proto-planetary disks surrounding the T-Tauri stars which are estimated to be 1$-$6$\\times$10$^{-3}$~km~s$^{-1}$~pc \\citep{Simon00,Kurtovic18,Perez18}. It could suggest an increase in the angular momentum of the disk during its evolution. In this case, the mean value of $j(r)$ in the inner envelope would be lower in the less evolved than in the more evolved Class~0 protostars, and it would increase with time until reaching the value contained in the T-Tauri disks. In this scenario, L1448-NB would be one of the most evolved objects in the sample. However, the borderline Class~0\/I protostar L1527, which is the most evolved object of the CALYPSO sample, has a specific angular momentum of $\\sim$6 $\\times$ 10$^{-4}$~km~s$^{-1}$~pc at the inner envelope scales (see Table \\ref{table:rayon-disque-brot}). In the same way, L1448-C has a specific angular momentum less than one order of magnitude lower than the values observed in the Class~II disks while \\cite{Maret20} suggest the presence of a Keplerian disk in the inner envelope. As most of the CALYPSO inner protostellar envelopes have an order of magnitude less specific angular momentum than in Class II disks, we discuss below several possible explanations:  \\\\\n(i) a part of the angular momentum inherited by the T-Tauri disks may not come from the rotating matter contained in the inner envelope accreted during the Class~0 phase. During the Class~I phase, the mass accreted could come from regions further away from the envelope ($r \\gg$1600~au) with a possibly higher specific angular momentum.\\\\\n(ii) disks may expand with time due to the transfer of angular momentum from their inner regions to their outer ones. Unfortunately, the specific angular momentum does not contain information about the mass. Large values of $j(r)$ may be carried by low masses at the outer disk radius but may remain difficult to quantify. To this day, the mechanisms at work in disk evolution remain an open question. Some studies, for example, suggest that viscous friction may be responsible for the disk expansion \\citep{Najita18}. \\\\\n(iii) the specific angular momentum of the proto-planetary disks may be biased toward high values from historical, large and massive disks. A new population of small T-Tauri disks with radii between 10 and 30~au has been observed thanks to PdBI and ALMA \\citep{Pietu14, Cieza19}. Assuming a small rotationally-supported disks around a stellar object including a total mass of 0.1$-$1~$M_{\\odot}$ \\citep{Pietu14}, one expects a specific angular momentum between 10$^{-5}$~km~s$^{-1}$~pc and 10$^{-4}$~km~s$^{-1}$~pc, values which are similar to those we obtained in the inner Class~0 protostellar envelopes with the CALYPSO sample. However, to this day, no resolved observations of gas kinemactics of these small Class II disks allow us to estimate observationally their specific angular momentum.\\\\\n\n\n\n\n\n\n\\subsection{Origin of the velocity gradients at $r>$1600~au} \\label{sec:velocity-gradient-1000au}\nAt outer envelope scales, we detect velocity gradients ($\\sim$2~km~s$^{-1}$~pc$^{-1}$ at $\\sim$10000~au, see Table \\ref{table:gradient-velocity-fit}) in the CALYPSO single-dish maps. They may not be directly related to rotational motions of the envelopes but rather to other mechanisms. Indeed, we observe in the CALYPSO dataset a systematic evolution of the orientation of the gradients between the inner and outer scales in the envelope (see Table \\ref{table:gradient-velocity-fit}). Figure \\ref{fig:evolution-theta-plot} shows the orientation of the mean velocity gradient observed at different scales of the envelope with respect to the position angle of the gradient observed at scales $\\sim$100~au. The clear dispersion ($\\sim$100$^{\\circ}$ on average, see Fig. \\ref{fig:evolution-theta-plot}) of gradient position angle across scales within individual objects may be due to a change of dominant mechanisms responsible for the observed gradients from inner to outer scales of the envelope. \nFrom the literature, velocity gradients are often measured in the outer protostellar envelopes along the equatorial axis and they are interpreted as due to rotational motions or infall from a filamentary structure at scales of 1500$-$10000~au \\citep{Ohashi97, Belloche02, Tobin11}. In this section, we explore the possible origins of the velocity gradients found at scales of $r>1600$~au {and used to build the |$j_\\mathrm{app}$|$(r)$ profiles (see Fig. \\ref{fig:profil-j-CALYPSO-tous-gradients}).}\n\n\n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.45,angle=0,trim=6.8cm 0cm 6.8cm 3cm,clip=true]{figures-discussion\/evolution-velocity-gradients.pdf}\n\\caption{Evolution of the orientation of the mean velocity gradient in the different datasets used to build the PV$_\\mathrm{rot}$ diagrams and angular momentum distributions with respect to the PA of the velocity gradient observed at small scales $\\Theta_\\mathrm{small}$ (PdBI C$^{18}$O emission). The error bars of the orientation $\\Theta$ are given in Table \\ref{table:gradient-velocity-fit}. They are smaller than $10^{\\circ}$ except for 7 of the 67 gradient measurements. For these 7 measurements, the large error bars are generally due to the absence of a clear gradient on either side of the central position of the source. Gradient measurements with large error are indicated by an empty circle. A typical error of $\\pm 10^{\\circ}$ is shown on the first point of the plot.}\n\\label{fig:evolution-theta-plot}\n\\end{figure}\n\n\n\n\\subsubsection{Questioning the interpretation of counter-rotation}\n\\label{subsec:Counter-rotation}\nSix sources in the sample show a clear reversal of the orientation of the mean velocity gradient ($| \\Theta - \\Theta_\\mathrm{small} | > 130^{\\circ}$) from the inner to the outer envelope scales: IRAS4A, IRAS4B, L1527, IRAM04191, L1157, and GF9-2. We note that the kinematics at scales where we observed reversed velocity gradients ($r>$1600~au) with respect to the small scales were not taken into account to build the PV$_\\mathrm{rot}$ diagrams in Fig. \\ref{fig:PV-diagrams-1}, or the specific angular momentum profiles shown for the full sample in panel (b) of Fig. \\ref{fig:diagramme-j-Belloche13+CALYPSO}. Indeed, these profiles were aimed at characterizing the rotational motions in the envelopes and the angular momentum due to this rotation: a reversal of the rotation, if real, would require a more complex model than the power-law ($\\mathrm{v} \\propto r^{\\alpha}$) model we adopted in Sects. \\ref{subsec:diagram-PV-construction} to \\ref{sec:j-inner-envelopes}. In this section, we discuss these complex patterns in more detail.\n\nIn IRAM04191, we observed velocity gradients at outer envelope scales of $r>$1600~au consistent with those observed previously by \\cite{Belloche02} and \\cite{Lee05} ($\\Theta \\sim$100$^{\\circ}$, see Table \\ref{table:gradient-velocity-fit}). However, in the inner envelope, we noticed a velocity gradient with a direction of $\\Theta =$-83$^{\\circ}$ (see bottom middle panel in Fig. \\ref{fig:velocity-maps-IRAM04191} and Table \\ref{table:gradient-velocity-fit}). In L1527, we found small-scale velocity gradients ($\\Theta \\sim$0$^{\\circ}$ at $r \\sim$ 1000~au) consistent with those previously observed by \\cite{Tobin11} which are in the opposite direction compared to the large-scale one ($r \\sim$8000~au, \\citealt{Goodman93}). \\cite{Tobin11} interpret this reversal of velocity gradients as counter-rotation but it also could be due to infalling motions that dominated the velocity field at the outer envelope scales \\citep{Harsono14}. \n\nOur study suggests that reversals of velocity gradients are common in Class~0 protostellar envelopes. However, the asymmetrical velocity gradients (for IRAS4B, GF9-2), the filamentary structures traced by the integrated intensity at scales of $r>$2000~au (for IRAS4A, IRAS4B, L1527, and GF9-2), and a strong external compression of the cloud hosting IRAS4A and IRAS4B \\citep{Belloche06} lead us to exclude the observed reversed gradients as counter-rotation of the envelope. \nMoreover, only MHD models with Hall effect succeed to form envelopes in counter-rotation. These models form a thin layer of counter-rotating envelopes at the outer radius of the disk ($r \\sim$50-200~au; \\citealt{Tsukamoto17}). This envelope layer is in counter-rotation compared to the formed disk and the protostellar envelope at $r>$200~au as a consequence of the Hall effect generated by the rotation of the disk which changes the angular momentum of the gas at the disk outer radius. Therefore, these models cannot explain the inversions of rotation in the different layers of the envelope at scales of $r>$3000~au as observed in our sample.\nHistorically, the gradients observed from single-dish mapping at $r>$3000~au have been used to quantify the amplitude for the angular momentum problem. However, incorrectly interpreted as pure rotational motions in the envelope, the resulting angular momentum measurements and the expected disk radii would be significantly overestimated. \n\nRecent studies on the angular momentum of the protostellar cores from hydrodynamical simulations of star formation are questioning the standard model of star formation from a collapsing core initially in solid-body rotation \\citep{Kuznetsova19, Verliat20}. They show that the angular momentum of synthetic protostellar cores is not directly related to the initial rotation of the synthetic cloud, and Keplerian disks can be formed from a simple non-uniform perturbation in the initial density distribution.\nIn this scenario, the angular momentum observed in inner protostellar envelopes and disks may not been inherited from larger-scale initial conditions but generated during the collapse itself.\n\n\n\\subsubsection{Contribution of infalling motions and core-forming motions}\n\nThe misalignments between the gradients observed in the envelopes at inner and outer envelope scales suggest a change of dominant mechanisms at $r>$1600~au. At large scales, infalling motions of the envelope can dominate rotational motions. In the hypothesis of a flattened infalling envelope, infall motions are expected to produce a velocity gradient projected in the plane of the sky that is oriented along the minor axis of the envelope, namely at the same position angle as the outflow. \nIn L1448-NB, SVS13-B and L1527, we detect velocity gradients aligned with the outflow axis at $r>$3000~au while at small scales the gradients are consistent with the equatorial axis (see Table \\ref{table:gradient-velocity-fit}). These three sources could be good candidates of the transition from collapse to rotation between large and small scales.\nThis scenario is also suggested in the study of \\cite{Ohashi97b}. They suggested that at outer envelope scales of $r \\sim$2000~au, the protostellar envelope L1527 is not rotationally supported ($\\mathrm{v}_\\mathrm{rot} \\sim$0.05~km~s$^{-1}$) but is dominated by the collapse ($\\mathrm{v}_\\mathrm{inf} \\sim$0.3~km~s$^{-1}$). \n\nCurrently, there are very few constraints on the infall velocities at scales of $r>$1600~au in the CALYPSO protostellar envelopes. \\cite{Belloche02} estimate an infall velocity of $\\mathrm{v}_\\mathrm{inf} \\sim$0.15~km~s$^{-1}$ at $r \\sim$1000~au from radiative transfer modeling of CS and C$^{34}$S emission in IRAM04191. In the dense core L1544, \\cite{Tafalla98} suggest also an infall velocity of $\\sim$0.1~km~s$^{-1}$ at scales $>$3000~au. The velocity offset, with respect to the systemic velocity assumed\nfor each source, found along the equatorial axis at $>$1000~au with CALYPSO is reported in Table \\ref{table:velocity-100-1000au}.\nFor most sources, we find typical velocity offsets $\\lesssim$0.3~km~s$^{-1}$ at scales of 1600~au (see Table \\ref{table:velocity-100-1000au}), consistent with infall velocities found in IRAM04191 and L1544, except for IRAS4A.\nIRAS4A harbors a velocity of $\\sim$0.5~km~s$^{-1}$ at $r \\sim$1000~au. This result is consistent with those of \\cite{Belloche06} at $\\sim$2000~au. They suggest that a fast collapse is triggered by an external compression from the cloud in which the source is embedded.\nThus, for all sources in our sample, the velocity gradient misalignment could be due to a change of mechanism dominating the velocities projected on the line of sight. This suggests either rotational velocities much smaller than infall velocities or a non axisymmetric geometry of the kinematics at outer envelope scales.\n\n\n\nMoreover, \\textit{Herschel} observations have shown that most solar-type prestellar cores and protostars form in filaments \\citep{Andre14}. Indeed, the column density maps of the Herschel Gould Belt Survey program\\footnote{See \\url{http:\/\/gouldbelt-herschel.cea.fr\/archives}} \\citep{Andre10} reveal that the CALYPSO protostars are embedded in or lie in the immediate vicinity of filamentary structures with $N_{H_2} >$10$^{21}$~cm$^{-2}$. Thus, the large-scale kinematics in protostellar envelopes could be contaminated or dominated by the kinematics of the filaments. \n\\cite{Kirk13} studied the velocity field of Serpens-South in the Aquila molecular cloud and showed a complex kinematics with longitudinal collapse along the main filament, radial contractions, and accretion streams from the cloud to the main filament. The longitudinal collapse of the filament could be responsible for the large-scale gradients observed in our protostellar envelopes as observed in the Serpens-Main region by \\cite{Dhabal18}. \nSeveral studies also highlighted transverse velocity gradients perpendicular to the main filament that suggested the material may be accreting along perpendicular striations \\citep{Palmeirim13,Dhabal18,Arzoumanian18, Shimajiri19}.\n\\cite{Palmeirim13} estimate the velocity of the infalling material to be ~0.5$-$1~km~s$^{-1}$ at $r \\sim$0.4~pc in the B211\/L1495 region. In our velocity maps at 10000~au along the equatorial axis, we measure typical velocities $<$0.3~km~s$^{-1}$ in most of the sources (see Table \\ref{table:velocity-100-1000au}) except in L1448-2A, IRAS4A, and IRAS4B. These three sources exhibit velocities of 0.5$-$1~km~s$^{-1}$ consistent with infall velocities estimated at filamentary scales. In these cases, host-filament motions could dominate the kinematics in outer protostellar envelopes ($r>$1600~au). \n\n\n\n\n\\begin{table}[!ht]\n\\centering\n\\caption{Value of velocity offset, with respect to the systemic velocity assumed for each source, at 100, 1000, and 10000~au along the equatorial axis from the velocity maps and considering all the gradients observed, even those not consistent with rotational motions.}\n\\label{table:velocity-100-1000au}\n\\input{tables\/Table-velocity-100-1000au}\n\\end{table}\n\n\n\n\n\n\\subsubsection{Contamination by turbulent motions from cloud scales}\n\nAll sources of the CALYPSO subsample (except L1448-NB) reveal a steep increase in apparent specific angular momentum with the radius at $\\sim$1600$-$10000~au scales, with an average trend of $j_\\mathrm{app} \\propto r^{1.6 \\pm 0.2}$ (see Table \\ref{table:chi2-fit-profil-japp} and Fig. \\ref{fig:profil-j-CALYPSO-tous-gradients}). This trend is similar to that observed in prestellar cores and clumps at scales $>$10000~au (see Figs. \\ref{fig:diagramme-j-Belloche13+CALYPSO} and \\ref{fig:profil-j-CALYPSO-tous-gradients}). Indeed, \\cite{Goodman93}, \\cite{Caselli02} and \\cite{Tatematsu16} show a trend between the size of prestellar cores and their observed mean angular momenta at scales on the order of 0.1~pc: $j(r)$ distribution scaling as $r^{1.2-1.7}$. From this dependency of $j$ with core radius and the linewidth-size relation, \\cite{Tatematsu16} suggest that non-thermal motions (turbulence) are related to the origin of angular momentum observed in these 0.1~pc cores.\n\n\n\\cite{Burkert00} studied numerical models of turbulent molecular clouds with a symmetric density profile and a Gaussian or random velocity field. They showed that 0.1~pc cores with random motions exhibit most of the time velocity gradients that, interpreted as rotation, would have specific angular momentum values of $j \\sim 3 \\times 10^{-3}$~km~s$^{-1}$~pc (10$^{21}$~cm$^2$~s$^{-1}$) and would scale as $j \\propto r^{1.5}$. This is in good agreement with observed values at 0.1~pc from the literature \\citep{Goodman93,Caselli02,Tatematsu16}. Our observations showing a trend of $j \\propto r^{1.6}$ at scales $\\sim$1600-5000~au could be either a signature of the turbulent cascade from the large-scale ISM propagating with subsonic properties to 1600~au envelope scales, or gravitationally-driven turbulence due to large-scale collapse motions at the interface between filaments and cores \\citep{Kirk13}.\n\n\n\nFrom the analysis of the gas velocity dispersion in molecular line observations, \\cite{Goodman98} and \\cite{Pineda10} identify the dense cores at a typical scale of 0.1~pc as the first velocity-coherent structures decoupled from the turbulent cloud. In this case, we would expect a quiescent structure with subsonic motions at radii $<$0.1~pc and the interpretation of ISM turbulent cascade with supersonic motions as a consequence of the steep increase of $j_\\mathrm{app}$ at scales $<10000$~au would no longer be valid.\nExcept L1448-2A, IRAS4A, and IRAS4B which exhibit velocities 0.5$-$1~km~s$^{-1}$ consistent with supersonic turbulent motions, all sources in our sample show typical velocities $\\lesssim$0.3~km~s$^{-1}$ consistent with subsonic-transonic turbulent motions.\nThis could suggest that the power-law behavior of $j_\\mathrm{app} \\propto r^{1.6}$ observed in the outer envelopes ($r>$1600~au) could be a scaling law due to the tail of a low velocity subsonic-transonic turbulent cascade.\n\n\nAt scales of $r>$1600~au, we observe typical velocity linewidths $\\lesssim$1~km~s$^{-1}$ (see Fig. \\ref{fig:line-width-diagrams}). We note that the linewidths tend to decrease from $\\sim$1600~au to larger scales in the outer envelopes and they do not show scaling laws with the radius as expected from turbulent motions in the ISM (\\citealt{Larson81}; see Fig. \\ref{fig:line-width-diagrams} and Table \\ref{table:chi2-fit-profil-dV}), but the velocity structure at these scales seems to show multiple components in velocity for several sources (L1527, L1448-C, IRAS2A, SVS13-B, IRAS4A, IRAS4B). As we can not disentangle them and identify exactly which component comes from the outer envelope or the host cloud for example, we need either a more elaborate model than a Gaussian or a HFS model to analyze the spectra or a more suitable tracer to determine more robustly the linewidths of the outer envelopes.\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=9cm]{fig-L14482A\/plot-distribution-dV-L1448-2A.pdf}\n\\includegraphics[width=9cm]{fig-L1448N\/plot-distribution-dV-L1448-NB2.pdf}\n\\includegraphics[width=9cm]{fig-L1448C\/plot-distribution-dV-L1448C.pdf}\n\\includegraphics[width=9cm]{fig-IRAS2A\/plot-distribution-dV-IRAS2A.pdf}\n\\includegraphics[width=9cm]{fig-SVS13B\/plot-distribution-dV-SVS13B.pdf}\n\\includegraphics[width=9cm]{fig-IRAS4A\/plot-distribution-dV-IRAS4A.pdf}\n\\caption{Linewidth along the equatorial axis of the CALYPSO protostellar envelopes. Blue and red dots show the linewidths at positions that have blue- and red-shifted velocities, respectively. Dots and large dots show the C$^{18}$O and N$_{2}$H$^{+}$ data, respectively. The dashed curve shows the best fit with a power-law model leaving the index $\\gamma$ as a free parameter ($D \\mathrm{v} \\propto r^{\\gamma}$) in the outer envelope (see Appendix \\ref{sec:details-Dv-distribution}). The radius of the outer envelope is given by the break radius of the $j(r)$ or $j_\\mathrm{app}(r)$ profiles (see Tables \\ref{table:chi2-fit-profil-moment-ang} and \\ref{table:chi2-fit-profil-japp}) or the radius where we observe a reversal of the velocity gradients with respect to the inner envelope. The vertical dashed lines show the transition radii between the different datasets (PdBI, combined, and 30m) and the two tracers as given in Table \\ref{table:radius-lines}.\n}\n\\label{fig:line-width-diagrams}\n\\end{figure*}\n\n\n\n\\begin{figure*}[!ht]\n\\addtocounter{figure}{-1}\n\\centering\n\\includegraphics[width=9cm]{fig-IRAS4B\/plot-distribution-dV-IRAS4B.pdf}\n\\includegraphics[width=9cm]{fig-IRAM04191\/plot-distribution-dV-IRAM04191.pdf}\n\\includegraphics[width=9cm]{fig-L1521F\/plot-distribution-dV-L1521F.pdf}\n\\includegraphics[width=9cm]{fig-L1527\/plot-distribution-dV-L1527.pdf}\n\\includegraphics[width=9cm]{fig-L1157\/plot-distribution-dV-L1157.pdf}\n\\includegraphics[width=9cm]{fig-GF92\/plot-distribution-dV-GF92.pdf}\n\\caption{Continued.\n}\n\\end{figure*}\n\n\n\n\n\n\n\n\\section{Conclusions}\nIn the framework of the CALYPSO survey, we analyzed the kinematics of Class~0 protostellar envelopes. The main results of our study are listed below.\n\\begin{enumerate}\n\\item We identify differential rotation motions in 11 sources in a sample of 12 Class~0 protostellar envelopes. The only exception is IRAS4A : the motions reported in the PV$_\\mathrm{rot}$ and modeled by a power-law function are consistent with a solid-body rotation, but the velocity gradient is not uniform in the inner envelope at $r<$2000~au as would be expected.\n\n\\item This is the first time that the specific angular momentum distribution as a function of envelope radius is determined homogeneously for a large sample of 11 Class~0 protostars. The high angular resolution and the high dynamic range of the CALYPSO observations allow us to identify two distinct regimes: the apparent specific angular momentum decreases as $j_\\mathrm{app} \\propto r^{1.6 \\pm 0.2}$ down to $\\sim$1600~au and then tends to become relatively constant around $\\sim$6 $\\times$ 10$^{-4}$~km~s$^{-1}$~pc down to $\\sim$50~au. \n\n\\item The values of specific angular momentum measured in the inner Class~0 envelopes suggest that material directly involved in the star formation process ($<$1600~au) typically encloses the same order of magnitude in specific angular momentum as what is inferred in small T-Tauri disks ($r \\sim$10~au). The constant values of $j$ at 50$-$1600~au allow us to determine good estimates of the centrifugal radius in the Class~0 protostars of the CALYPSO sample, which compare well with the disk radii estimated from the dust continuum \\citep{Maury18}. This suggests that the specific angular momentum is conserved during the accretion on the stellar embryo, resulting in disk formation. \n\n\n\\item At scales of $r>$1600~au, we conclude that the velocity gradients observed in the outer envelope with respect to small scales are not due to pure rotational motions or counter-rotation motions but related to other mechanisms. Historically, the gradients observed from single-dish mapping at $r>$3000~au have been interpreted as rotation and used to quantify the amplitude of the angular momentum problem for star formation. Thus, if the gradients are incorrectly interpreted as rotation, the angular momentum problem for star formation and the expected disk radii may have been significantly over-estimated. Moreover, we find no robust hints that envelopes are rotating with typical velocities higher than the sound speed at scales of $r>$1600~au. This suggests that the origin of angular momentum in the outer protostellar envelopes could be the gravitationally-driven turbulence due to large-scale collapse motions at the interface between filaments and cores, or the dissipation of the large-scale ISM turbulent cascade propagating with subsonic velocities to 1600~au envelope scales.\n\\end{enumerate}\n\n\n\n\n\\begin{acknowledgements}\nWe thank the IRAM staff for their support carrying out the CALYPSO observations. \nThis work has benefited from the support of the European Research Council under the European Union's Seventh Framework Programme (Advanced Grant ORISTARS with grant agreement no. 291294 and Starting Grant MagneticYSOs with grant agreement no. 679937). M.G. thanks the Max-Planck Institute for Radio Astronomy for its support toward the end of this work. We would like to thank Cecilia Cecarelli for comments and suggestions on the estimation of column density, and Nagayoshi Ohashi and Jaime E. Pineda for valuable discussions on the interpretation. We also thank the referee for the useful comments which helps to improve this paper.\n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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}#1)}\n\\newcommand{\\fpx}[0]{\\mathbb F_p^\\times}\n\\newcommand{\\lr}[1]{\\left(#1\\right)}\n\\newcommand{\\flr}[1]{\\left\\lfloor #1\\right\\rfloor}\n\\newcommand{\\ceil}[1]{\\left\\lceil#1\\right\\rceil}\n\\newcommand{\\supp}[0]{\\text{supp}}\n\\newcommand{\\Ker}[0]{\\text{Ker}}\n\\newcommand{\\qu}[1]{``#1''}\n\n\n\n\n\\newcommand\\ir{\\mathrm{Irr}} \n\\newcommand\\irr{\\mathrm{Irr}_1}\n\\newcommand\\lin{\\mathrm{Lin}}\n\\newcommand\\Ir{\\mathrm{I}}\n\\newcommand\\Irr{\\mathrm{I}_1} \n\\newcommand\\cd{\\mathrm{c. d.}} \n\\newcommand\\gc{\\gcd}\n\\newcommand\\be{\\begin{eqnarray*}}\n\\newcommand\\ee{\\end{eqnarray*}}\n\\newcommand\\beq{\\begin{equation}}\n\\newcommand\\eeq{\\end{equation}}\n\\newcommand{\\mathbb{E}}{\\mathbb{E}}\n\\newcommand{\\Cal}[1]{\\mathcal{#1}}\n\n\\newcommand\\ben{\\begin{eqnarray}}\n\\newcommand\\een{\\end{eqnarray}}\n\\newcommand\\ord{\\mathrm{ord}}\n\n\n\\begin{document}\n\n\n\\baselineskip=17pt\n\n\n\n\\title{On iterated product sets with shifts II}\n\n\\author[B. Hanson]{Brandon Hanson} \\address{Pennsylvania State University\\\\\nUniversity Park, PA, USA}\n\\email{bwh5339@psu.edu}\n\\author[O. Roche-Newton]{Oliver Roche-Newton} \\address{Johann Radon Institute for Computational and Applied Mathematics\\\\\nLinz, Austria}\n\\email{o.rochenewton@gmail.com}\n\\author[D. Zhelezov]{Dmitrii Zhelezov} \\address{Alfr\\'{e}d R\\'{e}nyi Institute of Mathematics \\\\ \nHungarian Academy of Sciences, Budapest, Hungary }\n\\email{dzhelezov@gmail.com}\n\\date{}\n\n\\begin{abstract} The main result of this paper is the following: for all $b \\in \\mathbb Z$ there exists $k=k(b)$ such that\n\\[ \\max \\{ |A^{(k)}|, |(A+u)^{(k)}|  \\}  \\geq |A|^b, \\]\nfor any finite $A \\subset \\mathbb Q$ and any non-zero $u \\in \\mathbb Q$. Here, $|A^{(k)}|$ denotes the $k$-fold product set $\\{a_1\\cdots a_k : a_1, \\dots, a_k \\in A \\}$.\n\n\nFurthermore, our method of proof also gives the following $l_{\\infty}$ sum-product estimate. For all $\\gamma >0$ there exists a constant $C=C(\\gamma)$ such that for any $A \\subset \\mathbb Q$ with $|AA| \\leq K|A|$ and any $c_1,c_2 \\in \\mathbb Q \\setminus \\{0\\}$, there are at most $K^C|A|^{\\gamma}$ solutions to\n\\[ c_1x + c_2y =1 ,\\,\\,\\,\\,\\,\\,\\, (x,y) \\in A \\times A.\n\\]\nIn particular, this result gives a strong bound when $K=|A|^{\\epsilon}$, provided that $\\epsilon >0$ is sufficiently small, and thus improves on previous bounds obtained via the Subspace Theorem.\n\nIn further applications we give a partial structure theorem for point sets which determine many incidences and prove that sum sets grow arbitrarily large by taking sufficiently many products.\n\n\n\n\n\\end{abstract}\n\n\n\\maketitle\n\n\\section{Introduction}\n\n\\subsection{Background and statement of main results}\n\n\nLet $A$ be a finite set of rational numbers and let $u\\in \\QQ$ be non-zero. In this article we wish to investigate the sizes of the $k$-fold product sets\n\\[A^{(k)}:=\\{a_1\\cdots a_k:a_1,\\ldots,a_k\\in A\\}\\]\nand\n\\[(A+u)^{(k)} =\\{(a_1+u)\\cdots (a_k+u):a_1,\\ldots,a_k\\in A\\}.\\]\nThis is an instance of a sum-product problem. Recall that the Erd\\H{o}s-Szemer\\'{e}di \\cite{ES} sum-product conjecture states that, for all $\\epsilon >0$ there exists a constant $c(\\epsilon)>0$ such that\n\\[ \\max \\{|A+A|, |AA| \\} \\geq c(\\eps) |A|^{2-\\eps} \\]\nholds for any $A \\subset \\mathbb Z$. Here $A+A:=\\{a+b : a,b \\in A \\}$ is the \\textit{sum set} of $A$, and $AA$ is another notation for $A^{(2)}$. Erd\\H{o}s and Szemer\\'{e}di also made the more general conjecture that for any finite $A \\subset \\mathbb Z$,\n\\[\n\\max \\{|kA|,|A^k|\\} \\geq c(\\epsilon)|A|^{k-\\epsilon},\n\\]\nwhere $kA:= \\{a_1+ \\dots +a_k : a_1,\\dots, a_k \\in A\\}$ is the \\textit{$k$-fold sum set}. Both of these conjectures are wide open, and it is natural to also consider them for the case when $A$ is a subset of $\\mathbb R$ or indeed other fields. The case when $k=2$ has attracted the most interest. See, for example, \\cite{KS}, \\cite{KS2}, \\cite{S}, \\cite{TV} and the references contained therein for more background on the original Erd\\H{o}s-Szemer\\'{e}di sum-product problem.\n\nMost relevant to our problem is the case of general (large) $k$. Little is known about the Erd\\H{o}s-Szemer\\'{e}di conjecture in this setting, with the exception of the remarkable series of work of Chang \\cite{C} and Bourgain-Chang \\cite{BC}. This culminated in the main theorem of \\cite{BC}: for all $b \\in \\mathbb R$ there exists $ k = k(b) \\in \\mathbb Z$ such that\n\\begin{equation} \\label{BCmain}\n\\max \\{|kA|,|A^k|\\} \\geq |A|^b\n\\end{equation}\nholds for any $A \\subset \\mathbb Q$. On the other hand, it appears that we are not close to proving such a strong result for $A \\subset \\mathbb R$.\n\nIn the same spirit as the Erd\\H{o}s-Szemer\\'{e}di conjecture, it is expected that an additive shift will destroy  multiplicative structure present in $A$. In particular, one expects that, for a non-zero $u$, at least one of $|A^{(k)}|$ or $|(A+u)^{(k)}|$ is large. The $k=2$ version of this problem was considered in \\cite{GS} and \\cite{JRN}. The main result of this paper is the following analogue of the Bourgain-Chang Theorem.\n\\begin{Theorem} \\label{thm:mainmain}\nFor all $b \\in \\mathbb Z$, there exists $k=k(b)$ such that for any finite set $A \\subset \\mathbb Q$ and any non-zero rational $u$,\n\\[  \\max \\{ |A^k|, |(A+u)^k| \\} \\geq |A|^b . \\]\n\\end{Theorem}\n\nThis paper is a sequel to \\cite{HRNZ}, in which the main result was the following.\n\\begin{Theorem} \\label{thm:usold} For any finite set $A \\subset \\mathbb Q$ with $|AA| \\leq K|A|$, any non-zero $u \\in \\mathbb Q$ and any positive integer $k$,\n$$| (A+u) ^{(k)}| \\geq \\frac{|A|^k}{(8k^4)^{kK}}. $$\n\\end{Theorem}\nThe proof of this result was based on an argument that Chang \\cite{C} introduced to give similar bounds for the $k$-fold sum set of a set with small product set. Theorem \\ref{thm:usold} is essentially optimal when $K$ is of the order $c\\log|A|$, for a sufficiently small constant $c=c(k)$. However, the result becomes trivial when $K$ is larger, for example if $K=|A|^{\\epsilon}$ and $\\eps>0$. The bulk of this paper is devoted to proving the following theorem, which gives a near optimal bound for the size of $(A+u)^{(k)}$ when $K=|A|^{\\eps}$,\nfor a sufficiently small but positive $\\eps$.\n\\begin{Theorem} \\label{thm:main1}\nGiven $0<\\gamma < 1\/2$, there exists a positive constant $C=C(\\gamma, k)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$ and any non-zero rational $u$,\n\\[ | (A+u) ^{(k)}|  \\geq \\frac{|A|^{k(1-\\gamma)-1}}{K^{Ck}}.\\]\n\\end{Theorem}\n\nIn fact, we prove a more general version of Theorem \\ref{thm:main1} in terms of certain weighted energies and so-called $\\Lambda$-constants (see Theorem \\ref{thm:lambda} for the general statement that implies Theorem \\ref{thm:main1} - see sections \\ref{sec:energy} and \\ref{sec:lambda} for the relevant definitions of energy and $\\Lambda$-constants). This more general result is what allows us to deduce Theorem \\ref{thm:mainmain}.\n\n\n\n\\subsection{A subspace type theorem -- an $l_\\infty$ sum-product estimate}\n\nIt appears that Theorem \\ref{thm:mainmain}, as well as the forthcoming generalised form of Theorem \\ref{thm:main1}, lead to some interesting new applications. To illustrate the strength of these sum-product results, we present three applications in this paper.\n\nOur main application concerns a variant of the celebrated Subspace Theorem by Evertse, Schmidt and Schlikewei \\cite{evertse2002linear} which, after quantitative improvements by Amoroso and Viada \\cite{amoroso2009small}, reads as follows.\n\nSuppose $a_1, \\ldots,  a_k \\in \\mathbb{C}^*$, $\\alpha_1,\\ldots,\\alpha_r \\in \\mathbb C^*$ and define\n$$\n\\Gamma = \\{\\alpha_1^{z_1} \\cdots \\alpha_r^{z_r}, z_i \\in \\mathbb{Z} \\},\n$$\nso $\\Gamma$ is a free multiplicative group\\footnote{The original theorem is formulated in a more general setting, namely for the division group of $\\Gamma$, but we will stick to the current formulation for simplicity.} of rank $r$. Consider the equation\n\\begin{equation} \\label{eq:subspace_eq}\na_1x_1 + a_2x_2 + \\cdots + a_kx_k = 1 \n\\end{equation}\nwith $a_i \\in \\mathbb{C}^*$ viewed as fixed coefficients and $x_i \\in \\Gamma$ as variables. A solution $(x_1, \\ldots, x_k)$ to (\\ref{eq:subspace_eq}) is called \\emph{nondegenerate} if\nfor any non-empty $J \\subsetneq \\{1, \\ldots, k \\}$\n$$\n\t\\sum_{i \\in J} a_ix_i \\neq 0.\n$$\n\n\\begin{Theorem}[The Subspace Theorem, \\cite{evertse2002linear} \\cite{amoroso2009small} ] \\label{thm:subspace}\nThe number $A(k, r)$ of nondegenerate  solutions to (\\ref{eq:subspace_eq}) satisfies the bound\n\\begin{equation} \\label{subspace_thm_ineq}\nA(k, r) \\leq {(8k)}^{4k^4(k + kr + 1)}.\n\\end{equation}\n\\end{Theorem} \n\n\n\n\n\n\nThe Subspace Theorem dovetails nicely to the following version of the Freiman Lemma. \n\\begin{Theorem} \\label{thm:FR-lemma}\nLet $(G, \\cdot)$ be a torsion-free abelian group and \n$A \\subset G$ with $|AA| < K|A|$. Then $A$ is contained in a subgroup $G' < G$ of rank at most $K$.\n\\end{Theorem}\n\nNow assume for simplicity that $A \\subset \\mathbb{Q}$ and $|AA| \\leq K|A|$. \n Let us call such sets (this definition generalizes of course to an arbitrary ambient group) $K$-\\emph{almost subgroups} \\footnote{One could've used a more general framework of $K$-\\emph{approximate subgroups} introduced by Tao. We decided to introduce a simpler definition in order to avoid technicalities. However, in the abelian setting the definitions are essentially equivalent.}.\n \n\nWe now show that it is natural to expect that the Subspace Theorem generalises to $K$-almost subgroups with $K$ taken as a proxy for the group rank. A straightforward corollary of Theorem~\\ref{thm:FR-lemma} and Theorem~\\ref{thm:subspace} is as follows. \n\n\\begin{Corollary}[Subspace Theorem for $K$-almost subgroups] \\label{corr:subspace_almost_subgroups}\n  Let $A$ be a $K$-almost subgroup. Then the number $A(k, K)$ of non-degenerate solutions $(x_1, x_2, \\ldots, x_k) \\in A^k$ to\n$$\nc_1x_1 + c_2x_2 + \\ldots c_kx_k = 1\n$$  \nwith fixed coefficients $c_i \\in \\mathbb{C^*}$ is bounded by \n$$ \nA(k, K) \\leq {(8k)}^{4k^4(k + kK + 1)}.\n$$\n  \n\\end{Corollary}\n\nSimilarly to Theorem~\\ref{BCmain}, the bound of Corollary~\\ref{corr:subspace_almost_subgroups} becomes trivial when $A$ is large and $K$ is larger than $c\\log |A|$ for some small $c > 0$. \n\nWe conjecture that a much stronger polynomial bound holds.\n\n\\begin{Conjecture} \\label{conj:Ksubspace}\n There is a constant $c(k)$ such that Corollary~\\ref{corr:subspace_almost_subgroups} holds with the bound\n $$\n  A(k, K) \\leq K^{c(k)}.\n $$\n\\end{Conjecture}\n\n\nWe can support Conjecture~\\ref{conj:Ksubspace} with a special case $k = 2$ and $A \\subset \\mathbb{Q}, c_i \\in \\mathbb{Q}$ and a somewhat weaker estimate, which we see as a proxy for the Beukers-Schlikewei Theorem \\cite{beukers1996equation}.\n\n\\begin{Theorem}[Weak Beukers-Schlikewei for $K$-almost subgroups] \\label{thm:BS_almost_subgroups}\n For any $\\gamma > 0$ there is $C(\\gamma) > 0$ such that for any $K$-almost subgroup $A \\subset \\mathbb{Q}$ and fixed non-zero $c_1, c_2 \\in \\mathbb{Q}$ the number $A(2, K)$ of solutions $(x_1, x_2) \\in A^2$ to \n $$\n c_1x_1 + c_2 x_2 = 1\n $$\n is bounded by \n $$\n A(2, K) \\leq |A|^\\gamma K^C.\n $$\n\n\\end{Theorem}\nOne can view Theorem \\ref{thm:BS_almost_subgroups} as an $l_{\\infty}$ version of the weak Erd\\H{o}s-Szemer\\'{e}di sum-product conjecture. The \\textit{weak Erd\\H{o}s-Szemer\\'{e}di conjecture} is the statement that, if $|AA| \\leq K|A|$ then $|A+A| \\geq K^{-C}|A|^2$ for some positive absolute constant $C$. For $A \\subset \\mathbb Z$, this result was proved in \\cite{BC}, but the conjecture remains open over the reals. \n\nA common approach to proving sum-product estimates is to attempt to show that, for a set $A$ with small product set,\nthe \\textit{additive energy} of $A$, which is defined as the quantity\n\\[E_+(A):= |\\{ (a,b,c,d) \\in A^4 : a+b=c+d \\}|, \\]\nis small. Indeed, this was the strategy implemented in \\cite{C} and \\cite{BC}, the latter of which showed\\footnote{This is something of an over-simplification, as \\cite{BC} in fact proved a much more general result which bounded the multi-fold additive energy with weights attached.} that, for all $\\gamma >0$, there is a constant $C=C(\\gamma)$ such that for any $A \\subset \\mathbb Q$ with $|AA| \\leq K|A|$,\n\\begin{equation} \nE_+(A) \\leq K^C|A|^{2+ \\gamma}. \n\\label{eq:weakES}\n\\end{equation}\nSince there are at least $|A|^2$ trivial solutions when $\\{a,b\\}=\\{c,d\\}$, this bound is close to best possible. It then follows from a standard application of the Cauchy-Schwarz inequality that\n\\[|A+A| \\geq \\frac{|A|^{2-\\gamma}}{K^C}. \\]\nDefining the representation function $r_{A+A}(c)=|\\{(a_1,a_2) \\in A \\times A : a_1+a_2=c\\}|$, it follows that\n\\[E_+(A)= \\sum_x r_{A+A}(x)^2,\\]\nand so bounds for the additive energy can be viewed as $l_2$ estimates for this representation function.\n\nTheorem \\ref{thm:BS_almost_subgroups} gives the stronger $l_{\\infty}$ estimate: it says that, if $|AA| \\leq K|A|$ then $r_{A+A}(c) \\leq K^C|A|^{\\gamma}$ for all $c \\neq 0$. This implies \\eqref{eq:weakES}, and thus in turn the weak Erd\\H{o}s-Szemer\\'{e}di sum-product conjecture. We prove Theorem~\\ref{thm:BS_almost_subgroups} in Section~\\ref{sec:conclusion}.\n\n\n\\begin{Remark}\n   It is highly probable that our method can be combined with the ideas of \\cite{bourgain2009sum} which would generalize Theorem~\\ref{thm:BS_almost_subgroups} to $K$-almost subgroups consisting of algebraic numbers of degree at most $d$ (though not necessarily contained in the same field extension). The upper power $C$ is going to depend on $d$ then, so the putative bound (using the notation of Theorem~\\ref{thm:BS_almost_subgroups}) is \n   $$\n      A(2, K) \\leq C'(d)|A|^\\gamma K^{C(\\gamma, d)}\n   $$\nwith some $C, C' > 0$.\nWe are going to consider this matter in detail elsewhere. Note, however, that proving a similar statement with no dependence on $d$ seems to be a significantly harder problem. \n\\end{Remark}\n\n\\subsection{Further applications}\n\n\\subsubsection{An inverse Szemer\\'{e}di-Trotter Theorem} Theorem \\ref{thm:BS_almost_subgroups} can be interpreted as a partial inverse to the Szemer\\'{e}di-Trotter Theorem. The Szemer\\'{e}di-Trotter Theorem states that, if $P$ is a finite set of points and $L$ is a finite set of lines in $\\mathbb R^2$, then the number of incidences $I(P,L)$ between $P$ and $L$ satisfies the bound\n\\begin{equation}\nI(P,L):= |\\{(p,l) \\in P \\times L : p \\in l \\}| =O(|P|^{2\/3}|L|^{2\/3} +|P| + |L| ).\n\\label{eq:ST}\n\\end{equation}\nThe term $|P|^{2\/3}|L|^{2\/3}$ above is dominant unless the sizes of $P$ and $L$ are rather imbalanced. The Szemer\\'{e}di-Trotter Theorem is tight, up to the multiplicative constant.  \n\nIt is natural to consider the inverse question: for what sets $P$ and $L$ is it possible that $I(P,L) = \\Omega (|P|^{2\/3}|L|^{2\/3})$? The known constructions of point sets which attain many incidences appear to all have some kind of lattice like structure. This perhaps suggests the loose conjecture that point sets attaining many incidences must always have some kind of additive structure, although such a conjecture seems to be far out of reach to the known methods.\n\nHowever, with an additional restriction that $P=A \\times A$ with $A \\subset \\mathbb Q$, Theorem \\ref{thm:mainmain} leads to the following partial inverse theorem, which states that if $A$ has small product set then $I(P,L)$ cannot be maximal.\n\n\\begin{Theorem} \\label{thm:STinverse} For all $\\gamma \\geq 0$ there exists a constant $C=C(\\gamma)$ such that the following holds. Let $A$ be a finite set of rationals such that $|AA| \\leq K|A|$ and let $P= A \\times A$. Then, for any finite set $L$ of lines in the plane, $I(P,L)\\leq 3 |P| + |A|^{\\gamma}K^C|L|$.\n\\end{Theorem}\n\nIn fact, not only does this show that $I(A \\times A,L)$ cannot be maximal when $|AA|$ is small, but better still the number of incidences is almost bounded by the trivial linear terms in \\eqref{eq:ST}. The insistence that the point set is a direct product is rather restrictive. However, since many applications of the Szemer\\'{e}di-Trotter Theorem make use of direct products, it seems likely that Theorem \\ref{thm:STinverse} could be useful. The proof is given in Section \\ref{sec:applications}.\n\n\\subsubsection{Improved bound for the size of an additive basis of a set with small product set} \n\nTheorem~\\ref{thm:BS_almost_subgroups} also yields the following application concerning the problem of bounding the size of an additive basis considered in \\cite{shkredov2016additive}. We can significantly improve the bound in the rational setting, pushing the exponent in (\\ref{eq:basis_bound}) from $1\/2 + 1\/442 - o_\\epsilon(1)$ to $2\/3 - o_\\epsilon(1)$ in the limiting case $K = |A|^{\\epsilon}$. \n\n\n\\begin{Theorem} \\label{thm:additivebasis}\nFor any $\\gamma > 0$ there exists $C(\\gamma)$ such that\nfor an arbitrary $A \\subset \\mathbb{Q}$ with $|AA| = K|A|$ and $B, B' \\subset \\mathbb{Q}$,\n$$\nS := \\left|\\{(b, b') \\in B \\times B' : b + b' \\in A\\} \\right| \\leq 2|A|^\\gamma K^C \\min \\{|B|^{1\/2}|B'| + |B|, |B'|^{1\/2}|B| + |B'| \\}. \n$$\nIn particular, for any $\\gamma > 0$ there exists $C(\\gamma)$ such that if $A \\subset B + B$ then\n\n\\begin{equation} \\label{eq:basis_bound}\n|B| \\geq |A|^{2\/3 - \\gamma}K^{-C}.\n\\end{equation}\n\n\\end{Theorem}\n\nThe proof of Theorem \\ref{thm:additivebasis} is given in Section \\ref{sec:applications}.\n\\begin{Remark}\n During the preparation of the manuscript we became aware that Cosmin Pohoata has independently proved Theorem~\\ref{thm:additivebasis} using an earlier result of Chang and by a somewhat different method.\n\\end{Remark}\n\n\\begin{comment}\n\\begin{proof}\nWe will prove that \n\\begin{equation}\nS \\leq 2|A|^\\gamma K^C(|B'|^{1\/2}|B| + |B'|).\n\\label{eq:case1}\n\\end{equation}\nSince the roles of $B$ and $B'$ are interchangeable, \\eqref{eq:case1} also implies that $S \\leq 2|A|^\\gamma K^C(|B|^{1\/2}|B'| + |B|)$, and thus completes the proof.\n\nLet $\\gamma > 0$ and $C(\\gamma)$, given by Theorem~\\ref{thm:BS_almost_subgroups}, be fixed. Without loss of generality assume that $S \\geq 2|B'|$ as otherwise the claimed bound is trivial. \n\nFor each $b \\in B$ define \n$$\nS_b := \\{ b' \\in B' : b + b' \\in A\\},\n$$\nand similarly for $b' \\in B'$\n$$\nT_{b'} := \\{b \\in B: b' + b \\in A \\}.\n$$\nIt follows from Theorem~\\ref{thm:BS_almost_subgroups} that for $b_1,b_2 \\in B$ with $b_1 \\neq b_2$ \n$$\n|S_{b_1} \\cap S_{b_2}| \\leq |A|^\\gamma K^C\n$$\nsince each $x \\in S_{b_1} \\cap S_{b_2}$ gives a solution $(a, a') := (b_1 + x, b_2 + x)$ to\n$$\na - a' = b_1 - b_2\n$$\nwith $a, a' \\in A$.\n\nOn the other hand, by double-counting and the Cauchy-Schwarz inequality,\n$$\n\\sum_{b \\in B} |S_b| + \\sum_{b_1,b_2 \\in B : b_1 \\neq b_2} |S_{b_1} \\cap S_{b_2}| =\n\\sum_{b' \\in B'} |T_{b'}|^2 \\geq |B'|^{-1}(\\sum_{b' \\in B'} |T_{b'}|)^2 = |B'|^{-1}S^2.\n$$\nTherefore,\n$$\n\\sum_{b_1,b_2 \\in B : b_1 \\neq b_2} |S_{b_1} \\cap S_{b_2}| \\geq |B'|^{-1}S^2 - \\sum_{b \\in B} |S_b| = |B'|^{-1}S^2 - S \\geq \\frac{1}{2} |B'|^{-1}S^2\n$$\nby our assumption.\n\n\nThe left-hand side is at most $ |B|^2|A|^\\gamma K^C$, and so\n$$\n\tS \\leq (2|A|^\\gamma K^C)^{1\/2} |C|^{1\/2}|B'|,\n$$\nwhich completes the proof.\n\n\n\n\\end{proof}\n\\end{comment}\n\n\n\n\n\\subsubsection{Unlimited growth for products of difference sets} It was conjectured in \\cite{BRNZ} that for any $b \\in \\mathbb R$ there exists $k=k(b) \\in \\mathbb N$ such that for all $A \\subset \\mathbb R$\n\\[|(A-A)^k| \\geq |A|^b .\\]\nIn another application of Theorem \\ref{thm:mainmain}, we give a positive answer to this question under the additional restriction that $A \\subset \\mathbb Q$. In fact, we prove the following stronger statement.\n\n\n\\begin{Theorem} \\label{thm:proddiff}\nFor any $b \\in \\mathbb R$ there exists $k=k(b) \\in \\mathbb N$ such that for all $A \\subset \\mathbb Q$ and $B \\subset \\mathbb Q$ with $|B| \\geq 2$,\n\\[|(A+B)^k| \\geq |A|^b .\\]\n\n\\end{Theorem}\nThe proof is given in Section \\ref{sec:applications}.\n\n\\subsection{The structure of the rest of this paper}\n\nIn section \\ref{sec:energy}, we introduce a new kind of mixed energy, and establish some initial bounds on this energy which are strong when the multiplicative doubling $K$ is of the order $c\\log |A|$ for a sufficiently small constant $c$. The structure of these arguments are similar to those introduced by Chang in \\cite{C}, and also used by the authors in \\cite{HRNZ}. We also introduce the notion of separating constants in section \\ref{sec:energy}, which generalises that of the aforementioned mixed energy.\n\nSection \\ref{sec:lambda} begins by stating the crucial Theorem \\ref{thm:goodsubset}, which states that is $|AA|$ is small then there is a large subset $A' \\subset A$ with a good separating constant. The rest of the section introduces the language of $\\Lambda$-constants and some of their crucial properties. These properties are then used in section \\ref{sec:conclusion} to conclude the proofs of the main results of this paper, Theorems \\ref{thm:mainmain}, \\ref{thm:main1} and \\ref{thm:BS_almost_subgroups}, using Theorem \\ref{thm:goodsubset} as a black box.\n\nIt then remains to prove Theorem \\ref{thm:goodsubset}. This is a long and technical proof, where we need to amplify the bounds obtained in section \\ref{sec:energy} in several stages. This process happens in sections \\ref{sec:fibers}, \\ref{sec:iterationscheme}, \\ref{sec:betterpair}, \\ref{sec:strongpair} and \\ref{sec:conclusion2}, and closely follows the exposition in \\cite{Z}.\\footnote{We recommend that the reader consult \\cite{Z} for more information about the proof of the Bourgain-Chang Theorem, and particularly the early parts of \\cite{Z}, where an attempt is made to outline some heuristics of the proof.} Finally, in section \\ref{sec:applications}, we give proofs of further applications of our main results.\n\n\n\n\n\n\\section{A Chang-type bound for the mixed energy} \\label{sec:energy}\n\nDifferent kinds of energies play a pivotal role in the work of Chang \\cite{C} and Bourgain-Chang \\cite{BC}, as well as \\cite{HRNZ}. In \\cite{C}, it was proved that, for any finite set of rationals $A$ with $|AA| \\leq K|A|$, the \\textit{k-fold additive energy}, which is defined as the number of solutions to\n\\begin{equation} \\label{changenergy}\na_1 + \\cdots + a_k = a_{k+1} + \\cdots a_{2k}, \\,\\,\\,\\,\\,\\,\\,\\, (a_1,\\dots,a_{2k}) \\in A^{2k}, \n\\end{equation}\nis at most $(2k^2-k)^{kK}|A|^{k}$. A simple application of the Cauchy-Schwarz inequality then implies that the \\textit{$k$-fold sum set} satisfies the bound\n\\[ |kA| \\geq \\frac{|A|^k}{(2k^2-k)^{kK}} .\\]\nBound \\eqref{changenergy} is close to optimal when $K=c \\log |A|$, but becomes trivial when $K=|A|^{\\eps}$. In \\cite{BC}, (a weighted version of) this bound was used as a foundation, and developed considerably courtesy of some intricate decoupling arguments, in order to prove a bound for the $k$-fold additive energy which remains very strong when $K$ is of the order $|A|^{\\eps}$.\n\nIn \\cite{HRNZ}, we followed a similarly strategy to that of \\cite{C}, proving that for any finite set of rationals $A$ with $|AA| \\leq K|A|$ and any non-zero rational $u$, the \\textit{k-fold multiplicative energy} of $A+u$, which is defined as the number of solutions to\n\\begin{equation}\n(a_1 + u) \\cdots  (a_k+u) = (a_{k+1} + u) \\cdots (a_{2k} +u ), \\,\\,\\,\\,\\,\\,\\,\\, (a_1,\\dots,a_{2k}) \\in A^{2k}, \n\\label{changprequel}\n\\end{equation}\nis at most $(Ck^2)^{kK}|A|^{k}$. Unfortunately, in adapting the approach of \\cite{C} in order to bound the number of solutions to \\eqref{changprequel} in \\cite{HRNZ}, we encountered some difficulties with dilation invariance which made the argument rather more complicated, and we were unable to marry our methods with those of \\cite{BC} to obtain a strong bound when $K$ is of order $|A|^{\\eps}$.\n\nIn this paper, we modify the approach of \\cite{HRNZ} by working with a different form of energy. Consider the following representation function:\n\\[r_k(x,y)=|\\{(a_1,\\ldots,a_k)\\in A^k  :a_1\\cdots a_k=x,\\ (a_1+u)\\cdots(a_k+u)=y\\}|.\\]\nThen, because $r_k$ is supported on $A^{(k)}\\times (A+u)^{(k)}$, it follows from the Cauchy-Schwarz inequality that\n\\begin{equation}\n|A|^{2k}=\\left(\\sum_{(x,y)\\in A^{(k)}\\times (A+u)^{(k)} }r_k(x,y)\\right)^2\\leq |A^{(k)}||(A+u)^{(k)}|\\sum_{(x,y)\\in A^{(k)}\\times (A+u)^{(k)}} r_k(x,y)^2.\n\\label{CSbasic}\n\\end{equation}\nThe innermost sum is the quantity\n\\[\\tilde E_k(A;u):=\\left|\\left\\{(a_1,\\ldots,a_k,b_1,\\ldots,b_k)\\in A^{2k} :\\prod_{i=1}^ka_i=\\prod_{i=1}^kb_i,\\ \\prod_{i=1}^k(a_i+u)=\\prod_{i=1}^k(b_i+u)\\right\\}\\right|.\\]\n\nWe summarise this in the following lemma.\n\\begin{Lemma} \\label{lem:CSbasic}\nFor any finite set $A\\subset \\mathbb R$, any  $u \\in \\mathbb R \\setminus \\{0\\}$ and any integer $k \\geq 2$, we have\n\\[|A|^{2k} \\leq |A^{(k)}||(A+u)^{(k)}| \\tilde E_k(A;u) .\\]\nIn particular,\n\\[\\frac{|A|^k}{\\tilde E_k(A;u)^{1\/2}}\\leq \\max\\{|A^{(k)}|,|(A+u)^{(k)}|\\}.\\]\n\\end{Lemma}\n\nOur goal is to estimate this energy and to show that, at least for sets of rationals, it cannot ever be too big. \n\n\nIn this section we seek to give an initial upper bound for $\\tilde E_k(A;u)$. The strategy is close to that of Chang \\cite{C}. There are also clear similarities with the prequel to this paper \\cite{HRNZ}.\n\nTo do this, as in \\cite{HRNZ}, we will write $\\tilde E_k(A;u)$ in terms of Dirichlet polynomials. In this case, our Dirichlet polynomials will be functions of the form \\[F(s_1,s_2)=\\sum_{(a,b)\\in\\QQ^2}\\frac{f(a,b)}{a^{s_1}b^{s_2}}\\] where $f:\\QQ^2\\to\\CC$ is some function of finite support. It will also be more convenient to count weighted energy. For $w_a$ a sequence of non-negative weights on $A$, let\n\\[\\tilde E_{k,w}(A;u)=\\sum_{\\substack{a_1\\cdots a_k=b_1\\cdots b_k\\\\ (a_1+u)\\cdots(a_k+u)=(b_1+u)\\cdots(b_k+u)}}w_{a_1}\\cdots w_{a_k}w_{b_1}\\cdots w_{b_k}\\]\n\n\\begin{Lemma} \\label{lem:direnergy}\nLet $A$ be a finite set of rational numbers and let $u$ be a non-zero rational number. Then, for any integer $k \\geq 2$, we have\n\\[\\tilde E_{k,w}(A;u)=\\lim_{T\\to \\infty}\\frac{1}{T^2}\n\\int_0^T\\int_0^T\\left|\\sum_{a\\in A} w_aa^{it_1}(a+u)^{it_2}\\right|^{2k}dt_1dt_2.\\]\n\\end{Lemma}\n\\begin{proof}\nExpanding, the double integral on the right hand side is equal to\n\\begin{multline*}\n\\sum_{a_1,\\ldots,a_k\\in A}\\sum_{b_1,\\ldots,b_k\\in A}w_{a_1}\\cdots w_{a_k}w_{b_1}\\cdots w_{b_k}\\cdot\\\\\n\\cdot\\int_0^T (a_1\\cdots a_kb_1^{-1}\\cdots b_k^{-1})^{it_1}dt_1\\int_0^T((a_1+u)\\cdots (a_k+u)(b_1+u)^{-1}\\cdots (b_k+u)^{-1})^{it_2}dt_2.\\end{multline*}\nNow\n\\[\\frac{1}{T}\\int_0^T (u\/v)^{it}dt=\\begin{cases}1&\\text{ if }u=v,\n\\\\ O_{u,v}(T^{-1})&\\text{ if }u\\neq v.\\end{cases}\\]\nFrom this, the lemma follows.\n\\end{proof}\n\nLet $\\|\\cdot \\|_{2k}$ be the standard norm in $L^{2k}[0, T]^2$, normalised such that $\\| 1\\|_{2k} = 1$. So,\n\\[\n\\| f \\|_{2k} := \\left( \\frac{1}{T^2} \\int_0^T \\int_0^T|f(t)|^{2k} dt \\right)^{1\/2k}. \n\\]\n\n\\begin{Lemma} \\label{lem:split}\nLet $\\cJ$ be a set of integers and decompose it as $\\cJ=\\cJ_1 \\cup \\cdots \\cup\\cJ_N$. For each $j \\in \\cJ$ let $f_j:\\RR\\times \\RR\\to \\CC$ be a function belonging to $L^{2k}\\lr{\\RR^2}$ for every integer $k\\geq 2$. Then, for every integer $k \\geq 2$,\n\\begin{multline}\\label{split}\\lim_{T\\to \\infty}\\lr{\\frac{1}{T^2}\\int_0^T\\int_0^T\\left|\\sum_{j\\in\\cJ} f_j(t_1,t_2)\\right|^{2k}dt_1dt_2}^{1\/k} \\\\\\leq N\\sum_{n=1}^N \\lim_{T\\to \\infty}\\lr{\\frac{1}{T^2}\\int_0^T\\int_0^T\\left| \\sum_{j\\in \\cJ_n}f_j(t_1,t_2)\\right|^{2k}dt_1dt_2}^{1\/k}.\n\\end{multline}\n\\end{Lemma}\n\n\\begin{proof} \nIt suffices to prove the inequality for all sufficiently large $T$, which we assume fixed for now.\nThen\n\\begin{equation} \\label{eq:Lknormsum}\n\\lr{\\frac{1}{T^2}\\int_0^T\\int_0^T\\left|\\sum_{j\\in\\cJ} f_j(t_1,t_2)\\right|^{2k}dt_1dt_2}^{1\/k} =\n \\left(\\left\\| \\sum_{n=1}^N\\sum_{j \\in \\cJ_n}  f_j \\right\\|_{2k} \\right)^2\\leq\\left(\\sum_{n=1}^N\\left\\| \\sum_{j \\in \\cJ_n}  f_j \\right\\|_{2k} \\right)^2,\n\\end{equation}\nby the triangle inequality. By the Cauchy-Schwarz inequality, (\\ref{eq:Lknormsum}) is bounded by \n\\begin{equation}\nN\\sum_{n=1}^N\\left\\| \\sum_{j \\in \\cJ_n}  f_j \\right\\|_{2k}^2.\n\\end{equation}\nLetting $T \\to \\infty$ we get the claim of the lemma.\n\\end{proof}\n\\begin{Corollary}\\label{Split}\nLet $A$ be a finite set of rational numbers, partitioned as $A=A_1\\cup\\cdots\\cup A_N$, let $w$ be a set of non-negative weights, and let $u$ be a non-zero rational number. Then for any integer $k \\geq 2$\n\\[\\tilde E_{k,w}(A;u)^{1\/k}\\leq N\\sum_{j=1}^N\\tilde E_{k,w}(A_j;u)^{1\/k}.\\]\n\\end{Corollary}\nNow let $p$ be a fixed prime. For $a \\in \\mathbb Q$, let $v_p(a)$ denote the $p$-adic valuation of $a$. For a set $A$ of rational numbers and an integer $t$, we let \n$A_t=\\{a\\in A:v_p(a)=t\\}$.\n\n\\begin{Lemma} \\label{thm:basecase}\nLet $p$ be a prime number. Suppose $A$ is a finite set of rational numbers and let $u$ be a non-zero rational number.\nThen for any $w$, a set of non-negative weights on $A$, and any integer $k \\geq 2$,\n\\[\\tilde E_{k,w}(A;u)^{1\/k}\\leq 2\\binom{2k}{2}\\sum_{d\\in \\ZZ}\\tilde E_{k,w}(A_d;u)^{1\/k}.\\]\n\\end{Lemma}\n\n\n\\begin{proof}\nFirst, let $A=A_+\\cup A_-$ where $A_+=\\{a\\in A:v_p(a)\\geq v_p(u)\\}$ and $A_-=\\{a\\in A:v_p(a)<v_p(u)\\}$. By Corollary \\ref{Split}, we have\n\\begin{equation}\n\\tilde E_{k,w}(A;u)^{1\/k}\\leq 2\\tilde E_{k,w}(A_+;u)^{1\/k}+2\\tilde E_{k,w}(A_-;u)^{1\/k}.\n\\label{+-split}\n\\end{equation}\nThese two terms will be dealt with in turn, starting with $E_{k,w}(A_+;u)^{1\/k}$. \n\\begin{comment}Suppose we have a solution\n\\[(a_1+u)\\cdots(a_k+u)=(b_1+u)\\cdots(b_k+u)\\]\nand so\n\\begin{equation}\n(a_1u^{-1}+1)\\cdots(a_ku^{-1}+1)=(b_1u^{-1}+1)\\cdots(b_ku^{-1}+1).\n\\label{eq:normal}\n\\end{equation}\nWe claim that $v_p$ cannot have a unique minimal value among the $a_i$ and $b_i$. Indeed, since $v_p(a_iu^{-1})\\geq 0$ and $v_p(b_iu^{-1})\\geq 0$, expanding out both sides of \\eqref{eq:normal} and simplifying\n\\[u^{-1}(a_1+\\cdots+a_k)+\\text{higher terms}=u^{-1}(b_1+\\cdots+b_k)+\\text{higher terms},\\] \nand if $v_p$ achieves a unique minimum, then the left hand side and the right hand side are divisible by distinct powers of $p$, a contradiction.\n\\end{comment}\nTo do this, we first set up some more notation. For an integer $d$, define the function\n\\[f_d(t_1,t_2):= \\sum_{a \\in A_d} w_a a^{it_1} (a+u)^{it_2} .\\]\nThen, by Lemma \\ref{lem:direnergy}\n\\[\\tilde E_{k,w}(A_+;u) = \\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T \\left|\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right|^{2k} dt_1 dt_2 .\\]\nExpanding this expression, as in the proof of Lemma \\ref{lem:direnergy}, we obtain that $\\tilde E_{k,w}(A_+;u)$ is equal to\n\\begin{equation}\n\\sum_{d_1,\\dots,d_{2k} \\geq v_p(u)}\\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  f_{d_1}(t_1,t_2)\\cdots f_{d_k}(t_1,t_2) \\bar{f_{d_{k+1}}(t_1,t_2)}\\cdots \\bar{f_{d_{2k}}(t_1,t_2)} dt_1 dt_2 .\n\\label{eq:long}\n\\end{equation}\n\nFor fixed $d_1, \\dots, d_{2k}$, the quantity\n\\[\\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  f_{d_1}(t_1,t_2)\\cdots f_{d_k}(t_1,t_2) \\bar{f_{d_{k+1}}(t_1,t_2)}\\cdots \\bar{f_{d_{2k}}(t_1,t_2)} dt_1 dt_2 .\\]\ngives a weighted count of the number of solutions to the system of simultaneous equations\n\\begin{align}\na_1 \\cdots a_k &= a_{k+1} \\cdots a_{2k} \\label{basiceq1}\n\\\\ (a_1+u) \\cdots (a_k+u) &= (a_{k+1}+u) \\cdots (a_{2k}+u),\n\\label{basiceq2}\n\\end{align}\nsuch that $a_i \\in A_{d_i}$.\n\nWe claim that there are no solutions to \\eqref{basiceq2}, and thus also no solutions to the above system, if all of the $d_i$ are distinct. Indeed, suppose we have a solution\n\\[(a_1+u)\\cdots(a_k+u)=(a_{k+1}+u)\\cdots(a_{2k}+u)\\]\nand so\n\\begin{equation}\n(a_1u^{-1}+1)\\cdots(a_ku^{-1}+1)=(b_{k+1}u^{-1}+1)\\cdots(b_{2k}u^{-1}+1).\n\\label{eq:normal}\n\\end{equation}\n\nSince $v_p(a_iu^{-1})\\geq 0$, expanding out both sides of \\eqref{eq:normal} and simplifying gives\n\\begin{equation}\nu^{-1}(a_1+\\cdots+a_k)+\\text{higher terms}=u^{-1}(b_{k+1}+\\cdots+b_{2k})+\\text{higher terms}.\n\\label{eq:contradiction}\n\\end{equation} \nIf all of the $d_i$ are distinct, then there is some unique smallest $d_i$, and thus a unique smallest value of $v_p(a_i)$. But then the left hand side and the right hand side are divisible by distinct powers of $p$, a contradiction.\n\nSo returning to \\eqref{eq:long}, we need only consider the cases in which one or more of the $d_i$ are repeated. There are three kinds of ways in which this can happen.\n\n\\begin{enumerate}\n\\item $d_i=d_i'$ with $1 \\leq i \\leq k$ and $k+1 \\leq i' \\leq 2k$. There are $k^2$ possible positions for such a pair $(i,i')$,\n\\item $d_i=d_i'$ with $1 \\leq i, i' \\leq k$. There are $\\binom{k}{2}$ possible positions for such a pair $(i,i')$,\n\\item $d_i=d_i'$ with $k+1 \\leq i, i' \\leq 2k$. There are $\\binom{k}{2}$ possible positions for such a pair $(i,i')$.\n\\end{enumerate}\n\nSuppose we are in situation (1) above. Specifically, suppose that $d_1=d_{2k}$. The other $k^2-1$ cases can be dealt with by the same argument. Then these terms in \\eqref{eq:long} can be rewritten as\n\n\\begin{multline}\n\\sum_{d_1 \\geq v_p(u)}\\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  f_{d_1}(t_1,t_2)  \\bar{f_{d_1}(t_1,t_2)} \n\\\\ \\sum_{d_2,\\dots,d_{2k-1} \\geq v_p(u)} f_{d_2}(t_1,t_2) \\cdots f_{d_k}(t_1,t_2) \\bar{f_{d_{k+1}}(t_1,t_2)}\\cdots \\bar{f_{d_{2k-1}}(t_1,t_2)} dt_1 dt_2 \n\\\\= \\sum_{d \\geq v_p(u)}\\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  |f_{d}(t_1,t_2)   |^2 \\left |\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right |^{2(k-1)} dt_1 dt_2.\n\\end{multline}\n\nSuppose we are in situation (2). Specifically, suppose that $d_1=d_{2}$. The other $ \\binom{k}{2}-1$ cases can be dealt with by the same argument. Then these terms in \\eqref{eq:long} can be rewritten as\n\n\\begin{align*} & \\sum_{d_1 \\geq v_p(u)}  \\lim_{T\\to \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  f_{d_1}^2(t_1,t_2)  \\sum_{d_3,\\dots, d_{2k} \\geq v_p(u)} f_{d_3}(t_1,t_2) \\cdots f_{d_k}(t_1,t_2)  \\bar{f_{d_{k+1}}(t_1,t_2)} \\cdots \\bar{f_{d_{2k}}(t_1,t_2)} dt_1 dt_2 \n\\\\& \\leq  \\sum_{d \\geq v_p(u)} \\lim_{T\\to \\infty}\\frac{1}{T^2}\\int_0^T\\int_0^T   |f_d(t_1,t_2)|^2\\left|\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right|^{k-2} \\left|\\sum_{d} \\bar{f_d(t_1,t_2)}\\right|^{k}  dt_1dt_2\n\\\\&= \\sum_{d \\geq v_p(u)} \\lim_{T\\to \\infty}\\frac{1}{T^2}\\int_0^T \\int_0^T   |f_d(t_1,t_2)|^2\\left|\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right|^{2(k-1)}   dt_1dt_2.\n\\end{align*}\n\nThe same argument also works in case (3). Returning to \\eqref{eq:long}, we then have\n\\begin{align*}\n\\tilde E_{k,w}(A_+;u)& \\leq \n \\binom{2k}{2}\\sum_{d \\geq v_p(u)} \\lim_{T\\to \\infty}\\frac{1}{T^2}\\int_0^T \\int_0^T   |f_d(t_1,t_2)|^2\\left|\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right|^{2(k-1)}   dt_1dt_2\\\\\n&\\leq \\binom{2k}{2}\\sum_{d\\geq v_p(u)}\\tilde E_{k,w}(A_d;u)^{1\/k}E_{k,w}(A_+;u)^{1-1\/k},\n\\end{align*}\nthe last inequality being H\\\"older's. It therefore follows that\n\\begin{equation}\n\\tilde E_{k,w}(A_+;u)^{1\/k} \\leq \\binom{2k}{2} \\sum_{d\\geq v_p(u)}\\tilde E_{k,w}(A_d;u)^{1\/k}  .\n\\label{plusbound}\n\\end{equation}\nNow we proceed to $E_{k,w}(A_-;u)^{1\/k}$. For any solution to the pair of equations\n\\begin{align*}\na_1\\cdots a_k&=a_{k+1}\\cdots a_{2k}\\\\ (a_1+u)\\cdots (a_k+u)&=(a_{k+1}+u)\\cdots (a_{2k}+u)\n\\end{align*}\nwe have a solution to the equation\n\\[(1+ua_1^{-1})\\cdots (1+ua_k^{-1})=(1+ua_{k+1}^{-1})\\cdots (1+ua_{2k}^{-1}).\\]\nAgain, we expand and simplify, using this time that $v_p(ua_i^{-1})$ is positive, and get\n\\[u(a_1^{-1}+\\cdots a_k^{-1})+\\text{higher terms}=u(a_{k+1}^{-1}+\\cdots a_{2k}^{-1})+\\text{higher terms}.\\]\nAs in the previous case\n\\footnote{Note that here we have used the information that $a_1 \\cdots a_k=a_{k+1} \\cdots a_{2k}$, whereas we did not use this when bounding $\\tilde E_{k,w}(A_+;u)$.}\n, we cannot have a unique smallest $v_p(ua_i^{-1})$. We can therefore repeat the arguments that gave us \\eqref{plusbound} in order to deduce that\n\\begin{equation}\n\\tilde E_{k,w}(A_-;u)^{1\/k} \\leq \\binom{2k}{2} \\sum_{d < v_p(u)}\\tilde E_{k,w}(A_d;u)^{1\/k}  .\n\\label{minusbound}\n\\end{equation}\nInserting \\eqref{plusbound} and \\eqref{minusbound} into \\eqref{+-split} completes the proof.\n\\end{proof}\n\n\n\nNext, this is used as a base case to give an analogous result with more primes.\n\n\\begin{Lemma} \\label{thm:chang1}\nLet $p_1,\\dots,p_K$ be a prime numbers. Suppose $A$ is a finite set of rational numbers and let $u$ be a non-zero rational number. For a vector $\\dd=(d_1,\\dots,d_K)$, define\n\\[A_{\\dd}=\\{a\\in A:v_{p_1}(a)=d_1, \\dots, v_{p_k}(a)=d_k\\}.\\]\nThen for any $w$, a set of non-negative weights on $A$, and for any integer $k \\geq 2$,\n\\[\\tilde E_{k,w}(A;u)^{1\/k}\\leq \\left(2\\binom{2k}{2} \\right)^K\\sum_{\\dd\\in \\ZZ^K}\\tilde E_{k,w}(A_{\\dd};u)^{1\/k}.\\]\n\\end{Lemma}\n\n\\begin{proof}\n\n\nThe aim is to prove that\n\\begin{multline}\n\\lim_{T\\to \\infty}\\left (\\frac{1}{T^2}\\int_0^T \\int_0^T\\left|\\sum_{\\dd\\in \\mathbb Z^K} \\sum_{a \\in A_{\\dd}}w_a a^{it_1}(a+u)^{it_2} \\right|^{2k}dt_1 dt_2 \\right)^{1\/k}\n\\\\ \\leq \\left(2\\binom{2k}{2} \\right)^K\\sum_{\\dd \\in \\mathbb Z^K}\\lim_{T\\to \\infty}\\lr{\\frac{1}{T^2}\\int_0^T \\int_0^T\\left| \\sum_{a \\in A_{\\dd}} w_a a^{it_1}(a+u)^{it_2}    \\right|^{2k}dt_1 dt_2}^{1\/k}.\n\\end{multline}\n\nWe proceed by induction on $K$, the base case $K=1$ being given by Lemma \\ref{thm:basecase}. Then\n\\begin{align*}\n&\\lim_{T\\to \\infty} \\left(\\frac{1}{T^2} \\int_0^T \\int_0^T \\left|\\sum_{\\dd \\in \\mathbb Z^K }\\sum_{a \\in A_{\\dd}}w_a a^{it_1}(a+u)^{it_2} \\right|^{2k} dt_1dt_2\\right)^{1\/k}\n\\\\&=\\lim_{T\\to \\infty} \\left(\\frac{1}{T^2} \\int_0^T \\int_0^T  \\left|\\sum_{d_K\\in \\mathbb Z} \\left(\\sum_{\\dd'\\in \\mathbb Z^{K-1} }\\sum_{a \\in A_{(\\dd',d)}}w_a a^{it_1}(a+u)^{it_2} \\right) \\right|^{2k} dt_1 dt_2\\right)^{1\/k}\n\\\\&\\leq 2\\binom{2k}{2}  \\sum_{d_K\\in \\mathbb Z} \\lim_{T\\to \\infty} \\left(\\frac{1}{T^2} \\int_0^T \\int_0^T \\left|\\sum_{\\dd'\\in \\mathbb Z^{K-1} }\\sum_{a \\in A_{(\\dd',d)}}w_a a^{it_1}(a+u)^{it_2}  \\right|^{2k} dt_1 dt_2\\right)^{1\/k}\n\\\\& \\leq 2\\binom{2k}{2}  \\sum_{d_K\\in \\mathbb Z} \\left( 2\\binom{2k}{2}  \\right)^{K-1} \\sum_{\\dd'\\in \\mathbb Z^{K-1} } \\lim_{T\\to \\infty} \\left(\\frac{1}{T^2} \\int_0^T \\int_0^T \\left| \\sum_{a \\in A_{(\\dd',d)}}w_a a^{it_1}(a+u)^{it_2}  \\right|^{2k} dt_1 dt_2\\right)^{1\/k}\n\\\\&=\\left( 2\\binom{2k}{2}  \\right)^{K} \\sum_{\\dd \\in \\mathbb Z^K}\\lim_{T\\to \\infty}\\lr{\\frac{1}{T^2}\\int_0^T \\int_0^T\\left| \\sum_{a \\in A_{\\dd}}w_a a^{it_1}(a+u)^{it_2}  \\right|^{2k}dt_1 dt_2}^{1\/k}.\n\\end{align*}\nThe first inequality above follows from an application of Lemma \\ref{thm:basecase}. The second inequality follows from the induction hypothesis.\n\\end{proof}\n\n\n\n\n\n\n\\begin{comment}\nIn particular, this implies the following version of Chang's Theorem, giving a strong estimate for the energy $\\tilde E_{k,w}(A)$ in the case when the multiplicative doubling constant $K$ is of size $c \\log |A|$.\n\n\\begin{Theorem} \\label{thm:chang} Let $A \\subset \\mathbb Q$ be finite with $|AA|=K|A|$. Then, for any non-zero rational\n\n\\end{Theorem}\n\\end{comment}\n\n\\subsection{Separating constants} \\label{sec:separatingconstants}\n\nThe following definition, which follows the terminology used in \\cite{Z}, is central to this paper. Let $\\psi$ be an arbitrary real number. A set $X \\subset \\mathbb Q$ is said\\footnote{Strictly speaking, we should perhaps include $k$ in this definition and say that a set is $(\\psi,k)$-separating if it satisfied the stated conditions. In order to simplify the notation we do not do this. Instead, we can think of $k \\geq 2$ as a fixed integer throughout the remainder of the paper, unless stated otherwise. Henceforth, this condition on $k$ will be omitted from statements of results.} to be \\textit{$\\psi$-separating} if for any non-zero $u \\in \\mathbb Q$, any set finite $Z \\subset \\mathbb Q$ of the form\n\\[Z=\\bigcup_{x \\in X} xY_x \\]\nsuch that $(x,Y_{x'}) =1$ for all $x,x' \\in X$, and any set of weights $w$ on $Z$\n\\[\\tilde E_{k,w}(Z;u)^{1\/k}\\leq \\psi \\sum_{x \\in X}\\tilde E_{k,w}(xY_x;u)^{1\/k}.\\]\n\nA first observation about separating constants comes in the form of the following claim.\n\\begin{Claim} \\label{cor:trivialseparation}\nAny finite $A \\subset \\mathbb Q$ is $|A|$-separating.\n\\end{Claim}\nThis claim follows immediately from Corollary \\ref{Split}. Combining this new definition with Lemma \\ref{thm:basecase}, we can also record the following corollary.\n\n\\begin{Corollary} \\label{cor:oneprimeseparation}\nLet $p$ be a prime number. Suppose $A$ is of the form $A=\\{p^h : h \\in H \\}$ for some finite set $H \\subset \\mathbb Z$. Then $A$ is $2 \\binom{2k}{2}$-separating.\n\\end{Corollary}\n\nWith this definition of the separating constant, we can use Lemma \\ref{thm:chang1} to get a first bound for the separating constant of a set with small product set. Once again, this bound is good when $K \\leq c \\log |A|$ for a sufficiently small constant $c$. \n\nTo do this, we recall an argument of Chang \\cite{C} which uses Freiman's Lemma to show that a set of rationals with small product set is determined by a small number of prime factors. Let $A$ be a set of rationals and let \n\\[\\cP:= \\{p: p\\text{ is prime and there exists }  a \\in A, v_p(a) \\neq 0\\} = \\{p_1,\\ldots,p_t\\}\\] \nbe the set of primes dividing some element of $A$. Abusing notation slightly, we define a map $\\cP:A\\to\\ZZ^t$ where $\\cP(a)=(v_{p_1}(a),\\ldots,v_{p_t}(a))$. Denoting by $\\cP(X)$ the image of a set $X$ under $\\cP$, observe that $\\cP(AA)=\\cP(A)+\\cP(A)$. We define the \\textit{multiplicative dimension} of $A \\subset Q$ to be the least dimension of an affine space $L$ containing $\\cP(A)$.\n\n\\begin{Theorem}[Freiman's Lemma] \\label{thm:FreimanLemma}\nLet $A\\subset \\RR^m$ be a finite set not contained in a proper affine subspace. Then \\[|A+A|\\geq (m+1)|A|-O_m(1).\\]\n\\end{Theorem}\n\n\\begin{Theorem} \\label{thm:chang2} Let $A \\subset \\mathbb Q$ be finite with $|AA|=K|A|$. Then, $A$ is $\\left(2\\binom{2k}{2} \\right)^K$-separating. \n\\end{Theorem}\n\n\\begin{proof} It follows from Freiman's Lemma that if $|AA|\\leq K|A|$ with $|A|$ sufficiently large, then $A$ has multiplicative dimension at most $K$.\n\nThis means that there is a set of $\\{p_1,\\dots,p_K\\}$ of primes and a set of vectors $\\cJ \\subset \\mathbb Z^K$ such that\n\\[A= \\bigcup_{\\jj=(j_1,\\dots,j_K) \\in \\cJ} p_1^{j_1}\\cdots p_K^{j_K} x_{\\jj}, \\]\nwhere each $x_{\\jj}$ is a rational number coprime\\footnote{We say that two rational numbers $a$ and $b$ are comprime if at least one of $v_p(a)$ and $v_p(b)$ is zero for all prime $p$. As with the case of integers, we write $(a,b)=1$.} to $p_1\\cdots p_K$. For $\\jj=(j_1,\\dots,j_K) \\in \\cJ$, write $a_{\\jj}=p_1^{j_1}\\cdots p_K^{j_K} x_{\\jj}$.\n\nNow, let\n\\[Z= \\bigcup_{\\jj \\in \\cJ} a_{\\jj}Y_{\\jj}  \\subset \\mathbb Q \\]\nwith the $(Y_{\\jj}, a_{\\jj'})=1$ for all $\\jj,\\jj' \\in \\cJ$. In particular, $Y_{\\jj}$ is coprime to $p_1\\cdots p_K$. Therefore, in the notation of Lemma \\ref{thm:chang1}\n\\[Z_{\\jj}= a_{\\jj} Y_{\\jj}. \\]\nThen, by Lemma \\ref{thm:chang1},\n\\[\\tilde E_{k,w}(Z;u)^{1\/k}\\leq \\left(2\\binom{2k}{2} \\right)^K\\sum_{\\jj\\in \\ZZ^K}\\tilde E_{k,w}(a_{\\jj}Y_{\\jj};u)^{1\/k}.\\]\n\n\n\\end{proof}\n\nWe recall now the Pl\\\"{u}nnecke-Ruzsa Theorem. See \\cite{P} for a simple inductive proof. Following convention, we state it using additive notation, although it will be used in the multiplicative setting.\n\n\\begin{Theorem} \\label{thm:Plun} Let $A$ be a subset of a commutative additive group $G$ with $|A+A| \\leq K|A|$. Then for any $h \\in \\mathbb N$,\n\\[ |hA| \\leq K^h|A|. \\]\n\\end{Theorem}\n\nOne may think of the separating constant of $X$ as a generalisation of the notion of the mixed energy $\\tilde E_k(X;u)$. Indeed, if $X$ is $\\psi$-separating then take $Y_x=\\{1\\}$ for all $x \\in X$ and $w(x)=1$ for all $x \\in X$. Then it follows that $\\tilde E_k(X;u) \\leq \\psi^k|X|^k$. In particular, Theorem \\ref{thm:chang2} implies the following result.\n\n\\begin{Theorem} \\label{thm:chang3} Let $A$ be a finite set of rational numbers and let $u \\in \\mathbb Q$ be non-zero. Suppose that $|AA| \\leq K|A|$. Then, for any integer $k \\geq 2$,\n\\[|(A+u)^k| \\geq \\frac{|A|^{k-1}}{K^k\\left( 2 \\binom{2k}{2} \\right)^{Kk}}\n\\]\n\\end{Theorem}\n\n\\begin{proof}\nBy Theorem \\ref{thm:chang2}, \n\\[ \\tilde E_k(A;u) \\leq \\left(2\\binom{2k}{2} \\right)^{Kk}|A|^k.\\] \nAlso, by Theorem \\ref{thm:Plun}, $|A^{(k)}| \\leq K^k|A|$. Inserting these two bounds into Lemma \\ref{lem:CSbasic} completes the proof.\n\\end{proof}\n\nA stronger version of Theorem \\ref{thm:chang3} was the main result of \\cite{HRNZ}, which used the standard $k$-fold multiplicative energy of the set $A+u$. \n\nA key goal of this paper is to amplify this approach in order to give a good bound for the case when $K=|A|^{\\eps}$. The advantage of working with this generalised notion of energy is that it has a crucial ``chaining property\" which will be important in the forthcoming analysis for pushing to get results for larger $K$.\n\n\\begin{Lemma} \\label{lem:chain} Let $A$ be a finite set of rationals which can be decomposed as a disjoint union\n\\[A= \\bigcup_{b \\in B} bC_b\\]\nand with $(b,C_{b'})=1$ for all $b,b' \\in B$. Assume also that $B$ is $\\psi_1$-separating and that each $C_b$ is $\\psi_2$-separating. Then $A$ is $\\psi_1\\psi_2$-separating.\n\\end{Lemma}\n\n\\begin{proof}\nLet $Z$ be a set of rationals which decomposes as\n\\[Z = \\bigcup_{a \\in A} aY_a \\]\nwith $(a,Y_{a'})=1$ for all $a,a' \\in A$. Then for any $u \\in \\mathbb Q$ and weights $w$ on $Z$,\n\\[ \\tilde E_{w,u}(Z;u)^{1\/k} = \\tilde E_{w,u} \\left (\\bigcup_{b \\in B} b \\left ( \\bigcup_{c \\in C_b} cY_{bc} \\right ) ;u \\right )^{1\/k} \\leq  \\psi_1 \\sum_{b \\in B} \\tilde E_{w,u} \\left ( b \\left ( \\bigcup_{c \\in C_b} cY_{bc} \\right );u \\right )^{1\/k}. \\]\nIn the inequality above we have used the fact that $B$ is $\\psi_1$ separating and that \n\\[ \\left (b,\\bigcup_{c \\in C_{b'}} cY_{b'c}\\right )=1\\]\nfor any $b,b' \\in B$. Indeed, take an arbitrary product $cy$ with $c \\in C_{b'}$ and $y \\in Y_{b'c}$. Then $b$ is coprime to $c$ by the hypothesis of the lemma. Also, $y$ is coprime to each element of $A$ by the definition of $Z$, which implies that $y$ is coprime to $b$ by the hypothesis of the lemma.\n\nWe therefore have\n\\begin{align*} \\tilde E_{w,u}(Z;u)^{1\/k} \\leq \\psi_1 \\sum_{b \\in B} \\tilde E_{w,u}  \\left ( \\bigcup_{c \\in C_b} c(bY_{bc}) ;u \\right ) ^{1\/k} & \\leq \\psi_1 \\sum_{b \\in B} \\psi_2 \\sum_{c \\in C_b} \\tilde E_{w,u}  \\left (  c(bY_{bc}) ;u \\right ) ^{1\/k}\n\\\\&= \\psi_1 \\psi_2 \\sum_{a \\in A} \\tilde E_{w,u}  \\left (  aY_a ;u \\right ) ^{1\/k}.\n\\end{align*}\nThe inequality above uses fact that $C_b$ is $\\psi_2$-separating and that $(c,bY_{bc'})=1$ for any $c,c' \\in C_b$. This can be verified in the same way as the previous inequality.\n\\end{proof}\n\n\\section{Lambda-constants} \\label{sec:lambda}\n\nWe will soon begin the process of amplifying Theorem \\ref{thm:chang2} from the previous section in order to get a better separating factor which leads to strong bound when $K=|A|^{\\eps}$. At the conclusion of this process we will prove the following result.\n\n\\begin{Theorem} \\label{thm:goodsubset} Given $0<\\tau, \\gamma < 1\/2$, there exist positive constants $C_1=C_1(\\tau,\\gamma,k)$ and $C_2=C_2(\\tau,\\gamma,k)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$, there exists $A' \\subset A$ with $|A'| \\geq K^{-C_1}|A|^{1-\\tau}$ such that $A'$ is $K^{C_2}|A|^{\\gamma}$-separating.\n\\end{Theorem}\n\nIn fact, one can check that the proof of Theorem \\ref{thm:goodsubset} goes through in a more general setting. Let $S: 2^{\\mathbb{Q}} \\to \\mathbb{R}$ be a function defined on rational sets with the following properties:\n\n\\begin{enumerate}\n   \\item (Trivial bound) For an arbitary set $A \\subset \\mathbb{Q}$ \n   $$\n   S(A) \\leq |A|.\n   $$\n   \n   \\item (Stability) If $A' \\subset A$ then\n   $$\n   S(A') \\leq S(A).\n   $$\n   \n   \\item ($p$-adic separation) There is an absolute constant $s \\geq 0 $ such that for any prime $p$ and $I \\subset \\mathbb{Z}$,\n   \n   $$\n    S(\\bigcup_{i \\in I} p^i) \\leq s.\n   $$\n   \n   \\item (Nesting) Let $A \\subset \\mathbb{Q}$ and $\\{ B_a \\}_{a \\in A}$ is a collection of sets such that $(a, B_{a'}) = 1$ for any $a, a' \\in A$. Further assume that $aB_a, a \\in A$ are pairwise disjoint. \n   Then\n   $$\n   S(\\bigcup_{a \\in A} aB_a) \\leq S(A) \\max_{a \\in A} S(aB_a).\n   $$\n\\end{enumerate}\n\nNote that our definition of the separating constant satisfies (1)-(4). \n\n\\begin{Theorem} \\label{thm:goodsubset_general} Let $S$ be a function with the properties above. Given $0<\\tau, \\gamma < 1\/2$, there exist positive constants $C_1=C_1(\\tau,\\gamma, s)$ and $C_2=C_2(\\tau,\\gamma, s)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$, there exists $A' \\subset A$ with $|A'| \\geq K^{-C_1}|A|^{1-\\tau}$ such that \n$$\nS(A') \\leq K^{C_2}|A|^{\\gamma}.\n$$\n\\end{Theorem}\n\nThe proof of Theorem~\\ref{thm:goodsubset}  is essentially borrowed from \\cite{BC}. We present here a  proof adapted to our setting to make the paper self-contained. The same proof applies to Theorem~\\ref{thm:goodsubset_general} with cosmetic modifications, but we expect that it might be useful to have such a general `black-box' version for future use.\n\nBefore we begin the lengthy proof of Theorem \\ref{thm:goodsubset}, we will take some time to see how it implies the two main theorems of this paper. To do this, it will be convenient to use the language of \\textit{$\\Lambda$-constants}, and to introduce some of their key properties. The main motivation behind  $\\Lambda$-constants is the stability property given by the forthcoming Corollary~\\ref{corr:stability}, which is absent in the non-weighted version of the energy. \n\nWe also encourage the interested reader to consult our preceding paper \\cite{HRNZ} for a slightly more gentle introduction to $\\Lambda$-constants in the setting of Dirichlet polynomials and more in-depth motivation behind this concept. \n\n\nLet $A \\subset \\mathbb Q$ be a finite set and let $u$ be a non-zero rational. Define\n\\[\\Lambda_k(A;u):= \\max \\tilde E_{k,w}(A;u)^{1\/k},\\]\nwhere the maximum is taken over all weights $w$ on $A$ such that\n\\begin{equation}\n\\sum_{a \\in A} w(a)^2 =1. \n\\label{eq:weightsnorm}\n\\end{equation}\nAn equivalent definition is\n\\[\\Lambda_k(A;u):= \\max \\lim_{T\\to \\infty}\n\\left \\|\\sum_{a\\in A}w_aa^{it_1}(a+u)^{it_2} \\right \\|_{2k}^2.\\]\nwhere the maximum is taken over the same range of weights.\n\n\n\n\\begin{Lemma} \\label{lm:anyweights}\n Let $A \\subset \\mathbb Q$ be a finite set with some non-negative real weights $w_a$ assigned to each element $a \\in A$ and let $u$ be a non-zero rational. Then\n \\begin{equation} \\label{eq:lambdaanyweights}\n \\left \\| \\sum_{a \\in A} w_a a^{it_1}(a+u)^{it_2} \\right\\|^2_{2k} \\leq \\Lambda_k(A;u) \\left(\\sum_{a \\in A} w_a^2 \\right) + o_{T\\to\\infty}(1).\n \\end{equation}\n\\end{Lemma}\n\\begin{proof}\nIf $\\sum_{a \\in A} w^2_a = 0$ the claim of the lemma is trivial. Otherwise, define new weights\n\\[\nw'_a := \\frac{w_a}{(\\sum_{a \\in A} w_a^2)^{1\/2}}\n\\]\nwhich satisfy (\\ref{eq:weightsnorm}). It thus suffices to show that\n\\[\n\\left \\| \\sum_{a \\in A} w'_a a^{it_1}(a+u)^{it_2} \\right\\|^2_{2k} \\leq \\Lambda_k(A;u) + o_{T\\to\\infty}(1),\n\\]\nwhich is a straightforward consequence of our definition of $\\Lambda_k(A;u)$.\n\\end{proof}\n\nWe will use the following stability property of $\\Lambda$-constants which helps us to work with subsets.\n\n\\begin{Corollary} \\label{corr:stability}\nSuppose that $A \\subset \\mathbb Q$, that $u$ is a non-zero rational and $A' \\subset A$. Then\n\\[\n\\Lambda_k(A'; u) \\leq \\Lambda_k(A; u).\n\\]\nIn particular,\n\\[\n\\tilde E^{1\/k}_k(A';u) \\leq \\Lambda_k(A;u)|A'|.\n\\]\nand\n\\[\n\\tilde E_k(A;u) \\leq \\Lambda^k_k(A;u)|A|^k.\n\\]\n\\end{Corollary}\n\\begin{proof}\n The first claim follows from the observation that any set of weights $\\{w_a \\}_{a \\in A'}$ with $\\sum w^2_a = 1$ can be trivially extended to a set of weights $\\{ w_a \\}_{a \\in A}$ by assigning zero weight to the elements in $A \\setminus A'$. Next observe that $E_k$ is just $E_{k, w}$ with all the weights being one and apply Lemma~\\ref{lm:anyweights}.\n\\end{proof}\n\nThe next lemma records that any set with small separating factors also has a small $\\Lambda$-constant.\n\\begin{Lemma} \\label{lem:lambdasep}\nLet $A \\subset \\QQ$ be $\\psi$-separating. Then for any $u \\in \\QQ \\setminus \\{0\\}$\n\\[ \\Lambda_k(A;u) \\leq \\psi .\\]\n\\end{Lemma}\n\n\\begin{proof} Let $w$ be any set of weights on $A$ that satisfy \\eqref{eq:weightsnorm}. Write\n\\[A= \\bigcup_{a \\in A} a Y_a \\]\nwith $Y_a= \\{1\\}$ for all $a \\in A$. Then by the definition of $\\psi$-separating, it follows that for any non-zero $u \\in \\QQ$\n\\[ \\tilde E_{k,w}(A;u) ^{1\/k} \\leq  \\psi \\sum_{a \\in A} \\tilde E_{k,w} (\\{a \\};u)^{1\/k} = \\psi \\sum_{a \\in A} (w(a)^{2k})^{1\/k} = \\psi.\\]\n\\end{proof}\n\n\n\\begin{Lemma} \\label{lem:subset}\nLet $A \\subset \\mathbb Q$ be a finite set with $|AA| \\leq K|A|$ and let $u$ be a non-zero rational number. Suppose that $A' \\subset A$ and $A'$ is $\\psi$-separating. Then\n\\[ \\Lambda_k(A;u) \\leq K^4 \\left(\\frac{|A|}{|A'|-1}\\right)^2 \\psi. \\]\n\\end{Lemma}\n\n\\begin{proof}\n\nLet $w$ be an arbitrary set of weights on $A$ such that $\\sum_{a \\in A} w(a)^2 =1$. We seek a suitable upper bound for\n\\[ \\left\\|  \\sum_{a\\in A}w_aa^{it_1}(a+u)^{it_2} \\right \\|_{2k}^2 .\\]\nFor a fixed $z \\in A\/A'$, define a set of weights $w^{(z)}$ on $zA'$ by taking $w^{(z)}(za')=w(za')$ if $za' \\in A$ and $w^{(z)}(za')=0$ otherwise. Define \n\\[R_{(A\/A'),A'}(x):=|\\{(s,a) \\in (A\/A') \\times A' : sa=x \\}| \\]\nand note that $R_{(A\/A'),A'}(x) \\geq |A'|-1$ for all $x \\in A$. This is because, for all non-zero $a' \\in A'$, $x=(\\frac{x}{a'})a'$. Therefore,\n\\begin{align*} \\left\\| \\sum_{z \\in A\/A'} \\sum_{a' \\in A'} w^{(z)}(za')(za')^{it_1}(za'+u)^{it_2} \\right \\|_{2k}\n&= \\left\\| \\sum_{a \\in A}R_{(A\/A'),A'} (a) w(a)a^{it_1}(a+u)^{it_2} \\right \\|_{2k}\n\\\\& \\geq |A'| \\left\\|  \\sum_{a\\in A}w_aa^{it_1}(a+u)^{it_2} \\right \\|_{2k}.\n\\end{align*}\nOn the other hand, by the triangle inequality and Lemma \\ref{lm:anyweights}\n\\begin{align*}\n\\left\\| \\sum_{z \\in A\/A'} \\sum_{a' \\in A'} w^{(z)}(za')(za')^{it_1}(za'+u)^{it_2} \\right \\|_{2k}\n& \\leq \\sum_{z \\in A\/A'} \\left\\|  \\sum_{a' \\in A'} w^{(z)}(za')(za')^{it_1}(za'+u)^{it_2} \\right \\|_{2k}\n\\\\& \\leq  \\sum_{z \\in A\/A'} \\Lambda_k(zA'; u)^{1\/2} + o_{T \\rightarrow \\infty}(1).\n\\end{align*}\nSince $A'$ is $\\psi$-separating, it follows from Lemma \\ref{lem:lambdasep} that $\\Lambda_k(zA';u) = \\Lambda_k(A';u\/z) \\leq \\psi $. We also have\n\\[|A\/A'| \\leq |A\/A| \\leq \\frac{|AA|^2}{|A|} \\leq K^2 |A|, \\]\nby the Ruzsa Triangle Inequality (see \\cite{TV}). It therefore follows that\n\\[ \\left\\|  \\sum_{a\\in A}w_aa^{it_1}(a+u)^{it_2} \\right \\|_{2k} \\leq K^2 \\left( \\frac{|A|}{|A'|-1} \\right ) \\psi^{1\/2} + o_{T \\rightarrow \\infty} (1),\n\\]\nand the result follows.\n\\end{proof}\n\nCombining this with Theorem \\ref{thm:goodsubset} gives the following, which is our main result concerning $\\Lambda$-constants.\n\n\\begin{Theorem} \\label{thm:lambda}\nGiven $0<\\gamma < 1\/2$, there exists a positive constants $C=C(\\gamma, k)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$ and any non-zero rational $u$,\n\\[ \\Lambda_k(A;u) \\leq K^C|A|^{\\gamma} .\\]\n\\end{Theorem}\n\n\\section{Concluding the proofs} \\label{sec:conclusion}\n\nIn this section we conclude the proof of Theorem \\ref{thm:mainmain}, which is the main theorem of this paper, and Theorem~\\ref{thm:BS_almost_subgroups} announced in the introduction. Both theorems are restated below for the convenience of the reader.\n\n\\begin{Theorem} \\label{thm:mainmainagain}\nFor all $b \\in \\mathbb Z$, there exists $k=k(b)$ such that for any finite set $A \\subset \\mathbb Q$ and any non-zero rational $u$,\n\\[  \\max \\{ |A^{(k)}|, |(A+u)^{(k)}| \\} \\geq |A|^b  \\]\n\\end{Theorem}\n\n\\begin{proof} Fix $b$ and assume that\n\\[|A^{(k)}| < |A|^b \\]\nfor some sufficiently large $k=2^l$. The value of $l$ (and thus also that of $k$) will be specified at the end of the proof.\nSince $|A^{(2^l)}| < |A|^b$, it follows that\n\\[ \\frac{|A^{(2^l)}|}{|A^{(2^{l-1})}|} \\frac{|A^{(2^{l-1})}|}{|A^{(2^{l-2})}|} \\cdots \\frac{|A^{(2)}|}{|A|} < |A|^{b-1}\\]\nand thus there is some integer $l_0 \\leq l$ such that\n\\[ \\frac{|A^{(2^{l_0+1})}|}{|A^{(2^{l_0})}|} < |A|^{\\frac{b-1}{l}} .\\]\nTherefore, writing $k_0=2^{l_0}$ and $B=A^{(k_0)}$, we have\n\\[|BB| < |B| |A|^{\\frac{b-1}{l}}. \\]\nAlso, for any non-zero $\\lambda \\in \\mathbb Q$, $|(\\lambda B)(\\lambda B)| <|B||A|^{\\frac{b-1}{l}}$. Therefore, by Theorem \\ref{thm:lambda},\n\\[ \\Lambda_h(\\lambda B;u) \\leq |A|^{C\\frac{b-1}{l}} |B|^{\\gamma} \\leq |A|^{C\\frac{b-1}{l}+ \\gamma b}  \\]\nwhere $C=C(h, \\gamma)$ and $h, \\gamma$ will be specified later.\n\nNow, for some $\\lambda \\in \\mathbb Q$, we have $A \\subset \\lambda B$, and thus by Corollary \\ref{corr:stability} and Lemma \\ref{lem:CSbasic}\n\\[ \\frac{|A|^2}{\\max \\{|A^{(h)}|,|(A+u)^{(h)}| \\}^{2\/h}} \\leq \\tilde E_h^{1\/h}(A;u) \\leq |A| \\Lambda_h(\\lambda B;u) \\leq |A|^{1+C\\frac{b-1}{l}+ \\gamma b} .\\]\nThis rearranges to\n\\[ \\max \\{|A^{(h)}|,|(A+u)^{(h)}| \\} \\geq |A|^{\\frac{h}{2}(1-C\\frac{b-1}{l}- \\gamma b)} .\\]\nChoose $\\gamma=1\/100b$ and $h=4b$. Then $C=C(h,\\gamma)=C(b)$ and we have\n\\[ \\max \\{|A^{(h)}|,|(A+u)^{(h)}| \\} \\geq |A|^{\\frac{h}{2}(99\/100-C(b)\\frac{b-1}{l})} .\\]\nThen choose $l=(b-1)4C$ to get\n\\[ \\max \\{|A^{(h)}|,|(A+u)^{(h)}| \\} \\geq |A|^{\\frac{h}{4}}=|A|^b .\\]\nNote that the choice of $l$ depends only on $b$ and thus $k=2^{4C(b-1)}=k(b)$.\nIn particular, since $k > h$, we conclude that\n\\[ \\max \\{|A^{(k)}|,|(A+u)^{(k)}| \\} \\geq |A|^b ,\\]\nas required. \n\n\\end{proof}\n\nTheorem \\ref{thm:lambda} also implies Theorem \\ref{thm:main1}. The statement is repeated below for the convenience of the reader.\n\n\\begin{Theorem}\nGiven $0<\\gamma < 1\/2$ and any integer $k \\geq 2$, there exists a positive constant $C=C(\\gamma, k)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$ and any non-zero rational $u$,\n\\[ | (A+u) ^{(k)}|  \\geq \\frac{|A|^{k(1-\\gamma)-1}}{K^{Ck}}.\\]\n\\end{Theorem}\n\n\\begin{proof} Define $w(a)= 1 \/ |A|^{1\/2}$ for all $a \\in A$ and note that \\eqref{eq:weightsnorm} is satisfied. Furthermore, for this set of weights $w$,\n\\begin{equation}\n\\tilde E_{k,w}(A;u) = \\frac{\\tilde E_k(A;u)}{|A|^{k}} \\geq \\frac{|A|^{k}}{|A^{(k)}||(A+u)^{(k)}|},\n\\label{eq:trivialweights}\n\\end{equation}\nwhere the inequality comes from Lemma \\ref{lem:CSbasic}. It follows from Theorem \\ref{thm:lambda} that there exists a constant $C=C(\\gamma,k)$ such that for any $u \\in \\mathbb Q \\setminus \\{0\\}$, $\\Lambda_{k}(A;u) \\leq K^C|A|^{\\gamma}$. Consequently, by the definition of $\\Lambda_{k}(A;u)$,\n\\[ \\tilde E_{k,w}(A;u) \\leq K^{Ck}|A|^{\\gamma k}. \\]\nCombining this with \\eqref{eq:trivialweights}, it follows that\n\\begin{equation}\n|A^{(k)}||(A+u)^{(k)}| \\geq \\frac{|A|^{k(1-\\gamma)}}{K^{Ck}}.\n\\label{eq:energybound}\n\\end{equation}\nFinally, since $|AA| \\leq K|A|$, it follows from the Pl\\\"{u}nnecke-Ruzsa Theorem that $|A^{(k)}| \\leq K^k|A|$. Inserting this into \\eqref{eq:energybound} completes the proof.\n\n\n\n\n\n\\end{proof}\n\n\nWe now turn to the proof of Theorem~\\ref{thm:BS_almost_subgroups}. Recall its statement.\n\n\\begin{Theorem} For any $\\gamma > 0$ there is $C(\\gamma) > 0$ such that for any $K$-almost subgroup $A \\subset \\mathbb{Q}$ and fixed non-zero $c_1, c_2 \\in \\mathbb{Q}$ the number $A(2, K)$ of solutions $(x_1, x_2) \\in A^2$ to \n $$\n c_1x_1 + c_2 x_2 = 1\n $$\n is bounded by \n $$\n A(2, K) \\leq |A|^\\gamma K^C.\n $$\n\n\\end{Theorem}\n\\begin{proof}\n\tLet $S \\subset A$ be the set of $x_1 \\in A$ such that $c_1x_1 + c_2x_2 = 1$ for some $x_2 \\in A$. Since the projection $(x_1, x_2) \\to x_1$ is injective, it suffices to bound the size of $S$.\n    \n    Since $S \\subset A$, by Theorem~\\ref{thm:lambda} and\n    Corollary~\\ref{corr:stability} for any non-zero $u$ \n$$\n\\tilde E_k(S; u) \\leq K^{k C(\\gamma', k)}|A|^{k\\gamma'} |S|^k\n$$\nwith the parameters $0 < \\gamma' < 1\/2, k \\geq 2$ to be taken in due course.\n\nIn particular, by Lemma~\\ref{lem:CSbasic}\n$$\n|S|^k \\leq \\left(K^{k C(\\gamma', k)}|A|^{k\\gamma'} |S|^k \\right)^{1\/2} \\max \\{ |S^k|, |(S-1\/c_1)^k| \\}.\n$$\n\nOn the other hand, $S \\subseteq A$ and $(S - 1\/c_1) \\subseteq (c_2\/c_1)A$, so by the Pl\\\"{u}nnecke-Ruzsa inequality\n$$\n\\max \\{ |S^k|, |(S-1\/c_1)^k| \\} \\leq |A^{(k)}| \\leq K^k|A|.\n$$\n\nWe then have\n$$\n|S| \\leq |A|^{\\gamma' + 2\/k} K^{C + 2},\n$$\nand taking $k = \\lfloor 2\/\\gamma' \\rfloor + 1$ and $\\gamma'= \\gamma\/2$, the claim follows.\n\n\\end{proof}\n\n\\section{Graph Fibering} \\label{sec:fibers}\nSuppose $Z_1$ and $Z_2$ abelian groups, with finite subsets $A,B\\subset Z_1\\times Z_2$. We will write $z_1\\oplus z_2$ for an element of $Z_1\\times Z_2$. We will write, for $x\\in X\\subset Z_1\\times Z_2$ with $\\pi_1(x)=x_1$, \\[X_2(x_1)=\\{x_2\\in\\pi_2(X):x_1\\oplus x_2\\in X\\}.\\] Suppose $G\\subset A\\times B$. Denote by $\\pi_1$ and $\\pi_2$ the projections onto the first and second coordinates of $Z_1\\times Z_2$ respectively. The set $G$ is interpreted as a bipartite graph on $A$ and $B$, and it can be decomposed into a union by considering the fibers of $\\pi_1$. Indeed, let \\[G_1=\\{(\\pi_1(a),\\pi_1(b)):(a,b)\\in G\\}\\] and for $(a_1,b_1)\\in G_1$, let \\[G_2(a_1,b_1)=\\{(a_2,b_2):(a_1\\oplus a_2,b_1\\oplus b_2)\\in G\\}\\subset \\pi_2(A)\\times\\pi_2(B).\\]\nRecall the notation \\[A+_G B=\\{a+b:(a,b)\\in G\\}.\\]\n\nOne of the primary reasons for decomposing a graph this way is that it behaves nicely with addition along the graph.\n\\begin{Lemma}\\label{FiberGraphSum}\nSuppose $A$ and $B$ are finite subsets of $Z_1\\times Z_2$. Then for $G\\subset A\\times B$ we have\n\\[|A+_G B|\\geq |\\pi_1(A)+_{G_1}\\pi_1(B)|\\min_{(a_1,b_1)\\in \\pi_1(A)\\times \\pi_1(B)}|A_2(a_1)+_{G_2(a_1,b_1)}B_2(b_1)|.\\]\n\\end{Lemma}\n\\begin{proof}\nWrite \n\\[A+_G B\\supseteq \\bigcup_{s\\in \\pi_1(A+_G B)} \\bigcup_{\\substack{(a_1\\oplus a_2,b_1\\oplus b_2)\\in G\\\\ a_1+b_1=s}}\\{(s\\oplus (a_2+b_2))\\}.\\]\nNext, from the observation that the first union above is disjoint, and the fact that $\\pi_1(A+_G B)=\\pi_1(A)+_{G_1}\\pi_1(B)$, we have\n\\[|A+_G B|\\geq\\sum_{s\\in \\pi_1(A)+_{G_1}\\pi_1(B)}\\big|\\bigcup_{\\substack{(a_1\\oplus a_2,b_1\\oplus b_2))\\in G\\\\ a_1+b_1=s}}\\{(s\\oplus (a_2+b_2))\\}\\big|.\\]\nSince, for fixed $a_1,b_1$,\n\\[\\bigcup_{\\substack{(a_1\\oplus a_2,b_1\\oplus b_2)\\in G\\\\ a_1+b_1=s}}\\{a_2+b_2\\}\\supseteq A_2(a_1)+_{G_2(a_1,b_1)} B_2(b_1)\\]\nthe lemma follows.\n\\end{proof}\n\n\\begin{Lemma}[Regularized decomposition]\\label{lm:FiberingLemma}\nLet $Z_1$ and $Z_2$ be abelian groups and let $A,B\\subset Z_1\\times Z_2$ be finite sets. Suppose that $\\delta>0$, $K\\geq 1$ and $G\\subset A\\times B$ are such that \\[|G|\\geq \\delta |A||B|,\\] and\n\\[|A+_G B|\\leq K(|A||B|)^{1\/2}.\\] There are absolute constants $c,C>0$, subsets $A'\\subset A$ and $B'\\subset B$, and a subset $G'\\subset A'\\times B'$ with the following properties.\n\\begin{enumerate}\n\\item(Uniform fibers) If \n\\ben \\label{eq:M_1M_2definition}\n\t\tM_A=|\\pi_1(A)|,\\ M_B=|\\pi_1(B)|\n\\een \nthen there are numbers $m_A$ and $m_B$ satisfying \n\\ben \n\tM_Am_A&\\geq c\\delta^2(\\log(K\/\\delta))^{-1}|A|,\\label{eq:FiberingLemmaSetSizes}\\\\ \n\tM_Bm_B&\\geq c\\delta^2(\\log(K\/\\delta))^{-1}|B|,\\label{eq:FiberingLemmaSetSizes2}\\\\\n\tm_A,m_B&\\geq c\\delta^{10}K^{-4}\\max_{a_1\\in\\pi_1(A),b_1\\in\\pi_1(B_1)}\t(|A_2(a_1)|+|B_2(b_1)|),\n\\een\nand such that we have approximately uniform fibers: \t\t\\ben\\label{eq:FiberingLemmaFiberSizes}\n|(A')_2(a_1)|\\approx m_A,\\ |(B')_2(b_1)|\\approx m_B\n\\een \nfor $a_1\\in \\pi_1(A')$ and $b_1\\in \\pi_1(B')$.\n\\item(Uniform graph fibering) For some $\\delta_1,\\delta_2>0$ satisfying\n\\ben \\label{eq:FiberingLemmaDeltaSizes}\n\t\\delta_1\\delta_2>c(\\log (K\/\\delta))^{-3}\\delta\n\\een we have that the first coordinate subgraph is dense: \n\\ben \\label{eq:FiberingLemmaBaseGraphSize}\n\t|G_1'|\\geq \\delta_1M_AM_B,\n\\een and that the subgraph has dense fibers: for each $(a_1,b_1)\\in G'_1$ we have\n\\ben \\label{eq:FiberingLemmaFiberGraphSize}\n\t|G'_2(a_1,b_1)|\\geq \\delta_2m_Am_B.\n\\een\n\\item(Bounded doubling) For some $K_1,K_2>0$ with\n\\ben\\label{eq:FiberingLemmaDoublingConstants}\n\tK_1K_2\\leq C\\delta^{-2}(\\log K)K\n\\een\nwe have \n\\ben\\label{eq:FiberingLemmaK1}\n\t|\\pi_1(A')+_{G'_1}\\pi_1(B')|= K_1(M_AM_B)^{1\/2},\n\\een\nand for each $(a_1,b_1)\\in G_1'$,\n\\ben \\label{eq:FiberingLemmaK2}\n\t|\\pi_2(A')+_{G'_2(a_1,b_1)}\\pi_2(B')|\\approx K_2(m_Am_B)^{1\/2}.\n\\een\n\\end{enumerate}\n\\end{Lemma}\n\\subsection{Proof of Theorem \\ref{lm:FiberingLemma}}\nWe will produce the sets $A'$ and $B'$ after a sequence of refinements. One such refinement comes from the following lemma. Here, and in what follows, when $G\\subseteq A\\times B$ we write $\\deg_G a$ (respectively, $\\deg_G b$) for the size of $\\{b'\\in B:(a,b')\\in G\\}$ (respectively, the size of $\\{a'\\in A:(a',b)\\in G\\}$).\n\\begin{Lemma} \\label{lemma:Step0}\nLet $A$ and $B$ be finite sets and $G\\subseteq A \\times B$ of size $\\delta |A||B|$. Then there exist $A' \\subset A, B' \\subset B$ and $G' \\subset G \\cap (A' \\times B')$ such that \n\\begin{itemize}\n\\item $\\deg_{G'} a\\geq \\frac{\\delta}{4} |B|$,\n\\item $\\deg_{G'} b\\geq \\frac{\\delta}{4} |A|$,\n\\item $|A'| \\geq \\frac{\\delta}{2}|A|$,\n\\item $|B'| \\geq \\frac{\\delta}{2}|B|$, and \n\\item $|G'| \\geq \\frac{\\delta}{2} |A||B|$\n\\end{itemize}\nfor any $a \\in A', b \\in B'$.\n\\end{Lemma}\n\\begin{proof}\nRemove from $A$ (respectively, $B$) one by one all vertices with degree less than $\\delta|A|\/4$ (respectively, $\\delta|B|\/4$), until both $A$ and $B$ contain only vertices of degree at least $\\delta|A|\/4$ (respectively, $\\delta|B|\/4$) in the remaining graph. At the end of this process, we cannot have removed more than $\\delta|A||B|\/2$ edges. Indeed, we remove at any stage at most $\\delta|B|\/4$ edges adjacent to a vertex in $|A|$ (and we can remove at most $|A|$ such vertices) or else at most $\\delta|A|\/4$ edges adjacent to a vertex in $B$ (and we can remove at most $|B|$ such vertices). Take $A'$ and $B'$ to be the sets of survived vertices in $A$ and $B$ respectively and $G' := G \\cap (A' \\times B')$.\n\\end{proof}\n\nNow, set $|A|=N_A$ and $|B|=N_B$. In view of the above lemma, and passing to subsets if necessary, we may assume \\[|A|\\geq \\frac{1}{2}\\delta N_A,\\ |B|\\geq \\frac{1}{2}\\delta N_B,\\] \\[|G|\\geq \\frac{1}{2}\\delta N_AN_B\\] and that for any $a\\in A$ and $b\\in B$ we have\n\\[\\deg_G a\\geq \\frac{1}{4}\\delta|B|,\\ \\deg_G b\\geq \\frac{1}{4}\\delta|A|.\\]\nFirst, we may assume without loss of generality that \\[n_A=\\max_{a_1\\in\\pi_1(A)}|A_2(a_1)|\\geq\\max_{b_1\\in\\pi_1(B)}|B_2(b_1)|\\]\nIt is also useful to observe that, if $a\\in A$ then $|\\{a\\}+_G B|=\\deg_G a$, where $\\deg_G a$ is the number of neighbours of $a$ in $G$. So, \n\\[\\delta N_B\\leq \\frac{1}{N_A}\\sum_{a\\in A}\\deg_G a\\leq |A+_G B|\\leq K(N_AN_B)^{1\/2}.\\]\nWe can apply the same argument, reversing the roles of $A$ and $B$, and we have proved\n\\begin{equation}\\label{SizeComparison}\n\\delta N_B^{1\/2}\\leq KN_A^{1\/2},\\ \\delta N_A^{1\/2}\\leq KN_B^{1\/2}.\n\\end{equation}\nHaving assumed this, our first order of business is to establish property (1) for $B'$.\n\n\\subsubsection{Regularization of $B$}\n\nLet $a_1\\in\\pi_1(A)$ be such that $|A_2(a_1)|=n_A$. Then $a_1\\oplus A_2(a_1)$ consists of $n_A$ elements of $A$ each with at least $\\frac{1}{4}\\delta|B|$ neighbours in $B$. Thus\n\\[|(a_1\\oplus A_2(a_1)\\times B)\\cap G|\\geq \\frac{1}{4}\\delta n_A|B|.\\] Let\n\\[B'=\\left\\{b\\in B:|\\{a_2:(a_1\\oplus a_2,b)\\in G\\}|\\geq \\frac{1}{8}\\delta n_A\\right\\}\\]\n\\begin{equation}\\label{Fibre1}\n|B'|\\geq \\frac{1}{8}\\delta |B|\\geq\\frac{1}{16}\\delta^2 N_B\n\\end{equation} and such that for each $b\\in B'$ we have\n\\[|((a_1\\oplus A_2(a_1))\\times\\{b\\})\\cap G|\\geq \\frac{1}{8}\\delta n_A.\\]\nMoreover, since every element in $B$ has at least $\\frac{1}{4}\\delta N_A$ neighbours in $A$, we have\n\\[|(A\\times B')\\cap G|\\geq\\frac{1}{4}\\delta N_A|B'|.\\] If $k=|\\pi_1(B')|$ then there are elements $b_1\\oplus b_1',\\ldots,b_k\\oplus b_k'$ with the $b_i$ distinct, and for each of them the sets\n\\[a_1\\oplus A_2(a_1)+b_i\\oplus b_i'\\] are disjoint, since their first coordinates are $a_1+b_i$ and are distinct. Each of these sets contains at least $\\frac{1}{8}\\delta n_A$ distinct elements of $A+_G B$ since each element of $B'$ has that many neighbours in $G$. From this it follows that\n\\[|a_1\\oplus A_2(a_1)+_G B'|\\geq \\frac{1}{8}\\delta n_A|\\pi_1(B')|\\] and so\n\\[\\frac{1}{8}\\delta n_A|\\pi_1(B')|\\leq |A+_G B|\\leq K(N_AN_B)^{1\/2}\\leq \\frac{K^2}{\\delta}N_B.\\] Here we have used the inequality (\\ref{SizeComparison}). Next, we define\n\\begin{equation}\\label{Fibre2}\nB''=\\bigcup_{\\substack{1\\leq i\\leq k\\\\ |B'_2(b_i)|\\geq 10^{-4}\\delta^5K^{-2}n_A}}b_i\\oplus B'_2(b_1).\n\\end{equation}\nBy (\\ref{Fibre2}) and (\\ref{Fibre1}),\n\\[|B'\\setminus B''|\\leq |\\pi_1(B')|10^{-4}\\delta^5K^{-2}n_A\\leq 10^{-3}\\delta^3N_B\\leq \\frac{\\delta}{10}|B'|.\\]\nNow, we have already assumed that $\\max_{b_1\\in\\pi_1(B)}|B_2(b_1)|\\leq n_A$, so applying a dyadic partition to the range $10^{-4}\\delta^5K^{-2}n_A\\leq m\\leq n_A$, we find a value of $m_B$ in this range and a subset \n\\[B'''=\\bigcup_{\\substack{b_1\\in\\pi_1(B')\\\\m_B\\leq |B''_2(b_1)|\\leq 2m_B}}b_1\\oplus B'_2(b_1)\\] which has size $|B'''|\\gg\\log(K\/\\delta)^{-1}|B''|$.\nThus\n\\[|B'''|\\gg\\frac{|B''|}{\\log(K\/\\delta)}\\gg \\frac{|B'|}{\\log(K\/\\delta)}\\gg \\frac{\\delta^2}{\\log(K\/\\delta)}N_B.\\]\nSince each element of $B$ has at $\\frac{1}{8}\\delta N_A$ neighbours in $G$, we further have\n\\[|(A\\times B''')\\cap G|\\geq \\frac{1}{8}\\delta N_A|B'''|.\\]\nIf $M_B=|\\pi_1(B''')|$, then because each element of $\\pi_1(B''')$ has about $m_B$ fibers, we have\n\\[|B'''|\\approx m_BM_B.\\]\nRedefine $B'=B'''$ and $N_B'=|B'|$. Then we have shown that \n\\[N_B'\\gg \\frac{\\delta^2}{\\log(K\/\\delta)}N_B.\\]\n\n\\subsubsection{Regularization of $A$}\n\nLet \\[A'=\\bigcup_{\\substack{a_1\\in\\pi_1(A)\\\\|A_2(a_1)|\\geq 10^{-5}\\delta^3K^{-2}m_B}}a_1\\oplus A_2(a_1).\\] \nWe first estimate $|A\\setminus A'|$. We write $A''=A\\setminus A'$, so that for each $a\\in A''$ we have \n\\begin{equation}\\label{A''}\n|A_2(a_1)|<10^{-5}\\delta^3K^{-2}m_B\n.\\end{equation} We will show $|(A''\\times B')\\cap G|\\leq \\frac{\\delta}{40} N_AN_B'$. To see why, assume the contrary. Then there is a $b_1\\in \\pi_1(B')$ with \\[|(A''\\times b_1\\oplus B'_2(b_1))\\cap G|\\geq\\frac{\\delta}{100}N_Am_B.\\] Indeed, each of the vertex sets $b_1\\oplus B'_2(b_1)$ are disjoint and have size $m_B$ up to a factor of $2$. Now let $A'''\\subset A''$ be the set of those $a$ for which\n\\[|(\\{a\\}\\times b_1\\oplus B'_2(b_1))\\cap G|\\geq \\frac{\\delta}{200}m_B.\\] From the definition, it follows that\n\\begin{equation}\\label{A'''}\n|A'''|\\geq \\frac{\\delta}{200}N_A.\n\\end{equation}\nLet \\[M=\\max_{a_1\\in\\pi_1(A''')}|A'''_2(a_1)|.\\]\nWe have\n\\[|A'''+_G(b_1\\oplus B'_2(b_1)))|\\leq |A+_G B|\\leq K(N_AN_B)^{1\/2}\\leq \\frac{K^2}{\\delta}N_A.\\] Because every element of $A'''$ has at least $(\\delta\/200)m_B$ neighbours in $b_1\\oplus B'_2(b_1)$, and because for each $a_1\\in \\pi_1(A''')$ the sets $(a_1\\oplus A_2'''(a_1))+_G(b_1\\oplus B'_2(b_1))$ are disjoint, we get\n\\[|A'''+_G(b_1\\oplus B'_2(b_1)))|\\geq (\\delta\/200) m_B|\\pi_1(A''')|.\\]\nIn view of (\\ref{A'''}) \\[|\\pi_1(A''')|\\geq \\frac{|A'''|}{M}\\geq \\frac{\\delta}{200M}N_A,\\] we obtain the bound\n\\[\\frac{\\delta^2}{4\\cdot 10^4M}N_Am_B\\leq \\frac{K^2}{\\delta}N_A\\] whence \n\\[M>\\frac{\\delta^3m_B}{10^5 K^2},\\] which contradicts (\\ref{A''}) and the definition of $M$.\nBy what we have just shown,\n\\[|(A'\\times B')\\cap G|\\geq\\frac{\\delta}{8}N_AN_B'.\\] Now, for each $a\\in A'$, we certainly have\n\\[|A_2(a_1)|\\leq n_A\\leq 10^4m_B \\delta^{-5}K^2\\] the final estimate coming from the bounds on the range range of $m_B$. Thus we partition the range \n\\[10^{-5}\\delta^3K^{-2}m_B\\leq |A_2(a_1)|\\leq 10^4m_B \\delta^{-5}K^2\\] dyadically, to find an $m_A$ in this range such that\n\\[A''''=\\bigcup_{\\substack{a_1\\in\\pi_1(A)\\\\m_A\\leq |A_2(a_1)|\\leq 2m_A}}a_1\\oplus A_2(a_1)\\]\nsatisfies\n\\[|(A''''\\times B')\\cap G|\\gg\\frac{\\delta}{\\log(K\/\\delta)}N_AN_B'.\\] Moreover, since $|(A''''\\times B')\\cap G|\\leq |A''''|N_B'$ we have $|A''''|\\gg \\delta(\\log(K\/\\delta))^{-1}N_A$. If we define $M_A=|\\pi_1(A'''')|$ then we have\n\\[|A''''|\\approx M_Am_A\\] as needed. We relabel $A'=A''''$ and $N_A'=|A'|$, observing that\n\\[N_A'\\gg \\frac{\\delta}{\\log(K\/\\delta)}N_A\\] and we are ready to proceed to the next step.\n\n\\subsubsection{Regularizing the graph fibers}\n\nSo far we have found subsets $A'$ and $B'$, and an absolute constant $c>0$, satisfying \n\\[|(A'\\times B')\\cap G|\\geq c\\frac{\\delta}{\\log(K\/\\delta)}|A'||B'|,\\]\n\\[|A'|\\approx m_AM_A\\geq c\\frac{\\delta}{\\log(K\/\\delta)}N_A,\\]\nand\n\\[|B'|\\approx m_BM_B\\geq c\\frac{\\delta^2}{\\log(K\/\\delta)}N_B.\\]\nFurthermore, each of $A'$ and $B'$ have fibers above $\\pi_1$ of size roughly $m_A$ and $m_B$ respectively. Recall that for $(a_1,b_1)\\in\\pi_1(A')\\times \\pi_1(B')$ we have the graph\n\\[G_2(a_1,b_1)=\\{(a_2,b_2)\\in A'_2(a_1)\\times B'_2(b_1):(a_1\\oplus a_2,b_1\\oplus b_2)\\in G\\}.\\] Because we have regularized the fibers of $A'$ and $B'$, each of these graphs has cardinality obeying \\[|G_2(a_1,b_1)|\\leq 4m_Am_B.\\] By a slight abuse of notation, we let\n\\[G_1=\\{(\\pi_1(a),\\pi_1(B)):(a,b)\\in (A'\\times B')\\cap G\\}\\] and define\n\\[G_1'=\\left\\{(a_1,b_1)\\in \\pi_1(A')\\times \\pi_1(B'):|G_2(a_1,b_1)|\\geq \\frac{c\\delta}{16\\log(K\/\\delta)}m_Am_B\\right\\}.\\]\nSince\n\\[\\sum_{(a_1,b_1)\\in\\pi_1(A')\\times\\pi_1(B')}|G_2(a_1,b_1)|=|(A'\\times B')\\cap G|\\geq c\\frac{\\delta}{\\log(K\/\\delta)}|A'||B'|\\]\nit follows that\n\\[\\sum_{(a_1,b_1)\\in G_1'}|G_2(a_1,b_1)|\\geq c\\frac{\\delta}{2\\log(K\/\\delta)}|A'||B'|.\\]\nBy a dyadic pigeon-holing for $\\delta'$ in the range $c\\delta(\\log(K\/\\delta))^{-1}\\leq \\delta '\\leq 4$, we can find $\\delta'\\gg \\delta (\\log(K\/\\delta))^{-1}$ such that\n\\[G_1''=\\left\\{(a_1,b_1)\\in G_1':\\delta'm_Am_B\\leq|G_2(a_1,b_1)|\\leq2\\delta'm_Am_B\\right\\}\\]\ncertainly satisfies\n\\[\\sum_{(a_1,b_1)\\in G_1''}|G_2(a_1,b_1)|\\gg c\\frac{\\delta}{(\\log(K\/\\delta))^2}|A'||B'|.\\]\nFrom this estimate, it also follows that\n\\[|G_1''|\\gg \\frac{\\delta}{\\delta'(\\log(K\/\\delta))^2}M_AM_B.\\]\nLet us relabel $G_1''$ as $G_1'$ and set \n\\[G'=\\{(a,b)\\in A'\\times B':(\\pi_1(a),\\pi_1(b))\\in G_1'\\}.\\] We move on to the final step of the lemma.\n\n\\subsubsection{Regularizing the doubling constant}\n\nFor $(a_1,b_1)\\in \\pi_1(A')\\times\\pi_1(B')$ we define\n\\[K_+(G_2(a_1,b_1))=\\frac{|A'_2(a_1)+_{G_2(a_1,b_1)}B'_2(b_1)|}{(|A'_2(a_1)||B'_2(b_1)|)^{1\/2}}.\\]\nThis quantity measure the growth of sumsets on the fibres lying above a pair $(a_1,b_1)$. Now define\n\\[H=\\{(a_1,b_1)\\in G_1':K_+(G_2(a_1,b_1))>C(\\log(K\/\\delta))^3\\delta^{-10}K\\}.\\]\nProvided $C$ is large enough we have $H\\leq \\frac{1}{10}|G_1'|$. To see this, first observe the trivial bound\n\\begin{equation}\\label{HSumset}|\\pi_1(A')+_H\\pi_1(B)|\\geq\\frac{|H|}{\\min\\{|\\pi_1(A')|,|\\pi_1(B')|\\}}\\geq \\frac{|H|}{(M_AM_B)^{1\/2}}.\n\\end{equation}\nLet \\[G_H=\\{(a_1,a_2)\\in G:(\\pi_1(a_1),\\pi_1(a_2))\\in H\\}\\subset G.\\]\nAlso, for $(a_1,b_1)\\in H$ we have\n\\[(G_H)_2(a_1,b_1)=G_2(a_1,b_1)\\] so that by Lemma \\ref{FiberGraphSum}\n\\[|A'+_G B'|\\geq |\\pi_1(A')+_H\\pi_1(B')|\\min_{(a_1,b_1)\\in H}(|A'_2(a_1)+_{G_2(a_1,b_1)}B'_2(b_1)|).\\]\nBy the definition of $H$ and (\\ref{HSumset}) we see\n\\[K(N_AN_B)^{1\/2}\\geq |A'+_G B'|\\geq C\\frac{|H|}{(M_AM_B)^{1\/2}}(\\log(K\/\\delta))^3\\delta^{-10}K(m_Am_B)^{1\/2}.\\]\nUsing our estimates for $m_AM_A$, $m_BM_B$ and $G_1'$, the right hand side is \n\\[C\\frac{|H|}{M_AM_B}(\\log(K\/\\delta))^3\\delta^{-10}K(M_Am_AM_Bm_B)^{1\/2}\\geq cCK(N_AN_B)^{1\/2}\\frac{|H|}{|G_1'|}.\\] Thus for $C$ sufficiently large in terms of $c$ (which was absolute), we have $|H|\\leq \\frac{1}{10}|G_1'|$. \nNow let $G_1''=G_1'\\setminus H$. We perform yet another dyadic pigeon-holing to find $K'\\leq C(\\log(K\/\\delta))^3\\delta^{-10}K$ such that\n\\[G_1'''=\\{(a_1,b_1)\\in G_1'':K'\\leq K_+(G_2(a_1,b_1))\\leq 2K'\\}\\] has cardinality \\[|G_1'''|\\gg \\frac{|G_1'|}{\\log(K\/\\delta)}.\\]\nNow, by Lemma \\ref{FiberGraphSum} along the subgraph of $G$ with first projection equal to $G_1'''$ we have\n\\[K(N_AN_B)^{1\/2}\\geq |\\pi_1(A')+_{G_1'''}\\pi_1(B')|K'(m_Am_B)^{1\/2}=K_+(G_1''')K'(M_Am_AM_Bm_B)^{1\/2},\\]\nwhere $K_+(G_1''')=|\\pi_1(A')+_{G_1'''}\\pi_1(A_2')|(M_AM_B)^{-1\/2}$. By the established bounds on $m_AM_A$ and $m_BM_B$, we get\n\\[K(N_AN_B)^{1\/2}\\gg K_+(G_1''')K'\\delta^{3\/2}\\log(K\/\\delta)(N_AN_B)^{1\/2}.\\]\nFrom this we see\n\\[K_+(G_1''')K'\\ll K\\log (K\/\\delta)\\delta^{3\/2}\\ll K\\log (K)\\delta^{2}.\\]\nNow let $G'=\\{(a,b)\\in A'\\times B':(\\pi_1(a),\\pi_1(b))\\in G_1'''\\}.$ Define $K_1=K_+(G_1''')$ and $K_2=K'$. Let $\\delta_2=\\delta'$ and $\\delta_1=c\\delta(\\delta_2(\\log(K\/\\delta))^3)^{-1}$. One then verifies that with these parameters, the claims of the lemma have all been justified.\n\n\n\\section{Iteration scheme} \\label{sec:iterationscheme}\n\nIn this section we will use Lemma \\ref{lm:FiberingLemma} in order to setup an iteration scheme. At each step we have a pair of sets $(\\mathcal{A}, \\mathcal{B})$ which correspond to a pair of additive sets $(A, B) := (\\mathcal{P}(\\mathcal{A}), \\mathcal{P}(\\mathcal{B}))$ and a graph $G$ on $A \\times B$, together with the data $(N, \\delta, K)$ such that:\n\\begin{enumerate}\n \t\\item $|A||B| = N \\label{eq:setupNGKdelta1}$\n\t\\item $|A +_G B| \\leq KN^{1\/2}$ \\label{eq:setupNGKdelta2} \n\t\\item $|G| \\geq \\delta N$ \\label{eq:setupNGKdelta3}.\n\\end{enumerate}\n\n\n\nApart from that, the setup above is equipped with a pair of functions $\\psi(N, \\delta, K)$, $\\phi(N, \\delta, K)$ (which are called \\emph{admissible} in \\cite{BC}). These functions are technical aids to carry out an induction type argument.\n\n\\begin{Definition}[Admissible pair of functions] \\label{def:admissible_pairs}\nA pair of functions $\\psi(N, \\delta, K)$, and $\\phi(N, \\delta, K)$ is said to be \\emph{admissible} if for arbitrary sets $A, B \\subset \\mathbb{Z}^{[n]}$ and a graph $G$ on $A \\times B$ satisfying (\\ref{eq:setupNGKdelta1})-(\\ref{eq:setupNGKdelta3}) the following holds.\n\nThere is a graph $G' \\subseteq G$ such that\n\n\\begin{enumerate}[(i)]\n\n\\item[(G)]{Graph size is controlled by $\\phi$:}\n$$\n\t|G'| \\geq \\phi(N, \\delta, K)\n$$\n\n\\item[(S)]{Separation of $G'$-neigborhoods is controlled by $\\psi$:\\\\} \nFor any $a \\in A$ (resp. $b \\in B$) the $\\mathcal{P}$-preimage of the $G'$-neighborhood \n$$\n\\mathcal{P}^{-1}\\left[G'(a)\\right] := \\mathcal{P}^{-1}\\left[\\{b \\in B: (a, b) \\in G' \\} \\right].\n$$ (resp. of $G'(b)$) is $\\psi(N, \\delta, K)$-separating.\n\\end{enumerate}\nFurthermore, we will assume that the following technical conditions hold for $\\phi(N, \\delta, K), \\psi(N, \\delta, K)$:\n\\begin{enumerate}\n \\item[(A1)] $\\phi, \\psi$ are non-decreasing in $N$\n \\item[(A2)] $\\phi$ is non-decreasing in $\\delta$, non-increasing in $K$ and for each $\\delta$ and $K$, we have $\\phi(N, \\delta, K) \\leq N$.\n \\item[(A3)] $\\psi$ is non-decreasing in $K$\n \\item[(A4)] If $N \\geq M$ then\n $$\n \\frac{\\phi(N, \\delta, K)}{N} \\leq \\frac{\\phi(M, \\delta, K)}{M}\n $$\n\\end{enumerate}\n\\end{Definition}\n\n\nNote that, by Claim \\ref{cor:trivialseparation}, \nthe pair $\\psi(N, \\delta, K) := N; \\phi(N, \\delta, K) := \\delta N$ is trivially admissible with much room to spare.\n\nThe following lemma gives a Freiman-type pair of admissible functions which is better than trivial in the regime $K=o(\\log N)$, and will be used later to bootstrap the argument. \n\n\\begin{Lemma}[Freiman-type admissible functions] \\label{lm:FreimanAdmissiblePair}\nThere is an absolute constant $C > 0$ such that the pair of functions\n\\begin{enumerate}\n\t\t\\item $\\psi(N, \\delta, K) := \\min \\left\\{ (2k^2)^{\\left(\\frac{K}{\\delta}\\right )^C}, N \\right\\}$ \\label{eq:Freimanadmissiblepsi} \n\t\t\\item $\\phi(N, \\delta, K) := \\left( \\frac {\\delta}{K} \\right)^C N$  \\label{eq:Freimanadmissiblephi}\n\\end{enumerate}\nis admissible.\n\\end{Lemma} \n\\begin{proof}\nThis pair is easily seen to satisfy (A1) through (A4). Thus it remains to check (G) and (S). By the setup, we are given two sets $\\mathcal{A}$ and $\\mathcal{B}$ of sizes $N_A$ and $N_B$ respectively, and a graph $G$ of size $\\delta N_AN_B$ such that\n\\begin{equation}\n |A +_G B| \\leq K \\sqrt{N_AN_B} \\label{eq:A_1A_2Gdoubling}\n\\end{equation}\n\nAssume without loss of generality that $N_A \\geq N_B$ and take $X = A \\cup B$, which is of size  $ \\approx N_A$. Since by (\\ref{eq:A_1A_2Gdoubling})\n$$\n\\frac{K^2}{\\delta^2} N_B \\geq N_A\n$$\nwe have \n$$\n|G| \\gg \\frac{\\delta^3}{K^2}|X|^2\n$$\nand\n$$\n|X +_G X| \\ll K |C|. \n$$\nBy a variant of the Balog-Szemer\\'edi-Gowers theorem (see e.g. \\cite{TV}, Exercise 6.4.10) there is $X' \\subseteq X$ such that $|X' + X'| < K'|X'|$ and $|G \\cap (X' \\times X')| > \\delta' N_A^2$ with\n\\begin{align}\n\t \\delta' &> \\left( \\frac{\\delta}{K}\\right)^C \\label{eq:deltaboundFreiman}\n\\\\\t K' &< \\left( \\frac{K}{\\delta} \\right)^C. \\label{eq:KboundFreiman}\n\\end{align}\n\nBy Theorem \\ref{thm:FreimanLemma} any subset of $X$ has rank at most $K'$ and by Theorem \\ref{thm:chang2}, the $\\mathcal{P}$-preimage of any subset of $X'$ is at most $(2k^2)^{K'^C}$-separating for some $C > 0$.  Thus, taking $G' := G \\cap (X' \\times X')$ by (\\ref{eq:deltaboundFreiman}) and (\\ref{eq:KboundFreiman}) we verify that the pair (\\ref{eq:Freimanadmissiblepsi}), (\\ref{eq:Freimanadmissiblephi}) is admissible.\n\\end{proof}\n\nThe goal is to find a better pair of admissible functions. The lemma below implements the `induction on scales' approach, which allows one to cook up a new pair $\\phi_*(N, \\cdot, \\cdot), \\psi_*(N, \\cdot, \\cdot)$ from a given pair of admissible functions, but taken at the smaller scale  $\\approx N^{1\/2}$.\n\n\n\n\n \n\\begin{Lemma} \\label{lm:InductionStepLemma}\n\nLet $\\psi$ and $\\phi$ be an admissible pair of functions. Then for some absolute constant $C > 0$ the pair of functions\n\\ben \\label{eq:admissiblepairinduction}\n\t\t\\psi_{*}(N, \\delta, K) &:=& Ck^2 \\max  \\psi (N', \\delta', K') \\psi  (N'', \\delta'', K'') \\\\\n\t\t\\phi_{*}(N, \\delta, K) &:=& \\min \\phi(N', \\delta', K') \\phi(N'', \\delta'', K'') \\label{eq:admissiblepairinduction2}\n\\een\nis admissible.\n\n Here $\\min$ and $\\max$ is taken over the data $(N', \\delta', K'), (N'', K'', \\delta'')$ such that\n\\ben  \\label{eq:parameterconstraints}\n\t\\lr{c\\frac{\\delta^9}{\\log^{22} (K\/\\delta)}} N &\\leq& N'N'' \\leq N\\\\\n    N' + N'' &\\leq& \\lr{C \\frac{K^{11}}{\\delta^{45}}}N^{1\/2}\\\\\n\tK'K'' &\\leq& \\lr{C \\frac{\\log^{15} K}{\\delta^{20}}}K\\\\\n\t\\delta' \\delta'' &\\geq& \\lr{c \\frac{1}{\\log^{6} (K\/\\delta)}} \\delta. \n\\een\n\n\\end{Lemma}\n\n\\begin{proof}\n\tLet us first check that $(\\phi_*, \\psi_*)$ given by (\\ref{eq:admissiblepairinduction}) and (\\ref{eq:admissiblepairinduction2}) indeed satisfy (A1) through (A4). Assume $N_1 < N_2$ and $\\delta, K$ are fixed. Then $\\psi_*(N_1, \\cdot, \\cdot) < \\psi_*(N_2, \\cdot, \\cdot)$ since for $\\psi_*(N_2, \\cdot, \\cdot)$ the maximum is taken over the larger range of parameters \n    $$\n    N'N'' \\leq N_2, \\,\\,\\,\\, N' + N'' \\leq C \\delta^{-45} K^{11}N_2^{1\/2}.\n    $$\n  Similarly, \n  $$\n  \\phi_*(N_1, \\cdot, \\cdot) < \\phi_*(N_2, \\cdot, \\cdot)\n  $$ \n  since the minimum is now taken over the smaller set \n  $$\n  c \\delta^9 \\log^{-22} (K\/\\delta) N_2 \\leq N'N''.\n  $$\n Note, that here we have used the fact that $\\phi$ and $\\psi$ are both increasing. This proves (A1).\n \n    In order to prove (A2) it suffices to note that when $\\delta$ increases (resp. $K$ decreases) the range of parameters $N', N'', \\delta', \\delta'', K', K''$ over which the minimum in $\\phi_*$ is taken is getting more narrow. Similarly, when $K$ increases the maximum in $\\psi_*$ is taken over a larger set which proves (A3). \n    \n\tIt remains to verify (A4). Let $M, \\delta, K$ be fixed and $M', M'', \\delta', \\delta'', K', K''$ be such that the minimum for $\\phi_*(M, \\delta, K)$ in (\\ref{eq:parameterconstraints}) is achieved. Let $c > 0$ be a parameter. Then $cM', cM'', \\delta', \\delta'', K', K''$ are in the admissible range for $\\phi_*(c^2M, \\delta, K)$ so\n    \\begin{align*}\n    \\phi_*(c^2M, \\delta, K) &\\leq \\phi(cM', \\delta', K')\\phi(cM'', \\delta'', K'') \\\\\n    &\\leq c^2\\phi(M', \\delta', K')\\phi(M'', \\delta'', K'') \\\\\n    &= c^2\\phi_*(M, \\delta, K).\n\t\\end{align*}\n Taking $c$ such that $c^2M = N$ we get (A4).\n \n\tLet $A, B\\subset \\mathbb{Z}^{n}$ of sizes $N_A, N_B$ respectively, $G \\subseteq A\\times B$ and suppose that the conditions (\\ref{eq:setupNGKdelta1})-(\\ref{eq:setupNGKdelta3}) are satisfied with parameters $(N, \\delta, K)$ where $N=N_AN_B$. Our ultimate goal is to find a subgraph of $G$ of size at least \n    \\[ \\phi(N', \\delta', K') \\phi(N'', \\delta'', K'') \\] \nsuch that the $\\mathcal{P}$-preimage of any its neighbourhoods is \n\\[Ck^2 \\psi (N', \\delta', K') \\psi  (N'', \\delta'', K'')-\\text{separating},\\] for some $N',N'',K',K'',\\delta ', \\delta ''$ satisfying \\eqref{eq:parameterconstraints}. Once this is done, the proof will be complete.  In order to achieve this goal, we will apply Lemma \\ref{lm:FiberingLemma} and then use the hypothesis that the pair $\\psi, \\phi$ is admissible for much smaller sets.\t\n \t\nDefine a function $f(t)$ for $0 \\leq t \\leq n$ as\n$$f(t)=\\max_{(a_1,b_1)\\in \\pi_{[t]}(A)\\times\\pi_{[t]}(B)}\\{|A_2(a_1)|+|B_2(b_1)|\\},$$ where $\\pi_{[t]}$ is the projection onto the first $t$ coordinates, and $A_2(a_1)$ and $B_2(b_1)$ are the fibres above $a_1$ and $b_1$ respectively.\nNote that $f$ is decreasing, $f(0)=|A|+|B|\\geq N^{1\/2}$, and $f(n) = 0$. Thus there is $t'$ such that\n \t\\ben \\label{eq:FirstcoordinateSplit}\n\t\tf(t') \\geq N^{1\/4} \t\n \t\\een\n\tbut\n\t\\ben \\label{eq:FirstcoordinateSplit2}\n\t\tf(t'+1) < N^{1\/4} \t.\n   \\een\t\n   \n\tWe use the $t'$ defined above for the decomposition $\\ZZ^{n}=\\ZZ^{t'}\\times \\ZZ^{n-t'}$ and let $\\pi_1$ and $\\pi_2$ denote the projection onto the first and second factor respectively. We now apply Lemma \\ref{lm:FiberingLemma} and get sets $A'\\subseteq A$ and $B'\\subseteq B$ together with a graph $G' \\subseteq G\\cap(A'\\times B')$ such that \n \t\\ben\n \t\tA' &=& \\bigcup_{a_1\\in \\pi_1(A') } a_1\\oplus A'_2(a_1) \\\\\n \t\tB' &=& \\bigcup_{b_1\\in \\pi_1(B') } b_1\\oplus B'_2(b_1)\n \t\\een\n and the fibers $A'_2(a_1), B'_2(b_1)$ together with the fiber graphs $G'_2(a_1,b_1)$ are uniform as defined in the statement of Lemma \\ref{lm:FiberingLemma}. Note that it is possible that $t' = 0$, in which case the sets split trivially with $\\pi_1(A') = \\pi_1(B') = \\{0\\}$. \n \n Using the notation of Lemma \\ref{lm:FiberingLemma}  we have \n \\ben\n \t|\\pi_1(A')+_{G'_1} \\pi_1(B)| \\leq K_1(M_AM_B)^{1\/2}.\n \\een\nSince $\\phi, \\psi$ is an admissible pair, there is $G''_1 \\subseteq G'_1$ of size at least $\\phi(M_1M_2, \\delta_1, K_1)$ such that all $\\mathcal{P}$-preimages of its vertex neighbourhoods are $\\psi(M_1M_2, \\delta_1, K_1)$-separating.\nNext, since $G_1''\\subseteq G_1'$, for each edge $(a_1, b_1) \\in G''_1$, there is a graph $G'_2(a_1, b_1)\\subseteq A'_2(a_1) \\times B'_2(b_1)$ such that $|G'_2(a_1, b_1)| \\geq \\delta_2m_Am_B$ and\n\\ben\n\t|A'_2(a_1) +_{G'_2(a_1, b_1)} B'_2(b_1)| \\leq K_2(m_Am_B)^{1\/2}.\n\\een\nAgain, by admissibility of $\\phi, \\psi$, there is $G''_2(a_1, b_1) \\subseteq G'_2(a_1, b_1)$ of size at least $\\phi(m_Am_B, \\delta_2, K_2)$ such that all $\\mathcal{P}$-preimages of its vertex neighbourhoods are $\\psi(m_Am_B, \\delta_2, K_2)$-separating.\n \t\nNow define $G'' \\subseteq G \\cap (A' \\times B')$ as \n$$\n\tG'' := \\{(a_1 \\oplus a_2, b_1 \\oplus b_2): (a_1, a_2) \\in G''_1, (a_2, b_2) \\in G''_2(a_1,b_1)\\}.\n$$\nIt is clear by construction that indeed all vertices of $G''$ belong to $A'$ and $B'$ respectively. Moreover, we have\n\\ben \\label{eq:firstreduction}\n\t|G''| \\geq \\phi(M_AM_B, \\delta_1, K_1)\\phi(m_Am_B, \\delta_2, K_2).\n\\een\n \t\nNow let's estimate the separating constant for the $\\mathcal{P}$-preimage of a neighbourhood $\\mathcal{P}^{-1}[G''(u)]$ of some $u \\in V(G'')$. Without loss of generality assume that $n \\in B'$ and $b = b_1 \\oplus b_2$. We can write\n\\ben\n\t\tG''(b) = \\bigcup_{a_1 \\in G''_1(b_1)} \\bigcup_{a_2 \\in G''_2(a_1, b_1)}\\{a_1 \\oplus a_2\\} .\n\\een\nThus,\n\\ben \n\\mathcal{P}^{-1}[G''(b)] = \\bigcup_{a_1\\in G''_1(b_1)} p_1^{a_1} \\cdot \\left\\{ \\bigcup_{a_2 \\in G''_2(a_1, b_1)} p_2^{a_2} \\right\\}.\n\\een\nHere we are using the notation $q^{r}=q_1^{r_1}\\cdots q_l^{r_l}$ for a vector $q$ of primes and a vector $r$ of integers, and $p_1$ and $p_2$ are respectively the first $t$ primes from the map $\\cP$ and the remaining primes. Now, since $G''_1(b_1)$ and $G''_2(a_1, b_1)$ are orthogonal as linear sets we conclude that $(p_1^{a_1}, p_2^{a_2}) = 1$.  Thus, by Lemma \\ref{lem:chain} and the admissibility of $\\phi, \\psi$ applied to $G''_1$ and $G''_2(a_1,b_1)$ we conclude that $\\mathcal{P}^{-1}[G''(b)] $ is at most $\\psi(M_AM_B, \\delta_1, K_1)\\psi(m_Am_B, \\delta_2, K_2)$-separating.\n\nWe now record the bounds for the various parameters following from Lemma \\ref{lm:FiberingLemma}. We have \n\\ben\n\t\\delta_1\\delta_2 &\\geq& \\lr{c \\frac{1}{\\log^{3}(K\/\\delta)}}\\delta. \\label{eq:delta_2}\\\\\n\tK_1 K_2 &\\leq&  \\lr{C\\frac{\\log K }{\\delta^{2}}}K   \\label{eq:K_2} \\\\\n\tM_Am_A &\\geq & \\lr{c \\frac{\\delta^2}{\\log(K\/\\delta)}}N_A \\label{eq:M_1m_1} \\\\\n\tM_Bm_B &\\geq & \\lr{c \\frac{\\delta^2}{\\log(K\/\\delta)}} N_B \\label{eq:M_2m_2} \\\\\n\tm_A, m_B &\\geq & \\lr{c \\frac{\\delta^{10}}{K^{4}}} N^{1\/4} \\label{eq:m_ibig}\n\\een\n\nIn particular, we have\n\\ben \\label{eq:M_1M_2upperbound}\n M_AM_B < \\frac {N_AN_B}{m_Am_B} < \\lr{c \\frac{K^8}{\\delta^{20}}}N^{1\/2}.\n\\een\nAs a first attempt, we set $N'=M_AM_B$ and $N''=m_Am_B$, $\\delta'=\\delta_1$, $K'=K_1$, $\\delta''=\\delta_2$ and $K''=K_2$.\nIf $N''=m_Am_B$ is less than $N^{1\/2}$, one can verify that all of the above bounds comply with the statement of this lemma, and we can stop. If $N''$ is too big, we will apply Lemma \\ref{lm:FiberingLemma} again.\n\nTo further reduce the size we apply Lemma \\ref{lm:FiberingLemma} again for each pair of sets $(A'_2(a_1), B'_2(b_1))$ such that $(a_1, b_1) \\in G'_1$, stripping off only a single coordinate as explained below. Assume the base point $(a_1, b_1)$ is fixed henceforth. \n\n\\begin{comment}\nThe reader may keep in mind the following model case: $N_1 = N_2 = N^{1\/2}$ and $A_1, A_2$ are three-dimensional arithmetic progressions\n$$\n\\{ [0,N_1^\\alpha]e_1 + [0, N_1^{1-2\\alpha}]e_2 +  [0, N_1^\\alpha]e_3  \\},\n$$\nfor some small $\\alpha > 0$. In this case $t'=1, M_1 M_2 \\approx N^\\alpha$  and $m_1m_2 \\approx N^{1-\\alpha}$. However, the fibers $A_1(x)$ and $A_2(y)$ are heavily concentrated on the second coordinate which makes the separating constant of the $\\mathcal{P}$-preimage small.\n\\end{comment}\n\\vskip 1em\n\nWe split the coordinates $\\{t'+1, \\ldots, n \\}$ as  $\\ZZ\\times \\ZZ^{n-t'-2}$. We apply Lemma \\ref{lm:FiberingLemma}, this time with to the pair of sets $A'_2(a_1)$ and $B'_2(b_1)$ and the graph $G'_2(a_1,b_1)$. To ease notation, let us set $U=A'_2(a_1)$, $V=B'_2(b_1)$, and $H=G'_2(a_1,b_1)$. Here, it is worth noting that $U,V$ and $H$ depend on the base point $(a_1,b_1)$. This time, we have the estimates\n\\[|U|\\approx m_A, |V|\\approx m_B\\] and\n\\[|U+_HV|\\leq K_2\\lr{|U||V|}^{1\/2}\\]\nwhere $|H|\\geq \\frac{\\delta_2}{4}|U||V|$.\nWe will again denote by $\\pi_1$ the projection onto the first coordinate, and by $\\pi_2$ the projection onto the remaining $n-t'-2$ coordinates. \nWe then get \n$$\nU' \\subseteq  U, \\,\\,\\, V'\\subseteq V\n$$ such that\n\\ben\n\tU' &=& \\bigcup_{u_1 \\in \\pi_{1}(U')} u_1\\oplus U'_2(u_1) \\\\\n\tV' &=& \\bigcup_{y_1 \\in \\pi_{1}(V')} v_1\\oplus V'_2(v_1)\n\\een\nand the fibers $U'_2(u_1)$ and $V'_2(v_1)$ are of approximately the same size, say $m_U$ and $m_V$ respectively. We also write $M_U=|\\pi_1(U)|$ and $M_V=|\\pi_1(V)|$. Note again that, for instance,  the fiber $U'_2(u_1)$ may be trivial (i.e. $\\{ 0\\}$), which simply means that $m_U \\approx 1$. By (\\ref{eq:FiberingLemmaSetSizes}), (\\ref{eq:FiberingLemmaSetSizes2}) we have the estimates\n\\[M_Um_U\\geq c\\delta_2^2(\\log(K_2\/\\delta_2))^{-1}|U|,\\ M_Vm_V\\geq c\\delta_2^2(\\log(K_2\/\\delta_2))^{-1}|V|\\]\n\nNext, we have a graph \\[H'\\subseteq (U'\\times V')\\cap H\\] with uniform fibers as defined in Lemma \\ref{lm:FiberingLemma}. The graph $H'$ splits into the base graph $H'_{1} \\subset \\pi_{1}(U') \\times \\pi_{1}(V')$ such that\t\n\\[|\\pi_1(U')+_{H'_1}\\pi_1(V')|\\leq K_3(M_UM_V)^{1\/2},\\]\nand fiber graphs $H'_2(u_1, v_1)$ such that for $(u_1, v_1) \\in H_{1}'$ \n\\ben\n|U'_2(u_1) +_{H'_2(u_1,v_1)} V'_2(v_1)| \\leq K_4(m_Um_V)^{1\/2},\n\\een\nwith\n\\ben\n|U'_2(u_1)| &\\approx& m_U \\\\\n|V'_2(v_1)| &\\approx& m_V \\\\\n|H'_2(u_1, v_1)| &\\geq& \\delta_4 m_Um_V.\n\\een\n\nThe parameters $m_U, m_V, \\delta_3, \\delta_4, K_3, K_4$ as well as the sizes of $H_1'$ and $H'_2(u_1,v_1)$ are controlled by Lemma \\ref{lm:FiberingLemma}. By the assumption that the original pair $(\\phi,\\psi)$ is admissible, for each such a graph $H'_2(u_1,v_1)$ there is a subgraph $H_2''(u_1,v_1) \\subseteq H_2'(u_1,v_1)$ with\n\\ben\n|H_2''(u_1,v_1)| \\geq \\phi(m_Um_V, \\delta_4, K_4)\n\\een\nsuch that the $\\mathcal{P}$-preimage of each neighborhood of $H_2''(u_1,v_1)$ is $\\psi(m_Um_V, \\delta_4, K_3)$-separating. Define $H'' \\subset H'$ as\n\\ben\n\tH''= \\{(u_1 \\oplus u_1, v_1 \\oplus v_2): (u_1, v_1) \\in H'_1, (u_2,v_2)\\in H_2''(u_1,v_1) \\}. \n\\een\nThe size of $H''$ is at least $|H_1'|\\phi(m_Um_V, \\delta_4, K_4)$. Next, the set of vertices of $H_1'$ all lie in a one-dimensional affine subspace, so combining Corollary \\ref{cor:oneprimeseparation} and Lemma \\ref{lem:chain} one concludes that the $\\mathcal{P}$-preimage of each neighborhood of $H''$ is \n$Ck^2\\psi(m_Um_V, \\delta_4, K_4)$-separating with some absolute constant $C > 0$. Putting together all of the details, we conclude\nthat, for $G_2'(a_1,b_1)\\subset A'(a_1) \\times B'(b_1)$, there is a subgraph $H'' \\subseteq G_2'(a_1,b_1)$ of size at least\n\\ben \\label{eq:step2psiphi}\n\t\\phi_{a_1, b_1} := |H_1'|\\phi(m_Um_V, \\delta_4, K_4)  \n\\een\nsuch that the $\\mathcal P$-preimage of each neighbourhood in $H''$ is $\\psi_{a_1,b_1}$-separating, where\n\\[\\psi_{a_1, b_1} := Ck^2\\psi(m_Um_V, \\delta_4, K_4).\\]\nSince the the graph $H''$ depends on the pair $(a_1,b_1)$, we now rename this graph $H''_{a_1,b_1}$.\n\nIn turn, substituting $\\psi_{a_1, b_1}$ and $\\phi_{a_1, b_1}$ into the argument leading to (\\ref{eq:firstreduction}) and Lemma \\ref{lem:chain}, we construct a graph\n\\[G''':= \\{(a_1 \\oplus a_2, b_1 \\oplus b_2) :(a_1,b_1) \\in G_1', (a_2,b_2) \\in H''_{a_1,b_1}\\}.\\] \nThe graph $G'''$ has size at least\n\\begin{equation}\n\\phi(M_AM_B, \\delta_1, K_1) \\cdot \\min_{(a_1, b_1) \\in G'_1} \\phi_{a_1, b_1},\n\\label{eq:admissiblepairs2}\n\\end{equation}\nand the separating factors are at most\n\\begin{equation}\n\\psi(M_AM_B, \\delta_1, K_1) \\cdot   \\max_{(a_1, b_1) \\in G'_1} \\psi_{a_1, b_1},\n\\label{eq:admissiblepairs22}\n\\end{equation}\n \nWith $G'''$ we have now found a large subgraph with good separating factors. In the remaining calculations, we show that the existence of this $G'''$ is good enough to imply the theorem. Essentially it remains to check that the quantities (\\ref{eq:admissiblepairs2}) and \\eqref{eq:admissiblepairs22} can indeed be bounded respectively by \\eqref{eq:admissiblepairinduction2} and (\\ref{eq:admissiblepairinduction}).\nNote that the quantities (\\ref{eq:admissiblepairs2}) and \\eqref{eq:admissiblepairs22} do depend on the structure of $A$ and $B$. We are going to show, however, that they are uniformly bounded by \\eqref{eq:admissiblepairinduction2} and (\\ref{eq:admissiblepairinduction}) which are functions of $(N, \\delta, K)$ only. We remark here that we will make use of the following fact: if $|X+_GY|\\leq K(|X||Y|)^{1\/2}$ for some $G\\subset X\\times Y$ of size at least $\\delta|X||Y|$, then $K\/\\delta\\geq 1$. \n\n\nFirst, since $(a_1, b_1) \\in G'_1$ we have by (\\ref{eq:FiberingLemmaDeltaSizes})\n\\begin{equation}\n\t \\delta_4\\geq\\delta_3\\delta_4 >c\\log^{-3}(K_2\/\\delta_2) \\delta_2.\n     \\label{eq:delta3}\n\\end{equation}\nBy (\\ref{eq:FiberingLemmaDoublingConstants}) and (\\ref{eq:FiberingLemmaDeltaSizes})\n\\ben\n\t\\frac {K_2} {\\delta_2}\\leq \\frac{K_1K_2}{\\delta_1\\delta_2} < \\frac{CK \\log(K) \\log^3(K\/\\delta)} {\\delta^{3}}\n\\een\n and so\n\\begin{equation}\n \\log (K_2\/\\delta_2) < C\\log(K\/\\delta).\n \\label{eq:Koverdelta}\n\\end{equation}\nConsequently,\n\\ben \\label{eq: delta_1delta_3}\n \\delta_1 \\delta_4 \\stackrel{\\eqref{eq:delta3}, \\eqref{eq:Koverdelta}}{>} c \\log^{-3} (K\/\\delta) \\delta_1 \\delta_2 \\stackrel{\\eqref{eq:delta_2}}{>} c \\log^{-6} (K\/\\delta) \\delta.\n\\een\nNext, by (\\ref{eq:FiberingLemmaDoublingConstants})\n\n\\begin{equation}\n K_4  \\leq \\frac{K_3K_4}{\\delta_3\\delta_4}\\leq C K_2 \\log^2 (K_2) \\delta_2^{-4}\n \\label{eq:K3K2}\n\\end{equation}\nand by (\\ref{eq:FiberingLemmaDeltaSizes})\n\\ben\n\\delta_2 &>& c \\log^{-3}(K\/\\delta) \\delta \\label{eq:delta2delta}\\\\\nK_2 &<& C  \\delta^{-4} K \\log^2 K .\n\\een\nTherefore\n\\ben\n\\log^2(K_2) \\delta_2^{-4} &\\leq& C \\log^{14} (K\/\\delta) \\delta^{-4} \\label{eq:logK2delta2}  \\\\\n\t\t\t\t\t\t\t\t\t&=& C (\\delta^{14} \\log^{14} (K\/\\delta)) \\delta^{-18} < C (\\log^{14}K) \\delta^{-18} \\nonumber\n\\een\nand\n\\ben \\label{eq:K_1K_3bound}\nK_1 K_4 \\stackrel{\\eqref{eq:K3K2}}{\\leq} C K_1 K_2 \\log^2 (K_2) \\delta_2^{-4} \\stackrel{(\\ref{eq:K_2}), (\\ref{eq:logK2delta2})}{\\leq} C \\frac{K \\log^{15} K}{\\delta^{20}}.\n\\een\nFinally, we have by (\\ref{eq:FiberingLemmaSetSizes}), (\\ref{eq:FiberingLemmaDeltaSizes}), (\\ref{eq:FiberingLemmaBaseGraphSize}) and (\\ref{eq:FiberingLemmaFiberGraphSize}) that\n\\ben\n|H_1'|m_Um_V &\\geq& c \\log^{-3}(K_2\/\\delta_2) \\delta_2 (\\delta_2^4 \\log^{-2}(K_2\/\\delta_2)) |A_2(a_1)||B_2(b_1)| \\nonumber \\\\\n\t\t\t\t\t&\\geq& c\\log^{-5} (K\/\\delta) \\delta_2^5 m_Am_B \\stackrel {(\\ref{eq:delta2delta})}{\\geq} c \\log^{-20} (K\/\\delta) \\delta^5 m_Am_B  \\label{eq:K_Il_1l_2} .\n\\een\n\nDefine \n\\ben \\label{eq:N''def}\n N'' := \\min \\{ N^{1\/2},  \\max\\{m_Um_V, c \\log^{-20} (K\/\\delta) \\delta^5 m_Am_B\\} \\}.\n\\een\nBy our choice of $t'$ it follows that $m_Um_V \\leq N''$. By (A4) we have\n\\ben\n\\frac{m_Um_V}{N''}\\phi(N'', \\delta_3, K_3) \\leq  \\phi(m_Um_V, \\delta_4, K_4).\n\\een\nDefining \n\\ben\nN' := \\frac{M_AM_Bm_Um_V}{N''}|H_1'|,\n\\een\nwe have by (\\ref{eq:K_Il_1l_2}) and (\\ref{eq:N''def}) that $M_AM_B \\leq N'$, so by (A4) again\n\\ben\n\\frac{M_AM_B}{N'}\\phi(N', \\delta_1, K_1) \\leq  \\phi(M_AM_B, \\delta_1, K_1),\n\\een \nso\n\\ben \\label{eq:phiNprimesbound}\n\t\\phi(N', \\delta_1, K_1) \\phi (N'', \\delta_3, K_3) &\\leq& \\frac{N'}{M_AM_B}\\phi(M_AM_B, \\delta_1, K_1) \\frac{N''}{m_Um_V}\\phi(m_Um_V, \\delta_3, K_3) \\nonumber \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t &\\stackrel {(\\ref{eq:step2psiphi})}{=}&  \\phi(M_AM_B, \\delta_1, K_1) \\phi_{a_1,b_1}.\n\\een\nOn the other hand, \n\\ben \\label{eq:Nprimesproduct}\nN'N'' &=& M_AM_Bm_Um_V |H_1'| \\\\\n&\\stackrel {(\\ref{eq:K_Il_1l_2})}{\\geq}& c\\log^{-20} (K\/\\delta) \\delta^5 M_AM_Bm_Am_B \\\\\n\t\t&\\stackrel {(\\ref{eq:FiberingLemmaSetSizes}), (\\ref{eq:FiberingLemmaSetSizes2})}{\\geq}& c  \\delta^9 \\log^{-22} (K\/\\delta) N.\n\\een\nAlso, since\n\\ben\n\tm_Am_B \\stackrel{\\eqref{eq:m_ibig}}{>} c \\delta^{20} K^{-8} N^{1\/2},   \\nonumber\n\\een\nit follows from the definition of $N''$ in \\eqref{eq:N''def} that\n\\ben\n\tc \\delta^{45} K^{-11}N^{1\/2} \\leq N'' \\leq N^{1\/2} . \\nonumber\n\\een\nThen, since $N'' N' \\leq N$,\n\\ben\n  N' \\leq C \\delta^{-45} K^{11}N^{1\/2}, \\nonumber\n\\een\nand so\n\\ben \\label{eq:Nprimessum}\n N' + N'' \\leq C \\delta^{-45} K^{11}N^{1\/2}.\n\\een\n\nWe now have all the estimates to finish the proof. The bounds (\\ref{eq: delta_1delta_3}), (\\ref{eq:K_1K_3bound}), (\\ref{eq:Nprimesproduct}), (\\ref{eq:Nprimessum}) verify that the parameters \n\\ben \n\\delta' &:=& \\delta_1, \\,\\,\\, \\delta'' := \\delta_4 \\nonumber \\\\\nK' &:=& K_1, \\,\\,\\, K'' := K_4 \\nonumber \\\\\n\\een\nand $N', N''$ indeed satisfy the constraints (\\ref{eq:parameterconstraints}).  Recall that by (A1)  $\\psi(\\cdot, \\delta, K)$ is increasing in the first argument, so by (\\ref{eq:M_1M_2upperbound}) and (\\ref{eq:FirstcoordinateSplit2})\n\\ben \n\t\t\\psi_{*}(N, \\delta, K) \t&\\geq& Ck^2 \\psi\\lr{\\max\\left\\{N^{1\/2}, \\frac{N}{m_Am_B} \\right\\}, \\delta_1, K_1}  \\psi\\lr{\\min \\{ N^{1\/2}, m_Am_B \\}, \\delta_4, K_4} \\nonumber \\\\\n\t\t\t\t\t&\\geq&  \\psi(M_AM_B, \\delta_1, K_1) \\psi_{x, y} . \\label{eq:psistargeneralbound}\n\\een\nIn the previous inequality, we have used monotonicity (A1) and the information that $\\frac{N}{m_Am_B} \\geq M_AM_B $, $N^{1\/2} \\geq m_Um_V$, $m_A \\geq m_U$ and $m_B \\geq m_V$.\n\n\nAlso, (\\ref{eq:phiNprimesbound}) and (\\ref{eq:admissiblepairs2}) verify that \n\\ben \n\\phi_{*}(N, \\delta, K) &\\leq& \\phi(N', \\delta_1, K_1) \\phi (N'', \\delta_4, K_4)  \\nonumber \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t &\\leq& \\phi(M_AM_B, \\delta_1, K_1) \\phi_{a_1, b_1}. \\label{eq:phistargeneralbound}\n\\een\n It follows that the pair $(\\psi_{*}, \\phi_{*})$ is indeed admissible since (\\ref{eq:psistargeneralbound}) and (\\ref{eq:phistargeneralbound}) hold for all base points $(a_1, b_1) \\in G'_1$ and thus uniformly bound \\eqref{eq:admissiblepairs22} and (\\ref{eq:admissiblepairs2}) respectively. \n\\end{proof}\n\n\\section{A better admissible pair} \\label{sec:betterpair}\n\nWith Lemma \\ref{lm:InductionStepLemma} at our disposal we can start with the data $(N, \\delta, K)$ and  reduce the problem to the case of smaller and smaller $N$ and $K$ with reasonable losses in $\\delta$. The process can be described by a binary a tree where each node with the data $(N, \\delta, K)$ splits into two children with the attached data being approximately equal to $(N^{1\/2}, \\delta', K')$ and $(N^{1\/2}, \\delta', K'')$, with $K'K''$ roughly equal to $K$ and $\\delta'\\delta''$ roughly equal to $\\delta$. Thus, when the height of the tree is about $\\log \\log K$, the $K$'s in the most of the nodes should be small enough so that Lemma \\ref{lm:FreimanAdmissiblePair} becomes non-trivial. Going from the bottom to the top we then recover an improved admissible pair of functions at the root node. \n\n\\begin{Lemma} \\label{lm:beteradmissiblepair}\nFor any $\\gamma > 0$ there exists $C(\\gamma) > 0$ such that the pair\n\\ben\n\t\\phi(N, \\delta, K) := \\left( \\frac{\\delta}{K}\\right)^{C \\log \\log (K\/\\delta)} N \\label{eq:betterphi}\\\\\n\t\\psi(N, \\delta, K) := k^{ \\log (K\/\\delta)^{C\/\\gamma}} N^{\\gamma}  \\label{eq:betterpsi}\n\\een  \nis admissible. \n\\end{Lemma}\n\n\\begin{proof}\nLet $N, \\delta, K$ be fixed. Take an integer $t = 2^l$ to be specified later ($l$ is going to be the height of the tree and $t$ the total number of nodes). \n\nLet $(\\phi_0, \\psi_0)$ be the Freiman-type admissible pair given by Lemma~\\ref{lm:FreimanAdmissiblePair}. We apply recursively Lemma~\\ref{lm:InductionStepLemma} and obtain admissible pairs for $i = 1, \\ldots, l$ as follows\n\n\\ben \\label{eq:admissiblepairinductionleveli}\n\t\t\\psi_{i} &:=& \\max Ck^2 \\psi_{i-1} (N', \\delta', K') \\psi_{i-1}  (N'', \\delta'', K'') \\\\\n\t\t\\phi_{i} &:=& \\min \\phi_{i-1}(N', \\delta', K') \\phi_{i-1}(N'', \\delta'', K''),\n\\een\n(with the $\\max$ and $\\min$ taken over the set of parameters constrained by (\\ref{eq:parameterconstraints})). Thus, at the root node we have the admissible pair $\\psi := \\psi_{l-1}, \\phi := \\phi_{l-1}$ given by\n\\ben\n\t\\psi(N, \\delta, K) &:=& (Ck^2)^{2^l} \\prod_{\\nu \\in \\{0, 1 \\}^l} \\psi_0(N'_\\nu, \\delta'_\\nu, K'_\\nu)\t\\label{eq:betterpsiproductformula}\\\\\n\t\\phi(N, \\delta, K) &:=& \\prod_{\\nu \\in \\{0, 1 \\}^l} \\phi_0(N_\\nu, \\delta_\\nu, K_\\nu) \\label{eq:betterphiproductformula} \n\\een\nfor some data $(N_\\nu, \\delta_\\nu, K_\\nu)$ and (possibly different) $(N'_{\\nu}, \\delta'_{\\nu}, K'_{\\nu})$ at the leaf nodes of the tree which attain the respective maxima and minima. For intermediate tree nodes $\\nu$, denoting by $\\{\\nu, 0\\}$ and $\\{\\nu, 1\\}$ the left and right child of $\\nu$ respectively, one has\n\\begin{align}  \\label{eq:parameterconstraintslevel}\n   c_1 \\delta_\\nu^9 \\log^{-22} (K_\\nu\/\\delta_\\nu) N_\\nu &\\leq N_{\\nu, 0}N_{\\nu, 1} \\leq N_\\nu \\\\ \n   N_{\\nu, 0} + N_{\\nu, 1} &\\leq C_1 \\delta_\\nu^{-45} K_\\nu^{11}N_\\nu^{1\/2}\t \\label{eq:Nsquarerootsplitting} \\\\\n   K_{\\nu, 0}K_{\\nu, 1} &\\leq C_1 \\frac{\\log^{15} K_\\nu }{\\delta_\\nu^{20}} K_\\nu \\label{eq:Ksquarerootsplitting}  \\\\ \n   \\delta_{\\nu, 0} \\delta_{\\nu, 1} &\\geq c_1 \\log^{-6} (K_\\nu\/\\delta_\\nu) \\delta_\\nu, \\label{eq:deltasquarerootsplitting}\n\\end{align}\nand similarly for $(N'_{\\nu}, \\delta'_{\\nu}, K'_{\\nu})$. The absolute constants $c_1$ and $C_1$ are exactly those given in the statement of Lemma \\ref{lm:InductionStepLemma} as $c$ and $C$ respectively. They have been relabelled here in an attempt to distinguish them.\n\nIn what follows we assume that $N$ is large enough so that $\\log K_\\nu > C$ and $\\log(\\delta_\\nu^{-1}) > c^{-1}$ and the constants $C, c$ can be swallowed by an extra power of $\\log (K\/\\delta)$.\n\nWe have\n\\[\n \\log \\frac{K_{\\nu, 0}}{\\delta_{\\nu, 0}} +  \\log \\frac{K_{\\nu, 1}}{\\delta_{\\nu, 1}} < 20 \\log \\frac{K_\\nu}{\\delta_\\nu}\n\\]\nso for an arbitrary $1 < l' \\leq l$\n\\beq \\label{eq:Kdeltanubound}\n   \\max_{  \\nu \\in \\{0, 1\\}^{l'} } \\log \\frac{K_\\nu}{\\delta_\\nu}\n   \\leq \\sum_{  \\nu \\in \\{0, 1\\}^{l'} } \\log \\frac{K_\\nu}{\\delta_\\nu}\n   <  20^{l'} \\log \\frac{K}{\\delta}.\n\\eeq\nNext, it follows from \\eqref{eq:deltasquarerootsplitting} and (\\ref{eq:Kdeltanubound}) that \n\\begin{align} \n \\prod_{\\nu \\in \\{0, 1\\}^{l'}} \\delta_\\nu = \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} \\delta_{\\nu,0} \\delta_{\\nu,1} & \\geq  \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} c_1 \\left(\\log \\frac{K_\\nu}{\\delta_\\nu} \\right)^{-6} \\delta_\\nu \\nonumber\n \\\\&> \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} c_1 \\left(20^{l'} \\log \\frac{K}{\\delta}\\right)^{-6} \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} \\delta_\\nu \\nonumber\n \\\\& = \\left(\\frac{20}{c_1}\\right)^{-3l'\\cdot 2^{l'}} \\left(\\log \\frac{K}{\\delta}\\right)^{-3 \\cdot 2^{l'}} \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} \\delta_\\nu . \\label{eq:deltatoiterate}\n\\end{align}\nApplying \\eqref{eq:deltatoiterate} iteratively then yields\n\\begin{equation}\n \\prod_{\\nu \\in \\{0, 1\\}^{l'}} \\delta_\\nu > \\left (\\frac{20}{c_1} \\right)^{-6l' \\cdot 2^{l'}} \\left (\\log \\frac{K}{\\delta}\\right)^{-6\\cdot 2^{l'}} \\delta .\n\\label{eq:deltanuproductbound}\n\\end{equation}\nUsing similar arguments, we obtain the following bounds:\n\\beq \\label{eq:Knuproductbound}\n  \\prod_{\\nu \\in \\{0, 1\\}^{l'}} K_\\nu < \\left(\\frac{20C_1}{c_1}\\right)^{280 \\cdot l' 2^{l'}} \\left( \\log \\frac{K}{\\delta}\\right )^{280 \\cdot 2^{l'}} \\delta^{-20l'}K\n\\eeq\nand\n\\beq \\label{eq:Nnuproductbound}\n\t\\prod_{\\nu \\in \\{0, 1\\}^{l'}} N_\\nu > \\left(\\frac{20}{c_1}\\right)^{-160 \\cdot l' 2^{l'}} \\left(\\log \\frac{K}{\\delta} \\right)^{-160  \\cdot 2^{l'}} \\delta^{9 l'} N.\n\\eeq\nFor more details on how these bounds are obtained, see \\cite[p. 492]{BC}.\n\nSubstituting (\\ref{eq:deltanuproductbound}), (\\ref{eq:Knuproductbound}), (\\ref{eq:Nnuproductbound}) into (\\ref{eq:betterphiproductformula}) and Lemma \\ref{lm:FreimanAdmissiblePair} (\\ref{eq:Freimanadmissiblephi}) we get\n\\begin{align*} \n \\phi (N, \\delta, K) = \\prod_{\\nu \\in \\{0, 1\\}^{l}} \\phi_0(N_\\nu, \\delta_\\nu, K_\\nu)  \n &= \\prod_{\\nu \\in \\{0, 1\\}^{l}} \\left (\\frac{\\delta_\\nu}{K_\\nu}\\right )^C N_\\nu \n \\\\& \\geq  e^{-C'l2^l} \\left( \\log \\frac{K}{\\delta} \\right)^{-C'2^l} \\delta^{lC'} K^{-C'}N,\n\\end{align*}\nfor some suitable $C' > 0$. Taking\\footnote{Strictly speaking we should ensure that $l$ is an integer by taking $l := \\lfloor \\log \\log (K\/\\delta) \\rfloor$. In order to simplify calculations and avoid adding further multiplicative constants, we assume that $l$ as defined here is already an integer.}\n$$\nl := \\log \\log (K\/\\delta) \n$$ \nwe obtain\n\\[ \\phi (N, \\delta, K) \\geq \\left( \\frac{\\delta}{K}\\right)^{C \\log \\log (K\/\\delta)} N\\]\nfor some suitable $C>0$.\n\n\nWe now turn to $\\psi$. For the sake of notation we use again $(N_\\nu, \\delta_\\nu, K_\\nu)$ instead of $(N'_\\nu, \\delta'_\\nu, K'_\\nu)$. The bounds above, however, still hold.\n\nBy \\eqref{eq:betterpsiproductformula} and Lemma \\ref{lm:FreimanAdmissiblePair}\n\\begin{equation}\n\\psi(N,\\delta,K) =(Ck^2)^{2^l}\\prod_{\\nu \\in \\{0,1 \\}^l} \\min \\left\\{ (2k^2)^{\\left(\\frac{K_\\nu}{\\delta\\nu}\\right )^C}, N_\\nu \\right\\}. \n\\label{eq:psibound}\n\\end{equation}\n\n\nIn order to bound the quantity of the right hand side effectively, we will need a suitable uniform bound for individual $N_\\nu$, which we deduce below.\n\nIt follows from \\eqref{eq:deltasquarerootsplitting} that \n\\begin{equation}\n\\delta_{\\nu,0}, \\delta_{\\nu,1} \\geq c_1\\left(\\log \\frac{K_\\nu}{\\delta_\\nu}\\right)^{-6}\\delta_\\nu.\n\\label{eq:deltatriv}\n\\end{equation}\nApplying this bound as well as \\eqref{eq:Kdeltanubound}, it follows that for any $1 \\leq l' \\leq l$ and $\\nu \\in \\{0,1\\}^{l'}$,\n\\begin{equation} \\label{eq:anotherdeltabound}\n\\delta_\\nu = \\delta_{\\nu',\\cdot} \\geq c_1 \\left(\\log \\frac{K_{\\nu'}}{\\delta_{\\nu '}}\\right)^{-6}\\delta_{\\nu '} \\geq  (20C)^{-6l'} \\left(\\log \\frac{K}{\\delta}\\right)^{-6}\\delta_{\\nu '}.\n\\end{equation}\nIteratively applying \\eqref{eq:anotherdeltabound} yields\n\\begin{equation} \\label{eq:yetanotherdeltabound}\n\\delta_\\nu \\geq (20C)^{-6l'^2} \\left(\\log \\frac{K}{\\delta}\\right)^{-6l'}\\delta.\n\\end{equation}\n\nSimilarly, since $K_\\nu \\geq \\delta_\\nu$, it follows from \\eqref{eq:Ksquarerootsplitting} and \\eqref{eq:deltatriv} that\n\\[ K_{\\nu,0} \\leq C_1 \\frac{K_\\nu\\log^{15} K_\\nu}{\\delta_\\nu^{20}\\delta_{\\nu,1}}\n\\leq C_1' \\frac{\\left(\\log \\frac{K_\\nu}{\\delta_\\nu} \\right)^{21}}{\\delta_\\nu^{21}}K_\\nu.\n\\]\nThe same argument implies that $ K_{\\nu,1} \\leq C_1' \\frac{K_\\nu \\left(\\log \\frac{K_\\nu}{\\delta_\\nu} \\right)^{21}}{\\delta_\\nu^{21}}$. Therefore, by applying \\eqref{eq:Kdeltanubound} and \\eqref{eq:yetanotherdeltabound}, it follows that for any $\\nu \\in \\{0,1\\}^{l'}$,\n\\begin{equation} \\label{eq:anotherKbound}\nK_\\nu = K_{\\nu',*} \\leq C_1' \\frac{K_{\\nu'} \\left(\\log \\frac{K_{\\nu'}}{\\delta_{\\nu'}} \\right)^{21}}{\\delta_{\\nu'}^{21}} \\leq \\frac{(20C)^{147l'^2} \\left( \\log \\frac{K}{\\delta}\\right)^{147l'}}{\\delta^{21}} K_{\\nu'} .\n\\end{equation}\nIterating \\eqref{eq:anotherKbound} yields\n\\begin{equation}\nK_\\nu  \\leq \\frac{(20C)^{147l'^3} \\left( \\log \\frac{K}{\\delta}\\right)^{147l'^2}}{\\delta^{21l'}} K.\n\\label{eq:yetanotherKbound}\n\\end{equation}\n\nTo bound $N_\\nu$, first note that \\eqref{eq:Nsquarerootsplitting}, \\eqref{eq:yetanotherdeltabound} and \\eqref{eq:yetanotherKbound} together imply that for any $\\nu' \\in \\{0,1\\}^{l'} $,\n\\[ N_{\\nu, 0} + N_{\\nu, 1} \\leq C_1 \\delta_\\nu^{-45} K_\\nu^{11}N_\\nu^{1\/2}\t\\leq \\frac{(20C_1)^{1887 l'^3} \\left( \\log \\frac{K}{\\delta} \\right)^{1887l'^2}K^{11}}{\\delta^{276l'}}N_\\nu^{1\/2} .\\]\n Applying this bound iteratively yields (with some rather crude estimates)\n \\begin{equation}\n N_\\nu \\leq \\frac{(20C_1)^{4000 l'^3} \\left( \\log \\frac{K}{\\delta} \\right)^{4000 l'^2}K^{22}}{\\delta^{4000 l'}}N^{\\frac{1}{2^{l'}}}.\n \\label{eq:yetanotherNbound}\n \\end{equation}\n\nBefore inserting \\eqref{eq:yetanotherNbound} into \\eqref{eq:psibound}, we split the data $(N_\\nu, \\delta_\\nu, K_\\nu)$ into two parts, $I \\cup J = \\{0, 1 \\}^l$, such that\n$$\n\tI= \\left\\{  \\nu : \\frac{K_\\nu}{\\delta_\\nu} < T \\right \\}\n$$\nand\n$$\n\tJ= \\left \\{ \\nu:  \\frac{K_\\nu}{\\delta_\\nu} \\geq T \\right \\},\n$$\nwith the threshold $T$ specified later. \n\nBy (\\ref{eq:deltanuproductbound}) and (\\ref{eq:Knuproductbound}) we see that $|J|$ is rather small:\n\\ben \\label{eq:AJbound}\n T^{|J|} \\leq \\prod_{\\nu \\in \\{0, 1 \\}^l}  \\frac{K_\\nu}{\\delta_\\nu} < \\lr{\\frac{20C_1}{c_1}}^{286 \\cdot l 2^{l}} \\log (\\frac{K}{\\delta})^{286 \\cdot 2^{l}} \\delta^{-21l}K.\n\\een\nSet $t := 2^l$, so it follows from (\\ref{eq:AJbound}) that for an appropriate constant $C_2$,\n$$\n|J| \\log T  \\leq C_2 l t.\n$$\nChoose\n\\begin{equation}\n \\log T := C_2\\gamma^{-1} l = \\frac{C_2 \\log \\log (K\/\\delta)}{\\gamma}.\n\\label{eq:Adefn}\n\\end{equation}\nThus \n\\begin{equation}\n\\frac{|J|}{t} \\leq \\frac{C_2 l}{\\log T} = \\gamma.\n\\label{eq:Jbound}\n\\end{equation}\n\nWe are finally ready to put everything together:\n\\begin{align*}\n\\psi(N,\\delta,K) &\\stackrel{\\eqref{eq:psibound}}{=}(Ck^2)^{2^l}\\prod_{\\nu \\in \\{0,1 \\}^l} \\min \\left\\{ (2k^2)^{\\left(\\frac{K_\\nu}{\\delta\\nu}\\right )^C}, N_\\nu \\right\\}. \n\\\\& \\leq (Ck^2)^{2^l}\\prod_{\\nu \\in I} (2k^2)^{T^C} \\prod_{\\nu \\in J} N_\\nu\n\\\\ & \\stackrel{\\eqref{eq:yetanotherNbound}}{\\leq} (C'k^2)^{tT^C}   \\left(\\frac{(20C_1)^{4000 l^3} \\left( \\log \\frac{K}{\\delta} \\right)^{4000 l^2}K^{22}}{\\delta^{4000 l}}N^{\\frac{1}{t}}\\right)^{|J|}\n\\\\& \\stackrel{\\eqref{eq:Jbound}}{\\leq} k^{ \\left( \\log \\frac{K}{\\delta} \\right)^{\\frac{C''}{\\gamma}}}N^{\\gamma}.\n\\end{align*}\n\n\n\\end{proof}\n\n\\section{A strong admissible pair} \\label{sec:strongpair}\n\nFinally, in this section we will use Lemma \\ref{lm:beteradmissiblepair} to get an even better pair of admissible functions. \n\\begin{Lemma} \\label{lm:finaldamissiblepairs}\nGiven $0 < \\tau, \\gamma < 1\/2$ there exist positive constants $\\alpha_i(\\tau, \\gamma,k), \\beta_i (\\tau, \\gamma,k), i= 1, 2, 3$ such that for all sufficiently large $N$, the pair\n\\ben\n  \\phi (N,\\delta,K):= K^{-\\alpha_1} \\delta^{\\alpha_2 \\log \\log N} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau} \\label{eq:bestphi} \\\\\n  \\psi (N, \\delta, K) :=   K^{\\beta_1} \\delta^{-\\beta_2 \\log \\log N} e^{-\\beta_3 (\\log \\log N)^2 } N^{\\gamma} \\label{eq:bestpsi}\n\\een\nis admissible. \n\n\\end{Lemma}\n\\begin{proof}\n\nThe strategy of the proof is as follows. We start with the already not-so-bad admissible pair given by Lemma \\ref{lm:beteradmissiblepair} and improve it by repeated application of Lemma \\ref{lm:InductionStepLemma}. \n\nLet $P_N[\\phi, \\psi]$ be the predicate that the pair $(\\phi, \\psi)$ given by (\\ref{eq:bestphi}) and (\\ref{eq:bestpsi}) is admissible in the sense of Definition \\ref{def:admissible_pairs} for all graphs of size at most $N$ and at least $N^{1\/2}$. \n\nWe are going to prove that \n\\begin{enumerate} \n   \\item The base case: $P_{N_0}[\\phi, \\psi]$ is true for some $N_0(\\tau, \\gamma)$.\n   \n   \\item The inductive step: $P_N[\\phi, \\psi] \\Rightarrow P_{N^{3\/2}}[\\phi, \\psi]$.\n\\end{enumerate}\nThe exponent $3\/2$ is of little importance here and is taken with much room to spare. Lemma \\ref{lm:finaldamissiblepairs} will then follow by induction, for all $N\\geq N_0$.\n\nIn order to prove (1) it suffices to find a fixed threshold $N_0(\\tau, \\gamma)$ such that the pair (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) is either trivial or worse than that given by Lemma \\ref{lm:beteradmissiblepair} if $N \\leq N_0$. One can achieve this by fine-tuning the constants $\\alpha_1, \\beta_1$, which we now explain. \n\n\n\n\n\n\n\n\n\nApply Lemma \\ref{lm:beteradmissiblepair} with $\\gamma= \\gamma \/4$ to obtain an admissible pair given by (\\ref{eq:betterphi}), (\\ref{eq:betterpsi}). We seek to choose $\\alpha_1, \\beta_1$ and  $N_0(\\delta, \\gamma)$ such that for each $N$ in the range $N_0^{1\/2} \\leq N \\leq N_0$\n\\ben\n\t\\left( \\frac{\\delta}{K}\\right)^{C(\\gamma) \\log \\log (K\/\\delta)} N \\geq K^{-\\alpha_1} \\delta^{\\alpha_2 \\log \\log N} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau} \\label{eq:bestphi2} \\\\\n \\min \\{N,\t\\exp\\lr{\\log k \\cdot \\log (K\/\\delta)^{C(\\gamma)\/\\gamma}} N^{\\gamma\/4} \\} \\leq K^{\\beta_1} \\delta^{-\\beta_2 \\log \\log N} e^{-\\beta_3 (\\log \\log N)^2 } N^{\\gamma} . \\label{eq:bestpsi2} \n\\een\nTo ensure (\\ref{eq:bestphi2}) holds it is sufficient to take $\\alpha_1 = \\frac{C(\\gamma)}{2}\\log \\log N_0$ with  $C(\\gamma) > 0$ from Lemma~\\ref{lm:beteradmissiblepair} and to take $\\alpha_2 = C_2 \\alpha_1$ and $\\alpha_3=C_3 \\alpha_1$ for some absolute constants $C_2,C_3 \\geq 1$. Indeed,\n\\begin{align*} K^{-\\alpha_1} \\delta^{\\alpha_2 \\log \\log N} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau} & \n\\leq \\left(\\frac{\\delta}{K} \\right)^{\\alpha_1} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau}\n\\\\& \\leq \\left(\\frac{\\delta}{K} \\right)^{C(\\gamma) \\log \\log N} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau}\n\\\\& \\leq \\left(\\frac{\\delta}{K} \\right)^{C(\\gamma) \\log \\log N}  N,\n\\end{align*}\nwhere the last inequality holds as long as we take $N_0$ sufficiently large (and thus also $N$ is sufficiently large). Inequality \\eqref{eq:bestphi2} then follows since the inequality $N \\geq \\frac{K}{\\delta}$ holds by definition of $N,\\delta$ and $K$.\n\n\nEnsuring (\\ref{eq:bestpsi2}) is more involved, as later on want to impose the further constraint $\\beta_3 > \\beta_2 > \\beta_1$.\nFor now, it suffices to guarantee that \n\\ben \\label{eq:Kboundexponencial}\n \\log k \\cdot \\log \\left( \\frac{K}{\\delta} \\right)^{\\frac{ C}{\\gamma}} < \\frac{\\gamma}{4} \\log N\n\\een\nand\n\\ben \\label{eq:B_3boundexponenial}\ne^{\\beta_3 (\\log \\log N)^2} < N^{\\frac{\\gamma}{2}}.\n\\een\nHowever, the bound (\\ref{eq:Kboundexponencial}) fails only if $K\/\\delta$ is rather large, namely\n$$\n\\frac{K}{\\delta} > e^{\\log^{c\\gamma} N}\n$$\nfor some $c(C, \\gamma, k) > 0$.  In this case it suffices to take $\\beta_1$ so large that \n\\[K^{\\beta_1} \\delta^{-\\beta_2 \\log \\log N} e^{-\\beta_3 (\\log \\log N)^2 } N^{\\gamma} > N\n\\]\nand thus (\\ref{eq:bestpsi2})  holds. To this end, we set\n$$\n\\beta_1 := (\\log N_0)^{1 - c\\gamma}\n$$\nand make the constraint that, say,\n$$\n\\beta_3, \\beta_2 < 10 \\beta_1  \\log \\log {N_0}.\n$$ \nMoreover, this constraint on $\\beta_3$ also ensures that \\eqref{eq:B_3boundexponenial} holds for $N$ sufficiently large.\n\nSumming up, we have found some fixed threshold $N_0(\\tau, \\gamma)$  at which (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) become admissible with fixed $\\alpha_1, \\beta_1$ and still some freedom to define the constants $\\alpha_2, \\beta_2, \\alpha_3$, and $\\beta_3$.\n\nWe now turn to part (2) of the induction scheme, the inductive step. Assuming that $N', N''$ are at the scale so that (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) are admissible with the data $(N', \\delta', K'); (N'', \\delta'', K'')$ we will show that  (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) are also admissible for the data $(N, \\delta, K)$ with $N \\approx N'N''$.\n\nAssuming $\\beta_1$ (or $N_0$) is large enough we may assume that \n\\ben \\label{eq:Kdeltasmall}\n\\frac{K}{\\delta} < N^{10^{-4}},\n\\een\nas otherwise (\\ref{eq:bestpsi})  $ > N$ which is trivially admissible.  \n\nWe need to estimate \n$$\n \\psi (N', \\delta', K') \\psi  (N'', \\delta'', K'') \n$$\nfrom above and \n$$ \n \\phi(N', \\delta', K') \\phi(N'', \\delta'', K''),\n$$\nfrom below in order to verify that (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) are admissible for $(N, \\delta, K)$. By (\\ref{eq:Kdeltasmall}), the constraints (\\ref{eq:parameterconstraints}) can be relaxed to\n\\ben\nN \\geq N'N'' &>& N\\left( \\frac{\\delta}{\\log N}\\right)^{40} > N^{99\/100} \\label{eq:N_1N_2simple} \\\\\nN' + N'' &<& N^{1\/2} \\left(\\frac{K}{\\delta} \\right)^{45} < N^{1\/2 + 1\/40}  \\label{eq:N_1plusN_2simple}\\\\\n\\delta' \\delta'' &>& \\frac{\\delta}{\\log^6 N} \\label{eq:deltadeltaprimesimple}\\\\\nK'K'' &<& \\delta^{-20} (\\log N)^{15}K \\label{eq:KprimeKprime} .\n\\een \n\nFrom (\\ref{eq:N_1N_2simple}) and (\\ref{eq:N_1plusN_2simple}) we have (with room to spare)\n\\beq\nN^{1\/2 - 1\/20} < N', N'' < N^{1\/2 + 1\/20}\n\\eeq\nand so assuming $N$ is large enough\n\\beq\n\\frac{99}{100} \\log \\log N < \\log \\log N',\\ \\log \\log N'' < \\log \\log N - \\log \\frac{20}{11}.\n\\eeq\nWith the constraints above, it suffices to verify (writing $l l$ for $\\log \\log $ as in \\cite{BC}) that\n\\ben \\label{eq:bestphisubstitution}\n(K'K'')^{-\\alpha_1} (\\delta')^{\\alpha_2 l l N'} (\\delta'')^{\\alpha_2 l l N''} e^{\\alpha_3[(l l N')^2 + (l l N'')^2]} (N' N'')^{1-\\tau}\n\\een\nis indeed always bounded below by (\\ref{eq:bestphi}).\nWe can bound (\\ref{eq:bestphisubstitution}) by\n\\ben\nK^{-\\alpha_1} \\delta^{\\alpha_2 l l N} e^{\\alpha_3 (l l N)^2} N^{1-\\tau} u \\cdot v\n\\een\nwhere\n\\ben\nu &=& (\\log N)^{-15\\alpha_1 - 6\\alpha_2 l l N - 40} e^{\\frac{9}{10}\\alpha_3 (l l N)^2} \\\\\nv &=& \\delta^{20\\alpha_1 - \\log \\frac{20}{11} \\alpha_2 + 40}.\n\\een\nFor suitable choices of $\\alpha_2, \\alpha_3 > \\alpha_1$ both $u, v > 1$ so  (\\ref{eq:bestphi}) is admissible. \n\nSimilarly for (\\ref{eq:bestpsi}) we have\n\\ben\n(K'K'')^{\\beta_1} (\\delta')^{-\\beta_2 l l N'} (\\delta'')^{-\\beta_2 l l N''} e^{-\\beta_3[(l l N')^2 + (l l N'')^2]} (N' N'')^{\\gamma} \\\\\n\t\t\t\t\t<K^{\\beta_1} \\delta^{-\\beta_2 l l N} e^{-\\beta_3 (l l N)^2} N^{\\gamma} u \\cdot v\n\\een\nwith\n\\ben\nu &=& (\\log N)^{15 \\beta_1 + 6\\beta_2 l l N} e^{-\\frac{9}{10}\\beta_3 (l l N)^2} \\\\\nv &=& \\delta^{-20\\beta_1 + \\log \\frac{20}{11} \\beta_2 }.\n\\een\nAgain, by taking suitable $\\beta_3 > \\beta_2 > \\beta_1$ we make $u, v < 1$ so (\\ref{eq:bestpsi}) is admissible. This closes the induction on scales argument and finishes the proof.\n\n\n\\end{proof}\n\n\\section{Concluding the proof of Theorem \\ref{thm:goodsubset} } \\label{sec:conclusion2}\nWe are finally ready to finish the proof of Theorem \\ref{thm:goodsubset}. Recall that the aim is to show that, given $0 < \\tau, \\gamma < 1\/2$, there are positive constants $C_1=C_1(\\tau,\\gamma,k)$ and $C_2=C_2(\\tau,\\gamma,k)$, such that for any $A \\subset \\mathbb Q$ with $|AA| \\leq K|A|$, there exists $A' \\subset A$ with $|A'| \\geq K^{-C_1}|A|^{1-\\tau}$, such that $A'$ is $K^{C_2}|A|^{\\gamma}$-separating.\n\nSince $|AA| \\leq K|A|$, after applying the prime evaluation map, we have $|\\mathcal P(A) +\\mathcal P(A)| \\leq K|\\mathcal P(A)|$. Fix $\\gamma'=\\gamma\/2$, $\\tau'=\\tau\/2$, and apply Lemma \\ref{lm:finaldamissiblepairs} for this choice of $\\gamma', \\tau'$, with the full graph $G= \\mathcal P(A) \\times \\mathcal P(A)$. It follows that there is a subgraph $G' \\subset G$ such\n\\[|G'| \\geq K^{-\\alpha_1} e^{\\alpha_3(\\log\\log |A|)^2} |A|^{2-2 \\tau'} \\geq K^{-\\alpha_1}  |A|^{2-2 \\tau'} \\]\nand such that for each $v \\in V(G)$ the $\\mathcal P$-preimages of $N_{G'}(v)$ is\n\\[K^{\\beta_1} e^{-\\beta_3(\\log\\log |A|)^2} |A|^{2\\gamma'} \\leq K^{\\beta_1} |A|^{2\\gamma'} \\]\nseparating.\\footnote{Note here that we have discarded the extra information coming from the terms of the form $e^{\\pm C (\\log\\log |A|)^2}$.} \n\nThen, by the pigeonhole principle, there is a vertex $v \\in V(G)$ such that $|N_{G'}(v)| \\geq |A|^{1-2 \\tau'}$. Write $A'=\\mathcal P^{-1}(N_{G'}(v))$ for the preimage of the neighbourhood of $v$. Then this is a subset of $A$ with the required properties.\n\n\\section{Further Applications} \\label{sec:applications}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:STinverse}] Recall that Theorem \\ref{thm:STinverse} is the following statement. For all $\\gamma \\geq 0$ there exists a constant $C=C(\\gamma)$ such that for any finite $A\\subset \\mathbb Q$ with $|AA| \\leq K|A|$ and any finite set $L$ of lines in the plane, $I(P,L)\\leq 3 |P| + |A|^{\\gamma}K^C|L|$, where $P= A \\times A$.\n\nFirst of all, observe that horizontal and vertical lines contribute a total of at most $2|P|$. This is because each point $p \\in P$ can belong to at most one horizontal and one vertical line. Similarly, lines through the origin contribute at most $|P| +|L|$ incidences, since each point aside from the origin belongs to at most one such line, and the origin itself may contribute $|L|$ incidences.\n\nIt remains to bound incidences with lines of the form $y=mx+c$, with $m,c \\neq 0$. Let $l_{m,c}$ denote the line with equation $y=mx+c$. Note that, if $m \\notin \\mathbb Q$ then $l_{m,c}$ contains at most one point from $P$. Indeed, suppose $l_{m,c}$ contains two distinct points $(x,y)$ and $(x',y')$ from $P$. In particular, since $A \\subset \\mathbb Q$, $x,y,x',y' \\in \\mathbb Q$. Then $l_{m,c}$ has direction $m= \\frac{y-y'}{x-x'}$. Therefore, lines $l_{m,c}$ with irrational slope $m$ contribute at most $|L|$ incidences.\n\nNext, suppose that $m \\in \\mathbb Q$ and $c \\notin \\mathbb Q$. Then $l_{m,c}$ does not contain any points from $P$, since if it did then we would have a solution to $y=mx+c$, but the left hand side is rational and the right hand side is irrational.\n\nIt remains to consider the case when $m,c \\in \\mathbb Q^*$. An application of Theorem \\ref{thm:BS_almost_subgroups} implies that $|l_{m,c} \\cap P| \\leq K^C|A|^{\\gamma}$. Therefore, these lines contribute a total of at most $|L|K^C|A|^{\\gamma}$ incidences.\n\nAdding together the contributions from these different types of lines completes the proof.\n\n\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:additivebasis}]\nRecall that Theorem \\ref{thm:additivebasis} states that, for any $\\gamma > 0$ there exists $C(\\gamma)$ such that\nfor an arbitrary $A \\subset \\mathbb{Q}$ with $|AA| = K|A|$ and $B, B' \\subset \\mathbb{Q}$,\n$$\nS := \\left|\\{(b, b') \\in B \\times B' : b + b' \\in A\\} \\right| \\leq 2|A|^\\gamma K^C \\min \\{|B|^{1\/2}|B'| + |B|, |B'|^{1\/2}|B| + |B'| \\}. \n$$\n\nWe will prove that \n\\begin{equation}\nS \\leq 2|A|^\\gamma K^C(|B'|^{1\/2}|B| + |B'|).\n\\label{eq:case1}\n\\end{equation}\nSince the roles of $B$ and $B'$ are interchangeable, \\eqref{eq:case1} also implies that $S \\leq 2|A|^\\gamma K^C(|B|^{1\/2}|B'| + |B|)$, and thus completes the proof.\n\nLet $\\gamma > 0$ and $C(\\gamma)$, given by Theorem~\\ref{thm:BS_almost_subgroups}, be fixed. Without loss of generality assume that $S \\geq 2|B'|$ as otherwise the claimed bound is trivial. \n\nFor each $b \\in B$ define \n$$\nS_b := \\{ b' \\in B' : b + b' \\in A\\},\n$$\nand similarly for $b' \\in B'$\n$$\nT_{b'} := \\{b \\in B: b' + b \\in A \\}.\n$$\nIt follows from Theorem~\\ref{thm:BS_almost_subgroups} that for $b_1,b_2 \\in B$ with $b_1 \\neq b_2$ \n$$\n|S_{b_1} \\cap S_{b_2}| \\leq |A|^\\gamma K^C\n$$\nsince each $x \\in S_{b_1} \\cap S_{b_2}$ gives a solution $(a, a') := (b_1 + x, b_2 + x)$ to\n$$\na - a' = b_1 - b_2\n$$\nwith $a, a' \\in A$.\n\nOn the other hand, by double-counting and the Cauchy-Schwarz inequality,\n$$\n\\sum_{b \\in B} |S_b| + \\sum_{b_1,b_2 \\in B : b_1 \\neq b_2} |S_{b_1} \\cap S_{b_2}| =\n\\sum_{b' \\in B'} |T_{b'}|^2 \\geq |B'|^{-1}(\\sum_{b' \\in B'} |T_{b'}|)^2 = |B'|^{-1}S^2.\n$$\nTherefore,\n$$\n\\sum_{b_1,b_2 \\in B : b_1 \\neq b_2} |S_{b_1} \\cap S_{b_2}| \\geq |B'|^{-1}S^2 - \\sum_{b \\in B} |S_b| = |B'|^{-1}S^2 - S \\geq \\frac{1}{2} |B'|^{-1}S^2\n$$\nby our assumption.\n\n\nThe left-hand side is at most $ |B|^2|A|^\\gamma K^C$, and so\n$$\n\tS \\leq (2|A|^\\gamma K^C)^{1\/2} |C|^{1\/2}|B'|,\n$$\nwhich completes the proof.\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:proddiff}]\nRecall that Theorem \\ref{thm:proddiff} states that for all $b$ there exists $k$ such that for all $A,B \\subset \\mathbb Q$ with $|B| \\geq 2$, $|(A+B)^{k}| \\geq |A|^b$.\n\nSince $|B| \\geq 2$, there exist two distinct elements $b_1, b_2 \\in B$. Apply Theorem \\ref{thm:mainmain} to conclude that for all $b$ there exists $k=k(b)$ with\n\\[ |(A+B)^k| \\geq  \\max \\{|(A+b_1)^k|, |((A+b_1)+(b_2-b_1))^{k} |\\} \\geq |A|^b.\\]\n\n\n\\end{proof}\n\n\\section*{Acknowledgements}\nOliver Roche-Newton was partially supported by the Austrian Science Fund FWF Project P 30405-N32. Dmitrii Zhelezov was supported by the Knut and Alice Wallenberg Foundation Program for Mathematics 2017. \n\nWe thank Brendan Murphy, Imre Ruzsa and Endre Szemer\\'edi for helpful conversations.\n\n\n\n\n\n\\AtEndEnvironment{thebibliography}{\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe asymptotic behavior of the ratio $R=F_2\/F_1$ of Pauli and Dirac electromagnetic nucleon form factors (FFs) has recently arised much interest from experimental and theoretical point of view. The last experimental data in space-like (SL) region \\cite{Jo00}, about the momentum transfer squared ($q^2=-Q^2) dependence$\\footnote{In the following text we will use the notation $t=q^2$ in TL region} of the ratio of the Sachs electric and magnetic FFs, $\\mu G_{Ep}(Q^2)\/ G_{Mp}(Q^2)$ ($\\mu$ is the proton magnetic moment), which has been measured with the polarization transfer method \\cite{Re68},  changed the belief that the QCD asymptotic behavior of $F_2\/F_1\\simeq 1\/Q^2$ \\cite{Ma73} had already been reached  for $Q^2\\ge 2$ GeV$^2$ \\cite{Le80}. \n\nThe recent data suggested a different behavior of this ratio: $F_2\/F_1\\simeq 1\/\\sqrt{Q^2}$. Such dependence has been justified in framework of different theoretical approaches \\cite{Ra03,Ra02,Mi02,Fr96,Ho96,Ca00,Wa01,Bo02}. Another approach, confirming the QCD $1\/Q^2$ behavior, discovered the importance of logarithmic corrections, \n$R\\simeq ln^2(Q^2\/\\Lambda^2) \/Q^2 $ \\cite{Be02}, where $\\Lambda$ is the soft scale related to the size of the nucleon. Note that the unexpected behavior of the ratio $\\mu G_{Ep}(Q^2)\/ G_{Mp}(Q^2)$ was predicted before the experiment took place, by a particular VDM model \\cite{Ia73} and also in framework of a soliton model \\cite{Ho96}.\n\nThe assumption of the analyticity of FFs \\cite{Dr61} allows to connect the nucleon FFs in SL and time-like (TL) regions and to study the behavior of the ratio $F_2\/F_1$ in TL region. The analyticity of FFs, which has been discussed  for example in Ref. \\cite{Ga66}, allows to extend a parametrization of FFs available in one kinematical region to the other kinematical region.\n\nDispersion relation approaches \\cite{Ho76,Du03,Ha04}, which are based essentially on the analytical properties of nucleon electromagnetic FFs, can be considered a powerful tool for the description of the $Q^2$ behavior of FFs in the entire kinematical region.\n\nThe VDM model \\cite{Ia73}, after appropriate treatment of the $\\rho$ contribution, can be also extrapolated from the SL region to the TL region \\cite{Ia03,Ia04,Bi04}.\n\nThe quark-gluon string model \\cite{Ka00}  allowed firstly to find the $Q^2$ dependence of the electromagnetic FFs in TL region, in a definite analytical form, which can be continued in the SL region.\n\nOne of the problems concerning FFs of pions and nucleons is the large difference in the absolute values in SL and TL regions. For example, at $q^2$=18 GeV$^2$,  the largest value at which proton TL FFs have been measured \\cite{An03}, the corresponding values in TL and SL regions differ by a factor of two. The analyticity of FFs allows to apply the Phragm\\`en-Lindel\\\"of \ntheorem \\cite{Ti39} which gives a rigorous prescription for the asymptotic behavior of \nanalytical functions: \n\\begin{equation}\n\\lim_{t\\to -\\infty} F^{(SL)}(t) =\\lim_{t\\to \\infty} F^{(TL)}(t).\n\\label{eq:eq1}\n\\end{equation}\nThis means that, asymptotically, FFs have the following constraints: \n\\begin{enumerate}\n\\item The imaginary part of FFs, in TL region, vanishes: $ Im F_i(t)\\to 0,$ as $t\\to \\infty $;\n\\item The real part of FFs, in TL region, coincides with the \ncorresponding value in SL region: $ Re  F_i^{(TL)}(t) [t\\to \\infty ]= F_i^{(SL)}(t) [t\\to -\\infty ]$, because FFs are real functions in SL region, due to the hermiticity of the corresponding electromagnetic Hamiltonian.\n\\end{enumerate}\nThe existing experimental data violate the \nPhragm\\`en-Lindel\\\"of theorem, even at $t$ values as large as 18 GeV$^2$ \\cite{ETG01}. In order to test the two requirements stated above, the knowledge of the differential cross section for $e^++e^-\\leftrightarrow p+\\bar{p}$ is not sufficient, and polarization phenomena have to be studied also. In this respect, T-odd polarization observables, which are determined by $Im F_1F_2^*$, are especially interesting. The simplest of these observables is the $P_y$ component of the proton polarization in $e^++e^-\\to p+\\bar{p}$ that in general does not vanish, even in collisions of unpolarized leptons \\cite{Du96}, or the asymmetry of leptons produced in $p+\\bar{p}\\to e^++e^-$, in the collision of unpolarized antiprotons with a transversally polarized proton target (or in the collision of transversally polarized antiprotons on an unpolarized proton target) \\cite{Bi93}. \n\nThese observables are especially sensitive to different possible parametrizations of the ratio $R$, suggested by QCD and VDM models. Calculations have been done up to $t\\simeq $ 40 GeV$^2$ and show that the $P_y$ component remains large in absolute value \\cite{Br04}. For example, QCD inspired parametrizations, which fit reasonably well the data in the SL region, predict $|P_y|\\simeq$ 35\\% up to $t\\simeq $ 40 GeV$^2$. Such behavior has to be considered an indication that the corresponding asymptotics are very far, in agreement with the estimations of the quark-gluon string model \\cite{Ka00} and VDM approach \\cite{Ia03}.\n\nNote another important property of QCD inspired predictions for nucleon FFs: the corresponding $Im F_i(t)$, $t\\ge 4m^2$, $i=1,2$ ($m$ is the nucleon mass), either vanish or have a definite sign in the TL region. The previously quoted parametrizations can not apply in the whole TL region: the asymptotic pQCD behavior follows $F_1(t)\\simeq t^{-2}$ and $F_2(t)\\simeq t^{-3}$ at large $t$, according to the quark counting rules \\cite{Ma73}. The superconvergent conditions:\n\\begin{equation}\n\\int_{t_0}^{\\infty} Im F_i(t)dt=0,~i=1,2\n\\label{eq:eq2}\n\\end{equation}\nhas to be satisfied, where the lower limit corresponds to $t_0=4m_{\\pi}^2$, for isovector FFs, and $t_0=9m_{\\pi}^2$ for isoscalar FFs, where $m_{\\pi}$ is the pion mass.\n\nThis implies that the nonzero QCD-contribution to Eq. (\\ref{eq:eq2}) has to be compensated by the corresponding non-perturbative contribution of opposite sign. We can expect that such contribution mainly arises from the special region of $t$: $t_0\\le t\\le 4m^2$, which is unphysical for the process $e^+ + e^-\\leftrightarrow p+\\bar{p}$. The contribution from the different vector mesons (with different masses) is expected to be very important here. We can say that the superconvergent condition (\\ref{eq:eq2}) can be interpreted as a manifestation of the special duality between pQCD, from one side, and the vector meson contribution, from another side \\cite{Gr74}. In principle such duality is similar to the well known Gilman-Bloom duality \\cite{Bl71}, concerning the electromagnetic properties of the nucleons in SL region, when the deep inelastic electron nucleon scattering is dual to the excitation of different nucleonic resonances in $e^-+N\\to e^-+N^*$. Also one can mention the duality in hadron physics relating the high energy behavior of the amplitudes of hadron-hadron scattering, from one side, to the resonance physics, on the other side.\n\nReturning to the unphysical region, $t_0\\le t\\le 4m^2$, we recall, for completeness, that another interesting physical effect has to be taken into account here: a specific $\\bar{N}N$ bound states, or even gluon states with $J^{PC}=1^{--}$ quantum numbers. And, due to the analyticity of FFs, these effects should appear in the SL region of momentum transfer, and should be correlated with the asymptotic behavior of FFs.\n\nOur main aim here is to discuss the asymptotic behavior of the existing parametrizations for $F_2\/F_1$ in TL region, from the point of view of the Phragm\\`en-Lindel\\\"of theorem. In particular we will analyze the behavior of $Im (F_2\/F_1)$, its convergence to zero and study more particularly the asymptotic behavior  of the $P_y$-component of the proton polarization in $e^+ +e^-\\to \\bar{p} +p$, which contains equivalent information. For completeness we will also consider the behavior of the ratio ${\\cal R}= |F_2\/F_1|_{TL}\/|F_2\/F_1|_{SL}$\nwhich should converge asymptotically to one, following the Phragm\\`en-Lindel\\\"of theorem.\n\nIn order to have a quantitative estimation of the corresponding value of the relevant variable, we will use the following prescription, from Ref. \\cite{Ia03}: {\\it \"A function  $f(z)$ is said to be x\\% scaled when its value is x\\% of the asymptotic value  $f_{as}(z)$. The value at which this condition is met is the solution of the equation $|f(z)|=x |f_{as}(z)|$ \"}. For the cases considered here, this definition translates into the following three equations\\footnote{When $f_{as}(z)=0$, we take $|f(z)|=x$.}:\n\\begin{eqnarray}\n{\\cal F}&=&|Im (F_2\/F_1)|\/|Re (F_2\/F_1)|=\\Delta \\label{eq:eqf}\\\\\n|P_y|&=&\\Delta~ \\label{eq:eqp}\\\\\n{\\cal R}&=&|F_2\/F_1|_{TL}\/|F_2\/F_1|_{SL}=1+\\Delta\n\\label{eq:eqr}\n\\end{eqnarray}\nwhere we will take $\\Delta=0.1$  and  $\\Delta=0.05$ in order to characterize the deviations from the asymptotic predictions of the Phragm\\`en-Lindel\\\"of theorem.\n\nFor this aim, we use the following three different parametrizations, which apply in the SL region:\n\\begin{equation}\n\\displaystyle\\frac{F_2}{F_1}=\\displaystyle\\frac{a}{\\sqrt{(-t)}},~a=1.25\\mbox{~GeV}\\mbox{~from~Ref.~\\protect\\cite{Br04}},\n\\label{eq:eq5}\n\\end{equation}\n\\begin{equation}\n\\displaystyle\\frac{F_2}{F_1}=0.17[\\mbox{~GeV}^2] \\displaystyle\\frac{ln^2(-t\/\\Lambda^2)}{(-t)}\\mbox{~with~} \\Lambda=0.3\\mbox{~GeV} \\mbox{~from~Ref.~ \\protect\\cite{Be02}}, \n\\label{eq:eq6}\n\\end{equation}\nand the VDM inspired  parametrization from Ref. \\protect\\cite{Ia04}:\n\\begin{equation}\n\\displaystyle\\frac{F_2}{F_1}=\\displaystyle\\frac{F_2^{(S)}+F_2^{(V)}}{F_1^{(S)}+F_1^{(V)}}\n\\label{eq:eq7}\n\\end{equation}\nwhere\n\\begin{eqnarray*}\nF_1^{(S)}(Q^2)&=&\n\\displaystyle\\frac{g(Q^2)}{2}\n\\left[(1-\\beta_\\omega-\\beta_\\phi)+\\beta_\\omega\\displaystyle\\frac{\\mu_\\omega^2}{\\mu_\\omega^2+Q^2}+\\beta_\\phi\n\\displaystyle\\frac{\\mu_\\phi^2}{\\mu_\\phi^2+Q^2}\\right],\\\\\nF_1^{(V)}(Q^2)&=&\\displaystyle\\frac{g(Q^2)}{2}\n\\left[(1-\\beta_\\rho)+\\beta_\\rho\n\\displaystyle\\frac{\\mu_\\rho^2+8\\Gamma_\\rho\\mu_\\pi\/\\pi}\n{(\\mu_\\rho^2+Q^2)+(4\\mu_\\pi^2+Q^2)\\Gamma_\\rho\\alpha(Q^2)\/\\mu_\\pi}\\right],\\\\\nF_2^{(S)}(Q^2)&=&\n\\displaystyle\\frac{g(Q^2)}{2}\n\\left[(\\mu_p+\\mu_n-1-\\alpha_\\phi)\n\\displaystyle\\frac{\\mu_\\omega^2}\n{\\mu_\\omega^2+Q^2}+\\alpha_\\phi\\displaystyle\\frac{\\mu_\\phi^2}{\\mu_\\phi^2+Q^2}\n\\right],\\\\\nF_2^{(V)}(Q^2)&=&\\displaystyle\\frac{g(Q^2)}{2}\n\\left[(\\mu_p-\\mu_n-1)\n\\displaystyle\\frac{\\mu_\\rho^2+8\\Gamma_\\rho\\mu_\\pi\/\\pi}{(\\mu_\\rho^2+Q^2)+\n(4\\mu_\\pi^2+Q^2)\n\\Gamma_\\rho\\alpha(Q^2)\/\\mu_\\pi}\\right],\n\\end{eqnarray*}\nwhere $g(Q^2)=\\displaystyle\\frac{1}{(1+\\gamma Q^2)^2}$ \nand $\\alpha(Q^2)=\\displaystyle\\frac{2}{\\pi}\n\\sqrt{\\displaystyle\\frac{Q^2+4\\mu_\\pi^2}{Q^2}}\nln\\left[\\displaystyle\\frac{\\sqrt{(Q^2+4\\mu_\\pi^2)}+\\sqrt{Q^2}}{2\\mu_\\pi}\\right]$,\n with the standard values of the masses $m=0.939$~GeV, $\\mu_{\\rho}=0.77$~GeV, $\\mu_\\omega=0.78$~GeV, $\\mu_\\phi=1.02$~GeV, $\\mu_\\pi=0.139$~GeV and the $\\rho$ width $\\Gamma_\\rho=0.112$~GeV. $\\mu_p$ and $\\mu_n$ are the magnetic moments of proton and neutron, respectively, whereas $\\gamma=0.25$ GeV$^{-2}$, $\\beta_{\\rho}=0.672$, $\\beta_{\\omega}=1.102$, $\\beta_{\\phi}=0.112$, and $\\alpha_{\\phi}=-0.052$ are parameters fitted on the data.\n\nThis paper is organized as follows. In Section II we analyze the $t$-behavior of the imaginary part of the $F_2\/F_1$ ratio for different approaches, and estimate the corresponding value of $t$ for deviations of the order of  $\\Delta$ from the expected asymptotic values. Then we give the expressions for the  polarization observables accessible through the reaction $e^++e^-\\to p\\overline p$ in terms of the ratio $F_2\/F_1$ and analyze in particular the $P_y$ component of the proton polarization, which depends on the imaginary part of this ratio (Section III). In Section IV we study how the ratio ${\\cal R}$ approaches to one, that is the expected value for the asymptotic regime.\n\n\\section{ Imaginary part of the nucleon electromagnetic form factors}\n\nLet us recall here the definition of the Phragm\\`en-Lindel\\\"of theorem, which will be the basis of the following discussion. Following \\cite{Ti39}: {\\it \nif $f(z)\\to a $ as $z\\to \\infty$ along a straight line, and $f(z)\\to b$ as $z\\to \\infty$ along another straight line, and $f(z)$ is regular and bounded in the angle between, then $a=b$ and $f(z)\\to a$ uniformly in the angle\"}. For the problem considered here, we identify the variable $z$ with the momentum transfer squared $t$. So one of these straight lines can be chosen along the $x$-axis, in the positive direction (in the complex $z$-plane), i.e., for $t$ values corresponding to the TL region, and the other line with negative $x$ direction, with $t$ in the SL region. Assuming the analyticity of FFs, $F_i(t)$, $i=1,2$, in the upper part of the $z$-plane, we satisfy the necessary conditions for the application of the Phragm\\`en-Lindel\\\"of theorem,  for all nucleon FFs, $F_{1,2}(t)$. More exactly, it holds also for the four independent FFs $F_{1,2}^{(S)}(t)$ and $F_{1,2}^{(V)}(t)$, where the upper indices $(S)$ or $(V)$ correspond to isoscalar or isovector electromagnetic FFs of the nucleon. Note that the analytical properties of $F_i(t)$, $i=1,2$, should be discussed namely for the isoscalar and the isovector FFs, and not for proton and neutron, because the unitarity conditions (which allow to calculate the imaginary part of FFs) have the simplest and most transparent form for the isotopic FFs. More exactly, isoscalar(isovector) FFs are determined by intermediate states with odd(even) number of pions.\n\nSo, finally, one can write the following four independent relations:\n\\begin{equation}\n\\lim_{t\\to +\\infty} F_{1,2}^{(S,V)}(t) =\\lim_{t\\to -\\infty} F_{1,2}^{(S,V)}(t) \n\\label{eq:eq1a}\n\\end{equation}\nas a consequence of the Phragm\\`en-Lindel\\\"of theorem. \n\nThis theorem has other applications in particle physics, such as, for example, the well known theorem of Pomeranchuk \\cite{Lo65}, concerning the asymptotic behavior of the total cross sections for $a+b$ and $\\bar a +b$ collisions ($a$ and $b$ any hadrons): $\\sigma_T(ab)=\\sigma_T(\\bar a b)$. However, to be rigorous, the applicability of this theorem to FFs, which seems evident, has not been proved up to now\\footnote{In Ref. \\protect\\cite{Go94} one can read {\\it \"There is, a priori, no general constraint to ensure that the limit of some observable, such as a form factor, should be the same in every direction in the complex plane\".}}.\n\nUnfortunately, this theorem does not allow to indicate the physical value of $t$, starting from which it is working at some level of precision. For this aim one needs some additional dynamical information.\n\nIn our considerations about nucleon electromagnetic FFs, such information is contained in the parametrizations of FFs. More precisely, we discuss the ratio $F_2\/F_1$ for the proton and use those parametrizations which work well in the SL region, where the available precise experimental data allow to constrain the necessary parameters. It is possible to continue analytically such parametrizations to the TL region, using the following prescription \\cite{Br04}:\n\\begin{equation}\nln(-t)=ln(t)-i\\pi,~t> 0.\n\\label{eq:eq4}\n\\end{equation}\nEvidently, the choice of sign for the imaginary part\\footnote{Note that in Ref. \\protect\\cite{Bi93} another sign has been taken: $ln(-t)=ln(t)+i\\pi$, whereas in Ref. \\protect\\cite{Ba00} the formula $ln(-t)=ln(t)\\pm i\\pi$ has been applied for the analytical continuation from SL to TL region.}, in Eq. (\\ref{eq:eq4}), results in strong physical consequences concerning the calculations of any T-odd polarization observable for $e^+ +e^-\\leftrightarrow N+\\bar N$.\n\nLet us firstly discuss the $t$-behavior of $Im (F_2\/F_1)$ in TL region, using the QCD inspired and VDM parametrizations. Following the Phragm\\`en-Lindel\\\"of theorem, the ratio ${\\cal F}=|Im(F_2\/F_1)|\/|Re(F_2\/F_1)|$ should converge to zero as $t\\to \\infty$. And the value of $t$, corresponding to the solution of the equation: ${\\cal F}=\\Delta$, $\\Delta\\ll 1$ characterizes how ${\\cal F}$ approaches to zero. \n\nAfter analytical continuation in TL region, one can see that parametrization\n(\\ref{eq:eq5}) gives $R\\to \\infty$, because it reduces in TL region to:\n\\begin{equation}\n\\displaystyle\\frac{F_2}{F_1}=i\\displaystyle\\frac{1.25\\mbox{~GeV}}{\\sqrt{t}},~t>0 ~ \\mbox{\\protect\\cite{Br04}}.\n\\label{eq:eq8}\n\\end{equation}\nSuch parametrization definitely contradicts the Phragm\\`en-Lindel\\\"of theorem because both form factors can not be real at the same time.\n\nThis situation is not changed, after a modification suggested to normalize FFs  at $t=0$ \\cite{Br04}:\n$$\\displaystyle\\frac{F_2}{F_1}\\to\\left [\\displaystyle\\frac{1}{\\kappa_p^2} +\\displaystyle\\frac{t}{(1.25)^2\\mbox{~GeV}^2}\\right ]^{-1\/2},\n$$\nwhere $\\kappa_p$ is the proton anomalous magnetic moment.\n \nParametrization (\\ref{eq:eq6}) results in the following formula for the relative size of the imaginary to the real part ${\\cal F}$: \n\\begin{equation}\n{\\cal F}=\\displaystyle\\frac{2\\pi~ln (t\/\\Lambda^2)}{ln^2 (t\/\\Lambda^2)-\\pi^2},~t>0, \n\\label{eq:eq9}\n\\end{equation}\nwhich implies ${\\cal F}\\to 0$, if $t\\to \\infty$, but very slowly. Quantitatively, the condition ${\\cal F}=\\Delta$ has two solutions:\n\\begin{equation}\nx_{\\pm}=ln\\displaystyle\\frac{t}{\\Lambda^2}=\n\\displaystyle\\frac{\\pi}{\\Delta}\\left (1\\pm\\sqrt{1+\\Delta^2}\\right ).\n\\label{eq:eq10}\n\\end{equation}\nFor the $x_+$ solution, which should be considered as the physical solution for the TL region, we obtain: \n$$\\sqrt{t}\\simeq 10^{13} \\mbox{~GeV},\\mbox{~for~} \\Delta=0.1,$$\nwhich represents a very large energy, not far from the Planck scale, $\\sqrt{t}=10^{19}$ GeV. This last value corresponds to a deviation of  6.5\\% from the expected asymptotical zero value. \n\nIn the model \\cite{Ia73}, the isoscalar FFs, $F_{1,2}^{(S)}$, are real in all the kinematical range. Only the isovector FFs, $F_{1,2}^{(V)}$, have non vanishing imaginary part, induced by the $\\rho$-meson contribution, which is, however,  one order of magnitude smaller than the real part. The individual FFs are shown in Fig. \\ref{Fig:fig1}. A singularity appears in the TL region, for all FFs, due to the dipole term and in $F_1^{(S)}$, due to a compensation of the $\\omega$ and $\\phi$ contributions.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=17cm]{fig1.eps}\n\\caption{\\label{Fig:fig1} Isoscalar and isovector FFs in SL and TL regions. \n(a) and (b): $F_1^{(S)}$ and $F_2^{(S)}$ in SL region (dashed line) and in TL region (solid  line), (c) and (d) $F_1^{(V)}$ and $F_2^{(V)}$ in SL region (dashed line) and in TL region: real part (solid line), imaginary part (dotted line) and absolute value (dash-dotted line) which overlaps almost everywhere with the real part.}\n\\end{center}\n\\end{figure} \nTaking the parameters from Ref. \\cite{Ia73}, one can find:\n\\begin{equation}\n {\\cal F}=\\displaystyle\\frac{19.36}\n{ [1+0.512 ln (\\sqrt{t}\/m_{\\pi})]^2-12.3 ln (\\sqrt{t}\/m_{\\pi})-23.5}\n\\label{eq:eq11}\n\\end{equation}\nwith a faster decreasing, proportional to $[ln(\\sqrt{t}\/m_{\\pi})]^{-2}$, relatively to the previously considered  parametrizations. Such asymptotic behavior leads to $\\Delta$=0.1(0.05) for $\\sqrt{t}=10^{11}(10^{15})$ GeV, again very far from the region experimentally accessible. Note that the contribution which is linear in logarithm as well as the constant terms in the denominator are important, as they are responsible for the zero of $Re (F_2\/F_1)$ at $ln(\\sqrt{t}\/ m_{\\pi})\\simeq 45$, which results in a number larger than the asymptotic value. \n\n\nRecently, the model \\cite{Ia73} has been modified with respect to a common factor for all FFs \\cite{Ia03,Bi04}:\n$$(1-\\gamma t)^{-2}\\to(1-\\gamma e^{i\\theta}t)^{-2},$$\nwhere $\\theta $ has been taken equal to 53$^0$ and $\\gamma$=0.25 GeV$^{-2}$. This term moves the corresponding singularity $t=1\/{\\gamma}$ to $t= 1\/\\gamma e^{i\\theta}\\simeq 4e^{-i\\theta}$ GeV$^2$ from the physical region of TL momentum transfer. Such  factor does not modify polarization phenomena as it cancels out. However, such substitution has some shortcomings as it violates the Schwartz reflection symmetry, in the following relation: $F^*(z)=F(z^*)$, and does not satisfy the Phragm\\`en-Lindel\\\"of theorem, because this factor induces: $Im F(t)\/Re F(t)\\simeq -\\tan 2\\theta\\simeq 3.5$, i.e., a nonzero value in the asymptotic region. \n\n\\section{Polarization observables and asymptotic behavior of the T-odd observable $P_y$}\n\n\nLet us analyze the polarization observables related to the process $e^++e^-\\to p+\\bar{p}$ and their asymptotic behavior for the considered parametrizations of $F_2\/F_1$. As the $\\cos\\theta$ dependence is not relevant for the following considerations, for the numerical calculations we take $\\theta=45^o$. The $\\cos\\theta$ dependence is well known in framework of one photon exchange \\cite{Du96}, therefore, its measurement can be useful to check the validity of this mechanism at large $Q^2$. It is straightforward to derive the expressions for the polarization observables in terms of $F_2\/F_1$ following the formalism   derived in Ref. \\cite{Du96}. The reference system is taken as follows: the $z$ axis along the direction of the colliding electron, the $y$ axis normal to the scattering plane, defined by the direction of the electron and of the outgoing proton, and the $x$ axis to form a left-handed coordinate system.\n\nIn case of unpolarized beam and target, only a single spin polarization observable does not vanish, the component of the polarization of the scattered proton which is normal to the scattering plane, $P_y$:\n\\begin{equation}\nP_y=-\\displaystyle\\frac{\\tau-1}{\\sqrt{\\tau}}\\displaystyle\\frac{Im F_2\/F_1}{D},\n\\label{eq:py}\\\\\n\\end{equation}\nwhere $\\tau=t\/(4m^2)$ and \n$$\nD=\\displaystyle\\frac{3}{2}\\left | 1+\\displaystyle\\frac{F_2}{F_1}\\right |^2+\n\\displaystyle\\frac{1}{2\\tau}\n\\left | 1+\\tau\\displaystyle\\frac{F_2}{F_1}\\right |^2=\n\\displaystyle\\frac{1}{2}\\left [3+8Re \\displaystyle\\frac{F_2}{F_1}\n+\\displaystyle\\frac{1}{\\tau}+(\\tau+3)\\left |\\displaystyle\\frac{F_2}{F_1}\\right |^2\\right ].$$\nThe double spin coefficients, which do not vanish due to parity and C conservations, are:\n\\begin{eqnarray}\nA_{xx}&=&\\displaystyle\\frac{1}{2D}\\left[1+\\displaystyle\\frac{1}{\\tau}+4Re \\displaystyle\\frac{F_2}{F_1}+ (1+\\tau) \\left |\\displaystyle\\frac{F_2}{F_1}\\right |^2\\right ],\n\\label{eq:Axx}\\\\\nA_{yy}&=&\\displaystyle\\frac{1-\\tau}{2\\tau D}\\left[1-\n\\tau \\left |\\displaystyle\\frac{F_2}{F_1}\\right |^2\\right ],\n\\label{eq:Ayy}\\\\\nA_{xz}&=&\\displaystyle\\frac{1}{\\sqrt{\\tau}D}\n\\left[1+(1+\\tau)Re \\displaystyle\\frac{F_2}{F_1} +\\tau \\left |\\displaystyle\\frac{F_2}{F_1}\\right |^2\\right ],\n\\label{eq:Axz}\\\\\nA_{zz}&=&\\displaystyle\\frac{1}{2D}\n\\left[3-\\displaystyle\\frac{1}{\\tau}+\n4Re \\displaystyle\\frac{F_2}{F_1}\n+(3-\\tau )\\left |\\displaystyle\\frac{F_2}{F_1}\\right |^2\\right ],\n\\label{eq:Azz}\n\\end{eqnarray}\nand they depend on the real part and\/or on the modulus of $F_2\/F_1$.\nThe observable $P_y$, which contains the imaginary part of the FFs ratio, can bring information for the comparison of  SL and TL asymptotic behavior.\n\nThe following formula for $P_y$, at $\\tau\\gg 1$, holds for the parametrization (\\ref{eq:eq5}):\n$$P_y=-\\displaystyle\\frac{\\left(1-\\displaystyle\\frac{1}{\\tau}\\right )\n\\displaystyle\\frac{a}{m}}{3+\\displaystyle\\frac{1}{\\tau}+\\left (1+\\displaystyle\\frac{3}{\\tau}\\right ) \\displaystyle\\frac{a^2}{4m^2}}\n\\to P_{y,as}=-\\displaystyle\\frac{a\/m}{3+a^2\/(4m^2)}\\simeq -0.387.\n$$\nThis parametrization results in non vanishing (negative) asymptotics $P_y$, with large absolute value, in contradiction with the Phragm\\`en-Lindel\\\"of theorem. The behavior of $P_y$ for $1\/\\tau\\ll 1$ can be approximated by:\n$$P_y=P_{y,as}\\left (1-\\displaystyle\\frac{p}{\\tau}\\right ),~\np=1+\\displaystyle\\frac{3+4m^2\/a^2}{1+12m^2\/a^2}=1.67.$$\nThis implies that a 10\\%(5\\%) difference from asymptotics appears at $\nt=58.8(117.6)\\mbox{~GeV}^2.$\n\nFor the logarithmic parametrization (\\ref{eq:eq6}), the asymptotic behavior of $P_y$ is described by:\n$$P_y\\to -0.19 \\displaystyle\\frac{ln(t\/\\Lambda^2)}{\\sqrt{\\tau}}.$$\nOne can see that the absolute value decreases with $t$, and one finds $P_y=-10$\\%(-5)\\% at $t\\simeq 350(6000)$ GeV$^2$, still too large to be achieved by experiments.\n\nFinally, the asymptotic behavior of the $P_y$ polarization in the model \\cite{Ia73,Ia03} can be described by the following formula:\n$$P_y\\to \\displaystyle\\frac{3.5\/\\sqrt{\\tau}}\n{\\left [1+0.51 ln  (\\sqrt{t}\/m_{\\pi} )\n\\right ]^2}\\to \n\\displaystyle\\frac\n{13.5}{\\sqrt{\\tau} ln^2  (\\sqrt{t}\/m_{\\pi})} $$\nwith a faster decreasing with $t$. Note that this polarization is positive in TL region. Moreover the constant term in the denominator is important at large $t$, for example, a value of $P_y$=0.02 is reached at $t=2\\cdot 10^6$ GeV$^2$, which corresponds to very far asymptotics.\n\nFor the cases discussed above, the large value of $|P_y|$ arises questions about the asymptotic trend of electromagnetic FFs. According to the prescriptions of Phragm\\`en-Lindel\\\"of theorem, $P_y$ should vanish.\n \n\\section{Difference between the absolute values of $F_2\/F_1$ in SL and TL regions}\n\nWe mentioned above that the measured values of the magnetic proton FFs are different in TL and SL regions of momentum transfer, up to $t=18$ GeV$^2$, where the TL values of $|G_{Mp}|^2$ exceed by a factor of two the corresponding values in SL region. These values should approach the same number at asymptotic values of $t$. But which number?\n\nLet us analyze the behavior of $|F_2\/F_1|$ in TL region, using, again, the considered parametrizations. \nThe parametrization (\\ref{eq:eq5}) gives $|F_2\/F_1|_{SL}=|F_2\/F_1|_{TL}$, at any value of $t$. Furthermore, this parametrization gives a specific behavior of the ratio $|G_E|^2\/|G_M|^2$ in TL region. One finds:\n\\begin{equation}\n\\displaystyle\\frac{ |G_E|^2}{|G_M|^2}\n=\\displaystyle\\frac{1+\\tau\\displaystyle\\frac{a^2}{4m^2}}\n{1+\\displaystyle\\frac{a^2}{4m^2\\tau}}\\to\\tau \\displaystyle\\frac{a^2}{4m^2}=0.44\\tau.\n\\label{eq:eqa}\n\\end{equation}\nNote, in this respect, that up to now the separation of the electric and magnetic contributions to the differential cross section in the TL region has not been realized, yet. The analysis of the experimental data is currently based on two assumptions: either $G_E=0$ or $|G_M|=|G_E|$. The extracted values for $G_M$ according to these prescriptions differ at most by 20\\%. \n\nHowever, Eq. (\\ref{eq:eqa}) suggests another possible relation between $G_E$ and $G_M$, that leads to comparable contributions of the electric and magnetic terms to the cross section, independently on the $t$-value. The resulting value for $G_M$ is 10\\% lower than the value corresponding to $|G_M|=|G_E|$, and still does not compensate the observed difference of FFs in SL and TL regions.\n  \nThe parametrization (\\ref{eq:eq6}) gives the following relation:\n$$\n{\\cal R}=\\displaystyle\\frac{|F_2\/F_1|_{TL}}{|F_2\/F_1|_{SL}}=1 +\\displaystyle\\frac{ \\pi^2}{ln^2(t\/\\Lambda^2)}.\n$$\nA deviation of ${\\cal R}$ from 1 by 10\\%(5\\%) is reached at $\\sqrt{t}\\simeq 43 (337)$ GeV.\n\n\n\\section{Conclusions}\nWe have analyzed the asymptotic behavior of recently suggested, pQCD inspired,  parametrizations of the ratio of the Dirac and Pauli FFs, $F_2\/F_1$. We have based our study on the requirements given by Phragm\\`en-Lindel\\\"of theorem, in particular the equality of FFs in SL and TL regions. As FFs are real in SL region and complex in TL region, this implies that the imaginary part of FFs in TL region vanishes, as well as the polarization of the emitted proton, in the annihilation reaction $e^++e^-\\leftrightarrow p+\\bar{p}$ (when the colliding particles are unpolarized).\n\nWe have shown that the considered parametrizations do not satisfy the asymptotic conditions suggested by the Phragm\\`en-Lindel\\\"of theorem or they do so only for very large values of $Q^2$, well beyond the experimentally accessible range. In particular, the $1\/\\sqrt{Q^2}$ behavior of this ratio, which reproduces the recent measurements in the SL region, is certainly not compatible with an asymptotic regime, showing that the presently measurable data should be better interpreted in frame of classical nucleon degrees of freedom.\n\nConcerning the double logarithmic parametrization, it has been pointed out long ago \\cite{Mi82}, that a suppression to Sudakov type contributions could take place.\n\nThe dipole-like formulas for FFs do satisfy the Phragm\\`en-Lindel\\\"of theorem. But such parametrization has the following evident problems:\n\\begin{itemize}\n\\item the threshold condition: $G_{EN}(4m^2)=G_{MN}(4m^2)$ is not satisfied,\n\\item the unitarity conditions for all nucleon FFs are strongly violated, as one should have a branching point at $t=4m_{\\pi}^2$ for isovector FFs and  $t=9m_{\\pi}^2$ for isoscalar FFs,\n\\item the prediction in TL region underestimates the experimental data.\n\\end{itemize}\n\nThe analytical continuation of nucleon electromagnetic FFs, presently used to describe the main properties of nucleon structure in SL region of momentum transfer squared (in some models), results as a rule, in an essential imaginary part in TL region. Moreover, the relative value (with respect to the real part) is a very slowly decreasing function of $t$. Such behavior, of course, is in agreement with the Phragm\\`en-Lindel\\\"of theorem, but the corresponding asymptotic regime corresponds to very large values of $t$.\n \nThe asymptotic regime defined by the prescriptions of the considered models and the asymptotic properties derived from the analyticity of form factors act at a different level. Phragm\\`en-Lindel\\\"of theorem defines the asymptotic conditions without direct connection with QCD. The most evident application of the  Phragm\\`en-Lindel\\\"of theorem in physics is the Pomeranchuk theorem - which relates the asymptotic behavior of the total cross section for $NN$ and $N\\bar N$ interaction. This is not QCD regime, because such theorem applies for $t=0$, i.e., to evidently non perturbative physics, despite the fact that the Mandelstam variable $s$ is very large.  So the connection between QCD asymptotics and asymptotics from Phragm\\`en-Lindel\\\"of theorem, from the point of view of hadron FFs is non trivial, as this theorem seems to work for elastic $NN$ and $N\\bar N$ amplitudes in the kinematical region where QCD does not apply.\n\nWe can consider the present results as an indirect indication of the importance of non perturbative contributions to the physics of the nucleon electromagnetic structure.\n\\section{Acknowledgments}\n\nThe authors acknowledge Prof. M. P. Rekalo for many interesting discussions and ideas, without which this paper would not have been realized in the present form. Thanks are due to Prof. E. Kuraev for his interest to this subject and to C. Duterte for contributing to the earlier stage of this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the community of animation, comics and games, the majority of artistic works can be created from sketches, which consumed a lot of time and effort. \\textbf{If there is a method to apply the style of a painting to a half-finished work in form of sketches, many redundant work will be saved}, e.g. using an existing picture of a specific character as a style map and then apply the style to a sketch of the character. The neural algorithm of artistic style \\cite{Gatys2015A} can produce amazing and perfect images combining of content maps and style maps, but it lacks the ability to deal with sketches. The paintschainer \\cite{pc}, as well as pix2pix \\cite{Isola2016Image}, can turn sketches into paintings directly and the result can be even perfect with pointed hints added, but it cannot take the advantage of existing paintings. We investigated residual U-net and auxiliary classifier GAN (AC-GAN) as a solution and our network can directly generate a combination of sketch and style image. Our model is fully feed-forward so as to synthetize paintings at a high speed. Furthermore, we find out that U-net and conditional GAN (cGAN) relatively declines in performance with absence of a balanced quantity of information of paired input and output, and we propose using residual U-net with two guide decoders. In addition, we compare many kinds of GANs and find that conditional GAN is not suitable for this task, resorting to the AC-GAN finally.\n\n\\textbf{Our contributions are}:\n\\begin{itemize}\n\\item{A feed-forward network to apply the style of a painting to a sketch.}\n\\item{An enhanced residual U-net capable of handling paired images with unbalanced information quantity.}\n\\item{An effective way to train residual U-net with two additional loss.}\n\\item{A discriminator modified from AC-GAN suitable to deal with paintings of different style.}\n\\end{itemize}\n\n\\section{Related Works}\n\n\\textbf{Neural Style Transfer} \\cite{Gatys2015A,Gatys2016Image,Gatys2016Preserving,Johnson2016Perceptual,Ulyanov2016Texture} can synthesize admirable image with art style from an image and content from another, based on an algorithm that minimize the difference in gram matrixes of deep convolution layers. Nevertheless, our objective is to combine a style image and a sketch. Unfortunately, the neural style transfer is not capable of this kind of task. In fact, the results of neural style transfer on sketches from style images can be really strange, far from a proper painting.\n\n\\textbf{Pix2Pix} \\cite{Isola2016Image} and some other paired image2image transfers based on conditional GAN \\cite{Mirza2014Conditional} are accomplished in transformation between paired images. In our research, the quality of the outputs of networks based on cGAN depends on the \"information gap degree\" between the inputs and outputs. That is to say, if the input and output are similar in the quantity of information, the result can be reliable. In the experiment of Pix2Pix's edge2cat, based on users' input edges, small circles always means eyes, triangles regarded as ears, and closed figures should always be filled with cat hair. If we shuffle the datasets of cat2edge, bag2edge, and even more like house, bicycle, tree, dog and so on, the quality of outputs will decline accordingly. Our task is to transfer sketches correspondingly to paintings, which is far more complicated than cats or bags, and the generator of cGAN needs to learn semantic features and low-level features simultaneously. Actually, a conditional discriminator can easily leads the generator to focus too much on the relationship between sketches and paintings, thus, to some extent, ignore the composition of a painting, leading to ineluctable overfitting.\n\n\\textbf{Paintschainer} \\cite{pc} has abandoned cGAN and resorted to an unconditional discriminator due to the same reason as above, and obtained remarkable and impressive achievements. It becomes a powerful and popular tool for painters. Users only need to input a sketch to get a colorized painting. They can add pointed color hints for better results. With massive existing finished paintings, though the demand for a method to colorize a sketch according to a specific style is extremly high, there is no reliable and mature solution for it. \n\n\\section{Methods}\n\nWe combine an enhanced residual U-net generator and an auxiliary classifier discriminator as our adversarial network. We feed the generator with sketch maps and style maps. Our discriminator can tell whether its input is real or fake, and classify corresponding style simultaneously.\n\n\\begin{figure}\n\\centering\\includegraphics[width=.5\\textwidth]{f8.pdf}\n\\caption{The architecture of our adversarial network. The discriminator is modified from AC-GAN, which not only has the ability to reveal whether the map is real or fake, but also tell the classification. We notice that the global style hint can be regarded as a low-level classification result with 2048 or 4096 classes.}\\label{fig:discriminator}\n\\end{figure}\n\n\\begin{figure}\n\\centering\\includegraphics[width=.35\\textwidth]{f4.pdf}\n\\caption{In the experiment of copying images, the mid-level layers of U-net receives no gradients.}\\label{fig:copy}\n\\end{figure}\n\n\\subsection{Architecture and Objective of Generator}\nThe detailed structure is in fig.~\\ref{fig:generator}. We are not the first to add the global style hint to mid-level layers of a encoder-decoder or a U-net\\cite{Ronneberger2015U}. In \\emph{Iizuka et.al's} \\cite{Iizuka2016Let} and \\emph{Richard Zhang et.al's} \\cite{Zhang2017Real} the global hint is extracted from a high-level layer of a classification network as a high dimension vector and then added to the colorization network. For photograph colorization, the shadow, material and texture is known variables in inputs and the network only need a spot of information to analyse the color distribution. In \\emph{Iizuka et.al's} \\cite{Iizuka2016Let}, they only use vectors of 1$\\times$1$\\times$256 as global hints. However, it is far from enough for the network to paint sketches into paintings with various unique styles. Therefore, We integrate style hint at size of 1$\\times$1$\\times$4096 or 1$\\times$1$\\times$2048. But we failed to train such a generator directly and finally found some \"nature\" of the well-known structure U-net. The failure result is in fig.~\\ref{fig:fail}.\n\n\\textbf{The U-net is \"lazy\".} That is to say if the U-net find itself able to handle a problem in low-level layers, the high-level layers will not bother to learn anything. If we train a U-net to do a very simple work \"copying image\" as in fig.~\\ref{fig:copy}, where the inputs and outputs are same, the loss value will drop to 0 immediately. Because the first layer of encoder discovers that it can simply transmit all features directly to the last layer of the decoder by skiping connection to minimize the loss. In this case, no matter how many times we train the U-net, the mid-level layers will not get any gradient.\n\nFor each layer in U-net's decoder, features can be acquired from higher layers or from skip connected layers. In each iteration of a training process, these layers select other layers' outputs with nonlinear activations in order to minimize the loss. In the experiment of copying image, when the U-net is initialized with Gaussian random numbers, the output of the first layer in encoder is informative enough to express the full input map while the output of second-to-last layer in decoder seems very noisy. Thus the \"lazy\" U-net gives up the relatively noisy features. \n\n\\begin{figure}\n\\centering\\includegraphics[width=.35\\textwidth]{f6.pdf}\n\\caption{The ideal architecture of a residual U-net. Additional losses are applied to all mid-level layers so as to provide gradients for these layers.}\\label{fig:iealunet}\n\\end{figure}\n\nBecause a hint of 1$\\times$1$\\times$256 is far from enough for sketch painting, we resorted to a hint of 1$\\times$1$\\times$4096, from the output of VGG 19's \\emph{fc1}, without the \\emph{Relu} activation. The 4096 hint vector is very informative and powerful. However, for a newly initialized U-net, the output of the mid-level layers can be extremely noisy if we add the vector of 4096 directly to the layer. As mentioned above, the noisy mid-level layer is given up by U-net and these layers cannot receive any gradient as a consequence.\n\nInspired by LeNet \\cite{LecunLeNet} and GooLeNet \\cite{Szegedy2014Going}, We resort to residual networks in fig.~\\ref{fig:iealunet}. If we attach additional loss to each layer which is possible to be \"lazy\", no matter how noisy the output of a mid-level layer is, the layer will never be given up by the U-net and all layers will get stable gradient in the whole training process. Thus, it is possible to add a very informative and to some extent noisy hint to the mid-level layers. We implemented two additional loss, in the \"Guide decoder 1\" and \"Guide decoder 2\", to avoid the gradient disappearance in mid-level layers. The loss trend with or without the two Guide Decoders can been seen in fig.~\\ref{fig:wo}. Significant difference of the networks' prediction can be seen in fig.~\\ref{fig:fail}.\n\nThe loss is defined as:\n\\begin{equation}\n\\label{eqn3_2}\n\\begin{aligned}\nL_{l1}(V,G_{f,g_{1},g_{2}})=\\mathbb{E}_{x,y\\sim P_{data}(x,y)}[||y-G_{f}(x,V(x))||_{1}+\\\\\\alpha||y-G_{g_{1}}(x)||_{1}+\\beta||y-G_{g_{2}}(x,V(x))||_{1}]\n\\end{aligned}\n\\end{equation}\n\nWhere the $x$, $y$ is the paired domain of sketches and paintings, and $V(x)$ is the output of VGG 19's $fc1$ without $Relu$, and $G_{f}(x,V(x))$ is the final output of U-net, the $G_{g_{1}}(x)$ and $G_{g_{2}}(x,V(x))$ are outputs of the two guide decoders at the entry and the exit of mid-level layers accordingly. The recommended value of $\\alpha$ and $\\beta$ is \\emph{0.3} and \\emph{0.9}. We also find out that the distribution of color can be improved by feeding the Guide Decoder located at the entry of mid-level layers with grayscale maps, so the final loss is as below, where $T(y)$ can transform y into grayscale image.\n\\begin{equation}\n\\label{eqn3_2}\n\\begin{aligned}\nL_{l1}(V,G_{f,g_{1},g_{2}})=\\mathbb{E}_{x,y\\sim P_{data}(x,y)}[||y-G_{f}(x,V(x))||_{1}+\\\\\\alpha||T(y)-G_{g_{1}}(x)||_{1}+\\beta||y-G_{g_{2}}(x,V(x))||_{1}]\n\\end{aligned}\n\\end{equation}\n\n\\begin{figure}\n\\centering\\includegraphics[width=.5\\textwidth]{f7.pdf}\n\\caption{Loss with or without the Guide Decoder}\\label{fig:wo}\n\\end{figure}\n\n\\begin{figure}\n\\centering\\includegraphics[width=.48\\textwidth]{f10.pdf}\n\\caption{The style map and sketch map are same with fig.~\\ref{fig:tit}. The neural style algorithm fails to transfer a sketch to a painting. Though normal U-net can predict meaningful paintings when the global hint is 1$\\times$1$\\times$256, the result can be rather disappointing when the hint of 1$\\times$1$\\times$4096 is applied. Our residual U-net with two Guide Decoders is very capable of handling such an informative global style hint.}\\label{fig:fail}\n\\end{figure}\n\n\\subsection{Architecture and Objective of Discriminator}\n\nAs we mentioned, the traditional cGAN is not suitable for our project. Painting is a complicated work and needs human artists to take color-selection, composition and fine-tuning into consideration, and all these need an artist to focus on the global manner of the painting. However, a conditional discriminator has always a tendency to focus much more on the relationship between sketch line and color than the global information. In our experiments with Pix2Pix, if a conditional discriminator is applied, the generator will resist fiercely and the color surrounding the line can be extremely over-colorized. Tuning parameters is not enough to reach a balance.\n\nFurthermore, it can be appreciated if the discriminator has the ability to tell the style and provide gradient accordingly, in order to fit the main task: style transfering. We finally integrate AC-GAN and our discriminator has exactly 4096 outputs, which will all be minimized to 0 when the input is fake and approach to the same value of VGG 19's \\emph{fc1} when the input is real in fig.~\\ref{fig:discriminator}. \n\nThe final loss is defined as:\n\\begin{equation}\n\\label{eqn3_2}\n\\begin{aligned}\nL_{GAN}(V,G_{f},D)=\\mathbb{E}_{y\\sim P_{data}(y)}[Log(D(y)+(1-V(y)))]+\\\\\\mathbb{E}_{x\\sim P_{data}(x)}[Log(1-D(G_{f}(x,V(x))))]\n\\end{aligned}\n\\end{equation}\n\nUnfortunately, to minimize the loss, it requires a large memory size in GPU. We also employed the traditional discriminator of DCGAN for fast speed and large batch size. We fine-tune our networks by shifting the two loss functions.\n\\begin{equation}\n\\label{eqn3_2}\n\\begin{aligned}\nL_{GAN}(V,G_{f},D)=\\mathbb{E}_{y\\sim P_{data}(y)}[Log(D(y))]+\\\\\\mathbb{E}_{x\\sim P_{data}(x)}[Log(1-D(G_{f}(x,V(x))))]\n\\end{aligned}\n\\end{equation}\n\nThe final objective is:\n\\begin{equation}\n\\label{eqn3_2}\n\\begin{aligned}\nG^{*}= \\arg \\min \\limits_{G_{f}} \\max \\limits_{D}L_{GAN}(V,G_{f},D) + \\lambda L_{l1}(V,G_{f,g_{1},g_{2}})\n\\end{aligned}\n\\end{equation}\n\n\\begin{figure*}\n\\centering\\includegraphics[width=\\textwidth]{f12.pdf}\n\\caption{Additional results of our network. All the reference paintings are from Pixiv. The creditions for all artists are available in the github page \"style2paints\".}\\label{fig:4}\n\\end{figure*}\n\n\\section{Limitations and Discussions}\n\nOur network is capable of drawing on sketches according to style maps given by users, depending on the classification ability of the VGG. However, the pretrained VGG is for ImageNet photograph classification, but not for paintings. In the future, we will train a classification network only for paintings to achieve better results. Furthermore, due to the large quantity of layers in our residual network, the batch size during training is limited to no more than 4. It remains for future study to reach a balance between the batch size and quantity of layers.\n\n\\nocite{*}\n\n\\bibliographystyle{plain}%\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nLet $X$ be a complex manifold. We denote by $\\mathcal{O}_X$ (resp. $\\mathcal{D}_X$) the sheaf of holomorphic functions (resp. holomorphic differential operators) on $X$. For any reduced divisor $Z$ in $X$, we denote by $\\mathcal{O}_X[\\ast Z]$ the sheaf of meromorphic functions with poles along $Z$ at most. Given a regular holonomic $\\mathcal{D}_X$-module $\\mathcal{M}$ and a meromorphic function $g$ on $X$ with poles along $Z$, we define the $\\mathcal{D}_X$-module $\\mathcal{M}e^g$ as being the $\\mathcal{O}_X$-module $\\mathcal{M}[\\ast Z]:=\\mathcal{O}_X[\\ast Z]\\otimes_{\\mathcal{O}_X}\\mathcal{M}$ with connection $\\nabla_g$ defined as $\\nabla+dg$, where $\\nabla$ is the connection on $\\mathcal{M}[\\ast Z]$ given by its left $\\mathcal{D}_X$-module structure. It is known that $\\mathcal{M}e^g$ is a holonomic $\\mathcal{D}_X$-module. It is irregular along $Z$. We will say $\\mathcal{M}e^g$ has \\emph{exponential type}.\n\n\nIn the algebraic setting, if $U$ is a smooth algebraic variety over $\\mathbb{C}$, we use an exponent ``$\\mathrm{alg}$'' in the notation of the corresponding sheaves and we define the notion of a holonomic $\\mathcal{D}_X^{\\mathrm{alg}}$-module of exponential type similarly.\n\nLet $\\mathcal{M}e^g$ be a holonomic $\\mathcal{D}_X$-module of exponential type. The cohomology sheaves of its direct image by any proper map $f:X\\to C$ to a complex curve are holonomic $\\mathcal{D}_C$-modules with possibly irregular singularities. The main result of this article consists in the computation of the formal invariants of these cohomology modules at their singularities.\n\nThe basic computation takes place in the following setting: the complex manifold $X$ is the product $D\\times \\mathbb{P}^1$ of a small disc $D$ centered at the origin in $\\mathbb{C}$ by the Riemann sphere, the meromorphic function $g$ is the projection $p_2:X\\to\\mathbb{P}^1$ with polar divisor $Z=D\\times\\{\\infty\\}$ and the map $f$ is the projection $p_1:X\\to D$. We will denote by $t$ a coordinate on $D$. As above, $\\mathcal{M}$ is a regular holonomic $\\mathcal{D}_X$-module. The only interesting  cohomology module to consider is $\\mathcal N=\\mathcal{H}^0p_{1+}(\\mathcal{M}e^{p_2})$ and we assume that $D$ is small enough so that $0$ is the only singular point of $\\mathcal{N}$.\n\nIn a neighbourhood $U$ of $(0,\\infty)\\in D\\times\\mathbb{P}^1$, the singular support of $\\mathcal{M}$ is an analytic curve $S$. We denote by $\\{S_\\ell\\mid\\ell\\in\\Lambda\\}$ the set consisting of the local irreducible components of $S$ which are distinct of $\\{0\\}\\times\\mathbb{P}^1$ or $D\\times\\{\\infty\\}$. Set $U^{\\ast}=U\\setminus S$. To any $S_\\ell$ at $(0,\\infty)$ we associate:\n\\begin{figure}[h]\n\\includegraphics[height=2.4cm]{Zlbis.eps}\n\\end{figure}\n\\begin{itemize}\n\\item\n$m_\\ell\\in\\mathbb{N}^*$:  the multiplicity of the conormal space $T^*_{S_\\ell}X$ in the characteristic cycle of $\\mathcal{M}$.\n\\item\n$p_\\ell,q_\\ell\\in\\mathbb{N}^*$: the intersection multiplicity at $(0,\\infty)$ of $S_\\ell$ with $\\{0\\}\\times \\mathbb{P}^1$ and $D\\times \\{\\infty\\}$, respectively.\n\\item\n$Q_\\ell\\subset\\mathbb{R}^2$: the convex hull in $\\mathbb{R}^2$ of the union of $Q:=\\{(u,v)\\in\\mathbb{R}^2\\mid u\\leq0,\\,v\\geq0\\}$ and $(m_\\ell p_\\ell,m_\\ell q_\\ell)+Q$.\n\\item $\\alpha_\\ell\\in\\tau^{-1}\\mathbb{C}[\\tau^{-1}]$ and $\\delta_\\ell\\in\\mathbb{C}\\{\\tau\\}$: the polar part and the holomorphic part of a Puiseux parametrization of $S_\\ell$ at $(0,\\infty)$. Let $u_{\\ell}\\in\\mathbb{C}\\{\\tau\\}$, $u_\\ell(0)\\neq 0$, such that $\\gamma(\\tau):=(\\tau^{p_\\ell},\\tau^{-q_\\ell}u_\\ell)$ is a parametrization of $S_\\ell$. $\\alpha_\\ell$ is the polar part of $\\tau^{-q_\\ell}u_\\ell$ and $\\delta_\\ell$ is its holomorphic part ($\\tau^{-q_\\ell}u_\\ell=\\alpha_\\ell+\\delta_\\ell$).\n\\end{itemize}\n\n\n\n\nThe irregularity number of $\\mathcal N_0$ is computed in [9]. But, we can go further in computing formal invariants of $\\mathcal N_0$.\n\n\\begin{theorem}\\label{1}\\mbox{}\\par\n$(1)$ After a suitable translation, the Newton polygon of the $\\mathcal D_{D,0}$-module $\\mathcal N_0$ is the Minkowski sum $\\sum_{\\ell\\in\\Lambda}Q_\\ell$. In particular, the set of slopes of $\\mathcal N_0$ is $\\{q_\\ell\/p_\\ell\\mid\\ell\\in\\Lambda\\}$ and the irregularity number of $\\mathcal N_0$ is $\\sum_{\\ell\\in\\Lambda}m_\\ell q_\\ell$.\n\n$(2)$ Let $p=\\mathrm{lcm}\\{p_\\ell\\mid\\ell\\in\\Lambda\\}$. After the ramification $\\rho:D^{'}\\to D$, $\\rho(\\tau)=t=\\tau^p$, the formal irregular part of $\\rho^*\\mathcal N_0$ decomposes as $\\oplus_{\\alpha\\in\\Gamma}R_\\alpha e^\\alpha$ where:\n\\begin{itemize}\n\\item\n$\\Gamma\\subset\\tau^{-1}\\mathbb C[\\tau^{-1}]$ is a finite subset and $\\alpha\\in\\Gamma$ if and only if there exist $\\ell\\in\\Lambda$ and $\\xi\\in\\mathbb{C}^{\\ast}$ with $\\xi^{p_\\ell}=1$ such that\n$$\n\\alpha(\\tau)=\\alpha_\\ell(\\xi\\tau^{p\/p_\\ell});\n$$\nthe set of such $\\ell\\in\\Lambda$ is denoted by $\\Lambda_\\alpha$.\n\\item\n$R_\\alpha$ is a regular holonomic $\\mathbb C[[\\tau]]\\langle\\partial_\\tau\\rangle$-module, $R_\\alpha=R_\\alpha[\\tau^{-1}]\\neq0$ and the rank of $R_\\alpha$ is $\\sum_{\\ell\\in\\Lambda_\\alpha}m_\\ell$.\n\\end{itemize}\n\\end{theorem}\n\n\nBut we can also compute the characteristic polynomial of the monodromy of $R_{\\alpha}$ under the assumption $(\\ast)$: for any $\\ell,\\ell^{'}\\in\\Lambda$ and for any $\\xi_\\ell,\\xi_{\\ell^{'}}\\in\\mathbb{C}^{\\ast}$ with $\\xi_\\ell^{p_\\ell}=1$ and $\\xi_{\\ell^{'}}^{p_{\\ell^{'}}}=1$, \n$$(\\ell,\\xi_\\ell)\\neq(\\ell^{'},\\xi_{\\ell^{'}})\\Longrightarrow \\alpha_\\ell(\\xi_\\ell\\tau^{\\sfrac{p}{p_\\ell}})+\\delta_{\\ell}(0)\\neq\\alpha_{\\ell^{'}}(\\xi_{\\ell^{'}}\\tau^{\\sfrac{p}{p_{\\ell^{'}}}})+\\delta_{\\ell^{'}}(0).$$\n\nWe remark that $\\rho^{'}=(\\rho,id):D^{'}\\times\\mathbb{P}^{1}\\to D\\times\\mathbb{P}^1$ is a normalization of all the $S_{\\ell}$'s. For any $\\ell\\in\\Lambda$, $\\rho^{'-1}(S_{\\ell})$ is the union of some smooth irreductible analytic curves $S_{\\ell}^i$.  \n\nThen to any $S_{\\ell}$ we associate $\\zeta_{\\ell}\\in\\mathbb{C}[\\lambda]$: the characteristic polynomial of the monodromy of $\\Phi_{h_{\\ell}}(DR~\\mathcal{M})$ around $(0,0)$ in a normalization of $S_{\\ell}$, where $h_{\\ell}=0$ is an equation of $S_{\\ell}$. Here, $\\Phi_{h_{\\ell}}$ is the vanishing cycle functor along $S_{\\ell}$ for bounded complex with constructible cohomology (cf. \\cite{MeMai}, $\\S 1$) and $\\DR$ is the de Rham functor (cf. Definition $2.6.4$ p. $27$ in \\cite{Me}). $\\Phi_{h_{\\ell}}(DR~\\mathcal{M})$ has support included in $S_{\\ell}$ and $\\Phi_{h_{\\ell}}(DR~\\mathcal{M})_{|S_{\\ell}\\setminus\\{(0,0)\\}}$ is just a local system. Then, $\\zeta_{\\ell}$ is the characteristic polynomial of the monodromy of the local system  $\\rho^{-1}(\\Phi_{h_{\\ell}}(DR~\\mathcal{M}))_{|S_{\\ell}^i\\setminus\\{(0,0)\\}}$ around $(0,0)$ (do not confuse with the monodromy around $S_{\\ell}$). As $S_{\\ell}^i\\simeq S_{\\ell}^j$, for $i,j\\in\\{1,\\ldots,p_{\\ell}\\}$, $\\zeta_{\\ell}$ is independant of $i$.\n\n\n\\begin{theorem}\\label{zeta}\nUnder the assumption $(\\ast)$, the characteristic polynomial of the monodromy of $R_\\alpha$ is equal to $\\prod_{\\ell\\in\\Lambda_\\alpha}\\zeta_{\\ell}$.\n\\end{theorem}\n\n\n\n\n\nConversely, we prove in Theorem \\ref{realisation} that given any formal holonomic $\\mathbb{C}[[t]]\\langle\\partial_t\\rangle$-module $\\widehat{\\mathcal{N}}$, there exist a ramification $\\rho:\\tau\\to t=\\tau^p$ and a regular holonomic $\\mathcal D_{D\\times\\mathbb P^1}$-module $\\mathcal{M}$, such that $\\rho^{\\ast}(\\widehat{\\mathcal{N}})$ is isomorphic to the formalization of $\\rho^{\\ast}(\\mathcal H^0p_{1+}(\\mathcal{M}e^{p_2}))_0$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn the algebraic setting, let us assume that $U$ is affine and that $f$ and $g$ are regular functions on $U$. Let $\\mathcal{M}$ be a holonomic $\\mathcal{D}^{\\mathrm{alg}}_U$-module which has regular singularities included at infinity. The direct images $\\mathcal{N}^k:=\\mathcal{H}^k f_+(\\mathcal{M}e^g)$ are holonomic $\\mathbb C[t]\\langle\\partial_t\\rangle$-modules with singularities at finite distance on the affine line $\\mathbb{A}^1$ and at infinity. We will reduce the computation of the formal invariants of $\\mathcal{N}^k$ at each of its singularities $c\\in\\mathbb{A}^1\\cup\\{\\infty\\}$ to the situation of Theorem \\ref{1} through the diagram\n\\[\nU\\xrightarrow{~(f,g)~}\\mathbb{A}^1\\times\\mathbb{A}^1\\stackrel{i}{\\to} \\mathbb{P}^1\\times\\mathbb{P}^1\\stackrel{i_c}{\\leftarrow} D_c\\times\\mathbb{P}^1.\n\\]\nLet $\\mathcal{P}_c$ denote the direct image $i_c^{\\ast}(i_+(f,g)_+\\mathcal{M})^{\\mathrm{an}}$. It consists in a complex of $\\mathcal{D}_{D_c\\times\\mathbb{P}^1}$-modules with regular holonomic cohomology modules $\\mathcal{H}^k\\mathcal{P}_c$.\n\\begin{theorem}\\label{2}\nThe germs $\\mathcal{N}^k_c$ and $\\mathcal{H}^0p_{1+}(\\mathcal{H}^k\\mathcal{P}_ce^{p_2})_c$ have the same formal irregular part.\n\\end{theorem}\n\n\\begin{figure}[h]\n\\includegraphics[height=2.5cm]{Zlalg.eps}\\\\\nSingular support of $\\mathcal{H}^k(i_+(f,g)_+\\mathcal{M})^{\\mathrm{an}}$.\n$c'$ is a regular singularity of $\\mathcal{N}^k$. $c$ and $\\infty$ are irregular singularities.\n\\end{figure}\n\n\n\n\n\\begin{example}\nLet $U$ be a smooth affine surface. Suppose $f,g:U\\to\\mathbb{A}^1$ are algebraically independant and $\\mathcal{M}=\\mathcal{O}_U^{\\mathrm{alg}}$. The only interesting cohomology module to consider is $\\mathcal{N}=\\mathcal{H}^0f_+(\\mathcal{O}_U^{\\mathrm{alg}}e^g)$; the others cohomology modules have punctual support (cf. Proposition 5.1 of \\cite{Ro1}). \n\nLet $c\\in\\mathbb{A}^1\\cup\\{\\infty\\}$ and $S$ be the singular support of $\\mathcal{H}^0i_+(f,g)_+\\mathcal{O}_U$ in the neighbourhood of $(c,\\infty)$. Theorem \\ref{1} and Theorem \\ref{2} allow us to describe the Newton polygon of $\\mathcal{N}_c$, up to a suitable translation, using the set $\\{S_{\\ell}~|~\\ell\\in\\Lambda\\}$ of irreducible components of $S$  which are distinct of $\\{c\\}\\times\\mathbb{P}^1$ or $\\mathbb{P}^1\\times\\{\\infty\\}$ and the multiplicity $m_{\\ell}$ of $T^{\\ast}_{S_{\\ell}}(\\mathbb{P}^1\\times\\mathbb{P}^1)$ in the characteristic cycle of $\\mathcal{H}^0i_+(f,g)_+\\mathcal{O}_U$.\n\nBut we can give a geometrical interpretation of the $S_{\\ell}$'s and the $m_{\\ell}$'s.\nLet $\\mathbb{X}$ be a smooth compactification of $U$ such that there exists two holomorphic functions $F,G:\\mathbb{X}\\to\\mathbb{P}^1$  which extend $f$ and $g$. Let $\\Gamma$ be the critical locus of $(F,G)$ and $D$ be the divisor $\\mathbb{X}\\setminus U$. \nWe denote by $\\Delta_1$ (resp. $\\Delta_2$) the cycle in $\\mathbb{P}^1\\times\\mathbb{P}^1$ which is the closure of $(F,G)(\\Gamma)\\cap (\\mathbb{A}^1\\times\\mathbb{A}^1)$ (resp. $(F,G)(D)\\cap (\\mathbb{A}^1\\times\\mathbb{A}^1)$), where the image is counted with multiplicity. If $c\\in\\mathbb{A}^1$ (resp. $c=\\infty$), the $S_{\\ell}$'s are the irreducible components of the germ $\\Delta_1\\setminus(\\{c\\}\\times\\mathbb{P}^1)$ (resp. $\\Delta_1\\cup\\Delta_2$) at $(c,\\infty)$; $m_{\\ell}$ is the multiplicity of $S_{\\ell}$ in $\\Delta_1\\setminus(\\{c\\}\\times\\mathbb{P}^1)$ (resp. $\\Delta_1\\cup\\Delta_2$). This computation can be found in \\cite{Ro2} (Theorem 6.4.1).\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe shall assume that the reader is familiar with the definitions and the properties of the specialization functor and the nearby cycles functor for $\\mathcal D$-modules along a hypersurface and we refer to \\cite{Ka}, \\cite{Ma2}, \\cite{MeSa}, \\cite{LaMa} and \\cite{MeMai} for details. Given $Y\\subset X$ a smooth hypersurface of $X$ defined by a global equation $h=0$ and given $\\mathcal{M}$ a specializable $\\mathcal{D}_X$-module, we will denote by $\\spe_Y\\mathcal{M}$ the specialization of $\\mathcal{M}$ along $Y$ and by $\\Psi_h\\mathcal{M}$ the nearby cycle module of $\\mathcal{M}$ along $Y$. If $V_{\\bullet}\\mathcal{M}$ is the canonical V-filtration with respect to the lexical order on $\\mathbb{C}\\simeq\\mathbb{R}\\oplus i\\mathbb{R}$, $\\spe_Y\\mathcal{M}=\\oplus_{\\beta\\in\\mathbb{C}}\\sfrac{V_{\\beta}\\mathcal{M}}{V_{<\\beta}\\mathcal{M}}$ and \n$\\Psi_h\\mathcal{M}=\\oplus_{-1\\leq\\beta<0}\\sfrac{V_{\\beta}\\mathcal{M}}{V_{<\\beta}\\mathcal{M}}$. Same definitions hold for $Y$  hypersurface with normal crossings.\nWe will also denote by $\\Psi_h$ the nearby cycle functor for bounded complexes with constructible cohomology (cf \\cite{MeMai} \\S 1).\n\nLet us explain the method we use to compute the formal invariants. We note that if a summand $R_\\alpha e^\\alpha$ shows up in the formal decomposition of a $\\mathbb{C}\\{t\\}\\langle\\partial_t\\rangle$-module $\\mathcal{N}_0$ after a ramification $\\rho$, then $R_\\alpha$ is the formal regular part of $((\\rho^*\\mathcal{N})e^{-\\alpha})_0$. This formal regular part can be recovered from the specialization of $(\\rho^*\\mathcal{N})e^{-\\alpha}$ at the origin (cf. Example 5.2.1 in \\cite{LaMa}). Moreover, as $R_{\\alpha}$ is localized at the origin, the rank of $R_{\\alpha}$ and the characteristic polynomial of the monodromy of $R_{\\alpha}$ can be computed using the nearby cycles module of $(\\rho^*\\mathcal{N})e^{-\\alpha}$ at the origin.\nOne of the main results we use is the commutation of the specialization functor (resp. nearby cycles functor) with proper direct images (cf. Theorem $9.4.1$ in \\cite{LaMa} for complexes of $\\mathcal{D}$-modules and Theorem $4.8-1$ in \\cite{MeSa} for $\\mathcal{D}$-modules). This property reduces the proof of Theorem \\ref{1} to local analytic computations on a suitable blowing-up space of $D\\times\\mathbb P^1$ above $(0,\\infty)$. It is also the main argument in the proof of Theorem \\ref{2}.\n\n\n\n\n\n\n\\section{Direct image of holonomic $\\mathcal{D}$-module of exponential type: the case of projections}\n\nThis section is devoted to the proof of Theorems \\ref{1} and \\ref{zeta}. We first note that Theorem \\ref{1} $(2)$ induces the construction of the Newton polygon (Theorem \\ref{1} $(1)$).\n\nLet $s\\in\\mathbb{N}^{\\ast}$. Denote by $\\Gamma_s$ the set of $\\alpha\\in\\Gamma$ with pole of order $s$ and by $\\Lambda_s$ the set of $\\ell\\in\\Lambda$ such that $\\sfrac{pq_{\\ell}}{p_{\\ell}}=s$. The Newton polygon of $\\oplus_{\\alpha\\in\\Gamma_s}R_{\\alpha}e^{\\alpha}$ has just one slope, equal to $s$; its height is the product of the rank of $\\oplus_{\\alpha\\in\\Gamma_s}R_{\\alpha}$ by the slope. According to Theorem \\ref{1} $(2)$, the slope is equal to $\\sfrac{pq_{\\ell}}{p_{\\ell}}$, for all $\\ell\\in\\Lambda_s$, and the height is  $\\sum_{\\alpha\\in\\Gamma_s}\\sum_{\\ell\\in\\Lambda_{\\alpha}}\\sfrac{pq_{\\ell}m_{\\ell}}{p_{\\ell}}=\\sum_{\\ell\\in\\Lambda_s}pq_{\\ell}m_{\\ell}$. If $\\widetilde{Q_{\\ell}}$ denotes the convex hull in $\\mathbb{R}^2$ of the union of $Q$ and $(m_{\\ell}p_{\\ell},pm_{\\ell}q_{\\ell})+Q$, the Newton polygon of $\\rho^{\\ast}\\mathcal{N}_0$ is, after a suitable translation, the Minkowski sum $\\sum_{\\ell\\in\\Lambda}\\widetilde{Q_{\\ell}}$. We deduce the Newton polygon of $\\mathcal{N}_0$ from the one of $\\rho^{\\ast}\\mathcal{N}_0$ by a dilation of the vertical axis in a ratio $\\sfrac{1}{p}$ (cf. Lemma 5.4.3 p. $34$ of \\cite{Sa}).\n\nWe focus now on the proof of Theorem \\ref{1} $(2)$ and Theorem \\ref{zeta}.\n\n\\subsection{Ramification}\n\nThe choice of the ramification enables us to reduce the proof of Theorem \\ref{1} and \\ref{zeta} to the case where the $p_l$'s are equal to $1$; it is the non ramified case ($p=1$).\n\nWe begin with a base change formula. Let $p_1^{'}:D^{'}\\times\\mathbb{P}^1\\to D^{'}$ and $p_2^{'}:D^{'}\\times\\mathbb{P}^1\\to\\mathbb{P}^1$ be the canonical projections. Consider the cartesian diagram:\n$$\\xymatrix{\nD^{'}\\times\\mathbb{P}^1\\ar[r]^{\\rho^{'}}\\ar[d]^{p_1^{'}}&D\\times\\mathbb{P}^1\\ar[d]^{p_1}\\\\\nD^{'}\\ar[r]^{\\rho}&D.}$$\n\n\n\\begin{lemma}\\label{comm}\n$\\rho^{\\ast}p_{1+}(\\mathcal{M}e^{p_2})=p_{1+}^{'}(\\rho^{'\\ast}(\\mathcal{M})e^{p_2^{'}})$.\n\\end{lemma}\n\n\\begin{proof}\nDenote by $~^p\\DR_{D\\times\\mathbb{P}^1\/D}$ (resp. $~^p\\DR_{D^{'}\\times\\mathbb{P}^1\/D^{'}}$) the relative de Rham functor of $p_1$ (resp. $p_1^{'}$). We adopt the convention that the relative de Rham complexes are concentrated in negative degrees. We have isomorphisms of complexes of $\\mathcal{D}_{D^{'}}$-modules:\n$$\n\\begin{array}{ll}\n\\rho^{\\ast}p_{1+}&(\\mathcal{M}e^{p_2})=\\\\\n&=\\mathcal{D}_{D^{'}\\to D}\\otimes_{\\rho^{-1}\\mathcal{D}_D}\\rho^{-1}\\mathbb{R}p_{1\\ast}\n~^p\\DR_{D\\times\\mathbb{P}^1\/D}(\\mathcal{M}e^{p_2})\\text{ (by definition),}\\\\\n&=\\mathcal{D}_{D^{'}\\to D}\\otimes_{\\rho^{-1}\\mathcal{D}_D}\\mathbb{R}p^{'}_{1\\ast}\\rho^{'-1}\n~^p\\DR_{D\\times\\mathbb{P}^1\/D}(\\mathcal{M}e^{p_2}),\\\\\n&\\text{(Proposition 2.6.7 in \\cite{KaSc}),}\\\\\n&=\\mathbb{R}p^{'}_{1\\ast}(\\mathcal{D}_{D^{'}\\times\\mathbb{P}^1\\to D\\times\\mathbb{P}^1}\\otimes_{\\rho^{'-1}\\mathcal{D}_{D\\times\\mathbb{P}^1}}\\rho^{'-1}\n~^p\\DR_{D\\times\\mathbb{P}^1\/D}(\\mathcal{M}e^{p_2})),\\\\\n&\\text{ (Proposition 2.6.6 in \\cite{KaSc}),}\\\\\n&=\\mathbb{R}p^{'}_{1\\ast}~^p\\DR_{D^{'}\\times\\mathbb{P}^1\/D^{'}}((\\mathcal{D}_{D^{'}\\times\\mathbb{P}^1\\to D\\times\\mathbb{P}^1}\\otimes_{\\rho^{'-1}\\mathcal{D}_{D\\times\\mathbb{P}^1}}\\rho^{'-1}\n(\\mathcal{M}))e^{p_2^{'}}),\\\\\n&=p_{1+}^{'}(\\rho^{'\\ast}(\\mathcal{M})e^{p_2^{'}}).\n\\end{array}$$\n\n\\end{proof}\n\nWe deduce that $\\rho^{\\ast}\\mathcal{N}=\\mathcal{H}^0p_{1+}^{'}(\\rho^{'\\ast}(\\mathcal{M})e^{p_2^{'}})$.\n\nThen we remark that the set $\\{S_{\\ell}^i~|~\\ell\\in\\Lambda,~i=1,\\ldots,p_{\\ell}\\}$ consists of the irreducible components of the singular support $S^{'}$ of $\\rho^{'\\ast}(\\mathcal{M})$ in a neighbourhood $U^{'}$ of $(0,\\infty)$, which are distinct of $\\{0\\}\\times\\mathbb{P}^1$ or $D^{'}\\times\\{\\infty\\}$. Let $\\xi_1,\\ldots,\\xi_{p_{\\ell}}$ be the $p_{\\ell}$-roots of the unity. Up to change of indices $i$, a parametrization of $S_{\\ell}^i$ is given by $\\gamma(\\tau)=(\\tau,\\alpha_{\\ell}(\\xi_i\\tau^{\\sfrac{p}{p_{\\ell}}})+\\delta_{\\ell}(\\xi_i\\tau^{\\sfrac{p}{p_{\\ell}}}))$; in particular, the intersection multiplicity \n$p_{\\ell}^i$ at $(0,\\infty)$ of $S_{\\ell}^i$ with $\\{0\\}\\times\\mathbb{P}^1$ is $1$.\n\nIn the next section, we will prove Theorem \\ref{1} $(2)$ and Theorem \\ref{zeta} in the case where all the $p_{\\ell}$'s are equal to $1$. Then, we may apply it to $\\rho^{'\\ast}(\\mathcal{M})$. At last, we have to compute the data associated with each $S_{\\ell}^i$ using data associated with the $S_{\\ell}$'s.\nBut we have:\\\\\n\n$\\bullet$ the polar part of a Puiseux parametrization of $S_{\\ell}^i$ at $(0,\\infty)$ is $\\alpha_{\\ell}(\\xi_i\\tau^{\\sfrac{p}{p_{\\ell}}})$. Moreover, if the assumption $(\\ast)$ is fulfilled for $\\mathcal{M}$ it is also fulfilled for $\\rho^{'\\ast}(\\mathcal{M})$, \\\\\n\n$\\bullet$ $m_{\\ell}$ is the multiplicity of  $T^{\\ast}_{S_{\\ell}^i}(D^{'}\\times\\mathbb{P}^1)$ in the characterstic cycle of $\\rho^{'\\ast}(\\mathcal{M})$,\\\\\n\n$\\bullet$ $\\zeta_{\\ell}$ is the characteristic polynomial of the monodromy of the local system  $\\Phi_{h_{\\ell}^i}\\DR~(\\rho^{'\\ast}(\\mathcal{M}))_{|S_{\\ell}^i\\setminus\\{(0,0)\\}}$ around $(0,0)$, where $h_{\\ell}^i=0$ is an equation of $S_{\\ell}^i$. \n\n\n\n\n\n\n\n\n\\subsection{Non ramified case: $p_{\\ell}=1$, for all $\\ell\\in\\Lambda$}\nIf we prove that the rank of $R_{\\alpha}$ is equal to $\\sum_{\\ell\\in\\Lambda_{\\alpha}}m_{\\ell}$, the irregularity number at $0$ of $\\mathcal{N}$ and of $\\oplus_{\\alpha\\in\\Gamma}R_{\\alpha}e^{\\alpha}$ are both equal to $\\sum_{\\ell\\in\\Lambda}m_{\\ell}q_{\\ell}$ (cf. \\cite{Ro} for $\\mathcal{N}$). Then it is sufficient to prove that each summand $R_{\\alpha}e^{\\alpha}$, with $\\alpha\\in\\Gamma$, shows up in the formal decomposition of $\\mathcal{N}_0$ and to compute the rank and the characteristic polynomial of the monodromy of each $R_{\\alpha}$. \n\n\nLet $\\alpha\\in\\Gamma$. We denote by $\\spe_0(\\mathcal{N}e^{-\\alpha})$ the specialization of $\\mathcal{N}e^{-\\alpha}$ at the origin. $\\spe_0(\\mathcal{N}e^{-\\alpha})_0$ is a $\\mathbb{C}[t]\\langle\\partial_t\\rangle$-module. As announced in the introduction, $R_{\\alpha}$ is the formal regular part of $(\\mathcal{N}e^{-\\alpha})_0$. It can be recovered using the isomorphism $R_{\\alpha}=\\mathbb{C}[[t]]\\otimes_{\\mathbb{C}[t]}\\spe_0(\\mathcal{N}e^{-\\alpha})_0$ (cf. Example $5.2.1$ of \\cite{LaMa}).\nLet $\\Psi_t(\\mathcal{N}e^{-\\alpha})$ be the nearby cycles module of $\\mathcal{N}e^{-\\alpha}$ at the origin. $\\Psi_t(\\mathcal{N}e^{-\\alpha})_0$ is a finite dimensional $\\mathbb{C}$-vector space equipped with an endomorphism of monodromy induced by $\\exp(-2i\\pi t\\partial_t)$.\nAs $\\mathcal{N}e^{-\\alpha}$ is localized at the origin, the rank of $R_{\\alpha}$ is the dimension of $\\Psi_t(\\mathcal{N}e^{-\\alpha})_0$ and the characteristic polynomials of their monodromies are equal.\n\nBut we have, $\\Psi_t(\\mathcal{N}e^{-\\alpha})_0=R\\Gamma(\\{0\\}\\times\\mathbb{P}^1,\\DR~\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1})[+1])$. Indeed,\n\n$$\\begin{array}{ll}\n\\mathcal{N}e^{-\\alpha}&=\\mathcal{O}_D[\\frac{1}{t}]e^{-\\alpha}\\otimes_{\\mathcal{O}_D}\\mathcal{H}^0p_{1+}(\\mathcal{M}e^{p_2}),\\\\\n&=\\mathcal{O}_D[\\frac{1}{t}]e^{-\\alpha}\\otimes^{\\mathbb{L}}_{\\mathcal{O}_D}\\mathcal{H}^0p_{1+}(\\mathcal{M}e^{p_2}),\\text{ (flatness),}\\\\\n&=(\\mathcal{O}_D[\\frac{1}{t}]e^{-\\alpha}\\otimes_{\\mathcal{O}_D}\\mathcal{D}_D)\\otimes^{\\mathbb{L}}_{\\mathcal{D}_D}\\mathcal{H}^0p_{1+}(\\mathcal{M}e^{p_2}),\\\\\n&=\\mathcal{H}^0p_{1+}((\\mathcal{O}_{D\\times\\mathbb{P}^1}[\\frac{1}{t}]e^{-\\alpha\\circ p_1}\\otimes_{\\mathcal{O}_{D\\times\\mathbb{P}^1}}\\mathcal{D}_{D\\times\\mathbb{P}^1})\\otimes_{\\mathcal{D}_D}^{\\mathbb{L}}\\mathcal{M}e^{p_2}),\\\\\n&\\text{ (Prop 2.6.6 in \\cite{KaSc}),}\\\\\n&=\\mathcal{H}^0p_{1+}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1}).\n\\end{array}$$\n\nLet $\\overline{p_1}:\\{0\\}\\times\\mathbb{P}^1\\to\\{0\\}$. According to the commutation of the nearby cycle functor with proper direct image (cf. Theorem 4.8-1 p. $226$ of \\cite{MeSa}),\n$$\\begin{array}{ll}\n\\Psi_t(\\mathcal{N}e^{-\\alpha})_0&=\\mathcal{H}^0\\overline{p_1}_+(\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1}))_0,\\\\\n&=R\\Gamma(\\{0\\}\\times\\mathbb{P}^1,\\DR~\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1})[+1]), \\text{ (cf. \\cite{Ma3} p.$5$).}\n\\end{array}$$\n\nThen, Theorem \\ref{1} $(2)$ and Theorem \\ref{zeta} follows from the proposition:\n\\begin{proposition}\\label{11}\n$\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1})$ has support included in $(0,\\infty)$ and the Euler characteristic of  $\\DR~\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1})_{(0,\\infty)}$ is $-\\sum_{\\ell\\in\\Lambda_{\\alpha}}m_{\\ell}$.\\\\\nMoreover, under the assumption $(\\ast)$, the zeta function of its monodromy is $\\prod_{\\ell\\in\\Lambda_\\alpha}\\zeta_{\\ell}^{-1}$.\n\\end{proposition}\n\nThe proof is given in the next sections.\n\n\n\\begin{remark}\nIn the non ramified case, the assumption $(\\ast)$ is equivalent to:\n$$\\forall\\ell,\\ell^{'}\\in\\Lambda_{\\alpha},~\\ell\\neq\\ell^{'}\\Longrightarrow\\delta_\\ell(0)\\neq\\delta_{\\ell^{'}}(0).$$\n\\end{remark}\n\n\n\n\n\\subsection{Local computations of nearby cycles modules}\nIn this section, we prepare the proof of Proposition \\ref{11} by giving some computations of nearby cycles modules along a normal crossing. Let $V$ be a neighbourhood of $(0,0)$ in $\\mathbb{C}^2$ and $(u,v)$ be some coordinates on $V$. Let $\\mathcal{M}$ be a regular holonomic $\\mathcal{D}_V$-module and $m,n,k,l\\in\\mathbb{N}$. We are interested in nearby cycles modules of the type $\\Psi_{u^mv^n}(\\mathcal{M}e^{\\sfrac{1}{u^kv^l}})$.\n\n\\subsubsection{Nearby cycles module along $u^mv^n=0$ of a module of exponential type}\\label{nota}\nSuppose that the singular support of $\\mathcal{M}$ is included in $uv=0$. Let $V^{\\ast}=V\\setminus\\{uv=0\\}$. $\\mathcal{M}_{|V^{\\ast}}$ is a holomorphic connection; we denote by $r$ its rank. Thus $(\\DR~\\mathcal{M})_{|V^{\\ast}}$ is a local system. For $P\\in V^{\\ast}$, we consider the monodromy $T_P$ of $(\\DR~\\mathcal{M})_P$ around $u=0$. As $V^{\\ast}$ is connected and the singular support of $\\mathcal{M}$ is included in a normal crossing, the characteristic polynomial of $T_P$ does not depend on $P$ (cf. I.2.2 p. 55 in \\cite{MeNa}). We denote it by $\\zeta_r$.\n\n\n\\begin{lemma}\\label{B}\n\\begin{enumerate}\n\\item If $k,m\\geq1$, $\\Psi_{u^m}(\\mathcal{M}e^{\\sfrac{1}{u^k}})=0$.\n\\item If $k,l,m,\\geq1$ and $n\\geq 0$, $\\Psi_{u^mv^n}(\\mathcal{M}e^{\\sfrac{1}{u^kv^l}})=0$.\n\\item If $n\\geq 1$, the Euler characteristic of  $\\DR~\\Psi_{uv^n}(\\mathcal{M}[\\frac{1}{uv}]e^{\\sfrac{1}{v}})_{(0,0)}$ is equal to $-r$ and the zeta function of its monodromy is $\\zeta_r^{-1}$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}\nThe proof of the last two points is given in Lemma $4.5.10$ of \\cite{Sa2} and its proof. The shift in the third point comes from the convention on the de Rham complexes. The first point is also proved in \\cite{Sa2} in the case where $\\mathcal{M}=\\mathcal{M}[\\frac{1}{v}]$. To conclude, it is sufficient to prove it in the case where $\\mathcal{M}$ has support in $v=0$. We have $\\mathcal{M}=i_+Li^{\\ast}(\\mathcal{M})[+1]$, where $i:\\{v=0\\}\\hookrightarrow V$,\nand $Li^{\\ast}(\\mathcal{M})[+1]$ is just a module.\nAs $\\Psi_{u^m}(Li^{\\ast}(\\mathcal{M})[+1]e^{\\sfrac{1}{u^k}}))=0$ (computation in one variable), we deduce that $\\Psi_{u^m}(\\mathcal{M}e^{\\sfrac{1}{u^k}})=0$.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Nearby cycles module along $u=0$ of a regular $\\mathcal{D}_V$-module}\n\nLet $\\widetilde{S}$ be the singular support of $\\mathcal{M}$ and $\\{\\widetilde{S_{\\ell}}~|~\\ell\\in\\Lambda\\}$ be the set consisting of the irreducible components of $\\widetilde{S}$ which are distinct of $u=0$. We will assume that {\\bf the intersection\nmultiplicity at $(0,0)$ of $\\widetilde{S_{\\ell}}$ with $u=0$ is equal to $1$}. \n\nWe denote by $r$ the rank of $\\mathcal{M}$. To any $\\widetilde{S_{\\ell}}$, we associate $m_{\\ell}$, the multiplicity of the conormal space $T^{\\ast}_{\\widetilde{S_{\\ell}}}V$ in the characteristic cycle of $\\mathcal{M}$.\n\n\n\\begin{lemma}\\label{A}\nThe Euler characteristic of $\\DR~\\Psi_u(\\mathcal{M}[\\frac{1}{u}])_{(0,0)}$ is equal to  $r-\\sum_{\\ell\\in\\Lambda}m_{\\ell}$.\n\\end{lemma}\n\n\\begin{proof}\nAs $\\mathcal{M}$ is regular holonomic, \n$$\\begin{array}{ll}\\DR~\\Psi_u(\\mathcal{M}[\\frac{1}{u}])&=\\DR~\\Psi_u(\\mathcal{M})\\text{ (Proposition 2.4-3 in \\cite{MeMai}),}\\\\\n&=\\Psi_u(\\DR~\\mathcal{M})\\text{ (Theorem $4.10.1$ p. 233 in \\cite{MeSa}).}\\end{array}$$ \n\n\nLet $\\epsilon$ and $\\eta$ small enough such that any $\\widetilde{S_{\\ell}}$ intersects\n$X_{\\epsilon,\\eta}=B(0,\\epsilon)\\cap\\{u=\\eta\\}$ at a unique point $P_{\\ell}$ (by assumption) and \n$\\Psi_u(\\DR~\\mathcal{M})_{(0,0)}=\\mathbb{R}\\Gamma(X_{\\epsilon,\\eta},\\DR~\\mathcal{M})$ (cf. Remark $1.1-7$ in \\cite{MeMai}).\nDenote by\n$X_{\\epsilon,\\eta}^c=X_{\\epsilon,\\eta}\\setminus\\{P_{\\ell},~\\ell\\in\\Lambda\\}$.  Let $B_{\\ell}$, $\\ell\\in\\Lambda$, be some disjoint balls centered at $P_{\\ell}$ of radius small enough. \n\n According to the index theorem of Kashiwara (Theorem 6.3.1 in \\cite{Ka1}), the Euler characteristic of $(\\DR~\\mathcal{M})_P$, $P\\in X_{\\epsilon,\\eta}^c$, (resp. $(\\DR~\\mathcal{M})_{P_{\\ell}}$) is equal to $r$ (resp. $r-m_{\\ell}$).\nAs $(\\DR~\\mathcal{M})_{|X_{\\epsilon,\\eta}}$ is constructible\nwith respect to the stratification $\\{X_{\\epsilon,\\eta}^c,\nP_{\\ell},~\\ell\\in\\Lambda\\}$, the Euler characteristic of  $\\mathbb{R}\\Gamma(X_{\\epsilon,\\eta},\\DR~\\mathcal{M})$ may be computed using the Mayer-Vietoris Theorem. We obtain:\n$$\\begin{array}{lll}\n\\chi(\\mathbb{R}\\Gamma(X_{\\epsilon,\\eta},\\DR~\\mathcal{M}))&=&\n\\chi(\\mathbb{R}\\Gamma(X_{\\epsilon,\\eta}^c,\\DR~\\mathcal{M}))+\\sum_{\\ell\\in\\Lambda}\\chi((\\DR~\\mathcal{M})_{P_{\\ell}})\\\\\n&&-\\sum_{\\ell\\in\\Lambda}\\chi(\\mathbb{R}\\Gamma(B_{\\ell}\\setminus\\{P_{\\ell}\\},\\DR~\\mathcal{M})),\\\\\n&=&\\chi(X_{\\epsilon,\\eta}^c).r+\\sum_{\\ell\\in\\Lambda}(r-m_{\\ell})\\\\\n&&-\\sum_{\\ell\\in\\Lambda}\\chi(B_{\\ell}\\setminus\\{P_{\\ell}\\}).r,\\\\\n&=&r-\\sum_{\\ell\\in\\Lambda}m_{\\ell}.\n\\end{array}$$\n\\end{proof}\n\nThe zeta function of the monodromy of $\\DR~\\Psi_u(\\mathcal{M}[\\frac{1}{u}])_{(0,0)}$ can not be compute with this method in the general case. Nevertheless, we can do it when the singular support of $\\mathcal{M}$ is included in the normal crossing $uv=0$. We adopt the notations of the section \\ref{nota}. Let $V_0=\\{v=0\\}\\setminus\\{(0,0)\\}$. We denote by $\\zeta$ the characteristic polynomial of the monodromy of the local system $\\phi_v(\\DR~\\mathcal{M})_{|V_0}$ around $(0,0)$.\n\n\\begin{lemma}\\label{C}\nIf the singular support of $\\mathcal{M}$ is included in a normal crossing $uv=0$, the zeta function of the monodromy of $\\DR~\\Psi_u(\\mathcal{M}[\\frac{1}{u}])_{(0,0)}$ is equal to $\\zeta_r\\zeta^{-1}$.\n\\end{lemma}\n\n\\begin{proof}\nWe have $\\DR~\\Psi_u(\\mathcal{M}[\\frac{1}{u}])_{(0,0)}=\\underset{\\underset{\\epsilon,\\eta}{\\leftarrow}}{\\lim}R\\Gamma(X_{\\epsilon,\\eta},\\DR~\\mathcal{M})$.\n\nAs the singular support of $\\mathcal{M}$ is included in $uv=0$ and as $\\DR~\\mathcal{M}$ is constructible with respect to a stratification such that $uv=0$ is a union of strata, we have $R\\Gamma(X_{\\epsilon,\\eta},\\DR~\\mathcal{M})=(\\DR~\\mathcal{M})_P$, where $P\\in V_0$ is the intersection point of $X_{\\epsilon,\\eta}$ with $v=0$. Then, the zeta function of the monodromy of $\\DR~\\Psi_u(\\mathcal{M}[\\frac{1}{u}])_{(0,0)}$ coincide with the zeta function of the monodromy of the complex of local systems $(\\DR~\\mathcal{M})_{|V_0}$ around $(0,0)$.\n\nConsider the triangle of complexes of local systems:\n$$(\\DR~\\mathcal{M})_{|V_0}\\to\\Psi_v(\\DR~\\mathcal{M})_{|V_0}\\to\\phi_v(\\DR~\\mathcal{M})_{|V_0}\\overset{+1}{\\to}.$$\n\nAs the zeta function of the monodromy of $\\Psi_v(\\DR~\\mathcal{M})_{|V_0}$ around $(0,0)$ is $\\zeta_r$, Lemma \\ref{C} follows.\n\\end{proof}\n\n\n\n\\subsection{Proof of Proposition \\ref{11}}~\\\\\nFirst we have to prove that $\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1})_{(0,c)}=0$, for any $c\\neq \\infty$. But in a neighbourhood $U$ of $(0,c)$ in $D\\times\\mathbb{P}^1$, there exist coordinates $(x,y)$ such that $p_1(x,y)=x$, $p_2(x,y)=y$ and $\\alpha\\circ p_1\\sim\\frac{1}{x^q}$, for some $q\\in\\mathbb{N}^{\\ast}$. Therefore,  $\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1})_{(0,c)}=\\Psi_x(\\mathcal{M}_{|U}e^{\\frac{1}{x^q}})_{(0,0)}$. As the singular support of $\\mathcal{M}_{|U}$ is not necessarily a normal crossing at $(0,0)$, we can not apply directly Lemma \\ref{B} 1.\n\nThe idea is to use a resolution  $\\pi=(\\pi_1,\\pi_2):\\mathbb{X}\\to U$, with exceptional locus $E$, such that the singular support of $\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\pi_1^{-1}(0)]$ has normal crossings in a neighbourhood of any point of $E$. As $\\mathcal{M}$ is holonomic and $\\pi$ is a proper isomorphism out of $E$, \n$\\mathcal{M}_{|U}e^{-\\frac{1}{x^q}}=\\pi_+(\\pi^{\\ast}(\\mathcal{M}_{|U})e^{-\\frac{1}{\\pi_1^q}})$ (cf. Proposition 7.4.5 of \\cite{Me}).\nUsing the commutation of the proper direct image with $\\Psi$ and $\\DR$ (Theorem 4.8.1 p. 226 in \\cite{MeSa} and Theorem 5.4.3 p. 77 in \\cite{Me}), we deduce that \n\\label{9}\n$$(\\ast\\ast)~\\DR ~\\Psi_{x}(\\mathcal{M}_{|U}e^{-\\frac{1}{x^q}})_{(0,0)}=R\\Gamma(E,\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})e^{-\\frac{1}{\\pi_1^q}})).$$\n\\begin{itemize}\n\\item If $P$ is a smooth point of $\\pi^{-1}_1(0)$, we have $\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})e^{-\\frac{1}{\\pi_1^q}})_P=0$ (cf. Lemma \\ref{B} 1). \n\\item If $P$ is a normal crossing point of $\\pi^{-1}_1(0)$, according to Lemma \\ref{B} 2, $\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})e^{-\\frac{1}{\\pi_1^q}})_P=0$. \n\\end{itemize}\n\nThen $\\Psi_{x}(\\mathcal{M}_{|U}e^{-\\frac{1}{x^q}})_{(0,0)}=0$ and $\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1})$ has support included in $(0,\\infty)$.\n\n\n\n\\vspace{0.5cm}\n\nLet us now focus on $\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1})_{(0,\\infty)}$. In a neighbourhood $U$ of $(0,\\infty)$ in $D\\times\\mathbb{P}^1$, there exist coordinates $(x,y)$ such that $p_1(x,y)=x$, $p_2(x,y)=\\sfrac{1}{y}$ and $p_2-\\alpha\\circ p_1(x,y)=\\sfrac{1}{y}-\\alpha(x)$. To simplify notation, we will set $g=\\sfrac{1}{y}-\\alpha(x)$. Then we have:\n$$\n\\Psi_{p_1}(\\mathcal{M}e^{p_2-\\alpha\\circ p_1})_{(0,\\infty)}=\\Psi_{x}(\\mathcal{M}_{|U}[\\frac{1}{xy}]e^{g})_{(0,0)}.$$\n\nWe have to compute the Euler characteristic and the zeta function of the monodromy of $\\DR~\\Psi_{x}(\\mathcal{M}_{|U}[\\frac{1}{xy}]e^g)_{(0,0)}$.\nWe can not use directly the local computations because $g$ is not equivalent to a function of the type $\\sfrac{1}{x^ky^l}$ at $(0,0)$. But we will be led to this situation with the help of a resolution. The choice of a good resolution is relevant for the end of the proof.\n\n\n\\subsubsection{Choice of a resolution}\n\n\\begin{lemma}\\label{resolution}\nThere exists a resolution $\\pi=(\\pi_1,\\pi_2):\\mathbb{X}\\to U$ with exceptional locus $E$ (actually it is a finite composition of the blow-up of the point $(0,0)\\in U$ and some blow-up of points in its exceptional locus) which satisfies the following conditions:\n\\begin{enumerate}\n\\item Let $\\widetilde{E}=\\pi^{-1}(\\{xy=0\\})$. There exists a unique irreducible component $E_d$ of $E$ which intersects $\\overline{\\widetilde{E}\\setminus E_d}$ in a unique point $P$. Moreover, in a neighbourhood of any point $Q\\in E$, we can choose local coordinates $(u,v)$ on $\\mathbb{X}$ such that:\n\\begin{enumerate}\n\\item If $Q\\notin E_d$, \n\\begin{itemize}\n\\item $\\pi_1(u,v)=u^m$ and $g\\circ\\pi(u,v)\\sim\\sfrac{1}{u^k}$, with $m,k\\geq 1$,\n\nor \n\\item $\\pi_1(u,v)=u^mv^n$ and $g\\circ\\pi(u,v)\\sim\\sfrac{1}{u^kv^l}$, with $k,l,m\\geq 1$, $n\\geq 0$.\n\\end{itemize}\n\\item If $Q=P$,\n\\begin{itemize}\n\\item $u=0$ (resp. $v=0$) is an equation of $E_d$ (resp. of the other component of $E$),\n\\item $\\pi_1(u,v)=uv^n$ and $g\\circ\\pi(u,v)\\sim\\sfrac{1}{v}$, with $n\\geq 1$.\n\\end{itemize}\n\\item If $Q\\in E_d\\setminus\\{P\\}$, \n\\begin{itemize}\n\\item $u=0$ is an equation of $E_d$,\n\\item $\\pi_1(u,v)=u$ and $g\\circ\\pi(u,v)=a+v$, with $a\\in\\mathbb{C}$.\n\\end{itemize}\n\\end{enumerate}\n\\item The singular support of $\\pi^{\\ast}(\\mathcal{M}_{|U}[\\frac{1}{xy}])=\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]$ has\nat most normal crossings at any point of\n$\\overline{E\\setminus E_d}$. \n\\end{enumerate}\n\\end{lemma}\n\n\\newpage\n\n\\begin{figure}[h]\n\\includegraphics[height=3cm]{exclocus.eps}\n\\caption{Exceptional locus $E$}\n\\end{figure}\n\n\\begin{proof}\nBy reducing to the same denominator, we have $g=\\sfrac{(x^q-y\\beta)}{x^qy}$, with $\\beta\\in\\mathbb{C}[x]$ and $\\beta(0)\\neq 0$.\n\nLet us begin by proving the first point. We will perform $2q$ blow-up of points. The resolution tree we will obtain has just one branch and the component $E_d$ will be the exceptional locus of the last blow-up.\n\nThe first $q$ blow-up allow us to decrease the degree on $x$ in the numerator. We obtain a resolution $\\pi=(\\pi_1,\\pi_2)$ composition of $q$ blow-up \n$$\n\\begin{array}{ccccccc}\nX^{(q)}&\\overset{\\pi^{(q)}}{\\to}&\\ldots&\\to&X^{(1)}&\\overset{\\pi^{(1)}}{\\to}&U\\\\\n\\cup&&&&\\cup&&\\\\\nE_q&&&&E_1&&\n\\end{array},$$\nwith exceptional locus $E=E_1\\cup\\ldots\\cup E_q$, such that $g\\circ\\pi$ can be written locally as in the case (a) except at a point $I\\in E_q\\setminus E_{q-1}$. In a neighbourhood of this point, we can choose local coordinates $(u,v)$ such that $\\pi_1(u,v)=u$ and $g\\circ\\pi(u,v)=\\sfrac{(u\\gamma(u)-v\\beta(u))}{u^qh(v)}$, with $h(v)=v+\\sfrac{1}{\\beta(0)}$ and $\\gamma(u)=\\sfrac{(\\beta(0)-\\beta(u))}{\\beta(0)u}$. \n\nNow the $q-1$ next blow-up allow us to decrease the degree in $u$ in the denominator and the degree in $u$ of $\\gamma(u)$. We obtain a new resolution $\\pi=(\\pi_1,\\pi_2)$, composition of $2q-1$ blow-up, such that $g\\circ\\pi$ can be written locally as in the case (a) except at a point $J\\in E_{2q-1}\\setminus E_{2q-2}$. In some suitable local coordinates $(u,v)$ in a neighbourhood of $J$,  $\\pi_1(u,v)=u$ and $g\\circ\\pi(u,v)=\\sfrac{(cu-v\\beta(u))}{uh(u,v)}$, with $c$ constant and $h(u,v)$ invertible.\n\nAt last, we blow-up $J$. The exceptional locus of this last blow up is $E_d$ and we can verify the local behaviour of $g$ on $E_d$. The details are left to the reader.\n\nTo prove the second point, it is sufficient to perform additional blow-up of points in $\\overline{E\\setminus E_d}$.\n\\end{proof}\n\n\n\n\n\n\n\n\nWe denote by $\\widetilde{S_{\\ell}}$ the strict transform of $S_{\\ell}$.\n\\begin{corollary}\\label{stricttransform}\\mbox{}\\par\n$(1)$ Let $\\ell\\in\\Lambda$. $\\widetilde{S_{\\ell}}\\cap E_d\\neq\\emptyset$ if\nand only if $\\ell\\in\\Lambda_{\\alpha}$.\n\n$(2)$ The assumption $(\\ast)$ is equivalent to:\n$\\forall \\ell,\\ell^{'}\\in\\Lambda_{\\alpha},~\\ell\\neq\\ell^{'}\\Longrightarrow\\widetilde{S_{\\ell}}\\cap E_d\\neq\\widetilde{S_{\\ell^{'}}}\\cap E_d$.\n\\end{corollary}\n\n\\begin{proof}\n$(1)$ According to the choice of the resolution,\n$\\widetilde{S_{\\ell}}\\cap E_d\\neq\\emptyset$ if and only if\n$\\widetilde{S_{\\ell}}\\cap (E_d\\setminus\\{P\\})\\neq\\emptyset$. Let $Q_{\\ell}\\in\\widetilde{S_{\\ell}}\\cap (E_d\\setminus\\{P\\})$ and  $(u,v)$ be some local coordinates in the\nneighbourhood of $Q_{\\ell}$ such that $f\\circ\\pi(u,v)=a+v$ and\n$\\pi_1(u,v)=u$, with $a\\in\\mathbb{C}$.\n\nAs the\nintersection multiplicity of $S_{\\ell}$ with $\\{0\\}\\times\\mathbb{P}^1$ is $1$ and\nas $\\pi_1(u,v)=u$, the intersection multiplicity of\n$\\widetilde{S_{\\ell}}$ with $E_d$ is also $1$. Then:\n\\begin{displaymath}\n\\begin{split}\n&\\widetilde{S_{\\ell}}\\cap E_d\\neq\\emptyset,\\\\\n&\\Longleftrightarrow\\exists\\delta(s)\\in\\mathbb{C}\\{s\\}\\text{ such that }a+v=\\delta(u) \\text{ is an equation of }\\widetilde{S_{\\ell}},\\\\\n&\\Longleftrightarrow\\exists\\delta(s)\\in\\mathbb{C}\\{s\\}\\text{ such that }f\\circ\\pi(u,v)=\\delta\\circ\\pi_1(u) \\text{ is an equation of }\\widetilde{S_{\\ell}},\\\\\n&\\Longleftrightarrow\\exists\\delta(s)\\in\\mathbb{C}\\{s\\}\\text{ such that }f(x,y)=\\delta(x) \\text{ is an equation of }S_{\\ell},\\\\\n&\\Longleftrightarrow\\exists\\delta(s)\\in\\mathbb{C}\\{s\\}\\text{ such that }\\sfrac{1}{y}=\\alpha(x)+\\delta(x) \\text{ is an equation of }S_{\\ell},\\\\\n&\\Longleftrightarrow \\ell\\in\\Lambda_{\\alpha}.\n\\end{split}\n\\end{displaymath}\n\n$(2)$ Here, we want to understand the behaviour of $S_{\\ell}$ after the resolution $\\pi$. We will keep the same notations as in the proof of Lemma \\ref{resolution}.\n\nLet $\\ell\\in\\Lambda_{\\alpha}$. Then $S_{\\ell}$ has equation $h(x,y)=(\\beta(x)+x^q\\delta_{\\ell}(x))y-x^q=0$. \n\nAfter the first $q$ blow-up, in the neighbourhood of the point $I\\in E_q\\setminus E_{q-1}$ with local coordinates $(u,v)$, \n$$h\\circ\\pi(u,v)=u^q\\big((\\beta(u)+u^q\\delta_{\\ell}(u))v+u(-\\gamma(u)+u^{q-1}\\sfrac{\\delta_{\\ell}(u)}{\\beta(0)})\\big).$$ \nAfter the next $(q-1)$ blow-up, in the neighbourhood of the point $J$, we can choose local coordinates $(u,v)$ such that \n$$h\\circ\\pi(u,v)=u^{2q-1}\\big((\\beta(u)+u^q\\delta_{\\ell}(u))v+u(-c+\\sfrac{\\delta_{\\ell}(u)}{\\beta(0)}+u\\widetilde{\\delta}(u))\\big),$$\nwhere $\\widetilde{\\delta}(u)\\in\\mathbb{C}\\{u\\}$ and $c$ is a constant which depends only on $\\alpha$.\n\nAt last, we blow-up $J$ and obtain that in the chart $u=s$, $v=st$, \n$$h\\circ\\pi(s,t)=s^{2q}\\big((\\beta(s)+s^q\\delta_{\\ell}(s))t-c+\\sfrac{\\delta_{\\ell}(s)}{\\beta(0)}+s\\widetilde{\\delta}(s)\\big).$$\nThen, we see that the strict transform $\\widetilde{S_{\\ell}}$ of $S_{\\ell}$ interesct $E_d$ at a point which depends only on $\\alpha$ and $\\delta_{\\ell}(0)$. The assertion $(2)$ follows.\n\\end{proof}\n\n\n\n\\subsubsection{Use of the local computations}\n\nThere exists an isomorphism: \n$$\\begin{array}{ll}\n\\DR~\\Psi_{x}(\\mathcal{M}_{|U}[\\frac{1}{xy}]e^g)_{(0,0)}\n&=R\\Gamma(E,\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U}[\\frac{1}{xy}])e^{g\\circ\\pi})),\\\\\n&\\text{(Same proof as for the isomorphism $(\\ast\\ast)$ p. \\pageref{9}),}\\\\\n&=R\\Gamma(E,\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})).\n\\end{array}$$\n\n\nLet us first prove that the complex $\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_{|E}$ has support included in $E_d$. We have two cases to consider:\n\n\\begin{enumerate}\n\\item In a neighbourhood $V\\subset\\mathbb{X}$ of a smooth point $Q$ in $E\\setminus E_d$, according to Lemma \\ref{resolution}, there exist coordinates $(u,v)$ on $V$ such that $\\pi_1(u,v)=u^m$, $g\\circ\\pi(u,v)\\sim\\sfrac{1}{u^k}$ and the singular support of $\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]$ is included in $uv=0$. Then, according to Lemma \\ref{B} 1, $$\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_Q=\\Psi_{u^m}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{\\frac{1}{u^k}})_{(0,0)}=0.$$\n\n\\item In a neighbourhood $V\\subset\\mathbb{X}$ of a normal crossing point $Q$ in $E\\setminus E_d$, according to Lemma \\ref{resolution}, there exist coordinates $(u,v)$ on $V$ such that $\\pi_1(u,v)=u^mv^n$, $g\\circ\\pi(u,v)\\sim\\sfrac{1}{u^kv^l}$ and the singular support of $\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]$ is included in $uv=0$. Then, according to Lemma \\ref{B} 2, $$\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_Q=\\Psi_{u^mv^n}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{\\frac{1}{u^kv^l}})_{(0,0)}=0.$$\n\\end{enumerate}\n\nThen $\\DR~\\Psi_{x}(\\mathcal{M}_{|U}[\\frac{1}{xy}]e^g)_{(0,0)}=R\\Gamma(E_d,\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi}))$.\n\n\\vspace{0.5cm}\n\nNow we want to examine the complex $\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})$ locally on $E_d$. Let $T$ be a tubular neighbourhood of $E_d$. According to Corollary \\ref{stricttransform} (1), the singular support $\\widetilde{S_T}$ of $(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}])_{|T}$ is $\\big(E\\cup(\\cup_{\\ell\\in\\Lambda_{\\alpha}}\\widetilde{S_\\ell})\\big)\\cap T$.\n\n\\vspace{0.5cm}\n\n\\noindent\n{\\bf Computation of the Euler characteristic:} We denote by $Q_1,\\ldots,Q_k$ the intersection point of $\\cup_{\\ell\\in\\Lambda_{\\alpha}}\\widetilde{S_\\ell}$ with $E_d$.\n\nLet $r$ be the rank of $\\mathcal{M}$. As $\\pi$ is an isomorphism out of $E$, the rank of $\\pi^{\\ast}(\\mathcal{M}_{|U})$ is $r$ and the multiplicity of the conormal space $T_{\\widetilde{S_\\ell}}^{\\ast}\\mathbb{X}$ in the characteristic cycle of $\\pi^{\\ast}(\\mathcal{M}_{|U})$ is $m_\\ell$.\n\n\\begin{figure}[h]\n\\includegraphics[height=2.5cm]{euler.eps}\n\\end{figure}\n\n\\begin{enumerate}\n\\item In a neighbourhood $V\\subset\\mathbb{X}$ of the point $P$, according to Lemma \\ref{resolution}, there exist coordinates $(u,v)$ on $V$ such that $\\pi_1(u,v)=uv^n$, $g\\circ\\pi(u,v)\\sim\\sfrac{1}{v}$ and the singular support of $\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]$ is included in $uv=0$. Then,\n$$\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_P=\\Psi_{uv^n}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\frac{1}{uv}]e^{\\frac{1}{v}})_{(0,0)}.$$ \n\nApplying Lemma \\ref{B} 3., we conclude that the Euler characteristic of the complex $\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_P$ is equal to $-r$.\n\n\n\n\\item In a neighbourhood $V\\subset\\mathbb{X}$ of a point $Q_i$, $g\\circ\\pi$ is holomorphic and $\\pi^{-1}_1(0)$ is smooth. Then we have\n$\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_{Q_i}=\\DR~\\Psi_u(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\frac{1}{u}])_{(0,0)}$\nand, according to Lemma \\ref{A}, the Euler characteristic of the complex $\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_{Q_i}$ is $r-\\sum_{\\widetilde{S_{\\ell}}\\cap E_d=\\{Q_i\\}}m_{\\ell}$.\n\n\\item In a neighbourhood of a point $Q\\in E_d\\setminus\\{P,Q_1,\\ldots,Q_k\\}$, as in the previous point, we prove that the Euler characteristic of the complex $\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_Q$ is $r$ (because $\\{\\ell\\in\\Lambda~|~\\widetilde{S_\\ell}\\cap E_d=\\{Q\\}\\}=\\emptyset$).\n\\end{enumerate}\n\nUsing the Mayer-Vietoris Theorem, we deduce that the Euler characteristic of the complex $R\\Gamma(E_d,\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi}))$ is equal to\n\n\\begin{displaymath}\n\\begin{split}\n-r+\\sum_{i=1}^k\\bigg(r-\\sum_{\\widetilde{S_{\\ell}}\\cap E_d=\\{Q_i\\}}m_{\\ell}\\bigg)+\\chi(E_d\\setminus\\{P,Q_1,\\ldots,Q_k\\}).r&=\n-\\sum_{\\widetilde{S_{\\ell}}\\cap E_d\\neq\\emptyset}m_{\\ell}\\\\\n&=-\\sum_{\\ell\\in\\Lambda_{\\alpha}}m_{\\ell}.\n\\end{split}\n\\end{displaymath}\n\n\n\n\n{\\bf Computation of the zeta function of the monodromy:} Assume that $(\\ast)$ is fulfilled. The fact that for any $\\ell\\in\\Lambda$, $p_\\ell=1$ and the assumption $(\\ast)$ imply that $\\widetilde{S_T}$ is a divisor with normal crossing (cf. Corollary \\ref{stricttransform} (2)).\n\nWe denote by $T^{\\ast}$ the complement of $\\widetilde{S_T}$ in $T$ and by $Q_{\\ell}$ the intersection point of one $\\widetilde{S_\\ell}$ with $E_d$, $\\ell\\in\\Lambda_{\\alpha}$.\n\n\n\\begin{figure}[h]\n\\includegraphics[height=2.5cm]{zeta.eps}\n\\end{figure}\n\n\\begin{itemize}\n\\item[$\\bullet$] $(\\DR~\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}])_{|T^{\\ast}}$ is a local system. We can define the monodromy endomorphism around $E_d$, $T_P:(\\DR~\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}])_{P}\\to (\\DR~\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}])_{P}$, for any $P\\in T^{\\ast}$. As $T^{\\ast}$ is connected and $\\widetilde{S_T}$ has normal crossings, the characteristic polynomial of $T_P$ does not depend on $P\\in T^{\\ast}$. We denote it by $\\zeta_r$. \n\\item[$\\bullet$] To any $\\ell\\in\\Lambda_{\\alpha}$, we associate the characteristic polynomial $\\zeta_\\ell$ of the monodromy of the local system $\\phi_{\\widetilde{h_\\ell}}(\\DR~\\pi^{\\ast}(\\mathcal{M}_{|U}))_{|\\widetilde{S_\\ell}\\setminus\\{(0,0)\\}}$ around $(0,0)$, where $\\widetilde{h_\\ell}$ is an equation of $\\widetilde{S_\\ell}$. If $h_\\ell$ is an equation of $S_\\ell$, as $\\pi$ is an isomorphism out of $E$, it coincides with the characteristic polynomial of the monodromy of $\\phi_{h_\\ell}(\\DR~\\mathcal{M})_{|S_\\ell\\setminus\\{(0,0)\\}}$ around $(0,0)$.\n\\end{itemize}\n\nUsing the local behaviour of $\\pi_1$ and $g\\circ\\pi$ on $E_d$ given by Lemma \\ref{resolution}, we have:\n\n\\begin{enumerate}\n\\item In a neighbourhood of the point $P$, according to Lemma \\ref{B} 3., the zeta function of the monodromy of $\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_P$ is equal to $\\zeta_r^{-1}$.\n\\item In a neighbourhood of a point $Q_\\ell$, $\\ell\\in\\Lambda_{\\alpha}$, according to Lemma \\ref{C}, the zeta function of the monodromy of $\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_{Q_\\ell}$ is equal to $\\zeta_r\\zeta_\\ell^{-1}$.\n\\item In a neighbourhood of a point $Q\\in E_d\\setminus\\{P,Q_\\ell,\\ell\\in\\Lambda_{\\alpha}\\}$, the zeta function of the monodromy of $\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi})_Q$ is equal to $\\zeta_r$.\n\\end{enumerate}\n\nThen, using the Mayer-Vietoris Theorem, we deduce that the zeta function of the monodromy of $R\\Gamma(E_d,\\DR~\\Psi_{\\pi_1}(\\pi^{\\ast}(\\mathcal{M}_{|U})[\\ast\\widetilde{E}]e^{g\\circ\\pi}))$ is equal to\n$$\\zeta_r^{-1}.\\prod_{\\ell\\in\\Lambda_{\\alpha}}(\\zeta_r\\zeta_\\ell^{-1}).\\zeta_r^{\\chi(E_d\\setminus\\{P,Q_\\ell,\\ell\\in\\Lambda_{\\alpha}\\})}=\\prod_{\\ell\\in\\Lambda_{\\alpha}}\\zeta_\\ell^{-1}.$$ \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Realization of  $\\mathbb{C}[[t]]\\langle\\partial_t\\rangle$-modules}\\label{realization}\nLet $\\widehat{\\mathcal{N}}$ be a formal holonomic $\\mathbb{C}[[t]]\\langle\\partial_t\\rangle$-module.\n\n\\begin{theorem}\\label{realisation}\nThere exist a ramification $\\rho:\\tau\\to t=\\tau^p$ and a regular holonomic $\\mathcal{D}_{D\\times\\mathbb{P}^1}$-module $\\mathcal{M}$ such that $\\rho^{\\ast}(\\widehat{\\mathcal{N}})$ is isomorphic to the formalization of $\\rho^{\\ast}(\\mathcal{H}^0p_{1+}(\\mathcal{M}e^{p_2}))_0$.\n\\end{theorem}\n\n\\begin{proof}\nWe recall that we identify $\\mathcal{D}_{D,0}$ with $\\mathbb{C}\\{t\\}\\langle\\partial_t\\rangle$, by choosing a coordinate $t$ on $D$. We begin by stating two lemmas.\n \n\\begin{lemma}\\label{BB} \n\\begin{enumerate}\n\\item $p_{1+}(\\mathcal{O}_{D\\times\\mathbb{P}^1}e^{p_2})=\\mathcal{O}_{D}$.\n\\item $p_{1+}(\\mathcal{O}_{D\\times\\mathbb{P}^1}[t^{-1}]e^{p_2})=\\mathcal{O}_{D}[t^{-1}]$.\n\\item Let $(p,\\alpha)\\in\\mathbb{N}^{\\ast}\\times(\\tau^{-1}.\\mathbb{C}[\\tau^{-1}])$. Let $\\rho:D^{'}\\to D$, defined by $\\rho(\\tau)=t=\\tau^p$. We denote by $Z$ the curve in $D\\times\\mathbb{P}^1$ parametrized by $\\tau\\to(\\tau^p,\\alpha)$ and by $\\widetilde{Z}$ the union of $Z$, $\\{0\\}\\times\\mathbb{P}^1$ and $D\\times\\{\\infty\\}$.\n$$\\rho^{\\ast}p_{1+}(\\mathcal{O}_{D\\times\\mathbb{P}^1}[\\ast \\widetilde{Z}]e^{p_2})=\\underset{\\xi^p=1}{\\oplus}\\mathcal{O}_{D^{'}}[\\tau^{-1}]e^{\\alpha(\\xi\\tau)}\\oplus \\mathcal{O}_{D^{'}}[\\tau^{-1}].$$ \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe just prove the third point. The first two points can be proved similary.\n\nLet $p_1^{'}:D^{'}\\times\\mathbb{P}^1\\to D^{'}$ and $p_2^{'}:D^{'}\\times\\mathbb{P}^1\\to\\mathbb{P}^1$ be the canonical projections and $\\rho^{'}=(\\rho,id):D^{'}\\times\\mathbb{P}^1\\to D\\times\\mathbb{P}^1$.\n\n\nFirst note that $\\rho^{\\ast}p_{1+}(\\mathcal{O}_{D\\times\\mathbb{P}^1}[\\ast \\widetilde{Z}]e^{p_2})=p^{'}_{1+}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})]e^{p_2^{'}})$ (cf. Lemma \\ref{comm}).\n\nDenote by $\\mathcal{K}^{\\bullet}:=~^pDR_{D^{'}\\times\\mathbb{P}^1\/D^{'}}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})]e^{p_2^{'}})$ the relative de Rham complex of $p_1$ concentrated in negative degrees. It consists in the complex $$\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})]\\overset{d}{\\to}\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})],$$\nwith $d(h)=(\\partial_{\\mathbb{P}^1}(h)+h\\partial_{\\mathbb{P}^1}(p_2^{'}))$.\n\nBy definition, $p_{1+}^{'}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})]e^{p_2^{'}})=\\mathbb{R}p_{1\\ast}^{'}(\\mathcal{K}^{\\bullet})$. We consider the spectral sequence $\\mathcal{H}^{i-k}(R^kp_{1\\ast}^{'}(\\mathcal{K}^{\\bullet}),d)\\Longrightarrow \\mathbb{R}^ip_{1\\ast}^{'}(\\mathcal{K}^{\\bullet})$, where $(R^kp_{1\\ast}^{'}(\\mathcal{K}^{\\bullet}),d)$ denotes the complex $R^kp_{1\\ast}^{'}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})])\\overset{d}{\\to}R^kp_{1\\ast}^{'}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})])$.\n\nWe claim that $R^kp_{1\\ast}^{'}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})])=0$, for $k\\neq 0$. Indeed, let $j:(D^{'}\\times\\mathbb{P}^1)\\setminus\\rho^{'-1}(\\widetilde{Z})\\hookrightarrow D^{'}\\times\\mathbb{P}^1$ and $P\\in D^{'}$:\n$$\\begin{array}{ll}\nR^kp_{1\\ast}^{'}&(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})])_P=\\\\\n&=R^kp_{1\\ast}^{'}Rj_{\\ast}j^{-1}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1})_P,\\\\\n&=R^k(p_1^{'}\\circ j)_{\\ast}j^{-1}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1})_P,\\\\\n&=\\varinjlim_{U\\ni P open}R\\Gamma(j^{-1}(U\\times\\mathbb{P}^1),j^{-1}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1})).\\\\\n\\end{array}$$\n\nAs $j^{-1}(U\\times\\mathbb{P}^1)=(U\\times\\mathbb{C})\\setminus\\rho^{'-1}(Z\\cup\\{0\\}\\times\\mathbb{C})$ is Stein, we conclude that  $R^kp_{1\\ast}^{'}(\\mathcal{O}_{D^{'}\\times\\mathbb{P}^1}[\\ast \\rho^{'-1}(\\widetilde{Z})])=0$, for $k\\neq 0$.\n\nNow $(R^0p_{1\\ast}^{'}(\\mathcal{K}^{\\bullet}),d)$ is the complex:\n$$\\mathcal{O}_{D^{'}}[y]\\bigg[\\frac{1}{\\tau\\underset{\\xi^p=1}{\\prod}(y-\\alpha(\\xi\\tau))}\\bigg]\\overset{d}{\\to} \\mathcal{O}_{D^{'}}[y]\\bigg[\\frac{1}{\\tau\\underset{\\xi^p=1}{\\prod}(y-\\alpha(\\xi\\tau))}\\bigg],$$\nwith $d(h)=(\\partial_y(h)+h)$.\n\n\\begin{itemize}\n\\item Computation of $\\mathcal{H}^{-1}(R^0p_{1\\ast}^{'}(\\mathcal{K}^{\\bullet}),d)$:\\\\\nLet $h\\in\\mathcal{O}_{D^{'}}[y]\\bigg[\\frac{1}{\\tau\\underset{\\xi^p=1}{\\prod}(y-\\alpha(\\xi\\tau))}\\bigg]$ such that $d(h)=0$. Thus $h=Ce^{-y}$, with $C\\in\\mathcal{O}_{D^{'}}[\\frac{1}{\\tau}]$ and then $h=0$. Thus, $\\mathcal{H}^{-1}(R^0p_{1\\ast}^{'}(\\mathcal{K}^{\\bullet}),d)=0$.\n\\item Computation of $\\mathcal{H}:=\\mathcal{H}^{0}(R^0p_{1\\ast}^{'}(\\mathcal{K}^{\\bullet}),d)$:\\\\\nWe note first that $\\mathcal{H}$ is generated over $\\mathcal{O}_{D^{'}}[\\frac{1}{\\tau}]$ by \\\\$\\{y^m,\\frac{y^m}{(y-\\alpha(\\xi\\tau))^n};~m\\geq 0,~n\\geq 1 \\text{ and }\\xi^p=1\\}$.\n\nMoreover, if $m,n>0$, we have:\n$\\frac{y^m}{(y-\\alpha(\\xi\\tau))^n}=\\frac{y^{m-1}}{(y-\\alpha(\\xi\\tau))^{n-1}}+\\frac{y^{m-1}\\alpha(\\xi\\tau)}{(y-\\alpha(\\xi\\tau))^n}$ and $y^m\\equiv -my^{m-1}$.\n\nThen $\\mathcal{H}$ is generated over $\\mathcal{O}_{D^{'}}[\\frac{1}{\\tau}]$ by $\\{1,\\frac{1}{(y-\\alpha(\\xi\\tau))^n};~n\\geq 1 \\text{ and }\\xi^p=1\\}$. But we have $\\frac{1}{(y-\\alpha(\\xi\\tau))^n}\\equiv\\frac{n}{(y-\\alpha(\\xi\\tau))^{n+1}}$. Then, $\\mathcal{H}$ is generated over $\\mathcal{O}_{D^{'}}[\\frac{1}{\\tau}]$ by $\\{1,\\frac{1}{(y-\\alpha(\\xi\\tau))};~\\xi^p=1\\}$.\n\nThen we have  $\\mathcal{H}=\\underset{\\xi^p=1}{\\oplus}\\mathcal{O}_{D^{'}}[\\tau^{-1}].\\frac{1}{(y-\\alpha(\\xi\\tau))}\\oplus \\mathcal{O}_{D^{'}}[\\tau^{-1}]$. \n\nAs $\n\\partial_{\\tau}(\\frac{1}{y-\\alpha(\\xi\\tau)})=\\partial_{\\tau}(\\alpha(\\xi\\tau)).\\frac{1}{(y-\\alpha(\\xi\\tau))^2}\\equiv\\partial_{\\tau}(\\alpha(\\xi\\tau)).\\frac{1}{y-\\alpha(\\xi\\tau)}$, we have an isomorphism of $\\mathcal{D}_{D^{'}}$-modules\n$$\\mathcal{O}_{D^{'}}[\\tau^{-1}].\\frac{1}{(y-\\alpha(\\xi\\tau))}=\\mathcal{O}_{D^{'}}[\\tau^{-1}]e^{\\alpha(\\xi\\tau)}.$$\n\nLemma \\ref{BB} follows.\n\\end{itemize}\n\n\n\n\\end{proof}\n\n\n\\begin{lemma}\\label{AA}\nLet $\\mathcal{N}$ be a holonomic $\\mathcal{D}_{D}$-module and $\\mathcal{M}$ be a holonomic $\\mathcal{D}_{D\\times\\mathbb{P}^1}$-module. Suppose that $\\mathcal{M}$ is $\\mathcal{O}_{D\\times\\mathbb{P}^1}$-flat. We have an isomorphism: $$p_{1+}(\\mathcal{M}\\otimes_{\\mathcal{O}_{D\\times\\mathbb{P}^1}}p_1^{\\ast}(\\mathcal{N}))=p_{1+}(\\mathcal{M})\\otimes_{\\mathcal{O}_{D}}^{\\mathbb{L}}\\mathcal{N}.$$\n\\end{lemma}\n\n\\begin{proof}\nFirst, we remark that $\\mathcal{M}\\otimes_{\\mathcal{O}_{D\\times\\mathbb{P}^1}}p_1^{\\ast}\\mathcal{N}=\\mathcal{M}\\otimes_{p_1^{-1}\\mathcal{O}_D}p_1^{-1}\\mathcal{N}$, with the structure of $\\mathcal{D}_{D\\times\\mathbb{P}^1}$-module given by $\\xi(m\\otimes n)=\\xi(m)\\otimes n+\\xi(p_1)m\\otimes \\partial_t(n)$, where $\\xi$ is a local section of the sheaf of vector fields on $D\\times\\mathbb{P}^1$.\n\nDenote by $~^p\\DR_{D\\times\\mathbb{P}^1\/D}$ (resp. $\\Omega^{\\bullet+1}_{D\\times\\mathbb{P}^1\/D}$) the relative de Rham functor of $p_1$ (resp. the complex of relative forms of $p_1$). The differential $d$ of the complex $~^p\\DR_{D\\times\\mathbb{P}^1\/D}(\\mathcal{M}\\otimes_{p_1^{-1}\\mathcal{O}_D}p_1^{-1}\\mathcal{N})=\\Omega^{\\bullet+1}_{D\\times\\mathbb{P}^1\/D}\\otimes_{\\mathcal{O}_{D\\times\\mathbb{P}^1}}(\\mathcal{M}\\otimes_{p_1^{-1}\\mathcal{O}_D}p_1^{-1}\\mathcal{N})$ is defined by:\n$$\\begin{array}{lll}\nd(w\\otimes(m\\otimes n))&=&dw\\otimes(m\\otimes n)+dp_2\\wedge w\\otimes\\partial_{\\mathbb{P}^1}(m\\otimes n),\\\\\n&=&dw\\otimes(m\\otimes n)+dp_2\\wedge w\\otimes(\\partial_{\\mathbb{P}^1}(m)\\otimes n),\n\\end{array}$$\nand the action of $p_1^{-1}\\mathcal{D}_D$ is given by:\n$$\\partial_t(w\\otimes(m\\otimes n))=w\\otimes(\\partial_tm\\otimes n)+w\\otimes(m\\otimes\\partial_tn).$$\n\nThen, we have an isomorphism of $p_1^{-1}\\mathcal{D}_D$-modules:\n$$\n~^p\\DR_{D\\times\\mathbb{P}^1\/D}(\\mathcal{M}\\otimes_{p_1^{-1}\\mathcal{O}_D}p_1^{-1}\\mathcal{N})=~^p\\DR_{D\\times\\mathbb{P}^1\/D}(\\mathcal{M})\\otimes_{p_1^{-1}\\mathcal{O}_D}p_1^{-1}\\mathcal{N}.$$\n\nWe conclude with:\n$$\\begin{array}{ll}\np_{1+}(\\mathcal{M}&\\otimes^{\\mathbb{L}}_{\\mathcal{O}_{D\\times\\mathbb{P}^1}}p_1^{\\ast}(\\mathcal{N}))=\\\\\n&=\\mathbb{R}p_{1\\ast}(~^p\\DR_{D\\times\\mathbb{P}^{1}\/D}(\\mathcal{M}\\otimes_{\\mathcal{O}_{D\\times\\mathbb{P}^1}}p_1^{\\ast}(\\mathcal{N}))),\\\\\n&=\\mathbb{R}p_{1\\ast}(~^p\\DR_{D\\times\\mathbb{P}^{1}\/D}(\\mathcal{M})\\otimes_{p_1^{-1}\\mathcal{O}_D}p_1^{-1}(\\mathcal{N})),\\\\\n&=\\mathbb{R}p_{1\\ast}(~^p\\DR_{D\\times\\mathbb{P}^{1}\/D}(\\mathcal{M})\\otimes^{\\mathbb{L}}_{p_1^{-1}\\mathcal{O}_D}p_1^{-1}(\\mathcal{N})),\\\\\n&(~^p\\DR_{D\\times\\mathbb{P}^{1}\/D}(\\mathcal{M})\\text{ is }p_1^{-1}\\mathcal{O}_D\\text{-flat),}\\\\\n&=\\mathbb{R}p_{1\\ast}(~^p\\DR_{D\\times\\mathbb{P}^{1}\/D}(\\mathcal{M})\\otimes^{\\mathbb{L}}_{p_1^{-1}\\mathcal{D}_D}(p_1^{-1}\\mathcal{D}_D\\otimes_{p_1^{-1}\\mathcal{O}_D}p_1^{-1}(\\mathcal{N}))),\\\\\n&=\\mathbb{R}p_{1\\ast}(~^p\\DR_{D\\times\\mathbb{P}^{1}\/D}(\\mathcal{M}))\\otimes^{\\mathbb{L}}_{\\mathcal{D}_D}(\\mathcal{D}_D\\otimes_{\\mathcal{O}_D}\\mathcal{N}),\\\\\n&\\text{(Proposition 2.6.6 in \\cite{KaSc}),}\\\\\n&p_{1+}(\\mathcal{M})\\otimes^{\\mathbb{L}}_{\\mathcal{O}_D}\\mathcal{N}.\n\\end{array}$$\n\\end{proof}\n\n\n\nAccording to the structure of a formal $\\mathbb{C}[[t]]\\langle\\partial_t\\rangle$-module (cf. \\cite{Ma}), we are led to prove the theorem in the cases of regular modules and purely irregular modules.\n\\begin{enumerate}\n\\item $\\underline{\\text{Regular case:}}$ Suppose $\\widehat{\\mathcal{N}}$ regular.\nThere exist a small disc $D$ in $\\mathbb{C}$ centered at the origin and a regular holonomic $\\mathcal{D}_D$-module $\\mathcal{N}$, such that the formalization of $\\mathcal{N}_0$ is $\\widehat{\\mathcal{N}}$ (cf. Theorem 5.3 p. 38 in \\cite{Ma}).\nLet $\\mathcal{M}$ be the regular holonomic $\\mathcal{D}_{D\\times\\mathbb{P}^1}$-module $p_1^{\\ast}(\\mathcal{N})$. We have:\n$$\\begin{array}{ll}\np_{1+}(\\mathcal{M}e^{p_2})&=p_{1+}(\\mathcal{O}_{D\\times\\mathbb{P}^1}e^{p_2}\\otimes_{\\mathcal{O}_{D\\times\\mathbb{P}^1}}p_1^{\\ast}(\\mathcal{N})),\\\\\n&=p_{1+}(\\mathcal{O}_{D\\times\\mathbb{P}^1}e^{p_2})\\otimes_{\\mathcal{O}_D}^{\\mathbb{L}}\\mathcal{N},\\text{ (Lemma \\ref{AA}),}\\\\\n&=\\mathcal{O}_D\\otimes_{\\mathcal{O}_D}^{\\mathbb{L}}\\mathcal{N},\\text{ (Lemma \\ref{BB}),}\\\\\n&=\\mathcal{N}.\n\\end{array}$$ \n\n\n\n\n\\item $\\underline{\\text{Purely irregular case:}}$ Let $(p,\\alpha)\\in \\mathbb{N}^{\\ast}\\times(\\tau^{-1}.\\mathbb{C}[\\tau^{-1}])$ and $\\widehat{R_{\\alpha}}$ be a regular holonomic $\\mathbb{C}[[\\tau]]\\langle\\partial_{\\tau}\\rangle$-module, such that $\\widehat{R_{\\alpha}}[\\tau^{-1}]=\\widehat{R_{\\alpha}}$. Let $\\rho:\\tau\\to t=\\tau^p$ and suppose $\\rho^{\\ast}(\\widehat{\\mathcal{N}})=\\underset{\\xi^p=1}{\\oplus}\\widehat{R_{\\alpha}}e^{\\alpha(\\xi\\tau)}$.\n\n\nThere exist a small disc $D^{'}$ in $\\mathbb{C}$ centered at the origin and a regular holonomic $\\mathcal{D}_{D^{'}}$-module $\\widetilde{R_{\\alpha}}$ such that the formalization of $(\\widetilde{R_{\\alpha}})_0$ is $\\widehat{R_{\\alpha}}$ (cf. Theorem 5.3 p. 38 in \\cite{Ma}).But there exists a regular holonomic $\\mathcal{D}_{D}$-module $R_{\\alpha}$ such that $\\widetilde{R_{\\alpha}}=\\rho^{\\ast}(R_{\\alpha})$, where $\\rho:D^{'}\\to D$, $\\rho(\\tau)=t=\\tau^p$. Indeed, as $\\rho^{\\ast}$ is exact and $\\widetilde{R_{\\alpha}}$ is regular holonomic with $\\widetilde{R_{\\alpha}}[\\tau^{-1}]=\\widetilde{R_{\\alpha}}$, we can reduce the proof to the case where \n$\\widetilde{R_{\\alpha}}=\\sfrac{\\mathcal{D}_{D^{'}}}{\\mathcal{D}_{D^{'}}((\\tau\\partial_{\\tau}-\\alpha)^r)}$, with $r\\in\\mathbb{N}^{\\ast}$ and $\\alpha\\in\\mathbb{C}\\setminus(-\\mathbb{N})$ (cf. Corollary 19 in \\cite{BrMai}). Then, we consider $R_{\\alpha}=\\sfrac{\\mathcal{D}_D}{\\mathcal{D}_D((t\\partial_t-\\frac{\\alpha}{p})^r)}$.\n\n\nUsing notations of Lemma \\ref{BB}, we denote by $\\mathcal{M}$ the regular holonomic  $\\mathcal{D}_{D\\times\\mathbb{P}^1}$-module $\\sfrac{p_1^{\\ast}(R_{\\alpha})[\\ast \\widetilde{Z}]}{p_1^{\\ast}(R_{\\alpha})[t^{-1}]}$. \nAccording to Lemmas \\ref{AA} and \\ref{BB}, we have two isomorphisms:\n$$\\begin{array}{l}\n\\rho^{\\ast}p_{1+}(p_1^{\\ast}(R_{\\alpha})[\\ast \\widetilde{Z}]e^{p_2})=\\underset{\\xi^p=1}{\\oplus}\\widetilde{R_{\\alpha}}e^{\\alpha(\\xi\\tau)}\\oplus \\widetilde{R_{\\alpha}}\\\\\n\\rho^{\\ast}p_{1+}(p_1^{\\ast}(R_{\\alpha})[t^{-1}]e^{p_2})=\\widetilde{R_{\\alpha}}.\n\\end{array}$$\n\nThen the formalization of $\\rho^{\\ast}p_{1+}(\\mathcal{M}e^{p_2})$ is $\\underset{\\xi^p=1}{\\oplus}\\widehat{R_{\\alpha}}e^{\\alpha(\\xi\\tau)}$.\n\\end{enumerate} \n\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Direct image of holonomic $\\mathcal{D}$-module of exponential type: the algebraic case}\\label{algebraic}\n\nIn this section, we prove Theorem \\ref{2}.\nWe consider $\\mathcal{N}^k:=\\mathcal{H}^kj_+f_+(\\mathcal{M}e^g)$, the extension of $\\mathcal{H}^kf_+(\\mathcal{M}e^g)$ at infinity, where $j:\\mathbb{A}^1\\hookrightarrow\\mathbb{A}^1\\cup\\{\\infty\\}\\simeq\\mathbb{P}^1$. Let $c\\in\\mathbb{P}^1$. The proof will be divided in two steps.\n\\begin{enumerate}\n\\item We consider the following diagrams\n$$\\xymatrix{\\mathbb{C}^2\\ar@{^(->}[r]^-i\\ar[d]^-{\\pi_1}&\\mathbb{P}^1\\times\\mathbb{P}^1\\ar[d]^-{p_1}&B_c\\times\\mathbb{P}^1\\ar@{_(->}[l]_-{i_c}\\ar[d]^{p_1}\n&\\mathbb{C}^2\\ar@{^(->}[r]^-i\\ar[d]^-{\\pi_2}&\\mathbb{P}^1\\times\\mathbb{P}^1\\ar[d]^-{p_2}&B_c\\times\\mathbb{P}^1\\ar@{_(->}[l]_-{i_c}\\ar[d]^{p_2}\\\\\n\\mathbb{C}\\ar@{^(->}[r]^-j&\\mathbb{P}^1&B_c\\ar@{_(->}[l]_-{j_c},&\\mathbb{C}\\ar@{^(->}[r]^-j&\\mathbb{P}^1&\\mathbb{P}^1\\ar@{=}[l],}$$\nwhere $\\pi_1$, $\\pi_2$ (resp. $p_1$, $p_2$) are the two canonical\nprojections. \n\nThen we have:\n$$\\begin{array}{lll}\nj_c^{\\ast}(j_+f_+(\\mathcal{M}e^g))^{\\mathrm{an}}&=&j_c^{\\ast}(j_+\\pi_{1+}(f,g)_+(\\mathcal{M}e^g))^{\\mathrm{an}},\\\\\n&&\\text{(Prop. 6.4 in \\cite{Bo} and }f=\\pi_1\\circ(f,g)),\\\\\n&=&j_c^{\\ast}(p_{1+}i_+(f,g)_+(\\mathcal{M}e^g))^{\\mathrm{an}},\\\\\n&&\\text{(Prop. 6.4 in \\cite{Bo} and }j\\circ\\pi_1=p_1\\circ i),\\\\\n&=&j_c^{\\ast}p_{1+}(i_+(f,g)_+(\\mathcal{M}e^g))^{\\mathrm{an}},\\\\\n&&\\text{(Prop 8.2.2 p. 179 in \\cite{Me})},\\\\\n&=&p_{1+}i_c^{\\ast}((i_+(f,g)_+(\\mathcal{M}))e^{p_2})^{\\mathrm{an}},\\\\\n&&(i_c\\text{ is an open inclusion}),\\\\\n&=&p_{1+}(\\mathcal{P}_ce^{p_2}).\n\\end{array}$$\n\nAs the formalization of the germ $\\mathcal{N}_c^k$ and of its analytization are equal, $\\mathcal{N}_c^k$ and $\\mathcal{H}^kp_{1+}(\\mathcal{P}_ce^{p_2})_c$ have the same formal irregular part.\n\n\n\\item Then we claim that $\\mathcal{H}^kp_{1+}(\\mathcal{P}_ce^{p_2})_c$ and $\\mathcal{H}^0p_{1+}(\\mathcal{H}^k\\mathcal{P}_ce^{p_2})_c$ have the same formal irregular part. According to the formal decomposition Theorem (cf. Theorem 1.2 p. 43 and Theorem 2.3 p. 51 in \\cite{Ma}), there exists a ramification $\\rho:D^{'}_c\\to D_c$, $\\rho(\\tau)=\\tau^p=t$, such that the modules $\\rho^{\\ast}(\\mathcal{H}^kp_{1+}(\\mathcal{P}_ce^{p_2}))_c$ and $\\rho^{\\ast}(\\mathcal{H}^0p_{1+}(\\mathcal{H}^k\\mathcal{P}_ce^{p_2}))_c$ are isomorphic to the finite direct sum of some formal modules of exponential type. Then, it is sufficient to prove that, given an element $\\alpha\\in\\tau^{-1}.\\mathbb{C}[\\tau^{-1}]$, the regular part of  $(\\rho^{\\ast}(\\mathcal{H}^kp_{1+}(\\mathcal{P}_ce^{p_2}))_c)e^{-\\alpha}$ and  $(\\rho^{\\ast}(\\mathcal{H}^0p_{1+}(\\mathcal{H}^k\\mathcal{P}_ce^{p_2}))_c)e^{-\\alpha}$ are isomorphic. Thus, we are led to prove that the specialization at $c$ of  $(\\rho^{\\ast}(\\mathcal{H}^kp_{1+}(\\mathcal{P}_ce^{p_2})))e^{-\\alpha}$ and  $(\\rho^{\\ast}(\\mathcal{H}^0p_{1+}(\\mathcal{H}^k\\mathcal{P}_ce^{p_2})))e^{-\\alpha}$ are isomorphic (cf. Example 5.2.1 in \\cite{LaMa}).\n\nLet $\\rho^{'}=(\\rho,id):D^{'}_c\\times\\mathbb{P}^1\\to  D_c\\times\\mathbb{P}^1$. We denote by $p_1^{'}:D_c\\times\\mathbb{P}^1\\to D_c$ and $p_2^{'}:D_c\\times\\mathbb{P}^1\\to\\mathbb{P}^1$ the canonical projection. Let $\\overline{p}_1^{'}:\\{c\\}\\times\\mathbb{P}^1\\to\\{c\\}$. We have:\n$$\\begin{array}{ll}\n\\spe_c((\\rho^{\\ast}&(\\mathcal{H}^kp_{1+}(\\mathcal{P}_ce^{p_2})))e^{-\\alpha})_c\\\\\n&=\\spe_c(\\mathcal{H}^kp_{1+}^{'}(\\rho^{'\\ast}(\\mathcal{P}_c)e^{p_2^{'}})e^{-\\alpha})_c,\\\\\n&\\text{(Lemma \\ref{comm} extended to complexes of $\\mathcal{D}$-modules),}\\\\\n&=\\spe_c(\\mathcal{H}^kp_{1+}^{'}(\\rho^{'\\ast}(\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}}))_c,\\\\\n&=\\mathcal{H}^k\\overline{p}_{1+}^{'}\\spe_{\\{c\\}\\times\\mathbb{P}^1}(\\rho^{'\\ast}(\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}})_c,\\\\\n&\\text{(Theorem 9.4.1 in \\cite{LaMa}),}\\\\\n&=R^k\\Gamma(\\{c\\}\\times\\mathbb{P}^1,\\DR~\\spe_{\\{c\\}\\times\\mathbb{P}^1}(\\rho^{'\\ast}(\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}})[+1]),\\\\\n&\\text{(p.5 in \\cite{Ma3}).}\n\\end{array}$$\n\nAccording to Proposition \\ref{11} extended to complexes of $\\mathcal{D}$-modules, the support of  $\\spe_{\\{c\\}\\times\\mathbb{P}^1}(\\rho^{'\\ast}(\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}})$ is included in $(c,\\infty)$. Then,\n\n$$\\begin{array}{ll}\n\\spe_c&((\\rho^{\\ast}(\\mathcal{H}^kp_{1+}(\\mathcal{P}_ce^{p_2})))e^{-\\alpha})_c=\\\\\n&=(\\mathcal{H}^k\\DR~\\spe_{\\{c\\}\\times\\mathbb{P}^1}(\\rho^{'\\ast}(\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}})[+1])_{(c,\\infty)},\\\\\n&=(\\DR~\\mathcal{H}^k\\spe_{\\{c\\}\\times\\mathbb{P}^1}(\\rho^{'\\ast}(\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}})[+1])_{(c,\\infty)},\\\\\n&=R^0\\Gamma(\\{c\\}\\times\\mathbb{P}^1,\\DR~\\mathcal{H}^k\\spe_{\\{c\\}\\times\\mathbb{P}^1}(\\rho^{'\\ast}(\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}})[+1]),\\\\\n&=R^0\\Gamma(\\{c\\}\\times\\mathbb{P}^1,\\DR~\\spe_{\\{c\\}\\times\\mathbb{P}^1}(\\rho^{'\\ast}(\\mathcal{H}^k\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}})[+1]),\\\\\n&=\\mathcal{H}^0\\overline{p}_{1+}^{'}\\spe_{\\{c\\}\\times\\mathbb{P}^1}(\\rho^{'\\ast}(\\mathcal{H}^k\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}})_c,\\\\\n&\\text{(p.5 in \\cite{Ma3}),}\\\\\n&=\\spe_c(\\mathcal{H}^0p_{1+}^{'}(\\rho^{'\\ast}(\\mathcal{H}^k\\mathcal{P}_c)e^{p_2^{'}-\\alpha\\circ p_1^{'}}))_c,\\\\\n&\\text{(Theorem 4.8.1 p. 226 in \\cite{MeSa}),}\\\\\n&=\\spe_c(\\mathcal{H}^0p_{1+}^{'}(\\rho^{'\\ast}(\\mathcal{H}^k\\mathcal{P}_c)e^{p_2^{'}})e^{-\\alpha})_c,\\\\\n&=\\spe_c((\\rho^{\\ast}(\\mathcal{H}^0p_{1+}(\\mathcal{H}^k\\mathcal{P}_ce^{p_2})))e^{-\\alpha})_c,\\\\\n&\\text{(Lemma \\ref{comm}).}\n\\end{array}$$\n\\end{enumerate}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe subdwarf B (sdB) stars are core helium-burning objects with envelopes that\nare too thin to sustain nuclear burning \\citep[e.g.\\ ][]{Heber86}. They can be\nidentified with models of Extreme Horizontal Branch (EHB) stars; in other\nwords, they have masses $\\sim$0.5\\,$M_\\odot$ and will evolve directly to the\nwhite dwarf cooling curve, bypassing the Asymptotic Giant Branch. While their\nfuture evolution seems secure, the formation of these stars is uncertain. In\nrecent years radial velocity surveys have shown that a large fraction of sdBs\nare in short-period binary systems \\citep{MHMN01}, it appears certain that\nbinary interaction plays a significant role, however many objects are\napparently single. A binary population synthesis study by \\citet{HanII} has\nfound that three channels can give rise to the observed characteristics of\nsdBs: one or two phases of common envelope evolution; stable Roche lobe\noverflow; and the merger of two helium-core white dwarfs. The latter scenario\ncould explain the population of single stars.\n\nThe possibility of pulsations in sdB stars was theoretically predicted by\n\\citet{CFB96} at around the same time they were observed by\n\\citet{ECpaperI}. The more than 30 known pulsators (officially known as\nV361\\,Hya stars) have $T_{\\mathrm{eff}}=29\\,000-35\\,000$\\,K and\n$\\log g=5.2-6.0$, periods of 1-10 minutes and amplitudes less than 60\\,mmag\n\\cite[see ][for a review]{Kilkenny2002}. The driving mechanism of the\noscillations is believed to be related to the ionisation of iron and other\nheavy elements at the base of the photosphere \\citep{CFB97a}. As is the case\nfor other types of pulsators \\citep[e.g.\\ the PG~1159 stars,][]{QFB04} there is\nan overlap in the ($T_{\\mathrm{eff}}, \\log g$) plane between pulsators and\nnon-pulsators \\citep{ECpaperXII}. Diffusion calculations by \\citet{CFB97a}\nsuggest that the surface iron abundance of pulsators should be higher than\nthat of non-pulsators, however studies by \\citet{EHN06}, \\citet{HE04} and\n\\citet[][ hereafter HRW]{HRW00} find that iron has approximately solar\nabundance in most sdBs.\n\nFor this reason we set out to determine if any correlation exists between\nsurface abundances of iron-group elements for pulsators and\nnon-pulsators. Since elements such as nickel, manganese and chromium are not\nnormally accessible through ground-based optical spectra, it was necessary to\nacquire high-resolution UV echelle spectra with the \\emph{Space Telescope\n  Imaging Spectrograph} onboard the \\emph{Hubble Space Telescope}\n(\\emph{HST\/STIS}). If it is not possible to separate\ngroups based on abundances, perhaps differing mass-loss rates contribute\nsignificantly, as suggested by \\citet{FC97}. Regardless of this, our abundance\nmeasurements will be extremely useful for testing diffusion theory. Previous\nstudies of sdBs using UV spectra from the \\emph{International Ultraviolet\n  Explorer} (\\emph{IUE}) suffered from mediocre S\/N as well as\nfrom poor, or a lack of, atomic data; this was especially the case for the\niron group \\citep[e.g.\\ ][]{BHS82,BSK82}.\n\n\\begin{table*}\n\\caption{Observations of six sdB stars using \\emph{HST\/STIS}. Note that\n  CPD~$-64^\\circ 481$\\ was observed only in the FUV and using the E140H grating.}\n\\label{tab:obs}\n\\begin{center}\n\\begin{tabular*}{1.00\\textwidth}{l|cccccccccc}\nTarget & $\\alpha$ & $\\delta$ & $V$ & $T_{\\mathrm{eff}}$ & $\\log g$ & log(He\/H)\n& Ref. & Pulsator? & $T_{exp}^{NUV}$ & $T_{exp}^{FUV}$ \\\\\n       & (2000)   & (2000)   & (mag) & (K) & & & & & (s) & (s) \\\\\n\\hline\nPG~1219$+$534 & 12 21 29.1 & +53 04 37 & 13.2 & 33\\,500 & 5.87 & $-$1.6\n& 4 & yes & 3100\/3100 & 2160\/3100 \\\\\nFeige~48     & 11 47 14.5 & +61 15 32 & 13.5 & 29\\,500 & 5.54 & $-$2.9\n& 2 & yes & 3500\/3511 & 3600\/3600 \\\\\nCPD~$-64^\\circ 481$ & 05 47 59.3 & -64 23 03 & 11.3 & 27\\,500 & 5.60 & $-$2.5 & 3 & no & --- &\n1440 \\\\\nFeige~66     & 12 37 23.5 & +25 03 60 & 10.6 & 34\\,500 & 5.83 & $-$1.6 & 4\n& no & 824 & 875 \\\\\nCD~$-24^\\circ 731$       & 01 43 48.6 & $-$24 05 10 & 11.0 & 35\\,400  & 5.90 & $-$2.9 & 4\n& no & 800 & 878 \\\\\n\\hline\n\\end{tabular*}\n\\end{center}\nReferences: (1) \\citet{HRW99}; (2) \\citet{HRW00}; (3) \\citet{OJF05}; (4) This\nwork.\n\\end{table*}\n\nThe  two pulsators we have chosen to observe are \nFeige~48 \\citep{ECpaperXI},  and\nPG~1219$+$534\\ \\citep{ECpaperXII}. \nQuantitative analysis of the first two objects using optical\nspectra was carried out by HRW and the latter by \\citet{HRW99}. As comparison\nobjects we have chosen Feige~66 and CD~$-24^\\circ 731$\\ (alias SB~707); a high-resolution \\emph{IUE}\nspectrum of the former was analysed by \\citet{BHS82}. A very high-resolution\nspectrum of the sdB CPD~$-64^\\circ 481$\\ was found in the \\emph{HST} archive;\nsince its temperature is close to that of Feige~48, and it is not pulsating\n(Koen, private communication), we decided to use it as a comparison\nstar to Feige~48. Note that these two stars, along with CD~$-24^\\circ 731$, are in\nclose binaries. Feige~48 and CPD~$-64^\\circ 481$\\ have periods of 0.376\\,d and\n0.2772\\,d respectively, while CD~$-24^\\circ 731$\\ has a period of 5.85\\,d\n\\citep{OHB04,EHA05}. The companions of Feige~48 and CD~$-24^\\circ 731$\\ are both\nmost likely white dwarfs, while the nature of the companion of\nCPD~$-64^\\circ 481$\\ is uncertain.\n\n\nIn this paper we present a detailed abundance analysis of each of these\nobjects\nbased on UV echelle\nspectra obtained using \\emph{HST\/STIS}. We discuss the possible\ncorrelation between iron-group abundances and pulsation, the\nabundances of heavy elements in the context of radiative acceleration\nand the trends discussed by \\citet{SJOT04}, and\na solution to the temperature discrepancy seen between Balmer line fitting and\nhelium ionisation equilibrium.\n\n\\section{Observations}\n\\label{sec:obs}\n\nObservations were made using the \\emph{Space Telescope Imaging Spectrograph}\n(\\emph{STIS}) onboard the \\emph{Hubble Space Telescope} (\\emph{HST}). We\nobserved five stars, three pulsators and two non-pulsators for comparison. One\nof the non-pulsators, Feige 66 is a spectroscopic twin of one of the pulsator\nPG~1219$+$534. Details of the observations are shown in Table \\ref{tab:obs}.\nThe stars were observed in the near- and far-UV (NUV and FUV) using the medium\nresolution echelle E140M and E230M grisms. The NUV spectra each have a central\nwavelength of 1978\\,\\AA, covering the wavelength range 1700-2370\\,\\AA, while\nthe FUV spectra are centered on 1425\\,\\AA, covering the 1160-1730\\,\\AA\\\nrange. For all spectra a slit width of 0.2''$\\times$0.06'' was\nused. Each pulsator was observed in time-tag mode, where the\narrival times of each photon is recorded; the two non-pulsators were observed\nin histogram mode. The spectra of Feige 48 have yielded one piece of\nserendipity -- the discovery of velocity variations indicating a binary\ncompanion to the star \\citep{OHB04}.\nAdditionally, we found very high-resolution spectra\nof Feige~66 and CPD~$-64^\\circ 481$, taken using the E140H grism with a\n0.2''$\\times$0.2'' slit covering the range 1163-1363\\,\\AA. All of the\nspectra are sharp-lined as expected from the low projected rotation\nvelocities seen in sdBs \\citep[][and HRW]{OHB04}.\n\n\n\n\n\\section{Synthetic spectra}\n\\label{sec:synth}\nIn hot subdwarfs, most of the spectral lines due to iron-group\nelements (V, Cr, Mn, Fe, Co, Ni) are found blueward of $\\sim$2300\\,\\AA,\nand in many cases continuum definition is difficult at the resolution\nof our spectra. The methods we used to get around these problems are\ndiscussed in more detail in Section \\ref{sub:contdef}.\n\nAs input for our spectrum synthesis, we used a metal line-blanketed LTE model\natmosphere with solar metalicity and Kurucz' ATLAS6 Opacity Distribution\nFunctions. The spectra were synthesised using Michael Lemke's version of the\nLINFOR program (originally developed by Holweger, Steffen, and\nSteenbock at Kiel University). Oscillator strengths were taken from\nthe Kurucz line list, as were damping constants for all metal\nlines. Only lines that have been observed experimentally were used, since we\nrequired the most accurate wavelengths possible. In the case of species\nheavier than Zn, values were taken from the resonance line lists\nof \\citet{Morton2000,Morton2003}. For the partition functions of\nGa\\,\\textsc{iii}, Ge\\,\\textsc{iv}, Sn\\,\\textsc{iv} and Pb\\,\\textsc{iv}\nwe used these ions' ground state statistical weight, since no\npublished data is available. This is a good approximation at\ntemperatures of 30\\,000-35\\,000\\,K. The Cu\\,\\textsc{iii} atomic data were\ntaken from the tables of \\citet{HH95}. Oscillator strengths for\nTi\\,\\textsc{iii} lines were taken from \\citet{RU97}. Lines\nwith $\\lambda>2000$\\,\\AA\\ were converted from air to vacuum wavelengths\nusing the formula of \\citet{Edlen1966}.\n\n\\subsection{Atmospheric parameters}\n\\label{sec:atmospar}\n\nThe determination of atmospheric parameters for the two pulsators to\nbe discussed here was presented by HRW (the values are shown in Table\n\\ref{tab:obs}).\nWe selected Feige~66 and CD~$-24^\\circ 731$\\ as potential comparison stars for the\npulsator PG~1219$+$534\\ because they are not known to pulsate and\npublished temperatures and gravities are similar to those of the latter.\nWhile PG~1219$+$534\\ has $T_{\\mathrm{eff}}=33\\,500$\\,K, $\\log g=5.85$\nfrom Balmer line analysis by HRW, Feige~66 has\n$T_{\\mathrm{eff}}=33\\,400$\\,K, $\\log g=6.2$ \\citep{SBK94}\nand CD~$-24^\\circ 731$\\ has $T_{\\mathrm{eff}}=34\\,000$\\,K, $\\log g=6.0$ \\citep{HHJ84}.\n\n\nFor this work we have reanalysed the Keck-HIRES spectrum of PG~1219$+$534\\ from\nHRW. A high resolution (0.1\\AA) optical spectrum of CD~$-24^\\circ 731$\\ was kindly\nprovided by M.\\ Altmann and H.\\ Edelmann. It was taken with the FEROS\nspectrograph at the ESO 2.2m telescope. A low resolution (5\\AA)\nspectrum of Feige~66 taken with the CAFOS spectrograph was provided by\nM.\\ Altmann. The spectral analysis of both spectra\nis described below. The parameters for CPD~$-64^\\circ 481$\\ have already been\ndetermined by \\citet{OJF05}.\n\n\n\n\n\n\\subsection{The hot sdB temperature discrepancy}\n\\label{sub:tempfix}\n\nAs has been noted by HRW, there is a discrepancy between temperatures\nderived from Balmer line fitting and helium ionisation equilibrium. For\nPG~1219$+$534, the difference in $T_{\\mathrm{eff}}$ is 2000\\,K.\n\n\nAfter the initial discovery of strongly supersolar abundances of heavy\nmetals  in three of the programme stars (Feige 66, CD~$-24^\\circ 731$\\ \\&\nPG~1219$+$534), we  investigated how this affects the determination of\nthe atmospheric  parameters. HRW used metal-line blanketed models with\nsolar metal content. Using the same model\natmospheres described above but with metals scaled by a factor of 10\n([M\/H]=1.0), we have recalculated both our abundances and stellar\nparameters ($T_{\\mathrm{eff}}, \\log g$). There are currently no\nopacity distribution functions with higher metalicities. Also,\ndespite the fact that iron itself is show roughly solar abundances for\nall our targets, in many cases the nickel abundances is 1-2 dex above\nsolar. It is possible that in these cases nickel may become the\ndominant source of line opacity.\n\n\nIn Figure \\ref{fig:pgp1fit} we show a fit with these models to the\noptical spectrum of PG~1219$+$534. The Balmer lines and the\nHe\\,\\textsc{ii} can now be matched simultaneously. The parameters we\nderive with our metal-enhanced models using all lines are closest\n(within errors) to those found by\nfitting the Balmer lines only with a solar-metalicity model. The\nHe\\,\\textsc{i} lines agree reasonably well, although the line cores of\nthe strongest lines match poorly. We expect ultimately the solution\nwill be found using opacity sampled models that can allow for enhanced\nabundances relative to iron.\n\n\n\\begin{figure}\n\\vspace{9cm}\n\\begin{center}\n    \\special{psfile=3948fig1.eps hscale=48 vscale=48 hoffset=-25\n    voffset=-90 angle=360}\n\\end{center}\n\\caption{Line profile fit for PG~1219$+$534\\ using solar metalicity\n  models (\\emph{top panel}) and metal-rich models (10 times solar,\n  \\emph{bottom panels}). The Balmer lines and He\\,\\textsc{ii} at\n  4686\\,\\AA\\ match simultaneously when using the metal rich models\n  (see text).}\n\\label{fig:pgp1fit}\n\\end{figure}\n\n\\begin{figure}\n\\vspace{9cm}\n\\begin{center}\n    \\special{psfile=3948fig2.eps hscale=48 vscale=48 hoffset=-25\n    voffset=-90 angle=360}\n\\end{center}\n\\caption{Same as Figure \\ref{fig:pgp1fit} but for CD~$-24^\\circ 731$.}\n\\label{fig:p1fit}\n\\end{figure}\n\nFor CD~$-24^\\circ 731$\\ we found the same mismatch of the helium ionisation equilibrium\nas for PG~1219$+$534\\ when solar metalicity models were used (see Figure\n\\ref{fig:p1fit}).  Again this\nproblem is remedied by using metal-enriched models. In this case\na significantly higher effective temperature was derived from\nthe metal-enriched models than from the solar metalicity ones,\ni.e. $T_{\\mathrm{eff}}=35\\,400$\\,K compared with\n$T_{\\mathrm{eff}}=33\\,800$\\,K. CD~$-24^\\circ 731$\\ is a rather helium poor star\ncompared to Feige~66 and PG~1219$+$534, which might explain the different\nmetallicity dependence of the results. In addition, CD~$-24^\\circ 731$\\ resides in a\nclose binary system ($P$=5.85\\,d), whereas Feige~66\nis not known to be variable (E. Green, private communication). PG~1219$+$534\\ is\nalso not known to be a radial velocity variable.\n\nThe low resolution spectrum of Feige~66 was analysed in the same way with\nthe metal-enriched models and we derived $T_{\\mathrm{eff}}=34500$\\,K,\n$\\log g=5.83$, log(He\/H)=$-$1.6. In this case the parameters\nagree with those found using non-LTE models; however, the He\\,\\textsc{ii}\n4686\\AA\\ line does not agree with observations in the enhanced\nmetallicity case, but does in the non-LTE case. A high-resolution\nspectrum is urgently required to resolve this issue.\n\nThe results of the new spectral analyses of PG~1219$+$534, CD~$-24^\\circ 731$\\ and Feige~66\nrevealed that their surface gravities are identical. The helium abundances\nof PG~1219$+$534\\ and Feige~66 are also identical, whereas helium abundance of\nCD~$-24^\\circ 731$\\ is much lower than for the other two.\nTheir effective temperatures, however, are not identical as suggested by\nthe earlier spectral analyses. Feige~66 and CD~$-24^\\circ 731$\\ are hotter than\nPG~1219$+$534\\ by 1000K and 1900K, respectively. \n\nThe temperatures we derive here match those determined using metal-free,\nnon-LTE models by HRW. Recently, \\citet{CFB05} used \\textsc{Tlusty} and\n\\textsc{Synspec} NLTE models and confirm the NLTE results of HRW for\nPG~1219$+$534\\ to within 500\\,K and 0.04 dex.\nSince the heavy metal abundances of Feige~48 and\nCPD~$-64^\\circ 481$\\ are much closer to solar than those of the former, a\nreanalysis of these stars is not necessary. The abundances shown in\nTable \\ref{tab:4abu} are derived from models with these parameters.\n\n\\section{Line identification}\n\\label{sec:lineID}\n\nAs has already been noted, previous studies of high resolution UV\nspectra of hot subdwarfs have suffered from poor, or a lack of, atomic\ndata. While the current situation is by no means perfect, we have\nendeavoured to identify as many spectral lines as possible, and then\nfrom these we have chosen suitable lines to measure\nequivalent widths or carry out spectrum fitting. Our method is outlined\nbelow, beginning with a brief discussion of the problems encountered\nwith continuum definition.\n\n\\subsection{Defining the continuum}\n\\label{sub:contdef}\n\n\\begin{figure}\n\\vspace{6cm}\n\\begin{center}\n    \\special{psfile=3948fig3.eps hscale=48 vscale=48 hoffset=10\n    voffset=-10 angle=360}\n  \\end{center}\n  \\caption{Comparison of a 15\\,\\AA\\ wide section of the near- and far-UV\n  spectrum of PG~1219$+$534. In the NUV it is possible to measure the continuum\n  level, while in the FUV it must be estimated. Note also the higher flux in\n  the FUV.}\n  \\label{fig:compspec}\n\\end{figure}\n\nIn each of our spectra, the continuum is well-defined in the NUV,\nespecially redward of 2000\\,\\AA, allowing equivalent widths to be\nmeasured. An example of the difference between crowded and\nwell-defined is shown in Figure \\ref{fig:compspec}. There are also some regions\nbelow 2000\\,\\AA\\ where equivalent width measurements are possible. In\norder to define the continuum in crowded regions, we measured\nequivalent widths of all unblended lines and then generated synthetic\nspectra using the abundances derived from this analysis; the continuum\nof the crowded regions was set by matching the synthetic spectra to\nthe observations using a ``chi-by-eye'' fit. This was not always\nstraightforward however, since many lines still suffer from imprecise\natomic data, and some resonance line profiles are not well reproduced\nin our assumed LTE atmosphere (see Section \\ref{sub:nlte} for more discussion\nand examples of this effect). Nevertheless there were almost always\nenough useful lines for our method to be successful. Note that we are\nnot relying on precise knowledge of the star's $U$, $B$, or $V$\nmagnitude -- which is not always available; PG~1219$+$534\\ is one case\n-- to calibrate the model flux.\n\n\\subsection{Unblended lines}\n\\label{sub:unblend}\n\nThere are various line lists available for the analysis of stellar\nspectra; most commonly used is the Kurucz list, although during our\nanalysis we found that often more accurate oscillator strengths and\nwavelengths are available in other databases (e.g.\\ \\citet{HH95}). In\nparticular we found that several Co lines had poor term\nidentifications in the Kurucz list, Fe oscillator strengths were often\nout of date, while oscillator strengths for Ti\\,\\textsc{iii} were\ncompletely inconsistent with our observations.\n\nTo determine which lines in our spectra are not blended, we generated\nsynthetic spectra using only those lines in the Kurucz list that have\nbeen observed experimentally and compared these directly with our\nobservations. If the continuum was clear on either side of the line\nwings, and our line list showed only one possible match, we considered\nit to be unblended. Lines that were visibly blended (e.g.\\ with overlapping\nwings) were ruled out for equivalent width measurement where the blend\ncould not be be identified.\n\n\\subsection{Individual elements}\n\\label{sub:indiel}\n\nIn many regions of our spectra, line blending is extremely severe,\nwhich means that continuum definition is difficult (as discussed\nabove) and\/or that many features are identified with more than one\nion. In this section we discuss the ions we have identified.\n\n\\subsubsection{C and N}\n\n\\begin{figure}\n\\vspace{6cm}\n\\begin{center}\n    \\special{psfile=3948fig4.eps hscale=50 vscale=50 hoffset=-10 voffset=-10\n    angle=0}\n\\end{center}\n\\caption{Spectrum fit to the C\\,\\textsc{iii} multiplet at\n  1174-1177\\,\\AA for Feige~48. The model is shown as the thick line. Other\n  strong lines due to Cr\\,\\textsc{iv} and Fe\\,\\textsc{iv} are marked.}\n\\label{fig:f48c3}\n\\end{figure}\n\nPG~1219$+$534\\ and CD~$-24^\\circ 731$\\ do not show any carbon lines. In\nCPD~$-64^\\circ 481$\\ the C\\,\\textsc{iii} lines at\n1247 and 2297\\,\\AA\\ are present, along with the 1176\\,\\AA\\ multiplet,\nwhile in Feige~48 and Feige~66 these C\\,\\textsc{iii} lines and the\nC\\,\\textsc{iv} resonance lines around 1550\\,\\AA\\ are seen. Figure\n\\ref{fig:f48c3} shows the 1176\\,\\AA\\ multiplet of Feige~48 together\nwith a model spectrum fit; results from spectrum fitting are discussed\nfurther in Section \\ref{sec:abu}.\n\nAs found in almost all other sdBs, nitrogen lines are strong in our\nspectra, although there are fewer lines than seen in optical\nspectra. In our two cooler targets, N\\,\\textsc{iii} lines are visible\nat 1183-1184\\,\\AA\\, while the N\\,\\textsc{iv} line at\n1718\\,\\AA\\ is present in Feige~48. In\nFeige~66, PG~1219$+$534\\ and CD~$-24^\\circ 731$, lines of N\\,\\textsc{iii} and\nN\\,\\textsc{iv}, as well as the N\\,\\textsc{v} resonance lines can be\nseen.\n\n\n\\subsubsection{Al and Si}\n\nAs has been noted by several authors \\citep[e.g.][]{MontrealIII,MontrealV},\nthe abundance of silicon in sdB stars drops sharply at\n$\\sim$32\\,000\\,K. This is also the case for aluminum, although the\navailable upper limits on abundances are not as strict as for silicon.\n\nSilicon in hot subdwarfs has been discussed in more detail recently by\n\\citet{SJOT04} and the objects studied here follow previously seen trends. The\ndetection of the Si\\,\\textsc{iv} resonance line in PG~1219$+$534\\ is discussed\nfurther in Section \\ref{sub:comparison}. The Al\\,\\textsc{iii}\nresonance lines are only detectable in Feige~48, although for\nCPD~$-64^\\circ 481$\\ the relevant wavelength range is not covered by our spectra.\n\n\n\\begin{figure}\n\\vspace{6cm}\n\\begin{center}\n    \\special{psfile=3948fig5.eps hscale=48 vscale=48 hoffset=10\n    voffset=-10 angle=360}\n\\end{center}\n\\caption{The S\\,\\textsc{iii} resonance lines for CPD~$-64^\\circ 481$, showing an\n  excellent match. The line at $\\sim$1194.5\\,\\AA is one of the Si\\,\\textsc{ii}\n  resonance lines, while the strong line at $\\sim$1194.15\\,\\AA is unidentified.}\n\\label{fig:SIII}\n\\end{figure}\n\n\\begin{figure}\n\\vspace{6cm}\n\\begin{center}\n    \\special{psfile=3948fig6.eps hscale=48 vscale=48 hoffset=10\n    voffset=-10 angle=360}\n\\end{center}\n\\caption{A region of spectrum showing blended iron-group lines for\n  Feige~48, PG~1219$+$534\\ and Feige~66. The line at $\\sim$1858.2\\,\\AA\\ is unidentified.}\n\\label{fig:fe1860}\n\\end{figure}\n\n\\subsubsection{S, Ar, Ca}\n\nThe higher resolution of our spectrum of CPD~$-64^\\circ 481$\\ allows us to clearly\ndistinguish the S\\,\\textsc{iii} resonance lines (shown in Figure\n\\ref{fig:SIII}). Because of severe crowding and lower resolution, this\nis not possible in the case of Feige~48. The crowding is less severe\nfor PG~1219$+$534, Feige~66 and CD~$-24^\\circ 731$\\ -- although in the latter the\nlines are weak -- so it is possible to make out the S\\,\\textsc{iii}\nlines.\n\nAr\\,\\textsc{iii} lines are difficult to measure as they are all quite weak,\nand often blended with lines due to iron-group elements. Despite this, we\ncould measure four lines in Feige~66 and one in PG~1219$+$534; the\nother stars have only upper limits.\n\n\n\nIn the case of calcium, we were able to measure lines of\nCa\\,\\textsc{iii} for Feige~66, CD~$-24^\\circ 731$\\ and PG~1219$+$534. In\nthe cooler pair the Ca lines are either very weak or blended, making\nmeasurements difficult.\n\n\\subsubsection{The iron group}\n\nLines of the iron group elements often lie at similar\n  wavelengths, making it difficult to measure equivalent widths, and\n  necessitating an analysis by spectrum synthesis. An example of this\n  can be seen in the 4\\,\\AA-wide spectrum slices shown in Figure\n  \\ref{fig:fe1860}.\n\n\\textit{Scandium}: There are very few Sc\\,\\textsc{iii} lines available to\nmeasure in our spectra, making abundances very difficult to measure.\nIt was only possible to measure the\nresonance lines at 1603.064\\,\\AA\\ and 1610.194\\,\\AA\\ for Feige~66 and\nCD~$-24^\\circ 731$. For Feige~48 and PG~1219$+$534\\ we could only place upper\nlimits, and our spectrum of CPD~$-64^\\circ 481$\\ does not cover these\nlines.\n\n\\textit{Titanium}: The Ti\\,\\textsc{iii} resonance lines are present in\nthe spectrum of each star. Some subordinate lines are also\nmeasurable. We note here that the oscillator strengths found in the\nKurucz line list are inconsistent with our observations. Therefore we\nhave used the values given by \\citet{RU97}, which match significantly better.\n\n\\textit{Vanadium}: Lines of V\\,\\textsc{iii} are typically blended or\nweak; no lines could be detected at all for Feige~48, allowing only\nupper limits to be set.\n\n\\begin{figure}\n\\vspace{12cm}\n\\begin{center}\n    \\special{psfile=3948fig7a.eps hscale=48 vscale=48 hoffset=10\n    voffset=165 angle=360}\n    \\special{psfile=3948fig7b.eps hscale=48 vscale=48 hoffset=10\n    voffset=-10 angle=360}\n\\end{center}\n\\caption{Determination of microturbulent velocity of CD~$-24^\\circ 731$\\ and\n  PG~1219$+$534 using Cr\\,\\textsc{iii} lines. We find\n  $\\xi=5^{+2}_{-1}$\\,km\\,s$^{-1}$ for the former, while the latter is\n  consistent with $\\xi=0$\\,km\\,s$^{-1}$.}\n\\label{fig:vmicro}\n\\end{figure}\n\n\n\\textit{Chromium}: All of our spectra contain many lines of both\nCr\\,\\textsc{iii} and Cr\\,\\textsc{iv} of varying strengths. These\nlines, along with those of iron, have been used to determine the\nmicroturbulence discussed in Section \\ref{sub:micro}.\n\n\\textit{Manganese}: Of all the iron-group elements, manganese causes\nthe largest difficulties, since it appears that many wavelengths given\nin the Kurucz list are not accurate (note that the \\citet{HH95} list\nuses the same wavelengths). We have examined the list of \\citet{UR97},\nhowever they only provide improved oscillator strengths, and also use\nthe Kurucz wavelengths.\n\n\\textit{Iron}: In all spectra there are many Fe\\,\\textsc{iii} and\nFe\\,\\textsc{iv} lines measurable. In the two hottest objects in our\nsample, Fe\\,\\textsc{v} lines are also present, although\nthe accuracy of oscillator strengths is uncertain, since the\nabundances derived from some of them differ by several orders of magnitude.\n\n\\textit{Cobalt}: Lines of Co\\,\\textsc{iii} and \\textsc{iv} are visible\nin our spectra, although often only in crowded regions. It appears\nthat the hotter objects show more strong lines.\n\n\\textit{Nickel}: There are very many nickel lines, particularly of\nNi\\,\\textsc{iii} and \\textsc{iv}. A few Ni\\,\\textsc{v} lines are also\nvisible in the spectra of PG~1219$+$534\\ and Feige~66, however once\nagain the accuracy of oscillator strengths is very uncertain.\n\n\\textit{Copper}: Cu\\,\\textsc{iii} and \\textsc{iv} are both\npresent. Neither ion is included in the Kurucz list, so wavelengths\nwere taken from \\citet{HH95}. Oscillator strengths are only available\nfor Cu\\,\\textsc{iii}. There are no published oscillator strengths of\nCu\\,\\textsc{iv}.\n\n\\textit{Zinc}: Zn\\,\\textsc{iii} and \\textsc{iv} lines are visible;\nhowever, the latter ion has no published oscillator strengths.\n\n\\subsubsection{Gallium, Germanium, Tin and Lead}\n\nThe discovery of lines of Ga, Ge, Sn and Pb was made by \\cite{SJOT04}. Lines\nof Ge\\,\\textsc{iv} and Sn\\,\\textsc{iv} have also been seen in hot DA white\ndwarfs by Vennes, Chayer \\& Dupuis (2005).\nAbundances for our four targets are almost all done using spectrum synthesis.\nResonance lines of Ga\\,\\textsc{iii}, Ge\\,\\textsc{iv}, Sn\\,\\textsc{iv}, and\nPb\\,\\textsc{iv} are present in all spectra. In stars with higher Ga\nabundances, Ga\\,\\textsc{iii} subordinate lines are also present. The\nresonance line of Sn\\,\\textsc{iii} at 1251.387\\,\\AA\\ is strong in\nFeige~66, and is present in all stars, but is blended.\n\n\nOne of the resonance lines of Pb\\,\\textsc{iv} is present in each\nobject; the other line lies in the wings of Ly$\\beta$, which is not\ncovered by our spectra (but is by \\emph{FUSE}).\n\n\\begin{table*}\n\\caption{Abundances for each ion for all five targets.}\n\\label{tab:4abu}\n\\begin{center}\n\\begin{tabular}{l||cr|cr|cr|cr|cr}\n & Feige 66 & & PG~1219$+$534\\ & & CD~$-24^\\circ 731$\\ & & Feige 48 & & CPD~$-64^\\circ 481$\n  & \\\\\n\\hline\nIon  &  $\\log\\epsilon$ & $n$ & $\\log\\epsilon$ & $n$ & $\\log\\epsilon$ & $n$ &\n$\\log\\epsilon$ & $n$ & $\\log\\epsilon$ & $n$ \\\\\n\\hline\nC\\,\\textsc{iii} &  6.93$\\pm$0.28\\rlap{$^*$} &   4 & & & & & 7.22$\\pm$0.10 & 3 &\n  7.54$\\pm$0.22 & 2 \\\\\nC\\,\\textsc{iv}\\rlap{$^*$} & 6.5$\\pm$0.5 & 2 & $<$3.1 & & $<$2.7 & &\n6.9$\\pm$0.5 & & &  \\\\\nN\\,\\textsc{ii} & & & & & & & & & 7.87$\\pm$0.14 & 3 \\\\\nN\\,\\textsc{iii} &  7.69$\\pm$0.23 &   2 & 7.64$\\pm$0.42 &  4 & 7.60$\\pm$0.47 &\n 6 & 7.62$\\pm$0.38 & 3 & 7.40$\\pm$0.00 & 2 \\\\\nN\\,\\textsc{iv} & 7.65$\\pm$0.00 & 1 & & & & & & \\\\\nN\\,\\textsc{v}   & 7.60$\\pm$0.11 & 2 & 7.58$\\pm$0.21 &  2 & 8.26$\\pm$0.15 & 2\n & & \\\\\nAl\\,\\textsc{iii} & $<$3.5 & & $<$3.8 & & $<$3.6 & & 5.51$\\pm$0.10 & 3 \\\\\nSi\\,\\textsc{iii} & & & & & & & 6.43$\\pm$0.31 & 8 & 6.38$\\pm$0.37 & 3 \\\\\nSi\\,\\textsc{iv}\\rlap{$^*$}  & $<$2.0 & & 3.63$\\pm$0.17 &  2 & $<$2.0 & & 6.16$\\pm$0.25 &\n 3 \\\\\nP\\,\\textsc{iii} & & & & & & & 4.41$\\pm$0.00 & 1 & 4.81$\\pm$0.09 & 3 \\\\\nP\\,\\textsc{iv} & 3.95$\\pm$0.00 & 1 & & & & & & \\\\\nS\\,\\textsc{iii} & 7.69$\\pm$0.46 & 4 & 7.00$\\pm$0.11 &  3 &\n5.7$\\pm$0.00 & 1 & & & 6.29$\\pm$0.27 & 3 \\\\\nAr\\,\\textsc{iii} &  7.86$\\pm$0.24 &   4 &  7.02$\\pm$0.00 &  1 & $<$7.0\n& & $<$7.0 & & & \\\\\nCa\\,\\textsc{iii} &  8.09$\\pm$0.20 &  20 &  7.73$\\pm$0.23 & 10 & 7.50$\\pm$0.22\n& 8 & 7.31$\\pm$0.00 & 1 & 6.97$\\pm$0.00 & 1 \\\\\nSc\\,\\textsc{iii} & 5.17$\\pm$0.51 & 2 & $<$5.0 & & 4.61$\\pm$0.00 & 1 & $<$3.0 & & & \\\\\nTi\\,\\textsc{iii} &  6.98$\\pm$0.22 &   10   &    6.51$\\pm$0.20 &  2 &\n5.77$\\pm$0.06 & 3 & 5.16$\\pm$0.20 & 4 & 5.68$\\pm$0.37 & 6 \\\\\nTi\\,\\textsc{iv}  &  6.91$\\pm$0.18 &   5   &    6.73$\\pm$0.00 &  1 &\n5.87$\\pm$0.15 & 3 & 5.21$\\pm$0.18 & 2 & 5.75$\\pm$0.07 & 2 \\\\\nV\\,\\textsc{iii} &  6.42$\\pm$0.22 &  11   &    5.80$\\pm$0.00 &  1  & &\n& & & 4.48$\\pm$0.15 & 4 \\\\\nV\\,\\textsc{iv}  &  6.29$\\pm$0.14 &   7   &    6.37$\\pm$0.34 &  2 &\n5.77$\\pm$0.37 & 9 & & & & \\\\\nCr\\,\\textsc{iii} &  7.29$\\pm$0.19 &  65   &    7.23$\\pm$0.17 & 44 &\n6.96$\\pm$0.23 & 44 & 5.90$\\pm$0.18 & 29 & 6.20$\\pm$0.30 & 33 \\\\\nCr\\,\\textsc{iv}  &  7.31$\\pm$0.23 &  25\t&   7.46$\\pm$0.14  & 17 &\n7.24$\\pm$0.47 & 28 & 6.41$\\pm$0.28 & 7 & 6.54$\\pm$0.20 & 10 \\\\\nMn\\,\\textsc{iii} &  6.02$\\pm$0.19 &  12\t&   7.25$\\pm$0.22 & 46  &\n6.89$\\pm$0.36  & 19 & 5.62$\\pm$0.21 & 13 & 5.54$\\pm$0.21 & 15 \\\\\nMn\\,\\textsc{iv}  &  5.99$\\pm$0.51 &   2\t&   7.72$\\pm$0.11 &  3  &\n7.12$\\pm$0.54  & 7 & & & 5.86$\\pm$0.18 & 2 \\\\\nFe\\,\\textsc{iii} &  6.46$\\pm$0.17 &  26\t&   7.16$\\pm$0.22 & 69  &\n6.66$\\pm$0.26  & 23 & 7.68$\\pm$0.19 & 107 & 7.38$\\pm$0.33 & 20 \\\\\nFe\\,\\textsc{iv}  &  6.40$\\pm$0.10 &  2\t&   7.43$\\pm$0.22 & 23  &\n7.03$\\pm$0.25  & 23 & 7.73$\\pm$0.21 & 10 & 7.52$\\pm$0.23 & 11 \\\\\nFe\\,\\textsc{v}   & & & 7.38$\\pm$0.41 &  2  & &  & & & & \\\\\nCo\\,\\textsc{iii} &  6.46$\\pm$0.33 &  39 &  6.73$\\pm$0.20 & 20  &\n6.13$\\pm$0.39  & 20 & 5.73$\\pm$0.25 & 5 & & \\\\\nCo\\,\\textsc{iv}  &  6.32$\\pm$0.22 &   20 &  6.72$\\pm$0.13 & 7  &\n6.28$\\pm$0.21  & 11 & 6.05$\\pm$0.25 & 3 & & \\\\\nNi\\,\\textsc{iii} &  6.86$\\pm$0.15 &  32 &  7.39$\\pm$0.18 & 32  &\n6.20$\\pm$0.48  & 12 & 6.63$\\pm$0.22 & 28 & 6.39$\\pm$0.25 & 6 \\\\\nNi\\,\\textsc{iv}  & 6.81$\\pm$0.25 & 32 &  7.41$\\pm$0.27 & 19  & 6.28$\\pm$0.28 &\n14 & 6.75$\\pm$0.21 & 7 & 6.50$\\pm$0.26 & 9 \\\\\nNi\\,\\textsc{v}   & 6.91$\\pm$0.55 & 10 &  7.60$\\pm$0.14 & 5  & & & & & & \\\\\nCu\\,\\textsc{iii} & 6.27$\\pm$0.38 & 8 & 6.36$\\pm$0.36 & 5 & & &5.25$\\pm$0.40 &\n3 & & \\\\\nZn\\,\\textsc{iii} &  6.64$\\pm$0.35 &   15 &  6.68$\\pm$0.36 &  5  &\n5.97$\\pm$0.50 & 2 & 5.30$\\pm$0.22 & 6 & 5.33$\\pm$0.34 & 6 \\\\\nGa\\,\\textsc{iii} & 5.53$\\pm$0.06 & 2 & 5.75$\\pm$0.04 & 2 & 5.23$\\pm$0.21 & 2 &\n4.70$\\pm$0.00 & 1 & 3.63$\\pm$0.00 & 1 \\\\\nGe\\,\\textsc{iv}  & 5.21$\\pm$0.05 & 2 & 4.99$\\pm$0.03 & 2 & 4.98$\\pm$0.01 & 2 &\n4.02$\\pm$0.08 & 2 & 3.57$\\pm$0.10 & 2 \\\\\nSn\\,\\textsc{iv}  & 4.12$\\pm$0.00 & 1 & 4.08$\\pm$0.00 & 1 & 3.11$\\pm$0.00 & 1 &\n2.94$\\pm$0.00 & 1 & 2.86$\\pm$0.00 & 1 \\\\\nPb\\,\\textsc{iv}  & 4.7$\\pm$0.00 & 1 & 4.6$\\pm$0.00 & 1 & 4.3$\\pm$0.00 & 1 &\n3.8$\\pm$0.00 & 1 & 3.43$\\pm$0.00 & 1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\hspace{2cm}\n$^*$Based on non-LTE-affected resonance lines; see Section\n\\ref{sub:nlte}.\n\\end{table*}\n\n\\section{Abundance analysis}\n\\label{sec:abu}\n\n\\begin{figure}\n\\vspace{6cm}\n\\begin{center}\n    \\special{psfile=3948fig8.eps hscale=50 vscale=50 hoffset=-10 voffset=-10\n    angle=0}\n\\end{center}\n\\caption{Spectrum fit to the C\\,\\textsc{iv} resonance lines at 1550\\,\\AA\\ for\n  Feige~66. The model is shown as the thick line. The line cores of the model\n  do not match the observations, and the abundance is set around 0.8 dex lower\n  than that derived from C\\,\\textsc{iii} lines. See text for details.}\n\\label{fig:f66c4}\n\\end{figure}\n\n\nIn this section we comment on results and trends for individual or groups of\nelements. The abundances are given in Table \\ref{tab:4abu} and plotted\nrelative to the sun in Figure \\ref{fig:4abu}.  The errors\n  given were determined using a simple mean and standard deviation\n  based on the individual measurement for each line. An error of zero\n  represents a measurement taken from only one line. The solar values\nwere taken from \\citet{GS98}. The O, Ne and Mg abundances of the\nprogramme stars are taken from HRW for PG~1219$+$534\\ and Feige 48, while for\nCD~$-24^\\circ 731$\\ and CPD~$-64^\\circ 481$, they are from Edelmann (private communication), along\nwith the Al abundance for the latter star.\n\nBefore discussing our derived abundances in more detail, however, it\nis necessary to first consider non-LTE effects and the effect of\nmicroturbulence on the absorption lines in our spectra.\n\n\\subsection{NLTE effects}\n\\label{sub:nlte}\n\n\\begin{figure*}\n\\vspace{22cm}\n\\begin{center}\n    \\special{psfile=3948fig9.eps hscale=98 vscale=98 hoffset=10\n      voffset=-60 angle=360}\n\\end{center}\n\\caption{Abundances measured for our five targets. Magenta symbols\n  represent values determined using singly ionised lines, green\n  represents doubly ionised lines, red\n  represents triply ionised lines, blue denotes quadruply ionised\n  lines and cyan represents upper limits. Note the generally excellent\nagreement between different ionisation stages.}\n\\label{fig:4abu}\n\\end{figure*}\n\n\nMost of the spectral lines we are dealing with in this paper are subordinate,\ni.e. they are between excited levels and do not involve the ground \nlevel. The\nresonance lines (transitions from the ground level) of some ions are present\nhowever, and it is these lines that are most sensitive to departures from our\nassumption of local thermodynamic equilibrium (LTE). Ions of\niron-group elements do not appear to be affected, but\nspecies such as C\\,\\textsc{iv} and Si\\,\\textsc{iv}, there are noticeable\neffects. In Figure \\ref{fig:f66c4} we show the C\\,\\textsc{iv}\nresonance lines at 1548\\,\\AA\\ and 1550\\,\\AA\\ of Feige~66 as an\nexample. In this star and Feige~48 the line cores of these lines are\nmatched by the abundances derived from the C\\,\\textsc{iii} lines,\nhowever the wings are much too broad. Reducing the C abundance results\nin an improved fit in the wings but a core that is too shallow. This\nis most likely a non-LTE effect. Detailed model atom calculations are\nneeded to confirm this, but are beyond the scope of this paper. There\nmay also be an interstellar component present, at least in Feige~66,\nwhich has a radial velocity of only $\\sim -6$\\,km\\,s$^{-1}$; however,\nthis is not the case for Feige~48, and the effect is still present.\n\n\n\nIn the case of C\\,\\textsc{iii} all three lines\/multiplets can be\nmatched well with a model spectrum at the listed abundance for\nFeige~48, however for Feige~66 things are not so simple. The\n1174-1177\\,\\AA\\ multiplet is apparently matched well; the\n1247\\,\\AA\\ line is not, while the 2297\\,\\AA\\ line matches in the wings\nbut not in the line core. Because Feige~66 is around 5000\\,K hotter\nthan Feige~48, we again attribute these problems to non-LTE effects.\n\nThe iron-group lines we have used for our abundance analysis are not\nresonance lines (with the exception of Ti\\,\\textsc{iii} where some\nresonance lines were used), so are\nnot likely to be strongly affected by non-LTE effects.\n\n\\subsection{Microturbulence}\n\\label{sub:micro}\n\nSince we have measured the equivalent widths of many lines for several ions,\nwe can investigate the microturbulence $\\xi$ for each star. Both of the\npulsating sdBs, along with CPD~$-64^\\circ 481$, are consistent with $\\xi=0$\\,km\\,s$^{-1}$;\nhowever, the two other non-pulsators both have $\\xi>0$\\,km\\,s$^{-1}$. The top\npanels of Figure \\ref{fig:vmicro} shows the difference between\n$\\xi=0$\\,km\\,s$^{-1}$ and the value we derive $\\xi=5^{+2}_{-1}$\\,km\\,s$^{-1}$\nfor CD~$-24^\\circ 731$; the bottom panels show the effect of non-zero microturbulence for\nPG~1219$+$534. For Feige~66 we find $\\xi=2\\pm1$\\,km\\,s$^{-1}$. These values have been\nderived using both Cr\\,\\textsc{iii} and Fe\\,\\textsc{iii} lines.\n\n\n\\subsection{Abundances}\n\\label{sub:comparison}\n\n\n\nBefore looking at the differences in iron-group abundances of our\ntargets, we first search for and examine trends amongst the lighter\nelements.\n\n\\subsubsection{Light metals} \n\nAs has been found by many previous studies, carbon abundances range\nfrom virtually none at all to slightly below the solar value. For\nPG~1219$+$534\\ and CD~$-24^\\circ 731$\\ the absence of the C\\,\\textsc{iv} resonance\nlines indicates carbon depletion by 10$^5$ or more. Because we cannot\nat the moment account for NLTE effects (see \\ref{sub:nlte} above)\nthese have to be regarded as order of magnitude estimates. We note\nhere that simple NLTE calculations were done by \\citet{HHH84} for the\nsdO star Feige~110 ($T_{\\mathrm{eff}}=40\\,000$\\,K) and a\nsimilar upper limit was derived.\n\nIn the case of nitrogen, the abundances are slightly below the solar\nvalue; this is also in keeping with previous analyses of sdB optical\nspectra.  We note in passing that we could not easily fit the\nN\\,\\textsc{iv} line at 1718.551\\,\\AA\\ in all of our sample because of\nsevere crowding -- only in Feige~66 is the line isolated -- however\nthe strength of the line is approximately consistent with the\nabundances derived from the other N ionisation stages. Note that the\nnitrogen abundances from three different ionisation stages are fully\nconsistent for Feige~66.  \n\nFor Feige~48 it was not possible to measure the sulfur abundance because the\nresonance lines are blended with a large number of metal lines that are not in\nthe Kurucz list of observed lines or the Hirata \\& Horaguchi list. This effect\nis seen to a lesser extent in PG~1219$+$534, making a spectrum fit possible,\nand hardly seen at all in Feige~66. Because of the absence of heavy crowding\nin the latter star, we suggest these may be iron lines. We examined the\neffect of including the ``computed'' (but not observed) lines in the Kurucz\nlist: the match is better for Feige~48, but it is still not possible to\nmeasure abundances using the S\\,\\textsc{iii} lines.\n\n\\subsubsection{Iron group}\n\nAs can be seen from Table \\ref{tab:4abu}, plenty of lines of doubly\nionised atoms (in particular Cr\\,\\textsc{iii}, Fe\\,\\textsc{iii} and\nNi\\,\\textsc{iii}) have been used for the abundances analysis, for Feige~48\nmore than 100 Fe\\,\\textsc{iii} lines were utilised. Moreover, triply\nionised atoms can be used to determine abundances for all iron group\nelements as well (except for V\\,\\textsc{iv} and Mn\\,\\textsc{iv} in\nFeige~48 and V\\,\\textsc{iv}, and Co\\,\\textsc{iv} in CPD~$-64^\\circ 481$), which in\nsome stars are also quite numerous (e.g. 32 Ni\\,\\textsc{iv} lines in\nFeige 66). Even four times ionised atoms have been used for analysis\nof nickel in Feige~66 and PG~1219$+$534\\ and of iron in PG~1219$+$534. It is worthwhile\nnoting that these ionisation equilibria are very well matched,\ne.g. the three ionisation stages of Ni  agree to within 0.1 dex for\nFeige 66 and to within 0.2 dex for PG~1219$+$534. For other ions of the iron\ngroup, we find:\n\n\\begin{itemize}\n\\item Fe\\,\\textsc{iii} and Fe\\,\\textsc{iv} as well as Ni\\,\\textsc{iii} and\n  Ni\\,\\textsc{iv}, respectively, agree very well (0.1~dex, typically) for\n  Feige~48 and CPD~$-64^\\circ 481$\\ and to within error limits in CD~$-24^\\circ 731$\\ (0.37~dex for iron).\n\n\\item Cr\\,\\textsc{iii} and Cr\\,\\textsc{iv} agree to better than 0.3~dex except\n  for Feige~48 (0.5~dex).\n\n\\item Mn\\,\\textsc{iii} and Mn\\,\\textsc{iv} are in perfect agreement for Feige\n  66, differ by about 0.3~dex in CD~$-24^\\circ 731$\\ and CPD~$-64^\\circ 481$. Only for PG~1219$+$534\\ they deviate\n  beyond the adopted error ranges. \n\n\\item Co\\,\\textsc{iii} and Co\\,\\textsc{iv} also match very well, to better\n  than 0.2~dex, except for Feige~48 which has much less Co lines than the\n  other programme stars.\n\n\\item Although only few titanium lines could be used for the analysis, the\n  results from Ti\\,\\textsc{iii} and Ti\\,\\textsc{iv} match extremely\n  well for all programme stars.\n\\end{itemize}\n\nWe regard the very good match of many ionisation equilibria as evidence that\nsystematic errors of the metal abundances and of the effective \ntemperature are small. This can also be seen in the good match between\nsynthetic spectrum and observations for three objects shown in Figure\n\\ref{fig:fe1860}.\n\nWhile iron is found to be nearly solar (PG~1219$+$534, Feige~48, CPD~$-64^\\circ 481$),\nslightly depleted in CD~$-24^\\circ 731$\\ and subsolar in Feige~66 by a factor of ten, all\nother elements of the iron group are enhanced by between 0.5 and 2.5 dex with\nrespect to solar values. The enhancements are large in Feige~66 and PG~1219$+$534, but\nmild for the three others.\n\n\\begin{figure}\n\\vspace{6cm}\n\\begin{center}\n    \\special{psfile=3948fig10.eps hscale=50 vscale=50 hoffset=-10 voffset=-10\n    angle=0}\n\\end{center}\n\\caption{Spectrum fit to the Ge\\,\\textsc{iv} resonance lines at 1189\\,\\AA\\ for\n  Feige~66. The model is shown as the thick line. }\n\\label{fig:f66ge4}\n\\end{figure}\n\n\\subsubsection{Gallium, Germanium, Tin and Lead} \n\nThe heavy metals Ga, Ge, Sn and Pb are all enriched with\nrespect to the Sun in all stars, reaching as high as 2.9~dex for Ga in \nPG~1219$+$534\\ or 2.75~dex for Pb in Feige~66. An example of a fit of the\nGe\\,\\textsc{IV} 1189\\,\\AA\\ line is shown in Figure\n\\ref{fig:f66ge4}. There are several strong lines also present that are\neither not identified or have no atomic data. Possible identifications\nare shown in the figure.\n\n\n\n\n\\subsection{Comparison with previous work}\n\\label{sub:previous}\n\nAs mentioned earlier, many of the objects we have observed for this project\nhave been studied with optical spectra before -- PG~1219$+$534\\ and\n  Feige~48 \\citep{HRW00}, and CPD~$-64^\\circ 481$\\ and CD~$-24^\\circ 731$\\ (Edelmann, private\n  communication) -- or with a noisy \\emph{IUE} spectrum and poor atomic data\n-- Feige~66 \\citep{BHS82}. It is useful here to compare our results with these\nprevious studies, as well as with any general trends seen amongst sdBs as a\ngroup.\n\n\nFirstly, for the two pulsators PG~1219$+$534\\ and Feige~48, our results compare\nvery well with those of HRW for the limited cross-over that exists\nbetween elements. All of HRW's abundances are, within errors,\nconsistent with ours, and because we could use resonance lines of\nseveral ions, we have been able to place stronger upper limits on C, Al\nand Si for PG~1219$+$534. In the case of\nFeige~48 the abundances we derive agree well with those measured by\nHRW, and are within $\\pm0.04$\\,dex.\n\nThe only element we have had difficulty\nderiving an abundance for is sulfur, for the reasons discussed earlier.\n\nIn the case of Feige~66, \\citet{BHS82} used \\emph{IUE} spectra with\nlower resolution than ours, and obtained somewhat different results to\nours. It is difficult to compare the two sets of abundances since\n\\citeauthor{BHS82} do not give error estimates explicitly; however, if\nwe assume the same errors as those given for another sdB, HD\\,149382,\nin the same paper, our values are not so discrepant after all. At the\ntime of \\citeauthor{BHS82} study, the quality of atomic data was lower\nthan it is at present (although in many cases it is still insufficient\nor of low accuracy), so their errors are very large, particularly for\nthe iron group.\n\nRecently, \\citet{EHN06} has carried out an analysis\nof CPD~$-64^\\circ 481$\\ and CD~$-24^\\circ 731$\\ based on high-resolution optical echelle\nspectra. For both stars their abundances are consistent with\nthose determined here.\n\nFinally, we can also compare any apparent trends seen here with the work of\n\\citet{HE04}, especially for the lighter elements plus iron.\nBoth our results here and the work of \\citet{HE04} suggests that aluminum\nfollows the same trend as silicon. \n\n\n\\subsection{Heavy element abundance trends?}\n\\label{sub:fetrend}\n\n\n\n\\begin{figure}\n\\vspace{7.5cm}\n\\begin{center}\n    \\special{psfile=3948fig11.eps hscale=53 vscale=53 hoffset=10\n    voffset=-10 angle=360}\n\\end{center}\n\\caption{Trends in heavy element abundances with iron abundance.}\n\\label{fig:fetrend}\n\\end{figure}\n\nAn interesting result not immediately obvious from the abundances in Table\n\\ref{tab:4abu} is a possible anti-correlation between heavy element abundance\nand iron abundance. The spread in iron abundance is 1.25 dex from the most\niron-poor (Feige~66) to the most iron-rich star (Feige~48), while the variation\n(X\/Fe) is between 2 and 3 dex (except for Mn and Ni). Therefore this trend\ndoes not simply reflect the trend in iron abundance. \nFour examples of this effect are\nshown in Figure \\ref{fig:fetrend} for Ca\/Fe, Ti\/Fe, Ni\/Fe and Pb\/Fe while\nTable \\ref{tab:fetrend} shows data for all elements; where there are two or\nmore ionisation stages, a weighted average is presented. The same basic trend\nis seen for all elements in the iron-group, except for manganese and perhaps\nnickel, and it is also seen for the heavier elements gallium, germanium, tin\nand lead. Caution must\nbe applied here however, since we have only five objects in our sample. A more\ndetailed analysis may be possible for at least titanium, since several sdB\nstars show Ti\\,\\textsc{iii} lines in their optical spectra (H. Edelmann,\nC. Karl, private communication). If these trends are real we believe they are\ncaused by a combination of radiative levitation, gravitational settling and a\nweak stellar wind, and therefore would help constrain diffusion models. We\nurge more theoretical work in this area.\n\n\\begin{table}\n\\caption{Average iron abundance and ratio of heavy element abundances to iron\n  for the five targets stars. The columns are as follows: F66 = Feige~66; CD =\n  CD~$-24^\\circ 731$; PG = PG~1219$+$534; CPD = CPD~$-64^\\circ 481$; F48 = Feige~48. Abbreviations: n-v = non-variable,\n  var. = variable.}\n\\label{tab:fetrend}\n\\begin{center}\n\\begin{tabular}{l|rrr|rr}\n & F66 & CD & PG & CPD & F48 \\\\\n & n-v & n-v & var. & n-v & var\\\\\n\\hline\nFe & 6.46 & 6.85 & 7.23 & 7.43 & 7.68 \\\\\n\\hline\nCa\/Fe & 1.63    & 0.65    & 0.50     &  $-$0.46  &  $-$0.37 \\\\\nTi\/Fe & 0.50    & $-$1.03 & $-$0.65  &  $-$1.74  &  $-$2.50\\\\\nV\/Fe  & $-$0.09 & $-$1.08 & $-$1.08  &  $-$2.95  &   ---   \\\\ \nCr\/Fe & 0.84    & 0.23    & 0.06     &  $-$1.15  &  $-$1.68 \\\\\nMn\/Fe & $-$0.44 & 0.10    & 0.05     &  $-$1.85  & $-$2.06 \\\\\nCo\/Fe & $-$0.05 & $-$0.69 & $-$0.50  &    ---    & $-$1.83 \\\\\nNi\/Fe & 0.39    & $-$0.61 & $-$0.19  &  $-$0.97  & $-$1.03 \\\\\nCu\/Fe & $-$0.19 &  ---    & $-$0.87  &    ---    & $-$2.43\\\\\nZn\/Fe & 0.18    & $-$0.88 & $-$0.59  &  $-$2.10  & $-$2.38\\\\\nGa\/Fe & $-$0.93 & $-$1.62 & $-$1.70  &  $-$3.80  & $-$2.98 \\\\\nGe\/Fe & $-$1.25 & $-$1.87 & $-$2.02  &  $-$3.86  & $-$3.66\\\\\nSn\/Fe & $-$2.34 & $-$3.74 & $-$3.11  &  $-$4.57  & $-$4.74 \\\\\nPb\/Fe & $-$1.76 & $-$2.55 & $-$2.53  &  $-$4.00  & $-$3.88 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Comparison of abundances of pulsator\/non-pulsator pairs}\n\\label{sub:pnpcomp}\n\n The results of our spectral analysis  \nallow us to compare the abundance\npatterns of four pairs of stars, having similar effective temperature:\n(i) the ``cool'' pulsator\/non-pulsator pair Feige~48\/CPD~$-64^\\circ 481$, \n(ii) the ``hot'' pulsator\/non-pulsator \npair PG~1219$+$534\/CD~$-24^\\circ 731$, (iii) the ``hot'' pulsator\/non-pulsator pair PG~1219$+$534\/Feige~66 and \n(iv) the pair of\ntwo ``hot'' non-pulsators Feige~66\/CD~$-24^\\circ 731$.\n We plot the relative iron group abundances for all four combinations \nin Fig.~\\ref{fig:ironcomp}).\n\n(i) If we compare Feige~48\nand CPD~$-64^\\circ 481$\\ (the bottom panel of Figure \\ref{fig:ironcomp}), we\nfind basically no difference. Not only are the abundances of \nthe iron group elements similar, but the lighter and\nheavier elements also agree reasonably well.\nThe only exception is gallium which is ten times higher in Feige~48 than in\nCPD~$-64^\\circ 481$.  Also looking at abundance ratios of iron group elements \nwith respect to iron (X\/Fe, Table \\ref{tab:fetrend}) reveals, that the \npatterns for Feige~48 and CPD~$-64^\\circ 481$\\ match each other reasonably well. \nThis would indicate that there\nis no difference in iron group abundances of a pulsator and a non-pulsator.\n\n(ii) The comparison of ``hot'' pulsator PG~1219$+$534\\ to the non-pulsator CD~$-24^\\circ 731$\\ reveals \nlarge differences for some light elements, silicon and sulphur being much\nless in CD~$-24^\\circ 731$\\ (more than 1.6 dex and 1.3 dex, respectively). Significant\ndifferences are also detected for the iron group as well as for the\nheavy elements. They, however, vanish almost if the difference in iron\ncontent is accounted for (except for Ni\/Fe and Sn\/Fe, see\nTable~\\ref{tab:fetrend}). Since the iron group elements \nare probably the drivers for pulsators, the comparison would indicate as in \nthe previous case that there is little difference (if at all) between\na pulsator and a non-pulsator.\n\n(iii) The comparison of the ``hot'' pulsator PG~1219$+$534\\ to the non-pulsator Feige~66\nyields a completely different picture. Abundances of several iron group elements\ndiffer (see Table~\\ref{tab:4abu}). Even if we take into account the large \ndifference in iron abundance (factor of 7) between the two stars, the \nabundance patterns remain dissimilar (see Table~\\ref{tab:fetrend}). Amongst the \nlight elements there is a huge difference in carbon\nabundance. PG~1219$+$534\\ has less than 5000 times as much C as Feige~66\nhas. The abundances of the heavy elements (Ga, Ge,\nSn, and Pb), however, are quite similar. In contrast to the previous comparisons\nin (i) and (ii) this would imply that indeed pulsators and non-pulsators have \n\\emph{different} iron group abundances.     \n\n(iv) The comparison of the two ``hot'' non-pulsators Feige~66 and CD~$-24^\\circ 731$\\ also \nreveals large differences in the abundances of iron group elements which \npersist if we account for the different iron abundance \n(see Table~\\ref{tab:fetrend}). Amongst the light elements as well as among the \nheavy elements there are large differences as well, which persist if we account for the \ndifferent iron abundances. \n          \n\nThe last finding is discouraging. If the difference among non-variable \nstars of similar $T_{\\mathrm{eff}}$ and  $\\log g$ are as large as we observe, \nthe comparison between a pulsator and a non-pulsator is rendered arbitrary. \n\n\\begin{figure}\n\\vspace{11.5cm}\n\\begin{center}\n    \\special{psfile=3948fig12.eps hscale=50 vscale=50 hoffset=0\n    voffset=-30 angle=360}\n\\end{center}\n\\caption{Comparison of iron-group abundances for our\n  pulsator\/non-pulsator pairs (F48\\,=\\,Feige~48; CPD\\,=\\,CPD~$-64^\\circ 481$; \n  PG\\,=\\,PG~1219$+$534;\n  CD\\,=\\,CD~$-24^\\circ 731$; F66\\,=\\,Feige~66). The dotted lines denote\n  equal abundances and the difference in iron abundance. The\n  differences between  PG~1219$+$534, CD~$-24^\\circ 731$\\ and Feige~66 are quite large\n  -- particularly for Fe, Ni and Mn -- while the pairs \n  PG~1219$+$534\\ $\\&$ CD~$-24^\\circ 731$\\ and Feige~48 $\\&$ CPD~$-64^\\circ 481$\\ show\n  quite similar abundances, in particular when the differences in iron abundance\n  are accounted for (see text).}\n\\label{fig:ironcomp}\n\\end{figure}\n\nThe theory of \\citet{CFB97a} does not specifically rule out iron-group\nelements other than iron itself as responsible for pulsation\ndriving. In particular Ni must be considered as it has substantially\nmore UV and FUV lines than iron. Indeed, in the case of PG~1219$+$534\\ the nickel \nabundance is high\nenough that it could make a significant contribution to the opacity,\nat least as much as iron. When we consider Feige~48 and CPD~$-64^\\circ 481$, we find\nthat perhaps the difference in the \\emph{combined} opacity of iron and\nnickel between the two stars is significant enough to discriminate\nbetween pulsator and non-pulsator. This leads us to ask: could we\ntheoretically have a pulsator with lower iron, but much higher\nnickel or other iron-group abundance than a non-pulsator?\n\n\n\n\\section{Further discussion}\n\\label{sec:disc}\n\nAs seen in Section \\ref{sub:tempfix}, enhanced metal abundances can\nhave an effect on $T_{\\mathrm{eff}}$ and $\\log g$ determination,\nespecially for stars showing both neutral and singly ionised\nhelium. It is also clear that simply scaling models from solar\nmetalicity ODFs is insufficient; opacity sampling is required for more\naccurate measurements, since while Fe abundances are approximately\nsolar, elements such as Ni and Mn -- which have a significant opacity\ncontribution -- are almost always enhanced. A preliminary study has\nbeen done in this direction by \\citet{BJ06} and \\citet{PNE06}.\n\nThe effect of these abundance patterns may be apparent for atmosphere\nmodels, but what about stellar evolution and pulsation models? We urge\nevolution theorists to investigate the effect of non-solar opacity\ndistributions on hot subdwarf evolution. In a similar vein, interior models\nthat include differential internal rotation and\/or magnetic fields should also\ninvestigate their effect on diffusion.\n\nIn a study of elements beyond the iron group in hot subdwarfs,\n\\citet{SJOT04} proposed a solution to  the silicon problem discussed in\nSection 4.3.2. Triply ionised elements of the same group in the\nPeriodic Table as silicon -- germanium, tin and lead -- are present in\nalmost all sdB spectra at \\emph{all} temperatures. Si\\,\\textsc{iv} on the\nother hand, almost completely disappears above 32\\,000\\,K. This shows\nthat arguments where silicon is ionised to  nobe gas configuration, \ni.e. to Si\\,\\textsc{v}, and then\nsinks deeper into the atmosphere, cannot be correct. If this were the\ncase, these heavier elements, which should feel the same  low radiative\nforces as silicon, should also sink. Instead, \\citet{SJOT04} suggested\nthat above $\\sim$32\\,000\\,K silicon could be carried away by a weak\nfractionated stellar wind, whereas the heavier elements should stay\nbehind. Indeed, \\citet{Unglaub06} has found that a one-component,\nuniform wind is inconsistent with observations, and that multicomponent\ncalculations are required, although this depends on\n  surface gravity and mass-loss rate. When we examine the abundances\nof Ge, Sn and Pb, we find that they are all higher in the hot stars\n(Feige~66, CD~$-24^\\circ 731$, and PG~1219$+$534) that show little or no silicon, than in the\ncooler stars (Feige~48, CPD~$-64^\\circ 481$) that do. This is what one would\nqualitatively expect from the hypothesis of \\citet{SJOT04}. Using\n\\emph{FUSE} spectra, \\citet{CFF06} found many other sdBs with similar\nproperties, but also some exceptions. The case of C\\,\\textsc{iv} is\nnot at all straightforward, since it is very difficult to explain two\nspectroscopically similar stars, one with measureable silicon but no\ncarbon (PG~1219$+$534), and the other with measureable carbon but no silicon\n(Feige~66). CD~$-24^\\circ 731$\\ has no carbon and no silicon, compounding the\nproblem. The reason for this remains a mystery, and presents\n  a challenge to the fractionated wind hypothesis, especially since\n  carbon is also in the same group as silicon, germanium, tin and\n  lead, and should in principle feel the same radiative forces.\n\nIt is also worth noting that PG~1219$+$534, one of our pulsators, shows measurable\ntraces of silicon, despite being hotter than 32\\,000\\,K, while the\nspectroscopically similar  stars Feige 66 and CD~$-24^\\circ 731$\\ do not. Could\nthis be an indicator of the age of the sdB? If the star has a weak\nfractionated wind, as suggested by \\citet{SJOT04}, then perhaps one of\nthe differences between PG~1219$+$534\\ and Feige~66\/CD~$-24^\\circ 731$\\ is age. In order to\ndetermine this, however, diffusion models including silicon are required.\n\nIt seems clear that radiative acceleration is bringing heavy elements to the\nsurface in these sdBs, and perhaps even expelling them through a stellar\nwind. We can at least qualitatively understand this in the case of the\niron-group and heavier elements: in the hotter stars heavy elements are much\nmore enhanced than in the cooler stars. We must also consider, however, that\ntwo of our ``hot'' stars are apparently single, while both of our ``cool'' \nstars are in close binary systems. Does this have any effect on the \nabundance patterns?\nIt is unclear what the cause  \nfor the differences  among the ``hot'' stars, Feige~66, CD~$-24^\\circ 731$, and PG~1219$+$534, is, \nbut binarity is one possibility. \nOur small\nnumber statistic  is the main limiting factor for this discussion, so we must\nlook forward to the abundance analyses of Edelmann et al. (2006), who\nhave studied a larger number of binaries and single stars,\nalbeit with optical spectra, which restricts the available heavier\nelements to iron and sometimes titanium.\n\n\\section{Summary and Conclusions}\n\\label{sec:conc}\n\nWe have analysed high-resolution UV echelle spectra of five hot\nsubdwarf B stars, two of which are member of the short-period, pulsating\nV361\\,Hya class. \n Abundances of no less than 25 elements including the iron group and \n  even \n  heavier elements such as tin and lead have been determined using \n  LTE curve-of-growth and spectrum \n  synthesis techniques. \n  Our investigation was initiated to test the hypothesis that a\ncorrelation exists between the abundances of iron-group elements in\nsdB stars and pulsation. We have compared a hot pulsator (PG~1219$+$534) with a\n two non-pulsators with similar stellar parameters (Feige~66 and CD~$-24^\\circ 731$) \nand a cooler pulsator\n(Feige~48) with a similar non-pulsator (CPD~$-64^\\circ 481$), and found no consistent\ndifferences between the members of each pair. \n The heavy element abundance pattern of CD~$-24^\\circ 731$\\ comes close to that\nobserved for PG~1219$+$534\\ except for its low iron and nickel. Feige~66 has an even lower\niron abundance, but its heavy metal abundance pattern does not match that of\nPG~1219$+$534\\ at all. In other words the abundance patterns of two non-variable stars \nof similar temperature and gravity are too dissimilar for a conclusive\ncomparison with a pulsator. This result  \nleads us to suspect\nthat there must be another, as yet unknown, discriminating factor between\npulsating and non-pulsating sdB stars. \n\nMore generally, we have uncovered a potential solution to the discrepant\neffective temperatures from Balmer lines and helium ionisation equilibria,\nwith a significant improvement found using supersolar metallicity\nmodels. Opacity sampling in place of distribution functions will sure lead to\nall model line profiles matching observations. Additionally, our spectra show\nlight element abundance patterns typical for sdBs: carbon varies from\nvirtually none to around 1 dex below solar, while nitrogen is within 0.5 dex\nof the solar value. There is evidence to support the fractionated weak stellar\nwind hypothesis of \\citet{SJOT04}, as the heavy element abundances increase\nwith temperature. We also find an interesting anti-correlation between the\nabundance of iron and the heavy element abundance relative to iron. More\nobservations are needed to confirm this trend.\n\nSo somewhat frustratingly we must conclude that we cannot provide any\ninsights into pulsation in sdB stars based on our spectroscopic\nmeasurements. On a more encouraging note, the results presented here will\nprovide a valuable resource for theoretical work into diffusion in sdB\nstars. It will be important to study the diffusion of all iron group\nand heavier elements individually in order to explain the trend we\nhave uncovered. We end then with a question and a challenge: how can we\nfind the reason some sdB pulsate and some don't if not by\nspectroscopic means?\n\n\n\\begin{acknowledgements}\nWe thank the referee for constructive comments that have improved the\nmanuscript. We would like to thank  Michael Lemke for his support with the\nLINFOR code and Norbert Przybilla for useful\ndiscussions on the quality of atomic data. SJOT is supported by the\nDeutsches Zentrum f\\\"ur Luft- und Raumfahrt (DLR) through grant\nno.\\ 50-OR-0202.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s:intro}\nThe human microbiome consist of trillions of microorganisms including bacteria, archaea, viruses, and fungi living in and on the human body and play important roles in our health (\\citealp{Turnbaugh2007}; \\citealp{Consortium2012}; \\citealp{Lloyd-Price2017}). Microbial dysbiosis has been linked to a variety of diseases including asthma, infection, and allergy in children (\\citealp{Chen2007};  \\citealp{Madan2012}; \\citealp{Hoen2015}), as well as cancer (\\citealp{Reikvam2011}; \\citealp{Castellarin2012}) and obesity (\\citealp{Turnbaugh2006}; \\citealp{Trasande2013}). To quantitatively study the assocation of human microbiome with exposure variables and clinical outcomes, sequencing technologies such as 16s ribosomal RNA gene sequencing \\citep{Cole2009} and shotgun metagenomic sequencing \\citep{Tringe2005} are employed to quantify the microbiome composition of a sample (e.g., stool, saliva), and then numerical measures for the association of interest can be derived with statistical and computational methods \\citep{Lee2015}. Because sequencing data is collected from a sample representing a small proportion of the ecosystem (e.g., gut), the raw sequencing count (i.e., absolute abundance) of a microbial taxon in the sample may not serve as a good estimate for its absolute abundance (AA) in the ecosystem \\citep{ancom}. The target of inference is very often the relative abundances (RA) which measure the fractions of microbial taxa in the ecosystem that can be approximated by the observed fractions in the sample (\\citealp{Lozupone2005}; \\citealp{LaRosa2012}; \\citealp{Chen2013}; \\citealp{Tang2018}).\n\nMaking inference on RA is challenging because perturbation in the abundance of one taxon will cause changes in fractions of all taxa due to  change in the common denominator (CD) for calculating all fractions. We will refer to this as the CD problem hereafter. The CD problem is also related to the compositional structure of RA's under which they are negatively correlated since an increase in one RA will necessarily result in a decrease in another one. Those false changes could generate false positive results or mask the true changes which then lead to false negative results. Existing methods have not been able to adequately resolve this issue. \n\nAnother well-known challenge for making inference on RA comes from the zero-inflated structure of the sequencing data which is also a general challenge for analyzing microbiome data. Many existing methods (\\citealp{Chen2013}; \\citealp{zig}; \\citealp{Lin2014};  \\citealp{ancom}) require imputing zero-valued sequencing counts with a positive number such as the Pseudocount of 0.5 or another number which could lead to biased estimates of the RA's. Because the log transformation over the interval (0,1) ranges from negative infinity to zero, this bias can be exaggerated to a surprisingly large value on the commonly used natural-log scale. For example, when RA changes from 0.1 to 0.00001 which corresponds to approximately 1-fold decrease in terms of magnitude on the original RA scale, the log-value of RA changes from $-2.30$ to $-11.51$ corresponding to a 5-fold change in terms of magnitude. Imputation of the zero counts could also be problematic when the sequencing depth (i.e., library size) is a confounder of the association of interest \\citep{Weiss2017}. Sequencing depth has a strong correlation with the diversity of microbiome community observed in a sample. For instance, the number of detected OTUs and sequencing depth are highly correlated (with an r-square of 0.92) in the Human Microbiome Project (\\citealp{Turnbaugh2007}; \\citealp{Consortium2012}) as shown in \\cite{zig}. So when comparing two groups, it is possible that one group has some taxa with more zero-valued RA due to smaller library sizes, and consequently imputation of the zero reads could create an artificial bias for the group difference when the true difference is null. \n\nTo address the above challenges, we propose a novel approach to draw inference on the AA of the ecosystem instead of the RA. The new approach will avoid the aforementioned CD problem associated with RA and get rid of the issue induced by the compositional structure of RA's. Unlike many existing methods, this new method does not require imputing zero although it can be used to analyze microbiome data after zero values are imputed with a pseudo count or any other number. The new algorithm consists of two phases with Phase 1 to identify the taxa whose AA are associated with the covariates of interest and Phase 2 to estimate the association parameters. Both phases utilize the ratios of non-zero AA observed in the samples. The advantage of using the ratios is that it can remove the impact of sequencing depth in the model because the sequencing depth is cancelled out in calculating the ratios. The key idea of phase 1 is that the ratio of two taxa should be independent of the covariates of interest if the two taxa are both independent of the covariates, and the ratio should be associated with the covariates if any one of the two taxa is associated with the covariates. This will allow for identification of the set of taxa (set A) that are associated with the covariates as well as the other set of taxa (set B) that are not associated with the covariates, and then in phase 2 we are able to quantify the associations between AA and the covariates for all taxa in set A with point estimates and confidence intervals by using a reference taxon that is indepdent of the covariates. Our approach can also remove the confounding effect of sequencing depth because the ratio of two taxa abundances does not depend on the sequencing depth, and thus it can not be a confounder in the model. By incorporating regularization methods, our approach can handle high-dimensional microbiome data as well as high-dimensional covariates data.\n\nWe organize this paper as follows. Model and notations are presented in Section \\ref{s:model}. Algorithms for identifying sets A and B and for parameter estimation are provided in Section \\ref{s:estimation} followed by an extensive simulation study under different scenarios to assess the performance of our approach in comparison with other established existing approaches in Section \\ref{s:simu}. We showcase the application of our new approach to two real studies in comparison with existing approaches in Section \\ref{s:app} followed by the discussion in Section \\ref{s:discu}.\n\n\\section{Model and Notation}\\label{s:model}\n\\subsection{Multivariate zero-inflated log-normal distribution}\\label{ss:ZILoN}\nSuppose there are $N$ subjects and $K+1$ taxa of interest. Let $\\mathcal{Y}_i=(\\mathcal{Y}^1_i,\\mathcal{Y}^2_i,...,\\mathcal{Y}^K_i,\\mathcal{Y}^{K+1}_i)$ denote the true microbial taxa absolute abundance (ie, counts) in the ecosystem (eg, gut) of the $i$th subject, $i=1,...,N$. The subject index $i$ will be suppressed for simplicity in this section. To describe the microbial abundance distribution, we propose a multivariate zero-inflated log-normal distribution that can account for the zero-inflated structure. It is a two-part distribution with a discrete part and a continuous part. The discrete part provides the probabilities governing the probabilities of taxa abundance being zero or non-zero: \n\\begin{align*}\nP(\\mathcal{Y}^1>0,\\mathcal{Y}^2=0,...,\\mathcal{Y}^K=0,\\mathcal{Y}^{K+1}=0)=p_1 \\hspace{0.3cm}\\\\\nP(\\mathcal{Y}^1=0,\\mathcal{Y}^2>0,...,\\mathcal{Y}^K=0,\\mathcal{Y}^{K+1}=0)=p_2  \\hspace{0.3cm}\\\\\n\\vdots \\hspace{5cm}\\\\\nP(\\mathcal{Y}^1=0,\\mathcal{Y}^2=0,...,\\mathcal{Y}^K=0,\\mathcal{Y}^{K+1}>0)=p_{K+1} \\\\\n\\vdots \\hspace{5cm}\n\\end{align*}\n\\begin{align*}\nP(\\mathcal{Y}^1=0,...,\\mathcal{Y}^{k_1-1}=0,\\mathcal{Y}^{k_1}>0,\\mathcal{Y}^{k_1+1}=0,...,\\mathcal{Y}^{k_L}>0,...,\\mathcal{Y}^{K+1}=0)=p_{k_1k_2...k_L}\\\\\n\\vdots\\hspace{7.8cm} \n\\end{align*}\n\\begin{align*}\nP(\\mathcal{Y}^1>0,\\mathcal{Y}^2>0,...,\\mathcal{Y}^{K+1}>0)=p_{12...K+1}\\\\\n\\sum_{\\substack{1\\leq k_1<k_2<...k_L\\leq K+1 \\\\ 1\\leq L\\leq K+1}} p_{k_1k_2...k_L}=1,\\hspace{1cm}\n\\end{align*}\nwhere $p_{k_1,k_2,...k_L}$ is the probability of the elements ${k_1,k_2,...,k_L}$ of the vector $\\mathcal{Y}_i$ being non-zero and the rest being zero. Notice that we don't consider the case of all taxa abundance being 0 because it would not be included in the analysis. In other words, a subject has to have at least one non-zero taxa to be included in the model. This is equivalent to a $(K+1)-$dimensional Bernoulli distribution conditional on at least one Bernoulli variable being 1. The discrete part of this distribution is essentially the same as the discrete part of the multivariate zero-inflated logistic-normal (MZILN) distribution described in  \\cite{MZILN}. Notice that a distribution without any zero values can be treated as a special case of a zero-inflated distribution with $p_{12...K+1}=1$ and all other $p_{k_1k_2...k_L}$ being 0 in the discrete part. With that feature, this model can be also directly applied to data sets where zero-valued data points are imputed by a Pseudocount or another positive number.\n\nNow we define the continuous part of the two-part distribution. Conditional on a subgroup of taxa being nonzero and the rest being zero as defined in the discrete part, the joint pdf function of the continuous part is defined as:\n\\begin{equation*}\n  f(y)=\\begin{cases}\n    p_1g_1\\big(\\log(y^1)\\big), & y=(\\log(y^1),0,...,0)^T.\\\\\n   \\hspace{0.5cm} \\vdots & \\hspace{1cm} \\vdots\\\\\n    p_{k_1...k_L}g_{k_1...k_L}\\big(\\log(y^{k_1}),...,\\log(y^{k_L})\\big), & \\hspace{-0.2cm}y=(0,.,\\log(y^{k_1}),0,.,0,\\log(y^{k_L}),.,0)^T.\\\\   \n   \\hspace{0.5cm} \\vdots & \\hspace{1cm} \\vdots\\\\\n   p_{1...K+1}g_{1...K+1}\\big(\\log(y^1),..,\\log(y^{K+1})\\big), & y=(\\log(y^1),..,\\log(y^{K+1}))^T,\n  \\end{cases}\n\\end{equation*}\nwhere $g_{k_1...k_L}(\\cdot)$ is the density function of a $L$-dimensional multivariate normal distribution with mean $\\mathcal{A}\\mu$ and variance matrix $\\mathcal{A}\\Sigma\\mathcal{A}^T$. Here $\\mathcal{A}$ is a $L\\times (K+1)$ matrix with the $l$th row,  $l=1,...,L$, equal to the $ k_l$th row of the $(K+1)\\times (K+1)$ identity matrix, $\\mu=(\\mu^1,...,\\mu^{K+1})^T$ is an unknown $(K+1)-$vector of means and $\\Sigma$ is a $(K+1)\\times (K+1)$ variance matrix. In other words, the mean vector of $\\big(\\log(\\mathcal{Y}^{k_1}),...,\\log(\\mathcal{Y}^{k_L})\\big)$ is $(\\mu^{k_1},...,\\mu^{k_L})$ which is the subvector of $\\mu$ indexed by $(k_1,k_2,...,k_L)$ and its variance matrix is the submatrix of $\\Sigma$ with the rows and columns indexed by $(k_1,k_2,...,k_L)$. The density function $f(y)$ includes the term $p_{k_1k_2...k_L}$ because it is essentially a density function conditional on  $\\big(\\mathcal{Y}^{k_1},...,\\mathcal{Y}^{k_L}\\big)$ being non-zero the rest of all taxa being zero. With the above definition of discrete and continuous parts, we complete the describtion of the two-part distribution which involves quite a lot of parameters including the mean vector $\\mu$, the variance matrix $\\Sigma$ and the discrete probability mass parameters  $p_{k_1k_2...k_L},1\\leq k_1<k_2<...<k_L\\leq K+1, 1\\leq L\\leq K+1$. The number of $p_{k_1k_2...k_L}$'s could be as many as $2^{K+1}-2$ because it needs to cover all possible scenarios of any subset of $\\mathcal{Y}_i$ being non-zero. Unlike the MZILN distribution for a compositional vector in the standard simplex space $s^K$ \\citep{MZILN}, the vector $\\mathcal{Y}_i$ here is not in the simplex space.\n\n\\subsection{Parameters of interest}\nOftentimes, the goal of a study is to investigate the associations of microbiome abundance with other covariates such as the environmental exposures that could change microbiome composition. Suppose there are $Q$ covariates of interest, denoted by the $Q$-dimensional vector $X_i$, for the association test. Our approach can allow a large number of covariates in the model (i.e., $Q>N$). Suppose there are other $S$ covariates (e.g., confounders) that will also be included in the model, but their associations with microbiome will not be examined. The number of confounders can also be large (e.g., $S>N$). These potential confounders are denoted by $W_i$, a $S$-dimensional vector. In this paper, we are focusing on the association between $X_i$ and the microbial abundance conditional on presence. Based on the previous two-part distribution, we use the following equations to model the association:\n\\begin{equation}\\label{eq:1b}\n\\log(\\mathcal{Y}_i^k)|\\mathcal{Y}_i^k>0=\\beta^{0k}+X_i^T\\beta^k+W_i^T\\gamma^k+Z_i^Tb_i+\\epsilon_i^k,\\hspace{0.2cm}k=1,...,K+1,\n\\end{equation}\nwhere the vertical line \"$\\vert$\" means \"conditional on\" since the natural-log function $\\log(\\cdot)$ can not be applied to 0 (which will be suppressed herein for simplicity), $b_i$  are the random effects that can address the heterogeneity (e.g., biological variation) across subjects on top of the random error $\\epsilon_i^k$. Here $Z_i$ is the design matrix for random effects $b_i$ which has a normal distribution with mean $\\mathbf{0}$ and its variance matrix does not have to be specified. In a later section we will see that the assumption of normal distribution for $b_i$ can be relaxed. The vector (or matrix) $\\mathbf{0}$ denotes a vector (or matrix) of $0$'s with appropriate dimension(s). Let $\\sigma_k$ denote the standard deviation of $\\epsilon_i^k$. This model can be also considered as a mixture model since the marginal distribution of $\\log{(\\mathcal{Y}_i^k)}$ is a linear mixture of normal distributions (of $\\epsilon_i^k$) over another normal distribution (of $b_i$). Conditional on presence of taxon $k$, the parameter vector $\\beta^k$ quantifies average change in the abundance of taxon $k$ on log scale given one unit change in covariates contained in $X_i$. Notice that model (\\ref{eq:1b}) implies that \n\\begin{align*}\n&\\mu_i^k=\\beta^{0k}+X_i^T\\beta^k+W_i^T\\gamma^k,\\hspace{0.1cm}i=1,\\dots,N;\\hspace{0.1cm}k=1,\\dots,K+1,\\\\\n&\\Sigma_i=\\text{diag}(\\sigma_1^2,\\dots,\\sigma_{K+1}^2)+{\\bf 1}_{K+1}Z_i^TVar(b_i)Z_i {\\bf 1}_{K+1}^T,\\hspace{0.1cm}i=1,\\dots,N,\n\\end{align*}\nwhere $\\mu_i^k$ and $\\Sigma_i$ were defined in Section \\ref{ss:ZILoN}, $\\text{diag}(\\sigma_1^2,\\dots,\\sigma_{K+1}^2)$ is the diagonal matrix with $\\sigma_1^2,\\dots,\\sigma_{K+1}^2$ being the diagonal elements, $Var(b_i)$ is the variance matrix of the random effect $b_i$ and ${\\bf 1}_{K+1}$ is the $(K+1)-$dimensional vector of one's.\n\n\\section{Parameter estimation}\\label{s:estimation}\nOur target of inference is $\\beta^k,k=1,\\dots, K+1$. In real studies, the true taxa abundances in an ecosystem (e.g., gut), denoted by $\\mathcal{Y}_i$ previously, usually cannot be observed because only a small portion of the ecosystem (e.g., stool sample) is used to produce the sequencing data. So what can be observed for the $i$th subject and $k$th taxon is $Y_i^k=C_i\\mathcal{Y}^k_i$ where $C_i$ is the unknown proportion and takes value between 0 and 1. It is straightforward to see that $C_i$ is directly related to sequencing depth (i.e., library size). Let $Y_i=(Y_i^1,...,Y_i^{K+1})^T$ denote the observed vector. The unknown variable $C_i$ could cause at least two challenges, the first of which is its confounding effect \\citep{Weiss2017}. This can be seen by plugging the observed abundance $Y_i^k$ into equation (\\ref{eq:1b}) and the resulted equation becomes:\n\\begin{equation*}\n\\log(Y_i^k)=\\log(C_i)+\\log(\\mathcal{Y}_i^k)=\\log(C_i)+\\beta^{0k}+X_i^T\\beta^k+W_i^T\\gamma^k+Z_i^Tb_i+\\epsilon_i^k,\\hspace{0.2cm}k=1,...,K+1,\n\\end{equation*} \nwhere (log-transformed) $C_i$, as a covariate in the regression equation, could be a confounder for the association of (log-transformed) $Y_i^k$ with $X_i$ when $C_i$ is associated with $X_i$ which would be true if sequencing depth is associated with $X_i$. Without appropriately accounting for the effect of $C_i$, the estimate of $\\beta^k$ could be distorted toward overestimation which leads to high false positive rate or underestimation which leads to high false negative rate. The second challenge due to $C_i$ is data dispersion. It could be overdispersion or underdispersion depending on the distribution of $C_i$. For example, in the case that $\\mathcal{Y}^k_i$ and $C_i$ are independent (or weakly dependent), it is straightforward to show (See Appendix for proof) that\n\\begin{align}\\label{eq:dispersion}\n\\text{var}(C_i)\\Big(\\text{var}(\\mathcal{Y}^k_i)+\\big(E(\\mathcal{Y}^k_i)\\big)^2\\Big)\\leq \\text{var}(Y^k_i)\\leq E(C^2_i)\\Big(\\text{var}(\\mathcal{Y}^k_i)+\\big(E(\\mathcal{Y}^k_i)\\big)^2\\Big).\n\\end{align}\nOverdispersion happens because of the left-hand side of the inequality. For example, $\\text{var}(Y^k_i)$ will be larger than $\\text{var}(\\mathcal{Y}^k_i)$ when $E(\\mathcal{Y}^k_i)\\ge \\text{var}(\\mathcal{Y}^k_i)$ and $\\text{var}(C_i)E(\\mathcal{Y}^k_i)>1$. This could explain the enormous variation of total sequencing reads across subjects commonly observed in real studies. From the right-hand side of the above inequality, we can see that $\\text{var}(Y^k_i)$ could be much smaller than $\\text{var}(\\mathcal{Y}^k_i)$ when $E(C^2_i)$ is very small and severe underdispersion could happen in such cases. For example, when $E(C^2_i)$ is extremely small which implies that the value of $C_i$ is likely to be extremely small, $Y_i^k$ will take value close to zero and it will be difficult to observe positive abundance of $Y_i^k$ which can explain why there are so many 0's in real datasets and some taxa have nearly zero dispersion. \n\n\\subsection{Known reference taxon}\nWe propose a novel method that can handle both confounding and data dispersion issues caused by unknown $C_i$. This approach involves identifying an optimal reference taxon whose log-transformation is (conditionally) independent of the covariates of interest conditional on the presence of the taxon. For illustration, let's first assume that we know there is such a taxon independent of $X_i$ and it is set to be the reference taxon. Without loss of generality, we label this reference taxon as $K+1$. We will explain the case with unknown reference taxon later. By taking the log-ratio of a taxon, say taxon $k$, over the reference taxon, we have:\n\\begin{align*}\n\\log(Y_i^k\/Y_i^{K+1})&=\\log(Y_i^k)-\\log(Y_i^{K+1}) \\\\\n&=\\log(C_i)+\\log(\\mathcal{Y}_i^k)-\\log(C_i)-\\log(\\mathcal{Y}_i^{K+1} )\\\\\n&=\\beta^{0k}-\\beta^{0,K+1}+X_i^T(\\beta^k-\\beta^{K+1})+W_i^T(\\gamma^k-\\gamma^{K+1})+\\epsilon_i^k-\\epsilon_i^{K+1},\n\\end{align*}\nwhere $\\log(C_i)$ is canceled out, and thus the impact of the unobserved $C_i$ is limited in our model. Notice that $Z_i^Tb_i$ is also canceled out and thus the distribution of $b_i$ does not have to be specified and it can have a non-normal distribution. Because the (log) reference taxon is independent of $X_i$, we have $\\beta^{K+1}=\\mathbf{0}$. The above equation becomes:\n\\begin{equation}\\label{eq:lr}\n\\log(Y_i^k\/Y_i^{K+1})=\\beta^{0k}-\\beta^{0,K+1}+X_i^T\\beta^k+W_i^T(\\gamma^k-\\gamma^{K+1})+\\epsilon_i^k-\\epsilon_i^{K+1}.\n\\end{equation}\nFrom equation (\\ref{eq:lr}), we can see that the log-ratio transformed data can be used to estimate the re-parameterized parameter vector $((\\beta^{0k}-\\beta^{0,K+1})^T,(\\beta^k)^T,(\\gamma^k-\\gamma^{K+1})^T)^T$ from which the estimate of $\\beta^k$ can be extracted. Equation (\\ref{eq:lr}) also shows that $\\log(Y_i^k\/Y_i^{K+1})$ follows a normal distribution conditional on both $Y_i^k$ and $Y_i^{K+1}$ being non-zero because the two error terms $\\epsilon_i^k$ and $\\epsilon_i^{K+1}$ are independent and have normal distributions. Actually the vector $\\big(\\log(Y_i^1\/Y_i^{K+1}), \\log(Y_i^2\/Y_i^{K+1}),...,\\linebreak\\log(Y_i^K\/Y_i^{K+1})\\big)$ follows a multivariate normal distribution conditional on all $Y_i^k$'s, $k=1,...,K+1$, being non-zero. Notice that \n\\begin{align*}\n\\log(Y_i^k\/Y_i^{K+1})=\\log\\bigg(\\frac{Y_i^k}{\\sum_{j=1}^{K+1}Y_i^j}\\bigg\/\\frac{Y_i^{K+1}}{\\sum_{j=1}^{K+1}Y_i^j}\\bigg),k=1,...,K\n\\end{align*} \nThe right-hand side of the above equation is actually the ratio of the two compositional proportions for taxa $k$ and $K+1$. Taken together, the composition vector $\\bigg(\\frac{Y_i^k}{\\sum_{j=1}^{K+1}Y_i^j},...,\\frac{Y_i^{K+1}}{\\sum_{j=1}^{K+1}Y_i^j}\\bigg)$  follows a multivariate zero-inflated logistic normal (MZILN) distribution as described in \\cite{MZILN}. Therefore, the parameter vectors $\\beta^k,k=1,...,K$ can be estimated with the approach proposed in \\cite{MZILN} where standard regularization approaches such as LASSO \\citep{Tibshirani2011a}, MCP \\citep{Zhang2010a} and SCAD \\citep{Fan2001} for association selection, and high-dimensional inference approaches (\\citealp{Javanmard2014};\\citealp{Zhang2014};\\citealp{Cai2017};\\citealp{HDCI}) can be incorporated to provide valid point estimates and confidence intervals for the parameters. \n\n\\subsection{Unknown reference taxon}\nIn practice, we do not know which taxa are independent of which covariates. We will refer to those taxa independent of all covariates contained in $X_i$ as independent taxa and those taxa associated with any covariate in $X_i$ as associated taxa hereafter. We assume there are at least two independent taxa among all the taxa of interest. Later we will see that the independent taxon is not identifiable if there is only one such taxon. If we are able to identify an independent taxon, we can proceed with estimating the parameters as described in the previous section, and thus the task becomes to find an independent taxon that can be as the reference taxon. Taxa can be divided into two sets based on the association with $X_i$: we call the set of associated taxa (with any covariate in $X_i$) set A, and the set of independent taxa set B. It is unknown which taxon belongs to which set. So there are two possible scenarios for randomly selecting a reference taxon: it is either from set A or set B. It is obvious that $\\beta^k=\\mathbf{0}$ for taxa in set B, and thus the log-ratio of any two taxa in set B is independent of $X_i$. We also know that the log-ratio of any two taxa in set A is not independent of $X_i$ and the log-ratio between a taxon in set A and a taxon in set B is not independent of the covariates. So in an ideal setting with no noise, if the reference taxon is from set B for implementing the MZILN \\citep{MZILN} approach with MCP, then all taxa in set B should not be selected for the association (with any covariate in $X_i$) and all taxa in set A should be selected. On the other hand, if the reference taxon is from set A, then all taxa in sets A and B should be selected for the association. Let $m_A$ and $m_B$ denote the set sizes (number of taxa) for the two sets respectively. The set sizes $m_A$ and $m_B$ are unknown, but we know that  $m_A+m_B=K+1$ since there are $K+1$ taxa in total. If we were to run the MZILN approach with MCP $K+1$ times and each time we use a different taxon as the reference taxon, then each taxa in set B should be selected $m_A$ times for the association and each taxon in set A should be selected $K$ times. If $m_A$ and $K$ are very different, i.e., the difference $K-m_A=m_B-1$ is big, we can differentiate set A and set B by simply counting the times of each taxon being selected for the association with $X_i$. The approach will not be able to differentiate sets A and B if $m_B=1$ in which case $K-m_A=0$. This is why we need to assume there are at least two independent taxa. It is straightforward to see that the bigger $m_B$, the better for our approach. If cycling through all the taxa for choosing the reference taxon, it will be very time consuming to run the MZILN approach $K+1$ times since $K$ could be very large. A more effective approach is to randomly pick $R$ different reference taxa, say $R=40$, and then run the MZILN approach with each of the picked taxa as reference taxon. This way the MZILN is implemented only $R$ times. Each taxon in set B is expected to be selected $Rm_A\/(K+1)$ times for the association and each taxon in set A is expected to be selected $k_A$ times which can be calculated as follows:\n\\begin{align*}\nk_A=\\frac{{K\\choose R-1}}{{K+1\\choose R}}(R-1)+\\frac{{K\\choose R}}{{K+1\\choose R}}R=\\frac{R}{K+1}(R-1)+\\frac{K-R+1}{K+1}R=\\frac{KR}{K+1}\n\\end{align*}\nwhere $\\cdot\\choose \\cdot$ is the binomial coefficient function, and ${K\\choose R-1}\/{K+1\\choose R}$ and ${K\\choose R}\/{K+1\\choose R}$ are the probabilities of each taxon in set A being chosen as one of reference taxa and not chosen as one of reference taxa respectively. The mean difference of selection times will be $k_A-\\frac{Rm_A}{K+1}=\\frac{(m_B-1)R}{K+1}$. So $R$ should be chosen big enough for the difference $\\frac{(m_B-1)R}{K+1}$ to be detectable. For example, if it is expected that about half of the taxa should be independent of $X_i$ (i.e., $(m_B-1)\\approx (K+1)\/2$), then choosing $R=40$ will give a mean difference approximately of $1\/2\\times 40=20$ which could be big enough to differentiate sets A and B. However, it might be challenging to choose $R$ without knowing the true value of $m_B$ which could lead to unacceptable misclassification of set A. We propose to use a permutation test to control the family-wise error rate (FWER) which automatically controls false discovery rate (FDR) because FWER is always larger than or equal to FDR. More details are provided in the following algorithm to select taxa in association with $X_i$.\n\\begin{center}\n\\captionof{algorithm}{Association identification and parameter estimation}\\label{alg:overall}\n\\begin{algorithmic}[1]\n\\Require Family wise error rate (FWER) for taxa selection $\\alpha$, number of randomly picked reference taxa $R$, number of permutations $P$\n\\Algphase{Phase 1a - Association identification}\n\\Ensure To obtain the count of each taxon being selected for the association with $X_i$.\n\\State\\label{alg:taxa1}\nRandomly pick $R$ taxa as the reference taxa set. These taxa may or may not be associated with $X_i$. Let $(Y^{T_1},\\dots,Y^{T_R}), 1\\le T_1<T_2<\\dots<T_R\\le K+1$ denote these taxa.\n\\State\\label{alg:taxa2}\nSet $r=1$ and the initial $(K+1)-$dimensional count vector $Z=(0,...,0)$ where all elements are zero. \n\\State\\label{alg:taxa3}\nSet $Y^{T_r}$ as the reference taxon and implement the MZILN approach with MCP using the selected reference taxon. This gives sparse estimates of the parameters of interest: $\\hat\\beta^1,\\dots, \\hat\\beta^{K+1}$ where $\\hat\\beta^{T_r}=\\mathbf{0}$ because $Y^{T_r}$ is the reference taxon.\n\\State\\label{alg:taxa4}\nRecord the taxa selection with the vector $Z_r$ given by $Z_r=\\bigg(1_{(\\hat\\beta^1\\neq\\mathbf{0})},...,1_{(\\hat\\beta^{K+1}\\neq\\mathbf{0})}\\bigg)$ where $\\beta^k$'s are vectors when $Q>1$ and scalars when $Q=1$ which corresponds to the case when only one covariate is of interest for the association examination.\n\\State\\label{alg:taxa5}\nSet $r=r+1$ and $Z=Z+Z_r$, and then repeat steps \\ref{alg:taxa3} and \\ref{alg:taxa4} until $r$ reaches $R$ (e.g., $R=40$). The  vector $Z$ contains the count of each taxon being selected for the association with $X_i$.\n\\Algphase{Phase 1b - Association identification}\n\\Ensure Permutation to find a threshold to divide the counts in $Z$ in order to identify set A.\n\\State\\label{alg:permu1}\nSet $p=1$.\n\\State\\label{alg:permu2}\nRandomly permute the rows of the matrix consisting of only the $X$ covariates in the data set. \n\\State\\label{alg:permu3}\nRepeat steps \\ref{alg:taxa2}-\\ref{alg:taxa5} by using the same reference taxa set selected previously in Phase 1a, and then find the maximum value of the vector $Z$ and denote it by $C_p^m$.\n\\State\nSet $p=p+1$ and repeat the above steps \\ref{alg:permu2} and \\ref{alg:permu3} until $p$ reaches $P$ (e.g., $P=40$). And then find the $100(1-\\alpha)$th percentile of the vector $(C_1^m,...,C_P^m)$ and denote it by $C^\\alpha$ which is the threshold to differentiate sets A and B.\n\\State\\label{alg:permu4}\nThose taxa with counts in the vector $Z$ larger than or equal to $C^\\alpha$ belong to set A and the others will be considered to belong to set B. \n\\Algphase{Phase 2 - Parameter estimation}\n\\State\\label{alg:param1}\nPick an independent taxon in set B as the final reference taxon, for example, a taxon with the smallest count in vector $Z$. One can also establish some criteria (see Section \\ref{ss:criteria} in the Appendix for example) to choose a good independent reference taxon.\n\\State\\label{alg:param2}\nWith the chosen reference taxon from step \\ref{alg:param1}, implement MZILN along with a high-dimensional inference approach \\citep{HDCI} to obtain the final estimates and confidence intervals (CI) for $\\beta^1,\\dots,\\beta^{K+1}$. \n\\end{algorithmic}\n\\end{center}\n\n\\section{Simulation}\\label{s:simu}\n\\subsection{Association identification}\nExtensive simulations were carried out to assess the performance of our approach in comparison with five established existing approaches: ANCOM \\citep{ancom}, DESeq2 \\citep{deseq2}, edgeR \\citep{edge}, Wilcoxon rank sum test and ZIG \\citep{zig} where DESeq2 and edgeR are popular approaches for analyzing RNA-seq data and they can be generalized to analyze microbiome data \\citep{holmes,Weiss2017}. To demonstrate the robustness of our approach with respect to mis-specification of our model (\\ref{eq:1b}), the simulation data was generated under the same setting as in the paper that proposed the ANCOM approach \\citep{ancom}. The only change we made is that the variables $C_i$ become associated with the group assignment such that $C_i$ is a confounder of the association of interest. In our simulation, 100 data sets were generated. In each data set, there are 50 subjects divided into two groups with each group having approximately 25 subjects. This corresponds to a univariate covariate variable $X$ (ie, $Q=1$) following a Bernoulli distribution with the probability parameter being 0.5. $W$ is empty since there are no other covariates except the group variable in the model. $500$ taxa were generated in each data set and $25\\%$ are assumed to have different mean abundances across the two groups. The true taxa abundance of each taxon in group 1 was generated using a Poisson distribution with the Poisson mean parameter $\\lambda_j,j=1,...,500,$ generated from a gamma distribution $\\Gamma (a,1)$. The parameter $a$ has three possible values: $50$, $200$ and $10000$ to represent low, medium and high abundance taxa. To mimic a real data scenario, the data was generated such that $10\\%$ of the taxa had high abundance, $30\\%$ medium abundance, and $60\\%$ low abundance. For group 2, those taxa that have the same mean abundance as group 1 were generated with the same distribution as in group 1. Those taxa that have different means than group 1 were generated with Poisson distributions having means equal to $\\lambda_j+\\lambda_j^*$ where $\\lambda_j^*$ was the difference of mean between group 1 and 2 and generated from a uniform distribution over the interval $(u_1, u_2)$ which is chosen to be $(100, 150)$, $(200, 400)$ or $(10000, 15000)$ to represent low, medium and high difference respectively. Among those taxa that have different means between the two groups, $60\\%$, $30\\%$ and $10\\%$ were set to have low, medium and high differences respectively. The parameter values for $(\\lambda_j,\\lambda_j^*),j=1,...,500,$ were fixed for the data generation across the 100 data sets.\n\nAfter the true taxa abundance $\\mathcal{Y}_i^k, k=1,...,K+1$ was generated for each subject $i$ as described above, we still need to generate $C_i$ to obtain the observed abundance, $Y^k_i=[C_i\\mathcal{Y}_i^k]$ where $[\\cdot]$ means extracting the integer part of the number. The variable $C_i$ is allowed to be associated with the group variable which is the only difference between our setting and the setting in the ANCOM paper \\citep{ancom} where $C_i$ has the same distribution across the two groups. We set $C_i$ to be a constant value within each group for simplicity. Let $C^1$ and $C^2$ denote its values in groups 1 and 2 respectively. Five scenarios were considered: $\\text{Scenario 1:}\\hspace{0.2cm}(C^1=1\/30,C^2=1\/30)$, $\\text{Scenario 2:}\\hspace{0.2cm}(C^1=1\/30,C^2=1\/90)$, $\\text{Scenario 3:}\\hspace{0.2cm}(C^1=1\/18,C^2=1\/90)$, $\\text{Scenario 4:}\\hspace{0.2cm}(C^1=1\/9,C^2=1\/90)$ and $\\text{Scenario 5:}\\hspace{0.2cm}(C^1=1\/6,C^2=1\/90)$. We use the ratio $C^1\/C^2$ as a measure of the association between $C_i$ and the group variable $X$ and it is equal to 1, 3, 5, 10 and 15 for the five scenarios respectively. This ratio would be equal to the ratio of average library size if there are no difference in terms of total abundance between the two groups. So these ratios can cover a wide range of scenarios including very uneven (10X) library sizes between groups that have been studied in the literature \\citep{Weiss2017}. Notice that the strength of the association increases from Scenario 1 to 5 where Scenario 1 corresponds to no association (i.e., no confounding) and Scenario 5 has the strongest association (i.e., strongest confounding). We studied the performance of our approach and others under the five scenarios. Four indices were used to evaluate the performance: Recall, Precision, F1 and Type I error rate (Type1) that were calculated as follows:\n\\begin{align*}\n\\text{Recall}=\\frac{TP}{TP+FN}, \\hspace{0.5cm}\\text{Precision}=\\frac{TP}{TP+FP},\\hspace{0.5cm} \\text{F1}=\\frac{2}{\\frac{1}{\\text{recall}}+\\frac{1}{\\text{precision}}},\n\\hspace{0.5cm}\\text{Type1}=\\frac{FP}{FP+TN}\n\\end{align*}\nwhere $TP$, $FP$, $FN$ and $TN$ denote true positive, false positive, false negative and true negative respectively. Recall is a measure of statistical power, the higher the better. Precision has an inverse relationship with false discovery rate (FDR) which is equal to (1-Precision), and thus the higher the Precision, the lower the FDR. F1 is the Harmonic mean \\citep{Hmean} of Recall and Precision that measures the overall performance in terms of Recall and Precision. The targeted FDR level is set to be 20\\% for all approaches. When implementing IFAA, we choose the FWER to be $\\alpha=20\\%$ such that FDR$\\le$20\\%, the number of random reference taxa $R=40$ and the number of permutations $P=40$. For implementing the ANCOM approach, the stringent correction option was used in the ANCOM R package throughout this paper. \n\nWe plotted the four performance measures against confounding strength as shown in Fig.\\ref{simuComparison}. When there was no confounding effect (i.e., Scenario 1), all approaches had Precision rates (Fig.\\ref{simuComparison}B) above or around 80\\% with DESeq2 and edgeR having the lowest Precision rates of (79.2\\%, 76.8\\%) that translate to FDR of (20.8\\%, 23.1\\%) which were a little higher than the targed FDR of 20\\%. All approaches had good Recall rates ($>$92\\%) and good type I error rates ($<$0.1) with ANCOM and our approach (IFAA) having the smallest type I error rates when there was no confounding. As the confounding strength increases, Precision rates (Fig.\\ref{simuComparison}B) dropped dramatically for all approaches except IFAA. Although the Precision rate of IFAA dropped to 79\\% at Scenario 1, it stayed higher than 80\\% across all other scenarios and thus achieved the desired FDR of 20\\% even for Scenario 5 that had the strongest confounding effect of $C_i$. Precision rates of all other approaches dropped to below 67\\% at Scenario 2, below 47\\% at Scenario 3, below 43\\% at Scenario 4 and below 41\\% at Scenario 5 which translates to $>$59\\% FDR rate that almost tripled the desired FDR of 20\\%. ZIG and Wilcoxon rank sum test had the worst performance in terms of Precision rate which dropped to below 26\\% starting form Scenario 2 and that translates to FDR$>$74\\%. The Recall rate of IFAA (Fig.\\ref{simuComparison}A) dropped from 0.93 to 0.81 at Scenario 2 and further dropped to 0.72 and remained stable after departing from Scenario 2. F1 score, the measure of overall performance in terms of Recall and Precision, of IFAA (Fig.\\ref{simuComparison}C) had the best values in the presence of confounding and outperformed all the other approaches by a big margin starting from Scenario 3. DESeq2 ranked number 2 in terms of F1 score in the presence of confounding effects.  As the confounding strength increases, ZIG had the worst F1 score because of its lowest Precision rate and big drops of Recall rate at Scenarios 3 and 4. ZIG showed a strange behavior of Recall rate. Its Recall rate dropped to 42\\% at Scenatio 3 and then bounced back to 82\\% at Scenario 5. We also examined the type I error rate in relation with the confounding strength (Fig.\\ref{simuComparison}D). IFAA had the lowest type I error rate ($<$0.12) for all scenarios with confounding effects. All other approaches had highly inflated type I error rates as the confounding strength increases. Some even had type I error rate inflated to above 0.95 at Scenario 5 such as Wilcoxon rank sum test, ZIG and ANCOM. Type I error rates of DESeq2 and edgeR were inflated to 0.43 and 0.69 respectively at Scenario 5. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width =0.7\\textwidth,angle=0]{simuComparison.png}\n\\caption{Comparison with ANCOM, DESeq2, edgeR, Wilcoxon rank sum test and ZIG}\n\\label{simuComparison}\n\\end{figure}\n\n\\subsection{Parameter estimation}\nOnce sets A and B were identified, we chose a taxon from set B that had the smallest count in vector $Z$ as the final reference taxon to obtain parameter estimates in Phase 2 of the Algorithm. As far as we know, there is no existing approach that can provide association parameter estimates regarding AA, so we did not have any existing approaches to compare with. We checked the estimation bias of IFAA for those truly non-zero values of $\\beta^k,k=1,\\dots,K+1$ (see table \\ref{Tab:EP}). The true parameter value for $\\beta^k$ was calculated as the $E\\big(\\log(\\mathcal{Y}^k)|X=1,\\mathcal{Y}^k>0\\big)-E\\big(\\log(\\mathcal{Y}^k)|X=0,\\mathcal{Y}^k>0\\big)$. Results showed that the mean magnitude of all biases stayed fairly stable across all scenarios including the case with strongest confounding effect. The estimates were expected to be biased because model (\\ref{eq:1b}) was severely mis-specified in the data generation. This performance was not too bad given that the results were fairly robust with respect to different confounding effects.\n\n\\begin{table}[H]\n\\caption{Estimation performance}\\label{Tab:EP}\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|c|} \n \\hline\n Confounding strength&mean of true parameter values & mean magnitude of biases & Bias\\% \\\\\n \\hline\n 1 & 1.74 & 0.21 & 11.95 \\\\\n \\hline\n 3 & 1.74 & 0.24 & 14.06 \\\\\n \\hline\n 5 & 1.74 & 0.18 & 10.57 \\\\\n \\hline\n 10 & 1.74 & 0.18 & 10.11\\\\\n \\hline\n 15 & 1.74 & 0.20 & 11.55\\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Real study applications}\\label{s:app}\n\\subsection{New Hampshire Birth Cohort Study (NHBCS)}\nThe NHBCS is a large NIH-funded ongoing longitudinal epidemiological project to study the health impacts of environmental exposures such as arsenic in mothers and their children \\citep{Farzan2013}. Pregnant mothers were recruited to the study at approximately 24 to 28 weeks of gestational age and longitudinal data are collected from both mothers and babies at followed up time points. We applied our approach in the NHBCS study to examine the association between {\\it in utero} arsenic exposure measured by maternal urinary arsenic concentrations \\citep{Farzan2016} during pregnancy and the infant gut microbiome. In our analysis, the natural log-transformed total {\\it in utero} arsenic level \\citep{Nadeau2014} was the exposure variable $X$ and gut microbiome of infants at 6 weeks of age was the outcome variable. Delivery mode (vaginal VS. C-Section) and feeding type (Breast fed VS. others) were adjusted as potential confounders in the model (i.e., $W_i$ in equation (\\ref{eq:1b})). The gut microbiome data was measured in DNA extracted from infant stool samples using 16S rRNA sequencing of the V4-V5 hypervariable regions (\\citealp{Madan2016};  \\citealp{MZILN}). Sequencing reads were quality checked and clustered into operational taxonomic units as described previously \\citep{Madan2016}. After quality control and data cleaning, there were 182 subjects and 218 genera available in the data set. About 85\\% of the microbiome data points were zero. AA of genera were analyzed as the outcome variables. Our model found two genera: {\\it Collinsella} and {\\it Serratia} that were significantly associated with {\\it in-utero} arsenic concentrations. FWER was controlled at 30\\%, 40 permutations were used and 40 reference taxa were randomly chosen in Algorithm \\ref{alg:overall} (i.e., $\\alpha=0.30$, $P=40$, $R=40$). It took about 73 minutes to finish running the analysis on a 8-core Windows 10 machine. The regression coefficients estimated from IFAA were -1.17 and 1.06 respectively meaning that one unit increase on the log-scale of {\\it in-utero} arsenic exposure level would lead to 69\\% reduction in the absolute abundance of {\\it Collinsella} and 1.9-fold increase in the absolute abundance of {\\it Serratia} on average in the entire gut conditional on presence of these genera. The 95\\% CI calculated with a Bootstrap Lasso + Partial Ridge method \\citep{HDCI} for the regression coefficients were (-1.42, -0.10) and (-0.18, 0.79) respectively without multiple testing correction. While {\\it Collinsella} is an innovative finding, {\\it Serratia} has been linked to arsenic in the literature \\cite{Lukasz2014}. To give a full picture of all associations, a heatmap (Figure \\ref{heatNHBCS}) was also constructed to show the number of times each genus was selected for the association with arsenic level in Phase 1 of the algorithm. These selection counts can be considered as measures of the strengths of associations. For comparison, we analyzed the data with the ANCOM method as well. Since the ANCOM R package does not allow adjusting for potential confounders, the raw associations between the arsenic variable and the gut microbiome were tested using ANCOM. It did not find any genera that are statistically significantly associated with the arsenic variable at the same FDR rate of 30\\%. We also applied the nonparametric Spearman correlation for testing the raw correlations between the arsenic variable and RA and it did not identify any taxa which suggests that the signal-to-noise ratio in this dataset might be weak (which could be due to the high data sparsity with 85\\% zeros) since simple nonparametric tests tend to overidentify associated taxa but it did not detect any assoicated taxa in this dataset. The Spearman correlation test for correlations between the arsenic variable and AA did not result in any significant associations either. We did not compare with DESeq2, EdgeR and ZIG in this application because they were developed for differential abundance analysis between two groups whereas the exposure variable here, {\\it in-utero} arsenic level, is a continuous variable. \n\n \\begin{figure}[H]\n  \\begin{center}\n  \\includegraphics[width=0.9\\textwidth,angle=0]{Heatmap_NHBCS.png}\n  \\end{center}\n  \\caption{Assocation heatmap for the NHBCS study. Blue and orange denote positive and negative associations with arsenic level respectively. Selection count from Phase 1 of the algorithm determine the darkness of the colors for all genera. Negative sign means negative association. Absence is coded as 0. Genera selected less than 5 times are not included. Genera are labeled on the vertical axis and samples are labeled on the horizontal axis.}\n  \\label{heatNHBCS}\n\\end{figure}\n\n\\subsection{VSL\\#3 mouse model}\nVSL\\#3 is a commercially available probiotic cocktail (Sigma-Tau Pharmaceuticals, Inc.) of eight strains of lactic acid-producing bacteria. In a mouse model, Arthur et al. \\citep{Arthur2013} studied the ability of VSL\\#3 to alter the colonic microbiota and decrease inflammation-associated colorectal cancer when administered as interventional therapy after the onset of inflammation. In this study, there were totally 23 mice of which 10 were treated with VSL\\#3 and 13 served as control. Gut microbiome data were collected from stools at the end of the study with 16S rRNA sequencing \\citep{lzgMediation}. There were 362 OTUs in total in the data sets after quality control and data cleaning. About 40\\% of the OTU abundance data points were zero. In this application, we are interested in the association between the gut microbiome and the dysplasia score (the higher the worse) which is a continuous variable measuring the abnormality of cell growth. AA of OTUs were analyzed as the $Y$ variable in the model. The treatment variable was adjusted as a potential confounder for this association in the analysis (i.e., $W_i$ in equation (\\ref{eq:1b})). Again, FWER was controlled at 30\\%, 40 permutations were used and 40 reference taxa were randomly chosen in Algorithm \\ref{alg:overall} (i.e., $\\alpha=0.30$, $P=40$, $R=40$). It took about 125 minutes to finish the analysis on a 8-core Windows 10 machine. Two OTUs were found to be significantly associated with the dysplasia score with one OTU assigned to the kingdom Bacteria and and the other OTU assigned to family S24-7 within the order Bacteroidales. The regression coefficients for the two OTUs were -1.18 (95\\% CI: -1.04, -0.12) and -0.87 (95\\% CI: -1.75, -0.78) respectively where the CI's were calculated using the Bootstrap LPR method \\citep{HDCI}. The negative associations suggest that these OTUs are associated with reduced dysplasia score and, on average, one unit increase of the dysplasia score is associated with 65\\% and 58\\% reduction in the absolute abundance of the two OTUs in the entire gut conditional on the presence of these OTUs. These findings are consistent with associations of Bacteroidales and S24-7 with intestinal tumorigenesis reported in the literature  (\\citealp{Braten2017}; \\citealp{Rudi2017}). To give a full picture of all associations, a heatmap (Figure \\ref{heatMouse}) was also constructed to show the number of times each OTU was selected for the association with dysplasia score in Phase 1 of the algorithm. We applied the ANCOM approach to test the raw associations between the dysplasia score and microbiome since its R package does not allow adjusting for potential confounders. ANCOM did not identify any OTUs at the same FDR rate of 30\\%. The nonparametric Spearman correlation test identified 68 taxa AA which is likely to be an overidentification. When testing the correlations of RA with the dysplasia score using Spearman correlation test, 61 taxa RA were identified. Again, we did not compare with DESeq2, EdgeR and ZIG in this application because the dysplasia score a continuous variable. \n \\begin{figure}[H]\n  \\begin{center}\n  \\includegraphics[width =0.8\\textwidth,angle=0]{Heatmap_Mouse.png}\n  \\end{center}\n  \\caption{Assocation heatmap for the VSL\\#3 study. Selection count from Phase 1 of the algorithm determine the darkness of the colors for all OTUs. Negative sign means negative association. Absence is coded as 0. OTUs selected less than 15 times are not included in the figure. Taxonomic assignment is labeled on the vertical axis and the unlabeled OTUs belong to the taxon on its top. Samples are labeled on the horizontal axis.}\n  \\label{heatMouse}\n\\end{figure}\n\n\\section{Discussion}\\label{s:discu}\nWe developed a novel approach (IFAA) that can draw inferences directly on the absolute abundance (AA) of microbial taxa in an ecosystem and provide point estimates and confidence intervals for the associations of AA with other covariates. By making inference on AA, IFAA circumvents the issues induced by the features of RA such as the CD problem and the compositional structure. IFAA can also address the possible confounding effect of sequencing depth that has been a challenging problem in the literature of microbiome research \\citep{Weiss2017}. IFAA identifies microbial taxa associated with the covariates of interest (set A) and the other taxa that are not associated with the covariates (set B) with a desired false positive rate in set A in Phase 1 of the alrogithm where permutation method is used to control FDR by controlling FWER since FWER can serve as an upper bound of FDR. In Phase 2, a reference taxon from set B that is independent of the covariates is picked for the model to generated valid estimates of the associations of all taxa in set A with the covariates. When using IFAA, one does not need to impute zero sequencing reads with a Pseudocount or any other number for the analysis which can avoid bias caused by the imputation. Although imputation of zero-valued reads is not required, IFAA can still be directly applied to data sets containing imputed values for investigators who are comfortable with imputation. When there are no zeros in an imputed data set, it corresponds to the  zero-inflated log-normal distribution with $p_{12\\dots K+1}=1$ and all other masses are zero in the discrete part as shown in Section \\ref{ss:ZILoN}. Normalization methods such as rarefaction \\citep{Weiss2017} can also be allowed in our approach to normalize the data for analysis. IFAA can also be directly applied to RA data as well because the ratio of two RA's is the same as the their AA's. This could be helpful for investigators who want to draw inference on AA with RA data. Our approach can be applied to different settings including two-group comparisons and regressions with continuous exposure variables where confounders can be adjusted in the model. IFAA can handle high-dimensional microbiome data as well as high-dimensional covariates data by incorporating regularization methods. An R package to implement IFAA can be installed directly from the github website (https:\/\/github.com\/gitlzg\/IFAA). \n\nWe started with assuming normal distributions for the random errors $\\epsilon_i^k$, but this assumption is not required as long as the distributions have mean of zero because the parameters are estimated using estimating equations \\citep{MZILN}. This property ensures the robustness of our approach (as demonstrated in the simulation) for a broad range of distributions that could be encountered in practice under different study settings with different study populations. Although we did not study batch effect on the method in this paper, we expect it to have good performance in the presence of batch effects because the ratio of two taxa abundances does not depend on library size, and therefore it should generate robust results with respect to batch effects on library size. This is similar to the phenomenon of controlling for the confounding effect of library size as presented in the simulation study. Most existing approaches including those for RNA-seq and microarray data use a normalization procedure to deal with batch effects (\\citealp{Chen2011}; \\citealp{Ritchie2015}; \\citealp{Gibbons2018}) and some incorporate the batch effect adjustment in the regression model for final analysis \\citep{Dai2018}. We will study the performance of our approach in comparison with existing approaches in a future project.\n\nAnother implicit assumption, which is also needed in the ANCOM method \\citep{ancom}, is that the observed abundance $Y^k_i$ is equal to $C_i\\mathcal{Y}^k_i$ which might not be true in practice because $C_i\\mathcal{Y}^k_i$ is probably not an integer most of the time. This assumption is important for using the ratios of abundances because $C_i$ can be canceled out in the ratios under this assumption. What is observed in practice is an integer, so it might make more sense to assume that $Y^k_i=[C_i\\mathcal{Y}^k_i]$ where $[x]$ denotes extracting the integer part of $x$. However, it can be shown that the difference, $\\log(C_i\\mathcal{Y}^k_i)-\\log\\big([C_i\\mathcal{Y}^k_i]\\big)$ (given $[C_i\\mathcal{Y}^k_i]\\ge 1$), is bounded by $1\/[C_i\\mathcal{Y}^k_i]$ (see Appendix for proof), and thus the impact of this difference on the estimation of $\\beta^k$'s is likely to be limited since the estimation for $\\beta^k$'s is conditional on non-zero observation of the abundance. This paper focuses on studying the association of non-zero taxa with exposures. The presence\/absence analysis of the microbial taxa can be treated as nuisance to the analysis of non-zero taxa \\citep{MZILN} and warrants future research as a separate project. \n\nIFAA is flexible in terms of choosing the high-dimensional inference method in Phase 2 to obtain point estimates and confidence intervals for the parameters of interest. In this paper, we used a Bootstrap Lasso + Partial Ridge method \\citep{HDCI} that requires less assumptions and can be readily applied using the R package ``HDCI'', but many other such methods can be employed in Phase 2 as well including (\\citealp{Javanmard2014}; \\citealp{Zhang2014}; \\citealp{Cai2017}). It warrants further investigation to select an optimal high-dimensional inference approach in combination with MZILN in Phase 2 for analyzing microbiome data that have complex inter-taxa correlation structure. When there are more than one good independent reference taxa available in Phase 2 for parameter estimation, an alternative way to obtain the parameter estimates could be implementing the steps in Phase 2 for all good independent reference taxa one by one and then take the average of all estimates for the final estimates. This will likely generate more stable estimates at the cost of increased computational burden.\n\n\\section{Appendix}\n\\subsection{Proof for equation (3): the dispersion equation}\nWhen $C_i$ and $\\mathcal{Y}^k_i$ are independent (or weakly dependent), we prove the following equation:\n\\begin{align*}\n\\text{var}(C_i)\\Big(\\text{var}(\\mathcal{Y}^k_i)+\\big(E(\\mathcal{Y}^k_i)\\big)^2\\Big)\\leq \\text{var}(Y^k_i)\\leq E(C^2_i)\\Big(\\text{var}(\\mathcal{Y}^k_i)+\\big(E(\\mathcal{Y}^k_i)\\big)^2\\Big).\n\\end{align*}\n{\\bf Proof:} We first show the inequality on the right-hand side:\n\\begin{align*}\n&\\text{var}(Y^k_i)=\\text{var}(C_i\\mathcal{Y}^k_i)\\\\\n&=E(C^2_i(\\mathcal{Y}^k_i)^2)-E(C_i\\mathcal{Y}^k_i)^2\\\\\n&\\le E(C^2_i(\\mathcal{Y}^k_i)^2)=E(C^2_i)E((\\mathcal{Y}^k_i)^2)\\\\\n&=E(C^2_i)(\\text{var}(\\mathcal{Y}^k_i)+E(\\mathcal{Y}^k_i)^2)\n\\end{align*}\nFor the left-hand side, we have\n\\begin{align*}\n&\\text{var}(Y^k_i)=\\text{var}(C_i\\mathcal{Y}^k_i)\\\\\n&=E(C^2_i(\\mathcal{Y}^k_i)^2)-E(C_i)^2E(\\mathcal{Y}^k_i)^2\\\\\n&=E(C^2_i)E((\\mathcal{Y}^k_i)^2)-E(C_i)^2E(\\mathcal{Y}^k_i)^2\\\\\n&=E(C^2_i)E((\\mathcal{Y}^k_i)^2)-E(C^2_i)E(\\mathcal{Y}^k_i)^2+E(C^2_i)E(\\mathcal{Y}^k_i)^2-E(C_i)^2E(\\mathcal{Y}^k_i)^2\\\\\n&=E(C^2_i)\\text{var}(\\mathcal{Y}^k_i)+\\text{var}(C_i)E(\\mathcal{Y}^k_i)^2\\\\\n&\\ge\\text{var}(C_i)\\text{var}(\\mathcal{Y}^k_i)+\\text{var}(C_i)E(\\mathcal{Y}^k_i)^2\\\\\n&=\\text{var}(C_i)\\Big(\\text{var}(\\mathcal{Y}^k_i)+E(\\mathcal{Y}^k_i)^2\\Big)\n\\end{align*}\n\n\\subsection{Proof for the bound of the difference: $\\log(C_i\\mathcal{Y}^k_i)-\\log\\big([C_i\\mathcal{Y}^k_i]\\big)$}\nFor $[C_i\\mathcal{Y}^k_i]\\ge1$ which is the case we consider in the paper, let $\\delta=C_i\\mathcal{Y}^k_i-[C_i\\mathcal{Y}^k_i]$ and thus $\\delta\\in[0,1)$. We have\n\\begin{align*}\n0\\le\\log(C_i\\mathcal{Y}^k_i)-\\log\\big([C_i\\mathcal{Y}^k_i]\\big)&=\\log([C_i\\mathcal{Y}^k_i]+\\delta)-\\log\\big([C_i\\mathcal{Y}^k_i]\\big)\\\\\n&=\\log\\bigg(1+\\frac{\\delta}{[C_i\\mathcal{Y}^k_i]}\\bigg)\\\\\n&\\le\\frac{\\delta}{[C_i\\mathcal{Y}^k_i]}\\\\\n&<\\frac{1}{[C_i\\mathcal{Y}^k_i]}.\n\\end{align*}\nThe first inequality is because $\\log(1+x)\\le x$ for any non-negative number $x$. So the difference could become very small when the observed absolute abundance $[C_i\\mathcal{Y}^k_i]$ is large.\n\n\\subsection{Suggestive criteria for identifying the final reference taxon}\\label{ss:criteria}\nSince a final independent taxon is needed in Phase 2 of the algorithm to obtain parameter estimates, it might be helpful to have some criteria in place for finding a good independent taxon in set B. The following are some criteria that might be useful.\n\\begin{titlemize}{\\textit{Suggestive criteria for identifying the final reference taxon:}}\n\\item The final reference taxon has $10\\%$ or more non-zero abundances observed among those subjects who have two or more observed non-zero taxa.\n\\label{cri:1}\n\n\\item When making inference on the associations with a binary covariate, the final reference taxon has $10\\%$ or more non-zero abundances observed in each group indicated by the binary covariate among those subjects who have two or more observed non-zero taxa.\n\\label{cri:2}\n\n\\item The final reference taxon has a small (if not zero) count contained in the vector $Z$ as calculated in step \\ref{alg:taxa4} of Algorithm \\ref{alg:overall}. The first tertile of the counts for all taxa in set A can be used as the threshold for good independent reference taxa. The cut at first tertile can be customized depending on the distribution of the counts in vector $Z$.\n\\label{cri:3}\n\n\\item The final reference taxon has enough variation for observed abundances caused by the variation of library size. For example, a taxon with sequencing reads equal to 1 in all subjects is not a good final reference taxon because its variance is 0.\n\\label{cri:4}\n\\end{titlemize}\n\n\\begin{remark}\nThe first two criteria are only relevant when there are zero-valued sequencing reads. If the method is applied to data sets where all zeros have been imputed by a Pseudocount or another number, these two criteria are not needed.  \n\\end{remark}\n\n\\begin{remark}\nThe reasons we only consider ``subjects who have two or more observed non-zero taxa'' are because our approach is the based on the log-ratio transformation of the taxa abundance which requires at least two non-zero taxa to calculate a ratio.\n\\end{remark}\n\n\\section{Funding}\nThis work was supported in part by US NIH grants R01GM123014, UH3OD023275, P01ES022832, P20GM104416 and U.S. EPA grant RD 83544201.\n\n\\vspace{0.1cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{I. Introduction} \nMachine-learning has progressed significantly over the last decade\nin various areas of physical sciences~\\cite{Snyder_2012,Gabbard_2018, Mills_2018} \nafter some theoretical works in the area of neural networks (See~\\cite{Hornik_1989,Cybenko_1989} for examples.)\\\\\n\\indent\nIn fluid dynamics area~\\textcite{Ling_2016} presents a method of using deep neural networks to learn a model for the Reynolds stress anisotropy tensor from high-fidelity simulation data (see also~\\cite{Kutz_2018}). \n\\textcite{Gamahara_2017} uses an artificial neural network to find a new subgrid model of the \nsubgrid-scale stress in large-eddy simulation.\nBy using ``Long Short-Term Memory~(LSTM)''~\\cite{Hochreiter_1997}, \\textcite{Wan_2018} studies a data-assisted reduced-order modeling of extreme events in various dynamics including the Kolomogorov flow of the two-dimensional incompressible Navier--Stokes equation.\nSee also~\\textcite{Vlachas_2018} for the result on the barotropic climate model.\\\\\n\\indent \nIt is recently reported that reservoir computing, brain-inspired machine-learning framework that employs a data-driven dynamical system,\n is effective in the inference of a future such as time-series, frequency spectra \nand the Lyapunov spectra~\\cite{Verstraeten_2007,Inubushi_2017, Zhixin_2017, Pathak_2017,Ibanez_2018,Pathak_2018,Antonik_2018}.\n\\textcite{Pathak_2017} exemplifies using the Lorenz system and the Kuramoto-Sivashinsky system that the model obtained by reservoir computing can generate an arbitrarily long time-series whose Lyapunov exponents approximate those of the input signal. \\\\\n\\indent A reservoir is a recurrent neural network whose internal parameters are not adjusted to fit the data in the training process. \nWhat is done is to train the reservoir by feeding it an input time-series and fitting a linear function of the reservoir state variables \nto a desired output time-series. \nDue to this approach of reservoir computing we can save a great amount of computational costs, \nwhich enables us to deal with a complex deterministic behavior. \nThe framework was proposed as Echo-State Networks~\\cite{Jaeger_2001,Jaeger_2004} and Liquid-State Machines~\\cite{Maass_2002}.\\\\\n\\indent \nIt is known that an inference of a fluid flow is difficult but important in both physical and industrial aspects. \nIn this paper, we infer variables of a chaotic fluid flow by applying the method of reservoir computing without a prior knowledge of physical process. \\\\\n\\indent\nAfter introducing the method of reservoir computing in Section II and a fluid flow in Section III, \nwe explain how to apply the method to the inference of fluid variables, \nand show that inferences of both microscopic and macroscopic behaviors are successful in Sections IV and V, respectively. \nIn Section VI, we exemplify that a time-series inference of high-dimensional dynamics is possible by using  delay coordinates,  even when  the number of measurements is smaller than the Lyapunov dimension of the attractor. \nDiscussions and remarks are given in Section VII.\n\\section{II. Reservoir computing} \nReservoir computing is recently used in the inference of complex dynamics~\\cite{Zhixin_2017, Pathak_2017,Pathak_2018,Ibanez_2018,Lu_2018}. \nThe reservoir computing focuses on the determination of a translation matrix from reservoir state variables to variables to be inferred \n(see eq.~(\\ref{eq:output2})).\nHere we review the outline of the method \\cite{Jaeger_2004,Zhixin_2017}. We consider a dynamical system \n$$\\frac{d\\mathbf{\\phi}}{dt}=\\mathbf{f}(\\mathbf{\\phi}),$$\ntogether with a pair of $\\phi$-dependent, vector valued variables \n\\begin{equation*}\n\\mathbf{u}=\\mathbf{h}_1(\\mathbf{\\phi})\\in \\mathbb{R}^{M} \n~~\\text{and}~~\n\\mathbf{s}=\\mathbf{h}_2(\\mathbf{\\phi})\\in \\mathbb{R}^{P}.\\label{eq:input}\n\\end{equation*} \nWe seek a method for using the continued knowledge of \n$\\mathbf{u}$ to determine an estimate of $\\mathbf{s}$ as a function of time when direct measurement of $\\mathbf{s}$ \nis not available, which we call the {\\bf partial-inference}. \nWe also consider the {\\bf full-inference} for which we have a knowledge $\\mathbf{u}$ only for $t\\le T$.\nConcerning the algorithm, this is just a variant of the partial-inference~\\cite{Pathak_2017,Pathak_2018}, and will be explained later.\\\\\n\\indent \nThe dynamics of the reservoir state vector \n$$\\mathbf{r}\\in \\mathbb{R}^{N}~(N \\gg M),$$\n is defined by\n\\begin{equation}\n\t\\mathbf{r}(t+\\Delta t)=(1-\\alpha)\\mathbf{r}(t)+\\alpha \\tanh(\\mathbf{A}\\mathbf{r}(t)+\\mathbf{W}_{\\text{in}}\\mathbf{u}(t)\n\t),\\label{eq:reservoir}\n\\end{equation}\nwhere $\\Delta t$ is a relatively short time step.\nThe matrix $\\mathbf{A}$ is a weighted adjacency matrix of the reservoir layer, and the $M$-dimensional \ninput $\\mathbf{u}(t)$ is fed in to the $N$ reservoir nodes via a linear input weight matrix denoted by $\\mathbf{W}_{\\text{in}}$.\nThe parameter $\\alpha$ ($0<\\alpha\\le 1$) in eq.~(\\ref{eq:reservoir}) adjusts the nonlinearity of the dynamics of $\\mathbf{r}$, \nand is chosen depending upon the complexity of the dynamics of measurements and the time step $\\Delta t$. \\\\\n\\indent Each row of $\\mathbf{W}_{\\text{in}}$ has one nonzero element, chosen from a uniform distribution on $[-\\sigma,\\sigma]$.\nThe matrix $\\mathbf{A}$ is chosen from a sparse random \nmatrix in which the fraction of nonzero matrix elements is $(D_1+D_2)\/N$, \nso that the average degree of a reservoir node is $D_1+D_2$. \nThe $D_1$ non-zero components are chosen from a uniform distribution on $[-1, 1]$, and $D_2$ from that on $[-\\gamma, \\gamma]$ for $\\gamma~(\\ll 1)$, \nwhere $D_2$ non-zero components are introduced to reflect weak couplings among components of $\\mathbf{r}$. \nThen we uniformly rescale all the elements of $\\mathbf{A}$ so that the largest value of the magnitudes of its eigenvalues becomes $\\rho$. \\\\\n\\indent The output, which is a $P$-dimensional vector, is taken to be a linear function of the reservoir state $\\mathbf{r}$:\n\\begin{equation}\n\t\\Hat{\\mathbf{s}}(t)=\\mathbf{W}_{out}\\mathbf{r}(t)+\\mathbf{c}.\\label{eq:output}\n\\end{equation}\nThe reservoir state $\\mathbf{r}$ evolves following eq.~(\\ref{eq:reservoir}) with input \n$\\mathbf{u}(t)$, \nstarting from random initial state $\\mathbf{r}(-\\tau)$ whose elements are chosen from $(0, 1]$ in order not to diverge, \nwhere \n$\\tau\/\\Delta t~(\\gg 1)$  is the transient time.\nWe obtain $L=T\/\\Delta t$ steps of reservoir states $\\{\\mathbf{r}(l\\Delta t)\\}_{l=1}^{L}$ by eq.~(\\ref{eq:reservoir}). \nMoreover, we record  the actual measurements of the state variables $\\{\\mathbf{s}(l\\Delta t)\\}_{l=1}^{L}$. \\\\\n\\indent We train the network by determining $\\mathbf{W}_\\text{out}$ and $\\mathbf{c}$ \nso that the reservoir output approximates the measurement for $0< t \\le T$~(training phase), which is the main part of this computation.\nWe do this by minimizing the following quadratic form with respect to $\\mathbf{W}_\\text{out}$ and $\\mathbf{c}$:\n\\begin{equation}\n\\displaystyle\\sum^{L}_{l=1} \\|(\\mathbf{W}_\\text{out}\\mathbf{r}(l\\Delta t)+\\mathbf{c})-\\mathbf{s}(l\\Delta t)\\|^2\n+\\beta[Tr(\\mathbf{W}_\\text{out}\\mathbf{W}^{T}_\\text{out})],\\label{eq:minimize}\n\\end{equation}\nwhere $\\|\\mathbf{q}\\|^2=\\mathbf{q}^T \\mathbf{q}$ for a vector $\\mathbf{q}$,\nand the second term is a regularization term introduced to avoid overfitting \n$\\mathbf{W}_\\text{out}$ for $\\beta \\ge 0$.\n\\color{black}{}\nWhen the training is successful, $\\Hat{\\mathbf{s}}(t)$ should approximate the desired unmeasured quantity $\\mathbf{s}(t)$ for $t>T$~(inference phase). \nFollowing eq.~(\\ref{eq:output}), we obtain \n\\begin{equation}\n\t\\Hat{\\mathbf{s}}(t)=\\mathbf{W}^*_\\text{out}\\mathbf{r}(t)+c^*,  \\label{eq:output2}\n\\end{equation}\nwhere $\\mathbf{W}^*_\\text{out}$ and $c^*$ denote the solutions for the minimizers \nof the quadratic form~(\\ref{eq:minimize})~(see~\\cite{Lukosevicius_2009}~P.140 for details):\n\\begin{align*}\n\t\\mathbf{W}^*_\\text{out}&=\\delta\\mathbf{S}\\delta\\mathbf{R}^{T}(\\delta\\mathbf{R}\\delta\\mathbf{R}^{T}+\\beta\\mathbf{I})^{-1},\\\\\n\tc^*&=-[\\mathbf{W}^*_\\text{out}\\overline{\\mathbf{r}}-\\overline{\\mathbf{s}}],\n\\end{align*}\t\nwhere $\\overline{r}=\\sum^{L}_{l=1} \\mathbf{r}(l \\Delta t)$\/L, \n$\\overline{s}=\\sum^{L}_{l=1} \\mathbf{s}(l \\Delta t)\/L$,\nand $\\mathbf{I}$ is the $N \\times N$ identity matrix, $\\delta\\mathbf{R}$ (respectively, $\\delta\\mathbf{S}$) is the matrix \nwhose $l$-th column is $\\mathbf{r}(l\\Delta t)-\\overline{r}$ (respectively, $\\mathbf{s}(l\\Delta t)-\\overline{s}$).\\\\\n\\begin{table*}[htb]\n\\normalsize\n\t\t\\begin{tabular}{|l|l|r|r|r|} \n\t\t\t\\hline  \t\n           \t\t  \\multicolumn{2}{|c|}{parameter}& (a) & (b) & (c)\\\\ \\hline\n  \t\t\t~$\\tau$&transient time& 1000  &2500 &2350 \\\\ \\hline\n \t\t\t~$T$&training time&10000   &20000  &20000\\\\ \\hline\n  \t\t\t~$M$&dimension of measurements&270  &9 &36  \\\\ \\hline\n  \t\t\t~$P$&dimension of inferred variables &2 &9  &36 \\\\ \\hline\n\t  \t\t~$N$&number of reservoir nodes&6400 &3200 &3200\\\\ \\hline\n \t  \t\t~$D_1$&parameter of determining elements of $\\mathbf{A}$&60 &320 &120  \\\\ \\hline\n   \t\t\t~$D_2$&parameter of determining elements of  $\\mathbf{A}$&60 &0  &0\\\\ \\hline\n   \t\t\t~$\\gamma$&scale of input weights in $\\mathbf{A}$ &0.1 &0  &0\\\\ \\hline\n\t \t \t~$\\rho$&maximal eigenvalue of $\\mathbf{A}$ &1.0 &0.5 &0.5 \\\\  \\hline\n   \t\t\t~$\\sigma$&scale of input weights in $\\mathbf{W}_{in}$&0.4 &0.3 &0.5 \\\\ \\hline\n\t   \t\t~$\\alpha$&nonlinearity degree of reservoir dynamics&0.7 &0.3 &0.4  \\\\ \\hline\n   \t\t\t~$\\Delta t$&time step for reservoir dynamics &0.1 &0.25 &0.5 \\\\  \\hline\n \t\t\t~$\\beta$&regularization parameter &0 &0.01  &0.1\\\\ \\hline\n \\end{tabular}\n \t\t\\caption{{\\bf Sets of parameters for our reservoir computing.} The set (a) is used for the partial-inference of microscopic Fourier variables, whereas the set (b) is for the full-inference of macroscopic variables of energy functions and energy spectrum, \n \t\tand the set (c) is for the full-inference from only one measurement.\n \t\t}\n \t\t\\label{tab:parameter}\n\\end{table*}\n\\indent \nIn order to consider the effect of all the variables equally, we take the normalized value $\\tilde{X}(t)$ for each variable $X(t)$, which will be used throughout the whole procedure of our reservoir computing:\n  $$\\tilde{X}(t)=[X(t)-X_1]\/X_2,$$\nwhere $X_1$ is the mean value and $X_2$ is the variance. \nWhen  we reconstruct $X(t)$ in the inference phase from $\\tilde{X}(t)$, \nwe employ $X_1$ and $X_2$ obtained in the training phase.\n Due to the normalization we can avoid adjustments of $\\sigma$. \\\\\n\\begin{figure}[]\n \\includegraphics[width=1.\\columnwidth,height=0.705\\columnwidth]{fig1a.eps}\n  \\includegraphics[width=1.\\columnwidth,height=0.705\\columnwidth]{fig1b.eps}\n\\caption{ {\\bf \nPartial-inference of time-series of microscopic variables in Fourier space of a fluid flow.}\nFourier variables ${\\tilde{a}}_{\\eta_1=(1, 3,3,3)}$~(top) and ${\\tilde{a}}_{\\eta_2=(1, 2,3,4)}$~(bottom) are inferred \nby using measured variables $\\tilde{a}_{\\eta}$ for $\\eta \\in S$ as well as the \npast time-series data for all the measured variables $\\tilde{a}_{\\eta}$ for $\\eta \\in S\\cup\\{\\eta_1, \\eta_2\\}$.\nWe can observe that the inferred time-series almost coincide with the actual ones obtained by the direct numerical simulation of the Navier--Stokes equation even after sufficiently large time has passed since the training phase finished.\nThe inference errors in $l^1$-norm averaged over $t-T\\in[0,2000]$ are $1.8 \\%$ and $3.5 \\%$ \nfor $\\tilde{a}_{\\eta_1}$ and $\\tilde{a}_{\\eta_2}$, respectively.\n}\\label{fig:partial-micro}\n\\end{figure}\n\\section{III. Fluid flow} \nIn order to generate measurements of the reservoir computing, \nwe employ the direct numerical simulation of the incompressible three-dimensional Navier--Stokes equation \nunder periodic boundary conditions:\n\t\\begin{align*}\n  \\begin{cases}\n\t\t\\partial_t v -\\nu \\Delta v\n\t\t\t+(v \\cdot \\nabla) v+\\nabla \\pi=f,~\\nabla \\cdot v=0, ~\\mathbb{T}^3\\times(0,\\infty),\\\\\nv\\big| _{t=0}=v_0\\quad \\text{with $\\nabla \\cdot v_0=0$}, ~~~~~~~~~~~~~~~~~~~~\\mathbb{T}^3,\n  \\end{cases}\n\t\\end{align*}\nwhere ${\\mathbb{T}}=[0,2\\pi)$, $\\nu>0$ is viscosity parameter, $\\pi (x,t)$ is pressure, and $v(x,t)= (v_1(x,t),v_2(x,t),v_3(x,t))$ is velocity.\nWe use the Fourier spectral method~\\cite{ishioka_1999} with $N_0(=9)$ modes in each direction, meaning that the system is approximated by \n$2(2 N_0+1)^3~(=13718)$-dimensional ordinary differential equations (ODEs).\nThe ODEs are integrated by the 4th-order Runge--Kutta method, and the forcing is input into the low-frequency variables at each time step so as to preserve the energy of the low-frequency part.\nThat is, both the real and the imaginary parts of the Fourier coefficient of the vorticity $\\omega~(=\\text{rot}~v)$, \n$$\n\t\\mathcal{F}_{[\\omega_{\\zeta}]}(\\kappa,t):= \\dfrac{1}{(2\\pi)^3} \\displaystyle\\int_{\\mathbb{T}^3}\n\t\t\t\\omega_{\\zeta}(x, t)\n\t\te^{-i(\\kappa\\cdot x)}dx, \n$$\n are kept constant  for $\\zeta=1,2$, $\\kappa = (1,0,0), (0,1,0)$. \n We use  an initial condition, which  has energy only in the low-frequency variables. \nSee~\\cite{ishioka_1999} for the details.  \n\\section{IV. Partial-inference of microscopic variables: Fourier variables of velocity.}\nWe consider the absolute value of Fourier variables of velocity $\\mathcal{F}_{[v_{\\zeta}]}(\\kappa,t)$ as the representative microscopic variables:\n\t\\begin{align}\n\t\ta_{\\eta} (t)= \\left| \\mathcal{F}_{[v_{\\zeta}]}(\\kappa,t) \\right|\n\t\t:= \\left| \\dfrac{1}{(2\\pi)^3} \\int_{\\mathbb{T}^3}\n\t\t\tv_{\\zeta}(x, t)\n\t\te^{-i(\\kappa\\cdot x)}dx  \\right|, \\label{eq:fourier-coefficient}\n\t\\end{align}\n\twhere \n\t\t$\\eta = (\\zeta, \\kappa)\\in S_0:= \\{(\\zeta, \\kappa_1, \\kappa_2, \\kappa_3) \\in \\mathbb{Z}^4 |~ \\zeta\\in \\{1,2,3\\}, \\kappa_1, \\kappa_2, \\kappa_3 \\in [-N_0,N_0] \\}$.\nSince $v$ is real, $a_{(\\cdot, \\kappa_1, \\kappa_2, \\kappa_3)}=a_{(\\cdot, -\\kappa_1, -\\kappa_2, -\\kappa_3)}$. \nThe reason why we take the absolute value in eq.~(\\ref{eq:fourier-coefficient}) is to kill the rotational invariance of a complex variable and to make an inference possible.\nWe  choose a chaotic parameter $\\nu=0.05862$, \nand set $\\mathbf{u}(t)$ as the time-series of $M=270$ Fourier variables $\\tilde{a}_{\\eta}$, \nwhere $\\eta \\in S:= \\{(2, \\pm  \\kappa_1, \\kappa_2, \\kappa_3) \\in \\mathbb{Z}^4 |~1 \\leq \\kappa_1 \\leq N_0, \\kappa_1 \\leq \\kappa_2 \\leq \\kappa_3 \\leq \\kappa_1+4 \\}$ and each component is taken $\\bmod~N_0$, \n that is, \n\t\\begin{align*}\n\t\\mathbf{u}(t)&=(\\{ \\tilde{a}_{\\eta} \\}_{\\eta \\in S})^{t}.\n\t\\end{align*}\nWe also set \n\t\\begin{align*}\n\t\\mathbf{s}(t)&=(\\tilde{a}_{(1, 3,3,3)}, \\tilde{a}_{(1, 2,3,4)})^{t},   \n\t\\end{align*}\nwhere $(1, 3,3,3),  (1, 2,3,4) \\notin S$. \nUnder the set of parameters in TABLE \\ref{tab:parameter}~(a)\nwe infer the time-series $\\mathbf{s}(t)$,  \nwhich is successful for quite a long time (see Fig.~\\ref{fig:partial-micro}).\\\\\n\\indent \nThe choice of variables to be trained is not very significant in this study, because the attractor does not show a homogeneous \nisotropic turbulence, and it has less symmetries.\nWe can see from the Poincar\\'e section of the microscopic variables that the flow is not isotropic and \nindeterminacy in inference due to the continuous symmetry does not appear. \nHowever, by training variables with different types of behaviors, we can construct a reservoir model \nin less computational costs with lower dimension $N$ of the reservoir system. \nIn fact, we confirmed that we can infer some other fluid variables including both low-frequency and high-frequency variables \nfrom some other training variables.\nWe found that an inference of a high-frequency variable tends to be more difficult, maybe because of the stronger intermittency.\nRemark that $D_2$ is useful to represent non-local relatively weak interactions among microscopic variables in the partial inference. \n\\begin{figure}\n\\includegraphics[width=1.0\\columnwidth,height=0.705\\columnwidth]{fig2a.eps}\n\\includegraphics[width=1.0\\columnwidth,height=0.705\\columnwidth]{fig2b.eps}\n\\includegraphics[width=1.0\\columnwidth,height=0.705\\columnwidth]{fig2c.eps}\n\\caption{ {\\bf \nFull-inference of time-series of macroscopic variables of a fluid flow.}\nTime-series of energy function ${\\tilde{E}}(k,t)$ for $k=4$ (top) and $9$ (middle)  are inferred from the reservoir system \nin comparison with that of a reference data obtained by the direct numerical simulation of the Navier--Stokes equation.\nThe inference error defined by $\\varepsilon(t)=\\sum^{N_0}_{k=1}|\\tilde{E}(k,t)-\\hat{\\tilde{E}}(k,t)|\/N_0$ $(N_0=9)$ is shown to grow exponentially with time up to $t-T=100$ (bottom), which is inevitable for a chaotic behavior of a fluid flow.\nThe growth of error within a short time highly depends on the direction of the perturbation vector\n $\\{\\tilde{E}(\\cdot,T+\\Delta t)-\\hat{\\tilde{E}}(\\cdot,T+\\Delta t)\\}$, \n \\color{black}{}\n and its slope can vary in different settings.\n}\\label{fig:full-macro}\n\\end{figure}\n\\begin{figure}\n   \\includegraphics[width=1.0\\columnwidth, height=0.705\\columnwidth]{fig3.eps}\t\n\\caption{\n{\\bf \nEnergy spectrum $\\overline{E}(k)$ reproduced from the reservoir computing.}\nThe spectrum is obtained from the full-inference of an energy function $E(k,t)$, which is compared with that for a reference data obtained by the direct numerical simulation of the Navier--Stokes equation.\nThe coincidence of the two energy spectra implies that the reservoir system captures the dynamics of a fluid flow in statistical sense, \neven after the time-series inference has failed due to the chaotic property (see Fig.~\\ref{fig:full-macro}). \nThe Kolmogorov $-5\/3$ law of the energy spectrum is shown as a reference.\nThe relative error of inferred variable $\\overline{\\hat{E}}(k)$ from $\\overline{E}(k)$ $(k=1,\\cdots,9)$ is up to $1.3\\%$. \n}\\label{fig:full-macro-average}\n\\end{figure}\n\\section{V. Full-inference of macroscopic variables: Energy function and Energy spectrum} \nWe study an energy function as the representative of a macroscopic variable. \nWe set $\\nu=0.058$ for which the flow is more turbulent than the previous case.\nHowever, the complexity of the dynamics is much less \nthan that for a microscopic variable for the same viscosity.  \nThis is because the energy function can be thought of as an averaged quantity of many microscopic variables. \nThe energy function $E_0(k, t)$ for wavenumber $k \\in \\mathbb{N}$ is defined by \n\t\\begin{align*}\n\t\tE_0(k, t):= \\dfrac{1}{2} \\int_{D_k} {\\sum_{\\zeta=1}^{3}}\n\t\t\t\\left| \n\t\t\t\t\\mathcal{F}_{[v_{\\zeta}]}(\\kappa, t)\n\t\t\t\\right|^2  d\\kappa, \n\t\\end{align*}\nwhere $D_k:=\\{ \\kappa \\in \\mathbb{Z}^3| k-0.5 \\leq |\\kappa| < k+0.5 \\}$. \nSee eq.~(\\ref{eq:fourier-coefficient}) for the expression of $\\mathcal{F}_{[v_{\\zeta}]}(\\kappa, t)$.\nIn order to get rid of the high-frequency fluctuation, we take the\nshort-time average \n$$E(k,t)=\\sum_{s=t-99\\Delta s}^{t}E_0(k,s)\/100,$$ \nwhere $\\Delta s=0.05$ is the time step of the integration of the Navier--Stokes equation.  \nThis helps us to obtain essential low-frequency dynamics of an energy function and infer its time-series \nwith less computational costs with lower dimension $N$ of the reservoir vectors. \nThe averaged energy function $E(k,t)$ will be called an energy function hereafter.\\\\\n\\indent In the training phase for $t \\in (0,T]$, $\\mathbf{W}^{*}_\\text{out}$ and $c^{*}$ are determined by settin\n\t\\begin{align*}\n\t\\mathbf{u}(t)&=(\\tilde{E}(1,t), \\tilde{E}(2,t),\\cdots,\\tilde{E}(9,t))^{t},  \\\\\n\t\\mathbf{s}(t)&=(\\tilde{E}(1,t), \\tilde{E}(2,t),\\cdots,\\tilde{E}(9,t))^{t}, \n\t\\end{align*}\nand by following the same procedure as the partial-inference.\nIn the inference phase for $t>T$, eq.(\\ref{eq:reservoir}) is written as \n\\begin{equation*}\n\\mathbf{r}(t+\\Delta t)=(1-\\alpha)\\mathbf{r}(t)+\\alpha \\tanh(\\mathbf{A}\\mathbf{r}(t)+\\mathbf{W}_{\\text{in}}\\Hat{\\mathbf{s}}(t)),\\label{eq:full-reservoir}\n\\end{equation*}\nby setting $\\mathbf{u}(t)$ as\n\t\\begin{align*}\n\t\\Hat{\\mathbf{s}}(t)=(\\hat{\\tilde{E}}(1,t), \\hat{\\tilde{E}}(2,t),\\cdots,\\hat{\\tilde{E}}(9,t))^{t} \n\t\\end{align*}\n\t obtained from eq.~(\\ref{eq:output2}).\nA set of parameters employed here is shown in TABLE \\ref{tab:parameter}~(b). \\\\\n\\indent We found that an inference of energy functions is successful \nfor some time after finishing training $9$-dimensional time-series data of energy functions. \nThe two cases for $\\tilde{E}(4,t)$ and $\\tilde{E}(9,t)$ are shown in Fig.~\\ref{fig:full-macro}~(top)(middle). \nThe failure in the long-term time-series inference is inevitable just due to the sensitive dependence on initial condition of a chaotic property of the fluid flow. \nIn fact, the growth rate of error in the energy functions is shown to be exponential for $t-T\\lesssim 100$ in Fig.~\\ref{fig:full-macro}~(bottom).\nHowever, the energy spectrum $\\overline{E}(k)=\\langle E(k,t) \\rangle$, the time average of an energy function $E(k,t)$,  can be reproduced from the inferred time-series data for $1000<t-T<2000$ (Fig.~\\ref{fig:full-macro-average}).\nThis implies that the reservoir system constructed without the knowledge of microscopic variables captures statistical property correctly,\nand that the obtained system can be understood as a chaotic dynamical system describing a behavior of energy functions.\\\\\n\\indent\n\\section{VI. Full-inference of Macroscopic variable from only one measurement using delay coordinates} \nIn various experiments and observations of high-dimensional complex phenomena, \nthere are usually much smaller number of measurements than the Lyapunov dimensions of the attractor.\nEven in such cases we can infer a time-series data by generating high-dimensional input data $\\mathbf{u}$ for the reservoir computation through the delay-coordinate embedding method~\\cite{Takens,Embedology}. \\\\\n\\indent\nHere we exemplify a full-inference of an energy function $E(4,t)$ for the same flow as in Section V, \nby assuming that the accessible measurement is limited to only one variable $E(4,t)$ \namong $9$ measurements $E(k,t)~(k=1,\\cdots,9)$ used in Section V.\nIn order to overcome  the lack of sufficiently large number of  measurements,\nwe introduce $36$-dimensional delay-coordinate function  with a time delay  $\\Delta\\tau=2.5$, that is, \n\\begin{align*}\n\\mathbf{u}(t)&=(\\tilde{E}(4,t), \\tilde{E}(4,t-\\Delta\\tau),\\cdots,\\tilde{E}(4,t-35\\Delta\\tau))^{t}, \\\\\n\\mathbf{s}(t)&=(\\tilde{E}(4,t), \\tilde{E}(4,t-\\Delta\\tau),\\cdots,\\tilde{E}(4,t-35\\Delta\\tau))^{t} .\n\\end{align*}\nAn inferred time-series of $\\tilde{E}(4,t)$ is shown in Fig.~\\ref{fig:energy_function_delay}, which is as successful \nas the case when there are $9$ measurements in Fig.~\\ref{fig:full-macro}~(top). \nA set of parameters employed here is shown in TABLE \\ref{tab:parameter}~(c). \\\\\n\\begin{figure}[]\n \\includegraphics[width=1.\\columnwidth,height=0.705\\columnwidth]{fig4a.eps}\n  \\includegraphics[width=1.\\columnwidth,height=0.705\\columnwidth]{fig4b.eps}\n\\caption{ \n{\\bf \nFull-inference of a macroscopic variable using the delay coordinates of only one measurement.\n\nWe infer an energy function ${\\tilde{E}}(4,t)$ for the same time range as in Fig.~\\ref{fig:full-macro}~(top)\nfrom only one measurement ${\\tilde{E}}(4,t)$. \nThe inferred time-series of  ${\\tilde{E}}(4,t)$ is shown together with a reference data obtained \nby the direct numerical simulation of the Navier--Stokes equation~(top).  \nErrors for the inference $\\varepsilon_1(t)=\\sum^{35}_{n=0}|\\tilde{E}(4,t-n \\Delta \\tau)-\\hat{\\tilde{E}}(4,t-n \\Delta \\tau)|^2\/36$  and $\\varepsilon_2(t)=|\\tilde{E}(4,t)-\\hat{\\tilde{E}}(4,t)|$ \nare shown~(bottom).\n}\\label{fig:energy_function_delay}\n\\end{figure}\n\\section{VII. Discussion and remarks}\nWe have succeeded in inferring time-series of both microscopic and macroscopic variables of a \nthree-dimensional \nfluid flow by machine-learning technique using reservoir computing.  \nThe method is especially useful in generating an arbitrarily long time-series data of macroscopic variables as well as a statistical property \n with small computational costs. \n That is, in order to generate a time-series data of a macroscopic variable of a fluid flow, \nwe do not need to refer microscopic behaviors. \nIt takes roughly $1\/80$ of time to obtain a time-series of the energy functions $E(k)$ with the same time-lengths, when we use the model constructed by the reservoir computation. The Navier--Stokes equation is calculated by 13718-dimensional ODEs with the $4$-stage Runge--Kutta method (time step $0.05$), whereas the model is calculated by 3200-dimensional map whose iterate corresponds to the time step $0.25$. \\\\\n\\indent The difficulty in the construction of a reservoir model can vary mainly depending on the viscosity $\\nu$. \nAs the degree of turbulence increases by decreasing $\\nu$,  longer training time $T$ and higher dimension $N$ of the reservoir state vector $\\mathbf{r}\\in \\mathbb{R}^{N}$ are required. However, for macroscopic variables the construction is relatively easy, even when the flow is turbulent.\nBecause the degree of instability of a macroscopic behavior is relatively low in comparison with that of a microscopic behavior. \\\\\n\\indent It is expected that our procedure will work, even if a high-frequency noise is added to the training data, because even in our current computation we have applied a low-pass filter for the inference of macroscopic variables.\nAlthough our approach focuses on constructing a model for a fluid flow with a fixed parameter $\\nu$, it will be very interesting to consider a framework of the construction of a model with a parameter. \\\\\n\\indent When we do numerical computation of the Navier--Stokes equation, we employ some discretized expressions using \nFourier spectrum method, finite difference method and finite element method. \nThe obtained reservoir system constructed from data can be understood as one of such expressions, describing a macroscopic (or a microscopic) dynamics of a  fluid flow.\\\\\n\\indent It is known that there is a difficulty in obtaining a closed form equation of macroscopic behavior of a fluid flow \nfrom the Navier--Stokes equation analytically, so called a ``closure problem''. That is, in order to express the dynamics of the $n$-th moment variables, the $n+1$-th moment variables are required for any positive integer $n$.\nOur study on the data-driven modeling may give us insights on this kind of problem.\nFor a relatively large value of $\\nu$ considered in our paper, $\\{E(k)\\}_{k=1}^{K}$ seems to be enough for\nrepresenting the dynamics of $E(k)$, whereas $\\{E(k)\\}_{k=1}^{K}$ will not be enough \nfor more turbulent case with a smaller value of $\\nu$, even if $K$ is chosen large enough. \nIn such a case time-delay variables can be used for generating high-dimensional input data as are used in Section VI. \n\\section{Acknowledgements} \nWe would like to thank anonymous referees for the critical reading of the manuscript and giving us some insightful comments to improve our paper.\nKN was supported by the Leading Graduate Course for Frontiers of Mathematical Sciences and Physics (FMSP) at the University of Tokyo.\nYS was supported by the JSPS KAKENHI Grant No.17K05360 and JST PRESTO JPMJPR16E5.\nPart of the computation was supported by the Collaborative Research Program for Young $\\cdot$ Women Scientists of ACCMS and IIMC, Kyoto University.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet us consider some smooth manifold $M$ with coordinates $x_1,\\ldots,x_m$ and~a~dynamical system\ndef\\\/ined by the following equations of motion\n\\begin{gather*}\n\\dot{x}_i=X_i,\\qquad i=1,\\ldots,m.\n\\end{gather*}\nWe can identify this system of ODE's with the vector f\\\/ield\n\\begin{gather*}\nX=\\sum X_i\\dfrac{\\partial}{\\partial x_i},\n\\end{gather*}\nwhich is a~linear operator on a~space of the smooth functions on $M$ that encodes the\ninf\\\/initesimal evolution of any quantity\n\\begin{gather*}\n\\dot{F}=X(F)=\\sum X_i\\dfrac{\\partial F}{\\partial x_i}.\n\\end{gather*}\nIn Hamiltonian mechanics one of the fundamental axiom is what we can call the energy paradigm that\ncan be stated as follows:\n``For every mechanical system there is a~function def\\\/ined on its space of states, called\nmechanical energy or Hamiltonian $H$ of the system, containing all its dynamical information''.\n\nAccording to this paradigm any function $H$ on $M$ generates vector f\\\/ield $X$ describing\na~dynamical system\n\\begin{gather*}\nX=X_H=PdH.\n\\end{gather*}\nHere $dH$ is a~dif\\\/ferential of $H$, and~$P$ is a~bivector on the phase space $M$.\nBy adding some other assumptions we can prove that $P$ is a~Poisson bivector.\nIn fact, it is enough to add energy conservation\n\\begin{gather*}\n\\dot{H}=X_H(H)=(PdH,dH)=0\n\\end{gather*}\nand compatibility of dynamical evolutions associated with two functions $H_{1,2}$\n\\begin{gather*}\nX_{H_1}(X_{H_2}(F))=X_{H_2}(X_{H_1}(F))+X_{X_{H_1}(H_2)}(F),\n\\end{gather*}\nsee~\\cite{ib96,jo64} and references therein.\n\nIn bi-Hamiltonian mechanics~\\cite{mag97} we are looking for another decomposition of the given\nvector f\\\/ield $X$,\n\\begin{gather*}\nX=PdH=f_1P'dH_1+\\cdots+f_mP'dH_m,\n\\end{gather*}\nby commuting Hamiltonian vector f\\\/ields\n\\begin{gather*}\nX_k=P'dH_k\n\\end{gather*}\ngenerated by integrals of motion $H_1,\\ldots,H_m$ and~some Poisson bivector $P'$ compatible with\n$P$.\nThese Poisson bivectors $P'$ can be divided into two groups of trivial and~nontrivial deformations\nof canonical Poisson bivector $P$, see~\\cite{al11,mag05,lih77,mag97,ts12a} and~references therein.\n\nSupposing that $P'$ is a~trivial deformation of canonical Poisson bivector $P$ completely def\\\/ined\nby Hamiltonian $H$ we can join the geometry of the phase space $M$ and~the energy paradigm.\nIn this case, second Poisson bivector\n\\begin{gather}\n\\label{poi-2}\nP'=\\mathcal L_{Y} P\n\\end{gather}\nis a~Lie derivative of the canonical bivector $P$ along the Liouville vector f\\\/ield\n\\begin{gather}\n\\label{y1}\nY=\\operatorname{Ad} H,\n\\end{gather}\nwhere $A$ is a~2-tensor f\\\/ield acting on the dif\\\/ferential of the Hamiltonian.\nRemind, that the Lie derivative $P'$~\\eqref{poi-2} is a~trivial deformation because it is\n2-coboundary and~simultaneously 2-cocycle in the Poisson--Lichnerowicz cohomology def\\\/ined by\ncanonical Poisson bivector $P$~\\cite{mag05,lih77}.\n\nThe main aim of this note is to show 2-tensor f\\\/ields $A$ associated with some well-known\nintegrable systems and~to prove that these tensor f\\\/ields may be useful to construct new\nintegrable systems.\nFor example, we discuss a~new trivial family of integrable noncommutative three dimensional\nsystems, which includes deformations of the rational Calogero--Moser system with three particle\ninteraction.\n\n\\section{Integrable systems on cotangent bundles}\n\\label{section2}\n\nLet us consider canonical Poisson bivector on the symplectic manifold $M=T^*Q$\n\\begin{gather}\n\\label{can-p}\nP=\\sum_{i=1}^n \\frac{\\partial}{\\partial q_i}\\wedge \\frac{\\partial}{\\partial p_i},\n\\end{gather}\nwhich is the one mostly used in Hamiltonian mechanics~\\cite{am78,arn89}.\nHere $q_i$ are local coordinates describing a~point $q$ on a~smooth manifold $Q$, and~$p_i$, the\ncanonical conjugate momenta, are local coordinates describing covectors on such manifold,\ni.e.\\;points $p$ in the cotangent bundle $T^*Q$ of $Q$.\nThe corresponding Poisson bracket looks like\n\\begin{gather}\n\\label{can-br}\n\\{p_i,q_j\\}=\\delta_{ij},\\qquad \\{q_i,q_j\\}=\\{p_i,p_j\\}=0.\n\\end{gather}\n\nIn local coordinates $x$ on $M$ the Lie derivative of a~bivector $P$\nalong a~vector f\\\/ield $Y$ reads as\n\\begin{gather*}\n\\bigl(\\mathcal L_{Y}P\\bigr)_{ij}=\\sum\\limits_{k=\n1}^{\\dim M}\\left(Y_k\\dfrac{\\partial P_{ij}}{\\partial x_k}\n-P_{kj}\\dfrac{\\partial Y_i}{\\partial x_k}-P_{ik}\\dfrac{\\partial Y_j}{\\partial x_k}\\right)\n\\end{gather*}\nand the Schouten bracket $[A,B]$ of two bivectors $A$ and~$B$\nis a~trivector with the following entries\n\\begin{gather}\n\\label{sh-br}\n[A,B]_{ijk}=-\\sum\\limits_{m=1}^{\\dim M}\\left(B_{mk}\\dfrac{\\partial A_{ij}}{\\partial x_m}\n+A_{mk}\\dfrac{\\partial B_{ij}}{\\partial x_m}+\\mathrm{cycle}(i,j,k)\\right).\n\\end{gather}\nIn our case $M=T^*Q$, $\\dim M=2n$ and~$x=(q,p)$.\n\nLet us consider natural Hamilton functions on $M=T^*Q$\n\\begin{gather}\n\\label{nat-h}\nH=T(q,p)+V(q),\n\\end{gather}\nwhich are the sum of the geodesic Hamiltonian $T$ and~potential energy $V(q)$.\nAccording to~\\cite{gts11,mpt11,ts10,ts11s}, for natural Hamiltonians there is other representation\nfor the vector f\\\/ield $Y$~\\eqref{y1}\n\\begin{gather*}\nY=\\left(\n\\begin{matrix}\n\\Lambda&0\n\\\\\n0&\\Pi\n\\end{matrix}\n\\right)\\left(\n\\begin{matrix}\ndq\n\\\\\ndp\n\\end{matrix}\n\\right).\n\\end{gather*}\nHere we use matrix notation of tensor objects in which, for instance, canonical Poisson bi\\-vec\\-tor~$P$~\\eqref{can-p} looks like\n\\begin{gather}\n\\label{can-p2}\nP=\\left(\n\\begin{matrix}\n0 & E\n\\\\\n-E & 0\n\\end{matrix}\n\\right).\n\\end{gather}\nHere $E$ is a~unit matrix.\n\nIf $\\Pi=0$ and~$\\Lambda$ is the conformal Killing tensor of gradient type or Yano--Killing tensor on~$Q$, one gets second Poisson bivector $P'=\\mathcal L_YP$\nassociated with Hamilton functions separable in orthogonal coordinate systems on~$Q$.\nIn this case eigenvalues of the\nthe Nijenhuis operator\n\\begin{gather}\n\\label{rec-op}\nN=P'P^{-1},\n\\end{gather}\nwhich is also called the hereditary or recursion operator, are variables of separation.\nIn order to get integrals of motion $H_k$ from $N$\nwe have to extend the initial phase space~\\cite{imm00} or to f\\\/ix separated relations~\\cite{ts10}.\n\nThe new idea is that we can substitute the arbitrary Hamilton function $H$~\\eqref{nat-h}\nand the 2-tensor f\\\/ield $A$ into the def\\\/inition~\\eqref{poi-2} and\ntry to f\\\/ind the Poisson bivectors $P'$ solving the equation\n\\begin{gather}\n\\label{sch-eq}\n[P',P']=[\\mathcal L_{\\operatorname{Ad} H}P,\\mathcal L_{\\operatorname{Ad} H}P]=0,\n\\end{gather}\nwhere $[\\cdot,\\cdot]$ is the Schouten bracket def\\\/ined by~\\eqref{sh-br}.\nIn this case $P'$ will be the Poisson bivector compatible with $P$ and~we will say that $M=T^*Q$ is\nthe bi-Hamiltonian manifold~\\cite{mag97}.\nThe next step is a~search of integrals of motion for this Hamilton function $H$.\n\nIf the recursion operator\n$N$ at every point has $n$ distinct functionally\nindependent eigenvalues, we can say that $M$ is a~regular bi-Hamiltonian manifold.\nIf the recursion operator $N$ does\nnot have this property then we can say that bi-Hamiltonian manifold $M$ is\n{irregular}~\\cite{mpt11, ts10}.\nSo, there are three dif\\\/ferent cases:\n\\begin{enumerate}\\itemsep=0pt\n\\item[1)] recursion operator produces the necessary number of integrals of motion;\n\\item[2)] recursion operator generates variables of separation instead of integrals of motion;\n\\item[3)] recursion operator produces only part of the integrals of motion or variables of\nseparation.\n\\end{enumerate}\nIn the third case we have to complement the recursion operator with some additional information in\norder to get integrals of motion.\nNamely this property allows us to get noncommutative\nintegrable systems, which will be considered in Section~\\ref{section3}.\n\nNow let us show a~collection of tensor f\\\/ields $A$ associated with some well-known integrable\nsystems.\n\n\\subsection{Toda lattice}\n\\label{section2.1}\n\nLet us consider the following tensor f\\\/ield $A$ depending only on $q$ variables\n\\begin{gather*}\nA=\\left(\n\\begin{matrix}\nB+2D-B^\\top&0\n\\\\\n0&B-B^\\top\n\\end{matrix}\n\\right)\nP,\n\\end{gather*}\nwhere $B$ is a~strictly upper diagonal matrix\n\\begin{gather}\n\\label{b-mat}\nB=\\left(\n\\begin{matrix}\n0 & 1 & 1 & \\cdots & 1\n\\\\\n0 & 0 & 1 & \\cdots & 1\n\\\\\n\\vdots & & & \\ddots & \\vdots\n\\\\\n& & &0 & 1\n\\\\\n0 & & \\ldots & & 0\n\\end{matrix}\n\\right)=\n\\sum_{i>j}^n e_{ij},\n\\end{gather}\nand $D$ is a~diagonal matrix\n\\begin{gather*}\nD=\\operatorname{diag}(q_1,q_2,\\ldots,q_n)=\\sum_{i=1}^n q_i e_{ii}.\n\\end{gather*}\nMatrices $ e_{ij}$ are $n\\times n$ with only one non zero $(ij)$ entry, which equals to unit.\n\nSubstituting this tensor f\\\/ield $A$ and~a~natural Hamilton function\n\\begin{gather*}\nH=\\sum_{i=1}^n p_i^2+V(q)\n\\end{gather*}\ninto the def\\\/inition of $P'$ one gets a~system of equations~\\eqref{sch-eq} on $V(q)$.\nOne of the partial solutions of this system is the Hamilton function for the open Toda lattice\nassociated with $\\mathcal A_n$ root system\n\\begin{gather*}\nH=\\sum_{i=1}^n p_i^2+a\\sum_{i=1}^{n-1}e^{q_i-q_{i+1}},\\qquad a\\in\\mathbb R.\n\\end{gather*}\nTraces of powers of the corresponding recursion operator $N$~\\eqref{rec-op}\n\\begin{gather}\n\\label{tr-ham}\nH_k=\\operatorname{tr} N^k, \\qquad k=1,\\ldots,n,\n\\end{gather}\nare functionally independent constants of motion in bi-involution\nwith respect to both Poisson brackets\n\\begin{gather*}\n\\{H_i,H_j\\}=\\{H_i,H_j\\}'=0.\n\\end{gather*}\nThis Poisson bivector $P'$ was found by Das, Okubo and~Fernandes~\\cite{das89,fern93}.\n\nIn generic case we can use a~more complicated tensor f\\\/ield\n\\begin{gather*}\n\\widetilde{A}=\\left(\n\\begin{matrix}\n\\widetilde{B}+\\widetilde{D}&0\n\\\\\n0&\\widetilde{C}\n\\\\\n\\end{matrix}\n\\right)P,\n\\end{gather*}\nwhere entries of $\\widetilde{D}$ are linear on $q_i$ and~$\\widetilde{B}$ and~$\\widetilde{C}$ are\nnumerical matrices.\nHere $\\widetilde{B}$ is an arbitrary matrix, whereas $\\widetilde{A}$ and~$\\widetilde{B}$ satisfy to\nalgebraic equations which may be obtained from~\\eqref{sch-eq} at $V(q)=0$.\n\nUsing this tensor f\\\/ield $\\widetilde{A}$ we can get the recursion operators which produce either\nintegrals of motion for the periodic Toda lattice~\\cite{gts06} or variables of separation for the\nToda lattice~\\cite{ts07}.\nIn similar manner we can consider the Toda lattices associated with other classical root\nsystems~\\cite{ts10}.\n\n\\subsection{Relativistic Toda lattice}\n\\label{section2.2}\n\nIf we substitute the Hamilton function $H(q,p)$ and~the following tensor f\\\/ield $A$\n\\begin{gather*}\nA=\\left(\n\\begin{matrix}\n-B^\\top&0\n\\\\\n-E&0\n\\end{matrix}\n\\right)\nP=\\left(\\begin{matrix}\n0&-B^\\top\n\\\\\n0&-E\n\\end{matrix}\n\\right),\n\\end{gather*}\nwhere $E$ is a~unit matrix and~$B$ is given by~\\eqref{b-mat},\ninto the def\\\/inition of $P'$~\\eqref{poi-2} we will obtain a~system of equations on $H$.\nOne of the solutions is the Hamiltonian of the open discrete Toda lattice associated with $\\mathcal\nA_n$ root system\n\\begin{gather}\n\\label{rtl-ham}\nH=\\sum_{i=1}^n \\bigl( c_i+d_i \\bigr),\n\\end{gather}\nwhere $c_i$ and~$d_i$ are the so-called Suris variables\n\\begin{gather*}\nc_i=\\exp(p_i-q_i+q_{i+1}),\\qquad d_i=\\exp(p_i),\\qquad\nq_0=-\\infty,\\qquad q_{n+1}=+\\infty.\n\\end{gather*}\nTraces of powers of the corresponding recursion operator $N$~\\eqref{tr-ham} are integrals of\nmotion in bi-involution with respect to both Poisson brackets.\nNamely this Poisson bivector~$P'$~\\eqref{poi-2} is discussed in~\\cite{rag89,sur93}.\n\nRemind, that according to~\\cite{sur93} there is an equivalence between the relativistic Toda lattice\nand the discrete time Toda lattice.\nNamely, substituting\n\\begin{gather*}\np_j =\\theta_j +\\frac12 \\ln\\left(\n{\\frac{1+\\exp({q}_j-{q}_{j-1} )}{1+\\exp({q}_{j+1} -{q}_j )}}\\right)\n\\end{gather*}\nin~\\eqref{rtl-ham} one gets standard Hamiltonian for the\nrelativistic Toda lattice\n\\begin{gather*}\nH=\\sum_{j=1}^{n-1} \\exp (\\theta_j )\n\\Bigl[\\bigl[1+\\exp({q}_j -{q}_{j-1} ) ]\n[ 1+\\exp({q}_{j+1} -{q}_j ) \\bigr]\\Bigr]^{1\/2}.\n\\end{gather*}\nTransformation $(\\theta_j\\cdot q_j )\\rightarrow\n(p_j\\cdot q_j )$ is a~canonical transformation.\n\nAs above, two numerical matrices $\\widetilde{B}$ and~$\\widetilde{C}$ in the tensor f\\\/ield\n\\begin{gather*}\nA=\\left(\n\\begin{matrix}\n0&\\widetilde{B}\n\\\\\n0&\\widetilde{C}\n\\\\\n\\end{matrix}\n\\right)\n\\end{gather*}\nallow us to get recursion operators $N=P'P^{-1}$ which generate either integrals of motion\nfor the periodic relativistic Toda lattice or variables of separation~\\cite{kuz94}.\n\n\\subsection[Henon-Heiles system]{Henon--Heiles system}\n\\label{section2.3}\n\nAt $n=2$ we can introduce the following linear in momenta tensor f\\\/ield $A$\n\\begin{gather}\n\\label{a-hh}\nA=\\left(\n\\begin{matrix}\nB & 0\n\\\\\n0 & C\n\\\\\n\\end{matrix}\n\\right)P=\\left(\n\\begin{matrix}\n0 & B\n\\\\\n-C & 0\n\\\\\n\\end{matrix}\n\\right),\n\\end{gather}\nwhere\n\\begin{gather*}\nB=\\left(\n\\begin{matrix}\n2q_1p_1&q_1p_2\n\\\\\nq_1p_2&q_2p_2\n\\\\\n\\end{matrix}\n\\right),\\qquad C=\\left(\n\\begin{matrix}\nf_1(q)p_1+f_2(q)p_2&0\n\\\\\n0&f_3(q)p_1+f_4(q)p_2\n\\\\\n\\end{matrix}\n\\right).\n\\end{gather*}\nSubstituting this tensor f\\\/ield $A$ and~a~natural Hamilton function\n\\begin{gather*}\nH_1=p_1^2+p_2^2+V(q)\n\\end{gather*}\ninto the def\\\/inition of $P'$ one gets a~system of equations~\\eqref{sch-eq} on $V(q)$ and~functions\n$f_k(q)$.\nThe resulting system of PDE's has two partial polynomial solutions\n\\begin{gather*}\nV(q)={c_1} q_2\\big(3q_1^2+16q_2^2\\big)+c_2 \\left(2q_2^2\n+\\dfrac{q_1^2}{8}\\right)+c_3q_2,\\qquad c_k\\in\\mathbb R,\n\\end{gather*}\nand\n\\begin{gather*}\nV(q)={c_1}\\big(q_1^4+6q_1^2q_2^2+8q_2^4\\big)\n+{c_2}\\big(q_1^2+4q_2^2\\big)+\\dfrac{c_3}{q_2^2}.\n\\end{gather*}\nSecond integrals of motion $H_2 =\\operatorname{tr} N^2$ are fourth order polynomials in momenta.\n\nSo, one gets the Henon--Heiles potential and~the fourth order potential~\\cite{gdr84} as\nparticular polynomial solutions of the equations~\\eqref{sch-eq} associated with tensor\nf\\\/ield~\\eqref{a-hh}.\n\nUsing slightly deformed tensor f\\\/ield $A$ we can get the same systems with singular\nterms~\\cite{gts11} and\ntheir three-dimensional counterparts~\\cite{ts10}.\n\n\\subsection[Rational Calogero-Moser model]{Rational Calogero--Moser model}\n\\label{section2.4}\n\nFollowing~\\cite{mpt11} let us consider tensor f\\\/ield $A$, which is proportional to $P$\n\\begin{gather*}\nA=\\rho(q,p)P,\n\\end{gather*}\nwhere $\\rho(q,p)$ is a~function on $M$.\nIf\n\\begin{gather}\n\\label{a-cal}\nA=(p_1q_1+\\cdots+p_nq_n)P, \\qquad \\rho=p_1q_1+\\cdots+p_nq_n,\n\\end{gather}\nthen equations~\\eqref{sch-eq} have the following partial solution\n\\begin{gather}\n\\label{ham-cal}\nH=\\dfrac{1}{2}\\sum_{i=1}^{n}p_{i}^{\\;2}+\\dfrac{g^2}{2}\\sum_{i\\neq\nj}^n\\frac{1}{(q_{i}-q_{j})^{2}},\n\\end{gather}\nwhere $g$ is a~coupling constant.\nIt is the Hamilton function of the $n$-particle rational Calogero--Moser model associated with the\nroot system $\\mathcal A_n$.\n\nThe corresponding recursion operator $N$~\\eqref{rec-op} generates only a~Hamilton function\n\\begin{gather*}\n\\operatorname{tr}N^k=2H^{k},\\qquad k=1,\\ldots,n,\n\\end{gather*}\nthat allows us to identify our phase space $M=\\mathbb R^{2n}$ with the irregular bi-Hamiltonian\nmanifold.\n\nIn this case~\\cite{mpt11,ts10} integrals of motion are polynomial solutions of the equations\n\\begin{gather}\n\\label{cal-int}\nP dH=-\\frac{1}{k} P' d\\ln H_k,\\qquad k=1,\\ldots,n,\n\\end{gather}\nwhich have two functionally independent solutions for any $k\\geq2$.\nIt is easy to see that the functions\n\\begin{gather}\n\\label{caz-cal}\nC_{km}=\\frac{H_m^{-1\/m}}{H_k^{-1\/k}}\n\\end{gather}\nare Casimir functions of $P'$, i.e.~$P'dC_{km}=0$.\n\nSome solutions of equations~\\eqref{cal-int} coincide with the well-known integrals of motion\n\\begin{gather*}\nJ_{n-m}\\equiv\\frac{1}{m!}\\underbrace{\\bigg\\{\\sum_{i=1}^n q_{i}\\cdots\\bigg\\{\n}_{m~\\text{times}}\\sum_{i=1}^n q_{i},J_{m}\\bigg\\}\\cdots\\bigg\\},\\qquad m=1,\\dots,n-1,\n\\end{gather*}\nobtained from the conserved quantity\n\\begin{gather}\nJ_{n}\\equiv \\exp\\left(-\\frac{g^{2}}{2}\\sum_{i\\neq j}\n\\frac{1}{(q_{i}-q_{j})^{2}}\\frac{\\partial^{2}}{\\partial\np_{i}\\partial p_{j}}\\right)\\prod_{k=1}^{n}p_{k}\n\\nonumber\n\\end{gather}\nby taking its successive Poisson brackets with $\\sum\\limits_{i=1}^{n}q^{i}$~\\cite{gon01}.\nThese $n$ solutions, including $J_2=H$, are in involution with respect to the Poisson\nbrackets~\\eqref{can-br}.\n\nOther $n-1$ functionally independent solutions of~\\eqref{cal-int},\n\\begin{gather*}\nK_{m}=m g_{1}J_{m}-g_{m}J_{1},\\qquad\ng_{m}=\\frac{1}{2}\\left\\{\\sum_{i=1}^{n}q_{j}^{\\;2},J_{m}\\right\\},\\qquad\nm=2,\\dots,n,\n\\end{gather*}\nare not in involution with respect to the canonical Poisson bracket def\\\/ined\nby~\\eqref{can-p}~\\cite{gon01}.\n\n\\subsection[Rational Ruijsenaars-Schneider model]{Rational Ruijsenaars--Schneider model}\n\\label{section2.5}\n\nLet us consider tensor f\\\/ield $A$, which is proportional to canonical bivector $P$\n\\begin{gather*}\nA=(q_1+\\cdots+q_n)P,\\qquad\\rho=q_1+\\cdots+q_n.\n\\end{gather*}\nIn this case equations~\\eqref{sch-eq} have the following partial solutions\n\\begin{gather}\n\\label{tr-rtl}\nJ_k=\\frac{1}{k!}\\operatorname{tr} L^k,\\qquad k=\\pm 1,\\pm2,\\ldots,\\pm n,\\end{gather}\nwhere $L$ is the Lax matrix of the Ruijsenaars--Schneider model\n\\begin{gather*}\nL=\\sum_{i,j=1}^n\\frac{\\gamma}{q_i-q_j+\\gamma} b_j e_{ij},\\qquad\nb_k=e^{p_k}\\prod_{j\\neq k}\\left(1-\\frac{\\gamma^2}{(q_k-q_j)^2}\\right)^{1\/2}.\n\\end{gather*}\nAs above recursion operator produces only the Hamilton function.\nIt is easy to prove that traces of powers of the Lax matrix $L$\n\\eqref{tr-rtl} satisfy to the following relations\n\\begin{gather}\n\\label{rs-rel}\nP dJ_{\\pm 2}=-\\frac{1}{k} P' d\\ln J_k,\\qquad k=\\pm 1,\\ldots,\\pm n,\n\\end{gather}\ninstead of the standard Lenard--Magri relations~\\cite{mag97,tt12}.\nMoreover, similar to the Calogero--Moser system, there are other solutions $K_m$ of these\nequations~\\eqref{rs-rel}, which are described in~\\cite{afg12}.\n\nRemind, that the so-called principal Ruijsenaars--Schneider Hamiltonian has the form\n\\begin{gather*}\nH_{RS}=\\frac{1}{2}(J_1+J_{-1})=\\sum_{k=\n1}^n(\\cosh2p_k)\\prod_{j\\neq k}\\left(1-\\frac{\\gamma^2}{(q_k-q_j)^2}\\right)^{1\/2}\n\\end{gather*}\nand that the rational Ruijsenaars--Schneider system is in duality with the corresponding variant of\nthe trigonometric Sutherland system, see~\\cite{afg12} and  references therein.\n\nWe want to highlight that for all integrable systems listed in~\\cite{mpt11,tt12,ts10,ts11s}\nthe second Poisson bivector $P'$~\\eqref{poi-2} is a~Lie derivative of the canonical Poisson\nbivector $P$ along the vector f\\\/ield\n$Y=\\operatorname{Ad}H$~\\eqref{y1}, where tensor f\\\/ield $A$ usually has a~very simple form.\n\nIn the next section we show that such simple tensor f\\\/ields $A$ may be useful to search for new\nintegrable systems.\n\n\\section{Noncommutative integrable systems}\n\\label{section3}\n\nThe extreme rarity of integrable dynamical systems makes the quest for them all the more exciting.\nWe want to apply tensor f\\\/ields $A$ to partial solution of this problem.\nBelow we present a~method to construct a~new family of three dimensional noncommutative integrable\nsystems.\n\nLet us consider natural Hamilton function on $M=\\mathbb R^{2n}$\n\\begin{gather*}\nH=\\sum_{i=1}^n p_i^2+V(q_1,\\ldots,q_n)\n\\end{gather*}\nand bivector $A$ associated with the rational Calogero--Moser system~\\eqref{a-cal}\n\\begin{gather*}\nA=(p_1q_1+\\cdots+p_nq_n)P,\n\\end{gather*}\nwhere $P$ is canonical Poisson bivector~\\eqref{can-p}, \\eqref{can-p2}.\n\nIn previous section we have discussed partial solutions of the equations~\\eqref{sch-eq}, here we\nwant to discuss their complete solution.\n\\begin{proposition}\nThe Lie derivative of $P$~\\eqref{can-p} along the vector field $Y$\n\\begin{gather}\n\\label{p-p}\nP'=\\mathcal L_Y P,\\qquad Y=(p_1q_1+\\cdots+p_nq_n)PdH\n\\end{gather}\nis a~Poisson bivector compatible with $P$ if and~only if\n\\begin{gather}\n\\label{m-pot}\nH=\\sum_{i=1}^n p_i^2+\\frac{1}{q_1^2}\nF\\left(\\frac{q_2}{q_1},\\frac{q_3}{q_1},\\ldots,\\frac{q_n}{q_1}\\right).\n\\end{gather}\nHere $F$ is an arbitrary homogeneous function of zero degree function depending on the homogeneous\ncoordinates\n\\begin{gather*}\nx_1=\\frac{q_2}{q_1},\\quad x_2=\\frac{q_3}{q_1},\\quad\\ldots,\\quad x_{n-1}=\\frac{q_n}{q_1}.\n\\end{gather*}\n\\end{proposition}\n\nThe definition of the homogeneous coordinates may be found in~\\cite{gr94}.\nProof is a~straightforward calculation of the Schouten bracket~\\eqref{sch-eq}.\n\nIt is easy to see that some Hamilton functions separable in spherical coordinates and~Hamilton\nfunctions for the rational Calogero--Moser systems associated with the $A_n$, $B_n$, $C_n$\nand~$D_n$ root systems have the form~\\eqref{m-pot}.\n\nWe got accustomed to believing that the notion of two compatible Poisson structures $P$ and~$P'$\nallows us to get the appropriate integrable systems~\\cite{gts11,mag97,tt12,ts10,ts11s}.\nIn our case recursion operator $N=P'P^{-1}$ reproduces only the Hamilton function\n\\begin{gather*}\n\\operatorname{tr}N^k=2(2H)^{k}.\n\\end{gather*}\nIt allows us to identify our phase space $M=\\mathbb R^{2n}$ with the irregular bi-Ha\\-mil\\-to\\-nian\nmani\\-fold~\\mbox{\\cite{mag97,ts10}}, but simultaneously it makes the use of standard constructions of the\nintegrals of motion impossible.\n\nWe do not claim that all the Hamilton functions~\\eqref{m-pot} are integrable because we do not have\nan explicit construction of the necessary number of integrals of motion.\nNevertheless, even in generic case there is one additional integral of motion.\n\\begin{proposition}\nThe following second order polynomial in momenta\n\\begin{gather*}\nC=(p_1q_1+\\cdots+p_nq_n)^2-(q_1^2+\\cdots+q_n^2)H\n\\end{gather*}\nis a~Casimir function of $P'$, i.e.~$P'dC=0$.\n\\end{proposition}\nConsequently we have\n\\begin{gather*}\n\\{H,C\\}=0.\n\\end{gather*}\nIt is enough for integrability at $n=2$ when we get Hamilton functions\n\\begin{gather*}\nH=p_1^2+p_2^2+\\frac{1}{q_1^2} F\\left(\\dfrac{q_2}{q_1}\\right)\n\\end{gather*}\nseparable in polar coordinates on the plane.\n\nAt $n>3$ we can make some assumptions on the form of the additional integrals of motion.\nFor instance,\nlet us postulate that our dynamical system is invariant with respect to translations, i.e.\\;that\nthere is a~linear in momenta integral of motion\n\\begin{gather*}\nH_{\\rm post}=p_1+\\cdots+p_n,\\qquad\\{H,H_{p\\rm ost}\\}=0.\n\\end{gather*}\nIt leads to the additional restriction on the form of the proper Hamilton functions~\\eqref{m-pot}\n\\begin{gather*}\nH=\\sum_{i=1}^n p_i^2+\\frac{1}{(q_2-q_1)^2} G\\left(\\frac{q_3-q_2}{q_2-q_1},\\frac{q_4-q_3}{q_2-q_1},\\ldots,\\frac{q_n-q_{n-1}}{q_1-q_2}\\right),\n\\end{gather*}\nwhich generate bi-Hamiltonian vector f\\\/ields\n\\begin{gather}\n\\label{vf-biham}\nX=PdH=P'd\\ln H_{\\rm post}^{-1}\n\\end{gather}\nequipped with the four integrals of motion\n\\begin{gather}\n\\label{4-int}\nH_1=H_{\\rm post},\\qquad H_2=H,\\qquad H_3=C,\\qquad H_4=\\{H_1,C\\}\n\\end{gather}\nwith the linearly independent dif\\\/ferentials $dH_i$.\nAccording to the Euler--Jacobi theorem~\\cite{jac36} it is enough for integrability by quadratures\nat $n=3$.\n\nRemind that the Euler--Jacobi theorem~\\cite{jac36} states that a~system of $N$ dif\\\/ferential\nequations\n\\begin{gather}\n\\label{eqm}\n\\dot{x}_i=X_i(x_1,\\ldots,x_N),\\qquad i=1,\\ldots,N,\\end{gather}\npossessing the last Jacobi multiplier $\\mu$ (invariant measure) and~$N-2$ independent f\\\/irst\nintegrals is integrable by quadratures.\nIn our case $N=6$, we have four independent integrals of motion~\\eqref{4-int} and~$\\mu=1$.\n\n{\\sloppy So, at $n=3$ the following Hamilton functions\n\\begin{gather}\n\\label{3g-ham}\nH_2=p_1^2+p_2^2+p_3^3+\\frac{1}{(q_2-q_1)^2} G\\left(\\frac{q_3-q_2}{q_2-q_1}\\right)\n\\end{gather}\nlabelled by functions $G$ generate integrable by quadratures Hamiltonian equations of\nmotion~\\eqref{vf-biham}--\\eqref{eqm}.\nBecause\n\\begin{alignat}{4}\n& \\{H_1,H_2\\}=0,\\qquad&& \\{H_1,H_3\\}=H_4,\\qquad &&\\{H_1,H_4\\}=2H_1^2-6H_2, &\n\\nonumber\\\\\n& \\{H_2,H_3\\}=0,\\qquad && \\{H_2,H_4\\}=0,\\qquad &&\\{H_4,H_3\\}=4H_1H_3 &\\label{alg-int}\n\\end{alignat}\nwe have noncommutative integrable systems with respect to the canonical\nPoisson bracket, see, for instance,~\\cite{kh10} and~references therein.\n\n}\n\nOf course, in the center of momentum frame, the total linear momentum of the system is zero $H_1=0$\nand~we have three integrals of motion $H_2$, $H_3$ and~$H_4$ in the involution that is enough for\nintegrability at $n=3$ and~$n=4$.\n\nOn the other hand, Hamilton functions~\\eqref{3g-ham} def\\\/ine superintegrable systems in the\nLiouville sense\n\\begin{gather*}\n\\{H_i,H_j\\}'=0,\\qquad i,j=1,\\ldots,4,\n\\end{gather*}\nwith respect to the second Poisson bracket $\\{\\cdot,\\cdot\\}'$ associated with the Poisson tensor\n$P'$~\\eqref{p-p}.\nIf we put\n\\begin{gather*}\nG(x)=g^2\\left(1+\\frac{1}{x^2}+\\frac{1}{(1+x)^2}\\right),\n\\end{gather*}\nwe can obtain a~well-known Hamiltonian for the rational Calogero--Moser system~\\eqref{ham-cal}\n\\begin{gather*}\nH_{\\rm CM}=\\sum_{i=\n1}^{3}p_{i}^{2}+\\frac{g^2}{(q_2-q_1)^2}+\\frac{g^2}{(q_3-q_2)^2}+\\frac{g^2}{(q_3-q_1)^2}.\n\\end{gather*}\nIn this case there are other polynomial integrals of motion~\\eqref{cal-int} and~other Casimir\nfunctions of~$P'$~\\eqref{caz-cal}.\nThis system was separated by Calogero~\\cite{cal69} in cylindrical coordinates in $\\mathbb R^3$.\nBeing a~superintegrable system it is actually separable in four other types of coordinate\nsystems~\\cite{bcr00}.\nThese variables of separation may be easily found using either the generalised Bertrand--Darboux\ntheo\\-rem~\\cite{rw03} or methods of the bi-Hamiltonian geometry~\\cite{gts05}.\nRemind, that variables of separation are eigenvalues of the Killing tensor $K$ satisfying equation\n\\begin{gather}\\label{killing-eq}\nKdV=0,\n\\end{gather}\nwhere $V$ is potential part of the Hamiltonian $H$.\n\nAny additive deformation of this function $G(x)$ leads to the integrable additive deformation of\nthe rational Calogero--Moser system, for instance, if\n\\begin{gather*}\n\\widetilde{G}(x)=G(x)+\\frac{a}{x},\n\\end{gather*}\nthen one gets an integrable system with the three-particle interaction\n\\begin{gather*\n\\widetilde{H}_{\\rm CM}=H_{\\rm CM}+\\frac{a}{(q_1-q_2)(q_2-q_3)}.\n\\end{gather*}\nFor this Hamilton function we couldn't f\\\/ind any polynomial in momenta\nintegrals of motion except $H_1$, $H_3$ and~$H_4$~\\eqref{4-int}).\nMoreover, we couldn't get variables of separation using the standard (regular) methods such as\ngeneralised Bertrand--Darboux theorem~\\cite{rw03} and~bi-Hamiltonian algorithm discussed\nin~\\cite{gts05}.\nNamely, in contrast with the case $a=0$ at $a\\neq0$ the Killing ten\\-sor~$K$\nsatisfying~\\eqref{killing-eq} has only functionally dependent eigenvalues.\n\nAt $n=3$ in order to get rational Calogero--Moser systems associated with other classical root\nsystems and~their deformations we can postulate an existence of the fourth order integral of motion\n\\begin{gather*}\nH_{\\rm post}=\\sum_{i\\neq j}^n p_i^2p_j^2+\\sum_{k}f_k(q)p_k^2+g(q),\n\\end{gather*}\nwith some unknown functions $f_k(q)$ and~$g(q)$.\nHowever we do not have\nan exhaustive classif\\\/ication as of yet.\n\nIn generic case at $n\\geq3$ we can use other hypotheses about\nadditional integrals of motion commuting with $H$~\\eqref{m-pot}.\n\nOf course, construction of such integrable systems is trivial and~closely related with construction\nof the\ngroup invariant solutions of partial dif\\\/ferential equations through imposing side\nconditions~\\cite{or87,ov82}.\nRemind, we can look for solution $W(q)$ of the Hamilton--Jacobi equation\n\\begin{gather*}\nH(p,q)=\\sum_{i=1}^n\\left(\\frac{\\partial W}{\\partial q_i}\\right)^2+W(q_1,\\ldots,q_n)=\n\\mathcal E,\\qquad p_j=\\dfrac{\\partial W}{\\partial q_j},\n\\end{gather*}\nup to the side condition\n\\begin{gather*}\n\\mathcal S(q,p)=0.\n\\end{gather*}\nIf $\\{H,\\mathcal S\\}=f(q,p)\\mathcal S$ then this side condition is consistent with $H$ and~the\ncorresponding integrals of motion are def\\\/ined modulo $\\mathcal S=0$, i.e.\n\\begin{gather*}\n\\{H,H_k\\}=g_k(q,p)\\mathcal S.\n\\end{gather*}\nHere $f$ and~$g_k$ are some functions on phase space and~$W(q)$ is the so-called characteristic\nHamilton function.\n\nIn our case the side condition is related with the transition to the center of momentum frame\n\\begin{gather*}\n\\mathcal S=H_{\\rm post}\\equiv p_1+p_2+p_3=0,\n\\end{gather*}\nwhich is always consistent with the Hamilton function~\\eqref{3g-ham} and~we have three integrals of\nmotion~$H_2$,~$H_3$ and~$H_4$~\\eqref{4-int} in involution by modulo $\\mathcal S=0$~\\eqref{alg-int}.\nConstruction of the variables of separation for the Hamilton--Jacobi equation with a~side\nconditions is discussed in~\\cite{mil12}.\n\nIn quantum case we can consider the Schr\\\"{o}dinger equation\n\\begin{gather*}\nH\\Psi=\\mathcal E\\Psi,\\qquad H=\\Delta+V(q_1,\\ldots,q_n),\n\\end{gather*}\nwhere $\\Delta$ is the Laplace--Beltrami operator on $M=\\mathbb R^{2n}$, and~study solution of this\nequation that also satisf\\\/ies a~side condition\n\\begin{gather*}\n\\mathcal S\\Psi=0.\n\\end{gather*}\nThe consistency condition for the existence of nontrivial solutions $\\Psi$ is a~standard\n\\begin{gather*}\n[H,\\mathcal S]=f\\mathcal S.\n\\end{gather*}\nIn this case linear dif\\\/ferential operator $K$ will be a~symmetry operator for $H$ modulo\n$\\mathcal S\\Psi=0$ if\n\\begin{gather*}\n[H,K]=g\\mathcal S.\n\\end{gather*}\nHere $f$ and~$g$ are some linear partial dif\\\/ferential operators, see examples and~discussion\nin~\\cite{mil12}.\n\nWe assume that the quantum counterpart of $H$~\\eqref{3g-ham} could\nbe embedded in this generic scheme.\n\n\\section{Conclusion}\n\\label{section4}\n\nWe have demonstrated that the trivial deformations of the canonical Poisson bracket\nassociated with the well-known integrable systems have a~very simple form def\\\/ined\nby some 2-tensor f\\\/ield $A$ acting on the dif\\\/ferential of the Hamilton function.\nWe have shown\na collection of examples and~also proven that such tensor f\\\/ields may be useful for searching new\nintegrable by quadratures dynamical systems.\nFor example, we have proven the\nnoncommutative integrability of a~new generalisation of the\nrational Calogero--Moser system with three particle interaction.\n\nIn fact, we propose a~new form for the old content and~believe that this unif\\\/ication\nis a~next step in creating the invariant and~rigorous geometric theory of integrable systems on\nregular and\nirregular bi-Hamiltonian manifolds.\n\n\\subsection*{Acknowledgements}\nWe would like to thank E.G.~Kalnins, W.~Miller, Jr.\\\nand G.~Rastelli for useful discussion on noncommutative integrable systems.\nThe study was supported by the ministry of education and~science of Russian Federation, project\n07.09.2012 no.~8501, grant no.~2012-1.5-12-000-1003-016.\n\n\\pdfbookmark[1]{References}{ref}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\\begin{figure}\n  \\includegraphics[width=\\linewidth]{figures\/venice2.png}\n  \\caption{\n  Optimized 3D reconstruction of the largest \\emph{venice} BAL dataset with 1778 cameras, around one million landmarks, and five million observations.\n  For this problem, the proposed square root bundle adjustment ($\\sqrt{BA}$) solver is 42\\% faster than the best competing method at reaching a cost tolerance of 1\\%.\n  }\n  \\label{fig:teaser}\n\\end{figure}\n\nBundle adjustment (BA) is a core component of many 3D reconstruction algorithms and structure-from-motion (SfM) pipelines.\nIt is one of the classical computer vision problems and has been investigated by researchers for more than six decades \\cite{brown1958solution}.\nWhile different formulations exist, the underlying question is always the same:\ngiven a set of approximate point (landmark) positions that are observed from a number of cameras in different poses, what are the actual landmark positions and camera poses?\nOne can already compute accurate 3D positions with only a few images; however, with more available images we will get a more complete reconstruction.\nWith the emergence of large-scale internet photo collections has come the need to solve bundle adjustment on a large scale, i.e., for thousands of images and hundreds of thousands of landmarks.\nThis requires scalable solutions that are still efficient for large problems and do not run into memory or runtime issues.\n\nModern BA solvers usually rely on the Schur complement (SC) trick that computes the normal equations of the original BA problem and then breaks them down into two smaller subproblems, namely (1) the solution for camera poses and (2) finding optimal landmark positions.\nThis results in the drastically smaller reduced camera system (RCS), which is also better-conditioned~\\cite{agarwal2010bundle} than the original normal equations.\nThe use of the Schur complement has become the de facto standard for solving large-scale bundle adjustment and is hardly ever questioned.\n\nIn this work, we challenge the widely accepted assumption of SC being the best choice for solving bundle adjustment, and provide a detailed derivation and analysis of an alternative problem reduction technique based on QR decomposition.\nInspired by similar approaches in the Kalman filter literature~\\cite{yang2017null}, we factorize the landmark Jacobian $J_l$ such that we can project the original problem onto the nullspace of $J_l$.\nThus, we circumvent the computation of the normal equations and their system matrix $H$, and are able to directly compute a matrix square root of the RCS while still solving an algebraically equivalent problem.\nThis improves numerical stability of the reduction step, which is of particular importance on hardware optimized for single-precision floats.\nFollowing terminology for nullspace marginalization on Kalman filters, we call our method \\emph{square root bundle adjustment}, short $\\sqrt{BA}$.\nIn particular, our contributions are as follows:\n\\setlist{nolistsep}\n\\begin{itemize}[noitemsep]\n    \\item We propose nullspace marginalization as an alternative to the traditional Schur complement trick and prove that it is algebraically equivalent.\n    \\item We closely link the very general theory of nullspace marginalization to an efficient implementation strategy that maximally exploits the specific structure of bundle adjustment problems.\n    \\item We show that the algorithm is well parallelizable and\n    that the favorable numerical properties admit computation in single precision,\n    resulting in an additional twofold speedup.\n    \\item We perform extensive evaluation of the proposed approach on the Bundle Adjustment in the Large (BAL) datasets and compare to the state-of-the-art optimization framework Ceres. \n    \\item We release our implementation and complete evaluation pipeline as open source to make our experiments reproducible and facilitate further research:\\\\\n    \\url{https:\/\/go.vision.in.tum.de\/rootba}.\n\\end{itemize}\n\n\\section{Related work}\n\nWe propose a way to solve large-scale bundle adjustment using QR decomposition, so we review both works on bundle adjustment (with a focus on large-scale problems) and works that use QR decomposition for other tasks in computer vision and robotics.\nFor a general introduction to numerical algorithms \n(including QR decomposition and iterative methods for solving linear systems),\nwe refer to~\\cite{bjorck1996numerical,golub13}.\n\n\\paragraph{(Large-scale) bundle adjustment}\nA detailed overview of bundle adjustment in general can be found in~\\cite{triggs1999bundle}, including an explanation of the Schur complement (SC) reduction technique~\\cite{brown1958solution} to marginalize landmark variables.\nByr\u00f6d and \u00c5str\u00f6m use the method of conjugate gradients (CG) on the normal equations~\\cite{hestenes1952methods, bjorck1979accelerated} to minimize the linearized least squares problem without the Schur complement~\\cite{byrod2010conjugate}.\nThey also QR-decompose the Jacobian, but only for block preconditioning without marginalizing landmarks.\nAgarwal et~al.\\ have proposed preconditioned CG on the RCS after SC \nto solve the large-scale case~\\cite{agarwal2010bundle}, and Wu et~al.\\ further extend these ideas to a formulation which avoids explicit computation of the SC matrix~\\cite{wu2011multicore}.\nA whole number of other works have proposed \nways to further improve efficiency, accuracy and\/or robustness of BA~\\cite{engels2006bundle, konolige2010sparse, jeong2011pushing, zach2014robust, natesan2017distributed}, all of them using the Schur complement.\nMore recently, in Stochastic BA~\\cite{zhou2020stochastic} the reduced system matrix is further decomposed into subproblems to improve scalability.\nSeveral open source BA implementations are available, e.g., the SBA package~\\cite{lourakis2009sba}, the g2o framework~\\cite{kummerle2011g}, or Ceres Solver~\\cite{ceres-solver}, which has become a standard tool for solving BA-like problems in both academia and industry.\n\n\\paragraph{Nullspace marginalization, square root filters, and QR decomposition}\nThe concept of nullspace marginalization has been used in contexts other than BA, e.g., for the multi-state constraint Kalman filter~\\cite{mourikis2007} and earlier in~\\cite{bayard2005estimation}.\n\\cite{yang2017null} proves the equivalence of nullspace marginalization and the Schur complement in the specific case of robot SLAM.\nSeveral works explicitly point out the advantage of matrix square roots in state estimation~\\cite{maybeck1982stochastic,bierman2006factorization,dellaert2006square,wu2015square}, but to the best of our knowledge matrix square roots have not yet been used for problem size reduction in BA.\nThe QRkit~\\cite{svoboda2018qrkit} emphasizes the benefits of QR decomposition for sparse problems in general and also mentions BA as a possible application, but the very specific structure of BA problems and the involved matrices is not addressed.\nThe orthogonal projector used in the Variable Projection (VarPro) method~\\cite{okatani11, hong17} is related to the nullspace marginalization in our approach.\nHowever, VarPro focuses on separable non-linear least squares problems, which do not include standard BA. While~\\cite{hong17} mentions the use of QR decomposition to improve numeric stability, we take it one step further by more efficiently multiplying in-place with $Q_2^\\top$  rather than explicitly with $I - Q_1Q_1^\\top$ (further discussed in Section~\\ref{sec:nullspace-marginalization}). This also enables our very efficient way to compute landmark damping (not used in VarPro).\nOur landmark blocks can be seen as a specific instance of Smart Factors proposed in~\\cite{carlone2014eliminating}, where nullspace projection with explicit SVD decomposition is considered. Instead of the re-triangulation in~\\cite{carlone2014eliminating}, we suggest updating the landmarks with back substitution. The factor grouping in~\\cite{carlone2014mining} allows to mix explicit and implicit SC. This idea is orthogonal to our approach and could be considered in future work.\n\n\n\\section{QR decomposition}\n\nWe briefly introduce the QR decomposition, which can be computed using Givens rotations (see appendix).\nFor further background, we refer the reader to textbooks on least squares problems (e.g.,~\\cite{bjorck1996numerical}).\nLet $A$ be an $m\\times n$ matrix of full rank with $m \\geq n$, i.e., $\\operatorname{rank}(A) = n$.\n$A$ can be decomposed into an $m \\times m$ orthogonal matrix $Q$ and an $m \\times n$ upper triangular matrix $R$.\nAs the bottom ($m - n$) rows of $R$ are zeros, we can partition $R$ and $Q$:\n\\begin{align}\n  A = QR = Q \\begin{pmatrix} R_1 \\\\ 0 \\end{pmatrix}\n    = \\begin{pmatrix} Q_1 & Q_2 \\end{pmatrix} \\begin{pmatrix} R_1 \\\\ 0 \\end{pmatrix}\n    = Q_1 R_1\\,,\n\\end{align}\nwhere $R_1$ is an $n \\times n$ upper triangular matrix, $Q_1$ is $m \\times n$, and $Q_2$ is $m \\times (m-n)$.\nNote that this partitioning of $Q$ directly implies that the columns of $Q_2$ form the left nullspace of $A$, i.e., $Q_2^\\top A =0$.\nSince $Q$ is orthogonal, we have\n\\begin{align}\n\\label{eq:q_orth}\n    Q^\\top Q &= \\id_m = QQ^\\top\\,,\n\\end{align}\nwhere $\\id_m$ is the $m \\times m$ identity matrix.\nFrom \\eqref{eq:q_orth} we derive:\n\\begin{gather}\n    Q_1^\\top Q_1 = \\id_n\\,, \\quad\n    Q_2^\\top Q_2 = \\id_{m-n}\\,, \\quad\n    Q_1^\\top Q_2 = 0\\,, \\\\\n\\label{eq:qqt}\n   Q_1Q_1^\\top = \\id_m - Q_2Q_2^\\top\\,.\n\\end{gather}\n\n\n\\section{Square root bundle adjustment}\n\nWe assume a very general form of bundle adjustment, similar to~\\cite{agarwal2010bundle}:\nlet $\\vect{x}$ be a state vector containing all the optimization variables.\nWe can subdivide $\\vect{x}$ into a pose part $\\vect{x}_p$ containing extrinsic and possibly intrinsic camera parameters for all $n_p$ images (index $i$), and a landmark part $\\vect{x}_l$ consisting of the 3D coordinates of all $n_l$ landmarks (index $j$).\nThe total bundle adjustment energy is a sum of squared residuals\n\\begin{align}\n\\label{eq:ba_energy}\n    E(\\vect{x}_p,\\vect{x}_l) = \\sum_{i}\\sum_{j\\in O(i)}{r_{ij}(\\vect{x}_p,\\vect{x}_l)^2} = \\Vert \\vect{r}(\\vect{x}_p,\\vect{x}_l) \\Vert^2\\,,\n\\end{align}\nwhere $j\\in O(i)$ means that landmark $j$ is observed in frame $i$ and $\\vect{r}(\\vect{x})$ is the concatenation of all residuals $r_{ij}$ into one vector.\nWe call the total number of residuals $N_r$.\nFor a pose dimensionality $d_p$, the length of the total state vector $(\\vect{x}_p,\\vect{x}_l)$ is $d_pn_p+3n_l=:N_p+3n_l$.\nTypically, $d_p=6$ if only extrinsic camera parameters need to be estimated, and $d_p=9$ if intrinsic calibration is also unknown.\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{figures\/sparsity2.pdf}\n  \\caption{Dense landmark blocks. (a)~Sparsity structure of the pose Jacobian is fixed during the optimization. Non-zero elements shown in blue, potentially non-zero elements after Givens QR shown in gray, and elements that will always stay zero shown in white. (b)~Dense storage for the outlined (red) landmark block that efficiently stores all Jacobians and residuals for a single landmark. (c)~Same landmark block after in-place marginalization. As Givens rotations operate on individual rows,  marginalization can be performed for each landmark block separately, possibly in parallel.}\n  \\label{fig:sparsity}\n\\end{figure*}\n\n\n    \n\\subsection{Least squares problem}\n\nThe energy in \\eqref{eq:ba_energy} is usually minimized by the Levenberg-Marquardt algorithm, which is based on linearizing $\\vect{r}(\\vect{x})$ around the current state estimate $\\vect{x}^0=(\\vect{x}_p^0,\\vect{x}_l^0)$ and then solving a damped linearized problem\n\\begin{align}\n\\begin{aligned}\n\\label{eq:linearized_residual}\n \\min_{\\Delta\\vect{x}}&\\left\\Vert \\begin{pmatrix}\\vect{r}\\\\0\\\\0\\end{pmatrix} + \\begin{pmatrix}J_p & J_l \\\\ \\sqrt{\\lambda}D_p & 0 \\\\ 0 & \\sqrt{\\lambda}D_l\\end{pmatrix}\\begin{pmatrix}\\Delta\\vect{x}_p \\\\ \\Delta\\vect{x}_l\\end{pmatrix}\\right\\Vert^2 = \\\\\n    \\min_{\\Delta\\vect{x}_p, \\Delta\\vect{x}_l}&\\bigg(\\Vert \\vect{r} + \\begin{pmatrix}J_p & J_l\\end{pmatrix}\\begin{pmatrix}\\Delta\\vect{x}_p \\\\ \\Delta\\vect{x}_l\\end{pmatrix}\\Vert^2  \\\\\n    & + \\lambda \\Vert D_p \\Delta\\vect{x}_p \\Vert^2 + \\lambda\\Vert D_l \\Delta\\vect{x}_l \\Vert^2\\bigg) \\,,\n\\end{aligned}\n\\end{align}\nwith $\\vect{r}=\\vect{r}(\\vect{x}^0)$, $J_p = \\left.\\frac{\\partial \\vect{r}}{\\partial \\vect{x}_p}\\right\\vert_{\\vect{x}^0}$, $J_l = \\left.\\frac{\\partial \\vect{r}}{\\partial \\vect{x}_l}\\right\\vert_{\\vect{x}^0}$, and $\\Delta\\vect{x}=\\vect{x}-\\vect{x}^0$.\nHere, $\\lambda$ is a damping coefficient and $D_p$ and $D_l$ are diagonal damping matrices for pose and landmark variables (often $D^2=\\text{diag}(J^\\top J)$). \n\nTo simplify notation, in this section we consider the undamped problem (i.e., $\\lambda=0$) and discuss the efficient application of damping in Section \\ref{sec:lm_damping}.\nThe undamped problem in \\eqref{eq:linearized_residual} can be solved by forming the corresponding normal equation\n\\begin{align}\n\\label{eq:normal_eq}\n\\begin{pmatrix}\nH_{pp} & H_{pl} \\\\\nH_{lp} & H_{ll}\n\\end{pmatrix}\n\\begin{pmatrix}\n-\\Delta \\vect{x}_{p}  \\\\\n-\\Delta \\vect{x}_{l}\n\\end{pmatrix}\n &= \n \\begin{pmatrix}\n\\vect{b}_{p}  \\\\\n\\vect{b}_{l}\n\\end{pmatrix},\n\\end{align}\nwhere\n\\begin{align}\nH_{pp} &= J_p^\\top J_p\\,, \\quad\nH_{ll} = J_l^\\top J_l\\,, \\\\\nH_{pl} &= J_p^\\top J_l = H_{lp}^\\top\\,, \\\\\n\\vect{b}_{p} &= J_p^\\top \\vect{r}\\,, \\quad\n\\vect{b}_{l} = J_l^\\top \\vect{r}\\,.\n\\end{align}\nThe system matrix $H$ of this problem is of size $(N_p+3n_l)^2$, which can become impractically large (millions of rows and columns) for problems like those in~\\cite{agarwal2010bundle} (see Figure~\\ref{fig:teaser} for an example).\n\n\\subsection{Schur complement (SC)}\n\nA very common way to solve \\eqref{eq:normal_eq} is by applying the Schur complement trick (see e.g.,~\\cite{brown1958solution,agarwal2010bundle,wu2011multicore}):\nwe form the RCS\n\\begin{align}\n\\label{eq:red_normal_eq}\n\\tilde{H}_{pp} (- \\Delta\\vect{x}_p) = \\tilde{\\vect{b}}_p\\,,\n\\end{align}\nwith\n\\begin{align}\n\\tilde{H}_{pp} &= H_{pp} - H_{pl} H_{ll}^{-1} H_{lp}\\,, \\\\\n\\tilde{\\vect{b}}_p &= \\vect{b}_p - H_{pl} H_{ll}^{-1}\\vect{b}_l\\,.\n\\end{align}\nThe solution $\\Delta\\vect{x}_p^*$ of \\eqref{eq:red_normal_eq} is the same as the pose component of the solution of \\eqref{eq:normal_eq}, but now the system matrix has a much more tractable size of $N_p^2$, which is usually in the order of thousands $\\times$ thousands.\nNote that as $H_{ll}$ is block-diagonal with blocks of size $3\\times 3$, the multiplication with $H_{ll}^{-1}$ is cheap.\nGiven an optimal pose update $\\Delta\\vect{x}_p^*$, the optimal landmark update is found by back substitution\n\\begin{align}\n\\label{eq:backsub_sc}\n    - \\Delta\\vect{x}_{l}^* &=  H_{ll}^{-1} (\\vect{b}_{l} - H_{lp} (-\\Delta\\vect{x}_{p}^*))\\,.\n\\end{align}\n\n\n\\subsection{Nullspace marginalization (NM)}\n\\label{sec:nullspace-marginalization}\n\nUsing QR decomposition on $J_l=QR$, and the invariance of the L2 norm under orthogonal transformations, we can rewrite the term in \\eqref{eq:linearized_residual}:\n\\begin{align}\n\\label{eq:q_linearized_residual}\n\\begin{aligned}\n &\\Vert \\vect{r} + \\begin{pmatrix}J_p & J_l\\end{pmatrix}\\begin{pmatrix}\\Delta\\vect{x}_p \\\\ \\Delta\\vect{x}_l\\end{pmatrix}\\Vert^2 \\\\\n &= \\Vert Q^\\top\\vect{r} + \\begin{pmatrix}Q^\\top J_p & Q^\\top J_l\\end{pmatrix}\\begin{pmatrix}\\Delta\\vect{x}_p \\\\ \\Delta\\vect{x}_l\\end{pmatrix}\\Vert^2 \\\\\n &= \\Vert Q_1^\\top\\vect{r} + Q_1^\\top J_p\\Delta\\vect{x}_p + R_1\\Delta\\vect{x}_l\\Vert^2 \\\\\n &\\qquad + \\Vert Q_2^\\top\\vect{r} + Q_2^\\top J_p\\Delta\\vect{x}_p\\Vert^2\\,.\n\\end{aligned}\n\\end{align}\nAs $R_1$ is invertible, for a given $\\Delta\\vect{x}_p^*$, the first term can always be set to zero (and thus minimized) by choosing\n\\begin{align}\n\\label{eq:backsub_nm}\n    \\Delta\\vect{x}_l^* = -R_1^{-1} (Q_1^\\top \\vect{r} + Q_1^\\top J_p \\Delta\\vect{x}_{p}^*)\\,.\n\\end{align}\nSo \\eqref{eq:linearized_residual} reduces to minimizing the second term in \\eqref{eq:q_linearized_residual}:\n\\begin{align}\n\\min_{\\Delta\\vect{x}_p}{\\Vert Q_2^\\top \\vect{r} + Q_2^\\top J_p \\Delta \\vect{x}_{p} \\Vert^2}\\,.\n\\label{eq:reduced}\n\\end{align}\nAgain, this problem is of significantly smaller size than the original one.\nHowever, as opposed to the (explicit) Schur complement trick, we do not explicitly have to form the Hessian matrix.\n\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{figures\/damping3.pdf}\n  \\caption{Illustration of the landmark damping in the Levenberg-Marquardt optimization. (a) We add three zero rows with diagonal damping for landmarks to the marginalized landmark block. (b) With 6 Givens rotations we eliminate the values on diagonal, which gives us a new landmark block with marginalized out landmark. (c) By applying the transposed rotations in reverse order and zeroing out the diagonal we can bring the landmark block to the original state. Zero entries of the landmark block are shown in white, parts that change are shown in green, and parts that stay unchanged are shown in blue.}\n  \\label{fig:damping}\n\\end{figure*}\n\n\n\\subsection{Equivalence of SC and NM}\n\nWith the QR decomposition $J_l = Q_1 R_1$ used in the last paragraphs, we get\n\\begin{align}\nH_{pp} &= J_p^\\top J_p\\,, \\\\\nH_{pl} &= J_p^\\top Q_1 R_1\\,, \\\\\nH_{ll} &= R_1^\\top Q_1^\\top Q_1 R_1 = R_1^\\top R_1\\,, \\\\\n\\vect{b}_{p} &= J_p^\\top \\vect{r}\\,, \\\\\n\\vect{b}_{l} &= R_1^\\top Q_1^\\top \\vect{r}\\,. \\\n\\end{align}\nUsing this, we can rewrite the Schur complement matrix $\\tilde{H}_{pp}$ and vector $\\tilde{\\vect{b}}_p$ and simplify with \\eqref{eq:qqt}:\n\\begin{align}\n&\\begin{aligned}\n\\label{eq:red_H}\n\\tilde{H}_{pp} &= H_{pp} - J_p^\\top Q_1 R_1 (R_1^\\top R_1)^{-1} R_1^\\top Q_1^\\top J_p \\\\\n&= H_{pp} - J_p^\\top (\\id_{N_r} - Q_2 Q_2^\\top) J_p \\\\\n&= J_p^\\top Q_2 Q_2^\\top J_p\\,,\n\\end{aligned}\\\\\n&\\begin{aligned}\n\\label{eq:red_b}\n\\tilde{\\vect{b}}_p &= \\vect{b}_p - J_p^\\top Q_1 R_1 (R_1^\\top R_1)^{-1} R_1^\\top Q_1^\\top \\vect{r} \\\\\n&= J_p^\\top Q_2 Q_2^\\top \\vect{r}\\,.\n\\end{aligned}\n\\end{align}\nThus, the SC-reduced equation is nothing but the normal equation of problem \\eqref{eq:reduced}, which proves the algebraic equivalence of the two marginalization techniques. \nAdditionally, we can show that the equations for back substitution for Schur complement (\\ref{eq:backsub_sc}) and nullspace marginalization (\\ref{eq:backsub_nm}) are also algebraically equivalent:\n \\begin{align}\n \\begin{aligned}\n  \\Delta\\vect{x}_{l}^* &=  -H_{ll}^{-1} (\\vect{b}_{l} + H_{lp} \\Delta\\vect{x}_{p}^*) \\\\\n  &=  -(R_1^\\top R_1)^{-1} (R_1^\\top Q_1^\\top \\vect{r} + (J_p^\\top Q_1 R_1)^\\top \\Delta\\vect{x}_{p}^*) \\\\\n  &=  -R_1^{-1} (Q_1^\\top \\vect{r} + Q_1^\\top J_p \\Delta\\vect{x}_{p}^*)\\,.\n \\end{aligned}\n \\end{align}\n\nNote that the above arguments also hold for the damped problem \\eqref{eq:linearized_residual}, the difference being that the Hessian will have an augmented diagonal and that the matrices in the QR decomposition will have a larger size.\n\n\n\\section{Implementation details}\n\nBundle adjustment is a very structured problem, so we can take advantage of the problem-specific matrix structures to enable fast and memory-efficient computation.\n\n\\subsection{Storage}\n\nWe group the residuals by landmarks, such that $J_l$ has block-sparse structure, where each block is $2k_j\\times 3$ with $k_j$ the number of observations for a particular landmark, see Figure~\\ref{fig:sparsity} (a).\nAs each landmark is only observed by a subset of cameras, the pose Jacobian $J_p$ is also sparse.\n\nWe group the rows corresponding to each landmark and store them in a separate dense memory block, which we name a \\emph{landmark block}.\nWe store only the blocks of the pose Jacobian that correspond to the poses where the landmark was observed, because all other blocks will always be zero. For convenience we also store the landmark's Jacobians and residuals in the same landmark block, as shown in Figure~\\ref{fig:sparsity} (b).\nThis storage scheme can be applied both to the undamped and the damped Jacobian (see Section \\ref{sec:lm_damping} for damping).\n\n\n\\subsection{QR decomposition}\n\nApplying a sequence of Givens rotations in-place transforms the landmark block to the marginalized state shown in Figure~\\ref{fig:sparsity} (c). The bottom part corresponds to the reduced camera system, and the top part can be used for back substitution. This transformation can be applied to each block independently, possibly in parallel.\nWe never have to explicitly store or compute the matrix $Q$; we simply apply the sequence of Givens rotations to the landmark block one by one, as they are computed.\nNote that alternatively we can use three Householder reflections per landmark block, with which we noticed a minor improvement in runtime.\n\n\\subsection{Levenberg-Marquardt damping}\n\\label{sec:lm_damping}\n\nThe augmentation of the Jacobians by diagonal matrices as used in \\eqref{eq:linearized_residual} consists of two parts that we treat differently to optimally exploit the nature of the BA problem in our implementation.\n\n\\paragraph{Landmark damping}\nWe first look at damping the landmark variables:\nrather than actually attaching a large diagonal matrix $\\sqrt{\\lambda}D_l$ to the full landmark Jacobian $J_l$, we can again work on the landmark block from Figure~\\ref{fig:sparsity} (b) and only attach a $3\\times 3$ sub-block there, \nsee Figure~\\ref{fig:damping} (a) and (b).\nTo simplify the expressions in figures, we slightly abuse notation when considering a single landmark and denote the corresponding parts of $J_p$, $J_l$ and $r$ in the landmark block by the same symbols.\nThe matrices involved in the QR decomposition of the undamped system are $Q_1$, $Q_2$, $R_1$ and those for the damped system are marked with a hat.\nNote that $Q$ and $\\hat{Q}$ are closely related; the additional three diagonal entries in the damped formulation can be zeroed using only six Givens rotations, such that\n\\begin{align}\n\\hat{Q} = \\begin{pmatrix}\\hat{Q}_1 & \\hat{Q}_2  \\end{pmatrix} = \\begin{pmatrix}Q_1 & Q_2  & 0 \\\\ 0 & 0 & \\id_{3} \\end{pmatrix}Q_\\lambda\n\\,,\n\\end{align}\nwhere $Q_\\lambda$ is a product of six Givens rotations.\nThus, applying and removing landmark damping is computationally cheap: we apply the Givens rotations one by one and store them individually (rather than their product $Q_\\lambda$) to undo the damping later.\nFigure~\\ref{fig:damping} illustrates how this can be done in-place on the already marginalized landmark block.\nThis can speed up LM's backtracking, where a rejected increment is recomputed with the same linearization, but with increased damping.\nBy contrast, for an explicit SC solver, changing the damping would mean recomputing the Schur complement from scratch.\n\n\\paragraph{Pose damping}\nMarginalizing landmarks using Givens rotations in the damped formulation of \\eqref{eq:linearized_residual} does not affect the rows containing pose damping.\nThus, it is still the original diagonal matrix $\\sqrt{\\lambda}D_p$ that we append to the bottom of $\\hat{Q}^\\top J_p$:\n\\begin{align}\n    \\hat{J}_p = \n    \\begin{pmatrix}\n    \\hat{Q}_1^\\top J_p \\\\ \\hat{Q}_2^\\top J_p \\\\ \\sqrt{\\lambda}D_p\n    \\end{pmatrix}\\,.\n\\end{align}\nIn practice, we do not even have to append the block, but can simply add the corresponding summand when evaluating matrix-vector multiplication for the CG iteration \\eqref{eq:cg_mult}.\n\n\\subsection{Conjugate gradient on normal equations}\n\nTo solve for the optimal $\\Delta x_p^*$ in small and medium systems, we could use dense or sparse QR decomposition of the stacked $Q_2^{\\top} J_p$ from landmark blocks to minimize the linear least squares objective $\\Vert Q_2^{\\top} J_p \\Delta x_p + Q_2^{\\top} r\\Vert^2$. However, for large systems this approach is not feasible due to the high computational cost.\nInstead, we use CG on the normal equations as proposed in~\\cite{agarwal2010bundle}.\nOther iterative solvers like LSQR~\\cite{paige1982lsqr} or LSMR~\\cite{fong2011lsmr} that can be more numerically stable than CG turned out to not improve stability for the case of bundle adjustment~\\cite{byrod2010conjugate}.\n\nCG accesses the normal equation matrix $\\tilde{H}_{pp}$ only by multiplication with a vector $\\vect{v}$, which we can write as\n\\begin{align}\n\\label{eq:cg_mult}\n\\tilde{H}_{pp} ~ v = (\\hat{Q}_2^\\top J_p)^\\top (\\hat{Q}_2^\\top J_p ~ v) + \\lambda D^2_p ~ v\n\\,.\n\\end{align}\nThis multiplication can be efficiently implemented and well parallelized using our array of landmark blocks.\nThus, we do not need to explicitly form the normal equations for the reduced least squares problem.\n\nStill, the CG part of our solver has the numerical properties of the normal equations (squared condition number compared to the marginal Jacobian $\\hat{Q}_2^\\top J_p$). To avoid numeric issues when using single-precision floating-point numbers, we scale the columns of the full Jacobian to have unit norm and use a block-Jacobi preconditioner, both standard procedures when solving BA problems and both also used in the other evaluated solvers.\nWe also note that with the Levenberg-Marquardt algorithm, we solve a strictly positive definite damped system, which additionally improves the numeric stability of the optimization.\n\nStoring the information used in CG in square root form allows us to make sure that $\\tilde{H}_{pp}$ is always strictly positive definite. As we show with our experiments (see Section \\ref{sec:analysis}), for many sequences small round-off errors during SC (explicit or implicit) render $\\tilde{H}_{pp}$ to be numerically indefinite with single-precision floating-point computations.\n\nWith the computed $\\Delta\\vect{x}_{p}^*$ we can do back substitution for each individual landmark block independently and in parallel. We already have all the necessary information ($\\hat{Q}_1^\\top J_p$, $\\hat{R}$, $\\hat{Q}_1^\\top r$) stored in the landmark blocks \nafter marginalization.\n\n\\subsection{Parallel implementation}\nAs pointed out above, the linearization, marginalization, and back substitution can be computed independently for each landmark block. There is no information shared between landmark blocks, so we can use a simple \\emph{parallel for} loop to evenly distribute the workload between all available CPU cores.\nThe matrix-vector multiplications that constitute the most computationally expensive part of CG can also be efficiently parallelized. In this case, multiplication results of individual landmark blocks have to be summed, so we employ the common \\emph{parallel reduce} pattern.\nHow effective these simple parallelization schemes are is underlined by our evaluation, which shows excellent runtime performance of the square root bundle adjustment implementation, compared to both our custom and Ceres' SC solvers.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.98\\textwidth]{results\/performance_profile_all.pdf}\n    \\caption{Performance profiles for all BAL datasets show percentage of problems solved to a given accuracy tolerance $\\tau$ with increasing relative runtime $\\alpha$.\n    Our proposed $\\sqrt{BA}$ solver outperforms other methods across all accuracy tolerances.\n    In single precision, the solver is about twice as fast as with double, but does not lose in accuracy, underpinning the favorable numerical properties of the square root formulation.\n    In contrast, while the SC solver in double precision is equally accurate, this is not the case for the single-precision variant. Keep in mind that Ceres is not as tailored to the exact problem setup as our custom implementation, possibly explaining its slower performance. Note that the performance profiles are cut off at the right side of the plots to show only the most relevant parts.\n    }\n    \\label{fig:performance_all}\n\\end{figure*}\n\n\n\n\\section{Experimental evaluation}\n\n\\subsection{Algorithms and setup}\n\n\\begin{table}\n\\setlength{\\tabcolsep}{0.3em}\n\\centering\n\\begin{tabular}{l|c;{1pt\/1pt}c;{1pt\/1pt}c;{1pt\/1pt}c|c;{1pt\/1pt}c|}\n& \\rotatebox{90}{\\emph{$\\sqrt{BA}$-32} (ours)} \n& \\rotatebox{90}{\\emph{$\\sqrt{BA}$-64} (ours)} \n& \\rotatebox{90}{\\emph{explicit-32}} \n& \\rotatebox{90}{\\emph{explicit-64}}\n& \\rotatebox{90}{\\emph{ceres-implicit}}\n& \\rotatebox{90}{\\emph{ceres-explicit}}\n\\\\\n\\hline\nsolver implementation  & \\multicolumn{4}{c|}{custom} & \\multicolumn{2}{c|}{Ceres} \\\\\n\\hline\nfloat precision  & s & d & s & d & \\multicolumn{2}{c|}{d} \\\\\n\\hline\nlandmark marginalization  & \\multicolumn{2}{c;{1pt\/1pt}}{NM} & \\multicolumn{2}{c|}{SC} & \\multicolumn{2}{c|}{SC} \\\\\n\\hline\nRCS storage  & \\multicolumn{2}{c;{1pt\/1pt}}{LMB} & \\multicolumn{2}{c|}{H} & -- & H\\\\\n\\hline\n\\end{tabular}\n\\vspace{0.3cm}\n\\caption{The evaluated solvers---proposed and baseline---are implemented either completely in our custom code base, or using Ceres, with single (s) or double (d) floating-point precision, using Nullspace (NM) or Schur complement (SC)-based marginalization of landmarks, and storing the reduced camera system sparsely in landmark blocks (LMB), sparsely as a reduced Hessian (H), or not at all (--).\n}\n\\label{tab:solver_properties}\n\\end{table}\n\n\n\nWe implement our $\\sqrt{BA}$ solver in C++ in single (\\emph{$\\sqrt{BA}$-32}) and double (\\emph{$\\sqrt{BA}$-64}) floating-point precision\nand compare it to the methods proposed in \\cite{agarwal2010bundle}\nas implemented in Ceres Solver \\cite{ceres-solver}.\nThis solver library is popular in the computer vision and robotics community, \nsince it is mature, performance-tuned, and offers many linear solver variations.\nThat makes it a relevant and challenging baseline to benchmark our implementation against.\nWhile Ceres is a general-purpose solver,\nit is very good at exploiting the specific problem structure as\nit was built with BAL problems in mind.\nOur main competing algorithms are Ceres' sparse Schur complement solvers, \nwhich solve the RCS iteratively by either explicitly saving $\\tilde{H}_{pp}$ in memory as a block-sparse matrix (\\emph{ceres-explicit}), \nor otherwise computing it on the fly during the iterations (\\emph{ceres-implicit}).\nIn both cases, the same block-diagonal of $\\tilde{H}_{pp}$ that we use in $\\sqrt{BA}$ is used as preconditioner.\nAs the bottleneck is not computing Jacobians, but marginalizing points and the CG iterations, we use analytic Jacobians for our custom solvers and Ceres' exact and efficient autodiff with dual numbers.\nFor an even more direct comparison,\nwe additionally implement the sparse iterative explicit Schur complement solver without Ceres, sharing much of the code with our $\\sqrt{BA}$ implementation. While Ceres always uses double precision, \nwe use our custom implementation to evaluate numerical stability by considering single (\\emph{explicit-32}) and double (\\emph{explicit-64}) precision.\nTable~\\ref{tab:solver_properties} summarizes the evaluated configurations.\n\n\nFor Ceres we use default options, unless otherwise specified.\nThis includes the scaling of Jacobian columns to avoid numerical issues~\\cite{agarwal2010bundle}.\nJust like in our custom implementation, \nwe configure the number of threads to be equal to the number of (virtual) CPU cores. \nOur Levenberg-Marquardt loop is in line with Ceres: \nstarting with initial value $10^{-4}$, we update the damping factor $\\lambda$ according to the ratio of actual and expected cost reduction,\nand run it for at most 50 iterations, terminating early if a relative function tolerance of $10^{-6}$ is reached. \nIn the inner CG loop we use the same forcing sequence as Ceres, with a maximum of 500 iterations and no minimum.\nWe run experiments on an Ubuntu 18.04 desktop with 64GB RAM and an Intel Xeon W-2133 with 12 virtual cores at 3.60GHz.\nIn our own solver implementation we rely on Eigen \\cite{eigenweb} for dense linear algebra and TBB\\cite{tbbweb} for \nsimple \\emph{parallel for} and \\emph{parallel reduce} constructs.\n\n\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{c@{\\hskip 0.005\\textwidth}c@{\\hskip 0.005\\textwidth}c@{\\hskip 0.005\\textwidth}c}\n        \\includegraphics[width=0.24\\textwidth]{results\/ladybug138.png} & \n        \\includegraphics[width=0.24\\textwidth]{results\/trafalgar170.png} & \n        \\includegraphics[width=0.24\\textwidth]{results\/venice1158.png} & \n        \\includegraphics[width=0.24\\textwidth]{results\/final4585-1.png} \\\\\n        \\includegraphics[width=0.24\\textwidth]{results\/bal_cost_convergence_ladybug138.pdf} & \n        \\includegraphics[width=0.24\\textwidth]{results\/bal_cost_convergence_trafalgar170.pdf} & \n        \\includegraphics[width=0.24\\textwidth]{results\/bal_cost_convergence_venice1158.pdf} & \n        \\includegraphics[width=0.24\\textwidth]{results\/bal_cost_convergence_final4585.pdf}\n    \\end{tabular}\n    \\caption{Convergence plots from small to large problems and rendered optimized landmark point clouds. The $y$-axes show the total BA cost (log scale), and the horizontal lines indicate cost thresholds for the tolerances $\\tau \\in \\{ 10^{-1}, 10^{-2}, 10^{-3}\\}$.}\n    \\label{fig:convergence}\n\\end{figure*}\n\n\n\\subsection{Datasets}\n\nFor our extensive evaluation we use all 97 bundle adjustment problems from the BAL \\cite{agarwal2010bundle} project page.\nThese constitute initialized bundle adjustment problems and come in different groups: the \\emph{trafalgar}, \\emph{dubrovnik}, and \\emph{venice} problems originate from successive iterations in a skeletal SfM reconstruction of internet image collections \\cite{svoboda2018qrkit}. \nThey are combined with additional leaf images, which results in the thus denser \\emph{final} problems.\nThe \\emph{ladybug} sequences are reconstructions from a moving camera, but despite this we always model all camera intrinsics as independent, \nusing the suggested \\emph{Snavely} projection model with one focal length and two distortion parameters. Figure~\\ref{fig:convergence} visualizes some exemplar problems after they have been optimized.\n\nAs is common, we apply simple gauge normalization as preprocessing:\nwe center the point cloud of landmarks at the coordinate system origin and rescale to median absolute deviation of 100.\nInitial landmark and camera positions are then perturbed with small Gaussian noise.\nTo avoid close-to-invalid state, we remove all observations with a small or negative $z$ value in the camera frame, \ncompletely removing landmarks with less than two remaining observations.\nWe additionally employ the Huber norm with a parameter of 1~pixel for residuals (implemented with IRLS as in Ceres).\nThis preprocessing essentially follows Ceres' BAL examples. It is deterministic and identical for all solvers by using a fixed random seed and being computed on state in double precision, regardless of solver configuration.\n\n\\subsection{Performance profiles}\n\nWhen evaluating a solver, we are primarily interested in accurate optimization results.\nSince we do not have independent ground truth of correct camera positions, intrinsics, and landmark locations,\nwe use the bundle adjustment cost as a proxy for accuracy. Lower cost in general corresponds to better solutions.\nBut depending on the application, we may desire in particular low runtime, which can be a trade-off with accuracy.\nThe difficulty lies in judging the performance across several orders of magnitudes in problem sizes, cost values, and runtimes (for the BAL datasets the number of cameras $n_p$ ranges from 16 to 13682).\nAs proposed in prior work \\cite{kushal2012visibility,dolan2002benchmarking}, we therefore use performance profiles to evaluate both accuracy and runtime jointly.\nThe performance profile of a given solver maps the relative runtime~$\\alpha$ (relative to the fastest solver for each problem and accuracy) to the percentage of problems solved to accuracy~$\\tau$.\nThe curve is monotonically increasing, \nstarting on the left with the percentage of problems for which the solver is the fastest, \nand ending on the right with the percentage of problems on which it achieves the accuracy~$\\tau$ at all.\nThe curve that is more to the left indicates better runtime and the curve that is more to the top indicates higher accuracy.\nA precise definition of performance profiles is found in the appendix.\n\n\n\\subsection{Analysis}\n\\label{sec:analysis}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{results\/memory_ladybug.pdf}\n    \\caption{Memory consumption for the relatively sparse \\emph{ladybug} problems grows linearly with the problem size. The number of cameras here ranges from 49 to 1723.}\n    \\label{fig:memory_ladybug}\n\\end{figure}\n\nFigure~\\ref{fig:performance_all} shows the performance profiles with all BAL datasets for a range of tolerances $\\tau \\in \\{ 10^{-1}, 10^{-2}, 10^{-3}\\}$.\nWe can see that our proposed square root bundle adjustment solver \\emph{$\\sqrt{BA}$-64} is very competitive, yielding better accuracy than some SC-based iterative solvers, and often at a lower runtime.\n\\emph{$\\sqrt{BA}$-32} is around twice as fast and equally accurate, which highlights the good numerical stability of the square root formulation. It clearly outperforms all other solvers across all tolerances.\nThe fact that \\emph{\\mbox{explicit-32}} does not always reach the same accuracy as its double precision counterpart \\emph{explicit-64} indicates that SC-based solvers do not exhibit the same numerical stability. We also do not observe the same twofold speedup, which is related to the fact that \\emph{\\mbox{explicit-32}} does have to backtrack in the LM loop significantly more often to increase damping when the Schur complement matrix becomes indefinite due to round-off errors. This happens at least once for 84 out of the 97 problems with \\emph{explicit-32} and even with \\emph{explicit-64} for 7 of the problems. With $\\sqrt{BA}$, we have never encountered this.\n\nA similar conclusion can be drawn from the convergence plots in Figure~\\ref{fig:convergence}, which show a range of differently sized problems.\nFor the small \\emph{ladybug} as well as the medium and large \\emph{skeletal} problems, our solver is faster. \nEven on the large and much more dense \\emph{final4585}, the square root solver is competitive. In the square root formulation memory and thus to some degree also runtime grows larger for denser problems---in the sense of number of observations per landmark---since a specific landmark block grows quadratically in size with the number of its observations. This is in contrast to density in the sense of number of cameras co-observing at least one common landmark, as for the SC.\nStill, across all BAL datasets, only for the largest problem, \\emph{final13682}, where the landmarks have up to 1748 observations, does \\emph{$\\sqrt{BA}$-32} run out of memory.\nFor sparse problems, such as \\emph{ladybug}, one can see in Figure~\\ref{fig:memory_ladybug} that the memory grows linearly with the  number of landmarks, and for \\emph{$\\sqrt{BA}$-32} is similar to Ceres' iterative SC solvers.\nIn summary, while for very small problems we expect direct solvers to be faster than any of the iterative solvers, and for very large and dense problems implicit SC solvers scale better due to their memory efficiency~\\cite{agarwal2010bundle}, the proposed \\emph{$\\sqrt{BA}$} solver outperforms alternatives for medium to large problems, i.e., the majority of the BAL dataset.\n\n\n\\section{Conclusion}\n\nWe present an alternative way to solve large-scale bundle adjustment that marginalizes landmarks without having to compute any blocks of the Hessian matrix.\nOur square root approach $\\sqrt{BA}$ displays several advantages over the standard Schur complement, in terms of speed, accuracy, and numerical stability.\nWe have combined a very general theoretical derivation of nullspace marginalization with a tailored implementation that maximally exploits the specific structure of BA problems.\nExperiments comparing our solver to both a custom SC implementation and the state-of-the-art Ceres library show how $\\sqrt{BA}$ can handle single-precision floating-point operations much better than the Schur complement methods, outperforming all evaluated competing approaches. We see great potential in $\\sqrt{BA}$ to benefit other applications that play up to its strong performance on sparse problems, for example incremental SLAM.\n\n\\balance\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\n\\section{Additional details}\n\\label{sec:appendix_details}\n\n\\subsection{Givens rotations}\n\nThe QR decomposition of a matrix $A$ can be computed using Givens rotations.\nA Givens rotation is represented by a matrix $G_{ij}(\\theta)$ that is equal to identity except for rows and columns $i$ and $j$, where it has the non-zero entries\n\\begin{align}\n    \\begin{pmatrix}\n    g_{ii} & g_{ij} \\\\g_{ji} & g_{jj}\n    \\end{pmatrix}\n    =\n    \\begin{pmatrix}\n    \\cos\\theta & \\sin\\theta \\\\ -\\sin\\theta & \\cos\\theta\n    \\end{pmatrix}\\,.\n\\end{align}\n$G_{ij}(\\theta)$ describes a rotation by angle $\\theta$ in the $ij$-plane.\nThe multiplication of a matrix $A$ with $G_{ij}(\\theta)$ changes only two rows in $A$, leaving all other rows unchanged.\n$\\theta$ can be chosen such that the element $(i,j)$ of $G_{ij}(\\theta)A$ is zero:\n\\begin{align}\n    \\cos\\theta = \\frac{a_{jj}}{\\sqrt{a_{jj}^2+a_{ij}^2}}\\,, \\quad\n    \\sin\\theta = \\frac{a_{ij}}{\\sqrt{a_{jj}^2+a_{ij}^2}}\\,.\n\\end{align}\nBy subsequent multiplication of $A$ with Givens matrices, all elements below the diagonal can be zeroed (see~\\cite[p. 252]{golub13} for the full algorithm).\nAs all $G_{ij}(\\theta)$ are orthogonal by construction, their product matrix $Q$ is also orthogonal.\n\n\n\\subsection{Performance profiles}\n\nLet $\\mathcal{P}$ be a set of BA problems and $\\mathcal{S}$ be the set of evaluated solvers which we run according to the termination criteria (maximum number of iterations and function tolerance).\nFor a given problem $p \\in \\mathcal{P}$ and solver $s \\in \\mathcal{S}$, \nwe define the minimal cost value achieved by that solver after time $t$ as $f(p, s, t)$.\nThe smallest achieved cost by any solver for a specific problem is denoted by $f^*(p) := \\min_{s,t} f(p,s,t)$, \nwhich we use to define for a chosen accuracy tolerance $0 < \\tau < 1$ the cost threshold\n\\begin{align}\n  f_\\tau(p) := f^*(p) +  \\tau (f_0(p) - f^*(p)),\n\\end{align}\nwhere $f_0(p)$ is the initial cost of problem $p$.\nThe runtime for solver $s$ to achieve that threshold is\n\\begin{align}\n  t_\\tau(p, s) := \\min \\; \\{ t \\; | \\; f(p, s, t) \\leq f_\\tau(p) \\} \\cup \\{ \\infty \\}.\n\\end{align}\nWith this, the performance profile of a solver~$s$ is\n\\begin{align}\n  \\rho_\\tau(s, \\alpha) := 100 \\frac{|\\{ p \\; | \\; t_\\tau(p, s) \\leq \\alpha \\min_s t_\\tau(p, s) \\}|}{|\\mathcal{P}|}.\n\\end{align}\nIn other words, $\\rho_\\tau(s, \\alpha)$ maps the relative runtime~$\\alpha$ to the percentage of problems that $s$ has solved to accuracy~$\\tau$.\nThe curve is monotonically increasing, \nstarting on the left with $\\rho_\\tau(s, 1)$, the percentage of problems for which solver~$s$ is the fastest, \nand ending on the right with $\\max_{\\alpha} \\rho_\\tau(s, \\alpha)$, \nthe percentage of problems on which the solver~$s$ achieves the cost threshold $f_\\tau(p)$ at all.\nComparing different solvers, the curve that is more to the left indicates better runtime \nand the curve that is more to the top indicates higher accuracy.\n\n\n\\section{Algorithm complexities}\n\\label{sec:complexities}\n\n\\begin{table*}[t]\n    \\centering\n    \\renewcommand{\\arraystretch}{1.4}\n    \\begin{tabular}{l | l l l}\n         & $\\sqrt{BA}$ (ours) & explicit SC & implicit SC \\\\\n         \\hline\n        \\textbf{outer iterations} \\\\\n        Jacobian computation &\n        $\\mathcal{O}(\\mu_on_l)$ & $\\mathcal{O}(\\mu_on_l)$ & $\\mathcal{O}(\\mu_on_l)$ \\\\\n        Hessian computation &\n        $0$ & $\\mathcal{O}(\\mu_on_l)$ & $0$ \\\\\n        QR &\n        $\\mathcal{O}((\\mu_o^2+\\sigma_o^2)n_l)$ & $0$ & $0$ \\\\\n        \\textbf{middle iterations} \\\\\n        damping &\n        $\\mathcal{O}(\\mu_on_l)$ & $\\mathcal{O}(n_l+n_p)$ & $0$ \\\\\n        SC &\n        $0$ & $\\mathcal{O}((\\mu_o^2+\\sigma_o^2)n_l)$ & $0$ \\\\\n        preconditioner & $\\mathcal{O}((\\mu_o^2+\\sigma_o^2)n_l+n_p)$ & $\\mathcal{O}(n_p)$ & $\\mathcal{O}(\\mu_on_l+n_p)$ \\\\\n        back substitution & $\\mathcal{O}(\\mu_on_l)$ & $\\mathcal{O}(\\mu_on_l)$ & $\\mathcal{O}(\\mu_on_l)$ \\\\\n        \\textbf{inner iterations} \\\\\n        PCG & $\\mathcal{O}((\\mu_o^2+\\sigma_o^2)n_l+n_p)$ & $\\mathcal{O}(n_p^2)$ (worst case) & $\\mathcal{O}(\\mu_on_l+n_p)$ \\\\        \n    \\end{tabular}\n    \\caption{Complexities of the different steps in our $\\sqrt{BA}$ solver compared to the two SC solvers (explicit and implicit), expressed only in terms of $n_l$, $n_p$, $\\mu_o$, and $\\sigma_o$.\n    We split the steps into three stages:\n    outer iterations, i.e., everything that needs to be done in order to setup the least squares problem (once per linearization); middle iterations, i.e., everything that needs to be done within one Levenberg-Marquardt iteration (once per outer iteration if no backtracking is required or multiple times if backtracking occurs); inner iterations, i.e., everything that happens within one PCG iteration.}\n    \\label{tab:complexities}\n\\end{table*}\n\nIn Table~\\ref{tab:complexities}, we compare the theoretical complexity of our QR-based solver to the explicit and implicit Schur complement solvers in terms of number of poses $n_p$, number of landmarks $n_l$, and mean $\\mu_o$ and variance $\\sigma_o^2$ of the number of observations per landmark.\nNote that the total number $n_o$ of observations equals $\\mu_on_l$.\nWhile most of the entries in the table are easily determined, let us briefly walk through the not so obvious ones:\n\n\\paragraph{$\\sqrt{BA}$ (our solver)}\nAssume landmark $j$ has $k_j$ observations.\nWe can express the sum over $k_j^2$ by $\\mu_o$ and $\\sigma_o^2$:\n\\begin{gather}\n    \\sigma_o^2=\\operatorname{Var}(\\{k_j\\}) = \\frac{1}{n_l}\\sum_{j=1}^{n_l}{k_j^2} - (\\frac{1}{n_l}\\sum_{j=1}^{n_l}{k_j})^2\\,, \\\\\n    \\Rightarrow \\sum_{j=1}^{n_l}{k_j^2} = n_l(\\mu_o^2 + \\sigma_o^2)\\,.\n\\end{gather}\nThe sum over $k_j^2$ appears in many parts of the algorithm, as the dense landmark blocks $\\begin{pmatrix}\\hat{J}_p & \\hat{J}_l & \\hat{r}\\end{pmatrix}$ after QR decomposition have size $(2k_j+3)\\times(d_pk_j+4)$, where the number of parameters per camera $d_p$ is 9 in our experiments.\nIn the QR step, we need $6k_j$ Givens rotations per landmark (out of which 6 are for the damping), and we multiply the dense landmark block by $\\hat{Q}^\\top$, so we end up having terms $\\mathcal{O}(\\sum_j{k_j})$ and $\\mathcal{O}(\\sum_j{k_j^2})$, leading to the complexity stated in Table~\\ref{tab:complexities}.\nFor the block-diagonal preconditioner, each landmark block contributes summands to $k_j$ diagonal blocks, each of which needs a matrix multiplication with $\\mathcal{O}(k_j)$ flops, thus we have a complexity of $\\mathcal{O}(\\sum_jk_j^2)$.\nPreconditioner inversion can be done block-wise and is thus $\\mathcal{O}(n_p)$.\nIn the PCG step, we can exploit the block-sparse structure of $\\hat{Q}_2^\\top \\hat{J}_p$ and again have the $k_j^2$-dependency.\nBecause of the involved vectors being of size $2n_o+d_pn_p$ (due to pose damping), we additionally have a dependency on $2n_o+d_pn_p$.\nFinally, for the back substitution, we need to solve $n_l$ upper-triangular $3\\times 3$ systems and then effectively do $n_l$ multiplications of a $(3\\times d_pk_j)$ matrix with a vector, which in total is of order $\\mathcal{O}(\\sum_jk_j)$.\n\n\\paragraph{Explicit SC}\nThe first step is the Hessian computation.\nAs each single residual only depends on one pose and one landmark, the Hessian computation scales with the number of observations\/residuals.\nDamping is a simple augmentation of the diagonal and contributes terms $\\mathcal{O}(n_l)$ and $\\mathcal{O}(n_p)$.\nMatrix inversion of $H_{ll}$ for the Schur complement scales linearly with $n_l$, while the number of operations to multiply $H_{pl}$ by $H_{ll}^{-1}$ scales with the number of non-zero sub-blocks in $H_{pl}$, and thus with $n_o$.\nThe multiplication of this product with $H_{lp}$ involves matrix products of sub-blocks sized $(d_p\\times 3)$ and $(3\\times d_p)$ for each camera pair that shares a given landmark, i.e., $\\mathcal{O}(k_j^2)$ matrix products for landmark $j$.\nThe preconditioner can simply be read off from $\\tilde{H}_{pp}$, and its inversion is the same as for $\\sqrt{BA}$.\nThe matrices and vectors involved in PCG all have size $d_pn_p(\\times d_pn_p)$.\nThe sparsity of $\\tilde{H}_{pp}$ is not only determined by $n_p$, $n_l$, $\\mu_o$, and $\\sigma_o$, but would require knowledge about which cameras share at least one landmark. In the worst case, where each pair of camera poses have at least one landmark they both observe, $\\tilde{H}_{pp}$ is dense.\nThus, assuming $\\tilde{H}_{pp}$ as dense we get quadratic dependence on $n_p$.\nBack substitution consists of matrix inversion of $n_l$ blocks, and a simple matrix-vector multiplication.\n\n\\paragraph{Implicit SC}\nSince the Hessian matrix is not explicitly computed, we need an extra step to compute the preconditioner for implicit SC.\nFor each pose, we have to compute a $d_p\\times d_p$ block for which the number of flops scales linearly with the number of observations of that pose, thus it is $\\mathcal{O}(n_o)$ in total.\nPreconditioner damping contributes the $n_p$-dependency.\nAs no matrices except for the preconditioner are precomputed for the PCG iterations, but sparsity can again be exploited to avoid quadratic complexities, this part of the algorithm scales linearly with all three numbers (assuming the outer loop for the preconditioner computation is a \\emph{parallel for} over cameras, rather than a \\emph{parallel reduce} over landmarks).\nLastly, back substitution is again only block-wise matrix inversion and matrix-vector multiplications.\nWhile avoiding a dependency on $(\\mu_o^2+\\sigma_o^2)n_l$ in the asymptotic runtime seems appealing, the implicit SC method computes a sequence of five sparse matrix-vector products in each PCG iteration in addition to the preconditioner multiplication, making it harder to parallelize than the other two methods, which have only one (explicit SC) or two ($\\sqrt{BA}$) sparse matrix-vector products.\nThus, the benefit of implicit SC becomes apparent only for very large problems. \nAs our evaluation shows, for medium and large problems, i.e.\\ the majority in the BAL dataset, our $\\sqrt{BA}$ solver is still superior in runtime.\n\n\n\\section{Problem sizes}\n\\label{sec:problem_sizes}\n\nTable~\\ref{tab:problem-size} in Section~\\ref{sec:problem_sizes_table}\ndetails the size of the bundle adjustment problem for each instance in the BAL dataset (grouped into \\emph{ladybug}, the skeletal problems \\emph{trafalgar}, \\emph{dubrovnik}, and \\emph{venice}, as well as the \\emph{final} problems). \nBesides number of cameras $n_p$, number of landmarks $n_l$, and number of observations $n_r = \\frac{N_r}{2}$, \nwe also show indicators for \\emph{problem density}: \nthe average number of observations per camera \\emph{\\#obs \/ cam} (which equals $n_r \/ n_p$), \nas well as the average number of observations per landmark \\emph{\\#obs \/ lm}, including its standard deviation and maximum over all landmarks. \n\nIn particular, a high mean and variance of \\emph{\\#obs \/ lm} indicates that our proposed $\\sqrt{BA}$ solver may require a large amount of memory (see for example \\emph{final961}, \\emph{final1936}, \\emph{final13682}), \nsince the dense storage after marginalization in a landmark block is quadratic in the number of observations of that landmark.\nIf on the other hand the problems are sparse and the number of observations is moderate, the memory required by $\\sqrt{BA}$ grows only linearly in the number of observations, similar to SC-based solvers (see Figure~6 in the main paper).\n\n\n\\section{Convergence}\n\\label{sec:convergence}\n\n\\balance\n\nIn Section~\\ref{sec:convergence_plots}, each row of plots corresponds to one of the 97 bundle adjustment problems and contains from left to right a plot of optimized cost by runtime (like Figure~5 in the main paper) and by iteration, trust-region size (inverse of the damping factor~$\\lambda$) by iteration, number of CG iterations by (outer) iteration, and peak memory usage by iteration. \nThe cost plots are cut off at the top and horizontal lines indicate the cost thresholds corresponding to accuracy tolerances $\\tau \\in \\{10^{-1}, 10^{-2}, 10^{-3}\\}$ as used in the performance profiles. The plot by runtime is additionally cut off on the right at the time the fastest solver for the respective problem terminated.\n\nWe make a few observations that additionally support our claims in the main paper: all solvers usually converge to a similar cost, but for most problems our proposed $\\sqrt{BA}$ solver is the fastest to reduce the cost. On the other hand, memory use can be higher, depending on problem size and density (see Section~\\ref{sec:problem_sizes}).\nMissing plots indicate that the respective solver ran out of memory, which for example for \\emph{$\\sqrt{BA}$-32} happens only on the largest problem \\emph{final13682}, where the landmarks have up to 1748 observations.\nOur single precision solver \\emph{$\\sqrt{BA}$-32} runs around twice as fast as its double precision counterpart, since it is numerically stable and usually requires a comparable number of CG iterations.\nThis is in contrast to our custom SC solver, where the twofold speedup for single precision is generally not observed. \nThe good numeric properties are further supported by the evolution of the trust-region size approximately following that of the other solvers in most cases.\nFinally, for the smallest problems (e.g., \\emph{ladybug49}, \\emph{trafalgar21}, \\emph{final93}), the evolution of cost, trust-region size, and even number of CG iterations is often identical for all solvers for the initial 5 to 15 iterations, before numeric differences become noticeable.\nThis supports the fact that the different marginalization strategies are algebraically equivalent and that our custom solver implementation uses the same Levenberg-Marquardt strategy and CG forcing sequence as Ceres.\n\n\n\\onecolumn\n\n\\section{Problem sizes table}\n\\label{sec:problem_sizes_table}\n\n{\n\\setlength{\\LTcapwidth}{0.99\\textwidth}\n\\begin{longtable}{l r r r r r r r}%\n\\label{tab:problem-size}\n\\endfirsthead\n\\endhead\n\\toprule\n&\\multicolumn{1}{c}{\\#cam}&\\multicolumn{1}{c}{\\#lm}&\\multicolumn{1}{c}{\\#obs}&\\multicolumn{1}{c}{\\#obs \/ cam}&\\multicolumn{3}{c}{\\#obs \/ lm} \\\\\n&\\multicolumn{1}{c}{$(n_p)$}&\\multicolumn{1}{c}{$(n_l)$}&\\multicolumn{1}{c}{$(n_r)$}& \\multicolumn{1}{c}{mean} & \\multicolumn{1}{c}{mean} & \\multicolumn{1}{c}{std-dev} & \\multicolumn{1}{c}{max}\\\\\n\\midrule\n\\input{results_supplementary\/problem_size.tex}\n\\bottomrule\n\\caption{Size of the bundle adjustment problem for each instance in the BAL dataset.}\n\\end{longtable}\n}\n\n\n\\section{Convergence plots}\n\\label{sec:convergence_plots}\n\n\\subsection{Ladybug}\n\\label{sec:ladybug}\n\n\\begin{center}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug49}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug73}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug138}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug318}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug372}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug412}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug460}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug539}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug598}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug646}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug707}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug783}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug810}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug856}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug885}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug931}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug969}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1064}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1118}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1152}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1197}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1235}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1266}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1340}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1469}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1514}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1587}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1642}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1695}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_ladybug1723}\n\\end{center}\n\n\\subsection{Trafalgar}\n\\label{sec:trafalgar}\n\n\\begin{center}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar21}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar39}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar50}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar126}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar138}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar161}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar170}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar174}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar193}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar201}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar206}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar215}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar225}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_trafalgar257}\n\\end{center}\n\n\\subsection{Dubrovnik}\n\\label{sec:dubrovnik}\n\n\\begin{center}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik16}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik88}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik135}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik142}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik150}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik161}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik173}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik182}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik202}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik237}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik253}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik262}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik273}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik287}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik308}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_dubrovnik356}\n\\end{center}\n\n\\subsection{Venice}\n\\label{sec:venice}\n\n\\begin{center}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice52}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice89}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice245}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice427}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice744}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice951}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1102}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1158}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1184}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1238}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1288}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1350}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1408}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1425}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1473}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1490}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1521}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1544}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1638}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1666}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1672}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1681}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1682}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1684}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1695}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1696}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1706}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1776}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_venice1778}\n\\end{center}\n\n\\subsection{Final}\n\\label{sec:final}\n\n\\begin{center}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_final93}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_final394}\\\\\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_final871}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_final961}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_final1936}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_final3068}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_final4585}\n\\includegraphics[width=\\textwidth]{results_supplementary\/bal_cost_convergence_final13682}\n\\end{center}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Nomenclature}\n\\addcontentsline{toc}{section}{Nomenclature}\n\\subsection{Constant parameters}\n\\begin{IEEEdescription}[\\IEEEusemathlabelsep\\IEEEsetlabelwidth{${\\underline P _{mn}}$,${\\overline P _{mn}}$}]\n\\item[$r_{ij}, \\chi_{ij}$]   Resistance and reactance of line $i\\rightarrow j$.\n\\item[$\\underline p_i, \\overline p_i$]  Controllable active power limit at node $i$.\n\\item[$\\underline q_i, \\overline q_i$]  Controllable reactive power limit at node $i$.\n\\item[$\\underline v_i, \\overline v_i$]  Voltage safety limit at node $i$.\n\\item[$\\overline \\ell_{ij}$]   Current safety limit on line $i\\rightarrow j$.\n\\item[]\n\\item[$A_f, B_f, A_s$] Constant matrices in the feasibility problem.\n\\item[$\\gamma_f, \\gamma_s$] Constant vectors in the feasibility problem.\n\\item[$A_{y},\\!b_{y},\\!c_{q},\\!\\gamma_{q}$] Constant matrices and vectors in SOCP.\n\\item [$\\underline w, \\overline w$] Bounds for the initial polytope in Algorithm \\ref{alg:approximate-socp}. \n\\item [$\\delta,\\eta, \\eta'$] Positive parameters used in Algorithm \\ref{alg:remove-inexact}. \n\\end{IEEEdescription}\n\n\n\\subsection{Variables}\n\\begin{IEEEdescription}[\\IEEEusemathlabelsep\\IEEEsetlabelwidth{${\\underline P _{mn}}$,${\\overline P _{mn}}$}]\n\\item[$p_i, q_i$] Controllable power injection at node $i$.\n\\item[$w_i$] Renewable active power generation at node $i$.\n\\item[$v_i$] Squared voltage magnitude at node $i$.\n\\item[$\\ell_{ij}$] Squared current magnitude on line $i \\rightarrow j$.\n\\item[$P_{ij}, Q_{ij}$] Active and reactive power flows onto line $i \\rightarrow j$.\n\\item[]\n\\item[$x$] Vector of state variables $x:=(p,q,v,\\ell,P,Q)$.\n\\item[$z_s$, $z_q$, $\\tilde z_q$] Vectors of nonnegative slack variables.\n\\item[$y_{ij} \\in \\mathbb{R}^3$] Auxiliary variables in SOCP for line $i\\rightarrow j$. \n\\item[$\\mu_f, \\mu_y$] Dual variables for equality constraints. \n\\item[$\\lambda_s, \\lambda_q$] Dual variables for inequality constraints.\n\\end{IEEEdescription}\n\n\n\\subsection{Optimization problems, values, sets}\n\\begin{IEEEdescription}[\\IEEEusemathlabelsep\\IEEEsetlabelwidth{${\\underline P _{mn}}$,${\\overline P _{mn}}$}]\n\\item[$\\mathrm{FP}(w)$] Feasibility problem for renewable generation $w$.\n\\item[$\\mathrm{fp}(w)$] Minimum objective value of $\\mathrm{FP}(w)$.\n\\item[$\\mathcal{W}$] Dispatchable region of $w$, in which $\\mathrm{fp}(w)=0$.\n\\item[$\\mathrm{FP}'(w)$] SOCP relaxation of $\\mathrm{FP}(w)$.\n\\item[$\\mathrm{fp}'(w)$] Minimum objective value of $\\mathrm{FP}'(w)$.\n\\item[$\\mathcal{W}'$] SOCP-relaxed dispatchable region of $w$.\n\\item[]\n\\item[$\\mathrm{DP}'(w)$] Dual problem of  $\\mathrm{FP}'(w)$, also an SOCP.\n\\item [$D_w(\\mu,\\lambda)$] Dual objective function. \n\\item [$\\mathrm{dp}'(w)$] Maximum objective value of $\\mathrm{DP}'(w)$.\n\\item[$\\mathrm{DP}''(w,\\delta)$] Dual SOCP with feasible set tightened by $\\delta$.\n\\item[$\\mathrm{dp}''(w,\\delta)$] Maximum objective value of $\\mathrm{DP}''(w,\n\\delta)$.\n\\item[]\n\\item[$\\mathcal{W}'_{poly}$] Polytopic approximation of $\\mathcal{W}'$, by Algorithm \\ref{alg:approximate-socp}.\n\\item[$\\mathcal{\\tilde W}$] SOCP-inexact region of $w$.\n\\item[$\\mathcal{\\tilde W}_d$] Approximation of $\\mathcal{\\tilde W}$ using dual SOCP.\n\\item[$\\mathcal{\\tilde W}_{poly}$] Polytopic approximation of $\\mathcal{\\tilde W}$, by Algorithm \\ref{alg:remove-inexact}.\n\\end{IEEEdescription}\n\n\\section{Introduction}\n\\IEEEPARstart{W}{ith} the benefits of near-zero carbon emissions and low operating costs, distributed renewable power is experiencing tremendous expansion in recent years \\cite{ahmad2018dynamic}. Meanwhile, its volatile and intermittent features pose great challenges to electric grid operation, especially the distribution system. Unlike the bulk system, distribution system has few controllable units and a stronger coupling of active and reactive power flows due to its high resistance to reactance ratio \\cite{shen2022admissible}. This makes it even harder to accommodate the fluctuating renewable generations. Therefore, characterizing the renewable power capacities that can be safely hosted by a distribution network prior to its actual operation is vital. This necessitates finding all renewable power outputs that can ensure \\emph{solvability} of the power flow equations and satisfaction of \\emph{safety limits}.\n\nThe first requirement is \\emph{solvability} of the power flow equations. For a transmission network modeled by direct-current (DC) power flow, solvability is easy to check since a closed-form solution can be obtained \\cite{soroudi2015stochastic}. However, for distribution networks, the lossless DC model is not accurate enough since the distribution lines have higher resistance to reactance ratios. Some literature proved sufficient conditions under which the alternating current (AC) power flow equations are solvable, by utilizing Banach fixed-point theorem for contraction mappings \\cite{bolognani2015existence, wang2016explicit} or Brouwer fixed-point theorem for continuous mappings over compact convex sets \\cite{dvijotham2017solvability, simpson2017theory}. Nonetheless, those methods cannot be readily applied to output the dispatchable region since they are based on power flow equations and can hardly deal with inequality safety constraints.\n\nTo further take into account the second requirement, i.e., satisfaction of safety limits, optimization based methods were developed. Two well-known concepts are the do-not-exceed limit (DNEL) \\cite{zhao2014variable} and the dispatchable region \\cite{wei2014dispatchable}. The DNEL provides an allowable power interval for each renewable generator based on robust optimization. Data-driven approach \\cite{qiu2016data} and topology control \\cite{korad2015enhancement} were incorporated to improve the accuracy of DNEL. The correlation between different renewable generators is ignored in DNEL, so the obtained capacity regions can be conservative. The dispatchable region further considers those correlations and provides the exact region consisting of all renewable power outputs that can be accommodated. An adaptive constraint generation algorithm was proposed to generate the dispatchable region \\cite{wei2015real}. The interaction between different prosumers with renewable generators was considered in \\cite{chen2021energy}. Similarly, dispatchable region can be applied to quantify the allowable variation of loads based on Fourier\u2013Motzkin elimination \\cite{abiri2016loadability}. The above studies are based on DC power flow models.\n\nAs mentioned above, AC power flow model is a must for a distribution network. Reference \\cite{chen2018convex} solved nonlinear programs to get a set of boundary points that each make a different safety limit binding, and then built a dispatchable region heuristically as the convex hull of those boundary points. Linearized models were used to approximate the real dispatchable region under AC power flow \\cite{wan2016maximum, liu2019real}. However, there is no guarantee that all scenarios inside the obtained region are feasible. Reference \\cite{shen2022admissible} used the intersection of the dispatchable regions generated from two linearized models to output a more accurate approximation. To guarantee feasibility, certified inner approximations of dispatchable regions were solved from convex programs based on a tightened-relaxed second-order cone approximation \\cite{nick2017exact} or refined linear approximations \\cite{nguyen2018constructing, nazir2019convex} to AC power flow. \nHowever, such estimation typically only works for a specific objective function that merely explores the dispatchable region towards a single direction or with a specific shape of the renewable generation vector. Moreover, all the aforementioned regions are convex, while the actual dispatchable region can be nonconvex due to the AC power flow constraints.\n\nIn this paper, we propose an alternative method to complement the literature above. Our main contributions are two-fold:\n\n\\begin{itemize}\n\\item[1)] \\emph{Accurate Dispatchable Region of the Second-Order Cone (SOC) Relaxed Model.} A nonlinear Dist-Flow model based optimization problem is developed to characterize the dispatchable region, which is hard to solve due to its nonconvexity. Therefore, we first relax the problem to a convex second-order cone program (SOCP). Then, unlike reference \\cite{shen2022admissible} that further linearized the SOC constraint using polyhedral approximation \\cite{ben2001polyhedral}, we generate the dispatchable region directly without further linearization. To be specific, the dual problem of the SOCP is derived and strong duality holds as proven in Proposition \\ref{prop:strong-duality}. We then propose a projection algorithm (Algorithm \\ref{alg:approximate-socp}) to construct a polytopic approximation of the SOCP-relaxed dispatchable region. We prove that the approximation is accurate under certain conditions.\n\n\\item[2)] \\emph{Removal of SOCP-Inexact Regions.} The other inaccuracy lies in the possible inexactness of the SOC relaxation. In fact, the actual dispatchable region may be nonconvex, but the algorithms developed in previous studies can only generate convex regions. Distinctly, we propose a heuristic method to find out the SOCP-inexact regions by requiring the corresponding dual variables to be larger than some small positive values. Removing the SOCP-inexact regions from the region generated by Algorithm 1 from the SOC relaxed model, we can build a tighter approximation of the actual dispatchable region. The proposed method provides an innovative idea for constructing an accurate dispatchable region as the difference of several convex sets. Numerical results show that the proposed method can approximate the complicated dispatchable region with a simple polytope after moderate computation, while preserving relatively good accuracy. It can also reach a satisfactory balance between ensuring safety and reducing conservatism.\n\\end{itemize}\n\nThe rest of this paper is organized as follows. Section \\ref{sec:model} introduces the power network model we use. Section \\ref{sec:problem} defines the dispatchable region and the optimization problem to characterize it. Section \\ref{sec:methods} elaborates our method to approximate the dispatchable region. Section \\ref{sec:numerical} reports numerical experiments, and Section \\ref{sec:conclusion} concludes the paper.   \n\n\\section{Power network model}\\label{sec:model}\n\nConsider the single-phase equivalent model of a distribution network, which is a radial graph with a set $\\mathcal{N}$ of nodes and a set $\\mathcal{L}$ of lines. Index the nodes as $\\mathcal{N} = \\{0, 1,\\dots,N\\}$, where $0$ represents the root node (slack bus). \nFor convenience, we treat the lines as directed; for example, if a line connects nodes $i,j \\in \\mathcal{N}$, where node $i$ is closer to the root than node $j$, then the line directs from $i$ to $j$ and is denoted by $i \\rightarrow j$. The power flow in the network at a particular time instant can be modeled by the classic Dist-Flow equations purely in real numbers \\cite{baran1989optimal, farivar2013branch}, elaborated as follows. \n\nAt each node $i\\in \\mathcal{N}$: let $v_i$ denote the squared voltage magnitude; aggregate all the \\emph{controllable} power sources and loads into a complex power injection $p_i + \\mathrm{j} q_i$; denote the \\emph{uncontrollable} active power generation of a renewable energy source as $w_i$. \nLet $\\ell_{ij}$ denote the squared current magnitude through each line $i\\rightarrow j$. \nLet $P_{ij}$ and $Q_{ij}$ denote the \\textit{net} active and \\textit{net} reactive power, respectively, that are sent by node $i$ onto line $i\\rightarrow j$; they are different from the net power arriving at node $j$ due to power loss, and are negative if node $i$ receives power from line $i\\rightarrow j$. \nLet $r_{ij}$, $\\chi_{ij}$ denote the constant resistance and reactance of line $i\\rightarrow j$, respectively.\nThe Dist-Flow equations are:\n\\begin{subequations}\\label{eq:dist-flow}\n\\begin{IEEEeqnarray}{rrCl}\n\\forall i\\rightarrow j: \n&P_{ij} -  r_{ij} \\ell_{ij} - \\sum_{k: j\\rightarrow k} P_{jk}+ p_j+ w_j &=& 0  \\label{eq:dist-flow:p} \\\\\n&Q_{ij} - \\chi_{ij} \\ell_{ij} - \\sum_{k: j\\rightarrow k} Q_{jk}+ q_j &=& 0  \\label{eq:dist-flow:q} \\\\\nv_{i} -  v_{j}  & - 2(r_{ij} P_{ij} + \\chi_{ij} Q_{ij}) + (r_{ij}^2 + \\chi_{ij}^2) \\ell_{ij} &=& 0 \\label{eq:dist-flow:v} \\\\\n& P_{ij}^2 + Q_{ij}^2 - v_i \\ell_{ij} &=& 0. \\label{eq:dist-flow:s}\n\\end{IEEEeqnarray}\n\\end{subequations}\n\nSuppose renewable energy sources only exist at a subset of nodes $\\mathcal{N}_w \\subseteq \\mathcal{N}\\backslash \\{0\\}$, whose cardinality is $W:=|\\mathcal{N}_w|$. For nodes $i\\notin \\mathcal{N}_w$, set constant $w_i\\equiv 0$.  \nThe variables in Dist-Flow equations \\eqref{eq:dist-flow} are grouped as follows:\n\\begin{itemize}\n\\item Renewable power generation $w:=(w_i,~i\\in \\mathcal{N}_w)\\in \\mathbb{R}^W$, which is treated as input to the system;\n\\item State variables $x:=(p,q, v, \\ell, P, Q)$, where each of $p$, $q$, $v$, $\\ell$, $P$, $Q$ is a column vector indexed by $\\{1,...,N\\}$.      \n\\end{itemize}\n\n\\textit{Remark:} Without loss of generality, we assume there is only one node, indexed as node $1$, connected to the root node $0$. In this case, the power exchange between the distribution network and the upper grid at node $0$ is $p_0 + \\mathrm{j} q_0 = P_{01} +\\mathrm{j} Q_{01}$, so that it is just considered as part of $(P,Q)$, not $(p,q)$. \nAs customary, assume $v_0$ is a given constant and thus not in state variable $v$.  \nThe radial network has $N$ lines, where each line $i\\rightarrow j$ can be uniquely indexed by its destination node $j$, so that we can index line variables $\\ell$, $P$, $Q$ by $\\{1,...,N\\}$.\n\nAssume known capacity limits of controllable power:\n\\begin{subequations}\\label{eq:pq_limits}\n\\begin{IEEEeqnarray}{rCl}\n\\underline p_i \\leq p_i \\leq \\overline p_i, &\\quad& \\forall  i =1,...,N \\label{eq:pq_limits:p} \\\\\n\\underline q_i \\leq q_i \\leq \\overline q_i, && \\forall i =1,...,N \\label{eq:pq_limits:q} \n\\end{IEEEeqnarray}\n\\end{subequations}\nAt any node $i$ where there are only fixed (or zero) power injections, the constant limits can be set as $\\underline p_i = \\overline p_i$ ($=0$) and\/or $\\underline q_i = \\overline q_i$ ($=0$).\nIn addition, power system operations require the following safety limits to be satisfied:\n\\begin{subequations}\\label{eq:vl_limits}\n\\begin{IEEEeqnarray}{rCl}\n\\underline v_i \\leq v_i \\leq \\overline v_i, &\\quad& \\forall  i =1,...,N \\label{eq:vl_limits:v} \\\\\n0\\leq \\ell_{ij} \\leq \\overline \\ell_{ij}, && \\forall i\\rightarrow j  \\label{eq:vl_limits:ell} \n\\end{IEEEeqnarray}\n\\end{subequations}\nwhere the voltage limits $\\underline v_i$, $\\overline v_i$ for all nodes $i$ and the current limits $\\overline \\ell_{ij}$ for all lines $i\\rightarrow j$ are given as positive constants.\n\nWith the model above, we next define and analyze the dispatchable region of renewable power generation. \n\n\\section{Dispatchable region and relaxation}\\label{sec:problem}\n\nIn this paper, the \\emph{dispatchable region} is the region of renewable power generation $w$, for which there is a feasible dispatch. Its formal definition is provided below.\n\n\n\\begin{definition}\\label{def:feasibility}\nA vector of renewable power generation $w\\in\\mathbb{R}^W$ has a \\textbf{feasible dispatch} if there exists $x=(p,q,v,\\ell,P,Q) \\in \\mathbb{R}^{6N}$ such that $(w,x)$ satisfies power flow equations \\eqref{eq:dist-flow}, capacity limits \\eqref{eq:pq_limits}, and safety limits \\eqref{eq:vl_limits}. The \\textbf{dispatchable region} of renewable power generation is defined as:\n\\begin{IEEEeqnarray}{rCl}\\nonumber\n\\mathcal{W}&:=& \\left\\{w \\in \\mathbb{R}^W~|~w~\\text{has a feasible dispatch.}\\right\\}\n\\end{IEEEeqnarray}\n\\end{definition}\n\n\nFor conciseness, we rewrite the linear part \\eqref{eq:dist-flow:p}--\\eqref{eq:dist-flow:v} of Dist-Flow equations as\n$A_f x \\!+\\! B_{f} w \\!+\\! \\gamma_f = 0$ and affine inequalities \\eqref{eq:pq_limits}--\\eqref{eq:vl_limits} as $A_s x + \\gamma_s  \\leq 0$, where both equality and inequality are element-wise, and constant matrices and vectors $A_f$, $B_f$, $\\gamma_f$, $A_s$, $\\gamma_s$ are provided in Appendix-A.\nGiven any $w$, we introduce the following optimization to check its feasibility\n\\begin{subequations}\\label{eq:opt-feasibility}\n\\begin{IEEEeqnarray}{rCl}\n\\mathrm{FP}(w):~\\min &~& 1^\\intercal \\tilde z \\label{eq:opt-feasibility:obj}\n\\\\ \n\\text{over} && x=(p,q,v,\\ell,P,Q), ~\\tilde z=(z_s, z_q, \\tilde z_q)\\geq 0 \\nonumber\n\\\\\n\\text{s. t.} && A_f x + B_f w + \\gamma_f = 0 \\label{eq:opt-feasibility:lin-dist-flow}\n\\\\ &&  A_s x +\\gamma_s \\leq z_s  \\label{eq:opt-feasibility:limits}\n\\\\ && P_{ij}^2 + Q_{ij}^2 - v_i \\ell_{ij} \\leq z_{q,ij}, ~\\forall i\\rightarrow j \\label{eq:opt-feasibility:nonlinear-small}\n\\\\ && v_i \\ell_{ij}-(P_{ij}^2 + Q_{ij}^2)\\leq \\tilde z_{q,ij}, ~\\forall i\\rightarrow j \\label{eq:opt-feasibility:nonlinear-big}\n\\end{IEEEeqnarray} \n\\end{subequations}\nwhere $1^\\intercal$ in objective \\eqref{eq:opt-feasibility:obj} is a row vector of all ones. Any element of the slack variable $\\tilde z$ can increase as needed to satisfy the corresponding inequality constraint, but only $\\tilde z=0$ can guarantee feasibility in terms of \\eqref{eq:dist-flow}--\\eqref{eq:vl_limits}.   \nTherefore, denoting the minimum objective value of $\\mathrm{FP}(w)$ as $\\mathrm{fp}(w)$, the dispatchable region in Definition \\ref{def:feasibility} is equivalently:\n\\begin{IEEEeqnarray}{rCl}\\nonumber\n\\mathcal{W}&=& \\left\\{w \\in \\mathbb{R}^W~|~\\mathrm{fp}(w)=0 \\right\\}.\n\\end{IEEEeqnarray}\n\nDue to the nonconvex quadratic inequality constraint \\eqref{eq:opt-feasibility:nonlinear-big}, problem $\\mathrm{FP}(w)$ is nonconvex and thus hard to analyze. \nBy removing \\eqref{eq:opt-feasibility:nonlinear-big} and rewriting \\eqref{eq:opt-feasibility:nonlinear-small}, we relax $\\mathrm{FP}(w)$ to a convex \\emph{second order cone program (SOCP)}:\n\\begin{subequations}\\label{eq:opt-feasibility-relaxed}\n\\begin{IEEEeqnarray}{rCl}\n\\mathrm{FP}'(w):~\\min &~& 1^\\intercal z \\label{eq:opt-feasibility-soc:obj}\n\\\\ \n\\text{over} && x,~y,~z=(z_s, z_q)\\geq 0 \\nonumber\n\\\\\n\\text{s. t.} && \\text{\\eqref{eq:opt-feasibility:lin-dist-flow}--\\eqref{eq:opt-feasibility:limits}} \\nonumber \\\\\n&& y = A_{y} x + b_{y} \\label{eq:opt-feasibility:soc-substitute}\\\\\n \\| y_{ij} \\|_2 &&\\leq c_{q,ij} x + \\gamma_{q,ij}+z_{q,ij},~  \\forall i\\rightarrow j \\label{eq:opt-feasibility:soc}\n\\end{IEEEeqnarray} \n\\end{subequations}\nwhere $y\\in \\mathbb{R}^{3N}$, $A_y \\in\\mathbb{R}^{(3N)\\times(6N)}$, and $b_y \\in \\mathbb{R}^{3N}$ vertically stack $y_{ij}\\in\\mathbb{R}^3$,  $A_{y,ij} \\in \\mathbb{R}^{3\\times(6N)}$, and $b_{y,ij}\\in \\mathbb{R}^3$ respectively for all lines $i\\rightarrow j$. Row vector $c_{q,ij} \\in \\mathbb{R}^{1\\times(6N)}$ and scalar number $\\gamma_{q,ij} \\in \\mathbb{R}$ are also stacked vertically for all $i\\rightarrow j$ as $c_q \\in \\mathbb{R}^{N\\times (6N)}$ and $\\gamma_q \\in \\mathbb{R}^{N}$. The constant matrices and vectors $A_{y}$, $b_{y}$, $c_{q}$, $\\gamma_{q}$ are provided in Appendix-B, which make: \n\\begin{IEEEeqnarray}{rCl}\nA_{y,ij} x + b_{y,ij} &=& [2P_{ij}, ~2Q_{ij}, ~ v_i \\!-\\! \\ell_{ij}]^\\intercal,\\quad \\forall i\\rightarrow j \\nonumber \\\\\nc_{q,ij} x + \\gamma_{q,ij} &=& v_i + \\ell_{ij}, \\qquad\\qquad\\qquad\\quad \\forall i\\rightarrow j  \\nonumber\n\\end{IEEEeqnarray}\nand thus make \\eqref{eq:opt-feasibility:soc-substitute}--\\eqref{eq:opt-feasibility:soc} equivalent to \\eqref{eq:opt-feasibility:nonlinear-small}.\\footnote{Given $x$, the values of $z_q$ in \\eqref{eq:opt-feasibility:nonlinear-small} and \\eqref{eq:opt-feasibility:soc} are generally not equal, but we do not differentiate notation due to their identical role as slack variables.} \n\nProblem $\\mathrm{FP}'(w)$ facilitates the definition of an \\textit{SOCP-relaxed} dispatchable region:\n\\begin{IEEEeqnarray}{rCl}\\nonumber\n\\mathcal{W}'&:=& \\left\\{w \\in \\mathbb{R}^W~|~\\mathrm{fp}'(w)=0 \\right\\}\n\\end{IEEEeqnarray}\nwhere $\\mathrm{fp}'(w)$ is the minimum objective value of $\\mathrm{FP}'(w)$. It is obvious that~$\\mathcal{W} \\subseteq \\mathcal{W}'$, i.e., $\\mathcal{W}'$ is a relaxation of $\\mathcal{W}$.\n\nA common practice to further simplify the dispatchable-region characterization is to outer approximate the second-order cone \\eqref{eq:opt-feasibility:soc} with a polytopic cone, which can achieve arbitrary precision by constructing sufficiently many planes tangent to the surface of the second-order cone \\cite{ben2001polyhedral,chen2018energy}. Consequently, $\\mathrm{FP}'(w)$ is relaxed to a linear program, and then the algorithm in \\cite{wei2014dispatchable,wei2015real, chen2021energy} can be employed to get a convex polytopic outer approximation of $\\mathcal{W}'$. \nIn this work, we propose an alternative method that does not rely on such linearization. Instead, we work directly on the SOCP $\\mathrm{FP}'(w)$ and its dual problem to preserve the intrinsic nonlinearity of the AC power flow model and hence the accuracy of our characterization.\n\n\\section{Polytopic approximation algorithms}\\label{sec:methods}\n\nTo offer a closed-form approximation of dispatchable region $\\mathcal{W}$, we first develop a convex polytopic approximation of its relaxation $\\mathcal{W}'$ via the dual problem of SOCP $\\mathrm{FP}'(w)$. \nWe then develop a heuristic method to approximately remove the renewable generations that make the SOCP relaxation inexact, resulting in a tighter approximation of $\\mathcal{W}$.   \n\n\\subsection{Dual SOCP}\n\nLet $\\mu:=(\\mu_f, \\mu_y)$ denote the dual variables for the equality constraints in problem $\\mathrm{FP}'(w)$, with $\\mu_f \\in \\mathbb{R}^{3N}$ for \\eqref{eq:opt-feasibility:lin-dist-flow} and $\\mu_y\\in \\mathbb{R}^{3N}$ for \\eqref{eq:opt-feasibility:soc-substitute} vertically stacking $\\mu_{y,ij}\\in \\mathbb{R}^3,~\\forall i\\rightarrow j$.  \nLet $\\lambda:=(\\lambda_s, \\lambda_q)$ denote the dual variables for the inequality constraints, with $\\lambda_s \\in \\mathbb{R}^{8N}$ for \\eqref{eq:opt-feasibility:limits} and $\\lambda_q=(\\lambda_{q,ij},~\\forall i\\rightarrow j) \\in \\mathbb{R}^N$ for \\eqref{eq:opt-feasibility:soc}. Then the Lagrangian of $\\mathrm{FP}'(w)$ is:\n\\begin{IEEEeqnarray}{rCl}\\label{eq:lagrangian}\nL_u &=& 1^\\intercal z \\ + \\  \\mu_f^\\intercal (A_f x + B_f w + \\gamma_f) \\nonumber \\\\\n&& + \\lambda_s^\\intercal (A_s x + \\gamma_s - z_s) +  \\mu_{y}^\\intercal \\left(y -A_{y} x - b_{y}\\right) \\nonumber \\\\\n&&+  \\sum_{i\\rightarrow j} \\lambda_{q,ij} \\left(\\| y_{ij} \\|_2 -  c_{q,ij} x - \\gamma_{q,ij} - z_{q,ij}\\right) \\nonumber \\\\ \n&=&  z^\\intercal (1-\\lambda) + \\sum_{i\\rightarrow j} \\left(y_{ij}^\\intercal \\mu_{y,ij} +\\| y_{ij} \\|_2\\lambda_{q,ij}\\right)\\nonumber \\\\\n&&  + x^\\intercal \\left(A_f^\\intercal \\mu_f + A_s^\\intercal \\lambda_s  - A_{y}^\\intercal \\mu_{y} - c_{q}^\\intercal \\lambda_{q}\\right)   \\nonumber \\\\\n&&  + \\mu_f^\\intercal (B_f w + \\gamma_f)+\\lambda_s^\\intercal \\gamma_s  -  \\mu_{y}^\\intercal b_y -\\lambda_{q}^\\intercal \\gamma_q. \\label{eq:Lagrangian}\n\\end{IEEEeqnarray}\n\nThrough $\\min_{z\\geq 0, x, y} L_u(x,y,z; \\mu,\\lambda)$ we can get the dual objective function.\nBy \\eqref{eq:Lagrangian}, $L_u$ can only attain a finite minimum over $(z\\geq 0, x,y)$ when the dual variables satisfy: \n\\begin{subequations}\\label{eq:dual-feasibility}\n\\begin{IEEEeqnarray}{rCl}\n0 \\leq &~\\lambda~ & \\leq 1 \\label{eq:dual-feasibility:z} \\\\\nA_f^\\intercal \\mu_f + A_s^\\intercal \\lambda_s  &=&  A_y^\\intercal \\mu_y + c_q^\\intercal \\lambda_q \\label{eq:dual-feasibility:x}\\\\\n \\|\\mu_{y,ij}\\|_2 &\\leq& \\lambda_{q,ij},\\quad\\forall i\\rightarrow j\\label{eq:dual-feasibility:y}\n\\end{IEEEeqnarray}\n\\end{subequations}\nNote that $\\lambda\\geq 0$ in \\eqref{eq:dual-feasibility:z} is a general requirement for all the dual variables associated with inequality constraints, and \\eqref{eq:dual-feasibility:y} must hold by noticing\n\\begin{IEEEeqnarray}{rCl}\ny_{ij}^\\intercal \\mu_{y,ij} +\\| y_{ij} \\|_2\\lambda_{q,ij} &\\geq& \\left(\\lambda_{q,ij} - \\|\\mu_{y,ij}\\|_2 \\right)~ \\|y_{ij}\\|_2. \\nonumber\n\\end{IEEEeqnarray}\nWhen \\eqref{eq:dual-feasibility} is satisfied, all the terms containing $(x,y,z)$ in \\eqref{eq:Lagrangian} attain their minimum value zero, and hence we obtain the dual problem for $\\mathrm{FP}'(w)$, which is also an SOCP:\n\\begin{IEEEeqnarray}{rCl}\n\\mathrm{DP}'(w):~\\max_{\\mu,\\lambda} &\\quad& \\mu_f^\\intercal (B_f w + \\gamma_f) + \\lambda_s^\\intercal \\gamma_s -  \\mu_{y}^\\intercal b_y -\\lambda_{q}^\\intercal \\gamma_q \\nonumber\n\\\\\n\\text{s. t.} && \\text{\\eqref{eq:dual-feasibility}.} \\nonumber\n\\end{IEEEeqnarray} \n Let $D_w(\\mu,\\lambda)$ denote the objective function and $\\mathrm{dp}'(w)$ denote the maximum objective value of $\\mathrm{DP}'(w)$.\nThe following result lays the foundation for approximating the SOCP-relaxed dispatchable region $\\mathcal{W}'$ via the dual SOCP $\\mathrm{DP}'(w)$.\n\n\\begin{proposition}\\label{prop:strong-duality}\nFor all $w\\in\\mathbb{R}^W$, strong duality holds between $\\mathrm{FP}'(w)$ and $\\mathrm{DP}'(w)$, i.e., their optimal values $\\mathrm{fp}'(w) = \\mathrm{dp}'(w)$. \n\\end{proposition}\n\\begin{IEEEproof}\nConsider an arbitrary $w\\in\\mathbb{R}^W$. Since problem $\\mathrm{FP}'(w)$ is convex, it is sufficient to prove Slater's condition \\cite[Section 5.2.3]{boyd2004convex}, i.e., existence of $(z\\geq 0,x,y)$ that satisfies affine constraints \\eqref{eq:opt-feasibility:lin-dist-flow}\\eqref{eq:opt-feasibility:limits}\\eqref{eq:opt-feasibility:soc-substitute} and strictly satisfies \\eqref{eq:opt-feasibility:soc}. \n\nIndeed, it is adequate to find a point $x=(p,q,v,\\ell, P, Q)$ to satisfy \\eqref{eq:opt-feasibility:lin-dist-flow}, i.e., \\eqref{eq:dist-flow:p}--\\eqref{eq:dist-flow:v}; then one can explicitly determine $y$ by \\eqref{eq:opt-feasibility:soc-substitute} and always find large enough $z$ to make \\eqref{eq:opt-feasibility:limits}\\eqref{eq:opt-feasibility:soc} (strictly) feasible, satisfying Slater's condition. \nSuch a point $x$ can be easily found as follows: set $p=q=\\ell=0 \\in \\mathbb{R}^N$; determine $(P,Q)$ backward from the leaves to the root of the radial network, using \\eqref{eq:dist-flow:p}--\\eqref{eq:dist-flow:q}; then determine $v$ forward from the root to the leaves, using \\eqref{eq:dist-flow:v}. This completes the proof.\n\\end{IEEEproof}\n\nBy Proposition \\ref{prop:strong-duality}, the relaxed region $\\mathcal{W}'$ is equivalently:\n\\begin{IEEEeqnarray}{rCl}\n&&\\mathcal{W}'= \\left\\{w \\in \\mathbb{R}^W~|~\\mathrm{dp}'(w)=0 \\right\\}  \\nonumber\\\\\n&=& \\left\\{w \\in \\mathbb{R}^W~|D_w(\\mu,\\lambda)\\leq 0,~\\forall (\\mu,\\lambda)~\\text{satisfying \\eqref{eq:dual-feasibility}}\\right\\} \\label{eq:relaxed-region}\n\\end{IEEEeqnarray}\nwhere the second equality holds because $D_w(\\mu,\\lambda)=0$ can always be attained at the dual feasible point $(\\mu,\\lambda)=0$.\n\n\\begin{proposition}\\label{proposition:convexityU}\n$\\mathcal{W}'$ is a convex set.\n\\end{proposition}\n\\begin{IEEEproof}\nConsider arbitrary $w_1, w_2 \\in \\mathcal{W}'$ and $t \\in [0,1]$. Denote $w_t:=t w_1 + (1-t) w_2$. Then for every $(\\mu,\\lambda)$ satisfying \\eqref{eq:dual-feasibility}, we have:\n\\begin{IEEEeqnarray}{rCl}\nD_{w_t}(\\mu, \\lambda) \n&=& t D_{w_1} (\\mu,\\lambda)  + (1-t)D_{w_2} (\\mu,\\lambda) \\nonumber \\\\\n&\\leq& t\\cdot 0 + (1-t) \\cdot 0 = 0 \\nonumber\n\\end{IEEEeqnarray}\nwhere the first equality is due to linearity of $D_w(\\mu,\\lambda)$ with respect to $w$ when $(\\mu, \\lambda)$ is fixed, and the inequality holds because $w_1, w_2 \\in \\mathcal{W}'$. Therefore $w_t \\in \\mathcal{W}'$. By the definition of a convex set, $\\mathcal{W}'$ is convex.\n\\end{IEEEproof}\n\n\\subsection{Approximating SOCP-relaxed dispatchable region}\n\n\n\n\\begin{algorithm}[t]\\label{alg:approximate-socp}\n\t\\caption{Approximate $\\mathcal{W}'$}\n\t1. \\textbf{Initialization:} $\\mathcal{W}'_{poly} = \\left\\{w \\in \\mathbb{R}^W~|~\\underline w \\leq w \\leq \\overline w\\right\\}$ for sufficiently low $\\underline w$ and high $\\overline w$; $\\mathcal{V}_{safe}=\\emptyset$; $c=0$.\n\t\n\t2. Update vertex set $vert\\left(\\mathcal{W}'_{poly}\\right)$. Let $\\mathrm{dp}'_{max}=0$; \n\t\n\t\\For{$w\\in \\mbox{vert}\\left(\\mathcal{W}'_{poly}\\right)$ and $w\\notin \\mathcal{V}_{safe}$}{\n\t\tsolve $\\mathrm{DP}'(w)$ to obtain an optimal solution $(\\mu^*, \\lambda^*)$ and maximum objective value $\\mathrm{dp}'(w)$;\n\t\t\n\t\t\\lIf{$\\mathrm{dp}'(w)>\\mathrm{dp}'_{max}$}{\n\t\t   \n\t\t   \\qquad $\\mathrm{dp}'_{max} \\leftarrow \\mathrm{dp}'(w)$;\n\t\t\n\t\t   \\qquad $(\\mu_{max},\\lambda_{\\max}) \\leftarrow (\\mu^*,\\lambda^*)$\n\t\t}\n\t\t\\lElseIf{$\\mathrm{dp}'(w)\\leq 0$}{\n\t\t   $\\mathcal{V}_{safe} = \\mathcal{V}_{safe} \\cup \\{w\\}$\n\t\t}\n\t}\n\t\\eIf{$\\mathrm{dp}'_{max} = 0$ or $c=C_{max}$}{\n\t    return $\\mathcal{W}'_{poly}$.\n\t}{\n\t    add to $\\mathcal{W}'_{poly}$ a cutting plane:\n\t    $\\mu_{f,max}^\\intercal (B_f w + \\gamma_f) + \\lambda_{s,max}^\\intercal \\gamma_s \\leq  \\mu_{y,max}^\\intercal b_y +\\lambda_{q,max}^\\intercal \\gamma_q$;\n\t    \n\t    $c \\leftarrow c+1$;\n\t    \n\t    go back to Line 2;\n\t}\n\\end{algorithm}\n\nWe propose Algorithm \\ref{alg:approximate-socp} to approximate $\\mathcal{W}'$ defined in \\eqref{eq:relaxed-region}. It starts with a region $\\mathcal{W}'_{poly}$ that is large enough to contain $\\mathcal{W}'$. Then it solves the dual SOCP $\\mathrm{DP}'(w)$ for every vertex $w$ of polytope $\\mathcal{W}'_{poly}$, records the vertex that most severely violates the condition in \\eqref{eq:relaxed-region}, and adds a corresponding cutting plane to remove that vertex from $\\mathcal{W}'_{poly}$. Meanwhile, all the vertices that satisfy the condition in \\eqref{eq:relaxed-region} are added to $\\mathcal{V}_{safe}$ and never checked again. \n\n\\begin{proposition}\\label{prop:polytopic-outer-approximation}\nThe output $\\mathcal{W}'_{poly}$ in an arbitrary iteration of Algorithm \\ref{alg:approximate-socp} is an outer approximation of $\\mathcal{W}'$.  \n\\end{proposition}\n\\begin{IEEEproof}\nNote the initial $\\mathcal{W}'_{poly}$ contains $\\mathcal{W}'$. \nWe next prove that any cutting plane added to $\\mathcal{W}'_{poly}$ would not remove any point in $\\mathcal{W}'$.\nTo show that, consider an arbitrary $w$ removed by a cutting plan whose coefficients are $(\\mu_{max},\\lambda_{max})$. Then there must be $D_w(\\mu_{max},\\lambda_{max}) > 0$. Since $(\\mu_{max},\\lambda_{max})$ is dual feasible satisfying \\eqref{eq:dual-feasibility}, we have $w\\notin \\mathcal{W}'$ by \\eqref{eq:relaxed-region}. \n\\end{IEEEproof}\n\n Unlike \\cite{wei2014dispatchable, chen2021energy} that based on linear programs, the SOCP-relaxed dispatchable region $\\mathcal{W}'$ may not be the intersection of a finite number of cutting planes (i.e., a convex polytope). \n Therefore, Algorithm \\ref{alg:approximate-socp} may not guarantee $\\mathrm{dp}'(w)= 0$ for all vertices $w\\in vert\\left(\\mathcal{W}'_{poly}\\right)$ in a finite number of iterations. However, if it does so, as what always happens in our numerical experiments, it will produce a nice result as follows.\n \n\\begin{proposition}\\label{prop:polytopic-approximation}\nIf Algorithm \\ref{alg:approximate-socp} terminates with $\\mathrm{dp}'_{max} = 0$ in a finite number of iterations, it returns the accurate SOCP-relaxed dispatchable region, i.e., $\\mathcal{W}'_{poly}=\\mathcal{W}'$.  \n\\end{proposition}\n\\begin{IEEEproof}\nProposition \\ref{prop:polytopic-outer-approximation} has shown $\\mathcal{W}' \\subseteq \\mathcal{W}'_{poly}$.\nIf Algorithm \\ref{alg:approximate-socp} terminates with $\\mathrm{dp}'_{max} = 0$ after adding a finite number of cutting planes, then it returns a convex polytope $\\mathcal{W}'_{poly}$. Moreover, all the vertices $w\\in vert\\left(\\mathcal{W}'_{poly}\\right)$ satisfy $\\mathrm{dp}'(w)= 0$, therefore, $w\\in \\mathcal{W}'$ by \\eqref{eq:relaxed-region}. \nThis fact, together with the convexity of $\\mathcal{W}'$ shown in Proposition \\ref{proposition:convexityU}, implies $\\mathcal{W}'_{poly} \\subseteq \\mathcal{W}'$. Thus we have proved $\\mathcal{W}'_{poly}=\\mathcal{W}'$.\n\\end{IEEEproof}\n\nAn immediate corollary of Proposition \\ref{prop:polytopic-approximation} is that if $\\mathcal{W}'$ is not a polytope, then Algorithm \\ref{alg:approximate-socp} cannot terminate in a finite number of iterations with $\\mathrm{dp}'_{max} = 0$. \nIf that happens, one can terminate Algorithm \\ref{alg:approximate-socp} when reaching the maximum number of iterations $C_{max}$, to obtain a convex polytopic outer approximation of $\\mathcal{W}'$.\nIn this sense, the outcome of Algorithm \\ref{alg:approximate-socp} serves as a posterior indicator of the structure of $\\mathcal{W}'$. \n\n\n\\subsection{Removing SOCP-inexact renewable generations}\n     \nRemember our goal is to characterize the dispatchable region $\\mathcal{W}$, whereas $\\mathcal{W}'$ studied so far is just a SOCP-relaxation of $\\mathcal{W}$. To overcome this drawback, we design a heuristic to approximately remove the SOCP-inexact region $\\mathcal{\\tilde W}:=\\mathcal{W}' \\backslash \\mathcal{W}$ from $\\mathcal{W}'$.\nThe renewable generations $w \\in \\mathcal{\\tilde W}$ are feasible in terms of the SOCP relaxation $\\mathrm{FP}'(w)$ but infeasible in terms of $\\mathrm{FP}(w)$, as formally defined below.\n\n\n\n\n\n\\begin{definition}\nA vector of renewable power generation $w\\in\\mathcal{W}'$ is \\textbf{SOCP-inexact}, if every optimal solution of $\\mathrm{FP}'(w)$ satisfies: \n\\begin{IEEEeqnarray}{rCl}\n\\| y_{ij} \\|_2 &<& c_{q,ij} x + \\gamma_{q,ij}\\quad\\text{for some}~ i\\rightarrow j. \\nonumber\n\\end{IEEEeqnarray}\nThe \\textbf{SOCP-inexact region} of $w$ is defined as:\n\\begin{IEEEeqnarray}{rCl}\n \\mathcal{\\tilde W} &=& \\left\\{w \\in \\mathcal{W}'~|~w~\\text{is SOCP-inexact}\\right\\}. \\nonumber\n\\end{IEEEeqnarray}\n\\end{definition}\n\nOur next focus is to build an approximation of $\\mathcal{\\tilde W}$. \nFor that, we consider the following set defined on the dual SOCP:\n\\begin{IEEEeqnarray}{rCl}\n\\mathcal{\\tilde W}_d &:=& \\{w \\in \\mathcal{W}'~|~\\text{Every optimal solution of}~\\mathrm{DP}'(w) \\nonumber \\\\\n&&\\qquad\\qquad\\quad \\text{satisfies}~\\lambda_{q,ij}=0~\\text{for some}~i\\rightarrow j \\}. \\nonumber\n\\end{IEEEeqnarray}\nBy complementary slackness \\cite[Section 5.5.2]{boyd2004convex}, for every primal-dual optimal of $\\mathrm{FP}'(w)$ and $\\mathrm{DP}'(w)$, there is:\n\\begin{IEEEeqnarray}{rCl}\n\\lambda_{q,ij}\\left(\\| y_{ij} \\|_2 - c_{q,ij} x - \\gamma_{q,ij}\\right)&=&0, \\quad \\forall i\\rightarrow j. \\nonumber\n\\end{IEEEeqnarray}\nThis implies $\\mathcal{\\tilde W}\\subseteq \\mathcal{\\tilde W}_d$. Although $\\mathcal{\\tilde W}= \\mathcal{\\tilde W}_d$ may not hold, their difference can only occur under rare circumstances where $\\lambda_{q,ij} = \\| y_{ij} \\|_2 \\!-\\! c_{q,ij} x \\!-\\! \\gamma_{q,ij} = 0$ at a primal-dual optimal. Hence we focus on $\\mathcal{\\tilde W}_d$ as an approximation of $\\mathcal{\\tilde W}$. \n\nGiven an arbitrary $w \\in \\mathcal{\\tilde W}_d \\subseteq \\mathcal{W}'$, the maximum objective value of $\\mathrm{DP}'(w)$ is $\\mathrm{dp}'(w)=0$ but with some $\\lambda_{q,ij}=0$ so the SOC relaxation is inexact (except for some very rare case). To approximate $\\tilde{\\mathcal{W}}_d$, first we add the following constraint to tighten the dual feasible set \\eqref{eq:dual-feasibility}:\n\\begin{IEEEeqnarray}{rCl}\\label{eq:add-dual-constraint}\n\\lambda_{q} &\\geq& \\delta\n\\end{IEEEeqnarray}\nwhere the inequality is element-wise and $\\delta\\in\\mathbb{R}_+^{9N}$ is a vector of all \\textit{strictly positive} parameters, whose design will be elaborated later. Consider the tightened dual SOCP:\n\\begin{IEEEeqnarray}{rCl}\n\\mathrm{DP}''(w,\\delta):~\\max_{\\mu,\\lambda} &\\quad&  \\mu_f^\\intercal (B_f w + \\gamma_f) + \\lambda_s^\\intercal \\gamma_s - \\mu_{y}^\\intercal b_y - \\lambda_{q}^\\intercal \\gamma_q \\nonumber\n\\\\\n\\text{s. t.} && \\text{\\eqref{eq:dual-feasibility}, \\eqref{eq:add-dual-constraint}} \\nonumber\n\\end{IEEEeqnarray} \nand let $\\mathrm{dp}''(w,\\delta)$ denote its maximum objective value. For $w\\in\\mathcal{\\tilde W}_d$, there must be $\\mathrm{dp}''(w,\\delta) < 0$, because otherwise $\\mathrm{DP}'(w)$ would have an optimal solution that satisfies \\eqref{eq:add-dual-constraint}, contradicting the definition of $\\mathcal{\\tilde W}_d$. \nActually $\\mathrm{dp}''(w,\\delta) \\leq -\\eta$ for some $\\eta>0$ that depends on $w$ and $\\delta$.\n\n\\begin{algorithm}[t]\\label{alg:remove-inexact}\n\t\\caption{Approximate $\\mathcal{\\tilde W}_d$ (or SOCP-inexact $\\mathcal{\\tilde W}$)}\n\t1. \\textbf{Initialization:} $\\mathcal{\\tilde W}_{poly} = \\mathcal{W}'_{poly}$ returned by Alg. \\ref{alg:approximate-socp}. Given positive $\\delta$, $\\eta$, $\\eta'$; $\\mathcal{V}_{safe}=\\emptyset$; $c=0$;\n\t\n\t2. Update vertex set $vert\\left(\\mathcal{\\tilde W}_{poly}\\right)$. Let $\\mathrm{dp}''_{max}= -\\eta$;\n\t\n\t\\For{$w\\in vert\\left(\\mathcal{\\tilde W}_{poly}\\right)$ and $w\\notin \\mathcal{V}_{safe}$}{\n\t\tsolve $\\mathrm{DP}''(w,\\delta)$ to obtain an optimal solution $(\\mu^*, \\lambda^*)$ and maximum objective value $\\mathrm{dp}''(w,\\delta)$;\n\t\t\n\t\t\\lIf{$\\mathrm{dp}''(w,\\delta)>\\mathrm{dp}''_{max}$}{\n\t\t   \n\t\t   \\qquad $\\mathrm{dp}''_{max} \\leftarrow \\mathrm{dp}''(w,\\delta)$;\n\t\t\n\t\t   \\qquad $(\\mu_{max},\\lambda_{\\max}) \\leftarrow (\\mu^*,\\lambda^*)$\n\t\t}\n\t\t\\lElseIf{$\\mathrm{dp}''(w,\\delta)\\leq -\\eta$}{\n\t\t   $\\mathcal{V}_{safe} = \\mathcal{V}_{safe} \\!\\cup\\! \\{w\\}$\n\t\t}\n\t}\n\t\\eIf{$\\mathrm{dp}''_{max} = -\\eta$ or $c=C_{max}$}{\n\t    return $\\mathcal{\\tilde W}_{poly}$.\n\t}{\n\t    add to $\\mathcal{\\tilde W}_{poly}$ a cutting plane:\n\t    $\\mu_{f,max}^\\intercal  (B_f w +  \\gamma_f) + \\lambda_{s,max}^\\intercal \\gamma_s \\leq \\mu_{y,max}^\\intercal b_y +\\lambda_{q,max}^\\intercal \\gamma_q  -  \\eta'$; \n\t    \n\t    $c\\leftarrow c+1$;\n\t   \n\t    go back to Line 2;\n\t}\n\\end{algorithm}\n\nThe idea above inspires us to approximate $\\mathcal{\\tilde W}_d$ (or $\\mathcal{\\tilde W}$) by\n\\begin{IEEEeqnarray}{rCl}\n\\tilde{\\mathcal{W}}_d \\approx \\left\\{w \\in \\mathbb{R}^W|D_w(\\mu,\\lambda)\\leq -\\eta,\\forall (\\mu,\\lambda)~\\text{satisfying \\eqref{eq:dual-feasibility},\\eqref{eq:add-dual-constraint}}\\right\\}  \\nonumber\n\\end{IEEEeqnarray}\nTo this end, Algorithm \\ref{alg:remove-inexact} can be designed using a similar procedure to Algorithm \\ref{alg:approximate-socp}.\nAlgorithm \\ref{alg:remove-inexact} returns a convex polytope $\\mathcal{\\tilde W}_{poly} \\subseteq \\mathcal{W}'_{poly}$ that guarantees $\\mathrm{dp}''(w,\\delta) \\leq -\\eta < 0$ for all $w \\in \\mathcal{\\tilde W}_{poly}$, which is an approximation of $\\tilde{\\mathcal{W}}_d$ (or $\\tilde{\\mathcal{W}}$). Removing $\\tilde{\\mathcal{W}}_{poly}$ from $\\mathcal{W}'_{poly}$, we can obtain an approximation $\\mathcal{W}_{poly}=\\mathcal{W}'_{poly} \\backslash \\tilde{\\mathcal{W}}_{poly}$ of the actual dispatchable region $\\mathcal{W}$. To make Algorithm \\ref{alg:remove-inexact} more robust, we may choose $\\eta'> \\eta$ for the added cutting plane in each iteration.\n\n\n\\textit{Remark:} The parameters $\\delta$ and $\\eta$ are essential for Algorithm \\ref{alg:remove-inexact}. A general guideline is that (1) given $\\delta$, choosing a smaller $\\eta$ and (2) given $\\eta$, choosing a bigger $\\delta$ will both make $\\mathcal{\\tilde W}_{poly}$ bigger and lead to a smaller (more conservative) approximation of $\\mathcal{W} = \\mathcal{W}' \\backslash \\mathcal{\\tilde{W}}$. Moreover, sometime it is difficult for Algorithm \\ref{alg:remove-inexact} to use a single convex polytope $\\mathcal{\\tilde W}_{poly}$ to accurately approximate the most likely nonconvex $\\mathcal{\\tilde{W}}$.\nTo deal with this difficulty, we propose to run Algorithm \\ref{alg:remove-inexact} multiple times with different vectors $\\delta$. As a result, we obtain multiple convex polytopes whose union serves as a better approximation of $\\mathcal{\\tilde{W}}$. Those vectors $\\delta$ can be selected in the following way. We traverse the vertices of $\\mathcal{W}'_{poly}$, select one vertex $w$, and solve the dual SOCP $\\mathrm{DP}'(w)$ to get an optimal solution $(\\mu^*,\\lambda^*)$. Then $\\delta$ is constructed by keeping all the strictly positive elements of $\\lambda_q^*$ as they are, and add a small positive perturbation to all the zero elements.  \n\n\n\\begin{table}[t]\n        \\renewcommand{\\arraystretch}{1.3}\n        \\renewcommand{\\tabcolsep}{1em}\n        \\centering\n        \\caption{Summary of different regions}\n        \\label{tab:summary}\n        \\begin{tabular}{m{1.2cm}<{\\centering}m{1.5cm}<{\\centering}m{0.1cm}<{\\centering}m{1.2cm}<{\\centering}m{0.1cm}<{\\centering}m{1.5cm}<{\\centering}}\n                \\hline \n           & SOCP-relaxed Region & = & SOCP-exact Region & + & SOCP-inexact Region \\\\\n           \\hline\n          Actual &  $\\mathcal{W}'$ & = & $\\mathcal{W}$ & + & $\\tilde{\\mathcal{W}} \\approx \\tilde{\\mathcal{W}}_d$ \\\\\n          Approx. & $\\mathcal{W}'_{poly}$ & =  & $\\mathcal{W}_{poly}$ & + & $\\tilde{\\mathcal{W}}_{poly}$ \\\\\n          Method & Algorithm \\ref{alg:approximate-socp} & & $\\mathcal{W}'_{poly} \\backslash \\tilde{\\mathcal{W}}_{poly}$ & &  Algorithm \\ref{alg:remove-inexact}\\\\\n                \\hline\n        \\end{tabular}\n\\end{table}\n\n\\textit{Summary}. The relationship of different regions mentioned in this paper is summarized in TABLE \\ref{tab:summary}. As discussed, the dispatchable region $\\mathcal{W}=\\mathcal{W}' \\backslash \\mathcal{\\tilde W}$, where $\\mathcal{W}'$ is the SOCP-relaxed dispatchable region and $\\mathcal{\\tilde W}$ is the SOCP-inexact region. \nWe develop Algorithm \\ref{alg:approximate-socp} to get $\\mathcal{W}'_{poly}$, a convex polytopic approximation of $\\mathcal{W}'$; and Algorithm \\ref{alg:remove-inexact} to get $\\mathcal{\\tilde W}_{poly}$, a convex polytopic approximation of $\\mathcal{\\tilde W}$. Algorithm \\ref{alg:remove-inexact} can run multiple times to obtain a more accurate approximation of nonconvex $\\mathcal{\\tilde W}_d$ (or $\\mathcal{\\tilde W}$). The outputs of multiple runs of Algorithm \\ref{alg:remove-inexact} are then removed from $\\mathcal{W}'_{poly}$ to obtain a generally nonconvex polytopic approximation of $\\mathcal{W}$.\n\n\n    \n\n\n\\section{Case Studies} \\label{sec:numerical}\n\\begin{figure}\n\t\\includegraphics[width=0.35\\textwidth]{network.pdf}\n\t\\centering\n\t\\caption{IEEE 33-node network model from \\cite{chen2018energy}.}\n\t\\label{fig:network}\n\\end{figure}\nIn this section, we conduct numerical experiments on the IEEE 33-bus system whose topology is in Fig. \\ref{fig:network}. The proposed algorithms are implemented to approximate the dispatchable region of renewable generation $(w_1, w_2)$ at nodes 13 and 29, respectively. Then, we test the impact of several factors and compare with other approaches.\n\\subsection{Benchmark} \\label{sec:numerical-1}\n\\begin{table}[t]\n        \\renewcommand{\\arraystretch}{1.3}\n        \\renewcommand{\\tabcolsep}{1em}\n        \\centering\n        \\caption{Parameters of generators in Benchmark}\n        \\label{tab:generator}\n        \\begin{tabular}{cccccc}\n                \\hline \n                Generator & Location & $\\underline{p}_i$ (p.u.) & $\\overline{p}_i$ (p.u.)\\\\\n                \\hline\n                G1 & node 10 & 0.4 & 0.6 \\\\\n                G2 & node 18 & 0.3 & 0.4\\\\\n                G3 & node 23 & 0.4 & 0.6\\\\\n                G4 & node 25 & 0.3 & 0.5\\\\\n                G5 & node 33 & 0.4 & 0.6\\\\\n                \\hline\n        \\end{tabular}\n\\end{table}\n\\begin{figure}[t]\n        \\centering\n        \\includegraphics[width=0.8\\columnwidth]{actual_relax.pdf}\n        \\caption{The SOCP-relaxed region $\\mathcal{W}'$ (outside) and the actual dispatchable region $\\mathcal{W}$ obtained by checking sampled points in the $(w_1,w_2)$ space.}\n        \\label{fig:actual-relax}\n\\end{figure}\n\n\\begin{figure}[t]\n        \\centering\n        \\includegraphics[width=1.0\\columnwidth]{algorithm1.pdf}\n        \\caption{The output of Algorithm \\ref{alg:approximate-socp} in different iterations (solid line) to approximate the SOCP-relaxed dispatchable region $\\mathcal{W}'$ (dashed line).}\n        \\label{fig:alg1}\n\\end{figure}\nIn the IEEE-33 bus system, there are 5 controllable generators whose parameters are given in TABLE \\ref{tab:generator}. Two renewable generators are connected to nodes 13 and 29, respectively.\nFor comparison, the actual SOCP-relaxed region $\\mathcal{W}'$ and the actual dispatchable region without relaxation $\\mathcal{W}$ are generated as in Fig. \\ref{fig:actual-relax}. This can be done by checking the feasiblity of a nonlinear optimization with \\eqref{eq:dist-flow}-\\eqref{eq:vl_limits} as its constraints, over sample points $w$ in the $(w_1,w_2)$ space using the nonlinear solver IPOPT. As we can see from Fig. \\ref{fig:actual-relax}, the actual dispatchable region $\\mathcal{W}$ can be nonconvex and the SOCP-relaxed region is not accurate enough. In the following, we apply the proposed algorithms to output a more accurate region.\n\nFirst, we test the performance of Algorithm \\ref{alg:approximate-socp}. We observe that algorithm terminates with $\\mathrm{dp}'_{max} = 0$ in 25 iterations, taking about 289.84s. The output regions $\\mathcal{W}'_{poly}$ in the 2nd, 5th, 10th, and final iterations are given in Fig. \\ref{fig:alg1}. The Algorithm \\ref{alg:approximate-socp} removes the nondispatchable regions iteratively (the blue region is becoming smaller), and finally returns a convex polytope $\\mathcal{W}'_{poly}$ exactly the same as the actual SOCP-relaxed region $\\mathcal{W}'$ (dashed line). This validates Proposition \\ref{prop:polytopic-approximation}.\n\n\\begin{figure}[t]\n        \\centering\n        \\includegraphics[width=0.8\\columnwidth]{algorithm2.pdf}\n        \\caption{The gray polytope $\\mathcal{W}_{poly}$ is an approximation of $\\mathcal{W}$ (red dash line). It is obtained by removing the output $\\tilde{\\mathcal{W}}_{poly}$ of Algorithm \\ref{alg:remove-inexact} (white polytope) from the output $\\mathcal{W}'_{poly}$ of Algorithm \\ref{alg:approximate-socp} (the  outside blue dash line).}\n        \\label{fig:algorithm2}\n\\end{figure}\n\n\nEven though Algorithm 1 can output the accurate SOCP-relaxed dispatchable region, as we can see in Fig. \\ref{fig:actual-relax}, there is still a gap between the actual dispatchable region $\\mathcal{W}$ and the relaxed one $\\mathcal{W}'$. If the renewable generator output $(w_1,w_2)$ lies in the gap area, there is actually no feasible dispatch that satisfies power flow equation \\eqref{eq:dist-flow} and safety limits \\eqref{eq:pq_limits}-\\eqref{eq:vl_limits}. Thus, using the SOCP-relaxed region as a guidance will threaten power system security. In this paper, Algorithm \\ref{alg:remove-inexact} is developed to further remove the nondispatchable points. As in Fig. \\ref{fig:algorithm2}, the $\\tilde{\\mathcal{W}}_{poly}$ (white area) generated by Algorithm \\ref{alg:remove-inexact} is removed and the resulting region $\\mathcal{W}_{poly}$ (grey area) is closer to the actual region (red dash line). This shows the great potential of the proposed algorithm in improving the accuracy of dispatchable region in a distribution system. The operational risk under the obtained region $\\mathcal{W}_{poly}$ and the SOCP-relaxed region $\\mathcal{W}'$ will be compared later in TABLE \\ref{tab:compare}.\n\n\\subsection{Impact of different factors} \\label{sec:numerical-2}\nIn the following, we test the impact of two factors (adjustable capability of controllable generators $[\\underline{p}_i,\\overline{p}_i],\\forall i$ and current limit $\\overline \\ell$) on the shape of the dispatchable region and the performance of the proposed algorithm.\n\n\\begin{table}[t]\n        \\renewcommand{\\arraystretch}{1.3}\n        \\renewcommand{\\tabcolsep}{1em}\n        \\centering\n        \\caption{Parameters of generators in Cases L and H}\n        \\label{tab:generator2}\n        \\begin{tabular}{ccccc}\n                \\hline \n                Generator &  \\multicolumn{2}{c}{Case L} & \\multicolumn{2}{c}{Case H} \\\\ No. & $\\underline{p}_i$ (p.u.) & $\\overline{p}_i$ (p.u.) & $\\underline{p}_i$ (p.u.) & $\\overline{p}_i$ (p.u.)\\\\\n                \\hline\n                G1 & 0.4 & 0.5 & 0 & 0.6\\\\\n                G2 & 0.3 & 0.4 & 0 & 0.4\\\\\n                G3 & 0.4 & 0.5 & 0 & 0.6\\\\\n                G4 & 0.4 & 0.5 & 0 & 0.5\\\\\n                G5 & 0.4 & 0.5 & 0 & 0.6\\\\\n                \\hline\n        \\end{tabular}\n\\end{table}\n\\begin{figure}[t]\n        \\centering\n        \\includegraphics[width=1.0\\columnwidth]{Comparison.pdf}\n        \\caption{Left: the $\\mathcal{W}'_{poly}$ returned by Algorithm \\ref{alg:approximate-socp} for Case L (subfigure (a)) and Case H (subfigure (c)). Right: the $\\mathcal{W}_{poly}$ returned by Algorithm \\ref{alg:remove-inexact} for Case L (subfigure (b)) and Case H (subfigure (d)).}\n        \\label{fig:comparison}\n\\end{figure}\n\\begin{figure}[t]\n\t\\includegraphics[width=0.8\\columnwidth]{Converge.pdf}\n\t\\centering\n\t\\caption{The change of $\\mathrm{dp}'_{max}$ over iterations of Algorithm \\ref{alg:approximate-socp}.}\n\t\\label{fig:dpmax}\n\\end{figure}\nTo show how $[\\underline{p}_i,\\overline{p}_i]$ influences the dispatchable region, we test three cases: (1) \\textbf{Benchmark}, which has the same setting as in Section \\ref{sec:numerical-1}. (2) \\textbf{Case L}, where the generators have less adjustable capability than the benchmark. (3) \\textbf{Case H}, where the generators have more adjustable capability than the benchmark. The parameters of the Cases L and H are given in TABLE \\ref{tab:generator2}. The regions returned by Algorithm \\ref{alg:approximate-socp} ($\\mathcal{W}'_{poly}$) and Algorithm \\ref{alg:remove-inexact} ($\\mathcal{W}_{poly}$) are given in Fig. \\ref{fig:comparison}. Subfigures (a), (b) are for Case L and subfigures (c), (d) are for Case H. The changes of $\\mathrm{dp}'_{max}$ under three cases are recorded in Fig. \\ref{fig:dpmax}.\n\nAs shown in Figs. \\ref{fig:alg1} and \\ref{fig:comparison}, Algorithm \\ref{alg:approximate-socp} can always output the accurate SOCP-relaxed dispatchable region, i.e., $\\mathcal{W}'_{poly}=\\mathcal{W}'$. The final dispatchable regions (grey area) returned by the proposed algorithms are much closer to the actual ones compared with the SOCP-relaxed regions. In addition, as the adjustable capability of generators decreases, the system's ability to accommodate volatile renewable power becomes weaker, and thus, the dispatchable region becomes smaller. We also find that with a weaker adjustable capability, the actual dispatchable region $\\mathcal{W}$ is more likely to be nonconvex and to differ more from the SOCP-relaxed region. The difference between the red dash line and the blue dash line in Fig. \\ref{fig:comparison}(b) is more significant than that in Fig. \\ref{fig:comparison}(d).\nIn the future power systems, more renewable generators are replacing the controllable generators, so the use of an SOCP-relaxed dispatchable region is not accurate enough.\nTherefore, the proposed Algorithms \\ref{alg:approximate-socp}-\\ref{alg:remove-inexact} to remove the nondispatchable points will be helpful. \n\n\\begin{table}[t]\n        \\renewcommand{\\arraystretch}{1.3}\n        \\renewcommand{\\tabcolsep}{1em}\n        \\centering\n        \\caption{Comparison of three cases.}\n        \\label{tab:compare}\n        \\begin{tabular}{cccccc}\n                \\hline \n                 & \\textbf{FR}($\\mathcal{W}'$) & \\textbf{FR}($\\mathcal{W}_{poly}$) & Reduction & Time(s) \\\\\n                 \\hline\n                 Benchmark & 10.4\\% & 4.5\\% & 56.73\\% & 289.84 \\\\\n                 Case L & 15.7\\% & 8.7\\% & 44.59\\% & 291.01\\\\\n                 Case H & 3.5\\% & 2.5\\% & 28.57\\% & 724.60\\\\\n                \\hline\n        \\end{tabular}\n\\end{table}\n\nIn Fig. \\ref{fig:dpmax}, the $\\mathrm{dp}'_{max}$ under all three cases decrease towards zero when Algorithm \\ref{alg:approximate-socp} terminates. The computational times are 289.84s (Benchmark), 291.01s (Case L), and 724.60s (Case H), respectively, showing that our algorithm is efficient. Moreover, we randomly generate 2000 points $(w_1,w_2)$ in the SOCP-relaxed dispatchable region $\\mathcal{W}'$ and the final obtained region $\\mathcal{W}_{poly}=\\mathcal{W}'_{poly}  \\backslash \\tilde{\\mathcal{W}}_{poly}$, and calculate the failure rate defined as\n\\begin{align}\\label{eq:failurerate}\n    \\textbf{FR}(\\mathcal{S})=\\frac{\\mbox{No. of points}~w \\in \\mathcal{S}~\\mbox{that is nondispatchable}}{\\mbox{No. of points}~w\\in \\mathcal{S}}\n\\end{align}\nThe failure rates under three cases are summarized in TABLE \\ref{tab:compare}. In all three cases, the proposed method can greatly reduce the failure rate, and the reduction is more than 50\\% under benchmark. This can help better ensure system security. Moreover, we can find that in a system with relatively small adjustable capability, the reduction is more significant.\n\nFurthermore, we test the impact of current limit by running two other cases where we halve and double the $\\overline \\ell$, respectively. The obtained region are shown in Fig. \\ref{fig:current}. The computation times are both less than 500s, which is acceptable. A more stringent line-flow limit results in a smaller dispatchable region and also a greater deviation between the relaxation region $\\mathcal{W}'$ and the exact region $\\mathcal{W}$.\n\\begin{figure}[t]\n\t\\includegraphics[width=1.0\\columnwidth]{Current.pdf}\n\t\\centering\n\t\\caption{The $\\mathcal{W}_{poly}$ returned by Algorithm 2 under two current limits.}\n\t\\label{fig:current}\n\\end{figure}\n\n\n\\subsection{Comparison with other methods}\n\\begin{figure*}[t]\n        \\centering\n        \\includegraphics[width=1.8\\columnwidth]{DifferentMethod.pdf}\n        \\caption{The regions returned by Linearized DistFlow model (left), the proposed algorithms (middle), and the polyhedral approximation of SOCP-relaxed model (right).}\n        \\label{fig:differentmethod}\n\\end{figure*}\nWe then compare the performance of the proposed algorithms with two well-known approaches based on (1) Linearized DistFlow model \\cite{baran1989optimal} (denoted as \\textbf{LinDistFlow}); (2) Polyhedral approximation of the SOCP-relaxed model \\cite{chen2018energy} (denoted as \\textbf{SOCP-Linear}). These two models are linear programs so that the adaptive constraint generation algorithm in \\cite{wei2015real} can be applied to generate the region. The results are shown in Fig. \\ref{fig:differentmethod}. Theoretically, the LinDistFlow region can be an inner\/outer approximation or a region that intersects the actual dispatchable region. In the benchmark case, the LinDistFlow \nregion is very small and conservative. The SOCP-Linear region is always an outer approximation of the actual region. We can see that it is very close to the SOCP-relaxed region $\\mathcal{W}'$. To better illustrate the results under three approaches, we calculate the failure rate \\eqref{eq:failurerate} and the missing rate (\\textbf{MR}) defined below.\n\\begin{align}\n    \\textbf{MR}(\\mathcal{S})=\\frac{\\mbox{No. of points}~ w \\in \\mathcal{W}~\\mbox{but}~\\notin \\mathcal{S}}{\\mbox{No. of points}~w \\in \\mathcal{W}}\n\\end{align}\nThe failure rate and missing rate under three approaches are compared in TABLE \\ref{tab:threemethod}. We can find that, in this simulation case, the LinDistFlow region is an inner approximation so its failure rate is zero. However, it has a very high missing rate, meaning that the region is too conservative. The SOCP-Linear region is always an outer approximation so its missing rate is zero, but its failure rate is high. The proposed method can achieve a good balance between ensuring security and reducing conservatism.\n\\begin{table}[t]\n        \\renewcommand{\\arraystretch}{1.3}\n        \\renewcommand{\\tabcolsep}{1em}\n        \\centering\n        \\caption{Comparison of three methods.}\n        \\label{tab:threemethod}\n        \\begin{tabular}{cccccc}\n                \\hline \n                 & \\textbf{FR}($\\mathcal{W}_{poly}$) & \\textbf{MR}($\\mathcal{W}_{poly}$) \\\\\n                 \\hline\n                 LinDistFlow & 0\\% & 91.1\\% \\\\\n                 Proposed Method & 4.5\\% & 2.7\\% \\\\\n                 SOCP-Linear & 11.8\\% & 0\\%\\\\\n                \\hline\n        \\end{tabular}\n\\end{table}\n\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper, we develop an improved approximation of the renewable generation dispatchable region in radial distribution networks. First, a nonconvex optimization problem is formulated to describe the dispatchable region. The nonconvex problem is then relaxed to a convex SOCP. An SOCP-based projection algorithm (Algorithm 1) is proposed to generate the accurate SOCP-relaxed dispatchable region under certain conditions. In addition, a heuristic method (Algorithm 2) is developed to remove the SOCP-inexact region from the region obtained above. Therefore, the final region can better approximate the actual nonconvex dispatchable region. Our main findings are:\n\\begin{itemize}\n    \\item The proposed method can reduce the operational risk (quantified by failure rate) by more than 50\\% compared with the SOCP-relaxed region.\n    \\item The proposed method has a greater potential in the future power system with fewer controllable units and thus weaker adjustable capability.\n    \\item Compared with existing approaches (LinDistFlow and SOCP-Linear), the proposed method achieves a better tradeoff between security and conservatism.\n\\end{itemize}\n\nThis paper provides an innovative perspective for constructing the dispatchable region: While the existing literature can only generate convex regions, the proposed algorithm can generate nonconvex approximations. For future work, we aim to improve the accuracy of the proposed algorithms by properly setting the initial points for heuristic searching.\n\n\n\\section*{Appendix. Constant parameters}\n\nThis appendix provides in full detail the constant matrices, vectors, and numbers used in Section \\ref{sec:methods}. \n\n\\subsection{Equation \\eqref{eq:opt-feasibility}: $A_f$, $B_f$, $\\gamma_f$, $A_s$, $\\gamma_s$}\n\nThe vector $x=(p,q, v,\\ell, P,Q)$ is arranged in the order explained in Section \\ref{sec:model}. Let $C\\in \\{-1,0,1\\}^{(N+1)\\times N}$ be the incidence matrix of the radial network, with its element at the $k$-th row, $j$-th column:\n\\begin{IEEEeqnarray}{rCl}\nC_{kj} &=& \\begin{cases}\n1,\\qquad\\text{if}~k=i~\\text{for line}~i\\rightarrow j \\\\\n-1,\\quad \\text{if}~k=j~\\text{for line}~i\\rightarrow j \\\\\n0,\\qquad \\text{otherwise.}\n\\end{cases}\\nonumber\n\\end{IEEEeqnarray}\nRemoving the first row of $C$, we get the reduced incidence matrix $\\overline C \\in \\{-1,0,1\\}^{N\\times N}$. Define diagonal matrices $R:=\\text{diag}(r_{ij}, \\forall i\\rightarrow j)$ and $X:=\\text{diag}(x_{ij}, \\forall i\\rightarrow j)$. Denote the $N \\times N$ all-zero matrix as $\\mathbf{O}_{N}$, identity matrix as $I_N$, and $N$-dimensional all-zero column vector as $0_N$. We have: \n\\begin{IEEEeqnarray}{rCl}\nA_f &=& \\begin{bmatrix}\nI_N &\\mathbf{O}_{N}&\\mathbf{O}_{N} & -R & -\\overline C & \\mathbf{O}_{N} \\\\\n\\mathbf{O}_N &I_N&\\mathbf{O}_{N} &  -X & \\mathbf{O}_{N} & -\\overline C \\\\\n\\mathbf{O}_N &\\mathbf{O}_N &\\overline C^\\intercal & \\left(R^2 \\!+\\!X^2\\right) & -2R & -2X\n\\end{bmatrix}\\nonumber\\\\\n\\nonumber\\\\\n\\gamma_f &=& \\left[0_N^\\intercal,~\n0_N^\\intercal,~\nv_0,~ 0_{N-1}^{\\intercal}\\right]^\\intercal.\\nonumber\n\\end{IEEEeqnarray}\nMoreover, we define: \n\\begin{IEEEeqnarray}{rCl}\nB'_f &=&\\left[\nI_N,~\\mathbf{O}_{N},~\\mathbf{O}_{N} \\right]^\\intercal\n\\nonumber\n\\end{IEEEeqnarray}\nand let $B_f$ be a submatrix of $B'_f$ that contains only the columns corresponding to the nodes $i$ with nonzero renewable generation $w_i$. \nDefine column vectors $\\overline v := (\\overline v_i, ~\\forall i=1,...,N)$, $\\underline v := (\\underline v_i, ~\\forall i=1,...,N)$, similarly $\\overline p$, $\\underline p$, $\\overline q$, $\\underline q$, and $\\overline \\ell =(\\overline \\ell_{ij}, ~\\forall i\\rightarrow j)$. To write inequalities \\eqref{eq:pq_limits}\\eqref{eq:vl_limits} as $A_s x + \\gamma_s \\leq 0$, we need: \n\\begin{IEEEeqnarray}{rCl}\nA_s &=& \\begin{bmatrix}\nI_N & \\mathbf{O}_{N} & \\mathbf{O}_{N}& \\mathbf{O}_{N} & \\mathbf{O}_{N} & \\mathbf{O}_{N} \\\\\n-I_N & \\mathbf{O}_{N} & \\mathbf{O}_{N}&  \\mathbf{O}_{N} & \\mathbf{O}_{N} & \\mathbf{O}_{N}\\\\\n \\mathbf{O}_{N} & I_N &\\mathbf{O}_{N}& \\mathbf{O}_{N} & \\mathbf{O}_{N} & \\mathbf{O}_{N} \\\\\n \\mathbf{O}_{N} &-I_N & \\mathbf{O}_{N}&  \\mathbf{O}_{N} & \\mathbf{O}_{N} & \\mathbf{O}_{N}\\\\\n  \\mathbf{O}_{N} & \\mathbf{O}_{N}& I_N &\\mathbf{O}_{N} & \\mathbf{O}_{N} & \\mathbf{O}_{N} \\\\\n \\mathbf{O}_{N} &\\mathbf{O}_{N}&-I_N &   \\mathbf{O}_{N} & \\mathbf{O}_{N} & \\mathbf{O}_{N}\\\\\n\\mathbf{O}_{N}& \\mathbf{O}_{N} & \\mathbf{O}_{N} & I_N & \\mathbf{O}_{N} & \\mathbf{O}_{N} \\\\\n\\mathbf{O}_{N}& \\mathbf{O}_{N} & \\mathbf{O}_{N} & -I_N & \\mathbf{O}_{N} & \\mathbf{O}_{N}\n\\end{bmatrix},~ \n\\gamma_s =\\begin{bmatrix}\n-\\overline p \\\\\n\\underline p \\\\\n-\\overline q \\\\\n\\underline q \\\\\n-\\overline v \\\\\n\\underline v \\\\\n-\\overline \\ell \\\\\n0_N\n\\end{bmatrix}.\\nonumber\n\\end{IEEEeqnarray}\n\n\\subsection{Equation \\eqref{eq:opt-feasibility-relaxed}: $A_{y}$, $b_{y}$, $c_{q}$, $\\gamma_{q}$}\n\nTo make \\eqref{eq:opt-feasibility:soc-substitute}--\\eqref{eq:opt-feasibility:soc} the same as: \n\\begin{IEEEeqnarray}{rCl}\n \\left\\| \\begin{bmatrix}\n2P_{ij}  \\\\\n2Q_{ij} \\\\\nv_i - \\ell_{ij}\n\\end{bmatrix} \\right\\|_2\n&\\leq& v_i + \\ell_{ij} +z_{q,ij}, \\quad  \\forall i\\rightarrow j  \\nonumber\n\\end{IEEEeqnarray}\nwe need $A_{y}$, $b_{y}$, $c_{q}$, $\\gamma_{q}$ as follows:\n\\begin{itemize}\n    \\item For all $i \\rightarrow j$, $A_{y,ij}$ is $3\\times (6N)$ sparse matrix with all elements zero except its element at the first row, $(4N+j)$-th column equal to $2$; at the second row, $(5N+j)$-th column equal to 2; at the third row, $(2N+i)$-th column equal to $1$ (if $i\\neq 0$), and $(3N+j)$-th column equal to $-1$.   \n    \\item For all $i\\rightarrow j$ except $0\\rightarrow 1$, $b_{y,ij}$ is a three-dimensional column vector of all zeros; $b_{y,01} = \\left[0, 0, v_0 \\right]^\\intercal$.\n    \n    \\item For all $i\\rightarrow j$, $c_{q,ij}$ is a $(6N)$-dimensional row vector of all zeros except its $(2N+i)$-th (if $i\\neq 0$) and $(3N+j)$-th elements both equal to $1$. \n    \\item $\\gamma_{q,ij}=0$ for all $i\\rightarrow j$ except $0\\rightarrow 1$; $\\gamma_{q,01} = v_0$. \n\\end{itemize}\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n\\newpage\n\\fi\n        \n        \\bibliographystyle{IEEEtran}\n        ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s:intro}\n\nThe study of galaxy kinematics as a function of cosmic time provides important insights into the evolution of galactic mass budgets and structure \\citep[e.g.][]{Sofue01, FS20}. Different kinematic tracers like molecular gas, ionised gas, or stars move in the same galactic potential and allow for estimates of the galactic dark matter content.\nThe kinematic signatures of different tracers vary due to their different nature: stars are collision-less while gas is dissipative; different gas phases have different temperatures, turbulent velocities, and might be affected by outflows. The various tracers often have different spatial distributions and probe different regions of the overall potential. Nonetheless, dynamical models based on complementary tracers should give the same mass estimates for systems in equilibrium.\n\nIn the local Universe, large interferometric and IFS surveys provide spatially resolved kinematics of stars, and atomic, molecular, and ionised gas. \nComparative studies of baryonic kinematics at $z=0$, in particular from the EDGE-CALIFA survey \\citep{Sanchez12, Bolatto17} and the ATLAS$^{\\rm 3D}$ project \\citep{Cappellari11}, brought forth the following general results:\n(i) rotation velocities are highest and velocity dispersions are lowest for cold gas, followed by ionised gas, and then stars \\citep[e.g.][]{Davis13, Martinsson13a, Bolatto17, Levy18, CrespoGomez21, Girard21}; \n(ii) modelling of stellar kinematics (e.g.\\ with axisymmetric Jeans Anisotropic Multi-Gaussian Expansion models; JAM; \\citealp{Cappellari08}) produces circular velocity curves that match observed cold molecular gas rotation velocities in regularly rotating galaxies \\citep[e.g.][]{Davis13, Leung18}, suggesting that the mass estimates from molecular gas and stars are in agreement;\n(iii) the mis-alignment of gas and stellar kinematic major axes with each other and with the morphological major axis is small for the majority of non-interacting systems without strong bars, but generally higher for Early-Type Galaxies (ETGs) \\citep[e.g.][]{FalconBarroso06, Sarzi06, Davis11, BarreraBallesteros14, BarreraBallesteros15, Serra14, Bryant19}. \n\nIn contrast, our knowledge of galaxy kinematics at $1<z<3$, when massive galaxies assemble most of their stellar mass \\citep[e.g.][]{Moster20}, is dominated by ionised gas observations. Efficient multiplexing near-infrared spectrographs such as KMOS \\citep[the K-band Multi-Object Spectrograph;][]{Sharples04, Sharples13} and MOSFIRE \\citep[the Multi-Object Spectrometer For Infra-Red Exploration][]{McLean10, McLean12}, trace strong rest-frame optical emission lines in several thousand galaxies up to $z\\sim3.5$ \\citep[see][for an overview]{FS20}. \nLarger surveys of molecular gas with few hundred $1<z<3$ galaxies focus on integrated quantities \\citep[e.g.][and references therein]{Tacconi13, Tacconi18, Freundlich19}, with some exceptions of spatially resolved kinematics for individual galaxies \\citep[e.g.][]{Tacconi10, Swinbank11, Genzel13, Uebler18}.\nStellar kinematic observations at $z>1$ were initially obtained almost exclusively for quiescent galaxies, and from slit spectroscopy, focusing on integrated quantities due to signal-to-noise (S\/N) considerations \\citep[but see][]{Newman15, Newman18, Toft17, Mendel20}. \nThe LEGA-C \\citep[Large Early Galaxy Astrophysics Census;][]{vdWel16, vdWel21, Straatman18} survey brought a step change in stellar kinematics of distant systems. Thanks to its deep uniform integrations and large sample size, spatially resolved kinematic analyses and modelling have become feasible for few hundred galaxies of all types at $0.6<z<1$ \\citep{Bezanson18, vHoudt21, Straatman22, vdWel22}, and integrated kinematic measures for few thousands \\citep{vdWel21}.\n\nMulti-tracer observations of galaxy kinematics at $z>0$ are sparse, but overall indicate similar trends as $z=0$ studies. Molecular disc velocity dispersions are lower relative to ionised gas at $z\\sim0.2$ (\\citealp{Cortese17}, see also \\citealp{Molina20}). There are indications that this trend prevails out to $z\\sim2$ (\\citealp{Girard19, Uebler19}; Liu et al., subm.), while some individual galaxies have comparable dispersions \\citep{Genzel13, Uebler18, Molina19}. \nTentative trends of higher stellar disc velocity dispersions compared to ionised gas are seen in the data by \\cite{Guerou17} of 17 galaxies at $z\\sim0.5$.\n\nA recent study by \\cite{Straatman22} compares dynamical mass estimates based on slit observations of ionised gas and stars for 157 galaxies at $0.6<z<1$ from the LEGA-C survey. Here, dynamical masses from stellar kinematics are inferred from JAM models \\citep{vHoudt21}, and dynamical masses from ionised gas are inferred from pressure-supported disc models, where the latter are found to be systematically lower by 0.15~dex. No correlations of this discrepancy with galaxy properties were found. \nThe authors find a similar offset when comparing JAM estimates to mass measurements based on integrated emission line widths.\n\nWe revisit the comparison of stellar and ionised gas dynamical mass models with IFS observations of the H${\\alpha}$ emission line from the KMOS$^{\\rm 3D}$ survey \\citep{Wisnioski15, Wisnioski19}. IFS data provide knowledge of the kinematic major axis of a galaxy through velocity and velocity dispersion maps. This allows for the extraction of major axis kinematics, which can be fed directly into dynamical models. However, the kinematic major axis is typically unknown for long-slit observations, and is usually assumed to coincide with the photometric position angle. Further corrections are required if the slit orientation is different from the photometric position angle, as is largely the case for the LEGA-C observations.\n\nIn this paper, we present a comparison of kinematics and dynamical mass estimates from IFS observations of the H${\\alpha}$ emission line from KMOS$^{\\rm 3D}$, and long-slit spectroscopic observations of stars from LEGA-C. \nThis paper is organized as follows. Our sample is described in Section~\\ref{s:data}. In Section~\\ref{s:kinex} we discuss the extraction of kinematic profiles from both surveys, including careful matching of instrumental effects, and the construction of dynamical mass models in Section~\\ref{s:modelling}. We discuss results based on stellar and ionised gas kinematics in Section~\\ref{s:compasobs}. In Section~\\ref{s:mdyn} we compare dynamical mass estimates from the different tracers and methods, and investigate correlations of dynamical mass offset with physical, structural, and kinematic properties of the galaxies. We conclude in Section~\\ref{s:conclusions}.\n\n\n\\section{Data}\\label{s:data}\n\\subsection{The Sample}\n\nFor our analysis we select galaxies observed within both the KMOS$^{\\rm 3D}$\\, and LEGA-C surveys.\nThere are 26 unique targets common to both surveys, all located in the Cosmic Evolution Survey (COSMOS) field \\citep{Scoville07}. Ten of the LEGA-C galaxies have been observed twice with separate mask designs, and nine galaxies have LEGA-C longslit observations oriented in E-W direction in addition to or instead of the default N-S observations. \nThe sample spans a range in stellar mass, star-formation rate, and size, as illustrated in Figure~\\ref{f:msmr}. Most galaxies are located at the massive end of the $z\\sim1$ main sequence and follow the mass-size relation, but a few log$(M_*\/M_{\\odot})<10$ systems are included, as well as some passive systems. \nThe quality of data for both KMOS$^{\\rm 3D}$\\, and LEGA-C varies across the sample, primarily due to integration time (for KMOS$^{\\rm 3D}$\\,) and observing conditions.\n\nFor the comparison of line-of-sight (LOS) kinematics in this work, we focus on 16 galaxies with at least five measurements of velocity and velocity dispersion along the (pseudo-)slit in both surveys, and for the dynamical mass comparison we utilise 10 galaxies for which dynamical masses can be measured from both surveys (see Sections~\\ref{s:kinex} and \\ref{s:modelling} for details). We list all galaxies discussed in this work with redshifts and KMOS$^{\\rm 3D}$\\, integration times in Table~\\ref{t:sample}.\n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.49\\textwidth]{figures\/MS-eps-converted-to.pdf}\n\t\\includegraphics[width=0.49\\textwidth]{figures\/MR-eps-converted-to.pdf}\n\t\\caption{Location of our parent sample (blue symbols) in the $M_{\\star}-$ SFR (left) and $M_{\\star}-R_e$ (right) planes on top of the underlying galaxy population in the COSMOS field at $0.6<z<1.2$ taken from the 3D-HST catalogue (grey points). Duplicate observations in LEGA-C are indicated by black hexagons, and LEGA-C observations in E-W orientation are indicated by a horizontal bar. Symbols with white centres indicate galaxies formally undetected in H$\\alpha$ in the KMOS$^{\\rm 3D}$\\, data release. Symbols with orange\/red centres indicate galaxies in our kinematics sample, and symbols with red centres indicate galaxies in our dynamical mass sample. Our galaxies span a range in stellar masses, SFRs, and sizes, but most objects are located at the massive end of the $z\\sim1$ main sequence and follow the mass-size relation.}\n\t\\label{f:msmr}\n\\end{figure*}\n\n\\begin{table*}\n\t\\centering\n\t\\caption{KMOS$^{\\rm 3D}$\\, ID with field and 3D-HST v4 catalog object ID \\citep{Skelton14}; KMOS$^{\\rm 3D}$\\, redshift; KMOS$^{\\rm 3D}$\\, on-source integration time in minutes; LEGA-C Mask-ID with mask number and UltraVista catalog object ID \\citep{Muzzin13}; second LEGA-C Mask-ID in case of duplicate observations; LEGA-C redshift. The integration times for the LEGA-C observations are all approximately 1200 minutes. LEGA-C Mask-IDs with mask number M101 refer to longslit observations in E-W orientation.}\n\t\\label{t:sample}\n\t\t\\begin{tabular}{lrrllr}\n\t\tKMOS$^{\\rm 3D}$\\, ID & KMOS$^{\\rm 3D}$\\, $z$ & KMOS$^{\\rm 3D}$\\, $t_{\\rm int}$ & LEGA-C Mask-ID & second LEGA-C Mask-ID & LEGA-C $z$ \\\\\n\t\t\\hline\n    COS4\\_03493 &      0.679 &       285 & M4\\_121150 & - & 0.678\\\\\n    COS4\\_13901 &      0.684 &       625 & M1\\_131104 & M2\\_131104 & 0.683\\\\\n    COS4\\_04943 &      0.758 &       230 & M3\\_122667 & - & 0.758\\\\\n    COS4\\_16227 &      0.796 &       335 & M1\\_131985 & - & 0.796\\\\\n    COS4\\_09601 &      0.834 &       230 & M5\\_126127 & M101\\_126127 & 0.834 \\\\\n    COS4\\_19648 &      0.889 &       315 & M1\\_134839 & - & 0.889\\\\\n    COS4\\_25353 &      0.897 &       255 & M1\\_139825 & M101\\_139825 & 0.897\\\\\n    COS4\\_17628 &      0.907 &       100 & M5\\_133199 & M101\\_133199 & 0.907\\\\\n    COS4\\_06487 &      0.907 &       370 & M3\\_123575 & M101\\_123575 & 0.908\\\\\n    COS4\\_05296 &      0.926 &       230 & M4\\_122584 & - & 0.925\\\\\n    COS4\\_10860 &      0.927 &       230 & M1\\_127387 & - & 0.927\\\\\n    COS4\\_05238 &      0.936 &       230 & M7\\_122836 & - & 0.937\\\\\n    COS4\\_09156 &      0.938 &       285 & M3\\_125617 & M101\\_125617 & 0.939\\\\\n    COS4\\_03700 &      0.956 &       230 & M3\\_121631 & M4\\_121631 & 0.956\\\\\n    COS4\\_08096 &      0.979 &       230 & M7\\_125257 & - & 0.980\\\\\n    COS4\\_12699 &      1.003 &       320 & M101\\_128834 & - & 1.005\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\n\n\\subsection{The KMOS\\texorpdfstring{$^{\\rm 3D}$}{3D} Survey}\\label{s:k3d}\n\nThe KMOS$^{\\rm 3D}$\\, survey is a 75-night GTO survey with the multi-IFS KMOS at the VLT, targeting the H$\\alpha$+[NII] line emission in 739 $\\log(M_*\/M_{\\odot})>9$ galaxies at $0.6<z<2.7$. The survey is presented by \\cite{Wisnioski15, Wisnioski19}, to which we refer the reader for details.\n\nThe KMOS$^{\\rm 3D}$\\, galaxies were drawn from the {\\it Hubble Space Telescope} ({\\it HST}) 3D-HST Treasury Survey \\citep{Brammer12, Skelton14, Momcheva16}, providing secure spectroscopic or grism redshift for optimal avoidance of skyline contamination at the location of H$\\alpha$. A $\\log(M_*\/M_{\\odot})>9$ and $K < 23$~mag selection function was chosen to obtain a population-wide census reducing biases in SFR or colors. Targets are located in COSMOS, GOODS-S (Great Observatories Origins Deep Survey) and UDS (Ultra Deep Survey). High-resolution Wide Field Camera 3 (WFC3) near-IR and Advanced Camera for Surveys (ACS) optical imaging is available from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey \\citep[CANDELS;][]{Grogin11, Koekemoer11, vdWel12}, and further multi-wavelength coverage from X-ray through optical, near- to far-infrared, and radio is accessible from e.g.\\ \\cite{Ueda08, Lutz11, Xue11, Civano12, Magnelli13, Skelton14}.\n\nThe publicly released data cubes have a spatial sampling in $x-y-$direction of $0.2\\arcsec$ which corresponds to $\\sim1.6$ kpc at $z=1$. The wavelength sampling in $z-$direction is 1.7~\\AA. The typical near-IR seeing of the KMOS$^{\\rm 3D}$\\, data has a FWHM of $0.5\\arcsec$, corresponding to $\\sim4.0$ kpc at $z=1$. Point-spread function (PSF) images representing the observing conditions for each combined data cube individually are included in the data release, together with both a Gaussian and Moffat parametrization. \nThe average spectral resolution for KMOS$^{\\rm 3D}$\\, observations in the $YJ$ filter is $R=\\lambda\/\\Delta\\lambda=3515$, corresponding to an average instrumental broadening of 36~km\/s. However, the line-spread function (LSF) of each galaxy is determined individually as a function of wavelength, and encoded in the fits header keywords as described by \\cite{Wisnioski19}. The average on-source integration time for $z\\sim1$ targets in KMOS$^{\\rm 3D}$\\, is 5 hours.\n\nStellar masses $M_*$ and star formation rates (SFRs) for all galaxies are derived from SED fitting following \\cite{WuytsS11a}, assuming a \\cite{Chabrier03} initial mass function, using \\cite{Bruzual03} models with solar metallicity, the reddening law by \\cite{Calzetti00}, and constant or exponentially declining star formation histories. \nStructural parameters such as the effective radius $R_e$, the S\u00e9rsic index $n_S$, the axis ratio $q=b\/a$, the morphological position angle PA$_{\\rm morph}$, and for some galaxies the bulge-to-total ratio $B\/T$ and corresponding radii and S\u00e9rsic indices, are constrained from single-S\u00e9rsic or double-S\u00e9rsic {\\sc{galfit}} \\citep{Peng10} models to the CANDELS F160W imaging as presented by \\cite{vdWel12, Lang14}.\n\n\n\\subsection{The LEGA-C Survey}\\label{s:lgc}\n\nThe LEGA-C survey is a 1107-hour public survey with the Visible Multi-Object Spectrograph (VIMOS) at the VLT \\citep{LeFevre03}, targeting 3741 galaxies at $0.6<z<1.5$. The survey is presented by \\cite{vdWel16, vdWel21, Straatman18}, to which we refer the reader for details.\n\nThe LEGA-C targets were drawn from the COSMOS\/UltraVISTA $K-$band selected catalogue with photometry \\citep{Muzzin13}, and the primary sample consists of $0.6<z<1.0$ galaxies with $K < 20.7 - 7.5\\cdot \\log[(1 + z)\/1.8]$~mag. ACS imaging is available for most targets \\citep{Scoville07}.\n\nLEGA-C slits are $1\\arcsec$ ($\\sim8.0$~kpc at $z\\sim1$) wide and typically at least $8\\arcsec$ long, oriented in N-S direction, with the exception of one mask having slits oriented in E-W direction. The sampling in the spatial direction is $0.205\\arcsec$, and in the wavelength direction 0.6~\\AA. The average PSF FWHM measured from Moffat fits is $0.8\\arcsec$.\nThe effective spectral resolution is $R\\sim3500$, and therefore comparable to the average KMOS $YJ$ instrumental resolution of 36~km\/s. The typical on-source integration time is 20 hours.\n\nStructural parameters such as $R_e$, $n_S$, $q$, and PA$_{\\rm morph}$, are constrained from single-S\u00e9rsic {\\sc{galfit}} models to ACS F814W imaging as presented by \\cite{vdWel16, vdWel21}. F814W is chosen because it has the largest overlap with the LEGA-C footprint, and therefore all dynamical mass estimates for LEGA-C discussed below use F814W-based structural parameters (although near-IR WFC3 imaging is also available for the subset studied in this work). \n\nWe note that the main analysis in this work uses the default imaging of the two surveys. We have tested using common imaging information, and we discuss those results where appropriate (see Section~\\ref{s:model_diff} and Appendix~\\ref{a:f814w_modelling}). Our main conclusions are not affected by this choice.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.935\\textwidth]{figures\/OverviewFig2_new.pdf}\n\t\\caption{Illustration of the available kinematic and ancillary data for galaxies common to the KMOS$^{\\rm 3D}$\\, and LEGA-C surveys.\n\tTop left: IJH {\\it HST} color-composite image, with white dashed lines indicating the $1\\arcsec$ wide VIMOS slit positions from LEGA-C, orange and yellow dash-dotted lines indicating the morphological position angles derived from the F814W and F160W passbands, respectively, and the violet dashed line indicating the kinematic position angle determined form the KMOS$^{\\rm 3D}$\\, H$\\alpha$ IFS data. \n\tTop middle, right: H$\\alpha$ velocity and velocity dispersion maps as derived from the KMOS$^{\\rm 3D}$\\, data cubes.\n\tMiddle left: 2D PV diagrams extracted in N-S direction around the [OII] emission line from LEGA-C (top) and the H$\\alpha$ emission line from KMOS$^{\\rm 3D}$\\, (bottom). Here, the VIMOS data are rebinned in wavelength direction to match the coarser KMOS sampling, and the KMOS data are convolved to match the typically worse VIMOS PSF.\n\tMiddle right: 1D integrated LEGA-C spectrum collapsed along the N-S slit (top; smoothed for illustrative purposes) with pink lines indicating low quality regions, and 1D velocity and velocity dispersion profiles (bottom) extracted along the N-S slit from KMOS$^{\\rm 3D}$\\, (filled blue circles) and LEGA-C (stars: filled golden stars; gas: open green diamonds); dotted and dashed vertical lines indicate the major axis effective radii derived from the F814W and F160W passbands.\n\tBottom panels: same as middle panels, but now for data extracted in E-W direction.\n\tWhite vertical bars indicate $1\\arcsec$ in the kinematic maps and PV diagrams.\n    This galaxy has a comparable 1D rotation pattern in the KMOS$^{\\rm 3D}$\\, and LEGA-C data, however the LEGA-C velocity dispersions appear asymmetric compared to the H$\\alpha$ extractions. Differences in the brightness distribution of the [OII] {\\it vs.\\ }H$\\alpha$ emission are apparent in the PV diagrams.}\n\t\\label{f:obsprof}\n\\end{figure*}\n\n\n\\section{Kinematic extractions and Dynamical models}\\label{s:methods}\n\n\\subsection{Fixed-slit kinematic extractions}\\label{s:kinex}\n\n\\subsubsection{Extractions for KMOS$^{3D}$}\\label{s:kmoskin}\n\nFor an adequate comparison of LOS kinematics, we process the KMOS$^{\\rm 3D}$\\, data as follows. First, we match the PSFs for each observation, then we extract position-velocity (PV) diagrams matching the orientation and slit width of the LEGA-C data, and finally we extract from those the 1D kinematic profiles by means of Gaussian fits to the H$\\alpha$ line profile in each spatial row.\n\nTo match the PSFs, we create for each galaxy a convolution kernel based on the Moffat parametrizations of the PSFs from each pair of observations.\\footnote{We use the Photutils \\citep{photutils} subpackage for PSF matching {\\tt{photutils.psf.matching}} with a top hat window, and verify the convolution based on the model PSFs.} For duplicate observations in LEGA-C, we create separate convolution kernels per observation and galaxy. These kernels are then applied to the KMOS$^{\\rm 3D}$\\, data cubes (which have smaller PSF FWHMs for all galaxies in our sample; see also Sections~\\ref{s:k3d}, \\ref{s:lgc}).\n\nTo extract PV diagrams, we place a pseudo-slit with width $1\\arcsec$ on the KMOS$^{\\rm 3D}$\\, cubes, oriented in N-S and\/or E-W direction, as appropriate.\nSlit centering is based on the location of the synthesized continuum maps created from the KMOS data cubes. \nNote that the exact VIMOS slit positioning is uncertain, but \\cite{vHoudt21} conclude that the typical offset from the galaxy center should be no more than $1-2$ pixels ($0.2-0.4\\arcsec$), based on an analysis of asymmetric light profiles. \nFor the purpose of our comparison, we assume that the position of galaxies in the LEGA-C slits aligns well with the KMOS$^{\\rm 3D}$\\, continuum center (but see Section~\\ref{s:multiple} for discussion of an outlier).\n\nFrom the 2D PV diagrams we extract spectra for each row, and determine the H$\\alpha$ velocity and velocity dispersion from Gaussian fits to the line profile at the H$\\alpha$ position. For the velocity dispersion profiles, we subtract the LSF appropriate for the H$\\alpha$ line position in quadrature, to remove instrumental broadening.\nAn example of the resulting profiles is shown in Figure~\\ref{f:obsprof}.\n\n\n\\subsubsection{Extractions for LEGA-C}\\label{s:lgckin}\n\nVelocity and velocity dispersion profiles for the stellar and ionised gas in LEGA-C are derived as described in detail by \\cite{Bezanson18} and \\cite{vdWel21}: the 2D PV diagrams are fit in each row (with median S\/N~$>2$ per pixel) with pPXF \\citep{Cappellari04, Cappellari17} by combining a high-resolution stellar population template and an emission line template. These templates are allowed to shift and broaden independently, delivering independent stellar and ionised gas velocity and velocity dispersion profiles. This procedure takes into account the LSF and removes instrumental broadening from the velocity dispersion profiles. An example for one galaxy is shown in Figure~\\ref{f:obsprof}. \nNote that the template for the [OII] doublet consists of two emission lines centred at $\\lambda=3727$~\\AA\\ and $\\lambda=3730$~\\AA.\n\nFor the purpose of our study, we make a few adjustments to the above methodology for individual objects: \nfor two observations in our sample we repeat the above fitting procedure with a relaxed S\/N-cut in order to obtain 1D kinematic profiles. \nAnother two galaxies show strong [NeV]$\\lambda3347$, [NeV]$\\lambda3427$ and [NeIII]$\\lambda3870$ emission. The high-ionisation [NeV]$\\lambda3427$ line is a tell-tale signature of a harder ionising radiation field than produced by pure star formation, indicative of AGNs or shocks \\citep[e.g.][]{Mignoli13, Feltre16, Vergani18, Kewley19}. Where present in the LEGA-C spectra in our sample, it is kinematically decoupled from other emission lines and centrally concentrated. To extract ionised gas kinematics for these galaxies, we mask the corresponding spectral regions and repeat the above fitting procedure.\nThe effect on the extracted gas kinematics is substantial, with differences in individual velocity and velocity dispersion measurements of up to 200~km\/s (see Appendix~\\ref{a:nev} for an example). \nThe stellar kinematic measurements are virtually unaffected by this procedure.\n\nFor visual comparison of the 2D PV diagrams we further resample the LEGA-C spectra to the (coarser) KMOS wavelength steps. Note that we do not resample in spatial direction due to the very small difference in the KMOS and VIMOS pixel scales of $0.005\\arcsec$.\n\n\n\\subsubsection{Measurements}\\label{s:measure}\n\nDue to the different radial coverage of the data it is not straight-forward to compare the gas and stellar kinematics in these systems even after matching the observing conditions. To quantify how well the KMOS$^{\\rm 3D}$\\, and LEGA-C kinematic data compare to each other, we define the following 1D measurements based on the LOS kinematic profiles (see Figure~\\ref{f:obsprof}):\n\n\\begin{itemize}\n    \\item $v_{\\rm max}$ is the maximum observed absolute velocity (uncorrected for inclination), and $r_{\\rm vmax}$ is the corresponding radius.\n    \\item $v_{\\rm rmax,both}$ is the (mean) velocity at the outermost radius covered by both the KMOS$^{\\rm 3D}$\\, and LEGA-C data, and $r_{\\rm max,both}$ is the corresponding radius.\n    \\item $\\sigma_{\\rm out}$ is the weighted mean observed velocity dispersion of the four outermost measured values (outer two on each side of the profile). Note that this measurement may still be affected by beam smearing, especially for smaller systems.\n    \\item $\\sigma_{\\rm rmax,both}$ is the (mean) observed velocity dispersion at $r_{\\rm max,both}$.\n    \\item $v_{\\rm rms}$ is an approximation of a classical root mean square velocity, via $v_{\\rm rms}^2=v_{\\rm max}^2 + \\sigma_{\\rm out}^2$.\n    \\item $v_{c,\\rm max}$ is an approximation of a circular velocity (here without corrections for inclination and beam-smearing), via $v_{c,\\rm max}^2=v_{c}^2(r_{\\rm vmax})=v_{\\rm max}^2+2\\sigma_{\\rm out}^2\\cdot r_{\\rm vmax}\/R_d$ \\citep[see][for details]{Burkert10}, where we assume that the disc scale length $R_d=R_e\/1.68$, with $R_e=R_{e,\\rm F160W}$.\n    \\item S$_{0.5,\\rm max}$ is an approximation of the total integrated velocity dispersion, via S$_{0.5, \\rm max}^2=0.5\\cdot v_{\\rm max}^2+\\sigma_{\\rm out}^2$ \\citep[see][for details]{Weiner06a, Kassin07}.\n\n\\end{itemize}\nFor the LEGA-C measurements, the above quantities are measured for ionised gas (primarily [OII] and\/or H$\\beta$) and stellar kinematics individually.\n\nWe stress that the above quantities are not derived from modelling, but are based on the LOS kinematics, which for the KMOS$^{\\rm 3D}$\\, galaxies have been extracted mimicking the LEGA-C observing conditions and setup.\nTherefore, while the LSF is accounted for, the measurements do not include corrections for inclination or beam-smearing. This is to say, intrinsic maximum velocities would be larger, and intrinsic velocity dispersions would be smaller. However, due to our matching of the PSFs differences in the effects of beam-smearing in the original observations are accounted for, and gas kinematics in KMOS$^{\\rm 3D}$\\, and LEGA-C should match if the emission lines trace the same ISM components. \n\nFurthermore we emphasize that the above observed `maximum' velocities do not necessarily represent the true observed maximum velocities of the galaxies, due to the kinematic major axes generally not being aligned with the slit orientations (see Section~\\ref{s:mdynk3d} for KMOS$^{\\rm 3D}$\\, kinematic extractions along the kinematic major axis).\n\nThe measurements described above cannot be meaningfully performed for all galaxies and observations in the sample. We exclude from the subsequent comparison galaxies for which there are less than five extractions of velocity and velocity dispersion possible along the (pseudo-)slit for either the KMOS$^{\\rm 3D}$\\, or LEGA-C data. We further exclude one LEGA-C observation for which the resolved kinematic extractions are contaminated through a secondary object in the slit. \nThe final sample includes 16 galaxies, resulting in 20 pairs of observations, including four duplicate observations from LEGA-C with the correspondingly different slit- and PSF-matched extractions from the KMOS$^{\\rm 3D}$\\, data cubes.\n\n\n\\subsection{Dynamical modelling}\\label{s:modelling}\n\nFor our comparison of dynamical mass measurements, we use the data at their native spatial and spectral resolutions, without matching observing conditions between the KMOS$^{\\rm 3D}$\\, and LEGA-C surveys. For KMOS$^{\\rm 3D}$\\, we build mass models which we fit to the H$\\alpha$ major axis kinematics (see Section~\\ref{s:mdynk3d}), and for LEGA-C we use published dynamical masses from JAM models (see Section~\\ref{s:mdynlgc}) and those computed from integrated stellar velocity dispersions. \nDue to the varying data quality across the sample, robust dynamical models cannot be constructed for all galaxies. Our dynamical mass comparison includes ten galaxies, four of which have two estimates based on integrated stellar velocity dispersion from LEGA-C due to duplicate observations, and six have LEGA-C estimates based on both integrated stellar velocity dispersion and JAM models (see Section~\\ref{s:mdynlgc}).\n\n\n\\subsubsection{Modelling for KMOS$^{\\rm 3D}$\\,}\\label{s:mdynk3d}\n\nFor KMOS$^{\\rm 3D}$\\, we exploit the 3D information available from the IFS data cubes to build 3D mass models to determine dynamical masses. Specifically, we place a pseudo-slit of width equal to the near-IR PSF FWHM on the continuum-subtracted cube along the kinematic major axis, which is well defined from the 2D projected velocity fields. From the 2D PV diagrams we then extract 1D profiles of velocity and velocity dispersion by summing rows spanning the PSF FWHM (or half PSF FWHM), and by fitting a Gaussian to the H$\\alpha$ line position (see Section~\\ref{s:kmoskin}). \n\nWe forward-model the H$\\alpha$ major axis kinematics using {\\sc{dysmal}} \\citep{Cresci09, Davies11, WuytsS16, Uebler18, Price21}, a code that allows for a flexible number of mass components, accounts for finite scale heights and flattened spheroidal potentials \\citep{Noordermeer08}, includes effects of pressure support from the turbulent interstellar medium \\citep{Burkert10, WuytsS16}, and consistently incorporates the observation-specific PSFs and LSFs.\n\nDue to the heterogeneous data quality in our sample, we consider two basic mass models for the baryonic component: a single S\u00e9rsic profile and a bulge-to-disc decomposition. Assuming mass follows light, we fix the structural parameters, specifically $i_{\\rm F160W}$, $R_{e,\\rm F160W}$ (or $R_{e,\\rm F160W,bulge}$ and $R_{e,\\rm F160W,disc}$), and $n_{S,\\rm F160W}$ (or $n_{S,\\rm F160W,bulge}$ and $n_{S,\\rm F160W,disc}$), to measurements from {\\sc{galfit}} models to the CANDELS F160W imaging as presented by \\cite{vdWel12, Lang14, WuytsS16} (see  Section~\\ref{s:k3d}). Here, we infer the galaxy inclination $i_{\\rm F160W}$ from $q_{\\rm F160W}=b\/a$ by assuming an intrinsic ratio of scale height to scale length of $q_0=0.2$ \\citep[see][]{vdWel14b, WuytsS16, Straatman22}. If including a bulge, we assume an axis ratio of 1 for this component.\nWe estimate the total baryonic mass $M_{\\rm bar}$ by adding the stellar mass $M_{\\star}$ from SED modeling and the gas mass $M_{\\rm gas}$ based on $M_{\\star}$, SFR, and redshift of each galaxy, by utilising the gas mass scaling relations by \\cite{Tacconi20}. This estimate is used to center a Gaussian prior with standard deviation 0.2~dex on the logarithmic total baryonic mass.\nThe intrinsic velocity dispersion $\\sigma_0$ is assumed to be isotropic and constant throughout the disc, supported by deep adaptive optics assisted observations of SFGs at this redshift (see \\citealp{Genzel06, Genzel08, Genzel11, Genzel17, Cresci09, FS18, Uebler19}; Liu et al., subm.). The value of $\\sigma_0$ is a free parameter in our modelling.\n\nAll our dynamical models include an NFW \\citep{NFW96} dark matter halo. Its total mass $M_{\\rm halo}$ is inferred from the dark matter mass fraction within the effective (disc) radius, $f_{\\rm DM}(<R_{e,\\rm F160W})$, which is a free parameter in our modelling. However we fix the halo concentration parameter $c$ following the fitting functions derived by \\cite{Dutton14}, by first assuming a typical dark matter halo mass based on the stellar mass and redshift of each galaxy following \\cite{Moster18}. Typical values for our sample are $c\\sim6-7$.\n\nIn total, we have three free parameters in our basic mass models: the total baryonic mass $M_{\\rm bar}$, the intrinsic velocity dispersion $\\sigma_0$, and the central dark matter fraction $f_{\\rm DM}$. The focus of our study is the enclosed dynamical mass, which we take to be the sum of the best-fit baryonic and dark matter masses as a function of radius: $M_{\\rm dyn}(<r)=M_{\\rm bar}(<r)+M_{\\rm DM}(<r)$. $M_{\\rm dyn}(<r)$ is calculated within spherical apertures from the non-spherically symmetric potential of the 3D mass distribution \\citep[see][]{Price22}.\nWe emphasize that the enclosed dynamical mass is relatively insensitive to the detailed partitioning of baryonic and dark matter mass \\citep[see also][]{WuytsS16, Price21}.\n\nWe fit using Markov Chain Monte Carlo (MCMC) techniques as implemented in {\\sc{dysmal}} through the {\\sc{emcee}} package \\citep{ForemanMackey13}. We use 300 walkers, a burn-in phase of 100 steps, and a run phase of 200 steps (greater than ten times the maximum auto-correlation time of the individual parameters). We adopt the maximum a posteriori values of the fit parameters as our best-fit values, based on a joint analysis of the posteriors for all free parameters \\citep[see][]{Price21}.\n\nFor two galaxies in our sample with very high H$\\alpha$ data quality, COS4\\_17628 and COS4\\_06487, more sophisticated dynamical models with a higher number of free parameters exist, in particular including fitting of structural parameters \\citep{Nestor22}. The dynamical masses at the radii considered in this analysis agree among the various mass models within 0.06~dex.\n\n\n\\subsubsection{Modelling for LEGA-C}\\label{s:mdynlgc}\n\nFor the LEGA-C dynamical mass estimates, we use the recently published values by \\cite{vdWel21} and \\cite{vHoudt21} based on the LEGA-C stellar kinematics. We refer the reader to those papers for details, but summarize the most relevant information here.\n\n\\cite{vHoudt21} construct axisymmetric Jeans anisotropic models to the stellar rms velocity for LEGA-C galaxies for which the morphological position angle is not misaligned with the slit by more than $45^{\\circ}$, that are not mergers or irregular galaxies, that are not mid-IR or X-ray-identified AGN\\footnote{\nSuch galaxies are identified by \\texttt{FLAG\\_SPEC}=1 in the LEGA-C data release \\citep[see][]{vdWel21}. Narrow-line or radio-identified AGN are not flagged.}, do not have flux calibration issues, and that have S\/N~$>10$ in at least three spatial resolution elements. The models consist of two mass components, a stellar component assuming mass follows light based on F814W imaging, and an NFW dark matter halo. The halo concentration $c$ is tied to the halo mass following \\cite{Dutton14}. Free fit parameters are the stellar velocity anisotropy, the stellar mass-to-light ratio $M\/L$, the circular velocity of the dark matter halo, the galaxy inclination, and the slit centering.\nHere, the inclination is constrained by a $q-$dependent prior assuming an intrinsic thickness distribution ${\\mathcal{N}}(0.41,0.18)$ which is constrained from the full primary LEGA-C sample. \\cite{vHoudt21} note that the inclination and the slit centering are typically unconstrained by the data.\n\nBased on the JAM results, dynamical masses are provided out to 20~kpc and\/or $2R_{e,\\rm F814W}$, if supported by the data, where $R_{e,\\rm F814W}$ is the semimajor axis effective radius determined from single-S\u00e9rsic {\\sc{galfit}} models to the F814W imaging as presented by \\cite{vdWel16, vdWel21}. The total JAM dynamical mass is defined as twice the mass within $R_{e,\\rm F814W}$.\n\nAs described above, Jeans anisotropic models can be built only for a subset of the LEGA-C survey. However, the existing models are used to calibrate more accessible virial mass estimators based on the integrated stellar velocity dispersion. The details of this calibration are described by \\cite{vdWel22}. In short, virial masses are computed as\n$$M_{\\rm vir}=K(n_{S,\\rm F814W})\\frac{\\sigma^2_{\\star,\\rm vir}R_{e,\\rm F814W}}{G},$$\nwhere $n_{S,\\rm F814W}$ and $R_{e,\\rm F814W}$ are derived from F814W imaging (see Section~\\ref{s:lgc}) and $K(n_{S})=8.87-0.831n_{S}+0.0241n_{S}^2$ following \\cite{Cappellari06}, $\\sigma_{\\star,\\rm vir}$ is the inclination- and aperture-corrected, integrated stellar velocity dispersion (measured from collapsed 1D spectra), and $G$ is the gravitational constant. \nThe correction for $\\sigma_{\\star,\\rm vir}$ is derived by calibration to the JAM dynamical masses.\n$M_{\\rm vir}$ corresponds to twice the mass within $R_{e,\\rm F814W}$.\n\nThe main focus of our dynamical mass comparison is between the KMOS$^{\\rm 3D}$\\, IFS H$\\alpha$ models and the LEGA-C JAM and $M_{\\rm vir}$ measurements based on stellar kinematics. Two galaxies in our dynamical mass sample are also part of the emission line modelling analysis by \\cite{Straatman22}, and we include their results where appropriate. \\cite{Straatman22} build a kinematic model where the rotation curve is parametrized by an arctan function, assuming the ionised emission originates from a thick, exponential distribution constrained from the F814W imaging, with a constant and isotropic intrinsic velocity dispersion. The model accounts for beam smearing and misalignment between the slit and PA$_{\\rm F814W}$, and the dynamical mass is calculated from the model rotation curve including a pressure support correction. See \\cite{Straatman22} for further details.\n\n\n\\subsubsection{Notable differences between the dynamical models}\\label{s:model_diff}\n\nOur H$\\alpha$ dynamical mass models use structural parameters from F160W imaging, while the stellar dynamical mass measurements use structural parameters from F814W imaging (see Sections~\\ref{s:k3d}, \\ref{s:lgc}, and Appendix~\\ref{a:structure} for further discussion).\nTo quantify the impact of different structural measurements for our sample, we repeat the dynamical modelling for the KMOS$^{\\rm 3D}$\\, galaxies, this time utilizing the $i-$band (F814W) based values for a single-S\u00e9rsic baryonic component. We construct two additional sets of dynamical models adopting $R_e$, $n_S$, and $q$ from F814W imaging. For the first set we assume an intrinsic thickness of $q_0=0.2$ (as for our fiducial models). For the second set we still assume an intrinsic thickness of our baryonic distribution of $q_0=0.2$, but infer the galaxy inclination assuming $q_0=0.41$. This second set attempts a closer match to the assumptions entering the LEGA-C JAM modelling, where the inclination is inferred from a $q-$dependent prior, assuming an intrinsic thickness distribution centered on $q_0=0.41$ \\citep{vHoudt21}. \nOverall, the impact on our dynamical mass estimates is minor, and the corresponding results are presented in Appendix~\\ref{a:f814w_modelling}: for the first set of alternative models we find an average increase in $M_{\\rm dyn}$ of 0.02~dex (standard deviation 0.26~dex); for the second set of alternative models we find an average decrease in $M_{\\rm dyn}$ of 0.03~dex (standard deviation 0.25~dex).\n\nAnother difference lies in the explicit assumption of mass components. As described in Section~\\ref{s:mdynk3d}, for the modelling of the KMOS$^{\\rm 3D}$\\, data we estimate total baryonic mass by including a cold gas component derived from the scaling relations by \\cite{Tacconi20}. For our sample, such derived gas-to-baryonic-mass fractions are between 2 and 70 per cent, with a mean value of $f_{\\rm gas}=0.27$. The LEGA-C JAM modelling assumes only a stellar and a dark matter component (see Section~\\ref{s:mdynlgc}). However, although they assume mass follows light, $M\/L$ is a free parameter in their fit. The combination of a free $M\/L$ and an explicit dark matter halo component therefore allows for an (unconstrained) contribution from gas (following stars) as well. \nThe more simplistic $M_{\\rm vir}$ calculation does not make any assumptions on the involved mass components, however uses the (corrected) integrated 1D stellar velocity dispersion as a tracer of dynamical mass. As the movement of stars is dictated by the full potential, this includes any contribution from all stars, gas, and dark matter.\n\n\n\n\\section{Stellar and ionised gas kinematics}\\label{s:compasobs}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.245\\textwidth]{figures\/lvmax_psfmatched_all_newfits_noNe-eps-converted-to.pdf}\n\t\\includegraphics[width=0.245\\textwidth]{figures\/lvmax_both_psfmatched_all_newfits_noNe-eps-converted-to.pdf}\n\t\\includegraphics[width=0.245\\textwidth]{figures\/lsout_psfmatched_all_newfits_noNe-eps-converted-to.pdf}\n\t\\includegraphics[width=0.245\\textwidth]{figures\/lsout_both_psfmatched_all_newfits_noNe-eps-converted-to.pdf}\n\t\\caption{Comparison of LOS kinematic quantities from PSF-matched fixed-slit extractions, as defined in Section~\\ref{s:measure}. From left to right: maximum observed absolute rotation velocity, $v_{\\rm max}(r_{\\rm vmax})$; (average, interpolated) rotation velocity at the outermost radius covered by both KMOS$^{\\rm 3D}$\\, and LEGA-C data, $v_{\\rm rmax,both}(r_{\\rm max,both})$; weighted mean outer velocity dispersion, $\\sigma_{\\rm out}$; (average, interpolated) velocity dispersion at the outermost radius covered by both KMOS$^{\\rm 3D}$\\, and LEGA-C data, $\\sigma_{\\rm rmax,both}(r_{\\rm max,both})$. \n\tGolden filled stars compare KMOS$^{\\rm 3D}$\\, H$\\alpha$ measurements with LEGA-C stellar measurements, and blue filled circles with LEGA-C gas measurements. Green open circles indicate the presence of prominent Balmer lines in the LEGA-C spectra. Larger symbols indicate galaxies for which a dynamical modelling of both the KMOS$^{\\rm 3D}$\\, and LEGA-C data is possible (typically higher-quality, more extended data, excluding mergers). \n    The shaded region around the 1:1 line indicates a constant interval of $\\pm0.1$~dex in all plots, to highlight differences in scatter between the comparisons.\n    Due to duplicate observations in LEGA-C, galaxies can appear multiple times in each panel.\n\tOn average, velocities are larger and velocity dispersions lower for H$\\alpha$ compared to stars.\n    }\n\t\\label{f:measures1}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.245\\textwidth]{figures\/lvmaxsout_psfmatched_all_newfits_noNe-eps-converted-to.pdf}\n\t\\includegraphics[width=0.245\\textwidth]{figures\/lvrms_psfmatched_all_newfits_noNe-eps-converted-to.pdf}\n\t\\includegraphics[width=0.245\\textwidth]{figures\/lvcirc_psfmatched_all_newfits_noNe-eps-converted-to.pdf}\n\t\\includegraphics[width=0.245\\textwidth]{figures\/lS05_psfmatched_all_newfits_noNe-eps-converted-to.pdf}\n\t\\caption{Comparison of LOS kinematic quantities from PSF-matched fixed-slit extractions, defined as in Section~\\ref{s:measure}. From left to right: rotational support, $v_{\\rm max}\/\\sigma_{\\rm out}$; root mean square velocity, $v_{\\rm rms}$; approximation of the circular velocity, $v_{c,\\rm max}(r_{\\rm vmax})$; approximation of the total integrated velocity dispersion, $S_{0.5,\\rm max}$.\n\tSymbols are as in Figure~\\ref{f:measures1}. \n    The shaded region around the 1:1 line indicates a constant interval of $\\pm0.1$~dex in all plots, to highlight differences in scatter between the comparisons. \n\tOn average, the rotational support measured from H$\\alpha$ emission is larger than in stellar and gas measurements from the LEGA-C spectra.\n    }\n\t\\label{f:measures2}\n\\end{figure*}\n\n\nWe now compare the stellar (from LEGA-C) and ionised gas (from KMOS$^{\\rm 3D}$) LOS kinematics from matched observing setups (Section~\\ref{s:kinex}), that is from fixed slits after matching the individual PSFs for each pair of observations. We also compare different measurements of the ionised gas observed kinematics using the H$\\alpha$ line from the KMOS$^{\\rm 3D}$\\, observations, and the emission line fits to the full LEGA-C spectrum, typically dominated by [OII] emission.\n\nIn Figure~\\ref{f:obsprof} we show an example of 2D and 1D kinematic extractions from KMOS$^{\\rm 3D}$\\, and LEGA-C data along both a N-S and an E-W (pseudo-)slit of width $1\\arcsec$ for one galaxy. For this galaxy, both the ionised gas and stellar velocities from LEGA-C and the H$\\alpha$ velocities from KMOS$^{\\rm 3D}$ qualitatively agree, with stellar velocities reaching somewhat lower amplitudes. The velocity dispersion profiles are often dissimilar, with asymmetric profiles for the ionised gas and stars from LEGA-C compared to the KMOS$^{\\rm 3D}$\\, profile that is centrally peaked as expected for rotating disc kinematics uncorrected for beam-smearing.\n\nIt has been shown that [OII] emission does not only trace star formation, but can be related to AGN activity and low-ionisation nuclear emission line regions \\citep[LINERS; e.g.][]{Yan06, Yan18, Lemaux10, DaviesRL14, Maseda21}. The differences in shape and intensity between the H$\\alpha$ and [OII] emission we see in the 2D position-velocity diagrams and the extracted 1D profiles further suggests that not all [OII] emission is originating from the co-rotating ISM. However, also line blending of the doublet emission complicates the extraction of kinematic information (due to degeneracies between the line amplitudes, widths, and centroids, when no other prominent emission lines are present, as is typical for LEGA-C galaxies at $z>0.9$). \n\n\n\\subsection{Velocities}\\label{s:comp_v}\nIn the left panels of Figure~\\ref{f:measures1} we compare maximum velocities ($v_{\\rm max}$) and velocities at the outermost common radius ($v_{\\rm rmax,both}$) for the KMOS$^{\\rm 3D}$\\, and LEGA-C samples, measured from the `observed' kinematics. Corresponding numbers for galaxies for which also a dynamical modelling is possible are listed in Table~\\ref{t:statobs}.\nOn average, the velocities measured from the KMOS$^{\\rm 3D}$\\, H$\\alpha$ data are larger compared to LEGA-C stars (golden stars in Figure~\\ref{f:measures1}) by $\\sim40$ per cent. This is similar when comparing to LEGA-C gas (blue circles), but here we also note that the gas velocity measurements agree well for the three non-interacting galaxies where the LEGA-C spectrum includes strong Balmer lines (large symbols with green circles).\nBased on a two-sample Kolmogorov-Smirnov statistic, only the maximum velocity of stars is different from the H$\\alpha$ $v_{\\rm max}$ by more than $1\\sigma$.\nIn general, lower amplitudes in rotation velocity for stars compared to gas are expected based on $z=0$ data (see Section~\\ref{s:intro}).\n\n\\subsection{Velocity dispersions}\\label{s:comp_s}\nIn the right panels of Figure~\\ref{f:measures1} we show the corresponding plots for the outer weighted mean observed velocity dispersion ($\\sigma_{\\rm out}$), and the (mean) velocity dispersion at the outermost radius common to both data sets ($\\sigma_{\\rm rmax,both}$). On average, the H$\\alpha$ dispersion measurements are lower than the LEGA-C measurements. \nIn particular the stellar velocity dispersions are larger by about a factor of two relative to H$\\alpha$, and show a significantly different distribution towards higher values by more than $2\\sigma$ based on a two-sample Kolmogorov-Smirnov statistic (see also Table~\\ref{t:statobs}). \nIn addition, there is some indication that, when measured at the same radius, the difference between stellar and H$\\alpha$ velocity dispersions is higher for systems with higher stellar velocity dispersion.\nIn general, higher disc velocity dispersions for stars compared to gas are also expected based on $z=0$ data, as discussed in Section~\\ref{s:intro}.\n\n\n\\begin{table}\n    \\centering\n    \\caption{Mean difference of the logarithm, log(KMOS$^{\\rm 3D}$\/LEGA-C), and corresponding standard deviation for various kinematic quantities, comparing KMOS$^{\\rm 3D}$\\, H$\\alpha$ to LEGA-C stars and gas, respectively, averaged over galaxies for which a dynamical modelling is possible. The quantities are defined in Section~\\ref{s:measure} and individual measurements are shown in Figures~\\ref{f:measures1} and \\ref{f:measures2} (large symbols).}\n    \\label{t:statobs}\n\\begin{tabular}{lcccc}\n    & \\multicolumn{2}{c}{stars} & \\multicolumn{2}{c}{gas} \\\\\n    Quantity [dex] & mean & std. dev. & mean  & std. dev. \\\\\n\\hline\n    $v_{\\rm max}$            & 0.13 & 0.24 & 0.13 & 0.12 \\\\\n    $v_{\\rm rmax,both}$      & 0.16 & 0.40 & 0.21 & 0.46 \\\\\n    $\\sigma_{\\rm out}$       & -0.30 & 0.20 & -0.15 & 0.14 \\\\\n    $\\sigma_{\\rm rmax,both}$ & -0.26 & 0.27 & -0.09 & 0.17 \\\\\n    $v_{\\rm max}\/\\sigma_{\\rm out}$  & 0.42 & 0.25 & 0.28 & 0.20 \\\\\n    $v_{\\rm rms}$            & -0.05 & 0.20 & 0.02 & 0.10 \\\\\n    $v_{\\rm c,max}$          & -0.08 & 0.19 & 0.02 & 0.11 \\\\\n    $S_{0.5, \\rm max}$       & -0.10 & 0.18 & -0.01 & 0.10 \\\\\n \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Rotational support}\\label{s:comp_vs}\nIn the left panel of Figure~\\ref{f:measures2} we plot the ratio of maximum LOS velocity and outer LOS velocity dispersion ($v_{\\rm max}\/\\sigma_{\\rm out}$), as defined in Section~\\ref{s:measure}. We stress again that these values are not derived from modelling, but have been measure from fixed slits after PSF matching. I.e., the LSF is accounted for but not any inclination effects or PSF effects, although the latter should be effectively the same for our KMOS$^{\\rm 3D}$\\, and LEGA-C extractions, as discussed in Section~\\ref{s:kmoskin}. \nOverall the KMOS$^{\\rm 3D}$\\, measurements suggest a stronger rotational support in the star-forming ionised gas phase, possibly indicating that the H$\\alpha$ line emission is originating from a more disc-like structure, or that it is less affected by non-circular motions (e.g.\\ compared to [OII]).\nThe difference between the H$\\alpha$ measurements and the stellar measurements is statistically significant by more than $2\\sigma$, and the H$\\alpha$ measurements and the LEGA-C gas measurements by more than $1\\sigma$.\n\nIn addition, we compare several combinations of observed velocity and observed velocity dispersion in the right-hand panels of Figure~\\ref{f:measures2}, as described in Section~\\ref{s:measure}. The combination of velocity and velocity dispersion into a common probe of the galactic potential results in more similar estimates on average between the KMOS$^{\\rm 3D}$\\, and LEGA-C data: both the average offsets and the scatter are reduced (see also Table~\\ref{t:statobs}).\nWe find the best average agreement between stellar and H$\\alpha$ data for $v_{\\rm rms}$ (second panel in Figure~\\ref{f:measures2}). \nWe note that \\cite{Bezanson18b} find comparable {\\it integrated} velocity dispersion for ionised gas and stars within the LEGA-C survey, in qualitative agreement with our result. \n\n\n\\subsection{Implications for compilations of ionised gas kinematics}\\label{s:compilations}\nThe differences in ionised gas velocities and velocity dispersions between the KMOS$^{\\rm 3D}$\\, and LEGA-C extractions, where the latter are mostly dominated by [OII] emission, serve as a caution in the combination of samples with different emission lines. The somewhat lower velocities and higher velocity dispersions measured from [OII] compared to H$\\alpha$ or other Balmer lines might motivate a revision of literature compilations for the study of galaxy gas kinematics evolution, such as the Tully-Fisher relation \\citep{Tully77}, or gas velocity dispersion. \n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.33\\textwidth]{figures\/COS3_05062_mcum_sph_JAMfine-eps-converted-to.pdf}\n\t\\includegraphics[width=0.33\\textwidth]{figures\/COS4_06487_mcum_sph_JAMfine-eps-converted-to.pdf}\n\t\\includegraphics[width=0.33\\textwidth]{figures\/COS4_03493_mcum_sph_JAMfine-eps-converted-to.pdf}\n\t\\includegraphics[width=0.33\\textwidth]{figures\/COS3_05444_mcum_sph_JAMfine-eps-converted-to.pdf}\n\t\\includegraphics[width=0.33\\textwidth]{figures\/COS3_18434_mcum_sph_JAMfine-eps-converted-to.pdf}\n\t\\includegraphics[width=0.33\\textwidth]{figures\/COS3_26546_mcum_sph_JAMfine-eps-converted-to.pdf}\n\t\\caption{Comparison of cumulative mass profiles from H$\\alpha$ dynamical mass models and JAM. The JAM estimates by \\citet{vHoudt21} are shown as golden diamonds every kiloparsec out to 10~kpc, and at $R_{e,\\rm F814W}$, and the KMOS$^{\\rm 3D}$ best-fit enclosed total mass is shown as a blue line, with lighter shading indicating one and two standard deviations as constrained by the full MCMC chains (60000 realisations). The vertical dashed grey line marks $R_{e,\\rm F814W}$, and the golden and blue arrows indicate the projected extent of the stellar and H$\\alpha$ kinematic data, respectively. Overall, the agreement between the JAM model estimates and the KMOS$^{\\rm 3D}$ model estimates is very good, demonstrating that the total mass distribution can be robustly inferred from different modelling techniques and data sets, as long as data quality allows. For galaxy COS4\\_03493-M4\\_121150 (top right), the JAM model overestimates the dynamical mass (see Section~\\ref{s:mcum} for details.)}\n\t\\label{f:mcum}\n\\end{figure*} \n\nIn fact, the discrepancy in zero-point offset of the stellar mass Tully-Fisher relation at $z\\sim1$ among \\cite{Miller11, Miller12} and \\cite{Tiley16, Uebler17} could be partly due to the use of different tracers.\\footnote{\nIn addition, other important reasons for such differences have been identified \\citep[see e.g.\\ Appendix~A by][]{Uebler17}: inclusion of galaxies where the peak velocity is not reached by the data, lack of a correction for beam-smearing, or sample selection effects based on e.g.\\, $v_{\\rm rot}\/\\sigma_0$ cuts.}\nThe measurements by \\cite{Miller11, Miller12} include or are based on [OII] emission, while \\cite{Tiley16, Uebler17} target H$\\alpha$. A zero-point difference of about $-0.3$~dex is found between those studies, with a corresponding offset in velocity of $\\sim0.1$~dex. Our results suggest that for a velocity of 50~km\/s (200~km\/s), a systematic velocity offset of up to 0.2~dex (0.05~dex) could be solely due to the use of different gas tracers, potentially resolving the disagreement between those studies.\n\nConsidering the recent literature compilation by \\cite{Uebler19} of the redshift evolution of intrinsic ionised gas velocity dispersion, our findings indicate that the difference in average disc velocity dispersion at fixed redshift found between some surveys could be due to the use of different emission line tracers. Any systematic difference between ionised gas velocity dispersion measured from H$\\alpha$ {\\it vs.\\,} [OII] or also [OIII], which is known to typically have a higher excitation contribution from narrow-line AGN than H$\\alpha$ \\citep[e.g.][]{Kauffmann03, DaviesRL14}, could affect evolutionary trends.\\footnote{\nWe note that studies of nearby giant HII regions typically find lower velocity dispersions for [OIII] compared to H$\\alpha$, and this has been linked to [OIII] originating from denser regions more deeply embedded in HII regions \\citep[e.g.][]{Hippelein86}. \\cite{Law22} find a correlation of $\\sigma_{\\rm [OIII]}\/\\sigma_{\\rm H\\alpha}$ with SFR for galaxies in the MaNGA survey \\citep{Bundy15}. In their work, velocity dispersions measured from [OIII] are higher relative to H$\\alpha$ for SFR$\\gtrsim1~M_{\\odot}\/yr$. Massive main-sequence galaxies at $z\\sim1-2$ have typical SFRs of $10-100~M_{\\odot}\/yr$ \\citep[e.g.][]{Whitaker14}. \\cite{Law22} also find [OII] velocity dispersions to be systematically higher compared to measurements based on H$\\alpha$.}\nIndeed, several surveys including [OII] or [OIII] emission lines have average intrinsic velocity dispersion values above the relation derived by \\cite{Uebler19}. However for the more challenging measurement of the velocity dispersion the situation is further complicated by different methodologies in accounting for beam-smearing in those studies. \nIndeed, several surveys including [OII] or [OIII] emission lines have average intrinsic velocity dispersion values above the relation derived by \\cite{Uebler19}. However for the more challenging measurement of the velocity dispersion the situation is further complicated by different methodologies in accounting for beam-smearing in those studies.\\\\\n\nIn general, considering the full 1D profiles, we find that (i) stellar velocities reach lower amplitudes and average disc stellar velocity dispersions are higher compared to the ionised gas kinematics, reminiscent of local Universe findings; (ii) stellar velocity dispersions and ionised gas velocity dispersion dominated by [OII]$\\lambda\\lambda3726,3729$ emission are often more asymmetric compared to H$\\alpha$; (iii) the correspondence between the KMOS$^{\\rm 3D}$\\, H$\\alpha$ data and the LEGA-C emission line data is better for LEGA-C spectra including Balmer lines (green circles in Figures~\\ref{f:measures1} and \\ref{f:measures2}).\n\nOverall, more high-quality data would be beneficial to characterise the differences in ionised gas kinematics provided by different tracers for the same galaxies. \nUpcoming data from {\\it James Webb Space Telescope} ({\\it JWST}) enabling H$\\alpha$ studies up to $z\\sim7$ and extensions of $z<3$ ground-based kinematic studies of multiple emission lines with IFUs such as ERIS, MUSE, and KMOS will provide important references.\n\n\n\n\\section{Dynamical Masses}\\label{s:mdyn}\n\nWe now proceed with a comparison of dynamical mass measurements from the KMOS$^{\\rm 3D}$\\, H$\\alpha$ data and the LEGA-C stellar kinematic data. \nIn contrast to the previous section, where we have matched the observing conditions between KMOS$^{\\rm 3D}$\\, and LEGA-C data, we now use the native spatial and spectral resolution of the data to build the best possible dynamical models based on H$\\alpha$ and stars. \n\n\\subsection{Cumulative total mass profiles based on \\texorpdfstring{H$\\alpha$}{Halpha} and stars} \\label{s:mcum}\n\nWe begin with a comparison of cumulative mass profiles from JAM models and our best-fit H$\\alpha$ dynamical mass models (see Sections~\\ref{s:mdynk3d} and \\ref{s:mdynlgc}). Figure~\\ref{f:mcum} shows the cumulative mass profiles of the six galaxies in our sample for which the data quality is high enough in both stars and H$\\alpha$ to construct spatially resolved mass models. The JAM measurements (golden diamonds with error bars indicating one standard deviation) are shown every kiloparsec out to 10~kpc, and at $R_{e,\\rm F814W}$, and the KMOS$^{\\rm 3D}$\\, models are shown as blue lines, with lighter shading indicating one and two standard deviations, respectively.\n\nIt is remarkable to see that for most cases, despite the different techniques, model inputs, and tracers, the constraints on the enclosed mass and its shape are in agreement. This comparison shows that both stars and H$\\alpha$ at $z\\sim1$ constrain the same total mass distribution over a large range of radii, when high-quality data suitable for dynamical modelling are available. \nWe note that the uncertainties for the H$\\alpha$ and JAM models are not directly comparable, since the former have fewer free parameters. However, also the extent out to which the model is constrained by the data is larger for H$\\alpha$ for all cases discussed here (golden and blue arrows in Figure~\\ref{f:mcum} for LEGA-C and KMOS$^{\\rm 3D}$, respectively), further reducing uncertainty in the model.\n\nFor four of the six galaxies (left panels), the dynamical models agree within their uncertainties from 1~kpc to (at least) 10~kpc, covering a range of $1-2.2~R_{e,\\rm F814W}$ ($1.8-2.6~R_{e,\\rm F160W}$). \nFor one galaxy, COS4\\_25353-M1\\_139825 (bottom right), the models agree within their uncertainties from 4~kpc to (at least) 10~kpc, while in the central 3~kpc the stellar model yields higher dynamical masses relative to the H$\\alpha$ model. This galaxy is seen almost face on, with a difference between the H$\\alpha$ kinematic major axis and the F814W position angle of $26.5^{\\circ}$. This is also one of the objects with strong [NeV] emission in the central region. We speculate that emission from the AGN could bias the light-weighted estimates of the central density for both models.\n\nThere is only one galaxy for which the JAM estimates and the KMOS$^{\\rm 3D}$\\, estimates are significantly different over a large range in radius, COS4\\_03493-M4\\_121150 (top right). At $R_{e,\\rm F814W}$, the JAM measurement is higher by $\\Delta M_{\\rm dyn}=0.35$~dex compared to the H$\\alpha$ model. For this highly inclined galaxy ($i\\approx68-84^\\circ$), the kinematic major axis and the F814W and F160W position angles all align within $1^{\\circ}$. However, the F814W structural parameters indicate a high S\u00e9rsic index ($n_{S,\\rm F814W}=5.1$) and a large disc ($R_{e,\\rm F814W}=8.2$~kpc). Yet, adopting structural parameters from F814W imaging for the H$\\alpha$ model has negligible effect on the dynamical mass constraints. Instead, it is likely that JAM fits a high $M\/L$ to the bright and higher-$S\/N$ bulge component, leading to an overestimate of the mass in the extended disc (see also discussion in Section~\\ref{s:multiple}). \n\n\n\\subsection{Comparison to measurements at \\texorpdfstring{$R_{e,\\rm F814W}$}{Re,F814W} based on integrated stellar velocity dispersion}\\label{s:compmdyn}\n\nThe agreement between H$\\alpha$ dynamical mass models and measurements based on integrated stellar velocity dispersion is not as good. As described in Section~\\ref{s:mdynlgc}, \\cite{vdWel21, vdWel22}, have utilised the JAM results for LEGA-C to re-calibrate virial mass measurements based on the integrated stellar velocity dispersion, which is available for a larger number of galaxies. In Figure~\\ref{f:mdyn} we compare dynamical masses at $R_{e,\\rm 814W}$ from our H$\\alpha$ models to LEGA-C estimates based on $\\sigma_{\\star,\\rm vir}$ (purple symbols) for ten galaxies. We choose this radius as it allows for a straight-forward comparison of the KMOS$^{\\rm 3D}$\\, models to the LEGA-C $M_{\\rm vir}$ values, where $M_{\\rm dyn,LEGA-C}(<R_{e,\\rm F814W})=0.5\\cdot M_{\\rm vir}$.\\footnote{\nWe remind the reader that for our KMOS$^{\\rm 3D}$\\, models, $M_{\\rm dyn}(<R_{e,\\rm F814W})$, {\\it does not} correspond to the dynamical mass enclosed within the adopted $R_e$ of the best-fit model, as here the fiducial structural parameters are based on the F160W measurements ($R_{e,\\rm F814W}$ is typically larger than $R_{e,\\rm F160W}$ due to its higher sensitivity for substructure, see discussion in Appendix~\\ref{a:structure}).} \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figures\/comp_mdyn_test-eps-converted-to.pdf}\n\t\\caption{Comparison of dynamical mass estimates within $R_{e,\\rm F814W}$ using mass models based on H$\\alpha$ for KMOS$^{\\rm 3D}$\\, and integrated stellar velocity dispersion \\citep[purple;][]{vdWel21} and JAM \\citep[teal;][]{vHoudt21} for LEGA-C.\n\tThe direction of triangles indicates the slit orientation of the LEGA-C data from which the estimates were derived. Uncertainties for LEGA-C measurements from integrated velocity dispersion are $1\\sigma$, and for JAM models the $16^{\\rm th}$ and $84^{\\rm th}$ percentiles. For KMOS$^{\\rm 3D}$\\, measurements, uncertainties are the $16^{\\rm th}$ and $84^{\\rm th}$ percentiles, with lower ceiling uncertainties of 0.1~dex.\n    Dynamical mass estimates from integrated stellar velocity dispersion are as accurate as the JAM models, with a standard deviation of 0.24~dex when comparing to the KMOS$^{\\rm 3D}$\\, H$\\alpha$ measurements.}\n\t\\label{f:mdyn}\n\\end{figure}\n\nSeveral objects in Figure~\\ref{f:mdyn} have multiple estimates based on the LEGA-C $\\sigma_{\\star,\\rm vir}$ data through duplicate observations, and we indicate the slit orientation of the LEGA-C data through down-wards (N-S) and right-wards (E-W) triangles. We also show JAM measurements within $R_{e,\\rm 814W}$ (teal symbols) for the six objects discussed in Section~\\ref{s:mcum}.\n\nFor the majority of measurement pairs, the dynamical mass estimates based on LEGA-C integrated stellar velocity dispersions are larger than the KMOS$^{\\rm 3D}$ estimates, with an average offset of $\\Delta M_{\\rm dyn}=0.12$~dex. However, in our sample we also find differences in the dynamical mass estimates from the two surveys for individual galaxies of up to 0.55~dex -- amounting to more than a factor of three.\nThe average offset of $\\Delta M_{\\rm dyn}=0.12$~dex is comparable to uncertainties typically attributed to $z\\sim1-2$ dynamical mass measurements from ionised gas \\citep[e.g.][]{WuytsS16}. \nIt is somewhat smaller than the average offset between stellar- and emission line-based dynamical mass estimates within the effective radius found within the LEGA-C survey alone. \\cite{Straatman22} find an average offset of 0.15~dex when comparing 149 LEGA-C galaxies with log(sSFR\/yr)~$>-11$ and $n_S\\leq2.5$. \n\nDue to the small sample size and the duplicate observations and two methods shown for LEGA-C, we concentrate on standard deviations between the various measurement sets to further quantify our results, as listed in Table~\\ref{t:stddev}. Considering all observational pairs for which LEGA-C estimates based on integrated velocity dispersion exist, we find a standard deviation of 0.24~dex between the LEGA-C and KMOS$^{\\rm 3D}$\\, dynamical mass estimates. The agreement between KMOS$^{\\rm 3D}$\\, and LEGA-C JAM is better, with a standard deviation of 0.13~dex (however for a sample of six). \nWe note that the reduction in scatter from 0.24 to 0.13 dex is marginally significant ($1.2\\sigma$). \nIf we further exclude the JAM measurement of galaxy COS4\\_03493-M4\\_121150 (see Section~\\ref{s:mcum}), we find a standard deviation of 0.07~dex when comparing to the KMOS$^{\\rm 3D}$\\, estimates. In this case, the reduction in scatter relative to the comparison of KMOS$^{\\rm 3D}$\\, models and LEGA-C models based on integrated velocity dispersion has a significance of $2.3\\sigma$.\n\nOverall, the discrepancy between KMOS$^{\\rm 3D}$\\, and LEGA-C $M_{\\rm dyn}$ estimates based on integrated stellar velocity dispersion in our sample is larger than the independent estimate of uncertainties from LEGA-C duplicate observations ($\\sigma_{\\rm Mvir,dupl}=0.14$), but comparable to the independent estimate of uncertainties from different methods within LEGA-C to determine dynamical mass ($\\sigma_{\\rm Mvir~vs~JAM}=0.24$). \nFor the full LEGA-C survey, the scatter among $M_{\\rm vir}$ and JAM measurements for SFGs is lower than our value, with $\\sigma_{\\rm Mvir~vs~JAM}=0.16$ \\citep{vdWel22}. This suggests that our sample includes some outliers in the $M_{\\rm vir}$-to-$M_{\\rm JAM}$ calibration by \\cite{vdWel22}.\n\n\n\\begin{table}\n\t\\centering\n\t\\caption{Standard deviation of dynamical mass discrepancy $\\Delta M_{\\rm dyn}$ for various subsets of KMOS$^{\\rm 3D}$\\, and LEGA-C data.}\n\t\\label{t:stddev}\n\t\\begin{tabular}{lcl} \n\t\t\\hline\n\t\tcomparison & std. dev.  & sample size\\\\\n\t\t& [dex] & \\\\\n\t\t\\hlin\n\t\tLEGA-C$_{\\rm JAM}$, KMOS$^{\\rm 3D}$\\, & 0.13 & 6 \\\\\n\t\tLEGA-C$_{\\rm Mvir}$, KMOS$^{\\rm 3D}$\\, & 0.24 & 14 (incl. duplicates)\\\\\n\t\tLEGA-C$_{\\rm Mvir}$, duplicates & 0.14 & 4 \\\\\n\t\tLEGA-C$_{\\rm Mvir}$, LEGA-C$_{\\rm JAM}$ & 0.24 & 6 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\\subsection{Notes on LEGA-C duplicate observations and \\texorpdfstring{$M_{\\rm dyn}$}{Mdyn} estimates from multiple techniques}\\label{s:multiple}\n\nFigure~\\ref{f:mdyn} shows for several objects multiple LEGA-C measurements of dynamical mass. \nFour galaxies in our dynamical mass sample have been observed with two different masks in LEGA-C, three of which with a different slit orientation. \nThe $M_{\\rm vir}$ estimates of these duplicate observations agree with each other within the uncertainties for all but one case. In this latter case, the duplicate observations have comparable $S\/N$, but the observation which is also in agreement with the KMOS$^{\\rm 3D}$\\, measurement is better aligned with the kinematic major axis. For all other cases, the duplicate observation with higher $S\/N$ is in better agreement with the KMOS$^{\\rm 3D}$\\, measurement. \nThis conforms to the expectation that in the absence of asymmetric motions, $S\/N$ is more important than alignment for integrated measurements, which are centrally weighted.\nThis is encouraging not only for existing ground-based surveys, but also for upcoming data from {\\it JWST} NIRSpec Multi Shutter Array observations.\n\nFor most cases, the JAM measurements are in better agreement with the KMOS$^{\\rm 3D}$\\, modelling than the $M_{\\rm dyn}$ estimates based on $\\sigma_{\\star,\\rm vir}$. As discussed in Section~\\ref{s:mcum}, for galaxy \nCOS4\\_03493-M4\\_121150 JAM predicts a too large dynamical mass within $R_{e,\\rm F814W}$ compared to the KMOS$^{\\rm 3D}$\\, model ($\\Delta M_{\\rm dyn}=0.35$~dex), but also compared to the LEGA-C measurement from $\\sigma_{\\star,\\rm vir}$ ($\\Delta M_{\\rm dyn}=0.26$~dex). \nFor this galaxy, spatially resolved modelling of the ionised gas from the LEGA-C survey exists as well \\citep{Straatman22}. Prominent emission lines in this LEGA-C slit spectrum are H$\\beta$ and [OIII], and the correspondence of the 2D KMOS$^{\\rm 3D}$\\, H$\\alpha$ pseudo-slit data and the LEGA-C H$\\beta$ data is good. The $M_{\\rm dyn}(<R_{e,\\rm F814W})$ estimate by \\cite{Straatman22} agrees with the KMOS$^{\\rm 3D}$\\, estimate, further supporting the interpretation that the JAM model overpredicts the dynamical mass in this case.\n\nA second object in our dynamical mass sample has an ionised gas-based dynamical mass estimate from LEGA-C data by \\cite{Straatman22}, COS4\\_04943-M3\\_122667. This is the only object in our sample with a significantly higher dynamical mass measurement from KMOS$^{\\rm 3D}$\\, compared to the $M_{\\rm dyn}$ measurement from the LEGA-C integrated stellar velocity dispersion ($\\Delta M_{\\rm dyn}=0.43$~dex). This time, the estimate from the LEGA-C ionised gas data is comparable to the estimate from integrated stellar velocity dispersion, with a difference of only $\\Delta M_{\\rm dyn}=0.04$~dex. Closer inspection of the LEGA-C data and the KMOS$^{\\rm 3D}$\\, pseudo-slit extractions reveals that the kinematic centre is offset by about $0.4\\arcsec$ ($\\sim2$ pixels) from the central pixel in the LEGA-C 2D data (see Figure~\\ref{f:COS3_05062}). While this cannot typically be tested, the shape of the ionised gas velocity and velocity dispersion profiles, here constrained through H$\\beta$ emission, can be aligned with the KMOS$^{\\rm 3D}$\\, H$\\alpha$ kinematic profiles. The resulting spatial shift of the LEGA-C profiles shows that the velocity gradients in the LEGA-C data are underestimated in both ionised gas and stars, providing a plausible explanation for their lower $M_{\\rm dyn}$ estimates. The JAM measurement agrees within the uncertainties with the KMOS$^{\\rm 3D}$\\, value, likely due to its flexibility of fitting a different centre position (see discussion in Appendix~\\ref{a:misalignment}). \n\n\n\\subsection{Correlations with dynamical mass discrepancy}\\label{s:corr}\n\nWe explore trends in dynamical mass discrepancy with structural parameters, kinematic, and physical properties of our galaxies. Due to the sample size and the comparison of multiple observations or estimates for the same objects, we cannot expect significant correlations. However, we can note a few informative trends that we discuss in this section. \nFor this investigation, we exclude the integrated dispersion measurement for galaxy COS4\\_04943-M3\\_122667, and the JAM measurement for galaxy COS4\\_03493-M4\\_121150 (see Section~\\ref{s:multiple}).\n\nWe list Spearman rank correlation coefficients and their significance between the dynamical mass discrepancy and various quantities in Table~\\ref{t:corr}.\n\n\\begin{table}\n    \\centering\n    \\caption{Spearman rank correlation coefficients $\\rho_S$ and their significance $\\sigma_\\rho$ between dynamical mass mismatch $\\Delta M_{\\rm dyn}$ and various structural, kinematic, and global physical properties of the galaxies.}\n    \\label{t:corr}\n\\begin{tabular}{lcccc}\n \\hline\n & \\multicolumn{2}{c}{F814W} & \\multicolumn{2}{c}{F160W} \\\\\n Quantity & $\\rho_S$ & $\\sigma_\\rho$ & $\\rho_S$ & $\\sigma_\\rho$ \\\\\n \\hline\n    $R_e$ [kpc] & -0.35 & 1.4 & 0.26 & 1.1 \\\\\n    $q$       & 0.54 & 2.2 & 0.81 & 3.3 \\\\\n    $n_S$       & -0.07 & 0.3 & 0.03 & 0.1 \\\\\n    PA [deg]    & -0.13 & 0.5 & 0.19 & 0.8 \\\\\n \\hline\n  & \\multicolumn{2}{c}{$\\rho_S$} & \\multicolumn{2}{c}{$\\sigma_\\rho$} \\\\\n\\hline\n    $\\Psi_{\\rm F814W,kin}$ & \\multicolumn{2}{c}{0.42} & \\multicolumn{2}{c}{1.7} \\\\\n    $\\Psi_{\\rm kin,slit}$ & \\multicolumn{2}{c}{0.24} & \\multicolumn{2}{c}{1.0} \\\\\n    $\\Psi_{\\rm F814W,slit} - \\Psi_{\\rm kin,slit}$ & \\multicolumn{2}{c}{0.22} & \\multicolumn{2}{c}{0.9} \\\\\n\\hline\n    $v_{\\rm circ,H\\alpha}(r=R_{e,{\\rm F814W}})$ [km\/s] & \\multicolumn{2}{c}{-0.78} & \\multicolumn{2}{c}{3.2} \\\\\n    $v_{\\rm rot,H\\alpha}(r=R_{e,{\\rm F814W}})\/\\sigma_{0,\\rm H\\alpha}$ & \\multicolumn{2}{c}{-0.51} & \\multicolumn{2}{c}{2.1} \\\\\n    $v_{\\rm rot,H\\alpha}(r=R_{e,{\\rm F814W}})$ [km\/s] & \\multicolumn{2}{c}{-0.74} & \\multicolumn{2}{c}{3.0} \\\\\n    $\\sigma_{0,\\rm H\\alpha}$ [km\/s] & \\multicolumn{2}{c}{0.30} & \\multicolumn{2}{c}{1.2} \\\\\n \\hline\n    SFR [$M_\\odot$\/yr] & \\multicolumn{2}{c}{0.20} & \\multicolumn{2}{c}{0.8} \\\\    \n    sSFR [1\/Gyr] & \\multicolumn{2}{c}{0.24} & \\multicolumn{2}{c}{1.0} \\\\\n    $\\Sigma_{\\rm SFR}$ [$M_\\odot$\/yr\/kpc$^2$] & \\multicolumn{2}{c}{0.32} & \\multicolumn{2}{c}{1.3} \\\\\n    $\\log(M_{\\star}\/M_\\odot)$ & \\multicolumn{2}{c}{-0.08} & \\multicolumn{2}{c}{0.3} \\\\\n    $\\log(M_{\\rm bar}\/M_\\odot)$ & \\multicolumn{2}{c}{-0.09} & \\multicolumn{2}{c}{0.4} \\\\\n \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{Structural parameters}\\label{s:corr_struct}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.31\\textwidth]{figures\/comp_mdyn_re_lgc_test-eps-converted-to.pdf}\n \\hspace{3mm}\n\t\\includegraphics[width=0.31\\textwidth]{figures\/comp_mdyn_ba_lgc_test-eps-converted-to.pdf}\n \\hspace{3mm}\n\t\\includegraphics[width=0.31\\textwidth]{figures\/comp_mdyn_pa_test-eps-converted-to.pdf}\n\t\\caption{Difference in dynamical mass measurements from LEGA-C stars and KMOS$^{\\rm 3D}$\\, H$\\alpha$, $\\Delta M_{\\rm dyn}=\\log(M_{\\rm dyn,LGC}\/M_{\\rm dyn,K3D})$, as a function of effective radius $R_{e,\\rm F814W}$ (left), projected axis ratio $q_{\\rm F814W}$ (middle), and kinematic misalignment $\\Psi_{\\rm F814W,kin}$ (right). The symbols are the same as in Figure~\\ref{f:mdyn}, and symbol size scales with $q_{\\rm F814W}$ (left, right) and $R_{e,\\rm F814W}$ (middle), as indicated in the panels. \n    We find larger mismatches between dynamical mass estimates for galaxies with stronger kinematic misalignment. These are systems that are seen more face-on.}\n\t\\label{f:dmdyn1}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.31\\textwidth]{figures\/comp_mdyn_vcirc_test-eps-converted-to.pdf}\n \\hspace{3mm}\n\t\\includegraphics[width=0.31\\textwidth]{figures\/comp_mdyn_vrotsig0_test-eps-converted-to.pdf}\n\t \\hspace{3mm}\n    \\includegraphics[width=0.31\\textwidth]{figures\/comp_mdyn_sig0_test-eps-converted-to.pdf}\n    \\caption{Difference in dynamical mass measurements from LEGA-C stars and KMOS$^{\\rm 3D}$\\, H$\\alpha$, $\\Delta M_{\\rm dyn}=\\log(M_{\\rm dyn,LGC}\/M_{\\rm dyn,K3D})$, as a function of H$\\alpha$ circular velocity $v_{\\rm circ}(R_{e,\\rm F814W})$ (left), rotational support $v_{\\rm rot}(R_{e,\\rm F814W})\/\\sigma_0$ (middle), and intrinsic velocity dispersion $\\sigma_0$ (right). The symbols are the same as in Figure~\\ref{f:mdyn}. We find larger mismatches between dynamical mass estimates primarily for galaxies with lower circular velocities and lower rotational support, as constrained from our best-fit models to the H$\\alpha$ data.}\n\t\\label{f:dmdyn3} \n\\end{figure*}\n\nDynamical mass discrepancies are larger for smaller galaxies that are seen more face-on, as illustrated in the left panels of Figure~\\ref{f:dmdyn1}. The measured effective radius $R_e$ and the projected axis ratio $q$ are themselves correlated: in the presence of surface brightness limitations, large face-on systems may be missing from a sample, thus imprinting an artificial $R_e$-$q$ correlation \\citep[e.g.][]{Driver07, Graham08, Mowla19}. In general, dynamical mass estimates for smaller galaxies are less certain since their kinematics are constrained by fewer independent resolution elements. For more face-on systems it is more difficult to infer robust inclination corrections to the anyhow shallow velocity gradients. Therefore, we interpret the larger discrepancy of H$\\alpha$-based and stellar-based $M_{\\rm dyn}$ estimates for smaller, more face-on galaxies as being due to less robust measurements.\n\nIn the middle panel of Figure~\\ref{f:dmdyn1} we also see that those $M_{\\rm dyn}$ estimates from LEGA-C that are lower than the KMOS$^{\\rm 3D}$\\, estimates have $q<0.6$. This could be a possible effect of underestimated inclinations in the LEGA-C dynamical models due to a larger intrinsic thickness prior (Section~\\ref{s:model_diff}), translated to the $M_{\\rm dyn}$ measurements from integrated velocity dispersions via the calibration based on the JAM models. We caution that assumptions on intrinsic thickness and related inclination estimates can introduce systematic effects in dynamical mass measurements (see also discussion in Appendix~\\ref{a:structure}).\n\nFor face-on galaxies, it is also harder to determine the position angle. It was shown by \\cite{Wisnioski15} that the majority of misalignments between kinematically (H$\\alpha$) and morphologically (F160W) determined position angles for $z\\sim1-2$ SFGs are found at $q>0.6$. These trends suggest that apparent misalignments between ionised gas kinematics and stellar light are not primarily due to intrinsic physical differences between the warm gas and stellar distributions in galaxies, a possible consequence of e.g.\\ misaligned accretion, but are largely due to limitations of photometric measurements for face-on systems. We find a comparable trend in our sample. \n\nThis motivates us to explore in more detail possible correlations between dynamical mass discrepancy and measures of position angle. Here, following \\cite{Wisnioski15} and the misalignment diagnostic $\\Psi$ by \\cite{Franx91}, we define $\\sin(\\Psi_{\\rm F814W,kin}) = |\\sin({\\rm PA_{F814W}}-{\\rm PA_{\\rm kin}})|$, where PA$_{\\rm kin}$ is measured from the H$\\alpha$ IFS data. We find that those galaxies with larger mismatches in their dynamical mass estimates also have stronger kinematic misalignments.\n\nFor $M_{\\rm dyn}$ measurements based on JAM models (or the spatially resolved models by \\citealt{Straatman22}), we would expect a trend such that kinematic PAs that are more inclined with respect to the slit orientation than the photometric PA would result in underestimated dynamical masses from the stellar data, whereas kinematic PAs that are closer to the slit orientation than the photometric PA would result in overestimated dynamical masses from the stellar data. We find no indication for a corresponding trend based on the JAM measurements only. \nConsidering all measurement pairs we find no significant correlation ($\\rho_S=0.22$; $\\sigma_\\rho=0.9$).\n\nFurther, we find at most a weak correlation between mismatches of the H$\\alpha$ kinematic major axis and the LEGA-C slit position for the $M_{\\rm dyn}$ discrepancies based on integrated stellar velocity dispersion measurements ($\\rho_S=0.31$; $\\sigma_\\rho=1.1$).\nWe also find no (significant) correlation with $M_{\\rm dyn}$ discrepancy and the F160W or F814W S\u00e9rsic index measurements,  their difference, or estimates of the central 1~kpc stellar surface density.\n\n\n\\subsubsection{Kinematic properties}\\label{s:corr_kin}\n\nWe consider correlations of dynamical mass discrepancy with kinematic quantities, specifically the H$\\alpha$ circular velocity at $R_{e, \\rm F814W}$ and the rotational support $v_{\\rm rot}\/\\sigma_0$. These quantities are based on the best-fit dynamical models of the H$\\alpha$ data from the KMOS$^{\\rm 3D}$\\, survey. We find a trend with circular velocity, as illustrated in the left panel of Figure~\\ref{f:dmdyn3}. This shows that the difference in $M_{\\rm dyn}$ estimates from stars and gas is lower for galaxies with higher circular velocities and stronger dynamical support from rotation in the ionised gas phase (middle panel). Possible explanations could be that systems with higher rotational support are closer to dynamical equilibrium (dynamical equilibrium is the base assumption for all dynamical modelling discussed in this work), or that the pressure support corrections for the H$\\alpha$ data are underestimated (cf.\\ right panel). The pressure support correction chosen in this work following the self-gravitating disc description by \\cite{Burkert10} is stronger than other corrections adopted in the literature, so the latter explanation is unlikely \\citep[see e.g.][]{Bouche22, Price22}.\n\nNo corresponding significant correlations have been found in the study by \\cite{Straatman22} comparing slit-based estimates for both ionised gas and stars, where the kinematic major axis is unknown.\n\nWe note that for two galaxies we can only robustly constrain upper limits on the intrinsic velocity dispersion $\\sigma_0$ from our models due to the spectral resolution of KMOS \\citep[see Section 3.4 by][for details]{Uebler19}. If we remove these two galaxies from our calculation of the correlation coefficients, we find $\\rho_S=0.18$ and $\\sigma_\\rho=0.4$ for the correlation between $\\Delta M_{\\rm dyn}$ and $\\sigma_0$, and $\\rho_S=-0.90$, $\\sigma_\\rho=2.2$ and $\\rho_S=-0.68$, $\\sigma_\\rho=1.7$ for the correlations with $v_{\\rm circ}(r=R_{e,{\\rm F814W}})$ and $v_{\\rm rot}(r=R_{e,{\\rm F814W}})\/\\sigma_0$, respectively.\n\n\\subsubsection{Global physical properties}\n\nConsidering physical properties related to feedback strength such as SFR, sSFR, $\\Sigma_{\\rm SFR}$, AGN activity, or outflow signatures, we find no (significant) correlations with dynamical mass offset. This suggests that feedback does not play a major role in systematically affecting the dynamical mass estimates differently for H$\\alpha$ and stars for galaxies in our sample \\citep[see also][]{Straatman22}. However, we remind the reader about the substantial effect of AGN tracer emission line species such as [NeV] on the ionised gas galaxy kinematics extracted from the LEGA-C spectra.\n\nWe also caution about the potential impact of the line broadening in integrated line emission spectra induced by the presence of strong outflows (and of important disc velocity dispersion), as discussed by \\cite{Wisnioski18} for compact massive SFGs. This underscores the benefit of spatially-resolved emission line kinematic modeling as performed here.\n\nAlthough (circular) velocity and galaxy mass are connected through the TFR, we find no significant correlation with $\\Delta M_{\\rm dyn}$ and galaxy mass (see Table~\\ref{t:corr}). To some extent, this can be explained by the scatter in the $z\\sim1$ TFR \\citep[e.g.][]{Uebler17}. However, in our sample the lack of correlation is also driven by the massive, compact galaxy COS4\\_08096 having a large dynamical mass discrepancy. If we exclude this galaxy from the calculations, we still find only weak trends, but more along the expected direction for $\\log(M_{\\star})$ ($\\rho_S=-0.28$, $\\sigma_\\rho=1.1$) and for $\\log(M_{\\rm bar})$ ($\\rho_S=-0.29$, $\\sigma_\\rho=1.2$).\\\\\n\nIn summary, despite the small sample size our investigation of correlations with dynamical mass discrepancy reveals interesting trends, which should be followed up in future studies.\nWe find mild correlations in particular with effective radius, projected axis ratio, dynamical support in the ionised gas phase, and with kinematic misalignment. This confirms on the one hand the expectation that it is more difficult to constrain robust dynamical masses for galaxies that are smaller, more face-on, and with higher dispersion support \\citep[see][for a detailed study]{Wisnioski18}. On the other hand it stresses the importance of spatially-resolved kinematic information to build accurate mass models.\n\n\n\n\\section{Discussion and Conclusions}\\label{s:conclusions}\n\nWe have compared kinematics and inferred dynamical masses from ionised gas and stars in 16 star-forming galaxies at $z\\sim1$, common to the KMOS$^{\\rm 3D}$\\, \\citep{Wisnioski15, Wisnioski19} and LEGA-C \\citep{vdWel16, vdWel21, Straatman18} surveys. Our main conclusions are as follows:\n\n\\begin{itemize}\n\n    \\item Comparing stellar and H$\\alpha$ kinematic profiles, we find that on average rotation velocities are higher by $\\sim45$ per cent and velocity dispersions are lower by a factor of two for H$\\alpha$ relative to stars, reminiscent of trends observed in the local Universe (Sections~\\ref{s:comp_v} and \\ref{s:comp_s}).\\\\\n    \n    \\item We measure higher rotational support in H$\\alpha$ compared to [OII]. This could explain systematic differences found in literature studies of e.g.\\ the Tully-Fisher relation when based only on $v_{\\rm rot}$ without accounting for pressure support (Sections~\\ref{s:comp_vs} and \\ref{s:compilations}).\\\\\n    \n    \\item We find excellent agreement between cumulative total mass profiles constrained from our {\\sc{dysmal}} models using H$\\alpha$ kinematics and from JAM models to the stellar kinematics, out to at least 10~kpc for five of six galaxies (average $\\Delta M_{\\rm dyn}(R_{e,\\rm F814W})<0.1$~dex, standard deviation 0.07~dex; Section~\\ref{s:mcum}). This shows that dynamical masses at $z\\sim1$ can be robustly measured from modelling spatially resolved observations, either of stellar or ionised gas kinematics. \\\\\n\n    \\item Simpler dynamical mass estimates based on integrated stellar velocity dispersion are less accurate (standard deviation 0.24~dex; Section~\\ref{s:compmdyn}). \\\\\n    \n    \\item We investigate correlations of dynamical mass offset with galaxy properties and find larger offsets e.g.\\ for galaxies with stronger misalignments of photometric and H$\\alpha$ kinematic position angles (Section~\\ref{s:corr}). This highlights the value of 2D spatially resolved kinematic information in inferring dynamical masses. \n            \n\\end{itemize}\n\nOur comparison of the kinematics of stars and ionised gas reveals differences in their resolved velocities and velocity dispersions that are marginally significant.\nLower rotational support, lower LOS disc velocities, and higher LOS disc velocity dispersions of stars relative to the star-forming gas phase are also seen in modern cosmological simulations (\\citealp{Pillepich19}; C. Lagos, priv. comm.; E.~Jimenez et al., in prep.).\nA possible scenario explaining lower rotational support and higher dispersion in the stellar component is that the observed stars have been born {\\it in-situ} from gas with higher velocity dispersions \\citep[e.g.][]{Bird21}. \nThe redshift evolution of molecular gas disc velocity dispersion is still poorly constrained through available data \\citep[see][]{Uebler19}. However, if the mild evolution of ionised gas disc velocity dispersion with redshift is an indication also for the evolution of molecular gas velocity dispersion, processes in addition to secular evolution are likely required to explain the stellar velocity dispersion for the majority of galaxies in our sample.\nThis is further supported by the tentative evidence of a larger difference in H$\\alpha$ and stellar velocity dispersion for systems with higher stellar disc velocity dispersion, indicating a process that affects the stellar dynamics differently, and plausibly at earlier times, than the ionised gas.\n\nIn general, the collision-less nature of stars allows for a variety of non-circular orbital motions. A higher fraction of low-angular momentum box orbits or $x$-tubes (rotation around the minor axis) can reduce the LOS velocity of stars \\citep[e.g.][]{Roettgers14}. The origin of such motions is plausibly connected to assembly history, where more frequent mergers in the past reduce the net angular momentum of the stellar component, in particular if their baryon content is dominated by stars \\citep[e.g.][]{Naab14}. \nSuch interactions could also contribute to disc heating, further increasing the velocity dispersion of the stars \\citep[e.g.][]{Grand16}. \nThis is in agreement with our finding of higher $v_{\\rm max}\/\\sigma_{\\rm out}$ measured from H$\\alpha$ compared to stars. \n\nTheoretically, the misaligned smooth accretion of gas can also result in different kinematics of gas and stars \\citep[e.g.][]{Sales12, Aumer13, Aumer14, Uebler14, vdVoort15, Khim21}. However, such processes typically reduce initially only the net angular momentum of the gas phase. This would correspond to a reduction of $v_{\\rm max}\/\\sigma_{\\rm out}$ measured from the star-forming gas relative to the full stellar population, which is not observed in our data set.\n\nDeviating kinematic signatures in gas {\\it vs.\\ }stars could also be caused by feedback. \nThe imprints of stellar- and AGN-driven winds on galaxy emission line spectra at $z\\sim1-2$ are routinely observed \\citep[see e.g.][for an analysis including the KMOS$^{\\rm 3D}$\\, survey]{FS19}. Such feedback can bias disc kinematic measurements due to the difficulty of disentangling e.g.\\ galaxy rotation from outflows in low spectral resolution observations \\citep[see also][]{Wisnioski18}. \nAt least three galaxies in our sample show signatures of outflows in their H$\\alpha$ spectra, but we could not identify a systematic effect on the spatially-resolved disc kinematic measurements of H$\\alpha$ and stars presented in Section~\\ref{s:compasobs}. One of these galaxies (COS\\_19648-M1\\_134839) shows indication of a counter-rotating disc in the stellar {\\it vs.} ionised gas components; the difference in $\\log(M_{\\rm dyn})$ for this object is $\\sim0.15$~dex. \nHowever, we clearly see the impact of feedback processes on the kinematic signatures of specific emission lines (especially [NeV]) deviating from the main disc rotation in the LEGA-C spectra of two objects (see Section~\\ref{s:lgckin} and Appendix~\\ref{a:nev}). \n\nOur results on dynamical mass estimates show that data quality and methods play a role for existing differences in dynamical mass estimates of $z\\sim1$ galaxies. \nThe fact that we find better agreement between the KMOS$^{\\rm 3D}$\\, {\\sc{dysmal}} and the LEGA-C JAM dynamical mass estimates, as compared to the LEGA-C estimates based on integrated velocity dispersion, demonstrates the advantage of detailed dynamical models leveraging the full structural information available over more approximate estimators. \nThe remarkable agreement between spatially resolved dynamical mass estimates from stars and H$\\alpha$, and from independent data sets, provides great confidence in our ability to probe the gravitational potential of $z\\sim1$ galaxies.\nIt further suggests that our implementation of the pressure support correction accounting for the turbulent motions in the ionised gas phase is adequate.\n\nAt the same time, the residual trends between KMOS$^{\\rm 3D}$\\, dynamical mass estimates with {\\sc{dysmal}} and LEGA-C dynamical mass estimates from integrated velocity dispersions, particularly with the major axis misalignment of F814W photometry and H$\\alpha$ kinematics, could be interpreted as signs of physical processes disturbing global equilibrium for some galaxies. \nA difference in position angle of gas and stars could stem from misaligned smooth accretion, but also from a disruptive merger event in the past \\citep[see e.g.][]{Khim21}. If the system has not yet reached a new equilibrium, this could be reflected in deviating dynamical mass estimates from the differently affected baryonic components.\nHowever, galaxies with large misalignment ($\\Delta{\\rm PA}>20^{\\circ}$) in our sample are also seen relatively face-on, indicating that the photometric PA measurements are more uncertain and any intrinsic misalignment is likely smaller. \n\nOverall, the dynamical mass measurements from LEGA-C stellar kinematics tend to be larger than the measurements from the KMOS$^{\\rm 3D}$\\, H$\\alpha$ kinematics by 0.12~dex on average \\citep[see also][]{Straatman22}. If dynamical mass measurements from stellar kinematics are systematically overestimated, this would reduce mass-to-light ratios inferred from such data and impact conclusions on the initial mass function of galaxies. It could also potentially impact the evolutionary study of the Fundamental Plane \\citep{Djorgovski87, Dressler87}. Larger comparison samples at $z>0$ are required to quantify any potential effect.\n\nLarger samples will also be necessary for a statistical assessment of the impact of physical processes on galaxy dynamics at this cosmic epoch. Of further interest would be the extension of our sample towards lower masses, where the shallower potential wells of haloes would allow feedback and accretion processes to have a larger impact on the host galaxy properties. Due to the smaller size of lower-mass galaxies, this would require higher spatial resolution observations than the data presented in this work. This could be achieved with instruments such as ERIS\/VLT, and in the future with HARMONI\/ELT, or GMTIFS\/GMT. Similarly, higher-resolution imaging providing better constrained structural parameters would help in building more accurate dynamical models.\nAt higher redshifts, higher accretion rates and shallower potential wells may cause larger and more frequent kinematic misalignments. This can be investigated through a combination of {\\it JWST}\/NIRCam imaging and {\\it JWST}\/NIRSpec IFS observations.\n\nFor a comprehensive assessment of baryonic kinematics and dynamics at $z\\sim1$, the high-quality data from the KMOS$^{\\rm 3D}$\\, and LEGA-C surveys would ideally be complemented by spatially resolved observations of another independent dynamical tracer, such as CO. With potentially lower disc velocity dispersion than stars and ionised gas, and unaffected by extinction, dynamical masses inferred from molecular gas kinematics could help to determine realistic uncertainties on dynamical masses, and improve our understanding of the role of corrections factors and modelling assumptions required to infer dynamical masses from other baryonic phases. \n\n\n\\section*{Acknowledgements}\n\nH{\\\"U} thanks Sirio Belli, Luca Cortese, Claudia Lagos, and Joop Schaye for insightful discussions on aspects of this work.\nWe thank Amit Nestor for sharing dynamical mass estimates based on the work by \\cite{Nestor22}. We thank Claudia Lagos for sharing a comparison of LOS velocity dispersion from stars and star-forming gas based on calculations described in \\cite{Lagos18} utilising the {\\sc{eagle}} simulations \\citep{Schaye15}. \nH{\\\"U} gratefully acknowledges support by the Isaac Newton Trust and by the Kavli Foundation through a Newton-Kavli Junior Fellowship.\nCMSS acknowledges support from Research Foundation - Flanders (FWO) through Fellowship 12ZC120N.\nThis research made use of \\href{http:\/\/www.astropy.org}{Astropy}, a community-developed core Python package for Astronomy \\citep{astropy13, astropy18}, and of Photutils, an Astropy package for detection and photometry of astronomical sources \\citep{photutils}.\n\n\\section*{Data Availability}\n\nThe KMOS$^{\\rm 3D}$\\, data cubes used in this research are publicly available and accessible at \\url{http:\/\/www.mpe.mpg.de\/ir\/KMOS3D} \\citep{Wisnioski19}. The 1D spectra and $M_{\\rm vir}$ measurements from the LEGA-C survey are published by \\cite{vdWel21}. JAM models for the LEGA-C data are published by \\cite{vHoudt21}.\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\n\n\nThe density of states (DoS):  $d(E) = \\sum_i \\delta(E-E_i)$ of a quantum\nsystem -in other words, the density of eigenstates $E_i$ at a given energy $E$- plays a\nkey role in the field of ``quantum chaos''.  Gutzwiller \\cit{Gut90} found\na semiclassical approximation for the oscillatory part of the DoS, in\nterms of the periodic orbits of the (chaotic) classical system.  However\nthe DoS, generally, is not probed directly in experiments, as they\nmeasure an observable $I(E)$.  Often the latter can be related to the DoS\nby a sort of Fermi golden rule:\n  \\begin{equation} I(E) = \\sum_i \\langle \\psi_i |\n\\hat{\\cal A} | \\psi_i \\rangle \\delta(E-E_i) \\com{.} \n \\label{mat} \n \\end{equation} \n In other words, the measured quantity $I(E)$ is the DoS weighted by the\nexpectation value of an observable $\\hat{\\cal A}$ over the eigenstate $|\n\\psi_i \\rangle$ with eigenenergy $E_i$. \n\nExperiments probing such a weighted DoS include ,among others,\nspectroscopic studied of atoms in static fields \\cit{Hol88}, molecular\n(Franck-Condon) transitions \\cit{SchEn90} and electronic transport in\nmicrocavities \\cit{BJS93}.  While the Periodic Orbits (POs) have offered a\npowerful tool for understanding these experiments, a quantitative\nanalysis requires one to go beyond the Gutzwiller formula.  Semiclassical\nexpressions for the ``matrix elements''  may also be investigated\nusing the stationary phase approximation (SPA).\n\nThe first semiclassical approximation for matrix elements \\cit{EFMW92}\ninvolved POs, and in essence reduced to the Gutzwiller trace formula\n(GTF) of the density of states, weighted by the Wigner transform $A_W$ of\nthe observable $\\hat{\\cal A}$, evaluated along each PO.\nThis result was found by assuming that $A_W$ is smooth in phase space,\nand by neglecting it in the SPA.\nOn the other hand, the photoabsorption rate of the hydrogen atom was\nexpressed in terms of closed orbits (COs) passing through the nucleus\n\\cit{DuDe88}, because the observable $\\hat{\\cal A}$ is very localized.\nAnother semiclassical formula, for the conductance fluctuations in\nmicrostructures,  involved ``angle orbits''\ndefined by an angle related to the width of the leads \\cit{BJS93}.\n\nHence the the type of contributing trajectories  depends\nstrongly on the relative smoothness of $\\hat{\\cal A}$ and the\nsemiclassical Green's function used to represent the DoS.  Further, the\ntype of semiclassical expressions also depended on the level of\napproximation used in the SPA integrations.  For the semiclassical matrix\nelements \\cit{EFMW92}, the variations of $A_W$ in both the SPA condition\nand integrations were neglected.  A more refined version developed for\nmolecular transitions \\cit{ZoAl93} also neglected them in the SPA\ncondition (which yielded periodic orbits), but included them in the\nintegrations.  Finally, ``angle orbits'' were obtained by considering\n{\\em both} the observable and the semiclassical Green's function when\napplying the SPA.\n\nA broad range of situations involve Gaussian matrix elements, given\nby an observable which contains a projector on a Gaussian:\n molecular excitation from a low vibrational state \\cit{ZoAl93}, in the conductance of microcavities with\nparabolic leads \\cit{Nari98} as well as the tunnelling diode experiments\nwhich form the subject of the present study. \nThe Gaussian matrix elements, conveniently, yield fairly simple integrals, \nand also have a very clear localization length scale.\n\nThe resonant tunneling diode in tilted fields (RTD) ---which is a\nmesoscopic realization of quantum wells with tunneling barriers--- was\nintroduced recently as a new probe of quantum chaos  \\cit{From94,Mul95}.\nThe oscillations of the measured current (as a function of the applied\nvoltage) were linked to unstable \\cit{From94} and stable \\cit{Mul95}\nperiodic oscillations, following a heuristic application of the\nGutzwiller trace formula \\cit{Gut90} and taking into account the\naccessibility of the periodic orbits (POs) to the tunneling electrons.\n\nThe RTD experiments generated considerable interest and prompted the\ndevelopment of a series of semiclassical theories.  One study proposed\ntwo separate formulae (obtained by two different level of approximation\nin the integrations) using normal orbits starting and finishing\nperpendicular to the barrier \\cit{BR98}.  Another study proposed a\nformula using periodic orbits \\cit{NSB98}, similar to the approach of\nZobay and Alber \\cit{ZoAl93}.  However, it was shown \\cit{SMR98} that the\nnormal and periodic orbits could not give satisfactory results in all\nexperimental regimes, and that a new type of complex and non-periodic\ntrajectories (the saddle orbits) \\cit{SM98b} provided an accurate\nsemiclassical description.  Subsequently, Closed orbits, as well as\norbits having a minimum transverse momentum were proposed \\cit{NS99} in\norder to achieve similar results using only real trajectories.\nHere we will review and also extend our study of RTD problem, in order\nto clarify previous work and to prescribe the best\nsemiclassical description for this type of problem.  \n\nIn\nsection \\ref{sc} we derive a general semiclassical expression for\nGaussian matrix elements, expressed in terms of an {\\em arbitrary} type\nof orbits [eq.  \\rf{gammagen}]. We outline the RTD experiments \nand show that they are a realization of Gaussian matrix elements.\n\n\nIn section \\ref{discu} we explain\ndifferent reasonable assumptions one can make regarding the localization of the\nGaussian observable in position or phase space. We show that different\nassumptions yield various types of contributing trajectories \n(POs, COs, normal orbits, etc.), and correspondingly\ndifferent expressions for the current formula.  We approach the problem\nfrom three levels of approximation:  $(i)$ no approximation, yielding\n{\\em saddle orbits} (SOs); $(ii)$ the intermediate formulae, where we\nneglect one term in the SPA condition; $(iii)$ the ``hard limit'' level,\nwhere we neglect one term in both the SPA condition and the SPA\nintegrations (for POs, this corresponds to the result of Eckhardt {\\em et al.}\n\\cit{EFMW92}).Our general formula enable us to reproduce easily any of\nthe five semiclassical theories of the current in the RTD that have been\nproposed in the literature \\cit{BR98,NSB98,NS99,SM98b}.  We also propose\na new type of trajectories, we term {\\em minimal orbits},selected by\nrequiring that the gradient of the argument of the exponential is minimal\n(instead of zero as in the standard SPA). \n\nIn section \\ref{comp} we test  the different formulae against quantum\nmechanical calculations and experimental results (obtained from M\\\"uller \\al\n\\cit{Mul95} and analyzed in \\cit{SM98}).  We focus on the most\ninteresting regimes, beyond the scope of standard PO theory:  regimes\nwhere there is no real contributing PO (``ghost'' regions), or where\nnon-isolated POs give non-separable contributions; in these cases the\nsaddle orbits succeed while the PO formula fails.  Here we find that the\nclosed orbit formula \\cit{NS99} represents an improvement over the PO\nformulae, but requires a complicated strategy where one switches between\ndifferent types of real trajectories in different regimes.  On the other\nhand, the minimal orbits achieve the goal of reproducing the accuracy of\nthe complex SOs ---across the whole transition from regularity to\nchaos--- by using only real dynamics.  We also investigate\nbifurcation-type phenomena for saddle orbits, where their contribution\npeaks sharply, somewhat reminiscent of the effect of bifurcations on the\nGTF. More detail on this work can be found in \\cit{Sara99}.\n\n\n\\section{THEORY: Derivation of general semiclassical formula\n and the RTD as an example of Gaussian matrix elements}\n\\label{sc}\n\n\\subsection{Semiclassical Gaussian matrix elements}\n\nWe wish to find a semiclassical approximation to the  quantity\n\\begin{equation}\nI(E) = \\sum_i \\langle \\psi_i | \\hat{\\cal A} | \\psi_i \\rangle\n\\delta(E-E_i)\n=-\\frac{1}{\\pi}  {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\: {\\rm Trace} \\:[ \\hat{\\cal A} \\hat{G} ]\n=-\\frac{1}{\\pi}  {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\: \\int d\\bfm{q} \\int d\\bfm{q'} {\\cal A}(\\bfm{q},\\bfm{q'})\nG(\\bfm{q}',\\bfm{q}) \n\\com{,}\n\\label{semimat}\n\\end{equation}\nwhere $| \\psi_i \\rangle$ is an eigenstate of the system with\neigenenergy $E_i$.\nWe also introduced the position matrix elements of the observable,\n${\\cal A}(\\bfm{q},\\bfm{q'})= \n\\langle \\bfm{q}| \\hat{\\cal A} | \\bfm{q}' \\rangle$, and of the energy\nGreen's function \n$G(\\bfm{q'},\\bfm{q})= \\langle \\bfm{q'}| \\hat{G} | \\bfm{q} \\rangle =\n\\lim_{\\eta\\to 0_+} \\langle \\bfm{q'}| (E+i\\eta-\\hat{H})^{-1} | \\bfm{q} \\rangle$.\nWe consider here a closed system with two degrees of freedom \n$\\bfm{q} =(x,z)$, and described by a Hamiltonian $\\hat{H}$.\nWe consider here the special case of {\\em Gaussian} matrix elements, i.e., \nprojectors on a Gaussian: \n\\begin{equation}\n\\hat{\\cal A}=| \\Phi \\rangle \\langle \\Phi | \\com{,}\n\\Phi(x,z)=\\phi(z) \\chi(x) \\com{,}\n \\phi(z) =\\left( \\frac{\\beta}{\\pi\n\\hbar} \\right)^{1\/4} e^{\\textstyle -\\frac{\\beta}{2 \\hbar}z^2}\n\\com{.}\n\t\\label{init}\n\\end{equation} \n$\\chi(x)$ can also be a Gaussian (in Franck-Condon transitions) or a\ncut $\\delta(x)$ (in  microcavities or the RTD).\nTo derive the semiclassical approximation, one proceeds in two steps.\nFirst, one uses the semiclassical expression of the Green's function\ninvolving all classically allowed trajectories going from $\\bfm{q}$ to\n$\\bfm{q'}$ with energy $E$ \\cit{Gut90}:\n\\begin{equation}\nG(\\bfm{q'},\\bfm{q})  \\stackrel{\\hbar \\to 0}{\\simeq} \n\\frac{2 \\pi}{(2 \\pi i \\hbar)^{3\/2}} \\sum_{\\bfm{q} \\to \\bfm{q'}}\n\\frac{m}{\\sqrt{p_x p'_x m_{12}}} \ne^{i S(\\bfm{q},\\bfm{q'})\/\\hbar } \\com{,}\n\t\\label{semigreen}\n\\end{equation}\nwhere $S$ is the action of the trajectories, $p_x$  the\nmomentum in $x$, and $m_{12}=\\pder{z'}{p_z}$.\nNote  that we define our monodromy matrix \n$M=\\partial (z',p_z')\/(\\partial z, \\partial p_z)$ with respect to $z$\nand not to \nthe coordinate perpendicular to the trajectory as usual.\nWe did not include the phase $(-i\\mu \\pi\/2)$ arising from the number $\\mu$\nof conjugate (or focal) points along the trajectory.\n\nThe second step requires the use of the stationary phase approximation \n(SPA) in eq. \\rf{semimat}.\nThis states that an exponential integral can be approximated by\na quadratic expansion of the argument of the exponential around the\npoint where the argument is stationary.\nDepending on the relative smoothness of $\\hat{\\cal A}$ and $\\hat{G}$,\ndifferent {\\em further} approximations can be made.\n\n\\subsection{Description of the RTD}\n\nA resonant tunneling diode (RTD) can be constructed by adding different\nlayers of semiconductors and applying a voltage $V$ between the\nemitter (where the electrons \naccumulate before entering the well) and the collector.\nIn effect, this will create a wide quantum well between two tunneling\nbarriers (see Fig. \\ref{rtdexpt}). \nOne also applies a magnetic field $\\bfm{B}$ at tilt angle $\\theta$ in the\n$X-Z$ plane, which creates instability in the classical dynamics in the \nwell.\n\n\n\\begin{figure}[!]\n\\vspace{-0cm}\n\\centerline{\\psfig{figure=fig1.eps,angle=0.,height=7.cm,width=12.cm}}\n\\vskip 1.cm \n\\caption{Schematic diagram of the RTD.\nWe show the experimental setup (not to scale), with the conduction\nband profile (effective voltage).\nBelow is the 3-$D$ coordinates axis; the magnetic field $\\bfm{B}$ is at\ntilt angle $\\theta$ with the electric field $-\\bfm{F}$ in the $X-Z$ plane.\nWe also show a representation of the distribution of the electrons in\nthe emitter setback: a Gaussian\ndistribution $\\phi$ in $Z$ due to the magnetic field $B \\cos\\theta$, and an\nAiry function $\\chi$ in $X$ due to the triangular well.\nThe width of the well is $L=120 \\: {\\rm nm} =2267  \\:{\\rm a.u.} $.\n}\n\\label{rtdexpt}\n\\end{figure}\n\n\nThe Hamiltonian describing the motion of the electrons in the well can \nbe reduced to two dimensions \\cit{NS98b,BR98} and reads\n\\begin{equation}\nH(\\bfm{p},\\bfm{q}) = \\frac{1}{2 m} (p_x^2 + p_z^2) -  F x  + \n\\frac{B^2}{2 m} (x \\sin\\theta-z \\cos\\theta)^2 \\com{,}\n\t\\label{2dham1}\n\\end{equation}\nwhere we used atomic units ($e=m_e=\\hbar=1$), $F=V\/L$ and the\neffective mass of the electron is $m=0.067$.\nThe length of the well is $L=120 \\: {\\rm nm} = 2267  \\:{\\rm a.u.} $.\nWe consider the barriers to be of infinite height; the classical\nelectrons will undergo specular bounces ($p_x \\to -p_x$) at the\nbarriers ($x=0,L$), while the quantum wave function $\\psi_i$\nof the (isolated) well has vanishing boundary conditions \n[$\\psi_i(x=0,z)=\\psi_i(x=L,z)=0$].\n\n\t\\subsection{Bardeen expression for the RTD current}\n\nThe current $I(E)$ can be calculated using the assumption of weak\ntunneling across the emitter barrier, the collector playing no \nimportant role as all the sites are free for the outgoing electrons\n(which are accelerated by the voltage drop).\nIn that case one can use the\nBardeen \\cit{Bar61} formalism, which is a sort of first-order\nperturbation theory for a high barrier \\cit{BR98}:\n\\begin{eqnarray}\nJ &=& \\frac{2 \\pi}{\\hbar} \n\\sum_i |{\\cal M}_i|^2 \\delta(E-E_i) \n\\com{,}\n{\\cal M}_i = \\frac{\\hbar^2}{2 m} \n\\int d\\bfm{q} \\left\\{ \\Phi^*(\\bfm{q}) \\pder{\\psi_i}{x}(\\bfm{q}) -\n\\pder{\\Phi^*}{x}(\\bfm{q}) \\psi_i(\\bfm{q}) \\right\\} \\delta(x) \n\\com{,} \\bfm{q}=(x,z) \\com{.}\n\t\t\\label{bard1}\n\\end{eqnarray}\nIn essence, this is an overlap between $\\psi_i$ and the ``initial\nstate'' $\\Phi$, which is the state of the electron in the emitter\nregion prior to tunneling.\nThe overlap is made on a cut taken on the emitter barrier at $x=0$.\nIt was shown \\cit{MDFB97}\nthat the use of finite or infinite barriers does not change\nsignificantly the important features of the current.\nNote that the Bardeen expression is formally a matrix element\nlike eq. \\rf{semimat}, if one writes\n${\\cal M}_i =i \\langle \\Theta | \\psi_i \\rangle$ with\n$ | \\Theta \\rangle = \\hbar\/(2 m) | \\big ( \\hat{p}_x\n\\hat{\\delta}_x  +  \\hat{\\delta}_x \\hat{p}_x \\big ) \\otimes  1 \\hspace{-1.2mm}  1 _z |\n\\Phi \\rangle , \\quad \\hat{\\delta}_x =|x=0 \\rangle \\langle x= 0|.$\n\nIn the RTD, one can assume \\cit{BR98} a separable form for $\\Phi$:\nan Airy function $\\chi(x)$ induced by the triangular well, and a Landau state\n$\\phi(z)$ induced by the effective magnetic field $\\beta:=B \\cos\\theta$.\nIn the experiments under consideration and for small $\\theta$, one\ncan further assume that only the lowest Landau level [eq. \\rf{init}] is occupied\n \\cit{From94}.\n\nIt is well known that the imaginary part of the Green's function can\nreproduce the sum of the delta functions of the energies, as \nwritten in eq. \\rf{semimat}.\nThe current is therefore given by\n\\begin{eqnarray}\nJ &=& -\\frac{\\hbar^3}{2 m^2}  {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\: \\int dz \\int dz' \\left\\{\n\\Phi(\\bfm{\\bar{q}}) \\Phi^*(\\bfm{\\bar{q}'}) \n\\partial^2_{x x'} G(\\bfm{\\bar{q}'},\\bfm{\\bar{q}}) -\n\\partial_x \\Phi(\\bfm{\\bar{q}}) \\Phi^*(\\bfm{\\bar{q}'}) \n\\partial_{x'} G(\\bfm{\\bar{q}'},\\bfm{\\bar{q}}) \n\\right. \n\\nonumber \\\\ && \\left. \n-\\Phi(\\bfm{\\bar{q}}) \\partial_{x'} \\Phi^*(\\bfm{\\bar{q}'}) \n\\partial_x G(\\bfm{\\bar{q}'},\\bfm{\\bar{q}}) +\n\\partial_x \\Phi(\\bfm{\\bar{q}}) \\partial_{x'} \\Phi^*(\\bfm{\\bar{q}'}) \nG(\\bfm{\\bar{q}'},\\bfm{\\bar{q}})\n\\right\\} \\com{.}\n\t\t\\label{bard2}\n\\end{eqnarray}\nThen one can use the semiclassical expression \\rf{semigreen} for the\nGreen's function.\nBecause of the Bardeen cut, only trajectories going from and to the\nleft barrier $\\bfm{\\bar{q}}=(x=0,z)$ can contribute to the current.\nNote that the derivatives of $\\psi_i$ in eq. \\rf{bard1} will be\ntransfered to derivatives of the Green's function, yielding  factors\n$-p_x=\\partial S\/\\partial x$ and $p_x'=\\partial S\/\\partial x'$.\nIt has been shown \\cit{NS99,BR98} that the Green's function, as well\nas its first derivatives, vanish at the hard barrier (this can be\nunderstood by the Dirichlet conditions for $\\psi_i$).\nHence only the first term in eq. \\rf{bard2} contributes, and one is\nleft with:\n\\begin{equation}\nJ = -\\frac{2 |\\chi(0)|^2}{m \\sqrt{2 \\pi \\hbar}}  {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\:  i^{-3\/2} \n{\\cal I} \\com{,}\n{\\cal I}= \\int dz \\int dz'\n\\sum_{(x=0,z) \\to (x'=0,z')} \\sqrt{\\frac{p_x p'_x}{-m_{12}}}\n\\phi(z) \\phi^*(z') e^{i S(x=0,z;x'=0,z')\/\\hbar} \t\\com{.}\n\t\t\\label{bard3}\n\\end{equation}\n\n\\subsection{Derivation of the general semiclassical formula}\n\nFirst we rewrite eq. \\rf{bard3} using the Gaussian form of the initial \nstate:\n\\begin{eqnarray}\n{\\cal I} & =&  \\sqrt{\\frac{\\beta}{\\pi \\hbar}} \\sum_{\\ell}\n\\int \\! \\!\\ \\!  \\! \\int_{\\Omega_{\\ell}} \\! \\!  dz \\: dz'\n\\sqrt{\\frac{p_x p'_x}{-m_{12}}} e^{\\varphi(z,z')\/\\hbar} \n\\com{with} \n\\varphi(z,z') := i S(z,z') - \\frac{\\beta}{2}(z^2 +z'^2) \\com{.}\n\t\t\\label{varphi}\n\\end{eqnarray}\nWe have also taken the sum over trajectories $(x=0,z) \\to (x'=0,z')$\noutside  the\nintegrals where it becomes a sum over all the different ``families''\n$\\{ \\ell \\}$  of trajectories existing in a domain $\\{ (z,z') \\in\n\\Omega_{\\ell} \\}$. \n\nThe tool used for this kind of integration is the {\\em stationary phase \napproximation} (SPA).\nBriefly, it states that only trajectories $z_0 \\to z_0'$\nhaving a stationary phase\n[$\\partial \\varphi(z_0,z_0')\/\\partial z=\n\\partial \\varphi(z_0,z_0')\/\\partial z'=0$] \ncontribute to the integrals, all the others being either damped\nby the Gaussian or destroyed by the random cancellations of the\noscillations due to the action.\nThen one has to expand the phase quadratically around the contributing \ntype of orbit and integrate, pushing the limit of the integration \n$\\Omega \\to  {\\rm I} \\hspace{-0.9mm} {\\rm R} ^2$.\n\nWe saw in the introduction that a variety of contributing orbits have\nbeen proposed in the case of the RTD.\nTherefore, we shall not specify the type of orbits yet, but\nrather develop a general formula valid for {\\em any} type, and discuss \nthe different choices in section \\ref{discu}.\n\nOne can relate the second derivatives of the action to the monodromy\nmatrix $M$ \\cit{Gut90}, and the quadratic expansion of the action reads:\n\\begin{eqnarray}\nS(z,z') & \\simeq & S(z_0,z'_0) + \n\\delta z \\pder{S}{z}(z_0,z'_0) + \\delta z' \\pder{S}{z'}(z_0,z'_0)\n\\nonumber \\\\\n&& +  \\frac{1}{2} \\left[ \n\\delta z^2 \\frac{\\partial^2 S}{\\partial z^2}(z_0,z'_0) +\n2 \\delta z \\delta z' \\frac{\\partial^2 S}{\\partial z \\partial z'}(z_0,z'_0)\n+\\delta z'^2 \\frac{\\partial^2 S}{\\partial z'^2}(z_0,z'_0)\n\\right]  \\\\\n & = & S_0 - p^0_z \\delta z + p^0_{z'} \\delta z' + \n\\frac{1}{2 m_{12}} \\left [ \n\\delta z^2 m_{11} - 2 \\delta z \\delta z' + \\delta z'^2 m_{22} \\right] \\\\\n& =: & {\\cal S}_2(\\delta z, \\delta z';z_0,z'_0)\n\\com{,}\t\t\\label{quadaction}\n\\end{eqnarray}\nwith $\\delta z=z-z_0$ and $\\delta z' = z'-z_0'$.\nThe ``phase'' of the initial state is already quadratic.\nWe follow the techniques of Bogomolny and Rouben \\cit{BR98}, and\ncomplete the square: \n\\begin{eqnarray}\n& {\\cal I} & \\stackrel{\\rm SPA}{\\simeq} \n\\sqrt{\\frac{\\beta}{\\pi \\hbar}} \\sum_{\\ell_0} \n\\int_{ {\\rm I} \\hspace{-0.9mm} {\\rm R} ^2} d\\bfm{\\gamma} \\sqrt{\\frac{p_x p'_x}{-m_{12}}}\n e^{\\varphi_2(\\bfm{\\gamma})\/\\hbar} \n\t\t\\label{integral2} \\\\\n& \\varphi_2(\\bfm{\\gamma}) & := \ni {\\cal S}_2(\\bfm{\\gamma};z_0,z'_0) - \\frac{\\beta}{2}(z^2 +z'^2) \\\\\n&& \\stackrel{\\rf{quadaction}}{=} i S_0 - \\frac{\\beta}{2}(z_0^2 +z_0'^2) + \n\\bfm{\\xi}^{\\rm T} \\bfm{\\gamma} + \n\\frac{1}{2}\\bfm{\\gamma}^{\\rm T} {\\cal H} \\bfm{\\gamma} \n\t\t\\label{lint}\t\\\\\t\t\n&& = i S_0 - \\frac{\\beta}{2}(z_0^2 +z_0'^2) - \n\\frac{1}{2} \\bfm{\\xi}^{\\rm T} {\\cal H}^{-1} \\bfm{\\xi} + \n\\frac{1}{2} (\\bfm{\\gamma} + \\bfm{\\gamma}_1)^{\\rm T} {\\cal H} \n(\\bfm{\\gamma} + \\bfm{\\gamma}_1)\n\\label{varphi2}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&& \\bfm{\\gamma}=(\\delta z, \\delta z') = (z-z_0,z'-z'_0) \\com{,} \\\\\n&& \\bfm{\\xi} = (-\\beta z_0 - i p_z^0, -\\beta z'_0 + i p_{z'}^0) \\com{,}\n\\bfm{\\gamma}_1= {\\cal H}^{-1} \\bfm{\\xi}   \\com{,} \\\\\n&& {\\cal H} = \\left ( \\begin{array}{cc}\n\\displaystyle - \\beta + i \\frac{m_{11}}{m_{12}} &\n\\displaystyle -i \\frac{1}{m_{12}} \\\\ \\\\\n\\displaystyle -i \\frac{1}{m_{12}} &\n\\displaystyle - \\beta + i \\frac{m_{22}}{m_{12}} \n\\end{array} \\right) \\com{.}\n\\end{eqnarray} \n$\\ell_0$ denotes the different contributing trajectories.\nWith the change of variables $\\bfm{\\gamma'} = \\bfm{\\gamma} +\n\\bfm{\\gamma}_1$, the last term in eq. \\rf{varphi2} gives a pure two-dimensional\nGaussian, equal to $2 \\pi \\hbar\/\\sqrt{\\det {\\cal H}}$.\nThe final result is\n\\begin{equation}\n{\\cal I} = \\sum_{\\ell_0 \\atop (z_0 \\to z'_0)} \n2 \\sqrt{\\frac{\\beta \\pi \\hbar p_x p_x'}{- \\cal D}} \ne^{\\left[ i S_0 + \\Gamma(z_0,z'_0)\n\\right] \/\\hbar }  \\com{,}\n\t\t\\label{generalformula}\n\\end{equation}\nwith \n\\begin{eqnarray}\n{\\cal D} &=& -m_{21} - i \\beta {\\rm Tr} M + \\beta^2 m_{12}\n\\label{cald}\t\t\\\\\n\\Gamma (z,z') \n&=& \n -\\frac{\\beta}{2 {\\cal D}} \\times \\Bigg\\{\nz^2 \\left[ -m_{21} - i \\beta m_{11} \\right] +\nz'^2 \\left[ -m_{21} - i \\beta m_{22} \\right] +\n2 i \\beta z z'            \t   \\nonumber  \\\\ \n& + & \n \\frac{1}{\\beta^2}p_z^2 \\left[ - i \\beta m_{22} + \\beta^2 m_{12} \\right] +\n\\frac{1}{\\beta^2}p_{z'}^2\\left[ - i \\beta m_{11} + \\beta^2 m_{12} \\right] +\n2 i \\frac{1}{\\beta} p_z p_{z'}        \\nonumber   \\\\ \n& + & \n 2 \\frac{i}{\\beta} z p_z \\left[ i \\beta m_{22} - \\beta^2 m_{12} \\right] +\n2 \\frac{i}{\\beta} z' p_{z'} \\left[ -i \\beta m_{11} + \\beta^2 m_{12} \\right] +\n2 z p_{z'} - 2 z' p_z \n\\Bigg\\} \\: \\com{.}\n\t\t\t\t\t\\label{gammagen}\n\\end{eqnarray}\n\n\nThe above formula describes the {\\em oscillatory} part of the \ncurrent.\nWe do not consider here the ``smooth'' part, obtained by considering\nzero-length trajectories \\cit{BR98};\nthis part varies slowly with the energy, and corresponds to the Weyl\nterm in Gutzwiller's theory of the density of states \\cit{Gut90}.\n\nThe loss of coherence due to phonon scattering is {\\em not} considered \nin the formalism presented here.\nIt can be modeled  by adding an exponential factor $\\exp(-T\/\\tau)$ in\nthe sum, where $T$ is the real part of the total time of each trajectory\nand $\\tau$ is the damping (decoherence) time.\nWe shall proceed the other way round, canceling the effects of the \ndamping in the experiments.\n\nThe formula is valid only for {\\em isolated} expansion orbits. \nAlso, we did not write explicitly the phase arising from \nthe number $\\mu$ of conjugate points along the\ntrajectory, as we shall primarily consider individual contributions\nfrom isolated trajectories.  \n\n\\section{THEORY: Semiclassical formulae for specific types of trajectories}\n\t\t\t\t\t\\label{discu}\n\nAt this stage one should go back to the SPA applied to \neq. \\rf{varphi}, and examine which choices of expansion orbits have\nbeen or can be made. \nFirst we mention one \nremarkable feature of the RTD, which is that  for {\\em any starting} $z$, \nthere exists, generically, a starting momentum $\\check{p}_z$ for which the \ntrajectory is a time-reversed duplicate of itself and therefore closed. \nWe  call such trajectories {\\bf time-symmetric (TS)}.\nThey are defined by\n\\begin{equation}\n\\bom{TS}  (z,\\check{p}_z) \\to (z',p_z') = (z,-\\check{p}_z) \n\t\t\\label{condsrco}\n\\end{equation}\nand satisfy the important relation $m_{11} = m_{22}$. \nThe existence of TS orbits is a consequence of the fact that for\nany starting $z$, one can find a starting $p_z$\nso that there is either a perpendicular bounce on a wall\n[$p_z(x=0 \\:{\\rm or} \\:L)=0$], or a turning point on the energy surface\nat a point where $\\bfm{p}=0$.\nNote that some self-retracing trajectories with $p_z^0 \\neq 0$ or\n$z' \\neq z$ are not TS.\n\nOne actually finds that each choice of expansion orbit shown below\ncontains a subset which \nis time-symmetric (TS), and that {\\em in almost all cases only the TS\nsubset contribute to the current}. \nThis is the reason why we  first write \\rf{gammagen} for \ntime-symmetric (TS) orbits. \nUsing \\rf{condsrco} and $m_{11}=m_{22}$, one finds\n\\begin{equation}\n\\Gamma_{\\rm TS}(z_0)= - \\frac{\\beta}{1-\\delta} \\Bigg\\{\nz_0^2 - \\left[ \\frac{1}{\\beta^2} (p_z^0)^2 - \n2 \\frac{i}{\\beta}z_0 p_z^0 \\right] \\delta \\Bigg\\} \n\t\t\\label{gammasrco}\n\\end{equation}\nwhere we have defined\n\\begin{equation}\n\\quad \\delta=-i \\beta \\frac{m_{12}}{m_{11}-1}  \\com{.}\n\t\t\\label{delta}\n\\end{equation}\nWe now consider different possible choices for the expansion points.\n\t\\subsection{Saddle orbits (SOs)}\n\nThe first expansion orbits we consider here are given by the strict\napplication of the stationary phase condition on \\rf{varphi}:\n\\begin{equation} \\bom{SO}\n\\left\\{ \\begin{array}{ll}\n& p_z^0=i \\beta z_0 \\\\\n& p_{z'}^0= -i \\beta z'_0 \\com{.} \\end{array} \\right. \n\t\t\\label{condso}\n\\end{equation}\nWe call such trajectories {\\bf saddle orbits (SOs)} \\cit{SM98b}.\nInserting eq. \\rf{condso} in eq. \\rf{gammagen}, one finds \n$\\Gamma_{\\rm SO}(z_0,z'_0) = -\\frac{\\beta}{2} \\left( z_0^2 + z_0^{'2}\n\\right)$, i.e.\n\\begin{equation}\n{\\cal I}_{\\rm SO} = \\sum_{{\\rm SOs} \\atop (z_0 \\to z'_0)} \n2 \\pi \\hbar \\sqrt{\\frac{ p_x p_x'}{- \\cal D}} \ne^{i S_0\/\\hbar} \\phi(z_0) \\phi(z_0')\n  \\com{.}\n\t\t\\label{gammaso}\n\\end{equation}\nIn the case where SOs are TS {\\bf (TSSOs)} one has\n$\\Gamma_{\\rm TSSO}(z_0) = -\\beta z_0^2 $.\nWe shall show in section \\ref{comp} that the SOs are the most\nsuccessful types of orbits for a semiclassical description of the\nquantum current. \nOne difficulty with SOs is the fact that they are {\\em complex}. \nThis is the reason why Bogomolny and Rouben \\cit{BR98} decided to\navoid them (and considered real trajectories),\nalthough they were well aware of the fact that the stationary phase \napproximation yields SOs.\n\nThe SOs are {\\em non-periodic}; this means that one cannot \nlook at repetitions of a ``primitive'' SO, as it would not satisfy the \nSO condition.\nInstead one must search for an SO with a higher period.\nContrary to complex POs, the complex conjugate of an SO is {\\em not} an SO.\n\n\t\\subsection{Normal orbits (NOs)}\n\t\\label{normorb}\n\nTo obtain real trajectories, one has to make a further approximation\nand neglect one term in the SPA condition \\rf{condso}.\nBogomolny and Rouben \\cit{BR98} considered that the dynamics in the\nwell are very \nchaotic; in that case one should \nexpect the oscillations due to the action to dominate the Gaussian damping\nof the initial state. \nFormally, this corresponds to taking the limit $\\beta \\to 0$ in\n\\rf{condso}, and  \nyields {\\bf normal orbits (NOs)}:\n\\begin{equation} \\bom{NO}\n\\left\\{ \\begin{array}{ll}\n& p_z^0=0  \\\\\n& p_{z'}^0=0 \\com{.} \\end{array} \\right. \n\t\t\\label{condno}\n\\end{equation}\nEssentially, this states that the contributing trajectories are determined\nsolely by the oscillations of the Green's function: they must cancel\nthe variations of the action, however small their accessibility to the\ninitial state is.\nMoreover, TS normal orbits {\\bf (TSNOs)} have $z_0'=z_0$. \nThis implies that TSNOs are actually a special subset of periodic orbits,\nthat are time-symmetric (TSPO) and start with $p_z$=0.\nIn the non-TS case ($z_0'\\neq z_0$), the {\\em second} repetition of the NO\nis actually a TSPO; \nthe first repetition of a non-TS NO is in a sense a ``half PO''. \nOne finds\n\\begin{equation}\n\\Gamma_{\\rm NO}(z_0,z'_0) = -\\frac{\\beta}{2 {\\cal D}} \\Bigg\\{ \nz_0^2 \\left[ -m_{21}- i \\beta m_{11} \\right] + \n(z_0')^2 \\left[ -m_{21} - i \\beta m_{22} \\right] + 2 i \\beta z_0 z'_0 \\Bigg\\} \n\\com{,} \n\t\t\\label{gammano}\n\\end{equation}\nwhich is equation (109) of Bogomolny and Rouben (1999) \\cit{BR98}.\nIn the TS case, one has\n$\\Gamma_{\\rm TSNO}(z_0) = -\\frac{\\beta}{1-\\delta}z_0^2$, and \n the current can be written\n\\begin{equation}\n{\\cal I}_{\\rm TSNO}=\\sum_{{\\rm NOs} \\atop (z_0 \\to z'_0)} \n2 \\sqrt{\\frac{ \\beta \\pi \\hbar p_x p_x'}{- |{ \\cal D}|} } \ne^{-\\beta z_0^2 \\left( 1-\\gamma \\right)\/\\hbar }\ne^{ i (S+ \\Delta S  -\\arg{\\cal D}\/2 )\/\\hbar } \\com{,}\n\\gamma = \\frac{|\\delta|^2}{1+|\\delta|^2} \\com{,}\n\\Delta S = \\beta z_0^2 \\frac{|\\delta|}{1+|\\delta|^2}\n\\com{.} \\label{gammasrno}\n\\end{equation}\t\t\nWe shall call this expression the {\\em PO\/NO formula}.\nNote the shift $\\Delta S$ of the frequency of the oscillations\nfrom the action $S$ of the PO.\nAlso, the term $\\gamma$ can reduce the Gaussian damping due to the\ninitial state, for trajectories with large $z_0$.\nIn fact, this formula takes into\naccount torus quantization effects {\\em \\`a la} Miller \\cit{Mil75}\noccurring in large islands of stability surrounding stable POs.\nIt has been shown \\cit{BR98} that it is\nanalytically equivalent to a model \\cit{SM98} building the current as\nan overlap between the initial Gaussian and the wavefunction in the well\napproximated by the harmonic oscillator state corresponding to Miller tori.\n\n\n\nIt is also interesting to consider the case when $\\beta$ is {\\em very} small.\nThe first terms of an expansion of $\\Gamma$ and ${\\cal D}$ in\npowers of $\\beta$ yield\n\\begin{equation}\n{\\cal D}_{\\rm HLNO} \\to -m_{21} \\com{,} \n\\Gamma_{\\rm HLNO} \\to -\\frac{\\beta}{2} \\left[z_0^2 + (z_0')^2 \\right]\n\\quad \\Rightarrow \\quad\n{\\cal I}_{\\rm HLNO}=\\sum_{{\\rm NOs} \\atop (z_0 \\to z'_0)} \n2 \\pi \\hbar \\sqrt{\\frac{ p_x p_x'}{m_{21}}} \ne^{i S\/ \\hbar} \\phi(z_0) \\phi(z_0')\n\t\t\\label{hlno}\n\\end{equation}\nWe  refer to this kind of expansion as {\\bf ``hard limit''\n(HL)}.\nThis was the first formula proposed by Bogomolny and Rouben \\cit{BR98}.\nIt is justified in the case of extremely chaotic dynamics, where the\noscillations of the Green's function  are\nsupposed to be much stronger than the Gaussian decay of the\ninitial state.\nIn the case of a TS orbit, the precise condition for the validity of\nthis theory is \\cit{BR98} \n\\begin{equation}\n\\frac{ \\left| \\frac{\\partial^2 S}{\\partial z^2} \\right| }{\\beta} =\n\\left| \\frac{ m_{11}-1}{\\beta m_{12}} \\right|  =\n\\frac{1}{\\left| \\delta \\right|} \\gg 1\n\\com{.}\t\\label{stabfact}\n\\end{equation}\n\nThe HL result corresponds to the SPA method applied to the Green's\nfunction only.\nAs $\\beta$ is supposed to be {\\em very} small, the initial state\nfunction is {\\em neglected} in the integrations \\rf{integral2}, and is \ntaken out of them; \nit is evaluated at the NO and gives the simple Gaussian factor\n$\\Gamma_{\\rm HL}$. \nThe integral is carried out only over the Green's function, \nbringing in a prefactor $m_{21}$ and the exponential of the pure action.\nNote that this is the usual way of proceeding with the SPA method,\nwhile considering the variation of both functions in the integral\n\\rf{integral2} is not standard.\n\n\t\\subsection{Central closed orbits (CCOs)}\n\nNarimanov and Stone \\cit{NS99} proposed a semiclassical approach\\footnote{This was a\ncomplement to their periodic orbit formula presented in Narimanov \\al \\cit{NSB98}\nand discussed below in subsection \\ref{po}.} which \neffectively amounts to consideration of the other extreme case, where \nthe Gaussian damping dominates the action oscillations; \nthis assumption can be justified for fairly regular dynamics.\nThis corresponds to taking the limit $\\beta \\to \\infty$ in \\rf{condso}, \nand yields {\\bf central closed orbits (CCOs)}:\n\\begin{equation} \\bom{CCO}\n\\left\\{ \\begin{array}{ll}\n& z_0=0  \\\\\n& z'_0=0 \\com{.} \\end{array} \\right. \n\t\t\\label{condcco}\n\\end{equation}\nHere the contributing trajectories give maximal accessibility to\nthe initial state, while they do not cancel the variations of the action.\nIn this case one finds\n\\begin{equation}\n\\Gamma_{\\rm CCO}(p_z^0,p_{z'}^0) = -\\frac{1}{2 \\beta {\\cal D}} \\Bigg\\{ \n(p_z^0)^2 \\left[ -i \\beta m_{22} + \\beta^2 m_{12} \\right] + \n(p_{z'}^0)^{2} \\left[ - i \\beta m_{11} + \\beta^2 m_{12} \\right] \n+ 2 i \\beta p_z^0 p_{z'}^0 \\Bigg\\} \n\\com{.} \n\t\t\\label{gammacco}\n\\end{equation}\nThis formula is equivalent to equation (14) of Narimanov and Stone\n\\cit{NS99}. (They derived a general formula for any number of excited \nLandau levels in the initial state.)\nTS central closed orbits {\\bf (TSCCOs)} have $p_{z'}^0=-p_z^0$, and\ngive\n$\\Gamma_{\\rm TSCCO}(z_0) = \\frac{1}{\\beta} \\frac{\\delta}{1-\\delta}(p_z^0)^2$.\n\n\n\nOne can also consider the hard limit, i.e. the first order expansion\nin ${\\cal O}(1\/\\beta)$:\n\\begin{equation}\n{\\cal D}_{\\rm HLCCO} \\to \\beta^2 m_{12} \\com{,}\n\\Gamma_{\\rm HLCCO} \\to -\\frac{1}{2 \\beta} \\left[ (p_z^0)^2 + (p_{z'}^0)^2 \\right]\n\\quad \\Rightarrow \\quad\n{\\cal I}_{\\rm HLCCO}=\\sum_{{\\rm CCOs} \\atop (z_0 \\to z'_0)} \n2 \\pi \\hbar \\sqrt{\\frac{ p_x p_x'}{-m_{12}}} \ne^{i S\/ \\hbar} \\tilde{\\phi}(p_z^0) \\tilde{\\phi}(p_{z'}^0)\n\\com{.} \t\\label{hlcco}\n\\end{equation}\nHere we introduced the initial state (i.e., the observable) in\nmomentum representation: $\\tilde{\\phi}(p_z)=(\\beta \\pi \\hbar)^{-1\/4}\n\\exp[-p_z^2\/(2 \\beta \\hbar) ]$.\nThe hard limit  is equivalent to neglecting the quadratic \nterm of the action in the\nintegral \\rf{integral2}.\nThe integration of the linear term with the\ninitial state is in effect a Fourier transform, and brings in the\nvalue of the\ninitial state in momentum representation at the CCO.\nAlternatively, one can express the Green's function and the initial\nstate in momentum representation, and argue that the latter is smooth and\ncan be taken out of the integral by stationary phase approximation.\nA similar expression in terms of ``closed orbits at the nucleus'' and\ninvolving a weighting by $m_{12}^{-1\/2}$ was found in the\nsemiclassical theory of the photoabsorption spectra of a hydrogen atom\nin external fields \\cit{DuDe88}.\nThe similarity is somewhat limited, as the\nexpression for the photoabsorption spectra is much more complicated\nthan mere momentum wave functions (it involves matching the\nsemiclassical Green's function to a quantum one in the vicinity of the \nnucleus). \n\n\t\t\\subsection{Periodic orbits (POs)}\n\t\t\\label{po}\n\t\t\nPeriodic orbits are a natural choice, as it follows the expansion around POs\nfound in the derivation of the Gutziller trace formula  for\nthe density of states. \nA discussion of this choice is more adequately made using a\nphase space formalism, described in appendix \\ref{pssc}.\nAlternatively, a more direct route is to \ndefine ``average-difference'' coordinates\n\\begin{equation} \\left\\{ \\begin{array}{ll}\n& \\bar{z}=\\frac{1}{2}(z'+z)  \\\\\n& \\Delta z=z'-z \\end{array} \\right. \\com{,}\n\\left\\{ \\begin{array}{ll}\n& \\bar{p_z}=\\frac{1}{2}(p_z'+p_z)  \\\\\n& \\Delta p_z=p_z'-p_z \\end{array} \\right. \n\\com{,}\t\t\\label{variable}\n\\end{equation}\nwhich one uses to write the $z$-observable $\\hat{A}=| \\phi \\rangle \\langle \n\\phi |$ in position space as:\n\\begin{equation}\n\\bar{A}(\\bar{z},\\Delta z) = \n \\phi(\\bar{z}-\\frac{1}{2} \\Delta z) \\phi^*(\\bar{z}+\\frac{1}{2}\\Delta z)=\n \\sqrt{\\frac{\\beta_{\\bar{z}}^{1\/2}\\beta_{\\Delta z}^{1\/2}}{\\pi \\hbar}}\n e^{\\textstyle -\\frac{\\beta_{\\bar{z}}}{\\hbar} \\bar{z}^2 -\n\\frac{\\beta_{\\Delta z}}{4 \\hbar}  \\Delta z^2 } \n\\com{,}\t\t\t\\label{initdiff}\n\\end{equation}\nwhile the Wigner transform is defined by\n\\begin{equation}\nW(\\bar{z},\\bar{p}_z)=\\frac{1}{2 \\pi \\hbar} \\int d \\Delta z \\:\ne^{i \\bar{p}_z \\Delta z \/\\hbar} \\bar{A}(\\bar{z},\\Delta z) =\n\\frac{1}{\\pi \\hbar} \ne^{\\textstyle -\\frac{\\beta_{\\bar{z}}}{\\hbar} \\bar{z}^2 - \n\\frac{1}{\\beta_{\\Delta z} \\hbar} \\bar{p}^2} \n\t\\label{wign} \\com{.}\n\\end{equation}\nHere we have written two different Gaussian widths $\\beta_{\\bar{z}}$\nand $\\beta_{\\Delta z}$ for respectively $\\bar{z}$ and $\\Delta z$.\nOf course in reality we have $\\beta_{\\bar{z}} = \\beta_{\\Delta z} = \\beta$,\nbut retaining the distinction clarifies the following discussion.\nThe action is\n$\\bar{S}(\\bar{z},\\Delta z)= S(\\bar{z}-\\Delta z\/2,\\bar{z}+\\Delta\/2)$, \nand its quadratic expansion around a point $(\\bar{z}_0,\\Delta\nz_0)$ reads:\n\\begin{eqnarray}\n\\bar{{\\cal S}}_2(\\bar{z},\\Delta z) &=&\n\\bar{S}_0 +  \\Delta p_z^0 \\delta \\bar{z} + \\bar{p}_z^0 \\delta \\Delta z\n  \\nonumber \\\\ &+&\n\\frac{1}{2 m_{12}} \\left [ \n\\delta \\bar{z}^2 ({\\rm Tr} M-2) + \\delta \\bar{z} \\delta \\Delta z (m_{22}-m_{11})\n + \\frac{1}{4} \\delta \\Delta z^2 ({\\rm Tr} M+2) \\right] \\com{} \\label{s2phase}\n\\end{eqnarray}\nwith $\\delta \\bar{z}= \\bar{z}-\\bar{z}_0$ and \n$\\delta \\Delta z=\\Delta z -\\Delta z_0$.\n\nThe idea is to apply the SPA method to the integral\n$ {\\cal I} \\propto \\int \\Delta z \\int \\bar{z} \\:\n\\bar{A}(\\bar{z},\\Delta z) \n\\exp [i \\bar{{\\cal S}}_2(\\bar{z},\\Delta z)\/\\hbar ]$, i.e., \nwith respect to the variables \\rf{variable}.\nThe SPA condition reads:\n\\begin{equation} \\left\\{ \\begin{array}{ll}\n& \\bar{p}_z^0 =- \\frac{1}{2} i \\beta_{\\Delta z} \\Delta z_0 \\\\\n& \\Delta p_z^0= -2 i \\beta_{\\bar{z}} \\bar{z}_0 \\com{.} \\end{array} \\right.\n\t\t\\label{condsovar}\n\\end{equation}\n\nIn Eckhardt {\\em et al.} \\cit{EFMW92}, one assumes the Wigner transform to be\nsmooth as a function of $(\\bar{z},\\bar{p}_z)$.\nThis corresponds to the case $\\beta_{\\bar{z}} \\to 0$ and \n$\\beta_{\\Delta z} \\to \\infty $, which gives for \\rf{condsovar}:\n\\begin{equation} \\bom{POs} \\left\\{ \\begin{array}{ll}\n& \\Delta z_0 =0  \\\\\n& \\Delta p_z^0= 0 \\com{,} \\end{array} \\right.\n\t\t\\label{condpo2}\n\\end{equation}\nthat is, {\\bf periodic orbits (POs)}.\nHence POs arise naturally when one consider a smooth Wigner\ntransform; as $\\bar{A}(\\bar{z}, \\Delta z)$ \n is its Fourier transform [see eq. \\rf{awign}],    \nit is smooth in $\\bar{z}$, but {\\em localized} in $\\Delta z$.\nThis corresponds to a ``local'' operator, in the sense that\n$\\bar{A}(\\bar{z},\\Delta z) \\sim \\bar{a}(\\bar{z}) \\delta(\\Delta z)$.\nThis also enables one to recover the Gutzwiller trace formula, via \n$\\hat{A} \\to  1 \\hspace{-1.2mm}  1  \\Rightarrow \\bar{A}(\\bar{z},\\Delta z) \\to \\delta(\\Delta z)$.\n\nThings are different for the Gaussian matrix elements, which are \nwritten as a projector over a Gaussian state.\nThey are the product of two functions depending separately on $z$ and $z'$, and\n{\\em cannot} have the property of being simultaneously smooth in\n$\\bar{z}$ and localized in $\\Delta z$: either it is localized in both, or it\nis smooth in both.\nOne cannot change $\\beta_{\\bar{z}}$ and $\\beta_{\\Delta z}$ independently.\nThis fact was noted by Zobay and Alber \\cit{ZoAl93} in their work on\nFranck-Condon \nmolecular transitions, which involved very similar equations.\n\nNevertheless, it is still fruitful to consider periodic orbits for the\nRTD.\nPutting $z'=z,p_z'=p_z$ in \\rf{gammagen}, one finds\n\\begin{eqnarray}\n\\Gamma_{\\rm PO}(z^0,p_{z}^0) &=& -\\frac{\\beta}{2 {\\cal D}} \\Bigg\\{ \nz_0^2 \\left[ -2 m_{21} - i \\beta ({\\rm Tr} M-2) \\right] +\n\\frac{1}{\\beta^2}(p_z^0)^2 \\left[ -i \\beta ({\\rm Tr} M-2) + 2 \\beta^2 m_{12}\n\\right]  \\nonumber \\\\\n&& \\hspace{2cm} - 2 z_0 p_z^0 \\left[ m_{22}-m_{11} \\right]\n\\Bigg\\}  \\com{.} \n\t\t\\label{gammapo}\n\\end{eqnarray}\nThis formula is equivalent to equation (19) of Narimanov and Stone \\cit{NS99}.\nAn important subset of POs are TS, and have $p_z^0=0$ {\\bf (TSPOs)}.\nAs mentioned above, TSPOs are identical to TSNOs, and therefore  give \nthe same contribution \\rf{gammasrno}.\n\n\n\n\nFor the hard limit, an expansion in $\\beta_{\\bar{z}}$ and\n$1\/\\beta_{\\Delta z}$  gives\n\\begin{equation}\n{\\cal D}_{\\rm HLPO} \\to \\beta ({\\rm Tr} M-2)\/2 i \\com{,} \n\\Gamma_{\\rm HLPO} \\to -\\beta_{\\bar{z}} \\bar{z}_0^2- \n\\frac{1}{\\beta_{\\Delta z}} (\\bar{p}_z^0)^2\n\\quad \\Rightarrow \\quad\n{\\cal I}_{\\rm HLPO}=\\sum_{{\\rm POs} \\atop (z_0 \\to z'_0)} \n2 \\pi \\hbar \\sqrt{\\frac{ 2 \\pi i \\hbar  p_x p_x'}{{\\rm Tr} M-2}} \ne^{i S\/ \\hbar} W(\\bar{z_0}, \\bar{p}_z^0)\n\\com{.}\t\t\\label{hlpo}\n\\end{equation}\nThis corresponds to the formula for semiclassical matrix elements \nproposed in Eckhardt {\\em et al.} \\cit{EFMW92}, that was derived for an\nobservable which is  \nsmooth in phase space.\nThis formula is basically the Gutzwiller trace formula (GTF) weighted\nby the Wigner transform \ncalculated for each PO.\\footnote{The result in Eckhardt {\\em et al.}\n\\cit{EFMW92} contains \nthe {\\em average} of the Wigner function taken along the path of the PO. \nIn our case, the Bardeen cut at $x=x'=0$  means that we need to\nevaluate the Wigner function only at the starting and final points of the PO.}\nTo get the hard limit directly from the integration \\rf{integral2},\none neglects the quadratic variation of $S$ due to $\\Delta z$ \n, and neglects the variations of $e^{-\\beta \\bar{z}^2\/\\hbar}$ over the integral\n(i.e., one uses its value at the PO).\nThe integration of $e^{-\\beta \\Delta z^2\/(4 \\hbar)}$ with\nthe linear term in\n$S$ due to $\\Delta z$ is a Fourier transform, which gives the Wigner\npart $e^{-(\\bar{p}_z^0)^2\/(\\beta \\hbar)}$.\nThe integration over the variations of the action due to $\\bar{z}$\ngives the ${\\rm Tr} M-2$ prefactor, as in the GTF.\nAlternatively, one can work in phase space and \napply the SPA to \\rf{phspint}, neglecting the\nvariations of the Wigner function $W$ in the integral.\nNote finally that the hard limit formula for POs expresses in some\nsense the heuristic approach that was first used to interpret\nsemiclassically the current in the RTD, where one considered the\neffects of the stability of POs on the density of states\n given by the GTF, while taking\ninto account the accessibility of the PO to the tunneling electrons\n\\cit{From94,Mul95}.\n\n\t\\subsection{Minimal orbits (MOs)}\n\t\\label{miniorb}\n\nNarimanov and Stone \\cit{NS99} proposed the CCOs   in order to extend the PO formula to\nregions  where one has no real POs (also called ``ghost regions'', see\nsection \\ref{comp}),  \nwhile avoiding the problem of complex dynamics raised by SOs.\nThey also proposed an extension of the CCO formula in terms of\ntime-symmetric\norbits which had a minimal momentum transfer $\\Delta p_z= -2 p_z$, \ni.e., $\\partial(\\Delta p_z)\/\\partial z=0$.\nThe argument was that the Wigner transform in the PO formula\nhas a Gaussian damping\nthat kills the contribution of trajectories with $p_z$ that are not\nsmall.\nThis is the only proposed semiclassical formula for the RTD that we\nhave not tested by comparison with quantum calculations.\nInstead, we will propose and test another formula which is based on a similar\nidea.\n\nThe SPA method applied on \\rf{varphi} prescribes\nfinding an expansion point\nwhich makes the function $\\varphi_2(z,z')$ stationary.\nThis can be achieved if one can find a point $(z_0,z'_0)$ such that\nthe linear term $\\bfm{\\xi}^{\\rm T} \\bfm{\\gamma}$ in \\rf{lint}\nvanishes {\\em for all} $\\bfm{\\gamma} = (z-z_0,z'-z'_0)$, i.e.,\n$\\bfm{\\xi}(z_0,z'_0) = (-\\beta z_0 - i p_z^0, -\\beta z'_0 + i\np_{z'}^0) = (0,0)$.\nAs already mentioned, this requires complex trajectories (the SOs).\nThe idea here is to find the {\\em real} trajectory which \n{\\em minimizes} $\\bfm{\\xi}^{\\rm T}(z_0,z'_0)$ and therefore should\ngives the {\\em minimal linear term} in \\rf{varphi2}.\nThis is in some\nsense the best {\\em real approximation} of the complex saddle point.\nDefining\n\\begin{equation}\n{\\cal L}(z,z') = \\left| \\bfm{\\xi}^{\\rm T} \\bfm{\\xi}(z,z') \\right|^2 =\n\\beta^2 \\left[ z^2 + z'^2 \\right] + p_z^2 + p_{z'}^2 \\com{,} \n\\end{equation}\none will look for\n\\begin{eqnarray}\n0 &=& \\frac{1}{2} \\pder{\\cal L}{z} = \n\\beta^2 z - \\frac{p_z m_{11} + p_z'}{m_{12}}  \\\\\n0 &=& \\frac{1}{2} \\pder{\\cal L}{z'} = \n\\beta^2 z' + \\frac{p_z + p_z' m_{22}}{m_{12}}   \\com{.}\n\\end{eqnarray}\nThis prescription defines {\\bf minimal orbits (MOs)}:\n\\begin{equation} \\bom{MO}\n \\left\\{ \\begin{array}{ll}\n\\displaystyle & z'_0 = - m_{22}z_0 + \\frac{m_{21}}{\\beta^2} p_z^0  \\\\\n\\displaystyle & p_{z'}^0 = \\beta^2 m_{12}z_0 - m_{11} p_z^0 \\com{.}\n\\end{array} \\right. \n\t\t\\label{condmo}\n\\end{equation}\nThe contribution of MOs to the current will be given by using \\rf{gammagen}\nwith the $\\{z_0,p_z^0;z'_0,p_{z'}^0\\}$ of the MO.\nAgain, one finds that the most important MOs are TS.\n{\\bf TSMOs} have\n$p_z^0 = z_0 \\beta^2 \\frac{m_{12}}{m_{11}-1} = z_0 \\beta |\\delta|$.\nTheir contribution will be calculated with \\rf{gammasrco}.\n\n\t\\subsection{Summary of the formulae}\n\nFor the sake of completeness, we mention here the last possibility for\nneglecting one element in \\rf{condso}.\nOne considers the case of a Wigner transform that is very localized\nin $\\bar{z}$ and $\\bar{p}_z$.\nThis corresponds to  $\\beta_{\\bar{z}} \\to \\infty$ and \n$\\beta_{\\Delta z} \\to 0 $, which gives for \\rf{condsovar}:\n\\begin{equation} \\bom{AO}\n\\left\\{ \\begin{array}{ll}\n& \\bar{z}_0 =0  \\\\\n& \\bar{p}_z^0= 0 \\com{.} \\end{array} \\right. \n\t\t\\label{condao}\n\\end{equation}\nWe call such trajectories {\\bf average orbits (AOs)}, but shall\nnot write nor study their contribution.\nIt is interesting to note that for TSAOs, one has \n$z_0=z_0'=0$, which means that TSAOs are identical to TSCCOs.\n\nWe show in Fig. \\ref{conc} a schematic representation of the different \norbits and their related formulae.\nWe classify them according to the level of approximation of the SPA\nmethod used in the Gaussian integrations, that is: \n$(i)$ no approximation, which gives the saddle orbits (SOs)\nand the related formula eq. \\rf{gammaso}; \n$(ii$) approximation in the SPA condition (but none in the\nintegration), which gives the normal [NO, eq. \\rf{gammasrno}], \ncentral closed [CCO, eq. \\rf{gammacco}],\nperiodic [PO, eq. \\rf{gammapo}] or average (AO) orbits [and their\nrelated formulae];\n$(iii)$ approximation in both the SPA condition and integrations, \nwhich give the hard limit formula (HLPO, HLNO, HLCCO).\nThen we classify them according to the underlying hypotheses regarding \nthe predominance of the Green's function $\\hat{G}$ or the observable\n$\\hat{A}$ in determining the contributing trajectories.\nThis is linked to their relative smoothness in position or \nphase space.\nNote that the SOs correspond in this classification to the angle orbits found\nfor the conductance of microcavities \\cit{BJS93}, in the sense that\nboth types of orbits \nare derived without any approximation of the SPA condition.\n\n\\begin{figure}[htb!]\n\\centerline{\\psfig{figure=fig2.eps,angle=0.,height=8.cm,width=9cm}}\n\\vskip 1cm \\caption{Schematic representation of the different\nsemiclassical formulae.\nThe vertical axis describes the three different levels of\napproximation: one can neglect one function ($A$ or $G$) \nin the saddle point\ncondition [COND], also in the integrations [COND + INT], or in none [NONE].\nThe horizontal plane describes the relative localization\/oscillations\nlength scales of $\\hat{A}$ and $\\hat{G}$ (e.g., the left end means\nthat the oscillations of $\\hat{G}$  \nin position space are on a much smaller scale than the localization of\n$\\hat{A}$, etc.). \n}\n\t\t\\label{conc}\n\\end{figure}\n\n\n\\section{Comparison between semiclassical results\n  and  quantum calculations and experiments;analysis of results.}\n\\label{comp}\n\n\n\t\\subsection{Scaling  the classical dynamics}\n\nIn our comparisons between classical\/semiclassical\/quantal \ndynamics we  exploit an\n important property of the RTD : its Hamiltonian\n can be scaled with respect to the magnetic field \\cit{MD96}.\nThen, the classical dynamics  depends only on the ratio $\\epsilon=F\/B^2$ instead of\nthe independent values of $F$ and $B$ (the ratio $R=E\/V$ is roughly constant\nin the experiments) .\nThe experimental regime \\cit{Mul95} corresponds to the interval \n$1000 < \\epsilon <100000$. The classical dynamics in this range\nevolve from chaotic (low $\\epsilon$) to regular dynamics\n(high $\\epsilon$) \\cit{SM98}.\nIt is preferable  to scale the action not with respect to $B$, but\nrather with respect the action of the $ \\theta=0^\\circ, B=0, R \\gg 1$ problem: \n$S_0 \\simeq 2 L \\sqrt{2 m V (R+1\/2) }$.\nIn this integrable case, the number of oscillations of the wave\nfunction approximated by the WKB method \\cit{Gut90} is given by \n\\begin{equation}\n{\\cal N} := \\frac{S_0}{2 \\pi} = \\frac{L}{\\pi} \\sqrt{2 m V (R+1\/2)}\n\\com{,}\t\t\\label{nofv}\n\\end{equation}\nwhich we shall consider as a measure of the ``effective $\\hbar^{-1}$'' in \nthe general case as well.  \nWe define the scaled action by $\\hat{S}(\\epsilon) := S\/S_0$.\nThis definition is convenient as the three types of experimental\noscillations \\cit{Mul95} then correspond to trajectories with $\\hat{S}\n\\simeq 1, 2$ or $3$. We term these period-one, period-two and \nperiod-three trajectories respectively.\nAlso, $\\hat{S}$ depends only on $\\epsilon$, but is roughly constant as\n$\\epsilon$ varies.\nWe called the important period-one trajectories $t$ and the most important \nprimitive period-two trajectories $s$ \\cit{SM98}.\n\nWe can obtain the period of the voltage oscillations generated by a\ngiven trajectory.\nThe frequency of the oscillations of the semiclassical current \nis given by the imaginary part of the argument of the exponential in\neq. \\rf{generalformula}:\n$\\Sigma =  {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  S +  {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\: \\Gamma$.\nWe can also define its scaled version $\\hat{\\Sigma} = \\Sigma\/ 2 \\pi {\\cal N}$.\nThen one has two consecutive \nmaxima $\\{V, V+\\Delta V\\}$ in the current-voltage trace \n(neglecting the variation of $\\arg{\\cal D}$) when \n\\begin{equation}\n2 \\pi = \\Sigma(V+\\Delta V)- \\Sigma(V) = \\Delta [\\Sigma] = \n\\Delta \\left[2 \\pi \\hat{\\Sigma}(\\epsilon) {\\cal N}(V) \\right]\n= 2 \\pi \\hat{\\Sigma}\\frac{\\cal N}{2 V} \\Delta V\n\\quad \\Rightarrow \\quad\n\\Delta V = \\frac{2 V}{{\\cal N} \\hat{\\Sigma}}\n  \\com{.} \n\t\t\\label{2pi}\n\\end{equation}\nThis can be contrasted to the heuristic interpretation based on the\nDoS that was used before \\cit{From94,Mul95}, where the voltage oscillations\nwere directly obtained from the energy oscillations given by the GTF:\n$\\Delta V = \\Delta E\/ R = 2 \\pi \\hbar\/ R T$, where $T$ is the period\nof the contributing PO. \nThis relation is not exactly correct for two reasons: the current oscillations \nare not given by the action $S$ but by $\\Sigma$, and the period $T$\narises in the GTF from $\\partial S(E,B,V)\/\\partial E =T$ taken\nat constant $B$ and $V$, while in our case $V$ varies with $E$ \nthrough the constant $R$.\n\n\n\\subsection{The different types of orbits}\n\nBecause of the decoherence induced by phonon scattering, only the\nshortest trajectories contribute to the current in the RTD we analyzed\n(which has a width $L=120 \\:{\\rm nm} = 2267  \\:{\\rm a.u.} $).\nWe compare in Fig. \\ref{plot} the shape of the different types of\ncontributing  (all of the $t$-type).\nExamples of plots of the other important class of orbits,\nthe $s$-type orbits may be found elsewhere eg ~\\cite{SM98b,NS99}.\nIn (a) we present the traversing periodic orbit (PO) $t_0$, which\nmakes one bounce on each wall, and is responsible for the broad\nexperimental voltage oscillations \\cit{SM98,From94}.\nIt is perpendicular on the left wall ($p_z^0=0$), and is therefore\ntime-symmetric (TS) and also a TS normal orbit (NO).\nThere are two period-two POs, born in two pitchfork bifurcations\naround $\\epsilon=13000$.\n$t_v$ is self-retracing but non-perpendicular;\nhence it is not TS: reversing the momentum\nat the end of the trajectory (on the left wall), \none does not find oneself on the portion of \nthe trajectory on which the orbit started.\n$t_u$ is TS and therefore also a TS NO; there is also another non-TS\nNO hidden: it is ``half'' the PO  $( z_0 = -80 \\to z_0' =550)$ and\nwill be\ndenoted by $t_u\\!-\\!\\! NO$.\n\n\nAs $\\epsilon$ decreases towards the chaotic regime, $t_0$ disappears with\nan unstable partner $t_0^-$ in a tangent bifurcation at $\\epsilon=6500$.\nThey are replaced by a pair of POs which are complex conjugates of each other; \nthe one giving a physical contribution is called ``ghost''\nand is denoted by $t_{\\rm gh}$.\nAt $\\epsilon=3000$ (b), a new real PO $t_1$ has appeared.\nWe also show the saddle orbit (SO) and minimal orbit (MO) that are\nrelated to the $t$-type trajectories; they do not disappear in any\nbifurcations and are linked to all the three POs ($t_0, t_{\\rm gh}$) and \n$t_1$ as $\\epsilon$ decreases.\nIn this region the SO and MO are between $t_{\\rm gh}$ and $t_1$.\n\nIn (c) we illustrate the non-periodicity of SOs.\nWe show $t_0 \\!-\\!\\! SO$, related to the primitive PO $t_0$, and the very\ndifferent $2t_0 \\!-\\!\\! SO$, that is related to the second repetition of\nthe PO.\nBecause SOs (as well as MOs and CCOs) are not periodic, one cannot continue\nthe propagation of a primitive orbit, but one has to look for another\norbit with the adequate action, period and number of bounces so that it \ncorresponds to the repetition of a primitive PO.\nNote that in some cases (e.g., $t_0\\!-\\!\\! SO$ at $ \\theta=27^\\circ, \\epsilon < 17000$),\none cannot find an SO corresponding to the second repetition of a PO,\nalthough one has the SO linked to the primitive PO.\n\n\\begin{figure}[htb!]\n\\centerline{\\psfig{figure=fig3.eps,angle=270.,height=8.cm,width=12.cm}}\n\\vskip 5mm \\caption{\nShape of different types of orbits of the $t$-type\nat $ \\theta=11^\\circ$.\nThe right wall is at $x= L = 120 \\: {\\rm nm} = 2267  \\:{\\rm a.u.}  $.\nFor complex trajectories, we show the real part $( {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  x,  {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  z)$.\n(a) $\\epsilon=14000$; the periodic orbit (PO) \n$t_v$ is self-retracing but not time-symmetric (TS), as\nit not perpendicular on the left wall ($p_z^0 \\neq 0$); \n$t_0$ and $t_u$ are TS POs and therefore also TS NOs;\nthe ``half PO'' $t_u \\!-\\!\\! NO$ $( z_0 = -80 \\to z_0' =550)$ is a\nnon-TS normal orbit (NO).\n(b) $\\epsilon=3000$; here we have the complex ghost $t_{\\rm gh}$ which has \nappeared in the tangent bifurcation of $t_0\\!-\\!\\! PO$ at $\\epsilon=6500$;\nwe show the related saddle orbit $t_0\\!-\\!\\! SO$ and minimal\norbit $t_0 \\!-\\!\\! MO$;\nwe also have a real PO $t_1$.\n(c) $\\epsilon=18000$; we show the saddle orbit $t_0 \\!-\\!\\! SO$ corresponding\nto the primitive $t_0\\!-\\!\\! PO$, as well as the saddle orbit $2t_0\\!-\\!\\! SO$ \ncorresponding to the second repetition of the PO.\n(d) $\\epsilon=10000$; the central closed orbit $t_0\\!-\\!\\! CCO$ is defined by $z_0=0$. \n}\n\\label{plot}\n\\end{figure}\nAn example of a central closed orbit (CCO) is shown in (d): \nit looks very different from the related PO.\n\n\n\t\\subsection{  $ \\theta=11^\\circ$ theory and experiments: ghost regions and torus\nquantization} \n\nThe method used to compare semiclassical, quantum and experimental\nresults was explained by Saraga and Monteiro \\cit{SM98}.\nFor each $\\epsilon$, we generate a scaled quantum (QM) and semiclassical\ncurrent that oscillates with ${\\cal N}$, in the range \n$12 < {\\cal N} <42 $ ,corresponding to the experimental $B$ range read at constant $V=0.5\n \\:{\\rm V} $.\nWe Fourier transform the current with respect to the pair ${\\cal N}, \\hat{S}$\nand get a power spectrum that has peaks at certain values of the action.\nThe height of the peaks gives the amplitude of the oscillation,\nwhile their position indicates their type: period-one oscillations\nwhen $\\hat{S} \\simeq 1$, period-two when $\\hat{S} \\simeq 2$, etc.\nFor the experiments, we  read the amplitude of the oscillations \ndirectly from $I-V$ traces provided by G Boebinger \\cit{Mul95}, \nthat we analyzed and presented in \\cit{SM98}. \nWe correct the experimental $\\epsilon$ by $30 \\%$ to take into account\neffect of the voltage dependence of the mass. \nTo allow for damping due to phonon scattering, we scale  the\namplitudes by $\\exp[T\/\\tau]$, where $T$ is the (real part of the) \ntotal time of\nthe contributing classical orbit and $\\tau \\sim 0.1\\:{\\rm ps}$ \\cit{From94}\nis a decoherence\ntime. We use here the value $\\tau \\simeq 0.115 \\:{\\rm ps}$ \nestimated by comparing the maximal values of the \nquantum and experimental amplitudes. \nWe normalize all amplitudes to the\namplitudes at $ \\theta=0^\\circ$, where for the semiclassics we have $ \\Gamma =0$ and \n$|{\\cal D} ( \\theta=0^\\circ)|= 2 B, \\forall \\epsilon$.\n\n\nFig. \\ref{amp11} presents the amplitudes of the different\nsemiclassical formulae at $ \\theta=11^\\circ$.\nHere we study period-one oscillations, which correspond to the broad\nvoltage oscillations seen in the experiments\\cit{Mul95} and called\n``$t$'' series \\cit{From94}.\nThey arise from trajectories making one bounce on each wall;\n( the $t_0$ orbit at high $\\epsilon$ and $t_1$ at low $\\epsilon$).\n\nIn (a) we see that the quantum calculation based on the Bardeen model\n reproduces quantitatively the experimental behaviour, over a\nlarge range of parameters (corresponding to $3  \\:{\\rm T}  < B < 12  \\:{\\rm T} $).\nWe did not read experimental amplitudes for $\\epsilon>25000$, because of\nthe presence of period-two oscillations.\n\n\\begin{figure}[htb!]\n\\centerline{\\psfig{figure=fig4.eps,angle=270.,height=10.cm,width=14.cm}}\n\\vskip 5mm \\caption\n{Amplitude of the semiclassical formulae for period-one oscillations\nat $ \\theta=11^\\circ$, compared with quantum mechanical calculations [QM, dotted\nline with squares] as the dynamical parameter $\\epsilon$ varies.\n(a) Comparison between experimental results [EXP] and quantum\nmechanical results [QM].\n(b)Quantal results and semiclassical results for the PO\/NO formula, \nto which three POs contribute: $t_0$, the complex\nghost $t_{\\rm gh}$ and $t_1$.\nWe also show the contribution of $t_u\\!-\\!\\! NO$, which is not a PO.\n(c) Quantal results and the semiclassical CCO formula.\n(d) Quantal results and the semiclassical SO and  MO formulae.\nThe figure shows that while both the SO and the MO formulae\ngive good agreement over the whole range, the PO\/NO\/CCO formulae\ngive agreement only over a partial range.}\n\\label{amp11}\n\\end{figure}\n\nThe periodic orbit theory is compared to quantum calculations in (b).\nFor $t_0, t_{\\rm gh}$ and $t_1$ we use the PO\/NO formula\n\\rf{gammasrno}, which is the common formula given by POs and NOs.\nThe semiclassical formula is accurate in both the chaotic\n($\\epsilon<3000$) and regular ($\\epsilon>10000$) regions.\nIn the latter, the semiclassical contribution can be understood by\nMiller torus quantization \\cit{Mil75}.\nThe large stable island of $t_0$ supports quantum states that are\napproximately harmonic oscillators (HO) functions in the plane\nperpendicular to the orbit; this will be discussed in more detail below.\nWe also see that the contribution of $t_u\\!-\\!\\! NO$ to the NO formula\nseems unrelated to the quantum behavior.\nNote the spike at $\\epsilon=6500$; this corresponds to the tangent\nbifurcation where $t_0$ and $t_0^-$ disappear.\nIt is {\\em not} a divergence, as the complex ${\\cal D}$ does not vanish.\nThe spike is due to the rapid variation of $\\Gamma$ near the bifurcation.\n\n\nThe most interesting region is $3000 < \\epsilon <6500$, where there is no\nreal contributing PO (the ``ghost'' region).\nWe included the contribution of the complex ghost PO $t_{\\rm gh}$, but \nwe see that its contribution is too small \\cit{SMR98}.\nA detailed view is shown in Fig. \\ref{det11} (a).\nThere we see that the ghost contribution is too small by roughly a\nfactor three compared to the QM results.\nThe saddle orbit (SO), on the other hand, describes accurately the QM\namplitudes all across the tangent bifurcation, the ghost region and\nthe region where $t_1\\!-\\!\\! PO$ takes over [see also Fig. \\ref{amp11} (d)].\nFinally, we see in Fig. \\ref{det11} (a) that the unstable partners of\nthe tangent bifurcations $t_0^-$ and $t_1^-$ do not contribute to the\ncurrent.\nThis is a general feature that we observed at any angle $\\theta$: only a\nvery small subset of trajectories give a significant semiclassical\ncontribution.\nA study of the amplitudes of period-two oscillation (not shown here) shows \nthat other POs like $t_v$ and $t_u$ are not related to the QM results\nalthough their contribution to the PO\/NO formula is significant.\n\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig5a.eps,angle=270.,height=5.cm,width=7.cm}\n\\psfig{figure=fig5b.eps,angle=270.,height=5.cm,width=7.cm}}\n\\vskip 5mm \\caption\n{\n(a) Details of the period-one amplitudes at $ \\theta=11^\\circ$ in the \n low $\\epsilon$ region.\nThis is the region where no real PO exist, and where the contribution\nfrom the complex ghost PO $t_{\\rm gh}$\nis too small, while the SO contribution is\naccurate and joins up the contribution from the POs $t_0$ and $t_1$.\nWe also show the contribution of the unstable POs $t_0^-$ and $t_1^-$, \nwhich are not seen in the QM behaviour.\n(b) Comparison of experimental voltage periods (line with\ncrosses) with the semiclassical period\ngenerated by the SO $t_0$ (solid line).\n}\n\\label{det11}\n\\end{figure}\n\nThe contributions of central closed orbits (CCOs) are shown in\nFig. \\ref{amp11} (c).\nThe main objective of the CCO theory as presented by Narimanov and Stone \\cit{NS99} was to\ncomplement their PO theory in the absence of real PO (the ``ghost''\nregion).\nIt partially succeeds, as its amplitude for $3000< \\epsilon < 5000$\ncorresponds to the quantum one.\nHowever, it is clearly inaccurate for $\\epsilon>5000$ and gets worse in the \nregular region, where one could have expected the \nassumptions underlying the theory to be respected (in this regular\nregime, the oscillations of the Green's function should be smooth\ncompared to the localization of the initial state).\nSimilarly, the CCO theory is not very accurate in the chaotic region\n(low $\\epsilon$).\n\nWe show in (d) the result of the saddle (SO) and\nminimal orbit (MO) formulae.\nThere is only one SO and one MO corresponding to the three POs $t_0,\nt_{\\rm gh}$ and $t_1$.\nBoth theories are very accurate and reproduce the quantum amplitudes\nacross the whole transition from regular to chaotic dynamics.\nActually, the MO contribution is even more precise than the SO at very \nlow $\\epsilon$.\n\nFinally, we study in Fig. \\ref{det11} (b) the frequencies of the\noscillations via their voltage period $\\Delta V$.\nWe show the semiclassical period calculated with eq. \\rf{2pi} from the \nsaddle orbit $t_0$.\nWe do not show quantum periods, which are accurately described by the\nsemiclassics.\nThe theoretical periods underestimate the experimental values by some $10 \n\\%$.\nThis however is a confirmation of the fact that $t_0$ orbits are indeed\nlinked to the broad experimental oscillations.\n\n\nTorus quantization is illustrated in Fig. \\ref{torquant}, where we\npresent Wigner and wave functions of quantum states\ncontributing to the current.\nThe wave functions are approximately separable into a HO\n state and a WKB-type wave function along the trajectory.\n${\\cal N}_i$ gives roughly the number of longitudinal oscillations,\nwhile the number of perpendicular oscillations corresponds to the\npseudo quantum number $k$ of the HO state.\nOne can use this assumption to build a current as the overlap between\nthe initial state and the HO state \\cit{SM98}.\nThis is valid for stable POs, and it has been shown to be equivalent to \nthe PO \\cit{NSB98} and NO \\cit{BR98} formula in the case of\ntime-symmetric orbits.\nThe Wigner distributions show the ring structure associated with HO states.\n\n\n\n\\begin{figure}[tb!]\n\\centerline{\n\\psfig{figure=fig6a.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6b.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6c.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6d.eps,angle=0.,height=4.4cm,width=3.752cm}\n}\n\\vspace{5mm}\n\\centerline{\n\\psfig{figure=fig6e.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6f.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6g.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6h.eps,angle=0.,height=4.3cm,width=3.75cm}\n}\n\\vskip 5mm \\caption{\nQuantum state contributing to the current \nin the torus quantization regime at $ \\theta=11^\\circ, \\epsilon=15000$, labelled by\ntheir eigenvalue ${\\cal N}_i$ . \nFor the Wigner distributions on the emitter barrier (top row),\n the vertical axis is $z$ and the horizontal axis is $-p_z$; \nthe range is adapted to the size of the (classically allowed) surface\nof section. \nFor the wave functions (bottom rows) the vertical axis is \n$x \\in [0,L=2267]  \\:{\\rm a.u.} $ and the horizontal axis is $-z \\in [-2000,2000]  \\:{\\rm a.u.} $.\nFor ${\\cal N}_i=20.894$ we also show in solid lines the classical\nstructures: $t_0$ for the wave function, and the main features of\nthe Poincar\\'e surface of section (points representing trajectories\nhitting the left wall)\nfor the Wigner distribution.\nIn the first two rows, the torus numbers are (left to right):\n$k=1,3,2$ and $0$.\n}\n\\label{torquant}\n\\newpage\n\\end{figure}\n\n\nThe hard limit formulae for POs and NOs (not shown here; see\n\\cit{SM98} for a test of the HLNO formula)\ngreatly underestimate the\ncontribution of off-center POs, because they cannot take into account torus \nquantization, which gives some accessibility to the PO via outer tori\nwhich can extend to the $z=0$ region at the center of the initial\nstate.\nFinally, torus quantization effects can explain ``jumps'' in the\nexperimental current, when the dominant torus changes with the\nmagnetic field \\cit{SM98}.\n\t\t\\subsection{Comparison at $ \\theta=27^\\circ$: non-isolated orbits}\n\nWe present in Fig. \\ref{amp27} \namplitudes of period-two oscillations at $ \\theta=27^\\circ$ (they were called\n``peak-doubling'' regions by M\\\"uller \\al \\cit{Mul95} as there is a secondary\nvoltage oscillation compared to the broad period-one oscillation).\nThe quantum model describes qualitatively the rather broad\nexperimental period-two signal (a). \nHowever, it overestimates it by some $20 \\%$.\nThis would be consistent with uncertainties in the\nestimate of the decoherence time. \n$\\tau$.\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig7.eps,angle=270.,height=10.cm,width=14.cm}}\n\\vskip 5mm \\caption\n{Amplitude of the semiclassical formulae for period-two oscillations\nat $ \\theta=27^\\circ$, compared with quantum mechanical calculations [QM, dotted\nline with squares] as the dynamical parameter $\\epsilon$ varies.\n(a) Experimental results [EXP]; they have been multiplied by\n$\\exp(T\/\\tau)$, where $T$ is the (real part of the) total time of\nthe contributing classical orbit and $\\tau \\simeq 0.115 \\:{\\rm ps}$ \nis a decoherence time associated with phonon scattering. \n(b) PO\/NO formula,\nto which three POs contribute: $s'$, $2t_0$ and $s_1$; \nnote the gap $3500 < \\epsilon <7700$ (``ghost'' region)\n between the take over of $s''$ and\nthe tangent bifurcation where $s'$ appears ($s'$ disappears at\n$\\epsilon=18000$ in a ``cusp bifurcation''.\n(c) CCO formula; note the extension of the semiclassical amplitude\ndown to $\\epsilon=4400$ in the ghost region, and the inaccurate\ncontribution of $2t_0$.\n(d) SO and MO formula; note how both formulae describe accurately the\nquantum amplitude from regular to chaotic across the ghost region\n(there is only one SO\/MO related to both POs $s'$ and $s''$).\n}\n\\label{amp27}\n\\end{figure}\n\nTwo types of orbit contribute to the period-two current, which have\nroughly double action and period than $t$-type orbits.\nFirst, we have the second repetition of $t$, that is a $2\\!\\!:\\!\\! 2$ orbit\nmaking two bounces on each barrier.\nSecondly, there are orbits of the $s$-type $(1\\!\\!:\\!\\! 2$), making one\nbounce on the left wall and two bounces on the right wall, with a\nturning point (where the particle runs out of kinetic energy)\nin-between.\n\nFirst we test the PO\/NO formula in (b).\nThe peak of the contribution of $2t_0$ corresponds to two successive\nperiod-doubling pitchfork bifurcations ($\\epsilon=12600$ and $14000$) of\nthe primitive PO $t_0$.\nThis peak can be understood via torus quantization effects: at the\nbifurcations,\nthe winding angle of the stable $t_0$ reaches the value $\\pi$; hence\ntwo period-one torus series corresponding to two successive $k$ numbers become\nexactly $\\pi$ out-of-phase and create a strong period-two signal.\n\nThere is a very large contribution from another PO ($s'$) in the same\nregion; it also describes qualitatively well the\nquantum behavior, as does $2t_0$.\nBoth POs  give a current with very similar scaled frequencies:\n$\\hat{\\Sigma} \\simeq 1.9054 $ \nfor $s'$ and $\\hat{\\Sigma} \\simeq 1.9154$ for $2t_0$.\nHence, one should not consider their independent amplitudes, as is\ndone in Fig. \\ref{amp27} (b), but instead consider the coherent superposition\nof their oscillatory current.\nThis will be done below in Fig. \\ref{det27}; however it seems that\nthese two competing contributions cannot be easily separated.\n\n$s'$ appears at $\\epsilon=18000$ in a ``cusp bifurcation'' \\cit{NS98b}, due to \nthe discontinuity of the hard bounce on the left wall (increasing\n$\\epsilon$ makes the trajectory hit the left wall instead of having a\nturning point on the energy surface).\nThen it\nundergoes a synchronous pitchfork bifurcation at $\\epsilon=13600$, with\nthe non self-retracing PO $s_*$; there is no \neffect on the semiclassical current.\nFinally it disappears at $\\epsilon=7700$ in a tangent bifurcation, below\nwhich there is no real PO able to explain the quantum and experimental \nsignal until $\\epsilon=4000$, where another PO of the same type ($s''$)\ngives a significant contribution.\nHence one has another ``ghost'' region between  $\\epsilon=4000$ and $7700$.\nThe low $\\epsilon$ quantum peak ($\\epsilon \\simeq 2000$) is well described by\n$s_1$ (a $1\\!\\!:\\!\\! 2$ PO which has one more cyclotron rotation than $s'$\nand $s''$).\nNote that the large spikes are not divergences.\n\nThe central closed orbit (CCO) formula is shown in Fig. \\ref{amp27}\n(c).\nThe low $\\epsilon$ is well described by the CCO $s_1$, with no spike\nas the one found in the PO\/NO formula.\nThe $s'$ contribution to the high $\\epsilon$ peak is significantly\nextended by the CCO formula, and\nfollows accurately the quantum amplitude down to $\\epsilon=4700$, where it has an\nunphysical spike.\nThe CCO theory also separates the contribution of $s'$ and $2t_0$, as\nthe latter appears where the former disappears, at $\\epsilon=21000$.\nHowever, the contribution of $2t_0$ is inaccurate.\n\nFinally, we show the saddle and minimal orbit formulae in (d).\nFor both theories, $s'$ describes accurately the quantum results all the way\nthrough the ghost region down to $\\epsilon=4000$, where there is no such spike\nas the one found with the CCO.\nThey also separate $2t_0$ and $s'$ in an accurate way, as $2t_0$\ndescribes precisely the quantum results for high $\\epsilon$ too.\nAs for $ \\theta=11^\\circ$, the SOs and MOs provide the best semiclassical\ndescription.\nWhile the success of the SOs is expected (as they are the correct\nsaddle point of the SPA integrations), the efficiency of the MOs is\nagain rather surprising.\nNote also the consequences of the non-periodicity of SOs, MOs and\nCCOs, for which the $2t_0$ orbit only appears around $\\epsilon \\simeq\n20000$, while the $t_0$ orbits (not shown here) exist at lower $\\epsilon$.\n\n\nBoth POs $2t_0$ and $s'$ give important contributions for the peak of the\nperiod-two signal at $ \\theta=27^\\circ$.\nTo build the coherent superposition\nof their oscillatory current, one needs to take into account the constant phase\n$(- i\\mu \\pi\/2)$ given by the number $\\mu$ of conjugate points [shown in\nFig. \\ref{det27} (a)].\nThe amplitude of their\ncollective contribution is given by the height of the peak of the Fourier\ntransform of the current, and is shown in Fig. \\ref{det27} (b).\n\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig8a.eps,angle=270.,height=5.cm,width=7.cm}\n\\psfig{figure=fig8b.eps,angle=270.,height=5.cm,width=7.cm}}\n\\vskip 5mm \\caption\n{\n(a) Maslov index and (b) coherent superposition\n(``$2t_0+s'$'',solid line with dots, )\nof the $2t_0$  and $s'$  contributions. \nWe also show the contribution of the non self-retracing $s_*$ PO, and the \nQM results. \n(c) Experimental (line with crosses) and semiclassical (solid line) \nvoltage periods.\n}\n\\label{det27}\n\\end{figure}\n\nThe coherent superposition [$2t_0+s'$] is much larger than the\nquantum amplitudes, because the individual isolated  contributions\nare already as high as the QM results.\nNote the discontinuous change at $\\epsilon=13600$; it is due to the\ndiscontinuous change of $\\mu$ for $s'$ at the pitchfork bifurcation.\nIt is clear that the coherent superposition of $2t_0$ and $s'$ cannot, \nwhatever their relative phases, describe accurately the quantum\nresults.\nNote that these POs are {\\em not} involved together in a bifurcation;\nthis is not the usual breakdown of semiclassics near a bifurcation,\nthat one could solve with the use of normal forms (cubic expansions of \nthe action).\nAlso, $2t_0$ and $s'$ seem to be well separated in position space\n(their starting \npositions are $z_0 \\simeq 0$ for $s'$ and $z_0 \\simeq 600$ for\n$2t_0$).\n\nLooking at quantum states contributing to the current reveals that\nthe two POs are, in some sense, not isolated.\nA Wigner distribution and the related wave function are shown in \nFig. \\ref{noniso-27}.\n\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig9a.eps,angle=0.,height=6.cm,width=6.cm}\n\\psfig{figure=fig9b.eps,angle=0.,height=6.cm,width=6.cm}\n}\n\\vskip 5mm \\caption{ \nLeft: Wigner distribution (color plot) at $ \\theta=27^\\circ,\n\\epsilon=16000$ and classical Poincar\\'e surface of section (dots), in the\n$(-p_z,z)$  plane on the emitter wall.\nThe square on lower-right corner represents $\\hbar$.\nRight: corresponding wave function in the $(-z,x)$ plane, \nwith the POs $t_0$ and $s'$.\nThe scales are the same as in Fig. \\ref{torquant}.}\n\\label{noniso-27}\n\\end{figure}\n\nThe Wigner distribution has the familiar ring structure of a $k=2$ quantized \ntorus in the stable island surrounding $t_0$.\nThe ring is nevertheless distorted in some way and is also localized\non the (here stable) PO  $s'$.\nSimilarly, both POs are within the region of the\nlocalization of the wave function in position space.\nWe can conclude that the quantum state cannot ``distinguish''\nthe two POs, and that the POs are hence\nnon-isolated: they contribute collectively to the quantum state and to \nthe current. The use of either SO or MO orbits, however, circumvents \nthis problem and yields good results throughout.\n\t\t\\subsection{Comparison at $ \\theta=20^\\circ$: divergence of the\nsaddle orbit formula} \n\nFig. \\ref{amp20} presents amplitudes for the period-two signal at $ \\theta=20^\\circ$.\nThe situation is similar to $ \\theta=27^\\circ$; we do not present here the CCO formula.\nThe low $\\epsilon$ quantum peak can be described rather well by the PO\n(a), SO (b) or MO (c) formulae.\n\nThe quantum model can describe qualitatively the shape of the\nexperimental peak at high $\\epsilon$ (c), over a large range of\nparameters ($10000 < \\epsilon <40000$).\nThe $15 \\%$ discrepancy is probably due to a small inaccuracy in the\nestimate of the decoherence time $\\tau$.\nSemiclassically, we have the same two competing orbits as at $ \\theta=27^\\circ$.\nAlthough its contribution is important, it seems that $s'$ does not\ninfluence the quantum amplitudes.\nNote the difference between the $s'$ contribution given by the PO\/NO\nformula from the one given by the SO and MO formulae.\n\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig10.eps,angle=270.,height=10.cm,width=14.cm}}\n\\vskip 5mm \\caption\n{The different semiclassical theories for P2 amplitudes at $ \\theta=20^\\circ$.\nThe broad maximum is related to $2t_0$ and $s'$ orbits,\nwhile the lower maximum is given by $s_1$ orbits.\nWe show QM results and experimental readings [EXP].\n(a) PO\/NO formula.\n(b) SO formula.\nThe inset is in a larger scale.\n(c) MO formula and experimental results.\n}\n\\label{amp20}\n\\end{figure}\nThe contribution of $2t_0$ to the PO\/NO formula is good, but the\nposition of the peak is not very accurate.\nThe SO formula yields unexpected results: it has a very large peak (see \ninset), which we shall investigate below.\nThe MO formula gives the correct position for the peak, but the height \nis not as\nprecise as could be expected.\n\n\n\n\nWe investigate in Fig. \\ref{det20} the large spike of the amplitude of\nthe contribution of $2t_0\\!-\\!\\! SO$ to the saddle orbit formula.\nWe see in (a) that the reason for this is the fact that the\ndeterminant $\\cal D$ [eq. \\rf{cald}] of the quadratic expansion used\nin the SPA integration almost vanishes around $\\epsilon =17500 - 17800$.\nBoth its real (solid line) and imaginary (dashed line)\npart simultaneously approach zero.\nThis is a remarkable  coincidence, as $\\cal D$ is a complex\nfunction, that one expects to vanish only if one can vary a parameter in the \n{\\em complex} plane (i.e., two real parameters).\nIn this case, varying $\\epsilon$ only on the real axis approaches very\nclosely the zero of $\\cal D$, which should be reached for\na value of $\\epsilon$ with a small imaginary component.\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig11.eps,angle=270.,height=8.cm,width=12.cm}}\n\\vskip 5mm \\caption\n{\nClassical characteristics of the SO $2t_0$ around the spike at $\\epsilon\n\\sim 17000-18000,  \\theta=20^\\circ$. The solid line indicates the real component\nand the dashed line indicates the imaginary component.\n(a) Scaled determinant $\\tilde{\\cal D}={\\cal D}\/B$ [eq. \\rf{cald}] \nof the quadratic\nexpansion used in the SPA integration.\n(b) Scaled action $\\hat{S}$.\n(c) Starting position $z_0$.\n(d) Real shape $( {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  x,  {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  z)$ of $2t_0 \\!-\\!\\! SO$ at $\\epsilon=17476$;\n for fig. (d) the dashed line indicates\nthe limit of the region allowed by ({\\em real dynamics}) \nenergy conservation. \n}\n\\label{det20}\n\\end{figure}\n\nThe classical characteristics of $2t_0\\!-\\!\\! SO$ in that region are not\nsmooth.\nThe real part of the action (b) reaches a maximum\nvalue at $\\epsilon=17588$, while the imaginary part changes abruptly over\nthe range $17000 < \\epsilon <18000$.\nThe imaginary part of the starting position (c) behaves similarly.\nThe real part of the starting position reaches the minimum value\n$ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  z_0=-89  \\:{\\rm a.u.} $ at $\\epsilon=17476$.\n\nWe show in (d) the real shape of $2t_0\\!-\\!\\! SO$ at $\\epsilon=17476$.\nThe outer leg (with high $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  z$) hits the left wall perpendicularly \nat a point which is very close (less than $20  \\:{\\rm a.u.} $) to the limit\nsurface of\nthe region accessible by trajectories defined by real dynamics.\nThis limit is where some self-retracing trajectories (like $s'$ and\n$s_1$) have a turning point (i.e., have zero momentum). \nIt is impossible to find trajectories with the same bounce structure\n($2 \\!\\!:\\!\\! 2$) for $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  z_0 < -100$, because they would miss the\nintermediate bounce and go back to the right wall via the limit surface.\nThis is similar to the ``cusp bifurcation'' of the PO $s'$; it is also \nthe only observed mechanism that can remove an SO, MO or CCO as $\\epsilon$ \nchanges.\n\nAs $\\epsilon$ increases, $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  z_0$ decreases until it reaches $-89  \\:{\\rm a.u.} $.\nIf it evolved smoothly, one would expect the SO at higher $\\epsilon$ to\nstart with a lower $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  z_0$ --which seems impossible as we have\nseen that such starting condition did not allow the correct bounce\nstructure.\nHence, one would expect ``a cusp disappearance'' of $2t_0\\!-\\!\\! SO$.\nThis nevertheless does not happen: one can still find the SO for\nhigher $\\epsilon$, as abruptly increasing $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\:  z_0$ satisfies the SO\ncondition.\nHence, it seems that we have a ``failed'' cusp disappearance of $2t_0\\!-\\!\\! \nSO$, precisely in the region where the quadratic expansion becomes\nalmost degenerate and where the classical characteristics of the orbit \nare not smooth.\n\nThis is altogether reminiscent of a bifurcation of periodic orbits,\nwhere two POs coalesce as the dynamical parameter ($\\epsilon$) is varied, \nand where the quadratic expansion of the action (used e.g. in the GTF) \nbecomes degenerate.\nHowever, PO bifurcations always involve more than one POs, while in\nthis case we have not observed any neighboring SO that could coalesce \nwith $2t_0\\!-\\!\\! SO$.\n\nNote that, strictly, one should not integrate \\rf{integral2} on the\nwhole $(z,z')$ plane, but only on the domain $\\Omega$\n where one does find the proper type of trajectories ($2\\!\\!:\\!\\! 2$).\nHence one should cut the integral for $z < -100$; this would yield\nError functions. \n\nThis situation raises very interesting questions about saddle\norbits.\n{\\em Do they undergo a type of bifurcation?} Apparently no, as a\nbifurcation should involve several SOs, which we have not seen.\n{\\em What is the origin the quasi-degeneracy?}\nIt comes from a failed cusp disappearance. \n{\\em Does it has an effect on the semiclassical current?}\nYes, the current shows a strong enhancement.\n{\\em Do quantum results show any sign of it?} Apparently no, as\nthe QM results are smooth in that region.\n{\\em What techniques can be employed to solve that problem?}\nWe tried a cubic expansion of the action, which removed most of the\nenhancement but did not show good agreement with quantum calculations. \nOne could try a cut-off in the integral.\nThe first task would be to locate the complex value of $\\epsilon$ for which\n${\\cal D} =0$.\n\n\n\n\t\t\\section{Conclusion}\n\nThe general semiclassical formula \\rf{generalformula} that we have\ndeveloped summarizes in a compact way all the theories that have been\nproposed for the current in the RTD (excluding the changes required by\nthe inclusion of excited Landau states and a shift of the Gaussian\n\\cit{NSB98,Sara99}).\nIt also shows clearly how the different assumptions on the smoothness\nof the Gaussian affect which type of orbits contribute to the current.\nWe found the types giving the best semiclassical description: the\ncomplex saddle orbits (SOs) and the real minimal orbits (MOs).\nIt appears that the more standard periodic or closed orbits do not\nsucceed in all the situations, in particular in the ghost regions and\nwhere one has non-isolated contributions.\n\nThe (near) divergence of the saddle orbit formula at $ \\theta=0^\\circ$ raises interesting\nquestions about the SOs, namely the existence of\na bifurcation-type phenomenon for trajectories defined by a pair of\ncondition \\rf{condso} as restrictive as the one defining POs (i.e.,\ngiving a discrete number of orbits).\nThe complex dynamics we implemented here\nare very strongly restricted by the definition of the hard\nbounces on the barriers.\nOne could obviously avoid this point by\nmodeling the barriers by soft exponential walls.\nA preliminary study shows that the ``cusp'' bifurcations are replaced\nby standard tangent bifurcations; however the search for complex POs\nand SOs with soft barriers appears to be more difficult and again\nraises interesting questions about the high-dimensional search for\ncomplex POs and SOs.\n\nThe techniques used in this work could  easily be applied to other\nsystems involving Gaussian matrix elements, in particular Franck-Condon\ntransitions and the conductance of microcavities with parabolic leads\n(where one further assumes that the lowest sub-band is occupied \\cit{Nari98}).\nThe importance of saddle or minimal orbits would then depend on the\nlocalization scale of the Gaussian.\n\nThe authors would like to thank E Bogomolny and D Rouben for invaluable\nhelp with the semiclassics and the complex dynamics, E Narimanov for\ncommunication of results prior to publication, and G Boebinger\nfor unpublished experimental data.\nT S M acknowledges financial support from the EPSRC. D S S was\nsupported by a TMR scholarship from the Swiss National Science Foundation.\n\n\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nGiven a multiset $M$ of $n$ positive integers, a word on $M$ is a\nsequence of positive integers $w = w_1w_2 \\cdots w_n$ that reorders\n$M$. A {\\em statistic on words} is an association of an element of\n$\\mathbb{N}$ to each word. A fundamental statistic that has been\nrediscovered in many guises is the {\\em inversion number} of a word,\ndefined as the number of pairs of indices $(i<j)$ such that $w_i >\nw_j$. A {\\em descent} of a word is an index $i$ such that $w_i >\nw_{i+1}$. In 1913, Major P. MacMahon \\cite{MacMahon1913} introduced an\nimportant statistic, now called the {\\em major index} in his honor,\ndefined as the sum over the descents of a word. Using generating\nfunctions, MacMahon \\cite{MacMahon1916} proved the remarkable fact\nthat the major index has the same distribution as the inversion\nnumber. Precisely, he showed that for $W_M$ the set of words on a\nfixed multiset $M$,\n\\begin{displaymath}\n  \\sum_{w \\in W_M} q^{\\ensuremath\\mathrm{maj}(w)} = \\sum_{w \\in W_M} q^{\\ensuremath\\mathrm{inv}(w)},\n\\end{displaymath}\nwhere $\\ensuremath\\mathrm{maj}(w)$ denotes the major index of $w$ and $\\ensuremath\\mathrm{inv}(w)$ denotes\nthe inversion number of $w$. Any statistic that is equidistributed\nwith the major index, i.e. a statistic satisfying the above equation,\nis called {\\em Mahonian}. MacMahon then raised the question to find a\nbijective proof that the inversion number is Mahonian. This question\nwas first resolved by Foata \\cite{Foata1968}, who constructed a\nbijection on words with the property that the major index of a word\nequals the inversion number of its image.\n\nIn this paper, we introduce a statistic called the $k$-major index\nwhich interpolates between the major index and inversion number. More\nprecisely, the $1$-major index is MacMahon's major index, and the\n$n$-major index of a word of length $n$ is the inversion number. By\nconstructing bijections on words with a recursive structure similar to\nFoata's bijection, we give a bijective proof that the $k$-major index\nis Mahonian for all $k$. Looking back through the literature, this\nsame statistic was discover by Kadell \\cite{Kadell1985} who also gave\na bijective proof that the distribution is Mahonian. Whereas Kadell's\nbijections in fact refine Foata's original bijection, the family of\nbijections defined herein is not the same as Kadell's and, when taking\nthe major index to the inversion number, give a bijection different\nfrom that of Foata. The $k$-major index statistic is defined in\n\\refsec{stats}, and the bijections and proof that the distribution is\nMahonian are given in \\refsec{bijections}.\n\nIt is also natural to define a major index statistic on standard Young\ntableaux, which are central objects in the study of symmetric\nfunctions. Recently, Haglund and Stevens \\cite{HaSt2006} defined an\ninversion number on tableaux. Their construction generalizes Foata's\nbijection to tableaux and shows that the inversion number and major\nindex are equidistributed over standard Young tableaux of a fixed\nshape. Motivated by this, we use the bijections presented here to\nextend the notion of the $k$-major index to standard Young tableaux,\nfor $k \\leq 3$. The hope is that this method might be used to build a\ncomplete family of statistics interpolating between major index and\ninversion number on tableaux. This exploration takes place in\n\\refsec{tableau}.\n\nOur discovery of the $k$-major index and the family of bijections\npresented here came about through the study of Macdonald polynomials\n\\cite{Assaf2007-2}. In \\refsec{macdonald}, we elaborate on this\nconnection and present a conjecture for yet another family of\nbijections sharing many of the same properties that would have the\nfurther consequence of providing a remarkably simple combinatorial\nproof of Macdonald positivity.\n\n\n\\section{Definitions and notation}\n\\label{sec:stats}\n\nAt times it will be convenient to consider a slightly more general\ndefinition for a word $w$, where $w_i$ is allowed to be either a\npositive integer or an $\\emptyset$. In this case, $\\emptyset$'s should\nbe regarded as incomparable to other letters, so that they are simply\na way of spacing out the nonempty letters of $w$.  This idea will be\nespecially important in connection with Macdonald polynomials\ndiscussed in \\refsec{macdonald}.\n\n\\begin{definition}\n  For $w$ a word, $k$ a positive integer, define the {\\em $k$-descent\n    set of $w$}, denoted $\\ensuremath\\mathrm{Des}_k(w)$, by\n  \\begin{displaymath}\n    \\ensuremath\\mathrm{Des}_k(w) = \\{ (i,i+k) \\; | \\; w_i > w_{i+k} \\} ,\n  \\end{displaymath}\n  and define the {\\em $k$-inversion set of $w$}, denoted $\\ensuremath\\mathrm{Inv}_k(w)$,\n  by\n  \\begin{displaymath}\n    \\ensuremath\\mathrm{Inv}_k(w) = \\{ (i,j) \\; | \\; k > j-i > 0 \\; \\mbox{and} \\; w_i >\n    w_j \\} .\n  \\end{displaymath}\n\\end{definition}\n\nFor example, for $w = 986173245$ and $k=3$ we have\n\\begin{eqnarray*}\n  \\ensuremath\\mathrm{Des}_3(9 \\; 8 \\; 6 \\; 1 \\; 7 \\; 3 \\; 2 \\; 4 \\; 5) & = & \\{\n  (1,4), (2,5), (3,6), (5,8)\\}, \\\\ \n  \\ensuremath\\mathrm{Inv}_3(9 \\; 8 \\; 6 \\; 1 \\; 7 \\; 3 \\; 2 \\; 4 \\; 5) & = & \\{ (1,2),\n  (1,3), (2,3), (2,4), (3,4), (5,6), (5,7), (6,7)\\}.\n\\end{eqnarray*}\n\nIn fact, it is enough to define $k$-descents since $k$-inversions may\nbe recovered from the observation\n\\begin{equation}\n  \\ensuremath\\mathrm{Inv}_k(w) \\; = \\; \\bigcup_{j < k} \\ensuremath\\mathrm{Des}_j(w) .\n\\label{eqn:alt-Invk}\n\\end{equation}\n\nNote that when $k=1$, $\\ensuremath\\mathrm{Des}_k$ gives the usual descent set for a\nword. Similarly, when $N \\geq n$, $\\ensuremath\\mathrm{Inv}_N$ gives the usual set of\ninversion pairs for a word of length $n$. We interpolate between the\ncorresponding statistics, $\\ensuremath\\mathrm{maj}$ and $\\ensuremath\\mathrm{inv}$, with the following\nstatistic depending on the parameter $k$.\n\n\\begin{definition}\n  Given a word $w$ and a positive integer $k$, define the {\\em\n    $k$-major index of $w$} by\n  \\begin{displaymath}\n    \\ensuremath\\mathrm{maj}_k (w) \\; = \\; \\left| \\ensuremath\\mathrm{Inv}_k(w) \\right| + \\sum_{(i,i+k) \\in\n      \\ensuremath\\mathrm{Des}_k(w)} i .\n  \\end{displaymath}\n\\end{definition}  \n\nFor the same example, we have $\\ensuremath\\mathrm{maj}_3(9 \\; 8 \\; 6 \\; 1 \\; 7 \\; 3 \\; 2\n\\; 4 \\; 5) = 8 + 1 + 2 + 3 + 5 = 19$. For a word $w$ of length $n \\leq\nN$, the previous observations show that\n\\begin{eqnarray*}\n  \\ensuremath\\mathrm{maj}_1(w) & = & \\ensuremath\\mathrm{maj}(w), \\\\\n  \\ensuremath\\mathrm{maj}_N(w) & = & \\ensuremath\\mathrm{inv}(w).\n\\end{eqnarray*}\nThe statistic $\\ensuremath\\mathrm{maj}_k$ was first defined by Kadell \\cite{Kadell1985},\nwho gives a bijective proof that this statistic is Mahonian. Kadell's\nbijections take $\\ensuremath\\mathrm{inv}$ to $\\ensuremath\\mathrm{maj}_k$, with the extreme case from $\\ensuremath\\mathrm{inv}$\nto $\\ensuremath\\mathrm{maj}$ corresponding precisely to the inverse of Foata's bijection\n\\cite{Foata1968}. In \\refsec{bijections}, we give a different family\nof bijections, taking $\\ensuremath\\mathrm{maj}_{k-1}$ to $\\ensuremath\\mathrm{maj}_k$, which, when composed\nappropriately, give a different bijection from $\\ensuremath\\mathrm{maj}$ to $\\ensuremath\\mathrm{inv}$.\n\nIn the case when $w$ is a permutation (possibly with $\\emptyset$s), we\nwill also be interested in the descent set of the inverse permutation,\ndenoted $\\ensuremath\\mathrm{iDes}$, defined by\n\\begin{equation}\n  \\ensuremath\\mathrm{iDes}(w) = \\ensuremath\\mathrm{Des}(w^{-1}) = \\{ i \\ | \\ \\mbox{$i$ appears to the left\n    of $i+1$ in $w$} \\} .\n\\end{equation}\nFor example, $\\ensuremath\\mathrm{iDes}(9 \\; 8 \\; 6 \\; 1 \\; 7 \\; 3 \\; 2 \\; 4 \\; 5) =\n\\{2,5,7,8\\}$. \n\nRecall that a {\\em partition} $\\lambda$ is a weakly decreasing\nsequence of positive integers:\n$\\lambda=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_m)$, $\\lambda_1 \\geq\n\\lambda_2 \\geq \\cdots \\geq \\lambda_m > 0$. A partition $\\lambda$ may\nbe identified with its {\\em Young diagram}: the set of points $(i,j)$\nin the $\\mathbb{Z}_+ \\times\\mathbb{Z}_+$ lattice quadrant such that $1\n\\leq i \\leq \\lambda_j$. We draw the diagram so that each point $(i,j)$\nis represented by the unit cell southwest of the point. A {\\em\n  standard Young tableau of shape $\\lambda$} is a labelling of the\ncells of the Young diagram of $\\lambda$ with the numbers $1$ through\n$n$, where $n = \\sum_i \\lambda_i$, such that the entries increase\nalong rows and up columns. For example, see \\reffig{tableau}.\n\n\\begin{figure}[ht]\n  \\begin{center}\n    \\begin{displaymath}\n      \\tableau{8 \\\\ 2 & 5 & 6 \\\\ 1 & 3 & 4 & 7}\n    \\end{displaymath}\n    \\caption{\\label{fig:tableau}A standard Young tableau of shape\n      $(4,3,1)$.}\n  \\end{center}\n\\end{figure}\n\nFor a standard Young tableau $T$, recall the {\\em descent set of $T$},\ndenoted $\\ensuremath\\mathrm{Des}(T)$, defined by\n\\begin{equation}\n  \\ensuremath\\mathrm{Des}(T) = \\{ (i,i+1) \\; | \\; i \\; \\mbox{lies strictly south of} \\;\n  i+1 \\; \\mbox{in} \\; T \\}.\n\\label{eqn:DesT}\n\\end{equation}\nCompletely analogous to the case with words, define the {\\em major\n  index of $T$}, denoted $\\ensuremath\\mathrm{maj}(T)$, by\n\\begin{equation}\n  \\ensuremath\\mathrm{maj}(T) = \\sum_{(i,i+k) \\in \\ensuremath\\mathrm{Des}(T)} i .\n\\label{eqn:majT}\n\\end{equation}\nFor the example in \\reffig{tableau}, $\\ensuremath\\mathrm{Des} = \\{(1,2), (4,5), (7,8)\\}$\nand so $\\ensuremath\\mathrm{maj} = 1+4+7 = 12$. The descent set for tableaux corresponds\nto the descent set of permutations in the sense that for a fixed set\n$D$,\n\\begin{displaymath}\n  \\# \\{ w \\in \\mathcal{S}_n \\; | \\; \\ensuremath\\mathrm{Des}(w) = D \\} =\n  \\sum_{\\lambda} f^{\\lambda} \\cdot  \\# \\{ T \\in \\ensuremath\\mathrm{SYT}(\\lambda) \\; | \\;\n  \\ensuremath\\mathrm{Des}(T) = D \\},\n\\end{displaymath}\nwhere $\\ensuremath\\mathrm{SYT}(\\lambda)$ denotes the set of standard Young tableaux of\nshape $\\lambda$ and $f^{\\lambda} = |\\ensuremath\\mathrm{SYT}(\\lambda)|$. This identity can\nbe proved using the Robinson-Schensted-Knuth correspondence which\nbijectively associates each permutation $w$ with a pair of standard\ntableaux $(P,Q)$ of the same shape such that $\\ensuremath\\mathrm{iDes}(w) = \\ensuremath\\mathrm{Des}(Q)$. We\npostpone the definition of $\\ensuremath\\mathrm{Des}_k$ and $\\ensuremath\\mathrm{maj}_k$ for \\refsec{tableau}.\n\n\n\\section{A family of bijections on words}\n\\label{sec:bijections}\n\nFor $k \\geq 2$, we will construct bijections $\\phi^{(k)}$ on words of\nlength $n$ such that \n\\begin{equation}\n  \\ensuremath\\mathrm{maj}_{k-1}(w) = \\ensuremath\\mathrm{maj}_k(\\phi^{(k)}(w)) .\n\\label{eqn:dist}\n\\end{equation}\nAs noted earlier, these bijections are not equivalent to those defined\nby Kadell, and the appropriate composition does not give Foata's\nbijection. That said, the construction below follows the idea of\n\\cite{Foata1968} in that $\\phi^{(k)}$ will be defined recursively\nusing an involution $\\gamma_{j}^{(k)}$ which permutes the letters of a\ngiven word. \n\nLet $x,a,b$ be (not necessarily distinct) integers. Say that $x$\n{\\em splits} the pair $a,b$ if $a \\leq x < b$ or $b \\leq x < a$. Let\n$w$ be a word of length $n$. For $k \\geq 2$ and $j \\leq n$, define a\nset of indices $\\Gamma_{j}^{(k)}$ of $w$ by\n\\begin{equation}\n  j-k \\in \\Gamma_{j}^{(k)}(w) \\;\\; \\mbox{if} \\;\\; w_j\n  \\;\\mbox{splits the pair} \\; w_{j-k}, w_{j-k+1},\n\\label{eqn:Gamma-i}\n\\end{equation}\n  and if $i \\in \\Gamma_{j}^{(k)}(w)$, then\n\\begin{equation}\n  i-k \\in \\Gamma_{j}^{(k)}(w) \\;\\; \\mbox{if exactly one of} \\; w_{i}\n  \\;\\mbox{or}\\; w_{i+1} \\;\\mbox{splits the pair}\\; w_{i-k},w_{i-k+1}.\n\\label{eqn:Gamma-r}\n\\end{equation}\nFor our running example, we have $\\Gamma_{8}^{(3)}(9 \\; 8 \\; 6 \\; 1 \\;\n7 \\; 3 \\; 2 \\; 4 \\; 5) = \\{ 5, 2\\}$.\n\nLet permutations act on words by permuting the indices, i.e. $\\tau\n\\cdot w \\; = \\; w_{\\tau(1)} w_{\\tau(2)} \\cdots w_{\\tau(n)}$. Define a\nmap $\\gamma_{j}^{(k)}$ by\n\\begin{equation}\n  \\gamma_{j}^{(k)}(w) \\; = \\; \\left( \\prod_{i \\in \\Gamma_{j}^{(k)}(w)} (i,\n  i+1) \\right) \\cdot w .\n\\label{eqn:gammak}\n\\end{equation}\nThat is to say, $\\gamma_{j}^{(k)}(w)$ is the result of interchanging\n$w_{i}$ and $w_{i+1}$ for all $i \\in \\Gamma_{j}^{(k)}(w)$. Back to our\nrunning example, we have $\\gamma_{8}^{(3)}(9 \\; \\mathbf{8} \\;\n\\mathbf{6} \\; 1 \\; \\mathbf{7} \\; \\mathbf{3} \\; 2 \\; 4 \\; 5) = 9 \\;\n\\mathbf{6} \\; \\mathbf{8} \\; 1 \\; \\mathbf{3} \\; \\mathbf{7} \\; 2 \\; 4 \\;\n5$.\n\nFor $w$ a word of length $n$, define $\\phi^{(k)}$ by\n\\begin{equation}\n  \\phi^{(k)} (w) \\; = \\; \\gamma_{n}^{(k)} \\circ \\gamma_{n-1}^{(k)}\n  \\circ \\cdots \\circ \\gamma_{1}^{(k)} (w).\n\\label{eqn:phik}\n\\end{equation}\nSince $\\gamma_j^{(k)}$ is the identity for $j \\leq k$, these terms may\nbe omitted from equations \\ref{eqn:phik} and \\ref{eqn:psik}.\n\nFor example, for $w=6 \\; 9 \\; 3 \\; 8 \\; 1 \\; 7 \\; 2 \\; 4 \\; 5$,\n$\\phi^{(3)}(w)$ is computed as follows.\n\\begin{displaymath}\n  \\begin{array}{rcccccccccc}\n    w \\phantom{)} & = & \n      6 & 9 & 3 & 8 & \\rnode{m3}{1} & 7 & 2 & 4 & 5 \\\\\n    \\gamma_4^{(3)}(w) & = & \n      9 & 6 & 3 & 8 & \\rnode{m4}{1} & 7 & 2 & 4 & 5 \\\\\n    \\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n      9 & 6 & 3 & 8 & \\rnode{m5}{1} & 7 & 2 & 4 & 5 \\\\\n    \\gamma_{6}^{(3)}\\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n      9 & 6 & 8 & 3 & \\rnode{m6}{1} & 7 & 2 & 4 & 5 \\\\\n    \\gamma_{7}^{(3)}\\gamma_{6}^{(3)}\\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n      9 & 6 & 8 & 1 & \\rnode{m7}{3} & 7 & 2 & 4 & 5 \\\\\n    \\gamma_{8}^{(3)}\\gamma_{7}^{(3)}\\gamma_{6}^{(3)}\\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n      9 & 8 & 6 & 1 & \\rnode{m8}{7} & 3 & 2 & 4 & 5 \\\\\n    \\phi^{(3)}(w) = \\gamma_{9}^{(3)}\\gamma_{8}^{(3)}\\gamma_{7}^{(3)}\\gamma_{6}^{(3)}\\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n      9 & 8 & 6 & 1 & \\rnode{m9}{7} & 3 & 2 & 4 & 5\n  \\end{array}\n  \\psset{nodesep=3pt,linewidth=.1ex}\n  \\everypsbox{\\scriptstyle}\n\\end{displaymath}\nNotice that for this example $\\ensuremath\\mathrm{maj}_2(w) = 19 =\n\\ensuremath\\mathrm{maj}_3(\\phi^{(3)}(w))$. Before proving \\refeq{dist} in general, we\ntake note of a few important properties that $\\phi^{(k)}$ shares with\nFoata's bijection (for Foata, properties (i) and (ii) are shown in\n\\cite{Foata1968}, and property (iii) is shown in \\cite{FoSc1978}).\n\n\\begin{proposition}\n  For each $k \\geq 2$, we have\n  \\begin{itemize}\n  \\item[(i)] the map $\\phi^{(k)}$ is a bijection on words on $M$ with\n    fixed $\\emptyset$ positions;\n  \\item[(ii)] for $w$ a word of length $n$, $w_{n-k+1} > w_n$ if and\n    only if $\\phi^{(k)}(w)_{n-k} > \\phi^{(k)}(w)_n = w_n$;\n  \\item[(iii)] for $w$ a permutation, $\\ensuremath\\mathrm{iDes}(w) = \\ensuremath\\mathrm{iDes}(\\phi^{(k)}(w))$.\n  \\end{itemize}\n\\label{prop:props}\n\\end{proposition}\n\n\\begin{proof}\n  Since $\\Gamma_{j}^{(k)}(\\gamma_{j}^{(k)}(w)) = \\Gamma_{j}^{(k)}(w)$,\n  $\\gamma_{j}^{(k)}$ is an involution on words of length $n$ for all\n  $j \\leq n$ and $k \\geq 2$. Therefore $\\phi^{(k)}$ is a bijection on\n  words of length $n$ for all $k \\geq 2$ with inverse given by\n  \\begin{equation}\n    \\psi^{(k)} (w) \\; = \\; \\gamma_{1}^{(k)} \\circ \\cdots \\circ\n    \\gamma_{n-1}^{(k)} \\circ \\gamma_{n}^{(k)} (w) .\n  \\label{eqn:psik}\n  \\end{equation}\n  It is clear from the definition of $\\gamma_j^{(k)}$ that\n  $\\phi^{(k)}$ in fact fixes the last $k-1$ letters of a word, so\n  indeed the last letter is fixed for every $k$. Let $u =\n  \\gamma_{n-1}^{(k)} \\cdots \\gamma_{1}^{(k)}(w)$, $u_{j} = w_{j}$ for\n  $j \\geq n-k+1$. If $u_{n-k}$ and $u_{n-k+1}$ compare the same with\n  $u_{n}$, then $u = \\phi^{(k)}(w)$ and (ii) clearly holds; otherwise,\n  these two letters are interchanged by $\\gamma_{n}^{(k)}$, again\n  showing that (ii) is satisfied. Also note that $\\phi^{(k)}$ may be\n  defined recursively by\n  \\begin{equation}\n    \\phi^{(k)}(wx) = \\gamma_{n+1}^{(k)} \\left( \\phi^{(k)}(w) \\right) x ,\n  \\label{eqn:recursivePhi}\n  \\end{equation}\n  which completely parallels Foata's original construction.  Finally,\n  since consecutive letters cannot be split, in the sense of\n  $\\Gamma_j^{(k)}$, they may never be interchanged by\n  $\\gamma_j^{(k)}$. Thus the inverse descent set is preserved.\n\\end{proof}\n\nTo prove \\refeq{dist}, we follow the strategy of \\cite{Foata1968}. The\nkey, therefore, lies in the following lemma.\n\n\\begin{lemma}\n  For $k \\geq 2$, $w$ a word of length $n$ and $j \\leq n$,\n  \\begin{displaymath}\n    \\ensuremath\\mathrm{maj}_k \\left( \\gamma_{j}^{(k)}(w_1 \\cdots w_{j-1}) \\right) \n    \\; = \\; \\ensuremath\\mathrm{maj}_k(w_1 \\cdots w_{j-1}) + \n    \\left\\{ \\begin{array}{rl}\n      1 & \\mbox{if} \\;\\; w_{j-k} > w_j \\geq w_{j-k+1}, \\\\\n     -1 & \\mbox{if} \\;\\; w_{j-k+1} > w_j \\geq w_{j-k}, \\\\\n      0 & \\mbox{otherwise}.\n    \\end{array} \\right.\n  \\end{displaymath}\n\\label{lem:gamma}\n\\end{lemma}\n\n\\begin{proof}\n  If neither of the first two cases holds, then $j-k \\not\\in\n  \\Gamma_{j}^{(k)}(w)$, so $\\gamma_{j}^{(k)}(w) = w$ and the result is\n  immediate. Assume, then, that $j-k \\in \\Gamma_{j}^{(k)}(w)$, and set\n  $u = w_{j-k} w_{j-k+1} \\cdots w_{j-1}$. Then\n  \\begin{equation}\n    \\ensuremath\\mathrm{maj}_k\\left(\\gamma_{j}^{(k)}(u)\\right) = \n    \\ensuremath\\mathrm{maj}_k(u) + \\left\\{ \\begin{array}{rl}\n        1 & \\mbox{if} \\;\\; w_{j-k} > w_j \\geq w_{j-k+1} , \\\\\n        -1 & \\mbox{if} \\;\\; w_{j-k+1} > w_j \\geq w_{j-k} .\n      \\end{array} \\right.\n  \\label{eqn:majk-u}\n  \\end{equation}\n  For $i \\in \\Gamma_{j}^{(k)}(w)$, let $u = w_{i} w_{i+1} \\cdots\n  w_{j-1}$, and, by induction, assume that \\refeq{majk-u} holds for\n  $u$.  Let $u' = w_{i-k} w_{i-k+1} \\cdots w_{j-1}$. We will show that\n  $u'$ also satisfies \\refeq{majk-u} by considering the contribution\n  to $\\ensuremath\\mathrm{maj}_k$ of $w_{i-k},w_{i-k+1}, \\ldots, w_{i-1}$. For $i-k+1 < h\n  < i$, $k$-inversions and $k$-descents involving $w_h$ are the same\n  for $u'$ and $\\gamma_{j}^{(k)}(u')$, so we need only consider\n  contributions from the potential $k$-inversions $(i-k,i-k+1)$ and\n  $(i-k+1,i)$, and the potential $k$-descents $(i-k,i)$ and\n  $(i-k+1,i+1)$.\n  \n  First suppose that $i-k \\in \\Gamma_{j}^{(k)}(w)$.  \n  In all eight possible scenarios for $w_{i-k},w_{i-k+1},w_{i},w_{i+1}$,\n  we have\n  \\begin{eqnarray*}\n    (i-k,i-k+1) \\in \\ensuremath\\mathrm{Inv}_k(w) & \\Leftrightarrow & (i-k,i-k+1) \\not\\in\n    \\ensuremath\\mathrm{Inv}_k\\left(\\gamma_{j}^{(k)}(w)\\right), \\\\\n    (i-k,i) \\in \\ensuremath\\mathrm{Des}_k(w) & \\Leftrightarrow & (i-k+1,i+1) \\in\n    \\ensuremath\\mathrm{Des}_k\\left(\\gamma_{j}^{(k)}(w)\\right), \\\\\n    (i-k+1,i+1) \\in \\ensuremath\\mathrm{Des}_k(w) & \\Leftrightarrow & (i-k,i) \\in\n    \\ensuremath\\mathrm{Des}_k\\left(\\gamma_{j}^{(k)}(w)\\right).\n  \\end{eqnarray*}\n  \n  If both or neither of $(i-k,i)$ and $(i-k+1,i+1)$ are $k$-descents\n  of $w$, then the same holds for $u'$ and $\\gamma_{j}^{(k)}(u')$. In\n  this case \n  exactly one of $(i-k,i-k+1)$ and $(i-k+1,i)$ is a $k$-inversion for\n  $w$, and\n  \\begin{eqnarray*}\n    (i-k+1,i) \\in \\ensuremath\\mathrm{Inv}_k(u') & \\Leftrightarrow & \n    (i-k+1,i) \\not\\in \\ensuremath\\mathrm{Inv}_k\\left(\\gamma_{j}^{(k)}(u')\\right).\n  \\end{eqnarray*}\n  The lemma now follows. On the other hand, if exactly one of\n  $(i-k,i)$ and $(i-k+1,i+1)$ is a $k$-descent of $w$, \n  then the difference in the contribution to $\\ensuremath\\mathrm{maj}_k$ from the\n  potential $k$-descents is offset by the difference from the\n  potential $k$-inversion $(i-k,i-k+1)$. Furthermore,\n  \\begin{eqnarray*}\n    (i-k+1,i) \\in \\ensuremath\\mathrm{Inv}_k(u') & \\Leftrightarrow & \n    (i-k+1,i) \\in \\ensuremath\\mathrm{Inv}_k\\left(\\gamma_{j}^{(k)}(u')\\right),\n  \\end{eqnarray*}\n  thereby establishing the result.\n  \n  To complete the proof, note that when $i-k \\not\\in\n  \\Gamma_{j}^{(k)}(w)$, either $w_{i}$ and $w_{i+1}$ compare the same\n  with $w_{i-k}$ and also with $w_{i-k+1}$ and so the $k$-inversions\n  and $k$-descents beginning with $i-k$ or $i-k+1$ are unchanged, or\n  the $k$-descent at $(i-k+1,i+1)$ is exchanged for a $k$-descent at\n  $(i-k,i)$ along with a $k$-inversion at $(i,i+1)$. In both cases the\n  contribution to the $k$-major index is preserved.\n\\end{proof}\n\n\\begin{proposition}\n  For $k \\geq 2$ and $w$ a word, $\\displaystyle{\\ensuremath\\mathrm{maj}_{k-1} (w) =\n    \\ensuremath\\mathrm{maj}_k \\left( \\phi^{(k)}(w) \\right)}$.\n\\label{prop:majk}\n\\end{proposition}\n\n\\begin{proof}\n  The result is clear for a words of length $\\leq k$. We proceed by\n  induction, assuming the result for words of length $n-1$.  Let $w$\n  be a word of length $n-1$ and $x$ a letter. To simplify notation,\n  let\n  $$\n  u = \\gamma_{n}^{(k)} \\left( \\phi^{(k)}(w) \\right).\n  $$\n  By expanding the definition of $\\ensuremath\\mathrm{maj}_k$ and applying\n  \\reflem{gamma}, we have\n  \\begin{displaymath}\n    \\begin{array}{l}\n    \\ensuremath\\mathrm{maj}_k \\left(\\phi^{(k)}(wx)\\right) \\\\\n    \\hspace{2em} = \\ensuremath\\mathrm{maj}_k(ux) \\\\ \n    \\hspace{2em} = \\ensuremath\\mathrm{maj}_k(u) + \\# \\{ i > n-k \\; | \\; u_i > x \\}\n        + \\left\\{ \\begin{array}{rl}\n        0 & \\mbox{if} \\; x \\geq u_{n-k} \\\\\n        n\\!-\\!k & \\mbox{if} \\; u_{n-k} > x \\end{array} \\right. \\\\\n    \\hspace{2em} = \\ensuremath\\mathrm{maj}_k(u) + \\# \\{i>n-k+1 \\; | \\; u_i > x\\} \n        + \\left\\{ \\begin{array}{rcrl}\n        n\\!-\\!k + 1 & \\mbox{if} & u_{n-k} > x,    & u_{n-k+1} > x \\\\\n        0       + 0 & \\mbox{if} & x \\geq u_{n-k}, & x \\geq u_{n-k+1} \\\\\n        n\\!-\\!k + 0 & \\mbox{if} & u_{n-k} > x,    & x \\geq u_{n-k+1} \\\\\n        0       + 1 & \\mbox{if} & x \\geq u_{n-k}, & u_{n-k+1} > x\n      \\end{array} \\right. \\\\\n    \\hspace{2em} = \\ensuremath\\mathrm{maj}_k \\left( \\gamma_{n}^{(k)}(u) \\right)\n    + \\# \\{i>n-k+1 \\; | \\; u_i > x\\} \n        + \\left\\{ \\begin{array}{rl}\n        n\\!-\\!k\\!+\\!1 + 0 & \\mbox{if} \\; u_{n-k},u_{n-k+1} > x \\\\\n        0             + 0 & \\mbox{if} \\; x \\geq u_{n-k},u_{n-k+1} \\\\\n        n\\!-\\!k\\      + 1 & \\mbox{if} \\; u_{n-k} > x \\geq u_{n-k+1} \\\\\n        1             - 1 & \\mbox{if} \\; u_{n-k+1} > x \\geq u_{n-k}\n      \\end{array} \\right. \\\\\n    \\hspace{2em} = \\ensuremath\\mathrm{maj}_{k-1} \\left( \\phi^{(k)}(w) \\right) \n          + \\# \\{i>n-k+1 \\; | \\; u_i > x\\}\n        + \\left\\{ \\begin{array}{rl}\n        0 & \\mbox{if} \\; x \\geq u_{n-k} \\\\\n        n\\!-\\!k\\!+\\!1 & \\mbox{if} \\; u_{n-k} > x\n      \\end{array} \\right.\n    \\end{array}\n  \\end{displaymath}\n\n  Recall from \\refprop{props} that for $i \\geq n-k+2$, $u_i = w_i$,\n  and so\n  $$\n  \\{i>n-k+1 \\; | \\; u_i > x\\} \\; = \\; \\{i>n-k+1 \\; | \\; w_i > x\\} .\n  $$\n  Furthermore, since $\\phi^{(k)}(w)_{n-k+1} = w_{n-k+1}$, we also have\n  $$\n  u_{n-k} \\leq x \\; \\Leftrightarrow \\; w_{n-k+1} \\leq x .\n  $$\n\n  Continuing from the above equation using these two facts and the\n  inductive hypothesis, we have\n  \\begin{displaymath}\n    \\ensuremath\\mathrm{maj}_k \\left(\\phi^{(k)}(wx)\\right) \n    = \\ensuremath\\mathrm{maj}_{k-1}(w) \n    + \\# \\{i>n-k+1 \\; | \\; w_i > x\\} + \\left\\{ \\begin{array}{rl}\n        0 & \\mbox{if} \\; x \\geq w_{n-k+1} \\\\\n        n\\!-\\!k\\!+\\!1 & \\mbox{if} \\; w_{n-k+1} > x\n      \\end{array} \\right.\n  \\end{displaymath}\n  which is exactly $\\ensuremath\\mathrm{maj}_{k-1} (wx)$, as desired.\n\\end{proof}\n\nFor $1 \\leq h < i$, we can compose these bijections to form the\nbijection\n\\begin{equation}\n  \\phi^{[i,h]} = \\phi^{(i)} \\circ \\cdots \\circ \\phi^{(h+1)}\n\\end{equation}\nsatisfying $\\ensuremath\\mathrm{maj}_h (w) = \\ensuremath\\mathrm{maj}_i \\left( \\phi^{[i,h]}(w) \\right)$.  In\nparticular, $\\phi^{[k,1]}$ provides a bijective proof of the\nfollowing.\n\n\\begin{theorem}\n  Let $W_M$ be the set of words on a multiset $M$ with a fixed\n  $\\emptyset$ positions. Then for $k \\geq 1$,\n  \\begin{eqnarray*}\n    \\sum_{w \\in W_M} q^{\\ensuremath\\mathrm{maj}(w)} & = &\n    \\sum_{w \\in W_M} q^{\\ensuremath\\mathrm{maj}_k(w)} .\n  \\end{eqnarray*}\n  That is to say, the $k$-major index has Mahonian distribution.\n\\label{thm:mahonian}\n\\end{theorem}\n\n\n\\section{Extending the $k$-major index to tableaux}\n\\label{sec:tableau}\n\nIn \\cite{HaSt2006}, Haglund and Stevens define an inversion number for\nstandard tableaux which is equidistributed with the major\nindex. Therefore it is natural to try to extend the $k$-major index\nstatistic to tableaux in a similar manner. However, to do this, we\nmust first define $\\ensuremath\\mathrm{Des}_k$ for standard Young tableaux.\n\nConsider the possible relative positions of $i$ and $i+k$ in a\nstandard Young tableau $T$.  Since $i < i+k$, $i$ must lie strictly\nwest or strictly south of $i+k$. If $i$ lies strictly west and weakly\nnorth of $i+k$, then the pair $(i,i+k)$ should not count as a\n$k$-descent.  Conjugately, if $i$ lies strictly south and weakly east\nof $i+k$, then the pair $(i,i+k)$ should count as a $k$-descent. The\ndifficulty arises in how to resolve the situation where $i$ lies\nstrictly southwest of $i+k$. The approach given in \\cite{HaSt2006} is\nquite involved as it is based on {\\em inversion paths} which must be\ncomputed iteratively. In most cases, interchanging even two\nconsecutive entries in a tableau completely alters the inversion paths\nin an opaque way. Therefore we begin at the other extreme, though\nbelow we succeed only up to $k=3$.\n\nFor $k=2$, the ambiguous case when $i$ lies strictly southwest of\n$i+2$ cannot arise in a standard tableaux. However, for $k=3$ we must\ndecide whether $(i-3,i)$ is a $3$-descent when $i-3,i-2,i-1,i$ appear\nin a $2\\times 2$ box in $T$. For reasons that will be made clear, we\nresolve the situations as indicated in \\reffig{3Des}.\n\n\\begin{figure}[ht]\n  \\begin{center}\n    \\begin{displaymath}\n      \\begin{array}{\\ccii\\ccii}\n        \\bigtableau{i\\!\\!-\\!\\!1 & i \\\\ i\\!\\!-\\!\\!3 & i\\!\\!-\\!\\!2} & \n        \\bigtableau{i\\!\\!-\\!\\!2 & i \\\\ i\\!\\!-\\!\\!3 & i\\!\\!-\\!\\!1} \\\\[2\\vsp]\n        (i-3,i) \\in \\ensuremath\\mathrm{Des}_3 & (i-3,i) \\not\\in \\ensuremath\\mathrm{Des}_3\n      \\end{array}\n    \\end{displaymath}      \n    \\caption{\\label{fig:3Des} Ambiguous cases for whether $(i-3,i)$\n      should constitute a $3$-descent.}\n  \\end{center}    \n\\end{figure}\n\nTo simplify notation, we introduce the following terminology. For\n$i<n$, say that {\\em i attacks n} if $i$ lies strictly south and\nweakly east of $n$ or if $i$ lies strictly southwest of $n$ and $i+1$\nattacks $n$. \n\n\\begin{definition}\n  For $T$ a standard tableau, $k \\leq 3$, define the {\\em $k$-descent\n    set of $T$}, denoted $\\ensuremath\\mathrm{Des}_{k}(T)$, by\n  \\begin{displaymath}\n    \\ensuremath\\mathrm{Des}_k(T) = \\{ (i,i+k) \\; | \\; \\mbox{$i$ attacks $i+k$} \\}, \n  \\end{displaymath}  \n  define the {\\em set of $k$-inversions of $T$}, denoted\n  $\\ensuremath\\mathrm{Inv}_k(T)$, by\n  \\begin{displaymath}\n    \\ensuremath\\mathrm{Inv}_k(T) = \\bigcup_{j < k} \\ensuremath\\mathrm{Des}_j(T) ,\n  \\end{displaymath}\n  and finally define the {\\em $k$-major index of $T$}, denoted\n  $\\ensuremath\\mathrm{maj}_k(T)$, by\n  \\begin{displaymath}\n    \\ensuremath\\mathrm{maj}_k (T) \\; = \\; \\left| \\ensuremath\\mathrm{Inv}_k(T) \\right| + \\sum_{(i,i+k) \\in\n      \\ensuremath\\mathrm{Des}_k(T)} i .\n  \\end{displaymath}\n\\label{defn:Tstats}\n\\end{definition}\n\nNote that for defining $k$-inversions we made use of the alternate\ndescription of $k$-inversions for words given in \\refeq{alt-Invk}. For\nthe example in \\reffig{tableau}, we have $\\ensuremath\\mathrm{Des}_2 = \\{(3,5), \\ (4,6), \\\n(6,8)\\}$, $\\ensuremath\\mathrm{Inv}_2 = \\{(1,2), \\ (4,5), \\ (7,8)\\}$ and so $\\ensuremath\\mathrm{maj}_2 = 3 +\n3 + 4 + 6 = 16$.\n\nParallel to \\refsec{bijections}, we aim to generalize\n\\refthm{mahonian} to tableaux by constructing bijections $\\Phi^{(k)}$,\n$k=2,3$, on standard Young tableaux of fixed shape such that\n\\begin{equation}\n\\ensuremath\\mathrm{maj}_{k-1}(T) = \\ensuremath\\mathrm{maj}_k(\\Phi^{(k)}(T)) .\n\\end{equation}\n\nThe first task, then, is to define the set $\\Gamma_{j}^{(k)}$. Here\ncare must be taken when determining when the ``splitting'' condition\nis satisfied. As a minimum requirement, since the intention is to\ninterchange $i$ and $i+1$, we must ensure that we do this only if $i$\nand $i+1$ do not appear in the same row or column. This motivates the\ndecision in \\reffig{3Des} as well as the following definitions. \n\nSay that $n$ {\\em splits} $a,b$ if exactly one of $a,b$ attacks\n$n$. For $k=2,3$, define $\\Gamma_{j}^{(k)}$ by\n\\begin{displaymath}\n  j-k \\in \\Gamma_{j}^{(k)}(T) \\;\\; \\mbox{if} \\;\\; \\mbox{$j$ splits the pair\n    $j-k,j-k+1$},\n\\end{displaymath}\nand if $i \\in \\Gamma_{j}^{(k)}(T)$, then\n\\begin{displaymath}\n  i-k \\in \\Gamma_{j}^{(k)}(T) \\;\\; \\mbox{if} \\;\\; \\mbox{exactly one\n    of}\\; i,i+1 \\; \\mbox{splits the pair}\\; i-k,i-k+1.\n\\end{displaymath}\n\nBy the definition of attacking, both or neither $i,i+1$ attack $n$\nwhenever $i$ is strictly southwest of $n$. Therefore in order for $n$\nto split $i,i+1$, one must lie strictly south and weakly east of $n$,\nand the other must lie weakly north of $n$. It follows, then, that if\n$i$ and $i+1$ lie in the same row or column of $T$, then $n$ does not\nsplit $i,i+1$ for any $n$. \n\nLet permutations act on standard fillings of a Young diagram by\npermuting the entries. While this is not, in general, a well-defined\naction on tableaux, the following application in fact is. For $k=2,3$,\ndefine $\\gamma_{j}^{(k)}$ by\n\\begin{equation}\n  \\gamma_{j}^{(k)}(T) \\; = \\; \\left( \\prod_{i \\in \\Gamma_{j}^{(k)}(T)}\n    (i, i+1) \\right) \\cdot T .\n\\label{eqn:gammak-T}\n\\end{equation}\nThat is, $\\gamma_{j}^{(k)}$ interchanges $i$ and $i+1$ for all $i \\in\n\\Gamma_{j}^{(k)}(T)$. As before, $\\gamma_j^{(k)}$ is the identity for\n$j \\leq k$.\n\nIf $i$ is strictly southwest of $n$, then $n$ cannot split the pair\n$i,i+1$. It follows that $\\Gamma^{(k)}_j(T) =\n\\Gamma^{(k)}_j(\\gamma^{(k)}_j(T))$; in particular, $\\gamma^{(k)}_j$ is\nan involution. For $k=2,3$, define a bijection $\\Phi^{(k)}$ on\ntableaux of a fixed shape by\n\\begin{equation}\n  \\Phi^{(k)} (T) \\; = \\; \\gamma_{n}^{(k)} \\circ \\gamma_{n-1}^{(k)}\n  \\circ \\cdots \\circ \\gamma_{1}^{(k)} (T).\n\\label{eqn:phi2-T}\n\\end{equation}\nFor the example in \\reffig{Phi2}, observe that $\\ensuremath\\mathrm{maj}_1(T) = 16 =\n\\ensuremath\\mathrm{maj}_2(\\Phi^{(2)}(T))$.\n\n\\begin{figure}[ht]\n  \\begin{center}\n    \\begin{displaymath}\n      \\begin{array}{\\cciii\\cciii c}\n        T \\ = \\ \\rnode{l}{\\tableau{8 \\\\ 2 & 4 & 6 \\\\ 1 & 3 & 5 & 7}} &\n        \\rnode{c}{\\tableau{8 \\\\ 3 & 4 & 6 \\\\ 1 & 2 & 5 & 7}} &\n        \\rnode{r}{\\tableau{8 \\\\ 2 & 5 & 6 \\\\ 1 & 3 & 4 & 7}} \\ = \\ \\Phi^{(2)}(T)\n      \\end{array}\n      \\psset{nodesepA=3pt,nodesepB=5pt,linewidth=.1ex}\n      \\ncline{->}  {l}{c} \\naput{\\gamma^{(2)}_{4}}\n      \\ncline{->}  {c}{r} \\naput{\\gamma^{(2)}_{6}}\n    \\end{displaymath}\n    \\caption{\\label{fig:Phi2}An example of $\\Phi^{(2)}$; here\n      $\\gamma^{(2)}_{j} = \\mathrm{id}$ for $j\\neq 4,6$.}\n  \\end{center}    \n\\end{figure}\n\nSimilar to before, the inverse of $\\Phi^{(k)}$ is given by composing\nthe maps $\\gamma^{(k)}_j$ in the reverse order. This establishes the\nanalogue of the property (i) of \\refprop{props}, and the analogue of\nproperty (ii) is that the largest letter of $T$ is fixed by\n$\\Phi^{(k)}$. As property (iii) has no real analogue in this setting, we\nmove on to the more important statement observed in the example,\nnamely the analogue of \\refprop{majk} below.\n\n\\begin{proposition}\n  For $T$ a standard Young tableau and $k=2,3$, we have\n  $\\displaystyle{\\ensuremath\\mathrm{maj}_{k-1} (T) \\; = \\; \\ensuremath\\mathrm{maj}_k \\left( \\Phi^{(k)}(T)\n    \\right)}$.\n\\label{prop:majk-T}\n\\end{proposition}\n\n\\begin{proof}\n  We use the proofs of \\reflem{gamma} and \\refprop{majk}. For this to\n  make sense, we make the substitution that for $i < n$, $w_i > w_n$\n  should be interpreted as ``i attacks n'' and similarly $w_i \\leq\n  w_n$ should be interpreted as ``i does not attack n''. In order for\n  the arguments to remain valid under this translation, interchanging\n  entries using $\\gamma_j^{(k)}$ may not change $k$-inversions or\n  $k$-descents between unmoved entries. The only potential violation\n  of this is the potential $3$-descent between $i-3$ and $i$ in the\n  situations depicted in \\reffig{3Des}. However, in either case $i-2\n  \\not\\in \\Gamma^{(3)}_j$ since neither $i+1$ nor $i+2$ can split the\n  pair $i-2, i-1$. Therefore, with this translation, the proofs carry\n  through verbatim.\n\\end{proof}\n\n\\begin{theorem}\n  For $\\lambda$ a partition, we have\n  \\begin{equation}\n    \\sum_{T \\in \\ensuremath\\mathrm{SYT}(\\lambda)} q^{\\ensuremath\\mathrm{maj}(T)} =\n    \\sum_{T \\in \\ensuremath\\mathrm{SYT}(\\lambda)} q^{\\ensuremath\\mathrm{maj}_{2}(T)} =\n    \\sum_{T \\in \\ensuremath\\mathrm{SYT}(\\lambda)} q^{\\ensuremath\\mathrm{maj}_{3}(T)} .\n  \\end{equation}\n  \\label{thm:mahonian-T}\n\\end{theorem}\n\nUnfortunately, \\refthm{mahonian-T} is the best we can do towards\nextending \\refthm{mahonian} using this direct analogue of\n$\\phi^{(k)}$. This technique breaks down at $k=4$ for the shape\n$(2,2,2)$. In this case, the $6$ must lie in the northeast corner and\nwill necessarily interchange the $2$ and $3$ if they both lie in the\nfirst two rows. Then if $1,2,3,4$ occupy the first two rows, this\nchanges whether $(1,4)$ is a $3$-descent ($4$-inversion). In order to\novercome this shortfall, either we must adopt a more dynamic notion of\n$k$-inversions as in the Haglund-Stevens approach or a more\ncomplicated bijection. \n\n\n\\section{Connections with Macdonald polynomials}\n\\label{sec:macdonald}\n\nThe $k$-major index statistic was rediscovered in the author's study\nof Macdonald polynomials. In this section we connect the results of\n\\refsec{bijections} back to Macdonald polynomials.\n\nIn \\cite{HHLRU2005}, Bylund and Haiman introduced the $k$-inversion\nnumber of a $k$-tuple of tableaux to be the number of inversions\nbetween certain entries, and it is shown that this statistic may be\nused to give an alternative definition for Lascoux-Leclerc-Thibon\npolynomials. In particular, when each shape of the $k$-tuple is a\nribbon, i.e. contains no $2 \\times 2$ block, the $k$-inversion number\nof the $k$-tuple is exactly $|\\ensuremath\\mathrm{Inv}_k(w)|$ where $w$ is a certain\nreading word of the $k$-tuple. In further study of these objects\n\\cite{Assaf2007-2}, it became natural to associate to each $k$-tuple\nnot only the $k$-inversion number, but also a $k$-descent set. Again,\nwhen the shapes of the $k$-tuple in question are all ribbons, this is\nexactly given by $\\ensuremath\\mathrm{Des}_k(w)$ for the same reading word $w$. Here it is\nessential that $w$ be allowed to contain $\\emptyset$'s in order to\ncorrectly space the entries of the $k$-tuple.\n\nThe case when the $k$-tuple consists entirely of ribbons is an\nimportant special case in light of \\cite{Haglund2004,HHL2005} where it\nis shown that the Macdonald polynomials are in fact positive sums of\nLLT polynomials where the shapes are ribbons. In this context, the\nindex $k$ is given by the number of columns of the indexing partition\nof the Macdonald polynomial. The Macdonald Positivity Theorem,\nconjectured by Macdonald in 1988 \\cite{Macdonald1988}, was first\nproved by Haiman using algebraic geometry \\cite{Haiman2001}, and more\nrecently by Grojnowski and Haiman using Kazhdan-Lusztig theory\n\\cite{GrHa2007} and the author using a purely combinatorial argument\n\\cite{Assaf2007-2}. This latter proof, while purely combinatorial,\nrelies on new combinatorial machinery, namely {\\em dual equivalence\n  graphs}, involving rather technical proofs of the main\ntheorems. Below we suggest how Macdonald positivity may be recovered\nin a completely elementary way using bijections similar to\n$\\phi^{(k)}$.\n\nThe main idea behind \\cite{Assaf2007-2} is to group together terms of\na Macdonald polynomial which contribute to a single Schur function and\nhave the same associated statistics. This is done in three steps; for\ncomplete details, see \\cite{Assaf2007-3}. First, quasisymmetric\nfunctions are used to reduce to standard words, i.e. permutations, and\nit is here that the inverse descent set of a permutation is\nrelevant. Next, for a given $k$, the permutations are divided into\nequivalence classes (in the language of \\cite{Assaf2007-2}, connected\ncomponents of a graph) using the following involutions.\n\nFor $i \\geq 2$, define involutions $d_i$ and $\\tilde{d}_i$ on permutations\nwhere $i$ does not lie between $i-1$ and $i+1$ by\n\\begin{eqnarray}\n  d_i    (\\cdots\\;   i  \\;\\cdots\\;i\\pm 1\\;\\cdots\\;i\\mp 1\\;\\cdots ) \n  & = &  \\cdots\\;i\\mp 1\\;\\cdots\\;i\\pm 1\\;\\cdots\\;  i   \\;\\cdots \\; ,\n  \\label{eqn:d} \\\\\n  \\tilde{d}_i(\\cdots\\;   i  \\;\\cdots\\;i\\pm 1\\;\\cdots\\;i\\mp 1\\;\\cdots ) \n  & = &  \\cdots\\;i\\pm 1\\;\\cdots\\;i\\mp 1\\;\\cdots\\;  i   \\;\\cdots \\; ,\n  \\label{eqn:dwig}\n\\end{eqnarray}\nwhere all other entries remain fixed. Combining these, define\n$D^{(k)}_i$ by\n\\begin{equation}\n  D^{(k)}_i(w) \\; = \\; \\left\\{\n    \\begin{array}{ll}\n      d_i(w) & \\mbox{if} \\;\\; \\mathrm{dist}(i-1,i,i+1) > k \\\\\n      \\tilde{d}_i(w) & \\mbox{if} \\;\\; \\mathrm{dist}(i-1,i,i+1) \\leq k\n    \\end{array} \\right. ,\n  \\label{eqn:Dk}\n\\end{equation}\nwhere $\\mathrm{dist}(i-1,i,i+1)$ is the maximum distance between the positions\nof $i-1,i,i+1$ in $w$. The $\\emptyset$'s, or spacers, in $w$ are\nessential for this step as they adjust the relative distance of the\nletters of $w$.\n\n\\begin{definition}\n  Call two permutations $w$ and $u$ {\\em $k$-equivalent}, denoted\n  $w \\sim_{k} u$, if $w = D^{(k)}_{i_1} D^{(k)}_{i_2} \\cdots\n  D^{(k)}_{i_m}(u)$ for some sequence $i_1,i_2,\\ldots,i_m \\geq 2$.\n\\end{definition}\n\n\\begin{figure}[ht]\n  \\begin{center}\n    \\begin{displaymath}\n      \\begin{array}{\\ccr \\ccr \\ccr \\ccr c}\n        \\mbox{$1$-classes:} &\n        \\left\\{ 1 \\ 2 \\ 3 \\right\\}; &\n        \\left\\{ 2 \\ 1 \\ 3 \\ , \\ 3 \\ 1 \\ 2 \\right\\}; &\n        \\left\\{ 2 \\ 3 \\ 1 \\ , \\ 1 \\ 3 \\ 2 \\right\\}; &\n        \\left\\{ 3 \\ 2 \\ 1 \\right\\} \\\\[.3\\vsp]\n        \\mbox{$2$-classes:} &\n        \\left\\{ 1 \\ 2 \\ 3 \\right\\}; &\n        \\left\\{ 2 \\ 1 \\ 3 \\ , \\ 1 \\ 3 \\ 2 \\right\\}; &\n        \\left\\{ 2 \\ 3 \\ 1 \\ , \\ 3 \\ 1 \\ 2 \\right\\}; &\n        \\left\\{ 3 \\ 2 \\ 1 \\right\\}\n      \\end{array}\n    \\end{displaymath}      \n    \\caption{\\label{fig:classes} Equivalence classes of permutations\n      of length $3$.}\n  \\end{center}\n\\end{figure}\n\n\\begin{remark}\n  Note that the $1$-equivalence classes are exactly the {\\em dual\n    equivalence classes} for partitions; see \\cite{Haiman1992}. In\n  particular, the sum of the quasisymmetric functions associated to\n  the permutations in a $1$-equivalence class is a Schur function.\n\\label{rmk:dec}\n\\end{remark}\n\nA key observation in \\cite{Assaf2007-2} is that $\\ensuremath\\mathrm{Des}_k(w) =\n\\ensuremath\\mathrm{Des}_k\\left( D^{(k)}_{i} (w) \\right)$ and $|\\ensuremath\\mathrm{Inv}_k(w)| = |\\ensuremath\\mathrm{Inv}_k\\left(\n  D^{(k)}_{i} (w) \\right)|$.  In particular, $\\ensuremath\\mathrm{Des}_k$ and $|\\ensuremath\\mathrm{Inv}_k|$\nare constant on $k$-equivalence classes. Therefore, the third and\nfinal step toward establishing the Macdonald Positivity Theorem is to\nprove that the sum over the quasisymmetric functions associated to a\ngiven $k$-equivalence class is Schur positive. By \\refrmk{dec}, a\nnatural approach is to relate $k$-classes to $1$-classes. Indeed, the\nproof presented in \\cite{Assaf2007-2} does this by showing that a\nconnected component of the graph for $k$-columns (a $k$-equivalence\nclass) may be broken into a union of connected dual equivalence graphs\n($1$-equivalence classes). It is for this step that the proof becomes\nquite technical and involved, and so the idea is to bypass the\nmachinery of dual equivalence graphs altogether. The following\nproposition achieves this for the $2$-column\/$2$-equivalence class\ncase.\n\n\\begin{proposition}\n  For $w$ a permutation such that $i$ does not lie between $i-1$ and\n  $i+1$, we have\n  \\begin{equation}\n    \\phi^{(2)} \\left( D^{(1)}_i(w) \\right) = D^{(2)}_i \\left(\n      \\phi^{(2)}(w) \\right) .\n  \\end{equation}\n\\label{prop:2equiv}\n\\end{proposition}\n\n\\begin{proof}\n  First note that $D^{(1)}_i = d_i$. Furthermore, $D^{(2)}_i(w) =\n  d_i(w)$ unless $i-1,i,i+1$ are adjacent in $w$. Without loss of\n  generality, we may assume that $w_r = i+1$, $w_s = i-1$ and $w_t =\n  i$ for some indices $r<s<t$. Set $\\widetilde{w} = d_1(w)$, so that $\\widetilde{w}_r =\n  i$, $\\widetilde{w}_s = i-1$ and $\\widetilde{w}_t = i+1$. We aim to show that\n  $D^{(2)}_i(\\phi^{(2)}(w)) = \\phi^{(2)}(\\widetilde{w})$.\n\n  For notational convenience, we write $\\gamma_j$ for $\\gamma^{(2)}_j$\n  and $\\Gamma_j$ for $\\Gamma^{(2)}_j$. Since the definition of\n  $\\Gamma_j$ depends only on the relative orders of letters, it\n  follows that $\\Gamma_{j} (w) = \\Gamma_{j}(\\widetilde{w})$ for $j < t$. Along\n  the same lines, $\\Gamma_{t}(w) \\neq \\Gamma_{t}(\\widetilde{w})$ if and only if\n  $r,s = t-2,t-1$. If this is not the case, then $\\gamma_t \n  \\cdots  \\gamma_1 (w) = d_i \\left( \\gamma_t  \\cdots \n    \\gamma_1 (\\widetilde{w}) \\right)$ and indeed $\\mathrm{dist}(i-1,i,i+1) > 2$ in\n  $\\gamma_t  \\cdots  \\gamma_1 (w)$. In the affirmative case,\n  $\\Gamma_{t}(w) = \\{t-2\\}$ since both or neither $i-1,i+1$ splits any\n  pair of preceding letters and $\\Gamma_{t}(\\widetilde{w}) =\n  \\emptyset$. Therefore $\\gamma_t  \\cdots  \\gamma_1 (w) =\n  \\tilde{d}_i \\left( \\gamma_t  \\cdots  \\gamma_1 (\\widetilde{w})\n  \\right)$ as desired since $\\mathrm{dist}(i-1,i,i+1) = 2$ in $\\gamma_t \n  \\cdots  \\gamma_1 (w)$.\n\n  Now consider the effect of $\\gamma_j$ for $j > t$. For the same\n  reasons as before, $\\Gamma_{j}(w) \\neq \\Gamma_{j}(\\widetilde{w})$ if and only\n  if $i-1,i,i+1$ are adjacent either before or after $\\gamma_j$ is\n  applied. For $\\widetilde{w}$, the relative positions of $i-1,i,i+1$ will never\n  change. Moreover, the position of $i+1$ in $\\widetilde{w}$ tracks the position\n  of $i$ in $w$, and the positions of $i,i-1$ in $\\widetilde{w}$ are the\n  positions of $i-1,i+1$ in $w$ (though not necessarily\n  respectively). For $w$, each time $i$ moves between adjacent and\n  nonadjacent to $i-1,i+1$, the difference between $\\Gamma_j$ for $w$\n  and $\\widetilde{w}$ is exactly that the former contains the index of the\n  leftmost of $i-1,i+1$ and the latter does not. Comparing $d_i$ with\n  $\\tilde{d}_i$, this is exactly the difference between the two\n  involutions, i.e. $i-1$ and $i+1$ interchange positions. Therefore\n  in the end, $\\tilde{d}_i(\\phi^{(2)}(w)) = \\phi^{(2)}(\\widetilde{w})$ if\n  $i-1,i,i+1$ are adjacent in $\\phi^{(2)}(w)$, and $d_i(\\phi^{(2)}(w))\n  = \\phi^{(2)}(\\widetilde{w})$ otherwise.\n\\end{proof}\n\nRecall that the sum over of an equivalence class is determined by the\nquasisymmetric functions associated to the permutation of the\nclass. Since the quasisymmetric function associated to a permutation\nis determined by the inverse descent set of the permutation,\n\\refprop{props} (iii) and \\refrmk{dec} establish the following\ncorollary to \\refprop{2equiv}.\n\n\\begin{corollary}\n  Macdonald polynomials indexed by partitions with $2$ columns are\n  Schur positive.\n\\label{cor:2pos}\n\\end{corollary}\n\nFor $k \\geq 3$, it is not possible for $\\phi^{(k)} \\left(\n  D^{(k-1)}_i(w) \\right) = D^{(k)}_i \\left( \\phi^{(k)}(w) \\right)$ in\ngeneral. The reason for this is that the sizes of the $k$-equivalence\nclasses increase with $k$. For permutations of length $n$, the\n$n$-equivalence classes have a nice description given in\n\\cite{Assaf2007-2} which allows us to prove the following.\n\n\\begin{proposition}\n  For $w$ a permutation such that $i$ does not lie between $i-1$ and\n  $i+1$, we have\n  \\begin{equation}\n    \\phi^{[1,n]} (w) \\sim_{n} \\phi^{[1,n]} \\left( D^{(1)}_i(w) \\right) .\n  \\end{equation}\n\\label{prop:nequiv}\n\\end{proposition}\n\n\\begin{proof}\n  For this case, $\\ensuremath\\mathrm{maj}_n = \\ensuremath\\mathrm{inv}$ in the usual sense and there are no\n  $n$-descents to consider. As already noted, $\\ensuremath\\mathrm{inv}$ is constant on\n  $n$-equivalence classes and, since $D^{(n)}_i \\equiv \\tilde{d}_i$,\n  $w_1 > w_n$ for some $w$ in an $n$-class if and only if $w_1 > w_n$\n  for every $w$ in an $n$-class. Furthermore, it is not difficult to\n  show that these two properties completely characterize $n$-classes.\n  Since $D^{(1)}_i \\equiv d_i$, $w_{n-1} > w_n$ for some $w$ in a\n  $1$-class if and only if $w_{n-1} > w_n$ for every $w$ in a\n  $1$-class.  By \\refprop{majk}, $\\ensuremath\\mathrm{maj}(w) = \\ensuremath\\mathrm{inv}(\\phi^{[1,n]} (w))$,\n  and by \\refprop{props} (ii), $w_{n-1} > w_n$ if and only if\n  $\\phi^{[1,n]} (w)_1 > \\phi^{[1,n]} (w)_n$. Therefore if $w \\sim_1\n  u$, then $\\ensuremath\\mathrm{inv}(\\phi^{[1,n]} (w)) = \\ensuremath\\mathrm{inv}(\\phi^{[1,n]} (u))$ and\n  $\\phi^{[1,n]} (w)_1 > \\phi^{[1,n]} (w)_n$ if and only if\n  $\\phi^{[1,n]} (u)_1 > \\phi^{[1,n]} (u)_n$. The result now follows.\n\\end{proof}\n\n\\begin{corollary}\n  Macdonald polynomials indexed by a single row are Schur positive.\n\\label{cor:npos}\n\\end{corollary}\n\nGiven this, one might still hope to express each $k$-equivalence class\nas a union of the images of certain $k-1$-equivalence classes under an\nappropriate map. However, for $k \\geq 3$, neither $\\phi^{(k)}$ nor the\ncorresponding composition of Kadell's bijections accomplishes\nthis. There is, however, considerable evidence suggesting that such a\nfamily of bijections does exist, and so we conclude with the following\nconjecture which, as a corollary, would yield a simple proof of\nMacdonald positivity.\n\n\\begin{conjecture}\n  There exists a family of bijections $\\theta^{(k)}$ on permutations\n  satisfying Propositions \\ref{prop:props} and \\ref{prop:majk} such\n  that if $w \\sim_{k-1} u$ then $\\theta^{(k)}(w) \\sim_{k}\n  \\theta^{(k)}(u)$.\n\\label{conj:theta}\n\\end{conjecture}\n\n\n\n\n\\bibliographystyle{amsalpha}\n\n\n\n\n\n\n\\bibliographystyle{abbrv} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHD~140283 (V = 7.21; Casagrande et al. 2010) is a subgiant, metal-poor \nstar in the solar neighbourhood, which is analysed extensively in the literature, \nand has historical importance in the context of the existence of \nmetal-deficient stars (Chamberlain \\& Aller 1951). \nIn recent decades, element abundances in HD~140283 were studied \nby several authors, and a bibliographic compilation up to 2010 can be found \nin the PASTEL catalogue (Soubiran et al. 2010). Most recently, Frebel \\& Norris (2015) \nstressed the importance of this star to the history of the discovery of the most metal-poor \nstars in the halo. More recently, hydrodynamical \n3D models and NLTE computations were applied for lithium lines, for example, by \nLind et al. (2012) and Steffen et al. (2012). \nThe fractions of odd and even barium isotopes in HD~140283 have been the subject of \nintense debate, given that the even-Z isotopes are only produced by the \nneutron capture s-process, whereas the odd-Z isotopes are produced by both \nthe s- and r-processes (Gallagher et al. 2010, 2012, 2015). \nBased on UV spectra, the molybdenum abundance in HD~140283 was derived by \nPeterson (2011). Roederer (2012) also used UV lines to obtain abundances of \nzinc (\\ion{Zn}{II}), arsenic (\\ion{As}{I}), and selenium (\\ion{Se}{I}), \namong other elements, in addition to upper limits for germanium (\\ion{Ge}{I}) \nand platinum (\\ion{Pt}{I}). Siqueira-Mello et al. (2012), hereafter Paper I, \nanalysed the origin of heavy elements in HD~140283 deriving the europium \nabundance and making the case for an r-process contribution in this star. \n\nBond et al. (2013) derived an age of 14.46$\\pm$0.31 Gyr for HD~140283, using \na trigonometric parallax of 17.15$\\pm$0.14 mas measured with the Hubble \nSpace Telescope, making this object the oldest known star for which a \nreliable age has been determined. Bond et al. employed evolutionary tracks \nand isochrones computed with the University of Victoria code (VandenBerg et \nal. 2012), with an adopted helium abundance of Y$=$0.250, and including \neffects of diffusion, revised nuclear reaction rates, and enhanced \noxygen abundance. More recently, VandenBerg et al. (2014) presented a \nrevised age of 14.27$\\pm$0.38 Gyr. This age is slightly larger than \nthe age of the universe of 13.799$\\pm$0.038 Gyr  based on the cosmic \nmicrowave background (CMB) radiation as given by the \nPlanck collaboration (Adam et al. 2015). \nAccording to VandenBerg et al. (2014), uncertainties, particularly in the oxygen \nabundance and model temperature to observed colour relations, can explain \nthis difference, but the remote possibility that this object is older than \n14 Gyr cannot be excluded.  HD~140283 is therefore a very old \nstar that must have formed soon after the Big Bang. In the future, \nasteroseismology could help to verify the age of HD~140283.\n\nIn this work, we carry out a detailed analysis and abundance derivation \nfor HD~140283. The main motivation for this study was triggered by the \ncontroversy discussed above about the interpretation of barium isotopic abundances, \nand the possibility of testing whether heavy elements \nare produced by the r- or s-process. With this purpose, we \nobtained a seven-hour exposure high-S\/N spectrum, with a wavelength coverage \nin the range 3700~$<\\lambda({\\rm \\AA})<$~10475 for this star.\n\nIn Sect. 2 the observations are reported. \nIn Sect. 3 the atmospheric parameters are derived. \nIn Sect. 4 the abundances computed in LTE and NLTE are presented. \nIn Sect. 5 the results are discussed, and final conclusions are drawn in Sect. 6.\n\n\\section {Observations and reductions}\n\nHD~140283 was observed in programme 11AB01 (PI: B. Barbuy) \nat the CFHT telescope with the spectrograph \nESPaDOnS in Queue Service Observing (QSO) mode, to obtain a spectrum \nin the wavelength range 3700-10475~{\\AA} with a resolving power of \nR~$=$~81 000. The observations were carried out in 2011, June 12, 14, 15, \nand 16, and July 8. The total number of 23 individual spectra with 20 min \nexposure each produced a total exposure time of more than seven hours. \nThe data reduction was performed using the software \nLibre-ESpRIT, a new release of ESpRIT (Donati et al. 1997), running within \nthe CFHT pipeline Upena\\footnote{http:\/\/www.cfht.hawaii.edu\/Instruments\/Upena\/index.html}. \nThis package facilitates reducing all exposures automatically, and further fits  continua \nand normalizes to 1. The co-added spectrum was obtained after radial velocity correction and \na S\/N ratio of 800~$-$~3400 per pixel was obtained. Three spectra were \ndiscarded because of their lower quality as compared with the average.\n\n\\section{Atmospheric parameters}\n\n\n\\subsection{Measurement of equivalent widths}\n\\label{EW}\n\nTo derive the atmospheric parameters and abundances, \nwe measured the equivalent widths (EWs) of several iron and titanium \nlines in their neutral and ionized states using a semi-automatic code, \nwhich traces the continuum and uses a Gaussian profile to fit the \nabsorption lines, as described in Siqueira-Mello et al. (2014). The \ncode can deal with blends on the wings, excluding the parts of the \nline from the computations. \n\nTo check the reliability of the implemented code, the results were \ncompared with those obtained using the routine for the automatic \nmeasurement of line equivalent widths in stellar spectra ARES \n(Sousa et al. 2007), and only the lines identified by ARES were used to achieve \nthe best confidence in the final results. A very good agreement between \nthe two measurements is shown in Fig. \\ref{EW_compara}. We find a mean \ndifference of $\\hbox{EW(this work)}-\\hbox{EW(ARES)}=-0.22\\pm0.43$~m{\\AA}, \nwhich can be considered negligible in terms of EWs.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{EW.eps}\n\\caption{Comparison of EWs measured for a set of \\ion{Fe}{I} and \n\\ion{Fe}{II} lines in HD 140283 using the code by Siqueira-Mello et al.\n(2014) and EWs measured using ARES.}\n\\label{EW_compara}\n\\end{figure}\n\nThe complete list of Fe and Ti lines used is shown in Table \\ref{EW_measurements}, \nwhich includes wavelength ({\\AA}), excitation potential (eV), \\loggf~values from \nthe VALD and NIST\\footnote{http:\/\/physics.nist.gov\/PhysRefData\/ASD\/lines$_-$form.html} \ndatabases, the measured EWs (m{\\AA}), and the derived abundances. Using the errors \ngiven for the parameters of the Gaussian profile obtained from the fitting \nprocedure, the uncertainties $\\sigma$EW of the equivalent widths were \ncomputed based on standard error propagation. \n\n\n\\subsection{Calculations}\n\nIron and titanium abundances were derived using equivalent widths, \nas usual. All other element abundances  were derived from fits of \nsynthetic spectra to the observed spectrum of HD~140283. The OSMARCS \n1D model atmosphere grid  was employed (Gustafsson et al. 2008). \n\nWe used the spectrum synthesis code Turbospectrum (Alvarez \\& Plez 1998), \nwhich includes treatment of scattering in the blue and UV domain, \nmolecular dissociative equilibrium, and collisional broadening by H, \nHe, and H$_{2}$, following Anstee \\& O'Mara (1995), Barklem \\& O'Mara (1997), \nand Barklem et al. (1998). The calculations used the Turbospectrum molecular \nline lists (Alvarez \\& Plez 1998), and atomic line lists from the VALD \ncompilation (Kupka et al. 1999)\\footnote{http:\/\/vald.astro.univie.ac.at\/~vald3\/php\/vald.php}. \nWhen available, new experimental oscillator strengths were adopted from \nliterature. In addition, hyperfine structure (HFS) splitting and isotope \nshifts were implemented when needed and available.\n\nWe also performed NLTE calculations for the following ions: \\ion{C}{I}, \n\\ion{O}{I}, \\ion{Na}{I}, \\ion{Mg}{I}, \\ion{Al}{I}, \\ion{K}{I}, \\ion{Ca}{I}, \n\\ion{Sr}{II}, and \\ion{Ba}{II}. Atomic models for these species were used \nin combination with the NLTE MULTI code (Carlsson 1986; Korotin et al. 1999), \nwhich facilitates a very good description of the radiation field. The updated \nversion of the MULTI code includes opacities from ATLAS9 (Castelli \\& Kurucz 2003), \nwhich modify the intensity distribution in the UV region.\n\nThe lines studied in NLTE are often blended \nwith lines of other species. Proper comparison of the synthesized and \nobserved profiles thus requires a multi-element synthesis. To accomplish this, \nwe fold the NLTE (MULTI) calculations into the LTE synthetic spectrum \ncode SYNTHV (Tsymbal 1996). With these two codes, we calculate synthetic \nspectra for each region in the vicinity of the line of interest taking \n(in LTE) all the blending lines located in  this region and listed in \nthe VALD database into account. \nFor the lines of interest (treated in NLTE), the corresponding departure \ncoefficients (so-called $b$-factors: $b = n_{\\rm i}\/n^{*}_{\\rm i}$, the \nratio of NLTE to LTE atomic level populations) obtained with MULTI code are \nthe input to SYNTHV code, where they are used in the calculation of the line \nsource function, and then for the NLTE line profile. Other possible blending \nlines are treated in LTE. \n\n\n\\subsection{Stellar parameters}\n\nFollowing Paper I, we adopted the stellar \nparameters \\Teff~$=5750\\pm100$~K, $\\hbox{[Fe\/H]}\\footnote{\n[X\/H]~$=$~A(X)$_{star}-$~A(X)$_{\\odot}$}=-2.5\\pm0.2$ \nand $\\xi=1.4\\pm0.1$~\\kms~from Aoki et al. (2004) and \n\\logg~$=3.7\\pm0.1$~[g in cgs] from Collet et al. (2009). \nUsing the newly measured EWs, we obtained the iron \nabundances A(\\ion{Fe}{I})\\footnote{We adopted the notation \nA(X) = log~(X) = log~n(X)\/n(H)~+~12, with n = number \ndensity of atoms.}~$=+4.91\\pm0.07$ and A(\\ion{Fe}{II})~$=+4.96\\pm0.07$, \nor $\\hbox{[Fe I\/H]}=-2.59\\pm0.08$ and $\\hbox{[Fe II\/H]}=-2.54\\pm0.08$, \nusing the solar abundance A(Fe)$_{\\odot}=+7.50\\pm0.04$ \nfrom Asplund et al. (2009). The results are in very good agreement with the \nliterature, as in Gallagher et al. (2010), where [Fe\/H]$=-2.59\\pm0.06$ was obtained. \nFor titanium, we found A(\\ion{Ti}{I})~$=+2.71\\pm0.01$ \nand A(\\ion{Ti}{II})~$=+2.69\\pm0.03$, or $\\hbox{[Ti I\/H]}=-2.24\\pm0.05$ \nand $\\hbox{[Ti II\/H]}=-2.26\\pm0.06$, using the solar abundance of \ntitanium A(Ti)$_{\\odot}=+4.95\\pm0.05$ from Asplund et al. (2009). \nIn Table \\ref{summary} we summarize the atmospheric parameters \nadopted, together with the iron and titanium abundances.\n\nFigure \\ref{avalia} shows the dependence of [\\ion{Fe}{I}\/H], [\\ion{Fe}{II}\/H], \n[\\ion{Ti}{I}\/H], and [\\ion{Ti}{II}\/H] on log~$(EW\/\\lambda)$, and on the \nexcitation potential of the lines obtained for HD~140283. \nThe blue solid lines represent the average \nabundances in each case. The excitation and ionization equilibria of \nFe and Ti lines, resulting from the set of atmospheric parameters adopted \nfor HD~140283, confirm the robustness of our choice. However, it should be \nnoted that the surface gravity value may be affected by NLTE effects and \nby uncertainties in the oscillator strengths of the Fe and Ti lines.\n\nThe broadening parameters in HD~140283 were carefully analysed by \nseveral authors. We adopted a Gaussian profile \nto take the effects of macroturbulence, rotational, and \ninstrumental broadening into account.\n\n\\begin{table}\n\\caption{Atmospheric parameters adopted for HD~140283.}            \n\\label{summary}     \n\\scalefont{1.0}\n\\centering                         \n\\begin{tabular}{cc}       \n\\hline\\hline                \n\\noalign{\\smallskip}\n\\hbox{} & \\hbox{Values} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{\\Teff}           & $5750\\pm100$~K \\\\\n\\hbox{\\logg}           & $3.7\\pm0.1$~[g in cgs] \\\\\n\\hbox{[Fe\/H]$_{model}$}& $-2.5\\pm0.2$ \\\\\n\\hbox{$\\xi$}           & $1.4\\pm0.1$~\\kms \\\\\n\\hbox{[\\ion{Fe}{I}\/H]} & $-2.59\\pm0.08$ \\\\\n\\hbox{[\\ion{Fe}{II}\/H]}& $-2.54\\pm0.08$ \\\\\n\\hbox{[\\ion{Ti}{I}\/H]} & $-2.24\\pm0.05$ \\\\\n\\hbox{[\\ion{Ti}{II}\/H]}& $-2.26\\pm0.06$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{avalia.eps}\n\\caption{Excitation and ionization equilibria of Fe and Ti lines, \nresulting from the set of atmospheric parameters for HD~140283. \nThe black dots are the abundances obtained from \\ion{Ti}{I} and \n\\ion{Fe}{I} lines, the red squares are those from \\ion{Ti}{II} \nand \\ion{Fe}{II} lines, and the blue solid lines represent the \naverage abundances.}\n\\label{avalia}\n\\end{figure}\n\n\\subsection{Uncertainties in the derived abundances}\n\nAs described in Paper I, the adopted atmospheric parameters present \ntypical errors of $\\Delta$T$_{\\rm eff}=\\pm100$~K, $\\Delta$\\logg~$=\\pm0.1$~[g in cgs], \nand $\\Delta\\xi=\\pm0.1$~\\kms. Since the stellar parameters are not \nindependent from each other, the quadratic sum of the various sources \nof uncertainties is not the best way to estimate the total error budget, \notherwise it is mandatory to include the covariance terms in this calculation, \nand an estimated correlation matrix may introduce uncontrollable error sources.\n\nTo compute the total error budget in the abundance analysis arising \nfrom the stellar parameters, we created a new atmospheric model with a 100 K \nlower temperature, determining the corresponding surface gravity and \nmicroturbulent velocity with the traditional spectroscopic method. \nRequiring that the iron abundance derived from \\ion{Fe}{I} and \\ion{Fe}{II} \nlines be identical, we determined the respective \\logg~value, and the microturbulent \nvelocity was found requiring that the abundances derived for individual \\ion{Fe}{I} \nlines be independent of the equivalent width values. \nThe result is a model with T$_{\\rm eff}=5650$~K, \\logg~$=3.3$~[g in cgs], \nand $\\xi=1.2$~\\kms. The abundance analysis was carried out with this new model, \nand the difference in comparison with the nominal model should represent the \nuncertainties from the atmospheric parameters. \n\nObservational errors were estimated using the standard deviation of the abundances \nfrom the individual lines for each element, and taking into account the uncertainties \nin defining the continuum, fitting the line profiles, and in the oscillator strengths. \nFor the elements with only one line available, we adopted the observational error from \niron as a representative value. The adopted total error budget is the quadratic sum of \nuncertainties arising from atmospheric parameters and observations. \n\nTo estimate the uncertainties in the individual abundances due to the uncertainties \nin the EWs $\\sigma$EW, we recomputed the abundances using $\\hbox{EW}+\\sigma\\hbox{EW}$, \nand the differences with respect to the nominal values were adopted as the errors. \nThe errors line-by-line are also shown in Table \\ref{EW_measurements}. \n\n\n\\begin{table}\n\\caption{Abundance uncertainties due to stellar parameters $\\Delta_{par}$, \nobservational errors $\\Delta_{obs}$, and adopted total error budget $\\Delta_{total}$ \nfor LTE abundances.}            \n\\label{finalabund}     \n\\scalefont{1.0}\n\\centering                         \n\\begin{tabular}{ccrr}       \n\\hline\\hline                \n\\noalign{\\smallskip}\n\\hbox{Element} & \\hbox{$\\Delta_{par}$} & \\hbox{$\\Delta_{obs}$} & \\hbox{$\\Delta_{total}$}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{[Fe\/H]}  & $-$0.07 & 0.07 & 0.10 \\\\\n\\hbox{[Li\/H]}  & $-$0.08 & 0.07 & 0.11 \\\\\n\\hbox{[C\/Fe]}  & $+$0.01 & 0.07 & 0.07 \\\\\n\\hbox{[N\/Fe]}  & $-$0.10 & 0.07 & 0.12 \\\\\n\\hbox{[O\/Fe]}  & $-$0.15 & 0.07 & 0.17 \\\\\n\\hbox{[Na\/Fe]} & $-$0.04 & 0.08 & 0.09 \\\\\n\\hbox{[Mg\/Fe]} & $-$0.04 & 0.07 & 0.08 \\\\\n\\hbox{[Al\/Fe]} & $-$0.01 & 0.07 & 0.07 \\\\\n\\hbox{[Si\/Fe]} & $-$0.08 & 0.01 & 0.08 \\\\\n\\hbox{[K\/Fe]}  & $-$0.06 & 0.07 & 0.09 \\\\\n\\hbox{[Ca\/Fe]} & $-$0.03 & 0.04 & 0.05 \\\\\n\\hbox{[Sc\/Fe]} & $-$0.07 & 0.06 & 0.09 \\\\\n\\hbox{[Ti\/Fe]} & $-$0.07 & 0.03 & 0.08 \\\\\n\\hbox{[V\/Fe]}  & $-$0.07 & 0.08 & 0.11 \\\\\n\\hbox{[Cr\/Fe]} & $-$0.08 & 0.04 & 0.09 \\\\\n\\hbox{[Mn\/Fe]} & $-$0.07 & 0.04 & 0.08 \\\\\n\\hbox{[Co\/Fe]} & $-$0.07 & 0.11 & 0.13 \\\\\n\\hbox{[Ni\/Fe]} & $-$0.07 & 0.05 & 0.09 \\\\\n\\hbox{[Zn\/Fe]} & $-$0.07 & 0.07 & 0.10 \\\\\n\\hbox{[Sr\/Fe]} & $-$0.02 & 0.07 & 0.07 \\\\\n\\hbox{[Y\/Fe]}  & $-$0.10 & 0.06 & 0.12 \\\\\n\\hbox{[Zr\/Fe]} & $-$0.11 & 0.05 & 0.12 \\\\\n\\hbox{[Ba\/Fe]} & $-$0.09 & 0.10 & 0.13 \\\\\n\\hbox{[Ce\/Fe]} & $-$0.07 & 0.18 & 0.19 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{Abundance derivation}\n\nTable \\ref{linelist} shows the line list of elements other than Fe and \nTi used in this work, including wavelength ({\\AA}), excitation \npotential (eV), adopted oscillator strength, and abundance derived \nfrom each line. The final LTE abundances derived in HD~140283 for all \nthe analysed species are shown in Table \\ref{finalabund}. The adopted \nsolar abundances from Asplund et al. (2009) are also listed in \nTable \\ref{finalabund}. We discuss below each element in terms of \nlines used, HFS splitting, and abundances adopted. In Table \n\\ref{fractions} we report the solar isotopic fractions adopted \nfrom Asplund et al. (2009) that are relevant for HFS computations, \nand in Table \\ref{HFSconstants} we summarize the hyperfine coupling \nconstants adopted for the lines retained in this analysis.\n\n\\begin{table}\n\\caption{LTE abundances. Column 2 gives the solar abundances from Asplund et al.\n(2009), columns 3, 4, and 5 give the absolute abundance\nwith respect to A(H)=12.0 and the usual logarithmic ratio notation\n with respect to H and to Fe, respectively.}            \n\\label{finalabund}     \n\\scalefont{1.0}\n\\centering                         \n\\begin{tabular}{ccrrr}       \n\\hline\\hline                \n\\noalign{\\smallskip}\n\\hbox{Ion} & \\hbox{A(X)$_{\\odot}$} & \\hbox{A(X)} & \\hbox{[X\/H]} & \\hbox{[X\/Fe]} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{\\ion{Fe}{I}}  & 7.50$\\pm$0.04 & $+$4.91 & $-$2.59 & ---  \\\\\n\\hbox{\\ion{Fe}{II}} & 7.50$\\pm$0.04 & $+$4.96 & $-$2.54 & ---  \\\\\n\\hbox{\\ion{Li}{I}}  & 1.05$\\pm$0.10 & $+$2.14 & $+$1.09 & ---  \\\\\n\\hbox{C(CH)}        & 8.43$\\pm$0.05 & $+$6.30 & $-$2.13 & $+$0.46 \\\\\n\\hbox{\\ion{C}{I}}   & 8.43$\\pm$0.05 & $+$6.44 & $-$1.99 & $+$0.60 \\\\\n\\hbox{N(CN)}        & 7.83$\\pm$0.05 & $+$6.30 & $-$1.53 & $+$1.06 \\\\\n\\hbox{[\\ion{O}{I}]} & 8.69$\\pm$0.05 & $+$6.95 & $-$1.74 & $+$0.85 \\\\\n\\hbox{\\ion{O}{I}}   & 8.69$\\pm$0.05 & $+$7.11 & $-$1.58 & $+$1.00 \\\\\n\\hbox{\\ion{Na}{I}}  & 6.24$\\pm$0.04 & $+$3.62 & $-$2.62 & $-$0.04  \\\\\n\\hbox{\\ion{Mg}{I}}  & 7.60$\\pm$0.04 & $+$5.27 & $-$2.33 & $+$0.26  \\\\\n\\hbox{\\ion{Mg}{II}} & 7.60$\\pm$0.04 & $+$5.66 & $-$1.95 & $+$0.64  \\\\\n\\hbox{\\ion{Al}{I}}  & 6.45$\\pm$0.03 & $+$2.96 & $-$3.50 & $-$0.91 \\\\\n\\hbox{\\ion{Si}{I}}  & 7.51$\\pm$0.03 & $+$5.30 & $-$2.21 & $+$0.38  \\\\\n\\hbox{\\ion{Si}{II}} & 7.51$\\pm$0.03 & $+$5.35 & $-$2.16 & $+$0.43  \\\\\n\\hbox{\\ion{K}{I}}   & 5.03$\\pm$0.09 & $+$2.98 & $-$2.05 & $+$0.54 \\\\\n\\hbox{\\ion{Ca}{I}}  & 6.34$\\pm$0.04 & $+$4.03 & $-$2.31 & $+$0.27 \\\\\n\\hbox{\\ion{Ca}{II}} & 6.34$\\pm$0.04 & $+$4.43 & $-$1.91 & $+$0.68 \\\\\n\\hbox{\\ion{Sc}{I}}  & 3.15$\\pm$0.04 & $+$0.58 & $-$2.58 & $+$0.01 \\\\\n\\hbox{\\ion{Sc}{II}} & 3.15$\\pm$0.04 & $+$0.75 & $-$2.40 & $+$0.18 \\\\\n\\hbox{\\ion{Ti}{I}}  & 4.95$\\pm$0.05 & $+$2.71 & $-$2.24 & $+$0.34 \\\\\n\\hbox{\\ion{Ti}{II}} & 4.95$\\pm$0.05 & $+$2.69 & $-$2.26 & $+$0.32 \\\\\n\\hbox{\\ion{V}{I}}   & 3.93$\\pm$0.08 & $+$1.44 & $-$2.49 & $+$0.09 \\\\\n\\hbox{\\ion{V}{II}}  & 3.93$\\pm$0.08 & $+$1.70 & $-$2.23 & $+$0.36  \\\\\n\\hbox{\\ion{Cr}{I}}  & 5.64$\\pm$0.04 & $+$2.95 & $-$2.69 & $-$0.11 \\\\\n\\hbox{\\ion{Cr}{II}} & 5.64$\\pm$0.04 & $+$3.32 & $-$2.32 & $+$0.26 \\\\\n\\hbox{\\ion{Mn}{I}}  & 5.43$\\pm$0.04 & $+$2.56 & $-$2.87 & $-$0.29 \\\\\n\\hbox{\\ion{Co}{I}}  & 4.99$\\pm$0.07 & $+$2.69 & $-$2.30 & $+$0.29 \\\\\n\\hbox{\\ion{Ni}{I}}  & 6.22$\\pm$0.04 & $+$3.76 & $-$2.46 & $+$0.12 \\\\\n\\hbox{\\ion{Ni}{II}} & 6.22$\\pm$0.04 & $+$3.88 & $-$2.34 & $+$0.25 \\\\\n\\hbox{\\ion{Zn}{I}}  & 4.56$\\pm$0.05 & $+$2.22 & $-$2.34 & $+$0.25  \\\\\n\\hbox{\\ion{Sr}{II}} & 2.87$\\pm$0.07 & $+$0.10 & $-$2.77 & $-$0.18 \\\\\n\\hbox{\\ion{Y}{II}}  & 2.21$\\pm$0.05 & $-$0.78 & $-$2.99 & $-$0.40 \\\\\n\\hbox{\\ion{Zr}{II}} & 2.58$\\pm$0.04 & $-$0.07 & $-$2.65 & $-$0.07 \\\\\n\\hbox{\\ion{Ba}{II}} & 2.18$\\pm$0.09 & $-$1.22 & $-$3.40 & $-$0.81 \\\\\n\\hbox{\\ion{La}{II}} & 1.10$\\pm$0.04 & $<-$1.85 & $<-$2.95 & $<-$0.36 \\\\\n\\hbox{\\ion{Ce}{II}} & 1.58$\\pm$0.04 & $-$0.83 & $-$2.41 & $+$0.18 \\\\\n\\hbox{\\ion{Eu}{II}} & 0.52$\\pm$0.04 & $-$2.35 & $-$2.87 & $-$0.28 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nIn Table \\ref{finalabundNLTE} we present the complete line list analysed in NLTE, \nwith the respective individual abundances. Below we also give element-by-element \ninformation about sources of the NLTE atomic models we used, and we also present \nthe graphical results of the NLTE line synthesis. \n\n\\begin{table}\n\\caption{Isotopic abundance fractions in the solar system from \nAsplund et al. (2009), relevant for HFS computations adopted in \nthe present work.}            \n\\label{fractions}     \n\\scalefont{1.0}\n\\centering                         \n\\begin{tabular}{ccr}       \n\\hline\\hline                \n\\noalign{\\smallskip}\n\\hbox{Elements} & \\hbox{Isotopes} & \\hbox{\\%} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{Sodium}    & \\hbox{$^{23}$Na} & 100.000 \\\\\n\\hbox{Aluminum}  & \\hbox{$^{27}$Al} & 100.000 \\\\\n\\hbox{Potassium} & \\hbox{$^{39}$K}  & 93.132 \\\\\n\\hbox{}          & \\hbox{$^{40}$K}  & 0.147 \\\\\n\\hbox{}          & \\hbox{$^{41}$K}  & 6.721 \\\\\n\\hbox{Scandium}  & \\hbox{$^{45}$Sc} & 100.000 \\\\\n\\hbox{Vanadium}  & \\hbox{$^{50}$V}  & 0.250 \\\\\n\\hbox{}          & \\hbox{$^{51}$V}  & 99.750 \\\\\n\\hbox{Manganese} & \\hbox{$^{55}$Mn} & 100.000 \\\\\n\\hbox{Zinc}      & \\hbox{$^{64}$Zn} & 48.630 \\\\\n\\hbox{}          & \\hbox{$^{66}$Zn} & 27.900 \\\\\n\\hbox{}          & \\hbox{$^{67}$Zn} & 4.100 \\\\\n\\hbox{}          & \\hbox{$^{68}$Zn} & 18.750 \\\\\n\\hbox{}          & \\hbox{$^{70}$Zn} & 0.620 \\\\\n\\hbox{Barium}    & \\hbox{$^{130}$Ba}& 0.106 \\\\\n\\hbox{}          & \\hbox{$^{132}$Ba}& 0.101 \\\\\n\\hbox{}          & \\hbox{$^{134}$Ba}& 2.417 \\\\\n\\hbox{}          & \\hbox{$^{135}$Ba}& 6.592 \\\\\n\\hbox{}          & \\hbox{$^{136}$Ba}& 7.854 \\\\\n\\hbox{}          & \\hbox{$^{137}$Ba}& 11.232 \\\\\n\\hbox{}          & \\hbox{$^{138}$Ba}& 71.698 \\\\\n\\hbox{Lanthanum} & \\hbox{$^{138}$La}& 0.091 \\\\\n\\hbox{}          & \\hbox{$^{139}$La}& 99.909 \\\\\n\\hbox{Europium}  & \\hbox{$^{151}$Eu}& 47.81 \\\\\n\\hbox{}          & \\hbox{$^{153}$Eu}& 52.19 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{NLTE abundances derived in HD~140238 using NLTE equivalent \nwidth fitting (EW) and NLTE line profile fitting (PF).}            \n\\label{finalabundNLTE}     \n\\scalefont{1.0}\n\\centering                         \n\\begin{tabular}{ccrrr}       \n\\hline\\hline                \n\\noalign{\\smallskip}\n\\hbox{Ion} & \\hbox{$\\lambda$~(\\AA)} & \\hbox{(X\/H)+12} & \\hbox{[X\/H]} & \\hbox{Method} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{\\ion{C}{I}}  & 8335.14 & $+$6.00 & $-$2.43 & PF \\\\ \n\\hbox{\\ion{C}{I}}  & 9061.43 & $+$5.97 & $-$2.46 & PF \\\\\n\\hbox{\\ion{C}{I}}  & 9062.48 & $+$6.03 & $-$2.40 & PF \\\\\n\\hbox{\\ion{O}{I}}  & 7771.94 & $+$7.04 & $-$1.67 & PF \\\\\n\\hbox{\\ion{O}{I}}  & 7774.16 & $+$6.98 & $-$1.73 & PF \\\\\n\\hbox{\\ion{O}{I}}  & 7775.39 & $+$7.00 & $-$1.71 & PF \\\\\n\\hbox{\\ion{O}{I}}  & 8446.36 & $+$7.04 & $-$1.67 & PF \\\\\n\\hbox{\\ion{Na}{I}} & 5889.95 & $+$3.37 & $-$2.88 & PF \\\\\n\\hbox{\\ion{Na}{I}} & 5895.92 & $+$3.37 & $-$2.88 & PF \\\\\n\\hbox{\\ion{Na}{I}} & 8194.82 & $+$3.37 & $-$2.88 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 4167.27 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 4571.10 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 4702.99 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 5172.68 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 5183.60 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 5528.40 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 5711.09 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 8806.76 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Al}{I}} & 3944.01 & $+$3.70 & $-$2.73 & PF \\\\\n\\hbox{\\ion{Al}{I}} & 3961.52 & $+$3.65 & $-$2.78 & PF \\\\\n\\hbox{\\ion{K}{I}}  & 7698.96 & $+$2.78 & $-$2.33 & PF \\\\\n\\hbox{\\ion{Ca}{I}} & 4226.73 & $+$4.04 & $-$2.27 & PF \\\\\n\\hbox{\\ion{Ca}{I}} & 4283.01 & $+$4.21 & $-$2.10 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4289.37 & $+$4.14 & $-$2.17 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4302.53 & $+$4.15 & $-$2.16 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4318.65 & $+$4.09 & $-$2.22 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4425.44 & $+$4.18 & $-$2.13 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4434.96 & $+$4.07 & $-$2.24 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4435.68 & $+$4.17 & $-$2.14 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4454.78 & $+$4.04 & $-$2.27 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5188.84 & $+$4.20 & $-$2.11 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5261.70 & $+$4.15 & $-$2.16 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5265.56 & $+$4.20 & $-$2.11 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5349.46 & $+$4.22 & $-$2.09 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5512.98 & $+$4.09 & $-$2.22 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5581.96 & $+$4.22 & $-$2.09 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5588.75 & $+$4.11 & $-$2.20 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5590.11 & $+$4.22 & $-$2.09 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5594.46 & $+$4.19 & $-$2.12 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5857.45 & $+$4.17 & $-$2.14 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6102.72 & $+$4.14 & $-$2.17 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6122.22 & $+$4.14 & $-$2.17 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6162.17 & $+$4.05 & $-$2.26 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6169.56 & $+$4.17 & $-$2.14 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6439.07 & $+$4.08 & $-$2.23 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6462.56 & $+$4.06 & $-$2.25 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6471.66 & $+$4.08 & $-$2.23 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6493.78 & $+$4.10 & $-$2.21 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6499.65 & $+$4.23 & $-$2.08 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 7148.15 & $+$4.19 & $-$2.12 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 7326.14 & $+$4.19 & $-$2.12 & EW \\\\\n\\hbox{\\ion{Ca}{II}}& 3933.68 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Ca}{II}}& 3968.47 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Ca}{II}}& 8498.02 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Ca}{II}}& 8542.09 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Ca}{II}}& 8662.14 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Sr}{II}}& 4077.72 & $+$0.02 & $-$2.90 & PF \\\\\n\\hbox{\\ion{Sr}{II}}& 4215.52 & $+$0.07 & $-$2.85 & PF \\\\\n\\hbox{\\ion{Sr}{II}}&10327.31 & $+$0.00 & $-$2.92 & PF \\\\\n\\hbox{\\ion{Ba}{II}}& 4554.03 & $-$1.07 & $-$3.24 & PF \\\\\n\\hbox{\\ion{Ba}{II}}& 6141.70 & $-$1.07 & $-$3.24 & PF \\\\\n\\hbox{\\ion{Ba}{II}}& 6496.92 & $-$1.01 & $-$3.18 & PF \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular} \n\\end{table}\n\n\\subsection{Light elements}\n\n$Lithium$. \nWe derived the LTE Li abundance A(Li)~$=+2.14$ based on \nthe \\ion{Li}{I} lines located at 6103~{\\AA} and 6707~{\\AA} \n(see Fig. \\ref{Li_fig}). The wavelength and oscillator strength values \nwere adopted from NIST, based on calculations by \nYan \\& Drake (1995). NLTE corrections are given in Asplund et al. (2006) \nfor the two Li lines in HD~140283: $+0.09$~dex for 6103~{\\AA} and \n$+0.03$ for 6707~{\\AA}. The corrected Li abundance \nA(Li)~$=+2.20$ is in excellent agreement with the NLTE \nabundance from Asplund et al. (2006).\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{Li.eps}}\n\\caption{LTE lithium abundance in HD~140283 from the \\ion{Li}{I} lines \nlocated at 6103~{\\AA} (upper panel) and 6707~{\\AA} (lower panel). \nObservations (crosses) are compared with synthetic spectra computed with \ndifferent abundances (blue dotted lines), as well as with the adopted \nabundances (red solid lines).}\n\\label{Li_fig}\n\\end{figure}\n\n\n$Carbon$. \nIn Paper I we analysed the CH A-X electronic transition band (G band) \nin HD~140283, showing a very good agreement between the observed spectrum \nand the computation using the LTE carbon abundance A(C)~$=+6.30$ adopted from \nHonda et al. (2004a). In fact, several CH lines were analysed and all of \nthem presented a good fit, leading us to assume that they are properly \ntaken into account.\n\nIt was also possible to use three \\ion{C}{I} lines: 8335.14~{\\AA}, \n9061.43~{\\AA}, and 9062.48~{\\AA}. They are free of telluric and \nother atomic or molecular lines. The lines of \\ion{C}{I} in the \nvisual part of the spectrum are not detectable. The LTE carbon \nabundance derived from these atomic transitions A(C)~$=+6.44$ is \nslightly higher in comparison with the result from molecular bands. \n\nThe same \\ion{C}{I} lines were analysed in NLTE (see upper left panel \nin Fig. \\ref{NLTE_O} for the 9062.48~{\\AA} line). To perform this work, \nwe used the carbon atomic model first proposed by Lyubimkov et al. (2015). \nOur calculations show that NLTE effects are very strong in the analysed lines, \nand act towards a strengthening of their equivalent widths: the ratio between NLTE \nand LTE EWs reaches a factor around 3. This means that simply analysing the program \ncarbon lines in LTE approximation results in the derived abundance overestimate \nby about $0.4-0.5$~dex (as it should be for the generally weak lines where equivalent \nwidth linearly depend upon the number of absorbing atoms). The same conclusion was \ngiven by Asplund  (2005), who noted that strong NLTE effects in high-excitation neutral \ncarbon lines are due to the decrease of source function (compared to the Planck function). \nIn particular this is valid for near-infrared \\ion{C}{I} lines. Fabbian et al. (2006) \nalso predict strong NLTE corrections in the carbon abundances in extremely metal-poor \nstars of  $\\sim$$0.4$~dex, in good agreement with the present result.\n\nWe find a difference between the C abundance deduced from CH and that \nfrom \\ion{C}{I} lines computed in LTE and NLTE \nof $\\Delta($C(CH)~$-$~C(\\ion{C}{I})$_{LTE})$~$=-0.14$~dex, and \n$\\Delta($C(CH)~$-$~C(\\ion{C}{I})$_{NLTE})$~$=+0.30$~dex, respectively. \nThe CH lines forming in the upper atmosphere might be affected \nin a 3D modelling calculation (Asplund 2005).\n\n$Nitrogen$. \nThe bandhead of CN(0,0) B$_2$$\\Sigma$~$-$~X$_2$$\\Sigma$ at 3883~{\\AA} gives A(N)~$=+6.30$, \nassuming the abundances A(C)~$=+6.30$ and A(O)~$=+7.00$. In Fig. \\ref{NO_fig} \n(upper panel) we present the synthetic spectra fitting to the CN bandhead. \nThis profile is located in the blue wing of the H8 line of the Balmer series, \ntaken into account in the calculations.\n\n\n$Oxygen$.\nThe forbidden [\\ion{O}{I}]~6300.31~{\\AA} line is free from NLTE effects (Kiselman 2001), \nand therefore it is the best oxygen abundance indicator. The line was inspected for telluric \nlines in each of the original exposures. From our series of observations, four of the spectra \nobserved in July had to be discarded, given that the telluric lines were masking the oxygen line. \nWe then proceeded to co-add the other 19 spectra, and were able to obtain a weak, but measurable line \n(see lower panel in Fig. \\ref{NO_fig}). We derive an abundance of A(O)~$=+6.95$, considered in \nthis calculation A(C)~$=+6.30$ (C derived from CH lines) and A(N)$=+6.30$.\n\nThe triplets at 7771-7775~{\\AA} and 8446~{\\AA} were also checked to compute the LTE O abundance \nA(O)~$=+7.11$ in HD~140283. Adopting an average of different models presented in Behara et al. (2010) \nfor the NLTE corrections to be applied on the triplet 7771-7775~{\\AA} lines \n($-0.13$~dex, $-0.10$~dex, and $-0.10$~dex, respectively), we obtain A(O)~$=+6.97$ from this triplet, \nin excellent agreement with the result derived using the forbidden line.\n\nThe triplets at 7771-7775~{\\AA} (see Fig. \\ref{NLTE_O}) and 8446~{\\AA} \nwere also analysed in NLTE. Our NLTE model of this atom \nwas first described in Mishenina et al. (2000), and then updated by \nKorotin et al. (2014). An updated model was applied to study the NLTE \nabundance of this element in cepheids, and its distribution in the \nGalactic disc (see, for instance, Korotin et al. 2014; Martin et al. 2015). \nWe obtain a NLTE oxygen abundance A(O)~$=+7.02$ (or [O\/Fe]~$=+0.90$), \nin agreement with the derived LTE abundances corrected for NLTE effects.\n\n\n\\subsection{$\\alpha$-elements: Mg, Si, S, and Ca}\n\n\n$Magnesium$. \nAfter checking the \\ion{Mg}{I} lines used by Zhao et al. (1998) in \nthe solar spectrum and the bluer lines used by Cayrel et al. (2004) \nfor metal-poor stars, we retained 12 lines from an initial list of 21 \nprofiles to derive the LTE abundance A(\\ion{Mg}{I})~$=+5.27$. \nThe oscillator strengths were adopted from the critical compilation \npresented in Kelleher \\& Podobedova (2008a). The individual abundances \nare consistent among the lines (see upper panels in Fig. \\ref{Mg_fig}), \nwith the exception of \\ion{Mg}{I}~8806.76~{\\AA}, which is $0.13$~dex higher \nthan the average abundance (see lower panel in Fig. \\ref{Mg_fig}).\n\nIn addition, it was possible to use two \\ion{Mg}{II} lines with the \nspectrum of HD~140283: 4481.13~{\\AA} and 4481.15~{\\AA}. The derived LTE \nabundance A(\\ion{Mg}{II})~$=+5.66$ is higher than the value obtained \nfrom the neutral species, but both \\ion{Mg}{II} lines are blended and \nthe result should be used with caution.\n\nSeven magnesium lines were selected for NLTE \nabundance determination. All these lines have very well-defined \nprofiles, which provide a robust NLTE abundance of this element (see\nFig. \\ref{NLTE_Mg}). To generate \\ion{Mg}{I} line profiles, \nwe used the NLTE \\ion{Mg}{I} atomic model described in detail \nby Mishenina et al. (2004). This model was used in several studies, in \nparticular, in the determination of the magnesium abundance in a sample \nof metal-poor stars (Andrievsky et al. 2010).\n\n\n\n\n$Silicon$. \nThe main Si abundance indicators are the \\ion{Si}{I} lines at 3905.52~{\\AA} \nand 4102.94~{\\AA}. The line at 3905~{\\AA} is blended with CH lines, but these \nmolecular lines are weak enough in a rather hot subgiant such that the \\ion{Si}{I} line can \nbe used. On the other hand, the line at 4102~{\\AA} is located in the red wing of the \nH$\\delta$ line, and therefore it was necessary to take the hydrogen line \nin the spectrum synthesis into account. Fig. \\ref{Si_fig} shows the LTE synthetic spectra \nused for these lines in HD~140283. It was possible to use another three weaker \n\\ion{Si}{I} lines located at 5645.61~{\\AA}, 5684.48~{\\AA}, and 7034.90~{\\AA}, with \nindividual abundances in good agreement with the previous transitions. \nThe final LTE Si abundance obtained was A(\\ion{Si}{I})$=+5.30$. \nBecause of the good quality of our data, it was possible to derive the LTE silicon abundance \nfrom the ionized species A(\\ion{Si}{II})$=+5.35$, using the \\ion{Si}{II}~6371.37~{\\AA} line. \n\n\n$Sulfur$. \nThree \\ion{S}{I} lines, which are potentially available for the abundance \ndetermination in metal-poor stars, are located at 9212.86~{\\AA}, \n9228.09~{\\AA}, and 9237.54~{\\AA}. Unfortunately, the first \nline is severely blended with telluric lines, and the other two lines \nare on a small gap in the spectrum. As a consequence, a reliable sulfur\nabundance is not presented here for HD~140283.\n\n\n\n$Calcium$. \nTo derive the LTE Ca abundance, 36 \\ion{Ca}{I} lines were used to obtain \nA(\\ion{Ca}{I})$=+4.03$. We adopted the \\loggf~values \nfrom Spite et al. (2012). Because of the high-quality spectrum, even weak and \nblended lines are useful, as shown in Fig. \\ref{Ca_fig}. \n\nWe inspected a few \\ion{Ca}{II} lines to evaluate the LTE calcium abundance \nfrom the ionized species. The presence of a strong NLTE effect, however, \ndoes not permit us to use most of these transitions. \nThe LTE abundance A(\\ion{Ca}{II})$=+4.43$ was derived based only on the \n\\ion{Ca}{II}~8927.36~{\\AA} line. We obtained a difference of \n$+0.40$~dex in comparison with the abundance derived from transitions \nof the neutral element. Mashonkina et al. (2007) analysed HD~140283 in \ntheir study of neutral and singly-ionized calcium in late-type stars, \nand the LTE results show a difference of $+0.30$~dex. Their NLTE \nabundances are [\\ion{Ca}{I}\/Fe]~$=+0.29\\pm0.06$ and [\\ion{Ca}{II}\/Fe]~$=\n+0.24$. Using similar atmospheric parameters for this star, Spite et al. \n(2012) found the NLTE abundances A(\\ion{Ca}{I})~$=+4.12\\pm0.04$ and \nA(\\ion{Ca}{II})~$=+4.08\\pm0.05$. Our LTE abundance shows that the \ndeparture from LTE is not strong for neutral calcium in this star.\n\nThe NLTE abundance A(Ca)~$=+4.14$ was determined \nfrom the average of 35 lines, including: i) the EW analysis of \n30 \\ion{Ca}{I} lines with 10~$<$~EW(m{\\AA})~$<$~40; \nii) the profile of five other lines, namely the strongest line of this \natom at 4226.73~{\\AA} (see upper \npanel in Fig. \\ref{NLTE_Ca}); the H and K \\ion{Ca}{II} lines \n(see lower  panel in Fig. \\ref{NLTE_Ca}); and another three \n\\ion{Ca}{II} lines located in the red. The NLTE atomic model was \ndescribed in Spite et al. (2012), where it was used for the study of \nhalo metal-poor stars. \n\n\n\n\\subsection{Odd-Z elements: Na, Al, and K}\n\n$Sodium$. \nTo derive the LTE sodium abundance we used an initial \nline list based on Baum\\\"uller et al. (1998) with \nupdated oscillator strengths from Kelleher \\& Podobedova (2008a). After \nchecking 13 \\ion{Na}{I} lines, the final result A(\\ion{Na}{I})~$=+3.62$ \nis based on five sodium lines. $^{23}$Na is the only stable isotope \nrepresenting the sodium abundance, with nuclear spin I~$=3\/2$\n\\footnote{Adopted from the Particle Data Group (PDG) collaboration: \nhttp:\/\/pdg.lbl.gov\/} and therefore exhibiting HFS. The hyperfine \ncoupling constants are adopted from Das \\& Natarajan (2008) \nand Safronova et al. (1999). When not available, these constants were assumed to be null. \nThe HFS for each line were computed by employing a code described and  \nmade available by McWilliam et al. (2013). Fig. \\ref{Na_fig} shows in \nthe upper panel the adopted synthetic spectrum for \\ion{Na}{I}~5895.92~\n{\\AA} and \\ion{Na}{I}~8194.82~{\\AA} lines as example of typical fitting \nprocedures. \n\n\n\nA NLTE atomic model of this element was presented for the first time in \nKorotin \\& Mishenina (1999) and then updated by Dobrovolskas et al. \n(2014). We analysed three \\ion{Na}{I} lines: D$_{1}$, D$_{2}$, and the \nline at 8194.82~{\\AA}. Synthesized NLTE profiles fitted to the observed \nsodium line profiles are shown in Fig. \\ref{NLTE_Na}, and the final \nNLTE abundance A(\\ion{Na}{I})~$=+3.37$ was adopted.\n\n\n\n\n$Aluminum$. \nThe only stable isotope for aluminum is $^{27}$Al and to derive \nthe Al abundance we used the resonance doublet \\ion{Al}{I}~3944.01~{\\AA} \nand \\ion{Al}{I}~3961.52~{\\AA}, taking the CH line blending \nwith the first line into account. The local continuum around the \\ion{Al}{I}~3961.52~{\\AA} \nline is defined by the blue wings of the H$\\varepsilon$ and H \\ion{Ca}{II} lines, \nwhich were taken into account in the spectrum synthesis. \n$^{27}$Al has nuclear spin I~$=5\/2$ and we adopted \nthe hyperfine coupling constants from Nakai et al. (2007) and Brown \\& Evenson (1999), \nwith updated oscillator strengths from Kelleher \\& Podobedova (2008b). \nFig. \\ref{Al_K_fig} shows in the upper panel the LTE synthetic \nspectra used for \\ion{Al}{I}~3961.52~{\\AA} as an example. The adopted LTE \nabundance is A(\\ion{Al}{I})$=+2.96$ and must be corrected for NLTE effects. \n\n\nThe NLTE \\ion{Al}{I} atomic model presented in Andrievsky et al. (2008) was \nemployed and the result of the profile fitting for \\ion{Al}{I}~3961.52~{\\AA} \nline is shown in Fig. \\ref{NLTE_AlK} (upper panel). The final NLTE abundance \nA(\\ion{Al}{I})$=+3.68$ is $0.72$~dex higher than the LTE result.\n\nAs it was stated above, in some cases the NLTE profile synthesis should \nbe combined with the LTE synthesis, which takes the blending lines in \nthe vicinity of the studied line into account. For instance, we would get \nthe wrong result if we derived the NLTE abundance from UV \\ion{Al}{I} lines \nonly using the pure MULTI NLTE profiles, since these lines are located in the \nwings of very strong H and K \\ion{Ca}{II} lines. \nTherefore it is absolutely necessary to take continuum distortion in their \nvicinity into account. This is made through a combination\nof calculations with the codes MULTI NLTE and SYNTHV LTE, which provides \na correct aluminum abundance. \n\n\n\n\n$Potassium$. \nThe final LTE potassium abundance A(\\ion{K}{I})~$=+2.98$ was only derived \nfrom the \\ion{K}{I}~7698.96~{\\AA} line, as shown in the lower \npanel of Fig. \\ref{Al_K_fig}. The other red doublet component at 7665~{\\AA} \nis strongly blended with telluric lines and was excluded. For this \nelement, we adopted the isotopic abundance fractions in the solar system, \nas described in Asplund et al. (2009) and reproduced in Table \n\\ref{fractions}. The HFS was computed taking into account the major \npotassium isotope $^{39}$K, which has nuclear spin I~$=3\/2$. We adopted \nhyperfine coupling constants from Belin et al. (1975) and Falke et al. \n(2006), with updated oscillator strength from Sansonetti (2008).\n\nA NLTE \\ion{K}{I} atomic model was presented in Andrievsky et al. (2010), \nwhere it was first employed to derive potassium NLTE abundances in a \nsample of extremely metal-poor halo stars. This model was used to \nsynthesize the profile of the \\ion{K}{I}~7698.96~{\\AA} line in HD 140283\n(see lower panel in Fig. \\ref{NLTE_AlK}), which gives \nA(\\ion{K}{I})~$=+2.78$ as the final abundance.\n\n\n\n\\subsection{Iron-peak elements: Sc, Ti, V, Cr, Mn, Fe, Co, Ni, and Zn}\n\n\n\n$Scandium$. \nWe were able to use two \\ion{Sc}{I} lines to derive the scandium \nabundance. The lines located at 4020.39~{\\AA} and 4023.68~{\\AA} present low \nexcitation potential and the average abundance A(\\ion{Sc}{I})$=+0.58$ should \nsuffer from over-ionization via NLTE effects (Zhang et al. 2008). \n$^{45}$Sc is the only stable scandium isotope, with nuclear spin I~$=7\/2$,\nand to compute the HFS we adopted hyperfine coupling constants from Childs \n(1971), Siefart (1977), and Ba\\c{s}ar et al. (2004). On other hand, the \nionized species presented 19 useful lines, covering a wider range in \nwavelengths and excitation potential values. The average result A(\\ion{Sc}{II})\n$=+0.75$ is higher in comparison with the abundance from the neutral species \nand it is not affected strongly by NLTE effects, at least at solar metallicity \n(see Asplund et al. 2009). The hyperfine coupling constants were adopted \nfrom Villemoes et al. (1992), Mansour et al. (1989), and Scott et al. \n(2015). In Fig. \\ref{Sc_fig} we show the fitting procedures used for the \n\\ion{Sc}{II}~4246.82~{\\AA} and \\ion{Sc}{II}~5526.79~{\\AA} lines as \nexamples. \n\n\n\n$Titanium$. \nFor titanium, \nwe applied equivalent widths to derive the final abundances \nA(\\ion{Ti}{I})~$=+2.71$ and A(\\ion{Ti}{II})~$=+2.69$ (Sect. \\ref{EW}).\n\n\n\n\n\n\n$Vanadium$. \nIn Paper I we analysed five \\ion{V}{I} \nlines and seven \\ion{V}{II} lines to derive the vanadium \nabundances A(\\ion{V}{I})~$=+1.35\\pm0.10$ and A(\\ion{V}{II})~$=+1.72\\pm0.10$, \nrespectively. The average result A(V)~$=+1.56\\pm0.11$ is in good \nagreement with A(V)~$=1.55$ from Honda et al. (2004a), derived in their \nanalysis from three lines (see Table 3 in Siqueira-Mello et al. 2012 \nfor details). We adopted seven \\ion{V}{I} and eight \n\\ion{V}{II} lines, including HFS based on hyperfine coupling \nconstants from Unkel et al. (1989), \nPalmeri et al. (1995), Armstrong et al. (2011), Gyzelcimen et al. (2014), \nand Wood et al. (2014), with nuclear spin I~$=7\/2$. We only used the major \nV isotope in the computations (see Table \\ref{fractions}). \nThe oscillator strengths were adopted from Whaling et al. (1985) for \\ion{V}\n{I} and from Wood et al. (2014) for \\ion{V}{II}. We obtained A(\\ion{V}{I})~$=\n+1.44$ and A(\\ion{V}{II})~$=+1.70$, in agreement with the previous results. \nIn addition, the final abundances derived from \\ion{V}{I} and \\ion{V}{II} \nlines are more consistent in the present analysis. \nIn Fig. \\ref{V_fig} we show the \n\\ion{V}{I}~4379.230~{\\AA} line (upper panel) and the \\ion{V}{II}~4023.378~\n{\\AA} line (lower panel) as examples. The differences measured between the \ntwo ionization stages should be explained by strong NLTE effects expected \nfor \\ion{V}{I} lines.\n\n\n\n\n$Chromium$. \nWe derived individual chromium abundances for 28 \\ion{Cr}{I} lines,\nbut we excluded the transitions located at 4756.11~{\\AA} and 5237.35~{\\AA} \nbecause of the higher abundances in comparison with results from other lines, \nand we obtained the abundance \nA(\\ion{Cr}{I})~$=+2.95$ based on seven \\ion{Cr}{I} lines. The fitting \nprocedure used for \\ion{Cr}{I}~4254.33~{\\AA} line is shown in Fig. \n\\ref{Cr_Mn_fig}. In addition, it was possible to use seven \n\\ion{Cr}{II} lines in HD~140283 to derive the abundance A(\\ion{Cr}{II})~$=\n+3.32$, higher by 0.37 dex than the result obtained from the neutral species.\n\n$Manganese$. \nThe only manganese stable isotope is $^{55}$Mn, \nwith nuclear spin I~$=5\/2$. \nIn addition to the three lines belonging \nto the resonance triplet (\\ion{Mn}{I}~4030.75~{\\AA}, 4033.06~{\\AA}, and \n4034.48~{\\AA}), it was also possible to measure 13 subordinate lines.\nWe took the HFS properly into account based on the hyperfine \nstructure line component patterns from Den Hartog et al. (2011), \nwhich also present the sources for hyperfine coupling constants. \nIt is well known that the abundances derived from the triplet\nresonance lines are \nsystematically lower in comparison with the results from subordinate lines \nin very metal-poor giant stars. Indeed, the result we obtained \nfor HD~140283 using the triplet A(\\ion{Mn}{I})~$=+2.31$ is lower than \nA(\\ion{Mn}{I})~$=+2.56$ derived from the subordinate lines. The resonance \ntriplet lines are more susceptible to NLTE effects, \nand for this reason we did not include them in the average\nMn abundance. The line \\ion{Mn}{I}~4041.35~{\\AA} is \nshown in Fig. \\ref{Cr_Mn_fig} (lower panel).\n\n\n$Iron$.\nWe used equivalent widths to derive the final \nabundances A(\\ion{Fe}{I})~$=+4.91$ and A(\\ion{Fe}{II})~$=+4.96$, as described\nin Sect. \\ref{EW}. \n\n$Cobalt$.\nIndividual cobalt abundances were derived from 17 \n\\ion{Co}{I} lines. The final abundance A(\\ion{Co}{I})~$=+2.69$ was \ncomputed excluding the lines located at 3861.16~{\\AA} and 3873.11~{\\AA} \nbecause of the higher differences in comparison with the average abundance. \n\n$Nickel$. \nFor this element \nwe derived the abundance A(\\ion{Ni}{I})~$=+3.76$ \nbased on 43 \\ion{Ni}{I} lines. In Fig. \\ref{Ni_Zn_fig} (upper panel) we show \nthe \\ion{Ni}{I}~3858.29~{\\AA} line. We were also able to use \nthe \\ion{Ni}{II}~3769.46~{\\AA} line in the present spectrum to derive the \nabundance A(\\ion{Ni}{II})~$=+3.88$ in this star. \n\n$Zinc$. \nThe isotopic structure of zinc is complex (see Table \\ref{fractions}),  \nbut HFS is not needed to be accounted for, therefore, we assumed \nZn as having a unique isotope with wavelengths dominated by the $^{64}$Zn. \nThe abundance A(\\ion{Zn}{I})~$=+2.21$ was derived from the \\ion{Zn}{I}~\n4722.15~{\\AA} and \\ion{Zn}{I}~4810.53~{\\AA} (see Fig. \\ref{Ni_Zn_fig}, lower \npanel) lines, with oscillator strengths adopted from Roederer \\& Lawler (2012).\n\n\n\n\n\\subsection{Neutron-capture elements}\n\n\n\n$Strontium$. \nIn HD~140283 the Sr abundance was derived based on three \\ion{Sr}{II} \nlines located at 4077.72~{\\AA}, 4215.52~{\\AA}, and 10327.31~{\\AA}. \nThese transitions show HFS effects, but hyperfine coupling constants \nonly exist for $^{87}$Sr (Borghs et al. 1983), which accounts for \nless than 7\\% of Sr (Asplund et al. 2009). In addition, the atomic lines from \nthe even isotopes $^{84}$Sr, $^{86}$Sr, and $^{88}$Sr appear as single lines \ndue to the small isotopic splitting for Sr (Hauge 1972). In conclusion, we treated \neach Sr line as a single component, with oscillator strengths adopted from \nGratton \\& Sneden (1994), which gives an average abundance \nof A(\\ion{Sr}{II})~$=+0.10$. In the upper panel of Fig. \\ref{Sr_Y_fig}, \nwe show the fitting procedure used for the \\ion{Sr}{II}~4077.72~{\\AA} line.\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{Sr_Y.eps}}\n\\caption{LTE strontium abundance in HD~140283 from \\ion{Sr}{II}~4077.72~{\\AA} \nline (upper panel) and yttrium abundance from \\ion{Y}{II}~3774.33~{\\AA} line \n(lower panel). Symbols are the same as in Fig. \\ref{Li_fig}.}\n\\label{Sr_Y_fig}\n\\end{figure}\n\nOur NLTE strontium atomic model was described in Andrievsky et al. (2011), \nwhere it was applied to a sample of metal-poor stars. \nWe analysed the same two blue \\ion{Sr}{II} lines at 4077.72~{\\AA} and \n4215.52~{\\AA} (see upper panel in Fig. \\ref{NLTE_Sr}), and a third \nline in the near-infrared located at 10327.31~{\\AA} (see lower panel in Fig. \n\\ref{NLTE_Sr}), with a final NLTE abundance A(\\ion{Sr}{II})~$=+0.03$ \nas the adopted result.\n\nIn Andrievsky et al. (2011) the NLTE corrections were calculated for the \n\\ion{Sr}{II}~4077.72~{\\AA}, 4215.52~{\\AA}, besides near-infrared lines for different \ntemperatures and gravities. Considering Fig. 7 of Andrievsky et al. (2011),\n at \\Teff~$=5750$~K and \\logg~$=3.7$ the correction \nshould be  small and positive. We obtained a NLTE strontium \nabundance that is slightly lower than the LTE abundance. We suggest that\nmain reasons for this discrepancy can be the result of: a) the LTE results \nfrom Turbospectrum may use slightly different atomic constants; and b) this \nstar is more metal-rich ([Fe\/H]~$=-2.59$) than the calculations for [Fe\/H]~$=-3.0$ \ngiven in Andrievsky et al. (2011).\n\n\n$Yttrium$. \nFor yttrium it was possible to check in LTE five \\ion{Y}{II} lines, \nusing oscillator strengths adopted from Hannaford et al. (1982) and \nGrevesse et al. (2015), with A(\\ion{Y}{II})~$=-0.78$ as the final \nabundance. The synthetic profile adopted for \\ion{Y}{II}~3774.33~{\\AA} \nis shown in Fig. \\ref{Sr_Y_fig} (lower panel), which takes the continuum \naffected by the H11 line from the Balmer series into account.\n\n\n\n$Zirconium$. \nAfter inspecting the spectrum, we decided to retain only the three best \n\\ion{Zr}{II} lines available, located at 3836.76~{\\AA}, 4208.98~{\\AA} \nand 4443.01~{\\AA}, to derive the LTE zirconium abundance. The \\loggf~values \nwere adopted from Bi\\'emont et al. (1981), with final LTE abundance of \nA(\\ion{Zr}{II})~$=-0.07$ (see Fig. \\ref{Zr_fig}). \n\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{Zr.eps}}\n\\caption{LTE zirconium abundance in HD~140283 from \\ion{Zr}{II}~4208.98~{\\AA} \n(upper panel) and \\ion{Zr}{II}~4443.01~{\\AA} (lower panel) lines. \nSymbols are the same as in Fig. \\ref{Li_fig}.}\n\\label{Zr_fig}\n\\end{figure}\n\n\n$Barium$. \nThe LTE barium abundance was previously analysed in Paper~I, based on \nthe \\ion{Ba}{II}~4554.03 and \\ion{Ba}{II}~4934.08~{\\AA} lines, \nwith oscillator strengths adopted from Gallagher (1967) and hyperfine structure \nline component patterns from McWilliam (1998). \nWe added two other \\ion{Ba}{II} lines, located \nat 6141.71~{\\AA} and 6496.90~{\\AA}, with \noscillator strengths and HFS from Barbuy et al. (2014). \nWith nuclear spin I~$=3\/2$, we took the Ba isotopic nuclides into account \naccording to Table \\ref{fractions}.\n\n\nFigure \\ref{Ba_fig} shows the synthetic profiles computed for the \n\\ion{Ba}{II} lines. The blue wing of the \\ion{Ba}{II}~4934.08~{\\AA} \nline is blended with \\ion{Fe}{I}~4934.01~{\\AA}, which we took properly \ninto account using \\loggf~$=-0.59$, adjusted to describe the \nobserved spectrum. The individual abundance agrees with the result \nderived from the clear \\ion{Ba}{II}~4554.03{\\AA} line. To compute the \nprofile for \\ion{Ba}{II}~6141.71~{\\AA}, it is important to include a \nblend with \\ion{Fe}{I}~6141.73~{\\AA} for which we adopted \n\\loggf~$=-1.60$ (Barbuy et al. 2014). For \\ion{Ba}{II}~6496.90~\n{\\AA} there is a telluric line located in the blue wing in the present \nspectrum. The individual abundance agrees with that derived from \n6141.71~{\\AA} line, but it is slightly higher in comparison with the \nresults from the first two Ba lines. We decided to adopt \nA(\\ion{Ba}{II})~$=-1.22$ as the final LTE abundance.\n\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{Ba.eps}}\n\\caption{LTE barium abundance in HD~140283 from \\ion{Ba}{II}~4554.03~{\\AA} \n(upper left panel), \\ion{Ba}{II}~4934.08~{\\AA} (upper right panel), \n\\ion{Ba}{II}~6141.71~{\\AA} (lower left panel), and \n\\ion{Ba}{II}~6496.90~{\\AA} (lower right panel) lines. \nSymbols are the same as in Fig. \\ref{Li_fig}.}\n\\label{Ba_fig}\n\\end{figure}\n\n\nA NLTE atomic model of \\ion{Ba}{II} was presented in Andrievsky et al.\n(2009). Three \\ion{Ba}{II} lines were analysed in HD~140283: \n4554.00~{\\AA}, 6141.70~{\\AA}, and 6496.92~{\\AA} (see Fig. \\ref{NLTE_Ba}). \nWe applied the odd-to-even isotopic ratio 50:50, which is applicable to old \nmetal deficient stars, to synthesize the \\ion{Ba}{II}~4554.00~{\\AA} line.\n\n\n\n$Lanthanum$. \nFor lanthanum, available transitions are too weak to enable us to derive \na robust value for the La abundance, but the upper limit of LTE \nabundance A(\\ion{La}{II})~$<-1.85$ was estimated from the \n\\ion{La}{II}~4123.22~{\\AA} line (see upper panel in Fig. \\ref{La_Ce_fig}). \nThis profile is located in the red wing of the H$\\delta$ line, \naccounted for in the spectrum synthesis. \nWe only used the major La isotope, with nuclear spin I~$=7\/2$, \nand the hyperfine coupling constants adopted basically from Lawler \net al. (2001), but also from Furmann et al. (2008) and Honle et al. \n(1982) when not in the basic reference. We also adopted experimental \noscillator strengths from Lawler et al. (2001). \n\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{La_Ce.eps}}\n\\caption{LTE lanthanum abundance in HD~140283 from \\ion{La}{II}~4123.22~{\\AA} \nline (upper panel) and cerium abundance in from \\ion{Ce}{II}~4083.22~{\\AA} line \n(lower panel). Symbols are the same as in Fig. \\ref{Li_fig}.}\n\\label{La_Ce_fig}\n\\end{figure}\n\n$Cerium$. \nFor cerium, we used the improved laboratory transition probabilities \npresented in Lawler et al. (2009) to derive the LTE abundance \nA(\\ion{Ce}{II})~$=-0.83$, based on two \\ion{Ce}{II} lines located \nat 4083.22~{\\AA} (lower panel in Fig. \\ref{La_Ce_fig}) and 4222.60~{\\AA}. \nThese are well known as good abundance indicators (e.g. Hill et al. 2002). \nThe local continuum around \\ion{Ce}{II}~4083.22~{\\AA} is \ndefined by the blue wing of the H$\\delta$ line. \nThese lines are weak in HD~140283, but still clearly\ndetectable as a result of the high quality of the spectrum. \nThese two lines give [Ce\/Fe]~$=+0.36$ and $+0.01$, respectively, \nand a corresponding mean overbundance of cerium [Ce\/Fe]~$=+0.18$. \nThe overabundance is therefore to be taken with caution.\n\n\n$Europium$. \nPaper~I was dedicated to derive the LTE europium abundance in HD~140283, \nusing the isotopic fractions in the solar material (Table \\ref{fractions}) \nwith nuclear spin I~$=5\/2$. The final abundance A(\\ion{Eu}{II})~$=\n-2.35\\pm0.07$ was obtained from \\ion{Eu}{II}~4129.70~{\\AA}, which is \nconsistent with the upper limit A(\\ion{Eu}{II})~$<-2.39$ estimated from \nthe \\ion{Eu}{II}~4205.05~{\\AA} line.\n\n\n\n\\section{Discussion}\n\n\\begin{table}\n\\caption{Abundances in HD~140283 computed in LTE, NLTE, and \nfinal abundances adopted. Iron is only computed in LTE.\n }            \n\\label{adopted_abundances}     \n\\scalefont{1.0}\n\\centering                         \n\\begin{tabular}{c@{}r@{}r@{}r@{}r@{}r}       \n\\hline\\hline                \n\\noalign{\\smallskip}\n\\hbox{Element} & \\phantom{-}\\phantom{-}\\hbox{[X\/Fe]$_{H04}$} &\n\\phantom{-}\\phantom{-}\\hbox{[X\/Fe]$_{LTE}$} & \n\\phantom{-}\\phantom{-}\\hbox{[X\/Fe]$_{NLTE}$} & \n\\phantom{-}\\phantom{-}\\hbox{[X\/Fe]$_{adopted}$}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{Fe}   & $-$2.53 & $-$2.59 &   ----  & $-$2.59 \\\\\n\\hbox{Li}   &    ---- & $+$2.14 & $+$2.20$^\\dagger$& $+$2.20  \\\\\n\\hbox{C(CH)}& $+$0.47 & $+$0.46 &   ----  & $+$0.46  \\\\\n\\hbox{C(CI)}&    ---- & $+$0.60 & $+$0.16 & $+$0.16\\\\\n\\hbox{N(CN)}&    ---- & $+$1.06 &  ----   & $+$1.06 \\\\\n\\hbox{O}    &    ---- & $+$0.97 & $+$0.90 & $+$0.90\\\\\n\\hbox{Na}   &    ---- & $-$0.04 & $-$0.29 & $-$0.29\\\\\n\\hbox{Mg}   & $+$0.25 & $+$0.26 & $+$0.43 & $+$0.43\\\\\n\\hbox{Al}   & $-$0.94 & $-$0.91 & $-$0.17 & $-$0.17\\\\\n\\hbox{Si}   & $+$0.34 & $+$0.38 &    ---- & $+$0.38\\\\\n\\hbox{K}    &    ---- & $+$0.54 & $+$0.26 & $+$0.26\\\\\n\\hbox{Ca}   & $+$0.33 & $+$0.27 & $+$0.42 & $+$0.42\\\\\n\\hbox{Sc}   & $+$0.10 & $+$0.10 &    ---- & $+$0.10\\\\\n\\hbox{Ti}   & $+$0.36 & $+$0.33 &    ---- & $+$0.33\\\\\n\\hbox{V}    & $+$0.21 & $+$0.22 &    ---- & $+$0.22\\\\\n\\hbox{Cr}   & $+$0.30 & $+$0.08 &    ---- & $+$0.08\\\\\n\\hbox{Mn}   & $-$0.25 & $-$0.29 &    ---- & $-$0.29\\\\\n\\hbox{Co}   & $+$0.25 & $+$0.29 &    ---- & $+$0.29\\\\\n\\hbox{Ni}   & $+$0.13 & $+$0.12 &    ---- & $+$0.12\\\\\n\\hbox{Zn}   &    ---- & $+$0.25 &    ---- & $+$0.25\\\\\n\\hbox{Ge}   &    ---- &    ---- &    ---- &$<-$0.46\\\\\n\\hbox{As}   &    ---- &    ---- &    ---- & $+$0.38$^1$\\\\\n\\hbox{Se}   &    ---- &    ---- &    ---- & $-$0.15$^1$\\\\\n\\hbox{Sr}   & $-$0.31 & $-$0.18 & $-$0.30 & $-$0.30\\\\\n\\hbox{Y}    & $-$0.46 & $-$0.40 &    ---- & $-$0.40\\\\\n\\hbox{Zr}   & $-$0.14 & $-$0.07 &    ---- & $-$0.07\\\\\n\\hbox{Mo}   &    ---- &    ---- &    ---- & $+$0.19$^2$\\\\\n\\hbox{Ru}   &    ---- &    ---- &    ---- & $<+$0.99$^2$\\\\\n\\hbox{Ba}   & $-$0.96 & $-$0.81 & $-$0.63 & $-$0.63\\\\\n\\hbox{La}   &    ---- &$<-$0.36 &   ----  &$<-$0.36 \\\\\n\\hbox{Ce}   &    ---- & $+$0.18 &   ----  & $+$0.18\\\\\n\\hbox{Eu}   &    ---- & $-$0.28 &   ----  & $-$0.28\\\\\n\\hbox{Pt}   &    ---- &   ---- &    ----  & $<+$0.37$^1$\\\\\n\\hbox{Pb}   &    ---- &   ---- &    ----  & $<+$1.54$^1$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\tablebib{H04: Honda et al. (2004a,b); LTE: this work, in LTE; \nNLTE: this work, in NLTE. $\\dagger$: based on the NLTE corrections from Asplund et al. (2006). \nReferences: 1 Roederer (2012); 2 Peterson (2011).}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{abgfe-oamix3.eps}}\n\\caption{Comparison of the LTE [O\/Fe] ratio obtained for the giant stars \nin the Large Programme ``First Stars'' based on the forbidden \n[\\ion{O}{I}]~6300.31~{\\AA} line (blue filled dots) and in HD~140283 (red star). \nThe blue open circles represent the two componnents of the turnoff binary \nCS~22876-032.}\n\\label{Ocompare_fig}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{pattern.eps}}\n\\caption{Abundance pattern of HD~140283 based on the present \nresults (red circles) and literature (blue circles). \nThe dotted line is the abundance pattern of CS~31082-001, \nan r-II EMP star, representing the expected\nabundances for an r-rich star, \nrescaled to match the Eu abundance in HD~140283.}\n\\label{pattern}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{hd122563-140283-31082.eps}}\n\\caption{Comparison of the abundance pattern of HD~140283, based on \nthe present results and literature (blue squares), with the \npattern of HD~122563 (green circles) and CS~31082-001 (black dots). \nThe solid line is the \nresidual solar r-process pattern.}\n\\label{pattern2}\n\\end{figure}\n\n\nIn Table \\ref{adopted_abundances} we present the abundances \nderived in LTE for 25 elements and an upper limit for lanthanum \nalong with abundances derived in NLTE for 9 elements, \nand the final abundances adopted. This table also includes abundances \nfrom Honda et al. (2004a) and abundances, for As, Se, Pt, Pb, from \nRoederer (2012) and, Mo and Ru, from Peterson (2011). \nAll of these are scaled to the metallicity and solar abundances \nadopted in this analysis. \n\nWe adopted as final abundances those analysed with NLTE models \n(Table \\ref{finalabundNLTE}), for the nine elements studied \nin NLTE, otherwise we adoted the LTE results. \nThe LTE abundances for Sc, V, and Cr are \nthe average from neutral and ionized lines. For \\ion{Mg}{II}, \\ion{Si}{II}, \nand \\ion{Ca}{II}, for which individual lines\nare reported in Table \\ref{linelist},\n the number of useful lines are too small in comparison with \nthe neutral species, so we did not include them in the average results.\n\n\\subsection{Comparison with the literature}\n\n\nThe forbidden [\\ion{O}{I}]~6300.31~{\\AA} line is considered \nthe best oxygen abundance indicator, given that it is not\naffected by NLTE effects. We derive an\nabundance of A(O)~$=+6.95$. The triplet lines at 7774-7~{\\AA} and 8446~{\\AA} \ncomputed in NLTE give an oxygen abundance A(O)~$=+7.02$, and [O\/Fe]~$=+0.90$ \nwas derived for HD~140238. Note that Fe was only computed in LTE. \nMel\\'endez et al. (2006) found a difference\nof $\\Delta$Fe$_{NLTE-LTE}=+0.08$~dex, which would reduce\nthe oxygen overabundance to [O\/Fe]~$=+0.82$. \nThis absolute oxygen abundance is in rather good agreement with previous NLTE derivation\nby Mel\\'endez et al. (2006) of A(O)~$=+6.97$, and [O\/Fe]~$=+0.57$, where\nthe difference in oxygen to iron is due to their\nmetallicity of [Fe\/H]~$=-2.25$ value (in NLTE),\nhigher by $0.34$~dex than the present value of [Fe\/H]~$=-2.59$. \nValues of [Fe\/H]~$=-2.3$, [O\/H]~$=-1.67$ and [O\/Fe]~$=+0.7$ were adopted \nby Bond et al. (2013).\n\nIn Fig. \\ref{Ocompare_fig} we present a comparison of the \nLTE [O\/Fe] abundance ratios obtained for the giant stars \nanalysed in the Large Programme ``First Stars'' \n(Cayrel et al. 2004; Spite et al. 2005) based on the \nforbidden [\\ion{O}{I}]~6300.31~{\\AA} line, \nwith the result derived for HD~140283. \nWe computed [O\/Fe] in HD~140283, adopting the solar oxygen abundance \nA(O)~$=+8.77$ for compatibility with the Large Programme. \nThe [O\/Fe] ratio obtained in the two components of the well-known \nspectroscopic turnoff binary CS~22876-032 \n(Gonz\\'alez Hern\\'andez et al. 2008) from the OH band, which includes \ncorrections for 3D effects, are also presented.\n\nThe high nitrogen abundance of [N\/Fe]~$=+1.06$, and a consequent \nhigh [C+N+O\/Fe] abundance ratio confirms previous findings by Barbuy (1983). \n\nThe results from Honda et al. (2004a) reported in Table \\ref{adopted_abundances} \nwere derived from a high-resolution and high-S\/N spectrum obtained with the Subaru High \nDispersion Spectrograph. The Cr abundance is the only result \nsignificantly different from ours for which the present value is $0.22$~dex lower \nin comparison with their result. In both cases the abundance is the average \nof \\ion{Cr}{I} and \\ion{Cr}{II} lines. Honda et al. (2004a) adopted a \n\\Teff~120~K lower and a gravity \\logg~0.2 dex lower with respect to \nour set of parameters, whereas they used the same value for \nmicroturbulence, which is the most important error source in Cr abundance. However, \ntheir Cr abundance was derived only based on three lines, in comparison with \n33 lines in the present work, which we consider a more reliable final \nabundance.\n\nRoederer (2012) analysed the zinc abundance in HD 140283 based on \ntwo \\ion{Zn}{I} \nlines also used in the present work, with a spectrum obtained with HARPS\/ESO,\n deriving a \nresult $0.08$~dex lower in comparison with our abundance. However, the signal-to-noise \nratio achieved in the present work is much higher than the value from Roederer (2012) \n(250 at 4500~{\\AA}), leading us to adopt the present result as the final Zn abundance. \nIn addition, Roederer (2012) was able to use the UV \\ion{Zn}{II} line located at \n2062.00~{\\AA} with a spectrum obtained with STIS\/HST to derive A(\\ion{Zn}{II})~$=+2.27$, \nor [\\ion{Zn}{II}\/Fe]~$=+0.30$, which is higher than the result from neutral transitions. \n\nFor strontium, the LTE result of [\\ion{Sr}{II}\/Fe]~$=-0.17$ derived here agrees with \nthe result from Roederer (2012), which used the same two \\ion{Sr}{II} lines, \nbut with \\loggf~$=+0.15$ for \\ion{Sr}{II}~4077.7~{\\AA}, slightly lower than the adopted value. \nOn the other hand, this value is higher in comparison with the LTE results from \nHonda et al. (2004a). Our adopted NLTE analysis gives [\\ion{Sr}{II}\/Fe]~$=-0.30$. \n\nThe yttrium abundance was derived in Honda et al. (2004a,b) \nbased on three \\ion{Y}{II} lines,\n located at 3788.70~{\\AA}, 3950.35~{\\AA}, and 4883.69~{\\AA}. \nWe added another two \\ion{Y}{II} lines and \nthe derived abundance [\\ion{Y}{II}\/Fe]~$=-0.40$ \nis slightly higher than the result from Honda et al. (2004a,b), and it is in \nvery good agreement with the abundance derived in Peterson (2011). \n\nThe zirconium abundance we derived is $0.1$~dex higher \nthan the result \nfrom Honda et al. (2004a,b), where two \\ion{Zr}{II} lines located \nat 3836.76~{\\AA} \nand 4208.98~{\\AA} were analysed. We adopted the same values of oscillator \nstrengths, and also used an extra \\ion{Zr}{II} line located at 4443.01~{\\AA}, \nwith individual abundances in agreement with the result from the 4208.98~{\\AA} \nline. Roederer (2012) also obtained a Zr abundance $0.1$~dex lower than the \npresent result, using the \\ion{Zr}{II} lines at \n3998.96~{\\AA}, 4149.20~{\\AA}, and \n4208.98~{\\AA}, in common with our analysis. For the latter line, Roederer \n(2012) obtained A(\\ion{Zr}{II})~$=-0.09$, only $0.04$~dex lower than the present \nresult. In our spectrum the lines 3998.96~{\\AA} and 4149.20~{\\AA} appear \nseverely blended and they were excluded. In addition, Peterson (2011) derived \nan intermediate Zr abundance between the present value and that from Roederer \n(2012), in agreement with our result within error bars.\n\n\n\n\\subsection{Source of neutron-capture elements}\n\nTruran (1981) suggested an early enrichment of heavy elements uniquely \nthrough the r-process, arguing that there would be no time for \nthe s-process to operate before the formation of the oldest stars. \nIn this scenario, even dominantly s-process elements\nsuch as Sr, Y, Zr, and Ba, would have been produced by the r-process in their\nrelatively reduced amounts. \n\nGiven the recent discussions on the r-process or s-process origin\nof the heavy elements in HD~140283, below we try to understand the\nresults concerning this star. Since HD~140283 is one of the oldest \nstars known so far, formed shortly after the Big Bang, \nit is a natural test star for studies of early heavy element \nformation.\n\nAs described in Paper~I, the barium isotopic abundances in HD~140283 are subject \nto intense debate in the literature. In a 1D LTE study of barium isotopes, \nGallagher et al. (2010, 2012) found  isotope ratios close to the s-process-only \ncomposition. This is supported by Collet et al. (2009), using 3D hydrodynamical \nmodels, where a maximum fraction of 15$\\pm$34\\% contribution of the r-process \nto the isotopic mix in HD~140283 is derived. More recently, Gallagher et al. (2015) \ncarried out 3D calculations of the barium isotopic lines in HD~140283, \nand the authors now favour a dominant r-process signature imprinted in the \nbarium isotopes in HD~140283. Gallagher et al. (2015) suggest that further work is \nneeded to improve the line formation in 3D, and that NLTE has \nto be taken into account.\n\nThe [Eu\/Ba] ratio is the unambigous indicator as to whether the heavy elements in \nHD~140283 are dominantly r- or s-process. In Paper I we concluded that the \n[Eu\/Ba]~$=+0.58\\pm0.15$ value indicates that r-process is the dominant nucleosynthesis process. \nWe found an LTE Ba abundance A(Ba)~$=-1.22$, in agreement \nwithin errors with A(Ba)~$=-1.28$ from 1D calculations in LTE derived by \nGallagher et al. (2015). With the newly derived Ba abundance, the ratio [Eu\/Ba]~$=+0.53\\pm0.18$ \nconfirms the previous conclusion concerning the dominant r-process origin. \n\nAn NLTE Ba abundance A(Ba)~$=-1.05$, or [\\ion{Ba}{II}\/Fe]~$=-0.63$ is derived here, \nand a lower [Eu\/Ba]~$=+0.34$ ratio is obtained in this case. \nIf an NLTE Ba abundance is considered, NLTE Eu also has to be considered: \nMashonkina et al. (2012) presented NLTE abundance corrections for the \n\\ion{Eu}{II}~4129~{\\AA} line in cool stars, showing that the corrections \nare small (lower than 0.1 dex) and positive for this element and these types of stars, \nwhich would turn the present [Eu\/Ba]$\\approx$0.44.\nA further ingredient is 3D vs. 1D calculations: Gallagher et al. (2015) \nreported A(Ba)~$=-1.43$ in HD~140283 from LTE 3D calculations, therefore, with \na $0.15$~dex lower value for the 3D with respect to 1D calculations, \nand in this case LTE [Eu\/Ba(3D)]~$=+0.74$ is obtained, or else\nadding the 3D effect to [Eu\/Ba]$\\approx$0.44 above, gives [Eu\/Ba]$\\approx$0.59.\nIn Table \\ref{eubaratio} we try to summarize all [Eu\/Ba] values given above.\n\nAccording to Simmerer et al. (2004), \na pure r-process contribution gives [Eu\/Ba]$_{r}=+0.698$, whereas \n[Eu\/Ba]$_{s}=-1.621$ is the abundance pattern due to the pure s-process. \nRecent models for the solar s-process abundances (Bisterzo et al. 2014) \npredict the same contribution for Ba in comparison with Simmerer et al. (2004), \nand a slightly higher contribution for the Eu abundance: \n2.7\\% from Simmerer et al. (2004); 6.0$\\pm$0.4\\% from Bisterzo et al. (2014). \nIn conclusion, the abundance ratios [Eu\/Ba] described above do not change \nsignificantly.\n\nIf a  lower [Eu\/Ba] value relative to Paper I is confirmed,\nit may indicate that the contribution from the \ns-process to the heavy elements is not so small.\n Table \\ref{eubaratio} shows that no clear conclusion can be reached,\nbut that conclusions from Paper I are still favoured.\n This discussion indicates \nthat a robust 3D+NLTE synthesis is needed to enable further conclusions.\n\n\n\\begin{table}\n\\caption{[Eu\/Ba] abundance ratios expected from the r- and s-process, \nand derived in HD~140283.}            \n\\label{eubaratio}     \n\\scalefont{1.0}\n\\centering                         \n\\begin{tabular}{cr}       \n\\hline\\hline                \n\\noalign{\\smallskip}\n\\hbox{Source} & \\hbox{[Eu\/Ba]} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{$^\\dagger$pure r-process} & $+$0.698 \\\\\n\\hbox{$^\\dagger$pure s-process} & $-$1.621 \\\\\n\\hbox{Eu$_{1D+LTE}$~$+$~Ba$_{1D+LTE}$}   & $+$0.53 \\\\\n\\hbox{Eu$_{1D+LTE}$~$+$~Ba$_{1D+NLTE}$}  & $+$0.34 \\\\\n\\hbox{Eu$_{1D+LTE}$~$+$~$^\\triangle$Ba$_{3D+LTE}$}   & $+$0.74 \\\\\n\\hbox{$^\\diamondsuit$Eu$_{1D+NLTE}$~$+$~Ba$_{1D+NLTE}$} & $+$0.44 \\\\\n\\hbox{$^\\diamondsuit$Eu$_{1D+NLTE}$~$+$~$^\\triangle$Ba$_{3D+NLTE}$} & $+$0.59 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\tablebib{$\\dagger$: Simmerer et al. (2004); $\\triangle$: 3D correction \nfrom Gallagher et al. (2015); $\\diamondsuit$: NLTE correction from \nMashonkina et al. (2012).}\n\\end{table}\n\n\n\n\\subsubsection{Abundance pattern}\n\nFigure \\ref{pattern} shows the abundance pattern of HD~140283, based on \nvalues derived in the present work (red circles), as well as results \nfrom the literature (blue circles), where upper limits are also indicated. \nThe upper panel presents the elements from carbon to zinc, where \ncarbon from both CH and \\ion{C}{I} lines are indicated. \nThe neutron-capture elements are shown in the lower panel. \nAs a reference abundance pattern, we used the r-element enhanced EMP (r-II) \nstar CS~31082-001 (Barbuy et al. 2011; Siqueira-Mello et al. 2013) \nfor comparison (dotted line), rescaled to match the\ndominantly r-process europium abundance in HD~140283. \nThis figure shows the overabundance of the lighter heavy elements \n(including Sr, Y, and Zr) in comparison with expected values from an r-rich \nstar. \n\nThe LTE abundances of Ba and Sr result in [Sr\/Ba]~$=+0.63$ for HD~140283. \nIf we take the LTE 3D Ba correction from Gallagher et al. (2015) into account, \nthis value would be [Sr\/Ba]~$=+0.78$, even more overabundant. The newly \nderived NLTE abundances of Ba and Sr give [Sr\/Ba]~$=+0.33$, or \n[Sr\/Ba]~$=+0.48$ if the 3D Ba correction is applied. \nOverabundances are also obtained for Y and Zr, reaching values of \n[Y\/Ba]~$=+0.41$ and [Zr\/Ba]~$=+0.74$ in the present work, \nusing only the LTE results. With the NLTE Ba abundance these ratios \ndecrease to [Y\/Ba]~$=+0.23$ and [Zr\/Ba]~$=+0.56$ in HD~140283, but the \n3D Ba correction enhances these values to [Y\/Ba]~$=+0.38$ and [Zr\/Ba]~$=+0.71$. \nIn Table \\ref{sryzrbaratio} these values are summarized, showing that it is \nimportant to account for NLTE and 3D effects to evaluate these \nabundance ratios.\n\n\n\\begin{table}\n\\caption{[Sr, Y, Zr\/Ba] abundance ratios derived in HD~140283.}            \n\\label{sryzrbaratio}     \n\\scalefont{1.0}\n\\centering                         \n\\begin{tabular}{cc}       \n\\hline\\hline                \n\\noalign{\\smallskip}\n\\hbox{Source} & \\hbox{Abundance} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{Sr$_{1D+LTE}$~$+$~Ba$_{1D+LTE}$}     & [Sr\/Ba]~$=+$0.63 \\\\\n\\hbox{Sr$_{1D+NLTE}$~$+$~Ba$_{1D+NLTE}$}   & [Sr\/Ba]~$=+$0.33 \\\\\n\\hbox{Sr$_{1D+LTE}$~$+$~$^\\triangle$Ba$_{3D+LTE}$}     & [Sr\/Ba]~$=+$0.78 \\\\\n\\hbox{Sr$_{1D+NLTE}$~$+$~$^\\triangle$Ba$_{3D+NLTE}$}   & [Sr\/Ba]~$=+$0.48 \\\\\n\\hbox{Y$_{1D+LTE}$~$+$~Ba$_{1D+LTE}$}      & [Y\/Ba]~$=+$0.41 \\\\\n\\hbox{Y$_{1D+LTE}$~$+$~Ba$_{1D+NLTE}$}     & [Y\/Ba]~$=+$0.23 \\\\\n\\hbox{Y$_{1D+LTE}$~$+$~$^\\triangle$Ba$_{3D+NLTE}$}     & [Y\/Ba]~$=+$0.38 \\\\\n\\hbox{Zr$_{1D+LTE}$~$+$~Ba$_{1D+LTE}$}     & [Zr\/Ba]~$=+$0.74 \\\\\n\\hbox{Zr$_{1D+LTE}$~$+$~Ba$_{1D+NLTE}$}    & [Zr\/Ba]~$=+$0.56 \\\\\n\\hbox{Zr$_{1D+LTE}$~$+$~$^\\triangle$Ba$_{3D+NLTE}$}    & [Zr\/Ba]~$=+$0.71 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\tablebib{$\\triangle$: 3D correction from Gallagher et al. (2015).}\n\\end{table}\n\n\nSeveral authors found high abundance ratios of first peak elements\nSr, Y, Zr, with respect to Ba: [Sr,Y,Zr\/Ba]~$>0$, in very metal-poor stars \n(e.g. Honda et al. 2004a,b; Honda et al. 2006; Fran\\c{c}ois et al. 2007; \nCowan et al. 2011). According to Simmerer et al. (2004), in the solar material these \nelements are mainly produced by the s-process, with fractions of \n89\\%, 72\\%, and 81\\%, respectively, but recent results described in \nBisterzo et al. (2014) are different for some elements (Sr: 68.9$\\pm$5.9\\%; \nY: 71.9$\\pm$8.0\\%; Zr: 66.3$\\pm$7.4\\%). Therefore, \nwhile a mechanism responsible for a r-II pattern is claimed to \nexplain the \nnucleosynthesis of the heaviest trans-iron elements, an extra \nmechanism (such as truncation or other) should act to produce \nthe enhancement of the lightest heavy elements relative to the \nheaviest elements, which must occur very early in the history of the Universe. \n\nSeveral models in the literature find that supernova explosion explain the \noverabundances of the first peak elements in metal-poor stars. \nMontes et al. (2007) provided evidence for the existence \nof a light element primary process that contributes to the nucleosynthesis of \nmost elements in the Sr to Ag range, producing early high [Sr,Y,Zr\/Ba] \nratios. The astrophysical scenarios in neutrino-driven winds are claimed \nas promising sources of light trans-iron elements (Wanajo 2013, Arcones et al. \n2013). See also the LEPP (e.g. Bisterzo et al. 2014).\n\nHansen et al. (2014) show that the observed patterns may be obtained \nby combining an r- and an s-pattern, but the 's' has to be (slowly) \nproduced in another generation  (AGB?), not very compatible with the great \nage of the star. Yet, a full understanding of core collapse supernova explosion \nusing 3D hydrodynamical modelling is needed.\n\n\nA possibility that HD~140283 formed in a very early dispersed cloud \ncould also lead to a previous early chemical enrichment or pollution \nby massive AGB stars, which overproduce the first peak s-process elements \n(Bisterzo et al. 2010). In this scenario, a previous enrichment in Fe seeds is needed, \nand the timescale of the whole process is too long and not \nideal for such an old star.\n\nIn Fig. \\ref{pattern2} we show the abundance pattern in HD~140283 \n(blue squares) compared with the values obtained in HD~122563 (green circles), \na metal-poor star well-known for its excesses of light neutron-capture elements. \nThe abundances of the heavy elements in HD~122563 have been taken from Honda et al. (2006), \nCowan et al. (2005), and Roederer et al. (2010). As references, we included \nthe abundances of CS~31082-001 (black dots) and the residual solar r-process \npattern, following the deconvolution by Simmerer et al. (2004). The data were rescaled \nto match the europium abundances. The overabundance \nof first peak elements derived in HD~140283 are only slighly lower than\n the values observed  in HD~122563. \n\nIn conclusion, Figures \\ref{pattern} and \\ref{pattern2} show overabundances \nof the first-peak heavy elements, and, at a lower level, also of Ba, La and Ce,\n with respect to CS~31082-001 and the residual solar r-process.\nIt is not clear if the extra mechanism claimed to explain the higher \nabundances of light neutron-capture elements might also produce \nheavier elements, but less efficiently.\n\nThe overabundance of cerium [Ce\/Fe]~$=+0.18$, a \ndominantly s-process element (83.5$\\pm$5.9\\% in the solar material, \nfrom Bisterzo et al. 2014),\nwas also found in other few stars. \nIn Fran\\c{c}ois et al. (2007), the star CS~30325-094 also showed a \nhigh Ce overabundance [Ce\/Fe]~$=+0.43$, however, with deficient values \nfor other s-elements like Ba ([Ba\/Fe]~$=-1.88$) and Sr ([Sr\/Fe]~$=-2.24$). \nIn Fig. \\ref{pattern} and \\ref{pattern2} the Ce abundance in \nHD~140283 is clearly higher than the r-process pattern, and the \nsame behaviour is observed in HD~122563. Because of the difficulty in explaining \nthe abundance pattern of HD~122563 as a combination of \nthe r-process and the main s-process, Honda et al. (2006) \nsuggested a single process that is responsible for the enhancement of the \nlight neutron-capture elements and the production of heavy elements \nin this star. A truncation process in the initial supernova \n(Aoki et al. 2013) could be a possible solution. A new model of hypernova \nshows that the explosion correctly produces the abundances of the elements \nobserved in HD~122563, therefore, explaining the so-called weak r-process \n(Fujibayashi et al. 2015) as well as the similar pattern derived in HD~140283.\n\n\n\n\\section{Conclusions}\n\nWe used a high-S\/N and high-resolution spectrum of HD~140283, obtained\nwith a seven hour exposure with the ESPaDOnS spectrograph at the CFHT telescope,\nto provide a line list for metal-poor subgiant stars of which HD~140283\nis a template. We carried out a detailed derivation of abundances, using\nboth LTE and NLTE calculations, based on as many as possible reliable lines \navailable.\n\nIn Paper I we concluded that the derived europium abundance was indicative\nof an r-process origin for europium. The present LTE [Eu\/Ba]~$=+0.53\\pm0.18$ confirms \nthat conclusion. Combining the newly derived NLTE Ba abundance with NLTE \ncorrections for Eu and 3D corrections for Ba from recent literature, the \nabundance ratio [Eu\/Ba]~$=+0.59\\pm0.18$ also indicates a small contribution (if any) \nfrom the main s-process to the neutron-capture elements in HD~140283.\n\nAn extra mechanism is claimed to explain the overabundances \nof lighter heavy elements, in addition to an r-II abundance pattern \nresponsible for the heavier elements, and possible astrophysical scenarios are \ndiscussed. The high Ba, La, and Ce abundances derived \nin HD 140283 are similar to those in HD 122563,\n and these two stars may be  excellent examples of abundances dominated \nby the weak r-process.\n\n\n\n\n\n\n\n\\begin{acknowledgements}\nBased on observations obtained with Brazilian time, provided by\na contract of the Laborat\\'orio Nacional de Astrof\\'{\\i}sica (LNA\/MCTI)\nand the Canada-France-Hawaii Telescope (CFHT), which is operated by the \nNational Research Council of Canada, the Institut National des Sciences \nde l'Univers of the Centre National de la Recherche Scientifique of France,  \nand the University of Hawaii.\nCS and BB acknowledge grants from CAPES, CNPq, and FAPESP. \nSMA is thankful to FAPESP for financial support and IAG for \nhospitality during his visit to Universidade de S\\~ao Paulo. \nMS and FS acknowledge the support of CNRS (PNCG and PNPS). \nSAK acknowledges the SCOPES grant No. IZ73Z0-152485 for financial support. \nWe thank the referee for his\/her useful comments. \nThis work has made use of the VALD database, operated at \nUppsala University, the Institute of Astronomy RAS in Moscow, \nand the University of Vienna.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nAutomatic meeting\/spoken-conversation analysis is one of essential fundamental technologies required \nfor realizing, e.g. communication robots that can follow and respond to our conversations. \nThe meeting analysis comprises several tasks, namely (a) diarization, i.e.,  determining who is speaking when, (b) source counting, i.e., estimating the number of speakers in the conversation, \n(c) source separation, and (d) automatic speech recognition (ASR), i.e., transcribing the separated streams corresponding to each person.\nWhile ideally these tasks should be jointly accomplished to realize optimal meeting analysis,\nmost studies focus on one of the aforementioned tasks,\nsince each task itself is already very challenging in general.\n\nFor example, a considerable number of research has been carried out for developing reliable diarization systems \\cite{Diarization_review, DIHARD_data, AMI_data}.\nMost of the conventional diarization approaches perform block or block-online processing with the following two steps \\cite{Diarization_review, Araki_ICASSP2007, Araki_HSCMA_2008, DIHARD_BUT, DIHARD_JHU}.\nFirst, at each block, they perform source separation (if necessary) and obtain speaker identity features concerning each speaker, \nin the form of, e.g. i-vector \\cite{i-vector}, x-vector \\cite{x-vector}, or spatial signature \\cite{Araki_ICASSP2007, Araki_HSCMA_2008, Drude_ICASSP2018}.\nThen, the correct association of speaker identity information among blocks, i.e., diarization results, is estimated by\nclustering these features by using e.g. agglomerative hierarchical clustering \\cite{DIHARD_JHU}.\nAlthough these conventional algorithms can achieve reasonable diarization performance,\nthe results are not guaranteed to be optimal,\nbecause the steps concerning speaker identity feature extraction and clustering are done independently.\nFocusing on this limitation, \\cite{Fujita_IS2019} recently proposed a neural network (NN)-based diarization approach \nthat directly outputs diarization results (without any clustering stage), \nand showed that it can outperform a conventional two-stage approach \\cite{DIHARD_JHU} in CALLHOME task \nwhere two people speak over a phone channel.\nNote that, although some diarization systems can deal with overlapped speech, it does not mean that they can perform speech enhancement, i.e., separation and denoising, which is\neventually required for the meeting analysis.\n\nAnother key challenge for automatic meeting analysis is the separation of overlapped speech. \nPerhaps surprisingly, even in professional meetings, the percentage of overlapped speech, \ni.e., time segments where more than one person is speaking, \nis in the order of 5 to 10\\%, \nwhile in informal get-togethers \nit can easily exceed 20\\% \\cite{onlineRSAN_ICASSP2019}.\nTo address the source separation problem, recently, many NN-based single-channel approaches have been proposed,\nsuch as Deep Clustering (DC)\\cite{Hershey_ICASSP16}, \nand Permutation Invariant Training (PIT)\\cite{Yu2016, Kolbaek2017}.\nDC can be viewed as two-stage algorithms, where in the first stage embedding vectors are estimated for each time-frequency (T-F) bin. \nIn the second stage, these embedding vectors are clustered to obtain separation masks, given the correct number of clusters, i.e., sources.\nPIT, on the contrary, is a single-stage algorithm, because it lets NNs directly estimate source separation masks.\nIn PIT, however, the NN architecture depends on the maximum number of sources to be extracted. \n\nTo lift this constraint on the number of sources, \nwe proposed Recurrent Selective Attention Network (RSAN) that is a purely NN-based mask estimator\ncapable of separating an arbitrary number of speakers and simultaneously counting the number of speakers in a mixture \\cite{RSAN}.\nIt extracts one source at a time from a mixture and recursively performs this process until all sources are extracted.\nThe RSAN framework is based on a recurrent NN (RNN) \nwhich can learn and determine how many computational steps\/iterations\nhave to be performed \\cite{Graves_2016_arxiv_adaptiveRNN}.\nFollowing the idea of the recursive source extraction, \\cite{Takahashi_Interspeech2019} proposed to incorporate a time-domain audio separation network (TasNet) \\cite{luo2019conv} into the RSAN framework.\n\nTo go one step further toward ideal meeting analysis that deals with aforementioned tasks (a)(b)(c) simultaneously,  \nin \\cite{onlineRSAN_ICASSP2019}, RSAN was extended to an all-neural block-online approach (hereafter, online RSAN) \nthat simultaneously performs source separation, source number counting and even speaker tracking from a block to a block, \ni.e., performing the diarization-like process.\nIt was shown that online RSAN can handle well-controlled scenarios such as clean (noiseless and anechoic) simulated dialog-like data \nof 30 seconds, and outperform conventional systems in terms of source separation and diarization performance.\n\n\nHowever, it was not clear from our past studies \nwhether such all-neural approach, i.e., online RSAN, can be generalized \nto realistic meeting scenarios containing more spontaneously-speaking speakers, \nsevere noise and reverberation, and how it performs in comparison with the state-of-the-art systems.\nTo this end, this paper focuses on the application of the online RSAN model \nto the realistic situations, and its evaluation in comparison with state-of-the-art diarization algorithms.\nWe first review the online RSAN model (in Section~\\ref{sec:prop})\nand introduces practical techniques to increase robustness against real meeting data,\nsuch as a decoding scheme that mitigates over-estimation error in source counting (in Section~\\ref{sec:decode}).\nWe then carry out experiments with real and simulated meeting data involving up to 6 speakers, \ncontaining a significant amount of noise and reverberation.\nThen, finally, it will be shown that, even in such difficult scenarios, \nonline RSAN can still perform effective speech enhancement, i.e., source separation and denoising, \nand simultaneously outperform a state-of-the-art diarization system \\cite{dihard19} developed \nfor a recent challenge \\cite{DIHARD_data}.\nThis finding is the main contribution of this paper. \nBefore concluding this paper, we also discuss the challenges that remain.\n\n\\section{Overall Structure of online RSAN}\n\\label{sec:prop}\n\n\nFigure \\ref{fig:overview} summarizes how online RSAN works on the first 2 blocks of an example mixture containing three sources.\nSince the model works in a block-online manner,\nwe first split the input magnitude spectrogram $\\mat{Y}$ into $B$ consecutive blocks of equal time length, $[\\mat{Y}_1,\\ldots,\\mat{Y}_b,\\ldots,\\mat{Y}_B]$, before feeding it to the system.\n\nOnline RSAN estimates a source extraction mask $\\hat{\\mathbf{M}}_{b,i}$ in each $b$-th block recursively to extract all source signals therein, \nwhile judging at each $i$-th source extraction iteration whether or not to proceed to the next iteration to extract more source signals.\nThe same neural network ``NN'' is repeatedly used for each block and iteration.\nAt every iteration in $b$-th block, NN receives three inputs, \na residual mask $\\mat{R}_{b,i}$ (res-mask in the figure), \nan input spectrogram $\\mat{Y}_b$, \nand an auxiliary feature $\\vect{z}_{b-1,i}$ (speaker embed.~vec. in the figure) \nto estimate a mask for a specific speaker $\\hat{\\mathbf{M}}_{b,i}$ \nand a speaker embedding vector representing that specific speaker $\\vect{z}_{b,i}$ as:\n\\begin{equation}\n    \\hat{\\mathbf{M}}_{b,i}, \\vect{z}_{b,i} = \\textrm{NN}(\\mat{Y}_b, \\mathbf{R}_{b,i}, \\vect{z}_{b-1,i}).\n\\end{equation}\nThe residual mask can be seen as an attention map that informs the network\nwhere to attend to extract a speaker that was not extracted in previous iterations in the current block.\nAt every first iteration in the $b$-th block, the residual mask is initialized with $\\mat{R}_{b,0}=\\mat{1}$.\n\n\\begin{figure}[t]\n \\begin{center}\n  \\includegraphics[width=90mm]{.\/fig\/network_architecture_ver2.jpg}\n     \\end{center}\n   \\caption{Overview of online RSAN framework}\n \\label{fig:overview}\n\\end{figure}\n\n\nIn the first processing block, $b=1$, where we have two sources, online RSAN performs two source extraction iterations \nto extract all sources.\nSince it is the first block, no speaker information is available from the previous block. \nTherefore, the input speaker information is set to zero, $\\vect{z}_{0,i}=\\vect{0}$. \nWithout guidance, the network decides on its own in which order it extracts the source signals.\nFig.~\\ref{fig:overview} shows a situation where source 1 is extracted at the first iteration.\nThen, the first iteration is finished after generating another residual mask for the next iteration \nby subtracting the estimated source extraction mask from the input residual mask as $\\mathbf{R}_{1,1} = \\mathbf{R}_{1,0}-\\hat{\\mathbf{M}}_{1,1}$.\nAt the second iteration, the network receives the residual mask $\\mathbf{R}_{1,1}$ \nand input spectrogram $\\mathbf{Y}_1$ to estimate a mask for another source.\nThen, it follows the same procedure as the first iteration.\nThe network decides to stop the iteration \nby examining how empty the updated residual mask is.\nSpecifically, the source extraction process is stopped in iteration $i$ if\n$\\frac{1}{TF} \\sum_{tf} \\mathbf{R}_{i,tf} < t_{\\text{res-mask}}$ \nwhere $T$ and $F$ correspond to the total number of time frames and frequency bins in the block,\nand $t_{\\text{res-mask}}$ is an appropriate threshold.\n\n\nNote that the speaker embedding vector $\\vect{z}_{b,i}$ is passed to the next time block, $b+1$, \nand guides the $i$-th iteration on that block to extract the same speaker as in $(b,i)$. \nIn the figure, it can be seen that the green source (source 1) is always extracted in the first, \nthe blue in the second and the pink in the third iterations. \nIf a source happens to be silent in a particular block (see the 2nd iteration in block 2), \nthe estimated mask is to be filled with zeros ($\\hat{\\mat{M}}_{b,i} = \\vect{0}$), \nand the residual mask is to stay unmodified ($\\mat{R}_{b,i} = \\mat{R}_{b,i-1}$).\n\nAt the second iteration of block 2, \nthe criterion to stop the source extraction iteration is not met because the residual mask is not empty.\nIn such a case, the model increases the number of iterations to extract any new speaker\nuntil the stopping criterion is finally met.\nOverall, by having this structure, the model can perform jointly source separation, counting and speaker tracking.\n\nTo deal with background noise,\nin this study, we force online RSAN to estimate a mask for the noise always at the first source extraction iteration \nin each block (see Fig. \\ref{fig:overview}).\nWith this scheme, we can easily identify signals corresponding to background noise among separated streams \nand discard them if necessary.\n\n\\section{Summary of practical techniques \\\\ required to handle real meeting data}\n\\label{sec:practical}\nThis section summarizes techniques that we incorporated into online RSAN\nto cope with noisy reverberant real meeting data.\nThey are divided into categories of input feature, training schemes, and decoding scheme\nand summarized in the following dedicated subsections. \n\n\\subsection{Input feature: Multichannel feature}\nAs in much past literature, e.g. \\cite{Yoshioka_ICASSP2018},\nour preliminary experiments confirmed that a multichannel feature as an additional input \nhelps improve separation performance in reverberant environments.\nIn this study, therefore, the inter-microphone phase difference (IPD) feature proposed in \\cite{Yoshioka_ICASSP2018} \nis concatenated by default to the magnitude spectrogram and used as input to online RSAN.\n\n\\subsection{Training scheme}\nTo train online RSAN,\nwe used training data comprising a pair of noisy reverberant meeting-like mixtures,\nand corresponding noiseless reverberant single-speaker signals.\n\n\\subsubsection{Cost function for model training}\n\\label{sec:prop:training}\nDuring training, the network is unrolled over multiple blocks and iterations and \nwas trained with back-propagation using the following multi-task cost function:\n\\begin{equation}\n\\mathcal{L} = {\\L^{(\\text{MMSE})}} + \\alpha {\\L^{(\\text{res-mask})}} + \\beta {\\L^{(\\text{TRIPLET})}}\n\\end{equation}\nIn the following, we explain each term on the right-hand side of the above equation,\nstarting from ${\\L^{(\\text{MMSE})}}$.\n\nAt each iteration, the network is required to output a mask for a certain source, but the order of source extraction when they first appear is not predictable.\nIn such a case, a permutation-invariant loss function is required.\nOnce a source was extracted and the permutation was chosen to minimize the error on its first appearance, \nits position in the iteration process is fixed for any following blocks, as the embedding vectors are passed and thus the desired output order is known.\nConsequently, a {\\it partially} permutation-invariant utterance-level mean square error (MSE) loss can be used for online RSAN as:\n\\begin{equation}\n{\\L^{(\\text{MMSE})}} = \\frac{1}{IB} \\sum_{i,b} |\\hat{\\mat{M}}_{i,b}\\odot \\mat{Y} - \\mat{A}_{\\phi_b}|^2,\n\\end{equation}\nwhere $\\mat{A}_{\\phi_b}$ is a target reverberant single-speaker signal.\n\\footnote{To handle reverberant mixture, it was found in our preliminary investigation \nthat the target signal $mat{A}_{\\phi_b}$ has to be magnitude spectrum of reverberant (not anechoic clean) speech}.\nWhen a source was active before, but is silent on the current block, a silent signal $\\mat{A}_{b,i}=\\mat{0}$ are inserted as a target.\nThe permutation $\\phi_b$ for $b$-th block is formed by concatenating the permutation used for the last block $\\phi_{b-1}$ with the permutation $\\phi^*_b$ that minimizes the utterance-level separation error for the\nnewly discovered sources in block $b$.\nTo force the network to estimate a mask for the noise always at the first source extraction iteration,\nwe always inserted noise magnitude spectrogram as a target at the first iteration,\nand from the second iteration, we used the above partially permutation-invariant loss.\n\n${\\L^{(\\text{res-mask})}}$ is a cost function related closely to the source counting performance.\nTo meet the speaker counting and iteration stopping criterion when all sources are extracted from a mixture, \nwe minimize this cost and pushes the values of the residual mask to $0$ if no speaker is remaining (see \\cite{RSAN} for more details).\nwe minimize the following ${\\L^{(\\text{res-mask})}}$ and pushes the values of the residual mask to $0$ if no speaker is remaining.\n\\begin{equation}\n{\\L^{(\\text{res-mask})}} = \\sum_{b,tf} \\left[\\max\\left(1-\\sum_{i}\\hat{\\mat{M}}_{b,i}, 0\\right)\\right]_{tf}\n\\end{equation}\n\n${\\L^{(\\text{TRIPLET})}}$ is a triplet loss that is shown to help increase speaker discrimination capability,\nby ensuring the cosine similarity between each pair of embedding vectors for the same speaker \nis greater than for any pair of vectors of differing speakers.\nIt is formed by first choosing one anchor vector $\\vect{a}$, a positive vector $\\vect{p}$ belonging to the same speaker as $\\vect{a}$,\nand a negative vector $\\vect{n}$ belonging to a different speaker from $\\vect{a}$, \nfrom a set of speaker embedding vectors within one minibatch.\nBased on the cosine similarities between the anchor and negative vectors $s_i^{an}$,\nand the anchor and the positive vectors $s_i^{ap}$, \nthe triplet loss for $N$ triplets can be calculated as \\cite{Li2017}: \n\\begin{equation}\n{\\L^{(\\text{TRIPLET})}} = \\sum_{i=0}^{N} \\max(s_i^{an} - s_i^{ap} + \\delta, 0).\n\\end{equation}\nwhere $\\delta$ is a small positive constant.\nInterestingly, in this study, this loss was found to improve not only speaker discrimination capability\nbut also speaker tracking capability of online RSAN.\nWhen training with meeting-like data where people speaks intermittently,\none minibatch is usually formed with only a part of a meeting, in which very often certain speaker speaks only one time\nand remains silent to the end of this meeting excerpt (although he\/she may start speaking again in a later part of the meeting).\nIf we do not use the triplet loss, the network is not encouraged to keep remembering such a person to the end of the meeting,\nand eventually, it starts estimating a speaker embedding vector irrelevant to the speaker.\nTo circumvent such issue and make the network ready always for a situation when he\/she starts speaking again,\nwe can use the triplet loss and make the network always output speaker embedding vectors that are consistent over frames\nno matter whether the speaker is speaking or not.\n\n\n\n\\subsubsection{Noise mask}\nTo deal with background noise in the mixture, we let the network to estimate a mask for background noise \nalways at the first source extraction iteration in each block.\nTo force such behavior to the network, we used the following permutation variant loss (as oppose to permutation\ninvariant loss) for the mask for background noise:\n\\begin{equation}\n{\\L^{(\\text{MMSE})}} = \\frac{1}{B} |\\hat{\\mat{M}}_{1,b}\\odot \\mat{Y} - \\mat{A}_b^{(\\textrm{noise})}|^2,\n\\end{equation}\nwhere $\\mat{A}_b^{(\\textrm{noise})}$ is the magnitude spectrum of background noise.\nWith this scheme, we can easily find and discard signals corresponding to background noise from enhancement results.\n\n\\subsubsection{Teacher forcing for iterative source extraction}\nDuring training, when calculating residual mask $\\mathbf{R}_{b,i+1}$ at $b$-th block for the $(i+1)$~-th iteration,\ninstead of using the estimated source extraction mask at the $i$-th iteration,\nwe can calculate it by using an oracle source extraction mask $\\hat{\\mathbf{M}}_{b,i}^{\\textrm{(oracle)}}$ \nbased on the estimated permutation~\\cite{RSAN},\ni.e., $\\mathbf{R}_{b,i+1} = \\mathbf{R}_{b,i}-\\mathbf{M}_{b,i}^{\\textrm{(oracle)}}$.\nIn RNN literature, this form of training is known as teacher forcing.\nWhile the effectiveness of this scheme was not so clear in the previous study \\cite{RSAN},\nwe found it mandatory when we cope with noise and many speakers, \nboth of which are inevitable during training with meeting-like data.\n\n\\subsection{Decoding scheme: decoding with consistency check}\n\\label{sec:decode}\nReal meeting data contains a lot of unexpected sound events that are hardly observed in the training data,\nsuch as laughing sounds, a sudden change in tone of voice, coughing sound, \nand rustling sounds from e.g. papers, to name a few.\nAll of these sounds can be a cause for online RSAN to mistakenly detect a new spurious speaker and increase the number of source count.\nWhen it increases the source count by mistake because of such unexpected unseen sounds, \nit tends to wrongly split a speaker into two; the new embedding vector \nand an embedding vector already associated with the speaker.\nIt causes over-estimation errors in the source counting,\nand degrades embedding vector representation of the speaker, leading to degradation in overall performance.\n\nLet us denote the speaker embedding vector set\nat $b'$-th block as $\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$\nwhere $I_{b'}$ corresponds to the total number of iteration in $b'$-th block.\nThen, to reduce such over-estimation error in source counting, \nwe propose to perform the following consistency check for the speaker embedding vector set, \n$\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$, when online RSAN increases the speaker count.\nSpecifically, we propose to decode all the past blocks with the embedding vector set $\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$.\nThen, if masks estimated with $\\vect{z}_{b',I_{b'}}$, $\\{\\hat{\\mat{M}}_{b,I_{b'}}\\}_{1 \\leq b \\leq b'}$, does not exceed $t_{\\text{res-mask}}$,\nit indicates that a speaker corresponding to $\\vect{z}_{b',I_{b'}}$ did not indeed appear \nin the past blocks and appeared for the first time at $b'$-th block. \nAnd thus, we accept the increase in the source count and keep using $\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$ for further process.\nOtherwise, we do not accept the increase in the speaker count,\nand discard $\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$ and replace the set of embedding vectors \nwith ones from the previous block $\\{\\vect{z}_{b'-1,i}\\}_{1 \\leq i \\leq I_{b'-1}}$ for further process.\n\n\\section{Experiments}\n\\label{sec:exp}\nIn this section, we evaluate online RSAN in comparison with state-of-the-art diarization methods,\nand shows its effectiveness. \n\n\\subsection{Experimental conditions}\n\\subsubsection{Data}\nWe generated three sets of noisy reverberant multi-speaker datasets for training;\n(dataset A) 20000 mixtures each of which is 10 seconds in length, and contains 1 or 2 speakers' speech signals, \n(dataset B) 10000 mixtures each of which is 60 seconds in length, and contains 1 to 6 speakers' speech signals,\nand (dataset C) 2372 mixtures each of which is 60 seconds in length, and contains 1 to 6 speakers' speech signals. \nTo all dataset, we added CHiME4 noise with SNR of 10 to 20~dB, and reverberation of $\\textrm{RT}_{\\textrm{60}}$ of 300 to 700~ms.\nUtterances for dataset A and B are taken from WSJ0 \\cite{WSJ0}, i.e., read speech, \nwhile those for dataset C are taken from headset recordings of real meetings recorded in our office, i.e., spontaneous speech. \nIn dataset A, each mixture was generated such that the first \\SI{5}{\\second} \ncontain one or two speakers with a probability of \\SI{50}{\\percent} each,\nwhile the second half contains zero, one  or two speakers \nwith a probability of \\SI{15}{\\percent}, \\SI{55}{\\percent} and \\SI{30}{\\percent}, respectively.\nSimilarly, in dataset B, \nthe first \\SI{5}{s} of the test utterance contains zero or one speaker with a probability of \\SI{50}{\\percent} each,\nwhile the mixture in the remaining time is generated such that it contains zero, \none, two or three speakers with a probability of \\SI{5}{\\percent}, \\SI{75}{\\percent} and \\SI{15}{\\percent} and \\SI{5}{\\percent} respectively.\n\n\nEvaluation was done with two datasets; (1) simulated meeting-like data\ncomprising 1000 mixtures similar to dataset B but with unseen speakers,\nand (2) real meeting data recorded at our office with a distant microphone-array \\cite{ArakiHSCMA2017}.\nReal meeting dataset consists of 8 meetings, each of which is 15 to 20 minutes in length.\nThe number of meeting participants varies from 4 to 6, all of who are unseen during training.\nThe meeting recording contains a significant amount of babble noise (SNR of 3 to 15~dB),\nand reverberation of $\\textrm{RT}_{\\textrm{60}}$ of 500~ms.\nThe percentage of overlapped speech in these meetings is found to be $25.7$~\\% on average.\n\n\n\\subsubsection{Implementation details of online RSAN}\nNN architecture and hyper-parameters for online RSAN was same as \\cite{onlineRSAN_ICASSP2019}.\nIt consists of one fully connected layer on top of two BLSTM layers.\nMultichannel input feature was calculated based on signals observed at 2 microphones,\nand thus overall online RSAN model in this study is a 2-channel system.\nFor the evaluation based on the simulated meeting-like data, \nthe online RSAN model was first trained with the training dataset A for 300 epochs,\nand then further trained with dataset B for 50 epochs.\nThen, to cope with real meeting data, the model was further trained with dataset C for 5 epochs.\nThe block size of online RSAN was set at 10 seconds.\n$t_{\\text{res-mask}}$ was set at 0.2.\nTo obtain diarization results with online RSAN,\nwe performed power-based voice activity detection (VAD) on extracted streams \nbased on a threshold value common to one meeting.\n\n\n\\subsubsection{Methods to be compared with}\nIn the evaluation with the simulated meeting-like data, \nthe performance of online RSAN was compared with a system similar to a top-performing system \\cite{DIHARD_JHU} in DIHARD-1 challenge \\cite{DIHARD_data}.\nFor this, we used off-the-shelf implementation and model from \\cite{dihard19}.\nSince it is based on clustering of x-vectors \\cite{x-vector},\nit will be referred to as ``x-vector clustering\" hereafter.\nIt is a single-channel system.\n\nFor the real meeting data evaluation, the performance of online RSAN was compared with ``x-vector clustering\"\nand a multi-channel diarization method based on online clustering of Time-Difference-Of-Arrival (TDOA) feature \\cite{Hori_TASLP2011},\nwhich will be referred to as ``TDOA clustering\". \nThe TDOA feature was calculated based on signals observed at 8 microphones.\nDiarization performance was evaluated in terms of diarization error rate (DER) \\cite{der} including speaker overlapped segments,\nwhile the speech enhancement performance was evaluated in terms of signal-to-distortion ratio (SDR) in BSSeval \\cite{BSSeval}.\nThe sampling frequency was 8k~Hz for all the methods.\n\n\\subsection{Experiment 1: Evaluation with simulated meeting-like data}\nBefore proceeding to evaluation with real meeting data,\nwe briefly examine the performance of online RSAN and whether it is ever possible to\ncope with noisy reverberant mixtures containing many speakers.\nTable \\ref{tbl:results_sim1} shows DERs of online RSAN and x-vector clustering, averaged over 1000 mixtures.\nIt was found that online RSAN works for noisy reverberant data, and outperformed the state-of-the-art x-vector clustering.\nSDR improvement obtained with online RSAN was 10.01~dB, which we believe is reasonably high.\n\n\\begin{table}[t]\n \\centering\n  \\caption{DERs for simulated meeting-like data (\\%)}\n \\label{tbl:results_sim1}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\begin{tabular}{c} x-vector clustering \\end{tabular} & \\begin{tabular}{c} Online RSAN \\end{tabular}  \\\\\n\\hline\n44.39 & \\textbf{33.7}     \\\\  \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}[t]\n \\begin{center}\n  \\includegraphics[width=130mm]{.\/fig\/specgram_ver2.jpg}\n     \\end{center}\n   \\caption{Spectrograms of 3 minute excerpt from real meeting data: observed signal (a), headset recordings of each speaker (b,c,d), and signals estimated by online RSAN (b',c',d')}\n \\label{fig:spectrogram}\n\\end{figure*}\n\n\\subsection{Experiment 2: Real meeting data}\nNow we evaluate the performance of online RSAN with real data.\nWe first evaluate effect of consistency-checking decoding (proposed in section \\ref{sec:decode}).\nAlso, since it may not be clear how much and what kind of training data would be necessary \nfor online RSAN to cope with real data,\nwe examined the effect of training with dataset C, i.e., spontaneous speech.\nTable \\ref{tbl:results_real1} suggests that both training with spontaneous speech \nand decoding with the consistency check is important. \nNote that, although one may think that DERs here are very high, \nit is in a similar range as DIHARD2 as you can see in \\cite{dihard19}.\n\n\\begin{table}[t]\n\\centering\n \\caption{Effect of model training with spontaneous speech, and decoding with consistency check when dealing with real meeting data}\n  \\label{tbl:results_real1}\n   \\begin{tabular}{l c}\n    \\hline\n    system & DER (\\%) \\\\ \\hline\n    Online RSAN trained only with dataset A \\& B       & 73.9 \\\\\n    \\ \\ + dataset C                                &  56.3 \\\\\n    \\ \\ \\ \\ +decoding with consistency checking   &  \\textbf{49.6} \\\\ \\hline\n  \\end{tabular}\n\\end{table}\n\nNow, let us compare, with the other methods, the performance of online RSAN model trained \non dataset A, B, and C\nand decoded with the consistency check.\nWhile we use a different threshold for power-based VAD for each meeting for table \\ref{tbl:results_real1}\nto take a closer look at the difference between each setting,\nwe reran the experiments with a common threshold for all meetings \nto make a fair comparison with the other methods.\nThe threshold was determined based on the validation dataset.\nTable \\ref{tbl:results_each} shows DERs of online RSAN, \nx-vector clustering and TDOA clustering.\nWhile there are some sessions where deep learning-based algorithms, i.e. x-vector clustering and online-RSAN did not work well,\non average, even in the real meeting scenario, online RSAN can outperform these conventional approaches.\n\nFigure \\ref{fig:spectrogram} shows spectrograms of unprocessed real meeting data \n(3-minute excerpt from one of the meetings used for the evaluation), \ncorresponding headset recording of each speaker, and signals estimated by online RSAN. \nNoise signals estimated by online RSAN are omitted from the figure.\nAudio examples corresponding to the spectrograms are available on our web-page \\cite{demo_page}.\nAs it can be seen, the headset recordings and the estimated signals look quite similar,\nwhich suggests that online RSAN extracted each speaker's voice clearly, and counted the number of speakers correctly. \nNote that, as the third speaker did not speak for the first 2 minutes, online RSAN did not output masks for that speaker in that period, \nand once that speaker starts speaking, it correctly increased the number of source extraction iteration to 4 (i.e., 3 speakers + noise) \nand started tracking that speaker.\n\n\n\\begin{table*}[]\n\\centering\n\\label{tbl:results_each}\n\\caption{DERs for noisy reverberant real meetings for each system}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\begin{tabular}{c} Meeting \\\\ ID \\end{tabular} & \\# of spk. & \\begin{tabular}{c} x-vector \\\\ clustering \\end{tabular} & \\begin{tabular}{c} TDOA \\\\ clustering \\end{tabular} & \\begin{tabular}{c} Online \\\\ RSAN \\end{tabular}  \\\\ \\hline\n1 & 6 &  51.2 & 46.8          & \\textbf{41.4}    \\\\ \\hline\n2 & 6 &  61.8 & 64.6          & \\textbf{58.4}    \\\\ \\hline\n3 & 6 &  73.5 & 62.6          & \\textbf{49.0}   \\\\ \\hline\n4 & 5 &  57.6 & \\textbf{23.8} & 55.7            \\\\ \\hline\n5 & 5 &  64.2 & \\textbf{47.5} & 72.5             \\\\ \\hline\n6 & 6 &  71.4 & 67.2          & \\textbf{40.3}     \\\\ \\hline\n7 & 4 &  68.1 & 73.6          & \\textbf{45.5}     \\\\ \\hline\n8 & 4 &  72.4 & 70.9          & \\textbf{48.8}     \\\\ \\hline \\hline\n  & Average  &  65.0 & 57.1   & \\textbf{51.4}     \\\\  \\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Discussion}\nAlthough online RSAN could cope with real meeting data to some extent, \nits performance was far from ``perfect\".\nWe found that most errors in the online RSAN process are due to its insufficient performance in source separation and tracking.\nIt sometimes confidently extracts or tracks two different speakers' signals \nwith one speaker embedding vector $\\{\\vect{z}_{b,i}\\}_{1 \\leq b \\leq B}$,\nprobably because their voice characteristics are similar from the system's point of view.\nThis type of error should be reduced by, for example, \nemploying more advanced NN architecture \\cite{Takahashi_Interspeech2019}, \nand increasing the number of speakers in training data like we propose in \\cite{Delcroix_ICASSP2020}.\n\n\n\n\n\\section{Conclusion}\nThis paper proposed several practical techniques required for all-neural diarization, source separation and \nsource counting model called online RSAN to cope with real meeting data.\nIt was shown that incorporation of the proposed consistency-checking decoding \nand training with spontaneous speech is effective.\nBased on the experiments with real meeting recordings, \nonline RSAN was shown to perform effective speech enhancement, and simultaneously outperform state-of-the-art diarization systems.\nOur future work includes incorporation of advanced source separation NNs into online RSAN,\nand evaluation in terms of ASR accuracy.\n\n\n\n\n\n\n\n\\bibliographystyle{.\/bibliography\/IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{First-principles calculations}\\label{sec:DFT_details}\n\nAll first-principles calculations were performed using the projector augmented wave (PAW) method~\\cite{PhysRevB.50.17953}\nas implemented in the Vienna \\emph{Ab initio} Simulation Package (VASP)~\\cite{PhysRevB.47.558, PhysRevB.54.11169}.\nFor density functional theory (DFT) calculations, the standard PAW potentials ({\\tt Zr\\_sv} and {\\tt O}) and a plane wave cutoff of 600 eV were employed.\nThe electronic interactions were described using the Perdew-Burke-Ernzerhof functional (PBE)~\\cite{PhysRevLett.77.3865}\nand the strongly constrained appropriately normed functional (SCAN)~\\cite{PhysRevLett.115.036402}.\nA $\\Gamma$-centered $k$-point grid with a spacing of 0.31 $\\AA^{-1}$ between $k$ points\n(corresponding to a 2$\\times$2$\\times$2 $k$-point grid for a 96-atom cell) was employed.\nThe Gaussian smearing method with a smearing width of 0.05 eV was used.\nWhenever ground state structures were required, the electronic optimization was performed until the total energy difference between two iterations was less than 10$^{-6}$ eV.\nThe structures were optimized until the forces were smaller than 5 meV\/\\AA.\nThe phonon dispersions were calculated by finite displacements using the Phonopy code~\\cite{PhysRevB.78.134106}.\nFor all phonon calculations, a 96-atom supercell and a $2\\times2\\times2$ $k$-point grid were used.\nTo account for the long-range dipole-dipole force constants, the nonanalytic contribution\nto the dynamical matrix was treated using the method of Ref.~\\cite{PhysRevB.55.10355}.\nThe static dielectric tensor and atomic Born effective charges were calculated using the PBEsol functional~\\cite{PhysRevLett.100.136406}.\n\n\nFor the random phase approximation (RPA) calculations, the GW PAW potentials ({\\tt Zr\\_sv\\_GW} and {\\tt O\\_GW})\nwere used. These GW PAW potentials are constructed by using additional projectors above the vacuum level,\nand therefore, they describe well the high-energy scattering properties of the atoms and are more accurate for the polarizability-dependent RPA calculations.\nAs shown in Ref.~\\cite{PhysRevResearch.2.043361}, the RPA is capable to simultaneously describe well both the structural and electronic properties of ZrO$_2$.\nDue to the large computational cost involved in RPA calculations, a reduced plane wave cutoff of 520 eV and\na `bcc'-like generalized regular grid with 8 $k$ points in the full Brillouin zone\nwere used. The RPA energies and forces are calculated using an efficient low-scaling algorithm~\\cite{PhysRevB.90.054115,PhysRevB.94.165109,PhysRevLett.118.106403}.\nThe energy cutoff for the response function was chosen to be the same as  the plane wave cutoff (520 eV)\nand the number of imaginary time\/frequency points ($N_{\\omega}$) was set to 10,\nwhich is sufficient to ensure convergence of RPA energies (see Fig.~\\ref{RPA_energy_wrt_NOMEGA16}) and forces (see Fig.~\\ref{RPA_force_wrt_NOMEGA16}).\nThe stress tensor $\\sigma_{\\alpha\\beta}$ at the RPA level was calculated  via finite differences of the RPA total energies using twelve slightly distorted structures through\n\\begin{equation}\\label{eq:stress_tensor}\n\\sigma_{\\alpha\\beta}\\approx\n -\\frac{1}{\\Omega} \\frac{E(e_{\\alpha\\beta}=+\\delta)-E(e_{\\alpha\\beta}=-\\delta)}{2\\delta},\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ represent the Cartesian coordinate indices, $\\Omega$ is the system volume, and $e_{\\alpha\\beta}$ is the strain tensor.\nHere, $\\delta=0.02$ was adopted. This value on the one hand ensures the convergence of the stress tensors for PBE calculations\nto within 2.5 kbar (see Fig.~\\ref{stress_vs_displacement}), and\non the other hand is sufficiently large to minimize the shell effects (plane wave $\\lG$-vectors moving in and out of the cutoff sphere) for RPA calculations. We note that the stress tensors calculated by the finite difference method is less prone to the Pulay stress than those calculated internally within VASP. The latter often suffers from basis set incompleteness errors because of the fixed plane wave basis set when the cell is distorted. This is the reason why we chose to use a relatively large plane wave cutoff of 600 eV for the DFT calculations, whose stress tensors are directly calculated by using the VASP internal routines. Even then the diagonal components of the stress tensor are corrected by the calculated Pulay stress before generating the machine-learned force fields (MLFFs). The phonon frequencies at $\\Gamma$ predicted by RPA were calculated by finite differences, with\nthe long-range dipole-dipole interactions calculated at the level of PBEsol.\n\n\n\n\n\\section{MLFF training}\\label{sec:training_details}\n\nOur MLFFs were initially trained on-the-fly during FP molecular dynamics (MD) simulations\nbased on the Bayesian linear regression~\\cite{PhysRevB.100.014105,book_Bishop}.\nFor a comprehensive description of the on-the-fly MLFF generation implemented in VASP, we refer to Refs.~\\cite{PhysRevLett.122.225701,PhysRevB.100.014105,doi:10.1063\/5.0009491}.\nA concise summary of this method can be found in Refs.~\\cite{doi:10.1021\/acs.jpclett.0c01061,Peitao_2020}.\nFor the on-the-fly training, the PBEsol functional~\\cite{PhysRevLett.100.136406} was used,\nsince it predicts accurate lattice parameters for all the three phases of ZrO$_2$~\\cite{Carla_2021} on par with SCAN, but it is cheaper.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth, trim = {0 0.05cm 0.2cm  0.2cm}, clip]{FigS1_RPA_energy_wrt_NOMEGA16.eps}\n\\end{center}\n\\caption{Calculated RPA energies (in meV\/atom) of a monoclinic structure of ZrO$_2$ with 24 atoms in the cell\nas a function of the employed number of imaginary\/frequency points $N_{\\omega}$.\nNote that the RPA energy obtained using $N_{\\omega}$=16 is taken as reference.\nThe RPA energy obtained using $N_{\\omega}=10$ converges to within 0.012 meV\/atom.\n }\n\\label{RPA_energy_wrt_NOMEGA16}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth, trim = {0 0.05cm 0.2cm  0.2cm}, clip]{FigS2_RPA_force_wrt_NOMEGA16.eps}\n\\end{center}\n\\caption{Calculated RPA forces (in meV\/\\AA) of each atom in a monoclinic structure of ZrO$_2$ with 24 atoms in the cell\nreferenced to the ones obtained using $N_{\\omega}=16$.\nThe results achieved by three $N_{\\omega}$ (12, 10, and 8) are shown. One can observe that the forces obtained using $N_{\\omega}=10$\nare converged to within 1 meV\/\\AA.\n }\n\\label{RPA_force_wrt_NOMEGA16}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth, trim = {0 0.05cm 0.2cm  0.2cm}, clip]{FigS3_stress_vs_displacement.eps}\n\\end{center}\n\\caption{Stress tensors (in kbar) calculated by finite differences at a plane wave cutoff of 520 eV\n referenced to the ones calculated internally by VASP using a plane wave cutoff of 1600 eV\n[($\\sigma_{xx}$, $\\sigma_{yy}$, $\\sigma_{zz}$, $\\sigma_{xy}$, $\\sigma_{yz}$, $\\sigma_{zx}$)=(79.15,  132.58,  58.07,  $-$5.52,  $ -$22.18,  49.09) kbar]\nas a function of strain displacement. The very large plane wave cutoff of 1600 eV was chosen to minimize the Pulay stress.\nA finite temperature MD snapshot of m-ZrO$_2$ with 24 atoms in the cell was employed for this test.\nA similar behavior was observed for the tetragonal as well as cubic phase of ZrO$_2$ (not shown). }\n\\label{stress_vs_displacement}\n\\end{figure}\n\n\nThe FPMD simulations were performed in the isothermal-isobaric (NPT) ensemble at ambient pressure using a Langevin thermostat~\\cite{book_Allen_Tildesley}\ncombined with the Parrinello-Raman method~\\cite{PhysRevLett.45.1196}. The time step was set to 2.5 fs.\nFor the MLFF generation, the separable descriptors~\\cite{doi:10.1063\/5.0009491} were used.\nThe cutoff radius for the three-body descriptors and the width of the Gaussian functions used for broadening the atomic\ndistributions of the three-body descriptors were set to 6 $\\AA$ and 0.4 $\\AA$, respectively. The number of radial basis\nfunctions and maximum three-body momentum quantum number of the spherical harmonics used to expand the atomic distribution\nfor the three-body descriptors were set to 15 and 4, respectively. The parameters for the two-body descriptors were\nthe same as those for the three-body descriptor.\n\nAs mentioned in the main text, two MLFFs were constructed on the fly.\nThe first MLFF was trained on a 96-atom cell, whereas the second one was trained on a smaller 24-atom cell.\nWe note that the second MLFF was not used and this training step was just used to collect a new reference dataset,\nfrom which a subset of structures were extracted using a singular value decomposition (SVD) rank compression of the kernel matrix using a CUR algorithm~\\cite{Mahoney697,PhysRevB.100.014105}\nand then recalculated by high-level QM calculations.\nFor the first MLFF, the detailed training procedures can be found in Ref.~\\cite{Carla_2021}, and in the end, 592 structures were collected in the training dataset.\nFor the second MLFF trained on smaller unit cells, the following training strategy was employed.\n($i$) We first trained the force field by heating the monoclinic ZrO$_2$ from 0 K to 1800 K using 20 000 MD steps starting from the DFT relaxed structure.\n($ii$) Then, we continued training the tetragonal phase by a heating run from 1700 K to 2600 K using 10 000 MD steps.\n($iii$) The force field was further trained  by heating the cubic phase from 2500 K to 2800 K using 3333 MD steps.\nNote that in steps ($ii)$ and ($iii$), the initial structures were obtained by equilibrating the tetragonal and cubic structures\nat 1700 K and 2500 K, respectively, using the on-the-fly MLFF scheme, but the thus generated MLFFs were discarded.\n($iv$) To include the ideal tetragonal and cubic structures at 0 K, additional short heating runs from 0 K to 10 K using 100 MD steps\nwere performed starting from the DFT relaxed  tetragonal and cubic structures, respectively.\nEventually, only 1275 FP calculations were performed out of 33 533 MD steps, i.e., nearly 96.2\\%  of the FP calculations were bypassed.\n\n\n\\begin{table*}\n  \\caption{ Zero-temperature structural parameters of all three phases of ZrO$_2$ and energy differences between\n  phases predicted by different exchange-correlation functionals and MLFFs. $d_z$ represents the displacement of the oxygen (in unit of $c$) atoms along the $z$ direction with respect\n to ideal cubic position, and $\\beta$ is the angle between the lattice vectors $\\bf a$ and   $\\bf c$ in the monoclinic phase.\n The values in parentheses are RPA predicted energy differences for the structures that are optimized by MLFF-RPA$^{\\Delta}$.\nThe experimental structural parameters and volumes are extrapolated to 0 K~\\cite{PhysRevB.49.11560,https:\/\/doi.org\/10.1111\/j.1151-2916.1985.tb15247.x}.}\n  \\begin{ruledtabular}\n \\begin{tabular}{lccccccccc}\n             & PBE &   MLFF-PBE & SCAN & MLFF-SCAN & MLFF-SCAN$^{\\Delta}$ & MLFF-RPA$^{\\Delta}$& Expt. \\\\\nMonoclinic                            &       &           &        &       &        &         &       \\\\\n$a$ (\\AA)                             & 5.194 &  5.189    &  5.150 & 5.152 & 5.155  &  5.146  & 5.151 \\\\\n$b$ (\\AA)                             & 5.248 &  5.251    &  5.225 & 5.223 & 5.220  &  5.219  & 5.212 \\\\\n$c$ (\\AA)                             & 5.381 &  5.378    &  5.326 & 5.329 & 5.335  &  5.313  & 5.317 \\\\\n$\\beta$ (deg.)                        & 99.66 &  99.68    &  99.35 & 99.40 & 99.40  &  99.39  & 99.23 \\\\\nVolume ($\\AA^3$\/f.u.)                 & 36.15 &  36.11    &  35.35 & 35.37 & 35.41  &  35.20  & 35.22 \\\\\n\\\\\nTetragonal                            &       &           &        &       &        &         &       \\\\\n$a$ (\\AA)                             & 3.624 &  3.625    &  3.600 & 3.600 & 3.601  &  3.592  & 3.571 \\\\\n$c$ (\\AA)                             & 5.284 &  5.290    &  5.220 & 5.230 & 5.231  &  5.189  & 5.182 \\\\\n$c\/a$                                 & 1.458 &  1.459    &  1.450 & 1.453 & 1.453  &  1.445  & 1.451 \\\\\n$d_z$                                 & 0.057 &  0.058    &  0.052 & 0.054 & 0.054  &  0.048  & 0.047 \\\\\nVolume ($\\AA^3$\/f.u.)                 & 34.70 &  34.76    &  33.82 & 33.90 & 33.91  &  33.47  & 33.01 \\\\\n$\\Delta E_{t-m}$ (eV\/f.u.) & 0.110 &  0.109    &  0.074 & 0.074 & 0.075  &  0.067 (0.069)  & --- \\\\\n\\\\\nCubic                                 &       &           &        &       &        &         &       \\\\\n$a$(\\AA)                              & 5.120 &  5.126    &  5.088 & 5.090 & 5.091  &  5.076  & --- \\\\\nVolume ($\\AA^3$\/f.u.)                 & 33.56 &  33.68    &  32.92 & 32.97 & 32.99  &  32.70  & --- \\\\\n$\\Delta E_{c-t}$ (eV\/f.u.) & 0.102 &  0.100    &  0.085 & 0.083 & 0.083  &  0.053 (0.047)  & --- \\\\\n \\end{tabular}\n \\end{ruledtabular}\n \\label{tab:lattice_constants}\n\\end{table*}\n\n\nAt the end of the on-the-fly training, we determined the final regression\ncoefficients\nby using a factor of 10 for the relative weight of the energy equations with respect to the equations for the forces and stress tensors,\nand a SVD to solve the least-squares problem.\nThese were found to improve the overall accuracy of MLFFs~\\cite{Peitao_2020,Carla_2021}.\n\n\nFinally, all the structures contained in $T^{(1)}$ and $T^{(3)}$ were recalculated by PBE and SCAN,\nwhereas RPA calculations were performed only for the structures in $T^{(3)}$ (including 168 structures of 24 atoms, see the main text).\nMLFF-$\\Delta$ was obtained by machine learning the differences in energies, forces and stress tensors between high-level and low-level QM calculations\nusing the separable descriptors~\\cite{doi:10.1063\/5.0009491} with low spatial resolution (0.8 $\\AA$) and a small number of radial basis functions (8) for\nboth the radial and angular parts.\n\nThe MLFFs directly trained using PBE and SCAN are referred to as MLFF-PBE and MLFF-SCAN, respectively.\nThe MLFFs indirectly trained using SCAN and RPA via the $\\Delta$-machine learning ($\\Delta$-ML) approach are denoted as MLFF-SCAN$^\\Delta$\nand MLFF-RPA$^\\Delta$, respectively.\nWe note that although it is possible to generate the MLFF-RPA$^\\Delta$  based on the MLFF-PBE (constructed using a plane wave cutoff of 600 eV and standard PAW potentials), the RPA calculations in particular for the forces are rather demanding using such a large cutoff energy for structures of 24 atoms.\nTherefore, it is expedient to generate a second MLFF-PBE using a reduced plane wave cutoff of 520 eV and the GW PAW potentials as in the RPA calculations,\non which the MLFF-RPA$^\\Delta$ is built. We note that the changes in the cutoff energy (from 600 eV to 520 eV) and the potentials (from PAW to GW PAW) for ZrO$_2$\nhave negligible effects on the accuracy of the resulting MLFF-PBE if the stress tensors calculated by the VASP internal routines were corrected for the Pulay stress.\nFor generating MLFF-$\\Delta$ where $\\Delta=$RPA$-$PBE, both PBE and RPA calculations were performed on the structures in $T^{(3)}$\nusing the plane wave cutoff of 520 eV and the GW PAW potentials.\n\n\n\n\n\\section{MLFF validation}\\label{sec:validation_details}\n\n\n\n\nThe MLFFs including MLFF-PBE, MLFF-SCAN, and MLFF-SCAN$^\\Delta$\nhave been validated on a test dataset containing 120 structures of 96 atoms\n(40 monoclinic structures at $T$=1000 K, 40 tetragonal structures at $T$=2000 K,  and 40 cubic structures at $T$=3000 K).\nHowever, for MLFF-RPA$^\\Delta$,\ndue to the high computational cost for the RPA calculations, a reduced test dataset containing 60 structures of 24 atoms (20 monoclinic structures\nat $T$=1000 K, 20 tetragonal structures at $T$=2000 K,  and 20 cubic structures at $T$=3000 K) was used.\nAll the structures in the test datasets were generated from MD simulations using the NPT ensemble and MLFF-SCAN.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth, clip]{FigS4_phonon_MLSCAN_SCAN.eps}\n\\end{center}\n\\caption{Phonon dispersion relations of (a) monoclinic, (b) tetragonal, and (c) cubic  ZrO$_2$  at 0 K predicted by SCAN (grey lines), MLFF-SCAN (blue lines),\nand MLFF-SCAN$^{\\Delta}$ (red lines). Almost no difference is observed between MLFF-SCAN$^{\\Delta}$ and MLFF-SCAN for all three phases,\nindicating their comparable accuracies.\n}\n\\label{fig:phonon_MLSCAN_SCAN}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth, clip]{FigS5_phonon_MLRPA_SCAN.eps}\n\\end{center}\n\\caption{Phonon dispersion relations of (a) monoclinic, (b) tetragonal, and (c) cubic  ZrO$_2$  at 0 K predicted by MLFF-RPA$^\\Delta$ (red lines).\nThe results from SCAN (grey lines) are also given for comparison, showing very similar phonon dispersions with MLFF-RPA$^\\Delta$ for all three phases.\n}\n\\label{fig:phonon_MLRPA_SCAN}\n\\end{figure*}\n\n\nAs shown in Table I of the main text, all generated MLFFs are very accurate with small validation errors.\nThe errors for MLFF-SCAN are slightly larger than those for MLFF-PBE.\nThis is likely due to the poor numerical performance of the SCAN functional~\\cite{doi:10.1063\/1.5094646,doi:10.1021\/acs.jpclett.0c02405}.\nThe MLFF-SCAN$^\\Delta$ derived by the $\\Delta$-ML approach\nexhibits almost comparable accuracy as MLFF-SCAN that was directly trained by SCAN.\nThe good accuracies of the obtained MLFF-PBE and MLFF-SCAN\n have also been showcased in their good predictions of structural parameters,\nenergy differences between different phases (Table~\\ref{tab:lattice_constants}), and phonon dispersion relations\n(Fig.~\\ref{fig:phonon_MLSCAN_SCAN}) as compared to their respective DFT counterparts.\nAs expected, PBE overestimates the lattice constants and the energy differences between the phases.\nSCAN improves upon PBE because of the improved treatment of intermediate Van der Waals interactions.\nOur DFT results are overall in good agreement with Ref.~\\cite{PhysRevResearch.2.043361}.\nFrom Table~\\ref{tab:lattice_constants} one can also observe that MLFF-SCAN$^\\Delta$\nperforms almost equally well as MLFF-SCAN in the prediction of structural parameters and energy differences between the phases,\nconsistent with the error analysis shown in Table I of the main text..\nThis is also true in predicting phonon dispersion relations, as demonstrated in Fig.~\\ref{fig:phonon_MLSCAN_SCAN}.\nAlmost no difference is observed between MLFF-SCAN$^{\\Delta}$ and MLFF-SCAN for all three phases,\nvalidating the feasibility of the $\\Delta$-ML approach.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth, clip]{FigS6_Str_plot_ZrO2.eps}\n\\end{center}\n\\caption{Crystal structures of (a) cubic, (b) tetragonal, and (c) monoclinic ZrO$_2$. Large and small spheres denote Zr and O atoms, respectively.\nThe red and purple colors in the tetragonal structure (b) are used to distinguish O atoms that are displaced\nupwards (red arrows) and downwards (purple arrows) as compared to the cubic structure, and in the monoclinic structure (c) they indicate the two nonequivalent O atoms.\n Structural models were generated using VESTA~\\cite{Momma:db5098}.\n }\n\\label{fig:Str_plot_ZrO2}\n\\end{figure*}\n\nThe MLFF-RPA$^\\Delta$ is validated on a reduced test dataset of small unit cells, i.e., 60 structures of 24 atoms,\nbecause of the large computational cost of the RPA calculations. The validation errors in energies, forces and stress\ntensors calculated by MLFF-RPA$^\\Delta$ are given in Table II of the main text. For comparison, the validation errors\nfor MLFF-PBE and MLFF-SCAN are also shown.\nFirst, one can observe that MLFF-RPA$^\\Delta$  shows comparable validation errors as MLFF-PBE and MLFF-SCAN,\nimplying their comparably good accuracy. The slightly larger validation errors calculated by MLFF-RPA$^\\Delta$\narise from the relatively noisy nature of RPA. Second, one may notice that the validation errors of energies and\nstress tensors on small unit cells of 24 atoms for all the MLFFs (not just limited to MLFF-RPA ) are larger than those\non larger unit cells of 96 atoms, whereas the RMSEs for the forces remain almost  unchanged (compare Table II and Table I in the main text).\nThis difference in fact originates from the definition of RMSE and can be understood from the error propagation with respect to the system size.\nSpecifically, according to basic statistics, the variance of the difference between the DFT and MLFF\npredicted energy Var$(E^{\\rm DFT}-E^{\\rm MLFF})$ is expected to be proportional to the system size $N$,\ni.e., Var$(E^{\\rm DFT}-E^{\\rm MLFF})\\sim N$, if the errors in the predicted local energies are statistically independent.\nThe RMSE of the energy per atom (in meV\/atom) are calculated as\n\\begin{equation}\\label{eq:RMSE_E}\n{\\rm RMSE}(E)= \\sqrt{\\frac{1}{M} \\sum_i^M  \\Big[(E^{\\rm DFT}_i - E^{\\rm MLFF}_i)\/N_i\\Big]^2},\n\\end{equation}\nwhere $M$ is the number of structures and  $N_i$ is the number of atoms in the structure $i$.\nFrom this definition, one can readily show that\n\\begin{equation}\\label{eq:RMSE_E2}\n{\\rm RMSE}(E)= \\sqrt{\\frac{{\\rm Var}(E^{\\rm DFT}-E^{\\rm MLFF})}{N^2}} \\sim \\frac{1}{\\sqrt{N}},\n\\end{equation}\nThis means that if the system becomes $N$ times larger, then the RMSE of the energy per atom becomes  $1\/\\sqrt{N}$ times smaller.\nThis explains why all the MLFFs predict a larger RMSE for the energy per atom for the 24-atom cells than for the 96-atom cells.\nThe same error analysis holds for the stress tensor, since by definition it is calculated as the derivative\nof the energy with respect to the strain and then divided by the system volume.\nNevertheless, this error propagation rule does not apply for the RMSE for the forces,\nsince the force is an intensive property that is independent of system size.\n\n\n\n\\section{Phase transitions of zirconia within the quasi-harmonic approximation}\\label{sec:phase_transition_details}\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth, clip]{FigS7_QHA_free_energy.eps}\n\\end{center}\n\\caption{The free energy difference (per formula unit) between the tetragonal and\nmonoclinic phase as a function of temperature predicted by the MLFFs within the quasi-harmonic approximation (QHA)\nusing the quantum Bose-Einstein (BE) statistics. We note that the results obtained from the BE statistics\nand classical Maxwell-Boltzmann statistics are almost identical for temperatures above 200 K (not shown),\nindicating that the quantum effect can be neglected at high temperatures.\n }\n\\label{fig:QHA}\n\\end{figure}\n\n\nThe primitive cells of the cubic, tetragonal, and monoclinic structures of\nzirconia contain one, two, and four ZrO$_2$ formula units respectively.\nA unit cell of 12 atoms can thus be used to accommodate all three phases (see Fig.~\\ref{fig:Str_plot_ZrO2}).\n\n\nAs shown in the phonon dispersions (Fig.~\\ref{fig:phonon_MLSCAN_SCAN}),\nboth monoclinic and tetragonal phases are dynamically stable at 0 K. In addition, no soft phonons are observed for both phases\nwhen the volume is changed within 5\\%. This allows us to estimate the $T_c$ using the quasi-harmonic approximation (QHA).\nFig.~\\ref{fig:QHA} shows the free energy difference between the tetragonal and monoclinic phase\nas a function of temperature predicted by the MLFFs within the QHA using the quantum Bose-Einstein statistics.\nAccordingly, MLFF-PBE predicts a value of 1511 K for $T_c$.\nMLFF-SCAN$^\\Delta$ and MLFF-SCAN yield close values of 1148 K and 1164 K, respectively,\nwhich are very close to the one obtained by the MLFF-PBEsol (1178 K) in Ref.~\\cite{Carla_2021}.\nIn addition, for nearly the entire temperature range, MLFF-SCAN$^\\Delta$  predicts very close free energies as compared to MLFF-SCAN.\nThis is expected, since both MLFF-SCAN$^\\Delta$  and MLFF-SCAN overall show similar validation errors\nas well as lattice parameters, energy differences between the phases, and phonon dispersions.\nAs compared to MLFF-SCAN, MLFF-RPA$^\\Delta$  predicts a slightly smaller value of $T_c$ (about 1117 K).\nIn general, we find that within the QHA the predicted $T_c$ is correlated to the calculated energy differences between the two phases at 0 K (see Table~\\ref{tab:lattice_constants}).\n\n\n\n\n\\section{SVD rank compression}\\label{sec:SVD_CUR}\n\nIn order to select few most representative structures for machine-learning the differences from a large pool of dataset,\nwe employed SVD rank compression of the kernel matrix based on the leverage-score CUR algorithm~\\cite{Mahoney697,PhysRevB.100.014105}.\nIn the following, the CUR algorithm is briefly introduced.\n\nWe denote $\\mathbf{K}$ as a kernel matrix calculated in the feature space. It is a squared matrix whose elements $K_{ij}$ measure the similarity between two local reference configurations $i$ and $j$. The formulation of the CUR algorithm starts from the diagonalization of the matrix $\\mathbf{K}$:\n\\begin{align}\n\\mathbf{U}^{\\textit{T}}\\mathbf{K}\\mathbf{U}&=\\mathbf{L}=\\mathrm{diag}\\left(l_{\\mathrm{1}},...,l_{N}\\right), \\label{equation_diag}\n\\end{align}\nwhere $N$ is the dimension of the matrix $\\mathbf{K}$ and $\\mathbf{U}$ is the eigenvector matrix defined as\n\\begin{align}\n\\mathbf{U}&=\\left(\\mathbf{u}_{\\mathrm{1}},...,\\mathbf{u}_{N}\\right), \\label{equation_U}\\\\\n\\mathbf{u}_{j}&=\\left(u_{1j},...,u_{Nj}\\right)^{\\textit{T}}, \\label{equation_u}\n\\end{align}\nEq.~(\\ref{equation_diag}) can be rewritten as\n\\begin{align}\n\\mathbf{k}_{j}&=\\sum\\limits_{\\xi = 1}^{N} \\left(u_{j \\xi} l_{\\xi}\\right) \\mathbf{u}_{\\xi}, \\label{equation_kj}\n\\end{align}\nwhere $\\mathbf{k}_{j}$ denotes the $j$th column vector of the matrix $\\mathbf{K}$.\nIn our implementation, we have adopted a modified version of the original CUR algorithm~\\cite{Mahoney697}\nsuch that the columns of $\\mathbf{K}$ that are strongly correlated with the $N_{low}$ eigenvectors $\\mathbf{u}_{\\xi}$ with the smallest eigenvalues $l_\\xi$  are disregarded.\nThis is achieved by defining the leverage scoring for each column of $\\mathbf{K}$\n\\begin{align}\n\\omega_{j}&=\\frac{1}{N_{low}} \\sum\\limits_{\\xi=1}^{N} \\gamma_{\\xi j}, \\label{equation_ls1}\\\\\n\\gamma_{\\xi j}&= \\begin{cases} u_{j\\xi}^{2}, & \\mbox{if } l_{\\xi}\/l_{\\rm max}< \\epsilon  \\\\ 0, & \\mbox{otherwise} \\end{cases} \\label{equation_ls2}\n\\end{align}\nwhere $l_{\\rm max}$ is the maximum eigenvalue of $\\mathbf{K}$ and $\\epsilon$ is a parameter used to define the degree of rank compression.\nBy defining Eq.~\\eqref{equation_ls1}, the $N_{low}$ columns of $\\mathbf{K}$ and the local reference configurations corresponding to those columns\nwith largest leverage scorings are disregarded. The remaining ones are deemed to be the most important ones.\nOur final representative structures are then those structures that contribute to these most important local reference configurations.\n\nUsing the SVD rank compression introduced above, we selected a subset of structures ($T^{(3)}$) from the $T^{(2)}$ dataset (including 1275 structures of 24 atoms).\nWe employed pair descriptors with low spatial resolution (0.8 $\\AA$) and a small number of radial basis functions (8) to construct the kernel.\nThe actual number of SVD compressed structures depends on the parameter $\\epsilon$ in Eq.~\\eqref{equation_ls2}.\nFor instance, the resulting subset $T^{(3)}$ contains 168 structures, when $\\epsilon=10^{-10}$.\nThis number will be reduced when $\\epsilon$ is increased. As shown in Table~\\ref{tab:error_MLFF-DFT_SM}, in addition to\nthe MLFF-SCAN$^\\Delta$ that is discussed in the main text, we generated another two new MLFFs (denoted as MLFF-SCAN$^\\Delta$-2 and MLFF-SCAN$^\\Delta$-3)\nvia the $\\Delta$-ML approach. These two MLFFs used just 102 and 72 structures for generating the MLFF-$\\Delta$.\nOne can observe that even 72 structures are sufficient to machine-learn the differences and the resulting MLFF-SCAN$^\\Delta$-3\nis still accurate, showing only slightly larger validation errors in energies as compared to MLFF-SCAN$^\\Delta$.\nMoreover, we find that MLFF-SCAN$^\\Delta$-3 predicts very similar lattice parameters, energy differences between the phases (not shown),\nas well as the phonon dispersion relations (see Fig.~\\ref{fig:phonon_two_ML_SCAN_delta}) as compared to MLFF-SCAN$^\\Delta$.\nThe same observations also apply when generating MLFF-RPA$^\\Delta$, so that in practice, similar results as in the main text could be obtained\nwith just half the number of RPA calculations.\n\n\n\\begin{table*\n\\caption {The validation root-mean-square errors (RMSE) in energies per atom (meV\/atom), forces (eV\/\\AA) and stress tensors (kbar)\nfor MLFF-SCAN, MLFF-SCAN$^\\Delta$ and MLFF-SCAN$^\\Delta$-$k$ ($k=$2 and 3).\nThe latter three MLFFs are obtained by the $\\Delta$-ML approach and\ndiffer in the level of SVD rank compression when constructing the $T^{(3)}$ dataset.\nHere, the parameter $\\epsilon$ defines the degree of rank compression [see Eq.~\\eqref{equation_ls2}]  and $N_{\\rm str}$\nrepresents the number of SVD compressed structures used to machine-learn the differences.\nThe test dataset includes 120 structures of 96 atoms.\n}\n\\begin{ruledtabular}\n\\begin{tabular}{lccccc}\n & Energy & Force & Stress & $\\epsilon$ & $N_{\\rm str}$ in $T^{(3)}$ \\\\\n               \\hline\nMLFF-SCAN$^\\Delta$ &  2.37 & 0.139 & 2.30 & 1E-10 & 168  \\\\\nMLFF-SCAN$^\\Delta$-2 &  2.45 & 0.139 & 2.28 & 1E-08 & 102  \\\\\nMLFF-SCAN$^\\Delta$-3 &  2.46 & 0.139 & 2.29 & 1E-07 &  72  \\\\\nMLFF-SCAN            &  2.49 & 0.139 & 2.38 &       &      \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tab:error_MLFF-DFT_SM}\n\\end{table*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth, clip]{FigS8_phonon_two_MLSCAN-delta.eps}\n\\end{center}\n\\caption{Phonon dispersion relations of (a) monoclinic, (b) tetragonal, and (c) cubic  ZrO$_2$  at 0 K predicted by MLFF-SCAN$^\\Delta$ (blue lines)\nand MLFF-SCAN$^\\Delta$-3 (red lines).\n}\n\\label{fig:phonon_two_ML_SCAN_delta}\n\\end{figure*}\n\n\n\n\\newpage\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{#1}\\setcounter{equation}{0}}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\newcommand{\\ben}{\\begin{eqnarray}}\n\\newcommand{\\een}{\\end{eqnarray}}\n\\newcommand{\\addref}{[References???]}\n\\newcommand{\\ex}[1]{e \\left( #1\\right)}\n\n\n\\def\\MA#1{{\\bf MA: #1}}\n\\def\\BH#1{{\\bf BH: #1}}\n\\def\\MH#1{{\\bf MH: #1}}\n\\def\\AK#1{{\\bf AK: #1}}\n\\def\\MR#1{{\\bf MR: #1}}\n\\def\\TW#1{{\\bf TW: #1}}\n\n\n\\title{Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes}\n\\author{Murad Alim$^{1}$,  Babak Haghighat$^{2,3}$, Michael Hecht$^4$, Albrecht Klemm$^3$, Marco Rauch$^3$ and Thomas Wotschke$^3$\n\\\\\n\\\\ $^1$Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA\n\\\\ $^2$Institute for Theoretical Physics and Spinoza Institute, Utrecht University, 3508 TD Utrecht, The Netherlands\n\\\\ $^3$Bethe Center for Theoretical Physics, University of Bonn, Nussallee 12, D-53115 Bonn, Germany\n\\\\ $^4$Arnold Sommerfeld Center for Theoretical Physics, LMU, Theresienstr. 37, D-80333 Munich, Germany\n}\n\n\n\n\n\\abstract{Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of $\\mathcal{N}=4$ topological Yang-Mills theory on $\\mathbbm{P}^2$ and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-holomorphic part is related to the contribution originating from bound-states of singly wrapped M5-branes on the divisor. We show in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculate on a natural extension of the anomaly to higher D4-brane charge.}\n\n\\preprint{BONN-TH-2010-13\\\\LMU-ASC 102\/10}\n\n\n\n\n\\begin{document}\n\n\n\\section{Introduction}\n\nThe study of background dependence of physical theories has been a rich source of insights. Understanding the change of correlators as the background parameters are \nvaried supplemented by boundary data can be sufficient to solve the theory. A class of theories where the question of background dependence can be sharply stated \nare topological field theories. Correlators in topological theories typically have holomorphic expansions near special values of the background moduli. The expansion \ncoefficients can be given precise mathematical meaning as topological invariants of the geometrical configuration contributing to the topological non-trivial \nsector of the path integral. Physically the expansion often captures information of the degeneracies of BPS states of theories related to the same geometry.\n\nAn example of this is the topological A-model \\cite{Witten:1988xj}, with a Calabi-Yau three-fold (CY) $X$ as target space, which in a large volume limit counts holomorphic maps from the world-sheet into $H_2(X,\\mathbbm{Z})$ and physically captures the degeneracies of BPS states coming from an M-theory compactification on $X$ \\cite{Gopakumar:1998ii,Gopakumar:1998jq}. Another example is the modified elliptic genus of an M5-brane wrapping a complex surface $P$,\\footnote{In the following we will use the terms surface, divisor and four-cycle (of a CY) interchangeably when the context is clear.} which was related in ref.~\\cite{Minahan:1998vr} to the partition function of topologically twisted $\\mathcal{N}=4$ Yang-Mills theory \\cite{Vafa:1994tf}, which computes generating functions of Euler numbers of moduli spaces of instantons.  This same quantity was shown in ref.~\\cite{Gaiotto:2006wm} to capture the geometric counting of degeneracies of systems of D4-D2-D0 black holes associated to the MSW string \\cite{Maldacena:1997de}.\n\nIn both cases the topological theories enjoy  duality symmetries. $T$-duality acting on the K\\\"ahler moduli on $X$ in the topological string case and $S$-duality for the $\\mathcal{N}=4$ SYM theory acting on the gauge coupling $\\tau=\\frac{4 \\pi i}{g^2} + \\frac{\\theta}{2 \\pi}$. The former\nsymmetry extends by mirror symmetry and both might extend to $U$-duality groups. Both symmetries can be conveniently expressed in the language of modular forms.\n\nThe holomorphic expansions of the topological string correlators are given in the moduli spaces of families of theories. Fixing a certain background corresponding to a certain point in the moduli space, the topological correlators are expected to be holomorphic expansions. In refs.~\\cite{Bershadsky:1993ta,Bershadsky:1993cx} holomorphic anomaly equations governing topological string amplitudes were derived showing that this is not the case and hence the correlators suffer from background dependence.\\footnote{The anomaly relates correlators at a given genus to lower genera thus providing a way to solve the theory. Using a polynomial algorithm \\cite{Yamaguchi:2004bt,Grimm:2007tm,Alim:2007qj} and boundary conditions \\cite{Huang:2006si} this can be used to compute higher genus topological string amplitudes on compact CY \\cite{Huang:2006hq} manifolds and solve it on non-compact CY \\cite{Haghighat:2008gw,Alim:2008kp}.} In ref.~\\cite{Witten:1993ed} a background independent meaning was given to the correlators, stating that the anomaly merely reflects the choice of polarization if the partition function is considered as a wave function only depending on half of the variables of some phase space which has a natural geometric meaning in this context.\n\nThis anomaly is also manifest in a failure of target space duality invariance of the holomorphic expansion which can only be restored at the expense of holomorphicity as shown in ref.~\\cite{Aganagic:2006wq}.\\footnote{Following the anomaly reformulation of refs.~\\cite{Dijkgraaf:2002ac,Verlinde:2004ck}, see also \\cite{Gunaydin:2006bz}.}  A similar story showed up in $\\mathcal{N}=4$ topological $U(2)$ SYM theory on $\\mathbbm{P}^2$ \\cite{Vafa:1994tf}, where it was shown that different sectors of the partition function need a non-holomorphic completion which was found earlier in ref.~\\cite{Zagier:1975} in order to restore $S$-dualtiy invariance.  An anomaly equation describing this non-holomorphicity was expected \\cite{Vafa:1994tf} in the cases where $b_{2}^+(P)=1$. In these cases holomorphic deformations of the canonical bundle are absent. The non-holomorphic contributions were associated with reducible connections $U(n) \\rightarrow U(m)\\times U(n-m)$ \\cite{Vafa:1994tf,Minahan:1998vr}. In ref.~\\cite{Minahan:1998vr} this anomaly was furthermore related to an anomaly appearing in the context of E-strings \\cite{Minahan:1997ct}. These strings arise from an M5-brane wrapping a del Pezzo surface  $\\mathcal{B}_9$, also called $\\frac{1}{2}$K3. The anomaly in this context was related to the fact that $n$ of these strings can form bound-states of $m$ and $(n-m)$ strings. Furthermore, the anomaly could also be related to the one appearing in topological string theory.\n\t\nThe anomaly thus follows from the formation of bound-states. Although the holomorphic expansion would not know about the contribution from bound-states, the restoration of duality symmetry forces one to take these contributions into account. The non-holomorphicity can be understood physically as the result of a regularization procedure. The path integral produces objects like theta-functions associated to indefinite quadratic forms which need to be regularized to avoid divergences. This regularization breaks the modular symmetry, restoring the symmetry gives non-holomorphic objects. The general mathematical framework to describe these non-holomorphic completions is the theory of mock modular forms developed by Zwegers in ref.~\\cite{Zwegers:2002}.\\footnote{See \\cite{Zagier:2007,MR2555930} and app.~\\ref{sec:mock} for an introduction and overview.} A mock modular form $h(\\tau)$ of weight $k$ is a holomorphic function which becomes modular after the addition of a function $g^*(\\tau)$, at the cost \nof losing its holomorphicity. Here, $g^*(\\tau)$ is constructed from a modular form $g(\\tau)$ of weight $2-k$, which is referred to as shadow.\n\nAnother manifestation of the background dependence of the holomorphic expansions of the  topological theories are wall-crossing phenomena associated to the enumerative content of the expansions. Mathematically, it is known that Donaldson-Thomas invariants jump on surfaces with $b_2^+(P)=1$, see \\cite{Gottsche:1998} and references therein, for related physical works see for example refs.~\\cite{Moore:1997pc,Losev:1997tp}. On the physics side wall-crossing refers to the jumping of the degeneracies of BPS states when walls of marginal stability are crossed. These phenomena were observed in the jumps of the soliton spectrum of two-dimensional theories  \\cite{Cecotti:1992rm} and were an essential ingredient of the work of Seiberg and Witten \\cite{Seiberg:1994rs} in four-dimensional theories. Recent progress\nwas triggered by formulae relating the degeneracies on both sides of\nthe walls, which were given from a supergravity analysis in\nrefs.~\\cite{Denef:2000nb,Denef:2007vg} and culminated in a mathematical rigorous formula of Kontsevich and Soibelman (KS) \\cite{KS:2008}, which could also be derived \nfrom continuity of physical quantities in refs.~\\cite{Gaiotto:2008cd,Cecotti:2009uf} (See also refs.~\\cite{Gaiotto:2009hg,Gaiotto:2010be,Cecotti:2010fi}). The \nfact that the holomorphic anomaly describes how to transform the counting functions when varying the background moduli, which\nin turn changes the degeneracy of BPS states, suggests that\nnon-holomorphicity and wall-crossing are closely related. In\nfact the failure of holomorphicity can be traced back to the boundary\nof the moduli space of the geometrical configuration, where the\nlatter splits in several configurations with the same\ntopological  charges. Mock modularity was used in a physical context studying the wall-crossing of degeneracies of $\\mathcal{N}=4$ dyons\\footnote{See for example ref.~\\cite{Dabholkar:2007zz} and references therein for more details.} in ref.~\\cite{Dabholkar:2010}. In the context of $\\mathcal{N}=2$ supersymmetric theories the application of ideas related to mock modularity was initiated in ref.~\\cite{Manschot:2009ia} and further pursued in refs.~\\cite{Manschot:2010xp,Manschot:2010nc,Bringmann:2010sd}. These motivated parts of our work.\\footnote{Further physical appearances of mock modularity can be found for example in refs.~\\cite{Eguchi:2008gc,Eguchi:2009cq,Eguchi:2010ej,Cheng:2010pq,Troost:2010ud}.}\n\nIn this paper we study the relation between wall-crossing and non-holomorphicity and relate the appearance of the two. A central role is played by a wall-crossing formula by G\\\"ottsche \\cite{Gottsche:1999}, where the K\\\"ahler moduli dependence of a generating function of Euler numbers of stable sheaves is given in terms of an indefinite theta-function due to G\\\"ottsche and Zagier \\cite{Gottsche:1998}. We show that this formula is equivalent to wall-crossing formulae of D4-D2-D0 systems in type IIA. The latter can be related to the (modified) elliptic genus of multiple M5-branes wrapping a surface. Rigid surfaces are subject to G\\\"ottsche's wall-crossing formula. Using ideas of Zwegers \\cite{Zwegers:2002}, we translate the latter into a holomorphic anomaly equation for two M5-branes wrapping the surface\/divisor. We show that this anomaly equation is the equation which was found in the context of $\\mathcal{N}=4$ SYM \\cite{Vafa:1994tf} and E-strings \\cite{Minahan:1997ct,Minahan:1998vr}. We further propose the generalization of the anomaly equation for higher wrappings and comment on its implications for the wall-crossing of multiple D4-branes.\n\nThe organization of this work is as follows. In section \\ref{sectwo} we review the effective description of physical theories obtained from wrapping $n$ M5-branes on rigid divisors. Depending on the perspective, this is either described in terms of the MSW CFT with (0,4) world-sheet supersymmetry \\cite{Maldacena:1997de} or by $U(n)$ $\\mathcal{N}=4$ topological SYM \\cite{Vafa:1994tf}. Both cases admit a decomposition into theta-functions, the latter carry a dependence on the chosen K\\\"ahler class which determines the split into right- and left-movers. We recall the equivalent type IIA D4-D2-D0 brane description of the BPS states of the M5-branes. We continue with outlining the $\\mathcal{N}=4$ Vafa-Witten theory and recall how bound-states of several E-strings cause an anomaly of the corresponding partition function.\n\nIn section 3 we first show the equivalence of the Kontsevich-Soibelman wall-crossing formula and a formula found by G\\\"ottsche in terms of an indefinite theta-function which captures the wall-crossing of the generating function of Euler numbers of moduli spaces of stable sheaves on a complex surface $P$ with $b_2^+(P)=1$. Physically this corresponds to the D4-D2-D0 bound-state description. We remedy the non-modularity of the indefinite theta-function using the ideas of Zwegers \\cite{Zwegers:2002}. With the results at hand we prove a holomorphic anomaly equation for two M5-branes wrapping the divisor.\n\nIn section 4 we apply the wall-crossing formula to compute the elliptic genus for several examples of surfaces with $b_2^+(P)=1$. Moreover, we provide the form of the holomorphic anomaly equation in the higher charge case that is compatible with the simple form appearing in the context of E-strings. We furthermore explore the possibility of the anomaly having its origin in a choice of contour while doing the Fourier expansion of a meromorphic Jacobi form paralleling the reasoning in $\\mathcal{N}=4$ dyon wall-crossing \\cite{Dabholkar:2010}.\n\nSection 5 presents our conclusions, points out open problems and suggestions for future work. In the appendices we summarize several details.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Effective descriptions of wrapped M5-branes}\\label{sectwo}\n\nIn this section we review the effective descriptions of M5-branes wrapping a complex surface $P$ as well as previous appearances of the holomorphic anomaly which will be derived in the next section.  The world-volume theory of M5-branes can have either a two-dimensional CFT description in terms of the (MSW) CFT \\cite{Maldacena:1997de} or a four-dimensional description giving the $\\mathcal{N}=4$ topologically twisted Yang-Mills theory of Vafa and Witten \\cite{Vafa:1994tf}. In the latter theory it was observed \\cite{Vafa:1994tf} that a non-holomorphicity \\cite{Zagier:1975} had to be introduced in order to restore $S$-dualtiy, the resulting holomorphic anomaly was related in ref.~\\cite{Minahan:1998vr} to an anomaly~\\cite{Minahan:1997ct} appearing in the context of E-strings. The anomaly was conjectured to take into account contributions coming from reducible connections in $\\mathcal{N}=4$ SYM theory. In ref.~\\cite{Minahan:1998vr} it was related to the curve counting anomaly \\cite{Bershadsky:1993cx} and was given the physical interpretation of taking into account the bound-state contribution of E-strings. Later we will show that the contributions from bound-states as a cause for non-holomorphicity will persist more generally for the class of surfaces we will be studying. In our work we investigate the (generalized\/modified) elliptic genus which captures the content of the CFT description of the M5-branes \\cite{Minahan:1998vr,Maldacena:1999bp} and its relation to D4-D2-D0 systems \\cite{Gaiotto:2006wm,deBoer:2006vg,Kraus:2006nb,Denef:2007vg,Gaiotto:2007cd,Manschot:2007ha} and the associated counting of black holes which has been intensively studied (e.g.~in ref.~\\cite{Dabholkar:2005dt}).  Our goal is to show that wall-crossing in D4-D2-D0 systems leads to an anomaly equation which coincides with the anomalies found before and hence our work complements in some sense this circle of ideas.\n\n\\subsection{The elliptic genus and D4-D2-D0 branes}\n\\label{sec:mswcft}\nIn the following we will start with the $2d$ CFT perspective of the M5-brane world-volume theory.\nWe want to study BPS states that arise in the context of an M-theory compactification on a Calabi-Yau manifold $X$ with $r$ M5-branes wrapping a complex surface (or a four-cycle) $P$,  and extended in $\\mathbbm{R}^{1,3} \\times S^1$. Considering $P$ to be small compared to the M-theory circle, the reduction of the world-volume theory of the M5-brane is described by a $(1+1)$-dimensional $(0,4)$ MSW CFT \\cite{Maldacena:1997de}.\\footnote{The target space sigma model description of which was given in ref.~\\cite{Minasian:1999qn}, for more details see ref.~\\cite{Guica:2007wd} and references therein. In the following we will be concerned with the natural extension of the analysis of the degrees of freedom to $r$ M5-branes.} The BPS states associated to the string that remains after wrapping the M5-branes on $P$ are captured by a further compactification on a circle. They are counted by the partition function of the world-volume theory of the M5-branes on $P\\times T^2$ \\cite{Minahan:1998vr}. The effective CFT description will thus exhibit invariance under the full SL$(2,\\IZ)$ symmetry of the $T^2$. Furthermore, excitations of the M5-branes will induce M2-brane charges corresponding to the flux of the self-dual field strength of the M5-brane world-volume theory. In addition, the momentum of the M2-branes along the M-theory circle will give rise to a further quantum number. As a result BPS states of the effective two-dimensional description will be labeled by the class of the divisor the M5-branes wrap, the M2-brane charges and by the momentum along $S^1$. In a type IIA setup, $r$ times the class of the divisor will correspond to D4-brane charge, the induced M2-brane charge corresponds to D2-brane charge and the momentum to D0-brane charge. Choosing a basis $\\Sigma_A\\, , A=1,\\dots, b_4(X)$ of $H_4(X,\\IZ)$, the charge vector will be given by\\footnote{See appendix \\ref{Dbound} for details.}\n\\be\n\\Gamma = (Q_6,Q_4,Q_2,Q_0) = r(0,p^A, q_A, q_0),\n\\ee\nwhere the $Q_p$ are the D$p$-brane charges and $r$ is the number of coincident M5-branes wrapping the divisor specified by $p^A$.  A priori the set of all possible induced D2-brane charges, or equivalently of $U(1)$ fluxes of the world-volume of the M5-brane would be in one-to-one correspondence with $\\Lambda_P = H^2(P,\\mathbbm{Z})$ which is generically a larger lattice than $\\Lambda=i^*H^2(X,\\IZ)$, where $ i: P \\hookrightarrow X$, however the physical BPS states are always labeled by the smaller lattice $\\Lambda$. The metric $d_{AB}$ on $\\Lambda$ is given by\n\\be\nd_{AB} = - \\int_P \\alpha_A \\wedge \\alpha_B,\n\\ee\nwhere $\\alpha_A$ is a basis of two-forms in $\\Lambda$, which is the dual basis to $\\Sigma_A$ of $H_4(X,\\IZ)$.  In order to obtain a generating series of the degeneracies of those BPS states one has to sum over directions along $\\Lambda^{\\perp}$ which is the orthogonal complement to $\\Lambda$ in $\\Lambda_P$ w.r.t.~$d_{AB}$ \\cite{Gaiotto:2006wm}. \\footnote{In general, the lattice $\\Lambda \\oplus \\Lambda^{\\perp}$ is only a sublattice of $H^{2}(P,\\mathbbm{Z})$, because $\\det d_{AB}\\neq1$ in general, see for example ref.~\\cite{Minasian:1999qn} and ref.~\\cite{Denef:2007vg} for a more recent exposition. However, we will only be concerned with divisors $P$ with $b_2^+(P)=1$, such that $\\det d_{AB}=1$.}\n\nThe partition function of the MSW CFT counting the BPS states is given by the modified elliptic genus\\footnote{We follow the mathematics convention of not writing out explicitly the dependence on $\\tb$ which will be clear in the context. Moreover, we denote $q= e^{2\\pi i \\tau}$ and $\\tau=\\tau_1+i\\tau_2$. To avoid confusion without introducing new notation we will denote the charge vector of D2-brane charges by $\\underline{q}$, its components by $q_A$.} \\cite{Minahan:1998vr,Maldacena:1999bp}\n\\begin{equation} \\label{zprime}\n{Z'}_{\\!\\!\\!P}^{(r)}(\\tau,z)= \\textrm{Tr}_{\\mathcal{H}_{\\rm{RR}}} \\, (-1)^{F_{{\\rm R}}} \\, F_{{\\rm R}}^2\\, q^{L'_0-\\frac{c_{\\rm L}}{24}}\\, \\bar q^{\\bar{L}'_0 -\\frac{c_{\\rm R}}{24}}e^{2\\pi i z\\cdot Q_{2}},\n\\end{equation}\nwhere the trace is taken over the RR Hilbert space. Furthermore, vectors are contracted w.r.t.~the metric $d_{AB}$, i.e. $x \\cdot y = x^A y_A = d_{AB}x^A y^B$. For a single M5-brane it was shown in ref.~\\cite{deBoer:2006vg} that ${Z'}_{\\!\\!\\!P}^{(1)}(\\tau,z)$ transforms like a SL$(2,\\IZ)$ Jacobi form of bi-weight $(0,2)$ due to the insertion of $F_{{\\rm R}}^2$, we demand that the same is true for all $r$.\n\nFollowing ref.~\\cite{deBoer:2006vg} the center of mass momentum $\\vec{p}_{\\rm cm}$ for the system of $r$ M5-branes can be integrated out. In this way $L_0'$ and $\\bar{L}_0'$ can be written in the form\n\\begin{equation}\n  L_0' = \\frac{1}{2} \\vec{p}^{\\, 2}_{\\rm cm} + L_0, \\quad \\bar{L}_0' = \\frac{1}{2} \\vec{p}^{\\, 2}_{\\rm cm} + \\bar{L}_0.\n\\end{equation}\nThis allows one to split up the center of mass contribution and rewrite formula (\\ref{zprime}) as\n\\begin{eqnarray}\n  {Z'}_{\\!\\!\\!P}^{(r)}(\\tau,z)\n  & =    & \\int d^3 p_{\\rm cm} (q\\bar{q})^{\\frac{1}{2}\\vec{p}_{\\rm cm}^{\\, 2}} Z^{(r)}_{P}(\\tau,z) \\nonumber \\\\\n  & \\sim & (\\tau_2)^{-\\frac{3}{2}}\\, Z^{(r)}_{P}(\\tau,z),\n\\end{eqnarray}\nwhere $Z^{(r)}_{P}(\\tau,z)$ is now a Jacobi form of weight $(-\\frac{3}{2},\\frac{1}{2})$ which we simply call elliptic genus for short in the following. \n\n\n\n\\subsubsection*{\\it The decomposition of the elliptic genus}\nThe elliptic genus $Z_P^{(r)}(\\tau, z)$ and equivalently the generating function of D4-D2-D0 BPS degeneracies is subject to a theta-function decomposition, which has been studied in many places, see for example refs.~\\cite{Dabholkar:2005dt,Gaiotto:2006wm,deBoer:2006vg,Kraus:2006nb,Denef:2007vg}. This is ensured by two features of the superconformal algebra of the (0,4) CFT. One of these is that the $\\tb$ contribution entirely comes from BPS states $| \\qu \\rangle$ satisfying\n\\begin{equation} \\label{heatop}\n    \\left(\\ov{L}_0-\\frac{c_{\\rm R}}{24}-\\frac{r}{2}q_{{\\rm R}}^2 \\right) |\\qu\\rangle =0,\n\\end{equation}\nthe other one is the spectral flow isomorphism of the ${\\cal N}=(0,4)$ superconformal algebra, which we want to recall for $r$ M5-branes here, building on refs.~\\cite{deBoer:2006vg,Weist:2009}, see also \\cite{Dabholkar:2005dt}. Proposition 2.9 of ref.~\\cite{Weist:2009} describes the spectral flow symmetry by an isomorphism between moduli spaces of vector bundles on complex surfaces. The complex surface here is the divisor $P$ and the vector bundle configuration describes the bound-states of D4-D2-D0 branes. Within this setup the result of \\cite{Weist:2009} translates for arbitrary $r$ to a symmetry under the transformations\n\\begin{eqnarray} \\label{spflow}\n    q_0 & \\mapsto & q_0 - k \\cdot \\qu - \\frac{1}{2} k \\cdot k, \\nonumber \\\\\n    \\qu   & \\mapsto & \\qu + k,\n\\end{eqnarray}\nwhere $k \\in \\Lambda$. Physically these transformations correspond to monodromies around the large radius point in the moduli-space of the Calabi-Yau manifold \\cite{Dabholkar:2005dt}. Denote by $\\Lambda^*$ the dual lattice of $\\Lambda$ with respect to the metric $r d_{AB}$. Keeping only the holomorphic degrees of freedom one can write\n\\begin{eqnarray}\n    Z^{(r)}_{P}(\\tau,z) & = & \\sum_{Q_0; Q_A}\\, d(Q,Q_0)\\,e^{-2\\pi i \\tau Q_0}\\, e^{2\\pi i z\\cdot Q_{2}} \\nonumber \\\\\n    ~                     & = & \\sum_{q_0; \\qu \\in \\Lambda^* + \\frac{[P]}{2}} \\,d(r,\\qu,-q_0)\\,e^{-2\\pi i \\tau r q_0}\\, e^{2\\pi i r z\\cdot \\qu},\n\\end{eqnarray}\nwhere $d(r,\\qu,-q_0)$ are the BPS degeneracies and the shift\\footnote{In components, $[P]$ is given by $d_{AB}p^A$.} $\\frac{[P]}{2}$ originates from an anomaly \\cite{Freed:1999vc,Minasian:1997mm}. Now, spectral flow symmetry predicts \\cite{deBoer:2006vg}\n\\begin{equation} \\label{spflow2}\n    d(r,\\qu,-q_0) = (-1)^{r p \\cdot k} d(r,\\qu+k,-q_0+k\\cdot \\qu + \\frac{k^2}{2}).\n\\end{equation}\nMaking use of this symmetry and the following definition\n\\begin{equation}\n \\qu = k + \\mu + \\frac{[P]}{2}, \\qquad \\mu \\in \\Lambda^*\/\\Lambda, \\qquad k \\in \\Lambda,\n\\end{equation}\none is led to the conclusion that the elliptic genus can be decomposed in the form\n\\begin{eqnarray}\n    Z^{(r)}_{P}(\\tau,z) & = & \\sum_{\\mu \\,\\in\\, \\Lambda^*\/\\Lambda} f^{(r)}_{\\mu,J}(\\tau) \\theta^{(r)}_{\\mu,J}(\\tau,z), \\label{thetadecomp}\\\\\n    f^{(r)}_{\\mu,J}(\\tau) & = & \\sum_{r \\hat{q}_0\\,\\geq\\,-\\frac{c_{\\rm L}}{24}} d^{(r)}_{\\mu}(\\hat{q}_0) e^{2\\pi i\\tau r \\hat{q}_0}, \\label{fdef}\\\\\n    \\theta^{(r)}_{\\mu,J}(\\tau,z)\n   & =& \\sum_{k\\,\\in\\,\\Lambda+\\frac{[P]}{2}} (-1)^{r p \\cdot (k+\\mu)}e^{2\\pi i \\bar{\\tau} r \\frac{(k+\\mu)_+^2}{2}}\n      e^{2\\pi i \\tau r \\frac{(k+\\mu)_-^2}{2}} e^{2\\pi i r z \\cdot (k+\\mu)}\\label{SNTheta},\n\\end{eqnarray}\nwhere $J\\in\\cC(P)$ and $\\cC(P)$ denotes the K\\\"ahler cone of $P$ restricted to $\\Lambda\\otimes\\IR$ and $\\hat{q}_0 = -q_0 - \\frac{1}{2} \\qu^2$ is invariant under the spectral flow symmetry. The subscript $+$ refers to projection onto the sublattice generated by the K\\\"ahler form $J$ and $-$ is the projection to its orthogonal complement, i.e.\n\\be\nk_+^2=\\frac{(k\\cdot J)^2}{J\\cdot J},\\quad k_-^2=k^2-k_+^2.\n\\ee\n\nThere are two issues here for the case of rigid divisors with $b_2^+(P) = 1$ on which we want to comment as this class of divisors is the\nfocus of our work. First of all note, that $q_0$ contains a contribution of the form\\footnote{See appendix \\ref{Dbound} for details.} $\\frac{1}{2} \\int_P F \\wedge F$ where $F \\in \\Lambda_P$. Now, $F$ can be decomposed into $F = \\qu + \\qu_{\\perp}$ with $\\qu_{\\perp} \\in \\Lambda^{\\perp}$, which allows us to write \\be \\hat{q}_0 = \\tilde{q}_0 + \\frac{1}{2}\\qu_{\\perp}^2.\\ee For $b_2^+(P) =1$ and $r=1$, the degeneracies $d(r,\\mu,\\tilde{q}_0)$ are independent of the choice of $\\qu_{\\perp}$ and moreover it was shown by G\\\"ottsche  \\cite{Gottsche:1990} that\n\\be\n\\sum_{\\tilde{q}_0} d(1,\\mu,\\tilde{q}_0) \\, e^{2\\pi i \\tau \\tilde{q}_0} = \\frac{1}{\\eta^{\\chi(P)}}.\n\\ee\nThen, for $r=1$ (\\ref{fdef}) becomes\n\\be \\label{frigid}\n    f^{(1)}_{\\mu,J}(\\tau)\n=\\frac{\\vartheta_{\\Lambda^{\\perp}}(\\tau)}{\\eta^{\\chi(P)}(\\tau)}, \\qquad\n\\vartheta_{\\Lambda^{\\perp}}(\\tau)  =  \\sum_{\\qu_{\\perp} \\in \\Lambda^{\\perp}} e^{i\\pi\\tau \\qu_{\\perp}^2}.\n\\ee\n\nThe second subtlety is concerned with the dependence on a K\\\"ahler class $J$. Due to wall-crossing phenomena we will find that $f^{(r)}_{\\mu,J}(\\tau)$ also depends on $J$. We expect that it has the following expansion ($\\tilde{q}_0=\\frac{d}{r}-\\frac{c_{\\rm L}}{24}$)\n\\begin{equation} \\label{fconj}\n    f^{(r)}_{\\mu,J}(\\tau) = (-1)^{r p \\cdot \\mu} \\,\\sum_{d\\,\\geq\\,0}\\bar{\\Omega}(\\Gamma; J) \\, q^{d - \\frac{r\\chi(P)}{24}}.\n\\end{equation}\nHere, the factor $(-1)^{r p \\cdot \\mu}$ is inserted to cancel its counterpart in the definition of $\\theta^{(r)}_{\\mu,J}$, which was only included to make the theta-functions transform well under modular transformations. The invariants $\\bar{\\Omega}(\\Gamma;J)$ are rational invariants first introduced by Joyce \\cite{Joyce:2006pf,Joyce:2008} and are defined as follows\n\\begin{equation} \\label{bomega}\n    \\bar{\\Omega}(\\Gamma; J) = \\sum_{m|\\Gamma} \\frac{\\Omega(\\Gamma\/m; J)}{m^2},\n\\end{equation}\nwhere $\\Omega(\\Gamma,J)$ is an integer-valued index of BPS degeneracies, given by \\cite{Dabholkar:2005by}\n\\be\n\\Omega(\\Gamma, J) = \\frac{1}{2} \\tr (2J_3)^2 (-1)^{2J_3},\n\\ee\nwhere $J_3$ is a generator of the rotation group ${\\rm Spin}(3)$.  Note, that for a single M5-brane $\\bar{\\Omega}$ and $\\Omega$ become identical and independent of $J$.\n\n\\subsection{$\\mathcal{N}=4$ SYM, E-strings and bound-states}\\label{n4ym}\nIn the following we recall the relation \\cite{Minahan:1998vr} of the elliptic genus of M5-branes to the $\\mathcal{N}=4$ topological SYM theory of Vafa and Witten \\cite{Vafa:1994tf}. Our goal is to relate the holomorphic anomaly equation which we will derive from wall-crossing in the next section to the anomalies appearing in the ${\\cal N}=4$ context. We review moreover the connection of the anomaly to the formation of bound-states given in ref.~\\cite{Minahan:1998vr}.\n\nThe $\\mathcal{N}=4$ topological SYM arises by taking a different perspective on the world-volume theory of $n$ M5-branes on $P\\times T^2$ considering the theory living on $P$ which is the $\\mathcal{N}=4$ topologically twisted SYM theory described in ref.~\\cite{Vafa:1994tf}. The gauge coupling of this theory is given by\n\\be\n\\tau=\\frac{4 \\pi i }{g^2}+ \\frac{\\theta}{2 \\pi},\n\\ee\nand is geometrically realized by the complex structure modulus of the $T^2$.  The partition function of this theory counts instanton  configurations by computing the generating functions of the Euler numbers of moduli spaces of gauge instantons \\cite{Vafa:1994tf}.  $S$-dualtiy translates to the modular transformation properties of the partition function. The analogues of D4-D2-D0 charges are the rank of the gauge group, different flux sectors and the instanton number. \n\nIn ref.~\\cite{Minahan:1998vr} the relation is made between this theory and the geometrical counting of BPS states of exceptional strings obtained by wrapping M5-branes around a del Pezzo surface $\\mathcal{B}_9$, also called $\\frac{1}{2}$K3. This string is dual to the heterotic string with an $E_8$ instanton of zero size \\cite{Ganor:1996mu,Seiberg:1996vs} and is therefore called E-string. In F-theory this corresponds to a\n$\\mathbbm{P}^1$ shrinking to zero size\n\\cite{Morrison:1996na,Morrison:1996pp,Witten:1996qb}. \nThe geometrical study of the BPS states of this non-critical string was initiated in ref.~\\cite{Klemm:1996hh} and further pursued in refs.~\\cite{Lerche:1996ni,Minahan:1997ch,Minahan:1997ct}. In\nref.~\\cite{Minahan:1998vr} the counting of BPS states of the \nexceptional string with increasing winding $n$ was related to \nthe $U(n)$, $\\mathcal{N}=4$ SYM partition functions. \n\nIn the following we will use the geometry of ref.~\\cite{Klemm:1996hh} which is an elliptic fibration over the Hirzebruch surface $\\mathbbm{F}_1$, which in turn is a $\\mathbbm{P}^1$ fibration over $\\mathbbm{P}^1$.\\footnote{The toric data of this geometry is summarized in appendix \\ref{toricdata}.} We will denote by $t_E, t_F$ and $t_D$ the K\\\"ahler parameters of the elliptic fiber, the fiber and the base of $\\mathbbm{F}_1$, respectively and enumerate these by $1,2,3$ in this order. We further introduce $\\tilde q_a=e^{2\\pi i \\tilde{t}_a}\\, , \\,a=1,2,3$ the exponentiated K\\\"ahler parameters appearing in the instanton expansion of the A-model at large radius, which are also the counting parameters of the BPS states. \n\nWithin this geometry we will be interested in the elliptic genus of M5-branes wrapping two different surfaces, one is a K3 corresponding to wrapping the elliptic fiber and the fiber of $\\mathbbm{F}_1$, the resulting string is the heterotic string. The other possibility is to wrap the base of $\\mathbbm{F}_1$ and the elliptic fiber corresponding to $\\frac{1}{2}$K3 and leading to the E-string studied in refs.~\\cite{Klemm:1996hh,Lerche:1996ni,Minahan:1997ch,Minahan:1997ct,Minahan:1998vr}. The two possibilities are realized by taking the limits $t_D,t_F\\rightarrow i \\infty$, respectively. The resulting surface in both cases is still elliptically fibered which allows one to identify the D4-D0 charges $n$ and $p$ with counting curves wrapping $n$-times the base and $p$-times the fiber of the elliptic fibration \\cite{Minahan:1998vr}. The multiple wrapping is hence encoded in the expansion of the prepotential $F_0(\\tilde q_1,\\tilde q_2,\\tilde q_3)$ of the geometry. In order to get a parameterization inside the K\\\"ahler cone of the K3 in which the corresponding curves in $H_2({\\rm K3},\\IZ)$ intersect with the standard metric of the hyperbolic lattice $\\Gamma^{1,1}$, we define $t_1=\\tilde{t}_1\\, ,t_2=\\tilde{t}_2-\\tilde{t}_1$ and $t_3=\\tilde{t}_3$ as well as the corresponding $q_1 = \\tilde q_1$, $q_2 = \\tilde q_2\/\\tilde q_1$ and $q_3=\\tilde q_3$. Taking $q_2$ or $q_3 \\rightarrow 0$, the multiple wrapping of the base is expressed by\n\\begin{equation}\nF_0(t_1,t_a)=\\sum_{n\\geq 1}Z^{(n)}(t_1) q_a^n\\, , \\quad a=2\\, \\textrm{ or } \\,3. \n\\end{equation}\nThe $Z^{(n)}$ can be identified with the elliptic genus of $n$ M5-branes wrapping the corresponding surface after taking a small elliptic fiber limit \\cite{Minahan:1998vr}. In this limit the contribution coming from the theta-functions (\\ref{SNTheta}) reduce to $\\tau_2^{-3\/2} \\left(\\tau_2^{-1\/2}\\right)$ for the K3($\\frac{1}{2}$K3) cases, these are the contributions of 3(1) copies of the lattice $\\Gamma^{1,1}$ appearing in the decomposition of the lattices of K3($\\frac{1}{2}$K3). Omitting these factors gives the $Z^{(n)}$ of weight $(-2,0)$ in both cases. The elliptic genera of wrapping $n$ M5-branes corresponding to $n$ strings are in both cases related recursively to the lower\nwrapping. The nature of the recursion depends crucially on the ability of the strings to form bound-states. \n\n\n\\subsubsection*{\\it The heterotic string, no bound-states} \\label{hetstring}\nThe heterotic string is obtained from wrapping an M5-brane on the K3 by taking the $q_3\\rightarrow 0$ limit. The heterotic string does not form bound-states and the recursion giving the higher wrappings in this case is the Hecke transformation\\footnote{For a review on Hecke transformations see Zagier's article in \\cite{1-2-3-modular}.} of $Z^{(1)}$ as proposed in ref.~\\cite{Minahan:1998vr}. The formula for the Hecke transformation in this case is given by\n\\begin{equation}\\label{eq:Hecke}\nZ^{(n)} (t)= n^{w_{\\rm L}-1} \\sum_{a,b,d} d^{-w_L} Z^{(1)}\\left(  \\frac{a t+b}{d}\\right) \\, ,\n\\end{equation}  \nwith $ad=n$ and $b < d$ and $a,b,d \\ge 0$. Which specializes for  $w_{\\rm L}=-2$ and $n=p$, where $p$ is prime to\n\\begin{equation} \nZ^{(p)} (t)= \\frac{1}{p^3} Z^{(1)} (p t) + \\frac{1}{p} \\left[Z^{(1)}\\left(\\frac{t}{p}\\right) + Z^{(1)}\\left(\\frac{t}{p} +\\frac{1}{p}\\right) + \\dots + Z^{(1)}\\left(\\frac{t}{p} +\\frac{p-1}{p}\\right) \\right] \\, .\n\\end{equation}\nFor example the partition functions for $n=1,2$ obtained from the instanton part of the prepotential of the geometry read\n\\begin{equation}\nZ^{(1)}=-\\frac{2 E_4 E_6 }{\\eta^{24}} ,\\quad Z^{(2)}=-\\frac{E_4 E_6 \\left(17 E_4^3+7 E_6^2\\right)}{96 \\eta^{48}}\\, ,\n\\end{equation}\nand are related by the Hecke transformation. Further examples of higher wrapping are given in the appendix \\ref{app:K3}. The fact that the partition functions of higher wrappings of the M5-brane on the K3, which correspond to multiple heterotic strings, are given by the Hecke transformation was interpreted \\cite{Minahan:1998vr} by the absence of bound-states. Geometrically, multiple M5-branes on a K3 can be holomorphically deformed off one another. This argument fails for surfaces with $b_2^+=1$ and in particular for $\\frac{1}{2}$K3.\n\n\nOne reason that the higher $Z^{(n)}$ can be determined in such a simple way from $Z^{(1)}$ can be understood in topological \nstring theory from the fact that the BPS numbers on K3 depend only on the intersection of a curve ${\\cal C}^2=2 g-2$~\\cite{Yau:1995mv}, and not on their class in $H_2(\\text{K3},\\mathbb{Z})$. \nThis allows to prove (\\ref{eq:Hecke}) to all orders in the limit of the topological string partition function under consideration \nby slightly modifying the proof in~\\cite{MKPS}. Using the Picard-Fuchs system of the elliptic fibration one shows in the limit \n$q_3\\rightarrow 0$ the first equality in the identity \n\\begin{equation} \n\\begin{split}\n\\frac{1}{2}\\left(\\frac{\\partial}{\\partial{t_2}}\\right)^3 F_0|_{q_3\\rightarrow 0}\n&=\\frac{ E_4(t_1) E_6(t_1) E_4(t_2)}{\\eta(t_1)^{24} (j(t_1)-j(t_2))}\\\\\n&= \\frac{q_1}{q_1-q_2}+ E_4(t_2)-\\sum_{d,l,k>0} l^3 c(k l) \nq_1^{kl} q_2^{ld}\\ ,\n\\end{split}\n\\end{equation}\nwhere $j=E_4^3\/\\eta^{24}$ and $c(n)$ are defined as \\be-\\frac{1}{2}Z^{(1)}=\\sum_n c(n) q^n.\\ee This equations shows \ntwo things. The BPS numbers inside the K\\\"ahler cone of K3 depend only on ${\\cal C}^2=k l$ and all  \n$Z^{(n)}$ are given by one modular form. The second fact can be  used as in \\cite{MKPS} to establish that \n\\be\\frac{1}{2}\\left(\\frac{\\partial}{\\partial{t_2}}\\right)^3 F_0|_{q_3\\rightarrow 0}=\n\\sum_{n=0}^\\infty F_n(t_1) q_2^n ,\\ee where $F_n$ is the Hecke transform of $F_1$, i.e.~$n^3 F_n=F_1| T_n$. Using Bol's identity and restoring the $n^3$ factors yields (\\ref{eq:Hecke}).\n\n\\subsubsection*{\\it E-strings and bound-states}\nThe recursion relating the higher windings of the E-strings to lower winding,  developed in \\cite{Minahan:1997ch,Minahan:1997ct,Minahan:1998vr} in contrast reads\n\\begin{equation}\n\\frac{\\partial Z^{(n)}}{\\partial E_2}=\\frac{1}{24} \\sum_{s=1}^{n-1} s(n-s) Z^{(s)}\\,Z^{(n-s)}\\, ,\n\\end{equation}\nwhich becomes an anomaly equation, when $E_2$ is completed into a modular object  $\\wh{E}_2$ by introducing a non-holomorphic part (see appendix \\ref{sec:mock}). The anomaly reads:\n\\begin{equation}\n\\partial_{\\bar{t}_1} \\wh{Z}^{(n)}=\\frac{i (\\textrm{Im}\\, t_1)^{-2}}{16 \\pi} \\, \\sum_{s=1}^{n-1} s(n-s) \\wh{Z}^{(s)} \\wh{Z}^{(n-s)}\\, ,\n\\end{equation}\nand was given the interpretation \\cite{Minahan:1998vr} of taking into account the contributions from bound-states. \nStarting from~\\cite{Klemm:1996hh}\n\\begin{equation} \nZ^{(1)} =\\frac{E_4 \\sqrt{q}}{\\eta^{12}},\n\\end{equation}\nand using the vanishing of BPS states of certain charges one obtains recursively all $Z^{(n)}$ \\cite{Minahan:1997ch,Minahan:1997ct,Minahan:1998vr}. E.g. the $n=2$ the \ncontribution reads:\n\\begin{equation}\n  \\wh{Z}^{(2)}=\\frac{q E_4 E_6}{12 \\eta^{24}} + \\frac{q \\wh{E_2} E_4^2}{24 \\eta^{24}}\\, ,\n\\end{equation}\nwhere the second summand has the form $\\wh{E}_2 \\left(Z^{(1)}\\right)^2$ and takes into account the contribution from bound-states of singly wrapped M5-branes. \n\nA relation to the anomaly equations appearing in topological string theory \\cite{Bershadsky:1993cx} was pointed out in ref.~\\cite{Minahan:1998vr} and proposed for arbitrary genus in refs.~\\cite{Hosono:1999qc,Hosono:2002xj}. The higher genus generalization reads \\cite{Hosono:1999qc,Hosono:2002xj}:\n\\begin{equation}\\label{eq:highergenus}\n\\frac{\\partial Z^{(n)}_{g}}{\\partial E_2}=\\frac{1}{24} \\sum_{g_1+g_2=g}\\sum_{s=1}^{n-1} s(n-s) Z^{(s)}_{g_1}\\, Z^{(n-s)}_{g_2}\\,+ \\frac{n(n+1)}{24} Z^{(n)}_{g-1}\\, \\, ,\n\\end{equation}\nwhere the instanton part of the A-model free energies at genus $g$ is denoted by\n$F_g(q_1,q_2,q_3)$, and $F_g(q_1,q_2\\rightarrow 0,q_3)= \\sum_{n\\ge1} Z^{(n)}_g q_3^n$.\nThe $Z^{(n)}_{g}$ have the form \\cite{Hosono:2002xj}\n\\begin{equation}\nZ^{(n)}_{g}=P^{(n)}_{g}(E_2,E_4,E_6) \\frac{q_1^{ n\/2}}{\\eta^{12 n}} \\, ,\n\\end{equation}\nwhere $P^{(n)}_{g}$ denotes a quasi-modular form of weight $2g+6n-2$.\n\n\n\n\n\\subsection{Generating functions from wall-crossing}\nIn the last section we have argued that the partition function of $\\cN=4$ $U(r)$ Super-Yang-Mills theory suffers from a holomorphic anomaly\nfor divisors with $b_2^+(P) = 1$. In fact there exists another way to see the anomaly which\nis also intimately related to the computation of BPS degeneracies encoded in the elliptic genus\nand will be the subject of this section.\nThis method relies on wall-crossing formulas and originally goes back to G\\\"ottsche and Zagier \\cite{Gottsche:1998,Gottsche:1999}.\nIn the physics context it has also been employed in \\cite{Manschot:2010xp,Manschot:2010nc}.\nIt will be used in section \\ref{sec:wcmm} to derive the elliptic genus for BPS states and their anomaly rigorously.\nIn the following presentation we will be very sketchy as we merely want to stress the main ideas.\nWe refer to section \\ref{sec:wcmm} for details.\n\nThe starting point is the Kontsevich-Soibelman formula \\cite{KS:2008} which describes the wall-crossing\nof bound-states of D-branes. Specifying to the case of two M5-branes and taking the equivalent\nD4-D2-D0 point of view the Kontsevich-Soibelman formula reduces to the primitive wall-crossing formula\n\\begin{equation} \\label{pwc1}\n    \\Delta \\Omega(\\Gamma;J\\rightarrow J')  = \\Omega(\\Gamma; J') - \\Omega(\\Gamma;J)= (-1)^{\\langle \\Gamma_1,\\Gamma_2\\rangle -1} \\langle \\Gamma_1,\\Gamma_2 \\rangle\\, \\Omega(\\Gamma_1) \\,\\Omega(\\Gamma_2),\n\\end{equation}\nwhich describes the change of BPS degeneracies of a bound-state with charge vector $\\Gamma = \\Gamma_1 + \\Gamma_2$,\nonce a wall of marginal stability specified by $J_W$ is crossed. The symplectic charge product $\\langle \\cdot, \\cdot \\rangle$ is defined by\n\\be\n\\langle\\Gamma_1,\\Gamma_2\\rangle=-Q_6^{(1)}Q_0^{(2)}+Q_4^{(1)}\\cdot Q_2^{(2)}-Q_2^{(1)}\\cdot Q_4^{(2)}+Q_0^{(1)}Q_6^{(2)}.\n\\ee\nHence, for D4-D2-D0 brane configurations $\\langle\\Gamma_1,\\Gamma_2\\rangle$ is independent of the D0-brane charge.\nFurther, in eq.~(\\ref{pwc1}) $\\Gamma_1$ and $\\Gamma_2$ are primitive charge vectors such that $\\Omega(\\Gamma_i)$ do not depend on the moduli.\nThus, the $\\Gamma_i$ can be thought of as charge vectors with $r=1$ whereas $\\Gamma$ corresponds to a charge vector with $r=2$.\nAssuming, that the wall of marginal stability does not depend on the D0-brane charge, formula (\\ref{pwc1}) can be translated into a generating series $\\Delta f^{(2)}_{\\mu,J\\rightarrow J'}$ defined by\n\\be\n    \\Delta f_{\\mu,J \\rightarrow J'}^{(2)} = \\sum_{d\\geq 0} \\Delta \\bar{\\Omega}(\\Gamma;J\\rightarrow J') \\, q^{d-\\frac{\\chi(P)}{12}}.\n\\ee\nAssuming that there exists a reference chamber $J'$ such that $\\bar\\Omega(\\Gamma;J)=0$, this gives us directly an expression for $f^{(2)}_{\\mu,J}$.\n\nAs it will turn out in the next section, $\\Delta f^{(2)}_{\\mu,J\\rightarrow J'}$ is given in terms of an indefinite theta-function $\\Theta^{J,J'}_{\\Lambda,\\mu}$, which contains the information about the decays due to wall-crossing as one moves from $J$ to $J'$. Indefinite theta-functions were analyzed by Zwegers in his thesis \\cite{Zwegers:2002}. One of their major properties is that they are not modular as one only sums over a bounded domain of the lattice $\\Lambda$ specified by $J$ and $J'$. However, Zwegers showed that by adding a non-holomorphic completion the indefinite theta-functions have modular transformation behavior and fall into the class of mock modular forms.\\footnote{We review some notions in appendix \\ref{sec:mock}.} Every mock modular form $h$ of weight $k$ has a shadow $g$, which is a modular form of weight $2-k$, such that the function\n\\begin{equation} \\label{mock}\n    \\hat{h}(\\tau) = h(\\tau) + g^*(\\tau)\n\\end{equation}\ntransforms as a modular form of weight $k$ but is not holomorphic. Here, $g^*$ is a certain transformation of the function $g$ that introduces a non-holomorphic dependence. Taking the derivative of $\\hat{h}$ with respect to $\\bar\\tau$ yields a holomorphic anomaly given by the shadow\n\\begin{equation}\n    \\frac{\\p \\hat{h}}{\\p \\bar{\\tau}} = \\frac{\\p g^*}{\\p \\bar{\\tau}} = \\tau_2^{-k} \\overline{g(\\tau)},\n\\end{equation}\nwhere $\\tau_2=\\text{Im}(\\tau)$.\n\nAs described in sections \\ref{sec:mswcft} and \\ref{n4ym} the (MSW) CFT and the ${\\cal N}=4$ $U(r)$ Super-Yang-Mills partition functions should behave covariantly under modular transformations of the SL$(2,\\IZ)$ acting on $\\tau$. Thus, the modular completion outlined above will effect the generating functions $f^{(2)}_{\\mu,J}$ through their relation to the indefinite theta-function $\\Theta^{J,J'}_{\\Lambda,\\mu}$, which needs a modular completion to transform covariantly under modular transformations, i.e. \\be\\Theta^{J,J'}_{\\Lambda,\\mu} \\mapsto \\widehat{\\Theta}^{J,J'}_{\\Lambda, \\mu} \\ee and consequently $f^{(2)}_{\\mu,J}$ is replaced by $\\hat{f}^{(2)}_{\\mu,J}$. Due to eq.~(\\ref{mock}) the counting function of BPS invariants $\\hat{f}^{(2)}_{\\mu,J}$ and thus the elliptic genus $Z^{(2)}_P$ are going to suffer from a holomorphic anomaly, to which we turn next. \n\n\n\n\\section{Wall-crossing and mock modularity}\\label{sec:wcmm}\n\nIn this section we derive an anomaly equation for two M5-branes wound on a rigid surface\/divisor $P$ with $b_2^+(P)=1$, inside a Calabi-Yau manifold $X$. We begin by reviewing D4-D2-D0 bound-states in the type IIA picture and their wall-crossing in the context of the Kontsevich-Soibelman formula. Then we proceed by deriving a generating function for rank two sheaves from the Kontsevich-Soibelman formula which is equivalent to G\\\" ottsche's formula \\cite{Gottsche:1999}. This generating function is an indefinite theta-function, which fails to be modular. As a next step we apply ideas of Zwegers to remedy this failure of modularity by introducing a non-holomorphic completion. This leads to a holomorphic anomaly equation of the elliptic genus of two M5-branes that we prove for rigid divisors $P$.\n\n\\subsection{D4-D2-D0 wall-crossing}\\label{sec:wc}\nIn the following we take on the equivalent type IIA point of view, adapting the discussion of refs. \\cite{Diaconescu:2007bf,Manschot:2010xp,Manschot:2010nc} to describe the relation to the Kontsevich-Soibelman wall-crossing formula \\cite{KS:2008}. We restrict our attention to the D4-D2-D0 system on the complex surface $P$ and work in the large volume limit with vanishing $B$-field.\n\nLet us recall that a generic charge vector with D4-brane charge $r$ is given by (see appendix \\ref{Dbound} for details)\n\\begin{equation} \\label{chv}\n    \\Gamma = (Q_6,Q_4,Q_2,Q_0) = r \\left(0,\\, [P],\\, i_* F(\\cE),\\, \\frac{\\chi(P)}{24} + \\int_P \\half F(\\cE)^2 - \\Delta(\\cE) \\right),\n\\end{equation}\nwhere $\\cE$ is a sheaf on the divisor $P$. Further, we define\n\\begin{equation}\n    \\Delta(\\cE)=\\frac{1}{r(\\cE)} \\left(c_2(\\cE) -\\frac{r(\\cE)-1}{2 r(\\cE)}c_1(\\cE)^2\\right)\\, , \\quad \\mu(\\cE)=\\frac{c_1(\\cE)}{r(\\cE)}\\, ,\n    \\quad F(\\cE) = \\mu(\\cE) + \\frac{[P]}{2}.\n\\end{equation}\nWe recall that in the large volume regime the notion of D-brane stability is equivalent to $\\mu$-stability, see \\cite{Diaconescu:2007bf} and appendix~\\ref{Dbound}. Given a choice of $J\\in\\cC(P)$, a sheaf $\\cE$ is called $\\mu$-semi-stable if for every sub-sheaf $\\cE'$\n\\begin{equation}\\label{stability}\n    \\mu(\\cE')\\cdot J \\le \\mu({\\cE})\\cdot J.\n    \\end{equation}\nMoreover, a wall of marginal stability is a co-dimension one subspace of the K\\\"ahler cone $\\cC(P)$ where the following condition is satisfied\n\\begin{equation} \\label{muwall}\n    (\\mu(\\cE_1)-\\mu(\\cE_2))\\cdot J=0,\n\\end{equation}\nbut is non-zero away from the wall. Across such a wall of marginal stability the configuration (\\ref{chv}) splits into two configurations with charge vectors\n\\begin{eqnarray}\n    \\Gamma_1 & = & r_1 \\left(0,\\, [P],\\, i_* F_1,\\, \\frac{\\chi(P)}{24} + \\int_P \\half F_1^2\n              - \\Delta(\\mathcal{E}_1)\\right), \\nonumber \\\\\n    \\Gamma_2 & = & r_2 \\left(0,\\, [P],\\, i_* F_2,\\, \\frac{\\chi(P)}{24} + \\int_P \\half F_2^2\n              - \\Delta(\\mathcal{E}_2)\\right), \\label{chv12}\n\\end{eqnarray}\nwhere $r_i = \\text{rk}(\\mathcal{E}_i)$ and $\\mu_i=\\mu(\\cE_i)$. By making use of the identity\n\\begin{eqnarray}\\label{c1split}\n      r \\Delta & = & r_1 \\Delta_1 + r_2 \\Delta_2 + \\frac{r_1 r_2}{2r} \\left(\\frac{c_1(\\mathcal{E}_1)}{r_1} - \\frac{c_1(\\mathcal{E}_2)}{r_2}\\right)^2,\n\\end{eqnarray}\none can show that $\\Gamma = \\Gamma_1 + \\Gamma_2$. Therefore, charge-vectors as defined in (\\ref{chv}) form a vector-space which will be\nessential for the application of the Kontsevich-Soibelman formula.\n\nBefore we proceed, let us note, that the BPS numbers and the Euler numbers of the moduli space of sheaves are related as follows. Denote by $\\mathcal{M}_{J}(\\Gamma)$ the moduli space of semi-stable sheaves characterized by $\\Gamma$. Its dimension reads \\cite{Maruyama:1977}\n\\begin{equation}\n    \\textrm{dim}_{\\mathbbm{C}} \\mathcal{M}_J(\\Gamma)= 2 r^2 -r^2 \\chi(\\mathcal{O}_P)+1.\n\\end{equation}\nThe relation between BPS invariants and the Euler numbers of the moduli spaces $\\mathcal{M}_J(\\Gamma)$\nis then given by \\cite{Diaconescu:2007bf}\n\\begin{equation}\n    \\Omega(\\Gamma,J)= (-1)^{\\textrm{dim}_{\\mathbbm{C}} \\mathcal{M}_J(\\Gamma)} \\chi(\\mathcal{M}(\\Gamma),J)\\, .\n\\end{equation}\nMoreover, for the system of charges we have specified to, the symplectic pairing of charges simplifies to \\cite{Diaconescu:2007bf}\n\\begin{equation}\\label{eq:chargeproduct}\n    \\langle \\Gamma_1,\\Gamma_2\\rangle = r_1 r_2 (\\mu_2-\\mu_1) \\cdot \\left[ P\\right].\n\\end{equation}\n\nThe holomorphic function $f^{(r)}_{\\mu,J}(\\tau)$ appearing in eq.~(\\ref{thetadecomp}) can now be identified with the generating function of BPS invariants of moduli spaces of semi-stable sheaves. Its wall crossing will be described in the following.\n\n\\subsubsection*{\\it Kontsevich-Soibelman wall-crossing formula}\nKontsevich and Soibelman \\cite{KS:2008} have proposed a formula which determines the jumping behavior of BPS-invariants $\\Omega(\\Gamma; J)$ across walls of marginal stability.  The wall-crossing formula is given in terms of a Lie algebra defined by generators $e_{\\Gamma}$ and a basic commutation relation\n\\begin{equation}\n  \\left[e_{\\Gamma_1}, e_{\\Gamma_2}\\right] = (-1)^{\\langle \\Gamma_1, \\Gamma_2 \\rangle} \\langle \\Gamma_1, \\Gamma_2 \\rangle e_{\\Gamma_1 + \\Gamma_2}.\n\\end{equation}\n\nFor every charge $\\Gamma$ an element $U_{\\Gamma}$ of the Lie group can be defined by\n\\begin{equation}\n  U_{\\Gamma} = \\textrm{exp}\\left(-\\sum_{n \\geq 1} \\frac{e_{n\\Gamma}}{n^2} \\right).\n\\end{equation}\n\nThe Kontsevich-Soibelman wall-crossing formula states that across a wall of marginal stability the following formula holds\n\\begin{equation} \\label{kswc}\n  \\prod_{\\Gamma: Z(\\Gamma; J) \\in V}^{\\curvearrowright} U_{\\Gamma}^{\\Omega(\\Gamma; J_+)}\n  = \\prod_{\\Gamma: Z(\\Gamma; J) \\in V}^{\\curvearrowright} U_{\\Gamma}^{\\Omega(\\Gamma; J_-)},\n\\end{equation}\nwhere $J_+$ and $J_-$ denote K\\\"ahler classes on the two sides of the wall. Further, $V$ is a region in $\\IR^2$ bounded by two rays starting at the origin and $\\curvearrowright$ denotes a clockwise ordering of the factors in the product with respect to the phase of the central charges $Z(\\Gamma; J)$, that are defined in eq.~(\\ref{eq:centralcharge}).\n\nRestricting to the case $r=2$ and $r_1 = r_2 = 1$, (\\ref{kswc}) can be truncated to\n\\begin{equation}\n    \\prod_{Q_{0,1}} U_{\\Gamma_1}^{\\Omega(\\Gamma_1)} \\prod_{Q_0} U_{\\Gamma}^{\\Omega(\\Gamma; J_+)}\n    \\prod_{Q_{0,2}} U_{\\Gamma_2}^{\\Omega(\\Gamma_2)}\n    =\n    \\prod_{Q_{0,2}} U_{\\Gamma_2}^{\\Omega(\\Gamma_2)} \\prod_{Q_0} U_{\\Gamma}^{\\Omega(\\Gamma; J_-)}\n    \\prod_{Q_{0,1}} U_{\\Gamma_1}^{\\Omega(\\Gamma_1)},\n\\end{equation}\nwhere $Q_0$ is the D0-brane charge of $\\Gamma$ and the $Q_{0,i}$ are the D0-brane charges belonging to $\\Gamma_i$, respectively.\nThe above formula has been derived by setting all Lie algebra elements with D4-brane charge greater than two to zero.\nTherefore, the element $e_{\\Gamma}$ is central, using the Baker-Campbell-Hausdorff formula $e^X e^Y = e^Y e^{[X,Y]} e^X$ and the fact that the symplectic product is independent of the D0-brane charge,\none finds the following change of BPS numbers across a wall of marginal stability \\cite{Manschot:2010xp, Gaiotto:2008cd}\n\\begin{equation} \\label{pwc}\n    \\Delta \\Omega(\\Gamma) = (-1)^{\\langle \\Gamma_1, \\Gamma_2 \\rangle - 1} \\langle \\Gamma_1, \\Gamma_2 \\rangle\n                            \\sum _{Q_{0,1}+Q_{0,2}=Q_0} \\Omega(\\Gamma_1)\\, \\Omega(\\Gamma_2).\n\\end{equation}\nMoreover, one can deduce that the rank one degeneracies $\\Omega(\\Gamma_1)$ and $\\Omega(\\Gamma_2)$ do not depend on the modulus $J$.\n\n\\subsection{Relation of KS to G\\\"ottsche's wall-crossing formula}\\label{sec:KSGwallcrossing}\nG\\\"ottsche has found a wall-crossing formula for the Euler numbers of moduli spaces of rank two sheaves in terms of an indefinite theta-function in ref.~\\cite{Gottsche:1999}.\nIn this section we want to derive a modified version of this formula from the Kontsevich-Soibelman wall-crossing formula associated to D4-D2-D0 bound-states with D4-brane charge equal to two.\n\nWe use the short notation $\\Gamma=(r,\\mu,\\Delta)$ to denote a rank $r$ sheaf with the specified Chern classes that is associated to the D4-D2-D0 states. For rank one sheaves the generating function has no chamber dependence and we have already seen that it is given by (\\ref{frigid}).\nFollowing the discussion of our last section, higher rank sheaves do exhibit wall-crossing phenomena and therefore do depend on the chamber in\nmoduli space, i.e.~on $J\\in\\cC(P)$.  \n\nOur aim now is to determine the generating function of the D4-D2-D0 system using the primitive wall-crossing formula derived from the KS wall-crossing formula. From now on we restrict our attention to rank two sheaves $\\mathcal{E}$. They can split across walls of marginal stability into rank one sheaves $\\mathcal{E}_1$\nand $\\mathcal{E}_2$ as outlined in section \\ref{sec:wc}. Using relation (\\ref{c1split}) we can write\n\\begin{equation} \\label{Deltarec}\n    d = d_1 + d_2 + \\xi\\cdot\\xi,\n\\end{equation}\nwhere $\\xi= \\mu_1-\\mu_2$ and $d=2\\Delta$. Further, a wall is given by (\\ref{muwall}), i.e.~the set of walls given a split of charges $\\xi$ reads\n\\begin{equation}\n    W^{\\xi} = \\left\\{J \\in \\cC(P)\\,|\\, \\xi\\cdot J = 0\\right\\}.\n\\end{equation}\nNow, consider a single wall $J_W\\in W^{\\xi}$ determined by a set of vectors $\\xi \\in \\Lambda + \\mu$. Let $J_+$ approach $J_W$ infinitesimally close from one side and $J_-$ infinitesimally close from the other side. Thus, in our context the primitive wall-crossing formula (\\ref{pwc}) becomes\n\\begin{equation} \\label{cswc}\n    \\bar{\\Omega}(\\Gamma;J_+) - \\bar{\\Omega}(\\Gamma;J_-)\n    = \\sum_{Q_{0,1}+Q_{0,2}=Q_0} (-1)^{2\\xi\\cdot[P]}\\, 2\\,  (\\xi\\cdot[P])\\, \\Omega(\\Gamma_1)\\, \\Omega(\\Gamma_2),\n\\end{equation}\nwhere we  have used the identity \\eqref{eq:chargeproduct}. Note, that $Q_{0,i}$ and $Q_0$ are determined in terms of $\\Gamma$ and $\\Gamma_i$ through (\\ref{chv}) and (\\ref{chv12}). Now, we can sum over the D0-brane charges to obtain a generating series. This yields \n\\begin{eqnarray}\n    & ~ & \\sum_{d\\,\\geq\\,0} (\\bar{\\Omega}(\\Gamma;J_+) - \\bar{\\Omega}(\\Gamma;J_-)) q^{d - \\frac{\\chi(P)}{12}} \\nonumber \\\\\n    ~ ~ & = & \\sum_{d_1,d_2\\,\\geq\\,0,\\,\\xi} (-1)^{2 \\xi\\cdot [P]}\\,  (\\xi\\cdot [P])\\, \\Omega(\\Gamma_1) \\Omega(\\Gamma_2)\n            q^{d_1 + d_2 + \\xi^2 - \\frac{2\\chi(P)}{24}} \\nonumber \\\\\n    ~ ~ & = & (-1)^{2 \\mu\\cdot[P]-1}\\frac{\\vartheta_{\\Lambda^\\perp}(\\tau)^2}{\\eta(\\tau)^{2\\chi(P)}}\\,\\sum_{\\xi} (\\xi\\cdot[P])\\, q^{\\xi^2},\n\\end{eqnarray}\nwhere for the first equality use has been made of the identities (\\ref{Deltarec}, \\ref{cswc}), and for the second equality\nthe identity (\\ref{frigid}) has been used. The last line can be rewritten as\n\\begin{equation}\n    (-1)^{2 \\mu\\cdot[P]-1} \\frac{1}{2}\\frac{\\vartheta_{\\Lambda^\\perp}(\\tau)^2}{\\eta(\\tau)^{2 \\chi(P)}} \\textrm{Coeff}_{2 \\pi i y} (\\Theta^{J_+,J_-}_{\\Lambda,\\mu}( \\tau, [P] y)),\n\\end{equation}\nwhere we have introduced the indefinite theta-function\n\\begin{equation} \\label{eq:indefinite}\n    \\Theta^{J,J'}_{\\Lambda,\\mu}(\\tau,x) :=\\frac{1}{2} \\sum_{\\xi \\in \\Lambda + \\mu} (\\sgn\\langle J,\\xi\\rangle-\\sgn\\langle J', \\xi\\rangle)\\, e^{2 \\pi i \\langle \\xi, x\\rangle}\\, q^{Q(\\xi)},\n\\end{equation}\nwith the inner product\\footnote{Note, that this is not the symplectic product of D-brane charges defined before.} defined by $\\langle x, y\\rangle = 2 d_{AB} x^A y^B$ and the quadratic form $Q(\\xi)=\\frac{1}{2} \\langle \\xi, \\xi \\rangle$. As these theta-functions obey the cocycle condition \\cite{Gottsche:1998}\n\\begin{equation}\n    \\Theta^{F,G}_{\\Lambda,\\mu} + \\Theta^{G,H}_{\\Lambda,\\mu} = \\Theta^{F,H}_{\\Lambda,\\mu},\n\\end{equation}\nwe finally arrive at the beautiful relation between the BPS numbers in an arbitrary chamber $J$ and those in a chamber $J'$ first found by G\\\"ottsche in the case $\\Lambda=H^2(P,\\IZ)$:\n\\be\\label{eq:thm41}\nf^{(2)}_{\\mu,J'}(\\tau)-f^{(2)}_{\\mu,J}(\\tau)=\\frac{1}{2}\\frac{\\vartheta_{\\Lambda^\\perp}(\\tau)^2}{\\eta^{2\\chi(P)}(\\tau)}\\,\\text{Coeff}_{2\\pi i y}(\\Theta^{J,J'}_{\\Lambda,\\mu}( \\tau,[P] y)).\n\\ee\n\n\n\\subsection{Holomorphic anomaly at rank two}\nIn this subsection we discuss the appearance of a holomorphic anomaly at rank two and give a proof of it by combing our previous results with results of Zwegers \\cite{Zwegers:2002}.\n\n\\subsubsection{Elliptic genus at rank two and modularity}\nAn important datum in eq.~(\\ref{eq:thm41}) is the choice of chambers $J,J'\\in\\cC(P)$, which are any points in the K\\\"ahler cone of $P$. As a consequence, the indefinite theta-series does not transform well under SL$(2,\\IZ)$ in general. However, from the discussion of sect.~\\ref{sec:mswcft} we expect, that the generating series $f^{(r)}_{\\mu,J}(\\tau)$ transforms with weight $-\\frac{r(\\Lambda)+2}{2}$ in a vector-representation under the full modular group, where $r(\\Lambda)$ denotes the rank of the lattice $\\Lambda$. Hence, there is a need to restore modularity. The idea is as follows.\n\nFollowing Zwegers \\cite{Zwegers:2002}, it turns out that the indefinite theta-function can be made modular at the cost of losing its holomorphicity. From the definition (\\ref{eq:indefinite}) Zwegers smoothes out the sign-functions and introduces a modified function as\n\\be\\label{eq:mocktheta}\n\\widehat{\\Theta}^{c,c'}_{\\Lambda,\\mu}(\\tau,x)=\\frac{1}{2}\\sum_{\\xi\\,\\in\\,\\Lambda+\\mu}\\left(\\!\\!E\\!\\left(\\frac{\\langle c, \\xi+\\frac{\\text{Im}\\,(x)}{\\tau_2}\\rangle\\sqrt{\\tau_2}}{\\sqrt{-Q(c)}}\\right)-E\\!\\left(\\frac{\\langle c',\\xi+\\frac{\\text{Im}\\,(x)}{\\tau_2}\\rangle\\sqrt{\\tau_2}}{\\sqrt{-Q(c')}}\\right)\\!\\!\\right)e^{2\\pi i \\langle \\xi, x\\rangle} q^{Q(\\xi)},\n\\ee\nwhere $E$ denotes the incomplete error function\n\\be\nE(x)=2\\int_0^x e^{-\\pi u^2}du.\n\\ee\nNote, that if $c$ or $c'$ lie on the boundary of the K\\\"ahler cone, one does not have to smooth out the sign-function. Zwegers shows, that the non-holomorphic function $\\widehat{\\Theta}^{c,c'}_{\\Lambda,\\mu}(\\tau,x)$ satisfies the correct transformation properties of a Jacobi form of weight $\\frac{1}{2}r(\\Lambda)$. Due to the non-holomorphic pieces it contains mock modular forms, that we want to identify in the following. In order to separate the holomorphic part of $\\widehat{\\Theta}^{c,c'}_{\\Lambda,\\mu}(\\tau,x)$ from its shadow we recall the following property of the incomplete error function\n\\be\nE(x)=\\text{sgn}(x)(1-\\beta_{\\frac{1}{2}}(x^2)),\n\\ee\nwhich enables us to split up $\\widehat{\\Theta}^{c,c'}_{\\Lambda,\\mu}(\\tau,x)$ into pieces. Here, $\\beta_{k}$ is defined by\n\\begin{equation}\n\\beta_k(t)=\\int_t^{\\infty} u^{-k} e^{-\\pi u} du.\n\\end{equation}\nHence, one can write eq.~\\eqref{eq:mocktheta} as\n\\be\n\\widehat{\\Theta}^{c,c'}_{\\Lambda,\\mu}(\\tau,x)=\\Theta^{c,c'}_{\\Lambda,\\mu}(\\tau,x)-\\Phi^{c}_{\\mu}(\\tau,x)+\\Phi^{c'}_{\\mu}(\\tau,x),\n\\ee\nwith\n\\be\n\\Phi^{c}_{\\mu}(\\tau,x)=\\frac{1}{2}\\sum_{\\xi\\,\\in\\,\\Lambda+\\mu}\\left[\\text{sgn}\\langle \\xi, c\\rangle-E\\left(\\frac{\\langle c , \\xi+\\frac{\\text{Im}\\,(x)}{\\tau_2}\\rangle \\sqrt{\\tau_2}}{\\sqrt{-Q(c)}}\\right)\\right]e^{2\\pi i \\langle \\xi,  x\\rangle} q^{Q(\\xi)}.\n\\ee\nIf $c$ belongs to ${\\cal C}(P)\\cap \\mathbb{Q}^{r(\\Lambda)}$, we may write\n\\be\n\\Phi^{c}_{\\mu}(\\tau,x)=R(\\tau,x)\\theta(\\tau,x),\n\\ee\nwhere we decomposed the lattice sum into contributions along the direction of $c$ and perpendicular to $c$ given by $R$ and $\\theta$, respectively. Hence, $\\theta$ is a usual theta-series associated to the quadratic form $Q|\\langle c\\rangle^\\perp$, i.e.~of weight $(r(\\Lambda)-1)\/2$. $R$ is the part which carries the non-holomorphicity. It transforms with a weight $\\frac{1}{2}$ factor and therefore $\\text{Coeff}_{2\\pi iy}(R(\\tau,[P]y))$ is of weight $\\frac{3}{2}$. Following the general idea of Zagier \\cite{Zagier:2007} that we recapitulate in appendix \\ref{sec:mock}, we should encounter the $\\beta_{\\frac{3}{2}}$ function in the $2\\pi iy$-coefficient of $\\Phi$. Indeed one can prove the following identity\n\\be\n\\text{Coeff}_{2\\pi iy} \\Phi^{c}_{\\mu}(\\tau,[P]y)=-\\frac{1}{4\\pi}\\frac{\\langle c,[P]\\rangle}{\\langle c,c\\rangle}\\sum_{\\xi\\,\\in\\,\\Lambda+\\mu}|\\langle c,\\xi\\rangle|\\, \\beta_{\\frac{3}{2}}\\left(\\frac{\\tau_2 \\langle c,\\xi\\rangle^2}{-Q(c)}\\right)q^{Q(\\xi)}.\n\\ee\nTaking the derivative with respect to $\\bar \\tau$ in order to obtain the shadow we arrive at the following final expression\n\\be\\label{aniholomder}\n\\partial_{\\bar \\tau} \\text{Coeff}_{2\\pi i y} \\Phi^{c}_\\mu (\\tau, [P] y)=-\\frac{\\tau_2^{-\\frac{3}{2}}}{8 \\pi i} \\frac{c\\cdot[P]}{\\sqrt{-c^2}}\\,(-1)^{4{\\mu}^2}\\, \\theta^{(2)}_{\\mu-\\frac{[P]}{2},c}(\\tau,0),\n\\ee\nwhere we define the Siegel-Narain theta-function $\\theta^{(r)}_{\\mu,c}(\\tau,z)$ as in eq.~(\\ref{SNTheta}). For more details on the transformation properties of the indefinite theta-functions we refer the reader to appendix \\ref{sec:mock}.\n\nNow, these results can be used to compute the elliptic genus for two M5-branes wrapping the divisor $P$. Consider\n\\be\nf^{(2)}_{\\mu,J}(\\tau)=f_{\\mu,J'}(\\tau)-\\frac{1}{2}\\frac{\\vartheta_{\\Lambda^\\perp}(\\tau)^2}{\\eta^{2\\chi(P)}}\\,\\text{Coeff}_{2\\pi i y}\\Theta^{J,J'}_{\\Lambda,\\mu}(\\tau,[P]y),\n\\ee\nwhere $f_{\\mu,J'}(\\tau)$ is a holomorphic ambiguity given by the generating series in a reference chamber $J'$, which we choose to lie at the boundary of the K\\\"ahler cone $J'\\,\\in\\,\\p\\cC(P)$. In explicit computations it may be possible to choose $J'$ such that the BPS numbers vanish. In general, however, such a vanishing chamber might not always exist, but since $J'$ is at the boundary of the K\\\"ahler cone, $f_{\\mu,J'}(\\tau)$ has no influence on the modular transformation properties, nor on the holomorphic anomaly. We write the full M5-brane elliptic genus as\n\\be\nZ^{(2)}_{P}(\\tau,z)=\\sum_{\\mu\\,\\in\\,\\Lambda^*\/\\Lambda}\\hat{f}^{(2)}_{\\mu,J}(\\tau)\\theta^{(2)}_{\\mu,J}(\\tau,z),\n\\ee\nwhere $\\hat{f}^{(2)}_{\\mu,J}$ denotes the modular completion as outlined above. We can show using Zwegers' results \\cite{Zwegers:2002}, that the M5-brane elliptic genus transforms like a Jacobi form of bi-weight $(-\\frac{3}{2},\\frac{1}{2})$. Again, we refer the reader to appendix \\ref{sec:mock} for further details.\n\n\\subsubsection{Proof of holomorphic anomaly at rank two}\\label{sec:holanomalyrantwo}\nNow, we are in position to prove the holomorphic anomaly at rank two for general surfaces $P$ with $b_2^+(P)=1$.  The holomorphic anomaly takes the following form\n\\be\\label{eq:provenholan}\n{\\cal D}_2 Z^{(2)}_{P}(\\tau,z)=\\tau_2^{-3\/2}\\frac{1}{16\\pi i}\\frac{J\\cdot[P]}{\\sqrt{-J^2}}\\left(Z^{(1)}_{P}(\\tau,z)\\right)^2,\n\\ee\nwhere the derivative ${\\cal D}_k$ is given as\n\\be\n{\\cal D}_k=\\p_{\\bar\\tau}+\\frac{i}{4\\pi k}\\p^2_{z_+},\n\\ee\nand $z_+$ refers to the projection of $z$ along a direction $J\\in{\\cal C}(P)$. For the proof, ${\\cal D}_2 Z^{(2)}_{P}$ can be computed explicitly. Using \\eqref{aniholomder} we obtain directly\n\\be\n{\\cal D}_2 Z^{(2)}_{P}(\\tau,z)=\\tau_2^{-3\/2}\\frac{1}{16\\pi i}\\frac{J\\cdot [P]}{\\sqrt{-J^2}}\\frac{\\vartheta_{\\Lambda^\\perp}(\\tau)^2}{\\eta(\\tau)^{2\\chi}}\\,\\sum_{\\mu\\,\\in\\,\\Lambda^*\/\\Lambda}(-1)^{4\\mu^2}\\theta^{(2)}_{\\mu-\\frac{[P]}{2},J}(\\tau,0)\\theta_{\\mu,J}^{(2)}(\\tau,z).\n\\ee\nSince the following identity among the theta-functions $\\theta_{\\mu,J}$ holds\n\\be\\label{theta-identity1}\n\\left(\\theta_{0,J}^{(1)}(\\tau,z)\\right)^2=\\sum_{\\mu\\,\\in\\,\\Lambda^*\/\\Lambda}(-1)^{4\\mu^2}\\theta^{(2)}_{\\mu-\\frac{[P]}{2},J}(\\tau,0)\\theta_{\\mu,J}^{(2)}(\\tau,z),\n\\ee\nwe have proven the holomorphic anomaly equation at rank two for general surfaces $P$.\n\n\n\n\n\n\n\\section{Applications and extensions}\nIn the following we want to apply the previous results to several selected examples. Before doing so, we explain two mathematical facts which will help to fix the ambiguity \n$f_{\\mu,J'}(\\tau)$, which are the blow-up formula and the vanishing lemma. After discussing the examples, we turn our attention to a possible extension to higher rank. This leads us to speculations about mock modularity of higher depth and wall-crossing having its origin in a meromorphic Jacobi form.\n\nFor the modular forms used in this section, we refer the reader to appendix \\ref{sec:mock}.\n\n\n\\subsection{Blow-up formulae and vanishing chambers}\nThere is a universal relation between the generating functions of stable sheaves on a surface $P$ and on its blow-up $\\tilde P$ \\cite{Vafa:1994tf, Yoshioka:1995, Yoshioka:1996, Li:1999, Gottsche:1999}. Let $P$ be a smooth projective surface and $\\pi:\\tilde{P}\\rightarrow P$ the blow-up at a non-singular point with $E$ the exceptional divisor of $\\pi$. Let $J\\in\\cC(P)$, $r$ and $\\mu$ such that gcd$(r,r\\mu\\cdot J)=1$. Then, the generating series $f^{(r)}_{\\mu,J}(\\tau;P)$ and $f^{(r)}_{\\mu,J}(\\tau;\\tilde P)$ are related by the blow-up formula\n\\be\\label{blowupformula}\nf^{(r)}_{\\pi^*(\\mu)-\\frac{k}{r} E,\\pi^*(J)}(\\tau;\\tilde P)=B_{r,k}(\\tau)f^{(r)}_{\\mu,J}(\\tau;P),\n\\ee\nwith $B_{r,k}$ given by\n\\be\nB_{r,k}(\\tau)=\\frac{1}{\\eta^r(\\tau)}\\sum_{a\\,\\in\\,\\IZ^{r-1}+\\frac{k}{r}}q^{\\sum_{i\\leq j}a_i a_j}.\n\\ee\n\nThe second fact states that for a class of semi-stable sheaves on certain surfaces the moduli space of the sheaves is empty. We refer to this fact as the vanishing lemma \\cite{Gottsche:1999}. For this let $P$ be a rational ruled surface $\\pi:P\\rightarrow\\IP^1$ and $J$ be the pullback of the class of a fiber of $\\pi$. Picking a Chern class $\\mu$ with $r\\mu\\cdot J$ odd, we have\n\\be\n{\\cal M}((r,\\mu,\\Delta),J)=\\emptyset\n\\ee\nfor all $d$ and $r\\geq 2$.\n\n\n\n\\subsection{Applications to surfaces with $b_2^+=1$}\\label{sec:surfaceswithb+=1}\nThe surfaces we are going to consider are $\\mathbb{P}^2$, the Hirzebruch surfaces $\\mathbb{F}_0$ and $\\mathbb{F}_1$, the del Pezzo surfaces $\\mathcal{B}_8$ and $\\mathcal{B}_9$.\n\n\n\\subsubsection*{\\it Projective plane $\\mathbb{P}^2$}\nThe projective plane $\\mathbb{P}^2$ has been discussed quite exhaustively in the literature. The rank one result was obtained by G\\\"ottsche \\cite{Gottsche:1990}\n\\be\nZ^{(1)}_{\\mathbb{P}^2}=\\frac{\\vartheta_1(-\\bar\\tau,-z)}{\\eta^{3}(\\tau)}.\n\\ee\nThe generating functions of the moduli space of rank two sheaves or $SO(3)$ instantons of Super-Yang-Mills theory on $\\mathbb{P}^2$ were written down by \\cite{Yoshioka:1994,Yoshioka:1995,Vafa:1994tf} and are given by\n\\be\\label{expansionforp2}\n\\begin{split}\nf_0(\\tau)&=\\sum_{n=0}^\\infty\\chi(\\cM((2,0,n),J))q^{n-\\frac{1}{4}}=\\frac{3h_0(\\tau)}{\\eta^6(\\tau)},\\\\\nf_1(\\tau)&=\\sum_{n=0}^\\infty\\chi(\\cM((2,1,n),J))q^{n-\\frac{1}{2}}=\\frac{3h_1(\\tau)}{\\eta^6(\\tau)}.\\\\\n\\end{split}\n\\ee\nHere, $h_j(\\tau)$ are mock modular forms given by summing over Hurwitz class numbers $H(n)$\n\\be\nh_j(\\tau)=\\sum_{n=0}^\\infty H(4n+3j)q^{n+\\frac{3j}{4}},\\qquad (j=0,1).\n\\ee\nTheir modular completion is denoted by $\\hat{h}_j(\\tau)$, where the shadows are given by $\\vartheta_{3-j}(2\\tau)$ \\cite{Zagier:1975}. Explicitly, we have\n\\be\n\\partial_{\\bar\\tau}\\hat{h}_j(\\tau)=\\frac{\\tau_2^{-\\frac{3}{2}}}{16\\pi i}\\vartheta_{3-j}(-2\\bar\\tau).\n\\ee\nNote, that these results are valid for all K\\\"ahler classes $J\\in H^2(\\IP^2,\\IZ)$ as there is no wall crossing in the K\\\"ahler moduli space of $\\IP^2$. This leads directly to the following elliptic genus of two M5-branes wrapping the $\\mathbb{P}^2$ divisor\n\\be\nZ^{(2)}_{\\mathbb{P}^2}(\\tau,z)=\\hat{f}_0(\\tau)\\vartheta_2(-2\\bar\\tau,-2z)-\\hat{f}_1(\\tau)\\vartheta_3(-2\\bar\\tau,-2z).\n\\ee\nDenoting by ${\\cal D}_2=\\p_{\\bar\\tau}+\\frac{i}{8\\pi}\\p_{z}^2$ one finds the expected holomorphic anomaly equation at rank two, given by\\footnote{This result has already been derived in \\cite{Bringmann:2010sd}.}\n\\be\n{\\cal D}_2\\, Z^{(2)}_{\\mathbb{P}^2}(\\tau,z)=-\\frac{3}{16\\pi i}\\tau_2^{-\\frac{3}{2}}\\left(Z^{(1)}_{\\mathbb{P}^2}(\\tau,z)\\right)^2,\n\\ee\nwhich can be derived directly from the simple fact that\n\\be\n\\vartheta_1(\\tau,z)^2=\\vartheta_2(2\\tau)\\vartheta_3(2\\tau,2z)-\\vartheta_3(2\\tau)\\vartheta_2(2\\tau,2z).\n\\ee\n\nFurther note, that the $q$-expansion of $f_0$, eq.~(\\ref{expansionforp2}), has non-integer coefficients. It was explained in \\cite{Manschot:2010xp} that this is due to the fact that the generating series involves the fractional BPS invariants $\\bar\\Omega(\\Gamma)$, which we encountered before.\n\n\\subsubsection*{\\it Hirzebruch surface $\\mathbb{F}_0$}\nOur next example is the Hirzebruch surface $P=\\mathbb{F}_0$. We denote by $F$ and $B$ the fiber and the base $\\IP^1$'s respectively. For an embedding into a Calabi-Yau manifold one may consult app.~\\ref{toricdata}. Let us choose $J=F+B$, $J'=B$ and Chern class $\\mu=F\/2$. The choice $\\mu=B\/2$ can be treated analogously and leads to the same results. The other sectors corresponding to $\\mu=0$ and $\\mu=(F+B)\/2$ require a knowledge of the holomorphic ambiguity at the boundary and will not be treated here. One obtains\n\\be\n\\begin{split}\nf^{(2)}_{\\mu,F+B}(\\tau)&=\\frac{1}{2\\eta^8(\\tau)}\\text{Coeff}_{2\\pi iy}(\\Theta^{F+B,B}_{\\Lambda,\\mu}(\\tau,[P] y))\\\\ &=q^{-\\frac{1}{3}}\\left(2q+22q^2+146q^3+742q^4+\\dots\\right),\n\\end{split}\n\\ee\nwhere we denote by $\\mu$ either $B\/2$ or $F\/2$. This exactly reproduces the numbers obtained in \\cite{Kool}.\n\nWe want to compute the shadow of the completion given by adding $\\Phi^{F+B}_{\\mu}$ and $\\Phi^{B}_{\\mu}$ to the indefinite theta-series $\\Theta^{F+B,B}_{\\Lambda,\\mu}$. Since $B$ is chosen at the boundary, $\\Phi^{B}_{\\mu}$ vanishes for $\\mu=F\/2,B\/2$. The only relevant contribution has a shadow proportional to $\\vartheta_2(\\tau)$. Precisely, we obtain\n\\be\n\\partial_{\\bar\\tau}f_{\\mu,F+B}^{(2)}(\\tau)=-\\tau_2^{-3\/2}\\frac{1}{4\\pi i\\sqrt{2}}\\,\\frac{\\overline{\\vartheta_2(\\tau)}\\vartheta_2(\\tau)}{\\eta^8(\\tau)}\\qquad (\\mu=\\frac{F}{2},\\frac{B}{2}).\n\\ee\n\n\n\n\n\n\\subsubsection*{\\it Hirzebruch surface $\\mathbb{F}_1$}\nThe next example is the Hirzebruch surface $\\mathbb{F}_1$, which is a blow-up of $\\mathbb{P}^2$. Again we denote by $F$ and $B$ the fiber and base $\\IP^1$'s. The $\\mathbb{P}^2$ hyperplane is given by the pullback of $F+B$ and $B$ is the exceptional divisor. This example is particularly nice, since we can check our results against the blow-up formula \\eqref{blowupformula} or use the results known from $\\mathbb{P}^2$ to write generating functions in sectors which are not accessible through the vanishing lemma. Notice, that the holomorphic expansions have been already discussed in ref.~\\cite{Manschot:2010nc}. From the general discussion one sees that there are four different choices for the Chern class $\\mu\\in\\{\\frac{B}{2},\\frac{F+B}{2},\\frac{F}{2},0 \\}$.\n\nFirst, we choose $J=F+B$, $J'=F$ and Chern class $\\mu=B\/2$. We then obtain\n\\be\\label{eqF+BF1}\n\\begin{split}\nf^{(2)}_{\\mu,F+B}(\\tau)&=\\frac{1}{2\\eta^8(\\tau)}\\text{Coeff}_{2\\pi iy}(\\Theta^{F+B,F}_{\\Lambda,B}(\\tau,[P] y))\\\\ &=q^{-\\frac{1}{12}}\\left(-\\frac{1}{2} - q + \\frac{15}{2} q^2 + 91 q^3 + 558 q^4+\\dots\\right).\n\\end{split}\n\\ee\nA check of this result against the blow-up formula \\eqref{blowupformula} applied to $\\mathbb{P}^2$ yields\n\\be\n\\frac{3h_0(\\tau)}{\\eta^6(\\tau)}\\frac{\\vartheta_2(2\\tau)}{\\eta^2(\\tau)}=q^{-\\frac{1}{12}}\\left(-\\frac{1}{2} - q + \\frac{15}{2} q^2 + 91 q^3 + 558 q^4+\\dots\\right)=f^{(2)}_{\\mu,F+B}(\\tau).\n\\ee\nFurther, we calculate the shadow by differentiating $\\hat{f}^{(2)}$ with respect to $\\bar\\tau$\n\\be\n\\partial_{\\bar\\tau}\\hat{f}^{(2)}_{\\mu,F+B}(\\tau)=\\frac{3}{16\\pi i}\\tau_2^{-3\/2}\\frac{\\overline{\\vartheta_3(2\\tau)}\\vartheta_2(2\\tau)}{\\eta^8(\\tau)},\n\\ee\nwhich also is in accord with the blow-up formula. Note, that \\eqref{eqF+BF1} has half-integer expansion coefficients, since $J=B+F$ lies on a wall for the Chern class $\\mu=B\/2$.\\\\\n\nAs a second case we choose $J=F+B$, $J'=F$ and Chern class $\\mu=(F+B)\/2$ and obtain\n\\be\n\\begin{split}\nf^{(2)}_{\\mu,F+B}(\\tau)&=\\frac{1}{2\\eta^8(\\tau)}\\text{Coeff}_{2\\pi iy}(\\Theta^{F+B,F}_{\\Lambda,F+B}(\\tau,[P]y))\\\\ &=q^{-\\frac{7}{12}}\\left(q + 13 q^2 + 93 q^3 + 496 q^4+\\dots\\right),\n\\end{split}\n\\ee\nwhich we again can check against the blow-up formula \\eqref{blowupformula} for $\\IP^2$\n\\be\n\\frac{3h_1(\\tau)}{\\eta^6(\\tau)}\\frac{\\vartheta_3(2\\tau)}{\\eta^2(\\tau)}=q^{-\\frac{7}{12}}\\left(q + 13 q^2 + 93 q^3 + 496 q^4+\\dots\\right)=f^{(2)}_{\\mu,F+B}(\\tau).\n\\ee\nCalculating the shadow yields\n\\be\n\\partial_{\\bar\\tau}\\hat{f}^{(2)}_{\\mu,F+B}(\\tau)=\\frac{3}{16\\pi i}\\tau_2^{-3\/2}\\frac{\\overline{\\vartheta_2(2\\tau)}\\vartheta_3(2\\tau)}{\\eta^8(\\tau)},\n\\ee\nwhich is also in accord with the blow-up formula.\\\\\n\nThe last two sectors $\\mu=F\/2,0$ are not accessible via the vanishing lemma. However, using a blow-down to $\\IP^2$ we observe, that the above two cases reproduce correctly the two Chern classes in the cases of rank two sheaves on $\\IP^2$. Using the blow-up formulas once more we finally arrive at\n\\be\n\\begin{split}\nf^{(2)}_{(0,0),J}(\\tau)&=\\frac{3h_0(\\tau)}{\\eta^6(\\tau)}\\frac{\\vartheta_3(2\\tau)}{\\eta^2(\\tau)},\\\\\nf^{(2)}_{(\\frac{1}{2},0),J}(\\tau)&=\\frac{3h_1(\\tau)}{\\eta^6(\\tau)}\\frac{\\vartheta_2(2\\tau)}{\\eta^2(\\tau)},\\\\\nf^{(2)}_{(0,\\frac{1}{2}),J}(\\tau)&=\\frac{3h_0(\\tau)}{\\eta^6(\\tau)}\\frac{\\vartheta_2(2\\tau)}{\\eta^2(\\tau)},\\\\\nf^{(2)}_{(\\frac{1}{2},\\frac{1}{2}),J}(\\tau)&=\\frac{3h_1(\\tau)}{\\eta^6(\\tau)}\\frac{\\vartheta_3(2\\tau)}{\\eta^2(\\tau)},\n\\end{split}\n\\ee\nwhere $J=F+B$ and $\\mu=(a,b)=aF+bB$. Note, that in the cases $f^{(2)}_{(0,0),J}$ and $f^{(2)}_{(0,\\frac{1}{2}),J}$ the blow-up formula is not valid since we violate the gcd-condition, as $\\pi_{*} \\mu=0$ in these cases. However, for rank two sheaves on ${\\mathbb F}_1$ the blow-up formula seems to work anyway, since the generating series using the blow-up procedure and the indefinite theta-function description coincide for the Chern class $\\mu=(0,\\frac{1}{2})$.\n\n\n\n\\subsubsection*{\\it Del Pezzo surface $\\mathcal{B}_8$}\nAs in \\cite{Gaiotto:2006wm} we embed the surface $\\mathcal{B}_8$ in a certain free $\\IZ_5$ quotient\\footnote{The only freely acting group actions for the quintic are a $\\IZ_5^2$ and the above $\\IZ_5$.} of the Fermat quintic $\\tilde{X}=\\{\\sum_{i=1}^5 x_i^5=0\\}$ in $\\mathbb{P}^4$. The action of the group $G=\\IZ_5$ on the projective coordinates of the ambient space is given by $x_i \\sim \\omega^i x_i $, where $\\omega=e^{2\\pi i \/5}$. For the hyperplane section, denoted $P$, we observe that $P^3=1$, as for the Fermat quintic the five points of intersection of three hyperplanes $\\{x_i=x_j=x_k=0\\}$ are identified under the action of the group $G$. The Euler character of the hyperplane is given by $\\chi(P) = 11$. It can be shown that the divisor $P$ is rigid and has $b_2^+=1$. We observe that  $H^2(P,\\IZ) = \\IZ \\oplus (-E_8)$ as is explained in \\cite{Gaiotto:2006wm}. The elliptic genus of a single M5-brane is then fixed by the modular weights\n\\ben\nZ^{(1)}_{P}(\\tau,z)= \\frac{E_4(\\tau)}{\\eta^{11}(\\tau)}\\,\\vartheta_1(-\\overline{\\tau},-z).\n\\een\nThe form of $Z^{(2)}_{P}$ can now be calculated as for $\\IP^2$ and is given by\n\\ben\nZ^{(2)}_{P}(\\tau,z) \\sim \\frac{E_4(\\tau)^2}{\\eta(\\tau)^{22}}\\,(\\hat{h}_0(\\tau)\\,{\\vartheta}_2(-2\\bar\\tau,-2z)- \\hat{h}_1(\\tau)\\,{\\vartheta}_3(-2\\bar\\tau,-2z)).\n\\een\nThe holomorphic anomaly equation fulfilled by $Z^{(2)}_{P}(\\tau,z)$ can be obtained as in the $\\IP^2$ case\n\\ben\n{\\cal D}_2\\, Z^{(2)}_{P}(\\tau,z)\\sim\\frac{\\tau_2^{-\\frac{3}{2}}}{16\\pi i}\\left(Z^{(1)}_{P}(\\tau,z)\\right)^2.\n\\een\n\n\n\\subsubsection*{\\it Del Pezzo surface $\\mathcal{B}_9$, the $\\frac{1}{2}$K3}\nWe end our examples by returning and commenting on $\\frac{1}{2}$K3 or $\\mathcal{B}_9$ which was the example of section (\\ref{n4ym}), as M5-branes wrapping on it give rise to the multiple E-strings. The  $\\mathcal{B}_9$ surface can be understood as a $\\IP^2$ blown up at nine points (see appendix \\ref{geometry} for details) or a rational elliptic surface. This case is interesting as one can map via T-duality along the elliptic fibration the computation of the modified elliptic genus to the computation of the partition function of topological string theory on the same surface \\cite{Minahan:1998vr}. The middle dimensional cohomology lattice of $\\mathcal{B}_9$ is given by $H^2(\\mathcal{B}_9,\\IZ)=\\Gamma^{1,1}\\oplus E_8$ and the Euler number can be computed to $\\chi(\\mathcal{B}_9)=12$. Modularity then fixes the form of the elliptic genus at rank one to\n\\ben\nZ^{(1)}_{\\mathcal{B}_9}(\\tau,z)=\\frac{E_4(\\tau)}{\\eta(\\tau)^{12}}\\,\\theta^{(1)}_{0,J}(\\tau,z),\n\\een\nwhere $\\theta^{(1)}_{0,J}(\\tau,z)$ is the theta-function associated to the lattice $\\Gamma^{1,1}$ with standard intersection form\n\\be\n(-d_{AB})=\\begin{pmatrix}\n          0&1\\\\ 1&0\n         \\end{pmatrix}\n.\n\\ee\nChoosing the K\\\"ahler form $J=(R^{-2},1)^T$, where $(1,0)^T$ is the class of the elliptic fiber, one can show that\n\\be\n\\theta^{(1)}_{0,J}(\\tau,0)\\rightarrow \\frac{R}{\\sqrt{\\tau_2}}\\quad \\text{as}\\quad R\\rightarrow\\infty.\n\\ee\nIn this limit of small elliptic fiber one recovers the results of sect.~\\ref{n4ym}. The factor $E_4(\\tau)$ is precisely the theta-function of the $E_8$ lattice. The results obtained from the anomaly for higher wrappings of refs.~\\cite{Minahan:1997ct,Minahan:1998vr} were proven mathematically for double wrapping in ref.~\\cite{Yoshioka:1999}. In this analysis the Weyl group of the $E_8$ lattice was used to perform the theta-function decomposition.\n\n\n\\subsection{Extensions to higher rank and speculations}\nIn the following sections we want to discuss the extension of our results to higher rank. Partial results for rank three can be found already in the literature \\cite{Kool,Manschot:2010nc,Weist:2009,Yoshioka:2008,Klyachko}. Thereafter, we discuss a possible generalization of mock modularity and speculate about a contour description which stems from a relation to a meromorphic Jacobi form.\n\n\n\n\\subsubsection{Higher rank anomaly and mock modularity of higher depth}\nWe want to focus on the holomorphic anomaly equation at general rank as conjectured in \\cite{Minahan:1998vr}. We recall that its form is given by\n\\be\\label{eq:anomalygeneral}\n{\\cal D}_r Z^{(r)}_{P} (\\tau, z) \\sim  \\sum_{n=1}^{r-1} n(r-n) Z^{(n)}_{P}(\\tau, z) Z^{(r-n)}_{P}(\\tau, z),\n\\ee\nwhere $Z^{(r)}_{P}(\\tau, z)$ can be decomposed into Siegel-Narain theta-functions as described in section \\ref{sec:mswcft}. One may thus ask the question what it implies for the functions $\\hat f^{(r)}_{\\mu,J} (\\tau)$ for general $r$. In order to extract this information we want to compare the coefficients in the theta-decomposition on both sides of (\\ref{eq:anomalygeneral}). For this we need a generalization of the identity (\\ref{theta-identity1}). A computation shows that\n\\be\n\\theta^{(n)}_{\\nu,J}(\\tau,z)\\,\\theta^{(r-n)}_{\\lambda,J}(\\tau,z)=\\sum_{\\mu\\,\\in\\,\\Lambda^*\/\\Lambda} c^\\mu_{\\nu\\lambda}(\\tau)\\, \\theta^{(r)}_{\\mu,J}(\\tau,z),\n\\ee\nwhere $c^\\mu_{\\nu\\lambda}$ are Siegel-Narain theta-functions themselves given by\n\\be\nc^\\mu_{\\nu\\lambda}(\\tau)=\\delta_g(\\mu)\\sum_{\\xi\\,\\in\\,\\Lambda+\\mu+\\frac{g}{r}(\\nu-\\lambda)}\\bar q^{-\\frac{rn(r-n)}{2g^2}\\xi_+^2}q^{\\frac{rn(r-n)}{2g^2}\\xi_-^2}\n\\ee\nwith $g=\\text{gcd}(n,r-n)$ and $\\delta_g(\\mu)$ yields one if $r\\mu$ is divisible by $g$ and vanishes otherwise. With this input one finds\n\\be\\label{eq:holanomalyfmu}\n\\partial_{\\bar\\tau} \\hat{f}^{(r)}_{\\mu,J} (\\tau)  \\sim \\sum_{n=1}^{r-1}n(r-n)\\sum_{\\nu,\\lambda\\,\\in\\Lambda^*\/\\Lambda} \\hat{f}^{(n)}_{\\nu,J}(\\tau) \\hat{f}^{(r-n)}_{\\lambda,J}(\\tau) c^\\mu_{\\nu\\lambda}(\\tau),\n\\ee\nwhich sheds some light into the question about the modular properties of generating functions at higher rank as follows.\n\n\n\nThe structure of eq.~(\\ref{eq:holanomalyfmu}) indicates, that an appropriate description of the generating function $\\hat{f}^{(r)}_{\\mu,J}$ needs a generalization of the usual notion of mock modularity. This results from the fact, that on the right hand side of the anomaly equation (\\ref{eq:holanomalyfmu}), mock modular forms appear, such that the shadow of $\\hat{f}^{(r)}_{\\mu,J}$ is a mock modular form itself. \nTherefore, it is also subject to a holomorphic anomaly equation. This would lead to the notion of mock modularity of higher depth \\cite{Zwegers:higherdepth}, similar to the case of \nalmost holomorphic modular forms of higher depth. These are functions like $\\widehat{E}_2(\\tau)$ and powers thereof, which can be written as a polynomial in $\\tau_2^{-1}$ with coefficients \nbeing holomorphic functions.\n\nA further motivation for this comes from the observation that the generating functions $\\hat{f}^{(r)}_{\\mu,J}$ could be obtained from an indefinite theta-function as in the case of two M5-branes. The lattice, however, that is summed over in these higher rank indefinite theta-functions will be of higher signature. In the case of $r$ M5-branes one would expect a signature $(r-1, (r-1)(r(\\Lambda)-1))$ due to the $r-1$ relative D2-brane charges of the possible $r$ decay products of D4-D2-D0 bound-states \\cite{Manschot:2009ia,Manschot:2010xp}. However, a complete discussion of the modular properties of such functions and their relation to mock modular forms of depth is beyond the scope of this work. We would like to come back to this question in future research.\n\n\n\n\\subsubsection{The contour description}\n\\label{sec:contour}\nThe elliptic genus of $r$ M5-branes wrapping $P$ is denoted by $Z^{(r)}_{P}(\\tau,z)$, where we don't indicate any dependence of $Z^{(r)}_{P}$ on a K\\\"ahler class\/ chamber $J\\in\\cC(P)$. The basic assumption is that the elliptic genus does not depend on such a choice. We simply think about $Z^{(r)}_{P}$ as being a \\textit{meromorphic} Jacobi form, which has poles as a function of the elliptic variable $z$. We assume, that it is of bi-weight $(-\\frac{3}{2},\\frac{1}{2})$. In the following we want to exploit the implications of this statement.\n\nIt is known that a Jacobi form has an expansion into theta-functions with coefficients being modular forms. Since Zwegers \\cite{Zwegers:2002}, we also know that a meromorphic Jacobi form with one elliptic variable has a similar expansion, where the coefficients are mock modular. Using our Siegel-Narain theta-function $\\theta_{\\mu,J}^{(r)}(\\tau,z)$, eq.~(\\ref{SNTheta}), we conjecture the following expansion\n\\be\\label{decompositionZ}\nZ^{(r)}_{P}(\\tau,z)=\\sum_{\\mu\\,\\in\\,\\Lambda^*\/\\Lambda} f^{(r)}_{\\mu,J}(\\tau)\\theta_{\\mu,J}^{(r)}(\\tau,z)+\\text{Res},\n\\ee\nwith $J$ a point in the K\\\"ahler cone which is related to a point $z_J\\in\\Lambda_\\IC$ where the decomposition is carried out. Note, that in eq.~(\\ref{decompositionZ}) the term ``Res'' should be given as a finite sum over the residues of $Z^{(r)}_{P}(\\tau,z)$ in the fundamental domain $z_J+e\\tau+e$ with $e=[0,1]^{r(\\Lambda)}$.\n\nLet's see how the dependence on $J$ comes about. Doing a Fourier transform we can write\n\\be\nf^{(r)}_{\\mu,J}(\\tau)=(-1)^{r\\mu\\cdot[P]}\\bar{q}^{\\frac{r}{2}\\mu_+^2}q^{-\\frac{r}{2}\\mu_-^2}\\int_{{\\cal C}_J}Z^{(r)}_{P}(\\tau,z)e^{-2\\pi ir(\\mu+\\frac{[P]}{2})\\cdot z}dz,\n\\ee\nwhere ${\\cal C}_J$ is a contour which has to be specified since $Z^{(r)}_{P}$ is meromorphic. Due to the periodicity in the elliptic variable ${\\cal C}_J$ can be given as $z_J+e$ for some point $z_J$. Now, suppose we have a parallelogram ${\\cal P}=z_J+ez_{J'}+e$ and that there is a single pole of $Z^{(r)}_{P}$ inside $\\cal P$, say at $z=z_0$. Then, we obtain by integrating over the boundary of $\\cal P$\n\\be\\label{eq:jumpingresidues}\nf^{(r)}_{\\mu,J}(\\tau)-f^{(r)}_{\\mu,J'}(\\tau)=2\\pi i\\,\\alpha_\\mu(\\tau)\\,\\underset{z=z_0}{\\text{Res}}\\left(Z^{(r)}_{P}(\\tau,z)e^{-2\\pi ir(\\mu+\\frac{[P]}{2})\\cdot z}\\right),\n\\ee\nwhere we abbreviate\n\\be\n\\alpha_\\mu(\\tau)=(-1)^{r\\mu\\cdot[P]}\\bar{q}^{\\frac{r}{2}\\mu_+^2}q^{-\\frac{r}{2}\\mu_-^2}.\n\\ee\nThat is, the coefficients of the Laurent expansion of the elliptic genus encode the jumping of the BPS numbers across walls of marginal stability and the walls are in one-to-one correspondence with the positions of the poles of $Z^{(r)}_{P}$. An analogous dependence on a contour of integration for wall-crossing of ${\\cal N}=4$ dyons was introduced in refs.~\\cite{Sen:2007vb,Cheng:2007ch}.\n\nMoreover, the shadow of $f^{(r)}_{\\mu,J}$ should be determined in terms of the residues of $Z^{(r)}_{P}$, since a generalizations of the ideas of \\cite{Zwegers:2002} should show, that it is contained in the factor ``Res'' of eq.~(\\ref{decompositionZ}). Thus, combining this result with the interpretation of eq.~(\\ref{eq:jumpingresidues}) one expects, that the shadow not only renders $f^{(r)}_{\\mu,J}$ modular, but also encodes the decay of bound-states and hence knows about the jumping of BPS invariants across walls of marginal stability.\n\nIt is tempting to speculate even further. When comparing our results to the case of dyon state counting in ${\\cal N}=4$ theories \\cite{Dabholkar:2007zz,Dabholkar:2010} \none might suspect that there is an analog of the Igusa cusp form $\\phi_{10}$ in our setup. In \nthe ${\\cal N}=4$ dyon case there are meromorphic Jacobi forms, often denoted $\\psi_m$, which are summed up \nto give $\\phi_{10}$. In analogy, it may be useful to introduce another parameter $\\rho\\in{\\cal H}$ \nand to study the object\n\\be\n\\phi_{P}^{-1}(\\tau,\\rho,z)=\\sum_{r\\geq1}Z^{(r)}_{P}(\\tau,z)e^{2\\pi ir\\rho}.\n\\ee\n\n\n\n\\section{Conclusions}\n\nIn this paper we investigated background dependence of theories that originate from  \n$r$ M5-branes wrapping a smooth (semi-)rigid divisor $P$ in a Calabi-Yau three-fold background.  \nSuch divisors $P$ have $b_2^+=1$ and (semi-)positive anti-canonical class. \nIn this case the wrapped M5-brane can be studied locally in the Calabi-Yau manifold\nusing an effective description of the M5-brane theory on $P\\times T^2$ \nby a twisted $U(r)$ ${\\cal N}=4$ Super-Yang-Mills theory on $P$. \n\nThe main object of interest was the partition function $Z_{P}^{(r)}$ of \nthe twisted gauge theory and its modular and holomorphic properties. \nThis partition function can be related to the modified elliptic genus \nof the ${\\cal N}=(0,4)$ sigma model description of \nthe M5-brane. Using the spectral flow symmetry one \nestablishes for all $r$ a decomposition of the partition function \ninto vector-valued modular forms $\\hat f^{(r)}_{\\mu, J}(\\tau)$ \nw.r.t.~the $S$-duality group of ${\\cal N}=4$ Super-Yang-Mills \nand Siegel-Narain theta-functions $\\theta^{(r)}_{\\mu,J}(\\tau,z)$. \n\nOur main result is a rigorous proof of a holomorphic anomaly equation \nof the partition function valid for rank two on all $P$ described above. \nThe proof in section \\ref{sec:holanomalyrantwo} relies on the large \nvolume wall-crossing formula of G\\\"ottsche~\\cite{Gottsche:1999} for \ninvariants associated to sheaves on $P$, which are \nrelated to integer BPS invariants. By summing the change of the invariants  across \nall intermediate walls one can express the difference of the \ngenerating function of the invariants $f^{(r)}_{\\mu, J}(\\tau)$  in \ntwo arbitrary chambers $J$ and $J'$ in the K\\\"ahler cone in terms of \nan indefinite theta-function $\\Theta^{J,J'}_{\\Lambda,\\mu}(\\tau,z)$~\\cite{Gottsche:1998}. \nThis theta-function is regularized by cutting out the negative \ndirections of the quadratic form on the homology lattice, \na procedure which renders the result in general not modular. \nThe spoiled $S$-duality invariance can be regained following \nthe work of Zwegers by smoothing out the cutting procedure with \nthe non-holomorphic error function. The non-holomorphicity introduced \nby this procedure completes the mock modular forms $f^{(r)}_{\\mu, J}(\\tau)$ \nto non-holomorphic modular forms $\\hat f^{(r)}_{\\mu, J}(\\tau)$. \nThe non-holomorphicity of the Siegel-Narain theta-functions on the other \nis trivial since it is annihilated by the  non-holomorphic \nheat operator. This allows to write a concise holomorphic \nanomaly equation for the partition function (\\ref{eq:provenholan}).\n\n\nWe check this holomorphic anomaly equation and its implications \nfor the counting of invariants of sheaves on $\\mathbb{P}^2$, \n$\\mathbb{F}_0$, $\\mathbb{F}_1$ and ${\\cal B}_8$ in section~\\ref{sec:surfaceswithb+=1}.\nThe anomaly equation (\\ref{eq:provenholan}) is in particular \ncompatible with the form of a holomorphic anomaly that has been \nconjectured in the context of E-strings on $\\frac{1}{2}$K3 for all $r$ \nand checked for certain classes using the duality to the genus zero topological \nstring partition function~\\cite{Minahan:1998vr}. Since the \nnon-holomorphicity of the $\\hat f^{(r)}_{\\mu, J}(\\tau)$ for $r>1$ is \nrelated in an intriguing way to mock modularity and wall-crossing, \nwe analyzed the decomposition for arbitrary rank and give a general \nform of the conjectured general anomaly equation at the level of the \n$\\hat f^{(r)}_{\\mu, J}(\\tau)$ in equation (\\ref{eq:holanomalyfmu}), which \nindicates a theory of mock modular forms of higher depth \\cite{Zwegers:higherdepth}.\nThe holomorphic limit of the $\\hat f^{(r)}_{\\mu, J}(\\tau)$ \nyield generating functions for invariants associated to sheaves \nof rank $r$. However, it is in general difficult to provide\nboundary conditions, which fix the holomorphic ambiguity.\n\n\nThe wall-crossing of G\\\"ottsche, which induces in the steps \ndescribed above the non-holomorphicity of the $\\hat f^{(2)}_{\\mu, J}(\\tau)$, \ncan be rederived using the Kontsevich-Soibelman wall-crossing formula, \nas we did in section \\ref{sec:KSGwallcrossing}. As the wall-crossing formula takes a primitive form at rank two, one can rewrite the generating function of BPS differences in terms of an indefinite theta-function. \nThe Kontsevich-Soibelman formula can be used for arbitrary rank to determine \nthe counting functions $f^{(r)}_{\\mu, J}(\\tau)$ for all sectors \n$\\mu$ in all chambers, if it is known in one chamber for \nall $\\mu$, e.g.~by a vanishing lemma or use of the \nblow-up formula. This was studied for rank 3 by~\\cite{Manschot:2010nc}, \nwhere it was also shown that the rank three wall-crossing formula \nis primitive. In general if the wall-crossing formula is primitive,\nthe sum over walls induce lattice sums of \nsignature $(r-1)(b_2^+,b_2^-)$ with similar regularization \nrequirements as for the rank two case. It is an interesting \nquestion if the program of Zwegers to build modular objects \ncan be extended to the higher rank situation and leads upon \nnon-holomorphic modular completion to the conjectured form of \nholomorphic anomaly equation and a precise notion of the\nmock modular forms of higher depth. \n\nThe problem of providing boundary conditions at least \nin one chamber for the del Pezzo surfaces (except for \nthe Hirzebruch surface $\\mathbbm{F}_0$) can in principle be solved by \nusing the blow formula in both directions in connection \nwith the wall-crossing formula before and after the \nblow-up. However, the blow-up formula in the literature \napply only if $r$ and $c_1\\cdot J$ have no common \ndivisor. This restriction forbids in general to \nprovide boundary conditions for all sectors.\n\n\nThe higher genus information discussed in equation \n(\\ref{eq:highergenus}) gives finer information \nabout the cohomology of moduli spaces of sheaves \nthan its Euler number. Namely, an elliptic genus \nobtained by tracing over the right $j^3_R$ quantum \nnumbers of the Lefshetz decomposition in the cohomology \nof the moduli space. On rigid surfaces it can be further \nrefined to include the general $\\Omega$ background parameters \nof Nekrasov~\\cite{Nekrasov:2002qd}, which capture the individual $(j^2_L,j^3_R)$ \nquantum numbers~\\cite{Iqbal:2007ii}. For rank two such refined \npartition functions have been considered in \\cite{Goettsche:2007}\nand it should be possible to extend the consideration \nabove to the refined BPS numbers. Furthermore, the relation between D6-D2-D0 brane systems as counted by topological string theory and D4-D2-D0 brane systems associated to black hole state counting is the hallmark of the OSV conjecture \\cite{Ooguri:2004zv}, which has been intensively studied. Wall-crossing issues in combination with this conjecture have been studied in ref.~\\cite{Denef:2007vg}  and more recently from an M-theory perspective for example in ref.~\\cite{Aganagic:2009kf}. It would be interesting to examine the implications of the anomaly equation in these contexts.\n\n\n\nA conceptually very interesting but at this point more speculative \napproach is to consider the elliptic genus as a $J$ independent \nmeromorphic Jacobi form, as we did in section~\\ref{sec:contour}. As shown by Zwegers such meromorphic Jacobi \nforms have an expansion in theta-functions whose coefficients \nare mock modular forms, just as holomorphic Jacobi forms have \nan expansion in theta-functions with holomorphic modular forms as\ncoefficients. This formalism relates the changes in the BPS numbers \nacross walls of marginal stability to the different choices of the contour in the \ndefinition of $f^{(r)}_{\\mu, J}(\\tau)$ as a Fourier integral of \n$Z_{P}^{(r)}$, i.e.~to the poles in $Z_{P}^{(r)}$, like in the ${\\cal N}=4$ case \\cite{David:2006ru, Cheng:2007ch}. \n\n\n\n\n\n\n\n\\acknowledgments\nWe would like to thank Frederik Denef, Lothar G\\\"ottsche, Jan Manschot, Sameer Murthy, Cumrun Vafa, Stefan Vandoren, Bernard De Wit, Xi Yin and Don Zagier for valuable discussions and comments. We would also like to thank the organizers of the workshop ``Interfaces and wall-crossing'', 2009 at the Arnold Sommerfeld Center in Munich and the organizers of the workshop on ``Automorphic forms, Kac-Moody algebras and strings'' at the Max Planck Institute of Mathematics in Bonn for stimulating research on these topics. M.A.~and M.H.~are grateful to Peter Mayr for proposing the connection of E-strings and the M5-brane partition function and for many discussions. M.A.~and M.H.~would furthermore like to thank Jean Dominique L\\\"ange for joint work on related projects.  M.A.~is supported by the Hausdorff Center for Mathematics and DFG fellowship\nAL 1407\/1-1. The work of M.H.~is supported by the program ``Origin and Structure of the Universe''. The work of M.R.~and T.W.~is supported by the program ``Bonn-Cologne Graduate School of Physics and Astronomy\".\n\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\n\tSince the beginning of computing, the computing paradigm itself, \"\\textit{the implicit hardware\/software  contract}~\\cite{AsanovicParallelCACM:2009}\", defined how mathematics-based theory and its science-based implementation must cooperate.\n\tComputing uses the general operating model, that data is delivered to the input of the processing element, and after some processing time, the resulting data is delivered to some store.  Mathematics, however, considers only the \\textit{logical dependencies} \n\tbetween its operands. It assumes that\n\tthe needed operands are instantly available, including that the result becomes an operand for the next operation immediately,\n\tonly after performing the actual operation. \n\t\n\tThat is, computing science considers that performing operations, delivering operands\n\tto and from processing units, is as kind of engineering imperfectness.\n\tAt the time when von Neumann proposed his famous abstraction\\footnote{He proposed a structure (functional grouping) of the \"computing organs\", and NO architecture}, both time of processing\n\tand time of accessing data (including those on a mass storage devices) were in the millisecond region,\n\twhile the physical data transfer time\n\twas in the range of microseconds, i.e., three orders of magnitude smaller.\n\t\\textit{It was a plausible assumption to consider that the total time of processing\n\t\tcomprises only the time of performing the computation plus the time of accessing the required data\n\t\tby the operating unit; the physical data delivery time was neglected}.\n\t\t\n\tFor today, however, the technical development changed the relations between those timings to their exact opposite. Today the data transfer time (and their access time) is much larger than the time needed to process them.\n\tBesides, the relative weight of the data transfer time has grown tremendously because of several reasons.\n\tFirstly, miniaturizing the processors to sub-micron size, while keeping the rest of the needed components (such as buses) above the centimeter scale. Secondly, the single-processor performance stalled~\\cite{GameOverYelick:2011}, because they reached the limits,\n\tthe laws of nature enable~\\cite{LimitsOfLimits2014}. Thirdly, making truly parallel computers failed~\\cite{AsanovicParallelCACM:2009}, and that goal had to be replaced with \\textit{parallelized sequential computing},\n\tdisregarding that the operating rules of that different kind of computing~\\cite{ScalingParallel:1993}\\textbf{\\cite{VeghReevaluate:2020,VeghHowMany:2020}} sharply differ from those of the segregated processors.\n\tFourthly, the mode of utilization (mainly multitasking) forced out using \\gls{OS}s, which imitate a \"new processor\" for a new task, at serious time expenses.\n\tFinally, the idea of \"real-time connected everything\"  introduced geographically vast distances with the corresponding several millisecond data delivery times and assumes that \"everything is in cache\", which represents a drastically different memory subsystem.\n\t\n\tThe theory of computing kept the idea of \"instant delivery\"; although even within the core, wiring has an increasing role~\\cite{LimitsOfLimits2014}. Even, since the technology broke through the $0.2~\\mu m$ technology bound, the wiring delay dominates the apparent performance~\\cite{WiringDominance:2019} (over the gate delay).  The idea of non-temporal behavior, however, was confirmed by accepting \"weak scaling\"~\\cite{Gustafson:1988}, \n\tsuggesting that \\textit{all housekeeping times, such as organizing the joint work of the parallelized serial processors, sharing resources, using exceptions and \\gls{OS} services, delivering data between processing units and data storage units, even through slow networks and satellite relay stations, are negligible}.\n\t\n\t\n\tVast computing systems can cope with their tasks with growing difficulty, enormously decreasing computing efficiency, enormously growing energy consumption and heat production.\n\tBeing not aware of that the collaboration between processors \n\tneeds a different approach (another paradigm),\n\tresulted \n\tin demonstrative failures already known (such as the supercomputers Gyoukou and Aurora'18, or the brain simulator SpiNNaker)\\footnote{The explanations are quite different: Gyoukou was withdrawn after its first appearance; Aurora failed: retargeted and delayed; Despite the failure of SpiNNaker1, the SpiNNaker2 is also under construction~\\cite{SpiNNaker2:2018};\n\t\t\"Chinese decision-makers decided to withhold the country's newest Shuguang supercomputers even though they operate more than 50 percent faster than the best current US machines\".} and many more  may  follow:  such as Aurora'21~\\cite{DOEAurora:2017},\n\tthe China mystic supercomputers\\footnote{https:\/\/www.scmp.com\/tech\/policy\/article\/3015997\/china-has-decided-not-fan-flames-super-computing-rivalry-amid-us} and\n\tthe EU planned supercomputers\\footnote{https:\/\/ec.europa.eu\/newsroom\/dae\/document.cfm? doc\\_id =60156}. \n\tEven the successful supercomputers do not develop, and the new (as of June, 2020) supercomputer $Fugaku$ stalled at \n\t40\\% of its planned capacity.\n\t\n\tGeneral-purpose computing systems comprising \"only\" millions of processors already show the issues, and brain-like systems want to comprise at least four orders of magnitude higher number of computing elements. When targeting neuromorphic features such as \"deep learning training\", the issues start to manifest at just a couple of dozens of processors~\\cite{DeepNeuralNetworkTraining:2016}\\textbf{\\cite{VeghAIperformance:2020}}. The scaling is nonlinear~\\textbf{\\cite{VeghReevaluate:2020,VeghScalingANN:2020}}, strongly depending on the workload type, and the \\gls{AI}-class workload is one of the worst workloads~\\textbf{\\cite{VeghAIperformance:2020,VeghScalingANN:2020}} one can run on conventional architectures.\\footnote{ https:\/\/www.nextplatform.com\/2019\/10\/30\/cray-revamps-clusterstor-for-the-exascale-era\/ :\n\t\t\\textit{artificial intelligence, \\dots it's the most disruptive workload from an I\/O pattern perspective}} \n\t\n\t\n\t\n\t\"\\textit{Successfully addressing these challenges [of neuromorphic computing] will lead to a new class of computers and systems architectures}\"~\\cite{NeuromorphicComputing:2015}. However,\n\tthe roundtable concentrated \\textit{only} on finding new materials\n\tand different gate devices.\n\t\\textit{They did not even mention that for such systems new computing paradigm may also be needed}.   \n\tThe result was that, as noticed by judges of the Gordon\n\tBell Prize,  \\textit{\"surprisingly, [among the winners of the supercomputer competition] there have been no brain-inspired massively parallel specialized computers\"}~\\cite{GordonBellPrize:2017}.\n\tDespite the vast need and investments, furthermore the concentrated and coordinated efforts, just because of the important bottleneck:\\textbf{ \\textit{the missing theory}}.\n\t\n\tWhen thinking about \\textbf{Re-Booting} Computing, it is worth to re-read the assumptions made when \\textbf{Booting} was made. In his famous publication~\\cite{EDVACreport1945}, section 6.3, von Neumann discusses\\footnote{As von Neumann formulated: \"6.3 At this point the following observation is necessary. In the human nervous system the conduction\n\t\ttimes along the lines (axons) can be longer than the synaptic delays, hence our above procedure of\n\t\tneglecting them aside of $\\tau$ [the processing time] would be unsound. In the actually intended vacuum tube interpretation,\n\t\thowever, this procedure is justified: $\\tau$ is to be about a microsecond, an electromagnetic impulse\n\t\ttravels in this time 300 meters, and as the lines are likely to be short compared to this, the conduction\n\t\ttimes may indeed by neglected. (It would take an ultra high frequency device -- $\\approx 10^{-8}$ seconds\n\t\tor less -- to vitiate this argument.)\"~\\cite{EDVACreport1945}}\nthat, \\textit{since the processor frequency exceeded 0.1~GHz, it is surely \\textit{unsound} to use von Neumann's  computing paradigm in its unchanged form.} For rebooting, the plausible first step is to consider the timely behavior of computing components, including wires. Other questions,\nsuch as which technology, new physical effects\/devices, can follow only when the computing paradigm is upgraded and is state-of-the-art.\n\t\n\t\n\t\n\t\\begin{figure*}[t!]\n\t\t\\centering\n\t\t\\begin{subfigure}[t]{0.39\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphic\n\t\t\t{RelativisticComputation}\n\t\t\t\\caption{The computing operation in time-space approach. The processing operators can be gates, processors, neurons or networked computers.\\label{fig:RelativisticComputation}}\n\t\t\\end{subfigure}%\n\t\t~ \n\t\t\\begin{subfigure}[t]{0.61\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphic\n\t\t\t{EffDependence2020Log.pdf}\n\t\t\t\\caption{The surface and the figure marks show at what efficiency the top supercomputers run the\n\t\t\t\t'best workload' benchmark \\gls{HPL}, and the 'real-life load' \\gls{HPCG}.~\\cite{VeghHowMany:2020}\n\t\t\t\tThe right bottom part displays the expected efficiency~\\cite{VeghBrainAmdahl:2019} of running\n\t\t\t\tneuromorphic calculations on \\gls{SPA} computers.\\label{fig:EfficiencyIdleTime}}\n\t\t\\end{subfigure}\n\t\t\\caption{The origin of \"idle waiting time\" and its effect on the efficiency on parallelized sequential processing systems}\n\t\t\n\t\t\\vspace{-\\baselineskip}\n\t\\end{figure*}\n\t\n\t\n\t\\section{Introducing time to computing}\n\t\n\t\n\tAs suspected by many experts, the computing paradigm itsel\n\tis responsible for the experienced issues:\n\t\"\\textit{No current programming model is able to cope with this development [of processors], though, as they essentially still follow the classical van Neumann model}\"~\\cite{SoOS:2010}.\n\tWhen thinking about \"advances beyond 2020\", the solution was expected from the \"\\textit{more efficient implementation of the von Neumann architecture}\"~\\cite{DeBenedictis_supercomputing:2014}, however.\n\tEven when speaking about \n\tbuilding up computing from scratch (\"rebooting the model\"~\\cite{RebootingComputingModels:2019}), only implementing different gating technology for \\textit{the same computing model} is meant. However, the paradigm prevents building large systems (among others, neuromorphic systems), too.\n\t\n\tThere are many analogies between science and computing~\\textbf{\\cite{VeghModernParadigm:2019}}; among others, how they handle time.\n\tBoth classic science and classic computing assume instant (infinitely fast) interaction between its objects.\n\tAn event happening at any location\n\tcan be instantly seen at all other locations: the time has no specific role, and an\n\tevent has an immediate effect on all other considered objects.\n\tIn science, inventing that the speed of light is insurmountable, led to introducing the \\textit{four-dimensional space-time}.\n\tSpecial relativity introduces a 'fourth space dimension', and \\textit{we calculate that coordinate of the Minkowski space from the time as the distance the light traverses in a given time}.\n\t\n\tTo introduce a \\textit{temporal logic} into computing, the reverse of that transformation is required. In computing, distances get defined during the fabrication of components and assembling the system. In biological systems, nature defines the neuronal distances, and in 'wet' neuro-biology, signal timing rather than axon length is the right (measurable) parameter. To describe the temporal operation of computing systems correctly, \\textit{we need to find out how much later a component notices that an event occurred in the system}.\n\tThat is, we need to use a special four-vector, where all coordinates are time values:\n\tthe first three are the corresponding local coordinates (distances from the location of the event, \n\tdivided by the speed of the interaction) having time dimension,\n\tand the fourth coordinate is the time itself; that is,\n\twe introduce a \\textit{four dimensional time-space} system. The resemblance with the Minkowski space is obvious,  and the name difference signals the different aspects of utilization.\n\t\n\t\n\tFigure~\\ref{fig:RelativisticComputation} (essentially a light cone in 2D space plus a time dimension) shows \\textit{why time must be considered explicitly in all kinds of computing}. The figure shows (for visibility) a 3-dimensional coordinate system:\n\thow an event behaves in a two-dimensional space (the concept is more comfortable to visualize with the number of spatial dimensions reduced from three to two).\n\tIn the figure, the direction 'y' is not used, but enables us to place observers at the same distance\n\tfrom the event without locating them in the same point.\n\tThe event happens at the point (0,0,0), the observers are located on the 'x' axis; the vertical\n\tscale corresponds to the time.\n\t\n\tIn the classic physical hypothetical experiment, we switch on a light in the origo, and the observer\n\tswitches his light when notices that\n\tthe first light was switched on. \n\tIf we graph the growing circle around the vertical axis of the graph representing time,\n\tthe result is a cone, known as the \\textit{future light cone}.\n\tBoth light sources have some \"processing time\",\n\tthat passes between noticing the light (receiving the instruction)  and switching the light on (performing the instruction).\n\tThat is, the instruction is received at the origo, at the bottom of the green arrow.\n\tThe light goes on at the head of the arrow (i.e., at the same location, but at a later time),\n\tafter the 'processing time' $T_p$ passed. Following that, the light propagates\n\tin the two spatial dimensions as a circle around axis \"t\".\n\tObservers at a larger distance notice the light at a later time:\n\ta 'transmission time' $T_t$ is needed.\n\tIf the \"processing time\" of the light source of the first event were zero,\n\tthe light would propagate along the gray surface at the origo.\n\tHowever, because of the finite processing time, the light will propagate along the\n\tblueish cone surface, at the head of the green arrow. \n\t\n\t\n\t\\begin{figure*}\n\t\t\\includegraphics[width=\\textwidth]\n\t\t\tFP16vsFP64HPCG}\n\t\t\\caption{The role of non-payload contribution in defining $HPL$ and $HPCG$ efficiencies,\n\t\t\tfor double and half precision floating operation modes, using different architectural concepts. For visibility, a hypothetic efficiency ratio $E_{HPL}\/E_{HPCG}$=10 assumed. The housekeeping (including transfer time and cache misses)\n\t\t\tdominates, the length of operands has only marginal importance.\\label{fig:FP16vsFP64HPCG}}\n\t\\end{figure*}\n\t\n\t\n\tA circle denotes the position of our observer on the axis \"x\".\n\tWith zero \"transmission time\", the second gray conical surface (at the head of the green dotted arrow)\n\twould describe his light. However, its \"processing time\" can only begin when the observer notices the light at his position: when the dotted red arrow hits the blueish surface.\n\tAt that point begins the \"processing time\" of the second light source;\n\tthe yellowish conical surface describes the second light propagation.\n\tThe horizontal (green dotted) arrow describes the physical distance of the observer (as a time coordinate),\n\tthe vertical (red dotted) arrow describes the time delay of the observer light.\n\tIt comprises two components: the $T_t$ transmission time to the observer and its $T_p$ processing time. The light cone of the observer starts at $t=2*T_p+T_t$.\n\t\n\tThe red arrow represents the resulting \\textit{apparent processing time} $T_A$: \n\tthe longer is the red vector; the slower is the system.\n\tAs the vectors are in the same plane,\n\t$T_A = \\sqrt{T_t^2+(2\\cdot T_p+T_t)^2}$, that is  $T_A = T_p\\cdot \\sqrt{R^2+(2+ R)^2}$.\n\tThis means, that \\textit{the apparent time is a non-linear function \n\t\tof both of its component times} and \\textit{their ratio $R$}.\n\tIf more computing elements are involved, $T_t$ \n\tdenotes the longest transmission time. (Similar statement is valid if the $T_p$ times are different) The effect is significant:  if $R=1$, the apparent execution time of performing the two computations is more than 3 times longer than the processing time.\n\tTwo more observers are located on the axis 'x', in the same position.  For visibility, their timings are displayed at points '1' and '2', respectively.  Their results illustrate the influence of the transmission speed (and\/or the ratio $R$).\n\tIn their case \\textit{the transmission speed differs by a factor of two} compared to that displayed at point '0'; in this way,\n\tthree different $R=T_t \/ T_p$ ratios are displayed.\n\t\n\tNotice that at half transmission speed (the horizontal green arrow is twice as long as that in the origo)\n\tthe vector is considerably longer, while at\n\tdouble transmission speed, the decrease of the time\n\tis much less expressed\\footnote{\\cite{VeghHowMany:2020} discusses this phenomenon in detail.}.\n\t\n\t\n\t\n\t\n\t\\section{Idle times in computing}\n\tSpecialized standard (benchmarking) programs are used to characterize the payload performance of a computing system. Those benchmarks programs express the performance of the system in such a way that the known number of the executed \"payload\" instructions are\n\tdivided by the total measurement time.\n\tIn this way, all activities, made by the computing system, are included in the total measurement time.\n\tThe \"idle time\" is present in the measurement,\n\tbut given that the classic computing assumes instant interaction, it is assumed to be zero.\n\tBecause of this, it is commonly believed that the housekeeping activity (including looping,\n\taddressing, branching) is negligible (or maybe it\n\tcan be done, at least partly,  in parallel with the payload computing).\n\tHowever, the calculation cannot begin until the operand\n\treaches the input of the processing unit, and similarly, the result\n\tcannot be transferred until the computation finished:\n\tthe transfer and computation operations mutually\n\tblock each other.\n\t\n\tFrom the point of view of\n\tcomputation, $T_p$ is a \"payload time\", and \n\t$T_t$ is a \"non-payload (although absolutely necessary) time\".\n\tOnly the \"payload computing\" contributes to\n\t\"payload time\". The time needed to deliver\n\tthe data from one place to another, with the limiting speed of interaction, contributes to the \"non-payload\" time, see their projection to the time axis. That is, even single-processor performance\n\tmeasurements include some amount of idle time.\n\tThe measurable single-processor payload performance depends on the underlying architectures and the measured instruction mix (the benchmark program).\n\tThe different benchmarks represent a different instruction mix, to which different effects, such as cache utilization, are attached (and so: they represent a different workload). In this way, the same computing systems show \n\tdifferent payload performances when measured by\n\tdifferent benchmarks~\\cite{DifferentBenchmarks:2017},~\\textbf{\\cite{VeghHowMany:2020}}. This difference gets remarkable,\n\twhen benchmarking systems, comprising parallelized single processors.\n\t\n\tIn our simple case, payload and non-payload times are interpreted for single computing events, but the same\n\tinterpretation is valid for the average transfer times  $T_A$.\n\tThe blocking nature of transfer persists.\n\tUnder different conditions, say, when the average number of cache failures changes, the apparent execution time \n\t$T_A$ (and so: the apparent computing performance) changes.\n\tThat is, an idle time naturally belongs to a computing \n\toperation, and its ratio to the payload operation, defines the computing efficiency of the computing system.\n\t\n\tNotice one more important aspect:\n\t\\textit{the $T_p$ transmission time is an 'idle time'} (the orange arrow on the figure) for the observer:\n\tit is ready to run, takes power, but does no useful work.\n\tDue to their finite physical size and limited interaction speed (both neglected in the classic paradigm), \\textit{temporal operation of computing systems \n\t\tresults inherently in an idle time of their\n\t\tprocessing units\\footnote{it can be a crucial factor of inefficiency of general-purpose chips~\\cite{InefficiencyHameed2010}}}, and -- since it sensitively depends on many factors and conditions -- \\textit{can be a significant contributor to\n\t\tnon-payload portion of their processing time}. \n\tOther major contributors originate from their technical implementation of the components, and these \"idle waiting\" times sharply decrease the payload performance of the systems. Figure~\\ref{fig:EfficiencyIdleTime} depicts\n\thow the efficiencies of recent supercomputers depend~\\textbf{\\cite{VeghHowMany:2020}} on the number of single-threaded processors in the system and the\n\tparameter $(1-\\alpha)$, describing the non-payload portion of the corresponding benchmark task. It is known for decades that \"\\textit{this decay in performance is not a fault of the\n\t\tarchitecture, but is dictated by the limited parallelism}\"~\\cite{ScalingParallel:1993}; in excessive systems of modern \\gls{HW}, \\textit{is also dictated by the laws of nature}~\\textbf{\\cite{VeghModernParadigm:2019}}. \n\t\n\t\\begin{figure*}[t!]\n\t\t\\centering\n\t\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\t\\hspace{-2cm}\n\t\t\t\\includegraphics[width=1.2\\textwidth]\n\t\t\t{Adder1bit}\n\t\t\t\\caption{Temporal dependence diagram of a 1-bit adder. The second XOR gate is at (-1,0).\n\t\t\t\t\\label{fig:Adder1bit}\n\t\t\t}\n\t\t\\end{subfigure}%\n\t\t~ \n\t\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\t\\hspace{-2cm}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.2\\textwidth]\n\t\t\t{Adder1bitPlus}\n\t\t\t\\caption{Temporal dependence diagram of a 1-bit adder. The second XOR gate is at (+1,0).\t\t\t\n\t\t\t\t\\label{fig:Adder1bitPlus}\n\t\t\t}\n\t\t\\end{subfigure}\n\t\t\\caption{Temporal diagram of a one-bit adder in time-space system. The diagram shows the logical equivalent of the SystemC source code of Listing~\\ref{lst:OneBitAdder}.\n\t\t\t\\textit{The time from axis $x$ to the bottom of green arrows} signals \"idle waiting\" time (undefined gate output). Notice how changing the position of a gate affects signal timing.\\label{fig:OneBitAdder}}\n\t\n\t\\end{figure*}\n\t\n\t\\subsection{Half versus double precision}\n\tOne of the most clean conditions to demonstrate\n\tthe impact of  idle time on the computing\n\tperformance of the system, is to use different operand\n\tlengths in the computing operations, but otherwise using\n\tthe same underlying architecture, number of operations, and housekeeping.\n\tFrom our point of view, only the amounts of payload and non-payload operations are different.\n\t\n\tThe so-called \\textit{HPL-AI} benchmark used Mixed Precision\\footnote{Both names are rather inconsequent. On one side, the test itself has not much to do with AI, just uses the operand length common in \\gls{AI} tasks (\\gls{HPL}, similarly to \\gls{AI}, is a workload type). On the other side, Mixed Precision is Half Precision: it is natural that for multiplication twice as long operands are used temporarily.  It is a different question that the operations are contracted.}\n\t\\cite{MixedPrecisionHPL:2018} rather than Double Precision \n\tcalculations. Recent top supercomputers  $Fugaku$~\\cite{DongarraFugakuSystem:2020} and $Summit$~\\cite{MixedPrecisionHPL:2018} provided also their \\gls{HPL} performance \n\tfor both 64-bit and 16-bit operands. Of course, the performance seems to be much better with shorter operand length\n\t(much shorter time for the same number of operations).\n\tOne can expect that their performance shall be four times higher, using four times shorter operands.\n\tThe power consumption data~\\cite{MixedPrecisionHPL:2018} underpin the expectations:\n\tthe power consumption is about four-fold lower.\n\tThe computing performance, however, shows a slighter performance enhancement only: 3.01 for $Summit$,\n\t3.42 for $Fugaku$, because of the needed housekeeping. \n\t\n\tUnfortunately, \\textit{the achievement comes from accessing less data in memory and using quicker operations on the  shorter operands\n\t\trather than reducing the communication intensity}.\n\tIn other words, it reduces proportionally only the payload performance, but much less the non-payload performance.\n\tAs discussed in detail in~\\textbf{\\textbf{~\\cite{VeghHowMany:2020}}},\n\tbenchmark \\gls{HPL} is computing-bound, so the longer (averaged) data delivery time\n\tblocks the operation of the floating point unit. \n\tAs seen in the figure, under \\gls{HPL} workload, the idle time is smaller than the\n\tcomputing time, so using shorter operands leads to a significant difference\n\tin the payload computing performance of the system.\n\tIn a different workload, where the non-payload portion is much higher, the non-payload contribution\n\tdominates; in the case of \\gls{HPCG} workload, the difference between the lengths of the operands becomes marginal.\n\t\n\t\n\t\n\tFig.~\\ref{fig:FP16vsFP64HPCG} illustrates the role of non-payload performance with respect to operand length.\n\tIn the figure (for visibility) a hypothetic ratio 10 of efficiencies measured by benchmarks $HPL$ and $HPCG$\n\twas assumed; in reality, it is about 200. The non-payload contribution (the data transfer) blocks the operation of the floating-point unit and defines the system's payload performance.\n\tThe dominant role of the non-payload contribution also means that \n\tit is of little importance if double or half precision operands are used in the computation in real-life tasks.\n\tThe blue vectors essentially represent the case of $Summit$. The red vector represents\n\tthe $FP_0$ value of $Fugaku$ (transformed to scaling of $Summit$).\n\tThe difference between their $HPL$ performances is attributed to their different\n\t$FP_0$ values. This difference, however, gets marginal as the workload approaches real-life conditions.\n\tUnder \\gls{AI} workload, this effect is even more substantial: for \\gls{ANN} architectures and workloads,\n\tusing shorter operands means no change in their payload performance.\n\t\n\t\\subsection{Different workloads}\n\tUnder different workloads a different amount of non-payload contribution is\n\tattached to the payload contribution, so the resulting apparent payload performance is different. \n\tWhat is more, the efficiency (and so: the payload performance)  depends also\n\ton the number of processing units in the system~\\textbf{\\cite{VeghScalingANN:2020}}.\n\tThe efficiency sharply decreases with the growth of the number of processing units,\n\tso some supercomputers use all their available cores when running benchmark \\gls{HPL},\n\tbut only a fraction of them when running \\gls{HPCG}. Even, some of them (typically cloud-like architectures) do not provide \n\ttheir \\gls{HPCG} efficiency. Figure~\\ref{fig:EfficiencyIdleTime} reveals how supercomputers' efficiencies depend on their number of processors and the workload they run. Moreover, it also demonstrates why is more advantageous to use only a fraction of their available cores (compare the efficiency values for $Fugaku$ and $Taihulight$). \n\t\n\t\n\t\\begin{lstlisting}[float,caption=The essential lines of source code of the one-bit adder implemented in \n\tSystemC,label=lst:OneBitAdder]\n\t\/\/We are making a 1-bit addition\n\taANDb = a.read() & b.read();\n\taXORb = a.read() ^ b.read();\n\tcinANDaXORb = cin.read() & aXORb;\n\t\n\t\/\/Calculate sum and carry out\n\tsum = aXORb ^ cin.read();\n\tcout = aANDb | cinANDaXORb;\n\t\\end{lstlisting}\n\t\\subsection{Programing as a source of idle time}\n\tFig.~\\ref{fig:RelativisticComputation} shows how the physical implementation (the finite size of computing components) introduces idle time into computing. The \"von Neumann-style\" programming~\\cite{BackusNeumannProgrammingStyle} also adds its contribution. Although the corresponding code shown in Listing~\\ref{lst:OneBitAdder} assumes  only logical dependence between the\n\tgate operations, their physical implementation converts that\n\tlogical dependence to a temporal one, as shown in Fig.~\\ref{fig:OneBitAdder} for the case of a one-bit adder. Changing \n\tthe \\textit{position} of one single gate changes the operating time of the gate.\n\t\n\t\n\\begin{figure*}\n\t\\centering\n\t\\hspace{-.2cm}\\begin{subfigure}[t]{.55\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]\n\t\t{RelativisticBus}\n\t\t\\caption{The operation of the sequential bus, in time-space coordinate system system. Near to axis \\textit{t}, \\textit{the lack of vertical arrows} signals \"idle waiting\" time\\label{fig:Relativisticbus}}\n\t\\end{subfigure}%\n\t\\hspace{.25cm}\\begin{subfigure}[t]{.43\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]\n\t\t{RelativisticDistributed}\n\t\t\\caption{The parallelized sequential operation as described in the proposed time-space system\n\t\t\t\\label{RelativisticDistributed}}\n\t\\end{subfigure}\n\n\t\\caption{The operation of two popular technical implementations in the time-space coordinate system \\label{fig:TechnicalImplementations}}\t\n\t\\vspace{-\\baselineskip} \n\\end{figure*} \n\n\t\n\t\n\t\\subsection{Technical implementations}\n\tThe idle time, the effect of which is depicted in Fig.~\\ref{fig:EfficiencyIdleTime},\n\tis \"including, but not limited to,\" the idle times shown in Fig.~\\ref{fig:RelativisticComputation}. In parallel with the computing paradigm,\n\tits technical implementations have also been inherited from the former solutions.\n\tFig.~\\ref{fig:Relativisticbus} depicts a common implementation of a \n\t\"high-speed bus\", connecting computing components. \n\tThe fundamental issue with using a shared medium is that -- for the time of the \n\tactual transfer --  the bus must be private, i.e., the components connected to the bus\n\tmust compete for the right of \"owing\" the medium.\n\tAs displayed, the overwhelming majority of the operating time is spent with\n\tcompeting for the bus, and the advantage of the \"high speed\" can be used\n\tin the little portion of the total time. In the case of \\gls{ANN}s, \n\tall neurons (in the layer) want to use that shared bus at the same time.\n\tIn this way, the role of the high speed becomes marginal, as discussed in~\\textbf{\\cite{VeghScalingANN:2020}}.\n\t\n\tFig.~\\ref{RelativisticDistributed} displays the temporal diagram of\n\tdistributed (parallelized sequential) processing. The idea that\n\tone of the processing units organizes the joint work works fine only\n\tuntil the number of the joint processing units is relatively low.\n\tAs the number of fellow processing units grows, so increases the idle time of the organizing unit.\n\tFor a detailed (and quantitative) discussion see~\\textbf{\\cite{VeghHowMany:2020}}.\n\t\n\t\n\t\\subsection{Impact of temporal behavior on ANNs}\n\tThe technical implementation may have \n\tvital impact on the operation of \\gls{ANN}s.\n\tThe results of the neuronal calculations\n\tare serialized, and indefinitely delayed, \n\tas discussed in~\\textbf{\\cite{VeghScalingANN:2020}}.\n\tThe good solution,\n\tthe biology applies, that all results are \n\tsent on the corresponding axons in parallel, that is, at the same time (or at least within a well-defined time window).\n\tA proper technical solution could be to make sure that all\n\tneeded neuronal operations are performed, that would need a synchronized operation.\n\tHowever, that solution would slow down the operation.\n\tThe present (wrong) technical solution is that\n\tthe vector and matrix operations are performed on their data without synchronization, as fast as the HW can perform the operations. The processed data is derived from the outputs of\n\tartificial neurons. Unlike their biological equivalents, those technical neurons produce \"continuous spikes\", and there is no way to find out which input signals were used to compute that output signal. During training, the computations' results are continuously used to make feedback, i.e., to adjust the weights, the neuronal calculations use.\n\tAt the beginning of the computations, \n\tthose vectors are not yet initialized:\n\ttheir correct input value is waiting in the queue. The calculation with those uninitialized vectors (and matrices) are performed,\n\tand the feedback adjusts the weights to those fake inputs. This temporal behavior makes it significantly harder to learn something,\n\tbecause the weights may be adjusted tendentiously to wong sets.\n\tIronically enough, the more neurons and\n\tthe faster (maybe, tensorial) math units, \n\tthe harder is to arrive at a reasonable solution: not considering the temporal behavior of their implementation,\n\tmisleads the neuronal networks and leads to weeks-long training times.\n\t\n\tA further issue is discussed in detail in~\\textbf{\\cite{BiologySpatioTemporal:2020}}: the time is a crucial attribute in learning (and training). Given that in artificial networks the time as such is not considered at all (both in math discussion and technical implementation), the produced behavior is practically unrelated to the biological systems they attempt to mimic.\n\t\n\\begin{figure*}\n\t\\centering\n\t\\hspace{-1cm}\\begin{subfigure}[t]{.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1.2\\textwidth]\n\t\t{RelativisticMemorySlow}\n\t\t\\caption{Normal speed cache memory. Two different cache memories, with the same physical cache sped, but at different internal on-chip cache position\\label{fig:RelativisticMemorySlow}}\n\t\\end{subfigure}%\n\t\\hspace{.5cm}\\begin{subfigure}[t]{.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1.2\\textwidth]\n\t\t{RelativisticMemorySuper}\n\t\t\\caption{Super-quick (10 times quicker) cache memory. Assumes new material\/physical mechanism. Two different cache memories, with the same physical cache sped, but at different internal on-chip cache position \\label{fig:RelativisticMemorySuper}}\n\t\\end{subfigure}\n\n\t\\caption{The performance dependence of an on-chip cache memory, at different cache operating times, in the same topology.\n\t\tThe cores at x=-0.5 and x=0.5 positions access on-chip cache at y=0.5 and y=1.0, respectively. The vertical orange arrows represent the physical cache operating time, and the vertical green arrows the apparent access time. The physical operating speed of the cache memory of the right subfigure is ten times better. Compare the apparent access times to the corresponding physical ones: the time ratio is better only about a factor of two. Notice also that the apparent operating speed is more sensitive to the position rather than to the speed of the cache memory\\label{fig:CachePerformance}}\t\n\t\\vspace{-\\baselineskip} \n\\end{figure*} \n\n\t\n\t\\subsection{New effects and\/or materials}\n\tGiven that the \\textit{apparent processing time} $T_A$ defines the performance of the system, $T_p$ (physical processing time,  a vector perpendicular to the XY plane) and $T_t$ (transfer time, a vector between different planes) must be concerted.\n\tTheir temporal behavior defines the limitations of researching\n\tfor new materials\/effects, etc.\n\tIn a complex system, \\textit{it is not reasonable to fabricate smaller components without decreasing \n\t\ttheir processing time proportionally; and similarly, replacing a \\gls{PU} with a very much quicker one\n\t\t(such as proposed in~\\cite{RecipeMemristor:2020,NatureBuildingBrain:2020}, and \\textit{may be proposed using any future new physical effect and\/or material}) \n\t\thas only a marginal effect, if the physical distance\n\t\tof the \\gls{PU}s cannot be reduced proportionally, at the same time.}\n\t\n\t\n\tFig.~\\ref{fig:CachePerformance} demonstrates why: two different topologies and two different \n\tphysical cache operating speeds are used in the figure. \n\tTwo cores are in positions (-0.5,0) and (0.5,0), furthermore two cache memories are located at (0,0.5) and (0,1). \n\tThe signal, requesting to access cache, propagates along the dotted green vector\n\t(it changes both its time and position coordinates), the cache starts to operate only when the green dotted arrow hits its position. After its operating time (the vertical orange arrow), the result is delivered back to the requesting core.\n\tThis time can also be projected back to the \"position axes\", and their sum (thin red arrow) can be calculated.\n\t\n\tThe physical delivery of the fetched value begins at the bottom of the lower vertical green arrows, includes waiting\n\tand finishes at the head of\n\tthe upper vertical green arrows; their distance defines the \\textit{apparent cache access time} $T_A$. The physical cache access time (the vertical orange arrow) begins when the\n\tsignal reaches the cache. Till that time, the cache is idle waiting. \n\tThe core is also idle waiting until the requested content arrives.\n\tNotice that \\textit{the apparent processing time is a monotonic function of the\n\t\tphysical processing time, but because of the included -- fixed time -- 'transmission times'\n\t\tdue to the physical distance of the respective elements, their dependence is far from being linear}.\n\tRepeated operation of course can change the idle\/to active ratio. \n\tHowever, one must consider the resources the signal delivery uses and the blocking effect discussed above.\n\t\n\tThe apparent processing time (represented by the distance of the vertical green arrows) is only slightly affected by the physical speed of the cache memory (represented by vertical orange arrows).\n\tThe right subfigure assumes that some new material\/technology\/effect decreased the\n\taccess time to one-tenth of the time assumed on the left subfigure.\n\tIn the figure, the technology (at considerable expenses) improved the physical access time by a factor of ten, but the apparent access speed has improved only by a factor of less than two.\n\t\\textit{Even if the physical cache time could be reduced to zero, the apparent access time cannot be\n\t\treduced below the time defined by the respective distances\/interaction times.}\n\tAs discussed theoretically in~\\cite{LimitsOfLimits2014} and in terms of practical design~\\cite{WiringDominance:2019}, in the today's processor technology wiring defines performance.\n\t\n\t\\section{Summary}\n\tAll fields of computing benefit from introducing temporal behavior for the components,\n\tfrom explaining the need of \"in-memory computing\" to reasoning the low efficiency of \\gls{GPU}s in general-purpose applications. We presented some possible case studies.\n\t\\textit{Neglecting their temporal behavior limits the utility of any new method, component, material or technology, if they are designed\/developed\/used in the spirit of the old (timeless) paradigm.}\n\tAs Neumann pointed out, it is \\textit{\\textbf{unsound}} to establish\n\t\tsupercomputing, different accelerators, ANNs, memristor, FPGA, quantum-processor, etc. based computing on the classic paradigm that is valid (in a quasi-strict mathematical sense) for the technological age of vacuum tubes.\n\t\tThe drastically increased weight of the transfer time compared to the computing time needs a modern paradigm.\n\t\n\t\\input{ICRC_time_2012_V1.bbl}\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}I Introduction}\n\nThe mechanism of heavy meson decays is one of the most interesting and challenging fields in particle physics, it involves both strong and weak interactions. Nowadays strong interaction in the non-perturbative region is still an unsolved problem. Compared with hadronic decays, leptonic decay is simpler. Strong interaction only occurs within the initial particle. Pure-leptonic decay of heavy meson can be used to determine the decay constant, which describes the possibility-amplitude for the quark-untiquark emerging at the same point. The pure-leptonic decay is helicity suppressed. The decay branching ratio of a pseudoscalar meson $P$ with quark content $\\bar{Q}q$ within the standard model is\n\\begin{equation}\nB(P\\to l\\bar{\\nu})=\\frac{G_F^2|V_{Qq}|^2}{8\\pi}\\tau_Pf_P^2m_l^2m_P(1-\\frac{m_l^2}{m_P^2})^2,\n\\end{equation}\nwhere $G_F$ is the Fermi coupling constant, $V_{Qq}$ the Cabibbo-Kobayashi-Maskawa (CKM) matrix element, $\\tau_P$ the life time of the meson $P$, and $m_P$ and $m_l$ the masses of the meson $P$ and lepton $l$, respectively. The decay rate is proportional to the lepton mass squared $m_l^2$ is the consequence of the helicity suppression. However, the presence of one photon in the final state can compensate the helicity suppression. As a result, the radiative leptonic decay can be as large as, or even larger than the pure-leptonic decay mode. It thus opens a window for detecting the dynamics of strong interaction in the heavy meson or studying the effect of strong interaction in the decay.\n\nThe radiative leptonic decay rates of the charged $B$ and $D$ mesons have been studied with various methods in the literature. In Ref. \\cite{workbefore11}, $B$ and $D_s\\to l\\bar{\\nu}\\gamma $ are calculated in a non-relativistic quark model, the branching ratios of the order of $10^{-4}$ for $D_s\\to l\\bar{\\nu}\\gamma$ and $10^{-6}$ for $B\\to l\\bar{\\nu}\\gamma$ are found. In Ref. \\cite{workbefore12} with perturbative QCD approach, it is found that the branching ratio of $D_{s}^+\\to e^+\\nu\\gamma$ is of the order of $10^{-3}$ and $D^{+}\\to e^+\\nu\\gamma$ of the order of $10^{-4}$, while the branching ratio of $B^{+}\\to e^+\\nu\\gamma$ is at the order of $10^{-6}$. On the other hand, a smaller branching ratio is obtained for $D_{(s)}\\to l\\bar{\\nu}\\gamma $ within the light front quark model \\cite{workbefore13}. Smaller result for $D_{(s)}\\to l\\bar{\\nu}\\gamma$ is also obtained in Ref. \\cite{workbefore2} within the non-relativistic constituent quark model, which gives that the branching ratio of $D^{-}\\to l\\bar{\\nu}\\gamma$ is of the order of $10^{-6}$ and $D^{-}_s\\to l\\bar{\\nu}\\gamma$ of the order of $10^{-5}$. The problem of factorization in QCD for $B\\to l\\nu\\gamma$ is studied in Ref. \\cite{Sachrajda}.\n\nIn this work, we study the radiative leptonic decays of the charged $B$, $D$ and $D_s$ mesons  to $l\\bar{\\nu}\\gamma$ including both the short and long-distance contributions. The short-distance contribution is considered at tree level. The wave function of the heavy meson used here is obtained in the relativistic potential model previously \\cite{sdbase}. The long-distance contribution is estimated by using the idea of the vector meson dominance (VMD) \\cite{VMD1,VMD2,VMD3,VMD4,VMD5} followed by the transition of the vector meson to a photon. We find that the long-distance contribution can enhance the decay rates seriously.\n\nThe remaining part of this paper is organized as follows. In Sec.II, we present the short-distance amplitude. In Sec.III, the long-distance contribution is considered. The numerical results and discussion are given in Sec.IV. Sec.V is a brief summary.\n\n\\subsection{\\label{sec:level2}II The Short-Distance Contribution}\n\nWe use $P$ to denote the pseudoscalar meson which is composed of a heavy anti-quark $\\bar{Q}$ and a light quark $q$, such as $B$ and $D$ mesons. There are four Feynman diagrams contributing to the radiative decays $P^-\\to l \\bar{\\nu}\\gamma$ at tree level, which are shown in Fig.~\\ref{fig:feynsd}. However the contribution of Fig.~\\ref{fig:feynsd} (d) is suppressed by a factor of $1\\left\/M_w^2\\right.$, it can be neglected for simplicity. The effective Hamiltonians corresponding to the other three diagrams in Fig.~\\ref{fig:feynsd} can be written as:\n\\begin{figure}\n\\includegraphics[scale=0.6]{fig1.eps}\n\\caption{\\label{fig:feynsd} Feynman diagrams of short-distance contribution at tree level.}\n\\end{figure}\n\n\\begin{equation}\n\\begin{split}\n&\\mathcal{H}_a=\\frac{- i e G_F V_{Qq}}{\\sqrt{2}}Q_Q\\bar{Q}\\slashed A \\frac{\\slashed p_{\\gamma }-\\slashed p_Q+m_Q}{2\\left(p_{\\gamma }{\\cdot}p_Q\\right)}P_L^{\\mu } q\\left(lP_{L\\mu}\\bar{\\nu}\\right),\\\\\n&\\mathcal{H}_b=\\frac{- i e G_F V_{Qq}}{\\sqrt{2}}Q_q\\bar{Q}P_L^{\\mu }\n\\frac{\\slashed p_q-\\slashed p_{\\gamma }+m_q}{2\\left(p_{\\gamma }{\\cdot}p_q\\right)}\\slashed A q\\left(lP_{L\\mu}\\bar{\\nu}\\right),\\\\\n&\\mathcal{H}_c=\\frac{- i e G_F V_{Qq}}{\\sqrt{2}}Q_Q\\bar{Q}P_L^{\\mu } q\\left(l\\slashed A\\frac{\\slashed p_{\\gamma }+\\slashed p_l+m_l}{2\\left(p_{\\gamma }{\\cdot}p_l\\right)}P_{L\\mu}\\bar{\\nu}\\right)\n,\\label{eq:one}\n\\end{split}\n\\end{equation}\nwhere $P_L^{\\mu }$ is defined as $\\gamma ^{\\mu }\\left(1-\\gamma _5\\right)$, and $V_{Qq}$ represents for the CKM matrix elements. $Q_Q$ and $Q_q$ are the electric charges of the quarks  $Q$ and $q$, respectively. $A$ is the electro-magnetic field.\n\n$\\mathcal{H}_a$, $\\mathcal{H}_b$ and $\\mathcal{H}_c$ can be divided into two terms for convenience, according to the numerator of the fermion propagator. For example, $\\mathcal{H}_a$ can be written as\n\\begin{equation}\n\\mathcal{H}_a=\\frac{- i e Q_Q G_F V_{Qq}}{2\\sqrt{2}}\\left(\\mathcal{M}_{\\text {a1}}+2\\mathcal{M}_{\\text {a2}}\\right)^{\\mu}\\left(lP_{\\text {L$\\mu $}}\\bar{\\nu}\\right)\n\\label{eq:two},\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\n\\mathcal{M}_{a1}^{\\mu}&=\\bar{Q}\\slashed A\\frac{\\slashed p_{\\gamma }}{p_{\\gamma }{\\cdot}p_Q}\\gamma ^{\\mu }\\left(1-\\gamma _5\\right) q.\\\\\n&=- i \\epsilon ^{\\alpha \\beta \\mu \\sigma }A_{\\alpha}p_{\\gamma\\beta}v_{a\\sigma}+p_{\\gamma }^{\\mu }\\left(A{\\cdot}v_a\\right)-A^{\\mu}\\left(p_{\\gamma }{\\cdot}v_a\\right),\\\\\n\\mathcal{M}_{a2}^{\\mu}&=\\frac{1}{2}\\bar{Q}\\slashed A\\frac{m_Q-\\slashed p_Q}{p_{\\gamma }{\\cdot}p_Q}\\gamma ^{\\mu }\\left(1-\\gamma _5\\right) q = - A _{\\alpha} t_a^{\\alpha \\mu },\n\\label{eq:three}\n\\end{split}\n\\end{equation}\nwith\n\\begin{equation}\n\\begin{split}\n&v_a^{\\mu }=\\bar{Q}\\frac{1}{p_{\\gamma }{\\cdot}p_Q}\\gamma ^{\\mu }\\left(1-\\gamma _5\\right) q,\\\\\n&t_a^{\\alpha \\mu }=\\bar{Q}\\frac{p_Q^{\\alpha }}{p_Q{\\cdot}p_{\\gamma }}\\gamma ^{\\mu}\\left(1-\\gamma _5\\right)q\n\\label{eq:four}.\n\\end{split}\n\\end{equation}\nThe amplitude of the radiative leptonic decay can be obtained by inserting the operator of the effective Hamiltonian between the initial and final particle states. For example, the contribution of Fig.~\\ref{fig:feynsd} (a) is\n\\begin{equation}\n\\begin{split}\n\\mathcal{A}_a&=<\\gamma \\bar{\\nu} l\\mid\\mathcal{H}_a\\mid P>\\\\\n&= \\frac{- i e Q_Q G_F V_{Qq}}{2\\sqrt{2}}  \\left(u_l P_{\\text {L$\\mu $}}v_{\\bar{\\nu}}\\right)\\\\\n&\\times \\left(<\\gamma \\mid \\mathcal{M}_{\\text {a1}} \\mid P>+2 <\\gamma \\mid \\mathcal{M}_{a2} \\mid P>\\right) \\\\\n&= \\frac{- i e Q_Q G_F V_{Qq}}{2\\sqrt{2}}  \\left(u_l P_{\\text {L$\\mu $}}v_{\\bar{\\nu}}\\right)\\\\\n&\\times \\left(- i \\epsilon ^{\\alpha \\beta \\mu \\sigma }\\varepsilon _{\\gamma\\alpha}^*p_{\\gamma\\beta}<0\\mid v_{a\\sigma}\\mid P>+p_{\\gamma }^{\\mu }\\varepsilon _{\\gamma }^*{\\cdot}<0\\mid v_a\\mid P>\\right.\\\\\n&\\left.-\\varepsilon _{\\gamma }^{\\mu *}p_{\\gamma }{\\cdot}<0\\mid v_a\\mid P> -2 \\varepsilon _{\\gamma \\alpha}^*<0\\mid t_a^{\\alpha \\mu }\\mid P>\\right)\n\\label{eq:five}.\n\\end{split}\n\\end{equation}\n\nThe matrix elements $<0\\mid v_a^{\\mu }\\mid P>$ and $<0\\mid t_a^{\\alpha \\mu }\\mid P>$ only depend on the momenta $p_P$ and $p_{\\gamma}$. According to their Lorentz structure, they can be decomposed as a linear combination of two terms of $p_P$ and $p_{\\gamma}$\n\\begin{equation}\n\\begin{split}\nV_a^{\\mu}&=<0 \\mid \\bar{Q}\\frac{1}{p_{\\gamma }{\\cdot}p_Q}\\gamma ^{\\mu }\\left(1-\\gamma _5\\right) q \\mid P>\\\\\n&=\\frac{m_P}{p_P{\\cdot}p_{\\gamma }}\\left(A_{a1}p_P^{\\mu }+B_{a1}\\frac{m_P^2}{p_P{\\cdot}p_{\\gamma }}p_{\\gamma }^{\\mu }\\right)\\\\\nT_a^{\\alpha \\mu }&=<0 \\mid \\bar{Q}\\frac{p_Q^{\\alpha }}{p_Q{\\cdot}p_{\\gamma }}\\gamma ^{\\mu}\\left(1-\\gamma _5\\right)q \\mid P>\\\\\n&= \\frac{m_P}{p_P{\\cdot}p_{\\gamma }}\\left[A_{a2}p_P^{\\alpha }p_P^{\\mu }+\\frac{m_P^2}{p_P{\\cdot}p_{\\gamma }}\\left(B_{a2}p_P^{\\alpha }p_{\\gamma }^{\\mu }+C_{a2}p_P^{\\mu }p_{\\gamma }^{\\alpha }\\right)\\right. \\\\ & \\left.\n+\\frac{m_P^4}{\\left(p_P{\\cdot}p_{\\gamma }\\right)^2}D_{a2}p_{\\gamma }^{\\alpha }p_{\\gamma }^{\\mu }+E_{a2}m_P^2g^{\\alpha \\mu }\\right. \\\\\n& \\left. +F_{a2}\\frac{m_P^2}{p_P{\\cdot}p_{\\gamma }}\\epsilon ^{\\alpha \\mu \\rho \\sigma}p_{P\\rho }p_{\\gamma \\sigma }\\right]\n\\label{eq:six}.\n\\end{split}\n\\end{equation}\nThe coefficients $A_{a1}$, $B_{a1}$, $A_{a2}$, $B_{a2}$, $C_{a2}$, $D_{a2}$, $E_{a2}$ and $F_{b2}$ are all dimensionless constants. The terms of $B_{a1}$, $C_{a2}$ and $D_{a2}$ do not contribute to the decay amplitude $\\mathcal{A}_a$ when substituting the above decomposition into eq.\\ref{eq:five}. Therefore these terms can be dropped. The coefficients can be obtained by the treatment in the following. Multiplying $V_a^{\\mu}$ with $p_{\\gamma \\mu}$, we can obtain $A_{a1}$ as\n\\begin{equation}\nA_{a1}=\\frac{1}{m_P}<0\\mid \\bar{Q}\\frac{1}{p_{\\gamma }{\\cdot}p_Q}\\slashed p_{\\gamma}\\left(1-\\gamma _5\\right) q \\mid P>\n\\label{eq:seven}.\n\\end{equation}\nSimilarly, multiplying $T_a^{\\alpha \\mu }$ with $p_{\\gamma \\alpha}$, we have\n\\begin{equation}\n\\begin{split}\n&B_{a2}+E_{a2}=0,\\\\\n&A_{a2}=\\frac{1}{m_P^3}<0 \\mid \\bar{Q}\\slashed p_P\\left(1-{\\gamma }_5\\right)q\\mid P>\n\\label{eq:eight}.\n\\end{split}\n\\end{equation}\nMultiplying $T_a^{\\alpha \\mu }$ with $p_{\\gamma \\mu}p_{P\\alpha}$ and $g_{\\alpha\\mu}$, and using $B_{a2}=-E_{a2}$, one can get\n\\begin{equation}\n\\begin{split}\n&E_{a2}=-B_{a2}\n=\\frac{1}{2 m_P^3}<0\\mid \\bar{Q}\\left[\\left(p_P\\cdot p_{\\gamma}\\right)\\slashed p_Q \\right.\\\\\n&\\left.- \\left(p_P\\cdot p_Q\\right)\\slashed p_{\\gamma} \\right]\\frac{\\left(1-{\\gamma }_5\\right)}{p_Q \\cdot p_{\\gamma}}q\\mid P>\n\\label{eq:nine}.\n\\end{split}\n\\end{equation}\nFinally $F_{a2}$ can obtained by multiplying $T_a^{\\alpha \\mu }$ with $\\epsilon_{\\alpha \\mu \\beta \\nu}p_{\\gamma}^{\\nu}p_P^{\\beta}$:\n\\begin{equation}\n\\begin{split}\nF_{a2}=\\frac{p_{\\gamma}^{\\nu}p_P^{\\beta}}{2 m_P^3}<0\\mid \\bar{Q}\\frac{\\epsilon _{\\alpha\\mu\\beta\\nu}p_Q^{\\alpha}}{p_Q\\cdot p_{\\gamma}}\\gamma ^{\\mu}\\left(1-\\gamma _5\\right)q \\mid P>\n\\label{eq:ten}.\n\\end{split}\n\\end{equation}\nThe amplitude $\\mathcal{A}_b$ can be treated in the same way with some coefficients defined as follows\n\\begin{equation}\n\\begin{split}\n&<0\\mid\\bar{Q}\\frac{1}{p_{\\gamma }\\cdot p_q}\\gamma ^{\\mu }\\left(1-\\gamma _5\\right) q\\mid P>=\\\\\n&\\frac{m_P}{p_P\\cdot p_{\\gamma }}\\left(A_{\\text {b1}}p_P^{\\mu }+B_{\\text {b1}}\\frac{m_P^2}{p_P\\cdot p_{\\gamma }}p_{\\gamma }^{\\mu }\\right),\\\\\n&<0\\mid \\bar{Q}\\frac{p_q^{\\alpha}}{p_{\\gamma}{\\cdot}p_q}\n{\\gamma}^{\\mu}\\left(1-{\\gamma}_5\\right)q\\mid P>=\\\\\n&\\frac{m_P}{p_P{\\cdot}p_{\\gamma}}\n\\left[A_{b2}p_P^{\\alpha}p_P^{\\mu}+\n\\frac{m_P^2}{p_P{\\cdot}p_{\\gamma}}\n\\left(B_{b2}p_P^{\\alpha}p_{\\gamma}^{\\mu}+C_{b2}p_P^{\\mu}p_{\\gamma}^{\\alpha}\\right)+\\right.\\\\\n&\\left.\\frac{m_P^4}{\\left(p_P{\\cdot}p_{\\gamma}\\right)^2}D_{b2}p_{\\gamma}^{\\alpha}p_{\\gamma}^{\\mu}+\nE_{b2}m_P^2 g^{\\alpha\\mu}\\right]\n\\label{eq:11}.\n\\end{split}\n\\end{equation}\nUsing the matrix element $A_{a2} m_P p_P^{\\mu}=<0\\mid \\bar{Q} {\\gamma}^{\\mu}\\left(1-\\gamma _5\\right) q \\mid P>$, the amplitude $\\mathcal{A}_c$ can be treated simlarly. Finally, the total amplitude can be expressed as\n\\begin{equation}\n\\begin{split}\n&\\mathcal{A}_a =\\frac{- i e G_F V_{Qq}}{2\\sqrt{2}}\\left(u_l P_{\\text {L$\\mu $}}v_{\\bar{\\nu}}\\right)\\left\\{ -\\frac{Q_Qm_PA_{a1}}{p_P{\\cdot}p_{\\gamma}}i \\epsilon ^{\\alpha \\beta \\mu \\sigma}p_{\\gamma \\beta }\\varepsilon ^*_{\\gamma \\alpha}p_{P\\sigma}\\right.\\\\\n&\\left.-\\left[\\frac{Q_Q m_P A_{a1}}{p_P\\cdot p_{\\gamma}}+\\frac{2 Q_Q E_{a2} m_P^3}{\\left(p_P\\cdot p_{\\gamma}\\right)^2}\\right]\\left[\\left(p_P{\\cdot}p_{\\gamma}\\right)\\varepsilon _{\\gamma}^{{\\mu}*}-\\left(p_P{\\cdot}\\varepsilon\\right)p_{\\gamma}^{\\mu}\\right]\n\\right.\\\\\n&\\left.-2\\left(Q_Q\\frac{p_P{\\cdot}\\varepsilon}{p_P{\\cdot}p_{\\gamma}}\\right)A_{a2}m_P p_P^{\\mu}\\right\\},\\\\\n&\\mathcal{A}_b =\\frac{- i e G_F V_{Qq}}{2\\sqrt{2}}\\left(u_l P_{\\text {L$\\mu $}}v_{\\bar{\\nu}}\\right)\\left\\{ -\\frac{Q_qm_PA_{b1}}{p_P{\\cdot}p_{\\gamma}}i \\epsilon ^{\\alpha \\beta \\mu \\sigma}p_{\\gamma \\beta }\\varepsilon ^*_{\\gamma \\alpha}p_{P\\sigma}\\right.\\\\\n&\\left.+\\left[\\frac{Q_q m_D A_{b1}}{p_P\\cdot p_{\\gamma}}+\\frac{2 Q_q E_{b2} m_P^3}{\\left(p_P\\cdot p_{\\gamma}\\right)^2}\\right]\\left[\\left(p_P{\\cdot}p_{\\gamma}\\right)\\varepsilon _{\\gamma}^{{\\mu}*}-\\left(p_P{\\cdot}\\varepsilon\\right)p_{\\gamma}^{\\mu}\\right]\n\\right.\\\\\n&\\left.+2\\left(Q_q\\frac{p_P{\\cdot}\\varepsilon}{p_P{\\cdot}p_{\\gamma}}\\right)A_{a2}m_P p_P^{\\mu}\\right\\},\\\\\n&\\mathcal{A}_c =\\frac{- i e G_F V_{Qq}}{2\\sqrt{2}}\\left(u_l P_{\\text {L$\\mu $}}v_{\\bar{\\nu}}\\right)\\left\\{ \\frac{m_PA_{a2}}{p_l{\\cdot}p_{\\gamma}}i \\epsilon ^{\\alpha \\beta \\mu \\sigma}p_{\\gamma \\beta }\\varepsilon^* _{\\gamma \\alpha}p_{P\\sigma}\\right.\\\\\n&\\left.+\\frac{m_P A_{a2}}{p_l\\cdot p_{\\gamma}}\\left[\\left(p_P{\\cdot}p_{\\gamma}\\right)\\varepsilon _{\\gamma}^{{\\mu}*}-\\left(p_P{\\cdot}\\varepsilon\\right)p_{\\gamma}^{\\mu}\\right]\n\\right.\\\\ &\\left.\n+\\frac{p_l{\\cdot}\\varepsilon}{p_l{\\cdot}p_{\\gamma}}A_{a2}m_P p_P^{\\mu}\\right\\}\n\\label{eq:add1}.\n\\end{split}\n\\end{equation}\nThe above equations show that the contribution of each diagram in Fig. \\ref{fig:feynsd} is not gauge invariant separately, but the sum of them is indeed gauge invariant, which is given in the following\n\\begin{equation}\n\\begin{split}\n\\mathcal{A}_{a+b+c} =& \\frac{- i e G_F V_{Qq}}{2\\sqrt{2}}\\left\\{i V\\epsilon ^{\\alpha \\beta \\mu \\sigma}p_{\\gamma \\beta }\\varepsilon^*_{\\gamma \\alpha}p_{P\\sigma}\n\\right.\\\\ &\\left.\n+A\\left[\\left(p_P{\\cdot}p_{\\gamma}\\right)\\varepsilon _{\\gamma}^{{\\mu}*}-\\left(p_P{\\cdot}\\varepsilon\\right)p_{\\gamma}^{\\mu}\\right]\n\\right.\\\\ &\\left.\n+2\\left[\\left(Q_q-Q_Q\\right)\\frac{p_P{\\cdot}\\varepsilon}{p_P{\\cdot}p_{\\gamma}}+\\frac{p_l{\\cdot}\\varepsilon}{p_l{\\cdot}p_{\\gamma}}\\right]A_{a2}m_D p_P^{\\mu}\\right\\}\\\\\n&\\times \\left(u_l P_{\\text {L$\\mu $}}v_{\\bar{\\nu}}\\right).\n\\label{eq:12}\n\\end{split}\n\\end{equation}\nThis equation clearly shows that the sum of the contributions of all the diagrams in Fig. \\ref{fig:feynsd} is\ngauge invariant.  In eq.(\\ref{eq:12})  the factors $V$ and $A$ are\n\\begin{equation}\n\\begin{split}\nV=&-\\frac{Q_Q m_P A_{a1}}{p_P{\\cdot}p_{\\gamma}}-\\frac{Q_q m_P A_{b1}}{p_P{\\cdot}p_{\\gamma}}+\\frac{m_P A_{a2}}{p_l{\\cdot}p_{\\gamma}}\\\\\n&+\\frac{2\\left(Q_q F_{b2}-Q_Q F_{b1}\\right)m_P^3}{\\left(p_P{\\cdot}p_{\\gamma }\\right)^2},\\\\\nA=&-\\frac{Q_Q m_P A_{a1}}{p_P\\cdot p_{\\gamma}}+\\frac{Q_q m_P A_{b1}}{p_P{\\cdot}p_{\\gamma}}+\\frac{m_P A_{a2}}{ p_l{\\cdot}p_{\\gamma}}\\\\\n&+\\frac{2\\left(Q_q E_{b2}-Q_Q E_{b1}\\right)m_P^3}{\\left(p_P{\\cdot}p_{\\gamma }\\right)^2}\n\\label{eq:13}.\n\\end{split}\n\\end{equation}\nNext we shall calculate the coefficients $A_{a1}$, $A_{a2}$, $E_{a2}$ and $F_{a2}$.\n\nThe pseudoscalar meson state can be written in terms of the quark-antiquark creation and annihilation operators\n\\begin{equation}\n\\begin{split}\n&{\\mid}P\\left(P=0\\right)>= \\frac{1}{\\sqrt{3}}\\sum _{\\substack i}\\int d^3k {\\Psi _0}(|\\vec{k}|)\\\\\n& \\times\\frac{1}{\\sqrt{2}}\\left[b_Q^{i+}(\\vec k,\\uparrow)d_q^{i+}(-\\vec k,\\downarrow)-b_Q^{i+}(\\vec k,\\downarrow)d_q^{i+}(-\\vec k,\\uparrow)\\right]\\mid 0>,\n\\label{eq:14}\n\\end{split}\n\\end{equation}\nwhere $i$ is the color index, the factor $1\/\\sqrt{3}$ the normalization factor for color indices, and $\\vec{k}$ the 3-momentum of the quarks in the rest frame of the heavy meson. The wave function ${\\Psi _0}(|\\vec{k}|)$ has been calculated in the relativistic potential model previously, the numerical solution of the wave function can be fitted in the exponential form \\cite{col,sdbase}\n\\begin{equation}\n\\Psi _0\\left(|\\vec{k}|\\right)=4{\\pi}{\\sqrt{\\lambda ^3 m_P}}e^{-\\lambda |\\vec{k}|}.\n\\label{eq:15}\n\\end{equation}\nThe numerical solutions of the parameter $\\lambda$ for $D$, $D_s$ and $B$ mesons are quoted from Ref. \\cite{sdbase} recently\n\\begin{displaymath}\n\\begin{array}{cc}\n\\lambda_D = 3.4{\\rm GeV}^{-1},& \\lambda_{D_s} = 3.2{\\rm GeV}^{-1}, \\\\\n\\lambda_B=2.8{\\rm GeV}^{-1}.&\n\\end{array}\n\\end{displaymath}\nWith the meson state given in eq. (\\ref{eq:14}), the matrix element $<0\\mid \\bar{Q}\\Gamma q\\mid P>$ can be calculated straightforwardly, the result is\n\\begin{equation}\n\\begin{split}\n&<0\\mid \\bar{Q}\\Gamma q\\mid P>=\\sqrt{\\frac{3}{2}}\\frac{1}{8\\pi^3}\\int dkd{\\Omega} k^2 {\\Psi _0}\\left(k\\right)\\\\\n&\\times tr\\left[M \\cdot \\Gamma \\right]\\sqrt{\\frac{m_Q m_q}{p_Q^0 p_q^0}},\n\\label{eq:16}\n\\end{split}\n\\end{equation}\nwhere $M=u_q(\\vec k,\\uparrow)\\bar{v}_Q(-\\vec k,\\downarrow)-u_q(\\vec k,\\downarrow)\\bar{v}_Q(-\\vec k,\\uparrow)$ is the Dirac spinner for the pseudoscalar meson, it can be obtained in the Dirac-Representation as\n\\begin{equation}\nM=\\frac{-\\frac{1}{2}\\left(\\slashed p_q + m_q\\right)\\left(1+{\\gamma}^0\\right)\\left(\\slashed p_Q + m_Q\\right){\\gamma}^5}{2\\sqrt{m_Q^0 m_q^0\\left(p_Q^0+m_Q\\right)\\left(p_q^0+m_q\\right)}}\n\\label{eq:17}.\n\\end{equation}\n\\subsection{\\label{sec:level3}III The Long-Distance Contribution}\n\nIn this section, we estimate the contributions of long-distance physics. According to the spirt of vector meson dominance model \\cite{VMD1,VMD2,VMD3,VMD4,VMD5}, we consider the resonance process $P^-\\to l\\bar{\\nu} V\\to l\\bar{\\nu}\\gamma$, where the intermediate vector resonance $V$ can be $\\rho$, $\\omega$ and $\\phi$. The contribution comes from the semileptonic intermediate $l\\bar{\\nu} V$, followed by the vector resonance turning to an on-shell photon $V\\to \\gamma$, which is shown in Fig.~\\ref{fig:ld}. The amplitude of the long-distance contribution can be written as\n\\begin{figure}\n\\includegraphics[scale=0.7]{fig2.eps}\n\\caption{\\label{fig:ld} The LD contribution.}\n\\end{figure}\n\\begin{equation}\n\\begin{split}\n\\mathcal{A}_{LD}=&\\frac{eG_F V_{Qq}Q_q}{\\sqrt{2}}u_l{\\gamma}^{\\mu}\\left(1-{\\gamma}_5\\right)v_{\\bar{\\nu}}\\\\\n&\\times\\sum _{\\substack {V}}\\varepsilon _{\\gamma}^{\\alpha*}<0\\mid \\bar{q}{\\gamma ^{\\alpha}}q\\mid V>\\frac{i e^{\\phi _V}}{p_V^2-m_V^2+i m_V {\\Gamma }_V}\\\\\n&\\times<V \\mid \\bar{Q}{\\gamma}^{\\mu}\\left(1-{\\gamma}_5\\right)q\\mid P >\n\\label{eq:19}.\n\\end{split}\n\\end{equation}\nFor the decays of $D$ and $B$ mesons, $V$ represents for $\\rho$ and $\\omega$ mesons, while for $D_s$ meson decay, $V$ represents for $\\phi$ meson. $\\phi _V$ is the relative phase between the long and short-distance contributions. The matrix element $<0 \\mid \\bar{q}{\\gamma}^{\\mu}q\\mid V >$ is used to define the decay constant of the vector meson\n\\begin{equation}\n<0 \\mid \\bar{q}{\\gamma}^{\\mu}q\\mid V >=C_{qV}f_{V}\\varepsilon _{V}^{\\alpha *}\n\\label{eq:20},\n\\end{equation}\nwhere the factor $C_{qV}$'s are\n\\begin{equation}\nC_{u\\rho}=C_{u\\omega}=C_{d\\omega}=\\frac{1}{\\sqrt{2}},C_{d\\rho}=-\\frac{1}{\\sqrt{2}},C_{s\\phi}=1\n\\label{eq:21}.\n\\end{equation}\nThe vector's decay constant $f_V$ can be derived from the decay rate of $V\\to e^+ e^-$. After a short calculation, the decay constants can be related to the vector meson's leptonic decay widths\n\\begin{equation}\n\\begin{split}\n&f_{\\rho}^2=\\frac{3 m_{\\rho}^3}{4 \\pi {\\alpha}^2}\\frac{2}{\\left(Q_u-Q_d\\right)^2}{\\Gamma }_{\\rho\\to e^+ e^-},\\\\\n&f_{\\omega}^2=\\frac{3 m_{\\omega}^3}{4 \\pi {\\alpha}^2}\\frac{2}{\\left(Q_u+Q_d\\right)^2}{\\Gamma }_{\\omega\\to e^+ e^-},\\\\\n&f_{\\phi}^2=\\frac{3 m_{\\phi}^3}{4 \\pi {\\alpha}^2}\\frac{1}{Q_s^2}{\\Gamma }_{\\phi\\to e^+ e^-},\n\\label{eq:22}\n\\end{split}\n\\end{equation}\nwhere $\\alpha$ is the electromagnetic fine structure constant, $Q_u$, $Q_d$ and $Q_s$ are the charges of the quarks $u$, $d$ and $s$, respectively.\n\nThe hadronic matrix element $<V \\mid J^{\\mu}\\mid P >$ in eq.~(\\ref{eq:19}) can be decomposed according to its Lorentz structure as \\cite{bsw1,bsw2}\n\\begin{equation}\n\\begin{split}\n&<V \\mid J_{V-A\\mu}\\mid P >=\\frac{2}{m_P+m_V}\\epsilon _{\\mu\\nu\\rho\\sigma} \\varepsilon _V^{\\nu}p_P^{\\rho}p_V^{\\sigma}V\\left(q^2\\right)\\\\\n&+i\\left\\{\\varepsilon _{V\\mu}\\left(m_P+m_V\\right)A_1\\left(q^2\\right)\\right.\\\\\n&\\left.-\\frac{\\varepsilon _V\\cdot q}{m_P+m_V}\\left(p_P+p_V\\right)_{\\mu}A_2\\left(q^2\\right)\\right.\\\\\n&\\left.-\\frac{\\varepsilon _V\\cdot q}{q^2}2m_Pq_{\\mu}A_3\\left(q^2\\right)\\right\\}\n+i\\frac{\\varepsilon _V\\cdot q}{q^2}2m_Vq_{\\mu}A_0\\left(q^2\\right),\n\\label{eq:23}\n\\end{split}\n\\end{equation}\nwhere $q=p_P-p_V$, and $V$, $A_0$, $A_1$, $A_2$ and $A_3$ are form factors.\n\nTo obtain the amplitude gauge invariant, we take the trick used in Ref.\\cite{dht} in treating the long-distance contribution in the precess $b\\to s{\\gamma}$ via the resonance $J\/\\Psi$. With the Lorentz decomposition of the hadronic matrix element in eq.(\\ref{eq:23}), the product of the two matrix elements in eq.(\\ref{eq:19}) can be calculated to be\n\\begin{equation}\n\\begin{split}\n&\\varepsilon _{\\gamma\\alpha}^*<0 \\mid \\bar{q}{\\gamma}^{\\alpha}q\\mid V ><V \\mid J_{V-A}^{\\mu}\\mid P >=\\\\\n&e Q_q C_{qV}f_V \\left\\{\\frac{2}{m_P+m_V}\\epsilon^{\\alpha\\beta\\mu\\sigma}p_{P\\sigma}p_{V\\beta}\\varepsilon _{\\gamma\\alpha}^*V\\left(q^2\\right)\\right.\\\\\n&\\left.-iA_1\\left(q^2\\right)\\left[\\varepsilon _{\\gamma}^{\\mu *}\\left(m_P+m_V\\right)-\\frac{\\left(m_P+m_V\\right)\\left(p_P{\\cdot}\\varepsilon _{\\gamma}\\right)}{p_P{\\cdot}p_V}p_V^{\\mu}\\right]\\right.\\\\\n&\\left.-i\\left(p_P{\\cdot}\\varepsilon _{\\gamma}\\right)\\left[\\frac{\\left(m_P+m_V\\right)p_V^{\\mu}}{p_P{\\cdot}p_V}A_1\\left(q^2\\right)\\right.\\right.\\\\\n&\\left.\\left.-\\frac{\\left(p_P+p_V\\right)^{\\mu}}{m_P+m_V}A_2\\left(q^2\\right)+\\frac{2 m_V q^{\\mu}}{q^2}\\left(A_3\\left(q^2\\right)-A_0\\left(q^2\\right)\\right)\\right]\\right\\}\n\\label{eq:24}.\n\\end{split}\n\\end{equation}\nIn the rest-frame of the heavy meson $P$, the product of the four-momentum of the meson $P$ and the polarization vector of the photon satifies $p_P{\\cdot}\\varepsilon = 0$. Then the last term in the above equation can be dropped. With $p_V=p_{\\gamma}$, we obtaine\n\\begin{equation}\n\\begin{split}\n\\mathcal{A}_{LD}=&\\sum _{\\substack {V}}\\frac{e G_F V_{Qq} Q_q}{\\sqrt{2}}u_l{\\gamma}^{\\mu}\\left(1-{\\gamma}_5\\right)v_{\\bar{\\nu}}\\\\\n&\\times \\left\\{i V_{LD}^V \\epsilon _{\\alpha \\beta \\mu \\sigma}p_{\\gamma }^{\\beta }\\varepsilon _{\\gamma }^{{\\alpha}*}p_P^{\\sigma}\n\\right.\\\\ &\\left.\n+A_{LD}^V\\left[\\left(p_P{\\cdot}p_{\\gamma}\\right)\\varepsilon _{\\gamma}^{{\\mu}*}-\\left(p_P{\\cdot}\\varepsilon\\right)p_{\\gamma}^{\\mu}\\right]\\right\\},\n\\label{eq:25}\n\\end{split}\n\\end{equation}\nwhere:\n\\begin{equation}\n\\begin{split}\n&V_{LD}^V=\\frac{e^{i\\phi _V}C_{qV}f_V}{-m_V^2+i m_V {\\Gamma}_V}\\frac{2}{m_P+m_V}V(q^2)\\\\\n&A_{LD}^V=\\frac{e^{i\\phi _V}C_{qV}f_V}{-m_V^2+i m_V {\\Gamma}_V}\\frac{m_P+m_V}{p_P{\\cdot}p_{\\gamma}}A_1(q^2)\n\\label{eq:26}.\n\\end{split}\n\\end{equation}\n\n\\subsection{\\label{sec:level4}IV Numerical Result and Discussion}\n\nThe numerical calculation is performed in the center-of-mass frame of the heavy meson, and the momentum of the photon is taken as $\\left(E_{\\gamma},0,0,-E_{\\gamma}\\right)$. For the input parameters, the masses of the quarks are taken as\n\\begin{displaymath}\n\\begin{array}{ll}\nm_u = m_d=0.08{\\rm GeV},& m_s = 0.30{\\rm GeV}, \\\\\nm_b = 4.98{\\rm GeV},& m_c = 1.54{\\rm GeV}\n\\end{array}\n\\end{displaymath}\nwhich are taken to be consistent with that used to derive the wave function of the heavy meson in the relativistic potential model in Ref. \\cite{sdbase}.\n\nThe electron and neutrinos are taken to be massless, the masses of the other leptons are taken from PDG \\cite{particaldatagroup}.  The Cabibbo-Kobayashi-Maskawa (CKM) matrix elements are:\n\\begin{displaymath}\nV_{cd}=0.2259, V_{cs}=0.974, V_{ub}=0.00389.\\\\\n\\end{displaymath}\n\nThe infrared divergence appears as the energy of the real photon goes to soft limit or the momentum of the photon is parallel to the momentum of a massless lepton. This divergence can be canceled when the decay width of the radiative leptonic decay is added with the pure leptonic decay width, in which one-loop radiative corrections are included \\cite{irdiv}. The radiative leptonic decay with the energy of the photon lower than the experimental resolution can not be distinguished from the pure leptonic decay. Only photons with the energy larger than the experimental resolution can be distinguished. Therefore the radiative leptonic decay width depends on the photon energy resolution. The photon energy resolution can be a few MeV in experiment \\cite{alice}. The dependence of the decay width on the resolution $\\Delta E_{\\gamma}$ is shown in Table \\ref{tab:IRDIV} and Fig.~\\ref{fig:irdiv}. For example, if taking $\\Delta E_{\\gamma}=10\\;{\\rm MeV}$, the decay width and branching ratio of $D\\to \\gamma e \\bar{\\nu}_e$ are\n\\begin{equation}\n\\begin{split}\n&\\Gamma (D\\to \\gamma e \\bar{\\nu}_e)=1.98\\times 10^{-18}{\\rm GeV},\\\\\n&Br(D\\to \\gamma e \\bar{\\nu}_e)=3.29\\times 10^{-6}\n\\label{eq:27}.\n\\end{split}\n\\end{equation}\nIn the following all the numerical calculation is performed by taking the resolution $\\Delta E_{\\gamma}=10\\;{\\rm MeV}$.\n\n\\begin{figure}\n\\includegraphics[scale=0.35]{fig3.eps}\n\\caption{\\label{fig:irdiv} Decay widths of $D^-\\to \\gamma e \\bar{\\nu}$ as a function of $\\Delta E_{\\gamma}$.}\n\\end{figure}\n\\begin{table}\n\\caption{\\label{tab:IRDIV}The dependence of the decay width of $D^-\\to \\gamma e \\bar{\\nu}$ on the photon\nresolution  $\\Delta E_{\\gamma}$.}\n\\begin{tabular}{cccc}\\hline\\hline\n$\\Delta E_{\\gamma}$&$\\Gamma _{D^-\\to \\gamma l \\bar{\\nu}}$&$\\Delta E_{\\gamma}$&$\\Gamma _{D^-\\to \\gamma l \\bar{\\nu}}$\\\\\n(MeV) & $(\\times 10^{-18}{\\rm GeV})$ & (MeV) & $(\\times 10^{-18}{\\rm GeV})$\\\\\n\\hline\n$5 $ & $2.30$ & 55  & $1.25 $\\\\\n$10$ & $1.98$ & 60  & $1.22 $\\\\\n$15$ & $1.80$ & 65  & $1.19 $\\\\\n$20$ & $1.67$ & 70  & $1.16 $\\\\\n$25$ & $1.57$ & 75  & $1.14 $\\\\\n$30$ & $1.49$ & 80  & $1.12 $\\\\\n$35$ & $1.43$ & 85  & $1.10 $\\\\\n$40$ & $1.38$ & 90  & $1.08 $\\\\\n$45$ & $1.33$ & 95  & $1.06 $\\\\\n$50$ & $1.29$ & 100 & $1.04 $\\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\nWe also checked that the decay width is not sensitive to the cutoff of the angle between the momentum of the photon and the electron. We find taking different values for the angle cutoff $\\Delta \\theta$, the result changes extremely slowly.\n\nTo show the contribution of each diagram in Fig.\\ref{fig:feynsd}, we list each diagram's contribution to the decay width in Table \\ref{tab:SDResult}. It is shown that, although the amplitude of the diagram with the photon emitted from the heavy quark is suppressed by a factor of the heavy quark mass in the dominator of the propagator, which is like $\\frac{i}{\\slashed p_{\\gamma}-\\slashed p_Q - m_Q}$, for the cases of $D$ and $D_s$ mesons, the mass of $c$ quark is not large enough, so the contributions of diagrams $a$, $b$ and $c$ are all at the same order, but for the case of $B$ decay, the suppression is large, the contribution of diagram $b$ dominates. It can also be shown that, the contributions of the diagrams  in Fig.\\ref{fig:feynsd} interfere destructively, especially in the case of $D$ and $D_s$ mesons, this is consistent with Ref.  \\cite{workbefore2}.\n\\begin{table}\n\\caption{\\label{tab:SDResult} The decay width contributed by each diagram in Fig. \\ref{fig:feynsd}. In the first column, $\\Gamma_a$, $\\Gamma_b$ and $\\Gamma_c$ means the contributions of diagrams $a$, $b$ and $c$, respectively, $\\Gamma_{a+b}$ the contribution of the diagrams $a$ and $b$, $\\Gamma_{a+b+c}$ the total contribution of diagrams $a$, $b$ and $c$.}\n\\begin{tabular}{cccc}\\hline\\hline\nWidth&$D^-\\to \\gamma e \\bar{\\nu}_e$&$D_s^-\\to \\gamma e \\bar{\\nu}_e$&$B^-\\to \\gamma e \\bar{\\nu}_e$ \\\\\n(GeV) & & &  \\\\\n\\hline\n$\\Gamma_a$ & $1.37\\times 10^{-17}$ & $3.06\\times 10^{-16}$ & $9.14\\times 10^{-21}$\\\\\n$\\Gamma_b$ & $1.35\\times 10^{-17}$ & $1.94\\times 10^{-16}$ & $1.01\\times 10^{-18}$\\\\\n$\\Gamma_c$ & $9.78\\times 10^{-18}$ & $3.23\\times 10^{-16}$ & $5.83\\times 10^{-20}$\\\\\n$\\Gamma_{a+b}$ & $9.86\\times 10^{-18}$ & $3.08\\times 10^{-16}$ & $0.96\\times 10^{-18}$\\\\\n$\\Gamma_{a+b+c}$ & $1.98\\times 10^{-18}$ & $8.46\\times 10^{-18}$ & $0.82\\times 10^{-18}$\\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nWe present the branching ratios with only the short-distance contributions for all the decay modes in TABLE.\\ref{tab:allSDResult}. The branching ratio for $D^-\\to \\gamma \\tau \\bar{\\nu}_{\\tau}$ is very small, because the mass of $\\tau$ is very large, the phase space for this decay mode is too small.\n\\begin{table}\n\\caption{\\label{tab:allSDResult}Branching ratios of radiative leptonic decays of $B$ and $D$ mesons.\n$BR_{SD}$ is the branching ratios with only the short-distance contribution. $BR_{LD}$ stands for branching ratio with only long-distance contribution. For $D$ and $B$ mesons,\n$BR_{LD1}$ is the branching ratios of long-distance contribution via $\\rho$ meson, while\nfor $D_s$ meson, the long-distance contribution is via $\\phi$ meson. $BR_{LD2}$ is the branching ratios\nof long-distance contribution via $\\omega$ meson.}\n\\begin{tabular}{cccc}\\hline\\hline\nModes&$BR_{SD}$&$BR_{LD1}$ &$BR_{LD2}$\\\\\n\\hline\n$D^-\\to \\gamma e \\bar{\\nu}_e$ & $3.3\\times 10^{-6}$ & $7.5\\times 10^{-6}$ & $6.3\\times 10^{-6}$\\\\\n$D^-\\to \\gamma \\mu \\bar{\\nu}_{\\mu}$ & $1.6\\times 10^{-5}$ & $7.3\\times 10^{-6}$ & $6.1\\times 10^{-6}$\\\\\n$D^-\\to \\gamma \\tau \\bar{\\nu}_{\\tau}$ & $5.5\\times 10^{-9}$ & $9.1\\times 10^{-10}$ & $7.6\\times 10^{-10}$\\\\\n\\hline\n$D_s^-\\to \\gamma e \\bar{\\nu}_e$ & $6.8\\times 10^{-6}$  & $1.0\\times 10^{-4}$ & -\\\\\n$D_s^-\\to \\gamma \\mu \\bar{\\nu}_{\\mu}$ & $2.0\\times 10^{-4}$ & $1.0\\times 10^{-4}$ & -\\\\\n$D_s^-\\to \\gamma \\tau \\bar{\\nu}_{\\tau}$ & $1.1\\times 10^{-6}$ & $6.5\\times 10^{-8}$ & -\\\\\n\\hline\n$B^-\\to \\gamma e \\bar{\\nu}_e$ & $2.1\\times 10^{-6}$  & $5.8\\times 10^{-7}$ & $4.0\\times 10^{-7}$\\\\\n$B^-\\to \\gamma \\mu \\bar{\\nu}_{\\mu}$ & $2.0\\times 10^{-6}$ & $5.8\\times 10^{-7}$ & $4.0\\times 10^{-7}$\\\\\n$B^-\\to \\gamma \\tau \\bar{\\nu}_{\\tau}$ & $1.6\\times 10^{-6}$ & $3.8\\times 10^{-7}$ & $2.7\\times 10^{-7}$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe short-distance branching ratios obtained in this work is slightly smaller than the previous works \\cite{workbefore2}. More details about the quark momentum distribution are included in this work by using wave function obtained in the relativistic potential model.\n\n\nTo calculate the long-distance contribution, the transition amplitude $V\\to {\\gamma}$ is needed. The transition amplitude is related to the decay constant defined in eq.(\\ref{eq:20}). Using the data on the decay rate of $V\\to l^+l^-$ given in PDG \\cite{particaldatagroup}, the decay constants of the vector mesons can be extracted as\n\\begin{equation}\n\\begin{split}\n&f_{\\rho}=0.169869{\\rm GeV}, \\\\\n&f_{\\omega}=0.154631{\\rm GeV},\\\\\n&f_{\\phi}=0.231784{\\rm GeV}.\n\\end{split}\n\\label{eq:28}\n\\end{equation}\nThe $q^2$ dependence of the form factors defined in the hadronic matrix element $<V|J^{\\mu}_{V-A}|P>$ are taken to be the usual pole approximation as\n\\begin{displaymath}\nV(q^2)=\\frac{h_v}{1-\\frac{q^2}{M_V^2}}, \\;A_1(q^2)=\\frac{h_{a1}}{1-\\frac{q^2}{M_{A1}^2}}.\n\\end{displaymath}\nThe parameters in the form factors for $D\\to V$ and $B \\to V$ transitions are quoted from Refs. \\cite{pball} and \\cite{pball2}, they are\n\\begin{displaymath}\n\\begin{array}{ll}\nh_{v_{D\\to \\rho}}=1.0,& h_{a1_{D\\to \\rho}}=0.5;\\\\\nM_{V_{D\\to\\rho}}=2.5\\;{\\rm GeV};& M_{A1_{D\\to\\rho}}=2.5\\;{\\rm GeV}; \\\\\nh_{v_{B\\to \\rho}}=0.323,& M_{V_{B\\to\\rho}}^2=38.34\\;{\\rm GeV}^2;\\\\\nh_{a1_{B\\to \\rho}}=0.242,& M_{A1_{B\\to\\rho}}^2=37.51\\;{\\rm GeV}^2;\\\\\nh_{v_{B\\to\\omega}}=0.293,& M_{V_{B\\to \\omega}}^2=37.45\\;{\\rm GeV}^2;\\\\\nh_{a1_{B\\to \\omega}}=0.219,& M_{A1_{B\\to\\omega}}^2=37.01\\;{\\rm GeV}^2.\n\\end{array}\n\\end{displaymath}\nFor the form factor of $D\\to \\omega$, we assume it is the same as that of $D\\to \\rho$. The form factors of $D_s\\to \\phi$ is taken from \\cite{dsform} as\n\\begin{displaymath}\nh_{v_{D_s\\to \\phi}}=1.21,\\; h_{a1_{D_s\\to \\phi}}=0.55,\\; M_{V_{D_s \\to \\phi}}=2.08\\;{\\rm GeV}.\n\\end{displaymath}\nThe form factor $A_1$ for $D_s\\to \\phi$ transition is approximated as a constant because the $q^2$ dependence of $A1$ is very weak \\cite{dsform}.\n\nWith the parameters given above, the long-distance contributions to the decay width for each decay mode can be estimated, they are listed in Table~\\ref{tab:allSDResult}, where the short-distance and long-distance contributions to the branching ratios of each decay mode are presented separately .  It shows that for decays of $D$ and $D_s$ mesons, long-distance contributions are as large as or even larger than the short-distance contributions, while for the case of $B$ decays, short-distance contributions dominate, long-distance contributions are roughly $4\\sim 5$ times smaller than short-distance contributions.\n\nTo get the total decay width, including both the short and long-distance contributions, one has to know the relative phase between the long and short-distance amplitudes.  Unfortunately we do not konw the relative phases exactly upto now. We have to leave the relative phases as free parameters.   To show how the decay width depends on the relative phases, we show the decay widths of $B$, $D$  and $D_s\\to \\gamma e\\bar{\\nu}_e$  decays in Fig.~\\ref{fig:allampl} as an example. In the case of $D\\to \\gamma e\\bar{\\nu}_e$ decay, because the long-distance contributions are as important as the short-distance contributions, the relative phases between the long and short-diatance contributions can affect the decay widths considerably, the decay widths can change several times.  For $D_s\\to \\gamma e\\bar{\\nu}_e$ decay, the long-distance contribution dominates (see Table \\ref{tab:allSDResult}), the dependence of the total branching ratio on the relative phase is weak.  While for $B\\to\\gamma e\\bar{\\nu}_e$ meson decay, the amplitudes of the short and long-distance contributions are at the same order, therefore the decay width still depends on the relative phase severely. For the case of the other decay modes, the dependence of the total branching ratios on the relative phases are illustrated numerically in Table \\ref{tab:AllResult}. The situation is similar to the $B$, $D$ and $D_s\\to\\gamma e\\bar{\\nu}_e$ decay modes. The contribution of the long-distance physics is important, in general the branching ratio heavily depends on the relative phase between the long and short-distance contributions. Some decay modes can be greatly enhanced by the long-distance contributions. The branching ratio of $D_s\\to\\gamma e\\bar{\\nu}_e$ decay can be enhanced from the order $10^{-6}$ to $10^{-4}$.\n\n\n\\begin{figure}\n\\includegraphics[scale=0.5]{fig4.eps}\n\\caption{\\label{fig:allampl}From up to down, they are decay widths (${\\rm GeV}$) of $D^-$, $B^-$ and $D_s^-\\to\\gamma e^- \\bar{v}_e$ decays, including both the short and long-distance contributions, as functions of $\\phi _{\\rho}$ and $\\phi _{\\omega}$ in the cases of $D^-$ and $B^-$ meson decays, and $\\phi _{\\phi}$ in the case of $D_s^-$ meson decay. }.\n\\end{figure}\n\n\n\\begin{table*}\n\\caption{\\label{tab:AllResult}Total branching ratios of the radiative leptonic decays including both the short and\nlong-distance contributions. In case of $D$ meson decays, for illustration, the relative phase $\\phi=\\phi _{\\omega}$ and\n$\\phi _{\\rho}=0$ are taken. While for $B$ meson decays, $\\phi=\\phi _{\\rho}=\\phi _{\\omega}$ is taken, and\nin case of $D_s$ meson decays, $\\phi$ is $\\phi _{\\phi}$.}\n\\begin{tabular}{ccccccc}\\hline\\hline\nModes& $BR_{tot}(\\phi=0^o)$ & $BR_{tot}(\\phi=30^o)$ & $BR_{tot}(\\phi=60^o)$ & $BR_{tot}(\\phi=90^o)$ & $BR_{tot}(\\phi=120^o)$ & $BR_{tot}(\\phi=150^o)$\\\\\n\\hline\n$D\\to \\gamma e \\bar{\\nu }_e$ & $4.0\\times 10^{-6}$ & $1.1\\times 10^{-5}$ & $3.0\\times 10^{-5}$ & $4.1\\times 10^{-5}$ & $3.4\\times 10^{-5}$ & $1.5\\times 10^{-5}$\\\\\n$D\\to \\gamma \\mu \\bar{\\nu }_{\\mu}$ &$2.5\\times 10^{-5}$&$3.1\\times 10^{-5}$&$4.8\\times 10^{-5}$&$5.9\\times 10^{-5}$&$5.2\\times 10^{-5}$&$3.5\\times 10^{-5}$\\\\\n$D\\to \\gamma \\tau \\bar{\\nu }_{\\tau}$ &$5.7\\times 10^{-9}$&$7.5\\times 10^{-9}$&$1.2\\times 10^{-8}$&$1.4\\times 10^{-8}$&$1.2\\times 10^{-8}$&$8.0\\times 10^{-9}$\\\\\n\\hline\n$B\\to \\gamma e \\bar{\\nu }_e$ &$6.7\\times 10^{-6}$&$8.9\\times 10^{-6}$&$1.3\\times 10^{-5}$&$1.4\\times 10^{-5}$&$1.2\\times 10^{-5}$&$8.2\\times 10^{-6}$\\\\\n$B\\to \\gamma \\mu \\bar{\\nu }_{\\mu}$ &$6.8\\times 10^{-6}$&$8.9\\times 10^{-6}$&$1.3\\times 10^{-5}$&$1.4\\times 10^{-5}$&$1.2\\times 10^{-5}$&$8.3\\times 10^{-6}$\\\\\n$B\\to \\gamma \\tau \\bar{\\nu }_{\\tau}$ &$5.3\\times 10^{-6}$&$6.7\\times 10^{-6}$&$9.2\\times 10^{-6}$&$1.0\\times 10^{-5}$&$8.8\\times 10^{-6}$&$6.3\\times 10^{-6}$\\\\\n\\hline\n$D_s\\to \\gamma e \\bar{\\nu }_e$ &$1.5\\times 10^{-4}$&$1.3\\times 10^{-4}$&$0.9\\times 10^{-4}$&$0.7\\times 10^{-4}$&$0.9\\times 10^{-4}$&$1.3\\times 10^{-4}$\\\\\n$D_s\\to \\gamma \\mu \\bar{\\nu }_{\\mu}$ &$4.5\\times 10^{-4}$&$4.4\\times 10^{-4}$&$4.1\\times 10^{-4}$&$3.9\\times 10^{-4}$&$4.1\\times 10^{-4}$&$4.4\\times 10^{-4}$\\\\\n$D_s\\to \\gamma \\tau \\bar{\\nu }_{\\tau}$ &$1.0\\times 10^{-6}$&$1.0\\times 10^{-6}$&$1.2\\times 10^{-6}$&$1.3\\times 10^{-6}$&$1.2\\times 10^{-6}$&$1.0\\times 10^{-6}$\\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{\\label{sec:level5}V Summary}\n\nThe radiative leptonic decays of $B$, $D$ and $D_s \\to l\\bar{\\nu}\\gamma$ are studied in this work. The short-distance contribution is calculated by using the wave functions of the heavy mesons obtained in the relativistic potential model, more details about the quark-momentum distribution are included in this work. In addition to the short-distance contribution, the long-distance contribution is also estimated in the vector meson dominance model. The study shows that the long-distance contributions can seriously affect the decay rates. The branching ratio of $D_s^-\\to \\gamma e \\bar{\\nu_e}$ can be enhanced to the order of $10^{-4}$, which should only be at the order of $10^{-6}$ if only considering the short-distance contribution.\n\nThis work is supported in part by the\nNational Natural Science Foundation of China under contracts Nos.\n10575108, 10975077, 10735080, 11125525 and by the Fundamental Research Funds for the\nCentral Universities No. 65030021.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAmong the most intriguing features of fundamental theories of extended objects are novel types of symmetries  and concomitant generalized notions of geometry. Particularly interesting examples of these symmetries are T-duality in closed string theory and the equivalence of commutative\/noncommutative descriptions in open string theory. These symmetries have their natural settings in generalized geometry and noncommutative geometry. Low energy effective theories link the fundamental theories to potentially observable phenomena in (target) spacetime. Interestingly, the spacetime remnants of the stringy symmetries can fix these effective theories essentially uniquely without the need of actual string computations: ``string theory with no strings attached.''\n\nThe main objective of this paper is to study this interplay of symmetry and geometry in the case of higher dimensional extended objects (branes). \nMore precisely, we intended to extend, clarify and further develop the construction outlined in \\cite{Jurco:2012yv} that tackles the quest to find an all-order effective action for a system of multiple $p$-branes ending on a $p'$-brane.\nThe result for the case of open strings ending on a single D-brane is well known: The Dirac-Born-Infeld action provides an effective description to all orders in $\\alpha'$ \\cite{Fradkin:1985ys,Leigh:1989jq,Tseytlin:1999dj}. The way that this effective action has originally been derived from first principles in string theory is rather indirect: The effective action is determined by requiring that its equations of motion double as consistency conditions for an anomaly free world sheet quantization of the fundamental string. A more direct target space approach can be based on T-duality arguments.\nMoreover, there is are equivalent commutative and non-commutative descriptions \\cite{Seiberg:1999vs}, where the equivalency condition fixes the action essentially uniquely \\cite{Cornalba:2000ua,Jurco:2001my}. This ``commutative-noncommutative duality\" has been used also to study the non-abelian DBI action \\cite{Terashima:2000ej,Cornalba:2000ua}. In the context of the M2\/M5 brane system a generalization has been proposed in\\cite{Chen:2010br}. \n\nIn this paper, we focus only on the bosonic part of the action. The main idea of \\cite{Jurco:2012yv}, inspired by \\cite{Chen:2010br}, was to introduce open-closed membrane relations, and a Nambu-Poisson map which can be used to relate ordinary higher gauge theory to a new Nambu gauge theory \\cite{Ho:2013iia,Jurco:2014aza,Ho:2013opa,Ho:2013paa}. See also the work of P.-M. Ho et al. \\cite{Ho:2007vk,Ho:2008nn,Ho:2008ve,Ho:2009zt} and K. Furuuchi et al. \\cite{Furuuchi:2009zx,Furuuchi:2010sp} on relation of M2\/M5 to Nambu-Poisson structures.\nIt turns out that the requirement of ``commutative-noncommutative duality\" determines the bosonic part of the effective action essentially uniquely. Interesting open problems are to determine, in the case of a M5-brane, the form of the full supersymmetric action and to check consistency with $\\kappa$-symmetry and (nonlinear) selfduality.\n\nNambu-Poisson structures were first considered by Y. Nambu already in 1973 \\cite{1973PhRvD...7.2405N}, and generalized and axiomatized more then 20 years later by L. Takhtajan \\cite{Takhtajan:1993vr}. The axioms of Nambu-Poisson structures, although they seem to be a direct generalization of Poisson structures, are in fact very restrictive. This was already conjectured in the pioneering paper \\cite{Takhtajan:1993vr} and proved three years later in \\cite{decomposability,Gautheron}. For a modern treatment of Nambu-Poisson structures see \\cite{hagiwara,2011ScChA..54..437B,2010arXiv1003.1004Z}.\n\nMatrix-model like actions using Nambu-Poisson structures are a current focus of research (see e.g. \\cite{Park:2008qe,Sato:2010ca,DeBellis:2010sy, Chu:2011yd}) motivated by the works of \\cite{Basu:2004ed,SheikhJabbari:2005mf,PhysRevD.75.045020,Bagger:2007jr,Gustavsson:2007vu} and others. See also \\cite{Maldacena:2002rb, SheikhJabbari:2004ik} for further reference. \nAmong the early approaches, the one closest to ours is the one of \\cite{Cederwall:1997gg,Bao:2006ef}, which uses $\\kappa$-symmetry as a guiding principle and features a non-linear self-duality condition. It avoids the use of an auxiliary chiral scalar \\cite{Pasti:1997gx} with its covariance problems following a suggestion of \\cite{Witten:1996hc}. For these and alternative formulations, e.g., those of \\cite{Howe:1996yn}, based on superspace embedding and $\\kappa$-symmetry, we refer to the reviews \\cite{Sorokin:1999jx,Simon:2011rw}.\n\nGeneralized geometry was introduced by N. Hitchin in \\cite{Hitchin:2004ut,2005math......8618H,2006CMaPh.265..131H}. It was further elaborated in \\cite{Gualtieri:2003dx}. Although Hitchin certainly recognized the possible importance for string backgrounds, and commented on it in \\cite{Hitchin:2004ut}, this direction is not pursued there.\nRecently, a focus of applications of generalized geometry, is superstring theory and supergravity. Here we mention closely related work \\cite{Coimbra:2011nw,Coimbra:2012af}. The role of generalized geometries in M-theory was previously examined by C.M. Hull in \\cite{Hull:2007zu}.\nA further focus is the construction of the field theories based on objects of generalized geometry. This is mainly pursued in \\cite{Kotov:2004wz,2005JGP....54..400B} and in \\cite{Kotov:2010wr}, see also \\cite{Zucchini:2005rh}. Generalized geometry (mostly Courant algebroid brackets) was also used in relation to worldsheet algebras and non-geometric backgrounds. See, for example, \\cite{Alekseev:2004np,Bonelli:2005ti,2011JHEP...03..074E} and \\cite{Halmagyi:2009te,Halmagyi:2008dr}. One should also mention the use of generalized geometry in the description of T-duality, see\\cite{2011arXiv1106.1747C}, or the lecture notes \\cite{Bouwknegt:2010zz}. An outline of the relation of T-duality with generalized geometry can be found in \\cite{Grana:2008yw}. Finally, there is an interesting interpretation of D-branes in string theory as Dirac structures of generalized geometry in \\cite{Asakawa:2012px,Asakawa:2014eva}.\nFinally, in \\cite{Jurco:2013upa}, we have used generalized geometry to describe the relation between string theory and non-commutative geometry.\n\nThis paper is organized as follows:\n\nIn section \\ref{sec_membrane}, we review  basic facts concerning classical membrane actions. In particular, we recall how gauge fixing  can be used to find a convenient form of the action.  We show that the corresponding Hamiltonian density is a fiberwise metric on a certain vector bundle. We present background field redefinitions, generalizing the well-known open-closed relations of Seiberg and Witten.\n\nIn section \\ref{sec_nambu}, we describe the sigma model dual to the membrane action. It is a straightforward generalization of the non-topological Poisson sigma model of the $p=1$ case.\n\nSection \\ref{sec_geometry} sets up the geometrical framework for the field redefinitions of the previous sections. An extension  of  generalized geometry is used to describe open-closed relations as an orthogonal transformation of the generalized metric on the vector bundle $TM \\oplus \\TM{p} \\oplus T^{\\ast}M \\oplus \\cTM{p}$. Compared to the $p=1$ string case, we find the need for a second ``doubling'' of the geometry. The split in $TM$ and $\\TM{p}$ has its origin in  gauge fixing of the auxiliary metric on the \\mbox{$p+1$}-dimensional brane world volume and the two parts are related to the temporal and spatial worldvolume directions. To the best of our knowledge, this particular structure $W \\oplus W^*$ with $W = TM \\oplus \\TM{p}$ has not been considered in the context of M-theory before.\n\nIn section \\ref{sec_gaugeF}, we introduce the $(p+1)$-form gauge field $F$ as a fluctuation of the original membrane background. We show that this can be viewed as an orthogonal transformation of the generalized metric describing the membrane backgrounds. On the other hand, the original background can equivalently be described in terms of open variables and this description can be extended to include fluctuations. Algebraic manipulations are used to identify the pertinent background fields. The construction requires the introduction of a target manifold diffeomorphism, which generalizes the (semi-classical) Seiberg-Witten map from the string to the $p>1$ brane case.\n\nThis map is explicitly constructed in section \\ref{sec_SWmap} using a generalization of Moser's lemma. The key ingredient is the fact that $\\Pi$, which appears in the open-closed relations, can be chosen to be a Nambu-Poisson tensor. \nAttention is paid to a correct mathematical formulation of the analogue of a symplectic volume form for Nambu-Poisson structures.\n\nBased on the results of the previous sections, we prove in  section \\ref{DBI} the equivalence of a commutative and semiclassically noncommutative DBI action. We present various forms of the same action using determinant identities of block matrices.  Finally, we compare our action to existing proposals for the M5-brane action.\n\nIn section \\ref{sec_BIG}, we show that the Nambu-Poisson structure $\\Pi$ can be chosen to be the pseudoinverse of the $(p+1)$-form background field $C$. In analogy with the $p=1$ case, we call this choice ``background independent gauge''. However, for $p>1$ we have to consider both algebraic and geometric properties of $C$ in order to obtain a well defined Nambu-Poisson tensor $\\Pi$. The generalized geometry formalism developed in section \\ref{sec_geometry} is used to derive the results in a way that looks formally  identical to the much easier $p=1$ case. (This is a nice example of the power of generalized geometry.)\n\nIn section \\ref{sec_NCdirections}, we introduce a convenient splitting of the tangent bundle and rewrite all membrane backgrounds in coordinates adapted to this splitting using a block matrix formalism. We introduce an appropriate generalization of the double scaling limit of \\cite{Seiberg:1999vs} to cut off the series expansion of the effective action.\n\nIn the final section \\ref{sec_matrix} of the paper, we use background independent gauge, double scaling limit, and coordinates adapted to the non-commutative directions to expand the DBI action up to first order in the scaling parameter. It turns out that this double scaling limit cuts off the infinite series in a physically meaningful way. We identify a possible candidate for the generalization of a matrix model.\nFor a discussion of the underlying Nambu-Poisson gauge theory we refer to \\cite{Jurco:2014aza}.\n\n\\section{Conventions}\n\nThorough the paper, $p>0$ is a fixed positive integer. Furthermore, we assume that we are given a  $(p+1)$-dimensional compact orientable\nworldvolume $\\Sigma$ with local coordinates\n$(\\sigma^{0}, \\dots, \\sigma^{p})$. We may interpret $\\sigma^{0}$ as a time parameter. Integration over all coordinates is\nindicated by $\\int d^{p+1}\\sigma$, whereas the integration over\nspace coordinates $(\\sigma^{1}, \\dots, \\sigma^{p})$ is indicted\nas $\\int d^{p}\\sigma$. Indices corresponding to the worldvolume\ncoordinates are denoted by Greek characters $\\alpha, \\beta, \\dots$, etc.\nAs usual, $\\partial_{\\alpha} \\equiv \\frac{\\partial}{\\partial\n\\sigma^{\\alpha}}$.\nWe assume that the $n$-dimensional target manifold $M$ is equipped\nwith a set of local coordinates $(y^{1},\\dots,y^{n})$. We denote the\ncorresponding indices by lower case Latin characters $i,j,k, \\dots$, etc.\nUpper case Latin characters $I,J,K, \\dots$, etc. will denote strictly\nordered $p$-tuples of indices corresponding to $(y)$ coordinates,\ne.g., $I = (i_{1}, \\dots, i_{p})$ with $1\\leq i_{1} < \\dots <\ni_{p}\\leq n$. We use the shorthand notation\n$\\partial_{J} \\equiv \\ppy{}{j_{1}} \\^ \\dots \\^ \\ppy{}{j_{p}}$ and\n$dy^{J} = dy^{j_{1}} \\^ \\dots \\^dy^{j_{p}}$. The degree $q$-parts of the exterior\nalgebras of vector fields $\\vf{}$ and forms  $\\Omega(M)$ are denoted by $\\vf{q}$ and $\\df{q}$, respectively.\n\nWhere-ever a metric $g$ on $M$ is introduced, we assume that it is positive definite, i.e.,  $(M,g)$ is a Riemannian manifold. With this choice we will find a natural interpretation of membrane backgrounds in terms of generalized geometry. For any metric tensor $g_{ij}$, we denote, as usually, by $g^{ij}$\nthe components of the inverse contravariant tensor.\n\nWe use the following convention to handle $(p+1)$-tensors on $M$.\nLet $B \\in \\df{p+1}$ be a $(p+1)$-form on $M$. We define the\ncorresponding vector bundle map $B_{\\flat}: \\TM{p} \\rightarrow\nT^{\\ast}M$ as $B_{\\flat}(Q) = B_{iJ} Q^{J} dy^{i}$, where $Q = Q^{J}\n\\partial_{J}$. We do not distinguish between vector bundle morphisms\nand the induced $C^{\\infty}(M)$-linear maps of smooth sections. We\nwill usually use the letter $B$ also for the $\\binom{n}{p} \\times n$\nmatrix of $B_{\\flat}$ in the local basis $\\partial_{J}$ of $\\vf{p}$\nand $dy^{i}$ of $\\df{1}$, that is $(B)_{i,J} = \\< \\partial_{i},\nB_{\\flat}(\\partial_{J}) \\>$. Similarly, let $\\Pi \\in \\vf{p+1}$; the\ninduced map $\\Pi^{\\sharp}: \\cTM{p} \\rightarrow TM$ is defined as\n$\\Pi^{\\sharp}(\\xi) = \\Pi^{iJ} \\xi_{J} \\partial_{i}$ for $\\xi =\n\\xi_{J} dy^{J}$. We use the letter $\\Pi$ also for the $\\binom{n}{p}\n\\times n$ matrix of $\\Pi^{\\sharp}$, that is $(\\Pi)^{i,J} = \\<dy^{i},\n\\Pi^{\\sharp}(dy^{J})\\>$. Clearly, with these conventions $(B)_{i,J} =\nB_{iJ}$ and $(\\Pi)^{i,J} = \\Pi^{iJ}$.\n\nLet $X: \\Sigma \\rightarrow M$ be a smooth map. We use the notation\n$X^{i} = y^{i} \\circ X$, and correspondingly $dX^{i} = d(X^{i}) =\nX^{\\ast}(dy^{i})$. Similarly, $dX^{J} = X^{\\ast}(dy^{J})$. We reserve\nthe symbol $\\px{J}$ for spatial components of the $p$-form $dX^{J}$,\nthat is, $\\px{J} = (dX^{J})_{1 \\dots p}$. We define the generalized\nKronecker delta $\\delta_{i_{1} \\dots i_{p}}^{j_{1}\n\\dots j_{p}}$ to be  $+1$ whenever the top $p$-index constitutes an\neven permutation of the bottom one, $-1$ if for the odd permutation,\nand $0$ otherwise. In other words, $\\delta_{i_{1} \\dots i_{p}}^{j_{1} \\dots j_{p}} = p! \\cdot \\delta_{[i_{1}}^{[j_{1}} \\dots \\delta_{i_{p}]}^{j_{p}]}$. We use the convention $\\epsilon_{i_{1} \\dots\ni_{p}} \\equiv \\epsilon^{i_{1} \\dots i_{p}} \\equiv \\delta_{i_{1}\n\\dots i_{p}}^{1 \\dots p} \\equiv \\delta_{1 \\dots p}^{i_{1} \\dots\ni_{p}}$. Thus, in this notation we  have $\\px{I} =\n\\partial_{l_{1}}X^{i_{1}} \\cdots\n\\partial_{l_{p}}X^{i_{p}} \\epsilon^{l_{1} \\dots l_{p}}$.\n\n\n\n\n\\section{Membrane actions} \\label{sec_membrane}\n\nThe most straightforward generalization of the relativistic string\naction to higher dimensional world volumes is the Nambu-Goto $p$-brane action, simply measuring the\nvolume of the $p$-brane:\n\\begin{equation} \\label{def_ngaction}\nS_{NG}[X] = T_{p} \\int d^{p+1}\\sigma \\sqrt {\\det{( \\partial_{\\alpha} X^{i} \\partial_{\\beta} X^{j} g_{ij} )}},\n\\end{equation}\nwhere $g_{ij}$ are components of the positive definite target  space metric $g$, and\n$X: \\Sigma \\rightarrow M$ is the $n$-tuple of scalar fields\ndescribing the $p$-brane. In a similar manner as for the string action,\none can introduce an auxiliary Riemannian metric $h$ on $\\Sigma$ and\nfind the classically equivalent Polyakov action of the $p$-brane:\n\n\\begin{equation} \\label{def_polyakov}\n S_{P}[X,h] = \\frac{T'_{p}}{2} \\int d^{p+1}\\sigma \\sqrt{h} \\Big( h^{\\alpha \\beta} \\partial_{\\alpha}X^{i} \\partial_{\\beta}X^{j} g_{ij} - (p-1)\\lambda \\Big),\n \\end{equation}\nwhere $\\lambda > 0$ can be chosen arbitrarily (but fixed), and $T'_{p}\n= \\lambda^{\\frac{p-1}{2}} T_{p}$. Using the equations of motion for\n$h^{\\alpha \\beta}$'s:\n\\begin{equation} \\label{eq_eqmhab}\n\\frac{1}{2} h_{\\alpha \\beta} \\big( h^{\\gamma \\delta} g_{\\gamma \\delta} - (p-1)\\lambda \\big) = g_{\\alpha \\beta},\n\\end{equation}\nwhere $g_{\\alpha \\beta} = [X^{\\ast}(g)]_{\\alpha \\beta} \\equiv\n\\partial_{\\alpha}X^{i} \\partial_{\\beta}X^{j} g_{ij}$, in $S_{P}$,\none gets back to (\\ref{def_ngaction}).  In the rest of the paper, we will\nchoose $T_{p} \\equiv 1$. Using reparametrization invariance, one\ncan always (at least locally) choose coordinates $(\\sigma^{0},\n\\dots, \\sigma^{p})$ such that $h_{00} = \\lambda^{p-1} \\det\nh_{ab}$, $h_{0a} = 0$, where $h_{ab}$ denotes the space-like\ncomponents of the metric. \nIn this gauge, the first term in\naction (\\ref{def_polyakov}) splits into two parts, one of them\ncontaining only the spatial derivatives of $X^{i}$ and the spatial\ncomponents of the metric $h$. Using now the equations of motion for\n$h_{ab}$, one gets the gauge fixed Polyakov action\\footnote{The gauge constraints on $h_{a0}$, $h_{0b}$ and $h_{00}$ imply an energy-momentum tensor with vanishing components $T_{a0} = T_{0a}$ and $T_{00}$. These constraints must be considered along with the equations of motion of the action (\\ref{def_polyakovgf}), to ensure equivalence with the actions (\\ref{def_ngaction}) and (\\ref{def_polyakov}). As discussed in \\cite{Bars}, the subgroup of the diffeomorphism symmetries that remains after gauge fixing is a symmetry of the gauge-fixed p-brane action (\\ref{def_polyakovgf}) and also transforms the pertinent components of the energy-momentum tensor into one another (even if they are not set equal to zero). The constraints can thus be consistently imposed at the level of\nstates.}\n\\begin{equation} \\label{def_polyakovgf}\nS_{P}^{gf}[X] = \\frac{1}{2} \\int d^{p+1}\\sigma\n\\big\\{\\partial_{0}X^{i} \\partial_{0}X^{j} g_{ij} +\n\\det{(\\partial_{a}X^{i} \\partial_{b}X^{j} g_{ij})} \\big\\}.\n\\end{equation}\nThe second term can be rewritten in a more convenient form once we\ndefine\n\\begin{equation} \\label{def_tensorofg}\n\\~g_{IJ} = \\sum_{\\pi \\in \\Sigma_{p}} sgn(\\pi) g_{i_{\\pi(1)} j_{1}}\n\\dots g_{i_{\\pi(p)} j_{p}} \\equiv  \\delta^{k_{1} \\dots k_{p}}_{I}\ng_{k_1 j_1} \\dots g_{k_p j_p}.\n\\end{equation}\nUsing this notation, one can write\n\\begin{equation} \\label{def_polyakovgf2}\nS_{P}^{gf}[X] = \\frac{1}{2} \\int d^{p+1}\\sigma \\big\\{ \\partial_{0}X^{i} \\partial_{0}X^{j} g_{ij} +\n\\px{I} \\px{J} \\~g_{IJ} \\big\\}.\n\\end{equation}\nFrom now on, assume that $g$ is a positive definite metric on $M$.\nNote that from the symmetry of $g$ it follows that $\\~g_{IJ} =\n\\~g_{JI}$.  We can view $\\~g$ as a fibrewise bilinear form on the vector\nbundle $\\TM{p}$. Moreover, at any $m \\in M$, one can define the\nbasis $(E_{I})$ of $\\Lambda^{p} T_{m}M$ as $E_{I} = e_{i_{1}} \\^\n\\dots \\^ e_{i_{p}}$, where $(e_{1}, \\dots, e_{n})$ is the\northonormal basis for the quadratic form $g(m)$ at $m \\in M$. In this\nbasis one has $\\~g(m)(E_{I},E_{J}) = \\delta_{I,J}$, which shows that $\\~g$ is a positive definite fibrewise metric on $\\TM{p}$.\n\nFor any $C \\in \\df{p+1}$, we can add the following coupling term to the action:\n\\begin{equation}\nS_{C}[X] = - i \\int_{\\Sigma} X^{\\ast}(C) = -i \\int d^{p+1}\\sigma \\partial_{0}X^{i} \\px{J} C_{iJ}.\n\\end{equation}\nThe resulting gauge fixed Polyakov action $S_{P}^{tot}[X] = S_{P}^{gf}[X] + S_{C}[X]$ has the form\n\\begin{equation} \\label{def_actiontot}\nS_{P}^{tot}[X] = \\frac{1}{2} \\int d^{p+1}\\sigma \\big\\{ \\partial_{0}X^{i}\n\\partial_{0}X^{j} g_{ij} +\n\\px{I} \\px{J} \\~g_{IJ} - 2i \\partial_{0}X^{i} \\px{J} C_{iJ} \\big\\}.\n\\end{equation}\nThis can be written in the compact matrix form by defining an ($n +\n\\binom{n}{p}$)-row vector \n\\[\n\\Psi = \\bv{i\\partial_{0}X^{i}}{\\px{J}} .\n\\]\nThe action then has the block matrix form\n\\begin{equation} \\label{def_actiontotmatrix}\nS_{P}^{tot}[X] = \\frac{1}{2} \\int d^{p+1}\\sigma \\{ \\Psi^{\\dagger} \\bm{g}{C}{-C^{T}}{\\~g} \\Psi \\}.\n\\end{equation}\n\nFrom now on, unless explicitly mentioned, we may assume that $\\~g$\nis not necessarily of the form (\\ref{def_tensorofg}), i.e., $\\~g$\ncan be any positive definite fibrewise metric on $\\TM{p}$. Any\nfurther discussions will, of course, be valid also for the special\ncase (\\ref{def_tensorofg}). Since $g$ is non-degenerate, we can pass\nfrom the Lagrangian to the Hamiltonian formalism and vice versa. The\ncorresponding Hamiltonian has the form\n\\begin{equation} \\label{def_polyakovham}\nH_{P}^{tot}[X,P] = -\\frac{1}{2} \\int d^{p}\\sigma \\bv{iP}{\\px{}}^{T}\n\\bm{g^{-1}}{-g^{-1}C}{-C^{T}g^{-1}}{\\~g + C^{T}g^{-1}C}\n\\bv{iP}{\\px{}}.\n\\end{equation}\nThe expression  $\\~g + C^{T}g^{-1}C$ in the Hamiltonian and a similar expression $g + C\\~g^{-1}C^{T}$ play the role of ``open membrane metrics'' and\nfirst appeared in the work of Duff and Lu \\cite{dufflu} already in~1990. Hamilton densities for membranes have also been discussed around that time, see e.g. \\cite{Bars}.\\footnote{We believe that the Hamiltonian (\\ref{def_polyakovham}) has been known, in this or a similar form, to experts for a long time but we were not able to trace it in even older literature, cf. \\cite{Duff:1989tf} for the string case. More recently, the Hamiltonian as well as the open membrane metrics appeared, e.g., in \\cite{Berman:2010is}. We thank D. Berman for bringing this paper to our attention.} The block matrix in the Hamiltonian can be viewed as  positive\ndefinite fibrewise metric $\\mathbf{G}$ on $T^{\\ast}M \\oplus \\TM{p}$\ndefined on sections as\n\n\\begin{equation}\\label{GenMet}\n\\mathbf{G}(\\alpha + \\bi{Q}, \\beta + \\bi{R}) = \\bv{\\alpha}{\\bi{Q}}^{T}\n\\bm{g^{-1}}{-g^{-1}C}{-C^{T}g^{-1}}{\\~g + C^{T}g^{-1}C}\n\\bv{\\beta}{\\bi{R}},\n\\end{equation}\nfor all $\\alpha, \\beta \\in \\df{1}$ and $\\bi{Q},\\bi{R} \\in \\vf{p}$. For $p=1$\nand $\\~g = g$, one gets exactly the inverse of the generalized metric corresponding\nto a Riemannian metric $g$ and a $2$-form $C$. Note that, analogously to the $p=1$ case,\n$\\mathbf{G}$ can be written as a product of  block lower triangular,\ndiagonal and upper triangular matrices:\n\\begin{equation} \\label{eq_gCdecomposition}\n\\mathbf{G} = \\bm{1}{0}{-C^{T}}{1} \\bm{g^{-1}}{0}{0}{\\~g} \\bm{1}{-C}{0}{1}.\n\\end{equation}\n\nBefore we proceed with our discussion of the corresponding Nambu\nsigma models, let us introduce another parametrization of the\nbackground fields $g$ and $C$. In analogy with the $p=1$ case, we\nshall refer to $g$ and $C$ as to the closed background fields. Let\n$\\mathbf{A}$ denote the matrix in the action\n(\\ref{def_actiontotmatrix}), that is,\n\\begin{equation}\\label{A}\n\\mathbf{A} = \\bm{g}{C}{-C^{T}}{\\~g}.\n\\end{equation}\nThis matrix is always invertible, explicitly:\n\\begin{equation} \\label{eq_Ainverse}\n\\mathbf{A}^{-1} = \\bm{(g + C\\~g^{-1}C^{T})^{-1}}{-(g +\nC\\~g^{-1}C^{T})^{-1}C\\~g^{-1}}{\\~g^{-1}C^{T}(g +\nC\\~g^{-1}C^{T})^{-1}}{(\\~g + C^{T}g^{-1}C)^{-1}}.\n\\end{equation}\nFurther, let us assume an arbitrary but fixed $(p+1)$-vector $\\Pi\n\\in \\mathfrak{X}^{p+1}(M)$ and consider a matrix $\\mathbf{B}$ of the form\n\n\\begin{equation}\n\\begin{split} \\label{def_Bmatrix}\n\\mathbf{B}   & = \\bm{G}{\\Phi}{-\\Phi^{T}}{\\~G}^{-1} + \\bm{0}{\\Pi}{-\\Pi^{T}}{0} \\\\\n             & = \\bm{(G+\\Phi \\~G^{-1} \\Phi^{T})^{-1}}{-(G+\\Phi \\~G \\Phi^{T})^{-1}\n             \\Phi \\~G^{-1} + \\Pi}{\\~G^{-1}\\Phi^{T} (G + \\Phi \\~G^{-1} \\Phi^{T})^{-1} -\n             \\Pi^{T}}{(\\~G + \\Phi^{T}G^{-1}\\Phi)^{-1}}\n\\end{split}\n\\end{equation}\nsuch that the equality $\\mathbf{A}^{-1}=\\mathbf{B}$, i.e.,\n\\begin{equation} \\label{eq_occorrespondece0}\n\\bm{g}{C}{-C^{T}}{\\~g}^{-1}=\\bm{G}{\\Phi}{-\\Phi^{T}}{\\~G}^{-1} +\n\\bm{0}{\\Pi}{-\\Pi^{T}}{0}\n\\end{equation}\nholds. This generalization was introduced and used in \\cite{Jurco:2012yv}. Again, in analogy with the case $p=1$, we will refer to $G$\nand $\\Phi$ as to the open backgrounds. More explicitly, we have the\nfollowing set of open-closed relations:\n\\begin{equation} \\label{eq_occorrespondence1}\ng + C\\~g^{-1}C^{T} = G + \\Phi \\~G^{-1} \\Phi^{T},\n\\end{equation}\n\\begin{equation} \\label{eq_occorrespondence2}\n\\~g + C^{T} g^{-1} C = \\~G + \\Phi^{T} G^{-1} \\Phi,\n\\end{equation}\n\\begin{equation} \\label{eq_occorrespondence3}\ng^{-1}C = G^{-1}\\Phi - \\Pi(\\~G + \\Phi^{T}G^{-1}\\Phi),\n\\end{equation}\n\\begin{equation} \\label{eq_occorrespondence4}\n\\Phi \\~G^{-1} = C\\~g^{-1} + (g + C\\~g^{-1}C^{T})\\Pi.\n\\end{equation}\n\nFor fixed $\\Pi$, given $(g,\\~g,C)$ there exist unique\n$(G,\\~G,\\Phi)$ such that the above relations are fulfilled, and vice\nversa. The explicit expressions are most directly seen from the\nequality $\\mathbf{A}= \\mathbf{B}^{-1}$, again using the formula for\nthe inverse of the block matrix $\\mathbf{B}$. In particular,\n\n\\begin{equation}\ng^{-1} = (1 - \\Phi \\Pi^{T})^{T} G^{-1} (1 - \\Phi \\Pi^{T}) +\n\\Pi \\~G \\Pi^{T},\n\\end{equation}\n\\begin{equation}\n\\~g^{-1} = (1 - \\Phi^{T} \\Pi)^{T} \\~G^{-1} (1 - \\Phi^{T} \\Pi)\n+ \\Pi^{T} G \\Pi,\n\\end{equation}\nand the explicit expression for $C$ can be found straightforwardly.\nObviously, the inverse relations are obtained simply by\ninterchanging $g \\leftrightarrow G$,  $\\~g \\leftrightarrow \\~G$, $C\n\\leftrightarrow \\Phi$, and $\\Pi \\leftrightarrow -\\Pi$.\nUsing these relations, we can write the action\n(\\ref{def_actiontotmatrix}) equivalently in terms of the open\nbackgrounds $G$, $\\Phi$ and the (so far auxiliary) $(p+1)$-vector\n$\\Pi$.\n\nIn terms of the corresponding Hamiltonian (\\ref{def_polyakovham}),\nthe above open-closed relations give just another factorization of\nthe matrix $\\mathbf{G}$. This time we have\n\\begin{equation} \\label{def_Gprime}\n\\mathbf{G} = \\bm{1}{\\Pi}{0}{1} \\bm{1}{0}{-\\Phi^{T}}{1}\n\\bm{G^{-1}}{0}{0}{\\~G} \\bm{1}{-\\Phi}{0}{1} \\bm{1}{0}{\\Pi^{T}}{1}.\n\\end{equation}\n\nIn the sequel it will be convenient to distinguish the respective\nexpressions of  above introduced matrices $\\mathbf{A}$ and\n$\\mathbf{G}$ in the closed and open variables. For the former we we\nshall use $\\mathbf{A_c}$ and $\\mathbf{G_c}$ and for the latter we\nintroduce $\\mathbf{A}_o$ and $\\mathbf{G}_o$, respectively. Hence the\nopen-closed relations can be expressed either way:\n$\\mathbf{A} \\equiv \\mathbf{A}_c = \\mathbf{A}_o \\equiv \\mathbf{B}^{-1}$ or\n$\\mathbf{G}_c=\\mathbf{G}_o$. Note, that the latter form is just\nequivalent to the statement about the decomposability of a 2x2 block\nmatrix with the invertible upper left block as a product of lower\ntriangular, diagonal, and upper triangular block matrices, the\ntriangular ones having unit matrices on the diagonal. Note that for\n$p=1$ and $\\~g = g$, the open-closed relations (see \\cite{Seiberg:1999vs}) are usually written\nsimply as\n\\begin{equation} \\label{eq_p1occorrespondence}\n\\frac{1}{g+C} = \\frac{1}{G+\\Phi} + \\Pi.\n\\end{equation}\nTo conclude this section, note that taking the determinant of the matrix $\\mathbf{A}_{c}$ , we may prove the useful identity:\n\\begin{equation} \\label{eq_detidentity}\n\\det{(\\~g + C^{T} g^{-1} C)} = \\frac{\\det{\\~g}}{\\det{g}} \\det{(g + C\\~g^{-1}C^{T})}.\n\\end{equation}\nTo show this, just note that $\\mathbf{A}_{c}$ can be decomposed in two different ways, either\n\\[ \\mathbf{A}_{c} = \\bm{1}{0}{-C^{T}g^{-1}}{1} \\bm{g}{0}{0}{(\\~g + C^{T}g^{-1}C)} \\bm{1}{g^{-1}C}{0}{1}, \\]\nor as\n\\[ \\mathbf{A}_{c} = \\bm{1}{C\\~g^{-1}}{0}{1} \\bm{(g+C\\~g^{-1}C^{T})}{0}{0}{\\~g} \\bm{1}{0}{-\\~g^{-1}C^{T}}{1}. \\]\nTaking the determinant of both expressions and comparing them yields (\\ref{eq_detidentity}).\n\\section{Nambu sigma model} \\label{sec_nambu}\nIn analogy with the $p=1$ case, we may ask whether there is a Nambu\nsigma model classically equivalent to the action\n(\\ref{def_actiontotmatrix}). To see this, introduce new auxiliary\nfields $\\eta_{i}$ and $\\~\\eta_{J}$, which transform according to\ntheir index structure under a change of coordinates on $M$. Define an\n$(n + \\binom{n}{p})$-row vector $\\Upsilon =\n\\bv{i \\eta_{i}}{\\~\\eta_{J}}$. The corresponding (non-topological)\nNambu sigma model then has the form:\n\\begin{equation} \\label{def_actionnsm}\nS_{NSM}[X,\\eta,\\~\\eta] = -\\int d^{p+1}\\sigma \\big\\{ \\frac{1}{2}\n\\Upsilon^{\\dagger} \\mathbf{A}^{-1} \\Upsilon + \\Upsilon^{\\dagger} \\Psi \\big\\},\n\\end{equation}\nwhere $\\mathbf{A}$ can be either of $\\mathbf{A_o}$ and\n$\\mathbf{A_c}$, supposing that the open-closed relations\n$\\mathbf{A_o}=\\mathbf{A_c}$ hold. Using the equations of motion for\n$\\Upsilon$, one gets back the Polyakov action\n(\\ref{def_actiontotmatrix}). For the detailed treatment of Nambu sigma models see \\cite{Jurco:2012gc}.\n\nYet another parametrization of $\\mathbf{A}^{-1}$ -- using new\nbackground fields $G_{N},\\~G_{N},\\Pi_{N}$, which we refer to as Nambu background\nfields\\footnote{Here, instead of fixing $\\Pi$ and finding open\nvariables in terms of closed ones, we fix $\\Phi$ to be zero and\nfind, again using the open-closed relations, unique $G_{N},\\~G_{N},\\Pi_{N}$ as\nfunctions of $\\mathfrak{g}, \\~g$ and $C$, or vice versa.} -- can be introduced\n\\begin{equation} \\label{def_Ainvnewfield}\n\\mathbf{A}^{-1} = \\bm{G_{N}^{-1}}{\\Pi_{N}}{-\\Pi_{N}^{T}}{\\~G_{N}^{-1}}.\n\\end{equation}\nWe will denote as $\\mathbf{A}_N$ the matrix $\\mathbf{A}$ expressed\nwith help of Nambu background fields $G_{N},\\~G_{N},\\Pi_{N}$. Using\n(\\ref{eq_Ainverse}), one gets the correspondence between closed and\nNambu sigma background fields:\n\n\\begin{equation} \\label{eq_correspondence1}\nG_{N} = g + C\\~g^{-1}C^{T},\n\\end{equation}\n\\begin{equation} \\label{eq_correspondence2}\n\\~G_{N} = \\~g + C^{T}g^{-1}C,\n\\end{equation}\n\\begin{equation} \\label{eq_correspondence3}\n\\Pi_{N} = -(g + C\\~g^{-1}C^{T})^{-1}C\\~g^{-1} = -g^{-1}C(\\~g + C^{T} g^{-1}C)^{-1}.\n\\end{equation}\n\nClearly, $G_{N}$ is a Riemannian metric on $M$ and $\\~G_{N}$ is a fibrewise\npositive  definite metric on $\\TM{p}$. It is important to note that\n in general, for $p>1$, $\\Pi_{N}: \\cTM{p} \\rightarrow TM$ is not\nnecessarily induced by a $(p+1)$-vector on $M$. This also means that it is not in general a Nambu-Poisson tensor. However; for $p=1$, it is easy to show that $\\Pi_{N}$ is a bivector.\n\nAlso note that even if $\\~g$ is a skew-symmetrized tensor product of\n$g$'s  (\\ref{def_tensorofg}), $\\~G_{N}$ is not in general the\nskew-symmetrized tensor product of $G_N$'s.\n\nThe converse relations are:\n\\begin{equation}\ng = (G_{N}^{-1} + \\Pi_{N} \\~G_{N} \\Pi_{N}^{T})^{-1},\n\\end{equation}\n\\begin{equation}\n\\~g = (\\~G_{N}^{-1} + \\Pi_{N}^{T} G_{N} \\Pi_{N})^{-1},\n\\end{equation}\n\\begin{equation}\nC = -(G_{N}^{-1} + \\Pi_{N} \\~G_{N} \\Pi_{N}^{T})^{-1} \\Pi_{N} \\~G_{N} = -G_{N} \\Pi_{N} (\\~G_{N}^{-1} +\n\\Pi_{N}^{T} G_{N} \\Pi_{N})^{-1}.\n\\end{equation}\n\nAgain, it is instructive to pass to the corresponding Hamiltonians.\nFirst, find the canonical Hamiltonian to (\\ref{def_actionnsm}), that\nis\n\\[ H^{c}_{NSM}[X,P,\\~\\eta] = \\int d^{p}\\sigma P_{i} \\partial_{0}X^{i} - \\mathcal{L}[X,P,\\~\\eta]. \\]\nSecond, use the equations of motion to get rid of $\\~\\eta$. In\nanalogy with the $p=1$ case, one expects that resulting  Hamiltonian\n$H_{NSM}$ coincides with (\\ref{def_polyakovham}), that is\n\\[ H_{NSM}[X,P] = H_{P}^{tot}[X,P].\\] Indeed, we get\n\\begin{equation} \\label{eq_nsmham}\nH_{NSM}[X,P] = -\\frac{1}{2} \\int d^{p}\\sigma \\bv{iP}{\\px{}}^{T} \\bm{G_{N}^{-1} + \\Pi_{N} \\~G_{N} \\Pi_{N}^{T}}{\\Pi_{N} \\~G_{N}}{\\~G_{N} \\Pi_{N}^{T}}{\t\\~G_{N}} \\bv{iP}{\\px{}}.\n\\end{equation}\nIf one plugs (\\ref{eq_correspondence1} - \\ref{eq_correspondence2})\nto (\\ref{eq_nsmham}), one obtains exactly the Hamiltonian\n(\\ref{def_polyakovham}). The matrix $\\mathbf{G}$ can be thus written\nas\n\\begin{equation} \\label{def_Gdualfields}\n\\mathbf{G} = \\bm{1}{\\Pi_{N}}{0}{1} \\bm{G_{N}^{-1}}{0}{0}{\\~G_{N}}\n\\bm{1}{0}{\\Pi_{N}^{T}}{1}\n\\end{equation}\nwhen using the Nambu background fields, in which case we shall\nintroduce the notation $\\mathbf{G}_N$ for it. This shows that to any $g,\\~g,C$ one can uniquely find $G_{N},\\~G_{N},\\Pi_{N}$ and\nvice versa, since they both come from the respective unique\ndecompositions of the matrix $\\mathbf{G}$.\n\nNote that for $p=1$ and $\\~g = g$, relations\n(\\ref{eq_correspondence1} - \\ref{eq_correspondence3})  are usually\nwritten simply as\n\\begin{equation} \\label{eq_p1correspondence}\n\\frac{1}{g+C} = \\frac{1}{G_{N}} + \\Pi_{N}.\n\\end{equation}\n\n\n\nWe will refer to the Poisson sigma model, when expressed -- using\n$\\Pi$ -- in open variables $(G,\\~G,\\Phi)$ as to augmented Poisson sigma model.\n\\section{Geometry of the open-closed brane relations} \\label{sec_geometry}\nFor $p=1$, the open-closed relations (\\ref{eq_p1occorrespondence}) can naturally be explained using the language of generalized geometry. We have developed this point of view in \\cite{Jurco:2013upa}. One expects that similar observations apply also for $p>1$ case. In the previous section we have already mentioned the possibility to define the generalized metric on the vector bundle $TM \\oplus \\cTM{p}$ by the inverse of the matrix \\eqref{eq_gCdecomposition}. Here we discuss an another approach to a generalization of the generalized geometry starting from equation (\\ref{eq_occorrespondece0}). Denote $W = TM \\oplus \\TM{p}$.\n\nThe main goal of this section is to show that we can without any additional labor adapt the whole formalism of \\cite{Jurco:2013upa} to the vector bundle $W \\oplus W^{\\ast}$.\n\nDefine the maps $\\mathcal{G}$, $\\mathcal{B}: W \\rightarrow W^{\\ast}$ using block matrices as\n\\begin{equation} \\label{eq_bigGbigB}\n \\mathcal{G} \\cv{V}{\\bi{P}} = \\bm{g}{0}{0}{\\~g} \\cv{V}{\\bi{P}}, \\ \\ \\mathcal{B} \\cv{V}{\\bi{P}} = \\bm{0}{C}{-C^{T}}{0} \\cv{V}{\\bi{P}},\n\\end{equation}\nfor all $V + \\bi{P} \\in \\Gamma(W)$. Next, define the map $\\Theta: W^{\\ast} \\rightarrow W$ as\n\\begin{equation} \\label{eq_Pbigdef}\n \\Theta \\cv{\\alpha}{\\Sigma} = \\bm{0}{\\Pi}{-\\Pi^{T}}{0} \\cv{\\alpha}{\\Sigma},\n\\end{equation}\nfor all $\\alpha + \\Sigma \\in \\Gamma(W^{\\ast})$. Then define $\\mathcal{H},\\Xi: W \\rightarrow W^{\\ast}$ as in (\\ref{eq_bigGbigB}) using the fields $G,\\~G,\\Phi$ instead of $g,\\~g,C$. The open-closed relations (\\ref{eq_occorrespondece0}) can be then written as simply as\n\\begin{equation} \\label{eq_ocalaSW}\n \\frac{1}{\\mathcal{G} + \\mathcal{B}} = \\frac{1}{\\mathcal{H} + \\Xi} + \\Theta.\n\\end{equation}\nWe see that they have exactly the same form as (\\ref{eq_p1occorrespondence}) for $p=1$. The purpose of this section is to obtain these relations from the geometry of the vector bundle $W \\oplus W^{\\ast}$.\n\nWe define an inner product $\\<\\cdot,\\cdot\\>: \\Gamma(W \\oplus W^{\\ast}) \\times \\Gamma(W \\oplus W^{\\ast}) \\rightarrow C^{\\infty}(M)$ on $W \\oplus W^{\\ast}$ to be the natural pairing between $W$ and $W^{\\ast}$, that is:\n\\[ \\< V + \\bi{P} + \\alpha + \\Sigma, W + \\bi{Q} + \\beta + \\Psi \\> = \\beta(V) + \\alpha(W) + \\Psi(\\bi{P}) + \\Sigma(\\bi{Q}), \\]\nfor all $V,W \\in \\vf{}$, $\\alpha,\\beta \\in \\df{1}$, $\\bi{P},\\bi{Q} \\in \\vf{p}$, and $\\Sigma,\\Psi \\in \\df{p}$. Note that this pairing has the signature $(n+\\binom{n}{p},n+\\binom{n}{p})$.\n\nNow, let $\\mathcal{T}: W \\oplus W^{\\ast} \\rightarrow W \\oplus W^{\\ast}$ be a vector bundle endomorphism squaring to identity, that is, $\\mathcal{T}^{2} = 1$. We say that $\\mathcal{T}$ is a generalized metric on $W \\oplus W^{\\ast}$, if the fibrewise bilinear form\n\\[ (E_{1},E_{2})_{\\mathcal{T}} \\equiv \\< E_{1}, \\mathcal{T}(E_{2})\\>, \\]\ndefined for all $E_{1},E_{2} \\in \\Gamma(W \\oplus W^{\\ast})$, is a positive definite fibrewise metric on $W \\oplus W^{\\ast}$. It follows from definition that $\\mathcal{T}$ is orthogonal and symmetric with respect to the inner product $\\<\\cdot,\\cdot\\>$. Moreover, it defines two eigenbundles $V_{\\pm} \\subset W \\oplus W^{\\ast}$, corresponding to eigenvalues $\\pm 1$ of $\\mathcal{T}$. It follows immediately from the properties of $\\mathcal{T}$, that they are both of rank $n + \\binom{n}{p}$, orthogonal to each other, and thus\n\\[ W \\oplus W^{\\ast} = V_{+} \\oplus V_{-}. \\]\nMoreover, $V_{+}$ and $V_{-}$ form the positive definite and negative definite subbundles of $\\<\\cdot,\\cdot\\>$, respectively. From the positive definiteness of $V_{+}$ it follows that $V_{+}$ has zero intersection both with $W$ and $W^{\\ast}$, and is thus a graph of a unique vector bundle isomorphism $\\mathcal{A}: W \\rightarrow W^{\\ast}$. The map $\\mathcal{A}$ can be written as a sum of a symmetric and a skew-symmetric part with respect to $\\<\\cdot,\\cdot\\>$: $\\mathcal{A} = \\mathcal{G} + \\mathcal{B}$. From the positive definiteness of $V_{+}$, it follows that $\\mathcal{G}$ is a positive definite fibrewise metric on $W$. From the orthogonality of $V_{+}$ and $V_{-}$ we finally obtain that:\n\\[ V_{\\pm} = \\{ (V + \\bi{P}) + ( \\pm \\mathcal{G} + \\mathcal{B} )(V + \\bi{P}) \\ | V+ \\bi{P} \\in W \\}. \\]\nThe map $\\mathcal{T}$, or equivalently the fibrewise metric $(\\cdot,\\cdot)_{\\mathcal{T}}$ can be reconstructed using the data $\\mathcal{G}$ and $\\mathcal{B}$ to get\n\\[ ( V + \\bi{P} + \\alpha + \\Sigma, W + \\bi{Q} + \\beta + \\Psi)_{\\mathcal{T}} = \\cv{V + \\bi{P}}{\\alpha + \\Sigma}^{T} \\bm{ \\mathcal{G} - \\mathcal{B} \\mathcal{G}^{-1} \\mathcal{B} }{\\mathcal{B} \\mathcal{G}^{-1}}{-\\mathcal{G}^{-1} \\mathcal{B}}{\\mathcal{G}^{-1}} \\cv{W+\\bi{Q}}{\\beta + \\Psi}. \\]\nNote that the above block matrix can be decomposed as a product\n\\[ \\bm{ \\mathcal{G} - \\mathcal{B} \\mathcal{G}^{-1} \\mathcal{B} }{\\mathcal{B} \\mathcal{G}^{-1}}{-\\mathcal{G}^{-1} \\mathcal{B}}{\\mathcal{G}^{-1}} = \\bm{1}{\\mathcal{B}}{0}{1} \\bm{\\mathcal{G}}{0}{0}{\\mathcal{G}^{-1}} \\bm{1}{0}{-\\mathcal{B}}{1}. \\]\nThe maps $\\mathcal{G},\\mathcal{B}$ can be parametrized as\n\\[ \\mathcal{G} \\cv{V}{\\bi{Q}} = \\bm{g}{D}{D^{T}}{\\~g} \\cv{V}{\\bi{Q}}, \\]\n\\[ \\mathcal{B} \\cv{V}{\\bi{Q}} = \\bm{B}{C}{-C^{T}}{\\~B} \\cv{V}{\\bi{Q}}, \\]\nwhere $g$ is a symmetric covariant $2$-tensor on $M$, $C,D: \\TM{p} \\rightarrow T^{\\ast}M$ are vector bundle morphisms, $B \\in \\Omega^{2}(M)$, and $\\~g$ and $\\~B$ are symmetric and skew-symmetric fibrewise bilinear forms on $\\TM{p}$, respectively. The fields $g,\\~g,D$ are not arbitrary, since $\\mathcal{G}$ has to be a positive definite fibrewise metric on $W$. One immediately gets that $g,\\~g$ have to be positive definite. The conditions imposed on $D$ can be seen from the equalities\n\\[\n\\begin{split}\n\\bm{g}{D}{D^{T}}{\\~g} & = \\bm{1}{0}{D^{T}g^{-1}}{1} \\bm{g}{0}{0}{\\~g - D^{T}g^{-1}D} \\bm{1}{g^{-1}D}{0}{1} \\\\\n& = \\bm{1}{D\\~g^{-1}}{0}{1} \\bm{g - D\\~g^{-1}D^{T}}{0}{0}{\\~g} \\bm{1}{0}{\\~g^{-1}D^{T}}{1}.\n\\end{split}\n\\]\nWe see that there are two equivalent conditions on $D$: the fibrewise bilinear form $\\~g - D^{T}g^{-1}D$, or $2$-tensor $g - D\\~g^{-1}D^{T}$ have to be positive definite. Inspecting the action (\\ref{def_actiontotmatrix}), we see that only the case when $B = \\~B = D = 0$ is relevant for our purpose.\n\nNow, let us turn our attention to the explanation of the open-closed relations. For this, consider the vector bundle automorphism $\\mathcal{O}: W \\oplus W^{\\ast} \\rightarrow W \\oplus W^{\\ast}$, orthogonal with respect to the inner product $\\<\\cdot,\\cdot\\>$, that is,\n\\[ \\< \\mathcal{O}(E_{1}), \\mathcal{O}(E_{2}) \\> = \\< E_{1}, E_{2} \\>, \\]\nfor all $E_{1},E_{2} \\in \\Gamma(W \\oplus W^{\\ast})$. Given a generalized metric $\\mathcal{T}$, we can define a new map $\\mathcal{T}' = \\mathcal{O}^{-1} \\mathcal{T} \\mathcal{O}$. It can be easily checked that $\\mathcal{T}'$ is again a generalized metric. Obviously, the respective eigenbundles $V_{+}$ are related using $\\mathcal{O}$, namely:\n\\begin{equation} \\label{eq_relgraphs}\n V_{+}^{\\mathcal{T}'} = \\mathcal{O}^{-1} ( V_{+}^{\\mathcal{T}} ).\n\\end{equation}\nWe have also proved that every generalized metric $\\mathcal{T}$ corresponds to two unique fields $\\mathcal{G}$ and $\\mathcal{B}$. This means that to given $\\mathcal{G}$ and $\\mathcal{B}$, and an orthogonal vector bundle isomorphism $\\mathcal{O}$, there exists a unique pair $\\mathcal{H}$, $\\Xi$ corresponding to $\\mathcal{T}' = \\mathcal{O}^{-1} \\mathcal{T} \\mathcal{O}$. We will show that open-closed relations are a special case of this correspondence. Also, note that $(\\cdot,\\cdot)_{\\mathcal{T}}$ and $(\\cdot,\\cdot)_{\\mathcal{T}'}$ are related as\n\\begin{equation} \\label{eq_GMogtransform}\n (\\cdot,\\cdot)_{\\mathcal{T}'} = (\\mathcal{O}(\\cdot),\\mathcal{O}(\\cdot))_{\\mathcal{T}}.\n\\end{equation}\nNow, consider an arbitrary skew-symmetric morphism $\\Theta: W^{\\ast} \\rightarrow W$, that is\n\\[ \\< \\alpha + \\Sigma, \\Theta(\\beta + \\Psi) \\> = - \\< \\Theta(\\alpha + \\Sigma), \\beta + \\Psi \\>, \\]\nfor all $\\alpha, \\beta \\in \\df{1}$, and $\\Sigma,\\Psi \\in \\df{p}$. It can easily be seen that the vector bundle isomorphism $e^{\\Theta}: W \\oplus W^{\\ast} \\rightarrow W \\oplus W^{\\ast}$, defined as\n\\[ e^{\\Theta} \\cv{ V+\\bi{Q}}{\\alpha + \\Sigma} = \\bm{1}{\\Theta}{0}{1} \\cv{V + \\bi{Q}}{\\alpha + \\Sigma}, \\]\nfor all $V + \\bi{Q} + \\alpha + \\Sigma \\in \\Gamma(W \\oplus W^{\\ast})$, is orthogonal with respect to the inner product $\\<\\cdot,\\cdot\\>$. Its inverse is simply $e^{-\\Theta}$. Let $\\mathcal{T}$ be the generalized metric corresponding to $\\mathcal{G} + \\mathcal{B}$. Note that $V_{+}^{\\mathcal{T}}$ can be expressed as\n\\[ V_{+}^{\\mathcal{T}} = \\{ (\\mathcal{G} + \\mathcal{B})^{-1}(\\alpha + \\Sigma) + (\\alpha + \\Sigma) \\ | \\ (\\alpha + \\Sigma) \\in W^{\\ast} \\}. \\]\nUsing the relation (\\ref{eq_relgraphs}), we obtain that\n\\[ V_{+}^{\\mathcal{T}'} = e^{-\\Theta} V_{+}^{\\mathcal{T}} = \\{ \\big( (\\mathcal{G} + \\mathcal{B})^{-1} - \\Theta \\big)(\\alpha + \\Sigma) + (\\alpha + \\Sigma) \\ | \\ (\\alpha + \\Sigma) \\in W^{\\ast} \\}. \\]\nWe see that the vector bundle morphism $\\mathcal{H} + \\Xi$ corresponding to $\\mathcal{T}'$ satisfies\n\\[ (\\mathcal{H} + \\Xi)^{-1} = (\\mathcal{G} + \\mathcal{B})^{-1}  - \\Theta. \\]\nBut this is precisely the relation (\\ref{eq_ocalaSW}). We also know how to handle this relation on the level of the positive definite fibrewise metrics $(\\cdot,\\cdot)_{\\tau}$ and $(\\cdot,\\cdot)_{\\tau'}$. From (\\ref{eq_GMogtransform}) we get the relation\n\\[ \\bm{\\mathcal{H} - \\Xi \\mathcal{H}^{-1} \\Xi}{\\mathcal{B} \\mathcal{H}^{-1}}{-\\mathcal{H}^{-1} \\Xi}{\\mathcal{H}^{-1}} = \\bm{1}{0}{-\\Theta}{1} \\bm{\\mathcal{G} - \\mathcal{B} \\mathcal{G}^{-1} \\mathcal{B}}{\\mathcal{B} \\mathcal{G}^{-1}}{-\\mathcal{G}^{-1} \\mathcal{B}}{\\mathcal{G}^{-1}} \\bm{1}{\\Theta}{0}{1}. \\]\nUsing the decomposition of the matrices, we can write this also as\n\\[ \\bm{1}{\\Xi}{0}{1} \\bm{\\mathcal{H}}{0}{0}{\\mathcal{H}^{-1}} \\bm{1}{0}{-\\Xi}{0} = \\bm{1}{0}{-\\Theta}{1}  \\bm{1}{B}{0}{1} \\bm{\\mathcal{G}}{0}{0}{\\mathcal{G}^{-1}} \\bm{1}{0}{-B}{1} \\bm{1}{\\Theta}{0}{1}. \\]\nComparing both expressions, we get the explicit form of open-closed relations:\n\\begin{equation} \\label{eq_ocrelationsbig1}\n\\mathcal{H} - \\Xi \\mathcal{H}^{-1} \\Xi = \\mathcal{G} - \\mathcal{B} \\mathcal{G}^{-1} \\mathcal{B},\n\\end{equation}\n\\begin{equation} \\label{eq_ocrelationsbig2}\n\\Xi \\mathcal{H}^{-1} = (\\mathcal{G} - \\mathcal{B} \\mathcal{G}^{-1}\\mathcal{B})\\Theta + \\mathcal{B}\\mathcal{G}^{-1},\n\\end{equation}\n\\begin{equation} \\label{eq_ocrelationsbig3}\n\\mathcal{H}^{-1} = (1 + \\Theta \\mathcal{B})\\mathcal{G}^{-1}(1 - \\mathcal{B} \\Theta) - \\Theta \\mathcal{G} \\Theta.\n\\end{equation}\nWe have proved that for given $\\mathcal{G},\\mathcal{B}$ and any $\\Theta$, $\\mathcal{H}$ and $\\Xi$ can be found uniquely. Inverse relations can be obtained by interchanging $\\mathcal{G} \\leftrightarrow \\mathcal{H}$, $\\mathcal{B} \\leftrightarrow \\Xi$ and $\\Theta \\leftrightarrow -\\Theta$. Note that, actually, the last equation follows from the first two. Now let us turn our attention to the case of $\\mathcal{G} + \\mathcal{B}$ in the form (\\ref{eq_bigGbigB}). One has\n\\[ \\mathcal{G} - \\mathcal{B} \\mathcal{G}^{-1} \\mathcal{B} = \\bm{g + C\\~g^{-1}C^{T}}{0}{0}{\\~g+C^{T}g^{-1}C}, \\]\n\\[ \\mathcal{B} \\mathcal{G}^{-1} = \\bm{0}{C\\~g^{-1}}{-C^{T}g^{-1}}{0}, \\ \\ \\mathcal{G}^{-1} = \\bm{g^{-1}}{0}{0}{\\~g^{-1}}. \\]\nParametrize $\\Theta$ as\n\\[ \\Theta = \\bm{\\pi}{\\Pi}{-\\Pi^{T}}{\\~\\pi}, \\]\nwhere $\\pi \\in \\vf{2}$, $\\Pi: \\cTM{p} \\rightarrow TM$, and $\\~\\pi$ is skew-symmetric fibrewise bilinear form on $\\cTM{p}$. Right-hand side of (\\ref{eq_ocrelationsbig2}) is then\n\\[\n\\begin{split}\n\\bm{g+C\\~g^{-1}C^{T}}{0}{0}{\\~g+C^{T}g^{-1}C} \\bm{\\pi}{\\Pi}{-\\Pi^{T}}{\\~\\pi} + \\bm{0}{C\\~g^{-1}}{-C^{T}g^{-1}}{0} = \\\\\n= \\bm{ (g + C\\~g^{-1}C^{T})\\pi }{(g+C\\~g^{-1}C^{T})\\Pi + C\\~g^{-1}}{-(\\~g + C^{T}g^{-1}C)\\Pi^{T} - C^{T}g^{-1}}{(\\~g + C^{T}g^{-1}C)\\~\\pi  }.\n\\end{split}\n\\]\nWe see that to obtain a generalized metric where $\\mathcal{H}$ is block diagonal, and $\\Xi$ is block off-diagonal, we have to choose $\\pi = \\~\\pi = 0$. This means that we choose $\\Theta$ to be of the form (\\ref{eq_Pbigdef}). Defining\n\\[ \\mathcal{H} = \\bm{G}{0}{0}{\\~G}, \\ \\Xi = \\bm{0}{\\Phi}{-\\Phi^{T}}{0}, \\]\nit is now straightforward to see that the set of equations (\\ref{eq_ocrelationsbig1} - \\ref{eq_ocrelationsbig3}) gives exactly the open-closed relations (\\ref{eq_occorrespondence1} - \\ref{eq_occorrespondence4}). The relations between the open membrane variables and Nambu fields $G_{N},\\~G_{N},\\Pi_{N}$ can be explained in a similar fashion. Indeed, note that the map $\\mathcal{G} + \\mathcal{B}$ is invertible, and its inverse, the vector bundle morphism from $W^{\\ast}$ to $W$, can be split into symmetric and skew-symmetric part:\n\\begin{equation} \\label{eq_openNambubig}\n (\\mathcal{G} + \\mathcal{B})^{-1} = \\mathcal{H}_{N}^{-1} + \\Theta_{N},\n\\end{equation}\nwhere $\\mathcal{H}_{N}$ is a fibrewise positive definite metric on $W$, and $\\Theta_{N}$ is a skew-symmetric fibrewise bilinear form on $W^{\\ast}$. Parametrizing them as\n\\[ \\mathcal{H}_{N} = \\bm{G_{N}}{0}{0}{\\~G_{N}}, \\ \\ \\Theta_{N} = \\bm{0}{\\Pi_{N}}{-\\Pi_{N}^{T}}{0}, \\]\nand expanding (\\ref{eq_openNambubig}), we obtain exactly the set of equations (\\ref{eq_correspondence1} - \\ref{eq_correspondence3}).\n\\section{Gauge field $F$ as transformation of the fibrewise metric} \\label{sec_gaugeF}\nIn this section, we would like to develop the equalities required in the discussion of DBI actions. In the previous sections we have shown how the closed and open membrane actions are related using the generalized geometry point of view. One expects that it is also true for their versions taking into account the fluctuations. The following paragraphs show that it is true ``up to an isomorphism\", fluctuated backgrounds cannot be related simply by open-closed relations in the form (\\ref{eq_occorrespondence1} - \\ref{eq_occorrespondence4}).\n\nWe also show that corresponding open backgrounds are essentially uniquely fixed, there is no ambiguity at all. For $p=1$, we have already used this observation in \\cite{Jurco:2013upa}.\n\n\nThe idea is the following: Suppose that we would like to add a fluctuation $F$ to the\n$(p+1)$-form $C$. At this point we consider  $F$ to be\ndefined globally on the entire manifold $M$, although everything works also in the case when\n$F$ is defined only on a some submanifold of $M$.\\footnote{Later,\nthis submanifold will correspond to a $p'$-brane, $p'\\geq p$, where\n$p$-branes can end.}\n\nGoing from $C$ to $C+F$ corresponds to\nreplacing $\\mathbf{G}_c$ in the Hamiltonian (\\ref{def_polyakovham})\nwith $\\mathbf{G}_c^{F}$, defined as\n\\begin{equation} \\label{def_GFmatrix}\n\\mathbf{G}_c^{F} = \\bm{1}{0}{-F^{T}}{1} \\mathbf{G}_c\n\\bm{1}{-F}{0}{1} \\equiv \\bm{1}{0}{-(C+F)^{T}}{1}\n\\bm{g^{-1}}{0}{0}{\\~g} \\bm{1}{-(C+F)}{0}{1}.\n\\end{equation}\nThe matrix $\\bm{1}{-F}{0}{1}$ corresponds to an endomorphism of $T^{\\ast}M\n\\oplus \\TM{p}$, which we denote as $e^{-F}$. Note that unlike in the\n$p=1$ case, $e^{-F}$ is not orthogonal with respect to the canonical\npairing (valued in $\\vf{p-1}$) on $T^{\\ast}M \\oplus \\TM{p}$, defined\nas:\n\\[ \\<\\alpha+\\bi{Q},\\beta+ \\bi{R}\\> = \\mathit{i}_{\\alpha}\\bi{R} + \\mathit{i}_{\\beta}\\bi{Q}, \\]\nfor all $\\alpha,\\beta \\in \\df{1}$ and $\\bi{Q},\\bi{R} \\in \\vf{p}$. It can be\nshown that any orthogonal $F$ has to be identically $0$. On the other\nhand, its transpose map, $(e^{-F})^{T} \\equiv e_{-F}$, which is an\nendomorphism of $TM \\oplus \\cTM{p}$, is orthogonal with respect to\nthe canonical pairing (valued in $\\df{p-1}$) on $TM \\oplus \\cTM{p}$\niff $F$ is a $(p+1)$-form in $M$. This pairing is defined as\n\\[ \\<V + \\Sigma, W + \\Xi\\> = \\mathit{i}_{V}\\Sigma + \\mathit{i}_{W}\\Xi,  \\]\nfor all $V,W \\in \\vf{}$ and $\\Sigma,\\Xi \\in \\df{p}$. In this notation, the transformation\n(\\ref{def_GFmatrix}) can be written   as\n\\begin{equation}  \\label{eq_Gffactorization}\n\\mathbf{G}_c^{F} = e_{-F} \\mathbf{G}_c e^{-F} \\equiv (e^{-F})^{T}\n\\mathbf{G}_c e^{-F}.\n\\end{equation}\nWe know that $\\mathbf{G}$ can be rewritten as $\\mathbf{G}_o$ in the open\nvariables $(G,\\~G,\\Phi)$,  corresponding to augmented Nambu sigma\nmodel. If we define the automorphism $e^{\\Pi}$ of $T^{\\ast}M\n\\oplus \\TM{p}$ as\n\\[ e^{\\Pi} \\bv{\\alpha}{\\bi{Q}} = \\bm{1}{0}{\\Pi^{T}}{1} \\bv{\\alpha}{\\bi{Q}}, \\]\nwe can express $\\mathbf{G}_o$ as\n\\begin{equation}\n\\mathbf{G}_o = e_{\\Pi} \\bm{1}{0}{-\\Phi^{T}}{1}\n\\bm{G^{-1}}{0}{0}{\\~G}  \\bm{1}{-\\Phi}{0}{1} e^{\\Pi},\n\\end{equation}\nwhere $e_{\\Pi} = (e^{\\Pi})^{T}$. Dually to the previous\ndiscussion, $e^{\\Pi}$ is an orthogonal transformation of\n$T^{\\ast}M \\oplus \\TM{p}$; although $e_{\\Pi}$, for non-zero\n$\\Pi$, is never orthogonal on $TM \\oplus \\cTM{p}$.\n\nNow, it is natural to ask whether to the gauged closed variables\n$(g,\\~g,C+F)$ there correspond some open variables and hence an\naugmented Nambu sigma model, described by  some $\\Pi'$ and\n$(G,\\~G,\\Phi + F')$, where $F'$ describes a fluctuation of the background\n$\\Phi$. More precisely, we ask whether one can write\n$\\mathbf{G}_o^{F}$ in the form\n\\begin{equation}\n\\mathbf{G}_o^{F} \\stackrel{?}{=} e_{\\Pi'} \\bm{1}{0}{-(\\Phi +\nF')^{T}}{1} \\bm{G^{-1}}{0}{0}{\\~G} \\bm{1}{-(\\Phi + F')}{0}{1}\ne^{\\Pi'}.\n\\end{equation}\nTranslated into the language of the corresponding automorphisms of $T^{\\ast}M \\oplus\n\\TM{p}$, this boils down to the question\n\\begin{equation} \\label{eq_ortequality}\ne^{\\Pi} e^{-F}\\stackrel{?}{=} e^{-F'} e^{\\Pi'},\n\\end{equation}\nfor some $\\Pi'$ and $F'$. In general, this is not possible. Explicitly the\nequation (\\ref{eq_ortequality}) reads\n\\[ \\bm{1}{-F}{\\Pi^{T}}{1 - \\Pi^{T}F} \\stackrel{?}{=}\n\\bm{1 - F'\\Pi'}{-F'}{\\Pi'^{T}}{1}. \\] This implies $\\Pi^{T}\nF = 0$, which, of course, in general is not satisfied. The\ndecomposition on the right-hand side therefore has to contain a\nblock-diagonal term. Note that $e^{-F'}$ is upper triangular,\nwhereas $e^{\\Pi'}$ is lower triangular. For a matrix to have a\ndecomposition into a product of a block upper triangular, diagonal\nand lower triangular matrix, it has to have an invertible bottom\nright block, that is $1 - \\Pi^{T}F$. Hence, we assume that $1 -\n\\Pi^{T} F$ is an invertible $\\binom{n}{p} \\times \\binom{n}{p}$\nmatrix. We are now looking for a solution of the equation\n\\begin{equation} \\label{eq_ortequality2}\n e^{\\Pi}e^{-F} = e^{-F'} \\bm{M}{0}{0}{N} e^{\\Pi'},\n\\end{equation}\nwhere $M: T^{\\ast}M \\rightarrow T^{\\ast}M$ and $N: \\TM{p}\n\\rightarrow \\TM{p}$ are (necessarily) invertible vector bundle\nmorphisms.\n\nWe can decompose $e^{\\Pi} e^{-F}$ as\n\\begin{equation}\n\\bm{1}{-F(1-\\Pi^{T}F)^{-1}}{0}{1}\n\\bm{1 + F(1 - \\Pi^{T}F)^{-1}\\Pi^{T}}{0}{0}{1 - \\Pi^{T}F}\n\\bm{1}{0}{(1 - \\Pi^{T}F)^{-1}\\Pi^{T}}{1}.\n\\end{equation}\nFrom this we see that $F' = F(1 - \\Pi^{T} F)^{-1}$, $\\Pi' =\n\\Pi (1 - F^{T}\\Pi)^{-1} $ and $N = 1 - \\Pi^{T}F$. To find\nan alternative description of $F'$, $\\Pi'$ and $M$, examine the\ninverse of the equation (\\ref{eq_ortequality2}):\n\\begin{equation}\ne^{F} e^{-\\Pi} = e^{-\\Pi'}  \\bm{M^{-1}}{0}{0}{N^{-1}} e^{F'}.\n\\end{equation}\nThe left hand side of this equation is\n\\[ e^{F} e^{-\\Pi} = \\bm{1 - F\\Pi^{T}}{F}{-\\Pi^{T}}{1}, \\]\nwhich shows that $1 - \\Pi^{T}F$ is invertible iff $1 -\nF\\Pi^{T}$ is invertible. The decomposition of $e^{F}e^{-\\Pi}$\nreads\n\\begin{equation}\n\\bm{1}{0}{-\\Pi^{T}(1 - F\\Pi^{T})^{-1}}{1}\n\\bm{1 - F\\Pi^{T}}{0}{0}{1 + \\Pi^{T}(1 - F \\Pi^{T})^{-1}F}\n\\bm{1}{(1-F\\Pi^{T})^{-1}F}{0}{1}.\n\\end{equation}\nWe thus get that $F' = (1-F\\Pi^{T})^{-1}F$, $\\Pi' = (1 -\n\\Pi F^{T})^{-1} \\Pi$ and $M = (1 - F\\Pi^{T})^{-1}$.\n\nWe can conclude that the fields $F'$, $\\Pi'$, and vector bundle\nmorphisms $M,N$ in the decomposition (\\ref{eq_ortequality2}) have\none of the following equivalent forms:\n\n\\begin{equation} \\label{eq_primed1}\nF' = F(1-\\Pi^{T}F)^{-1} = (1-F\\Pi^{T})^{-1}F,\n\\end{equation}\n\\begin{equation} \\label{eq_primed2}\n\\Pi' = \\Pi(1 - F^{T}\\Pi)^{-1} = (1 - \\Pi F^{T})^{-1} \\Pi,\n\\end{equation}\n\\begin{equation} \\label{eq_primed3}\nM = 1 + F(1 - \\Pi^{T}F)^{-1}\\Pi^{T} = 1 + F'\\Pi^{T} =\n(1 - F\\Pi^{T})^{-1},\n\\end{equation}\n\\begin{equation} \\label{eq_primed4}\nN = 1 - \\Pi^{T}F = \\big( 1 + \\Pi^{T}(1 - F\\Pi^{T})^{-1}F\n\\big)^{-1} = (1 + \\Pi'^{T}F)^{-1}.\n\\end{equation}\n\nThus, we have found a factorization of $\\mathbf{G}_o^{F}$ in the\nform\n\\begin{equation}\n\\mathbf{G}_o^{F} = e_{\\Pi'} \\bm{M^{T}}{0}{0}{N^{T}} e_{-(\\Phi +\nF')} \\bm{G^{-1}}{0}{0}{\\~G} e^{-(\\Phi + F')} \\bm{M}{0}{0}{N}\ne^{\\Pi'}.\n\\end{equation}\nComparing this to  $\\mathbf{G}_c^{F}$, in particular comparing\nthe respective bottom right blocks, we get the important identity\n\\begin{equation}\n\\~g + (C+F)^{T}g^{-1}(C+F) = N^{T}\\big(\\~G + (\\Phi + F')^{T}G^{-1}\n(\\Phi + F')\\big)N.\n\\end{equation}\nSimilarly, comparing the top left blocks of the inverses, one gets\n\\begin{equation}\\label{openclosedF}\ng + (C+F)\\~g^{-1}(C+F)^{T} = M^{-1}\\big(G + (\\Phi + F')\\~G^{-1}\n(\\Phi + F'\\big)^{T}M^{-T}.\n\\end{equation}\n\nEquivalently, one can gauge the matrix $\\mathbf{A}_c$, i.e., set\n\\begin{equation}\\label{ACF}\n{\\mathbf{A}}_c^F=\\begin{pmatrix} g & (C+F )\\\\\n-(C+F)^T& \\~g\\end{pmatrix}.\n\\end{equation} To express this matrix in\nopen variables we introduce the following notation: $\\bar G^{-1}\n:=M^TG^{-1}M$, $\\bar{\\~G}=N^T\\~G N$, $\\bar \\Phi :=M^{-1}\\Phi N$ and\n$\\bar F' :=M^{-1}F'N$. If we now put\n\\begin{equation}\n{\\mathbf{A}}_o^F=\\begin{pmatrix} \\bar G &  (\\bar\\Phi + \\bar F') \\\\\n-(\\bar \\Phi +\\bar F')^T & \\bar{\\~G}\\end{pmatrix}^{-1}\n +\n\\begin{pmatrix} 0 & \\Pi' \\\\\n-\\Pi'^T & 0\\end{pmatrix}, \\end{equation} the (gauged) open-closed\nrelations are equivalent to ${\\mathbf{A}}_c^F={\\mathbf{A}}_o^F$. As in the previous sections, using the matrices ${\\mathbf{A}}_c^F$,\n${\\mathbf{A}}_o^F$, ${\\mathbf{G}}_c^F$ and ${\\mathbf{G}}_o^F$, one\ncan write down the corresponding Polyakov or (augmented) Nambu sigma\nmodels, i.e.,\n\\begin{equation} \\label{def_actiontotmatrixF}\nS_{P}^{tot,F}[X] = \\frac{1}{2} \\int d^{p+1}\\sigma \\{ \\Psi^{\\dagger} \\mathbf{A}^F_c \\Psi \\}=\\frac{1}{2} \\int d^{p+1}\\sigma \\{ \\Psi^{\\dagger} \\mathbf{A}^F_o \\Psi \\},\n\\end{equation}\n\\begin{equation} \\label{def_actionNambumatrixF}\nS_{NSM}^{F}[X,\\eta,\\eta'] = - \\int d^{p+1}\\sigma \\{ \\Upsilon^{\\dagger} {\\mathbf{A}^F_c}^{-1} \\Upsilon  + \\Upsilon^{\\dagger} \\Psi \\}=- \\int d^{p+1}\\sigma \\{ \\Upsilon^{\\dagger} {\\mathbf{A}^F_c}^{-1} \\Upsilon + \\Upsilon^{\\dagger} \\Psi \\},\n\\end{equation}\n\\begin{equation} \\label{def_polyakovhamF}\n\\begin{split}\nH_{P}^{tot,F}[X,P] = H_{NSM}^F[X,P] & = -\\frac{1}{2} \\int d^{p}\\sigma \\bv{iP}{\\px{}}^{T} \\mathbf{G_c^F}\n\\bv{iP}{\\px{}} \\\\\n& =  -\\frac{1}{2} \\int d^{p}\\sigma \\bv{iP}{\\px{}}^{T} \\mathbf{G_o^F}\n\\bv{iP}{\\px{}}.\n\\end{split}\n\\end{equation}\n\n\n\n\\section{Seiberg-Witten map} \\label{sec_SWmap}\nIn the previous section, we have developed the correspondence between closed and open fields, including their respective fluctuations. However, they are not related simply by open-closed relations. Instead, the discussion brings new vector bundle isomorphisms $M$ and $N$, defined by (\\ref{eq_primed3}, \\ref{eq_primed4}), respectively, into the picture. The determinant of the left-hand side of (\\ref{openclosedF}) seems to be a likely candidate to appear in the  ``commutative\" membrane DBI action, whereas the determinant on the right-hand side of (\\ref{openclosedF}) seems to contain as a factor a likely candidate to appear in its ``noncommutative\" counterpart.\n\nThis observation suggests that we should look for a change of coordinates on the manifold $M$, the Jacobian of which could cancel the $\\det{}^{2}(N)$ factor coming under the determinant from the right-hand side of (\\ref{openclosedF}).\nThe resulting diffeomorphism will be called a Seiberg-Witten map in analogy to the string $p=1$ case. We use a direct generalization of the semi-classical construction used first in \\cite{Jurco:2001my}. The most intriguing part will be to define carefully a substitute for a determinant of a Nambu-Poisson $(p+1)$-vector.\n\nIn the following, let $\\Pi$ be a Nambu-Poisson $(p+1)$-vector\n(see appendix \\ref{sec_NPstructures}) on $M$. We can examine the\n$F$-gauged tensor $\\Pi' = (1 - \\Pi F^{T})^{-1} \\Pi$.\\footnote{We assume that $1 - \\Pi F^{T}$ is invertible. In a more formal approach we also could treat $\\Pi'$ as a formal power series in $\\Pi$.\\label{invertible}}\nWe will now show that for $p>1$ this tensor is always a Nambu-Poisson $(p+1)$-vector, whereas for\n$p=1$ it is a Poisson bivector if $F$ is closed.\n\nFirst, for $p > 1$, one can see that\n\\begin{equation}  \\label{eq_thetaprimetrace}\n\\Pi' = (1 - \\frac{1}{p+1}\\< \\Pi, F\\>)^{-1} \\Pi,\n\\end{equation}\nwhere $\\<\\Pi,F\\> = \\Pi^{iJ} F_{iJ} \\equiv Tr(\\Pi F^{T})$.\nFor this, one has to prove that\n\\begin{equation} \\label{TwoTheta_prime_expressions}\\Pi = (1 - \\Pi F^{T}) (1 - \\frac{1}{p+1}\\<\\Pi,F\\>)^{-1} \\Pi.\n\\end{equation}\nThis can easily be checked in coordinates $(x^{1}, \\dots, x^{n})$ in which (\\ref{eq_Pidecomposition}) holds, and hence, for $\\Pi$ with components $\\Pi^{iJ} = \\epsilon^{iJ}$. Now, using (\\ref{eq_thetaprimetrace}) and lemma\n\\ref{lem_nptensormultiple}, we see that $\\Pi'$ is again a\nNambu-Poisson tensor.\n\nTo include the $p=1$ case:\nFor $p \\geq 1$, and $F$ closed, we can use the fact that\n$G_{\\Pi'} = e_{-F} G_{\\Pi}$, where $G_{\\Pi}$ and\n$G_{\\Pi'}$ are graphs of the maps $\\Pi^{\\sharp}$ and\n$\\Pi'^{\\sharp}$, respectively (see lemma \\ref{lem_nplemma}). This is easily\nverified using (\\ref{eq_primed2}). It can be seen that the Dorfman\nbracket (\\ref{def_dorfman}) satisfies\n$[e_{-F}(V+\\xi),e_{-F}(W+\\eta)]_{D} = e_{-F}[V+\\xi,W+\\eta]_{D}$,\nwhenever $F$ is closed. But this implies that $G_{\\Pi'}$ is\nclosed under the Dorfman bracket, which is according to\n\\ref{lem_nplemma} equivalent to the Nambu-Poisson fundamental\nidentity. On the other hand, note that for $p>1$, $F'$ is not necessarily a\n$(p+1)$-form.\n\nNext, see that the scalar function in front of $\\Pi$ in (\\ref{eq_thetaprimetrace}) is related to the determinant of the vector bundle isomorphism $1 - \\Pi F^{T}$. For $p>1$, any Nambu-Poisson tensor and any $(p+1)$-form $F$, its holds\n\n\\begin{equation} \\label{detFprimeasfunction}\n\\det{(1 - \\Pi F^{T})} = (1 - \\frac{1}{p+1} \\<\\Pi,F\\>)^{p+1}.\n\\end{equation}\nTo prove this identity, note  that both sides are scalar functions. We may therefore use any local coordinates on $M$. Again, use those in which $(\\ref{eq_Pidecomposition})$ holds. The rest of the proof is straightforward.\n\nFurther on, assume that $F$ is closed, that is at least locally $F =\ndA$ for a $p$-form $A$. Define a $1$-parametric family of tensors\n$\\Pi'_{t} := (1 - t\\Pi F^{T})^{-1} \\Pi$, cf. Footnote \\ref{invertible}. This is obviously\nchosen so that $\\Pi'_{0} = \\Pi$ and $\\Pi'_{1} = \\Pi'$.\nDifferentiation of $\\Pi'_{t}$ with respect to $t$ gives:\n\\begin{equation}\n\\partial_{t} \\Pi'_{t} = \\Pi'_{t} F^{T} \\Pi'_{t}.\n\\end{equation}\n\nThis equation can be rewritten as\n\\begin{equation} \\label{eq_difffortheta}\n\\partial_{t} \\Pi'_{t} = -\\mathcal{L}_{A^{\\sharp}_{t}} \\Pi'_{t},\n\\end{equation}\nwhere the time-dependent vector field $A^{\\sharp}_{t}$ is defined as\n$A^{\\sharp}_{t} = {\\Pi'}^{\\sharp}_{t}(A)$. To see this, note that\n$\\Pi'_{t}$ is, using similar arguments as above, a Nambu-Poisson\ntensor. Then recall the property (\\ref{npfieq1}), and choose $\\xi =\nA$ and $\\eta = dy^{J}$. Contracting the resulting vector field\nequality with $dy^{i}$ gives exactly $\\mathcal{L}_{A^{\\sharp}_{t}}\n\\Pi'_{t} = - \\Pi'_{t} F^{T} \\Pi_{t}'$. Equation\n(\\ref{eq_difffortheta}) states precisely that the flow $\\phi_{t}$\ncorresponding to $A_{t}^{\\sharp}$, together with condition\n$\\Pi'_{0} = \\Pi$, maps $\\Pi_{t}$ to $\\Pi$, that is,\n\\begin{equation}\n\\phi_{t}^{\\ast}(\\Pi'_{t}) = \\Pi.\n\\end{equation}\nWe have thus found the map $\\rho_{A} \\equiv \\phi_{1}$, which gives\n$\\rho_{A}^{\\ast}(\\Pi') = \\Pi$. This is the $p \\geq 1$ analogue\nof the well known semiclassical Seiberg-Witten map. Obviously, it preserves the singular foliation defined by $\\Pi$.  We emphasize the\ndependence of this map on the $p$-form $A$ by an explicit addition\nof the subscript $A$.\n\nDenote ${J^{i}}_{k} = \\frac{\\partial \\widehat X^{i}}{\\partial x^{k}}$, with $\\widehat X^{i}:=\\rho_A^{\\ast}(x^i)$ being {\\it covariant} coordinates. We\nhave\n\\begin{equation} \\label{JJJ} \\rho_A^{\\ast}( {\\Pi'}^{j_1,\\ldots j_{p+1}} ) = J^{j_1}_{i_1}\\ldots J^{j_{p+1}}_{i_{p+1}} \\Pi^{i_1\\ldots i_{p+1}}.\n\\end{equation}\n\nFurther, denote by $|J|$ the determinant of $J^{i}_{k}$ in some (arbitrarily) chosen local coordinates $(x^{1}, \\dots, x^{n})$ on $M$. One can choose, for instance, the special coordinates $(\\~x^{i}, \\dots \\~x^{n})$ on $M$ in which (\\ref{eq_Pidecomposition}) holds. We will use the notation $|\\~J|$ for the determinant of the matrix $\\~J^{i}_{k} = \\frac{\\partial \\~x^{i}(\\rho_{A}(x))}{\\partial \\~x^{k}}$. From now, for any function $\\varphi$ (e.g., a matrix component, determinant, etc.), the symbol $\\widehat{\\varphi}$ will always denote the function defined as $\\widehat{\\varphi}(x) \\equiv \\rho_{A}^{\\ast}(\\varphi)(x) = \\varphi(\\rho_{A}(x))$. Recall now the definition (\\ref{def_pidensity}) of the density $|\\Pi(x)|$\\footnote{For $p=1$, one can (around every regular point of the characteristic distribution) define $|\\Pi(x)|$ to be the Jacobian of the transformation to the Darboux-Weinstein coordinates. This gives a good definition even if $\\Pi$ is degenerate.}. By definition of $|J|$, we then have\n\\begin{equation} \\label{eq_JastildJ}\n |J| = |\\~J| \\frac{|\\widehat{\\Pi}(x)|^{\\frac{1}{p+1}}}{|\\Pi(x)|^{\\frac{1}{p+1}}}\n\\end{equation}\nThe Jacobian $|\\~J|$ can easily  be calculated using (\\ref{eq_thetaprimetrace}) and (\\ref{JJJ}). Indeed, the equation (\\ref{JJJ}) can be, in $(\\~x)$ coordinates, rewritten as\n\\[ (1 - \\frac{1}{p+1} \\<\\widehat{\\Pi},\\widehat{F}\\>)^{-1} \\epsilon^{j_{1} \\dots j_{p+1}} = \\epsilon^{j_{1} \\dots j_{p+1}} \\~J_{i_{1}}^{1} \\dots \\~J_{i_{p+1}}^{p+1} \\epsilon^{i_{1} \\dots i_{p+1}}. \\]\nTo justify this, note that Seiberg-Witten map acts nontrivially only in the directions of the first $(p+1)$-coordinates. The Jacobi matrix $\\~J$ of $\\rho_{A}$ in $(\\~x)$ coordinates is thus a block upper triangular with identity matrix in the bottom right block. Moreover, the determinant of $\\~J$ is then equal to the determinant of the top left block. We can divide both sides with $\\epsilon^{j_{1} \\dots j_{p+1}}$. We thus remain with the equation\n\\[ (1 - \\frac{1}{p+1} \\<\\widehat{\\Pi},\\widehat{F}\\>)^{-1} = \\~J_{i_{1}}^{1} \\dots \\~J_{i_{p+1}}^{p+1} \\epsilon^{i_{1} \\dots i_{p+1}}= |\\~J|. \\]\nPutting this back into (\\ref{eq_JastildJ}), we obtain the useful relation\n\\begin{equation}\n|J|^{p+1} = (1 - \\frac{1}{p+1} \\< \\widehat{\\Pi}, \\widehat{F} \\>)^{-(p+1)} \\frac{|\\widehat{\\Pi}(x)|}{|\\Pi(x)|},\n\\end{equation}\nor using (\\ref{detFprimeasfunction})\\footnote{For $p=1$, one can derive this relation by calculating $|\\~J|$ in Darboux-Weinstein coordinates directly from (\\ref{JJJ}) and the definition of $\\Pi'$, and then use (\\ref{eq_JastildJ}).}\n\\begin{equation} \\label{Jacobian}\n|J|^{p+1} = \\det{(1 - \\widehat{\\Pi} \\widehat{F}^{T})}^{-1} \\frac{|\\widehat{\\Pi}(x)|}{|\\Pi(x)|}.\n\\end{equation}\n\nNote that this expression does not depend on the choice of the Darboux coordinates in which the densities $|\\Pi(x)|$ are calculated. We discuss this subtlety in the appendix \\ref{sec_NPstructures} under (\\ref{eq_Piintwocoordinates}). We see that $|\\Pi(x)|$  itself transforms as in (\\ref{eq_Pidensitytransform}). Fortunately, the determinant of the block $M$ in (\\ref{eq_Piintwocoordinates}) does not depend on the coordinates $(\\~x^{1}, \\dots \\~x^{p+1})$. Since these are the only coordinates changed by the Seiberg-Witten map, we get $(\\det{M})(x) = (\\det{M})(\\rho_{A}(x))$. In other words, these determinants cancel out in the fraction $|\\widehat{\\Pi}(x)| \/ |\\Pi(x)|$, as expected.\n\nThe following observation is in order: The Nambu-Poisson tensor $\\Pi_t$ does not depend on the\nchoice of the gauge $p$-potential $A$. As already mentioned, the Nambu-Poisson map $\\rho_{A}$ does: An infinitesimal gauge transformation $\\delta A = d \\lambda$ -- with a\n$(p-1)$-form gauge transformation parameter $\\lambda$ -- induces a change in the flow, which is generated by the vector field $X_{[\\lambda, A]} = \\Pi^{iJ} d\\Lambda_J\\partial_i$, where\n\\begin{equation}\n\\Lambda= \\sum_{k=0}^{\\infty} \\frac{(\\mathcal{L}_ {A_{t}^\\sharp} +\n\\partial_t)^k(\\lambda)}{(k+1)!}\\Big| _{t=0}  \\,,\n\\end{equation}\nis the semiclassically noncommutative $(p-1)$-form gauge parameter.\nThis is the $p$-brane analog of the exact Seiberg-Witten map for the gauge transformation parameter. It is straightforwardly obtained by application of the BCH formula to $\\rho^\\ast_{A+d\\lambda}(\\rho^\\ast_A)^{-1}$.\nFinally, in analogy with the $p=1$ case, we define the (components of the) semiclassically noncommutative field strength to be\n\\begin{equation}\\label{NCF}\n\\widehat F'_{i_1,\\ldots, i_{p+1}} =\\rho^{\\ast}_A F'_{i_1,\\ldots, i_{p+1}},\n\\end{equation}\ni.e., the components of $F'$ evaluated in the covariant coordinates. Infinitesimally, components of $\\hat F$ transform as\n\\begin{equation}\\label{NCF2}\n\\delta\\widehat F' =\\Pi^{iJ} d\\Lambda_J\\partial_i \\hat F',\n\\end{equation}\nwhich justifies the adjectives ``semiclassically noncommutative\".\n\n\\section{Nambu gauge theory; Equivalence of commutative and semiclassically\nnoncommutative DBI action} \\label{DBI}\n\nHere we consider a system of multiple open M2 branes ending on an M5 brane. We would like to describe this system by an\neffective action that is exact, for slowly varying fields,  to all orders in the coupling constant. Since we focus only on the bosonic part of this action, we do not need to restrict ourselves to the values $p=2$ and $p'=5$ and our construction is valid for arbitrary values of $p$ and $p'$ such that $p\\leq p'$.\nOur goal is thus the construction of an effective action for a $p'$-brane with open $p$-branes ending on it while being submerged in a  $C_{p+1}$-background.\nThe construction is based on two guiding principles: Firstly, this effective action should have dual descriptions similar to the commutative and non-commutative ones of the D-brane and open strings\\footnote{Actually, our exposition so far closely followed our previous work \\cite{Jurco:2012gc}, where the role of generalized geometry was emphasized.} and secondly, it should feature expressions that also appear in the $p$-brane action (\\ref{def_actiontotmatrixF}).\n\nDenote the $p'$-brane submanifold as $N$. We shall now clarify the geometry underlying the following discussion. Originally, $g,\\~g,C$ were assumed to be the closed membrane backgrounds in the ambient background manifold $M$. Hereafter, we denote by the same characters their pullbacks to the $p'$-brane $N$. This makes sense since all of them are covariant tensor fields on $M$. Little subtlety comes with the Nambu-Poisson tensor $\\Pi$. We have basically two options. First, we would like to restrict some Nambu-Poisson tensor in $M$ to the $p'$-brane. This in fact requires $N$ to be a Nambu-Poisson submanifold of $M$. The latter option is to \\emph{choose} the Nambu-Poisson tensor $\\Pi$ on $N$ after we restrict the other backgrounds to $N$. The open membrane variables $G,\\~G,\\Phi$, calculated using the membrane open-closed relations (\\ref{eq_occorrespondence1} - \\ref{eq_occorrespondence4}), are assumed to be calculated entirely on $N$, using the pullbacks of closed variables. Finally, the field $F$ is assumed to be a $(p+1)$-form defined and having components only in $N$. All the discussion related to Seiberg-Witten map in the previous section is assumed to take place on the submanifold $N$.\n\nThe open-closed membrane relations (\\ref{openclosedF}) immediately imply\n\\begin{multline} \\label{miracle}\n\\det[g + (C+F) \\tilde g^{-1} (C+F)^T] =  \\det{}^2[1-F \\Pi^{T}]\\cdot \\det[G+(\\Phi + F') \\tilde G^{-1} (\\Phi + F')^T] \\,,\n\\end{multline}\nwhere $F' = (I - F \\Pi^T)^{-1} F$. Obviously, in order get a sensible action we have to form an integral density, which can be integrated over the world volume of the larger $p'$-brane. And, in order to obtain a noncommutative action from the right hand side of \\eqref{miracle}, we have to apply  the Seiberg-Witten map $\\rho^{\\ast}_A$ to it. It would be tempting to take the square root of the identity \\eqref{miracle} to construct the action. But, recall \\eqref{Jacobian} and notice the factor $\\det{}^{-(p+1)}[1-F\\Pi^T]$ appearing in it upon the application of the Seiberg-Witten map. Hence, not the square root but the $2(p+1)$-th root of \\eqref{miracle} is the most natural choice to enter the effective action that we look for. As we already said, the Lagrangian density  must be an integral density, and therefore we need to multiply that piece of the action by a proper power of the determinant of the pullback of the target space metric. These considerations fix the action essentially uniquely and we postulate \\vspace{-1ex}\n\\begin{equation} \\label{pDBI}\nS_\\text{$p$-DBI} = -\\int d^{p'+1} x \\, \\frac{1}{g_m} \\det{}^\\frac{p}{2(p+1)}(g) \\cdot \\det{}^{\\frac{1}{2(p+1)}}\\big[g + (C+F) \\tilde g^{-1} (C+F)^T \\big] \\,,\n\\end{equation}\nwhere $g_m$ is a ``closed membrane'' coupling constant. The integration is over the $p'$-brane and the fields $g$, $\\tilde g$, and $C$ in this expression are the pull-backs of the corresponding background target space fields to this $p'$-brane.\nAsking for\n\\begin{multline} \\label{MI}\n\\frac{1}{g_m} \\det{}^\\frac{p}{2(p+1)}g \\cdot \\det{}^{\\frac{1}{2(p+1)}}\\big[g + (C+F) \\tilde g^{-1} (C+F)^T \\big] \\\\\n= \\frac{1}{G_m} \\det{}^\\frac{p}{2(p+1)}(G)\\det{}^{\\frac{1}{(p+1)}}[1-\\Pi F^T] \\cdot \\det{}^{\\frac{1}{2(p+1)}}\\big[G + (\\Phi+F') \\tilde G^{-1} (\\Phi+F')^T \\big],\n\\end{multline}\nit follows from \\eqref{miracle} that the closed and open coupling constants $g_m$ and $G_m$ must be related as\n\\begin{equation}\nG_m = g_m \\left(\\det G\/\\det g\\right)^{\\frac{p}{2(p+1)}} \\,.\n\\end{equation}\n\nAs desired, the action (\\ref{pDBI}) is exactly equal to its ``noncommutative'' dual\n\\vspace{-1ex}\n\\begin{equation}\\label{pNCDBI}\nS_\\text{$p$-NCDBI} =  -\\int d^{p'+1} x \\, \\frac{1}{\\widehat G_m} \\,\\frac{\\widehat{|\\Pi|}^\\frac{1}{p+1}}{|\\Pi|^\\frac{1}{p+1}}  \\det{}^{\\frac{p}{2(p+1)}} \\widehat G\n \\cdot \\det{}^{\\frac{1}{2(p+1)}}\\big[ \\widehat G + (\\widehat \\Phi + \\widehat F') \\widehat{\\tilde G}{}^{-1} (\\widehat \\Phi + \\widehat F')^T  \\big] \\,,\n\\end{equation}\nwhere as before $\\,\\widehat{\\;}\\,$ denotes objects evaluated at covariant coordinates\\footnote{Let us emphasize that this is not a coordinate transformation of a tensor. We just evaluate the component functions in different coordinates.} and $\\widehat F'$ is the Nambu (NC) field strength \\eqref{NCF}.\nThis follows from integrating of \\eqref{MI} followed by the change of integration variables on its right hand side according to the Seiberg-Witten map.\n\nThe factor involving the quotient of $\\widehat{|\\Pi|}$ and $|\\Pi|$ vanishes for constant $|\\Pi|$, but it is essential for the gauge invariance of (\\ref{pNCDBI}) in all other cases.\n\nLet us give two alternative, but equivalent, expressions for the action \\eqref{pDBI}, which might turn out to be useful when looking for supersymmetric generalizations. The first one is obvious:\n\\begin{equation} \\label{pDBI1}\nS_\\text{$p$-DBI} = -\\int d^{p'+1} x \\, \\frac{1}{g_m} \\det{}^\\frac{1}{2}(g) \\cdot \\det{}^{\\frac{1}{2(p+1)}}\\big[1 + g^{-1}(C+F) \\tilde g^{-1} (C+F)^T \\big] \\,.\n\\end{equation}\nA very similar expression can be found using (\\ref{eq_detidentity})\n\\begin{equation} \\label{pDBI1b}\nS_\\text{$p$-DBI} = -\\int d^{p'+1} x \\, \\frac{1}{g_m} \\det{}^\\frac{1}{2}(g) \\cdot \\det{}^{\\frac{1}{2(p+1)}}\\big[1 + \\~g^{-1}(C+F)^{T} g^{-1} (C+F) \\big] \\,.\n\\end{equation}\n\nFor the second one, let us note that $\\det \\tilde g= \\det^{p'\\choose p-1}g$, in the case of factorizable $\\~g$.\nHence, in this case:\n\\begin{equation} \\label{pDBI2}\nS_\\text{$p$-DBI} = - \\int d^{p'+1}x \\frac{1}{g_m}  \\det{}^{\\frac{p-{p'\\choose p-1}}{2(p+1)}}g\\cdot \\det{}^{\\frac{1}{2(p+1)}} \\bm{g}{(C+F)}{-(C+F)^{T}}{\\~g}.\n\\end{equation}\n\nLet us note that in the case of a D-brane, i.e., $p=1$, we get indeed the DBI D-brane action. In the other extreme case, $p=p'$, we get\\footnote{The notation $S_M$ will be justified later.}\n\n\\begin{equation} \\label{S_M}\nS_M = - \\int d^{p+1}x \\frac{1}{g_m} \\det{}^{\\frac{1}{2(p+1)}} \\bm{g}{(C+F)}{-(C+F)^{T}}{\\~g}.\n\\end{equation}\n\nNow we can compare our action, e.g, to the DBI part of the M5-brane action in equation (2.9) of \\cite{Cederwall:1997gg}, \\cite{Bao:2006ef}. Their action is, up to conventions,\n\\begin{equation}\\label{Cederwall}\nS'=-\\int d^6x\\, \\sqrt{\\det g} \\sqrt{1 + \\frac{1}{3} { \\rm tr}k -  \\frac{1}{6} { \\rm tr}{k}^2  +\n \\frac{1}{18} ({ \\rm tr}\\,{k})^2}\\, ,\n\\end{equation}\nwhere $k^i_j=(dA+C)^{ikl}(dA +C)_{jkl}$ is the modified field strength.\n(See also \\cite{Bergshoeff:1996ev}, for an early proposal with a similar index structure.)\nThe form of the polynomial in $k$ in the action has been determined by lengthy computation based on $\\kappa$-symmetry and the requirement of\nnon-linear self-duality, the self-duality relations being consistently decoupled from the background. More precisely, in \\cite{Cederwall:1997gg}, \\cite{Bao:2006ef}, it is shown that consistency of the non-linear self-duality is restrictive enough that demanding $\\kappa$-symmetry gives its explicit form, which can be obtained without a priori specifying the form of the polynomial in the action. At the same time the projector specifying the $\\kappa$-symmetry and the form of the polynomial are determined.  \n\nTo our surprise, we found that this action $S'$ can be interpreted as a\nlow-energy (second order in $k$) approximation of our $p$-DBI action (\\ref{pDBI}). Indeed,for $p=2$ and $p'=5$ we have $d^{p'+1}x = d^6 x$, $\\frac{1}{2(p+1)} = \\frac{1}{6}$ and\n\\[\ndet^\\frac{1}{6}(1 + k) = \\sqrt{1 + \\frac{1}{3} { \\rm tr}k -  \\frac{1}{6} { \\rm tr}k^2  +\n \\frac{1}{18} ({ \\rm tr}\\,k)^2 + \\ldots}\\,.\n\\]\nThe fact that two very different approaches (one based on non-linear self-duality and $\\kappa$-symmetry, the other on commutative\/non-commutative duality) give rise to the same action in the low energy limit is very encouraging and seems to indicate that our proposal can indeed be extended to a full supersymmetric action.\n\nFinally, let us mention that noncommutative structures in the context of the M5 brane have previously been discussed, for example, in \\cite{Berman:2001rka} and \\cite{Berman:2004jv}. However, the type of noncommutativity discussed in these earlier papers is the well-known deformation of the commutative point-wise multiplication along a (constant) Poisson tensor that already appeared in the $p=1$ string theory case. This is very different from the notion of noncommutativity that we argue to be pertinent for $p>1$ and in particular for the  $p=2$ case relevant for the M5 brane: For $p>1$, we do not deform the commutative product -- our ``noncommutativity'' has rather to be understood in the Nambu-Poisson sense as explained in detail above, cf. the remark at the end of the previous section.\n\n\n\n\\section{Background independent gauge} \\label{sec_BIG}\n\nFor $p=1$, assuming that the pullback of the background $2$-form $C$ to the $p'$-brane $N$ is non-degenerate and closed (that is symplectic), one can choose the bivector $\\Pi$ to be the inverse of $C$ (that is a Poisson bivector corresponding to the symplectic structure $C$).\nSolving the open-closed relations then gives\n\\begin{equation} \\label{eq_classicBIG} G = -Cg^{-1}C , \\  \\Phi = -C. \\end{equation}\nThis is known as the background independent gauge \\cite{Seiberg:1999vs}. Our aim is to generalize this construction for $p \\geq 1$, even giving milder assumptions on $C$ for $p=1$.\n\nLet us start on the level of linear algebra first. Assume that $V$ is a finite-dimensional vector space. Let $g$ be an inner product on $V$, and $C \\in \\Lambda^{2}V^{\\ast}$ a $2$-form. Let $P: V \\rightarrow V$ denote a projector orthogonal with respect to $g$, such that\n\\[ \\ker(C) = \\ker(P), \\]\nwhere $C$ is viewed as a map $C: V \\rightarrow V^{\\ast}$. Then there exists a unique bivector $\\Pi \\in \\Lambda^{2}V$, satisfying\n\\begin{equation} \\label{def_thetapinv}\n \\Pi C = P \\ , \\ P \\Pi = \\Pi.\n\\end{equation}\nThe reader can find the proof of this statement in proposition \\ref{tvrz_2formpseudoinverse} of appendix \\ref{sec_BIGsupplement}.\n\nRecall that open-closed relations for $p=1$ have the form\n\\begin{equation}\n\\frac{1}{g+C} = \\frac{1}{G + \\Phi} + \\Pi.\n\\end{equation}\nThis equality can be rewritten as\n\\begin{equation}\nG + \\Phi = (1 - (g+C)\\Pi)^{-1}(g+C).\n\\end{equation}\nUsing (\\ref{def_thetapinv}), one gets\n\\[ G + \\Phi = P'^{T}gP' - Cg^{-1}C - C, \\]\nwhere $P' = 1-P$. From this we can read of the symmetric and skew-symmetric part to get\n\\begin{equation} \\label{eq_BIGocsolution} G = P'^{T}gP' - Cg^{-1}C \\ , \\ \\Phi = -C. \\end{equation}\nWe can view this as a generalization of (\\ref{eq_classicBIG}), not assuming a non-degenerate $C$. See that $G$ is again a positive definite metric, and $G + \\Phi$ is thus invertible. Note that we are now on the level of a single vector space $V$, not discussing any global properties of $\\Pi$ yet.\n\nWe would like to generalize this procedure to $p \\geq 1$ case. Our goal is to find a suitable choice for $\\Pi$, such that $\\Phi = -C$. Assume that $C: \\Lambda^{p}V \\rightarrow V^{\\ast}$ is a linear map, $g$ is an inner product on $V$, and $\\~g$ is an inner product on $\\Lambda^{p}V$. The key is to keep in mind the open-closed relations (\\ref{eq_ocalaSW}). We see that by defining\n\\[ \\mathcal{G} = \\bm{g}{0}{0}{\\~g} \\ , \\ \\mathcal{B} = \\bm{0}{C}{-C^{T}}{0}, \\]\nwe get an inner product $\\mathcal{G}$ on $W \\equiv V \\oplus \\Lambda^{p}V$, and a bilinear skew-symmetric form $\\mathcal{B} \\in \\Lambda^{2} W^{\\ast}$.\n\nThe situation is thus analogous to the previous one, if we replace $V$ by $W$, the metric $g$ by $\\mathcal{G}$, and the $2$-form $C$ by $\\mathcal{B}$.  If we define $\\mathcal{P}$ to be an orthogonal projector with respect to $\\mathcal{G}$ with $\\ker(\\mathcal{P}) = \\ker(\\mathcal{B})$, we may again apply  proposition \\ref{tvrz_2formpseudoinverse} to see that there exists a unique $\\Theta \\in \\Lambda^{2}W$, such that\n\\begin{equation} \\label{eq_Pg1BIGPinv}\n \\Theta \\mathcal{B} = \\mathcal{P} \\ , \\mathcal{P} \\Theta = \\Theta.\n\\end{equation}\nNow we can solve the open-closed relations (\\ref{eq_ocalaSW}) for this choice of $\\Theta$, using the same calculation as we did in order to obtain (\\ref{eq_BIGocsolution}). One gets\n\\begin{equation} \\label{eq_Pg1BIGsolutions}\n\\mathcal{H} = \\mathcal{P}'^{T} \\mathcal{G} \\mathcal{P}' - \\mathcal{B} \\mathcal{G}^{-1} \\mathcal{B} \\ , \\ \\Xi = -\\mathcal{B},\n\\end{equation}\nwhere $\\mathcal{P}' = 1 - \\mathcal{P}$. Exploring what $\\mathcal{B}$ and $\\Xi$ are, leads to $\\Phi = -C$, as intended. However, we do not know whether $\\mathcal{H}$ and $\\Theta$ obtained by this procedure are of the suitable form, that is whether $\\mathcal{H}$ is block-diagonal and $\\Theta$ block-off-diagonal. This can be easily proved by examining the projector $\\mathcal{P}$. Clearly, one has\n\\[ \\ker{\\mathcal{B}} = \\ker{C^{T}} \\oplus \\ker{C} \\subseteq V \\oplus \\Lambda^{p}V. \\]\nTherefore we have that $\\Img(\\mathcal{P}) = \\ker{\\mathcal{B}}^{\\perp} = (\\ker{C}^{T})^{\\perp(g)} \\oplus (\\ker{C})^{\\perp(\\~g)}$. This proves that in a block form, we have\n\\[ \\mathcal{P} = \\bm{P}{0}{0}{\\~P}, \\]\nwhere $P: V \\rightarrow V$ is an orthogonal projector with respect to $g$, and $\\~P: \\Lambda^{p}V \\rightarrow \\Lambda^{p}V$ is an orthogonal projector with respect to $\\~g$. This and the relation (\\ref{eq_Pg1BIGsolutions}) imply that $\\mathcal{H}$ is block-diagonal.\nThe second equality in (\\ref{eq_Pg1BIGPinv}) then proves that $\\Theta$ is block-off-diagonal, that is\n\\[ \\Theta = \\bm{0}{\\Pi}{-\\Pi^{T}}{0}, \\]\nwhere $\\Pi: \\Lambda^{p}V^{\\ast} \\rightarrow V$.\nWe can now simply extract all the relations from (\\ref{eq_Pg1BIGPinv}). The equality $\\Theta \\mathcal{B} = \\mathcal{P}$ gives\n\\[ \\bm{0}{\\Pi}{-\\Pi^{T}}{0} \\bm{0}{C}{-C^{T}}{0} = \\bm{P}{0}{0}{\\~P}, \\]\nwhich translates into\n\\begin{equation} \\label{eq_Cpinv}\n\\Pi C^{T} = -P \\ , \\ \\Pi^{T}C = -\\~P.\n\\end{equation}\nRewriting the equation $\\mathcal{B} \\mathcal{P} = \\mathcal{B}$, we get\n\\[ \\bm{0}{C}{-C^{T}}{0} \\bm{P}{0}{0}{\\~P} = \\bm{0}{C}{-C^{T}}{0}, \\]\nwhich translates into\n\\begin{equation}\nC \\~P = C \\ , \\ C^{T}P = C^{T}.\n\\end{equation}\nAlso see that $\\ker(\\~P) = \\ker(C)$, and $\\ker(P) = \\ker(C^{T})$. The equality $\\mathcal{P} \\Theta = \\Theta$ gives\n\\[  \\bm{P}{0}{0}{\\~P} \\bm{0}{\\Pi}{-\\Pi^{T}}{0} = \\bm{0}{\\Pi}{-\\Pi^{T}}{0}, \\]\nand thus\n\\begin{equation} \\label{eq_ThPcoop}\n P \\Pi = \\Pi \\ , \\ \\~P \\Pi^{T} = \\Pi^{T}.\n\\end{equation}\nFinally, we may examine (\\ref{eq_Pg1BIGsolutions}) to find\n\\begin{equation} \\label{eq_OCsoltionsBIG}\n G = P'^{T}gP' + C \\~g^{-1} C^{T} \\ , \\ \\~G = \\~P'^{T}\\~g\\~P' + C^{T}g^{-1}C \\ , \\ \\Phi = -C.\n\\end{equation}\nWe have thus shown that, corresponding to the orthogonal projectors $P$ and $\\~P$ and the linear map $C: \\Lambda^{p}V \\rightarrow V^{\\ast}$, there exists a unique linear map $\\Pi: \\Lambda^{p}V^{\\ast} \\rightarrow V$, such that (\\ref{eq_Cpinv}) and (\\ref{eq_ThPcoop}) hold. Plugging this $\\Pi$ into open-closed relations (\\ref{eq_ocalaSW}) gives (\\ref{eq_OCsoltionsBIG}).\n\nTo use this for our purposes, we have to impose conditions on $C$ to ensure that $\\Pi$ is a Nambu-Poisson tensor.\n\nFor $p>1$, first observe that the linear map $\\Pi: \\Lambda^{p}V^{\\ast} \\rightarrow V$ induced (at a chosen point on $M$) by a Nambu-Poisson tensor  has rank either $0$ or $p+1$. Since $\\Pi$  always has the same rank as $C$, we get the first assumption on the linear map $C$.\n\nThere will always arise problems with the smoothness of $\\Pi$ at points $x \\in N$, where $C(x) = 0$. If this set has measure zero, we can change the area of integration in DBI action from $N$ to an open submanifold $N'$, where $C(x) \\neq 0$. If not, we cannot go to the background-independent gauge. Let us hereafter assume that $C(x) \\neq 0$ for all $x \\in N$, and therefore that $\\mbox{rank}(C) = p+1$.\n\nNow assume that the linear map $C$ is induced by a $(p+1)$-form $C \\in \\Lambda^{p+1}V^{\\ast}$. Note that in this case, we always have the estimate $\\mbox{rank}(C) \\geq p+1$.\n\nLet $D \\subseteq V$ denote the non-degenerate subspace of $C^{T}$ orthogonal (with respect to $g$) to its kernel, that is $D = \\ker(C^{T})^{\\perp}$. Assumption on the rank of $C$ thus means that $\\dim(D) = p+1$. From the skew-symmetry of $C$, we have that $C \\in \\Lambda^{p+1}D^{\\ast}$. It is thus a top-level form on $D$. Choose now an orthonormal basis $(e_{1}, \\dots, e_{p+1})$ of $D$. We see that\n\\begin{equation} \\label{eq_CinONframe}\n C = \\lambda \\cdot e^{1} \\^ \\dots \\^ e^{p+1},\n\\end{equation}\nwhere $\\lambda \\neq 0$. Now, choosing an arbitrary complementary basis $(f_{1}, \\dots, f_{p'-p})$ of $\\ker(C^{T}) \\equiv D^{\\perp}$, one can find counterexamples to the assumption that, for a general $\\~g$, the map $\\Pi$ is a $(p+1)$-vector (although it has a correct rank). We thus have to add the second assumption: $\\~g$ has to be of the special skew-symmetrized tensor product form (\\ref{def_tensorofg}).\n\nIn this case we find that $\\Lambda^{p}D$ is spanned by orthonormal basis of the form $e_{1} \\^ \\dots \\^ \\hat{e}_{r} \\^ \\dots \\^ e_{p+1}$. This allows us to write $\\Pi$ explicitly as\n\\begin{equation} \\label{eq_Thetacalculated}\n\\Pi = - \\frac{1}{\\lambda} \\cdot e_{1} \\^ \\dots \\^ e_{p+1}.\n\\end{equation}\nIt is easy to show that such a $\\Pi$ indeed satisfies (\\ref{eq_Cpinv}) and (\\ref{eq_ThPcoop}), and since such a $\\Pi$ is unique, this is the one. We can thus conclude that for $\\mbox{rank}(C) = p+1$, and $\\~g$ in the form (\\ref{def_tensorofg}), $\\Pi$ is a $(p+1)$-vector, more precisely $\\Pi \\in \\Lambda^{p+1} D$.\n\nWe now turn our attention to global properties. If we assume that $C(x) \\neq 0$ on the $p'$-brane, we can define the subspace $D$ at every point, defining a smooth subbundle (it is an orthogonal complement to the kernel of constant rank vector bundle morphism $C^{T}$). Around any point, we can choose a local orthonormal frame $(e_{1}, \\dots, e_{p+1})$, forming a local basis for the sections of $D$. The expression (\\ref{eq_Thetacalculated}) proves that $\\Pi$ is a smooth $(p+1)$-vector on the $p'$-brane, since $\\frac{1}{\\lambda}$ is a smooth function.\n\nFinally, we have to decide under which conditions $\\Pi$ forms a Nambu-Poisson tensor. In the view of lemma \\ref{lem_nptoplevel}, we see that the sufficient and necessary condition is that the subbundle $D$ defines an integrable distribution in $N$. This distribution has to be regular, and thus, this condition is  equivalent to the involutivity of $D$ under vector field commutator: $[D,D] \\subseteq D$.\n\nOne can find a simple equivalent criterion for $C$ to define an integrable distribution $D$.  In order to do so, assume now that $(e_{1}, \\dots, e_{p+1}, f_{1}, \\dots f_{p'-p})$ is a positively oriented orthonormal local frame for $N$, such that $(e_{1}, \\dots, e_{p+1})$ is a local orthonormal frame for $D$. The metric volume form $\\Omega_{g}$ is then by definition\n\\[ \\Omega_{g} = e^{1} \\^ \\dots \\^ e^{p+1} \\^ f^{1} \\^ \\dots \\^ f^{p'-p}. \\]\nHaving a volume form, one can form the Hodge dual of $C$. Using (\\ref{eq_CinONframe}) we get\n\\[ \\ast C = \\lambda \\cdot f^{1} \\^ \\dots \\^ f^{p'-p}. \\]\nWe see that $D = \\ker(\\ast C)^{T}$, $(\\ast C)^{T}: TN \\rightarrow \\Lambda^{p'-p-1} T^{\\ast}N$. But forms with integrable kernel distribution have their own name, they are called integrable forms, see Appendix \\ref{sec_BIGsupplement} for the definition and basic properties. We can conclude that $\\Pi$ is a Nambu-Poisson $(p+1)$-vector if and only if $\\ast C$ is an integrable everywhere non-vanishing $(p' - p)$-form on $N$. Note that the Hodge star is defined with respect to the induced metric on $N$.\n\nThere exists a nice sufficient integrability condition: If $C$ is a $(p+1)$-form of rank $p+1$, such that $\\delta C = 0$, then $\\ast C$ is integrable. By $\\delta$ we denote the codifferential defined using the Hodge duality. Note that $\\delta C = 0$ are the non-homogeneous charge free Maxwell equations for the field strength $C$. Also, note that in the whole discussion, we do not need the integrability of the distribution $D^{\\perp}$. Since $C$ is already a non-vanishing $(p+1)$-form of rank $p+1$, the sufficient condition for integrability of $D^{\\perp}$ is $dC = 0$. Interestingly, both $D$ and $D^{\\perp}$ are integrable regular distributions if $C$ is a $(p+1)$-form of rank $p+1$, satisfying the Maxwell equations $dC = 0$, $\\delta C = 0$.\n\nFor $p=1$, the discussion is very similar, except that the rank of $C$ can be any nonzero even integer not exceeding $n$. This adds another condition on $dC$. In particular, the necessary and sufficient condition on $C$ to define a Poisson tensor $\\Pi$ is the integrability of the  regular smooth distribution $D$, and a condition $dC|_{\\Gamma(D)} = 0$.\n\\section{Non-commutative directions, double scaling limit} \\label{sec_NCdirections}\nBy the construction of the preceding section, we have the decompositions\n\\[ TM = D \\oplus D^{\\perp}, \\ \\TM{p} = \\~D \\oplus \\~D^{\\perp}, \\]\nwhere $\\~D = \\Lambda^{p}D$. We say that tangent vectors contained in $D$ point in ``non-commutative\" directions. Because $D$ is integrable, around each point there are coordinates  such that $D$ is spanned by coordinate tangent vectors corresponding to first $p+1$ of these coordinates. These local coordinates are accordingly called ``non-commutative\" coordinates. This terminology comes from the fact that for $p=1$, we have $\\{x^{i},x^{j}\\} = \\Pi^{ij}$. The right-hand side is non-vanishing when both $x^{i}$ and $x^{j}$ correspond to $D$. This gives non-vanishing quantum-mechanical commutator of these coordinates.\n\nWe can thus write all involved quantities in the block matrix form corresponding to this decomposition. From the orthogonality of respective subspaces, the matrices of $g$ and $\\~g$ will be block diagonal:\n\\[ g = \\bm{g_{\\bullet}}{0}{0}{g_{\\circ}}, \\ \\~g = \\bm{\\~\\mathfrak{g}_{\\bullet}}{0}{0}{\\~g_{\\circ}}, \\]\nwhere $g_{\\bullet}$ is a positive definite fibrewise metric on $D$, $\\mathfrak{g}_{\\circ}$ is a positive definite fibrewise metric on $D^{\\perp}$ and $\\~g_{\\bullet}$ and $\\~g_{\\circ}$ are positive definite fibrewise metrics on $\\~D$ and $\\~D^{\\perp}$, respectively. In the same fashion we obtain\n\\[ C = \\bm{C_{\\bullet}}{0}{0}{0}, \\ \\Pi = \\bm{\\Pi_{\\bullet}}{0}{0}{0} \\ , \\  F = \\bm{F_{\\bullet}}{F_{{\\text{\\bfseries{\\scriptsize{I}}}}}}{F_{\\text{\\bfseries{\\scriptsize{II}}}}}{F_{\\circ}}. \\]\nExamine how the $F$-gauged tensor $\\Pi'$ looks like in this block form.\nWe have\n\\[ 1 - F^{T} \\Pi = \\bm{1 - F_{\\bullet}^{T} \\Pi_{\\bullet}}{0}{-F_{{\\text{\\bfseries{\\scriptsize{I}}}}}^{T} \\Pi_{\\bullet}}{1}. \\]\nHence\n\\[ \\Pi' \\equiv \\Pi (1 - F^{T}\\Pi)^{-1} = \\bm{ \\Pi_{\\bullet} (1 - F_{\\bullet}^{T} \\Pi_{\\bullet})^{-1}}{0}{0}{0}. \\]\nDenote $\\Pi_{\\bullet}' = \\Pi_{\\bullet}(1 - F_{\\bullet}^{T} \\Pi_{\\bullet})^{-1}$. We also have $\\Pi_{\\bullet}' = (1 - \\Pi_{\\bullet} F_{\\bullet}^{T})^{-1} \\Pi_{\\bullet}$. Also, note that in this formalism $P$ and $\\~P$ are simply given as\n\\[ P = \\bm{1}{0}{0}{0} \\ , \\ \\~P = \\bm{1}{0}{0}{0}. \\]\nHence, the defining equations of $\\Pi$ can be written as\n\\begin{equation} \\label{eq_tildCinv}\n \\Pi_{\\bullet} C_{\\bullet}^{T} = -1 \\ , \\ \\Pi_{\\bullet}^{T} C_{\\bullet} = -1.\n\\end{equation}\n\nHaving this in hand, recall that for $p=1$, the background independent gauge could be obtained in a completely different way. It was obtained by Seiberg and Witten in \\cite{Seiberg:1999vs} as a following limit of the relation (\\ref{eq_p1correspondence}). Reintroducing the Regge slope $\\alpha'$ into description, the relation between closed variables $g$, $C$ and Nambu fields $G_{N}$, $\\Pi_{N}$ is explicitly\n\\[ G_{N} = g - (2\\pi \\alpha')^{2} Cg^{-1}C^{T}, \\ \\frac{1}{2 \\pi \\alpha'} \\Pi_{N} = -(2 \\pi \\alpha')g^{-1}C \\big(g - (2\\pi \\alpha')^{2}Cg^{-1}C\\big)^{-1}. \\]\nNow one would like to do the zero slope limit $\\alpha' \\rightarrow 0$ in a way such that $G_{N}$ and $\\Pi_{N}$ remain finite. This clearly requires the simultaneous scaling of the metric $g$. Scaling the $g$ as a whole will not work, since the resulting $G_{N}$ will not be a metric. The correct answer is given by scaling the non-commutative part $g_{\\bullet}$ and commutative part $g_{\\circ}$ of the metric $g$ differently. The resulting maps $G_{N}$ and $\\Pi_{N}$ also split accordingly as\n\\[ G_{N \\bullet} = g_{\\bullet} - (2\\pi \\alpha')^{2} C_{\\bullet} g_{\\bullet}^{-1} C_{\\bullet}^{T}, \\ G_{N \\circ} = g_{\\circ}, \\]\n\\[ \\frac{1}{2 \\pi \\alpha'}\\Pi_{N \\bullet} =  -(2\\pi \\alpha') g_{\\bullet}^{-1} C_{\\bullet} (g_{\\bullet} - (2 \\pi \\alpha')^{2}C_{\\bullet} g_{\\bullet}^{-1}C_{\\bullet})^{-1}. \\]\nNow, scaling $g_{\\bullet} \\propto \\epsilon$, $g_{\\circ} \\propto 1$, $\\alpha' \\propto \\epsilon^{\\frac{1}{2}}$ as $\\epsilon \\mapsto 0$ gives in this limit\n\\[ G_{N \\bullet} = - C_{\\bullet} g_{\\bullet}^{-1} C_{\\bullet}^{T}, \\ G_{N \\circ} = g_{\\circ}, \\]\n\\[ \\Pi_{N \\bullet} = C_{\\bullet}^{-1}. \\]\nReplacing $\\Pi_{N}$ by $\\Pi$ and $G_{N}$ by $G$ is exactly the background independent gauge. This double scaling limit was then used to determine which terms should be kept in the expansion of the DBI action. We would like to find an analogue of this in our $p>1$ case.\\footnote{See \\cite{Chen:2010br} for a previous discussion of the double scaling limit in the context of the M2\/M5 system that came to different conclusions regarding the appropriate powers of $\\epsilon$.} We immediately see that first naive answer would be wrong. One of the relations is\n\\[ G_{N \\bullet} = g_{\\bullet} + C_{\\bullet} \\~g_{\\bullet}^{-1} C_{\\bullet}^{T}. \\]\nNote that $\\~g_{\\bullet}$ is again a skew-symmetrized $p$-fold tensor product of $g_{\\bullet}$. This suggests that if $g_{\\bullet} \\propto \\epsilon$, then $\\~g_{\\bullet} \\propto \\epsilon^{p}$. This would imply that $C_{\\bullet} \\propto \\epsilon^{\\frac{p}{2}}$ in order to keep $G_{N \\bullet}$ finite (we have included $\\epsilon$ into $C$). But the second relation is\n\\[ \\~G_{N \\bullet} = \\~g_{\\bullet} + C_{\\bullet}^{T} g_{\\bullet}^{-1} C. \\]\nThis shows that $\\~G_{N} \\rightarrow 0$ as $\\epsilon \\rightarrow 0$. This is clearly not very plausible. However, this can still be fixed by using the remaining gauge fixing freedom of the Polyakov action (\\ref{def_polyakov}) by scaling also the ratio between $g$ and $\\~g$. The biggest issue comes with the fact that $\\~g_{\\circ}$ is not a tensor product of $g_{\\circ}$'s only. In fact, every component $(\\~g_{\\circ})_{IJ}$ contains as many $g_{\\bullet}$'s as the number of ``commutative\" indices in $I$ (or $J$) is. This means that every component of $\\~g_{\\circ}$ should scale differently. We must thus abandon the idea of scaling just $g$, we have to scale $\\~g$ independently! The correct answer is given by the geometry of the vector bundle $W= TM \\oplus \\Lambda^p TM$ again. We immediately see that scaling $\\mathcal{G}_{\\bullet} \\propto \\epsilon$, $\\mathcal{G}_{\\circ} \\propto 1$ and $\\mathcal{B} \\propto \\epsilon^{\\frac{1}{2}}$ gives in limit $\\epsilon \\rightarrow 0$ the background independent gauge. This corresponds to\n\\begin{equation} \\label{def_doublescaling}\n g_{\\bullet} \\propto \\epsilon, \\ \\~g_{\\bullet} \\propto \\epsilon, \\ g_{\\circ} \\propto 1, \\ \\~g_{\\circ} \\propto 1, \\ C_{\\bullet} \\propto \\epsilon^{\\frac{1}{2}}.\n\\end{equation}\nLet us note that in the case of an M5 brane a scaling treating directions differently was described in \\cite{Bergshoeff:2000ai} and \\cite{Bergshoeff:2000jn}. It would be interesting to compare the scaling in these papers with the one introduced here. \n\\section{Matrix model} \\label{sec_matrix}\n\nNow we will apply the previous generalization of the background independent gauge.\nWe will use the double scaling limit to cut off the power series expansion of the DBI action. It turns out that we find an action describing a natural $p>1$ (semi-classical) analogue of a matrix model with higher brackets and an interacting with the gauge field $F$. It will be of order $2(p+1)$ in the matrix variables $\\widehat{X}^a$,  and at most quadratic in  $F$. The term of order $2(p+1)$ in $\\widehat{X}^a$'s and constant in $F$ gives a possible $p>1$ analogue of the semiclassical pure matrix model.\n\nAssume that $C$ satisfies all the conditions required for $\\Pi$ to be a Nambu-Poisson tensor on $N$.\nFrom (\\ref{pDBI1}), we have that Lagrangian of the commutative $p$-DBI action has the form\n\\[ \\mathcal{L}_{p-\\text{DBI}} = - \\frac{1}{g_{m}} \\det{}^{\\frac{1}{2}}(g) \\cdot \\det{}^{\\frac{1}{2(p+1)}}[ 1 + g^{-1}(C+F)\\~g^{-1}(C+F)^{T}]. \\]\nNote that the second determinant is the determinant of the vector bundle endomorphism $X: TM \\rightarrow TM$, where $X = 1 + g^{-1}(C+F)\\~g^{-1}(C+F)^{T}$.\nIn the block form $X: D \\oplus D^{\\perp} \\rightarrow D \\oplus D^{\\perp}$, we have\n\\[ X = \\bm{1 + g_{\\bullet}^{-1}(C_{\\bullet}+F_{\\bullet})\\~g_{\\bullet}^{-1}(C_{\\bullet}+F_{\\bullet})^{T} + g_{\\bullet}^{-1}F_{{\\text{\\bfseries{\\scriptsize{I}}}}}\\~g_{\\circ}^{-1}F_{{\\text{\\bfseries{\\scriptsize{I}}}}}^{T}}{g_{\\bullet}^{-1}(C_{\\bullet}+F_{\\bullet})\\~g_{\\bullet}^{-1}F_{\\text{\\bfseries{\\scriptsize{II}}}}^{T} + g_{\\bullet}^{-1}F_{{\\text{\\bfseries{\\scriptsize{I}}}}}\\~g_{\\circ}^{-1}F_{\\circ}^{T}}{g_{\\circ}^{-1}F_{\\text{\\bfseries{\\scriptsize{II}}}}\\~g_{\\bullet}^{-1}(C_{\\bullet}+F_{\\bullet})^{T} + g_{\\circ}^{-1}F_{\\circ}\\~g_{\\circ}^{-1}F_{{\\text{\\bfseries{\\scriptsize{I}}}}}^{T}}{1 + g_{\\circ}^{-1}F_{\\text{\\bfseries{\\scriptsize{II}}}}\\~g_{\\bullet}^{-1}F_{\\text{\\bfseries{\\scriptsize{II}}}}^{T} + g_{\\circ}^{-1} F_{\\circ} \\~g_{\\circ}^{-1} F_{\\circ}^{T}}. \\]\nHere we have used the following notations for the blocks of $F$\n\\[ F = \\bm{F_{\\bullet}}{F_{{\\text{\\bfseries{\\scriptsize{I}}}}}}{F_{\\text{\\bfseries{\\scriptsize{II}}}}}{F_{\\circ}}. \\]\nThis can be decomposed as a product\n\\[ X = \\bm{g_{\\bullet}^{-1}(C_{\\bullet} + F_{\\bullet})}{0}{0}{1} Y \\bm{\\~g_{\\bullet}^{-1}(C_{\\bullet} + F_{\\bullet})^{T}}{0}{0}{1}, \\]\nwhere the vector bundle endomorphism $Y: \\~D \\oplus D^{\\perp} \\rightarrow \\~D \\oplus D^{\\perp}$ is\n\\[ Y = \\bm{1 + \\Pi_{\\bullet}'^{T}(g_{\\bullet} + F_{{\\text{\\bfseries{\\scriptsize{I}}}}} \\~g_{\\circ}^{-1} F_{{\\text{\\bfseries{\\scriptsize{I}}}}}^{T}) \\Pi_{\\bullet}' \\~g_{\\bullet}}{\\~g_{\\bullet}^{-1}(F_{\\text{\\bfseries{\\scriptsize{II}}}}^{T} - \\~g_{\\bullet} \\Pi_{\\bullet}'^{T}F_{{\\text{\\bfseries{\\scriptsize{I}}}}} `\\~g_{\\circ}^{-1} F_{\\circ}^{T})}{g_{\\circ}^{-1}(F_{\\text{\\bfseries{\\scriptsize{II}}}} - F_{\\circ} \\~g_{\\circ}^{-1}F_{{\\text{\\bfseries{\\scriptsize{I}}}}}^{T} \\Pi_{\\bullet}' \\~g_{\\bullet})}{1 + g_{\\circ}^{-1}F_{\\text{\\bfseries{\\scriptsize{II}}}}\\~g_{\\bullet}^{-1}F_{\\text{\\bfseries{\\scriptsize{II}}}}^{T} + g_{\\circ}^{-1}F_{\\circ}\\~g_{\\circ}^{-1}F_{\\circ}^{T}}. \\]\nWriting $Y$ in block form as\n\\[ Y = \\bm{Y_{\\bullet}}{Y_{{\\text{\\bfseries{\\scriptsize{I}}}}}}{Y_{\\text{\\bfseries{\\scriptsize{II}}}}}{Y_{\\circ}}, \\]\nnote that $Y_{\\bullet}$ is an invertible matrix. This is true because it is a top left block of the matrix $Y$ coming from positive definite matrix $g + (C+F)\\~g^{-1}(C+F)$ by multiplying it by invertible block-diagonal matrices. Hence, we can write\n\\begin{equation} \\det{(Y)} = \\det{(Y_{\\bullet})} \\det{ (Y_{\\circ} - Y_{{\\text{\\bfseries{\\scriptsize{I}}}}} Y_{\\bullet}^{-1} Y_{\\text{\\bfseries{\\scriptsize{II}}}})}. \\label{detY1}\\end{equation}\nThe second matrix has the form\n\\[\n\\begin{split}\nY_{\\circ} - Y_{{\\text{\\bfseries{\\scriptsize{I}}}}} Y_{\\bullet}^{-1} Y_{\\text{\\bfseries{\\scriptsize{II}}}} &= 1 + g_{\\circ}^{-1} F_{\\text{\\bfseries{\\scriptsize{II}}}} (1 - Y_{\\bullet}^{-1})\\~g_{\\bullet}^{-1} F_{\\text{\\bfseries{\\scriptsize{II}}}}^{T} + g_{\\circ}^{-1} F_{\\circ} \\~g_{0}^{-1} F_{\\circ}^{T} + g_{\\circ}^{-1} F_{\\text{\\bfseries{\\scriptsize{II}}}} Y_{\\bullet}^{-1} \\Pi_{\\bullet}'^{T} F_{{\\text{\\bfseries{\\scriptsize{I}}}}} \\~g_{\\circ}^{-1} F_{\\circ}^{T} \\\\\n& + g_{\\circ}^{-1} F_{\\circ} \\~g_{\\circ}^{-1} F_{{\\text{\\bfseries{\\scriptsize{I}}}}}^{T} \\Pi_{\\bullet}' \\~g_{\\bullet} Y_{\\bullet}^{-1} \\~g_{\\bullet}^{-1} F_{\\text{\\bfseries{\\scriptsize{II}}}}^{T} - g_{\\circ}^{-1} F_{\\circ} \\~g_{\\circ}^{-1} F_{{\\text{\\bfseries{\\scriptsize{I}}}}}^{T} \\Pi_{\\bullet}' \\~g_{\\bullet} Y_{\\bullet}^{-1} \\Pi_{\\bullet}'^{T} F_{{\\text{\\bfseries{\\scriptsize{I}}}}} \\~g_{\\circ}^{-1} F_{\\circ}^{T}.\n\\end{split}\n\\]\nAt this point, we will employ the double scaling limit introduced above. Namely, in the $\\det^{\\frac{1}{2(p+1)}}(Y)$, we wish to keep only the terms scaling at most as $\\epsilon^{1}$. Note that $(Y_{\\bullet} - 1) \\propto \\epsilon$. Also, $Y_{\\bullet}^{-1} = 1 - (Y_{\\bullet} - 1) + o(\\epsilon^{2})$. Using this, we can write\n\\[\n\\begin{split}\nY_{\\circ} - Y_{{\\text{\\bfseries{\\scriptsize{I}}}}} Y_{\\bullet}^{-1} Y_{\\text{\\bfseries{\\scriptsize{II}}}} = 1 + g_{\\circ}^{-1} \\Big( F_{\\text{\\bfseries{\\scriptsize{II}}}} \\Pi_{\\bullet}'^{T} g_{\\bullet} \\Pi_{\\bullet}' F_{\\text{\\bfseries{\\scriptsize{II}}}}^{T} +  \\big( F_{\\text{\\bfseries{\\scriptsize{II}}}} \\Pi_{\\bullet}'^{T} F_{I} + F_{\\circ} \\big) \\~g_{\\circ}^{-1} \\big( F_{\\text{\\bfseries{\\scriptsize{II}}}} \\Pi_{\\bullet}'^{T} F_{I} + F_{\\circ} \\big)^{T} \\Big) + o(\\epsilon^{2}).\n\\end{split}\n\\]\nThe whole term in parentheses after $g_{0}^{-1}$ is of order $\\epsilon^{1}$. Therefore, we have\n\\[\n\\begin{split}\n\\det{}^{\\frac{1}{2(p+1)}}(Y_{\\circ} - Y_{{\\text{\\bfseries{\\scriptsize{I}}}}} Y_{\\bullet}^{-1} Y_{\\text{\\bfseries{\\scriptsize{II}}}}) & = 1 + \\frac{1}{2(p+1)} \\tr ( g_{\\circ}^{-1} F_{\\text{\\bfseries{\\scriptsize{II}}}} \\Pi_{\\bullet}'^{T} g_{\\bullet} \\Pi_{\\bullet}' F_{\\text{\\bfseries{\\scriptsize{II}}}}^{T} ) \\\\\n& +  \\frac{1}{2(p+1)} \\tr \\Big( g_{\\circ}^{-1} \\big( F_{\\text{\\bfseries{\\scriptsize{II}}}} \\Pi_{\\bullet}'^{T} F_{I} + F_{\\circ} \\big) \\~g_{\\circ}^{-1} \\big( F_{\\text{\\bfseries{\\scriptsize{II}}}} \\Pi_{\\bullet}'^{T} F_{I} + F_{\\circ} \\big)^{T} \\big) \\Big) + o(\\epsilon^{2}).\n\\end{split}\n\\]\nFor the first factor in (\\ref{detY1}), we have\n\\[ \\det{}^{\\frac{1}{2(p+1)}}(Y_{\\bullet}) = 1 + \\frac{1}{2(p+1)} \\tr\\big( \\Pi_{\\bullet}'^{T} (g_{\\bullet} + F_{{\\text{\\bfseries{\\scriptsize{I}}}}} \\~g_{\\circ}^{-1} F_{{\\text{\\bfseries{\\scriptsize{I}}}}}^{T}) \\Pi_{\\bullet}' \\~g_{\\bullet} \\big) + o(\\epsilon^{2}). \\]\nPutting all together, we obtain\n\\begin{equation} \\label{eq_detYexpansion}\n\\begin{split}\n\\det{}^{\\frac{1}{2(p+1)}}(Y) & = 1 + \\frac{1}{2(p+1)} \\tr\\big( \\Pi_{\\bullet}'^{T} (g_{\\bullet} + F_{{\\text{\\bfseries{\\scriptsize{I}}}}} \\~g_{\\circ}^{-1} F_{{\\text{\\bfseries{\\scriptsize{I}}}}}^{T}) \\Pi_{\\bullet}' \\~g_{\\bullet} \\big) + \\frac{1}{2(p+1)} \\tr ( g_{\\circ}^{-1} F_{\\text{\\bfseries{\\scriptsize{II}}}} \\Pi_{\\bullet}'^{T} g_{\\bullet} \\Pi_{\\bullet}' F_{\\text{\\bfseries{\\scriptsize{II}}}}^{T} ) \\\\\n& +  \\frac{1}{2(p+1)} \\tr \\Big( g_{\\circ}^{-1} \\big( F_{\\text{\\bfseries{\\scriptsize{II}}}} \\Pi_{\\bullet}'^{T} F_{I} + F_{\\circ} \\big) \\~g_{\\circ}^{-1} \\big( F_{\\text{\\bfseries{\\scriptsize{II}}}} \\Pi_{\\bullet}'^{T} F_{I} + F_{\\circ} \\big)^{T} \\Big) + o(\\epsilon^{2}).\n\\end{split}\n\\end{equation}\nNow, comparing the definitions of scalar densities corresponding to $\\Pi$ and  $\\Pi'$, it is clear that\n\\[ \\det(C_{\\bullet} + F_{\\bullet}) = \\pm \\det(1 - \\Pi F^{T}) \\cdot |\\Pi(x)|^{-(p+1)}. \\]\nHere we assume that one chooses the basis of $\\Lambda^{p}D$ induced by the basis of $D$. The sign $\\pm$ depends on the ordering of that basis. Next, see that $\\det(\\~g_{\\bullet}) = \\det{}^{\\binom{p}{p-1}}(g_{\\bullet}) = \\det{}^{p}(g_{\\bullet})$. This shows that\n\\[ S_\\text{$p$-DBI} = \\mp \\int d^{p'+1}x \\frac{1}{g_{m}} \\frac{\\det{}^{\\frac{1}{p+1}}(1 - \\Pi F^{T})}{|\\Pi(x)|^{\\frac{1}{p+1}} \\det{}^{\\frac{1}{2}}(g_{\\bullet})} \\det{}^{\\frac{1}{2}}(g) \\det{}^{\\frac{1}{2(p+1)}}(Y). \\]\nChanging the coordinates according to Seiberg-Witten map, we get the noncommutative DBI action in the form:\n\\[ S_\\text{$p$-NCDBI} = \\mp \\int d^{p'+1}x \\frac{1}{\\widehat{g}_{m}} \\frac{\\det{}^{\\frac{1}{2}}(\\widehat{g})}{|\\Pi(x)|^{\\frac{1}{p+1}} \\det{}^{\\frac{1}{2}}(\\widehat{g}_{\\bullet})}  \\det{}^{\\frac{1}{2(p+1)}}(\\widehat{Y}). \\]\n\nIn the last part of the discussion assume that the distribution $D^{\\perp}$ is also integrable, so we can use the set of local coordinates $(x^{1}, \\dots, x^{p+1}, x^{p+2}, \\dots, x^{p'+1})$ on $N$, such that $(\\ppx{}{1}, \\dots \\ppx{}{p+1})$ span $D$, and $(\\ppx{}{p+2}, \\dots, \\ppx{}{p'+1})$ span $D^{\\perp}$. All quantities with indices in $D^{\\perp}$ are now assumed to be in this coordinate basis. Under this assumptions, the integral density in the action can be written as\n\\[ \\det{}^{\\frac{1}{2}}(g) = \\det{}^{\\frac{1}{2}}(g_{\\bullet}) \\cdot \\det{}^{\\frac{1}{2}}(g_{\\circ}). \\]\n\nFinally, to distinguish the noncommutative and commutative coordinates, we reserve the letters $(a,b,c)$ for labeling the coordinates $(x^{1}, \\dots, x^{p+1})$, $(i,j,k)$ for labeling the coordinates $(x^{p+2},\n\\dots, x^{p'+1})$, $(A,B,C)$ for $p$-indices containing only noncommutative indices (thus $p$-indices labeling $\\~D$) and $(I,J,K)$ for $p$-indices containing at least one commutative index (thus $p$-indices labeling $\\~D^{\\perp})$.  Also, note that from the definition of $\\rho_{A}$, we have\n\\[ \\widehat{\\Pi}^{'aB} = \\{ \\widehat{X}^{a}, \\widehat{X}^{b_{1}}, \\dots, \\widehat{X}^{b_{p}} \\}, \\]\nwhere $\\{\\cdot,\\dots,\\cdot\\}$ is the Nambu-Poisson bracket corresponding to $\\Pi$, $\\widehat{X}^{a} = \\rho_{A}^{\\ast}(x^{a})$, and $B = (b_{1}, \\dots, b_{p})$. To simplify the expressions, we shall also use the shorthand notation $\\{\\cdot,\\widehat{X}^{A}\\} \\equiv \\{\\cdot, \\widehat{X}^{a_{1}}, \\dots, \\widehat{X}^{a_{p}} \\}$. Finally, we also introduce usual index raising\/lowering conventions, for example, $\\Fud{k}{A} = \\sum_{n=1}^{p'+1} \\widehat{g}^{kn} \\widehat{F}_{nA} = \\widehat{g}^{kl} \\widehat{F}_{lA}$, or  $\\Fdu{k}{A} = \\widehat{\\~g}^{AB} \\widehat{F}_{kB}$ for multiindices. Note that since both $g$ and $\\~g$ are block diagonal, no confusion concerning range of summation appears. Implementing this notation, we can write\n\\[\n\\begin{split}\nS_\\text{$p$-NCDBI} = \\mp \\int d^{p'+1}x \\frac{1}{\\widehat{g}_{m}} \\frac{\\det{}^{\\frac{1}{2}}(\\widehat{g}_{\\circ})}{|\\Pi(x)|^{\\frac{1}{p+1}}} \\Big( 1 + \\frac{1}{2(p+1)}  \\{ \\widehat{X}^{a}, \\widehat{X}^{A}\\} \\{ \\widehat{X}_{a},\\widehat{X}_{A} \\} \\\\\n+ \\frac{1}{2(p+1)} \\{\\widehat{X}^{a}, \\widehat{X}^{A} \\} \\Fdu{a}{I} \\widehat{F}_{bI} \\{\\widehat{X}^{b}, \\widehat{X}_{A} \\} + \\frac{1}{2(p+1)} \\{\\widehat{X}^{a}, \\widehat{X}^{A}\\} \\widehat{F}_{kA} \\Fud{k}{B} \\{ \\widehat{X}_{a}, \\widehat{X}^{B} \\} \\\\\n+ \\frac{1}{2(p+1)}(\\widehat{F}_{kA} \\{ \\widehat{X}^{a}, \\widehat{X}^{A} \\} \\widehat{F}_{aJ} + \\widehat{F}_{kJ}) (\\Fud{k}{B} \\{ \\widehat{X}^{b}, \\widehat{X}^{B} \\} \\Fdu{b}{J} + \\widehat{F}^{kJ}) \\Big) + \\cdots.\n\\end{split}\n\\]\nNote that the first non-cosmological term $\\{\\widehat{X}^{a},\\widehat{X}^{A}\\} \\{ \\widehat{X}_{a}, \\widehat{X}_{A}\\}$ can be rewritten as\n\\begin{equation}\n \\{ \\widehat{X}^{a}, \\widehat{X}^{A}\\} \\{ \\widehat{X}_{a},\\widehat{X}_{A} \\} = \\frac{1}{p!} \\widehat{g}_{a_{1}b_{1}} \\dots \\widehat{g}_{a_{p+1} b_{p+1}} \\{ \\widehat{X}^{a_{1}}, \\dots, \\widehat{X}^{a_{p+1}} \\} \\{ \\widehat{X}^{b_{1}}, \\dots, \\widehat{X}^{b_{p+1}} \\},\n\\end{equation}\nwhere summation now goes over all (not strictly ordered) $(p+1)$-indices $(a_{1}, \\dots, a_{p+1})$ and $(b_{1}, \\dots, b_{p+1})$. Here, we have used the fact that $\\~g_{\\bullet}$ is a skew-symmetrized $p$-fold tensor product of $g_{\\bullet}$. We can even drop the restriction of the summations to noncommutative directions, since the Nambu-Poisson bracket takes care of this automatically. This term corresponds to a $p>1$ generalization of the matrix model.\nNote that using the double scaling limit for the expansion of (\\ref{eq_detYexpansion}) leads to a series in positive integer powers of $\\epsilon$, automatically truncating higher-order powers in $F$. This gives an independent justification of the independent scaling of $\\~g_{\\bullet}$ and $\\~g_{\\circ}$ in (\\ref{def_doublescaling}).\n\n\\section{Conclusions and Discussion}\nIn this paper we have extended, clarified and further developed the construction outlined in \\cite{Jurco:2012yv}. We discussed in detail the bosonic part of an all-order effective action for a system of multiple $p$-branes ending on a $p'$-brane. The leading principle was to have an action allowing, similarly to the DBI action, for two mutually equivalent descriptions: a commutative and a ``noncommutative'' one. As explained in the main body of the paper, the noncommutativity means a semicalssical one, in which the Poisson tensor is replaced by a Nambu-Poisson one.\\footnote{Let us notice, that in our approach to noncommutativity of fivebrane, the ordinary point-wise product remains undeformed}  It turned out that this requirement determines the bosonic part of the effective action essentially uniquely. \n\nIn our derivation of the action, generalized geometry played an essential role. All key ingredients, have their origin in the generalized geometry. It already has been  appreciated in the literature that the presence of a (p+1)-form leads to a generalized tangent space $TM\\oplus \\Lambda^p T^*M$. Although, this observation perfectly applies also in our situation, we found it very useful to double it, i.e., to consider the the extended\/doubled generalized tangent space $W\\oplus W^*$, with $W=TM\\oplus \\Lambda^p TM$. \n\nLet us comment on this more: In the string case, $p$=1, the sum of the background fields $g+B$ plays a prominent role. It enters naturally the Polyakov action, the DBI action, Buscher's rules, etc. In generalized geometry, one way define a generalized metric, is to give a subbundle of the generalized tangent bundle $TM\\oplus T^*M$ of maximal rank, on which the natural (+) pairing on generalized tangent bundle is positive definite. Such a subbundle can be characterized as a graph of the map from $TM\\to T^*M$ defined by the sum $g+B$. Therefore, it is quite natural to look for a formalism which would allow for a natural ``sum\" of a metric and a higher rank $(p+1)$-form. What this sum should be is indicated by the Polyakov type membrane action in its matrix form (\\ref{def_actiontotmatrix}). From here it is just a small step to recognize the doubled generalized tangent bundle as a right framework for a meaningful interpretation of the ``sum\" of the metric and a higher rank $(p+1)$-form. This observation is further supported by the form of the open closed relations in the doubled form (\\ref{eq_occorrespondece0}) and the matrix form of the Nambu sigma model (\\ref{def_actionnsm}). Finally, the corresponding Hamiltonian (\\ref{def_polyakovham}), cf. also (\\ref {eq_nsmham}), tells us what the relation to the generalized metric on $TM\\oplus\\Lambda^p T^\\ast M$ is. Hence, at the end, we do not really use the full doubled generalized tangent bundle, we use it only for a nice embedding of the generalized tangent bundle $TM\\oplus\\Lambda^p T^\\ast M$.\\footnote{The doubled generalized geometry formalism can also be introduced for the $p$=1 string case and allows an elegant formulation of the theory. For any $p$, the appearance of $T M$ and $\\Lambda^p T M$ (and similarly of $T^\\ast M$ and $\\Lambda^p T^\\ast M$) is related to the split into one temporal and $p$~spatial world-sheet directions.}\n\nNevertheless, we found the doubled generalized geometry quite intriguing. Extending on the above comments: Since on the doubled generalized tangent bundle there is a natural function-valued non-degenerated  pairing $\\langle.,.\\rangle$, we can mimic the standard constructions with $TM\\oplus T^*M$. For instance, one can speak of the orthogonal group, define the generalized metric using an involutive endomorphism $\\mathcal{T}$ on $W\\oplus W^*$, such that $\\langle \\mathcal{T},.\\rangle$ defines a fibre-wise metric on the doubled generalized tangent bundle, etc. \n\nHowever, we are still facing a problem; We lack a canonical Courant algebroid structure. The reason lies basically in very limited choices for the anchor map $\\rho: W\\oplus W^* \\rightarrow TM$, which leave us only with a projection onto the tangent bundle $TM$. The map $\\rho$ is therefore ``too simple\" to control the symmetric part of any bracket. However, we can still consider Leibniz algebroid structures on $W \\oplus W^{\\ast}$. \nThere are several possibilities to do this. To choose the one suitable for $p$-brane backgrounds, one can consider the action of the map $e^{\\mathcal{B}}: W \\oplus W^{\\ast} \\rightarrow W \\oplus W^{\\ast}$, where $\\mathcal{B}$ is a general section of $\\Lambda^{2}W$, viewed as a map from $W$ to $W^{\\ast}$, and extended to $\\mbox{End}(W \\oplus W^{\\ast})$ by zeros. The map $e^{\\mathcal{B}}$ is thus an analogue of the usual $B$-field transform of generalized geometry $TM \\oplus T^{\\ast}M$. It turns our that there is a Leibniz algebroid, such that the condition for $e^{\\mathcal{B}}$ to be an isomorphism of the bracket forces $\\mathcal{B}$ to take the block off-diagonal form (\\ref{eq_bigGbigB}), with $C \\in \\Omega^{p+1}_{closed}(M)$. This bracket coincides with the one defined by Hagiwara in \\cite{hagiwara} to study Nambu-Dirac manifolds. Moreover, Nambu-Poisson manifolds appear naturally as its Nambu-Dirac structures. Interestingly, its full group of orthogonal automorphisms can be calculated, giving (for $p>1$) a semi-direct product $\\Diff(M) \\ltimes ( \\Omega^{p+1}_{closed} \\rtimes G)$, where $G$ is the group of locally constant non-zero functions on $M$. Notably, this coincides with the group of all automorphisms of higher Dorfman bracket, see e.g. \\cite{2011ScChA..54..437B}. \n\nRelating our approach, based on the generalized geometry on the vector bundle $W \\oplus W^{\\ast}$, with the usual generalized geometries in $M$-theory and supergravity \\cite{Hull:2007zu, Berman:2010is, Coimbra:2011nw, Coimbra:2012af}, we notice the following. A choice of a generalized geometry is subject to the field content one wants to describe. In principle, one can double each of of them and use the advantages of having a natural function-valued pairing as we did for our case of interest in this paper.  However, the field content coming with such a doubled generalized geometry is much bigger then we started with and we have to reduce it accordingly. \n\nFinally, let us again notice the striking similarity with the result of \\cite{Cederwall:1997gg}, \\cite{Bao:2006ef} -- based on a very different approach -- and  discussed after equation (\\ref{Cederwall}). We find worth to pursue a deeper understanding of this similarity in the future.\n\n\n\n\n\\section*{Acknowledgement}\n\nIt is a pleasure to thank Tsuguhiko Asakawa, Peter Bouwknegt, Chong-Sun Chu, Pei-Ming Ho, Petr Ho\\v rava, Dalibor Kar\\'asek, Noriaki Ikeda, Matsuo Sato, Libor \\v Snobl, and Satoshi Watamura for helpful discussions. B.J. and P.S. appreciate the hospitality of the Center for Theoretical Sciences, Taipei, Taiwan, R.O.C. B.J. thanks CERN for hospitality.\nWe gratefully acknowledge financial support by the grant GA\\v CR P201\/12\/G028 (B.J.), by the Grant Agency of the Czech Technical University in Prague, grant No. SGS13\/217\/OHK4\/3T\/14 (J.V.), and by the DFG within the Research Training Group 1620 ``Models of Gravity'' (J.V., P.S.).\nWe thank the DAAD (PPP) and ASCR \\& MEYS (Mobility) for supporting our collaboration. We also thank the referee for his comments which helped to improve the manuscript.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFor more than a decade transport equation coupled with fluid flow model attracted the attention of researchers due to its significant role in modelling various real life problems ranging from environmental issues to physiological importance. For instance contemporary world is tackling with the challenges of ground water pollution due to diffusion of pollutant transported through rivers, use of drug-eluting stents to remove stenosis in human arteries after implanting stents into them etc. Authors of $\\cite{RefJ}-\\cite{RefN}$ have studied various coupled systems involving different fluid flow models and transport equation. Whereas $Vassilev$ and $Yotov$ in $\\cite{RefJ}$ have presented a mixed finite element analysis of coupled Stokes-Darcy-Transport model, $Hui$ and $Jhang$ in $\\cite{RefM}$ have discussed about a stabilized mixed finite element method for coupled transient Stokes-Darcy flows with transport. $Cesmelio\\breve{g}lu$ $et$ $al.$  have studied continuous and discontinuous finite element methods for coupled Navier-Stokes\/Darcy and transport problems in $\\cite{RefK}$ and $Cesmelio\\breve{g}lu$ together with  $Rivi\\grave{e}re$  in $\\cite{RefL} $ have presented a mathematical analysis of existence and uniqueness of coupled NS-Darcy- unsteady Transport equation. Recently $Chowdhury$ and $Kumar$ $\\cite{RefN}$ have considered to study subgrid scale stabilized finite element analysis of coupled Stokes-Brinkman-Transport problem. Importantly authors in $\\cite{RefL}$-$\\cite{RefN}$ have considered the viscosity of fluid flow problem dependent on concentration of the solute transported into the fluid. In few recent works authors in $\\cite{RefW}$-$\\cite{RefX}$ have focused on studying advection-diffusion transport equation coupled with incompressible Navier-Stokes equation. Whereas $Du$ and $Liu$ in $\\cite{RefW}$ have worked with lattice Boltzmann model, $Yua$ $et$ $al.$ ($\\cite{RefX}$) have studied finite difference method for the coupled model. Both of these studies have considered constant viscosity coefficient which indicates an one-way or weak coupling between fluid flow model and transport equation. In this paper we have presented a stabilized finite element analysis of transient Navier-Stokes ($NS$) fully-coupled with unsteady Advection-diffusion-reaction equation with variable coefficients ($VADR$). $Subgrid$ $multiscale$ ($SGS$), a most general finite element stabilization technique, has been employed to study this coupled system. We have considered concentration dependent viscosity in the fluid flow model as well as spatially variable diffusion coefficients in transport equation. These considerations make this coupling  not only two-sided or strong but also more efficient to model the contemporary real life challenges accurately. The previous studies on coupled $NS$-Transport model have neither considered variable viscosity and diffusion coefficients nor discussed about any error estimation for the method applied to study the model. In this study we have elaborately carried out both $apriori$ and $aposteriori$ error estimations for a general finite element stabilization scheme to study strongly coupled transient $NS$-$VADR$ model. \\vspace{1mm}\\\\\nIt is well known fact that lack of stability in standard Galerkin finite element method has driven researchers to introduce stabilized methods such as Streamline upwind\/Petrov-Galerkin ($SUPG$) formulation, Galerkin\/least-squares ($G$ $LS$) method, characteristic-based split ($CBS$) method, Subgrid Scale ($SGS$) method, bubble stabilization etc. Over last four decades huge developments have been taken place in the study of various stabilization techniques. Starting with the works of $Brooks$ and $Hughes$ $\\cite{RefA}$ on $SUPG$; $Hughes$, $Franca$ and $Hulbert$ $\\cite{RefB}$ on $GLS$; $Hughes$ introducing $SGS$ in $\\cite{RefC}$; $Russo$ $\\cite{RefH}$ explaining bubble stabilization method for the linearized incompressible $NS$ equations, the stabilization schemes have been growing through the studies of $Hannani$ $et$ $al.$ $\\cite{RefD}$ on comparison between $SUPG$ and $GLS$ formulation for steady state incompressible $NS$ equations; $Codina$ and $Zienkiewicz$ $\\cite{RefI}$ on comparison of $CBS$ and $GLS$; $Codina$ $et$ $al.$ on numerical comparison of $CBS$ and $SGS$ method of the incompressible $NS$ equations in $\\cite{RefE}$; $Russo$ $\\cite{RefG}$ on comparison of $SUPG$ and residual-free bubbles($RFB$); $Kirk$ and $Carey$ $\\cite{RefF}$ on development and validation of $SUPG$ for compressible $NS$ equations etc. $Codina$ in $\\cite{RefO} $ has experimentally established that for solving $ADR$ equation $SGS$ method performs better than other stabilized methods, such as $SUPG$, $GLS$, $Taylor$-$Galerkin$ etc. and in fact it is the most general method amongst them. \\vspace{1mm}\\\\\n Generally two approaches of $SGS$ stabilized formulation, namely algebraic approach, abbreviated as $ASGS$ and orthogonal projection approach, known as $OSGS$ method,  have been studied. Though few studies $\\cite{RefA1}$- $\\cite{RefA3}$ are there applying only general form of $SGS$ method instead of working with one specific approach, but authors in $\\cite{RefB1}$-$\\cite{RefB4}$ have employed $ASGS$ method, whereas authors of $\\cite{RefC1}$- $\\cite{RefC5}$ have worked with $OSGS$ method. Again $Badia$ and $Codina$ in $\\cite{RefV}$ have studied both the approaches for unified Stokes-Darcy fluid flow problem and experimentally established equally well performances of both the stabilized formulations. In this paper we have considered algebraic approach of approximating subscales which implies the stabilization parameters are of algebraic forms. This stabilization method begins with division of weak solution space into the spaces of the known finite element space and an unknown subgrid or unresolvable scale space and finally the stabilized formulation has been reached through expressing the element of subgrid scales in terms of the element of resolvable finite element space. For time discretization fully implicit schemes have been chosen. A detailed derivation of $apriori$ error estimation has been carried out for this stabilized variational form of the coupled system. Further more residual based $aposteriori$ error estimate too is derived elaborately. First order convergence in space has been established with respect to complete norms on all the variables. This paper also establishes the accuracy of the stabilized method through various numerical results, which include all possible combinations of cases containing small and large Reynolds numbers as well as cases involving concentration dependent viscosity. In every numerical example $ASGS$ performs consistently well in compared to standard Galerkin finite element method.\\vspace{1mm}\\\\\nThis paper is organised as follows: Section 2 introduces the coupled system along with important assumptions. In the next section we have introduced weak formulation, stabilized formulation and finally fully-discrete formulation after applying time-discretization rule. This section  also contains stability analysis of the fully-discrete stabilized form. Section 4 has elaborately described the derivations of $apriori$ and $aposteriori$ error estimations for this stabilized formulation. Before concluding the article section 5 presents numerical results to verify accuracy of the method.\n\n\\section{Model problem}\nIn this section we introduce the  flow problem described through transient $Navier$-$Stokes$ equations coupled with unsteady transport model over an open bounded domain $\\Omega \\subset$ $\\mathbb{R}^d$, d=2,3 with piece-wise smooth boundary $\\partial\\Omega$. Let us first present the $Navier$-$Stokes$ fluid flow model in the following: \nFind velocity $\\textbf{u}$: $\\Omega$ $\\times$ (0,T) $\\rightarrow \\mathbb{R}^d$ and pressure $p$: $\\Omega \\times$ (0,T) $\\rightarrow \\mathbb{R}$ of the fluid such that,\n\\begin{equation}\n\\begin{split}\n\\rho \\frac{\\partial \\textbf{u}}{\\partial t}+\\rho ( \\textbf{u} \\cdot \\bigtriangledown ) \\textbf{u} - \\mu(c) \\Delta \\textbf{u} + \\bigtriangledown p & = \\textbf{f} \\hspace{2mm} in \\hspace{2mm} \\Omega \\times (0,T) \\\\\n\\bigtriangledown \\cdot \\textbf{u} &= \\textbf{0} \\hspace{2mm} in \\hspace{2mm} \\Omega \\times (0,T) \\\\\n\\textbf{u} &= \\textbf{0} \\hspace{2mm} on \\hspace{2mm} \\partial\\Omega \\times (0,T) \\\\\n\\textbf{u} &= \\textbf{u}_0 \\hspace{2mm} at \\hspace{2mm} t=0 \\\\\n\\end{split}\n\\end{equation} \nwhere $\\rho$ is the density of the fluid, $\\mu(c)$ is the dynamic viscosity of the fluid depending on concentration $c$ of the dispersing mass of the solute,  $\\textbf{f}$ is body force and $\\textbf{u}_0$ is the initial velocity. \\vspace{2 mm}\\\\\nThis above flow problem is fully-coupled with the following transient advection-diffusion-reaction equation with variable coefficients($VADR$), representing the transportation of solute in $\\Omega$.\\\\\nFind the concentration $c$: $\\Omega \\times$ (0,T) $\\rightarrow \\mathbb{R}$ of the solute such that,\n\\begin{equation}\n\\begin{split}\n \\frac{\\partial c}{\\partial t}- \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} c + \\textbf{u} \\cdot \\bigtriangledown c + \\alpha c & = g \\hspace{2mm} in \\hspace{2mm} \\Omega \\times (0,T) \\\\\nc &= 0 \\hspace{2mm} on \\hspace{2mm}\\partial \\Omega \\times (0,T) \\\\\nc & = c_0 \\hspace{2mm} at \\hspace{2mm} t=0\\\\\n\\end{split}\n\\end{equation}\nwhere the notation, $\\tilde{\\bigtriangledown}: = \\sum_{i=1}^d D_i \\frac{\\partial}{\\partial x_i} e_i $ for $d=2,3$ and $\\{e_i \\}_{i=1}^d$ is standard basis of $\\mathbb{R}^d$. $D_i$ are variable diffusion coefficients, $\\alpha$ is the reaction coefficient and $g$ denotes the source of solute mass and $c_0$ is the initial concentration of the solute.  \\vspace{1.0 mm}\\\\ \nLet us consider a notation \\textbf{U}= (\\textbf{u},p,c) and the system of equations can be written in the following operator form,\n\\begin{equation}\nM\\partial_t \\textbf{U} + \\mathcal{L}(\\textbf{u}; \\textbf{U}) = \\textbf{F}\n\\end{equation}\nwhere M, a matrix = diag($\\rho$,$\\rho$,0,1), $\\partial_t \\textbf{U} = (\\frac{\\partial \\textbf{u}}{\\partial t}, \\frac{\\partial p}{\\partial t}, \\frac{\\partial c}{\\partial t})^T$ \\\\\n\\[\n\\mathcal{L} (\\textbf{u}; \\textbf{U})=\n  \\begin{bmatrix}\n  \\rho ( \\textbf{u} \\cdot \\bigtriangledown ) \\textbf{u}  - \\mu(c) \\Delta \\textbf{u} + \\bigtriangledown p\\\\\n    \\bigtriangledown \\cdot \\textbf{u} \\\\\n    - \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} c + \\textbf{u} \\cdot \\bigtriangledown c + \\alpha c \n  \\end{bmatrix}\n\\]\n and \\[\n\\textbf{F}=\n  \\begin{bmatrix}\n    \\textbf{f} \\\\\n    0 \\\\\n    g\n  \\end{bmatrix}\n\\]\nLet us introduce the adjoint $\\mathcal{L}^*$ of $\\mathcal{L}$ as follows,\n\\[\n\\mathcal{L}^* (\\textbf{u}; \\textbf{U})=\n  \\begin{bmatrix}\n  -\\rho ( \\textbf{u} \\cdot \\bigtriangledown ) \\textbf{u} - \\mu(c) \\Delta \\textbf{u}  - \\bigtriangledown p\\\\\n    -\\bigtriangledown \\cdot \\textbf{u} \\\\\n    - \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} c - \\textbf{u} \\cdot \\bigtriangledown c + \\alpha c \n  \\end{bmatrix}\n\\]\nNow we assume suitable conditions on the coefficients mentioned above, which will be useful to conclude the results further. \\vspace{1mm} \\\\\n\\textbf{(i)} The fluid viscosity $\\mu(c)= \\mu \\in C^0(\\mathbb{R}^+; \\mathbb{R}^+)$, the space of positive real valued functions defined on positive real numbers and we will have two positive real numbers $\\mu_l$ and $\\mu_u$ such that \n\\begin{equation}\n0 < \\mu_l \\leq \\mu(x) \\leq \\mu_u \\hspace{2mm} for \\hspace{2mm} any \\hspace{2mm} x\\in \\mathbb{R}^+\n\\end{equation}\n\\textbf{(ii)} $D_i= D_i(\\textbf{x},t) \\in C^0(\\mathbb{R}^d \\times (0,T);\\mathbb{R})$ (for $i=1,...,d$) where $ C^0(\\mathbb{R}^d \\times (0,T);\\mathbb{R})$ is the space of real valued continuous function defined on $\\mathbb{R}^d$ for fixed $t \\in (0,T)$. Both are bounded quantity that is we can find lower and upper bounds for both of them.\\vspace{1mm} \\\\\n\\textbf{(iii)} $\\rho$ and $\\alpha$ are positive constants. \\vspace{1mm}\\\\\n\\textbf{(iv)} The spaces of continuous solution $(\\textbf{u},p,c)$ are assumed as: \\\\ $\\textbf{u} \\in L^{\\infty}(0,T;(H^2(\\Omega))^d)\\bigcap C^{0}(0,T; H_0^1(\\Omega))$ and \\\\\n$p \\in L^{\\infty}(0,T;H^1(\\Omega))\\bigcap C^{0}(0,T;L^2_0(\\Omega)) $,\n $c \\in L^{\\infty}(0,T;H^2(\\Omega))\\bigcap C^0(0,T;H^1_0(\\Omega))$ \\vspace{0.1mm}\\\\\n\\textbf{(v)} Additional assumptions on continuous velocity and concentration solutions are: $\\textbf{u}_{tt}, \\textbf{u}_{ttt}, c_{tt}, c_{ttt}$ all are taken to be bounded functions on $\\Omega$ for each $t \\in (0,T)$. \\vspace{1mm}\\\\\n\\textbf{Weak formulation}: Assuming the body force $\\textbf{f} \\in L^2(0,T; (L^2(\\Omega))^d)$ and the source term $g \\in L^2(0,T; L^2(\\Omega))$ let us consider the spaces suitable to define the weak form as $V=H^1_0(\\Omega)$ and $Q=L^2(\\Omega)$ and J= (0,T).\\vspace{1mm}\\\\\nNow denoting the space $V^d \\times Q \\times V$ (for $d=2,3$) by $\\textbf{V}_F$ the weak formulation of (3) is to find \\textbf{U}= (\\textbf{u},p,c): J $ \\rightarrow \\textbf{V}_F$ such that $\\forall$ \\textbf{V}=(\\textbf{v},q,d) $\\in  \\textbf{V}_F$\n\\begin{equation}\n\\begin{split}\n(M\\partial_t \\textbf{U},\\textbf{V}) + B(\\textbf{u}; \\textbf{U}, \\textbf{V}) = L(\\textbf{V})  \n\\end{split}\n\\end{equation} \nwhere \n$B(\\textbf{u}; \\textbf{U}, \\textbf{V}) = c(\\textbf{u},\\textbf{u},\\textbf{v})+ a_{NS}(\\textbf{u},\\textbf{v})- b(\\textbf{v},p)+ b(\\textbf{u},q)+ a_T(c,d)$ \\vspace{1mm}\\\\\n$(M \\partial_t \\textbf{U}, \\textbf{V})= \\rho \\int_{\\Omega} \\frac{\\partial \\textbf{u}}{\\partial t} \\cdot \\textbf{v}+\\int_{\\Omega}\\frac{\\partial c}{\\partial t} d$ and $L(\\textbf{V})= l_{NS}(\\textbf{v})+ l_T(d) $ \\vspace{1mm}\\\\\nwhere the notations are defined in the following:\\\\\n $c(\\textbf{u},\\textbf{v},\\textbf{w})=\\rho \\int_{\\Omega} ((\\textbf{u} \\cdot \\bigtriangledown)\\textbf{v})\\cdot \\textbf{w}+ \\frac{\\rho}{2} \\int_{\\Omega} (\\bigtriangledown \\cdot \\textbf{u})\\textbf{v} \\cdot \\textbf{w} $\\vspace{1 mm}\\\\\n$a_{NS}(\\textbf{u},\\textbf{v})= \\int_{\\Omega} \\mu(c) \\bigtriangledown \\textbf{u}:\\bigtriangledown \\textbf{v}$ \\vspace{1 mm}\\\\\n $b(\\textbf{v},q)= \\int_{\\Omega} (\\bigtriangledown \\cdot \\textbf{v}) q$ \\vspace{1 mm} \\\\\n $a_T(c,d) = \\int_{\\Omega} \\tilde{\\bigtriangledown}c \\cdot \\bigtriangledown d + \\int_{\\Omega} d \\textbf{u} \\cdot \\bigtriangledown c + \\alpha\\int_{\\Omega}cd $ \\vspace{1 mm} \\\\\n $l_{NS} (\\textbf{v})= \\int_{\\Omega} \\textbf{f} \\cdot \\textbf{v}$ and  $l_T(d)= \\int_{\\Omega} gd$ \\vspace{1 mm} \\\\\nThe modified trilinear form $c(\\cdot,\\cdot,\\cdot)$ considered here is equivalent to the original trilinear form obtained from the non-linear convective term in (1). By the virtue of this modified form the trilinear term introduces the following important property. \\vspace{1mm}\\\\\n\\textbf{(a)} For any $\\textbf{u} \\in V^d$, $c(\\textbf{u},\\textbf{v},\\textbf{v})= 0$  $\\forall \\textbf{v} \\in V^d$ \\vspace{1mm}\\\\\nBesides the trilinear form $c(\\cdot,\\cdot,\\cdot)$ has the another property \\cite{RefR} too. \\vspace{1mm}\\\\\n\\textbf{(b)} For any $\\textbf{u}, \\textbf{v}, \\textbf{w}$ $\\in V^d$\n \\begin{equation}\n  c(\\textbf{u},\\textbf{v},\\textbf{w})\\leq \\begin{cases}\n    C \\|\\textbf{u}\\|_1 \\|\\textbf{v}\\|_1 \\|\\textbf{w}\\|_1 & \\\\\n    C \\|\\textbf{u}\\|_0 \\|\\textbf{v}\\|_2 \\|\\textbf{w}\\|_1 & \\\\\n    C \\|\\textbf{u}\\|_2 \\|\\textbf{v}\\|_1 \\|\\textbf{w}\\|_0 & \\\\\n    C \\|\\textbf{u}\\|_0 \\|\\textbf{v}\\|_1 \\|\\textbf{w}\\|_{L^{\\infty}(\\Omega)} \n  \\end{cases}\n\\end{equation}\nwhere $C$ is a constant and $\\| \\cdot \\|_j$ for $ j$=0,1,2 denote standard $L^2, H^1, H^2$ full norms respectively.\nIn the other forms the bilinear forms $a_{NS}(\\cdot, \\cdot)$ is $coercive$ $\\cite{RefR}$ and $a_T(\\cdot,\\cdot)$ is also $continuous$ and $coercive$ $\\cite{RefS}$. Again $b(\\cdot,\\cdot)$ satisfies $inf$-$sup$ condition $\\cite{RefR}$ too.\\vspace{1mm}\\\\\n\n\\section{Discrete formulation}\n\\subsection{Semi-discrete formulation}\nIn this section we introduce the finite element space discretization for the variational formulation (5) followed by a stabilized finite element formulation for the same.\\vspace{1 mm} \\\\\nLet the domain $\\Omega$ be discretized into finite numbers of subdomains $\\Omega_k$ for k=1,2,...,$n_{el}$, where $n_{el}$ is the total number element subdomains. Let $h_k$ be the diameter of each subdomain $\\Omega_k$ and h= $\\underset{k=1,2,...n_{el}}{max} h_k$ \\vspace{1 mm}\\\\\nLet $\\tilde{\\Omega}= \\bigcup_{k=1}^{n_{el}} \\Omega_k$ be the union of interior elements.\\vspace{1 mm}\\\\\nLet $V^h= \\{ v \\in V: v(\\Omega_k)= \\mathcal{P}^k(\\Omega_k)\\} $ and \n$Q^h= \\{ q \\in Q : q(\\Omega_k)= \\mathcal{P}^k(\\Omega_k)\\}$ \\vspace{1 mm}\\\\\nwhere $V^h$ and $Q^h$ be finite dimensional subspaces of $V$ and $Q$ respectively and $\\mathcal{P}^k(\\Omega_k)$  denotes complete polynomial of order $k$ over each $\\Omega_k$ for k=1,2,...,$n_{el}$. For regular partitions the functions belonging to finite dimensional spaces satisfy the following inverse inequalities:\\\\\n\\begin{center}\n$\\|\\Delta v_h\\| \\leq C_I h^{-1} \\|\\bigtriangledown v_h\\|_{0,k}$ \\hspace{2mm} and \\hspace{2mm} $\\|\\bigtriangledown v_h\\|_{0,k} \\leq C_I h^{-1} \\|v_h\\|_{0,k}$\n\\end{center}\nConsidering similar notation $\\textbf{V}_F^h$, denoting $\\textbf{V}_F^h=(V^h)^d \\times Q^h \\times V^h $ the \\textbf{finite element formulation}  of the variational form (5) in the finite dimensional space $\\textbf{V}_F^h$ is to\nfind $\\textbf{U}_h $= $(\\textbf{u}_h,p_h,c_h)$: J $ \\rightarrow \\textbf{V}_F^h$ such that $\\forall$ $\\textbf{V}_h=(\\textbf{v}_h,q_h,d_h)$ $\\in \\textbf{V}_F^h$\n\\begin{equation}\n(M\\partial_t \\textbf{U}_h,\\textbf{V}_h) + B(\\textbf{u}_h; \\textbf{U}_h, \\textbf{V}_h) = L(\\textbf{V}_h)   \n\\end{equation}\nwhere $(M\\partial_t \\textbf{U}_h,\\textbf{V}_h)$= $ \\rho (\\frac{\\partial \\textbf{u}_h}{\\partial t}, \\textbf{v}_{h})+ (\\frac{\\partial c_h}{\\partial t}, d_h)$ \\\\\n $B(\\textbf{u}_h; \\textbf{U}_h, \\textbf{V}_h)$ = $ c(\\textbf{u}_h,\\textbf{u}_h,\\textbf{v}_h)+ a_{NS}(\\textbf{u}_h,\\textbf{v}_h)- b(\\textbf{v}_h,p_h)+ b(\\textbf{u}_h,q_h)+ a_T(c_h,d_h)$ \\\\\n and $L(\\textbf{V}_h)= l_{NS}(\\textbf{v}_h)+ l_T(d_h) $ \\vspace{2mm}\\\\\n In addition let us consider the initial conditions $(\\textbf{u}_h, \\textbf{v}_h)\\mid_{t=0}=(\\textbf{u}_0, \\textbf{v}_h)$ $\\forall \\textbf{v}_h \\in (V^h)^d $ and $(c_h,d_h)\\mid_{t=0}= (c_0,d_h)$ $\\forall d_h \\in V^h$. \\vspace{1mm}\\\\\nNow we are going to introduce $subgrid$ $multiscale$ stabilized finite element method with algebraic approximation of the subscales of (5).\nIt involves decomposition of the weak solution space $\\textbf{V}_F$ into the spaces of resolvable scales and unresolvable or subgrid scales. The finite element space $\\textbf{V}_F^h$ is chosen to be the space of resolvable scales and in literature one of the ways of choosing the space of subgrid scales is the space that completes $\\textbf{V}_F^h$ in $\\textbf{V}_F$. Then the final form of subgrid formulation will be arrived while the elements of subgrid scales will be expressed in the terms of elements of resolvable scales. \\vspace{1 mm} \\\\\nThe \\textbf{stabilized algebraic subgrid multiscale ($ASGS$) formulation} for this coupled equation to \nfind $\\textbf{U}_h $= $(\\textbf{u}_h,p_h,c_h)$: J $ \\rightarrow \\textbf{V}_F^h$ such that $\\forall$ $\\textbf{V}_h=(\\textbf{v}_h,q_h,d_h)$ $\\in \\textbf{V}_F^h$ \n\\begin{equation}\n(M\\partial_t \\textbf{U}_h,\\textbf{V}_h) + B_{ASGS}(\\textbf{u}_h ; \\textbf{U}_h, \\textbf{V}_h)  = L_{ASGS}(\\textbf{V}_h)  \n\\end{equation}\nwhere $B_{ASGS}(\\textbf{u}_h ;\\textbf{U}_h, \\textbf{V}_h)= B(\\textbf{u}_h; \\textbf{U}_h, \\textbf{V}_h)+ \\sum_{k=1}^{n_{el}} (\\tau_k'(M\\partial_t \\textbf{U}_h + \\mathcal{L}(\\textbf{u}_h ;\\textbf{U}_h)-\\textbf{d}), -\\mathcal{L}^*(\\textbf{u}_h;\\textbf{V}_h))_{\\Omega_k}- \\sum_{k=1}^{n_{el}}((I-\\tau_k^{-1}\\tau_k')(M\\partial_t \\textbf{U}_h + \\mathcal{L}(\\textbf{u}_h;\\textbf{U}_h)), \\textbf{V}_h)_{\\Omega_k} \\\\\n-\\sum_{k=1}^{n_{el}} (\\tau_k^{-1}\\tau_k' \\textbf{d}, \\textbf{V}_h)_{\\Omega_k}$ \\vspace{2 mm}\\\\\n$L_{ASGS}(\\textbf{V}_h)= L(\\textbf{V}_h)+ \\sum_{k=1}^{n_{el}}(\\tau_k' \\textbf{F}, -\\mathcal{L}^*(\\textbf{u}_h;\\textbf{V}_h))_{\\Omega_k}- \\sum_{k=1}^{n_{el}}((I-\\tau_k^{-1}\\tau_k')\\textbf{F}, \\textbf{V}_h)_{\\Omega_k}$  \\vspace{1 mm} \\\\\nwhere the stabilization parameter $\\tau_k$ is in matrix form as \n\\[\n\\tau_k= diag(\\tau_{1k},\\tau_{1k},\\tau_{2k},\\tau_{3k}) =\n  \\begin{bmatrix}\n    \\tau_{1k} I_{d \\times d} & 0 & 0 \\\\\n    0 & \\tau_{2k} & 0 \\\\\n    0 & 0 & \\tau_{3k}\n  \\end{bmatrix}\n\\]\nand \n\\[\n\\tau_k'= (\\frac{1}{dt}M+ \\tau_k^{-1})^{-1} =\n  \\begin{bmatrix}\n    \\frac{\\tau_{1k} dt}{dt+ \\rho \\tau_{1k}}I_{d \\times d} & 0 & 0 \\\\\n    0 & \\tau_{2k} & 0 \\\\\n    0 & 0 & \\frac{\\tau_{3k} dt}{dt+ \\tau_{3k}}\n  \\end{bmatrix}\\\\\n  = diag (\\tau_{1k}',\\tau_{1k}',\\tau_{2k}',\\tau_{3k}') \n\\]\n$I_{d \\times d}$ is an identity matrix for $d=2,3$.\\vspace{1 mm}\\\\\n$\\textbf{d}$= $\\sum_{i=1}^{n+1}(\\frac{1}{dt}M\\tau_k')^i(\\textbf{F} -M\\partial_t \\textbf{U}_h - \\mathcal{L}(\\textbf{u}_h ;\\textbf{U}_h))$ =$[\\textbf{d}_1,d_2,d_3]^T$ \\vspace{1 mm}\\\\\nIt can be easily observed that $d_2$ is always 0 due to the matrix M. \\vspace{2 mm}\\\\\nWe have the forms of the stabilization parameters $\\tau_{1k}, \\tau_{2k}$ for $Navier$-$Stokes$ equation in $\\cite{RefU}$ and $\\tau_{3k}$ for $VADR$ equation $\\cite{RefT}$ and for each k=1,2,...,$n_{el}$ all the coefficients $\\tau_{ik}$ coincide with $\\tau_{i}$ for i=1,2,3 and choosing the parameters $c_1,c_2,c_3$ suitably that $\\tau_{i}$'s are as follows:\n\n\\begin{equation}\n\\begin{split}\n\\tau_{1k} &= \\tau_{1}= (c_1 \\frac{\\mu_{\\textbf{u}}}{h^2}+  c_2 \\frac{\\rho \\| \\textbf{u}_h \\|}{h})^{-1} \\\\\n\\tau_{2k} &=\\tau_{2}=\\frac{h^2}{c_1 \\tau_{1}} \\\\\n\\tau_{3k} & = \\tau_{3}= c_3(\\frac{9D}{4h^2} + \\frac{3 \\| \\textbf{u}_h \\|}{2h} + \\alpha )^{-1}\n\\end{split}\n\\end{equation}\nwhere $ \\textbf{u}_h $ is the computed velocity.\n\\begin{remark}\nConsidering continuity of the solutions at the inter-element boundaries, we have not encountered with any jump term in the above stabilized formulation.\n\\end{remark}\n\n\\subsection{Fully-discrete formulation}\nBefore introducing time discretization, some notations  have been introduced: for $dt$= $\\frac{T}{N}$, where $N$ is a positive integer, $t_n= n dt$ and for given $0 \\leq \\theta \\leq 1$,\n\\begin{equation}\n\\begin{split}\nf^n & = f(\\cdot , t_n) \\hspace{4 mm} for \\hspace{2 mm} 0 \\leq n \\leq N\\\\\nf^{n,\\theta} &= \\frac{1}{2} (1 + \\theta) f^{(n+1)} + \\frac{1}{2} (1- \\theta) f^n \\hspace{4mm} for \\hspace{2mm} 0\\leq n \\leq N-1\n\\end{split}\n\\end{equation}\nLater we will see for $\\theta=0$ the discretization follows Crank-Nicolson formula and for $\\theta=1$ it is backward Euler discretization rule.\\vspace{1mm}\\\\\nFor sufficiently smooth function $f(t)$, using the Taylor series expansion about t= $t^{n,\\theta}$, we will have \\vspace{1mm}\\\\\n\\begin{equation}\n\\begin{split}\nf^{n+1} & = f(t^{n,\\theta})+ \\frac{(1-\\theta)  dt}{2} \\frac{\\partial f}{\\partial t}(t^{n,\\theta}) + \\frac{(1-\\theta)^2 dt^2}{8} \\frac{\\partial^2 f}{\\partial t^2} (t^{n,\\theta}) + \\mathcal{O}(dt^3)\\\\\nf^{n} & = f(t^{n,\\theta})- \\frac{(1+\\theta) dt}{2} \\frac{\\partial f}{\\partial t}(t^{n,\\theta}) + \\frac{(1+\\theta)^2 dt^2}{8} \\frac{\\partial^2 f}{\\partial t^2}(t^{n,\\theta}) + \\mathcal{O}(dt^3)\n\\end{split}\n\\end{equation}\nWe have considered here $t^{n,\\theta}- t^n= \\frac{(1+\\theta) \\Delta t}{2}$\\\\\nMultiplying the above first and second sub-equations in (14) by $\\frac{1+\\theta}{2}$ and $\\frac{1-\\theta}{2}$ respectively and then adding them we will have the following\\\\\n\\begin{equation}\nf^{n,\\theta} = f(t^{n,\\theta}) + \\frac{1}{8} (1+\\theta)(1-\\theta) dt^2 \\frac{\\partial^2 f}{\\partial t^2}(t^{n,\\theta}) + \\mathcal{O}(dt^3)\n\\end{equation} \nLet $\\textbf{u}^{n,\\theta},p^{n,\\theta},c^{n,\\theta}$ be approximations of $\\textbf{u}(\\textbf{x},t^{n,\\theta}), p(\\textbf{x},t^{n,\\theta}),c(\\textbf{x},t^{n,\\theta})$ respectively. Now by Taylor series expansion \\cite{RefQ},we have \n\\begin{equation}\n\\begin{split}\n\\frac{\\textbf{u}^{n+1}-\\textbf{u}^n}{dt} & = \\frac{\\partial \\textbf{u}}{\\partial t}(\\textbf{x},t^{n,\\theta}) + \\textbf{TE}_1\\mid_{t=t^{n,\\theta}} \\hspace{4mm} \\forall \\textbf{x} \\in \\Omega  \\\\\n\\frac{c^{n+1}-c^n}{dt} & = \\frac{\\partial c}{\\partial t}(\\textbf{x},t^{n,\\theta}) + TE_2\\mid_{t=t^{n,\\theta}} \\hspace{5mm} \\forall \\textbf{x} \\in \\Omega\n\\end{split}\n\\end{equation}\nwhere the truncation error $TE\\mid_{t=t^{n,\\theta}}$ $\\simeq$ $TE^{n,\\theta}$ depends upon time-derivatives of the respective variables and $dt$.\n\\begin{equation}\n\\begin{split}\n\\|\\textbf{TE}_1^{n,\\theta}\\| & \\leq\n      \\begin{cases}\n      C' dt \\|\\textbf{u}_{tt}^{n,\\theta}\\|_{L^{\\infty}(t^n,t^{n+1},L^2)} & if \\hspace{1mm} \\theta=1 \\\\\n    C'' dt^2 \\|\\textbf{u}_{ttt}^{n,\\theta}\\|_{L^{\\infty}(t^n,t^{n+1},L^2)} & if \\hspace{1mm} \\theta=0\n      \\end{cases}\n\\end{split}\n\\end{equation}\nThe above relation holds for $TE_2$ in similar manner.\nNow applying assumption $\\textbf{(v)}$ on $\\textbf{u}_{tt}$ and $\\textbf{u}_{ttt}$ we will have another property as follows:\n\\begin{equation}\n\\begin{split}\n\\|\\textbf{TE}_1^{n,\\theta}\\| & \\leq\\begin{cases}\n      C' dt  & if \\hspace{1mm} \\theta=1 \\\\\n    C'' dt^2 & if \\hspace{1mm} \\theta=0\n      \\end{cases}\n\\end{split}\n\\end{equation}\nSimilarly\n\\begin{equation}\n\\begin{split}\n\\| TE_2^{n,\\theta}\\| & \\leq\\begin{cases}\n      C' dt  & if \\hspace{1mm} \\theta=1 \\\\\n    C'' dt^2 & if \\hspace{1mm} \\theta=0\n      \\end{cases}\n\\end{split}\n\\end{equation}\nAfter introducing all the required definitions finally the fully-discrete formulation of $subgrid$ form is as follows: For given $\\textbf{U}_h^n = (\\textbf{u}_h^n,p_h^n,c_h^n)\\in \\textbf{V}_F^h$ find $\\textbf{U}_h^{n+1}= (\\textbf{u}_h^{n+1},p_h^{n+1},c_h^{n+1}) \\in \\textbf{V}_F^h $  such that , $\\forall \\hspace{1mm} \\textbf{V}_h=(\\textbf{v}_h,q_h,d_h) \\in \\textbf{V}_F^h $\n\\begin{equation}\n(M\\frac{(\\textbf{U}_h^{n+1}-\\textbf{U}_h^n)}{dt}, \\textbf{V}_h)+ B_{ASGS}(\\textbf{u}_h^{n} ;\\textbf{U}_h^{n,\\theta}, \\textbf{V}_h) = L_{ASGS}(\\textbf{V}_h) + (\\textbf{TE}^{n,\\theta},\\textbf{V}_h)  \n\\end{equation}\nAgain for the exact solution we will have the discrete formulation as follows: \\\\\nFor given $\\textbf{U}^n = (\\textbf{u}^n,p^n,c^n)\\in \\textbf{V}_F$ find $\\textbf{U}^{n+1}= (\\textbf{u}^{n+1},p^{n+1},c^{n+1}) \\in \\textbf{V}_F $  such that , $\\forall \\hspace{1mm} \\textbf{V}_h=(\\textbf{v}_h,q_h,d_h) \\in \\textbf{V}_F^h$\n\\begin{equation}\n(M\\frac{(\\textbf{U}^{n+1}-\\textbf{U}^n)}{dt}, \\textbf{V}_h)+ B(\\textbf{u}^{n} ;\\textbf{U}^{n,\\theta}, \\textbf{V}_h) = L(\\textbf{V}_h) + (\\textbf{TE}^{n,\\theta},\\textbf{V}_h) \n\\end{equation}\n\n\n\n \n\\section{Error estimates}\nWe start this section with introducing the projection operator corresponding to each unknown variable followed by notation of error and it's component wise splitting. Later we go to derive $ apriori$ and $aposteriori$ error estimates.\n\n\\subsection{Projection operators : Error splitting}\nLet us introduce the projection operator for each of these error components.\\vspace{1 mm}\\\\\n(i)For any $\\textbf{u} \\in (H^2(\\Omega))^d  $  we assume that there exists an interpolation $I^h_{\\textbf{u}}:  (H^2(\\Omega))^d \\longrightarrow  (V^h)^d $ satisfying $b(\\textbf{u}-I^h_{\\textbf{u}}\\textbf{u}, q_h)=0$ \\hspace{2mm} $\\forall q_h \\in Q^h$ \\vspace{2mm}\\\\\n(ii) Let $I^h_p: H^1(\\Omega) \\longrightarrow Q^h$ be the $L^2$ orthogonal projection given by \\\\ $\\int_{\\Omega}(p-I^h_pp)q_h=0$  \\hspace{1mm} $\\forall q_h \\in Q^h$ and for any $p \\in H^1(\\Omega)$ \\vspace{2mm}\\\\\n(iii)Similarly let $I^h_{c}: H^2(\\Omega) \\longrightarrow V^h$ be the $L^2$ orthogonal projection given by \\\\ $\\int_{\\Omega}(c-I^h_c c)d_h=0$ \\hspace{1mm} $ \\forall d_h \\in V^h$ and for any $c \\in H^2(\\Omega)$ \\vspace{2mm}\\\\\nLet $\\textbf{e}=(e_{\\textbf{u}},e_p,e_c)$ denote the error where the components are $e_{\\textbf{u}}=\\textbf{u}-\\textbf{u}_h, e_p= (p-p_h)$ and $e_c=(c-c_h)$. Now each component of the error can be split into two parts interpolation part, $E^I$ and auxiliary part, $E^A$ as follows: \\vspace{1mm}\\\\\n$e_{\\textbf{u}}=(\\textbf{u}-\\textbf{u}_{h})=(\\textbf{u}-I^h_{\\textbf{u}}\\textbf{u})+(I^h_{\\textbf{u}}\\textbf{u}-\\textbf{u}_{h})= E^{I}_{\\textbf{u}}+ E^{A}_{\\textbf{u}}$ \\vspace{1mm}\\\\\nSimilarly\n$e_{p}=E^{I}_{p}+ E^{A}_{p}$, and \n$e_{c}=E^{I}_{c}+ E^{A}_{c}$ \\vspace{2mm}\\\\\nAt this point let us mention the standard \\textbf{interpolation estimation} result \\cite{RefQ} in the following: for any exact solution with regularity upto (m+1)\n\\begin{equation}\n\\|v-I^h_v v\\|_l = \\|E^I_v\\|_l \\leq C(p,\\Omega) h^{m+1-l} \\|v\\|_{m+1} \n\\end{equation}\nwhere l ($\\leq m+1$) is a positive integer and C is a constant depending on m and the domain. For l=0 and 1 it  implies standard $L^2(\\Omega)$ and $H^1(\\Omega)$ norms respectively. For simplicity we will use $\\| \\cdot \\|$ instead of $\\| \\cdot \\|_0$ to denote $L^2(\\Omega)$ norm.\nNow we put some results using the properties of projection operators and these results will be used in error estimations.\n\\begin{result}\n\\begin{equation}\n(\\frac{\\partial}{\\partial t} E^{I,n}, v_h)=0 \\hspace{2mm} v_h \\in V^h\n\\end{equation}\n\\end{result}\n\n\\begin{result}\nFor any given auxiliary error $E^{A,n}$ and unknown $E^{A,n+1}$\n\\begin{equation}\n(\\frac{\\partial}{\\partial t} E^{A,n}, E^{A,n,\\theta}) \\geq  \\frac{1}{2 dt} (\\|E^{A,n+1}\\|^2- \\|E^{A,n}\\|^2)\n\\end{equation}\n\\end{result}\n\n\\begin{remark}\nThe proof of the results have been discussed in  \\cite{RefN} elaborately.\n\\end{remark}\n \n\\subsection{Apriori error estimate}\nIn this section we will find $apriori$ error bound, which depends on the exact solution. Here we first estimate $auxiliary$ error bound and later using that we will find $apriori$ error estimate. Before deriving error estimations let us define norms required for error estimations. Let us consider the space $\\tilde{\\textbf{V}} $ := $L^2(0,T; V_s)\\bigcap L^{\\infty}(0,T; Q_s)$ and it's associated norm is denoted by $\\tilde{\\textbf{V}}$-norm. For the functions $g_1, g_2, g_3$ belonging to the spaces $L^2(0,T; L^2(\\Omega))$, $L^2(0,T; H_0^1(\\Omega))$,  $\\tilde{\\textbf{V}}$ respectively norms over these spaces, abbreviated as $L^2(L^2)$, $L^2(H^1)$, $\\tilde{\\textbf{V}}$ are defined in the following\n\\begin{equation}\n\\begin{split}\n\\|g_1\\|_{L^2(L^2)}^2 &= \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\int_{\\Omega} \\mid g_1^{n,\\theta} \\mid^2 dt \\\\\n\\|g_2\\|_{L^2(H^1)}^2 &= \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} (\\int_{\\Omega} \\mid g_2^{n,\\theta} \\mid^2  +  \\int_{\\Omega} \\mid \\frac{\\partial g_2}{\\partial x}^{n,\\theta} \\mid^2  +  \\int_{\\Omega} \\mid \\frac{\\partial g_2}{\\partial y}^{n,\\theta} \\mid^2 )dt \\\\\n\\|g_3\\|_{\\tilde{\\textbf{V}}}^2 & = \\underset{0\\leq n \\leq N}{max} \\|g_3^n\\|^2 + \\|g_3\\|_{L^2(H^1)}^2\\\\\n\\end{split}\n\\end{equation}\n\\begin{theorem} \\textbf{(Auxiliary error estimate)} \nFor computed velocity $\\textbf{u}_h$, pressure $p_h$ and concentration $c_h$ belonging to $(V^h)^d \\times Q^h \\times V^h$ satisfying (31)-(32), assume $dt$ is sufficiently small and positive, and sufficient regularity of exact solution in equations (1)-(2). Then there exists a constant C, depending upon $\\textbf{u},p,c$ such that\n\\begin{equation}\n\\|E^A_{\\textbf{u}}\\|^2_{\\tilde{\\textbf{V}}} + \\|E^A_p\\|_{L^2(L^2)}^2  + \\|E^A_{c}\\|^2_{\\tilde{\\textbf{V}}} \\leq C (h^2+ dt^{2r})\n\\end{equation}\nwhere\n\\begin{equation}\n    r=\n    \\begin{cases}\n      1, & \\text{if}\\ \\theta=1 \\\\\n      2, & \\text{if}\\ \\theta=0\n    \\end{cases}\n  \\end{equation}\n\\end{theorem}\n\\begin{proof} In first part we will find bound for auxiliary error part of velocity $\\textbf{u}$ and concentration c with respect to ${\\tilde{\\textbf{V}}}$-norm and in the second part we will estimate auxiliary error for pressure term with respect to $Q$ norm and finally combining them we will arrive at the desired result. \\vspace{2mm} \\\\\n\\textbf{First part} Subtracting (17) from (18) and then simplifying the terms, we have $\\forall \\hspace{1mm} \\textbf{V}_h \\in (V^h)^d \\times Q^h \\times V^h$\n\n\\begin{multline}\n(M\\frac{(\\textbf{U}^{n+1}-\\textbf{U}^{n+1}_{h})- (\\textbf{U}^{n}-\\textbf{U}^{n}_{h})}{dt},\\textbf{V}_{h}) + B(\\textbf{u}^{n};\\textbf{U}^{n,\\theta}, \\textbf{V}_h)-B(\\textbf{u}_h^{n};\\textbf{U}^{n,\\theta}_h, \\textbf{V}_h)\\\\\n+ \\sum_{k=1}^{n_{el}}(\\tau_k'(M\\partial_t (\\textbf{U}^n-\\textbf{U}^n_h)+ \\mathcal{L}(\\textbf{u}^{n} ;\\textbf{U}^{n,\\theta})-\\mathcal{L}(\\textbf{u}^{n}_h ;\\textbf{U}^{n,\\theta}_h)),-\\mathcal{L}^* (\\textbf{u}_h;\\textbf{V}_h))_{\\Omega_k} \\\\\n+\\sum_{k=1}^{n_{el}}((I-\\tau_k^{-1}\\tau_k')(M \\partial_t(\\textbf{U}^{n}-\\textbf{U}^{n}_h) + \\mathcal{L}(\\textbf{u}^{n} ;\\textbf{U}^{n,\\theta})-\\mathcal{L}(\\textbf{u}^{n}_h ;\\textbf{U}^{n,\\theta}_h)), -\\textbf{V}_h)_{\\Omega_k}\\\\\n+ \\sum_{k=1}^{n_{el}} (\\tau_k^{-1}\\tau_k' \\textbf{d}, \\textbf{V}_h)_{\\Omega_k}+\\sum_{k=1}^{n_{el}}(\\tau_k' \\textbf{d},-\\mathcal{L}^*(\\textbf{u}_h; \\textbf{V}_h))_{\\Omega_k}=(\\textbf{TE}^{n,\\theta}, \\textbf{V}_h)\\\\\n\\end{multline}\nwhere $\\textbf{d}$= $(\\sum_{i=1}^{n+1}(\\frac{1}{dt}M\\tau_k')^i)(M\\partial_t (\\textbf{U}^n-\\textbf{U}^n_h) + \\mathcal{L}(\\textbf{u}^{n,\\theta} ;\\textbf{U}^{n,\\theta})-\\mathcal{L}(\\textbf{u}_h^{n,\\theta} ;\\textbf{U}_h^{n,\\theta}))$ \\vspace{2mm}\\\\\nNow after applying error splitting for each of the terms and later using the result obtained in (31) and properties of projection operators we have rearranged the above equation (36) as follows: $\\forall  \\textbf{V}_h \\in (V^h)^d \\times Q^h \\times V^h$\n\\begin{multline}\n\\rho (\\frac{E^{A,n+1}_{\\textbf{u}}-E^{A,n}_{\\textbf{u}}}{dt}, \\textbf{v}_h) + (\\frac{E^{A,n+1}_c- E^{A,n}_c}{dt}, d_h) + \\int_{\\Omega} \\mu(c^n) \\bigtriangledown E^{A,n,\\theta}_{\\textbf{u}} :  \\bigtriangledown \\textbf{v}_h +\\\\\n \\int_{\\Omega} \\mu(c^n) E^{A,n,\\theta}_{\\textbf{u}} \\cdot \\textbf{v}_h +\\int_{\\Omega} \\tilde{\\bigtriangledown} E^{A,n,\\theta}_c \\cdot \\bigtriangledown \\textbf{v}_h + \\alpha \\int_{\\Omega} E^{A,n,\\theta}_c d_h = \\int_{\\Omega} (\\bigtriangledown \\cdot \\textbf{v}_h) (E^{I,n,\\theta}_p + E^{A,n,\\theta}_p)\\\\\n -\\int_{\\Omega}  (\\bigtriangledown \\cdot E^{A,n,\\theta}_{\\textbf{u}}) q_h - \\int_{\\Omega} \\mu(c^n)  \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}} :  \\bigtriangledown \\textbf{v}_h - \\sigma \\int_{\\Omega} E^{I,n,\\theta}_{\\textbf{u}} \\cdot \\textbf{v}_h - \\int_{\\Omega} \\tilde{\\bigtriangledown} E^{I,n,\\theta}_c \\cdot \\bigtriangledown d_h- \\\\\n \\alpha \\int_{\\Omega} E^{I,n,\\theta}_c d_h - \\int_{\\Omega} d_h \\textbf{u}^{n} \\cdot \\bigtriangledown E^{I,n,\\theta}_c - \\int_{\\Omega} d_h \\textbf{u}^{n} \\cdot \\bigtriangledown E^{A,n,\\theta}_c - \\int_{\\Omega} d_h E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown c_h^{n,\\theta}- \\\\\n \\int_{\\Omega} d_h E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown c_h^{n,\\theta} -c(\\textbf{u}^{n},E^{I,n,\\theta}_{\\textbf{u}}, \\textbf{v}_h)- c(\\textbf{u}^{n},E^{A,n,\\theta}_{\\textbf{u}}, \\textbf{v}_h) - c(E^{I,n}_{\\textbf{u}}, \\textbf{u}^{n, \\theta}_h,\\textbf{v}_h) \\\\\n - c(E^{A,n}_{\\textbf{u}}, \\textbf{u}^{n, \\theta}_h,\\textbf{v}_h) + \\int_{\\Omega} \\mu(c^n) E^{A,n,\\theta}_{\\textbf{u}} \\cdot \\textbf{v}_h -I_1- I_2 -I_3 - I_4 - (\\textbf{TE}^{n,\\theta}, \\textbf{V}_h) \\hspace{5mm} \n\\end{multline}\nApplying various properties of the projection operators we have the final expression of $I_1$ above.\n\n\\begin{equation}\n\\begin{split}\nI_1 &=\\sum_{k=1}^{n_{el}}(\\tau_k'(M\\partial_t (\\textbf{U}^n-\\textbf{U}^n_h)+ \\mathcal{L}(\\textbf{u}^{n} ;\\textbf{U}^{n,\\theta})-\\mathcal{L}(\\textbf{u}^{n}_h ;\\textbf{U}^{n,\\theta}_h)),-\\mathcal{L}^* (\\textbf{u}_h;\\textbf{V}_h))_{\\Omega_k} \\\\\n& = \\sum_{k=1}^{n_{el}} [( \\tau_{1k}' I_{d \\times d} \\{\\rho \\partial_t E^{I,n}_{\\textbf{u}}+ \\rho E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown \\textbf{u}^{n,\\theta} + \\rho \\textbf{u}_h^{n} \\cdot \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}} - \\mu(c^n) \\Delta E^{I,n,\\theta}_{\\textbf{u}} \\\\\n& \\quad + \\bigtriangledown E^{I,n,\\theta}_p \\}, \\rho(\\textbf{u}_h \\cdot \\bigtriangledown) \\textbf{v}_h + \\mu(c) \\Delta \\textbf{u}_h + \\bigtriangledown p_h)_{\\Omega_k} + ( \\tau_{1k}' I_{d \\times d} \\{\\rho \\partial_t E^{A,n}_{\\textbf{u}}+ \\rho \\\\\n& \\quad E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown \\textbf{u}^{n,\\theta} + \\rho \\textbf{u}_h^{n} \\cdot \\bigtriangledown E^{A,n,\\theta}_{\\textbf{u}} - \\mu(c^n) \\Delta E^{A,n,\\theta}_{\\textbf{u}} + \\bigtriangledown E^{A,n,\\theta}_p \\}, \\rho(\\textbf{u}_h \\cdot \\bigtriangledown) \\textbf{v}_h \\\\\n& \\quad + \\mu(c) \\Delta \\textbf{u}_h + \\bigtriangledown p_h)_{\\Omega_k} + (\\tau_{2k}' \\bigtriangledown \\cdot (E^{I,n,\\theta}_{\\textbf{u}}+E_{\\textbf{u}}^{A,n,\\theta}), \\bigtriangledown \\cdot \\textbf{v}_h)_{\\Omega_k}  + (\\tau_{3k}' \\{ \\partial_t E^{I,n}_{c} \\\\ \n & \\quad - \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} E^{I,n,\\theta}_{c}+( E_{\\textbf{u}}^{I,n} \\cdot \\bigtriangledown )c^{n,\\theta} + (\\textbf{u}_h^{n} \\cdot \\bigtriangledown) E^{I,n,\\theta}_{c}+\\alpha E_{c}^{I,n,\\theta} \\}, \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} d_h \\\\\n& \\quad + \\textbf{u}_h \\cdot \\bigtriangledown d_h - \\alpha d_h )_{\\Omega_k} + (\\tau_3' \\{ \\partial_t E^{A,n}_{c} -\\bigtriangledown \\cdot \\tilde{\\bigtriangledown} E^{A,n,\\theta}_{c}+( E_{\\textbf{u}}^{A,n} \\cdot \\bigtriangledown )c^{n,\\theta}   \\\\\n& \\quad + (\\textbf{u}_h^{n}\\cdot \\bigtriangledown) E^{A,n,\\theta}_{c}+\\alpha E_{c}^{A,n,\\theta} \\},\\bigtriangledown \\cdot \\tilde{\\bigtriangledown} d_h + \\textbf{u}_h \\cdot \\bigtriangledown d_h - \\alpha d_h )_{\\Omega_k}]\\\\\n& = I_1^1 +I_1^2+I_1^3+I_1^4+ I_1^5 \\hspace{2mm} (say)\n\\end{split} \n\\end{equation} \nwhere $I_1^i$ for $i=1,2,...,5$ are five terms of $I_1$ which we will discuss in the later part of the proof and since $(1- \\tau_2^{-1}\\tau_2')=0$ the next term will take the following form \n\\begin{equation}\n\\begin{split}\nI_2 & = \\sum_{k=1}^{n_{el}}((I-\\tau_k^{-1}\\tau_k')(M \\partial_t(\\textbf{U}^{n}-\\textbf{U}^{n}_h) + \\mathcal{L}(\\textbf{u}^{n} ;\\textbf{U}^{n,\\theta})-\\mathcal{L}(\\textbf{u}^{n}_h ;\\textbf{U}^{n,\\theta}_h)), -\\textbf{V}_h)_{\\Omega_k} \\\\\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n& = \\sum_{k=1}^{n_{el}} [(\\frac{\\rho \\tau_{1k}}{dt+\\rho \\tau_{1k}}I_{d \\times d} \\{\\rho \\partial_t E^{I,n}_{\\textbf{u}}+ \\rho E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown \\textbf{u}^{n,\\theta} + \\rho \\textbf{u}_h^{n} \\cdot \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}} - \\mu(c^n) \\\\\n& \\quad  \\Delta E^{I,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{I,n,\\theta}_p \\}, - \\textbf{v}_h)_{\\Omega_k} + (\\frac{\\rho \\tau_{1k}}{dt+\\rho \\tau_{1k}}I_{d \\times d} \\{\\rho \\partial_t E^{A,n}_{\\textbf{u}} + \\rho E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown \\textbf{u}^{n,\\theta}\\\\\n& \\quad  + \\rho \\textbf{u}_h^{n} \\cdot \\bigtriangledown E^{A,n,\\theta}_{\\textbf{u}} - \\mu(c^n) \\Delta E^{A,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{A,n,\\theta}_p \\}, - \\textbf{v}_h)_{\\Omega_k} + (\\frac{\\tau_{3k}}{dt+\\tau_{3k}}\\{ \\partial_t E^{I,n}_{c} \\\\\n& \\quad - \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} E^{I,n,\\theta}_{c}+( E_{\\textbf{u}}^{I,n} \\cdot \\bigtriangledown )c^{n,\\theta} + (\\textbf{u}_h^{n} \\cdot \\bigtriangledown) E^{I,n,\\theta}_{c} +\\alpha E^{I,n,\\theta}_{c}\\}, -d_h)_{\\Omega_k} +\\\\\n& \\quad (\\frac{\\tau_{3k}}{dt+\\tau_{3k}}\\{ \\partial_t E^{A,n}_{c}- \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} E^{A,n,\\theta}_{c}+( E_{\\textbf{u}}^{A,n} \\cdot \\bigtriangledown )c^{n,\\theta} + (\\textbf{u}_h^{n} \\cdot \\bigtriangledown) E^{A,n,\\theta}_{c} + \\\\\n& \\quad \\alpha E^{A,n,\\theta}_{c}\\}, -d_h)_{\\Omega_k}] \\\\\n& = I_2^1 +I_2^2+I_2^3+ I_2^4 \\hspace{2mm} (say)\n\\end{split}\n\\end{equation} \nThe next terms of $LHS$ in (36) are as follows:\n\\begin{equation}\n\\begin{split}\nI_3 & =\\sum_{k=1}^{n_{el}} (\\tau_k^{-1}\\tau_k' \\textbf{d}, \\textbf{V}_h)_{\\Omega_k} \\\\\n&= \\sum_{k=1}^{n_{el}} (\\{\\frac{dt}{dt+ \\rho \\tau_{1k}} I_{d \\times d} \\textbf{d}, \\textbf{v}_h)_{\\Omega_k} + (\\frac{dt}{dt+\\tau_{3k}} d_4, d_{h})_{\\Omega_k} \\} \\hspace{30mm} \\\\\n\\end{split}\n\\end{equation}\nand \n\\begin{equation}\n\\begin{split}\nI_4 & = \\sum_{k=1}^{n_{el}}(\\tau_k' \\textbf{d},-\\mathcal{L}^*(\\textbf{u}_h; \\textbf{V}_h))_{\\Omega_k} \\\\\n& =  \\sum_{k=1}^{n_{el}}\\{ (\\tau_{1k}' I_{d \\times d} \\textbf{d},\\rho (\\textbf{u}_h \\cdot \\bigtriangledown) \\textbf{v}_h +\\mu(c) \\Delta \\textbf{v}_h  + \\bigtriangledown q_h)_{\\Omega_k}+ (\\tau_{3k}' d_3, \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} d_h +\\\\\n& \\quad \\textbf{u}_h \\cdot \\bigtriangledown d_h - \\alpha d_h )_{\\Omega_k}\\} \\\\\n\\end{split}\n\\end{equation}\nNow we will treat each term separately to find out the estimate. Our procedure contains finding two bounds: one is lower bound of $LHS$ and the other one is upper bound for the terms in $RHS$ and combining those bounds in the equation (36) we will finally obtain the required estimate. Before further proceeding let us mention an important consideration: since the above equation holds for all $\\textbf{V}_h \\in V_s^h \\times V_s^h \\times Q_s^h \\times V_s^h$, therefore in each term we replace $v_{1h},v_{2h},q_h,d_h$ by $E^{A,n,\\theta}_{u1},E^{A,n,\\theta}_{u2},E^{A,n,\\theta}_{p},E^{A,n,\\theta}_{c}$ respectively as these auxiliary part of the errors belonging to their respective finite element spaces. From now onwards we will start derivation of each expression after considering the replacements directly.\\vspace{2mm}\\\\\nApplying the result obtained in (32) on the first term of $LHS$ and taking out the $infimum$ of the coefficients of the remaining terms we can easily see that\n\\begin{multline}\n\\frac{\\rho}{2 dt} (\\|E^{A,n+1}_{\\textbf{u}}\\|^2-\\|E^{A,n}_{\\textbf{u}}\\|^2)+ \\frac{1}{2 dt}(\\|E^{A,n+1}_{c}\\|^2-\\|E^{A,n}_{c}\\|^2)+ \\mu_l \\| E^{A,n,\\theta}_{\\textbf{u}}\\|_1^2  \\\\\n+D_l \\mid E^{A,n,\\theta}_c \\mid_1^2  +  \\alpha \\|E^{A,n,\\theta}_{c}\\|^2  \\leq LHS = RHS\\\\\n\\end{multline}\nwhere $D_l$= min $\\{ \\underset{\\Omega}{inf} D_1, \\underset{\\Omega}{inf} D_2  \\}$. \\\\\nNow we will find upper bounds of each of the terms in the $RHS$ of the equation (37). We usually use $Cauchy-Schwarz$ and $Young's$ inequality to reach at the desired bounds. We have already estimated the bounds of few terms on $RHS$ in $\\cite{RefN}$. Therefore we only mention the results here and the estimation of the remaining terms are shown later in details.\n\\begin{equation}\n\\begin{split}\n\\int_{\\Omega}(\\bigtriangledown \\cdot E^{A,n,\\theta}_\\textbf{u})E^{I,n,\\theta}_{p} \n& \\leq  \\epsilon_1 C^2 h^2 (\\frac{1+\\theta}{2}\\| p^{n+1}\\|_1+ \\frac{1-\\theta}{2}\\| p^n \\|_1)^2 +\\\\\n& \\quad \\frac{1}{2 \\epsilon_1}\\mid E^{A,n,\\theta}_{\\textbf{u}} \\mid_1^2\\\\\n-\\int_{\\Omega} \\mu(c^n)\\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}} : \\bigtriangledown E^{A,n,\\theta}_\\textbf{u} & \\leq \\epsilon_2 \\mu_u  C^2 h^2(\\frac{1+\\theta}{2}\\| \\textbf{u}^{n+1}\\|_2+ \\frac{1-\\theta}{2} \\| \\textbf{u}^n\\|_2)^2 +\\\\\n& \\quad  \\frac{\\mu_u}{2 \\epsilon_2} \\mid E^{A,n,\\theta}_{\\textbf{u}} \\mid_1^2 \\\\\n- \\int_{\\Omega} \\tilde{\\bigtriangledown}E^{I,n,\\theta}_{c} \\cdot \\bigtriangledown E^{A,n,\\theta}_c & \\leq    \\frac{D_m \\epsilon_3}{2} C^2 h^2 (\\frac{1+\\theta}{2} \\| c^{n+1} \\|_2+\\frac{1-\\theta}{2}\\| c^n \\|_2)^2 + \\\\\n& \\quad \\frac{D_m}{2 \\epsilon_3}(\\|\\frac{\\partial E^{A,n,\\theta}_{c}}{\\partial x}\\|^2 + \\|\\frac{\\partial E^{A,n,\\theta}_{c}}{\\partial y}\\|^2)\\\\\n- \\sigma \\int_{\\Omega} E^{I,n,\\theta}_{\\textbf{u}} \\cdot E^{A,n,\\theta}_{\\textbf{u}} & \\leq \\frac{\\epsilon_4}{2}\\sigma h^4 (\\frac{1+\\theta}{2} \\|\\textbf{u}^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|\\textbf{u}^{n}\\|_2)^2 + \\\\\n& \\quad \\frac{\\sigma}{ 2\\epsilon_4} \\|E^{A,n,\\theta}_{\\textbf{u}}\\|^2 \\\\\n-\\alpha \\int_{\\Omega} E^{I,n,\\theta}_c E^{A,n,\\theta}_c & \\leq \\frac{\\epsilon_5}{2} \\alpha h^4 (\\frac{1+\\theta}{2} \\|c^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|c^n\\|_2)^2 + \\\\\n& \\quad \\frac{\\alpha}{2 \\epsilon_5} \\|E^{A,n,\\theta}_c \\|^2\n\\end{split}\n\\end{equation}  \nwhere $D_m$= max $\\{ \\underset{\\Omega}{sup} D_1, \\underset{\\Omega}{sup} D_2 \\}$ \\\\\nNow applying $Poincare$  inequality in the following we have:\n\\begin{equation}\n\\begin{split}\n\\int_{\\Omega} \\mu(c^n)\\{(E^{A,n,\\theta}_{u1})^2+(E^{A,n,\\theta}_{u2})^2 \\} & \\leq \\mu_u (\\|E^{A,n,\\theta}_{u1}\\|^2+\\|E^{A,n,\\theta}_{u2}\\|^2)\\\\\n& \\leq \\mu_u C_P (\\mid E^{A,n,\\theta}_{u1}\\mid_1^2+\\mid E^{A,n,\\theta}_{u2}\\mid_1^2)\n\\end{split}\n\\end{equation}  \nwhere $C_P$ is the $Poincare$ constant.  The next term is estimated following $\\cite{RefN}$\n\\begin{equation}\n\\begin{split}\n-\\int_{\\Omega}E^{A,n,\\theta}_c \\textbf{u}^{n} \\cdot \\bigtriangledown E^{A,n,\\theta}_{c} &  \\leq 2 \\bar{C}_1^2 (\\| \\frac{\\partial E^{A,n,\\theta}_c}{\\partial x} \\|^2 + \\| \\frac{\\partial E^{A,n,\\theta}_c}{\\partial y} \\|^2) \\\\\n-\\int_{\\Omega}E^{A,n,\\theta}_c \\textbf{u}^{n} \\cdot \\bigtriangledown E^{I,n,\\theta}_{c} &  \\leq \\bar{C}_1^2 C^2 h^2 \\epsilon_6 (\\frac{1+\\theta}{2}\\| c^{n+1} \\|_2 + \\frac{1-\\theta}{2} \\| c^n \\|_2)^2+ \\\\\n& \\quad \\frac{\\bar{C}_1^2}{\\epsilon_6} \\| E^{A,n,\\theta}_c\\|^2 \n\\end{split}\n\\end{equation}\nNow we estimate the trilinear term $c(\\cdot,\\cdot,\\cdot)$ using it's properties \\textbf{(a)} and \\textbf{(b)} given in section 2. Let us start the estimation with the first trilinear term on $RHS$ of  (37) as follows:\n\\begin{equation}\n\\begin{split}\n-c(\\textbf{u}^{n},E^{I,n,\\theta}_{\\textbf{u}}, E^{A,n,\\theta}_{\\textbf{u}}) & \\leq C \\|\\textbf{u}^{n}\\|_2 \\|E^{I,n,\\theta}_{\\textbf{u}}\\|_1 \\|E^{A,n,\\theta}_{\\textbf{u}}\\| \\\\\n& \\leq  \\frac{C_2 \\epsilon_7}{2} \\|E^{I,n,\\theta}_{\\textbf{u}}\\|_1^2 + \\frac{C_2}{2 \\epsilon_7} \\|E^{A,n,\\theta}_{\\textbf{u}}\\|^2 \\\\\n& \\leq  \\frac{C_2 \\epsilon_7}{2} (\\frac{1+\\theta}{2}\\|E^{I,n+1}_{\\textbf{u}}\\|_1 + \\frac{1-\\theta}{2}\\|E^{I,n}_{\\textbf{u}}\\|_1)^2 + \\frac{C_2}{2 \\epsilon_7} \\|E^{A,n,\\theta}_{\\textbf{u}}\\|^2 \\\\\n& \\leq  \\frac{ \\epsilon_7}{2} C_2 h^2 (\\frac{1+\\theta}{2} \\|\\textbf{u}^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|\\textbf{u}^{n}\\|_2)^2 + \\frac{C_2}{2 \\epsilon_7} \\|E^{A,n,\\theta}_{\\textbf{u}}\\|^2\n\\end{split}\n\\end{equation}\nBy the property \\textbf{(a)} of trilinear case for both the linear and non-linear cases:\n\\begin{equation}\n-c(\\textbf{u}^{n},E^{A,n,\\theta}_{\\textbf{u}}, E^{A,n,\\theta}_{\\textbf{u}})=0 \n\\end{equation}\nThe next term,\n\\begin{equation}\n\\begin{split}\n-c(E^{I,n}_{\\textbf{u}},\\textbf{u}^{n,\\theta}_h, E^{A,n,\\theta}_{\\textbf{u}}) & = c(E^{I,n}_{\\textbf{u}},E^{I,n,\\theta}_{\\textbf{u}}, E^{A,n,\\theta}_{\\textbf{u}})+ c(E^{I,n}_{\\textbf{u}},E^{A,n,\\theta}_{\\textbf{u}}, E^{A,n,\\theta}_{\\textbf{u}}) -\\\\\n& \\quad c(E^{I,n}_{\\textbf{u}},\\textbf{u}^{n,\\theta}, E^{A,n,\\theta}_{\\textbf{u}}) \\\\\n& \\leq C \\|E^{I,n}_{\\textbf{u}}\\| \\|E^{I,n,\\theta}_{\\textbf{u}}\\|_1 \\|E^{A,n,\\theta}_{\\textbf{u}}\\|_1 +C \\|E^{I,n,\\theta}_{\\textbf{u}}\\| \\|\\textbf{u}^{n,\\theta}\\|_2 \\|E^{A,n,\\theta}_{\\textbf{u}}\\|_1 \\\\\n& \\leq h^4 \\frac{C_2 \\epsilon_8 }{2} \\|\\textbf{u}^{n}\\| (\\frac{1+\\theta}{2} \\|\\textbf{u}^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|\\textbf{u}^{n}\\|_2)^2 +\\frac{C_2}{ \\epsilon_8} \\|E^{A,n,\\theta}_{\\textbf{u}}\\|_1^2 \\\\\n& \\quad +h^4 \\frac{C_2 \\epsilon_8}{2 } (\\frac{1+\\theta}{2} \\|\\textbf{u}^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|\\textbf{u}^{n}\\|_2)^2 \n\\end{split}\n\\end{equation}\nand \n\\begin{equation}\n\\begin{split}\n-c(E^{A,n}_{\\textbf{u}},\\textbf{u}^{n,\\theta}_h, E^{A,n,\\theta}_{\\textbf{u}}) & = c(E^{A,n}_{\\textbf{u}},E^{I,n,\\theta}_{\\textbf{u}}, E^{A,n,\\theta}_{\\textbf{u}}) + c(E^{A,n}_{\\textbf{u}},E^{A,n,\\theta}_{\\textbf{u}}, E^{A,n,\\theta}_{\\textbf{u}})  \\\\\n& \\quad - c(E^{A,n}_{\\textbf{u}},\\textbf{u}^{n,\\theta}, E^{A,n,\\theta}_{\\textbf{u}}) \\\\\n& \\leq C  \\|E^{A,n,\\theta}_{\\textbf{u}}\\| \\|E^{A,n,\\theta}_{\\textbf{u}}\\|_1 \\{ \\|E^{I,n,\\theta}_{\\textbf{u}}\\|_2  + \\|\\textbf{u}^{n,\\theta}\\|_2  \\}\\\\\n& \\leq C \\|E^{A,n,\\theta}_{\\textbf{u}}\\|_1^2 (\\frac{1+\\theta}{2}\\|E^{I,n+1}_{\\textbf{u}}\\|_2 + \\frac{1-\\theta}{2}\\|E^{I,n}_{\\textbf{u}}\\|_2) + \\\\\n& \\quad C_2 \\|E^{A,n,\\theta}_{\\textbf{u}}\\|_1^2\\\\\n& \\leq C \\|E^{A,n,\\theta}_{\\textbf{u}}\\|_1^2 \\{ (\\frac{1+\\theta}{2}\\|\\textbf{u}^{n+1}\\|_2 + \\frac{1-\\theta}{2}\\|\\textbf{u}^n\\|_2)+ C_2 \\}\\\\\n& \\leq C_2' \\|E^{A,n,\\theta}_{\\textbf{u}}\\|_1^2\n\\end{split}\n\\end{equation}\nNow we will find bounds for each remaining term of $I_1$ separately. Before going to further calculations let us mention an important observation:\nBy the virtue of the choices of the finite element spaces $V^h$ and $Q^h$, we can clearly say that over each element sub-domain $\\Omega_k$ every function belonging to that spaces and their first and second order derivatives all are bounded functions. We can always find positive finite real numbers to bound each of the functions over element sub-domain. We will use this fact for several times further. \\vspace{1mm}\\\\\nLet us start with $I_1^1$\n\\begin{equation}\n\\begin{split}\nI_1^1 & = \\sum_{k=1}^{n_{el}} (\\rho \\tau_{1k}' I_{d \\times d}  \\partial_t E^{I,n}_{\\textbf{u}},  \\rho(\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}} + \\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}} + \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k} +   \\\\\n& \\quad \\sum_{k=1}^{n_{el}} ( \\tau_{1k}' I_{d \\times d} \\{ \\rho E^{I,n,\\epsilon \\theta}_{\\textbf{u}} \\cdot \\bigtriangledown \\textbf{u}^{n,\\theta} + \\rho \\textbf{u}_h^{n,\\theta} \\cdot \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}} - \\mu(c^n) \\Delta E^{I,n,\\theta}_{\\textbf{u}}  + \\bigtriangledown E^{I,n,\\theta}_p \\},  \\\\\n& \\quad  \\rho(\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}} + \\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}} + \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k} \\\\\n& = I_1^{11}+ I_1^{12} \\hspace{2mm} (say)\n\\end{split}\n\\end{equation}\nNow we present the estimations of these two terms separately in details. According to the above observation we can find bounds on each of the terms $\\textbf{u}_h, E^{A,n,\\theta}_{\\textbf{u}},  E^{A,n,\\theta}_p$ and their first and second order derivatives over each sub-domain $\\Omega_k$.  Applying these bounds in the following we will have\n\\begin{equation}\n\\begin{split}\n I_1^{11} & = \\int_{\\Omega'} \\rho \\tau_{1k}'I_{d \\times d} \\frac{E^{I,n+1}_{\\textbf{u}}-E^{I,n}_{\\textbf{u}}}{dt} \\{ \\rho(\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}} + \\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}} + \\bigtriangledown E^{A,n,\\theta}_{p}\\} \\\\\n & \\leq \\rho \\frac{\\mid \\tau_{1}'\\mid}{dt} \\{\\sum_{k=1}^{n_{el}} D_{B_{1k}} \\} (\\|E^{I,n+1}_{\\textbf{u}}\\|+ \\|E^{I,n}_{\\textbf{u}}\\|)\\\\\n& \\leq \\rho Ch^2 \\frac{\\mid \\tau_{1}\\mid}{\\mid dt+ \\rho \\tau_1 \\mid}\\{\\sum_{k=1}^{n_{el}} D_{B_{1k}}\\} (\\|\\textbf{u}^{n+1}\\|_2+\\|\\textbf{u}^n\\|_2)  \\\\\n& \\leq  h^2 \\frac{\\rho C C_{\\tau_1}}{(T_0-\\rho C_{\\tau_{1}})} \\{\\sum_{k=1}^{n_{el}} D_{B_{1k}}\\}(\\|\\textbf{u}^{n+1}\\|_2+\\|\\textbf{u}^n\\|_2)  \\\\\n\\end{split}\n\\end{equation}\nwhere the constant $D_{B_{1k}}$ is obtained after imposing bounds on the above bracketed terms over each sub-domain $\\Omega_k$. $C_{\\tau_1}$and $T$ are upper bounds on respectively $\\tau_1$ and $dt$. Since $dt$ is a non-zero positive real number, let $T_0$ is lower bound on $dt$. In order to make $(T_0-\\rho C_{\\tau_{1}})$ positive we have to take $h$ very small.\\vspace{1mm}\\\\\nNow the estimation of the second term is as follows:\n\\begin{equation}\n\\begin{split}\nI_1^{12} & = \\sum_{k=1}^{n_{el}} ( \\tau_{1k}' I_{d \\times d} \\{ \\rho E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown \\textbf{u}^{n,\\theta} + \\rho \\textbf{u}^{n} \\cdot \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}} - \\rho E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}}- \\\\\n& \\quad  \\rho E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}}- \\mu(c^n) \\Delta E^{I,n,\\theta}_{\\textbf{u}}  + \\bigtriangledown E^{I,n,\\theta}_p \\}, \\rho(\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}} +  \\\\\n& \\quad \\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}} + \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k} \\\\\n& \\leq \\mid \\tau_{1k}' \\mid [ \\sum_{k=1}^{n_{el}} D_{B_{1k}} \\{ \\sum_{i=1}^d (\\rho \\|E^{I,n}_{ui} \\| \\| \\frac{\\partial u_i^{n,\\theta}}{\\partial x_i} \\| + \\rho \\|u_i^{n}\\| \\|\\frac{\\partial E^{I,n,\\theta}_{ui}}{\\partial x_i}\\| + \\| E^{I,n}_{ui}\\|  \\\\\n& \\quad \\rho \\| \\frac{\\partial E^{I,n,\\theta}_{ui}}{\\partial x_i}\\| + \\rho\\|E^{A,n}_{ui}\\|_k \\| \\frac{\\partial E^{I,n,\\theta}_{ui}}{\\partial x_i}\\|+ \\mu(c) \\| \\Delta E^{I,n,\\theta}_{ui}\\| + \\| \\frac{\\partial E^{I,n,\\theta}_p}{\\partial x_i}\\| )\\}]\\\\\n & \\leq \\frac{\\mid \\tau_{1}\\mid T}{(T_0- \\rho C_{\\tau_1})} C (\\sum_{k=1}^{n_{el}} D_{B_{1k}}) [\\sum_{i=1}^d \\{\\rho h^2 \\|\\frac{\\partial u_i^{n, \\theta}}{\\partial x_i}\\| + \\rho h \\|u_i^{n,\\epsilon\\theta}\\| + \\rho h B^i_{1k} + \\mu_u  \\\\\n& \\quad  +h^3 (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2)\\} (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2) + \\\\\n& \\quad h (\\frac{1+\\theta}{2} \\|p^{n+1}\\|_1 + \\frac{1-\\theta}{2} \\|p^n\\|_1) ]\n\\end{split}\n\\end{equation} \nwhere $B^i_{1k}$ are bounds on $E^{A,n,\\theta}_{ui}$ for $i=1,...,d$. Now the next term $I_1^2$ can be estimated by dividing it into two terms $I_1^{21}$ and $ I_1^{22}$ as above. Therefore we directly start the estimation here with the first term of $I_1^2$ denoting that by $I_1^{21}$.\n\\begin{equation}\n\\begin{split}\nI_1^{21} & =  \\int_{\\Omega'} \\rho \\tau_{1k}'I_{d \\times d} \\frac{E^{A,n+1}_{\\textbf{u}}-E^{A,n}_{\\textbf{u}}}{dt} \\{ \\rho(\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}} + \\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}} + \\bigtriangledown E^{A,n,\\theta}_{p}\\} \\\\\n& \\leq  \\frac{\\rho T C_{\\tau_1}}{dt (T_0- \\rho C_{\\tau_1})} (\\sum_{k=1}^{n_{el}} D_{B_{1k}}) \\{ \\|E^{A,n+1}_{\\textbf{u}} \\|^2 - \\| E^{A,n}_{\\textbf{u}}\\|^2 \\}\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n\\begin{split}\nI_1^{22} & =  \\sum_{k=1}^{n_{el}} ( \\tau_{1k}' I_{d \\times d} \\{ \\rho E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown \\textbf{u}^{n,\\theta} + \\rho \\textbf{u}^{n} \\cdot \\bigtriangledown E^{A,n}_{\\textbf{u}} - \\rho E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown E^{A,n,\\theta}_{\\textbf{u}}-\\rho E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown  \\\\\n& \\quad E^{A,n,\\theta}_{\\textbf{u}} - \\mu(c^n) \\Delta E^{A,n,\\theta}_{\\textbf{u}}  + \\bigtriangledown E^{A,n,\\theta}_p \\}, \\rho(\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}} +  \\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}} + \\\\\n& \\quad \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k} \\\\\n& \\leq \\mid \\tau_1' \\mid  [ \\sum_{k=1}^{n_{el}} D_{B_{1k}} \\{ \\sum_{i=1}^d (\\rho  \\|E^{A,n}_{ui}\\|_k \\| \\frac{\\partial u_i^{n}}{\\partial x_i} \\| + \\rho \\|u_i^{n,\\theta} \\| \\| \\frac{\\partial E^{A,n,\\theta}_{ui}}{\\partial x_i}\\|_k + \\rho  \\| E^{I,n}_{ui}\\| \\\\\n& \\quad  \\| \\frac{\\partial E^{A,n,\\theta}_{ui}}{\\partial x_i}\\|_k +  \\| E^{A,n}_{ui}\\|_k \\| \\frac{\\partial E^{A,n,\\theta}_{ui}}{\\partial x_i}\\|_k + \\mu_u \\| \\Delta E^{A,n,\\theta}_{ui} \\|_k + \\| \\frac{\\partial E^{A,n,\\theta}_p}{\\partial x_i} \\|_k) \\} ] \n\\end{split}\n\\end{equation}\nApplying bounds on the functions belonging to $V^h$ and $Q^h$ on the above equation and denoting that bound by $\\bar{D}_{B_{1k}}$ we have \n\\begin{equation}\nI_1^{22} \\leq \\frac{\\mid \\tau_1 \\mid T}{(T_0- \\rho C_{\\tau_1})} (\\sum_{k=1}^{n_{el}} D_{B_{1k}} \\bar{D}_{B_{1k}})\n\\end{equation}\nNow expanding out the next term of $I_1$ we can proceed to estimate that in the following way:\n\\begin{equation}\n\\begin{split}\nI_1^3 & =\\sum_{k=1}^{n_{el}} \\int_{\\Omega_k}\\tau_2 \\{ ( \\bigtriangledown \\cdot E^{I,n,\\theta}_{\\textbf{u}}) \\cdot ( \\bigtriangledown \\cdot E^{A,n,\\theta}_{\\textbf{u}}) + ( \\bigtriangledown \\cdot E^{A,n,\\theta}_{\\textbf{u}})^2 \\} \\\\\n& \\leq C_{\\tau_2}\\{ \\sum_{i=1}^d (\\| \\frac{\\partial E^{I,n,\\theta}_{ui}}{\\partial x_i}\\|^2 + C_1 \\| \\frac{\\partial E^{A,n,\\theta}_{ui}}{\\partial x_i}\\|^2) \\}\\\\\n& \\leq C_{\\tau_2} \\{ h^2 \\sum_{i=1}^d (\\frac{1+\\theta}{2}\\|u_i^{n+1}\\|_2+ \\frac{1-\\theta}{2}\\|u_i^n\\|_2)^2+ \\mid E^{A,n,\\theta}_{\\textbf{u}}\\mid_1^2 \\}\n\\end{split}\n\\end{equation}\nwhere $C_{\\tau_2}$ is the maximum numerical value of $\\tau_2$ over $\\Omega$. Now the remaining terms of $I_1$ associated with the variable $c$ representing concentration are estimated as follows:\n\\begin{equation}\n\\begin{split}\nI_1^4 & = \\sum_{k=1}^{n_{el}} \\tau_{3k}'(\\partial_t E^{I,n}_c, \\bigtriangledown \\cdot\\tilde{\\bigtriangledown} E^{A,n,\\theta}_c+\\textbf{u}_h \\cdot \\bigtriangledown E^{A,n,\\theta}_c -\\alpha E^{A,n,\\theta}_c)_{\\Omega_k}  +\\\\\n& \\quad \\sum_{k=1}^{n_{el}} \\tau_{3k}'(-\\bigtriangledown \\cdot\\tilde{\\bigtriangledown} E^{I,n,\\theta}_c + (E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown)c^{n,\\theta}+ (\\textbf{u}_h^{n} \\cdot \\bigtriangledown)E^{I,n,\\theta}_c + \\alpha E^{I,n,\\theta}_c ,\\\\\n& \\quad \\bigtriangledown \\cdot\\tilde{\\bigtriangledown} E^{A,n,\\theta}_c+\\textbf{u}_h \\cdot \\bigtriangledown E^{A,n,\\theta}_c -\\alpha E^{A,n,\\theta}_c)_{\\Omega_k} \\\\\n& \\leq C h^2 \\mid \\tau_3' \\mid \\{ \\sum_{k=1}^{n_{el}} D_{B_{3k}} \\} (\\|c^{n+1}\\|_2+\\|c^n\\|_2) + \\mid \\tau_3' \\mid [\\sum_{k=1}^{n_{el}} D_{B_{3k}} \\{ \\sum_{i=1}^d (\\|E^{I,n}_{ui}\\|  \\\\\n& \\quad \\|\\frac{\\partial c^{n,\\theta}}{\\partial x_i}\\| + D_{im} \\| \\frac{\\partial^2 E^{I,n,\\theta}_c}{\\partial x_i^2}\\|+ \\bar{D}_{im} \\|\\frac{\\partial E^{I,n,\\theta}_c}{\\partial x_i}\\|+ (\\|u_i^{n}\\|+\\|E^{I,n}_{ui}\\|+ \\|E^{A,n}_{ui}\\|)\\\\\n& \\quad \\|\\frac{\\partial E^{I,n,\\theta}_c}{\\partial x_i}\\|+\\mid \\alpha \\mid \\|E^{I,n,\\theta}_c\\| )\\}] \\\\\n& \\leq h^2 \\frac{C C_{\\tau_3}}{(T_0- C_{\\tau_3})} \\{ \\sum_{k=1}^{n_{el}} D_{B_{3k}} \\} (\\|c^{n+1}\\|_2+\\|c^n\\|_2) + \\frac{C \\mid \\tau_3 \\mid}{(T_0- C_{\\tau_3})}  [\\sum_{k=1}^{n_{el}} D_{B_{3k}} \\sum_{i=1}^d \\{h^2 \\\\\n& \\quad (\\frac{1+\\theta}{2}\\|u_i^{n+1}\\|_2+\\frac{1-\\theta}{2}\\|u_i^n\\|_2)\\|\\frac{\\partial c^{n,\\theta}}{\\partial x_i}\\| + (D_{im}+h \\bar{D}_{im}+ \\mid \\alpha\\mid  h^2+ h\\|u_i^{n,\\theta}\\| \\\\\n& \\quad +hB_{1k}^i+ h^3 (\\frac{1+\\theta}{2}\\|u_i^{n+1}\\|_2+\\frac{1-\\theta}{2}\\|u_i^n\\|_2))(\\frac{1+\\theta}{2}\\|c^{n+1}\\|_2+\\frac{1-\\theta}{2} \\|c^n\\|_2)\\}]\n\\end{split}\n\\end{equation}\nLet $D_{B_{3k}}$ be the summation of the bounds imposed on the elements $\\frac{\\partial^2 E^{A,n,\\theta}_c}{\\partial x_i^2}$, $\\frac{\\partial E^{A,n,\\theta}_c}{\\partial x_i}$, $ E^{A,n,\\theta}_c$ belonging to finite element space $V^h$ and $D_{im}, \\bar{D}_{im}$ be the $supremum$ of $D_i$ and $\\frac{\\partial D_i}{\\partial x_i}$ respectively over each sub-domain $\\Omega_k$ for $i=1,...,d$. Now the estimation of  the last term $I_1^5$ follows the same way as above and considering $\\bar{D}_{B_{3k}}$ as an expression to denote the estimated result briefly the derivation of the bound of $I_1^5$ is in the following:\n\\begin{equation}\n\\begin{split}\nI_1^{5} & =\\sum_{k=1}^{n_{el}}\\tau_{3k}'(\\partial_t E^{A,n}_c, \\bigtriangledown \\cdot\\tilde{\\bigtriangledown} E^{A,n,\\theta}_c+\\textbf{u}_h \\cdot \\bigtriangledown E^{A,n,\\theta}_c -\\alpha E^{A,n,\\theta}_c)_{\\Omega_k}  +\\\\\n& \\quad \\sum_{k=1}^{n_{el}} \\tau_{3k}'(-\\bigtriangledown \\cdot\\tilde{\\bigtriangledown} E^{A,n,\\theta}_c + (E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown)c^{n,\\theta}+ (\\textbf{u}_h^{n} \\cdot \\bigtriangledown)E^{A,n,\\theta}_c + \\alpha E^{A,n,\\theta}_c ,\\\\\n& \\quad \\bigtriangledown \\cdot\\tilde{\\bigtriangledown} E^{A,n,\\theta}_c+\\textbf{u}_h \\cdot \\bigtriangledown E^{A,n,\\theta}_c -\\alpha E^{A,n,\\theta}_c)_{\\Omega_k} \\\\ \n& \\leq \\frac{T C_{\\tau_3}}{dt (T_0-C_{\\tau_3})} \\{ \\sum_{k=1}^{n_{el}} D_{B_{3k}} \\} (\\|E^{A,n+1}_c \\|^2 -\\|E^{A,n}_c \\|^2) + \\\\\n& \\quad \\frac{C \\mid \\tau_3 \\mid}{(T_0-C_{\\tau_3})} \\{\\sum_{k=1}^{n_{el}} \\bar{D}_{B_{3k}} D_{B_{3k}} \\}\n\\end{split}\n\\end{equation}\nThis completes estimation of $I_1$ finally. Now we see that the terms $I_2^i$ (for $i=1,...,4$) are same as that of $I_1$. Therefore we only mention the results here for each of them as follows:\n\\begin{equation}\n\\begin{split}\nI_2^{1} & = \\sum_{k=1}^{n_{el}}( \\frac{\\rho\\tau_{1k}}{dt+\\rho\\tau_{1k}} I_{d \\times d}  \\partial_t E^{I,n}_{\\textbf{u}}, -E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k} +  \\sum_{k=1}^{n_{el}} (\\frac{\\rho\\tau_{1k}}{dt+\\rho\\tau_{1k}} I_{d \\times d} \\{ \\rho E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown \\textbf{u}^{n,\\theta}   \\\\\n& \\quad + \\rho \\textbf{u}^{n} \\cdot \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}}- \\rho E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}}- \\rho E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown E^{I,n,\\theta}_{\\textbf{u}} - \\mu(c^n) \\Delta E^{I,n,\\theta}_{\\textbf{u}}\\\\\n&\\quad  + \\bigtriangledown E^{I,n,\\theta}_p \\},-E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k}\\\\\n& \\leq  h^2 \\frac{\\rho C C_{\\tau_1}}{(T_0- \\rho C_{\\tau_{1}})}[ \\sum_{i=1}^d \\{\\sum_{k=1}^{n_{el}} B_{1k}^i \\}(\\|u_i^{n+1}\\|_2+\\|u_i^n\\|_2)] +  \\frac{ \\mid \\tau_{1}\\mid}{(T_0- \\rho C_{\\tau_1})} \\sum_{i=1}^d (\\sum_{k=1}^{n_{el}} B_{1k}^i) \\\\\n& \\quad  [\\{\\rho h^2 \\|\\frac{\\partial u_i^{n, \\theta}}{\\partial x_i}\\| +\\rho h \\|u_i^{n}\\| +\\rho h B^i_{1k} + \\mu_u + h^3 (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2)\\}\\\\\n& \\quad   (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2) +  h (\\frac{1+\\theta}{2} \\|p^{n+1}\\|_1 + \\frac{1-\\theta}{2} \\|p^n\\|_1)  ]\n\\end{split}\n\\end{equation}\nand \n\\begin{equation}\n\\begin{split}\nI_2^2 & = \\sum_{k=1}^{n_{el}}( \\frac{\\rho\\tau_{1k}}{dt+\\rho\\tau_{1k}} I_{d \\times d}  \\partial_t E^{A,n}_{\\textbf{u}}, -E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k} +  \\sum_{k=1}^{n_{el}} (\\frac{\\rho\\tau_{1k}}{dt+\\rho\\tau_{1k}} I_{d \\times d} \\{ \\rho E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown \\textbf{u}^{n,\\theta}   \\\\\n& \\quad + \\rho \\textbf{u}^{n} \\cdot \\bigtriangledown E^{A,n,\\theta}_{\\textbf{u}}- \\rho E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown E^{A,n,\\theta}_{\\textbf{u}}- \\rho E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown E^{A,n,\\theta}_{\\textbf{u}} - \\mu(c^n) \\Delta E^{A,n,\\theta}_{\\textbf{u}}\\\\\n& \\quad  + \\bigtriangledown E^{A,n,\\theta}_p \\}, -E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k}\\\\\n& \\leq  \\frac{\\rho T C_{\\tau_1}}{dt (T_0- \\rho C_{\\tau_1})}[\\sum_{i=1}^d (\\sum_{k=1}^{n_{el}} B^i_{1k}) ]\\{ \\|E^{A,n+1}_{\\textbf{u}} \\|^2 - \\| E^{A,n}_{\\textbf{u}}\\|^2 \\} +\\\\\n& \\quad \\frac{\\mid \\tau_1 \\mid T}{(T_0- \\rho C_{\\tau_1})} [\\sum_{i=1}^d \\sum_{k=1}^{n_{el}} \\bar{D}_{B_{1k}} B^i_{1k} ]\n\\end{split}\n\\end{equation}\nFrom these results it is clear that estimations of the remaining terms of $I_2$ follow the same path as we have done for $I_1^4$ and $I_1^5$. Hence considering $B_{2k}$ as bound for $E^{A,n,\\theta}_c$ over each sub-domain $\\Omega_k$ we are skipping the repetition in mentioning the similar kind of results, though they will be added up  in the final stage of combining all the results. \\vspace{1mm}\\\\\nNow the job is to estimate next part denoted by $I_3$ and $I_4$ of the equation (43) which contain the matrix $\\textbf{d}$. Earlier we have mentioned that $d_2$ is $zero$. Let us look at the other three terms explicitly.\n\\begin{equation}\n\\begin{split}\n\\textbf{d}_1 & = \\{\\sum_{i=1}^{n+1}(\\frac{\\rho}{dt}\\tau_1')^i\\} I_{d \\times d}[\\rho \\partial_t(\\textbf{u}^n-\\textbf{u}_h^n)+ \\rho ((\\textbf{u}^{n}-\\textbf{u}_h^{n}) \\cdot \\bigtriangledown) \\textbf{u}^{n,\\theta}+\\rho (\\textbf{u}_h^{n} \\cdot \\bigtriangledown) \\\\\n& \\quad (\\textbf{u}^{n,\\theta}-\\textbf{u}_{h}^{n,\\theta})-\\mu(c^n) \\Delta(\\textbf{u}^{n,\\theta}-\\textbf{u}_{h}^{n,\\theta}) + \\bigtriangledown (p^{n,\\theta}- p_h^{n,\\theta})] \\\\\n& \\leq \\{\\sum_{i=1}^{\\infty}(\\frac{\\rho}{dt}\\tau_1')^i\\}  I_{d \\times d} [\\rho \\partial_t (E^{I,n}_{\\textbf{u}}+E^{A,n}_{\\textbf{u}})+ \\rho ((E^{I,n}_{\\textbf{u}} + E^{A,n}_{\\textbf{u}})\\cdot \\bigtriangledown) \\textbf{u}^{n,\\theta} + \\\\\n& \\quad \\rho (\\textbf{u}_h^{n} \\cdot \\bigtriangledown)(E^{I,n,\\theta}_{\\textbf{u}}+E^{A,n,\\theta}_{\\textbf{u}})-\\mu(c^n) \\Delta(E^{I,n,\\theta}_{\\textbf{u}}+E^{A,n,\\theta}_{\\textbf{u}})+ \\bigtriangledown E^{I,n,\\theta}_p \\\\\n& \\quad + \\bigtriangledown E^{A,n,\\theta}_p ] \\\\\n& = \\frac{\\rho \\tau_1'}{dt-\\rho \\tau_1'} I_{d \\times d}[ \\{\\rho \\partial_t E^{I,n}_{\\textbf{u}}+  \\rho (E^{I,n}_{\\textbf{u}} \\bigtriangledown)\\textbf{u}^{n,\\theta}+ \\rho (\\textbf{u}_h^{n} \\cdot \\bigtriangledown)E^{I,n,\\theta}_{\\textbf{u}}-\\mu(c^n)  \\\\\n& \\quad \\Delta E^{I,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{I,n,\\theta}_p \\} +  \\{\\rho \\partial_t E^{A,n}_{\\textbf{u}}+ \\rho (E^{A,n}_{\\textbf{u}} \\bigtriangledown)\\textbf{u}^{n,\\theta}+\\rho (\\textbf{u}_h^{n} \\cdot \\bigtriangledown)E^{A,n,\\theta}_{\\textbf{u}}  \\\\\n& \\quad  -\\mu(c^n) \\Delta E^{A,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{A,n,\\theta}_p \\}]\n\\end{split}\n\\end{equation}\nSince $\\frac{\\rho \\tau_1}{dt+ \\rho \\tau_1} < 1$, which implies $\\frac{\\rho \\tau_1'}{dt} < 1$ and therefore the series $\\sum_{i=1}^{\\infty}(\\frac{\\rho \\tau_1'}{dt})^i$ converges to $\\frac{\\rho \\tau_1'}{dt-\\rho \\tau_1'}$. \\vspace{1mm}\\\\\nSimilar to $\\textbf{d}_1$, the other component $d_3$ is as follows:\n\\begin{equation}\n\\begin{split}\nd_3 & \\leq \\frac{\\tau_3'}{dt- \\tau_3'} [  \\partial_t E^{I,n}_{c}-\\bigtriangledown \\cdot \\tilde{\\bigtriangledown} E^{I,n,\\theta}_{c} + (\\textbf{u}_h^{n} \\cdot \\bigtriangledown) E^{I,n,\\theta}_{c} + (E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown) c^{n,\\theta}+\\alpha E^{I,n,\\theta}_{c} \\\\\n& \\quad  +\\partial_t E^{A,n}_{c}-\\bigtriangledown \\cdot \\tilde{\\bigtriangledown} E^{A,n,\\theta}_{c} + (\\textbf{u}_h^{n} \\cdot \\bigtriangledown) E^{A,n,\\theta}_{c} + (E^{A,n}_{\\textbf{u}} \\cdot \\bigtriangledown)c^{n,\\theta} +\\alpha E^{A,n,\\theta}_{c} ]\n\\end{split}\n\\end{equation}\nIt is clearly seen in the expansion of $\\textbf{d}_1$ and $d_3$ that the terms in $I_3$ and $I_4$ exactly match with the terms in  $I_2$ and $I_1$ respectively. Hence their estimations also follow the same way as we have done earlier. Therefore skipping the repetition of presenting same results, here we have mentioned the estimated results only for one term from each of $I_3$ and $I_4$ in the following. Denoting first term of $I_3$ by the notation $I_3^{1}$ we have the estimated result as follows:\n\\begin{equation}\n\\begin{split}\nI_3^1 & = \\sum_{k=1}^{n_{el}} (\\frac{\\rho \\tau_{1k}}{dt+\\rho \\tau_{1k}} I_{d \\times d} \\{\\rho \\partial_t E^{I,n}_{\\textbf{u}}+  \\rho (E^{I,n}_{\\textbf{u}} \\bigtriangledown)\\textbf{u}^{n,\\theta}+ \\rho (\\textbf{u}_h^{n} \\cdot \\bigtriangledown)E^{I,n,\\theta}_{\\textbf{u}}  \\\\\n& \\quad -\\mu(c^n) \\Delta E^{I,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{I,n,\\theta}_p \\}, E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k}\\\\\n& \\leq  h^2 \\frac{\\rho^2 C C_{\\tau_1}}{(T_0- \\rho  C_{\\tau_{1}})}[ \\sum_{i=1}^d \\{\\sum_{k=1}^{n_{el}} B_{1k}^i \\}(\\|u_i^{n+1}\\|_2+\\|u_i^n\\|_2)] +  \\frac{ \\rho \\mid \\tau_{1}\\mid}{(T_0- \\rho C_{\\tau_1})} \\sum_{i=1}^d (\\sum_{k=1}^{n_{el}} B_{1k}^i) \\\\\n& \\quad  [\\{\\rho h^2 \\|\\frac{\\partial u_i^{n, \\theta}}{\\partial x_i}\\| +\\rho h \\|u_i^{n}\\| +\\rho h B^i_{1k} + \\mu_u + h^3 (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2)\\}\\\\\n& \\quad   (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2) +  h (\\frac{1+\\theta}{2} \\|p^{n+1}\\|_1 + \\frac{1-\\theta}{2} \\|p^n\\|_1)  ]\n\\end{split}\n\\end{equation}\nand denoting first term of $I_4$ by $I_4^1$ we have the estimated result in the following\n\\begin{equation}\n\\begin{split}\nI_4^1 & =   \\sum_{k=1}^{n_{el}} (\\frac{\\rho \\tau_{1k}^2}{dt+\\rho \\tau_{1k}} I_{d \\times d} \\{\\rho \\partial_t E^{I,n}_{\\textbf{u}}+  \\rho (E^{I,n}_{\\textbf{u}} \\cdot \\bigtriangledown)\\textbf{u}^{n,\\theta}+ \\rho (\\textbf{u}_h^{n} \\cdot \\bigtriangledown)E^{I,n,\\theta}_{\\textbf{u}} -\\mu(c^n)  \\\\\n& \\quad \\Delta E^{I,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{I,n,\\theta}_p \\},  \\rho(\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}} + \\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}} + \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k}\\\\\n& \\leq  h^2 \\frac{\\rho^2 C C_{\\tau_1}^2}{(T_0-\\rho C_{\\tau_{1}})} (\\sum_{k=1}^{n_{el}} D_{B_{1k}}) \\{\\|\\textbf{u}^{n+1}\\|_2+\\|\\textbf{u}^n\\|_2 \\}  +  \\frac{ \\rho \\mid \\tau_{1}\\mid^2}{(T_0- \\rho C_{\\tau_1})} (\\sum_{k=1}^{n_{el}} D_{B_{1k}}) [ \\sum_{i=1}^d\\\\\n& \\quad  \\{\\rho h^2 \\|\\frac{\\partial u_i^{n, \\theta}}{\\partial x_i}\\| +\\rho h \\|u_i^{n}\\| +\\rho h B^i_{1k} + \\mu_u + h^3 (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2)\\}\\\\\n& \\quad   (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2) +  h (\\frac{1+\\theta}{2} \\|p^{n+1}\\|_1 + \\frac{1-\\theta}{2} \\|p^n\\|_1)  ]\n\\end{split}\n\\end{equation}\nNow the estimations of the remaining terms are quite obvious. Therefore we directly add those results while combining them into (43) at last. Finally the last term containing truncation error can be estimated as follows:\n\\begin{equation}\n\\begin{split}\n(\\textbf{TE}^{n,\\theta}, E^{A,n,\\theta}_{\\textbf{U}}) & =(\\textbf{TE}_1^{n,\\theta}, E^{A,n,\\theta}_{\\textbf{u}})+ (TE_2^{n,\\theta}, E^{A,n,\\theta}_{c})\\\\\n& \\leq \\frac{ \\epsilon_9}{2}( \\|\\textbf{TE}^{n,\\theta}_1\\|^2+ \\|TE^{n,\\theta}_2\\|^2) + \\frac{1}{2 \\epsilon_9} (\\|E^{A,n,\\theta}_{\\textbf{u}}\\|^2 +\\|E^{A,n,\\theta}_{c}\\|^2)\n\\end{split}\n\\end{equation}\nThis completes estimation of all the terms in the $RHS$ of (43).  Now we start with putting all the bounds, obtained for each of the terms in the right hand side of (43). Then we take out few common terms in the left hand side and consequently we have left with those terms multiplied by $h^2, \\mid \\tau_1 \\mid$ and $ \\mid \\tau_3 \\mid$. Now we multiply both sides by 2 and taking integration over $(t^n,t^{n+1})$ for $n$=0,1,...,$(N-1)$ to both the sides. Finally we have (43) as follows:\n\\begin{multline}\n\\{1- \\frac{2 T C_{\\tau_3}(1-C_{\\tau_3})}{T_0+\\rho C_{\\tau_3}}\\sum_{k=1}^{n_{el}}D_{B_{3k}}- \\frac{4 T C_{\\tau_3}}{T_0-\\rho C_{\\tau_3}} \\sum_{k=1}^{n_{el}} B_{2k}\\} \\sum_{n=0}^{N-1} (\\|E^{A,n+1}_{c}\\|^2 -\\|E^{A,n}_{c}\\|^2)\\\\\n + \\rho \\{ 1- \\frac{2(1+\\rho C_{\\tau_1}) T C_{\\tau_1}}{T_0-\\rho C_{\\tau_1}} \\sum_{k=1}^{n_{el}} D_{B_{1k}} - \\frac{2(1+\\rho)T C_{\\tau_1}}{T_0-\\rho C_{\\tau_1}} \\sum_{i=1}^d  \\sum_{k=1}^{n_{el}} B_{1k}^i \\} \\sum_{n=0}^{N-1} (\\|E^{A,n+1}_{\\textbf{u}}\\|^2\\\\\n  -\\|E^{A,n}_{\\textbf{u}}\\|^2)  +  \\{ 2\\mu_l - \\frac{1}{\\epsilon_1}- \\frac{\\mu_u}{\\epsilon_2}- \\frac{4 C_2}{\\epsilon_8}- 4C_2' -2 C_{\\tau_2}\\}  \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\mid E^{A,n,\\theta}_{\\textbf{u}} \\mid_1^2 dt \\\\\n   +\\{2 \\sigma - \\frac{\\sigma}{\\epsilon_4}- \\frac{2 C_2}{\\epsilon_7}- \\frac{4 C_2}{\\epsilon_8} -4 C_2'-\\frac{1}{\\epsilon_9} \\} \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\| E^{A,n,\\theta}_{\\textbf{u}}\\|^2 dt   \\\\\n+\\{2D_l- \\frac{D_m}{\\epsilon_2}- 4 \\bar{C}_1^2\\} \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\mid E^{A,n,\\theta}_c \\mid_1^2 dt +\\\\\n\\end{multline}\n\\begin{multline}\n  \\{ 2 \\alpha - \\frac{\\alpha}{\\epsilon_5} - \\frac{4 \\bar{C}_1^2}{\\epsilon_6}-\\frac{1}{\\epsilon_9} \\} \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\| E^{A,n,\\theta}_c\\|^2 dt \\\\\n \\leq  h^2 \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}}[ 2 C^2 \\epsilon_1 (\\frac{1+\\theta}{2} \\|p^{n+1}\\|_2 +\\frac{1-\\theta}{2} \\|p^{n}\\|_2)^2 + \\{ 2C^2 \\mu_u \\epsilon_2 + h^2 \\sigma \\epsilon_4 +  2C_2 \\epsilon_7 \\\\\n+3 C_2 h^2 \\epsilon_8 + C_{\\tau_2} \\} (\\frac{1+\\theta}{2} \\|\\textbf{u}^{n+1}\\|_2 +\\frac{1-\\theta}{2} \\|\\textbf{u}^{n}\\|_2)^2 + \\{ C^2 D_m \\epsilon_3+ h^2 \\alpha \\epsilon_5 + \\bar{C}_1^2 C^2 \\epsilon_6 \\}  \\\\\n(\\frac{1+\\theta}{2} \\|c^{n+1}\\|_2 +\\frac{1-\\theta}{2}  \\|c^{n}\\|_2)^2 + 2\\frac{\\rho C C_{\\tau_1}}{T_0- \\rho C_{\\tau_1}} \\{ (1+ \\rho C_{\\tau_1}) (\\sum_{k=1}^{n_{el}} D_{B_{1k}}) + (1+ \\rho)  \\\\\n(\\sum_{i=1}^d \\sum_{k=1}^{n_{el}}B_{1k}^i) \\}(\\|\\textbf{u}^{n+1}\\|_2 + \\|\\textbf{u}^n\\|_2)  +\\frac{2 C C_{\\tau_3}}{T_0-  C_{\\tau_3}} \\{\\sum_{k=1}^{n_{el}} D_{B_{3k}}+(1+C_{\\tau_3})\\sum_{k=1}^{n_{el}} B_{2k} \\}\\\\\n(\\|c^{n+1}\\|_2+\\|c^n\\|_2)] dt  + 2 \\mid \\tau_1 \\mid \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} [ \\frac{T C + \\rho C_{\\tau_1}}{T_0- \\rho C_{\\tau_1}} \\sum_{k=1}^{n_{el}} D_{B_{1k}} + \\frac{1+\\rho}{T_0- \\rho C_{\\tau_1}} \\sum_{i=1}^d \\\\\n \\sum_{k=1}^{n_{el}} B_{1k}^i] [\\{\\rho h^2 \\|\\frac{\\partial u_i^{n, \\theta}}{\\partial x_i}\\| +\\rho h \\|u_i^{n,\\epsilon\\theta}\\| +\\rho h B^i_{1k} + \\mu_u + h^3 (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2)\\}\\\\\n   (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|u_i^n\\|_2) +  h (\\frac{1+\\theta}{2} \\|p^{n+1}\\|_1 + \\frac{1-\\theta}{2} \\|p^n\\|_1)  ] dt + \\\\\n \\mid \\tau_1 \\mid  \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\frac{ 2T}{T_0-\\rho C_{\\tau_1}} [(1+\\rho) \\sum_{k=1}^{n_{el}} D_{B_{1k}} \\bar{D}_{B_{1k}}+ (1+\\rho C_{\\tau_1}) \\sum_{i=1}^d  \\sum_{k=1}^{n_{el}}  \\bar{D}_{B_{1k}} B_{1k}^i] dt \\\\\n+2 \\mid \\tau_3 \\mid \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} [ \\frac{2C}{T_0- C_{\\tau_3}} \\sum_{k=1}^{n_{el}} D_{B_{3k}} + \\frac{1+ C_{\\tau_3}}{T_0- C_{\\tau_3}} \\sum_{k=1}^{n_{el}} B_{2k}] [ \\sum_{i=1}^d \\{h^2  \\|\\frac{\\partial c^{n,\\theta}}{\\partial x_i}\\|\\\\\n  (\\frac{1+\\theta}{2}\\|u_i^{n+1}\\|_2+\\frac{1-\\theta}{2}\\|u_i^n\\|_2) + (D_{im}+h \\bar{D}_{im}+ \\mid \\alpha\\mid  h^2+ h\\|u_i^{n,\\theta}\\| + hB_{1k}^i\\\\\n+ h^3  (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2+\\frac{1-\\theta}{2}\\|u_i^n\\|_2))(\\frac{1+\\theta}{2}\\|c^{n+1}\\|_2+\\frac{1-\\theta}{2} \\|c^n\\|_2)\\}] dt \\\\\n+ \\mid \\tau_3 \\mid  \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\frac{2C}{T_0 - C_{\\tau_3}} [\\sum_{k=1}^{n_{el}} \\bar{D}_{B_{3k}} D_{B_{3k}} + (1+ C_{\\tau_3}) \\sum_{k=1}^{n_{el}} \\bar{D}_{B_{3k}} B_{2k}] dt \\\\\n+  \\epsilon_9 \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}}( \\|\\textbf{TE}^{n,\\theta}_1\\|^2+ \\|TE^{n,\\theta}_2\\|^2) dt \\hspace{42mm}\n\\end{multline}\nWe can choose the values of the arbitrary parameters in such a manner that we can make all the coefficients in the left hand side positive. In order to satisfy such condition it is inevitable to choose the characteristic lengths small. Now after taking minimum of all the coefficients in left hand side, let us divide both the sides with that minimum, which turns out to be a positive real number. Applying assumption $\\textbf{(iv)}$ it can be seen that $\\|\\textbf{u}^n\\|_2$, $\\|p^n\\|_1$ and $\\|c^n\\|_2$ are bounded for $n=0,1,2,...,N$.  The initial conditions considered in section 3.1, imply $\\|E^{A,0}_{\\textbf{u}}\\|=0$ and $\\|E^{A,0}_c\\|=0$. \\vspace{1mm}\\\\\nAfter performing all these intermediate steps and applying the properties (15)-(16) on truncation errors  we finally arrive at the following expression since $\\tau_1$ and $\\tau_3$ are of order $h^2$:\n\\begin{multline}\n\\|E^{A,N}_{\\textbf{u}}\\|^2 +\\|E^{A,N}_{c}\\|^2 +\\sum_{n=0}^{N-1}  \\int_{t^n}^{t^{n+1}}   \\|E^{A,n,\\theta}_{\\textbf{u}} \\|_1^2 dt+ \\sum_{n=0}^{N-1}  \\int_{t^n}^{t^{n+1}}  \\|E^{A,n,\\theta}_{c} \\|^2_1 dt\\\\\n \\leq C(T,\\textbf{u},p,c) (h^2+dt^{2r}) \\hspace{30mm}\n\\end{multline}\nThis implies\n\\begin{equation}\n\\boxed{\\|E^{A}_{\\textbf{u}}\\|_{\\tilde{\\textbf{V}}}^2  + \\|E^{A}_{c}\\|_{\\tilde{\\textbf{V}}}^2 \\leq C(T,\\textbf{u},p,c) (h^2+dt^{2r})}\n\\end{equation}\nwhere\n\\begin{equation}\n    r=\n    \\begin{cases}\n      1, & \\text{if}\\ \\theta=1 \\\\\n      2, & \\text{if}\\ \\theta=0\n    \\end{cases}\n  \\end{equation}\nWe have used the fact that $\\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} M dt \\leq M T  $. This completes the first part of the proof. \\vspace{2mm}\\\\\n\\textbf{Second part} Using this above result we are going to estimate auxiliary error part of pressure. We will use inf-sup condition to find estimate for $E_p^A$. Applying Galerkin orthogonality only for variational form of Navier-Stokes flow problem we have obtained \\\\\n\\begin{multline}\n(\\frac{\\partial (\\textbf{u}-\\textbf{u}_h)}{\\partial t}, \\textbf{v}_{h})+ c(\\textbf{u},\\textbf{u},\\textbf{v}_h)- c(\\textbf{u}_h,\\textbf{u}_h,\\textbf{v}_h)+ \n a_{NS} (\\textbf{u}-\\textbf{u}_h, \\textbf{v}_h)- b(\\textbf{v}_h, p-p_h)  =0 \\\\ \n \\end{multline}\nSplitting of the errors implies the following \n\\begin{multline}\nb(\\textbf{v}_h, p-I_hp)+ b(\\textbf{v}_h, I_hp-p_h)  = (\\partial_t E^A_{\\textbf{u}}, \\textbf{v}_{h})+ c(E^I_{\\textbf{u}},\\textbf{u},\\textbf{v}_h)+c(E^A_{\\textbf{u}},\\textbf{u},\\textbf{v}_h)+\\\\\n c(\\textbf{u}_h,E^I_{\\textbf{u}},\\textbf{v}_h)+ c(\\textbf{u}_h,E^A_{\\textbf{u}},\\textbf{v}_h)+ a_{NS}(E^I_\\textbf{u}, \\textbf{v}_h)+  a_{NS}(E^A_\\textbf{u}, \\textbf{v}_h)\n\\end{multline}\nWithout loss of generality considering the inclusion $\\bigtriangledown \\cdot V^h \\subset Q^h$ and the property of the $L^2$ orthogonal projection of $I^h_p$ we have \n\\begin{equation}\nb(\\textbf{v}_h, p-I_hp)= \\int_{\\Omega}(p-I_hp)(\\bigtriangledown \\cdot \\textbf{v}_h)=0\n\\end{equation}\nNow according to inf-sup condition we will have the following expression\n\\begin{equation}\n\\begin{split}\n\\|I_hp-p_h\\|_{L^2(L^2)}^2& = \\|E_p^A\\|_{L^2(L^2)}^2 \\\\\n& = \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\|E_p^{A,n,\\theta}\\|^2 dt \\\\\n& \\leq  \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\underset{\\textbf{v}_h}{sup} \\frac{b(\\textbf{v}_h, E_p^{A,n,\\theta})}{\\|\\textbf{v}_h\\|_1} dt\n\\end{split}\n\\end{equation}\nUsing (74) on (73) we will have\n\\begin{equation}\n\\begin{split}\n\\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} b(\\textbf{v}_h, E_p^{A,n,\\theta}) dt & = \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\{ (\\frac{E^{A,n+1}_{\\textbf{u}}-E^{A,n}_{\\textbf{u}}}{dt},\\textbf{v}_h) + a_{NS}(E^{I,n,\\theta}_{\\textbf{u}},\\textbf{v}_h)\\\\\n& \\quad + c(E^{I,n}_{\\textbf{u}},\\textbf{u}^{n,\\theta},\\textbf{v}_h)+ c(E^{A,n}_{\\textbf{u}},\\textbf{u}^{n,\\theta},\\textbf{v}_h)+ \\\\\n& \\quad c(\\textbf{u}_h^{n}, E^{I,n,\\theta}_\\textbf{u},\\textbf{v}_h)+ c(\\textbf{u}_h^{n}, E^{A,n,\\theta}_\\textbf{u},\\textbf{v}_h)+ \\\\\n& \\quad  a_{NS}(E^{A,n,\\theta}_{\\textbf{u}},\\textbf{v}_h)\\}dt\n\\end{split}\n\\end{equation}\nNow applying the results obtained in the previous part we will have,\n\\begin{equation}\n\\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} b(\\textbf{v}_h, E_p^{A,n,\\theta}) dt \\leq C(T,\\textbf{u},p,c)(h^2+ dt^{2r}) \\|\\textbf{v}_h\\|_1\n\\end{equation}\nUsing this above result into (75), we will have the estimate for the pressure term\n\\begin{equation}\n\\|I_hp-p_h\\|_{L^2(L^2)}^2 \\leq C(T,\\textbf{u},p,c) (h^2+dt^{2r})\n\\end{equation}\nNow combining the results obtained in the first and second part we have finally arrived at the following auxiliary error estimate:\n\\begin{equation}\n\\|E^A_{\\textbf{u}}\\|^2_{\\tilde{\\textbf{V}}} + \\|E^A_p\\|_{L^2(L^2)}^2  + \\|E^A_c\\|^2_{\\tilde{\\textbf{V}}} \\leq C(T,\\textbf{u},p,c) (h^2+ dt^{2r})\n\\end{equation}\nwhere\n\\begin{equation}\n    r=\n    \\begin{cases}\n      1, & \\text{if}\\ \\theta=1 \\\\\n      2, & \\text{if}\\ \\theta=0\n    \\end{cases}\n  \\end{equation}\nThis completes the proof.\n\\end{proof} \n \n\\begin{theorem}\\textbf{(Apriori error estimate)}\nAssuming the same condition as in the previous theorem, \n\\begin{equation}\n\\|\\textbf{u}-\\textbf{u}_h\\|_{\\tilde{\\textbf{V}}}^2+\\|p-p_h\\|_{L^2(L^2)}^2 + \\|c-c_h\\|^2_{\\tilde{\\textbf{V}}} \\leq C' (h^2+ dt^{2r})\n\\end{equation}\nwhere $C'$ depends on T, $\\textbf{u}$,p,c and\n\\begin{equation}\n    r=\n    \\begin{cases}\n      1, & \\text{if}\\ \\theta=1 \\\\\n      2, & \\text{if}\\ \\theta=0\n    \\end{cases}\n  \\end{equation}\n\\end{theorem}\n\\begin{proof}\nBy applying triangle inequality, the interpolation inequalities and the result of the previous theorem we will have,\n\\begin{multline}\n\\|\\textbf{u}-\\textbf{u}_h\\|_{\\tilde{\\textbf{V}}}^2+\\|p-p_h\\|_{L^2(L^2)}^2 +\\|c-c_h\\|^2_{\\tilde{\\textbf{V}}} \\\\\n = \\|E^I_{\\textbf{u}}+E^A_{\\textbf{u}}\\|_{\\tilde{\\textbf{V}}}^2 +  \\|E^I_{p}+E^A_{p}\\|_{L^2(L^2)}^2+ \\|E^I_{c}+E^A_{c}\\|_{\\tilde{\\textbf{V}}}^2  \\hspace{31mm} \\\\\n \\leq \\bar{C} (\\|E^I_{\\textbf{u}}\\|_{\\tilde{\\textbf{V}}}^2 +\\|E^I_{p}\\|_{L^2(L^2)}^2 +  \\|E^I_{c}\\|_{\\tilde{\\textbf{V}}}^2+ \n \\|E^A_{\\textbf{u}}\\|_{\\tilde{\\textbf{V}}}^2+ \\|E^A_{p}\\|_{L^2(L^2)}^2+\\|E^A_{c}\\|_{\\tilde{\\textbf{V}}}^2) \\\\\n \\leq C'(T,\\textbf{u},p,c)(h^2+ dt^{2r}) \\hspace{65mm}\n\\end{multline}\nThis completes $apriori$ error estimation.\n\\end{proof} \n\n\\subsection{Aposteriori error estimation}\nIn this section we are going to derive residual based aposteriori error estimation. This estimation is also comprised of two parts similar to the auxiliary apriori error estimate derived in the earlier section. \\vspace{1mm} \\\\\n\\begin{theorem} \nFor computed velocity $\\textbf{u}_h$, pressure $p_h$ and concentration $c_h$ belonging to $(V^h)^d \\times Q^h \\times V^h$ satisfying (15)-(16), assume $dt$ is sufficiently small and positive, and sufficient regularity of exact solution in equations (1)-(2). Then there exists a constant $\\bar{C}$, independent of $\\textbf{u},p,c$ and depending on the residual such that\n\\begin{equation}\n\\|\\textbf{u}-\\textbf{u}_h\\|^2_{\\tilde{\\textbf{V}}} + \\|p-p_h\\|_{L^2(L^2)}^2  + \\|c-c_h\\|^2_{\\tilde{\\textbf{V}}} \\leq \\bar{C}(\\textbf{R}) (h^2+ dt^{2r})\n\\end{equation}\nwhere $\\textbf{R}$ is the residual vector and\n\\begin{equation}\n    r=\n    \\begin{cases}\n      1, & \\text{if}\\ \\theta=1 \\\\\n      2, & \\text{if}\\ \\theta=0\n    \\end{cases}\n  \\end{equation}\n\\end{theorem}\n\\begin{proof}\nWe estimate $aposteriori$ error by dividing the procedure into two parts. In the first part we find error bound corresponding to $velocity$ and $concentration$ followed by the second part estimating error associated with the $pressure$ term. Let us first introduce the residual vector corresponding to each equations \n\\[\n\\textbf{R}=\n  \\begin{bmatrix}\n \\textbf{f}-\\{ \\rho \\frac{\\partial \\textbf{u}_h}{\\partial t} + \\rho (\\textbf{u}_h \\cdot \\bigtriangledown) \\textbf{u}_h - \\mu(c) \\Delta \\textbf{u}_h  + \\bigtriangledown p_h \\} \\\\\n    -\\bigtriangledown \\cdot \\textbf{u}_h \\\\\n g-(\\frac{\\partial c_h}{\\partial t} - \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} c_h + \\textbf{u} \\cdot \\bigtriangledown c_h + \\alpha c_h )\n  \\end{bmatrix}\n   = \n  \\begin{bmatrix}\n \\textbf{R}_1\\\\\n  R_2 \\\\\n  R_3\n  \\end{bmatrix}\n\\]\n\\textbf{First part}: We have $\\forall \\textbf{V} \\in \\textbf{V}_F$\n\\begin{equation}\n\\mu_l \\mid \\textbf{v} \\mid_1^2 + D_{\\alpha} \\|d\\|_1^2 \\leq B(\\textbf{u};\\textbf{V},\\textbf{V})= a_{NS}(\\textbf{v},\\textbf{v})+a_T(d,d)\n\\end{equation} \nSince $\\textbf{e} \\in \\textbf{V}_F$ we substitute the errors $e_{\\textbf{u}},e_c$ into the above relation:\n\\begin{equation}\n\\mu_l \\|e_{\\textbf{u}}\\|_1^2+ D_{\\alpha} \\|e_c\\|_1^2  \\leq a_{NS}(e_\\textbf{u},e_\\textbf{u})+a_T(e_c,e_c)+ \\mu_l \\|e_{\\textbf{u}}\\|^2\\\\\n\\end{equation} \nBy adding few terms in both sides the above equation becomes\n\\begin{multline}\n (\\frac{\\partial e_{\\textbf{u}}}{\\partial t},e_{\\textbf{u}})+(\\frac{\\partial e_{c}}{\\partial t},e_{c})+\\mu_l \\|e_{\\textbf{u}}\\|_1^2+  D_{\\alpha} \\|e_c\\|_1^2 \\\\\n \\leq   (\\frac{\\partial e_{\\textbf{u}}}{\\partial t},e_{\\textbf{u}})+(\\frac{\\partial e_{c}}{\\partial t},e_{c})+c(\\textbf{u}, e_\\textbf{u},e_\\textbf{u})+a_{NS}(e_\\textbf{u},e_\\textbf{u})+a_T(e_c,e_c) \\hspace{10mm} \\\\\n +b(e_\\textbf{u},e_p)-b(e_\\textbf{u},e_p)+ \\mu_l \\|e_{\\textbf{u}}\\|^2\n\\end{multline} \nNow first we will find a lower bound of $LHS$ and then upper bound for $RHS$ and finally combining them we will get $aposteriori$ error estimate. To find the lower bound the $LHS$ can be written as\n\\begin{multline}\nLHS = (\\frac{e^{n+1}_{\\textbf{u}}-e^{n}_{\\textbf{u}}}{dt},e_{\\textbf{u}}^{n,\\theta})+ (\\frac{e_{c}^{n+1}-e_{c}^n}{dt},e_{c}^{n,\\theta})+ \n\\mu_l \\|e_{\\textbf{u}}^{n,\\theta}\\|_1^2 +    D_{\\alpha} \\|e_{c}^{n,\\theta}\\|^2_1\n\\end{multline}\nApplying (32) on first two terms of $LHS$ we have the following relations\n\\begin{equation}\n\\begin{split}\n(\\frac{e_{\\textbf{u}}^{n+1}-e_{\\textbf{u}}^n}{dt},e_{\\textbf{u}}^{n,\\theta}) & \\geq \\frac{1}{2 dt}(\\|e_{\\textbf{u}}^{n+1}\\|^2-\\|e_{\\textbf{u}}^n\\|^2) \\\\\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n\\begin{split}\n(\\frac{e_{c}^{n+1}-e_{c}^n}{dt},e_{c}^{n,\\theta}) & \\geq \\frac{1}{2 dt}(\\|e_{c}^{n+1}\\|^2-\\|e_{c}^n\\|^2) \\\\\n\\end{split}\n\\end{equation}\nHence\n\\begin{multline}\n \\frac{1}{2 dt}(\\|e_{\\textbf{u}}^{n+1}\\|^2-\\|e_{\\textbf{u}}^n\\|^2)+\\frac{1}{2 dt}(\\|e_{c}^{n+1}\\|^2-\\|e_{c}^n\\|^2)+\\mu_l \\|e_{\\textbf{u}}^{n,\\theta}\\|_1^2 +   D_{\\alpha}\\|e_{c}^{n,\\theta}\\|^2_1\\\\\n  \\leq LHS \\leq RHS\n\\end{multline}\nNow our job is to find upper bound for $RHS$ and to reach at the desired estimates let us divide it into two broad parts by splitting errors in each of the terms in the following way:\n\\begin{equation}\n\\begin{split}\nRHS &= [(\\frac{e_{\\textbf{u}}^{n+1}-e_{\\textbf{u}}^n}{dt},E_{\\textbf{u}}^{I,n,\\theta}) +(\\frac{e_c^{n+1}-e_c^n}{dt},E^{I,n,\\theta}_c)+c(\\textbf{u}^{n},e_\\textbf{u}^{n,\\theta}, E^{I,n,\\theta}_\\textbf{u}) +\\\\\n& \\quad a_{NS}(e^{n,\\theta}_\\textbf{u},E^{I,n,\\theta}_\\textbf{u})+ \nb(e^{n,\\theta}_\\textbf{u},E^{I,n,\\theta}_{p})-  b(E^{I,n,\\theta}_\\textbf{u},e^{n,\\theta}_p) +a_T(e^{n,\\theta}_c,E^{I,n,\\theta}_{c}) ] + \n\\\\\n& \\quad [(\\frac{e_{\\textbf{u}}^{n+1}-e_{\\textbf{u}}^n}{dt},E_{\\textbf{u}}^{A,n,\\theta}) + (\\frac{e^{n+1}_{c}-e_c^n}{ dt},E^{A,n,\\theta}_{c})+c(\\textbf{u}^{n},e_\\textbf{u}^{n,\\theta}, E^{A,n,\\theta}_\\textbf{u}) +\n \\\\\n& \\quad a_{NS}(e^{n,\\theta}_\\textbf{u},E^{A,n,\\theta}_\\textbf{u})+ b(e^{n,\\theta}_\\textbf{u} ,E^{A,n,\\theta}_{p})-  b(E^{A,n,\\theta}_\\textbf{u},e^{n,\\theta}_p) +a_T(e^{n,\\theta}_c,E^{A,n,\\theta}_{c})] \\\\\n& \\quad + \\mu_l \\|e_{\\textbf{u}}^{n,\\theta}\\|^2 \\\\\n&= RHS^I + RHS^A  + \\mu_l \\|e_{\\textbf{u}}^{n,\\theta}\\|^2\\\\\n\\end{split}\n\\end{equation}\nOur aim is to bring residual into context and for this purpose $RHS^I$ involving interpolation error terms can be written as follows:\n\\begin{equation}\n\\begin{split}\nRHS^I & = [\\rho (\\frac{e_{\\textbf{u}}^{n+1}-e_{\\textbf{u}}^n}{dt},E^{I,n,\\theta}_\\textbf{u}) + (\\frac{e_c^{n+1}-e_c^n}{dt}, E^{I,n,\\theta}_c) + c(\\textbf{u}^{n}, \\textbf{u}^{n,\\theta},E^{I,n,\\theta}_\\textbf{u})-\\\\\n& \\quad c(\\textbf{u}^{n}_h, \\textbf{u}^{n,\\theta}_h,E^{I,n,\\theta}_\\textbf{u}) + a_{NS}(e^{n,\\theta}_{\\textbf{u}}, E^{I,n,\\theta}_\\textbf{u}) - b(E^{I,n,\\theta}_{\\textbf{u}},e^{n,\\theta}_{p})+ b(e^{n,\\theta}_{\\textbf{u}},E^{I,n,\\theta}_p) \\\\\n& \\quad + a_{T}(e_c^{n,\\theta}, E^{I,n,\\theta}_c) ]-c(e_\\textbf{u}^{n}, \\textbf{u}_h^{n,\\theta}, E^{I,n,\\theta}_\\textbf{u})\\\\\n& =RHS^I_1 - c(e_\\textbf{u}^{n}, \\textbf{u}_h^{n,\\theta}, E^{I,n,\\theta}_\\textbf{u}) \n\\end{split}\n\\end{equation}\nThe bracketed term in the above equation is denoted by $RHS^I_1$.  $RHS^A$ involving auxiliary part of error can also be decomposed in the similar manner as above and let us denote the alike term corresponding to $RHS^A$ by $RHS^A_1$. Therefore\n\\begin{equation}\nRHS^A =RHS^A_1 - c(e_\\textbf{u}^{n}, \\textbf{u}_h^{n,\\theta}, E^{A,n,\\theta}_\\textbf{u}) \\\\\n\\end{equation}\nHence combining these above two results (93) becomes\n\\begin{equation}\n\\begin{split}\nRHS & = RHS^I_1 + RHS^A_1 - c(e_\\textbf{u}^{n}, \\textbf{u}_h^{n,\\theta}, e_\\textbf{u}^{n,\\theta})+ \\mu_l \\|e_{\\textbf{u}}^{n,\\theta}\\|^2\\\\\n& =   RHS^I_1 + RHS^A_1 - c(e_\\textbf{u}^{n}, \\textbf{u}^{n,\\theta}, e_\\textbf{u}^{n,\\theta}) + c(e_\\textbf{u}^{n}, e_\\textbf{u}^{n,\\theta}, e_\\textbf{u}^{n,\\theta})+ \\mu_l \\|e_{\\textbf{u}}^{n,\\theta}\\|^2\\\\\n& =   RHS^I_1 + RHS^A_1 - c(e_\\textbf{u}^{n}, \\textbf{u}^{n,\\theta}, e_\\textbf{u}^{n,\\theta})+ \\mu_l \\|e_{\\textbf{u}}^{n,\\theta}\\|^2\n\\end{split}\n\\end{equation}\nProperty \\textbf{(a)} of trilinear term $c(\\cdot, \\cdot, \\cdot)$ implies $c(e_\\textbf{u}^{n,\\epsilon \\theta}, e_\\textbf{u}^{n,\\theta}, e_\\textbf{u}^{n,\\theta})=0$. \\\\\nIn the most general way for $Navier$-$Stokes$ flow problem we have for all $\\textbf{v} \\in (V)^d$\n\\begin{multline}\n\\rho (\\frac{e_{\\textbf{u}}^{n+1}- e_{\\textbf{u}}^{n}}{dt}, \\textbf{v})+ c(\\textbf{u}^{n}, \\textbf{u}^{n,\\theta}, \\textbf{v})- c(\\textbf{u}^{n}_h, \\textbf{u}^{n,\\theta}_h, \\textbf{v})+ a_{NS}(e^{n,\\theta}_{\\textbf{u}},\\textbf{v})-b(\\textbf{v}, e_p^{n,\\theta})\\\\\n= \\int_{\\Omega} \\textbf{R}_1^{n,\\theta} \\cdot \\textbf{v} \\hspace{90mm}\\\\\nSimilarly \\hspace{2mm} \\int_{\\Omega} (\\bigtriangledown \\cdot e_\\textbf{u}^{n,\\theta})q = \\int_{\\Omega} R_2^{n,\\theta} q \\hspace{2mm} \\forall q \\in Q \\hspace{47mm}\\\\\n\\int_{\\Omega} (\\frac{ e_c^{n+1}-e_c^{n}}{dt} d + \\tilde{\\bigtriangledown} e_c^{n,\\theta} \\cdot \\bigtriangledown d + d \\textbf{u}^{n} \\cdot \\bigtriangledown e_c^{n,\\theta} + \\alpha e_c^{n,\\theta} d)= \\int_{\\Omega} R_3^{n,\\theta} d \\hspace{3 mm} \\forall d \\in V\n\\end{multline}\nNow substituting $\\textbf{v} ,q,d$ in the above expressions by $E^{I,n,\\theta}_{\\textbf{u}}, E^{I,n,\\theta}_{p}, E^{I,n,\\theta}_{c}$ respectively, we have the $RHS^I_1$ as,\n\\begin{equation}\n\\begin{split}\nRHS^I_1 & = \\int_{\\Omega} (\\textbf{R}_1^{n,\\theta} \\cdot E^{I,n,\\theta}_{\\textbf{u}}+ R_2^{n,\\theta} E^{I,n,\\theta}_{p}+ R_3^{n,\\theta} E^{I,n,\\theta}_{c} )\\\\\n& \\leq h^2 \\{ \\|\\textbf{R}_1^{n,\\theta}\\| (\\frac{1+\\theta}{2} \\| \\textbf{u}^{n+1} \\|_2 + \\frac{1-\\theta}{2} \\| \\textbf{u}^n \\|_2) + C_2 \\|R_2^{n,\\theta}\\| (\\frac{1+\\theta}{2} \\|p^{n+1}\\|_1  \\\\\n& \\quad  + \\frac{1-\\theta}{2} \\|p^n\\|_1) + \\|R_3^{n,\\theta}\\|  (\\frac{1+\\theta}{2}\\|c_{n+1}\\|_2 +\\frac{1-\\theta}{2} \\|c^n\\|_2)\\}\\\\\n& \\leq h^2( \\bar{C}_1 \\|\\textbf{R}_1^{n,\\theta}\\| + \\bar{C}_2 \\|R_2^{n,\\theta}\\|+ \\bar{C}_3 \\|R_3^{n,\\theta}\\|) \\\\\n\\end{split}\n\\end{equation}\nThe parameters $\\bar{C}_i$, for i=1,2,3,4, are coming from imposing assumption  \\textbf{(iv)}. Now we are going to estimate of $RHS^A_1$. For that we employ $subgrid$ formulation (8). Subtracting (8) from the variational finite element formulation satisfied by the exact solution we have $\\forall \\textbf{V}_h \\in \\textbf{V}_F^h $\n\\begin{multline}\n\\rho (\\frac{e_{\\textbf{u}}^{n+1}-e_{\\textbf{u}}^n}{dt}, \\textbf{v}_h)+ (\\frac{e_c^{n+1}-e_c^{n}}{dt},d_h) + c(\\textbf{u}^{n},\\textbf{u}^{n,\\theta},\\textbf{v}_h)- c(\\textbf{u}^{n}_h,\\textbf{u}^{n,\\theta}_h,\\textbf{v}_h) \\\\\n+a_{NS}(e_{\\textbf{u}}^{n,\\theta},\\textbf{v}_h)- b(\\textbf{v}_h, e^{n,\\theta}_p)+ b(e_{\\textbf{u}}^{n,\\theta}, q_h) + a_{T}(e_c^{n,\\theta},d_h)\\\\\n = \\sum_{k=1}^{n_{el}} \\{(\\tau_k'(\\textbf{R}^{n,\\theta}+\\textbf{d}), -\\mathcal{L}^* \\textbf{V}_h)_{\\Omega_k} - ((I-\\tau_k^{-1}\\tau_k) \\textbf{R}^{n,\\theta}, \\textbf{V}_h)_{\\Omega_k} + (\\tau_k^{-1}\\tau_k \\textbf{d}, \\textbf{V}_h)_{\\Omega_k} \\} \\\\\n  +(\\textbf{TE}^{n,\\theta}_1, \\textbf{v}_{h})+(TE^{n,\\theta}_2,d_h)\\\\\n\\end{multline}\n\\begin{multline}\n= \\sum_{k=1}^{n_{el}} [( \\tau_1' I_{d \\times d} \\{\\textbf{R}_1^{n,\\theta}+\\textbf{d}_1 \\}, \\rho (\\textbf{u}_h \\cdot \\bigtriangledown) \\textbf{v}_{h}+\\mu(c) \\Delta \\textbf{v}_{h}+ \\bigtriangledown q_h)_{\\Omega_k} + \\tau_2'(R_2^{n,\\theta}, \\bigtriangledown \\cdot \\textbf{v}_h)_{\\Omega_k} \\\\\n\\quad   + \\tau_3'(R_3^{n,\\theta}+d_3, \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} d_h + \\textbf{u}_h \\cdot \\bigtriangledown d_h - \\alpha d_h)_{\\Omega_k}+ ((1-\\tau_1^{-1}\\tau_1') I_{d \\times d} \\textbf{R}_1^{n,\\theta},\\textbf{v}_{h})_{\\Omega_k}\\\\\n +( (1-\\tau_3^{-1}\\tau_3') R_3^{n,\\theta},d_{h})_{\\Omega_k} +(\\tau_1^{-1}\\tau_1' I_{d \\times d} \\textbf{d}_1,\\textbf{v}_{h})_{\\Omega_k}+\\tau_3^{-1}\\tau_3' (d_3,d_{h})_{\\Omega_k}]  \\\\\n+ (\\textbf{TE}_1^{n,\\theta},\\textbf{v}_{h}) +(TE_2^{n,\\theta},d_h) \\hspace{55mm}\n\\end{multline} \nNow substituting $\\textbf{V}_h$ by $(E^{A,n,\\theta}_{\\textbf{u}},  E^{A,n,\\theta}_{p}, E^{A,n,\\theta}_{c})$ in the above equation we have $RHS^A_1$ as follows \n\\begin{equation}\n\\begin{split}\nRHS^A_1 & = \\sum_{k=1}^{n_{el}}[( \\tau_1' I_{d \\times d} \\{\\textbf{R}_1^{n,\\theta}+\\textbf{d}_1 \\}, \\rho (\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}}+\\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k}  \\\\\n& \\quad + \\tau_2'(R_2^{n,\\theta}, \\bigtriangledown \\cdot E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k} + \\tau_3'(R_3^{n,\\theta}+d_4, \\bigtriangledown \\cdot \\tilde{\\bigtriangledown} E^{A,n,\\theta}_{c} + \\textbf{u}_h \\cdot \\bigtriangledown E^{A,n,\\theta}_{c} \\\\\n& \\quad - \\alpha E^{A,n,\\theta}_{c})_{\\Omega_k} + ((1-\\tau_1^{-1}\\tau_1') I_{d \\times d} \\textbf{R}_1^{n,\\theta}, E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k} + (1-\\tau_3^{-1}\\tau_3') (R_3^{n,\\theta}, \\\\\n& \\quad E^{A,n,\\theta}_{c})_{\\Omega_k}+(\\tau_1^{-1}\\tau_1'I_{d \\times d}\\textbf{d}_1,E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k}+\\tau_3^{-1}\\tau_3' (d_3,E^{A,n,\\theta}_{c})_{\\Omega_k}] + \\\\\n& \\quad (\\textbf{TE}_1^{n,\\theta},E^{A,n,\\theta}_{\\textbf{u}})  +(TE_2^{n,\\theta},E^{A,n,\\theta}_{c})\n\\end{split}\n\\end{equation}\nNow we estimate each term separately. We use the results mentioned earlier during $apriori$ error estimation.\n\\begin{equation}\n\\begin{split}\n \\sum_{k=1}^{n_{el}}( \\tau_1' I_{d \\times d} \\textbf{R}_1^{n,\\theta}, \\rho (\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}}+\\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k} \\\\\n  \\leq \\frac{\\mid \\tau_1 \\mid T}{T_0- \\rho C_{\\tau_1}} (\\sum_{k=1}^{n_{el}} D_{B_{1k}})\\|\\textbf{R}_1^{n,\\theta}\\| \\hspace{49mm} \\\\\n\\sum_{k=1}^{n_{el}} \\tau_3'(R_3^{n,\\theta},\\bigtriangledown \\cdot \\tilde{\\bigtriangledown} E^{A,n,\\theta}_{c} + \\textbf{u}_h \\cdot \\bigtriangledown E^{A,n,\\theta}_{c} - \\alpha E^{A,n,\\theta}_{c} )_k \\hspace{14mm} \\\\\n \\leq \\frac{\\mid \\tau_3 \\mid T}{T_0-  C_{\\tau_3}} (\\sum_{k=1}^{n_{el}} D_{B_{3k}}) \\|R_3^{n,\\theta}\\| \\hspace{50mm}\\\\ \n\\end{split}\n\\end{equation}\nand the other set of terms can be estimated as follows:\n\\begin{equation}\n\\begin{split}\n\\sum_{k=1}^{n_{el}}((1-\\tau_1^{-1} \\tau_1')I_{d \\times d} \\textbf{R}_1^{n,\\theta}, E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k} & \\leq \\frac{\\rho \\mid \\tau_1 \\mid}{T_0-\\rho C_{\\tau_1}} (\\sum_{i=1}^d \\sum_{k=1}^{n_{el}}B_{1k}^i) \\|\\textbf{R}_1^{n,\\theta}\\| \\\\\n\\sum_{k=1}^{n_{el}}(1-\\tau_3^{-1} \\tau_3')(R_3^{n,\\theta}, E^{A,n,\\theta}_{c})_{\\Omega_k} & \\leq \\frac{ \\mid \\tau_3 \\mid}{T_0- C_{\\tau_3}} (\\sum_{k=1}^{n_{el}}B_{2k}) \\|R_3^{n,\\theta}\\| \\\\\n\\end{split}\n\\end{equation} \nlet us look into the form of the column vector $\\textbf{d}$ which has components $\\textbf{d}_1,d_2,d_3$. \\vspace{1mm} \\\\\n$\\textbf{d}$= $\\sum_{i=1}^{n+1}(\\frac{1}{dt}M\\tau_k')^i(\\textbf{F} -M\\partial_t \\textbf{U}_h - \\mathcal{L}\\textbf{U}_h)=\\sum_{i=1}^{n+1}(\\frac{1}{dt}M\\tau_k')^i \\textbf{R}$ \\vspace{2mm}\\\\\nHence we have the components $\\textbf{d}_1= (\\sum_{i=1}^{n+1}(\\frac{1}{dt} \\tau_1')^i) I_{d \\times d} \\textbf{R}_1^{n,\\theta}$, $d_2=0$ and $d_3=(\\sum_{i=1}^{n+1}(\\frac{1}{dt} \\tau_3')^i) R_3^{n,\\theta}$ \\vspace{2mm} \\\\ \nNow the terms containing the components of $\\textbf{d}$ can be estimated in the following way:\n\\begin{multline}\n\\sum_{k=1}^{n_{el}} (\\tau_1' I_{d \\times d} \\textbf{d}_1,\\rho (\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}}+\\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k} \\\\\n = \\sum_{k=1}^{n_{el}}( \\tau_1' \\{\\sum_{i=1}^{n+1}(\\frac{1}{dt} \\tau_1')^i\\} I_{d \\times d}\\textbf{R}_1^{n,\\theta},  \\rho (\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}}+\\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k}\\\\\n \\leq \\sum_{k=1}^{n_{el}}(\\tau_1' \\{\\sum_{i=1}^{\\infty}(\\frac{1}{dt} \\tau_1')^i\\} I_{d \\times d} \\textbf{R}_1^{n,\\theta}, \\rho (\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}}+\\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k}\\\\\n = \\sum_{k=1}^{n_{el}}( \\frac{\\rho \\tau_1^2}{dt+\\rho \\tau_1} I_{d \\times d} \\textbf{R}_1^{n,\\theta},  \\rho (\\textbf{u}_h \\cdot \\bigtriangledown) E^{A,n,\\theta}_{\\textbf{u}}+\\mu(c) \\Delta E^{A,n,\\theta}_{\\textbf{u}}+ \\bigtriangledown E^{A,n,\\theta}_{p})_{\\Omega_k} \\hspace{8mm} \\\\\n \\leq \\frac{\\mid \\tau_1 \\mid}{T_0-\\rho C_{\\tau_1}} \\rho C_{\\tau_1} (\\sum_{k=1}^{n_{el}} D_{B_{1k}}) \\|\\textbf{R}_1^{n,\\theta}\\| \\hspace{53mm}\n\\end{multline} \nSimilarly the next few terms will follow the same way as above.\n\\begin{multline}\n \\sum_{k=1}^{n_{el}} \\tau_3'(d_3,\\bigtriangledown \\cdot \\tilde{\\bigtriangledown} E^{A,n,\\theta}_{c} + \\textbf{u}_h \\cdot \\bigtriangledown E^{A,n,\\theta}_{c} - \\alpha E^{A,n,\\theta}_{c} )_k \\hspace{35mm}\\\\\n \\leq \\frac{\\mid \\tau_3 \\mid}{T_0- C_{\\tau_3}}  C_{\\tau_3} (\\sum_{k=1}^{n_{el}} D_{B_{3k}}) \\|R_3^{n,\\theta}\\| \\hspace{63mm} \\\\\n \\sum_{k=1}^{n_{el}} (\\tau_1^{-1}\\tau_1' I_{d \\times d} \\textbf{d}_1,E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k} \\leq \\frac{\\rho \\mid \\tau_1 \\mid}{T_0- \\rho C_{\\tau_1}} (\\sum_{i=1}^d \\sum_{k=1}^{n_{el}} B_{1k}^i) \\|\\textbf{R}_1^{n,\\theta}\\| \\hspace{20mm} \\\\\n \\sum_{k=1}^{n_{el}} \\tau_3^{-1}\\tau_3'(d_3,E^{A,n,\\theta}_{c})_{\\Omega_k} \\leq \\frac{ \\mid \\tau_3 \\mid}{T_0-  C_{\\tau_3}} (\\sum_{k=1}^{n_{el}} B_{2k}) \\|R_3^{n,\\theta} \\| \\hspace{19mm}\n \\end{multline} \nThe terms containing truncation errors already have been estimated earlier during apriori error estimation. Now we estimate the remaining terms as follows:\n\\begin{equation}\n\\begin{split} \n& \\sum_{k=1}^{n_{el}} \\tau_2'(R_3^{h,n,\\theta}, \\bigtriangledown \\cdot E^{A,n,\\theta}_{\\textbf{u}})_{\\Omega_k}  \\\\\n& =  \\sum_{k=1}^{n_{el}} \\tau_2' (R_3^{h,n,\\theta}, \\bigtriangledown \\cdot e^{n,\\theta}_{\\textbf{u}})_{\\Omega_k} -  \\sum_{k=1}^{n_{el}} \\tau_2' (R_3^{h,n,\\theta}, \\bigtriangledown \\cdot E^{I,n,\\theta}_{\\textbf{u}})_{\\Omega_k}  \\\\\n& =  \\sum_{k=1}^{n_{el}} \\tau_2'  (\\bigtriangledown \\cdot e^{n,\\theta}_{\\textbf{u}}, \\bigtriangledown \\cdot e^{n,\\theta}_{\\textbf{u}})_{\\Omega_k} -  \\sum_{k=1}^{n_{el}} \\tau_2' ( \\bigtriangledown \\cdot e^{n,\\theta}_{\\textbf{u}}, \\bigtriangledown \\cdot E^{I,n,\\theta}_{\\textbf{u}})_{\\Omega_k}  \\\\\n \\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split} \n& \\leq \\sum_{k=1}^{n_{el}} \\tau_2 \\int_{\\Omega_k} ( \\bigtriangledown \\cdot e^{n,\\theta}_{\\textbf{u}})^2 +  \\sum_{k=1}^{n_{el}} \\tau_2 \\int_{\\Omega_k} \\mid  (\\bigtriangledown \\cdot e^{n,\\theta}_{\\textbf{u}} ) (\\bigtriangledown \\cdot E^{I,n,\\theta}_{\\textbf{u}})  \\mid  \\\\ \n& \\leq  C_{\\tau_2} (\\sum_{i=1}^d \\|\\frac{\\partial e^{n,\\theta}_{ui}}{\\partial x_i}\\|)^2 +  C_{\\tau_2} (\\sum_{i=1}^d \\|\\frac{\\partial e^{n,\\theta}_{ui}}{\\partial x_i}\\|) (\\sum_{i=1}^d \\|\\frac{\\partial E^{I,n,\\theta}_{ui}}{\\partial x_i}\\| )  \\\\\n & \\leq 2 C_{\\tau_2} \\sum_{i=1}^d \\|\\frac{\\partial e^{n,\\theta}_{ui}}{\\partial x_i}\\|^2 + \\epsilon_1' C_{\\tau_2} \\sum_{i=1}^d \\|\\frac{\\partial e^{n,\\theta}_{ui}}{\\partial x_i}\\|^2  +  \\frac{C_{\\tau_2}}{\\epsilon_1'} \\sum_{i=1}^d \\|\\frac{\\partial E^{I,n,\\theta}_{ui}}{\\partial x_i}\\|^2   \\\\\n& \\leq C_{\\tau_2}[ (2+ \\epsilon_1')  \\sum_{i=1}^d \\| e^{n,\\theta}_{ui}\\|_1^2  + \\frac{h^2}{\\epsilon_1'} \\sum_{i=1}^d (\\frac{1+\\theta}{2} \\|u_i^{n+1}\\|_2+ \\frac{1-\\theta}{2} \\|u_i^{n}\\|_2)^2 ]  \\\\\n& \\leq   (2+ \\epsilon_1') C_{\\tau_2} \\| e^{n,\\theta}_{\\textbf{u}}\\|_1^2 + h^2 \\frac{C_{\\tau_2}}{\\epsilon_1'} \\bar{C}_5 \n \\end{split}\n\\end{equation}\nwhere the parameter $\\bar{C}_5$ comes for applying assumption $\\textbf{(iv)}$. Now the terms involving trancation error can be estimated in slightly different way as we have done in the previous section. Let us present here a detailed derivation of one term only and the other follows the same way.\n\\begin{equation}\n\\begin{split}\n(TE_2^{n,\\theta}, E^{A,n,\\theta}_{c}) & = (TE_2^{n,\\theta}, e^{n,\\theta}_c) - (TE_2^{n,\\theta}, E^{I,n,\\theta}_c) \\\\\n& \\leq \\mid (TE_2^{n,\\theta}, e^{n,\\theta}_c) \\mid + \\mid (TE_2^{n,\\theta}, E^{I,n,\\theta}_c) \\mid \\\\\n& \\leq \\|TE_2^{n,\\theta} \\| (\\|e^{n,\\theta}_c\\| + \\|E^{I,n,\\theta}_c \\|)\\\\\n& \\leq \\frac{1}{\\epsilon_2'} \\|TE_2^{n,\\theta}\\|^2 + \\frac{\\epsilon_2'}{2} (\\|e^{n,\\theta}_c\\|^2 + \\|E^{I,n,\\theta}_c \\|^2) \\\\\n& \\leq  \\frac{1}{\\epsilon_2'} \\|TE_2^{n,\\theta}\\|^2 +  \\frac{\\epsilon_2'}{2} \\{ \\|e^{n,\\theta}_c\\|^2 + h^4 (\\frac{1+\\theta}{2} \\|c^{n+1}\\|_2 + \\frac{1-\\theta}{2} \\|c^n\\|_2)^2 \\} \\\\\n& \\leq \\frac{1}{\\epsilon_2'} \\|TE_2^{n,\\theta}\\|^2 +  \\frac{\\epsilon_2'}{2} \\|e^{n,\\theta}_c\\|^2_1 +  h^4 \\frac{\\epsilon_2'}{2} \\bar{C}_6\n\\end{split}\n\\end{equation}\nSimilarly the estimated result for the remaining term is\n\\begin{equation}\n(\\textbf{TE}_1^{n,\\theta}, E^{A,n,\\theta}_{\\textbf{u}}) \\leq  \\frac{1}{\\epsilon_2'} \\|\\textbf{TE}_1^{n,\\theta}\\|^2 +  \\frac{\\epsilon_2'}{2} \\|e^{n,\\theta}_{\\textbf{u}}\\|^2_1 +  h^4 \\frac{\\epsilon_2'}{2} \\bar{C}_7\n\\end{equation}\n and this completes estimating all the terms of $RHS^A_1$. Now the $trilinear$ term in $RHS$ in (96) can be estimated as follows using property \\textbf{(b)} of the $trilinear$ term as follows:\n\\begin{equation}\n\\begin{split}\nc(e_\\textbf{u}^{n},\\textbf{u}^{n,\\theta}, e_\\textbf{u}^{n,\\theta}) & \\leq C \\|e_\\textbf{u}^{n}\\| \\|\\textbf{u}^{n,\\theta}\\|_2 \\|e_\\textbf{u}^{n,\\theta}\\|_1\\\\\n& \\leq \\bar{C}_8 \\|e_\\textbf{u}^{n}\\| \\|e_\\textbf{u}^{n,\\theta}\\|_1\\\\\n& \\leq  \\bar{C}_8 \\|e_\\textbf{u}^{n,\\theta}\\|^2\n\\end{split}\n\\end{equation}\nThe term $\\|\\textbf{u}^{n,\\theta}\\|_2$ is bounded by the virtue of assumption \\textbf{(iv)}\nand applying $Poincare$ inequality on last term in $RHS$ in (96) we have\n\\begin{equation}\n\\mu_l \\|e_{\\textbf{u}}^{n,\\theta}\\|^2 \\leq \\mu_l C_P \\mid e_{\\textbf{u}}^{n,\\theta} \\mid_1^2 \\leq \\mu_l C_P \\|e_{\\textbf{u}}^{n,\\theta}\\|^2\n\\end{equation}\nNow this completes finding bounds for each term in the $RHS$ of (93). Putting common terms all together in the left hand side and multiplying them by $2$ and  then integrating both sides over $(t^n,t^{n+1})$ for $n=0,...,(N-1)$ , we will finally have \n\\begin{multline}\n\\sum_{n=0}^{N-1} ( \\|e^{n+1}_{\\textbf{u}}\\|^2-\\|e^{n}_{\\textbf{u}}\\|^2)+\\sum_{n=0}^{N-1} ( \\|e^{n+1}_c\\|^2-\\|e^{n}_c\\|^2) + \\{ 2\\mu_l-  2(2+\\epsilon_1') C_{\\tau_2}  \\\\\n\\quad - 2 C_8 - \\mu_l C_P -\\epsilon_2' \\} \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\|e^{n,\\theta}_{\\textbf{u}}\\|_1^2 dt +(2 D_{\\alpha}- \\epsilon_2') \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\|e^{n,\\theta}_c\\|^2_1 dt \\\\\n\\leq h^2 \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\{ \\bar{C}_1 \\|\\textbf{R}_1^{n,\\theta}\\| + \\bar{C}_2 \\|R_2^{n,\\theta}\\|+ \\bar{C}_3 \\|R_3^{n,\\theta}\\|+ \\frac{2 C_{\\tau_2}}{\\epsilon_1'} \\bar{C}_5 + h^2 \\epsilon_2' (\\bar{C}_6 + \\bar{C}_7) \\}dt \\\\\n\\quad  \n+ 2 \\frac{\\mid \\tau_1 \\mid }{T_0- \\rho C_{\\tau_1}} [ (T+ \\rho C_{\\tau_1}) (\\sum_{k=1}^{n_{el}} D_{B_{1k}}) + 2 \\rho \\sum_{i=1}^d \\sum_{k=1}^{n_{el}} B_{1k}^i ]  \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\|\\textbf{R}_1^{n,\\theta}\\|^2 dt \\\\\n+ 2 \\frac{\\mid \\tau_3 \\mid }{T_0- \\rho C_{\\tau_3}}[(T+ C_{\\tau_3}) \\sum_{k=1}^{n_{el}}D_{B_{3k}}+ 2 \\sum_{k=1}^{n_{el}} B_{2k}] \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\|R_3^{n,\\theta}\\|^2 dt +\\\\\n\\frac{2}{\\epsilon_2'} \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}}( \\|\\textbf{TE}^{n,\\theta}_1\\|^2+ \\|TE^{n,\\theta}_2\\|^2) dt \\hspace{20mm}\n\\end{multline}\nChoose the arbitrary parameters including $C_{\\tau_2}$ and the $Poincare$ constant $C_P$ in such a way that all the coefficients in the left hand side can be made positive.\nThen taking minimum over the coefficients in the left hand side let us divide both sides by them. Using properties (15)-(16) associated with both implicit time discretisation scheme  and the fact that $\\tau_1, \\tau_3$ are of order $h^2$, we have arrived at the following relation:\n\\begin{equation}\n\\boxed{\\|\\textbf{u}-\\textbf{u}_h\\|_{\\tilde{\\textbf{V}}}^2  + \\|c-c_h\\|_{\\tilde{\\textbf{V}}}^2 \\leq C'(\\textbf{R}) (h^2+dt^{2r})}\n\\end{equation}\nwhere\n\\begin{equation}\n    r=\n    \\begin{cases}\n      1, & \\text{if}\\ \\theta=1 \\hspace{1mm} for  \\hspace{1mm} backward  \\hspace{1mm} Euler  \\hspace{1mm} rule \\\\\n      2, & \\text{if}\\ \\theta=0  \\hspace{1mm} for  \\hspace{1mm} Crank-Nicolson  \\hspace{1mm} scheme\n    \\end{cases}\n  \\end{equation}\n  This only completes one part of $aposteriori$ estimation and in the next part we combine the corresponding pressure part.\\vspace{2mm}\\\\\n\\textbf{Second part}: Using the result (74) we can rewrite (73) in the following form:\n\\begin{equation}\nb(\\textbf{v}_h, I_hp-p_h)  = (\\frac{\\partial e_{\\textbf{u}}}{\\partial t}, \\textbf{v}_{h})+  c(e_{\\textbf{u}},\\textbf{u},\\textbf{v}_h)+\n c(\\textbf{u}_h, e_{\\textbf{u}},\\textbf{v}_h)+ a_{NS}(e_{\\textbf{u}},\\textbf{v}_h)\n\\end{equation}\nIntegrating both sides with respect time\n\\begin{equation}\n\\begin{split}\n\\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} b(\\textbf{v}_h, E_p^{A,n,\\theta}) dt & = \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\{ (\\frac{e_{\\textbf{u}}^{n+1}-e_{\\textbf{u}}^n}{dt},\\textbf{v}_{h})+ c(e_\\textbf{u}^{n}, \\textbf{u}^{n,\\theta},\\textbf{v}_h) +\\\\\n& \\quad  c(\\textbf{u}_h^{n},e_{\\textbf{u}}^{n,\\theta}, \\textbf{v}_h)+ a_{NS}(e_{\\textbf{u}}^{n,\\theta},\\textbf{v}_h) \\}dt\\\\\n& = \\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} \\{ (\\frac{e_{\\textbf{u}}^{n+1}-e_{\\textbf{u}}^n}{dt},\\textbf{v}_{h})+ c(e_\\textbf{u}^{n}, \\textbf{u}^{n,\\theta},\\textbf{v}_h) +\\\\\n& \\quad  c(\\textbf{u}^{n}, \\textbf{u}^{n,\\theta},\\textbf{v}_h)- c(e_\\textbf{u}^{n}, e_\\textbf{u}^{n,\\theta},\\textbf{v}_h)+  a_{NS}(e_{\\textbf{u}}^{n,\\theta},\\textbf{v}_h)\\}dt\n\\end{split}\n\\end{equation} \nNow applying $Cauchy-Schwarz$'s inequality, $Young$'s inequality, property \\textbf{(b)} of the $trilinear$ form $c(\\cdot, \\cdot, \\cdot)$ and the above result (113) on (116) we have\n\\begin{equation}\n\\sum_{n=0}^{N-1} \\int_{t^n}^{t^{n+1}} b(\\textbf{v}_h, E_p^{A,n,\\theta}) dt \\leq \\bar{C}'(\\textbf{R})(h^2+ dt^{2r}) \\|\\textbf{v}_h\\|_1\n\\end{equation}\nApplying this result on (75) we have\n\\begin{equation}\n\\|I_hp-p_h\\|^2_{L^2(L^2)} \\leq \\bar{C}''(\\textbf{R})(h^2+ dt^{2r})\n\\end{equation} \nNow combining the results obtained in the first and second part and applying interpolation estimate on pressure interpolation term $E^I_p$, we finally arrive at the following\n\\begin{equation}\n\\boxed{\\|\\textbf{u}-\\textbf{u}_h\\|^2_{\\tilde{\\textbf{V}}} + \\|p-p_h\\|_{L^2(L^2)}^2  + \\|c-c_h\\|^2_{\\tilde{\\textbf{V}}} \\leq \\bar{C}(\\textbf{R}) (h^2+ dt^{2r})}\n\\end{equation} \nNow this finally completes derivation of $aposteriori$ error estimation. \n\\end{proof}\n\\begin{remark}\nThese estimations clearly imply that the scheme is $first$ order convergent in space with respect to total norm, whereas in time it is $first$ order convergent for backward Euler time discretization scheme and $second$ order convergent for Crank-Nicolson method.\n\\end{remark}\n\n\\section{Numerical Experiment}\nIn this section we verify the credibility of $ASGS$ method for this coupled transient $Navier$-$Stokes$-$VADR$ model through several numerical examples. Here we present a comparative study between standard Galerkin and $ASGS$ finite element method. We have considered two broad cases based on one way coupling and two-way or strong coupling. First case is further divided into three sub-cases consisting of different values Reynolds number and in the later one the viscosity of the fluid is taken to be dependent upon concentration of the solute and variable diffusion coefficients have been considered. This case too consists of two sub-cases involving different viscosity coefficients. \\vspace{1mm}\\\\\nLet us take $\\Omega$ to be a square bounded domain (0,1) $\\times$ (0,1).   Piecewise continuous linear finite element(P1) space is considered for approximating velocity, pressure and concentration. Now renaming the error in the following way we have examined the performances of both Galerkin and $ASGS$ methods.  \n\\begin{center}\n$Total$ $error$ =  $\\{\\|\\textbf{u}-\\textbf{u}_h\\|^2_{\\tilde{\\textbf{V}}} + \\|p-p_h\\|_{L^2(L^2)}^2  + \\|c-c_h\\|^2_{\\tilde{\\textbf{V}}}\\}^{\\frac{1}{2}}$\n\\end{center}\n The exact solutions for all the cases are taken as follows: \\vspace{1mm}\\\\\n$\\textbf{u}=(e^{-t} x^2(x-1)^2y(y-1)(2y-1), -e^{-t}x(x-1)(2x-1)y^2(y-1)^2 )$, \\vspace{1mm} \\\\\n $p=e^{-t}(3x^2+3y^2-2)$ and $c=e^{-t} x y (x-1)(y-1)$ \\vspace{1mm} \\\\\n \n \\begin{table}[]\n\\centering\n\\begin{tabular}{| *{6}{c|} }\n    \\hline\n Time &  Grid     & \\multicolumn{2}{c|}{Galerkin method}\n            & \\multicolumn{2}{c|}{ASGS method}\\\\\n      \\hline       \n  step & size     & Total error & RoC & Total error & RoC \\\\\n \\hline\n0.1& 10 $\\times$ 10 & 0.158556 &  & 0.158435 & \\\\\n   \\hline\n 0.05&    20 $\\times$ 20 & 0.0833 & 0.928605  & 0.0833011 & 0.927481\\\\  \n     \\hline\n  0.025&   40 $\\times$ 40 & 0.0430609 & 0.95194  & 0.0430864 & 0.951103\\\\   \n    \\hline         \n 0.0125&   80 $\\times$ 80 & 0.0219347 & 0.973161  & 0.0219556 & 0.972645\\\\  \n     \\hline         \n 0.00625&   160 $\\times$ 160 & 0.0110526 & 0.98883  & 0.011068 & 0.988194\\\\    \n    \\hline      \n\\end{tabular}\n\\caption{Total error and Rate of convergence(RoC) under both Galerkin and $ASGS$ method for small Reynolds number(Re=50) at $T=1$}\n    \\end{table}\n \n  \\begin{table}[]\n\\centering\n\\begin{tabular}{| *{6}{c|} }\n    \\hline\n Time &  Grid     & \\multicolumn{2}{c|}{Galerkin method}\n            & \\multicolumn{2}{c|}{ASGS method}\\\\\n      \\hline       \n  step & size     & Total error & RoC & Total error & RoC \\\\\n \\hline\n0.1& 10 $\\times$ 10 & 0.170253 &  & 0.158437 & \\\\\n   \\hline\n 0.05&    20 $\\times$ 20 & 0.0871451 & 0.966187  & 0.0833212 & 0.92715\\\\  \n     \\hline\n  0.025&   40 $\\times$ 40 & 0.043821 & 0.991797  & 0.0431014 & 0.950949\\\\   \n    \\hline         \n 0.0125&   80 $\\times$ 80 & 0.022057 & 0.990389  & 0.0219237 & 0.975243\\\\  \n     \\hline         \n 0.00625&   160 $\\times$ 160 & 0.011189 & 0.979155  & 0.011076 & 0.985054\\\\    \n    \\hline      \n\\end{tabular}\n\\caption{Total error and Rate of convergence(RoC) under both Galerkin and $ASGS$ method for small Reynolds number(Re=500) at $T=1$}\n    \\end{table}\n \n \\begin{table}[]\n\\centering\n\\begin{tabular}{| *{6}{c|} }\n    \\hline\n Time &  Grid     & \\multicolumn{2}{c|}{Galerkin method}\n            & \\multicolumn{2}{c|}{ASGS method}\\\\\n      \\hline       \n  step & size     & Total error & RoC & Total error & RoC \\\\\n \\hline\n0.1& 10 $\\times$ 10 & 0.226209 &  & 0.158438 & \\\\\n   \\hline\n 0.05&    20 $\\times$ 20 & 0.164603 & 0.458663  & 0.0833293 & 0.927026\\\\  \n     \\hline\n  0.025&   40 $\\times$ 40 & 0.0822173 & 1.00148  & 0.0431091 & 0.950832\\\\   \n    \\hline         \n 0.0125&   80 $\\times$ 80 & 0.0310324 & 1.37098  & 0.0219345 & 0.974791\\\\  \n     \\hline         \n 0.00625&   160 $\\times$ 160 & 0.022146 & 0.486729  & 0.0110245 & 0.992489\\\\    \n    \\hline      \n\\end{tabular}\n\\caption{Total error and Rate of convergence(RoC) under both Galerkin and $ASGS$ method for small Reynolds number(Re=5000) at $T=1$}\n    \\end{table}\n \n  \\begin{table}[]\n\\centering\n\\begin{tabular}{| *{6}{c|} }\n    \\hline\n Time &  Grid     & \\multicolumn{2}{c|}{Galerkin method}\n            & \\multicolumn{2}{c|}{ASGS method}\\\\\n      \\hline       \n  step & size     & Total error & RoC & Total error & RoC \\\\\n \\hline\n0.1& 10 $\\times$ 10 & 0.159204 &  & 0.158826 & \\\\\n   \\hline\n 0.05&    20 $\\times$ 20 & 0.0834547 & 0.931812  & 0.0834583 & 0.928321\\\\  \n     \\hline\n  0.025&   40 $\\times$ 40 & 0.0430917 & 0.953584  & 0.043141 & 0.951997\\\\   \n    \\hline         \n 0.0125&   80 $\\times$ 80 & 0.021942 & 0.973715  & 0.0219748 & 0.973209\\\\  \n     \\hline         \n 0.00625&   160 $\\times$ 160 & 0.011122 & 0.980278  & 0.011022 & 0.993703\\\\    \n    \\hline      \n\\end{tabular}\n\\caption{Total error and Rate of convergence(RoC) under both Galerkin and $ASGS$ method for variable viscosity and diffusion coefficients (first sub-case) at $T=1$}\n    \\end{table}\n \n  \\begin{table}[]\n\\centering\n\\begin{tabular}{| *{6}{c|} }\n    \\hline\n Time &  Grid     & \\multicolumn{2}{c|}{Galerkin method}\n            & \\multicolumn{2}{c|}{ASGS method}\\\\\n      \\hline       \n  step & size     & Total error & RoC & Total error & RoC \\\\\n \\hline\n0.1& 10 $\\times$ 10 & 0.236613 &  & 0.161085 & \\\\\n   \\hline\n 0.05&    20 $\\times$ 20 & 0.201906 & 0.22884  & 0.0855817 & 0.912444\\\\  \n     \\hline\n  0.025&   40 $\\times$ 40 & 0.128248 & 0.654755  & 0.0445594 & 0.941572\\\\   \n    \\hline         \n 0.0125&   80 $\\times$ 80 & 0.0495898 & 1.37082  & 0.0227193 & 0.971811\\\\  \n     \\hline         \n 0.00625&   160 $\\times$ 160 & 0.041146 & 0.269291  & 0.011148 & 1.027133\\\\    \n    \\hline      \n\\end{tabular}\n\\caption{Total error and Rate of convergence(RoC) under both Galerkin and $ASGS$ method for variable viscosity and diffusion coefficients (second sub-case) at $T=1$}\n    \\end{table}\n\n$\\textbf{(I)First case:}$ Here we have considered constant viscosity coefficient and therefore the coupled system becomes an one-way coupling. The importance behind considering this case is here that we want to verify the performance of $ASGS$ method for different Reynolds number. Here diffusion coefficients are also taken constant. \\vspace{1mm}\\\\\n$\\textbf{(a)Small Reynolds number}$ The exact solutions remain same. The values of Reynolds number $Re$=50, diffusion coefficient $D$=2 and reaction coefficient $\\beta$=0.01. \\vspace{1mm}\\\\\nTable 1 presents total errors and rates of convergence (RoC) of the coupled system for this case under Galerkin and $ASGS$ methods for different time steps $dt$ and grid sizes. It is clearly seen that both Galerkin and $ASGS$ method performs equally well for small Reynolds number. We can conclude the order of convergence for each of the methods is 1.\\vspace{1mm}\\\\\n$\\textbf{(b) Medium Reynolds number:}$ For this case the values of coefficients are taken as $Re$=500, $D$=2 and $\\beta$=0.01. Similar to the previous case table 2  represents the total errors and rates of convergence of the coupled system for this case under Galerkin and $ASGS$ methods  for different time steps $dt$ and grid sizes. In this case though both Galerkin and $ASGS$ method perform equally well and retain the desired first order convergence, but total error obtained in $ASGS$ method is less compared to that of Galerkin method. \\vspace{1mm}\\\\\n$\\textbf{(c)Large Reynolds number}$ The values of the coefficients are considered as $Re$=5000, $D$=2 and $\\beta$=0.01. Table 3 presents the total errors and rates of convergence of the coupled system for this case under Galerkin and $ASGS$ methods. It can be observed that Galerkin method behaves in somewhat oscillatory manner and it is not possible to conclude a definite order of convergence for this case, whereas $ASGS$ method performs consistently well at every time steps and grid sizes and rate of convergence in this case again turns out to be 1. \\vspace{1mm}\\\\\n$\\textbf{(II)Second case:}$ Here we consider the viscosity to be dependent upon concentration and hence $Navier$-$Stokes$ and Transport equations are coupled in two-way manner. The proposed expression of concentration dependent viscosity is taken from \\cite{RefP} and depending upon different viscosity coefficients we have divided this case into two sub-cases. In this case we have considered variable diffusion coefficients as follows: \\vspace{1mm}\\\\\n$D_1$= $e^{-t}y^2(y-1)^2(2y-1)^2x^4(x-1)^4$ and $D_2$= $e^{-t}x^2(x-1)^2(2x-1)^2y^4(y-1)^4$ \\\\\nand the reaction coefficient $\\beta$=0.01.\\vspace{1mm}\\\\\n$\\textbf{(a) First sub-case:}$ The viscosity coefficient is $\\mu(c)=0.00954 e^{27.93 \\times 0.028 c}$. Table 4 presents the total errors and rates of convergence of the coupled system for this case under Galerkin and $ASGS$ methods for different time steps $dt$ and grid sizes. Both  Galerkin and $ASGS$ method performs equally well and order of convergence for both the methods is 1. \\vspace{1mm}\\\\\n$\\textbf{(b) Second sub-case:}$ Here we have considered slightly small viscosity coefficient $\\mu(c)=0.0000954 e^{27.93 \\times 0.028 c}$. Table 5 presents the total errors and rates of convergence of the coupled system for this case under Galerkin and $ASGS$ methods. This table shows that Galerkin method performs poorly, whereas the $ASGS$ method performs far better and obtains the desired first order convergence.\n\n\\section{Conclusion}\nThis paper presents algebraic $subgrid$ $multiscale$ stabilized finite element analysis of transient $Navier$-$Stokes$ fluid flow equation strongly coupled with unsteady $VADR$ transport problem. Consideration of concentration dependent viscosity makes this time dependent coupling more accurate to model real life based contemporary problems. To ensure the efficiency of the stabilized finite element method for this model, both $apriori$ and $aposteriori$ error estimates have been derived in detail. It is essential to mention that the norm employed for error estimation consists of the full norms corresponding to each variable belonging to their respective spaces. Therefore it provides a wholesome information about convergence of the method. Theoretically the rate of convergence for both $apriori$ and $aposteriori$ error estimations is $O(h)$ in space and first and second order convergences have come out  for two implicit time discretization schemes viz. backward Euler and Crank-Nicolson methods respectively. The accuracy of the stabilized method has been numerically tested through considering two different kind of examples and various possible combinations among them. Numerical results both in tabular and figure representations show better performance of the stabilized $ASGS$ method than standard $Galerkin$ finite element method and verify theoretically established results too.\\vspace{1cm}\\\\\n{\\large \\textbf{Acknowledgement}} \\vspace{2mm}\\\\\nThis work has been supported by grant from Innovation in Science Pursuit for Inspired Research (INSPIRE) programme sponsored and managed by the Department of Science and Technology(DST), Ministry of Science and Technology, Govt.of India.\n\n\n\n\n\n \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{intro}\n\nLet $M$ be a differentiable manifold.  The question of whether there is a `best'\nRiemannian metric on $M$ is intriguing.  A great deal of deep results in\nRiemannian geometry have been motivated, and even inspired, by this single natural\nquestion. For several good reasons, an Einstein metric is a good candidate, if\nnot the best, at least a very distinguished one (see \\cite[Chapter 0]{Bss}).  A\nRiemannian metric $g$ on $M$ is called {\\it Einstein} if its Ricci tensor $\\ricci_g$\nsatisfies\n\\begin{equation}\\label{eeq}\n\\ricci_g=cg, \\qquad \\mbox{for some}\\; c\\in\\RR.\n\\end{equation}\n\nThis notion can be traced back to \\cite{Hlb}, where Einstein metrics emerged as\ncritical points of the total scalar curvature functional on the space of all metrics\non $M$ of a given volume. Equation (\\ref{eeq}) is a non-linear second order PDE\n(recall that the number of parameters is $\\tfrac{n(n+1)}{2}$ on both sides,\n$n=\\dim{M}$), which also gives rise to some hope, but a good understanding of the\nsolutions in the general case seems far from being attained.  A classical reference for Einstein\nmanifolds is the book \\cite{Bss}, and some updated expository articles are\n\\cite{And}, \\cite{LbrWng}, \\cite[III,C.]{Brg1} and \\cite[11.4]{Brg2}.\n\nThe Einstein condition (\\ref{eeq}) is very subtle, even when restricted to almost\nany subclass of metrics on $M$ one may like.  It is too strong to allow general\nexistence results, and sometimes even just to find a single example, and at the same\ntime, it is too weak to get obstructions or classification results.\n\nBut maybe the difficulty comes from PDEs, so let us `algebrize' the problem\n(algebra is always easier for a geometer ...).  Let us consider homogeneous\nRiemannian manifolds.  Indeed, the Einstein equation for a homogeneous metric is\njust a system of $\\tfrac{n(n+1)}{2}$ algebraic equations, but unfortunately, a quite\ninvolved one, and the following main general question is still open in both compact\nand noncompact cases:\n\n\\begin{quote}\nWhich homogeneous spaces $G\/K$ admit a $G$-invariant Einstein Riemannian metric?\n\\end{quote}\n\nWe refer to \\cite{BhmWngZll} and the references therein for an update in the compact\ncase.  In the noncompact case, the only known examples until now are all of a very\nparticular kind; namely, simply connected solvable Lie groups endowed with a left\ninvariant metric (so called {\\it solvmanifolds}). According to the following long\nstanding conjecture, these might exhaust all the possibilities for noncompact\nhomogeneous Einstein manifolds.\n\n\\begin{quote}\n{\\bf Alekseevskii's conjecture} \\cite[7.57]{Bss}.  If $G\/K$ is a homogeneous\nEinstein manifold of negative scalar curvature then $K$ is a maximal compact\nsubgroup of $G$ (which implies that $G\/K$ is a solvmanifold when $G$ is a linear\ngroup).\n\\end{quote}\n\nThe conjecture is wide open, and it is known to be true only for $\\dim \\leq 5$, a\nresult which follows from the complete classification in these dimensions given in\n\\cite{Nkn}.  One of the most intriguing facts related to this conjecture, and maybe\nthe only reason so far to consider Alekseevskii's conjecture as too optimistic, is\nthat the Lie groups $\\Sl_n(\\RR)$, $n\\geq 3$, do admit left invariant metrics of\nnegative Ricci curvature, as well as does any complex simple Lie group (see\n\\cite{DttLt}, \\cite{DttLtMtl}).  However, an inspection of the eigenvalues of the\nRicci tensors in these examples shows that they are far from being close to each\nother, giving back some hope.\n\nLet us now consider the case of left invariant metrics on Lie groups.  Let $\\ggo$ be\na real Lie algebra.  Each basis $\\{ X_1,...,X_n\\}$ of $\\ggo$ determines structural\nconstants $\\{ c_{ij}^k\\}\\subset\\RR$ given by\n$$\n[X_i,X_j]=\\sum_{k=1}^nc_{ij}^kX_k, \\qquad 1\\leq i,j\\leq n.\n$$\nThe left invariant metric on any Lie group with Lie algebra $\\ggo$ defined by the\ninner product given by $\\la X_i,X_j\\ra=\\delta_{ij}$ is Einstein if and only if the\n$\\tfrac{n^2(n+1)}{2}$ numbers $c_{ij}^k$'s satisfy the following $\\tfrac{n(n+1)}{2}$\nalgebraic equations for some $c\\in\\RR$:\n\\begin{equation}\\label{condi}\n\\sum_{kl}-\\unm c_{ik}^lc_{jk}^l +\\unc c_{kl}^ic_{kl}^j-\\unm c_{jk}^lc_{il}^k+\\unm\nc_{kl}^lc_{ki}^j +\\unm c_{kl}^lc_{kj}^i = c\\delta_{ij}, \\quad 1\\leq i\\leq j\\leq n.\n\\end{equation}\n\nIn view of this, one may naively think that the classification of Einstein left\ninvariant metrics on Lie groups is at hand.  However, the following natural\nquestions remain open:\n\n\\begin{itemize}\n\\item[ ]\n\\item[(i)] Is any Lie group admitting an Einstein left invariant metric either solvable or compact?\n\n\\item[ ]\\item[(ii)] Does every compact Lie group admit only finitely many Einstein left invariant metrics up to isometry and scaling?\n\n\\item[ ]\\item[(iii)]  Which solvable Lie groups admit an Einstein left invariant metric?\n\n\\item[ ]\n\\end{itemize}\n\nWe note that question (i) is just Alekseevskii Conjecture restricted to Lie groups,\nand question (ii) is contained in \\cite[7.55]{Bss}.  The only group for which the\nanswer to (ii) is known is $\\SU(2)$, where there is only one (see \\cite{Mln}).  For\nmost of the other compact simple Lie groups many Einstein left invariant metrics\nother than minus the Killing form are explicitly known (see \\cite{DtrZll}).\n\nEven if one is very optimistic and believes that Alekseevskii Conjecture is true, a\nclassification of Einstein metrics in the noncompact homogeneous case will depend on\nsome kind of answer to question (iii).  The aim of this expository paper is indeed\nto give a report on the present status of the study of Einstein solvmanifolds.\n\nPerhaps the main difficulty in trying to decide if a given Lie algebra $\\ggo$ admits\nan Einstein inner product is that one must check condition (\\ref{condi}) for any\nbasis of $\\ggo$, and there are really too many of them.  In other words, there are\ntoo many left invariant metrics on a given Lie group, any inner product on the\nvector space $\\ggo$ is playing. This is quite in contrast to what happens in\nhomogeneous spaces $G\/K$ with not many different $\\Ad(K)$-irreducible components in\nthe decomposition of the tangent space $\\tang_{eK}(G\/K)$.  Another obstacle is how to\nrecognize your Lie algebra by just looking at the structural constants $c_{ij}^k$'s.\nThough even in the case when we have two solutions to (\\ref{condi}), and we know they define\nthe same Lie algebra, to be able to guarantee that they are not isometric, i.e. that we\nreally have two Einstein metrics, is usually involved.\n\nIf we fix a basis $\\{ X_1,...,X_n\\}$ of $\\ggo$, then instead of varying all possible\nsets of structural constants $\\{ c_{ij}^k\\}$'s by running over all bases, one may\nact on the Lie bracket $\\lb$ by $g.\\lb=g[g^{-1}\\cdot,g^{-1}\\cdot]$, for any\n$g\\in\\Gl(\\ggo)$, and look at the structural constants of $g.\\lb$ with respect to the\nfixed basis $\\{ X_1,...,X_n\\}$.  This give rises to an orbit $\\Gl(\\ggo).\\lb$ in the\nvector space $V:=\\Lambda^2\\ggo^*\\otimes\\ggo$ of all skew-symmetric bilinear maps\nfrom $\\ggo\\times\\ggo$ to $\\ggo$, which parameterizes, from a different point of\nview, the set of all inner products on $\\ggo$. Indeed, if $\\ip$ is the inner product\ndefined by $\\la X_i,X_j\\ra=\\delta_{ij}$ then\n\n\\begin{quote}\n$(\\ggo,g.\\lb,\\ip)$ is isometric to $(\\ggo,\\lb,\\la g\\cdot,g\\cdot\\ra)$ for any\n$g\\in\\Gl(\\ggo)$.\n\\end{quote}\n\nThe subset $\\lca\\subset V$ of those elements satisfying the Jacobi condition is\nalgebraic, $\\Gl(\\ggo)$-invariant and the $\\Gl(\\ggo)$-orbits in $\\lca$ are precisely\nthe isomorphism classes of Lie algebras.  $\\lca$ is called the {\\it variety of Lie\nalgebras}. Furthermore, if $\\Or(\\ggo)\\subset\\Gl(\\ggo)$ denotes the subgroup of\n$\\ip$-orthogonal maps, then two points in $\\Gl(\\ggo).\\lb$ which lie in the same\n$\\Or(\\ggo)$-orbit determine isometric left invariant metrics, and the converse holds\nif $\\ggo$ is completely solvable (see \\cite{Alk2}).\n\nThis point of view is certainly a rather tempting invitation to try to use geometric\ninvariant theory in any problem which needs a running over all left invariant\nmetrics on a given Lie group, or even on all Lie groups of a given dimension.  We\nshall see throughout this article that indeed, starting in \\cite{Hbr}, the approach `by varying\nLie brackets' has been very fruitful in the study of Einstein solvmanifolds\nduring the last decade.\n\nThe latest fashion generalization of Einstein metrics, although they were introduced\nby R. Hamilton more than twenty years ago, is the notion of {\\it Ricci soliton}:\n\\begin{equation}\\label{rseq}\n\\ricci_g=cg+L_Xg, \\qquad\\mbox{for some}\\; c\\in\\RR, \\quad X\\in\\chi(M),\n\\end{equation}\n\n\\noindent where $L_Xg$ is the usual Lie derivative of $g$ in the direction of the\nfield $X$.  A more intuitive equivalent condition to (\\ref{rseq}) is that $\\ricci_g$\nis tangent at $g$ to the space of all metrics which are homothetic to $g$ (i.e.\nisometric up to a constant scalar multiple).  Recall that Einstein means\n$\\ricci_{g}$ tangent to $\\RR_{>0}g$.  Ricci solitons correspond to solutions of the\nRicci flow\n$$\n\\ddt g(t)=-2\\ricci_{g(t)},\n$$\nthat evolves self similarly, that is, only by scaling and the action by\ndiffeomorphisms, and often arise as limits of dilations of singularities of the\nRicci flow.  We refer to \\cite{soliton}, \\cite{GntIsnKnp}, \\cite{libro} and the\nreferences therein for further information on the Hamilton-Perelman theory of Ricci\nflow and Ricci solitons and the role played by nilpotent Lie groups in the story.\n\nA remarkable fact is that if $S$ is an Einstein solvmanifold, then the metric\nrestricted to the submanifold $N:=[S,S]$ is a Ricci soliton, and conversely, any\nRicci soliton left invariant metric on a nilpotent Lie group $N$ (called {\\it\nnilsolitons}) can be uniquely `extended' to an Einstein solvmanifold.  This\none-to-one correspondence is complemented with the uniqueness up to isometry of\nnilsolitons, which finally turns the classification of Einstein solvmanifolds into a\nclassification problem on nilpotent Lie algebras.  These are not precisely good\nnews.  Historically, as the literature and experience shows us, any classification\nproblem involving nilpotent Lie algebras is simply a headache.\n\n\\vs \\noindent {\\it Acknowledgements.}  I am grateful to the Scientific\nCommittee for the invitation to give a talk at the `Sixth Workshop on Lie Theory and\nGeometry', November 13-17, 2007, Cruz Chica, C\\'ordoba, Argentina.  I wish to thank Yuri Nikolayevsky and Cynthia Will for very useful comments on a first version of the paper, and to Roberto Miatello for going over the manuscript.  I also wish to express my gratitude to the young collaborators Alejandra \\'Alvarez, Adri\\'an\nAndrada, Gast\\'on Garc\\'{\\i}a and Emilio Lauret for the invaluable help they generously provided to the organization of the workshop.\n\n\n\n\n\n\n\n\n\n\n\\section{Structure and uniqueness results on Einstein solvmanifolds}\\label{pre}\n\nA {\\it solvmanifold} is a simply connected solvable Lie group $S$ endowed with a\nleft invariant Riemannian metric.  A left invariant metric on a Lie group $G$ will\nbe always identified with the inner product $\\ip$ determined on the Lie algebra\n$\\ggo$ of $G$, and the pair $(\\ggo,\\ip)$ will be referred to as a {\\it metric Lie\nalgebra}.  If $S$ is a solvmanifold and $(\\sg,\\ip)$ is its metric solvable Lie\nalgebra, then we consider the $\\ip$-orthogonal decomposition\n$$\n\\sg=\\ag\\oplus\\ngo,\n$$\nwhere $\\ngo:=[\\sg,\\sg]$ is the derived algebra (recall that $\\ngo$ is nilpotent).\n\n\\begin{definition}\\label{stan}\nA solvmanifold $S$ is said to be {\\it standard} if\n$$\n[\\ag,\\ag]=0.\n$$\n\\end{definition}\n\nThis is a very simple algebraic condition, which may appear as kind of technical,\nbut it has nevertheless played an important role in many questions in homogeneous\nRiemannian geometry:\n\n\\begin{itemize}\n\\item \\cite{GndPttVnb} K$\\ddot{{\\rm a}}$hler-Einstein noncompact homogeneous manifolds are all standard solvmanifolds.\n\n\\item\\cite{Alk,Crt} Every quaternionic K$\\ddot{{\\rm a}}$hler solvmanifold (completely real) is standard.\n\n\\item \\cite{AznWls} Any homogeneous manifold of nonpositive sectional curvature is a standard solvmanifold.\n\n\\item \\cite{Hbr2} All harmonic noncompact homogeneous manifolds are standard solvmanifolds (with $\\dim{\\ag}=1$).\n\\end{itemize}\n\nPartial results on the question of whether Einstein solvmanifolds are all standard\nwere obtained for instance in \\cite{Hbr} and \\cite{Sch}, who gave several sufficient\nconditions. The answer was known to be yes in dimension $\\leq 6$ (see \\cite{NktNkn})\nand followed from a complete classification of Einstein solvmanifolds in these\ndimensions.  On the other hand, it is proved in \\cite{Nkl0} that many classes of\nnilpotent Lie algebras can not be the nilradical of a non-standard Einstein\nsolvmanifold.\n\n\n\\begin{theorem}\\cite{standard}\\label{stand}\nAny Einstein solvmanifold is standard.\n\\end{theorem}\n\nAn idea of the proof of this theorem will be given in Section \\ref{proof}.  Standard\nEinstein solvmanifolds were extensively investigated in \\cite{Hbr}, where the\nremarkable structural and uniqueness results we next describe are derived.  Recall\nthat combined with Theorem \\ref{stand}, all of these results are now valid for any\nEinstein solvmanifold.\n\n\\begin{theorem}\\label{u}\\cite[Section 5]{Hbr} {\\rm (}{\\bf Uniqueness}{\\rm )}\nA simply connected solvable Lie group admits at most one standard Einstein left\ninvariant metric up to isometry and scaling.\n\\end{theorem}\n\nA more general result is actually valid: if a noncompact homogeneous manifold $G\/K$\nwith $K$ maximal compact in $G$ admits a $G$-invariant metric $g$ isometric to an\nEinstein solvmanifold, then $g$ is the unique $G$-invariant Einstein metric on $G\/K$\nup to isometry and scaling.  This is in contrast to the compact homogeneous case,\nwhere many pairwise non isometric $G$-invariant Einstein metrics might exist (see\n\\cite{BhmWngZll} and the references therein), although it is open if only finitely\nmany (see \\cite[7.55]{Bss}).\n\nIn the study of Einstein homogeneous manifolds, the compact case is characterized by\nthe positivity of the scalar curvature and Ricci flat implies flat (see\n\\cite{AlkKml}).  The following conditions on an Einstein solvmanifold $S$ are\nequivalent:\n\n\\begin{itemize}\n\\item[(i)] $\\sg$ is unimodular (i.e. $\\tr{\\ad{X}}=0$ for all $X\\in\\sg$).\n\n\\item[(ii)] $S$ is Ricci flat (i.e. $\\scalar(S)=0$).\n\n\\item[(iii)] $S$ is flat.\n\\end{itemize}\n\nWe can therefore consider from now on only nonunimodular solvable Lie algebras.\n\n\\begin{theorem}\\label{ror}\\cite[Section 4]{Hbr} {\\rm (}{\\bf Rank-one reduction}{\\rm )}\nLet $\\sg=\\ag\\oplus\\ngo$ be a nonunimodular solvable Lie algebra endowed with a\nstandard Einstein inner product $\\ip$, say with $\\ricci_{\\ip}=c\\ip$.  Then $c<0$\nand, up to isometry, it can be assumed that $\\ad{A}$ is symmetric for any $A\\in\\ag$.\nIn that case, the following conditions hold.\n\\begin{itemize}\n\\item[(i)] There exists $H\\in\\ag$ such that the eigenvalues of $\\ad{H}|_{\\ngo}$ are all positive integers without a common divisor.\n\n\\item[(ii)] The restriction of $\\ip$ to the solvable Lie algebra $\\RR H\\oplus\\ngo$ is also Einstein.\n\n\\item[(iii)] $\\ag$ is an abelian algebra of symmetric derivations of $\\ngo$ and the inner product on $\\ag$ must be given by $\\la A,A'\\ra=-\\tfrac{1}{c}\\tr{\\ad{A}\\ad{A'}}$ for all $A,A'\\in\\ag$.\n\\end{itemize}\n\\end{theorem}\n\nThe Ricci tensor for these solvmanifolds has the following simple formula.\n\n\\begin{lemma}\\label{fr}\nLet $S$ be a standard solvmanifold such that $\\ad{A}$ is symmetric and nonzero for\nany $A\\in\\ag$. Then the Ricci tensor of $S$ is given by\n\\begin{itemize}\n\\item[(i)] $\\ricci(A,A')=-\\tr{\\ad{A}\\ad{A'}}$ for all $A,A'\\in\\ag$.\n\n\\item[(ii)] $\\ricci(\\ag,\\ngo)=0$.\n\n\\item[(iii)] $\\ricci(X,Y)=\\ricci_{\\ngo}(X,Y) -\\la\\ad{H}(X),Y\\ra$, for all $X,Y\\in\\ngo$, where $\\ricci_{\\ngo}$ is the Ricci tensor of $(\\ngo,\\ip|_{\\ngo\\times\\ngo})$ and $H\\in\\ag$ is defined by $\\la H,A\\ra=\\tr{\\ad{A}}$ for any $A\\in\\ag$.\n\\end{itemize}\n\\end{lemma}\n\nThe natural numbers which have appeared as the eigenvalues of $\\ad{H}$ when\n$(\\sg,\\ip)$ is Einstein play a very important role.\n\n\\begin{definition}\\label{et}\nIf $d_1,...,d_r$ denote the corresponding multiplicities of the positive integers\nwithout a common divisor $k_1<...<k_r$ given by Theorem \\ref{ror}, (i), then the\ntuple\n$$\n(k;d)=(k_1<...<k_r;d_1,...,d_r)\n$$\nis called the {\\it eigenvalue type} of the Einstein solvmanifold $(\\sg,\\ip)$.\n\\end{definition}\n\nWe find here the first obstruction: if a solvable Lie algebra admits an Einstein\ninner product then the nilpotent Lie algebra $\\ngo=[\\sg,\\sg]$ is $\\NN$-{\\it graded},\nthat is, there is a decomposition $\\ngo=\\ngo_1\\oplus...\\oplus\\ngo_r$ such that\n$[\\ngo_i,\\ngo_j]\\subset\\ngo_{i+j}$ for all $i,j$ (recall that some of the $\\ngo_i$'s\nmight be trivial).  This is precisely the decomposition into eigenspaces of the derivation\nwith positive integer eigenvalues $\\ad{H}$.  Another important consequence of\nTheorem \\ref{ror} is that to study Einstein solvmanifolds, it will be enough to consider\n{\\it rank-one} (i.e. $\\dim{\\ag}=1$) metric solvable Lie algebras, since every higher\nrank Einstein solvmanifold will correspond to a unique rank-one Einstein\nsolvmanifold and certain abelian subalgebra $\\ag$ of derivations of $\\ngo$\ncontaining $\\ad{H}$.  Recall that how to extend the inner product is determined by\nTheorem \\ref{ror}, (iii).\n\nLet $\\mca$ be the moduli space of all the isometry classes of Einstein solvmanifolds\nof a given dimension with scalar curvature equal to $-1$, endowed with the\n$C^{\\infty}$-topology.  Notice that any $n$-dimensional solvmanifold $S$ is\ndiffeomorphic to the euclidean space $\\RR^n$, and so any $S$ can be viewed as a\nRiemannian metric on $\\RR^n$ (which is in addition invariant by some transitive\nsolvable Lie group of diffeomorphisms of $\\RR^n$).\n\n\\begin{theorem}\\label{m}\\cite[Section 6]{Hbr} {\\rm (}{\\bf Moduli space}{\\rm )}\nIn every dimension, only finitely many eigenvalue types occur, and each eigenvalue\ntype $(k;d)$ determines a compact path connected component $\\mca_{(k;d)}$ of $\\mca$,\nhomeomorphic to a real semialgebraic set.\n\\end{theorem}\n\nResults on the topology and `dimension' of the moduli spaces $\\mca_{(k;d)}$ near a\nrank-one symmetric space are obtained in \\cite[Section 6.5]{Hbr}.  This has also\nbeen done for many other symmetric spaces (rank $\\geq 2$) in \\cite{GrdKrr}, where\neven explicit examples are exhibited to describe a neighborhood.  The moduli spaces\n$\\mca_{(1<2;q,p)}$ are studied in detail in \\cite{Ebr} and \\cite{Nkl3}.\n\nIn the light of Theorem \\ref{ror} and Lemma \\ref{fr}, it is reasonable to expect\nthat Einstein solvmanifolds are actually completely determined by their nilpotent\nparts.  Let us describe this more precisely.\n\n\\begin{definition}\nGiven a metric nilpotent Lie algebra $(\\ngo,\\ip)$, a metric solvable Lie algebra\n$(\\sg=\\ag\\oplus\\ngo,\\ip')$ is called a {\\it metric solvable extension} of\n$(\\ngo,\\ip)$ if $\\ngo$ is an ideal of $\\sg$, $[\\ag,\\ag]\\subset\\ngo$ and\n$\\ip'|_{\\ngo\\times\\ngo}=\\ip$.\n\\end{definition}\n\nIt turns out that for each $(\\ngo,\\ip)$ there exists a unique rank-one metric\nsolvable extension of $(\\ngo,\\ip)$ which stands a chance of being an Einstein\nmanifold.  Indeed, since the inner product on $\\ag$ is determined by Theorem\n\\ref{ror}, (iii), the only datum we need to recover $(\\sg,\\ip')$ from $(\\ngo,\\ip)$\nis the way $\\ag$ is acting on $\\ngo$ by derivations.  Recall that $\\dim{\\ag}=1$.  It\nfollows from Lemma \\ref{fr} that if $A\\in\\ag$ satisfies $||A||^2=\\tr{\\ad{A}}$ and we\nset $D:=\\ad{A}|_{\\ngo}$, then the Ricci operator $\\Ric_{\\ip}$ of $(\\ngo,\\ip)$ equals\n$$\n\\Ric_{\\ip}=cI+D.\n$$\nBut $\\Ric_{\\ip}$ is orthogonal to $D$ and actually to any symmetric derivation of\n$(\\ngo,\\ip)$ (see \\cite[(2)]{critical}), thus\n\\begin{equation}\\label{pE}\n\\tr{DA}=-c\\tr{A}, \\qquad c=\\tfrac{\\tr{\\Ric_{\\ip}^2}}{\\tr{\\Ric_{\\ip}}},\n\\end{equation}\n\n\\noindent for any symmetric derivation $A$ of $(\\ngo,\\ip)$.  This determines $D$ in\nterms of $(\\ngo,\\ip)$ (recall that if $\\ngo$ is nonabelian then $\\tr{\\Ric_{\\ip}}<0$,\nand in the abelian case both $\\Ric_{\\ip}$ and $D$ equal zero).  We have therefore\nseen that a rank-one Einstein solvmanifold is completely determined by its (metric)\nnilpotent part.  This fact turns the study of rank-one Einstein solvmanifolds into a\nproblem on nilpotent Lie algebras.\n\n\\begin{definition}\\label{enil}\nA nilpotent Lie algebra $\\ngo$ is said to be an {\\it Einstein nilradical} if it\nadmits an inner product $\\ip$ such that there is a metric solvable extension of\n$(\\ngo,\\ip)$ which is Einstein.\n\\end{definition}\n\nSuch a solvable extension must satisfy $[\\ag,\\ag]=0$ by Theorem \\ref{stand} and it\ncan be assumed that $\\dim{\\ag}=1$ by Theorem \\ref{ror}, (ii).  In other words,\nEinstein nilradicals are precisely the {\\it nilradicals} (i.e. the maximal nilpotent\nideal) of Lie algebras of Einstein solvmanifolds.\n\nGiven a nilpotent Lie algebra $\\ngo$, an inner product $\\ip$ on $\\ngo$ can be\nextended to construct an Einstein solvmanifold if and only if any of the following\nequivalent conditions hold, which shows us that these left invariant metrics on\nnilpotent Lie groups are very special from many other points of view:\n\n\\begin{itemize}\n\\item[ ]\n\\item[(i)] $\\Ric_{\\ip}=cI+D$ for some $c\\in\\RR$ and\n$D\\in\\Der(\\ngo)$, where $\\Ric_{\\ip}$ is the Ricci operator of $(\\ngo,\\ip)$ and\n$\\Der(\\ngo)$ is the space of all derivations of $\\ngo$.\n\n\\item[ ]\n\\item[(ii)] $\\ip$ is a {\\it Ricci soliton} metric: the solution $\\ip_t$\nwith initial point $\\ip_0=\\ip$ to the Ricci flow\n$$\n\\ddt\\ip_t=-2\\ricci_{\\ip_t},\n$$\nremains isometric up to scaling to $\\ip$, that is, $\\ip_t=c_t\\vp_t^*\\ip$ for some\none parameter group of diffeomorphisms $\\{\\vp_t\\}$ of $N$ and $c_t\\in\\RR$ (see for\ninstance \\cite{libro}), where $N$ denotes the simply connected nilpotent Lie group\nwith Lie algebra $\\ngo$.\n\n\\item[ ]\n\\item[(iii)] $\\ip$ is a {\\it quasi-Einstein} metric:\n$$\n\\ricci_{\\ip}=c\\ip+L_X\\ip\n$$\nfor some $C^{\\infty}$ vector field $X$ in $N$ and $c\\in\\RR$, where $L_X\\ip$ denotes\nthe usual Lie derivative.  This class of metrics were actually first considered in\ntheoretical physics (see \\cite{Frd} and \\cite{ChvVln}).\n\n\\item[ ]\n\\item[(iv)] $\\ip$ is a {\\it minimal} metric:\n$$\n||\\ricci_{\\ip}||=\\min \\left\\{ ||\\ricci_{\\ip'}|| :\n\\scalar(\\ip')=\\scalar(\\ip)\\right\\},\n$$\nwhere $\\ip'$ runs over all left invariant metrics on $N$ and $\\scalar(\\ip)$ denotes\nscalar curvature of $(\\ngo,\\ip)$ (see \\cite{praga}).  A nilpotent Lie group $N$ can\nnever admit an Einstein left invariant metric, unless it is abelian, and a way of\ngetting as close as possible to satisfy the Einstein condition is to have a minimal\nmetric. Indeed,\n$$\n||\\ricci_{\\ip}-\\tfrac{\\scalar(\\ip)}{n}\\ip||^2=||\\ricci_{\\ip}||^2\n-\\tfrac{\\scalar(\\ip)^2}{n}.\n$$\n\n\\item[ ]\n\\end{itemize}\n\n\n\\begin{definition}\\label{nilsoliton}\nA left invariant metric on a nilpotent Lie group (or equivalently an inner product\non a nilpotent Lie algebra) is called a {\\it nilsoliton} if it satisfies any of the\nabove conditions (i)-(iv).\n\\end{definition}\n\n\\begin{theorem}\nA nilpotent Lie algebra $\\ngo$ is an Einstein nilradical if and only if $\\ngo$\nadmits a nilsoliton metric.\n\\end{theorem}\n\nIt is then reasonable to expect that uniqueness for nilsolitons should hold, as it\ndoes for standard Einstein solvmanifolds (see Theorem \\ref{u}).  This is actually\ntrue, and even a very similar proof worked out.\n\n\\begin{theorem}\\label{un}\\cite{soliton}\nThere is at most one nilsoliton metric on a nilpotent Lie group up to isometry and\nscaling.\n\\end{theorem}\n\nWe therefore obtain the following picture of one-to-one correspondences on the\nclassification problem:\n\n$$\n\\begin{array}{c}\n\\hline \\\\ \\\\\n\n\\left\\{\\mbox{Rank-one Einstein solvmanifolds}\\right\\} \/ \\mbox{isometry and scaling} \\\\ \\\\\n\\updownarrow \\\\ \\\\\n\n\\left\\{\\mbox{Ricci soliton (simply connected) nilmanifolds}\\right\\} \/ \\mbox{isometry and scaling} \\\\ \\\\\n\\updownarrow \\\\ \\\\\n\n\\left\\{\\mbox{Einstein nilradicals}\\right\\} \/ \\mbox{isomorphism} \\\\ \\\\\n\\hline \\\\\n\\end{array}\n$$\n\nThus the classification of Einstein solvmanifolds reduces to a completely\n`algebraic' problem; namely, the classification of nilpotent Lie algebras which are\nEinstein nilradicals. This problem will be treated in Section \\ref{clas}.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Technical background}\\label{Tb}\n\nIn this section, we fix the notation and give all the definitions and elementary\nresults we need to use throughout the paper.  All this mainly concerns the vector\nspace where the Lie algebras of a given dimension live and the action determining\nthe isomorphism relation between them.\n\nLet us consider the space of all skew-symmetric algebras of dimension $n$, which is\nparameterized by the vector space\n$$\nV=\\lam=\\{\\mu:\\RR^n\\times\\RR^n\\longrightarrow\\RR^n : \\mu\\; \\mbox{bilinear and\nskew-symmetric}\\}.\n$$\nThen\n$$\n\\nca=\\{\\mu\\in V:\\mu\\;\\mbox{satisfies Jacobi and is nilpotent}\\}\n$$\nis an algebraic subset of $V$ as the Jacobi identity and the nilpotency condition\ncan both be written as zeroes of polynomial functions.  $\\nca$ is often called the\n{\\it variety of nilpotent Lie algebras} (of dimension $n$).  There is a natural\naction of $\\G$ on $V$ given by\n\\begin{equation}\\label{action}\ng.\\mu(X,Y)=g\\mu(g^{-1}X,g^{-1}Y), \\qquad X,Y\\in\\RR^n, \\quad g\\in\\G,\\quad \\mu\\in V.\n\\end{equation}\n\nRecall that $\\nca$ is $\\G$-invariant and the Lie algebra isomorphism classes are\nprecisely the $\\G$-orbits. The action of $\\g$ on $V$ obtained by differentiation of\n(\\ref{action}) is given by\n\\begin{equation}\\label{actiong}\n\\pi(\\alpha)\\mu=\\alpha\\mu(\\cdot,\\cdot)-\\mu(\\alpha\\cdot,\\cdot)-\\mu(\\cdot,\\alpha\\cdot),\n\\qquad \\alpha\\in\\g,\\quad\\mu\\in V.\n\\end{equation}\n\n\\noindent We note that $\\pi(\\alpha)\\mu=0$ if and only if $\\alpha\\in\\Der(\\mu)$, the\nLie algebra of derivations of the algebra $\\mu$, which is actually the Lie algebra\nof $\\Aut(\\mu)$, the group of automorphisms of the algebra $\\mu$.  Recall that\n$\\Aut(\\mu)$ is the isotropy subgroup at $\\mu$ for the action (\\ref{action}), and so\n$\\dim{\\G.\\mu}=n^2-\\dim{Der(\\mu)}$.\n\nThe canonical inner product $\\ip$ on $\\RR^n$ determines an $\\Or(n)$-invariant inner\nproduct on $V$, also denoted by $\\ip$, as follows:\n\\begin{equation}\\label{innV}\n\\la\\mu,\\lambda\\ra= \\sum\\limits_{ij}\\la\\mu(e_i,e_j),\\lambda(e_i,e_j)\\ra\n=\\sum\\limits_{ijk}\\la\\mu(e_i,e_j),e_k\\ra\\la\\lambda(e_i,e_j),e_k\\ra,\n\\end{equation}\n\n\\noindent and also the standard $\\Ad(\\Or(n))$-invariant inner product on $\\g$ given\nby\n\\begin{equation}\\label{inng}\n\\la \\alpha,\\beta\\ra=\\tr{\\alpha \\beta^{\\mathrm t}}=\\sum_{i}\\la\\alpha e_i,\\beta e_i\\ra\n=\\sum_{ij}\\la\\alpha e_i,e_j\\ra\\la\\beta e_i,e_j\\ra,, \\qquad \\alpha,\\beta\\in\\g.\n\\end{equation}\n\n\\begin{remark} We have made several abuses of notation concerning inner products.  Recall\nthat $\\ip$ has been used to denote an inner product on $\\sg$, $\\ngo$, $\\RR^n$, $V$\nand $\\g$.\n\\end{remark}\n\nWe note that $\\pi(\\alpha)^t=\\pi(\\alpha^t)$ and $(\\ad{\\alpha})^t=\\ad{\\alpha^t}$ for\nany $\\alpha\\in\\g$, due to the choice of canonical inner products everywhere.\n\nLet $\\tg$ denote the set of all diagonal $n\\times n$ matrices.  If $\\{\ne_1',...,e_n'\\}$ is the basis of $(\\RR^n)^*$ dual to the canonical basis $\\{\ne_1,...,e_n\\}$ then\n$$\n\\{ v_{ijk}=(e_i'\\wedge e_j')\\otimes e_k : 1\\leq i<j\\leq n, \\; 1\\leq k\\leq n\\}\n$$\nis a basis of weight vectors of $V$ for the action (\\ref{action}), where $v_{ijk}$\nis actually the bilinear form on $\\RR^n$ defined by\n$v_{ijk}(e_i,e_j)=-v_{ijk}(e_j,e_i)=e_k$ and zero otherwise.  The corresponding\nweights $\\alpha_{ij}^k\\in\\tg$, $i<j$, are given by\n\\begin{equation}\\label{alfas}\n\\pi(\\alpha)v_{ijk}=(a_k-a_i-a_j)v_{ijk}=\\la\\alpha,\\alpha_{ij}^k\\ra v_{ijk},\n\\quad\\forall\\alpha=\\left[\\begin{smallmatrix} a_1&&\\\\ &\\ddots&\\\\ &&a_n\n\\end{smallmatrix}\\right]\\in\\tg,\n\\end{equation}\n\n\\noindent where $\\alpha_{ij}^k=E_{kk}-E_{ii}-E_{jj}$ and $\\ip$ is the inner product\ndefined in (\\ref{inng}).  As usual $E_{rs}$ denotes the matrix whose only nonzero\ncoefficient is $1$ at entry $rs$.  The structural constants $\\mu_{ij}^k$ of an\nalgebra $\\mu\\in V$ are then given by $$ \\mu(e_i,e_j)=\\sum_{k=1}^n\\mu_{ij}^ke_k,\n\\qquad\\mbox{or}\\quad \\mu=\\sum\\mu_{ij}^kv_{ijk}, \\quad i<j.\n$$\nLet $\\tg^+$ denote the Weyl chamber of $\\g$ given by\n\\begin{equation}\\label{weyl}\n\\tg^+=\\left\\{\\left[\\begin{smallmatrix} a_1&&\\\\ &\\ddots&\\\\ &&a_n\n\\end{smallmatrix}\\right]\\in\\tg:a_1\\leq...\\leq a_n\\right\\}.\n\\end{equation}\n\nFor $\\alpha\\in\\tg^+$ we define the parabolic subgroup\n\\begin{equation}\\label{para}\nP_{\\alpha}=B\\G_{\\alpha},\n\\end{equation}\n\n\\noindent where $B$ is the subgroup of $\\G$ of all lower triangular invertible\nmatrices and\n$$\n\\G_{\\alpha}=\\{ g\\in\\G:g\\alpha g^{-1}=\\alpha\\}.\n$$\nIn general, for any $\\alpha'\\in\\g$ which is diagonalizable over $\\RR$, we let\n$P_{\\alpha'}:=gP_{\\alpha}g^{-1}$ if $\\alpha'=g\\alpha g^{-1}$, $\\alpha\\in\\tg^+$. This\nis well defined since $h\\alpha h^{-1}=g\\alpha g^{-1}$ implies that\n$h^{-1}g\\in\\G_{\\alpha}\\subset P_{\\alpha}$ and so\n$h^{-1}gP_{\\alpha}g^{-1}h=P_{\\alpha}$.\n\nThere is an ordered basis of $V$ with respect to which the action of $g$ on $V$ is\nlower triangular for any $g\\in B$, and furthermore the eigenvalues of $\\pi(\\alpha)$\nare increasing for any $\\alpha\\in\\tg^+$.\n\nGiven a finite subset $X$ of $\\tg$, we denote by $\\CH(X)$ the convex hull of $X$ and\nby $\\mcc(X)$ the {\\it minimal convex combination of} $X$, that is, the (unique)\nvector of minimal norm (or closest to the origin) in $\\CH(X)$.  If\n$X=\\{\\alpha_1,...,\\alpha_r\\}\\subset\\tg$ and $\\beta:=\\mcc(X)$, then there exist\n$c_i\\geq 0$, $i=1,...,r$, such that $\\sum\\limits_{i=1}^rc_i=1$ and\n$\\beta=\\sum\\limits_{i=1}^rc_i\\alpha_i$. Since $\\la\\beta,\\alpha_i\\ra\\geq ||\\beta||^2$\nfor all $i$ (why?), we have that\n$$\n||\\beta||^2=\\sum_{i=1}^rc_i\\la\\beta,\\alpha_i\\ra\\geq ||\\beta||^2,\n$$\nfrom which follows that $\\la\\beta,\\alpha_i\\ra=||\\beta||^2$ for all $i$ such that $c_i>0$.  We\ncan therefore assume that $\\la\\beta,\\alpha_i\\ra=||\\beta||^2$ for all $i$, and also\nthat $\\beta=\\mcc(\\{\\alpha_1,...,\\alpha_s\\})$, where $\\{\\alpha_1,...,\\alpha_s\\}$ is a\nlinearly independent subset of $X$.  Thus the $s\\times s$ matrix\n$U:=\\left[\\la\\alpha_i,\\alpha_j\\ra\\right]$ is invertible and satisfies\n\\begin{equation}\\label{rat}\nU\\left[\\begin{smallmatrix} c_1 \\\\ \\vdots \\\\ c_s\\end{smallmatrix}\\right]=\n||\\beta||^2\\left[\\begin{smallmatrix} 1 \\\\ \\vdots \\\\ 1\\end{smallmatrix}\\right].\n\\end{equation}\n\nIn particular, if all the entries of $\\alpha_i$ are in $\\QQ$ for any $i=1,...,r$,\nthen also the entries of $\\beta:=\\mcc(X)$ are all in $\\QQ$.  Indeed,\n$\\tfrac{c_i}{||\\beta||^2}\\in\\QQ$ for all $i$ and so their sum\n$\\tfrac{1}{||\\beta||^2}\\in\\QQ$, which implies that $c_i\\in\\QQ$ for all $i$ and\nconsequently $\\beta$ has all its coefficients in $\\QQ$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Variational approach to Einstein solvmanifolds}\\label{va}\n\nEinstein metrics are often considered as the nicest, or most privileged ones on a\ngiven differentiable manifold (see for instance \\cite[Introduction]{Bss}).  One of\nthe justifications is the following result due to Hilbert (see \\cite{Hlb}): the\nEinstein condition for a compact Riemannian manifold $(M,g_{\\circ})$ of volume one\nis equivalent to the fact that the total scalar curvature functional\n$$\n\\scalar : g\\mapsto\\int_M\\scalar(g)\\mu_g\n$$\nadmits $g_{\\circ}$ as a critical point on the space of all metrics of volume one\n(see also \\cite[4.21]{Bss}).  This variational approach still works for $G$-invariant\nmetrics on $M$, where $G$ is any compact Lie group acting transitively on $M$ (see\n\\cite[4.23]{Bss}).\n\nOn the other hand, it is proved in \\cite{Jns} that in a unimodular $n$-dimensional\nLie group, the Einstein left invariant metrics are precisely the critical points of\nthe scalar curvature functional on the set of all left invariant metrics having a\nfixed volume element.  However, this fails in the non-unimodular case.  For\ninstance, if $\\sg$ is a solvable non-unimodular Lie algebra, then the scalar\ncurvature functional restricted to any leaf\n$F=\\{t\\}\\times\\Sl(\\sg)\/\\SO(\\sg)\\subset\\pca$ of inner products, has no critical\npoints (see \\cite[3.5]{Hbr}).  Thus, the approach to study Einstein solvmanifolds by\na variational method should be different.\n\nIn this section, we shall describe the approach proposed in the introduction: to\nvary Lie brackets rather than inner products.  Recall that when $\\ngo$ is an\n$n$-dimensional nilpotent Lie algebra, then the set of all inner products on $\\ngo$\nis very nice, it is parameterized by the symmetric space $\\Gl_n(\\RR)\/\\Or(n)$.\nHowever, isometry classes are precisely the orbits of the action on\n$\\Gl_n(\\RR)\/\\Or(n)$ of the group of automorphisms $\\Aut(\\ngo)$, a group mostly\nunknown, hard to compute, and far from being reductive, that is, ugly from the point\nof view of invariant theory.  If we instead vary Lie brackets, isometry classes will\nbe given by $\\Or(n)$-orbits, a beautiful group.  But since nothing is for free in\nmathematics, the set of left invariant metrics will now be parameterized by a\n$\\Gl_n(\\RR)$-orbit in the variety $\\nca$ of $n$-dimensional nilpotent Lie algebras,\na terrible space.\n\nWe fix an inner product vector space\n$$\n(\\sg=\\RR H\\oplus\\RR^n,\\ip),\\qquad \\la H,\\RR^n\\ra=0,\\quad \\la H,H\\ra=1,\n$$\nsuch that the restriction $\\ip|_{\\RR^n\\times\\RR^n}$ is the canonical inner product\non $\\RR^n$, which will also be denoted by $\\ip$.  A linear operator on $\\RR^n$ will\nbe sometimes identified with its matrix in the canonical basis $\\{ e_1,...,e_n\\}$ of\n$\\RR^n$.  The metric Lie algebra corresponding to any $(n+1)$-dimensional rank-one\nsolvmanifold, can be modeled on $(\\sg=\\RR H\\oplus\\ngo,\\ip)$ for some nilpotent Lie\nbracket $\\mu$ on $\\RR^n$ and some $D\\in\\Der(\\mu)$, the space of derivations of\n$(\\RR^n,\\mu)$.  Indeed, these data define a solvable Lie bracket $[\\cdot,\\cdot]$ on\n$\\sg$ by\n\\begin{equation}\\label{solv}\n[H,X]=DX,\\qquad [X,Y]=\\mu(X,Y), \\qquad X,Y\\in\\RR^n,\n\\end{equation}\n\n\\noindent and the solvmanifold is then the simply connected Lie group $S$ with Lie\nalgebra $(\\sg,[\\cdot,\\cdot])$ endowed with the left invariant Riemannian metric\ndetermined by $\\ip$.  We shall assume from now on that $\\mu\\ne 0$ since the case\n$\\mu=0$ (i.e. abelian nilradical) is well understood (see \\cite[Proposition\n6.12]{Hbr}).  We have seen in the paragraph above Definition \\ref{enil} that for a\ngiven $\\mu$, there exists a unique  symmetric derivation $D_{\\mu}$ to consider if we\nwant to get Einstein solvmanifolds.  We can therefore associate to each nilpotent\nLie bracket $\\mu$ on $\\RR^n$ a distinguished rank-one solvmanifold $S_{\\mu}$,\ndefined by the data $\\mu,D_{\\mu}$ as in (\\ref{solv}), which is the only one with a\nchance of being Einstein among all those metric solvable extensions of $(\\mu,\\ip)$.\n\nWe note that conversely, any $(n+1)$-dimensional rank-one Einstein solvmanifold is\nisometric to $S_{\\mu}$ for some nilpotent $\\mu$. Thus the set $\\nca$ of all\nnilpotent Lie brackets on $\\RR^n$ parameterizes a space of $(n+1)$-dimensional\nrank-one solvmanifolds\n$$\n\\{ S_{\\mu}:\\mu\\in\\nca\\},\n$$\ncontaining all those which are Einstein in that dimension.\n\nConcerning the identification\n$$\n\\mu\\longleftrightarrow (N_{\\mu},\\ip),\n$$\nwhere $N_{\\mu}$ is the simply connected nilpotent Lie group with Lie algebra\n$(\\RR^n,\\mu)$, the $\\G$-action on $\\nca$ defined in (\\ref{action}) has the following\ngeometric interpretation: each $g\\in\\G$ determines a Riemannian isometry\n\\begin{equation}\\label{id}\n(N_{g.\\mu},\\ip)\\longrightarrow (N_{\\mu},\\la g\\cdot,g\\cdot\\ra)\n\\end{equation}\n\n\\noindent by exponentiating the Lie algebra isomorphism\n$g^{-1}:(\\RR^n,g.\\mu)\\longrightarrow(\\RR^n,\\mu)$.  Thus the orbit $\\G.\\mu$ may be\nviewed as a parametrization of the set of all left invariant metrics on $N_{\\mu}$.\nBy a result of E. Wilson, two pairs $(N_{\\mu},\\ip)$, $(N_{\\lambda},\\ip)$ are\nisometric if and only if $\\mu$ and $\\lambda$ are in the same $\\Or(n)$-orbit (see\n\\cite[Appendix]{minimal}), where $\\Or(n)$ denotes the subgroup of $\\G$ of orthogonal\nmatrices. Also, two solvmanifolds $S_{\\mu}$ and $S_{\\lambda}$ with\n$\\mu,\\lambda\\in\\nca$ are isometric if and only if there exists $g\\in\\Or(n)$ such\nthat $g.\\mu=\\lambda$ (see \\cite[Proposition 4]{critical}).  From (\\ref{id}) and the\ndefinition of $S_{\\mu}$ we obtain the following result.\n\n\\begin{lemma}\\label{enilmu}\nIf $\\mu\\in\\nca$ then the nilpotent Lie algebra $(\\RR^n,\\mu)$ is an Einstein\nnilradical if and only if $S_{g.\\mu}$ is Einstein for some $g\\in\\G$.\n\\end{lemma}\n\nRecall that being an Einstein nilradical is a property of a whole $\\G$-orbit in\n$\\nca$, that is, of the isomorphism class of a given $\\mu$.\n\nFor any $\\mu\\in\\nca$ we have that the scalar curvature of $(N_{\\mu},\\ip)$ is given\nby $\\scalar(\\mu)=-\\unc ||\\mu||^2$, which says that normalizing by scalar curvature\nand by the spheres of $V$ is actually equivalent.  The critical points of any\nscaling invariant curvature functional on $\\nca$ appear then as very natural\ncandidates to be distinguished left invariant metrics on nilpotent Lie groups.\n\n\\begin{theorem}\\label{crit}\\cite{soliton,critical,einsteinsolv}\nFor a nonzero $\\mu\\in\\nca$, the following conditions are equivalent:\n\\begin{itemize}\n\\item[(i)] $S_{\\mu}$ is Einstein.\n\n\\item[(ii)] $(N_{\\mu},\\ip)$ is a nilsoliton.\n\n\\item[(iii)] $\\mu$ is a critical point of the functional $F:V\\longrightarrow\\RR$ defined by\n$$\nF(\\mu)=\\tfrac{16}{||\\mu||^4}\\tr{\\Ric_{\\mu}^2},\n$$\nwhere $\\Ric_{\\mu}$ denotes the Ricci operator of $(N_{\\mu},\\ip)$.\n\n\\item[(iv)] $\\mu$ is a minimum of $F|_{\\G.\\mu}$ (i.e. $(N_{\\mu},\\ip)$ is minimal).\n\n\\item[(v)] $\\Ric_{\\mu}\\in\\RR I\\oplus\\Der(\\mu)$.\n\\end{itemize}\nUnder these conditions, the set of critical points of $F$ lying in $\\G.\\mu$ equals\n$\\Or(n).\\mu$ (up to scaling).\n\\end{theorem}\n\nThus another natural approach to find rank-one Einstein solvmanifolds would be to\nuse the negative gradient flow of the functional $F$.  It follows from \\cite[Lemma\n6]{critical} that if $\\pi$ is the representation defined in (\\ref{actiong}) then\n$$\n\\grad(F)_{\\mu}=\\tfrac{16}{||\\mu||^6}\\left(||\\mu||^2\\pi(\\Ric_{\\mu})\\mu-4\\tr{\\Ric_{\\mu}^2}\\mu\\right).\n$$\nSince $F$ is invariant under scaling we know that $||\\mu||$ will remain constant in\ntime along the flow. We may therefore restrict ourselves to the sphere of radius\n$2$, where the negative gradient flow $\\mu=\\mu(t)$ of $F$ becomes\n\\begin{equation}\\label{flow}\n    \\ddt\\mu=-\\pi(\\Ric_{\\mu})\\mu+\\tr{\\Ric_{\\mu}^2}\\mu.\n\\end{equation}\n\nNotice that $\\mu(t)$ is a solution to this differential equation if and only if\n$g.\\mu(t)$ is so for any $g\\in\\Or(n)$, according to the $\\Or(n)$-invariance of $F$.\nThe existence of $\\lim\\limits_{t\\to\\infty}\\mu(t)$ is guaranteed by the compactness\nof the sphere and the fact that $F$ is a polynomial (see for instance \\cite[Section\n2.5]{Sjm}).\n\n\\begin{lemma}\\label{strataflow}\\cite{einsteinsolv}\nFor $\\mu_0\\in V$, $||\\mu_0||=2$, let $\\mu(t)$ be the flow defined in {\\rm\n(\\ref{flow})} with $\\mu(0)=\\mu_0$ and put $\\lambda=\\lim\\limits_{t\\to\\infty}\\mu(t)$.\nThen\n\\begin{itemize}\n    \\item[(i)] $\\mu(t)\\in\\G.\\mu_0$ for all $t$.\n    \\item[(ii)] $\\lambda\\in\\overline{\\G.\\mu_0}$.\n    \\item[(iii)] $S_{\\lambda}$ is Einstein.\n   \\end{itemize}\n\\end{lemma}\n\nPart (i) follows from the fact that $\\ddt\\mu\\in\\tang_{\\mu}\\G.\\mu$ for all $t$ (see\n(\\ref{flow})), and part (ii) is just a consequence of (i).  Condition (ii) is often\nreferred in the literature as the Lie algebra $\\mu_0$ {\\it degenerates} to the Lie\nalgebra $\\lambda$.  Some interplays between degenerations and Riemannian geometry of\nLie groups have been explored in \\cite{inter}, by using the fact that for us, the\norbit $\\G.\\mu_0$ is the set of all left invariant metrics on $N_{\\mu_0}$.  We note\nthat if the limit $\\lambda\\in\\G.\\mu_0$, then $\\mu_0$ is an Einstein nilradical.  We\ndo not know if the converse holds.  Since $\\lambda$ is a critical point of $F$ and\n$\\lambda\\in\\nca$ by (ii) and the fact that $\\nca$ is closed, we have that part (iii)\nfollows from Theorem \\ref{crit}.\n\nIn geometric invariant theory, a moment map for linear reductive Lie group actions\nover $\\CC$ has been defined in \\cite{Nss} and \\cite{Krw1} (see Appendix).  In our\nsituation, it is an $\\Or(n)$-equivariant map\n$$\nm:V\\smallsetminus\\{ 0\\}\\longrightarrow\\sym(n),\n$$\ndefined implicitly by\n\\begin{equation}\\label{defmm}\n\\la m(\\mu),\\alpha\\ra=\\tfrac{1}{||\\mu||^2}\\la\\pi(\\alpha)\\mu,\\mu\\ra, \\qquad \\mu\\in\nV\\smallsetminus\\{ 0\\}, \\; \\alpha\\in\\sym(n).\n\\end{equation}\n\n\\noindent We are using $\\g=\\sog(n)\\oplus\\sym(n)$ as the Cartan decomposition for the\nLie algebra $\\g$ of $\\G$, where $\\sog(n)$ and $\\sym(n)$ denote the subspaces of\nskew-symmetric and symmetric matrices, respectively.\n\nRecall that $\\nca\\subset V$ and each $\\mu\\in\\nca$ determines two Riemannian\nmanifolds $S_{\\mu}$ and $(N_{\\mu},\\ip)$. A remarkable fact is that this moment map\nencodes geometric information on $S_{\\mu}$ and $(N_{\\mu},\\ip)$; indeed, it was\nproved in \\cite{minimal} that\n\\begin{equation}\\label{mmR}\nm(\\mu)=\\tfrac{4}{||\\mu||^2}\\Ric_{\\mu}.\n\\end{equation}\n\nThis allows us to use strong and well-known results on the moment given in\n\\cite{Krw1} and \\cite{Nss}, and proved in \\cite{Mrn} for the real case (see the\nAppendix for an overview on such results).  We note that the functional $F$ defined\nin Theorem \\ref{crit}, (iii) is precisely $F(\\mu)=||m(\\mu)||^2$, and so the\nequivalence between (iii) and (iv) in Theorem \\ref{crit} follows from Theorem\n\\ref{marian}, (i).  It should be pointed out that actually most of the results in\nTheorem \\ref{crit} follow from general results on the moment map proved in\n\\cite{Mrn}.  For instance, the last sentence about uniqueness of critical points of\n$F$ (see Theorem \\ref{marian}, (ii)), is easily seen to be equivalent to the\nuniqueness of standard Einstein solvmanifolds (see Theorem \\ref{u}) and nilsolitons\n(see Theorem \\ref{un}).\n\nIn Section \\ref{st}, we shall see that one can go further in the application of\ngeometric invariant theory to the study of Einstein solvmanifolds, by considering a\nstratification for $\\nca$ intimately related to the moment map.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{On the classification of Einstein solvmanifolds}\\label{clas}\n\nAs we have seen in Section \\ref{pre}, the classification of Einstein solvmanifolds\nis essentially reduced to the rank-one case.  There is a bijection between the set\nof all isometry classes of rank-one Einstein solvmanifolds and the set of isometry\nclasses of certain distinguished left invariant metrics on nilpotent Lie groups\ncalled nilsolitons, and the uniqueness up to isometry of nilsolitons finally\ndetermines a new bijection with the set of all isomorphism classes of Einstein\nnilradicals.  For better or worse, what we get in the end is then a classification\nproblem on nilpotent Lie algebras.\n\nRecall that a nilpotent Lie algebra $\\ngo$ is an Einstein nilradical if and only if\n$\\ngo$ admits a nilsoliton, that is, an inner product $\\ip$ such that the\ncorresponding Ricci operator $\\Ric_{\\ip}$ satisfies\n$$\n\\Ric_{\\ip}=cI+D, \\qquad \\mbox{for some}\\; c\\in\\RR,\\; D\\in\\Der(\\ngo).\n$$\nTherefore, in order to understand or classify Einstein nilradicals, a main problem\nwould be how to translate this condition based on the existence of an inner product\non $\\ngo$ having a certain property into purely Lie theoretic conditions on $\\ngo$.\nThe following questions also arise:\n\n\\begin{itemize}\n\\item[(A)] Besides the existence of an $\\NN$-gradation, is there any other neat structural obstruction for a nilpotent Lie algebra to be an Einstein nilradical?\n\n\\item[ ]\\item[(B)] Is there any algebraic condition on a nilpotent Lie algebra which is sufficient to be an Einstein nilradical?\n\n\\item[ ]\\item[(C)] An $\\NN$-graded nilpotent Lie algebra can or can not be an\nEinstein nilradical, what is more likely?\n\\end{itemize}\n\nLet us now review what we do know on the classification of Einstein nilradicals.\n\nAny nilpotent Lie algebra of dimension $\\leq 6$ is an Einstein nilradical (see\n\\cite{Wll}). There are $34$ of them in dimension $6$,  giving rise to $29$ different\neigenvalue-types (there are $5$ eigenvalue-types with exactly two algebras).  In\ndimension $7$, the first nilpotent Lie algebras without any $\\NN$-gradation appear,\nbut also do the first examples of $\\NN$-graded Lie algebras which are not Einstein\nnilradicals.  The family of $7$-dimensional nilpotent Lie algebras defined for any\n$t\\in\\RR$ by\n\\begin{equation}\\label{ex7}\n\\begin{array}{lll}\n[X_1,X_2]_t=X_3, & [X_1,X_5]_t=X_6, & [X_2,X_4]_t=X_6, \\\\ \\\\\n\n[X_1,X_3]_t=X_4, & [X_1,X_6]_t=X_7, &  [X_2,X_5]_t=tX_7, \\\\ \\\\\n\n[X_1,X_4]_t=X_5, & [X_2,X_3]_t=X_5, &  [X_3,X_4]_t=(1-t)X_5,\n\\end{array}\n\\end{equation}\n\n\\noindent is really a curve in the set of isomorphism classes of algebras (i.e.\n$\\lb_t\\simeq\\lb_s$ if and only if $t=s$) and $\\lb_t$ turns to be an Einstein\nnilradical if and only if $t\\ne 0,1$ (see \\cite{einsteinsolv}).  Recall that all of\nthem admit the gradation $\\ngo=\\ngo_1\\oplus\\ngo_2\\oplus...\\oplus\\ngo_7$, $\\ngo_i=\\RR\nX_i$ for all $i$.  This example in particular shows that to be an Einstein\nnilradical is not a property which depends continuously on the structural constants\nof the Lie algebra.\n\nPerhaps the nicest source of examples of Einstein nilradicals is the following.\n\n\\begin{theorem}\\cite{Tmr}\nLet $\\ggo$ be a real semisimple Lie algebra.  Then the nilradical of any parabolic\nsubalgebra of $\\ggo$ is an Einstein nilradical.\n\\end{theorem}\n\nIf we add to this that H-type Lie algebras and any nilpotent Lie algebra admitting a\nnaturally reductive left invariant metric are Einstein nilradicals, one may get the\nimpression that any nilpotent Lie algebra which is special or distinguished in some\nway, or just has a `name', will be an Einstein nilradical.  This is contradicted by\nthe following surprising result, which asserts that free nilpotent Lie algebras are\nrarely Einstein nilradicals.\n\n\\begin{theorem}\\cite{Nkl1}\nA free $p$-step nilpotent Lie algebra on $m$ generators is an Einstein nilradical if\nand only if\n\\begin{itemize}\n\\item $p=1,2$;\n\\item $p=3$ and $m=2,3,4,5$;\n\\item $p=4$ and $m=2$;\n\\item $p=5$ and $m=2$.\n\\end{itemize}\n\\end{theorem}\n\nA nilpotent Lie algebra $\\ngo$ is said to be {\\it filiform} if $\\dim{\\ngo}=n$ and\n$\\ngo$ is $(n-1)$-step nilpotent.  These algebras may be seen as those which are as\nfar as possible from being abelian along the class of nilpotent Lie algebras, and in\nfact most of them admit at most one $\\NN$-gradation.  Several families of filiform\nalgebras which are not Einstein nilradicals have been found in \\cite{Nkl2}, as well\nas many isolated examples of non-Einstein nilradicals belonging to a curve of\nEinstein nilradicals as in example (\\ref{ex7}).  In \\cite{Arr}, a weaker version of\nTheorem \\ref{Upos} given in \\cite{Nkl2} is used to get a classification of\n$8$-dimensional filiform Einstein nilradicals.\n\nThe lack of $\\NN$-gradations is not however the only obstacle one can find for\nEinstein nilradicals.  Several examples of non-Einstein nilradicals are already\nknown in the class of $2$-step nilpotent Lie algebras (i.e. $[\\ngo,[\\ngo,\\ngo]]=0$),\nthe closest ones to being abelian and so algebras which usually admit plenty of\ndifferent $\\NN$-gradations.\n\n\\begin{definition} A $2$-step nilpotent Lie algebra $\\ngo$ is said to be of {\\it type} $(p,q)$ if $\\dim{\\ngo}=p+q$ and $\\dim{[\\ngo,\\ngo]}=p$.\n\\end{definition}\n\nIn \\cite{einsteinsolv}, certain $2$-step nilpotent Lie algebras attached to graphs\nare considered (of type $(p,q)$ if the graph has $q$ vertices and $p$ edges) and it\nis proved that they are Einstein nilradicals if and only if the graph is positive\n(i.e. when certain uniquely defined weighting on the set of edges is positive).  For\ninstance, any regular graph and also any tree such that any of its edges is adjacent\nto at most three other edges is positive.  On the other hand, a graph is not\npositive under the following condition: there are two joined vertices $v$ and $w$\nsuch that $v$ is joined to $r$ vertices of valency $1$, $w$ is joined to $s$\nvertices of valency $1$, both are joined to $t$ vertices of valency $2$ and\n$(r,s,t)$ is not in a set of only a few exceptional small triples. This provides a\ngreat deal of $2$-step non-Einstein nilradicals, starting from types $(5,6)$ and\n$(7,5)$, and any dimension $\\geq 11$  is attained.\n\nMany other $2$-step algebras of type $(6,5)$ and $(7,5)$ which are not Einstein\nnilradicals have appeared from the complete classification for types $(p,q)$ with\n$q\\leq 5$ and $(p,q)\\ne (5,5)$ carried out in \\cite{Nkl3}.\n\nCuriously enough, at this point of the story, with so many examples of non-Einstein\nnilradicals available, a curve was still missing.  In each fixed dimension,\nonly finitely many nilpotent Lie algebras which are not Einstein nilradicals have\nshowed up.  But this potential candidate to a conjecture has recently been dismissed\nby the following result.\n\n\\begin{theorem}\\cite{Wll2}\nLet $\\ngo_t$ be the $9$-dimensional Lie algebra with Lie bracket defined by\n$$\n\\begin{array}{lll}\n\n[X_5,X_4]_t=X_7, & [X_1,X_6]_t=X_8, &  [X_3,X_2]_t=X_9, \\\\ \\\\\n\n[X_3,X_6]_t=tX_7, & [X_5,X_2]_t=tX_8, &  [X_1,X_4]_t=tX_9, \\\\ \\\\\n\n[X_1,X_2]_t=X_7. &&\n\\end{array}\n$$\n\nThen $\\ngo_{t},$  $t\\in (1,\\infty)$, is a curve of pairwise non-isomorphic $2$-step\nnilpotent Lie algebras of type $(3,6)$, none of which is an Einstein nilradical.\n\\end{theorem}\n\nThe following definition is motivated by (\\ref{pE}), a condition a rank-one solvable\nextension of a nilpotent Lie algebra must satisfy in order to have a chance of being\nEinstein.\n\n\\begin{definition}\\label{preE} A derivation $\\phi$ of a real Lie algebra $\\ggo$ is called {\\it pre-Einstein} if it is diagonalizable over $\\RR$ and\n$$\n\\tr{\\phi\\psi}=\\tr{\\psi}, \\qquad\\forall\\; \\psi\\in\\Der(\\ggo).\n$$\n\\end{definition}\n\nThe following result is based on the fact that $\\Aut(\\ggo)$ is an algebraic group.\n\n\\begin{theorem}\\cite{Nkl3}\\label{eupreE}\nAny Lie algebra $\\ggo$ admits a pre-Einstein derivation, which is unique up to\n$\\Aut(\\ggo)$-conjugation and has eigenvalues in $\\QQ$.\n\\end{theorem}\n\nLet $\\ngo$ be a nilpotent Lie algebra with pre-Einstein derivation $\\phi$.  We note\nthat if $\\ngo$ admits a nilsoliton metric, say with $\\Ric_{\\ip}=cI+D$, then $D$\nnecessarily equals $\\phi$ up to scaling and conjugation (see (\\ref{pE})), and thus\nthe eigenvalue-type of the corresponding Einstein solvmanifold is the set of\neigenvalues of $\\phi$ up to scaling.  In particular, $\\phi>0$.  It is proved in\n\\cite{Nkl3} that also $\\ad{\\phi}\\geq 0$ as long as $\\ngo$ is an Einstein nilradical.\nThese conditions are not however sufficient to guarantee that $\\ngo$ is an Einstein\nnilradical (see \\cite{Nkl0}).  In order to get a necessary and sufficient condition\nin terms of $\\phi$ we have to work harder.\n\nLet us first consider\n\\begin{equation}\\label{gphi}\n\\ggo_{\\phi}:=\\{\\alpha\\in\\glg(\\ngo):[\\alpha,\\phi]=0, \\quad\\tr{\\alpha\\phi}=0,\n\\quad\\tr{\\alpha}=0\\}\n\\end{equation}\n\n\\noindent and let $G_{\\phi}$ be the connected Lie subgroup of $\\Gl(\\ngo)$ with Lie\nalgebra $\\ggo_{\\phi}$.  Recall that the Lie bracket $\\lb$ of $\\ngo$ belongs to the\nvector space $\\Lambda^2\\ngo^*\\otimes\\ngo$ of skew-symmetric bilinear maps from\n$\\ngo\\times\\ngo$ to $\\ngo$, on which $\\Gl(\\ngo)$ is acting naturally by\n$g.\\lb=g[g^{-1}\\cdot,g^{-1}\\cdot]$.\n\n\n\\begin{theorem}\\cite{Nkl3}\\label{madreN}\nLet $\\ngo$ be a nilpotent Lie algebra with pre-Einstein derivation $\\phi$. Then\n$\\ngo$ is an Einstein nilradical if and only if the orbit $G_{\\phi}.\\lb$ is closed\nin $\\Lambda^2\\ngo^*\\otimes\\ngo$.\n\\end{theorem}\n\nThis is certainly the strongest general result we know so far concerning questions\n(A) and (B) above, and of course it has many useful applications, some of which we will now\ndescribe (see also Theorem \\ref{madre} for a turned to be equivalent result).\n\n\\begin{definition}\\label{nice}  Let $\\{ X_1,...,X_n\\}$ be a basis for a nilpotent Lie algebra $\\ngo$,\nwith structural constants $c_{ij}^k$'s given by\n$[X_i,X_j]=\\sum\\limits_{k=1}^{n}c_{ij}^kX_k$.  Then the basis $\\{ X_i\\}$ is said to\nbe {\\it nice} if the following conditions hold:\n\\begin{itemize}\n    \\item for all $i<j$ there is at most one $k$ such that $c_{ij}^k\\ne 0$,\n    \\item if $c_{ij}^k$ and $c_{i'j'}^k$ are nonzero then either $\\{ i,j\\}=\\{\n    i',j'\\}$ or $\\{ i,j\\}\\cap\\{ i',j'\\}=\\emptyset$.\n\\end{itemize}\n\\end{definition}\n\nA nice property a nice basis $\\{ X_i\\}$ has is that the Ricci operator $\\Ric_{\\ip}$\nof any inner product $\\ip$ for which $\\{ X_i\\}$ is orthogonal diagonalizes with\nrespect to $\\{ X_i\\}$ (see \\cite[Lemma 3.9]{einsteinsolv}).  Uniform bases\nconsidered in \\cite{Dlf} and \\cite{Wlt} are nice.  The existence of a nice basis for\na nilpotent Lie algebra looks like a strong condition, although we do not know of\nany example for which we can prove the non-existence of a nice basis.  Not even an\nexistence result for such example is available.\n\nLet $(\\ngo,\\ip)$ be a metric nilpotent Lie algebra with orthogonal basis $\\{ X_i\\}$\nand structural constants $[X_i,X_j]=\\sum\\limits_{k=1}^{n}c_{ij}^kX_k$.  If we fix an\nenumeration of the set $\\left\\{\\alpha_{ij}^k:c_{ij}^k\\ne 0\\right\\}$ (see Section\n\\ref{Tb}), we can define the symmetric matrix\n\\begin{equation}\\label{defU}\nU=\\left[\\left\\langle\\alpha_{ij}^k,\\alpha_{i'j'}^{k'}\\right\\rangle\\right],\n\\end{equation}\n\n\\noindent and state the following useful result.\n\n\\begin{theorem}\\cite{Pyn}\\label{tracy}\nAssume that $(\\ngo,\\ip)$ satisfies $\\Ric_{\\ip}\\in\\tg$.  Then $(\\ngo,\\ip)$ is a\nnilsoliton if and only if\n$$\nU\\left[(c_{ij}^k)^2\\right]=c [1], \\qquad c\\in\\RR,\n$$\nwhere $\\left[(c_{ij}^k)^2\\right]$ is meant as a column vector in the same order used\nin {\\rm (\\ref{defU})} for defining $U$ and $[1]$ is the column vector with all\nentries equal to $1$.\n\\end{theorem}\n\nIt turns out that equations $U\\left[(c_{ij}^k)^2\\right]=c[1]$ are precisely those given by the\nLagrange method applied to find critical points of the functional $F$ in Theorem\n\\ref{crit}.  In \\cite{Pyn}, a Cartan matrix is associated to $U$ and the theory of\nKac-Moody algebras is applied to analyze the solutions space of such a linear\nsystem.\n\nRecall that $\\mcc(X)$ denotes the unique vector of minimal norm in the convex hull\n$\\CH(X)$ of a finite subset $X$ of $\\tg$.\n\n\\begin{theorem}\\cite{Nkl3}\\label{Upos}\nA nonabelian nilpotent Lie algebra $\\ngo$ with a nice basis $\\{ X_i\\}$ and\nstructural constants $[X_i,X_j]=\\sum\\limits_{k=1}^{n}c_{ij}^kX_k$ is an Einstein\nnilradical if and only if any of the following equivalent conditions hold:\n\\begin{itemize}\n\\item[(i)] $\\mcc\\{\\alpha_{ij}^k:c_{ij}^k\\ne 0\\}$ lies in the interior of $\\CH\\left(\\{\\alpha_{ij}^k:c_{ij}^k\\ne 0\\}\\right)$.\n\n\\item[(ii)] Equation $U[x_{ij}^k]=[1]$ has a positive solution $[x_{ij}^k]$.\n\\end{itemize}\n\\end{theorem}\n\nThis is a non-constructive result, in the sense that it is in general very difficult\nto explicitly find the nilsoliton metric.  The absence of an inner product in its\nstatement (compare with Theorem \\ref{tracy}), however, makes of Theorem \\ref{Upos},\nquite a useful result.\n\n\n\\begin{theorem}\\cite{Nkl3}\nLet $\\ngo_1$, $\\ngo_2$ be real nilpotent Lie algebras which are isomorphic as\ncomplex Lie algebras (i.e. they have isomorphic complexifications\n$\\ngo_{i}\\otimes\\CC$).  Then $\\ngo_1$ is an Einstein nilradical if and only if\n$\\ngo_2$ is so, and in that case, they have the same eigenvalue-type.\n\\end{theorem}\n\nThis turns our classification of Einstein nilradicals into a problem on complex\nnilpotent Lie algebras, with all the advantages an algebraically closed field has if\nwe want to use known classifications in the literature or results from algebraic\ngeometry and geometric invariant theory.\n\nThe following result reduces the classification of Einstein nilradicals to those\nwhich are {\\it indecomposable} (i.e. non-isomorphic to a direct sum of Lie\nalgebras).\n\n\\begin{theorem}\\cite{Nkl3}\\label{suma}\nLet $\\ngo=\\ngo_1\\oplus\\ngo_2$ be a nilpotent Lie algebra which is the direct sum of\ntwo ideals $\\ngo_1$ and $\\ngo_2$. Then $\\ngo$ is an Einstein nilradical if and only\nif both $\\ngo_1$ and $\\ngo_2$ are Einstein nilradicals.\n\\end{theorem}\n\nAny $2$-step nilpotent Lie algebra of type $(p,q)$ can be identified with an element\nin the vector space $V_{q,p}:=\\Lambda^2(\\RR^q)^*\\otimes\\RR^p$, and it is easy to see\nthat two of them are isomorphic if and only if they lie in the same\n$\\Gl_q(\\RR)\\times\\Gl_p(\\RR)$-orbit.\n\n\\begin{theorem}\\label{generic}\\cite{Ebr, Nkl3}\nIf $(p,q)\\ne (2,2k+1)$, then the\nvector space $V_{q,p}$ contains an open and dense subset of Einstein nilradicals of\neigenvalue-type $(1<2;q,p)$.\n\\end{theorem}\n\nIn view of this result, one may say that for most types, a $2$-step nilpotent Lie\nalgebra of type $(p,q)$ is typically, or generically, an Einstein nilradical of\neigenvalue-type $(1<2;q,p)$.  This is no doubt an important indicator related to\nquestion (C) above, but we must go carefully. What Theorem \\ref{generic} is actually\nasserting is that if one throws a dart on $V_{q,p}$, then, with probability one, the\ndart will hit at a Lie bracket $\\lb\\in V_{q,p}$ which is an Einstein nilradical of\neigenvalue-type $(1<2;q,p)$.  Recall that each algebra of type $(p,q)$ is identified\nwith a whole $\\Gl_q(\\RR)\\times\\Gl_p(\\RR)$-orbit in $V_{q,p}$, not with a single\npoint, and some of these orbits can be much thicker than others.\n\nLet us consider a simple example to illustrate this phenomenon.  There are exactly\n$7$ algebras up to isomorphism in the vector space $V_{4,2}$, including the abelian\none.  Only two of them are Einstein nilradicals of type $(1<2;4,2)$; namely, the\nH-type Lie algebra $\\hg_3\\otimes\\CC$ (i.e. the complexification of $\\hg_3$ viewed as\nreal) and $\\hg_3\\oplus\\hg_3$, where $\\hg_3$ denotes the $3$-dimensional Heisenberg\nalgebra (see \\cite[Table 4]{Wll}).  If we fix basis $\\{ X_1,...,X_4\\}$ and $\\{\nZ_1,Z_2\\}$ of $\\RR^4$ and $\\RR^2$, respectively, then each $\\lb\\in V_{4,2}$ is\ndetermined by $12$ structural constants as follows:\n$$\n[X_i,X_j]=c_{ij}^1Z_1+c_{ij}^2Z_2, \\qquad c_{ij}^k\\in\\RR, \\quad 1\\leq i<j\\leq 4,\n\\quad k=1,2.\n$$\nIf we take variables $x,y$ and define the skew-symmetric matrix $J$ with $ij$ entry,\n$i<j$, given by $c_{ij}^1x+c_{ij}^2y$, then $\\det{J}$ is a $4$-degree homogeneous\npolynomial on $(x,y)$ with a `square root' $f(x,y)$, a $2$-degree homogeneous\npolynomial called the Pfaffian form of $\\lb$ (see \\cite[Section 2]{ratform}).  Thus\nthe Hessian of $f$ is a real number $h(\\lb)$ which depends polynomially on the\n$c_{ij}^k$'s.  This defines a polynomial function $h:V_{4,2}\\longrightarrow\\RR$,\nwhich turns to be $\\Sl_4(\\RR)\\times\\Sl_2(\\RR)$-invariant.\n\nIt is not hard to see that $h(\\lb)\\ne 0$ if and only if $\\lb$ is isomorphic to\neither $\\hg_3\\otimes\\CC$ ($h>0$) or $\\hg_3\\oplus\\hg_3$ ($h<0$).  This implies that\nthe union of the two $\\Gl_4(\\RR)\\times\\Gl_2(\\RR)$-orbits corresponding to\n$\\hg_3\\oplus\\CC$ and $\\hg_3\\oplus\\hg_3$, which coincides with the set of all\nEinstein nilradicals of eigenvalue type $(1<2;4,2)$ in $V_{4,2}$, is open and dense\nin $V_{4,2}$.  However, recall that the net probability of being an Einstein\nnilradical of eigenvalue type $(1<2;4,2)$ in $V_{4,2}$ is $\\tfrac{2}{7}$.\n\nOne may try to avoid this by working on the quotient space\n$V_{q,p}\/\\Gl_q(\\RR)\\times\\Gl_p(\\RR)$, where Theorem \\ref{generic} is by the way also\ntrue, but the topology here is so ugly that an open and dense subset can never be\ntaken as a probability one subset.  In fact, there could be a single point set which\nis open and dense.  On the other hand, the coset of $0$ is always in the closure of\nany other subset, which shows that this quotient space is far from being $T_1$.\n\nIt has very recently appeared in \\cite{Nkl4} a complete classification for $2$-step\nEinstein nilradicals of type $(2,q)$ for any $q$.  In \\cite{Jbl}, a construction\ncalled concatenation of $2$-step nilpotent Lie algebras is used to obtain Einstein\nnilradicals of type $(1<2;q,p)$ from smaller ones, as well as many new examples of\n$2$-step non-Einstein nilradicals.\n\n\n\n\n\n\n\n\n\n\\section{Known examples and non examples}\\label{ene}\n\nAs far as we know, the following is a complete chronological list of nilpotent Lie\nalgebras which are known to be Einstein nilradicals,  or equivalently, of known\nexamples of rank-one Einstein solvmanifolds:\n\\begin{itemize}\n\n\\item[ ]\n\\item \\cite{Cart} The Lie algebra of an Iwasawa $N$-group: $G\/K$ irreducible symmetric space of\nnoncompact type and $G=KAN$ the Iwasawa decomposition.\n\n\\item[ ]\n\\item\\cite{GndPttVnb} Nilradicals of normal $j$-algebras (i.e. of\nnoncompact homogeneous K$\\ddot{{\\rm a}}$hler Einstein spaces).\n\n\\item[ ]\n\\item\\cite{Alk, Crt} Nilradicals of homogeneous quaternionic K$\\ddot{{\\rm\na}}$hler spaces.\n\n\\item[ ]\n\\item\\cite{Dlf} Certain $2$-step nilpotent Lie algebras for which there\nis a basis with very uniform properties (see also \\cite[1.9]{Wlt}).\n\n\\item[ ]\n\\item\\cite{Bgg} $H$-type Lie algebras (see also \\cite{Lnz}).\n\n\\item[ ]\n\\item\\cite{EbrHbr, manus} Nilpotent Lie algebras admitting a naturally reductive left invariant metric.\n\n\\item[ ]\n\\item\\cite{Hbr} Families of deformations of Lie algebras of Iwasawa $N$-groups in the rank-one\ncase.\n\n\\item[ ]\n\\item\\cite{Fan1,Fan2} Certain $2$-step nilpotent Lie algebras\nconstructed via Clifford modules.\n\n\\item[ ]\n\\item\\cite{GrdKrr} A $2$-parameter family of $2$-step nilpotent Lie algebras of type $(3,6)$ and certain\nmodifications of the Lie algebras of Iwasawa $N$-groups (rank $\\geq 2$).\n\n\\item[ ]\n\\item\\cite{finding} Any nilpotent Lie algebra with a codimension one abelian ideal.\n\n\\item[ ]\n\\item\\cite{finding} A curve of $6$-step nilpotent Lie algebras of dimension $7$, which\nis the lowest possible dimension for a continuous family.\n\n\\item[ ]\n\\item\\cite{Mor} (and Yamada), Certain $2$-step nilpotent Lie algebras defined from\nsubsets of fundamental roots of complex simple Lie algebras.\n\n\\item[ ]\n\\item\\cite{finding} Any nilpotent Lie algebra of dimension $\\leq 5$.\n\n\\item[ ]\n\\item\\cite{Wll} Any nilpotent Lie algebra of dimension $6$.\n\n\\item[ ]\n\\item\\cite{inter} A curve of $2$-step nilpotent Lie algebras of type $(5,5)$.\n\n\\item[ ]\n\\item\\cite{Krr} A $2$-parameter family of deformations of the nilradical of the $12$-dimensional quaternionic hyperbolic space.\n\n\\item[ ]\n\\item\\cite{Pyn} Any filiform (i.e. $n$-dimensional and $(n-1)$-step nilpotent) Lie algebra with at least two linearly independent semisimple derivations.\n\n\\item[ ]\n\\item\\cite{einsteinsolv} Certain $2$-step nilpotent Lie algebras attached to graphs as soon as a uniquely defined weighting on the graph is positive.  Regular graphs and trees without any edge adjacent to four or more edges are positive.\n\n\\item[ ]\n\\item\\cite{Nkl1} The free $p$-step nilpotent Lie algebras $\\fg(m,p)$ on $m$ generators for $p=1,2$; $p=3$ and $m=2,3,4,5$; $p=4$ and $m=2$; $p=5$ and $m=2$.\n\n\\item[ ]\n\\item\\cite{Nkl2} Several families of filiform Lie algebras.\n\n\\item[ ]\n\\item\\cite{Tmr} The nilradical of any parabolic subalgebra of a semisimple\nLie algebra.\n\n\\item[ ]\n\\item\\cite{Nkl3} Any $2$-step nilpotent Lie algebra of type $(p,q)$ (i.e. $p+q$-dimensional and $p$-dimensional derived algebra) with $q\\leq 5$ and $(p,q)\\ne (5,5)$, with the only exceptions of the real forms of six complex algebras of type $(6,5)$ and three of type $(7,5)$.\n\n\\item[ ]\n\\end{itemize}\n\n\nWe now give an up to date list of $\\NN$-graded nilpotent Lie\nalgebras which are not Einstein nilradicals, that is, they do not admit any\nnilsoliton metric.\n\n\\begin{itemize}\n\\item[ ]\n\\item\\cite{einsteinsolv} Three $6$-step nilpotent Lie algebras of dimension $7$, and\ncertain $2$-step nilpotent Lie algebras attached to graphs in any dimension $\\geq\n11$ (only finitely many in each dimension).\n\n\\item[ ]\n\\item\\cite{Nkl1} The free $p$-step nilpotent Lie algebras $\\fg(m,p)$ on $m$ generators\nfor $p=3$ and $m\\geq 6$; $p=4$ and $m\\geq 3$; $p=5$ and $m\\geq 3$; $p\\geq 6$.\n\n\\item[ ]\n\\item\\cite{Nkl2} Many filiform Lie algebras starting from dimension $8$ (see also \\cite{Arr}).\n\n\\item[ ]\n\\item\\cite{Nkl3} Real forms of six complex $2$-step nilpotent Lie algebras of\ntype $(6,5)$ and three of type $(7,5)$.\n\n\\item[ ]\n\\item\\cite{Wll2} Two curves of $2$-step nilpotent Lie algebras of type $(3,6)$.\n\n\\item[ ]\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A stratification for the variety of nilpotent Lie algebras}\\label{st}\n\nIn this section, we define a $\\G$-invariant stratification for the representation\n$V=\\lam$ of $\\G$ by adapting to this context the construction given in \\cite[Section\n12]{Krw1} for reductive group representations over an algebraically closed field.\nThis construction, in turn, is based on some instability results proved in\n\\cite{Kmp} and \\cite{Hss}.  We decided to give in \\cite[Section 2]{standard} a\nself-contained proof of all these results, bearing in mind that a direct application\nof them does not seem feasible (see also \\cite{strata}).\n\nWe shall use the notation given in Section \\ref{Tb}.  For any $\\mu\\in V$ we have\nthat\n$$\n\\lim\\limits_{t\\to\\infty}e^{tI}.\\mu= \\lim\\limits_{t\\to\\infty}e^{-t}\\mu=0,\n$$\nand hence $0\\in\\overline{\\G.\\mu}$, that is, any element of $V$ is unstable for our\n$\\G$-action (see Appendix).  Therefore, in order to distinguish two elements of $V$\nfrom the point of view of geometric invariant theory, we would need to measure in\nsome sense `how' unstable each element of $V$ is.  Maybe the above is not the\noptimal way to go to $0$ along the orbit starting from $\\mu$.\n\nLet us consider $\\mu\\in V$ and $\\alpha\\in\\dca$, where $\\dca$ denotes the set of all\n$n\\times n$ matrices which are diagonalizable, that is,\n$$\n\\dca=\\bigcup_{g\\in\\G}g\\tg g^{-1}.\n$$\nThus $\\pi(\\alpha)$ is also diagonalizable (see (\\ref{actiong})), say with\neigenvalues $a_1,...,a_r$ and eigenspace decomposition  $V=V_1\\oplus...\\oplus V_r$.\nThis implies that if $\\mu\\ne 0$ and $\\mu=\\mu_1+...+\\mu_r$, $\\mu_i\\in V_i$, then\n$$\ne^{-t\\alpha}.\\mu= \\sum_{i=1}^r e^{-ta_i}\\mu_i,\n$$\nand so $e^{-t\\alpha}.\\mu$ goes to $0$ when $t\\to\\infty$ if and only if $\\mu_i=0$ as\nsoon as $a_i\\leq 0$.  Moreover, in that case, the positive number\n$$\nm(\\mu,\\alpha):=\\min\\{ a_i : \\mu_i\\ne 0\\},\n$$\nmeasures the degree of instability of $\\mu$ relative to $\\alpha$, in the sense that\nthe train has not arrived until the last wagon has.  Indeed, the larger\n$m(\\mu,\\alpha)$ is, the faster $e^{-t\\alpha}.\\mu$ will converge to $0$ when\n$t\\to\\infty$.  Recall that for an action in general the existence of such $\\alpha$\nfor any unstable element is guaranteed by Theorem \\ref{RS}, (iv).\n\nNotice that $m(\\mu,c\\alpha)=c m(\\mu,\\alpha)$ for any $c>0$.  We can therefore\nconsider the most efficient directions (up to the natural normalization) for a given\n$\\mu\\in V$, given by\n$$\n\\Lambda(\\mu):=\\left\\{\\beta\\in\\dca: m(\\mu,\\beta)=1=\\sup\\limits_{\\alpha\\in\\dca}\\left\\{\nm(\\mu,\\alpha):\\tr{\\alpha^2}=\\tr{\\beta^2}\\right\\}\\right\\}.\n$$\nA remarkable fact is that $\\Lambda(\\mu)$ lie in a single conjugacy class, that is,\nthere exists an essentially unique direction which is `most responsible' for the\ninstability of $\\mu$.  All the parabolic subgroups $P_{\\beta}$ of $\\G$\nnaturally associated to any $\\beta\\in\\Lambda(\\mu)$ defined in (\\ref{para}) coincide, and\nhence they define a unique parabolic subgroup $P_{\\mu}$ which acts transitively on\n$\\Lambda(\\mu)$ by conjugation. A very nice property $P_{\\mu}$ has is that\n\\begin{equation}\\label{aut}\n\\Aut(\\mu)\\subset P_{\\mu}.\n\\end{equation}\n\nSince\n$$\n\\Lambda(g.\\mu)=g\\Lambda(\\mu)g^{-1}, \\qquad \\forall\\mu\\in V, \\quad g\\in\\G,\n$$\nwe obtain that $\\Lambda(g.\\mu)$ will meet the Weyl chamber $\\tg^+$ for some\n$g\\in\\G$, and the intersection set will consist of a single element $\\beta\\in\\tg^+$\n(see (\\ref{weyl})).\n\nSummarizing, we have been able to attach to each nonzero $\\mu\\in V$, and actually to\neach nonzero $\\G$-orbit in $V$, a uniquely defined $\\beta\\in\\tg^+$ which comes from\ninstability considerations.\n\n\\begin{definition}\nUnder the above conditions, we say that $\\mu\\in\\sca_{\\beta}$ and call the subset\n$\\sca_{\\beta}\\subset V$ a {\\it stratum}.\n\\end{definition}\n\nWe note that $\\sca_{\\beta}$ is $\\G$-invariant for any $\\beta\\in\\tg^+$ and\n$$\nV\\smallsetminus\\{ 0\\}=\\bigcup\\limits_{\\beta\\in\\tg^+}\\sca_{\\beta},\n$$\na disjoint union.  An alternative way to define $\\sca_{\\beta}$ is\n$$\n\\sca_{\\beta}=\\G.\\left\\{\\mu\\in V:\\tfrac{\\beta}{||\\beta||^2}\\in\\Lambda(\\mu)\\right\\},\n$$\nwhich actually works for any $\\beta\\in\\tg$.  From now on, we will always denote by\n$\\mu_{ij}^k$ the structure constants of a vector $\\mu\\in V$ with respect to the\nbasis $\\{ v_{ijk}\\}$:\n$$\n\\mu=\\sum\\mu_{ij}^kv_{ijk}, \\qquad \\mu_{ij}^k\\in\\RR, \\qquad {\\rm i.e.}\\quad\n\\mu(e_i,e_j)=\\sum_{k=1}^n\\mu_{ij}^ke_k, \\quad i<j.\n$$\nEach nonzero $\\mu\\in V$ uniquely determines an element $\\beta_{\\mu}\\in\\tg$ given by\n$$\n\\beta_{\\mu}=\\mcc\\left\\{\\alpha_{ij}^k:\\mu_{ij}^k\\ne 0\\right\\}.\n$$\nRecall that $\\mcc(X)$ denotes the unique element of minimal norm in the convex hull\n$\\CH(X)$ of a subset $X\\subset\\tg$, and thus $\\beta_{\\mu}$ has rational coefficients\n(see (\\ref{rat})).  We also note that $\\beta_{\\mu}$ is always nonzero since\n$\\tr{\\alpha_{ij}^k}=-1$ for all $i<j$ and consequently $\\tr{\\beta_{\\mu}}=-1$. If for\n$\\mu\\in V$ we define $\\Lambda_T(\\mu)$ as above but by replacing $\\dca$ with the set\nof diagonal matrices $\\tg$, then one can prove that\n$$\n\\Lambda_T(\\mu)=\\left\\{\\tfrac{\\beta_{\\mu}}{||\\beta_{\\mu}||^2}\\right\\},\n$$\nthat is, $\\beta_{\\mu}$ is the (unique) `most responsible' direction for the\ninstability of $\\mu$ with respect to the action of the torus $T$ with Lie algebra\n$\\tg$ on $V$.  Another equivalent definition for the stratum $\\sca_{\\beta}$,\n$\\beta\\in\\tg$, is given by\n$$\n\\sca_{\\beta}=\\Big\\{\\mu\\in V\\smallsetminus\\{ 0\\}:\\beta\\;\\mbox{is an element of\nmaximal norm in}\\;\\{\\beta_{g.\\mu}:g\\in\\G\\}\\Big\\}.\n$$\nIf $\\mu$ runs through $V$, there are only finitely many possible vectors\n$\\beta_{\\mu}$, and consequently the set $\\{\\beta\\in\\tg:\\sca_{\\beta}\\ne\\emptyset\\}$\nis finite.  We furthermore get from this new description that if $\\beta\\in\\tg$\nsatisfies $\\sca_{\\beta}\\ne \\emptyset$ then $\\beta$ has rational coefficients and\n\\begin{equation}\\label{trb}\n\\tr{\\beta}=-1.\n\\end{equation}\n\n\\begin{remark}\nA very illustrative exercise is to consider the action given in Example\n\\ref{hompol1}, draw the nice picture of its weights, detect all possible\n$\\beta_{p}$'s and try to figure out which of them actually determine a nonempty\nstratum (i.e. $\\sca_{\\beta_p}\\ne\\emptyset$).\n\\end{remark}\n\nRecall from Section \\ref{va} that the moment map $m$ for the $\\G$-representation $V$\nplays a fundamental role in the study of Einstein solvmanifolds and nilsolitons, as\n$m(\\mu)=-\\tfrac{1}{\\scalar(\\mu)}\\Ric_{\\mu}$, where $\\Ric_{\\mu}$ and $\\scalar(\\mu)$\ndenote the Ricci operator and the scalar curvature of $(N_{\\mu},\\ip)$, respectively.\nThe square norm functional $F(\\mu)=||m(\\mu)||^2$ therefore provides a natural\ncurvature functional on the space $\\nca$ of all left invariant metrics on\n$n$-dimensional nilpotent Lie groups whose critical points are precisely nilsoliton\nmetrics (see Theorem \\ref{crit}).\n\nWe have collected in the following theorem some relationships between $m$, $F$ and\nthe strata. Let $p_{\\tg}(\\alpha)$ denote the orthogonal projection on $\\tg$ of an\n$\\alpha\\in\\sym(n)$ (i.e. the diagonal part of $\\alpha$).\n\n\\begin{theorem}\\label{strataF}\\cite{einsteinsolv}\nLet $\\mu=\\sum\\mu_{ij}^kv_{ijk}$ be a nonzero element of $V$.\n\\begin{itemize}\n\\item[(i)] $p_{\\tg}(m(\\mu)) =\\tfrac{2}{||\\mu||^2}\\sum\\limits_{i<j}(\\mu_{ij}^k)^2\n\\alpha_{ij}^k\\in\\CH\\left\\{\\alpha_{ij}^k:\\mu_{ij}^k\\ne 0\\right\\}$.\n\n\\item[(ii)] $F(\\mu)\\geq ||\\beta||^2$ for any $\\mu\\in\\sca_{\\beta}$.\n\n\\item[(iii)] If $\\inf{F(\\G.\\mu)}=||\\beta_{\\mu}||^2$ then $\\mu\\in\\sca_{\\beta_{\\mu}}$.\n\n\\item[(iv)] If $\\mu\\in\\nca$, $m(\\mu)\\in\\tg$ and $S_{\\mu}$ is Einstein then $\\mu\\in\\sca_{m(\\mu)}$.\n\n\\item[(v)] For $\\mu\\in\\sca_{\\beta}\\cap\\nca$, the following conditions are equivalent:\n\\begin{itemize}\n\\item[(a)] $S_{\\mu}$ is Einstein.\n\n\\item[(b)] $S_{\\mu}$ is Einstein of eigenvalue-type $\\beta+||\\beta||^2I$ (up to a positive multiple).\n\n\\item[(c)] $m(\\mu)$ is conjugate to $\\beta$.\n\n\\item[(d)] $F(\\mu)=||\\beta||^2$.\n\\end{itemize}\n\\end{itemize}\n\\end{theorem}\n\nIt follows from part (v) in the above theorem that the stratum $\\sca_{\\beta}$ to\nwhich $\\mu$ belongs determines the eigenvalue type of a potential Einstein\nsolvmanifold $S_{g.\\mu}$, $g\\in\\G$ (if any), and so the stratification provides a\nconvenient tool to produce existence results as well as obstructions for nilpotent\nLie algebras to be an Einstein nilradical.  Thus $\\beta$ plays a role similar to\nthe one played by the pre-Einstein derivation $\\phi$ (see Definition \\ref{preE}).  The subtle\nrelationship between $\\beta$ and $\\phi$ will be explained in Section \\ref{smadre}.\n\nWe will now give a description of the strata in terms of semistable vectors (see\nAppendix).  For each $\\beta\\in\\tg$ consider the sets\n$$\n\\begin{array}{l}\nZ_{\\beta}=\\{ \\mu\\in V: \\la\\beta,\\alpha_{ij}^k\\ra=||\\beta||^2,\\quad\\forall \\mu_{ij}^k\\ne 0\\}, \\\\ \\\\\n\nW_{\\beta}=\\{\\mu\\in\nV:\\la\\beta,\\alpha_{ij}^k\\ra\\geq||\\beta||^2,\\quad\\forall\\mu_{ij}^k\\ne 0\\}. \\\\ \\\\\n\nY_{\\beta}=\\{ \\mu\\in W_{\\beta}:\\la\\beta,\\alpha_{ij}^k\\ra=||\\beta||^2,\\quad\\mbox{for\nat least one}\\; \\mu_{ij}^k\\ne 0\\}.\n\\end{array}\n$$\n\nNotice that $Z_{\\beta}$ is actually the eigenspace of $\\pi(\\beta)$ with eigenvalue\n$||\\beta||^2$, and so $\\mu\\in Z_{\\beta}$ if and only if\n$\\beta+||\\beta||^2I\\in\\Der(\\mu)$.  We also note that $W_{\\beta}$ is the direct sum\nof all the eigenspaces of $\\pi(\\beta)$ with eigenvalues $\\geq ||\\beta||^2$, and\nsince $Z_{\\beta}\\subset Y_{\\beta}\\subset W_{\\beta}$, they are all\n$\\G_{\\beta}$-invariant, where $\\G_{\\beta}$ is the centralizer of $\\beta$ in $\\G$.\n\nLet $\\g_{\\beta}$ denote the Lie algebra of $\\G_{\\beta}$, that is,\n$$\n\\g_{\\beta}=\\{\\alpha\\in\\g:[\\alpha,\\beta]=0\\},\n$$\nand let $G_{\\beta}$ be any reductive subgroup of $\\G$ with Lie algebra\n$\\ggo_{\\beta}$, the orthogonal complement of $\\beta$ in $\\g_{\\beta}$.\nThus\n\\begin{equation}\\label{descgb}\n\\g_{\\beta}=\\ggo_{\\beta}\\oplus\\RR\\beta,\n\\end{equation}\n\n\\noindent is an orthogonal decomposition,\n$$\n\\ggo_{\\beta}=(\\sog(n)\\cap\\ggo_{\\beta})\\oplus(\\sym(n)\\cap\\ggo_{\\beta})\n$$\nis a Cartan decomposition and\n$\\tg\\cap\\ggo_{\\beta}=\\{\\alpha\\in\\tg:\\la\\alpha,\\beta\\ra=0\\}$ is a maximal abelian\nsubalgebra of $\\sym(n)\\cap\\ggo_{\\beta}$.  Recall that if $\\sca_{\\beta}\\ne\\emptyset$\nthen $\\beta$ is rational and so such a reductive group $G_{\\beta}$ does exist.\n\n\\begin{definition}\nA vector $\\mu\\in V$ is called $G_{\\beta}$-{\\it semistable} if\n$0\\notin\\overline{G_{\\beta}.\\mu}$.\n\\end{definition}\n\n\\begin{theorem}\\label{zybeta}\\cite{standard}\nFor any $\\beta\\in\\tg$, the $\\G_{\\beta}$-invariant subsets\n$Z_{\\beta}^{ss}:=Z_{\\beta}\\cap\\sca_{\\beta}$ and\n$Y_{\\beta}^{ss}:=Y_{\\beta}\\cap\\sca_{\\beta}$ satisfy:\n\\begin{itemize}\n\\item[(i)] $\\sca_{\\beta}=\\Or(n).Y_{\\beta}^{ss}$.\n\n\\item[(ii)] $Y_{\\beta}^{ss}=\\{ \\mu\\in\\sca_{\\beta}:\\beta_{\\mu}=\\beta\\}$.\n\n\\item[(iii)] $Z_{\\beta}^{ss}$ is the set of $G_{\\beta}$-semistable vectors in $Z_{\\beta}$.\n\n\\item[(iv)] $Y_{\\beta}^{ss}$ is the set of $G_{\\beta}$-semistable vectors in $W_{\\beta}$.\n\\end{itemize}\n\\end{theorem}\n\nWe summarize in the following theorem the main properties of the $\\G$-invariant\nstratification of the vector space $V$ given above.\n\n\\begin{theorem}\\label{strata}\\cite{standard}\nThere exists a finite subset $\\bca\\subset\\tg^+$, and for each $\\beta\\in\\bca$ a\n$\\G$-invariant subset $\\sca_{\\beta}\\subset V$ (a {\\it stratum}) such that\n$$\nV\\smallsetminus\\{ 0\\}=\\bigcup_{\\beta\\in\\bca}\\sca_{\\beta} \\qquad \\mbox{(disjoint\nunion)}.\n$$\nIf $\\mu\\in\\sca_{\\beta}$ then\n\\begin{equation}\\label{adbeta}\n\\left\\la[\\beta,D],D\\right\\ra\\geq 0 \\qquad\\forall\\; D\\in\\Der(\\mu)\n\\qquad(\\mbox{equality holds}\\;\\Leftrightarrow [\\beta,D]=0)\n\\end{equation}\n\n\\noindent and\n\\begin{equation}\\label{betapos}\n\\beta+||\\beta||^2I \\quad\\mbox{is positive definite for all}\\; \\beta\\in\\bca\n\\;\\mbox{such that}\\; \\sca_{\\beta}\\cap\\nca\\ne\\emptyset.\n\\end{equation}\n\nIf in addition $\\mu\\in Y_{\\beta}^{ss}$, i.e.\n\\begin{equation}\\label{cond}\n\\min\\left\\{\\la\\beta,\\alpha_{ij}^k\\ra:\\mu_{ij}^k\\ne 0\\right\\}=||\\beta||^2, \\quad\n\\mbox{or equivalently}\\quad 0\\notin\\overline{G_{\\beta}.\\mu},\n\\end{equation}\n\n\\noindent then\n\\begin{equation}\\label{betaort}\n\\tr{\\beta D}=0 \\quad\\forall\\; D\\in\\Der(\\mu),\n\\end{equation}\n\n\\noindent and\n\\begin{equation}\\label{delta}\n\\left\\la\\pi\\left(\\beta+||\\beta||^2I\\right)\\mu,\\mu\\right\\ra\\geq 0 \\qquad\n(\\mbox{equality holds}\\;\\Leftrightarrow \\beta+||\\beta||^2I\\in\\Der(\\mu)).\n\\end{equation}\n\nMoreover, condition {\\rm (\\ref{cond})} is always satisfied by some $g.\\mu$ with\n$g\\in\\Or(n)$, for any $\\mu\\in\\sca_{\\beta}$.\n\\end{theorem}\n\n\n\\begin{remark} We note that (\\ref{betapos}) is actually the only result stated in this section where we really need $\\mu$ to be\na nilpotent Lie algebra, and not just any vector in $V$.  It is known for instance\nthat semisimple Lie algebras lie in the stratum $\\sca_{\\beta}$ for\n$\\beta=-\\tfrac{1}{n}I$, and consequently $\\beta+||\\beta||^2I=0$ (see \\cite{strata}).\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The stratification and the standard condition}\\label{proof}\n\nWe now apply the stratification described in Section \\ref{st} to prove that Einstein\nsolvmanifolds are all standard.\n\nLet $S$ be a solvmanifold, that is, a simply connected solvable Lie group endowed\nwith a left invariant Riemannian metric.  Let $\\sg$ be the Lie algebra of $S$ and\nlet $\\ip$ denote the inner product on $\\sg$ determined by the metric.  We consider\nthe orthogonal decomposition $\\sg=\\ag\\oplus\\ngo$, where $\\ngo=[\\sg,\\sg]$. Recall\nthat $S$ is called standard if $[\\ag,\\ag]=0$.  The mean curvature vector of $S$ is\nthe only element $H\\in\\ag$ which satisfies $\\la H,A\\ra=\\tr{\\ad{A}}$ for any\n$A\\in\\ag$.  If $B$ denotes the symmetric map defined by the Killing form of $\\sg$\nrelative to $\\ip$ then $B(\\ag)\\subset\\ag$ and $B|_{\\ngo}=0$ as $\\ngo$ is contained\nin the nilradical of $\\sg$.  The Ricci operator $\\Ricci$ of $S$ is given by (see for\ninstance \\cite[7.38]{Bss}):\n\\begin{equation}\\label{ricci}\n\\Ricci=R-\\unm B-S(\\ad{H}),\n\\end{equation}\n\n\\noindent where $S(\\ad{H})=\\unm(\\ad{H}+(\\ad{H})^t)$ is the symmetric part of\n$\\ad{H}$ and $R$ is the symmetric operator defined by\n\\begin{equation}\\label{R}\n\\la Rx,y\\ra=-\\unm\\displaystyle{\\sum\\limits_{ij}}\\la [x,x_i],x_j\\ra\\la [y,x_i],x_j\\ra\n+\\unc\\displaystyle{\\sum\\limits_{ij}}\\la [x_i,x_j],x\\ra\\la [x_i,x_j],y\\ra,\n\\end{equation}\n\n\\noindent for all $x,y\\in\\sg$, where $\\{ x_i\\}$ is any orthonormal basis of\n$(\\sg,\\ip)$.\n\nIt is proved in \\cite[Propositions 3.5, 4.2]{minimal} that $R$ is the only symmetric\noperator on $\\sg$ such that\n\\begin{equation}\\label{Rmm}\n\\tr{RE}=\\unc\\la \\pi(E)[\\cdot,\\cdot],[\\cdot,\\cdot]\\ra, \\qquad\\forall E\\in\\End(\\sg),\n\\end{equation}\n\n\\noindent where we are considering $[\\cdot,\\cdot]$ as a vector in\n$\\Lambda^2\\sg^*\\otimes\\sg$, $\\ip$ is the inner product defined in (\\ref{innV}) and\n$\\pi$ is the representation given in (\\ref{actiong}) (see the notation in Section\n\\ref{Tb} and replace $\\RR^n$ by $\\sg$).  This is equivalent to saying that\n$$\nm([\\cdot,\\cdot])=\\tfrac{4}{|| [\\cdot,\\cdot]||^2}R,\n$$\nwhere $m:\\Lambda^2\\sg^*\\otimes\\sg\\longrightarrow\\sym(\\sg)$ is the moment map for the\naction of $\\Gl(\\sg)$ on $\\Lambda^2\\sg^*\\otimes\\sg$ (see (\\ref{defmm})).  Thus the\n`anonymous' tensor $R$ in formula (\\ref{ricci}) for the Ricci operator is precisely\nthe value of the moment map at the Lie bracket $[\\cdot,\\cdot]$ of $\\sg$ (up to\nscaling).\n\nWe therefore obtain from (\\ref{ricci}) and (\\ref{Rmm}) that $S$ is an Einstein\nsolvmanifold with $\\Ricci=cI$, if and only if, for any $E\\in\\End(\\sg)$,\n\\begin{equation}\\label{einstein}\n\\tr{\\left(cI+\\unm B+S(\\ad{H})\\right)E}= \\unc\\la\n\\pi(E)[\\cdot,\\cdot],[\\cdot,\\cdot]\\ra.\n\\end{equation}\n\nLet $S$ be an Einstein solvmanifold with $\\Ricci=cI$.  We can assume that $S$ is not\nunimodular by using \\cite{Dtt}, thus $H\\ne 0$ and $\\tr{\\ad{H}}=||H||^2>0$.  By\nletting $E=\\ad{H}$ in (\\ref{einstein}) we get\n\\begin{equation}\\label{c}\nc=-\\tfrac{\\tr{S(\\ad{H})^2}}{\\tr{S(\\ad{H})}}<0.\n\\end{equation}\n\nIn order to apply the results in Section \\ref{st}, we identify $\\ngo$ with $\\RR^n$\nvia an orthonormal basis $\\{ e_1,...,e_n\\}$ of $\\ngo$ and we set\n$\\mu:=[\\cdot,\\cdot]|_{\\ngo\\times\\ngo}$.  In this way, $\\mu$ can be viewed as an\nelement of $\\nca\\subset V$. If $\\mu\\ne 0$ then $\\mu$ lies in a unique stratum\n$\\sca_{\\beta}$, $\\beta\\in\\bca$, by Theorem \\ref{strata}, and it is easy to see that\nwe can assume (up to isometry) that $\\mu$ satisfies (\\ref{cond}), so that one can\nuse all the additional properties stated in the theorem.  In particular, the\nfollowing crucial technical result follows.  Consider $E_{\\beta}\\in\\End(\\sg)$\ndefined by\n$$\nE_{\\beta}=\\left[\\begin{smallmatrix} 0&0\\\\\n0&\\beta+||\\beta||^2I\\end{smallmatrix}\\right],\n$$\nthat is, $E|_{\\ag}=0$ and $E|_{\\ngo}=\\beta+||\\beta||^2I$.\n\n\\begin{lemma}\\label{pie}\nIf $\\mu\\in\\sca_{\\beta}$ satisfies {\\rm (\\ref{cond})} then\n$\\la\\pi(E_{\\beta})[\\cdot,\\cdot],[\\cdot,\\cdot]\\ra\\geq 0$.\n\\end{lemma}\n\nWe then apply (\\ref{einstein}) to $E_{\\beta}\\in\\End(\\sg)$ and obtain from Lemma\n\\ref{pie}, (\\ref{c}), (\\ref{trb}) and (\\ref{betaort}) that\n$$\n\\tr{S(\\ad{H})^2}\\tr{E_{\\beta}^2}\\leq (\\tr{S(\\ad{H})E_{\\beta}})^2,\n$$\na `backwards' Cauchy-Schwartz inequality.  This turns all inequalities which\nappeared in the proof of Lemma \\ref{pie} into equalities, in particular:\n$$\n\\unc\\sum_{rs}\\la(\\beta+||\\beta||^2I)[A_r,A_s],[A_r,A_s]\\ra=0,\n$$\nwhere $\\{ A_i\\}$ is an orthonormal basis of $\\ag$.  We finally get that $\\ag$ is\nabelian since $\\beta+||\\beta||^2I$ is positive definite by (\\ref{betapos}).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The stratification and Einstein solvmanifolds via closed orbits}\\label{smadre}\n\nWe shall describe in this section some other applications of the strata defined in\nSection \\ref{st} to the study of Einstein solvmanifolds.\n\nLet $\\ngo$ be a nonabelian nilpotent Lie algebra of dimension $n$.  We fix any basis\n$\\{ X_1,...,X_n\\}$ of $\\ngo$ and consider the corresponding structural constants:\n$$\n[X_i,X_j]=\\sum_{k=1}^n c_{ij}^kX_k, \\qquad 1\\leq i<j\\leq n.\n$$\nLet $\\beta$ denote the unique element of minimal norm in the convex hull of the set\n$\\{\\alpha_{ij}^k:c_{ij}^k\\ne 0\\}$, where $\\alpha_{ij}^k$ is the diagonal $n\\times n$\nmatrix $-E_{ii}-E_{jj}+E_{kk}$.  Notice that $\\tr{\\beta}=-1$, and so $\\beta$ is\nalways nonzero.  We define the Lie algebra\n$$\n\\ggo_{\\beta}=\\{\\alpha\\in\\End(\\ngo):[\\alpha,\\beta]=0, \\quad\\tr{\\alpha\\beta}=0\\},\n$$\nand take any reductive subgroup $G_{\\beta}$ of $\\Gl(\\ngo)$ with Lie algebra\n$\\ggo_{\\beta}$ (existence is guaranteed by rationality of $\\beta$).\n\nRecall that the Lie bracket $\\lb$ of $\\ngo$ belongs to the vector space\n$\\Lambda^2\\ngo^*\\otimes\\ngo$ of skew-symmetric bilinear maps from $\\ngo\\times\\ngo$\nto $\\ngo$, on which $\\Gl(\\ngo)$ is acting naturally by\n$g.\\lb=g[g^{-1}\\cdot,g^{-1}\\cdot]$.\n\n\\begin{theorem}\\label{madre}\\cite{einsteinclosed}\nLet $\\ngo$ be a nonabelian nilpotent Lie algebra and for any basis $\\{\nX_1,...,X_n\\}$ of $\\ngo$ consider $\\beta$ and $G_{\\beta}\\subset\\Gl(\\ngo)$ as defined\nabove.\n\n\\begin{itemize}\n\\item[(i)] If the orbit $G_{\\beta}.\\lb$ is closed in $\\Lambda^2\\ngo^*\\otimes\\ngo$ then $\\ngo$ is an Einstein nilradical and $\\beta+||\\beta||^2I\\in\\Der(\\ngo)$.\n\n\\item[(ii)] If $\\ngo$ is an Einstein nilradical, $\\beta+||\\beta||^2I\\in\\Der(\\ngo)$ and $0\\notin\\overline{G_{\\beta}.\\lb}$, then the orbit $G_{\\beta}.\\lb$ is closed in $\\Lambda^2\\ngo^*\\otimes\\ngo$ .\n\\end{itemize}\n\\end{theorem}\n\n\n\\begin{remark}\nThe following example shows that condition $\\beta+||\\beta||^2I\\in\\Der(\\ngo)$ is\nnecessary in part (ii) of Theorem \\ref{madre}.  Let $\\ngo$ be the $4$-dimensional\n$3$-step nilpotent Lie algebra with Lie bracket given by\n$$\n[X_1,X_2]=X_3+X_4, \\qquad [X_1,X_3]=X_4.\n$$\nIt is easy to see that $\\beta=(-1,-\\unm,0,\\unm)$ and\n$0\\notin\\overline{G_{\\beta}.\\lb}$.  If $\\lambda$ is defined by\n$$\n\\lambda(X_1,X_2)=X_3, \\qquad \\lambda(X_1,X_3)=X_4,\n$$\nthen $\\lambda\\in\\Gl_4(\\RR).\\lb$, $m(\\lambda)=\\beta$ and\n$\\beta+||\\beta||^2I\\in\\Der(\\lambda)$, from which follows that $\\lambda$ is a\nnilsoliton and so $\\ngo$ is an Einstein nilradical.  However,\n$$\n\\lambda=\\lim_{t\\to\\infty} e^{-t\\alpha}.\\lb\\in\\overline{G_{\\beta}.\\lb}, \\qquad\n\\mbox{for}\\quad\\alpha=(1,0,1,2),\n$$\nand thus $G_{\\beta}.\\lb$ is not closed.  Indeed, $\\lambda\\notin G_{\\beta}.\\lb$ since\n$\\beta+||\\beta||^2I\\in\\Der(\\lambda)$ and $\\beta+||\\beta||^2I\\notin\\Der(\\ngo)$.\n\\end{remark}\n\nRecall that $\\beta$ has entries in $\\QQ$ and so if $\\beta\\in\\tg^+$ and has\neigenvalues $b_1<...<b_r$ with multiplicities $n_1,...,n_r$, respectively, then one can for instance take the reductive group $G_{\\beta}$ given by\n$$\nG_{\\beta}=\\left\\{\\left[\\begin{smallmatrix} g_1&& \\\\ &\\ddots&\\\\\n&&g_r\\end{smallmatrix}\\right]:\\det{g_1}^{mb_1}...\\det{g_r}^{mb_r}=1, \\quad\ng_i\\in\\Gl_{n_i}(\\RR)\\right\\},\n$$\nwhere $m$ is the least common multiple of the denominators of the $b_i$'s.  If\n$\\sigma$ is a permutation of $\\{ 1,...,n\\}$ then the new basis $\\{\nX_{\\sigma(1)},...,X_{\\sigma(n)}\\}$ of $\\ngo$ has structural constants\n$$\n[X_{\\sigma(i)},X_{\\sigma(j)}]=\\sum_{k=1}^n\nc_{\\sigma(i)\\sigma(j)}^{\\sigma(k)}X_{\\sigma(k)}, \\qquad 1\\leq i<j\\leq n,\n$$\nand so the new $\\beta$ has eigenvalues $b_{\\sigma^{-1}(1)},...,b_{\\sigma^{-1}(n)}$\nwith respective eigenvectors $X_{\\sigma(1)},...,X_{\\sigma(n)}$ (see the beginning of\nthe proof of \\cite[Theorem 2.10]{standard}).  Therefore, we can always assume that\n$\\beta\\in\\tg^+$, up to just a permutation of the basis $\\{ X_i\\}$.  Otherwise, if\none insists on keeping the original basis, one may take as $G_{\\beta}$ the group\n$h^{-1}G_{h\\beta h^{-1}}h$, where $h\\in\\Gl(\\ngo)$ is a permutation matrix such that\n$h\\beta h^{-1}\\in\\tg^+$.\n\nThe following two results show the interplay between the stratum $\\beta$ and the pre-Einstein derivation $\\phi$ (see Definition \\ref{preE}), providing in particular a new method to compute $\\phi$.\n\n\\begin{lemma}\\label{madre2}\\cite{einsteinclosed}\nLet $\\ngo$ be a nonabelian nilpotent Lie algebra and for any basis $\\{\nX_1,...,X_n\\}$ of $\\ngo$ consider $\\beta$ and $G_{\\beta}\\subset\\Gl(\\ngo)$ as defined\nabove.\n\n\\begin{itemize}\n\\item[(i)] $\\lb\\in\\sca_{\\beta}$ if and only if $0\\notin\\overline{G_{\\beta}.\\lb}$.\n\n\\item[(ii)] If $\\beta+||\\beta||^2I\\in\\Der(\\ngo)$ and $0\\notin\\overline{G_{\\beta}.\\lb}$ then\n$$\n\\phi:=\\tfrac{1}{||\\beta||^2}(\\beta+||\\beta||^2I)\n$$\nis a pre-Einstein derivation of $\\ngo$ and $G_{\\phi}.\\lb$ is closed if and only if\n$G_{\\beta}.\\lb$ is closed.\n\\end{itemize}\n\\end{lemma}\n\n\n\\begin{remark}\nWe conclude from Lemma \\ref{madre2} that Theorem \\ref{madreN} and Theorem\n\\ref{madre} are equivalent.\n\\end{remark}\n\n\\begin{remark}\nThe stratum a given nilpotent Lie algebra belongs to provides useful information on\nits automorphism group.  Indeed, let $\\ngo$ be a nilpotent Lie algebra and for any\nbasis $\\{ X_1,...,X_n\\}$ of $\\ngo$ consider $\\beta$ as defined above. If\n$\\lb\\in\\sca_{\\beta}$, then $\\Aut(\\ngo)\\subset P_{\\beta}$ by (\\ref{aut}).\n\\end{remark}\n\n\\begin{lemma}\\label{madre3}\\cite{einsteinclosed}\nLet $\\ngo$ be a nonabelian nilpotent Lie algebra and let $\\phi$ be a pre-Einstein\nderivation of $\\ngo$ with basis of eigenvectors $\\{ X_1,...,X_n\\}$ and define\n$$\n\\beta:=\\tfrac{1}{n-\\tr{\\phi}}(\\phi-I).\n$$\nThen $\\lb\\in\\sca_{\\beta}$ if and only if $0\\notin\\overline{G_{\\beta}.\\lb}$, and in\nthat case, $\\beta=\\mcc(\\{\\alpha_{ij}^k:c_{ij}^k\\ne 0\\})$.\n\\end{lemma}\n\nIt follows from Lemma \\ref{madre3} that if $\\phi$ is a pre-Einstein derivation of\n$\\ngo$ then $\\phi>0$ (see (\\ref{betapos})) and $\\ad{\\phi}\\geq 0$ (see\n(\\ref{adbeta})) are necessary conditions in order to have\n$0\\notin\\overline{G_{\\beta}.\\lb}$ (i.e. $\\lb\\in\\sca_{\\beta}$).  These conditions are not however sufficient (compare with the paragraph below Theorem \\ref{eupreE}).  For instance, any free nilpotent Lie algebra which is not an Einstein nilradical provides a counterexample (see \\cite[Remark 2]{Nkl3}).  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Open problems}\\label{op}\n\nLet $\\ngo$ be an $\\NN$-graded nilpotent Lie algebra.\n\n\\begin{enumerate}\n\\item {\\bf Obstructions}. To find algebraic necessary conditions on $\\ngo$ to be an Einstein nilradical.\n\n\\item {\\bf Existence}. Are there algebraic conditions on $\\ngo$ which are sufficient to be an Einstein nilradical?\n\n\\item Does the assertion `$\\ngo$ is an Einstein nilradical' have probability $1$ in some sense?\n\n\\item Does the assertion `$\\ngo$ is not an Einstein nilradical' have probability $1$ in some sense?\n\n\\item Assume that $\\ngo$ is an Einstein nilradical with Lie bracket $\\mu_0\\in\\nca$, and consider the flow $\\mu(t)$ defined in (\\ref{flow}) with $\\mu(0)=\\mu_0$.  Does $\\lambda=\\lim\\limits_{t\\to\\infty}\\mu(t)$ necessarily belong to $\\G.\\mu_0$? (this would provide a nice obstruction).\n\n\\item To exhibit an explicit example or prove the existence of a nilpotent Lie algebra which does not admit a nice basis (see Definition \\ref{nice}).\n\n\\item Are there only finitely many $\\NN$-graded filiform Lie algebras which are not Einstein nilradicals in each dimension?\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix: Real geometric invariant theory}\\label{git}\n\n\nLet $G$ be a real reductive group acting linearly on a finite dimensional real\nvector space $V$ via $(g,v)\\mapsto g.v$, $g\\in G,v\\in V$. The precise definition of\nour setting is the one considered in \\cite{RchSld}. We also refer to \\cite{EbrJbl},\nwhere many results from geometric invariant theory are adapted and proved over\n$\\RR$.\n\nThe Lie algebra $\\ggo$ of $G$ also acts linearly on $V$ by the derivative of the\nabove action, which will be\n denoted by $(\\alpha,v)\\mapsto\\pi(\\alpha)v$, $\\alpha\\in\\ggo$, $v\\in V$. We consider\n a Cartan decomposition\n$\\ggo=\\kg\\oplus\\pg$, where $\\kg$ is the Lie algebra of a maximal compact subgroup\n$K$ of $G$. Endow $V$ with a fixed from now on $K$-invariant inner product $\\ip$\nsuch that $\\pg$ acts by symmetric operators, and endow $\\pg$ with an\n$\\Ad(K)$-invariant inner product $\\ipp$.\n\nThe function $m:V\\smallsetminus\\{ 0\\}\\longrightarrow\\pg$ implicitly defined by\n$$\n(m(v),\\alpha)=\\isn\\la\\pi(\\alpha)v,v\\ra, \\qquad \\forall\\alpha\\in\\pg,\\; v\\in V,\n$$\nis called the {\\it moment map} for the representation $V$ of $G$.  Since\n$m(cv)=m(v)$ for any nonzero $c\\in\\RR$, we also may consider the moment map on the\nprojective space of $V$, $m:\\PP V\\mapsto\\pg$, with the same notation and definition\nas above for $m([v])$, $[v]$ the class of $v$ in $\\PP V$.  It is easy to see that\n$m$ is $K$-equivariant: $m(k.v)=\\Ad(k)m(v)$ for all $k\\in K$.\n\nIn the complex case (i.e. for a complex representation of a complex reductive\nalgebraic group), under the natural identifications $\\pg=\\pg^*=(\\im\\kg)^*=\\kg^*$,\nthe function $m$ is precisely the moment map from symplectic geometry, corresponding\nto the Hamiltonian action of $K$ on the symplectic manifold $\\PP V$ (see for\ninstance the survey \\cite{Krw2} or \\cite[Chapter 8]{Mmf} for further information).\nFor real actions, this nice interplay with symplectic geometry is lost ($\\PP V$\ncould even be odd dimensional), but the moment map is nevertheless a very natural\nobject attached to a real representation encoding a lot of information on the\ngeometry of $G$-orbits and the orbit space $V\/G$.\n\nLet $\\mca=\\mca(G,V)$ denote the set of {\\it minimal vectors}, that is,\n$$\n\\mca=\\{ v\\in V: ||v||\\leq ||g.v||\\quad\\forall g\\in G\\}.\n$$\nFor each $v\\in V$ define\n$$\n\\rho_v:G\\mapsto\\RR, \\qquad \\rho_v(g)=||g.v||^2.\n$$\nIn \\cite{RchSld}, it is shown that the nice interplay between closed orbits and\nminimal vectors discovered in \\cite{KmpNss} for actions of complex reductive\nalgebraic groups, is still valid in the real situation.\n\n\\begin{theorem}\\cite{RchSld}\\label{RS}\nLet $V$ be a real representation of a real reductive group $G$, and let $v\\in V$.\n\\begin{itemize}\n\\item[(i)] The orbit $G.v$ is closed if and only if $G.v$ meets $\\mca$.\n\n\\item[(ii)] $v\\in\\mca$ if and only if $\\rho_v$ has a critical point at $e\\in G$.\n\n\\item[(iii)] If $v\\in\\mca$ then $G.v\\cap\\mca=K.v$.\n\n\\item[(iv)] The closure $\\overline{G.v}$ of any orbit $G.v$ always meets $\\mca$.  Moreover, there always exists\n$\\alpha\\in\\pg$ such that $\\lim\\limits_{t\\to\\infty}\\exp(-t\\alpha).v=w$ exists and\n$G.w$ is closed.\n\n\\item[(v)] $\\overline{G.v}\\cap\\mca$ is a single $K$-orbit, or in other words, $\\overline{G.v}$ contains a unique\nclosed $G$-orbit.\n\\end{itemize}\n\\end{theorem}\n\nAs usual in the real case, classical topology of $V$ is always considered rather\nthan Zarisky topology, unless explicitly indicated.\n\nLet $(\\dif\\rho_v)_e:\\ggo\\mapsto\\RR$ denote the differential of $\\rho_v$ at the\nidentity $e$ of $G$. It follows from the $K$-invariance of $\\ip$ that\n$(\\dif\\rho_v)_e$ vanishes on $\\kg$, and so we can assume that\n$(\\dif\\rho_v)_e\\in\\pg^*$, the vector space of real-valued functionals on $\\pg$.  If\nwe identify $\\pg$ and $\\pg^*$ by using $\\ipp$, then it is easy to see that\n$$\nm(v)=\\tfrac{1}{2||v||^2}(\\dif\\rho_v)_e.\n$$\nThe moment map at $v$ is therefore an indicator of the behavior of the norm along\nthe orbit $G.v$ in a neighborhood of $v$.  It follows from Theorem \\ref{RS}, (ii)\nthat\n$$\n\\mca\\smallsetminus\\{ 0\\} = \\{ v\\in V\\smallsetminus\\{ 0\\}: m(v)=0\\}.\n$$\nThus if we consider the functional square norm of the moment map\n\\begin{equation}\\label{norm}\nF:V\\smallsetminus\\{ 0\\}\\mapsto\\RR, \\qquad  F(v)=||m(v)||^2,\n\\end{equation}\nwhich is a $4$-degree homogeneous polynomial times $||v||^{-4}$,  $\\mca\\setminus\\{ 0\\}$\ncoincides with the set of zeros of $F$. It then follows from Theorem \\ref{RS}, parts\n(i) and (iii), that a nonzero orbit $G.v$ is closed if and only if $F(w)=0$ for some\n$w\\in G.v$, and in that case, the set of zeros of $F|_{G.v}$ coincides with $K.v$.\nRecall that $F$ is scaling invariant and so it is actually a function on any sphere\nof $V$ or on $\\PP V$.\n\nA natural question arises: what is the role played by the remaining critical points\nof $F$ (i.e. those for which $F(v)>0$) in the study of the $G$-orbit space of the\naction of $G$ on $V$?. This was independently studied in \\cite{Krw1} and \\cite{Nss}\nin the complex case, who have shown that non-minimal critical points still enjoy\nmost of the nice properties of minimal vectors stated in Theorem \\ref{RS}. In the\nreal case, the analogues of some of these results have been proved in \\cite{Mrn}.\n\nWe endow $\\PP V$ with the Fubini-Study metric defined by $\\ip$ and denote by\n$x\\mapsto \\alpha_x$ the vector field on $\\PP V$ defined by $\\alpha\\in\\ggo$ via the\naction of $G$ on $\\PP V$, that is, $\\alpha_x=\\ddt|_0\\exp(t\\alpha).x$. We will also\ndenote by $F$ the functional $F:\\PP V\\longrightarrow\\RR$, $F([v])=||m([v])||^2$.\n\n\n\\begin{lemma}\\cite{Mrn}\\label{marian2}\nThe gradient of the functional $F:V\\setminus\\{ 0\\}\\longrightarrow\\RR$ is given by\n$$\n\\grad(F)_{v}=\\tfrac{4}{||v||^2}\\Big(\\pi(m(v))v-||m(v)||^2v\\Big), \\qquad v\\in\nV\\setminus\\{ 0\\},\n$$\nand for $F:\\PP V\\longrightarrow\\RR$ we have that\n$$\n\\grad(F)_{[v]}=4m([v])_{[v]}, \\qquad [v]\\in\\PP V.\n$$\n\\end{lemma}\n\nTherefore, $v$ is a critical point of $F$ (or equivalently, of $F|_{G.v}$) if and\nonly if $v$ is an eigenvector of $\\pi(m(v))$, and $[v]$ is a critical point of $F$\n(or equivalently, of $F|_{G.[v]}$) if and only if $\\exp{tm([v])}$ fixes $[v]$.\n\n\\begin{theorem}\\cite{Mrn}\\label{marian}\nLet $V$ be a real representation of a real semisimple Lie group $G$.\n\\begin{itemize}\n\\item[(i)] If $x\\in\\PP V$ is a critical point of $F$ then the functional $F|_{G.x}$ attains its minimum value at\n$x$.\n\n\\item[(ii)] If nonempty, the critical set of $F|_{G.x}$ consists of a single $K$-orbit.\n\\end{itemize}\n\\end{theorem}\n\n\n\\begin{definition}\\label{stable}\nA nonzero vector $v\\in V$ is called {\\it unstable} if $0\\in\\overline{G.v}$, and {\\it\nsemistable} otherwise.  If a semistable vector has in addition compact isotropy\nsubgroup then it is called {\\it stable}.\n\\end{definition}\n\nIf the orbit of a nonzero $v\\in V$ is closed then $v$ is clearly semistable. More\ngenerally, $v\\in V$ is semistable if and only if the unique (up to $K$-action) zero\nof $F$ which belongs to $\\overline{G.v}$ is a nonzero vector. On the contrary, any\ncritical point of $F$ which is not a zero of $F$ is unstable.  Indeed, if\n$\\pi(m(v))v=cv$, $c=||m(v)||^2>0$ (see Lemma \\ref{marian2}), then\n$$\n\\lim_{t\\to\\infty}\\exp(-tm(v)).v=\\lim_{t\\to\\infty}e^{-tc}v=0,\n$$\nand so $0\\in\\overline{G.v}$.  Thus the study of critical points of $F$ other than\nzeroes gives useful information on the orbit space structure of the subset of all\nunstable vectors, often called the {\\it nullcone} of $V$.\n\n\n\\begin{example}\\label{hompol1}\nLet us consider the example of $G=\\Sl_3(\\RR)$ and $V=P_{3,3}(\\RR)$, the vector space\nof all homogeneous polynomials of degree $3$ on $3$ variables.  The action is given\nby a linear change of variables on the left\n$$\n(g.p)(x_1,x_2,x_3)=p\\left(g^{-1}\\left[\\begin{smallmatrix} x_1\\\\ x_2\\\\ x_3\n\\end{smallmatrix}\\right]\\right), \\qquad \\forall g\\in\\Sl_3(\\RR),\\quad p\\in P_{3,3}(\\RR).\n$$\nIt follows that $\\ggo=\\slg_3(\\RR)$, $K=\\SO(3)$, $\\kg=\\sog(3)$ and $\\pg=\\sym_0(3)$ is\nthe space of traceless symmetric $3\\times 3$ matrices.  As an $\\Ad(K)$-invariant\ninner product on $\\pg$ we take $(\\alpha,\\beta)=\\tr{\\alpha\\beta}$, and it is easy to\nsee that the inner product $\\ip$ on $V$ for which the basis of monomials\n$$\n\\{ x^D:=x_1^{d_1}x_2^{d_2}x_3^{d_3}:d_1+d_2+d_3=3,\\; D=(d_1,d_1,d_3)\\}\n$$\nis orthogonal and\n$$\n||x^D||^2=d_1!d_2!d_3!, \\qquad\\forall D=(d_1,d_2,d_3),\n$$\nsatisfies the required conditions.  Let $E_{ij}$ denote as usual the $n\\times n$\nmatrix whose only nonzero coefficient is a $1$ in the entries $ij$.  Since\n$$\n\\pi(E_{ij})p=\\ddt|_0p(e^{-tE_{ij}}\\cdot )=-x_j\\tfrac{\\partial p}{\\partial x_i},\n$$\nwe obtain that the moment map $m:P_{3,3}(\\RR)\\longrightarrow\\sym_0(3)$ is given by\n$$\nm(p)=I-\\tfrac{1}{||p||^2}\\left[\\la x_j\\tfrac{\\partial p}{\\partial x_i},p\\ra\\right].\n$$\nWe are using here that $\\la x_j\\tfrac{\\partial p}{\\partial x_i},p\\ra=\\la\nx_i\\tfrac{\\partial p}{\\partial x_j},p\\ra$ for all $i,j$.\n\nIt is also easy to see that the action of a diagonal matrix $\\alpha\\in\\slg_3(\\RR)$\nwith entries $a_1,a_2,a_3$ is given by\n\\begin{equation}\\label{diagact}\n\\pi(\\alpha)x^D=-\\left(\\sum_{i=1}^3a_id_i\\right)x^D, \\qquad\\forall D=(d_1,d_2,d_3).\n\\end{equation}\nA first general observation is that any monomial is a critical point of $F$. Indeed,\n$$\nm(x^D)=\\left[\\begin{smallmatrix} 1-d_1&&\\\\ &1-d_2&\\\\ &&1-d_3\n\\end{smallmatrix}\\right],\n$$\nand so $x^D$ is an eigenvector of $m(x^D)$ with eigenvalue $F(x^D)=\\sum d_i^2-1$\n(see Lemma \\ref{marian2}).  It follows that $m(p)=0$ for $p=x_1x_2x_3$, that is, $p$\nis a minimal vector and its $\\Sl_3(\\RR)$-orbit is therefore closed.  We also have in\nsuch case that $p_1=p+x_1^3$ is a semistable vector whose orbit is not closed.\nIndeed, by acting by diagonal elements with entries $t,\\tfrac{1}{t},1$ we get that\n$p+t^dx_1^3\\in\\Sl_3(\\RR).p$ for all $t\\ne 0$ and so $p\\in\\overline{\\Sl_3(\\RR).p_1}$\n(recall that $p$ and $p_1$ can never lie in the same orbit since they have\nnon-isomorphic isotropy subgroups).\n\nFor the vector $q=x_1^2x_3+x_1x_2^2$ we have that\n$$\nm(q)=\\left[\\begin{smallmatrix} -\\tfrac{1}{2}&&\\\\ &0&\\\\\n&&\\tfrac{1}{2}\n\\end{smallmatrix}\\right].\n$$\nIt follows from (\\ref{diagact}) that $\\pi(m(q))q=\\tfrac{1}{2}q$ proving that $q$ is\na critical point of $F$ with critical value $F(q)=\\tfrac{1}{2}>0$.  On the other\nhand, the family $p_{a,b}=ax_1^2x_3+bx_2^3$, $a,b\\ne 0$, lie in a single orbit and\n$$\nm(p_{a,b})=I-\\tfrac{1}{2a^2+6b^2} \\left[\\begin{smallmatrix} 4a^2&&\\\\ &18b^2&\\\\\n&&2a^2\n\\end{smallmatrix}\\right].\n$$\nIt is then easy to see by using (\\ref{diagact}) that $p_{a,b}$ is a critical point\nif and only if $5a^2=27b^2$ and the critical value equals $\\tfrac{155}{49}-3$, a\nnumber smaller than $\\unm$. In particular, $p_{a,b}$ can not be in the closure of\nthe orbit of $q$ by Theorem \\ref{marian}, (i).\n\\end{example}\n\n\n\n\n\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Autographic Visualization}\nWe present autographic visualization as a counter-model to data visualization, focusing on material rather than data encoded in numbers and symbols. It is a counter-model not in the sense that it excludes data visualization---there is considerable overlap between the two models---but in the sense that it clarifies the characteristics of each model through this juxtaposition. \n\nWe refer to autographic visualization as a set of techniques for revealing material phenomena as visible traces and guiding their interpretation. Designing an autographic display means setting the conditions that allow a trace to emerge. We understand a trace as any transient or persistent configuration of matter presenting itself to the senses.\n\nA central goal of autographic visualization is to make environmental information legible and the processes of data collection and their underlying causalities experiential and accountable. Since a data set is the outcome rather than the starting point (Fig.~\\ref{fig:sketch}, middle), autographic visualization cannot rely on the representation of data. It is non-representational: rather than re-presenting something absent, the phenomenon presents itself. Autographic visualizations can be accidental, such as the desire paths across grass areas in the city or the uneven traces of wear on a staircase or a computer keyboard. But in general, they are the outcome of design operations that aim to reveal, isolate, amplify, conserve, and present material traces as records of past processes and events. For example, the display of the sundial~\\cite{Peters2015Marvelous} is a product of a natural phenomenon untouched by human intervention. At the same time, it is a computational device designed to calculate not only the time of day but also month and season. Its display often incorporates a calendar---a classic form of data visualization---geometrically aligned with the path of the sun in the particular location.\n\n\nAutographic visualization techniques draw from a long history of epistemic and material cultures that deal with the visual interpretation of traces, symptoms, or signatures as forms of material evidence. Its practices range from scientific experimentalism to ancient techniques of hunting, navigating, and healing. This paper is based on two premises. First, the diverse space of practices engaging with traces can be generalized into several distinct design operations. And second, these visual operations of autographic visualization are closely related to the modes of exploration facilitated by information visualization.\n\nWhile the interpretation of medical symptoms, the design of experimental systems, or the design of shape-changing materials are usually considered in isolation, autographic visualization identifies common visual strategies across all of these practices. Table~\\ref{infovsauto} summarizes the main differences between InfoVis and autographic visualization. To avoid confusion, we use \\textit{symbolic data} to refer to digital data.\n\n\\begin{table}\n\t\\caption{Comparison between InfoVis and Autographic Visualization}\\label{infovsauto}\\centering\\scriptsize\\medskip\n\t\\begin{tabular}\n\t\t{L{1.8cm} L{2.7cm} L{3cm}} & InfoVis & Autographic Visualization\\\\\n\t\t\\midrule Role of symbolic data & Begins with data & Ends with data\\medskip\\\\\n\t\tFocus & Inwards: reveals patterns within data & Outwards: reveals the process of data generation\\medskip\\\\\n\t\tRole of representation & Representational: visual marks stand for a phenomenon & Non-representational: the phenomenon presents itself\\medskip\\\\\n\t\tRole of design & Mapping data to visual variables \\& layouts & Elucidating qualities of a phenomenon\\\\\n\t\\end{tabular}\n\\end{table}\n\nDespite these differences, there is a close kinship between autographic and information visualization\u2014both are rooted in the same visual culture and take advantage of similar perceptual mechanisms~\\cite{Noe2004Action,Haroz2006Natural}. Foundational literature in visualization and HCI frequently invokes natural phenomena as metaphor or inspiration. Whether charts and graphs should be viewed as abstractions of natural phenomena based on shared organizational principles or as metaphorical references will not be elaborated here. However, it is worth noting that both InfoVis and autographic visualization were at one point considered to be the same approach. Etienne-Jules Marey's late 19\\textsuperscript{th}~century \\emph{Methode Graphique} encompasses both the charting of statistical information and the construction of self-registering devices for recording blood pressure, the flight of birds, or the turbulence of air. In pursuit of his declared goal to capture \"the language of the phenomena themselves,\" Marey's pioneering work included autographic devices such as the wind tunnel and, most prominently, his invention of chronophotography~\\cite{Marey1878La}, inspiring other non-mimetic uses of photography~\\cite{felton_photoviz:_2016}.\n\nAnalog information visualizations are often at the same time physical traces. Mechanically excited by seismic movements, a simple seismometer produces a classic line chart. This dual role creates a conceptual ambiguity that blurs the boundary between InfoVis and autographic visualization. If the line chart produced by the seismometer is a physical trace, what about a satellite image, what about the electrical charge generated by a digital sensor connected to a computer? The difference between an analog and a digital medium is not relevant for the underlying causality since both devices operate in a deterministic way. From this perspective, many symbolic datasets indeed share the character of a material trace; the material aspects of data collection inscribe themselves, sometimes unintentionally, into the data set~\\cite{Dourish2013Media}. \n\nThis can be illustrated through a public data set of GPS traces of drop-off and pick-up locations of NYC taxis. Plotting the data set in Cartesian space yields, unsurprisingly, a figure that resembles a map of the city. Some areas on this map, however, appear blurrier than others: an artifact of diminished GPS reception between tall buildings. In other words, the two-dimensional geographic datum contains hidden information about the three-dimensional shape of the city. But this latent information is only accessible if the materiality of GPS is understood and considered. A material reading that takes advantage of such artifacts, or \"dust\" in the data~\\cite{Loukissas2016Taking}, differs from a classic approach of cleaning the data set by excluding obvious errors, e.g., points that fall into the ocean or within buildings. While information- and autographic visualization may differ in the length of the causal chains that link phenomenon and representation, the autographic perspective can to some extent also be applied to digital information, further explored in Section~\\ref{performative}.\n\n\\begin{figure*}\n\t[t] \\centering \n\t\\includegraphics[width=18cm]{collage.jpg} \n\t\\caption{Autographic visualizations and their design operations (Table~\\ref{designop}), top-left to bottom-right: (a)~Cyanometer, a device for measuring the blueness of the sky, \\emph{framing} and \\emph{encoding}~\\cite{Saussure1791Description}; (b)~Mercury-in-glass thermometer, \\emph{constraining} and \\emph{encoding}; (c)~filter for sampling airborne particulate matter, \\emph{aggregating}; (d) southern blot for DNA electrophoresis, \\emph{separating, registering}; (e)~EJ Marey's smoke machine to visualize airflow, \\emph{coupling}; (f)~Campbell-Stokes sunshine recorder, \\emph{registering, encoding}; (g)~Chladni figure revealing sound waves, \\emph{coupling, registering}; (h)~a planning diagram for neuro-surgery, \\emph{annotating}~\\cite{Wulfingen2017Traces:}; (i)~pedocomparator for sampling and comparing soil samples, \\emph{aggregating, encoding}~\\cite{Latour1999Pandora's}; (j)~reagent strips for ozone detection, \\emph{registering, encoding}.}\\label{collage}\n\\end{figure*}\n\n\\section{Theoretical Perspectives on Material Traces}\nClassic semiology, in many ways foundational for information design and visualization~\\cite{Bertin1983Semiology,Robinson1975Map,Cleveland1987Graphical}, offers a framework for analyzing physical traces. Philosopher Charles Sanders Peirce differentiates \\emph{symbol}, \\emph{icon}, and \\emph{index} as three kinds of (non-exclusive) relationships a sign can have with a corresponding object or concept in the world. The symbol is linked to its object based on arbitrary convention; the icon is based on a relationship of resemblance; and in the case of the index, the relationship is an existential connection such as the causal link between a footprint and the person that left it~\\cite{Peirce1998What}. While \\emph{icons} (Peirce includes diagrams in this category) and \\emph{symbols} play a prominent role in information design and visualization, indexical signs appear only implicitly; for example, as patterns and signals in data sets. \n\nIndexical phenomena have been explored in HCI, ubiquitous computing, and to a lesser extent, information visualization~\\cite{Moere2009Analyzing,Offenhuber2012Kuleshov's,Schofield2013Indexicality,Offenhuber2015Indexical}. The application of indexicality to physical traces, however, is somewhat limited by the central role of linguistic concepts in semiotic theory. Peirce, for example, describes a pointing finger, a physical trace, and the word \"there\" as equivalent examples of indexical signs. By relying on the sign as the universal vehicle of meaning, semiotic perspectives reduce the trace to its role as a signifier. Scholars have critiqued the semiotic model of representation, in which meaning is conveyed by signs that stand for concepts in the world. To paraphrase historian Lorraine Daston, the proposition of a one-to-one correspondence between a sign and its object turned out to be as useless as the Borgesian 1-to-1 map that fully covers the territory~\\cite{Coopmans2014Representation}.\n\nWhen considering all the processes, actions, and material conditions involved in exploring traces, it is not always useful to make explicit what exactly constitutes a sign and how it is used to generate meaning. Scholars in science, technology, and society (STS) have formulated alternative perspectives that focus on the performative and embodied modes of cognition with regard to the roles of traces and trace-making in the history of science. Bruno Latour describes data, traces, and visualizations as \\emph{immutable mobiles}: aspects of the world that have been stabilized, flattened, and made mobile to support arguments in scientific discourse~\\cite{Latour1990Visualisation,Latour1999Pandora's}. In a similar vein, Hans-J\u00f6rg Rheinberger speaks about epistemic things: objects manipulated in the laboratory that should not just be regarded as samples collected from the world, but as materializations of research questions and scientific models that are embodied in the countless transformations applied to them~\\cite{Rheinberger1997Toward}.\n\nThe notion of the trace as objective evidence and science as a process of trace-making has blossomed in the 19\\textsuperscript{th}~century paradigm of \\emph{mechanical objectivity}. Charting the history of objectivity through scientific atlases and visualization, historians Daston and Galison describe the paradigm as a pursuit to develop modes of inscription that create pure and objective visualizations without human intervention, even if just for the sake of removing dirt and imperfections~\\cite{Daston2007Objectivity}. Culminating in the work of E. J. Marey, mechanical objectivity still resonates in contemporary efforts to develop canonical visualization principles based on scientific criteria. \n\nMechanical objectivity in its purest ambition of tracing \"nature's pencil,\" however, was bound to fail due to the indispensability of narrative explanation and the ambiguous nature of the trace. Historian Carlo Ginzburg describes the interpretation of traces, clues, and symptoms as a method of conjecture rather than computation~\\cite{Ginzburg1979Clues}. Philosopher Sybille Kr\u00e4mer locates traces \"at the seam of where the meaningless becomes meaningful,\" embodying meaning through material configuration rather than verbal attribution. In her understanding, traces are not found, but constructed in the act of reading: a trace is whatever is recognized as a trace~\\cite{Kraemer2007Spur}. Contemporary thinkers under the umbrella of \\emph{new materialism}, however, do not insist on the centrality of the human observer~\\cite{VanderTuin2012New}. Distinct from both \\emph{realist} (focusing on the external world) and \\emph{anti-realist} (focusing on the relationships among signs) perspectives, Karen Barad's concept of \\emph{agential realism} considers the human subject as a part, but not the center of an external phenomenon~\\cite{Barad2007Meeting}. Avoiding any dualism between objects in the world and their representations, Barad understands a phenomenon as an ongoing process of what she describes as intra-actions rather than a fixed set of objects and their relationships.\n\nTranslated to the subject at hand, this implies that autographic visualizations are not stable artifacts whose correct interpretation is just a matter of visual literacy, but phenomena that emerge from a recipients' extensive engagement with the world and with the knowledge of others, like a hunter who learns to spot latent animal tracks that are not just invisible but non-existent for an unskilled person. Philosopher Michael Polanyi aptly describes how a complex trace can depend on theoretical concepts and language~\\cite{Polanyi1998Personal}:\n\\begin{quote}\n\tThink of a medical student attending a course in the X-ray diagnosis of pulmonary diseases. He watches, in a darkened room, shadowy traces on a fluorescent screen placed against a patient's chest, and hears the radiologist commenting to his assistants, in technical language, on the significant features of these shadows. At first, the student is completely puzzled. [\\dots] The experts seem to be romancing about figments of their imagination- he can see nothing that they are talking about. Then, as he goes on listening for a few weeks, looking carefully at ever-new pictures of different cases, a tentative understanding will dawn on him; he will gradually forget about the ribs and begin to see the lungs. And eventually, if he perseveres intelligently, a rich panorama of significant details will be revealed to him (p. 106) \n\\end{quote}\n\n\\section{Autographic Visualization Neighbors}\nAutographic visualization shares a space with other visualization models concerned with physical information displays, embedded in physical environments and contexts of action~\\cite{willett_embedded_2017}. They can be seen in the tradition of ubiquitous computing and its explorations of tangible media, ambient and situated displays~\\cite{Weiser1991computer,Weiser1996Designing,IshiiTangible,Wisneski1998Ambient}. \n\nWithin the information visualization discourse, the field of data physicalization is closest to the concept of autographic visualization. Data physicalization investigates three-dimensional physical embodiments of information and their possible advantages for data communication and exploration~\\cite{jansen_opportunities_2015}. Unlike autographic visualization, however, physicalization (or physical visualization) is a data-first approach. As Jansen et al.\\ explain: \"Traditional visualizations map data to pixels or ink, whereas physical visualizations map data to physical form\"~\\cite{Jansen2013Evaluating}. Data physicalization aims to take advantage of the cognitive processes involved in examining, manipulating, and constructing three-dimensional objects that may not be accessible through visual observation of two-dimensional representations. The goal of data physicalization is therefore epistemological---supporting data analysis---while autographic visualization emphasizes ontological questions such as what constitutes a datum and how it relates to the world.\n\nBased on the Peircean concept of the index, indexical visualization presents a design space spanned by the dimensions of symbolic and causal distance~\\cite{Offenhuber2015Indexical}; the former describes the amount of symbolic mediation used to transform a phenomenon into a display, the latter the number of transformations in the causal chain. Despite its short causal distance, a simple seismometer involves a high degree of symbolic mediation; its line chart can no longer be connected to the phenomenon without knowledge of the process that created it. Conversely, an ambient display that mimics the outdoor sky based on weather data would have a short symbolic distance, but a long causal distance because of the complexity of the mediating apparatus. In place of these two dimensions, the concept of qualitative displays elegantly presents a one-dimensional measure of \"directness,\" describing the degree of intervention by a designer~\\cite{Lockton2017Exploring}. This dimension spans five different levels ranging from visual phenomena that are their own visualization to highly artificial data physicalizations at both extremes of the scale. The authors argue that visualization, so far, has been biased towards quantitative information while neglecting qualitative aspects.\n\nIndexical visualization and qualitative displays both are motivated by a gap in existing frameworks: the neglect of the index compared to icons and symbols in the former, the neglect of qualitative information in the latter. Both emphasize the embeddedness of visualizations in the physical world~\\cite{willett_embedded_2017}. Neither, however, fully capture the nature of analog visualizations of material information: Indexicality requires adhering to a semiotic framework that insists on explicating visual codes. The term qualitative display, on the other hand, seems overly broad as a descriptor of material displays. The term \\emph{autographic} addresses the main difference to information design, InfoVis, and data physicalization: the self-inscribing nature of material displays, in which the designer creates the apparatus that lets traces emerge rather than explicitly defining symbolic mappings. Areas of intersection exist: for example, data visualization software that generates and displays its own data from user interaction and therefore assumes autographic qualities, or projects such as \\emph{Dear Data}, when the signature of the author is considered as a trace~\\cite{lupi_dear_2016}.\n\nAutographic visualization continues the explorations into self-illustrating phenomena, first presented by Pat Hanrahan in his 2004 IEEE Vis capstone talk~\\cite{Hanrahan2004Self-illustrating}. Referencing a concept from the history of scientific representation~\\cite{Robin1992scientific}, Hanrahan focused on scientific experiments rather than the broader cultural field of visual practices. Autographic displays, however, are not limited to science but can be found throughout history and culture. The term autographic not only reflects the process of visualization and the role of the designer in this process but is also historically accurate, since the term was widely used during the late 19\\textsuperscript{th} and early 20\\textsuperscript{th}~century to describe self-inscribing mechanisms~\\cite{Siegel2014Forensic}. As reflected by a Google n-gram search, the terms \"autographic\" and \"self-registering\" saw their peak in the early 20\\textsuperscript{th}~century, where they show up in many patent applications for mechanical visualization devices and photographic techniques, before losing popularity later in the 20\\textsuperscript{th}~century.\\footnote{See~\\url{https:\/\/books.google.com\/ngrams\/graph?content=autographic\\%2Cself-registering&year_start=1800}}\n\n\\section{Autographic Design Operations} \n\n\\begin{table}\n\t\\caption{Overview of autographic design operations}\\label{designop}\\scriptsize\\centering\\medskip\n\t\\begin{tabular}\n\t\t{L{2cm} L{2cm} L{3.2cm}} Objective&Operations&Description\\\\\n\t\t\\midrule Establishing perceptual space~\\cite{Latour1990Visualisation,Lynch1985Discipline} & Framing& Establishing a perceptual context to isolate a phenomenon~\\cite{Bateson1972Steps,Berger2008Ways}\\medskip\\\\\n\t\t&Constraining&Isolating a single quality by constraining other qualities~\\cite{Rheinberger1997Toward,Peirce1998Essential}\\medskip\\\\\n\t\tTuning scale and intensity&Aggregating&Making visible by aggregating material~\\cite{Rheinberger1997Toward,Latour1999Pandora's}\\medskip\\\\\n\t\t&Separating&Making visible by separating material~\\cite{Rheinberger1997Toward,Daston2007Objectivity}\\medskip\\\\\n\t\tTrace-making& Coupling & Making visible by allowing the phenomenon to interact with another substance~\\cite{Rheinberger1997Toward,Daston2007Objectivity}\\medskip\\\\\n\t\t&Registering&Creating a persistent trace~\\cite{Latour1990Visualisation,Daston2007Objectivity,Siegel2014Forensic,Goodman1968Language}\\medskip\\\\\n\t\tMeasuring and Interpretation& Annotating & Adding graphical elements to guide the interpretation~\\cite{Bredekamp2015technical,Wulfingen2017Traces:}\\medskip\\\\\n\t\t& Encoding & Adding a scale for discretizing a phenomenon~\\cite{Goodman1968Language,Cole2002Suspect}\\\\\n\t\\end{tabular}\n\\end{table}\n\nThe production of interpretable traces is facilitated by cultural techniques that involve various degrees of intervention. In the most simple case, an environmental trace presents itself to a skilled observer. At the other end of the spectrum, traces are the product of a complex experimental apparatus involving many transformations. Along this continuum, the engagement with traces can be articulated as a design process that comprises a set of operations to turn a phenomenon into encodable data. The designer has to decide which aspect of a phenomenon can be used as an indicator and proceed to apply different operations that make this indicator legible. \n\nThe visual vocabulary of information visualization is formalized in schemata ranging from the foundational concept of visual variables to the grammar of graphics, organized by data structure and user needs~\\cite{Bertin1983Semiology,Shneiderman1996eyes,Card1997structure,Wilkinson2005grammar,Wickham2010Layered}. A taxonomic approach that categorizes trace-phenomena into visual variables seems impractical and would introduce another level of symbolic representation. Instead, our approach focuses on the design operations involved in autographic design (Fig.~\\ref{collage}). Table~\\ref{designop} provides an overview of these operations, grouped by the kinds of transformations they achieve. Literature categorizing traces exists in domain-specific areas, from the forensic analysis of crash skid marks~\\cite{Struble2013Automotive} to the identification of animal tracks~\\cite{Liebenberg1990art}. But to our knowledge, there are no overarching accounts that generalize the visual operations of trace-making across disciplines. The following taxonomy is an attempt to this effect. \n\nThe construction of the proposed autographic design space involved multiple steps. The fundamental concepts were drawn from theoretical literature, including history of science~\\cite{Daston2007Objectivity}, theory of scientific representation~\\cite{Latour1990Visualisation,Coopmans2014Representation}, and other perspectives on the ontology and epistemology of the trace~\\cite{derrida_grammatology_2016,Kraemer2007Spur,Rheinberger2011Infra-experimentality}. We also included  professional literature from fields concerned with preparing traces, especially in medicine and the forensic sciences~\\cite{Struble2013Automotive,Weizman2017Forensic,Cole2002Suspect}. \n\nThe reviewed literature made clear that the goals of trace-making diverge across fields. While the natural sciences are interested in the generalization of the phenomenon behind the trace, forensic science is striving for individualization: finding what differentiates a particular object from all other things in the world, e.g., the gun that fired a bullet. However, despite these different objectives, the practices of identifying, preparing, and transcribing a trace share more similarities than differences. The review, therefore, focused on practices more than the underlying intent. \n\nThe next step involved the collection and analysis of 800 examples of traces and techniques of trace-making in the broadest sense.\\footnote{For reference~\\url{https:\/\/www.pinterest.com\/dietmaro\/autographic-visualization}} These examples were examined considering the theoretical concepts identified earlier and used to reflect on these concepts. To include autographic devices such as the sundial or the Cyanometer (Fig.~\\ref{collage}a), we expanded the definition of the trace from the narrow meaning of a persistent imprint~\\cite{Kraemer2007Spur} to a broader definition that includes ephemeral phenomena such as shadows or sound.\n\nThe resulting design space is grouped into four sections that loosely correspond to the steps involved in trace-making: 1.\\ frame the context in which the phenomenon can emerge, 2.\\ adjust the intensity of a phenomenon to make it intelligible, 3.\\ register the trace phenomenon and make it persistent, and 4.\\ annotate and encode it into data. The steps are not strictly sequential (steps 2 and 3 are sometimes skipped), so the term \"pipeline\" does not seem appropriate, but they are generally traversed in one direction. The four steps synthesize and simplify operations discussed in the literature under technical image production and \\emph{chains of representation}~\\cite{Bredekamp2015technical,Latour1999Pandora's}.\n\n\\subsection{Establishing Perceptual Space}\n\nIn an environment saturated with latent information, the first step involves defining the space and context for the autographic visualization and thus offering a scaffolding for its reception. This problem rarely emerges in InfoVis, where charts are usually recognized as such and appear in familiar contexts such as newspapers, websites, or exhibitions. But to facilitate decoding, also traditional visualizations have to define a spatial reference system and clarify the domain covered by the data~\\cite{Lynch1985Discipline}. Framing and constraining are two families of operations that focus the attention, isolate the phenomenon from its background, and offer a reference for comparison.\n\n\\paragraph{Framing}As the most fundamental autographic operation, framing circumscribes the perceptual space of the visualization. Framing guides the attention to a particular quality of a phenomenon while masking the many other qualities that are not considered relevant. Framing manipulates the context of a phenomenon without touching it. Nevertheless, framing determines how the phenomenon presents itself, shaping the qualities of the display. Framing can be illustrated through the Cyanometer, a historical device for measuring the blueness of the sky consisting of a numbered color scale with a hole at the center~\\cite{Saussure1791Description}. The frame separates the color as the quality of interest from its surrounding context and simultaneously allows constructing a different context that allows comparison or measurement~(Fig.~\\ref{collage}a). Framing is also a rhetorical strategy and, as such, omnipresent in information design and visualization. In communication theory, framing describes a form of meta-communication that places a message into an existing interpretative context~\\cite{Bateson1972Steps}. \n\n\\paragraph{Constraining}As a stronger form of framing, constraining involves physically manipulating the phenomenon. Constraining isolates a particular quality of a phenomenon from all others, but unlike framing it makes it observable by physically limiting the degrees of freedom of other qualities and behaviors. As an example, a glass thermometer allows a liquid to expand with temperature, but only in a single direction and by amplifying the expansion through the diameter of the tube~(Fig.~\\ref{collage}b). As a sentinel species, the proverbial canary in the coal mine is constrained in a cage, so that its demise can alert miners of dangerous gases in the surrounding atmosphere. Constraining can be compared to similar interaction techniques in data visualization that control for changes in a specific variable by keeping the others constant.\n\n\\subsection{Tuning Scale and Intensity} \nAfter a perceptual space is established, the phenomenon might still be invisible because it is too faint or too intense, too large or too small, too fast or too slow to be perceived. The second group of autographic operations is therefore aimed at tuning the scale, speed, and intensity of a phenomenon to the gamut of human perception. Hans-J\u00f6rg Rheinberger goes as far as describing compression and dilatation as the two fundamental procedures of scientific experimentation~\\cite{Rheinberger2011Infra-experimentality}. In the following, aggregation refers to operations that compress a phenomenon in space, time, and magnitude, while separation achieves the opposite. In data visualization, the two operations are equivalent, usually accomplished by tweaking visual variables. In autographic space, however, aggregation and separation are quite different in terms of operations and levels of complexity. We therefore discuss them separately.\n\n\\paragraph{Aggregation}A straightforward way to amplify the visual intensity of a material substance is to aggregate the substance over time until visual differences become apparent. Air-borne particulate matter is invisible but reveals itself in the filter of a dust mask worn over an extended period in polluted air. An example of spatial compression involves the collection of soil samples from a larger territory, which can then be organized into a grid that serves as a compact visualization of the territory~(Fig.~\\ref{collage}i). Aggregating material under controlled conditions is at the core of many methods of sensing and measurement, such as the gravimetric measurement of particulate matter~(Fig.~\\ref{collage}c). InfoVis methods designed to reveal patterns based on aggregation include scatter plots and heat map displays.\n\n\\paragraph{Separation}Often the opposite is necessary, untangling a material mixture and spatially separating it into its components based on their physical properties. A prism separates white light into its different wavelengths. In paper chromatography, the components of an ink blot on a piece of paper dipped into water are separated by the force of capillary action. DNA Electro-chromatography works by the same principle, separating fragments of DNA embedded in a gel driven by the force of an electric field~(Fig.~\\ref{collage}d). The analytical separation of multiple correlated variables is a central task in InfoVis, addressed in methods such as scatterplot matrices, parallel coordinate displays, or faceting.\n\n\\subsection{Trace-making techniques}\nMany phenomena of interest are non-visual but can be visualized through their interaction with certain substances and processes. Other visual phenomena are ephemeral; operations of marking and tracing can help to preserve traces and create a persistent record. The operations under this rubric include most analog visualization techniques, which are historically and visually linked to the contemporary languages of data visualization. \n\n\\paragraph{Coupling} links an invisible phenomenon to a second phenomenon that serves as a visible indicator or proxy. Wind itself may be invisible, but reveals itself in the movement of grass; smoke injected into a wind tunnel serves the same purpose~(Fig.~\\ref{collage}e). Many archaic skills of farming, hunting, or navigating rely on observing such proxy indicators linked with the phenomenon of interest. Coupling can involve adding a tracer substance, such as color dyes to visualize the movement of liquids. 18\\textsuperscript{th}~century experimentalist Ernst Chladni visualized the shape of sound-waves by adding sand on a metal plate struck by a violin bow~(Fig.~\\ref{collage}g). Mechanical instruments can be designed to respond to a phenomenon with visible changes, as illustrated by pressure gauges or meteorological instruments. By translating the phenomenon into changes in a defined visual layout, such instruments provide a bridge into the space of symbolic representation. \n\n\\paragraph{Registering or marking} is a stronger form of coupling that aims to create a permanent trace---a cast from a footprint, the groove of a vinyl record, or the photochemical reaction in an exposed photograph. Tracer substances can also be used to create a permanent trace, including dyes to reveal structure in biological specimen, radioactive tracers used in radiology, or powder to reveal latent fingerprints. Campbell-Stokes' sunshine recorder serves its literal purpose through a spherical lens that burns a linear trace into a paper strip~(Fig.~\\ref{collage}f). Registration creates not only a spatial but also a temporal record of a phenomenon. James Watt's indicator mechanism, producing a two-dimensional line chart of steam pressure over piston displacement, became the first of many self-registering devices, including the black boxes used in aviation~\\cite{Siegel2014Forensic}. Real-time data visualizations from sensor input are, to some extent, autographic visualizations.\n\n\\subsection{Measuring and Interpretation}\\label{encode}\nThe last step in the design of autographic visualizations involves the interpretation of the trace and to make it comparable to other traces. This step can involve many different forms, from manual annotations of traces during the analysis, visual guides for non-expert viewers, to scales for encoding the trace into symbolic data.\n\n\\paragraph{Annotating} Graphical annotations are frequently found where traces and records of measurements are interpreted, creating a hybrid of graphic and autographic displays~(Fig.~\\ref{collage}h). Such scribbles can themselves be seen as traces of a thought process or collective discourse. Annotations and legends can also guide the attention and support the discovery of traces where implicit framing is not sufficient. Museum displays of archeological artifacts often include abstracted representations that point out significant features of the object. In all of these examples, annotations serve largely identical purposes in information design and autographic visualization.\n\n\\paragraph{Encoding} represents the last step in the process of translating a phenomenon into data. Encoding begins by marking different conditions over time, registering observations~\\cite{lupi_dear_2016}. A systematic application of such marks becomes a scale that allows encoding the phenomenon into discrete elements~(Fig.~\\ref{collage}j). Fingerprints became a viable means of identification only after an efficient system of encoding their intricate appearance into a sparse sequence of symbols was developed~\\cite{Cole2002Suspect}. In the operation of encoding, an analog system becomes a digital system. Nelson Goodman describes a pressure gauge as an analog system if the marks on the gauge face are used for mere orientation, but it becomes a digital system, once only the discrete intervals are considered~\\cite{Goodman1968Language}\n\n\\section{Autographic Systems} \nThe design operations described in the previous section form the basic vocabulary of autographic visualization. Revealing a phenomenon, however, typically requires the application of several operations, combined in an \\emph{autographic system}. Most examples given earlier, when applied in practice, form autographic systems of various complexity, comprising a system of operations for framing, tuning, and recording a trace and making it measurable. A set of design operations can either be deployed in parallel to create a controlled environment to observe a phenomenon in isolation, or sequentially as a series of material transformations to make an invisible phenomenon accessible to the senses. While desire paths may appear accidental and free of design intent, complex autographic systems can be highly artificial displays that represent through analogies---as in the case of MONIAC, a hydraulic model meant to simulate economic flows~\\cite{bissell_historical_2007}. In the following, three common types of autographic systems will be discussed. The list in Table~\\ref{autosys}, neither exclusive nor exhaustive, includes experimental apparatuses, analog visualizations and simulations, hybrid systems using digital and analog components, and new materials with intentionally designed properties and behaviors.\n\n\\begin{table}\n\t\\caption{Autographic systems}\\label{autosys}\\scriptsize\\centering \n\t\\begin{tabular}\n\t\t{L{3cm} L{5cm}} Autographic systems&\\\\\n\t\t\\midrule Autographic environments&Systems that combine operations to generate traces under controlled conditions\\medskip\\\\\n\t\tDigital\/physical systems&Coupling analog and digital systems\\medskip\\\\\n\t\tAutographic materials&Encoding behavior into smart materials or synthetic organisms\\\\\n\t\\end{tabular}\n\\end{table}\n\n\\subsection{Autographic Environments} \nAutographic environments are environments in which a phenomenon can unfold, largely isolated from external influences. Operations such as \\emph{framing, constraining, coupling,} and \\emph{registering} are mobilized to turn a phenomenon into a usable trace under controlled conditions. The wind tunnel is a simulated environment isolated from its surroundings. It includes mechanisms for producing traces (e.g., through smoke or silk strings), for observing and recording them. Another example is the large bacterial growth area in Fig.~\\ref{antibiotic} to study the adaptation of bacteria to antibiotic environments. Autographic environments are analog computers and visualization systems, allowing us to perform the same visualization tasks on different inputs and observe the results.\n\n\\begin{figure}\n\t[htb] \\centering \n\t\\includegraphics[width=3.45in]{antibiotic.png} \n\t\\caption{Autographic environment: gel with graduated antibiotic presence to observe microbial evolution towards antibiotic tolerance~\\cite{Baym2016Spatiotemporal}}\\label{antibiotic}\n\\end{figure}\n\n\\subsection{Digital\/Analog Systems} \nDigital\/analog systems are a special case of autographic environments, which utilize both digital and analog forms of computation for controlling a phenomenon. The digital\/analog coupling can happen in three different ways. First, the physical conditions in the autographic environment can be computationally controlled to achieve different results. Another possibility is to digitize the outputs of the apparatus and subject them to further computational analysis. The third possibility is to couple digital and analog processes in a dynamic feedback loop, creating a hybrid, autopoietic system. Since, as discussed earlier, digital sensors and circuits process material information, the line between digital and analog components, discrete logic and continuous feedback, is somewhat blurry. Fig.~\\ref{cloud} shows an example of an autopoietic digital\/analog system, a cloud chamber controlled by a digital algorithm aiming to sculpt the shape of generated clouds.\n\n\\begin{figure}\n\t[htb] \\centering \n\t\\includegraphics[width=3.45in]{cloud.png} \n\t\\caption{Digital\/physical system: Clemens Winkler. Per-forming clouds. 2018 \u2013 art project attempting to create rectangular clouds~\\cite{Winkler2018Per-Forming}}\\label{cloud}\n\\end{figure}\n\n\\subsection{Autographic Materials} \nThe last group of autographic systems avoids the complexity of digital\/analog apparatuses and aims to develop materials and biological organisms with truly autographic properties~\\cite{telhan_designature_2016}. The concept of \\emph{4D printing} investigates geometries and materials that dynamically respond to environmental changes or possess the capacity for self-assembly~\\cite{tibbits_4d_2014}. Autographic materials also include work in synthetic biology with modified bacteria that react to environmental changes with visible changes of, for example, color or smell (Fig.~\\ref{echromi}).\n\\begin{figure}\n\t[htb] \\centering \n\t\\includegraphics[width=3.45in]{chromi.png} \n\t\\caption{Autographic material: Daisy Ginsberg, James King, E.Chromi, 2009 \u2013 color changing bacteria to detect gut diseases, a speculative design project~\\cite{Ginsberg2009E-Chromi}}\\label{echromi}\n\\end{figure}\n\n\\section{Rhetoric Techniques of Autographic Visualization} \nConsidering the pervasive availability of digital information and the extent to which our experience is shaped by it, are there compelling reasons, beyond nostalgia for the analog, to engage with the slow, ambiguous, and bespoke domain of material displays? The beginning of this paper presented the argument that autographic visualizations allow to experience the phenomenon behind the data and render legible the circumstances of data collection. Traces are not representations; they present themselves. This argument, however, deserves more scrutiny. Traces are often equated with incontestable evidence. As unintentional side-products of past processes and events, material traces are considered trustworthy, and their display can achieve a persuasive effect that is difficult to attain with digital representations. However, as discussed earlier, traces neither speak for themselves nor are recognized by everyone. The persuasiveness of traces is shaped by social and cultural processes, as illustrated by the slow acceptance of fingerprinting and DNA identification~\\cite{Cole2002Suspect}. Traces require interpretation, and their interpretation relies on assumptions and often speculation. The most apparent patterns are often misleading. Interpretation, therefore, requires a rhetoric scaffolding for integrating material displays into a framing argument. The following section discusses examples of autographic design principles as they are used in practice. In all of these cases, material traces are used as rhetorical devices; as visual arguments that support specific claims. \n\n\\subsection{Evidentiary Aesthetics in Citizen Science}\nTufte's imperative of information visualization to \"above all else show the data\" becomes \"above all else move closer to the phenomenon\" in autographic visualization. The tactic to enable the sensory experience of causality is often found in the domain of citizen science~\\cite{Snyder2017Vernacular}. Especially groups who investigate issues of environmental pollution are often met with skepticism of the data they collect\u2014whether their methods are rigorous, their instruments accurate, or their biases reflected in data collection. Many grassroots scientists lack institutional affiliations and scientific credentials, making their data sets often seem less trustworthy in public perception. In such an environment, the evidentiary aesthetics of a raw trace can be a more effective way to generate trust than a well-designed chart using canonical visualization principles.\n\n\\begin{figure}\n\t[ht] \\centering \n\t\\includegraphics[width=3.45in]{wylie.png} \n\t\\caption{Map of photo strips tarnished by H2S emanation on the field site, Public Lab. Registering, annotating~\\cite{Wylie2017Materializing}}\\label{h2s}\n\\end{figure}\n\nThe Public Lab is a grassroots science collective investigating environmental pollution and the social impacts of oil spills and fracking; issues that in their view do not receive enough attention by environmental agencies~\\cite{Wylie2017Materializing}. The group has developed a DIY method using photo paper to detect harmful hydrogen sulfide gas emanating from the ground in proximity to fracking sites. The maps presenting their results incorporate an arrangement of small pieces of the original, stained photo papers over an abstracted representation of the landscape (Fig.~\\ref{h2s}). The grid of samples not only shows a visual pattern similar to a heat map indicating the locations of highest exposure, it also suggests physical circumstances to the uninitiated viewer: that a chemical reaction that stains photo paper is associated with these places, implying the papers were actually exposed at the indicated locations (a rhetorical claim that cannot be verified using the map). In combination with an encoded data set, the trace map serves as an illustration and justification of the method. But the autographic map is not only directed outwards towards a skeptical audience but also inwards at the own collaborators. The modes of data collection are often participatory and depend on the engagement of volunteers and members of the affected community. To this end, data collection is staged as a public experiment; the physical traces serve as rhetorical devices to make the nature of pollution tangible for collaborators, and the results of their voluntary efforts visible. It may be a lucky accident that the pollutant causes an noticeable and reproducible trace to visualize environmental harms. Autographic design involves discovering such opportunities and building an explanatory framework around a suitable indicator. The first rhetoric strategy involves choosing a presentation that suggests immediacy and direct causal connection over more abstract representations. In the case of Public Lab's grassroots science, the presentation of causality is not just concerned with data veracity, but, more importantly, with the methods and practices of its researchers. \n\n\\subsection{Performative Mapping and Annotated Walkthroughs}\\label{performative}\nThe second rhetoric strategy involves guiding the audience through the causal chain, helping them to \"connect the dots\" and perform the analysis themselves. Traces imply causality, but the immediate cause is absent from the trace\u2014what is left is an imprint. Annotated walkthroughs put the traces that are considered relevant next each other and allow the recipient to explore the latent connections. \n\nThe example to illustrate this approach is based on digital information---satellite images, video footage, and other media formats read through a material-forensic lens, highlighting the indexical and material \"residues\" in digital data. In recent years, an active community of conflict mappers and amateur forensic experts has emerged who analyze and compare social media content from conflict regions and gained prominence during the Syrian civil war~\\cite{Kurgan2017Conflict,Weizman2017Forensic}. In the first major military conflict in which social media played a decisive role, all adversaries made extensive use of platforms such as YouTube or Twitter to disseminate footage from the frontlines recorded by drones, smartphones, and body-cams. Social media not only served propagandistic purposes but also as a backchannel for reporting military success to foreign donors, which in some cases even involved staging fake battles~\\cite{Atwan2015Islamic}.\n\n\\begin{figure}\n\t[htb] \\centering \n\t\\includegraphics[width=3in]{syria.png} \n\t\\caption{Amateur visual forensics, Institute for United Conflict Analysts \u2013 framing, annotating (IUCA)~\\cite{IUCA2016Syria}}\\label{syria}\n\\end{figure}\n\nThe community of conflict mappers took it upon themselves to verify claims of the warring parties by geo-referencing buildings and locations shown in the videos and placing the events into a temporal sequence in order track the shifting frontlines~\\cite{Offenhuber2017Maps}. As in the case of the grassroots scientists, the conflict mappers are not represented by certified institutions, and therefore choose their visual displays to address questions of credibility. For their frontline analysis, conflict mappers have created a specific format of display. It does not synthesize information from individual sources into a consistent cartographic language but arranges snippets from the raw sources. Elements in this tableau are annotated with simple shapes, indicating the same building or location from various angles in smartphone footage and satellite images. In the example featured in Fig.~\\ref{syria}, conflict mappers countered the claim of the Syrian military to have captured a particular city by demonstrating that the shown location is not inside the city, but rather on its outskirts. To interpret these displays, the recipient is forced to do the work of the cartographer, judge the likelihood of whether the highlighted elements are the same building or whether they are shot at the same time of day. This rhetoric strategy has been described as non-representational or performative cartography~\\cite{Kitchin2007Rethinking}. The same strategy of guiding the viewer by juxtaposing elements, implying connection through spatial proximity, and highlighting relevant aspects can be equally applied to material traces, is used in museum displays of archaeological artifacts. The displays of the conflict mappers try to persuade not by encoding information into visual variables, but by emphasizing the authenticity of the sources and inviting the viewers to \"see for themselves.\" Their use of publicly available tools such as Google Earth without bothering to modify their visual defaults underscores this invitation as if to say that anyone can conduct this investigation and arrive at the same conclusion. \n\n\\subsection{Sensory Accountability---Exploring Data Materiality}\nThe third rhetoric strategy contextualizes digital data and computational models with material displays to explore their grounding in physical reality. A considerable number of artists have taken on visualizing the materiality of climate change and environmental pollution through situated and material displays. Proxy data sources play, implicitly or explicitly, a major role in this genre: from visceral explorations of the rich material qualities of ice core samples and arctic ice to the instrumentalization of bioindicators and sentinel species---plants and animals that are especially sensitive to particular conditions and can serve as environmental sensors. Many of these projects not only comment on environmental phenomena such as climate change, but also on the aesthetic and ontological dimensions of scientific measurement: what is it that is measured, which qualities are captured or overlooked, and what are the political underpinnings of how a problem is articulated and operationalized. \n\n\\begin{figure}\n\t[htb] \\centering \n\t\\includegraphics[width=3.45in]{staub.png} \n\t\\caption{D. Offenhuber, Staubmarke \u2013 reverse graffiti washed into concrete, calling attention to air pollution, aggregating, encoding~\\cite{Offenhuber2018Dust}}\\label{staub}\n\\end{figure}\n\nIn public controversies around air pollution stemming from particulate matter and ozone, the discourse among actors with conflicting positions often quickly converges on technicalities of measurement such as appropriate threshold values and exposure times. The work of grassroots science initiatives thus has a strong political dimension: challenging conventional protocols of measurement by revealing what is not shown by them. Their perspective of data collection values richness and completeness over accuracy; aims to capture the implications of pollution on the lives of individuals and communities even if the accuracy of the cheap sensors used in these projects is \"good enough\" rather than perfect. In this context, material displays can be used to call attention to the basic assumptions of environmental sensing and their political implications. We use the term \"sensory accountability\" to describe approaches that call attention to the complex material qualities of a phenomenon that is often reduced to a single quantitative dimension. As an example, the project \"Staubmarke\" (Fig.~\\ref{staub}) visualizes air pollution by applying visual markers on urban surfaces to make accumulations of particulate matter legible. These markers are executed as \\emph{reverse graffiti}, which are based on selectively cleaning dirty surfaces rather than applying paint. With ongoing pollution, the markers will fade over time, starting with the most delicate textures in the pattern. The markers were applied at locations also surveyed by the sensors installed by a citizen sensing initiative, allowing the comparison between data values and physical appearance of the markers. Sensory accountability is a critical inquiry into sensing methods: what is captured by the sensor, what is ignored, and how do the recorded values correspond with sensory phenomena. Highlighting changes in the physical environment and contextualizing these changes with corresponding data offers an interesting space for visualization.\n\n\\section{Discussion---Reviving the Public Experiment}\nPhysical expressions of data currently enjoy burgeoning interest, from the popularity of data physicalization in design and education to the wide range of artistic projects focusing on data and the environment. One can make the argument that this is not merely a short-lived design trend, but an expression of a new sensibility for the relationship between digital data and the world around us. The current fascination with materiality in design and the humanities coincides with renewed critiques of the central role of data in society and a discomfort with the explicit encoding of the world into symbolic categories. Fields such as critical data studies probe the assumptions behind data collection, the politics of categorization, and the opacity of algorithmic decision systems. Research on the interpretability of machine learning and algorithmic decision-making models is currently thriving in information science and information visualization. Autographic visualization also resonates with the recent resurgence of analog computing, advances in microfluidics and synthetic biology~\\cite{Ulmann2013Analog,Whitesides2006origins}. The disconnect between science and public opinion in the issue of climate change and the role of climate data as a proxy-site to negotiate political positions further demonstrate the need to probe the material foundations of data generation. Disinformation and the phenomenon fake news also provide reasons why it is crucial to investigate data generation through a forensic-material lens, based on the premise that \"no two things in the physical world are ever exactly alike\"~\\cite{Kirschenbaum2008Mechanisms:}, which extends to digital camera sensors and hard drives. We argue that these issues also justify a shift in the agenda of visualization from the patterns inside data to the conditions of data generation. Surely, misleading patterns and false narratives can be sufficiently addressed with better explanations and more nuanced visualizations that are contextualized with scientific arguments. But the original impetus of visualization has always been to show rather than tell, to produce images that say more than a 1000 words. Autographic visualization continues this project by extending information visualization beyond the space of symbolic encodings into the spaces where data take shape. The practices of grassroots science offer an interesting model that connects to the history of public experiments during the enlightenment period, where natural phenomena were explained through spectacular public demonstrations~\\cite{Nieto-Galan2016Science}.\n\n\\section{Limitations of Autographic Visualization}\nMany trace-phenomena have charismatic qualities; they fascinate and invite exploration while remaining elusive. As Michelangelo Antonioni's movie \"Blow Up\" illustrates, as one gets closer to a trace, the meaning seems to disappear: through successive magnification, a candid photograph reveals a murder scene but eventually dissolves into ambiguous patterns.\n\nOne has to avoid a na\u00efve empiricism that uncritically elevates trace-reading over theoretical inquiry as a source of knowledge. Traces often seem to inspire attributions of meaning even when there is none. With regards to trace-reading, science and superstition are often uncomfortably close~\\cite{Ginzburg1979Clues,butz_superstition_2007,huxley_method_1880}. As previous sections have pointed out, the charisma of traces can be exploited for rhetorical purposes by selectively curating, framing, and guiding the process of interpretation, and autographic visualization is not different from any other visual practice in this regard.\n\nThe strongest limitations are encountered on the practical level. Compared to data visualization, the creation of autographic visualizations is slow and limited by the available material. Autographic visualizations lack the agility, versatility, and potential scale of computational analysis. Where data visualization is limited by the gap between data and physical phenomenon, autographic visualization cannot compete with the full scope of computational possibilities. \n\n\\section{Conclusion}\nThis paper introduces the concept of autographic visualization, which examines the discovery and preparation of material traces. It offers a preliminary systematization of the design space of autographic display including its design operations, their combination into composite autographic systems, and the rhetorical strategies of presenting material traces. Autographic visualization considers the structures in the physical world as a form of data and thus serves as a speculative counter-model to data visualization, which is limited to the space of symbolic representation. \n\nIn the tradition of Marey's \\emph{graphical method}, which encompasses both the production of traces and the graphical display of data, we argue that InfoVis and autographic visualization can complement each other. Expanding the scope of visualization beyond symbolic data would open fertile areas for research, addressing questions such as: how do we see traces, and how do these perceptions relate to individual knowledge and skills? J.J. Gibson's theory of affordances~\\cite{Gibson1979ecological} has been foundational for the entire field of HCI but has been so far mostly operationalized from a functional perspective, without further attention to the phenomenology of affordances. In this context, also the aesthetics of experience and its relationship to knowledge construction (\\emph{aisthesis}) deserve attention.\n\nBeyond academic questions, what is the place of autographic approaches in visualization practice? Artists and citizen scientists provide examples that show how the immediacy, directness, and richness of material information can be utilized. Material traces serve as visual means of evidence construction: public experiments turn data collection from a bureaucratic exercise into a sensory experience of causality; physical data proxies make abstract climate models and their predictions relatable and observable.\n\nAutographic visualization aims not just to bridge the gap between data and phenomenon, but also the one between observer and display. Designers can no longer blindly rely on normative conventions on how to visualize data for an idealized, data-literate audience. In the space of material information, the observer becomes an experimenter, having to actively construct evidence by connecting the dots. Autographic visualization is therefore a critical practice in the sense that it de-naturalizes the concept of data and its underlying assumptions.\n\n\\acknowledgments{My gratitude goes to my long-term collaborators Orkan Telhan and Gerhard Dirmoser for their ideas and inspiring discussions. I also would like to thank the reviewers for their close attention and helpful feedback. }\n\n\\bibliographystyle{abbrv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theory of modulated structures in macroscopic ferromagnets \\cite%\n{dzialoshin}, ferroelectrics \\cite{levaniuk} and metal alloys\n\\cite{hachatur} has a rich history. The number of works dedicated\nto a study of the modulated structures in small samples is\nsignificantly less \\cite{char}. The standard approach to the\ntheory of the modulated structures (including the incommensurate\nones) uses the constant order parameter amplitude function\napproximation \\cite{levaniuk}, \\cite{izium}. Our consideration of\nthe bounded sample (thin film, in particular) requires us to drop\nthis approximation: the order parameter must satisfy some boundary\nconditions and it is unlikely to find a solution with constant\norder parameter satisfying these conditions. That is why we have\nto consider a generic case of connected set of the nonlinear\nequations for the order parameter amplitude and phase functions.\nWe will show here that the solutions of this system are\nsignificantly more complicated, than in the constant order\nparameter approximation.\n\n\\section{Free energy and the equations for the order parameter for a\nferroelectric with the Lifshitz invariants}\n\n\\qquad The Landau free energy functional reads \\cite{izium}:%\n\\begin{eqnarray}\n\\Phi & =\\int dz\\left\\{\n-r(\\eta_{1}^{2}+\\eta_{2}^{2})+u_{1}(\\eta_{1}^{2}+\\eta_{2}^{2})^{2}+u_{2}(%\n\\eta_{1}^{2}\\eta_{2}^{2})\\right\\} +  \\notag \\\\\n& +\\int dz\\left\\{ \\sigma\\left( \\eta_{2}\\frac{\\partial\\eta_{1}}{\\partial z}%\n-\\eta_{1}\\frac{\\partial\\eta_{2}}{\\partial z}\\right) +\\gamma\\left[ \\left(\n\\frac{\\partial\\eta_{1}}{\\partial z}\\right) ^{2}+\\left( \\frac{\\partial\n\\eta_{2}}{\\partial z}\\right) ^{2}\\right] \\right\\} ,  \\label{free}\n\\end{eqnarray}\nwhere $\\eta_{1}$ and $\\eta_{2}$\\ are the order parameter components; we\nconsider only one-dimensional configurations; parameters $r,$\\ $\\sigma$, $%\nu_{1},u_{2}$ and $\\gamma$ are the Landau free energy expansion coefficient.\nIntroducing the amplitude and phase variables\n\n\\begin{equation*}\n\\eta_{1}=\\rho\\cos\\varphi,\\eta_{2}=\\rho\\sin\\varphi,\n\\end{equation*}\nwe obtain the following expression for the Landau free energy:%\n\\begin{eqnarray}\n\\Phi=\\int dz \\{ -r\\rho^{2}+u\\rho^{4}+w\\rho^{n}(1+\\cos n\\varphi)  \\notag \\\\\n-\\sigma\\rho^{2}\\frac{\\partial\\varphi}{\\partial z}+ \\gamma\\left[ \\left( \\frac{%\n\\partial\\rho}{\\partial z}\\right) ^{2}+\\rho^{2}\\left( \\frac {\\partial\\varphi}{%\n\\partial z}\\right) ^{2}\\right] \\} .  \\label{free2}\n\\end{eqnarray}\nVarying the free energy we obtain the equilibrium equations \\cite{izium}%\n\\begin{eqnarray}\n-r\\rho+2u\\rho^{3}+\\frac{n}{2}w\\rho^{n-1}(1+\\cos n\\varphi)+\\gamma\\rho (\\frac{%\n\\partial\\varphi}{\\partial z})^{2}-  \\notag \\\\\n\\gamma\\frac{\\partial^{2}\\rho }{\\partial z^{2}}-\\sigma\\rho\\frac{%\n\\partial\\varphi}{\\partial z}=0,   \\label{ampeq}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\gamma\\rho^{2}\\frac{\\partial^{2}\\varphi}{\\partial z^{2}}+2\\gamma\\rho \\frac{%\n\\partial\\rho}{\\partial z}\\frac{\\partial\\varphi}{\\partial z}-\\sigma \\rho\\frac{%\n\\partial\\rho}{\\partial z}+\\frac{n}{2}w\\rho^{n}\\sin n\\varphi=0.\n\\label{phaseeq}\n\\end{eqnarray}\n\nHere $n$ is an integer number describing the system symmetry. Now we\nintroduce the dimensionless variables:%\n\\begin{eqnarray}\n\\rho=\\sqrt{\\frac{r}{2u}}R,\\xi=z\\sqrt{\\frac{r}{\\gamma}},\\frac{dR}{dz}=\\frac {%\ndR}{d\\xi}\\sqrt{\\frac{r}{\\gamma}},  \\notag \\\\\n\\frac{\\sigma}{\\sqrt{\\gamma r}}=T,u^{1-\\frac{n}{2}}nwr^{\\frac{n}{2}-2}2^{-%\n\\frac{n}{2}}=K.   \\label{dimlessvariables}\n\\end{eqnarray}\nThen the equations (\\ref{ampeq}) and (\\ref{phaseeq}) take the form%\n\\begin{eqnarray}\nR^{\\prime\\prime}-R^{3}+(1-\\varphi^{\\prime2}+T\\varphi^{\\prime})R-  \\notag \\\\\nR^{n-1}K(\\cos n\\varphi+1)=0,  \\label{ampeqdimless}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\varphi^{\\prime\\prime}+2\\frac{R^{\\prime}}{R}\\varphi^{\\prime}-\\frac{R^{\\prime}%\n}{R}T+R^{n-2}K\\sin n\\varphi=0.   \\label{phaseeqdimless}\n\\end{eqnarray}\n\n\\section{Approximate analytic solution}\n\nLet us begin from the limit $K=0.$ Then the equation (\\ref{phaseeqdimless})\ncan be solved:%\n\\begin{eqnarray}\n\\varphi^{\\prime}\\equiv\\psi=\\frac{C_{0}}{R^{2}}+\\frac{T}{2},\n\\label{phasesolution}\n\\end{eqnarray}\nwhere $C_{0}=\\left[ \\psi\\left( 0\\right) -\\frac{T}{2}\\right] R\\left( 0\\right)\n^{2}$ is the integration constant, which is determined by the initial\nconditions $\\psi\\left( 0\\right) $ and $R\\left( 0\\right) $. Now we can\nsubstitute this expression for $\\varphi^{\\prime}$ into the equation (\\ref%\n{ampeqdimless}). We obtain in result a closed equation for the amplitude\nfunction $R:$%\n\\begin{eqnarray}\nR^{\\prime\\prime}-R^{3}+R(1+\\frac{T^{2}}{4})-\\frac{C_{0}^{2}}{R^{3}}=0.\n\\label{ampclosed}\n\\end{eqnarray}\n\nThis equation can be interpreted as a dynamics equation with the effective\npotential:%\n\\begin{eqnarray}\nU=\\frac{R^{2}}{2}(1+\\frac{T^{2}}{4})-\\frac{R^{4}}{4}+\\frac{C_{0}^{2}}{2R^{2}}%\n.   \\label{effpot}\n\\end{eqnarray}\n\nAn interesting feature of this potential is its dependence on the initial\nconditions via the constant $C_{0}^{2}.$ There exists a domain of\nparameters, where the potential has a minimum and, therefore, an oscillating\nsolution for the amplitude function can take place. A condition of the\nmaximum and minimum points merging into the inflection point with the\nhorizontal derivative $U^{\\prime}\\left( R\\right) =U^{\\prime\\prime}\\left(\nR\\right) =0$ gives the equation\n\n\\begin{equation*}\nT^{6}+12T^{4}+48T^{2}+64-432C_{0}^{2}=0\n\\end{equation*}\n\nHowever, a presence of a minimum is necessary but not sufficient condition:\nan oscillating solution will not exist if the initial point is situated\noutside the potential well.\n\nA border line for the domain, where oscillating solutions can exist is\ndepicted here:\n\n\\begin{center}\n\\includegraphics[\nheight=2in, width=1.5in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{cond.eps}\n\nFIG.1. A border line for the domain, where oscillating solutions can exist.\n\\end{center}\n\n\\bigskip\n\nIf we have a solution for the amplitude function $R(\\xi),$ the phase\nfunction $\\varphi(\\xi)$ is given by the equation (\\ref{phasesolution}). The\nphase function consists from two terms: a slow contribution stemming from\nthe second term in (\\ref{phasesolution}) and more or less diffused jumps due\nto the first term in this equation. It is important that the jump value\nequals exactly to $\\pi:$\n\nThe first integral of \\ref{ampeqdimless} reads $\\frac{dR}{d\\xi}=\\sqrt {2(E-U)%\n},$ so $d\\xi=\\frac{dR}{\\sqrt{2(E-U)}}.$\n\nNotice that the jump value $\\int\\frac{C_{0}}{R^{2}}d\\xi$ calculated in the\nvicinity of the $R(\\xi)$ minimum equals to\n\\begin{eqnarray}\n\\Delta\\varphi & =\\int\\frac{C_{0}}{R^{2}}d\\xi=\\int\\frac{C_{0}}{R^{2}}\\frac {dR%\n}{\\sqrt{2(E-U)}}\\simeq2\\int_{R_{0}}^{R_{e}}\\frac{C_{0}dR}{R^{2}\\sqrt{2(\\frac{%\nC_{0}^{2}}{2R_{0}^{2}}-\\frac{C_{0}^{2}}{2R^{2}})}}=  \\notag \\\\\n& =2\\int_{R_{0}}^{R_{e}}\\frac{dR}{R\\sqrt{\\frac{R^{2}}{R_{0}^{2}}-1}}%\n=2\\arccos\\left\\vert \\frac{R_{0}}{Re}\\right\\vert \\simeq\\pi,\n\\label{phasejump}\n\\end{eqnarray}\nwhere $R_{e}\\gg R_{0}.$ We have used the first integral of\n(\\ref{ampclosed}) above. Only leading terms of $U(R)$ were taken\ninto account. Note that the model (\\ref{free2}), which can be\nreduced to the sine-Gordon model in the limit of $R=const$, admits\nsolutions with jumps exactly equal to $\\pm\\pi$ (topological charge\n\\cite{soliton}). However, we have obtained a similar\nresult within the approximation $K=0$, i.e. neglecting the cosine term in (%\n\\ref{free2})! As it is seen from the formulae \\ref{phasejump}, these phase\njumps appear due to the excursion of the amplitude function near the\nsingularity $\\frac{C_{0}}{R^{2}}$.\n\nThe equation (\\ref{ampclosed}) can be solved analytically, but we\nwill present below numerical solutions both for $K=0$ and\n$K\\neq0.$\n\n\\section{Numerical solution for K = 0}\n\nSome results of calculation of the amplitude and phase functions spatial\ndependence for the case of $K=0$ are presented below. Figures 2 -- 5 and 6\n-- 9 differ only by initial values of the amplitude function: in the case of\nfigures 2 -- 5 we have small initial amplitude function and, therefore,\noscillations of the amplitude function are near to harmonic ones, while in\nthe case of figures 6 -- 9 the initial amplitude function is near to the\napex of the hump and, therefore, we have a train of solitons. \\newpage\n\n\\begin{center}\n\\includegraphics[\nheight=2in, width=1.5in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{U1.eps}\n\nFIG.2. Effective potential for $n=4,$ $K=0,$ $T=1,$ $R(0)=0.3,$ $R^{\\prime\n}(0)=0,$ $\\varphi(0)=0,$ $\\varphi^{\\prime}(0)=0.3$.\n\nVertical dash marks $R(0)$ value.\n\n\\includegraphics[\nheight=2in, width=1.5in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{R1.eps}\n\nFIG. 3. Spatial dependence of the amplitude function for $n=4,$ $K=0,$ $T=1,$\n$R(0)=0.3,$ $R^{\\prime}(0)=0,$ $\\varphi(0)=0,$ $\\varphi^{\\prime}(0)=0.3$\n\n\\includegraphics[\nheight=2in, width=1.5in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{F1.eps}\n\nFIG. 4. Spatial dependence of the phase function for $n=4,$ $K=0,$ $T=1,$ $%\nR(0)=0.3,$ $R^{\\prime}(0)=0,$ $\\varphi(0)=0$\n\n(a) thin line $\\varphi^{\\prime}(0)=0.3$\n\n(b) heavy line $\\varphi^{\\prime}(0)=0.7.$\n\n\\includegraphics[\nheight=1.8in, width=1.8in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{P1.eps}\n\nFIG. 5. Amplitude-phase polar diagram for $n=4,$ $K=0,$ $T=1,$ $R(0)=0.3,$ $%\nR^{\\prime}(0)=0,$ $\\varphi(0)=0$\n\n(a) thin line $\\varphi^{\\prime}(0)=0.3$\n\n(b) heavy line $\\varphi^{\\prime}(0)=0.7.$\n\\end{center}\n\n\\newpage\n\n\\begin{center}\n\\includegraphics[\nheight=2in, width=1.5in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{U2.eps}\n\nFIG. 6. Effective potential for $n=4,$ $K=0,$ $T=1,$ $R(0)=1.099,$ $%\nR^{\\prime }(0)=0,$ $\\varphi(0)=0,$ $\\varphi^{\\prime}(0)=0.3$\n\nVertical dash marks $R(0)$ value.\n\n\\includegraphics[\nheight=2in, width=1.5in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{R2.eps}\n\nFIG. 7. Spatial dependence of the amplitude function for $n=4,$ $K=0,$ $T=1,$\n$R(0)=1.099,$ $R^{\\prime}(0)=0,$ $\\varphi(0)=0,$ $\\varphi^{\\prime}(0)=0.3$\n\n\\includegraphics[\nheight=2in, width=1.5in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{F2.eps}\n\nFIG. 8. Spatial dependence of the phase function for $n=4,$ $K=0,$ $T=1,$ $%\nR(0)=1.099,$ $R^{\\prime}(0)=0,$ $\\varphi(0)=0$\n\n(a) thin line $\\varphi^{\\prime}(0)=0.3$\n\n(b) heavy line $\\varphi^{\\prime}(0)=0.7.$\n\n\\includegraphics[\nheight=1.8in, width=1.8in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{P2.eps}\n\nFIG. 9. Amplitude-phase polar diagram for $n=4,$ $K=0,$ $T=1,$ $R(0)=1.099,$\n$R^{\\prime}(0)=0,$ $\\varphi(0)=0$\n\n(a) thin line $\\varphi^{\\prime}(0)=0.3$\n\n(b) heavy line $\\varphi^{\\prime}(0)=0.7.$\n\\end{center}\n\n\\newpage\n\n\\section{Numerical solution for K}\n\nResults of the numerical solution of our equations in the case of $K\\neq0$\nare presented in the figures 10 -- 15. The following important distinctions\nfrom the case of $K=0$ must be mentioned:\n\n(i) The periodicity of the spatial dependence is broken.\n\n(ii) The frequency and amplitude modulation can be seen in Figs. 10 and 11.\n\n(iii) A direction of the phase function staircase spatial dependence may\nchange to the opposite after some number of steps. The number of steps\nbetween neighboring direction changes is random (see FIG. 15).\n\n(iv) There exists a parameter range, where the trajectories in the\npolar diagram show closed periodic movement: a synchronization\nwith the periodic contribution of the sine of the monotonously\nincreasing component of the phase function takes place.\n\n\\begin{center}\n\\includegraphics[\nheight=3in, width=2in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{R3.eps}\n\nFIG. 10. Spatial dependence of the amplitude function for $n=4,$ $K=1.6,$ $%\nT=1,$ $R(0)=0.3,$ $R^{\\prime}(0)=0,$ $\\varphi(0)=0,$ $\\varphi^{\\prime\n}(0)=0.3$\n\n\\includegraphics[\nheight=3in, width=2in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{R4.eps}\n\nFIG. 11. Spatial dependence of the amplitude function for $n=4,$ $K=1.6,$ $%\nT=1,$ $R(0)=0.3,$ $R^{\\prime}(0)=0,$ $\\varphi(0)=0,$ $\\varphi^{\\prime\n}(0)=0.75$: thin line\n\nSpatial dependence of the parameter $C^{2}$ (see below) in arbitrary units:\nheavy line.\n\n\\includegraphics[\nheight=3in, width=2in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{F34.eps}\n\nFIG. 12. Spatial dependence of the phase function for $n=4,$ $K=1.6,$ $T=1,$\n$R(0)=0.3,$ $R^{\\prime}(0)=0,$ $\\varphi(0)=0$\n\n(a) thin line $\\varphi^{\\prime}(0)=0.3$\n\n(b) heavy line $\\varphi^{\\prime}(0)=0.75.$\n\n\\includegraphics[\nheight=2.5in, width=2.5in,angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{P3.eps}\n\nFIG. 13. Amplitude-phase polar diagram for $n=4,$ $K=1.6,$ $T=1,$ $R(0)=0.3,$\n$R^{\\prime}(0)=0,$ $\\varphi(0)=0,\\varphi^{\\prime}(0)=0.3$\n\n\\includegraphics[\nheight=2.5in, width=2.5in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{P4.eps}\n\nFIG. 14. Amplitude-phase polar diagram for $n=4,$ $K=1.6,$ $T=1,$ $R(0)=0.3,$\n$R^{\\prime}(0)=0,$ $\\varphi(0)=0,\\varphi^{\\prime}(0)=0.75$\n\\end{center}\n\n\\newpage\n\n\\section{Discussion}\n\nThus, the main distinctions of the solution in the case of the\nnon-vanishing anisotropy parameter $K$ are a random change of the\njumps direction (see FIG. 12) and an amplitude modulation of the\namplitude function (see FIG. 11). Just these random direction\nchanges lead to the tangled trajectory in polar coordinates\npresented in FIG 14.\n\nIn the case of $K=0$ we introduced the integration constant $C_{0}$ (\\ref%\n{phasesolution}). Let us introduce the function:\n\n\\begin{equation*}\nC(\\xi)=\\left[ \\psi(\\xi)-\\frac{T}{2}\\right] R(\\xi)^{2}.\n\\end{equation*}\n\nDirect differentiation shows that this function reduces to a constant in the\ncase of vanishing anisotropy parameter $K$:%\n\\begin{eqnarray}\n\\frac{dC}{d\\xi}=2R\\frac{dR}{d\\xi}\\left[ \\psi-\\frac{T}{2}\\right] +R^{2}\\frac{%\nd\\psi}{d\\xi}=  \\notag \\\\\n=R^{2}(2\\frac{R^{\\prime}}{R}\\varphi^{\\prime} -T\\frac{R^{\\prime}}{R}%\n+\\varphi^{\\prime\\prime})=  \\notag \\\\\n=R^{2}(-R^{n-2}K\\sin n\\varphi)=-R^{n}K\\sin n\\varphi.\n\\end{eqnarray}\n\n\\begin{center}\n\\includegraphics[\nheight=3in, width=2in, angle=270, trim = 0.1in 0.1in 0.1in 0.1in, clip\n]{C4.eps}\n\nFIG. 15. Spatial dependence of the parameter $C$ for $n=4,$ $K=1.6,$ $T=1,$ $%\nR(0)=0.3,$ $R^{\\prime}(0)=0,$ $\\varphi(0)=0,\\varphi^{\\prime}(0)=0.75$\n\n(a) thin line $C^{\\prime}$\n\n(b) heavy line $C.$\n\\end{center}\n\nComparison of FIG. 15 and FIG. 12 confirms that changes of the direction\ntake place in the points where $C(\\xi)$ function changes its sign.\n\n\\section{Conclusion}\n\nWe have shown in this paper that the constant amplitude\napproximation gives a poor description of the real picture of the\nspatial evolution of the amplitude and phase functions for the\nmodel of the incommensurate ferroelectric with the Lifshitz\ninvariant.\n\n\n\n\\begin {thebibliography}{99}\n\n\\bibitem {dzialoshin}{ I. Dzyaloshinskii, Soviet Physics JETP 19, 960\n(1964).}\n\n\\bibitem{levaniuk}{A.P. Levaniuk and D.G. Sannikov, Fiz. Tverd. Tela 18,\n423 (1976).}\n\n\\bibitem{hachatur}{ A.G. Khachaturian, \\textit{Theory of Structural\nTransformations in Solids, Wiley, 1983}}\n\n\\bibitem{char}{  E.V. Charnaya, S.A. Ktitorov, O.S. Pogorelova,\nFerroelectrics, 297, 29 (2003).}\n\n\\bibitem{izium}{ Ju. A. Iziumov, V.N. Syromiatnikov, Phase Transitions and\nCrystal Symmetry, Moscow, Nauka, 1984.}\n\n\\bibitem{soliton}{Mark J. Ablowitz and Harvey Segur, Solitons\nand the Inverse Scattering Transform, SIAM, Philadelphia, 1981}\n\\end{thebibliography}\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAtherosclerotic plaque formation is today seen as a chronic inflammation of the arterial wall which grows over decades and may finally lead to a heart attack in case the artery is occluded or a thrombus is built through a rupture of the plaque. To understand the mechanisms of the chronic inflammation it was recently shown in \\cite{kuhlmann2012implantation} that genetically modified (apoE knockdown) mice with a cuff around their carotid develop atherosclerotic plaque formation up- and downstream of the cuff after they were fed with a Western diet. A low wall shear stress of the blood onto the arterial wall or highly oscillating blood flow was shown to be an important indicator for the development of plaque because it damages the endothelial layer. \n\nAt this point our mathematical model (cf. \\cite{ibragimov2005mathematical}) comes into play which we want to present in section 2: A dysfunction of the endothelial allows low-density lipoproteins (LDL) to enter the artery wall. Once inside the arterial wall, the LDL becomes oxidized which leads to a recruitment of immune cells, i.e. monocytes. Monocytes differentiate into active macrophages when inside the arterial wall starting continuously absorbing the oxidized LDL. Finally, the macrophages differentiate into foam cells, die and build a necrotic core. Smooth muscle cells (SMCs) from the outer regions of the arterial wall can migrate into the lesion and either become an apoptotic cell or migrate around the lesion to form a fibromuscular cap overlaying the plaque.   \n\nSection 3 describes the spatial and temporal discretization of the CDG2 method which was successfully tested for elliptic problems, scalar convection-diffusion equations and compressible Navier-Stokes equations in \\cite{brdar2012cdg2,klofkorn2011benchmark,dgimpl:12}.\n\nWe summarize our paper with some 2D and 3D benchmark tests. in section 4 and a conclusion in section 5.  \n\n\n\n\\section{Mathematical Model for Atherosclerotic Inflammation}\n\nA variety of mathematical models dealing with atherosclerotic plaque formation exist, \nsee  \\cite{calvez2010mathematical,ibragimov2005mathematical}. Here, we focus on six species: immune cells $n_1$ (we do not distinguish between monocytes and macrophages, here), SMCs $n_2$, debris $n_3$ (i.e. all dead or apoptotic cells), chemoattractant $c_1$ (immune cells and SMCs attract to), non oxidized $c_2$ and oxidized LDL $c_3$. Let $\\Omega\\subset\\mathbb{R}^d$, $d=2,3$ be the domain of the arterial wall, $\\Gamma_1$ the boundary between the arterial wall and the lumen and $\\Gamma_2$ the outer boundary of the arterial wall.\n\nLet us suppose that for all $x\\in\\Omega$ and $t>0$ the following system holds:\n\\begin{eqnarray}\\label{eq:01}\n\\partial_t n_1 &=& \\nabla\\cdot \\left( \\mu_1 \\nabla n_1 -\\chi(n_1,c_1,\\chi_{11}^0,\\chi_{11}^{th})\\nabla c_1 - \\chi(n_1,c_3,\\chi_{13}^0,\\chi_{13}^{th})\\nabla c_3 \\right) - d_1n_1,\\\\\n\\partial_t n_2 &=& \\nabla\\cdot \\left( \\mu_2 \\nabla n_2 -\\chi(n_2,c_1,\\chi_{21}^0,\\chi_{21}^{th})\\nabla c_1 + \\chi( n_2,n_1,\\chi_{21}^0,\\chi_{21}^{th})\\nabla n_1 \\right) - d_2n_2,\\\\\n\\partial_t n_3 &=& \\nabla\\cdot ( \\mu_3 \\nabla n_3 ) + d_1 n_1 + d_2 n_2 - F(n_3,c_3) n_1,\\\\\n\\partial_t c_1 &=& \\nabla\\cdot ( \\nu_1 \\nabla c_1 ) - \\alpha_1 n_1 c_1 - \\alpha_2 n_2 c_1 + f_1(n_3)n_3,\\\\\n\\partial_t c_2 &=& \\nabla\\cdot (\\nu_2 \\nabla c_2 ) - k c_2,\\\\\n\\partial_t c_3 &=& \\nabla\\cdot (\\nu_3 \\nabla c_3 ) + k c_2.\n\\end{eqnarray}\nIn our model we assume the motility coefficients $\\mu_1$, $\\mu_2$, $\\mu_3$, $\\nu_1$, $\\nu_2$ and $\\nu_3$ to be constant. The parameters $d_1$ and $d_2$ are also constant and describe the death rates of immune cells and SMCs. Chemoattractant is neutralized by immune cells and SMCs which is described by $\\alpha_1$ and $\\alpha_2$. The parameter $k$ describes how fast the native LDL becomes oxidized. \n\nThe functions $\\chi$ is called tactic sensitivity function.\nWe have chosen $\\chi(x,y,a,b) = a \\frac{x}{y+b}$ to mimic a high sensitivity of cells to the relative gradient $\\frac{\\nabla c}{c}$ of a chemoattractant (or other cells) $c$ on the one hand and a small penalization term to regularize the (chemo-)tactic movement for small concentrations $c$ on the other hand. A lot of other tactic sensitivity functions are possible as well. Our tactic sensitivity functions are defined by constants $\\chi_{ij}^0$ and $\\chi_{ij}^{th}$.\n\nFor a healthy immune system debris is degraded which is indicated by a general function $F>0$. We suppose $\\gamma:=F<0$ to be constant indicating a diseased state. The function $f_1$ is a production term which is debris dependent. \nWe allow LDL and immune cells to enter the arterial wall through the inner boundary and SMCs to enter through the outer arterial wall. The immune cell (SMC) inflow is triggered when a threshold $c_1^{*}$ ($c_1^{**}$) of chemoattractant is exceeded, i.e.\n\\begin{eqnarray}\\label{eq:03}\n\t\\partial_n n_1 &=& -\\beta_1 H(c_1 -c_1^{*} ) \\quad\\quad \\forall x\\in \\Gamma_1,\\:t>0,\\\\\n\t\\partial_n n_2 &=& -\\beta_2 H(c_1 -c_1^{**} ) \\quad\\quad \\forall x\\in \\Gamma_2,\\:t>0,\\\\\n\t\\partial_n c_2 &=& -\\sigma \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad \\forall x\\in \\Gamma_{1,in},\\:t>0,\n\\end{eqnarray}\nwith Heaviside function $H$ and a boundary $\\Gamma_{1,in}\\subset\\Gamma_1$ for the inflow of LDL. Here, $\\beta_1$, $\\beta_2$ and $\\sigma$ denote constant inflow rates for immune cells, SMCs and LDL, respectively. For all other boundary conditions we choose a no-flow condition.\nThe initial data is supposed to be given by $n_i(x,0)=n_i^0(x)$ and $c_i(x,0)=c_i^0(x)$, $i=1,2,3$, $x\\in\\Omega$.\n\nDefining a vector $U := (n_1, n_2, n_3, c_1, c_2, c_3 )$ and functions $\\mathcal{F}:\\mathbb{R}^{6}\\rightarrow\\mathbb{R}^{6\\times d}$, $\\mathcal{A}:\\mathbb{R}^{6}\\rightarrow\\mathbb{R}^{6\\times 6}$ and $S:\\mathbb{R}^{6}\\rightarrow\\mathbb{R}^{6}$ by\n\\begin{eqnarray*}\n\t\\mathcal{F}(U)&:=&(\\chi(n_1,c_1,\\chi_{11}^0,\\chi_{11}^{th})\\nabla c_1 + \\chi(n_1,c_3,\\chi_{13}^0,\\chi_{13}^{th})\\nabla c_3,\\\\\n      &&\\chi(n_2,c_1,\\chi_{21}^0,\\chi_{21}^{th})\\nabla c_1 - \\chi( n_2,n_1,\\chi_{21}^0,\\chi_{21}^{th})\\nabla n_1,0\\ldots0),\\\\\n\t\\mathcal{A}(U)&:=&\\mathrm{diag}(\\mu_1,\\mu_2,\\mu_3,\\nu_1,\\nu_2,\\nu_3 ),\\\\\n\tS(U)&:=&-(d_1n_1,d_2n_2,-d_1 n_1 - d_2 n_2 + \\gamma n_1,\\alpha_1 n_1 c_1+\\alpha_2 n_2 c_2 - f_1n_3, k c_2,-k c_2)\n\\end{eqnarray*}\nequation \\eqref{eq:01} can be written as \n\\begin{equation}\n\t\\partial_t U = - \\nabla\\cdot( \\mathcal{F}(U) - \\mathcal{A}(U)\\nabla U) + S(U).\n\\end{equation}\n\n\n\\section{Discretization}\n\\label{seq:discretization}\n\nThe considered discretization is based on the Discontinuous Galerkin (DG) approach and \nimplemented in \\textsc{Dune-Fem}\\xspace \\cite{dedner2010generic}  a module of the\n\\textsc{Dune}\\xspace framework \\cite{bastian2008generic}.\nThe current state of development allows for simulation of convection dominated \n(cf.~\\cite{limiter:11}) as well as viscous flow (cf.~\\cite{brdar2012cdg2}).\nWe consider the CDG2 method from \n\\cite{brdar2012cdg2} for various polynomial orders in space and 2nd (or 3rd) order \nin time for the numerical investigations carried out in this paper.\n\n\\subsection{Spatial Discretization}\n\nThe spatial discretization is derived in the following way. \nGiven a tessellation ${\\mathcal{T}_h}$ of the domain $\\Omega$ with \n$\\cup_{K \\in {\\mathcal{T}_h}} K = \\Omega$ the \ndiscrete solution $\\vecU_h$ is sought in the piecewise polynomial space \n\\begin{equation}\n\\label{eqn:vspace}\n    V_h = \\{\\vect{v}\\in L^2(\\Omega,\\mathbb{R}^{n_{spec}}) \\; \\colon\n    \\vect{v}|_{K}\\in[\\mathcal{P}_k(K)]^{n_{spec}}, \\ K\\in{\\mathcal{T}_h}\\}\n      \\quad\\textrm{for some}\\;k \\in {\\mathbbm N}, \\nonumber\n\\end{equation}\nwhere $n_{spec}$ is the number of species and $\\mathcal{P}_k(K)$ is a space containing polynomials up to degree\n$k$. \n\n\\newcommand{\\dual}[1]{\\langle \\vect{\\varphi}, #1 \\rangle\nWe denote with $\\Gamma_i$ the set of all intersections between two \nelements of the grid ${\\mathcal{T}_h}$ and accordingly with $\\Gamma$ the set of all\nintersections, also with the boundary of the domain $\\Omega$. \nThe following discrete form is not the most general but still\ncovers a wide range of well established DG methods. \nFor all basis functions $\\vect{\\varphi} \\in V_h$ we \ndefine \n\\begin{equation}\n\\label{convDiscr}\n\\dual { \\oper{L}_h(\\vecU_h) } := \\dual{ \\oper{K}_h(\\vecU_h) } + \\dual{ \\oper{I}_h(\\vecU_h) }\n\\end{equation}\nwith the element integrals  \n\\begin{eqnarray}\n\\label{eqn:elementint}\n  \n   \\dual{ \\oper{K}_h(\\vecU_h) } &:=&\n     \n      \\sum_{\\entity \\in {\\mathcal{T}_h}} \\int_{\\entity}\n      \\big( ( \\mathcal{F}(\\vecU_h) - \\mathcal{A}(\\vect{U}_h) \\nabla \\vecU_h ) : \\nabla\\vect{\\varphi} + S(\\df)\n      \\cdot \\vect{\\varphi} \\big),\n\\end{eqnarray}\nand the surface integrals (by introducing appropriate numerical fluxes \n$\\flux{\\mathcal{F}}_{\\isec}$, $\\flux{\\mathcal{A}}_{\\isec}$ for the convection and diffusion terms, respectively) \n\\begin{eqnarray}\n\\label{eqn:surfaceint}\n  \n   \\dual{ \\oper{I}_h(\\vecU_h) } &:=&\n      \\sum_{e \\in \\Gamma_i} \\int_e \\big(\n      \\vaver{\\mathcal{A}(\\vect{U}_h)^T\\nabla\\vect{\\varphi}} : \\vjump{\\vecU_h} +\n      \\vaver{\\mathcal{A}(\\vect{U}_h)\\nabla\\vecU_h} : \\vjump{\\vect{\\varphi}} \\big) \\nonumber \\\\\n    &-& \\sum_{e \\in \\Gamma} \\int_e \\big( \\flux{\\mathcal{F}}_{\\isec}(\\vecU_h) - \\flux{\\mathcal{A}}_{\\isec}(\\vecU_h)\\big) :\n      \\vjump{\\vect{\\varphi}},      \n\\end{eqnarray}\nwhere $\\vaver{ \\vect{V} } = \\frac{1}{2}( \\vect{V}^+ + \\vect{V}^- )$ denotes the average and \n$\\vjump{ \\vect{V} } = (\\boldsymbol{n}^+ \\otimes \\vect{V}^+  + \\boldsymbol{n}^-\\otimes \\vect{V}^-) $ the jump of the\ndiscontinuous function $\\vect{V}\\in V_h$ over element boundaries.\nFor matrices $\\sigma,\\tau\\in\\mathbb{R}^{m\\times n}$ we use standard notation\n$\\sigma : \\tau = \\sum_{j=1}^m\\sum_{l=1}^n\\sigma_{jl}\\tau_{jl}$. Additionally, for vectors\n$\\vect{v} \\in \\mathbb{R}^m,\\vect{w}\\in\\mathbb{R}^n$, we define $\\vect{v}\\otimes\\vect{w}\\in\\mathbb{R}^{m\\times n}$\naccording to $(\\vect{v}\\otimes\\vect{w})_{jl}=\\vect{v}_j \\vect{w}_l$ for $1\\leq j\\leq m$, $1\\leq l\\leq n$.\n\nThe convective numerical flux $\\flux{\\mathcal{F}}_{\\isec}$ can be any appropriate numerical flux known for\nstandard finite volume methods. \nFor the results presented in this paper we choose $\\flux{\\mathcal{F}}_{\\isec}$ to be the widely used\nlocal Lax-Friedrichs numerical flux function.\n\nA wide range of diffusion fluxes $\\flux{\\mathcal{A}}_{\\isec}$ can be found in the\nliterature, for a summary see \\cite{arnold2002unified}.\nWe choose the CDG2 flux\n\\begin{eqnarray}\n\\flux{\\mathcal{A}}_{\\isec}(\\vect{V}) := 2\\chi_e \\big(\\mathcal{A}(\\vect{V}){\\liftr_e}(\\vjump{\\vect{V}})\\big)|_{{K^-_e}}\n\\quad\\mbox{for } \\vect{V}\\in V_h,\n\\end{eqnarray}\nwhich was shown to be highly efficient for advection-diffusion equations (cf. \\cite{brdar2012cdg2}). \nBased on stability results, we choose ${K^-_e}$ to be the element adjacent to the edge $e$ with the smaller\nvolume. ${\\liftr_e}(\\vjump{\\vect{V}})\\in [V_h]^d$ is the lifting of the jump of $\\vect{V}$ defined by\n\\begin{eqnarray}\n    \\int_\\Omega {\\liftr_e}(\\vjump{\\vect{V}}) : \\boldsymbol{\\tau} = -\\int_e\n    \\vjump{\\vect{V}} : \\vaver{\\boldsymbol{\\tau}} \\quad\n    \\mbox{for all}\\;\\boldsymbol{\\tau}\\in [V_h]^d.\n\\end{eqnarray}\nFor the numerical experiments in this paper we use $\\chi_e= \\frac{1}{2}\\mathcal{N}_\\grid$,\nwhere $\\mathcal{N}_\\grid$ is the maximal number of intersections one element in the grid\ncan  have (cf. \\cite{brdar2012cdg2}). We use triangular elements \nwhere $\\chi_e=1.5$ for all $e \\in \\Gamma$, and tetrahedral elements \nwhere $\\chi_e=2$ for all $e \\in \\Gamma$.\n\n\\subsection{Temporal discretization}\n\\label{TimeDisc}\n\nThe discrete solution $\\vecU_h(t) \\in V_h$ \nhas the form $\\vecU_h(t,x) = \\sum_i \\vect{U}_i(t)\\vect{\\varphi}_i(x)$.\nWe get a system of ODEs for the coefficients of $\\vect{U}(t)$ which reads \n\\begin{eqnarray}\n  \\label{eqn:ode}\n  \\vect{U}'(t) &=& f(\\vect{U}(t),t)  \\mbox{ in } (0,T]\n\\end{eqnarray}\nwith $f(\\vect{U}(t),t) = M^{-1}\\oper{L}_h(\\vecU_h(t),t)$, $M$ being the mass matrix which is in\nour case block diagonal or even diagonal, depending on the choice of basis\nfunctions. $\\vect{U}(0)$ is given by the projection of $\\vect{U}_0$ onto $V_h$.\n\nFor the numerical results \nwe have chosen Diagonally Implicit Runge-Kutta (DIRK) \nsolvers of order $2$, $3$, or $4$ depending\non the polynomial order of the basis functions. The DIRK solvers are  \nbased on a Jacobian-free Newton-Krylov method (see \\cite{knoll:04}).\nThe Krylov method is chosen to be GMRES. \nThe implicit solver relies on a \\textbf{matrix-free} implementation of the discrete operator \n$\\oper{L}_h$. In a follow-up paper we will compare this approach to a fully assembled\napproach.\n\n\\section{Numerical Results}\nIn this section we present some benchmark tests for 2D and 3D focusing on parallelization and higher order DG schemes. Due to the lack of an exact solution $U$ we have computed the $L^2$-error between the discrete solution $U_h$ and a very fine, higher order solution $U_{h'}$. The quadrature order to compute $\\|U_h-U_{h'}\\|_{L^2(\\Omega)}$ was chosen to be $2k+4$, where $k$ denotes the order of the scheme. All computations are done on an unstructured, tetrahedral mesh.\n\n\\subsection{A 2D numerical experiment with six species}\n\n\n\\begin{table}[t]\n\\small\n\\caption{Accuracy of the CDG2 scheme with 32 threads}\n\\begin{tabular}{ll|lll|lll|lll}\n\\hline\\noalign{\\smallskip}\n & & & linear & & & quadratic & & & cubic & \\\\\nlevel & grid size & time$^a$ & $L^2$-error & EOC$^b$ & time$^a$ & $L^2$-error & EOC$^b$ & time$^a$ & $L^2$-error & EOC$^b$ \\\\\n\\hline\n0 & 80    & 5.72E-1& 2.42E-3 & ---     & 2.00E0 & 2.18E-3 & ---     & 6.52E0 & 1.96E-3 & --- \\\\\n1 & 320   & 5.56E0 & 2.10E-3 & 0.20311 & 2.33E1 & 1.82E-3 & 0.26650 & 8.63E1 & 1.50E-3 & 0.38074 \\\\\n2 & 1280  & 3.98E2 & 1.82E-3 & 0.21263 & 2.09E2 & 1.34E-3 & 0.43315 & 8.22E2 & 9.26E-4 & 0.69823 \\\\\n3 & 5120  & 3.33E3 & 1.39E-3 & 0.38944 & 2.21E3 & 7.92E-4 & 0.76429 & 9.12E3 & 4.32E-4 & 1.0993 \\\\\n4 & 20480 & 3.01E4 & 8.28E-4 & 0.74208 & 2.10E4 & 2.94E-4 & 1.4284  & 8.02E4 & 8.77E-5 & 2.3024 \\\\\n5 & 81920 & 2.67E5 & 3.21E-4 & 1.3659  & 1.93E5 & 7.26E-5 & 2.0193  & 6.96E5 & 2.33E-5 & 1.9122 \\\\\n\\hline\n\\end{tabular}\n$^a$ total CPU time, $^b$ experimental order of convergence, \n\\label{tab_CGD2}\n\\end{table}\n\n$U_{h'}$ was calculated using the 4th order CDG2 scheme on a grid with 81,920 elements (refinement level $5$), i.e. 7,372,800 degrees of freedom. For each $h$-refinement of the grid we bisect the time step size. Results for linear, quadratic and cubic DG schemes can be seen in table \\ref{tab_CGD2}. In figure \\ref{fig:eoc} (left picture) we compare on a log-log scale the total CPU time of all threads with the $L^2$-error. Although the convergence rate is not as high as from the theory for parabolic problems, we see better rates for higher order schemes. We assume that re-entrant corners \nare responsible for the reduced convergence rates, see re-entrant corners in left picture of figure \\ref{fig:2d}.\n\n\n\\begin{figure}[t]\n\\includegraphics[scale=0.215]{girke-2d.eps}\n\\caption{Left: The coarsest grid for the EOC calculations containing 80 elements visualising a re-entrant corner (blue). The angle of $171^{\\circ}$ stays fixed for all refinements. Middle: Initial distribution for the immune cells. Right: Solution for 6 species from left to right, up to down: Immune cells, SMCs, debris, chemoattractant, native LDL, oxidized LDL. (data visualisation: Paraview.)}\n\\label{fig:2d}\n\\end{figure}\n\nThe right picture of figure \\ref{fig:eoc} shows that the CDG2 is as good as the BR2 scheme and outperforms other DG schemes. \n\n\\begin{figure}[t]\n\\includegraphics[scale=0.17]{girke-eoc2.eps}\n\\caption{Plot CPU time vs. $L^2$-error: left: 1st, 2nd and 3rd order CDG2 scheme, right: 1st order CDG, CDG2, Baumann-Oden (BO), Bassy-Rebay (BR2), interior penalty (IP) scheme (Visualisation of graphs: gnuplot.)}\n\\label{fig:eoc}\n\\end{figure}\n\n\n\n\\subsection{A 3D numerical experiment with three species}\n\nFor the 3D benchmark we simplify our model and do our simulation only for immune cells, debris and  chemoattractant. This reduces the considered model to \n\\begin{eqnarray}\\label{eq:02}\n\\partial_t n_1 &=& \\nabla\\cdot \\left( \\mu_1 \\nabla n_1 -\\chi(n_1,c_1,\\chi_{11}^0,\\chi_{11}^{th})\\nabla c_1\\right),\\\\\n\\partial_t n_3 &=& \\nabla\\cdot ( \\mu_3 \\nabla n_3 ) + d_1 n_1 + d_2 n_2 - F(n_3,c_3) n_1,\\\\\n\\partial_t c_1 &=& \\nabla\\cdot ( \\nu_1 \\nabla c_1 ) - \\alpha_1 n_1 c_1 + f_1(n_3)n_3.\n\\end{eqnarray}\nWe cannot trigger the inflammation through an inflow of LDL anymore. Thus, we suppose that the inflammation is triggered by a local, high concentration of debris and keep all other boundary and initial data from the last section.\n\nIn the 3D benchmark we examine parallelization using MPI and present in table \\ref{tab_parallel} strong scaling results for a third order CDG2 scheme on a grid with 113,549 elements and 13,625,880 degrees of freedom. Figure \\ref{fig:3d} shows the distribution of the processors and a discrete solution of the chemoattractant calculated using first order CDG2.\n\n\\begin{table}[t]\n\\caption{CPU time for a parallel runs using the cubic CDG2 method for computation of\n$10$ time steps.}\n\\begin{tabular}{p{2.5cm} p{1.0cm}p{1.0cm}p{1.0cm}p{1.0cm}p{1.0cm}p{1.0cm}}\n\\hline\nprocessors      & 8      &   16   &   32   &   64   & 128    & 256 \\\\\nCPU time in sec & $1177$ & $528$  & $277$  & $142$  & $75$   & $39$\\\\\nspeedup         &  ---   & $2.23$ & $4.29$ & $8.29$ & $15.7$ & $30.18$\\\\\n\\hline\n\\end{tabular}\n\\label{tab_parallel}\n\\end{table}\n\n\n\n\n\\begin{figure}[t]\n\\includegraphics[scale=0.16]{girke-3d.eps}\n\\caption{3D cuff model. Left: Each colour denotes a processor in a parallel run with 32 processors, right: Isolines of the distribution of the chemoattractant after the inflammation has started}\n\\label{fig:3d}\n\\end{figure}\n\n\\section{Conclusion}\n\nWe have shown that Discontinuous Galerkin schemes are well suited for solving huge coupled reactive diffusion transport systems. Modern techniques, such as parallelization, help to handle large systems in an appropriate CPU time.\nFurthermore, we have shown that it is possible to model the early stages of atherosclerotic plaque formation. A lot of more work needs to be done: In a future paper we will model the wall shear stress and some more species to understand later stages of atherosclerosis. \n\n\\section*{Acknowledgement}\nThis work was supported by the Deutsche Forschungsgemeinschaft, Collaborative Research Center\nSFB 656 ``Cardiovascular Molecular Imaging'', project B07, M{\\\"u}nster, Germany. The scaling results were produced using the super computer Yellowstone\n(ark:\/85065\/d7wd3xhc) provided by NCAR's Computational and Information Systems\nLaboratory, sponsored by the National Science Foundation.\nRobert Kl\\\"ofkorn is partially funded by \nthe DEO program BER under award DE-SC0006959.\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\nIn this paper, we study a class of second-order ordinary differential equations (ODEs) of the form\n\\begin{align} \\label{eq: nonlinear ode}\nM \\ddot{x} + D \\dot{x} + f(x) = 0,\n\\end{align}\nand its corresponding first-order system\n\\begin{align} \\label{eq: nonlinear ode 1 order_intro}\n\\begin{bmatrix} \\dot{x} \\\\ \\dot{y}\n\\end{bmatrix}\n= \n\\begin{bmatrix}\n0 & I \\\\\n0 & -M^{-1}D\n\\end{bmatrix}\n\\begin{bmatrix} {x} \\\\ {y}\n\\end{bmatrix}\n- M^{-1} \n\\begin{bmatrix} 0 \\\\ f(x)\n\\end{bmatrix},\n\\end{align}\nwhere $f:\\mathbb{R}^n\\to \\mathbb{R}^n$ is a continuously differentiable function, the dot denotes differentiation with respect to the independent variable $t\\ge0$, the dependent variable $x\\in\\mathbb{R}^n$ is a vector of state variables, and the coefficients $M\\in\\mathbb{S}^n$ and $D\\in\\mathbb{S}^n$ are constant $n\\times n$ real symmetric matrices.\nWe refer to $M$ and $D$ as the inertia and damping matrices, respectively. \nWe restrict our attention to the case where $M$ is nonsingular, thereby avoiding differential algebraic equations, and $D \\in \\mathbb{S}^n_+$ is positive semi-definite (PSD). We also investigate and discuss the case where $M$ and $D$ are not symmetric.  \n\n\nAn important example of \\eqref{eq: nonlinear ode} is an electric power system with the set of interconnected generators $\\mathcal{N}=\\{1,\\cdots,n\\}, n\\in\\mathbb{N}$ characterized by the second-order system\n\\begin{align} \\label{eq: 2nd order swing-intro}\n\t\t\\frac{m_j}{\\omega_s} \\ddot{\\delta}_j(t)+ \\frac{d_j}{\\omega_s} {\\dot{\\delta}}_j(t) = P_{m_j} - \\sum \\limits_{k = 1}^n { V_j  V_k Y_{jk} \\cos \\left( \\theta _{jk} - \\delta _j + \\delta _k \\right)} && \\forall j \\in \\mathcal{N},\n\\end{align}\nwhere $\\delta\\in\\mathbb{R}^n$ is the vector of state variables. The inertia and damping matrices in this case\nare $M= \\frac{1}{\\omega_s}  \\mathbf{diag}(m_1,\\cdots,m_n)$ and $D=\\frac{1}{\\omega_s}\\mathbf{diag}(d_1,\\cdots,d_n)$. \nSystem \\eqref{eq: 2nd order swing-intro}, which is  known as the \\emph{swing equations}, describes the nonlinear dynamical relation between the power output and voltage angle of generators \\cite{1994-kundur-stability, 2020-fast-certificate}.\nThe first-order system associated with swing equations is also of the form \\eqref{eq: nonlinear ode 1 order_intro}, i.e.,\n\\begin{subequations} \\label{eq: swing equations -intro}\n\t\\begin{align}\n\t\t& \\dot{\\delta}_j(t) = \\omega_j(t) && \\forall j \\in \\mathcal{N},  \\label{eq: swing equations a-intro}\\\\\n\t\t&  \\frac{m_j}{\\omega_s} \\dot{\\omega}_j(t)+ \\frac{d_j}{\\omega_s} \\omega_j(t) = P_{m_j} - \\sum \\limits_{k = 1}^n { V_j  V_k Y_{jk} \\cos \\left( \\theta _{jk} - \\delta _j + \\delta _k \\right)} && \\forall j \\in \\mathcal{N}, \\label{eq: swing equations b-intro}\n\t\\end{align}\n\\end{subequations}\nwhere $(\\delta,\\omega)\\in\\mathbb{R}^{n+n}$ is the vector of state variables. Note that each generator $j$ is a second-order oscillator, which is coupled to other generators through the cosine term in \\cref{eq: swing equations b-intro} and the admittance $Y_{jk}$ encodes the graph structure of the power grid (see \\Cref{Sec: Power System Model and Its Properties} for full details on swing equations).\n\n\n\n\nAmong the various aspects of model \\eqref{eq: nonlinear ode}, the impact of damping matrix $D$ on the stability properties of the model is one of the most intriguing topics \\cite{1980-Miller-Asymptotic,2013-adhikari-structural-book, 2017-koerts-second-order}. Moreover, better understanding of the damping impact in swing equations \\eqref{eq: 2nd order swing-intro} is of particular importance to the stability analysis of electric power systems \\cite{2012-Dorfler-synchronization}.  \nUndamped modes and oscillations are the root causes of several blackouts, such as the WECC blackout on August 10, 1996 \\cite{1996-blackout} as well as the more recent events such as the forced oscillation event on January 11, 2019 \\cite{nerc2019} in the Eastern Interconnection of the U.S. power grid. In order to maintain system stability in the wake of unexpected equipment failures, many control actions taken by power system operators are directly or indirectly targeted at changing the effective damping of system \\eqref{eq: 2nd order swing-intro} \\cite{1994-kundur-stability,2019-Patrick-koorehdavoudi-input,2012-aminifar-wide-area-damping}. In this context, an important question is \\emph{how the stability properties of power system equilibrium points change as the damping of the system changes}. Our main motivation is to rigorously address this question for the general model \\eqref{eq: nonlinear ode} and show its applications in power system model \\eqref{eq: 2nd order swing-intro}.\n\n\n\n\n\n\n\n\n\\subsection{Literature Review}\nThe dynamical model \\eqref{eq: nonlinear ode} has been of interest to many researchers who have studied necessary and sufficient conditions for its local stability \\cite{1980-skar-nontrivial-transfer-conductance1, 2020-fast-certificate} or characterization of its stability regions \\cite{1988-chiang-stability-of-nonlinear}. \nWhen $f(x)$ is a linear function, this model coincides with the model of $n$-degree-of-freedom viscously damped vibration systems which are also extensively studied in the structural dynamics literature \\cite{1984-laub-controllability,2010-Ma-decoupling,2013-adhikari-structural-book}.\nEquation \\eqref{eq: nonlinear ode} is also the cornerstone of studying many physical and engineering systems such as an $n$-generator electric power system \\cite{2012-Dorfler-synchronization}, an $n$-degree-of-freedom rigid body \\cite{1988-chiang-stability-of-nonlinear}, and a system of $n$ coupled oscillators \\cite{2011-dorfler-critical-coupling, 2012-Dorfler-synchronization, 2005-kuramoto-review}, in particular Kuramoto oscillators with inertia \\cite{2016-Igor-Inertia-Kuramoto,2020-Igor-Inertia-Kuramoto}.\n\n\n\n\nRegarding damping effects in power systems, the results are sporadic and mostly based on empirical studies of small scale power systems. For example, it is known that the lossless swing equations (i.e., when the transfer\nconductances of power grid are zero, which corresponds to $\\nabla f(x)$ in \\eqref{eq: nonlinear ode} being a real symmetric matrix for all $x$) have no periodic solutions, provided that all generators have a positive damping value \\cite{1982-Arapostathis-global-analysis-periodicSol}. It is also shown by numerical simulation that subcritical and supercritical Hopf bifurcations, and as a consequence, the emergence of periodic solutions, could happen if the swing equations of a two-generator network are augmented to include any of the following four features:  variable damping, frequency-dependent electrical torque, lossy transmission lines, and excitation control \\cite{1981-Abed-Oscillations, 1986-Alexander-Oscillatory-Solutions}. Hopf bifurcation is also demonstrated in a three-generator undamped system as the load of the system changes \\cite{1989-Kwatny-Energy-Analysis-Flutter}, where several energy functions for such undamped lossy swing equations in the neighborhood of points of Hopf bifurcation are developed to help characterize Hopf bifurcation in terms of energy properties. Furthermore, a frequency domain analysis to identify the stability of the periodic orbits created by a Hopf bifurcation is presented in \\cite{1990-Kwatny-Frequency-Analysis_Hopf}.\nThe existence and the properties of limit cycles in power systems with higher-order models are also numerically analyzed in  \\cite{2007-Hiskens-Limit-cycle2,2005-Hiskens-limit-cycle1}.\n %\n\n\nAnother set of literature relevant to our work studies the role of power system parameters in the stability of its equilibrium points. For instance, the work presented in \\cite{2019-Patrick-koorehdavoudi-input} examines the dependence of the transfer functions on the system parameters in the swing equation model. In \\cite{2017-paganini-global}, the role of inertia in the frequency response of the system is studied. Moreover, it is shown how different dynamical models can lead to different conclusions. Finally, the works on frequency and transient stabilities in power systems \\cite{2013-Lee-Power-dynamics-stochastic,2014-Low-NaLi-stability,2016-Vu-framework,2005-ortega-transient,2018-Dorfler-robust} are conceptually related to our work.\n\n\n\n\n\n\n\n\n\n\\subsection{Contributions}\nThis paper presents a thorough theoretical analysis of the role of damping in the stability of model \\eqref{eq: nonlinear ode}-\\eqref{eq: nonlinear ode 1 order_intro}. Our results provide rigorous formulation and theoretical justification for the intuitive notion that damping increases stability. The results also characterize the hyperbolicity and Hopf bifurcation of an equilibrium point of \\eqref{eq: nonlinear ode 1 order_intro} through the inertia $M$, damping $D$, and  Jacobian $\\nabla f$ matrices. These general results are applied to swing equations \\eqref{eq: 2nd order swing-intro} to provide new insights into the damping effects on the stability of power grids.\n\nThe contributions and main results of this paper are summarized below.\n\\begin{enumerate}\n    \\item We show that increasing damping has a monotonic effect on the stability of equilibrium points in a large class of ODEs of the form \\eqref{eq: nonlinear ode} and \\eqref{eq: nonlinear ode 1 order_intro}. In particular, we show that, when $M$ is nonsingular symmetric, $D$ is symmetric PSD, and $\\nabla f(x_0)$ is symmetric at an equilibrium point $(x_0,0)$ of the first-order system \\eqref{eq: nonlinear ode 1 order_intro}, if the damping matrix $D$ is perturbed to $D'$ which is more PSD than $D$, i.e. $D'-D\\in\\mathbb{S}^n_+$, then the set of eigenvalues of the Jacobian of \\cref{eq: nonlinear ode 1 order_intro} at $(x_0,0)$ that have a zero real part will not enlarge as a set (\\cref{thrm: monotonic damping behaviour}). We also show that these conditions on $M, D, \\nabla f(x_0)$ cannot be relaxed. To establish this result, we prove that the rank of a complex symmetric matrix with PSD imaginary part does not decrease if its imaginary part is perturbed by a real symmetric PSD matrix (\\cref{thrm: rank}), which may be of independent interest in the matrix perturbation theory.\n   \n   \n   \n   \n\n    \\item\n    We propose a necessary and sufficient condition for an equilibrium point $(x_0,0)$ of the first-order system \\eqref{eq: nonlinear ode 1 order_intro} to be hyperbolic. Specifically, when $M$ and $\\nabla f(x_0)$ are symmetric positive definite and $D$ is symmetric PSD, then $(x_0,0)$ is hyperbolic if and only if the pair ($M^{-1}\\nabla f(x_0), M^{-1}D$) is observable (\\cref{thrm: nec and suf for pure imaginary lossless}). We extend the necessary condition to the general case where $M, D, \\nabla f(x_0)$ are not symmetric (\\cref{thrm: nec and suf for pure imaginary lossy}). Moreover, we characterize a set of sufficient conditions for the occurrence of Hopf bifurcation, when the damping matrix varies as a smooth function of a one dimensional bifurcation parameter (\\cref{coro: Hopf bifurcation} and \\cref{thm: fold and Hopf bifurcation}).\n   \n   \n    \n    \\item We show that the theoretical results have key applications in the stability of electric power systems. \n   \n    We propose a set of necessary and sufficient conditions for breaking the hyperbolicity in lossless power systems (\\cref{prop: nec and suf for pure imaginary lossless power system}). We prove that in a lossy system with two or three generators, as long as only one generator is undamped, any equilibrium point is hyperbolic (\\cref{prop:hyperbolicity n2 n3}), and as soon as there are more than one undamped generator, a lossy system with any $n\\ge 2$ generators may lose hyperbolicity at its equilibrium points (\\cref{prop: non-hyper example}).\n    Finally, we perform bifurcation analysis to detect Hopf bifurcation and identify its type based on two interesting case studies.\n   \n \\end{enumerate}\n\n\\subsection{Organization}\nThe rest of our paper is organized as follows. \\Cref{sec: Background} introduces some notation and provides the problem statement. In \\Cref{Sec: Monotonic Behavior of Damping}, we rigorously prove that damping has a monotonic effect on the local stability of a large class of ODEs. \\Cref{Sec: Impact of Damping in Hopf Bifurcation} further investigates the impact of damping on hyperbolicity and bifurcation and presents a set of necessary and\/or sufficient conditions for breaking the hyperbolicity and occurrence of bifurcations. \\Cref{Sec: Power System Model and Its Properties} introduces the power system model (i.e., swing equations), provides a graph-theoretic interpretation of the system, and analyzes the practical applications of our theoretical results in power systems. \\Cref{Sec: Computational Experiments} further illustrates the developed theoretical results through numerical examples, and finally, the paper concludes with \\Cref{Sec: Conclusions}.\n\\section{Background} \\label{sec: Background}\n\\subsection{Notations}\nWe use $\\mathbb{C_{-\/+}}$ to denote the set of complex numbers with negative\/positive real part, and $\\mathbb{C}_{0}$ to denote the set of complex numbers with zero real part.  $\\frak{i}=\\sqrt{-1}$ is the imaginary unit. If $A\\in\\mathbb{C}^{m\\times n}$, the transpose of $A$ is denoted by $A^\\top$, the real part of $A$ is denoted by $\\mathrm{Re}(A)$, and the imaginary part of $A$ is denoted by $\\mathrm{Im}(A)$. The conjugate transpose of $A$ is denoted by $A^*$ and defined by $A^* = \\Bar{A}^\\top$, in which $\\Bar{A}$ is the entrywise conjugate.\nThe matrix $A\\in\\mathbb{C}^{n \\times n}$ is said to be symmetric if $A^\\top = A$, Hermitian if $A^* = A$, and unitary if $A^*A = I$. The spectrum of a matrix $A\\in\\mathbb{R}^{n\\times n}$ is denoted by $\\sigma(A)$. We use $\\mathbb{S}^n$ to denote the set of real symmetric $n\\times n$ matrices, $\\mathbb{S}^n_+$ to denote the set of real symmetric PSD $n\\times n$ matrices, and $\\mathbb{S}^n_{++}$ to denote the set of real symmetric positive definite $n\\times n$ matrices. For matrices $A$ and $B$, the relation $B \\succeq A$ means that $A$ and $B$ are real symmetric matrices of the same size such that $B-A$ is PSD; we write $A \\succeq 0$ to express the fact that $A$ is a real symmetric PSD matrix. Strict version $B \\succ A$ of $B \\succeq A$ means that $B-A$ is real symmetric positive definite, and $A\\succ 0$ means that $A$ is real symmetric positive definite.  \n\n\n\\subsection{Problem Statement} \\label{subsec: Problem Statement}\nConsider the second-order dynamical system \\eqref{eq: nonlinear ode}.\nThe smoothness (continuous differentiability) of $f$ is a sufficient condition for the existence and uniqueness of solution. \nWe transform \\eqref{eq: nonlinear ode} into a system of $2n$ first-order ODEs of the form\n\\begin{align} \\label{eq: nonlinear ode 1 order}\n\\begin{bmatrix} \\dot{x} \\\\ \\dot{y}\n\\end{bmatrix}\n= \n\\begin{bmatrix}\n0 & I \\\\\n0 & -M^{-1}D\n\\end{bmatrix}\n\\begin{bmatrix} {x} \\\\ {y}\n\\end{bmatrix}\n- M^{-1} \n\\begin{bmatrix} 0 \\\\ f(x)\n\\end{bmatrix}.\n\\end{align}\nIf $f(x_0)=0$ for some $x_0\\in\\mathbb{R}^n$, then $(x_0,0)\\in\\mathbb{R}^{n+n}$ is called an equilibrium point. The stability of such equilibrium points can be revealed by the spectrum of the Jacobian of the $2n$-dimensional vector field in \\eqref{eq: nonlinear ode 1 order} evaluated at the equilibrium point. Note that $f:\\mathbb{R}^n\\to\\mathbb{R}^n$ is a vector-valued function, and its derivative at any point $x\\in\\mathbb{R}^n$ is referred to as the Jacobian of $f$ and denoted by $\\nabla f(x)\\in\\mathbb{R}^{n\\times n}$. This Jacobian of $f$ should not be confused with the Jacobian of the $2n$-dimensional vector field in right-hand side of \\eqref{eq: nonlinear ode 1 order}, which is \n\\begin{align}\\label{eq: J general case}\nJ(x) := \\begin{bmatrix}\n0 & I \\\\\n-M^{-1} \\nabla f(x)  & - M^{-1}D \\\\\n\\end{bmatrix} \\in\\mathbb{R}^{2n\\times 2n}.\n\\end{align} \nIf the Jacobian $J$ at an equilibrium point $(x_0,0)\\in\\mathbb{R}^{n+n}$\nhas all its eigenvalues off the imaginary axis, then we say that $(x_0,0)$ is a \\emph{hyperbolic} equilibrium point. \nAn interesting feature of hyperbolic equilibrium points is that they are either unstable or asymptotically stable. Breaking the hyperbolicity (say due to changing a parameter of the system), leads to bifurcation.\nAs mentioned before, we restrict our attention to the case where inertia matrix $M$ is nonsingular. Instead, we scrutinize the case where damping matrix $D$ is not full rank, i.e., the system is partially damped. This is a feasible scenario in real-world physical systems \\cite{2017-koerts-second-order}, and as will be shown, has important implications specially in power systems.\nNow, it is natural to ask the following questions:\n\\begin{enumerate}[(i)]\n\t\\item How does changing the damping matrix $D$ affect the stability and hyperbolicity of equilibrium points of system \\cref{eq: nonlinear ode 1 order}? \\label{Q1}\n\t\\item What are the conditions on $D$ under which an equilibrium point is hyperbolic? \\label{Q2}\n\t\\item When we lose hyperbolicity due to changing $D$, what kind of bifurcation happens? \\label{Q3}\n\\end{enumerate}\nNote that in these questions, the inertia matrix $M$ is fixed, and the bifurcation parameter only affects the damping matrix $D$.\nQuestions \\eqref{Q1}-\\eqref{Q3} will be addressed in the following sections, but before that, we present \\cref{lemma: relation between ev J and ev J11} \\cite{2020-fast-certificate} which provides some intuition behind the role of different factors in the spectrum of the Jacobian matrix $J$.\nLet us define the concept of matrix pencil \\cite{2001-Tisseur-Pencil}. Consider $n\\times n$ matrices $Q_0,Q_1,$ and $Q_2$. A quadratic matrix pencil is a matrix-valued function $P:\\mathbb{C}\\to\\mathbb{R}^{n \\times n}$ given by $\\lambda \\mapsto P(\\lambda)$ such that  $P(\\lambda) = \\lambda^2Q_2 + \\lambda Q_1 + Q_0$.\n\\begin{lemma} \\label{lemma: relation between ev J and ev J11}\n\tFor any $x\\in\\mathbb{R}^n$, $\\lambda$ is an eigenvalue of $J(x)$ if and only if the quadratic matrix pencil $P(\\lambda):= \\lambda^2 M + \\lambda D + \\nabla f(x)$ is singular.\n\\end{lemma}\n\\begin{proof}\n\tFor any $x\\in\\mathbb{R}^n$, let $\\lambda$ be an eigenvalue of $J(x)$ and $( v , u )$ be the corresponding eigenvector. Then\t\n\t\\begin{align} \\label{eq: J cha eq}\n\t\\begin{bmatrix}\n\t0 & I \\\\\n\t-M^{-1} \\nabla f(x)     &     - M^{-1}D \\\\\n\t\\end{bmatrix}   \\begin{bmatrix} v \\\\u  \\end{bmatrix}  = \\lambda   \\begin{bmatrix} v \\\\u  \\end{bmatrix},\n\t\\end{align}\n\twhich implies that $ u = \\lambda v$ and $-M^{-1} \\nabla f(x) v - M^{-1}D u  = \\lambda u$. Substituting the first equality into the second one, we get\n\t\\begin{align} \n\n\n\t\\left( \\nabla f(x) + \\lambda D +  \\lambda^2 M \\right) v = 0. \\label{eq: quadratic matrix pencil}\n\t\\end{align}  \n\tSince $v \\not = 0$ (otherwise $u = \\lambda \\times 0 = 0 $ which is a contradiction), equation \\eqref{eq: quadratic matrix pencil} implies that the matrix pencil $P(\\lambda)= \\lambda^2 M + \\lambda D + \\nabla f(x)$ is singular.\n\t\n\tConversely, for any $x\\in\\mathbb{R}^n$,\n\tsuppose there exists $\\lambda \\in \\mathbb{C}$ such that  $P(\\lambda)= \\lambda^2 M + \\lambda D + \\nabla f(x)$ is singular. Choose a nonzero $v \\in   \\ker(P(\\lambda))$ and let $ u := \\lambda v$. \n\tAccordingly, the characteristic equation \\eqref{eq: J cha eq} holds, and consequently, $\\lambda$ is an eigenvalue of $J(x)$.\n\\end{proof}\n\nTo give some intution, let us pre-multiply \\eqref{eq: quadratic matrix pencil} by $v^*$ to get the quadratic equation\n\\begin{align} \nv^* \\nabla f(x) v + \\lambda v^*Dv +  \\lambda^2 v^*Mv  = 0, \\label{eq: quadratic matrix pencil equation}\n\\end{align}\nwhich has roots\n\\begin{align} \n\\lambda_{\\pm} = \\frac{-v^*Dv \\pm \\sqrt{(v^*Dv)^2-4(v^*Mv)(v^* \\nabla f(x) v)}}{2v^*Mv}.\n\\label{eq: quadratic matrix pencil equation roots}\n\\end{align}\nEquation \\eqref{eq: quadratic matrix pencil equation roots} provides some insights into the impact of matrices $D$, $M$, and $\\nabla f(x)$ on the eigenvalues of $J$. For instance, when $D\\succeq0$, it seems that increasing the damping matrix $D$ (i.e., replacing $D$ with $\\hat{D}$, where $\\hat{D}\\succeq D$) will lead to more over-damped eigenvalues. However, this argument is not quite compelling because by changing $D$, the eigenvector $v$ would also change.     \nAlthough several researchers have mentioned such arguments about the impact of damping \\cite{2010-Ma-decoupling}, to the best of our knowledge, this impact has not been studied in the literature in a rigorous fashion. We will discuss this impact in the next section.\n\\section{Monotonic Effect of Damping}\n\\label{Sec: Monotonic Behavior of Damping}\nIn this section, we analytically examine the role of damping matrix $D$ in the stability of system \\eqref{eq: nonlinear ode}. Specifically, we answer the following question:\nlet System-I and System-II be two second-order dynamical systems \\eqref{eq: nonlinear ode} with partial damping matrices $D_I\\succeq 0$ and $D_{II}\\succeq 0$, respectively. Suppose the two systems are identical in other parameters (i.e., everything except their dampings) and $(x_0,0)\\in\\mathbb{R}^{2n}$ is an equilibrium point for both systems. Observe that changing the damping of system \\eqref{eq: nonlinear ode} does not change the equilibrium points. Here, we focus on the case where $M$ and $L:=\\nabla f(x_0)$ are symmetric (these are reasonable assumptions in many dynamical systems such as power systems). Now, if System-I is asymptotically stable, what kind of relationship between $D_I$ and $D_{II}$ will ensure that System-II is also asymptotically stable?   \nThis question has important practical consequences. \nFor instance, the answer to this question will illustrate how changing the damping coefficients of generators (or equivalently, the corresponding controller parameters of inverter-based resources) in power systems will affect the stability of equilibrium points. Moreover, this question is closely intertwined with a problem in matrix perturbation theory, namely \ngiven a complex symmetric matrix with PSD imaginary part, how does a PSD perturbation of its imaginary part affect the rank of the matrix? We answer the matrix perturbation question in \\Cref{thrm: rank}, which requires \\Cref{lemma: Autonne} to \\Cref{lemma: rank principal} and \\Cref{Prop: diagonal complex update nonsingularity}. Finally, the main result about the monotonic effect of damping is proved in \\Cref{thrm: monotonic damping behaviour}. \n\n\nThe following lemma on Autonne-Takagi factorization is useful.\n\n\\begin{lemma}[Autonne-Takagi factorization] \\label{lemma: Autonne}\nLet $S\\in \\mathbb{C}^{n\\times n}$ be a complex matrix. Then $S^\\top=S$ if and only if there is a unitary $U\\in \\mathbb{C}^{n\\times n}$ and a nonnegative diagonal matrix $\\Sigma\\in \\mathbb{R}^{n\\times n}$ such that $S=U\\Sigma U^\\top$. The diagonal entries of $\\Sigma$ are the singular values of $S$.\n\\end{lemma}\t\n\\begin{proof}\nSee e.g. \\cite[Chapter 4]{2013-Horn-matrix-analysis}.\n\\end{proof}\n\nWe also need the following lemmas to derive our main results. \\Cref{lemma: definiteness inverse imaginary part} generalizes a simple fact about complex numbers to complex symmetric matrices: a complex scalar $z\\in\\mathbb{C}, z\\ne0$ has a nonnegative imaginary part if and only if $z^{-1}$ has a nonpositive imaginary part.\n\\begin{lemma} \\label{lemma: definiteness inverse imaginary part}\n\tLet $S\\in \\mathbb{C}^{n\\times n}$ be a nonsingular complex symmetric matrix. Then $\\mathrm{Im}(S) \\succeq 0$ if and only if $\\mathrm{Im}(S^{-1})\\preceq 0$. \n\\end{lemma}\t\n\\begin{proof}\nSince $S$ is nonsingular complex symmetric, by Autonne-Takagi factorization, there exists a unitary matrix $U$ and a diagonal positive definite matrix $\\Sigma$ such that $S = U\\Sigma U^\\top$. The inverse $S^{-1}$ is given by $S^{-1} = \\bar{U}\\Sigma^{-1}U^*$. The imaginary parts of $S$ and $S^{-1}$ are \n\t\\begin{align*}\n\t& \\mathrm{Im}(S) = -\\frac{1}{2} \\frak{i} (U\\Sigma U^\\top - \\bar{U}\\Sigma U^*), \\\\\n\t& \\mathrm{Im}(S^{-1}) = -\\frac{1}{2} \\frak{i} (\\bar{U}\\Sigma^{-1} U^* - U\\Sigma^{-1} U^\\top).\n\t\\end{align*}\n\tThe real symmetric matrix $2\\mathrm{Im}(S^{-1})=\\frak{i} (U\\Sigma^{-1} U^\\top-\\bar{U}\\Sigma^{-1} U^*)$ is unitarily similar to the Hermitian matrix $\\frak{i}(\\Sigma^{-1}U^\\top U - U^*\\bar{U}\\Sigma^{-1})$ as\n\t\\begin{align*}\n\t U^*(2\\mathrm{Im}(S^{-1}))U &= U^*(\\frak{i}(U\\Sigma^{-1} U^\\top - \\bar{U}\\Sigma^{-1} U^*))U \\\\\n\t & = \\frak{i} (\\Sigma^{-1}U^\\top U - U^*\\bar{U}\\Sigma^{-1}),\n\t\\end{align*} \n\tand is *-congruent to $\\frak{i} (U^\\top U \\Sigma - \\Sigma U^*\\bar{U})$ as\n\t\\begin{align*}\n     \\Sigma U^*(2\\mathrm{Im}(S^{-1}))U\\Sigma &= \\frak{i}\\Sigma(\\Sigma^{-1}U^\\top U - U^*\\bar{U}\\Sigma^{-1})\\Sigma \\\\\n\t& = \\frak{i} (U^\\top U \\Sigma - \\Sigma U^*\\bar{U}).\n\t\\end{align*}\n\tNote that the latter transformation is a *-congruence because $U\\Sigma$ is nonsingular but not necessarily unitary. Hence, $2\\mathrm{Im}(S^{-1})$ has the same eigenvalues as $\\frak{i}(\\Sigma^{-1}U^\\top U - U^*\\bar{U}\\Sigma^{-1})$ and has the same inertia as $\\frak{i}(U^\\top U \\Sigma - \\Sigma U^*\\bar{U})$ by Sylvester's law of inertia. \n\tFurthermore, since $U^\\top U$ is unitary and $\\overline{U^\\top U}=U^*\\bar{U}$, then\n\t\\begin{align*}\n\t\\frak{i}(U^\\top U \\Sigma - \\Sigma U^*\\bar{U}) = (U^\\top U)(\\frak{i}(\\Sigma U^\\top U - U^*\\bar{U}\\Sigma))(U^*\\bar{U}),\n\t\\end{align*}\n\twhich implies that $\\frak{i}(U^\\top U \\Sigma - \\Sigma U^*\\bar{U})$ has the same eigenvalues as $\\frak{i}(\\Sigma U^\\top U - U^*\\bar{U}\\Sigma)$. Furthermore, since \n\n\t\\begin{align*}\n\tU(\\frak{i}(\\Sigma U^\\top U - U^*\\bar{U}\\Sigma))U^* = \\frak{i}(U\\Sigma U^\\top - \\bar{U}\\Sigma U^*) = -2\\mathrm{Im}(S),\n\t\\end{align*}\n\t$\\mathrm{Im}(S^{-1})$ and $-\\mathrm{Im}(S)$ have the same inertia, i.e., they have the same number of positive eigenvalues and the same number of negative eigenvalues. Therefore, $\\mathrm{Im}(S)\\succeq 0$ if and only if all eigenvalues of $\\mathrm{Im}(S^{-1})$ are nonpositive, that is, if and only if $\\mathrm{Im}(S^{-1})\\preceq0$.\n\\end{proof}\n\n\n\\Cref{lemma: rank-one imaginary PSD update} shows how rank-one perturbation to the imaginary part of a nonsingular complex matrix preserves its nonsingularity.\n\\begin{lemma} \\label{lemma: rank-one imaginary PSD update}\n\tLet $S\\in\\mathbb{C}^{n\\times n}$ be a nonsingular complex symmetric matrix. If $\\mathrm{Im}(S)\\succeq 0 $, then $S+\\frak{i} vv^\\top$ is nonsingular for any real vector $v\\in\\mathbb{R}^n$.\n\\end{lemma}\n\\begin{proof}\n\tWe use Cauchy's formula for the determinant of a rank-one perturbation \\cite{2013-Horn-matrix-analysis}:\n\t\\begin{align*}\n\t\\det(S + \\frak{i} vv^\\top) &= \\det(S) + \\frak{i} v^\\top\\mathrm{adj}(S)v\\\\\n\t&= \\det(S) + \\frak{i} v^\\top S^{-1}v \\det(S) \\\\\n\t&= \\det(S)(1 + \\frak{i} v^\\top S^{-1}v) \\\\\n\t&= \\det(S)(1 - v^\\top\\mathrm{Im}(S^{-1})v + \\frak{i}  v^\\top\\mathrm{Re}(S^{-1})v),\n\t\\end{align*}\n\twhere $\\mathrm{adj}(S)$ is the adjugate of $S$, which satisfies $\\mathrm{adj}(S)=(\\det(S))S^{-1}$. \n\n\tSince $\\det(S)\\ne0$, we only need to prove that the complex scalar $z:=(1 - v^\\top\\mathrm{Im}(S^{-1})v + \\frak{i}  v^\\top\\mathrm{Re}(S^{-1})v)$ is nonzero for any $v\\in\\mathbb{R}^n$.\n\tBy \\cref{lemma: definiteness inverse imaginary part}, $\\mathrm{Im}(S^{-1}) \\preceq 0$, thus $\\mathrm{Re}(z)=1-v^\\top\\mathrm{Im}(S^{-1})v \\ge 1$. This proves that any rank-one update on the imaginary part of $S$ is nonsingular.\n\\end{proof}\n\nNow, we extend \\cref{lemma: rank-one imaginary PSD update} to the case where the perturbation is a general real PSD matrix.\n\\begin{proposition} \\label{Prop: diagonal complex update nonsingularity}\n\tLet $S\\in\\mathbb{C}^{n\\times n}$ be a nonsingular complex symmetric matrix with $\\mathrm{Im}(S)\\succeq 0 $. Then, for any real PSD matrix $E\\in\\mathbb{S}^{n}_+$, $S+\\frak{i} E$ is nonsingular.\n\\end{proposition}\n\\begin{proof}\n\n\tSince $E$ is a real PSD matrix, its eigendecomposition gives $E=\\sum_{\\ell=1}^n v_\\ell v_\\ell^\\top$, where $v_\\ell$ is an eigenvector scaled by the $\\ell$-th eigenvalue of $E$. Now, we need to show that $S+\\sum_{\\ell=1}^n \\frak{i}  v_\\ell v_\\ell^\\top$ is nonsingular. According to \\cref{lemma: rank-one imaginary PSD update}, $\\tilde{S}_\\ell:=S + \\frak{i}  v_\\ell v_\\ell^\\top$ is nonsingular for each $\\ell \\in \\{1,\\cdots,n\\}$. Moreover, $\\tilde{S}_\\ell$ is a complex symmetric matrix with $\\textrm{Im}(\\tilde{S}_\\ell)\\succeq 0$. Therefore, \\cref{lemma: rank-one imaginary PSD update} can be consecutively applied to conclude that $S+\\frak{i} E$ is nonsingular.\n\\end{proof}\n\\begin{remark}\n\tThe assumption of $S$ being complex symmetric cannot be relaxed. For example, consider unsymmetric matrix\n\t\\begin{align*}\n\tS = \\begin{bmatrix}\n\t1+\\frak{i}  & \\sqrt{2} \\\\ -\\sqrt{2} & -1\n\t\\end{bmatrix}, E = \\begin{bmatrix} 0 & 0\\\\0 & 1\\end{bmatrix}, S+\\frak{i} E = \\begin{bmatrix}\n\t1+\\frak{i}  & \\sqrt{2} \\\\ -\\sqrt{2} & -1+\\frak{i} \n\t\\end{bmatrix}.\n\t\\end{align*}\n\tThen, $\\textrm{Im}(S)\\succeq 0$, $\\det(S)=1-\\frak{i} $, but $\\det(S+\\frak{i} E)=0$. Likewise, the assumption of $E$ being real PSD cannot be relaxed.\n\\end{remark}\nBefore proceeding further with the analysis, let us recall the concept of principal submatrix. For $A\\in\\mathbb{C}^{n\\times n}$ and $\\alpha \\subseteq \\{1,\\cdots,n\\}$, the (sub)matrix of entries that lie in the rows and columns of $A$ indexed by $\\alpha$ is called a principal submatrix of $A$ and is denoted by $A[\\alpha]$. We also need \\cref{lemma: rank principal} about rank principal matrices. In what follows, the direct sum of two matrices $A$ and $B$ is denoted by $A\\oplus B$.\n\\begin{lemma} [rank principal matrices] \\label{lemma: rank principal}\n\tLet $S\\in \\mathbb{C}^{n\\times n}$ and suppose that $n>\\mathop{\\bf rank}(S)=r \\ge 1$. If $S$ is similar to $B\\oplus0_{n-r}$ (so\n\t$B \\in  \\mathbb{C}^{r\\times r} $is nonsingular), then $S$ has a nonsingular $r$-by-$r$ principal submatrix, that is,\n\t$S$ is rank principal.\n\\end{lemma}\n\\begin{proof}\n    See \\cref{proof of lemma: rank principal}.\n\\end{proof}\nNow we are ready to state our main matrix perturbation result.\n\\begin{theorem} \\label{thrm: rank}\nSuppose $A\\in\\mathbb{S}^{n}$ is a real symmetric matrix and $D\\in\\mathbb{S}^{n}_+$ and $E\\in\\mathbb{S}^{n}_+$ are real symmetric PSD matrices. \nThen $\\mathop{\\bf rank}(A+\\frak{i} D)\\le \\mathop{\\bf rank}(A+\\frak{i} D+\\frak{i} E)$. \n\\end{theorem}\n\\begin{proof\nDefine $r:=\\mathop{\\bf rank}(A+\\frak{i} D)$ and note that if $r=0$, i.e., $A+\\frak{i} D$ is the zero matrix, then the rank inequality holds trivially. If $r\\ge1$, the following two cases are possible.\n\n\t\n\t\n\tFor $r=n$ : in this case $S:=A+\\frak{i} D$ is a nonsingular complex symmetric matrix with $\\mathrm{Im}(S)\\succeq 0 $, and according to \\cref{Prop: diagonal complex update nonsingularity}, $A+\\frak{i} D+\\frak{i} E$ is also nonsingular, i.e., $\\mathop{\\bf rank}(A+\\frak{i} D+\\frak{i} E)=n$. Thus, in this case, the rank inequality $\\mathop{\\bf rank}(A+\\frak{i} D)\\le \\mathop{\\bf rank}(A+\\frak{i} D+\\frak{i} E)$ holds with equality.\n\n\t\n\t For $1\\le r < n$: since $A+\\frak{i} D$ is complex symmetric, using Autonne-Takagi factorization in \\cref{lemma: Autonne}, $A+\\frak{i} D = U\\Sigma U^\\top$ for some unitary matrix $U$ and a diagonal real PSD matrix $\\Sigma$. Moreover, $r=\\mathop{\\bf rank}(A+\\frak{i} D)$ will be equal to the number of positive diagonal entries of $\\Sigma$. In this case, $A+\\frak{i} D$ is unitarily similar to $\\Sigma=B\\oplus0_{n-r}$, for some nonsingular diagonal $B\\in\\mathbb{R}^{r\\times r}$. According to \\cref{lemma: rank principal}, there exists a principal submatrix of $A+\\frak{i} D$ with size $r$ that is nonsingular, that is, there exists an index set $\\alpha \\subseteq \\{1,\\cdots,n\\}$ with $\\textrm{card}(\\alpha)=r$ such that $A[\\alpha]+\\frak{i} D[\\alpha]$ is nonsingular. Note that $A[\\alpha]+\\frak{i} D[\\alpha]$ is also complex symmetric. Now, using the same index set $\\alpha$ of  rows and columns, we select the principal submatrix $E[\\alpha]$ of $E$. Recall that if a matrix is PSD then all its principal submatrices are also PSD. Therefore, $D[\\alpha] \\succeq 0$ and $E[\\alpha] \\succeq 0$.\n\t %\n\t\n    \n    %\n     Using the same argument as in the previous case of this proof, we have $\\mathop{\\bf rank}(A[\\alpha]+\\frak{i} D[\\alpha]) = \\mathop{\\bf rank}(A[\\alpha]+\\frak{i} D[\\alpha]+\\frak{i} E[\\alpha])=r$. \n     On the one hand, according to our assumption, we have $\\mathop{\\bf rank}(A+\\frak{i} D)=\\mathop{\\bf rank}(A[\\alpha]+\\frak{i} D[\\alpha])=r$. On the other hand, we have\n\t\\begin{align} \\label{eq: rank ineq of rank thrm}\n\t\t \\mathop{\\bf rank}(A+\\frak{i} D+\\frak{i} E) \\ge \\mathop{\\bf rank}(A[\\alpha]+\\frak{i} D[\\alpha]+\\frak{i} E[\\alpha]) = r = \\mathop{\\bf rank}(A+\\frak{i} D).\n\t\\end{align}\n\tNote that the inequality in \\eqref{eq: rank ineq of rank thrm} holds because the rank of a principal submatrix is always less than or equal to the rank of the matrix itself. In other words, by adding more columns and rows to a (sub)matrix, the existing linearly independent rows and columns will remain linearly independent. Therefore, the rank inequality $\\mathop{\\bf rank}(A+\\frak{i} D)\\le \\mathop{\\bf rank}(A+\\frak{i} D+\\frak{i} E)$ also holds in this case.\n\\end{proof}\n\nWe now use \\cref{thrm: rank} to answer the question of how damping affects stability. In particular, \\cref{thrm: monotonic damping behaviour} shows a monotonic effect of damping on system stability. Namely, when $\\nabla f(x_0)$ is symmetric at an equilibrium point $(x_0,0)$, the set of eigenvalues of the Jacobian $J(x_0)$ that lie on the imaginary axis does not enlarge, as the damping matrix $D$ becomes more positive semidefinite. \n\\begin{theorem}[Monotonicity of imaginary eigenvalues in response to damping] \\label{thrm: monotonic damping behaviour}\n   \n    Consider the following two systems,\n    \\begin{align} \n         \\quad & M \\ddot{x} + D_{I}  \\dot{x} + f(x) = 0, \\tag{System-I} \\label{System-I} \\\\\n         \\quad & M \\ddot{x} + D_{II} \\dot{x} + f(x) = 0, \\tag{System-II} \\label{System-II}\n    \\end{align}\n    where $M\\in\\mathbb{S}^n$ is nonsingular and $D_{I},D_{II} \\in \\mathbb{S}^n_+$. Suppose $(x_0,0)$ is an equilibrium point of the corresponding first-order systems defined in \\eqref{eq: nonlinear ode 1 order}. Assume $L:=\\nabla f(x_0)\\in\\mathbb{S}^n$. Denote $J_I, J_{II}$ as the associated Jacobian matrices at $x_0$ as defined in \\eqref{eq: J general case}. Furthermore,\n    %\n   \n   \n    let $\\mathcal{C}_{I} \\subseteq \\mathbb{C}_{0} $ (resp. $\\mathcal{C}_{II} \\subseteq \\mathbb{C}_{0}$) be the set of eigenvalues of $J_{I}$ (resp. $J_{II}$) with a zero real part, which may be an empty set. Then the sets $C_I, C_{II}$ of eigenvalues on the imaginary axis satisfy the following monotonicity property, \n    \\begin{align}\n      D_{II} \\succeq D_{I} \\implies \\mathcal{C}_{II}  \\subseteq \\mathcal{C}_{I}.  \n    \\end{align}\n   \n\\end{theorem}\n\\begin{proof}\nRecall the Jacobian matrices are defined as\n\\begin{align*}\n J_I = \\begin{bmatrix}\n0 & I \\\\\n- M^{-1} L     & -M^{-1} D_{I}  \\\\\n\\end{bmatrix},\nJ_{II} = \\begin{bmatrix}\n0 & I \\\\\n- M^{-1} L   &   - M^{-1} D_{II} \\\\\n\\end{bmatrix},\n\\end{align*}\nat an equilibrium point $(x_0,0)$.\nAccording to \\cref{lemma: relation between ev J and ev J11}, $\\frak{i} \\beta\\in\\mathcal{C}_{I}$ if and only if the quadratic matrix pencil $P_I(\\frak{i} \\beta):= (L - \\beta^2 M) + \\frak{i} \\beta D_I$ is singular. The same argument holds for $\\mathcal{C}_{II}$. Since $J_I$ and $J_{II}$ are real matrices, their complex eigenvalues will always occur in complex conjugate pairs. Therefore, without loss of generality, we assume $\\beta\\ge0$. Note that for any $\\beta\\ge0$ such that $\\frak{i} \\beta\\notin\\mathcal{C}_{I}$ the pencil $P_I(\\frak{i} \\beta)= (L - \\beta^2 M) + \\frak{i} \\beta D_I$ is nonsingular. \nMoreover, $(L-\\beta^2M)\\in\\mathbb{S}^{n}$ is a real symmetric matrix and $\\beta D_I\\in\\mathbb{S}^{n}_+$ is a real PSD matrix. According to \\cref{thrm: rank}, \n\\begin{align*}\nr & =  \\mathop{\\bf rank}(L-\\beta^2 M +\\frak{i} \\beta D_I) \\\\\n  &\\le \\mathop{\\bf rank}(L-\\beta^2 M +\\frak{i} \\beta D_I + \\frak{i} \\beta(D_{II}-D_{I})) \\\\\n  & = \\mathop{\\bf rank}(L-\\beta^2  M +\\frak{i} \\beta D_{II}),\n\\end{align*}\nconsequently, $P_{II}(\\frak{i} \\beta)=L-\\beta^2M+\\frak{i} \\beta D_{II}$ is also nonsingular and $\\frak{i} \\beta\\notin\\mathcal{C}_{II}$. This implies that $\\mathcal{C}_{II}  \\subseteq \\mathcal{C}_{I}$ and completes the proof.\n\\end{proof}\n\\begin{remark}\n    In the above theorem, the assumption of $L=\\nabla f(x_0)$ being symmetric cannot be relaxed. For example, consider \n\t\\begin{align*}\n\tf(x_1,x_2) = \\begin{bmatrix}\n\t2x_1 + \\sqrt{2} x_2 \\\\ -\\sqrt{2} x_1\n\t\\end{bmatrix},\n\tD_{I} = \\begin{bmatrix} 1 & 0\\\\0 & 0\\end{bmatrix}, D_{II} = \\begin{bmatrix} 1 & 0\\\\0 & 1\\end{bmatrix}, M = \\begin{bmatrix} 1 & 0\\\\0 & 1\\end{bmatrix}.\n\t\\end{align*}\n\tHere, the origin is the equilibrium point of the corresponding first-order systems, and $L=\\nabla f(0,0)$ is not symmetric. The set of eigenvalues with zero real part in \\eqref{System-I} and \\eqref{System-II} are $\\mathcal{C}_I=\\emptyset$ and $\\mathcal{C}_{II}=\\{ \\pm \\frak{i}  \\}$. Accordingly, we have $D_{II}\\succeq D_{I}$, but $\\mathcal{C}_{II}  \\not \\subseteq \\mathcal{C}_{I}$.\n\\end{remark}\n\\section{Impact of Damping on Hyperbolicity and Bifurcation}\n\\label{Sec: Impact of Damping in Hopf Bifurcation}\n\\subsection{Necessary and Sufficient Conditions for Breaking Hyperbolicity}\n\\label{subsec: Sufficient and Necessary Conditions}\nWe use the notion of observability from control theory to provide a necessary and sufficient condition for breaking the hyperbolicity of equilibrium points in system \\eqref{eq: nonlinear ode 1 order} when the inertia, damping, and Jacobian of $f$ satisfy $M\\in\\mathbb{S}^{n}_{++}, D\\in\\mathbb{S}^n_+, \\nabla f(x_0)\\in\\mathbb{S}^n_{++}$ at an equilibrium point $(x_0,0)$ (\\cref{thrm: nec and suf for pure imaginary lossless}). We further provide a sufficient condition for the existence of purely imaginary eigenvalues in system \\eqref{eq: nonlinear ode 1 order} when $M, D, \\nabla f(x_0)$ are not symmetric (\\cref{thrm: nec and suf for pure imaginary lossy}). Such conditions will pave the way for understanding Hopf bifurcations in these systems. Observability was first related to stability of second-order system \\eqref{eq: nonlinear ode} by Skar \\cite{1980-Skar-stability-thesis}.\n\\begin{definition}[observability] \\label{def: Observability}\n\tConsider the matrices $A\\in\\mathbb{R}^{m\\times m}$ and $B\\in\\mathbb{R}^{n\\times m}$. The pair $(A,B)$ is observable if $Bx\\ne 0$ for every right eigenvector $x$ of $A$, i.e.,\n\t\\begin{align*}\n\t\\forall \\lambda \\in \\mathbb{C}, x \\in \\mathbb{C}^m, x\\ne 0  \\: \\text{ s.t. } Ax = \\lambda x \\implies Bx \\ne 0.\n\t\\end{align*}\n\\end{definition}\n\nWe will show that the hyperbolicity of an equilibrium point $(x_0,0)$ of system \\eqref{eq: nonlinear ode 1 order} is intertwined with the observability of the pair $(M^{-1}\\nabla f(x_0),M^{-1}D)$. Our focus will remain on the role of the damping matrix $D\\in\\mathbb{S}^n_+$ in this matter. Note that if the damping matrix $D$ is nonsingular, the pair $(M^{-1}\\nabla f(x_0),M^{-1}D)$ is always observable because the nullspace of $M^{-1}D$ is trivial. Furthermore, if the damping matrix $D$ is zero, the following lemma holds.\n\\begin{lemma} \\label{lemma: undamped systems}\n    In an undamped system (i.e., when $D=0$), for any $x\\in\\mathbb{R}^n$ the pair $(M^{-1}\\nabla f(x),M^{-1}D)$ can never be observable. Moreover, for any $x\\in\\mathbb{R}^n$\n    \\begin{align*}\n        \\lambda \\in\\sigma(J(x))\\iff \\lambda^2\\in\\sigma(-M^{-1}\\nabla f(x)).\n    \\end{align*}\n\\end{lemma}\n\\begin{proof}\nThe first statement is an immediate consequence of \\cref{def: Observability} and the second one follows from \\cref{lemma: relation between ev J and ev J11}.\n\\end{proof}\n\nThe next theorem yields a necessary and sufficient condition on the damping matrix $D$ for breaking the hyperbolicity of an equilibrium point.\n\\begin{theorem}[hyperbolicity in second-order systems: symmetric case] \\label{thrm: nec and suf for pure imaginary lossless}\n    Consider the second-order ODE system \\eqref{eq: nonlinear ode} with inertia matrix $M\\in\\mathbb{S}^n_{++}$ and damping matrix $D \\in\\mathbb{S}^n_+$. Suppose $(x_0,0)\\in \\mathbb{R}^{n+n}$ is an equilibrium point of the corresponding first-order system \\eqref{eq: nonlinear ode 1 order} with the Jacobian matrix $J\\in\\mathbb{R}^{2n\\times 2n}$ defined in \\eqref{eq: J general case} such that $L=\\nabla f(x_0)\\in \\mathbb{S}^n_{++}$. Then, the equilibrium point $(x_0,0)$ is hyperbolic if and only if the pair $(M^{-1}L,M^{-1}D)$ is observable.\n    %\n    %\n    %\n   \n   \n       \n       \n       \n       \n        \n       \n       \n       \n       \n   \n\\end{theorem}\n\\begin{proof}\n    According to \\cref{lemma: undamped systems}, if $D=0$, the pair $(M^{-1}L,M^{-1}D)$ can never be observable. \n\t%\n\tMoreover, $M^{-1}L=M^{-\\frac{1}{2}}  \\hat{L}  M^{\\frac{1}{2}}$, where $\\hat{L}:=M^{-\\frac{1}{2}} L M^{-\\frac{1}{2}}$. This implies that $M^{-1}L$ is similar to (and consequently has the same eigenvalues as) $\\hat{L}$. \n \tNote that $\\hat{L}$ is *-congruent to $L$. According to Sylvester's law of inertia, $\\hat{L}$ and $L$ have the same inertia. Since $L\\in \\mathbb{S}^n_{++}$, we conclude that $\\hat{L}\\in \\mathbb{S}^n_{++}$. Therefore, the eigenvalues of $M^{-1}L$ are real and positive, i.e., $\\sigma(M^{-1}L)\\subseteq \\mathbb{R_{++}}=\\{\\lambda\\in\\mathbb{R}:\\lambda>0\\}$.\n    %\n\n\tMeanwhile, when $D=0$, we have $\\mu\\in\\sigma(J)\\iff\\mu^2\\in\\sigma(-M^{-1}L)$, hence all eigenvalues of $J$ would have zero real parts, i.e., $\\sigma(J)\\subseteq\\mathbb{C}_0$, and consequently, the theorem holds trivially. In the sequel, we assume that $D\\not=0$.\n\n   \n\t\\emph{Necessity:}\n\tSuppose the pair $(M^{-1}L,M^{-1}D)$ is observable, but assume the equilibrium point is not hyperbolic, and let us lead this assumption to a contradiction. Since $L=\\nabla f(x_0)$ is nonsingular, \\cref{lemma: relation between ev J and ev J11} asserts that $0\\not\\in\\sigma(J)$. Therefore, there must exist $\\beta>0$ such that $\\frak{i} \\beta \\in \\sigma(J)$. \n\n\tBy \\cref{lemma: relation between ev J and ev J11}, $\\frak{i}\\beta \\in \\sigma(J)$ if and only if the matrix pencil $(M^{-1}L + \\frak{i}\\beta M^{-1}D - \\beta^2 I)$ is singular: \n\n\t\\begin{align*}\n\t\n\t& \\det \\left(  M^{-\\frac{1}{2}}(M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}} + \\frak{i} \\beta  M^{-\\frac{1}{2}}DM^{-\\frac{1}{2}} - \\beta^2 I) M^{\\frac{1}{2}} \\right) = 0,\n\t\\end{align*}\n\tor equivalently, $\\exists (x + \\frak{i} y) \\ne 0$ such that $x,y\\in\\mathbb{R}^n$ and\n\t\\begin{align} \\label{eq: proof of parially damped hopf bifurcation}\n\t\\notag & (M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}} + \\frak{i} \\beta  M^{-\\frac{1}{2}} D M^{-\\frac{1}{2}} - \\beta^2 I)(x+\\frak{i} y) = 0 \\\\\n\t& \\iff  \\begin{cases}\n\t(M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}}-\\beta^2 I)x - \\beta M^{-\\frac{1}{2}}DM^{-\\frac{1}{2}} y = 0, \\\\ \n\t(M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}}-\\beta^2 I)y + \\beta M^{-\\frac{1}{2}}DM^{-\\frac{1}{2}} x = 0.\n\t\\end{cases}\n\t\\end{align}\n\tLet $\\hat{L}:=M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}}$, $\\hat{D}:=M^{-\\frac{1}{2}}DM^{-\\frac{1}{2}}$, and observe that\n\t\\begin{align*}\n\t\\begin{cases}\n\ty^\\top (\\hat{L}-\\beta^2 I)x = y^\\top (\\beta \\hat{D} y) = \\beta y^\\top \\hat{D} y  \\ge 0,\n\t\\\\\n\tx^\\top (\\hat{L}-\\beta^2 I)y = x^\\top (-\\beta \\hat{D} x) = -\\beta x^\\top \\hat{D} x  \\le 0,\n\t\\end{cases}\n\t\\end{align*}\n    %\n\twhere we have used the fact that $\\hat{D}$ is *-congruent to $D$. According to Sylvester's law of inertia, $\\hat{D}$ and $D$ have the same inertia. Since $D\\succeq 0$, we conclude that $\\hat{D}\\succeq0$.\n\n\tAs $(\\hat{L}-\\beta^2 I)$ is symmetric, we have $x^\\top (\\hat{L}-\\beta^2 I)y = y^\\top (\\hat{L}-\\beta^2 I)x$. Therefore, we must have $x^\\top \\hat{D} x = y^\\top \\hat{D} y  =0$. Since $\\hat{D}\\succeq0$, we can infer that $x\\in\\ker(\\hat{D})$ and $y\\in\\ker(\\hat{D})$.\n\t%\n\n   \n   \n   \n   \n\tNow considering $\\hat{D} y = 0$ and using the first equation in \\eqref{eq: proof of parially damped hopf bifurcation} we get \n\t\\begin{align}\n\t( \\hat{L}-\\beta^2 I)x = 0 \\iff M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}} x=\\beta^2 x,\n\t\\end{align}\n\tmultiplying both sides from left by $M^{-\\frac{1}{2}}$ we get $M^{-1}L (M^{-\\frac{1}{2}} x) = \\beta^2 (M^{-\\frac{1}{2}} x)$. Thus, $ \\hat{x}:= M^{-\\frac{1}{2}} x$ is an eigenvector of $M^{-1}L$. Moreover, we have \n\t\\begin{align*}\n\t    M^{-1}D \\hat{x}  = M^{-1}DM^{-\\frac{1}{2}} x = M^{-\\frac{1}{2}} (  \\hat{D}  x) = 0,    \n\t\\end{align*}\n\twhich means that the pair $(M^{-1}L,M^{-1}D)$ is not observable; we have arrived at the desired contradiction.\n\n\n\n\n\n\t\\emph{Sufficiency:}\n\n\t%\n   \n   \n   \n   \n   \n   \n   \n\n\tSuppose the equilibrium point is hyperbolic, but assume that the pair~ $\\allowbreak (M^{-1}L, M^{-1}D)$ is not observable;  we will show that this assumption leads to a contradiction.\tAccording to \\cref{def: Observability}, $\\exists \\lambda \\in \\mathbb{C}, x \\in \\mathbb{C}^n , x\\ne 0$ such that\n\t\\begin{align} \\label{eq: observability in lossless general}\n\t M^{-1}Lx = \\lambda x \\text{ and } M^{-1}Dx = 0.\n\t\\end{align}\n   \n\tWe make the following two observations.\n\tFirstly, as it is shown above, we have $\\sigma(M^{-1}L)\\subseteq \\mathbb{R_{++}}$. \n\t%\n\tSecondly, since $L$ is nonsingular, the eigenvalue $\\lambda$ in \\eqref{eq: observability in lossless general} cannot be zero.\n\t%\n\n   \n   \n   \n   \n   \n    %\n\tBased on the foregoing two observations, when the pair $(M^{-1}L,M^{-1}D)$ is not observable, there must exist $\\lambda \\in \\mathbb{R_{+}}, \\lambda\\ne0$ and $x \\in \\mathbb{C}^n , x\\ne 0$ such that \\eqref{eq: observability in lossless general} holds.\n\n   \n   \n   \n\n\tDefine $\\xi=\\sqrt{-\\lambda}$, which is a purely imaginary number. The quadratic pencil $M^{-1}P(\\xi)=\\xi^2 I + \\xi M^{-1}D + M^{-1}L$ is singular because $M^{-1}P(\\xi)x = \\xi^2 x + \\xi M^{-1}Dx + M^{-1}Lx = -\\lambda x + 0 + \\lambda x = 0$. By  \\cref{lemma: relation between ev J and ev J11}, $\\xi$ is an eigenvalue of $J$. Similarly, we can show $-\\xi$ is an eigenvalue of $J$. Therefore, the equilibrium point is not hyperbolic, which is a desired contradiction.\n\\end{proof}\n\n\nAs mentioned above, if matrix $D$ is nonsingular, the pair $(M^{-1}\\nabla f(x_0),\\allowbreak  M^{-1}D)$ is always observable. Indeed, if we replace the assumption $D\\in\\mathbb{S}^n_+$ with $D\\in\\mathbb{S}^n_{++}$ in \\cref{thrm: nec and suf for pure imaginary lossless}, then the equilibrium point $(x_0,0)$ is not only hyperbolic but also asymptotically stable. This is proved in \\Cref{thrm: Stability of Symmetric Second-Order Systems with nonsing damping} in \\Cref{appendix: Stability of Symmetric Second-Order Systems with Nonsingular Damping}.\n\n\n\n\n\nAnother interesting observation is that when an equilibrium point is hyperbolic, \\cref{thrm: nec and suf for pure imaginary lossless} confirms the monotonic behaviour of damping in \\cref{thrm: monotonic damping behaviour}. Specifically, suppose an equilibrium point $(x_0,0)$ is hyperbolic for a value of damping matrix $D_I\\in\\mathbb{S}^n_+$. \\cref{thrm: nec and suf for pure imaginary lossless} implies that the pair $(M^{-1}\\nabla f(x_0),M^{-1}D_I)$ is observable. Note that if we change the damping to $D_{II}\\in\\mathbb{S}^n_+$ such that $D_{II}\\succeq D_I$, then the pair $(M^{-1}\\nabla f(x_0),M^{-1}D_{II})$ is also observable. Hence, the equilibrium point $(x_0,0)$ of the new system with damping $D_{II}$ is also hyperbolic. This is consistent with the monotonic behaviour of damping which is proved in \\cref{thrm: monotonic damping behaviour}.\n\n\n\n\n\n\n\n\n\n\nUnder additional assumptions, \\cref{thrm: nec and suf for pure imaginary lossless} can be partially generalized to a sufficient condition for the breakdown of hyperbolicity when $L$, $M$, and $D$ are not symmetric as in the following\n\\begin{theorem}[hyperbolicity in second-order systems: unsymmetric case]\\label{thrm: nec and suf for pure imaginary lossy}\nConsider the second-order ODE system \\eqref{eq: nonlinear ode} with nonsingular inertia matrix $M\\in\\mathbb{R}^{n\\times n}$ and damping matrix $D \\in\\mathbb{R}^{n\\times n}$. Suppose $(x_0,0)\\in \\mathbb{R}^{n+n}$ is an equilibrium point of the corresponding first-order system \\eqref{eq: nonlinear ode 1 order} with the Jacobian matrix $J\\in\\mathbb{R}^{2n\\times 2n}$ defined in \\eqref{eq: J general case} such that $L=\\nabla f(x_0)\\in \\mathbb{R}^{n\\times n}$. If $M^{-1}L$ has a positive eigenvalue $\\lambda$ with eigenvector $x$ such that $x$ is in the nullspace of $M^{-1}D$, then the spectrum of the Jacobian matrix $\\sigma(J)$ contains a pair of purely imaginary eigenvalues.\n\\end{theorem}\n\\begin{proof}\n    The proof is similar to that of \\Cref{thrm: nec and suf for pure imaginary lossless} and is given in \\cref{proof of thrm: nec and suf for pure imaginary lossy}.\n\\end{proof}\n\n\n\\subsection{Bifurcation under Damping Variations}\n\\label{subsec: Bifurcation under Damping Variations}\nIn \\Cref{subsec: Sufficient and Necessary Conditions}, we developed necessary and\/or sufficient conditions for breaking the hyperbolicity through purely imaginary eigenvalues. Naturally, the next question is: what are the consequences of breaking the hyperbolicity? To answer this question, consider the parametric ODE \n\\begin{align} \\label{eq: parameter-dependent second order system}\n     M \\ddot{x} + D(\\gamma) \\dot{x} + f(x) = 0,\n\\end{align}\nwhich satisfies the same assumptions as \\eqref{eq: nonlinear ode}. Suppose $D(\\gamma)$ is a smooth function of $\\gamma\\in\\mathbb{R}$, and $(x_0,0)\\in \\mathbb{R}^{n+n}$ is a hyperbolic equilibrium point of the corresponding first-order system at $\\gamma=\\gamma_1$ with the Jacobian matrix $J(x,\\gamma)$ defined as\n\\begin{align}\\label{eq: J general case hopd thrm proof}\n\tJ(x, \\gamma) = \\begin{bmatrix}\n\t0 & I \\\\\n\t-M^{-1} \\nabla f(x)  & - M^{-1} D(\\gamma) \\\\\n\t\\end{bmatrix} \\in\\mathbb{R}^{2n\\times 2n}.\n\\end{align} \nLet us vary $\\gamma$ from $\\gamma_1$ to $\\gamma_2$ and monitor the equilibrium point. There are two ways in which the hyperbolicity can be broken. Either a simple real eigenvalue approaches zero and we have $0\\in\\sigma(J(x_0,\\gamma_2))$, or a pair of simple complex eigenvalues approaches the imaginary axis and we have $\\pm \\frak{i}  \\omega_0 \\in \\sigma( J(x_0,\\gamma_2) )$ for some $\\omega_0>0$. The former corresponds to a fold bifurcation, while the latter is associated with a Hopf (more accurately, Poincare-Andronov-Hopf) bifurcation\\footnote{It can be proved that we need more parameters to create extra eigenvalues on the imaginary axis unless the system has special properties such as symmetry \\cite{2004-kuznetsov-hopf}.}. The next theorem states the precise conditions for a Hopf bifurcation to occur in system \\eqref{eq: parameter-dependent second order system}.\n\n\\begin{theorem} \\label{coro: Hopf bifurcation}\n\tConsider the parametric ODE \\eqref{eq: parameter-dependent second order system}, with inertia matrix $M\\in\\mathbb{S}^n_{++}$ and damping matrix $D(\\gamma) \\in\\mathbb{S}^n_+$. Suppose $D(\\gamma)$ is a smooth function of $\\gamma$, $(x_0,0)\\in \\mathbb{R}^{n+n}$ is an isolated equilibrium point of the corresponding first-order system, and $L:=\\nabla f(x_0)\\in \\mathbb{S}^n_{++}$. Assume the following conditions are satisfied:\n\t\\begin{enumerate}[(i)]\n\t\t\\item There exists $\\gamma_0\\in\\mathbb{R}$ such that the pair $(M^{-1}L, M^{-1}D(\\gamma_0) )$ is not observable, that is, $\\exists \\lambda\\in\\mathbb{C},v\\in\\mathbb{C}^n, v\\ne0$ such that \n\t\t\\begin{align} \\label{eq: hopf observ}\n\t\tM^{-1} L v = \\lambda v \\text{ and } M^{-1}D(\\gamma_0)v=0.\n\t\t\\end{align}\\label{coro: hopf conditions_1}\n\t\t\\vspace{-5mm}\n\t\t\\item $\\frak{i} \\omega_0$ is a simple eigenvalue of $J(x_0,\\gamma_0)$, where $\\omega_0=\\sqrt{\\lambda}$. \n\t\t\\label{coro: hopf conditions_3}\n       \n\t\n\t\n\t\t\\item $\\mathrm{Im}( q^* M^{-1} D'(\\gamma_0) v ) \\ne 0$, where $D'(\\gamma_0)$ is the derivative of $D(\\gamma)$ at $\\gamma=\\gamma_0$, $\\ell_0=(p,q)\\in\\mathbb{C}^{n+n}$ is a left eigenvector of $J(x_0, \\gamma_0)$ corresponding to eigenvalue $\\frak{i}  \\omega_0$, and $\\ell_0$ is normalized so that $\\ell_0^*r_0 = 1$ where $r_0=(v,\\frak{i} \\omega_0 v)$.\n\t\t\\label{coro: hopf conditions_transvers} \n\t\t%\n       \n\t\t%\n\t\n\t\t\\item $\\det({P(\\kappa)})\\ne0$ for all $\\kappa\\in\\mathbb{Z} \\setminus \\{-1,1\\}$, where $P(\\kappa)$ is the quadratic matrix pencil given by $P(\\kappa):= \\nabla f(x_0)-\\kappa^2\\omega_0^2 M + \\frak{i} \\kappa\\omega_0 D(\\gamma_0)$.\n\t\t\\label{coro: hopf conditions_4}\n        %\n\t\n\t\t%\n\t\n\t\\end{enumerate}\n\n\tThen, there exists smooth functions $\\gamma=\\gamma(\\varepsilon)$ and $T=T(\\varepsilon)$ depending on a parameter $\\varepsilon$ with $\\gamma(0)=\\gamma_0$ and $T(0)=2\\pi |\\omega_0|^{-1}$ such that there are nonconstant periodic solutions of \\eqref{eq: parameter-dependent second order system} with period $T(\\varepsilon)$ which collapses into the equilibrium point $(x_0,0)$ as $\\varepsilon\\to 0$.\n\\end{theorem}\n\\begin{proof}\n\n\tBy \\cref{thrm: nec and suf for pure imaginary lossless}, condition (\\ref{coro: hopf conditions_1}) implies that the Jacobian matrix \\eqref{eq: J general case hopd thrm proof} at $(x,\\gamma) = (x_0,\\gamma_0)$ possesses a pair of purely imaginary eigenvalues $\\pm \\frak{i}  \\omega_0$, where $\\omega_0=\\sqrt{\\lambda}$. Moreover, a right eigenvector of $\\frak{i}  \\omega_0$ is $(v, \\frak{i}  \\omega_0 v)$, where $v$ is from \\eqref{eq: hopf observ}. \n\t%\n\tAccording to condition (\\ref{coro: hopf conditions_3}), the eigenvalue $\\frak{i} \\omega_0$ is simple. Therefore, according to the eigenvalue perturbation theorem \\cite[Theorem 1]{2020-greenbaum-perturbation}, for $\\gamma$ in a neighborhood of $\\gamma_0$, the matrix $J(x_0,\\gamma)$ has an eigenvalue $\\xi(\\gamma)$ and corresponding right and left eigenvectors $r(\\gamma)$ and $\\ell(\\gamma)$ with $\\ell(\\gamma)^*r(\\gamma)=1$ such that $\\xi(\\gamma)$, $r(\\gamma)$, and $\\ell(\\gamma)$ are all analytic functions of $\\gamma$, satisfying $\\xi(\\gamma_0)=\\frak{i} \\omega_0$, $r(\\gamma_0)=r_0$, and $\\ell(\\gamma_0)=\\ell_0$. Let us differentiate the equation $J(x_0,\\gamma) r(\\gamma) = \\xi(\\gamma) r(\\gamma)$ and set $\\gamma=\\gamma_0$, to get\n\t\\begin{align}\n\t    J'(x_0,\\gamma_0) r(\\gamma_0) + J(x_0,\\gamma_0) r'(\\gamma_0) = \\xi'(\\gamma_0) r(\\gamma_0) + \\xi(\\gamma_0) r'(\\gamma_0).\n\t\\end{align}\n\tAfter left multiplication by $\\ell_0^*$, and using $\\ell_0^*r_0=1$, we obtain the derivative of $\\xi(\\gamma)$ at $\\gamma = \\gamma_0$:\n\t\\begin{align*}\n\t\\xi'(\\gamma_0) =  \n\t\\begin{bmatrix}\n\t p^{*} & q^{*}\n\t\\end{bmatrix}\n\t\\begin{bmatrix}\n\t0 & 0 \\\\\n\t0  & - M^{-1} D'(\\gamma_0) \\\\\n\t\\end{bmatrix} \n\t\\begin{bmatrix}\n\tv \\\\ \\frak{i} \\omega_0 v\n\t\\end{bmatrix} = - \\frak{i} \\omega_0 q^{*} M^{-1} D'(\\gamma_0) v.\n\t\\end{align*}\n\t%\n\t%\n   \n    %\n   \n    %\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    %\n\tNow, $\\mathrm{Im}( q^* M^{-1} D'(\\gamma_0) v ) \\ne 0$ in condition (\\ref{coro: hopf conditions_transvers}) implies that $\\mathrm{Re}(\\xi'(\\gamma_0)) \\ne 0$ which is a necessary condition for Hopf bifurcation. \n    %\n\n   \n    %\n\n\tTherefore, the results follow from the Hopf bifurcation theorem \\cite[Section 2]{1978-Schmidt-hopf}. Note that $J(x_0,\\gamma)$ is singular if and only if $\\nabla f(x_0)$ is singular. Thus, nonsingularity of $\\nabla f(x_0)$ is necessary for Hopf bifurcation.\n\\end{proof}\n\n\nIf one or more of the listed conditions in \\cref{coro: Hopf bifurcation} are not satisfied, we may still have the birth of a periodic orbit but some of the conclusions of the theorem may not hold true. The bifurcation is then called a degenerate Hopf bifurcation. For instance, if the transversality condition (\\ref{coro: hopf conditions_transvers}) is not satisfied, the stability of the equilibrium point may not change, or multiple periodic orbits may bifurcate \\cite{1978-Schmidt-hopf}. The next theorem describes a safe region for damping variations such that fold and Hopf bifurcations will be avoided.\n\n\\begin{theorem}\\label{thm: fold and Hopf bifurcation}\n Consider the parametric ODE \\eqref{eq: parameter-dependent second order system}, with a nonsingular inertia matrix $M\\in\\mathbb{R}^{n\\times n}$. Suppose the damping matrix $D(\\gamma)\\in\\mathbb{R}^{n\\times n}$ is a smooth function of $\\gamma$, $(x_0,0)\\in \\mathbb{R}^{n+n}$ is a hyperbolic equilibrium point of the corresponding first-order system at $\\gamma=\\gamma_0$, and $L:=\\nabla f(x_0)\\in\\mathbb{R}^{n\\times n}$. Then, the following statements hold:\n \\begin{enumerate}[(i)]\n     \\item Variation of $\\gamma$ in $\\mathbb{R}$ will not lead to any fold bifurcation.\n     %\n    \n     %\n     \\item Under the symmetric setting, i.e., when $M\\in\\mathbb{S}^n_{++}$, $D(\\gamma) \\in\\mathbb{S}^n_+$, and $L\\in \\mathbb{S}^n_{++}$,  variation of $\\gamma$ in $\\mathbb{R}$ will not lead to any Hopf bifurcation, as long as $D(\\gamma) \\succeq D(\\gamma_0)$. If in addition the equilibrium point is stable, variation of $\\gamma$ in $\\mathbb{R}$ will not make it unstable, as long as $D(\\gamma) \\succeq D(\\gamma_0)$.  \n     %\n    \n \\end{enumerate}\n  %\n \n\n\n\\end{theorem}\n\\begin{proof}\n    According to \\cref{lemma: relation between ev J and ev J11}, zero is an eigenvalue of $J$ if and only if the matrix $L=\\nabla f(x_0)$ is singular. Therefore, the damping matrix $D$ has no role in the zero eigenvalue of $J$. The second statement follows from \\Cref{thrm: monotonic damping behaviour}.\n\\end{proof}\nThe above theorem can be straightforwardly generalized to bifurcations having higher codimension.\n\n\n\n\n\n\n\n\n\\section{Power System Models and Impact of Damping}\n\\label{Sec: Power System Model and Its Properties}\nThe Questions \\eqref{Q1}-\\eqref{Q3} asked in \\Cref{subsec: Problem Statement} and the  theorems and results discussed in the previous parts of this paper arise naturally from the foundations of electric power systems. These results are useful tools for analyzing the behaviour and maintaining the stability of power systems. In the rest of this paper, we focus on power systems to further explore the role of damping in these systems.\n\\subsection{Power System Model} \\label{subsec: Multi-Machine Swing Equations}\nConsider a power system with the set of interconnected generators $\\mathcal{N}=\\{1,\\cdots,n\\}, n\\in\\mathbb{N}$. Based on the classical small-signal stability assumptions \\cite{2008-anderson-stability}, the mathematical model for a power system is described by the following second-order system:\n\\begin{align} \\label{eq: 2nd order swing}\n\t\t\\frac{m_j}{\\omega_s} \\ddot{\\delta}_j(t)+ \\frac{d_j}{\\omega_s} {\\dot{\\delta}}_j(t) = P_{m_j} - P_{e_j}(\\delta(t)) && \\forall j \\in \\mathcal{N}.\n\\end{align}\nConsidering the state space $\\mathcal{S}:= \\{    (\\delta,\\omega): \\delta \\in \\mathbb{R}^n,  \\omega \\in \\mathbb{R}^n \\}$, the dynamical system \\eqref{eq: 2nd order swing} can be represented as a system of first-order nonlinear autonomous ODEs, aka swing equations:  \n\\begin{subequations} \\label{eq: swing equations}\n\t\\begin{align}\n\t\t& \\dot{\\delta}_j(t) = \\omega_j(t) && \\forall j \\in \\mathcal{N},  \\label{eq: swing equations a}\\\\\n\t\t&  \\frac{m_j}{\\omega_s} \\dot{\\omega}_j(t)+ \\frac{d_j}{\\omega_s} \\omega_j(t) = P_{m_j} - P_{e_j}(\\delta(t)) && \\forall j \\in \\mathcal{N}, \\label{eq: swing equations b}\n\t\\end{align}\n\\end{subequations}\nwhere for each generator $j\\in\\mathcal{N}$, $P_{m_j}$ and $P_{e_j}$ are mechanical and electrical powers in per unit, $m_j$ is the inertia constant in seconds, $d_j$ is the unitless damping coefficient, $\\omega_{s}$ is the synchronous angular velocity in electrical radians per seconds, $t$ is the time in seconds, $\\delta_j(t)$ is the rotor electrical angle in radians, and finally $\\omega_j(t)$ is the deviation of the rotor angular velocity from the synchronous velocity in electrical radians per seconds. For the sake of simplicity, henceforth we do not explicitly write the dependence of the state variables $\\delta$ and $\\omega$ on time $t$. \nThe electrical power $P_{e_j}$ in \\eqref{eq: swing equations b} can be further spelled out: \n\\begin{align} \\label{eq: flow function}\n\tP_{e_j}(\\delta) & = \\sum \\limits_{k = 1}^n { V_j  V_k Y_{jk} \\cos \\left( \\theta _{jk} - \\delta _j + \\delta _k \\right)}\n\n\\end{align}\nwhere $V_j$ is the terminal voltage magnitude of generator $j$, and\n$Y_{jk}\\exp{({\\frak{i} \\theta_{jk}})}$ is the $(j,k)$ entry of the reduced admittance matrix, with $Y_{jk}\\in\\mathbb{R}$ and $\\theta_{jk}\\in\\mathbb{R}$. The reduced admittance matrix encodes the underlying graph structure of the power grid, which is assumed to be a connected graph in this paper.\nNote that for each generator $j\\in\\mathcal{N}$, the electrical power $P_{e_j}$ in general is a function of angle variables $\\delta_k$ for all $k\\in\\mathcal{N}$. Therefore, the dynamics of generators are interconnected through the function $P_{e_j}(\\delta)$ in \\eqref{eq: 2nd order swing} and \\eqref{eq: swing equations}.\n\\begin{definition}[flow function] \\label{def: flow function}\n\tThe smooth function $P_e:\\mathbb{R}^n \\to \\mathbb{R}^n$ given by $\\delta \\mapsto P_e(\\delta)$ in \\eqref{eq: flow function} is called the flow function.\n\\end{definition}\nThe smoothness of the flow function (it is $\\mathcal{C}^\\infty$ indeed) is a sufficient condition for the existence and uniqueness of the solution to the ODE \\eqref{eq: swing equations}. \nThe flow function is translationally invariant with respect to the operator $\\delta \\mapsto \\delta + \\alpha \\mathbf{1}$, where $\\alpha \\in \\mathbb{R}$ and $\\mathbf{1}\\in\\mathbb{R}^n$ is the vector of all ones. In other words, $P_e(\\delta + \\alpha \\mathbf{1})=P_e(\\delta)$. A common way to deal with this situation is to define a reference bus and refer all other bus angles to it. This is equivalent to projecting the original state space $\\mathcal{S}$ onto a lower dimensional space. We will delve into this issue in \\Cref{subsec: Referenced Model}.\n\\subsection{Jacobian of Swing Equations} \\label{SubSec: Graph Theory Interpretation and Spectral Properties}\nLet us take the state variable vector $(\\delta ,\\omega)\\in\\mathbb{R}^{2n}$ into account and note that the Jacobian of the vector field in \\eqref{eq: swing equations} has the form \\eqref{eq: J general case}\nwhere $M= \\frac{1}{\\omega_s}  \\mathbf{diag}(m_1,\\cdots,m_n)$ and $D=\\frac{1}{\\omega_s}\\mathbf{diag}(d_1,\\cdots,d_n)$. Moreover, $f=P_e-P_m$ and $\\nabla f =  \\nabla P_e (\\delta) \\in\\mathbb{R}^{n\\times n}$ is the Jacobian of the flow function with the entries:\n\\begin{align*}\n\t&  \\frac {\\partial P_{e_j}} {\\partial \\delta_j} = \\sum \\limits_{k \\ne j} { V_j V_k Y_{jk} \\sin \\left( {\\theta _{jk} - {\\delta _j} + {\\delta _k}} \\right) }, \\forall j \\in \\mathcal{N}\\\\\n\t%\n\t&  \\frac{\\partial P_{e_j} } {\\partial \\delta _k} =  - {V_j} {V_k}{Y_{jk}}\\sin \\left( {{\\theta _{jk}} - {\\delta _j} + {\\delta _k}} \\right),\\forall j,k \\in \\mathcal{N}, k \\neq j.\n\\end{align*}\nLet $\\mathcal{L}$ be the set of transmission lines of the reduced power system. We can rewrite $ {\\partial P_{e_j}}\/ {\\partial \\delta_j}=\\sum_{k=1, k\\ne j}^n w_{jk}$ and ${\\partial P_{e_j} }\/ {\\partial \\delta _k}=-w_{jk}$, where \n\\begin{align} \\label{eq: weights of digraph lossy}\nw_{jk} = \n\\begin{cases}\n {V_j} {V_k}{Y_{jk}}\\sin \\left( \\varphi_{jk} \\right)  & \\forall \\{ j,k \\} \\in \\mathcal{L} \\\\\n 0 & \\text{otherwise},\n\\end{cases}\n\\end{align}\nand $\\varphi_{jk} = {{\\theta _{jk}} - {\\delta _j} + {\\delta _k}}$. Typically, \nwe have $\\varphi_{jk} \\in ( 0,\\pi )$ for all $\\{j,k \\} \\in \\mathcal{L}$ \\cite{2020-fast-certificate}. Thus, it is reasonable to assume that the equilibrium points $(\\delta^{0},\\omega^{0})$ of the dynamical system \\eqref{eq: swing equations} are located in the set $\\Omega$ defined as\n\\begin{align*}\n\\Omega = \\left\\{ (\\delta,\\omega)\\in\\mathbb{R}^{2n} :   0 < \\theta_{jk}-\\delta_j+\\delta_k < \\pi , \\forall \\{j,k\\} \\in \\mathcal{L},\\omega = 0  \\right\\}.\n\\end{align*}\nUnder this assumption, the terms $w_{jk} > 0$ for all transmission lines $\\{j,k\\} \\in \\mathcal{L}$.\nConsequently, $ {\\partial P_{e_j}} \/ {\\partial \\delta_j}\\ge0, \\forall j \\in \\mathcal{N}$ and ${\\partial P_{e_j} }\/ {\\partial \\delta _k} \\le 0, \\forall j,k \\in \\mathcal{N}, k \\neq j$. Moreover, $\\nabla P_e (\\delta)$ has a zero row sum, i.e., $ \\nabla P_e (\\delta)\\mathbf{1} = 0 \\implies 0 \\in \\sigma(\\nabla P_e (\\delta))$. Given these properties, $\\nabla P_e (\\delta^0)$ turns out to be a singular M-matrix for all $(\\delta^{0},\\omega^{0})\\in\\Omega$ \\cite{2020-fast-certificate}. Recall that a matrix $A$ is an \\emph{M-matrix} if the off-diagonal elements of $A$ are nonpositive and the nonzero eigenvalues of $A$ have positive real parts \\cite{1974-M-matrices}. Finally, if the power system under study has a connected underlying undirected graph, the zero eigenvalue of $\\nabla P_e (\\delta^0)$ will be simple \\cite{2020-fast-certificate}. \n\n\n\n    In general, the Jacobian $\\nabla P_e (\\delta)$ is not symmetric. When the power system is \\emph{lossless}, i.e., when the transfer conductances of the grid are zero, then $\\theta_{jj}=-\\frac{\\pi}{2}, \\forall j \\in \\mathcal{N}$ and $\\theta_{jk}=\\frac{\\pi}{2}, \\forall \\{j,k\\} \\in \\mathcal{L}$. In a lossless system, $\\nabla P_e (\\delta)$ is symmetric. If in addition an equilibrium point $(\\delta^0,\\omega^0)$ belongs to the set $\\Omega$, then $\\nabla P_e (\\delta^0) \\in\\mathbb{S}^n_+$, because $\\nabla P_e (\\delta^0)$ is real symmetric and diagonally dominant \\cite{2021-gholami-sun-MMG-stability}. \n\n\\subsection{Referenced Power System Model}\n\\label{subsec: Referenced Model}\n\nThe translational invariance of the flow function $P_e$ gives rise to a zero eigenvalue in the spectrum of $\\nabla P_e (\\delta)$, and as a consequence, in the spectrum of $J(\\delta)$. This zero eigenvalue and the corresponding nullspace pose some difficulties in monitoring the hyperbolicity of the equilibrium points, specially during Hopf bifurcation analysis. As mentioned in \\Cref{subsec: Multi-Machine Swing Equations}, this situation can be dealt with by defining a reference bus and referring all other bus angles to it. Although this is a common practice in power system context \\cite{2008-anderson-stability}, the spectral and dynamical relationships between the original system and the referenced system are not rigorously analyzed in the literature. In this section, we fill this gap to facilitate our analysis in the later parts.\n\n\\subsubsection{Referenced Model}\nDefine $\\psi_j:=\\delta_j-\\delta_n, \\forall j \\in \\{1,2,...,n-1\\}$ and reformulate the swing equation model \\eqref{eq: swing equations} into the \\emph{referenced model}\n\\begin{subequations} \\label{eq: Swing Equation Polar referenced}\n\t\\begin{align}\n\t& \\dot{\\psi}_j = \\omega_j - \\omega_n \\: \\qquad \\qquad \\qquad \\qquad \\forall j \\in \\{1,...,n-1\\},  \\\\\n\t& \\dot{\\omega}_j = - \\frac{d_j}{m_j} \\omega_j + \\frac{\\omega_s}{m_j}  (P_{m_j} - P_{e_j}^r(\\psi)) \\: \\: \\: \\: \\forall j \\in \\{1,...,n\\},\n\t\\end{align}\n\\end{subequations}\nwhere for all $j$ in $\\{1,...,n\\}$ we have\n\t\\begin{align} \\label{eq: flow function referenced compact}\n\tP_{e_j}^r(\\psi) & = \\sum \\limits_{k = 1}^{n} { V_j  V_k  Y_{jk} \\cos \\left( \\theta _{jk} - \\psi_j + \\psi_k \\right)},\n\t\\end{align}\nand $\\psi_n\\equiv0$. The function $P_e^r: \\mathbb{R}^{n-1} \\to \\mathbb{R}^n$ given by \\eqref{eq: flow function referenced compact} is called the referenced flow function. \n\\subsubsection{The Relationship}\nWe would like to compare the behaviour of the two dynamical systems \\eqref{eq: swing equations} and \\eqref{eq: Swing Equation Polar referenced}. Let us define the linear mapping $\\Psi:\\mathbb{R}^n\\times\\mathbb{R}^n \\to \\mathbb{R}^{n-1} \\times\\mathbb{R}^n$ given by $(\\delta,\\omega)\\mapsto(\\psi,\\omega)$ such that\n\\begin{align*}\n   \\Psi(\\delta,\\omega) = \\left  \\{ (\\psi,\\omega) : \\psi_j:=\\delta_j-\\delta_n, \\forall j \\in \\{1,2,...,n-1\\}  \\right  \\}. \n\\end{align*}\nThis map is obviously smooth but not injective.\nIt can also be written in matrix form \n\\begin{align*}\n       \\Psi(\\delta,\\omega) = \\begin{bmatrix} T_1 & 0 \\\\ 0 & I_n \\end{bmatrix}\n       \\begin{bmatrix} \\delta \\\\ \\omega \\end{bmatrix}\n\\end{align*}\nwhere $I_n\\in\\mathbb{R}^{n\\times n}$ is the identity matrix, $\\mathbf{1}\\in\\mathbb{R}^{n-1}$ is the vector of all ones, and\n\\begin{align}\nT_1:=\\begin{bmatrix}\nI_{n-1}& -\\mathbf{1}\n\\end{bmatrix} \\in \\mathbb{R}^{(n-1)\\times n}.\n\\end{align}\n\n\nThe next proposition, which is proved in \\cref{proof of thrm: original model vs referenced model}, establishes the relationship between the original model \\eqref{eq: swing equations} and the referenced model \\eqref{eq: Swing Equation Polar referenced}.\n\\begin{proposition} \\label{thrm: original model vs referenced model}\nLet $(\\delta^0,\\omega^0)$ be an equilibrium point of the swing equation \\eqref{eq: swing equations} and $(n_-,n_0,n_+)$ be the inertia\\footnote{Inertia of a matrix (see e.g. \\cite{2013-Horn-matrix-analysis} for a definition) should not be confused with the inertia matrix $M$.} of its Jacobian at this equilibrium point. The following two statements hold:\n\\begin{enumerate}[(i)]\n    \\item $\\Psi(\\delta^0,\\omega^0)$ is an equilibrium point of the referenced model \\eqref{eq: Swing Equation Polar referenced}.\n    \\item $(n_-,n_0-1,n_+)$ is the inertia of the Jacobian of \\eqref{eq: Swing Equation Polar referenced} at $\\Psi(\\delta^0,\\omega^0)$.\n   \n\\end{enumerate}\n\\end{proposition}\n\n\n\\begin{remark}\n    Note that the equilibrium points of the referenced model \\eqref{eq: Swing Equation Polar referenced} are in the set \n    \\begin{align*}\n        \\Tilde{\\mathcal{E}} =  \\biggl\\{(\\psi,\\omega)\\in \\mathbb{R}^{n-1}\\times\\mathbb{R}^n : \\; & \\omega_j = \\omega_n \\:, \\forall j \\in \\{1,...,n-1\\}, \\\\ & P_{m_j} = P_{e_j}^r(\\psi) + d_j \\omega_n\/\\omega_s , \\: \\forall j \\in \\{1,...,n\\}   \\biggr\\}\n    \\end{align*}\n    where $\\omega_n$ is not necessarily zero. Therefore, the referenced model \\eqref{eq: Swing Equation Polar referenced} may have extra equilibrium points which do not correspond to any equilibrium point of the original model \\eqref{eq: swing equations}.\n \\end{remark}\n\n\n\n\n\\subsection{Impact of Damping in Power Systems}\n\\label{Sec: Impact of Damping in Power Systems}\nThe theoretical results in \\Cref{Sec: Monotonic Behavior of Damping} and \\Cref{Sec: Impact of Damping in Hopf Bifurcation} have important applications in electric power systems. For example, \\cref{thrm: monotonic damping behaviour} is directly applicable to lossless power systems, and provides new insights to improve the situational awareness of power system operators. Recall that many control actions taken by power system operators are directly or indirectly targeted at changing the effective damping of the system \\cite{1994-kundur-stability,2019-Patrick-koorehdavoudi-input,2012-aminifar-wide-area-damping}. In this context, \\cref{thrm: monotonic damping behaviour} determines how the system operator should change the damping of the system in order to avoid breaking the hyperbolicity and escaping dangerous bifurcations. \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    Now, consider the case where a subset of power system generators have zero damping coefficients. Such partial damping is possible in practice specially in inverter-based resources (as damping coefficient corresponds to a controller parameter which can take zero value). The next theorem and remark follow from \\cref{thrm: nec and suf for pure imaginary lossless}, and show how partial damping could break the hyperbolicity in lossless power systems.\n    %\n   \n    %\n    \n\n        \n\\begin{theorem}[purely imaginary eigenvalues in lossless power systems] \\label{prop: nec and suf for pure imaginary lossless power system}\n    Consider a lossless network \\eqref{eq: swing equations} with an equilibrium point $(\\delta^{0},\\omega^{0})\\in \\Omega$. Suppose all generators have positive inertia and nonnegative damping coefficients. Then, $\\sigma(J(\\delta^{0}))$ contains a pair of purely imaginary eigenvalues if and only if the pair $(M^{-1}\\nabla P_e (\\delta^0), \\allowbreak M^{-1}D)$ is not observable.\n\\end{theorem}\n\n\\begin{proof}\n    As mentioned above, we always assume the physical network connecting the power generators is a connected (undirected) graph. Under this assumption,\n    as mentioned in \\Cref{SubSec: Graph Theory Interpretation and Spectral Properties}, matrix $L:=\\nabla P_e (\\delta^0)$ has a simple zero eigenvalue with a right eigenvector $\\mathbf{1}\\in\\mathbb{R}^{n}$, which is the vector of all ones \\cite{2020-fast-certificate}.\n    Moreover, since the power system is lossless and $(\\delta^{0},\\omega^{0})\\in \\Omega$, we have $L\\in\\mathbb{S}^n_+$.\n    %\n    If $D=0$, the pair $(M^{-1}L,M^{-1}D)$ can never be observable. Using a similar argument as in the first part in the proof of \\cref{thrm: nec and suf for pure imaginary lossless}, it can be shown that $M^{-1}L$ has a simple zero eigenvalue and the rest of its eigenvalues are positive, i.e., $\\sigma(M^{-1}L)\\subseteq \\mathbb{R_{+}}=\\{\\lambda\\in\\mathbb{R}:\\lambda\\ge0\\}$. Meanwhile, when $D=0$, we have $\\mu\\in\\sigma(J)\\iff\\mu^2\\in\\sigma(-M^{-1}L)$. Notice that a power grid has at least two nodes, i.e. $n\\ge2$, and hence, $M^{-1}L$ has at least one positive eigenvalue, i.e., $\\exists \\lambda\\in\\mathbb{R}_+, \\lambda>0$ such that $\\lambda\\in\\sigma(M^{-1}L)$. Hence, $\\mu= \\sqrt{-\\lambda}$ is a purely imaginary number and is an eigenvalue of $J$. Similarly, we can show that $-\\mu$ is an eigenvalue of $J$. Consequently, the theorem holds in the case of $D=0$. In the sequel, we assume that $D\\not=0$.\n\t%\n\t\n\t\n   \n   \n\t\\emph{Necessity:}\n\tAssume there exists $\\beta>0$ such that $\\frak{i} \\beta \\in \\sigma(J)$. We will show that the pair $(M^{-1}L,M^{-1}D)$ is not observable.\n\t%\n\n\tBy \\cref{lemma: relation between ev J and ev J11}, $\\frak{i}\\beta \\in \\sigma(J)$ if and only if the matrix pencil $(M^{-1}L + \\frak{i}\\beta M^{-1}D - \\beta^2 I)$ is singular: \n\n\t\\begin{align*}\n\t\n\t& \\det \\left(  M^{-\\frac{1}{2}}(M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}} + \\frak{i} \\beta  M^{-\\frac{1}{2}}DM^{-\\frac{1}{2}} - \\beta^2 I) M^{\\frac{1}{2}} \\right) = 0,\n\t\\end{align*}\n\tor equivalently, $\\exists (x + \\frak{i} y) \\ne 0$ such that $x,y\\in\\mathbb{R}^n$ and\n\t\\begin{align} \\label{eq: proof of parially damped hopf bifurcation lossless swing}\n\t\\notag & (M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}} + \\frak{i} \\beta  M^{-\\frac{1}{2}} D M^{-\\frac{1}{2}} - \\beta^2 I)(x+\\frak{i} y) = 0 \\\\\n\t& \\iff  \\begin{cases}\n\t(M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}}-\\beta^2 I)x - \\beta M^{-\\frac{1}{2}}DM^{-\\frac{1}{2}} y = 0, \\\\ \n\t(M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}}-\\beta^2 I)y + \\beta M^{-\\frac{1}{2}}DM^{-\\frac{1}{2}} x = 0.\n\t\\end{cases}\n\t\\end{align}\n\tLet $\\hat{L}:=M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}}$, $\\hat{D}:=M^{-\\frac{1}{2}}DM^{-\\frac{1}{2}}$, and observe that\n\t\\begin{align*}\n\t\\begin{cases}\n\ty^\\top (\\hat{L}-\\beta^2 I)x = y^\\top (\\beta \\hat{D} y) = \\beta y^\\top \\hat{D} y  \\ge 0,\n\t\\\\\n\tx^\\top (\\hat{L}-\\beta^2 I)y = x^\\top (-\\beta \\hat{D} x) = -\\beta x^\\top \\hat{D} x  \\le 0,\n\t\\end{cases}\n\t\\end{align*}\n   \n    %\n\twhere we have used the fact that $\\hat{D}$ is *-congruent to $D$. According to Sylvester's law of inertia, $\\hat{D}$ and $D$ have the same inertia. Since $D\\succeq 0$, we conclude that $\\hat{D}\\succeq0$.\n\n\tAs $(\\hat{L}-\\beta^2 I)$ is symmetric, we have $x^\\top (\\hat{L}-\\beta^2 I)y = y^\\top (\\hat{L}-\\beta^2 I)x$. Therefore, we must have $x^\\top \\hat{D} x = y^\\top \\hat{D} y  =0$. Since $\\hat{D}\\succeq0$, we can infer that $x\\in\\ker(\\hat{D})$ and $y\\in\\ker(\\hat{D})$.\n\t%\n\n   \n   \n   \n   \n\tNow considering $\\hat{D} y = 0$ and using the first equation in \\eqref{eq: proof of parially damped hopf bifurcation lossless swing} we get \n\t\\begin{align}\n\t( \\hat{L}-\\beta^2 I)x = 0 \\iff M^{-\\frac{1}{2}}LM^{-\\frac{1}{2}} x=\\beta^2 x,\n\t\\end{align}\n\tmultiplying both sides from left by $M^{-\\frac{1}{2}}$ we get $M^{-1}L (M^{-\\frac{1}{2}} x) = \\beta^2 (M^{-\\frac{1}{2}} x)$. Thus, $ \\hat{x}:= M^{-\\frac{1}{2}} x$ is an eigenvector of $M^{-1}L$. Moreover, we have \n\t\\begin{align*}\n\t    M^{-1}D \\hat{x}  = M^{-1}DM^{-\\frac{1}{2}} x = M^{-\\frac{1}{2}} (  \\hat{D}  x) = 0,    \n\t\\end{align*}\n\twhich means that the pair $(M^{-1}L,M^{-1}D)$ is not observable.\n\n\n\n\n\t\n\n\t%\n   \n   \n   \n   \n   \n   \n   \n\n\t\n\t\\emph{Sufficiency:}\n\tSuppose the pair~ $\\allowbreak (M^{-1}L, M^{-1}D)$ is not observable. We will show that $\\sigma(J)$ contains a pair of purely imaginary eigenvalues.\n\n\tAccording to \\cref{def: Observability}, $\\exists \\lambda \\in \\mathbb{C}, x \\in \\mathbb{C}^n , x\\ne 0$ such that\n\t\\begin{align} \\label{eq: observability in lossless general swing}\n\t M^{-1}Lx = \\lambda x \\text{ and } M^{-1}Dx = 0.\n\t\\end{align}\n   \n\tWe make the following two observations.\n\tFirstly, as it is shown above, we have $\\sigma(M^{-1}L)\\subseteq \\mathbb{R_{+}}$. \n\t%\n\tSecondly, \n\n\t%\n\n   \n   \n    $L$ has a simple zero eigenvalue and a one-dimensional nullspace spanned by $\\mathbf{1}\\in\\mathbb{R}^n$. We want to emphasize that this zero eigenvalue of $L$ cannot break the observability of the pair $(M^{-1}L,M^{-1}D)$. Note that $\\ker(L)=\\ker(M^{-1}L)$ and $M^{-1}L \\mathbf{1} =0$ implies that $M^{-1}D \\mathbf{1} \\ne0$ because $D\\ne0$.\n   \n   \n    %\n\tBased on the foregoing two observations, when the pair $(M^{-1}L,M^{-1}D)$ is not observable, there must exist $\\lambda \\in \\mathbb{R_{+}}, \\lambda\\ne0$ and $x \\in \\mathbb{C}^n , x\\ne 0$ such that \\eqref{eq: observability in lossless general swing} holds.\n\n   \n   \n   \n\n\tDefine $\\xi=\\sqrt{-\\lambda}$, which is a purely imaginary number. The quadratic pencil $M^{-1}P(\\xi)=\\xi^2 I + \\xi M^{-1}D + M^{-1}L$ is singular because $M^{-1}P(\\xi)x = \\xi^2 x + \\xi M^{-1}Dx + M^{-1}Lx = -\\lambda x + 0 + \\lambda x = 0$. By  \\cref{lemma: relation between ev J and ev J11}, $\\xi$ is an eigenvalue of $J$. Similarly, we can show $-\\xi$ is an eigenvalue of $J$. Therefore, $\\sigma(J)$ contains the pair of purely imaginary eigenvalues $\\pm\\xi$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe following remark illustrates how \\cref{prop: nec and suf for pure imaginary lossless power system} can be used in practice to detect and damp oscillations in power systems.\n\\begin{remark}\n\n\tConsider the assumptions of \\cref{prop: nec and suf for pure imaginary lossless power system} and suppose there exists a pair of purely imaginary eigenvalues $\\pm \\frak{i}  \\beta \\in \\sigma(J(\\delta^0))$ which give rise to Hopf bifurcation and oscillatory behaviour of the system. This issue can be detected by observing the osculations in power system state variables (through phasor measurement units (PMUs) \\cite{2012-aminifar-wide-area-damping}). According to \\cref{prop: nec and suf for pure imaginary lossless power system}, we conclude that $\\beta^2\\in\\sigma(M^{-1}\\nabla P_e (\\delta^0))$. Let $\\mathcal{X}:= \\{x^1,...,x^k\\}$ be a set of independent eigenvectors associated with the eigenvalue $\\beta^2\\in\\sigma(M^{-1}\\nabla P_e (\\delta^0))$, i.e., we assume that the corresponding eigenspace is $k$-dimensional. According to \\cref{prop: nec and suf for pure imaginary lossless power system}, we have $M^{-1}Dx^\\ell=0,\\forall x^\\ell\\in\\mathcal{X}$, or equivalently, $Dx^\\ell=0,\\forall x^\\ell\\in\\mathcal{X}$. Since $D$ is diagonal, we have $d_jx_j^\\ell=0,\\forall j\\in\\{1,\\cdots,n\\}, \\forall x^\\ell\\in\\mathcal{X}$.\n\tIn order to remove the purely imaginary eigenvalues, we need to make sure that $\\forall x^\\ell\\in\\mathcal{X}, \\exists j \\in \\{1,\\cdots,n\\}$ such that $d_j x_j^\\ell\\ne0$. This can be done for each $x^\\ell\\in\\mathcal{X}$ by choosing a $j\\in\\{1,\\cdots,n\\}$ such that $x_j^\\ell\\ne0$ and then increase the corresponding damping $d_j$ from zero to some positive number, thereby rendering the pair $(M^{-1}\\nabla P_e (\\delta^0),M^{-1}D)$ observable.\n\\end{remark}\n\n\n\\Cref{prop: nec and suf for pure imaginary lossless power system} gives a necessary and sufficient condition for the existence of purely imaginary eigenvalues in a lossless power system with \\emph{nonnegative} damping and positive inertia. It is instructive to compare it with an earlier result in \\cite{2021-gholami-sun-MMG-stability}, which shows that when all the generators in a lossless power system have \\emph{positive} damping $d_j$ and positive inertia $m_j$, then any equilibrium point in the set $\\Omega$ is asymptotically stable.\nThis is also proved in \\Cref{thrm: Stability of Symmetric Second-Order Systems with nonsing damping} of \\Cref{appendix: Stability of Symmetric Second-Order Systems with Nonsingular Damping} for the general second-order model \\eqref{eq: nonlinear ode}.\n\n\n\n\n\n\n\nRecall that the simple zero eigenvalue of the Jacobian matrix $J(\\delta^0)$ in model \\eqref{eq: swing equations} stems from the translational invariance of\nthe flow function defined in \\Cref{def: flow function}. As mentioned earlier, we can eliminate this eigenvalue by choosing a reference bus and refer all other bus angles to it. According to \\Cref{thrm: original model vs referenced model}, aside from the simple zero eigenvalue, the Jacobians of the original model \\eqref{eq: swing equations} and the referenced model \\eqref{eq: Swing Equation Polar referenced} have the same number of eigenvalues with zero real part. Hence, \\cref{prop: nec and suf for pure imaginary lossless power system} provides a necessary and sufficient condition for breaking the hyperbolicity in the referenced lossless power system model \\eqref{eq: Swing Equation Polar referenced}.\n\n\n\n\n\nIn lossy power systems, matrix $\\nabla P_e (\\delta)$ may not be symmetric. In this case, \\cref{thrm: nec and suf for pure imaginary lossy} can be used for detecting purely imaginary eigenvalues. Meanwhile, let us discuss some noteworthy cases in more detail. \\Cref{prop:hyperbolicity n2 n3} asserts that in small lossy power networks  with only one undamped generator, the equilibrium points are always hyperbolic. The proof is provided in \\cref{proof of prop:hyperbolicity n2 n3}.\n\\begin{theorem} \\label{prop:hyperbolicity n2 n3}\nLet $n\\in\\{2,3\\}$ and consider an $n$-generator system with only one undamped generator. Suppose $(\\delta^{0},\\omega^{0})\\in \\Omega$ holds, the underlying undirected power network graph is connected, and $\\nabla P_e (\\delta)$. Then the Jacobian matrix $J(\\delta^0)$ has no purely imaginary eigenvalues.\nWe allow the network to be lossy, but we assume ${\\partial P_{e_j} }\/ {\\partial \\delta _k}=0$ if and only if ${\\partial P_{e_k} }\/ {\\partial \\delta _j}=0$. The lossless case is a special case of this.\n\\end{theorem}\n\n\n\t\n\t\n\n\n\n\t\n\t\n   \n\n\n\nThe following counterexample shows that as long as there are two undamped generators, the Jacobian $J(\\delta)$ at an equilibrium point may have purely imaginary eigenvalues.\n\\begin{proposition} \\label{prop: non-hyper example}\nFor any $n\\ge 2$, consider an $(n+1)$-generator system with $2$ undamped generators and the following $(n+1)$-by-$(n+1)$ matrices $L=\\nabla P_e (\\delta^0)$, $D$, and $M$:\n\\begin{align*}\nL = \\begin{bmatrix} \n1 & -\\frac{1}{n} & -\\frac{1}{n} & \\cdots & -\\frac{1}{n} \\\\\n-\\frac{1}{n} & 1 & -\\frac{1}{n} & \\cdots & -\\frac{1}{n} \\\\\n\\vdots & \\vdots  & \\vdots & \\ddots & \\vdots \\\\\n-\\frac{1}{n} & -\\frac{1}{n} & -\\frac{1}{n} & \\cdots & 1\n\\end{bmatrix},\\\\\nD= \\mathbf{diag} ([0,0,d_3,d_4,\\cdots,d_{n+1}]), \\: M =  I_{n+1}.\n\\end{align*}\nThen $\\pm \\frak{i} \\beta \\in \\sigma(J(\\delta^0))$, where $\\beta^2 = 1+\\frac{1}{n}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\beta^2 = 1+\\frac{1}{n}$ and observe that $\\mathop{\\bf rank}(L-\\beta^2 M)=1$ and $\\mathop{\\bf rank}(\\beta D)=(n+1)-2=n-1$. The rank-sum inequality \\cite{2013-Horn-matrix-analysis} implies that\n\\begin{align*}\n\\mathop{\\bf rank}(L-\\beta^2 M- \\frak{i} \\beta D) &\\le \\mathop{\\bf rank}(L-\\beta^2 M)+\\mathop{\\bf rank}(- \\frak{i}\\beta D) = 1+(n-1)=n,\n\\end{align*}\nthat is $\\det\\left( L + \\frak{i}  \\beta D - \\beta^2 M \\right)  = 0$. Now according to \\cref{lemma: relation between ev J and ev J11}, the latter is equivalent to $\\frak{i}  \\beta \\in \\sigma(J(\\delta^0))$. This completes the proof.\nAlso note that the constructed $L$ is not totally unrealistic for a power system.\n\\end{proof}\n\n\n\n\\section{Illustrative Numerical Power System Examples} \\label{Sec: Computational Experiments}\nTwo case studies will be presented to illustrate breaking the hyperbolicity and the occurrence of Hopf bifurcation under damping variations. Additionally, we adopt the center manifold theorem to determine the stability of bifurcated orbits. Note that using the center manifold theorem, a Hopf bifurcation in an $n$-generator network essentially reduces to a planar system provided that aside from the two purely imaginary eigenvalues no other eigenvalues have zero real part at the bifurcation point. Therefore, for the sake of better illustration we focus on small-scale networks.\n\\subsection{Case $1$} \\label{subsubsec: case 1}\nConsider a $3$-generator system with $D=\\mathbf{diag} ([\\gamma,\\gamma,1.5])$, $M=I_3$, $Y_{12}=Y_{13}=2Y_{23}=\\frak{i} $ p.u., $P_{m_1}=-\\sqrt{3}$ p.u., and $P_{m_2}=P_{m_3}=\\sqrt{3}\/2$ p.u. The load flow problem for this system has the solution $V_j=1$ p.u. $\\forall j$ and $\\delta_1=0$, $\\delta_2=\\delta_3=\\pi\/3$. Observe that when $\\gamma=0$, the pair $(M^{-1}\\nabla P_e (\\delta^0),M^{-1}D)$ is not observable, and \\cref{prop: nec and suf for pure imaginary lossless power system} implies that the spectrum of the Jacobian matrix $\\sigma(J)$ contains a pair of purely imaginary eigenvalues. Moreover, this system satisfies the assumptions of \\cref{prop: non-hyper example}, and consequently, we have $\\pm \\frak{i}  \\sqrt{1.5} \\in \\sigma(J)$. In order to eliminate the zero eigenvalue (to be able to use the Hopf bifurcation theorem), we adopt the associated referenced model using the procedure described in \\Cref{subsec: Referenced Model}. \nThe conditions (\\ref{coro: hopf conditions_1})-(\\ref{coro: hopf conditions_4}) of \\cref{coro: Hopf bifurcation} are satisfied (specifically, the transversality condition (\\ref{coro: hopf conditions_transvers}) holds because $\\mathrm{Im}( q^* M^{-1} D'(\\gamma_0) v ) = -0.5$), and accordingly, a periodic orbit bifurcates at this point. \nTo determine the stability of bifurcated orbit, we compute the \\emph{first Lyapunov coefficient} $l_1(0)$ as described in \\cite{2004-kuznetsov-hopf}. If the first Lyapunov coefficient is negative, the bifurcating limit cycle is stable, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical. In this example, we get $l_1(0)=-1.7\\times10^{-3}$ confirming that the type of Hopf bifurcation is supercritical and a stable limit cycle is born. Figs. (\\ref{fig:sfig1 case1 bifurcation})-(\\ref{fig:sfig3 case1 bifurcation}) depict these limit cycles when the parameter $\\gamma$ changes. Moreover, Fig. (\\ref{fig:sfig4 case1 bifurcation}) shows the oscillations observed in the voltage angles and frequencies when $\\gamma=0$.\n\\begin{figure}\n\\centering\n\\begin{subfigure}{0.33\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/case1\/y1y2_rev.pdf}\n  \\caption{}\n  \\label{fig:sfig1 case1 bifurcation}\n\\end{subfigure}%\n\\begin{subfigure}{0.35\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/case1\/w1w2w3.pdf}\n  \\caption{}\n  \\label{fig:sfig2 case1 bifurcation}\n\\end{subfigure}\n\\begin{subfigure}{0.34\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/case1\/w2w3d.pdf}\n  \\caption{}\n  \\label{fig:sfig3 case1 bifurcation}\n\\end{subfigure}\n\\begin{subfigure}{0.35\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/case1\/y1y2w1w2w3_rev.pdf}\n  \\caption{}\n  \\label{fig:sfig4 case1 bifurcation}\n\\end{subfigure}\n\\caption{Occurrence of supercritical Hopf bifurcation in Case $1$. (a)-(c) Projection of limit cycles into different subspaces as the parameter $\\gamma$ changes. (d) Oscillations of the voltage angles $\\psi$ in radians and the  angular frequency deviation $\\omega$ in radians per seconds when $\\gamma=0$. Note that $\\psi_3 \\equiv 0$.}\n\\label{fig: case1 bifurcation}\n\\end{figure}\n\\subsection{Case $2$} \\label{subsubsec: case 2}\nNext, we explore how damping variations could lead to a Hopf bifurcation in lossy systems. It is proved in \\cref{prop:hyperbolicity n2 n3} that a $2$-generator system with only one undamped generator cannot experience a Hopf bifurcation. To complete the discussion, let us consider a fully-damped (i.e., all generators have nonzero damping) lossy $2$-generator system here. Note also that the discussion about a fully-undamped case is irrelevant (see \\cref{lemma: undamped systems}). Suppose $M=I_2$, $D= \\mathbf{diag} ([\\gamma,1])$,  $Y_{12}=-1+\\frak{i} 5.7978$ p.u., $P_{m_1}=6.6991$ p.u., and $P_{m_2}=-4.8593$ p.u. The load flow problem for this system has the solution $V_j=1$ p.u. $\\forall j$ and $\\delta_1=1.4905$, $\\delta_2=0$. \nWe observe that $\\gamma=0.2$ will break the hyperbolicity and lead to a Hopf bifurcation  with the first Lyapunov coefficient $l_1(0.2)=1.15$. This positive value for for $l_1(0.2)$ confirms that the type of Hopf bifurcation is subcritical and an unstable limit cycle bifurcates for $\\gamma\\ge0.2$. Therefore, the situation can be summarized as follows:\n\\begin{itemize}\n    \\item If $\\gamma <0.2$, there exists one unstable equilibrium point.\n    \\item If $\\gamma =0.2$, a subcritical Hopf bifurcation takes place and a unique small unstable limit cycle is born.\n    \\item If $ \\gamma >0.2$, there exists a stable equilibrium point surrounded by an unstable limit cycle. \n\\end{itemize}\n\nFigs. (\\ref{fig:sfig1 case2 bifurcation})-(\\ref{fig:sfig3 case2 bifurcation}) depict the bifurcating unstable limit cycles when the parameter $\\gamma$ changes in the interval $[0.2,0.35]$. This case study sheds lights on an important fact: bifurcation can happen even in fully damped systems, provided that the damping matrix $D$ reaches a critical point (say $D_c$). When $D \\preceq D_c$, the equilibrium point is unstable. On the other hand, when $D \\succ D_c$, the equilibrium point becomes stable but it is still surrounded by an unstable limit cycle. As we increase the damping parameter, the radius of the limit cycle increases, and this will enlarge the region of attraction of the equilibrium point. Note that the region of attraction of the equilibrium point is surrounded by the unstable limit cycle. This also confirms the monotonic behaviour of damping proved in \\cref{thrm: monotonic damping behaviour}. Fig. (\\ref{fig:sfig4 case2 bifurcation}) shows the region of attraction surrounded by the unstable limit cycle (in red) when $\\gamma=0.25$. In this figure, the green orbits located inside the cycle are spiraling in towards the equilibrium point while the blue orbits located outside the limit cycle are spiraling out.\n\\begin{figure}\n\\centering\n\\begin{subfigure}{0.35\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/case2\/y1w1w2_rev.pdf}\n  \\caption{}\n  \\label{fig:sfig1 case2 bifurcation}\n\\end{subfigure}%\n\\begin{subfigure}{0.35\\textwidth}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{figures\/case2\/y1d_rev.pdf}\n  \\caption{}\n  \\label{fig:sfig2 case2 bifurcation}\n\\end{subfigure}\n\\begin{subfigure}{0.35\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/case2\/y1w1d_rev.pdf}\n  \\caption{}\n  \\label{fig:sfig3 case2 bifurcation}\n\\end{subfigure}\n\\begin{subfigure}{0.34\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/case2\/y1w1_rev.pdf}\n  \\caption{}\n  \\label{fig:sfig4 case2 bifurcation}\n\\end{subfigure}\n\\caption{Occurrence of subcritical Hopf bifurcation in Case $2$. (a) Unstable limit cycles as the parameter $\\gamma$ changes. (b)-(c) Projection of limit cycles into different subspaces as the parameter $\\gamma$ changes. (d) The region of attraction of the equilibrium point when $\\gamma=0.25$. The unstable limit cycle is shown in red, while the orbits inside and outside of it are shown in green and blue, respectively. Note that $\\psi_2 \\equiv 0$.}\n\\label{fig: case2 bifurcation}\n\\end{figure}\n\nAlthough both supercritical and subcritical Hopf bifurcations lead to the birth of limit cycles, they have quite different practical consequences. The supercritical Hopf bifurcation which occurred \\Cref{subsubsec: case 1} corresponds to a soft or noncatastrophic stability loss because a stable equilibrium point is replaced with a stable periodic orbit, and the system remains in a neighborhood of the equilibrium. In this case, the system operator can take appropriate measures to bring the system back to the stable equilibrium point. Conversely, the subcritical Hopf bifurcation in \\Cref{subsubsec: case 2} comes with a sharp or catastrophic loss of stability. This is because the region of attraction of the equilibrium point (which is bounded by the unstable limit cycle) shrinks as we decrease the parameter $\\gamma$ and disappears once we hit $\\gamma=0.2$. In this case, the system operator may not be able to bring the system back to the stable equilibrium point as the operating point may have left the region of attraction.\n\n\n\n\\section{Conclusions} \\label{Sec: Conclusions}\nWe have presented a comprehensive study on the role of damping in a large class of dynamical systems, including electric power networks. Paying special attention to partially-damped systems, it is shown that damping plays a monotonic role in the hyperbolicity of the equilibrium points. We have proved that the hyperbolicity of the equilibrium points is intertwined with the observability of a pair of matrices, where the damping matrix is involved. We have also studied the aftermath of hyperbolicity collapse, and have shown both subcritical and supercritical Hopf bifurcations can occur as damping changes. It is shown that Hopf bifurcation cannot happen in small power systems with only one undamped generator. In the process, we have developed auxiliary results by proving some important spectral properties of  the power system Jacobian matrix, establishing the relationship between a power system model and its referenced counterpart, and finally addressing a fundamental question from matrix perturbation theory. Among others, the numerical experiments have illustrated how damping can change the region of attraction of the equilibrium points.\nWe believe our results are of general interest to the community of applied dynamical systems, and provide new insights into the interplay of damping and oscillation in one of the most important engineering system, the electric power systems.\n\\section{Proof of Theorem \\ref{thrm: rank}}\n\\section{Proof of \\Cref{lemma: rank principal}} \\label{proof of lemma: rank principal}\n   \n   \n   \n   \n   \n\\begin{proof}\n   \n    \tAssume that all $r$-by-$r$ principal submatrices of $S$ are singular, and let us lead this assumption to a contradiction. Since $\\mathop{\\bf rank}(S)=r$, all principal submatrices of size larger than $r$ are also singular. Therefore, zero is an eigenvalue of every $m$-by-$m$ principal submatrix of $S$ for each $m\\ge r$. Consequently, all principal minors of $S$ of size $m$ are zero for each $m\\ge r$. Let $E_\\ell(S)$ denote the sum of principal minors of $S$ of size $\\ell$ (there are $n\\choose \\ell$ of them), and observe that we have $E_m(S) = 0, \\forall m \\ge r$. Moreover, thought of as a formal polynomial in $t$, let $p_S (t)=\\sum_{\\ell=0}^n a_\\ell t^\\ell$ with $a_n=1$ be the characteristic polynomial of $S$, and recall that the $k$-th derivative of $p_S(t)$ at $t=0$ is $p_S^{(k)}(0)=k!(-1)^{n-k}E_{n-k}(S), \\forall k\\in\\{0,1,\\cdots,n-1\\}$, and the coefficients of the characteristic polynomial are $a_k=\\frac{1}{k!}p_S^{(k)}(0)$. \n    \tIn this case, our assumption leads to $a_k=p_S^{(k)}(0)=0,\\forall k\\in\\{0,1,\\cdots,n-r\\}$, i.e., zero is an eigenvalue of $S$ with algebraic multiplicity at least $n-r+1$. But from the assumption of the lemma we know $S$ is similar to $B\\oplus0_{n-r}$, that is, zero is an eigenvalue of $S$ with algebraic multiplicity exactly $n-r$, and we arrive at the desired contradiction.\n\\end{proof}\n\n\n\n\\section{Stability of Symmetric Second-Order Systems with Nonsingular Damping}\n\\label{appendix: Stability of Symmetric Second-Order Systems with Nonsingular Damping}\n\n\n\n\n\n\\Cref{thrm: nec and suf for pure imaginary lossless} provides a necessary and sufficient condition for the hyperbolicity of an equilibrium point $(x_0,0)$ of the second-order system \\eqref{eq: nonlinear ode}, when the inertia, damping, and Jacobian of $f$ satisfy $M\\in\\mathbb{S}^{n}_{++}, D\\in\\mathbb{S}^n_+, \\nabla f(x_0)\\in\\mathbb{S}^n_{++}$. \nIn this section, we prove that if we replace the assumption $D\\in\\mathbb{S}^n_+$ with $D\\in\\mathbb{S}^n_{++}$, then the equilibrium point $(x_0,0)$ is not only hyperbolic but also asymptotically stable. This asymptotic stability is proved for lossless swing equations in \\cite[Theorem 1, Part d]{2021-gholami-sun-MMG-stability}.\nThe next theorem generalizes \\cite[Theorem 1, Part d]{2021-gholami-sun-MMG-stability} to the second-order system \\eqref{eq: nonlinear ode} where the damping and inertia matrices are not necessarily diagonal.\n\\begin{theorem}[stability in second-order systems: symmetric case] \\label{thrm: Stability of Symmetric Second-Order Systems with nonsing damping}\n    Consider the second-order ODE system \\eqref{eq: nonlinear ode} with inertia matrix $M\\in\\mathbb{S}^n_{++}$ and damping matrix $D \\in\\mathbb{S}^n_{++}$. Suppose $(x_0,0)\\in \\mathbb{R}^{n+n}$ is an equilibrium point of the corresponding first-order system \\eqref{eq: nonlinear ode 1 order} with the Jacobian matrix $J\\in\\mathbb{R}^{2n\\times 2n}$ defined in \\eqref{eq: J general case} such that $L=\\nabla f(x_0)\\in \\mathbb{S}^n_{++}$. Then, the equilibrium point $(x_0,0)$ is locally asymptotically stable.\n\\end{theorem}\n\n\n\n\\begin{proof}\nWe complete the proof in three steps:\\newline\n\\textbf{Step 1:} First, we show all real eigenvalues\nof $J$ are negative. Assume $\\lambda\\in\\mathbb{R}, \\lambda\\ge0$ is a nonnegative eigenvalue of $J(x_0)$, and let us lead this assumption to a contradiction. According to \\Cref{lemma: relation between ev J and ev J11},\n\\begin{align} \\label{eq: pencil det zero appendix}\n    \\det\\left(\\lambda^2 M + \\lambda D + L \\right)  = 0.\n\\end{align}\nSince all three matrices $L, D$, and $M$ are positive definite, the\nmatrix pencil $P(\\lambda)=\\lambda^2 M + \\lambda D + L $ is also a positive definite matrix for any nonnegative $\\lambda$. Hence $P(\\lambda)$ is nonsingular, contradicting \\eqref{eq: pencil det zero appendix}.\n\n\n\n\\noindent\n\\textbf{Step 2:} Next, we prove that the eigenvalues of $J$ cannot be purely imaginary. We provide two different proofs for this step.\nAccording to our assumption, the damping matrix $D$ is nonsingular, and the pair $(M^{-1}\\nabla f(x_0),M^{-1}D)$ is always observable because the nullspace of $M^{-1}D$ is trivial. Hence, according to\n \\Cref{thrm: nec and suf for pure imaginary lossless}, the  equilibrium point $(x_0,0)$ is hyperbolic, and $J(x_0)$ does not have any purely imaginary eigenvalue, so the first proof of this step is complete. For the second proof,\nlet $\\lambda \\in \\sigma(J(x_0))$, then according to \\Cref{lemma: relation between ev J and ev J11}, $\\exists v\\in\\mathbb{C}^n, v\\ne0$ such that $(\\lambda^2 M + \\lambda D + L) v = 0.$\n\tSuppose, for the sake of contradiction, that $\\lambda  = \\frak{i} \\beta \\in \\sigma(J(x_0))$ for some nonzero real $\\beta$. Let $v  =  x + \\frak{i} y$, then $((L - \\beta^2 M) + \\frak{i}\\beta D) (x+\\frak{i} y) = 0$,\n\twhich can be equivalently written as\n\t\\begin{align}\n\t\\begin{bmatrix}\n\tL - \\beta^2 M & -\\beta D \\\\\n\t\\beta D & L - \\beta^2 M\n\t\\end{bmatrix} \\begin{bmatrix}\n\tx\\\\y\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t0\\\\0\n\t\\end{bmatrix}.\n\t\\end{align}\n\tDefine the matrix \n\t\\begin{align} \\label{eq:M}\n\tH(\\beta) := \\begin{bmatrix}\n\t\\beta D &  L  - \\beta^2 M\\\\\n\t L  - \\beta^2 M & -\\beta D\n\t\\end{bmatrix}. \n\t\\end{align}\n\tSince $L \\in \\mathbb{S}^n_{++}$, $H(\\beta)$ is a symmetric matrix.\n\tNotice also that $H(\\beta)$ cannot be positive semidefinite due to the diagonal blocks $\\pm\\beta D$. \n\tSince $D\\in \\mathbb{S}^n_{++}$, the determinant of $H(\\beta)$ can be expressed using Schur complement as\n\t\\begin{align*}\n\t {\\det}(H(\\beta))  =  {\\det} (-\\beta D) {\\det} (\\beta D \n\t  + \\beta^{-1} ( L  - \\beta^2 M )D^{-1}(  L  - \\beta^2 M)).\n\t\\end{align*}\n\tSo we only need to consider the nonsingularity of the Schur complement.\n\tDefine the following matrices for the convenience of analysis: \n\t\\begin{align*}\n\t\tA(\\beta) &:=  L  - \\beta^2 M, \\\\\n\t\tB(\\beta) &:= D^{-\\frac{1}{2}}A(\\beta)D^{-\\frac{1}{2}},\\\\\n\t\tE(\\beta) &:= I + \\beta^{-2} B(\\beta)^2.\n\t\\end{align*}\n\tThe inner matrix of the Schur complement can be written as\n\n\n\t\t\\begin{align*}\n\t\t& \\beta D + \\beta^{-1} ( L  - \\beta^2 M )D^{-1}(  L  - \\beta^2 M) \\\\ \n\t & =  \\beta D^{\\frac{1}{2}} (I + \\beta^{-2} D^{-\\frac{1}{2}}A(\\beta)D^{-1}A(\\beta)D^{-\\frac{1}{2}})D^{\\frac{1}{2}} \\\\\n\t & =  \\beta D^{\\frac{1}{2}} (I + \\beta^{-2} B(\\beta)^2)D^{\\frac{1}{2}} = \\beta D^{\\frac{1}{2}} E(\\beta) D^{\\frac{1}{2}}.\n\t\t\\end{align*}\n\n\tNotice that {$E(\\beta)$ and $B(\\beta)$ have the same eigenvectors and the eigenvalues of $E(\\beta)$ and $B(\\beta)$ have a one-to-one correspondence: $\\mu$ is an eigenvalue of $B(\\beta)$ if and only if $1 + \\beta^{-2} \\mu^2$ is an eigenvalue of $E(\\beta)$}. Indeed, we have \n\t $E(\\beta)v = v + \\beta^{-2}B(\\beta)^2v = v + \\beta^{-2}\\mu^2 v = (1+\\beta^{-2}\\mu^2)v$ for any eigenvector $v$ of $B(\\beta)$ with eigenvalue $\\mu$. \n\t%\n\tSince $B(\\beta)$ is symmetric, $\\mu$ is a real number. Hence, $E(\\beta)$ is positive definite (because $1+\\beta^{-2}\\mu^2>0$), therefore $H(\\beta)$ is nonsingular for any real nonzero $\\beta$. Then, the eigenvector $v=x+\\frak{i} y$ is zero which is a contradiction. This proves that $J(x_0)$ has no eigenvalue on the punctured imaginary axis.\n\n\n   \n\n\n\t\n\t\\noindent \\textbf{Step 3:}\n\tFinally, we prove that any complex nonzero eigenvalue of $J(x_0)$ has a negative real part. \n\t%\n\t For a complex eigenvalue $\\alpha + \\frak{i} \\beta$ of $J(x_0)$ with $\\alpha\\ne 0, \\beta\\ne 0$, by setting $v  =  x + \\frak{i}  y$, the pencil singularity equation becomes\n\t\\begin{align*}\n\t( L  + (\\alpha + \\frak{i} \\beta) D + (\\alpha^2 - \\beta^2 + 2\\alpha\\beta \\frak{i} )M)(x+\\frak{i} y) = 0.\n\t\\end{align*}\n\tSimilar to Step 2 of the proof, define the matrix $H(\\alpha,\\beta)$ as\n\t\\begin{align*}\n\tH(\\alpha,\\beta):=\\begin{bmatrix}\n\t L  + \\alpha D + (\\alpha^2-\\beta^2) M & -\\beta(D + 2\\alpha M) \\\\\n\t\\beta(D+2\\alpha M) &  L  + \\alpha D + (\\alpha^2-\\beta^2) M\n\t\\end{bmatrix}.\n\t\\end{align*}\n\tWe only need to consider two cases, namely 1) $\\alpha > 0, \\beta > 0$ or 2) $\\alpha <0, \\beta > 0$. For the first case, $\\beta(D+{2}\\alpha M)$ is invertible and positive definite, therefore, we only need to look at the invertibility of the Schur complement\n\t\\begin{align*}\n\t& S(\\alpha,\\beta) + T(\\alpha,\\beta) S^{-1}(\\alpha,\\beta) T(\\alpha,\\beta),\n\t\\end{align*}\n\twhere $S(\\alpha,\\beta):= \\beta(D+{2}\\alpha M)$ and $T(\\alpha,\\beta):= L  + \\alpha D + (\\alpha^2-\\beta^2) M$.\n\tUsing the same manipulation as in Step 1 of the proof, we can see that the Schur complement is always invertible for any $\\alpha >0, \\beta>0$. This implies the eigenvector $v$ is 0, which is a contradiction. Therefore, the first case is not possible. So any complex nonzero eigenvalue of $J(x_0)$ has a negative real part.\n\\end{proof}\t\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of \\cref{thrm: nec and suf for pure imaginary lossy}}\n\\label{proof of thrm: nec and suf for pure imaginary lossy}\n\\begin{proof}\n    %\n   \n\n\tThere exist\t$\\lambda \\in \\mathbb{R_{+}}, \\lambda\\ne0$ and $x \\in \\mathbb{C}^n , x\\ne 0$ such that\n\t\\begin{align} \\label{eq: observability in lossy nonsymmetric}\n\t M^{-1}Lx = \\lambda x \\text{ and } M^{-1}Dx = 0.\n\t\\end{align}\n\n\tDefine $\\xi=\\sqrt{-\\lambda}$, which is a purely imaginary number. The quadratic matrix pencil $M^{-1}P(\\xi)=\\xi^2 I + \\xi M^{-1}D + M^{-1}L$ is singular because $M^{-1}P(\\xi)x = \\xi^2 x + \\xi M^{-1}Dx + M^{-1}Lx = -\\lambda x + 0 + \\lambda x = 0$. By  \\cref{lemma: relation between ev J and ev J11}, $\\xi$ is an eigenvalue of $J$. Similarly, we can show $-\\xi$ is an eigenvalue of $J$. Therefore, $\\sigma(J)$ contains a pair of purely imaginary eigenvalues.\n\\end{proof}\n\n\n\n\n\n\n\n\t\n\n\t\n\t\n\n\n\n\n\\section{Proof of \\cref{thrm: original model vs referenced model}}\n\\label{proof of thrm: original model vs referenced model}\n\n\nLet us first prove the following useful lemma.\n\\begin{lemma} \\label{lemma: pencil for referenced systems}\n    Let $(\\delta^0,\\omega^0)$ be an equilibrium point of the swing equation \\eqref{eq: swing equations} and $\\Psi(\\delta^0,\\omega^0)$ be the corresponding equilibrium point of the referenced model \\eqref{eq: Swing Equation Polar referenced}. Let $J^r$ denote the Jacobian of the referenced model at this equilibrium point.\n\tFor any $\\lambda \\ne 0$, $\\lambda$ is an eigenvalue of $J^r$ if and only if the quadratic matrix pencil $P(\\lambda):= \\lambda^2 M  + \\lambda D + \\nabla P_e (\\delta^0)$ is singular.\n\\end{lemma}\n\\begin{proof}\n The referenced model \\eqref{eq: Swing Equation Polar referenced} can be written as\n\\begin{align}\\label{eq:swing reduced}\n\\begin{bmatrix}\n\\dot {\\psi} \\\\\n\\dot \\omega \n\\end{bmatrix}\n=\n\\begin{bmatrix}\nT_1 \\omega \\\\\n- D M^{-1} \\omega + M^{-1} (P_m -P_e^r(\\psi))\n\\end{bmatrix}.\n\\end{align}\nNote that the Jacobian of the referenced flow function $\\nabla P^r_e(\\psi)$ is an $n \\times (n-1)$ matrix and we have $\\nabla P^r_e(\\psi^0) = \\nabla P_e (\\delta^0) T_2$, where\n\\begin{align}\nT_2:=\\begin{bmatrix}\nI_{n-1} \\\\ 0_{1\\times(n-1)}\n\\end{bmatrix} \\in \\mathbb{R}^{n\\times(n-1)}.\n\\end{align}\nAccordingly, the Jacobin of the referenced model \\eqref{eq: Swing Equation Polar referenced} is\n\\begin{align}\nJ^r=\\begin{bmatrix}\n0_{(n-1)\\times (n-1)} & T_1 \\\\\n-M^{-1} \\nabla P_e (\\delta^0) T_2 & -DM^{-1}\n\\end{bmatrix}.\n\\end{align}\n\t\\textit{Necessity:} Let $\\lambda$ be a nonzero eigenvalue of $J^r$ and $ (v_1 , v_2 ) $ be the corresponding eigenvector with $v_1\\in\\mathbb{C}^{n-1}$ and $v_2\\in\\mathbb{C}^{n}$. Then\t\n\t\\begin{align} \\label{eq: J cha eq referenced}\n\t\\begin{bmatrix}\n\t0_{(n-1)\\times (n-1)} & T_1 \\\\\n\t-M^{-1}\\nabla P_e (\\delta^0) T_2 & -DM^{-1}\n\t\\end{bmatrix}   \\begin{bmatrix} v_1 \\\\v_2  \\end{bmatrix}  = \\lambda   \\begin{bmatrix} v_1 \\\\v_2  \\end{bmatrix},\n\t\\end{align}\n\twhich implies that $ T_1 v_2 = \\lambda v_1$. Since $\\lambda \\ne 0$, we can substitute $ \\lambda^{-1} T_1 v_2 = v_1$ in the second equation to obtain \n\t\\begin{align} \n\t\\left(\\lambda^2 M + \\lambda D + \\nabla P_e (\\delta^0) T_2 T_1 \\right) v_2 = 0. \\label{eq: quadratic matrix pencil referenced}\n\t\\end{align}  \n\tSince the eigenvector $(v_1,v_2)$ is nonzero, we have $v_2 \\not = 0$ (otherwise $ v_1=\\lambda^{-1} T_1 0 = 0   \\implies (v_1,v_2) = 0 $), Eq. \\eqref{eq: quadratic matrix pencil referenced} implies that the matrix pencil $P(\\lambda)= \\lambda^2 M + \\lambda D + \\nabla P_e (\\delta^0) T_2 T_1$ is singular. Next, we show that $\\nabla P_e (\\delta^0) T_2 T_1 = \\nabla P_e (\\delta)$. Since $\\nabla P_e (\\delta^0)$ has zero row sum, it can be written as\n\t\\begin{align*}\n\t\\nabla P_e (\\delta^0)=\\begin{bmatrix}\n\tA&b\\\\c^\\top&d\n\t\\end{bmatrix}, \\text{ where } A\\mathbf{1}=-b, c^\\top \\mathbf{1}=-d.\n\t\\end{align*}\n\tTherefore, we have\n\t\\begin{align*}\n\t\\nabla P_e (\\delta^0) T_2 T_1 =  \\begin{bmatrix}\n\tA&b\\\\c^\\top&d\n\t\\end{bmatrix} \n\t\\begin{bmatrix}\n\tI_{n-1} \\\\ 0\n\t\\end{bmatrix}\n\t\\begin{bmatrix}\n\tI_{n-1}& -\\mathbf{1}\n\t\\end{bmatrix}\n\t= \n\t\\begin{bmatrix}\n\tA&-A\\mathbf{1}\\\\c^\\top&-c^\\top\\mathbf{1}\n\t\\end{bmatrix}\n\t=\n\t\\begin{bmatrix}\n\tA&b\\\\c^\\top&d\n\t\\end{bmatrix}. \n\t\\end{align*}\n\t\n\t\\textit{Sufficiency:}\t\n\tSuppose there exists $\\lambda \\in \\mathbb{C}, \\lambda \\ne 0$ such that  $P(\\lambda)= \\lambda^2 M + \\lambda D + \\nabla P_e (\\delta^0)$ is singular. Choose a nonzero $v_2 \\in \\ker (P(\\lambda))$ and let $v_1:=\\lambda^{-1} T_1 v_2$. \n\tAccordingly, the characteristic equation \\eqref{eq: J cha eq referenced} holds, and consequently, $\\lambda$ is a nonzero eigenvalue of $J^r$.\n\\end{proof}\n\nNow, we are ready to prove \\cref{thrm: original model vs referenced model}.\n\n\\begin{proof}\nAny equilibrium point $(\\delta^0,\\omega^0)$ of the swing equation model \\eqref{eq: swing equations} is contained in the set\n\\begin{align*}\n    \\mathcal{E}:=\\left \\{ (\\delta,\\omega)\\in \\mathbb{R}^{2n} : \\omega = 0, P_{m_j} = P_{e_j}(\\delta), \\quad \\forall j \\in \\{1,...,n\\} \\right \\}.\n\\end{align*}\nLet $(\\psi^0,\\omega^0)=\\Psi(\\delta^0,\\omega^0)$, and note that $\\omega^0=0$. From \\eqref{eq: flow function} and \\eqref{eq: flow function referenced compact}, we observe that $P_{e_j}(\\delta^0) = P_{e_j}^r(\\psi^0), \\forall j \\in \\{1,...,n\\}$ where $\\psi^0_n :=0$. Therefore, $(\\psi^0,\\omega^0)$ is an equilibrium point of the the referenced model \\eqref{eq: Swing Equation Polar referenced}. \n\\\\\nTo prove the second part, recall that $\\lambda$ is an eigenvalue of the Jacobian of \\eqref{eq: swing equations} at $(\\delta^0,\\omega^0)$ if and only if $\\det( \\nabla P_e (\\delta^0)  +\\lambda D + \\lambda^2 M)=0$. \nAccording to \\cref{lemma: pencil for referenced systems}, the nonzero eigenvalues $J$ and $J^r$ are the same. Moreover, the referenced model \\eqref{eq: Swing Equation Polar referenced} has one dimension less than the  swing equation model \\eqref{eq: swing equations}. This completes the proof.\n\\end{proof}\n\n\n\n\\section{Proof of \\Cref{prop:hyperbolicity n2 n3}}\n\\label{proof of prop:hyperbolicity n2 n3}\n\nWe prove the following lemmas first:\n\\begin{lemma} \\label{lemma: complex representation}\nLet $A,B\\in\\mathbb{R}^{n\\times n}$ and define \n$$C:=  \\small{ \\left[\\begin{array}{cc}A&-B\\\\B&A\\end{array}\\right] }.$$\nThen $\\mathop{\\bf rank}(C)=2\\mathop{\\bf rank}(A+\\frak{i} B)$ which is an even number.\n\\end{lemma}\n\\begin{proof}\nLet $V:= \\frac{1}{\\sqrt{2}} \\left[\\begin{array}{cc}I_n&\\frak{i} I_n\\\\\\frak{i} I_n&I_n\\end{array}\\right] $ and observe that $V^{-1}=\\bar{V}=V^*$, where $\\bar{V}$ stands for the entrywise conjugate and $V^*$ denotes the conjugate transpose of $V$. We have \n\\begin{align*}\nV^{-1}CV=\\left[\\begin{array}{cc}A-\\frak{i} B&0\\\\0&A+\\frak{i} B\\end{array}\\right] = (A-\\frak{i} B) \\oplus (A+\\frak{i} B).\n\\end{align*}\nSince rank is a similarity invariant, we have $\\mathop{\\bf rank}(C)=\\mathop{\\bf rank}((A-\\frak{i} B) \\oplus (A+\\frak{i} B))=2\\mathop{\\bf rank}(A+\\frak{i} B)$.\n\\end{proof}\n\\begin{lemma}  \\label{lemma: matrix form of pencil sinularity}\n$\\lambda = \\frak{i}  \\beta$ is an eigenvalue of $J$ if and only if\nthe matrix\n\\begin{align*}\n\\mathcal{M}(\\beta):= \\begin{bmatrix}\nL-\\beta^2 M & -\\beta D\\\\ \\beta  D & L-\\beta^2 M  \n\\end{bmatrix}\n\\end{align*}\nis singular. Here $L=\\nabla P_e (\\delta^0)$.\n\\end{lemma}\n\\begin{proof}\nAccording to \\Cref{lemma: relation between ev J and ev J11}, $\\frak{i}  \\beta \\in \\sigma(J)$ if and only if $\\exists x\\in\\mathbb{C}^n,x\\ne0$ such that\n    \\begin{align} \\label{eq: pencil matrix singularity theorem proof}\n        \\left( L - \\beta^2 M + \\frak{i}  \\beta D  \\right) x = 0.\n    \\end{align}\nDefine $A:=L - \\beta^2 M$, $B:=\\beta D$, and let $x=u+\\frak{i} v$. Rewrite \\eqref{eq: pencil matrix singularity theorem proof} as $(A+\\frak{i}  B)(u+\\frak{i} v) = (Au - Bv) + \\frak{i} (Av + Bu) = 0$,\nwhich is equivalent to \n\\begin{align*}\n\\begin{bmatrix}\nA & -B \\\\ B & A \n\\end{bmatrix} \\begin{bmatrix}\nu \\\\ v\n\\end{bmatrix} = 0.\n\\end{align*}\n\\end{proof}\nNow, we are ready to prove \\Cref{prop:hyperbolicity n2 n3}:\nAccording to \\Cref{lemma: matrix form of pencil sinularity}, $\\frak{i} \\beta\\in\\sigma(J)$ for some nonzero real $\\beta$ if and only if the matrix\n\\begin{align*} \n\\mathcal{M}(\\beta) := \\begin{bmatrix}\nL - \\beta^2 M & -\\beta D\\\\\n\\beta D & L - \\beta^2 M\n\\end{bmatrix} \n\\end{align*}\nis singular. Recall that $L:=\\nabla P_e (\\delta^0)$. In the sequel, we will show under the assumptions of  \\Cref{prop:hyperbolicity n2 n3}, $\\mathcal{M}(\\beta)$ is always nonsingular. First, we prove the theorem for $n=2$. In this case, \n$$\nL= \\left[ \\begin{array}{cc}a_{12}&-a_{12}\\\\-a_{21}&a_{21}\\end{array}\\right], a_{12}>0, a_{21}>0.\n$$\nAccording to \\Cref{lemma: complex representation}, we have $\\mathop{\\bf rank}(\\mathcal{M}(\\beta))=2\\mathop{\\bf rank}(L-\\beta^2M- \\frak{i}  \\beta D)$, and $L-\\beta^2M- \\frak{i}  \\beta D$ is full rank because \n\\begin{align*}\nL-\\beta^2M- \\frak{i}  \\beta D= \\left[ \\begin{array}{cc}a_{12}-\\beta^2 m_1&-a_{12}\\\\-a_{21}&a_{21}-\\beta^2 m_2- \\frak{i}  \\beta d_2\\end{array}\\right],\n\\end{align*}\nand $\\det(L-\\beta^2M- \\frak{i}  \\beta D)=(a_{12}-\\beta^2 m_1)(a_{21}-\\beta^2 m_2- \\frak{i}  \\beta d_2)-a_{12}a_{21}$. It is easy to see that the real part and imaginary parts of the determinant cannot be zero at the same time. Therefore, $ \\mathcal{M}(\\beta)$ is also nonsingular and a partially damped $2$-generator system cannot have any pure imaginary eigenvalues. \n\nNow, we prove the theorem for $n=3$. Let $A\\in\\mathbb{R}^{2n\\times 2n}$. For index sets $\\mathcal{I}_1\\subseteq\\{1,\\cdots,2n\\}$ and $\\mathcal{I}_2\\subseteq\\{1,\\cdots,2n\\}$, we denote by $A[\\mathcal{I}_1,\\mathcal{I}_2]$ the (sub)matrix of entries that lie in the rows of $A$ indexed by $\\mathcal{I}_1$ and the columns indexed by $\\mathcal{I}_2$. For a $3$-generator system, the matrix $L$ can be written as\n\\begin{align*}\n    L = \\begin{bmatrix}\n    a_{12}+a_{13} & -a_{12} & -a_{13}\\\\\n    -a_{21} &  a_{21}+a_{23}  &-a_{23}\\\\\n    -a_{31} & - a_{32} & a_{31} + a_{32}\n    \\end{bmatrix}\n\\end{align*}\nwhere $a_{jk}\\ge0, \\forall j,k\\in\\{1,2,3\\}, j\\ne k$ and $a_{jk}=0 \\iff a_{kj}=0$. Moreover, $M=\\mathbf{diag}(m_1,m_2,m_3)$ and $D=\\mathbf{diag}(0,d_2,d_3)$.\nWe complete the proof in three steps:\n\\begin{itemize}\n\\item Step $1$: We show that the first four columns of $\\mathcal{M}(\\beta)$ are linearly independent, i.e., $\\mathop{\\bf rank}(\\mathcal{M}(\\beta))\\ge4$.\\\\\nTo do so, we show that the equation \n\\begin{align*}\n    \\mathcal{M}(\\beta)\\left [ \\{1,...,6\\},\\{1,2,3,4\\}\\right ] \n    \\begin{bmatrix}\nx_1 \\\\ x_2 \\\\x_3 \\\\ x_4\n\\end{bmatrix} = 0\n\\end{align*}\nhas only the trivial solution.\n\n\\begin{enumerate}[(i)]\n\t\\item If $a_{12}+a_{13}-\\beta^2 m_1\\ne0$, then $x_4=0$. Moreover, we have $\\beta d_2 x_2 = 0$ and $\\beta d_3 x_3 = 0$ which imply $x_2=x_3=0$ because $\\beta, d_2$, and $d_3$ are nonzero scalars. Finally, the connectivity assumption requires that at least one of the two entries $a_{21}$ and $a_{31}$ are nonzero, implying that $x_1=0$.\n\t\n\t\\item If $a_{12}+a_{13}- \\beta^2 m_1=0$, then by expanding the fifth and sixth rows we get\n\t\\begin{align*}\n\t    & \\beta d_2x_2 -a_{21}x_4=0  \\implies x_2=\\frac{a_{21}}{\\beta d_2}x_4,\\\\\n\t\t& \\beta d_3x_3 -a_{31}x_4=0, \\implies x_3= \\frac{a_{31}}{\\beta d_3}x_4.\n\t\\end{align*}\n\tExpanding the first row and substituting $x_2$ and $x_3$ from above gives\n\t\\begin{align*}\n\t& -a_{12}x_2-a_{13}x_3=0 \\implies -\\frac{a_{12} a_{21}}{\\beta d_2}x_4 -\\frac{a_{13}a_{31}}{\\beta d_3}x_4 =0.\n\t\\end{align*}\n\tThe connectivity assumption (and the fact that $a_{kj}\\ge0, \\forall k\\ne j$ and $a_{kj}=0\\iff a_{jk}=0$) leads to $x_4=0$. This implies $x_2=x_3=0$ and further $x_1=0$ due to the connectivity assumption.\n\t\n\\end{enumerate}\n\n\\item Step $2$: We prove that the first five columns of $\\mathcal{M}(\\beta)$ are linearly independent, i.e., $\\mathop{\\bf rank}(\\mathcal{M}(\\beta))\\ge5$.\\\\\nTo do so, we show that the equation \n\n\n\\begin{align*}\n    \\mathcal{M}(\\beta) \\left [\\{1,...,6\\},\\{1,2,3,4\\} \\right ] \n    \\begin{bmatrix}\nx_1 \\\\ x_2 \\\\x_3 \\\\ x_4\n\\end{bmatrix} = \\begin{bmatrix}\n0 \\\\ -\\beta d_2 \\\\0 \\\\ -a_{12} \\\\ a_{21}+a_{23}- \\beta^2 m_2 \\\\ -a_{32}\n\\end{bmatrix}\n\\end{align*}\nhas no solution, i.e., the fifth column is not in the span of the first four columns. Based on the equation in the fourth row we consider the following situations:\n\\begin{enumerate}[(i)]\n\t\\item If $a_{12}+a_{13}- \\beta^2 m_1=0$ and $a_{12}\\ne0$, then there exists no solution.\n\t\n\t\\item If $a_{12}+a_{13}- \\beta^2 m_1=0$ and $a_{12}=0$, then $a_{13}=\\beta^2 m_1$. Expanding the first row yields $-a_{13}x_3=0\\implies x_3=0$. Expanding the second row provides $(a_{23}-\\beta^2 m_2)x_2=-\\beta d_2 \\implies x_2=- \\frac{\\beta d_2}{(a_{23}- \\beta^2 m_2)}$. Note that we assume $(a_{23}-\\beta^2 m_2)\\ne0$, since otherwise the system has no solution.\n\tFinally, we expand the fifth row and substitute $x_2$ into it:\n\t\\begin{align*}\n\t \\beta d_2 x_2 = a_{23}-\\beta^2 m_2  \n\t&  \\implies -\\frac{(\\beta d_2)^2}{(a_{23}-\\beta^2 m_2)} = a_{23}-\\beta^2 m_2 \\\\ \n\t&  \\implies -(\\beta d_2)^2 = (a_{23}-\\beta^2 m_2)^2\n\t\\end{align*}\n\twhich is a contradiction.\n\t\n\t\\item If $a_{12}+a_{13}-\\beta^2 m_1\\ne0$ and $a_{12}=0$, then $x_4=0$. By expanding the fifth and sixth rows we get\n\t\\begin{align*}\n\t& \\beta d_2x_2 = a_{23}- \\beta^2 m_2  \\implies x_2=\\frac{a_{23}-\\beta^2 m_2}{\\beta d_2},\\\\\n\t& \\beta d_3x_3 = -a_{32} , \\implies x_3= -\\frac{a_{32}}{\\beta d_3}.\n\t\\end{align*}\n\tExpanding the second row and substituting $x_2$ and $x_3$ from above gives\n\t\\begin{align*}\n\t (a_{23}-\\beta^2 m_2)x_2-a_{23}x_3 = -\\beta d_2 \n\t \\implies  \\frac{(a_{23}-\\beta^2 m_2)^2}{\\beta d_2} + \\frac{a_{23}a_{32}}{\\beta d_3} = -\\beta d_2\n\t\\end{align*}\n   which is a contradiction.\n   \n   \\item If $a_{12}+a_{13}-\\beta^2 m_1\\ne0$ and $a_{12}\\ne0$, then $x_4=\\frac{-a_{12}}{a_{12}+a_{13}- \\beta^2 m_1}$. By expanding the fifth and sixth rows and substituting $x_4$ we get\n   \\begin{align*}\n   & \\beta d_2x_2   + \\frac{a_{12}a_{21}}{a_{12}+a_{13}- \\beta^2 m_1} = a_{21}+a_{23}- \\beta^2 m_2,\\\\\n   & \\beta d_3x_3   + \\frac{a_{12}a_{31}}{a_{12}+a_{13}-\\beta^2 m_1} = -a_{32}.\n   \\end{align*}\n   Now we expand the first row to get $  x_1=\\frac{a_{12}x_2+a_{13}x_3}{a_{12}+a_{13}-\\beta^2 m_1}$. Finally, we expand the second row and substitute for $x_1, x_2$, and $x_3$:\n   \\begin{align*}\n    -a_{21}\\frac{a_{12}x_2+a_{13}x_3}{a_{12}+a_{13}-\\beta^2 m_1} + (a_{21}+a_{23}-\\beta^2 m_2)x_2 -a_{23}x_3=-\\beta d_2, \n   \\end{align*}\n   which implies\n   \\begin{align*}\n   & (  (a_{21}+a_{23}- \\beta^2 m_2) -\\frac{a_{12}a_{21}}{a_{12}+a_{13}-\\beta^2 m_1} ) x_2  \\\\\n   & - (a_{23}  +\\frac{a_{13}a_{21}}{a_{12}+a_{13}-\\beta^2 m_1}) x_3 =-\\beta d_2, \n   \\end{align*}\n   or equivalently\n   \\begin{align*}\n   &  \\frac{1}{\\beta d_2} (  (a_{21}+a_{23}- \\beta^2 m_2) -\\frac{a_{12}a_{21}}{a_{12}+a_{13}- \\beta^2 m_1} )^2 \\\\\n   & + \\frac{1}{\\beta d_3}  (a_{23}  +\\frac{a_{13}a_{21}}{a_{12}+a_{13}- \\beta^2 m_1})^2 =-\\beta d_2.\n   \\end{align*}\n   which is a contradiction.\n\\end{enumerate}\n\n\\item Step $3$: $\\mathop{\\bf rank}(\\mathcal{M}(\\beta))$ is an even number.\\\\\nFinally, \\Cref{lemma: complex representation} precludes the rank of $\\mathcal{M}(\\beta)$ from being equal to $5$. Therefore, $\\mathop{\\bf rank}(\\mathcal{M}(\\beta))=6$, i.e., $\\mathcal{M}(\\beta)$ is always nonsingular. This completes the proof.\n\\end{itemize}\n\n\n\n\n\n\n\n\\section{Introduction}\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","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\\section*{Acknowledgment}\n\n\n\n\n\\bibliographystyle{bibFiles\/IEEEbib}\n\n\\section*{Acknowledgment}\n\n\n\n\n\\bibliographystyle{unsrtnat}\n\n\\section{Introduction} \\label{sec:intro}\n\nComputing a distributed optimal controller in which subcontrollers have access to subsets of global system information is in general a computationally intractable problem.  Indeed, even when restricted to quadratic costs, Gaussian noise, and linear dynamics, the resulting optimal controller can be nonlinear and difficult to compute \\citep{witsenhausen1968counterexample}.  Nevertheless, significant progress has been made in distributed optimal controller synthesis over the past two decades by identifying structural assumptions on the underlying dynamics and information exchange structure such that the resulting distributed controller synthesis problem is convex. \n\nOne such structural assumption that has been shown to lead to tractable distributed optimal control problems is spatial invariance \\citep{bamieh2002distributed} (and other closely related notions of symmetry \\citep{massioni2009distributed}). Such systems are invariant under subsystem permutations, and have been shown to have optimal centralized controllers that are approximately distributed.  In particular, this allows for distributed controllers that enjoy stability and near-optimality guarantees to be computed by appropriately truncating the centralized controller.\n\n\\emph{Contributions:} In this paper, inspired by results from graph signal processing, we introduce the notion of Graph Symmetric Systems (GSSs), which are systems that are symmetric with respect to an underlying graph topology (formalized in \\S\\ref{sec:LQR}-A).  We show that for such systems, the resulting Linear Quadratic (LQ) centralized optimal controller admits an efficient message passing implementation in the form of a novel class of graph filters defined by transfer function filter taps.  We subsequently propose and analyze two complementary approaches to computing near-optimal distributed controllers by truncating the centralized optimal controller subject to stability constraints.  By leveraging tools from robust System Level Synthesis (SLS) \\citep{matni2017scalable, anderson_system_2019}, we show that these truncation algorithms can be solved via convex optimization, and that the resulting distributed controllers enjoy sub-optimality guarantees relative to the centralized optimal controller.  These results constitute an important first step towards bridging the complementary, but traditionally disparate, fields of distributed optimal control and graph signal processing.\n\n\\emph{Related work:} An alternative structural assumption for tractable distributed optimal control of linear systems can be specified in terms of the sparsity and delay patterns of the control system.  In particular, it is possible to characterize conditions on the sparsity and delay patterns of the information exchanged between subcontrollers relative to the propagation of signals through sparse and delayed distributed plants such that distributed optimal control is tractable.  The seminal paper \\cite{rotkowitz2005characterization} introduced the notion of quadratic invariance,\\footnote{We note that spatially invariant systems, as defined in~\\cite{bamieh2002distributed}, also satisfy quadratic invariance.  We show in Appendix B that graph symmetric systems and controllers lead to optimal control problems satisfying quadratic invariance.} which built upon and generalized funnel causality \\citep{bamieh2005convex}, showed that so long as subcontrollers could communicate as quickly as control signals propagated through the plant, then the resulting distributed optimal control problem could be solved via convex optimization.  This convex parameterization of sparse and delayed controllers has since been further generalized in the System Level Synthesis (SLS) \\citep{anderson_system_2019} and Input-Output Parameterization (IOP) \\citep{furieri2019input} frameworks, which allow for even richer classes of sparsity and delay patterns to be imposed on distributed controllers.\n\nA related class of distributed controllers are those based on Graph Neural Networks (GNNs).  GNNs can be viewed as graph filters followed by pointwise nonlinear activation functions \\citep{Ruiz2021-GNN}, and among other favorable properties, enjoy stability to graph perturbations \\citep{Gama2020-Stability}.  While recent use of GNNs for distributed control has shown promise \\citep{Gama2022-DistributedLQR, Gama2022-ControlGNN, yang2021communication}, such results currently lack strong guarantees of stability.  We believe the results in this paper are a first step towards addressing this gap in the literature, by explicitly connecting graph filters and distributed optimal controllers. The direct relationship between graph filters and GNNs suggests that understanding the former will give insight in the effects of the latter.\n\n\\textit{Notation}: We use upper- and lower-case letters such as $A$ and $x$ to denote matrices and vectors respectively, although lower-case letters might also be used for scalars or functions (the distinction will be apparent from the context). For both upper- and lower-case letters, we use \\textit{boldface} such as $\\mtLambda$ and $\\boldsymbol\\phi$  to denote transfer matrices or vector\/scalar transfer functions.\n\\section{The Linear Quadratic Regulator Problem for Graph Symmetric Systems} \\label{sec:LQR}\nConsider a discrete-time linear time-invariant (LTI) system composed of $N$ interconnected scalar subsystems, each with state $x_i(t) \\in \\fdR$, control input $u_i(t) \\in \\fdR$ and which evolves under the dynamics\n\\begin{equation}\\label{eq:subsys-dynamics}\n    x_i(t+1) = \\sum_{j=1}^{N}A_{ij}x_j(t) + \\sum_{j=1}^{N}B_{ij}u_j(t) + w_i(t),\n\\end{equation}\nfor suitable matrices $A_{ij}, B_{ij}$ describing the interaction between subsystems. Here $w_i(t)$ is an i.i.d. zero-mean noise. We can compactly express the dynamics of the full system in terms of the joint states $x(t) = [x_1(t), \\ldots, x_N(t)]^\\top$ and joint control actions $u(t) = [u_1(t), \\ldots,u_N(t)]^\\top$ as\n\\begin{equation} \\label{eq:linearDynamics}\n    x(t+1) = A x(t) + B u(t) + w(t),\n\\end{equation}\nwhere $(A,B)$ are defined such that the global dynamics \\eqref{eq:linearDynamics} are consistent with the subsystem dynamics \\eqref{eq:subsys-dynamics}.\n\nOur goal is to find a (potentially time-varying) state-feedback controller $K_t$ that minimizes the cost\n\\begin{equation} \\label{eq:quadraticCost}\n    J \\big( \\{x(t)\\}, \\{u(t)\\} \\big) := \\lim_{T \\to \\infty} \\frac{1}{T}\\sum_{t=0}^{T-1} \\xp_{w} \\big[ x(t)^{\\Tr} Q x(t) + u(t)^{\\Tr} R u(t) \\big]\n\\end{equation}\nwhere $u(t) = K_t(x(t))$ and $Q \\succeq 0, R \\succ 0$ are known symmetric $N \\times N$ matrices. The Linear Quadratic Regulator (LQR) problem is then given by\n\\begin{equation} \\label{eq:LQG}\n    \\min_{\\mtK} J\\big( \\{x(t)\\},\\{u(t)\\} \\big) \\quad \\text{s. t. } u(t) = K_t\\big(x(t)\\big).\n\\end{equation}\nIn the centralized setting where each subsystem has access to the global state, it is well-known that the controller that solves \\eqref{eq:LQG} is a linear static controller $u(t) = K^{\\opt} x(t)$ where $K^{\\opt} = -(R+ B^{\\top} P B)^{-1}B^{\\top} P A$ and $P$ is the unique solution to the discrete-time algebraic Ricatti equation:\n\\begin{equation} \\label{eq:Ricatti}\n    P = A^{\\top} P A - A^{\\top} P B \\big( R+ B^{\\top} P B \\big)^{-1} B^{\\top} P A + Q.\n\\end{equation}\nIn this work, we consider a distributed variant of \\eqref{eq:LQG} where each subsystem can only exchange information with a small subset of subsystems. Specifically, this communication constraint is encoded as a graph $\\graph = \\{\\stV, \\stE\\}$, where $\\stV = \\{\\lmv_{1},\\ldots,\\lmv_{N}\\}$ is the set of $N$ components (nodes) and where $\\stE \\subseteq \\stV \\times \\stV$ is the set of the corresponding interconnections (edges). It is assumed that the graph is undirected, i.e. $(\\lmv_{i},\\lmv_{j}) \\in \\stE$ if and only if $(\\lmv_{j},\\lmv_{i}) \\in \\stE$. As described in the introduction, general information exchange constraints can lead to non-convex optimal control problems \\citep{witsenhausen1968counterexample}. However, as we show later, under suitable graph symmetry assumptions on the dynamics matrices $A, B$ and the cost matrices $Q, R$, the optimal centralized controller admits a distributed message passing implementation allowing for a principled tradeoff between communication complexity and controller performance.\n\nIn the rest of this section, we borrow ideas from graph signal processing \\citep{Sandryhaila2013-DSPG} and introduce the notion of \\textit{graph symmetric systems}. First, we introduce a convenient way to define operations that respect the underlying communication graph structure via the graph matrix description (GMD) $S \\in \\fdR^{N \\times N}$.\nThe matrix $S$ is such that the $(i,j)^{\\text{th}}$ entry is zero whenever there is no connection between components $\\lmv_{i}$ and $\\lmv_{j}$, i.e. $[S]_{ij}=0$ if $i \\neq j $ and $(\\lmv_{i},\\lmv_{j}) \\notin \\stE$.\nNote that, since the graph is undirected, the matrix $S$ is symmetric.\nTherefore, it has an eigedecomposition in terms of an orthonormal basis of eigenvectors $S = V \\Lambda_{S} V^{\\top}$ where $\\Lambda_{S} \\in \\fdR^{N \\times N}$ is a diagonal matrix with elements $\\lambda_{S,i} \\in \\fdR$ such that $S v_{i} = \\lambda_{S,i} v_{i}$ for $v_{i} \\in \\fdR^{N}$ being the $i^{\\text{th}}$ column of $V$. We now introduce the notion of a graph symmetric system.\n\n\\begin{definition}[Graph Symmetric System]\\label{def:distributedLinear}\nGiven a GMD $S=V \\Lambda_{S} V^{\\top}$ for a graph $\\mathcal G$, a linear system \\eqref{eq:linearDynamics} is \\textit{graph symmetric} with respect to $\\mathcal G$ if the dynamics matrices $A, B$ are simultaneously diagonalized by $V$, i.e.,\n   \n    \\begin{equation} \\label{eq:distributedLinear}\n    \\begin{aligned}\n        A = V \\Lambda_{A} V^{\\top} \\quad &, \\quad B = V \\Lambda_{B} V^{\\top},\n    \\end{aligned}\n    \\end{equation}\nwhere $\\Lambda_{A}, \\Lambda_{B}$ are diagonal.\n\\end{definition}\n\nNote that Definition \\ref{def:distributedLinear} does not require the dynamics to be sparse. In fact, matrices $A$ and $B$ of the form in Definition \\ref{def:distributedLinear} can be arbitrarily dense, i.e., the evolution of a subsystem state $x_i(t+1)$ can depend on subsystems that $i$ cannot directly communicate with \\citep{Gama2019-LinearControl}. This is distinct from sparsity\/delay constraints used in \\cite{rotkowitz2005characterization, anderson_system_2019, furieri2019input}, and encodes a different notion of symmetry than that exploited in the distributed control of spatially invariant systems \\citep{bamieh2002distributed}.\n\nBy well-known results in graph signal processing \\citep{Sandryhaila2013-DSPG}, simultaneous diagonalizability of the system matrices $(A,B)$ and the GMD $S$ implies\\footnote{Under the assumption that $S$ corresponds to a finite graph and has all distinct eigenvalues. On a high level, the result follows directly from the Cayley-Hamilton theorem.} that they can be written as matrix polynomials of $S$ of degrees at most $N-1$,\n\\begin{equation} \\label{eq:polynomialOfS}\n    A = \\sum_{k=0}^{N-1}h_{A,k}S^k \\quad , \\quad B = \\sum_{k=0}^{N-1}h_{B,k}S^k.\n\\end{equation}\nMatrices that can be expressed in this matrix polynomial forms are called \\textit{graph filters} \\citep{Segarra2017-GraphFilterDesign} and the coefficients $h_{A,k}, h_{B,k}$ are referred to as the filter weights or \\textit{filter taps}.\n\nWe now give a message-passing interpretation of graph symmetric systems. First, it can be seen from the sparsity pattern of the GMD $S$ that the output of $Sx(t)$ can be computed entirely as a linear combination of the states in nodes $1$ hop away in $\\mathcal G$. To see this, consider the operation $S x(t)$ whose $i^{\\text{th}}$ entry yields\n\\begin{equation} \\label{eq:graphShift}\n    [S x(t)]_{i} = \\sum_{j:(\\lmv_{j},\\lmv_{i}) \\in \\stE} [S]_{ij} [x(t)]_{j}.\n\\end{equation}\nMore generally, when considering polynomials, it is observed that $S^{k}x(t)$ is equivalent to exchanging $k$ times information with one-hop neighbors.\nTherefore, if the system matrix $A$ and the control matrix $B$ are polynomials of $S$, then the evolution of the system can be computed entirely by means of exchanges with neighboring nodes. Hence, the system dynamics can be viewed as implementing distributed message passing \\citep{Ruiz2021-GNN}.\nExamples of such linear, distributed systems, include both discrete-time and continuous-time diffusions, solutions to the heat equation, among many others, see \\cite{Gama2019-LinearControl} and references therein.\n\nFor the rest of the paper, we also assume that the cost matrices for the LQR problem \\eqref{eq:LQG} can also be simultaneously diagonalized with the dynamics matrices. Formally, we make the following assumption.\n\\begin{assumption}\\label{assump:diagonalizableCost}\nThe system \\eqref{eq:linearDynamics} defines a graph symmetric system with respect to a fixed GMD $S$, and the cost matrices $(Q,R)$ defining the LQR problem \\eqref{eq:LQG} are graph symmetric with respect to $S$, i.e., they are simultaneously diagonalized by the ortho-bases $V$ satisfying $S=V\\Lambda_S V^\\top$.  In particular\n\\[\nQ = V \\Lambda_{Q} V^{\\top} \\quad, \\quad R = V \\Lambda_{R} V^{\\top},\n\\]\nwhere $\\Lambda_Q, \\Lambda_R$ are symmetric.\n\\end{assumption}\n\\section{Optimal Distributed Linear Controller via System Level Synthesis} \\label{sec:SLS}\n\nSLS provides a convex parameterization of achievable closed-loop system responses \\citep{wang_separable_2018,anderson_system_2019}, which can be leveraged to show that the optimal controller for graph symmetric systems under Assumption \\ref{assump:diagonalizableCost} can be written as a novel class of graph filters defined by transfer function valued filter taps.\n\n\\subsection{Background: System-Level Synthesis}\n\nAs noted in \\S4 of \\cite{anderson_system_2019}, we can compactly write the system dynamics \\eqref{eq:linearDynamics} in the frequency domain as\n\\[(zI - A)\\mathbf x = B \\mathbf u + \\mathbf w,\\]\nwhere $\\mathbf x = \\sum_{t=0}^{\\infty} z^{-t}x(t)$ is the signal $x(t)$ in the $z$-domain, and idem for $\\mathbf u$ and $\\mathbf w$. For a (dynamic) linear state-feedback controller $\\mathbf{u} = \\mathbf K \\mathbf x$, it follows immediately that\n\\begin{equation}\\label{eq:system-response}\n    \\begin{aligned}\n    \\vcx &= (zI - A - B\\mtK)^{-1} \\vcw =: \\mathbf\\Phi_{x}(z) \\vcw,\\\\\n    \\vcu &= \\mtK (zI - A - B\\mtK)^{-1} \\vcw =: \\mathbf\\Phi_u(z) \\vcw,\n    \\end{aligned}\n\\end{equation}\nwhere $\\resp_x(z) \\in \\fdC^{N \\times N}$ and $\\resp_u(z) \\in \\fdC^{N \\times N}$ are system responses that map the disturbance $\\vcw$ to state $\\vcx$ and control input $\\vcu$, respectively. The following SLS theorem states that all achievable responses lie in an affine subspace of strictly proper stable rational transfer functions $\\frac{1}{z} \\mathcal R \\mathcal H_\\infty$.\n\n\\begin{theorem} \\emph{\\cite[Thm. 4.1]{anderson_system_2019}}\nFor the LTI system evolving under the dynamics \\eqref{eq:linearDynamics} and control policy $\\vcu = \\mtK \\vcx$, the following statements are true:\n\\begin{enumerate}\n    \\item The affine subspace defined by\n    \\begin{equation}\\label{eq:achievability-constraint}\n        \\begin{bmatrix} zI-A & -B \\end{bmatrix} \\begin{bmatrix} \\mathbf\\Phi_x \\\\ \\mathbf\\Phi_u    \\end{bmatrix} = I, \\quad \\mathbf\\Phi_x, \\mathbf\\Phi_u \\in \\frac{1}{z}\\mathcal R \\mathcal H_\\infty\n    \\end{equation}\n    parameterizes all system responses from $\\vcw$ to $(\\vcx, \\vcu)$ as defined in \\eqref{eq:system-response}, achievable by an internally stabilizing state feedback controller $\\mtK$.\n    \\item For any transfer matrices $\\mathbf\\Phi_x, \\mathbf\\Phi_u$ satisfiying \\eqref{eq:achievability-constraint}, the controller $\\mtK = \\mathbf\\Phi_u \\mathbf\\Phi_x^{-1}$ is internally stabilizing and achieves the desired system response in \\eqref{eq:system-response}.\n\\end{enumerate}\n\\label{thm:SLS}    \n\\end{theorem}\n\nFor disturbance $w(t) \\overset{\\text{i.i.d.}}{\\sim} \\mathcal{N}(0,I)$, one can recast the optimization problem \\eqref{eq:LQG} in terms of system responses $\\mathbf\\Phi_{x}$ and $\\mathbf\\Phi_u$ as\n\\begin{equation} \\label{eq:SLSOptimization}\n\\begin{aligned}\n    \\underset{\\mathbf\\Phi_{x} , \\mathbf\\Phi_{u}}{\\text{minimize}} &\\ \\quad \n    \\htwonorm{\\mtQ^{1\/2}\\mathbf\\Phi_{x}}^{2} + \\htwonorm{\\mtR^{1\/2}\\mathbf\\Phi_{u}}^{2} \\\\\n    \\text{s.t. } &\\ \\quad \\text{constraint \\eqref{eq:achievability-constraint}.}\n\\end{aligned}\n\\end{equation}\nWith a slight abuse of notation, we define $J(\\resp_x, \\resp_u)$ to be the LQR cost achieved by $\\resp_x$ and $\\resp_u$ in the objective of \\eqref{eq:SLSOptimization} and $J(\\mtK)$ as the cost \\eqref{eq:quadraticCost} achieved by applying controller $\\mtK$.\n\n\\subsection{SLS for Graph Symmetric Systems}\nWe now proceed to show that under Assumption \\ref{assump:diagonalizableCost}, the optimal system response for a graph symmetric system that solves the LQR problem \\eqref{eq:SLSOptimization} can be written as a graph filter.\n\n\\begin{theorem} \\label{thm:optimalLinear}\nGiven a GMD $S = V \\Lambda_{S} V^{\\top}$, consider an instance of the LQR problem \\eqref{eq:SLSOptimization} where the underlying system and cost satisfy Assumption~\\ref{assump:diagonalizableCost}. Then, there exists a global optimum $(\\optresp_x, \\optresp_u)$ where both $\\optresp_x$ and $\\optresp_u$ are diagonalizable by $\\mtV$, i.e.,\n\\begin{equation*}\n\\begin{aligned}\n    \\optresp_x &= V \\mtLambda_x^* V^{\\top}, \\quad  \\optresp_u &= V \\mtLambda_u^* V^{\\top},\n\\end{aligned}\n\\end{equation*}\nwhere $\\mtLambda_x^*$ and $\\mtLambda_u^*$ are diagonal transfer matrices. Hence, the optimal controller $\\mtK^\\opt=(\\optresp_u)(\\optresp_x)^{-1}$ can also be diagonalized by $V$.\n\\end{theorem}\n\\begin{proof}\nSee Appendix.\n\\end{proof}\n\n\\begin{remark}\nNote that the elements defined by the diagonal responses $\\mtLambda^\\opt_x, \\mtLambda^\\opt_u$ are \\emph{transfer functions} $[\\mtLambda^\\opt_x]_{ii}(z), [\\mtLambda^\\opt_u]_{ii}(z)$. Thus the resulting graph filter taps are transfer functions as well, i.e., a transfer function $\\resp(z)$ that is simultaneously diagonalizable with the matrix $S$ can be written as:\n\\begin{equation} \\label{eq:optimalLinear}\n    \\resp(z) = \\sum_{k=0}^{N-1}\\boldsymbol\\phi_k(z)S^k.\n\\end{equation}\n\\end{remark}\n\nThe main implication of Theorem~\\ref{thm:optimalLinear} is that the optimal linear state-feedback controller for graph symmetric systems under Assumption \\ref{assump:diagonalizableCost} is a graph filter and can thus be implemented via distributed message passing. We note, however, that the above result implies that the resulting optimal system response $\\optresp := (\\optresp_x, \\optresp_u)$ could be dense, as $S^{N-1}$ is dense if $S$ defines a connected graph. This can be undesirable in practice, as it requires $N-1$ communication exchanges with one-hop neighbors, potentially causing significant delays if the size $N$ of the graph is large. In the next section, we leverage a robust variant of the SLS parameterization given in Theorem \\ref{thm:SLS} to restrict the optimal system responses to only the first $F\\ll N-1$ filter taps while guaranteeing stability and near optimal performance.\n\nWe end this section by noting that in the graph signal processing literature \\citep{Sandryhaila2013-DSPG}, a controller of the form of \\eqref{eq:optimalLinear} is known as a linear, shift-invariant (LSI) graph filter, and is analogous to an LTI filter.\nNote that $S \\optresp \\vcx = \\optresp S \\vcx$, hence the name.\nIn particular, Equation \\eqref{eq:optimalLinear} is a spatially finite impulse response (FIR) graph filter \\citep{Segarra2017-GraphFilterDesign} that is completely characterized by a finite set of $N$ filter taps that can be conveniently described by a collection of transfer functions $\\boldsymbol\\phi^{\\opt} = [\\boldsymbol\\phi_{0}^{\\opt},\\ldots,\\boldsymbol\\phi_{N-1}^{\\opt}]^{\\Tr} \\in \\fdR^{N}$.  We emphasize that the transfer functions themselves are not restricted to be temporally FIR.\nSpatially FIR graph filters are also known as convolutional graph filters \\citep{Ruiz2021-GNN} due to their sum-and-shift nature, understanding that the effect of the operation $S \\vcx$ is to shift the signal around the graph (thus, oftentimes, the GMD $S$ is referred to as the graph shift operator).\nFurthermore, spatially FIR graph filters satisfy the convolution theorem that indicates that a convolution in the vertex domain can be computed by means of an elementwise multiplication in the spectrum domain \\citep{Sandryhaila2013-DSPG}.\nFinally, in the context of finite graphs, it is observed that the space of FIR graph filters of the form \\eqref{eq:optimalLinear}, characterized by $N$ filter taps, is equivalent to the space of spatially infinite impulse response (IIR) graph filters as well as autoregressive, moving average (ARMA) graph filters \\citep{Isufi2017-ARMA}.\n\n\n\\section{Localized Approximations to the Optimal Distributed Linear Controller} \\label{sec:approx}\nIn this section, we discuss several methods to approximate the optimal dense system response in the form of \\eqref{eq:optimalLinear} with one that is localized and uses ${F \\ll N}$ filter taps. We start with a projection method based on graph filter design. We then present a robust SLS formulation of the approximation problem that guarantees the stability of the resulting localized controller. Finally, we show how these two can be combined into a robust projection method that also ensures stability.  In the following, we define $\\optresp := (\\optresp_x, \\optresp_u)$, and recall that $\\optresp$ can be written as a graph filter \\eqref{eq:optimalLinear} defined by transfer function filter taps $\\boldsymbol\\phi^\\opt_k(z)$, $k=0,...,N-1$.  We further recall that each transfer function filter tap $\\boldsymbol\\phi^\\opt_k(z)$ admits the following expansion in terms of its Markov parameters: $\\boldsymbol\\phi^\\opt_k(z) = \\sum_{i=1}^\\infty z^{-i}\\phi^\\opt_k[i]$.\n\n\\subsection{Naive Projection}\nWe propose an approach inspired by the graph signal processing literature, wherein we exploit the graph filter structure of the optimal system responses \\citep{Segarra2017-GraphFilterDesign}. More specifically, we project the optimal system responses $\\optresp$ onto graph filters of order $F$ in the $\\mathcal H_2$ norm by solving the following optimization problem\n\\begin{equation} \\label{eq:minGraphFilter}\n    \\min_{\\boldsymbol\\phi} \\htwonorm{ \\resp - \\optresp }^2.\n\\end{equation}\nHere $\\boldsymbol\\phi := [\\boldsymbol\\phi_{0}(z),\\ldots,\\boldsymbol\\phi_{F-1}(z)]^{\\top} \\in \\fdC^{F}$ collects the $F$ transfer function filter taps defining $\\resp := \\sum_{k=0}^{F-1} \\boldsymbol \\phi_k(z)S^k$. If we further restrict each transfer function filter tap  $\\phi_k(z)$ to be FIR of order $n$, i.e., if we write $\\boldsymbol\\phi_k(z) = \\sum_{i=1}^n z^{-i}\\phi_k[i]$, then this reduces to solving the following unconstrained quadratic program\n\\begin{equation}\\label{eq:minGraphFilterFIR}\n    \\min_{\\{\\phi_k[i]\\}} \\sum_{i=1}^n \\left\\|\\sum_{k=0}^{F-1}\\phi_k[i]S^k - \\sum_{l=0}^{N-1}\\phi^\\opt_l[i]S^l \\right\\|_F^2.\n\\end{equation}\n\n\n\n\\begin{proposition}[Approximating filter taps] \\label{prop:filterApproximation}\nIf the eigenvalues $\\{\\lambda_{i}\\}$ of the GMD $\\mtS$ are all distinct, then the filter taps that solve \\eqref{eq:minGraphFilter} are given by\n\\begin{equation}\n    \\phi_{k}[i] = \\phi_{k}^{\\opt}[i] + \\vceps_k[i]\n\\end{equation}\nwhere $\\vceps$ is the error vector computed as\n\\begin{equation}\n    \\vceps[j] = \\Big( \\sum_{i=1}^{N} \\vclambda_{iF} \\vclambda_{iF}^{\\Tr} \\Big)^{-1} \\Big( \\sum_{i=1}^{N} \\vclambda_{iF} \\vclambda_{i(N-F)}^{\\Tr} \\Big) \\boldsymbol\\phi_{N-F}^{\\opt}[j]\n\\end{equation}\nwith $\\vclambda_{iF} := [1,\\lambda_{i},\\lambda_{i}^{2},\\ldots,\\lambda_{i}^{F-1}] \\in \\fdR^{F}$ is the collection of the first $F$ powers of $\\lambda_{i}$, $\\vclambda_{i(N-F)} := [\\lambda_{i}^{F},\\ldots,\\lambda_{i}^{N-1}] \\in \\fdR^{N-F}$ is the collection of the remaining powers and $\\boldsymbol\\phi_{N-F}^{\\opt}[i] := [\\phi^\\opt_{F}[i],\\ldots,\\phi^\\opt_{N-1}[i]] \\in \\fdR^{N-F}$ collects the tail $N-F$ optimal filter taps.\n\\end{proposition}\n\\begin{proof}\n    It follows from using the convexity of \\eqref{eq:minGraphFilterFIR}, matrix calculus and properties of Vandermonde matrices.\n\\end{proof}\n\nProp.~\\ref{prop:filterApproximation} determines in closed-form how to compute the filter of order $F$ that best approximates the optimal linear distributed controller in the $\\mathcal{H}_2$ norm. It also shows that each filter tap transfer function $\\boldsymbol\\phi_{k}(z)$ is obtained as the optimal tap $\\boldsymbol\\phi_{k}^{\\opt}(z)$ with an added corrective term that accounts for the $N-F$ taps that could not be included.\n\nThis approach is easy to implement computationally, as it only requires solving a least squares problem to minimize the projection cost. However, the resulting controller cannot be guaranteed to be stabilizing. As we show later via numerical simulation, approximations with a small number of filter taps $F$ are often unstable. This motivates an approach that takes into account the stability of the resulting controller.\n\n\\subsection{Localized Approximations via Robust SLS}\nRobust SLS \\citep{matni2017scalable, anderson_system_2019} offers a systematic way to reason about \\emph{approximate system responses}, i.e., system responses that do not exactly satisfy the achievability constraint \\eqref{eq:achievability-constraint}. In particular, as shown in the following result, robust SLS allows for an explicit characterization of the effects of using approximate system responses for controller design.\n\n\\begin{theorem}[Corollary 4.4 of \\cite{anderson_system_2019}]\\label{thm:robustSLS}\nLet $(\\resp_x, \\resp_u, \\mathbf{\\Delta})$ be a solution to\n\\begin{equation}\\label{eq:robust-achievability-constraint}\n    \\begin{bmatrix} zI-A & -B  \\end{bmatrix} \\begin{bmatrix} \\resp_x \\\\ \\resp_u    \\end{bmatrix} = I + \\mathbf\\Delta, \\quad \\resp_x, \\resp_u \\in \\frac{1}{z}\\rhinfty.\n\\end{equation}\nThen if $\\lVert \\mathbf\\Delta \\rVert_{\\mathcal H_\\infty} < 1$ the controller $\\mathbf{K} = \\resp_u\\resp_x^{-1}$ stabilizes the system \\eqref{eq:linearDynamics}, and the actual system response that is achieved is given by\n\\[\n\\begin{bmatrix} \\vcx \\\\ \\vcu \\end{bmatrix} = \\begin{bmatrix} \\resp_x \\\\ \\resp_u \\end{bmatrix} (I + \\mathbf\\Delta)^{-1} \\vcw.\n\\]\n\\end{theorem}\n\nWe leverage this result to provide an upper-bound on the amount of truncation that can be applied to $\\optresp$ without destabilizing the system.\n\n\\begin{corollary}\\label{cor:robust-cost-bound}\nLet $(\\resp_x, \\resp_u, \\mathbf{\\Delta})$ be a solution to \\eqref{eq:robust-achievability-constraint} and assume that $\\hinfnorm{\\mathbf\\Delta} < 1$. Then the controller $\\mtK = \\resp_u\\resp_x^{-1}$ achieves an LQR cost \\eqref{eq:SLSOptimization} $J$ that can be bounded as\n\\begin{equation*}\n    J(\\mtK) \\leq \\frac{1}{1-\\hinfnorm{\\mathbf\\Delta}} \\htwonorm{\\begin{bmatrix} Q^{1\/2} & 0 \\\\ 0 & R^{1\/2} \\end{bmatrix} \\begin{bmatrix} \\resp_x \\\\ \\resp_u \\end{bmatrix}}.\n\\end{equation*}\n\\end{corollary}\n\\begin{proof}\nFirst, we note that by Theorem \\ref{thm:robustSLS}, the system responses $(\\tilderesp_x, \\tilderesp_u)$ achieved by $\\mtK$ are given by\n\\[\\begin{bmatrix} \\tilderesp_x \\\\ \\tilderesp_u \\end{bmatrix} = \\begin{bmatrix} \\resp_x \\\\ \\resp_u \\end{bmatrix} (I + \\mathbf\\Delta)^{-1}.\\]\nThus, the cost achieved by the controller $\\mtK$ is bounded by\n\\begin{align*}\n    J(\\mtK) &= J(\\tilderesp_x, \\tilderesp_u)\\\\\n    &= \\htwonorm{\\begin{bmatrix} Q^{1\/2} & 0 \\\\ 0 & R^{1\/2} \\end{bmatrix} \\begin{bmatrix} \\resp_x \\\\ \\resp_u \\end{bmatrix} (I + \\mathbf\\Delta)^{-1}}\\\\\n    &\\leq \\hinfnorm{(I + \\mathbf\\Delta)^{-1}} \\htwonorm{\\begin{bmatrix} Q^{1\/2} & 0 \\\\ 0 & R^{1\/2} \\end{bmatrix} \\begin{bmatrix} \\resp_x \\\\ \\resp_u \\end{bmatrix}}\\\\\n    &\\leq \\frac{1}{1-\\hinfnorm{\\mathbf\\Delta}} \\htwonorm{\\begin{bmatrix} Q^{1\/2} & 0 \\\\ 0 & R^{1\/2} \\end{bmatrix} \\begin{bmatrix} \\resp_x \\\\ \\resp_u \\end{bmatrix}},\n\\end{align*}\nwhere we used Cauchy-Schwarz in the first inequality and the small gain theorem and the fact that $\\hinfnorm{\\mathbf\\Delta} < 1$ in the last step.\n\\end{proof}\nCorollary \\ref{cor:robust-cost-bound} offers a way to synthesize \\textit{stable} truncated system responses.\nSpecifically, to synthesize a system response that uses only $F < N-1$ filter taps while guaranteeing both stability and performance, we propose the following optimization problem:\n\\begin{subequations} \\label{eq:robustSLSOptimization}\n\\begin{align}\n\\begin{split}\n    \\underset{\\boldsymbol\\phi_{x}, \\boldsymbol\\phi_{u}, \\gamma\\in(0,1)}{\\text{minimize}} &\\ \\quad \n    \\frac{1}{1-\\gamma} \\htwonorm{ \\begin{bmatrix}  Q^{1\/2} & 0 \\\\ 0 & R^{1\/2}    \\end{bmatrix}\n    \\begin{bmatrix}     \\resp_x \\\\ \\resp_u    \\end{bmatrix} }\n\\end{split}\\\\\n\\begin{split}\\label{eq:robustConstraint}\n    \\text{s.t. } & \\quad\n    \\begin{bmatrix}    zI - A & -B   \\end{bmatrix}\n    \\begin{bmatrix}     \\resp_x \\\\ \\resp_u    \\end{bmatrix} = I + \\mathbf\\Delta, \\\\\n    &\\ \\hinfnorm{\\mathbf\\Delta} \\leq \\gamma,\\\\\n    &\\ \\resp_x(z) = \\sum_{k=0}^{F-1} \\boldsymbol\\phi_{x,k}(z) S^k, \\, \\boldsymbol\\phi_{x,k}(z) \\in \\frac{1}{z}\\mathcal{RH}_\\infty,\\\\\n    &\\ \\resp_u(z) = \\sum_{k=0}^{F-1} \\boldsymbol\\phi_{u,k}(z) S^k, \\, \\boldsymbol\\phi_{u,k}(z) \\in \\frac{1}{z}\\mathcal{RH}_\\infty.\n\\end{split}\n\\end{align}\n\\end{subequations}\n\n\nBy Corollary \\ref{cor:robust-cost-bound}, the solution to \\eqref{eq:robustSLSOptimization} defines a controller that is stabilizing.  We further show in the next result that it enjoys guaranteed suboptimality bounds relative to the optimal controller defined by $\\optresp$.  We first introduce the following notation: for a system response of the form $\\resp = \\sum_{k=0}^{N-1} \\boldsymbol\\phi_k(z)S^k$, we define the $F$-truncation $P_F(\\resp)$ and the $F$-tail $P_{F\\perp}(\\resp)$ as\n\\begin{align*}\n    P_F(\\resp) := \\sum_{k=0}^{F-1} \\boldsymbol\\phi_k(z)S^k,\\\n    P_{F\\perp}(\\resp) := \\sum_{k=F}^{N-1} \\boldsymbol\\phi_k(z)S^k.\n\\end{align*}\n\\begin{theorem}\nLet $(\\tilderesp_x, \\tilderesp_u, \\tilde{\\gamma})$ be the optimal solution to the robust SLS problem \\eqref{eq:robustSLSOptimization}. Let $(\\optresp_x, \\optresp_u)$ be the optimal solution to the untruncated SLS problem \\eqref{eq:SLSOptimization}. Suppose that\n\\[\n\\hinfnorm{\\mathbf\\Delta^*} := \\hinfnorm{ (zI - A)P_{F\\perp}(\\optresp_x) - B P_{F\\perp}(\\optresp_u) } < 1.\n\\]\nThen, the controller $\\mtK = \\tilderesp_u \\tilderesp_x^{-1}$ is stabilizing and\n\\begin{equation}\\label{eq:suboptimalityBound}\n    J(\\mtK) \\leq \\frac{1}{1 - \\hinfnorm{\\mathbf\\Delta^*}} \\Big( J(\\optresp_x, \\optresp_u) + J(P_{F\\perp}(\\optresp_x), P_{F\\perp}(\\optresp_u))    \\Big)\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBy the constraints \\eqref{eq:robustConstraint} and Theorem \\ref{thm:robustSLS}, we immediately have that $\\mtK$ is stabilizing. To show the given suboptimality bound, we first note that there exist some $\\gamma$ such that $(P_F(\\optresp_x), P_F(\\optresp_u), \\gamma)$ is a feasible solution to the robust optimization problem \\eqref{eq:robustSLSOptimization}. To see this, observe that\n\\begin{align*}\n    & \\begin{bmatrix} zI-A & -B \\end{bmatrix} \\begin{bmatrix} P_F(\\optresp_x) \\\\ P_F(\\optresp_u) \\end{bmatrix} \\\\\n    =\\ &\\begin{bmatrix} zI-A & -B \\end{bmatrix} \\begin{bmatrix} \\optresp_x \\\\ \\optresp_u \\end{bmatrix} - \\begin{bmatrix} zI-A & -B \\end{bmatrix} \\begin{bmatrix} P_{F\\perp}(\\optresp_x) \\\\ P_{F\\perp}(\\optresp_u) \\end{bmatrix}\\\\\n    =\\ & I - \\mathbf\\Delta^*,\n\\end{align*}\nwhere the last step follows from the achievability of the optimal response $(\\optresp_x, \\optresp_u)$ and the definition of $\\mathbf\\Delta^*$. Thus, $(P_F(\\optresp_x), P_F(\\optresp_u), \\hinfnorm{\\mathbf\\Delta^*})$ is a feasible solution with our assumption that $\\hinfnorm{\\mathbf\\Delta^*} < 1$. Denote the robust SLS objective \\eqref{eq:robustSLSOptimization} as $\\tilde{J}(\\resp_x, \\resp_u, \\gamma)$. By the optimality of the solution $(\\tilderesp_x, \\tilderesp_u, \\tilde{\\gamma})$, we have that\n\\begin{align*}\n \\tilde{J}(\\tilderesp_x, &\\tilderesp_u, \\tilde{\\gamma}) \\leq \\tilde{J}(\\optresp_x, \\optresp_u, \\hinfnorm{\\mathbf\\Delta^*}) \\\\\n &\\leq \\frac{1}{1-\\|\\mathbf\\Delta^*\\|_{H_\\infty}} \\htwonorm{ \\begin{bmatrix}  Q^{1\/2} & 0 \\\\ 0 & R^{1\/2}    \\end{bmatrix}    \\begin{bmatrix}  P_F(\\optresp_x) \\\\ P_F(\\optresp_u)\\end{bmatrix}},\n\\end{align*}\nwhere we applied Corollary \\ref{cor:robust-cost-bound} in the second inequality. The desired result follows then from the fact that\n\\[\n\\begin{bmatrix}  P_F(\\optresp_x) \\\\ P_F(\\optresp_u)\\end{bmatrix} = \\begin{bmatrix}  \\optresp_x \\\\ \\optresp_u\\end{bmatrix} - \\begin{bmatrix}  P_{F\\perp}(\\optresp_x) \\\\ P_{F\\perp}(\\optresp_u)\\end{bmatrix}\n\\]\nand an application of the triangle inequality.\n\\end{proof}\nThis optimization problem is jointly quasi-convex and can be solved efficiently using bisection. Further, feasibility provides a stability certificate in the form of $\\hinfnorm{\\mathbf\\Delta}<1$.\n\n\\subsection{Robust Projection}\nLastly, we can combine the robustness constraints used in robust SLS with the signal-processing-based projection method. Specifically, we solve the following optimization problem\n\\begin{equation} \\label{eq:robustProjection}\n\\begin{aligned}\n    \\underset{\\boldsymbol\\Phi_x , \\boldsymbol\\Phi_{u}, \\gamma\\in(0,1)}{\\text{minimize}} &\\ \\quad\n    \\htwonorm{\\resp - \\resp^*}^2 \\\\\n    \\text{s.t. } &\\ \\quad (\\boldsymbol\\Phi_x, \\boldsymbol\\Phi_u, \\gamma)\\;\\text{satisfy constraint \\eqref{eq:robustConstraint}}.\n\\end{aligned}\n\\end{equation}\nWe note that solving this problem does not give an upper bound on the cost of the resulting controller, but the robustness constraint ensures that the resulting controller is stabilizing.\n\n\\subsection{Implementation}\nWe end this section by detailing practical implementation details for optimization problems \\eqref{eq:robustSLSOptimization} and \\eqref{eq:robustProjection}. First, we note that computationally, one cannot directly optimize for the IIR system responses as is written in \\eqref{eq:robustSLSOptimization}, \\eqref{eq:robustProjection}. In practice, we use an FIR approximation of the strictly proper transfer functions $\\boldsymbol\\phi_x(z)$ and $\\boldsymbol\\phi_u(z)$, i.e., $\\boldsymbol\\phi(z)$ is parameterized as\n\\[\n\\boldsymbol\\phi(z) = \\sum_{i=1}^{n} z^{-i}\\phi[i]\n\\]\nfor some given FIR order $n$. As shown in \\cite{anderson_system_2019}, the suboptimality incurred by such an FIR approximation decays exponentially in the horizon $n$.\n\nThe $\\mathcal H_\\infty$-norm constraints on the (also FIR) transfer matrix $\\mathbf\\Delta$ can then be enforced via semidefinite programming (see Theorem 5.8 in \\cite{dumitrescu2007positive}), potentially introducing a nontrivial computational burden. However, we note that one can replace the $\\mathcal H_\\infty$-norm constraint in optimization problems \\eqref{eq:robustSLSOptimization} and \\eqref{eq:robustProjection} with any induced norm constraint.  A particularly appealing option is the $\\ell_1 \\to \\ell_1$ induced norm (which defines the $\\mathcal{L}_1$-norm of the transpose system), as this norm decomposes columnwise.  As shown in \\cite{anderson_system_2019, wang_separable_2018}, the resulting robustness constraints are linear and embarrassingly parallelizable. We defer this extension to future work.\n\\section{Numerical Experiments} \\label{sec:sims}\nWe show that our approach offers a principled way to trade off performance and communication complexity through numerical experiments. We also demonstrate the importance of the robustness constraints in synthesizing stable distributed controllers and compare the performance of robust SLS and projection-based methods on synthesizing localized controllers. All code needed to reproduce the examples found in this section is available at \\url{https:\/\/github.com\/unstable-zeros\/graph-symmetric-systems}.\n\n\\subsection{Setup}\nIn the following experiments, we consider the distributed linear quadratic regulator (LQR) problem \\eqref{eq:LQG} over $N=10$ scalar subsystems. We generate the GMD $S$ and dynamic matrices $A$ and $B$ using a process similar to that in \\cite{Gama2022-DistributedLQR}. To generate a problem instance, we start by creating the communication network $\\mathcal{G}$ by randomly sampling $N$ numbers $\\{u_i\\}_{i=1}^N\\sim{}U[0,1]$, and creating a bi-directional link between $v_i$ and each of its $3$ nearest points as defined by the topology on the interval $[0,1]$ under the metric $d(v_i,v_j)=|u_i-u_j|$. We then take $S$ to be a symmetric matrix that shares the sparsity pattern of the Laplacian of $\\mathcal G$, with its entry values sampled independently from $\\mathcal N(0,1)$. The GMD $S$ is then normalized to have a spectral radius of $1$. We generate the dynamics matrices $A$ and $B$ to share the same eigenvectors as $S$, and sample their eigenvalues i.i.d.\\ from the standard normal distribution -- hence both $A$ and $B$ are symmetric matrices. We take the cost matrices $Q = R = I_{N}$. For both of the following experiments, we randomly generate $50$ problem instances using this process. We end by noting that $\\mathcal{G}$ generated this way have, on average, a diameter of $5.92$ hops.\n\nFor the implementation of the optimization problems, we approximate the transfer functions $\\boldsymbol\\phi(z)$ with an FIR horizon of $n=10$. Further, for the robust SLS problem \\eqref{eq:robustSLSOptimization}, instead of using bisection to determine the best value of $\\gamma$, we fix $\\gamma=0.98$, as empirically the value of $\\gamma$ does not significantly affect the cost achieved by the controllers.\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=0.5\\textwidth]{imgs\/percentage_10_main.png}}\n\\caption{The percentage of synthesized $F$-hop controllers that are stabilizing using naive projection and robust SLS-based synthesis across 50 random trials.  The dashed red line denotes the average graph diameter of systems.}\n\\label{fig:trunc10-successrate}\n\\end{figure}\n\n\\subsection{Importance of Stability Constraints}\nIn this experiment, we demonstrate the importance of the robust SLS-based stability constraints in synthesizing stable distributed controllers. We vary the number of allowed filter taps and apply both the naive projection \\eqref{eq:minGraphFilter} and robust SLS \\eqref{eq:robustProjection} methods to $50$ randomly generated problem instances. For the naive projection method, we report the percentage of resulting controllers that are stable. For robust SLS, we report the percentage of optimization problems \\eqref{eq:robustSLSOptimization} being feasible, as feasible solutions optimization problem \\eqref{eq:robustSLSOptimization} are guaranteed to be stabilizing. The results are shown in Figure~\\ref{fig:trunc10-successrate}.\n\nFirst, we observe that as expected, a higher number of filter taps result in a higher probability of synthesizing stable responses for both methods. However, the naive projection method has nonzero probability of resulting in an unstable controllers even with a large number of filter taps i.e., even when the projection error between the \\textit{stable} optimal system responses and the projected responses is small.\nOn the other hand, the percentage of stable solutions resulting from the robust SLS problem increases monotonically with the number of filter taps, which is expected for a principled way of synthesizing stable controllers. Robust SLS also generally achieves a higher percentage of certifiably stable responses than that of naive projection, except for in the extremely sparse case of $F=3$. This suggests that the robust constraint might be too restrictive for synthesizing extremely sparse responses. Combined with the low computation cost of naive projection, this suggests a potential benefit of applying both methods in the sparse regime.\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=0.5\\textwidth]{imgs\/performance_10_main.png}}\n\\caption{Cost achieved by $F$-hop controllers across 50 randomly generated systems. Solid lines denote the median cost achieved by each method, and the shaded regions show the 25-th and 75-th percentile of the costs.}\n\\label{fig:trunc10-perf}\n\\end{figure}\n\n\\subsection{Truncation Performance}\nIn this experiment, we compare the performance of robust SLS and robust projection for different filter tap numbers $F$ on the same $50$ randomly generated problem instances. We show the median (solid lines), $25$-th and $75$-th percentile (shaded regions) of the costs achieved by both methods in Figure~\\ref{fig:trunc10-perf}.\n\nFirst, we note that the median costs decreases monotonically for both methods as the number of hops increase. This shows that that the optimization problems can leverage the increase in expressivity of the graph filters to achieve better performance, which matches our intuition. \nSecond, we note that the robust SLS-based method achieves a lower cost than robust Projection over for all numbers of filter taps considered. We also note that to the left of $4$ hops, the upper boundary of the shaded region, which represents the $75$-th percentile of the cost, is infinite, indicating that at least $25\\%$ of the robust synthesis problems are infeasible. This again suggests a need to develop more flexible methods in the sparse regime.\n\\section{Conclusion}\nIn this works, we introduced the notion of graph symmetric systems and showed that for linear quadratic problems, the optimal system response for graph symmetric systems can be written as (potentially dense) graph filters. We then proposed three methods to approximate the optimal responses with localized responses and validated their performance in numerical simulation. Directions of future work include relaxing the GSS constraints, applying the results on $\\mathcal L_1$ norm to enable distributed computation, and understanding how this can better inform GNN-based controllers with nonlinear activation functions.\n\\subsection{Proof for Theorem \\ref{thm:optimalLinear}}\nWe show that for any optimal system response $\\optresp_x$, $\\optresp_u$ of the optimization problem \\eqref{eq:SLSOptimization} that is not diagonalizable with $V$, i.e.,\n\\begin{equation*}\n    \\begin{aligned}\n    \\mathbf\\Lambda^\\opt_x &= V^\\top \\optresp_x V, \\quad\n    \\mathbf\\Lambda^\\opt_u  = V^\\top \\optresp_u V\n    \\end{aligned}\n\\end{equation*}\nare not diagonal, we can construct a simultaneously diagonalizable system response $\\resp_x'$, $\\resp_u'$ that is equally optimal.  In particular, we construct such a system response as follows:\n\\begin{gather}\n    \\resp_x' = V \\mathbf\\Lambda_x' V^{\\top}, \\quad\n     \\resp_u' = V \\mathbf\\Lambda_u' V^{\\top}\\\\\n    [\\mathbf\\Lambda_x']_{ij} =\n    \\begin{cases}\n        [\\mathbf\\Lambda^\\opt_x]_{ij} & i=j \\\\\n        0, & \\text{o.w.}\n    \\end{cases},\\;\n    [\\mathbf\\Lambda_u']_{ij} =\n    \\begin{cases}\n        [\\mathbf\\Lambda^\\opt_u]_{ij} & i=j \\\\\n        0 & \\text{o.w.}\n    \\end{cases}.\\label{eq:diag_phi}\n\\end{gather}\n\nWe first show that $\\resp_x'$, $\\resp_u'$ are feasible solutions of \\eqref{eq:SLSOptimization}. From the achievability condition on $\\optresp_x, \\optresp_u$, we have that\n\\begin{equation}\n\\begin{aligned}\n    (zI - A) \\optresp_x - B \\optresp_u &= I.\n\\end{aligned}\n\\end{equation}\nUsing the simultaneous diagonalizability of $A$ and $B$, we have\n\\begin{equation}\n    V (zI - \\Lambda_{A}) V^{\\top} V \\mtLambda_x^\\opt V^{\\top} -\n    V \\Lambda_{B} V^{\\top} V \\mtLambda_u^\\opt V^{\\top} = I\n    \\implies (zI - \\Lambda_{A}) \\mtLambda^\\opt_x - \\Lambda_{B} \\mtLambda^\\opt_u = I.\n\\end{equation}\nSince the matrices $(zI - \\Lambda_{A})$, $\\Lambda_{B}$ and $I$ are diagonal, we have\n\\begin{equation}\n    (zI - \\Lambda_{A}) \\mtLambda_x' - \\Lambda_{B} \\mtLambda_u'\n    = \\diag\\Big[(zI - \\Lambda_{A}) \\mtLambda^\\opt_x - \\Lambda_{B} \\mtLambda^\\opt_u\\Big]\n    = I,\n\\end{equation}\nwhere $\\diag[\\mtM]$ projects a matrix $\\mtM$ onto its diagonal elements. Therefore, $\\resp_x'$, $\\resp_u'$ is also feasible.\n\nNow, we show that $\\resp_x'$, $\\resp_u'$ gives a cost at least as good as that of $\\optresp_x, \\optresp_u$. By the simultaneous diagonalizability of the matrices $Q$ and $R$, and the fact that the $\\mathcal H_2$-norm is invariant under unitary transformations, we have that\n\\begin{equation}\\label{eq:spectral-cost}\n\\begin{aligned}\n    &\\htwonorm{Q^{1\/2}\\resp_x'}^2 + \\htwonorm{R^{1\/2}\\resp_u'}^2 \\\\\n    =\\;& \\htwonorm{V \\Lambda_Q^{1\/2} V^\\top V\\mtLambda_x' V^\\top} + \\htwonorm{V \\Lambda_R^{1\/2} V^\\top V\\mtLambda_u' V^\\top}\\\\\n    =\\;& \\htwonorm{\\Lambda_{Q}^{1\/2} \\mtLambda_x'}^{2} + \\htwonorm{\\Lambda_{R}^{1\/2} \\mtLambda_u'}^{2}.\\\\\n\\end{aligned}\n\\end{equation}\nDenoting the $i$-th eigenvalue of $\\mtQ$ and $\\mtR$ with $\\lambda_{Q,i}, \\lambda_{R,i}$, respectively, we have the inequality\n\\begin{equation}\n\\begin{aligned}\n    &\\htwonorm{\\Lambda_{Q}^{1\/2} \\mtLambda_x'}^{2} + \\htwonorm{\\Lambda_{R}^{1\/2} \\mtLambda_u'}^{2}.\\\\\n    =\\;&\\sum_{i,j} \\lambda_{Q, i} \\htwonorm{[\\mtLambda_x']_{ji}}^2 + \\sum_{i,j} \\lambda_{R, i} \\htwonorm{[\\mtLambda_u']_{ji}}^2\\\\\n    \\leq\\;& \\sum_{i,j} \\lambda_{Q, i} \\htwonorm{[\\mtLambda_x^\\opt]_{ji}}^2 + \\sum_{i,j} \\lambda_{R, i} \\htwonorm{[\\mtLambda_u^\\opt]_{ji}}^2\\\\\n    =\\;&\\htwonorm{\\Lambda_{Q}^{1\/2} \\mtLambda_x^*}^{2} + \\htwonorm{\\Lambda_{R}^{1\/2} \\mtLambda_u^*}^{2},\\\\\n\\end{aligned}\n\\end{equation}\nwhich follows from the definition of $\\mtLambda_x'$ and $\\mtLambda_u'$ in equation \\eqref{eq:diag_phi}. Reversing the steps in \\eqref{eq:spectral-cost}, we see that $\\resp_x', \\resp_u'$ achieves a cost at least as good as that of $\\optresp_x, \\optresp_u$. We can thus conclude that there always exists an optimal simultaneously diagonalizable system response to the LQR problem \\eqref{eq:SLSOptimization}. The controller $\\mtK' = (\\resp_u')(\\resp_x')^{-1}$ is thus optimal and simultaneously diagonalizable by $V$.\n\n\\subsection{GSS and Controllers Satisfy Quadratic Invariance}\nHere we show that optimal control problems over graph symmetric systems and controllers satisfy quadratic invariance \\citep{rotkowitz2005characterization}.  Before proceeding, we remark that the analysis of LQR optimal control problem over GSSs does not require quadratic invariance.  In particular, in Theorem \\ref{thm:optimalLinear} we analyze the \\emph{unconstrained optimal control problem} and show that the resulting \\emph{unconstrained optimal controller} satisfies a corresponding notion of graph symmetry.  However, in the interest of completeness, we show that if such a constraint were imposed on the controller during synthesis, the resulting problem satisfies quadratic invariance.\n\nTo that end, the corresponding constrained controller synthesis problem can be stated as\n\\begin{equation*}\n    \\begin{aligned}\n        \\underset{\\mtK}{\\text{minimize}}&\\quad J(\\mtK)\\\\\n        \\text{subject to}&\\quad \\mtK\\ \\text{stabilizes system \\eqref{eq:linearDynamics}}\\\\\n        &\\quad \\mtK \\in \\mathcal{S}\n    \\end{aligned}\n\\end{equation*}\nwhere $\\mathcal{S} := \\{S \\in \\mathcal{RH}_{\\infty}\\ |\\ S\\ \\text{is diagonalizable by } V\\}.$ Denoting the plant input-output transfer function as \n$$\\mtG(z) = (zI-A)^{-1}B,$$\nwe have the following proposition.\n\\begin{proposition}\nThe set of graph symmetric controllers $\\mathcal{S}$ is quadratically invariant under $\\mtG$ if system \\eqref{eq:linearDynamics} is graph symmetric.\n\\end{proposition}\n\\begin{proof}\nThe proof follows directly from the definition of quadratic invariance \\cite[Def. 2]{rotkowitz2005characterization} by straightforward calculation. First, we note that\n$$\\mtG(z) = V(zI-\\mtLambda_A)^{-1}\\mtLambda_BV^\\top$$\nis diagonalizable by $V$. For any controller $\\mtK \\in \\mathcal{S}$, it then follows immediately that\n$$\\mtK \\mtG \\mtK \\in \\mathcal{S}$$\nas the product of simultaneously diagonalizable matrices is also simultaneously diagonalizable, proving the claim.\n\\end{proof}\n\n\\section{The Distributed Linear-Quadratic Problem} \\label{sec:LQR}\n\nConsider a linear system described by a state vector $x(t) \\in \\fdR^{N}$ and controlled by an action $u(t) \\in \\fdR^{N}$.\nThe system is determined by the system matrix $A \\in \\fdR^{N \\times N}$ and the control matrix  $B \\in \\fdR^{N \\times N}$ as follows:\n\\begin{equation} \\label{eq:linearDynamics}\n    x(t+1) = A \\vcx(t) + B \\vcu(t) + w(t)\n\\end{equation}\nfor $t=0,1,\\ldots$ and where $\\{w(t)\\}$ is a random, white noise sequence.\n\\red{FG: If we include the noise, isn't it a `LQG' problem?}\nThe linear-quadratic problem consists on finding the \\red{static} state-feedback controller $K:\\fdR^{N} \\to \\fdR^{N}$ that minimizes a given quadratic cost\n\\begin{equation} \\label{eq:quadraticCost}\n    \\fnJ \\big( \\{x(t)\\}, \\{u(t)\\} \\big) = \\lim_{T \\to \\infty} \\sum_{t=0}^{T-1} \\xp_{w} \\big[ x(t)^{\\Tr} Q x(t) + u(t)^{\\Tr} R u(t) \\big]\n\\end{equation}\nwhere $u(t) = K(x(t))$ and where $Q, R \\in \\fdR^{N \\times N}$ are known symmetric matrices that satisfy $Q \\succeq 0$ and $R \\succ 0$.\nThe controller that solves\n\\begin{equation} \\label{eq:LQG}\n    \\min_{\\fnH \\in \\fdH} \\fnJ\\big( \\{\\vcx(t)\\},\\{\\vcu(t)\\} \\big) \\quad \\text{s. t. } \\vcu(t) = \\fnH\\big(\\vcx(t)\\big)\n\\end{equation}\nis a linear static controller $\\vcu(t) = \\mtH^{\\opt} \\vcx(t)$ with linear operator $\\mtH^{\\opt} = -(\\mtR+ \\mtB^{\\Tr} \\mtP \\mtB)^{-1}\\mtB^{\\Tr} \\mtP \\mtA$ and where $\\mtP$ is a matrix obtained as the unique solution to the following discrete-time algebraic Ricatti equation:\n\\begin{equation} \\label{eq:Ricatti}\n    \\mtP = \\mtA^{\\Tr} \\mtP \\mtA - \\mtA^{\\Tr} \\mtP \\mtB \\big( \\mtR+ \\mtB^{\\Tr} \\mtP \\mtB \\big)^{-1} \\mtB^{\\Tr} \\mtP \\mtA + \\mtQ.\n\\end{equation}\n\nConsider now a linear system of interconnected components described by a graph $\\graph = \\{\\stV, \\stE\\}$, where $\\stV = \\{\\lmv_{1},\\ldots,\\lmv_{N}\\}$ is the set of $N$ components (nodes) and where $\\stE \\subseteq \\stV \\times \\stV$ is the set of the corresponding interconnections (edges).\nIt is assumed that the graph is undirected, i.e. $(\\lmv_{i},\\lmv_{j}) \\in \\stE$ if and only if $(\\lmv_{j},\\lmv_{i}) \\in \\stE$.\nIn this context, the state vector $\\vcx(t)$ is such that the $i^{\\text{th}}$ entry $[\\vcx(t)]_{i} = x_{i}(t)$ corresponds to the value of the state at the  $i^{\\text{th}}$ component $\\lmv_{i}$ at time $t$.\nLikewise, the $i^{\\text{th}}$ entry of the control action $[\\vcu(t)]_{i} = u_{i}(t)$ corresponds to the action taken by the $i^{\\text{th}}$ component at time $t$.\n\nControlling systems of interconnected components oftentimes demands distributed controllers.\nThese are controllers $\\fnH_{\\stG}:\\fdR^{N} \\to \\fdR^{N}$ whose operations are restricted to follow the connection patterns of the system.\nThat is, the only operations allowed are those that involve components that share a connection.\nThis is denoted by the subscript $\\stG$ in the controller notation $\\fnH_{\\stG}$,  indicating that a distributed controller has to respect the structure of the graph $\\stG$ describing the connections between components.\n\nA convenient way to be able to define operations that respect the graph structure is to leverage a graph matrix description (GMD) $\\mtS \\in \\fdR^{N \\times N}$.\nThe matrix $\\mtS$ is such that the $(i,j)^{\\text{th}}$ entry is zero whenever there is no connection between components $\\lmv_{i}$ and $\\lmv_{j}$, i.e. $[\\mtS]_{ij}=0$ if $i \\neq j $ and $(\\lmv_{i},\\lmv_{j}) \\notin \\stE$.\nNote that, since the graph is undirected, the matrix $\\mtS$ is symmetric.\nTherefore, it has en eigedecomposition in terms of an orthonormal basis of eigenvectors $\\mtS = \\mtV \\mtLambda_{S} \\mtV^{\\Tr}$ where $\\mtLambda_{S} \\in \\fdR^{N \\times N}$ is a diagonal matrix with elements $\\lambda_{S,i} \\in \\fdR$ such that $\\mtS \\vcv_{i} = \\lambda_{S,i} \\vcv_{i}$ for $\\vcv_{i} \\in \\fdR^{N}$ being the $i^{\\text{th}}$ column of $\\mtV$.\n\nIn a general linear system as in \\eqref{eq:linearDynamics}, the matrices $\\mtA$ and $\\mtB$ are not related to the structure of the underlying graph.\nThis means that the evolution of the system is not tied to the structure of connections, i.e. the vector $\\vcx(t+1)$ can be built by linear combinations of the entries in $\\vcx(t)$ that may not respect the interconnected nature of the system.\nTo be able to describe distributed, linear system dynamics, the GMD $\\mtS$ has to be leveraged.\nMore specifically, we define a distributed linear system as follows:\n\\begin{definition}[Distributed Linear System] \\label{def:distributedLinear}\n    Given a GMD $\\mtS=\\mtV \\mtLambda_{S} \\mtV^{\\Tr}$, a linear system as in \\eqref{eq:linearDynamics} is distributed if the system and control matrices satisfy\n   \n    \\begin{equation} \\label{eq:distributedLinear}\n        \\mtA = \\mtV \\mtLambda_{A} \\mtV^{\\Tr} \\quad , \\quad \\mtB = \\mtV \\mtLambda_{B} \\mtV^{\\Tr}.\n    \\end{equation}\n    %\n\\end{definition}\nDef.~\\ref{def:distributedLinear} requires that both the system matrix $\\mtA$ and the control matrix $\\mtB$ be simultaneously diagonalizable with the GMD $\\mtS$.\nIn practical terms (for finite graphs with GMDs that have all distinct eigenvalues), this means that $\\mtA$ and $\\mtB$ can be written as a polynomial of $\\mtS$,\n\\begin{equation} \\label{eq:polynomialOfS}\n    \\mtA = p_{A}(\\mtS) \\quad , \\quad \\mtB = p_{B}(\\mtS).\n\\end{equation}\nPolynomials $p_{A}:\\fdR \\to \\fdR$ and $p_{B}:\\fdR \\to \\fdR$ are of degree at most $N-1$ \\red{[Cite Cayley Hamilton]}.\nNote that, when $\\mtA$ and $\\mtB$ can be written as a polynomial of $\\mtS$, this implies that the dynamics evolving from $\\vcx(t)$ to $\\vcx(t+1)$ rely entirely on linear combinations carried out between components that share a connection, and hence, the system is considered distributed.\nTo see this, consider first the operation $\\mtS \\vcx(t)$ whose $i^{\\text{th}}$ entry yields\n\\begin{equation} \\label{eq:graphShift}\n    [\\mtS \\vcx(t)]_{i} = \\sum_{j:(\\lmv_{j},\\lmv_{i}) \\in \\stE} [\\mtS]_{ij} [\\vcx(t)]_{j}.\n\\end{equation}\nAs it can be seen, due to the sparsity pattern of the GMD $\\mtS$, the output of $\\mtS\\vcx(t)$ can be computed entirely as a linear combination of the states in neighboring nodes.\nMore generally, when considering polynomials, it is observed that $\\mtS^{k}\\vcx(t)$ is equivalent to exchanging $k$ times information with one-hop neighbors.\nTherefore, if the system matrix $\\mtA$ and the control matrix $\\mtB$ are polynomials of $\\mtS$, then the evolution of the system is computed entirely by means of exchanges with neighboring nodes.\nHence, Def.~\\ref{def:distributedLinear} is a proper definition of distributed, linear systems.\nExamples of such linear, distributed systems, include both discrete-time and continuous-time diffusions, solutions to the heat equation, among many others \\red{[Add references]}.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the context of AdS\/CFT, the entanglement entropy\\footnote{There exists a huge literature on entanglement entropy. For background and interesting applications see  \\cite{calabrese, myersme,  mukund, bum1, quantum,takatemp, New, bbkss, erd, AB}.} for a boundary field theory which is dual to Einstein gravity can be calculated using the well-known Ryu-Takayanagi proposal \\cite{ryu,rev}. This proposal states that the entanglement entropy  $S_{EE}$ of any region on the boundary of AdS can be calculated by evaluating the area of a minimal surface in the bulk which is homologous to this boundary region:\n\\begin{equation}\n\tS_{EE} = \\frac{{\\rm Area}}{4G_N} \\,.\n\\label{arealaw}\n\\end{equation}\nBuilding upon earlier attempts~\\cite{chm, head,  fursaev, fur1}, this proposal was recently proved in Ref.~\\cite{maldacena}, for a general entangling surface.\n\nThe entanglement entropy formula in \\equ{arealaw} is of the same form as the formula for calculating the entropy of a black hole. In the black hole case, there exists a simple generalization of this area law for calculating the entropy of a black hole in any general higher-derivative gravity theory, known as the Wald entropy \\cite{wald,Iyer, Kang}.\nIt is natural to ask then if one can generalize the Ryu-Takayanagi prescription to higher-derivative gravity theories by simply replacing the RHS of \\equ{arealaw} with the Wald entropy. However, this is known not to be the case \\cite{higher,higher2}. \n\nRecently, a general formula for calculating the holographic entanglement entropy (HEE) in higher-derivative gravity theories was proposed in Refs.~\\cite{dong,camps}. It was also conjectured that the minimal entangling surface can be determined by interpreting this formula as the entropy functional for the higher derivative gravity theory and extremizing it. At present there exists no general proof of this proposal. In this paper, we will carry out various tests to determine the validity of this conjecture. \n\nWe will first work with general four-derivative theory. The conjectured form of the holographic entropy functional for general R$^2$ theory first appeared in Ref.~\\cite{solo}. The formula of Refs.~\\cite{dong, camps} also reduces to this functional for general R$^2$ theory. For the purpose of this paper, we will refer to this functional as the FPS (Fursaev-Patrushev-Solodukhin) functional after the authors of the paper where it was first proposed.\nIn Ref.~\\cite{abs} it was shown that this entropy functional leads to the expected universal terms in the entanglement entropy for cylindrical and spherical entangling surfaces, so the FPS functional passes this basic first test. The obvious next step is to determine whether the surface equation of motion derived from extremizing this functional is the same as that derived using the generalized gravitational entropy method (which we will refer to as the LM method) of Ref.~\\cite{maldacena}. \n\nGeneral R$^2$ theory depends on three parameters: $\\lambda_1,\\lambda_2$ and $\\lambda_3$. Gauss-Bonnet gravity is a special point in this parameter space \\cite{gbholo} and the FPS functional reduces to the Jacobson-Myers functional at this point. For Gauss-Bonnet gravity, the question whether the surface equation of motion one gets from the Jacobson-Myers functional matches with the surface equation of motion derived using the LM method was addressed in Refs.~\\cite{aks,abs,Chen}. We will look at the Gauss-Bonnet case again in this paper to emphasize several interesting points for this theory. For this theory, the surface equation of motion that one gets from the Jacobson-Myers functional matches with what one gets from the LM method, provided that terms cubic in the extrinsic curvature are suppressed. In this paper, we will find that for general R$^2$ theory using a procedure similar to the Gauss-Bonnet case leads to a match in the leading-order terms on both sides, where we designate terms cubic in the extrinsic curvature as sub-leading. However, as we will show, in the case of R$^2$ theory, the LM method also yields an extra condition that cannot be satisfied at arbitrary points of the parameter space. The conclusion is, therefore, that for a general R$^2$ theory the conditions that follow from the LM method do not correspond exactly to the surface equation of motion derived from the FPS functional.  \n\nAn alternative method to demonstrate that the FPS functional is the correct entropy functional for R$^2$ theory is to show that it can be interpreted as the action of a cosmic brane. This method was employed in Ref.~\\cite{dong}, where it was referred to as the cosmic brane method. In this paper, we will re-examine this procedure for R$^2$ theory and show that the result we get is consistent with what we get using the LM method.\n\nWhat happens when we go to a six-derivative gravity theory? In this case, we consider quasi-topological gravity \\cite{quasi} which is again a special point in the parameter space of R$^3$ theories. We first construct the entropy functional for quasi-topological gravity using the formula proposed in Ref.~\\cite{dong, camps}. We then show that this functional reproduces the expected universal terms for this theory for the cylindrical and spherical entangling surfaces. This is in agreement with the result of Ref.~\\cite{miao} that the entropy functional proposed in Refs.~\\cite{dong,camps} leads to the correct universal terms for a general higher-derivative theory. We also find the minimal surface condition for this theory using the LM method and show that it deviates from what is expected from the HEE functional. \n\nOur paper is organized as follows. In Sec.~(\\ref{dongformula}) we review the general entropy functional proposed in \\cite{dong, camps}. Our main focus in this paper is general four-derivative gravity theory, for which the entropy functional is the FPS functional. In Sec.~(\\ref{R2}) we find the surface equation of motion for R$^2$ theory by extremizing the FPS functional on the codimension-2 surface. We then compare it with what we obtain using LM prescription. We also make some remarks on the Gauss-Bonnet case. We then investigate the cosmic-brane method of Ref.~\\cite{dong}. In Sec.~(\\ref{Quasi}), we repeat our analysis for quasi-topological gravity. Lastly, in Sec.~(\\ref{discussion}) we summarize our findings and discuss their implications.\n\n\n\n\n\\section{Proposed entropy functional for general theories of gravity \\label{dongformula}}\n\nIn this section we will review the general entropy formula proposed in \\cite{dong,camps}. First we summarize the argument leading up to this proposal, following \\cite{dong}. For details the reader is referred to \\cite{dong, camps, maldacena, solo}. Some applications of this entropy formula are in \\cite{extra}.\n\nIn field theory, the entanglement entropy $S_{EE}=-{\\rm Tr}[\\rho\\log\\rho]$ can be calculated as the $n\\to1$ limit of the R\\'enyi entropy.\nThe R\\'enyi entropy in turn can be computed as\n\\begin{equation}\\label{renyi}\nS_n= -\\frac{1}{n-1}(\\log Z_n-n\\log Z_1) \\,.\n\\end{equation}\nHere $Z_n$ is the partition function of the field theory on the manifold $M_n$ which is the $n$-fold cover of the  original spacetime manifold $M_1$. In the holographic dual theory one can construct a suitable bulk solution $B_n$ with boundary $M_n$. The manifold $M_n$ at integer $n$ has a $Z_n$ symmetry that cyclically permutes the $n$ replicas. In \\cite{maldacena} it was proposed that this replica symmetry extends to the bulk $B_n$. Orbifolding the bulk by this symmetry results in a space $\\hat B_n = B_n \/ Z_n \\,,$ that is regular except at the fixed points of the $Z_n$ action. The fixed points form a codimension 2 surface with a conical defect in the bulk. This is the surface that is ultimately identified with the minimal entangling surface in the $n\\rightarrow 1$ limit. \nFurther, one can use gauge-gravity duality \\cite{AdS} to identify the field theory partition function on $M_n$ with the on-shell bulk action on $B_n$ in the large-$N$ limit\n\\begin{equation}\\label{adscft}\nZ_n\\equiv Z[M_n]= e^{-S[B_n]}\\,.\n\\end{equation}                                                       \nIt is now straightforward to calculate the entanglement entropy. By construction, one can identify\n\\begin{equation}\\label{sorb}\nS[B_n] = n S[\\hat B_n]\n\\end{equation}\nat integer $n$, where $S[\\hat B_n]$ is the classical action for the bulk configuration $\\hat B_n$ not including any contribution from the conical defect.  By analytically continuing $\\hat B_n$ to non-integer $n$, \\equ{sorb} can be used to define $S[B_n]$.  Using Eqs.~\\eqref{renyi} and \\eqref{adscft} and expanding around $n=1$, one gets\n\\begin{equation}\\label{snbh}\nS_{EE} = \\lim_{n\\to1} \\frac{n}{n-1}\\left(S[\\hat B_n]-S[ B_1]\\right)  = \\left . \\partial_\\epsilon S[\\hat B_n]\\right|_{\\epsilon=0} \n\\end{equation}\nwhere $\\epsilon\\equiv n-1$.\nThe quantity $S[\\hat B_n]$ can be calculated for the bulk theory by writing the bulk metric locally around the surface in gaussian normal coordinates and introducing a conical defect. It can be shown \\cite{maldacena, dong} that $\\partial_\\epsilon S[\\hat B_n]$ gets a contribution entirely from the tip of the cone. To compute it, therefore, one employs a metric regularized at the tip of the cone.  \n\nThis calculation is similar to that employed in Ref.~\\cite{solod} for calculating the Wald entropy from a regularized cone metric. Indeed, evaluating $ S[\\hat B_n]$ for a bulk theory with the cone metric to linear order in $\\epsilon$ and using \\equ{snbh} will result in two terms. The first is $S_{\\rm Wald}$: the Wald entropy for the theory. However, as was noted in \\cite{solo}, there is a second way for a linear contribution to arise. A term in the bulk lagrangian that is of order $\\epsilon^2$ can get enhanced to order $\\epsilon$ after integrating over the transverse directions. Following \\cite{dong}, we label the contribution of such terms as $S_{\\rm Anomaly}$. At this point, the calculation of the form of $S_{EE}$ is basically finished. \\equ{snbh} can be used to find the entanglement entropy for any higher-derivative theory, including ones whose lagrangians involve derivatives of the Riemann tensor.  However, for a general higher-derivative theory it can be computationally difficult to compute $S_{\\rm Anomaly}$ directly using \\equ{snbh}.\n\nIn \\cite{dong,camps} a simpler prescription for calculating the holographic entanglement entropy for higher-derivative theories of gravity in $d+1$ dimensions, for which the lagrangian ($\\mathcal{L}$) contains only contractions of the Riemann tensor, was given. The formula is:\n\\begin{equation}\\label{eei}\nS_{EE}= 2\\pi\\int d^d y \\sqrt{h} \\left\\{ \\frac{\\partial \\mathcal{L}}{\\partial R_{z{\\bar z} z{\\bar z}}} + \\sum_\\alpha \\(\\frac{\\partial^2 \\mathcal{L}}{\\partial R_{zizj} \\partial R_{{\\bar z} k{\\bar z} l}}\\)_\\alpha \\frac{8 {\\mathcal K}_{zij} {\\mathcal K}_{{\\bar z} kl}}{q_\\alpha+1} \\right\\} \\,.\n\\end{equation}\nThe notation used in the above equation and also in the rest of the paper is as follows: We use Greek Letters $\\mu,\\,\\nu,\\,\\rho,\\,\\sigma,\\, \\cdots$ to label the bulk indices. We use the Latin letters $a,b, ...... m, n$ to label the indices of the codimension 2 surface, while reserving the letters $p,q,r,s$ to denote the indices of the transverse directions. In these directions, we use the complex coordinates $z$ and $\\bar z$. The bulk metric is denoted by $g_{\\mu\\nu}$.The metric on the codimension-2 entangling surface is denoted by $h_{ij}$ while the surface itself is denoted by $\\Sigma$. The bulk Riemann tensor is denoted by $R_{\\mu \\nu\\rho\\sigma}$ while the intrinsic Riemann tensor of the surface is denoted by ${\\mathcal R}_{ikjl}$. The extrinsic curvatures of the surface are denoted by ${\\mathcal K}_{rij}$, where the first index labels the extrinsic curvature in the transverse directions.  We follow the curvature conventions in Ref.~\\cite{Waldb}.\n\nThe first term in \\equ{eei} is the Wald entropy. The second term is the correction to the Wald entropy and is evaluated in the following way: The second derivative of the lagrangian $\\mathcal{L}$ is a polynomial in components of the Riemann tensor. We expand the components $R_{pqij}, R_{piqj}$ and $R_{ikjl}$ using\n\\ba\nR_{pqij} &= \\tilde R_{pqij} +  {\\mathcal K}_{pjk} {\\mathcal K}_{qi}^{k} - {\\mathcal K}_{pik} {\\mathcal K}_{qj}^{k} \\,,\\nonumber\\\\\nR_{piqj} &= \\tilde R_{piqj} +  {\\mathcal K}_{pjk} {\\mathcal K}_{qi}^{k} - {\\mathcal Q}_{pqij} \\,,\\nonumber\\\\\nR_{ikjl} &= {\\mathcal R}_{ikjl} + {\\mathcal K}_{pil} {\\mathcal K}_{pjk} - {\\mathcal K}_{pij} {\\mathcal K}_{pkl} \\,.\n\\label{rexpi}\n\\ea\nHere, ${\\mathcal Q}_{pqij} \\equiv \\frac{1}{2}\\partial_p \\partial_q g_{ij}|_\\Sigma$.  $\\tilde R_{pqij}$ and $\\tilde R_{piqj}$ can also be defined in terms of metric variables, but the exact definition is not needed here.  The variable $\\alpha$ is used to label the terms in the expansion.  For each term labelled by $\\alpha$, $q_\\alpha$ is defined as the number of ${\\mathcal Q}_{zzij}$ and $Q_{{\\bar z}\\zb ij}$, plus one half of the number of ${\\mathcal K}_{pij}$, $R_{pqri}$, and $R_{pijk}$. The final step is to sum over $\\alpha$ with weights $1\/(1+q_\\alpha)$.  The quantities $\\tilde R_{pqij}$, $\\tilde R_{piqj}$, and ${\\mathcal R}_{ikjl}$ can then be eliminated using \\equ{rexpi}, resulting in an expression involving only components of $R_{\\mu\\nu\\rho\\sigma}$, ${\\mathcal K}_{pij}$ and ${\\mathcal Q}_{pqij}$. \n\nTo yield the entanglement entropy, the formula in \\equ{eei} should be evaluated on the minimal entangling surface. This surface is supposed to be determined following the LM method. Refs.~\\cite{dong,camps} also contain the proposal that the minimal surface can be determined by extremizing $S_{EE}$ as given in \\equ{eei} ---  $S_{EE}$ therefore being the entanglement entropy functional for a general theory of gravity.\n\n\n\n\\section{Test of the entropy functional for R$^2$ theory \\label{R2}}\nIn this section we consider general $R^2$ theory in five dimensions. The lagrangian for this theory is\n\\begin{equation}\n\\mathcal{L}=\\mathcal{L}_1+\\mathcal{L}_2\\,,\n\\end{equation}\nwhere \n\\begin{equation}\n\\mathcal{L}_1=R+\\frac{12}{L^2} \\label{Ei} \n\\end{equation}\nis the usual Einstein-Hilbert lagrangian with the cosmological constant appropriate for five-dimensional AdS space and \n\\begin{equation} \n\\mathcal{L}_2= \\frac{L^2}{2}\\(\\lambda_1 R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}+\\lambda_2 R_{\\alpha\\beta}R^{\\alpha\\beta}+\\lambda_3 R^2 \\)\n\\label{Lr}\n\\end{equation}  \nis the R$^2$ lagrangian. \n\nThe proposed entropy functional for this theory is\n\\begin{equation} \n  S_{\\rm EE,\\, R^2}= S_{\\rm Wald, \\,R^2} + S_{\\rm Anomaly, \\,R^2}\\,,\n  \\label{area}\n\\end{equation}\nwhere \n\\begin{align}\n S_{\\rm Wald,\\, R^2} &= \\frac{2\\pi}{\\ell_P^{3}}\\int d^{3}x \\sqrt{h} \\big\\{1+\\tfrac{L^{2}}{{2}} \\left(2\\lambda_{3}R+\\lambda_{2} R_{\\mu\\nu}n^{\\nu}_{r}n^{r\\mu}+ 2\\lambda_{1} R_{\\mu\\nu\\rho\\sigma}n^{\\mu}_{r}n^{\\nu}_{s}n^{r\\rho}n^{s\\sigma}\\right) \\big\\}\\,,\n\\label{Wald}   \\\\ \\textrm{ and~~~ }\n S_{\\rm Anomaly, \\,R^2}&=\\frac{2\\pi}{\\ell_P^{3}}\\int d^{3}x \\sqrt{h} \\big\\{ \\tfrac{L^{2}}{{2}} \\big(-\\textstyle{\\frac{1}{2}} \\lambda_{2} {\\mathcal K}_{r}{\\mathcal K}^{r}- 2\\lambda_{1}{\\mathcal K}_{sij}{\\mathcal K}^{sij} \\big)\\big\\}\\,.\n\\label{Extra} \n\\end{align}\nAs mentioned earlier, this entropy functional leads to the correct universal terms. In this section, we will further test this entropy functional by determining whether the surface equation of motion one gets from extremizing this functional is the same as the surface equation of motion one gets following the LM method. In Sec.~(\\ref{variation}), we extremize the functional for R$^2$ theory. In this particular section, we will first find the surface equation of motion for this functional in a general spacetime background.  However, the Ryu-Takayanagi proposal and its generalizations are most precisely formulated in the AdS\/CFT context, so eventually we will specialize to the AdS background. In Sec.~(\\ref{LMmethod}) we find the surface equation of motion using the LM method. In this case, we will always assume that the bulk is AdS space. Since a variation of the LM method -- called the cosmic-brane method -- was used in Ref.~\\cite{dong} to formulate a proof that the FPS functional is the correct entropy functional for R$^2$ theory, we also investigate this method in Sec.~(\\ref{deltas}). \n\n\\subsection{Minimal surface condition from the entropy functional \\label{variation}}\n\nTo extremize the functional in \\equ{area}, we follow the methods of Refs.~\\cite{diffgeo, Stringcurv1, Stringcurv2}. We denote the surface we are going to extremize w.r.t the action in \\equ{area} by $\\Sigma$. The induced metric on $\\Sigma$ is \n\\begin{equation}\n  h_{ij} = e^{\\mu}_{i} e^{\\nu}_{j} g_{\\mu\\nu},\n\\end{equation}\nwhere $g_{\\mu\\nu}$ is the bulk metric and $e^{\\mu}_{i} \\equiv {\\partial}_{i}X^{\\mu}$ are the basis vectors tangent to the surface $\\Sigma$, $X^{\\mu}$ being the bulk coordinates. On the surface $\\Sigma$, the $g_{ir}$ component of the bulk metric vanishes. The two normals to the surface are denoted by $n^{\\mu}_{r}$ where $r=1,2$ are the transverse directions. The metric tensor in the tangent space spanned by the normal vectors (the metric tensor of the normal bundle of the sub-manifold $\\Sigma$) is the Kronecker delta:\n\\begin{equation}\n\t\\delta_{rs}  = \\epsilon \\, n_{r}^{\\mu} n^{\\nu}_s g_{\\mu\\nu}\n\\end{equation}\nWe work in Euclidean signature and set $\\epsilon=+1$. We use the inverse metric $\\delta^{rs}$, to raise indices in the normal directions: $n^{r \\mu } =\\delta^{rs} n^{\\mu}_{s}$. Note that, repeated $s$ indices always imply summation over the transverse directions: $n^{\\mu}_s n^{\\nu s} \\equiv n^{\\mu}_1 n^{\\nu}_1 + n^{\\mu}_2 n^{\\nu}_2$.\nIn this notation, the completeness relation relating $g^{\\mu\\nu}$, the inverse of the bulk metric, to $h^{ij}$, the inverse of the induced metric, is\n\\begin{equation}\n  g^{\\mu\\nu} = h^{ij}e^{\\mu}_{i}e^{\\nu}_{j} +  n^{\\mu}_s n^{\\nu s}. \n  \\label{completeness}\n\\end{equation}\n\n\nThe Gauss and Weingarten equations are \n\\begin{align} \\label{GaussW}\n\\nabla_{i} e^{\\mu}_{j}&=\\partial_{i} e^{\\mu}_{j} + {\\hat{\\Gamma}}^{\\mu}_{\\nu\\rho}e^{\\rho}_{i} e^{\\nu}_{j} -\\Gamma_{ij}^{k}e^{\\mu}_{k}= -{\\mathcal K}^{r}_{ij} n^{\\mu}_{r} \\nonumber \\\\\n\\nabla_{i} n^{\\mu}_{s}&= \\partial_{i} n^{\\mu}_{s} +{\\hat{\\Gamma}}^{\\mu}_{\\rho\\nu} e^{\\rho}_{i} n^{\\nu}_s-\\Gamma_{si}^{r}n_{r}^{\\mu} ={\\mathcal K}_{ s i}^{j} e^{\\mu}_{j}\\,.\n\\end{align}\nHere, $\\nabla$ is the Van der Waerden-Bortolotti covariant derivative~\\cite{diffgeo} which acts on a general tensor $T^{s\\cdots r}_{i\\cdots j}$ as\n\\begin{equation}\n   \\nabla_k T^{s\\cdots r}_{i\\cdots j} = \\tilde{\\nabla}_k T^{s\\cdots r}_{i\\cdots j} + {\\Gamma}^{s}_{p k} T^{p\\cdots r}_{i\\cdots j} + \\cdots +  {\\Gamma}^{r}_{p k} T^{s\\cdots p}_{i\\cdots j}  \\,,\n   \\label{defnabla}\n\\end{equation}\nwhere $\\tilde{\\nabla}$ is the usual covariant derivative associated with the surface Christoffel. This Christoffel is related to the bulk Christoffel ${\\hat{\\Gamma}}^{\\mu}_{\\sigma \\nu}$ as \n\\begin{equation}\n\t{\\Gamma}^{i}_{j k} = (\\partial_j e^{\\mu}_k  + {\\hat{\\Gamma}}^{\\mu}_{\\sigma \\nu} e^{\\sigma}_j e^{\\nu}_k ) e^{i}_{\\mu}\\,.\n\\end{equation}\nThe Chrisoffel ${\\Gamma}^{r}_{i s}$ is the Christoffel induced in the normal bundle. It is related to the bulk Christoffel as\n\\begin{equation}\n\t{\\Gamma}^{r}_{i s} = (\\partial_i n^{\\mu}_s  + {\\hat{\\Gamma}}^{\\mu}_{\\sigma \\nu} e^{\\sigma}_i n^{\\nu}_s) n^r_{\\mu}\\,.\n\t\\label{defCA}\n\\end{equation}\nThis Christoffel can be interpreted geometrically as the freedom to perform rotations of the normal frame. It is, therefore, equivalent to a gauge field ${\\mathcal A}_{k}$, commonly referred to as a twist potential. This field is defined as:\n\\begin{equation}\n  {\\mathcal A}_{k} \\equiv \\frac{1}{2} \\varepsilon^{rs} \\partial_r g_{ks}\\,,\\textrm{ so that } {\\Gamma}^{s}_{j r} =\\delta^{ps}\\varepsilon_{rp} {\\mathcal A}_{j} \\,,\n  \\label{defA}\n\\end{equation}\nwhere $\\varepsilon_{rs}$ is the Levi-Civita symbol. \n\nThe Gauss identity relating the bulk Riemann with all indices projected in the tangential directions with the surface Riemann is\n\\begin{equation} \\label{GaussRicci}\n\t\tR_{\\mu\\nu\\rho\\sigma}e^{\\mu}_{k}e^{\\nu}_{i}e^{\\rho}_{l}e^{\\sigma}_{j}= {\\mathcal R}_{kilj}-{\\mathcal K}^{r}_{kl} {\\mathcal K}_{rij} + {\\mathcal K}^{r}_{ik}{\\mathcal K}_{rjl}\\,.\n\\end{equation}\nThe Codazzi-Mainardi relation is\n\\begin{equation} \\label{codazzi}\n\t \\nabla_k {\\mathcal K}_{rij}  -  \\nabla_i {\\mathcal K}_{rkj}  =  R_{\\mu\\nu\\rho\\sigma}e^{\\mu}_{k}e^{\\nu}_{i}e^{\\sigma}_{j}n^{\\rho}_{r}\\,.\n\\end{equation}\nFrom \\equ{GaussRicci} we get the Gauss-Codazzi identity\n\\begin{equation}\n  {\\mathcal R} = R - 2 R_{\\mu\\nu}n^{\\nu r}n^{\\mu}_{r} + R_{\\mu\\nu\\rho\\sigma}n^{\\mu r}n^{\\nu s}n^{\\rho}_{r}n^{\\sigma}_{s} + {\\mathcal K}_r{\\mathcal K}^r - {\\mathcal K}^s_{ij}{\\mathcal K}_s^{ij}\\,.\n  \\label{GC}\n\\end{equation}\n\n\nWe now consider an infinitesimal variation of the surface $\\Sigma$ given by $X^{\\mu} \\longrightarrow  X^{\\mu} +\\delta X^{\\mu}$. The change $\\delta X^{\\mu}$ is\n\\begin{equation}\n  \\delta X^{\\mu} = {\\xi}^r n^{\\mu}_r +  {\\xi}^i e^{\\mu}_i\\,.\n  \\label{fullvariation}\n\\end{equation}\nwhere ${\\xi}^r$ and ${\\xi}^i$ are small parameters. \nFor deriving the equation describing the minimal surface we are only concerned with the variation in the normal direction, since the tangential variation leads to a constraint equation. The variation then reduces to\n\\begin{equation}\n  \\delta X^{\\mu} = {\\xi}^r n_{r}^{\\mu}\\,,\n\\end{equation} \n\nThe variation $\\delta X^{\\mu}$ in the surface will induce a variation in the basis vectors $e_{i}^{\\mu}$. This can be computed by finding the basis vectors at $X^{\\mu} +\\delta X^{\\mu}$ and parallel transporting them back to $X^{\\mu} $. Taking the difference between the parallel-transported quantity and the original basis vector at the coordinate $X^{\\mu}$, using the identities in \\equ{GaussW} and then restricting to normal variation results in \n\\begin{equation}\n  \\delta e_{i}^{\\mu} =  n^{\\mu}_{s} {\\nabla}_i \\xi^s  + e_{j}^{\\mu} {\\mathcal K}^{j}_{si}  \\xi^s\\,.\n  \\label{tangentvar}\n\\end{equation}\nThe details of this calculation are in Ref.~\\cite{diffgeo}.\nAs stated in \\equ{defnabla}, the covariant derivative $\\nabla$ acts on $\\xi_s$ as\n\\begin{equation}\n\t\\nabla_i \\xi^s = \\partial_i \\xi^s +  {\\Gamma}^{s}_{i r} \\xi^{r}\\,.\n\\end{equation}\n\nThe variation in any other tensor quantity can be calculated in a similar way, by parallel transporting the quantity at the new coordinate back to the old coordinate and taking the difference. This gives the variation in the bulk metric as zero. We write down the result for other variations. For details the reader is referred to \\cite{diffgeo}.\nThe variation of the induced metric is \n\\begin{align}\n  \\delta h_{ij} &=  2 {\\xi}^r {\\mathcal K}_{rij}\\,, \\nonumber \\\\\n  \\delta \\sqrt{h} &=  {\\xi}^r \\sqrt{h}  {\\mathcal K}_r\\,.\n  \\label{metricvar}\n\\end{align}\n\nThe variation of the extrinsic curvature is \n\\begin{align}\n  \\delta {\\mathcal K}^s_{ij} &= (- {\\nabla}_{i} {\\nabla}_{j} {\\delta}^{s}_{r} + {\\mathcal K}^{s}_{i k} {\\mathcal K}^{k}_{rj} + R_{\\mu\\nu\\rho\\sigma}n^{s\\mu}n^{\\sigma}_r e^{\\rho}_{i} e^{\\nu}_{j} ) {\\xi}^{r}\\,,\\nonumber \\\\\n\\delta {\\mathcal K}^s &= ( - {\\nabla}^{i} \\nabla_{i} {\\delta}^{s}_{r} - {\\mathcal K}^{s}_{i k} {\\mathcal K}^{ki}_{r} + h^{ij}R_{\\mu\\nu\\rho\\sigma}n^{s\\mu}n^{\\sigma}_r e^{\\rho}_{i} e^{\\nu}_{j} ) {\\xi}^{r}\\,.\n\\label{Kvar}\n\\end{align}\nThe covariant derivatives $\\nabla$ act all the way to the right.\n\nUsing these variations we can now compute the change in the action. For this we first rewrite the $R_{\\mu\\nu}n^{\\nu}_{r}n^{r\\mu}$ and $R_{\\mu\\nu\\rho\\sigma}n^{\\mu}_{r}n^{\\nu}_{s}n^{r\\rho}n^{s\\sigma}$ terms in the action given in Eq.~(\\ref{Wald}) as\n\\begin{align}\n  R_{\\mu\\nu}n^{\\nu r}n^{\\mu}_{r} &= R - h^{ij} R_{\\mu\\nu}e^{\\nu}_{i}e^{\\mu}_{j} \\, \\nonumber \\\\\n  R_{\\mu\\nu\\rho\\sigma}n^{\\mu r}n^{\\nu s}n^{\\rho}_{r}n^{\\sigma}_{s} &= R - 2 h^{ij} R_{\\mu\\nu}e^{\\nu}_{i}e^{\\mu}_{j} + h^{ik}h^{jl}R_{\\mu\\nu\\rho\\sigma}e^{\\mu}_{i}e^{\\nu}_{j}e^{\\rho}_{k}e^{\\sigma}_{l}\n\\end{align}\nusing the completeness relation in Eq.~(\\ref{completeness}). The variation of a term such as $h^{ij} R_{\\mu\\nu}e^{\\nu}_{i}e^{\\mu}_{j}$ is given by\n\\begin{equation}\n\t\\delta ( h^{ij} R_{\\mu\\nu}e^{\\nu}_{i}e^{\\mu}_{j} ) = ( \\delta h^{ij} )R_{\\mu\\nu}e^{\\nu}_{i}e^{\\mu}_{j} + 2  h^{ij} R_{\\mu\\nu} \\delta (e^{\\nu}_{i}) e^{\\mu}_{j} + h^{ij} \\delta (R_{\\mu\\nu}) e^{\\nu}_{i}e^{\\mu}_{j}\n\\end{equation}\nThe first two variations can be computed using Eqs.~(\\ref{metricvar}) and (\\ref{tangentvar}) respectively.  \nFor evaluating the last term we need the variation of the bulk Ricci Tensor\\footnote{We thank Joan Camps for pointing out that such terms will contribute to the total variation.} which is given by \n\\begin{equation}\n\t\\delta (R_{\\mu\\nu}) =  n^{\\sigma}_{r} {\\hat{\\nabla}_\\sigma}R_{\\mu\\nu}  {\\xi}^{r}\\,.\n\\end{equation}\nThe variation in the bulk Ricci scalar and Riemann tensor take a similar form. All these variations are under the integral sign in \\equ{Wald} and we  perform a integration by parts where needed, discarding the term that is a total variation. Then \nusing the variations given above we obtain:\n\\begin{alignat}{2} \n  \\delta(\\sqrt{h} R) &= && \\sqrt{h}~ {\\mathcal K}_{s} R \\,{\\xi}^s + n^{\\mu}_{s} \\sqrt{h} \\hat \\nabla_{\\mu}R\\,{\\xi}^{s}\\,, \\nonumber \\\\\n  \\delta(\\sqrt{h} R_{\\mu\\nu}n^{\\nu r}n^{\\mu}_{r}) &= && \\sqrt{h}~ {\\mathcal K}_{s} R_{\\mu\\nu}n^{\\nu r}n^{\\mu}_{r} {\\xi}^s + 2 \\sqrt{h} \\,\\nabla^{i}(R_{\\mu\\nu}n^{\\nu}_s e^{\\mu}_{i}) {\\xi}^s~-\\nonumber\\\\\n\t  &\\phantom{=}&&\\sqrt{h} \\,  n^{\\sigma}_{s}h^{ij} e^{\\mu}_{i}e^{\\nu}_{j}\\hat \\nabla_{\\sigma}R_{\\mu\\nu}{\\xi}^{s} + n^{\\mu}_{s} \\sqrt{h} \\,\\hat \\nabla_{\\mu}R\\,{\\xi}^{s}\\,, \\nonumber \\\\\n\t\\delta(\\sqrt{h} R_{\\mu\\nu\\rho\\sigma}n^{\\mu r}n^{\\nu s}n^{\\rho}_{r}n^{\\sigma}_{s}) &=  && \\sqrt{h}~ {\\mathcal K}_{s} R_{\\mu\\nu\\rho\\sigma}n^{\\mu r}n^{\\nu q}n^{\\rho}_{r}n^{\\sigma}_{q} {\\xi}^s-\n\t4 \\sqrt{h}  \\,\\nabla^{i}(R_{\\mu\\nu\\rho\\sigma} n^{\\mu}_s e^{\\nu}_{j}e^{\\rho}_{i}e^{\\sigma}_{k}h^{jk} ){\\xi}^s + \\nonumber\\\\\n\t  &\\phantom{=}&&4 \\sqrt{h}\\,\\nabla^{i}( R_{\\mu\\nu}n^{\\nu}_s e^{\\mu}_{i}) {\\xi}^s+\\sqrt{h}  h^{ik}h^{jl}e^{\\mu}_{i}e^{\\nu}_{j}e^{\\rho}_{k}e^{\\sigma}_{l} n^{\\alpha}_{s} \\hat \\nabla_{\\alpha} R_{\\mu\\nu\\rho\\sigma}{\\xi}^{s}  ~ -\\nonumber\\\\\n\t  &\\phantom{=}&& 2\\,n^{\\nu}_{s} h^{ij} e^{\\mu}_{i}e^{\\rho}_{j}\\hat \\nabla_{\\nu}R_{\\mu\\rho}{\\xi}^{s}+\\sqrt{h} n^{\\mu}_{s}\\hat \\nabla_{\\mu}R{\\xi}^{s} \\,.\n\t \\label{varR}\n\\end{alignat}\nSimilarly the variations for the terms present in the action in Eq.~(\\ref{Extra}) are\n\\begin{alignat}{2}\n  \\delta(\\sqrt{h}{\\mathcal K}^s{\\mathcal K}_{s}) &= &-& 2 \\sqrt{h}\\nabla_i\\nabla^i {\\mathcal K}_{r} {\\xi}^r +  \\sqrt{h} {\\mathcal K}_{r} {\\mathcal K}^s {\\mathcal K}_{s}  {\\xi}^r - 2 \\sqrt{h} {\\mathcal K}^{s} {\\mathcal K}_{sij}{\\mathcal K}_r^{ij}  {\\xi}^r - \\nonumber\\\\\n    &\\phantom{=} &&2 \\sqrt{h} {\\mathcal K}^{s} R_{\\mu\\nu\\rho\\sigma} h^{ij} n_r^{\\mu} e^{\\nu}_{i}n_s^{\\rho}e^{\\sigma}_{j}{\\xi}^r \\,, \\nonumber\\\\ \n  \\delta(\\sqrt{h}{\\mathcal K}_{sij}{\\mathcal K}^{sij}) &=&-&  2 \\sqrt{h} \\nabla_i\\nabla_j {\\mathcal K}_r^{ij} {\\xi}^r +  \\sqrt{h}{\\mathcal K}_{r} {\\mathcal K}_{sij}{\\mathcal K}^{sij}  {\\xi}^r - 2\\sqrt{h} {\\mathcal K}_{sij}{\\mathcal K}^{si}_{k} {\\mathcal K}_{r}^{kj}{\\xi}^r - \\nonumber\\\\\n    &\\phantom{=}&&  2 \\sqrt{h}{\\mathcal K}^{sij} R_{\\mu\\nu\\rho\\sigma}n^{\\mu}_r e^{\\nu}_{i}n^{\\rho}_s e^{\\sigma}_{j}{\\xi}^r \\,.\n  \\label{varK2}\n\\end{alignat}\nAdding these variations up with the appropriate factors will give us the equation for the minimal surface for the action in Eq.~(\\ref{area}) in a general spacetime background.\n\nAs a check of these equations we now demonstrate that the above results lead to the correct surface equation of motion in the Gauss-Bonnet case. For Gauss-Bonnet the entropy functional for general R$^2$ theory reduces to the Jacobson-Myers functional \n\\begin{equation}\n  S_{JM}=\\frac{2\\pi}{\\ell_P^3} \\int d^3 x \\sqrt{h}\\{(1+\\lambda L^2 (R - 2 R_{\\mu\\nu}n^{\\nu r}n^{\\mu}_{r} + R_{\\mu\\nu\\rho\\sigma}n^{\\mu r}n^{\\nu s}n^{\\rho}_{r}n^{\\sigma}_{s} + {\\mathcal K}_s{\\mathcal K}^s - {\\mathcal K}_{sij}{\\mathcal K}^{sij})\\}\\,.\n\\label{JMbig}\n\\end{equation}\nThis functional is valid in a general space-time background. Note that the integrand is equal to $\\sqrt{h}(1+\\lambda L^2 {\\mathcal R})$ on using the Gauss-Codazzi identity \\equ{GC}. The surface equation of motion for this theory using this form of the functional is\n~\\cite{abs},\n\\begin{equation}\n   {\\mathcal K} + \\lambda L^2 ({\\mathcal R} {\\mathcal K} - 2{\\mathcal R}^{ij}{\\mathcal K}_{ij}) =0\\,.\n   \\label{GBminimal}\n\\end{equation}\nWe now find the surface equation of motion by directly varying \\equ{JMbig}. Using the variation equations Eqs.~(\\ref{varR}--\\ref{varK2}) and simplifying using the identities in Eqs.~(\\ref{GaussRicci}--\\ref{codazzi}) we get \n\\begin{alignat}{2} \\label{GS2}\n\\sqrt{h}~{\\mathcal K}_s\\,{\\xi}^s+\\lambda L^2\\Big[&\\sqrt{h}~ {\\mathcal K}_{s} {\\mathcal R} \\,{\\xi}^s- 2\\,\\sqrt{h} {\\mathcal R}_{jk}{\\mathcal K}_s^{jk}\\, {\\xi}^s \\nonumber\\\\\\phantom{=}&+\\sqrt{h}  h^{ik}h^{jl}e^{\\mu}_{i}e^{\\nu}_{j}e^{\\rho}_{k}e^{\\sigma}_{l} n^{\\alpha}_{s} \\hat \\nabla_{\\alpha} R_{\\mu\\nu\\rho\\sigma}{\\xi}^{s} -\n\t2 \\sqrt{h}\\,\\nabla^{i}( R_{\\mu\\nu\\rho\\sigma} n^{\\mu}_s e^{\\nu}_{j}e^{\\rho}_{i}e^{\\sigma}_{k}h^{kj} ){\\xi}^s  \\nonumber\\\\\\phantom{=}&-2 \\sqrt{h} {\\mathcal K}^{r} R_{\\mu\\nu\\rho\\sigma} h^{ij} n_s^{\\mu} e^{\\nu}_{i}n_r^{\\rho}e^{\\sigma}_{j}{\\xi}^s +2 \\sqrt{h}{\\mathcal K}^{rij} R_{\\mu\\nu\\rho\\sigma}n^{\\mu}_s e^{\\nu}_{i}n^{\\rho}_r e^{\\sigma}_{j}{\\xi}^s  \\nonumber\\\\\\phantom{=}&+2 \\sqrt{h} R_{\\mu\\nu\\rho\\sigma}e^{\\mu}_{j}e^{\\nu}_{i}e^{\\rho}_{k}e^{\\sigma}_{l}h^{il}{\\mathcal K}_s^{jk} \\, {\\xi}^s\\Big]\\,.\n\\end{alignat}\nThe first three terms give precisely the equation of motion for Gauss-Bonnet theory. The rest of the terms add up to zero, as we show in the following.\nWe use the Bianchi identity on the $\\hat \\nabla_{\\alpha} R_{\\mu\\nu\\rho\\sigma}$ factor of the fourth term giving\n\\begin{equation}\n\\hat \\nabla_{\\alpha} R_{\\mu\\nu\\rho\\sigma}= -\\hat \\nabla_{\\sigma} R_{\\mu\\nu\\alpha\\rho}-\\hat\\nabla_{\\rho} R_{\\mu\\nu\\sigma \\alpha}\n\\end{equation}\nand then rewrite each of these terms as \n\\begin{equation} \\label{leftovera}\n  h^{ik}h^{jl}e^{\\mu}_{i}e^{\\nu}_{j}e^{\\rho}_{k}e^{\\sigma}_{l} n^{\\alpha}_{s} \\hat \\nabla_{\\sigma} R_{\\mu\\nu\\alpha\\rho}~=~ e^{\\sigma}_{l} \\hat \\nabla_{\\sigma}(h^{ik}h^{jl}e^{\\mu}_{i}e^{\\nu}_{j}e^{\\rho}_{k} n^{\\alpha}_{s} R_{\\mu\\nu\\alpha\\rho})- \ne^{\\sigma}_{l} \\hat \\nabla_{\\sigma}(h^{ik}h^{jl}e^{\\mu}_{i}e^{\\nu}_{j}e^{\\rho}_{k} n^{\\alpha}_{s}) R_{\\mu\\nu\\alpha\\rho}\\,.\n\\end{equation}\nThe expression in brackets in the first term of the R.H.S is a bulk scalar and therefore this term can be written as \n\\begin{align}\n \\partial_{l}(h^{ik}h^{jl}e^{\\mu}_{i}e^{\\nu}_{j}e^{\\rho}_{k} n^{\\alpha}_{s} R_{\\mu\\nu\\alpha\\rho}) =  -&\\nabla^{i}( R_{\\mu\\nu\\rho\\sigma} n^{\\mu}_s e^{\\nu}_{j}e^{\\rho}_{i}e^{\\sigma}_{k}h^{jk} )+ \\Gamma^{r}_{sl}h^{ik}h^{jl}e^{\\mu}_{i}e^{\\nu}_{j}e^{\\rho}_{k} n^{\\alpha}_{r} R_{\\mu\\nu\\alpha\\rho} + \\nonumber \\\\&\\Gamma _{jl}^{m}h^{ik}h^{jl}e^{\\mu}_{i}e^{\\nu}_{m}e^{\\rho}_{k} n^{\\alpha}_{s} R_{\\mu\\nu\\alpha\\rho}\\,,\n\\end{align}\nInserting these expressions in \\equ{GS2} after expanding the second term on the R.H.S of \\equ{leftovera} and using the identities in \\equ{GaussW} will lead to cancellation of all terms except for the terms in the first line of \\equ{GS2}.\n\n\\subsubsection*{AdS background \\label{ADSeom}}\n\nWe now specialize to the case of AdS background which is the background we will use while finding the equation of motion using the LM method. In AdS space the Riemann tensor is\n\\begin{equation}\n     R_{\\mu\\nu\\rho\\sigma} = -C(g_{\\mu\\rho}g_{\\nu\\sigma}-g_{\\mu\\sigma}g_{\\nu\\rho}) \\,,\n     \\label{ADSRiemann}\n    \n\\end{equation}\nwhere we have defined $C=f_{\\infty}\/{L}^2$. Here, $L$ is the length scale associated with the cosmological constant and is related to the AdS radius $\\tilde L$ as $L =\\tilde L \\sqrt{f_{\\infty}}$. The variable $f_{\\infty}$ satisfies the following equation for $R^2$ theory\n\\begin{equation}\n1-f_\\infty+\\frac{1}{3}f_\\infty^2 (\\lambda_1+2\\lambda_2+10\\lambda_3)=0\\,.\n\\end{equation}\n\nFor ease of comparison with later results, we also rewrite the variation in $\\sqrt{h}R$ given in Eq.~(\\ref{varR}) using the Gauss-Codazzi relation \\equ{GC}. The minimal surface equation is then\n\\begin{align}\n   {\\mathcal K}^{r} + L^2 \\{&\\lambda_3({\\mathcal R} {\\mathcal K}^r - 2{\\mathcal R}^{ij}{\\mathcal K}^{r}_{ij} + 2\\nabla^2 {\\mathcal K}^{r} - 2 \\nabla_i\\nabla_j {\\mathcal K}^{rij} -\\nonumber\\\\& ~~~~ {\\mathcal K}^{r} \\tilde{{\\mathcal K}}_2  + 2 {\\mathcal K}^{r}_{ij} {\\mathcal K}_{2}^{ij} +  {\\mathcal K}^{r} {\\mathcal K}_2 - 2 {\\mathcal K}_3^{r}  - 18 C {\\mathcal K}^{r} ) + \\nonumber \\\\\n   &\\lambda_2( \\tfrac{1}{2} \\nabla^2 {\\mathcal K}^{r}  - \\tfrac{1}{4} {\\mathcal K}^{r} \\tilde{{\\mathcal K}}_2 + \\tfrac{1}{2} {\\mathcal K}^{r}_{ij} {\\mathcal K}_{2}^{ij} - \\tfrac{11}{2} C {\\mathcal K}^{r} ) + \\nonumber \\\\\n   &\\lambda_1(2 \\nabla_i\\nabla_j {\\mathcal K}^{rij} -  {\\mathcal K}^{r} {\\mathcal K}_2 + 2 {\\mathcal K}_3^{r} - 4 C {\\mathcal K}^r ) \\} =0\\,,\n   \\label{R2minimalo}\n\\end{align}\nwhere we have defined ${\\mathcal K}_2= {\\mathcal K}_{sij}{\\mathcal K}^{sij},{\\mathcal K}_{2}^{ij} = {\\mathcal K}_{s}{\\mathcal K}^{sij}, \\tilde{{\\mathcal K}}_2 = {\\mathcal K}_s {\\mathcal K}^{s}$ and ${\\mathcal K}_3^{r}= {\\mathcal K}_{sij}{\\mathcal K}^{si}_k {\\mathcal K}^{rkj}$. Note that these are a set of two equations one for each of the extrinsic curvatures ${\\mathcal K}^1,{\\mathcal K}^2$.\n\nIn AdS space we can make a further simplification using \\equ{codazzi}. The R.H.S of this equation disappears on using \\equ{ADSRiemann}. We then get the identity $\\nabla^k \\nabla_k{\\mathcal K}_{r}= \\nabla^i \\nabla^j {\\mathcal K}_{rij} $ on taking a further covariant derivative of the L.H.S. As explained in Appendix~\\ref{K2}, in the LM method for a time-independent metric, we can set ${\\mathcal K}^{1}={\\mathcal K}^{2}={\\mathcal K}$. We, therefore, also drop the $r$ index and \\equ{R2minimalo} simplifies to\n\\begin{align}\n   {\\mathcal K} + L^2 \\{&\\lambda_3({\\mathcal R} {\\mathcal K} - 2{\\mathcal R}^{ij}{\\mathcal K}_{ij}  - {\\mathcal K}^3 + 3 {\\mathcal K}  {\\mathcal K}_2 - 2 {\\mathcal K}_3 - 18 C {\\mathcal K} ) + \\nonumber \\\\\n   &\\lambda_2( \\tfrac{1}{2} \\nabla^2 {\\mathcal K}  - \\tfrac{1}{4} {\\mathcal K}^3 + \\tfrac{1}{2} {\\mathcal K} {\\mathcal K}_{2} - \\tfrac{11}{2} C {\\mathcal K} ) + \\nonumber \\\\\n   &\\lambda_1(2 \\nabla^2 {\\mathcal K}  -  {\\mathcal K} {\\mathcal K}_2 + 2 {\\mathcal K}_3 - 4 C {\\mathcal K}) \\} =0\\,.\n   \\label{R2minimal}\n\\end{align}\nWe have also verified this equation by determining the bulk extremal surfaces for different types of boundary entangling regions (sphere, cylinder and slab).  \n\nFor the Gauss-Bonnet case: $\\lambda_1=\\lambda, \\lambda_2=-4\\lambda$ and $\\lambda_3=\\lambda$, this equation reduces to the known result in \\equ{GBminimal}. Note that terms cubic in the extrinsic curvature as well as the $C {\\mathcal K}$ terms are not present in that equation. The Gauss-Bonnet case is special in this sense. No such simplification occurs if we set the value for Weyl$^2$ theory, $\\lambda_1=\\lambda,\\lambda_2=-4\\lambda\/3$ and $\\lambda_3=\\lambda\/6$:\n\\begin{equation}\n   {\\mathcal K} + \\frac{\\lambda L^2}{6} ({\\mathcal R} {\\mathcal K} - 2{\\mathcal R}^{ij}{\\mathcal K}_{ij} + 8\\nabla^2 {\\mathcal K}  +   {\\mathcal K}^3   -7 {\\mathcal K} {\\mathcal K}_2  +  10 {\\mathcal K}_3  +2 C{\\mathcal K} )  =0 \\,.\n   \\label{Weyleom}\n\\end{equation}\nThe $C {\\mathcal K}$  term, in particular, stands out. If we trace the provenance of this term, it comes from terms of the form ${\\mathcal K}_{s} R_{\\mu\\nu\\rho\\sigma}n^{\\mu}_{r}n^{\\nu}_{q}n^{\\rho r}n^{\\sigma q}$ and ${\\mathcal K}^{ij}_{s} R_{\\mu\\nu\\rho\\sigma}n^{\\mu}_{r}e^{\\nu}_{i}n^{\\rho}_{s}e^{\\sigma}_{j}$ in Eqs.~(\\ref{varR}) and (\\ref{varK2}) --- terms that have components normal to the surface. Nevertheless, for AdS background these reduce to a term intrinsic to the surface. In fact, using the Gauss-Codazzi identity, Eq.~(\\ref{GC}), we can rewrite this $C {\\mathcal K}$  term as $\\sim {\\mathcal K}^3 + {\\mathcal R}{\\mathcal K}$. \n\nSo far, we have only considered normal variations of the surface. Considering tangential variations leads to a constraint equation. For R$^2$ theory this constraint equation is indistinguishable from the condition in \\equ{codazzi} which is the Codazzi-Mainardi relation.\n\n\n\\subsection{Minimal surface condition from the Lewkowycz-Maldacena method \\label{LMmethod}}\n\nWe will now derive the surface equation for R$^2$ using the LM method. As already mentioned, the main idea of Ref.~\\cite{maldacena} is that one can obtain the minimal surface condition by extending the replica trick to the bulk. \nThe bulk will then have a $Z_n$ symmetry. Orbifolding by this symmetry will lead to a spacetime in which the fixed points form a codimension-2 surface with a conical deficit. In the $n \\rightarrow 1$ limit this surface can be identified with the entangling surface. The metric of this surface can be parametrized in gaussian normal coordinates as follows: \n\\begin{align}\nds^2 =  e^{2\\rho(z,\\bar z)} \\{dz d{\\bar z}~ +~ &e^{2\\rho(z,\\bar z)} {\\Omega} ({\\bar z} dz-z d{\\bar z})^2 \\} + (g_{ij} + {\\mathcal K}_{rij} x^r + {\\mathcal Q}_{rsij} x^r x^s ) dy^i dy^j ~+ \\nonumber \\\\ \n&2 e^{2\\rho(z,\\bar z)} ({\\mathcal A}_i + {\\mathcal B}_{ri} x^r) ({\\bar z} dz-z d{\\bar z}) dy^i + \\cdots \\,.\n\\label{metric2}\n\\end{align}\nHere $\\rho(z,\\bar z)= -\\frac{\\epsilon}{2} \\ln( z\\bar z)$ and $\\epsilon=n-1$, while $x^1=z$ and $x^2=\\bar{z}$.\nThis is the most general form of the metric upto terms second order in $z({\\bar z})$~\\cite{solo,dong,camps, smolkin}. The $\\cdots$ denote higher-order terms. As we will see later, for $R^2$ theory we also need to include third-order terms in the metric expansion.\nThe quantity ${\\mathcal K}_{ij}$ in this metric is identified with the extrinsic curvature, while ${\\mathcal A}_i$ is identified with the twist potential. Both of these are standard quantities that characterize the embedding of the surface in the bulk. The quantities ${\\Omega}, {\\mathcal B}_{ri}$ and ${\\mathcal Q}$ in the second-order terms in the metric are not arbitrary, but can be written in terms of ${\\mathcal K}_{rij},{\\mathcal A}_i$ and the components of the curvature tensors.\n\nThe bulk equation of motion will now contain divergences arising out of the conical singularity of the form $\\frac{\\epsilon}{z},\\frac{\\epsilon}{z^2}$. However, the matter stress-energy tensor is expected to be finite. Therefore, we must set all divergences to zero. This condition fixes the location of the entangling surface. \n\nThe bulk equation of motion for general four-derivative theory is\n\\cite{Sinha:2010pm}:\n\\begin{equation}\nR_{\\alpha\\beta}-\\frac{1}{2}g_{\\alpha\\beta}R- \\frac{6}{L^2}g_{\\alpha\\beta}-\\frac{L^2}{2} H_{\\alpha\\beta}=0\\,,\n\\label{bulkeom}\n\\end{equation}\nwhere\n\\begin{align}\n  H_{\\alpha\\beta}~=~ &\\lambda_1 \\Big(\\frac{1}{2}g_{\\alpha\\beta} R_{\\delta \\sigma \\mu \n  \\nu} R^{\\delta\\sigma \\mu\\nu}-2 R_{\\alpha \\sigma \\delta \\mu}{ R_\\beta}{}^{\\sigma \\delta \n  \\mu}-4 \\hat \\nabla^2  R_{\\alpha \\beta}+2\\hat \\nabla_\\alpha \\hat \\nabla_\\beta  R~+~\\nonumber\\\\&~~~~~4  R_\\alpha^\\delta  R_{\\beta \\delta}+4 R^{\\delta \\sigma} R_{\\delta \\alpha \\beta \\sigma}\\Big) +  \\nonumber \\\\\n  &\\lambda_2\\Big(\\hat\\nabla_{\\alpha}\\hat \\nabla_{\\beta} R+2 R^{\\delta \\sigma} R_{\\delta \\alpha \\beta \\sigma}  - \\hat \\nabla^2  R_{\\alpha\\beta}+\\frac{1}{2} g_{\\alpha\\beta}  R_{\\delta \\sigma} R^{\\delta\\sigma}-\\frac{1}{2} g_{\\alpha\\beta} \\hat \\nabla^2  R\\Big) + \\nonumber \\\\ \n  &\\lambda_3\\Big(-2  R  R_{\\alpha\\beta}+2 \\hat \\nabla_\\alpha \\hat \\nabla_\\beta  R +\\frac{1}{2}g_{\\alpha\\beta} R^2 - 2 g_{\\alpha\\beta} \\hat \\nabla^2  R\\Big) \\,.\n  \\label{Hab}\n\\end{align}\n\n\\subsubsection{Gauss-Bonnet theory revisited}\nOur eventual goal is to find the surface equation of motion for general R$^2$ theory, but it is illuminating to look at the Gauss-Bonnet case first. The Gauss-Bonnet case was addressed in Refs.~\\cite{aks, abs, Chen} using a metric linear in $z({\\bar z})$. In this section, we will find the surface equation of motion for this theory using the metric in \\equ{metric2}. \n\nWe first show that the second-order metric in \\equ{metric2} suffices for Gauss-Bonnet theory and inclusion of higher-order terms in this conical metric will not affect the surface equation of motion that we find for this theory from the LM method. \nThe bulk equation of motion for Gauss-Bonnet theory can be obtained from \\equ{bulkeom} by setting $\\lambda_1=\\lambda,\\lambda_{2}=-4\\lambda$ and $\\lambda_{3}=\\lambda$ giving:\n\\begin{align}\n  H_{\\alpha\\beta}~=~&4  R_\\alpha^\\delta  R_{\\beta \\delta}-4 R^{\\delta \\sigma} R_{\\delta \\alpha \\beta \\sigma}-2  R  R_{\\alpha\\beta} -2 R_{\\alpha \\sigma \\delta \\mu}{ R_\\beta}{}^{\\sigma \\delta \n  \\mu} + \\nonumber \\\\\n& \\tfrac{1}{2}g_{\\alpha\\beta} (R_{\\delta \\sigma \\mu \n  \\nu} R^{\\delta\\sigma \\mu\\nu}-4R_{\\delta \\sigma} R^{\\delta\\sigma}+R^2) \\,.\n \\label{HabGB}\n\\end{align}\nThe surface equation of motion is derived by finding the divergences in this equation that arise on using the conical metric in the limit $z={\\bar z}\\rightarrow 0$. Terms higher than second-order in the metric will not contribute to the curvature tensors to zeroeth-order in $z({\\bar z})$, although they might contribute at higher order. This is because the curvature tensors are of dimension $1\/\\rm{Length}^2$ while third-order terms in the metric will be of order $1\/\\rm{Length}^3$. The explicit values of the curvature tensors are listed in Appendix~(\\ref{append}). These are calculated using a conical metric which is third-order in $z({\\bar z})$. Note also, that the curvature tensors contain at most divergences of order $1\/z$. In the above equation of motion all terms are the product of two curvature tensors. Since each curvature tensor can only contribute at most a $1\/z$ divergence and no third(or higher)-order term occurs at zeroeth order in any curvature tensor, third(and higher)-order  terms will be absent in the divergence equations.\n\nBy the same logic one can see that second-order terms will contribute to the divergence equations. However, in this case, cancellations between terms remove most of the second-order quantities, leaving only the quantities ${\\mathcal Q}_{zzij}$ and ${\\mathcal Q}_{{\\bar z} {\\bar z} ij}$ in the divergence equations.\n\nIn the $z={\\bar z}\\rightarrow 0$ limit ${\\mathcal K}^1={\\mathcal K}^2$ as explained in Appendix~\\ref{K2}, so we drop the index $r$ on ${\\mathcal K}^r$.\nThe divergence in the $zz$ component from $H_{\\alpha\\beta}$ term in the bulk eom is \n\\begin{align}\n~~~~~H_{zz} ~=~\\frac{\\epsilon }{ z}&\\Big[\\lambda ({\\mathcal R}{\\mathcal K}-2 {\\mathcal K}^{ij} {\\mathcal R}_{ij})\\Big]+\\frac{\\epsilon}{z}\\Big[e^{-2\\rho(z,\\bar{z})}\\lambda\\Big\\{-{\\mathcal K}^3+3{\\mathcal K}\\K_{2}-2{\\mathcal K}_3\\Big\\}\\Big] \\,.\n\\label{zzcomponentgb}\n\\end{align}\nSetting this divergence to zero should yield the condition for the extremal surface.\nThere is no divergence in the $z{\\bar z}$ component. The divergence in the $zi$ component is\n\\begin{align}\nH_{zi}~= ~\\frac{\\epsilon }{z}&\\Big[e^{-2\\rho(z,\\bar z)}\\lambda\\Big\\{2{\\mathcal K} \\nabla_i {\\mathcal K} -2{\\mathcal K}\\nabla_j{\\mathcal K}^j_i-2{\\mathcal K}^j_i\\nabla_j {\\mathcal K}+2{\\mathcal K}_{ij}\\nabla_k {\\mathcal K}^{kj}~-~\\nonumber\\\\&~~~~~~~~~~~~~~~2{\\mathcal K}_{kj}\\nabla_i {\\mathcal K}^{kj}+2 {\\mathcal K}_{jk}\\nabla^j {\\mathcal K}^{k}_i\\Big\\}\\Big]\\,.\n\\label{zicomponentgb}\n\\end{align}\nThis divergence is equivalent to the constraint equation one gets for the entropy functional (which doesn't have to be necessarily the Jacobson-Myers functional) using tangential variations of the surface and vanishes similarly by \\equ{codazzi}.\nFinally, the divergence in the $ij$ component is\n\\begin{align}\n~~~~H_{ij}~=~\\frac{4\\epsilon }{z}&\\Big[e^{-4\\rho(z,\\bar z)}\\lambda\\Big\\{2{\\mathcal K}_{ik}{\\mathcal K}^{kl}{\\mathcal K}_{lj}+h_{ij}{\\mathcal K}\\K_2-\n{\\mathcal K}_{ij}{\\mathcal K}_2-h_{ij}{\\mathcal K}_3-{\\mathcal K}\\K_{ik}{\\mathcal K}^{k}_{j}~-~4 h_{ij}{\\mathcal K} {\\mathcal Q}_{zz}\\nonumber\\\\&~~~~~~~~~~~~~~~+4 h_{ij}{\\mathcal K}_{kl}{\\mathcal Q}_{zz}^{kl}~-~\n8 {\\mathcal K}_{ki}{\\mathcal Q}^{k}_{zzj}+\n4 {\\mathcal K}_{ij}{\\mathcal Q}_{zz}~+~4{\\mathcal K}{\\mathcal Q}_{zzij}\\Big\\}\\Big]~+\\nonumber\\\\ \\frac{2\\epsilon^2 }{z^2}&\\Big[e^{-4\\rho(z,\\bar z)}\\lambda\\Big\\{2{\\mathcal K}_{ij}{\\mathcal K}-2{\\mathcal K}_{ik}{\\mathcal K}^{k}_{j}-h_{ij}{\\mathcal K}^2+h_{ij}{\\mathcal K}_2\n\\Big\\}\\Big]\\,.\\label{ijcomponentgb}\n\\end{align}\nIn the above equation we have set ${\\mathcal Q}_{zzij}={\\mathcal Q}_{{\\bar z} {\\bar z} ij}$. Using the value of the $R_{zizj}$ component of the Riemann tensor from Appendix~\\ref{append} and setting the background to AdS space, using \\equ{ADSRiemann}, we can show that ${\\mathcal Q}_{zzij}= \\frac{1}{4} {\\mathcal K}_{ik}{\\mathcal K}^{k}_{j}$ and as a result the $\\frac{\\epsilon}{z}$ divergence exactly vanishes. However the $\\frac{\\epsilon^2 }{z^2}$ divergence will remain. \n\nThe final step is to take the $\\epsilon, z \\rightarrow 0$ limit. Depending on the ordering one chooses, there are two ways to do this.  One way is to take $z\\rightarrow 0$ limit first. Physically, this corresponds to looking for a divergence in the bulk equation of motion while there is still a small but non-zero conical deficit parameter $\\epsilon$.  The second way is take $\\epsilon \\rightarrow 0$ first. The limit is, therefore, an iterated limit -- the final result depends on the order in which the limit is taken, so there is an inherent ambiguity in this procedure. In fact, this ambiguity can be made even larger in scope if we take the limit simultaneously in $\\epsilon$ and $z$. Mathematically, the divergence is a function of the two variables: $\\epsilon$ and $z$. In this $\\epsilon$-$z$ plane there are an infinite number of paths along which we can take the limit. At least on a mathematical level, there exists no reason why the limit should only be taken along the $z = 0$ or $\\epsilon = 0$ path.\n\nThe path $z = 0$ is, however, the simplest way to take the limit so as to obtain $\\frac{\\epsilon}{z} \\rightarrow \\infty$. In this case, all terms containing $\\frac{\\epsilon}{z}$ are leading divergences while terms containing $e^{-2\\rho(z,\\bar z)} \\epsilon\/z= (z{\\bar z})^\\epsilon \\epsilon\/z$ contribute to sub-leading divergences. Therefore, in this way of taking limits, setting the $H_{zz}$ divergence to zero will yield two different conditions for the minimal surface. The first condition which, after adding the Einstein term, corresponds to the surface equation of motion is\n\\begin{equation}\n\t{\\mathcal K} + L^2\\lambda({\\mathcal R}{\\mathcal K}-2 {\\mathcal K}^{ij} {\\mathcal R}_{ij})= 0\\,.\n\\label{GGEeom}\n\\end{equation}\nThis agrees with the surface equation that comes from the Jacobson-Myers functional. However, there will also be an extra constraint~\\cite{erd} of the form  \n\\begin{equation}\n\t-{\\mathcal K}^3+3{\\mathcal K}\\K_{2}-2{\\mathcal K}_3=0\\,,\n\t\\label{ExC}\n\\end{equation}\ncoming from the sub-leading divergence. The $H_{ij}$ divergence will also lead to a similar constraint. The above condition can only be true for very special surfaces and therefore is an over-constraint on the surface. In fact, if these two conditions were to be true simultaneously, the surface equation of motion we would end up getting is:\n\\begin{equation}\n  c\\,{\\mathcal K}  + \\alpha \\lambda L^2 ({\\mathcal K}^3-3{\\mathcal K}\\K_{2}+2{\\mathcal K}_3)\\, =0.\n\\end{equation}\nTo get this form of the equation, we have used the Gauss identities on AdS space. Here, $c = (1-2f_{\\infty}\\lambda)$ is proportional to the Weyl anomaly and $\\alpha$ is a variable that can take any arbitrary numerical value. The surface equation of motion corresponding to the Jacobson-Myers functional can be recovered if $\\alpha=1$. However, at present nothing within the LM method sets the value of this parameter to one. Note that if $\\alpha$ was zero, the minimal surface that we would get is the same as in the Einstein case. It also the minimal surface that would follow if one were minimizing just the Wald part of the entropy functional. \n\nIn the above paragraph we outlined one way in which the LM method could potentially give rise to the correct surface equation of motion. Let us now explore if we can change the limit-taking procedure itself to get the correct equation. This can be accomplished by choosing a different path in the $\\epsilon$-$z$ plane to take the limit. Taking the limit along the path $\\epsilon = 0$ will simply kill off all divergences; this is not surprising since physically this corresponds to turning off the conical deficit in the metric. \nHowever, we can pick a path in the $\\epsilon, z$ plane that will kill off the sub-leading divergence but preserve the leading divergence. For example, as was shown in Ref.~\\cite{aks}, taking any path of the form $(z)^{2\\epsilon}={(\\frac{z}{\\epsilon})}^{1+v}$, where $v$ is a number greater than one, will keep only the leading divergence. At this point, we can offer no justification of why one should choose this particular way of taking limits. We are merely demonstrating that there does exist a way to take limits in the $\\epsilon,z$ plane that leads to the correct surface equation of motion in the Gauss-Bonnet case. This way of taking limits is equivalent to discarding terms suppressed by $e^{-2\\rho(z,\\bar z)}$ and was also used in Ref.~\\cite{dong} to show that the LM method leads to the same surface equation of motion in the Lovelock case as can be derived from the entropy functional for Lovelock gravity~\\cite{dong,Kang,Sarkar}. This is the way of taking limits that we will use. However, unless one can specify a mechanism or a physical interpretation which reproduces this way of taking limits (which is possible if the metric itself is re-defined), the argument that the LM method reproduces the correct surface equation of motion for Gauss-Bonnet theory remains incomplete.\n\nThe same ambiguity in taking limits exists for general R$^2$ theory. To remain consistent with the Gauss-Bonnet point, for R$^2$ theory we will continue to take limits as stated in the paragraph above. However, in the general case this is not an ideal solution. As we will see, $\\sim {\\mathcal K}^3$ terms always occur with the $e^{-2\\rho(z,\\bar z)}$ factor in the divergence equations for R$^2$ theory. This means that if we use the above way of taking limits we will never get such terms at any point in the parameter space. As we saw in \\equ{R2minimal}, the surface equation of motion for R$^2$ theory does contain such terms. However, our goal for general R$^2$ theory is to see to what extent we are able to reproduce the surface equation of motion in \\equ{R2minimal}, while taking the limit in such a manner that the result at the Gauss-Bonnet point agrees with what comes from the Jacobson-Myers functional. \nIt is clear, though, that the question of taking limits in the LM method deserves more study.\n\n\\subsubsection{The general case \\label{newmetric}}\nWe now work out the divergence equations for the R$^2$ case.\nFor general R$^2$ theory all second-order quantities will enter into the divergences. We can anticipate the effect that terms containing $\\Omega$ and ${\\mathcal B}$ will have on the surface equation of motion coming from the LM method. Consider the following components of the bulk Riemann tensor around $z={\\bar z}=0$:\n\\begin{align}\nR_{pqrs}\\Big|_{(z=0,{\\bar z}=0)} &= -3 e^{4\\rho(z,\\bar z)} \\hat\\varepsilon_{pq} \\hat\\varepsilon_{rs} {\\Omega} \\,,\\nonumber\\\\\nR_{pqri}\\Big|_{(z=0,{\\bar z}=0)} &= 3 e^{2\\rho(z,\\bar z)} \\hat\\varepsilon_{pq} {\\mathcal B}_{ri} \\,,\n\\label{Riemanns1}\n\\end{align}\nwhere, $\\hat\\varepsilon_{ab}$ is defined as $\\hat\\varepsilon_{z{\\bar z}}=-\\hat\\varepsilon_{{\\bar z} z}= e^{-2\\rho(z,\\bar z)} g_{z{\\bar z}}$. The quantity $\\Omega$ is therefore equivalent to $-1\/3 R_{\\mu\\nu\\rho\\sigma}n^{\\mu}_{r}n^{\\nu}_{s}n^{\\rho}_{r}n^{\\sigma}_{s}$ evaluated at $z,{\\bar z}=0$. We can determine a numerical value for the quantities  ${\\mathcal B}^{r}_{i} $ and ${\\Omega}$ in the metric by demanding that the bulk Riemann tensor be the AdS solution at zeroeth order. Since for AdS space the Riemann tensor is given by \\equ{ADSRiemann}, we can write the components of the bulk Riemann tensor on the L.H.S of \\equ{Riemanns1} in terms of the components of the bulk metric. Expanding the metric using \\equ{metric2} and keeping only the zeroeth-order terms in $z,{\\bar z}$ we get\n\\begin{equation}\n\t{\\Omega} = -\\frac{1}{12} C \\;\\;\\; \\textrm{and}  \\;\\;\\; {\\mathcal B}_{ri} = 0\n\t\\label{Tvalue}\n\\end{equation}\nTherefore, ${\\mathcal B}^{r}_{i}$ can be set to zero. In writing the divergences, we also ignore\\footnote{As we saw for the Gauss-Bonnet case, these terms will be present in the divergences, but they will not change our conclusions for $R^2$ theory.} $Q^{zz}_{ij}$ and $Q^{{\\bar z}\\zb}_{ij}$, the remaining component being $Q_{ij}=Q^{{\\bar z} z}_{ij}=Q^{z{\\bar z}}_{ij}$. \n\nFor R$^2$ theory the derivative order of the equation of motion is four. That means we should include order $z^3$ terms in the metric, since they can contribute to the divergences. These terms can be parametrized as\n\\begin{equation}\nds^2 =  e^{4\\rho(z,\\bar z)} \\Delta_{pqrst} x^p x^q x^r dx^s dx^t + {\\mathcal W}_{rspij} x^r x^s x^p  dy^i dy^j + 2 e^{2\\rho(z,\\bar z)}  {\\mathcal C}_{rsi} x^r x^s ({\\bar z} dz-z d{\\bar z}) dy^i \\,.\n\\label{metric3}\n\\end{equation}\nThis is the most general form of the third-order terms in the metric. Here, we have written the $e^{2\\rho(z,\\bar z)} $ dependence of each term explicitly. As for the second-order quantities, the third-order quantities $ \\Delta_{pqrst}, {\\mathcal W}_{rspij} $ and ${\\mathcal C}_{rs i} $ can be found by calculating the curvature tensors, but to linear order in $z({\\bar z})$. Then, for example, $e^{4\\rho(z,\\bar z)}\\Delta_{pqrst} \\equiv -1\/6 \\partial_p ({R_{\\mu\\nu\\rho\\sigma}n^{\\mu}_{q}n^{\\nu}_{r}n^{\\rho}_{s}n^{\\sigma}_{t}})$ evaluated at $z={\\bar z}=0$. Note that the factor of $e^{4\\rho(z,\\bar z)}$ will cancel from both sides on using the AdS background. In fact, this particular term vanishes altogether in this background. On using the metric with the third-order terms listed above to find the divergences in the equation of motion we find that the ${\\mathcal C}_{rs i} $,  ${\\mathcal W}_{zzzij}$ and ${\\mathcal W}_{{\\bar z}\\zb{\\bar z} ij}$ do not contribute. The terms that are relevant are ${\\mathcal W}_{zz{\\bar z} ij}$ and ${\\mathcal W}_{{\\bar z}\\zb z ij}$, because as will show below they will lead to unsuppressed $C {\\mathcal K}$ terms. Without loss of generality, we can set them to be equal and denote this term as ${\\mathcal W}_{ij}$. \n\nFor general R$^2$ theory, the divergence in the $zz$ component from the $H_{\\alpha\\beta}$ term in the bulk eom is \n\\begin{align}\n~~~~~H_{zz} ~=~\\frac{\\epsilon }{ z}&\\Big[-\\tfrac{1}{2}(\\lambda_{2}+4\\lambda_{3})\\nabla ^2{\\mathcal K}+(2\\lambda_1+\\lambda_2+2\\lambda_3)\\nabla^i\\nabla^j{\\mathcal K}_{ij}+\\lambda_{3}({\\mathcal R}{\\mathcal K}-2 {\\mathcal K}^{ij} {\\mathcal R}_{ij})~+~\\nonumber\\\\&~~~ 4(-2\\lambda_1+3\\lambda_2+14\\lambda_3){\\mathcal K}_{ij}\\mathcal{A}^i\\mathcal{A}^j-6(\\lambda_2+4\\lambda_3){\\mathcal K} \\mathcal{A}^i\\mathcal{A}_i~+\\nonumber\\\\&~~~\n  8(3\\lambda_1+2\\lambda_2+5\\lambda_3){\\mathcal K} \\Omega  \\Big]~-\\nonumber\\\\\\frac{\\epsilon}{z^2}&\\Big[e^{-2\\rho(z,\\bar{z})}\\Big\\{(2\\lambda_{1}-\\lambda_{2} -6\\lambda_{3} ) {\\mathcal K}_2+ \\tfrac{1}{2}(\\lambda_{2}+4\\lambda_{3}){\\mathcal K}^2+2(\\lambda_2+4\\lambda_3) {\\mathcal Q}\\Big\\}\\Big]~+\\nonumber\\\\\n\\frac{\\epsilon}{z}&\\Big[e^{-2\\rho(z,\\bar{z})}\\Big\\{-\\lambda_{3}{\\mathcal K}^3+(\\lambda_{2}+7 \\lambda_{3}){\\mathcal K}\\K_{2}-2(3\\lambda_{1}+2\\lambda_{2}+6\\lambda_{3}){\\mathcal K}_3~+\\nonumber\\\\\n&~~~~~~~~~~~~~~~~(6\\lambda_{1}+5\\lambda_{2}+14\\lambda_{3}){\\mathcal K}_{ij}{\\mathcal Q}^{ij}-\\tfrac{3}{2}(\\lambda_{2}+4\\lambda_{3}){\\mathcal K}{\\mathcal Q}~-\\nonumber\\\\&~~~~~~~~~~~~~~~~~4(\\lambda_2 +4 \\lambda_3) {\\mathcal W}\\Big\\}\\Big] \\,.\n\\label{zzcomponentm2}\n\\end{align}\nThe divergences in the other components are\n\\begin{align}\n\\hspace{-.5cm}H_{z\\bar z}~= ~\\frac{2 \\epsilon}{z}&\\Big[ e^{-2\\rho(z,\\bar z)}\\Big\\{(\\lambda_{3}+\\tfrac{1}{4}\\lambda_2){\\mathcal K}^3+(\\lambda_1-\\tfrac{3}{2}\\lambda_2-7\\lambda_3){\\mathcal K}\\K_{2}\\nonumber~+\\\\&~~~~~~~~~~~~~2(-2\\lambda_1+\\lambda_2+6\\lambda_3){\\mathcal K}_{3}+(2\\lambda_1-3\\lambda_2-14\\lambda_3){\\mathcal K}_{kl}{\\mathcal Q}^{kl}\\nonumber~+\\\\&~~~~~~~~~~~~~\\tfrac{3}{2}(\\lambda_2+4\\lambda_3){\\mathcal K}{\\mathcal Q} +  8( \\lambda_2+4\\lambda_3) {\\mathcal W} \\Big\\}\\Big]\\,,\n\\label{zbzcomponentm2}\n\\end{align}\n\\begin{align}\n~~~~~H_{zi}~=~ \\frac{2\\epsilon }{z}&\\Big[e^{-2\\rho(z,\\bar z)}\\Big\\{-\\tfrac{1}{2}(2\\lambda_1+\\lambda_2){\\mathcal K}^{k}_{i}\\nabla_{k}{\\mathcal K}-(3\\lambda_1-\\lambda_{2}-6\\lambda_3){\\mathcal K}^{kl}\\nabla_{i}{\\mathcal K}_{kl}~-\\nonumber\\\\&~~~~~~~~~~~~~\\tfrac{1}{4}(3\\lambda_2+8\\lambda_3){\\mathcal K}\\nabla_i{\\mathcal K}+\n(5\\lambda_{1}+\\lambda_{2}){\\mathcal K}^{k}_{i}\\nabla_{l}{\\mathcal K}^{l}_{k}-\\lambda_{1}{\\mathcal K}\\nabla_{k}{\\mathcal K}^{k}_{i}\n~+\\nonumber\\\\&~~~~~~~~~~~~~~(9\\lambda_1+2\\lambda_2){\\mathcal K}^{kj}\\nabla_{k}{\\mathcal K}_{ji}-(\\lambda_{2}+4\\lambda_{3})\\nabla_{i}{\\mathcal Q}-(4\\lambda_{1}+\\lambda_{2})\\nabla_{k}{\\mathcal Q}^{k}_{i}\n~-~\\nonumber\\\\&~~~~~~~~~~~~~~(10\\lambda_1-2\\lambda_2-18\\lambda_3) \\mathcal{A}_i{\\mathcal K}_2-\\tfrac{1}{2}(3\\lambda_2+12\\lambda_3) \\mathcal{A}_i{\\mathcal K}^2~+~\\nonumber\\\\&~~~~~~~~~~~~~~8(4\\lambda_1+\\lambda_2){\\mathcal K}_{ij}{\\mathcal K}^{jk}\\mathcal{A}_{k}\n-2(\\lambda_2+4\\lambda_3)\\mathcal{A}_{i}{\\mathcal Q}\\Big\\}\\Big]\\,,\n\\label{zicomponent}\n\\end{align}\n\\begin{align}\n~~~H_{ij}~=~\\frac{4\\epsilon }{z}&\\Big[e^{-4\\rho(z,\\bar z)}\\Big\\{(\\tfrac{1}{3}\\lambda_{1}+\\tfrac{1}{4}\\lambda_{2}+\\tfrac{2}{3}\\lambda_3)h_{ij}{\\mathcal K}^3-(7\\lambda_1+2\\lambda_2+2\\lambda_3){\\mathcal K}\\K_{ik}{\\mathcal K}^{k}_{j}~+\\nonumber\\\\&~~~~~~~~~~~~~2(16\\lambda_{1}+4\\lambda_{2}+\\lambda_3){\\mathcal K}_{ik}{\\mathcal K}^{kl}{\\mathcal K}_{lj}-(\\lambda_1+3\\lambda_2+10\\lambda_3)h_{ij}{\\mathcal K}\\K_2~-\\nonumber\\\\&~~~~~~~~~~~~~\n(3\\lambda_1-2\\lambda_3){\\mathcal K}_{ij}{\\mathcal K}_2-\\tfrac{1}{3}(\\lambda_1-18\\lambda_2-70\\lambda_3)h_{ij}{\\mathcal K}_3~+\\nonumber\\\\&~~~~~~~~~~~~~2(4\\lambda_{1}+\\lambda_{2}){\\mathcal Q}_{ij}{\\mathcal K}+2(\\lambda_{2}+4\\lambda_{3})h_{ij}{\\mathcal K}{\\mathcal Q}-8(4\\lambda_{1}+\\lambda_2){\\mathcal K}_{ik}{\\mathcal Q}^{k}_{j}\n~-\\nonumber\\\\&~~~~~~~~~~~~~(\\lambda_{2}+4\\lambda_{3}){\\mathcal K}_{ij}{\\mathcal Q}-7(\\lambda_{2}+4\\lambda_{3})h_{ij}{\\mathcal K}_{kl}{\\mathcal Q}^{kl} + 32 (4\\lambda_1+\\lambda_2) {\\mathcal W}_{ij} + \\nonumber \\\\\n&~~~~~~~~~~~~~32(\\lambda_2+4\\lambda_3)h_{ij} {\\mathcal W} \\Big\\}\\Big]\\,.\n\\label{ijcomponent}\n\\end{align}\nWhether or not the divergences in the $ij$, $zi$ and $z{\\bar z}$ components vanish before taking the $\\epsilon\\rightarrow 0$ limit will depend upon the exact values of the second-order terms. The $zi$ divergence, in particular, should be equivalent to the constraint equation coming from the tangential variations and should vanish by the Codazzi relation in \\equ{codazzi}. As in the Gauss-Bonnet case, the divergence in the $ij$ component is not expected to fully vanish by itself. We therefore take the limit as prescribed in the last section. This reduces the divergences in the $zi$ and $ij$ components to zero. However, because of the presence of the ${\\mathcal W}$ term there still remains an unsuppressed divergence in the $z{\\bar z}$ component. This divergence can only go to zero if ${\\mathcal K}=0$ or the theory is at the Gauss-Bonnet point.\n\nWe now examine the divergences in the $zz$ component, to be able to compare it with the surface equation of motion derived using the FPS functional. First looking at the $1\/z$ divergence in that component, one can see that it contains the unsuppressed terms ${\\mathcal K}_{ij} {\\mathcal A}^i {\\mathcal A}^j$ and ${\\mathcal K} {\\mathcal A}^i {\\mathcal A}_i$ which are not present in \\equ{R2minimal}. However, these terms can be eliminated in favor of other variables. Consider the $R_{zi{\\bar z} j}$ component of the Riemann tensor expanded around $z=0,{\\bar z}=0$:\n\\begin{equation}\n  R_{zi{\\bar z} j}\\Big|_{(z=0,{\\bar z}=0)} = \\tfrac{1}{2} e^{2\\rho(z,\\bar z)} \\mathcal{F}_{ij} - 2 e^{2\\rho(z,\\bar z)} {\\mathcal A}_i {\\mathcal A}_j  + \\tfrac{1}{4} {\\mathcal K}_{zik} {\\mathcal K}^k_{{\\bar z} j} -\\tfrac{1}{2} {\\mathcal Q}_{z {\\bar z} ij} \\,.\\label{Riemanns21}\n\\end{equation}\nUsing \\equ{ADSRiemann} again and multiplying both sides by ${\\mathcal K}_{ij}$, we find that the $\\mathcal{A}_i \\mathcal{A}_j {\\mathcal K}^{ij}$ term can be written as $\\sim C{\\mathcal K} + e^{-2\\rho(z,\\bar z)} {\\mathcal K}^3 + e^{-2\\rho(z,\\bar z)} {\\mathcal Q}{\\mathcal K}$. The ${\\mathcal A}_i {\\mathcal A}^i {\\mathcal K}$  terms can be written in a similar fashion. Since only the $C{\\mathcal K}$ term is unsuppressed we find\n\\begin{align}\n\t\\mathcal{A}_i \\mathcal{A}_j {\\mathcal K}^{ij} &= \\frac{C{\\mathcal K}}{4} + \\cdots\\;\\; {\\rm and } \\nonumber \\\\\n\t{\\mathcal A}_i {\\mathcal A}^i {\\mathcal K} &= \\frac{3C{\\mathcal K}}{4} + \\cdots\\,,\n\t\\label{a2kvalue}\n\\end{align}\nwhere the dots denote the suppressed terms. \n\nNext looking at the $e^{-2\\rho(z,\\bar z) }\/z$ divergence we find that the ${\\mathcal W}$ term will contribute to the surface equation of motion, since this term contains a $e^{2\\rho(z,\\bar z)}$ factor that enhances the divergence to $1\/z$.\nThis term can be determined by using the following equation\n\\begin{equation}\n\t\\partial_z R_{{\\bar z} {\\bar z}}\\Big|_{(z=0,{\\bar z}=0)} = - {\\mathcal W} + 2  e^{2\\rho(z,\\bar z)} {\\mathcal K}^{ij}{\\mathcal A}_i {\\mathcal A}_j  - 2  e^{2\\rho(z,\\bar z)}\\Omega {\\mathcal K} + \\cdots.\n\\end{equation}\nThe R.H.S of this equation disappears in the AdS background. Using Eqs.~(\\ref{Tvalue}) and (\\ref{a2kvalue}) we find\n\\begin{equation}\n\t{\\mathcal W}= \\frac{2  e^{2\\rho(z,\\bar z)} C{\\mathcal K}}{3} + \\cdots\\,.\n\\end{equation}\nThe ${\\mathcal Q}$ terms that are also present in this divergence do not contribute since as we show below they are expected to contain only $\\sim {\\mathcal K}^2$ terms and therefore remain suppressed.\n\nSubstituting these values in \\equ{zzcomponentm2}, and adding the Einstein term we find that the $1\/z$ divergence of the $zz$ component gives rise to the following surface equation of motion:\n\\begin{equation}\n{\\mathcal K} + L^2\\{(2\\lambda_{1}+\\tfrac{1}{2}\\lambda_{2})\\nabla ^2{\\mathcal K}+\\lambda_{3}({\\mathcal R}{\\mathcal K}-2 {\\mathcal K}^{ij} {\\mathcal R}_{ij}) + \\lambda_1 C_1 {\\mathcal K} + \\lambda_2 C_2 {\\mathcal K} + \\lambda_3 C_3{\\mathcal K}\\}=0\\,\n\\label{GGEeom2}\n\\end{equation}\nwhere $C_1=-4C$, $C_2=-11 C\/2 $ and $C_3=-18 C $. The coefficients of the $\\nabla^2{\\mathcal K}, {\\mathcal R}{\\mathcal K}$, ${\\mathcal K}^{ij} {\\mathcal R}_{ij}$ and $C{\\mathcal K}$ terms in the above equation all match with those in \\equ{R2minimal}. Because of the way we are taking limits, the ${\\mathcal K}^3$ terms that are present in \\equ{R2minimal} are not present here.\n\nFinally we look at the $\\epsilon\/z^2$ divergence in the $zz$ component. For this divergence to vanish, we get the condition\n\\begin{equation}\n  (2\\lambda_{1}-\\lambda_{2} -6\\lambda_{3} ) {\\mathcal K}_2+ \\tfrac{1}{2}(\\lambda_{2}+4\\lambda_{3}){\\mathcal K}^2+2(\\lambda_2+4\\lambda_3) {\\mathcal Q} =0 \\,.\n  \\label{ExtraDiv}\n\\end{equation}\nTo satisfy this condition at arbitrary points of the parameter space, one has to demand that ${\\mathcal Q}$ be a function of $\\sim {\\mathcal K}^2$ terms, and also $\\lambda_1$, $\\lambda_2$ and $\\lambda_3$. Demanding that ${\\mathcal Q}$ be independent of  $\\lambda_1$, $\\lambda_2$ and $\\lambda_3$, will pick out a special point in the parameter space (apart from the Gauss-Bonnet point where this condition is trivially satisfied). \n\nTo summarize the results for R$^2$ theory:\n\\begin{enumerate}\n \\item Apart from the absence of $\\sim{\\mathcal K}^3$ terms, \\equ{GGEeom2} that we found using the LM method is exactly the surface equation of motion that results from the FPS functional.\n \\item There are some problematic extra divergences. The $z {\\bar z}$ component of the bulk equation of motion has a divergence that can only disappear at the Gauss-Bonnet point. There is also a second-order $1\/z^2$ divergence in the $zz$ component. This can be taken to fix the value of the term ${\\mathcal Q}$; however, it is not possible to do this in a way that is independent of the parameters of R$^2$ theory.\n\\end{enumerate}\n\n\n\\subsection{The stress-energy tensor from the brane interpretation \\label{deltas}}\nIn Ref.~\\cite{maldacena}, it was noted that a equation of motion of a cosmic string is the same as the equation for the minimal entangling surface. This is because a cosmic string produces a spacetime with a conical defect with a metric of the form in \\equ{metric2}. The equation of motion is given by minimizing its action. For Einstein gravity this is just the Nambu-Goto action and equation of motion of a cosmic string is\n\\begin{equation}\n    {\\mathcal K} = 0.\n\\end{equation}\nThis condition minimizes the surface area of the string as it sweeps through spacetime. The same thing holds for a cosmic brane.\n\nAs was done in Ref.~\\cite{dong}, where it was referred to as the cosmic brane method, this fact can be exploited to construct the entropy functional from the bulk equation of motion. In this section, we will check this construction of Ref.~\\cite{dong}. The idea is that the bulk equation of motion in \\equ{bulkeom} should lead to the cosmic brane as a solution, to linear order in $\\epsilon$. In particular, this means that L.H.S of \\equ{bulkeom} should be equal to the stress-energy tensor of the brane. Since the brane is a localized source, the stress-energy tensor will contain delta functions. Once we have found the stress-energy tensor we can identify the associated action via $T_{\\alpha \\beta} = \\frac{ \\delta{S}}{\\delta g_{\\alpha \\beta}}$. \n\nLet us see how this works in the Gauss-Bonnet case. In the bulk equation of motion, terms such as $\\partial_{\\bar{z}}\\partial_{z}\\rho(z,\\bar z)$ correspond to delta functions. We set $\\delta(z,\\bar z) = e^{-2\\rho(z,\\bar{z})} \\partial_{\\bar{z}}\\partial_{z}\\rho(z,\\bar z)$. Note that $\\delta(z,\\bar z)$ defined this way contains a factor of $\\epsilon$. \n\nThe delta divergences in the $ij$ component of the bulk equation of motion to linear order in $\\epsilon$ are then:\n\\begin{align}\nT_{ij} =\\delta(z,\\bar z)\\Big\\{  &-4 \\, \\lambda \\,(h_{ij} {\\mathcal R} -2 {\\mathcal R}_{ij}) +\\nonumber \\\\\n& -2 \\,\\lambda\\, e^{-2\\rho(z,\\bar{z})}(h_{ij}{\\mathcal K}_{2}- h_{ij}{\\mathcal K}^2+2{\\mathcal K}_{ij}{\\mathcal K}-2{\\mathcal K}_{ik}{\\mathcal K}^{k}_{j} )\\Big\\}\\,.\n\\end{align}\nTo identify this as the stress-energy tensor coming from the Jacobson-Myers functional (interpreted as a cosmic brane action), the second term should go to zero. This term carries a factor of $e^{-2\\rho(z,\\bar{z})}$ as compared to the first term and according to our way of taking limits is suppressed.  \nOur result is then in agreement with the claim in Ref.~\\cite{dong} that the cosmic-brane method can be used to show that the Jacobson-Myers functional is the right entropy functional for Gauss-Bonnet theory. \n\nHowever, as we will see there are problems for the general four-derivative theory. For R$^2$ theory, the delta divergences in the $ij$ component are\n\\begin{align}\n~T_{ij}~=~ &\\delta(z,\\bar z) \\Big[ -4 \\,\\lambda_3 \\,\\,(h_{ij} {\\mathcal R} -2 {\\mathcal R}_{ij}) - 16 (6\\lambda_1+11\\lambda_2+38\\lambda_3)h_{ij} \\Omega ~+\n\\nonumber\\\\&~~~~~~~ ~~~  e^{-2\\rho(z,\\bar z)}\\Big\\{-(\\lambda_2+2\\lambda_3) h_{ij}{\\mathcal K}^2+2(\\lambda_2+3\\lambda_3)h_{ij}{\\mathcal K}_2~-\n\\nonumber\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~2(12\\lambda_1+4\\lambda_2+4\\lambda_3){\\mathcal Q}_{ij}-2 (\\lambda_2+4\\lambda_3){\\mathcal Q} h_{ij}~-\n\\nonumber\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~2(4\\lambda_1+\\lambda_2+2\\lambda_3){\\mathcal K}_{ij}{\\mathcal K}+2(14\\lambda_1+4\\lambda_{2}+4\\lambda_3){\\mathcal K}_{kj}{\\mathcal K}^{k}_{j} )~+\n\\nonumber\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~16(20\\lambda_1+11\\lambda_2+24\\lambda_3)\\mathcal{ A}_{i}\\mathcal{A}_{j}\\Big\\}~ \\Big]\n+\\nonumber\\\\&e^{-2\\rho(z,\\bar z)}\\Big\\{\\partial_{z}\\delta(z,\\bar z)+\\partial_{\\bar z}\\delta(z,\\bar z)\\Big\\}\\Big\\{-2(2\\lambda_{1}+\\lambda_2+2\\lambda_{3}){\\mathcal K}_{ij}+(4\\lambda_3+\\lambda_2) h_{ij}{\\mathcal K}\\Big\\}~\n-\\nonumber\\\\&4e^{-2\\rho(z,\\bar z)}\\,\\partial_{z}\\partial_{\\bar z}\\delta(z,\\bar z)(\\lambda_2+4\\lambda_3)\\,.\n\\end{align}\nAgain, barring the term suppressed by $e^{-2\\rho(z,\\bar{z})}$, we have checked that the result for this component is of the same form as that produced on calculating the stress-energy tensor from an action equivalent to the FPS functional. The derivative of delta terms like $\\partial_{z}\\delta(z,\\bar z)$ are typical in the stress-energy tensor of actions containing terms that depend on the extrinsic curvature \\cite{Stringcurv2}.  \nHowever, the $zz$  and $z{\\bar z}$ components of the bulk equation of motion also contain delta divergences that are not suppressed:\n\\begin{equation}\nT_{zz}~=~-4\\partial_{z}^{2}\\delta(z,\\bar z)(2\\lambda_{1}+\\lambda_{2}+2\\lambda_{3})-2 \\partial_{z}\\delta(z,\\bar z)(4\\lambda_{1}+\\lambda_2){\\mathcal K}\n\\end{equation}\nand\n\\begin{align}\nT_{z\\bar z}~=~-2 &\\big\\{\\partial_{z}\\delta(z,\\bar z)+\\partial_{\\bar z}\\delta(z,\\bar z)\\big\\}\\big(2\\lambda_1+\\lambda_2+2\\lambda_3\\big){\\mathcal K}~+\\nonumber\\\\&4\\,\\partial_{z}\\partial_{\\bar z} \\delta (z,\\bar z)(2\\lambda_1+\\lambda_2+2\\lambda_3)\\,.\n\\end{align}\nTaking the delta divergences in all components into account, the $T_{\\mu \\nu}$ we have found does not look like the stress-energy tensor for a cosmic brane corresponding to a three-dimensional surface in the five-dimensional bulk. Note that the extra divergences all vanish for the Gauss-Bonnet theory. The Gauss-Bonnet result therefore stands. However, any attempt to use this method to show that the FPS functional is the correct entropy functional for R$^2$ theory should be able to account for these extra delta divergences.   \n\n\\section{Quasi-topological gravity \\label{Quasi}}\nThe lagrangian for quasi-topological gravity~\\cite{quasi} contains terms cubic in the Riemann tensor. It can be used to study a class of CFT's involving three parameters in four dimensions. It has many interesting features including the fact that its linearized equation of motion is two-derivative order. Unitarity for this theory was studied in Ref.~\\cite{Sisman}.\n\nIn Sec.~(\\ref{qef}), we find the HEE functional for quasi-topological theory using \\equ{eei} and compute the universal terms is Sec.~(\\ref{uef}). In Sec.~(\\ref{eomef}), we find the surface equation of motion for this theory using the LM method.\n\\subsection{The entropy functional \\label{qef}}\nThe action for quasi-topological theory in five dimensions is \n\\begin{equation}\nS_{QT} =-\\frac{1}{2\\ell_P^3}\\int d^5 x \\Big( \\mathcal{L}_1 + \\mathcal{L}_2 + \\nu \\,Z_{5}\\Big)\\,,\n\\end{equation}\nwhere $\\mathcal{L}_1$ is the Einstein-Hilbert action given in \\equ{Ei} and $\\mathcal{L}_2$ is the Gauss-Bonnet lagrangian as in \\equ{Lr} with $\\lambda_1=\\lambda_3=\\lambda\\,, \\lambda_2=-4\\lambda$.\nThe last term is the R$^3$ lagrangian:\n\\begin{align}\nZ_5=&\\mu_{0} R_{\\alpha\\beta}{}^{\\gamma\\delta}R_{\\gamma\\delta}{}^{\\mu\\nu}R_{\\mu\\nu}{}^{\\alpha\\beta}+\\mu_1 R_{\\alpha}{}^{\\beta}{}_{\\gamma}{}^\\delta \nR_{\\beta}{}^{\\eta}{}_{\\delta}{}^{\\zeta} R^{\\alpha}{}_{\\eta}{}^{\\gamma}{}_{\\zeta} +\n\\mu_2 R_{\\alpha \\beta \\gamma \\delta}R^{\\alpha \\beta \\gamma \\delta} R~+~\\nonumber \\\\&\\mu_3 R_{\\alpha \\beta \\gamma \n\\delta}R^{\\alpha\n\\beta \\gamma}{}_{\\eta}R^{\\delta \\eta}+\\mu_4 R_{\\alpha \\beta \\gamma \\delta} R^{\\alpha \n\\gamma}R^{\\beta \\delta}\n+\\mu_5 R_\\alpha{}^{\\beta}R_\\beta{}^{\\gamma}R_\\gamma{}^{\\alpha} +\\mu_6 \nR_\\alpha^{\\,\\,\\beta}R_\\beta^{\\,\\,\\alpha}R+\\mu_7 R^3 \\label{Lrr} \\,.\n\\end{align}\nThere are two different consistent R$^3$ theories. For the first theory\n\\begin{equation} \\label{th}\n\\mu_0=0\\,,~\n\\mu_{1}=1\\,,~\\mu_{2}=\\tfrac{3}{8}\\,,~\\mu_{3}=-\\tfrac{9}{7}\\,,~\\mu_{4}=\\tfrac{15}{7}\\,,~\n\\mu_{5}=\\tfrac{18}{7}\\,,~\\mu_{6}=-\\tfrac{33}{14}\\,,~\\mu_{7}=\\tfrac{15}{56}\n\\end{equation}\nand the coupling constant is $\\nu=\\frac{ 7 \\mu L^4}{4}$, while for the second theory\n\\begin{equation} \\label{th1}\n\\mu_0=1\\,,~\n\\mu_{1}=0\\,,~\\mu_{2}=\\tfrac{3}{2}\\,,~\\mu_{3}=-\\tfrac{60}{7}\\,,~\\mu_{4}=\\tfrac{72}{7}\\,,~\n\\mu_{5}=\\tfrac{64}{7}\\,,~\\mu_{6}=-\\tfrac{54}{14}\\,,~\\mu_{7}=\\tfrac{11}{14}\n\\end{equation}\nand the coupling constant $\\nu=\\frac{ 7 \\mu L^4}{8}\\,.$\n\nThe R$^3$ part for the HEE functional is\n\\begin{align} \n\\begin{split}\n\\label{R3}\nS_{\\rm EE,\\, R^3}&= \\frac{2\\pi \\nu}{\\ell_P^{3}}\\int d^{3}x \\sqrt{h}\\, \\left(\\mathcal{L}_{\\rm Wald,\\,R^3}+\\mathcal{L}_{\\rm Anomaly,\\, R^3}\\right)\\,,\n\\end{split}\n\\end{align}\nwhere\n\\beq\n\\mathcal{L}_{\\rm Wald, R^3}&=&6\\mu_0 R^{ z \\bar z \\alpha \\beta}R_{  z \\bar z \\alpha\\beta}+3 \\mu_1 \\big(R^{z \\alpha \\bar z}{}_{ \\beta}R_{z \\alpha \\bar z }{}^{\\beta }-R^{z \\alpha z}{}_{ \\beta}R_{ z \\alpha z}{}^\\beta\\big)\n+\\mu_2\\big(R_{\\alpha\\beta\\rho\\sigma}R^{\\alpha\\beta\\rho\\sigma}-\\nonumber \\\\\n&&4 R\\,R^{z \\bar z }{}_{\\bar z  z}\\big)+2\\mu_3 \\big(R_{\\alpha}{}^{z}{}_{\\bar z }{}^{\\bar z} R^{ \\alpha}{}_{  z}-R_{\\alpha}{}^{\\bar z}{}_{z}{}^{z} R^{ \\alpha}{}_{\\bar z}+ \\tfrac{1}{2} R_{\\alpha\\beta\\rho }{}^{\\bar z} R^{\\alpha\\beta\\rho }{}_{\\bar z}\\big) +\n\\mu_4 (2 R^{z}{}_{\\alpha}{}_{ z}{}_{ \\beta}R^{\\alpha\\beta}+\\nonumber\\\\\n&& \n(R^{z}{}_{ z})^2- R^{zz}R_{ z  z})+  3 \\mu_5 R^{z \\alpha}R_{z\\alpha}+\n\\mu_6 \\big (R_{\\alpha\\beta}R^{\\alpha\\beta}+2 R R^{z }{}_{z}\\big)+\n3 \\mu_7 R^2\\,.\n\\eeq\nThe symbols $z$ and $\\bar z$ in the above expression label the two orthogonal directions while the indices $\\alpha ,\\beta,...$ are the usual bulk indices. The expression for the anomaly part is\n\\begin{align}\n\\mathcal{L}_{\\rm Anomaly,\\,R^3}=&\\mu_0(12\\, {\\mathcal K}_2{}^{ij} {\\mathcal Q}_{ij}-6\\, {\\mathcal K}_4)-\\mu _1(\\tfrac{3}{2}\\,{\\mathcal K}_4-\\tfrac{3}{2}{\\mathcal K}_2{}^2+3\\,{\\mathcal K}_{ij}{\\mathcal K}_{kl}{\\mathcal R}^{ikjl})-\\nonumber\\\\&\\mu _2 (6\\, {\\mathcal K}_2{}^2-2\\, \n{\\mathcal K}_2\\,{\\mathcal K}^2-8\\, {\\mathcal K}_2\\,{\\mathcal Q}+4\\, {\\mathcal K}_2 {\\mathcal R})-\\nonumber\\\\&\\mu _3(2\\, {\\mathcal K}_4+\\tfrac{1}{2}\\, \n{\\mathcal K}_2{}^2-{\\mathcal K}_2\\,{\\mathcal Q}-2\\, {\\mathcal K}_2{}^{ij} {\\mathcal Q}_{ij}-2\\, {\\mathcal K}^{ij}{\\mathcal Q}_{ij}\\,{\\mathcal K}+2\n\\,{\\mathcal K}_2{}^{ij}{\\mathcal R}_{ij})-\\nonumber\\\\&\\mu _4(2\\, {\\mathcal K}_3 \\,{\\mathcal K}-{\\mathcal K}_2\\,{\\mathcal K}^2-2 \n{\\mathcal K}^{ij}{\\mathcal Q}_{ij}\\,{\\mathcal K}+2\\, {\\mathcal K}^{ij}{\\mathcal R}_{ij}\\,{\\mathcal K})-\\nonumber\\\\&\\mu _5 (\\tfrac{3}{4}\\, {\\mathcal K}_2 \n\\,{\\mathcal K}^2-\\tfrac{3}{2}\\, {\\mathcal K}^2 \\,{\\mathcal Q})-\\mu _6 (\\tfrac{3}{2} \\,{\\mathcal K}_2\\,{\\mathcal K}^2-\\tfrac{1}{2} \n{\\mathcal K}^4-2\\, {\\mathcal K}^2{\\mathcal Q}+ {\\mathcal K}^2{\\mathcal R})\\,.\n\\end{align}\nwhere ${\\mathcal K}_{4}={\\mathcal K}^{ij}{\\mathcal K}_{jl}{\\mathcal K}^{lk}{\\mathcal K}_{ki}\\,.$\nIn calculating the anomaly part from \\equ{eei}, we have used the value of ${\\mathcal B}^i=0$ that we found in Sec.~(\\ref{newmetric}). This is the reason that while terms involving ${\\mathcal B}^i$ are supposed to contribute to \\equ{eei}, the above equation does not contain any terms containing ${\\mathcal B}^i$. The full HEE functional has contributions from the Einstein and R$^2$ part also which are given in~Eqs.(\\ref{Wald}) and (\\ref{Extra}).\n\\subsection{Universal terms \\label{uef}}\nIn this section, we will demonstrate that our HEE functional for the quasi-topological gravity produces the correct universal terms. For the general structure and calculation of the universal term of the entanglement entropy in four dimensions, see \\cite{solod1}. These central charges can be easily calculated using the technique of Ref~\\cite{anom}. \n\nWe follow the procedure given in \\cite{abs} for R$^2$ theory. Here we sketch the main steps of this calculation. We will minimize \\equ{R3} for a bulk surface with a spherical and cylindrical boundary. We will carry out this procedure for the five-dimensional bulk AdS metric \n\\begin{equation} \\label{metric}\nds^{2}=\\frac{\\tilde L^{2}}{ z^{2}}(dz^{2}+d\\tau^{2}+h_{ij}dx^{i}dx^{j})\\,.\n\\end{equation}\nHere, $\\tilde L$ is  the AdS radius and $h_{ij}$ is a 3-dimensional boundary metric \ngiven below.  \nFor the calculation of EE for a spherical entangling surface we can write the boundary $h_{ij}$ in \nspherical polar coordinates as\n\\begin{equation}\n^{sphere}h_{ij}dx^{i}dx^{j}= d\\rho^{2}+\\rho^{2}d\\Omega_{2}^{2}\\,,\n\\end{equation} \nwhere $d\\Omega_{2}^2=d\\theta^{2}+\\sin^2 \\theta d\\phi^{2}$ is the metric of a \nunit two-sphere, $\\theta$ goes from $0$ to $\\pi$ and $\\phi$ goes from $0$ to $2\\pi$.\nSimilarly, for a cylindrical entangling surface\n\\begin{equation}\n^{cylinder}h_{ij}dx^{i}dx^{j}=du^{2}+d\\rho^{2}+\\rho^{2}d\\phi^{2}\\,.\n\\end{equation}\nHere, $u$ is the coordinate along the direction of the length of the cylinder. For a \ncylinder of length $H$, $u$ goes form $0$ to $H\\,.$\n\nWe set $\\rho=f(z), \\tau=0$ in the metric in \\equ{metric}  and \nminimize the entanglement entropy functional (whose R$^3$ part is given in \\equ{R3}) on this codimension 2 surface to find the Euler-Lagrange \nequation for $f(z)$. Using the solution for $f(z)$ we evaluate the entropy functional to \nget the EE.\n  \nFor the spherical boundary, we get  $f(z)=\\sqrt{f_{0}^{2}-z^{2}}$ which gives the EE as\n\\begin{equation}\nS_{EE}= -4 a \\ln(\\frac{f_{0}}{\\delta})\\,.\n\\end{equation}\nHere, $\\delta$ is the UV cut-off that comes from the lower limit of the $z$ integral and \n$f_{0}$ is the radius of the entangling surface. The value of $a$ is \n \\begin{align} \n \\label{a} a&= \\frac{\\pi^{2} \nL^{3}}{f_{\\infty}^{3\/2}\\ell_P^{3}}(1-6f_{\\infty}\\lambda+9 f_{\\infty}^2\\mu)\\,. \n\\end{align}\nFor this case, the entire contribution comes from the Wald entropy as the extrinsic curvatures are identically zero.\n\nFor the cylindrical boundary, we find $f(z)=f_{0}-\\frac{z^{2}}{4f_{0}}+...$ leading to\n\\begin{equation}\nS_{EE}= - \\frac{c H}{2 R} \\ln(\\frac{f_{0}}{\\delta})\\,. \n\\end{equation}\nThe  value of $c$ corresponding to the theory in \\equ{th} is\n \\begin{align} \n \\label{c1}\nc&= \\frac{\\pi^{2} \nL^{3}}{f_{\\infty}^{3\/2}\\ell_P^{3}}\\Big\\{1-2f_{\\infty}\\lambda+9 f_{\\infty}^2\\mu+ f_{\\infty}^2\\mu (42\\mu_{1} - 336 \\mu_2 - 56 \\mu_3)\\Big\\}\\,,\n\\end{align}\nwhile that corresponding to the theory in \\equ{th1} is\n\\begin{align} \n\\label{c2}\nc&= \\frac{\\pi^{2} \nL^{3}}{f_{\\infty}^{3\/2}\\ell_P^{3}}\\Big\\{1-2f_{\\infty}\\lambda+9 f_{\\infty}^2\\mu- f_{\\infty}^2\\mu (168 \\mu_2 + 28 \\mu_3)\\Big\\}\\,.\n\\end{align}\nThe $1+9 f_{\\infty}^2\\mu$ part is the usual Wald entropy contribution, while the remaining part comes from the anomaly part. After putting in the values of $\\mu$'s given in Eqs.(\\ref{th}) and (\\ref{th1}) we obtain\n\\begin{align} \n\\label{c}\nc&= \\frac{\\pi^{2} \nL^{3}}{f_{\\infty}^{3\/2}\\ell_P^{3}}(1-2f_{\\infty}\\lambda-3 f_{\\infty}^2\\mu )\n\\end{align}\nfor both theories.\n\nThese results for the universal terms agree with those calculated in Ref.~\\cite{abs} for the two quasi-topological theories. Note from Eqs.~(\\ref{c1}) and (\\ref{c2}) that only a few terms from $\\mathcal{L}_{\\rm Anomaly,\\, R^3}$ have contributed to the universal term. Terms of the form $\\sim{\\mathcal K}^4$ do not contribute to this calculation at all. Since ${\\mathcal Q} \\sim {\\mathcal K}^2$, terms of the form $\\sim{\\mathcal K}^2 {\\mathcal Q}$ also do not contribute. \n\n\\subsection{Minimal surface condition \\label{eomef}}\n\nWe now find the surface equation of motion for quasi-topological  gravity using the LM method. For ease of calculation, we set all second-order quantities and cross-components in the metric in \\equ{metric2} to zero. The bulk equation of motion for this theory is \\cite{Sinha:2010pm}:\n\\begin{equation}\nR_{\\alpha\\beta}-\\frac{1}{2}g_{\\alpha\\beta}R- \\frac{6}{L^2}g_{\\alpha\\beta}-\\frac{L^2}{2}H_{\\alpha\\beta}-\\nu F_{\\alpha\\beta}=0\\,,\n\\end{equation}\nwhere $F_{\\alpha\\beta}$ is defined in Ref.~\\cite{quasi, Sinha:2010pm}.\nThe $\\frac{\\epsilon}{z}$ divergence in the $zz$ component of the equation of motion coming from the $F_{\\alpha\\beta}$ term is\n\\begin{align}\nF^{1}_{zz}=\n&\\frac{\\epsilon}{z}\\Big[(\\tfrac{3}{2}\\mu_1-\\mu_2-\\mu_3-\\tfrac{3}{2}\\mu_5-4\\mu_6-12\\mu_7){\\mathcal R}^{ij}\\nabla^2{\\mathcal K}_{ij}-(\\tfrac{1}{2}\\mu_2+\\mu_6+6\\mu_7){\\mathcal R}^{ij}{\\mathcal K}_{ij}{\\mathcal R}~+~\\nonumber\\\\&~~~~(\\mu_2+\\tfrac{1}{6}\\mu_6+3\\mu_7){\\mathcal R}_{ij}{\\mathcal R}^{ij}{\\mathcal K}\n+\\tfrac{1}{2}(\\mu_6+\\tfrac{1}{2}\\mu_4){\\mathcal K}\\nabla^2{\\mathcal R}+(\\mu_4+\\mu_3+4\\mu_2)\\nabla_i\\nabla^i{\\mathcal K}~-~\\nonumber\\\\&~~~~(3\\mu_1-8\\mu-2-3\\mu_3-\\mu_4+\\tfrac{3}{2}\\mu_5)\\nabla^l{\\mathcal R}_{lijk}\\nabla^k{\\mathcal K}^{ij}-\\tfrac{1}{2}(\\mu_4+3\\mu_1){\\mathcal K}^{kl}\\nabla^i\\nabla^j{\\mathcal R}_{kijl}~-~\\nonumber\\\\&~~~~(\\tfrac{3}{4}\\mu_1-\\tfrac{5}{2}\\mu_2-\\mu_3-\\tfrac{1}{2}\\mu_4-\\tfrac{3}{4}\\mu_5-2\\mu_6-6\\mu_7){\\mathcal R}\\nabla^i\\nabla^j{\\mathcal K}_{ij}~+~\\nonumber\\\\&~~~~\\tfrac{1}{4}(\\mu_4+2\\mu_3+8\\mu_2){\\mathcal K}^{ij}\\nabla_{i}\\nabla_{j}{\\mathcal R}\\Big]\\,.\n\\end{align}\nWhile we haven't computed the surface equation of motion that one gets on minimizing the functional in \\equ{R3}, this is not very hard to do using the methods of Sec.~(\\ref{variation}) and Mathematica\\footnote{We have used the Xact package for a number of calculations in this paper}.  The main point is, however, that the surface equation of motion that one will get from the entropy functional will contain $~{\\mathcal K}^4$ terms that are absent in the above divergence.\n\nOther divergences are also present in the $zz$ component:\n\\begin{align}\nF^{2}_{zz}=&\\frac{\\epsilon}{z^2}\\Big[ e^{-2\\rho(z,\\bar z)}\\Big \\{\\tfrac{1}{2}( 3\\mu_1+19\\mu_2+2\\mu_3+\\tfrac{14}{3}\\mu_6){\\mathcal R}_{ijkl}{\\mathcal K}^{ik}{\\mathcal K}^{jl}-(\\tfrac{7}{2}\\mu_6+18\\mu_7) {\\mathcal R}{\\mathcal K}_2 ~+~\\nonumber\\\\&~~~~~~~~~~~~~~~~(2\\mu_2+\\tfrac{3}{2}\\mu_6+6\\mu_7){\\mathcal R}{\\mathcal K}^2 +(\\mu_4-\\tfrac{3}{2}\\mu_5){\\mathcal K}\\nabla^2{\\mathcal K}~-~\\nonumber\\\\&~~~~~~~~~~~~~~~~(\\tfrac{4}{3}\\mu_2-\\mu_4+\\tfrac{3}{2}\\mu_5+\\tfrac{4}{3}\\mu_6){\\mathcal K}^{ij}\\nabla_{i}\\nabla_{j}{\\mathcal K}-(\\mu_4-\\tfrac{3}{2}\\mu_5){\\mathcal K}\\nabla^i \\nabla^j {\\mathcal K}_{ij}~+~\\nonumber \\\\&~~~~~~~~~~~~~~~~(3\\mu_1-\\tfrac{2}{3}\\mu_2 -\\mu_3 + \\mu_4-\\tfrac{3}{2}\\mu_5-\\tfrac{2}{3}\\mu_6)\n\\nabla_{k}{\\mathcal K}^{ij}\\nabla^{k}{\\mathcal K}_{ij}~-~\\nonumber\\\\&~~~~~~~~~~~~~~~~(3\\mu_1+\\tfrac{2}{3}\\mu_2 +\\mu_3 +3 \\mu_4-\\tfrac{3}{2}\\mu_5+\\tfrac{2}{3}\\mu_6)\\nabla^i{\\mathcal K}_{ij}\\nabla^k{\\mathcal K}_{k\n}^j ~-~\\nonumber \\\\&~~~~~~~~~~~~~~~~(8\\mu_2-\\mu_4+\\tfrac{3}{2}\\mu_5){\\mathcal R}_{ij}{\\mathcal K}^{ij}{\\mathcal K}+(3\\mu_1+8\\mu_2-\\mu_4-2\\mu_6){\\mathcal R}_{ij}{\\mathcal K}^{jk}{\\mathcal K}_{k}^{i}\\Big\\}\\Big]~+~\\nonumber\\\\& \n\\frac{\\epsilon}{z^3}\\Big[e^{-4\\rho(z,\\bar z)}\\Big\\{(3\\mu_1-2\\mu_2-2\\mu_3){\\mathcal K}_3+(\\mu_4-3\\mu_5-2\\mu_6){\\mathcal K}\\K_2\\Big\\}\\Big]\\,.\n\\end{align}\nAs for R$^2$ theory, these divergences can be used to determine second and higher-order terms in the metric. At linear-order in the metric, divergences in all other components of the equation of motion go to zero if we take the limit as mentioned in Sec.~(\\ref{LMmethod}).\n\n\\section{Discussion \\label{discussion}}\n\nIn this paper, we found the surface equation of motion for general R$^2$ theory and quasi-topological gravity using the generalized gravitational entropy method of Ref.~\\cite{maldacena}. We found that these do not match exactly with what can be derived by extremizing the HEE functional for these theories -- the HEE functional being calculated using the formula proposed in Refs.~\\cite{dong,camps}.\n\nLet us summarize our findings regarding R$^2$ theory. First, the leading-order terms on both sides do match. In fact, barring $\\sim{\\mathcal K}^3$ terms, the surface equation of motion that follows from the LM method is precisely the surface equation of motion that follows from the FPS functional. \n\nThe main problem with the LM method is that there are divergences in components other than the $zz$ component, for a general higher-derivative theory. In the Gauss-Bonnet case, there are ways we can take the limit to set these divergences to zero. However, the effect of taking the limit in this way is to remove all $\\sim{\\mathcal K}^3$ divergences from all components of the equation of motion. This means that we do not get any $\\sim{\\mathcal K}^3$ terms in the surface equation of motion using the LM method. No matter what the HEE functional for R$^2$ theory is, it is unlikely that no $\\sim{\\mathcal K}^3$ terms will occur in its surface equation of motion at any point in its parameter space. Even after taking the limit as prescribed, for general $R^2$ theory, there remain extra divergences in the bulk equation of motion. It is impossible to set these divergences to zero at all points of the parameter space, although this can be done for specific points like the Gauss-Bonnet point. \n\nAs we discussed in the paper, the absence of $\\sim{\\mathcal K}^3$ terms is the R$^2$ equation of motion is an artifact of the way limits have to be taken in the LM method for the Gauss-Bonnet case. The limit can also be taken in such a way so as to preserve $\\sim{\\mathcal K}^3$ terms. It is worth recapitulating the results this way of taking the limit gives for Gauss-Bonnet theory. As we showed, using the second-order conical metric, the bulk equation of motion for Gauss-Bonnet theory, before taking the limit, has divergences only in the $zz$ and $ij$ components. There is no divergence in the $z{\\bar z}$ component, while the divergence in the $zi$ component turns out to be a constraint equation that vanishes by itself on using the Codazzi-Mainardi relation on AdS space. This same constraint equation results from the Jacobson-Myers functional, as well, on taking tangential variations of the surface. It is not clear what the relevance of the divergence in the $ij$ component is in the LM method. Were we to ignore this divergence, the surface equation of motion that would result from the $zz$ component for Gauss-Bonnet theory, after taking the limit, is $c\\, {\\mathcal K} =0$, where $c$ is proportional to the Weyl anomaly. This equation is clearly not what comes from the Jacobson-Myers functional. However, the resulting minimal surface is what one obtains on extremizing just the Wald entropy part of the functional. It would be interesting to check whether the $zz$ component of R$^2$ theory also leads to the same result. \n\nOne of the pending issues with the LM method is to fix the ambiguity present in the limit-taking procedure. However, fixing this by itself does not seem enough to simultaneously cure the two problems present for R$^2$ theory: the absence of $\\sim{\\mathcal K}^3$ terms and the presence of extra constraints; although, it can remove one of these problems from the list. The ambiguity in the limit-taking procedure is not unique to the LM method. Similar, though not exactly the same, issues occur in studies of co-dimension two branes in the context of brane-world gravity \\cite{Bostock}. It is possible that a further modification to the LM method will fix these problems; on the contrary, it may be that one cannot get rid of it in any way. The problem of extra divergences is related to the derivative order of the bulk equation of motion and seems to spring from the pathology of the general R$^2$ theory itself. In this sense, it is not surprising that we encounter it for general higher-derivative theories. Higher-derivative theories are known to suffer from problems regarding unitarity~\\cite{Deser, Unitary, Ali}. These problems seem to be manifested in the LM method in the inability to remove all divergences, that occur on using the conical metric, from the bulk equation of motion.\n\nWhat does our analysis say about the validity of the formula proposed in Refs.~\\cite{camps,dong} as the entanglement entropy functional? For general R$^2$ theory as we demonstrated the leading-order terms match on both sides, which stops short of being a validation of the proposal for this theory. This test, at present, is similar in scope in refining conjectured entropy functionals for higher-derivative theories as the test whether the entropy functional leads to the correct universal terms. As we showed in this paper, for quasi-topological theory the universal terms are not sensitive to terms of the form  $\\sim{\\mathcal K}^4$ in the entropy functional (similar statement applies for other higher-derivative theories) and one can change these terms and still have the universal terms come out to be correct.  \n\nThe LM method, therefore, in its current form has limitations that make it ineffective in testing proposed entropy functionals for generic higher-derivative gravity theories.\nThe fact that the LM method only works for specific theories may indicate one of two things. One possibility is that \nentropy functionals only exist for specific theories such as Lovelock theories, for which the result of the surface equation of motion from the existing entropy functional and the LM method coincide. The other possibility, as mentioned before, is that the LM method needs some modification. In this context, it is also desirable that alternate methods to test entropy functionals be developed.\n\n\n\\section*{Acknowledgements}\nWe are grateful to Aninda Sinha for valuable suggestions and  remarks. We also thank Joan Camps and Rajesh Gopakumar for discussions. A.B thanks the string theory group at Harish-Chandra Research Institute, Allahabad for hospitality during part of this work. A.B also thanks the members of  the department of particle physics, University of Santiago de Compostela, specially Jose Edelstein, the Max Planck Institute of Gravitational Physics, Golm specially Axel Kleinschmidt and the theory division of Max Planck Institute for Physics, Munich specially Johanna Erdmenger for hospitality and the opportunity to present part of this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\n\n\nIn the early seventies, Penrose \\cite{Penrose1972, Penrose1973}\nshowed that by combining several\ningredients of the ``establishment viewpoint'' of gravitational\ncollapse, an inequality of the form \n\\begin{eqnarray}\nM \\geq \\sqrt{\\frac{A}{16 \\pi}} \\label{PI1}\n\\end{eqnarray}\nfollows, where $M$ is the total mass and $A$ the area\nof a black hole. Cosmic censorship is one of the fundamental\ningredients of the argument, and by far the weakest one.\nThus, finding a counterexample\nof (\\ref{PI1}) would very likely involve a spacetime for which cosmic\ncensorship  fails to hold.\nIn fact, this was Penrose's original motivation to study the inequality. \nOn the other hand, a proof of a suitable version of (\\ref{PI1}) \nwould give indirect support to cosmic censorship. Inequalities of this\ntype are collectively termed ``Penrose inequalities'' \n(sometimes also ``isoperimetric inequality for black holes'' \\cite{Gibbons1972})\nand finding suitable versions thereof and trying to \nprove them has become a major task in mathematical relativity. After a first\nperiod of heuristic proofs and partial results, important\nbreakthroughs have been made in the last ten years. \nThe aim of this\nreview is to try to explain the problem and describe the main \napproaches that have been followed.\n\n\nThe first observation to be made is the necessity of replacing \nthe area of the black hole in (\\ref{PI1}) by the area of a suitable alternative surface.\nThis is because in order to determine whether a spacetime is a black hole, detailed knowledge of its global\nfuture behaviour is required. On the other hand, cosmic\ncensorship is precisely a statement on the global future evolution of a spacetime. \nIn order to have an inequality logically independent of cosmic censorship (although\nmotivated by it) the area\non the black hole must be replaced by the area of a surface which can be located\nindependently of the global future behaviour of the spacetime (for instance, directly in terms\nof the initial data) and which is guaranteed (or at least expected)\nto have less or equal area than the event\nhorizon that may eventually form during the evolution.\n\n\nThe global setup which supports the validity of (\\ref{PI1}) is well-known and, in rough terms, \ngoes as follows. Assume a spacetime $({\\cal M},g)$ which is asymptotically flat in the sense of being\nstrongly asymptotically predictable, admitting\na complete future null infinity ${\\mycal I}^{+}$ and satisfying $J^{-}({\\mycal I}^{+}) \\neq {\\cal M}$\n(see \\cite{Wald1984} for definitions). The event\nhorizon ${\\cal H}$ is the boundary of $J^{-} ({\\mycal I}^{+})$ and it is, therefore, a null hypersurface at least \nLipschitz continuous.\nAssume, moreover, that the spacetime admits an asymptotically flat partial Cauchy surface\nwith total ADM energy $E_{ADM}$ and which intersects ${\\cal H}$ on a cut $S$.\nIf ${\\cal H}$ is a smooth hypersurface then this cut\nis a smooth embedded surface which has a well-defined area\n$|S|$. For general event horizons, the area $|S|$ still makes sense\nprovided it is interpreted as its 2-dimensional Hausdorff measure (the Hausdorff measurability\nof $S$ is demonstrated in \\cite{Chrusciel2001}). Consider now any cut $S_1$ to the causal future\nof $S$ along the event horizon.\nThe black hole area law \\cite{Hawking1971,Hawking1972}\nstates $|S_1| \\geq |S|$ provided the null energy condition holds.\nThis area theorem was proven\nfor general event horizons in \n\\cite{Chrusciel2001} under much milder asymptotic\nconditions. From physical principles, the spacetime is expected to settle down to some\nequilibrium configuration. Assuming also that all the matter\nfields are swallowed by the black hole in the process (an external electromagnetic field would \nnot alter the conclusions, see Sect. \\ref{stronger}), the uniqueness theorems for stationary black holes\n(see e.g. \\cite{Heusler1996}) imply that the spacetime\nmust approach the Kerr metric (modulo several technical conditions that \nstill remain open, see \\cite{Chrusciel2008} for a recent account). For the\nKerr metric, the area of the event horizon  \n$A_{\\mbox{\\tiny Kerr}}$\nis independent of the cut  (as for any Killing horizon) and takes the value (in units\n$G=c=1$) $A_{\\mbox{\\tiny Kerr}} = 8 \\pi M \\left (M + \\sqrt{M^2 - L^2\/M^2}\n\\right ) \\leq 16 \\pi M^2$ where $M$ and $L$\nare respectively the total mass and total angular momentum of the spacetime\n(we do not use the more common term ``$J$'' for the angular momentum\nto avoid confusion with the energy flux used later). In particular,\n$M$ should be the asymptotic value of the Bondi mass along ${\\mycal I}^{+}$. Since\ngravitational waves carry positive energy, the Bondi mass cannot increase\nto the future \\cite{Bondi1962, Sachs1962}. Provided the Bondi mass approaches the ADM mass\n$M_{ADM}$ of the initial slice (which is only known under additional assumptions, see\n\\cite{Ashtekar1979, Hayward2003, Kroon2003, Zhang2006}), the inequality $M_{ADM} \\geq \\sqrt{|S|\/16 \\pi}$ follows.\nThis inequality is still global in the sense that the cut $S$ of the event horizon cannot be \ndetermined directly in terms of the initial data. Penrose's idea was to consider situations in which\none could estimate the area of the cut from below in terms of the area of some surface that could \nbe located without having to solve the whole future evolution.\n\nOne such situation occurs when the initial data set is asymptotically euclidean\nand contains a future trapped surface (see below for definitions).\nThen, the singularity theorems of Penrose \\cite{Penrose1965}, Hawking \\cite{Hawking1967}\nand others (see \\cite{Senovilla1998} for a review) state that, provided \nthe strong energy condition holds (in some cases the null energy condition suffices),\nthe maximal globally hyperbolic development of this data\nmust contain  a singularity, i.e. an incomplete inextendible causal geodesic.  \nVery little is known in general about the nature of the singularity that forms.\nMore specifically it is not known whether\nthe future development admits a complete ${\\mycal I}^+$ and therefore defines a black hole\nspacetime. The weak cosmic censorship conjecture, first \nproposed by Penrose \\cite{Penrose1969}, asserts that all singularities \nlie behind an event horizon  and therefore are invisible to an observer at infinity.\nNot much  is known about the general validity of this conjecture, which remains\na fundamental open problem in gravitational collapse physics, see \n\\cite{Wald1997}. Rigorous results are available only in the case of spherical symmetry, where\nthe conjecture has been proven for several matter models \\cite{Christodoulou1999, Dafermos2005}.\n\nUnder cosmic censorship, the initial data containing a future trapped\nsurface $S$ develops a black hole spacetime.\nA general result on black hole spacetimes \\cite{HawkingEllis, Claudel2000} \nis that  no future trapped surface can enter into the causal past of \n${\\mycal I}^{+}$. \nTherefore, the intersection of the event horizon and the initial data set $\\Sigma$ defines\na spacelike surface ${{\\cal H}}_{\\Sigma}$\nthat separates $S$ from the asymptotic region. In general, the\nlocation of ${\\cal H}_{\\Sigma}$ cannot be determined directly \nfrom the initial data. Moreover, ${\\cal H}_{\\Sigma}$ can have smaller area than  $S$, even though it lies in its exterior.\nNevertheless, under suitable\nrestrictions on $S$ (details will be given below) it makes sense to consider all surfaces\nin $\\Sigma$ enclosing $S$. The infimum of the areas of all such surfaces, denoted by $A_{\\min}(S)$,\nhas the obvious property that\n$|{\\cal H}_{\\Sigma} | \\geq A_{\\min}(S)$. Consequently, \nthe inequality\n\\begin{eqnarray}\nM_{ADM} \\geq \\sqrt{\\frac{A_{\\min}(S)}{16 \\pi}} \\label{LocalPI}\n\\end{eqnarray}\nfollows from Penrose's heuristic argument. \nThe need of using the minimum area enclosure of $S$ in (\\ref{LocalPI})\nwas first noticed by Jang\nand Wald  \\cite{JangWald1977} and the first example showing that $S$ may have greater area\nthan surfaces enclosing it is due to Horowitz \\cite{Horowitz1984}.\n\nInequality (\\ref{LocalPI}) involves objects\ndefined solely in terms of the local geometry of the initial data set and its validity \ncan therefore be addressed independently of whether weak cosmic censorship (or any other of the\ningredients entering into the argument) holds or not.\nIt is clear that, since $A_{\\min}(S)$ depends on the hypersurface $\\Sigma$\ncontaining $S$, we can still take the supremum of the right-hand side\nwith respect to all asymptotically flat hypersurfaces containing $S$, and the resulting inequality still follows \nfrom Penrose's heuristic argument. This gives a clearly stronger inequality. However,\nit depends on the piece of spacetime available and therefore looses the desirable\nproperty of depending solely on objects defined on the initial data set. \n\nIf the initial data set is asymptotically hyperbolic instead of asymptotically flat \n(i.e. such that it intersects future null infinity in its Cauchy development), the same\nheuristic argument gives (\\ref{LocalPI}) with the Bondi mass replacing the  ADM mass in the\nleft hand side. \n\n\nAnother setup where a local geometric inequality is implied by the global heuristic argument\nof gravitational collapse appears in the seminal \npaper by Penrose \\cite{Penrose1973} and consists of a null shell ${\\cal N}$\n(with compact cross sections)  of collapsing dust in \nMinkowski spacetime. As described in more detail below,\nfor any given shape of the shell (restricted to be convex at each instant of time in order\nto avoid shell crossings in the past) and any chosen cross section $S$ on the shell, the energy density of\nthe collapsing dust\ncan be adjusted so that $S$ is \na marginally trapped surface with respect to the geometry of the spacetime left after the shell has passed.\nSince a singularity will definitely form in this setup (the shell has self-intersections in its future),\ncosmic censorship predicts the formation of a black hole. Similarly as before,\nthe marginally trapped surface $S$ cannot enter the causal past of ${\\mycal I}^{+}$\n\\cite{ChruscielGalloway2008}. The intersection of the event horizon with the shell ${\\cal N}$ must therefore\nlie in the causal past of $S$. However, since the shell is convex and collapsing, the \nevent horizon cut automatically has at least the same area as $S$. Consequently, the heuristic collapse\nargument implies $M \\geq \\sqrt{|S|\/16\\pi}$, where $M$ is the mass of the shell. By energy conservation,\nthis can be computed directly on \n$S$ in terms of its geometry as a surface in Minkowski. Thus, a geometric inequality is obtained for\na class of spacelike surfaces in Minkowski spacetime. The status of this version of the\nPenrose inequality will be discussed in detail later on.\n\n\nAll versions of the Penrose inequality have the structure of a lower bound of \nthe total mass of the spacetime in terms of the area of suitably chosen surfaces. \nTherefore, they can be regarded as strengthenings of the positive mass theorem, which says\nthat the total mass of an asymptotically flat spacetime cannot be negative, under suitable energy and completeness\nconditions.\nThe positive mass theorem\nalso has a rigidity part, namely that the total mass vanishes \nonly for the Minkowski spacetime. The Penrose inequality conjecture also has a rigidity part, which in rough terms\nstates that equality will only happen for the Schwarzschild spacetime. Heuristically this can\nbe understood because, in the case of equality,\nthe final mass of the spacetime must coincide with the starting one. \nTherefore, no gravitational \nwaves are emitted in the process. This suggests that the whole configuration should be stationary.\nHowever, the only\nstationary, vacuum black hole is the Kerr spacetime, and equality happens for this metric only \nif the\nangular momentum is zero, i.e. if the metric is, in fact, the Schwarzschild spacetime. \n\n\nSince the original proposal by Penrose this topic has become an active area of research.\nHowever, the problem has proven\nto be a difficult one and relatively little progress was made during the first decades.\nThe most important\ncontribution in this period is due to Geroch \\cite{Geroch1973}, who observed that a suitable functional defined \non surfaces embedded in a spacelike hypersurface $(\\Sigma,\\gamma)$ is monotonically increasing\nif the curvature scalar of $\\gamma$ is non-negative and the surfaces are moved outwards\nat a speed which is inversely proportional to the mean curvature of the surface at each point.\nThis is the so-called {\\it inverse mean curvature flow}. This functional (now called Geroch mass) has the \nproperty of approaching the ADM energy of the hypersurface (provided this is asymptotically euclidean)\nif the surfaces become sufficiently spherical at infinity. \nGeroch's original idea was to prove the positive mass theorem by starting the inverse mean curvature\nflow at a point (where the Geroch mass vanishes). This idea was then adapted by Jang and Wald \\cite{JangWald1977},\nwho noticed that\nthe Geroch mass coincides exactly with the right-hand side of (\\ref{PI1}) if the starting surface is connected,\nof spherical topology and minimal (i.e. with vanishing mean curvature). If the flow existed globally and the leaves\ncould be seen to approach round spheres at infinity, then the monotonicity of the Geroch mass would imply\nthe Penrose inequality in the particular case of time-symmetric initial data sets. The flow\nhowever, generically develops singularities and therefore the argument could not be made\nrigorous at the time. The only case where the Penrose inequality could be proven to hold was in\nspherical symmetry (irrespectively of whether the initial data set was time-symmetric or not) using the so-called\nHawking mass, which is a generalization of the Geroch mass when the second fundamental form is not zero.\n\n\nIn the late nineties, however, two important breakthroughs were made. First of all, Huisken and Ilmanen \n\\cite{HuiskenIlmanen2001} were able to\nprove that Geroch's heuristic derivation could be turned into a rigorous proof. This required deep results\nin geometric analysis and geometric measure theory. These authors established therefore the validity of the\nPenrose inequality for asymptotically euclidean Riemannian manifolds of non-negative Ricci curvature and having\na boundary consisting of an outermost minimal surface (this is now called the {\\it Riemannian\nPenrose inequality}). Although this boundary was allowed to be disconnected,\nthe Penrose inequality could only be established for the area of any of its connected components. Shortly afterwards,\nBray \\cite{Bray2001} was able to prove the Riemannian Penrose inequality in full generality (i.e. in terms\nof the total area of the outermost minimal surface, independently of whether this is connected or not). \nBray's argument is completely different to the  previous one and uses a deformation of the given metric\nin such a way that all the non-trivial geometry gets swallowed inside the minimal surface while\nthe total mass of the space does not increase and the area of the horizon does not decrease. This process settles\ndown into an equilibrium state given by the Schwarzschild metric. Since the Penrose inequality is fulfilled\nin the final state (in fact, with equality), the Penrose inequality holds also for the original space. \nThese two fundamental results have boosted tremendously the interest in the Penrose inequality, which has become\na very important topic in mathematical relativity. Although the general case is still open, several ideas have been\nproposed to approach it. In particular, an important step forward has been made very recently by\nBray and Khuri \\cite{BrayKhuri2009}. Although several issues remain still open regarding this proposal (in particular,\nthe existence of solutions of certain PDE problems) the idea is indeed very promising.\n\nThe aim of this review is to present the most important developments in this field. I have chosen not to follow\na historical order, and rather I have tried to organize the presentation in a way which I consider logically\nconvenient. Since several approaches share some of the techniques, I have collected many preliminary results\nin one section (Sect. \\ref{definitions}). While this has the potential disadvantage that, in a first reading,\nit is not clear why and where such results are needed, it has the advantage of simplifying the location of the\nrequired tools. Readers wishing to enter straight into the topic of the Penrose inequality\nmay skip this section and refer back to it whenever necessary.\n\n\nAn important warning is in order. The topic ``Penrose inequalities'' is vast and has many\nramifications. Although I will try to cover the most important results in the field, I make\nno claim of exhaustivity. There are several other\nreview papers on this topic in the literature. For the early results the reader is advised\nto look at \\cite{Malec1991}. The breakthroughs of Huisken \\& Ilmanen and Bray triggered the publication \nof a number of interesting reviews, where the proofs of the Riemannian Penrose inequality\nwere treated in detail, see \n\\cite{HuiskenIlmanen1997}, \\cite{HuiskenIlmanen1998},\n\\cite{Herzlich2000}, \\cite{Bray2002}, \\cite{Bray2002-2}, \\cite{BrayChrusciel2004}, \n\\cite{Schoen2005}, \\cite{Mars2007}.\n\n\nThe structure of the review is as follows. In section \\ref{definitions} \nthe  basic facts\nthat will be needed later are introduced. This section is divided into five subsections. In subsection\n\\ref{codimension-two} the geometry of $(n-2)$-dimensional surfaces as submanifolds of \nan $n$-dimensional spacetime is described, and several types of surfaces are defined. In\nsubsection \\ref{variation_null_expansion} some relevant variational formulas for the null expansions are\nsummarized. In subsection \\ref{embedded}, the geometry of codimension-two surfaces as submanifolds\nof spacelike hypersurfaces is discussed and two important existence theorems for outermost surfaces\nwith suitable properties are  reviewed. In subsection \\ref{UEF}, the Hawking and Geroch quasi-local masses,\nfor surfaces in four dimensional spacetimes and their general variation formulas are discussed. Subsection\n\\ref{asymptoticflatness} recalls the notion of asymptotically euclidean. The following Section (Sect. \\ref{Formulations})\nis devoted to the discussion of the various formulations of the \nPenrose inequality that have been proposed. Section \\ref{Spher} deals with the spherically symmetric\ncase. In Section \\ref{Riemannian} the so-called Riemannian Penrose inequality is treated. \nThe main ideas behind the remarkable proofs of Huisken and Ilmanen and of Bray\nare discussed in subsections \n\\ref{HuiskenIlmanen} and \\ref{Braysproof} respectively. However, \nprevious approaches based on spinors and isoperimetric methods which admit \ninteresting generalizations  are also covered in subsections \\ref{spinors} and \\ref{isoperimetricprofile}. \nSection \\ref{hyperbolicSect} discusses the Penrose inequality when the \ninitial data set is asymptotically hyperbolic instead of asymptotically euclidean. Section \n\\ref{GeneralPI} is devoted to describing different attempts that have been\nproposed to address the \nPenrose inequality in the non time-symmetric case. Particular attention is paid to\nthe  very promising recent ideas put forward by Bray and Khuri \\cite{BrayKhuri2009}.\nSection \\ref{stronger} treats several strengthenings of the Penrose\ninequality when particular matter fields or symmetries are present. Section \\ref{applications} is devoted\nto describing some applications\nof the Penrose inequality. The paper finishes with a some concluding remarks in Section \\ref{concluding}.\n\n\n\n\n\\section{Basic facts and definitions}\n\\label{definitions}\n\nA spacetime $({\\cal M},g)$ is a connected Hausdorff \nmanifold endowed with a smooth metric\nof Lorentzian signature (with sign convention $\\{-,+,+ \\cdots\\}$). \nSmoothness is assumed for simplicity, many of the results below \nhold under  weaker differentiability assumptions. We further\ntake ${\\cal M}$ orientable and $({\\cal M},g)$ time-orientable and that an orientation\nfor both has been chosen. In this review we  will be mainly concerned\nwith four-dimensional spacetimes, although higher dimensional results will\nbe mentioned at some places. ${\\cal M}$ will always be four-dimensional unless \nexplicitly stated.\n\n\nThe Penrose inequality involves the area of codimension-two surfaces.\nLet us therefore start with some basic properties concerning their geometry.\n\n\\subsection{Geometry of codimension two surfaces}\n\\label{codimension-two}\n\n$S$ will denote a compact, embedded, oriented, codimension-two surface\nin an $n$-dimensional spacetime $({\\cal M},g)$ defined via\nan  embedding $\\Phi: S \\rightarrow {\\cal M}$ ($S$ will usually be identified \nwith its image). The induced first fundamental\nform on $S$, denoted by $h$, is assumed to be positive definite, i.e.\n$S$ is spacelike. Such an object will be called simply ``surface''. The\narea of $S$ is denoted by $|S|$.\n\nAt $p \\in S$ we denote by $T_p S$ and $N_p S$ the tangent and normal\nspaces of $S$. This implies $T_p {\\cal M} = T_p S \\oplus N_p S$. For any \nvector $\\vec{V}$ at $p$ this decomposition defines a parallel and a normal\nvector\naccording to $ \\vec{V} =\n\\vec{V}^{\\parallel} + \\vec{V}^{\\bot}$.\nThe second fundamental form vector $\\vec{K}$ \nof $S$ is defined, as usual, as $\\vec{K} (\\vec{X},\\vec{Y}) \n= - ( {\\nabla^{g}}_{\\vec{X}} \\vec{Y} )^{\\bot}$ where $\\vec{X}$ and $\\vec{Y}$\nare tangent to $S$ and ${\\nabla^{g}}$ is the covariant derivative on $({\\cal M},g)$.\nThis tensor is symmetric and its\ntrace $\\vec{H} = \\mbox{tr}_{h} \\vec{K}$ defines the mean curvature vector. \nThe trace-free part of $\\vec{K}$ will be called $\\vec{\\Pi}$ in the following.\n\n\n\n$S$ being oriented and the ambient spacetime being oriented and time oriented,\nit follows easily that there exist (globally on $S$) \ntwo linearly independent null vector fields $\\vec{l}^{+}$ and\n$\\vec{l}^{-}$ orthogonal to $S$. These vectors, which \nspan the normal bundle $NS = \\cup_p N_pS$, will always be chosen to be\nfuture directed and satisfying\n$(\\vec{l}^{+} \\cdot \\vec{l}^{-}) = -2$ \n(dot stands for scalar product with the spacetime metric $g$).\nThey are uniquely defined up to\nrescalings \n$\\vec{l}^{+}  \\rightarrow  F \\vec{l}^{+}$, $\\vec{l}^{-} \\rightarrow  \nF^{-1} \\vec{l}^{-}$ ($F>0$),\nplus interchange $\\vec{l}^{+} \\leftrightarrow \\vec{l}^{-}$. \nThe null expansions of $S$\nare defined as $\\theta_{\\pm} = (\\vec{H} \\cdot \\vec{l}^{\\pm} )$\nand they\ncontain the same information as $\\vec{H}$ since\n$\\vec{H} = - \\frac{1}{2} \\left ( \\theta_- \\vec{l}^{+} +\n\\theta_{+}\\vec{l}^{-} \\right )$.  \n\nA surface $S$ can be classified according to the\ncausal character of $\\vec{H}$. Unfortunately, there is no unique and generally accepted\nagreement on how to call the various types of surfaces which arise. In this paper, \nthe following notation will be used  (in our convention, the zero vector is both future null and\npast null):\n\n\n\n\\begin{itemize}\n\\item[] {\\bf Future trapped:} $\\vec{H}$ is timelike and future directed everywhere (equivalently\n$\\theta_{+}<0$ and $\\theta_{-}<0$)\n\\item[] {\\bf Weakly future trapped:} $\\vec{H}$ is causal and future directed everywhere\n($\\theta_{+} \\leq 0$, $\\theta_{-} \\leq 0$).\n\\item[] {\\bf Marginally future trapped:} $\\vec{H}$ is proportional to one of the null normals\n$\\vec{l}^{\\pm}$ with a non-negative proportionality factor ($\\theta_{+} =0$ and \n$\\theta_{-} \\leq 0$, or viceversa).\n\\end{itemize}\n{\\bf Past trapped}, {\\bf weakly past trapped} and {\\bf marginally past trapped} are defined\nby reversing all inequalities. \n\nAll these types of surfaces have a \nmean curvature vector of definite causal character and time orientation. For brevity, when no indication to future\nor past is given, then future should be understood (i.e. a ``trapped surface'' is meant to be a \nfuture trapped surface, and similarly for the other cases).\n\nIt is often useful to consider surfaces for which \none of the null normals can be geometrically selected. This preferred normal\nwill be called ``outer null normal'' and will always be denoted as $l^{+}$.\nNotice that ``outer'' here does not necessarily refer to\nany notion of exterior to $S$, so this definition must be used with\nspecial care. Nevertheless, when the null normals can be\ngeometrically distinguished, the corresponding null expansions are  also\ngeometrically distinct and the following definitions (which place no\nrestriction on $\\theta_{-}$) become of interest.\n\\begin{itemize}\n\\item[] {\\bf Weakly outer trapped:} $\\theta_{+} \\leq 0$.\n\\item[] {\\bf Marginally outer trapped or MOTS:} $\\theta_{+} =0$ (equivalently,\n$\\vec{H}$ points along the outer null normal).\n\\item[] {\\bf Weakly outer untrapped:} $\\theta_{+} \\geq 0$.\n\\item[] {\\bf Outer untrapped:} $\\theta_{+} > 0$.\n\\end{itemize} \n\nThese definitions depend not only on the choice of outer direction, but also on the time\norientation of the spacetime. If the time orientation is reversed (without \nmodifying the outer direction) then $-\\vec{l}^{-}$ becomes the future outer null\ndirection. Therefore by adding the word ``past'' to any of the four definitions above, the surface\nis meant to satisfy the same inequalities with $\\theta_{+}$ replaced by $-\\theta_{-}$. For instance,\na {\\bf past weakly outer trapped} surface satisfies $\\theta_{-} \\geq 0$.\n\n\n\n\nVery recently Bray and Khuri \\cite{BrayKhuri2009} have proposed a \nversion of the Penrose inequality which involves a type of surfaces\nwhose definition is insensitive to\nreversals of time direction. However, they still require a preferred\nouter direction.\n\\begin{itemize}\n\\item[] {\\bf Generalized trapped surface:} \nAt each point, either $\\theta_{+} \\leq 0$ or $\\theta_{-} \\geq 0$ (or both).\n\\item[] {\\bf Generalized apparent horizon:} At each point, either\n$\\theta_+=0$ and $\\theta_{-} \\leq 0$, or\n$\\theta_-=0$ and $\\theta_{+} \\geq 0$. Notice that generalized\napparent horizons are in particular generalized \ntrapped surfaces.\n\\end{itemize}\nIn the detailed classification of surfaces \naccording to its mean curvature vector presented in \\cite{Senovilla2007},\ngeneralized apparent horizons are termed {\\it null untrapped}, which,\nin my opinion, has the advantage of being  more descriptive. In fact, my preferred term for these surfaces \nwould be {\\it null outer untrapped} because this emphasizes the fact that they\nhave a  privileged ``outer'' direction. However,\nnotation is always a very personal matter and, in this review, I will stick to the name {\\it generalized\napparent horizon} put forward by Bray and Khuri in their new proposal of the Penrose inequality.\n\nIn terms of the mean curvature vector, a generalized trapped surface is one where\n$\\vec{H}$ is allowed to point everywhere except along a\nspacelike outer direction (defined to be a spacelike direction with positive scalar\nproduct with $\\vec{l}^{+}$). A generalized apparent horizon has \na mean curvature vector which is null everywhere and moreover, has non-negative\nscalar product with any outer spacelike direction. Notice that the second condition\nis ensured provided the scalar product with {\\it one} outer spacelike direction is non-negative.\n\nGeneralized trapped surfaces are indeed generalizations of the previous concepts. \nIn particular, weakly future trapped and weakly outer trapped surfaces are\nautomatically generalized trapped surfaces, and the same is true\nfor marginally outer trapped surfaces.\n\n\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|l|c|}\n\\hline\nName of surface & Null expansions \\\\\n\\hline\n\\hline\n{\\bf Future trapped}  &  $\\theta_{+}<0$ and $\\theta_{-}<0$ \\\\\n\\hline\n{\\bf Weakly future trapped} & $\\theta_{+} \\leq 0$ and $\\theta_{-} \\leq 0$ \\\\\n\\hline\n{\\bf Marginally future trapped} & ($\\theta_{+} =0$ and $\\theta_{-} \\leq 0$) or\n($\\theta_{+} \\leq 0$ and $\\theta_{+} = 0$) \\\\\n\\hline\n{\\bf Weakly outer trapped} & $\\theta_{+} \\leq 0$  \\\\\n\\hline\n{\\bf Marginally outer trapped (MOTS)} & $\\theta_{+} =0$  \\\\\n\\hline\n{\\bf Weakly outer untrapped} &  $\\theta_{+} \\geq 0$  \\\\\n\\hline\n{\\bf Outer untrapped} & $\\theta_{+} > 0$  \\\\\n\\hline\n{\\bf Generalized trapped surface} & At no point, $\\theta_{+} \\geq 0$ and $\\theta_{-} \\leq 0$ \\\\\n\\hline\n{\\bf Generalized apparent horizon} &  At each point, $\\theta_{+} \\theta_{-}=0$ and\n$ \\theta_{+} - \\theta_{-} \\geq 0$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Types of surfaces according to their null expansion(s).}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\\subsection{First order variations of the null expansions}\n\\label{variation_null_expansion}\n\nAn important technical tool that will be used often below is the first order\nchange of the null expansions\n$\\theta_{\\pm}$ under variations of the surface.\nIn this section we write down the variations of $\\theta_{\\pm}$ along the null\nnormals $\\vec{l}^{\\pm}$,  which we write \nas $\\pounds_{\\vec{l}^{\\pm}} \\theta_{\\pm}$. For the sake of completeness, let us first recall\nbriefly how  variations of a geometric object $F$ are defined. \nAny vector field $\\vec{\\xi}$ in a neighbourhood\nof $S$ defines a {\\it variation}\nof the surface as follows. For small enough $\\lambda \\in \\mathbb{R}$, let \n$\\Phi_{\\lambda}: S \\rightarrow {\\cal M}$ be the embedding defined by moving $p \\in  S$ \na parametric amount $\\lambda$ along the integral curve of $\\vec{\\xi}$\nstarting at $p$. We write $S_{\\lambda} = \\Phi_{\\lambda} (S)$. \nAssume, for definiteness,\nthat $F$ is a covariant tensor geometrically defined on any surface \nand take a variation $\\Phi_{\\lambda}$ of $S$.\nDefine $F_{\\lambda} (p) = \\Phi_{\\lambda}^{\\star} (F |_{\\Phi_{\\lambda}(p)})$,\ni.e. the pull-back of the tensor $F$ \nattached to $S_{\\lambda}$ at the point $\\Phi_{\\lambda} (p)$.  This defines\na curve of tensors at each point $p \\in S$. The geometric variation is simply\n$\\pounds_{\\vec{\\xi}} F = \\partial_{\\lambda} F_{\\lambda} (p) |_{\\lambda=0}$. This derivative only depends\non the values of $\\vec{\\xi}$ on $S$ for truly geometric objects. \nHowever, when\n$F$ requires additional structure for its definition, then derivatives of $\\vec{\\xi}$ on $S$\nmay also arise. This behaviour occurs for instance in $\\pounds_{\\vec{\\xi}\\,} \\theta_{+}$ which \nrequires making a specific choice of $\\vec{l}_{+}$ on each surface $S_{\\lambda}$.\n\nApplying this procedure to the area of $S$ gives (see e.g \\cite{Jost2001})\nthe {\\it first variation of area} \n$\\left . \\frac{d |S_{\\lambda}|}{d\\lambda} \\right |_{\\lambda=0} = \\int_S (\\vec{H} \\cdot \\vec{\\xi} ) \\bm{\\eta_S}$\nwhere $\\bm{\\eta_S}$ is the metric volume form of $S$ \nIn particular, the change of area \nof $S$ for variations along the null directions $\\vec{l}^{\\pm}$ are determined by the\nintegral of the null expansion $\\theta_{\\pm}$. \n\nLet us next write down $\\pounds_{\\vec{l}_{+}} \\theta_{\\pm}$. \nSince $\\vec{l}^+$ and $\\vec{l}^{-}$ are\ninterchangeable, the corresponding expression with also \nhold for variations along\n$\\vec{l}^-$ after making the substitution $+\n\\longleftrightarrow -$ everywhere.\nWe start with the variation of the  induced\nmetric, which is well-known to be\n\\begin{eqnarray*} \n\\vec{l}_{+} (h_{AB}) = 2 \\,  \\vec{l}_{+} \\cdot\n\\vec{K}_{AB},\n\\end{eqnarray*} \nwhere $A,B$ etc. are tensorial indices on $S$. This implies\n\\begin{eqnarray}\n\\vec{l}_{+} (\\bm{\\eta_S}) = \\theta_{+} \\bm{\\eta_{S}}\n\\label{FirstVarArea}\n\\end{eqnarray}\nfor the variation of the volume form on $S$. For the variation of $\\theta_{+}$ along $\\vec{l}^{+}$, it is necessary\nto relate the null normal $\\vec{l}_{+}$ on each one of the varied\nsurfaces $S_{\\lambda}$  to the corresponding null normal on the original\nsurface. As already stressed, this introduces a dependence on the\nfirst derivative of $\\vec{l}^{+}$ via $Q^{+} \\equiv -1\/2 (\\vec{l}^{-} \\cdot {\\nabla^{g}}_{\\vec{l}^{+}} \\vec{l}^{+} )\n|_{\\lambda=0}$ in the  resulting expression, which is the  well-known\nRaychaudhuri equation\n\\begin{eqnarray} \n\\pounds_{\\vec{l}_{+}} \\theta_{+} = Q^{+} \\theta_{+} -\nK^{+}_{AB} K^{+ \\, AB}  - \\mbox{Ric} \\left ( \\vec{l}_{+}, \\vec{l}_{+}\n\\right), \\label{lthetal}\n\\end{eqnarray} \nwhere $K^{\\pm}_{AB} = ( \\vec{l}^{\\pm} \\cdot\n\\vec{K}_{AB} )$ and   $\\mbox{Ric}$ is the Ricci tensor of $({\\cal M},g)$. A\nmore involved calculation gives the derivative of $\\theta_{-}$ along\n$\\vec{l}_{+}$  (see e.g. \\cite{AMS08})\n\\begin{eqnarray} \n\\pounds_{\\vec{l}_{+}} \\theta_{-} =  - Q^{+}\n\\theta_{-} - K^{+}_{AB} K^{- \\, AB}  - \\mbox{Ric} (\\vec{l}_{+},\n\\vec{l}_{-} ) + \\frac{1}{2} \\mbox{Riem}  ( \\vec{l}_{+}, \\vec{l}_{-},\n\\vec{l}_{+}, \\vec{l}_{-} ) + 2 \\left ( D_A S^A + S_A S^A \\right),\n\\label{lthetak}\n\\end{eqnarray} \nwhere $\\mbox{Riem}$ is the Riemann tensor of $({\\cal M},g)$,\n$D$ denotes the Levi-Civita covariant derivative of $(S, h)$ and the\none-form  $S_A$ is defined by\n\\begin{eqnarray} \nS (\\vec{X}) = - \\frac{1}{2} \\left ( \\vec{l}_{-} \\cdot\n{\\nabla^{g}}_{\\vec{X}} \\vec{l}_{+} \\right ),\n\\label{Svector}\n\\end{eqnarray} \nwhere $\\vec{X}$ is any tangent vector to $S$.\nThe Gauss identity for $S$ as a submanifold of $({\\cal M},g)$ implies\n\\begin{eqnarray*} \nR(g) = R(h) - ( \\vec{H} \\cdot \\vec{H}) + \\vec{K}_{AB} \\cdot\n\\vec{K}^{AB} - 2 \\mbox{Ric} (\\vec{l}_{+}, \\vec{l}_{-} ) + \\frac{1}{2}\n\\mbox{Riem} (\\vec{l}_{+}, \\vec{l}_{-}, \\vec{l}_{+}, \\vec{l}_{-} ),\n\\end{eqnarray*}\n where $R(g)$ is the scalar curvature of\n$({\\cal M},g)$. Thus, (\\ref{lthetak}) can be rewritten as\n\\begin{eqnarray} \n\\pounds_{\\vec{l}_{+}} \\theta_{-} =  - Q^{+}\n\\theta_{-} + \\mbox{Ein} (\\vec{l}_{+},\\vec{l}_{-} ) - \\left ( R (h) -\n(\\vec{H}\\cdot \\vec{H}) \\right )  +2  \\left ( D_A S^A + S_A S^A \\right ),\n\\label{lthetak2}\n\\end{eqnarray} \nwhere $\\mbox{Ein}$ is the Einstein tensor of $g$. These\nexpressions are valid in any dimension. The expression for\n$\\pounds_{\\vec{l}_{-}} \\theta_{+}$ follows from (\\ref{lthetak2}) by\ninterchanging $ + \\leftrightarrow -$ and substituting \n$S_A \\rightarrow -S_A$ (see formula (\\ref{Svector}) above).\n\n\n\n\\subsection{Surfaces embedded in a spacelike hypersurface}\n\\label{embedded}\n\nCodimension-two surfaces usually arise as codimension-one\nsurfaces embedded  in a\nspacelike hypersurface $\\Sigma$ of the spacetime ${\\cal M}$. \nThe induced metric on $\\Sigma$ will be denoted by $\\gamma_{ij}$ (Latin, lower case\nindices run from 1 to 3) and the second fundamental form\nwith respect to the unit future normal $\\vec{n}$ will be denoted by $A_{ij}$.\nThe  constraint equations relate the geometry of $\\Sigma$ with\nsome  components of the Einstein tensor\n\\begin{eqnarray}\nR(\\gamma) - A_{ij} A^{ij} + \n(\\mbox{tr}_{\\gamma} A)^2 = 16 \\pi \\rho, \\label{HamiltonianConst} \\\\\n\\nabla_{j} A^{j}_{l} - \\nabla_{l} \\mbox{tr}_{\\gamma} A = - 8 \\pi J_l, \\label{DiffConstraint}\n\\end{eqnarray}\nwhere $R(\\gamma)$ is the curvature scalar of $\\gamma$, $\\nabla_i$ is the  \ncovariant derivative in $(\\Sigma,\\gamma)$, the total energy density is defined\nas $8 \\pi \\rho \\equiv \\mbox{Ein} (\\vec{n}, \\vec{n} )$  and the\nenergy flux one-form $J_i$ is defined as\n$8 \\pi J_i X^i \\equiv - \\mbox{Ein} (\\vec{X}, \\vec{n} )$  on any \nvector $\\vec{X}$ tangent to $\\Sigma$. The initial data set satisfies the\n{\\it dominant energy condition} provided $\\rho \\geq |\\vec{J}|$, where $|\\vec{J}|$ is the norm\nof $J^i$ with respect to the metric $\\gamma_{ij}$. Notice that, despite their names,\n$\\rho$ and $\\vec{J}$ are defined directly in terms of the Einstein tensor. No field equations\nfor are therefore assumed  either here or elsewhere in this paper.\nThe initial data set\nis said to be  {\\it time-symmetric} whenever $A_{ij}=0$.\n\n\n\n\n\n\nAssume, as\nbefore, that $S$ is orientable and that a preferred unit normal $\\vec{m}$ tangent to  $\\Sigma$ \ncan be selected. In\nthis situation, the null normals will always be uniquely  chosen as\n$\\vec{l}^{\\pm} = \\vec{n} \\pm \\vec{m}$.  As a submanifold of $\\Sigma$,\n$S$ has second fundamental form $\\kappa_{AB}$ with respect to\n$\\vec{m}$.  Its trace $p$ is the mean curvature of $S$ in $\\Sigma$. The \ntrace-free part of $\\kappa_{AB}$ will be written as $\\Pi^{\\vec{m}}_{AB}$.\n\n\nThe decomposition of the second fundamental form $A_{ij}$ into\ntangential and normal components to $S$ will also play a role. We will\ndenote by $A^S_{AB}$ the projection of $A_{ij}$ on $S$,\n$\\Pi^{\\vec{n}}_{AB}$ the  trace-free part of this tensor and $q$ its\ntrace. Then, (see e.g.  \\cite{Jost2001})  $\\vec{K}_{AB} = - A^S_{AB}\n\\vec{n} + \\kappa_{AB} \\vec{m}$,  $\\vec{\\Pi}_{AB} = -\n\\Pi^{\\vec{n}}_{AB} \\vec{n} + \\Pi^{\\vec{m}}_{AB} \\vec{m}$ and  $\\vec{H}\n= - q \\vec{n} + p \\vec{m}$, which implies $\\theta_{\\pm} = \\pm p + q$.\n\nThe three remaining independent components of $A_{ij}$ can be encoded\nin the trace $\\mbox{tr}_{\\gamma} (A)$ and in the  normal-tangential components\n\\begin{eqnarray}\nS_A \\equiv A_{ij} m^i e^{j}_A. \\label{DefS_A}\n\\end{eqnarray}\nThis definition of $S_A$ is\nconsistent with (\\ref{Svector}) once  the choice of $\\vec{l}^{\\pm}$\ndescribed above is made. The general variation formulas \n(\\ref{lthetal}) and (\\ref{lthetak2}) can be rewritten in this context,\nafter a straightforward calculation which uses the constraint equation\n(\\ref{HamiltonianConst}), as\n\\begin{eqnarray}\n\\pounds_{\\vec{m}} p & = & - \\frac{1}{2} \\Pi^{\\vec{m}}_{AB} \\Pi^{\\vec{m} \\, AB} - \\frac{3}{4}  p^2\n- \\frac{1}{2} R(\\gamma) + \\frac{1}{2} R (h), \\label{mp}\\\\\n\\pounds_{\\vec{m}} q & = & p A_{ij} m^{i} m^j - A^{S}_{AB} \\kappa^{AB} + D_A S^A + 8 \\pi (\\vec{J} \\cdot \\vec{m} ).\n\\label{mq}\n\\end{eqnarray}\n\n\n\n\nIn this $2+1+1$ context, MOTS satisfy $p\n+ q =0$, generalized trapped surfaces satisfy $p \\leq |q|$ and\ngeneralized apparent horizons $p = |q|$. Notice that the latter always\nhave  non-negative mean curvature, which means that outward variations\ndo not increase the area. This property in fact generalizes to a\nglobal statement for so-called {\\it outermost}  generalized horizons,\nas follows.\n\n\nConsider an initial data set $(\\Sigma,\\gamma_{ij},A_{ij})$ with\n$\\Sigma$ a compact manifold with boundary and assume that the boundary\ncan be split into two disjoint components $\\partial^- \\Sigma$ and\n$\\partial^{+} \\Sigma$ (neither of which is necessarily connected). We\ndenote by  $\\Sigma^{\\circ}$ the interior of $\\Sigma$, so that $\\Sigma = \\Sigma^{\\circ}\n\\cup \\partial^- \\Sigma \\cup\n\\partial^+ \\Sigma$.  We want to\nthink of  $\\partial^{-} \\Sigma$ as the ``inner'' boundary, which means\nthat we will endow it with a unit normal $\\vec{m}^{-}$ pointing\ntowards $\\Sigma$. Similarly $\\partial^{+} \\Sigma$ is the ``outer''\nboundary, which we endow with the unit normal $\\vec{m}^{+}$ pointing\noutside $\\Sigma$ (see Figure \\ref{Bounding}).\n\n\\begin{figure}[h!]\n\\begin{center}\n\\psfrag{S}{$S$}\n\\psfrag{Sigma}{$(\\Sigma,\\gamma_{ij},A_{ij})$}\n\\psfrag{Outer}{$\\partial^{+} \\Sigma$}\n\\psfrag{Inner}{$\\partial^{-} \\Sigma$}\n\\psfrag{Omega}{$\\Omega$}\n\\psfrag{mm}{$\\vec{m}^{-}$}\n\\psfrag{mp}{$\\vec{m}^{+}$}\n\\includegraphics[width=8cm]{Bounding.eps}\n\\caption{Initial data set with an inner ($\\partial^{-} \\Sigma$) an outer boundary\n($\\partial^{+} \\Sigma$) and the corresponding choice of unit normals.\nA bounding surface $S$, and its exterior domain $\\Omega$, are shown.}\n\\label{Bounding}\n\\end{center}\n\\end{figure}\n\nIn many cases,  $\\Sigma$ will contain plenty of\nweakly outer trapped surfaces. An important question arises then: is there\nan outermost weakly outer trapped surface $S$? Intuitively, this means\na surface which encloses all others, or equivalently, such that no weakly outer\ntrapped surface can penetrate outside $S$. In order to make this precise, it\nis necessary to have a well-defined notion of ``outside'' of S. This can be\nachieved by restricting the class of surfaces\nto those which are homologous to the outer boundary.  More explicitly,\na surface $S^{\\prime}$ is called {\\it bounding} if it is contained in $\\Sigma^{\\circ} \\cup\n\\partial^- \\Sigma$ (in particular, it is disjoint with the outer\nboundary) and, together with $\\partial^{+} \\Sigma$ bounds an open\ndomain $\\Omega^{\\prime}$.  For such surfaces, the mean curvature $p$\nand the outer expansion $\\theta_{+}$ is calculated with respect to the\nnormal  pointing towards $\\Omega^{\\prime}$. A surface $S$ is outermost in some class\nif no other surface in the class enters the domain $\\Omega$ bounded by $S$ and \n$\\partial^{+} \\Sigma$. The existence of outermost\nMOTS in this context has been proven by Andersson and Metzger \n\\cite{AnderssonMetzger2007} assuming\nthe inner boundary to be weakly outer trapped ($\\theta_{+} \\leq 0$) and\nthe outer boundary to be outer untrapped ($\\theta_{+} >0$), so that they play\nthe role of barriers. The proof uses the Gauss-Bonnet theorem so the\nresult is valid in the 3+1 dimensional setting.  The precise statement\nis as follows\n\\begin{theorem}[Andersson \\& Metzger \\cite{AnderssonMetzger2007}]\n\\label{AnderssonMetzger} \nLet\n$(\\Sigma,\\gamma_{ij},A_{ij})$ by three-dimensional \nand have an inner and\nouter boundary as described in the previous paragraph.\nAssume that the inner boundary  has $\\theta_{+}\n(\\partial^- \\Sigma ) \\leq 0$ and the outer boundary $\\theta_{+}\n(\\partial^+ \\Sigma) >0$. Then there exists a unique smooth embedded\nbounding MOTS $S$, i.e. $S \\cup \\partial^{+} \\Sigma = \\partial \\Omega$\nand $\\theta_{+} (S) = 0$,  which is outermost: any other weakly outer\ntrapped surface $S'$ which together with $\\partial^{+} \\Sigma$ bounds\na domain $\\Omega^{\\prime}$ satisfies $\\Omega \\subset \\Omega^{\\prime}$.\n\\end{theorem} \nAn important issue refers to the topology of the outermost MOTS $S$.\nA classic result by Hawking \\cite{HawkingEllis} states that \nthe topology of each connected component of the outermost MOTS\nhas to be either toroidal or spherical, provided the initial data set satisfies the \ndominant energy condition.\nRecently, this result has been extended to \nhigher dimensions by Galloway and Schoen \\cite{SchoenGalloway2005}, where it is proven that,\nexcept for exceptional cases, the outermost MOTS in any  \nspacelike hypersurface of a spacetime satisfying the dominant energy condition\nmust by of positive Yamabe type. The exceptional cases have been ruled out by\nGalloway in \\cite{Galloway2007} thus leaving only the positive Yamabe case. In four\nspacetime dimensions, this implies that each connected component of the outermost MOTS must be\na sphere.\n\nAn existence result involving generalized trapped surfaces\nand generalized apparent horizons instead of weakly trapped\nsurfaces and MOTS has been proven recently by Eichmair\n\\cite{Eichmair2008}.\nThis result does not use  the Gauss-Bonnet theorem, so it it not\nrestricted to 3+1 dimensions. However, it relies on regularity of\nminimal surfaces,\nwhich restricts the dimension of $\\Sigma$ to be at\nmost seven. Outermost generalized apparent horizons have one\nfundamental advantage over MOTS; they are {\\it area outer minimizing},\ni.e. they have less or equal area than any other bounding surface fully contained in the\nclosure of the exterior region. This area minimizing property makes these surfaces\npotentially very interesting for the Penrose inequality, as first\ndiscussed by  Bray and Khuri \\cite{BrayKhuri2009}. This theorem of\nEichmair  answers in the  affirmative a conjecture due to H.L. Bray and\nT. Ilmanen \\cite{BrayKhuri2009} on the existence of outermost\ngeneralized apparent horizons and its area outer minimizing property.\n\\begin{theorem}[Eichmair \\cite{Eichmair2008}] \n\\label{Eichmair}\nLet $(\\Sigma,\\gamma_{ij},A_{ij})$ be $m$-dimensional,\nwith $3 \\leq m \\leq 7$ and have an inner and\nouter boundary as described above. Assume that\nthe inner boundary  is a generalized trapped surface $p \\leq |q|$ and\nthe outer boundary satisfies $p > |q|$. Then there exists a unique\n$C^2$ embedded  generalized apparent horizon $S$ ($p = |q|$) which is\nbounding, $S \\cup \\partial^{+} \\Sigma = \\partial \\Omega$, and outermost:\nany other generalized apparent horizon $S'$ which together with\n$\\partial^{+} \\Sigma$ bounds a domain $\\Omega^{\\prime}$ satisfies\n$\\Omega \\subset \\Omega^{\\prime}$.  Moreover $S$ is area outer\nminimizing with respect to variations in $\\overline{\\Omega}$.\n\\end{theorem}\n\nIn the purely Riemannian case ($A_{ij}=0$), MOTS and\ngeneralized apparent horizons are simply minimal surfaces (i.e. $p=0$) and \nthe two theorems above\nbecome statements on the existence of an outermost minimal surface on\na compact domain with barrier boundaries. This particular case\nwas already known before (see Sect. 4 of \\cite{HuiskenIlmanen2001} and references therein).\nIn fact, the existence of an\noutermost  minimal surface played an important role in the\nproof of the Riemannian Penrose inequality by G. Huisken and\nT. Ilmanen \\cite{HuiskenIlmanen2001}.\n\n\n\\subsection{Hawking mass, Geroch mass and their variation formulas}\n\\label{UEF}\n\n\n\nHuisken and Ilmanen's proof of the Riemannian Penrose inequality\nis based on a very interesting result by Geroch \\cite{Geroch1973} who found \na certain functional (now called Geroch mass) which is monotonic under so-called\ninverse mean curvature flows (IMCF). \nThis functional is defined for surfaces embedded\nin a Riemannian manifold $(\\Sigma,\\gamma)$. For codimension-two surfaces embedded in a spacetime,\nthe analogous to the Geroch mass is the so-called Hawking mass \n\\cite{Hawking1968} (both coincide when\nthe surface is embedded in a time-symmetric initial data set). Since these two masses have similar properties,\nit is conceivable  that the Hawking mass may also be useful to tackle\nthe general Penrose inequality (i.e. for non time-symmetric initial data sets). A natural first step is to\nanalyze whether the\nHawking mass is also monotonic under suitable flows. Monotonicity along null directions was first \nstudied by Hayward \\cite{Hayward1994}.\nMore recently, the rate of change of the Hawking mass along an IMCF\nin an arbitrary initial data set $(\\Sigma,\\gamma_{ij},A_{ij})$ was studied\nin \\cite{MalecMarsSimon2002}. The null and initial data set approaches can be unified and extended\nby using a spacetime flow formulation, where the two-dimensional surface\nis varied in spacetime. This is discussed in detail in \\cite{BrayHayward2007}. Let us briefly\ndescribe the main points, and how previous results fit into this framework.\n\nFor any closed spacelike surface  $S$ embedded in a four dimensional spacetime $({\\cal M},g)$,\nthe Hawking mass \\cite{Hawking1968}\nis defined by\n\\begin{eqnarray}\nM_{H}(S) = \\sqrt{ \\frac{\\left | S \\right |}{16 \\pi}} \\left (\n\\frac{\\chi(S)}{2} - \\frac{1}{16 \\pi} \\int_S \\left ( (\\vec{H}\\cdot \\vec{H}) + \\frac{4}{3} C \\right )\\bm{\\eta_S} \n\\right ),\n\\label{HawMass}\n\\end{eqnarray}\nwhere $\\chi(S)$ is the Euler \ncharacteristic of $S$. Hence $\\chi(S)=2$ for a connected surface with spherical\ntopology. For an arbitrary compact surface $\\chi(S) = \\sum 2 (1- g)$ where the sum is over the\nconnected components and $g$ is the genus. At this point, the parameter $C$ in (\\ref{HawMass})\nis a completely\narbitrary constant. Its inclusion\nis relevant for settings where the spacetime has a cosmological constant, or when dealing\nwith hyperbolic initial data sets in an asymptotically flat spacetime (see Sect. \\ref{hyperbolicSect} below).\nThis constant was first introduced\nin \\cite{BoucherGibbonsHorowitz1984} in the time-symmetric context.\n\n\n\nLet $\\vec{V}^{\\star}$ denote the Hodge dual operation on the normal\nspace $N_p S$ (which, under our assumptions, has a Lorentzian induced metric and it is orientable, so it\nadmits a canonical volume element once an orientation is chosen).\nThis operation is idempotent and transforms any vector into an orthogonal vector with opposite norm. \nWhenever $S$ admits a notion of ``outer'', the orientation of the normal bundle will be chosen\nso that  $\\vec{l}^{+ \\star} = \\vec{l}^{+}$ holds (this implies $\\vec{l}^{- \\star} = - \\vec{l}^{-}$).\n\nLet us also denote by\n$\\nabla^{\\bot}$ be the connection on the normal bundle (i.e.\nif $\\vec{V}$ is normal to $S$ and $\\vec{X}$ is tangent to $S$,\n$\\nabla^{\\bot}_{\\vec{X}} \\, \\vec{V} \\equiv (   {\\nabla^{g}}_{\\vec{X}} \\, \\vec{V} )^{\\bot})$. \nFor any\northogonal variation vector $\\vec{\\xi}$, the derivative of the Hawking mass along $\\vec{\\xi}$ reads\n\\cite{BrayHayward2007}\n\\begin{eqnarray}\n\\left . \n\\frac{d M_{H} (S_{\\lambda})}{d \\lambda} \\right |_{\\lambda=0} & = & \n\\frac{1}{8 \\pi} \\sqrt{ \\frac{ \\left | S \\right |}{16 \\pi}}  \\int_{S}\n\\left[ \\left ( \\mbox{Ein} (\\vec{H}^{\\star}, \\vec{\\xi}^{\\star} ) + C  (\\vec{H}^{\\star} \\cdot \\vec{\\xi}^{\\star} \\,  )\n\\right )\n +  \n8 \\pi \\Theta^T (\\vec{H}^{\\star},\\vec{\\xi}^{\\star} \\, ) + \\right. {}  \\nonumber \\\\\n & + & \\left.\n\\mbox{tr}_{S} \\left ( \\vec{H} \\cdot \\nabla^{\\bot} \\nabla^{\\bot} \\vec{\\xi} \\right )\n-  \\left ( \\frac{1}{2} R(h) -  \\frac{1}{4} (\\vec{H}\\cdot \\vec{H}) \\right )\n\\left[ \\left (\\vec{\\xi} \\cdot \\vec{H} \\right ) - a  \\right ] \\right ] \n\\bm{\\eta_{S}},\n\\label{dMH1}\n\\end{eqnarray}\nwhere $a$ is the constant obtained by averaging $(\\vec{\\xi} \\cdot \\vec{H})$ on $S$, i.e.\n\\begin{eqnarray*}\na  \\equiv  \\frac{\\int_{S}\n\\left ( \\vec{\\xi} \\cdot \\vec{H} \\right ) \\bm{\\eta_{S}}}{\\left | S\\right |},\n\\end{eqnarray*}\nand\n$8 \\pi \\Theta^T(\\vec{H}^{\\star},\\vec{\\xi}^{\\star}) \\equiv \n(\\vec{\\Pi}_{AB}\\cdot\\vec{H}^\\star ) (\\vec{\\xi}^\\star \n\\cdot \\vec{\\Pi}^{AB} \\, ) - \n\\frac{1}{2} (\\vec{\\Pi}_{AB}\\cdot\\vec{\\Pi}^{AB} ) (\\vec{\\xi}^\\star  \\cdot \\vec{H}^\\star \\,)$\nis the ``transverse part of the gravitational energy''. This object  has good positivity\nproperties \\cite{BrayHayward2007}. In particular is non-negative if \n$\\vec{H}$ and $\\vec{\\xi}$ are both achronal and have positive inner product. \n\n\nFrom expression (\\ref{dMH1}) is it straightforward to show that \n$d M_{H} (S_{\\lambda})\/d \\lambda \\geq 0$ whenever \n$(\\mbox{Ein} + C g)$ satisfies the\ndominant energy condition\n(recall that a covariant tensor satisfies this property when\nit is non-negative when acting on any combination of causal future directed vectors), \n$\\vec{H}$ is spacelike and\nthe variation vector takes the form\n\\begin{eqnarray}\n\\vec{\\xi} = \\frac{1}{(\\vec{H} \\cdot \\vec{H})} \\left ( a \\vec{H} + c  \\vec{H}^{\\star} \\right )\n\\quad \\Longleftrightarrow \\quad \\left (\\vec{\\xi} \\cdot \\vec{H} \\right ) = a, \n \\left (\\vec{\\xi} \\cdot \\vec{H}^{\\star} \\right ) = - c ,\n\\label{Flowvector}\n\\end{eqnarray}\nwhere $|c| \\leq a$ is an arbitrary constant. Following Hayward \n\\cite{Hayward1994} in the null case, these flows have been termed\n{\\it uniformly expanding}  in \\cite{BrayHayward2007}\nsince the mean curvature\nalong $\\vec{\\xi}$ and its dual $\\vec{\\xi}^{\\star}$ are both constant.\n\n\nThe case with $c = \\pm a$\ncorresponds to a null variation vector and corresponds exactly \nto the flow studied by Hayward  \\cite{Hayward1994}. \nThe case \n$c=0$ can be naturally called\n{\\it inverse mean curvature flow vector} (because $\\vec{\\xi} = \\frac{1}{(\\vec{H} \\cdot \\vec{H})} \\vec{H}$)\nand monotonicity of the Hawking mass in this\ncase was first mentioned\nin \\cite{HuiskenIlmanen2001} and studied in detail by Frauendiener \\cite{Frauendiener2001}.\nThe case $|c | < a$\ncorresponds to a spacelike\nflow vector and therefore can be rephrased as variations within an initial\ndata set, as follows.\n\nFor a surface $S$ embedded in the initial data\nset $(\\Sigma,\\gamma_{ij},A_{ij})$, the mean curvature\nvector of $S$ reads $\\vec{H} = - q \\vec{n} + p \\vec{m}$\nand the Hawking mass takes the form \n\\begin{eqnarray*}\n\\label{HawkingMassID}\nM_H(S) = \\sqrt{\\frac{|S|}{16\\pi}} \\left ( \\frac{\\chi(S)}{2} - \\frac{1}{16 \\pi}\n\\int_S \\left ( p^2-q^2 + \\frac{4}{3} C \\right ) \\bm{\\eta_S}  \\right ).\n\\end{eqnarray*}\nFor an outward pointing variation vector $\\vec{\\xi}$ tangent to\n$\\Sigma$, i.e. $\\vec{\\xi} = e^{\\psi} \\vec{m}$, the general variation formula (\\ref{dMH1})\ncan be rewritten in terms of the initial data geometry as\n\\begin{eqnarray}\n\\label{dMH2}\n\\left . \n\\frac{d M_{H} (S_{\\lambda})}{d \\lambda}  \\right |_{\\lambda=0}    =   \n\\frac{1}{8 \\pi} \\sqrt{ \\frac{ \\left | S \\right |}{16 \\pi}}  \\int_{S}\n\\left [ e^{\\psi} \\left [p \\left ( 8 \\pi \\rho - C \\right )\n+ 8 \\pi q J_i m^i) \\right ]\n+ \\frac{1}{2} e^{\\psi} \\left ( \\Pi^{\\vec{n}}_{AB} \\bullet \\Pi^{\\vec{m}\\, AB} \\right )\n\\right . \\nonumber \\\\ \n +  \\left . \ne^{\\psi} \\left ( S_A \\bullet D^A \\psi \\right )  \n+ e^{\\psi} q \\left (D_A S^A \\right) +\n\\left [ \\Delta_S \\psi - \\frac{1}{2} R(h)  +\n\\frac{1}{4} \\left ( p^2 - q^2 \\right ) \\right ] \n\\left (p e^{\\psi} - a \\right ) \n\\right ] \\bm{\\eta_{S}},\n\\end{eqnarray}\nwhere  the $\\bullet$ operation acts on two tensors of the same class\n$X_{A_1 \\cdots A_r}$, $Y_{A_1 \\cdots A_r}$ and gives\n\\begin{eqnarray*}\n\\left ( X_{A_1 \\cdots A_r} \\bullet Y^{A_1 \\cdots A_r} \\right ) = \np X_{A_1 \\cdots A_r} X^{A_1 \\cdots A_r}  - q \nX_{A_1 \\cdots A_r} Y^{A_1 \\cdots A_r}\n+ p Y_{A_1 \\cdots A_r} Y^{A_1 \\cdots A_r}. \n\\end{eqnarray*}\nThis quadratic expression is non-negative provided \n$|q| \\leq p$, or equivalently when $\\theta_{+} \\geq 0, \\theta_{-} \\leq 0$\nat each point, which is an ``untrappedness'' condition (note that this condition\nis basically the complementary at each point of the \ngeneralized trapped surface condition).\nIn the particular case of $C=0$ and IMCF ($p e^{\\psi} = a$),\nthe variation formula (\\ref{dMH2}) was obtained in \\cite{MalecMarsSimon2002}. \n\nThe first two terms\nin (\\ref{dMH2}) are non-negative provided $ 8 \\pi | \\vec{J}| \\leq 8 \\pi \\rho - C$ holds (this is automatically true\nif $\\mbox{Ein} + C g$ satisfies the dominant energy condition)\nand $|q| \\leq p$. The last two terms have no\nsign in general. However, the factor in round brackets on each of them integrates\nto zero. Thus, monotonicity can be ensured \\cite{BrayHayward2007} provided\nthe following conditions hold simultaneously: \n(i) $q e^{\\psi}  = c$ \\underline{or} $S^A$ is divergence-free and (ii)\n$\\Delta_S \\psi - \\frac{1}{2} R(h)  +\n\\frac{1}{4} \\left ( p^2 - q^2 \\right ) =\\alpha$\n\\underline{or} $e^{\\psi} p = a$, where $c$ and $\\alpha$ are constants. This leads to\nfour different alternatives for which the Hawking mass is monotonic. \nThe uniformly\nexpanding flow condition (\\ref{Flowvector}) with $|c| <a$ corresponds  to the\ncase $q e^{\\psi}  = c$, and $p e^{\\psi} = a$. This follows easily from the expressions\n$\\vec{H} = -q \\vec{n} + p \\vec{m}$ and its dual \n$\\vec{H}^{\\star} = -q \\vec{m} + p \\vec{n}$. How restrictive are any of these four\nalternatives on a given spacetime remains an open problem.\n\n\nWhile the Hawking mass is intrinsically a functional on surfaces in spacetime (although it obviously admits\na rewriting in 3+1 language), the Geroch mass\nis directly a functional on surfaces in a spacelike slice $(\\Sigma,\\gamma)$.\nIts definition is\n\\begin{eqnarray}\n\\label{GerochMass}\nM_G(S) = \\sqrt{\\frac{|S|}{16\\pi}} \\left ( \\frac{\\chi(S)}{2} - \\frac{1}{16 \\pi}\n\\int_S \\left (p^2  + \\frac{4}{3} C \\right ) \\bm{\\eta_S}  \\right ).\n\\end{eqnarray}\nThe Geroch mass is fully insensitive to the second fundamental form on the slice (in this\nsense, it is a purely Riemannian object) and always satisfies $M_G (S) \\leq M_H(S)$.\nIts variation can be obtained formally from expression (\\ref{dMH2}) simply by putting\nall terms depending on the second fundamental form equal to zero. In particular,\nthis requires substituting $8 \\pi \\rho \\rightarrow 1\/2 R(\\gamma)$, which follows from the \nGauss equation for $\\Sigma$ when the second fundamental form vanishes. Explicitly\n\\begin{eqnarray}\n\\left . \n\\frac{d M_{G} (S_{\\lambda})}{d \\lambda}  \\right |_{\\lambda=0}   & =  &  \n\\frac{1}{8 \\pi} \\sqrt{ \\frac{ \\left | S \\right |}{16 \\pi}}  \\int_{S}\n\\left [ e^{\\psi} p \\left ( \\frac{1}{2} R(\\gamma) - C \\right )\n+ \\frac{1}{2} e^{\\psi} p \\; \\Pi^{\\vec{m}}_{AB} \\Pi^{\\vec{m} \\, AB}\n+ e^{\\psi} p D_A \\psi D^a \\psi + \\right . \\nonumber \\\\\n& + & \\! \\! \\! \\left . \\frac{}{} \\left ( \\Delta_S \\psi - \\frac{1}{2} R(h)  +\n\\frac{p^2}{4} \\right ) \\left (p e^{\\psi} - a \\right ) \n\\right ]  \\label{dMG}\n\\bm{\\eta_{S}}.\n\\end{eqnarray}\nWe emphasize that this result is valid for any spacelike hypersurface of the spacetime,\nnot only for time-symmetric ones.\nFor vanishing $C$ this expression was first obtained by Geroch \\cite{Geroch1973}.\nIn the time-symmetric case, for arbitrary $C$ and imposing  IMCF, this\nexpression was derived in \\cite{BoucherGibbonsHorowitz1984}.\n\n\n\n\n\n\n\n\n\\subsection{Asymptotic flatness}\n\\label{asymptoticflatness}\n\n\n\n \nAn initial data set $(\\Sigma,\\gamma_{ij},A_{ij})$ is {\\it asymptotically euclidean} provided $\\Sigma$ is the disjoint\nunion of a compact set $K$ and a finite union of asymptotic ends $\\Sigma^{\\infty}_i$,\neach of which is  diffeomorphic to $\\mathbb{R}^3 \\setminus\nB$ where $B$ is a closed ball.  Moreover, in Cartesian coordinates $x^{i}$ \nin $\\Sigma^{\\infty}_i$\ninduced by this diffeomorphism, the metric and second fundamental forms have the\nfollowing asymptotic behaviour (we restrict to 3+1 dimensions for definiteness)\n\\begin{eqnarray}\n\\gamma_{ij} = \\delta_{ij} + O\\left (\\frac{1}{r}\\right ), \\quad \n\\partial \\gamma_{ij} = O \\left (\\frac{1}{r^2} \\right ), \\quad  \n\\partial \\partial \\gamma_{ij} = O \\left (\\frac{1}{r^3} \\right ),\n\\nonumber \\\\\nA_{ij} = O \\left (\\frac{1}{r^2} \\right ), \\quad \\partial A_{ij} = O \\left (\\frac{1}{r^3} \\right ), \\label{decay}\n\\end{eqnarray}\nwhere $r  = \\sqrt{\\delta_{ij} x^i x^j}$. \nThe Penrose inequality involves the total mass of a spacetime. In the asymptotically euclidean setting,\nthe total ADM energy-momentum vector \\cite{ADM} is defined through the coordinate expressions\n\\begin{eqnarray*}\nE _{ADM}= \\lim_{r \\rightarrow \\infty}\n\\frac{1}{16 \\pi} \\int_{S_r} \\left ( \\partial_j \\gamma_{ij} - \\partial_i \\gamma_{jj} \\right)\ndS^i,  \\\\\nP_{ADM \\, i} = \\lim_{r \\rightarrow \\infty}\n\\frac{1}{8\\pi} \\int_{S_r} \\left ( A_{ij} - \\gamma_{ij} \\mbox{tr}_{\\gamma} A \\right ) dS^j,\n\\end{eqnarray*}\nwhere $S_r$ is the surface at constant $r$ and \n$dS^i = n^i dS$ with $\\vec{n}$ being the outward unit normal to $S_r$ and $dS$\nthe surface element. This definition depends a priori on the choice of coordinates $x^i$.\nHowever, they can be shown to define geometric quantities on $(\\Sigma,\\gamma_{ij},A_{ij})$\nprovided the constraints satisfy the additional decay properties\n\\begin{eqnarray}\nR(\\gamma) = O\\left ( \\frac{1}{r^4}\\right ), \\quad \\nabla_{i} A^{i}_{j} - \\nabla_j A = O \\left (\\frac{1}{r^4} \\right ).\n\\label{RDA}\n\\end{eqnarray}\nIn fact, the ADM energy-momentum is shown in \\cite{Bartnik2005}, generalizing\nresults in \\cite{Bartnik1986},\nto be well-defined under much weaker conditions, where the metric and second fundamental forms\nbelong to appropriate weighted Sobolev spaces involving just two derivatives\nfor $\\gamma$ and one for $A_{ij}$, and that the left hand sides of (\\ref{RDA}) are integrable\non $\\Sigma$. The total ADM mass of an asymptotically euclidean initial data set is defined\nas $M_{ADM} = \\sqrt{E_{ADM}^2 - \\delta^{ij} P_{ADM \\, i} P_{ADM \\, j}}$.\n\n\n\nThe positive mass theorem of Schoen and Yau \\cite{SchoenYau1979, SchoenYau1981}\nstates that, provided the initial data satisfies the dominant energy condition,\n$\\rho \\geq |\\vec{J}|$, then $M_{ADM}$ is real and in fact \nstrictly positive except whenever $(\\Sigma,\\gamma_{ij},A_{ij})$ corresponds\nto a slice of Minkowski spacetime. So far, this theorem has been proven in any space-time\ndimension up to \n$n=8$. Furthermore, it also holds for spin manifolds of any dimension, for which \nWitten's spinorial proof \\cite{Witten1981,ParkerTaubes1982} of the positive mass theorem\ncan be applied.\n\n\n\n\n \n\\section{Formulations of the Penrose inequality}\n\\label{Formulations}\n\nThe heuristic derivation of the Penrose inequality for a surface $S$ relies on two fundamental facts.\nFirst and foremost, that the ``establishment viewpoint'' of gravitational collapse holds and second,\nthat $S$ is known to lie behind the event horizon\n in any situation where a black hole does indeed form. This second aspect makes it useful to \nconsider the class of weakly outer trapped surfaces $S$ in a given asymptotically euclidean slice.\nIn order to have a useful notion of outer direction in this case, one restricts\nthe surfaces to be boundaries of domains ${\\cal D}^{+}$\ncontaining the asymptotically euclidean end (if there is more than one end,\none should be selected from the outset). \nThe outer normal is chosen to point inside this domain. Such surfaces\nwill be called {\\bf weakly outer trapped boundaries} in the following. \nGiven two surfaces $S_1$ and $S_2$ (not necessarily weakly outer trapped)\nthat bound respectively exterior domains ${\\cal D}_1^{+}$ and\n${\\cal D}^{+}_2$, we shall say that $S_2$ encloses $S_1$ provided ${\\cal D}^{+}_1$ contains ${\\cal D}^{+}_2$.\nThe complementary of the outer domain ${\\cal D}^{+}$ in \n$\\Sigma$ is called ``interior region of $S$'' and denoted by ${\\cal D}^{-}$. It is also useful to define the\nouter trapped region ${\\cal T}^{+}_{\\Sigma}$ in $\\Sigma$ to be the union of interior regions of all weakly outer trapped\nboundaries.  An important fact is that in strongly asymptotically\npredictable spacetimes satisfying the null convergence condition (i.e. $\\mbox{Ric}(\\vec{l},\\vec{l}) \\geq 0$ \nfor each null vector $\\vec{l}$) the interior region of any weakly outer trapped boundary\nis contained in the black hole region $M \\setminus J^{-}({\\mycal I}^{+})$ (see Proposition 12.2.4\nin \\cite{Wald1984}, and Theorem 6.1 in \\cite{ChruscielGalloway2008}). Consequently, the same holds\nfor ${\\cal T}^{+}_{\\Sigma}$. Thus, this type of surfaces\nare good candidates for a Penrose inequality. As mentioned in the introduction, the area\nof the event horizon cut ${\\cal H}_{\\Sigma} = {\\cal H} \\cap \\Sigma$ may be smaller than the area of \nany such surface $S$. However, for spacetime dimensions $n \\leq 8$\nany surface $S$ that bounds an exterior domain, always has a {\\bf\nminimal area enclosure}, i.e. the outermost of all surfaces which enclose $S$ and have less\nor equal area than any other surface enclosing $S$.  We will denote by $A_{\\min} (S)$\nthe area of the minimal surface enclosure of $S$. For a weakly outer trapped boundary in a black hole\nspacetime it follows that $A_{\\min} (S) \\leq |{\\cal H}_{\\Sigma}|$, because ${\\cal H}_{\\Sigma}$ encloses $S$. Hence the heuristic\nargument by Penrose implies\n$M_{ADM} \\geq \\sqrt{A_{\\min}(S)\/16\\pi}$ where $S$ is any weakly outer trapped boundary. Therefore it also\nfollows\n\\begin{eqnarray}\nM_{ADM} \\geq \\sup_{S} \\sqrt{\\frac{A_{\\min}(S)}{16 \\pi}}, \\label{PIHeu}\n\\end{eqnarray}\nwhere the supremum is taken with respect to all\nweakly outer trapped boundaries $S$. This inequality can be rewritten in a much simpler way as follows. \n\n\nSince a sufficiently large coordinate sphere $S_r$ in the asymptotically euclidean region\nhas positive outer expansion, Theorem \\ref{AnderssonMetzger}\ncan be applied \\cite{AnderssonMetzger2007} to conclude that\n$\\partial  {\\cal T}^{+}_{\\Sigma}$ is a smooth \nmarginally outer trapped boundary, which by construction encloses all other weakly outer \ntrapped boundaries. It follows immediately that $A_{\\min} (\\partial {\\cal T}^{+}_{\\Sigma}) \\geq A_{\\min} (S)$\nfor any weakly outer trapped boundary $S$, and hence also\n$A_{\\min} (\\partial {\\cal T}^{+}_{\\Sigma}) \\geq \\sup_S A_{\\min} (S)$. On the other hand\n$\\partial {\\cal T}^{+}_{\\Sigma}$ is itself\na weakly outer trapped boundary, and hence \n$A_{\\min} (\\partial {\\cal T}^{+}_{\\Sigma}) \\leq \\sup_S A_{\\min} (S)$. Thus, they necessarily coincide and the Penrose\ninequality (\\ref{PIHeu}) can be written in the simpler form\n\\begin{eqnarray}\nM_{ADM} \\geq \\sqrt{\\frac{A_{\\min}(\\partial {\\cal T}^{+}_{\\Sigma})}{16 \\pi}}. \\label{PIHeuT+}\n\\end{eqnarray}\nAlthough the Penrose inequality, seen as a consequence of cosmic censorship, should\nbe expected to hold only for the minimum area enclosure of $\\partial {\\cal T}^{+}_{\\Sigma}$,\none can often find in the literature a version of the Penrose inequality \ninvolving the area of the outermost MOTS,\n\\begin{eqnarray}\nM_{ADM} \\geq \\sqrt{\\frac{|\\partial {\\cal T}^{+}_{\\Sigma}|}{16 \\pi}}. \n\\label{PIT+}\n\\end{eqnarray}\n\nThis is of course a stronger \nand simpler looking inequality. However, it is presently known not to be true. A counterexample has\nbeen found by  I. Ben-Dov \\cite{Ben-Dov2004} by considering a spherically symmetric spacetime\ncomposed of four regions. The innermost region is\na portion of a dust-filled closed FLRW spacetime. It is followed by a portion of the Kruskal spacetime of mass $M$\nwhich is then joined to another region made of dust, closed FLRW in such a way that the outermost region,\nwhich is again vacuum, has less mass than the intermediate Kruskal region.\nIn this context, there exists a slice for which the outermost MOTS lies in the intermediate Kruskal portion,\nwhich leads directly to a violation of (\\ref{PIT+}). This example is also a\ncounterexample to a version of the Penrose inequality suggested by Penrose himself in\n\\cite{Penrose1982}, which goes a follows. Suppose that an asymptotically euclidean initial data set contains\na weakly future trapped boundary $S$ and the spacetime evolves according to weak cosmic censorship.\nThen  $S$ is contained in the black hole region that forms. Consider the null hypersurface ${\\cal N}$ consisting of past directed\nnull geodesics emanating orthogonally from $S$ towards the outer direction, and let $\\vec{k}$ be the\ncorresponding null tangent vector. \nSince $S$ is weakly future trapped, we have $\\theta_k \\geq 0$ and the area of $S$ does not decrease initially\nalong ${\\cal N}$, and this remains to be true as long as $\\theta_k$ stays non-negative along the null hypersurface.\nSince the collapse is taking place in the future and ${\\cal N}$ extends towards the past and exterior \nregion, one expects from physical grounds that $\\theta_k$ does not change sign (this is because ${\\cal N}$\napproaches weaker gravitational fields and hence  outer past directed \nexpansions should be positive). Under these circumstances,\nthe intersection of ${\\cal N}$ with the event horizon has at least the same area than $S$, and hence the usual\nheuristic argument implies $M_{ADM} \\geq \\sqrt{|S|\/16 \\pi}$. The counterexample by Ben-Dov shows that this inequality\nis not generally true either. The difficulty with the argument is double. First of all,\nthe assumption that $\\theta_k$ never becomes negative along ${\\cal N}$ need not be true (notice that such \na property is {\\it not} implied by the  Raychaudhuri equation (\\ref{lthetal})).\nIn the example by Ben-Dov, this property is not fulfilled\nif the starting surface $S$ is any of the spherically symmetric future trapped surfaces\nlying either in the intermediate Kruskal region or in \nthe innermost closed dust FLRW region (see Figure 6 in \\cite{Ben-Dov2004}). Secondly, the argument above also relays\non the  implicit assumption that ${\\cal N}$ intersects the black hole event horizon. In the example\nin \\cite{Ben-Dov2004} these hypersurfaces end up in the\nwhite hole singularity and do not intersect the event horizon anywhere.\n\nWeakly outer trapped surfaces on a spacelike hypersurface satisfy $\\theta_+ = p + q \\leq0$ and the outermost\nsuch surface $\\partial {\\cal T}^{+}_{\\Sigma}$ is a MOTS (i.e. $\\theta_+ = p +q =0$).\nGiven an initial data set $(\\Sigma,\\gamma_{ij}, A_{ij})$, a new one\n$(\\Sigma,\\gamma^{\\prime}_{ij}, A^{\\prime}_{ij})$\ncan be obtained simply by changing the time orientation, i.e. by setting $\\gamma^{\\prime}_{ij} =\n\\gamma_{ij}$ and  $A^{\\prime}_{ij} = -A_{ij}$. \nThe heuristic argument to prove the Penrose inequality obviously applies equally well to this new initial data set.\nFor surfaces which are boundaries of exterior domains, the future outer null direction is now $\\vec{l}^{\\prime}_{+} =\n- \\vec{l}_{-}$, and therefore the weakly outer trapped surfaces satisfy $\\theta^{\\prime}_{+}= -\\theta_{-} \\leq 0$, \nor equivalently $p  - q \\leq 0$ (this is, in fact, obvious since the change in time orientation\nchanges $q \\rightarrow -q$). Similarly as before, one can construct the {\\it past trapped region}\n${\\cal T}^{-}_{\\Sigma}$ as the union of interiors of all such boundaries.\nAndersson and Metzger's result (Theorem \\ref{AnderssonMetzger}) implies again the existence of a\nunique  outermost surface $\\partial {\\cal T}^{-}_{\\Sigma}$ which satisfies $p -q =0$ (i.e. a \npast marginally outer trapped surface). Thus, Penrose's heuristic argument also supports\nthe inequality\n\\begin{eqnarray}\nM_{ADM} \\geq \\sqrt{\\frac{A_{\\min}(\\partial {\\cal T}^{-}_{\\Sigma})}{16 \\pi}}. \\label{PIHeuT-}\n\\end{eqnarray}\nIn a given initial data set, neither ${\\cal T}^{+}_{\\Sigma}$ contains ${\\cal T}^{-}_{\\Sigma }$ nor viceversa, in general.\nAlthough not supported\nby any heuristic argument, a version of the Penrose inequality along the lines of (\\ref{PIT+}) which has\nalso been proposed is\n\\begin{eqnarray}\nM_{ADM} \\geq \\sqrt{\\frac{|\\partial ({\\cal T}^{+}_{\\Sigma} \\cup {\\cal T}^{-}_{\\Sigma})|}{16 \\pi}}. \n\\label{PIT}\n\\end{eqnarray} \nSince ${\\cal T}^{+}_{\\Sigma} \\cup {\\cal T}^{-}_{\\Sigma}$ contains both trapped\nsets, the area of its boundary is not smaller than the minimal area enclosures $A_{\\min} (\\partial\n{\\cal T}^{+}_{\\Sigma})$\nor $A_{\\min} (\\partial {\\cal T}^{-}_{\\Sigma})$. Thus, (\\ref{PIT}) is in general\na stronger inequality than (\\ref{PIHeuT+}) or\n(\\ref{PIHeuT-}). So far no counterexample of (\\ref{PIT}) has been found and it is not clear\nwhether the inequality should hold or not. In the time-symmetric case \nthis inequality reduces to either of (\\ref{PIHeuT+}) or (\\ref{PIHeuT-}) and its validity has been\nproven in general, as discussed in Sect. \\ref{Riemannian} below. Inequality (\\ref{PIT}) also reduces\nto either (\\ref{PIHeuT+}) or (\\ref{PIHeuT-}) in any situation where $q$ is known to be identically\nzero on the outermost future and past MOTS. An example of this behaviour\nis given by a class of axially symmetric and conformally\nflat initial data sets presented in \\cite{Dain_et_al_2002}. For a subclass thereof, the validity of the inequality\n(\\ref{PIT}) has been verified numerically in \\cite{Jaramillo_Vasset2007}.\n\nIn the spherically symmetric case, both\n${\\cal T}^{+}_{\\Sigma}$ and ${\\cal T}^{-}_{\\Sigma}$ are obviously spherically symmetric and one of them\nnecessarily contains\nthe other. The Penrose inequality (\\ref{PIT}) is known to be true in that case, as we discuss in the following\nsection. Since the example by Ben-Dov deals with a spherically symmetric spacetime and a spherically symmetric\nslice, it cannot provide a  counterexample to (\\ref{PIT}). \nOn the other hand, very few explicit examples are known where (\\ref{PIT}) has been verified in\nsituations where it does not reduce\nto either (\\ref{PIHeuT+}) or (\\ref{PIHeuT-}).\nThe only cases outside spherical symmetry \nthat I am aware of involve a\nnumerical analysis of three special classes of  initial data sets \\cite{Karkowski-Malec2005}.\nIn all these examples the inequality (\\ref{PIT}) was numerically confirmed.\n\nStill another version of the Penrose inequality has been very recently proposed by H. Bray and M. Khuri. \nThe fundamental idea behind this proposal is to use generalized trapped surfaces instead of\nweakly outer trapped surfaces. Recall that, on an initial data set, a surface\nthat bounds an exterior domain is a generalized trapped surface provided $p \\leq  |q|$. This class\nof surfaces has two main advantages over weakly outer trapped surfaces: first, its definition is \ninsensitive to time reversals, so that one can get rid of the complications of dealing with \nseveral sets, like ${\\cal T}^{+}_{\\Sigma}$, ${\\cal T}^{-}_{\\Sigma}$ or their union. There is only one generalized\ntrapped set ${\\cal T}_{\\Sigma}$ (in a given initial data set) and this contains both\n${\\cal T}^{+}_{\\Sigma}$ and ${\\cal T}^{-}_{\\Sigma}$ (and, obviously, its union). Moreover, the boundary $\\partial {\\cal T}_{\\Sigma}$\nis a smooth generalized\napparent horizon, due to Eichmair's result (Theorem \\ref{Eichmair} above).  The second\nadvantage is that $\\partial {\\cal T}_{\\Sigma}$ is area outer minimizing, i.e.  $|\\partial {\\cal T}_{\\Sigma}| = A_{\\min} (\n\\partial {\\cal T}_{\\Sigma} )$. Consequently, a Penrose inequality involving generalized trapped surfaces can be \nstated in a much simpler form in terms of the area of $\\partial {\\cal T}_{\\Sigma}$, without the need of \ninvoking minimal area enclosures. The Penrose inequality proposed by Bray and Khuri thus reads\n\\begin{eqnarray*}\nM_{ADM} \\geq \\sqrt{\\frac{|\\partial {\\cal T}_{\\Sigma}|}{16 \\pi}}. \\label{PIKhuriBray}\n\\end{eqnarray*}\nThis version is automatically stronger than (\\ref{PIHeuT+}) or (\\ref{PIHeuT-}) because ${\\cal T}_{\\Sigma}$\nencloses the other trapped sets. Arguments in favor of this version will be given below, where we discuss\nBray and Khuri's approach in more detail. We should emphasize, however, that the heuristic argument\nby Penrose {\\it does not} support this version. The reason is that one of the key ingredients for\nthe heuristics to go through is that the surfaces under consideration lie behind the event horizon provided\na black hole does indeed form. However, this property is not generally true for generalized trapped surfaces. An example\ncan be constructed as follows. Consider \na Cauchy slice $\\Sigma$  of the Kruskal spacetime.\nThe portion of $\\Sigma$ lying in the domain of outer communications (i.e. in the exterior Schwarzschild region) has\na boundary which necessarily lies in the union of the black hole event horizon, the bifurcation surface and\nthe white hole event horizons. Select $\\Sigma$ such that this boundary has a non-empty intersection with \nboth the black hole and the white hole event horizons. Since we are considering the\nKruskal spacetime, the intersection of $\\Sigma$ with the black hole event horizon \nis a smooth MOTS and the intersection with the while hole event horizon is a smooth past MOTS,\nit follows that the outermost generalized apparent horizon $\\partial {\\cal T}_{\\Sigma}$ must contain both of them. However,\n$\\Sigma$ can be easily chosen so that the former intersect transversally (see Fig \\ref{Nonsmooth}).\nConsequently, some portion of\n$\\partial {\\cal T}_{\\Sigma}$ (which must be $C^2$) must lie strictly inside\nthe domain of outer communications, i.e. outside the black hole region.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=8cm]{Nonsmooth.eps}\n\\caption{Spacelike hypersurface $\\Sigma$ in the Kruskal spacetime which intersects both the black hole and white\nhole event horizons in such a way that the two surfaces defined by these intersections meet transversaly. \nIn the figure, only the portion of $\\Sigma$ lying outside the black hole and the white hole regions is\nshown. However, $\\Sigma$ is a Cauchy hypersurface, so the intersection with the\nblack hole event horizon is a compact surface without boundary, and the same holds\nfor the intersection with the white hole event horizon. Since both surfaces are generalized\ntrapped surfaces, the boundary $\\partial {\\cal T}_{\\Sigma}$ must enter the shaded region somewhere.}\n\\end{center}\n\\label{Nonsmooth}\n\\end{figure}\n\n\n\\section{The Penrose inequality in spherical symmetry}\n\\label{Spher}\n\nThe Penrose inequality has been proven to hold \nunder the assumption of spherical symmetry, i.e. when $(\\Sigma,\\gamma_{ij},A_{ij})$\nis invariant under an $SO(3)$ action with (generically) $S^2$ orbits. This inequality was first established by\nMalec and \\'O Murchadha \\cite{MalecMurchadha1994} assuming that the initial data is maximal, i.e. \n$\\mbox{tr}_{\\gamma} A =0$ and in full generality by Hayward \\cite{Hayward1996}. In either case the inequality\nbounds the ADM energy (not the ADM mass) in terms of the area of the outermost surface among \n$\\partial {\\cal T}^{+}_{\\Sigma}$ (the outermost MOTS) and $\\partial {\\cal T}^{-}_{\\Sigma}$ (the outermost past MOTS).\n\nBoth arguments are ultimately based on properties of the Misner-Sharp quasi-local energy\n\\cite{MisnerSharp1964}, although\nthis is made fully explicit only in \\cite{Hayward1996}. This quasi-local energy is exactly the Hawking\nmass specialized to spherical symmetry. Thus, the inequality in this case can be easily\nderived from the general expressions in subsection \\ref{UEF}.\n\nFor a spherically symmetric asymptotically euclidean initial data set, \nthe outer trapped region ${\\cal T}^{+}_{\\Sigma}$, being a geometrically defined set, is necessarily spherically symmetric.\nIts boundary $\\partial {\\cal T}^{+}_{\\Sigma}$ is therefore a metric sphere (if non-empty, which we assume from now on). \nNow, two different possibilities arise. Either the region outside $\\partial {\\cal T}^{+}_{\\Sigma}$ contains\nminimal surfaces (case (i))  or not (case (ii)).\nIn case (i) there is an  outermost minimal surface $S_m$ lying outside\n$\\partial {\\cal T}^{+}_{\\Sigma}$. This surface is necessarily a two-sphere and therefore encloses\n$\\partial {\\cal T}^{+}_{\\Sigma}$.\nLet us consider the region outside $S_m$ in case (i) and the region outside $\\partial {\\cal T}^{+}_{\\Sigma}$\nin case (ii). We denote this region by $\\Sigma_{ext}$ and\nthe corresponding first and second fundamental forms by\n$\\gamma_{ext}$ and $A_{ext}$ respectively. This region is free of minimal\nsurfaces, so that the metric can be written as $\\gamma_{ext}=\n\\frac{dr^2}{1- \\frac{2 m(r)}{r}} + r^2 \\left (d\\theta^2 + \\sin^2 \\theta d \\phi^2 \\right )$,\nwith $2m(r) > r$. At infinity \n$\\lim_{r \\rightarrow \\infty} m(r) = E_{ADM}$. \nFrom spherical symmetry, the second fundamental form can be written as\n$A_{ext} = W(r) dr^2 + Z(r) \\left (d\\theta^2 + \\sin^2 \\theta d \\phi^2 \\right ).$\n\nA direct calculation shows that the mean curvature $p$ and the trace $q$ of the second fundamental\nforms on the\nsurfaces $\\{r=\\mbox{const}\\}$ are $p=\\frac{2}{r}\\sqrt{1-\\frac{2m}{r}} > 0 $, $q= 2 Z\/r^2$. \nConsequently, the Hawking mass (\\ref{HawMass})\n(with $C=0$) of these\nspheres reads\n$M_H(r) \\equiv m(r) + Z^2\/(2r)$. Its radial derivative can be obtained immediately\nfrom (\\ref{dMH2}), using the fact that \n$\\partial_r = \\sqrt{1 - \\frac{2 m(r)}{r}} \\, \\vec{m}$, which implies $e^{\\psi} p = \nr \/2$. This gives\n\\begin{eqnarray}\n\\frac{d M_H}{dr} = 4 \\pi r^2 \\left (\\rho + \\frac{Z}{r} J_r \\right ),\n\\label{DerM_h}\n\\end{eqnarray}\nwhere $J_r = (\\vec{J} \\cdot \\partial_r)$. The dominant energy\ncondition $\\rho \\geq | \\vec{J} |$ becomes in this case $\\rho \\geq \\left |\nJ_r \\sqrt{1 - 2 m(r)\/r} \\right |$. Furthermore,\nasymptotically euclidean demands\n$W(r) = O(1\/r^2)$, $Z(r)=O(1)$ so that \n$\\lim_{r \\rightarrow \\infty} M_H(r) = E_{ADM}$.  \n\nLet us first deal with case (i): since $S_m$ is a minimal surface and cannot be\nweakly outer trapped (it lies outside $\\partial {\\cal T}^{+}_{\\Sigma}$), it must have $q >0$\nand hence it is past weakly outer trapped ($\\theta_{-} >0$).  The spheres\nin the asymptotic region have $\\theta_{-}=q - p <0$, and consequently\nthere  must exist an outermost sphere $\\partial {\\cal T}^{-}_{\\Sigma}$ with vanishing\n$\\theta_{-}$ (i.e. an outermost past MOTS). In the exterior \nof $\\partial {\\cal T}^{-}_{\\Sigma}$ we have $\\theta_+ = p +q >0$ and $\\theta{^-} = - p + q <0$. This implies\n$|q| < p$ and thus the bound  $| Z r^{-1} ( 1 - 2m \/r )^{-1\/2} | <1$ outside\n$\\partial {\\cal T}^{-}_{\\Sigma}$. This, together with\nthe dominant energy condition implies that $M_H(r)$ is non-decreasing\noutside $\\partial {\\cal T}^{-}_{\\Sigma}$\n(this is just a particular case of the monotonicity properties of $M_H$ discussed above).\nBeing $\\partial {\\cal T}^{-}_{\\Sigma}$ a past MOTS, we have\n$M_H(\\partial {\\cal T}^{-}_{\\Sigma}) = \\frac{r}{2} = \\sqrt{|\\partial {\\cal T}^{-}_{\\Sigma}|\/16 \\pi}$. The monotonicity\nof $M_H(r)$ from this surface to infinity and the fact that $S_{m}$ is area\nouter minimizing establishes the Penrose inequality\n\\begin{eqnarray*}\nE_{ADM} \\geq \\sqrt{\\frac{|\\partial {\\cal T}^{-}_{\\Sigma}|}{16 \\pi}} \\geq \n\\sqrt{\\frac{A_{\\min} (\\partial {\\cal T}^{+}_{\\Sigma} )}{16 \\pi}} \n\\end{eqnarray*}\nfor case (i). For case (ii) we have that $\\partial {\\cal T}^{+}$ is automatically area outer minimizing and\na similar argument applies: if there is an outermost  past MOTS $\\partial {\\cal T}^{-}_{\\Sigma}$\noutside $\\partial {\\cal T}^{+}_{\\Sigma}$, apply the monotonicity of $M_H$\nfrom $\\partial {\\cal T}^{-}_{\\Sigma}$ to infinity. If there is none, apply monotonicity of $M_H$\nfrom $\\partial {\\cal T}^{+}_{\\Sigma}$ to infinity. In either case one concludes (since\n$\\partial {\\cal T}^{+}_{\\Sigma}$ is area outer minimizing),\n\\begin{eqnarray}\nE_{ADM} \\geq \\sqrt{\\frac{|\\partial {\\cal T}^{+}_{\\Sigma}|}{16 \\pi}}. \\label{partialT} \n\\end{eqnarray}\n\nNotice that the Penrose inequality {\\it does not} state (\\ref{partialT})\nin  case (i). There, the minimum area needed to enclose $\\partial {\\cal T}^{+}_{\\Sigma}$ must be used.\nThis agrees with the discussion in Section \\ref{Formulations}. \nAs already mentioned there, Ben-Dov \\cite{Ben-Dov2004} \nhas found an explicit example in\nspherical symmetry where the inequality (\\ref{partialT}) is violated. On the other hand,\nthe argument above in fact proves the Penrose inequality for the outermost\nof the two surfaces $\\partial {\\cal T}^{+}_{\\Sigma}$ and $\\partial {\\cal T}^{-}_{\\Sigma}$ and thus\nalso for $\\partial ( {\\cal T}^{+}_{\\Sigma} \\cup {\\cal T}^{-}_{\\Sigma} )$, due to spherical symmetry.\n\nNotice also that the spherically symmetric Penrose inequality above\ninvolves the total energy of the slice. This is weaker than the expected Penrose inequality in\nterms of the total ADM mass. If the slice $(\\Sigma,\\gamma_{ij},A_{ij})$ is\nsuch that one can generate a piece of spacetime which admits another slice with vanishing\ntotal momentum, then the Penrose inequality in terms of the ADM mass also follows.  However, the existence\nof this piece of spacetime is not always obvious. In any case, it would be of interest\nto find a proof of the Penrose inequality in terms of the total ADM mass in spherical\nsymmetry directly in terms of the given data. This might give some new clues\non how the general Penrose inequality can be addressed.\n\n\n\n\\section{Riemannian Penrose inequality}\n\\label{Riemannian}\n\n\nAs already mentioned, the field has experienced a fundamental breakthrough\nin the last decade or so with the complete proof of the \nPenrose inequality in the time symmetric case, first by \nHuisken and Ilmanen \\cite{HuiskenIlmanen2001} \nfor a connected horizon and then by Bray \n\\cite{Bray2001} for an arbitrary horizon. Both papers dealt with the\nfour-dimensional case. However, while Huisken and Ilmanen's proof is\nvery specific to four dimensions (because of the use of the Geroch mass),\nBray's approach can be generalized to any spacetime dimension not bigger than eight,\nas recently shown by Bray and Lee \\cite{BrayLee2007}.\n\nBy definition, an initial data set is called time-symmetric whenever $A_{ij}=0$. This has\ntwo immediate consequences, namely that the ADM three-momentum vanishes identically, so that \n$M_{ADM}=  E_{ADM}$, and that only one constraint equation remains,\n\\begin{eqnarray*}\nR(\\gamma) = 16 \\pi \\rho\n\\end{eqnarray*}\nwhich gives $R(\\gamma)\\geq 0$ provided the dominant energy condition holds. In fact,\nthe weak energy condition (defined as $\\mbox{Ein} (\\vec{u}, \\vec{u} ) \\geq 0$ for all \ncausal vectors) suffices in this case.\n\n\nAnother immediate consequence is that $\\theta_{+} = - \\theta_{-} = p$ and, hence, the trapped region ${\\cal T}^{+}_{\\Sigma}$,\nthe past trapped region ${\\cal T}^{-}_{\\Sigma}$ and the generalized trapped\nregion ${\\cal T}_{\\Sigma}$ all coincide in this case. Its boundary $S_m$ is the outermost\nminimal surface, which is non-empty as soon as there is a bounding surface\nwith negative mean curvature (with respect to the normal pointing into the\nchosen asymptotically euclidean end). This is a corollary of Theorems \\ref{AnderssonMetzger} \nand \\ref{Eichmair} above. However, in this context this result is known\nto hold even in more generality. As discussed in \\cite{HuiskenIlmanen2001}, following\nclassic results on minimal surfaces \\cite{MeeksIIISimonYau1982}, it is sufficient\nto define the trapped region $K$ as the \nimage of all immersed minimal surfaces in $\\Sigma$ together with \nthe bounded connected components of its complementary. It follows that the boundary of this set is a\ncollection of smooth, embedded, minimal 2-spheres and that any connected component of $\\Sigma \\setminus K$\nis an ``exterior'' region, i.e. an asymptotically euclidean manifold free of minimal surfaces (even\nimmersed) and  with compact and minimal boundary composed of a finite union of \n2-spheres.  The Penrose inequality therefore becomes an inequality relating the total mass\nand the area of the outermost minimal surface $S_m$ in $(\\Sigma,\\gamma)$ with respect\nto the chosen asymptotically euclidean end. The corresponding inequality\n\\begin{eqnarray}\nM_{ADM} \\geq \\sqrt{\\frac{|S_m|}{16\\pi}}\n\\label{PIRiem}\n\\end{eqnarray}\nis usually termed  ``Riemannian Penrose inequality'' since it involves\ndirectly Riemannian manifolds, with no further structure coming from the\nambient Lorentzian manifold. \n\nAs already mentioned, the Penrose inequality has a rigidity part, namely that equality is achieved\nonly for slices of the Kruskal extension of the Schwarzschild metric. The time-symmetric\nslices of this spacetime define a manifold\n$(\\Sigma_{\\mbox{\\tiny{Sch}}} = \\mathbb{R}^3 \\setminus {0}, \\gamma_{\\mbox{\\tiny{Sch}}})$ with induced\nmetric\n\\begin{eqnarray}\n\\gamma_{\\mbox{\\tiny{Sch}}} = \\left ( 1 + \\frac{m}{2r} \\right )^4 \\left (dr^2 + r^2 d\\theta^2 + r^2 \\sin^2 \\theta d \\phi^2\n\\right ). \\label{gsch}\n\\end{eqnarray}\nThe surface $r= m\/2$ is minimal and separates the manifold into two isometric\npieces (corresponding to the two asymptotic ends of the Kruskal metric). The rigidity part\nof the Riemannian Penrose inequality states that if equality is achieved in (\\ref{PIRiem})\nthen the region in $(\\Sigma,\\gamma)$ outside its outermost minimal\nsurface $S_m$ is isometric to the domain $r > m\/2$ of $(\\Sigma_{\\mbox{\\tiny{Sch}}},\\gamma_{\\mbox{\\tiny{Sch}}})$ with $m= M_{ADM}$.\n\n\nBefore discussing the breakthroughs of Huisken \\& Ilmanen and Bray, let us\ndiscuss some previous attempts to address the Riemannian Penrose inequality.\n\n\\subsection{Spinor methods}\n\\label{spinors}\n\n\nShortly after Schoen and Yau proved the positive mass theorem, Witten \\cite{Witten1981}\nproposed a completely different method using spinors (see \\cite{ParkerTaubes1982} for a rigorous \nversion of Witten's ideas). It is natural to ask whether spinorial techniques can be also \napplied to the Penrose inequality. After all, the spinor techniques had been \nsuccessfully extended to prove the positive mass theorem in the presence of black holes (more precisely,\nmarginally trapped surfaces) in the initial data \\cite{Gibbons_Hawking_1983} (see \\cite{Herzlich1998} \nfor a rigorous proof). The main difficulty lies in finding suitable boundary conditions\non Witten's equation on the boundary of the black holes so that the boundary\nterm arising by integrating the Schr\\\"odinger-Lichnerowicz \\cite{Schrodinger1932,Lichnerowicz1963}\nidentity can be related to the area of the black hole. An interesting attempt to achieve this \nin the Riemannian case \nis due to M. Herzlich \\cite{Herzlich1997}, who obtained a Penrose-like inequality involving\nnot just the total mass and area of the minimal surface but also a \nSobolev type constant of the manifold. Depending on the space under consideration, Herzlich's\ninequality may turn out  to be stronger or weaker than the Penrose inequality (see\n\\cite{Malec-Roszkowski1998} for a limiting case where this inequality reduces simply to a positive mass statement).\nNevertheless, the inequality is still optimal in the sense that equality is achieved only for the \nSchwarzschild manifold.\n\n\nThe class of manifolds considered in \\cite{Herzlich1997} consists of asymptotically euclidean\n3-dimensional, orientable Riemannian manifolds $(\\Sigma,\\gamma)$ having an inner boundary\n$\\partial \\Sigma$ which  is topologically an $S^2$ and geometrically a minimal surface.\nNo assumption is made on whether this\nsurface is the  outermost minimal\nsurface in $(\\Sigma,\\gamma)$ or not. This already indicates that the inequality to be proven cannot be\nthe standard Penrose inequality because minimal surfaces with large area \ncan be shielded from an asymptotically euclidean region with small mass by \nan outermost minimal surface with sufficiently small area, \nwhile maintaining non-negative Ricci curvature everywhere. This is often called ``shielding effect''\nand explicit examples are easily constructed by cutting the Schwarzschild manifold (\\ref{gsch})\non a sphere at $r < m\/2$ and attaching a piece of $S^3$ of large radius with a cap \nnear the north pole removed (this manifold belongs to the class of initial data leading to the \nso-called Oppenheimer-Snyder spherical dust collapse \\cite{Oppenheimer-Snyder1939})\n\n\n\n\nThe inequality proven by Herzlich reads \\cite{Herzlich1997}\n\\begin{eqnarray}\nM_{ADM} \\geq \\frac{\\sigma}{2(1 + \\sigma)} \\sqrt{\\frac{|\\partial \\Sigma|}{\\pi}},\n\\label{HerzlichPI}\n\\end{eqnarray}\nwhere $\\sigma$ is a geometric, scale invariant, quantity on $(\\Sigma,\\gamma)$ defined\nas \n\\begin{eqnarray*}\n\\sigma = \\sqrt{\\frac{|\\partial \\Sigma|}{\\pi}} \n\\inf_{f \\in C^{\\infty}_c, f \\not\\equiv 0}\n\\frac{ \\int_{\\Sigma} \\left (df, df \\right )_{\\gamma} {\\bm{\\eta_{\\gamma}}} }{\\int_{\\partial \\Sigma}\nf^2 \\bm{\\eta_{\\partial \\Sigma}}},\n\\end{eqnarray*}\nwhere $C^{\\infty}_c$ denotes, as usual, the collection of smooth functions with compact support.\nEquality in (\\ref{HerzlichPI}) occurs if and only if $(\\Sigma,\\gamma)$ is isometric to the exterior\nof the Schwarzschild manifold (\\ref{gsch}) outside the minimal surface $r= m\/2$.\n\n\nThe proof is based on an improvement of the  positive mass theorem (also proven in \n\\cite{Herzlich1997}) valid for asymptotically euclidean Riemannian\nmanifolds $(\\overline{\\Sigma},\\overline{g})$ of non-negative scalar curvature \nhaving an inner boundary $\\partial \\overline{\\Sigma}$ which is topologically a sphere \nand which may have positive, but not too large, mean curvature (with respect to the  normal pointing\ntowards infinity). More precisely, \nthe mean curvature $p$ must satisfy the upper bound \n\\begin{eqnarray}\np \\leq 4 \\sqrt{\\frac{\\pi}{|\\partial \\overline{\\Sigma}|}}.\n\\label{InequalityMeanCurvature}\n\\end{eqnarray}\nThis positive mass theorem also has a rigidity part that states that equality is achieved if and only if\n$(\\overline{\\Sigma},\\overline{g})$ is the exterior of a ball in Euclidean space (this implies, \nin particular, that the only surface $S$ in $\\mathbb{R}^3$ which is topologically an $S^2$\nand which satisfies $p \\leq 4 \\sqrt{\\pi \/ |S|}$ is in fact a sphere). The proof of this\ntheorem involves finding appropriate boundary conditions\nfor the Witten spinor on $\\partial \\overline{\\Sigma}$ so that the boundary term in the Schr\\\"odinger-Lichnerowicz\nidentity on the inner boundary gives a non-negative contribution.\n\n\nThe Penrose-like inequality (\\ref{HerzlichPI}) is then proven by finding a conformally\nrescaled metric $\\overline{g} = f^4 g$, with $f \\rightarrow 1$ at infinity in such \na way that $\\overline{g}$ has vanishing scalar curvature and \nthe mean curvature of $\\partial \\Sigma$ with respect to the metric $\\overline{g}$ saturates\nthe inequality (\\ref{InequalityMeanCurvature}). Moreover, the conformal rescaling is shown to\ndecrease the mass at least by the amount given in the right-hand side of (\\ref{HerzlichPI}). \nHere is where the quantity $\\sigma$ arises. Since the mass after the rescaling is\nnon-negative due to the positive mass theorem above,  the Penrose-like inequality follows.\nThe rigidity part holds because, in the case of equality in (\\ref{HerzlichPI}),  $(\\Sigma,\\gamma)$\nis conformal to the flat metric outside a ball and, moreover, its curvature scalar vanishes (if this was non-zero, then\nthe decrease in mass due to the conformal\nrescaling would be {\\it larger} than the  right-hand side of (\\ref{HerzlichPI}), which cannot occur in the equality case).\nSince the only conformally flat, scalar flat and  asymptotically euclidean Riemannian manifold with a minimal \nsurface is the Schwarzschild space (\\ref{gsch}), the rigidity part follows.\n\nThis Penrose-like inequality has been generalized in three different ways. First, Herzlich \n\\cite{Herzlich2002} extended the results to spin manifolds of arbitrary dimension $n$. In this case the\nboundary $\\partial \\Sigma$ is assumed to be a compact and connected minimal surface\nof positive Yamabe type (i.e. such that it admits a metric of positive constant scalar curvature $R_0$).\nDenoting by ${\\cal Y}$ the Yamabe constant (i.e. ${\\cal Y} = R_0 |\\partial \\Sigma|^{\\frac{2}{n-1}}$ where \n$|\\partial \\Sigma|$ is the $(n-1)$-dimensional volume of the boundary), the Penrose-like inequality reads\n\\begin{eqnarray}\nM_{ADM} \\geq \\frac{1}{8 \\pi} \\sqrt{ \\frac{ {\\cal Y} (n-1)}{n-2} }\n\\frac{\\sigma }{\\sigma + 1} | \\partial \\Sigma|^{\\frac{n-2}{n-1}}\n\\label{PIHerzlichDimn}\n\\end{eqnarray}\nwhere $\\sigma$ is the analogous scale invariant quantity in higher dimensions\n\\begin{eqnarray*}\n\\sigma = \\sqrt{\\frac{4 (n-1)}{(n-2) {\\cal Y}}} \n|\\partial \\Sigma|^{\\frac{1}{n-1}} \\inf_{f \\in C^{\\infty}_c, f \\not\\equiv 0}\n\\frac{ \\int_{\\Sigma} \\left (df, df \\right )_{\\gamma} \\bm{\\eta_{\\gamma}} }{\\int_{\\partial \\Sigma}\nf^2 \\bm{\\eta_{\\partial \\Sigma}}}.\n\\end{eqnarray*}\nThe idea of the proof is similar to the three-dimensional case.\n\nThe second  generalization \\cite{Maerten2007}\ninvolves {\\it maximal} initial\ndata sets $(\\Sigma,\\gamma_{ij},A_{ij})$. Although this result\nis not a Riemannian Penrose inequality, the methods used are very similar to the\nprevious ones and it is therefore natural to include it here. The idea of the proof\ninvolves, again, a positive mass theorem\nfor manifolds with suitable boundary and \nfinding an appropriate conformal factor which transforms the data so that \nthe previous mass theorem can be applied while decreasing the mass by a certain amount. \n\nMore precisely, the class of manifolds $(\\Sigma,\\gamma_{ij},A_{ij})$ under consideration\nare asymptotically euclidean spin manifolds of arbitrary dimension $n$ satisfying the dominant energy\ncondition $\\rho \\geq |\\vec{J}|$.\nAs before, the boundary $\\partial \\Sigma$ is connected, compact and of positive Yamabe type.\nThe condition of being minimal is replaced by three conditions, namely  \n(i) the mean curvature $p$ (with respect to the\ndirection pointing towards infinity) is non-positive (ii) $q \\geq |p|$ (or alternatively $-q \\geq |p|$)\neverywhere, and \n\\begin{eqnarray*}\n\\mbox{(iii)} \\quad \\Theta \\equiv \\sqrt{\\frac{n-2}{(n-1) {\\cal Y}}} \\,\\, |\\partial \\Sigma|^{\\frac{1}{n-1}} \n\\sup_{\\partial \\Sigma} \\left ( p + \\sqrt{q^2 + S_AS^A} \\right ) < 1.\n\\end{eqnarray*}\n($S_A$ is defined in(\\ref{DefS_A})). Under these circumstances, the total energy satisfies the bound \\cite{Maerten2007}\n\\begin{eqnarray}\nE_{ADM} \\geq \\frac{1}{8 \\pi} \\sqrt{ \\frac{ {\\cal Y} (n-1)}{n-2} }\n\\frac{\\sigma \\left ( 1- \\Theta \\right ) }{\\sigma + 1 - \\Theta} | \\partial \\Sigma|^{\\frac{n-2}{n-1}}.\n\\label{MaertenPI}\n\\end{eqnarray}\nThe constant $\\Theta$ is always non-negative under assumptions (i)-(ii). Thus, inequality (\\ref{MaertenPI})\nis weaker than the corresponding one in the time-symmetric case (\\ref{PIHerzlichDimn}). The conditions on the boundary\n$\\partial \\Sigma$ are somewhat surprising. Conditions (i) and (ii) state that the mean curvature\nvector of the surface points inward (in\nthe sense that its product with the outer normal $\\vec{m}$ is non-positive) and is \ncausal past (future) everywhere. Thus, the boundary is indeed weakly past (future) trapped. However, these type of\nsurfaces are necessarily not area outer minimizing (except in the very special case\nthat their mean curvature vector vanishes identically).\nConsequently, the minimal area enclosure of the boundary lies, in general, inside $\\Sigma$. \nThus, this Penrose-like inequality is obtained for a surface for which the original Penrose\ninequality is not expected to hold. It is therefore interesting that such an inequality exists.\n\nThe third generalization is due to M. Khuri \\cite{Khuri2009} and again involves a non-vanishing second\nfundamental form. Since the method uses the Jang equation in a fundamental way, we postpone its discussion to\nthe end of Subsection \\ref{Jang1}.\n\n\n\n\\subsection{Isoperimetric surfaces}\n\\label{isoperimetricprofile}\n\nAn interesting attempt to proof the Riemannian Penrose inequality is discussed in H.L. Bray's\nPh.D thesis \\cite{BrayThesis}. The idea is to consider a special class of surfaces which interpolate\nbetween the outermost minimal surface $S_m$, tends to large round spheres at infinity and for which the Geroch\nmass is non-increasing. Although this sounds familiar with the inverse curvature flow argument of Geroch,\nthe idea exploited by Bray is in fact very different. Indeed, instead of using\nflows of surfaces, Bray considers, for any given volume,\nan area minimization problem (i.e. an isoperimetric problem).\nAssume that the outermost minimal surface $S_m$ is connected and consider the class of surfaces in the\nsame homology class as $S_m$. One can associate to each surface $S$ in this class the volume bounded between\n$S_m$ and $S$ (counted negatively in the portion where $S$ lies inside $S_m$ and positively where it lies\noutside). For a given value of $V \\geq 0$, consider the collection of surfaces which bound,\ntogether with $S_m$, precisely a volume $V$ and define $A(V)$ as the infimum of the corresponding areas.\nIf the infimum is attained on a surface $S_V$, then this surface is obviously of constant mean\ncurvature $p(V)$ because it is the solution of  an isoperimetric problem. Bray's idea is to show that\nthe Geroch mass is a non-increasing function of $V$.  \n\nLet us start by assuming that $S_V$ exists and\nthat the  function $A(V)$ is twice differentiable. Then, the first variation of area together with\nthe fact that the area is the first variation of  volume, gives\n$A'(V) = p(V)$ (prime denotes derivative with respect to $V$) and the Geroch mass (\\ref{GerochMass})\n becomes simply \n\\begin{eqnarray}\n  M_G (V) = \\sqrt{\\frac{A(V)}{16 \\pi}} \\left ( 1 - \\frac{1}{16 \\pi} A (V) A'(V)^2 \\right),\n\\label{GerochMassIsoperimetric}\n\\end{eqnarray}\nprovided the surface $S_V$ is connected and of spherical topology. Consider now a variation of $S_V$\nalong its outer unit normal $\\vec{m}$ with unit speed. If we denote by $S_V(t)$ the corresponding\nflow of surfaces (with $S_V(t=0)=S_V$), the\nsecond variation of area gives\n\\begin{eqnarray*}\n\\left . \\frac{d^2 |S_V(t)|}{dt^2} \\right |_{t=0}\n= \\int_{S_V} \\left ( \\pounds_{\\vec{m}} p + p ^2 \n\\right ) \\bm{\\eta_{S_V}} =\n\\int_{S_V}  \\left ( - \\frac{1}{2} \\Pi^{\\vec{m}}_{AB} \\Pi^{\\vec{m} \\, AB} + \\frac{1}{4}  p^2\n- \\frac{1}{2} R(\\gamma) + \\frac{1}{2} R (h) \\right ) \n\\bm{\\eta_{S_V}} \n\\end{eqnarray*}\nwhere we have used (\\ref{mp}) in the second equality. Using now the \nGauss-Bonnet theorem and the non-negativity of the curvature scalar of $(\\Sigma,\\gamma)$\nwe conclude $ \\frac{d^2 |S_V(t)|}{dt^2} |_{t=0} \\leq 4 \\pi +  \\frac{1}{4} p(V)^2 A(V)$. Denoting\nby $V(t)$ the volume bounded by $S_V(t)$, it follows that $A(V(t)) \\leq |S_V(t)|$ because \n$A(V)$ is the infimum of all areas bounding a volume $V$. Since these two functions touch at $t=0$, it \nfollows $\\frac{d^2 A(V(t))}{dt^2} |_{t=0}\\leq \\frac{d^2 |S_V(t)|}{dt^2} |_{t=0}$. Performing\na change of variables $t \\rightarrow V$, one concludes\n\\begin{eqnarray*}\nA''(V) \\leq \\frac{4\\pi}{A(V)^2} - \\frac{3 A'(V)^2}{4 A(V)} \\quad \\Longrightarrow \\quad M_G(V)' \\geq 0.\n\\end{eqnarray*}\nAlthough the argument just described assumes that $A(V)$ is differentiable,\nthe conclusion is still valid if a suitable weak (distributional) derivative\nis taken \\cite{BrayThesis}. \nThe other two assumptions that enter into the argument are (i) that $S_V$ is connected\nand of spherical topology and (ii) that the infimum of $A(V)$ is in fact attained i.e. that\nthe surface $S_V$ exists. The first condition turns out to be crucial and needs to be imposed\nas an assumption (in fact, the hypothesis can be relaxed to demand just that for each $V>0$\nif one or more minimizers of $A(V)$ exists then at least one of them is connected; this\nis termed ``Condition 1'' in \\cite{BrayThesis}). Most of the technical work in\n\\cite{BrayThesis} consists in showing that\ncondition (ii) (i.e. the existence of a minimizer for all $V \\geq 0$) imposes no extra\nrestriction. To that aim, Bray first argues that the class of metrics $(\\Sigma,\\gamma)$\ncan be chosen to be exactly Schwarzschild at infinity (a similar, but weaker, reduction is\nalso used in Bray's full proof of the Riemannian Penrose inequality summarized in subsection \\ref{Braysproof}\nand will be discussed in  more detail there). The heart of the proof of existence of the minimizer\n$S_V$ consists in proving that the isoperimetric surfaces in the Schwarzschild spacetime\nare given by the spherical orbits of the $SO(3)$ isometry group. Contrarily to what one may expect, proving\nthis fact is not a trivial matter. The fundamental idea behind the construction is to use a \ncomparison metric (in this case the flat metric on $\\mathbb{R}^3$)\nfor which one knows that the spheres are the solutions of the isometric problem. This argument,\nwhich was tailored for the Schwarzschild metric, has been simplified and substantially extended in \n\\cite{BrayMorgan2001}, where simple conditions are found on a given spherically symmetric metric which \nensure that the spheres are the minimizers of the isoperimetric problem. \nSummarizing, Bray proves in \\cite{BrayThesis} that the Penrose inequality (\\ref{PIRiem})\nholds for each asymptotically\neuclidean Riemannian  manifold which has a connected  outermost minimal surface $S_m$ and satisfies ``Condition 1''. \n\n\n\n\n\n\n\\subsection{Huisken and Ilmanen's proof}\n\\label{HuiskenIlmanen}\n\n\nThe heuristic idea behind the proof of Huisken and Ilmanen was first proposed by Geroch \n\\cite{Geroch1973} and is based on the observation  that the\nGeroch mass (\\ref{GerochMass}) (with $C=0$)\nis monotonically increasing if the surfaces are moved\nby inverse mean curvature (i.e. $p e^{\\psi}=1$) provided the scalar curvature of \n$(\\Sigma,\\gamma)$ is non-negative.  This fact is clear from \n(\\ref{dMG}) since $p e^{\\psi} = a=1$ and the derivative of $M_G(S_{\\lambda})$ is then\na sum of non-negative\nterms. Another immediate property of $M_G$ is that its value on any\nconnected, topologically $S^2$ minimal surface is $M_G = \\sqrt{|S|\/(16\\pi)}$. \nFor surfaces $S_r = {r = \\mbox{const}}$ in the asymptotically euclidean end $\\Sigma^{\\infty}$,\nthe asymptotic decay of $\\gamma$ implies $\\lim_{r \\rightarrow \\infty} M_G(S_r) = M_{ADM}$. Thus,\n$M_G$ indeed interpolates between the left and right-hand sides of the Penrose inequality.\n\nGeroch original idea was to prove the positive mass theorem by starting the inverse mean\ncurvature flow from a point so that $M_G=0$ initially (the point can be \napproached as the limit of very small coordinated spheres). The positivity of mass\nwould follow provided the inverse mean curvature flow remained smooth all the way to infinity\nand the flow approached large coordinate spheres in the asymptotically euclidean end. \nJang and Wald \\cite{JangWald1977} realized later on that the same argument could be used\nto prove the Penrose inequality by starting from the outermost minimal surface. In fact, $p$\nvanishes on this minimal surface, and hence the velocity $e^{\\psi}$ diverges there. However, by\nstarting the flow from surfaces which approximate the outermost minimal surface from outside, the\nPenrose inequality would follow.  However, it was immediately\nrealized that the flow will not remain smooth in general and that singularities\nwill develop. The first variation of area implies that, as long as the flow remains smooth,\nthe area of the surfaces increases exponentially. Consider now, \nas a very simple example, two disjoint balls\nin Euclidean space. By symmetry, each surface will evolve under IMCF to a larger sphere with \nexponentially increasing radius. Thus, the two surfaces will necessarily touch\nin finite time and the flow cannot remain smooth forever. A less trivial example, discussed\nin \\cite{HuiskenIlmanen1997}, consists of a thin torus in Euclidean space. The differential\nequation satisfied by $p$ under IMCF is of parabolic type and the maximum principle implies that\n$p$ stays bounded above in terms of its initial value (and the background geometry) as long as the flow remains\nsmooth. Thus, the torus will flow outwards at positive speed bounded away from zero and it will thicken.\nHowever, sufficiently thick torus have vanishing\nmean curvature at points on their inner rim. The speed becomes infinity there\nand the flows necessarily stops being smooth.\n\n\nThe presence of singularities made this idea dormant for decades. The only case were the\nmethod was made rigorous involved a particular case of metrics called {\\it quasi-spherical\nmetrics with divergence-free shear} \\cite{Bartnik1995}, where the flow was seen to remain\nsmooth all the way to infinity. Huisken and Ilmanen's \nfundamental contribution was to define the flow in a suitably weak sense so that\nthe singularities could be treated (and, in fact, basically avoided). \n\nAn important ingredient in Huisken and Ilmanen's approach is the use of a level set\nformulation for the IMCF (which is a\ngeometric parabolic flow). This means describing \nthe leaves of the flow as the level sets of a real function $u$ on $\\Sigma$. The IMCF \ncondition translates directly into the following \ndegenerate elliptic equation for $u$\n\\begin{equation}\n\\label{degel}\n\\mbox{div}_{\\gamma} \\left(\\frac{{\\nabla} u}{|{\\nabla} u|} \\right) = |{\\nabla} u|,\n\\end{equation}\nwhere $\\mbox{div}_{\\gamma} V$ is\nthe divergence of $V$ in $(\\Sigma,\\gamma)$. \nIn principle it is possible that  $u$\nremains constant on open sets, which has the immediate consequence that \nthe flow may jump across regions with positive measure. The fundamental idea is \nto use these jumps precisely to avoid the singularities that the smooth IMCF would\notherwise have. In order to achieve this, Huisken and Ilmanen find a variational\nformulation for (\\ref{degel}). This equation is not the Euler-Lagrange equation of any\nfunctional. However, by freezing the right-hand side to $|{\\nabla} u|$, the authors\nwrite down the functional (which now depends on $u$)\n\\begin{equation}\n\\label{funct1} \nJ_u(v) = \\int_{\\Sigma}\\left( |{\\nabla} v| + v |{\\nabla} u|\\right) \\bm{\\eta_{\\gamma}}.\n\\end{equation}\nThe critical points of this functional with respect to compactly supported variations\nof $v$ gives (\\ref{degel}) with $u$ replaced by $v$ on the left-hand side. One then looks for\nfunctions $u$ which minimize their own functional, thus giving (\\ref{degel}). \n\nThis variational formulation has a geometric counterpart which, in rough terms, implies that\neach of the level sets of $u$ (defined as $\\partial \\{u < t\\}$ for positive $t$) is area outer minimizing. \nThus, if we start from a smooth surface $S_0$ which is area outer minimizing and with positive\nmean curvature, the IMCF (which enjoys short time existence thanks to its\ngeometric parabolic character) will evolve \nthe surface smoothly for some ``time'' $\\lambda$. For small enough values of $\\lambda$, each level set\nwill be outer area minimizing. However, there may well exist a value for which $S_{\\lambda}$\nceases to be area outer minimizing. Take the smallest of such values, $\\lambda_1$. This means that \nthere exists a surface $S'_{\\lambda_1}$ which encloses $S_{\\lambda_1}$ and has less or equal area. \nIn fact, it must have equal area because if if had \nstrictly less area, it would also have less area than some close enough previous leaf (the flow\nis smooth up to $\\lambda_1$ and the area changes smoothly).  The surface $S'_{\\lambda_1}$ may \nhave common points with $S_{\\lambda_1}$. However, the surface $S'_{\\lambda_1}\n\\setminus S_{\\lambda_1}$\nmust be a minimal surface because otherwise it would admit a compactly supported\nvariation that would reduce its area while still remaining \noutside $S_{\\lambda_1}$ and enclosing it. However, this varied surface would\n(i) enclose all surfaces in the flow from $S_0$ up to $S_{\\lambda_1}$, (ii) have larger area than\n$S_0$ (because the latter is area outer minimizing) and (iii) have less area that $S_{\\lambda_1}$.\nSo, for some value of the flow parameter smaller than $\\lambda_1$ it would have the same area, and\nit would enclose it. This would contradict the definition of $\\lambda_1$ as being the smallest \nvalue with such property. Notice that, a priori, this outer minimizing prescription\ndoes not exclude that the flow may become singular already in the interval $0 < \\lambda < \\lambda_1$,\nwhere all the leaves remain area outer minimizing.\nThat this cannot happen \nis one of the several statements that Huisken and Ilmanen had to show, and which makes their proof\ntechnically difficult.\n\n\nIn the torus example before, the smooth flow would thicken the torus (in an axially\nsymmetric way) until a surface is obtained which has exactly the same area than the surface\nobtained by closing its hole by two horizontal planes. It is clear in this example\nthat the new pieces have vanishing mean curvature.\n\nSince the level set function $u$ is such that each level set is outer area minimizing, this means\nthat the IMCF evolves smoothly for as long as the leaves remains area outer minimizing until\na surface which is not area outer minimizing is eventually reached.\nAt this point, the surface jumps (meaning that $u$ remains constant between\nthe two surfaces). The flow should be continued from the new surface outwards. Since\nthe new surface has pieces with $p=0$, the IMCF cannot be defined there in a classical\nsense. However, the weak formulation in terms of level sets also takes care of this.\n\nA crucial condition for the monotonicity argument to go through in this formulation\nis that  the Geroch mass does not increase in the jumps. We know that the total area does not change. \nMoreover, the new pieces of the surface\nhave $p=0$, while the deleted pieces have $p>0$ because the flow was smooth up to and including\n$S_{\\lambda}$. Thus, across the jump the Geroch mass is non-decreasing provided the Euler\ncharacteristic of the surface does not decrease. The example with the torus \nshows that in general the topology of the surface may change across the jump, in that case\nchanging from a surface with genus one to a topological sphere. In this particular case,\nthe Euler characteristic increases and the Geroch mass would be monotonically increasing. The \nquestion is therefore whether this is a general property or not. The example with two \nspheres in Euclidean space shows that this cannot be true in general. There, the value of the Euler \ncharacteristic is initially four and after the jump a topological sphere forms, for which\n$\\chi=2$. A general result by Huisken and Ilmanen \\cite{HuiskenIlmanen2001} is that a\ntopologically $S^2$\nsurface in an asymptotically euclidean three-manifold $(\\Sigma,\\gamma)$ outside\nthe outermost minimal surface cannot jump, via the weak formulation of the IMCF, to\nanother surface with smaller Euler characteristic, i.e. no holes can appear on the \nsurface after the jump. Nevertheless, the outermost minimal surface $S$ in $(\\Sigma,\\gamma)$\nneeds not be connected. Consequently, for the Geroch mass to remain \nmonotonic under the weak IMCF, a connected component $S_i$ of \nthe outermost minimal surface must be chosen as initial surface. The Penrose inequality proven by\nHuisken and Ilmanen is, therefore,\n\\begin{eqnarray}\nM_{ADM} \\geq \\max_i \\sqrt{\\frac{|S_i|}{16 \\pi}}, \\label{PIHI}\n\\end{eqnarray}\nwhere $i$ runs over the connected components of the outermost minimal surface.\n\nAs mentioned before, the Riemannian Penrose inequality also has a rigidity part. In this setting, this states\nthat equality is achieved in (\\ref{PIHI}) if and only if \nthe region in $(\\Sigma,\\gamma)$ outside its outermost minimal\nsurface $S_m$ is isometric to the domain $r > m\/2$ of $(\\Sigma_{\\mbox{\\tiny{Sch}}},\\gamma_{\\mbox{\\tiny{Sch}}})$\n(\\ref{gsch})  with $m= M_{ADM}$.\nHeuristically, it is clear that some rigidity is to be expected already\nfrom the monotonicity property of the Geroch mass.\nIf equality is achieved, then the derivative of the Geroch mass is everywhere zero. In particular, in\nthe smooth part of the flow,  $R(\\gamma)=0$, $\\psi= \\mbox{const}$ and each leaf is totally\numbilical (i.e. $\\Pi^{\\vec{m}}_{AB}=0$) at each point. The IMCF condition gives $p = \\mbox{const.}$ on \neach leaf. Since $\\partial_{\\lambda} = e^{\\psi} \\vec{m} = \\frac{1}{p} \\vec{m}$, the\nvariation formula (\\ref{mp}) implies $\\frac{d p}{d \\lambda} = - \\frac{3}{4} p + \\frac{R(h)}{2p}$. \nThis implies that each leaf is a metric sphere (because $R(h) = \\mbox{const.}$). The exponential grow of\nthe area gives $|S_{\\lambda}| = |S_0 | e^{\\lambda}$. Defining a new variable $\\hat{r}= \\sqrt{|S_{\\lambda}|\/4\\pi}$, \nthe metric $\\gamma$ takes the form $\\gamma = \\frac{4}{p^2 \\hat{r}^2} d\\hat{r}^2 + \\hat{r}^2 d \\Omega$.\nThe expression for the Geroch\nmass and the fact that it takes a value $m$ independent of $\\lambda$  gives $\\hat{r}^2 p^2 = 4 (1 -\n\\frac{2 m}{\\hat{r}} )$, and\nhence $\\gamma$ is the Schwarzschild metric outside the minimal surface $\\hat{r} = 2 m$. This is, of course,\na heuristic derivation. Huisken and Ilmanen are able show the same results using only the weak flow.\n\n\nHuisken and Ilmanen's proof has interesting side consequences. First of all, the argument does not use\nthe positive mass theorem anywhere and therefore the method gives an independent proof of \nthis important result. In the presence of an outermost minimal surface, positivity of $M_{ADM}$\nis an obvious consequence of the theorem. If $\\Sigma$ is free of minimal surfaces, it suffices to start\nthe weak inverse mean curvature flow at one point, making rigorous the original idea of Geroch.\n\nAnother interesting consequence deals with the so-called Bartnik mass (or capacity)\n\\cite{Bartnik1989}, defined as follows. A 3-dimensional Riemannian manifold $(\\Omega,\\gamma)$ with no boundary\nand compact metric closure is called \nadmissible if it  can be isometrically embedded into a complete and connected\nasymptotically euclidean three-dimensional manifold $(\\Sigma,\\gamma)$ satisfying the following\nthree properties: (a) the curvature scalar is non-negative, (b) the boundary of $\\Sigma$\nis either empty or a minimal surface, and (c) $\\Sigma$ is free of any other minimal\nsurfaces (the original definition excludes minimal surfaces even at the boundary of $\\Sigma$, this \nmodification is due to Huisken and Ilmanen \\cite{HuiskenIlmanen2001}). $(\\Sigma,\\gamma)$ is called an admissible extension.\nThe Bartnik capacity is the infimum of the ADM masses of all possible extensions of \n$(\\Omega,\\gamma)$. By the positive mass theorem, it is immediately non-negative and it was conjectured\nin \\cite{Bartnik1989-2, Bartnik1995, Bartnik1997} to be positive if $(\\Omega,\\gamma)$ is not a subset of Euclidean space.\nHuisken and Ilmanen used the weak IMCF to reach a slightly weaker conclusion, namely\nthat if the Bartnik capacity is zero, then $(\\Omega,\\gamma)$ is locally isometric to Euclidean \nspace. The idea of the proof is to take a point $p$ in $\\Omega$ where the metric\nis non-flat and choose any admissible extension \n$(\\Sigma,\\gamma)$. Consider the weak IMCF starting at $p$. For some small enough value of the\nflow parameter the corresponding leaf $S_{\\lambda_0}$ \ncan be proven to have a positive Geroch mass, and to lie within $\\Omega$ independently of the extension (this means in\nparticular that $S_{\\lambda_0}$ is area outer minimizing independently of the extension). Thus, monotonicity\nof the IMCF gives the lower bound $M_{ADM} \\geq M_G (S_{\\lambda_0})$. Since this is independent of the extension, the result\nis established.\n\nAnother consequence of Huisken and Ilmanen's construction is the so-called ``exhaustion'' property of the Bartnik mass.\nThis can be formulated as follows. Consider an asymptotically euclidean Riemannian manifold $(\\Sigma, \\gamma)$ and any\nincreasing collection of bounded sets $\\Omega_i \\subset \\Sigma$ satisfying $\\cup_i \\Omega_i = \\Sigma$. Then\nthe Bartnik mass $m_B(\\Omega_i)$ tends to the ADM mass when $i \\rightarrow \\infty$. This is proven\nin \\cite{HuiskenIlmanen2001} by taking suitable coordinate balls within each $\\Omega_i$ in such a way that the coordinate\nradius $R_i$ tends to $\\infty$. It is shown, by using the IMCF, that the Bartnik mass of each $\\Omega_i$ is bounded below\nby the Geroch mass of the corresponding coordinate balls. The Geroch mass of the coordinate balls tends to the ADM \nmass when $R_i \\rightarrow \\infty$. Since the Bartnik mass of $\\Omega_i$ is obviously bounded above by $M_{ADM}$ of $\\Sigma$,\nthe exhaustion property $m_B (\\Omega_i) \\rightarrow M_{ADM}$ follows.\n\n\n\n\n\\subsection{Bray's proof}\n\\label{Braysproof}\n\n\nH.L. Bray was able \\cite{Bray2001} to prove the Riemannian Penrose inequality for arbitrary outermost\nminimal surfaces, with no restriction of connectedness. Bray's proof also uses geometric\nflows in an essential way, but instead of flowing surfaces in a fixed Riemannian \nmanifold, as in Huisken and Ilmanen's proof, it uses a flow of metrics.\nThe idea is to modify the metric with a one-parameter family of conformal factors  so that\nthe curvature scalar remains non-negative,  the new horizon with respect to the conformal metric\nhas the same area as the original one and the total ADM mass does not increase during the process.\nIf, moreover, the flow can be defined in such a way that, as $t\\rightarrow \\infty$, the Riemannian\nmanifold outside its horizon converges to the ${r > m\/2}$ portion of the Schwarzschild\nmanifold (\\ref{gsch}) in a suitable sense, then the Penrose inequality for the original manifold\nfollows because the inequality (in fact equality) holds in the limit. This conformal flow also allows\nBray to show the rigidity part of the Penrose conjecture, namely that equality \nholds if and only if the starting Riemannian manifold is isometric to the exterior region of\nSchwarzschild.\n\n\nBray's statement of the Penrose inequality deals with three-dimensional \nasymptotically euclidean, Riemannian manifolds $(\\Sigma,\\gamma)$ with one or several\nasymptotic ends (in the latter case, one of them is selected and the mass and the concept of ``outer''\nare taken with respect to this end) and which contain \nan area outer minimizing horizon $S_0$. By definition,\na horizon is a smooth, compact, minimal\nsurface which is a boundary of an open set $\\Omega^{-}$ containing all\npossible asymptotic ends on $(\\Sigma,\\gamma)$ except the one that has been selected.\nIn the case of several ends the existence of such a horizon is\nguaranteed, otherwise $(\\Sigma, \\gamma)$ is assumed to have one. An area outer minimizing horizon need not be\noutermost, although the outermost one (which\nalways exists) will have at least the same area as $S_0$ (by the \narea outer minimizing property of the former). As already said, $S_0$\nneed not be connected and  Bray's theorem states $M_{ADM} \\geq \\sqrt{\\frac{|S_0|}{16 \\pi}}$ with \nequality if and only if $(\\Sigma,\\gamma)$ is isometric to the exterior region of the Schwarzschild manifold.\n\n\nThe first step in Bray's proof is to reduce the class of metrics under consideration. \nTo that aim, Bray uses an interesting result due to Schoen and Yau \\cite{SchoenYau1981-1}\n(c.f. Theorem 7 in \\cite{BrayThesis}) which states that\ngiven an asymptotically euclidean, three-dimensional Riemannian\nmanifold $(\\Sigma,\\gamma)$  (see Proposition 4.1 in \\cite{Schoen1989} for a similar statement\nin any dimension)\nwith $R(\\gamma) \\geq 0$, and any positive number $\\epsilon$, there always exists an asymptotically\neuclidean metric ${\\gamma}_{\\epsilon}$ on $\\Sigma$, with $R(\\gamma_{\\epsilon}) \\geq 0$ everywhere\nand identically zero outside a compact set $K$, which is conformally flat outside $K$, i.e.\n${\\gamma}_{\\epsilon} = s_{\\epsilon}^4 \\delta$ on $\\Sigma^{\\infty} \\equiv \\Sigma \\setminus K$, where\n$s_{\\epsilon}: \\Sigma^{\\infty} \\rightarrow \\mathbb{R}$ is a positive function\napproaching a positive constant at infinity (on each one of the asymptotic ends). Moreover,\nthe following conditions are fulfilled,\n\\begin{eqnarray}\n\\left | M_{ADM} (\\gamma) - M_{ADM} ({\\gamma}_{\\epsilon}) \\right | & \\leq & \\epsilon, \\nonumber \\\\\n\\left | \\frac{{\\gamma}_{\\epsilon} ( \\vec{X}, \\vec{X} )}{\\gamma (\\vec{X},\n\\vec{X} )} - 1 \\right | & \\leq & \\epsilon, \\quad \\forall \\vec{X} \\in T_x \\Sigma\n\\mbox{\\,\\,\\,\\, and  \\,\\,\\,\\,} \\forall x \\in \\Sigma, \\label{epsilonchange}\n\\end{eqnarray}\ni.e. the total mass changes within $\\epsilon$ and the metric itself changes pointwise\nat most by $\\epsilon$ (in the sense that unit vectors change their norm at most by $\\epsilon$).\nThe condition that ${\\gamma}_{\\epsilon}$\nhas vanishing curvature scalar outside $K$ implies that $s_{\\epsilon}$ is harmonic (with respect\nto the flat metric). Therefore, the metric ${\\gamma}_{\\epsilon}$ \nis called {\\it harmonically flat}. This perturbation result has been strengthened in\nBray's Ph.D. thesis \\cite{BrayThesis} to show that $s_{\\epsilon}$ can in fact be taken to be\nspherically symmetric, in which case $\\gamma_{\\epsilon}$ is the Schwarzschild metric outside $K$.\nSuch a metric is called {\\it Schwarzschild at infinity}. This stronger version was needed to apply the\nisoperimetric methods described in Sect. \\ref{isoperimetricprofile}.\n\n\nSchoen and Yau's perturbation result implies that, in order to prove the Riemannian Penrose inequality,\nit is sufficient\nto consider harmonically flat metrics. Indeed, if there existed a counterexample to \nthe Penrose inequality, i.e. a metric with total ADM mass strictly smaller than\nthe Geroch mass of its outermost minimal surface, then we would be able to find\na harmonically flat metric ${\\gamma}_{\\epsilon}$ with ADM mass within $\\epsilon$ of the original one. \nSince the area of the outermost minimal surface also depends continuously on\n$\\epsilon$, it is clear that $\\epsilon$ can be arranged so that \nthe Penrose inequality would also \nbe violated for the harmonically flat metric ${\\gamma}_{\\epsilon}$. \n\n\n\nFor this  class of manifolds,\nBray defines a flow of metrics by a conformal rescaling ${\\gamma}_{t} =  u_{t}^4 \\gamma$ which depends\non a parameter $t \\geq 0$.  The function $u_t$ is defined via an elliptic equation for\n$v_t \\equiv \\frac{d u_t}{dt}$\ntogether with the initial value $u_0 = 1$ (so that ${\\gamma}_0 = \\gamma$). $v_t$ is defined by solving the \nDirichlet problem\n\\begin{eqnarray}\n\\left . \\begin{array}{rcl}\n\\Delta_{\\gamma} v_{t} & = &  0 \\mbox{\\,\\,\\, \\,\\,\\,\\,\\, on \\,\\,   } \\Sigma_{t}, \\\\\nv_t & = & 0 \\mbox{\\,\\,\\, \\,\\,\\,\\,\\, on \\,\\,   } \\Sigma \\setminus \\Sigma_{t}, \\\\\n\\left . v_{t}  \\right |_{S_{t}} & = &  0, \\\\\nv_{t} & \\rightarrow & -e^{-t} \\mbox{\\, \\,\nat \\, \\,} \\infty \n\\end{array} \\right \\},  \\label{BraysFlow}\n\\end{eqnarray} \nwhere $S_t$ is the outermost minimal area surface enclosing $S_0$ with respect to \nthe metric ${\\gamma}_t$ and $\\Sigma_t$ is the exterior of $S_t$. Since $S_t$\nis defined using the metric one is trying to construct, it is not obvious a \npriori that such problem admits a solution. Bray uses a discretization argument whereby\n$u_t(x)$ as a function of $t$ (i.e. for fixed value of $x$) is discretized\nas a continuous piecewise linear function $u^{\\epsilon}_t(x)$ (in $t$), where the jumps in the derivatives occur\nat fixed intervals of length $\\epsilon$. The slope of $v^{\\epsilon}_t = \\frac{d u^{\\epsilon}_t}{dt}$ on \nthe $k$-th interval, i.e. on $t \\in [(k-1) \\epsilon, k \\epsilon)$, $k \\in \\mathbb{N}$ \nis defined inductively in $k$. At $k=1$, $v^{\\epsilon}_t$ is zero inside \n$S_0$ and solves the Laplace equation $\\Delta_{{\\gamma}_0} v^{\\epsilon}_t =0$ with boundary data\n$v^{\\epsilon}_t =0$ on $S_0$ and $v^{\\epsilon}_t \\rightarrow -1$ at infinity. $S^{\\epsilon}_1$ is then defined\nas the outermost minimal area enclosure of $S_0$ with respect to the metric $(u^{\\epsilon}_{t=\\epsilon})^4 {\\gamma}_0$\n(the function $u^{\\epsilon}_{t=\\epsilon}(x)$ is known by continuity). \nNotice that since $S_0$ was not assumed to be the outermost minimal surface, $S_1$ may  be far away from\n$S_0$ even for arbitrarily small $\\epsilon$. Assume that the construction has been carried out for\n$k \\in \\mathbb{N}$, then the slope $v^{\\epsilon}_t$ on the interval $t \\in [k \\epsilon, (k+1) \\epsilon )$ is defined\nas being zero inside $S_k$ and solving the Laplace equation $\\Delta_{{\\gamma}_0} v^{\\epsilon}_t =0$ which\napproaches the constant $- (1- \\epsilon)^{k}$ at infinity and vanishes on $S_{k}$.\nThe surface $S_{k+1}$ is defined as the \noutermost minimal area enclosure of $S_k$ with respect to the metric $(u^{\\epsilon}_{t=(k+1)\\epsilon})^4 {\\gamma}_0$\n(again, the function $u^{\\epsilon}_{t=(k+1)\\epsilon}(x)$ is known by continuity).\n\nThus, while the metric is changed continuously in $t$, the boundaries $S^{\\epsilon}_{k}$ \nchange only at discrete values. Existence of $u^{\\epsilon}_t$ follows from the inductive construction.\nThe conformal flow of metrics depends on $\\epsilon$.\nIt is an important ingredient of Bray's proof to show that a suitable limit $\\epsilon \\rightarrow 0$\nexists, thus defining the flow of metrics in (\\ref{BraysFlow}).\nThe limit is performed in two ways, first the conformal factors are seen to converge\nwhen $\\epsilon \\rightarrow 0$ to a locally Lipschitz function $u_t$. This\nalready defines the metric ${\\gamma}_t$ and the surfaces $S_t$ as before. Extending the discrete\ncollection of $S^{\\epsilon}_k$ to a one-parameter family via $S^{\\epsilon}(t) \\equiv\nS^{\\epsilon}_k$ for $t \\in [k \\epsilon, (k+1)\\epsilon)$ the convergence of these surfaces when $\\epsilon \\rightarrow 0$ \n(for fixed $t$) is then considered. It turns out that the limit may depend\non the subsequence $\\epsilon_i \\rightarrow 0$. Bray denotes the collection of all limit surfaces\nas $\\{ S_{\\alpha} (t) \\}$, where $\\alpha$ identifies the element in the collection. Convergence is seen to be in the\nHausdorff distance sense and each element in $\\{S_{\\alpha}\\}$ is proven to be a smooth surface.\nThe surfaces $S_t$ and $\\{ S_{\\alpha} (t) \\}$ need not always coincide. However, they are\nclosely related: for any $t_2 > t_1 \\geq 0$, $S_{t_2}$ encloses $S_{\\alpha_1}(t_1)$ (for any \n$\\alpha_1$) and $S_{\\alpha_2} (t_2)$ encloses $S_{t_1}$ (for any $\\alpha_2$). Thus, they must coincide for any value where\n$S_t$ depends continuously on $t$. Bray then proves that $S_t$ is continuous except\nfor $t$ in a countable set $J$ (which may be empty). Furthermore, $S_t$ is everywhere upper-semicontinuous\nand the right limit always encloses the left limit. Moreover, for $t_2 > t_1$,\n$S_{t_2}$ encloses $S_{t_1}$. Thus, the flow of surfaces $S_{t}$ is everywhere outward\nand may jump, also outwards, at a countable number of places. \n\nIn addition, and this is crucial for the proof of the Penrose inequality, the area of $S_t$\nis constant for all $t$, even at the jumps. Outside the jumps, this constancy can be understood heuristically\nbecause $S_t$ is a closed minimal surface and hence its area does not change to first order with\nrespect to any variation, in particular the variation which transforms $S_t$ into\n$S_{t + \\delta}$ ($\\delta$ small) in the metric ${\\gamma}_t$.\nThe area also changes because the metric depends on $t$.\nHowever, since $v_t =0$ on $S_t$ by construction,\nthe metric ${\\gamma}_{t+ \\delta}$ coincides with ${\\gamma}_t$ on $S_t$  to first order  \nand therefore\nthe area of $S_t$ with the metric ${\\gamma}_t$ coincides to first order with its area\nwith respect to ${\\gamma}_{t+\\delta}$. This implies that $|S_t|$ is constant. At the jumps, the statement\nrequires a careful analysis of the various limiting procedures involved.\n\n\nThe second crucial ingredient in Bray's proof is the fact that the total mass $m(t)$\nof the conformally rescaled metric ${\\gamma}_t$ is a non-increasing function of $t$. The\nproof relies on two facts. First of all, the definition of $v_t$ in\n$(\\ref{BraysFlow})$ seems to depend on $\\gamma$ and on $t$. However, as a simple consequence\nof how the Laplacian changes under conformal rescalings, \nit follows that $v_t$ depends only on ${\\gamma}_t$ (i.e. $v_t$\nsolves a suitable Dirichlet problem with respect to the metric ${\\gamma}_t$,\nwith no reference to the original metric ${\\gamma}_0$). Therefore, proving that\n$\\frac{d m(t)}{dt} \\leq 0$ is equivalent to showing \n$\\left . \\frac{dm}{dt} \\right |_{t=0} \\leq 0$ because there is nothing that distinguishes ${\\gamma}_0$\nfrom any other metric ${\\gamma}_t$ in the flow. \nFor $t=0$, the function $v_0$ restricted to\n$\\Sigma_0$ (the exterior of $S_0$ in $\\Sigma$) is just minus the Green function of ${\\gamma}_0$, defined as\n\\begin{eqnarray*}\n\\left . \\begin{array}{rcl}\n\\Delta_{{\\gamma}_0} G & = &  0 \\mbox{\\,\\,\\, \\,\\,\\,\\,\\, on \\,\\,   } \\Sigma_0, \\\\\n\\left . G  \\right |_{S_0} & = &  0, \\\\\nG  & \\rightarrow & 1 \\mbox{\\, \\, at \\, \\,} \\infty \n\\end{array} \\right \\}.\n\\end{eqnarray*} \nThe total mass for a harmonically flat metric $u^4 \\delta$, can be computed directly from the \nbehaviour of $u$ at infinity. Namely, if $u = a +  b \/(2 r) + O(1\/r^2)$\nthen the ADM mass is $m= ab$. The metric ${\\gamma}_t$ is of the form $\\gamma_t\n= (1 - t G + O(t^2) )^4 {\\gamma}_0 = (1- t G + O(t^2))^4 {\\cal U}^4 \\delta$,\nwith ${\\cal U} = 1 - \\frac{M_{ADM}}{2r} + O(r^{-2})$, where $M_{ADM}$ is the ADM mass of ${\\gamma}_0$.\nExpanding the Green function $G$ in the asymptotic region\nas $G = 1 - c\/(2r) + O(1\/r^2)$, it follows from a direct calculation that \n$m(t)= M_{ADM} + t (c - 2 M_{ADM}) + o(t)$. Thus, the mass\nwill not increase provided $c \\leq 2 M_{ADM}$. This turns out to be a general\nproperty of the Green function $G$ of any asymptotically flat metric on a manifold with\ncompact minimal boundary $S_0$. This result, also due to Bray \\cite{Bray2001},\nhas independent interest and has been already used in another context by P. Miao \\cite{Miao2005}\nto characterize the Schwarzschild initial data as the only static initial data\nwith a non-empty minimal boundary. \n\nThe proof of $c \\leq 2M_{ADM}$ relies on the positive mass theorem and uses a technique first introduced by\nBunting and Masood-ul-Alam \\cite{BuntingMassood-ul-Alam1987}\nto prove uniqueness of the Schwarzschild black hole. The idea is to double\n$(\\Sigma_0,{\\gamma}_0)$ across its boundary $S_0$ (i.e. take two copies and identify the boundaries)\nto define a new manifold $\\hat{\\Sigma}$.\nDefine also a function $\\Phi(x) = \\frac{1+G(x)}{2}$\non one of the copies and $\\Phi(x) = \\frac{1-G(x)}{2}$ on the other copy. This defines a \nfunction which is harmonic away from $S_0$ (which remains a minimal surface) and which approaches\n$1$ on one asymptotic end and zero on the other asymptotic end.  This function is also $C^1$\neverywhere. Consider the conformally rescaled metric $\\hat{\\gamma} = \\Phi^4 {\\gamma}_0$ on\n$\\hat{\\Sigma}$. The asymptotic behaviour of $G$ near the infinity where it vanishes\nshows that $(\\hat{\\Sigma},\\hat{\\gamma})$ admits a one-point compactification there, thus\ndefining a complete asymptotically euclidean Riemannian manifold. Furthermore, the behaviour\nof the curvature scalar under conformal rescalings and the fact that $\\Phi$ is harmonic\nimplies that ${\\gamma}_0$ has non-negative curvature scalar away from $S_0$. If $(\\hat{\\Sigma}, \\hat{\\gamma})$\nwere smooth, the positive mass theorem could be invoked to conclude that the\nmass of $\\hat{\\gamma}$ is non-negative, and zero if and only if $(\\hat{\\Sigma},\\hat{\\gamma})$ is \nEuclidean space. However, the metric\n$\\hat{\\gamma}$ is not smooth across $S_0$ and Bray needs to use an approximation argument \nto reach the same conclusion. Consequently, the ADM mass of $(\\hat{\\Sigma},\\hat{\\gamma})$ is non-negative.\nThe mass of this manifold is just $M_{ADM} - c\/2$, from which $c \\leq 2M_{ADM}$ follows.\n\n\n\n\nUsing related techniques, Miao \\cite{Miao2002} has extended these results and\nhas proven that the positive mass theorem holds for complete, asymptotically euclidean Riemannian manifolds \n$(\\Omega,\\gamma)$ of dimension $n \\leq 7$ and non-negative curvature scalar, with a metric\nwhich is allowed to be non-smooth across  a compact \ncodimension one hypersurface $S$ provided (i) this hypersurface separates $\\Omega$ into two\npieces $\\Omega^{\\pm}$, with $\\Omega^{+}$ containing the chosen asymptotically euclidean end, (ii) \nthe induced metric on $S$ from both sides coincides and (iii) the mean curvature \nwith respect to the outer direction satisfies the inequality \n\\begin{eqnarray}\np^+(S) \\geq p^- (S), \\label{IneqH}\n\\end{eqnarray}\nwhere $p^{\\pm}(S)$ is computed with the geometry of $(\\Omega^{\\pm},\\gamma|_{\\Omega^{\\pm}} )$.\nIn the case considered by Bray, the metrics are $\\hat{\\gamma}_{\\pm} = [(1\\pm G)\/2]^4 {\\gamma}_0$. Under\na conformal rescaling $\\hat{\\gamma} = u^4 \\gamma$,\nthe mean curvature $p$ of a hypersurface changes according to\n$\\hat{p} = (p + 4 \\vec{n} (u))\/u^2$, where $\\vec{n}$ is the unit normal along\nwhich $p$ is calculated. Applying this to the metrics $\\hat{\\gamma}_{\\pm}$ and using that\n$S_0$ is minimal, it follows $p^{-}(S_0) = p^{+}(S_0)$ and the positive mass theorem \nproven by Bray is recovered.  It is worth mentioning that a related positive mass theorem has been proven\nusing spinor techniques by Shi and Tam \\cite{ShiTam2002} \non spin manifolds in any dimension (this \nentails no restriction in three dimensions, provided the manifold is orientable)\nfor metrics which are smooth on $\\overline{\\Omega^{+}}$ and\non $\\overline{\\Omega^{-}}$, only Lipschitz across $S$ and such that equality \nin (\\ref{IneqH}) holds.\n\n\nReturning to Bray's proof of the Penrose inequality, once the area of the horizons $S_t$ is known\nto be constant and that the mass does not increase, the final step is to \nshow that the metric outside $S_t$ approaches the Schwarzschild metric in a suitable sense. The\nsurfaces $S_t$ flow outwards in $\\Sigma$. In fact, Bray proves that, for \nsufficiently large $t$, $S_t$ encloses any bounded set of $\\Sigma_0$.\nThus, the shrinking manifold $\\Sigma_t$ behaves in such a way that any given point $p \\in \\Sigma_0$\nis left out for a sufficiently large $t$. It may seem, therefore, that this manifold\ndisappears in the limit. Here is where\nthe asymptotic condition $u_t \\rightarrow e^{-t}$ at infinity comes into play.\nFor very large $t$, the asymptotic value of the conformal factor $u_t$ also tends to zero. Thus, vectors\nthat were unit at large distances in the original metric ${\\gamma}_0$ have increasingly smaller\nlengths in the metric ${\\gamma}_t$. However, this metric is also asymptotically euclidean, so for a suitable choice \nof coordinates, the metric must approach the flat metric in Cartesian coordinates. In these coordinates,\nthe  region $\\Sigma_t$ shrinks more slowly (or it may even expand). In order to see why is this so, consider\nthe metric \ndefined by\n\\begin{eqnarray}\ng^{\\mbox{\\tiny{Euc}}}_t = e^{-4t} \\left (dr^2 + r^2 d\\theta^2 + r^2 \\sin^2 \\theta d\\phi^2 \\right)\n\\label{euclidean}\n\\end{eqnarray}\non the domain $r > R_0(t)$ with $R_0(t) \\rightarrow \\infty$ as $t \\rightarrow \\infty$. The domain\nis shrinking in the coordinates $\\{r,\\theta,\\phi\\}$. However, (\\ref{euclidean}) is \nthe Euclidean metric for any $t$. The transformation to the standard spherical coordinates $\\{ r^{\\prime},\n\\theta, \\phi \\}$ is given by $r^{\\prime} = e^{-2t} r$. In the new coordinates, the domain is\n$r^{\\prime} > e^{-2t} R_0(t)$ which is not shrinking if $R_0(t)$ grows at most as $e^{2t}$. \nThis is the type of behaviour that occurs to the conformal flow of metrics introduced by Bray. This can be seen explicitly\nby considering the flow in the particular case when the starting metric is the Schwarzschild manifold (\\ref{gsch}). Rotational\nsymmetry allows one to integrate the equations easily. The flow of metrics ${\\gamma}_t$ in the exterior domain $r \\geq m\/2$\nare, for $t \\geq 0$, \\cite{Bray2001}.\n\\begin{eqnarray}\n{\\gamma}_t = \\left \\{  \\begin{array}{lll}\n\\frac{4m^2}{r^2} \\left ( dr^2 + r^2 d\\theta^2 + r^2 \\sin^2 \\theta d \\phi^2 \\right )\n& \\frac{m}{2} \\leq r \\leq \\frac{m}{2} e^{2t} & \\mbox{interior domain,} \\\\\n\\left ( \\frac{m}{2 r} e^t + e^{-t} \\right )^4 \\left ( dr^2 + r^2 d\\theta^2 + r^2 \\sin^2 \\theta d \\phi^2 \\right )\n& r \\geq \\frac{m}{2} e^{2t} & \\mbox{exterior domain.} \\\\\n\\end{array} \\right .  \\label{FlowSchw}\n\\end{eqnarray} \nThe exterior domain indeed shrinks in these coordinates. However, performing the\ncoordinate transformation (or the diffeomorphism if an active point of view is preferred) $r^{\\prime} = r e^{-2t}$,\nwith $\\theta$ and $\\phi$ unchanged, the exterior region is transformed back into the original exterior\npart of the Schwarzschild manifold. The metric (\\ref{FlowSchw}) \nin the region already swapped by the surfaces $\\{S_t\\}$ is cylinder-like in the\nsense that each surface $r= \\mbox{const.}$ has the same area. This is a general property of Bray's flow, since each \nof the surfaces $S_t$ has the same area as the starting horizon.\nThe behaviour in the exterior part is also general. The idea is that since the surface $S_t$ grows unboundedly,\neventually it swallows all the interior geometric features of $(\\Sigma_0,{\\gamma}_0)$ and leaves only the asymptotic\ngeometry. Since the mass does not increase along the flow of metrics, its limit exists and is non-negative due to the \npositive mass theorem. Bray proves that the limit mass is, in fact, positive and hence the geometry in the\nexterior approximates Schwarzschild with increasing accuracy. The asymptotic\nbehaviour of the conformal factor $u_t \\rightarrow e^{-t}$ has the effect that this exterior remains unbounded,\nin the sense that after performing the appropriate diffeomorphism, all the surfaces $S_t$ stay within a bounded set. \nMore precisely, Bray shows that for each $\\epsilon >0$\nthere exists a $T$ so that for $t > T$ \nthere exists a diffeomorphism $\\Phi_t$ between\n$(\\Sigma_t, {\\gamma}_t)$  and the exterior region $r \\geq m_F\/2$ of a fixed Schwarzschild manifold \n$(\\Sigma_{\\mbox{\\tiny{Sch}}},\\gamma_{\\mbox{\\tiny{Sch}}})$ of mass $m_F$,\nsuch that both metrics are $\\epsilon$-close to each other\n(in the sense that the length of any unit vector of the metric ${\\gamma}_t$ has length \nin the interval $(1-\\epsilon,1+\\epsilon)$ for the metric $\\Phi^{\\star}_t ({\\gamma}_{sch})$). Moreover,\n$m(t)$ and $m_F$ also differ at most by $\\epsilon$. This concludes the proof\nof the inequality\n\\begin{eqnarray}\nM_{ADM} \\geq \\sqrt{\\frac{|S_0|}{16 \\pi}} \\label{BrayPI}\n\\end{eqnarray}\nfor the original metric. As Bray points out, no property from $(\\Sigma,{\\gamma}_0)$ inside\n$S_0$ is used in the proof, therefore establishing (\\ref{BrayPI}) also \nfor manifolds with boundary $S_0$,  provided this is an area outer minimizing horizon.\n\nWhile Huisken and Ilmanen's proof is strongly dependent on the dimension of the manifold, Bray's\napproach can be generalized to any dimension $n \\leq 7$,\nas proven by Bray and Lee in \\cite{BrayLee2007}. The limitation $n\\leq 7$ comes basically from two facts. First,\nregularity of minimal surfaces holds only for  dimensions $n \\leq 7$, and the outermost\nminimal surface enclosure of $S_0$, which is a crucial ingredient in the method, needs to be smooth\nfor the argument to go through. Secondly, the positive mass theorem for general asymptotically euclidean manifolds\n(not spin) is only known to hold so far for dimensions $n \\leq 7$ (the underlying reason is again regularity\nof minimal surfaces, as Schoen and Yau's proof uses this type of surfaces in a fundamental way). The statement of the \nRiemannian Penrose inequality in higher dimensions is, of course, different to (\\ref{BrayPI}), since the physical \ndimension of mass is not length in higher dimensions. The proper statement in (space) dimension $n$ is\n\\begin{eqnarray}\nM_{ADM} \\geq \\frac{1}{2} \\left ( \\frac{|S_0|}{\\omega_{n-1}} \\right )^{\\frac{n-2}{n-1}}\n\\label{RiemPIHiguerDim}\n\\end{eqnarray}\nwhere $\\omega_{n-1}$ is the area of an $(n-1)$-dimensional sphere of unit radius.\nThe rigidity statement says that\nequality in (\\ref{RiemPIHiguerDim}) occurs only for the higher dimensional Schwarzschild metric (first \ndiscussed by Tangherlini \\cite{Tangherlini1963})\n\\begin{eqnarray}\n\\gamma^{(n)}_{\\mbox{\\tiny{Sch}}} = \\left ( 1 + \\frac{m}{2 |x|^{n-2}} \\right )^{\\frac{4}{n-2}} \\delta_{ij} dx^i dx^j.\n\\label{Tanger}\n\\end{eqnarray}\nAs discussed in \\cite{BrayLee2007} several of the steps in Bray's proof extend easily to\nhigher dimensions: the definition and existence of the conformal flow,\nthe proof that the area of $S_t$ remains constant under the flow, and the fact that the ADM mass $m(t)$\ndoes not increase. However, the arguments required to show that\n$S_t$ eventually encloses all bounded sets,\nand that the flow of metrics converges to a Schwarzschild\nmetric after a suitable $t$-dependent diffeomorphism, do in fact depend strongly on the dimension. This is because\nthe\noriginal argument is based on\nthe Gauss-Bonnet theorem and a Harnack type inequality which is valid only in three-dimensions. \nMoreover, the convergence to Schwarzschild requires \na refinement of the rigidity part of the positive mass theorem, which roughly speaking gives a quantitative description\nof how the metric approaches the flat metric when the mass tends to zero. In Bray \\cite{Bray2001}, this\nis proven using spinor techniques. This result is extended\nby Lee \\cite{Lee2007} \nto arbitrary higher dimensional manifolds for which the positive mass theorem holds (in particular for dimensions\n$n \\leq 7$).   \n\nThe main technical work to establish (\\ref{RiemPIHiguerDim}) in \\cite{BrayLee2007} is therefore devoted to prove\nthat the conformal flow converges in a suitable sense to \nthe metric (\\ref{Tanger}). Besides being more general, the argument presented in \\cite{BrayLee2007} \nin fact simplifies and streamlines\nsome of the original steps in the three-dimensional case.\n\n\n\\section{Penrose inequality for asymptotically hyperbolic Riemannian manifolds}\n\\label{hyperbolicSect}\n\nDespite its difficulty, the Riemannian Penrose inequality is considerably simpler than\nthe general Penrose inequality because all the complications arising from the\nsecond fundamental form disappear. \nThe time-symmetric case, however, is not the only one where this kind of simplification\noccurs. Another example is given by\ninitial data sets $(\\Sigma, \\gamma_{ij}, A_{ij})$ which are umbilical, i.e.\nwhen the second fundamental form is proportional to the metric with a constant\nproportionality factor. Writing $A_{ij} =\n\\lambda \\gamma_{ij}$, the Hamiltonian constraint (\\ref{HamiltonianConst}) \nbecomes $R(\\gamma) = - 6 \\lambda^2 + 16 \\pi \\rho$ (or\n$R(\\gamma) = - n (n-1) \\lambda^2 + 16 \\pi \\rho$ if $\\Sigma$ is $n$-dimensional).\nIf the energy condition $\\rho \\geq 0$ holds, then $R(\\gamma) > - 6 \\lambda^2$\n\nIn an asymptotically flat spacetime, slices with this property and $\\lambda\n\\neq 0$  cannot reach\nspacelike infinity because the decay (\\ref{decay})\nis obviously not satisfied. This type of initial data are usually called ``asymptotically\nhyperbolic'' (also ``hyperboloidal'') since the simplest example is given by \nthe hyperboloid $t = \\sqrt{r^2 + \\lambda^{-2} }$ in Minkowski spacetime. \nAs a Riemannian\nmanifold, this hypersurface is just the standard three-dimensional \nhyperbolic space of radius $1\/\\lambda$ (i.e. $\\mathbb{R}^3$ endowed\nwith the metric of constant negative\ncurvature $-\\lambda^2$). Asymptotically hyperbolic initial data sets approach null infinity\nprovided $\\gamma_{ij}$ satisfies suitable \nasymptotic conditions. In the context of the Penrose inequality, the model example \nconsists of the  spherically symmetric slices $\\Sigma$ of the Kruskal spacetime\nsatisfying $A_{ij} = \\lambda \\gamma_{ij}$. It turns out that for $\\lambda>0$, $\\Sigma$\nis fully contained in the advanced Eddington-Finkelstein portion of the spacetime\nand satisfies the equation (in advanced coordinates $(v,r,\\theta,\\phi)$)\n\\begin{eqnarray*}\n\\frac{dv}{dr} = \\frac{\\lambda r - \\sqrt{1 - \\frac{2m}{r} + \\lambda^2 r^2} }{\\left (1- \\frac{2 m}{r}\n\\right ) \\sqrt{1 -\\frac{2m}{r} + \\lambda^2 r^2 }}.\n\\end{eqnarray*}\nThese hypersurfaces approach future null infinity\nfor $r \\rightarrow \\infty$ and \nintersects the white hole event horizon on the surface $S_1 \\subset \\{r=2m\\}$, which has\nmean curvature $ p =\n2 \\lambda$ (because $q = 2 \\lambda$ on any surface of $\\Sigma$ and $\\theta_{-} = - p + q$ vanishes\non the white hole event horizon). The induced metric on $\\Sigma$ is \n\\begin{eqnarray}\ndl^2 = \\frac{dr^2}{1- \\frac{2m}{r} + \\lambda ^2 r^2} + r^2 d \\Omega^2\n\\label{hyperbolic}\n\\end{eqnarray}\nand obviously satisfies $R(\\gamma) = -6 \\lambda^2$. This model space arises not only as an umbilic hypersurface\nof Kruskal, but also as time-symmetric hypersurfaces in spherically symmetric solutions\nof the vacuum Einstein field equations with cosmological\nconstant, i.e. satisfying $\\mbox{Ein}( g ) = - \\Lambda  g$.\nThe general spacetime solution is given by the so-called Kottler metric \n\\cite{Kottler1918} (discovered independently by Weyl \\cite{Weyl1919}) and reads\n\\begin{eqnarray}\nds^2= - \\left ( 1- \\frac{2m}{r} - \\frac{\\Lambda r^2}{3} \\right ) dt^2\n+ \\frac{dr^2}{ 1- \\frac{2m}{r} - \\frac{\\Lambda r^2}{3}} + r^2 d \\Omega^2\n\\label{Kottler}\n\\end{eqnarray}\nThis metric is often called \nSchwarzschild-de Sitter ($\\Lambda >0$) or Schwarzschild-anti de Sitter ($\\Lambda <0$) since it\ncontains both the Schwarzschild (when $\\Lambda=0$) and the de Sitter metrics (when $m=0$). The Kottler metric admits\nan interesting generalization (still satisfying the vacuum Einstein equations with cosmological\nconstant) whereby the spheres  $t=\\mbox{const}$, $r=\\mbox{const}$ are replaced  \nby any compact connected manifold $M^2$ without boundary and the metric is\n\\begin{eqnarray}\nds^2= - \\left ( k - \\frac{2m}{r} - \\frac{\\Lambda r^2}{3} \\right ) dt^2\n+ \\frac{dr^2}{ k - \\frac{2m}{r} - \\frac{\\Lambda r^2}{3}} + r^2 d \\Omega_k^2\n\\label{GeneralizedKottler}\n\\end{eqnarray}\nwhere $k = {1,0,-1}$ depending, respectively, on whether the genus of $M^2$ is zero, one or higher.\nHere $d \\Omega_k$ stands for the two-dimensional metric of constant curvature $k$.\nThe Kottler metric obviously corresponds to the case $k=1$.\nFor $\\Lambda <0$ and $m >0$, let $r_+$ be the only positive root of $k - 2m\/r - \\Lambda r^2 \/3 =0$. \nThe hypersurface $\\{t=0\\}$ of (\\ref{GeneralizedKottler}) \nrestricted to the range $r \\geq r_{+} >0$ is a manifold with boundary which admits a smooth \nconformal compactification (see \\cite{ChruscielSimon2001}, \\cite{ChruscielHerzlich2003})\nto a manifold with a boundary consisting of two\ncopies of $M^2$. One of the copies corresponds to the inner boundary $r=r_+$\nand the other one to the ``surface at infinity'', usually denoted by $\\partial^{\\infty} \\Sigma$. \n\nIn the case $k=1$ the induced metric on $\\{t=0\\}$ is exactly \n(\\ref{hyperbolic}) with $\\Lambda = -3 \\lambda^2$. \nIn this context, the surface $S_1 = \\{r = 2 m\\}$ plays no special\nrole regarding the Penrose inequality (because the surface is outer untrapped) and instead one has to look for minimal\nsurfaces (similarly as in any time-symmetric case). Assuming again $m >0$, the outermost minimal surface of (\\ref{hyperbolic}) \nis given by $S_2 = \\{r = r_{+}\\} $. Since $r_{+} < 2m$ we have that $S_1$ encloses $S_2$. \n\nIt follows from this discussion that the  Penrose inequality in this  context\nhas two flavors, one for which the surface of interest has $p=2 |\\lambda| $ and another\none where the surface to consider is minimal. The Riemannian manifolds $(\\Sigma,\\gamma)$ \nrelevant for this setting \nsatisfy $R(\\gamma) > - 6 \\lambda^2$,\nwhich is a consequence of (\\ref{HamiltonianConst}) both when $A_{ij} = \\lambda \\gamma_{ij}$ and $\\rho \\geq 0$ \nor when the spacetime satisfies $\\mbox{Ein} ( g) = - \\Lambda  g + 8 \\pi T$ provided $\\Lambda \\equiv - 3 \\lambda^2$\nand the energy-momentum $T$\nsatisfies the week energy condition.\n\nThe metric $\\gamma$ has to satisfy appropriate asymptotics so that a useful concept of mass can be defined.\nBoth the appropriate asymptotic behaviour and the definition of mass is considerably \nmore difficult than the corresponding definition in the\nasymptotically euclidean case. In the case when the boundary at infinity $\\partial^{\\infty} \\Sigma$ is a two-sphere,\nthe definition was first given in \\cite{Wang2001}. The definition for other topologies at infinity (and\nadmitting weaker asymptotic conditions) appears in \\cite{ChruscielHerzlich2003} (see also \n\\cite{ChruscielNagy2001} for a definition of mass from a spacetime viewpoint). For the $\\{t=0\\}$ slice of\n(\\ref{GeneralizedKottler}) the mass according to this definition is $m$. \n\nThe statement of the Penrose inequality in this setting should be expected to be different \nwhen it involves outermost minimal surfaces or outermost surfaces with mean curvature $p = 2 |\\lambda|$.\nIn the case of boundaries with several\nconnected components or when the topology of $\\partial^{\\infty} \\Sigma$ is not \nspherical, it is not clear how the precise statement of the inequality should read.\nHowever, when $\\partial^{\\infty} \\Sigma= S^2$ and the inner boundary $S$ is connected,\na natural version of the inequality reads\n\\begin{eqnarray}\nM \\geq (1- g_{S} ) \\sqrt{\\frac{|S|}{16 \\pi}} + \\frac{\\lambda^2}{2} (1- \\epsilon) \\left ( \\frac{|S|}{4 \\pi}\n\\right )^{\\frac{3}{2}},\n\\label{PIhyperbolic}\n\\end{eqnarray}\nwhere $M$ is the total mass, $g_{S}$ is the genus of $S$\nand $\\epsilon =0,1$ depending, respectively, on whether $S$ is outermost\nminimal or outermost with mean curvature $p = 2 |\\lambda|$. In the minimal case, this inequality has been proposed in \n\\cite{Gibbons1999} \nfor the case $g_S=0$ and in \\cite{ChruscielSimon2001} for arbitrary genus (in this reference, \nthe Penrose inequality in terms of the asymptotic value of the Geroch mass under smooth inverse mean curvature flow is \nalso discussed for $\\partial^{\\infty} \\Sigma$ of arbitrary genus). \nIn \\cite{Gibbons1999}, an inequality for $\\partial^{\\infty} \\Sigma$\nof arbitrary genus also appears. However, as noted in \n\\cite{ChruscielSimon2001}, this version fails for the slice $\\{t=0\\}$ of the generalized Kottler metric when $M^2$ has at\nleast genus three. In the case \nof surfaces with $p = 2 |\\lambda|$ the inequality (\\ref{PIhyperbolic}) was conjectured in \\cite{Wang2001}.\nSupport for the validity of (\\ref{PIhyperbolic}) comes from the fact that the $\\{t=0\\}$ slice \nof the Kottler metric gives equality. Moreover, by choosing the\nvalue $C = - 3 \\lambda^2$, the Geroch mass \n(\\ref{GerochMass}) evaluated on a surface $S$ with $p = 2 |\\lambda| \\epsilon$ gives precisely the \nright-hand side of (\\ref{PIhyperbolic}). This value of the constant $C$ is adapted to the inequality $R(\\gamma) \n\\geq  -6 \\lambda^2$ because then the Geroch mass is monotonic under smooth inverse mean curvature flow\n\\cite{BoucherGibbonsHorowitz1984, Gibbons1999}. This can be seen explicitly from the general formula (\\ref{dMG}).\n\nThus, it is tempting to try and adapt Huisken and Ilmanen's proof to the hyperbolic case, at least when the\nboundary $S$ is connected. However, a recent result by Neves \\cite{Neves2007} shows that this is not possible\nin general. The difficulty lies in the fact that the Geroch mass of a flow of surfaces\nmoved by inverse mean curvature does not necessarily approach  \nthe total mass of the asymptotically hyperbolic manifold. This is proven in \\cite{Neves2007} by showing that\nin the manifold $\\{r \\geq 2m\\} \\times S^2$ with metric \n(\\ref{hyperbolic}) there exist surfaces \nwith Geroch mass larger than $m$ and which can be flowed smoothly by inverse\nmean curvature all the way to infinity. Consequently,\nthe limit of the Geroch mass under the flow is still larger than $m$ due to the monotonicity of $M_G$. \nIn  this  example the inner boundary does not satisfy $p = 2 |\\lambda|$. \nHowever, the author is able to modify the construction so that the \nflow starts on a horizon ($p=2 |\\lambda| $) and remains smooth for all values of the\nparameter in such a way that the leaves of the flow do {\\it not} become\nrounder (in a precise sense) at infinity. Thus, according to the author, it becomes impossible \nto compare the limit value of  Geroch mass with the value of the total mass of the manifold, which makes the \ninverse mean curvature flow inconclusive for the Penrose inequality.\nDespite this failure of the inverse mean curvature method to prove the\ninequality (\\ref{PIhyperbolic}), its validity is still open. \n\n\n\n\nUsing \nisoperimeric surface methods \\cite{Corvino_et_al2007} (see \nSubsect. \\ref{isoperimetricprofile}), the inequality (\\ref{PIhyperbolic}) (with \n$\\epsilon=0$) has been proven in the special case that $(\\Sigma,\\gamma)$ \noutside a compact set is isometric to the Kottler metric (\\ref{hyperbolic}) outside a\nsphere $r = \\mbox{const}$ and the following two conditions are satisfied: (i) \n$(\\Sigma,\\gamma)$ contains a unique closed and connected surface $S_m$ with $p = 2 |\\lambda|$ and (ii) \nthe isoperimetric surfaces $S_V$ (with respect to $S_m$) are connected and coincide with the spheres\n$r=\\mbox{const.}$ in the asymptotic region for $V$ large enough.\n\nIt is also interesting to note that the Penrose inequality in the hyperbolic setting \nmay be a powerful tool to address the uniqueness problem of asymptotically hyperbolic\nstatic initial data sets \\cite{ChruscielSimon2001}. More precisely, \ndenoting by $U$\nthe square norm of the static Killing vector and by $M_G(U)$ the Geroch mass of the \nlevel sets of $U$, it is proven in \\cite{ChruscielSimon2001} that the validity of the\nPenrose inequality\n\\begin{eqnarray}\nM_G (U) \\geq (1- g_{S} ) \\sqrt{\\frac{|S|}{16 \\pi}} +\n\\frac{\\lambda^2}{2} \\left ( \\frac{|S|}{4 \\pi}\n\\right )^{\\frac{3}{2}} \\label{uniq}\n\\end{eqnarray}\nimplies a uniqueness theorem for the \ngeneralized Kottler metric in the case that $\\partial^{\\infty} \\Sigma$ if of genus larger than one and $S$\nis connected. The reason is that in the static setting, the Geroch mass $M_{G}(U)$ can be bounded\n{\\it above} in terms of the mass of the generalized Kottler solution \nwith the same surface gravity $\\kappa$ (provided this satisfies the inequality $0 < \\kappa \\leq | \\lambda|$).\nThe area $|S|$ is also bounded {\\it below}  by the area of the Killing horizon of the corresponding Kottler\nsolution. Combining these inequalities with (\\ref{uniq}), it follows that \nequality in (\\ref{uniq}) is in fact the only possibility. However, \nthe only static initial data set which saturates (\\ref{uniq}) \nturns out to be the $\\{t=0\\}$ slice of the Kottler metric.\nThus a proof of the Penrose inequality for $M_G(U)$\nwould establish a uniqueness result for static initial data sets in the hyperbolic setting.\n\n\n\n\n\n\n\\section{On the general Penrose inequality}\n\\label{GeneralPI}\n\n\nThe validity of the\nPenrose inequality for arbitrary initial data sets (with an arbitrary second fundamental form\nand without the assumption of spherical symmetry)\nis still open. The proofs by Huisken and Ilmanen and Bray of the Riemannian Penrose\ninequality involve manifolds $\\Sigma$ with a positive definite metric of non-negative curvature scalar \nadmitting an outermost minimal surface. Although primarily intended to \ncover the time-symmetric case, the proofs \nonly require the presence of a minimal surface and a Riemannian metric with non-negative\nscalar of curvature. Thus, the method also gives useful results in the case \nof  maximal hypersurfaces, $\\mbox{tr}_{\\gamma} A=0$ (assuming  the energy density $\\rho$\nin (\\ref{HamiltonianConst}) to be non-negative), provided\nthe Riemannian manifold contains at least a bounding minimal surface. The Riemannian arguments\nin the previous section, however, do\nnot settle the general Penrose inequality \nin the maximal hypersurface case for two reasons: firstly because they would give a lower bound \nfor the ADM energy instead of the ADM mass\nand secondly because the outermost minimal surface does not coincide, in general, with\nany of the minimal area enclosures arising in any of the  \nversions of the Penrose inequalities discussed in Sect. \\ref{Formulations} (except in the \ntime-symmetric case, of course).\n\n\nAlthough the general Penrose inequality remains open, several methods have been proposed to address it. We devote this section to discuss them.\n\n\n\\subsection{Null shells of dust}\n\n\nAs described in the Introduction, \nPenrose's original setup \\cite{Penrose1973}\nto test the validity of cosmic censorship consisted of a shell of matter moving inwards\nat the speed of light in a flat spacetime. The shell is assumed to have closed \nand connected cross sections and the matter within the shell is made of\nnull dust (meaning that the particles \ndefining the shell are massless and that \nall pressures vanish). \nIn order to have a flat metric inside the imploding shell, all points in their interior\nmust be causally disconnected (to the past) with all points on the shell. Choosing a\nMinkowskian time $t$ inside the shell, this demands that, to the past of some $t=t_0$, the null\nhypersurface ${\\cal N}$ swept by the incoming shell of matter\nmust be free of self-intersections. Since the matter within the shell\nis moving at the speed of light, the cross section\n$S_t \\equiv {\\cal N} \\cap \\{ t = \\mbox{const} \\}$ can be viewed, after \nthe natural identification of points in different instants of time, as the \nsurface  lying  at distance $t_0-t$ from $S_{t_0}$, where distance is positive\nto the exterior and negative to the interior (the fact that $S_{t_0}$ separates \nEuclidean space into an interior and an exterior is always true  \nas a consequence of the \nJordan-Brower separation theorem for smooth hypersurfaces in $\\mathbb{R}^n$, see e.g.\n\\cite{Lima1988}).\nThe distance level function from a given closed surface\nin Euclidean space is smooth everywhere in its exterior if and only if the surface is convex (i.e.\nit has non-negative principal curvatures). The setup, therefore, requires that the null hypersurface\nhas one cross section $S_{t_0}$ which is convex. This property is then true for all $t \\leq t_0$. \nTowards the future, \n${\\cal N}$ will become singular\nat the first focal point of the incoming null geodesics. A spacetime singularity \nwill form there.  Outside the null shell, the metric is no longer flat, in particular because\ngravitational waves may be emitted by the collapsing dust. \n\n\nPenrose devised this physical process as a potential counterexample to the inequality\n(\\ref{PI1}). The fundamental simplification of this problem is that the inequality can be\ntranslated into an inequality directly in Minkowski space, as follows \n\\cite{Penrose1973,Tod1985,Tod1992,Gibbons1997}:\n\nLet $\\vec{l}^{-}$ be the future directed null tangent \nto ${\\cal N}$ normalized to satisfy $\\vec{l}^{-} (t)=1$,  \nwhere, as above, $t$ is a Minkowskian time in the interior part of the shell. Take any closed, spacelike surface\nembedded in ${\\cal N}$ and let $\\vec{l}^{+}$ be its future null normal satisfying $(\\vec{l}^{+} \\cdot \\vec{l}^{-})=-2$.\nThe energy momentum of the spacetime is a distribution supported on ${\\cal N}$ which reads\n$T_{\\alpha\\beta} =  8 \\pi \\mu l^{-}_{\\alpha} l^{-}_{\\beta} {\\bm \\delta}$, where $\\mu$ is the energy\ndensity of the shell and the Dirac $\\bm{\\delta}$ is defined with\nrespect to  the volume form $d\\sigma$ induced by the normal $l^{-}_{\\alpha}$ to ${\\cal N}$, i.e.\n$l^{-}_{\\mu} d \\sigma = \\eta_{\\mu\\alpha\\beta\\gamma} e^{\\alpha}_1 e^{\\beta}_2 e^{\\gamma}_3$ where\n$e^{\\alpha}_i = \\frac{\\partial x^{\\alpha}}{\\partial y^i}$ and $y^i \\rightarrow x^{\\alpha}(y^i)$ is a\ncoordinate expression for the embedding of ${\\cal N}$ (see e.g. \\cite{MarsSenovilla1993} for details). The \nnull expansion $\\theta_{+}$ jumps across ${\\cal N}$. The jump can be determined using the Raychaudhuri\nequation (\\ref{lthetal}). One way\nof doing this is by extending the null vector $\\vec{l}_{+}$ to a geodesic null congruence and taking its divergence\non each side of the shell. This defines the null expansion $\\theta_{+}$ as a discontinuous function\non the spacetime.  A distribution can then be introduced as\n$\\bm{\\theta_{+}} = \\theta_{+}^{E} \\bm{\\Theta} + \\theta_{+}^{I} ( \\bm{1} - \\bm{\\Theta})$, where\n$\\bm{\\Theta}$ is th standard Heaviside distribution (it acts on tests functions by integration on the domain outside\nthe shell) and the superscript $I (E)$ stands\nfor interior (exterior) of the shell.\nSince $\\partial_{\\mu} \\bm{\\Theta} = - l^{-}_{\\mu} \\bm{\\delta}$, the derivative\nof $\\bm{\\theta_{+}}$ along $\\vec{l}_{+}$ gives \n\\begin{eqnarray*}\nl^{\\mu}_{+} \\partial_{\\mu}\\bm{\\theta_{+}} = l^{\\mu}_{+} \\partial_{\\mu}\\theta_{+}^E \\left (\n\\bm{1} - \\bm{\\Theta} \\right )\n+ l^{\\mu}_{+} \\partial_{\\mu}\\theta_{+}^I \\bm{\\Theta} + 2 \\left [ \\theta_{+} \\right ] \\bm{\\delta}, \n\\end{eqnarray*}\nwhere the jump $[ \\theta_{+}] \\equiv (\\theta^{E}_{+} - \\theta^{I}_{+}) |_{{\\cal N}}$. The Raychaudhuri equation (\\ref{lthetal}), which \nin this case is a distributional equation, has a singular part supported on ${\\cal N}$ only through the term\n$ - \\mbox{Ric} (\\vec{l}_{+},  \\vec{l}_{+} ) = - 32 \\pi \\mu \\bm{\\delta}$. The jump must therefore satisfy\n$\\theta^{E}_{+} |_{\\cal N} = \\theta^{I}_{+} |_{\\cal N} - 16 \\pi \\mu$. \n\n\nFor an arbitrary surface embedded within a null hypersurface, the expansion along the null direction tangent\nto the hypersurface\ncoincides with the null expansion of the hypersurface. This means, in particular, that \nit only depends on the point where it is calculated but not on the specific surface passing through that point. \nThis has as immediate consequence that the incoming null\nexpansion $\\theta_{-}$  of the shell is continuous\nacross the shell. On the surface $S_{t_0}$, the null expansion $\\theta_{-}$ coincides with the mean curvature \nof $S_{t_0}$  as a surface\nof Euclidean space with respect to the inner normal and it is therefore non-positive\n(since $S_{t_0}$ is convex) and not everywhere zero (since\n$S_{t_0}$ is closed). It follows from the Raychaudhuri\nequation that $\\theta_{-} \\leq 0$ everywhere on ${\\cal N}$. Consequently, \nright after the shell has passed, a spacelike surface $S \\subset {\\cal N}$ is\nmarginally outer trapped \n(i.e. $\\theta^{E}_{+} =0$) if and only if it is  \nmarginally future trapped.\n\nAssume such an $S$ exists along the shell. We want to show that the Penrose heuristic argument based on cosmic censorship\nthen implies \n$M_{B} \\geq \\sqrt{\\frac{|S|}{16 \\pi}}$ \nwhere $M_B$ is the Bondi mass (see e.g. \\cite{Wald1984} for its definition)\nof past null infinity at the cut defined \nby ${\\cal N}$. Indeed, under cosmic censorship the singularity that necessarily forms in the future\nis shielded from infinity by an event horizon. Since $S$ must be contained in the\nblack hole region, the intersection ${\\cal H}$ of the event horizon with ${\\cal N}$ must lie completely in the causal past of $S$. \nUsing the fact that the null expansion $\\theta_ {-}$ is non-positive, this implies $|{\\cal H}| \\geq |S|$. Since the\nstandard heuristic argument gives $M_{B} \\geq \\sqrt{\\frac{|{\\cal H}|}{16 \\pi}}$, the claim above follows.\nNotice that, in this case,\nthe inequality is expected to hold for $S$ irrespectively of whether \nor not this surface is area outer minimizing with respect\nto any spacelike slice. Using now the conservation equation ${\\nabla^{g}}_{\\alpha} T^{\\alpha\\beta}=0$ (which \nholds in distributional sense, see e.g. \\cite{MarsSenovilla1993})), it follows that \nthe integral $\\int_{\\hat{S}} \\mu \\bm{\\eta_{\\hat{S}}}$ does not depend on the cut $\\hat{S}$ of ${\\cal N}$.\nEvaluating this integral at past null infinity \ngives precisely the Bondi mass (this can be easily shown for instance using the Hawking mass introduced in Subsect. \\ref{UEF}).\nUsing now that $S$ is marginally outer trapped from the exterior, we have, after defining $\\theta_{+}\n\\equiv \\theta_+^I$,\n\\begin{eqnarray*}\n\\int_{S} \\theta_{+} \\bm{\\eta_S} = \n\\int_{S} \\theta^{I}_{+} \\bm{\\eta_S} = \n\\int_{S} 16 \\pi \\mu \\bm{\\eta_S} = \n16 \\pi M_B \\geq \\sqrt{16 \\pi S},\n\\end{eqnarray*}\nwhere the last step is precisely the Penrose inequality in this\nsetting. Thus, the Penrose inequality for incoming shells can  be rewritten as \n\\begin{eqnarray}\n\\int_{S} \\theta_{+} \\bm{\\eta_S} \\geq \\sqrt{16 \\pi |S|}, \\label{Shells}\n\\end{eqnarray}\nwhich  has the remarkable property of making no reference to the exterior geometry at all. Since the density \n$\\mu$ is freely specifiable, this inequality\nshould hold for any closed spacelike surfaces $S$ in Minkowski spacetime for which \nthe null hypersurface generated by past directed and outer null geodesics orthogonal to $S$ remains regular\neverywhere. A similar inequality can be derived in any spacetime dimension \\cite{Gibbons1997}.\n\nIn the particular case of a surface lying on the past null cone of a point $p$, the surface $S$ can be described\nby a single positive function $r$ which measures the distance to $p$ (after projection to a constant time hypersurface).\nA straightforward calculation gives the outer null expansion $\\theta_{+}$ and  \nthe inequality (\\ref{Shells}) becomes\n\\begin{eqnarray}\n\\int_{S^2} \\left ( r + \\frac{|d r |^2}{r} \\right )  \\bm{{\\eta}_{S^2}} \\geq\n\\sqrt{ 4 \\pi \\int_{S^2} r^2  \\bm{{\\eta}_{S^2}}},\n\\label{PIShellSpher}\n\\end{eqnarray}\nwhere all geometric objects refer to the standard metric of unit radius on the sphere.\nThis inequality already appears in \\cite{Penrose1973} and a more detailed derivation can be\nfound in \\cite{BarrabesIsrael1991}. Its validity (in fact of a stronger version)\nwas proven by Tod \\cite{Tod1985}\nas a consequence of the Sobolev inequality applied to a \nsuitable class of functions on $\\mathbb{R}^4$ (see\n\\cite{Tod1986} for a different proof which gives an even stronger inequality).\n\n\nIf  $S$ lies in a spacelike hyperplane in Minkowski, then $\\theta_{+}$ is the mean curvature\n$p$ of $S$ as a surface of Euclidean space and the inequality becomes\n\\begin{eqnarray}\n\\int_{S} p \\bm{\\eta_{S}} \\geq \\sqrt{16 \\pi |S|}. \\label{MinkowskiIneq}\n\\end{eqnarray}\nAs first noticed by Gibbons in his Ph.D. thesis, this inequality is exactly the Minkowski inequality for convex\nbodies in Euclidean space, see e.g. \\cite{BuragoZalgaller}. This settles the Penrose inequality when $S$\nlies on a constant time hyperplane \\cite{Gibbons1997}. The range of validity of the\nMinkowski inequality  (\\ref{MinkowskiIneq})  has been extended by Trudinger \\cite{Trudinger1994},\nto cover all mean convex bodies in Euclidean space, i.e. all closed surfaces \nwith $p \\geq 0$. Gibbons has used this result to claim the validity of the Penrose inequality\n(\\ref{Shells}) in the general case. The idea of the argument was to project $S$ orthogonally onto\na constant time hyperplane. The projection $\\hat{S}$ can be seen to have at least the same area as $S$,\ni.e. $|\\hat{S}| \\geq |S|$. Furthermore, by direct calculation, the author finds that \nthe mean curvature $\\hat{p}$ of the projected surface is non-negative and that \n$\\int_{S}{\\theta_{+}} \\bm{\\eta_{S}}  = \\int_{\\hat{S}} \\hat{p} \\bm{\\eta_{\\hat{S}}}$. Thus, the Penrose inequality would follow\nfrom Trudinger's version of the Minkowski inequality. Unfortunately, the calculation leading\nto $\\hat{p} \\geq 0$ and $\\int_{S}{\\theta_{+}} \\bm{\\eta_{S}}  = \\int_{\\hat{S}} \\hat{p} \\bm{\\eta_{\\hat{S}}}$ contains\nan error which invalidates the argument. Instead of going into the details of the derivation, it is simpler\nto just notice that one can easily construct a surface on a null hypersurface ${\\cal N}$ in Minkowski spacetime\nwhich has a projection $\\hat{S}$ which is not mean convex. Consider the past null cone of a point at $t=1$, \nand consider the sphere obtained as the cross section of ${\\cal N}$ with $\\{t=0\\}$. A function $s$ on $S^2$\ntaking values in $[0,1)$ defines a surface $\\hat{S}$ on the hyperplane $\\{t=0\\}$ simply by moving\nradially inwards each point of the sphere a distance $s$. It is easy to construct  surfaces $\\hat{S}$ which are\nnot mean convex. Consider, for instance,\na surface of revolution defined by $s(\\theta)$ \nwith equatorial symmetry \nand with a neck on the equator (i.e. such that $s(\\theta)$ is symmetric under\n$\\theta \\rightarrow \\pi - \\theta$ and $s_0= s(\\theta = \\pi\/2)$ is a local maximum).\nThe principal curvatures on a point on the equator are simply \n$1\/(1-s_0)$ and $1\/(1-s_0) + s''_0\/(1-s_0)^2$, where $s''_0$ is the second derivative of $s(\\theta)$\nat the equator. Thus, if $s''_0 < -2 (1-s_0)$ (i.e. the second derivative is\nnegative and sufficiently large in absolute value on the equator )\nthen the surface is not mean convex.  But $\\hat{S}$ is obviously the orthogonal projection\non $t=0$ of the surface on ${\\cal N}$ constructed by lifting each of its points a temporal amount $s$\n to the future, see Fig. \\ref{Projection}. \nThis example shows that the projection performed in \\cite{Gibbons1997} is not correct. \nThe validity of the Penrose inequality for null shells is therefore still open (in any spacetime\ndimension).\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\psfrag{S}{$S$}\n\\psfrag{hatS}{$\\hat{S}$}\n\\psfrag{P}{$p$}\n\\psfrag{T}{Hyperplane $t=0$}\n\\psfrag{sm}{$s(\\theta)$}\n\\includegraphics[width=10cm]{Projection.eps}\n\\caption{Surface $\\hat{S}$ in Euclidean space which is non-mean convex and which \ncan be obtained by orthogonal projection of a surface $S$ lying in the past null cone of a point $p$\nin Minkowski spacetime.}\n\\label{Projection}\n\\end{center}\n\\end{figure}\n\n\n\nAnalogously as in (\\ref{PIShellSpher}), the inequality (\\ref{Shells}) can be rewritten in terms of the \ngeometry of an arbitrary closed and convex surface $S_0$ in $\\mathbb{R}^3$\nand a function $s$ defined on $S_0$, as \n\\begin{eqnarray}\n\\int_{S_0} \\left ( P_0 - 2 s K_0 \\right )\n\\left ( 1 + g_0^{AB} \\partial_A s \\partial_B s \n\\frac{1 - 2 s P_0 + s^2 \\left ( P_0^2 - K_0 \\right )}{\n\\left ( 1- s P_0 + s^2 K_0 \\right )^2 }\n+ \\kappa_0^{AB} \\partial_A s \\partial_B s \n\\frac{s \\left ( 2- s P_0 \\right )}{ \n\\left ( 1- s P_0 + s^2 K_0 \\right )^2 } \\right ) \\bm{\\eta_{S_0}} \\geq \n\\nonumber \\\\\n\\geq  \\sqrt{ 16 \\pi \\int_{S_0} \\left (1 - s P_0 + s^2 K_0 \\right ) \\bm{\\eta_{S_0}}},\n\\hspace{5cm}\n\\label{PIshells_rewritten}\n\\end{eqnarray}\nwhere $g_0^{AB}$ and $\\kappa_{0}^{AB}$ are, respectively, the induced metric and second fundamental form of $S_0$\nrespect to the outer normal, $P_0$ and $K_0$ are the mean curvature and the Gauss curvature of $S_0$  or, \nin terms of the principal curvatures $\\lambda_1,\\lambda_2$ \nof the surface, $P_0 = \\lambda_1 + \\lambda_2$ and $K_0 = \\lambda_1 \\lambda_2$.\nThe smooth function $s$ must satisfy the inequalities\n$s < 1\/ \\lambda_1$ and $s < 1 \/\\lambda_2$, but is otherwise arbitrary. It defines the surface $\\hat{S}$ (and hence $S$)\nin a similar way as before.\nDetails of the derivation of (\\ref{PIshells_rewritten}) and some of its consequences will be given elsewhere\n\\cite{Mars2009}.\n\n\\subsection{Spinor techniques on null hypersurfaces: Ludvigsen-Vickers and Berg\\-qvist approaches}\n\n\nWitten's proof of the positive mass theorem is based on the properties of spinors satisfying a\nsuitable elliptic equation and which approach a constant spinor at spatial infinity. The same ideas\nhave been applied to prove positivity of the Bondi mass using asymptotically hyperbolic initial data sets (see\n\\cite{Chrusciel_et_al_2004} and references therein). Another possibility to\napproach the positivity of the Bondi mass is to use null hypersurfaces. This allowed\nLudvigsen and Vickers \\cite{LudvigsenVickers1982}\nto replace the elliptic equations for the spinor by much simpler transport equations.\nSimilar ideas (with different transport equations) allowed the same authors \\cite{LudvigsenVickers1983} \nto argue \nthat, in spacetimes satisfying the dominant energy condition,\nthe inequality \n$M_B \\geq \\sqrt{|S|\/16 \\pi}$ holds for any weakly future trapped (in particular marginally future\ntrapped) surface $S$ which has the property that\none of the two null hypersurfaces generated by past directed null geodesics normal to $S$ can be extended\nto past null infinity while remaining smooth everywhere. Although this is a global assumption on the \nspacetime, it makes no reference to the {\\it future} evolution of the spacetime and hence it is logically\nindependent of cosmic censorship. For instance, this assumption is automatically satisfied for the incoming\nnull shell of dust discussed in the previous section. However,  Bergqvist \\cite{Bergqvist1997} found \na gap in the proof and the range of validity of the argument remains, at present, open. We discuss this next.\n\nBergqvist  reformulated  Ludvigsen and Vickers' argument so that all spinors could\nbe completely dispensed of. The idea\nis, in some sense, analogous to the inverse mean curvature flow for the Geroch mass\nand is based on using a\nfunctional on spacelike closed surfaces $S_{\\mu}$ constructed as follows: start with \na closed spacelike surface $S$ and let $\\vec{l}$ and $\\vec{k}$ be {\\it past} directed\nnull normals to $S$ partially fixed by the normalization $(\\vec{l} \\cdot \\vec{k}) = -2$. Assume also\nthat $\\vec{l}$ points outwards of $S$ in the sense that the null geodesics starting on $S$ \nwith tangent vector $\\vec{l}$ extend to infinite values of the affine parameter and intersect\n${\\mycal I}^{-}$. Let $\\mu$ be the affine parameter of this geodesic with $\\mu = \\mu_0$ on $S$ where $\\mu_0$\nis a constant to be chosen later and assume that the surfaces $\\{\\mu = \\mbox{const}\\}$ are smooth for\nall $\\mu \\geq \\mu_0$. The collection of all these surfaces defines a null hypersurface\n${\\cal N}$ (the ``outer past null cone'' of $S$) which is assumed to intersect ${\\mycal I}^{-}$ on a smooth cut.\nThe Bergqvist mass  is defined on each leave $\\{S_{\\mu}\\}$ of this foliation as \n\\begin{eqnarray*}\nM_{b} (S_{\\mu}) = \\frac{1}{16 \\pi}\n\\int_{S_{\\mu}} \\theta_k \\bm{{\\eta}_{S_{\\mu}}} + 4 \\pi \\chi(S_{\\mu}) \\mu. \n\\end{eqnarray*}\nwhere $\\theta_k$ is the null expansion along $\\vec{k}$. \nUsing the expressions (\\ref{FirstVarArea}) and (\\ref{lthetak2}), it is easy to obtain \n\\begin{eqnarray*}\n\\vec{l} \\left ( \\int_{S_{\\mu}} \\theta_k \\bm{\\eta_{S_{\\mu}}} \\right ) =\n\\int_{S_{\\mu}} \\left ( \\mbox{Ein}(\\vec{l},\\vec{k}) + 2 S_A S^A  \\right ) \\bm{\\eta_{S_{\\mu}}} - 4 \\pi \\chi(S_{\\mu}),\n\\end{eqnarray*}\nwhere the Gauss-Bonnet theorem $\\int_S R(h) \\bm{\\eta_S} = 4 \\pi \\chi(S)$ has been used. We therefore get\n\\begin{eqnarray*}\n\\frac{ d M_b (S_{\\mu})}{d \\mu} = \\frac{1}{16 \\pi}  \n\\int_{S_{\\mu}} \\left ( \\mbox{Ein} (\\vec{l}, \\vec{k})  + 2 S_A S^A \\right )  \\bm{\\eta_{S_{\\mu}}} \\geq 0,\n\\end{eqnarray*}\nprovided the dominant energy condition is satisfied. Assuming that $\\mu_0$\ncan be chosen so that asymptotically $\\theta_{k} = -2\/\\mu + O(1\/\\mu^3)$, i.e. without a term in $\\mu^{-2}$,\nthen the Bergqvist mass can be seen to approach the Bondi mass when $\\mu \\rightarrow \\infty$.\nAssuming now that the initial surface $S$ is a marginally trapped surface, if the value of $M_B$\ncould be somehow related to the area of $S$, a Penrose inequality\nwould follow. Bergqvist \\cite{Bergqvist1997} shows that this can indeed be done but only under the\nassumption that the induced metric $g^{S_{\\mu}}_{AB}$ of the surface $S_{\\mu}$ becomes round at infinity \nin the sense that $\\lim_{\\mu \\rightarrow \\infty} \\mu^{-2} g^{S_{\\mu}}_{AB} \\rightarrow g^{S_2}_{AB}$, where\n$g^{S_2}_{AB}$ is the standard unit metric on the sphere. At present, it is not clear how restrictive \nis this requirement.\nThe idea however remains interesting and deserves further investigation.\n\n\n\\subsection{Uniformly expanding flows}\n\n\nThe success of proving the Riemannian Penrose inequality in the connected\nhorizon case using the monotonicity of the Geroch mass under the (weak) inverse mean curvature flow\nsuggests a possible scenario for approaching the Penrose inequality in the general case. As discussed\nin Subsect. \\ref{UEF}, the Hawking mass (\\ref{HawMass}) is a functional on surfaces which coincides\nwith the Geroch mass in the time symmetric context. Moreover, the Hawking mass (with $C=0$) takes the value\n$\\sqrt{|S|\/(16\\pi)}$ on any topological 2-sphere which is either a marginally outer trapped surface,\na past marginally outer trapped surface, or a generalized\napparent horizon (all of which have null mean curvature vector). Moreover, as discussed in Subsect. \\ref{UEF},\nunder suitable spacetime variations of \na given surface, the Hawking mass is monotonically increasing \\cite{BrayHayward2007}.\nSince its numerical value for large coordinate spheres\nin the asymptotically euclidean region approaches the ADM energy, a flow which interpolates between the horizon and infinity\nand falls into any of the four monotonicity cases discussed in Sect (\\ref{UEF}) would imply the general Penrose\ninequality between the ADM energy and the area of the horizon. Among the four cases, the closest one to the Riemannian inverse mean\ncurvature flow is the so-called ``uniformly expanding flow'', defined by (\\ref{Flowvector}). \n\nWith these monotonicity properties, the situation regarding the general Penrose inequality can be compared\nto the status of the Riemannian Penrose inequality after Geroch's heuristic argument. It is conceivable that the\nuniformly expanding flows might be useful for the proof of the general Penrose inequality. However, many issues\nremain open. For instance, in a spacetime formulation, the jumps that occurred in the Riemannian setting will remain,\nbut it is unclear how and when the jumps should take place, even from a purely heuristic point of\nview. Moreover, the inverse mean curvature flow is a parabolic equation in the Riemannian setting, so that\nlocal existence is guaranteed, but the situation is quite different for the uniformly expanding flows.\n\n\nLocal existence in this case has been proven only for null flows, i.e. $|c| = a$.\nA null flow will obviously not reach spacelike infinity. Nevertheless, assuming that\na sufficiently large portion of the spacetime is at hand, this null flow could be used to study the Penrose inequality \ninvolving the Bondi mass. The difficulty, however, is that the uniformly expanding null flow seems to have the tendency\nto deform the surface in such a way that the mean curvature vector becomes causal at some places, even if one starts with a \nsurface with spacelike mean curvature everywhere. This behaviour is observed in explicit examples in Minkowski spacetime\n(provided the surfaces do not lie on a constant time hyperplane and in fact\ncover a sufficiently large time interval).\nOne alternative that may still give interesting results is to use\na flow which is null and future directed for some interval and then continue with a null {\\it past directed}\nuniformly expanding flow, then with a {\\it future} directed null flow, and so on. This flow can be constructed in a tubular neighbourhood of the\ngiven initial data set, and therefore does not need strong global assumptions on the spacetime. Moreover, it is conceivable\nthat this broken flow can approach spacelike infinity. However, it is not clear which criterion should be used\nto stop the future flow and continue with the past one (and viceversa).\nMoreover, the construction must be such that the mean curvature vector remains spacelike\neverywhere\nin order to ensure monotonicity of the Hawking mass.\n\nRegarding the non-null case, $|c| < a$,  the flow equations  form\na so-called forward-backward parabolic system, for which no local existence theory is known (this was noted by\nHuisken and Ilmanen \\cite{HuiskenIlmanen2001} in the case $c=0$ and was extended to arbitrary $c$ in \\cite{BrayHayward2007}).\nThis is, of course, a major difficulty and addressing it would require a much better understanding of this type\nof partial differential equations. In Huisken and Ilmanen's work, a fundamental part of the analysis dealt with the\nlevel set formulation, which gives a degenerate elliptic equation. More specifically, the existence of a variational\nformulation turned out to be a fundamental ingredient to solve the problem of existence and to study the jumps. Remarkably, the\nuniformly expanding flows also admit a variational formulation \\cite{BrayHayward2007}. The new basic ingredient\nis that the field to be varied is not just the level set function $v$ (see (\\ref{funct1})) as\nin the Riemannian case, but also the\nspacelike hypersurface $\\Sigma$ where this function is defined. Whether this variational formulation      \ncan give useful hints on how to define the jumps remains an open and difficult problem. \n\nIt should be remarked in this context that the simplest spacelike uniformly expanding flow corresponds\nto $c=0$, i.e. $\\vec{\\xi} = \\frac{1}{(\\vec{H} \\cdot \\vec{H})} \\vec{H}$ ($a=1$ can be chosen without loss of generality). The flow vector is\ntherefore the inverse mean curvature vector.\nIn terms of initial data information $(\\Sigma,\\gamma_{ij},\nA_{ij})$ this corresponds to the case $q=0$ (this condition has been often termed ``polar gauge'' in the literature).\nAssuming this gauge condition and a second restriction which turns out to coincide with the inverse\nmean curvature flow condition $p e^{\\psi} = \\mbox{const.}$ Jezierski was able to prove \\cite{Jezierski1994.1,Jezierski1994.2}\nthe Penrose\ninequality for small (but non-linear) electrovacuum perturbations of the Reisner-Nordstr\\\"om time-symmetric initial data \noutside the black hole event horizon. His method was based on writing the Hamiltonian\nconstraint as a divergence term plus a non-positive reminder. The Penrose inequality is\nestablished by integrating this equation between the event horizon and infinity\nafter using the Gauss theorem to transform the divergence into a surface integral at infinity and on the horizon (the\nformer gives the ADM energy and the latter the area term in the inequality).\nAlthough Jezierski's argument does not use the monotonicity of the Hawking mass, the calculation does in fact\ncorrespond to the general identity (\\ref{dMH2}) specialized to the case at hand. With hindsight, one can therefore\nconclude that this monotonicity property of the Hawking mass is the \nunderlying reason why the argument works. Jezierski supports the plausibility of the gauge conditions $q=0$ and $p e^{\\psi} = 1$\nby studying linear perturbations of Reissner-Nordstr\\\"om, where he finds that the two equations decouple, one giving a\nparabolic equation that needs to be integrated radially outwards and another one\nalso parabolic but which needs\nto be integrated radially inwards. This is, of course, a manifestation of the forward-backward parabolic nature of the full  \nsystem. A similar existence result for linear, axial (i.e. odd) perturbations of maximal slices of the Schwarzschild spacetime\nhas been obtained in \\cite{RoszkowskiMalec2005}.\n\n\n\nAnother observation worth mentioning regarding the inverse mean curvature vector flow\n(i.e. $c=0$) is that\nthe monotonicity formula turns out to be  insensitive to the value of the \nenergy flux $\\vec{J}$. If follows that the weak energy condition is already sufficient to ensure monotonicity\nof the Hawking mass in this context. In principle this opens up the possibility that the full Penrose\ninequality might be true for spacetimes satisfying just the weak energy condition. This is not so, however.\nAn explicit counterexample for scalar field initial data has been constructed by V. Husain \\cite{Husain1999}.\n\n\n\n\\subsection{Jang equation}\n\\label{Jang1}\n\nSchoen and Yau's proof of the positive mass theorem proceeded in two steps. First, the purely Riemannian case\n(i.e. vanishing second fundamental form) was solved by using minimal surface techniques. In a second step,\nthe general case $(\\Sigma,\\gamma_{ij},A_{ij})$ was treated by modifying the metric $\\gamma_{ij}$ with a transformation\nintroduced by Jang \\cite{Jang1978}, namely \n\\begin{eqnarray}\n\\hat{\\gamma}_{ij} = \\gamma_{ij} + \\partial_i f \\partial_j f,\n\\label{JangTrans}\n\\end{eqnarray}\nwhere $f$ solves the so-called {\\it Jang equation}\n\\begin{eqnarray}\n\\label{JangEq}\n\\left ( \\gamma^{ij} - \\frac{\\nabla^i f \\nabla^j f}{\n1+ |df|^2_{\\gamma}} \\right ) \\left (\n\\frac{\\nabla_{i} \\nabla_j f }{\\sqrt{ 1 + |df|^2_{\\gamma}}} - A_{ij} \\right )=0.\n\\end{eqnarray}\nThis transformation has the property that\nthe curvature scalar of $\\hat{\\gamma}$ has a lower bound that allows to prove existence of a conformal\nfactor $\\Omega >0$ such that the conformally rescaled metric $\\Omega^2 \\hat{\\gamma}_{ij}$ has\nvanishing scalar curvature. This metric is still asymptotically euclidean and has at most the same mass as\nthe original metric.  The Riemannian positive mass theorem then gives the desired result. \nThe proof is however involved because the Jang equation does not\nadmit regular solutions when the initial data set $(\\Sigma,\\gamma_{ij},A_{ij})$ contains marginally trapped surfaces,\nbut the idea can nevertheless be made to work \\cite{SchoenYau1981}.\n\nA natural question, already asked in \\cite{Bray2001}, is whether a similar idea can be applied to prove\nthe general Penrose inequality. The Penrose inequality has already been proven in full generality in the\nRiemannian setting so the present status is similar to the situation of the proof of the positive mass theorem after its\nRiemannian proof. This idea has been analyzed by Malec and \\'O Murchadha \n\\cite{Malec-Murchadha2004}. Their argument is based on the observation that, \nif the Jang equation could be used to prove the general Penrose inequality, it should be able to do\nso in the particular case of spherical symmetry. Restricting to spherically symmetric functions $f$,\nthe Jang equation becomes a simple ODE and existence of regular solutions in the exterior region outside\nthe full trapped region ${\\cal T}^{+}_{\\Sigma} \\cup {\\cal T}^{-}_{\\Sigma}$ \ncan be easily shown.\nThe difficulty of the method is that along the\nprocess (Jang's transformation and subsequent conformal rescaling) the mass of the manifold should not\nincrease and the area\nof the outermost horizon $S = \\partial ({\\cal T}^{+}_{\\Sigma} \\cup {\\cal T}^{-}_{\\Sigma} )$ should not decrease.\nThis is because one wants to use the Riemannian Penrose inequality for the final manifold and \nconclude that the same inequality holds in the original initial data. It is also clear that $S$ must be transformed\ninto a minimal surface after the conformal rescaling. The simplest situation where this can be achieved is by \ndemanding that $S$ becomes minimal already for the Jang transformed metric $\\hat{\\gamma}_{ij}$. This requires that\nthe outer normal derivative of $f$ diverges to either $+\\infty$ or to $-\\infty$ on $S$. In the first case, it follows\nthat the metric $\\hat{\\gamma}$ has a cylindrical end near $S$. The conformal transformation is expected to compactify\nthis end with one point (this behaviour was found in \\cite{SchoenYau1981}, and this was important in order\nto apply the Riemannian positive mass theorem). Hence, the area of $S$ decreases in the process (it vanishes in the final\nmanifold) and nothing can be concluded. In the second case (outer normal derivative of $f$ diverging to \n$- \\infty$) the conformal factor $\\Omega$ \nis expected to have an interior local minimum. This would imply that the outer normal derivative of $\\Omega$\n is negative on $S$, and hence that this surface is not the outermost minimal surface in the final manifold.\nThus, the Riemannian Penrose\ninequality applied to the final manifold is again inconclusive for the original one.\nAlthough these arguments are not definitive\nin discarding the Jang equation method for the Penrose inequality, they indeed show that difficulties should be \nexpected for the method to work, at least when $S$ is required to transform to a minimal surface by the \nJang transformation. The situation where $S$ is minimal only after the final conformal transformation is not considered\nin \\cite{Malec-Murchadha2004}  and should be further investigated.\n\nVery recently, the Jang equation has been successfully applied to prove a Penrose-like inequality in the spirit\nof Herzlich's inequality discussed in Subsect. \\ref{spinors}, but allowing non-vanishing\nsecond fundamental form. The class of initial data \n$(\\Sigma,\\gamma_{ij},A_{ij})$ under consideration are such that  $\\Sigma = K \\cup \\Sigma^{\\infty}$ with $K$ \ncompact and $\\Sigma^{\\infty}$ an asymptotically euclidean end. The boundary $\\partial \\Sigma$ is compact and\nconsisting of a finite collection of future MOTS (with respect to the normal pointing towards $\\Sigma$).\nMoreover, no weakly future or past trapped surface strictly enclosing $\\partial \\Sigma$ is allowed\nto exist in $\\Sigma$. In other words, $\\partial \\Sigma$ is the outermost MOTS in $\\Sigma$\nand, moreover, no past weakly outer trapped boundary is allowed to exist in $\\Sigma$ except possibly $\\partial \\Sigma$ itself.\nAssuming also \nthat $(\\Sigma,\\gamma_{ij},A_{ij})$ satisfies the dominant energy condition $\\rho \\geq |\\vec{J}|$, Khuri\nhas recently proven \\cite{Khuri2009} that the Penrose-like inequality\n\\begin{eqnarray}\nE_{ADM} \\geq \\frac{\\hat{\\sigma}}{2 ( 1 + \\hat{\\sigma})} \\sum_{a=1}^{k} \\sqrt{\\frac{|\\partial_a \\Sigma|}{\\pi}}\n\\label{KhuriIneq}\n\\end{eqnarray}\nholds, where $k$ is the number of connected components of $\\partial \\Sigma$ and\nthe scale invariant quantity $\\hat{\\sigma}$ is defined as\n\\begin{eqnarray*}\n\\hat{\\sigma} = \\frac{1}{\\sum_{a=1}^{k} \\sqrt{4 \\pi |\\partial_a \\Sigma|}} \\inf_{v \\in C^{\\infty}} \n\\int_{\\Sigma} \\left (dv, dv \\right )_{\\overline{\\gamma}} \\bm{\\eta_{\\overline{\\gamma}}}. \n\\end{eqnarray*}\nThe infimum is taken with respect to functions $v$ with approach one at infinity and zero on $\\partial \\Sigma$.\nThe metric $\\overline{\\gamma}$ is constructed using the Jang transformation (\\ref{JangTrans})  and $f$ is a solution\nof the Jang equation approaching zero at infinity and blowing up to $+ \\infty$ on $\\partial \\Sigma$. The existence of \nsuch an $f$ has been established by Metzger in \\cite{Metzger2008}. The asymptotic behaviour of $f$ at infinity is such that \n$(\\Sigma,\\overline{\\gamma})$ is asymptotically euclidean and $E_{ADM}(\\gamma) = E_{ADM} (\\overline{\\gamma})$.\nSince $f \\rightarrow \\infty$ on $\\partial \\Sigma$, the level sets $S_T \\equiv \\{f =T\\}$, for $T$ large enough, \nform a foliation near $\\partial \\Sigma$ which converge to $\\partial \\Sigma$. The idea is to conformally transform\n$\\overline{\\gamma}$ outside $S_T$ in such a way that the conformally rescaled metric $\\gamma^T_{ij} = u_T^4 \\overline{\\gamma}_{ij}$ \nis still asymptotically euclidean, with vanishing\ncurvature scalar and such that each connected component $S_{T,a}$ of $S_T$\nsatisfies $p^T_a = 4 \\sqrt{\\pi\/|S_{T,a}|_{\\gamma^T}}$, where $p^T_a$ is the mean curvature of $S_{T,a}$\nwith respect to $\\gamma^T_{ij}$ (and the area is also calculated with this metric). Existence\nof $u_T$ is established as a consequence of the positivity properties of $R(\\overline{\\gamma})$  and the\nblowing up behavior of $f$ at $\\partial \\Sigma$.\nThe conformal rescaling is such that the ADM energy decreases by an amount which is at least\nequal to the right-hand side of (\\ref{KhuriIneq}) except for terms that vanish in the limit $T\\rightarrow \\infty$.\nThe only remaining step to conclude (\\ref{KhuriIneq})  is that the ADM energy of \n$\\gamma^T$ is non-negative. But this is precisely the content of the positive mass theorem\nproven by Herzlich \\cite{Herzlich1997} for asymptotically euclidean manifolds with a connected boundary of spherical\ntopology and satisfying (\\ref{InequalityMeanCurvature}). Khuri notices that this positive mass theorem still holds\nif the boundary has a finite number of connected components of spherical topology, each one satisfying the bound \n(\\ref{InequalityMeanCurvature}). In the case at hand, this inequality is satisfied (in fact, saturated) by construction and \nthe spherical topology is guaranteed by Galloway and Schoen's results\n\\cite{SchoenGalloway2005, Galloway2007} on the topology of outermost MOTS.\n\nComparing this Penrose-like inequality with the difficulties\ndescribed by Malec and \\'O Murchadha to use the Jang transformation to prove the\ngeneral Penrose inequality in spherical symmetry, the main difference is that this method ultimately relies\non a positive mass theorem, instead of on the Riemannian Penrose inequality. Thus, there is no need to obtain\nan outermost minimal boundary after the metric is modified by the Jang transformation and the subsequent \nconformal rescaling. \n\n\n\\subsection{Bray and Khuri approach}\n\nVery recently Bray and Khuri have made an important step forward towards establishing the \ngeneral Penrose inequality. As mentioned in Sect. \\ref{Formulations}, these authors propose to use generalized\napparent horizons as the appropriate surfaces for which the Penrose inequality should hold.\nAn important guiding principle that led Bray and Khuri to make this conjecture is related to the fact that,\nindependently of which method for proving the inequality is used, \nthe estimates involved must all\ngive equality whenever $(\\Sigma,\\gamma_{ij},A_{ij})$ is any of the slices of\nthe Kruskal spacetime for which equality holds. Therefore, a preliminary issue of importance is: for which slices of the Kruskal spacetime should equality be expected? Any spacelike Cauchy slice\n$\\Sigma$ of the Kruskal spacetime must intersect both \nthe black hole and the white\nhole event horizons. In Kruskal coordinates, the metric reads (c.f. \n\\cite{Wald1984})\n\\begin{eqnarray*}\nds^2 = \\frac{32 m^3}{r} e^{-\\frac{r}{2m}} dudv + r^2 \\left ( d\\theta^2 + \\sin^2 \\theta d \\phi^2 \\right ),\n\\end{eqnarray*}\nwhere $uv > -1$ and $r(uv)$ is defined by $uv = e^{\\frac{r}{2m}} ( \\frac{r}{2m} -1 )$ (our sign\nconvention for $u$ is different to that in \\cite{Wald1984}). The null vectors\n$\\partial_v$ and $-\\partial_u$ are future directed, the black hole event horizon is located at $u=0$, the white\nhole event horizon at $v=0$ and the domain of outer communications ${\\cal U}$ is located at $\\{ u >0, v>0 \\}$. If the \nboundary of $\\Sigma^{\\mbox{\\tiny{DOC}}} \\equiv \\Sigma \\cap {\\cal U}$ satisfies $u=0$ everywhere, then \n$\\partial \\Sigma^{\\mbox{\\tiny{DOC}}}$ is an area outer minimizing MOTS. Moreover,  since its area is $16 \\pi m^2$\nand there are no other weakly outer\ntrapped surfaces (future or past) in $\\Sigma^{\\mbox{\\tiny{DOC}}}$, it follows that\nthis slice satisfies any of the versions of the Penrose inequality (\\ref{PIHeuT+}), (\\ref{PIHeuT-})\nor (\\ref{PIT}). It also satisfies the inequality involving generalized apparent horizons\n(\\ref{PIKhuriBray}) provided there are no generalized trapped surfaces in $\\Sigma^{\\mbox{\\tiny{DOC}}}$\n(this is plausible although not yet proven, as far as I know). Similar things happen if $v=0$ everywhere\non the boundary of $\\Sigma^{\\mbox{\\tiny{DOC}}}$. However, it neither $u$ and $v$ are\nidentically zero on $\\partial\n\\Sigma^{\\mbox{\\tiny{DOC}}}$, the situation is quite different. In this case the boundary \n$\\partial \\Sigma^{\\mbox{\\tiny{DOC}}}$ is neither a future\nor past MOTS and, in most cases, it is not even smooth. Moreover, the intersection of $\\Sigma$ with the $\\{u=0\\}$\nhypersurface (which is always a MOTS and has area $16 \\pi m^2$) is not\narea outer minimizing because its mean curvature $p$ is negative whenever $v < 0 $ (i.e.\non the points lying in the white hole event horizon with respect to the second asymptotically flat \nspacetime region). Thus, its minimal area enclosure has strictly less\narea. Consequently, the version (\\ref{PIHeuT+}) of the Penrose inequality holds but\n{\\it not} with equality (this statement assumes\nthat $\\{u=0 \\} \\cap \\Sigma$ is the outermost MOTS of the slice, which is again plausible\nbut not yet proven, as far as I know). It follows\nthat not all slices of the Kruskal spacetime are automatically equality cases, at least\nfor the version (\\ref{PIHeuT+}). Obviously, the more slices of Kruskal satisfy equality, the sharper is the version of the Penrose inequality, in the sense of being capable of identifying the\nKruskal spacetime in a larger number of  cases. \nA version that gives equality for any slice of the Kruskal\nspacetime is (\\ref{PIT}), even when the boundary of $\\Sigma^{\\mbox{\\tiny{DOC}}}$ is non-smooth. Although, as\nalready mentioned, no counterexample of this version has been found, its validity would however come as\na surprise because the minimal area enclosure of\n$\\partial ({\\cal T}^{+}_{\\Sigma} \\cup {\\cal T}^{-}_{\\Sigma})$ may \na priori have much smaller area than itself.\nThe alternative put forward by Bray and Khuri involves\ngeneralized trapped surfaces.\nThis has the advantage that, as soon as $\\Sigma^{\\mbox{\\tiny{DOC}}}$\nhas a smooth boundary,\nthis is\na generalized trapped surface (in fact, a generalized apparent horizon). \nEichmair's result, see Subsect. \\ref{embedded}, implies that\nan outermost generalized apparent horizon must exist on $\\Sigma$.\nIt is highly plausible that\n$\\partial \\Sigma^{\\mbox{\\tiny{DOC}}}$ is in fact the outermost apparent horizon in this case. \nSince its area is $16 \\pi m^2$, any slice with smooth\n$\\partial \\Sigma^{\\mbox{\\tiny{DOC}}}$ would \nbelong to the equality case of the Penrose inequality (\\ref{PIKhuriBray}). \nIf the boundary \n$\\partial \\Sigma^{\\mbox{\\tiny{DOC}}}$ is not smooth, then it cannot be its own\nminimal area enclosure, and hence it cannot give equality neither in\n(\\ref{PIHeuT+}) nor in (\\ref{PIHeuT-}).\n\nThese considerations led Bray and Khuri\nto conjecture the following version of the Penrose inequality\n(recall that our definition of asymptotically euclidean requires in \nparticular that $(\\Sigma,\\gamma_{ij})$ is complete and that\nan initial data set is called {\\bf Schwarzschild at infinity} if outside a compact set, the metric \n$\\gamma$ is exactly Schwarzchild, c.f. the discussion after (\\ref{epsilonchange})).\n\n\n\\begin{conjecture}[Bray and Khuri \\cite{BrayKhuri2009}]\n\\label{BrayKhuriConjecture}\nSuppose that the initial data set $(\\Sigma,\\gamma_{ij},A_{ij})$\nis asymptotically euclidean and Schwarzschild at\ninfinity, with total mass $M_{ADM}$ (in a chosen end) and satisfying the dominant energy condition\n$\\rho \\geq |\\vec{J}|$. \nLet $S$ be a closed surface which bounds an open set $\\Omega$\ncontaining all the asymptotically euclidean ends\nexcept the chosen one. Assume that $S$  is a generalized trapped surface \n(with respect to the normal\npointing towards the chosen end). Then\n\\begin{eqnarray*}\nM_{ADM} \\geq \\sqrt{\\frac{|S_{\\min}|}{16 \\pi}},\n\\end{eqnarray*}\nwhere $S_{\\min} = \\partial \\Omega_{\\min}$ is the minimal area enclosure of $S$\n(i.e. $\\Omega \\subset \\Omega_{\\min}$ and $S_{\\min}$ has least area among all surfaces with\nthis property). Furthermore, equality occurs if\nand only if $(\\Sigma \\setminus \\Omega_{\\min}, \\gamma_{ij},\nA_{ij})$ is the induced data of an embedding of $\\Sigma \\setminus \\Omega_{\\min}$\ninto the Kruskal spacetime such that $S_{\\min}$ is mapped to a generalized apparent\nhorizon.\n\\end{conjecture}\nThe use of generalized apparent horizons is indeed a completely new idea for the\nPenrose inequality. This version is not supported by Penrose's heuristic argument of gravitational collapse\nbecause it is not true that all generalized apparent horizons lie inside the \nevent horizon in a black hole spacetime. In fact, little is known in general about this type\nof surfaces in general spacetimes. An important question regarding the plausibility of Conjecture\n\\ref{BrayKhuriConjecture} was posed by R. Wald \\cite{Wald2008} who asked whether\nthere are any generalized apparent horizons in Minkowski spacetime (the question of whether\nsurfaces with causal mean curvature can exist in Minkowski spacetime was already asked in\n\\cite{MarsSenovilla2003} in a somewhat different context). The existence of any generalized apparent\nhorizon embedded in a spacelike Cauchy slice of Minkowski and bounding a compact set would immediately falsify\nConjecture \\ref{BrayKhuriConjecture}. Khuri has been able to prove \\cite{Khuri2009-2}\nthat no such surfaces are present\nin Minkowski spacetime.\nIn fact he proves much more by showing that any asymptotically euclidean initial data set satisfying the\ndominant energy condition and with compact (non-empty)\nboundary consisting of finitely many generalized trapped surfaces \nsatisfies a strict positive mass theorem\n$E_{ADM} > |\\vec{P}_{ADM}|$. The proof is based on Witten's spinorial method for the positive mass.\n\nThe strategy that Bray and Khuri propose to address Conjecture \\ref{BrayKhuriConjecture}\nis related to\nthe Shoen and Yau's reduction of the general positive mass theorem\nto the time-symmetric case. Recall that this was based on \nthe Jang transformation  (\\ref{JangTrans}) of the metric, where the function $f$\nsolves the Jang equation. This equation is specifically\ntailored so that it always admits a solution for any initial data set\nin Minkowski spacetime (the solution is the height function of the slice\nover any constant time hyperplane). This is relevant because slices\nof Minkowski immediately give equality in the positive mass theorem.\nA fundamental observation of Bray and Khuri is that, since\nthe equality case of the Penrose inequality should correspond to the Schwarzschild metric,\nthe Jang equation\nshould be accordingly modified so that it holds identically for slices in this spacetime.\nMore generally, Bray and Khuri consider static spacetimes\n$({\\cal M},g) = ( \\mathbb{R} \\times \\Sigma, ds^2 = - \\phi^2 dt^2 + \\overline{\\gamma})$,\nwhere $\\phi >0$ and  $\\overline{\\gamma}$ is a Riemannian metric. The idea\nis to consider spacelike graphs $(t=f(x),x)$ for $x \\in \\Sigma$ and derive\nan equation for the graph function which \nholds identically in this case and which still makes sense\nfor an arbitrary initial data set $(\\Sigma,\\gamma_{ij},A_{ij})$.\n\n\n\nThe induced metric on the graph is \n\\begin{eqnarray*}\n\\gamma_{ij} = \\overline{\\gamma}_{ij} - \\phi^2 \\partial_i f \\partial_j f,\n\\end{eqnarray*}\nwhich is Riemannian provided the gradient of $f$ is not too large with respect\nto $\\overline{\\gamma}$,\nnamely if  $\\phi |df|_{\\overline{\\gamma}} < 1$. However, this expression can also be used\nto {\\it define} $\\overline{\\gamma}_{ij}$\ngiven an initial data set $(\\Sigma,\\gamma_{ij}, A_{ij})$ and two arbitrary functions $\\phi$ and $f$.  This implies that the\nRiemannian metric $\\gamma$ of any initial data set can be obtained\nas the induced metric of a hypersurface in a\nsuitably constructed static spacetime. With this point of view, the function\n$f$ becomes arbitrary (with no restriction on its gradient)\ndue to the identity\n$(1- \\phi^2 |df|^2_{\\overline{\\gamma}} ) ( 1 + \\phi^2 |df |^2_{\\gamma} ) = 1$. \nThe inverse metric is \n$\\overline{\\gamma}^{ij} = \\gamma^{ij} - v^i v^j$ with\n\\begin{eqnarray}\nv^i \\equiv  \\frac{\\phi \\nabla^i f}{\\sqrt{1 + \\phi^2 |df|^2_{\\gamma}}}\n\\label{vecv}\n\\end{eqnarray}\nwhere indices are lowered and raised with the metric $\\gamma_{ij}$ and its inverse.\nThe second fundamental form on the graph is \\cite{BrayKhuri2009}\n\\begin{eqnarray}\n\\overline{A}_{ij} = \\frac{\\phi \\nabla_i \\nabla_j f + \\nabla_i \\phi \\nabla_j f + \n\\nabla_j \\phi \\nabla_j f}{\\sqrt{ 1 + \\phi^2 |df|^2_{\\gamma}}}.\n\\label{NewSecForm}\n\\end{eqnarray}\nOf course, this second fundamental form has nothing to do a priori with the second fundamental form $A_{ij}$ of the given initial data set. Nevertheless,\n$(\\Sigma,\\gamma_{ij},\\overline{A}_{ij})$ is the induced geometry\nof a spacelike hypersurface in a static spacetime. The\nexistence of the isometry generated by\n$\\partial_t$ can be used to relate the geometry of this slice with the geometry of the\ncorresponding $\\{t=0\\}$ slice, \ni.e. of $(\\Sigma,\\overline{\\gamma})$. After an involved calculation,\nthis observation leads to the following remarkable identity\nfor the curvature scalar $R (\\overline{\\gamma})$ \\cite{BrayKhuri2009},\ncalled {\\it generalized Schoen-Yau identity}, \n\\begin{eqnarray}\nR (\\overline{\\gamma} ) & = &  16 \\pi \\left ( \\rho - J_i v^i \\right ) + || \\overline{A} - A ||^2_{\\overline{\\gamma}} \n+ 2 |\\vec{z}\\, |^2_{\\overline{\\gamma}} - \\frac{2}{\\phi} \\mbox{div}_{\\overline{\\gamma}} (\\phi \\vec{z} \\,) \\nonumber \\\\\n& + &  \\mbox{tr}_{\\overline{\\gamma}} (\\overline{A} - A)  \n\\left [ \\mbox{tr}_{\\overline{\\gamma}} (\\overline{A} + A ) + 2 A_{ij} v^i v^j \\right ] \n+ 2 v^i \\partial_{i} \\left ( \\mbox{tr}_{\\overline{\\gamma}} (\\overline{A} - A) \\right ), \n\\label{Generalized}\n\\end{eqnarray}\nwhere $z_i = ( A_{ij} - \\overline{A}_{ij} ) v^j$ and $\\vec{z}$ is obtained\nafter raising its index with the inverse of the metric $\\overline{\\gamma}_{ij}$.\nIn the case $\\phi=1$, this identity was obtained by Schoen and Yau \\cite{SchoenYau1981} and was used to\nshow that any solution $f$ of the Jang equation  defines\na metric $\\overline{\\gamma}_{ij}$ which admits a conformal rescaling with non-negative curvature scalar.\n\nBray and Khuri's approach aims at finding appropriate functions $\\phi$  and $f$ so that the\nPenrose inequality for $(\\Sigma,\\gamma_{ij},A_{ij})$ follows as a consequence\nof the  Riemannian Penrose inequality applied to the transformed data $(\\Sigma,\\overline{\\gamma})$.\nIn order to use\nthe Riemannian Penrose inequality, it is necessary that $R(\\overline{\\gamma}) \\geq 0$.\nThe dominant energy condition $\\rho \\geq |\\vec{J}|_{\\gamma}$ together with the fact that\nthe vector $\\vec{v}$ satisfies $|\\vec{v} |_{\\gamma} <1$ (see (\\ref{vecv}))\nimplies that the first three\nterms in (\\ref{Generalized}) are non-negative.  The remaining terms have no sign in general.\nMotivated by the structure of \n(\\ref{Generalized}), Bray and Khuri\nintroduce the {\\it generalized  Jang equation} $\\mbox{tr}_{\\overline{\\gamma}} ( \\overline{A} - A)=0$ or, explicitly,\n\\begin{eqnarray}\n\\label{GenJangEq}\n\\left ( \\gamma^{ij} - \\frac{\\phi^2 \\nabla^i f \\nabla^j f}{\n1+ \\phi^2 |df|^2_{\\gamma}} \\right ) \\left (\n\\frac{\\phi \\nabla_{i} \\nabla_j f + \\nabla_{i} \\phi \\nabla_j f +\n\\nabla_j \\phi \\nabla_i f }{\\sqrt{ 1 + \\phi^2 |df|^2_{\\gamma}}} - A_{ij} \\right )=0.\n\\end{eqnarray}\nThis obviously reduces to the original Jang equation (\\ref{JangEq}) when $\\phi=1$. In the present\ncase, however, this equation involves  two unknowns, $\\phi$ and $f$. \nThe Jang equation is known not to admit regular solutions \nwhen $\\Sigma$ contains a future or past MOTS. In a similar fashion, Bray and Khuri observe\nthat the generalized Jang equation may blow up on surfaces satisfying $|p|  = |q|$, which\nare generalized trapped surfaces.\nThis, combined with the existence of an outermost \ngeneralized apparent horizon on $(\\Sigma,\\gamma_{ij},A_{ij})$, led to authors to study the conjecture\n(\\ref{BrayKhuriConjecture})  in the particular case that $\\partial \\Sigma$ is a generalized apparent\nhorizon and that no further generalized trapped surfaces exist in $\\Sigma$.\nThe conjecture in \nthis setting is\n\\begin{conjecture}[Bray and Khuri \\cite{BrayKhuri2009}]\n\\label{BrayKhuriConjecture-2}\nSuppose that the initial data set $(\\tilde{\\Sigma},\\gamma_{ij},A_{ij})$ is asymptotically euclidean \nand Schwarzschild at\ninfinity, with total mass $M_{ADM}$ (in a chosen end) and satisfying the dominant energy condition $\\rho \\geq\n| \\vec{J}|$. \nLet $S$ be a closed surface which bounds \nan open set $\\Sigma^{\\mbox{\\tiny{int}}}$\ncontaining all the asymptotically euclidean ends\nexcept the chosen one and that \n$S$ is  an outermost generalized trapped surface with respect to the normal\npointing towards the chosen end, i.e. that $\\Sigma \\equiv \\tilde{\\Sigma} \\setminus\n\\overline{\\Sigma^{\\mbox{\\tiny{int}}}}$ contains no generalized trapped surfaces.\nThen \n\\begin{eqnarray}\nM_{ADM} \\geq \\sqrt{\\frac{|\\partial \\Sigma|}{16 \\pi}}.\n\\label{Ineq}\n\\end{eqnarray}\nFurthermore, equality occurs if\nand only if $(\\Sigma, \\gamma_{ij}, A_{ij})$ is the induced data of an embedding\nof $\\Sigma$\ninto the Kruskal spacetime such that $\\partial \\Sigma$ is mapped to a generalized apparent horizon.\n\\end{conjecture}\nUnder the conditions of this conjecture, \nit is plausible\nthat (\\ref{GenJangEq}) admits regular solutions in  $\\Sigma$. The boundary\nbehaviour is typically singular, as examples in the Kruskal spacetime show. The hope is that this singular\nbehaviour on the boundary can be adjusted so that the surface $\\partial \\Sigma$\nhas non-positive mean curvature with respect to the transformed metric $\\overline{\\gamma}_{ij}$. \nThis is useful because the area of any surface never decreases under \nthe transformation $\\gamma_{ij} \\rightarrow \\overline{\\gamma}_{ij}$  (due to the fact that\nthe volume form of $\\overline{\\gamma}_{ij}$ is $\\bm{\\eta_{\\overline{\\gamma}}} = ( 1 + \\phi^2 |df|^2_{\\gamma} ) \\bm{\\eta_{\\gamma}}$).\nConsequently, the minimal area enclosure $\\overline{S}_{\\min}$ of\n$\\partial \\Sigma $ in $(\\Sigma,\\overline{\\gamma})$ \nsatisfies\n\\begin{eqnarray}\n| \\overline{S}_{\\min} |_{\\overline{\\gamma}} \\geq |\\overline{S}_{\\min} |_{\\gamma} \\geq |\\partial \\Sigma |_{\\gamma},\n\\label{chain}\n\\end{eqnarray}\nwhere the subindex denotes which metric is used to calculate the area and the second inequality\nholds because $\\partial \\Sigma$\nis area outer minimizing in $(\\Sigma,\\gamma)$. Thus, \nan upper bound for  $|\\overline{S}_{\\min}|_{\\overline{\\gamma}}$\n(via the Riemannian Penrose inequality) implies an upper bound for\n$| \\partial \\Sigma |_{\\gamma}$, which is the type of information relevant for the\nfull Penrose inequality.\n\nShowing existence of solutions\nof the generalized Jang equation (\\ref{GenJangEq}) which satisfy this\ncriterion is a fundamental open issue in this approach.\nNevertheless, by construction there is an interesting particular\ncase where (\\ref{GenJangEq}) admits solutions, namely when the initial data set $(\\Sigma,\\gamma_{ij},A_{ij})$ is in fact a slice of a static spacetime,\ni.e. when there exist functions $\\phi>0$ and $f$ such that\n$A_{ij} = \\overline{A}_{ij}$. In this case not only the generalized\nJang equation holds trivially, but also  $\\vec{z} =0$. \nUsing (\\ref{Generalized}), this implies $R (\\overline{\\gamma}) \\geq 0$.\nIf $f$ decays fast enough at infinity (for instance if $f$ is of compact support) then $M_{ADM} (\\gamma)\n= M_{ADM} (\\overline{\\gamma})$. Assuming that $\\partial \\Sigma$\nis a generalized apparent horizon, it follows\nthat $\\partial \\Sigma$ is a minimal surface in $(\\Sigma,\\overline{\\gamma}_{ij})$ provided\n$\\phi$ and $f$ are smooth up to the boundary and $\\phi  |_{\\partial \\Sigma} =0$ (it should\nbe remarked that these two conditions\nare quite restrictive, for instance they hold for slices of the Kruskal spacetime\nonly if they intersect\nthe black hole event horizon precisely at the bifurcation surface $\\{u=v=0\\}$). The Riemannian Penrose inequality then\ngives $M_{ADM}(\\overline{\\gamma})  \\geq \\sqrt{ |\\overline{S}_{\\min} |_{\\overline{\\gamma}}\/(16 \\pi)}$, and hence\nthe Penrose inequality $M_{ADM} (\\gamma) \\geq \\sqrt{|\\partial \\Sigma|_{\\gamma} \/ (16 \\pi)}$ follows from\n(\\ref{chain}). \n\nReturning to the general case, whenever the generalized Jang equation is satisfied, \nthe curvature scalar of $\\overline{\\gamma}_{ij}$ reduces to\n\\begin{eqnarray}\nR (\\overline{\\gamma} ) & = &  16 \\pi \\left ( \\rho - J_i v^i \\right ) + || \\overline{A} - A ||^2_{\\overline{\\gamma}} \n+ 2 |\\vec{z}\\, |^2_{\\overline{\\gamma}} - \\frac{2}{\\phi} \\mbox{div}_{\\overline{\\gamma}} (\\phi \\vec{z}\\, ). \n\\label{RicciUnderJang}\n\\end{eqnarray}\nThis expression has no sign in general, so the Riemannian Penrose inequality cannot\nbe applied directly. However, the generalized Jang equation involves two\nunknowns, so it must be supplemented by a second condition in order to have\na determined problem. Bray and Khuri discuss\ntwo possibilities. \n\nThe simplest one consists in putting equal to zero\nthe  last summand in (\\ref{RicciUnderJang}), i.e. \n\\begin{eqnarray}\n\\mbox{div}_{\\overline{\\gamma}} (\\phi \\vec{z} \\,) =0.\n\\label{divergence}\n\\end{eqnarray}\nThe two equations (\\ref{GenJangEq})-(\\ref{divergence}) are called the {\\it Jang -  zero  divergence\nequations} in \\cite{BrayKhuri2009}. The equation (\\ref{divergence}) is third order in $f$. However,\nafter substracting suitable derivatives of (\\ref{GenJangEq}) it can be converted into a second order\nequation for $f$ (with quadratic second derivatives). The resulting system is degenerate\nelliptic. Bray and Khuri conjecture that the system\nadmits solutions with appropriate behaviour at infinity and such that \n$\\partial \\Sigma$ is a minimal surface with respect to $\\overline{\\gamma}_{ij}$.\nUnder this conjecture, the \nPenrose inequality as stated in (\\ref{BrayKhuriConjecture-2}) follows (modulo a\ntechnical point regarding the equality case, see Conjecture 7 in \\cite{BrayKhuri2009}).\n\nThe second possibility \nis based on the observation that, while (\\ref{GenJangEq})\ndoes not imply $R (\\overline{\\gamma})\\geq 0$, the integrated inequality\n$\\int_{\\Sigma} \\phi R(\\overline{\\gamma}) \\bm{\\eta_{\\overline{\\gamma}}} \n\\geq 0$ follows from (\\ref{RicciUnderJang})\nunder suitable decay conditions at infinity and restrictions on $\\partial \\Sigma$.\nBray and Khuri make the interesting observation that in any situation\n(irrespective of whether $R(\\overline{\\gamma}) \\geq 0$ or not) where the \nmass $M_{ADM} (\\overline{\\gamma})$ can be shown to satisfy \n\\begin{eqnarray}\nM_{ADM} (\\overline{\\gamma}) - \\sqrt{\\frac{|S_{\\min}|_{\\overline{\\gamma}}}{16 \\pi}} \\geq \\int_{\\Sigma} Q(x) R (\\overline{\\gamma}) \\bm{\\eta_{\\overline{\\gamma}}}\n\\label{GeneralType}\n\\end{eqnarray}\nwith some $Q(x) \\geq 0$, then the prescription\n$\\phi = Q$ implies $M_{ADM} (\\gamma) \\geq \\sqrt{|S_{\\min}|_{\\overline{\\gamma}}\/(16 \\pi)}$\nand hence the Penrose inequality (\\ref{Ineq}) provided $M_{ADM}(\\gamma) \\geq M_{ADM} (\\overline{\\gamma})$. \nThe question is, therefore, under which circumstances a general type inequality of the form (\\ref{GeneralType})\nholds. As the authors point out,\nany such  inequality would imply the Riemannian Penrose inequality as a particular case.\nIt is therefore natural to study whether the known proofs of the Riemannian Penrose inequality are capable of \nestablishing the validity of (\\ref{GeneralType}) for some $Q \\geq 0$.\nThe authors show explicitly that\nthis is indeed the case for the Huisken and Ilmanen method,\nprovided the second homology class of $\\Sigma$\nis trivial and $\\partial \\Sigma$ is connected. The idea is to integrate (\\ref{dMG}) with respect to \n$\\lambda$ and convert the double integral (in $\\lambda$ and on $S_{\\lambda}$) as a volume integral. \nBy using the weak formulation in terms of level sets, this can be accomplished even along the\njumps. The result is that $Q = | \\overline{\\nabla} u |_{\\overline{\\gamma}} \\sqrt{e^u\n|S_{\\min}|_{\\overline{\\gamma}}}\/(16 \\pi)^{3\/2}$\nwhere $u$ is the weak solution of the inverse mean curvature flow equal to\nzero on $\\partial \\Sigma$.\nThus, existence of appropriate solutions for the pair of equations (\\ref{GenJangEq}) and $\\phi = Q(x)$ implies the general\nPenrose inequality (\\ref{Ineq}) for connected $\\partial \\Sigma$ (assuming\nthat $\\Sigma$ has trivial second homology class).\nAs noted by the authors, existence in this case looks harder \nthan for the Jang-zero divergence system \nbecause $Q$ vanishes identically wherever\nthe inverse mean curvature flow jumps. \nThe equation therefore implies $\\phi =0$ on the jumps.\nHowever, if $f$ stays smooth, then $\\overline{A}_{ij} =0$  there (see (\\ref{NewSecForm})). \nBut then, the\ngeneralized Jang equation\n(\\ref{GenJangEq}) requires $\\mbox{tr}_{\\gamma} A=0$ along the jumps, which is a\ncondition on the initial data  and not an equation.\nThus, existence of classical solutions of the system \n$\\phi =Q$ and (\\ref{GenJangEq})\nshould not be expected in general.\nIt may be, however, that existence can be granted if $f$ is allowed to be unbounded (or even undefined)  in suitable regions.\n\nThe other existing method to prove the Riemannian Penrose inequality is the conformal flow of metrics\ndue to Bray \\cite{Bray2001}. As discussed in \\cite{BrayKhuri2009},\nthis method is also capable of producing an inequality of the form (\\ref{GeneralType}). In this case $Q$\nis expected to be continuous and strictly positive outside $S_{\\min}$. On the other hand, the\nresulting equation $\\phi = Q$\nis not local, in the sense that it does not define a local P.D.E. at each point. Existence of the\ncoupled system with the generalized Jang equations appears to be difficult in this case as well.\n\n\n\\section{Stronger versions of the Penrose inequality}\n\\label{stronger}\n\nThe Penrose inequality can be strengthened under some circumstances,  in particular when suitable\nmatter fields are present in the spacetime. In order to understand heuristically why this is to be expected,\nlet us return to the original argument by Penrose based on cosmic censorship. Assume that the \ncollapsing matter is electrically charged and that the end-state of the collapse is a black hole in\nequilibrium. In this situation, all the matter sources of the electromagnetic field are expected to lie within the\nevent horizon and the black hole is therefore electrovacuum in its exterior. According to the\nblack hole uniqueness result, the end-state is therefore a Kerr-Newman black hole. The area\nof any cut of the event horizon in this spacetime is given by\n\\begin{eqnarray}\n|S| = 4 \\pi \\left ( 2 M^2 - Q^2 + 2 M \\sqrt{M^2 - L^2\/M^2 - Q^2} \\right ),\n\\label{KN}\n\\end{eqnarray}\nwhere $L$ is the total angular momentum of the final state and $Q$ the total electric charge. \nFrom (\\ref{KN}) it follows immediately $|S| \\leq 4 \\pi ( M + \\sqrt{M^2 - Q^2} )^2$\nwhich makes no reference to the angular momentum. Since the total electric charge of the spacetime is\nconserved provided no charged matter escapes to infinity, the Penrose heuristic argument implies that any \nasymptotically euclidean electrovacuum initial data set \nshould satisfy the inequality\n\\begin{eqnarray}\n\\label{PICharged}\n\\sqrt{\\frac{A_{\\min} (\\partial {\\cal T}^{+}_{\\Sigma} )}{16 \\pi}} \\leq \n\\frac{1}{2} \\left (M_{ADM} + \\sqrt{M_{ADM}^2 - Q^2 } \\right ).\n\\end{eqnarray}\nIn the time  symmetric case, the electrovacuum initial data reduces to the triple $(\\Sigma,\\gamma_{ij},E_i)$, where \nthe electric field $\\vec{E}$ satisfies $\\mbox{div}_{\\gamma} \\vec{E} =0$. The total charge is defined\nas $4 \\pi Q = \\int_S (\\vec{E} \\cdot \\vec{m} ) \\bm{\\eta_{S}}$ where $S$ is homologous to any large sphere in the asymptotically euclidean end. The inequality (\\ref{PICharged}) simplifies to\n\\begin{eqnarray}\n\\label{PIChargedSym}\n\\sqrt{\\frac{|S_{m}|}{16 \\pi}} \\leq \n\\frac{1}{2} \\left (M_{ADM} + \\sqrt{M_{ADM}^2 - Q^2 } \\right ).\n\\end{eqnarray}\nwhere $S_{m}$ is the outermost minimal surface. For this inequality to make sense it\nis necessary that $M_{ADM} \\geq |Q|$ for all electrovacuum initial data. This is a strengthening\nof the positive mass theorem in the presence\nof electromagnetic\nfields and was first proven in \\cite{Gibbons_Hawking_1983} (see \\cite{GibbonsHull1982}\nfor a generalization including matter and \\cite{Chrusciel_Reall_Tod2006} for a rigorous statement).\nThe minimum value of the right-hand side of (\\ref{PIChargedSym}) is $|Q|\/2$. Thus, \nfor horizons of small area ($|S_m| < 4 \\pi Q^2$) the Penrose inequality (\\ref{PIChargedSym})\nreduces to the positive mass theorem $M_{ADM} \\geq |Q|$, with no reference to the area of the horizon. \nThis implies, in particular, that (\\ref{PIChargedSym}) does not admit an equality case (i.e.\na rigidity statement) for horizons of small area. On the\nother hand, the equality case in the Penrose inequality \n$|S_m| \\leq 16 \\pi M_{ADM}^2$  has the interesting\nconsequence of providing \na variational characterization of the Schwarzschild metric (\\ref{gsch}) as the absolute\nminimum of total mass among asymptotically euclidean, Riemannian manifolds of non-negative Ricci\nscalar and with an outermost minimal surface\nof given area $|S_m|$. This is similar to the variational characterization of Euclidean space as the absolute\nminimum of total mass among asymptotically euclidean Riemannian manifolds with $R(\\gamma) \\geq 0$. \n\nA natural question is whether there exists another version of the charged Riemannian Penrose inequality\nwhich is able to give a variational characterization (among metrics of fixed charge and fixed area of \nthe outermost minimal surface) of the Reisner-Nordstr\\\"om and Papapetrou-Majumdar metrics, which are the only\nstatic and electrovacuum  black holes (see \\cite{ChruscielTod2007} and references therein). A strengthening\nof (\\ref{PIChargedSym}) that has been proposed is\n\\begin{eqnarray}\nM_{ADM} \\geq \\frac{1}{2} \\left (\\sqrt{\\frac{|S_m|}{4 \\pi}}  + Q^2 \\sqrt{\\frac{4 \\pi}{|S_m|}} \\right ). \n\\label{PIChargedCon}\n\\end{eqnarray}\nThis inequality was first discussed and proven by Jang \\cite{Jang1979} in the case\nof asymptotically euclidean, electrovacuum initial data sets $(\\Sigma,\\gamma_{ij}, E_{i})$ with a {\\it\nconnected} outermost minimal surface $S_m$, provided the inverse mean curvature flow starting on $S_m$\nremains smooth. This last requirement is, in fact, unnecessary thanks to the weak formulation of the flow\nintroduced by Huisken and Ilmanen. This establishes (\\ref{PIChargedCon}) for connected $S_m$.\n\nInequality (\\ref{PIChargedCon}) is, however, not generally true when the outermost minimal \nsurface is allowed to have several connected components. A counterexample has been found by\nWeinstein and Yamada \\cite{Weinstein-Yamada2005}. Their basic idea \nwas  to realize that\nthe Papapetrou-Majumdar spacetime, which represents a static configuration of $N$\nblack holes of masses $m_i>0$ and charges $Q_i = \\epsilon m_i$, with $\\epsilon =\\pm 1$,  \nhas the property that the total area $|S|$ of the event horizon violates the inequality (\\ref{PIChargedCon}). Indeed, using $Q= M_{ADM}$, it follows\n\\begin{eqnarray*}\nM_{ADM} - \\frac{1}{2} \\left (\\sqrt{\\frac{|S|}{4 \\pi}}  \n+ Q^2 \\sqrt{\\frac{4 \\pi}{|S|}} \\right ) \n= - \\frac{1}{2} \\sqrt{\\frac{4 \\pi}{|S|}} \\left ( M_{ADM} - \\sqrt{\\frac{|S|}{4 \\pi}}  \\right )^2 \\leq 0,\n\\end{eqnarray*}\nirrespectively of the value of $|S|$. The only way how (\\ref{PIChargedCon}) could hold is $|S| = 4 \\pi M^2_{ADM}$. \nHowever, a simple computation gives $|S| = 4 \\pi \\sum_{i} ( m_i)^2$. Since the ADM mass is\n$M_{ADM} = \\sum_i m_i$, equality can only happen where there is only one black hole (i.e.\nwhen the spacetime is the extreme Reissner-Nordstr\\\"om black hole). For any configuration with two\nor more black holes, (\\ref{PIChargedCon}) is violated for the area of the event horizon.\nThis argument is however, not a proof that (\\ref{PIChargedCon}) fails to hold because the static \ninitial data set (i.e. the hypersurface\northogonal to the static Killing vector) in the Papapetrou-Majumdar spacetime does not contain any minimal\nsurface. Each connected component of the event horizon corresponds to an asymptotic cylinder. Thus, some engineering\nis required to construct an electrovacuum initial data set with a minimal surface and which violates\n(\\ref{PIChargedCon}). The method followed in \\cite{Weinstein-Yamada2005} consists in an adaptation of the \ngluing technique developed in \\cite{Isenberg_Mazzeo_Pollack_2002}. More specifically, it consists in\ntaking two copies of a static initial data set of the Papapetrou-Majumdar spacetime with two black holes of equal\nmass $m_1=m_2=m$, such that one of the copies has positive charges and the other one negatives charges. By modifying\nthe geometry far enough along the cylindrical ends, the two copies can be glued together to construct\nan initial data set with two asymptotically euclidean ends and a minimal surface with two connected components. The\nfinal step is to conformally transform the data so that the curvature  scalar vanishes.\nBy taking $m$\nsmall enough, the resulting manifold violates the inequality (\\ref{PIChargedCon}).\nAs the authors stress,  this is not a counterexample of\n(\\ref{PIChargedSym}), which is the inequality that follows from Penrose's heuristic argument.\n\nReturning to the question of whether the charged Riemannian Penrose inequality provides a variational characterization\nof the electrovacuum static black holes (see also \\cite{Gibbons1984} for a related discussion),\nwe notice that the inequality (\\ref{PIChargedSym})  can be written in the\nfollowing equivalent way\n\\begin{eqnarray}\n\\left \\{ \\begin{array}{ll}\n                  M_{ADM} \\geq |Q| & \\quad \\mbox{if} \\quad |S_m| \\leq 4 \\pi Q^2 \\quad \\mbox{(case (i))} \\\\\n                  M_{ADM} \\geq \\frac{1}{2} \\left (\\sqrt{\\frac{|S_m|}{4 \\pi}}  + Q^2 \\sqrt{\\frac{4 \\pi}{|S_m|}} \\right ) &\n\\quad \\mbox{if} \\quad |S_m|\\geq  4 \\pi Q^2  \\quad \\mbox{(case (ii))} \\\\\n\\end{array} \\right . \\label{split2}\n\\end{eqnarray}\nThe Reissner-Nordstr\\\"om black holes have event horizons (or equivalently outermost minimal surfaces in their\nstatic initial data) with satisfy $|S_m| \\geq 4 \\pi Q^2$ (see (\\ref{KN}) with $L=0$), with equality only for\nthe extreme Reisner-Nordstr\\\"om case ($M=|Q|$). So, these metrics belong to case (ii) above and,\nin fact, saturate the corresponding inequality. Similarly, \nthe event horizon of the Papapetrou-Majumdar spacetime has area \n$|S| = 4 \\pi \\sum_i m_i^2 \\leq 4 \\pi \\left ( \\sum_i m_i \\right )^2 = 4 \\pi Q^2$, with equality only if there is only\none black hole (i.e. the metric is extreme Reissner-Nordstr\\\"om again). So, this case belongs to case (i)\nand obviously the inequality is again saturated. Moreover, the Papapetrou-Majumdar static initial data\nis the only asymptotically euclidean, electrovacuum initial data $(\\Sigma,\\gamma_{ij},E_{i})$ satisfying $M=|Q|$ (see\nTheorem 1.2 in \\cite{Chrusciel_Reall_Tod2006} as well as the related previous work  \\cite{GibbonsHull1982},\n\\cite{Tod1983}). Thus, the formulation (\\ref{split2}) would in fact provide a\nvariational characterization of all static charged black holes provided the inequality can be proven\nin case (ii) with equality only for the Reissner-Nordstr\\\"om initial data.\n\n\nIn the non time-symmetric case, Gibbons conjectured \\cite{Gibbons1984}\nthe inequality (\\ref{PIChargedCon}) for connected\nand outermost future (or past) marginally outer trapped surfaces $S$.\nIn the non-connected case, the corresponding conjecture\ninvolves the sum over each connected component of the right-hand side\nof (\\ref{PIChargedCon}). Although no counterexample is explicitly known,\nsuch an inequality would reduce in vacuum to a stronger statement\nthan the standard Penrose inequality. As noted by\nWeinstein and Yamada \\cite{Weinstein-Yamada2005}, it seems that\nan initial data representing two Schwarzschild black holes sufficiently far\napart should violate this version of the inequality. \nIn the particular case of spherical symmetry (where\n$S$ is automatically connected), Gibbons conjecture has been proven by\nMalec  and \\'O Murchadha \\cite{MalecMurchadha1994} for maximal initial data sets and by Hayward \n\\cite{Hayward1998} in the general case.\n\n\nIn the discussion above we have dropped the angular momentum term in (\\ref{KN}) and have\nretained the charge. It is natural to ask what is the situation \nin the reverse case, i.e. when the charge is dropped\nand the angular momentum in kept. The inequality that results is\n$|S| \\leq  8 \\pi M ( M + \\sqrt{M^2 - L^2\/M^2 })$. However, contrarily to the electromagnetic case, the\ntotal angular momentum of the final end-state after the collapse has been completed need not coincide\nwith the initial one since gravitational waves carry angular momentum and this can be radiated away.\nAs first discussed in \\cite{Friedman_Mayer_1982} (see also \\cite{Horowitz1984}), there is one important\nsituation where angular momentum must be conserved along the evolution, namely in the axially symmetric case.\nUnder this restriction, the Penrose heuristic argument implies \n\\cite{Hawking1972},\\cite{Dain_et_al_2002},\n\\begin{eqnarray}\n\\label{PIAngularMomentum}\nA_{\\min} (\\partial {\\cal T}^{+}_{\\Sigma} ) \n\\leq \n8 \\pi M_{ADM} \\left (M_{ADM} + \\sqrt{M_{ADM}^2 - L^2\/M_{ADM}^2 } \\right ).\n\\end{eqnarray}\nSimilarly as before, this inequality only makes sense provided one can show that any asymptotically euclidean\nand axially symmetric initial data set satisfying the dominant energy condition \n$\\rho \\geq |\\vec{J}|$ satisfies the inequality $M_{ADM} \\geq \\sqrt{|L|}$.\nAgain, this is a strengthening of the positive mass theorem. This inequality is supported by a heuristic\nargument based on cosmic censorship and the conservation of angular momentum in the axially symmetric case\n\\cite{DainPRL2006} and its validity has been rigorously proven in \n\\cite{Dain2008} for any initial data set $(\\Sigma,\\gamma_{ij},A_{ij})$\nwhich is vacuum, maximal\n($\\mbox{tr}_{\\gamma} A=0$), contains one or more asymptotically euclidean ends \nas well as possibly additional\nasymptotically cylindrical ends (which correspond to degenerate horizons) and such that\nthe outermost MOTS is connected \n(see also \\cite{Chrusciel_Li_Weinstein_2008} for an extension which\nadmits furthermore non-negative energy-density and relaxes some of the technical requirements in\n\\cite{Dain2008}). Moreover, the case\nof equality $M_{ADM} = \\sqrt{|L|}$ occurs if and only if\nthe initial data is a slice of the extreme Kerr black hole.  This provides\na variational characterization of extreme Kerr. The inequality in the\ncase with an outermost MOTS with several connected components remains still open. Numerical \nevidences for its validity have been recently given in \\cite{DainOrtiz2009}.\n\n\nThe situation for the Penrose inequality with angular momentum \nis therefore similar to the charged\ncase. The inequality (\\ref{PIAngularMomentum}) is equivalent to \n\\begin{eqnarray}\n\\left \\{ \\begin{array}{ll}\n                  M^2_{ADM} \\geq |L| & \\quad \\mbox{if} \\quad |S| \\leq 8 \\pi |L| \\quad \\mbox{(case (i))} \\\\\n                  M^2_{ADM} \\geq \\frac{|S|}{16 \\pi}  + \\frac{4 \\pi L^2}{|S|} &\n\\quad \\mbox{if} \\quad |S|\\geq  8 \\pi |L|   \\quad \\mbox{(case (ii))} \\\\\n\\end{array} \\right . \\label{M-L-2}\n\\end{eqnarray}\nwhere $|S| = A_{\\min} (\\partial {\\cal T}^{+}_{\\Sigma} ) $. This version of the Penrose conjecture (for axially \nsymmetric initial data sets) admits a rigidity case which states that equality in case (ii) can only occur of the \ninitial data is a slice of the Kerr black hole. Again, this would provide a variational characterization\nof the Kerr metric. \n\n\n\\section{Some applications of the Penrose inequality}\n\\label{applications}\n\nIn this section we briefly mention some recent situations where the Penrose inequality has been \nexploited to derive new results. The list is probably not exhaustive, but it gives a hint on the potential\npower of the Penrose inequality as a geometric tool for adressing, a priori, completely unrelated problems.\n\n\nWe have already mentioned in Subsect. \\ref{HuiskenIlmanen} \nthat the Riemannian Penrose inequality has interesting consequences for the\nquasi-local definition of mass due to Bartnik. The Riemannian Penrose inequality has also allowed\nfor a dual definition of quasi-local mass due to Bray \\cite{Bray2001} (see also \\cite{BrayChrusciel2004}). \nHere, an asymptotically euclidean domain with non-negative curvature scalar and whose boundary $S$ is area outer minimizing\nis kept fixed, and  all possible ``fill in''s (with non-negative curvature scalar) are considered.\nThe {\\it inner mass} is defined as the supremum of $\\sqrt{|S|\/(16\\pi)}$\nwhere $S$ is the minimal area needed to enclose all the asymptotically euclidean ends, except the given one. \nAs a consequence of\nthe Riemannian Penrose inequality, the inner mass is always bounded above by the ADM mass \nof the given region.\nThe definition can also be extended to the case of non-zero second fundamental form.\n\nWe have also mentioned in Section \\ref{hyperbolicSect}\nanother application of the Penrose inequality for the uniqueness problem of\nstatic black holes with negative cosmological constant and topology at infinity of genus larger than one.\n\n\nThe Riemannian Penrose inequality (in fact, its proof using inverse mean curvature flow) has been applied\nrecently to obtain lower bounds of the so-called Brown-York energy for simply connected, compact, three-dimensional\ndomains $(\\Omega,\\gamma)$ with non-negative curvature scalar and with smooth boundary $\\partial \\Omega$ of positive\nGauss curvature and positive mean curvature  $p$ (with respect to the outer direction). The Brown-York mass\nis defined as\n\\begin{eqnarray*}\nM_{BY} (\\partial \\Omega ) = \\frac{1}{8 \\pi} \\int_{\\partial \\Omega} \\left ( p_0 - p \\right ) \\bm{\\eta_{\\partial \\Omega}},\n\\end{eqnarray*}\nwhere $p_0$ is the mean curvature of $\\partial \\Omega$ when this surface is embedded isometrically\nin $\\mathbb{R}^3$. This mass is proven to be non-negative in \\cite{ShiTam2002}. Using the inverse mean curvature flow,\nShi and Tam prove \\cite{ShiTam2007} (among other things)  that the inequality\n$M_{BY} (\\partial \\Omega) \\geq M_{G} (\\partial \\Omega)$ holds with equality if only if $\\Omega$ is a\nstandard ball in $\\mathbb{R}^3$. \n\nStill another\napplication of the Riemann Penrose inequality is due to J. Corvino \\cite{Corvino2005}, who has shown that \nasymptotically euclidean, 3-dimensional, Riemannian manifolds with non-negative curvature scalar and small mass cannot \ncontain any minimal surface and must be diffeomorphic to $\\mathbb{R}^3$. The required  ``small mass''\ncondition reads $2 M_{ADM} \\sqrt{K} \\leq 1$, where $K$ is a positive upper bound for all the sectional\ncurvatures of the manifold. The proof is based on the fact that\nany outermost minimal \nsurface $S$ must satisfy $|S| \\geq \\frac{4 \\pi}{K}$ as a consequence\nof the Gauss-Bonnet theorem. Therefore, the Penrose inequality implies\nthat no minimal surface can exist in these circumstances.\n\n\n\\section {Concluding remarks}\n\\label{concluding}\n\nIn this review I have discussed the present status of \nthe Penrose inequality. The emphasis has been put on trying to describe the\ntechniques involved in the various approaches to prove it and trying to place the results into the right context \nso that a clear picture emerges of how impressive the body of work in this field has already been\nand what are the open problems that still remain. Although I have tried to cover the main results in this\nfield, some aspects have been touched upon in less detail. For instance, I have concentrated mostly\non the four dimensional case, although several results in higher dimensions have been\ndiscussed at various places. Some further results in \nhigher dimensions can be found in the references \n\\cite{IdaMakao}, \\cite{BarrabesFrolov}, \\cite{GibbonsHolzegel}, \\cite{DafermosHolzegel}.\n\nNumerical work has also been important for a better understanding of the Penrose inequality. Outermost\nmarginally outer trapped surfaces (i.e. the boundary of the outer trapped region $\\partial {\\cal T}^{+}_{\\Sigma}$) are\nlocated routinely in numerical black hole evolutions in order to track the boundaries\nof the black holes. The numerical routines to do this job are collectively termed\n{\\it apparent horizon finders} (see \\cite{Thornburg2007} for a review) and obviously they can also be\nused to test the validity of the Penrose conjecture. They have been used to \ncheck  whether the Penrose inequality\nis fulfilled in explicit numerical examples, as well as for looking for counterexamples to\nsome of its versions. They can also serve as a test to make sure that the MOTS being located\nis, in fact, the outermost one \\cite{Jaramillo_Vasset2007}. In this review, numerical results have been\nmentioned only very tangentially. Further details can be found in\n\\cite{Karkowski_Malec_1993}, \\cite{Karkoswski_Koc_1994}, \\cite{Husain1998}, \n\\cite{Dain_et_al_2002}, \\cite{Karkowski-Malec2005},  \\cite{Karkowski2006}, \n\\cite{Jaramillo_Vasset2007},\n\\cite{Jaramillo_Kroon} \nand \\cite{Tippett2009}.\n\nIn a 3+1 evolution of a spacetime the outermost MOTS generates a tube of surfaces which is generally\nsmooth but may jump from time to time \\cite{Mars_et_al_2009}. In the smooth part, this tube has been called\nmarginally outer trapped tube \\cite{Booth2005}. If the foliation is by marginally trapped surfaces\n(instead of MOTS), the tube is usually called \ntrapping horizon \\cite{Hayward1994-2} or dynamical horizon \\cite{AshtekarKrishnan2002} (with slight differences\nin their definitions). A proper study of the evolution of these tubes, specially their late time\nbehaviour, is potentially a powerful method for establishing the Penrose inequality \\cite{Mars_et_al_2005}.\nThis is because it is expected that, at late times,\nthe tube approaches the event horizon of the spacetime. Furthermore, if the MOTS foliating the tube\nare in fact marginally trapped surfaces, then their area increases with time \\cite{AshtekarKrishnan2002}.\nSince this\narea is believed to approach that of the event horizon of the final black hole that forms (and this\nis greater than the initial ADM mass, as usual) the Penrose inequality would follow from a detailed\nunderstanding of the  late time evolution of the spacetime and of the outermost marginally\nouter trapped tube. This approach however, is very different in spirit to the ones discussed above\nbecause understanding the late time behaviour of the tube goes a long way towards establishing\nweak cosmic censorship. Thus, in essence, the Penrose inequality would follow because cosmic censorship\nwould hold. So, instead of looking at the Penrose inequality as an indirect test of cosmic\ncensorship, as originally envisaged by Penrose, it would become a remarkable corollary of a \nmuch stronger theorem \nestablishing weak cosmic censorship (or something very close to it).\nIn any case, studying the evolution of the outermost tube is an active area of research, which combines\nphysical properties, numerical simulations  and rigorous geometric results. The interested\nreader is referred to \\cite{AshtekarKrishnan2003}, \\cite{AshtekarGalloway2005},\n\\cite{Mars_et_al_2005}, \\cite{Andersson2006}, \\cite{Schnetter_et_al2006},\n\\cite{AshtekarKrishnan2007},\n\\cite{Jaramillo_Kroon}, \n\\cite{Gourgoulhon2008}, \n\\cite{Williams2008}, \n\\cite{AMS08}, \n\\cite{Hayward2009}, \n\\cite{Mars_et_al_2009} and references therein.\n\n\nTo conclude, as I have tried to show in this review, the Penrose conjecture is a very challenging problem\nthat requires techniques from geometric analysis, partial differential equations, Riemannian and\nLorentzian geometry, as well as physical intuition. The recent advances in this field have been \nimpressive and our understanding of the problem is now better than ever. Nevertheless, many open\nproblems remain and their study is likely to uncover new and unexpected features in the future.\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\n\nI am indebted to Hugh L. Bray, Alberto Carrasco, Jos\\'e Luis Jaramillo,\nMarkus Khuri, Miguel S\\'anchez, Jos\\'e M.M. Senovilla, Juan\nValiente Kroon and Ra\\\"ul Vera for \nuseful comments on a previous version of this paper.\nFinancial support under projects\nFIS2006-05319 of the Spanish MEC, SA010CO5 of the Junta de Castilla y Le\\'on \nand P06-FQM-01951 of the Junta de Andaluc\\'{\\i}a is gratefully acknowledged.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\subsection{Multi-modality in Machine Learning}\n\nOne of the major issues in machine learning is exploiting multi-modal data for various applications, such as data generation~\\cite{2013_VAE,2014_GAN, 2018_Handpose, 2017_Yoo_Regression}, retrieval~\\cite{2016_retrieval_survey} and recognition~\\cite{2016_Temporal,2011_MultimodalDeep}. \nThere are a lot of studies that extract modality independent features by finding the shared representation of multi-modal data~\\cite{2018_multimodal_survey}.\nThe shared representation is utilized in diverse applications such as handling a missing modality~\\cite{2016_Robotscene,2018_Handpose} or accomplishing better performance than models trained on single-modal data~\\cite{2016_Temporal,2011_MultimodalDeep}.\nThe research related to multi-modality can be categorized into two groups~\\cite{2018_multimodal_survey}.\n\n\nOne is a method mapping data from diverse modalities to the shared latent space.\n\\cite{2018_multimodal_weakly} proposes the extended version of a Variational auto-encoder~\\cite{2013_VAE} which combines distribution parameters from encoders and calculate integrated distribution parameters.\n\\cite{2018_Handpose} also a variant of Variational auto-encoder~\\cite{2013_VAE} for hand pose estimation with multi-modal data. The model proposed by \\cite{2018_Handpose} chooses the input modality and the output modality pair and train the corresponding encoder and decoder pair at every iteration.\n\\cite{2016_Robotscene} trains an auto-encoder that takes RGB images, depth images and semantic images as its network input, then the trained model can a generate complete depth image and semantic image from an RGB image and partial depth and semantic image.\n\\cite{2011_MultimodalDeep} builds a deep-belief network structure that maps audio data and lip images into the common hidden node for audio-visual speech recognition. \n\\cite{2016_Temporal} extends the RBM structure to reflect the sequential characteristic of a speech dataset.\n\n\nThe other group comprises methods that encode the corresponding data to the latent space of each modality but enforce similarity constraints to corresponded latent vectors.\n\\cite{2018_MixandMatch} trains domain specific encoders and decoders, allowing encoders and decoders from different modality to be combined, then, the model is able to generate an unseen data pair by combining the encoders and decoders.\n\\cite{2017_conditional_multimodal_embedding} extracts low-level representation from original data first. Then \\cite{2017_conditional_multimodal_embedding} trains auto-encoders for each modality and enforces similarity constraints to embedding spaces of each auto-encoders for correlated data pair.\nIn \\cite{2013_DeViSE}, a model is trained to maximize the similarity of an image feature and a vectorized label to infer a proper label for a given image.\n\n\n\n\n\\subsection{Associative Learning inspired by the Brain}\n\nThe artificial neural network is an engineering model inspired by the biological mechanism of the brain.\nParameters of those networks are usually updated by Hebbian learning rule where weight connections between firing nodes for input data are strengthened~\\cite{1961_Hebbian}.\nThe Hopfield network and Boltzmann machine are representative examples~\\cite{1985_Boltzmann_Machine}. \nThe Hopfield network models associative memory of human, thus network is trained to memorize specific patterns.\nEven if the input is incomplete, The Hopfield network can restore incomplete data through recurrent iteration.\nThe Boltzmann machine is a stochastic version of the Hopfield network, which can learn a latent representation for input data through its hidden nodes.\n\n\nThere have been many studies that investigate associative learning from the perspective of neuroscience~\\cite{1993_synaptic_memory, 1998_cortical_plasticity, 1999_neurogenesis_mice, 2004_auditory_cortex}.\nIn the recent study which tried to analyze associative learning at the cellular substrate level~\\cite{2018_associative_brain, 2017_associative_brain, 1997_elementary_associative}, they introduce the associative memory cells to describe brain neurons which are mainly involved in integration and storage of associated signals.\nA brain learns associated information by enhancing the strength of the synapses between co-activated associative memory cells activated by associated signals.\nIn this paper, we realize the cross-modal association mechanism recently proposed in \\cite{2018_associative_brain}, which assumes the comprehensive diagram based on the associative memory cells.\n\n\n\n\\subsection{Problem Statements}\n\\label{sec:problem_statement}\n\nAccording to recent studies~\\cite{2017_associative_brain, 2018_associative_brain}, associative learning process in the brain includes intra-modal and cross-modal association processes.\nThe intra-modal association process is to make humans familiar with single-modal sensory information.\nOn the other hand, the cross-modal association process is accomplished to enhance the strength of the synapses connecting multi-modal information to be associated.\nThe goal of this paper is to establish the Bayesian formulation of these two association processes and to realize them in a variational auto-encoder framework.\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{Graphical Model of Cross-Modal Association}\n\n\\input{figure_graphical_model.tex}\n\n\\subsubsection{Intra-Modal Association}\n\nIntra-modal association is the process of memorizing single-domain information.\nTo efficiently memorize a vast amount of information, the model needs to extract the expressive features of the data.\nOne way to make the encoding model remember the features of the data in an unsupervised manner, is to formulate a mathematical model reconstructing the original sensory data from the encoded information.\nFigure~\\ref{fig:graphical structure}.(a) shows Bayesian graphical model to formulate the intra-modal association to memorize a distribution of the latent variable $\\mathbf{z}_i$ associated with the input variable $\\mathbf{x}_i$ for an observation in modality $i$.\nIn the Bayesian framework, the objective is to infer model parameter $\\theta_i$ of posterior distribution $p_{\\theta_i}(\\mathbf{z}_i|\\mathbf{x}_i)$. \n\n\nOne of the most popular approaches to approximate an intractable posterior is the variational inference method.\nIn this method, the variational distribution $q_{\\phi_i}(\\mathbf{z}_i|\\mathbf{x}_i)$ approximates the true posterior $p_{\\theta_i}(\\mathbf{z}_i|\\mathbf{x}_i)$ by minimizing the Kullback-Leibler divergence, $D_{KL} \\left(q_{\\phi_i}(\\mathbf{z}_i|\\mathbf{x}_i) \\| p_{\\theta_i}(\\mathbf{z}_i|\\mathbf{x}_i)\\right)$.\nAccording to \\cite{2013_VAE}, the minimization of $D_{KL} \\left(q_{\\phi_i}(\\mathbf{z}_i|\\mathbf{x}_i) \\| p_{\\theta_i}(\\mathbf{z}_i|\\mathbf{x}_i)\\right)$ can be replaced with the maximization of the evidence lower bound, given by\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:vae_elbo}\n    \\mathcal{L}(q_{\\phi_i}(\\mathbf{z}_{i}|\\mathbf{x}_i)) = &-\n    D_{KL}(q_{\\phi_i}(\\mathbf{z}_i|\\mathbf{x}_i) \\| p_{\\theta_i}(\\mathbf{z}_i))\\\\\n    &+ \\mathbb{E}_{q_{\\phi_i}(\\mathbf{z}_i|\\mathbf{x}_i)}[\\log{p_{\\theta_i}(\\mathbf{x}_i|\\mathbf{z}_i)}], \\\\\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbb{E}_{q_{\\phi_i}(\\mathbf{z}_i|\\mathbf{x}_i)}$ indicates expectation over distribution $q_{\\phi_i}(\\mathbf{z}_i|\\mathbf{x}_i)$.\n\n\n\\input{figure_network.tex}\n\n\\subsubsection{Cross-Modal Association}\n\nIn this section,\nwe design a graphical model to represent the cross-modal association mechanism as in Figure~\\ref{fig:graphical structure}.(b) Without loss of generality, we consider a path from modality $i$ to $j$.\nFrom observations of an associated variable pair $(\\mathbf{x}_i, \\mathbf{x}_j)$, the distribution parameter $\\rho_{ji}$ is inferred to model the association between $\\mathbf{z}_i$ and $\\mathbf{z}_j$.\n \n \nFor a given observation pair $(\\mathbf{x}_i, \\mathbf{x}_j)$,\nthe cross-posterior distribution $p_{\\theta_{i}, \\rho_{ji}}(\\mathbf{z}_j|\\mathbf{x}_i)$ is defined by marginalization for $\\mathbf{z}_i$ as \n\\begin{equation}\n\\begin{aligned}\n\tp_{\\theta_{i}, \\rho_{ji}}(\\mathbf{z}_{j}|\\mathbf{x}_i) = \\int p_{\\rho_{ji}}(\\mathbf{z}_j|\\mathbf{z}_i) p_{\\theta_{i}}(\\mathbf{z}_i|\\mathbf{x}_i) \\: d\\mathbf{z}_i.\n\\end{aligned}\n\\label{eq:cross_posterior}\n\\end{equation}\nTo establish the cross-modal association model, we define a variational distribution for cross-posterior distribution\n$q_{\\phi_i, \\rho_{ji}}(\\mathbf{z}_{j}|\\mathbf{x}_i)$.\nThen, to infer the distribution parameters $(\\phi_i, \\rho_{ji})$, we minimize Kullback-Leibler divergence between $p_{\\theta_j}(\\mathbf{z}_j|\\mathbf{x}_j)$ and $q_{\\phi_i, \\rho_{ji}}(\\mathbf{z}_{j}|\\mathbf{x}_i)$. To avoid clutter, subscripts for the distribution parameters are omitted in the remainders of this section.\nKullback-Leibler divergence between $p(\\mathbf{z}_j|\\mathbf{x}_j)$ and $q(\\mathbf{z}_{j}|\\mathbf{x}_i)$ is given by\n\\begin{equation}\n\tD_{KL}(q(\\mathbf{z}_j|\\mathbf{x}_i) \\| p(\\mathbf{z}_j|\\mathbf{x}_j)) \n    = \\log{p(\\mathbf{x}_j)} - \\mathcal{L}(q(\\mathbf{z}_{j}|\\mathbf{x}_i)), \n\\end{equation}\nwhere\n\\begin{equation}\n    \\mathcal{L}(q(\\mathbf{z}_{j}|\\mathbf{x}_i)) = \\int q(\\mathbf{z}_{j}|\\mathbf{x}_i) \\log{\\frac{p(\\mathbf{x}_j)p(\\mathbf{z}_j|\\mathbf{x}_j)}{q(\\mathbf{z}_{j}|\\mathbf{x}_i)}} \\: d\\mathbf{z}_j.\n\\label{eq:cross1}\n\\end{equation}\nSince log-evidence $\\log{p(\\mathbf{x}_j)}$ is independent to the model parameter, the target problem is identical to maximizing the evidence lower bound $\\mathcal{L}(q(\\mathbf{z}_{j}|\\mathbf{x}_i))$.\nWith probabilistic tricks, $\\mathcal{L}(q(\\mathbf{z}_{j}|\\mathbf{x}_i))$ can be decomposed as following.\n\\begin{equation}\n\\begin{aligned}\n\t\\mathcal{L}(q(\\mathbf{z}_{j}|\\mathbf{x}_i)) =\n    &-D_{KL}(q(\\mathbf{z}_{j}|\\mathbf{x}_i) \\| p(\\mathbf{z}_j))\\\\\n    &+ \\mathbb{E}_{q(\\mathbf{z}_{j}|\\mathbf{x}_i)}[\\log{p(\\mathbf{x}_j|\\mathbf{z}_j)}].\n\\end{aligned}\n\\label{eq:cross2}\n\\end{equation}\nThe detail derivation is given in Appendix A of supplementary document.\nIn Eq.~\\eqref{eq:cross2}, the first term is a negative KL divergence term that leads $\\mathbf{z}_j$ given by $\\mathbf{x}_j$ to have similar distribution with a prior distribution of target modality.\nThe expectation term in Eq.~\\eqref{eq:cross2} minimizes the reconstruction error of decoded output from $\\mathbf{z}_j$ fired from $\\mathbf{x}_i$, which also promotes the inference for $\\rho_{ji}$.\nBy the similar steps, we can easily derive the opposite association from modality $j$ to modality $i$.\n\n\n\n\n\\subsection{Realization: Cross-Modal Association Network}\n\nWe accomplish a realization of the aforementioned intra-modal and cross-modal association models by extending the Variational Auto-Encoder framework (VAE)~\\cite{2013_VAE}.\nFigure~\\ref{fig:network} illustrates the proposed cross-modal association network for modality $i$ and $j$.\nAlthough only two modalities are considered in this paper, the proposed model can be applied to the association among three or more modalities also.\nIn the proposed structure, the {\\it encoder} produces the parameter of $q_{\\phi_i}(\\mathbf{z}_i|\\mathbf{x}_i={x}_i)$, and the {\\it  decoder} produces the parameter of $p_{\\theta_i}(\\mathbf{x}_i|\\mathbf{z}_i={z}_i)$. The encoder and decoder are realized by deep neural networks.\nLikewise, the latent space associating models $p_{\\rho_{ji}}(\\mathbf{z}_j|\\mathbf{z}_i=z_{i})$ and $p_{\\rho_{ij}}(\\mathbf{z}_i|\\mathbf{z}_j=z_{j})$ are also realized by deep neural networks, which are called by \\textit{associator}.\nThus, the intra-modal association network contains several auto-encoders, each of which considers one of the multiple modalities only.\nThe latent spaces of the auto-encoders are connected by {\\it associators} in a pairwise manner, which configure the cross-modal association network.\n\n\nThe proposed network is trained in the two phases: intra-modal training phase and cross-modal training phase. \nIn the intra-modal training phase, the auto-encoder in each modality is trained separately by minimizing the approximated version of the negative evidence lower bound in Eq.~\\eqref{eq:vae_elbo}~\\cite{2013_VAE}.\nAs derived in \\cite{2013_VAE}, variational distributions are assumed by the centered isotropic multivariate Gaussian distribution.\nFor a given observation sample $x_i$, the encoder $E_i$ produces the mean $\\mu_{\\phi_i}$ and the variance $\\sigma_{\\phi_i}$ for a Gaussian distribution of $q_{\\phi_i}(\\mathbf{z}_i|\\mathbf{x}_i=x_i)$. \nThen, the latent vector $z_i$ is\nsampled as $z_i=\\mu_{\\phi_i} + \\sigma_{\\phi_i}*\\epsilon$ and $\\epsilon \\backsim N(0, I)$.\nSimilarly, the decoder $D_i$ also produces the mean $\\mu_{\\theta_i}$ and the variance $\\sigma_{\\theta_i}$ for a Gaussian distribution of $p_{\\theta_i}(\\mathbf{x}_i|\\mathbf{z}_i=z_i)$. \nThen, the reconstruction vector $\\hat{x}_i$ is\nsampled as $\\hat{x}_i=\\mu_{\\theta_i} +   \\sigma_{\\theta_i}*\\epsilon$ and $\\epsilon \\backsim N(0, I)$. \n\n\nUsing the samples, the empirical loss for auto-encoder can be derived as\n\\begin{equation}\n\\label{eq:intra_loss}\n\\begin{aligned}\n\t\\mathcal{L}_{int}&(\\theta_i, \\phi_i; x_i) = -\\mathbb{E}_{q_{\\phi_i}(\\mathbf{z}_i|x_i)}[\\log{p_{\\theta_i}(x_i|\\mathbf{z}_i)}] \\\\\n    &+ \\lambda^{'}_{int} D_{KL}(q_{\\phi_i}(\\mathbf{z}_i|x_i)||p_{\\theta_i}(\\mathbf{z}_i)),\\\\\n   &=|| x_i-\\hat{x}_i||^2_2\\\\\n    &- \\lambda_{int} \\sum_k^H (1+\\log \\sigma_{\\phi_i(k)}^2 -\\mu_{\\phi_i(k)}^2 - \\sigma_{\\phi_i(k)}^2).\n\\end{aligned}\n\\end{equation}\nwhere $\\lambda_{int}$ is a user-defined parameter and $H$ is the dimension of the latent variable $\\mathbf{z}_i$.\n$\\mu_{\\phi_i(k)}$ and $ \\sigma_{\\phi_i(k)}^2$ denote the $k$-th element of $\\mu_{\\phi_i}$ and $ \\sigma_{\\phi_i}^2$.\nThe detail derivation presents in Appendix B of supplementary document.\n\n\nAfter the convergence of the intra-modal training phase, the following cross-modal training phase proceeds to train the associators while freezing the weights of the auto-encoders.\nIn the same way as in the intra-modal training phase, for a given observation pair $x_i$ and $x_j$, \nthe encoders $E_i$ and $E_j$ produce the latent vectors $z_i$ and $z_j$, respectively. \nIn addition, associators $A_{ji}$ and $A_{ij}$ produce the latent vectors $z_{ji}$ and $z_{ij}$ using inputs $z_{i}$ and $z_{j}$, respectively. Thereafter, the decoders $D_i$ and $D_j$ produce\nthe reconstruction vectors $\\hat{x}_{ij}$ and $\\hat{x}_{ji}$ from $z_{ij}$ and $z_{ji}$, respectively.\n\n\nUsing the samples, the empirical loss for $A_{ji}$ is designed according to Eq.~\\eqref{eq:cross2} as follows:\n\\begin{equation} \n\\label{eq:cross_loss}\n\\begin{aligned}\n\t\\mathcal{L}_{crs}&(\\rho_{ji};x_i, x_j) = \n    -\\mathbb{E}_{q_{\\rho_{ji}}(\\mathbf{z}_{ji}|x_i)}[\\log{p_{\\theta_{j}}(x_j|\\mathbf{z}_{ji})}] \\\\\n    &+ \\lambda^{'}_{crs} D_{KL}(q_{\\phi_i,\\rho_{ji}}(\\mathbf{z}_{ji}|x_i)||p_{\\theta_{j}}(\\mathbf{z}_{ji}))\\\\\n    &=|| x_j-\\hat{x}_{ji}||^2_2\\\\\n    &- \\lambda_{crs} \\sum_k^H (1+\\log \\sigma_{\\rho_{ji}(k)}^2 -\\mu_{\\rho_{ji}(k)}^2 - \\sigma_{\\rho_{ji}(k)}^2).\n\\end{aligned}\n\\end{equation}\nwhere $\\lambda_{crs}$ is a user-defined parameter and $H$ is the dimension of the latent variable $\\mathbf{z}_{ji}$.\n($\\mu_{\\rho_{ji}}$, $ \\sigma_{\\rho_{ji}}^2$) are parameters for Gaussian distribution $q_{\\phi_i,\\rho_{ji}}(\\mathbf{z}_{ji}|x_i)$  produced by $A_{ji}$.\nThe detail derivation presents in Appendix B of supplementary document.\n\n\nThe loss $\\mathcal{L}_{crs}(\\rho_{ij};x_i, x_j)$ for $A_{ij}$ is given in the same form except the index. Note that all $\\mu$'s and $\\sigma$'s in Eq.~\\eqref{eq:intra_loss} and Eq.~\\eqref{eq:cross_loss} are the functions of weights ($\\mathbf{w}$) in encoders, decoders, or associators. Hence, the weights of the proposed network are trained by the negative direction of the gradient of the losses with respect to the weights ($\\nabla_{\\mathbf{w}}\\mathcal{L}(\\cdot)$).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Advantages of Proposed Method}\n\nOwing to the newly introduced {\\it associator}, the proposed model can associate heterogeneous modalities effectively.\nReckless coalescence of heterogeneous data may have a fatal impact on \nassociative learning such as the problem that\nshared latent vectors can be biased to the dominant modality.\nHowever, in our model, the associator acts as a translator\nbetween heterogeneous modalities and thus the characteristics of each latent space are preserved.\nFurthermore, in contrast to the existing models which adopt a shared latent space for the different modalities~\\cite{2018_multimodal_weakly, 2018_Handpose,2016_Robotscene, 2011_MultimodalDeep}, our structure can provide a flexible dimensional encoding in each latent space depending on the complexity of each modality.\nThis provides better cross-modal data association results.\n\n\nThe proposed model easily incorporates additional modalities while maintaining the existing modalities.\nThat is, a new modality can be added via training of only a new associator between an existing auto-encoder and a new auto-encoder.\nThough the associator only associates the new modality with one of the existing modalities, the model can associate the new modality with the rest of the modality by passing through multiple associators.\n\n\nFinally, in contrast to existing models which always require paired data for cross-modal association, our structure can train the associator with the only small amount of paired data in a semi-supervised manner after learning each auto-encoder using unpaired data independently. Since obtaining paired data for cross-modal association is more expensive than obtaining unpaired data, our model is cost-effective. Furthermore, our model is plausible in that, \nwhen a person learns a cross-modal association, the paired examples are rarely given by a teacher after the person has become familiar with each modality via self-experience without a teacher. \n\n\nIn the following experiment section, we validate the aforementioned advantages of the proposed structure.\n\n\n\\subsection{Datasets}\n\\noindent\\textbf{Google Speech Commands (GSC)~\\cite{2018_GSC_dataset}}:\nAs the data for the auditory modality, we used the GSC dataset, which consists of 105,829 audio samples containing utterances of 35 short words.\nEach audio sample is one-second-long and encoded with a sampling rate of 16KHz.\nAmong 35 words, we chose 14 words, including words for each digit ('ZERO' to 'NINE') and four traffic commands ('GO,' 'STOP,' 'LEFT,' 'RIGHT').\nThe chosen set has 54,239 samples.\nWe extracted the Mel-Frequency Cepstral Coefficient (MFCC) from each audio clip to generate an audio feature.\nMFCC has been widely used in the processing of voice data because it reflects the human auditory perception mechanism well~\\cite{2010_MFCC_voicerecognition,2000_MFCC_music,2016_MFCC_heartsound}.\nThe resulting features are $40\\times101$ matrices.\nWe randomly divided the original dataset into training, validation and test sets at the ratio of 8:1:1.\n\n\n\\vspace{0.3cm}\n\\noindent\\textbf{MNIST~\\cite{1998_Lecun_MNIST}}:\nWe used the MNIST dataset as the corresponding visual data to the GSC for each digit.\nThe MNIST consists of center-aligned $28 \\times 28$ gray-scale images for handwritten digits from 0 to 9. The dataset contains 60k and 10k samples for the training set and testing set, respectively.\n\n\n\\vspace{0.3cm}\n\\noindent\\textbf{Fashion-MNIST (F-MNIST)~\\cite{2017_FashionMNIST}}:\nWe used the F-MNIST dataset as another visual modality which has a similar specification to MNIST dataset.\nThe F-MNIST consists of center-aligned $28 \\times 28$ gray-scale images assigned with a label from 10 kinds of clothing such as T-shirt, Trouser and Sneaker.\nThe dataset contains 60k and 10k samples for the training set and testing set, respectively.\n\n\n\\vspace{0.3cm}\n\\noindent\\textbf{German Traffic Sign Recognition Benchmark (GTSRB)~\\cite{2011_German_traffic}}:\nFor the visual data that correspond to the traffic commands in GSC, we used the GTSRB dataset, which consists of 51,839 RGB color images illustrating 42 kinds of traffic signals.\nIn particular, to evaluate the performance on pairs of traffic sign images and voice commands in GSC, we chose four pair sets, where each pair set has  similar semantic meaning, i.e., ('Ahead only,' 'GO'), ('No entry for vehicle,' 'STOP'), ('Turn left and ahead,' 'LEFT'), and ('Turn right and ahead,' 'RIGHT').\nThen, to prevent the four signs from occupying the entire latent space, we chose additional sign images such as 'No overtaking,' 'Entry to 30kph zone,' 'Prohibit overweighted vehicle,' 'No-waiting zone,' and 'Roundabout'.\nThe chosen set includes 10,709 samples.\nAll of the chosen signs have a circular backboard.\nThe size of each image varies from $15\\times15$ to $250\\times250$ pixels for each RGB channel in the original dataset.\nIn our experiments, we resized all images into $52\\times52$.\n\n\n\\input{tables\/cross_results}\n\\input{tables\/table_recog_vae.tex}\n\\subsection{Evaluation Metric}\n\nSince datasets in Section.4.1 have no direct matching relationships, we cannot measure cross-likelihood $p(\\mathbf{x}_1|\\mathbf{x}_2)$ for paired sample $(\\mathbf{x}_1, \\mathbf{x}_2)$ used in \\cite{2018_multimodal_weakly, 2016_joint_multimodal}.\nIn our work, we used the reconstruction accuracy as the evaluation metric of the association models. The quality of an image reconstructed by an association model can be a valid measure to evaluate the association model since the quality of the reconstructed image is acceptable to both the human and recognition model.\nThe reconstruction accuracy was measured by our own recognition networks trained with the original data used in our experiments. \nTable~\\ref{tabular:recog_vae} shows the performance of the recognition networks trained with the original data, which shows sufficient performance for evaluating the reconstructed output of the compared encoders by using the recognition models.\nFor the GSC dataset, we get performance comparable to the 88.2\\% in \\cite{2018_GSC_dataset}. \n\n\n\n\\subsection{Intra-Modal Association}\n\nAs mentioned in section~\\ref{sec:problem_statement}, it is essential for the cross-modal association to learn the intra-modal association that encodes single modal input data into the latent space.\nFor a fair comparison to existing works, we trained encoders and decoders for each dataset with the fixed dimension of latent space (dim$(\\mathbf{z})=64$).\nIn addition, to show the advantages of the proposed model where the dimension of the latent space can be flexibly designed according to the complexity of target modality, we trained additional auto-encoder whose latent space dimension is $256$ for GTSRB dataset.\n\n\nTable~\\ref{tabular:recog_vae} shows the reconstruction performance of the intra-modal association network implemented by VAE. As shown in the table, the voice data in the GSC dataset shows much degraded accuracy, which means that the voice data are hard to be reconstructed than other modalities.\nSince F-MNIST has confused classes such as pullover, coat and shirt, performance on F-MNIST dataset is also degraded.\n\n\n\\input{figure_experiment.tex}\n\n\\subsection{Cross-Modal Association}\nWe evaluated the proposed model on four scenarios: (1) Association between F-MNIST and MNIST, (2) MNIST and GSC, (3) F-MNIST and GSC, (4) GTSRB and GSC.\nScenario (1) is for association between datasets which have similar characteristics to each other.\nScenario (2) and (3) are for association between heterogeneous datasets, i.e. voice and image datasets.\nScenario (4) is a more practical case than the others.\nIn order to train cross-modal association, we used randomly paired training samples from each dataset belonging to the correlated class.\nFor example, we paired a randomly chosen sample in '0' class of MNIST dataset with a randomly chosen sample in 'ZERO' class of GSC dataset.\n\n\n\nTo evaluate the proposed associator, the following methods were compared:\n\\textbf{VAE} and \\textbf{VAE-CG} are variants of the standard VAE~\\cite{2013_VAE}. The direct concatenation of paired data is given to VAE as an input. \nTo allow VAE to acquire cross-generation capability, \\textbf{VAE-CG} is trained to generate the target modality sample via associator for given input sample from other modality.\nThis model has to be trained only by supervised data with input and output pairs.\nJoint Multimodal Variational Auto-Encoder (\\textbf{JMVAE})~\\cite{2016_joint_multimodal} has two kinds of latent spaces: one is for each modality and the other is for jointly encoding of two modalities. The joint latent space is shared for association between two modalities. The training for encoding in the joint latent space is done to  minimize Kullback-Leibler divergence between the latent vector of each encoder and the joint latent vector of the joint encoder. In comparison, the hyper-parameter $\\alpha$ was set to 0.01 for whole scenarios.\nCross-modal Variational Auto-Encoder (\\textbf{CVA})~\\cite{2018_Handpose} is an extension of VAE for cross-modal data. \nIn CVA, the latent space is shared between two modalities. In the training process, the selected sample pair are trained alternately throughout iteration.\nMultimodal Variational Auto-Encoder (\\textbf{MVAE})~\\cite{2018_multimodal_weakly} is also a variant of VAE for cross-modal data. MVAE uses the standard VAE for each modality, but each latent space is associated via a shared latent space expressing the unified distribution of the association modalities. \nWe trained MVAE by using the sub-sampled training paradigm presented in their paper.\n\n\nTo evaluate the flexibility of encoding dimension in our model, we have conducted an experiment where each modality is encoded in a different dimensional space from the other.\n\\textbf{ours-flex} has large dimension of latent space for GTSRB dataset ($dim(\\mathbf{z})=256$).\nExcept for ours-flex, \nall compared models use the same VAE of which the latent space dimension is 64.\n\n\nTable~\\ref{tabular:cross} shows the evaluation result of the proposed model and the compared models for the cross-modal association.\nThe proposed model accomplishes significant enhancement from the compared algorithms for most of the scenarios.\nInterestingly, in the challenging scenarios such as the association between heterogeneous modalities, for instance, between audio (GSC) and visual data (MNIST, GTSRB), the proposed model achieves a remarkable improvement compared to the existing models.\n\n\nFigure~\\ref{fig:experiment} shows the qualitative results of our model for images generated from GSC dataset.\nFigure~\\ref{fig:experiment} (a) and (b) show 3 generated images for each `number' command of GSC.\nFigure~\\ref{fig:experiment} (c) shows 5 generated images for each `traffic command' of GSC.\nThe proposed model successfully generates images with correct semantics though images converge to similar shape due to the random pairing between training samples.\n\n\n\n\n\n\\input{tables\/handpose_rebuttal.tex}\n\\subsection{Application: Hand pose estimation}\n\nWe have conducted additional experiments for hand pose estimation on Rendered Hand pose Dataset (RHD)~\\cite{2017_RHD_dataset}. \nRHD dataset provides $320 \\times 320$ RGB image, depth map, segmentation map and 21 keypoints for each hand.\nWe consider the case of generating 3D keypoints from the RGB image.\nEvaluation metric is the average End-Point-Error (EPE), which measures euclidean distance between ground truth keypoints and estimated keypoints, as CVA~\\cite{2018_Handpose} did.\nWe used the same encoders and decoders structure to CVA and added the associator.\nTable~\\ref{tabular:handpose} and Figure~\\ref{fig:handpose} show that our model outperforms previous works~\\cite{2018_Handpose, 2017_RHD_dataset}.\n\\input{figure_handpose.tex}\n\n\n\n\n\\subsection{Semi-supervised Learning}\nWe conducted an additional experiment to verify the effectiveness of the proposed associator in semi-supervised learning.\nFigure~\\ref{fig:semi_supervised} illustrates a trend of performance variation of the proposed associator depending on the proportion of paired data, from 100\\% to 1\\% in the three scenarios including F-MNIST $\\rightarrow$ MNIST, MNIST $\\rightarrow$ GSC and F-MNIST $\\rightarrow$ GSC case.\nThe result shows that the proposed associator can achieve good performance with only a small proportion of paired data (5\\%) in a semi-supervised manner.\n\\input{figure_semi_supervised.tex}\n\n\n\n\n\\subsection{Scalability}\n\nThe proposed structure can easily expand a new modality while maintaining the existing modalities. That is, a new modality can be added via training of only a new associator between an existing auto-encoder and the new auto-encoder.\nSince the associator connect only two latent spaces, if the existing network associates $N$ modality, $N$ associators need to be trained.\nIn our model, this inefficiency can be mitigated by cascading association through multiple associators.\nFor example, suppose that MNIST and F-MNIST datasets are connected by an associator, and GSC and F-MNIST are also connected by an associator. Then even if there is no direct associator between GSC and MNIST, the association between them can be made by the cascading of the two existing associators.\nTable~\\ref{tabular:scale} compares the results of cascading association and direct association.\nAlthough the cascading association has some performance degradation, it still has good performance compared to other algorithms presented in Table~\\ref{tabular:cross}.\n\\input{tables\/scalability_rebuttal.tex}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nPlease follow the steps outlined below when submitting your manuscript to\nthe IEEE Computer Society Press.  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All authors will benefit from reading\nMermin's description of how to write mathematics:\n\\url{http:\/\/www.pamitc.org\/documents\/mermin.pdf}.\n\n\n\\subsection{Blind review}\n\nMany authors misunderstand the concept of anonymizing for blind\nreview.  Blind review does not mean that one must remove\ncitations to one's own work---in fact it is often impossible to\nreview a paper unless the previous citations are known and\navailable.\n\nBlind review means that you do not use the words ``my'' or ``our''\nwhen citing previous work.  That is all.  (But see below for\ntechreports.)\n\nSaying ``this builds on the work of Lucy Smith [1]'' does not say\nthat you are Lucy Smith; it says that you are building on her\nwork.  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Why the previous paper\n   was accepted without this analysis is beyond me.\n\n   [1] Smith, L and Jones, C. ``The frobnicatable foo\n   filter, a fundamental contribution to human knowledge''.\n   Nature 381(12), 1-213.\n\\end{quote}\n\nIf you are making a submission to another conference at the same time,\nwhich covers similar or overlapping material, you may need to refer to that\nsubmission in order to explain the differences, just as you would if you\nhad previously published related work.  In such cases, include the\nanonymized parallel submission~\\cite{Authors14} as additional material and\ncite it as\n\\begin{quote}\n[1] Authors. ``The frobnicatable foo filter'', F\\&G 2014 Submission ID 324,\nSupplied as additional material {\\tt fg324.pdf}.\n\\end{quote}\n\nFinally, you may feel you need to tell the reader that more details can be\nfound elsewhere, and refer them to a technical report.  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Don't write ``We show how to\nimprove our previous work [Anonymous, 1968].  This time we tested the\nalgorithm on a lunar lander [name of lander removed for blind review]''.\nThat would be silly, and would immediately identify the authors. Instead\nwrite the following:\n\\begin{quotation}\n\\noindent\n   We describe a system for zero-g frobnication.  This\n   system is new because it handles the following cases:\n   A, B.  Previous systems [Zeus et al. 1968] didn't\n   handle case B properly.  Ours handles it by including\n   a foo term in the bar integral.\n\n   ...\n\n   The proposed system was integrated with the Apollo\n   lunar lander, and went all the way to the moon, don't\n   you know.  It displayed the following behaviours\n   which show how well we solved cases A and B: ...\n\\end{quotation}\nAs you can see, the above text follows standard scientific convention,\nreads better than the first version, and does not explicitly name you as\nthe authors.  A reviewer might think it likely that the new paper was\nwritten by Zeus \\etal, but cannot make any decision based on that guess.\nHe or she would have to be sure that no other authors could have been\ncontracted to solve problem B.\n\n\\noindent\nFAQ\\medskip\\\\\n{\\bf Q:} Are acknowledgements OK?\\\\\n{\\bf A:} No.  Leave them for the final copy.\\medskip\\\\\n{\\bf Q:} How do I cite my results reported in open challenges?\n{\\bf A:} To conform with the double blind review policy, you can report results of other challenge participants together with your results in your paper. For your results, however, you should not identify yourself and should not mention your participation in the challenge. Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\\medskip\\\\\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\fbox{\\rule{0pt}{2in} \\rule{0.9\\linewidth}{0pt}}\n  \n\\end{center}\n   \\caption{Example of caption.  It is set in Roman so that mathematics\n   (always set in Roman: $B \\sin A = A \\sin B$) may be included without an\n   ugly clash.}\n\\label{fig:long}\n\\label{fig:onecol}\n\\end{figure}\n\n\\subsection{Miscellaneous}\n\n\\noindent\nCompare the following:\\\\\n\\begin{tabular}{ll}\n \\verb'$conf_a$' &  $conf_a$ \\\\\n \\verb'$\\mathit{conf}_a$' & $\\mathit{conf}_a$\n\\end{tabular}\\\\\nSee The \\TeX book, p165.\n\nThe space after \\eg, meaning ``for example'', should not be a\nsentence-ending space. So \\eg is correct, {\\em e.g.} is not.  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The second and following pages should begin 1.0 inch (2.54 cm) from\nthe top edge. On all pages, the bottom margin should be 1-1\/8 inches (2.86\ncm) from the bottom edge of the page for $8.5 \\times 11$-inch paper; for A4\npaper, approximately 1-5\/8 inches (4.13 cm) from the bottom edge of the\npage.\n\n\\subsection{Margins and page numbering}\n\nAll printed material, including text, illustrations, and charts, must be kept\nwithin a print area 6-7\/8 inches (17.5 cm) wide by 8-7\/8 inches (22.54 cm)\nhigh.\n\n\n\n\\subsection{Type-style and fonts}\n\nWherever Times is specified, Times Roman may also be used. If neither is\navailable on your word processor, please use the font closest in\nappearance to Times to which you have access.\n\nMAIN TITLE. Center the title 1-3\/8 inches (3.49 cm) from the top edge of\nthe first page. 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Short captions should be centred.\n\n\\noindent Callouts should be 9-point Helvetica, non-boldface type.\nInitially capitalize only the first word of section titles and first-,\nsecond-, and third-order headings.\n\nFIRST-ORDER HEADINGS. (For example, {\\large \\bf 1. Introduction})\nshould be Times 12-point boldface, initially capitalized, flush left,\nwith one blank line before, and one blank line after.\n\nSECOND-ORDER HEADINGS. (For example, { \\bf 1.1. Database elements})\nshould be Times 11-point boldface, initially capitalized, flush left,\nwith one blank line before, and one after. If you require a third-order\nheading (we discourage it), use 10-point Times, boldface, initially\ncapitalized, flush left, preceded by one blank line, followed by a period\nand your text on the same line.\n\n\\subsection{Footnotes}\n\nPlease use footnotes\\footnote {This is what a footnote looks like.  It\noften distracts the reader from the main flow of the argument.} sparingly.\nIndeed, try to avoid footnotes altogether and include necessary peripheral\nobservations in\nthe text (within parentheses, if you prefer, as in this sentence).  If you\nwish to use a footnote, place it at the bottom of the column on the page on\nwhich it is referenced. Use Times 8-point type, single-spaced.\n\n\n\\subsection{References}\n\nList and number all bibliographical references in 9-point Times,\nsingle-spaced, at the end of your paper. When referenced in the text,\nenclose the citation number in square brackets, for\nexample~\\cite{Authors14}.  Where appropriate, include the name(s) of\neditors of referenced books.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|}\n\\hline\nMethod & Frobnability \\\\\n\\hline\\hline\nTheirs & Frumpy \\\\\nYours & Frobbly \\\\\nOurs & Makes one's heart Frob\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Results.   Ours is better.}\n\\end{table}\n\n\\subsection{Illustrations, graphs, and photographs}\n\nAll graphics should be centered.  Please ensure that any point you wish to\nmake is resolvable in a printed copy of the paper.  Resize fonts in figures\nto match the font in the body text, and choose line widths which render\neffectively in print.  Many readers (and reviewers), even of an electronic\ncopy, will choose to print your paper in order to read it.  You cannot\ninsist that they do otherwise, and therefore must not assume that they can\nzoom in to see tiny details on a graphic.\n\nWhen placing figures in \\LaTeX, it's almost always best to use\n\\verb+\\includegraphics+, and to specify the  figure width as a multiple of\nthe line width as in the example below\n{\\small\\begin{verbatim}\n   \\usepackage[dvips]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]\n                   {myfile.eps}\n\\end{verbatim}\n}\n\n\n\\subsection{Color}\n\nPlease refer to the author guidelines on the ICCV 2019 web page for a discussion\nof the use of color in your document.\n\n\\section{Final copy}\n\nYou must include your signed IEEE copyright release form when you submit\nyour finished paper. We MUST have this form before your paper can be\npublished in the proceedings.\n\n{\\small\n\\bibliographystyle{ieee}\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\input{1-Introduction_new.tex}\n\n\\section{Related Works}\n\\input{2-Related}\n\n\\section{Method}\n\\input{3-1-Method_intro}\n\\input{3-2-Method_bayes}\n\\input{3-3-Method_network}\n\\input{3-4-Method_compare}\n\n\\section{Experiment}\n\\input{4-Experiment}\n\n\\section{Conclusion}\n\\input{5-Conclusion}\n\n{\\small\n\\bibliographystyle{ieee}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe sixth-generation (6G) wireless networks are expected to support ubiquitous connectivity to a wide range of mobile terminals, spanning from autonomous cars to unmanned aerial vehicles (UAV), low-earth-orbit (LEO) satellites, and high speed-trains, etc.\nOne of the critical challenges for these services is to provide reliable communications in high-mobility environments.\nAdditionally, the spectrum congestion under 6 GHz creates a fundamental bottleneck for capacity improvement and sustainable system evolution.\nThe ultra-high data rate requirements of panoramic and holographic video streaming push mobile providers to utilize higher frequency bands, such as the millimeter wave (mmWave) bands, where a huge chunk of unused spectrum is available.\nIn general, wireless communications in high-mobility scenarios at high carrier frequencies are extremely challenging due to the hostile channel variations. \nRelying on adaptive coherent\/non-coherent detection \\cite{XuChaoCoherentNocoherent} is beneficial for attaining a certain degree of robustness against channel variations.\nNevertheless, recently an increasing amount of research attention has been dedicated to designing new modulation waveform and schemes for high-mobility communications of next-generation wireless networks.\n\nHigh-mobility communications operating at high carrier frequencies suffer from severe Doppler spreads, mainly caused by the relative motion between the transmitter, receiver, and scatterers.\nConventional orthogonal frequency-division multiplexing (OFDM) modulation, which has been widely used in both the fourth-generation (4G) and the emerging fifth-generation (5G) cellular systems as well as in WiFi networks, suffers in high-mobility scenarios.\nOFDM waveform is impaired by severe inter-carrier interference (ICI), which is aggravated by the fact that the highest and lowest subcarriers exhibit rather different normalized Doppler.\nHence, synchronization is also a challenge.\nRecently, a new two-dimensional (2D) modulation scheme, namely orthogonal time-frequency space (OTFS)\\cite{Hadani2018otfs}, has been proposed as a promising candidate for high-mobility communications.\n\n\n\n\nOTFS modulates information in the delay-Doppler (DD) domain rather than in the time-frequency (TF) domain of classic OFDM modulation, providing a strong delay- and Doppler-resilience, whilst enjoying the potential of \\textit{full diversity} \\cite{Hadani2018otfs}, which is the key for supporting reliable communications.\nAdditionally, OTFS modulation can transform a time-variant channel into a 2D quasi-time-invariant channel in the DD domain, where its attractive properties can be exploited.\nGiven that most of the existing wireless system designs have been conceived for low-mobility and low-carrier scenarios, OTFS introduces new critical challenges  in transceiver architecture and algorithmic designs.\nTo unleash the full potentials of OTFS, challenging fundamental research problems have to be addressed, including channel estimation, detection, as well as multi-antenna and multi-user designs.\n\n\n{In contrast to existing tutorial papers on OTFS \\cite{Hadani2018otfs,ramachandran2020otfs}, this article portrays OTFS modulation conceived for communications over high-mobility environments by providing an easy-reading overview of its fundamental concepts, highlighting the challenges and potential solutions as well as exploring new promising areas for future research.}\nThe rest of this article is organized as follows.\nThe next section introduces the fundamentals of OTFS.\nThe potential applications and research opportunities of OTFS are presented in Section III.\nThen, before concluding, we demonstrate the most important design challenges of OTFS and their potential solutions.\n\n\\section{Fundamentals of OTFS}\nA fundamental question to answer for motivating the research community and industry to investigate OTFS is why we shall perform modulation in the DD domain.\nHence, commencing from the channel characteristics of high-mobility communications, we present the basic concepts and properties of the DD domain channels, the DD domain multiplexing, the OTFS transceiver architecture and signal waveform, as well as the OTFS system design principle.\n\n\\subsection{From Time-Invariant to Time-Variant Channels}\nWireless channels can be modeled by a linear time-invariant (LTI) system, provided that the channel impulse response (CIR) is \\textit{time-invariant} or has a long coherence time, cf. Fig. \\ref{DoublySelectiveChannels}.\nIn the presence of multiple scatterers, the dispersive LTI channel's output is a temporal-smeared version of the transmitted signal, but again, the CIR is {time-invariant}.\nIn this case, a one-dimensional (1D) CIR in the delay domain $h\\left(\\tau\\right)$ is sufficient for characterizing the \\textit{time-dispersive} channel.\nThe Fourier transform (FT) of this CIR is a \\textit{frequency-selective} channel transfer function (CTF).\nWith increasing the CIR delay spread, the selectivity becomes more severe since the separation of frequency-domain (FD) fades is increasingly proportional to the CIR-length.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.3in]{Figures\/DoublySelectiveChannels4.pdf}\n\t\\caption{An illustration of frequency-selective, time-selective, and doubly-selective channel models.}\n\t\\label{DoublySelectiveChannels}\n\\end{figure}\n\nHowever, the assumption of having LTI CIRs may no longer hold in the face of increased user mobility and carrier frequency.\nTherefore, the linear \\textit{time-variant} (LTV) channel model \\cite{hlawatsch2011wireless} has attracted considerable research attention in high-mobility scenarios.\nLTV channels give rise to frequency shifts due to the Doppler effect, yielding a spectral-smeared version of the transmitted signal, i.e., \\textit{frequency-dispersive}.\nFrequency-dispersive channels are \\textit{time-selective} and the separation of the channel's time-domain (TD) fades is increasingly proportional to the Doppler spread.\nIn practice, LTV channels of high-mobility scenarios are often \\textit{doubly-dispersive} due to the joint presence of multipath propagation and Doppler effects, cf. Fig. \\ref{DoublySelectiveChannels}.\nThe transmitted signals suffer from dispersion both in the TD and FD.\nIn such scenarios, each tap of the CIR function is time-dependent, fluctuating according to the rate of $\\frac{\\lambda}{2v}$ between consecutive TD fades, cf. Fig. \\ref{TDDomainChannel}, where $\\lambda$ denotes the wavelength and $v$ is the relative speed between the transmitter and receiver.\nHence, this results in a 2D CIR function $h\\left(t, \\tau\\right)$ in the time-delay domain.\nIn contrast to the traditional way of treating TD and FD dispersion as undesired channel impairments, we can beneficially exploit the additional degrees of freedom (DoF) of doubly-dispersive channels for achieving reliable diversity-aided communications in high-mobility scenarios.\n\n\n\\subsection{LTV Channels in TF and DD Domains}\n\\begin{figure*}[!t]\n\t\\centering\n\t\\includegraphics[width=6.5in]{Figures\/Channels_Domains.pdf}\n\t\\caption{LTV channels in the time-delay, TF, and DD domains.}%\n\t\\label{TDDomainChannel}%\n\\end{figure*}\n\t\nApart from the time-delay domain, LTV channels can be equivalently described in either the TF or DD domain, cf. Fig. \\ref{TDDomainChannel}.\nTo emphasize the TF selectivity, the TF domain channel, $H\\left(t, f\\right)$, can be obtained by the FT of $h\\left(t, \\tau\\right)$ with respect to (w.r.t.)  delay $\\tau$.\nNote that $H\\left(t, f\\right)$ can be interpreted as the complex CTF coefficient at time instant $t$ and frequency $f$.\nDue to the limited coherence time and coherence bandwidth (coherence region in Fig. \\ref{TDDomainChannel}) of LTV channels, channel acquisition in the TF domain would be challenging and would impose a significant signaling overhead.\nFor instance, for an OFDM system having a carrier frequency of $f_c = 3.5$ GHz and a subcarrier spacing of $\\Delta f = 15$ kHz supporting a relative velocity of $v = 300$ km\/h, the maximum Doppler shift is $\\nu_{\\mathrm{max}} = 972.22$ Hz and the OFDM symbol duration including a 20\\% cyclic prefix (CP) is $80 \\;\\mu$s.\nAssuming that the channel's coherence time is $\\frac{1}{4\\nu_{\\mathrm{max}}} = 257.14 \\;\\mu$s, one channel coherence interval can only accommodate at most 3 OFDM symbols.\n\n\nApplying the FT to $h\\left(t, \\tau\\right)$ w.r.t. $t$ yields the DD domain channel (spreading function), $h\\left(\\tau,\\nu\\right)$.\nThe DD domain channel $h\\left(\\tau,\\nu\\right)$ characterizes the intensity of scatterers having a propagation delay of $\\tau$ and Doppler frequency shift of $\\nu$, which directly captures the underlying physics of radio propagation in high-mobility environments.\nMore importantly, the LTV channel in the DD domain exhibits beneficial features of separability, stability, compactness, and possibly sparsity, as illustrated in Fig. \\ref{TDDomainChannel} and detailed below, which can be exploited to facilitate efficient channel estimation and data detection.\n\\begin{itemize}\n\t\\item \\textbf{Separability}: Additionally introducing the Doppler domain of wireless channels allows us to separate the propagation paths experiencing an identical delay and thus fully discloses the available channel DoF.\n\t%\n\n\t\n\t\\item \\textbf{Stability}: Only the drastic change of propagation path lengths and moving speeds may cause channel variations in the DD domain.\n\t%\n\tConsequently, DD domain channels fluctuate much slower than time-delay domain or TF domain channels.\n\t\n\t\\item \\textbf{Compactness}: It is worth noting that in typical wireless channels, we have $4\\tau_{\\mathrm{max}}\\nu_{\\mathrm{max}} \\le 1$ \\cite{hlawatsch2011wireless}, where $\\tau_{\\mathrm{max}}$ indicates the maximum delay.\n\t%\n\tHence, $h\\left(\\tau,\\nu\\right)$ has a compact DD domain support within the intervals $\\left[0,\\tau_{\\mathrm{max}}\\right]$ and  $\\left[-\\nu_{\\mathrm{max}},\\nu_{\\mathrm{max}}\\right]$ along the delay and Doppler dimensions, respectively.\n\t\n\t\\item \\textbf{Potential sparsity}: When an open-space rural propagation environment having few moving scatterers is considered, the DD domain channel exhibits a sparse response\\cite{ShenOTFSCS}.\n\t%\n\n\\end{itemize}\n\n\nNote that the above discussions are mainly related to the deterministic description of LTV channels.\nMore details on the stochastic characterization of LTV channels can be found in \\cite{hlawatsch2011wireless,SteeleMobileChannel}.\nParameterizing channel with the aid of delay and Doppler is not new, which has been widely adopted in the areas of radar and sonar.\nThe main benefit of the OTFS waveform is the DD domain multiplexing, which will be detailed in the next section.\n\n\n\\subsection{From TF to DD Domain Multiplexing}\n\n\\begin{figure*}[!th]\n\t\\centering\n\t\\includegraphics[width=6.8in]{Figures\/GridV5.pdf}\n\t\\caption{OTFS transceiver architecture, the concept of DD domain data multiplexing and their coupling with DD domain channels.}%\n\t\\label{DDDomainMultiplexing}%\n\\end{figure*}\n\nThe classic modulation techniques typically multiplex data in the TD or FD, which are briefly introduced as follows.\n\\begin{itemize}\n\t\\item \\textbf{Time-division multiplexing (TDM)}: TDM carries information (QAM) symbols in unique, user-specific time-slots, within the same frequency band.\n\t\\item \\textbf{Frequency-division multiplexing (FDM)}: FDM multiplexes streams or users in dedicated FD slots occupied at the same time.\n\t\\item \\textbf{Code-division multiple access (CDMA)}: The information symbols (of different users) are carried over unique (user-specific) single-carrier TD, multi-carrier FD or multi-carrier time-frequency-domain spreading sequences.\n\t\\item \\textbf{OFDM}: OFDM transmits its information symbols by overlapping sinc-shaped orthogonal subcarriers in parallel.\n\t\\item \\textbf{Index modulation (IM)}: IM relies on the generalized ON\/OFF keying principle to map the information bits to the indices of spatial-, time- and frequency-domain resources.\n\\end{itemize}\nHowever, these traditional modulation techniques may not work well in the face of severe Doppler spread.\nFor instance, the popular OFDM modulation efficiently transforms a frequency-selective fading channel to multiple parallel frequency-flat subchannels, which allows a low-complexity single-tap FD equalization for transmission over LTI channels.\nHowever, the orthogonality of OFDM waveform erodes in high-mobility scenarios when conventional wireless transceivers are adopted.\n\n\n{In contrast to the existing 1D TD or FD modulation techniques, OTFS is a 2D modulation technique, where the data symbols are multiplexed in the DD domain and each symbol is spread right across the entire TF domain.\nThis property is desirable for attaining the maximum achievable diversity for transmission over doubly dispersive channels, provided that each TD and FD sample experiences independent fading.\nThe maximum attainable diversity order is determined by the number of independently fading resolvable paths in the DD domain.\nTo fully exploit the maximum attainable diversity order, advanced detection techniques are required that can efficiently combine the information conveyed by the different propagation paths to the OTFS receiver.\nAlthough the asymptotic diversity gain of the uncoded OTFS system can be as low as one in some special cases as revealed in \\cite{SurabhiDiversityAnalysisBER}, the probability of occurrence of these cases tends to zero for a moderate DD domain grid size \\cite{RavitejaEffectiveDiversity}.\nFor example, it becomes as low as $10^{(-150)}$ for a DD domain grid of $M = N =16$ \\cite{RavitejaEffectiveDiversity}.\nFurthermore, channel coding is capable of avoiding the occurrence of these cases and thus OTFS can achieve the full diversity order almost surely.}\n\n\n{As a benefit of DD domain multiplexing, the information symbols are carried by localized pulses \\cite{Hadani2018otfs,Mohammed2020derivation} of the DD domain.\nBy increasing the time duration and bandwidth of the transmission, the information-bearing pulses can be further localized, cf. Fig. \\ref{DDDomainMultiplexing}, demonstrating the orthogonality of the OTFS waveforms associated with different DD grid points.\nThis potentially perfect pulse localization capability experienced in the DD domain stands in sharp contrast to the properties of its dual pair, namely the TF domain, where the Heisenberg uncertainty principle prevents corresponding pulse localization.\nThe above-mentioned pulse-localization allows OTFS to enjoy the aforementioned DD domain channel properties.\n\nMoreover, adopting 2D localized pulses for conveying the information in the DD domain is beneficial for exploiting the underlying physical propagation properties. \nWe note that conventional wireless communication designs treat channel fading as an inevitable deleterious effect, aiming for combating or exploiting it while ignoring its basic underlying causes.\nIn more detail, the channel impairments of propagation delay and Doppler frequency shifts are modeled as a pair of operations imposed by wireless channels on the transmitted waveform. \nFurthermore, fading is viewed as a phenomenon that manifests itself at the channel's output imposed by the combined destructive effect of this pair of fundamental channel-induced phenomena.\nIn contrast, OTFS generates a complete orthogonal family of waveforms, which is closed under arbitrary combinations of the time delay and frequency shift.\nIn other words, upon transmitting a signal in this family, the received signal will remain within the family under arbitrary channel impairments. \nThe mathematical structure underlying this unique characteristic of the OTFS waveform family is the quasi-periodicity property of the DD domain signal representation, as discussed in \\cite{Hadani2018otfs}. \nThis property gives rise to a 2D (quasi-) circular localized inter-symbol interference (ISI) pattern in the DD domain, representing the channel impairments, cf. Fig. \\ref{DDDomainMultiplexing}. \nSince the time- and frequency-shifts are separated in the DD domain, the destructive effects of fading are substantially mitigated.}\n\n\n\\subsection{OTFS Transceiver Architecture and Waveform}\n\n\n\n\n\n\n\nThe OTFS transceiver is shown in Fig. \\ref{DDDomainMultiplexing}, where the modulated (pilot) symbols are firstly mapped to the DD domain.\nThen, an orthogonal 2D precoding, such as the inverse symplectic finite Fourier transform (ISFFT) and Walsh-Hadamard transform\\cite{ZemenOrthogonalPrecoding}, transplants the DD domain signal into the TF domain.\nThen, a multi-carrier modulator, such as OFDM or filterbank multicarrier (FBMC) modulator, is employed in each time slot for further transforming the TF domain signal to the TD before being transmitted over the channel. \nAt the receiver side, a cascaded combination of multi-carrier demodulation and 2D orthogonal decoding transforms the received signal back into the DD domain and retrieves the transmitted symbols in the DD domain using an appropriate channel estimator \\cite{RavitejaOTFSCE} and equalizer\\cite{RavitejaOTFS}. \nWe can observe that the OTFS transceiver can be implemented based on the conventional OFDM architecture by adding some pre-processing and post-processing blocks, thus making OTFS attractive from the perspective of implementation.\n{Nevertheless, as a block modulation scheme, OTFS systems suffer from a higher latency than classic OFDM systems.}\n\n{As illustrated in Fig. \\ref{DDDomainMultiplexing}, a single pulse representing a symbol at $\\left(\\tau,\\nu\\right)$ in the DD domain will be spread across the whole TF domain.\nThe resultant TD waveform is the fluctuating pulse train seen in Fig. \\ref{DDDomainMultiplexing}, where the fluctuation rate is determined by the Doppler frequency $\\nu$, while the pulse location within each time slot is determined by the delay $\\tau$.\nConsequently, the TD waveform of OTFS behaves locally like TDMA (localized pulses in the TD), globally like OFDM (localized pulses in the Doppler domain) and spreading like CDMA (2D spreading in the DD domain), thus inheriting their beneficial properties.\nFor example, OTFS exhibits resilience to narrow-band interference and it is eminently suitable for multi-user scenarios.\nAdditionally, OTFS transforms the violently fluctuating TF domain channel into a quasi-time-invariant channel in the DD domain, which can potentially be exploited for striking a compelling trade-off between the performance, computational complexity and signaling overhead.}\n\n\nApart from its potential of exploiting full diversity and Doppler-resilience, OTFS also has some further benefits over conventional modulation techniques.\nFor example, despite its multicarrier nature, the peak-to-average power ratio (PAPR) of OTFS is much lower than that of both OFDM and generalized frequency division multiplexing (GFDM), which is particularly beneficial for power-limited systems, such as the Internet-of-Things (IoT).\nAdditionally, the guard intervals are only required between consecutive OTFS frames, rather than between time slots and thus the associated idle time is significantly reduced.\nFurthermore, as a benefit of its Doppler-resilience, OTFS is more robust against the carrier frequency offset than OFDM.\nThese advantages render OTFS eminently suitable for high-mobility and high-carrier scenarios.\n\n\\subsection{OTFS System Design Principle}\n{According to Fig. \\ref{DDDomainMultiplexing}, the data rate of OTFS systems is determined by the amount of data symbols accommodated within a single OTFS frame, with its largest possible value being $MN$.\nSince a single OTFS frame occupies a time duration of $NT$ and a bandwidth of $M\\Delta f$, the spectral efficiency of OTFS systems is $(1-\\eta)R_{\\mathrm{c}}\\log_2\\mathcal{M} $ bit\/s\/Hz, where $R_{\\mathrm{c}}$ is the code rate, $\\mathcal{M}$ is the modulation order, and $\\eta$ captures the training overhead as well as the guard interval overhead.\nThe guard interval in the TD has to be longer than the channel's delay spread to avoid interference between OTFS frames.\nFurthermore, the code rate $R_{\\mathrm{c}}$ and the modulation order $\\mathcal{M}$ have to be carefully selected to guarantee communication reliability.\nAs such, a key technique of improving the spectral efficiency of OTFS systems is that of reducing the training overhead, as shown in Section IV-A.\nOn the other hand, the OTFS system parameters $M$, $N$, $\\Delta f$, and $T$ have to be chosen appropriately for its practical implementations.\nIn particular, the OTFS frame duration $NT$ has to be smaller than the tolerable application latency.\nFurthermore, as the computational complexity and PAPR of OTFS modulation are proportional to $N$ \\cite{RavitejaOTFS}, we prefer to choose a smaller $N$ with a longer slot duration $T$.\nMeanwhile, the available spectrum is granulated into more subcarriers, i.e. a higher $M$ is associated with a smaller subcarrier spacing $\\Delta f = \\frac{1}{T}$.}\n\n\n\\section{Potential Applications and Opportunities}\nOTFS modulation is envisioned to have diversified applications in next-generation wireless networks, as discussed in the following.\n\n\\subsection{Vehicular Networks}\nVehicular networks allow various vehicles to wirelessly exchange information with each other or with roadside units (RSUs) to provide a variety of benefits, including cooperative traffic management, road-safety improvements and the support of autonomous driving.\nOTFS can play a key role in future vehicular networks owing to its intrinsic advantages for communications over high-mobility channels.\nThe current standards of vehicular communications, such as IEEE 802.11bd and 5G NR V2X, mainly consider OFDM-based waveforms, where the impact of channel variations is mitigated either by inserting a midamble or by increasing the subcarrier spacing.\nBy contrast, the potential channel stability of OTFS exhibited in the DD domain enables prompt initial link setup, agile sidelink scheduling, as well as predictive resource scheduling.\nFurthermore, the OTFS waveform enjoys a lower PAPR than OFDM, allowing a better communication coverage for vehicular networks.\n\n\\subsection{Millimeter-wave Communications}\nThe millimeter-wave frequency band possesses a large amount of under-utilized spectrum and has the potential of offering giga-bit-per-second communication services in future wireless networks.\nThe Doppler effect becomes more severe upon increasing the carrier frequency even at a low\/medium velocity.\nAlthough increasing the subcarrier spacing to mitigate the resultant ICI is feasible, the TD symbol duration will be shorter and inserting a CP for guarding against ISI will introduce a significant overhead.\nThe excessive phase noise associated with high-frequency oscillators also results in a time-varying composite channel.\nOTFS provides a strong immunity to the oscillator phase noise, which is crucial for mmWave communications.\n\n\\subsection{Non-Terrestrial Networks}\nNon-terrestrial networks (NTN) provide a new telecommunication infrastructure based on airborne or spaceborne vehicles, such as satellites, UAVs or high altitude platforms (HAPs).\nThey are capable of supporting the terrestrial 5G networks in the provision of global coverage and mobility, as well as ubiquitous connectivity and enhanced network reliability.\nSince the airborne and spaceborne vehicles usually move fast, the high Doppler spread experienced by the NTN imposes new challenges on its air interface design.\nOTFS modulation has rich potentials in the NTN owing to its prominent capability of handling the Doppler effect.\nAdditionally, airborne and spaceborne vehicles have limited on-board power supply and computing capability, hence the low PAPR and low-complexity of OTFS are of pivotal importance.\nMoreover, the corresponding NTN communication links spanning to the ground terminals usually exhibit spatial channel sparsity in the DD domain, which allows OTFS to strike an attractive performance vs. complexity trade-off.\n\n\\subsection{Underwater Acoustic Communications}\nUnderwater acoustic (UWA) channels are regarded as one of the most challenging wireless channels, due to their high delay spread, limited bandwidth, and rapid time variations.\nSingle-carrier modulation using decision feedback equalizers (DFE), OFDM and orthogonal signal division multiplexing (OSDM) are the most popular schemes for UWA communications.\nHowever, they all transmit information in the TF domain, where both the ISI and ICI equalization become tedious tasks.\nBy contrast, OTFS is Doppler-resilient, hence transmitting information in the DD domain may outperform these TF domain modulation schemes in UWA channels.\nFurthermore, UWA channels tend to be sparse in the DD domain, where the equalization might be easier than that in the TF domain.\nMoreover, UWA communications are usually considered as wideband systems, since the ratio of acoustic signal bandwidth over the carrier frequency is typically much higher than that in terrestrial communications.\nHence, the potential multipath-scale diversity of the DD domain channel \\cite{hlawatsch2011wireless} can be beneficially exploited.\n\n\n\n\\section{Challenges and Solutions}\nAs a fledgling waveform, OTFS modulation unveils new opportunities but also has its own challenges.\nIn this section, we introduce three fundamental research problems of OTFS and their potential solutions.\n\n\n\\subsection{Channel Estimation}\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=3.5in]{Figures\/MSE_OTFS_V2.pdf}\n\t\\caption{Performance comparison of the DD and TF domain channel estimation with different training overheads. The number of paths in the DD domain is 5, $N = 32$, $M = 32$, $\\nu_{\\mathrm{max}}$ = $\\frac{2}{NT}$, and $\\tau_{\\mathrm{max}}$ = $\\frac{4}{M\\Delta f}$.}%\n\t\\label{CEOTFS}%\n\\end{figure}\n\nThe channel envelope fluctuates violently even in a short time period in high-mobility environments.\nAccurately estimating the CIR in OTFS systems is a challenging but vital requirement for reliable detection.\nThanks to the DD domain channel sparsity and quasi-stationarity, channel acquisition in the DD domain is more convenient than that in the TF domain, even for a lower training overhead, cf. Fig. \\ref{CEOTFS}.\nHowever, the DD domain channel may not always be sparse\\cite{WeiOTFS}, particularly in the presence of fractional Doppler\\cite{RavitejaOTFS}.\nFor example, when the exact Doppler frequency straddles a pair of finite-resolution bins in Fig. \\ref{DDDomainMultiplexing}, rather than falling exactly into the $1$-th bin, the DD domain channel is spread across all\nthe Doppler indices.\nDue to this channel spreading, a much larger guard space is needed around the pilot symbols to avoid the interference caused by unknown data symbols for channel estimation, which imposes a significant training overhead.\n\nA promising solution to address this issue is enhancing the channel sparsity via designing a bespoke TF domain window.\nIn particular, we proposed to apply a Dolph-Chebyshev (DC) window at the OTFS transmitter or receiver \\cite{WeiOTFS} to suppress the channel spreading.\nDue to the enhanced channel sparsity, the DC windowing achieves a much improved channel estimation accuracy over the conventional rectangular window \\cite{RavitejaOTFSCE}, cf. Fig. \\ref{CEOTFS}.\n{To further reduce the training overhead, we propose to use decision-directed channel estimation (DDCE), where the reliably detected data symbols can be used for refining channel estimates based on the classic decision-directed principle, rather than purely relying on the known pilot symbols.\n\t%\n\tThe refined channel estimates can again be used for OTFS symbol detection, which can in turn improve the channel estimation accuracy.\n\t%\n\tFig. \\ref{CEOTFS} demonstrates that the proposed DDCE scheme without guard space consumes only $0.1\\%$ training overhead while can achieve an order of magnitude reduction in channel estimation error than that of its conventional counterpart.}\n\n\n\\subsection{Efficient DD Domain Data Detection}\nThe output signal in the DD domain can be regarded as a 2D circular convolution of the input data symbols and the effective aggregate channel, cf. Fig \\ref{DDDomainMultiplexing}, which results in a rather specific interference pattern, where a pair of symbols far from each other in the DD domain may interfere with each other.\nMitigating this peculiar interference requires a bespoke receiver.\nAdopting the optimal maximum a posteriori (MAP) detector would indeed perfectly mitigate the interference between symbols, but at an excessive complexity, precluding its deployment in practical systems. \nHence, most OTFS detectors focused on the complexity reduction, based on the classic message passing algorithm (MPA) and its variants. \nThe main problem of MPA-based detection is its poor convergence behavior in the face of short cycles, which may lead to performance degradation.\n\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=3.5in]{Figures\/DetectionWithDifferentSpeeds.pdf}\n\t\\caption{Performance comparison of OTFS with different detectors, moving speeds, and waveforms. The number of paths in the DD domain is 4, $N = 8$, $M = 16$, $\\nu_{\\mathrm{max}}$ = $\\frac{3}{NT}$ for the velocity of $150$ km\/h, $\\nu_{\\mathrm{max}}$ = $\\frac{6}{NT}$ for the velocity of $300$ km\/h, and $\\tau_{\\mathrm{max}}$ = $\\frac{3}{M\\Delta f}$.}%\n\t\\label{Detection}%\n\\end{figure}\n\nA potent solution is to adopt the variational framework of\\cite{WeijieOTFS}, which can adaptively construct the distributions of OTFS symbols according to their interference patterns. \nBy appropriately constructing the distributions of OTFS symbols for variational purposes, we can design rapidly converging OTFS detection.\nOwing to its better convergence, the variational Bayes (VB) OTFS detector can achieve a modest performance gain over the MPA detector, cf. Fig. \\ref{Detection}.\nThe performance of MAP detection is also provided as the baseline, which has the best performance, albeit at the cost of an excessive complexity. \n{Another potential solution is the so-called cross-domain iterative detection (CDID), where a conventional linear minimum mean squared error (L-MMSE) estimator is adopted for equalization in the time domain and low-complexity symbol-by-symbol detection is utilized in the DD domain, while the extrinsic information is iteratively exchanged between the time domain and DD domain via the corresponding unitary transformation.\nThe proposed CDID detector is also capable of exploiting the time domain channel sparsity and the DD domain symbol constellation constraints, which can achieve a close-to-optimal performance at a much reduced computational complexity, cf. Fig. \\ref{Detection}.}\nFor all these detectors, the OTFS performance remains similar upon increasing the velocity from 150 km\/h to 300 km\/h.\nIn contrast, the detection performance of the MMSE OFDM detector remains poor due to the excessive ICI.\n\n\n\n\n\n\n\\subsection{Coded OTFS System}\n\n\nWhile OTFS has the potential of attaining the maximum achievable diversity gain, the channel codes have to be carefully designed for OTFS modulation. \nMoreover, perfect detection at near-capacity signal-to-noise ratios (SNRs) may not be attained for practical OTFS systems due to the associated poor channel conditions. \nHence, the channel decoder has to cope with the OTFS detector's residual errors, which would require iterative OTFS receivers.\nHowever, how to design such a receiver and how to choose the coding parameters for near-capacity joint detection and decoding remains an interesting open issue.}\nRecent research has unveiled a fundamental trade-off between the diversity gain and coding gain, which may shed light on code design of OTFS systems \\cite{ShuangyangOTFS}.\nIn particular, the diversity gain of OTFS systems improves with the number of independent resolvable channel paths in the DD domain, while the coding gain declines, cf. Fig. \\ref{Code_OTFS}.\n{This is not unexpected as the transmitted signal energy is distributed to multiple paths, as discussed in \\cite{ShuangyangOTFS}.}\nMoreover, both the coded and uncoded OTFS systems outperform the corresponding OFDM systems, which clearly shows the advantage of OTFS modulation. \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=3.5in]{Figures\/Code_design_pic1.pdf}\n\t\\caption{Performance comparison of coded\/uncoded OTFS and OFDM modulations, $N = 8$, $M = 16$, $\\nu_{\\mathrm{max}}$ = $\\frac{3}{NT}$, and $\\tau_{\\mathrm{max}}$ = $\\frac{5}{M\\Delta f}$.}%\n\t\\label{Code_OTFS}%\n\\end{figure}\n\n\n\n\n\n\n\\section{Outlook for OTFS}\nOTFS has great potential in providing reliable communications for high-mobility applications and its open facets are expected to stimulate new research, as discussed below.\n\n\n\\subsection{FDD or TDD?}\nAs demonstrated by Cohere \\cite{Hadani2018otfs}, the DD domain channels are capable of reaching a coherence time of 100 ms and a coherence bandwidth of 100 MHz.\nHence, accurate channel reciprocity holds in the DD domain and thus time-division duplexing (TDD) is a good option for OTFS.\nBesides, due to the significantly increased coherence bandwidth of the DD domain channel, we can potentially infer the downlink channel from the uplink channel even for frequency-division duplexing (FDD) OTFS systems, which significantly reduces the channel feedback overhead.\nHowever, the key is to find an accurate deterministic or statistical mapping between the uplink and downlink channels.\nFurthermore, it is also important to conduct a systematic comparative study of TDD and FDD-aided OTFS systems.\n\n\n\\subsection{Scalable Multiple Access Schemes for OTFS}\nHow to support a multiplicity of users in high-mobility environments is a very challenging issue.\nOTFS offers the opportunity to accommodate multiple users in the DD domain, where employing carefully designed user scheduling and guard spaces has the potential of avoiding multi-user interference.\nHowever, how to scale the systems for accommodating a large number of users without a significant overhead is an interesting open research problem.\nThe coexistence of promising multiple access schemes and OTFS, such as non-orthogonal multiple access, spatial-division multiple access and interleave-division multiple access, is worth further exploring.\n\n\\subsection{Joint Sensing and Communications using OTFS}\nSince the DD domain channel directly exploits the physics of propagation, relying on the distance, speed and scattering intensity, OTFS is eminently suitable for integrating sensing and communications solutions in a single platform.\nEfficient sensing algorithms to exploit the OTFS signal structure are still unknown.\nFinding the optimal trade-off between the sensing and communication performances remains an interesting open question.\nMoreover, as location and velocity can serve as beneficial side information for improving communication performance, sensing-based communications relying on OTFS is an exciting open topic to investigate.\n\n\\subsection{MIMO-OTFS}\nApplying OTFS in multi-antenna systems provides additional hitherto unexploited spatial DoF for multiplexing.\nIn contrast to TF domain channels, which may fluctuate dynamically for different antennas in different time slots and subcarriers, the DD domain channels tend to remain quasi-stationary both in the time and antenna domains, which may result in an efficient channel estimation and multi-input multiple-input multiple-output (MIMO) detection.\nHow to design sophisticated beamforming\/precoding to fully exploit all the available spatial DoFs and how to perform low-complexity detection for MIMO-OTFS constitute intriguing problems.\nMoreover, the analytical framework of MIMO-OTFS system performance versus the number of antennas is also unexplored in the open literature.\n\n\\subsection{Index Modulation for OTFS}\n{OTFS modulation maps the classic modulated symbols to the DD domain and spreads them across the whole TF domain grid for transmission.\nTherefore, IM can be carried out in the TF domain for improving the spectral efficiency, while slightly sacrificing the transmission diversity order.\nIn particular, the additional information bits can be mapped to the ON\/OFF states of the TF grid indices.\nAdditionally, we can also extend this to the spatial dimension, such as antenna indices or propagation paths, for incorporating IM into OTFS systems, where the information bits are transmitted by active antenna or path indices.\nThe intriguing error rate performance vs.  complexity and spectral efficiency trade-offs of IM-OTFS systems remain to be investigated.\nFurthermore, the design of efficient detectors for jointly demodulating the additional information bits and conventional constellation symbols of IM-OTFS systems is also a critical open research challenge.}\n\n\\section{Conclusions}\nIn a nutshell, OTFS constitutes a promising next-generation candidate.\nWe commenced with an overview of the fundamental concept of OTFS, including the main features of the DD domain channel, the DD domain multiplexing and OTFS transceiver architecture.\nThe critical challenges of OTFS, such as channel estimation, efficient data detection and coding\/decoding problems were highlighted and pertinent preliminary results were provided.\nThe potential applications of OTFS and several promising research directions were introduced.\nIt is hoped that this article will help inspire future research in this exciting new area and pave the way for designing next-generation networks.\n\n\\section{Acknowledgement}\nThe authors would like to thank the support from Telstra Corporation Ltd., particularly Dr. Paul G. Fitzpatrick, Dr. Taka Sakurai, and Mr. Paul Sporton for valuable discussions during this work.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nThank you for downloading this Scribner eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Scribner and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\n## Contents\n\n\"The Tradition\" by Jericho Brown\n\nIntroduction by Jesmyn Ward\n\nPART I: LEGACY\n\nHomegoing, AD by Kima Jones\n\nThe Weight by Rachel Kaadzi Ghansah\n\nLonely in America by Wendy S. Walters\n\nWhere Do We Go from Here? by Isabel Wilkerson\n\n\"The Dear Pledges of Our Love\": A Defense of Phillis Wheatley's Husband by Honor\u00e9e Fanonne Jeffers\n\nWhite Rage by Carol Anderson\n\nCracking the Code by Jesmyn Ward\n\nPART II: RECKONING\n\nQueries of Unrest by Clint Smith\n\nBlacker Than Thou by Kevin Young\n\nDa Art of Storytellin' (a Prequel) by Kiese Laymon\n\nBlack and Blue by Garnette Cadogan\n\nThe Condition of Black Life Is One of Mourning by Claudia Rankine\n\nKnow Your Rights! by Emily Raboteau\n\nComposite Pops by Mitchell S. Jackson\n\nPART III: JUBILEE\n\nTheories of Time and Space by Natasha Trethewey\n\nThis Far: Notes on Love and Revolution by Daniel Jos\u00e9 Older\n\nMessage to My Daughters by Edwidge Danticat\n\nAcknowledgments\n\nContributors\n\nPermissions\n\nAbout the Editor\nTo Trayvon Martin and the many other black men, women, and children who have died and been denied justice for these last four hundred years\n\n## The Tradition\n\n## JERICHO BROWN\n\nAster. Nasturtium. Delphinium. We thought\n\nFingers in dirt meant it was our dirt, learning\n\nNames in heat, in elements classical\n\nPhilosophers said could change us. Star Gazer.\n\nFoxglove. Summer seemed to bloom against the will\n\nOf the sun, which news reports claimed flamed hotter\n\nOn this planet than when our dead fathers\n\nWiped sweat from their necks. Cosmos. Baby's Breath.\n\nMen like me and my brothers filmed what we\n\nPlanted for proof we existed before\n\nToo late, sped the video to see blossoms\n\nBrought in seconds, colors you expect in poems\n\nWhere the world ends, everything cut down.\n\nJohn Crawford. Eric Garner. Mike Brown.\n\n## Introduction\n\n## JESMYN WARD\n\nAfter George Zimmerman shot and killed Trayvon Martin on February 26, 2012, I took to Twitter. I didn't have anywhere else to go. I wanted to hear what others, black writers and activists, were thinking about what happened in Sanford, Florida. Twitter seemed like a great social forum, a virtual curia, a place designed to give us endless voice in declarations of 140 characters or fewer.\n\nI found the community I sought there. I found so many people giving voice to my frustration, my anger, and my fear. We shared news and updates and photos, anything we could find about Trayvon. During that time, I was pregnant, and I was revising a memoir about five young black men I'd grown up with, who all died young, violent deaths. Every time I logged in or read another article about Trayvon, my unborn child and my dead brother and my friends sat with me. I imagined them all around me, our faces long with dread. Before Zimmerman was acquitted of second-degree murder and manslaughter in July 2013, I suspected Trayvon's death would be excused. During this period, I returned often to the photo of Trayvon wearing a pale hoodie. As I gazed on his face\u2014his jaw a thin blade, his eyes dark and serious, too big in the way that children's eyes are\u2014I saw a child. And it seemed that no one outside of Black Twitter was saying this: I read article after article that others shared on Twitter, and no major news outlet was stating the obvious. Trayvon Martin was a seventeen-year-old child, legally and biologically; George Zimmerman was an adult. An adult shot and killed a child while the child was walking home from a convenience store where he'd purchased Skittles and a cold drink. Everything, from Zimmerman stalking and shooting Trayvon to the way Trayvon was tried in the court of public opinion after his death, seemed insane. How could anyone look at Trayvon's baby face and not see a child? And not feel an innate desire to protect, to cherish? How?\n\nAnd then I realized most Americans did not see Trayvon Martin as I did. Trayvon's sable skin and his wide nose and his tightly coiled hair signaled something quite different for others. Zimmerman and the jury and the media outlets who questioned his character with declarations like He abused marijuana and He was disciplined at school for graffiti and possessing drug paraphernalia saw Trayvon as nothing more than a wayward thug. They didn't see him as an adult human being, either, but as some kind of ravenous hoodlum, perpetually at the mercy of his animalistic instincts. Although this was never stated explicitly, his marijuana use and adolescent mischief earned this hoodlum in a hoodie his death.\n\nI knew that myth. It was as familiar to me as my own eyes, my own nose, my own hair, my own fragile chest. It was as familiar to me as the air I grew up in, air as dense and heavy and close and hot as the air Trayvon breathed before Zimmerman shot him. I, too, grew up in a place that could sometimes feel as limiting and final as being locked in an airtight closet, the air humid and rank with one's own breath and panic. A place where for all the brilliant, sun-drenched summer days, there is sometimes only the absence of light: America, and the American South. A place where the old myths still hold a special place in many white hearts: the rebel flag, Confederate monuments, lovingly restored plantations, Gone with the Wind. A place where black people were bred and understood to be animals, a place where some feel that the Fourteenth Amendment and Brown v. Board of Education are only the more recent in a series of unfortunate events. A place where black life has been systematically devalued for hundreds of years.\n\nIn December 2002, my then senator, Trent Lott, attended a function honoring the outgoing Senator Strom Thurmond, who is famous for opposing the Civil Rights Act of 1957 so strenuously he conducted the longest lone filibuster ever, one that lasted twenty-four hours and eighteen minutes. At this event, Lott, who is from a small town on the Mississippi Gulf Coast around twenty-five miles from mine, said: \"We're proud of it [voting for Thurmond in the 1948 presidential election]. And if the rest of the country had followed our lead, we wouldn't have had all these problems over all these years, either.\" It was dismaying to hear this, to see what those in power thought of people like me, but it wasn't a surprise. After all, when I participated in Presidential Classroom in Washington, D.C., I, along with around five of my high school classmates, met Senator Trent Lott. My schoolmates were white. I was not. Trent Lott took a whip as long as a car off his office table, where it lay coiled and shiny brown, and said to my one male schoolmate who grinned at Lott enthusiastically: Let's show 'em how us good old boys do it. And then he swung that whip through the air and cracked it above our heads, again and again. I remember the experience in my bones.\n\nI know little. But I know what a good portion of Americans think of my worth. Their disdain takes form. In my head, it is my dark twin. Sometimes I wonder which of us will be remembered if I die soon, if I suffocate in that closet. Will I be a vicious menace, like Trayvon Martin? An unhinged menace, like Tamir Rice? A monstrous menace, like Mike Brown? An unreasonable menace, like Sandra Bland? A sly menace, like Emmett Till? I imagine I will be as black and fetid as the horde at Scarlett's heels, crowding her wagon, thundering to rip it apart, wheel by rivet.\n\nReplace ropes with bullets. Hound dogs with German shepherds. A gray uniform with a bulletproof vest. Nothing is new.\n\n\u2022 \u2022 \u2022\n\nI needed words. The ephemera of Twitter, the way the voices of the outraged public rose and sank so quickly, flitting from topic to topic, disappointed me. I wanted to hold these words to my chest, take comfort in the fact that others were angry, others were agitating for justice, others could not get Trayvon's baby face out of their heads. But I could not. The nature of the application, even the nature of the quality journalism of the time, with so much of it published online, meant that I couldn't go to one place for it all. I couldn't fully satisfy my need for kinship in this struggle, commiserate with others trying to find a way out of that dark closet. In desperation, I sought James Baldwin.\n\nI read Baldwin's essay \"Notes of a Native Son\" while I was in my mid-twenties, and it was a revelation. I'd never read creative nonfiction like Baldwin's, never encountered this kind of work, work that seemed to see me, to know I needed it. I read it voraciously, desperate for the words on the page. I needed to know that someone else saw the myriad injustices of living while black in this country, that someone so sharp and gifted and human could acknowledge it all, and speak on it again and again. Baldwin was so brutally honest. His prose was frank and elegant in turn, and I returned to him annually after that first impression-forming read. Around a year after Trayvon Martin's death, a year in which black person after black person died and no one was held accountable, I picked up The Fire Next Time, and I read: \"You can only be destroyed by believing that you really are what the white world calls a nigger. I tell you this because I love you, and please don't you ever forget it.\" It was as if I sat on my porch steps with a wise father, a kind, present uncle, who said this to me. Told me I was worthy of love. Told me I was worth something in the world. Told me I was a human being. I saw Trayvon's face, and all the words blurred on the page.\n\nIt was then that I knew I wanted to call on some of the great thinkers and extraordinary voices of my generation to help me puzzle this out. I knew that a black boy who lives in the hilly deserts of California, who likes to get high with his friends on the weekend and who freezes in a prickly sweat whenever he sees blue lights in his rearview, would need a book like this. A book that would reckon with the fire of rage and despair and fierce, protective love currently sweeping through the streets and campuses of America. A book that would gather new voices in one place, in a lasting, physical form, and provide a forum for those writers to dissent, to call to account, to witness, to reckon. A book that a girl in rural Missouri could pick up at her local library and, while reading, encounter a voice that hushed her fears. In the pages she would find a wise aunt, a more present mother, who saw her terror and despair threading their fingers through her hair, and would comfort her. We want to tell her this: You matter. I love you. Please don't forget it.\n\nThe Fire Next Time is roughly divided into two parts: a letter to Baldwin's nephew, which looks forward to the future, and an essay about religion and the Nation of Islam, which concerns itself with Baldwin's past and present. I initially thought that The Fire This Time would be divided into three parts, roughly inspired by Baldwin's chronological division: essays or poems about the past, deemed legacy, essays or poems about the present, labeled reckoning, and essays or poems about the future, or jubilee, and all of them would wrestle with the specters of race and history in America, and how those specters are haunting us now. But as the pieces of work my editor and I solicited came in, I realized that the structure I envisioned for the work would not be as tidy as I thought. But race in the United States is not a tidy matter. Only three of the submitted pieces explicitly referenced the future. Most of them were concerned with the past and the present. And that told me two things. First, it confirmed how inextricably interwoven the past is in the present, how heavily that past bears on the future; we cannot talk about black lives mattering or police brutality without reckoning with the very foundation of this country. We must acknowledge the plantation, must unfold white sheets, must recall the black diaspora to understand what is happening now. Second, it reveals a certain exhaustion, I think. We're tired. We're tired of having to figure out how to talk to our kids and teach them that America sees them as less, and that she just might kill them. This is the conversation we want to avoid. We're tired of feeling futile in the face of this ever-present danger, this omnipotent history, predicated as this country is, founded as this country was, on our subjugation. But the pieces in this work that do invoke the future\u2014Daniel Jos\u00e9 Older's letter to his wife and future child, Natasha Trethewey's poem about the many planes on which time exists, and Edwidge Danticat's essay exploring the idea that people of the black diaspora are refugees\u2014help me to believe that I might be able to have that conversation with my child in the future. These pieces give me words that I might use to push past the fear and exhaustion and speak to my daughter, my nieces and nephews. This work helps me to believe that this is worthwhile work, and that our troubling the water is worthy.\n\nIf I were smarter, perhaps I wouldn't say this, but I attest to this because I feel it: all these essays give me hope. I believe there is power in words, power in asserting our existence, our experience, our lives, through words. That sharing our stories confirms our humanity. That it creates community, both within our own community and beyond it. Maybe someone who didn't perceive us as human will think differently after reading Garnette Cadogan's essay on the black body in space, or after reading Emily Raboteau's work on urban murals. Perhaps after reading Kiese Laymon's essay on black artists and black love and OutKast, or after reading Mitchell S. Jackson's piece on composite fathers, a reader might see those like me anew. Maybe after reading Rachel Kaadzi Ghansah's essay on Baldwin or Kevin Young's hilarious essay about Rachel Dolezal and what it means to be black, a reader might cry in sympathy and then rise to laughter, and in doing so, feel kinship.\n\nAt the end of The Fire Next Time, Baldwin writes:\n\nThis past, the Negro's past, of rope, fire, torture, castration, infanticide, rape; death and humiliation; fear by day and night, fear as deep as the marrow of the bone; doubt that he was worthy of life, since everyone around him denied it; sorrow for his women, for his kinfolk, his children, who needed his protection, and whom he could not protect; rage, hatred, and murder, hatred for white men so deep that it often turned against him and his own, and made all love, all trust, all joy impossible\u2014this past, this endless struggle to achieve and reveal and confirm a human identity, human authority, yet contains, for all its horror, something very beautiful . . . people who cannot suffer can never grow up, can never discover who they are. . . . Everything now, we must assume, is in our hands; we have no right to assume otherwise. If we\u2014and now I mean the relatively conscious whites and the relatively conscious blacks, who must, like lovers, insist on, or create, the consciousness of others\u2014do not falter in our duty now, we may be able, handful that we are, to end the racial nightmare, and achieve our country, and change the history of the world. If we do not dare everything, the fulfillment of that prophecy, re-created from the Bible in song by the slave, is upon us: God gave Noah the rainbow sign, No more water, the fire next time!\n\nI hope this book makes each one of you, dear readers, feel as if we are sitting together, you and me and Baldwin and Trethewey and Wilkerson and Jeffers and Walters and Anderson and Smith and all the serious, clear-sighted writers here\u2014and that we are composing our story together. That we are writing an epic wherein black lives carry worth, wherein black boys can walk to the store and buy candy without thinking they will die, wherein black girls can have a bad day and be mouthy without being physically assaulted by a police officer, wherein cops see twelve-year-old black boys playing with fake guns as silly kids and not homicidal maniacs, wherein black women can stop to ask for directions without being shot in the face by paranoid white homeowners.\n\nI burn, and I hope.\n\n# PART I\n\n# LEGACY\n\n## Homegoing, AD\n\n## KIMA JONES\n\nHere's the down south story we didn't tell you: sixteen hours in and Jack can't feel her feet but we never stop. Our uncle asleep at the wheel and we that closer to death with each mile. Turned around again and again, before GPS, we learned North Carolina is a long state: tobacco taller than us, the fields and fields of it, no washing it out of our clothes, the air so wet and thick of it, choking us.\n\nJack won't fly. Full grown with a dead granddaddy and still she won't fly, she tells us I-95 has always been the way back home so we gun it. Straight through, no stopping, sixteen hours and Jack doesn't care how bad we need to pee, she says, Hold it. Sixteen hours till we saw the palmetto trees and smelled the paper mill and knew Savage Road was in sight.\n\nGeorgie 'n' em got Grandaddy laid out in the front room like a piece of furniture and ushers fanning the top of Grandmama's head. We couldn't find our place in the business of departing: hams out the oven, lemon cake iced, organ tuned, tea made, napkins folded, the children's black patent leather shoes set out for the dirt road come morning.\n\nHere's the down south story we didn't tell you: Leroy barking at us from the grill because when did everybody stop eating pork and why he got separate meat and when all the women become Nefertiti bangles and headwraps and all us named like Muslims. Our cousins who couldn't make it because he died on the wrong Friday, wadn't payday, and our cousins who did and their many children tearing up the front yard. Our decision to sneak into the woods with red cups, black and milds, Jim Beam, a blue lighter plucked from the card table, and Toya's gold cap kept in her change purse. The pot of greens we brought out with us and the mosquitoes keeping company like we wasn't down in the swamps to bury our dead.\n\nOur cousins know the dark and the heat, but we haven't been home in so long. Our back sweating and this old bra sticky so more and more from the red cup. Our cousin say, Lemme top it off for youse, so we oblige and when he said pull, we pulled and when he said blow, we blew smoke over our shoulder and then into his open mouth, giggling. Our cousin say, You know they found him in the bed, right? And we nod cuz sleep don't come easy no how. He say, Just like that. And our cousin clap when he say that and we think of Grandaddy setting his glasses down on the nightstand one last time. Our cousin say, You missed me? And we smile cuz his hand is on our hip and it's hot out and he smell good and it's the darkest Charleston has ever been. The dead of night is forgiving when you're kin. Grandaddy gone and we sitting up in the woods with brown liquor, necking, our cousin hard on our thigh. Toya say, Keep watch for them copperheads, but copperheads ain't never kill nobody\u2014we got our eyes trained for gators.\n\nWe think we can still outrun 'em.\n\nWho threw that rock at the gator?\n\nDon't know\n\nWhere Toya?\n\nY'all there?\n\nWe here.\n\nGator comin, boy, run\n\nDon't see no gator, cuh Well, gator see us, nigga\n\nRunnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn\n\nso we run\n\nfast\n\ncuz gator made for water but children born for land.\n\n## The Weight\n\n## RACHEL KAADZI GHANSAH\n\nIt was an acquaintance's idea to go there, to Baldwin's house. He knew from living in Paris that Baldwin's old place, the house where Baldwin died, was near an elegant and renowned hotel in the C\u00f4te d'Azur region of France. He said that both places were situated in a medieval-era walled town that was scenic enough to warrant the visit. He said we could go to Jimmy's house and then walk up the road for drinks at the hotel bar where Baldwin used to drink in the evening. He said we would make a day of it, that I wouldn't regret it.\n\nFor the first time in my life I was earning a bit of money from my writing, and since I was in London anyway for family obligations I decided to take the train over to Nice to meet him. But I remained apprehensive. Having even a tiny bit of disposable cash was very new and bizarre to me. It had been years since I had I bought myself truly new clothes, years since going to a cash machine to check my balance hadn't warranted a sense of impending doom, and years since I hadn't on occasion regretted even going to college because it was increasingly evident that I would never be able to pay back my loans. There were many nights where I lay awake turning over in my mind the inevitable, that soon Sallie Mae or some faceless, cruel moneylender with a blues-song-type name would take my mother's home\u2014she had cosigned for me\u2014and thus render my family homeless. In my mind, three generations of progress would be undone by my vain commitment to tell stories about black people in a country where the black narrative was a quixotic notion at best. If I knew anything about being black in America it was that nothing was guaranteed, you couldn't count on a thing, and all that was certain for most of us was a black death. In my mind, a black death was a slow death, the accumulation of insults, injuries, neglect, second-rate health care, high blood pressure and stress, no time for self-care, no time to sigh, and in the end, the inevitable, the erasing of memory. I wanted to write against this, and so I was writing a history of the people who I did not want to forget. For many years, I taught during the day and wrote at night\u2014long pieces, six thousand words for which I was paid two hundred bucks. I loved it; nothing else mattered because I was remembering, I was staving off death.\n\nSo I was in London when a check with one comma hit my account. It wasn't much but to me it seemed enormous. I decided if I was going to spend any money, something I was reluctant, if not petrified, to do, at the very least I would feel best about spending it on James Baldwin. After all, my connection to him was an unspoken hoodoo-ish belief that he had been the high priest in charge of my prayer of being a black person who wanted to exist on books and words alone. It was a deification that was fostered years before during a publishing internship. Basically, during a lonely week I had spent in the storeroom of a magazine's editorial office organizing the archives from 1870 to 2005, I had once found time to pray intensely at the altar of Baldwin. I had asked him to grant me endurance and enough fight so that I could exit that storeroom with my confidence intact. I told him what all writers chant to keep on, that I had a story to tell. But later, away from all of that, I quietly felt repelled by him\u2014as if he were a home I had to leave to become my own. Instead, I had spent years immersing myself in the books of Sergei Dovlatov, Vivian Gornick, Henry Dumas, Sei Sh\u014dnagon, John McPhee, and bell hooks. Baldwin didn't need my prayers, he had the praise of the entire world.\n\nI still liked Baldwin but in a divested way, the way that anyone who writes and aspires to write well does. When people asked me my opinion on him I told them the truth: that Baldwin had set the stage for every American essayist who came after him. One didn't need to worship him, or desire to emulate him, to know this and respect him for it. And yet, for me, there had always been something slightly off-putting about him\u2014the strangely accented, ponderous way he spoke in the interviews I watched; the lofty, precious way in which he appeared in an essay by Joan Didion as the bored, above-it-all figure that white people revered because he could stay collected while the streets boiled. What I resented about Baldwin wasn't even his fault. I didn't like how many men who only cared about Ali, Coltrane, and Obama praised him as the black authorial exception. I didn't like how every essay about race cited him. I didn't like that he and my grandfather were four years apart in age, but that Baldwin, as he was taught to me, had escaped to France and avoided his birth-righted fate whereas millions of black men his age had not. It seemed easy enough to fly in from France to protest, whereas it seemed straight hellish to live in it with no ticket out. It seemed to me that Baldwin had written himself into the world\u2014and I wasn't sure what that meant in terms of his allegiances to our interiors as an everyday, unglamorous slog.\n\nSo even now I have no idea why I went. Why I took that high-speed train past the sheep farms, and the French countryside, past the brick villages and stone aqueducts until the green hills faded and grew into Marseille's tall, dusky pink apartments and the bucolic steppes gave way to blue water where yachts and topless women with leather for skin were parked.\n\nIt was on that train that I had time to consider the first time I started to revere Baldwin, something that had occurred ten years earlier, when I was accepted as an intern at one of the oldest magazines in the country. I had found out about the magazine only a few months before. A friend who let me borrow an issue made my introduction, but only after he spent almost twenty minutes questioning the quality of my high school education. How could I have never heard of such an influential magazine? I got rid of the friend and kept his copy. But still I had no idea of what to expect.\n\nDuring my train ride into the city on my first day, I kept telling myself that I really had no reason to be nervous; after all, I had proven my capability not just once but twice. Because the internship was unpaid I had to decline my initial acceptance to instead take a summer job and then reapply. When I arrived at the magazine's offices, the first thing I noticed was the stark futuristic whiteness. The entire place was a brilliant white except for the tight, gray carpeting.\n\nThe senior and associate editors' offices had sliding glass doors and the rest of the floor was divided into white-walled cubicles for the interns and the assistant editors. The windows in the office looked out over the city, and through the filmy morning haze I could see the cobalt blue of the Manhattan Bridge and the water tanks that spotted some of the city's roofs. The setting, the height, and the spectacular view were not lost on me. I had never before had any real business in a Manhattan skyscraper.\n\nEach intern group consisted of four people; mine was made up of a recent Vassar grad, a hippie-ish food writer from California, and a dapper Princeton grad of Southeast Asian and Jewish descent. We spent the first part of the day learning our duties, which included finding statistics, assisting the editors with the magazine's features, fact-checking, and reading submissions. Throughout the day various editors stopped by and made introductions. Sometime after lunch the office manager came into our cubicle and told us she was cleaning out the communal fridge and we were welcome to grab whatever was in it. Eager to scavenge a free midday snack, we decided to take her up on the offer. As we walked down the hall the Princeton grad joked that because he and I were the only brown folks around we should be careful about taking any food because they might say we were looting. I had forgotten about the tragedy of that week, Hurricane Katrina, during the day's bustle, and somehow I had also allowed the fact that I am black to fade to the back of my thoughts, behind my stress and excitement. It was then that I was smacked with the realization that the walls weren't the only unusually white entities in the office\u2014the editorial staff seemed to be strangely all white as well.\n\nBecause we were interns, neophytes, we spent the first week getting acquainted with each other and the inner workings of the magazine. Sometime toward the end of my first week, a chatty senior editor approached me in the corridor. During the course of our conversation I was informed that I was (almost certainly) the first black person to ever intern at the magazine and there had never been any black editors. I laughed it off awkwardly, only because I had no idea of what to say. I was too shocked. At the time of my internship the magazine was more than one hundred and fifty years old. It was a real Guess Who's Coming to Dinner moment. Except that I, being a child of the eighties, had never watched the film in its entirety, I just knew it starred Sidney Poitier, as a young, educated black man who goes to meet his white fianc\u00e9e's parents. The film was set in the 1960s; I had been born in 1981.\n\nWhen my conversation with the talkative editor ended I walked back to my desk and decided to just forget about it. Besides, I reasoned, it was very possible that the editor was just absentminded. I tried to forget it but I could not, and finally I casually asked another editor if it was true. He told me he thought there had been an Algerian-Italian girl many years ago, but he was not certain if she really \"counted\" as black. I was also alerted later to there being one editor who was half Filipina, half white. But when I asked how this dismal situation could be possible I was told that the lack of diversity was due to the lack of applications from people of color. As awkward as these comments were, they were made in the spirit of oblivious commonwealth. It was office chatter meant to make me feel like one of the gang, but instead of comforting my concerns they made me feel like an oddity.\n\nOn good days, being the first black intern meant having my work done quickly and sounding extra witty around the water cooler; it meant I was chipping away at the glass ceiling that seemed to top most of the literary world. But on bad days I gagged on my resentment and furiously wondered why I was selected. I became paranoid that I was merely a product of affirmative action, even though I knew I wasn't. I had completed the application not once but twice and never did I mention my race. Still, I never felt like I was actually good enough. And with my family and friends so proud of me, I felt like I could not burst their bubble with my insecurity and trepidation.\n\nSo when I was the only intern asked by the deputy editor to do physical labor and reorganize all of the old copies of the magazine in the freezing, dusty storeroom, I fretted in private. Was I asked because of my race or because that was merely one of my duties as the intern-at-large? There was no way to tell. I found myself most at ease with the other interns and the staff that did not work on the editorial side of the magazine, the security guards, the delivery guys, the office manager, and the folks at the front desk; among them the United Nations was almost represented. With them, I did not have to worry that one word pronounced wrong or one reference not known would reflect not just poorly on me but also on any black person who might apply after me.\n\nI also didn't have to worry about that in that storeroom. There I could think. I realized three things spending a week in the back of that dismal storeroom. That yes, I was the only intern asked to do manual labor, but also that I was surrounded by two hundred years of the greatest American essays ever written, and I discovered that besides the physical archives and magazines stored there, the storeroom was also home to the old index card invoices that its writers used to file. In between my filing duties, I spent time searching those cards, and the one that was most precious to me was Baldwin's. In 1965, he was paid $350 for an essay that is now legend. The check went to his agent's office. There is nothing particularly spectacular about the faintly yellowed card except that its routineness suggested a kind of normalcy. It was human and it looped a great man back to the earth for me. And in that moment, Baldwin's eminence was a gift. Because he had made it out of the storeroom. He had taken a steamer away from being driven mad from maltreatment. His excellence had moved him beyond the realm of physical labor. He had disentangled himself from being treated like someone who was worthless or questioning his worth. And better yet, Baldwin was so good they wanted to preserve his memory. I would look at that card every day of that week. Baldwin joined the pantheon of black people who were from that instructional generation of civil rights fighters.\n\nOn that train to Baldwin's house I thought more about this generation and about the seemingly vast divide between Baldwin and my grandfather. They had very little in common, except they were of the same generation, the same race, and were both fearless men, which in black America says a lot. Whereas Baldwin spent his life writing against a canon, writing himself into the canon, he was a black man who was recording the Homeric legend of his life himself; my grandfather simply wanted to live with dignity.\n\nMy grandfather liked to look on the bright side. Even when I visited him in Los Angeles for one of the last times, he insisted things weren't so bad. He was eking out a living on the money we sent and social security. My mother asked him to come east to stay with us, but he refused. He had lived in this building for almost fifty years, but now the upstairs neighbors were what he called \"young bloods,\" guys who threatened to shoot him when he complained about their noise. The landlord wanted him out to raise the rent; he needed more money for the place. All my grandfather had were a few worn tracksuits and his rusted golf clubs. No one needed an eighty-year-old carpenter, no matter how clever he was: he'd worked hard but had made next to nothing. California had once been fertile ground for him, but in the end it, too, was bound to the country that had long seen him and us as subservient human beings. But my grandfather preferred not to focus on that sort of thing. What Baldwin understood is that to be black in America is to have the demand for dignity be at absolute odds with the national anthem.\n\nFrom the outside, Baldwin's house looks ethereal. The saltwater air from the Mediterranean acts like a delicate scrim over the heat and the horizon, and the dry, craggy yard is wide and long and tall with cypress trees. I had prepared for the day by watching clips of him in his gardens. I read about the medieval frescos that had once lined the dining room. I imagined the dinners he had hosted for Josephine Baker and Beauford Delaney under a trellis of creeping vines and grape arbors. I imagined a house full of books and life.\n\nI fell in love with Baldwin, because Baldwin didn't go to France because it was France. He was not full of na\u00efve, empty admiration for Europe; as he once said in an interview: \"If I were twenty-four now, I don't know if and where I would go. I don't know if I would go to France, I might go to Africa. You must remember when I was twenty-four there was really no Africa to go to, except Liberia. Now, though, a kid now . . . well, you see, something has happened which no one has really noticed, but it's very important: Europe is no longer a frame of reference, a standard-bearer, the classic model for literature and for civilization. It's not the measuring stick. There are other standards in the world.\"\n\nBaldwin left the States for the primary reason that all emigrants do\u2014because anywhere seems better than home. This freedom-seeking gay man, who deeply loved his sisters and brothers\u2014biological and metaphorical\u2014never left them at all. He preserved himself so he could stay alone. In France, I saw Baldwin didn't live the life of a wealthy man but he did live the life of a man who wanted to travel, to erect an estate that held a mood of his own design, where he could write as an outsider from the noise, alone in silence, with fearlessness.\n\nDecades after his death in 1987, what I found left behind in Baldwin's house was something similar to what we experienced when I waded through my grandfather's effects after his house had burned down. Two months later, my grandfather would die from shock and stress caused by the fire. Baldwin's death, too, came at him hard and fast. In both houses, I found mail strewn on dirt piles in rooms that no longer had doors or windowpanes, entryways nailed over to prevent trespassers like us. In each case, clearly someone had forced entry in order to drink beer. The scattered, empty beer cans were recent additions, as were the construction postings from the company that was tasked with tearing the house down. So that nothing remained. No remembrance of the past. Not even the sense that a great man had once lived there.\n\nJames Baldwin lived in this house for more than twenty-five years, and all that was left were half a dozen pink teacups and turquoise saucers buried by the house's rear wall, orange trees that were heavy with fruit, but the fruit was bitter and sharp to the taste. We see Baldwin's name in connection to the present condition more often than we see Faulkner's, Whitman's, or Thoureau's. We can visit houses and places where they lived and imagine how their geography shaped the authors and our collective vocabulary. By next year, Baldwin's house will just be another private memory for those who knew it.\n\nI do not know if I will ever see his house again. If I will be able to pull sour oranges from his trees and wonder if they were so bitter when he lived there. I do know that Baldwin died a black death.\n\nFor a while when I came back to the States, I started to send desperate e-mails to people who knew him that read:\n\nFor the last two days, I've basically found myself frantically, maniacally looking for everything that I could find about Baldwin's life there. To be honest, I'm not at all sure what I am looking for, but when I walked up that steep little hill, past the orange and cypress trees out onto the main road, and looked back at his house, I just felt a compulsion to start asking people who knew him about his life in that house. The compound is almost gone, as they are in the process of demolishing it, and yet something about it and him seemed to still be very much there.\n\nI sent those notes\u2014feeling as hopeless as I sounded\u2014because I wanted to save that building. Because I was scared that no one else would ever be able to see that Baldwin had a rainbow kitchen\u2014an orange sink and purple shelves\u2014in his guesthouse. I wanted someone else to wonder what he ate from this kitchen, who he loved, who stayed in this annex of his estate; did his love feel free in this kitchen, in this house where two men could embrace in private behind the ramparts of his home in another country? I wanted someone else to understand the private, black language found in one of Baldwin's last conversations with his brother David. Frail, sick, being carried to his deathbed in his brother's arms, what the world thought of him might as well have been an ocean away. In that moment, Baldwin didn't refer to French poets, or to the cathedrals of his genius, he instead returned to a popular Hollies song. He loved music, and he told his brother: So it is true what they say\u2014he is my brother and he's not heavy.\n\nBaldwin once wrote, \"Life is tragic simply because the earth turns and the sun inexorably rises and sets, and one day, for each of us, the sun will go down for the last, last time. Perhaps the whole root of our trouble, the human trouble, is that we will sacrifice all the beauty of our lives, will imprison ourselves in totems, taboos, crosses, blood sacrifices, steeples, mosques, races, armies, flags, nations, in order to deny the fact of death, the only fact we have. It seems to me that one ought to rejoice in the fact of death\u2014ought to decide, indeed, to earn one's death by confronting with passion the conundrum of life. One is responsible for life: It is the small beacon in that terrifying darkness from which we come and to which we shall return.\"\n\nBecause I am telling you this now, writing it all down, I am finding time to regard memory and death differently. I'm holding them up in the light and searching them, inspecting them, as they are not as I want them to be. On that hill, in Saint Paul de Vence, I wanted to alter fate, and preserve things. But why? They did not need me\u2014Baldwin seemed to have prepared himself well for his black death, his mortality, and even better, his immortality. Indeed, he bested all of them, because he wrote it all down. And this is how his memory is carried. On the scent of wild lavender like the kind in his yard, in the mouths of a new generation that once again feels compelled to march in the streets of Harlem, Ferguson, and Baltimore. What Baldwin knew is that he left no heirs, he left spares, and that is why we carry him with us. So now when people ask me about James Baldwin, I tell them another truth: He is my brother, he ain't heavy.\n\n## Lonely in America\n\n## WENDY S. WALTERS\n\nI have never been particularly interested in slavery, perhaps because it is such an obvious fact of my family's history. We know where we were enslaved in America, but we don't know much else about our specific conditions. The fact that I am descended from slaves is hard to acknowledge on a day-to-day basis, because slavery does not fit with my self-image. Perhaps this is because I am pretty certain I would not have survived it. I am naturally sharp-tongued, suffer from immobility when I am cold, and am susceptible to terrible sinus infections and allergies. My eyesight is poor. Most of the time I don't think about how soft the good fortune of freedom has made me, but if I were to quantify my weakness of body and character I would guess that at least half the fortitude my enslaved ancestors must have possessed has been lost with each generation in the family line, leaving me with little more than an obtuse and metaphorical relationship to that sort of suffering.\n\nI resist thinking about slavery because I want to avoid the overwhelming feeling that comes from trying to conceive of the terror, violence, and indignity of it. I do not like to think of it happening in my hometown, where I work, in my neighborhood, or near any of the places where I conduct my life. My cultural memory of slavery, which I don't think is so unlike that of many other Americans, suggests that it was primarily a Southern phenomenon, one confined to the borders of plantations, which, if they haven't been transformed into shopping complexes or subdivisions, exist now only as nostalgic, sentimentalized tourist attractions. The landscapes associated with slavery, however, extend far beyond the South.\n\n\u2022 \u2022 \u2022\n\nMy home is in New England and in the winter my house feels slight against the wind as its windows tremble with every blustery gust, which makes me want to stay in bed, though I am not at all the type of person who likes to linger there once awake, unless circumstances are such that I am not alone, and then, even in that rare case, I can be restless and ready to set forth at sunrise. In the winter of 2006, I was not working at my regular job, which might have been a good thing had I not been prone to a melancholy obsession over recent personal disappointments. I began to notice pains in my body I had never felt before: a tendon pulling across the length of my leg when I sat down, a sharp twinge in my side when I stood up, and sometimes when I'd shower my skin was so sore I could barely stand to feel the water on it. I knew these pains were likely psychosomatic, evidence of how deeply I was suffering from loneliness. Because I suspected that the hope of escaping my loneliness was adding to my discomfort, I had been trying to cure myself of optimism as a strategy to ward off future misery. The value of this approach was confirmed by a self-help book I kept on my nightstand. When I dared to open it, I could read only a single chapter at a sitting because each reiterated a simple point that I just could not seem to accept\u2014that to become free from disappointment one must acknowledge the obvious, then learn to live with it.\n\nBy mid-January, the United States' war with Iraq was coming to the end of its fourth year, the war in Afghanistan was intensifying again, and the shortcomings of the federal government that had been noted after Hurricane Katrina were fading from media attention, which was now absorbed with a surge of reports that, come summer, another movie star couple would be expecting their first biological child. I found myself momentarily enthralled in speculation: How long would this new relationship last? What did his ex-wife think of the sudden pregnancy of his new girlfriend? Who would take time off from their career to care for the family? These questions, though deeply irrelevant to my own life, served to distract me from the obvious fact that an unpopular war, entered into on misinformation, was showing no signs of ending. I studied the news reports on the radio every morning, which covered many subjects: planned highway construction projects, politics, movie stars, pop music stars, television stars, impending diseases, lying politicians, local sports, bank robberies, soldiers killed in Iraq. I suppose I was hoping the radio would serve as a kind of personal oracle, that stories of real human struggle might release me from solipsistic self-pity and show me how to leave my bungalow by entering the world with a sense of purpose, or at least a sense of direction.\n\nIt was with this ambition that I had gone to New Orleans to help my great-aunt Louise come to recognize that her home there had been destroyed, even though my gumption was clearly tainted with dread. Sitting in a cold house listening to the radio was painful enough, but the thought of actually walking through so much loss made me worry that I would have to face more of the obvious than I could be distracted from noticing. As our plane flew over the Gulf Coast it was hard to tell how bad things actually were on the ground. Muddy patches of brown and tan signaled the normally slow growth of a Southern winter. I saw the edge of Lake Pontchartrain, into which, during the early 1920s, my great-grandmother Susie had thrown her wedding ring when she needed to affirm a point that her husband would not accept. On our descent I began to see blue tarps stretched over large holes in people's roofs.\n\nWhen we arrived at Aunt Lou's tiny red-brick, railroad-style house, her nephew met us. Chester was a former longshoreman. He had returned to the city a few weeks after the water had been pumped out and had been living in a FEMA trailer while he gutted his own house. Even though he had warned my mother and Aunt Lou on the phone that the house was in very bad condition, he wanted to make sure we understood this before we entered it, because from the outside, there appeared to be little structural damage. About four feet from the ground, a black, bathtub-ring-like watermark circled the exterior. Garbage and a broken ladder lay across the front lawn. When we opened the front door, dirt, mud, debris, and seaweed covered the hardwood floors and the sofa, which had floated over to the opposite wall from where it had been set. The house looked like someone had picked it up and shook it hard before setting it back down onto its cinder-block frame. We put on face masks and gloves, and booties over our shoes. A chifforobe sitting in water for weeks had gently exploded and still-wet clothes poked out of holes in the sides. Black and brown mud blotted the wall next to an antique brass bed. In the back room, the ceiling had caved in and wires and other debris hung low from what was left of the roof, like snakes in trees.\n\nAunt Lou said, My house is tore up.\n\nRadio and television news reports about New Orleans mentioned that several of the city's cemeteries had been badly damaged by the flooding, and Chester's wife said coffins had been turning up all over the city. So I convinced my mother to drive over to Holt Cemetery, where our family crypt is situated, but she wouldn't get out of the car to check on it with me. Instead she shouted from its window, Watch out for water moccasins! as I walked through a rusty and twisted wrought-iron fence bolstered by rotten tree stumps into a field of tall, dead grasses and sun-bleached tombstones. Cypress trees sheltered the perimeter, branches reaching like veins across a heart.\n\nDespite the fact that most of the graves at Holt are belowground, unlike many cemeteries in New Orleans, it looked to me that Holt had kept hold of its dead better than the grave sites near the end of Canal Street, where mausoleums appeared stained and tumbled over by water. In the early 1900s a portion of Holt was used as a \"pauper's field\" for the poor and indigent, and during segregation it was one place where blacks could be buried. Our family crypt had been there longer than anyone could remember, but its precise location was unknown; its marker had been stolen in 1969, just weeks after my great-grandmother was laid to rest. Up close I could see new grass, slender and gold-green, appearing in short tufts at the foot of the headstones, most of which were pitched in one direction or another toward the ground. Handmade markers in wood or cement were adorned with bottle glass, sea shells, or not at all, with the names and dates of the deceased written in by hand. Some had been decorated with Mardi Gras beads and silk flowers. I walked down the dirt path to the part of the cemetery where most of the stones were missing and called out to my ancestors. I have no idea where you are. Tell me where you are. But I heard nothing.\n\nTen years ago, when I last visited Holt, the grounds had not been well tended, and even then many of the grave markers were missing, in disrepair, or toppled down. Back then I had only a piece of scrap paper with a row and plot number written on it to guide me to the crypt's location. The ground, deeply sunken where bodies were buried, looked as if waves were passing through it in slow motion. On that visit I worried that I could not read the names of the people I was walking over to apologize directly for the disrespect of having done so. Hastily I laid some flowers and a note where I thought the crypt should be and left without ever planning to return. On this visit, Holt felt strangely serene; unlike the rest of the city, it appeared not to have changed in any significant way. In fact, I might have wandered among the unknown dead for hours had I not heard my mother frantically shouting from the car. Wendy, come on! We've got to get to Constantinople before dark!\n\nWe picked up Aunt Lou from Chester's and drove over to Constantinople, a street in the Magazine district, to meet up with her childhood friend who had survived the flood trapped with her son in the attic of his house for a week before being rescued. Because none of the traffic lights were working, my mother was nervous and complained all the way there about how I had lingered too long in the cemetery. She chided me that one should let the dead rest. That's right, Aunt Lou said. I interrupted: I just wanted to make sure that our people hadn't floated away. They went quiet.\n\nBut I took a walk around, and it looked like everybody was still tucked in tight.\n\n\u2022 \u2022 \u2022\n\nI returned from New Orleans more miserable than when I left. As much as I had wanted to come back from that trip with a sense of conviction, inspired to action that would distract me from my loneliness, I could not find a singular source of outrage on which to fixate\u2014not poverty, racism, the failure of the federal government, a history of community self-destructiveness, a river, a lake, or a hurricane. Not a house without a roof, a felled tree across a path, a tumbled-down tombstone, or a wayward corpse. I was faced with too much that was obvious about the way class and race work in America. More than I wanted to see. More than I was capable of seeing.\n\nThis is when I realized my loneliness had deeper roots than I had initially suspected, and that, in addition to personal disappointments, it came from having a profound sense of disconnection from what I thought America was, and who, in that context, I knew myself to be. My post\u2013New Orleans loneliness seemed to emanate from a place that preceded my own memory and stretched across time into a future that extended far beyond my vision. It was as if I had been thrown overboard into the sea and was paralyzed by the shock of it. I could neither breathe nor drown. I could not sink or return to the surface.\n\nThen early one morning in January as I was listening to Boston's public radio station, I heard a story about the 2003 discovery of a grave site in Portsmouth, New Hampshire. As city workers attempted to dig a manhole near the corner of Court and Chestnut streets in the seaside town, a pine coffin was discovered with leg bones sticking out. An independent archaeological firm had been brought in immediately to lead an exhumation. Eight coffins and the remains of thirteen people were removed. The report noted that a combination of forensic evidence and DNA testing had confirmed that at least four of the remains in question were of African ancestry, most likely slaves buried there during the 1700s. The archaeologists' report had just been released to the city of Portsmouth, which was engaged in public discussion about the most appropriate and respectful way to deal with those exhumed, as well as the fact that as many as two hundred people might still be buried at the site.\n\nPerhaps if I had not already spent more than a couple of weeks being so down in the dumps, if talk about the expected duration of the wars in Iraq and Afghanistan suggested a time frame other than the interminable, if images from my trip to New Orleans were not so powerfully present to me, then maybe the NPR report would have floated past me that morning. But something about hearing that Africans are buried beneath a public street in a small, coastal New England town gave me a new context to reconsider what is obvious and how one might learn to live with it. I knew I had to go there to see the people, even if they were still tucked in tight, if I was ever going to start letting go of the expectation that I could someday feel less lonely in America.\n\n\u2022 \u2022 \u2022\n\nThe first time I drove the two hours north from Providence to Portsmouth I had no idea what I was going to do when I got there. It was a Sunday in late February, the day after a large snowfall had dumped about six inches of snow along New England's southern coast. By morning, the roads were no longer wet and the snowdrifts at the side of the road glowed while ghostly wisps of fine powder swirled in the winnows of eighteen-wheelers trying to close the distance on Monday. From the interstate, I saw a sign for the Strawbery Banke Museum, which had been mentioned in the radio report, and I followed its direction.\n\nThe museum turned out to be a neighborhood of restored colonial houses at the edge of the Piscataqua River. The main entrance was closed, so I followed an elderly white couple into Stoodley's Tavern, which served as the museum ticket office on weekends. An older white woman with silver-bobbed hair sat at a table covered with pamphlets advertising local tourist attractions. Are you here for the tour? she asked. I nodded yes. Ten dollars. Charles, our docent, chatted about the weather with the five of us who waited for the tour to begin: me, the senior couple from Kittery, Maine, and a very young, blond couple just recently moved to Vermont from Tahoe, Nevada.\n\nWe walked across the street into the original settlement founded in 1630, known as Puddle Dock. The Old Mainer wanted to know: Where were the borders of the marsh before the houses were built? Where had the water been pushed back to? He was wearing a cap that said USS Indianapolis. Charles asked him if he was on the ship during WWII and he said yes. Charles said, Were you on it when it went down? The Old Mainer told us that he had gone ashore at Pearl Harbor just before the ship had set sail for Guam. Charles enthusiastically told us the story of how the ship went down, as if its history illuminated an unseen aspect of the tour. On July 30, 1945, the ship, en route from Guam to the Gulf of Leyte, was torpedoed by the Japanese. More than nine hundred sailors were hurled into cold, choppy water. Although they radioed U.S. forces for help as they went down, no one came for four days. By August 8, at the end of the rescue effort, only 317 men of the 1,196 originally on board had survived. The rest had been picked off by sharks or drowned.\n\nAfter looking through a few of the houses in Puddle Dock, the Old Mainer, his wife, and I fell behind the guide and the young couple, who kept bragging about the beehive stove in an eighteenth-century farmhouse they were thinking of buying and restoring. They asked questions about the interior design of every home we toured. I took copious notes on Portsmouth's history and in this, felt my dour mood lightening. Details were comforting. Charles told us that Portsmouth was an Anglican, not Puritan, settlement and that among its original inhabitants were seventy-two Africans and eight Danes. Many of the wealthiest families in town made their fortunes in \"the trade\" first by shipping food, lumber, livestock, and other goods to British colonies in the West Indies and then by carrying captured Africans to the Caribbean, Virginia, and Portsmouth from the late seventeenth century through much of the eighteenth. Throughout the tour Charles occasionally used the word \"servant\" but never the word \"slave.\"\n\nIn an alcove at the top of a staircase in a house built in 1790, the Old Mainer said to me, I'd never live in one of these old houses. They're too cold. There were two pictures on the mantel over the fireplace in the dining room. One called An Emblem of Africa featured a black woman walking with a feathered headdress next to a tiger in the background. The other picture, An Emblem of Europe, featured a white woman with a globe at her feet holding a book and a horn of plenty filled with fruit and flowers at the crook of her arm.\n\nWhen the young couple asked about the role of the Native population in the development of Portsmouth, Charles explained that they were not a factor: Most died out before the town became sizable, after catching diseases from their contact with the Europeans, he said.\n\nAt the end of the tour, I returned to Stoodley's Tavern to ask for directions to the slave graves mentioned in the radio report. Charles told me, You can't see anything. There's nothing there. I thought he meant that the site had not been commemorated or officially rededicated, but his reaction made me wonder if there was even a historical marker indicating the graveyard's boundaries. The woman who had sold me a ticket said, They've been reinterred. I told them I still planned to go and asked if Chestnut Street was close, since Portsmouth's downtown area is quite small. Or should I drive? I said. She responded tersely, It doesn't matter. It's just an intersection.\n\n\u2022 \u2022 \u2022\n\nIt was sharply cold and the wind was picking up when I arrived at Chestnut Street near the corner of Court. Several restored colonials now serving as lawyers' and doctors' offices lined its east side. On the west side there was a beauty salon and a sign indicating a \"Drug Free School Zone.\" Other than these buildings, it seemed that there was nothing to see. As I rounded the corner at Chestnut and State, I noticed a brass plaque affixed to the clapboards of a house: In colonial Portsmouth, segregation applied in death as in life. City officials approved a plan in 1705 that set aside this city block for a 'Negro Burial Ground.' It was close to town but pushed to what was then its outer edge. By 1813, houses were built over the site. I got back in my car to write notes about what I found. This is when I realized my car was probably sitting on top of people. I knew I should feel something about that, but all I felt was a familiar loneliness creeping in on me.\n\n\u2022 \u2022 \u2022\n\nThe trip to Portsmouth had not elicited much outrage in me, even after I discovered that one of the oldest known grave sites of blacks in New England was neither green nor sacred space. I accepted the reality that the historic colonial houses\u2014now the business residences of attorneys, hairstylists, insurance agents, and doctors\u2014were considered by most people to be more valuable than the bodies down below them. But while I had thought that my lack of feelings while standing on people would allow me to forget that I had been standing on people, it didn't. I had no intuition about how these dead Africans might have felt about being paved over, no feelings of ancestral connection to those buried below, and I heard no discernible voices calling to me from the depths of that darkness. I wondered if the woman at the museum had been right. Maybe the corner was just an intersection.\n\nThe ambivalence the folks at the Strawbery Banke Museum expressed for those buried beneath Portsmouth's downtown was all the more surprising when I later learned that the first bodies exhumed from the African Burying Ground had been housed at the museum before they were transported to the temporary laboratory. I assumed that my own lack of feeling was due, in part, to the randomness with which I had selected Portsmouth as the place to try to make sense of the remains of slavery in America. I had no personal connection to New Hampshire, no familial bond to any of the people buried there, and I became certain that was the reason I couldn't feel anything while standing on those Africans. I thought maybe I needed to visit a slave grave site more closely related to my life if I was going to experience some true cathexis.\n\nSo once back in Rhode Island, I went to a talk given by Theresa Guzm\u00e1n Stokes at Newport's Redwood Library about that city's largest African burial ground, called God's Little Acre, a cemetery founded in 1747. For more than twenty years, without city support, she had been maintaining its grounds out of personal respect for those buried there, clearing away litter and weeds and eventually establishing a fund to protect it. She runs a website about the cemetery, and she and her husband, Keith Stokes, former executive director of the Newport County Chamber of Commerce, are writing a book on the subject.\n\nWhile introducing his wife, Stokes assured the small audience, We're not interested in slavery. It's emotional and it separates people. But the absurdity of slavery means it is practically impossible for anyone to contain all the contradictions that arise when speaking of it. So despite his promise seconds earlier to refrain from talk of slavery, Stokes started by explaining how often the term \"servant\" is used as a euphemism for \"slave\" in New England and how there is a presumption that Africans here were somehow \"smarter\" and treated better than those in the South. This misperception, he pushed, is because people don't want to remember the dehumanization. Without hesitating, he went on to say, Slavery is violent, grotesque, vulgar, and we are all implicated in how it denigrates humanity.\n\nAccording to a series of articles by Paul Davis running that same week in the Providence Journal, Newport was a hugely significant port in the North Atlantic slave trade, and from 1725 to 1807 more than a thousand trips were made to Africa in which more than a hundred thousand men, women, and children were forced into slavery in the West Indies and throughout the American colonies. Ms. Guzm\u00e1n Stokes explained how African people built many of the prominent colonial houses throughout New England, including those in Newport, and while many of those buildings remain restored in one form or another, just a handful of graves of Africans who made this contribution to the town's development can be found.\n\n\u2022 \u2022 \u2022\n\nOn my way to God's Little Acre, I came upon the tiny Newport Historical Cemetery #9, which Theresa Guzm\u00e1n Stokes had also mentioned during her talk, but I could not figure out which graves belonged to Africans and which belonged to whites. A white woman was taking pictures of stones, so I asked her if she knew. She pointed to two graves in the corner. These over here, she said and then explained she had looked for information on African graves on the Web before she left her home in Seattle. The woman told me she was originally from Connecticut, but when she decided to marry an African American man in the 1970s, her family disowned her. She had four children with him, none of whom ever met her parents. She had brought her youngest daughter back east to visit historical sites for a vacation and confessed that she was glad she no longer lived in New England. I couldn't take all of this \"in your face\" history. Like Thames Street, the blue stones, she said, referring to the pavers on a road that edges Newport's harbor. Each one of those stones represents an African. Every stone was from the ballast of a slave ship and was carried by a slave as he or she debarked. When I called the Newport Historical Society to confirm this, the reference librarian and genealogist Bert Lippincott III, C.G., insisted that stones like that were used as ballast on all ships coming into Newport, not just slave ships. He added, Many Newporters bankrolled ships in the trade, but Newport was not a major destination for slave ships. When I mentioned the article in the Providence Journal that claimed most Africans in colonial Newport were slaves, he said, Many were third-generation Americans. Most were skilled, literate, and worked as house servants.\n\nAt God's Little Acre on the edge of Newport, three stones stand erect, three others appear jackknifed into the ground at a forty-five-degree angle. One lies level to the ground. Only these seven tombstones remain in the graveyard that commemorates the contributions of Africans to the city's early history. While surrounded on three sides by larger, crowded cemeteries and an eight-foot wrought-iron fence facing Farewell Street, God's Little Acre is comparatively pastoral, and most of the grave markers are missing as a result of vandalism or landscaping contractors running tractor mowers through it for many years. The inscriptions on those few slate stones still standing are fading due to the way weather and pollution wear on them. Many are now just barely legible.\n\nA white woman with a backpack was taking pictures of the scant stones. She told me she teaches courses on American graveyards at a school in Connecticut. Pointing to one of the graves, she said, He must have been loved by his \"family\" because stones were very expensive back then. I wanted to say, So were people. And then I remembered reading an inventory from the estate of Joseph Sherburne, whose house has been preserved at the Strawbery Banke Museum. The linens were listed as worth forty dollars while the African woman who washed and pressed them had a line-item value of fifty dollars.\n\n\u2022 \u2022 \u2022\n\nMy trip to Newport made me realize that I knew almost nothing about the lives of blacks in Portsmouth during slavery and I wondered if that was the reason I was so unmoved by my visit. So I drove back up to New Hampshire to walk the Black Heritage Trail, put together by a retired schoolteacher and local historian, Valerie Cunningham, in order to learn about the experiences of Africans and African Americans in Portsmouth. Some of the sites on the Black Heritage Trail highlight historic accomplishments of blacks in Portsmouth such as the New Hampshire Gazette printing office where Primus, a skilled slave, operated a press for fifty years; the Town Pump and Stocks, where black leaders were elected in a ritual following loosely from the Ashanti festival tradition of Odwira; and St. John's Church, where the records indicate that Venus, most likely a poor but free black woman, received a gift of one dollar from the church in 1807 on Christmas Day.\n\nI sat on a bench overlooking the Memorial Bridge, which crosses the Piscataqua River from Kittery, Maine, to where captive Africans would have first encountered Portsmouth, the wharf at what is now Prescott Park. The first known African captive arrived in Portsmouth around 1645 from Guinea, and slave ships started landing regularly as early as 1680 carrying small loads of mostly male children and adolescents. I tried to imagine what it felt like to come into this swiftly moving river harbor after a long journey across the Atlantic in the cargo hold of a ship\u2014after having been starved, beaten, shackled, and covered in the feculence of the living and dead. Did seeing the flat, tidy fronts of buildings outlining this colonial settlement for the first time make them feel hopeful? So many rectangles. How far away the rest of the world must have seemed.\n\n\u2022 \u2022 \u2022\n\nI ended my walk at the Portsmouth Public Library, which held no significance on the trail, but, according to the first news story I heard about the burial ground, had in its collection a copy of the archaeologists' report on the burial site. When I asked a reference librarian if I could see it, she hesitated and wanted to know if I planned on making copies. I told her I was not sure if I wanted to make copies because I hadn't yet seen the report. She then consulted with the head reference librarian, who told me that the burial site is a very sensitive issue for the city and that he needed to consult with the city attorney's office before releasing it. He took down my information\u2014name, city of residence, and school affiliation\u2014then asked me to wait while he placed the call.\n\nThe librarian was worried about how I might represent Portsmouth in a piece on the subject, because he cared about the town. I liked the town, too. It is pretty, easy to navigate, and surprisingly friendly for New England. I felt guilty and ashamed about my affinity for the town because at the time I could not muster more than a diffuse intellectual identification with the people who were buried just a few streets over.\n\nBefore copying the report, I remembered how easy it was for me to ignore what was already obvious, so I wrote down some details to remind myself of what I shouldn't forget: people were carried like chattel on ships to America; they were sold to other people; they were stripped of their names, spiritual practices, and culture; they worked their entire lives without just compensation; they were beaten into submission and terrorized or killed if they chose not to submit; when they died they were buried in the ground at the far edge of town; and as the town grew, roads and houses were built on top of them as if they had never existed.\n\nI spent the long summer with my friends at the beach, drinking Bloody Marys and eating lobster rolls on the open-air deck of a clam shack in Galilee, Rhode Island, while the Block Island Ferry, serried with tourists, made its lethargic heave past the docked commercial fishing boats. Once school started, I turned my attention back to the spiritless tedium of lesson planning and grading papers. In all that time I did not once touch the archaeologists' report.\n\nI could make something up about why I let the report sit in a manila folder on my desk for nine months without ever once attempting to read it\u2014something about wanting to let the dead rest or about how loneliness swells and recedes\u2014but I won't. The reason is not clear to me even now. What I do know is that holding the copy I had made of the report near the Xerox machine by the dimly lit front door of the Portsmouth Public Library that previous spring made me feel more than I had felt during any of my grave-site visits, like a balloon in my chest was expanding and taking up all the space I normally used to breathe.\n\nIntense discomfort, I had thought. Maybe that's enough.\n\nBut by January I was driving back up to Portsmouth, irritated with myself for not reading the copy of the report I had already made but even more irritated with myself for not being able to let it go unread. The once tattered and gloomy public library had moved to a brilliant new building a few streets over, and as I walked around the landscapers installing the brick steps, I caught the sign on the door that said, \"Welcome to Your New Library.\" In the breezeway, three junior high school girls gathered around a computer terminal and giggled. A woman in a purple cardigan greeted me from behind the circulation desk with a smile and thin wave. Seduced by all of it, I thought, I love my new library.\n\nWhen I asked the reference librarian about the report, he told me it was now shelved in the local history section in the regular stacks. I thought, Now it's all out in the open. Now there's nothing to hide. I grabbed it off the wall, took a seat at one of the new blond reading tables, and thumbed through it lightly as if it were a mere tabloid magazine. I took notes from the acknowledgments, introduction, and background chapters, but when I got to the section describing the removal of the coffins\u2014those same pages I had copied nearly a year before\u2014a shrill noise came up from the back of my throat at the pitch of a full teakettle in a rolling-boil whistle. I cleared my throat and went back to reading, but my din started again. It was sharp enough for anyone to hear, so I decided I had better leave\u2014but not before making a fresh copy of the report to take with me.\n\n\u2022 \u2022 \u2022\n\nWhen a story is unpleasant, it is hard to focus on details that allow you to put yourself in the place of the subject, because the pain of distortion starts to feel familiar. Paying attention often requires some sort of empathy for the subject, or at the very least, for the speaker. But empathy, these days, is hard to come by. Maybe this is because everyone is having such a hard time being understood themselves. Or because empathy requires us to dig way down into the murk, deeper than our own feelings go, to a place where the boundaries between our experience and everyone else's no longer exist.\n\nArchaeologists removed the remains of thirteen people from beneath the intersection of Chestnut and State streets with the help of some machinery, but they did most of the digging by hand. Once in the laboratory, they used potters' tools and paintbrushes to remove excess soil from the bones and teeth. The exact dates associated with each burial remain unknown, but it is assumed that all were interred during the eighteenth century. Four males and one female could be identified by sex, but they found it impossible to determine the sex of the other eight, though most were believed to be in early adulthood, between the ages of twenty-one and forty years. Heads of the deceased generally faced west, suggesting a burial in the Christian tradition. In no cases were all the bones of an individual represented, perhaps due to the commingling of remains during previous installations of gas and sewer lines, the stacking of coffins, or a high water table in the soil. Thus no cause of death could be determined for any of those recovered. Archaeologists noted, however, that the lack of visible traumatic defects, cut marks, fresh or healed fractures does not rule out the presence of trauma. The teeth of each person, which in several cases constituted the entirety of the remains, appeared to be better preserved than their bones, which were found wet, free of flesh, colored gray or black, and, in the case of long bones, often missing the ends.\n\nPieces of the skull, portions of the upper and lower limbs, shoulder girdle, ribs, spine, and pelvis of a male person between the ages of twenty-one and thirty years represent Burial 1. An excavator operator noticed his leg bones sticking out from the bottom of his coffin, which was made of white pine and was hexagonal in shape. All of his mandibular and some of his maxillary teeth were present, but like most of those recovered at the site, his teeth exhibited traces of enamel hypoplasia, a sign of previous infection or nutritional stress. His bones revealed a calcified blood vessel in his right lower leg and prolonged shin splints. A pumpkin seed of unexplained significance was found in his coffin as well as a metal object, probably a shroud pin, suggesting he was naked at burial.\n\nIn Burial 2, the remains of another male person between twenty-one and twenty-six years of age were found in good condition despite the fact that part of his skull had been unintentionally crushed by the excavator, leaving only his mandible and several teeth. A gas line running through the foot portion of his coffin meant that many bones in his right foot also were missing. His body was slumped to the left side, probably due to his coffin being tipped during burial, and his hipbone was broken in several places. His right hand lay over his thigh. Further analysis of his bones showed signs of repetitive forearm rotation and possible inflammation of the right leg, presumably from heavy shoveling, lifting, or other strenuous work. Salt, either used as a preservative before burial or for some other ritual, and a single tooth of unknown origin found between his knees, further distinguished his remains.\n\nBurial 3 contained the remains of a person of indeterminate sex, thought to be approximately thirty to fifty years of age with the head facing east, perhaps toward Mecca. Archaeologists recovered only extremely fragile fragments of the cranium and major long bones. The part of the mandible that was still intact suggests participation in a West African puberty ritual as there is a long-healed-over gap where lower and lateral incisors would have been. Stains in the soil represented most of the coffin wood. Only thirty teeth, small fragments of bone, some twenty wood and coffin nails accounted for the person of twenty-one to forty years of age in Burial 4. Those remains were extremely damaged by erosion and the unintentional intrusion of the excavator.\n\nPipe laid around 1900 across the bottom of the coffin of the male person aged twenty-one to forty in Burial 5 eventually disintegrated his lower extremities. Shovel marks on the coffin base indicate where a crew member either hit his coffin accidentally or attempted to cut through it.\n\nThe head of the female person in Burial 6 was located under the sidewalk, which had to be caved in to allow for her removal. Only the upper portion of her coffin was found intact. Her lower legs, cut off where they intersected with a utility trench and a ceramic sewage pipe installed around 1900, revealed evidence of a bone infection and severe inflammation of the shins. Her left arm appeared to be laid across her torso, and her cranium, now missing the face, pointed to the right side of the coffin. Her upper central incisors were shaved, possibly according to a West African cultural tradition, and represent the earliest documented case of such dental modification in North America.\n\nThe person in Burial 7 was a child between the ages of seven and twelve, of unknown sex, whose remains were damaged by heavy rain and a redirected sewer line that flooded the grave shaft during excavation. Decades of a sewer pipe lying across the child's midsection also contributed to this poor state, despite the fact that the coffin was found to be in relatively good condition. Directly beneath that body were the remains of a male person between twenty-one and forty years of age in Burial 12 whose bones were very soft also due to the high water table of the soil. At present, it is impossible to tell if these two people were buried at the same time or possibly even generations apart. The coffinless remains of persons in Burials 2B, 3B, 4B, 5B, and 7B were discovered beneath the sidewalk. Dental fragments and hand bones from a person not presently attributed to Burial 2 but found nearby are all that exists of the person in Burial 2B. Twelve teeth represent the person in Burial 3B. One tooth each indicates persons in Burials 4B and 5B, and a femur shaft fragment resting atop the child's coffin in Burial 7 is all that was found of the person in Burial 7B.\n\n\u2022 \u2022 \u2022\n\nThe boundaries of Portsmouth's African Burying Ground are still a mystery, as they have been for more than one hundred years, but plans to build a formal memorial are under way. Public discussions led by the state's archaeologists have asked city residents to consider whether a part of either street should be closed to vehicular traffic. Some Portsmouth residents have submitted samples of their DNA to see if they are in any way related to those people whose remains, now stored in Ethafoam, 0.002 mil polybags, and acid-free archival storage boxes in a municipally provided laboratory space, await reinterment.\n\nBecause I worried that I would lose track of the archaeologists' report among the bills, magazines, and student papers that littered my desk, for many months I kept it beside my bed, on the floor beneath my nightstand. Each morning the radio woke me with news of the war, a pop star's addiction, dismal predictions for the American economy. Later, I put the report in my backpack, its pages flat against my spine. At some point, I am not sure when, I grew accustomed to its weight and stopped noticing I was carrying it around.\n\n## Where Do We Go from Here?\n\n## ISABEL WILKERSON\n\nBefore the summer of 2014, before we had seen Eric Garner dying on a Staten Island street and Michael Brown lifeless in the Missouri sun for hours, before the grand jury decisions and the die-ins that shut down interstates, we may have lulled ourselves into believing that the struggle was over, that it had all been taken care of back in 1964, that the marching and bloodshed had established, once and for all, the basic rights of people who had been at the bottom for centuries. We may have believed that, if nothing else, the civil rights movement had defined a bar beneath which we could not fall.\n\nBut history tells us otherwise. We seem to be in a continuing feedback loop of repeating a past that our country has yet to address. Our history is one of spectacular achievement (as in black senators of the Reconstruction era or the advances that culminated in the election of Barack Obama) followed by a violent backlash that threatens to erase the gains and then a long, slow climb to the next mountain, where the cycle begins again.\n\nThe last reversal of black advancement was so crushing that historians called it the Nadir. It followed the leaps African Americans made after enslavement, during the cracked window of opportunity known as Reconstruction. The newly freed people built schools and businesses and ascended to high office.\n\nBut a conservative counterreaction led to a gutting of the civil rights laws of that time and to the start of a Jim Crow caste system in the South that restricted every step an African American could make. Any breach of the system could mean one's life. African Americans were lynched over accusations of mundane infractions, such as stealing a hog or 75 cents, during a period that lasted into the 1940s.\n\nSix million African Americans fled that caste system, seeking asylum in the rest of the country during what would become the Great Migration. Denied the ballot, they voted with their bodies.\n\nTheir defection put pressure on the country, North and South, and freed them to pursue their dreams of self-determination. But in the North, they were met with hostility from the onset\u2014redlining, overpolicing, hypersegregation, the seeds of the disparities we see today. The past few months have forced us to confront our place in a country where we were enslaved for far longer than we have been free. Forced us to face the dispiriting erosion that we have witnessed in recent years\u2014from the birther assaults on a sitting black president to the gutting of the Voting Rights Act that we had believed was carved in granite.\n\nAnd now police assaults on black people for the most ordinary human behaviors\u2014a father tasered in Minnesota while waiting for his children; a motorist shot to death in North Carolina while seeking help after a car accident. It is as if we have reentered the past and are living in a second Nadir: It seems the rate of police killings now surpasses the rate of lynchings during the worst decades of the Jim Crow era. There was a lynching every four days in the early decades of the twentieth century. It's been estimated that an African American is now killed by police every two to three days.\n\nThe outcomes in Staten Island and Ferguson and elsewhere signal, as in the time of Jim Crow, that the loss of black life at the hands of authorities does not so much as merit further inquiry and that the caste system has only mutated with the times. From this, we have learned that the journey is far from over and that we must know our history to gain strength for the days ahead. We must love ourselves even if\u2014and perhaps especially if\u2014others do not. We must keep our faith even as we work to make our country live up to its creed. And we must know deep in our bones and in our hearts that if the ancestors could survive the Middle Passage, we can survive anything.\n\n## \"The Dear Pledges of Our Love\": A Defense of Phillis Wheatley's HusbandI\n\n## HONOR\u00c9E FANONNE JEFFERS\n\nAs a little girl in the seventies, I memorized the names of prominent African Americans for Black History Month. These were the images that my teachers would trace from construction paper, then tack to the bulletin boards in my school. How I loved those dark silhouettes.\n\nMy school was 99 percent segregated; the laws had changed in the South but custom had not. My teachers celebrated black \"firsts\" to shore up our self-esteem, to fortify us against our smaller and shabbier schools and a pervasive white unfriendliness from those outside our enclave.\n\nTo my teachers, the eighteenth-century poet Phillis Wheatley was the first of the firsts, a beacon for black children. My parents were teachers, too, and at home they filled in the details. Wheatley was a child stolen from across the Atlantic and enslaved. A young genius whose playthings were the poems of Homer and Terence, she was the first African woman on this side of the Atlantic to publish a book of poetry. Neither of my parents liked her poetry much, but that wasn't the point. The point was loyalty to the race, to African American men and women. This probably wasn't my first lesson about the responsibilities of being a black person, but it's the first one I remember and the most lasting.\n\nI don't recall my elementary school teachers or my parents ever mentioning Wheatley's husband. I believed she never had married before dying at the young age of thirty-four, and I found it heartbreaking that she did not have someone to love from her native land, someone who looked like her and shared the same cultural memories. All she had was the white people who used to own her. When I first encountered information about Wheatley's husband, John Peters, in my junior year of college, I was confronted with the dominant negative stereotypes of black men. Those thirty-five words describe him as an arrogant good-for-nothing who deserted his family.\n\nTalladega College, where I was studying at the time, was founded by former slaves. The campus is situated in a rural Alabama town, but smack in the middle of the 'hood. As I read the words portraying Peters, how he had abandoned Wheatley and their children, leaving them to poverty and then eventual death in the midst of squalor, images of black men came to me: the shiftless brothers who hung at the edge of campus, townies waiting for college girls. They sometimes shouted to us and promised pleasures of all kinds. These were the young men my doggedly middle-class mother had warned me about, in her alto, cigarette-tuned voice. My mother had fought her way up from backwoods poverty in rural Georgia and she cautioned me: One wrong step with a man could land me in perdition; living in a shack or a one-room apartment surrounded by my screaming, misbehaved progeny. As a formerly poor person, my mother looked down her nose at poor, black folk who had not escaped to tell the advisory tale, as she had.\n\nI thought of Wheatley, the \"Ethiop\" genius, taken in by Peters's charms, falling from her magnificent perch. I pictured her, beguiled by a man who whispered in her ear, told her lies to get into her starched, Good Negro bloomers.\n\n\u2022 \u2022 \u2022\n\nMuch of the information on Wheatley's personal life comes from Memoir and Poems of Phillis Wheatley, A Native African and a Slave, a book published in 1834, fifty years after the poet's death. The author, Margaretta Matilda Odell, identifies herself as a \"collateral descendant\" of Phillis's former mistress. In her biography of Wheatley, she pushes a well-meaning abolitionist message: Black folks do not deserve to be slaves, and someone like Wheatley is the example of what her brethren could be, if they only had a chance.\n\nAccording to Odell, the child-who-would-be-renamed-Phillis was \"supposed to have been about seven years old, at this time, from the circumstance of shedding her front teeth\" when she arrived in Boston. Susannah Wheatley, the wife of a merchant, was looking for a \"faithful domestic in her old age.\" Instead, she found \"the poor, naked child\" with a piece of cloth tied around her like a skirt. Once the child was taken home,\n\nA daughter of Mrs. Wheatley, not long after the child's first introduction to the family, undertook to learn her to read and write; and, while she astonished her instructress by her rapid progress, she won the good will of her kind mistress, by her amiable disposition and the propriety of her behavior.\n\nAnd with the help of Susannah, the smart and well-behaved Phillis began writing poetry. Odell doesn't give specific dates about Phillis's journey to freedom, but later biographers of the poet do. On May 8, 1773, she sailed to London (accompanied by the Wheatleys' grown son, Nathaniel) to promote her work, staying there six weeks. According to Wheatley scholar Vincent Carretta, on September 9, 1773, advertisements appeared for Phillis's only book of poetry, Poems on Various Subjects, Religious and Moral. We know that upon her return from London, her owners freed her, because Phillis mentions this fact in a letter dated October 18, 1773.\n\nWe can't know if Phillis hoped to return to her African homeland after receiving her freedom, but we do know that she retained a connection to the continent and its people. She developed a years-long friendship with Obour Tanner, a fellow kidnapped African who lived in Newport, Rhode Island. For several years, these two women exchanged fervent, spiritual thoughts in their letters. Phillis dedicated a poem to \"S.M., A Young African Painter,\" and she references Africa in several of her poems, too. \"To the Right Honorable William, Earl of Dartmouth\" contains lines about her involuntary\u2014and possibly violent\u2014migration from her native land:\n\nShould you, my lord, while you peruse my song,\n\nWonder from whence my love of Freedom sprung,\n\nWhence flow these wishes for the common good,\n\nBy feeling hearts alone best understood,\n\nI, young in life, by seeming cruel fate\n\nWas snatch'd from Afric's fancy'd happy seat:\n\nWhat pangs excruciating must molest,\n\nWhat sorrows labour in my parent's breast?\n\nOdell paints the Wheatleys as kind masters, and draws an especially sympathetic portrait of Susannah, who acted as an eighteenth-century stage mother to push forward Phillis's career. Susannah wrote to Selina, Countess of Huntingdon, a philanthropist and a leader in the Methodist movement, in an attempt to secure her patronage for the young poet. (The countess replied to this letter on May 13, 1773.) Phillis herself speaks lovingly of Susannah, and she continued to live with the Wheatleys after receiving her freedom. In a letter to Obour (dated March 21, 1774), Phillis writes of the death of her former mistress, and how a white woman treated Phillis as less a \"servant\" and more a \"child.\" But as we look back on this era, kindness must be viewed through a complex prism, for slavery was a scatological, morally bankrupt enterprise. Besides Phillis, the Wheatleys owned at least one other slave and they did not raise their voices publicly or act overtly against the institution of slavery. Odell seems to think that Phillis was living the kidnapped African's dream, however, and that, after the death of Susannah, that dream collapsed:\n\nAt this period of destitution, Phillis received an offer of marriage from a respectable colored man of Boston. The name of this individual was Peters. He kept a grocery in Court-Street, and was a man of very handsome person and manners; wore a wig, carried a cane, and quite acted out \"the gentleman.\" In an evil hour he was accepted; and he proved utterly unworthy of the distinguished woman who honored him by her alliance. He was unsuccessful in business, and failed soon after their marriage; and he is said to have been both too proud and too indolent to apply himself to any occupation below his fancied dignity. Hence his unfortunate wife suffered much from this ill-omened union.\n\nHaving written in great, flattering detail about the poet's years with the white Wheatleys, Odell uses her talents to draw a contemptuous likeness of John Peters. She informs us that Phillis never used the last name of her husband\u2014and, it's implied, we should assume that this decision had something to do with Peters's qualities as a mate. Odell accuses him of possible abuse, writing in delicate terms that while Phillis was in very bad health, she wouldn't have been \"unmindful . . . of her conjugal or matronly duties.\" In other words, Peters pressed his frail wife into sexual service when he shouldn't have, which resulted in (according to Odell) three pregnancies. From there, Odell opines, Wheatley's destruction was a foregone conclusion: There was terrible poverty. Each of the three children born to Wheatley fell sick in infancy and died. And at the age of thirty-three or -four, Wheatley herself died from an illness exacerbated by her \"extreme misery\" in living in \"a filthy apartment\" with a \"negligent\" husband.\n\nThis is the story branded into literary history.\n\n\u2022 \u2022 \u2022\n\nIn 2003, while working on my third book of poetry, I read an essay on Wheatley written by Henry Louis Gates, Jr., in The New Yorker. It was an excerpt from his soon-to-be-published book, a treatment of Wheatley juxtaposed against the racism of Enlightenment scholars such as Immanuel Kant, and more specifically, Thomas Jefferson. As someone who explored American history in my poetry, I found Gates's thesis fascinating: He believed Wheatley was important in dispelling derisive eighteenth-century notions about black humanity; her poetry had rebutted Kant's ordering of the nations with Africans down at the very bottom. Because of Wheatley's important symbolism for black humanity, Thomas Jefferson's negative response to Wheatley's poetry\u2014\"[t]he compositions published under her name are below the dignity of criticism\"\u2014was a symbol as well. It meant that the struggle for black equality on all fronts was not yet won. And thus, Gates argues, an intellectual movement was born, one that triggered a wave of eighteenth-century black literary and scholarly production, which persisted into the 1960s and continues into contemporary times.\n\nMy encounter with Gates's article started me on a Wheatley reading jag. For the next six years, I read everything I could find on her. I checked books out from the library, I downloaded scholarly articles, and I began to think deeply about her most (in)famous poem, \"On Being Brought from Africa to America.\" This eight-line poem begins with discordancy, with seeming racial self-hatred combined with religious fervor. The tone of these verses earned Wheatley sharp, ugly criticism from other black poets, most notably the Black Arts Movement poet Amiri Baraka (n\u00e9 LeRoi Jones):\n\n'Twas mercy brought me from my Pagan land,\n\nTaught my benighted soul to understand\n\nThat there's a God, that there's a Saviour too:\n\nOnce I redemption neither sought nor knew.\n\nAs a woman in my (then) thirties, I had a different take from Baraka. I thought about a little girl's pain at being torn from her parents in Africa, her trauma on board a slave ship. I thought of her mother's grief, her wondering what had become of her child. I thought about my own and other black folks' beliefs in a benevolent God, in spite of our history in this country, the brutality enacted against us. And in a burst of empathy, I wrote these lines:\n\nMercy: what Phillis claimed\n\nafter that sea journey.\n\nJourney.\n\nLet's call it that.\n\nLet's lie to each other.\n\nNot early descent into madness.\n\nNaked travail among filth and rats.\n\nWhat got Phillis over the sea?\n\nWhat kept a stolen daughter?\n\nPerhaps it was mercy,\n\nDear Reader.\n\nMercy,\n\nDear Brethren.\n\nHowever, until I traveled to Worcester, Massachusetts, to the American Antiquarian Society, I had no idea that the devastating picture of the naked, gap-toothed child wrapped in a carpet may have been Odell's imaginary reflection. It was 2009, and I was the recipient of the society's artist fellowship. Its archives house one of the largest collections of printed material from early colonial days through 1876, about the United States, the West Indies, and Canada. I was on a mission to write a series of poems based upon Wheatley's life, and I was in search of primary resources.\n\nAt the beginning of my fellowship, I was ready to get to work. Though I'd been conducting archival research for nearly twenty years, I wasn't formally trained as a historian, but as the research librarians remarked, I was quick and a self-starter. In only a matter of days, I found references to Odell's Memoir and Poems of Phillis Wheatley, A Native African and a Slave. Looking through the bibliographies in texts on Wheatley, I noticed that they either cited Memoir directly or they summarized Odell and listed a relative of the Wheatley family as a reference. There was no overt tracing of Odell's lineage, no proof of how she was related to the Wheatleys, no way to establish Odell's authority.\n\nIn July, around the middle of my fellowship term, I drove from Worcester to the Northeast National Archives in Waltham, Massachusetts. It was at the urging of my mentor at the society that I made the drive, even after she told me that records would be on microfiche; the very mention of microfiche made me sick to my stomach. I spent a couple hours looking through the census records, and as I feared, it was not the exhilarating process I'd hoped for. My eyeballs ached and the lobster roll from the day before threatened to repeat on me.\n\nI was ready to return to Worcester, when I saw John Peters on the 1790 census of Suffolk County, Massachusetts\u2014the city of Boston. He was listed as a free man of color.\n\nNo, it wasn't a mistake. There was Peters's name.\n\nSwallowing my nausea, I rechecked the entire census, just to be sure. There was no other black John Peters, the narcissistic man who had abandoned his wife off and on, and then\u2014as Odell had written\u2014supposedly had moved farther south after his wife's death. I looked the census over completely two more times and took pictures of the relevant pages.\n\nI sat there, confused. Rather than verifying facts about America's first black poet, which had been my intention, I realized literary history had entrusted the story of Wheatley and Peters to a white woman who may have made assumptions about Wheatley's husband that might not just be wrong, but also the product of racial stereotypes.\n\nWhy would Peters have moved farther south after the Revolution? This piece of it didn't make sense to me. Why would a free, black man in his natural, right mind move south, taking his body to slaveholding territory, where white men would be waiting to place him in chains?\n\nWheatley had died in 1784, but the census had been taken in 1790. It's possible, then, that Peters had been in Boston through that decade, which meant that Peters may have been in Boston when Wheatley died. What could this mean? Had the young couple been separated? Had he left her for another woman? Had she left him? Maybe they had remained together. Maybe he hadn't abandoned her. Maybe Odell misrepresented their relationship. And if Odell had misrepresented the relationship of Wheatley and Peters, maybe she had done the same about her relationship to the white Wheatleys.\n\nI drove back to the AAS and huddled with my mentor and the research librarian. I asked them what Odell meant in her book when she claimed to be a \"collateral descendant\" of the white Wheatleys? A cousin? A niece? An in-law married to a direct descendant? Both of them advised me to look up Odell in the New England Historic Genealogical Society. I did, and I found Margaretta Matilda Odell of \"Jamaica Plain, Massachusetts\"\u2014and that's all I found. Nothing else.\n\nI returned to the texts on Wheatley. In each, I double-checked the notes and indexes several times, sure that I had overlooked something. Every night, back in my fellow's room, I took hours to draft possible genealogies of the blood relatives and in-laws of Susannah and John Wheatley, and those of their twins, Mary Wheatley Lathrop and Nathaniel Wheatley. I uncovered no documentation connecting Odell to the white Wheatleys. There was no establishment of family bona fides. Rather, it appeared that the only proof that Odell had been related to Susannah Wheatley, the former mistress of Phillis Wheatley, was that she had said so.\n\n\u2022 \u2022 \u2022\n\nVincent Carretta, the author of Phillis Wheatley: Biography of a Genius in Bondage (2011)\u2014to date, the most comprehensive biography of the poet\u2014has unearthed a treasure trove of previously unpublished material on Phillis Wheatley and her husband: legal documents, newspaper notices, records in Boston's Taking Book, along with other important minutiae. Still, even Carretta doesn't know how or when Wheatley met Peters. There is a reference to a \"young man\" in a letter she wrote her friend Obour, on October 30, 1773, but we don't know his identity. We do know that in 1778, Phillis Wheatley and John Peters, \"free Negroes,\" married during the tumultuous period of the American Revolution. In a letter to Obour (dated May 10, 1779), the poet signs herself as \"Phillis Peters\"; thereafter, whenever she refers to herself in print, she always uses her married name.\n\nIn the years leading up to the Revolution and directly afterward, Massachusetts was the site of black political agitation. Just as I had heard the name of Phillis Wheatley in elementary school, so had I learned about Crispus Attucks, a biracial African American and the first to fall during what became known as the Boston Massacre. Over the years, I would learn the names of others, like Lemuel Haynes, a minister who had fought in the Revolution, and Prince Hall, who founded the African Masons. There was Belinda, who petitioned the Massachusetts Assembly for a pension in her old age. And I would read the words of Felix, an unidentified black man\u2014and presumably a slave\u2014who petitioned the same body, demanding his freedom and that of other African American men:\n\nWe have no Property. We have no Wives. No Children. We have no City. No Country. But we have a Father in Heaven, and we are determined, as far as his Grace shall enable us, and as far as our degraded contemptuous Life will admit, to keep all his Commandments. . . .\n\nWhen I view Peters through the lens of the eighteenth century, he fits in quite easily with his brethren. Carretta depicts him as a smart, hard worker, trying his hand in different business enterprises: law, commerce, real estate, even medicine. (The latter was not the profession that we know today, and required no specific schooling.)\n\nAs a white woman of the nineteenth century Odell fits in perfectly with her era, too. It doesn't take much speculation to deduce that she believed Peters to be an uppity Negro. He was a black man who had the nerve to possess high self-esteem, who cajoled Wheatley away from her white friends. Even though Odell dedicates her book to \"friends of the Africans,\" her tone ridicules his ambitions: \"[Peters] is said to have been both too proud and too indolent to apply himself to any occupation below his fancied dignity.\" In other words, how dare a black man want to be anything other than a day laborer with calluses on his hands? Who did he think he was, to desire property and not be property, to style himself as a business owner, to marry a high-status, accomplished woman of his own race?\n\nThere are other second- or third-hand, derisive accounts of Wheatley's husband, all by whites. Carretta quotes from Josiah Quincy, who claims to have met John Peters in court, and who didn't think much of the encounter. While doing my own research, I found a footnote in the November 1863 Proceedings of the Massachusetts Historical Society, which claims that an acquaintance of Obour Tanner told someone else that Tanner had told her\u2014keep up, now; this is getting complicated\u2014that Tanner did not like Peters, that Wheatley had \"let herself down\" by marrying him. But this same footnote giving this ostensible inside information also gets Wheatley's death date wrong, by ten years.\n\nEven if Wheatley did, as Odell claims, give birth to children who died in infancy\u2014and at present, there is no documentation for that\u2014infant mortality rates were disturbingly high during this period. It was not uncommon for parents to lose several offspring in infancy or childhood, even those parents who fed, clothed, and loved their children. There would have been nothing for Peters to forestall a child's death from a disease such as measles. As for Wheatley's passing and whether Peters had a direct or indirect hand in it, there is no proof that he pushed her into an early grave, either. In the eighteenth century, life spans were short for whites, and even shorter for African Americans. Wheatley died around the age of thirty-three, which, unfortunately, is in keeping with life expectancies for black women in America at that time.\n\nCarretta supplies evidence that, at the time of Wheatley's death, Peters was living in Massachusetts indeed, but he was in prison because he couldn't pay his bills. (In twenty-first-century terms, he had bad credit.) This is not a crime by our contemporary standards, but it was during Peters's time. There was an economic depression in New England, in the aftermath of the American Revolution; many people in Massachusetts, black and white, couldn't pay their bills or even afford food. Starvation was not unheard of.\n\nPeters was released from jail, and, according to Carretta, for the next sixteen years, he continued to aspire to the role of gentleman. And now we have definitive proof that Peters also kept trying to publish the second book of poetry that his late wife had written. In October 2015, I corresponded (by e-mail) with the assistant curator of manuscripts at the American Antiquarian Society. Knowing my interest in the Wheatley-Peters marriage, she shared with me an excerpt of a never-published letter (dated June 2, 1791) between the printers Ebenezer T. Andrews and Isaiah Thomas. In this letter, Andrews refers to a \"proposal\" for \"Wheatley's Poems,\" and that he had promised Peters that they would \"print them\" and split the \"neat profits with him.\" But bewilderingly, that second book never appeared.\n\nCarretta notes that Peters died in 1801; he was fifty-five at the time of his death. He never was able to pay off his debts, but he left some nice belongings behind. A horse, a desk, some leather-bottomed chairs. Books, which meant he not only was literate, but may also have enjoyed reading.\n\nWhen you look at Peters's life, okay, the brother \"did a couple bids,\" but at least he didn't leave behind any people that had to be sold to erase his debts, as Jefferson did. Families were broken up, auctioned off, and sifted like chaff. That would be the fate of the slaves of Monticello after Jefferson died. I can't help but wonder what Odell would have thought of his actions.\n\n\u2022 \u2022 \u2022\n\nI have continued my research on Wheatley. I regularly search for new information; I read new articles about her, and always, I check the notes and the bibliography. Periodically, I look for primary materials to see if any new information on her has emerged. When finances permit, I travel and do my primary research in person.\n\nIn the meantime, I publish poems based upon Wheatley research. In 2010, I published an essay on her as well, mentioning my sighting of Peters on that 1790 census, discussing the unproven connection between Odell and the Wheatley family:\n\n[It] is distressing that, in 176 years, scholars have not questioned Odell's right to speak for Phillis Wheatley. This blind trust continues the disturbing historical trend of African Americans, and black women in particular, needing white benefactors to justify their lives and history.\n\nThe thing is, I have fallen in love with Wheatley and I want to do right by her legacy. I want to get everything correct, but if I'm not the one to uncover new information, if someone else finds it, that isn't a problem for me. I just want it to be found. I have hoped that by pointing to the absence of documentation on Odell, researchers will take notice and renew the search for her genealogy. If no family records can be identified, then the responsible, professional cause of action would be to cease using Odell as a primary source for Wheatley's life. The other option would be to categorize Memoir as historical fiction, but whatever the categorization, someone must directly challenge Odell's authority to provide the most enduring depiction of Wheatley, and of her husband as a sycophant and a hustler.\n\nI've tried to be pragmatic when it comes to the work of Wheatley biographers and scholars. Research is hard. It's time-consuming and frustrating; I know that from personal experience. Furthermore, there often isn't much information to go on. For example, if Odell's Memoir were to be eliminated as a primary source for Wheatley's life, what else would be left to rely upon? Precious little. Yes, a chronicle that may not be fully accurate is more than exists for most eighteenth-century, formerly enslaved black folks, but really, these traces provide a pitiful tribute to the woman who is the mother of African American literature.\n\nNever mind that controversial, beginning line of her poem \"'Twas mercy brought me from my Pagan land . . .\" Wheatley is much more than that. She proved something to white people about us: that we could read and think and write\u2014and damn it, we could feel, no matter what the racists believed. We already knew those things about ourselves. I'm pretty positive about that, but during her time, philosophers were arranging the \"nations\" with Africans at the bottom, while other Europeans measured black people's skulls alongside those of orangutans to determine if the two species were kissing cousins. In the midst of these soul assaults, Wheatley's poems carried the weight for African people on this side of the Atlantic. As a result, Wheatley\u2014along with black soldiers and sailors who fought on the winning side of the American Revolution, black intellectuals and writers, and various individuals of African descent asserting their God-given rights of liberty\u2014helped to sway many white Americans and Europeans that slavery was wrong.\n\nYet I'm waiting for someone to write a more emotionally charged book about Wheatley, one that would take into account her pre-American existence. Although she was a little girl when she arrived in Boston, and although the Wheatleys were \"kind\" to her, she did have African birth parents. Her life did not begin in America or with slavery. She had a free lineage that did not include the Wheatleys. If nothing else, a treatment of precolonial West African history, along with the eighteenth-century culture of that region, would be an appropriate and respectful introduction to Wheatley's life in America.\n\nIn addition, I'm waiting for someone to include a compassionate, well-fleshed depiction of John Peters, which considers how he fit into African American intellectual, commercial, and activist life of the Revolutionary era. Perhaps I seem na\u00efve or silly, but I'd like scholars to view him as a natural occurrence in Wheatley's trajectory, instead of a low-down disruption that led to her demise. Oddly, no account that I've read of Peters gives the most obvious, commonsense reason for why Wheatley might have married him.\n\nMaybe he didn't trick her. She wasn't desperate or temporarily out of her mind. They married because they were deeply, passionately in love.\n\nIs that explanation so ridiculous? Why wouldn't they love each other? American people of African descent did fall in love back then, and, if allowed by local power structures, they legally married. They did this in the midst of war, slavery, economic chaos, and\/or posttraumatic stress over being torn from their homelands and sent over the horrific Middle Passage. I think it's logical to assume that many, many black folk fell in love with many, many other black folk. This assumption is a rational consequence of acknowledging black humanity.\n\nAt times, when I'm impatiently waiting for scholars to reexamine the complicated realities of these two people, I imagine Phillis and John, what their moments together might have been.\n\nMaybe Peters thought Wheatley was beautiful. He was drawn to her delicate face, to her very dark skin, her full lips, her tight, kinky hair, to the ring in her nose that might have been an ornament she carried from across the water. (Look very closely at that engraving in her book. Use a magnifying glass and you will see that nose ring.)\n\nAnd maybe Wheatley thought Peters handsome. He might have looked like her relatives, back in the Gambia that she wrote about. She and Peters might have shared a hankering for a place that lived only in their memories. He might have been born in America\u2014we probably will never know\u2014but in any case, he would have been of African heritage. Maybe at night, when they settled down together in their rickety bed, they talked in whispers, telling each other stories of that faraway place across the water. Folktales or proverbs that had been passed down.\n\nHe possessed ambitions, the same as she, and instead of stories, maybe they talked about the future, their hopes for his fledgling businesses and her new book of poetry, the glories that would be accomplished by their children. Anything was possible in that time, when messages of liberty abounded.\n\nMaybe he was a tender lover and they laughed and cried and clutched. The words they spoke after their passion were to be believed, even though they came from the mouths of black folk.\n\n* * *\n\nI. The title of this essay uses a line from Phillis Wheatley's \"To a Clergyman on the Death of His Lady,\" published in Poems on Various Subjects, Religious and Moral (1773).\n\n## White Rage\n\n## CAROL ANDERSON\n\nWhen we look back on what happened in Ferguson, Missouri, during the summer of 2014, it will be easy to think of it as yet one more episode of black rage ignited by yet another police killing of an unarmed African American male. But that has it precisely backward. What we've actually seen is the latest outbreak of white rage. Sure, it is cloaked in the niceties of law and order, but it is rage nonetheless.\n\nProtests and looting naturally capture attention. But the real rage smolders in meetings where officials redraw precincts to dilute African American voting strength or seek to slash the government payrolls that have long served as sources of black employment. It goes virtually unnoticed, however, because white rage doesn't have to take to the streets and face rubber bullets to be heard. Instead, white rage carries an aura of respectability and has access to the courts, police, legislatures, and governors, who cast its efforts as noble, though they are actually driven by the most ignoble motivations.\n\nWhite rage recurs in American history. It exploded after the Civil War, erupted again to undermine the Supreme Court's Brown v. Board of Education decision, and took on its latest incarnation with Barack Obama's ascent to the White House. For every action of African American advancement, there's a reaction, a backlash.\n\nThe North's victory in the Civil War did not bring peace. Instead, emancipation brought white resentment that the good ol' days of black subjugation were over. Legislatures throughout the South scrambled to reinscribe white supremacy and restore the aura of legitimacy that the antislavery campaign had tarnished. Lawmakers in several states created the Black Codes, which effectively criminalized blackness, sanctioned forced labor, and undermined every tenet of democracy. Even the federal authorities' promise of 40 acres\u2014land seized from traitors who had tried to destroy the United States of America\u2014crumbled like dust.\n\nInfluential white legislators such as Rep. Thaddeus Stevens (R-Pa.) and Sen. Charles Sumner (R-Mass.) tried to make this nation live its creed, but they were no match for the swelling resentment that neutralized the Thirteenth, Fourteenth, and Fifteenth amendments, and welcomed the Supreme Court's 1876 United States v. Cruikshank decision, which undercut a law aimed at stopping the terror of the Ku Klux Klan.\n\nNearly eighty years later, Brown v. Board of Education seemed like another moment of triumph\u2014with the ruling on the unconstitutionality of separate public schools for black and white students affirming African Americans' rights as citizens. But black children, hungry for quality education, ran headlong into more white rage. Bricks and mobs at school doors were only the most obvious signs. In March 1956, 101 members of Congress issued the Southern Manifesto, declaring war on the Brown decision. Governors in Virginia, Arkansas, Alabama, Georgia, and elsewhere then launched \"massive resistance.\" They created a legal doctrine, interposition, that supposedly nullified any federal law or court decision with which a state disagreed. They passed legislation to withhold public funding from any school that abided by Brown. They shut down public school systems and used tax dollars to ensure that whites could continue their education at racially exclusive private academies. Black children were left to rot with no viable option.\n\nA little more than half a century after Brown, the election of Obama gave hope to the country and the world that a new racial climate had emerged in America, or that it would. But such audacious hopes would be short-lived. A rash of voter-suppression legislation, a series of unfathomable Supreme Court decisions, the rise of stand-your-ground laws, and continuing police brutality make clear that Obama's election and reelection have unleashed yet another wave of fear and anger.\n\nIt's more subtle\u2014less overtly racist\u2014than in 1865 or even 1954. It's a remake of the Southern Strategy, crafted in the wake of the civil rights movement to exploit white resentment against African Americans, and deployed with precision by Presidents Richard Nixon and Ronald Reagan. As Reagan's key political strategist, Lee Atwater, explained in a 1981 interview: \"You start out in 1954 by saying, 'N\u2014\u2014-, n\u2014\u2014-, n\u2014\u2014-.' By 1968 you can't say 'n\u2014\u2014-'\u2014that hurts you. Backfires. So you say stuff like 'forced busing,' 'states' rights,' and all that stuff. You're getting so abstract now you're talking about cutting taxes, and all these things you're talking about are totally economic things, and a byproduct of them is blacks get hurt worse than whites. And subconsciously maybe that is part of it. I'm not saying that.\" (The interview was originally published anonymously, and only years later did it emerge that Atwater was the subject.)\n\nNow, under the guise of protecting the sanctity of the ballot box, conservatives have devised measures\u2014such as photo ID requirements\u2014to block African Americans' access to the polls. A joint report by the NAACP Legal Defense and Educational Fund and the NAACP emphasized that the ID requirements would adversely affect more than 6 million African American voters. (Twenty-five percent of black Americans lack a government-issued photo ID, the report noted, compared with only 8 percent of white Americans.) The Supreme Court sanctioned this discrimination in Shelby County v. Holder, which gutted the Voting Rights Act and opened the door to twenty-first-century versions of nineteenth-century literacy tests and poll taxes.\n\nThe economic devastation of the Great Recession also shows African Americans under siege. The foreclosure crisis hit black Americans harder than any other group in the United States. A 2013 report by researchers at Brandeis University calculated that \"half the collective wealth of African-American families was stripped away during the Great Recession,\" in large part because of the impact on home equity. In the process, the wealth gap between blacks and whites grew: Right before the recession, white Americans had four times more wealth than black Americans, on average; by 2010, the gap had increased to six times. This was a targeted hit. Communities of color were far more likely to have riskier, higher-interest-rate loans than white communities, with good credit scores often making no difference.\n\nAdd to this the tea party movement's assault on so-called Big Government, which despite the sanitized language of fiscal responsibility constitutes an attack on African American jobs. Public-sector employment, where there is less discrimination in hiring and pay, has traditionally been an important venue for creating a black middle class.\n\nSo when you think of Ferguson, don't just think of black resentment at a criminal justice system that allows a white police officer to put six bullets into an unarmed black teen. Consider the economic dislocation of black America. Remember a Florida judge instructing a jury to focus only on the moment when George Zimmerman and Trayvon Martin interacted, thus transforming a seventeen-year-old, unarmed kid into a big, scary black guy, while the grown man who stalked him through the neighborhood with a loaded gun becomes a victim. Remember the assault on the Voting Rights Act. Look at Connick v. Thompson, a partisan 5\u20134 Supreme Court decision in 2011 that ruled it was legal for a city prosecutor's staff to hide evidence that exonerated a black man who was rotting on death row for fourteen years. And think of a recent study by Stanford University psychology researchers concluding that when white people were told that black Americans are incarcerated in numbers far beyond their proportion of the population \"they reported being more afraid of crime and more likely to support the kinds of punitive policies that exacerbate the racial disparities,\" such as three-strikes or stop-and-frisk laws.\n\nOnly then does Ferguson make sense. It's about white rage.\n\n## Cracking the Code\n\n## JESMYN WARD\n\nWhen my father moved to Oakland, California, after Hurricane Camille wrecked the Mississippi Gulf Coast, in 1969, strangers he encountered from El Salvador and Mexico and Puerto Rico would spit rapid-fire Spanish at him, expecting a reply in kind. \"Are you Samoan?\" a Samoan asked him once. But my father, with his black, silky hair that curled into Coke-bottle waves at the ends, skin the color of milky tea, and cheekbones like dorsal fins breaking the water of his face, was none of these things. He attended an all-black high school in Oakland; in his class pictures, his is one of the few light faces. His hair is parted in the middle and falls away in a blowsy afro, coarsened to the right texture by multiple applications of relaxer.\n\nMy father was born in 1956 in Pass Christian, a small Mississippi town on the coast of the Gulf of Mexico, fifty miles from New Orleans. He grew up in a dilapidated single-story house: four rooms, with a kitchen tacked onto the back. It was built along the railroad tracks and shook when trains sped by; the wood of the sloped floor rotted at the corners. The house was nothing like the great columned mansions strung along the man-made beach just half a mile or so down the road, houses graced with front-facing balconies so that the wealthy white families who lived in them could gaze out at the flat pan of the water and the searing, pale sand, where mangrove trees had grown before they'd bulldozed the land.\n\nPut simply, my father grew up as a black boy in a black family in the deep South, where being black, in the sixties, was complicated. The same was true in the eighties, when I was growing up in DeLisle, a town a few miles north of Pass Christian. Once, when I was a teen, we stood together in a drugstore checkout line behind an older, blondish white woman. My father, an inveterate joker, kept shoving me between my shoulder blades, trying to make me stumble into her. \"Daddy, stop,\" I mouthed, as I tried to avoid a collision. I was horrified: Daddy's going to make me knock this white woman over. Then she turned around, and I realized that it was my grandaunt Eunice, my grandmother's sister\u2014that she was blood. \"I thought you were white,\" I said, and she and my father laughed.\n\nCoastal Mississippi is a place where Eunice\u2014a woman pale as milk, with blond hair and African heritage\u2014is considered, and considers herself, black. The one-drop rule is real here. Eunice wasn't allowed on the beaches of the Gulf Coast or Lake Pontchartrain until the early seventies. The state so fiercely neglected her education that her grandfather established a community school for black children. Once Eunice graduated, after the eighth grade, her schooling was done. She worked in her father's fields, and then as a cleaning woman for the white families in their mansions on the coast. On the local TV station, she watched commentators discuss what it meant to be a proper Creole, women who were darker than her asserting that true Creoles have only Spanish and French ancestry. Theirs was part of an ongoing attempt to write anyone with African or Native American heritage out of the region's history; to erase us from the story of the plantations, the swamps, the bayou; to deny that pla\u00e7age, those unofficial unions, during the time of antimiscegenation laws, between European men and women of African heritage had ever taken place.\n\nIt's impossible for most black Americans to construct full family trees. Official census records, used by so many genealogy enthusiasts to piece together their families' pasts, don't include our non-European ancestors. Both my mother's and my father's family name is Dedeaux (I bear my paternal grandmother's last name), and several relatives on my mother's side have traced their lineage through European Dedeauxs back to France, but building a family tree of people of color is far harder. I always understood my ancestry, like that of so many others on the Gulf Coast, to be a tangle of African slaves, free men of color, French and Spanish immigrants, British colonists, Native Americans\u2014but in what proportion, and what might that proportion tell me about who I thought I was?\n\nI was at a dinner with some professors from Spring Hill College in Mobile, Alabama, when one of them told me about the genetic-testing company 23andMe. It cost ninety-nine dollars\u2014that was my first surprise. I imagined that the price of such a service would be exorbitant, but evidently it wasn't. You order a kit online, the professor explained, and get it in the mail a week or so later, then register it on the company's website, spit into a test tube, seal it, and send it back in the provided box. Around six weeks later, you receive your results. The professor said that his girlfriend had spent hours poring over hers, fascinated by her genetically based health analysis. (Due to an FDA crackdown, 23andMe no longer provides that particular service.) But I was interested in genetic testing for a different reason.\n\nI ordered tests for my father, my mother, and myself. We submitted our samples, then waited for the company's scientists to decode the ancestral information in our DNA.\n\nMy mother and I were sitting at her kitchen table when her test came back. My father was at my sister's house, surrounded by his children, when he received his. Their results confirmed some of the notions we'd had about our ancestry, as passed down through family lore, and subverted others. My father, who'd always believed himself to have Native American heritage, and who had a strong affinity for Native American history and culture, found that he is 51 percent Native American, as well as nearly equal parts sub-Saharan African and European (British, Irish, Spanish, and Ashkenazi)\u201423.5 percent and 22.5 percent, respectively\u2014and just over 1 percent North African. My mother, who has told me story after story about her white great-grandparents taking their mixed-race children to visit their families in Kiln, Mississippi, only to hide the kids in the trunk of the car at the end of every visit when the sun set and it was no longer safe, found that she is 55 percent European\u2014a mixture of British, Irish, French, German, Scandinavian, and Iberian\u201441 percent sub-Saharan African, and 3.4 percent Native American.\n\nMy parents' results gave them the concrete proof of their ancestry that they'd always been denied. My father, a former member of the Black Panther party, proudly claimed his Native American heritage by registering with the Choctaw tribe of Slidell, Louisiana. My mother could at last make educated guesses about the parentage of her great-grandparents. It was as if 23andMe had taught them to read the language of their family histories, enabling them to finally understand the incomprehensible book of their ancestral pasts: to read what had been gibberish.\n\nYet I found my own results both surprising and troubling. I was raised in Mississippi, in a family and a community that identified as black, and I have the stories and the experiences to go with it. One of my great-great-grandfathers was killed by a gang of white Prohibition patrollers. My mother helped to integrate the local elementary school in the 1960s. My father was run out of segregated Pass Christian's beaches and the local park. I was the only black girl at my private high school in Pass Christian, the target of my classmates' backward ideas about race. Despite my parents' sense of their mixed roots, I had thought that my genetic makeup would confirm the identity that I'd grown up with\u2014one that located Africa as my ancestors' primary point of origin, and that allowed me to claim a legacy of black resistance and strength.\n\nSo it was discomfiting to find that my ancestry was 40 percent European\u2014a mixture of British, Irish, French, German, Scandinavian, Iberian, Italian, and Ashkenazi\u201432 percent sub-Saharan African, a quarter Native American, and less than 1 percent North African. For a few days after I received my results, I looked into the mirror and didn't know how to understand myself. I tried to understand my heritage through my features, to assign each one a place, but I couldn't. All I could see was my hair: hair that grows up and out instead of falling flat, like my father's; hair that refuses to be as smooth and tidy as my mother's but instead bushes and tangles and curls in all directions at once. Mine is a mane that bears the strongest imprint of my African ancestors, hair that my stylist combed out into a voluminous afro during one of my visits to New York City, so that I walked the streets with a ten-inch halo that repelled the rain and spoke of Africa to everyone who saw it.\n\nThat's how I remembered myself. I remembered that people of color from my region of the United States can choose to embrace all aspects of their ancestry, in the food they eat, in the music they listen to, in the stories they tell, while also choosing to war in one armor, that of black Americans, when they fight for racial equality. I remembered that in choosing to identify as black, to write about black characters in my fiction and to assert the humanity of black people in my nonfiction, I've remained true to my personal history, to my family history, to my political and moral choices, and to my essential self: a self that understands the world through the prism of being a black American, and stands in solidarity with the people of the African diaspora.\n\nThis doesn't mean that I don't honor and claim the myriad other aspects of my heritage. I do, in ways serious and silly. I read Philip Larkin and Seamus Heaney and love all things Harry Potter and Doctor Who. I study French and Spanish and attempt to translate the simplest poems by Pablo Neruda and Federico Garc\u00eda Lorca into English (and fail awfully). I watch obscure French movies with subtitles. I attend powwows and eat fry bread and walk along the outside of the dancing circles with a kind of wistful longing because I want to understand the singing so badly, because I want to stomp the earth in exultation and to belong in that circle, too. But I imagine that my ancestors from Sierra Leone and Britain, from France and the Choctaw settlements on the Mississippi bayou, from Spain and Ghana\u2014all those people whose genetic strands intertwined to produce mine\u2014felt that same longing, even as they found themselves making a new community here at the mouth of the Mississippi. Together, they would make new music, like blues and jazz and zydeco, and new dances, second lining through the streets. They would make a world that reflected back to them the richness of their heritage, and in doing so discover a new type of belonging.\n\n# PART II\n\n# RECKONING\n\n## Queries of Unrest\n\n## CLINT SMITH\n\nAfter Hanif Willis-Abdurraqib\n\nMaybe I come from the gap\n\nbetween my father's teeth.\n\nMaybe I was meant to see a little\n\nbit of darkness every time he smiled.\n\nMaybe I was meant to understand that\n\ndarkness magnifies the sight of joy.\n\nMaybe I come from where the sidewalk\n\nends, or maybe I just read that in a book once.\n\nIt can be hard to tell the difference sometimes.\n\nMaybe that's because when I was a kid\n\na white boy told me I was marginalized\n\nand all I could think of was the edge\n\nof a sheet of paper, how empty it is\u2014\n\nthe abyss I was told never to write into.\n\nMaybe I'm scared of writing another poem\n\nthat makes people roll their eyes\n\nand say, \"another black poem.\"\n\nMaybe I'm scared people won't think\n\nof the poem as a poem, but as a cry for help.\n\nMaybe the poem is a cry for help.\n\nMaybe I come from a place where people\n\nare always afraid of dying.\n\nMaybe that's just what I tell myself\n\nso I don't feel so alone in this body.\n\nMaybe there's a place where everyone is both\n\nin love with and running from their own skin.\n\nMaybe that place is here.\n\nMaybe that's why I'm always running from\n\nthe things that love me. Maybe I'm trying\n\nto save them the time of burying darkness\n\nwhen all they have to do is close their eyes.\n\n## Blacker Than Thou\n\n## KEVIN YOUNG\n\nIt was never easy for me. I was born a poor black child . . .\n\n\u2022 \u2022 \u2022\n\nThe beginning of Steve Martin's The Jerk still makes me laugh with its twist on Once Upon a Time. The dissonance between what we know of the white comedian Martin, his relative success, and his obviously false declaration sends up not only the tragic showbiz biography but the corny black one: in both, the worser, the better. It also suggests his character's transformation, his overcoming\u2014after all, he's clearly white now!\u2014not to mention his current lot in which he's as smudged, bummy, apparently destitute. His isn't blackface, but his face half-greased is certainly part of the effect\u2014it's a familiar one, in other words, to black people used to watching white people only claim blackness as a \"poor me\" stance.\n\nNow, why does this jerk remind me of Rachel Dolezal?\n\n\u2022 \u2022 \u2022\n\nThere's a long-standing American tradition of whites donning blackface, or redface, or any other colored mask they pretend is a face. Those who wear blackface reduce blackness to skin in order not to be white. The implication of course is that black people are just miscolored or extra-dark white people. Many a joke told for my benefit in my Kansas grade school reinforced the same. Know why black people's palms are white?\n\nBut if you are white but truly \"feel black\" then why do you have to look like it?\n\n\u2022 \u2022 \u2022\n\nMy next nonfiction book, Unoriginal Sin, is about hoaxers and impostors, plagiarists and phonies. I finished it last week and sent it in to my publisher, elated and relieved. Now I have to take time to write about Rachel Dolezal too?\n\nI can't decide if Dolezal, the woman revealed to have been merely pretending to be black, lecturing as such and even leading her local Oregon NAACP, is the natural extension of what I've been saying in my next book, or a distraction from this larger point: that quite regularly, faced with the paradox of race, the hoax rears its head. It turns out, I now know, it rears its rear too.\n\n\u2022 \u2022 \u2022\n\nWhen Rachel Dolezal first broke, and was simply a joke on Black Twitter, I identified some of my favorite Twitter titles for the inevitable, anticipated memoir: \"Their Eyes Were Watching Oprah\" (that one's mine); \"Imitation of Imitation of Life\" (from Victor LaValle); \"Blackish Like Me\" (mine too). Now things done got serious.\n\n\u2022 \u2022 \u2022\n\nWhen you are black, you don't have to look like it, but you do have to look at it. Or look around. Blackness is the face in the mirror, a not-bad-looking one, that for no reason at all some people uglify or hate on or wish ill for, to, about. Sometimes any lusting after it gets to be a drag too.\n\n\u2022 \u2022 \u2022\n\nEvery black person has something \"not black\" about them. I don't mean something white, because despite our easy dichotomies, the opposite of black is not white. This one likes European classical music; that one likes a little bit of country (hopefully the old stuff); this one is the first African American principal ballerina; this one can't dance. Black people know this\u2014any solidarity with each other is about something shared, a secret joy, a song, not about some stereotypical qualities that may be reproducible, imitable, even marketable. This doesn't mean there aren't similarities across black people or communities or better yet memory\u2014just that these aren't exactly about bodies and not really about skin at all, but culture.\n\n\u2022 \u2022 \u2022\n\nThere is a long tradition of passing\u2014of racial crossing the line, usually going from black to white. You could say it was started, like this country, by Thomas Jefferson.\n\n\u2022 \u2022 \u2022\n\nOne of the best things about being black is that, barring some key exceptions, it's not a volunteer position. You can't just wish on a dark star and become black. It's not paid either. It's more like a long internship with a chance of advancement.\n\n\u2022 \u2022 \u2022\n\nI've never seen the TV show Blackish all the way through. (I hear it's quite good now.) From what I've seen, Fresh Off the Boat, another of ABC's offerings, seems to me a more accurate portrayal of the complexity of racial identity, even black identity. (This is despite the worries of its creator, chef and author Eddie Huang.) The young Asian immigrant who's the main character identifies with hip-hop in order to be both American and remain and help explain being nonwhite. It's funny, and frequently brilliant: How do you become American?\n\nIs this the same as becoming black?\n\n\u2022 \u2022 \u2022\n\nTraditionally, pretend blackness was the fastest route to becoming white. This is true for Irish and Jewish immigrants, who adopted blackface in large numbers in the late nineteenth and early twentieth, and soon assimilated; and for Northerners, for whom blackface helped them imagine themselves a nation since blackface's advent in the 1830s.\n\nCue that f'in caricature of Jim Crow dancing.\n\n\u2022 \u2022 \u2022\n\nLike Rachel Dolezal, I too became black around the age of five. I first became a nigger at nine, so I had me a good run.\n\n\u2022 \u2022 \u2022\n\nThe problem isn't just that Rachel Dolezal can wash off whatever she's sprayed on herself (it just don't look right), or that blackness is a choice, but that what she's wearing isn't just bronzer, but blacker: a notion that blackness is itself hyperbolic, excessive, skin tone only. Well, and wigs.\n\nThis last, some black observers have praised.\n\n\u2022 \u2022 \u2022\n\nDid Dolezal really fool those black folks around her? I have a strange feeling she didn't, that many simply humored her. You have to do this with white people, from time to time.\n\n\u2022 \u2022 \u2022\n\nBlack people are constantly identifying and recognizing those who look like secret black folks\u2014many light-skinned people I know get identified as white by white people, but we know they're black. (This isn't passing, btw.) Most look like one of my aunties. Knowing they are black, it is hard to see them another way.\n\nIt's one of the advantages of my folks being from Louisiana\u2014there's lots of folks who don't \"look black\" but are (which of course should make us stop and reevaluate what \"looking black\" is). Because of the one-drop rule, though begun as a controlling race law, black people themselves adapted and even invented and accepted a broader blackness. In general this has made black people\u2014I am speaking for every single black person without exception here, of course\u2014wary yet accepting.\n\n\u2022 \u2022 \u2022\n\nThose surprised by a white lady darkening her skin and curling her hair haven't been out of the house or online in a while.\n\n\u2022 \u2022 \u2022\n\nThere was the rather white-looking bank manager in Athens, Georgia, who chatted me up one day and mentioned a couple key black striver things\u2014a black sorority here, the Links there\u2014that let me know she was black too. It was brilliant, and in no way calculated; hers was smart survival.\n\nIt was also a test to see if I was woke, or a striver, too.\n\n\u2022 \u2022 \u2022\n\nTeaching a class about blackness doesn't mean you are black. Blackness isn't a bunch of facts to memorize, or a set of stock behaviors; nor darker skin color neither. It's like the jazz heads I've seen, often white, who can tell you every sideman on every session, but seem in the daylight unable to find the beat. The beat is there always; doesn't mean you can always hear it.\n\nWhile black folks often hear the beat, and set it, doesn't mean when anyone else hears it, that she gets to be black.\n\n\u2022 \u2022 \u2022\n\nEvery church I know of had a white lady who arrived one day. Ours in Topeka did. After she hung around awhile, and proved herself she wasn't a tourist, \"Mrs. Pete\" was accepted and seen as part of the AME congregation, even singing in the choir (which was a high bar, as it were). But we never thought she was, or somehow became, black. She's good people, folks would say.\n\nShe did get herself a perm: I mean a white, curly one, instead of a straightened, black one; a clarification that's one more sign we're awfully mixed up. There's the joke: You didn't get yourself a perm, but a temporary.\n\n\u2022 \u2022 \u2022\n\nThere is the other, far rarer passing, which we may call reverse-passing, of whites living as black. The most prominent I know of may be Johnny Otis\u2014who was successful enough that many race women and men I know aren't aware he was actually born white. Or the Baseball Hall of Fame inductee, owner of a Negro League team who likely wasn't black herself. What's interesting is to wonder what the black people around them thought, usually accepting them\u2014not necessarily as what they said they were, but how they acted. It isn't that they weren't judged, just that when they were, they weren't found wanting.\n\n\u2022 \u2022 \u2022\n\nSo when the killer [name withheld] walked into Mother Emanuel Church in Charleston one week after the Dolezal story broke, I am not surprised that the black worshippers there welcomed him. Welcome is an integral part of the African American Christian tradition; it is especially so in the African Methodist Episcopal one, begun over two hundred years ago when the Methodist church prevented blacks, mostly freedmen and women, to pray beside its whites, even pulling them off their knees.\n\nHow long did [name redacted] sit there waiting, deciding to deny the evidence of humanity before him? Nothing, it appears, could have convinced him not to kill blacks, whom he believed\u2014and spewed hate about\u2014preyed on white people, especially women. One suspects he may not've known any women besides his family.\n\n\u2022 \u2022 \u2022\n\nThomas Jefferson hated black people but slept with one who bore his children, six of them. (Misery is often the parent of the most affecting touches in poetry.\u2014Among the blacks is misery enough, God knows, but no poetry, he wrote in Notes on Virginia.) That Sally Hemings was also his wife's half-sister neither stopped him nor did it make him reevaluate his stance toward black thought, which he saw as an impossible paradox.\n\nJefferson had black heirs who he, and for centuries his (sorta) white heirs and white defenders, denied. In our time, Strom Thurmond had him a black daughter out of wedlock; the only people surprised by this were the white voters he courted by vehement racist rhetoric. Of course, this behavior, demeaning blacks while desiring at least one, descends from slavery and is how we got most light-skinned folks who \"look white\" in the first place.\n\nWhy doesn't Rachel Dolezal seem to know that a white person can have a black child (see one-drop rule above)? (See Obama.) (See Hemings.) (See Jefferson.) See . . .\n\n\u2022 \u2022 \u2022\n\nBeing black is not a feeling. I don't always feel colored. Nor is it simply a state of mind.\n\nBlackness: a way of being.\n\n\u2022 \u2022 \u2022\n\nIt would be one thing, I think, if in her house, to her pillow or family, Dolezal said she felt black. I imagine many white households across the country don blackface and grab banjos and have themselves a good ol' time when no one else is around. It's when that somehow translates to what she does, when she teaches black studies as if she's a black person\u2014not a teacher, but a mind reader\u2014that it becomes a problem. She wears the mask not to hide but to gain authority over the very thing she claims to want to be. How very white of her!\n\n\u2022 \u2022 \u2022\n\nAfter Rachel Dolezal had mumbled her way through various news shows looking like Gilly from Saturday Night Live and answered the question of whether she was black or not with I don't understand the question, came the murders in cold blood at Mother Emanuel Church in Charleston. Both cases didn't seem just coincidental, but near-simultaneous misapprehensions not just of blackness but of whiteness too.\n\n\u2022 \u2022 \u2022\n\nAfter the killings in Charleston, several things happened: Dolezal's story went back to merely being ridiculous. Talk shows moved on to something else and those who somehow willed Dolezal sublime retreated. Flags flew at half staff\u2014except the Confederate flag on South Carolina statehouse grounds. It took a black woman to climb up and take that down.\n\nThey gave the assignment to a black man to raise the \"rebel flag,\" the stars and bars, back up. Like Sally Hemings, he might not have minded, but he certainly couldn't have refused.\n\n\u2022 \u2022 \u2022\n\nSoon the Confederate battle flag would be voted down by the state assembly, but flag sales would soar. Customers began to hoard them like guns once most major outlets suspended sales. Yet given the killer's postings of himself with Confederate flags and separatist slogans, easy slogans like \"heritage not hate\" stood naked. The proof here only increased as a pro-flag rally brought out the American Nazi flag, side by side and even mashed up with the Confederate one.\n\nIn a place like the South that loves its tall tales, why do people take their Confederate stories hyperseriously? As gospel? Everyone's a colonel, someone joked with me about the South when I was at the University of Georgia, where I taught for five years.\n\n\u2022 \u2022 \u2022\n\nIt was my first job, and I was regularly thought by strangers at the university to be passing for a student (and not a grad student). You look too young to be a professor, surprised interrogators would say, usually after asking what year in school I was. (It's true I was only twenty-five, but had a book already and a degree or two.) After a while, I began to translate the comment about looking young to be a more polite way of saying what they couldn't: You look too black to be a professor.\n\n\u2022 \u2022 \u2022\n\nMaybe blackness is only a look, one we're told cannot ever look back?\n\n\u2022 \u2022 \u2022\n\nFar more interesting and provocative than a white mother in blackface would be a white mother with black children. Wouldn't that provide a much more complex identity than any blackface? You get the feeling that, for Dolezal, blackness equals hiding.\n\nFor the deaconess at the church who had to make her way by cleaning white people's houses during the week, blackness don't mean hiding. Sunday meant rest, and a respite, wearing a different kind of white, black hair crowned by lace.\n\n\u2022 \u2022 \u2022\n\nBlackness too often veers between two poles in the public eye: opaqueness and invisibility. For [racist killer], blackness wasn't just opaque but conspicuous. It named an enemy and provided a uniform that allowed mass judgment\u2014and murder.\n\nRachel Dolezal could be conspicuously outraged all the time, filing lawsuits, marching, because she didn't have to save any energy for just being herself.\n\n\u2022 \u2022 \u2022\n\nDolezal's drama didn't just start recently. The persecution complex, the past lawsuits (when she was white) against a historically black institution like Howard University no less, seem like the whitest thing ever. It's like when you are with a white friend and they experience racism, likely for the first time, alongside you: they usually go wild, protesting no one and everyone; you shrug as much as shout. Some things are just part of the daily dose of being black. The cab will drive away with a white friend in it rather than drive you too. It's dealing with blackness that black people have perfected\u2014or at least gotten practiced at.\n\nRacism's daily injustices are almost an inoculation against it. Almost.\n\n\u2022 \u2022 \u2022\n\nWhenever I tell a white person about the injustices at the airport, or on the street, the daily snubs, or that my white neighbor's farewell to me as I was moving out of my apartment last year was Goodbye, nigger and that no one in the condos or its board, both painted white, did a thing about it, they too grow silent.\n\nPart of grief, I've found, is silence. Protest too, at times. What the no-longer-neighbor wanted from me most, I knew instantly, was a reaction. Bye, I said. Good riddance, I meant.\n\n\u2022 \u2022 \u2022\n\nI've heard even Dolezal's paintings of black faces while in school as a white graduate student at Howard were actually plagiarized. Our Dolezal didn't just want to disappear into blackness, but disappear. For her, blackness was not a private thing, which ultimately may be where blackness best tells us what it knows. It is this private, shifting, personal blackness that cannot be borrowed. What can be: wigs, tanning booth, rhetoric.\n\n\u2022 \u2022 \u2022\n\nDolezal's righteous rage looks more like self-righteousness\u2014or is it other-righteousness?\n\n\u2022 \u2022 \u2022\n\nWhat no one seems to say is just how Dolezal's actions, over many years, conform to the typical hoaxer's: a constant shifting set of stories to explain her identity (it's complicated), an array of attempts to be not just someone else as anyone might, but to be exotic, even in her birth (which she said was in a teepee or tipi). When asked directly on the teevee if she was born in a teepee, she answered, \"I wasn't born in a teepee,\" emphasis allowing that maybe, just maybe, she could later say she was born near or under one. The hoaxer is always leaving the pretend teepee door ajar.\n\n\u2022 \u2022 \u2022\n\nDolezal also says she was abused, and claimed to have lived in South Africa. It is true that her actual parents did live there, but not with her, only her siblings\u2014many of whom were actually adopted and black. She apparently earlier equated their alleged beatings (that several of them have denied) with slavery. Given her disproven lies, abuse does not so much provide an explanation for her behavior as much as a distraction: true or not, like her making slavery a mere metaphor it would seem part of a scenario of victimhood, which to her is also, inherently, black.\n\nBorrowed blackness and nativeness provide her the ultimate virtual victimhood.\n\n\u2022 \u2022 \u2022\n\nFinally the chief problem with racial impostors or blackface: it can be only, as James Weldon Johnson said of stereotypical black dialect, comic or tragic. Ultimately, it conforms to white views of \"the blacks\" themselves, offstage: as either a joke or a set of jailed youths and stooped old people.\n\nEven the president, who started up a Twitter feed weeks before the Dolezal incident, was inundated by racists posting pictures of nooses and equating him to a monkey or worse. It is only when one feels such stereotypes as real that one might find being in blackface freeing\u2014not because you believe the stereotypes, but because you want to establish other, corny ones.\n\n\u2022 \u2022 \u2022\n\nSinking feeling: blackfaced person always occupies a bigger public stage than a black one.\n\n\u2022 \u2022 \u2022\n\nStanding back, maybe it's true: not that being black is only comic or tragic, but that too often white thinking or acting out about it, as demonstrated in Dolezal's hoax and the Charleston murders, remains only polarized: comic or tragic. Both are nullifying.\n\nAmid the bewilderment and grief, for just a moment I wondered how onetime NAACP chapter leader Dolezal would've responded, as surely she would have sought to, had she not been unmasked. Where's our fearless leader now? I thought. Then I didn't think of her again.\n\n\u2022 \u2022 \u2022\n\nI came out as black as a teenager. Before then, I was simply a boy. After, I was sometimes, still.\n\n\u2022 \u2022 \u2022\n\nWhen President Obama broke into \"Amazing Grace\" at the funeral for those killed at Mother Emanuel, it was mere hours after the Supreme Court declared gay marriage legal, and barring it as unconstitutional. There it was strange yet strangely fitting to hear him sing that song written by the reformed slaver while at sea. I like to think the slaves who took the song over and made it a Negro spiritual were not the same kinds of wretch as its author.\n\n\u2022 \u2022 \u2022\n\nOf course you can see why anyone would want to be black: being black is fun. Don't tell nobody.\n\n\u2022 \u2022 \u2022\n\nThis morning I woke from a \"deep Negro sleep,\" as Senghor put it. I then took a black shower and shaved a black shave; I walked a black walk and sat a black sit; I wrote some black lines; I coughed black and sneezed black and ate black too. This last at least is literal: grapes, blackberries, the ripest plums.\n\nSummer 2015\n\n## Da Art of Storytellin' (a Prequel)\n\n## KIESE LAYMON\n\nFrom six in the morning until five in the afternoon, five days a week, for thirty years, my grandmama Catherine's fingers, palms, and wrists wandered deep in the bellies of dead chickens. Grandmama was a buttonhole slicer at a chicken plant in central Mississippi\u2014her job was to slice the belly and pull out the guts of thousands of chickens a day. Grandmama got up every morning around 4:30 a.m. She took her bath, then prepared grits, smoked sausage, and pear preserves for us. After breakfast, Grandmama made me take a teaspoon of cod liver oil \"for my vitamins,\" then she coated the area between her breasts in powder before putting on the clothes she had ironed the night before. I was ten, staying with Grandmama for the summer, and I remember marveling at her preparations and wondering why she got so fresh, so clean, just to leave the house and get dirty.\n\n\"There's layers to this,\" Grandmama often said, when describing her job to folks. She went into that plant every day, knowing it was a laboratory for racial and gendered terror. Still, she wanted to be the best at what she did\u2014and not just the best buttonhole slicer in the plant, but the best, most stylized, most efficient worker in Mississippi. She understood that the audience for her work was not just her coworkers or her white male shift managers, but all the Southern black women workers who preceded her and, most important, all the Southern black women workers coming next.\n\nBy the end of the day, when the two-tone blue Impala crept back into the driveway on the side of our shotgun house, I'd run out to welcome Grandmama home. \"Hey, baby,\" she'd say. \"Let me wash this stank off my hands before I hug your neck.\"\n\nThis stank wasn't that stink. This stank was root and residue of black Southern poverty, and devalued black Southern labor, black Southern excellence, black Southern imagination, and black Southern woman magic. This was the stank from whence black Southern life, love, and labor came. Even at ten years old, I understood that the presence and necessity of this stank dictated how Grandmama moved on Sundays. As the head of the usher board at Concord Baptist, she sometimes wore the all-white polyester uniform that all the other church ushers wore. On those Sundays, Grandmama was committed to out-freshing the other ushers by draping colorful pearls and fake gold around her neck, or stunting with some shiny shoes she'd gotten from my aunt Linda in Vegas. And Grandmama's outfits, when she wasn't wearing the stale usher board uniform, always had to be fresher this week than the week before.\n\nShe was committed to out-freshing herself, which meant that she was up late on Saturday nights, working like a wizard, taking pieces of this blouse from 1984 and sewing them into these dresses from 1969. Grandmama's primary audience on Sundays, her church sisters, looked with awe and envy at her outfits, inferring she had a fashion industry hookup from Atlanta, or a few secret revenue streams. Not so. This was just how Grandmama brought the stank of her work life into her spiritual communal life, in a way that I loved and laughed at as a kid.\n\nI didn't fully understand or feel inspired by Grandmama's stank or freshness until years later, when I heard the albums ATLiens and Aquemini from those Georgia-based artists called OutKast.\n\n\u2022 \u2022 \u2022\n\nOne day near the beginning of my junior year in college, 1996, I walked out of my dorm room in Oberlin, Ohio, heading to the gym, when I heard a new sound and a familiar voice blasting from the room of my friend John Norris, a Southern black boy from Clarksville, Tennessee.\n\nMy soliloquy may be hard for some to\n\nswallow\n\nBut so is cod liver oil.\n\nI went into John's room, wondering who was rapping about cod liver oil over reverbed bass, and asked him, \"What the fuck is that?\" It was \"Wheelz of Steel,\" from ATLiens. John handed me the CD. The illustrated cover looked like a comic book, its heroes standing back-to-back in front of a mysterious four-armed force: Big Boi in a letterman jacket with a Braves hat cocked to the right, and Andr\u00e9 in a green turban like something I'd only seen my Grandmama and Mama Lara rock. Big Boi's fingers were clinched, ready to fight. Andr\u00e9's were spread, ready to conjure.\n\nJohn and I listened to the record twice before I borrowed my friend's green Geo, drove to Elyria, and bought ATLiens for myself. Like Soul Food by Atlanta's Goodie Mob, another album I was wearing out at the time\u2014their song \"Thought Process,\" which featured Andr\u00e9, had nudged me through the sadness of missing Mississippi a year earlier\u2014ATLiens was unafraid of the revelatory dimensions of black Southern life. Like Soul Food, ATLliens explored the inevitability of death and the possibility of new life, new movement, and new mojo.\n\nBut something was different.\n\nI already knew OutKast; I loved their first album, Southernplayalisticadillacmuzik, in part because of the clever way they interpolated funk and soul into rap. ATLiens, however, sounded unlike anything I'd ever heard or imagined. The vocal tones were familiar, but the rhyme patterns, the composition, the production were equal parts red clay, thick buttery grits, and Mars. Nothing sounded like ATLiens. The album instantly changed not just my expectations of music, but my expectations of myself as a young black Southern artist.\n\nBy then, I already knew I was going to be a writer. I had no idea if I would eat off of what I wrote, but I knew I had to write to be a decent human being. I used ink and the page to probe and to remember through essays and sometimes through satire. I was imitating, and maybe interrogating, but I'm not sure that I had any idea of how to use words to imagine and really innovate. All my English teachers talked about the importance of finding \"your voice.\" It always confused me because I knew we all had so many voices, so many audiences, and my teachers seemed only to really want the kind of voice that sat with its legs crossed, reading The New York Times. I didn't have to work to find that cross-legged voice\u2014it was the one education necessitated I lead with.\n\nWhat my English teachers didn't say was that voices aren't discovered fully formed, they are built and shaped\u2014and not just by words, punctuation, and sentences, but by the author's intended audience, by the composition's form, and by subject. It was only after listening to ATLiens that I realized in order to get where I needed to go as a human being and an artist, in order to release my own spacey stank blues, I had to write fiction. Dre and Big showed me it was possible to create fake worlds wholly concerned with \"what if\" and \"maybe\" and \"what really was.\"\n\nI remember sitting in my tiny dorm room under my huge Black Lightning poster, next to my tiny picture of Grandmama. I was supposed to be doing a paper on \"The Cask of Amontillado,\" but I was thinking about OutKast's \"Wailin'.\" The song made me know that there was something to be gained, felt, and used in imitating sounds from whence we came, particularly in the minimal hook: the repeated moan of one about to wail. I'd heard that moan in the presence of older Southern black folk my entire life, but I'd never heard it connecting two rhymed verses. Art couldn't get any fresher than that.\n\n\u2022 \u2022 \u2022\n\nBy the mid-nineties, hip-hop was an established art form, foregrounding a wide, historically neglected audience in completely new ways. Never had songs had so many words. Never had songs lacked melodies. Never had songs pushed against the notion of a hook repeated every 45 seconds. Like a lot of Southern black boys, I loved New York hip-hop, although I didn't feel loved or imagined by most of it.\n\nWhen Andr\u00e9 said, \"The South got something to say and that's all I got to say,\" at the Source awards in 1995, I heard him saying that we were no longer going to artistically follow New York. Not because the artists of New York were wack, but because disregarding our particular stank in favor of a stink that didn't love or respect us was like taking a broken elevator down into artistic and spiritual death.\n\nWith OutKast, Dre and Big each carved out their own individual space, and along with sonic contrast\u2014Big lyrically fought and Andr\u00e9 lyrically conjured\u2014they gave us philosophical contrast. When Dre raps, \"No drugs or alcohol so I can get the signal clear as day,\" I remember folks suggesting there was a smidgen of shade being thrown on Big Boi, who on the same album rhymed, \"I got an ounce of dank and a couple of dranks, so let's crank up this session.\" If there was ever shade between them back then, I got the sense, they'd handle it like we Southern black boys did: they'd wrassle it out, talk more shit, hug, and come back ready to out-fresh each other, along with every artist who'd come before them in the making of lyrical art.\n\nOutKast created a different kind of stank, too: an urban Southern stank so familiar with and indebted to the gospel, blues, jazz, rock, and funk born in the rural black South. And while they were lyrically competing against each other on track after track, together Big and Dre were united, railing and wailing against New York and standing up to a post-civil-rights South chiding young Southern black boys to pull up our pants and fight white supremacy with swords of respectability and narrow conceptions of excellence. ATLiens made me love being black, Southern, celibate, sexy, awkward, free of drugs and alcohol, Grandmama's grandbaby, and cooler than a polar bear's toenails.\n\nRight out of Oberlin, I earned a fellowship in the MFA program at Indiana University, to study fiction. For the first time in my life, I was thinking critically about narrative construction in everything from malt liquor commercials to the Bible. It was around that time that Lauryn Hill gave my generation an elixir to calm, compete with, and call out a culture insistent on coming up with new ways to devalue black women. In The Miseducation of Lauryn Hill, I saw myself as the intimate partner doing wrong by Lauryn, and she made me consider how for all the differences between Andr\u00e9 and Big Boi, they shared in the same kind of misogynoir on their first two albums. (Particularly on the song \"Jazzy Belle\": \". . . even Bo knew, that you got poked \/ like acupuncture patients while our nation is a boat.\") Miseducation had me expecting a lot more from my male heroes. A month later, OutKast dropped Aquemini.\n\nDeep into the album, the song \"West Savannah\" ends with a skit. We hear a young black boy trying to impress his friend by calling a young black girl on the phone, three-way. When the girl answers, we hear a mama, an auntie, or a grandmama tell her to \"get your ass in here.\" The girl tells the boy she has to go\u2014and then the boy tells her that his friend wants some sex. The girl emphatically lets the boy know there is no way she's having sex with him, before hanging up in his face. This is where the next song, \"Da Art of Storytellin' (Pt. 1),\" begins.\n\nIn the first verse, Big rhymes about a sexual experience with a girl named Suzy Screw, during which he exchanges a CD and a poster for oral sex. In the second, Andr\u00e9 raps about Suzy's friend Sasha Thumper. As Andr\u00e9's verse proceeds, he and Sasha are lying on their backs \"staring at the stars above, talking bout what we gonna be when we grow up.\" When Dre asks Sasha what she wants to be, Sasha Thumper responds, \"Alive.\" The song ends with the news that Sasha Thumper has overdosed after partnering with a man who treats her wrong. Here was \"another black experience,\" as Dre would say to end another verse on the album.\n\nHip-hop has always embraced metafiction. In the next track\u2014\"Da Art of Storytellin' (Pt. 2)\"\u2014Big and Dre deliver a pair of verses about the last recording they'll ever create due to an environmental apocalypse. We've long had emcees rhyming about the potency of their own rhymes. But I have never heard a song attribute the end of the world to a rhyme. In the middle of Dre's verse, he nudges us to understand that there's something more happening in this song: \"Hope I'm not over your head, but if so you will catch on later.\"\n\nBig Boi alludes to the book of Revelation, mentions some ballers trying to unsuccessfully repent and make it to heaven, and then rhymes about getting his family and heading to the Dungeon, their basement studio in Atlanta\u2014the listener can easily imagine it as a bunker\u2014where he'll record one last song. The world is ending. He grabs the mic: \"I got in the booth to run the final portion.\" Of course, this ending describes the very track we're hearing, thus bringing the fictional apocalypse of the song into our real world.\n\nI was reading Octavia Butler's Kindred at the time Aquemini came out. Steeped in all that stank, I conceived of a book within a book within a book, written by a young Southern black girl whose parents disappear. \"I'm a round runaway character\" was the first sentence my narrator wrote. I decided that she would be an emcee, but I didn't know her name. I knew that she would tell the world that she was an ellipsis, a runaway ellipsis willing to do any and all things to stop her black Southern community from being written off the face of the earth. I scribbled these notes on the blank pages of Kindred while Aquemini kept playing in the background. By the time the song \"Liberation\" was done, Long Division, my first novel, was born.\n\n\u2022 \u2022 \u2022\n\nI thought about interviewing Andr\u00e9 and Big Boi for this piece. I was going to get them to spend the night at this huge house I'm staying in this year as the writer in residence at the University of Mississippi. I planned on inviting Grandmama, too. Between the four of us, I thought we could get to the bottom of some necessary stank, and maybe play a game of \"Who's Fresher?: Georgia vs. Mississippi.\" But the interviews fell through, and Grandmama refused to come up to Oxford because I'm the only black person she knows here, and she tends to avoid places where she doesn't know many black folks.\n\nI kept imagining the meeting, though, and I thought a lot about what in the world I would say to Big Boi and Andr\u00e9. As dope as they are, there's nothing I want to ask them about their art. I experienced it, and I'm thankful they extended the traditions and frequencies from whence we came. Honestly, the only thing I'd want to ask them would be about their grandmamas. I'd want to know if their grandmamas thought they were beautiful. I'd want to know how their grandmamas wanted to be loved. I'd want to know how good they were at loving their grandmamas on days when the world wasn't so kind.\n\nThe day that my grandmama came home after work without the stank of chicken guts, powder, perfume, sweat, and Coke-Cola, I knew that her time at the plant was done. On that day\u2014when her body wouldn't let her work anymore\u2014I knew I'd spend the rest of my life trying to honor her and make a way for her to be as fresh and remembered as she wants to be.\n\nDue to diabetes, Grandmama moves mostly in a wheelchair these days, but she's still the freshest person in my world. Visually, I'm not so fresh. I wear the same thing every day. But I am a Southern black worker, committed to building stank-ass art rooted in honesty, will, and imagination.\n\nThis weekend, I'm going to drive down to Grandmama's house in central Mississippi. I'm going to bring my computer. I'm going to ask her to sit next to me while I finish this essay about her artistic rituals of labor vis-\u00e0-vis OutKast. I'm going to play ATLiens and Aquemini on her couch while finishing the piece, and think of every conceivable way to thank her for her stank, and for her freshness. I'm going to tell Grandmama that because of her, I know what it's like to be loved responsibly. I'm going to tell her that her love helped me listen, remember, and imagine when I never wanted to listen, remember, or imagine again. I'm going to read the last paragraph of this piece to her, and when Grandmama hugs my neck, I'm going to tell her that when no one in the world believed I was a beautiful Southern black boy, she believed. I'm going to tell Grandmama that her belief is the only reason I'm still alive, that belief in black Southern love is why we work.\n\n## Black and Blue\n\n## GARNETTE CADOGAN\n\n\"My only sin is my skin.\n\nWhat did I do, to be so black and blue?\"\n\n\u2014Fats Waller, \"(What Did I Do to Be So) Black and Blue?\"\n\n\"Manhattan's streets I saunter'd, pondering.\"\n\n\u2014Walt Whitman, \"Manhattan's Streets I Saunter'd, Pondering\"\n\nMy love for walking started in childhood, out of necessity. No thanks to a stepfather with heavy hands, I found every reason to stay away from home and was usually out\u2014at some friend's house or at a street party where no minor should be\u2014until it was too late to get public transportation. So I walked.\n\nThe streets of Kingston, Jamaica, in the 1980s were often terrifying\u2014you could, for instance, get killed if a political henchman thought you came from the wrong neighborhood, or even if you wore the wrong color. Wearing orange showed affiliation with one political party and green with the other, and if you were neutral or traveling far from home you chose your colors well. The wrong color in the wrong neighborhood could mean your last day. No wonder, then, that my friends and the rare nocturnal passerby declared me crazy for my long late-night treks that traversed warring political zones. (And sometimes I did pretend to be crazy, shouting non sequiturs when I passed through especially dangerous spots, such as the place where thieves hid on the banks of a storm drain. Predators would ignore or laugh at the kid in his school uniform speaking nonsense.)\n\nI made friends with strangers and went from being a very shy and awkward kid to being an extroverted, awkward one. The beggar, the vendor, the poor laborer\u2014those were experienced wanderers, and they became my nighttime instructors; they knew the streets and delivered lessons on how to navigate and enjoy them. I imagined myself as a Jamaican Tom Sawyer, one moment sauntering down the streets to pick low-hanging mangoes that I could reach from the sidewalk, another moment hanging outside a street party with battling sound systems, each armed with speakers piled to create skyscrapers of heavy bass. These streets weren't frightening. They were full of adventure when they weren't serene. There I'd join forces with a band of merry walkers, who'd miss the last bus by mere minutes, our feet still moving as we put out our thumbs to hitchhike to spots nearer home, making jokes as vehicle after vehicle raced past us. Or I'd get lost in Mittyesque moments, my young mind imagining alternate futures. The streets had their own safety: Unlike at home, there I could be myself without fear of bodily harm. Walking became so regular and familiar that the way home became home.\n\nThe streets had their rules, and I loved the challenge of trying to master them. I learned how to be alert to surrounding dangers and nearby delights, and prided myself on recognizing telling details that my peers missed. Kingston was a map of complex, and often bizarre, cultural and political and social activity, and I appointed myself its nighttime cartographer. I'd know how to navigate away from a predatory pace, and to speed up to chat when the cadence of a gait announced friendliness. It was almost always men I saw. A lone woman walking in the middle of the night was as common a sight as Sasquatch; moonlight pedestrianism was too dangerous for her. Sometimes at night as I made my way down from hills above Kingston, I'd have the impression that the city was set on \"pause\" or in extreme slow motion, as that as I descended I was cutting across Jamaica's deep social divisions. I'd make my way briskly past the mansions in the hills overlooking the city, now transformed into a carpet of dotted lights under a curtain of stars, saunter by middle-class subdivisions hidden behind high walls crowned with barbed wire, and zigzag through neighborhoods of zinc and wooden shacks crammed together and leaning like a tight-knit group of limbo dancers. With my descent came an increase in the vibrancy of street life\u2014except when it didn't; some poor neighborhoods had both the violent gunfights and the eerily deserted streets of the cinematic Wild West. I knew well enough to avoid those even at high noon.\n\nI'd begun hoofing it after dark when I was ten years old. By thirteen I was rarely home before midnight, and some nights found me racing against dawn. My mother would often complain, \"Mek yuh love street suh? Yuh born a hospital; yuh neva born a street.\" (\"Why do you love the streets so much? You were born in a hospital, not in the streets.\")\n\n\u2022 \u2022 \u2022\n\nI left Jamaica in 1996 to attend college in New Orleans, a city I'd heard called \"the northernmost Caribbean city.\" I wanted to discover\u2014on foot, of course\u2014what was Caribbean and what was American about it. Stately mansions on oak-lined streets with streetcars clanging by, and brightly colored houses that made entire blocks look festive; people in resplendent costumes dancing to funky brass bands in the middle of the street; cuisine\u2014and aromas\u2014that mashed up culinary traditions from Africa, Europe, Asia, and the American South; and a juxtaposition of worlds old and new, odd and familiar: Who wouldn't want to explore this?\n\nOn my first day in the city, I went walking for a few hours to get a feel for the place and to buy supplies to transform my dormitory room from a prison bunker into a welcoming space. When some university staff members found out what I'd been up to, they warned me to restrict my walking to the places recommended as safe to tourists and the parents of freshmen. They trotted out statistics about New Orleans's crime rate. But Kingston's crime rate dwarfed those numbers, and I decided to ignore these well-meant cautions. A city was waiting to be discovered, and I wouldn't let inconvenient facts get in the way. These American criminals are nothing on Kingston's, I thought. They're no real threat to me.\n\nWhat no one had told me was that I was the one who would be considered a threat.\n\nWithin days I noticed that many people on the street seemed apprehensive of me: Some gave me a circumspect glance as they approached, and then crossed the street; others, ahead, would glance behind, register my presence, and then speed up; older white women clutched their bags; young white men nervously greeted me, as if exchanging a salutation for their safety: \"What's up, bro?\" On one occasion, less than a month after my arrival, I tried to help a man whose wheelchair was stuck in the middle of a crosswalk; he threatened to shoot me in the face, then asked a white pedestrian for help.\n\nI wasn't prepared for any of this. I had come from a majority-black country in which no one was wary of me because of my skin color. Now I wasn't sure who was afraid of me. I was especially unprepared for the cops. They regularly stopped and bullied me, asking questions that took my guilt for granted. I'd never received what many of my African American friends call \"The Talk\": No parents had told me how to behave when I was stopped by the police, how to be as polite and cooperative as possible, no matter what they said or did to me. So I had to cobble together my own rules of engagement. Thicken my Jamaican accent. Quickly mention my college. \"Accidentally\" pull out my college identification card when asked for my driver's license.\n\nMy survival tactics began well before I left my dorm. I got out of the shower with the police in my head, assembling a cop-proof wardrobe. Light-colored oxford shirt. V-neck sweater. Khaki pants. Chukkas. Sweatshirt or T-shirt with my university insignia. When I walked I regularly had my identity challenged, but I also found ways to assert it. (So I'd dress Ivy League style, but would, later on, add my Jamaican pedigree by wearing Clarks Desert Boots, the footwear of choice of Jamaican street culture.) Yet the all-American sartorial choice of white T-shirt and jeans, which many police officers see as the uniform of black troublemakers, was off-limits to me\u2014at least, if I wanted to have the freedom of movement I desired.\n\nIn this city of exuberant streets, walking became a complex and often oppressive negotiation. I would see a white woman walking toward me at night and cross the street to reassure her that she was safe. I would forget something at home but not immediately turn around if someone was behind me, because I discovered that a sudden backtrack could cause alarm. (I had a cardinal rule: Keep a wide perimeter from people who might consider me a danger. If not, danger might visit me.) New Orleans suddenly felt more dangerous than Jamaica. The sidewalk was a minefield, and every hesitation and self-censored compensation reduced my dignity. Despite my best efforts, the streets never felt comfortably safe. Even a simple salutation was suspect.\n\nOne night, returning to the house that, eight years after my arrival, I thought I'd earned the right to call my home, I waved to a cop driving by. Moments later, I was against his car in handcuffs. When I later asked him\u2014sheepishly, of course; any other way would have asked for bruises\u2014why he had detained me, he said my greeting had aroused his suspicion. \"No one waves to the police,\" he explained. When I told friends of his response, it was my behavior, not his, that they saw as absurd. \"Now why would you do a dumb thing like that?\" said one. \"You know better than to make nice with police.\"\n\n\u2022 \u2022 \u2022\n\nA few days after I left on a visit to Kingston, Hurricane Katrina slashed and pummeled New Orleans. I'd gone not because of the storm but because my adoptive grandmother, Pearl, was dying of cancer. I hadn't wandered those streets in eight years, since my last visit, and I returned to them now mostly at night, the time I found best for thinking, praying, crying. I walked to feel less alienated\u2014from myself, struggling with the pain of seeing my grandmother terminally ill; from my home in New Orleans, underwater and seemingly abandoned; from my home country, which now, precisely because of its childhood familiarity, felt foreign to me. I was surprised by how familiar those streets felt. Here was the corner where the fragrance of jerk chicken greeted me, along with the warm tenor and peace-and-love message of Half Pint's \"Greetings,\" broadcast from a small but powerful speaker to at least a half-mile radius. It was as if I had walked into 1986, down to the soundtrack. And there was the wall of the neighborhood shop, adorned with the Rastafarian colors red, gold, and green along with images of local and international heroes Bob Marley, Marcus Garvey, and Haile Selassie. The crew of boys leaning against it and joshing each other were recognizable; different faces, similar stories. I was astonished at how safe the streets felt to me, once again one black body among many, no longer having to anticipate the many ways my presence might instill fear and how to offer some reassuring body language. Passing police cars were once again merely passing police cars. Jamaican police could be pretty brutal, but they didn't notice me the way American police did. I could be invisible in Jamaica in a way I can't be invisible in the United States.\n\nWalking had returned to me a greater set of possibilities. And why walk, if not to create a new set of possibilities? Following serendipity, I added new routes to the mental maps I had made from constant walking in that city from childhood to young adulthood, traced variations on the old pathways. Serendipity, a mentor once told me, is a secular way of speaking of grace; it's unearned favor. Seen theologically, then, walking is an act of faith. Walking is, after all, interrupted falling. We see, we listen, we speak, and we trust that each step we take won't be our last, but will lead us into a richer understanding of the self and the world.\n\nIn Jamaica, I felt once again as if the only identity that mattered was my own, not the constricted one that others had constructed for me. I strolled into my better self. I said, along with Kierkegaard, \"I have walked myself into my best thoughts.\"\n\n\u2022 \u2022 \u2022\n\nWhen I tried to return to New Orleans from Jamaica a month later, there were no flights. I thought about flying to Texas so I could make my way back to my neighborhood as soon as it opened for reoccupancy, but my adoptive aunt, Maxine, who hated the idea of me returning to a hurricane zone before the end of hurricane season, persuaded me to come to stay in New York City instead. (To strengthen her case she sent me an article about Texans who were buying up guns because they were afraid of the influx of black people from New Orleans.)\n\nThis wasn't a hard sell: I wanted to be in a place where I could travel by foot and, more crucially, continue to reap the solace of walking at night. And I was eager to follow in the steps of the essayists, poets, and novelists who'd wandered that great city before me\u2014Walt Whitman, Herman Melville, Alfred Kazin, Elizabeth Hardwick. I had visited the city before, but each trip had felt like a tour in a sports car. I welcomed the chance to stroll. I wanted to walk alongside Whitman's ghost and \"descend to the pavements, merge with the crowd, and gaze with them.\" So I left Kingston, the popular Jamaican farewell echoing in my mind: \"Walk good!\" Be safe on your journey, in other words, and all the best in your endeavors.\n\n\u2022 \u2022 \u2022\n\nI arrived in New York City, ready to lose myself in Whitman's \"Manhattan crowds, with their turbulent musical chorus.\" I marveled at what Jane Jacobs praised as \"the ballet of the good city sidewalk\" in her old neighborhood, the West Village. I walked up past midtown skyscrapers, releasing their energy as lively people onto the streets, and on into the Upper West Side, with its regal Beaux Arts apartment buildings, stylish residents, and buzzing streets. Onward into Washington Heights, the sidewalks spilled over with an ebullient mix of young and old Jewish and Dominican American residents, past leafy Inwood, with parks whose grades rose to reveal beautiful views of the Hudson River, up to my home in Kingsbridge in the Bronx, with its rows of brick bungalows and apartment buildings nearby Broadway's bustling sidewalks and the peaceful expanse of Van Cortlandt Park. I went to Jackson Heights in Queens to take in people socializing around garden courtyards in Urdu, Korean, Spanish, Russian, and Hindi. And when I wanted a taste of home, I headed to Brooklyn, in Crown Heights, for Jamaican food and music and humor mixed in with the flavor of New York City. The city was my playground.\n\nI explored the city with friends, and then with a woman I'd begun dating. She walked around endlessly with me, taking in New York City's many pleasures. Coffee shops open until predawn; verdant parks with nooks aplenty; food and music from across the globe; quirky neighborhoods with quirkier residents. My impressions of the city took shape during my walks with her.\n\nAs with the relationship, those first few months of urban exploration were all romance. The city was beguiling, exhilarating, vibrant. But it wasn't long before reality reminded me I wasn't invulnerable, especially when I walked alone.\n\nOne night in the East Village, I was running to dinner when a white man in front of me turned and punched me in the chest with such force that I thought my ribs had braided around my spine. I assumed he was drunk or had mistaken me for an old enemy, but found out soon enough that he'd merely assumed I was a criminal because of my race. When he discovered I wasn't what he imagined, he went on to tell me that his assault was my own fault for running up behind him. I blew off this incident as an aberration, but the mutual distrust between me and the police was impossible to ignore. It felt elemental. They'd enter a subway platform; I'd notice them. (And I'd notice all the other black men registering their presence as well, while just about everyone else remained oblivious to them.) They'd glare. I'd get nervous and glance. They'd observe me steadily. I'd get uneasy. I'd observe them back, worrying that I looked suspicious. Their suspicions would increase. We'd continue the silent, uneasy dialogue until the subway arrived and separated us at last.\n\nI returned to the old rules I'd set for myself in New Orleans, with elaboration. No running, especially at night; no sudden movements; no hoodies; no objects\u2014especially shiny ones\u2014in hand; no waiting for friends on street corners, lest I be mistaken for a drug dealer; no standing near a corner on the cell phone (same reason). As comfort set in, inevitably I began to break some of those rules, until a night encounter sent me zealously back to them, having learned that anything less than vigilance was carelessness.\n\nAfter a sumptuous Italian dinner and drinks with friends, I was jogging to the subway at Columbus Circle\u2014I was running late to meet another set of friends at a concert downtown. I heard someone shouting and I looked up to see a police officer approaching with his gun trained on me. \"Against the car!\" In no time, half a dozen cops were upon me, chucking me against the car and tightly handcuffing me. \"Why were you running?\" \"Where are you going?\" \"Where are you coming from?\" \"I said, why were you running?!\" Since I couldn't answer everyone at once, I decided to respond first to the one who looked most likely to hit me. I was surrounded by a swarm and tried to focus on just one without inadvertently aggravating the others.\n\nIt didn't work. As I answered that one, the others got frustrated that I wasn't answering them fast enough and barked at me. One of them, digging through my already-emptied pockets, asked if I had any weapons, the question more an accusation. Another badgered me about where I was coming from, as if on the fifteenth round I'd decide to tell him the truth he imagined. Though I kept saying\u2014calmly, of course, which meant trying to manage a tone that ignored my racing heart and their spittle-filled shouts in my face\u2014that I had just left friends two blocks down the road, who were yes, sir, yes, officer, of course, officer, all still there and could vouch for me, to meet other friends whose text messages on my phone could verify that, it made no difference.\n\nFor a black man, to assert your dignity before the police was to risk assault. In fact, the dignity of black people meant less to them, which was why I always felt safer being stopped in front of white witnesses than black witnesses. The cops had less regard for the witness and entreaties of black onlookers, whereas the concern of white witnesses usually registered on them. A black witness asking a question or politely raising an objection could quickly become a fellow detainee. Deference to the police, then, was sine qua non for a safe encounter.\n\nThe cops ignored my explanations and my suggestions and continued to snarl at me. All except one of them, a captain. He put his hand on my back, and said to no one in particular, \"If he was running for a long time he would have been sweating.\" He then instructed that the cuffs be removed. He told me that a black man had stabbed someone earlier two or three blocks away and they were searching for him. I noted that I had no blood on me and had told his fellow officers where I'd been and how to check my alibi\u2014unaware that it was even an alibi, as no one had told me why I was being held, and of course, I hadn't dared ask. From what I'd seen, anything beyond passivity would be interpreted as aggression.\n\nThe police captain said I could go. None of the cops who detained me thought an apology was necessary. Like the thug who punched me in the East Village, they seemed to think it was my own fault for running.\n\nHumiliated, I tried not to make eye contact with the onlookers on the sidewalk, and I was reluctant to pass them to be on my way. The captain, maybe noticing my shame, offered to give me a ride to the subway station. When he dropped me off and I thanked him for his help, he said, \"It's because you were polite that we let you go. If you were acting up it would have been different.\"\n\n\u2022 \u2022 \u2022\n\nI realized that what I least liked about walking in New York City wasn't merely having to learn new rules of navigation and socialization\u2014every city has its own. It was the arbitrariness of the circumstances that required them, an arbitrariness that made me feel like a child again, that infantilized me. When we first learn to walk, the world around us threatens to crash into us. Every step is risky. We train ourselves to walk without crashing by being attentive to our movements, and extra-attentive to the world around us. As adults we walk without thinking, really. But as a black adult I am often returned to that moment in childhood when I'm just learning to walk. I am once again on high alert, vigilant.\n\nSome days, when I am fed up with being considered a troublemaker upon sight, I joke that the last time a cop was happy to see a black male walking was when that male was a baby taking his first steps. On many walks, I ask white friends to accompany me, just to avoid being treated like a threat. Walks in New York City, that is; in New Orleans, a white woman in my company sometimes attracted more hostility. (And it is not lost on me that my woman friends are those who best understand my plight; they have developed their own vigilance in an environment where they are constantly treated as targets of sexual attention.) Much of my walking is as my friend Rebecca once described it: A pantomime undertaken to avoid the choreography of criminality.\n\n\u2022 \u2022 \u2022\n\nWalking while black restricts the experience of walking, renders inaccessible the classic Romantic experience of walking alone. It forces me to be in constant relationship with others, unable to join the New York flaneurs I had read about and hoped to join. Instead of meandering aimlessly in the footsteps of Whitman, Melville, Kazin, and Vivian Gornick, more often I felt that I was tiptoeing in Baldwin's\u2014the Baldwin who wrote, way back in 1960, \"Rare, indeed, is the Harlem citizen, from the most circumspect church member to the most shiftless adolescent, who does not have a long tale to tell of police incompetence, injustice, or brutality. I myself have witnessed and endured it more than once.\" Walking as a black man has made me feel simultaneously more removed from the city, in my awareness that I am perceived as suspect, and more closely connected to it, in the full attentiveness demanded by my vigilance. It has made me walk more purposefully in the city, becoming part of its flow, rather than observing, standing apart.\n\n\u2022 \u2022 \u2022\n\nBut it also means that I'm still trying to arrive in a city that isn't quite mine. One definition of home is that it's somewhere we can most be ourselves. And when are we more ourselves but when walking, that natural state in which we repeat one of the first actions we learned? Walking\u2014the simple, monotonous act of placing one foot before the other to prevent falling\u2014turns out not to be so simple if you're black. Walking alone has been anything but monotonous for me; monotony is a luxury.\n\nA foot leaves, a foot lands, and our longing gives it momentum from rest to rest. We long to look, to think, to talk, to get away. But more than anything else, we long to be free. We want the freedom and pleasure of walking without fear\u2014without others' fear\u2014wherever we choose. I've lived in New York City for almost a decade and have not stopped walking its fascinating streets. And I have not stopped longing to find the solace that I found as a kid on the streets of Kingston. Much as coming to know New York City's streets has made it closer to home to me, the city also withholds itself from me via those very streets. I walk them, alternately invisible and too prominent. So I walk caught between memory and forgetting, between memory and forgiveness.\n\n## The Condition of Black Life Is One of Mourning\n\n## CLAUDIA RANKINE\n\nA friend recently told me that when she gave birth to her son, before naming him, before even nursing him, her first thought was, I have to get him out of this country. We both laughed. Perhaps our black humor had to do with understanding that getting out was neither an option nor the real desire. This is it, our life. Here we work, hold citizenship, pensions, health insurance, family, friends, and on and on. She couldn't, she didn't leave. Years after his birth, whenever her son steps out of their home, her status as the mother of a living human being remains as precarious as ever. Added to the natural fears of every parent facing the randomness of life is this other knowledge of the ways in which institutional racism works in our country. Ours was the laughter of vulnerability, fear, recognition, and an absurd stuckness.\n\nI asked another friend what it's like being the mother of a black son. \"The condition of black life is one of mourning,\" she said bluntly. For her, mourning lived in real time inside her and her son's reality: At any moment she might lose her reason for living. Though the white liberal imagination likes to feel temporarily bad about black suffering, there really is no mode of empathy that can replicate the daily strain of knowing that as a black person you can be killed for simply being black: no hands in your pockets, no playing music, no sudden movements, no driving your car, no walking at night, no walking in the day, no turning onto this street, no entering this building, no standing your ground, no standing here, no standing there, no talking back, no playing with toy guns, no living while black.\n\nEleven days after I was born, on September 15, 1963, four black girls were killed in the bombing of the 16th Street Baptist Church in Birmingham, Alabama. Now, fifty-two years later, six black women and three black men have been shot to death while at a Bible-study meeting at the historic Emanuel African Methodist Episcopal Church in Charleston, South Carolina. They were killed by a homegrown terrorist, self-identified as a white supremacist, who might also be a \"disturbed young man\" (as various news outlets have described him). It has been reported that a black woman and her five-year-old granddaughter survived the shooting by playing dead. They are two of the three survivors of the attack. The white family of the suspect says that for them this is a difficult time. This is indisputable. But for African American families, this living in a state of mourning and fear remains commonplace.\n\nThe spectacle of the shooting suggests an event out of time, as if the killing of black people with white-supremacist justification interrupts anything other than regular television programming. But Dylann Storm Roof did not create himself from nothing. He has grown up with the rhetoric and orientation of racism. He has seen white men like Benjamin F. Haskell, Thomas Gleason, and Michael Jacques plead guilty to, or be convicted of, burning Macedonia Church of God in Christ in Springfield, Massachusetts, just hours after President Obama was elected. Every racist statement he has made he could have heard all his life. He, along with the rest of us, has been living with slain black bodies.\n\nWe live in a country where Americans assimilate corpses in their daily comings and goings. Dead blacks are a part of normal life here. Dying in ship hulls, tossed into the Atlantic, hanging from trees, beaten, shot in churches, gunned down by the police, or warehoused in prisons: Historically, there is no quotidian without the enslaved, chained, or dead black body to gaze upon or to hear about or to position a self against. When blacks become overwhelmed by our culture's disorder and protest (ultimately to our own detriment, because protest gives the police justification to militarize, as they did in Ferguson), the wrongheaded question that is asked is, What kind of savages are we? Rather than, What kind of country do we live in?\n\nIn 1955, when Emmett Till's mutilated and bloated body was recovered from the Tallahatchie River and placed for burial in a nailed-shut pine box, his mother, Mamie Till Mobley, demanded his body be transported from Mississippi, where Till had been visiting relatives, to his home in Chicago. Once the Chicago funeral home received the body, she made a decision that would create a new pathway for how to think about a lynched body. She requested an open coffin and allowed photographs to be taken and published of her dead son's disfigured body.\n\nMobley's refusal to keep private grief private allowed a body that meant nothing to the criminal-justice system to stand as evidence. By placing both herself and her son's corpse in positions of refusal relative to the etiquette of grief, she \"disidentified\" with the tradition of the lynched figure left out in public view as a warning to the black community, thereby using the lynching tradition against itself. The spectacle of the black body, in her hands, publicized the injustice mapped onto her son's corpse. \"Let the people see what I see,\" she said, adding, \"I believe that the whole United States is mourning with me.\"\n\nIt's very unlikely that her belief in a national mourning was fully realized, but her desire to make mourning enter our day-to-day world was a new kind of logic. In refusing to look away from the flesh of our domestic murders, by insisting we look with her upon the dead, she reframed mourning as a method of acknowledgment that helped energize the civil rights movement in the 1950s and '60s.\n\nThe decision not to release photos of the crime scene in Charleston, perhaps out of deference to the families of the dead, doesn't forestall our mourning. But in doing so, the bodies that demonstrate all too tragically that \"black skin is not a weapon\" (as one protest poster read last year) are turned into an abstraction. It's one thing to imagine nine black bodies bleeding out on a church floor, and another thing to see it. The lack of visual evidence remains in contrast to what we saw in Ferguson, where the police, in their refusal to move Michael Brown's body, perhaps unknowingly continued where Till's mother left off.\n\nAfter Brown was shot six times, twice in the head, his body was left facedown in the street by the police officers. Whatever their reasoning, by not moving Brown's corpse for four hours after his shooting, the police made mourning his death part of what it meant to take in the details of his story. No one could consider the facts of Michael Brown's interaction with the Ferguson police officer Darren Wilson without also thinking of the bullet-riddled body bleeding on the asphalt. It would be a mistake to presume that everyone who saw the image mourned Brown, but once exposed to it, a person had to decide whether his dead black body mattered enough to be mourned. (Another option, of course, is that it becomes a spectacle for white pornography: the dead body as an object that satisfies an illicit desire. Perhaps this is where Dylann Storm Roof stepped in.)\n\nBlack Lives Matter, the movement founded by the activists Alicia Garza, Patrisse Cullors, and Opal Tometi, began with the premise that the incommensurable experience of systemic racism creates an unequal playing field. The American imagination has never been able to fully recover from its white-supremacist beginnings. Consequently, our laws and attitudes have been straining against the devaluation of the black body. Despite good intentions, the associations of blackness with inarticulate, bestial criminality persist beneath the appearance of white civility. This assumption both frames and determines our individual interactions and experiences as citizens.\n\nThe American tendency to normalize situations by centralizing whiteness was consciously or unconsciously demonstrated again when certain whites, like the president of Smith College, sought to alter the language of \"Black Lives Matter\" to \"All Lives Matter.\" What on its surface was intended to be interpreted as a humanist move\u2014\"aren't we all just people here?\"\u2014didn't take into account a system inured to black corpses in our public spaces. When the judge in the Charleston bond hearing for Dylann Storm Roof called for support of Roof's family, it was also a subtle shift away from valuing the black body in our time of deep despair.\n\nAnti-black racism is in the culture. It's in our laws, in our advertisements, in our friendships, in our segregated cities, in our schools, in our Congress, in our scientific experiments, in our language, on the Internet, in our bodies no matter our race, in our communities, and, perhaps most devastatingly, in our justice system. The unarmed, slain black bodies in public spaces turn grief into our everyday feeling that something is wrong everywhere and all the time, even if locally things appear normal. Having coffee, walking the dog, reading the paper, taking the elevator to the office, dropping the kids off at school: All of this good life is surrounded by the ambient feeling that at any given moment, a black person is being killed in the street or in his home by the armed hatred of a fellow American.\n\nThe Black Lives Matter movement can be read as an attempt to keep mourning an open dynamic in our culture because black lives exist in a state of precariousness. Mourning then bears both the vulnerability inherent in black lives and the instability regarding a future for those lives. Unlike earlier black-power movements that tried to fight or segregate for self-preservation, Black Lives Matter aligns with the dead, continues the mourning, and refuses the forgetting in front of all of us. If the Rev. Martin Luther King, Jr.'s civil rights movement made demands that altered the course of American lives and backed up those demands with the willingness to give up your life in service of your civil rights, with Black Lives Matter, a more internalized change is being asked for: recognition.\n\nThe truth, as I see it, is that if black men and women, black boys and girls, mattered, if we were seen as living, we would not be dying simply because whites don't like us. Our deaths inside a system of racism existed before we were born. The legacy of black bodies as property and subsequently three-fifths human continues to pollute the white imagination. To inhabit our citizenry fully, we have to not only understand this, but also grasp it. In the words of the playwright Lorraine Hansberry, \"The problem is we have to find some way with these dialogues to show and to encourage the white liberal to stop being a liberal and become an American radical.\" And, as my friend the critic and poet Fred Moten has written: \"I believe in the world and want to be in it. I want to be in it all the way to the end of it because I believe in another world and I want to be in that.\" This other world, that world, would presumably be one where black living matters. But we can't get there without fully recognizing what is here.\n\nDylann Storm Roof's unmediated hatred of black people; Black Lives Matter; citizens' videotaping the killings of blacks; the Ferguson Police Department leaving Brown's body in the street\u2014all these actions support Mamie Till Mobley's belief that we need to see or hear the truth. We need the truth of how the bodies died to interrupt the course of normal life. But if keeping the dead at the forefront of our consciousness is crucial for our body politic, what of the families of the dead? How must it feel to a family member for the deceased to be more important as evidence than as an individual to be buried and laid to rest?\n\nMichael Brown's mother, Lesley McSpadden, was kept away from her son's body because it was evidence. She was denied the rights of a mother, a sad fact reminiscent of pre\u2013Civil War times, when as a slave she would have had no legal claim to her offspring. McSpadden learned of her new identity as a mother of a dead son from bystanders: \"There were some girls down there had recorded the whole thing,\" she told reporters. One girl, she said, \"showed me a picture on her phone. She said, 'Isn't that your son?' I just bawled even harder. Just to see that, my son lying there lifeless, for no apparent reason.\" Circling the perimeter around her son's body, McSpadden tried to disperse the crowd: \"All I want them to do is pick up my baby.\"\n\nMcSpadden, unlike Mamie Till Mobley, seemed to have little desire to expose her son's corpse to the media. Her son was not an orphan body for everyone to look upon. She wanted him covered and removed from sight. He belonged to her, her baby. After Brown's corpse was finally taken away, two weeks passed before his family was able to see him. This loss of control and authority might explain why after Brown's death, McSpadden was supposedly in the precarious position of accosting vendors selling T-shirts that demanded justice for Michael Brown that used her son's name. Not only were the procedures around her son's corpse out of her hands; his name had been commoditized and assimilated into our modes of capitalism.\n\nSome of McSpadden's neighbors in Ferguson also wanted to create distance between themselves and the public life of Brown's death. They did not need a constant reminder of the ways black bodies don't matter to law enforcement officers in their neighborhood. By the request of the community, the original makeshift memorial\u2014with flowers, pictures, notes, and teddy bears\u2014was finally removed by Brown's father on what would have been his birthday and replaced by an official plaque installed on the sidewalk next to where Brown died. The permanent reminder can be engaged or stepped over, depending on the pedestrian's desires.\n\nIn order to be away from the site of the murder of her son, Tamir Rice, Samaria moved out of her Cleveland home and into a homeless shelter. (Her family eventually relocated her.) \"The whole world has seen the same video like I've seen,\" she said about Tamir's being shot by a police officer. The video, which was played and replayed in the media, documented the two seconds it took the police to arrive and shoot; the two seconds that marked the end of her son's life and that became a document to be examined by everyone. It's possible this shared scrutiny explains why the police held his twelve-year-old body for six months after his death. Everyone could see what the police would have to explain away. The justice system wasn't able to do it, and a judge found probable cause to charge the officer who shot Rice with murder, while a grand jury declined to indict any of the officers involved. Meanwhile, for Samaria Rice, her unburied son's memory made her neighborhood unbearable.\n\nRegardless of the wishes of these mothers\u2014mothers of men like Brown, John Crawford III, or Eric Garner, and also mothers of women and girls like Rekia Boyd and Aiyana Stanley-Jones, each of whom was killed by the police\u2014their children's deaths will remain within the public discourse. For those who believe the same behavior that got them killed if exhibited by a white man or boy would not have ended his life, the subsequent failure to indict or convict the police officers involved in these various cases requires that public mourning continue and remain present indefinitely. \"I want to see a cop shoot a white unarmed teenager in the back,\" Toni Morrison said in April. She went on to say: \"I want to see a white man convicted for raping a black woman. Then when you ask me, 'Is it over?' I will say yes.\" Morrison is right to suggest that this action would signal change, but the real change needs to be a rerouting of interior belief. It's an individual challenge that needs to happen before any action by a political justice system would signify true societal change.\n\nThe Charleston murders alerted us to the reality that a system so steeped in anti-black racism means that on any given day it can be open season on any black person\u2014old or young, man, woman, or child. There exists no equivalent reality for white Americans. We can distance ourselves from this fact until the next horrific killing, but we won't be able to outrun it. History's authority over us is not broken by maintaining a silence about its continued effects.\n\nA sustained state of national mourning for black lives is called for in order to point to the undeniability of their devaluation. The hope is that recognition will break a momentum that laws haven't altered. Susie Jackson; Sharonda Coleman-Singleton; DePayne Middleton-Doctor; Ethel Lee Lance; the Rev. Daniel Lee Simmons, Sr.; the Rev. Clementa C. Pinckney; Cynthia Hurd; Tywanza Sanders; and Myra Thompson were murdered because they were black. It's extraordinary how ordinary our grief sits inside this fact. One friend said, \"I am so afraid, every day.\" Her son's childhood feels impossible, because he will have to be\u2014has to be\u2014so much more careful. Our mourning, this mourning, is in time with our lives. There is no life outside of our reality here. Is this something that can be seen and known by parents of white children? This is the question that nags me. National mourning, as advocated by Black Lives Matter, is a mode of intervention and interruption that might itself be assimilated into the category of public annoyance. This is altogether possible; but also possible is the recognition that it's a lack of feeling for another that is our problem. Grief, then, for these deceased others might align some of us, for the first time, with the living.\n\n## Know Your Rights!\n\n## EMILY RABOTEAU\n\nOn the Saturday after the Charleston church massacre wherein nine worshippers at one of the nation's oldest black churches were slaughtered during Bible study by a white gunman hoping to ignite a race war, we dragged our kids to the east side to walk them over New York City's oldest standing bridge. It seemed as good a way as any to kill a weekend afternoon. The High Bridge, which was built with much fanfare in the mid nineteenth century as part of the Croton Aqueduct system and as a promenade connecting Upper Manhattan to the Bronx over the Harlem River, had recently\u2014and somewhat miraculously\u2014reopened after forty-odd years of disuse. I say \"miraculously\" because the bridge was an infrastructure most of us had come to accept as blighted, even as some civic groups had coalesced to resurrect it. In the back of our minds that summer of 2015, as an uprising and its violent suppression raged in Missouri, was the problem of when and how to talk to our children about protecting themselves from the police.\n\nAt what age is such a conversation appropriate? By what age is it critical? How could it not be despairing? And what, precisely, should be said? The boy was four then. The girl, just two.\n\nThe day was hot. En route to the bridge we felt no reprieve from the sun, just as we'd felt no relief from the pileup of bad news about blacks being murdered with impunity. When we learned of the terror at AME Emanuel in Charleston, we had not yet recovered from the unlawful death of Freddie Gray in Baltimore, nor the shooting of Mike Brown in Ferguson, nor the chokehold death of Eric Garner in Staten Island, nor the shooting of Trayvon Martin in Florida, nor the shooting of Tamir Rice in Cleveland, to name but a few triggers of civil unrest. We weren't surprised there were no indictments in these cases, sadly enough, but we were righteously indignant. The deaths seemed to be cascading in rapid succession, each one tripping a live wire, like the feet of Muybridge's galloping horse.\n\nThe picture we were getting, and not because it was growing worse, but because our technology now exposed it, was clear and mounting evidence of discriminatory systems that don't treat or protect our citizens equally, and escalating dissent was giving rise to a movement that insists what should be evident to everyone: Black Lives Matter. There were hashtag alerts for pop-up protests in malls, die-ins on roads, and other staged acts of civil disobedience such as disruptions of white people eating their brunch. Protesters against police brutality dusted off some slogans from the civil rights era, such as \"No justice\u2014no peace!\" but others were au courant: \"I can't breathe,\" \"Hands up, don't shoot!\" \"White silence is violence,\" and most poignant to me as a mother, \"Is my son next?\"\n\n\"It's too hot and my legs are too small,\" our son protested on the way to the bridge.\n\nThe boy was right\u2014it was hot and getting hotter. He was tall for four but still so little. When standing at our front door, his nose just cleared the height of the doorknob. He was the same size as the pair of boys depicted in a two-panel cartoon by Ben Sargent circulating widely on my Facebook feed that summer. Both panels depict a little boy at the threshold, on the verge of stepping outdoors. The drawings are nearly identical except that the first boy is white and the second, black. \"I'm goin' out, Mom!\" each boy calls to a mother outside of the frame. The white boy's mother simply replies, \"Put on your jacket.\" But the other mother's instructions comprise so intricate, leery, and vexed a warning that her words obstruct the exit: \"Put on your jacket, keep your hands in sight at all times, don't make any sudden moves, keep your mouth shut around police, don't run, don't wear a hoodie, don't give them an excuse to hurt you . . .\" and so on until the text in her speech bubble blurs, as in a painting by Glenn Ligon. The cartoon is titled, \"Still Two Americas.\"\n\nI didn't wish to be her, the mother who needed to say, \"Some people will read you as black and therefore X.\" Why should I be the fearful mother? Nor did I covet the white mother's casual regard. I wanted to be the mother who got to say to her children, \"Keep your eyes open for interesting details and take notes,\" as well as, \"Enjoy yourselves!\" on their way out the door.\n\nBut for now, I carried our sweaty girl down 173rd Street on my back while my husband led our stubborn son by the hand. You know the thermometer's popping in Washington Heights when there aren't any Dominicans out on the sidewalks playing dominoes. Nobody had yet cranked open the fire hydrants. The heat knocked out the girl as if it were a club. The boy was in a rotten mood. He demanded a drink then rejected the water we'd packed. He whined that the walk was too long, then challenged our authority in a dozen other hectoring ways until we at last arrived at Highbridge Park. There he refused to descend the hundred stairs to the bridge by flinging himself onto the asphalt with his arms and legs bent in the style of a swastika, not five feet from a dead rat. The kid's defiance bothered us for all the usual reasons a parent should find it irksome, but also because if allowed to incubate in the ghetto where we live, that defiance could get him killed.\n\nOur son was soon coaxed down the vertiginous stairs by the magical horn of an Amtrak train on the railway beneath the bridge. He has explained to me his fierce attraction to trains and boats and vehicles in general with irritation that I didn't already know the answer: \"They take you somewhere else.\" That's just it. From the time your children begin walking, they are moving away from you. This is as it should be, even when you can't protect them from harm with anything but the inadequate outerwear of your love.\n\nA sweet old man in seersucker shorts stopped us at the entrance to the bridge to make sure we appreciated the marvel of its rehabilitation. He was something of a history buff and spoke in a European accent\u2014Greek, I think. He could recall when the bridge was shut down after falling into long decline, and the time before that when miscreants and vandals tossed projectiles over the guardrail into the polluted water below or at the traffic on the Harlem River Drive. Thanks to him, I know that the bridge was a feat of engineering originally modeled after a Roman aqueduct, siphoning water from Westchester County through pipes beneath its walkway into the city, enabling New Yorkers to enjoy their first indoor plumbing (including the flush toilet). The old man never thought he'd live to see the day when the High Bridge was back in business, and was proud that the citizen-led campaign to reopen it had succeeded. \"This bridge changed everything,\" the old man said in wonderment, as if the relic was a truer paean to empire than the skyscrapers twinkling in the skyline far to the south of us\u2014the Chrysler Building, the former Citicorp Center, and the spire of the Empire State. Dutifully, we paraded across to the Bronx. Maybe it was because I so admired the old man's perspective, attuned as it was to a less conspicuous wonder of the world, that on our return trip home I noticed a mural I could have sworn had not been there before.\n\n\u00a9 Emily Raboteau\n\nArtist: Nelson Rivas, aka Cekis. Washington Heights, Upper Manhattan, Wadsworth Avenue and 174th Street, 2009. \"If you are detained or arrested by a police officer, demand to speak with an attorney and don't tell them anything until an attorney is present.\" \"Ud. no tiene que estar de acuerdo con un chequeo de si mismo, su carro o su casa. No trate f\u00edsicamente de parar la polic\u00eda. Solo diga que ud. no da permiso para el chequeo. Tienes el derecho de no aceptarlo.\" \"Ud. tiene el derecho de observar y filmar actividades policiales.\"\n\u2022 \u2022 \u2022\n\n\"KNOW YOUR RIGHTS!\" the mural trumpeted in capital letters. How had it escaped my attention? The artwork covered a brick wall abutting the twenty-four-hour Laundromat I passed every weekday morning on the walk to the children's day care. A vision of tropical blues, it splashed out from the gritty gray surroundings, creating an illusion of depth. My eyes drank it in.\n\nThis mural operates like a comic strip in panels marrying image and text. In the first panel, a youngster is carded by a law enforcement official. In the second, a goateed man in a baseball cap is being handcuffed. In the third, a group of citizens stare evenly outward. One of them wears a look of disgust, and a T-shirt that says, \"4th Amendment,\" a sly allusion to the part of our constitution that protects us against unreasonable search and seizure without probable cause. Another holds his cell phone aloft to record what is happening on the street. \"You have the right to film and observe police activity,\" the mural states in Spanish, appropriate for a neighborhood where Spanish is the dominant language and where young men of color are regularly stopped and frisked by the police. In the lower left-hand corner the Miranda rights are paraphrased in English.\n\nMy first instinct was to take a picture of the mural so that I could carry it with me in my pocket. I was grateful for it, not only as a thing of beauty on the gallery of the street, but also as a kind of answer to the question that had been troubling us\u2014how to inform our children about the harassment they might face. The mural struck me as an act of love for the people who would pass it by. I understood why it had been made, and why it had been made here in the hood next to a Laundromat as opposed to on Fifth Avenue next to Henri Bendel, Tiffany's, or Saks. It was armor against the cruelty of the world. It was also a salve, to reclaim physical and psychic space. I wondered who had done it.\n\nAfter some Internet sleuthing I discovered the painter was a Chilean artist who goes by the tag name Cekis, and that this mural was the first of several public artworks commissioned by a coalition of grassroots organizations called People's Justice for Community Control and Police Accountability. The other Know Your Rights murals were spread out across four of New York City's five boroughs (excluding Staten Island, where a great number of cops live) in poor neighborhoods most plagued by police misconduct. For the rest of that summer and into the fall, I photographed as many of them as I could, like a magpie collecting bright things for her nest.\n\n\u00a9 Emily Raboteau\n\nLead Artist: Sophia Dawson. Know Your Rights, Harlem, Upper Manhattan, 138th Street and Adam Clayton Powell Jr. Boulevard, 2013. \"Write down the officer's badge #, name, and\/or other identifying info.\" \"Get medical attention if needed and take pictures of injuries.\" \"You don't have to answer any questions from police. When they approach, say, 'Am I being detained, or am I free to go?' If they detain you, stay silent + demand a lawyer. A frisk is only a pat down. If police try to do more than that say loudly, 'I do not consent to this search.' \" \"You have the right to observe, photograph, record, and film police activity.\"\n\u2022 \u2022 \u2022\n\nThe second mural I shot was in central Harlem.\n\nAs with the mural in Washington Heights, I chose to capture a passerby in the frame, to give a sense of scale but with the intent to preserve the subject's anonymity. Thrown against a sharp white background, the man in Harlem appears in silhouette, his beard like Thelonious Monk's, his shadow extending from his feet, and the shadow of the fire escape above him slanting down against the mural like the bars of a cage. A woman depicted in the mural's foreground holds a bullhorn to her mouth. A portion of the text reads, \"Write down the officer's badge number, name, and\/or other identifying info. You don't have to answer any questions from police.\" Her advice is specifically targeted to those at risk of being stopped and frisked.\n\nStop-and-frisk policing was implemented in New York as part of an increased trend of enforcement that began in response to rising crime and the crack cocaine epidemic of the 1980s and '90s. The technique disproportionately affects young men of color. (From 2004 through 2012, African Americans and Hispanics were subject to nearly 90 percent of the 4.4 million stop-and-frisk actions despite constituting only about half of the city's population.) In black and Latino neighborhoods like Harlem and Washington Heights, residents often view the police, a force ostensibly there to protect them, with mistrust and fear. In 2013, the year the Harlem mural was made, a federal court judged the use of stop-and-frisk tactics to be excessive and unconstitutional. Since then, their use has declined. Critics of reducing the practice predicted a rise in crime. Instead, overall crime has dropped. I would like to believe these statistics mean it's growing slightly safer for my children to walk.\n\nYul-san Liem, who works for one of the activist organizations that makes up People's Justice, explained to me that the murals were financed by the Center for Constitutional Rights. \"Visual art communicates differently than the written or spoken word,\" she commented. \"By creating Know Your Rights murals, we seek to bring important information directly to the streets where it is needed the most, and in a way that is memorable and visually striking.\"\n\nPeople's Justice formed in 2007 in the wake of the NYPD killing of the unarmed black man Sean Bell the day before his wedding. \"It wasn't an isolated incident,\" Liem lamented, recalling the 1999 killing of Amadou Diallo, the unarmed black man shot forty-one times by police, and the assault of Abner Louima, who was sodomized by police with a broom handle in 1997, allegedly told to \"Take that, nigger!\" Liem said, \"Our original goal was to highlight the systemic nature of police violence in communities of color. We've taken a proactive approach to empowerment that includes organizing neighborhood-based Cop Watch teams and outreach that uses public art as a means of education. It's about shifting culture and creating hope.\"\n\nMaybe that's what I was scavenging for. Hope. I like how Emily Dickinson defined it\u2014\"the thing with feathers.\"\n\n\u00a9 Emily Raboteau\n\nArtist: Dasic Fern\u00e1ndez. Know Your Rights, Bushwick, Brooklyn, Irving Avenue and Gates Avenue, 2011. \"If you are harassed by police, write down the officer's badge number, name, and\/or other identifying information. Get medical attention if you need it and take pictures of any injuries.\" \"All students have the right to attend school in a safe, secure, non-threatening and respectful learning environment in which they are free from harassment.\" \"No tenant can be evicted from their apartment without being taken to housing court.\" \"Si ud. es detenido o arrestado por un polic\u00eda, pida hablar con un abogado immediatamente. No diga nada hasta que tiene un abogado presente.\" \"Owners are required by law to keep their buildings safe, well maintained and in good repair. If not, call 911.\"\n\u2022 \u2022 \u2022\n\nThe third mural that I shot was in Bushwick, Brooklyn. I had difficulty finding it, in part because Bushwick is a neighborhood of murals but also because Liem had given me bum directions. I lost myself in the rainbow spectacle of street art. There was Nelson Mandela on a wall overlooking the parking lot of a White Castle, but where was the mural I sought? I asked a group of kids in Catholic school uniforms if they knew where I could find it. They all claimed to know the Know Your Rights mural, but none could give me an exact address. Either it was somewhere down Knickerbocker Avenue or else it was located in the opposite direction past three or four schoolyards and a car wash. In the end, one girl kindly volunteered to walk me there. She wore a purple backpack, braces on her teeth, and a gold name necklace that said coincidentally (or not) \"Esperanza.\" Esperanza told me with excitement that she'd be getting an iPhone like mine for her thirteenth birthday. After perambulating for a half an hour we finally located the mural in an overgrown lot behind a chain-link fence. We'd had so many false sightings at that point that I sensed it was part of the girl's mental, rather than physical, landscape. It rose out of the weeds in pastel shades like an enormous Easter egg. \"I love this one,\" she confessed. \"It's so big.\"\n\nAs with the murals in Washington Heights and Harlem, the text of the Bushwick mural exhorts the viewer to watch and film police activities. This time the message is underscored by a figure in the foreground who points to her enormous eye as if to say, \"Watch out. Keep your eyes open.\" A man directly behind her uses his phone to film a police officer making an arrest in the mural's background. The phone is configured as a weapon for social change. The teenager that I photograph walking past the mural is also on the phone. Though she appears oblivious to the mural, she also appears, in the context of my photo, to be wielding a tool. That is, the phone distracts her from being present but she could also deploy its camera at any moment to record what's happening on the street.\n\n\u00a9 Emily Raboteau\n\nArtist: Dasic Fern\u00e1ndez. Know Your Rights, Long Island City, Queens, Thirty-fifth Avenue and Twelfth Street, 2012. \"If you are HARRASSED by police, write down the officer BADGE number, name and\/or other identifying information. Take PICTURES of any INJURIES.\" \"Ud no tiene que estar de aquerdo con un chequeo de si mismo, su carro o su casa. No pare f\u00edsicamente a la polic\u00eda. Diga que ud. no da permiso para el chequeo.\"\n\u2022 \u2022 \u2022\n\nThe fourth mural that I shot was painted on a corrugated fence in Long Island City, Queens, across from the Ravenswood housing projects. On my way through the projects I passed a barefoot lady in a church hat pushing a stroller full of cans. She was involved in a heated argument over a Metrocard with a man invisible to me. In the middle of an invective she stopped to tell me he was a lying thief. \"I believe you,\" I said, emphatically. \"He's a jerk.\" We smiled at each other. She returned to her dispute and I went on my way.\n\n\"If you are HARASSED by police . . .\" the Long Island City mural advises, \". . . take PICTURES of any INJURIES.\" Again, the mural is a backdrop to walking but this time, because it consists entirely of text, the message is even starker. A woman is about to cross the street. I don't know where she's going, or what she's looking at. She may be checking for oncoming traffic or reading the warning on the mural. Her braids swing across her back as her sneaker approaches the curb. My friend the writer Garnette Cadogan has said, \"Walking is among the most dignified of human activities.\" But here, the woman's simple dignified act of walking, whether home from work or school, or to the bodega for a carton of milk, is erupted by the somber memo that hangs in the background. The public space feels contested and even traumatic because of the public art. The intersection looks hazardous, like something is about to hit her.\n\n\u00a9 Emily Raboteau\n\nArtist: Trust Your Struggle (collective), Trust Your Struggle, Bedford Stuyvesant, Brooklyn, Marcus Garvey Boulevard and Macdonough Street, 2010. \"Justice or Just Us.\" \"LOVE\/HATE.\" \"Stay calm and in control. Don't get into an argument. Remember officer's badge and patrol car number. Don't resist, even if you believe you're innocent. You don't have to consent to be searched. Try to find a witness & get their name & contact. Anything you say can be used against you. Know Your Rights. Trust Your Struggle. Spread love. It's the Brooklyn way. Didn't pass the bar, but know a little bit; enough that you won't illegally search N.Y.\"\n\u2022 \u2022 \u2022\n\nThe fifth mural I shot was in Bedford-Stuyvesant, the swiftly gentrifying Brooklyn neighborhood made famous by Spike Lee's landmark film Do the Right Thing. In fact, the Bed-Stuy mural directly references that movie by depicting the character Radio Raheem. At the start of the movie, Radio Raheem blasts Public Enemy's \"Fight the Power\" from his boombox like a reveille. Near the movie's end, he's choked to death by a nightstick-wielding cop\u2014a pivotal plot point that incites a riot, much like the uprisings that followed the Rodney King verdict in Los Angeles, and the Freddie Gray verdict in Baltimore, and the Michael Brown verdict in Ferguson, which reverberated across the country like so many waves of heat.\n\nIn New York, I remember the Ferguson protesters took to the streets chanting, \"Whose streets? Our streets!\" I myself was drawn to the vortex of 125th Street, where I shot pictures of the crowd swarming toward the Triborough Bridge. I paused there at the edge of my own reason sometime before midnight to return to my children, but the mob pushed on as far as the tollbooths on the Manhattan side, succeeding in shutting the bridge down. It felt so logical an impulse, to act unruly in the face of misrule. Yet this impulse is what the Bed-Stuy mural admonishes against.\n\nRadio Raheem's fist is the focal point of the mural, adorned with its gold \"LOVE\" knuckleplate. The mural, dominated by the color red, cautions the viewer to \"Stay calm and in control. Don't get into an argument . . . Don't resist, even if you believe you're innocent.\"\n\nThe man I photograph walking past the love punch wears paint-splattered work boots, a headcloth over his dreadlocks, and earphones. I wonder what he's listening to. Perhaps because he's distracted by his music, he's unaware that I've shot him with my phone.\n\n\u00a9 Emily Raboteau\n\nArtist: Dasic Fern\u00e1ndez. Know Your Rights, Hunts Point, Bronx, Barretto Street and Garrison Avenue, 2012. \"You have the right to watch & film police activities.\" \"If you are detained or arrested by a police officer, demand to speak with an attorney and don't say anything until attorney is present.\"\n\u2022 \u2022 \u2022\n\nSo was the woman in the Bronx, where I took my sixth and final picture. She was too absorbed by the screen of her device to notice me, though if she looked my way, she would have seen that I too was operating my phone. My posture mirrored the person in the mural who films a plainclothes police officer cuffing a man over the hood of a car. I had to wait over an hour to get this shot because a belligerent drunk pissing on the sidewalk refused to get out of the frame. Finally, he zipped up and drifted off beneath the Bruckner Expressway. Of all the neighborhoods I traversed, Hunts Point felt the roughest. On the long walk to the Point from the elevated 2 train through the red light district, I was surveyed with interest. I felt that if I wasn't mindful, someone down on his luck might succeed in snatching my phone. Yet I stayed planted by the mural, looking for something concrete.\n\nThe phone in the Hunts Point mural is almost as tall as the woman walking beneath it, its screen the approximate size of her handbag. In the screenshot we see repeated the nested image of the plainclothes police officer cuffing a man over the hood of the car. The dizzying effect of the mural is to put the viewer in the perspective of the photographer.\n\nI have fallen into the mural or rather the mural has sucked me in. I am the third dimension; the watcher. I am the photographer with the phone in her hand. So, potentially, is the passerby, though in this context her posture is also a reminder that passivity has its cost. The woman is about to step out of my frame. For now she is caught, as in a web, by the shadows of power lines and trees. The text behind her echoes that of the first mural I shot on the streets of Washington Heights: \"If you are detained or arrested by a police officer, demand to speak with an attorney and don't say anything until the attorney is present.\"\n\nIt was as if the text were on a loop. I'd begun to feel I was moving in circles and so I stopped to take stock of my pictures, scrolling backward. Though the style of each mural was distinct, the message was the same. Somebody loves you enough to try to keep you safe by informing you of your rights. The murals' insistence on those rights, which the citizens of our nation don't yet equally enjoy, reminded me that like the High Bridge, the Constitution is just another lofty infrastructure in need of rehabilitation. Such changes do occur, it seems. Were it not for the fact that I shot them in different locales, I felt I could craft a zoetrope of the passersby to show my children. The many walkers would appear unified as one\u2014even if at times that walker was a woman or a man, or black, or brown, or old or young\u2014advancing toward one steady goal. \"Look how marvelous,\" I would say of the moving image. And if my children asked me where the walker was going, I would answer, \"To the bridge.\"\n\n\u00a9 Emily Raboteau\n\n## Composite Pops\n\n## MITCHELL S. JACKSON\n\nHow does a fatherless boy spell father?\n\nOne answer is in the video of a poet who monologues about a dream in which he's a child contestant in a spelling bee. For the win, he has to spell the word father. He proceeds to spell the word m-o-t-h-e-r. Then when the spellmaster says he's \"incorrect,\" he launches into a rant about absentee fathers and womanizing men and maternal strength . . .\n\nWhile plenty mothers in the world deserve the most huge hurrahs, what I want to say to this poet and other like minds is this: no matter how much we lambast men and high-note praise women, a woman maketh a father not.\n\nYes, ours is indeed a revolutionary era of gender fluidness and sexual equality and girls doubtless need dads too\u2014I repeat: girls need their dads. No way no how no day would I try to diminish or worse negate the role of a dad in his daughter's life. No one, and that includes humans, saints, and extraterrestrials, could convince me that my princess's life would be better off without me in it. However, just as there are some aspects of being a female that my daughter's mother is more equipped to guide her through, there are aspects of being a male that I hope I have helped my son navigate in a way that only I could.I\n\nThis is my beating heart: boys need fathers.\n\nBoys need fathers\u2014period, exclamation point.\n\nAnd if a boy is not blessed with a father or gifted with a dynamic stand-in then he must find ways to make one. He must identify the fatherish men in his life, find what he needs from them, and compose one.\n\nIt is an act of necessity, and I should know. My mother was not far along into her nineteenth year when she had me by a man who lived no more than a bike ride away but was absent for my first decade of life.II To say I had no father, though, is a half-truth. To say my mother was my father would be a sentimental-ass lie. I had a father, and I had one because I made one. Or rather I composed a father from the men at hand, brothers who kept me long before Obama made it a project.III\n\nThere was my mother's long-term boyfriend Big Chris, my maternal grandfather Sam, my maternal uncle Anthony, my paternal uncle Henry, and at long last my biological father, Wesley. If you asked me to spell father, I could turn their names into one long-ass portmanteau.\n\nOr I might just say \"p-o-p-s.\"\n\nPops was a group of men who provided a loving example of what it would soon enough mean to be a man. Pops nurtured me. Bestowed me with his wisdom. Pushed me to nuance the way I saw the world. He inspired me to dream. He tended my harms. He made sure I knew it was in me to surpass him.\n\n### BIG CHRIS\n\nFar as I knew growing up my biological father was a ghost by the time I was born. By the time I was a year old, my mother had been heart-throbbed by a man named Big Chris. Big Chris was a recent parolee\u2014bank robbery, what a dreamer!\u2014and a neophyte\/soon-to-be-prosperous pimp, but also a smart, witty, compassionate man whose jokes could give you stomachaches. My mother had two boys by Big Chris and stayed with him until just before I reached double digits. For years after he left, he would swing through trying to rekindle his and my mother's faded love or else connect with his boys, and without a doubt, whether Big Chris and my mom were an official couple or not, I was one of those boys. The man never treated me one bit different from the sons of his seed. The naysayers can knock how he hustled his bread and meat, but that don't change the fact that Big Chris was the one who showed me the value and impact of a father's love, that family often had nothing to do with genetics. This was a lesson he taught me in life and in death.\n\nIn September 2009, I got a call from Big Chris's daughter\u2014my oldest sister\u2014saying that he was sick and that I should fly out to Phoenix to see him. In the span of a few days, she went from prodding me to make it out soon, to imploring me to come ASAP if I wanted to see him alive. The next day I was on a flight, bracing for the worst and praying against it. My flight landed heartstroke hours later and while passengers were grabbing their bags, I turned on my cell phone to a fusillade of texts: from my mother, from my brothers, from my sister\u2014all warning me Big Chris had died. My big sister picked me up from the airport, and tried to console me with Dad's near-to-last words: \"I've got to hold on. I've got to hold on so I can see Mitch.\" The story didn't console me in the moment, but later, much later, when the grief begrudged me room to breathe, Dad's near-to-last words confirmed for me the bond that we'd shared, reaffirmed that I would forever be one of his boys, that our kinship was deeper than DNA.\n\n### SAM\n\nMy maternal grandfather, Sam Jackson, Jr., rose every day for thirty-plus years to go to the same job. He attends church every Sunday\u2014and arrives on time all the time. He pays his bills and his tithes. He represents at neighborhood rallies and community meetings. He bought and has lived in the same house since the '70s, lived there with his wife until she died, lives there with a new wife now. Granddad or Dad, as I call him, rescued us\u2014my mom and her boys\u2014countless times with funds because the electric company put an apartment of ours on eclipse or the rent had somehow vamoosed out of my mom's purse. Granddad moved me into his house for my last two years of high school, this after I ran away from my biological father's house, after I'd made it clear to all concerned adults that I couldn't be trusted under the charge of my half-paralyzed great-grandmother. Me, Granddad, and my cousin-brother Jesse ate breakfast together in his kitchen damn near every weekday.IV Granddad sat in the bleachers at my home and away high school hoop games and kept full stats. He chided me to mow the lawn and take out the trash and repaid me by spotting me the bucks I needed to hang with my homeboys on Friday nights. He never once bemoaned being my caretaker, as I imagined he had a right to, not even after he had to slap spit from me for the class-A house crime of sneaking girls into my basement bedroom.V Granddad has modeled what it means to be a stand-up dude, what it means to honor your commitments, what it means to shoulder your obligations and your burdens without gripe.\n\n### ANTHONY (ANT)\n\nMy maternal uncle Ant wore some version of a Jheri curl well past the great epoch of Jheri curls. Furthermore, Ant's held on to his almost-a-high-school-All-American story for generations, a legend I've heard told so often at family dinners, that sometimes I go ahead and tell it myself. Let Ant or me tell it, the judges clocked him at 9.7, 9.6, and 9.5 in the 100-yard sprint at a district track meet, but if they had given him an official time of 9.5 instead of 9.6, he \"would've been an All-American that year.\"\n\nAnt's story is a tendon to what happened to me in sixth grade, the only year I ever competed on a track team. That year, I'd taken second place in the district meet to a rival who had been putting a whooping on me all season. Ant attended that meet and was disappointed right along with me. He could've let me play a defeatist, but instead he took it upon himself to train me for the city championship, swooping me afternoons after school and teaching me to run on my toes and lean forward and lengthen my stride, drilling into my porous brain the idea that I could beat anyone as long as I used good form and believed. The championships rolled around a couple of weeks later and sure enough I was lined up against my rival in the 100-meter final. Pow! We took off and by midway through the race I was losing in slo-mo and heard my heart scream no, no, nooooooooooo. Then by some kind of Prefontaine magic I heard Ant screaming, \"REACH, nephew! REACH!\" above all other voices, and reach I did on the way to winning the race with a slight cushion. You should have seen Ant afterward, rejoicing as if, at once, I'd won Olympic gold and salved his All-American wound. Thanks to Ant, I had my first taste of being a champion in public, of realizing that with assiduousness and self-confidence, my impossible was possible. For sure he was a father that day, one who'd pushed me to succeed where he'd failed, to be bolder, bigger, stronger, best.\n\n### HENRY\n\nSomewhere in my random collection of family archives is a hubris-building copy of a news feature on my paternal uncle Henry titled, \"Superman in Solitary: Oregon's Biggest Dope Dealer Tells All.\" The story details Uncle Henry's 1970s to mid-'80s evolution from car thief and pimp to drug kingpin. The article was straight-up inspiration, though, full disclosure, I didn't know Uncle Henry at all during the days of him hustling enough funds to buy a plane and Rolls-Royce. In fact, we spent almost no time together until right after I graduated high school, which was the summer I decided that being a devoted part-time dope dealer was the best present way for me to make a living. Keep in mind, this was decades past Uncle Henry's gilded heyday, well into the age of him being a shyster and ardent addict, and though I knew about his fall, I was beguiled by the lore, was hungry to profit from secrets I was sure he owned. So one day my older brother\u2014a fellow neophyte dope dealer\u2014and me tracked down Uncle Henry at the apartment of another uncle and pressed him for what in effect was a session of Drug Dealing 101. Uncle Henry, ever the capitalist, obliged us a lesson for a few shards of our dope. Can't recall everything he said, but one point will stick with me till I'm dust: \"The fast nickel beats the slow twenty.\" My uncle went on to explain that while we waited forever for a twenty sale, we could've sold umpteen fives, which meant to me that what I dreamed of would not arrive in a windfall but would accrue one small sale at a time. It's easy to make the case that Uncle Henry was undermining my brother and me, but the way I see it, his advice had less to do with corrupting our youth or sabotaging our gleaming futures, and more to do with the munificence of exposing us to a maxim that had grand effect for him. Because he knew that no matter what we did, we would need to learn how to hustle\u2014to reimagine paths to success\u2014that hustling was vital to young black boys, that without it we were destined to be failed black men. Though I never made hundreds of thousands nor had the misfortune of being the local drug kingpin, Uncle Henry's lecture and legacy helped convince me that I had hustle in my blood, and please believe me when I tell you, I've been a hustler ever since.\n\n### WESLEY\n\nTen\u2014that's how old I was when I met my biological father. One of my most significant memories of him occurred not too long after when his wife, oldest son, two daughters, and I road-tripped to visit Disneyland, Sea World, and a few Californian family members. One of those relatives lived in an apartment complex with a pool\u2014a pool! We all changed into swimming gear and headed out to the pool, where my dad and my brother and sisters began having a grand old time swimming and playing in the water and goading me to get in while I gallivanted around the pool and at most teased my foot in the shallow end once or twice. My trepidation was for good reason as this was circa '85, arguably the height of a certain highwater-pants and rhinestone-glove-wearing pop star, and I had a Jheri curl befitting a kid who claimed to certain credulous classmates that I was a not-so-distant cousin. Or let me put it like this: young Mitch Jackson was not about to get his MJ-esque dew wet nor\u2014the extent of my swimming skills at the time was a hella-weak doggy paddle\u2014was I about to risk my life. But my biological father flexed contrary designs by creeping up behind me and scooping me in the air and tossing me in the pool. He didn't flinch while I flailed and screamed and gulped mouthfuls of overchlorinated water. He said something to me that I can't remember but that my subconscious must've heard because soon I calmed and got my curly head above the surface, and stayed in the pool and had a damn good day frolicking with my father, brother, and sisters\u2014aka the Johnsons. The message of that day took years to reveal itself to me: \"Troubled water or not, you best learn to swim. 'Cause when your young-ass get to drowning, I may not be moved to rescue.\" That message, by the way, I now count as an act of stern beneficence.\n\n\u2022 \u2022 \u2022\n\nNot one of the men I mentioned has existed in my life beyond the reach of critique. Oh yes, I comprehend flaws. But their foibles weren't the crux of what I used to build. I must say, too, that they were much more than mentors. Mentors teach you a skill. Fathers teach you to live. Your mentor's role can remain static. Your father's role must evolve. A mentor's direction might be free of deep feeling. A father's guidance must be rooted in love.\n\nWho I am now is who I must be: a flawed human striving to live in a state of becoming. Along the way I've discovered a thing or more about myself: that who and how I love is not dictated by law or blood, that being a constant presence is as much a part of being a man as almost anything else, that what I want must be earned, that I can win and win I will, that there's hustle in my genes, that either I swim or drown and there is no one more important to that outcome than me.\n\nNow here I am the father of two children, trying my all-out damnedest to mind the lessons of my beloved composite, all the while feeling encouraged by the fact I know they're rooting for me to best the job they did.\n\nThank. You. Pops.\n\n* * *\n\nI. Praise be to the gender politicians. By male and female I mean cisgendered male and female\u2014the Latin prefix cis means \"on the same side\"\u2014i.e., men and women whose gender identity is aligned with the gender they were assigned by birth.\n\nII. For my DNA dad's sake, I must note that the absoluteness of his early absence is a point of dispute.\n\nIII. Obama (BO) is the latest exemplar\u2014a total of twelve were either abandoned or lost their biological fathers when they were young\u2014of a president whose life confirms how efficacious it is to compose a composite. It's damn near folklore now, how Barack Hussein Obama, Sr., had bounced on his wife and BO by the time he was a toddler, how his mother spent time in Seattle, remarried in Hawaii, took young BO to live with her new husband in Indonesia, but sent him back to the Aloha State to live with her parents around the time he entered the fifth grade. One of BO's composites thereafter, if nothing else for the fact that he assumed the role of his long-term primary caretaker until he went off to college, was his maternal grandfather Stanley Dunham (no shade to Stanley's wife's role in co-parenting her grandson). Stanley was also the one who introduced BO to the man who just might own the title of Most Controversial of all presidential composites: a libertine, ex-journalist, poet, and Communist associate named Frank Marshall Davis, a man who became especially infamous during BO's first campaign when conspiracy theorists claimed Davis was his biological father. The truth, though, as confirmed by BO in his memoir, is that Davis helped shape his views on racial identity, race relations, and social justice. Davis was a part of BO's life but for a handful of years, but I'm calling him a composite for his impact. For example, though this next point may be a stretch (then again, so was a black man being elected the leader of the free world), remnants of Davis's radical thought can be found in the socialist-leaning legislation that is Obama Care. From the last to the first. George Washington (GW) lost his father, Augustine (Augustine's people called him Gus), when he was eleven. From that point, GW's older half-brother Lawrence Washington became his surrogate father. Answer me this: What would America look like if GW hadn't followed Lawrence into the military and politics (Lawrence fought in the War of Jenkins Ear and was later elected to Virginia's House of Burgesses)? Lawrence christened the Mount Vernon estate (or should we call it a plantation?), and GW paid homage to his beloved older brother when it was in his sole possession by hanging only his portrait in his study. GW and BO are notable for being the first and last, but the list between them includes Thomas Jefferson (TJ), who lost his father at thirteen and found a mentor in the philosophy professor William Small when he entered William and Mary College a few years later. Smith fostered in TJ a great appreciation for diverse disciplines and also a love of Enlightenment thinkers. He also introduced TJ to the politician and law professor George Wythe\u2014the man who became TJ's unofficial political and cultural mentor\u2014as much a composite as any man was for the future president. How amazing it must've been for an ambitious young TJ to sit around a supper table discussing politics and culture with Small, Wythe, and a governor. How fortunate TJ was to have been given the chance to later study law (there were no law schools in colonial America) with Wythe, and have that apprenticeship that included history, philosophy, and ethics. If you're looking for the lasting influence of TJ's composite, you need look no further than the ideals and language of the most important document in American history. The list of presidents who built composites also includes Gerald Ford (GF)\u2014he was born Leslie King, Jr.\u2014whose mother, Dorothy, divorced his biological father, Leslie King, Sr., on the grounds of \"extreme cruelty\" when her son was five months old. GF's biological father was the son of the millionaire businessman Charles Henry King, but that didn't stop him from bolting out of state (so much for broke pockets being the impetus for a deadbeat dad) and, as rumor had it, colluding with his father to skirt alimony and child-support judgments. Lucky for baby GF that Dorothy met Gerald Ford, Sr., a couple of years later. Ford Sr. wasn't no slouch. He became a successful businessman, was a church vestryman, a Mason, and later a local politician. He married Dorothy, adopted her young son, christened him a junior, and was, in GF's words, \"kind, fair, and firm.\" Ford Sr. and Dorothy, who had three more boys together, didn't mention to GF that Ford Sr. was not his biological father. GF didn't find that out until his biological father showed up at his high school job. But years and years later, in a letter GF dictated from the Oval Office, you can see how that visit did little to change his mind about his beloved composite: \"I loved and was guided in life by the only father I ever had\u2014Gerald R. Ford Sr. There was never any longing on my part to seek family outside of the one in which I was raised with such love, tenderness, and happiness.\"\n\nIV. When my cousin-brother Jesse's mother was murdered, he went to live with my great-grandparents for a time, but when he hit the first grade, he moved in with my granddad and lived with him until he became a legal adult. My granddad parenting Jesse is yet another hashmark in the ledger of why he deserves my love, respect, and admiration.\n\nV. The backstory: this occurred after I'd been caught on occasion with naked to half-naked girls in my basement bedroom. The scene: my granddad's house sits beside an alley, and there's a park bench at the opening of the alley. The action: this particular day my granddad came home early from work and spotted me sitting on that park bench with a girl whose name I couldn't name now if you paid me, but whose face I will never forget. By sitting I mean that I was leaned into her ear whispering the sweet nothings I hoped would lead to her knickknacks. She and I had not been in the house, though, so I was miffed when my granddad stopped his Buick in the alley and furied over to us. \"Hey, Dad, this is\u2014\" and before I could finish, he barked, \"What did I tell you?! What did I tell you about this?!\" and slapped the sound of a firework out of my cheek. He breathed over me for a moment or two and stomped back to his idling ride and meandered it into the garage. The girl's mouth was agape when I got up the courage to look at her. You should go, I said. She shook her head yes. You should go now, I said, and she rose and headed up the alley. She and I never spoke another word to each other after that day. These years later, I realize it was a testament to how much I love my granddad that it never dawned on me to curse him under my breath or consider running away like I damn sure would've if instead it was my DNA dad who'd struck me. Pretty sure I never snuck another girl into the basement either\u2014which spells mission accomplished for Sam Jackson, Jr., excuse me, mission accomplished for Dad.\n\n# PART III\n\n# JUBILEE\n\n## Theories of Time and Space\n\n## NATASHA TRETHEWEY\n\nYou can get there from here, though\n\nthere's no going home.\n\nEverywhere you go will be somewhere\n\nyou've never been. Try this:\n\nhead south on Mississippi 49, one-\n\nby-one mile markers ticking off\n\nanother minute of your life. Follow this\n\nto its natural conclusion\u2014dead end\n\nat the coast, the pier at Gulfport where\n\nrigging of shrimp boats are loose stitches\n\nin a sky threatening rain. Cross over\n\nthe man-made beach, twenty-six miles of sand\n\ndumped on the mangrove swamp\u2014buried\n\nterrain of the past. Bring only\n\nwhat you must carry\u2014tome of memory,\n\nits random blank pages. On the dock\n\nwhere you board the boat for Ship Island\n\nsomeone will take your picture:\n\nthe photograph\u2014who you were\u2014\n\nwill be waiting when you return.\n\n## This Far: Notes on Love and Revolution\n\n## DANIEL JOS\u00c9 OLDER\n\nAugust 2015\n\nDear Nastassian:\n\nYou told me to write this essay to our future children, but I'm writing to you instead. You said to tell them about how their mom worried, how she wasn't sure if it was a good idea bringing black life into a world that doesn't value it, but that she landed on hope amidst all the despair. Tell them, you said, about why their father does the work he does, what kind of world you hope to help build for them.\n\nAnd I will, love, I will. But this moment right now\u2014the night is quiet and I write while you sleep\u2014this moment with all its weight and responsibility, this turning point in the world and our lives, is ours, and these words are for you.\n\nThree weeks ago we rode through the midnight streets of Kingston, Jamaica, past shacks and gas stations, jerk chicken cookouts and quicky motels, to the airport and this new life together. Our Twitter feeds and the national news were filled with updates on Sandra Bland, the latest black life destroyed while in police custody, the most recent name to become a hashtag. Every time her deathlike mugshot flashed across the screen I felt an ache detonate in me. It's an ache many of us have become intimate with over the past year, as the hard work of protesters brings light to each new state-sanctioned murder. It recedes and then returns, compounded by the tragedy of how familiar it feels to mourn a stranger.\n\nIn college, I scribbled a quote from Eqbal Ahmad in the back of my notebook: \". . . this out-administration occurs when you identify the primary contradiction of your adversary and expose that contradiction . . . to the world at large.\" Ahmad wrote those words in reference to global struggles against empire, and trapped as I was just then, and probably always will be, in some wordy labyrinth between the future and past, the sentence settled somewhere in my brain and caught fire.\n\nIn a way, these words infer the same conclusion as the other quotes I'd copied around it: that art is a creator and a destroyer and no less a player in the great stage of the world than politics or violence. \"It is in the nexus of representation, words, and space,\" Michel Foucault wrote, \"that the destiny of peoples is silently formed.\" Or the Syrian poet Nizar Qabbani: \"Our deliverance is in drawing with words.\"\n\nBut Mr. Ahmad dispenses with the formality of arguing the power of representation, and jumps directly into strategy. Unlike so many of the texts we read in college, this passage is not concerned with making people comfortable or rehashing basic truths that are deemed controversial only because they agitate overprotected egos. \"I argued that armed struggle,\" Eqbal Ahmad writes at the beginning of that paragraph, \"is less about arms and more about organization, that a successful armed struggle proceeds to out-administer the adversary and not out-fight him.\" Ahmed is concerned with victory, which is to say, survival.\n\nIt's been a year since Officer Darren Wilson shot Michael Brown dead in the streets of Ferguson. (I was in an airport that day, too, waiting for a flight to Cuba and watching Twitter explode with tweets from the scene of the murder. You texted me then and many times since, that you weren't so sure about coming to a country that could do this to its people, a country that went out of its way to destroy black life.) It's been a year of politicians stumbling to declare that all lives matter and reinstill the illusion of justice to the justice system. It's been a year in which police took more than three hundred black lives as protestors shut down bridges and highways across the country to remind the world that those lives matter.\n\nI spent my twenties with a healthy distrust of the word revolution. When I was a kid, it was ancient American history or what Star Wars characters did\u2014something heroic and distant. But I'm the son of a survivor of how wrong revolutions can go, the nephew of a revolutionary turned counterrevolutionary turned political prisoner. And these days, you're more likely to see revolution on a car ad than anywhere meaningful. Words mean things, we say again and again, but overuse and abuse can wear those meanings down, render them pale parodies of what they once were. And revolution, it seemed, had long since lost its meaning.\n\nThe Ferguson uprising changed that. The movement for black lives spread from city to city, spurned on by social media and the long-pent-up feeling that no social movement in recent memory has done anything but tiptoe toward justice. You can't tiptoe toward justice. You can't walk up to the door all polite and knock once or twice, hoping someone's home. Justice is a door that, when closed, must be kicked in. \"No state,\" Baldwin wrote, \"has been able to foresee or prevent the day when their most ruined and abject accomplice\u2014or most expensively dressed prostitute\u2014will growl, 'This far and no further.' \" And maybe that day is more like a series of days, the whole year of protest that erupted between now and then, a culminating mass of days and nights, bodies laying down in intersections, symphony halls, strip malls, superhighways across this country, stopping traffic and business-as-usual, declaring by their very presence: \"No further,\" and again, \"No further.\"\n\nI texted you updates as we marched: \"Still safe and things are mostly calm. We've taken Columbus Circle. Helicopters overhead but cops can't seem to keep up with us or figure out where we're going next.\" They couldn't figure out where we were going next because we had no idea where we'd go next. We spun in an impossible, unruly snake through Midtown, spilled out into the streets and then the bridges and throughways. One night we shut down the Manhattan Bridge and pushed deep into Crown Heights, an army of flashing blue lights at our backs. With no coordination, no grant dictating our steps or signs, no leader, we marched in lockstep with hundreds of thousands of protestors across the United States and then the globe, and the simple, resonating demand that black lives matter laid bare the twin lies of American equality and exceptionalism. Even on the left, even in this age of exposed racial rifts, politicians still say with a straight face that this country was founded on principles of equality. Words mean things, we say again and again, but actions mean much more, and still as a nation, we worship the very slave owners who gave legal precedence to the notion of percentages of human beings. We scream equality and freedom while unabashedly modeling our actions on the fathers of genocide. The only way to rationalize this most American of contradictions is to devalue the lives of the slaughtered, as was done then, so it must be now, and so apologists remind us that those were the times, and they didn't know better, and on and on. But if those lives matter now then they mattered then, and the clapback stretches through history, unraveling all the creation myths this country has always held most sacred, toppling our many false idols and cleaning out our profaned temples.\n\nThere was a terrible hunger revealed in that ongoing funeral procession. So many showed up because so many must mourn, the trauma of bearing witness etched across the streets of America. And collective mourning became collective resistance, and the hunger born from so much witnessing and so little action over the years was the hunger to rebel. Revolution has sounded, as Tracy Chapman once sang, like a whisper. I heard it in my own writing on equality in publishing, demanding more than just reform, more than just diversity. Heard it in my friends and loved ones' quiet ferocity as we talked into the night. But suddenly it was a collective howl, it echoed through the streets and out across the world: \"This far and no further.\"\n\nIt's that hunger that I was trying to understand back in college when I jotted that quote down. It felt like tracing along the clues of a murder mystery: something was wrong. I couldn't identify the crime, but I was aware of it, inside of me and in the whole world around me, and both were deeply connected. That's why Eqbal Ahmad's idea about primary contradiction tattooed itself on my brain. I knew from very early on that I was an artist, that art was my own form of medicine, both for myself and the world: a tool that could create or destroy. And I knew I profited from the crime\u2014as a straight cis man, a Latino who isn't black, a citizen of this great disastrous nation. And I knew I suffered from this crime with no name, too, that it robbed everyone in its grasp of humanity and self, made us tools and killers and liars and suicides. There were so many myths to unravel, even just within my own heart, my own head, but mythology was something I could understand. There were myths that were lies and myths made from truth, and often the falsest ones were the most plausible and the truest filled with dragons and gods. Every journey is a crisis, a turning point, a shedding of myths, and mine began with the gnawing certainty that something did not add up. And in a way, this journey never ends, but in another sense, it ends where all great roads lead: to the discovery of voice.\n\nThis year, the desperate hunger born from so much mourning found its collective voice. It happened in the streets, but it also happened across the Internet, in journals, and late-night phone calls. The revolution wasn't televised\u2014there we saw only burning cars and concerned pundits\u2014but it was live-tweeted. And while my own revolution took place on the page and in the streets, yours was a much more personal one, profound and earthshaking in its own, very different, way. I watched you stand at the crossroads of despair, Nastassian. Watched fear wash over you, and uncertainty. As that ache detonated again and again, I know the temptation to shut down entirely loomed large. We arrived at Manley Airport in Kingston to return to New York, checked our bags and emptied our pockets into the plastic bins, took off shoes and belts and were patted down and X-rayed and then whisked up an escalator to the waiting area. Kingston was a distant smattering of lights across the bay and home seemed a long way off for both of us. Sandra Bland's face stared out from the television as broadcasters wondered through their phony cheer about her last moments on earth, languishing in that Texas jail cell. International travel is a closed circuit\u2014once you're in flight, there's no turning back and then you're vomited directly into the hands of U.S. immigration officials, passport control, customs, sniffing dogs, and the forever fallout of 9\/11. And even if you have nothing to hide, and we had nothing to hide, it feels like the cold machinery of the state closing around your neck. I thought about all the times I'd been \"randomly\" searched and quelled my own anxiety and turned to you, wondering if you would be freaking out, but you met me with a smile.\n\nTell them how their mother landed on hope amidst all the despair, you told me weeks later when I said I didn't know how to write this essay. And in that I saw a miracle: your own journey, your own revolution, unraveling beside me and mine and also separate, a whole country and sea away. You chose hope, and the night is quiet and I write while you sleep\u2014and this moment with all its weight and responsibility, this turning point in the world and our lives, is ours, and these words are for you.\n\n\u2014D\n\n## Message to My Daughters\n\n## EDWIDGE DANTICAT\n\nSoon after the one-year anniversary of the fatal shooting of Michael Brown by the Ferguson police officer Darren Wilson, I was in Haiti, at the southernmost end of the country's border with the Dominican Republic, where hundreds of Haitian refugees had either been deported or driven out of the Dominican Republic by intimidation or threats. Many of these men and women had very little warning that they were going to be picked up or chased away and most of them had fled with nothing but the clothes on their backs.\n\nIt was a bright sunny day, but the air was thick with dust. As some friends and I walked through the makeshift resettlement camps on the Haitian side of the border, in a place called Pak Kado, it felt as though we, along with the residents of the camps, were floating through clouds. Around us were lean-tos made of cardboard boxes and sheets. Dust-covered children walked around looking dazed even while playing with pebbles that stood in for marbles, or while flying plastic bags as kites. Elderly people stood on the edge of food and clothes distribution lines, some too weak to wade into the crowd. Later the elderly, along with pregnant women and the disabled, would be given special consideration by the priest and nuns who were giving out the only food available to the camp dwellers, but the food would always run out before they could get to everyone.\n\nA few days after leaving Haiti and returning to the United States, I read a Michael Brown anniversary opinion piece in The Washington Post written by Raha Jorjani, an immigration attorney and law professor. In her essay, Jorjani argues that African Americans living in the United States could easily qualify as refugees. Citing many recent cases of police brutality and killings of unarmed black men, women, and children, she wrote:\n\nSuppose a client walked into my office and told me that police officers in his country had choked a man to death over a petty crime. Suppose he said police fatally shot another man in the back as he ran away. That they arrested a woman during a traffic stop and placed her in jail, where she died three days later. That a 12-year-old boy in his country was shot and killed by the police as he played in the park.\n\nSuppose he told me that all of those victims were from the same ethnic community\u2014a community whose members fear being harmed, tortured or killed by police or prison guards. And that this is true in cities and towns across his nation. At that point, as an immigration lawyer, I'd tell him he had a strong claim for asylum protection under U.S. law.\n\nThis is not the first time that the idea of African Americans as internal or external refugees has been floated or applied. The six-million-plus African Americans who migrated from the rural south to urban centers in the northern United States for more than half a century during the Great Migration were often referred to as refugees, as were those people internally displaced by Hurricane Katrina.\n\nHaving now visited many refugee and displacement camps, the label \"refugee\" at first seemed an extreme designation to assign to citizens of one of the richest countries in the world, especially if it is assigned on a singular basis to those who are black. Still, compared to the relative wealth of the rest of the society, a particularly run-down Brooklyn public housing project where a childhood friend used to live had all the earmarks of a refugee camp. It occupied one of the least desirable parts of town and provided only the most basic necessities. A nearby dilapidated school, where I attended junior high, could have easily been on the edge of that refugee settlement, where the primary daily task was to keep the children occupied, rather than engaged and learning. Aside from a few overly devoted teachers, we were often on our own. We, immigrant blacks and African Americans alike, were treated by those who housed us, and were in charge of schooling us, as though we were members of a group in transit. The message we always heard from those who were meant to protect us: that we should either die or go somewhere else. This is the experience of a refugee.\n\nI have seen state abuses up close, both in Haiti, where I was born under a ruthless dictatorship, and in New York, where I migrated to a working-class and predominantly African, African American, and Caribbean neighborhood in Brooklyn at the age of twelve. In the Haiti of the 1970s and early '80s, the violence was overtly political. Government detractors were dragged out of their homes, imprisoned, beaten, or killed. Sometimes their bodies were left out in the streets, in the hot sun, for extended periods, to intimidate neighbors.\n\nIn New York, the violence seemed a bit more subtle, though no less pervasive. When I started riding the city bus to high school, I observed that a muffled radio message from an annoyed bus driver\u2014about someone talking too loud or not having the right fare\u2014was all it took to make the police rush in, drag a young man off the bus, and beat him into submission on the sidewalk. There were no cell phone cameras back then to record such abuse, and most of us were too terrified to demand a badge number.\n\nBesides, many of us had fled our countries as exiles, migrants, and refugees just to escape this kind of military or police aggression; we knew how deadly a confrontation with an armed and uniformed authoritarian figure could be. Still, every now and then a fellow traveler would summon his or her courage and, dodging the swaying baton, or screaming from a distance, would yell some variation of \"Stop it! This is a child! A child!\"\n\nOf course, not all of the police's victims were children. Abner Louima, a family friend, was thirty years old when he was mistaken for someone who had punched a police officer outside a Brooklyn nightclub, on August 9, 1997, sixteen years to the day before Michael Brown was killed. Abner was arrested, beaten with fists, as well as with police radios, flashlights, and nightsticks, and then sexually assaulted with the wooden handle of a toilet plunger or a broom inside a precinct bathroom. After Abner, there was Amadou Diallo, a Guinean immigrant, who was hit by nineteen of the forty-one bullets aimed at him as he retrieved his wallet from his pocket. Then there was Patrick Dorismond, the U.S.-born child of Haitian immigrants, who died trying to convince undercover cops that he was not a drug dealer.\n\nThese are only a few among the cases from my era that made the news. There was also sixty-six-year-old Eleanor Bumpurs, who, thirteen years before Abner's assault, was killed by police with a twelve-gauge shotgun inside her own apartment. I have no doubt there were many others. We marched for all of them in the Louima\/Diallo\/Dorismond decade. We carried signs and chanted \"No Justice! No Peace!\" and \"Whose Streets? Our Streets!\" even while fearing the latter would never be true. The streets belonged to the people with the uniforms and the guns. The streets were never ours. Our sons and brothers, fathers and uncles, our mothers and sisters, daughters and nieces, our neighbors were, and still are, prey.\n\nMy father, a Brooklyn cab driver, used to half joke that police did not beat him up because, at sixty-five years old, he was too skinny and too old, and not worth the effort. Every now and then, when he was randomly stopped by a police officer and deigned to ask why, rather than a beating, he would be given a handful of unwarranted traffic citations that would wipe out a few weeks' hard-earned wages. Today, one might generously refer to such acts as micro-aggressions. That is, until they turn major and deadly, until other unarmed black bodies, with nowhere to go for refuge, find themselves in the path of yet another police officer's or armed vigilante's gun.\n\nWhen it was announced that Darren Wilson would not be indicted for the killing of Michael Brown, I kept thinking of Abner Louima, whose assault took place when Michael Brown was just eighteen months old. Abner and I have known each other for years. Both our families have attended the same Creole-speaking church for decades, so I called him to hear his thoughts about Michael Brown's killer going free. If anyone could understand all those broken hearts, all the rage, all the desperation, the yearning for justice, what it is to be a member of a seemingly marooned and persecuted group, I thought, he would.\n\nAbner Louima, unlike Michael Brown, had survived. He went on with his life, moved from New York City to south Florida, started businesses. He has a daughter and two sons. One son was eighteen years old when we spoke, the same age Michael Brown was when he died.\n\nHow does he feel, I asked him, each time he hears that yet another black person was killed or nearly killed by police?\n\n\"It reminds me that our lives mean nothing,\" he replied.\n\nWe are in America because our lives meant nothing to those in power in the countries where we came from. Yet we come here to realize that our lives also mean nothing here. Some of us try to distance ourselves from this reality, thinking that because we are another type of \"other\"\u2014immigrants, migrants, refugees\u2014this is not our problem, nor one we can solve. But ultimately we realize the precarious nature of citizenship here: that we too are prey, and that those who have been in this country for generations\u2014walking, living, loving in the same skin we're in\u2014they too can suddenly become refugees.\n\nParents are often too nervous to broach difficult subjects with their children. Love. Sex. Death. Race. But some parents are forced to have these conversations early. Too early. A broken heart might lead to questions we'd rather not answer, as might an inappropriate gesture, the death of a loved one, or the murder of a stranger.\n\nEach time a black person is killed in a manner that's clearly racially motivated, either by a police officer or a vigilante civilian, I ask myself if the time has come for me to talk to my daughters about Abner Louima and the long list of dead that have come since. My daughters have met Abner, but I have never told them about his past, even though his past is a future they might have to face.\n\nWhy don't I tell them? My decision is about more than avoiding a difficult conversation. The truth is, I do not want my daughters to grow up as I did, terrified of the country and the world they live in. But is it irresponsible of me to not alert them to the potentially life-altering, or even life-ending horrors they might face as young black women?\n\nThe night President Barack Obama was first elected (would he too qualify for refugee status?), my oldest daughter was three and I was in the last weeks of my pregnancy with my second. When President Obama was inaugurated for the first time, I was cradling both my little girls in my arms.\n\nTo think, I remember telling my husband, our daughters will never know a world in which the president of their country has not been black. Indeed, as we watched President Obama's inaugural speech, my oldest daughter was shocked that no woman had ever been president of the United States. That day, the world ahead for my girls seemed full of greater possibility\u2014if not endless possibilities, then at least greater than those for generations past. Many more doors suddenly seemed open to my girls, and the \"joyous daybreak\" evoked by Martin Luther King, Jr., in his \"I Have a Dream\" speech, a kind of jubilee, seemed to have emerged. However, it quickly became clear that this one man was not going to take all of us with him into the postracial promised land. Or that he even had full access to it. Constant talk of \"wanting him to fail\" was racially tinged, as were the \"birther\" investigations, and the bigoted commentaries and jokes by both elected officials and ordinary folk. One of the most consistent attacks against the president, was that, like my husband and myself, he was born elsewhere and was not really American.\n\nLike Barack Obama's father, many of us had brought our black bodies to America from somewhere else. Some of us, like the president, were the children of such people. We are people who need to have two different talks with our black offspring: one about why we're here and the other about why it's not always a promised land for people who look like us.\n\nIn his own version of \"The Talk,\" James Baldwin wrote to his nephew James in \"My Dungeon Shook,\" \"You were born in a society which spelled out with brutal clarity, and in as many ways as possible that you were a worthless human being.\"\n\nThat same letter could have been written to a long roster of dead young men and women, whose dungeons shook, but whose chains did not completely fall off. Among these very young people are Oscar Grant, Aiyana Stanley-Jones, Rekia Boyd, Kimani Gray, Renisha McBride, Trayvon Martin, Michael Bell, Tamir Rice, Michael Brown, Sandra Bland, and counting. It's sad to imagine what these young people's letters from their loved ones may have said. Had their favorite uncle notified them that they could qualify for refugee status within their own country? Did their mother or father, grandmother or grandfather warn them to not walk in white neighborhoods, to, impossibly, avoid police officers, to never play in a public park, to stay away from neighborhood watchmen, to never go to a neighbor's house, even if to seek help from danger?\n\nI am still, in my own mind, drafting a \"My Dungeon Shook\" letter to my daughters. It often begins like this. Dear Mira and Leila, I've put off writing this letter to you for as long as I can, but I don't think I can put it off any longer. Please know that there will be times when some people might be hostile or even violent to you for reasons that have nothing to do with your beauty, your humor, or your grace, but only your race and the color of your skin. Please don't let this restrict your freedom, break your spirit, or kill your joy. And if possible do everything you can to change the world so that your generation of brown and black men, women, and children will be the last who experience all this. And please do live your best lives and achieve your full potential. Love deeply. Be joyful. In Jubilee, Mom.\n\nTo my draft of this letter, I often add snippets of Baldwin.\n\n\"I tell you this because I love you and please don't you ever forget it,\" Baldwin reminded his James. \"Know whence you came. If you know whence you came, there is no limit to where you can go.\"\n\n\"The world is before you,\" I want to tell my daughters, \"and you need not take it or leave it as it was when you came in.\"\n\nI want to look happily forward. I want to be optimistic. I want to have a dream. I want to live in jubilee. I want my daughters to feel that they have the power to at least try to change things, even in a world that resists change with more strength than they have. I want to tell them they can overcome everything, if they are courageous, resilient, and brave. Paradoxically, I also want to tell them their crowns have already been bought and paid for and that all they have to do is put them on their heads. But the world keeps tripping me up. My certainty keeps flailing.\n\nSo I took them to the border, the one between Haiti and the Dominican Republic, where hundreds of refugees were living, or rather existing. There they saw and helped comfort men, women, and children who look like them, but are stateless, babies with not even a bedsheet between them and a dirt floor, young people who may not be killed by bullets but by the much slower assault of disease.\n\n\"These are all our causes,\" I tried to both tell and show them, brown and black bodies living with \"certain uncertainty,\" to use Frantz Fanon's words, black bodies fleeing oppression, persecution, and poverty, wherever they are.\n\n\"You think your pain and your heartbreak are unprecedented in the history of the world, but then you read,\" James Baldwin wrote. Or you see. Or you weep. Or you pray. Or you speak. Or you write. Or you fight so that one day everyone will be able to walk the earth as though they, to use Baldwin's words, have \"a right to be here.\" May that day come, Mira and Leila, when you can finally claim those crowns of yours and put them on your heads. When that day of jubilee finally arrives, all of us will be there with you, walking, heads held high, crowns a-glitter, because we do have a right to be here.\n\n## Acknowledgments\n\nI'd like to thank all of the writers who contributed to this book. I wasn't asking an easy thing of them when I solicited their creative work; instead, I was asking them to write toward the hurt, to wrestle with the ugly truths that plague us in this country. Each of the writers did just that, and they did so beautifully. My editor, Kathryn Belden, believed in this book before I could even articulate why I wanted to work on it. She edited this collection tirelessly. She makes me a better writer, and I am so grateful to work with her. Her assistant, David Lamb, ushered the book through the first round of edits, communicated with the contributors about contracts, and aided in copyedits. I'm thankful for Jennifer Lyons, my agent, who has always been my advocate. I am indebted to Scribner, who welcomed this book into their catalog and me into their coterie of writers. Finally, I'd like to remember the sage, fierce artist who inspired this book, James Baldwin. I hope our work makes you proud.\n\n## Contributors\n\nCAROL ANDERSON is the Samuel Candler Dobbs Professor and chair of African American Studies at Emory University and the author of White Rage: The Unspoken Truth of Our Racial Divide and Eyes Off the Prize: The United Nations and the African American Struggle for Human Rights, 1944\u20131955, which was awarded the Myrna F. Bernath Book Award and the Gustavus Myers Outstanding Book Award and was selected as a finalist for the Truman Book Award and the W. E. B. Du Bois Book Award.\n\nJERICHO BROWN has published two poetry collections, Please and The New Testament, and has been the recipient of the Whiting Writers' Award, the American Book Award, a fellowship at the Radcliffe Institute for Advanced Study, and a National Endowment for the Arts grant. He is an associate professor of English and creative writing at Emory University in Atlanta.\n\nGARNETTE CADOGAN is editor-at-large of Nonstop Metropolis: A New York City Atlas (coedited by Rebecca Solnit and Joshua Jelly-Schapiro). He is currently a visiting fellow at the Institute for Advanced Studies in Culture at the University of Virginia, and a visiting scholar at the Institute for Public Knowledge at New York University. He writes about culture and the arts for various publications, and is at work on a book about walking.\n\nEDWIDGE DANTICAT, born in Haiti and raised in New York, has written both fiction and nonfiction for adults and children. Her memoir, Brother, I'm Dying, was awarded the National Book Critics Circle Award, and was finalist for the National Book Award, as was her short story collection Krik? Krak! She is a 2009 MacArthur Fellow.\n\nRACHEL KAADZI GHANSAH is an essayist and critic whose writing has appeared in the Believer, Rolling Stone, the Paris Review, Transition, and elsewhere. She was a finalist for the National Magazine Award, and she is a contributing writer for the New York Times Magazine. Her first book, The Explainers and Explorers, examines twenty-first-century America within the context of what it means to be black, brave, and self-defined, and it will be published by Scribner in 2017.\n\nMITCHELL S. JACKSON is the author of The Residue Years, which won the Ernest J. Gaines Award and was a finalist for the Center for Fiction's Flaherty-Dunnan First Novel Prize, the PEN\/Hemingway Award for Debut Fiction, and the Hurston\/Wright Legacy Award. He is a recipient of a Whiting Award and teaches writing at NYU, where he earned an MFA in creative writing.\n\nHONOR\u00c9E FANONNE JEFFERS is a poet, fiction writer, and critic. She is the author of four books of poetry and is at work on her first novel. Her fifth poetry book, in progress, The Age of Phillis, imagines the life and times of the eighteenth-century poet Phillis Wheatley, the first (known) black woman to publish a book. The recipient of fellowships from the National Endowment for the Arts and the Witter Bynner Foundation through the Library of Congress, Jeffers is an elected member of the American Antiquarian Society, an organization to which fourteen U.S. presidents have been elected. She teaches at the University of Oklahoma.\n\nKIMA JONES'S work has appeared in Guernica, NPR, PANK, Scratch Magazine, and The Rumpus, and she has received fellowships from PEN Center USA, Yaddo, and the MacDowell Colony.\n\nKIESE LAYMON is associate professor of English and Africana Studies at Vassar College and a recent Grisham Writer in Residence at the University of Mississippi. He is the author of the novel Long Division, which was selected as a best book of 2013 by Buzzfeed, The Believer, Salon, Guernica, Library Journal, and the Chicago Tribune, and an essay collection, How to Slowly Kill Yourself and Others in America. He is a columnist at The Guardian, and his forthcoming memoir, Heavy, will be published by Scribner.\n\nDANIEL JOS\u00c9 OLDER is the author of the Bone Street Rumba urban fantasy series (Roc Books, 2015 and 2016) and the Young Adult novel Shadowshaper (Scholastic, 2015), which was shortlisted for the Kirkus Prize in Young Readers' Literature. He coedited the Locus and World Fantasy Award\u2013nominated anthology Long Hidden: Speculative Fiction from the Margins of History. You can find Daniel's thoughts on writing, read dispatches from his decadelong career as an NYC paramedic, and hear his music at danieljoseolder.net and @djolder on Twitter.\n\nEMILY RABOTEAU is the author of The Professor's Daughter: A Novel and Searching for Zion: The Quest for Home in the African Diaspora, winner of the 2014 American Book Award and finalist for the Hurston Wright Legacy Award. Her distinctions include a Pushcart Prize, a literature fellowship from the National Endowment for the Arts, and the Chicago Tribune's Nelson Algren Award. She is a professor of English at the City College of New York, in Harlem, where she codirects the MFA program in creative writing. Her next novel is in the works.\n\nCLAUDIA RANKINE is the author of five collections of poetry including Citizen: An American Lyric, which won the National Book Critics Circle Award for Poetry in 2015, the PEN\/Open Book Award, the PEN Literary Award, and the NAACP Image Award, and was a finalist for the National Book Award. She lives in California, where she is the Aerol Arnold Professor of English at the University of Southern California.\n\nCLINT SMITH is a National Science Foundation Graduate Research Fellow and a doctoral candidate in education at Harvard University. He was a 2014 National Poetry Slam champion and has two popular TED Talks, The Danger of Silence and How to Raise a Black Son in America. His poems and essays have appeared in The New Yorker, The Guardian, and The American Literary Review. He is the author of the poetry collection Counting Descent.\n\nNATASHA TRETHEWEY is the author of four poetry collections, Domestic Work, Bellocq's Ophelia; Native Guard (which won the 2007 Pulitzer Prize) and Thrall, as well as a book of nonfiction, Beyond Katrina: A Meditation on the Mississippi Gulf Coast. Her column, \"Poem,\" appears weekly in the New York Times Magazine. She teaches at Emory University as the Robert W. Woodruff Professor of English and creative writing and has served for two terms as the United States Poet Laureate (2012\u201314).\n\nWENDY S. WALTERS is the author of Multiply\/Divide: On the American Real and Surreal; Troy, Michigan; Longer I Wait, More You Love Me; and a chapbook, Birds of Los Angeles. She is associate professor of creative writing and literature at The New School.\n\nISABEL WILKERSON is the first African American woman to win a Pulitzer Prize in journalism, awarded for her coverage of the 1993 Midwestern floods and for her profile of a ten-year-old boy caring for his four siblings. She is the author of the New York Times bestseller The Warmth of Other Suns, which won the National Book Critics Circle Award for Nonfiction in 2010.\n\nKEVIN YOUNG has edited five collections of poetry and published eight poetry collections of his own, including For the Confederate Dead, Book of Hours, and Jelly Roll, which was a finalist for the National Book Award for Poetry, as well as Blue Laws: Selected and Uncollected Poems, 1995\u20132015. He is also the author of The Grey Album: On the Blackness of Blackness, an encyclopedic nonfiction book examining through Jay Z's The Black Album and The Beatles's The White Album how African American culture is in many ways American culture, which won the Pen\/Open Book Award and the Graywolf Press Nonfiction Prize and was a finalist for the National Book Critics Circle Award for Criticism.\n\n## Permissions\n\nCarol Anderson, \"White Rage,\" from The Washington Post. Copyright \u00a9 2014. Reprinted by permission of the author.\n\n\"The Tradition\" by Jericho Brown. Copyright \u00a9 2016 by Jericho Brown.\n\nGarnette Cadogan, \"Black and Blue,\" from Freeman's: Arrival. Copyright \u00a9 2015. Reprinted by permission of the author.\n\n\"Message to My Daughters\" by Edwidge Danticat. Copyright \u00a9 2016 by Edwidge Danticat.\n\n\"The Weight\" by Rachel Kaadzi Ghansah. Copyright \u00a9 2016 by Rachel Kaadzi Ghansah.\n\n\"Composite Pops\" by Mitchell S. Jackson. Copyright \u00a9 2016 by Mitchell S. Jackson.\n\n\"The Dear Pledges of Our Love\" by Honor\u00e9e Fanonne Jeffers. Copyright \u00a9 2016 by Honor\u00e9e Fanonne Jeffers.\n\n\"Homegoing, AD\" by Kima Jones. Copyright \u00a9 2016 by Kima Jones.\n\n\"Da Art of Storytellin' (a prequel)\" by Kiese Laymon, first published in the Oxford American. Copyright \u00a9 2015 by Kiese Laymon.\n\n\"This Far: Notes on Love and Revolution\" by Daniel Jos\u00e9 Older. Copyright \u00a9 2016 by Daniel Jos\u00e9 Older.\n\n\"Know Your Rights!\" by Emily Raboteau. Copyright \u00a9 2016 by Emily Raboteau.\n\nClaudia Rankine, \"The Condition of Black Life Is One of Mourning,\" first published in The New York Times Magazine (June 22, 2015). Copyright \u00a9 2015 by Claudia Rankine and The New York Times Company. Reprinted by permission of the author.\n\n\"Queries of Unrest\" by Clint Smith. Copyright \u00a9 2016 by Clint Smith.\n\nNatasha Trethewey, \"Theories of Time and Space,\" from Native Guard: Poems. Copyright \u00a9 2016 by Natasha Trethewey. Reprinted by permission of the author.\n\n\"Lonely in America\" by Wendy S. Walters is reprinted from the work entitled Multiple\/Divide: On the American Real and Surreal. Copyright \u00a9 2014 by Wendy S. Walters. Reprinted with the permission of The Permissions Company, Inc., on behalf of PCI Books, Inc., www.PCIbooks.org.\n\n\"Cracking the Code\" by Jesmyn Ward, first published on NewYorker.com. Copyright \u00a9 2015 by Jesmyn Ward. Reprinted by permission of the author.\n\n\"Introduction\" by Jesmyn Ward. Copyright \u00a9 2016 by Jesmyn Ward.\n\nIsabel Wilkerson, \"Where Do We Go From Here?\" first published in Essence. Copyright \u00a9 2015 by Essence Communications, Inc. Reprinted by permission of the author.\n\n\"Blacker Than Thou\" by Kevin Young. Copyright \u00a9 2016 by Kevin Young.\n\n## ABOUT THE EDITOR\n\n\u00a9 KIM WELSH\n\nJesmyn Ward received her MFA from the University of Michigan and is currently an associate professor of creative writing at Tulane University. She is the author of the novels Where the Line Bleeds and Salvage the Bones, which won the 2011 National Book Award and was a finalist for the New York Public Library Young Lions Fiction Award and the Dayton Literary Peace Prize. She is also the author of the memoir Men We Reaped, which was a finalist for the National Book Critics Circle Award and the Hurston\/Wright Legacy Award and won the Chicago Tribune Heartland Prize and the Media for a Just Society Award.\n\nMEET THE AUTHORS, WATCH VIDEOS AND MORE AT\n\nSimonandSchuster.com\n\nauthors.simonandschuster.com\/Jesmyn-Ward\nAlso by Jesmyn Ward\n\nMen We Reaped: A Memoir\n\nSalvage the Bones: A Novel\n\nWhere the Line Bleeds: A Novel\nWe hope you enjoyed reading this Scribner eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Scribner and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\nSCRIBNER\n\nAn Imprint of Simon & Schuster, Inc.\n\n1230 Avenue of the Americas\n\nNew York, NY 10020\n\nwww.SimonandSchuster.com\n\nCompilation and Introduction copyright \u00a9 2016 by Jesmyn Ward\n\nAll rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever. For information address Scribner Subsidiary Rights Department, 1230 Avenue of the Americas, New York, NY 10020.\n\nFirst Scribner hardcover edition August 2016\n\nSCRIBNER and design are registered trademarks of The Gale Group, Inc., used under license by Simon & Schuster, Inc., the publisher of this work.\n\nFor information about special discounts for bulk purchases, please contact Simon & Schuster Special Sales at 1-866-506-1949 or business@simonandschuster.com.\n\nThe Simon & Schuster Speakers Bureau can bring authors to your live event. For more information or to book an event contact the Simon & Schuster Speakers Bureau at 1-866-248-3049 or visit our website at www.simonspeakers.com.\n\nInterior design by Erich Hobbing\n\nJacket design by Na Kim\n\nISBN 978-1-5011-2634-5\n\nISBN 978-1-5011-2636-9 (ebook)\n\nCopyright notices continued on page 225.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nGhost Moon\n\nHaving been thrown out onto the Edinburgh streets by her family, Maggie knows she must fight to survive. Many years later, the struggles she had to endure can be kept a secret no longer.\n\nSet mostly in post-war Britain and inspired by a real-life story, Ghost Moon is narrated with humour and compassion. A life-affirming read.\n\nPraise for Ron Butlin\n\n\"One of the most powerful and compelling pieces to emerge from the pen of this superb writer.\" \u2014ALEXANDER McCALL SMITH\n\n\"Poetic genius . . . Ron Butlin is the voice of Edinburgh.\" \u2014FringeReview.com\n\n\"Butlin is the best, the most productive Scottish poet of his generation.\" \u2014DOUGLAS DUNN\nGhost Moon\n\nWith an international reputation as a prize-winning novelist, Ron Butlin is also the Edinburgh Makar (poet laureate). In 2009 he was made the first ever Honorary Writing fellow (together with Ian Rankin) at Edinburgh University. Much of his poetry, as well as many of his novels and short stories have been broadcast and translated into over ten languages. In addition to his plays for BBC radio and theatre (most recently Sweet Dreams for Oran Mor in Glasgow), he has written five operas, two of them for Scottish Opera. \nBy the same author\n\nNOVELS\n\nThe Sound of My Voice\n\nNight Visits\n\nBelonging\n\nSHORT STORIES\n\nThe Tilting Room\n\nVivaldi and the Number\n\nNo More Angels\n\nPOETRY\n\nThe Wonnerfuu Warld o John Milton\n\nStretto\n\nCreatures Tamed by Cruelty\n\nThe Exquisite Instrument\n\nRagtime in Unfamiliar Bars\n\nHistories of Desire\n\nWithout a Backward Glance\n\nThe Magicians of Edinburgh\n\nDRAMA AND OPERA\n\nThe Music Box\n\nBlending In\n\nWe've Been Had\n\nSweet Dreams\n\nGood Angel \/ Bad Angel\n\nMarkheim\n\nFaraway Pictures\n\nThe Perfect Woman\n\nThe Money Man\n\nPublished by Salt Publishing Ltd\n\n12 Norwich Road, Cromer, Norfolk NR27 0AX\n\nAll rights reserved\n\nCopyright \u00a9 Ron Butlin, 2014\n\nThe right of Ron Butlin to be identified as the author of this work has been asserted by him in accordance with Section 77 of the Copyright, Designs and Patents Act 1988.\n\nThis book is in copyright. Subject to statutory exception and to provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Salt Publishing.\n\nSalt Publishing 2014\n\nCreated by Salt Publishing Ltd\n\nThis book is sold subject to the conditions that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.\n\nISBN 978 1 84471 998 3 electronic\nTo my wife, Regi, my sister Pam and my mother Elizabeth\n\nPART ONE\n\nSUNDAY\n\nTHE BLIZZARD WAS full-on \u2013 three hours instead of one with freezing fog, black ice and snow-snow-snow all the way. Penicuik, the Devil's Beef Tub, then tailgating the same Argos lorry over the Moffat Hills, snailing it behind Mr and Mrs Cautious down the M74.\n\nA one-man avalanche from Edinburgh \u2013 you've made it. You're here. That's what counts.\n\nAnd so . . .\n\nTime to get psyched up, get focused.\n\nBut first things first \u2014\n\nLog on. You've been driving for ever, so there's bound to be something. So many messages, so many puffs of oxygen to keep you breathing. It's not just you, of course \u2013 you can see it in people's faces when their mobiles ring, the relief that somebody wants them.\n\nC U @ 8 lol J x\n\nThe lovely Janice. You text her, confirming . . . and that's you back on solid ground. Another Sunday, another Mum visit, then a seventies download and the body-contour leather to carry you safely home. Second shave, second shower and into the second-best suit \u2013 for Janice, plus all the trimmings.\n\nTakes care of the day, takes care of you.\n\nYour phone snapped shut, you breathe easy once more.\n\nLife's good.\n\nWell, isn't it?\n\nZap-lock the car.\n\nThe path's needing shovelled clear and salted. A good six inches' worth. Fuck. If only you'd thought to bring your magic wand! But no probs. Fifteen minutes tops will see it \u2014\n\nNo.\n\nNo. No.\n\nThe front door's locked and her key's still in the mortice, inside. Jesus. How often have you told her about that key, about not leaving it in the lock? Maybe she's not been out all weekend? Maybe she's fallen? Can't get herself out of the bath?\n\nWhich'll mean breaking down the door.\n\nNot again.\n\nA good loud knock first, loud enough to wake the \u2014\n\nNo. Don't even think of it.\n\nThumping your fist big time. The snow's running ice-wet under your collar, the wind's razor-cutting your face. Stamp-stamp-stamping your feet on the front step to stop them turning into blocks of ice. You spoke on the phone only a few hours ago. She'd sounded fine, looking forward to seeing you.\n\nYou're freezing. Stamp-stamp-stamp. Thump-thump-thump . . .\n\nNinety years lying crumpled in a heap on the living room floor like she's \u2014\n\nDon't even think \u2014\n\nThank God.\n\nThe time it's taking her to turn the key in the lock . . .\n\n'Yes?' Her tone of voice, like she's never met you before. Not opening the door enough to let you in.\n\n'Mum? It's Sunday. I've come to \u2014 '\n\n'Your mother? Are you sure you've got the right house? I'm Mrs Stewart. Maggie Stewart.'\n\n'Let me in, Mum. I'm freezing out here. It's me. Tom.'\n\n'Tom? You know Tom?' Her face suddenly all smiles. Opening the door a little more. 'You've some news of him?'\n\n'News? It's me, I'm telling you.'\n\n'But you do know Tom?'\n\n'Of course I know \u2014 '\n\n'Then you'd better come in.'\n\nAt last. Into the cottage, into her sitting room \u2013 and a coal fire blazing in the grate. That's more like it.\n\n'Well then, and how is Tom?' She sits down.\n\n'But, Mum, can't you see it's me?'\n\n'I was told he'd be well looked after, so I hope he's fine. Mrs Saunders was most reassuring.'\n\n'Mrs Saunders? Who's Mrs \u2014 ?'\n\n'Tom won't remember her, of course. He was far too young. Between you and me, it's best he never hears her name. Best for everyone.'\n\nWhat the hell's she on about? The melted snow's dripping into your eyes, down your neck, your back. You want a towel. You want a seat. You want to get warmed up.\n\n'Don't you recognise me, Mum? Today's Sunday. I've driven down from Edinburgh same as always to see you \u2014 '\n\nNoticing you've glanced across at the tea trolley beside her, laid out with the usual straggled columns of playing cards \u2014\n\n'Learned to play patience during the war, and still keep it up,' she tells you. Like you didn't know already. 'Learned how to cheat then, too. The way you cheated turned into new rules so the game could go on. It had to, so you'd survive.'\n\nLike she's talking to a complete stranger.\n\n'Played it while waiting for the bombs to fall, sitting there in the black-out, waiting and waiting. Hearing the planes, the anti-aircraft guns in Leith . . .'\n\nBest to move things along. 'I'll make us some tea.'\n\nThrough to the kitchen. Water. Kettle. A towel for the hair.\n\nTea in the pot. Mugs, milk jug, plate of biscuits. Finish off the drying, set up the tray. Then through to get the afternoon back on course.\n\n'Here we are, Mum. I found a packet of HobNobs.'\n\nGetting everything back to a normal Sunday. Some tea, some talk. Fix what needs fixed. Clear her path, then hit the road in time to begin the arctic crawl that'll take \u2014\n\nBut . . . her photos? What's she done with the signed publicity shot of you as Mr Magic? It should be on the mantelpiece. And the lace curtains? What's she taken them down for? Might make the room seem brighter, but \u2014\n\n'. . . if the game worked out. We got bombed anyway. I was the only one in the family to survive. Nowhere to stay till the laird in the big house took pity. Lived here ever since. Sixty years and more, would you believe?'\n\nTalking like you'd never lived here yourself, weren't brought up here. Like she's never seen you before. Doesn't even know you.\n\nCan it happen that sudden?\n\nShe's half-rising from her chair as if to greet you for the first time. Such a warm, warm smile \u2013 you've not seen the like for months. So unexpected, so different from how she usually \u2014\n\n'You needn't worry, I'm really pleased to see you. I really am. I always knew you'd come.' She's almost in tears. Happy tears.\n\n'But I always do, Mum! I come every Sunday, don't I? When we spoke this morning I \u2014 '\n\n'I knew you'd come.' She's taken your hand and begun drawing it slowly down her face so your fingertips rest briefly on her eyes, her cheeks. Her lips.\n\nWhat the hell's all this?\n\n'I'm so very, very happy.'\n\nNext thing, she's led you across to the window where you stand side by side gazing out at her garden and the countryside beyond, a complete white-out of fields, woods and sky as far as you can see. Now the snow's easing off there's a few patches of faint blue, a last handful of tumbling flakes.\n\nGripping your arm: 'That's how I knew.' Pointing to what at first looks like a smudged fingerprint on the glass, a daytime moon: 'Everything'll be fine now, won't it, now you're here?'\n\nAnd next moment she's touching your face, your eyes, cheeks, lips. She smiles again, 'And you are here, aren't you?' She's so happy, happier than you've seen her in a very long time.\n\nQuite unexpectedly, she steps up close and kisses you on the mouth.\n\n'Mum?'\n\nAt the same moment she's put her arms round you, pressing herself against you: 'I'm so pleased, so \u2014 '\n\n'Stop, Mum! What're you doing?' You pull away from her. 'No. No. You can't \u2014 '\n\n'Michael! Michael, please \u2014 '\n\nMichael? Your father? She thinks that you're \u2014 ?\n\nYou take a step back. As gently as possible, holding her at arms' length.\n\n'It's me, Mum. Tom. It's Sunday, same as always, and I've come to see you.' Does she understand what you're saying? 'Mum? Don't you know who I \u2014 ?'\n\nThe utter desperation in her face now, the wretchedness.\n\nThen her anger, sudden and out of nowhere: 'Yes, I know who you are. I know all about you.'\n\nHer anger, then utter fury.\n\n'Get out of my house!' Screaming now, almost losing her balance as she staggers a couple of steps. 'Get out! Get out!' Her arm waving wildly towards the door. 'out!'\n\nYou can see what's going to happen next. About to fall, she's clutched at the trolley . . .\n\nWhich tips over, scattering playing cards everywhere . . .\n\nYou rush over and manage to catch her just in time.\n\nSaving the day.\n\nNice one. That's you \u2013 a safe pair of hands. Mr Magic, right enough. Forget the no-kids and marriage number three flushed down the pan with no regrets, you're all she's got. She knows it, too. Deep down. She must do. She loves you.\n\nAnd it's only a moment later \u2013 when you're helping your mother back to her seat, your arm holding her and keeping her steady \u2013 that she turns and spits in your face.\n\n1\n\nLISTEN, MAGGIE: AS the years continue to slip from your grasp, you can see your younger self in front of you \u2013 thirty-year-old Maggie Davies as was, walking the length of the ship and back again, alone, past oil drums and coils of rusted chain, past the covered lifeboat creaking on its ropes.\n\nNo one else on deck, not a breath of wind, and the June sun failing to break through the half-mist, half-drizzle cloud that hangs above the dull heaviness of sea.\n\nUp and down, up and down the wet boards, she goes . . .\n\nApart from the muffled chug-chug-chug of the ship's engine somewhere below, the only sounds are from seagulls screeching overhead and the propellers churning up the sullen stillness of the Minch. Maggie is glad of her woollen beret, her gabardine and scarf \u2013 these are all the outdoor clothes she's brought. The trip is not a holiday.\n\nLying here in the near-darkness of your bedroom in the care home, adrift between past and present, you're being carried to and fro by currents that run deeper than any measured time, currents that crisscross days, years and decades alike. Your days and years \u2013 until once again you're on that small ferry sailing from Mallaig to Stornoway. The salt-sea dampness in the air washes over you, the smell of paraffin almost turning your stomach whenever you pass too near the stern . . .\n\nHow clearly it's all coming back to you now, more vivid than any long-ago photograph could ever be. Mumble-singing to yourself: 'Speed bonnie boat, like a bird on the wing . . .'\n\nStraining to open your eyes, desperate for the reassurance of the familiar chest of drawers, the wardrobe and easy chair you've brought with you from your cottage to Rosehaven House, your toiletries arranged on the glass shelf above the wash hand basin, the tea trolley with its half-finished game of patience waiting for you, the portable DAB radio beside your bed . . .\n\nSixty years foreshortened at a glance \u2013 like when you'd stood at your living-room window that morning, gazing out across the countryside under snow . . . yesterday, was it? Last week? Last month? No matter. Overnight, a handful of precious stones had been scattered across the pane where they'd frozen solid \u2013 an arrested cascade of light. Too cold to go out, the side roads probably blocked, and the pearl-sheen sky hard as a sheet of ice. The path up to the big house would be impossible for walking. Across on Keir's Hill, you could see some of the village children stumbling about in the snow, trying to clear a run for their sledges \u2014\n\nThat's when you saw it, high above the village, and remembered the name you'd made up all those years ago.\n\nA ghost moon.\n\nSo comforting it seemed to you then, as you'd paused for a moment at your cottage window, and so familiar. You traced its outline, hardly daring to press your fingertip on the chill glass, afraid the sliver of unexpected light might melt to nothing at your touch.\n\nBut there's no need to be afraid any more, Maggie, and no need to pretend. When the past returns, it is already an act of pretence. There's no shame in this \u2013 for how else could anyone bear to go on living?\n\nClose your eyes and watch as Maggie Davies continues to pace the wooden deck. See, she's come to a sudden stop to stare down into the cold waters of the Outer Hebrides. A solitary woman, a mere silhouette pressed by chance against a backdrop of mist and sunless water . . .\n\nThe bedsheet feels slightly tight across your chest where it's been tucked in. They wanted you to be comfortable is all, and feel secure. The worst is over, Maggie, and the best just about to begin. Really.\n\nThough it's still daylight outside, the woman you call Boss Beryl has already been in to draw your curtains. She pulled the door shut behind her when she left, without a word.\n\nClose your eyes. Something wonderful is about to happen.\n\nListen \u2014\n\nWHEN THE FERRY docked in Stornoway, Maggie struggled down the swaying wooden gangway, her heavy suitcase bumping against her legs. No Waverley Station this with its grime and grit, no soot-blackened walls and layers of soiled daylight seeping down through the trapped, filthy air \u2013 the ship's arrival had been perfectly timed for her to see the mist being burned away to reveal a Hebridean sky so vast, so generous and light-filled it looked newly made. There was the clean smell of sea and of heaped cod, mackerel and salmon packed in ice \u2013 the morning's catch being winched ashore to be stacked on the quayside next to the creels of live lobster and crab. But as for the stench of gutted fish wafting in the hot sun \u2013 she had to move further away, and quickly.\n\nThe white-bricked harbour master's office was so spick-and-span it must surely have been built only the day before, ready to greet her. Two men who might have been father and son were lifting fish boxes into a van. Seagulls celebrated her safe arrival by strutting up and down the quayside, screeching in excitement, and rising into the air every few seconds only to settle again a yard or so nearer to the dripping crates. The older man looked up and addressed her in Gaelic. Maggie shook her head.\n\n'Grand day, missus!' His unfamiliar cadence, its singsong gentleness, made the commonplace greeting sound like an ancient psalm of welcome.\n\n'Really beautiful!' She smiled back at him.\n\nWhat energy surged through her! What hope! Coming here had cost her more than she could afford, but what choice had she? Nowhere to stay in Edinburgh, no job and her small savings soon to run out. Nobody knew her here, not really. The Isle of Lewis was a foreign country near enough, a new beginning surely.\n\nBy the time she'd found out what bus she should take, it had already departed and the next wasn't due to leave until the following morning.\n\nA taxi?\n\nShe'd no money for taxis.\n\nBut then she'd no money for hotels either.\n\nThe boarding house Maggie was directed to overlooked Stornoway harbour. The landlady, a woman who introduced herself as Mrs Stewart, ushered her into the sitting room, talking all the while about the promise of a good summer to come and asking if she'd be taking a cup of tea with them?\n\n'This is my son Michael,' she added, indicating a man in his thirties seated by the fire.\n\nMaggie was surprised when son Michael made no move to get up from his chair to greet her; instead he simply held out his hand, letting it waver slightly as he fumbled the empty air to make contact.\n\n'Pleased to meet you.' His voice had such unexpected warmth and assurance that she let her hand remain in his as he invited her to lean towards him. Next moment, he had reached up to pass his fingers over her face. It only took a few seconds, his fingertips were so gentle she hardly felt their touch. His eyes meanwhile remained still, seeming to be permanently awash with milk. She kept expecting him to blink to clear his vision, but he never did.\n\nWhile she and Mrs Stewart discussed terms, Maggie noticed that the glass on the mantelpiece clock had been removed. Hours later, lying sleeplessly upstairs, she was to imagine the blind man getting up from his chair, taking a few steps across the well-charted darkness to hold his hands up to the clock face in front of him, feeling for the time.\n\nThe cost of one night's lodging settled, she was shown up to the first-floor bedroom. Having pointed out the bathroom and WC at the end of the corridor, the older woman then lowered her voice to explain about her son returning from the war, blinded for life. 'But at least he came back. Not like my husband.' Adding, as if to remind herself: 'We count ourselves lucky.'\n\nDinner was fish and chips, sitting on a bench that overlooked the sea. Then Maggie took a walk round the harbour before returning to the boarding house. It was still light when she went to bed.\n\n'A grand summer's morning, sure enough. On holiday, are you?' Mrs Stewart had come through to offer more tea.\n\n'Just a short break.'\n\n'That'll be nice \u2013 a few days to yourself before your husband joins you.'\n\nThough her porridge hadn't had time to cool properly, Maggie at once doubled her spooning rate. 'I've never been to Lewis before and hear it's beautiful.'\n\n'Aye, when the rain's crossed to the mainland and the midges are safe in their beds, it has a beauty like nowhere else on earth. You could try visiting the west coast, over Bernera way.'\n\n'Well . . .'\n\n'You'll have folk to visit, more like?' The teapot was positioned on its protective cork mat, the tea-cosy replaced.\n\nHot spoonful after hot spoonful was being cleared rapidly under Mrs Stewart's steady gaze.\n\n'Can I get you some toast, Mrs Davies?'\n\n'No thank you, Mrs Stewart. I'd best be getting my things ready.' One last spoonful and she was finished. 'Thank you for the lovely breakfast \u2013 a great start to the day.' She got to her feet. 'I'll look in before I go, to settle up.'\n\nTen minutes later Maggie made her way to the terminus where the bus waited to take her to the Eye Peninsula.\n\nOnce out of Stornoway, the Portnaguran bus rattled and bumped along the single track road sounding its horn every few minutes to warn slow-moving horse-and-carts to pull into the nearest passing place. It stopped to let off passengers at small villages, at road ends and junctions, at single houses even. When it crossed the open stretch of causeway the bus seemed to fill with light and, on either side, there was a glitter of sun-splashed waters and endless sky. The peninsula itself was flat moorland, utterly treeless. Maggie began keeping watch for road signs announcing the next huddle of cottages and the occasional black house with its turf roof \u2013 the village of Knock . . . then Melbost . . . Sleebost . . .\n\nThe photograph showed a stone-built house set well back from the road. The Callanders were distant family. Maggie had never met them, but for as long as she could remember there'd been an exchange of Christmas cards between the two households. On taking over their croft a year or so before, they'd written to her parents inviting them to visit any time, adding that their daughter, if she was still living at home, might fancy coming over for a longer stay. She could help around the croft and with the peat-cutting, they'd suggested \u2013 and there were more than enough local men back from the war who were still looking for a wife! 'Maybe you'd have better luck in the Hebrides than in these awful dance halls,' her mother had remarked as she'd propped up the photograph on the mantelpiece, next to some postcards. 'John and Isobel \u2013 C\u00e9ad mile f\u00e1ilte' was scribbled on the back, 'a hundred thousand welcomes'.\n\nMaggie peered once more at the photograph, holding it up close to the bus window to see better the weathered-looking building with its storm windows set deep in the wall, the cement path with vegetable patch on one side, drying green on the other, and the moorland stretching beyond the fence. Not a tree in sight and hardly a bush even, nothing to relieve the emptiness of the landscape. While pacing the ship's deck she'd gone up that front path a score of times at least, trying to decide how she'd introduce herself \u2013 and she'd still no idea.\n\nMr and Mrs Callander, John and his wife Isobel, had done their best to take up a happy-family pose on the front step of their new property \u2013 arms round each other, the promise of kindness showing in their faces, and Callander squinting into the sun's glare with a hand raised to shield his eyes. Maggie tilted the photo to catch the best light.\n\n'clachtarvie!' The driver had to call out several times before she realised she'd come to her stop \u2013 she'd been far too engrossed in what she could make out of Mr Callander's face and in the cheerful smile his wife was giving to the camera.\n\nThe bus drove off leaving her in the middle of nowhere. There was a scattering of cottages, the peat bog, and a clear-sounding peewit, peewit from high up in the sky \u2013 no bird to be seen, however. Ahead lay the dazzling sheen of sunlight caught by the sea. Like the landscape in the black-and-white photograph but friendlier-looking, and with the pleasing warmth of the sun on her skin.\n\nTo her left an unpaved road led down towards a bay. There were no street signs, but this had to be the right direction. She began walking. The houses on either side stood a good fifty feet apart, each on its own patch of ground. She made her way down the street inspecting them as she passed. It was hot now, but with a chill undercurrent blowing in from the sea. At the last house on the right, she stopped. Yes, here was the place she knew so well from the photograph. Someone had made a start on pebble-dashing the front wall and small stones lay heaped nearby. Maybe she could offer to help them finish? A life-sized jigsaw where all the pieces were the same \u2013 easy! The wreck of a dark-blue car, wheel-less, with its axles up on blocks and one of the side doors missing, squatted over by the fence. Its bumper trailed in the uncut grass.\n\nPushing the gate so that it swung open to admit her . . . Waiting for it to close with a dull thwack of wood against wood . . .\n\nForcing herself the three, four, five, six steps up the path.\n\nThe door turned out to be varnished a dark brown. There was no bell. She put down her suitcase.\n\nA deep breath. Her hand lifted ready to knock . . .\n\nHer last chance to turn back.\n\nHer firm rap on the wood panel echoed inside the house. Such a dull hollowness was nothing like the cheerful tongue-and-clapper jingle made by a city tenement doorbell swinging on its wire to announce that she'd arrived at a friend's and was waiting downstairs, eager to be let in. Back home, in a decade that had seen parts of Edinburgh and Glasgow turned to rubble, the purely physical summons of bare knuckles battering on someone's door would have suggested urgency and alarm, a warning that something terrible was happening or had already taken place \u2013 a house bombed, the danger of fire, escaped gas or the building's imminent collapse. Here on the Outer Hebrides, however, her knock would hold no such threat. It was a friendly tap on a door, nothing more. This was how things were done here and always had been, she told herself, a commonplace gesture of neighbourliness. Having knocked once, she lifted her hand away . . . and took a step back.\n\nNo need to repeat the knock. Maggie could hear someone coming, calling ahead in a rush of Gaelic as they made their way from the back of the house. In time, she thought to herself, she'd probably have to learn the language.\n\nThe door swung open. A man stood in the half-darkness of the interior. John Callander, it had to be. Red hair, red face. Smaller than in the photograph, dressed in a collarless shirt and waistcoat. Slippers. What had probably been intended as words of welcome were broken off in mid-phrase.\n\nNow they were face to face, was he about to greet her, to smile and shake her hand?\n\nTo step aside, perhaps, ready to throw the door wide open?\n\nWas he about to take charge of her suitcase, and invite her in?\n\nWas he hell.\n\nJohn Callander stared at her and said nothing. There was a movement in the dimly lit hallway behind him, a suggestion of sweeping yellow hair and pale-coloured jersey. This would be Mrs Callander. Leaning against the inside of the door, she, too, seemed in no hurry to do anything.\n\nMaggie looked from one to the other and back again. Red man. Yellow woman. John and wife Isobel. Their combined silence blocking her entry.\n\nShe cleared her throat. 'Hello, I'm Maggie Davies. I've come from Edinburgh and \u2014 '\n\n'Yes, we know who you are . . .'\n\nThe contempt in his voice, the disgust.\n\nAs if a charge of electricity had found a hateful circuitry already in place inside her, she felt her body seize, her every muscle lock tight. She couldn't breathe even, the next few seconds swelling up in her chest, her throat \u2013 a solid, choking mass.\n\n'. . . and we know all about you.' Callander took a step back into his house. And slammed the door in her face.\n\nStumbling over to the derelict car to slump against its rusted bonnet, tears running down her face. Not even the strength to wipe them away.\n\nThe partly pebble-dashed stonework, the vegetables planted in their rows, the trackless moorland, the very sky itself \u2013 everything around her suddenly reduced to a meaningless slapdash.\n\nShe'd come to where the world stops.\n\nShe stayed there.\n\nIt was not until later, when she heard the sound of a vehicle going past on the main road, that she glanced up to see a van crossing the featureless landscape \u2013 she watched it getting smaller and smaller, its windscreen catching the sun's glare for a moment as the road curved. Finally it vanished. At one point a teenage boy wearing an oversized army coat came out of the cottage opposite. He took his bicycle from where it leant against the wall and wheeled it across the garden before mounting. A last wave to someone at the window before he set off across the peat bog, his too-long coat tails flapping with each pedal thrust. Like the car earlier, he too grew smaller and smaller as he headed further into the flat, empty landscape. Finally he too vanished.\n\nThe Callanders remained indoors all this time. What did they do while waiting for her to leave? Did they flick through the newspaper? Listen to the radio? Read their Bibles? Did they glance at each other every few minutes: Has she gone yet? Or did they sit completely at their ease, secure in their faith, confident that sooner or later they'd hear her footsteps retreat back down the path, followed by the thwack of their wooden gate as she took herself and the disgrace of her unwanted pregnancy out of their lives \u2013 helter-skeltering herself straight back to Hell where she belonged?\n\nWhen Maggie eventually managed to haul herself to her feet and stumble out onto the unpaved road, she was aware of being observed from behind the tight little window \u2013 no doubt the Callanders were making sure she hadn't left her suitcase behind.\n\nWith only the unseen peewit for company, she dragged herself and her suitcase all the way up to the main road. Four hours later, a bus appeared. She stared out the window all the way back to Stornoway, seeing nothing.\n\nHaving returned to the bench where she'd eaten her fish supper the night before, Maggie sat watching the comings and goings on the small ferry tied up at the dockside. From time to time smoke came in casual puffs from its funnel. Through the large window she could see two men on the bridge having a long conversation. Passengers weren't boarding yet. Every few minutes a red-headed sailor staggered up the gangway carrying a box or a crate; the bulkier-looking packages he balanced on his shoulders. Another sailor was coiling rope on the rear deck while a third looked on, smoking a cigarette. Everyone, it seemed, had something to do.\n\nShould she take the ferry back to the mainland, then the train to Edinburgh? She was in good time to board, if she wanted to.\n\nBut did she?\n\nBack home? Back to her parents?\n\nBack to Edinburgh, at least?\n\nStay on Lewis?\n\nFor what?\n\nBut . . . back to Edinburgh?\n\nAfter a while she felt like she'd been sitting there for ever, her suitcase at her feet.\n\nEventually the small ship cast off, and soon there was only a slow trail of smoke drifting above the bay. The faraway rumble of the engines grew fainter and fainter.\n\nWhy hadn't she gone on board? She didn't know.\n\nWhy had she stayed? She didn't know that either.\n\nShe got to her feet and made her way along to Mrs Stewart's boarding house.\n\nEarly next morning Maggie was woken by the sound of heavy rain. She climbed out of bed and closed her window. Standing barefoot on the cold linoleum, she watched the fishing boats manoeuvre towards the harbour mouth before passing, one by one, out into the open sea. The grey-white puff-puff-puffs from their smokestacks were flattened by the wind before being taken up, tossed into the air and shredded to nothing. The steady thud-thud-thud of the engines was blown towards the shore, and whenever an extra-strong gust rattled the glass in its frame she pressed her hand against the pane, and stilled it.\n\nSoon enough the storm seemed to become a live thing trying to force its way into the house. She could hear it battering its fists on the downstairs door and hurling itself at the walls, but she knew she was safe. Mrs Stewart's house was pre-war, and by a good couple of centuries. It had outlasted many storms and would see this one out, too, no problem.\n\nThe floor shook so much she almost expected to be lifted off her feet or else to see the walls billow in and out like the sails of a long-ago ship far out at sea . . .\n\nYes, far out at sea \u2013 that was where she really was. No land in sight and her only cargo her unborn child. Men, it seemed, always had some sort of harbour to make for. That was the nature of their world \u2013 a map of place names like Normandy, Amiens, Berlin. For men it was enough to identify aims and objectives, and then draw co-ordinates \u2013 that done, and with bayonets fixed, they marched, marched, marched into the future, whatever the cost. But for her, after that reception at the Callanders', what destination remained?\n\nEven though it was mainly the Leith docks that had been bombed, everyone's life and routine had been smashed beyond repair. Something had got into the works \u2013 the grit of countless deaths, perhaps \u2013 and a new kind of routine had taken over. She herself had worked long hours in a factory making bombs she knew would kill and maim, she'd packed ammunition that would cause death to someone's husband or son.\n\nThe last fishing boat having gone, she returned to bed. It was only five o'clock. Back under the blankets she lay listening to the storm. Even before getting herself pregnant, what had she been hoping for? During the last year of the war when their housemaid Annie had left to join the WRACs, Maggie had been expected to assist her mother in running the house. She'd cooked and cleaned, she'd scrubbed floors and brushed shoes, beaten carpets, polished brass work and mopped the outside steps. She was given pocket money for tram fare, for the pictures and for the odd evening out at the dancing. By the time peace was declared, she had turned twenty-five. Would this be the way of it from now on? she'd sometimes wondered as she passed through the hall on her way back to the kitchen, carrying the dirty dishes, her footsteps falling all too readily in step with the slow relentless tick . . . tick . . . tick . . . of the grandfather clock standing at the bottom of the stairs.\n\nEven when she joined her friends in the packed dance halls on Princes Street or at Fairley's Ballroom at the top of Leith Walk, she'd often been aware of that merciless tick . . . tick . . . tick . . . inside her, as if she herself had become the empty sounding-board for the hours and days and years being relentlessly sliced off her life. Twenty-five became twenty-six, became twenty-seven. Whatever the music, whatever the band, more and more she found herself alone at the edge of the floor, stranded there, waiting to be asked to dance and waiting in vain. Every man, of course, was guaranteed a partner. Ten times over. She was marking time merely.\n\nShe turned twenty-eight, twenty-nine, thirty . . .\n\nHe wore a tie-pin that was probably some kind of army crest. Having persuaded her to remain on the floor for a second dance, a slow waltz, he told her he was called Danny and invited her to have a drink with him at the bar. As they stood in the crush, he said how he'd sometimes pictured a girl like her when things had got difficult. Taking her hand, he murmured that he'd been lucky to get through it all and had come home hoping to find someone special \u2013 did she know what he meant? While he rubbed the back of her hand, almost nervously, unconsciously perhaps even, he'd added in a shy voice that he'd imagined someone who, in their turn, had been waiting and keeping themselves for someone like him. He didn't want one of these would-be glamorous girls who paraded about like they had film-star looks and were interested only in silk stockings, cigarettes and ration coupons. He leant nearer and whispered that he wanted a quiet girl, someone trusting, affectionate \u2013 someone to care for and who would care for him in return. This was what he told her as they stood sipping their drinks between dances that first evening.\n\nAnd, yes, she'd been exactly the kind of girl he'd wanted \u2013 a girl so desperate to get away from her parents' home that she'd make herself believe every single word he told her. That she'd trust him, totally.\n\nShe pushed aside the covers and placed both hands on her stomach. Just over three months' pregnant but nothing showed yet, not even the slightest bump. The Callanders must have been warned. Having thrown her out of the house, her parents must have noticed the photograph was missing and written to them, letting them know of their daughter's shameful 'condition' and expressing their own church-going views on the matter. Girls became women became wives became mothers \u2013 that was the proper way of it, the only way. If a girl couldn't wait, then she had to marry whoever made her pregnant. Call it divine intervention, call it Russian roulette.\n\nMaggie punched her pillow, then lay staring up at the ceiling. She'd been so ashamed. She'd thought only about leaving Edinburgh, running away as fast and as far as she could \u2013 it was as simple as that. Running away to where no one really knew her or knew anything about her.\n\nWhat had she been hoping for, coming here \u2013 the hundred thousand welcomes?\n\nAt seven she dressed and went down to breakfast.\n\nOut of the dining room window she could see the storm was \u00adeasing, but the sky remained weighted with dark clouds and a watery daylight that threatened more rain. Once again she seemed to be the only guest. The stiff-backed dining chair sighed as she sat down and the others stared blankly back at her, without passing remark. The dark mahogany dresser had clearly run out of conversation years ago.\n\nThis grim silent room. Outside, weathered to a complete indifference to her or anything about her, were the rain-lashed streets and buildings of Stornoway. In Edinburgh she'd have been getting ready to catch the tram to join the Saturday morning bustle of Princes Street, in and out of Jenners, PT's, J&R Allen's, Binn's. A pot of tea in Mackie's or Crawford's. Some chatter and gossip while soaking up the sun on the grassy slopes of the Gardens.\n\nNo. No. No.\n\nMaggie took a deep breath and tapped her knuckles on the polished table as though to call a meeting to order. She had propositions to consider, decisions to make.\n\nTop of the agenda \u2013 Item One: Should she remain on Lewis, or leave?\n\nStraightforward enough, surely.\n\nAs she ate her breakfast, she asked each of the empty chairs in turn for their advice and was about to consult the dresser when Mrs Stewart entered.\n\n'Another storm on its way, Mrs Davies. Some of the fishing boats are coming back so you can be sure it'll be a big one \u2013 these men need every penny they can haul out of the sea. We're in for it, right enough.'\n\n'And the ferry?'\n\n'Cancelled most likely. And tomorrow is the Sabbath.'\n\nSo that was that.\n\nBy noon the rain was coming down heavier than ever. There was a knock on her bedroom door. It was Mrs Stewart.\n\n'You're not telling me you're going out in that, Mrs Davies?' she said. 'If you don't mind eating in the kitchen, you're welcome to have a bite of lunch with us.'\n\n'That's very kind of you, but \u2014 ' Maggie started to protest.\n\n'I'll not take no for an answer.' Mrs Stewart smiled. 'See you downstairs in half an hour.'\n\nOver a bowl of broth with bread and cheese for afters, Mrs Stewart asked all the questions and Maggie did her best to make up the answers. No mention of her being pregnant, of course. She'd lost her husband a year ago almost to the day, she told them. A car accident. Her parents were no longer alive, killed when the family house was bombed during the war. Her brothers had been killed too. She announced this as an afterthought, to help keep things simple. She'd been the only survivor.\n\nOnce she got into her stride, her story seemed to tell itself; it flowed out easily and cheerfully almost. She'd come to Lewis for a short break to help get herself through the first anniversary of her loss. Her poor husband \u2013 she christened him Alfred for some reason and had difficulty keeping a straight face every time she mentioned his name \u2013 had been very brave. She described how Alfred had suffered, how Alfred had never complained, how Alfred had lingered for several months, needing her constant care. Before she could stop herself, she heard herself adding that Alfred had had a beard that just grew longer and longer as he lay there. She had to trim it so he'd not get himself tangled in the blankets when he turned over in bed. Seeing Mrs Stewart's rather puzzled look, she quickly went on to tell how Alfred had died in her arms. She spoke brokenly, following her words with a few moments' respectful silence.\n\nMrs Stewart was very sympathetic, even more so when she learned there had been no children.\n\nMaggie's offer to help clear the table was declined; instead she was urged to stay, enjoy another cup of tea and chat with son Michael.\n\nLater, when she was getting up to leave, Michael asked if he might 'read' her face again 'so's to help me remember, Mrs Davies.'\n\nShe let him, naturally.\n\nThe rain having turned into a steady drizzle, she borrowed an umbrella and went for a long walk. There was no ferry boat tied up at the quayside. Instead, a handwritten notice announced all sailings were cancelled till Monday. She felt strangely relieved. No need for a decision, not today anyway.\n\nOn her return to the boarding house she was told that, unless she had other plans, there would be a place laid for her at the kitchen table that evening. 'Nothing fancy, you understand, Mrs Davies. Simple fare.'\n\nThe meal of herring and potatoes, with tinned rice pudding for dessert, was rounded off with more questions.\n\nNo, she didn't own the flat where she now lived. When she returned to Edinburgh she'd have to start looking for somewhere smaller.\n\nShe would have to find a job, too.\n\nHad she ever worked?\n\nOnly making bombs.\n\nWhen Michael asked what colour her hair was and what she was wearing, Mrs Stewart interrupted the awkwardness of her reply to say that their guest was being far too modest.\n\n'She's a bonny lass. Nice face, nice figure and just coming into her mid-twenties (she winked at Maggie, who'd been about to protest, and put a finger to her lips). They'll not be letting her leave the island, you can be sure of that.'\n\n'Mrs Stewart, I \u2014 '\n\n'Never heed me, lass, just teasing . . . But maybe you'll be finding yourself staying on the island just that wee bit longer than you planned!'\n\n'Mother! Don't embarrass our guest. I must apologise, Mrs Davies, my mother can sometimes be rather \u2014 '\n\n'There's no problem, really. I can take a joke.' She glanced back across at Michael but, of course, he hadn't noticed her quick smile, and never would.\n\nLater, when Maggie said she would go upstairs, Michael asked if he might be allowed to pass his hands over her face one more time. 'It'll help me really picture you.'\n\n'So, you're coming to join us at the kirk, Mrs Davies?' Mrs Stewart greeted her at the foot of the stairs, while her son stood waiting at the front door. Both of them were dressed as for a funeral \u2013 their best black relieved only by the red edging on the older woman's bible and by the white of \u00adMichael's stick.\n\n'Church? Oh, I hadn't thought.' Then Maggie had an inspiration: 'The service will be in Gaelic, won't it?'\n\n'Aye, it will, but \u2014 '\n\n'Mrs Davies has no need to sit through two hours of boredom. It's a glorious summer's day that God has given her. She should go out and enjoy it.'\n\nWhich she did.\n\nNext morning, after breakfast, Mrs Stewart called up to say that she'd be off to the shops shortly and wouldn't be back till after twelve. 'Kitchen table's set for three.'\n\nBefore she could reply, Maggie heard the front door being shut.\n\nShe planned to take a walk into town to find out when the normal ferry sailings would be resumed and was getting herself ready to leave when she heard someone come out of the sitting room. Next, they were coming up the stairs.\n\nQuickly, she crossed to the mirror that was set in the wardrobe door.\n\nA light tap on the door. The slight hesitancy in his voice: 'Mrs Davies? . . . Maggie?'\n\nGiving her hair a pat, smoothing down the front of her blouse.\n\n'Maggie? I was wondering if you'd like to come down and join me for a cup of tea?'\n\nTrying to re-fasten the clasp of an awkward brooch in the mirror, wrong-handedly.\n\n'That'd be nice, Michael. Be with you in a few minutes.'\n\nStopping herself, hand poised in mid-air. What was she doing? Michael wouldn't notice if she wasn't looking her \u2013\n\nMaggie came into the kitchen to find Michael pouring out a cup of tea for her. The wooden table had been laid with teacups and saucers, side plates, milk jug, sugar bowl. He placed the teapot on the tea-stand and covered it with an embroidered tea-cosy.\n\n'I forgot the biscuits. Would you like one?'\n\n'That's kind of you. But, Michael, please don't trouble yourself to \u2014 '\n\n'It's no trouble. Have a seat. Help yourself to milk and sugar.'\n\nMaggie watched in fascination as Michael crossed to a cupboard, opened the door and, without any hesitation, selected and brought out a large tin. He returned to the table, took his seat, removed the lid. So surely did he move about the kitchen performing his various tasks that she would never have known he was blind.\n\n'There should be a couple of snowballs left in there as well as some digestives. One of them'll have your name on it.'\n\n'But how can you \u2013 ?' she began before she could stop herself.\n\n'Oh, I keep an eye on the snowballs!' he joked. 'But really, Maggie, just because I'm blind doesn't mean I'm completely helpless. I know this house, because I've learnt it. So long as things remain in the same place I'm fine. Quite independent really. In fact, when Mother had to keep to her bed for a few days last winter with a bad cold I managed to look after her perfectly well, and with no outside help. Cooking, washing, cleaning \u2013 the lot! You might say that my blindness didn't make a blind bit of difference! Slightly slower than the average home help perhaps, but no less thorough, so I'm told \u2013 at least for a man!' He laughed. 'I know the streets of Stornoway, too, and the shops. If you fancy, we can go for a walk through the town afterwards to let you get properly acquainted with the place.'\n\nHalf an hour later they had left the house and were walking along the quayside. It was turning into a beautiful summer's day and, for the first time in months, Maggie felt herself relax as they strolled along together, Michael's white stick tap-tap-tapping out a path for them.\n\nHe took her to the Town Hall, to the bank where he'd worked before the war; he showed her his favourite bar. Then, having come round almost full-circle, they returned to the harbour.\n\n'Along here is a bench where I sometimes go to sit in the mornings, to feel the sun on my face.'\n\nThe instant Michael had spoken, Maggie was certain he was about to lead her to the very same bench where she'd eaten fish and chips that first evening she'd arrived.\n\nWhich he did.\n\nOnce they'd sat down, Michael touched her lightly on the arm: 'Tell me what you see, Maggie \u2013 and give me colours, lots and lots of colours.'\n\n'Well, it really is a lovely day, hardly a cloud in the sky \u2013 in the blue, blue sky, I should say. The sea is flat, totally calm. The water looks greenish and shiny with the sun on it. There are two fishing boats at the quayside, one's natural wood with a yellow cabin, the other has a black-painted cabin. Just along from us, a fisherman in a dark brown jersey and with green wellingtons up to his knees is sitting next to a heap of lobster pots; he's doing something with a net, untangling it or mending it. There are white seagulls, a red van . . .'\n\nShe felt a sudden need to close her eyes. Like she was trying to imagine what Michael saw, sitting here beside her on the bench on this beautiful sunny day. Darkness. Pitch-black darkness, a night that for him went on and on and on, and that he awoke to every morning.\n\nShe just wanted to feel what it was like for him.\n\nDidn't she?\n\nNo.\n\nThat wasn't it. Not really.\n\nRather, it felt like \u2013\n\nLike she'd stepped into one of those screened confessionals for Catholics, where they speak to an unseen priest.\n\nShe'd closed her eyes because she wanted to confess, was that it? To tell him she was pregnant? Tell him, and be \u00adforgiven?\n\nShe wanted to tell him \u2013 yes. She needed to. All morning she'd felt that need get stronger and stronger, until it had become almost unbearable. Like a threat, it had now taken over everything she could see and hear \u2013 the harbour, the sky, their sitting together on the bench. It had built up inside her until her whole world seemed to shudder from moment to moment with the force of what remained unspoken.\n\nBut there was no reason to tell him. What would it matter to him that she was pregnant?\n\nShe had to, though. She couldn't help it.\n\nHer eyes tight shut, she was about to speak when \u2013\n\n'I was driving a British Army truck in a convoy across France. It was a summer's day, Maggie, just like this. Blue skies and hardly a cloud to be seen, exactly how you described it. Overhead the German fighters were strafing us, sometimes Stukas dive-bombed. My lorry got hit. Seems I was thrown clear; they found me crawling on my hands and knees along the ditch beside the road. Not that I remember anything, except waking up . . .'\n\nMaggie had to stop herself from reaching over to touch the back of his hand.\n\n'I'm so sorry, Michael. It must have been so . . . so terrible. I can't begin to imagine how you . . .'\n\n'It was seven years ago. I've had seven years more than the other men who were in the lorry, and I get the gift of an extra day every morning. Like today \u2013 and here I am sitting on this bench in the sun with you, and enjoying the blue sky, the red van, the fisherman's dark brown jersey and his green waders. All thanks to you!'\n\nWithout thinking about it, she gave his arm a comforting squeeze.\n\nWhen they returned, Michael asked if he could read her face again. 'To keep in touch,' he smiled.\n\nAfterwards he didn't step away.\n\n'Would you like to try? See what it feels like?'\n\nShe closed her eyes.\n\nThere was no tremble in his hand as he guided her fingers across his smoothly shaved cheek. But, as she stood there, I'm pregnant, I'm pregnant kept hammering over and over in her head. All she could feel was the effort it was taking her to stop from screaming the words out loud.\n\nIn bed that night she allowed herself to relive the touch of Michael's fingertips, their warmth as they'd traced out the smoothness of her forehead, her eyelids, cheeks, lips, the curve down to her neck.\n\nSuppose he had begun to stroke her hair, suppose he had taken her in his arms and kissed her? Pregnant. Pregnant. Pregnant.\n\nThe spell of good weather continued. A few days later, while they were enjoying a picnic of sandwiches and a thermos flask of tea on their bench, Maggie heard herself say: 'I don't know what's happening between us, Michael . . .'\n\nShe knew perfectly well, of course \u2013 what was happening to her, certainly. She was falling in love. She couldn't help herself.\n\n'. . . but it's good. It feels very good.'\n\nNext thing, he had fumbled for her hand and taken it. Raised it to his lips.\n\nThat night she lay awake for hours remembering what happened next. When they'd kissed, she'd longed \u2013 longed with a desperation she'd never known before \u2013 for his kiss to be all there was to her life.\n\nShe stood at the top of the stairs one morning, gazing around at the seascape print on the wall, the runner carpet, varnished floorboards, the view of the harbour through the small storm window, the arrangement of flowers in a vase . . . Was it possible that her days would begin with a glance like this out of the low window to check on the weather, with her noticing some mornings that the flowers looked a bit tired and could do with being replaced?\n\nMrs Stewart insisted on always setting a place for her at table. 'Don't embarrass me,' she'd say whenever Maggie brought up the subject of payment for her lodgings.\n\nOne afternoon she heard him whisper: 'I'm so happy we've found each other, Maggie.'\n\nYes, she answered into herself. Yes. Then she said it aloud: 'Yes, Michael. So am I.'\n\nI'm pregnant. I'm pregnant.\n\nNext moment, she'd blurted the words out. Told him how it had happened. Then closed her eyes, waiting for his reply. When it came, it was short.\n\n'We'll manage.'\n\nThe following afternoon they almost bumped into the Callanders. She and Michael were walking along the main street having been to the butcher's and greengrocer's \u2013 mince, potatoes, onions and carrots for the evening meal \u2013 and were making for the baker's, the last on their list.\n\nThe Callanders were approaching from the opposite direction \u2013 Mr Callander with a black and green tartan shopping bag in his right hand. Noticing her, they came to a halt right outside the shop. They stared. Not a word was spoken.\n\n'The baker's, Maggie, and then we're done,' said Michael.\n\n'It looks pretty busy in there, Michael.' She glanced across at the Callanders standing side by side only a few yards away. 'Let's go to our bench at the harbour instead. We'll get the bread later on.' She pulled at his arm to steer him back the way they'd come.\n\n'Carrying these bags? No chance.' Michael laughed. 'A few minutes' queuing won't matter. We can get ourselves a couple of doughnuts to have on the bench \u2013 my treat!' He tapped his way past the Callanders and went into the baker's. Maggie followed.\n\nWhen they came out a few minutes later, the Callanders were nowhere to be seen.\n\nMichael turned to her: 'Thought you said it was busy?'\n\n'I just wanted to go and sit in the sun with you.' She squeezed his arm. 'And now we've doughnuts to share as well!'\n\nThat evening she came downstairs to find Michael already seated at the kitchen table. There was no sign of Mrs Stewart. To her surprise, one of the place settings had been removed. Was the older woman, in her most encouraging way, leaving the two of them to enjoy an intimate dinner by themselves?\n\nBefore sitting down, she went round to him. As always, his gaze was fixed unwaveringly on nothing.\n\n'Hello, Michael.' She raised her hand and was about to read his face in greeting before kissing him, when she became aware of his hand fumbling to take hers.\n\n'Maggie?'\n\nHe grasped her fingers and began stroking them. 'Mother met a Clachtarvie woman today in the street.'\n\n'Your mother met \u2013 who?'\n\nHe remained staring straight at her, sightlessly. 'Oh Maggie. I'm so \u2013 so sorry.'\n\nThen she understood. The Callanders.\n\nFor several seconds they remained hand in hand, without speaking. 'Everybody knows everybody here, Maggie. And knows everything about everybody. We'll need to \u2014 '\n\n'We don't provide dinner for guests, Miss Davies.' Mrs Stewart was standing in the doorway. 'There's nothing for you here.'\n\n'Mother? Maggie and I are \u2014 '\n\n'This is my house, as well you know, and I'm the one who runs it \u2013 as well you know, too.' The older woman advanced into the kitchen. 'You're not welcome, Miss Davies \u2013 and I doubt you'll be welcome anywhere on the island, not any more. Coming here, abusing our trust.'\n\n'Mrs Stewart, I never meant to \u2014 '\n\n'Bringing your sinfulness into our house, bringing your shame.'\n\n'How dare you, I \u2014 '\n\n'But I know my Christian duty. The next ferry is tomorrow morning. Keep to your room till then. We'll be well rid of you, and cheap at the price.' Mrs Stewart turned away to see to a pot on the stove. 'Michael? If you're ready for your soup . . .'\n\nMichael got to his feet and came round from behind the table to stand beside Maggie, his hand on her shoulder. 'No, mother. Maggie and I are \u2014 '\n\n'Not a word, Michael. Nothing's changed. Nothing that I can see. Our life might not be easy but we manage, you and I. We don't need the likes of \u2014 ' She switched to Gaelic. Not singsong Gaelic.\n\n'But, Mrs Stewart. Michael and I \u2014 '\n\nThe Gaelic continued.\n\n'Please, Mrs Stewart \u2014 '\n\n'Your room, Miss Davies.' The older woman had taken a step towards her, soup ladle in one hand. 'You'll be getting your bill and I expect it paid in full before you leave.' Then the Gaelic was resumed.\n\n'But Mrs \u2014 '\n\n'Your room, I said.'\n\n'Don't you dare speak to me like \u2014 '\n\n'This is my house and I will speak in any way I choose.'\n\nThen Gaelic, Gaelic, Gaelic.\n\nTen minutes later, leaving mother and son to rage at each other, Maggie went upstairs. She stood at the window, staring out at the harbour where the early evening sun hung above the slow-lapping water. The argument down in the kitchen continued for over an hour. Lying on her bed, staring up at the ceiling, she listened to it all, to every last incomprehensible word.\n\nThen came silence. Then the rattle of pans, the rush of tap water.\n\nAt one point during the evening an envelope was pushed under her door. Her bill.\n\nIt was shortly after midnight when Maggie became aware of someone outside her room. She heard the handle being turned and the door softly opening. Then being closed.\n\nFootsteps were approaching the side of her bed \u2013\n\n'Maggie?'\n\nMichael was leaning over her. She could feel his breath warm on her cheek.\n\n'Maggie?'\n\nWithout speaking, she reached up to stroke his face in greeting. They kissed . . . and in the darkness Michael's blinded country became hers.\n\nMaggie got up early. Very, very early. She dried her eyes, washed her face and dressed as quickly as she could. After she'd finished packing, she put on her coat. Then, suitcase in one hand and shoes in the other, she left her room.\n\nHaving tiptoed along the corridor to the top of the stairs, she paused for a moment. Listened.\n\nQuite certain at last that the rest of the house was still asleep, she went downstairs.\n\nOn the hall table she left two single pound notes under her key, two pounds that she couldn't really afford. Then checked herself in the mirror, dabbed her eyes again. Took a deep breath.\n\nOnce outside on the doorstep she wanted to slam the front door behind her, slam it full-force \u2013 but managed to stop herself. Then, having pulled it shut, she wanted to thump her fists against its hardwood panels, and to kick and kick and kick.\n\nAgain she stopped herself.\n\nShe had to leave. She knew she had to leave. End of story.\n\nIn tears again, she made her way along the silent street to the ferry terminal. An hour later she was on board and heading back to the mainland.\n\nSUNDAY\n\nHIGH-BACKED ARMCHAIRS LINED up against the dayroom walls. Meals, meds, bath, bed. The bay window keeping everything else out \u2013 the front lawn, the bush, the small tree, the big tree, the red yellow blue flowers, the visitors' car park, the visitors' cars. The main street, the people, the traffic, shop-windows, tenement windows. The sky that's no longer yours. The TV that's always on.\n\nThe first time the man you think of as Tom's friend brought you here for a look round the place, you heard someone say it felt like a waiting room. Waiting for what? you joked. A joke you always remember when you come through after breakfast to find every single minute and every long hour of the day ahead already gathered here, waiting for you. The dayroom \u2013 the very name makes you shiver every time you take your seat.\n\nThursday? Monday? Saturday? Different names for the one same day that slides backwards and forwards along the one same week that never comes to an end, but keeps starting over keeps starting over keeps starting over . . .\n\nYou'll be safe now, Maggie. By day you have your own seat two places down from the Murray sisters, and at night you have your own bed. Here the corridors are all one day long. The same day.\n\nWere the Murrays twins from the start? Seated side by side, feet splayed out flat on the floor, hands crossed on their laps, coat-hanger shoulders, silver-thread hair. Staring, staring into space \u2013 rabbits caught in a set of headlights that no one else can see. Dorothy seated in the corner calling out Wait for me, Mother. Wait for me. mother? mother? Wait for me . . . all day long. The Murrays, Dorothy. No one speaks by name to any of the other women lined up in their chairs, and the women never speak back. Not sleeping, not waking \u2013 but dreaming. Yes, you hope they're dreaming.\n\nAnd the man of your dreams? You know that when he appears in the doorway, he's going to ask one of you to dance. In the end, that's why you're all here.\n\nFred Astaire top hat, bow-tie and tails like he's stepped out of a black-and-white film, he'll pause for a moment as if taking a good look round \u2013 searching, probably, for the next Ginger Rogers. Scouting out the talent, it used to be called at Fairley's Ballroom and the other dance halls on Leith Walk and along Princes Street. Sweeping his hat from his head, he gives the entire room a boulevardier bow of such well-practised elegance that you can imagine the intake of breath on all sides. As he drifts round the edge of the floor, his black patent leather shoes drip with the sunlight that's pooled here and there on the polished lino. Like he's wading through the lushness of strings coming from an unseen orchestra. Lingering before each one of you in turn, a tilt of the head, a knowing smile here, a few softly spoken words there \u2013 who will he choose to be his partner for this special once-in-a-lifetime dance?\n\nOne day, of course, he'll stop when he reaches you. And it'll be Michael. Yes, you know it will be him. You'll know by the touch of his fingertips upon your face, their gentleness, his sightless eyes brimming with \u2014\n\n'Med time, Mrs Stewart.'\n\nWhile murmuring your name he'll draw you up into his arms. Shut your eyes, Maggie, and then you'll feel him close, so very close. Keep them shut \u2013 his hand's resting lightly on your waist as he birls you across the room and back, up and down the floor until all the rag-tag, long-ago years whirl round and round the pair of you in a swirling blur of \u2014\n\n'Mrs Stewart.'\n\nFaster and faster you go, your feet no longer touching the ground \u2014\n\n'Med time, Mrs Stewart.'\n\nLike you're dancing on air, on sunlight itself \u2014\n\n'med time. mrs stewart.'\n\nNo music, no Michael, no being swept round in his \u2014\n\n'Not for me. I'm Davies. Maggie Davies. How many times do I have to tell you?'\n\n'Maggie, then. Meds. Drink now, Maggie. One . . . two . . . Drink. Today Sunday. Today Mr Magic come.'\n\n'Who?'\n\n'Mr Magic he called \u2013 yes? He come for you today. You, Maggie, you stay sit and he come. Like he say he come. No worries.'\n\n'Mr Magic?'\n\n'All the Sundays. When you first here he say he come all the Sundays and today Sunday.'\n\n'It's a mix-up calling me Mrs Stewart, you know. Real \u00admixter-maxter. Names really matter, Donna. Can't be too careful with meds. If you give me Mrs Stewart's meds, then what about her? What's poor Mrs Stewart getting?'\n\nWhenever you ask them to point Mrs Stewart out to you, they roll their eyes and shake their heads.\n\nThe only man here is called Slow Peter. They've told you another name, but you know that name's not right. So you keep to Slow Peter. You know Slow Peter. They've told you he'll fix leaks and come when your bedroom window's jammed or a light bulb's gone. You know his real name even if no one else does. You know all their names \u2013 Slow Peter, Donna, Mrs Saunders, Beryl . . . When you tell them, trying to help keep them right, they just smile and say Suit yourself, Maggie. So you do.\n\n'Hello, Mum.'\n\nSomeone's bent down to kiss your forehead, leaning too close for you to see their face. Calling you his mother, like he's lost his own.\n\nYou'd like to call him 'Michael', to say the name to someone, to hear it spoken out loud.\n\nHe's taken your hand like he wants to keep good hold while he talks.\n\nSo let him.\n\nTalk and talk and talk. And sometimes you talk back at him. About the Murrays and Dorothy, about Slow Peter and Mrs Saunders, about Donna and Beryl, about the things on TV \u2014\n\n'Yes, Mum, when I'm on TV I'm called Mr Magic, remember. Been going a lot longer than The X Factor! You've seen me sometimes \u2014 '\n\nYou manage to keep looking over his shoulder. 'That letter's made her cry. Look!' Pointing to what's happening on the screen: 'Look at that poor woman \u2013 imagine someone writing to her so as to make her cry. Like they were really wanting to make her cry.'\n\n'Mum \u2014 ' He's started to stroke your arm. 'Mum, please. It's only something on TV, like when you see me make things disappear, playing cards and flowers and even people sometimes. \u00adRemember me telling you how I created that illusion about the \u2014 '\n\n'Heartless it is. A disgrace. A real disgrace.'\n\n'Don't get upset. It's nothing. Would you like me to \u2014 ?'\n\n'The poor woman. But Jean's got a cake all ready and waiting, never fear. A slice of that would cheer her up and take her mind off things. They'll be bringing it soon. Cream and jam, layers and layers of cream and jam. Real cream too, mind \u2013 all the ration coupons she could get hold of. Had to be good if we were to \u2014 ' You peer over his shoulder again, half-afraid at what you might see. 'Will you look at that poor woman! Look. Look at her. look at her!'\n\nTears have started running down your cheeks. Tears you make no move to wipe away.\n\nInstead \u2014\n\nStruggling to get to your feet, pointing your finger at the screen: 'That poor, poor woman. can't somebody do something? can't somebody help her? help her help her help her help her!'\n\nNext moment it's all turned to horse-racing and a red-faced man talking into a microphone. Which is nothing much, so you sit down again.\n\nThey're closing your curtains. But it's not the woman you call Donna, who's grown up into a well-meaning lassie and always has time to stop for a word or two. It's Boss Beryl \u2013 you'd know her vicious tug-and-swish anywhere.\n\n'Leave them, please. It's too soon. He's away making us a pot of tea. Nice man \u2013 for a man. Brought me some biscuits, too, my favourites \u2014 '\n\n'And flowers, Maggie. Brought you flowers, too. Must have put them in a vase for you, too. See?'\n\n'Flowers?'\n\nThen like out of nowhere, it seems, there's suddenly a vase of flowers on the chest of drawers. You can feel the glow spreading across your face, beaming into a smile. A real grin. 'Michael? Michael's really come?'\n\n'That his name, was it? He's long gone.'\n\n'Don't be silly, Beryl. There's the flowers. How else would they have got here? Making us some tea. Back in a jiffy, he said.'\n\n'Went home ages ago, Maggie.'\n\n'HobNobs, my favourites. But you can have one, if you like.'\n\n'An hour and more's drive back up to Edinburgh. He had to go. It's late now, Maggie. I've brought you your hot drink.'\n\n'He was right here only a moment ago, Beryl. Right here. We were talking about . . . something. He's always talking. He's along the corridor making us both a cup of tea . . .'\n\n'That was this afternoon, Maggie. It's nearly night now.'\n\n'Afternoon? Night? How can it \u2014 ?'\n\n'You've had your first Sunday here and he says he'll come every Sunday afternoon just to see that you're \u2014 '\n\n'I'm not stupid, Beryl. I'm not one of the Murrays. I'm not Dorothy. Sunday afternoon. I know Sunday afternoon. I understand afternoon. I understand Sunday.'\n\n'Goodnight now, Maggie. It's your sleep time. Here's your \u2014 '\n\n'I know Sunday. I want Sunday! Sunday! sunday!'\n\n'You need to rest and \u2014\n\n'sunday! sunday! sunday!'\n\n2\n\nDIRTY YELLOWISH SMOKE hung over the tracks and platforms of Waverley Station, and once again Maggie found herself walking through cloud \u2013 a cloud of steam this time, of soot and perpetual twilight. Her eyes smarted and she could taste the coal dust coating her tongue. Someone jostled her, making her stumble against the bottom step of the driver's cab. Close to, a furnace-heat roar came from the engine. She breathed in the smell of hot metal and burnt oil. Steam hissed out from between the massive wheels.\n\n'And the same to you!' she hissed back at them.\n\nGripping her suitcase, she pushed her way towards the ticket barrier. To her right a guard's whistle blew \u2013 another train was leaving. From all sides came the clash and grind of metal on metal, the rumble of porters' wagons, the slamming of carriage doors, and passengers shouting at each other to be heard above the din. Here was no windswept, treeless desolation \u2013 but real-life noise and bustle welcoming her back to her home city.\n\nHaving handed over her ticket, she stepped into the crowd of friends and family come to greet the new arrivals. No one would be waiting for her, but she couldn't help glancing across whenever there was a sudden cry and someone rushed forward into the open arms of someone else \u2013 wife\/husband, girlfriend\/boyfriend, brother\/sister, friends. She'd turn away, but wasn't always quick enough to shut out the sight of other people's happiness. At the same time, she wanted to see it, to enjoy their pleasure at being together once more \u2013 and to snatch that glimpse of an embrace, a kiss . . . of a separation healed.\n\nMichael had asked for her photograph to keep and, even though he couldn't see, he said he could always 'read' it. At that very moment he and his mother would probably be finishing their lunch. Maggie toiled up the steep slope that led out of the station, her suitcase getting heavier at every step \u2013 they'd be sitting facing each other at the kitchen table, with its two settings. A blind man? Could she really have coped with a newborn baby, and being married to a blind man, and all the while trying to settle down to a life as Mrs Stewart number two in that house \u2013 for her mother-in-law would certainly have been number one? Should she have stood her ground? Should she have stayed and \u2014 ?\n\nIn time she'd surely have learned how to \u2014\n\nForget it. Taking her suitcase in both hands, she marched herself up the slope as best she could \u2013 Forget it . . . Forget it . . . Forget it \u2013 marched herself towards the street ahead \u2013 Forget it . . . Forget it . . . Forget it \u2013 past the never-ending line of taxis crawling down to drop off their passengers, their wheels thumping the cobbles, sounding their horns and belching out exhaust. As she strode along, she promised herself that when she reached the top of the slope she'd be stepping into the brightness of the very sky itself. Ahead, she could see the roof of the National Gallery outlined against a cloudless blue, then the sweeping curve of the Mound with the Bank of Scotland building so grand at the top and, beyond, the Castle itself with its upward tumble of walls and battlements shimmering in the haze. Edinburgh seemed shaped by the sky and the sky itself by the city, and, for a moment, everything seemed possible and the future hers. She was so very, very glad to be back.\n\nThe afternoon heat rose from the paving stones. A small child bumped into her, then ran off into the crowd of shoppers. She passed a comforting hand over her stomach.\n\nJust as when her train had been approaching the city, she again heard the rhythm of its wheels rattling over the tracks: Somewhere to stay Somewhere to stay Somewhere to stay . . .\n\nBut first, she thought, somewhere to eat.\n\nWith luck, she might have missed the lunchtime rush. Suitcase in hand, she hurried along Princes Street, heading to Mackie's Buttery.\n\nSo, it seemed, was everyone else.\n\nThe place was mobbed. Overcrowded. Packed. Heaving. Queue right to the door.\n\nAnd hot. Hot. Hot.\n\nQuarter of an hour later, she and her tray with its corned-beef sandwich and pot of tea had found a small round table set in a crowd of other small round tables over by the window. Having put her suitcase down beside her, she stared out at the Gardens, then up at the Castle. The Black Douglas and his men, shields tied to their backs, had actually climbed up the rock face to capture it, or was that Stirling Castle? The little she'd learnt about Scottish history when she was at school \u2013 Bruce and the spider, William Wallace being betrayed and beheaded, and then Mary Queen of Scots \u2013 was from such a long time ago. As she understood it, nothing much else had happened during the next five hundred years, except down in England. She kept meaning to read a Scottish history book, if there was such a thing. She couldn't remember ever hearing of \u2014\n\n'These seats taken?'\n\nShe shook her head. 'No, please \u2014 '\n\nThree people sat down and took over her table. Mr and Mrs Bicker and little Miss Bicker.\n\nFrom the start Mr Bicker was on the defence:\n\nThe choice was his, no? He'd given up the fags, but the cigarette coupons were still his. Weren't they?\n\nMrs Bicker said she needed shoes for work. Last year's had been soled and heeled twice \u2013 the soles and heels were now all there was. First drop of rain and she might as well go barefoot.\n\nNo one's asking you to go \u2014\n\nDo you want me to go barefoot?\n\nIt's not a question of \u2014\n\nSo you do want me to go barefoot? Soled and heeled twice, I'm telling you. First drop of rain and I might as well go barefoot. And as for Annie's shoes . . .\n\nMrs Bicker kept at it.\n\nMeanwhile Young Annie Bicker sat and stared, watching every single mouthful as Maggie finished her sandwich.\n\nOnly a few sips of tea remained.\n\nHaving remained politely silent so far, Young Annie now spoke up: 'You got any, missus?'\n\n'Pardon?'\n\n'Coupons for sweeties. I like sweeties.'\n\n'No, I'm sorry I can't help. A real shame if you're \u2014 '\n\nBut Young Annie had already turned away.\n\nMaggie finished her tea and mumbled a goodbye as she stood up. She grabbed her suitcase that was now wedged between Mr Bicker's chair and her own. She jerked it free. No one paid any attention. She might as well have been invisible.\n\nLeaving the Bickers to each other was surely a good start to the rest of the day.\n\nSomewhere to stay Somewhere to stay Somewhere to stay . . .\n\nShe was crossing Princes Street intending to buy an Evening News from the paper seller at the top of the steps leading down to the Gardens when \u2014\n\nnewhaven was written at the front of the tram.\n\nNext moment she found herself sitting downstairs in her favourite seat behind the driver, her suitcase stowed under the stairs. She was certain her parents would slam the door in her face, but, then again, they might not. They'd had time to cool down, to think better about things and about her. It would be different. There was surely more to the world than a corridor running between Stornoway and Edinburgh, a corridor with a slammed-shut door at either end? Yet here she was, going down its spiteful length once more as if she couldn't help herself. But Newhaven was her home. She'd been born there. She'd gone up and down the same front steps every day for the last thirty years near enough. Ten thousand times up and down. This would make it her ten thousandth and first time \u2013 and it was going to be different.\n\nIf she changed her mind, she could get off at any stop. Any stop at all.\n\nThe tram trundled along Princes Street towards the East End, past the Waverley Market and the North British Hotel before sweeping left down into Leith Street, past \u00adFairley's Ballroom where the squaddies and sailors still battled out the war with each other, not that she'd ever be going there again. Then past the Playhouse cinema where the organist on his platform rose up through the floor to play during the interval, past tree-lined Elm Row and into Leith Walk proper. Yes, this was her city. Here, at least, she knew she belonged.\n\nThe best way of not getting the front door slammed in her face . . . was to walk straight in.\n\nThrough the vestibule door's stippled glass she could see the familiar hall, the hat stand, the curved hall-table against the wall with the oval mirror above, the grandfather clock, the staircase going up to the bedrooms.\n\nFrom the sitting room straight across came the sounds of a football match on the radio. Her father would be in his armchair, following every kick of the game. Her mother would be in there, too, reading the Scotsman or knitting.\n\nEver so slowly Maggie inched open the vestibule door and stepped into the main hall. She took her time easing the snib back into place, making sure there was no tell-tale click.\n\nShe was now inside. She was back home.\n\nSo far so good.\n\nShe put down her suitcase at the bottom of the stairs.\n\nThe measured tick . . . tick . . . tick of the grandfather clock, the faint sounds of the football match, the smell of the freshly polished floor \u2013 this was her home on a Saturday afternoon.\n\nA last-minute check in the hall mirror. The light was poor. A quick comb through her hair and giving it a pat. Some fresh lipstick? But what if she smeared? She didn't need a wounded-looking mouth, a crazed and begging-looking mouth. What she wanted was a war-mouth, a blood-red snarl of a mouth, to show her parents she was no longer their dutiful daughter, no longer their little girl with no life of her own and no plans for her future but theirs. She'd be a mother herself by the end of the year. Like it or not, she was their equal.\n\nShe snap-shut her lipstick, pocketed it, then marched firmly across the parquet floor. Grasping the handle, she took a deep breath and threw open the sitting-room door, stepping into the comfortable predictability of her parents' Saturday afternoon.\n\nHer mother looked up, startled. Knitting needles click-clicking, click-clicking, click-clicking, click \u2014\n\nHer father's faraway gaze was suddenly bewildered, no longer at the Easter Road stadium but not yet back in his own sitting-room, and with his cup of tea arrested just a few inches from his lips . . .\n\nNot a word. Not a movement. The pair of them freeze-framed in shock. The radio commentary continued: Ormond's crossed high into the box, Turnbull's there and Laurie Reilly . . .\n\nShe crossed the room to take up position on the fireside rug.\n\n'Hello.' She didn't smile. Didn't step forward to hold out her hand. Didn't make as if she was about to embrace them.\n\nIn her unbuttoned coat with the belt hanging loose, she looked like someone who'd just stepped in for a flying visit, someone with a life of their own outside the confines of home and family, someone who decided for herself when she'd arrive and when she'd leave.\n\nShe glanced at each of them in turn.\n\nThe crowd was cheering, chanting, there was the whirring-ratchet clatter of football rattles.\n\nHer mother? A deeper sag to her shoulders and to her mouth.\n\nHer father? His eyes refusing to meet hers, the fingers of his left hand picking at the armrest of his chair, his half-raised cup of tea now in real danger of spilling. His unaccustomed awkwardness, his uncertainty. His gathering anger.\n\nThis was not the man who'd pushed her in the Victoria Park swings every Saturday morning, always so careful to use both hands to keep her safe, taking a step backwards and hauling the wooden seat up to his fullest stretch. 'Higher, Daddy! Higher!' She'd squealed with pleasure as he sent her hurtling forward, her back arched and legs thrust out in front. Arcing upwards and upwards, soaring into the open sky . . .\n\nHer father: 'That's them ahead now, Muriel, 3-2.'\n\nHer mother: click, clickety-click, clickety-click . . . (tugging at the length of wool to release a few more coils from the ball at her feet). 'Who's playing again?'\n\n'The Hibees, of course. Against Third Lanark. The Thirds drew with Rangers last season \u2013 one more goal and they'd have won us the League for the second year running. But this is just a friendly.'\n\n'Are they not always friendly?'\n\n'Ach, Muriel, you need to take more interest.'\n\nShe stepped forward until she was almost touching the knitting needles.\n\n'It's me, Mother.'\n\nWithout seeming to notice her own daughter standing in front of her, the older woman turned away and stooped to unhank her line of wool from where the ball had rolled under her seat. That done, she resumed knitting, keeping her head bent over her needles like a conscientious schoolgirl learning a new pattern. Fingering the individual stitches along the length of the needle, she counted them out under her breath. Eight, nine, ten . . .\n\nMaggie stared down at the uneven parting in the grey hair, a near-white dandruff-flecked line. As a little girl, she used to watch her mother check her appearance every morning in the hall mirror. Such grown-up neatness was part of an adult world she herself looked forward to belonging to one day. After a last few dabs of face powder the small puff-pad would be replaced in its sleek round tin with the mirror set inside the lid, then, returning to the large oval looking-glass, her mother would angle her head this way and that to check herself first in profile and then full on. Chin on chest and comb in hand, she'd lean forward to inspect the top of her head and smooth out any remaining loose strands, laying them to the right or left of her precisely combed parting.\n\n'But no one sees through your hat, Mum,' Maggie had said to her once.\n\nGod does, little one \u2013 and so do other women.\n\nThe parting she could see now, unprotected either by a hat or stray curls, jerked in abrupt lunges across the exposed skin like a piece of badly done stitchwork, a clumsy suturing of the thinning scalp. Her mother seemed suddenly much older. Her father too. As if they'd been fast-forwarded from their mid-fifties straight into premature old age, missing out the years in between.\n\n'Mum?' It was all she could do to stop herself placing a hand on her mother's head and letting it rest there.\n\n'But then I'd no want you ever coming along tae an actual game, Muriel. Fitba terraces are no place for a woman. Nae toilets for a start, less ye fancy joining us in the line, splashing up against the wall!'\n\n'Colin! How can you? Really, you are the very limit sometimes.'\n\nHer mother's shoulders had started to tremble. The light grey knitwork shook in her hands like an unfinished sail billowing in a sea-breeze, except there was no wind and no sailing boat \u2013 only an old woman's distress.\n\n'I'm standing here, Mother. Right in front of you.' How hard it was to resist reaching down to take the hands that had begun to shake, to force them to lay aside their knitting; and to make her mother look at her.\n\nHer father got to his feet, crossed over to the radio and turned up the volume:\n\n. . . can hold their lead for the last two minutes. what a front line. the famous five they've started calling them. smith, johnstone, reilly . . .\n\nAlmost cringing under the too-loud commentary now booming around the room, her mother seemed to shrink in her seat, dwindling further into old age.\n\nHer father sat down again.\n\nthe hibs supporters can hardly believe what's happened. keep this up and next year the league and cup double might . . .\n\nShe faced each of them in turn: 'Mother? Father?'\n\n. . . the famous five leading the march to victory. the whole of easter road's yelling for the referee to blow his . . .\n\nClenching her hands at her sides, gritting her teeth.\n\nWanting to march across to the radio and switch it off.\n\nWanting to rip the knitting out of her mother's hands and yell: look at me. mother! look at me!\n\nWanting to pick up her father's tea from the small side-table and throw it in his face . . .\n\nUpstairs, her bedroom was in near-darkness, the curtains pulled shut. No pictures on the walls. No books on the shelves. The mattress stripped. Empty fire grate. Cleared of all her ornaments and knicknacks, her dressing table had been polished to a hard shine. When she opened the wardrobe, the hangers made a wooden-sounding clack as they knocked against each other.\n\nAs a teenager she'd taken to tilting the dressing-table mirror this way and that, when checking her appearance. The instant she saw a hint of attractiveness in the sweep of her hair, in her practised smile and the elegance of her unclenched hands, she'd been ready to leave. Where else could she have found the courage to go out and face the world?\n\nAnd now?\n\nWas this really her reflection? There was no courage here, nor any sign of it \u2013 not in these wounded eyes, these clumsy hands, this slack hair.\n\nCome on, Maggie, come on, she urged the woman in the mirror.\n\nLike the room itself, had she, too, been emptied out, stripped bare? Her reflection half-raised a hand as if about to offer comfort, only to let it drop again. Then it clenched a fist, clenched and clenched until Maggie felt the fingernails digging into her palms.\n\nHer parents. Her own parents. Within minutes of her confessing what had happened to her, they'd told her to leave the house and never come back. They'd shouted at her and yelled. When she collapsed weeping on the couch, they'd simply got up and walked out the room. Later, in the hall, she'd found her suitcase packed and waiting. They'd ordered her to go, to go now. Pushed her into the vestibule. Pushed and pushed her onto the front step, onto the street. Locked the front door.\n\nShe glared at the woman in the mirror \u2013 why hadn't she made a scene? Why hadn't she rung the bell, punched and kicked the door? Screamed abuse at them?\n\nHer voice hardly above a whisper but charged with cold, cold fury: how could she have let herself be treated like that? Her whole life . . . like she'd never been born? She heard herself curse the house, curse it to a bombed-out rubble of shattered doorways and gaping windows. Curse her parents trapped in the shut-in self-righteousness of their sitting room.\n\nClump-clump-clumping back down the stairs as loudly as she could. Clump-clump-clump! Maggie came to a stop in the hall. Hesitating. The sitting-room door had been closed again \u2013 their chilling contempt beginning to freeze around her.\n\nBehind her, the grandfather clock's relentless tick . . . tick . . . tick . . . continued to fill the house. Tick . . . tick . . . tick . . . measuring out endurance to the lifeless furniture, to the carpets, doors, corridors, the staircase . . . and to her parents barricaded behind their silence.\n\nThen, and without stopping to think what she was doing, she went straight up to the grandfather clock and its hateful tick . . . tick . . . tick . . .\n\nShe opened its front case.\n\nShe reached in.\n\ntick . . . tick . . . tick . . . ti \u2014\n\nNo one paid any attention as Maggie laboured across to the very edge of Newhaven harbour. No one glanced over to see her stand her suitcase on the quayside, then pause for a moment, taking time to gaze down into the sunlit water that rippled-and-broke, rippled-and-broke almost soundlessly against the thick wooden posts.\n\nNo one watched her as she pulled back her arm, took aim \u2013 then hurled the pendulum as far and as high as she possibly could. Up into the air it rose, glittering as it arced briefly in the afternoon sun, suspended motionless for an instant before falling straight down into the blue-green depths.\n\nThere was hardly a splash.\n\nA wonderful moment, and Maggie savoured it. She gave herself a really big smile. As she stood there on the quayside she imagined the dead stillness now filling her parents' house, and pictured the pendulum sinking ever more deeply into the harbour's muddy ooze, its hateful tick . . . tick . . . tick . . . choked at last to a permanent silence.\n\nShe returned to Princes Street, getting off at the stop between the Scott Monument and the Galleries. It took her so long to pull her suitcase out from below the stairs that she had to struggle through the press of passengers already crowding to get on:\n\n'Out the way, please . . . Please, I'm trying to get off . . . Out of the way . . .'\n\nThen she was down. Once off the tram her suitcase seemed suddenly much heavier than before and she could manage only the clumsiest stumble-steps. Having reached the end of the platform she hesitated, not daring to leave the safety of the tram stop, the one small island of calm in the middle of the rushing street. The city centre was crowded, every department store and shop sure to be packed. Jenners, Fraser's, J&R Allen's, Patrick Thomson's, British Home Stores \u2013 the Saturday afternoon shoppers seemed to have melded together into one sweltering mass of summer frocks, hats, handbags, shopping bags and children squeezing in and out of doorways, jostling on the pavements, cramming themselves even more tightly together at junctions while waiting to cross.\n\nThe ground beneath shook with the rumble of tram after tram; overhead cables sparked their electric hiss-and-spit; bells dinged warnings to the men and women swarming across the rails. The heat was pounding at her. A few minutes to pull herself together and she'd be all right. Maybe a rest on one of the benches along the stretch of pavement in front of Princes Street Gardens?\n\nYes.\n\nGripping her suitcase even tighter, she stepped down onto the street \u2013 only to stumble almost at once on the cobbles, and nearly fall.\n\nIt was unbearable in the full sun, dizzying almost. Sweat trickled into her eyes and down the side of her face. She wiped it clear. Her blouse stuck to her back. An awkward dash across the tramlines . . .\n\nThe first bench was taken. So was the next. And the next.\n\nLifting and placing each foot . . .\n\nBreathing in, then breathing out . . .\n\nWiping the sweat from her eyes. Wiping again and again.\n\nAll but dragging her suitcase along . . . its arm-\u00adwrenching heaviness . . . thump, thump onto the pavement every few steps . . .\n\nSomewhere out of the sun. Somewhere to sit down. Mackie's was too far. Jenners Tearoom? Once inside the store she'd be able to take the lift all the way up to the top floor.\n\nOnly, it would mean crossing Princes Street again. It would mean trekking over the uneven cobbles, picking a safe path between lumbering trams that came in both directions. She'd have to struggle through more pedestrians, keep her balance on the shuddering tangle of tram lines that twisted and turned as they caught at her heels.\n\nA few minutes' rest, nothing more.\n\nThen she'd be fine.\n\nDashing across the front of a stationary tram, and behind another one heading in the opposite direction towards Calton Hill. A van hooted at her, but she paid no attention.\n\nShe stepped up onto the pavement. Jenners, at last. The large shop window glared back at her, a harsh cascade of blinding sunlight and raw colours \u2014\n\nPressing her forehead against the pane of sun-heated glass. Counting to twenty, slowly. Then counting again. Not until she felt she was again standing on solid ground did she continue her journey. The entrance was several yards ahead.\n\nStepping onto the next paving stone, and the next . . .\n\nShe had a special compass trembling and swaying inside her, of course. Doing its best to guide her. One more step, one more . . .\n\nBy the time she managed to stagger out of the sun and into Jenners doorway, it was all she could do to lean against the wall for a moment.\n\nThe red-brown marble felt so deliciously cool against her back, the solid stone taking the weight of her exhaustion.\n\nSweat was now streaming down her face, her neck, her back. Noise clamour brakes horns voices people people people. Surging in and out the doorway, shopping bags knocking against her, feet stumbling against hers, against her suitcase. Knocking it over. 'They'll have it in here . . . said we'd meet at four . . . and some linen for the . . .' Faces staring into hers: 'Are you all right, missus?'\n\nSo many rips in the near-transparent curtain that's fallen between her and the city she's known all her life.\n\nPeople people hectic sky staring sun dizzying blue . . .\n\nA hot wind, such a hot rushing wind springing up out of nowhere, tearing the blameless sky to shreds. Too sudden voices. Too abrupt laughter. Everything too close. Then too far away. The street rushing again rushing again rushing again . . . The hot wind wrenching the arched stonework above, the marbled pillars buckling first to one side, then to the other . . .\n\nThe air already sucked out of the next moment and the next . . .\n\nWhen Maggie opened her eyes, she found she was lying in Jenners' doorway, a man's shiny black shoes planted firmly on the ground right beside her cheek. Strangers brought to a standstill were gazing down at her. Over their shoulders she could see the top of the Scott Monument set against a calmness of sky.\n\n'White as a sheet, the poor woman.'\n\n'Some water, someone.'\n\nOne of the faces leant closer, becoming a patch of shadow shielding her from the sun's glare, asking if she was all right?\n\nDid she want to sit up, maybe?\n\nSomeone had taken her shoulder.\n\nNo. No. No, she screamed back at them, but couldn't make the words come out, not even a whisper.\n\nAsking if she wanted to put her head back, to lean against the wall?\n\nNo. No. No.\n\nAsking if she wanted to come inside where there was a chair?\n\nNo. No. No.\n\nSomeone took her arm and began helping her to her feet.\n\nno. no. no.\n\nIf only she could shout the words out loud. If only the people could hear her. If only she could remain peacefully stretched out on the floor of Jenners marbled entrance. Resting there, resting as on layer upon layer of the earth itself \u2013 each layer in turn bearing her weight and giving her the peace she longed for and wished could go on for ever.\n\nMeanwhile she'd been helped into the shop . . . helped into a heavily upholstered chair . . . a glass tumbler rattled against her teeth . . . water dribbled down her chin.\n\n'Take your time.' Such kindness in the man's voice, such concern \u2013 she could hardly hold back her tears. A woman had placed an arm around her shoulders.\n\n'You'll be all right in a few minutes, lass.'\n\nThe gentlest squeeze. 'Right as rain.' Another squeeze. A smile. 'If you've a phone at home we can call someone. Your husband, maybe?'\n\nShe jerked into sitting upright.\n\n'No phone in the house. I'll be fine. Just the heat.' She tried to smile. She didn't want the woman to withdraw her arm, not yet. So, so comforting even if only for a few more minutes.\n\n'You're a bit peely-wally though. Maybe coming down with \u2014 '\n\n'You're very kind. Both of you.'\n\n'Should be at home, feet up, with a magazine, the radio on and him bringing you tea and cake.'\n\nAround her, the noise had quickly built up until it was the same Saturday pell-mell rush all over again. Men and women, their bags and their children, pushing to get in, pushing to get out, pushing to reach the counters, the ringing tills.\n\nShe was helped out of the chair. 'Thank you.'\n\nShaky, but standing. She'd survived the fainting. She'd survived the kindness.\n\n'My man's at the football and \u2014 '\n\nShe told them she'd be getting a tram home. It wasn't far. Just down to Newhaven. She'd be fine now. Totally fine. Really.\n\n'Well, if you're sure you're okay, lass . . . ?'\n\nAs steadily as she could, she walked out of the department store and back into the glare of Princes Street, setting off in the direction of the tram stop opposite Waverley Steps.\n\nThen a miracle happened.\n\n'Maggie!'\n\nGetting a tram home? She had no home. She kept walking. When she reached the stop \u2013 what then? Somewhere to stay Somewhere to stay Somewhere to stay . . .\n\n'maggie!'\n\nShe looked round. Bleach-bottle blonde, pillar-box red lipstick, cheerful and friendly. It was her sister-in-law.\n\nThe older woman stood directly in front of her: 'Remember me?' She was joking, of course.\n\n'Oh, Jean! Hello there. I'm \u2013 I'm \u2014 '\n\nJean glanced down at the suitcase: 'Are you coming, or going?'\n\n'I don't know.'\n\nThe two women looked at each other.\n\n'It's true, Jean. I really don't know if I'm coming or going!'\n\nWhich made them both laugh out loud.\n\nTen minutes later Maggie was being treated to afternoon tea at the North British. Busy restaurant, chattering on all sides, waitresses in black starched uniforms, comfortable seat, a two-tiered cake stand \u2013 scones at the bottom and an upper layer of fancy cakes. There was a dish of glistening wet-yellow pats of butter, three kinds of jam, honey and clotted cream. No ration problems here. Two pots \u2013 one for tea, one for hot water. Linen napkins.\n\nHer sister-in-law had her own small business as a quality baker, making cakes and confectionery to order. Not until some years later, when Jean could finally afford to give up the rented one-room flat where she did her baking and lease proper shop premises in Haymarket \u2013 practically the West End, after all \u2013 was her professional status finally acknowledged by her mother-in-law. Until then she was always referred to as Jean, Billy's wife, who does us a nice cake when the pair of them come round. Her hair was too blonde, her lips too red and her accent . . . too Dalry.\n\n'Another brandy snap? Some cake? Let's hope it's just twae yer eating fer!' Jean nodded towards the few remaining crumbs on Maggie's empty plate.\n\n'Wonderful to see you again, Jean. You've no idea.'\n\nHer sister-in-law smiled. 'Feeling mair like yersel, are ye?'\n\n'Yes, thank you. A bit shaky back there with the heat and \u2014 '\n\n'Billy telt me whit happened. Ye poor woman! Throwing their ain daughter oot on the street. I'm scunnered, fair scunnered! Whit a shameless pair o \u2014 '\n\n'It's my own fault, Jean. If I hadn't let myself be \u2014 '\n\n'Dinna talk daft. That's the wey men talk, but we ken better. See, Maggie \u2014 '\n\n'I feel so \u2013 so ashamed. When he drove me back home \u2013 afterwards, you understand? \u2013 he hardly spoke a word, just dropped me at the end of my street,' she paused. 'After all his talk and his promises, suddenly I was nothing. That look on his face as he drove off . . . A smile right enough, but like he was glad to be rid of me . . . So much dirt he'd scraped off his \u2014 '\n\n'Maggie, dinna let yersel \u2014 !'\n\n'Could feel the shame of it burning into me as I went up our front steps.'\n\n'But, Maggie, ye shouldnae feel \u2014 '\n\n'Let me say it, Jean. I've not talked with anyone. Once I knew I was . . . well, you know what I mean . . . Anyway, when I tried to tell Mum and Dad . . . they were even worse. Their disgust, like I was the lowest of the low. I went to the YW that night and \u2014 '\n\n'Ye should've cam tae me. I'd hiv \u2014 '\n\n'I couldn't bear to see anyone. I wanted to be where no one knew me. Next morning I went to Lewis.'\n\n'Whit?' Jean's cry of surprise silenced the nearby tables. She continued in a low voice, 'Lewis? What in God's name possessed ye tae fetch yersel there?'\n\nMaggie shrugged. 'Relatives, but I didn't know them, never even met them before, and so I thought everything would be \u2014 ' She shook her head. 'Truth is, I wasn't thinking straight. I should've written to them first and saved myself the trip.' Managing not to cry, she told her sister-in-law about the hundred thousand welcomes she'd received.\n\nLong before she was finished, Jean's hand had reached across to cover hers. 'I tell ye, Maggie, I hope thae Callanders burn in Hell.'\n\n'That's an awful thing to say.'\n\nThe older woman shrugged and took out a packet of cigarettes. 'Smoke?'\n\n'No thanks.'\n\nJean lit up. 'Some family you're frae! Even my Billy \u2013 and he's the best of thon heartless brood \u2013 willnae want tae hear we've met. So I'll no be telling him.' She took a draw of her cigarette. 'Yer gang ahead wi it then?'\n\n'I suppose so.'\n\n'Hmm.' She paused to tap off some ash. 'Well, I'm glad tae hear yer no thinking o some back street butcher or of daein it yersel wi gin and knitting needles.'\n\nMaggie had managed to put away two slices of chocolate sponge, a nut-tasting cake with yellow icing and two scones with butter and strawberry jam. Feeling so much better than earlier, she licked the tip of her forefinger and dabbed at the cake crumbs.\n\n'Which reminds me, Maggie, when yer time comes, promise me ye'll no be seeking refuge in the airms o Christ?'\n\n'What?'\n\n'Promise me you'll book intae a proper nursing home. Promise me, Maggie. You're no gang tae be yin o those poor women doun on their hands and knees scrubbing the church flagstones right up tae the last minute, then getting tied tae a table tae gie birth. I've heard some awfae stories. We're no having merciful sisters saying you and yer sinful bairn are gang straight tae hell and gien ye both a taste of damnation in advance. We're no letting that happen. How are ye fer money, by the bye?'\n\n'Fine for the moment. Something in the Post Office.'\n\n'Good. And where are ye biding?'\n\n'Back at the YW, I suppose.'\n\n'Well, I know a place \u2013 in darkest, slummiest Dalry,' Jean said, imitating her mother-in-law's put-on posh. 'It'll dae fer the time being. All right?'\n\n'Jean, I can't \u2014 '\n\n'Not good enough fer you?'\n\n'No, I don't mean \u2014 '\n\n'That's that sorted then. And . . . the man?'\n\nMaggie shrugged. What was there to say?\n\nJean ground out her half-smoked cigarette. 'He can burn as weel.'\n\nJean's small bakery turned out to be an Aladdin's cave just off the Dalry Road. Maggie stepped from the grim, cobbled side street of stone-faced tenements straight into a one-room oriental palace where the oven-warm air was drenched with the scents of cinnamon, cinnabar and cloves, with the sweetness of melted chocolate and icing sugar. Like King and Queen, a large gas cooker and a generously deep kitchen sink ruled over an assembled court of shelf upon shelf of flour tins, spice racks, glass jars of raisins, currants, almonds, dried orange and lemon peel . . .\n\n'I'm a black-market baker,' she joked, 'either that or I'd be stuck daein scones and naethin but!'\n\nThe sitting room\/kitchen of Jean's small 'single end' flat was for her baking work only, but she'd fitted up the snug, cabin-sized boxroom at the back as a restroom, should she ever feel like putting her feet up. Instead of a bed there was the luxury of a Louis-the-Something chaise longue. Some blankets and a spare cushion were kept in a small trunk whose flat lid doubled as a side table. For Maggie, it was perfect.\n\nOnce Jean had left to go home, promising to be back first thing Monday morning, Maggie got herself settled in. A few days? A few weeks? She'd no idea, but for the time being this would be home. Her coat and scarf she hung on the back of the boxroom door, her best dress on a coat hanger that was nailed to the wall, creating a pleasing splash of blue and green against the pale-coloured wallpaper. Her two blouses went on another hanger to decorate the wall opposite. The rest of her clothes \u2013 a cardigan, two skirts, a jersey, her underwear and stockings \u2013 remained in her suitcase which she stowed under the couch. The chaise longue had gilt arm- and back-rests, and was upholstered in thick red velvet. Its remaining three legs were slender and curved gracefully upwards suggesting effortless, indeed miraculous, support; the fourth had been replaced by a weight from an old set of grocer's scales. The couch felt quite solid and secure, however. All in all, there was a certain elegance to her new sleeping arrangements, Maggie decided \u2013 and she intended to do her best to live up to it.\n\nThat evening as she shook out the first blanket, she heard her mother's voice: You've made your bed, now you have to lie on it.\n\n'I will,' she heard herself reply out loud, 'just watch me!'\n\nTwo days later Maggie saw the handwritten notice in the window of Fusco's Fish Restaurant on Gorgie Road, and went straight in. Perfect timing. She could hardly believe her luck \u2013 the usual assistant had just been sacked after turning up drunk and two hours late, yet again. If she could start at once, the job was hers \u2013 six-day week serving from 11 to 8, with two hours off in the afternoon. Haddock and chips, cod and chips, white pudding supper, black pudding supper, sausage and chips, steak pie and chips. Pickled eggs, pickled onions. At least she'd never go hungry. Tony Fusco had been an Italian POW who'd stayed on in Scotland and ended up marrying a girl from Leith.\n\n'She called Maggie like you! Good name, good woman \u2013 you work good, too!'\n\nHer nylon overalls were several sizes too big, which was a real plus as they'd help keep her 'situation' hidden for a good while longer. The Light Programme coming from the shelf above the till helped the minutes pass, if not the hours \u2013 Music While You Work, Housewives' Choice, Workers' Playtime . . . By the end of her first evening, the hours had slowed down to a crawl and she was so tired that she could hardly stop the chips from leaping off her scoop before they reached the newspaper, or the fish from nose-diving onto the floor as she carried the plates through to the sit-ins. Already she was looking forward to her first Sunday off \u2013 she'd stay in bed and enjoy a double-shift of deep-fried sleep.\n\nGrace, who worked at Fusco's from five in the afternoon till it closed at eleven, had helped her get started: 'Think of yer man and ram yer fish in the fryer fer a richt guid battering!' Slow Peter worked in the kitchen; he hardly spoke but instead offered the world a permanent grin.\n\nA few days after she'd started, Maggie happened to mention she was looking for a place to stay.\n\n'Yer in luck!' Grace grinned at her. 'Mrs McKenzie's lodger frae across oor landing, a widow woman, passed away last night. The room'll be going spare. If ye want, I'll let Mrs M ken you're interested. Ye can look in afore work the morn. If ye dinnae fancy it, no harm done. The auld soul'll be getting carried out first thing so ye can get yersel moved straight in, same day. I'm sure Tony'll let you off for a bit longer in the afternoon to get things sorted. A whole room tae yersel \u2013 I'm jealous already!'\n\nWhen Maggie called round after ten the following morning, Mrs McKenzie apologised, saying the undertakers hadn't turned up yet, but now that she was here Maggie might as well take a look round. 'Dinna fear \u2013 she's decent.'\n\nThey went into the room. Not only was it much larger than her boxroom at Jean's, but there was a window, and a door that she could lock. Perfect.\n\nMrs McKenzie nodded in the direction of the sheet-covered figure stretched out on the bed. 'She'll be out by this evening, Miss Davies.' Then, pointing to the gas meter beside the fire, she added, 'Ye canna say Isa was tight-fisted \u2013 she's left ye a good shilling's worth!'\n\nHaving discussed terms, Maggie agreed to come back with her suitcase during her afternoon break, and move in properly after work.\n\nWhen she returned at eight-thirty that evening the room was all hers. The dead woman's bed stood in the corner. Opposite was a family-sized tombstone of a wardrobe that reeked of shoes, old clothes and camphor. The top of the dresser was a clutter of postcards, photographs, some letters in a rack, a comb, nailfile, a Present from Dunbar ashtray; wisps of greyish hair were tangled among the bristles of the hairbrush.\n\nMaggie dumped the lot into an old tartan shopping bag she found hanging behind the door, to go out in the next bin collection. She felt sad for the widow woman dying here alone, but the less she knew about her the better she'd get on. The drawers were empty. Mrs McKenzie must have gone through them already.\n\nShe set about making the bed. A Louis-the-Something chaise longue was one thing, but a real bed with a mattress, clean-smelling sheets, thick blankets and a quilt that fitted neat as a pie-crust on top was quite another matter. After a full day on her feet, scooping and salting, she could hardly wait to climb in under that welcoming crust and get baked to sleep. She had a quick wash in the bathroom down the hall, undressed and got under the blankets. So snug and pie-warm!\n\nIt was only when she laid her head on the pillow that she realised she'd just made her bed without hearing her mother's usual comment. She grinned to herself in the dark. It was a good start.\n\nFor the next three months Maggie would sleepwalk out from her room at Mrs McKenzie's, down the stairs, along Fountainbridge, over the railway, past the graveyard, then sharp right, down to the junction, across to the start of Gorgie Road, under a railway bridge and straight on till she reached Fusco's \u2013 and sleepwalk back again nine hours later. Back and forth, back and forth, six times a week, like she was in a dream. Someone else's dream, not hers.\n\nEvery Saturday evening she went to Mrs Mackenzie's kitchen to pay her week's rent in advance, placing the ten-shilling note and loose coins onto her landlady's waiting palm. Back in her room she dropped whatever cash remained, uncounted, into the empty drawer of her bedside cabinet to join the previous weeks' earnings and the slither of pennies, threepenny bits, and occasional sixpences she'd been given as tips. If she needed to buy something, she took a handful of small change from the drawer. She spent very little. With Sunday as her only day off, she didn't have much opportunity \u2013 all the shops were shut and so were the cinemas and variety shows. She was at liberty to go to church, of course. Failing that, she was free to walk up and down the more or less deserted streets, or, if she found a park that was unlocked, she could stroll along the paths, past playgrounds with their swings, roundabouts and the witch's hat all chained up for the Sabbath. There was no question of visiting her brother Billy at home. It would be like visiting her parents all over again \u2013 except it'd make things really hard for her sister-in-law.\n\nThe best Sundays were when there was a light in Jean's shop, which meant she was having to work over the weekend. What a joy it was to walk into the comforting smell of cakes and baking, the sweetness of thick icing and melting chocolate, and the luxury of her sister-in-law's good humour and kindness. Here Maggie would sit and chat, and laugh, and tell Fusco stories about some of her customers, about Grace and her football-daft husband Norrie, storing up Jean's cheerfulness inside her to carry back to her room at Mrs McKenzie's.\n\n'Were you ever in the Hebrides?'\n\nWithout glancing up, Maggie continued scooping chips into the newspaper. 'Salt and sauce?'\n\n'The both, please. I'm wondering, were you ever in the Hebrides, on Lewis?'\n\nShe shook on the salt, then the sauce. 'Pickled onion?'\n\n'No thanks. Stornoway maybe? Can ye double-wrap? I'll be taking them back to the B&B.'\n\n'No, never been.' Maggie finished folding the two extra sheets of newspaper around the fish supper. Now she'd have to hand it over, to raise her eyes and look at the customer.\n\nHe was early thirties, red hair. Never seen him before. She took the man's half-crown and rang up the till.\n\n'Didn't you stay at Mrs Stewart's boarding house a few months back?'\n\nHer hand shook as she laid out the man's change on the counter.\n\n'Stewart? I don't know anybody called Stewart. Never been to Lewis either, like I said.' But even as she spoke she could feel herself blush. She drew her hand across her face as if to wipe away sweat. It suddenly seemed hotter than ever standing next to the fryer.\n\n'You've a perfect double then \u2013 Michael showed me a photograph. Not that he can see it, poor man, but he keeps it.'\n\nWhen Maggie made no reply, he continued. 'Well, maybe I'm wrong, but I'll give him your best. Greetings from a bonnie lass in Edinburgh! Michael'd like that!' A moment later the man was gone.\n\nFor the rest of that shift, Maggie let her hands carry on with the scooping, salting and wrapping while she herself stood again at the window of her room at Mrs Stewart's gazing out over the harbour, Michael's arms around her.\n\nA week later she came into the chip shop to find Grace grinning from ear to ear.\n\n'Ye've got fan mail!' Grace pointed to an envelope propped on the shelf next to the radio. 'Came this morning, Tony says.'\n\nMargaret Davies, c\/o Fusco's Fish Restaurant, Gorgie Road, Edinburgh. Firm handwriting, neat fountain pen.\n\n'Postmarked Stornoway. Whae'd ye ken in thon place?'\n\nMaggie stuck it in the pocket of her overalls. 'Relatives.'\n\n'In the land o the Wee Frees?'\n\nA second letter came in the afternoon post.\n\n'Mair relatives?' asked Grace.\n\nBack at Mrs McKenzie's, her coat still on, she sat on the edge of her bed and tore open the first envelope. She unfolded the light blue notepaper \u2013\n\nDearest Maggie, . . . I still think of it as your room, as our room really after all we shared together there . . . When I came downstairs in the morning to find you . . .\n\nBut she was hardly able to read the words, as written down by his friend Lachlan. Her tears seemed to have come from nowhere.\n\nShe wrote back to Michael an hour later.\n\nIt was the last Saturday in August. Since starting at eleven that morning she'd been scooping and salting for three hours solid while walking miles, it felt like, up and down the same stretch of lino behind the metal counter. The occasional trip to the back kitchen or to carry heaped plates into the side room had seemed like a relief. Her face, the inside of her mouth and even her lungs felt so clotted with grease that, by the time the lunch customers left, she felt she herself had been battered and basted to a turn. Forget staying indoors. She went out onto the back green. Putting her plate on the step, she eased herself down onto the sun-warmed stone and leant against the tenement wall. She slipped off her shoes. At last she could relax completely. Balancing her steak pie and chips on her knees, and with a cup of sweetened tea to hand, she was to all appearances a woman without a care in the world. Thanks to the loose-fitting overalls no one could notice that she was starting to show. She ate slowly while staring at the sheets and pillowcases, shirts, dresses and underwear strung out on the drying lines.\n\n'Maggie?'\n\nTony was standing in the doorway. She put down her fork.\n\n'Hi, Tony.'\n\n'Grace, she very tired. Like dog she say. Good you got more sense than have babies, Maggie. You stay more late this night, help me? Much peoples. I do money, you chips \u2013 easy.'\n\nGrace, too, was pregnant. During the slack periods Maggie was kept up to date with the expectant mother's state of health, her ever-changing views on baby clothes and baby care. The younger woman had slept well the previous night, or not so well. The baby had kicked. The baby had kicked again. Did Maggie think a newborn should be left to cry like that scientist said on the radio or picked up at once? Grace didn't want to spoil him. A girl was different. But it was going to be a boy because that's what Norrie wanted \u2013 someone to play for the Hearts. Should he be bottle- or breast-fed? She talked about cribs, prams, grandparents; she detailed how she coped with morning sickness, exhaustion, swollen ankles. Maggie was spared none of it and endured every last symptom and circumstance while unable to say anything of her own in return. Sometimes she felt like screaming.\n\n'Happy to help, Tony.' It would get her through another day.\n\n'Thanks, thanks. Real big help for me. For Grace too. For you extra money.'\n\n'Always welcome.' Maggie waved her fork to show how pleased she was.\n\n'More tips when pubs shut. Saturday night is different peoples. But I always here and Slow Peter. So no worry, Maggie.'\n\nDressed in his usual blue overalls and sandshoes without socks, Slow Peter was standing at the kitchen sink with his back to her, rinsing dishes. It was eight o'clock. The weather had broken and for the last half hour heavy rain had been battering the kitchen window.\n\nShe watched him run water over one side of a plate, turn it over, then rinse the other. Careful and methodical, he filled slot after slot on the drying rack. One plate . . . one plate . . . one plate. She sighed as she put down the fresh stack of dirty dishes she'd brought through.\n\n'You fine, Maggie?'\n\n'Fine, thanks, Peter. Bit tired.'\n\nSlow Peter lifted the top plate and began scraping the leftovers, some fish batter and mashed-up chips, into the metal waste bin next to his sink. It was a badly dented oil drum with Mobil lettered in red.\n\n'Pigs'll be grinning the night. Like Christmas dinner tae them.'\n\nThat rancid smell of fried fat \u2013 she'd need to wash it out of her hair before she could finally crawl into bed. She'd put on fresh sheets before leaving for work to be ready for tomorrow's all-day sleep. Would there be any hot water left? Had she a clean towel? Nearly three hours to go, more like three years.\n\n'My pigs is always pleased tae see me. Smiling and grinning they are. Cleans their bowls tae the last lick, not like them.' He jerked his head in the direction of the side room while tipping an uneaten fish cake into his drum.\n\nMaggie smiled, not a waitress-smile this time but the real thing.\n\n'We were taught to never waste food.' The instant she spoke she realised she was sounding like her mother, but couldn't seem to stop: 'Rationing, you understand, and the War. My mother . . .' she heard herself blundering on, 'she made soup out of anything. Even the cheese rinds went in. And bones too, of course.'\n\n'My pigs eats bones.'\n\n'We didn't actually eat the bones, Peter.' Her mother's voice again, her mother sitting in the Newhaven house, her knitting needles endlessly click-clicking, click-clicking, click-clicking . . .\n\n'You sure you're okay, Maggie?' Slow Peter was holding out a soapy wet hand as if offering to steady her.\n\nThirty-one years old, and over six months pregnant. No husband. No family. Spending her Saturday night standing in puddles of greasy kitchen water in the back room of a rain-lashed fish and chip shop on Gorgie Road \u2013 she was okay?\n\n'Fine thanks.'\n\nWhat else could she say to a man? Even to kindly Slow Peter? If she clammed up, if she did her best to ignore him, to ignore them all, there they'd still be, as always, the men of this world \u2013 like so many closed doors blocking her at every turn. Unless, of course, they wanted something enough to let her in. Afterwards the door was slammed shut . . . and she'd be shoved out into the street once more. Always the same closed door, the same street, the same man-made world. Not Michael, though, and his letters of long-distance affection . . . the words and phrases repeated over and over . . . hope . . . happiness . . . one day . . . you and me . . . They helped her blot out everything else. The photograph he'd sent her of him in his uniform stood propped up on her bedside table like a souvenir of another life, someone else's. Often, when she was about to write to him, she'd sit on the edge of her bed and let her fingertip trace the forehead, the cheeks and lips she could see in the photograph, like he'd shown her. No white stick, no milky eyes \u2013 but, yes, it was Michael, her Michael. Of course it was. And she loved him.\n\n'I'm fine, Peter. Thanks.' Another half-smile, a waitress one this time. 'Better get back through.'\n\n'Bye, bye, Maggie.'\n\nShe returned to find a queue had built up \u2013 a line of pig snouts and trotters at their counter-long feeding trough. Cod and chips no sauce \u2013 scoop and scoop, salt and wrap \/ pie supper with two pickled onions \u2013 scoop and scoop, salt and sauce and wrap \/ three sit-in suppers for a family guzzling at their own small trough in the side room. Back to the queue . . .\n\nThey started piling in at closing time. Piling in, staggering in, tumbling in, falling in, laughing, singing, joking, flirting; the queue soon became a multi-headed beast with no tail in sight. Some were drunk, some very drunk. Loud and cheerful, mostly. Tony had said to her, 'Like in pub \u2013 they at counter, they get served.'\n\nBy ten-thirty she was worn out, utterly.\n\nAt last the queue had dwindled to nothing and the counter was deserted except for a solitary man at the far end in drenched-through overalls, finishing his sixpenny bag. He stood, dripped and fed one chip after another into his mouth while staring out at the darkening street. He'd be leaving any minute. That was the good news. The bad news was that Norrie had turned up. Norrie was Grace's husband. Drunk but still on his feet, he'd come roistering in at closing time with some other men to mop up the night's beer with sit-in suppers all-round. The three men were in their thirties and heavily built. One wearing his cap with the brim pulled down low like a duck-bill, another with fair hair and a hesitant moustache. Norrie, in a blue check shirt, had his work jacket slung over the chair back. It seemed they'd just been paid off from building work on some new bungalows up the road. They seemed to fill the cafe with their noise.\n\nShe leant against the counter, giving the stainless steel a last wipe with a wet cloth.\n\n'A seat here for ye, Maggie. Rab's been keeping it warm for ye!' Norrie looked flushed with drink. 'He's eaten good and's fair raring for his afters!'\n\nDuck Bill roared with laughter, the Moustache gave Norrie a punch on the arm.\n\nShe managed a half nod in their direction while sidling herself and her cloth further off down the counter.\n\n'Been a bit pan-loafie, aren't ye, Maggie?' Norrie called out. No need to look over, she could imagine him sticking his nose in the air to mime a mock-polite sniff, and not for the first time.\n\n'Yer no frae Gorgie, are ye? Fancy yersel New Town? Morningside?' Duck Bill chipped in.\n\n'Where sex is what their coal gets delivered in!' shouted Norrie. More laughter.\n\nThe man in the overalls had now finished his chips and was holding out the scrunched-up wrapping paper for her to take. She was forced to return back along the counter, for the bin.\n\n'Naebody misses a slice off a cut loaf, pan or plain, eh?' The Moustache grinned at her and patted the empty seat beside him.\n\n'Rab's no being cheeky or nowt, Maggie.' Norrie stubbed out his cigarette in his half-finished black pudding supper as he called over to her: 'Sae lang as yer a woman's whit he means. An yer aa that!' Even at that distance his sun-beaten face showed the tiredness of several days' grey stubble.\n\n'Things okey-dokey, Maggie?' Tony had come through from the kitchen and was ringing up the till.\n\n'Fine.' She waitress-smiled and tucked a loose strand of hair behind her ear.\n\nSmell the cooking grease on it? She could feel it, a smeary slickness that stuck the separate strands together into a solid, larded hank.\n\nNorrie had started singing:\n\n'Rab kens a lassie, a bonnie, Scottish lassie,\n\nShe's sweet as the heather in the dell . . .'\n\nThe others joined in, clapping in time.\n\n'There's nane sae classie as Rab's bonnie lassie,\n\nMaggie, his Scots bluebell!'\n\nLoud cheers. Applause. Expectant faces turned in her direction. The windows were steamed up, but she could see the rain coming down harder than ever. The headlights of a passing car swept the darkness. If she was lucky she might still get the last tram home . . .\n\nBack through in the kitchen she ran herself a glass of cold water and dabbed her face and brow.\n\nSlow Peter straightened up from stacking plates in a cupboard. 'These pigs likes their windows closed tight. Gets choking hot.'\n\n'I'm okay now, thanks. Long day.'\n\n'Days here aye long.' Slow Peter picked up more plates. 'Sunday the morn. Shortest of the week, but aye the best.'\n\n'You said it, Peter.' She returned to the counter.\n\n'Here she comes. Rab's bonnie lassie!' called Norrie. 'Maggie, ye broke his hairt when ye went away there! Couldnae even finish his chips \u2013 we thocht he wis a goner.'\n\nAnother laugh.\n\n'Likely he'll be needing the kiss of life \u2013 and yer the only woman in the place. Job's all yours!'\n\nMore laughter, hands slapping the table top.\n\nThe kiss of life? She'd gone on only one date with the man she'd met at Fairley's \u2013 a drive down to Silverknowes in his Morris Minor. A romantic stroll along the seafront, he suggested. They'd parked at the top end away from the streetlights and that's when a sudden downpour started. The heavens emptied. He suggested they move into the back seat to be more comfortable. Again . . . and again . . . and again the slow sweep of a lighthouse beam from the island out in the Forth lit up his face, then drew total darkness over it: I want to show how much I care for you, Maggie. I want to show you. I want so much to \u2013\n\n'Kiss of life we're saying, Maggie!'\n\n. . . The smell of the leather seats, the heavy rain clattering onto the thin metal roof only inches above her head. The offer of whisky from his hip flask.\n\nNo, thanks.\n\nI'm drinking for two then. Cheers.\n\nThat peat-brown brackishness on the man's breath, making her pull away. A real stomach-churning \u2013\n\nThe memory made her want to gag, to turn away like she was still trying to avoid the man's lips, to squirm away from his touch. The beam of the lighthouse still searing her eyes . . .\n\nShe gripped the metal edge to steady herself, gripped it tight and held on. Doing her best to shut out the men in the cafe, to shut out the cafe itself, the darkness, the wind and rain, to shut out everything. Kiss of life? She wanted Michael's kiss, she wanted his arms around her, holding her safe and secure, shutting out the rest of the world . . .\n\nWithout any warning she felt a flutter deep inside her. So faint and seeming to have come from nowhere . . . Like nothing she'd ever . . .\n\nBut she knew at once what it was. She recognised it. Had been waiting for it . . .\n\nMaking her feel . . .? Making her feel that everything she'd ever known before, ever seen before, touched, heard . . .\n\nShe stood quite still, her hands resting on the counter, and suddenly nothing else mattered.\n\nNothing. Else. Mattered.\n\nFusco's was empty at last.\n\n'You go now, Maggie. I finish. Good work. Good work.' Tony handed her two half-crowns. 'This for you.'\n\n'Thanks. Like I said, happy to help.' Waitress words, waitress smile. 'Goodnight, Tony.'\n\nShe went through to the kitchen for her coat.\n\nSlow Peter must have left. Clean scoops and sieves hung from their hooks on the wall, dishcloths were spread out to dry over taps and racks, the sink and draining board gleamed. She felt she could have lain down on the main worktable there and then, and been asleep within seconds.\n\nTaking her coat out from the broom cupboard, she headed for the street door.\n\n'Good night, Maggie. You safe home now.'\n\nShe waved goodbye. 'See you Monday.'\n\nIt had stopped raining. Apart from the occasional car, its tyres hissing on the wet cobbles, and the few late-night stragglers making their way home, Gorgie Road was more or less deserted. She hurried along the pavement, side-stepping the puddles and the occasional gush of rainwater from a rone pipe. It was well after eleven o'clock. Fingers crossed there might still be a last tram.\n\nTime to step lively. Past the line of closed shops, past the Tynecastle street-end, quick-marching under the railway bridge, singing to herself to set a good pace: I love a lassie, a bonnie Scottish lassie . . .' A very brisk fifteen minutes and she'd be home.\n\nComing up to the Dalry Road junction. To left and right stretched a darkness of streetlamps spaced further and further apart, straight ahead lay an empty street of closed doors and tenement windows with their curtains pulled. No wind any more, the air damp and everything silent. A few parked cars and hardly a sound except for the brisk click-click-click of her heels on the pavement echoing back from the wall.\n\nSweet as the heather in the dell . . .\n\nFootsteps.\n\nSomeone else's, and coming close behind. Long steady strides. A man's.\n\nStarting to speed up: Nane sae classie-as-ma-bonnie-\u00adlassie . . .\n\n'Hey there!'\n\nShe felt herself freeze inside. Freeze and close up tight.\n\nThen abruptly walking even faster: A-bonnie-bonnie-\u00adlassie . . .\n\n'Hey there, Maggie, wait a moment!'\n\nIt was Norrie, and he'd almost caught up with her.\n\n'What are ye running for? Olympics were last year. Just saying hello. Friendly greeting, Maggie, that's all.' He was now level with her. 'You're yin fast woman right enough!'\n\nShe could smell the beer on him. 'That's because I want to get home.'\n\nKeeping her eyes fixed straight ahead, she accelerated to double-speed.\n\nAnother flutter, stronger than before . . .\n\n'Couldnae be better! The baith o us gang the same way, being neighbours like, on the same stair. Grand, eh!'\n\n'It's late.'\n\nFluttering like the very gentlest kick-kick-kick . . .\n\n'I'm walking ye back, seeing ye safe home.'\n\n'I'm fine thanks.'\n\nKick-kick-kick . . . Yes, she was fine \u2013 they were both fine. Needing nothing except to be left alone.\n\n'Ye dinnae like me?'\n\n'Go away, Norrie. I'm tired. Been on my feet all day and \u2014 '\n\nHis hand grasping her arm, his fingers digging into her elbow. Making her stop in mid-stride.\n\n'Ye dinna like me?'\n\n'Go away, I'm telling you.' Trying to shake herself free. 'Stop it.'\n\nHis grip was too strong. Forcing her to turn and look at him. His face only inches away, his unshaven cheeks looking dingy and raw.\n\n'Let go, you're hurting me.'\n\n'Then say ye like me.'\n\nSqueezing her arm harder.\n\nShe gritted her teeth. 'What do you mean \u2013 Say I like you? What about Grace?'\n\n'Grace isnae here.' Another squeeze. 'Come on.'\n\nSo tense now she could hardly manage to speak.\n\n'But Grace'll be waiting for you to come \u2014 '\n\n'say it, Maggie.'\n\n'I'm saying nothing. Let go of me.'\n\n'Yer a quiet yin, richt enough. Need encouraging. Am I right?' He passed his hand over her hair, stroking it lightly. 'Shy's right for a lady, ken? A proper lady. Makes them . . . special.' Stroking her cheek. 'Makes you special. Let's walk th'gither.'\n\nForced to walk at his side she remained rigid, her face turned away from him.\n\nA house light clicked off as they came to the beginning of Fountainbridge. Mrs McKenzie's was only five minutes away, five minutes at most.\n\nHe paused before crossing the street. 'Rare night, eh, Maggie? All the stars and everything. Peaceful. Like in the black-out. Mind hou bonnie the sky looked then?'\n\nJerking her elbow, making her stumble forward across the main road before abruptly wheeling her round into a side street. 'No far nou.'\n\n'But this isn't the way to \u2014 '\n\n'Shut it, Maggie. Or I'll shut you.' His arm had moved around her shoulder to hold her more firmly.\n\nLike she was resting her head on his shoulder, her face pressed up against the grit and greasiness of his work jacket. To anyone else they'd probably look like a courting couple making their way home \u2013 and if she screamed, folk would just think they were having a row . . .\n\nBut there was no one to see them. The street was deserted, the tenement windows curtained.\n\n'Grace hisnae let me near her in weeks. She's nae wife tae me. It's bairn this an bairn that.'\n\n'But, Norrie \u2014 '\n\n'Shut it, I said.' He marched her another twenty yards. 'Dinna be feart, Maggie. I'm no gang to harm ye. I like ye. I really do. Nothing bad's gang tae happen. I promise.'\n\nInto the next street.\n\n'Wanting tae show ye whit I'd done, is aa. Grace isnae \u00adinterested \u2013 just in the money I bring hame. But I'm proud o whit I've done an ken ye'll appreciate it. Ye've got class.' Another squeeze.\n\nA few steps later he halted outside the first in a line of new bungalows.\n\n'Folks'll be moving in Monday.' Hauling her up the steps to the front door. 'I kept us a key.'\n\nSUNDAY\n\nTHE WHITEBOARD IN the hall reads today is . . . monday.\n\nBoss Beryl's come up to you saying, 'What're you doing here, Maggie? Waiting for a bus? Won't come just because you're standing here.'\n\nNo point saying anything back to the likes of Boss Beryl, so no one ever does. Complete waste of time.\n\nYou mumble what might be a yes or might be a no and give her a shrug, making it look like you're really heading towards the dayroom . . .\n\nBut if you really do walk off, then you'll just have to turn and come all the way back once she's gone.\n\nSo you rein in your zimmer and give her a puzzled look. Puzzled, and yet hopeful.\n\n'The fact is, Beryl, I'm waiting for a number 26. Maybe you can help me? Do you have any idea when the next one's due? It'll take me right along Princes Street. Pity there's no timetable. Can't Mrs Saunders get one fixed up?'\n\nThe look on her torn face! She'll certainly not be hanging around much longer. She's got better things to do than . . .\n\nAnd so . . .\n\nNot even the hint of grin: 'Or, if a 7 turns up I'll ask for a two-penny transfer and get off in Princes Street. Some of the shops stay open over lunchtime and . . .'\n\nSuccess. Boss Beryl's given a snort, shaken her head and said that she's got better things to do than stand around passing the time of day.\n\nAfter waiting to make sure she's gone all the way down the corridor and disappeared round the corner, you execute a neat U-turn and zimmer yourself right up to the board. No one about \u2013 and so you grab hold of the marker pen . . .\n\nThe shaft of sunlight from the bay window has started inching its way across the lino, turning the chairs lined up against the walls into the hour marks scratched on a sundial. Anyone coming in the door can tell what time it is by how far the sun has travelled round the room and who it's pointing at. It's Dorothy o'clock now. The old woman doesn't move a muscle, doesn't even turn away to shield her eyes.\n\nYou changed what was written on the whiteboard, of course. Not that you believe today will end up any different because of that. You're not that far gone. But today is . . . sunday written up there for everyone to read, must mean something? The words are real words that make real sense, so they can't be completely wrong. They must be a little bit true. Surely?\n\nHere in the dayroom things move forward only when the sun carries them, and at meal times. You come in, take your seat. No one changes their seat or gets up and walks anywhere except on the TV screen where folk are always talking and waving their arms and having car chases with emergency blue lights and sirens and loud crashes. But there's no point trying to follow what's being said or \u2013\n\n'Your son's here, Maggie.'\n\n'Who? Someone's here today?'\n\nFor a moment it seems that Fred Astaire has appeared in the doorway looking for a new partner to dance with, but without his top hat. After pausing to check who's sitting where, he's come across the empty space in the middle of the room like he's wading through a pool of sunlight, splashing brightness on everyone. He's making straight for you.\n\nYou'd like to hold out your arms for him to take you and lift you onto your feet, then together the two of you'll go whirling round and round the dance floor.\n\n'But today's really called Monday.' Damn. You've gone and said it out loud before you could stop yourself. You don't want them to find out it was you who changed the board, do you? Of course not. But you can't have really made it Sunday . . . ?\n\nHe's standing in front of you. Will he reach up to touch the side of your face, let his fingertips pass slowly over your eyes and lips?\n\n'Happy Birthday!' His hands on your shoulders, he's bent forward to kiss you on the forehead. The briefest touch and he's already back in his seat. Too quick. All too quick and finished with.\n\n'My birthday?' You can hardly follow what he's saying. Doesn't he want to clasp you in his arms? Pass his hands over your face?\n\nCan he really see you?\n\n'Congratulations! Happy Birthday, Mum. Ninety \u2013 and looking fabulous! That's why I've driven down special to be here today. Didn't they have the cake at lunchtime that I ordered? And everyone singing Happy Birthday?'\n\n'Cake? Yes, I remember cake.'\n\nBefore you know what's happening, he's waving his hands in the air \u2013 making magic passes is what he calls it. A trick \u2013 he's doing a magic trick like he says he sometimes does on TV. Maybe he'll make you vanish in a puff of smoke? Or else the Murray twins will start talking? Or else Dorothy? Or . . . ?\n\nThere's the flick of a yellow silk handkerchief that he's pulled out of nowhere:\n\n'Let the sun shine and the earth whirl,\n\nFor our very own . . . birthday girl!\n\nHappy Birthday, Mum!'\n\nA small package has suddenly appeared on your lap, wrapped in red paper and tied with curly gold ribbon. He's helping you with the ribbon, the paper.\n\nYou mutter a thank-you, adding that it's very kind of him. What else can you say?\n\nIt's a book. A book with a hard red cover. You never asked for a book. He wouldn't know you stopped reading a long time ago. Sad stories, they always felt sad. All that living that just goes on and on until it tears the heart out of you, till you ache.\n\n'For you, Mum. I made it.'\n\nOnly there's no title. Nothing written on the front or on the back. No pages even, not the usual kind of pages anyway.\n\n'Well, come on, Mum, open it!'\n\nThe first page: A black and white photograph set in a plastic sleeve. Not a book at all, but photographs like in a family album. That's what it is \u2013 somebody's photo album. Out of politeness you flick through the opening pages while he tells you he'd come across a lot of photos in an envelope in a cupboard, along with your old typewriter. Halfway through, you stop at a snap of some couple or other posing outside their house \u2013\n\nComplete strangers.\n\nMakes you feel bad seeing them, whoever they are. Makes you feel ashamed, poking your nose into other people's lives. You give him the album back.\n\nBut all he does is show you another picture. This time . . . it's you! A photograph of you wearing a starched white blouse and looking very efficient, your hands poised above the keyboard of an ancient-looking typewriter. Your old Underwood.\n\n'Remember, Mum? How you used to do the laird's typing \u2013 that's how we lived, wasn't it? You did it for everybody in the village too, near enough \u2013 the school, the minister, Arnott's shop, the smiddy. And helped them with their accounts. You ran the place!'\n\nYou remember the typing all right. Good times! Battering away on the keys, the bell ringing at the end of every line, the carbon paper and inky ribbon, everything set up on the tea trolley with its extension flaps. Handwritten scraps of paper you could sometimes hardly read to the left of your machine, neatly typed sheets stacked on the right \u2013 like doing the ironing, you called it. You really enjoyed it, didn't you? Making order out of mess \u2013 and it certainly helped you quickly get to know everyone in the village. The farmers sometimes paying you in eggs, vegetables; Arnott's giving you groceries as well as a good discount; other folk giving you rabbits, fish out the Annan . . .\n\n'Mum, who were the Callanders?'\n\n'The who? Never heard of them.'\n\nHe's gone back to that photo of the man and woman on their doorstep. 'Their names are on the back. I don't think you ever mentioned them and \u2014 '\n\n'I don't know who they are, and I don't want to know.' You snatch the album from him, slam it shut and hand it back. If it's mostly people you don't know, you tell him, he might as well give the book to someone else.\n\nBut he won't stop. Next comes a photo of your cottage. Yes, that's where you live. It's yours. When the laird's estate got broken up in the eighties, Tom bought it for you so that you'd always be secure. That photo's worth keeping, you say.\n\n'It's where you brought me up, eh, Mum?'\n\n'Where you were brought up? What are you on about? Don't talk daft.'\n\nBut now he's started, there he goes talk talk talk talk talk.\n\nOf course, today's really Monday. Monday. Monday. monday.\n\nMonday's always washday. Take the week's washing with you in the morning when you go up to the factor's office, then use the laundry tub at the back of the house in the afternoon. Pretending Tom was with you, helping. Pretending so hard that sometimes he really seemed to be there at your side, the two of you singing as you worked:\n\n'Scrub a dub-dub, Three men in a tub . . . !'\n\n'What's that, Mum?'\n\nDirty, crumpled clothes into the sink, the cold water turning your hands red-raw, the stone floor puddled from the wash being lifted between the deep sink and the tub.\n\n'The butcher, the baker, the candlestick maker . . .'\n\n'The \u2013 who, Mum? I'm talking about the cottage. See, that's me and my new bike on the drive up to the big house.'\n\nYes, Tom's helping you today, both hands at once, barrel-organ-ing the mangle with his little-boy strength, sending the squeezed-out water splashing down into the tub. A pair of small dungarees will be first to go between the rollers, sodden, then dripping ice-cold water as it's forced through. All the way in . . . and out the other side it comes, crushed and flattened into your waiting arms.\n\n'. . . and there's the tree you were always telling me not to climb, and the laird's horse looking out its stable. Rusty he was called. Remember when they lifted me up onto Rusty's back so I could go for a . . . ?'\n\nTom celebrates by knocking on the side of the tub with his fist \u2013 Boom! Boom! Boom! Makes you both laugh. You take good hold of the dungarees by the shoulders and shake them, making them snap in the air, crack like a whip. Another Boom! Boom! Boom! Then you lay them down flat in the cane basket, the arms and legs dangling over the sides.\n\n'Here's the vegetable patch, you wearing your big wellies and me helping you with . . .'\n\nThen socks, underwear, pillowcases . . . till you've got the first basketful of the week's wash stacked and ready. Swinging it in time, off you go down to the gatehouse garden to peg everything out on the line. Winter days are fierce, the shirts and pillowcases freeze rigid in a few hours. Tuesday's always the ironing.\n\n'She looked so neat and nimble-o\n\nDarning with her thimble-o.\n\nDashing away with the smoothing iron . . .'\n\nLast thing at night you set up the wooden clothes-horse in front of what's left of the fire and, just before you turn out the light, you wait to see the clothes steam in the heat as if they're actually breathing. New life \u2013 which is always a good finish to the day!\n\nMost mornings, up to the big house and into the estate office to do the typing, the filing and accounts. Then the afternoons \u2013\n\nTuesday \u2013 the ironing.\n\nWednesday \u2013 the cleaning . . . and typing for people in the village.\n\nThursday. Friday. Saturday. Village typing.\n\nSunday.\n\nBut not any more \u2013 since coming here the same week's become the same day, the same moment. There's no weather, no date on the calendar, no time on the clock. And the years you've lived through? They're here, and always have been.\n\nListen \u2013\n\n3\n\nNORRIE PUSHED HER through the door of the bungalow and into a smell of new carpet, fresh paint and woodwork. There was a small bouquet of plastic flowers on the hall table. The living room was straight out of the Ideal Home Exhibition with modern-looking art hanging above the mantelpiece, wall lamps on either side, a beige three-piece suite, coffee-table with glossy magazines, white hearthrug, cream-coloured curtains.\n\n'This yin's the real thing, it's fer showing people, the others are just bare walls and floorboards. Top tae bottom electric \u2013 ye want something, ye press a button. I tell ye, Maggie, yince the hydro-electric really gets going, it'll be free electricity and buttons for us all. See this?' He turned a switch set into the wall next to the kitchen doorway. 'Central heating. Ye put it tae ony temperature ye want. Let's get oorsels nice an toasty, eh! 75 degrees!' He grabbed her by the shoulders. 'Ye'll no be needing this onymair!'\n\nRealising her coat was in danger of getting ripped, Maggie took it off and laid it on a chair next to the front door. Norrie threw his work jacket after it, but missed. Like an aggressive salesman showing how wonderful her brand-new life would be in this, her brand-new home, he then took her by the arm and hustled her through a tour of the rooms, stopping on the way to point out each labour-saving gadget, each clever new feature \u2013 the double-sink in the kitchen, the mixer tap, the waste disposal that ground up old food and garbage.\n\n'See this hatch intae the dining room, Maggie? \u2013 was me cut it and fitted it. This breakfast bar? \u2013 me. This formica top? \u2013 me.'\n\nIn the living room he pointed out the side lighting, the fitted carpets and matching curtains, the skirting and double glazing. A large wooden cabinet stood next to the fireplace. It had to be the biggest wireless set Maggie had ever seen. Norrie switched it on. The dial lit up.\n\n'Latest thing, this. Medium Wave, Long Wave, Short Wave \u2013 mair waves than the sea itsel.'\n\nOnce the valves had warmed up, he slewed through an electric storm of crackles and swoops until he came to \u2013\n\n. . . and gentlemen. Direct from the heart of London we bring you Saturday Night on the Light \u2013 with Max Jaffa and his orchestra.\n\n'A bit of music, eh. Set the mood fer us.'\n\nSteered her out and across the hall. 'The best is yet tae come.'\n\nPushed her into the room opposite. The bedroom.\n\n'See this!' He clicked a switch set in the wall beside the door, and turned on the faraway bedlights. 'Magic, eh!' He grinned. 'And there's anither yin next to the bed fer turning them off. Luxury! Built the whole fucking place near enough, so I did. Me an Grace deserve it \u2013 no? Or are we no guid enough tae live here? Think we're no guid enough, Maggie?'\n\nThere was a large double bed with blankets and a shiny quilt, one corner already turned down in invitation to the prospective owner. Having walked her over to the window, he tugged at a cord with his free hand, 'Let's make us nice and cosy, eh?' The curtains glided shut.\n\n'Fuckers that'll be moving in come Monday, that's what they'll think. That we're just working trash. Bairn on the way, and us still sharing wi Grace's parents in thon top floor slum. Running water in baith rooms, right enough \u2013 running doun the fucking walls.'\n\nHe pulled her over to the dressing table. 'Look at the pair o us!'\n\nWrenched her into position till they were facing the mirror, standing side by side. His reflection glared back at her, the bloodshot anger in his eyes:\n\n'Working till we drop, and fer what? Fuck's sake, Maggie, let's hae yin five-star night in our lives. Yin fucking night, eh!'\n\nFrom through in the living room came the sounds of the radio orchestra.\n\n'Glenn Miller. We could hae a wee dance, you an me.'\n\nKeeping firm hold of her he made as if to begin a waltz, then seemed to change his mind. 'Whaure's ma manners? First things, first. Get the lady a drink.'\n\nWith his free hand he drew a bottle from the side pocket of his jacket. He pulled the cork out with his teeth and spat it out. 'A wee toast tae us baith. Cheers.' Not taking his eyes off her, he took a deep swallow. 'Now you, ma lady.' He held the whisky up to her. She turned away.\n\n'No tak a drink wi me? Am a no guid enough fer ye?'\n\n'I don't want to drink, Norrie. I want to go home. If you let me leave now I won't say anything to Grace. Like this never \u2014 '\n\nNext moment he'd pushed her down onto the bed.\n\nThe pub stink of him, the unspoken threat \u2013\n\n'Get off me, Norrie. Stop this. Stop it before you go too far. You've had one drink too many is all. Let's just leave now and go home. Grace'll be \u2014 '\n\n'Like I said, Grace isnae here.' The menace in his voice: 'I'm no tellin, an I'm shair ye'll keep stumm if ye ken whit's guid fer ye. Right?'\n\nOver his shoulder, she could see the open door leading out to the hall and the living room beyond. They were playing 'Moonlight Serenade'.\n\n'A few drinks and a bit of fun \u2013 whit's yer problem, Maggie? Ye pan-loafie bitch! Easy seen whose side you're on.' The weight of him keeping her pinned down on the bed. He shoved the bottle at her:\n\n'Ye'll hae tae get catched up wi me, dram fer dram.'\n\nShe tried to twist her head from side to side, but he forced the bottle against her mouth, upending it. Some whisky slopped over and ran down her chin.\n\n'Dinna waste it.'\n\nClenching her jaw tighter shut.\n\nHis fingers wrenching her lips apart, and then her teeth \u2013 the tobacco taste nearly making her retch. Her mouth flooding with the harsh liquid till she almost choked. She had to swallow.\n\n'That's the style! Come on, yin mair tae get yersel real loosened up, eh?' Tilting the bottle again.\n\nShe struggled under him, trying to push back, to kick out, but he held her tight.\n\nHaving to swallow again. And again. His fingers in her mouth, forcing it open each time. Another swallow.\n\n'Come on, Maggie, guid stuff this. Better than mither's milk fer ye.'\n\nHis hand clamped so she couldn't spit out \u2013\n\nTilting the bottle again.\n\nHis loud whisper, his hoarseness: 'Guid lass. We'll hae some fun nou, you and me.' Leaning across her, he put the bottle on the side table.\n\nThe rawness of his unshaven chin, his unwashed sweat-smell. His roaring whisky breath \u2013\n\nPushing himself hard up against her \u2013\n\n'What the fuck!' The flat of his hand sliding down to press her stomach. 'What the fuck's this, Maggie? You're fucking in the club, aren't ye?'\n\nShe jerked away, pulled her knees tight up to her chest.\n\n'Fuck's sake, Maggie. Fuck's sake. Up the stick, an yer making me work fer it? Ye fucking keelie!'\n\nHe was going to hit her. That was coming next. She could see it. She reached across and grabbed for the bottle to defend herself.\n\nBut he didn't. Instead he half-rolled away from her and started muttering over and over to himself, 'Fucking keelie . . . Fucking keelie . . . Fucking keelie . . .'\n\nShe wrenched herself out from under him and clambered off the bed. She stood up. Whisky-dizzy.\n\nMeanwhile, the dance band music continued . . . it seemed to be playing right inside her now, inside her head, inside her whole body, like it was spinning her round. She stumbled away from the bed, the floor see-sawing under her feet. The whisky bottle still in her hand, she raised her arm as if that gesture could bring everything to a stop.\n\nNext moment she watched the bottle shatter against the wall only inches from Norrie's head. An explosion of glass and whisky that spattered everywhere, followed by the slow drip . . . drip . . . drip onto the headboard.\n\nHis voice was a whine: 'Could've killed me, ye bitch! Fucken bitch ye! Fucking hoor! Fuck \u2013 Fuck \u2014 '\n\nShe heard herself scream back: 'I hope you burn in hell, the whole bloody lot of you!'\n\nIt was after midnight when she fumbled her key into Mrs Mckenzie's door.\n\nHaving pulled off her coat and let it fall to the floor, she slumped onto her bed. She was shaking. If she'd not fought back, Norrie would have \u2013\n\nNext thing, she was sitting with her money drawer on her lap. The loose coins slid from side to side. Everything had started to blur.\n\nBack and forth she rocked herself, trying to blink her eyes clear, but they blurred again almost immediately. Through her tears the silver and copper glittered, with here and there the red of a crumpled ten shilling note.\n\nShe was still crying when she heard the downstairs street door bang shut. Someone had come into the close. She listened hard. The footsteps stumbled up to the first landing. Then stopped.\n\nOnly to carry on a moment later. Was that him?\n\nShe wiped away the tears with the back of her hand. And the next time she saw Norrie? And Grace?\n\nThe footsteps had almost reached the top floor. Norrie, for certain . . .\n\nHardly breathing even, steadying the drawer on her knees. Go away. Go away.\n\n'Maggie!'\n\nThe drunken fool was shouting through the letter box. He'd wake up the whole house, the whole stair.\n\n'maggie!'\n\nShe placed the drawer down beside her and lay flat on the bed. The instant she closed her eyes the room started to spin.\n\n'maggie, i'm sorry for causing ye bother, i'm sorry . . .'\n\n'Miss Davies? There seems to be something of yours out on the landing \u2013 kindly get rid of it!' Mrs McKenzie was standing in her open doorway.\n\n'sorry, maggie. sorry. sorry. sorry . . .'\n\n'That is Norrie Chalmers, isn't it? Frae next door?'\n\n'Yes, Mrs McKenzie.' Maggie had to concentrate to speak normally, distinctly. All by themselves the words she wanted kept slithering across her tongue. 'Foll'ed me long the street so's I \u2014 '\n\n'Get rid of him. Then get rid o yersel. A woman that's no married at your age \u2013 naethin but grief for us respectable folks.' She gave a sniff, then she settled for acid-polite English. 'Drinking, too, I notice. Well, not in my house.'\n\n'B' Msss McKenz \u2014 '\n\n'I'll take your key, if you please.'\n\nOnce Mrs McKenzie had gone, Maggie upended the contents of the drawer into her handbag. She stuffed her clothes into her suitcase.\n\nTwenty minutes later, she was letting herself into the bakery \u2013 thank goodness Jean had insisted she keep a key 'just in case'. Without bothering to wash or undress, she stretched out on the chaise longue and pulled the blankets tightly around her.\n\nYou've made your bed . . .\n\nThere was no question of returning to Fusco's, she told Jean. Plenty unskilled jobs in offices \u2013 filing, reception, answering the phone, making tea. Or else she could waitress in a respectable teashop, a city-centre restaurant or hotel.\n\nThe first employment agency she tried was in George Street. Up to their first-floor office, then across the small carpeted hall to the desk marked reception. As the girl sitting there was on the phone, Maggie stood and waited. And stood. And waited. At one point the girl (was Miss Snooty Junior really old enough to have left school?) glanced in her direction, gave her a nod, then carried on with her conversation. Miss Snooty Junior had an impressive telephone voice. The call seemed to be something about last month's records, which hadn't arrived somewhere or else hadn't been sent. They were supposed to have been posted in good time. Mrs Somebody would have weighed them herself and Mrs Somebody else should have taken them to the Post Office at Waterloo Place. Three days ago. No four.\n\nMaggie was about to leave when the call came to an end. She watched the receiver being set down in its cradle, the girl letting her hand linger on it for several seconds before glancing across.\n\n'That was Head Office.' Miss Snooty Junior had a Reception voice, too.\n\nMaggie was treated to a Reception smile and given an application form. Told to sit down.\n\nEasy questions first:\n\nname and address? She'd give Jean's bakery \u2013 just as well it wasn't a proper shop.\n\nage?\n\nschool?\n\nThen came the hard ones:\n\nqualifications?\n\nprevious jobs and experience?\n\nprevious positions of responsibility \/ authority?\n\nposition sought?\n\nFrom school onwards, Maggie's life was quickly reduced to a series of blanks. Snooty Junior glanced at her completed form, said thank you and repeated the smile. Someone would be in touch should anything suitable turn up.\n\nThree more agencies, three more Snooty Juniors \u2013 same voice, same blouse, same lipstick, same smile. They all said they'd let her know.\n\nLunch was a Scotch egg, a half-pint of milk and an apple while sitting on a bench in Princes Street Gardens just along from the Scott Monument. She needed her gabardine buttoned up to the chin to keep warm in a sun that, at this time of the year, was starting to get past its best. From across the sloping grass of the gardens came the occasional hoots of trains entering and leaving Waverley. But it was restful sitting in the park watching the people, the pigeons . . . and she could have happily remained there all afternoon, doing nothing, saying nothing, as if part of an unfinished painting: Edinburgh City Centre. Then she'd remember \u2013 and, all at once, the picture seemed to dissolve around her, leaving her sitting on her bench, alone. She got to her feet, brushed the crumbs from her lap, then headed back into Princes Street and to more agencies.\n\nThanks to Jenners, Patrick Thompson's and Forsyth's, she managed to get through the afternoon. Whenever she couldn't face another Snooty Junior or filling in another form, she made straight for the nearest department store to wallow in scarves, perfumes, hats, gloves, shawls, working her way along the rails and counters until she felt better. Trailing a silk scarf between her fingers was like dipping them into the coolness of running water; when trying on a hat, she'd let the veil drop and was able to relax behind it, if only for a moment; as she dabbed perfume onto her wrist, she'd close her eyes, breathe in deeply, and let her weariness dissolve into Chanel.\n\nIt was getting towards five o'clock when she toiled up the too-many flights of narrow, badly lit stairs to present herself at Superior Employment. Her arrival was perfectly timed \u2013 no receptionist in sight. Had this particular Snooty Junior left early? Was she playing with her dolls? Sitting on the boss's knee? Maggie didn't wait to find out. Ignoring the brass bell with its notice asking visitors to ring for attention, she made her way along a short corridor until she came to a door which stood invitingly half open.\n\nShe walked straight in.\n\nThe name plate said: Mr Wilson, and this was Mr Wilson himself, she presumed, seated behind the desk, head bent over a pile of paperwork. Unlike her mother's hair-parting, Mr Wilson's was ruler-straight and precise enough to look painted on. She had to resist the temptation to reach down and touch its jet-black glossiness to see if the paint was still wet. Without raising his head, Mr Wilson continued to scrutinise the form in front of him, ticking his way down the boxes of what was probably someone's job history.\n\n'One moment, please.'\n\nEach ponderous tick was accompanied by a 'humph' of approval whose seriousness reminded her of an elderly Recording Angel, one whose recommendation would be given great weight. Even though she couldn't read the form upside-down, it was clear that here was an applicant with no blanks in their life. Every box was so crammed that the handwritten details of their busy career had spilled over into the surrounding page.\n\nThe Recording Angel inscribed one final extra-large tick of approval before glancing up.\n\n'Yes? Can I. Help you?'\n\n'You are Mr Wilson?'\n\n'Yes.' His tone was cautious as if he might have been about to add but only on weekdays or only in this office.\n\n'My name is Miss Davies and I'm seeking employment.' She sat down. The wooden seat was hard and straight-backed, forcing her to lean forward as if she had difficulty catching what was being said.\n\nShe tried a smile. 'Your agency has come very highly recommended.'\n\n'Experience?'\n\nFrom somewhere out in the corridor came the clatter-clatter of a typewriter. The Recording Angel had his back to the window, which kept his face in shadow, reducing it to a mere suggestion of a face. His one-word question was expanded:\n\n'What. Experience. Have you?'\n\nHe wore glasses. Small, rodent-like eyes \u2013 she felt their gaze gnawing at her, felt it scampering along the cut of her blouse and jacket, teasing the creases she'd ironed out on Jean's travelling-trunk-cum-table. His face remained immobile, his lips parting no wider than the minimum necessary to allow his prepared words their exit. Between words, the mouth stayed firmly closed.\n\n'My experience?' She glanced beyond Mr Wilson's shoulder to the top storey and roof of the building opposite. 'I am a good worker. Reliable, honest and . . .'\n\n'Yes. Naturally. All applicants are.'\n\n'My mother always said I should be awarded an M.A. \u2013 Mother's Assistant \u2013 I was so good at helping her run the house.'\n\n'And. Outside the. House?'\n\n'Outside? Of course . . . I've been a waitress.'\n\nThe face arranged itself into what might have been intended as a smile of encouragement. 'Where?'\n\nShe said the first thing that came into her head. 'Lewis.'\n\n'Lewis?' Mr Wilson's mouth seemed to savour the unexpectedness of this place-name.\n\n'In the Outer Hebrides.'\n\n'Yes. Miss Davies. I do know. Where Lewis is.' Of its own accord his right hand picked up a pen from the desk and held it in readiness. 'Silver Service?'\n\nDoing her best to keep her thumbs out of the chips when serving the sit-in suppers was the closest she'd come to the niceties of Silver Service, but this was no time for hair-splitting. There was no need to burden her interviewer with unnecessary information.\n\n'Yes, I'm experienced in Silver Service. And also acting as cashier, when required.' Well, why not? \u2013 she handled money every day, after all.\n\n'Indeed. References?'\n\n'Yes, naturally . . .'\n\nShe'd completely forgotten about references\n\n'. . . They can be produced when asked for.' She and Jean could easily cobble something together, something that would include a glowing testament to her Silver Service skills.\n\n'Hmm. Well. Now, Miss . . . Davies. A few details. Full name?'\n\nIt was Snooty Junior's perfume, mashed roses, which entered the room first, closely followed by the familiar blouse, lipstick and Reception smile. The whole effect was topped off by a bleached perm. 'Excuse me, Mr Wilson . . .'\n\n'Margaret Davies, Miss. I have \u2014 '\n\n'. . . Excuse me, I hadn't realised you were occupied, Mr Wilson. It's the Caledonian.'\n\n'Thank you, Miss Webster.' The Recording Angel took the sheet of paper the receptionist was holding out to him. 'Seems you're in luck, Miss Davies. Perfect timing, in fact. The Caledonian Hotel is urgently looking for someone experienced in Silver Service and \u2014 '\n\nIndicating Maggie with a nod of her head, Snooty Junior gave an emphatic cough, then leant down to whisper something into her boss's ear. Mr Wilson listened, then followed her gaze.\n\n'Ah,' he nodded a moment later, and this time his 'humph' was one of disapproval.\n\nBoth Snooty Junior and the Recording Angel were now looking very closely at her, closely and in silence.\n\nMr Wilson was first to speak. 'Ah, yes. Indeed. Quite right. To bring it to. My attention. Thank you, Miss Webster.'\n\nSnooty Junior inclined her head in acknowledgement, but said nothing. She remained standing at her boss's side.\n\n'In these circumstances, Miss Davies, I'm afraid there is no position available for you.' His unexpected rush of words concluded: 'Nor need you put yourself to the trouble of returning here . . . afterwards. Good Day.'\n\nHaving pronounced sentence, the Recording Angel withdrew into even greater shadow than before.\n\nMaggie was hardly aware of coming down the four flights and returning to the end-of-day bustle of Hanover Street. What had she been hoping for? Nearly seven months pregnant and unmarried, did she really expect someone to give her a job?\n\nThe downward slope of the pavement carried her on to Princes Street. Across the road stood the Royal Scottish Academy looking more than ever like a Greek temple that had been left for too long out in the Scottish rain. Over the years, layer upon layer of soot from the nearby trains and the city chimneys had drifted onto its pillars and walls, to turn into black mould. The grime was so ingrained that the stonework looked like it was being eaten away from the inside. The nearby Scott Monument looked just as dingy. If she herself stood in Princes Street long enough \u2013 and what other plans did she have? \u2013 would that black, tarry grit settle on her and turn her into a statue? A memorial to the Unmarried Mother, with her swollen belly for everyone to see?\n\nShe could imagine them gathered round her plinth \u2013 the Snooty Juniors, the Wilsons, the Norries, the Callanders, her parents and the rest of them \u2013 so many faces glaring up at her, despising her. The whole city and beyond come to show their contempt.\n\nWell, to hell with them! Let them all burn, as Jean said. She wouldn't even lean down and spit on them to put out the flames.\n\nPART TWO\n\nMAGGIE GAVE UP looking for work and spent the remaining weeks helping her sister-in-law as much as she was able, making the local deliveries, washing the baking bowls and pots, sweeping and mopping the floor every night, cleaning the oven at the weekends. The letters that came from Michael were the high points. They seemed to be from another time, another world \u2013 did they tell of a past she was in danger of forgetting or of a future that was still waiting for her? Last thing at night, lying on her chaise longue in the boxroom, she would read them over and over, trying to bring them into the present, trying to take their reassuring words and promises along with her into the night ahead.\n\nWith Jean's help she found a discreet nursing home off Minto Street in the Southside, and booked herself in to stay overnight when her time came. She planned to give birth there, away from the snubs and sneers of a public ward, and without the shame of an empty chair by her hospital bed when the proud husbands came to visit with flowers.\n\nBut then what?\n\nThe last of her Fusco savings would soon run out. Then what?\n\nOne thing was sure, cosy though the boxroom was, she couldn't live there for ever \u2013 not with a newborn baby.\n\nThe children's home was called Woodstock House. It was a large Victorian townhouse that stood like a turreted galleon moored in a sea of green lawn while around it lay a scattered archipelago of hope \u2013 a neatly laid-out kitchen garden, a line of small brightly painted sheds along the back wall next to a greenhouse. This country mansion lookalike had been built as a trumpet blast of one man's infatuation with himself and his commercial rapacity, but with the captain of Scottish industry now long gone, so too were the finances necessary for the building's upkeep. According to Jean, the children's home was a private institution that only survived thanks to donations, mostly anonymous, and a dedicated staff. There might be a church involved in it somewhere, but she wasn't sure. Or else it might be some kind of charity place, like those houses for fallen women. Not that Maggie was one of those, her sister-in-law had quickly added.\n\nThe brass bell-pull slid stiffly back into the wall. The clang-clang and its echo tolled out emptily. Somewhere a child shouted, 'Ding-ding! Ding-ding!'\n\nThere was the sound of light, skipping footsteps. The front door opened.\n\n'Hello!'\n\nThe girl was a teenager, if that \u2013 an upturned face that was mostly grin, keen eyes and a tangle of unbrushed blonde curls. She shifted from foot to foot to unheard dance music while shaking her head and clicking her fingers to keep time.\n\nMaggie hesitated. 'Hello . . . I've come to see . . .'\n\n'Yes? Plenty to see in here. Come in.' The young girl did a half-twirl and pointed towards a coarse mat. 'This here's for the rain. Mrs Saunders doesn't like rain, or mud. I'm going to be a chorus girl.'\n\nMaggie wiped her feet.\n\nThe apprentice chorus girl high-kicked, birled herself quickly round, then faced-to again: 'Do you want to see her?'\n\n'I phoned and \u2014 '\n\n'Mrs Saunders sees people when people come. I'm just Donna.' She took a step back, then kicked out her right leg in a chorus-line of one. 'I'll take you.'\n\nThe vestibule had a tiled floor.\n\n'Thank you, Donna.'\n\nThe girl carefully pulled the front door behind her. 'We keep it shut, for the heat.'\n\nWith a soft-shoe shuffle, the young dancer led the way into a large hall that smelled of cooking. Dim light came through a glass cupola above, the walls were a pale green and hung generously with dusty-looking portraits in heavy frames. Maggie could feel a chill coming up through the linoleum. The only furniture was a small but elaborately carved wooden chair that stood at the bottom of the staircase and looked like a make-believe throne waiting to be claimed by the pretend king of a make-believe little country. Parked next to it was a cumbersome, old-fashioned pram, dark green with large spoked wheels reaching high up its sides like a paddle-steamer.\n\nThe girl pointed to it. 'That's the Tractor.'\n\n'It's big, right enough.'\n\nAnother flight of stairs disappeared into darkness below, presumably down to the basement.\n\nDonna peered at her: 'Are you a mother?'\n\n'I'm going to be \u2013 very soon.'\n\nThe young girl came to an abrupt stop in the middle of the hall, her arms arched above her head in a ballet pose. Her face turned in profile, she held the position for several seconds. 'It's not easy.'\n\nDid she mean being a mother wasn't easy, or that this particular pose was a strain to hold? Could the girl be implying that she herself was a mother, this slip of a lass?\n\nMaggie gazed round the uncarpeted space. 'I suppose not.'\n\nThe soft-shoe shuffle was then resumed until, with a sudden and unexpectedly grown-up sway of her hips, Donna halted outside a closed door marked private. She curtseyed.\n\n'In here.'\n\nIn one smooth unbroken action she knocked, turned the handle and pushed open the door. This done, she went tap-dancing off down the corridor, clicking her fingers in time.\n\n'Enter.' A firm voice.\n\nThe room was slightly warmer, with a dark brown carpet and rust-red curtains that sagged like a pair of comfortably slack stockings. The woman sitting behind the desk glanced up \u2013 'Be with you shortly' \u2013 then continued reading. A cigarette burned in the ashtray beside her.\n\nMaggie was back at school once again, being made to wait in the headmistress's office where she'd been sent to get shouted at for not paying attention in class, or for coming in late, or for walking around in a daze. Though the actual details of the offence usually varied, in essence the charge was always the same \u2013 she was getting punished for being herself. This time, however, the punishment was for being herself and for getting herself pregnant. She stared down at the floor.\n\nThe superintendent laid the sheet of paper aside. 'Yes?'\n\n'I'm Maggie Davies. I phoned.' The unspoken school-girl 'Miss' slid to the floor where it was immediately absorbed into the carpet. Even the young sprite Donna had seemed older and more mature than she herself felt at this moment.\n\n'Davies?' Mrs Saunders began leafing through one of the stacks of papers on her desk. 'Davies? Davies? Davies? . . .' She riffled through another stack. 'You phoned recently, you say?' Then started on stack number three. 'Ah yes, here you are. I remember now. You're due in a month and going to the nursing home in Queens Crescent. A good place, well worth the expense.'\n\nMrs Saunders' smile of approval at once cancelled out the schoolroom-strictness, replacing it with a feeling of warmth and unexpected kindness. Even the rain hitting the window seemed to ease off slightly.\n\n'Thank you, Mrs Saunders. I wanted to do the best I could.'\n\n'There's no current address given here. You're staying with your family, perhaps, or \u2014 ?' Mrs Saunders looked closely at her. 'You do have somewhere to live, don't you?'\n\n'Yes, I'm staying with a friend at the moment. I'll be starting a new job shortly, then getting my own place. Permanent. Somewhere near here, so that I can \u2014 '\n\n'That's fine, Miss Davies, thank you. So long as I have an address for my records.'\n\nMaggie nodded to show her willingness.\n\nMrs Saunders continued, 'You know the rules and conditions?'\n\n'Rules? Oh yes, I knew there'd be rules.'\n\nThe superintendent took a puff at her cigarette, blew out the smoke and asked her to sit down.\n\nWhile the rules and conditions were gone through, Maggie did her best to concentrate on what was being said and not let her mind drift to the ever-changing patterns the rain made as it streamed down the window. She wondered where Donna had sashayed herself off to . . . The cheerful yells she could hear, were they coming from a children's playroom somewhere nearby? Had the junior chorus girl actually been born here? She seemed almost like a ghost-child haunting the empty hall, the spirit of all the young lives who \u2013\n\n'. . . then sign here at the bottom,' Mrs Saunders was saying, 'where it's marked with a cross.'\n\nMaggie took the sheet of paper that had been pushed across the desk to her. She glanced down the form:\n\nMOTHER'S NAME \u2014\n\nMOTHER'S ADDRESS \u2014\n\n'Once the child is in our care, he will be well looked after. He will be our responsibility day and night. He will receive good food and all the comfort and concern one could wish for. He will be happy.'\n\nMOTHER'S OCCUPATION \u2014\n\nFATHER'S NAME (if known) \u2014\n\nFATHER'S OCCUPATION (if known) \u2014\n\n'You can leave the name of the child blank for the moment. Just sign.'\n\nCHILD'S NAME \u2014\n\nCHILD'S DATE OF BIRTH \u2014\n\nCHILD'S PLACE OF BIRTH \u2014\n\n'Once the child is in our care, as I say, you needn't give him a second thought \u2013 you can forget him. In fact, it's better that you do. In my experience, things always work out much better when mothers don't see the children at all. Only makes things harder. The more you visit, the more he'll become part of your life and you of his, and the more painful will be the final parting. Unbearably painful \u2013 for both of you.' Mrs Saunders allowed herself another deep drag of her cigarette and stared into the smoke wreathing between them.\n\n'I have to stress, Miss Davies, that when it comes, the parting will be final. You will not be given the address of your child's new home, nor the name of his new parents. Over the years I've learned to encourage new parents to let the child believe that they are his true parents. It's kinder that way, kinder for everyone.' Another puff, as if taking a bow.\n\n'What's this at the bottom about 'a limit of six calendar months?'\n\n'That? A formality. Of course, the earlier he's adopted the easier he's adopted, if you understand me.' The superintendent smiled. 'Rest assured, the adoptive parents' love and affection for their new child will follow in good time. It always does.' A second smile. 'I can have him placed within days, then he'll be free to get on with his new life and you can get on with yours. Best for everyone.' Smile number three.\n\n'But he won't be adopted just like that, will he? I'm not wanting him to be \u2014 '\n\n'No, of course, not, Miss Davies. The child's best interests always come first. He's our prime concern at all times. But you can rest assured that all new parents are carefully vetted. We make sure they are respectable people, church-going and financially secure. Home-owners. Pillars of the community.' With every quality listed, the pointed look in the superintendent's eyes emphasised her real meaning: We make sure they're everything the likes of you could never be.\n\n'And did I mention \u2013' Mrs Saunders gave a slight cough '\u2013 that the new parents frequently want to show their appreciation to the mother? You understand what I mean? Not that they will ever meet you, of course \u2013 that's naturally quite out of the question \u2013 but I will forward on to you any token of their appreciation. The amount can be quite considerable sometimes . . .'\n\nMaggie had reached the cross marking where she was to sign. Her child would be well looked after, it seemed. For the first six months any adoption needed her approval, which she naturally wouldn't give. 'And after the six months?'\n\n'Well, Miss Davies, let's cross that bridge when we come to it, shall we?' Having taken a final drag at her cigarette, the superintendent ground it out in the ashtray. 'Lots can happen before then, can't it?'\n\nIt certainly could \u2013 the moment she'd started her new job and found somewhere more suitable to stay, she'd be taking her baby back.\n\n'All in good time, Miss Davies.'\n\nHaving filled in the form, Maggie signed her name.\n\nWhen she looked up, Mrs Saunders was giving her a warm smile.\n\n'Thank you, Miss Davies, and speaking on behalf of everyone here at Woodstock House, let me say how much we all look forward to welcoming your newborn child and to caring for him. We have the address in Queen's Crescent and will see to all the arrangements.' She stood up. 'It's been a pleasure to meet you, Miss Davies.'\n\n'I'd like to see his room, please.'\n\n'His room?' Pen in hand, Mrs Saunders was already reaching towards the stack of papers. 'Where his cot'll be, you mean? I'm afraid that's not possible right now. Disrupts routine.' There was no smile this time. 'All in due course, Miss Davies. All in due course.'\n\nMaggie got back home to find Jean had a present for her.\n\n'The answer tae yer prayers,' the older woman explained.\n\nSitting on Jean's baking table, the shiny black typewriter fairly bristled with keys, knobs and levers.\n\nMaggie's obvious objection: 'But, Jean, I can't type.'\n\n'There's a book comes wi it, tellin ye whit tae dae an see ye stairted. The book maks it look easy enough.'\n\n'Books always do.'\n\n'Well, seems it's maistly practice. When the time's richt, ye'll get yersel a job, a guid job. No as a skivvie waitress or stuck in a pub fou o gropin auld drunks. Folks in offices are aye needin typists \u2013 aa thae tycoon businessmen, bankers an lawyers, you name it.' Her sister-in-law gave her a gentle punch in the arm. 'Play yer cards richt, Maggie, an a smairt-lookin lass like you micht land hersel mair than just a job!'\n\nSUNDAY\n\nTIME TO BRACE yourself on the Rosehaven doorstep, press the bell. Stand on the yellow cross marked on the doormat, show yourself to the CCTV angled above. Speak your name into the security grille: Tom Stewart.\n\nBuzzed into the overheated hall, into the combined smells of floral air-freshener, yesterday's macaroni cheese, urine, today's stew and vegetables, laundry, disinfectant.\n\nNot managing more than the first few steps along the corridor leading to the dayroom before it hits you \u2013 you're about to throw up there and then.\n\nReaching the visitors' toilet just in time.\n\nForget about what your ex-wife said about your having no feelings. What does she know? Too many feelings, more like \u2013 and always too keen to share them. You just keep trying and hoping. And getting hurt. The lovely Janice . . . and now Mandy. A steady girl, a caring girl. Maybe that's what you need?\n\nFeelings? Clearly you're deeply, deeply distressed about your mother's deteriorating condition. Visiting her at the cottage every Sunday without fail and now coming here to her care home is the hardest thing you've ever done . . .\n\nHours it seems like, leaning over the wash hand basin . . . one dry heave after another. Retching and retching. There's a slick of cold sweat on your cheeks. Your hands shake as they grip onto the porcelain rim. Your stomach's churning, but nothing comes. Never does. Hard work even to spit. You keep trying.\n\nThen stop.\n\nBecause, quite abruptly, you're feeling fine again. Back in top form. A1 and then some. Yes, you've got the magic touch, all right!\n\nNow, bin the paper towel \u2013 your face and your feelings at default setting once more, you're ready to go through and greet your mother. Bringing her a smile!\n\n'Hello, Mum. How are you today?' You sit down in the empty seat next to hers. A few minutes' chat to get things started, and then you'll suggest she zimmer herself through to her room \u2013 no old biddies there, talking to themselves, crying and the rest of it. There you can almost pretend everything's normal. At least you've got her into a good place \u2013 costs megabucks, but hey, she's your mother. It was the best of the homes you checked out, the very best, and every time you sit with her in the dayroom you do what you can to shut out the worst of it \u2013 the little accident that's not been mopped up, the spilled food, the helplessness, the calls going neglected, the residents' feeding, bathing and bedtime arranged for the convenience of the staff. Most of all, the total dependency. The locked doors. The closed windows. The smell.\n\nShe's calling you Michael again today. Better to ignore it and tell her instead about your drive down from Edinburgh, the weather, the traffic, why you prefer the Moffat road via the Beef Tub to the multi-lane racetrack of the M74, show her your new iPad, tell her how you didn't manage to leave Edinburgh till lunchtime as you'd had a late gig last night. Because she keeps nodding and smiling at the right bits, you hope she's following everything. Best to say nothing about getting the cottage ready for sale, of course, no sense in upsetting her. Instead, you describe a new trick you're working on and explain that nobody wants rabbits out of hats these days, not unless they're virtual rabbits, virtual hats and performed by a virtual magician! You're working on it, you joke. She doesn't always follow what you're saying, but you keep talking to keep things moving forward.\n\nThe drugs trolley's rattled up to her seat. It's Kylie on duty today, the small overweight woman, the one your mother calls Boss Beryl and who looks like a binbag that's not been fastened properly at the neck. Must be the Polish girl's day off. A real pity as she certainly brightens the place up.\n\n'Time for your meds, Mrs Stewart.'\n\n'I'm called Maggie, I keep telling you. Maggie, maggie davies.'\n\n'OK, Mum, okay. Don't get upset. It's all right.'\n\nPatches of red stand out on her cheeks. Clutching your arm so tightly you feel each separate bone in her fingers. 'Tell her. Tell Beryl there's no Mrs Stewart and there never was.'\n\nWhile lifting a small plastic cup of water from the trolley, you give Kylie a smile that's half-apology and half-embarrassment.\n\n'Here you are, Mum. Another green one, another sip, and you're done.'\n\nEmpty cup replaced, you thank the woman. She and her trolley move off to the next chair.\n\nWhen you showed the Polish girl \u2013 Mariella? Marietta? \u2013 a couple of simple tricks recently, making a pound coin appear out of her ear and then turning it into a shower of petals from her closed hand, she gave you a very big smile. Meaning that she liked how you'd touched her hair and her ear, and enjoyed the older-man confidence with which you'd opened out her cool fingers, one by one. She'd been impressed. She's mid-twenties at most. You could be her father? Her grandfather, more like! But so what? You can appreciate her, can't you? Those blonde curls gathered into a bunch at the back of her head, the loose strands framing her high cheekbones, that not-so-innocent glint in her blue eyes.\n\nFor several moments you and your mother sit in silence.\n\n'Would you like some tea, Mum? I can ask them to \u2014 '\n\n'It was cake. Jean baked a cake.'\n\nThat bloody cake again. If you've heard about it once, you've heard it a hundred times. 'Yes, Mum. Auntie Jean had a cake shop \u2013 in Haymarket you said, wasn't it?'\n\n'This cake wasn't for sale.'\n\n'No? Just for eating?'\n\n'Eating? Jean wasn't going to eat it, me neither.'\n\nThe red floods back into her cheeks, warm-red this time. Genuine pleasure. And she's grinning: 'What a cake it was! Three layers. A sponge with cream and chocolate and marzipan, slathered all over with icing and dusted with hundreds-and-thousands. Irresistible.'\n\n'Pity you never took a photo of it, eh, Mum! We could have put it into that album I \u2014 '\n\nHer sudden anger. 'For the last time: I don't want to see any photos. I don't want to see any people. And I don't want to see you, whoever you are. Coming here, asking questions. I don't want people coming, people not coming. I don't want questions . . .' The red in her cheeks has become like a burn mark. She struggles to raise herself out of her chair. 'I don't want . . .'\n\nA moment later she has calmed down and is perfectly still once more. Completely composed.\n\n'Nice of you to visit. What did you say your name was again?'\n\n'Tell me about the cake again, Mum.'\n\n'What cake?'\n\n'The one you said that Auntie Jean made.'\n\n'Jean made lots of cakes.'\n\n'All chocolate and marzipan, you told me, and \u2014 '\n\n'I don't want to talk about it.'\n\n'Who was it for?'\n\n'For Boss Beryl and the others, and Donna, too, of course. Who else?'\n\n'You mean the people who're looking after you here? I don't understand, Mum. Auntie Jean died . . . years ago. How could she have known the people here?'\n\n'Forget the cake. There was no cake. I must have dreamt it. All of it. I'm tired of your questions. I'm tired of you. Who do you think you are? Coming here and upsetting me \u2013 you're not Michael, you're not anyone. You make me feel like I've been doing the laundry all day, like you're squeezing and squeezing me to get the last of the \u2014 '\n\n'But Mum \u2014 '\n\n'Tom would have helped me. Sometimes I'd pretend he was standing there beside me on the three-legged stool, turning the handle while I fed the wet clothes through. We'd have had great fun together! The windows steamed up with the condensation so there was nothing outside, nothing in the whole world but the two of us . . . They'll be bringing tea soon. No cake today unless you brought some.'\n\n'I had a lovely slice last time I was \u2014 '\n\n'You had a slice? Not Jean's cake you didn't. She certainly wouldn't have given you a slice.'\n\n'But, Mum \u2014 '\n\n'She wasn't giving that cake to just anybody. And you \u2013 you're no one. Get out. get out!'\n\nBoss Beryl's abrupt tug-and-swish of the curtains.\n\n'Been looking through the family snaps, have we?' She's picked up the photo album that's still sitting on the chest of drawers. 'Mind if I \u2013?' Without waiting for a by-your-leave, she starts flicking through it.\n\nYou want to snatch it away. Whoever's pictures they are, they're not Boss Beryl's, that's for sure. You don't want that woman's hands all over them, nobody does. Her sweaty pawprints and snide remarks.\n\n'Someone's secretary, were you, Maggie? Very smart- looking. Weren't you the heartbreaker!'\n\nOnly my own heart. Keeping its jagged shards clutched to you for most of your life, keeping them for the touch of Michael's fingertips to melt away the pain.\n\n4\n\nAFTER THE EARLY morning dash in the taxi across the \u00adwintry-dark city, across the Meadows and down a completely \u00addeserted Minto Street to Queen's Crescent with Jean's hand in hers to grip hold of at each contraction, there came five hours of pain and then exhaustion, followed by more pain and more \u00adexhaustion, and people telling her to push-push-push.\n\nAfterwards, as she lay awash with sweat and still trying to catch her breath, she was told to look up for a moment. But only if she wanted to:\n\nThe tiniest mouth and nose, damp feather-light fair, hazy blue eyes. A boy.\n\nHeld up for only the briefest moment, held too far away for her to touch \u2013\n\nHardly the chance to catch a glimpse of each other \u2013\n\nThen whisked away out of sight. Through to another room to get cleaned and swaddled up.\n\nWoodstock House has been phoned, they said to her, and someone's on their way.\n\nMaggie knew this was going to happen, didn't she? Her son going straight to the children's home? She herself must have arranged it, they reminded her. With Mrs Saunders at Woodstock House. She and Mrs Saunders would've discussed all the details between them.\n\nGetting upset like this would only make things worse, they told her. Always best to be separated as soon as possible. It was easier that way. Easier for everyone.\n\nOf course he'd be well taken care of, they reassured her. Woodstock House had a good reputation. She needn't worry. She could be really proud of herself and it was time to let others take over now. She'd done all the hard work. He looked a bonnie wee baby and was going to be fine. Everything was going to be fine. She needed a proper rest now. Needed to get her strength back.\n\nMaggie struggled to sit up in bed, begging and begging to be allowed to see him one more time, to hold him just once before \u2013\n\nAll in good time, they said. All in good time.\n\nLet's open these curtains a little wider so you can see out.\n\nLet's straighten these covers and plump up these pillows. Let's get you comfortable.\n\nA cup of tea? \u2013 with plenty of sugar, if you fancy it.\n\nGetting upset like this wasn't helping anyone, least of all her, they said. Once she calmed down they'd leave her to have a good sleep. She'd feel better after that, they said. A good sleep. A good sleep. A good sleep. They could give her something if she wanted.\n\nTotally worn out, Maggie fell back onto the freshly arranged pillows. She turned her face to the window, away from their firm hands, away from their repertoire of comforting words and kindness. Better to stare out at the November afternoon, better to follow the tracks of the raindrops streaming down the glass . . .\n\nHer suitcase, a handful of painkillers, good luck and goodbye.\n\nJean had come to collect her and the taxi was waiting in the street.\n\nMaggie stood looking down the flight of stone steps, the same six steps she'd toiled up the day before, more like six hundred now, and pitiless every one of them. There was a handrail, at least. She held onto it, quite unable to start her descent.\n\nTaxi or not, she was in no rush.\n\nA cold easterly pulled at her knotted headscarf and made the loose ends flap against her cheeks; she could see the clouds being hurried on, driven forward, scattered across the ragged winter sky. The wind tugged and tugged at a leafless silver birch that stood so close to a wall its thinnest branches scraped to rawness against the stone. On the rooftop directly above where she stood, gust after gust set the grannies whirling in their chimney pots. They screeched at her to get herself down the steps and out of their sight. Now she'd given birth she no longer belonged in this mountain refuge of soft pillows and round-the-clock care. Down the stairs with you, they shrieked. Get back to where you came from.\n\nShe caught hold of her loosening scarf. Jean took her arm.\n\n'Yer gey peelie wallie lookin, Maggie. Shouldae stayed a bit langer, no?'\n\n'At \u00a39 a day?'\n\n'Come on then, let's get ye back hame.' Her sister-in-law guided her to the top step. 'Taxi's my treat.'\n\n'Jean, I \u2014 ' She stared down into the everyday lowlands far below \u2013 the Edinburgh streets, tenements, her couch in the back room of the bakery, the job she'd need to find, the room she'd need to rent, the visits to see Tom at Woodstock House. Would he recognise her? Would he come to love her?\n\n'Come on, Maggie, let's get you back home.'\n\n'My suitcase?'\n\n'I've got it. Ready? We'll take it easy. One step at a time.'\n\nHow shaky Maggie felt to be up and on her feet again. Her left hand clutching at empty air for better balance until she'd found the rail. Her right gripping Jean's arm, together they began their descent.\n\nTwo steps, three, four, five . . .\n\nThe driver stayed in his cab with his engine running for the heater. It was warm inside, thank goodness. Maggie slumped down in the corner. Jean stood the suitcase at her feet.\n\n'Are you all right, Maggie?'\n\n'Thanks, Jean. Taxi's a real kindness. A tram would have been beyond me.'\n\n'A tram! Away wi ye!' Jean gave the address and they drove off.\n\n'I was wanting to go and see Tom today, but \u2014 '\n\n'Tom?'\n\n'Hardly saw him for more than a few moments, but long enough to know he's called Tom. He knows it too, I could tell.'\n\n'A guid name. There's nae Tom in the family as far as I ken. He'll be stairting aff wi a clean slate.'\n\nMaggie leaned forward and turned to face her: 'I'll go and visit him first thing tomorrow.'\n\n'We'll see. Ye'll want tae get some colour back into yer cheeks first, or he'll be thinking his mither's a ghost.'\n\nA ghost? Maggie bit her lip, and sank back into her seat.\n\nThe door of Woodstock House was opened by the same tangle of blonde curls and smiles as before.\n\n'Hello.' The girl looked Maggie full in the face. 'You're Miss Davies, aren't you?'\n\n'That's right, Donna. Hello to you, too. Can I come in, please?'\n\n'They never said, so I don't know. Well . . . (a theatrical sigh), I suppose you're here now. Mrs Saunders will have heard the bell anyway.' She stood aside to let Maggie enter. 'No need for the mat today, the wind's blown away all the wet.' She pushed the door closed behind them.\n\n'How's your dancing coming on?'\n\n'Blisters. Thank you for asking. Some advice \u2013 never try doing the can-can barefoot. Too many splinters.'\n\n'Thank you. I'll remember.'\n\nThey crossed the hall. Though it was only a month since Maggie had been here, the linoleum seemed to have hardened to a sheet of permafrost and the varnished staircase become encrusted with ice. It was so cold she could see her breath.\n\n'Heat's kept for the children's rooms. No one hangs about in the hall, so why heat it? That's what Mrs Saunders says. Sometimes I do my dancing here because it's a good big space \u2013 but only if I've warmed up in the kitchen first, Mrs Saunders says, or else my muscles'll break in the cold or maybe even my arms and legs. He's a lovely wee boy.'\n\n'Yes. He's called Tom.'\n\n'Tom?' They had reached Mrs Saunders' room. 'A nice name. If you want, I'll look out for him.'\n\n'How do you mean?'\n\n'But only if you want me to.'\n\nDonna knocked and opened the door for her. 'Really suits the wee lad. Tom. Cheerio.' She shimmied off up the corridor.\n\n'Cheerio, Donna,' Maggie called after her.\n\nThe superintendent was again seated behind her desk, her cigarette sending up a thin line of smoke from the ashtray at her elbow. She was clearly involved in very important work and made no effort to look up when her visitor approached. Her fountain pen continued to scratch line after line onto a sheet of headed foolscap.\n\nHands by her side Maggie stood in front of the desk, and waited.\n\nTwo further lines were completed.\n\nShe coughed. 'Good afternoon, Mrs Saunders.'\n\nThe superintendent didn't look up. 'One moment, please.'\n\nWhere the surface of the desk wasn't covered in papers, there were ink stains, cup ring marks. The right-hand edge was scarred by a line of cigarette burns.\n\nMaggie was about to speak again, but stopped when the older woman put down her pen and reached for a sheet of blotting paper. Finally, the completed foolscap page was placed on top of a nearby pile. The superintendent laid her palms flat on the desk:\n\n'Yes?'\n\n'I've come to see my little boy. He was born yesterday, and \u2014 '\n\n'Name?'\n\nMaggie took a step forward, right to the front of the desk. She smiled. 'I'm going to call him \u2014 '\n\n'Your name?'\n\n'Me? Maggie Davies. I came a month ago to arrange for \u2014 '\n\n'Miss Davies? I wasn't told you had an appointment today.'\n\n'I'm sorry, I didn't realise I had to \u2014 '\n\n'Perhaps you'd like to sit down, Miss Davies? You must still be tired.'\n\nThe chair she'd sat in on her previous visit was standing against the wall beside a large green metal filing cabinet. Maggie dragged it over to the desk.\n\nMrs Saunders took a moment to finish her cigarette, stubbing it out in the ashtray.\n\n'Well, Miss Davies. How did it go?'\n\n'How did \u2013?'\n\n'The birth, Miss Davies. The birth of your child. Everything went satisfactorily, I believe. Smooth delivery and no forceps \u2013 yes?'\n\n'They said everything was fine.' Then she added, 'Afterwards I felt very \u2014 '\n\n'The baby certainly looks healthy enough. No harelip, webbed toes or fingers, thank goodness.'\n\n'I saw him for only a few seconds, he seemed . . . perfect. He looked lovely. I wanted to hold him, but they \u2014 '\n\n'Yes, best all round. No sense in causing unnecessary distress. Can't begin too early to get the child used to being without his \u2014 '\n\n'He's called Tom.'\n\n'Pardon?'\n\n'My son is called Tom.'\n\n'Is he? I hadn't realised.'\n\n'I want to see him.'\n\n'I understand your concern, Miss Davies. Only natural, and many would consider it does you credit. But I must repeat what I said when you were here before, and urge you to do no such thing.'\n\n'But he's my son and I \u2014 '\n\n'I advised you to leave him with us, if you remember \u2013 leave him here and forget all about him.'\n\n'But \u2014 '\n\n'You do understand?''\n\nMaggie said nothing.\n\n'Once more, Miss Davies, I strongly urge you to turn around and walk straight out the door. Now. This very minute. Out the door and don't look back.'\n\n'But Tom's all I have, and \u2014 '\n\n'Like I said, it's time to begin your new life, and let your new-born child begin his.' The superintendent half-rose from her seat as if preparing to show Maggie out of the room. 'The sooner he can be put up for adoption, the sooner he can be \u2014 '\n\n'I want to see Tom.'\n\nMrs Saunders sat down again. 'Adoption. This really is the best time. Everything can be arranged with the minimum of fuss and concluded in a matter of days. Like I say, best for everyone. Best for you and best for little . . . What did you say his name was?'\n\nMaggie made no response.\n\n'Best for your son.' The superintendent paused. 'Miss Davies?'\n\nMaggie sat and said nothing.\n\n'You're being extremely selfish, you know.'\n\nMaggie gripped the sides of her chair. She sat up straight, met the older woman's gaze and held it. 'I'm his mother.' She could feel the beginnings of tears behind her eyes.\n\nThe superintendent leant forward. 'I agree, Miss Davies, that you're his . . . mother.' Paused for, but unspoken, the word unmarried hung in the air between them.\n\n'Yes, Mrs Saunders, I am his mother . . .' Then, without waiting for the older woman's invitation, Maggie stood up. '. . . And, as I have already told you, my son is called Tom. I've come to see him. Now where is he?'\n\nMrs Saunders shook her head.\n\n'Miss Davies, you must understand that I don't mean to be hard. I know it may seem like that. In my years here I've seen so many children . . . and so much unhappiness.'\n\n'If you don't take me to see Tom this instant, I will leave and return with the police.'\n\nThe superintendent snorted: 'The police? You? An unmarried mother of no fixed abode? Do you really think they'd pay any heed to the likes of you?'\n\n'I only want to see my wee boy. That's not a crime.'\n\n'Your child is now my responsibility, Miss Davies \u2013 which means that I decide who sees him and who doesn't.'\n\n'I'm not going to harm him. I love him, and \u2014 '\n\n'Love him? You've hardly even seen him. You don't know anything about him. You wouldn't even recognise \u2014 '\n\nMaggie jumped to her feet. 'i love him! Can't you understand? He's my son, I'm all he's got.'\n\n'Not any more.'\n\n'Right, the police it is!' She started towards the door.\n\nThe superintendent stood up.\n\n'We don't want any trouble, Miss Davies. We don't want the children upset, do we?' For a moment she seemed to have finished speaking, but then added, 'You mothers can only do what you must, I suppose.' She shook her head. 'If you get to see him this time, you must promise never to come back here again? Will you?'\n\nMaggie didn't reply. She stood and she waited.\n\nFinally Mrs Saunders gave a sigh, crossed to the door and opened it. 'Come on, then.'\n\nA couple of small boys, aged about four and five, were down on their knees playing in the main hall with their Dinky cars, racing them up and down the floor and crashing them into the skirting. The smaller one had a harelip.\n\n'It's a bit cold here, boys, why don't you go through to the playroom?'\n\n'Yes, Mrs Saunders.' Dutifully, they got to their feet and trooped off.\n\n'They're brothers, these two.' The superintendent began to go up the stairs. 'Refused to be split up. Each time Andy \u2013 he's the older one, the one without the . . . (she touched her top lip) \u2013 was taken by a family, he acted up so badly he was always sent back. No one ever offered to take Bobby, of course, let alone both of them together. A real shame. Little Bobby's a delightful child really. They both are.'\n\n'And Donna?'\n\n'Her mother died when she was nine. Father said he was a working man and couldn't cope. No relatives, it seemed. Brought her here to see if she'd like a day visit. Then never came back. Turned out to be a false name and address. No paperwork because we thought it was just for a few hours. Certainly never made that mistake again. This is Tom's room.'\n\nA dozen or so mismatched cribs and cots were stood side by side along the walls. The centre of the room was taken up by a table covered with several feeding bottles, a stack of clean nappies, a stack of clean towels. There was a desk light and what looked like a diary or some kind of record book was laid open next to a half-filled cup of tea. An armchair and small foot stool relaxed in one corner while a deep sink with draining board occupied another. The room felt overheated and probably neither of the two large windows had been opened today. The smell of soiled nappies, bedding and powdered milk caught in her throat.\n\nA woman wearing a dark blue housecoat stood over by an open cupboard, checking through shelves of linen.\n\nEach cot had a handwritten number affixed to its head rail.\n\nMrs Saunders called over to the attendant: 'Beryl. Queen's Crescent \u2013 came in yesterday afternoon?'\n\nAged anything between thirty and fifty, Boss Beryl was small and stocky, her dark hair set in stiff curls. Standing with one hand resting on her hip and staring flatly across the room, she reminded Maggie of a rather squat and angry-looking petrol pump. 'Queen's Crescent?' There was a swift jerk of her head in the direction of the far corner. 'Number 11, and he's sleeping.'\n\n'Miss Davies won't wake him. She's just looking in for a short visit.' Then the superintendent left the room.\n\nWithout taking her eyes off this unwanted intrusion, the petrol pump lifted down a green blanket from one of the shelves and made as if to re-fold it.\n\nMaggie picked her way across the room between discarded towels and pillow-cases heaped here and there on the floor, then skirted round the central table.\n\nA baby started howling. Crib number 11?\n\nNo. It came from over by one of the windows, from some other cot, some other baby. She ignored it.\n\nBut then, as she hurried across to Tom's crib, she felt the unknown child's cry pierce her, felt it like a wound in the tip of her breast. Glancing down she saw a small damp patch on the front of her blouse.\n\nSide-stepping a slew of wet-looking sheets, she almost knocked over a pail of water with soiled nappies dripping over the side.\n\n'Watch where you're going!' shouted the petrol pump.\n\nThe pail stood next to an empty cot, the pillow almost small enough for a doll's bed. Crib number 10.\n\nTom's would be next.\n\nCrib number 11: a halo of feather-light fair hair, scrunched-up face, wrinkled skin, patches of reddish pink, impossibly small hands . . .\n\nShe reached into the crib. More than her uncertainty and her awkwardness was her joy. Overwhelming, heart-swelling joy.\n\nNervously, her fingers brushed the warm smoothness of his cheek.\n\nBoss Beryl? Mrs Saunders and her lecture about selfishness? \u2013 she didn't care about them, but was nervous suddenly, afraid almost. Afraid she was about to burst into tears. With Boss Beryl scrutinising her every movement, starting to cry would be the worst thing she could do. She daren't show any weakness, not here. If she picked Tom up too quickly, she'd be accused of trying to usurp Boss Beryl's authority. If she hesitated for too long, she'd be judged uncaring, confirming that she should be written off as an unfit mother. Which, being unmarried, she already was in everyone's eyes.\n\nShe straightened Tom's bright patchwork blanket, stroked his uncovered arm.\n\nPetrol pump Beryl took a step forward. One single step \u2013 and it was as if she'd been turned into a wild cat ready to attack.\n\nBut not even a dozen wild cats could have stopped Maggie now. In one smooth act of loving reclamation she reached down, lifted Tom out of his cot and took him into her arms.\n\n'Tom, Tom,' she swayed him back and forth.\n\n'Leave him where he is.' The wild cat was only inches away, hissing and spitting. 'You'll have him yelling the place down. Give him here.'\n\nMaggie ignored her. Moments later she was holding Tom to her breast and, for as long as he remained there with his small mouth clutching onto her and sucking, nothing else in the world seemed to matter.\n\nWhen he'd finished, she lifted him up close to the window so he could see the rain-streaked glass, the separate water-drops racing down the pane. 'Tom, Tom, hush-a-bye, Tom. Look!' she pointed to a clear drop that trembled, poised, holding itself together until it was ready to start its journey. 'Look \u2013 that's you!'\n\nShe brought him up close to her face to feel the warmth of his stubby little fingers against her cheeks and lips.\n\nAfter watching the curtain of rain break into a swirl of colours where the glass was flawed \u2013 'That's your very own rainbow, Tom!' \u2013 she whisked him past the petrol punp and out of the room, along the landing, down the stairs and across the empty hall, into the crowded playroom to show him to everyone, and to see the wind-up train set, the big toy-box, the stacks of wooden bricks. Then, his introduction into society complete, she brought him back upstairs again.\n\nThe instant his head touched the pillow, his tiny face scrunched up into a scream of such ferocity it hardly seemed possible to have come out of such a tiny mouth. His small body shook with tears. Howling, howling tears. Crib number 12 woke up, setting up a domino-effect of howls and shrieks.\n\nBoss Beryl was furious. A feral hiss: 'See what you've done, see what you've done . . .'\n\nMaggie had been pretty overwhelmed by the typewriter Jean had given her a month previously. It looked like a shrunk-down church organ, and was about as appealing. At first she'd ignored it. It might be the road to her salvation as Jean had said, it certainly looked unwelcoming enough. Like an accompanying bible, the instruction book was short on laughs and promised only duty and hard labour. Two days had passed before she'd finally put in a sheet of paper and turned the roller as directed in Chapter One: Getting Started. Then she'd taken the sheet out again. Smoothed it flat, re-inserted it straighter and tried a second time. She hit a key. Then a second key. A third. A fourth. She'd looked up at the paper: mivc. She'd tried again \u2013 michaelmagie took five attempts. michaelmaggiejean took nearly five hundred, it seemed like. That achieved, she'd started to work her away through the manual, exercise after exercise, till her fingers were sore.\n\n'Got a tune oot o it yet?' Jean had asked as she left the bakery with Maggie still battering away at the keys.\n\n'Getting there.'\n\nA few days later, she'd tried typewriting a letter to Michael. It took nearly the whole evening and used up a lot of paper. Her next had been handwritten. By the time she was ready to give birth, the road to salvation had brought her up to a hit-and-miss ten words a minute. Fewer and fewer mistakes, and each one swiftly erased.\n\nWith Tom safe and secure at Woodstock House for the time being, she now had to find a job \u2013 and as soon as possible\n\nIt was nine o'clock on the following Monday morning, the beginning of a new working week \u2013 and for Maggie, the beginning of her new working life. She hoped. She'd have to lie to them, of course, and not only about her typing skills. no unmarried mothers wanted was surely written in mile-high letters above the centre of Edinburgh.\n\nAfter a last adjustment to her collar, she inspected herself in the mirror \u2013 the ivory silk blouse, black pleated skirt, patent leather shoes. Shiny black handbag.\n\nFrom Jean: 'You're just the dab. Perfect.'\n\nThe mirror showed: respectable, trustworthy \u2013 a thirty-one-year-old woman who'd soon be turning thirty-two.\n\nBut a mother? \u2013 did she still look like an unmarried mother?\n\nDrawing her comb through her hair one last time. Pursing her red lipsticked lips and allowing herself another glance in the glass. No motherly looks, please. Think single. Think confidence. Think ten words a minute. Think pay-packets.\n\n'My seams straight?'\n\n'Aa the wey up an aa the wey doun. Keep smiling Maggie, like our army boys \u2013 chest oot and stomach in. Your coat, ma'am. And aye mind yer Rabbie Burns \u2013 A man's a man for aa that! A man's a meal ticket \u2013 plenty women marry fer it. Course, ye're welcome tae bide here as lang as ye want, Maggie, but \u2014 '\n\n'Thanks, but come my first pay packet, I'll be moving into a place of my own. Then I'll take back Tom and \u2014 '\n\n'\u2014 and then what? I'm telling you, yer only chance is to get yersel a man.'\n\n'Jesus!'\n\n'Or someone like him!'\n\nWhich made them both laugh.\n\nRaw winter light streaked the length of Dalry Road, cutting round the outlines of the tenements' edges and corners, trimming the roofs and chimneys exactly to size. The pavements seemed to have been polished overnight and now had a sheen as smooth as the Union Canal. As Maggie turned down Dalry Road, she felt she was walking on water . . .\n\nBut she wouldn't be walking all the way into town on foot, not today. Not if she wanted to remain looking neat, crisp and employable.\n\nThe tram rumbled down to the foot of Dalry Road, went clatter-clack, clatter-clack over the intersecting tramlines at Haymarket, then trundled along the sunlit valley of Atholl Crescent before entering Shandwick Place and the West End. Past the \u00adCaledonian station and hotel with its uniformed doorman on the steps and the taxis lined up in front, and then into Princes Street.\n\nThe thirty-one-year-old, highly experienced typist got off at the bottom of Frederick Street and strolled over to the nearest plate-glass window. Against a background blur of office staff and shop assistants hurrying to reach their morning's work on time, she checked off her credentials one by one: the serious glance, the spontaneous smile, the tilt of her head. The poise, the confidence. Professional. Reliable. Dependable. Single. Childless.\n\nBy mid-afternoon Maggie had had it with slogging in and out of offices and agencies, up and down narrow staircases, she'd had it with Snooty Juniors, their Reception voices and their forms. Coming out of an office in Castle Street she walked down to Princes Street and was in time to see a tram marked Morningside coming along from Waverley. Good enough. Next thing, she was on board and soon turning up Lothian Road towards Tollcross, Bruntsfield and Carluke Avenue.\n\nIgnoring the brass bell-pull, she let herself in through the unlocked front door of Woodstock House, slipped across the hall and went up the stairs without even a glance in the direction of the superintendent's office. Then along the top landing and into the dormitory, making straight for cot number 11. In one smooth sweeping gesture, she leant down and gathered Tom into her arms.\n\nMrs Saunders appeared in the doorway a few minutes later: 'You promised you wouldn't come back, Miss Davies.'\n\n'I promised nothing.'\n\n'She was here Saturday as well, and Sunday,' spoke up Beryl the snitch.\n\nMaggie could once again feel the pain piercing her breasts and the warm wetness of milk leaking from her. With her free hand she unbuttoned her blouse.\n\nThe superintendent took a step towards her. 'Stop! I won't allow \u2014 '\n\nMaggie held Tom in the crook of her arm, ready to suck.\n\n'Are you going to rip my child out of my arms?'\n\nThe two women glared at each other while the wild cat looked on. Paying no attention to anyone else, Tom fastened onto Maggie's nipple and began to suckle.\n\n'I'm warning you, Miss Davies.'\n\nWithout breaking eye-contact, Maggie shifted Tom's position to make him more comfortable.\n\nThe superintendent raised her voice: 'I said, I'm warning you, Miss Davies.'\n\n'She was like this at the weekend, too, Mrs Saunders. Thinks she's Lady Muck, but she's nothing more than \u2014 '\n\n'Get on with your work, Beryl. You see the trouble you cause, Miss Davies. Coming here, disturbing \u2014 '\n\n'I'm disturbing no one.' Maggie bent down to kiss the top of Tom's head . . .\n\n'You mothers are not expected to \u2014 '\n\n. . . and stroke his feathery hair. 'How's my wee boy. How's my wee boy, how's Tom? Have you missed me?'\n\nFor several seconds Mrs Saunders looked on and said nothing. Finally she let her hands drop to her sides. 'We don't want any trouble, remember. Just make sure you don't get in the way.' She turned on her heel and walked out the door.\n\nMaggie shifted Tom to her other breast.\n\nBy the end of the first week Maggie had a routine:\n\nUp at 6.30 for two hours' typing, then the tram into town. She'd get off at the Waverley end of Princes Street, walk up North Bridge to the Scotsman offices, to scan through the Situations Vacant columns as early as possible. Late morning, she'd return to get the first edition of the Evening News. Her days were an endless round of employment agencies, receptionists, application forms, waiting rooms, interviews. Queuing to use public phones, walking to offices in the New Town, taking trams to offices in Newington, in Leith, in Gorgie, Stockbridge. Come late afternoon she'd usually had enough rejections for the day and would take the next tram she saw going in the direction of Carluke Avenue, to be in time to give Tom his early evening feed. She'd wanted to breastfeed him, but after only a few days she'd had to ask to use one of the Woodstock House baby feeding bottles \u2013 her milk was already starting to dry up. She'd been given it, grudgingly.\n\nHer nights were spent at the bakery table, answering advertisements in her own best handwriting, and doing more typing practice. Last thing, and when she had the energy, she'd add a few sentences to her current letter to Michael. Trams, telephone calls, letters, newspapers. Being unemployed was a full-time job. And exhausting. And expensive.\n\nThe closer it came to Christmas, the fewer were the employers looking to take on new staff. But Maggie kept trying. Kept phoning and being put on hold, kept being told the vacancy was already filled, being told they were looking for someone younger, or someone older. Or else they wanted a man. A bloody man \u2013 the answer to everyone's problems, according to Jean. Doggedly, rain or shine, she tramped around the city centre \u2013 George St, Hanover St, Frederick St, Castle St, the West End, the New Town . . . She went in and out of wood-panelled offices, some with fresh cut flowers in their reception rooms and views over Queen St Gardens; elsewhere she laboured up and down narrow and uncarpeted stairs, found herself shown into forlorn offices with grubby skylights, plasterboard partitions and audible plumbing.\n\nChristmas brought a card and a small cake from Jean, and a card came from Michael \u2013 My Xmas wish is that we were together had been inscribed below the festive greetings in Lachlan's neat handwriting. She put his card next to the photograph he'd sent her in an earlier letter \u2013 a snap of himself as a soldier, standing next to an army lorry covered in mud except for where the sweep of its single wiper had kept the windscreen clear enough to see through. Beside him there was a road sign: BERLIN 867 Kms. The photograph always confused her, it wasn't the man she knew \u2013 his blindness, his dependence. Not even the handwriting on the card was his. Years ago, when the photograph was taken, she'd still have been living in her parents' house doing her best to get through the war. Re-reading Michael's letters, which she did, they often seemed written to another Maggie altogether, one who lived a completely different life. Briefly, as she read, she'd let herself become this other woman, allowing herself to feel loved and cherished and to believe that everything would end happily. This happy-ever-after Maggie didn't have to struggle through every day, there seemed to be no loneliness in her life, no exhaustion. Clearly she never wept.\n\nFor Hogmanay she was invited to Jean's home. At first she said no, thank you, she'd prefer to see the New Year in by herself. But Jean kept on insisting.\n\nIt was getting on for midnight when she left the bakery to make her way to her sister-in-law's flat in the nearby colonies, just off Haymarket. After the bells rang out the New Year the small flat began filling up with neighbours come to first foot. Then the dancing began. An hour after she'd arrived her brother Billy still hadn't acknowledged her, let alone spoken to her. She'd caught him glancing over a couple of times, only to see him immediately turn away. Finally he came across:\n\n'Well then?'\n\n'Happy New Year to you, too,' said Maggie.\n\n'I'm asking,' her brother continued, 'what do you think you're doing? Mother's heart-broken. Father'll not have your name mentioned in the house. A fair disgrace, he says.'\n\n'So? I can't help how they \u2014 '\n\n'You can marry the man, can't you? Tell us where he lives and we'll pay him a visit. Once you're married, nothing of this'll matter any more.'\n\n'Like it never happened?'\n\n'That's the ticket, and everyone'll be happy.'\n\n'That'll be nice for them.'\n\nShe left shortly after.\n\nTwice on her way home she was grabbed and given a New Year's kiss. Then, just as she was turning into her own side street, a group of first-footers called to her from the opposite pavement: 'Happy New Year! Happy New Year!'\n\n'Happy New Year to you, too!' She called back to them, and meant it. Forget the past. Forget brother Billy and her parents, forget the Callanders and Mrs Stewart. She had the New Year to look forward to. She had Tom . . . and she had Michael.\n\nSUNDAY\n\nTO MAKE THE best of seeing her, the best for both of you, you have to make the effort to block out the TV's over-cranked volume, block out the empty stare of the Murray twins, block out the whole depressing end-of-the-line feel of things, and cut straight to what concerns you \u2013 your mother. You can only manage a few hours every week so you want to make the most of it. No expense spared and coming as often as you can to spend time with her. The good days, and the not-so-good days . . .\n\nToday she's been dressed in a pink jersey and M&S slacks. Best to catch her eye before crossing over to take the empty seat next to her. Try to catch it, at least.\n\n'Hello, Mum!'\n\nNothing. Like she's morphed into Murray number three. Has she even noticed you've come into the room?\n\n'Hello, Mum. How are you today?'\n\nStill nothing.\n\nNot a good day. Sit down next to her, touching her lightly on the arm. 'Really good to see you again, Mum.'\n\n'When there's bubbles of soap, it needs another rinse. Another rinse and another good mangle.'\n\nDefinitely not a good day. Give her hand a squeeze, try to catch what she's saying so you'll both be on the same page, the two of you sharing a Sunday afternoon together in the nursing home. Making up for lost time, it feels like, all the caring and loving you want to give her before it's too \u2013\n\n'Can't abide that chemical smell of soap in clothes.'\n\n'Remember that big block of green soap and the washboard, Mum? We'd sing, \"Scrub-a-dub-dub, Three men in a tub\" . . .'\n\nNo response.\n\nFine.\n\nTV's even louder than usual, battering the dayroom and everyone in it. Better to take her through to her own room. Where's her zimmer?\n\n'You're not leaving already, are you? Stay with me. They'll be bringing round a cup of tea any minute.'\n\n'That's good.' Managing a hopeful-looking smile. 'Then we can \u2014 '\n\n'I'll need to iron it. And a chicken. You're a man, you can do that, eh?'\n\n'Do what, Mum? What do you want me to \u2013?'\n\n'One of the hens, of course. Kill it.'\n\nA really bad day. A few seats along, the sexy Polish girl is trying to get one of the Murray twins to drink out of an orange plastic cup with a spout. A safety lid and Bart Simpson on the side.\n\n'Need drink, Joan. Need meds.'\n\nThe Murray doesn't seem to notice her, the old woman's mouth remains closed and her hands have collapsed to a slackness on her lap.\n\n'Drink, Joan. Help meds work. Drink. Drink.' The girl takes the trembling hands and wraps them round Bart, raising the spout into position. 'Good, Joan, good. Drink.'\n\nBut the Murray's having none of it. Her gaze is fixed far in the distance like she's really somewhere else, like on another planet. Her eyes are wide, wide open \u2013 is it possible she doesn't even see the girl crouching down beside her? Doesn't even feel the plastic spout pressed against her own mouth?\n\n'One med for finish. Drink. One med for finish, Joan, then I go.'\n\nProbably the girl feels like ramming the spout full-force between the old woman's lips and yanking the Murray head back \u2013 and who could blame her? Whatever. She's a real stunner, and would make the perfect assistant. Even in these politically correct days a magician needs a pretty assistant to display the inside of the shiny magic box and show that it's empty, to let herself be lasered in half, or else to help you disappear in a puff of smoke.\n\nIt's far too hot in here, the sun's melted and is pouring out pure heat, the windows are sealed tight shut as always. What a place. Some of the Dorothys and Murrays are facing the TV screen and some aren't. Rosehaven social life.\n\nYour mother's turned to stare at the TV.\n\n'That letter's made her cry. The poor woman. Making her cry and cry.'\n\nYou look across at the screen. Not that again. It's a rerun of what had been on the first day you'd come to visit, the same US soap that had made your mother so upset you'd gone over and changed it to horse racing. Nobody said anything, or noticed even \u2013 but Kylie spoke to you afterwards, telling you to please not do it again. The residents might not say anything, she explained, but it would still upset them. If your mother gets upset again, she'd added, best to take her through to her own room.\n\n'can't someone do something? does no one care?'\n\n'We've seen this episode before, Mum, let's go to your room. It'll be quieter there and we can have a good chat together.' Taking her arm, ready to help her stand up. 'I'll make some tea, if you like. I brought us some hobnobs.'\n\nOut of the corner of your eye you can see the woman's now finished reading her letter and is about to let it slip from her hand. This was the moment in the scene when you switched channels last time, but you can guess what's going to happen anyway \u2013 there'll likely have been some cheesy direction telling the actress to give the sheet of paper a very slight flick of the wrist as she lets it drop, that way it'll flip over a couple of times on the way down to the floor.\n\nAnd sure enough, the camera follows its descent in slow-motion to show everyone that the woman's heart being turned over. Spelling out her grief \/ disappointment \/ regret \/ sense of loss. Whatever. A cheap trick. But effective.\n\nNo problem this time round \u2013 your mother's face has gone quite blank, as if she's been put into a trance. If only. Then you could keep her safe and suggest to her only the sort of things that'll make her happy. You want her so much to be happy.\n\nPerfectly on cue, she gives you a smile. 'Hello! Are you here for the cake?'\n\n'Cake?' You sit down again. 'Of course, I'd love some. Then we can go through to your room and \u2014 '\n\n'Because if you are, you'll need a name tag. Security.'\n\n'Security? I gave my name and walked in today same as usual. No problem, like every Sunday. I've been coming here for weeks now, Mum. Never seen a name tag. No one's wearing any.'\n\n'Only the ones that need to. The staff, Mrs Saunders, Donna \u2014 '\n\n'You've not got one.'\n\n'Mine's getting changed. Seems there was some kind of mix-up. They were going to give me Mrs Stewart's till I put them right. It's getting made up now, it's all on their computer. maggie davies it'll say. Mrs Saunders is organising it.'\n\n'That's nice of her.'\n\n'Otherwise poor Mrs Stewart'll be walking about with no name \u2013 might as well not exist, eh?'\n\n'Mmm, I suppose not. By the way, I meant to bring you some flowers same as usual, I'm really sorry. Bring you a bunch next time, picked from the cottage. Happy memories, eh. My childhood home, after all.'\n\n'My cottage your childhood home? What on earth are you talking about? I don't know who you are, I don't know where you came from, I don't know anything about you. You just keep talking talking talking. More sense in what they're showing on the TV.'\n\nNext moment she's turned back to the screen and probably won't even notice when you get up to say goodbye. Might as well take out your iPad.\n\n5\n\nJANUARY CAME AND went. Then February, March . . .\n\nShe owed Jean money for food, for laundry, for tram fare, for the telephone, for everything. She needed new shoes.\n\nDown to only one letter a week to Michael now, and not just because of the cost of the stamp. She didn't want his pity by return. Tramping the ice-hard, wind-hammered winter streets of Edinburgh day after day, looking for jobs that she'd no hope of getting, left her with no energy for evenings of writing bright, cheerful, hope-filled letters. Sometimes she felt like grabbing an office-warm, brittle-voiced, white-blouse-and-lipsticked Snooty Junior by the ankles and dangling the girl out the window to give her a taste of her day.\n\nTom had begun his stay in Woodstock House on the seventeenth of November, which meant the six months would be up on the seventeenth of May. After that she would probably lose him. She had to find a job. Then find a place to stay where they accepted children. She'd tell the landlord that her husband had died, or left her, or was stationed in Germany, or Malta, or somewhere far away. The details could be sorted out when the time came.\n\nBy the beginning of April Maggie was no nearer to finding a job, or somewhere to stay. Only about six weeks remained. It was a bitter afternoon \u2013 no spring showers these, no gentle breeze. Feeling she couldn't manage another step, she made for the poshest agency on Queen Street, overlooking the private gardens. Not that she expected to suddenly hear about a job, but she knew the place had a carpeted waiting room with soft seats and usually a glowing coal fire. She needed a rest, even if only for a few minutes.\n\nNo one at Reception. Perfect.\n\nShe collapsed into the armchair nearest the fire, undid her coat and gradually began to thaw out. Quarter of an hour later she was leafing through a copy of People's Friend when she heard a quiet cough.\n\nSnooty Junior had returned to her post.\n\nDamn. Damn. Damn.\n\n'Hello, Miss Davies. And how are you today?'\n\nThat I've-got-a-job brightness. Maggie glanced longingly over at the window. Then steeled herself to reply: 'Fine, thank you. I was just looking in for a moment to see if anything had . . .' Her usual query.\n\nTo her surprise, Snooty Junior smiled at her. 'Right time, right place.' By any chance, she added, did Miss Davies know Blair & Blair, the well-respected solicitors across in Abercrombie Place?\n\nMaggie nodded vaguely.\n\nWell then, continued Snooty Junior, she might just be in luck.\n\nA moment later the girl had looked out Maggie's application form, then shown her through to her boss's office to be assessed for suitability.\n\nMaggie's assessment took less than five minutes. Blair & Blair, she was told, had been on the phone less than quarter of an hour previously. They'd been badly let down by someone who was supposed to start that morning and simply hadn't turned up. They were desperate. Miss Davies' typing speed? Her references? Availability?\n\nCould she go straight round?\n\nBlair Jnr looked in his early forties going on fourteen. His three-piece suit struggled to contain the overspill of a cheerful schoolboy chubbiness that was topped by freckled skin and ginger hair. No mistaking the public school sheen of self-confidence, however. Also, there was no mistaking the no-wedding ring \u2013 which Maggie certainly wouldn't be reporting back to Jean, or she'd never hear the end of it.\n\nAfter all that time spent filling in forms and answering advertisements, she had her story word perfect. Her reference from Cavendish & Son (Realtors) in Vancouver, was excellent \u2013 Maggie, of course, had checked in the library to make sure that no such company existed.\n\n'First-rate, Miss Davies,' Mr Blair remarked as he glanced through the painstakingly crafted catalogue of her professional attainments supported by a list of her outstanding personal qualities. 'I congratulate you.'\n\nWas the cheerful schoolboy being sarcastic? She risked meeting his eye, and he smiled straight back at her. His public-school confidence seemed to shed its own glow upon the typewritten sheets, turning their inaccuracies into truths and their blatant deceptions into recovered innocence. Blessing them, almost.\n\nMaggie went on to explain that, sadly, Cavendish & Son had gone out of business when the Son turned sixty and decided to retire. It had been a small firm and her position as Mr Cavendish's PA had been most rewarding as well as prestigious. There had been no shortage of other job opportunities on offer, of course, but her parents were coming to that age when they needed . . . Well, she was sure Mr Blair understood what she meant?\n\nThe schoolboy solicitor certainly did. 'My own parents . . .' he began before trailing off to finish his sentence with a regretful shake of the head. He was now looking genuinely concerned.\n\nSo it had seemed best to return to Scotland, she continued. Having spent the first few weeks helping her mother and father settle into their new routine, she was ready and eager to return to work.\n\nAs rehearsed several times with Jean, Maggie now moved into full interview role.\n\n'It's . . . life, I suppose.' She looked away to cover a show of awkwardness, a hint of momentary embarrassment. After a well-timed pause, she managed to gain control of herself, swallowed discreetly, and proceeded: 'They've done their best for me, so now it's my turn to do my best for them.' She and Jean had debated whether there should be a hint of tears at this point, and decided she'd best play it by ear. She dabbed her nose, struggling bravely to carry on: 'It's only right that Mum and Dad should keep their independence for as long as possible. We'll see how things work out \u2013 when I've secured employment, I plan to move into accommodation that's close by.'\n\nMr Blair nodded sympathetically. Maggie was then sent through to the room next door to demonstrate her typing skills to Mrs Woodward, who was in charge of Blair & Blair's two-desk typing pool. While she showed off her by now reliable thirty words a minute, her prospective superior looked on, and smoked. A page and a half later, she was invited to hand over the sheets for Mrs Woodward's inspection.\n\n'Hmmm . . .' A pencil mark, a large puff of the cigarette. 'Hmmm . . .' Several more pencil marks and several more large puffs. 'Hmmm . . .' A final pencil mark. The cigarette stubbed out.\n\nMrs Woodward laid the typed sheets on her desk. 'Mr Blair prefers a one-line gap between paragraphs, and so do I. But good enough, I suppose.'\n\nMaggie was sent back to Mr Blair's office. The schoolboy beamed at her.\n\n'Now, Miss Davies, as I told the agency, we are very pressed at present. Could you possibly start first thing tomorrow?'\n\nFor the first time in months she couldn't wait to get home and write to Michael.\n\nIt was a busy week. As well as working full-time, learning to cope with Mrs Woodward \u2013 whom she soon took to thinking of as 'Old Woodbine' \u2013 and visiting Tom in the evenings, Maggie had to check the Accommodation Vacant columns in the Scotsman, the Evening News and Evening Dispatch. She went to see five possible rooms \u2013 but either they were too small, too dirty, too inconvenient or too expensive. One landlord was too friendly. On Friday she phoned about a place near the Meadows, at the edge of Tollcross. It sounded ideal. The rent was manageable and the location, midway on the direct tramline between Blair & Blair's and Woodstock House, would be perfect. The landlady was a Mrs McCann \u2013 could Mrs Stewart come round on Saturday?\n\nMrs Stewart. That was the name she gave. As she was planning to bring Tom to live with her as soon as possible, she had to be a Mrs. She was already called Miss Davies at work, but so long as she kept her stories straight, and separate, there would be no problem. And what about her husband, Mr Stewart? Well, she'd got till lunchtime next day to fit him in, somehow.\n\nThat night in her letter to Michael she told him about the possible room and asked him to keep his fingers crossed \u2013 for Mrs Stewart!\n\nLate morning on Saturday, the final hour of her first week at Blair & Blair's was being ticked off one slow-motion minute at a time. Maggie watched the hand of the large office clock on the wall opposite strain to make the next tick. Then strain ever harder, and take even longer, to gather its strength for the one after . . .\n\nOnly 11.30. She tap-tap-tapped through to the end of another near-incomprehensible letter. Original went into the red folder marked awaiting signature, copy into the blue folder marked carbon copies.\n\n'You still live with your parents, I believe?' Mrs Woodward lit a cigarette and blew a lungful of smoke towards the yellow-stained ceiling. 'They must be . . .' \u2013 with her fingertip she tapped the dead ash into the metal ashtray next to her own typewriter \u2013 '. . . getting on in years?'\n\nClatter-clatter, clatter-clatter, clatter-clatter . . . Ding!\n\nMaggie's notepad lay beside her, line after line bristling with constipated gobbets of legalise: 'heretofore', 'without prejudice', 'available to be relied upon' and 'our rights and pleas' and the like. It was soothing to let her fingers go tap-tap-tap, picking out unintelligible gibberish that required neither her interest nor her understanding.\n\n'You're having to take care of them, I suppose.'\n\nMaggie hammered out the next paragraph in one continuous burst:\n\nFollowing our letter of the fifth inst., you are hereby called upon to advise by return . . .\n\n'Can't be easy for you. A woman of your age, you naturally want a life of your own, a family perhaps and . . .'\n\n. . . your failure to comply with this will be founded upon . . .\n\n'I was spared all that, you might say. Father was killed at Ypres, Mother followed him a year later with the Spanish Flu.'\n\n. . . without prejudice to any rights and any pleas and any costs and recovery of said moneys.\n\n'I was nine.'\n\nMaggie stopped typing. 'I'm very sorry to hear that, Mrs Woodward \u2013 so terribly young too.'\n\n'Brought up by my granny. Ancient history. I was married at twenty and I've never looked back. Douglas came along at just the right time. When war broke out, he volunteered for the RAF and ended up a rear gunner. Came through the whole war without a scratch. He . . .' She paused.\n\nMaggie leant forward, waiting for the older woman to continue.\n\n'All those bombing raids, Jerry's Mescherschmitt fighters, barrage balloons, anti-arcraft fire \u2013 he survived the lot of them. Hamburg, Dresden, the Ruhr, Berlin and not even a scratch.' Mrs Woodward was no longer looking at her, but staring into space. 'Lucky, don't you think? Really lucky?'\n\n'Yes, he certainly was. You both were.'\n\nWith unexpected viciousness, Mrs Woodward stubbed out her cigarette, grinding it into the ashtray. 'That's what we thought too, at first. I go to visit him out at Gogarburn every Saturday afternoon \u2013 not that he notices.'\n\n'I'm so \u2013 so sorry.'\n\nMrs Woodward inserted a fresh sheet of foolscap into her machine and resumed work.\n\nMaggie's typing was up to date \u2013 her morning's carbon copies in the blue carbon copies folder and her morning's originals in the red awaiting signature folder. The appropriate envelopes were typed, stamped and stacked, all ready to be filled and sealed. Time had come to a complete standstill at eleven forty-seven. The clock hands just would not budge, and the final thirteen minutes seemed to have locked solid.\n\nHer filing too was up to date, her typewriter fitted with a brand-new spool of ribbon in readiness for the brand-new week starting on Monday. Her stationery drawers \u2013 foolscap, octavo and quarto \u2013 all brimmed in readiness.\n\nTick \u2014\n\n'Your first week at Blair & Blair is drawing to its close, Miss Davies.'\n\n'Yes, Mrs Woodward.'\n\n'I am pleased to note that you perform your tasks adequately and . . .'\n\n'Thank you, Mrs Woodward.'\n\nThe older woman looked directly at her. There was a pause as she drew on her cigarette, holding the smoke for several seconds before exhaling. '. . . and you know your place.'\n\nTick \u2014\n\nWithout meaning to, Maggie glanced over at the clock in time to catch the minute hand jerk forward to eleven forty-nine.\n\n'In a rush to go somewhere this afternoon, Miss Davies? Blair & Blair pays for your attendance up to midday today. Eleven full minutes still remain and it seems only fair that . . .'\n\nIn a rush? She certainly was. A number 10, 11, 15, 16 or 23 straight to Tollcross. She had to see Mrs McCann and her room. She had still to sort out the details of her Mrs Stewart story, and get them straight. Was there a Mr Stewart? If so, where was he? Same with baby Tom \u2013 if he existed, why wasn't he with her? She had to think and think hard. She didn't want to lose the room \u2013 it really did sound exactly what she was looking for.\n\nMeanwhile, Tom would be waiting for her.\n\nThe weather looked fine and if she wrapped him up warm . . . He loved going out in that Victorian-looking pram that stood in the hall . . . She'd bought him a new rattle yesterday and \u2014\n\n'. . . if you don't mind my asking?' Old Woodward had clearly continued talking and was now waiting for her reply to something.\n\n'Pardon?'\n\n'Your parents. You were telling Mr Blair about them. How are they accustoming themselves to your new routine, if you don't mind my asking?'\n\nHer parents? Another story.\n\n'They're fine. It's working out very well. I'm usually home in time to prepare their evening meal . . .'\n\n'They must be very proud.'\n\n'Proud? I . . . I don't understand.'\n\n'Of you, of course. Proud of their daughter. It's not everyone can secure a position with Blair & Blair.'\n\nTick \u2014\n\nEleven fifty-three.\n\n'Yes, I suppose they are.'\n\n'Well, Miss Davies, I'll be going out to Gogarburn this afternoon, same as always. He's still my husband, after all \u2013 and wears the name tag round his neck to prove it. They give us tea and biscuits. I read the Scotsman aloud to him. News, letters page, fashion. It's all one to him.' She took a long draw of her cigarette. 'Then we just sit.\n\n'When I get up to leave there's a kind of flicker in his eyes sometimes, like he knows I'm going away. I have to go, though. I have to. But let me tell you, Miss Davies, I occasionally wonder if he'd be better off without me visiting him at all \u2013 at least that way he'd not have the distress of me leaving him over and over every Saturday evening. Four years it's been now.'\n\nTick \u2014\n\nTick \u2014\n\nEleven fifty-five.\n\nAfter one final puff, Old Woodbine stubbed out her cigarette. 'Well, that's our weekends staring us in the face.' She almost smiled. 'I'm sure Mr Blair won't quibble over the last couple of minutes. We'll see you on Monday morning at your desk, Miss Davies. Nine o'clock sharp, remember. Goodbye.'\n\nCoat, hat, handbag, and Maggie was already halfway out of the door.\n\n'Goodbye, Mrs Woodward. See you on Monday.'\n\nRushing down to Princes Street, across to the Gardens side and wait for a tram.\n\nRight \u2013 now to go over her story. Her Mrs Stewart story.\n\nHer husband had died? Definitely. That was best and simplest. He'd died just before Christmas, leaving her all alone, a poor widow having to care for their wee baby. No calling him Alfred this time, and giving him a beard!\n\nBut what if everything worked out between her and Michael? She'd not be moving to Lewis so he might come to Edinburgh one day and \u2013\n\nA number 16. Take her straight to Tollcross.\n\nIt was packed, downstairs and up. Standing room only. Saturday shoppers, their shopping bags, their children. A dog. Two dogs.\n\nHow's she supposed to think?\n\nTurning into Lothian Road already. She needs to think \u2013 but how can she get her story straight with all this noise and people shoving her, wanting past to get off, and the clippie asking if she's not got change \u2014\n\nOne of the children's started crying, setting off one of the dogs. Yell-yell, bark-bark \u2014\n\nWhich sets off the other one. Passing the Usher Hall. think! She needs to think! Mr Stewart should still be alive. He has to be alive. Right. But they can't be divorced. Definitely not. Would cause even more problems. And so . . . ? She's all on her own, and she's \u2013 she's what? But if her husband is still alive why aren't they \u2014 ?\n\nGoing past the Tollcross clock already. She'll be arriving there any minute.\n\nThen it came to her in a flash.\n\nThank you, Mrs Woodward!\n\nShe got off at the stop just past the King's Theatre, walked about twenty yards and turned first left into Glengyle Terrace. A posh-looking street facing Bruntsfield Links, with railings and steps up from the pavement to posh-looking front doors. Not only that, but it was a main door flat.\n\nMrs McCann was in her late twenties. Cheerful. Welcoming. The room she was shown more than lived up to Maggie's expectations and would be large enough for when she brought Tom home. There was even a wash hand basin and mirror \u2013 and she'd share the McCann's bathroom and separate toilet. It turned out that Mrs McCann had a small boy called Douglas. Three years old, he played with some wooden bricks at his mother's feet while the various details were discussed over a cup of tea.\n\nThen came the questions. Friendly enough though. Concerned even.\n\nMrs Stewart's husband?\n\nHe'd come back from the war blinded, and badly wounded. Hospitalised at first, then discharged far too soon, like so many of them. Then, just before last Christmas, he'd had to be re-admitted to Gogarburn. The doctors had no idea how long he had to live. Sometimes he seemed to be on the road to recovery, but at other times . . .\n\nMaggie let the sadness in her voice finish the sentence.\n\nMrs McCann said it must be so very hard for her.\n\nYes, it was. And . . . For a moment Maggie seemed unable to go on, then she forced herself:\n\n. . . And they'd had a wee boy just the month before.\n\n'Poor, poor you,' said the landlady, 'having to cope with everything all on your own.' She shook her head. She'd no time for these women who got themselves in the family way without a family, if Maggie understood what she meant. 'But poor, poor you,' she said again.\n\nHe's called Tom. He's lovely.\n\nMrs McCann supposed that Tom must be a real consolation. But where was the wee lad?\n\nStaying with his granny for the time being.\n\nThe landlady looked puzzled.\n\nBecause, explained Maggie, things being as they were, she'd needed to take a job and couldn't look after Tom during the day.\n\nCouldn't Mrs Stewart move in with her mother?\n\nNo, it wasn't possible. Because . . .\n\nMaggie glanced out the window at Bruntsfield Links. To have the park so close would be perfect for when Tom came to live. Those grassy slopes, the walks, picnics in the shade of the trees . . .\n\nHer mother? Think. Think. Why couldn't she move in with her mother?\n\nHer mind had gone blank. Completely blank.\n\nA car hooted out on the main street. She caught sight of a green Eastern Scottish bus labouring up the hill to Bruntsfield on its way out of town . . .\n\nBecause . . . ? Because? She couldn't move in with her mother because . . . ?\n\nThen she had it. Because her mother lived out of the city, near Flotterstone . . . and it was too far for her to travel into work. So she went to see Tom there at weekends. Her husband \u2013 he was called Michael \u2013 she visited as often as she could during the week after work. Once she was settled, of course, she planned to bring Tom to live with her. Wherever she was. Would that be all right? She'd be able to find a minder in Tollcross surely? That way she'd be able to keep working. No idea when Michael would be able to join them. Her tone of voice hinting, sadly, that he might never come.\n\nMrs McCann said she was very sorry to hear of her troubles. War was a terrible thing. Then she went on. 'We'll give it a week or so to see how we get on, you and I. Maybe you can babysit Douglas one night, and if everything works out . . . ?'\n\nThey agreed that Maggie would move in the following afternoon, to be ready for the week ahead. As they stood up, she only just stopped herself in time from giving Mrs McCann a big, big hug.\n\nWhen Mrs McCann's front door closed behind her, Maggie lingered for a couple of moments before going down to the street. Heaven stretched out before her. She had a job. She had a room. Tom would be welcome. From now on, life would be one long walk in the park.\n\nOf course, she'd need to remember her story and keep it straight. On the tram to the children's home, she recapped: during the week when she visited Tom at Woodstock House she'd say she was out at Gogarburn visiting her invalid husband; at the weekends she'd say she was going to Flotterstone to see Tom. A bit complicated, but couldn't be helped. Depending on how things developed between her and Michael, she would say her husband was completely cured, or that he'd died.\n\nThat night's letter to Michael would be signed Mrs Stewart!\n\nIt was after three when she rushed up the front steps of Woodstock House. Donna was waiting for her, pushing Tom up and down the hall in the Tractor.\n\n'Afternoon, Miss Davies. Tom's all dressed and ready. I was putting in some pram practice for when I have a wee boy of my own. I told him you'd probably be taking him out. You are, aren't you?'\n\n'Well, yes. I've been looking forward to \u2014 '\n\nDonna parked the pram at the bottom of the staircase and rushed over to her. 'Mrs Saunder's camera's got one photo left and she said I could have it of me and Tom if you'd take it. Please. Please. Please.' She held out the Kodak Brownie. 'We'll do it outside and I'll just hold him. Maybe us standing next to the Tractor?'\n\nTen minutes later Maggie was manhandling the Victorian monstrosity down the steps and out through the garden gate into the street. At every bump Tom laughed and shook his new rattle.\n\n'Sunny day \u2013 Holiday!' she shouted. The sunlight was cold, but bright, bright, bright. A clear, crisp early spring day.\n\n'Left or right, Tom \u2013 which d'you fancy? Left'll take us to the shops and right to the canal.'\n\nHe shook his rattle loud enough to show approval of every possible option.\n\n'Or we could go to the moon?'\n\nFor there, above them, no more than a faint and almost transparent smudge against the ice-blue sky, was the moon. He rattled again and added a gurgle.\n\n'It's come out early \u2013 just for us, Tom. Doesn't show itself often, only on special days and only to special people. So let's give it a special name. A ghost moon, we'll call it. Come on, Tom, let's go to the ghost moon!'\n\nThe Tractor was a solid piece of engineering, all bulk and weight. She gave it an extra-hard shove and let go, 'whee . . . !' sending it and Tom trundling a few yards forward by themselves.\n\nShe caught up with them: 'Sunny Day . . . Holiday! Sunny Day . . . Holiday! We're going to the moon!'\n\nFour brisk steps . . . and another firm shove. 'whee!' Like she was already pushing him on the swings, a swing that reached all the way up into the sky.\n\nAt every push Tom shook his rattle like a champion. The pavement was deserted. She pushed and pushed . . .\n\n'When we \u2013 whee! \u2013 go round the next corner, we'll really start to soar \u2013 whee! \u2013 soar up into the sky. whee!' The Tractor was picking up speed now. 'Nearly there, Tom. Ready? Hold on tight . . . Here we go . . . !'\n\nTurning into the next street, she could see the moon directly ahead of them, set high above the roof of a large townhouse. Against the clear-cut outline of bricks and chimneys it looked like an unfinished sketch, a hastily drawn scribble of light that might dissolve at any moment. Faster and faster towards it they went, her feet no longer touching the pavement.\n\n'We're rising up now, Tom. Feel it?' Treading air now, she rose higher and higher. 'Don't worry. I've got good hold of you and won't let you go \u2013 ever.' The Tractor swayed from side to side like a ship riding high on invisible waves. The higher they went, the louder and clearer sounded his rattle. Maggie pointed over to Craiglockart Hill.\n\n'Look down there, do you see the tiny trees, the dolls' houses, the Matchbox cars and trams?'\n\nRattle-rattle, rattle-rattle.\n\n'Up here it's just you and me, Tom. No one to bother us, no one to tell us what we can and can't do. No one.'\n\nThe city was spread out below \u2013 a ruler-straight neatness of streets, avenues and crescents with dotted lines for houses, shops and shaded green for trees and parks. The moving dots were traffic. Down there were Mrs Saunders and old Woodbine, each puffing out little clouds of smoke at their tiny toy desks in their tiny toy offices and, a few inches over, just next to the splash of blue sea, was her parents' house . . . The Forth Road Bridge was a cat's cradle of red spanning a streak of silver paint spilled in the cold winter sun, and the Pentland Hills had been polished to a smoothness of moss-green.\n\n'There'll be no Boss Beryl, Tom. No Mrs Saunders. No Old Woodbine. No lawyer gibberish, no rubbled houses. No bombed Leith, no Coventry, no London.\n\n'The world's a ball that's got burst, Tom. We don't have to play with it any more. The ghost moon's so full of light it's almost see-through. Michael's waiting for us there \u2013 he's the man in the ghost moon. He's putting the kettle on for us, getting out ghost moon cakes and \u2014 '\n\n'Maggie!'\n\nWithin a split-second they'd tumbled back to Earth.\n\nA woman was calling to her from across the street. Coming over to greet her. 'Maggie, I thought it was you!'\n\nHer mother's neighbour.\n\n'Oh, hello, Mrs Melville.'\n\n'What a surprise! Where've you been keeping yourself? I heard you'd gone away for good. London, was it? This you back up for a wee visit?'\n\nMrs Melville. Iron-grey hair brushed to a hard shell, face powder cracking around her mouth, her lipstick framing the unspoken accusation: Your mother never said anything.\n\nShe looked down at the pram. 'What a lovely wee . . . boy, is he?'\n\n'Yes.'\n\nShe leaned closer. 'What's your name, wee man?'\n\n'He's called Tom.'\n\n'A lovely name. Hello, Tom!' She bent further down till she was almost under the pram hood. 'Come up to Scotland, have you? To see Granny Muriel and your grandpa?' She patted the top of his head. 'And how old are you, my wee lad?'\n\n'He's just a few months.'\n\n'Really?' Another pat. 'And is your daddy up visiting, too?'\n\n'Just me and Tom.'\n\n'That's nice. A bit of time on your own.' She gave Tom a final wave and straightened up again. 'I'll meet your daddy another time. Oh, you're such a bonny wee boy!' She turned to face Maggie. Her lipsticked concern: 'Everything all right, is it?'\n\nSuddenly Maggie could bear no more.\n\n'I'm sorry, Mrs Melville, but I've really got to go. I'm visiting a friend in the next street. I'm late already. Goodbye.' She took a step forward, pushing the Tractor. Then at once began setting a smart pace.\n\nMrs Melville had to hurry to remain at her side. 'Here's a thought. Are you free tomorrow afternoon, Maggie? . . . You and your mother'd be welcome to . . . to drop round for a cup of tea.' The older woman was soon gasping for breath, trying to keep up. 'Nice Dundee cake . . . I've been saving . . . A chance . . . a chance to hear all . . . your news and . . . if you've a photo of your \u2014 '\n\nMaggie was now a good dozen yards ahead. She called back: 'Goodbye, Mrs Melville. I really need to keep going.'\n\n'See you tomorrow, Maggie. I'll say to Muriel . . .' Maggie didn't catch the rest.\n\n'Vroom-vroom!' she twists the pram handle as if she's on a motorcycle \u2013 a sudden burst of acceleration sends her roaring full-throttle forward. 'Vroom-vroom-vroom!' Tom's rattle urging her faster and faster. Rattle-rattle, rattle-rattle . . .\n\nHe's gurgling and laughing fit to burst, his arms and his rattle going like windmills.\n\nInto third gear, into top.\n\nInto overdrive.\n\nCurve in the street coming up . . .\n\nLeaning over to take the corner on two wheels, the houses on either side blurring past . . .\n\nLeaving Mrs Melville far, far behind.\n\nAccelerating out into the straight.\n\nFull-tilt round another corner, the two of them hurtling along faster and faster, rising again into the air, soaring weightlessly up and up into the afternoon sunlight . . .\n\nMaggie raises her head to shout out loud: 'Ghost moon here we come!'\n\nSUNDAY\n\nROSEHAVEN AGAIN. THE yellow cross, the bell. CCTV, the security grille.\n\nBuzzed in.\n\nSomeone's singing to herself. That folksong about going to Skye. Bonnie Prince Charlie, wasn't it? Posh English words for something so Scottish. Elderly cracked-sounding voice, pleasant enough, but hardly X Factor.\n\nManaging a good half-dozen steps down the corridor until . . .\n\nRushing into the toilet before you throw up. Hanging over the basin, dry-heaving, your forehead in a cold sweat, your hands trembling. Retching, and retching.\n\nLike sea-sickness, but you're not sick. You never are. Hang in there.\n\nHang in there.\n\nIt'll pass. Like it always does. It will. Steady?\n\nSteady. Better now?\n\nRinse out your mouth and you'll be fine. No matter how rough you're feeling, the adrenaline of performance gets you through every time. Three cheers for Doctor Showbiz!\n\nA last wipe-down with the paper towel. Deep breath. Reality check in the mirror \u2013 colour flooding back into the cheeks, a cheerful smile, bright eyes. You want her to see a loving son, someone eager to visit and spend the afternoon with her. Someone who cares.\n\nThere's a genuine Mr Magic spring in your step as you stride along the corridor.\n\nBut your mother's not in the dayroom, not in her bedroom either. You find Kylie sorting out pills in the kitchen, planting them like seeds into their miniature plastic tubs.\n\n'Can ye not hear her? Entertaining us all through lunch, she wis. Widnae go back efter, so we just let her be. She's fine, though. Gang through and see for yersel.'\n\nThe dining room's at the rear of the care home \u2013 easy-wash flooring, plastic chairs, formica tables, white Venetian blinds, pale green walls, the day's menu and fire regulations pinned to a green felt noticeboard. There are half-a-dozen yellow-topped tables, each with a posy of artificial flowers in a small vase. Your mother's table stands in the middle of the room like a desert island adrift on a sea of blue linoleum.\n\n'Speed bonny boat like a bird on the wing . . .'\n\n'Mum? Are you all right?'\n\n'Carry the lad who's born to be king . . .'\n\nShe's far, far away in an elsewhere place that has no borders except for the table edge she's gripping as fiercely as if her life depended on it.\n\n'Mum?'\n\nHer knuckles have gone white. Under her blouse, her collar bone feels brittle-thin, mere skeleton.\n\n'Over the sea to Skye.'\n\n'I'll sit with you if you like.'\n\nYou take the seat directly opposite hers. There's no sign she's noticed you've sat down, no sign she's noticed that anyone's sat down.\n\n'What are you doing here, Mum? Lunch is over, the others have all gone back to the dayroom.'\n\n'Speed bonnie boat . . .'\n\nHas she been crying? If so, they were tears that have left no trace. But there's broken skin, a gouge mark as if she'd run her fingernails down her cheeks.\n\n'Mum, it's me. Tom. I've come to see you.'\n\n'Over the sea to Skye.'\n\n'That's a good song, Mum, I don't remember you ever singing it when I was \u2014 '\n\n'How the winds blow, how the storm roars . . .'\n\n'Hello, Mum. I've come to \u2014 '\n\n'Hardly a chance even to see him, only a few seconds because they're coming for him, because it's all been arranged with Mrs Saunders. What else can I do?'\n\n'Don't cry, Mum. Everything's fine. I'm here to see you and \u2014 '\n\n'What else can I do? The rain just came down and down at Silverknowes \u2013 I couldn't stop it. You can't stop the rain, can you?'\n\n'Everything's fine now, Mum. Everything's going to be all right, everything's \u2014 '\n\n'But I'll go and see him as soon as I can . . . They don't want me to, but \u2014 '\n\n'Don't cry, Mum, everything's \u2014 '\n\n'Mrs Saunders said he'll be well looked after. So don't worry.' She smiles and lets go of the table. 'I'll be visiting as often as I can, I told her.'\n\n'No need.' You make a joke of it. 'Here's me visiting you!'\n\nA joke? How many jokes could survive this room with its washed-out d\u00e9cor, its empty tables, Venetian blinds half-slatted to conceal the neighbouring brick wall?\n\n'Let's go through to the dayroom, Mum, it'll be brighter there. Or your own room if you'd rather.' You get to your feet.\n\n'My room's been cleared out. Nothing left.'\n\n'No, Mum, no. No one's done anything, really it's \u2014 '\n\n'My whole life like it's never been. How could they? My whole life, like I'd never been born.'\n\n'Mum, it's all right. No one's done anything to your room. I've just been there and it's the same as always. Let's go through and you can see for yourself. We'll get you settled.'\n\nShe'll need your hand under her arm to help her stand up. There's a puddle of spilt soup and flecks of scattered rice where she's been sitting.\n\nPutting your arm round her and taking gentle hold of her elbow to give support, it feels like you're cradling the fragility of a bird's egg in your hand. One slow step at a time, you begin the laborious journey to the door, guiding her between the tables set with unattended cutlery, with flowers that never need watering. So near-weightless she seems, that you feel she might rise up into the air at her next step and float away out of reach . . .\n\nShe stumbles into one of the plastic chairs \u2013\n\n'Take your time, Mum. No rush.'\n\n'No need to squeeze the life out of me,' she snaps. 'I can manage quite well, thank you. I'm going home, amn't I?' She's watching everything now, nervous and alert, treading with caution around another chair while clutching your arm.\n\n'We can stop whenever you want. There's no rush, Mum. We've all afternoon.'\n\n'All afternoon? Have we?'\n\n'Yes, of course. I'll be with you for hours yet, all afternoon like I say. And we can make it longer if you want and . . .'\n\nNot too long, though. You're seeing the lovely Mandy at Whigham's for drinks, then off to that new Italian along Shandwick Place.\n\nShe's come to a halt in the doorway, hesitating, not wanting to leave the dining room.\n\n'That's it, Mum. Just like you taught me \u2013 Look left, look right, then left again. We'll get there.'\n\nSafely out into the hall. To your right's the dayroom with the door standing wide open as usual and the TV unloading its noise over the elderly women propped up in their chairs, lining the walls. Not another afternoon in there. You couldn't bear it. You steer her to the left in the direction of her room.\n\n'Someone's expecting us?'\n\n'What's that, Mum? Who'd be expecting us? We'll be sitting together, just the two of us. I'll make us some tea. I've brought biscuits, chocolate digestives.' Keeping you both moving down the corridor.\n\n'Will they let us in?'\n\n'Who?'\n\nShe's stopped. Won't move a step. 'They know I'm coming. They've been told all about me. They'll be waiting. Suppose they slam the door in my face again? Suppose they \u2014 '\n\n'Your bedroom door? No one'll do that, don't worry. Anyway, I'm here and I'll sort out anyone who tries to \u2014 '\n\n'The handrail's here for when it starts to get rough, which it can do sometimes. I'll keep good hold, just in case.'\n\n'Come on, Mum, we can't just stop here in the middle of nowhere.'\n\n'So cold here.' She's started shivering. 'But at least I brought my coat and scarf. Why won't the sun come out? What if they don't let us in? We're so far from Edinburgh and \u2014 '\n\n'Listen, Mum. We're going to your room, that's all. No one else'll be there. Be just the two of us, believe me.'\n\n'No one else? You mean there'll be no one to let us in?' Her lips begin to tremble.\n\n'There'll be me, Mum. And you. We'll sit together and talk like usual. We'll have tea and chocolate digestives.'\n\n'But if no one's there how'll we get in?'\n\n'We're there already. See, the door's standing open for us. No problem.' You manage to help her in.\n\nStill clutching your arm for support, she works her way round the furniture in her room, running her fingers across the top of the chest of drawers, opening and closing the wardrobe door, reaching up to straighten a picture, a seascape that's mostly clouds, touching the playing cards on her tea-trolley. Finally she sits down in her chair. 'This is my room, isn't it! I'm so glad to be back here. Everything's going to be all right now. I know it.' The smile she gives you lights up her face.\n\nThe photo album's opened at that couple standing in front of their house. She couldn't remember anything last time, but maybe you'll have more luck today? They look friendly enough. Definitely not in the city, you point out. You slide it out of its plastic sleeve so she can have a closer look. The man and woman are both well into their fifties.\n\n'So, Mum. The mysterious Callanders \u2013 who are they? Really bleak-looking place, not a tree in sight, a bit like Orkney maybe or Lewis \u2014 '\n\n'Shut up! Shut up! You go on and on and on about them. I've never met them, don't want to meet them.' She pauses. 'I know what, let's get rid of them once and for all . . .' And before you can stop her, she's grabbed the photo and ripped it in half. Then in half again. Her hands and arms shaking, her cheeks flushed red. 'Burn them. Burn the pair of them, then maybe you'll shut up about them.'\n\nSome of the pieces have fallen on her lap, others on the floor at her feet. You're reaching down to gather them \u2013\n\n'Leave them. They're nothing, they never were,' she hisses. 'Nothing. Nothing. Nothing.' She starts ripping them into even smaller pieces.\n\n'It's all right, Mum. We can leave the photos.' You take her hands in yours to steady them. 'We can just sit and \u2014 '\n\nShe snatches her hand free. 'Go on, make yourself useful.' Spittle flies from her mouth: 'Get us some matches!'\n\nWhen you return a few minutes later with Kylie, you find your mother sitting straight up in her seat, a smile on her face.\n\n'Hello, Beryl.'\n\nThere's no sign of the torn up photograph, or the red album.\n\n'Whit've we been up tae, Maggie? Yer son said something aboot yer gettin upset an wantin matches? Ye ken we dinna allow \u2014 '\n\n'Matches? I'm quite warm enough, thank you, Beryl. We've central heating here \u2013 don't need fires. All that mess with soot and smoke and the grate needing cleaned out every morning.' She looks at the two of you in turn. 'I'm fine. Never felt better.'\n\nAnd it's true. You can see that she looks utterly content. A moment later she's taken your hands in hers and lifted them to her face. She draws them slowly across her eyelids, her cheeks, her lips . . .\n\n'It's so very good to see you again, Michael.'\n\nYour father again. She keeps thinking that you \u2013\n\n'Stay here with me, Michael. Please.'\n\n'Yes, I will.' What else can you say?\n\n6\n\nMAGGIE TYPED AND filed her way through the next week. She visited Tom. She wrote Michael several letters, double-sided sheets crammed with all sorts of hopes and possibilities, with her love for him, her longing for the three of them to be a family. It would happen. Their longed-for life together lay only just beyond this one same day that kept repeating itself over and over \u2013 a day of working, shopping, visiting Woodstock House, cooking her evening meal on the twin-ring electric Belling, keeping her bedsit clean. Soon, soon.\n\nSpring was in the air and she walked to work enjoying the morning sunshine and freshness, and to save money. She'd do her best to start to paying Jean back and also try to put at least ten shillings into a Post Office account every week. Best of all, Tom's six months at Woodstock House had another whole month to run \u2013 so she would be able to remove him in good time!\n\nShe babysat Douglas on the Wednesday evening. It went well. Mrs McCann was pleased. 'You must call me Sheila.'\n\nAfterwards, her landlady made the most unexpected and wonderful suggestion.\n\nTo be sure of catching Mrs Saunders in good time the following day, Maggie pretended she had an urgent dentist's appointment, and was already on a tram to Woodstock House by mid-afternoon.\n\nThe moment she'd taken her usual seat at the front she pictured Tom, fast asleep in cot number 11. How would he be today? He seemed to have changed a little each time she saw him, grown a little more, learned new gurgles and grins, new ways of grasping her fingers. When she left him, it felt like he was being wrenched afresh out of her body. Re-opening a birth wound that never had a chance to heal.\n\nBut not any more.\n\nShe imagined herself lifting him into her arms \u2013 and as she did so, the tram and the busy street outside dissolved around her until there were no other passengers, no windows, no solid steel floor, no metal rails beneath nor any sparking electrics overhead. Nothing mattered but the moment when Tom would open his eyes and see her \u2013 and, with every passing second, that moment was coming nearer and nearer. Soon it was no longer the tram that was swaying from side to side, but Maggie herself rocking him in her arms. Nothing else existed as she whispered his name over and over under her breath. This was going to be the best weekend ever.\n\nShe rushed into the children's home and straight to the superintendent's office. It turned out that Mrs Saunders was busy at present, but would see her in half an hour.\n\nHaving taken Tom for a short walk along the canal in the Tractor, Maggie returned him to his dormitory, settled him in his cot and went downstairs. For once, the superintendent was bound to be pleased with her \u2013 the six months would soon be up and here she was, preparing to take Tom back.\n\n'Really, Miss Davies?' Mrs Saunders took a cigarette from the packet on her desk, lit it and blew out the smoke in a slow, steady stream. For a moment they both watched it curl in the air between them. 'For the weekend, you say? Your landlady suggested it? A \"try-out\", like something you might get on approval from Jenners or PT's, is that what you mean?'\n\n'No, of course not. It's to see how he gets on so that the next time \u2014 '\n\n'The next time?' Another lengthy drag on her cigarette. 'Do you seriously think you can just waltz in here, tell me some story about your landlady and expect to be allowed to waltz straight back out the door, carrying Tom?'\n\n'No, it's not like \u2014 '\n\n'It certainly isn't. There are procedures.' Mrs Saunders looked her full in the face: 'And then, of course, there's you.'\n\n'Me? But what have I . . .?'\n\n'Well, Miss Davies, how can I put this without seeming to cause offence?' The superintendent paused. 'Whatever happens, or does not happen, will depend on whether we decide if you're a fit mother or not.'\n\n'But I love him.'\n\n'Love? That's the easy bit. Love's never enough and usually ends up causing more problems than it solves. In Tom's case we're well past the maternal love stage. Different when he was a newborn baby; then you were free to choose to look after him \u2013 and if you remember, you chose not to.'\n\n'But that's not what \u2014 '\n\n'We've cared for Tom, looked after him day and night. In normal circumstances he would've been adopted long ago, Miss Davies. But we've been very patient with you, letting you come and go as you please, letting you take him for a quick tour round the block . . .' she paused. 'And now your landlady's feeling in a good mood, here you are, telling me you fancy having him home for the weekend.'\n\n'But I'm his mother and \u2014 '\n\nMrs Saunders held up her hand for silence.\n\n'That, Miss Davies, remains to be seen. I grant that whenever you honour us with your presence he's very pleased to see you . . .'\n\n'Yes, he is, always, and \u2014 '\n\nAgain the hand was held up. 'That's only natural. But you know nothing of what goes on after you leave \u2013 how he cries and screams and won't let anyone hold him. Throws his toys at the other children, takes theirs and pulls them apart, bangs his head against the wall and \u2014 '\n\n'No. Tom's not like that. He \u2014 '\n\n'Don't you understand? Each time he sees you leaving, he really believes he'll never see you again. Never. You've abandoned him not once, but a hundred times.'\n\n'I've not abandoned him. I come as often as I can. It took ages to find a job, but now that I've managed to \u2014 '\n\n'You show up \u2013 and he's ecstatic. Naturally. It's like you've risen from the dead.' The older woman leant closer. 'Let me tell you, Miss Davies, an infant can take only so much ecstasy and grief, only so much loss. God only knows what he'll be like when he grows up.'\n\n'But that's why I'm wanting to \u2014 '\n\n'And the man?'\n\n'What man?'\n\n'Tom's father, naturally. Assuming, that is, you know who \u2014 '\n\n'How dare you! Of course I know who his father is!'\n\n'Had himself a very late war, did he? Only got demobbed a few days ago?'\n\nMaggie gritted her teeth. The woman was simply goading her.\n\n'Saving up for the tram fare to visit his son, is he?'\n\nMaggie gripped the edge of the chair to stop herself answering back. Whatever she said would be wrong.\n\nWith exaggerated calm Mrs Saunders turned to glance out of the window before continuing, 'I'm going on holiday soon. My husband and I are spending a long weekend with friends. Do you know Skye, Miss Davies?'\n\nSky? What had the sky to do with anything? For a moment Maggie pictured the superintendent and her friends drifting at their ease, perfectly at home among the clouds. That was Mrs Saunders' charmed life \u2013 holidays, friends, permanent sunshine.\n\n'No, I don't.'\n\n'Lovely place. But even if it wasn't, even if it was a total hell-hole \u2013 pardon my French \u2013 I know I'd still be having myself a ball. Why? Because I won't be here. I won't be having to deal with the likes of you.'\n\nAt once Maggie was half out of her seat, both hands on the superintendent's desk. 'What do you mean, the likes of \u2014 ?'\n\nSeeming not to notice the effect of her words, Mrs Saunders continued: 'The couple who adopt Tom \u2013 and I may as well tell you that there are several couples extremely keen to . . .'\n\nMaggie sat back down again. Several couples?\n\n'. . . will be a real couple. Married, settled, respectable. Able to give Tom a good home, eager to support our work here . . .'\n\nSeveral couples . . . a good home?\n\n'. . . and doubtless keen to express their gratitude to you.'\n\nMaggie glared at the superintendent. 'Yes, you told me all this the first time.'\n\n'Come in very handy for a smart new outfit and some high heels for the evening.'\n\n'You think I'd sell my son for a new pair of shoes!'\n\n'Not interested? Well, suit yourself. As a charitable institution we can't afford to be so sniffy. We need all the help we can get.' Mrs Saunders picked up her pen and resumed her paperwork.\n\n'Still here?' She laid the pen down again a moment later. 'Well, Miss Davies, perhaps you'd care to explain why you and your gentleman friend aren't getting married?'\n\nMaggie had run out of words.\n\n'No? Then let me guess . . .' The superintendent took a deep drag of her cigarette and blew out the smoke in a steady jet, then stubbed it out. 'He's already married, isn't he?'\n\nMrs Saunders waited for her reply, hand poised in mid-air as if ready to remove Maggie from Tom's life for ever with a single stroke of a pen.\n\nMaggie felt a wave of total exhaustion pass over her. 'The man's dead.' She stood up. 'Enjoy Skye.'\n\nShe left.\n\nThe following day, Maggie had to stay after work to make up for leaving early the day before, and so she arrived at Woodstock House much later than usual. For the first time, she found the front door locked.\n\nBut it couldn't be.\n\nShe tried the handle again.\n\nFirmly locked.\n\nShe reached for the bell pull.\n\nIts dull jangle-jangle tolled in the empty hall, to be answered almost at once by a scamper of footsteps coming to the door. Donna's welcoming smile was accompanied by a most elaborate curtsey.\n\n'Hello, Miss Davies. I'm practising for my first ball.'\n\n'Very good, Donna. I'm sure you'll have all the young men at your feet.' Maggie moved to go in.\n\nThe young chorus girl-cum-debutante didn't stand aside to let her pass. 'I'm sorry, but Mrs Saunders said that if you came, you were please to wait.'\n\nBefore Maggie realised what was happening, the door was closed again.\n\nWhat was she to wait for? She was here to see Tom, same as usual. Nothing was different. She'd come to take him for an evening walk like she'd done dozens of times before.\n\nOne good strong tug at the bell-pull would set it jangling like a demented Big Ben. No keeping her waiting after that!\n\nShe took hold of the bell-pull and was about to \u2013\n\nWhen she stopped herself.\n\nToo much noise and she'd probably wake the younger children. And as for Mrs Saunders . . . After their conversation yesterday, who knew what the old battle-axe might do if she got annoyed \u2013 maybe not let her see Tom at all?\n\nBut supposing he was ill? Mumps, chickenpox, measles . . .\n\nThe door opened again.\n\n'If you came they said to tell you that Tom has a bit of a cold today, but he'll be fine. Nothing to worry about, and this is for you.' Donna held out a sealed white envelope: miss margaret davies.\n\nMaggie ripped it open. Headed notepaper, stiff. Typed.\n\nDear Miss Davies . . . over six calendar months since . . . the contract dated 15th October 1949.\n\nShe had to start reading it again: Dear Miss Davies \u2013 two short paragraphs, and signed Yours faithfully, E Saunders (Superintendent).\n\nAnd again: Dear Miss Davies . . . 15th October . . . failed to take back your child . . . In absence of any formal application . . .\n\nOctober. The six months had been calculated from the day when the contract was signed. She'd not understood, that's all. It was just a mistake, a simple mistake. She could tell Mrs Saunders she'd got it wrong, explain to her that \u2013\n\n'Seeing Tom's not coming out today, can you take me with you instead?' Donna was tugging at her sleeve. 'We could go to the canal, if you like. See the ducks. Last week I saw four of them when \u2014 '\n\nShe'd thought it was six months from the day when Tom first came to the home. From October 1949, Tom wasn't even born then. How could they \u2013 ?\n\n'I need to see Mrs Saunders.'\n\n'Mrs Saunders isn't here. They said to say she's having dinner with the Government.'\n\nThe second paragraph was one short sentence. She had to read the words several times over: 'From the date of this letter, no further access to your child will be permitted.'\n\nNo further access . . .\n\nNo further access . . .\n\nNo further access . . .\n\nDonna was tugging at her sleeve. 'So can we go to the canal?'\n\n'Tom is here, isn't he? You've seen him?'\n\n'They said to tell you he's got a bit of a cold and he's getting better now, like I said. He's maybe sleeping.' Donna was trying to take hold of her hand now. 'The canal. Please, Miss Davies, please.'\n\nMaggie took a step forward and pushed at the part-opened door. It was being jammed from behind.\n\n'Donna. Indoors, now!' Boss Beryl stood in the doorway.\n\n'Bye, Miss Davies. Maybe we can see the ducks another time?' With a cheerful wave, Donna stretched up onto her tip-toes and then ballet-stepped gracefully back into the hall.\n\n'Mrs Saunders isn't here, Miss Davies, and the letter explains the situation.' Boss Beryl tried to push the door shut.\n\nMaggie stuck her foot in the gap. 'Is Tom all right?'\n\n'Please remove your foot, Miss Davies.'\n\n'Is Tom all right? I want to know. This letter says \u2014 '\n\n'Remove your foot and I'll tell you.'\n\nBeryl opened the door a crack wider . . . and the instant Maggie withdrew her foot the door was slammed in her face. She heard the key turn.\n\nMaggie stared at the locked door. If only she'd not spoken to the superintendent the day before. If only she'd not said a word to anyone, just taken Tom for his walk, same as usual, then simply kept on walking. He was her own child so it couldn't be stealing, surely?\n\nShe made a complete circuit of the house \u2013 all the windows were closed, all the doors locked. When she rapped on the playroom window, some of the older children saw her and waved back. The staff ignored her. There was no sign of Donna. Then, one by one, all the downstairs curtains were drawn shut. The building had been made into a fortified castle.\n\nTwice she went round it.\n\nHaving returned to the front door she gave the bell pull another tug \u2013 firmly but not too strong. Not wanting to wake the wee ones who'd already be in bed, not wanting to upset anyone, not wanting to make a scene. All she wanted was to see Tom, to know that he was safe and \u2013\n\nAnother tug at the bell.\n\nNot a sound this time.\n\nShe tugged again.\n\nAnd again.\n\nThe bell pull had gone completely slack, its brass handle no longer sliding smoothly back into the wall.\n\nHanging loosely on its wire.\n\nDisconnected? Did they really think a disconnected bell would would stop her? That she'd simply give up and walk away?\n\nShe crouched down and began calling through the letter-box, loudly as she dared: 'Beryl! Mrs Saunders!'\n\n'Stop that row!' Boss Beryl was right behind the door.\n\n'Let me see tom!'\n\nThe response was immediate. 'Stop your shouting! You'll frighten the children!'\n\n'Open the door then. Please.'\n\n'We'll call the police.'\n\n'No \u2013 I only want to see Tom, and to know he's okay. That's all. I don't want any trouble.'\n\nShe waited.\n\n'Talk to me face to face at least.'\n\nShe waited longer.\n\nThen, after several seconds, Beryl's voice came back to her: 'All right. But you have to stand clear first. Then I'll open it.'\n\nMaggie took an immediate step back. 'I'm standing away.'\n\nAs soon as she heard the key turning in the lock, she prepared herself.\n\nShe watched the door ease open an inch at a time.\n\nBoss Beryl faced her through the tiny gap. 'You're still too close, Miss Davies.'\n\nWithout taking her eyes off Beryl, Maggie took another step back. 'Far enough?'\n\nShe focused on the crack she could see widening between the door and its frame. She tensed herself. She was ready. She knew she would only have the one chance.\n\n'Is Tom with you?'\n\n'No, he's fast aslee \u2014 '\n\nLunging suddenly forward, her whole weight shouldering the door \u2013\n\nBoss Beryl tried to stand firm, but was too late.\n\nMaggie shoved her aside, then rushed across the hall and bounded up the stairs three steps at a time. A group of younger children, some holding hands, stood on the landing, one of them calling out, 'Mummy! Mummy!' at the top of his voice.\n\n'Sshhh! It's okay, it's okay. Shh! I'm here to see Tom.'\n\nBoss Beryl came hurrying after.\n\n'Stop! You can't just barge your way in. Mrs Saunders said that \u2014 '\n\nNext moment she was in Tom's dormitory, Boss Beryl calling behind her:\n\n'What do you think you're doing? Get back downstairs at once.'\n\nMaggie headed across to the far corner. Past Crib Number 8. Crib Number 9 . . .\n\n'Mrs Saunders said that if you came . . .'\n\n. . . Crib 10 . . . Crib 11 \u2013\n\n'. . . I was to tell you that \u2014 '\n\nCrib 11 was empty. Tom's crib was empty.\n\nShe snatched up the patchwork blanket, pressing it to her face. Tom's blanket.\n\n'Where is he? Where is he?'\n\nSUNDAY\n\n'FINISH, MRS STEWART?' Donna's come to take away your tray.\n\nThat lass had better take care she doesn't let herself go, helping herself to a chocolate biscuit every time she gets a chance . . . She'll never become a chorus girl gorging herself like that.\n\n'Where's all the photos, Mum? There's nothing here.'\n\nYou watch him turn over blank page after blank page until he comes to the only photograph left in the album.\n\n'Who's that?'\n\n'Donna, of course.' You smile at the would-be dancer as she lifts away the tray. 'Before she discovered chocolate biscuits. Am I right?'\n\nThe young Polish woman squints at the photo. 'Nice baby, Mrs Stewart. I come soon with meds. Bye, Mrs Stewart.'\n\n'But, Mum, you can't be meaning someone here, in Rosehaven? This is just a wee girl \u2013 it's an old black & white photo, must have been taken over fifty years ago! The pram's like something out of a museum. How can it be the Polish woman here, the one you call Donna? And what's happened to the rest of the photos?'\n\nYou shrug again. All these questions. Like someone turning the wringer, wanting to get all the water squeezed out. But there's hardly a drop left, is there?\n\nManaging without anyone's help, you slide the photo out of its plastic sleeve. 'That's my little boy. He's called Tom.'\n\n'Tom? But I'm \u2014 '\n\n'Ssh! It's a secret, remember. But he's being well looked after. Donna treats him like a wee brother.'\n\n'I don't understand, Mum.'\n\n'Sssh! I know what: you take it, you can look after it for me. Things vanish here sometimes.'\n\nHe keeps saying he doesn't understand and you keep insisting. Finally he gives you a nod. 'If you're sure that's what you want,' and he slips Tom's photo into his jacket pocket. 'It'll be safe now.'\n\nYou smile at each other. For once, something feels right.\n\nWith your visitor gone, the sun'll be on the move again. And so . . .\n\nZimmering full-speed out of your bedroom door, down the corridor, round the corner, across the hall and into the dayroom. Your usual seat in its usual place in the line against the wall. You sit down just in time. Closing your eyes, you feel the sun edging its way onto your cheek. Its warmth flowing across your face, your eyelids, your cheeks, touching your lips, your neck . . . soaking into you.\n\nMore. More. You want more. You want to feel that warmth flooding into you, drenching your whole body.\n\nTake care \u2013 if you open your eyes too soon, you'll find yourself back in the dayroom.\n\nThe sun is all that matters now, holding you safe and secure before it moves on. Till then, you can give yourself completely to its touch, its warm and loving touch. Like Michael's. These sighs of contentment are the thanks you offer in return.\n\n7\n\nSTANDING MOTIONLESS BESIDE Tom's empty cot, her face buried in his patchwork quilt, breathing in his baby-smell. Drawing in its sweetness, breathing it deep inside her . . .\n\nThen she felt the touch of someone's hand on her shoulder.\n\n'It's all right, Miss Davies. Tom's fine. Your wee boy's fine.' Boss Beryl had switched to professional calm \u2013 pitched low, her voice set at sympathy-and-concern tone. The reassuring smile seemed genuine: 'Don't worry, Miss Davies. Really. Everything's been taken care of and couldn't be better. Tom's fine and in good hands. No need to distress yourself.' Boss Beryl crossed to the nearest cot and picked up a swaddled bundle that was clearly on the point of turning into a swollen-faced, full-volume, red scream and began rocking the baby in her arms. 'There, there, sweetie. Nothing to worry about. Everyone's going sleepy-byes . . .'\n\nMaggie clutched the patchwork blanket to her chest. 'Where is he? Where's Tom?'\n\n'Arthur's Seat, I think they said.'\n\n'Arthur's Seat?' Maggie glanced towards the window. 'I don't understand. What's he doing there? Who's they, and why \u2014 ?'\n\n'It's okay, Miss Davies, really. Calm down. Lucky boy's been having himself a day out.'\n\n'He's coming back?'\n\n'Of course.'\n\n'When?'\n\n'Or maybe it was Princes Street Gardens they said? A chance for him to hear the band. A day out, like I said. He'll have had a whale of a time, you can count on it.' She reached to take the blanket. 'This his? I'll put it in the wash. Thanks.'\n\nMaggie's grip on the patchwork tightened. 'When's he coming back?'\n\n'Keep your voice down, Miss Davies.'\n\n'No one told me anything. Not a word.'\n\n'Maybe they should've. Bit of a shock, I suppose. But . . . well, it had to happen one day. Children don't stay here for ever.'\n\n'He is coming back?'\n\n'Yes. But Mrs Saunders says that seeing you've not managed to \u2014 '\n\n'I told her that this weekend I was \u2014 '\n\n'I'm very sorry, Miss Davies, and I wish I could help you.' Boss Beryl shook her head and again reached for the blanket. 'I really do.'\n\nMaggie's fingers kept firm hold of the patchwork. 'Mrs Saunders can't just give him away.'\n\n'Keep your voice down.'\n\n'Don't you tell me what to do. I've done everything I can. Everything. And now he's being stolen from \u2014'\n\nThe petrol pump took a step back. 'No one's stealing him. He'll be coming back, I tell you.'\n\n'When?'\n\n'Mrs Saunders says \u2014 '\n\n'I couldn't care less about Mrs Saunders.' She advanced on Boss Beryl. 'When? I asked you when will my son be brought back?'\n\n'No idea. I don't know. No one tells me. I don't do the office stuff.'\n\nNo idea? Without meaning to, Maggie ripped the small patchwork quilt in two and then, the torn pieces still in her hands, she all but collapsed against the empty cot. She was suddenly exhausted.\n\n'I'll wait for him.'\n\n'Suit yourself \u2013 but you'll not be waiting in here. Everything's okay, little one. Close your sleepy eyes and . . . Mrs Saunders is at a Governors' dinner this evening. Your best plan's to give her a phone first thing Monday morning.'\n\n'But what am I to do? That letter says that I'm not allowed to \u2014 '\n\n'Look, Miss Davies, I've got my hands full here. If it was up to me . . .' She shrugged. 'Drop her a line's my advice.' She laid the now sleeping child down on its mattress. 'Back to bed, sweetie. Sleepy heads, sleepy beds, sleepy, sleepy sleep . . .'\n\nArthur's Seat? Princes St Gardens? A chance to hear the band . . . ? Was the whole city of Edinburgh betraying her? Maggie turned and stumbled out of the room.\n\nSpending the day with strangers, and the evening too. Having a whale of a time with strangers. Strangers holding him in their arms. Strangers making him smile and laugh . . .\n\nThe wooden banister . . . then down the staircase, tread after wearying tread . . . the empty hall.\n\nShe sat down in the carved wooden chair, the torn pieces of Tom's patchwork blanket on her lap. Not even strength enough to cross her legs.\n\nQuarter-of-an-hour later Boss Beryl came down carrying a bundle of dirty sheets.\n\n'You still here?'\n\n'I'm waiting for Tom.'\n\nBoss Beryl snorted, and continued down the basement stairs. When she returned a short time later she crossed the hall to Mrs Saunders' office, closing the door behind her. Maggie heard her speaking on the phone but couldn't make out the words. A few moments later she was back.\n\n'I'm sorry, Miss Davies, but you'll have to wait outside. You're a disturbance.'\n\n'What? I'm not disturbing anybody. I'm not leaving till I see \u2014 '\n\n'And now you've received Mrs Saunders' letter, you're trespassing. That's the law. Do I have to call the police?'\n\nUp and down the short stretch of pavement outside the home, up and down in the gathering darkness. Eventually letting herself back into the garden to sit on the bench next to the sandpit with its tumbled remains of a bucket-and-spade castle, its trampled-down battlements. She was careful to keep watch on the street, but no cars stopped. No one came to the house, no one left. From time to time a tram rumbled past. At ten o'clock she saw the upstairs lights being switched on and off as the staff made their final rounds.\n\nAt ten-thirty the last light went out \u2013 everyone had gone to bed.\n\nNext day was Saturday. Come lunchtime, Maggie hurried out of Blair & Blair's and up the street to the Hanover Street PO phone box.\n\nNo reply \u2013 Jean probably wasn't at the bakery this weekend. She pressed button B and her pennies clattered back to her.\n\nWhich left her parents.\n\nHer parents. Their ignoring her, cutting her dead as she'd stood there in front them \u2013 the knitting needles' relentless click-clicking, the stripped-out bedroom upstairs. Like she'd never even existed.\n\nMrs Melville would have talked, though. Her mother would have listened. Tom was her own grandson, after all.\n\nIt was worth a try. She was going to need all the help she could get.\n\nShe dropped in the coins once more and dialled \u00adNewhaven.\n\nAs it rang, she pictured the telephone receiver on the hall table with the oval mirror above, and the silenced grandfather clock standing nearby.\n\nHer father answered. She hung up.\n\nMaggie made two trips to Woodstock House that afternoon and three on Sunday, each time finding the front door firmly locked. The bell hadn't been re-connected and the only response to her frantic knocking was to see either Boss Beryl or Donna appear at one of the windows. Boss Beryl would jerk her arm from side to side, gesturing at her to go away; Donna would give her a friendly wave and remain standing at the window until she left.\n\nFirst thing Monday morning before work, Maggie phoned Jean from the Hanover Street call box.\n\n'Lucky to catch me in, Maggie. I was picking up a cake I've made for the folks we'll be staying with. We're just off.'\n\n'Jean, they've stopped me seeing Tom. Mrs Saunders' letter says that \u2014 '\n\nJean was really, really sorry but she had to dash. She and Billy were going away for a few days. Been arranged ages ago. She'd be back late on Friday. After Maggie had finished work on Saturday they could go together to Woodstock House and \u2013\n\n'But they won't let us in. Saunders' letter says that I can't \u2014 '\n\nJean was so, so sorry. How could they? That was awful. Really terrible. But she had to go. She just had to. Billy was waiting for her on the platform and she was already late. If they missed that train . . . Maggie mustn't lose heart, mustn't give up. She would be in her thoughts all week. And Tom, too.\n\nThey said goodbye.\n\nMaggie checked her appearance in the mirror of her compact. She waited, dabbing her eyes. She now had to go to work. Nothing of the distress tearing her apart must show, nothing must betray her.\n\nFor the next few days Maggie hardly ate or slept. Before work, during her lunch break, and after work she'd ring the children's home. When Old Woodbine was out of the office, she sometimes even risked using the phone on the older woman's desk. The instant she heard her returning along the corridor, she hung up and hurried back to her own seat.\n\nShe was told that Mrs Saunders wasn't in.\n\nShe was told to please wait, Mrs Saunders would come to the phone shortly.\n\nShe was told that Mrs Saunders would return later in the morning \/ later in the afternoon \/ first thing next day.\n\nShe was told that Mrs Saunders knew she'd been trying to get hold of her and would be sure to be in touch.\n\nShe was always asked to leave her number.\n\nLeave her number? Forget it. For even if Mrs Saunders did phone her back \u2013 which Maggie was certain she never would \u2013 then Old Woodbine would take the call . . . and that would be that. Mr Blair was a kindly man \u2013 he might be the boss, but he wasn't in charge. Old Woodbine sharing an office with an unmarried mother? Blair & Blair letters typed on a Blair & Blair typewriter by an unmarried mother? Blair & Blair clients being exposed to the risk of having contact with an unmarried mother? No chance, not in a million years. The first hint of her having an illegitimate child hidden away somewhere, and Old Woodbine would have her sacked on the spot. She'd be shown the street door so fast her feet wouldn't even touch the stairs on the way down.\n\nShe wrote three letters to the superintendent, each more desperate than the last.\n\nHer first received an immediate response \u2013 a copy of the letter she'd already been given.\n\nHer second \u2013 no reply.\n\nHer third \u2013 no reply.\n\nShe wrote to Michael. She told him everything, said her life was falling apart, said she couldn't sleep for worrying she might lose Tom, might never see him again . . . They'd stopped her. Everything was locked. And he wasn't there. They were giving him to strangers. She was his mother. If that happened . . . she couldn't go on living . . . couldn't bear it. Page after page, it was a frantic outpouring of frustration and rage, of despair. She received Michael's reply by return, on the Saturday morning.\n\nWas Old Woodbine never going to move, was she going to remain glued to her chair all morning? Not until nearly 11.00 did the older woman get up and leave the room.\n\nThe instant she was alone Maggie crossed to the other desk and phoned Jean. Her sister-in-law was in.\n\n'Thank God, I've caught you, Jean. It's Michael. I'm at my wits' end. I don't know what I'm going to . . . Can I come round and \u2014 ?'\n\n'I'll be here all day, Maggie. Take it easy. You can tell me when you get \u2014 '\n\nHearing Old Woodbine coming back along the corridor, Maggie hung up.\n\nFrom then on, the hands of the office clock slowed down to quarter speed.\n\nAt last she was on the tram. Standing room only, all the way to Dalry Road. She leapt off well before it stopped, ran across the street, straight up to Jean's, and in her front door. Not even pausing to say hello \u2013\n\n'He's coming to Edinburgh.'\n\nOnly when she'd said the words out loud did she herself really take them in: Michael. In Edinburgh.\n\n'He's whit?'\n\n'Arriving late tonight.'\n\n'Tae rescue ye, like a knight on a white horse? A white stick mair like \u2014 '\n\n'Wants us to get married.'\n\nJean picked up the phone: 'Ring him. Tell him tae \u2014 '\n\n'Too late. He's already on his way. Morning ferry, then the train. Lachlan's bringing him here.'\n\n'Tae the bakery?'\n\n'He said he knew he couldn't just turn up at Mrs McCann's, so I'm to meet him here tonight at 11.' She trailed off in embarrassment: 'He wasn't sure how the trains would work out . . .'\n\n'He can stay a few days, Maggie, but that's aa. I canna dae mair, it's just no \u2014 '\n\n'I'm sorry, Jean. I'm so sorry, I'm sorry . . .' She collapsed onto the chair with her head in her hands. Then she told her sister-in-law about trying to phone Mrs Saunders and writing letters. 'I'm going to lose Tom. I'm going to lose him, Jean. What'll I do?'\n\n'Dinna tak on, lass. Dinna greet.'\n\nMaggie felt the older woman's arms round her, and her hand stroking her hair.\n\n'I'm sorry tae,' Jean whispered. 'Oh, Maggie \u2013 it's twae bairns ye'll hae nou.'\n\n'But I love him. I love them both.'\n\n'Richt enough, but Tom's the yin as needs ye maist. Tom needs ye. Forget Mrs Saunders, forget writing her letters and phoning and leaving messages. Forget playing by their rules, Maggie. Ye'll only lose \u2013 that's whit rules are fer.'\n\n'And so?'\n\n'Play tae yer strengths. We'll play tae oor strengths. Mrs Saunders will be away fer a lang weekend, ye said? Okay \u2013 it's nou or never.' Her sister-in-law stood up. 'Come on, we've work tae dae.'\n\nIt was evening when the two women climbed out of the taxi. Side by side they marched up the front steps of Woodstock House.\n\nOnce it was explained that they'd come to hand over a present for the staff, a token of Maggie's appreciation for all their good work when looking after Tom, they were let in.\n\nThe shiny pink box, tied with the showy gold ribbon that Jean had curled at the ends, was placed on the kitchen table. It looked a very special gift indeed. When it was unwrapped, there were genuine gasps as the lid was raised to reveal a triple-layered, mouth-watering masterpiece of chocolate, cream and marzipan that had been showered in multi-coloured hundreds-and-thousands. Quite clearly the ration book had been thrown out the window. thank you was spelled out in red icing. Thick slices were cut and handed round.\n\nEverything was going to plan, and so Maggie asked if she might go upstairs for one last look at her wee boy?\n\nThere was an embarrassed silence.\n\n'I won't wake him or pick him up.'\n\nMore silence.\n\nBoss Beryl finished swallowing down a mouthful of cake: 'I'm afraid that's not possible. Mrs Saunders has made it quite clear that \u2014 '\n\n'Mrs Saunders isn't here.'\n\n'I am her representative. Myself, I'd like to let you see him, Miss Davies, really I would, but \u2014 '\n\n'Come on,' interrupted Jean. 'Yin last look at her bairn, for God's sake. Just because yer boss is a wee Hitler disnae mak you yin. Maggie winna even touch him \u2013 aa richt? Ye can gang up wi her. I'll stay here \u2013 guard yer next slice for ye.'\n\nSeeing Tom lying there asleep in his crib, Maggie broke down and cried as if her heart would burst. While Boss Beryl looked on, she sobbed and sobbed. She couldn't help it. No pretence needed, these were real tears. Maggie knew that if things didn't work out this might very well be the last time she would see Tom. She was so upset she nearly had Boss Beryl in tears as well. Finally, though, she mastered herself and begged to be allowed to kiss him a final goodbye.\n\nAfterwards, wiping her eyes clear, she turned and stumbled out of the room.\n\nDown in the hall, Boss Beryl laid a hand on her arm, adding that she was so very sorry that everything had ended like this. Maggie should rest assured Tom would be well looked after, he'd be brought up in a good home with a loving family who could provide the best for him. Mrs Saunders was a really kind woman and always had the children's best interests at heart. Maggie nodded, then asked if she could use the toilet before they left. Boss Beryl remained waiting outside until Maggie had finished, and then escorted her back to the kitchen.\n\nFive minutes later Maggie and Jean said their goodbyes and left.\n\nNow they had to go their separate ways \u2013 Jean to the \u00adbakery to wait for Michael's arrival, Maggie to Glengyle Terrace to make the necessary preparations. By ten-thirty she had everything ready. It was too early to return to Woodstock House, but there wasn't enough time to go all the way to the bakery and be there to greet Michael.\n\nMichael. How she longed to see him. Nearly a whole year's longing. And in less than a couple of hours . . .\n\nThings would work out, they had to. She'd done \u00adeverything she could. Now she needed to wait. That was all.\n\nSo she put on her coat and outdoor shoes, and sat on the edge of her bed. Waiting. A few minutes later she was up again, pacing the room. Then back on the bed, only to get up and start pacing all over again . . .\n\n'It was nearly 11 when Maggie pulled the front door shut behind her, went down the steps and across to the park. Late April perhaps, but it looked and felt like the darkest night of the year. Not a star in the sky and a dampness and heaviness in the air that threatened rain, and lots of it. Good. It would keep people off the street.\n\nShe stepped lively \u2013 across Bruntsfield Links, past the unlit shops and silent tenements to Holy Corner. The only sound was her own footsteps.\n\nIf she wanted, she could turn round at any moment and go straight to the bakery instead. No harm done. She would marry Michael, set up home and then Mrs Saunders would surely let her keep Tom. She would have to. She would. Surely.\n\nBut by then it might all be too late.\n\nMaggie kept walking. The first drops of rain began to fall as she turned off Morningside Road. A glance at her watch, plenty of time.\n\nShe didn't trust Mrs Saunders. It was as simple as that.\n\nBy the time the outline of the children's home loomed up in the darkness, the rain was coming down in torrents. Not a single light showed. Maggie checked her watch again \u2013 she'd made good time and now had exactly thirty-five minutes. She pushed open the gate.\n\nOf course you can't come, too, Jean. I'll need someone to visit me in jail! she'd joked as they'd said goodbye. But it was no joke now.\n\nShe paused for a moment on the wet path, drenched through and almost blinded by the driving rain. This was her last chance to turn back. If she was caught and sent to prison, then Tom would be lost to her for ever.\n\nBut as things were, she'd already lost him. So what real choice had she?\n\nNext moment she was squelching across the lawn.\n\nHaving reached the rear of the building, she flashed her torch quickly round the cement yard till she found a pile of discarded wooden boxes stacked at the side of a small outhouse. An orange crate looked the sturdiest. More or less intact and the wood slimy with rain, it was probably strong enough to take her weight. She returned to the back wall \u2013 another flash of her torch located what she was looking for, the window with frosted glass. She positioned the box beneath it.\n\nHolding onto the window ledge for support, clambering up. Standing there for a moment to get her balance, wobbling slightly. A tentative half-rocking from side to side on her feet to test it. The box held.\n\nSo far so good.\n\nNow for the tricky part. Both hands pressed against the sides of the window frame, she pushed upwards.\n\nNothing happened. Not the slightest give.\n\nPushed again.\n\nShe paused to wipe the rain from her eyes, then pushed even harder . . .\n\nNot too hard though. The window had to be eased open ever so slowly. Eased open noiselessly.\n\nShe pushed . . . and pushed . . . and pushed. It was the toilet window all right, the one she'd unsnibbed a few hours earlier, but someone must have re-snibbed it.\n\nAfter all their careful planning . . . She stopped herself from screaming out loud.\n\nFor several seconds she remained standing on the box, hands on the wooden sill, struggling to hold back her tears, resisting putting her fist through the window.\n\nAt last she stepped down and carried the box along to the next window. If she had to, she'd work her way right round the house. Having come this far, what else could she do?\n\nClambering up again at the next window. Then the next after that. Her hands pressed against the frame . . .\n\nStarting to push upwards . . .\n\nThird time lucky! It slid open easily. The wood screeched. She froze.\n\nRemained standing motionless for as long as she could bear, the rain lashing down onto her head, shoulders and back . . . She forced herself not to move. Fist clenched, she counted under her breath, ready to jump clear and make a run for it at the first sign of a light going on, or at the sounds of someone coming.\n\nNineteen, twenty. She relaxed, inched the window up enough to let her climb through . . . Moments later she was standing in Mrs Saunders' office. Perfect.\n\nThere was even a towel provided, a pair of towels, in fact \u2013 rust-red, extra thick and long. Having wiped some of the rain from her hands and face, she closed the curtains and switched on her torch. A glance at her watch \u2013 twenty-two minutes left. She and Jean had debated what she should do next \u2013 secure the paperwork first, or go straight upstairs? There'd be masses of files and she'd no idea how long she'd take to find what she was looking for.\n\nShe had to speed up.\n\nThe metal filing cabinet wasn't locked. Good.\n\nTop drawer \u2013 Adoptions. The files each had a tab and were arranged by date. Good again. Most recent at the front \u2013 Montrose, Iris. 3\/12\/50. Next was Watson, James. 28\/11\/50. No mention of any adoption involving Tom. She breathed out a sigh of relief and slid the drawer back into place.\n\nThe one beneath was labelled Admissions.\n\nShe riffled through the files . . . 12\/12\/50 . . . 18\/11\/50 . . . 3\/10\/50 . . . 25\/9\/50 . . . to reach those further at the back. About three-quarters in, she came across the tab marked '17\/11\/49 \u2013 Davies, Tom.' Her hands shook as she pulled out the file.\n\nShe laid the folder on Mrs Saunders' desk and, by torchlight, turned over the few pages of Tom's short life. Even though she was in a hurry, she flicked through the sheets to check everything was there \u2013 his birth certificate, the letter from the Queen's Crescent nursing home and the doctor's notes, the detested contract she'd signed and the various other forms Mrs Saunders had made her complete. Her own details were also included: Jean's bakery address had been scored out and replaced by Glengyle Terrace. A handwritten note was attached: Father probably still married. There was a copy of the letter denying her access and her own unanswered replies, also the photo she herself had taken of Donna and Tom beside the Tractor. In addition, a sheaf of papers clipped together was headed: Adoption \u2013 Interest. Promising herself to burn these unread, she stuffed the complete file under her jersey and pushed the drawer shut. Good \u2013 there'd be no tell-tale paperwork left behind.\n\nAnother glance at her watch \u2013 only eighteen minutes left. She switched off the torch and stepped into the hall. It was in complete darkness. Above her, she could hear the rain hammering onto the glass cupola. Making sure she didn't blunder into the wooden throne or the Tractor, she crossed to the vestibule. A quick flash of the torch located the key, she turned it slowly to unlock the front door. No top and bottom bolts, thank goodness. Back to the hall. If it had been a clear night there would have been enough light to see her way upstairs. But it really was pitch black.\n\nKeeping the torch beam pointed down at her feet, she began climbing the stairs, holding on to the banister in the darkness.\n\nHalfway up, one of the boards creaked underfoot. She froze. Switched off her torch. Stopped breathing.\n\nBegan counting into herself as slowly as she dared. One, two, three . . .\n\nShe stood motionless, straining to hear the slightest sign that someone . . . eight . . . nine . . . ten . . . eleven . . .\n\nAt full alert, rigid, peering into the darkness . . . fourteen, fifteen . . .\n\nThen she relaxed.\n\nPlaced her foot down gradually, very, very gradually. Letting the next step take her weight slowly, steadily . . .\n\nAgain she held her breath.\n\nNothing.\n\nThe next tread. Pressing lightly as she could. A little more, a little more . . . The only sound was the rain gusting heavier every few seconds against the cupola. She continued up to the top.\n\nFifteen minutes left.\n\nFrom along the corridor came the rasp-and-snort of someone snoring. Boss Beryl? She certainly hoped so.\n\nTime began speeding up. Keeping the torch beam directed at the floor, she inched open the door and stepped into Tom's dormitory. Tiptoeing across the room, slowly, carefully. Watching out for the table that stood in the middle, and for any pails or \u2013\n\nOne of the babies was gurgling to itself, clearly awake. Another was standing up in its cot and stared at her as she passed.\n\n'Ssssh! Little one. Please don't . . . don't . . .'\n\nThe baby blinked, reached out its small hand and started to whimper.\n\n'No . . no. Please, please no. Sssssh!'\n\nThen, a moment later, it slid down flat again on its mattress and closed its eyes.\n\nMaggie let out a breath she hadn't realised she'd been holding in. Like crossing a minefield, she thought. She reached cot number 11.\n\nTom was fast asleep, wrapped in a new blanket and with one of Donkey Boy's floppy ears clutched in his tight little fist. He looked perfectly content.\n\nHe was usually wide awake when she visited. Not like this, lying here so peacefully, safe and secure in his familiar surroundings. She knew she was about to disrupt all that. She'd be wrenching him away from everything he had ever known in his brief life, destroying a carefully planned future. Other people's good intentions, other people's plans.\n\nShe could still turn back now, creep off down the stairs and walk out the front door. But then what?\n\nThe pendulum she'd thrown into the glass-green water and had imagined sinking deeper and deeper into the harbour \u2013 would that be her? Without Tom, what would the days and years ahead be but a hopeless choking slide into ever-deepening mud?\n\nNext moment she'd lifted him out of the crib.\n\nHe didn't even wake.\n\nShe closed the dormitory door behind her.\n\nOut in the corridor she paused. Something seemed different from before. The complete silence. It had stopped raining. Moonlight was beginning to filter down from the cupola.\n\nShe switched off the torch and, treading even more carefully, made her way along the corridor. Already she could make out the silhouettes of a fire bucket, a side table, picture frames. The top of the stairs only a few steps away.\n\n'Miss Davies?'\n\nShe stopped. Utterly. Rigid.\n\nA faint shimmer of whiteness up ahead. A whispered voice: 'Miss Davies?'\n\n'Ssh, Donna!' Maggie hissed and put a finger to her lips.\n\n'You've got Tom with you?'\n\n'Ssh!' Maggie stood beside her at the top of stairs. 'Donna, please. Ssh!'\n\n'Take me, too.'\n\nThe moonlight was getting stronger. The future chorus-girl was in a long white nightdress and bare feet, her hair sleep-tangled. 'Please, Miss Davies, please.'\n\nWhat if she just shoved Donna out of the way and rushed down the stairs without stopping, then ran out the front door and off up the street? Everything might still be all right. She had all the official records with her, and she had Tom. But she'd have to do it now.\n\nMaggie shook her head and tried to push past.\n\nThe girl didn't budge: 'If you don't take me, I'll scream as loud as I can and wake everyone.' She stretched her arms out wide, barring Maggie's way down.\n\nTom had started fidgeting in his sleep, his small feet pressed into her stomach and his head shifted from side to side as if he might wake at any moment.\n\n'Please, Donna. You don't understand. Let me past. I have to take Tom and \u2014 '\n\n'I don't care,' the young girl whispered. 'Take me \u2014 '\n\nA door opened at the end of the corridor. 'Who's there?'\n\nIt was Boss Beryl. Maggie clutched Tom to her chest.\n\n'It's just me, Mrs Ferguson.' Donna replied in her normal voice. 'Needed the toilet. Very sorry for waking you.'\n\n'I've told you before about drinking water last thing at night, Donna. Be as quiet as you can and then straight back to bed. Goodnight.' The door closed.\n\nMaggie whispered, 'Thank you, Donna.'\n\n'Give me Tom.'\n\n'What?'\n\n'I need to get dressed and I'm not letting you leave without me. Give him here.' She held out her hands. 'If you don't, I'll scream. I mean it.'\n\nMaggie hesitated.\n\n'I will, Miss Davies. I will. Really.'\n\nDonna occupied an attic room up a flight of narrow, uncarpeted stairs \u2013 more like a walk-in cupboard with a bed under the sloping ceiling. Maggie watched her get dressed. Only eight minutes remained.\n\n'Donna, you must understand that \u2014 '\n\n'I'm ready now, Miss Davies. Let's go.'\n\nWithout answering, Maggie laid Tom down carefully on the girl's bed.\n\n'Listen, Donna . . .'\n\nShe grabbed the girl by the shoulders and leant into her face. 'You can't come with us. You can't. Understand?'\n\nStruggling and kicking out to free herself, the girl jerked to one side as if she was about to call for help. Instantly Maggie clamped a hand over her mouth.\n\n'You try coming with us and I'll get rid of you,' she hissed into the child's ear. 'They'll think you ran off during the night, taking Tom with you.'\n\n'But that's not what \u2014 '\n\nMaggie clamped her hand harder.\n\n'The police'll hunt you down and when they find you alone and without Tom \u2013 he and I will be long gone by then \u2013 they'll think you killed him.'\n\n'Mmmmmm . . .'\n\n'People do that sometimes, you know, kill children.'\n\n'But I wouldn't \u2013 mmmmm . . .'\n\n'They'll hang you, Donna.\n\n'Nooo \u2013 mmmm . . .'\n\n'The rope'll go tighter and tighter round your neck until you can't breathe, till your eyes burst, till your tongue turns black. Understand?' She ignored the terror she could see in the girl's face. 'Do you want that?'\n\nDonna shook her head.\n\n'If you scream for help now, I'll run and I'll get away.' Maggie paused, then stared deep into the girl's eyes. 'I'll return one night soon, just like I did tonight. And I'll creep up these stairs, like I did tonight. I'll come here, right into your room while you're asleep . . . and snap your neck. Understand?'\n\nThe girl nodded. She'd started to cry.\n\n'So then, are you going to scream?'\n\nDonna shook her head.\n\n'Good girl.' Maggie began loosening her grip.\n\n'You won't take me with you \u2013 ever?'\n\n'I can't. I'm really sorry, Donna. I'd like to, but I just can't.' Then she added. 'Maybe once we're settled we can meet one day and \u2014 '\n\n'I hate you!' Wiping the tears with the back of her hand\n\n'Donna \u2014'\n\n'I hate you. I hate you.' The girl threw herself onto her bed, burying her face in the pillow.\n\nMaggie reached down to stroke the tangle of curls, then stopped herself. She sighed, picked up Tom, and left the room.\n\nClutching Tom in her arms, Maggie paced up and down the short stretch of pavement where the taxi was supposed to meet her. Past low stone walls stripped of their iron railings; past high stone walls still streaming wet in the streetlight; past gates locked for the night: past trees whose branches thrashed the darkness. Three, four times she walked to the red pillar box at the next corner, turned and came back again. Up and down, up and down with hardly a pause. She'd got there only a couple of minutes after midnight. Maybe the taxi had arrived and, seeing no one waiting, had driven off? But then where was Jean? Michael?\n\nAt every turn she glanced back down the street towards the children's home, terrified she'd see lights and hear people shouting. Woodstock House, however, remained in darkness. Remained silent. She was ashamed at the way she'd threatened Donna. But what else could she have done?\n\nHalting several times in mid-stride \u2013 was that the clanging of a police bell? Was a squad car about to come screeching round the corner?\n\nTwice she saw headlights approaching from Bruntsfield, only to watch the car drive straight past. The only sound was rainwater gushing out of a blocked gutter nearby, splashing down onto someone's front step.\n\nOnce again she walked to the pillar box and back, singing under her breath:\n\nMy Baby Bunting,\n\nDaddy's gone a-hunting,\n\nTo catch a baby rabbit's skin\n\nTo wrap our \u2013\n\nAt last \u2013 another set of headlights, and coming from the right direction. It was slowing down. A taxi. Their taxi. Had to be.\n\nThe black cab drew up beside her, the driver's window was lowered:\n\n'Sorry I'm late, Missus. Engine kept needing cranked up. Doesn't like getting up this early on a Sunday morning!'\n\nJean was sitting in the back. She was alone.\n\nMaggie climbed in. 'But where's \u2013?'\n\n'Dinna fear, Maggie, Michael's back at the bakery. He's waiting fer ye.'\n\nSUNDAY MORNING\n\nAN AXE TO the worn-out dining chairs. Slice and dice. Turn carpentry to kindling and haul the smashed woodwork in armfuls out to the garden and up the path to the heaped bonfire. Dragging the fifties' kitchen cabinet that even the charity shop didn't want. The three-legged stool you used to stand on when your mother did the washing, the rickety hat stand, wooden clothes-horse, bits of smashed shelving \u2013 worthless junk the lot of it, good only for burning.\n\nYou'll keep these photographs of your mother, of course \u2013 in her wellingtons boots digging up potatoes, dressed up smart for a trip to Edinburgh, and some others. The rest are a load of strangers and there's no one to tell you who they were. Back in the black-and-white past, people hoarded their photos. Not like today's disposable memories \u2013 the freeze-framed smiles, drunken moonies, drunken meals, the googled Earth itself. Everyone's life Photoshopped to perfection for posting on Facebook or YouTube. The moment's been saved . . . then so are you?\n\nNot really. But getting there.\n\nA nice one of Auntie Jean. That smoky voice of hers and her tobacco-smelling clothes \u2013 the only relative you ever met and then only for afternoon tea and cakes in Mackie's. Never visited her home. Seems there was an Uncle Billy, not that you ever met him. Never asked why. This soldier standing next to a signpost pointing to Berlin \u2013 is that your Uncle Billy? Your mother won't remember, that's for sure. And so \u2013 in goes Mr Berlin, along with all the others.\n\nBut you have tried. Making that album for her, hoping she'll talk you through the photos. Would be nice to learn more about her life and about your dad, more than that he was blind and got knocked down by an Edinburgh tram when you were very young. Talking about the past makes her sad, she says.\n\nWhich leaves that snap she gave you, the one of the little girl standing next to a big old-fashioned pram, holding somebody's baby in her arms. A neighbour? A girl in the village? No idea. Might as well go in, too.\n\nAnd so \u2013 pour on the paraffin, toss on the lighted match. Whoosh! Cremation of a sort.\n\nThey're gone in seconds.\n\nTime for a final walk through the cleared-out cottage. No curtains, no lampshades, empty grates; kitchen shelves and windowsills stripped bare. Without furniture, the walls look grimy, the woodwork's badly scuffed, the paintwork chipped. No retro charm here, the buyer'll start by ripping out that monstrosity of a pre-war fireplace. Ghosts are already haunting the place: the pale after-images on the walls where your mother's pictures hung, the outlines of her kitchen cabinet and dresser, her wardrobe. The lino's scarred where you dragged out the cooker, the fridge and washing machine \u2013 three veterans fit only for recycling at the Council dump. Easy to imagine hearing your mother's voice here \u2013 from the time when she still knew who you were, of course. Maybe her footsteps will echo in the empty rooms? Yours, too?\n\nNo . . . not a sound in the place. Your long-ago childhood home. Empty. Hollowed-out. You lock the door behind you.\n\nThe bonfire can be left to burn itself out.\n\n8\n\nTHE NEXT FORTNIGHT passed very quickly, and happily. Maggie explained to Sheila that she'd had a big row with her mother about turning up like that and wanting to take Tom away for the weekend. If everything worked out, she'd asked, could he now stay at Glengyle Terrace?\n\n'Let's see how we get on,' had been her landlady's response.\n\nLuckily they got on very well. For a little extra on the rent Sheila agreed to look after Tom during the day, which allowed Maggie to continue working at Blair & Blair's. On her part, Maggie was available evenings and weekends, if required, to look after both boys. They'd soon settled into a routine.\n\nIt would have been too much of a coincidence if Michael had turned up completely cured on the very same day she brought her baby home, so it was decided that he should remain at the bakery for a little longer. Maggie visited him as often as possible, Jean taking Tom out in the pram to give them privacy.\n\n'My husband's really starting to respond,' Maggie told Sheila one morning. 'The doctors say we might soon have him home.'\n\nEverything seemed to be working out. For the first few days Maggie had jumped to her feet in alarm whenever she heard the doorbell \u2013 perhaps in her haste at Woodstock House she'd overlooked something that could be used to trace her? Again and again she went over in her mind the paperwork she'd found in Tom's file. His birth certificate, her signed contract, various letters. That would be everything, surely? But were there duplicates somewhere?\n\nAfter a week, however, she began to relax. The week then became a fortnight.\n\nOn the first Saturday in June, nearly a year to the day she'd stepped off the ferry at Stornoway and been directed to Mrs Stewart's lodging house, Maggie hesitated outside the street door in Glengyle Terrace. Michael was beside her. Never had she felt so nervous.\n\n'It's blue,' she said. 'A shiny blue door, with a brass handle and a letter box.' She wanted to say how wonderful it was to have him here at last, how this was the beginning of their life together. 'Here's our key, I'll let us in.'\n\nShe went in and Michael followed. Once inside, she couldn't seem to stop.\n\n'There's a long hall with a table for letters and things. Mrs McCann's put some welcome flowers on it for you. Sun-yellow, pink and white. Ones you can smell. Freesias, I think.' She brought him up to the vase. 'As I said, Mrs McCann's really nice. She's taken Tom and her wee boy Douglas to visit her sister, to let us get settled.'\n\nMichael leant forward and breathed in. 'They're lovely. Really kind of her.'\n\nShe guided him across the corridor. 'This is us, here. Right opposite the table. The door's white.' She opened it. 'Our room.'\n\nHaving brought him in, she paused without meaning to . . . was she expecting to hear him offer a polite remark about how nice it looked? A split-second later, she added, 'I know we'll be very happy here.'\n\n'Thank you, Maggie. It feels fresh and inviting and . . .'\n\nShe closed the door behind them. 'We're home, Michael.'\n\nThey embraced and kissed. As he held her, she could feel the handle of his stick press into her back.\n\nShe turned him towards the window. 'We've a really big bay window. The park's just opposite. Grass that slopes all the way up to Bruntsfield, there's paths for folk walking with their kids. Lots of trees, too, really big some of them. Elms, I think. Our curtains come down to the floor, they've got a flower pattern on them, red and purple roses.' She guided him a couple of steps. 'Here's your armchair . . . and here's mine. They're light brown, high-backed and a bit old-fashioned looking, but comfy enough. We'll sit in front of a gas fire with a fine-looking wooden mantelpiece. It'll be cosy in the evenings while Tom plays on the rug.' She took his hand again and led him past the chairs and a small table. 'This . . . this is our bed.'\n\nMaggie had prepared a picnic for them to take across to the park.\n\nAs she shook out the tartan blanket to let it settle flat on the grass, she heard her mother's voice: You've made your bed, now you can lie on it.\n\nShe shrugged and smiled to herself as she unpacked the sandwiches wrapped in greaseproof paper \u2013 spam, mashed egg \u2013 the thermos of tea, the cups and plates, some milk in a brown medicine bottle, salt and pepper in little twists of paper. Laid out invitingly, the picnic meal looked good enough to be photographed for a magazine as a 'Family Day in the Park'. Yes, that's what it was, and what they were at last \u2013 a family. Maybe she'd write to her mother to let her know how well she was getting on, how happy she was. Maybe.\n\nFrom then on, when she came up the front steps back from work, she'd see Michael standing at the window waiting for her. And every time, without thinking, she'd wave to him. Inside their room, she'd be greeted by his placing his hands on her face. That was the moment when she knew she was really home \u2013 the first of the perfect moments in the rest of her day. Next was Tom's excitement when she collected him from Mrs McCann. The third perfect moment happened several times: whenever she paused during the evening and allowed herself to feel their togetherness. The last was when she switched off the light and entered Michael's darkness.\n\nA real family, yes \u2013 but they couldn't get married. Not yet \u2013 to do that would mean putting up banns in the name of Mr Stewart and Miss Davies, and the danger of making everything public. She felt like Mrs Stewart, she was known as Mrs Stewart. To everyone, apart from at Blair & Blair's, she was Mrs Stewart.\n\nTom had his first birthday: they shared their first Christmas together: then came Hogmanay \u2013 but not at Jean's. This time the New Year was seen in with Sheila and Gordon.\n\nOne freezing-cold day towards the end of January, Maggie was putting her key in the front door when \u2014\n\n'Oh, Miss Davies! Miss Davies! It's you, it's really you!'\n\nMaggie turned to see the familiar tangle of blonde curls and smiling face. It was Donna.\n\nSUNDAY AFTERNOON\n\nPULL INTO THE Rosehaven car park.\n\nTaking your usual five in the contour leather to log on, to update.\n\nMandy.\n\nYour agent.\n\nMandy again. Reliable girl.\n\nGR8 CU@8 Tx\n\nTwo voice messages.\n\nNo hang-ups. Hang-ups are a message in themselves \u2013 if you're not there to answer, then you're not needed. Like you're a hang-up nearer to being dead.\n\nBut not you. Three texts. Two messages. Mr Magic lives!\n\nTweet the cottage get-out. Tweet another Sunday, another Mum visit.\n\nHoping it's a good day.\n\nPsyched up, ready to rock'n'roll.\n\nOnce again you're standing on the yellow cross, your smile in place, flowers in one hand and the other raised in greeting for the CCTV. That's you on the screen \u2013 not looking so great. The loving son come to visit his elderly mother who's now seriously confused.\n\nYes, she's totally losing the place. She's like a book whose pages have fallen out of sequence \u2013 but who's going to put them back into their right order? There's no other copy to refer to.\n\nCue Mr Magic preparing to step onstage once more with his repertoire of well-tested tricks, his cheerful patter. Cue another Sunday. You'll make it a good one.\n\nYou're buzzed in. Heading straight down the corridor, on course to brighten up her \u2014\n\nSudden swerve for the usual pit-stop in the visitors' WC. Dry heave and spit. And spit again. Take a few minutes to get focused, get upbeat.\n\nBetter?\n\nYou and your feelings back on track, you splash your face. The paper towel's screwed up and binned. You're ready to breeze through to the dayroom. You want to cheer her up. Be all sunshine and smiles.\n\nThe afternoon sun flooding the bay window with light, freeze-framing the scene: the high backed chairs, the footstools and zimmers, coffee tables, the TV, the Murray twins, sad Dorothy calling and calling . . .\n\nBut where's your mother?\n\nHer chair's empty. Her zimmer's stainless steel tubing flashes its accusation: You're late.\n\nSomeone's taken her to the toilet?\n\nBut her zimmer's here. So how did she manage to \u2014 ?\n\n'Mr Stewart?' The young Polish girl's standing beside you. 'Mr Stewart? Please.'\n\nMariella? Marietta?\n\n'You come, please.' She's not smiling.\n\nTurning your back on the overheated hours burnt into the dayroom walls, you follow Mariella \/ Marietta out into the hall. Mariella. Yes, that's her name.\n\nShe's wearing a green housecoat today, jeans and trainers. When she's signed up as your assistant you'll dress her in something special, something sleek, black and Futuristic. Distraction on high heels. It's the right time to upgrade. Invest in a new generation of high-tech props and \u2014\n\n'Room. Please, we go Mrs Stewart.'\n\n'Mariella? Is my mother all right?'\n\nThe curtains have been pulled almost shut. A small lamp burns on the side table \u2013 its low watt glow washing over the room's stillness. You see your mother first in the dressing-table mirror and find yourself taking a half-step towards the reflected figure. Then immediately correct yourself.\n\nMariella's tilted the lampshade so the light falls more gently across your mother's face. A kindly girl . . . If she was dressed in sleek and shiny-black . . .\n\nThat time you watched her trying to get one of the Murray twins take her meds, you were shocked that the old woman could be so utterly unaware of the girl standing right in front of her, as if the Murray couldn't even see her. But what do you see now? Your ninety-year-old mother resting in her bed \u2013 do you really see her? Or is she becoming a fading memory already \u2013 the happiness in her voice whenever you phoned, her excited wave from the window as she saw you coming up the path?\n\n'I go now.'\n\nDon't go, Mariella, you want to say. Please.\n\nAt this moment you would give anything to be a genuine Mr Magic and make the glow return to her sunken cheeks, the glossy blackness to her hair, redness to her slack lips. No sleight of hand any more, no sleight of heart.\n\nNot any longer.\n\nYour mother's dying. You see it now. You're afraid you might burst into tears.\n\nThe chair beside her bed creaks as you sit down. It creaks again as you lean forward to take her hand.\n\nSomeone's just come into your room. You're aware of them sitting down in the chair beside your bed and can feel the warmth of their hand as it squeezes yours, the tenderness in their voice . . .\n\nA farewell? The first of a hundred thousand welcomes?\n\nListen \u2014\n\nWithin an hour of seeing Donna, the three of them had moved out of Glengyle Terrace. There was a family emergency, Maggie explained. Having thanked the McCanns for all their kindness, she promised she'd be in touch as soon as things settled. She didn't specify what things exactly. Her mother was mentioned. The rent was paid up for another fortnight, but Sheila should feel free to let the room from tomorrow.\n\nOne suitcase was all they took with them, and Michael's ex-army kitbag stuffed with Tom's clothes and bedding. What they couldn't carry, they left behind.\n\n'Family emergency, right enough,' thought Maggie as she lifted their luggage into the taxi that would take them round to Jean's bakery. She was really sorry to be leaving Glengyle Terrace, but Mrs Saunders might appear at any moment, perhaps even bringing the police. Blair & Blair's, in turn, would probably be contacted. Maggie felt ashamed at lying to Sheila, but they couldn't risk staying even a day longer.\n\nAnd so \u2013 no more Blair & Blair's and, most likely, no more \u00adEdinburgh.\n\nWhat happened next was like a gift from heaven, a once-in-a-lifetime stroke of good luck. The advert in the Scotsman announced: 'Part-time home help wanted in exchange for rent-free cottage. Rural location. Apply in own handwriting.'\n\nMaggie did, and received a reply by return, with a telephone number to phone. The cottage turned out to be the small gatehouse of an estate in the Borders. Learning that her husband was a veteran who'd been blinded in the war, the laird clearly took pity on them. Maggie said they could come immediately.\n\nThere's the sound of doors opening and closing, the tap of someone's walking stick on the linoleum out in the corridor, a woman saying she hopes there might be some cake.\n\nYour familiar room. Your wardrobe. Armchair. Wash hand basin and mirror. Tea trolley with playing cards laid out for the game of patience you will probably never finish \u2014\n\nA fortnight after they'd moved in, Maggie was seated at her small work table in the living room, grinning with pleasure as she inserted the first sheet of foolscap into her typewriter.\n\nNo wonder she was grinning \u2013 on learning that his new help could not only type but knew how to keep basic accounts, the laird had at once offered her the position of part-time secretary to the factor of his estate. And so \u2013 no more mops and pails, brushes and carpet beaters, no more dripping, heavy-wet bedsheets needing hung out and then ironed. If she wanted, she was even allowed to take in extra typing work from the village to do in her own time. Michael had soon learned his way about the cottage and within a few days was walking up to the big house and round the grounds all by himself.\n\nMaggie glanced over the top of her typewriter \u2013 there had been a real blizzard during the night and now a blue-skied, snow-silenced winter's day waited for her outside. The fields and hedgerows were white, the telegraph poles and their wires stretched in a line like so many pencil marks pointing out a hidden road to the village. Once she'd finished typing up Rev McKay's service for the following morning, the three of them would wrap up well and go for a walk down by the river, now surely frozen over.\n\nWinter gave way to spring. Mrs Stewart wrote regularly to Michael, far too regularly. The news and gossip from Stornoway always came heavily larded with complaints and veiled accusations \u2013 her bad back made running the lodging house more and more difficult, the struggle to live on her war widow's pension was getting harder and harder. She missed her son and needed him, needed him desperately.\n\nMichael would listen in grim silence, then apologise to Maggie saying his mother couldn't help herself, she was angry, that was all. Give her a while longer to accept things. Once they'd got married, which they'd do in secret, he'd ask if they could come and visit. It would mean waiting until the moment was right, of course . . .\n\nBy this time Maggie would have put the hateful letter back in its envelope and stuck it in the drawer with the others. The letters made her furious \u2013 the venom in the old woman's words was really directed at her. Not that she was ever mentioned, or Tom. It was like they didn't exist. Very soon she began to edit as she read. Maggie always typed the replies which Michael then took to the village post office next time he was doing the shopping.\n\nOn a May morning about four months after they'd moved in, there was a rap on the cottage door. Norman the Post. Michael was making a pot of tea. Having been to answer, Maggie returned to the living room:\n\n'A letter. Don't know the handwriting.'\n\nStruggling to push aside the duvet \u2013 as if you could somehow manage to get young Maggie's attention. You want to comfort her. You want to grab at the letter she's about to read, to snatch it out of her hand.\n\nBut then what?\n\nYou know what's going to happen. You can't stop it now, any more than you could have stopped it back then. You can do nothing but watch.\n\nCarefully as always, Michael places a plate of biscuits on the tea-trolley. 'Postmark?' he asks.\n\nPostmark. She ought to have checked first, before saying anything. But it's too late now.\n\n'Stornoway.'\n\n'It's from Lachlan \u2013 not heard from him in ages.'\n\nBut she knows it's not from Lachlan. She knows Lachlan's handwriting \u2013 all those letters from Michael she's kept in her treasure box.\n\nYou watch her pick up the letter knife. Watch her slit open the envelope.\n\nWere this a kinder world, Maggie would sense you beside her as she reads out the letter from Mrs Stewart's neighbour:\n\nDear Michael, I'm sorry to have to write to you with bad news but . . .\n\nMichael's mother had slipped in the street \u2013 it was a serious fall. She'd needed an ambulance. She was back home now and holding her own, so far. Could Michael come as soon as possible?\n\nThe soothing warmth of someone's hand as they continue to sit here holding yours. No need to say anything, no need to make the effort to open your eyes \u2014\n\nMaggie's spoken to the guard who helped her put Michael onto the train at Lockerbie station. Lockerbie, Carstairs, Glasgow Central and then the West Highland Line to Mallaig, the ferry across to Lewis \u2013 he'll be passed from guard to guard like a precious parcel to be delivered safely to his mother in Stornoway.\n\nYou are with Maggie now on the platform, giving her the courage she needs to send the man she loves on his way. Giving her the courage to kiss him goodbye.\n\nTogether you share the warmth of his fingertips as they pass over her eyes, her cheeks, her lips: 'So's I'll have \u00adsomething to remember you,' he whispers.\n\nThere's a belch-belch-belch of grey smoke as the engine shudders forward, its wheels gaining traction on the steel rails. Straining, it gradually picks up speed and pulls out of the station. When the train has disappeared from sight, you accompany Maggie and Tom back towards the exit, ready to return with them to the gatehouse, ready to help them through the coming days, weeks, months.\n\nNo longer having the strength to hold on, drifting to and fro, from moment to moment \u2014\n\nOnce again you are back with young Maggie Davies on her ferry trip across to Stornoway, once again you can feel the salt dampness soaking into your coat and scarf, its freshness tightening the skin on your face.\n\nThe handrail is ice-cold. Slate-coloured waves slide over each other, sealing in a stretch of water whose briefly churned-up surface always flattens afterwards to a dead calm. The small boat trails silence in its wake. Together you look down into the depths, and shudder with an abruptness that makes the blood stall in your heart, the breath catch in your throat . . .\n\nYour body trembles; your nerves seize. Held under the weight of the duvet, your arm again struggles to raise itself as though to grasp hold of an invisible clock's next tick and force it on \u2014\n\nIt is late September, a glorious autumn morning. The post has just been and you've returned indoors with another of Michael's weekly letters. You expect it will be filled with the usual words of love and reassurance, the usual hopes and promises. He's been away for nearly five months.\n\nYou sit down at your work table, slit open the envelope.\n\nThe doctor, he writes, now believes that though she will likely live for a good many years to come, his mother will never regain her full strength and mobility. It seems that she'll always need someone to look after her. She has never left the island and now she never will.\n\nI'm all she's got, he explains, and so I'll have to stay here with \u2014\n\nYou let the page drop. You know what's coming. With every passing week, you'd grown to dread Norman the Post's rap at the door. You'd been half-expecting this letter, but now that it's arrived, it's almost a relief. You glance over at Tom who's sitting on the floor next to the tea-trolley, completely absorbed in throwing your playing cards around the room. Seeing you look at him, he holds out a crushed fistful towards you.\n\nMichael goes on to say that he loves you and misses you. He misses you more and more each day. He misses Tom, too.\n\nAnd you miss him. You love him and know he will be the only man in your life. But what did Mrs Saunders say that time \u2013 love is the easy bit?\n\nWhen the time is right, he continues, he'll start telling his mother about their plans to get married. So far it's not been possible to say anything . . . she's very easily upset . . . but as soon as there's an opportunity . . . pick his moment carefully . . . take time . . . he's sure you understand . . . talk her into allowing you and Tom . . . because he can't leave her, not now she . . .\n\nHaving folded up the letter, you put it back into its envelope, place it on the table. For several minutes you sit with your hands in your lap, head bowed, staring down at your neatly written name and address, and at the slightly smudged Stornoway postmark . . .\n\nLetting your breathing ease back to normal, your heartbeat steady itself once more \u2014\n\nYou're feeling very tired suddenly. A very pleasant tiredness, as though you were already half-asleep \u2014\n\nAn hour later you've written your reply, sealed the envelope and stamped it. If you leave now, you will be in time to catch the post. The sooner the letter goes, the better.\n\nYou lift Tom into his pushchair, put on your coat, your hat.\n\nLast thing, you glance in the hall mirror to check your appearance. A dab of powder, a quick touch of lipstick.\n\nYou're ready.\n\nThe buttons of your coat fastened up and your hat straightened, you give yourself a smile. You've written to Michael for the last time. You've dried your tears. You have made your decision and will stick to it. You will manage, somehow. The door pulled shut, you start off down the front path.\n\nA gust of wind sets the fallen leaves swirling around your feet, and when you scoop up a handful to scatter over his head Tom tries grabbing them as they tumble all about him. He's laughing and squealing.\n\n'whee!' You send him and his pushchair several yards ahead of you along the road. A few steps and you catch up. 'whee . . .!' Pushing him again several yards ahead. Then hurrying after him to catch up. Above, the sky is a clear and cloudless blue. Small birds flit in and out of the hedgerows on either side and it feels like they're keeping you \u2014\n\nACKNOWLEDGEMENTS\n\nExcerpts of this novel have appeared in the Scotsman and Gutter 09. The author would like to gratefully acknowledge a Royal Literature Fund Fellowship at Edinburgh University (Office of Lifelong Learning), which allowed him time to write. Grateful thanks also to my wonderful editor Nick Royle, to my ultra-patient agent Lucy Luck, to Lesley Glaister, Andrew Greig, Moez Surani and Dora Staub for their valuable comments, and most of all to my wife Regi Claire for her insightful readings of the text through its many versions, and for her endless patience and support. This novel was written while I was the Edinburgh Makar \/ Poet Laureate and I would like to thank the City of Edinburgh Council and the Edinburgh UNESCO City of Literature for their support during this period, in particular Lynne Halfpenny, Denise Brace, Ali Bowden, Peggy Hughes and Sarah Morrison.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTo my friends Kate Atwood, Carleigh Pearson,  \nCole Pearson, Luke Pearson, and everyone else  \nwho has lost a parent too soon.\n\n\u2022 \u2022 \u2022\n\nAnd as always, to Carol Harmel.  \nI couldn't ask for a better mother.  \nI love you!\n\n# A BIG THANK-YOU\n\nTo Kate Atwood, the founder of Kate's Club in Atlanta, and to NFL quarterback Brian Griese, the founder of Judi's House in Denver. Both of you lost a parent too soon, and you've turned your grief into something that has helped thousands of children. Your mothers would be so very proud of you. I'm honored to have become a part of your world.\n\nTo the Pearsons: Susan, Carleigh, Cole, and Luke. This book, while not based on you, was inspired by the time I've spent with you. I'm so glad to call you friends; I feel like you're my Atlanta family! And I'm so impressed with all of you; you're all amazing, strong, kind people, and I can't wait to see what wonderful things life brings you.\n\nTo my wonderful editor, Wendy Loggia, who has once again helped beat a rough manuscript into shape. Your guidance is invaluable, and I'm glad to work with you.\n\nTo my amazing literary agent, Jenny Bent; her assistant, Chris Kondrich; my film agent, Andy Cohen; and the wonderful Delacorte Press family, including Elizabeth Zajac, Krista Vitola, and Angela Carlino.\n\nTo my _People_ magazine editor, Nancy Jeffrey, who allows me to work on the kind of stories that inspire me, move me, and let me share the heroism of good people with the world.\n\nTo my own family, especially Mom, Dad, Karen, and Dave, and to all my wonderful friends.\n\nTo all of my many writer friends: It's such a pleasure and honor to know all of you. Thanks especially to Megan Crane, Liza Palmer, Jane Porter, Melissa Senate, Sarah Mlynowski, Alison Pace, Lynda Curnyn, Brenda Janowitz, Lisa Daily, and Emily Giffin, who are truly wonderful people as well as wonderful writers.\n\nAnd to you, the reader. This book is about changing your own little corner of the world. I hope that you feel inspired. Thanks for reading!\n\n# prologue\n\nThe day my whole world changed started like any other Saturday.\n\n\"Lacey!\" my dad called. \"Are you coming? It's going to be dinnertime when we get there!\"\n\nI looked in the bathroom mirror and made a face. He said the same thing every Saturday morning\u2014but maybe that was because I took longer getting ready than anyone else.\n\n\"Why don't you just get up earlier?\" My brother Logan, who was eleven months older than me, appeared in the doorway and looked suspiciously at my reflection. I knew he'd been sent up to get me. I was putting on a coat of mascara and paused to glare at him.\n\n\"I need my beauty sleep,\" I said, trying to sound haughty.\n\nHe rolled his eyes. \"No kidding,\" he muttered. \"I think you need a little more.\"\n\nHe was gone by the time I threw a tube of toothpaste at him.\n\nFive minutes later, when I came downstairs, my dad, Logan, and my little brother, Tanner, were standing in the hallway, already bundled up in their coats and scarves. It was unusually cold that day, even though it was only November fifteenth. There had been an early freeze, and it hadn't worn off yet. My dad held out my pink puffer jacket, and as I stepped into the hallway and took it from him, he winked, one corner of his mouth jerking upward just a little. I knew he was trying to hide his amusement from Logan and Tanner.\n\n\"What the heck takes you so long anyway?\" Tanner said. \"I'm glad I'm not a girl.\"\n\nLogan high-fived him. My dad looked up at me. \"Is Your Royal Highness finally ready?\" he asked, bowing slightly.\n\nMy dad always called me that when I took a long time to get dressed. Even though he sometimes pretended to be as exasperated as Logan and Tanner, I think he secretly didn't mind.\n\n\"Where's my beautiful wife?\" Dad singsonged as I zipped up my jacket. Mom rounded the corner, dressed in the same ratty pink bathrobe she'd had for years, the one she would never throw away because it was the first gift Logan and I ever picked out for her, when Logan was four and I was three and Dad took us Christmas shopping. We'd bought her a new one last Christmas, but she refused to switch over.\n\nShe was in her usual state of morning messiness, with sleep-flattened reddish brown, shoulder-length curls flying every which way and her cheeks slightly blotchy before she made it to her vanity mirror and her tray full of makeup. I always wished that I had inherited her pretty hair and Dad's flawless complexion, but instead, it was the other way around. I had Dad's stick-straight dirty blond hair that always looked stringy if I didn't use a curling iron on the ends (which I hardly ever had time to do considering I shared a bathroom with two boys) and Mom's acne-prone skin. Thank goodness for Clearasil, but most of the time my face was sporting at least one major zit, usually in a totally unflattering location like the middle of my forehead or smack in the center of my chin.\n\n\"You're taking my family and leaving me?\" Mom asked dramatically, clutching her hands over her heart. \"Whatever will I do?\"\n\nMom said the same thing every Saturday when Dad took the three of us out to breakfast. He called it \"Dad time,\" and while we were out scarfing down pancakes at the Plymouth Diner, Mom was having her weekly \"Mom time,\" which apparently included sitting around in her robe, sipping a cup of coffee, and putting on a facial mask while she fast-forwarded through TiVoed episodes of _Grey's Anatomy_ and _CSI_ and whatever else she'd dozed off watching during the previous week.\n\n\"Your mom thinks we're giving her time alone,\" Dad would whisper to us while she pretended she couldn't hear, \"but really, it's just a good excuse for the four of us to hang out and eat greasy bacon and hash browns, right?\"\n\nIt had been our Saturday-morning routine for as long as I could remember. And it was the highlight of every week. Dad, Logan, Tanner, and I would sit at breakfast and talk about school and our friends and stuff, and Tanner, who wanted to be a comedian when he grew up, would always tell some silly joke he had just learned from his friends or the Internet that week, and when we'd get home, the house would always be a little cleaner, and Mom was always in a good mood. If we didn't have anything big to do, we'd all go out for a hike or a bike ride or to play tennis at the local country club, where Mom had insisted we needed a membership, against Dad's halfhearted protests.\n\nMom and Dad kissed goodbye, then she gave each of us a peck on the top of our heads, and we were off.\n\n\"Everyone have their seat belts on?\" Dad asked as he started the car. Logan climbed in beside him.\n\n\"Yes!\" the three of us answered in unison. Dad turned and grinned at Tanner and me in the back, buckled his own seat belt, and put the car in reverse. As we pulled out of the driveway, he beeped the horn at Mom and blew her a kiss.\n\n\"Cheesy!\" Logan and I chorused. Tanner laughed.\n\nMom smiled, waved from the doorway, and went inside.\n\nIt took three minutes for us to get out of our neighborhood, Plymouth Heights, and onto a main street. It's weird how normal everything still was in those final minutes. We saw Mrs. Daniels walking down her driveway to pick up the newspaper, and she waved at us as we passed. Dad and Logan waved back. I noticed Jay Cash and Anne Franklin, two kids from Tanner's grade, playing basketball in the Cashes' driveway. Anne tripped on her shoelace just before we passed, and I turned my head slightly to see if she'd start crying. She didn't. Logan was absorbed in flipping through the radio stations, finally settling on the classic-rock station, which was playing the Eagles' \"Hotel California,\" one of Dad's favorite songs. He started to sing along, and when the chorus ended and a guitar solo began, Dad glanced at Tanner and me in the rearview mirror and grinned.\n\n\"You guys would love California,\" he said. \"Maybe we'll go there someday and surf.\"\n\n\"I want to surf!\" Tanner exclaimed. At age eleven, he had just discovered skateboarding, and he had announced more than once at dinner that when he turned eighteen, he was going to move west, bleach his hair blond, and learn to catch waves. I had to admit, it was a fun fantasy to have in the middle of a Massachusetts winter.\n\n\"I know!\" Dad laughed as the light on Mayflower Avenue turned green and he eased his foot off the brake and onto the gas. He put on a fake surfer accent. \"Hang ten, dudes!\"\n\nIt was the last thing Dad ever said.\n\nI think I saw it an instant before it happened, but my throat closed up, and there wasn't time to open my mouth or even to scream before the Suburban plowed into the driver's side of the car, hitting us with such force that the whole side crumpled, pinning me up against Dad's seat. It was like everything was suddenly compressed into a much smaller space than it had been a second ago. I felt a terrible pain along the left side of my body, shooting from my upper leg, up my side, and down my shoulder into my arm. I screamed and felt Tanner grope for my right arm.\n\nThe world felt dark and hazy. I couldn't see anything, just shapes, and everything sounded muffled. I wondered for a second if I was dying. Far away, I could hear Logan yelling and Tanner crying. But I couldn't hear Dad. Why couldn't I hear Dad?\n\nMy throat felt like I'd swallowed cotton balls, and my mouth wasn't responding when I tried to make it work. I opened and closed it a few times, but I was only gurgling, not talking. I remember being terrified, and when I look back now, I think it was pure fear that kept me from being able to speak. When I finally did, there was only one thing that came out of my mouth.\n\n\"Daddy?\" I whispered weakly. I hadn't called him that since I was twelve.\n\nIt was the last thing I remember saying before everything went black.\n\n\u2022 \u2022 \u2022\n\nWhen I came to, I was in the hospital. I didn't know how much time had passed. But I think I already knew about Dad. I don't know how\u2014I didn't see him again after the Suburban hit us\u2014but maybe when you're that close to someone, you can feel it when they're not there anymore. That's what I think, anyhow.\n\nIt took me a few moments to focus on Mom's face as I gradually swam to the surface of consciousness. Her eyes were bloodshot and her face was blotchier than usual. I couldn't help noticing that she was still wearing the tattered pink bathrobe over her pajamas, which seemed strange and out of place in public. Mom was a lawyer in Boston, and she never left the house looking anything less than completely put-together.\n\nThe hospital room was white and almost uncomfortably bright under big fluorescent lights. I licked my lips and realized I couldn't feel my body.\n\nMom jumped up and leaned over me. She looked scared.\n\n\"You're going to be okay,\" she blurted out. \"You broke your left femur\u2014that's the big bone in your thigh\u2014in two places, and you have a few broken ribs and a broken left wrist, but they say all the bones should heal just fine.\"\n\n\"Where's Dad?\" I asked slowly, in a voice that sounded too thick to be my own.\n\nMom's lower lip quivered and she bit it, like it was the only way she could stop it from shaking. Her eyes filled with tears again.\n\n\"Lacey, baby,\" she said softly, sitting on the edge of my bed and reaching for my hands. I couldn't feel her. I couldn't feel anything. \"The accident was really bad.\"\n\nI stared at her for a minute. She hadn't answered me. \"Where's Dad?\" I repeated. \"Where are Tanner and Logan?\"\n\nShe blinked at me a few times. \"The boys are in the waiting room,\" she said. \"Uncle Paul's with them. They're going to be okay. Tanner broke his arm, and Logan had to get stitches, but they're fine.\"\n\nI remembered Tanner reaching for me just before everything went black. He must have been scared about me. \"But Dad?\" I asked again, my voice rising a little bit as panic began to set in.\n\n\"Dad...,\" Mom began, and then stopped. She took a big breath, glanced away, and then looked back at me with eyes that seemed foggy and lost. \"The car... hit right around the driver's seat,\" she said slowly. \"The doctors did everything they could, but...\" She stopped, unable to say it.\n\n\"Daddy died,\" I completed her sentence, feeling tears well up in my eyes. \"He died, didn't he?\"\n\nMom nodded. A pair of fresh tears rolled down her face, one for each cheek, like skiers racing to the bottom of the slopes. I remembered the last thing Dad had said, and tried to imagine her tears as graceful surfers instead, trying to ride a wave into shore. But then the tears dropped off her jawline and melted into her robe, and I had the sudden feeling that the imaginary surfers had fallen off the edge of the wave and disappeared forever. It was that image that finally made me burst into tears.\n\nMom wrapped her arms around me, and we sobbed together, with no more words to say.\n\nLater, after Logan and Tanner had come in to see me and Uncle Paul had taken them home, Mom sat by my bedside and told me that Dad had lost consciousness right away. The doctors said he probably didn't even see it coming and didn't feel scared, and that he was never awake to hurt. It was, they told her, the most painless way to go. One second, he was driving along happily on a Saturday morning with his three kids, and the next, it was all over. He never knew. He never had a chance to say goodbye.\n\nAfter a while, Mom asked if I had any questions. I said no, but of course, that was a lie.\n\nI wanted to ask what would have happened if I hadn't had to curl my hair or if I hadn't insisted on putting on mascara or if I hadn't purposely dragged my feet a little just to annoy Logan and Tanner. But I didn't need to ask. I knew what would have happened. We'd be sitting at home right now, trying to figure out whether to play Monopoly or Life or whether to watch a movie. Dad would be trailing his hand lazily down Mom's back in that affectionate way that sometimes made me and Logan smile and roll our eyes at each other. Mom would be getting up every few minutes to put dishes in the dishwasher or to start the washing machine. Logan and Tanner would be fighting over the remote control because Tanner wanted to watch a Pok\u00e9mon DVD and Logan wanted to watch sports.\n\nDad wouldn't be dead.\n\nAnd it wouldn't be my fault.\n\n# chapter 1\n\n**TEN MONTHS LATER**\n\n\"And then I told Willow that her shoes were totally the wrong color for that outfit and actually, the shirt is really hideous anyhow, and I couldn't believe she was going to actually go out in that, never mind go to the movies with me, and then Melixa said to me...\"\n\nSydney droned on and on from the front seat as I tried in vain to tune her out. Her high-pitched, squeaky voice made that pretty impossible, though. My best friend, Jennica, and I had decided that she must be trying to attract boys by sounding like a squeak toy, but until recently, I'd been sure that it only attracted dogs and whales and whatever else could hear such a high frequency.\n\nBut then she landed Logan, who apparently found squeakiness enticing. This pretty much meant that I was stuck with her, because she was now our official ride to school. Mom had refused to let me or Logan take our driver's tests since the accident, so it was either the school bus or hitching a ride with Logan's popularity-obsessed girlfriend.\n\n\"Uh-huh,\" Logan said patiently from the passenger seat, as if he were actually listening. As far as I could tell, Sydney was telling the longest story in the world about a bunch of senior cheerleaders who didn't matter to me at all.\n\n\"So what do you think?\" Sydney finally paused for what I was pretty sure was the first breath she had taken since picking us up ten minutes ago.\n\n\"Um...,\" Logan began, his voice trailing off. I hid a smile. He obviously hadn't been listening either. I watched in amusement as he struggled for words. \"What do I think?\" he said finally. \"I think you're the most beautiful girlfriend in the world.\"\n\nOh, gag me. I waited for Sydney to realize that he was completely copping out, but instead she giggled, turned a weird shade of pink, and glanced at me in the rearview.\n\n\"What do _you_ think, Lacey?\" she asked. \"Don't you think Summer was acting totally slutty? I mean, considering she's practically engaged to Rob Macavey?\"\n\nI sighed. \"I don't even really know her.\"\n\n_\"Everyone_ knows Summer Andrews,\" Sydney said, looking at me like I was a mental patient.\n\n\"Right.\" I bit my tongue. What I wanted to say was that everyone knew who Summer Andrews _was_ \u2014the cheerleading, BMW-driving, shiny-haired queen bee of our school\u2014but that there were few people she actually deigned to talk to. And I was not one of them. I was pretty popular in my own grade, but I was definitely more bookworm than beauty-pageant contestant, which meant that Summer and her crowd hardly knew I was alive.\n\nLogan was a different story. Since he and social-climbing Sydney had begun dating six months ago, he had come home more than once proudly reporting\u2014out of Mom's earshot, of course\u2014that he'd gotten drunk alongside Summer Andrews and her clones, Willow and Melixa, at parties. Like that was some major accomplishment.\n\nBut I refrained from saying any of this, because Logan would kill me if I did. He always seemed to be walking on thin ice around Sydney. I must have been making a face without meaning to, though, because Sydney glanced at me once more in the rearview and snorted.\n\n\"Oh come _on_ , Lacey,\" she said. \"Just because you're too busy making straight As and going to student council meetings and whatever else you think is so important doesn't mean that the rest of us can't have a social life.\"\n\nI simmered for a minute. I was good at shutting my mouth, pressing my feelings into a little lockbox inside, and turning the key. I took a deep breath, blinked a few times, and said, \"Wow, look at that! We're here already!\"\n\nBefore either of them could respond, I hopped out of the car and began striding across the junior lot toward the school building without bothering to look back. Somewhere behind me, Sydney was babbling about how she couldn't believe I'd jumped out of her car before she'd even had a chance to park.\n\n\u2022 \u2022 \u2022\n\nIt was the end of the third week of school, and already, it seemed to have turned to fall. Last summer, the heat had hung on for ages, taunting us cruelly from outside the classroom windows with persistent rays of sunshine. But this year, the New England dreariness had moved in early, bringing hulking gray clouds and winds with a chilly edge. The first leaves on the trees were turning, seemingly overnight, from muted greens to the deep reds, oranges, and golden yellows that always reminded me of a sunset. I wasn't ready for it to be autumn again, but the seasons seemed to march on without caring.\n\nForty-five minutes after hopping out of Sydney's car, I was in trig class, trying to pay attention, which was hard to do considering that Jennica, who sat beside me, kept trying to get my attention. I was attempting to ignore her.\n\nMath came easily to me. I had always wanted to be an architect when I grew up, like my dad. Plus, there was something about the clear-cut right and wrong of math equations that I found appealing. In math, there were no gray areas. There were rules, and I'd discovered that when you stayed inside the lines, life made a lot more sense.\n\n\"Psst!\" Jennica hissed. I glanced to my right, where she had angled her desk closer to mine and was holding out a folded square of paper.\n\nI glanced to the front of the room, where Mrs. Bost, our twentysomething teacher, was jotting a series of cosine problems on the board. In the few weeks we'd been in school, I'd already discovered that she had superhuman hearing. I suspected she could hear a note unfolding from miles away. So I coughed loudly to cover up the crinkling sound as I quickly unfolded Jennica's message.\n\n_You'll never believe this: Brian told me he LOVES ME last night!_ she'd written. I could feel Jennica's eyes on my face, so I was careful not to do anything inappropriate like, say, wrinkle my nose or stick out my tongue. It wasn't that I didn't like Brian. He was okay. But he and Jennica were so lovey-dovey with each other that I felt nauseated half the time I was around them. And much as I hated to admit it, I was a little jealous. _I_ was the one Jennica had done everything with and told all her secrets to since we met in the first grade. And now Brian was her constant companion, and I felt like the third wheel.\n\nIt was like I'd lost my best friend. But it was selfish to feel that way, so I told myself not to. I'd gotten good at deciding how I should and shouldn't feel. Sometimes I felt like the director of the movie of my own life, yelling _action_ in my head and then setting scenes in motion the way I'd decided they'd go.\n\nI pulled out my cell phone, checked to make sure Mrs. Bost wasn't looking, and quietly texted Jennica: great. I watched as she silently pulled her cell from her purse, read my text, and frowned. She thought for a second, and I tried to tune back in to Mrs. Bost while Jennica typed. But the lecture was boring, and I was tired of thinking about trig and boyfriends and all the other dumb stuff that went along with eleventh grade. I was itching to graduate and get out of this place, to move on to the next phase of my life and leave Plymouth East behind, but I had a year and nine more months to go. It was endless.\n\nThe new-message indicator lit up on my phone. i know u've never been in love b4 but this is a REALLY BIG DEAL, Jennica had written, complete with a smiley face at the end of the sentence, to let me know she wasn't trying to be mean. Still, the words stung. I _knew_ it was a big deal to her. But in my world, having a boy tell you he loved you wasn't exactly as earth-shattering as, say, your dad dying. it was when we were watching grey's antmy on dvd, the message continued. mcdrmy told mrdth he luved her & B turned 2 me & said, I luv u like derek luvs mer. sooo romantic, right?\n\nI was just about to write something back when the door to the classroom creaked open. Mr. Dorsett, the assistant principal, was standing there with someone behind him. Mrs. Bost smiled and set down the marker she'd been using.\n\n\"I'm sorry to bother you,\" Mr. Dorsett said. He glanced over the room and then back at Mrs. Bost. \"But we have a late addition to your class.\"\n\nTwenty-four pairs of eyes strained to see the tall guy in a faded leather jacket and dark jeans who followed Mr. Dorsett through the doorway, his eyes focused coolly above our heads. His hair was dark, and it looked like he needed a haircut\u2014or at least a comb. It stuck up wildly in some places and grazed his collar in others, making him look a bit like a mad scientist who forgot to go to the barber. His skin was tan, which made his pale green, thick-lashed eyes seem unusually bright.\n\nA buzz went around the classroom. Plymouth was a pretty small town, and most of us had gone to elementary school or junior high together, so it wasn't very often that we saw an unfamiliar face. Maybe he'd transferred from the Catholic high school. Sometimes we got new students from there.\n\n\"Who's that?\" Jennica whispered urgently, like everyone else in the room wasn't wondering the same thing. I shrugged without taking my eyes off the guy. I didn't usually notice things like this, but his eyes were unbelievable. They were almost the exact color of the ocean right before a storm. That had always been my favorite time to gaze out from the shore, while the wind whipped through my hair and the sky rumbled, getting ready to change the earth below it.\n\nWhile Mr. Dorsett held an inaudible conversation with Mrs. Bost, the new guy shifted from foot to foot and avoided looking at anyone. I couldn't figure out whether he thought he was too cool for us or whether he was just nervous.\n\n\"Okay,\" Mrs. Bost finally began, pulling away from Mr. Dorsett. He nodded once at us, clapped the new guy on the back awkwardly, and headed out the door.\n\n\"This is Samuel Stone,\" Mrs. Bost continued once Mr. Dorsett was gone. \"He'll be joining our class. I'd like you all to give him a warm welcome.\"\n\nJennica and I exchanged glances. The room was silent for a few seconds, then someone in the back started clapping slowly, and the rest of the class joined in. The new guy took a step forward and whispered something to Mrs. Bost.\n\n\"What?\" she asked. She glanced at us. \"Class! Shhh!\"\n\nWe all quieted down in time to hear him say more loudly, \"Sam.\"\n\nAll eyes were on the new guy, and suddenly I felt bad for him. I knew what that felt like. I'd been the subject of the same kinds of stares last fall, when I finally returned to school after the accident. It was the worst kind of attention; no one says anything; they just look and look, judging you. I blinked, cleared my throat, and shifted my gaze to the floor.\n\n\"Sam,\" he repeated, his voice sounding deeper than I'd expected it to. \"I go by Sam.\"\n\n\"Oh,\" Mrs. Bost said. \"I'm sorry. Welcome, Sam. There's an empty desk there, next to Lacey. Lacey, can you raise your hand?\"\n\nI looked up, startled. There seemed to be little need for me to put my hand in the air since Mrs. Bost was pointing straight at me, but I did anyhow, feeling my cheeks heat up as I did.\n\nSam began weaving through the rows full of students, who continued to stare like he was some kind of science project. I couldn't blame them. Not only was he new, but he was gorgeous. I mean, _really_ gorgeous.\n\n\"Hey,\" he said, settling into the seat next to mine.\n\n\"Hey,\" I replied. He scooted his desk closer to mine so that he could see my book, and as he leaned over to glance at the text, I could feel his warm breath on my arm. I looked up and was surprised to find him studying me.\n\nHis eyes locked with mine. I shifted my gaze down and fumbled with my book. When I snuck another glance, he was still looking at me.\n\nAnd for the first time since I'd seen him, Sam Stone cracked a small smile, and I felt a little tingle run up my spine. I smiled shyly back and looked away.\n\n# chapter 2\n\nSam Stone wound up in my sixth-period AP English class, too, and when he walked through the door and noticed me, he shot me a relieved look.\n\n\"Hey,\" he said, slipping into the empty seat beside me after yet another awkward, lengthy teacher introduction. \"You're in this class too.\"\n\nIt was the longest sentence I'd heard him speak all day. I merely nodded, wondering why I seemed incapable of stringing words together.\n\n\"Lacey, right?\" Sam asked, cracking another smile.\n\n\"Yeah,\" I said, my cheeks pinking.\n\n\"Cool name,\" he said, and for the first time, I noticed he had dimples. Not normal dimples, but almost vertical indentations along his cheeks, lines that made his face appear like it had been sculpted quite carefully by a really talented artist. \"I'm Sam.\"\n\n\"I know.\" I didn't know what else to say, so I didn't say anything. He probably thought I was rude. Or maybe just dumb. I wasn't sure which was worse.\n\nJennica came home with me after school to study for our trig quiz on Friday. Sydney and Logan were going to some homecoming planning committee meeting, so Jennica and I had to take the bus. She didn't have a car either, although she had a license, and her mom let her borrow her car sometimes on the weekends.\n\n\"How come you're so good at this, and I'm so terrible?\" Jennica grumbled as we sat down at the kitchen table and cracked our math books. Mom, who seemed to work 24-7, was still at the office, and Tanner had come home minutes after us and locked himself in his room, so we had the rest of the house to ourselves.\n\nI shrugged. \"You're not terrible,\" I said. \"I'm just good at math, the way you're good at swimming.\"\n\nJennica was the captain of our school's swim team, even though she was only a junior. She snorted. \"Yeah, because swimming is a real life skill,\" she said. \"I'll definitely be able to use that someday.\"\n\nI knew she was worried about getting into colleges, but I tried to laugh it off. \"You never know,\" I said. \"You could have to save a drowning child or something someday.\"\n\n\"Why does it always have be a drowning kid in these rescue fantasies?\" she asked with a smile. \"Can't it be a drowning movie star or something?\"\n\n\"Right,\" I said. \"I can just imagine you pulling Robert Pattinson out of the ocean.\"\n\n\"Or Shia LaBeouf,\" she said. She paused and giggled. \"It could happen.\"\n\n\"You'd probably have to give them mouth-to-mouth,\" I deadpanned. \"You know, to save them, of course.\"\n\n\"You're right. I should definitely go into a career as a celebrity rescue swimmer,\" Jennica said. She glanced down at the book. \"But until then, you'd better teach me about sines and cosines. Just so I have a backup plan if Rob and Shia don't wash up in Plymouth.\"\n\nI grinned, and for the next forty-five minutes, I slowly went through the equations and formulas we'd talked about in class, and sketched little diagrams to demonstrate everything to her. I was used to this; Jennica always had problems absorbing things in class, and she usually needed some extra explanation, especially in math and science. Her dad, Mr. Arroyo, had been calling me \"Miracle Worker Mann\" since I helped Jennica bring up a D-plus to a B-minus in seventh-grade earth science.\n\nBut I didn't mind at all. I kind of liked my role as her unofficial tutor, especially now, because it gave me some uninterrupted time with her, without Brian nibbling at her neck or trying to slip his arm protectively around her. It felt like it used to feel when it was just the two of us. I wished I could slow down time or freeze the frame so that I could savor it. But like everything good, the moment was fleeting and would be gone before I knew it.\n\n\"You got anything to eat?\" Jennica asked after she'd successfully completed a problem.\n\n\"I'll look.\" I crossed the kitchen and swung the refrigerator door open. \"Not really.\"\n\n\"You must have _something_ in there,\" Jennica protested. \"I'm starving.\"\n\nI frowned at the illuminated shelves. There were a quarter carton of expired milk, five Diet Cokes, three eggs, some carrots, and two slices of pizza left from Saturday night's dinner. Dad used to do the grocery shopping, and after the accident, Mom just forgot sometimes. She worked long hours in Boston, and most nights when she got home, she was too tired to cook.\n\nI'd thought it would get better in July, after the vehicular homicide trial ended. The woman who hit us had been high on drugs. The police couldn't figure out what she was doing in our neighborhood; she lived nine miles away, in North Carver. Mom had gone to the trial every day and had even spoken at the woman's sentencing, but she'd only gotten four years, a suspended license, and a fine. I couldn't believe that was all my dad's life was worth.\n\nI'd hoped that after the sentencing, Mom would have a little bit of closure and would go back to acting somewhat normal. But instead, she'd just started working even more. We hardly ever saw her. She had Pizza Hut, Papa John's, and Fung Wa Chinese in the #1, #2, and #3 spots on speed dial; most of the time, she called from the office to ask me to order food because she wouldn't be home in time for dinner.\n\nI cracked the pizza box and inspected the slices. No mold growing on them yet. I shrugged and pulled the box out. \"How about pizza?\" I asked Jennica. \"What kind?\"\n\nI checked out the slices more closely. \"Pepperoni and sausage, I think.\"\n\nShe wrinkled her nose. \"I don't eat meat anymore,\" she said. \"But I guess I could pick it off.\"\n\nI stared at her. \"You don't eat meat anymore?\"\n\n\"I'm trying to lose weight,\" she mumbled.\n\n\"Since when?\" I asked. Jennica had always had curves I was jealous of, and she stayed in great shape, thanks to swimming. I'd had enough Twizzler and Doritos binges at sleepovers with her to know that she'd never been concerned about stuff like that in the past.\n\nShe looked down. \"I just don't want Brian to think I'm fat.\"\n\n\"Did he say that?\"\n\n\"No.\"\n\nI paused, unsure what to say. \"So why are you worried?\"\n\nShe didn't say anything for a long time. Then, in a voice I could barely hear, she said, \"I don't know. What if that's why my dad left my mom? Because she got fat?\"\n\n\"Did your dad say that?\" I asked.\n\nShe shook her head. \"It's just my dad started dating Leanne, like, right away and she's super skinny. And now my mom's put on, like, thirty pounds, and Leanne keeps shrinking. And he's always talking about how beautiful she is.\"\n\nI took a deep breath. I knew it made me a terrible friend, but I had trouble hearing about Jennica's problems with her mom and dad. I felt bad for her that they had just gotten divorced\u2014they had separated just a month after the accident\u2014but the way Jennica talked about it drove me crazy. It was like her world was ending because her mom and dad no longer lived under the same roof.\n\nBut at least they were both alive.\n\nI didn't say that, though. I didn't tell her that her problems paled in comparison to mine. Because that would make me a really horrible friend, wouldn't it? So instead, I pasted on a smile. \"I'm sure that had nothing to do with your parents' divorce.\"\n\n\"How do you know?\" Jennica asked.\n\nI paused. \"I just do,\" I said. \"Besides, that has nothing to do with you and Brian. He's totally in love with you.\"\n\nJennica looked down again. \"Yeah,\" she said softly.\n\nI microwaved the pizza for Jennica. After she'd eaten it, dutifully picking off all traces of meat, we did some more sample questions for the trig quiz. She left around five; Logan came traipsing through the front door at six after making out with Sydney in the driveway; and Mom called around seven to say she wouldn't be home for a few hours and to go ahead and eat without her. Like that was anything new.\n\nI ordered fried rice, sweet-and-sour chicken, and beef with broccoli from Fung Wa, and Logan, Tanner, and I ate in silence, none of us making eye contact. After dinner, the boys retreated to their rooms, shutting the doors behind them. I cleaned up the kitchen table, put the leftovers in Tupperware, and loaded the dishwasher. Then I sat down to crack open my fortune cookie.\n\n_The one you love is closer than you think_ , the fortune read. At first I snorted, thinking it meant some guy I loved. And since I didn't love any guy, that was impossible. Then I wondered if it meant something else. I glanced at the ceiling, imagining Logan and Tanner in their rooms, with their stereos on, already entirely separated from the reality of our family. I thought of Mom, forty miles away in Boston and a thousand miles away emotionally.\n\nFinally, I thought of Dad. \"The one you love is closer than you think,\" I said aloud. I looked up and wondered why I didn't believe the words. Well-intentioned adults always told me that my dad was in heaven, watching over me and my mom and brothers. It was an easy thing to say, but if it was true, why couldn't I feel him anymore? Why couldn't I feel anything?\n\n\u2022 \u2022 \u2022\n\nI had just gotten Tanner to bed, and Logan was locked in his room talking on his cell phone, when Mom walked through the door later that evening. I noticed right away that her eyes were bloodshot.\n\n\"What are you doing still up?\" she asked, staring at me as she came in through the garage door.\n\nI was sitting in the kitchen, reading _The Great Gatsby_ for English class. I liked it way more than I'd expected to, and I'd read past what we were required to read for class this week. I glanced at the clock and realized it was just past eleven. \"I guess I lost track of time.\"\n\n\"You really need to get to bed at a reasonable hour, Lacey, or you're going to be tired for school. We've talked about this before. You can't be irresponsible.\"\n\nHearing her say that made my insides twist. Irresponsible was the last thing I was. But I knew the conversation wasn't really about me being up past eleven. \"Are you okay?\" I asked.\n\nShe looked away. \"I'm fine,\" she said. \"Is there some dinner left over?\"\n\nI hopped up. \"I'll make you a plate.\"\n\n\"I don't need\u2014\" Mom began, but I cut her off.\n\n\"Don't worry, I'll get it,\" I said. \"Just sit down and relax.\"\n\nShe opened her mouth to reply, but nothing came out. Instead, she sank slowly into a seat at the kitchen table, kicked off her heels, and sighed.\n\n\"So,\" I said as I scooped cold fried rice and sweet-and-sour chicken onto a plate, \"do you want to talk about it?\"\n\n\"Talk about what?\"\n\n\"Whatever's wrong,\" I said. I slid the plate into the microwave, set it for a minute thirty, and pushed Start. I turned and looked her in the eye. \"You've been crying.\"\n\n\"No, I haven't,\" Mom protested.\n\n\"Can you at least not lie to me?\" I said. She looked away. \"Is it about money?\"\n\n\"What would make you think that?\" she asked. \"You know Dad had a life insurance policy and that I'm making plenty. Why do you keep worrying about that?\"\n\nI shrugged. \"You always seem worried.\"\n\nShe didn't say anything. The microwave beeped. I pulled the plate out and slid it in front of her, along with a fork. I sat down beside her and tried a different tactic. \"You were at the office late today.\"\n\nMom didn't look at me as she speared a piece of chicken and took a bite. \"I had a lot to do,\" she said after she'd swallowed.\n\n\"Like what?\"\n\n\"I don't want to bore you with it,\" she said. \"Lawyer stuff.\" She took another bite.\n\nI knew that was code for _Stop asking me questions_ , so I changed the subject. \"Tanner has to do a diorama for school,\" I said. \"They're supposed to make scale models of their bedrooms. So he'll probably need some supplies.\"\n\n\"Okay,\" Mom said. \"If you e-mail me a list, I'll pick up the materials on my way home from work tomorrow.\"\n\n\"He'll probably need some help with it,\" I prompted. \"I don't think he's done a diorama before.\"\n\nMom took another bite and glanced up. \"Lacey, I've got a really busy week. My caseload is just unbelievable.\" She scooped up some rice and added, \"Maybe you can help him. You're good at that kind of thing.\"\n\n\"At dioramas?\" I couldn't resist asking.\n\nMom shrugged. \"You're more creative than me,\" she said. \"And you have more time. You'd be doing me a big favor, honey. Please?\"\n\n\"Yeah, okay.\" I paused and tried to decide how to phrase what I wanted to say. \"Look, maybe you could spend some time with Tanner this weekend or something, though. I'm really worried about him.\"\n\n\"Lacey, he's always been quiet. You can't keep worrying about everybody and everything.\"\n\n\"But if I don't,\" I said before I could think about it, \"who will?\"\n\nMom held my gaze. Then she stood up from the table and scraped the remainder of her food into the trash can. She put her plate in the sink and turned to me. \"I'm going to go to bed,\" she said. \"You should get some sleep too, honey.\"\n\nI watched her walk out of the kitchen. She reminded me a little of a ghost. She'd lost a lot of weight since the accident, and now, instead of walking with the purposeful stride of an attorney who knew what she wanted out of life, the way she used to, she seemed to shuffle from place to place, a vacant look on her pale face. I wondered whether she acted like this at her office, too, and if anyone noticed.\n\nI cleaned up the kitchen, rinsed Mom's plate, started the dishwasher, and walked upstairs to my room, wondering how it was possible to have an entire conversation without saying anything at all.\n\n# chapter 3\n\nBy Wednesday, Sam Stone had gotten his own textbooks, so he didn't have to share with me anymore in trig. And it wasn't like we had any other reason to talk to each other. The rumor was he had moved from somewhere out in western Massachusetts, but I didn't feel like it was my place to ask him why. If anyone knew what it felt like to be drilled with unwelcome questions, it was me.\n\nAt lunch that day, I was sitting with Jennica and Brian as usual. He had his right arm draped around her, which I figured must make it tough to eat.\n\n\"So that new guy, Sam?\" Jennica asked. I looked up, startled that she seemed to be reading my mind. \"You know,\" she continued, \"that guy from our trig class?\"\n\n\"Yeah,\" I said.\n\n\"So he's hot, huh?\" She leaned forward and grinned at me.\n\n\"Jennica!\" Brian exclaimed, feigning hurt as he pulled her closer.\n\n\"Aw, baby, he's not as hot as you,\" she said.\n\nBrian stuck out his bottom lip in a mock sulk. \"Really?\"\n\nJennica giggled. \"You're the hottest of the hot.\" She gave him a quick kiss on the lips.\n\n\"No, _you're_ the hottest of the hot,\" Brian said in an equally disgusting voice.\n\n\"No, you are,\" Jennica said, batting her eyelashes.\n\n\"No, you are, pookie,\" Brian said, leaning forward to kiss her.\n\n_Pookie?_\n\n\"I think I just threw up a little in my mouth,\" I muttered. I stood, and the two of them looked up from their love haze.\n\n\"What's wrong?\" Jennica asked, blinking at me.\n\n\"Nothing. I'm just not hungry anymore. I'll see you later.\" I grabbed my tray and waited for her to ask me where I was going\u2014after all, there weren't exactly a lot of exciting lunchtime options at Plymouth East\u2014but she had already turned back to Brian.\n\nI threw out my trash and headed out the door to the mostly empty halls. We were allowed to make brief trips to our lockers during lunchtime, but we got in trouble for hanging around too long, so I figured I'd just switch out my morning books for my afternoon ones and go outside. It was overcast, but it hadn't rained yet, and there was a bench under the big oak tree near the senior parking where I sat when I didn't feel like sitting with Romeo and Juliet in the cafeteria.\n\nI had just opened my locker and was digging around in the back, trying to find my compact mirror, when a deep voice coming from the other side of my locker door startled me.\n\n\"Hey.\"\n\nI swung the door closed and found myself face to face with Sam. He was leaning casually against the lockers, his hands jammed in his pockets. I blinked at him, then dropped the English textbook I was holding. It bounced off my backpack and hit me in the calf. I winced.\n\n\"Problem?\" Sam asked, glancing down at the textbook and then back at me.\n\n\"No,\" I said quickly.\n\nSam studied me and then smiled, the corners of his mouth creeping slowly upward like a stream of syrup spreading across a pancake. \"You sure?\" he asked.\n\n\"Positive.\" I felt a little short of breath.\n\nHe bent down and picked up my backpack and my textbook in one smooth motion. \"Here,\" he said, handing them to me. \"You might need these.\"\n\n\"Thanks.\" I stared at my feet, willing my face to stop flaming. What was wrong with me? I was reliable, mature Lacey Mann, who could be trusted to behave like a grown-up in any situation. And here I was acting like, well, Sydney.\n\n\"So.\" Sam put his hands back in his pockets. His warm green eyes met mine. \"What are you doing out here in the hallway? Shouldn't you be eating with that friend of yours in the cafeteria? Jennica?\"\n\n\"It's no big deal,\" I said. \"She's with her boyfriend. I just felt like walking around.\"\n\n\"Guess you don't want to watch her and her boyfriend all over each other,\" he said.\n\nI looked up sharply. \"What? No. That's not it.\"\n\nSam looked like he didn't believe me. \"It would bother me.\"\n\nI paused. \"Okay,\" I admitted. \"Maybe it bothers me a little.\" I cleared my throat, suddenly desperate to change the subject. \"So, um, your old school,\" I said. \"Where is it? I mean, where did you come from?\"\n\n\"Taunton.\"\n\n\"Oh,\" I said. \"I've been there.\" It was about thirty minutes away.\n\n\"Oh yeah?\"\n\nI nodded. \"My brother Logan played in a baseball tournament there a few years ago.\" _Back when he still played baseball_ , I added silently. _Back when things were normal_.\n\n\"Cool,\" Sam said. \"I used to play ball. Maybe I played against him. Are you a baseball fan?\"\n\n\"Definitely.\"\n\n\"Sox?\" he asked.\n\nI nodded again. \"My dad always takes my brothers and me to Fenway a few times each summer.\" Then I stopped abruptly, the words caught in my throat as I realized what I'd just said.\n\n\"Cool,\" Sam said, oblivious. \"I haven't met a lot of girls who like baseball. Did you guys make it to a lot of games this year?\"\n\nI swallowed hard. \"No,\" I said without elaborating.\n\nSam seemed to register that something was off. He un-slouched from the locker and drew himself up to his full height. He was taller than I had realized.\n\n\"You okay?\" he asked.\n\n\"Fine,\" I said.\n\n\"Okay,\" Sam said uncertainly. He gave me a half smile. \"I'll see you in class then, cool?\" He turned and walked away.\n\n\u2022 \u2022 \u2022\n\nThe rest of the week sped by, the way the weeks in the fall always did, when your new grade and new classes still felt fresh and exciting. Sam had begun smiling at me in class now and saying hi in the halls like we were friends. I always smiled back and then looked quickly away, as if locking eyes with him would be a dead giveaway that I was beginning to develop a crush.\n\nIt's not like it was wrong to feel that way about him. It was just that I figured I didn't have much of a chance. Why bother liking him if the chances of him liking me back as more than a friend were slim to none? Summer Andrews was already flirting with him, and on Thursday, I saw him sitting at her popular senior table for lunch. I wasn't an outsider\u2014I was on student council and played lacrosse in the spring, and people liked me just fine\u2014but I wasn't a cheerleader either. I was brainy, quiet Lacey who everyone thought of as sweet instead of sexy. And despite what my dad used to tell me freshman year, when I'd come home sometimes on the verge of tears, there wasn't a single guy at Plymouth East who would go for a nice girl over an easy one.\n\nOn Saturday morning, I was lying in bed, half-awake, trying to stop thinking about Sam, when the sounds from downstairs snapped my eyes open. I glanced over at the clock: 6:06. Too early for me to be awake. Too early for the TV in the living room to be on. But there was no such thing as too early or too late in our house anymore.\n\nI sat up and listened, wondering what Tanner was watching. It was a pretty safe bet that it was either a cartoon or something to do with animals. He was obsessed with animals. Sure enough, when I went down the stairs a few minutes later and rounded the corner into the dark living room, my little brother was sitting a foot from the TV, his face bathed in the glow from the screen. I could see a giraffe ambling through the wilderness.\n\n\"Good morning,\" I said casually, as if it were normal for him to be sitting there, looking like he wanted to climb inside the TV and escape into the wild himself. Tanner turned his head slightly and nodded before returning his attention to the screen.\n\nI went into the kitchen to make us some breakfast. I was determined to pretend that everything was normal until it actually was.\n\nAfter scanning the fridge to see if Mom had picked anything up on her way home last night\u2014she had\u2014I turned the stove on and slipped three pieces of wheat toast in the toaster. I pulled out a frying pan, put it on the burner, sprayed it with PAM, and cracked three eggs into it, making sure their edges didn't touch, the way Dad always used to when he made breakfast for us.\n\nA few minutes later, I scooped the eggs, their yolks still runny, out of the pan and onto the toast. When I walked back to the living room, Tanner accepted his plate without even looking up. He was riveted to the screen.\n\n\"So what are you watching?\" I asked after I'd set two juice glasses down and taken a bite of my toast. I knew it was _The Crocodile Hunter_ , one of Tanner's favorite shows, but I wanted him to say it. Ever since the accident, he had retreated further and further into himself, and now he hardly said a word, not even to his friend Jay, who came over to play video games once a week. Although, come to think of it, I hadn't seen Jay for a while now. I wondered if he'd finally given up on Tanner.\n\nNobody seemed to care but me. I had tried bringing it up with Mom, but she just shrugged and said that it wasn't all that abnormal and that Tanner would deal with things in his own time. But what did she know? She saw her legal assistant ten times more often than she saw her kids; Tanner was usually asleep by the time she got home. I had also tried talking about Tanner with Dr. Schiff, the psychologist my mom made us visit every other Saturday. But she had just told me that it wasn't my responsibility. \"You're just a kid,\" she would always say.\n\nIt always made my blood boil.\n\nAs usual, Tanner didn't answer my question. Instead, he grabbed the remote and hit the Info button until the name of the show appeared at the bottom of the screen. He shot me a look and returned his attention to the TV.\n\n\"This looks like a good one!\" I said enthusiastically, as if we were having a normal conversation. I fished for something else to say. \"I really like how he explains everything so well. And his accent is really cool. Don't you think?\"\n\nTanner nodded without taking his eyes off the screen. He took another bite of his toast. I pushed mine away. I didn't feel hungry anymore. I made some more cheerful, one-sided small talk before I gave up. Tanner obviously wasn't going to respond. And I had run out of things to say.\n\n\"Okay, Tanner,\" I said, feigning cheerfulness. \"I'm going to go hop in the shower.\"\n\nI had just crossed the living room into the kitchen when I heard Tanner's voice. I stopped and turned around.\n\n\"What?\" I asked.\n\nHe was silent for a minute, and I started to doubt that he'd said anything at all. Maybe I'd imagined it. But then he spoke again.\n\n\"You know, he died too,\" he said clearly, still staring at the TV screen. \"The Crocodile Hunter. A stingray got him.\" He looked at me, evidently expecting some kind of response.\n\nI gulped. \"Yeah, I know.\"\n\n\"No one saved him either,\" he said. Then, he turned the volume up. The conversation was over.\n\nI stood there, my heart thudding in my chest. Guilt and responsibility weighed down on me, squeezing me from the inside out.\n\nA hundred times a day, I thought about how different life would be if I hadn't insisted on taking those extra moments in the bathroom. Or if I had cried out to warn Dad, in that instant before the Suburban hit us. Instead, I hadn't reacted. It had been the only important thing I'd ever had to do in my life, and I'd failed. It was like Tanner said. Nobody saved the Crocodile Hunter.\n\nAnd I hadn't done anything to save my dad.\n\n\u2022 \u2022 \u2022\n\nBy the beginning of the next week, Sam Stone was the talk of the school. Summer Andrews had apparently decided that he was her big new love interest. It didn't seem to matter to anyone else that Summer actually _had_ a boyfriend, Rob Macavey, a senior with big arms, close-cropped dark hair, and eyes that were just a little too close together. Jennica and I agreed that he'd clearly been hit in the head one too many times on the football field.\n\nBut Summer, who didn't even have a class with Sam, had decided that she had a crush on him, and so the whole school knew she had called dibs. I couldn't understand why she couldn't just limit the number of guys she tried to pounce on.\n\nIn class, Sam and I were apparently friends now. I supposed it was because he didn't know anyone very well yet, and since he sat next to me in two classes, I was a logical person to strike up a conversation with. I was surprised to realize how much I liked talking to him, though. It started to be a routine that he would sit down in trig, grin at me, and rattle off the Red Sox score from the previous night, as well as some kind of commentary about a player who screwed things up\u2014even if the Sox ended up winning. I'd been a Sox fan for years and could practically recite the roster in my sleep, but I'd never known a boy before who would talk to me about sports like I knew what I was talking about. It was nice.\n\nAfter school on Wednesday, I was surprised to find Sam waiting for me by my locker. I'd had plans to go to the mall with Jennica and then study with her later, but she had texted me after the final bell to say that something had come up with her mom and she couldn't make it. Concerned, I'd texted her back to see what was wrong. Don't worry, she had written. Brian's with me.\n\n\"There you are,\" Sam said as I approached.\n\n\"Hey.\"\n\n\"So how's it going?\" Sam asked. Was it my imagination, or did he seem nervous?\n\n\"Good,\" I said. I couldn't for the life of me figure out what he wanted.\n\n\"So I hear you're really good at trig,\" he said. \"Right?\"\n\nI shrugged. \"I guess.\"\n\n\"I was just wondering if maybe you could help me study for the test on Monday,\" he said.\n\n\"Yeah, okay.\" I forced a smile. I was Lacey, the reliable study buddy. I just wasn't sure how he'd figured this out so quickly. \"I was going to study with Jennica tomorrow, so you can come over too, if you want.\"\n\n\"Cool, thanks.\" He paused. \"So, do you need a ride home or something?\"\n\n\"Now?\"\n\n\"Yeah.\"\n\nI glanced around. Jennica was gone. I'd probably already missed the bus. And riding with Sam would be preferable to riding with Logan and Sydney any day. \"Okay,\" I said. \"That would be great.\"\n\nI pulled a few books out of my locker and shut it. Sam surprised me by taking my backpack off my arm, slipping my books into it, and tossing it over his shoulder. \"C'mon,\" he said.\n\nI followed him outside. He opened the door of his Jeep for me and tossed our bookbags in the back. I told him how to get to my house, and soon we were cruising down Court Street. The silence between us was beginning to feel stifling.\n\nFinally, I blurted out, \"So are you going out with Summer Andrews?\" I felt like an idiot the moment the question left my mouth.\n\nSam looked at me in surprise. \"What?\"\n\n\"Nothing,\" I mumbled. It wasn't my business.\n\n\"Summer Andrews?\" he asked after a pause. \"That senior girl?\"\n\nI nodded.\n\n\"What would make you think that?\"\n\n\"I just heard she liked you.\"\n\nSam seemed to consider this for a minute. \"She seems nice enough,\" he said. \"But I barely know her.\"\n\n\"I'm sure that'll change.\"\n\nSam turned left on Samoset. \"She's not really my type.\"\n\n\"Really?\" I was baffled. Who was this new breed of boy, immune to Summer's powers? \"Oh.\"\n\n\"I like girls who are smart,\" Sam continued. \"You know, girls who don't flirt with every guy in the school. Girls who have a little substance to them. I get the feeling I'm not exactly describing Summer.\"\n\n\"You're right about that,\" I muttered.\n\nWe rode in silence for a few minutes as I tried to process what he'd said. He barely knew me either, but he'd sought me out in the hallway after school. Maybe it _wasn't_ just to study.\n\nI was just beginning to feel like maybe I'd gotten it all wrong, when we pulled up in front of my house and Sam turned to me. His eyes looked even brighter than ever, and even when he wasn't smiling, the vague indentations of his dimples remained.\n\n\"Listen,\" he said. He was definitely nervous now. \"I was thinking that maybe we could go out sometime. If you want to. I mean, it would be cool to hang out outside of class, you know?\"\n\nWas he asking me out? A smile rolled across my face before I could stop it. \"That sounds good.\"\n\nSam looked like he wanted to say something else. It was so nice, I thought in the silence, to finally have someone look at me for me, not as someone they had to feel sorry for or tiptoe around. Last winter, after the accident, several Plymouth East guys had messaged me on MySpace or stopped me in the halls, and I knew that it was just because I was a minor celebrity for a few weeks. That's when Sydney had first taken an interest in Logan too; it's when we became _somebodies_.\n\nAnd now, for the first time since the accident, I finally felt like someone was seeing me for something other than that-poor-girl-whose-dad-is-dead. Sam didn't know my history. He didn't know he was supposed to feel sorry for me or whisper about me behind my back or purposely avoid mentioning anything to do with fathers.\n\nAnd just when I was feeling good, Sam opened his mouth and ruined everything. \"I heard about your dad,\" he said.\n\nI could practically feel the walls coming up around me. The smile fell from my face, and everything went cold. I didn't say anything. I just stared at Sam.\n\nHe looked uncomfortable. \"Listen, I'm sorry.\"\n\n\"Yeah, well, it's old news.\" My voice was full of ice.\n\n\"If you ever want to talk about it...,\" Sam said, his voice trailing off.\n\n\"Look, I don't need some hero to make it all better, if that's what you're trying to do,\" I snapped. \"I'm _fine_. It happened a long time ago.\"\n\n\"I'm not trying to do that.\" Sam looked surprised. I could have sworn I saw hurt flicker across his face too, but I didn't care. Who was _he_ to be hurt? \"I just meant, well, I know how you feel,\" he added.\n\nI could taste bile in my mouth. I stared at him. Of all the things people said to me to try to make me feel better, I hated that sentence the most. Sam Stone didn't know how I felt. How could he? I was sick and tired of people who'd had a grandparent die and thought it was the same thing. Or even worse, people who'd had to bury a pet iguana or the dog they'd grown up with. Sure, I felt sad for them, but how could they possibly compare that to losing a parent?\n\n\"You have no idea how I feel,\" I said coldly. I reached into the backseat and grabbed my bookbag. I couldn't get out of the Jeep fast enough.\n\n\"But Lacey\u2014\"\n\n\"Forget it,\" I said firmly. I fumbled with the door handle and spilled out with my things. I could feel Sam watching me all the way to my front door, but I didn't turn around.\n\n\u2022 \u2022 \u2022\n\nI was overreacting. I knew it. But I couldn't melt the wall of ice that had formed around my heart in those last few minutes in Sam's Jeep. I hated it when people tried to help me, especially now. Couldn't they see I was dealing just fine? _I_ was the person holding my family together. I didn't need anyone's help or pity. Especially not some new guy's. I wondered if he had roved the halls of his old school too, looking for sad girls to save.\n\nSo I steadfastly ignored Sam, even when he tried to pass me a note the next morning in trig class, even when he threw a paper airplane at my arm to get my attention. I didn't want to talk to him. He wasn't the person I thought he was; he was nosy, just like the rest of them.\n\nThat was what I was thinking about when Brooke Newell arrived in the doorway with a note in her hand. She was one of the seniors who was community college-bound already and was taking an office-assistant class for credit. She handed Mrs. Bost the note, snuck a look around the classroom, waved to Krista Sivrich, and then hurried away.\n\nMrs. Bost unfolded the note and read it. When she looked up, she stared right at me.\n\n\"Miss Mann,\" she said, \"your presence is requested in Mr. Miller's office.\"\n\nA murmur went through the class, and I swallowed hard. Mr. Miller was the main principal. You didn't get sent to him unless something was really wrong. I certainly hadn't done anything to get myself in trouble, so my first thought was _Mom_. Had something happened to her? Or to Tanner? Could something have happened to Logan since I got out of the car thirty minutes ago?\n\nI stood up and stuffed my notebook and pen into my bag.\n\n\"Does it say why he wants to see me?\" I asked, hating that my voice sounded nearly as panicked as I felt. Someone in the back of the room snickered, and I heard someone else say, \"Ooh, she's in _trouble!\"_\n\n\"No,\" Mrs. Bost said. I glanced at Jennica, who looked worried. Then, just because I couldn't help it, I locked eyes with Sam.\n\n\"Want me to come with you?\" he asked, like it was the most normal question in the world. I opened my mouth to say no, but Mrs. Bost preempted me.\n\n\"I think Lacey is capable of finding the principal's office by herself,\" she said, giving Sam a look.\n\nSam glanced at me again and shrugged. I could feel my cheeks getting hot. I strode quickly into the hall before my throat could close up entirely.\n\n# chapter 4\n\nMr. Miller's secretary ushered me into his office right away, which only added to my already heightened sense of panic.\n\n\"Is my mom okay?\" I asked immediately, without bothering to say hello. \"And my brothers?\"\n\n\"Yes, yes,\" Mr. Miller said hastily. He looked a little confused. \"Of course. As far as I know.\"\n\nI felt the air I'd been holding in leave my body in a whoosh. \"Thank God,\" I said.\n\nMr. Miller was silent for a minute, as if waiting for me to say something else. He gestured to a chair facing his desk, and I sat down. He continued to stand, staring down at me. He was tall, well over six feet, and he had a comically thick shock of dark hair\u2014too uniformly brown for a man over the age of fifty\u2014that looked out of place on his egg-shaped head. \"He's had hair transplant surgery, for sure,\" Dad used to murmur to me whenever we'd see Mr. Miller at football games and school concerts.\n\nThat's what I was thinking about when Mr. Miller cleared his throat. \"Lacey, do you know Kelsi Hamilton?\" he asked.\n\n\"Yeah,\" I said. \"Her mom has cancer.\" The moment the words were out of my mouth, I hated myself a little bit for saying them. It was the way everyone identified me: by the sad thing that had happened in my life.\n\nI'd known Kelsi since elementary school, and I'd had a class with her last year, but she was quiet, and we hadn't sat near each other, so we barely ever talked. I knew as well as anyone else in the school that her mom had been diagnosed with lung cancer back in May. Bad news tended to travel fast, whispered near lockers between classes, until everyone was walking around with a piece of your life stuck in their back pocket like a trading card.\n\n\"Lacey, Kelsi's mother passed away last Saturday,\" Mr. Miller said.\n\n\"Oh no,\" I said, my heart sinking for Kelsi. \"That's awful.\"\n\n\"Yes,\" he said, sitting down. He pressed his hands together. \"Lacey, I need to ask you a favor. And please, feel free to say no.\"\n\n\"Okay.\"\n\n\"Kelsi is back in school today,\" he said. \"For the first time since her mother, um....\"\n\n\"Died,\" I filled in. It was sometimes hard for people to actually say the word. I had gotten used to filling it in, in awkward silences, like I was playing a constant game of Mad Libs with only one word to put in the blanks.\n\n\"Yes,\" Mr. Miller said. \"I was wondering whether you might... spend some time with her.\"\n\n\"What do you mean?\"\n\nMr. Miller cleared his throat. \"Kelsi's father called this morning, and of course she's still very upset. He was hesitant to send her back to school, but apparently she insisted. Now, last year, when your father passed...\" He paused awkwardly. \"Well, I know you had Logan to help you through. At school, anyhow.\"\n\nI resisted the urge to snort. What exactly had Logan done to help me?\n\n\"So I'd like to ask you, as a favor to me\u2014well, to Kelsi, really\u2014if you'd talk to her,\" Mr. Miller concluded.\n\n\"Talk to her?\" I echoed.\n\n\"You know. Just let her know that you're there for her.\"\n\n\"Oh. Of course,\" I said right away. After all, Kelsi had to know that I'd understand in a way other people couldn't. I wished I'd had someone like that when my dad died, instead of feeling like such an oddball. Sure, Cody Johnson's dad had died in Iraq when we were all in eighth grade, so I suppose he could identify with me when my dad died. But he never said anything. In fact, I could swear he deliberately avoided me, just like so many other people who didn't know how to act. I wished I could scream at people that I was the same person, that all they had to do was treat me normally. But apparently when you had a parent die, you became some sort of science experiment, to be poked and prodded and stared at.\n\n\"I've already spoken with your second-period teachers,\" Mr. Miller said. \"You and Kelsi are both good students, so they have no problem releasing you from class so you can have a chat. Maybe the two of you can take a walk or something.\"\n\nWell, that sounded supremely dorky. I suspected that Mr. Miller was imagining that when we came back from our stroll, Kelsi wouldn't be upset anymore. I didn't want to be the one to tell him that real life didn't exactly work that way.\n\n\"Sure,\" I said instead.\n\n\"Thank you, Lacey.\" Mr. Miller sighed and looked very relieved, like he had just had a great weight lifted off his slumped shoulders.\n\nI could feel the weight he'd just lifted settle inside my chest. \"No problem.\"\n\n\u2022 \u2022 \u2022\n\nBack in class, I pretended I didn't notice Jennica's raised eyebrows. I also pretended I didn't see Sam staring at me. Actually, pretty much _everyone_ was looking at me. I'm sure they were all wondering what I'd done wrong to be called into the principal's office.\n\nI escaped Jennica's questions after class by mumbling something about Logan being in trouble again. I knew I should have just told her the truth. But I figured that it wasn't my place to be telling people Kelsi's bad news. I knew that the rumor would be all over school in a few hours, but I didn't want to be one of the people to spread it.\n\nThirty minutes later, I was headed back to Mr. Miller's office with a hall pass, filled with a strange kind of trepidation. I wanted to help Kelsi, but I was almost paralyzed by the fear that I wouldn't know what to say or do. _Relax, Lacey_ , I told myself. _You're holding your family together. You can definitely figure out how to help this girl_.\n\nKelsi was already sitting in Mr. Miller's office when I got there. Her carrot-colored curls, which were usually cute and perky, were hanging limply, like she hadn't thought to wash or comb her hair in days. She looked thin. She was wearing old, faded jeans and a Plymouth East marching band shirt that was too big for her. I stared for a second, realizing this was what I must have looked like in the weeks after the accident, like I didn't care, didn't even realize that people were noticing my disheveled appearance.\n\n\"Hey,\" I said to Kelsi.\n\nKelsi looked up at me. \"Hey,\" she said. Her eyes looked tired, but not like she'd been crying. Maybe she'd run out of tears. It happened sometimes.\n\nI glanced at Mr. Miller and sat down in the other chair facing his desk. Kelsi was staring at her lap now. She looked like she wanted to disappear. My heart ached a little with the familiarity of it all.\n\n\"I'm sorry,\" I heard myself say after a minute. I hadn't meant to say it. In fact, I hated it when people said that to me. It wasn't like _they_ were the ones who had killed my dad. What were they sorry for? But the words escaped before I could stop them.\n\nKelsi looked up. \"Yeah,\" she said. It seemed like she was having trouble focusing on me.\n\nI glanced at Mr. Miller again. \"So,\" I said, \"do you want to take a walk or something?\"\n\nThe question sounded strange, and I expected Kelsi to react like I was crazy. But instead she just shrugged. \"Whatever.\" Without looking at me, she grabbed her bookbag. \"Let's go,\" she said. I followed her out of the office, thinking for the first time that I might be in over my head.\n\n\u2022 \u2022 \u2022\n\nOutside the school building, I had to jog to keep pace with Kelsi.\n\n\"Wait up,\" I said. This probably wasn't the bonding experience Mr. Miller had visualized, me speeding after Kelsi while she practically ran to escape me.\n\nBy the time we rounded the corner, I realized she was making a beeline for her car, a lime green VW Bug. She slid behind the wheel and slammed the door. I heard the engine turn on, and for half a second, as I stood in front of the car, I half expected her to lay on the gas pedal and run me over. Instead, she just sat there, staring at me. Finally, she rolled down her window. \"Well? Are you getting in or what?\"\n\nI glanced around. \"We could get in trouble,\" I said. We could get detention for sitting inside our cars during the school day, and suspended for leaving school grounds.\n\n\"You really think anyone's going to bust you and me?\" Kelsi asked. \"The girls with the dead parents?\"\n\nShe was right. Besides, Kelsi needed me. And my responsibility to help her outweighed the risk. I took a deep breath. \"Okay,\" I agreed.\n\nI opened the car door and slid in. \"So. Are we going somewhere?\"\n\nKelsi didn't look at me. \"No,\" she said. \"Unless there's somewhere you want to go.\"\n\n\"No,\" I said quickly.\n\nThe car engine continued to hum. The air conditioner was on high, even though it was in the fifties outside.\n\nJust as the silence was getting uncomfortable, I blurted out, \"Kelsi, I'm really sorry about your mom.\"\n\nMore silence. I could feel my cheeks flaming. Mr. Miller had obviously picked the wrong person to talk to Kelsi.\n\nThen Kelsi said softly, \"Thanks.\" She glanced at me. \"I'm sorry about your dad, too. I never told you that.\"\n\n\"Thank you.\" I was quiet for a moment. \"So are you okay? I mean, how are you?\"\n\nKelsi glanced back out the windshield. She squinted, like the answer to my question might be located on the brick wall of the school. \"It's not like it's a big deal or anything,\" she said finally, still not looking at me. Her words poured out in a rush, like she couldn't wait to get rid of them. \"I mean, she'd been sick for a while. We knew it was coming. I should have\u2014I should have been more prepared for it.\"\n\nI wondered what it was like to have time to say goodbye, to know the end was coming. Did you have fewer regrets? \"But it's not like that makes it any easier,\" I said.\n\n\"But it's supposed to,\" Kelsi mumbled. \"Isn't it?\"\n\nShe was looking at me like I had all the answers. The truth was, I wasn't even sure what the questions were anymore. \"I don't think so,\" I said finally.\n\nI tried to think of something else to say, the kind of thing I would have wanted someone to say to me. But nothing was coming to me. I sat back in the seat.\n\n\"Can you just go away now?\" Kelsi asked. \"I want to be alone.\"\n\nI looked at her, surprised. \"Um, yeah, sure,\" I said, hoping she wasn't depressed enough to do something stupid. \"Are you sure you're okay?\"\n\nShe glanced at me. \"What do _you_ think? Are _you_ okay?\"\n\nI was taken aback. \"Yeah,\" I said.\n\nShe snorted and looked away. \"Yeah. You're very convincing.\"\n\nHer words startled me. I was fine. I was happy. I had gone back to being normal. \"I _am_ okay,\" I insisted.\n\n\"Whatever,\" Kelsi said. \"But look, I really just want to be alone.\"\n\nI grabbed my bag and opened the car door. \"If you need anything, you can ask me, okay? I mean, I've been through this.\"\n\n\"I know,\" Kelsi said. She paused and then added, \"Thanks.\" The word was so soft I could barely hear it.\n\n# chapter 5\n\n\"You have to help me,\" I told Logan at lunch. As I plunked down beside him in the cafeteria, Sydney looked at me like I'd just arrived from outer space. In Plymouth East terms, maybe I had.\n\n\"Hi,\" Sydney said, glancing around, probably calculating how much my presence at the table was reducing her social status.\n\n\"I have to help you?\" Logan said. \"With what?\"\n\n\"With Kelsi Hamilton,\" I said.\n\n\"Oh yeah, I heard her mom died,\" Logan said casually, like it was no big deal. \"Bummer.\"\n\n\"It's all over school,\" Sydney chipped in. \"Did you only just hear about it? I've known since, like, nine this morning.\"\n\n\"Now it's a contest?\" I asked. I refrained from adding that I'd known earlier than that.\n\nSydney mumbled something and made a face. I turned back to Logan. \"Yeah, Lo, her mom died. It's horrible.\"\n\nHe shrugged. \"I guess. I mean, it's not like we really know her.\"\n\n\"I know her,\" I said.\n\nLogan raised an eyebrow. \"Since when?\"\n\n\"Since... always,\" I said. I didn't think what had happened this morning was any of his business. But I let myself gloat, just a little bit, that Mr. Miller had asked for my help, not his.\n\n\"Okay,\" Logan said. \"But what does that have to do with me?\"\n\nI took a deep breath and began to explain the idea I'd been thinking about since I'd gotten out of Kelsi's car two hours ago. \"It has everything to do with you. I thought maybe one day this week we could get Cody Johnson and the three of us could get together with Kelsi after school.\"\n\nLogan and Sydney stared at me like I'd suggested that we eat worms. \"Why would we do that?\" Logan asked.\n\n\"To show her that she's not alone.\"\n\nLogan rolled his eyes, and exchanged looks with Sydney. \"Lacey,\" he said slowly, like he was talking to a child, \"just because our dad died doesn't mean you have to fix everyone else who loses a parent.\"\n\nI stared back. \"I'm not doing that. Maybe I just want to help. What's wrong with that?\"\n\nLogan shook his head. I was surprised to see anger in his eyes. \"You know, Lacey, maybe for once you could just let things go, you know? Can't you just grow up and move on?\"\n\n\"What are you talking about?\" I demanded, suddenly aware that my voice had risen an octave.\n\n\"You know exactly what I'm talking about.\"\n\nI stood up abruptly. \"You know, Logan, I'm just asking for a little help,\" I said. \"But if you can't do that, forget I ever said anything.\"\n\n\u2022 \u2022 \u2022\n\nI was still simmering when Jennica met me at my locker after school. \"We still on for studying today?\" she asked.\n\n\"Of course,\" I said. \"Why wouldn't we be?\"\n\nShe cleared her throat. \"I don't know. I heard about Kelsi. I thought maybe you had to go talk to her or something. Is that why Mr. Miller called you in this morning?\"\n\nI averted my eyes.\n\n\"How come you couldn't tell me that when I asked you?\" she said. There was accusation in her voice.\n\n\"I don't know,\" I said. \"I guess I didn't feel like it was my business to talk about it.\"\n\n\"But I'm your best friend.\" She paused. \"Is it because you think I wouldn't understand?\"\n\n\"No,\" I said too quickly. \"Of course not.\"\n\n\"You know, Lacey, having someone die isn't the only way to lose a parent.\"\n\nI just looked at her. _Not again_ , said the voice in my head.\n\n\"It was hard for me when my parents got divorced,\" she went on. \"But you act like it's no big deal, just because my dad is still alive.\"\n\nI bit my tongue. Hard. I didn't want to get into this with her. I knew it bothered Jennica that I didn't ask her about her parents' divorce very often. And it wasn't that I didn't care. It was just that I couldn't compare a divorce to a death. She could tell her dad she loved him any time she decided to. My chances, on the other hand, were all gone. Forever.\n\n\"I'm sorry,\" I said finally.\n\nJennica sighed. \"I know.\"\n\nI was just about to say something else when I saw Sam approaching. I began shoving books from my locker into my bag. Jennica furrowed her brow. \"What's wrong with you?\"\n\n\"Nothing,\" I said, just as Sam walked up. Jennica looked at him, then at me, and stepped back.\n\n\"Hey,\" he said. He smiled at me. \"So, are you two still studying this afternoon?\"\n\nI shrugged.\n\n\"Can I still study with you?\" Sam tried again.\n\nI took a deep breath. I didn't want to care. But I did. \"I don't think that's a good idea.\"\n\n\"I don't even know what I said to make you upset,\" Sam said. He was standing so close that I could feel his breath on my hair. It gave me goose bumps. \"Look, can I talk to you for a minute? There's something I really need to tell you.\"\n\nI looked away. \"Maybe later,\" I said, trying to sound casual. \"Jennica and I are in a rush now. We've got to catch a ride with my brother and his girlfriend before they leave without us.\"\n\nI slammed my locker door shut, grabbed Jennica's arm, and walked away before Sam could say anything else.\n\n\u2022 \u2022 \u2022\n\nJennica waited to bring Sam up until we were sitting at my kitchen table forty-five minutes later with two Diet Cokes, a bag of microwave popcorn, some Twizzlers Jennica had brought, and our trig books open in front of us.\n\n\"So, are you going to explain what that was all about?\" she asked.\n\nI fiddled with the edge of the popcorn bag and then popped a few pieces in my mouth. \"It's nothing.\"\n\nJennica chomped on a piece of licorice. \"Try again.\"\n\nI sighed. \"Fine. He drove me home yesterday, and I actually thought for, like, a minute that maybe he liked me. Then he said he'd heard about my dad and that he knew how I felt.\" I made a face.\n\n\"Okay,\" Jennica said, waiting for me to go on. \"And?\"\n\nI shrugged. \"That's it.\"\n\n\"Let me get this straight,\" she said. \"The hot new guy, who every girl at school\u2014including Summer Andrews\u2014is into, offers you a ride, has clearly been asking around about you, and says something thoughtful. And this is a problem _why?\"_\n\n\"Jennica, there's a difference between liking someone and feeling sorry for them,\" I said. \"Don't you understand that? The last thing I need is some guy's pity.\"\n\n\"Okay,\" Jennica said slowly. \"Only, what if he doesn't pity you? What if he's just trying to be nice? Because he likes you?\"\n\n\"Well, I don't need someone telling me he knows how I feel,\" I grumbled. \"You know how much I hate that.\"\n\n\"Yeah, well, sometimes you don't give people a chance,\" Jennica said.\n\nI resisted the urge to snap at her that the only person she gave a chance to anymore was her boyfriend. I didn't want to sound jealous. \"Jennica,\" I began. I paused, unsure of what I wanted to say. I wanted to tell her that I missed her, that I missed this, that I missed _us_. I wanted to tell her that there was a huge gulf between us, and I didn't know how to cross it anymore. But before I had a chance to say anything, the doorbell rang.\n\nI swung the door open to find Jay Cash, Tanner's friend from down the street, standing there. \"Hi, Jay,\" I said, surprised to see him. His visits had been getting less and less frequent.\n\n\"Hey,\" he said. He had hit a growth spurt over the summer, but he still had the same goofy grin and wore the faded, dingy Red Sox cap he'd been wearing every day for the past three years. He was holding a baseball glove. \"Is Tanner home? I was wondering if he wanted to play catch or something.\"\n\n\"Yeah, hang on,\" I said. \"He's in his room.\"\n\n\"My mom sent me down,\" Jay added, shifting his weight uneasily from one foot to the other. He looked a little guilty. _That explains it_ , I thought.\n\nI invited Jay in and went upstairs to get my brother. I knocked, and when there was no reply, I pushed the door open.\n\nTanner was squatting in the corner of his room near the window, peering into the cage where his hamster, McGee, lived. Dad had always talked about letting us get a dog, but we'd never gotten around to it. After the accident, Tanner, who had vowed he'd become a vet one day to help save all the sick animals he could, had begged for a puppy. Mom had been firm on saying no; she said we had enough to worry about. But as Tanner's silence deepened, she finally broke down and agreed to get him a hamster, as long as it stayed in his room. He'd had McGee, a chubby puff of brown and white fur, since May.\n\n\"Hey, Tanner,\" I said as I walked in. \"Jay's here. He wants you to come out and play catch with him.\"\n\nTanner was silent for a minute. \"Why?\"\n\n\"Because you're friends,\" I said gently. \"Right?\"\n\nTanner glanced back at McGee, who was curled up in the corner of his cage, his little hamster chest rising and falling in sleep. \"I'm busy.\"\n\n\"Tanner,\" I said, \"you're not really that busy. McGee's just sleeping. Why don't you go play with Jay for a while? You guys haven't hung out in ages.\"\n\nTanner shrugged.\n\n\"Don't you hang out with him at school?\" Tanner shook his head.\n\n\"Why?\" I asked. \"Have you made other friends?\" These were the questions a parent should be asking, I knew. But Mom didn't seem to know how to talk to us anymore.\n\n\"Why not?\" I asked when he shook his head.\n\nNo reply.\n\n\"Is it because you feel sad? About Dad?\" He shook his head.\n\n\"Because the other kids at school tease you?\" I tried again.\n\nMore head shaking.\n\n\"Because you feel left out when people talk about their dads?\" I guessed again. I didn't know what else to say. \"Buddy,\" I said finally, \"I think you should go outside with Jay. Just for a little while.\" I paused, trying to think of what Dad would do. But then again, Tanner had never had this problem when Dad was around. Maybe Dad wouldn't understand it any better than I did. \"You can come home whenever you want to,\" I said.\n\nTo my surprise, Tanner slowly stood up. \"Okay,\" he said. He grabbed his baseball glove off the corner of his bookshelf.\n\nMy heart lurched a little. I put my hand on his shoulder as we walked out of the room and started down the stairs. \"Good,\" I said. \"It'll be fun.\"\n\nTanner stopped and looked up at me. \"It's not fair,\" he said.\n\n\"What's not fair?\"\n\n\"I'm not supposed to have fun,\" he said after a long pause. \"Dad doesn't get to.\" He was gone before I had a chance to open my mouth.\n\n# chapter 6\n\nThe whole next week, I avoided Sam's eyes, ignored his notes in class, and tried not to feel guilty when I noticed the C+ on his trig test. I knew that if I'd helped him study, he could have gotten a higher grade. But I couldn't take care of everyone. And there was someone far more important to pay attention to: Kelsi Hamilton. Knowing that I was the only one who knew how she felt _and_ cared about helping her weighed on me. I watched her walk through the hallways like a zombie, floating from one class to the next.\n\nThe whispers were what bothered me the most.\n\n\"Did you hear about Kelsi?\" was the most common refrain. Some people didn't even bother to whisper. But the worst, by far, were the students who tried to capitalize on her grief to win extra popularity points. People who wouldn't have given her a second look before her tragedy now wanted to be all buddy-buddy with her so that they could be at the center of attention when anyone asked about her.\n\n\"Why can't people just be normal to Kelsi?\" I exploded to Jennica in the cafeteria on Thursday. A hush had fallen over the room as Kelsi walked in. Dozens of pairs of eyes followed her as she sat down at a table by herself and pulled a brown bag out of her backpack without looking up. \"We should invite her to sit with us,\" I added.\n\nJennica looked at me. \"Then you're just acting like everyone else, aren't you?\" she asked. \"Trying to get a piece of her?\"\n\nI glared. Jennica knew very well that wasn't what I was doing. But she was grumpy today because Brian was home sick. Jennica had already asked several times whether I thought she should skip school and bring him chicken noodle soup. I had responded that I thought he was capable of opening a can of Campbell's on his own. She just gave me a look and began mumbling about how when you really loved someone, you shared everything, even germs.\n\nPersonally, I thought that was kind of gross.\n\n\"I'm not trying to get a piece of her,\" I said through gritted teeth. Jennica shrugged and took a bite of her tofu sandwich on whole wheat. She was still on her weight-loss kick, but I saw her staring lustily at the fries on my tray each day while she munched on carrot sticks.\n\nJust then, I noticed two of my least favorite people in the school, Tali Bonner and Tatiana Roseberg, approaching Kelsi's table, their matching raven-colored hair swinging behind them like pendulums. I stiffened. Tali and Tatiana, known collectively as the TaTas\u2014and not just because of the first letters of their names\u2014were senior cheerleaders and two of the most popular girls at Plymouth East, right beneath Summer on the social scale. Tatiana's romantic exploits with underclassmen were legendary, while Tali was rumored to only go for college guys.\n\nOf course, they were also the worst when it came to situations that might increase their popularity. In fact, it was because of them that I'd first realized that I was no longer just plain old Lacey Mann. I was Lacey Whose Dad Died. I had been eating in the cafeteria with Jennica on my third day back to school when they had sauntered up, arm in arm, smiling at me.\n\n\"Oh my God, you must be so depressed!\" Tali had started off without even a hello.\n\n_\"So_ depressed,\" Tatiana had chimed in. I looked around to make sure they were actually talking to me. They confirmed it by settling into two of the empty seats at our table.\n\n\"I mean, you must feel _so_ guilty,\" Tali had continued in the same tone of voice you'd use to compliment someone's outfit.\n\nI just looked at her. Fortunately, Tatiana jumped in to explain. \"Because you were with your dad in the car,\" she said.\n\n\"And you didn't save him,\" Tali added helpfully, a big smile plastered across her heavily made-up face.\n\nSuddenly, the tears that I'd been holding back so successfully were running down my face in rivers. Tatiana looked disgusted; Tali looked delighted. As I jumped up from the table, it felt like the whole cafeteria was watching me.\n\nIt was the last time I'd cried. Since then, the tears wouldn't come.\n\nThe TaTas never spoke to me or acknowledged me again, but I'd heard them telling people that their _friend_ Lacey was still _really_ depressed, and they were doing everything they could to help, because they cared so much. More than once, I'd heard people ask them how I was doing\u2014instead of asking me. The whole thing had made my blood boil.\n\nAnd now they were zeroing in on Kelsi.\n\n\"I gotta go,\" I said to Jennica. I jumped up, dropped my trash in the garbage, and slid into an empty seat at Kelsi's table just as the TaTas were getting started.\n\nI hadn't heard the opening of their conversation, but Kelsi was staring blankly at them. I wasn't sure if it was the blankness of someone who couldn't understand why two of the most popular cheerleaders in school were standing at her table, or the blankness of someone who didn't care about anything anymore, because nothing else mattered when one of your parents had just died.\n\n\"Hi, Kelsi!\" I chirped in my brightest voice, forcing a megawatt smile that could trump any cheerleader's. The TaTas turned to me, matching blank expressions on their faces.\n\nKelsi looked at me. Her eyes were bleary. \"Hi?\" she said. The TaTas were still staring at me, but their blankness had turned to annoyance.\n\n\"Kelsi, did you forget?\" I bubbled, making it up as I went. \"You told me you'd help me with my history homework. I'm so nervous about the test I have today.\"\n\n\"Homework?\" she repeated.\n\n\"Yeah, you know, the assignment you promised to help me with? But I, uh, get distracted with people around. No offense.\" I smiled fakely at the TaTas. \"Let's go outside so I can concentrate.\"\n\nKelsi glanced at the TaTas. \"Yeah, okay.\"\n\n\"What was that all about?\" Kelsi asked as we walked outside.\n\n\"Didn't you see the way they were looking at you?\" I asked, incredulous. \"It was like you were some kind of prize.\"\n\nKelsi shrugged. \"I didn't notice.\"\n\nWe were walking toward the parking lot now, and I wondered if we were going to have another bizarre rap session in Kelsi's car.\n\nSuddenly, Kelsi pulled a pack of cigarettes from her backpack and shook the box a little until one fell out. I watched, wide-eyed, as she lit up as if she'd done it a thousand times before.\n\n\"You _smoke?\"_ I asked, so surprised that I actually stopped in my tracks.\n\nKelsi took a long drag on the cigarette and then exhaled, the smoke forming a lingering cloud as it exited her mouth. \"So?\"\n\nI paused. \"But your mom died of _lung cancer.\"_\n\nThe words hung in the air between us, big and ugly.\n\nWhen Kelsi finally spoke, she didn't look at me, but there was something in her voice that hadn't been there before.\n\n\"The doctors lie,\" she said. The words were clipped, cold, and filled with something ugly that felt familiar to me. \"What do you mean?\"\n\nKelsi stared out toward the parking lot. \"My mom never smoked a cigarette in her life. So how does a person who hasn't smoked, and has hardly ever been around smokers, die of lung cancer?\"\n\nI looked at her, surprised. \"She never smoked?\"\n\n\"No.\" She took another drag off her cigarette. \"So what's the point anyhow? I mean, if you can do everything right and then still die of lung cancer, why bother?\"\n\nI wanted to tell her not to smoke, that it was bad for her, and stupid, too, but this hardly seemed like the time.\n\n\"And you know what the best part is?\" Kelsi continued. \"It could be hereditary. So yeah, there's a higher chance that I'll get it, because my mom had it. So what the hell?\"\n\n\"Yeah, but smoking doesn't seem like the answer,\" I said. I eyed her cigarette warily, trying to understand her mixed-up feelings. It was probably a little similar to having a dad who always buckled up and always drove the speed limit and then got killed in a car accident that never should have happened. \"Life just isn't fair sometimes,\" I added, more to myself than to her.\n\n\"Yeah, thanks for telling me,\" she said sarcastically. \"I hadn't noticed.\" She dropped her cigarette on the ground and stubbed it out with the toe of her sneaker. \"So,\" she said, \"you wanna skip or what?\"\n\nI stared at her. \"You mean, like, now?\"\n\nKelsi rolled her eyes. \"You can't be perfect _all_ the time,\" she said. \"Besides, do you really think we're going to get in trouble? Seriously. We're the kids all the teachers feel sorry for.\"\n\nKelsi had a point. All those teachers who had buzzed around me with fake, cheerful smiles, telling me that they wanted to help if there was anything they could do, would probably look the other way if I was caught, wouldn't they?\n\n\"Okay,\" I said after a minute. I took a deep breath and tried to tell myself this was something I had to do to help Kelsi. \"Let's go.\"\n\nAnd for the first time that week, a small smile appeared on Kelsi's face.\n\n# chapter 7\n\nAs we pulled out of the school lot, Kelsi fiddled with the radio distractedly. It made me a little nervous that she wasn't paying enough attention. Then again, I always felt uneasy in cars since the accident.\n\nOnce we'd driven for a few minutes, she rolled the windows down and turned the stereo up. The new Star Beck song was on. The wind whipped through my hair, getting colder as Kelsi picked up speed. We were approaching the interstate, and I wondered if we were going to get on. Cape Cod was just to the south, over the Bourne Bridge. If we headed north, we'd be in Boston in under an hour, and honestly, my mom would probably kill me if she found out.\n\n\"Where are we going?\" I finally shouted over the wind and music.\n\nKelsi didn't answer, and I wasn't sure she had heard me. Then she said, \"I don't know. Does it matter?\"\n\nI shrugged. The on-ramp for the interstate was coming up, but Kelsi zoomed right past it, shooting toward Milton Park, where my mom used to take me and Logan to play when we were kids. Kelsi pulled in, but instead of parking, she looped slowly around in the lot and headed back the way we came.\n\nWe drove a few minutes more in silence until we were back on Summer Street. I wasn't paying attention until Kelsi suddenly slowed and made a hard right into a parking lot I'd managed to avoid all year.\n\nI froze in my seat. \"What are we doing here?\" I asked, all my nerves on edge. I hoped that she was just turning around, like she had at Milton Park. But instead, she pulled neatly into a parking space in the nearly deserted lot.\n\nShe cut the ignition and climbed out of the car, pulling her cigarettes out of her pocket.\n\nI stayed in the car, glued to my seat. My limbs felt stiff and uncooperative, but I wasn't sure what I wanted them to do anyhow.\n\nWe were at St. Joseph's Cemetery, a place I hadn't set foot in since my dad's funeral. It was a pretty place, really, with lots of rolling green hills and chirping birds and squirrels running around like nothing was wrong. Sunlight trickled down beautifully in little patches through the leaves of the lush, overgrown oak trees that dotted the property. But it was impossible to see it as a nice, peaceful place. I hadn't wanted to see it at all, in fact, and deliberately turned away every time I passed it.\n\nMy mom went every Sunday morning to lay flowers on Dad's gravestone. Sometimes, early on, Logan had gone with her, although now he was usually too busy with Sydney. Tanner went occasionally too.\n\nBut I always refused. Seeing Dad's grave made it real. I wasn't delusional; I knew he was dead. But sometimes I could still wake up in the morning, and for those foggy few seconds before reality dawned, I'd have a fleeting instant of wondering what Dad was going to make for breakfast.\n\nI loved those moments. And I had the feeling that there would be fewer of them\u2014or that they would disappear entirely\u2014if I started visiting his grave. I didn't want to remember him as a cold piece of stone or an eight-by-four patch of green grass.\n\nI took a deep breath and scrambled out of the car. \"I don't want to be here,\" I announced, walking over to Kelsi, who was fiddling with her pack of cigarettes.\n\nShe leaned back against her car door and drew in a deep breath of smoke, which she exhaled suddenly with a sharp cough, leading me to wonder if she was really the experienced smoker she seemed to want me to think she was.\n\n\"You want one?\" she offered, holding the pack out.\n\n\"I don't smoke.\"\n\nKelsi rolled her eyes. \"Neither did I,\" she said. \"Things change.\"\n\nWe stood there in silence for a minute. I was trying very hard to forget where we were. I felt cold inside. I swallowed hard a few times, a weird pang in my chest.\n\n\"Give me one,\" I blurted out, surprising myself with the desperation in my voice. Kelsi looked at me with mild interest, then handed me the pack of cigarettes.\n\n\"You have to shake it,\" she said, smiling at my hesitation, \"to get one out.\"\n\nI nodded, feeling silly, and did as she said until a cigarette dropped into my hand. I had never smoked before. I'd had it drilled into me from an early age that smoking would kill you. But then again, so would driving in your own neighborhood on a Saturday morning with your kids.\n\nI tentatively put the cigarette between my lips and Kelsi took a step closer. She flicked the lighter a few times until it lit and held it to the tip of my cigarette. \"Inhale,\" she commanded.\n\nI did, watching as the cigarette ignited with the force of my breath. All of a sudden, my lungs filled with smoke\u2014sharp, dark, itchy smoke\u2014and I began to cough, hard at first and then even harder, unable to control myself. I dropped the cigarette and Kelsi quickly stubbed it out while I doubled over, coughing some more. It felt like I couldn't get the smoke out of my lungs. I gasped for air.\n\nKelsi shook her head. \"You suck at this.\" I coughed some more and glared at her. \"Shut up.\" Kelsi watched me hacking up my lungs, and to my surprise, she started to giggle, slowly at first and then harder. I looked down at my stubbed-out cigarette and lifted a hand to my cheek, which I knew was red from all the coughing. In a dark, weird way, I had to admit, it was a funny scenario.\n\nKelsi's laughter was contagious, and soon, I was giggling and then laughing too. Here we were, two Goody Two-shoes, smoking in the cemetery parking lot during school hours. The Plymouth East gossip mill would never have believed it.\n\nIn that moment, I felt closer to Kelsi than I had to anyone\u2014even Logan or Jennica or my mom\u2014in ten months. After all, Kelsi knew exactly how I felt, in a way that few people did.\n\nIt was also the first time I could remember laughing\u2014 _really_ laughing\u2014in a long time. I had to admit, it felt good.\n\nThen, I realized that the sound of Kelsi's laughter had changed. The giggles were coming in gasps, and they trembled on their way out.\n\nShe was crying. Hard. Tears were rolling down her cheeks. I had never seen anyone laugh and sob hysterically at the exact same time. I knew I was supposed to do something, but I didn't know what.\n\n\"Kelsi?\" I asked tentatively. I watched, feeling totally helpless, as she leaned back hard against her car and slid slowly down it, eventually collapsing to the ground, a puddle of limp limbs, still crying. The laughter was gone now, having given way to pure sobs that racked her whole body. My heart ached.\n\nSlowly, uncomfortably, I knelt down, intending to pull her into a hug, because it was all I could think to do. But when my fingertips touched her upper arm, she jerked angrily away, as if I had burned her with the tip of one of her cigarettes.\n\n\"Leave me alone!\" she barked. \"Just go away!\"\n\nShe drew her knees up to her chest and put her face down on them, working herself into a little ball. I didn't know what to do, so I sat down beside her, feeling miserable.\n\nEventually, Kelsi's sobbing slowed. I tentatively put my hand on her arm again, and when she didn't shake me off, I slipped it loosely around her back in a sort of half-hug, the best I could manage side by side on the ground. We sat like that for a while as Kelsi wiped at her eyes.\n\n\"It gets better,\" I said.\n\nKelsi shook my hand off her shoulders. \"Oh yeah?\n\nWhen?\"\n\nI didn't have an answer. _Did_ it get better? Sure, I wasn't the crying mess that I was for the first few weeks after Dad's death. I was fine now. I never cried anymore. But there was still an emptiness inside me that wouldn't go away. And sometimes, I was sure that the empty space was growing bigger and bigger, threatening to swallow me whole.\n\n\"So, do you want to go see your mom's grave or something?\" I asked.\n\n\"No,\" she snapped.\n\n\"Oh,\" I said, caught off guard and not sure what else to say.\n\nKelsi sighed. \"I just want to go home,\" she said finally, in a voice so quiet I could barely hear her. \"I just want to go back to the way it was before.\"\n\nI knew exactly what she meant. And she knew I knew.\n\nSilently, we stood up, dusted our jeans off, and got back into the car. As Kelsi started the engine and pulled out of her spot, I looked out the window, turning my face away from the rolling green cheerfulness of the darkest place I'd ever seen. I knew that as we pulled back on to Summer Street, if I looked toward the cemetery, I would see my father's grave. It was on the crest of a little hill midway into the cemetery. Although I'd only been there once, the day of his funeral, his grave's location was burned into my mind, and I knew it as well as I knew the location of my own fingers and toes.\n\n\u2022 \u2022 \u2022\n\nIt was nearly three-thirty by the time we turned onto Main Street on our way back to school. We were just a few blocks inland from the harbor, the same rocky, jagged jut of coastline the Pilgrims had landed on four hundred years ago. I could smell the salt in the air and feel it on my skin, the way I always could when the wind was blowing west. Today, even though the afternoon had warmed up, the breeze made me shiver.\n\nKelsi and I weren't talking, but it wasn't the same kind of weird, uncomfortable silence that had filled the car during our roundabout drive to the cemetery. Something had shifted between us.\n\n\"Hey, Kelsi?\" I said as we pulled up to a red light.\n\nShe looked at me, a question on her face.\n\n\"What if we did this more often?\" I ventured.\n\nKelsi laughed. \"Skip school so we can go smoke and cry in the parking lot of the cemetery?\"\n\n\"No.\" I smiled. _\"This_. I mean, it feels normal now, doesn't it? I mean, not _normal_ normal. But more normal than we feel at school, anyhow. What if we got together sometimes?\"\n\nThe light changed, and Kelsi eased her foot back onto the gas. She gave me a funny look. \"Why would we do that? It's not like we're even friends.\"\n\n_Ouch_ , I thought. But still, I pressed on. \"Because with me, you don't have to be Kelsi Whose Mom Died. And I don't have to be Lacey Whose Dad Died. You know?\"\n\nKelsi was silent for so long that I began to think she wasn't going to respond. Then, finally, in an almost inaudible voice, she said, \"Yeah. I know.\"\n\n\"Maybe we can see if Logan wants to come too,\" I said. As Kelsi turned left into the school parking lot, kids were pouring out of the buildings toward the cars. The final bell must have just rung.\n\n\"Whatever,\" she said casually, like she didn't care. But then she added, \"Maybe we should ask Mindy Rodriguez, too. She's a freshman. I heard her mom died last year.\"\n\n\"And Cody Johnson,\" I said.\n\nKelsi frowned. \"So you want to start, like, some kind of club for kids with dead parents or something?\"\n\n\"Not really.\" The plan was forming in my mind as I spoke, and I wasn't sure if it was stupid or not. \"What if it's just us getting together and hanging out sometimes without feeling like outcasts?\" I asked. \"I mean, we can talk about our parents if we want to. But we don't have to. We can feel like we did before.\"\n\nKelsi pulled into a parking spot, cut the ignition, and stared at her lap for a long time. Finally, she looked up at me. \"Okay,\" she said. \"I'm in.\"\n\n# chapter 8\n\nOnce I'd had the idea of getting us all together, I couldn't stop thinking about it. I thought about it at school. I thought about it at home. I lay in bed at night thinking about how I just might be able to help everyone who hurt the same way I did. I imagined scenarios in which the program was such a success, I would be asked to travel all around the country to talk to grown-ups about how to help kids who'd lost a parent.\n\nBut I was getting ahead of myself. I hadn't talked to anyone but Kelsi about it, and I hadn't even researched how to go about setting up an informal group of teens who got together to be with other people who didn't make them feel like outcasts. Still, I knew in my gut that it was something I had to do. I just had to figure out how.\n\nJennica came over after school on Friday to do our weekend trig homework, and then my mom drove us to Jennica's house. I'd told her we were having a sleepover, which wasn't exactly a lie, since I really was sleeping over at Jennica's place. But we were also going to a party at Brooke Newell's house, and I knew my mom would probably say no if I asked her. Ever since the accident, she'd been completely freaked out about anything that involved teenagers, cars, and possibly alcohol. Not that I blamed her. But it wasn't like we were going to drink and drive. I knew what could happen when you got in a car, even when alcohol wasn't a factor.\n\n\"It'll just be me and Tanner tonight at home,\" my mom said as she drove. She glanced in the rearview mirror at Tanner, who was sitting beside Jennica and gazing out the window.\n\n\"Why?\" I asked. \"Where's Logan going?\"\n\n\"Over to Will's house to play some video game, I think,\" Mom said absently. \"Or maybe to watch movies. He's having a sleepover too, like you girls.\"\n\nJennica and I exchanged looks. Will was Logan's friend last year, but they hardly ever talked anymore, thanks to the fact that Logan now spent all his time with Sydney. I doubted he was spending Friday night at Will's, but Will was the excuse he used most weekends to sneak out of the house. I hadn't blown his cover yet, although with the way he acted toward me sometimes, it was pretty tempting.\n\nI wondered if Logan would be at Brooke's party too. I'd always assumed that the Will lie was a cover for sneaking out with Sydney. But maybe my brother was going to more of these popular-crowd parties than I realized.\n\nMy mom dropped us at Jennica's, and after kissing me absently on the cheek, she drove away, back to the silent bubble of our house.\n\n\"She's really out of it, isn't she?\" Jennica said quietly.\n\nI sighed. \"It's been like that for a while.\"\n\nJennica nodded. \"Come on,\" she said as she started toward the door. \"We don't have much time to get you dressed. I told Brian we'd pick him up in an hour.\"\n\nI looked down at what I was wearing: my favorite pair of jeans, flip-flops, a pale pink tee, and a gray hoodie that I'd gotten at the Star Beck concert Jennica and I had gone to a year and a half ago in Boston. It had the redheaded singer's face emblazoned on the back and a handful of little sequined stars down the right front side. It was one of my favorite pieces of clothing, and I figured the sequins dressed it up enough to make it party appropriate. \"What's wrong with what I have on?\" I asked.\n\nJennica rolled her eyes. \"Everything,\" she said.\n\nShe led me inside, past the kitchen, where the fridge door was open, obscuring all but the feet of Jennica's mom, who was standing behind it, looking inside.\n\n\"Hi, girls!\" she said, straightening up with a smile. \"I was just about to throw something in for dinner. Hungry?\"\n\nShe shut the refrigerator door, and I couldn't help staring. I'd known Mrs. Arroyo for years now, and she'd always been the quintessential mom, so much so that I'd caught myself feeling jealous lots of times this year when my mom had retreated inside herself so much. I was used to seeing Mrs. Arroyo in jeans and a T-shirt, or covered up in an apron, with her hair tied back and little makeup on.\n\nBut today, she was wearing a denim miniskirt and a halter top. Her hair was loose and had been curled at the ends, and she had on way too much blush and lipstick.\n\nJennica audibly sighed. \"We'll be in my room, Mom,\" she said. Then, before I could say anything or react, she grabbed my hand and dragged me toward the staircase. She smiled an obviously fake smile at me. \"Divorce is fun!\" she said brightly.\n\nIt had been ages since I'd been to Jennica's house. She was always busy with Brian, and I supposed I'd been glad to have the distance between us; seeing her perfect family depressed me. But is this what had happened to all of them since I'd stopped paying attention?\n\nI followed Jennica upstairs. She pulled me into her room and shut the door behind us, then she flopped onto her bed.\n\n\"What's the deal with your mom?\" I asked, sitting down beside her.\n\n\"She thinks she's a teenager. Apparently, it's her plan to get her 'sexy' back.\" Jennica shrugged. \"I mean, good for her, I guess. But _I_ would never dress like that.\"\n\n\"Me neither,\" I said with a shudder. \"And my mom would kill me if I did.\"\n\nJennica snorted. \"Don't be so sure,\" she said. \"Your mom looks about as tuned in as mine does.\"\n\nI was silent. She was right.\n\nJennica shook her head and got up from the bed. \"Anyhow,\" she said. \"Let's get you dressed.\"\n\nFor the next twenty minutes, I felt like a Barbie doll as Jennica made me try on outfit after outfit. She had determined that we should both be wearing tight jeans and cleavage-baring tops tonight, like all the popular girls.\n\n\"Okay,\" I said slowly. \"That sounds good. But I have two problems.\"\n\nJennica simply raised an eyebrow and waited for me to go on.\n\n\"First, it's like forty degrees outside,\" I said. \"I think it's past the skimpy-top-wearing season.\"\n\nJennica rolled her eyes. \"Fine, so we'll wear skimpy tops with jackets. Happy?\"\n\nI smiled at her without answering. \"However, the second, and much more pressing, issue is that I _have_ no cleavage. So a cleavage-baring top is pretty much impossible.\"\n\n\"Girl!\" Jennica said, shaking her head. \"Don't you know how to work what you have?\"\n\nWhat I _had_ was an A-cup. But in Jennica's frighteningly capable hands, and with the help of pads from an old bra she'd grown out of a couple years ago and some double-sided tape, I suddenly looked a lot different than usual in a pair of long black pants, high heels, and a low-cut sparkly silver tank top of Jennica's that made me look much curvier than I really was.\n\n\"Voil\u00e0!\" she exclaimed, stepping back to admire her admittedly impressive handiwork. I just stared at myself in the mirror.\n\n\"How'd you do that?\" I asked in astonishment.\n\n\"I'm not done yet,\" Jennica said. It took her ten more minutes to apply bronzer, blush, lipstick, eyeliner, and mascara to my usually bare face. I looked in the mirror with trepidation, expecting to see something horrific (or at least something like her over-blushed mom). Instead, I looked... good.\n\n\"Hellooooo, hot mama!\" Jennica said, grinning at me in the mirror.\n\nI giggled. \"This is... different. You're like a miracle worker!\"\n\nJennica shrugged. \"Nah,\" she said. \"I didn't do anything. I just played up what you've already got!\"\n\nI stared at the mirror and shook my head in amazement. She was right; I did look like me. Just a prettier version.\n\n\u2022 \u2022 \u2022\n\nTen minutes later, Jennica had changed too\u2014into jeans, heels, and a sparkly purple tank (the difference being that she actually had real curves to fill it out)\u2014and I had persuaded her that I needed a cardigan so I wouldn't freeze to death. Grudgingly, she had handed one over. I intended to bring my Star Beck hoodie, too, because I figured it would be cold enough outside that I'd want to layer up.\n\nWe walked downstairs and found Jennica's scantily clad mom removing a pizza from the oven while Jennica's little sister, Anne, sat at the table, drinking a glass of milk.\n\n\"Just in time for dinner, girls!\" Mrs. Arroyo exclaimed. Jennica started to protest, but her mom turned firm. \"Jennica! I can't send you and Lacey out of here with empty stomachs, now can I?\" Without waiting for an answer, she added, \"Wash some salad mix and get the dressing out, will you? Lacey, what would you like to drink?\"\n\nI sat next to Anne and flashed her a smile. She looked up from her glass, and I tried not to giggle when I noticed her milk mustache.\n\n\"Hey, you,\" I said. \"How's it going?\"\n\n\"Fine,\" she answered gruffly. \"What's up with you?\" Anne was twelve and right in the middle of that tough phase when _you_ know you're grown up, but the rest of the world still treats you like a kid. I knew she was trying to sound as adult as possible. I played along.\n\n\"Not much,\" I said with a shrug.\n\n\"Got a boyfriend?\" she asked, turning her gaze back to her milk.\n\nI looked at her, surprised. \"Um, no,\" I said. \"Do you?\"\n\nShe glanced at Jennica, who was pouring salad into a bowl. Then she returned her attention to me. \"Yeah,\" she said casually, \"I got a few options.\"\n\nI looked over at Jennica in time to see her roll her eyes. It had always bugged her that Anne seemed to copy every move she made. Her younger sister had insisted she was \"playing the field\" when Jennica was single, but now that Jennica had Brian, Anne was always saying cryptic things about how she had lots of boyfriend options.\n\n\"Having a boyfriend isn't all it's cracked up to be, kiddo,\" Jennica muttered. I turned and glanced at her, wondering what that was all about.\n\nWe scarfed down our pizza and salad in the same kind of silence that pervaded my house. This surprised me. I'd just assumed that Jennica's family was just as it had always been.\n\nApparently, I was wrong.\n\nAfter Jennica and I had put our plates in the sink and wrapped the remaining slices of pizza in foil, Mrs. Arroyo stood up to give Jennica a hug and to pinch me on the cheek, which used to annoy me when I was a kid but which I now thought was kind of cute.\n\n\"Have a good time at the party, baby,\" she said to Jennica.\n\nJennica nodded. \"We will.\"\n\n\"Don't drink too much,\" her mother said, leaning in to kiss her on the cheek. \"And call me if you can't drive.\"\n\nI looked at Jennica in astonishment. But she merely nodded again, mumbled a goodbye, and grabbed my hand to drag me out the door.\n\n\"Bye!\" Anne yelled behind us.\n\nI waved, but I couldn't even muster the words to say goodbye. I was still in shock.\n\nAs soon as we got outside and the door was shut behind us, I exploded. \"Your mom _knows_ you drink?\"\n\nI knew Jennica sometimes drank beer when she was at parties or out with Brian, and I thought it was wrong. She could get in huge trouble! But she was always saying that everybody did it, so why shouldn't she?\n\n\"Yeah,\" Jennica muttered. She was looking at the ground. \"So?\"\n\n\"Soooo,\" I said, drawing the word out. \"Don't you think that's weird?\"\n\nJennica shrugged. \"Whatever,\" she said. \"She's cool, you know? She treats me like a grown-up now.\"\n\nI just stared at her. I didn't even know what to say. What had happened to the old maternal, strict Mrs. Arroyo?\n\nJennica paused. \"It's just been recently,\" she said. \"Since my dad started dating the Spandex Leech. It's like my mom suddenly turned sixteen again. I found her in my closet one day, trying on my clothes, when I got home from school.\"\n\n\"That's so... weird,\" I said.\n\nJennica shrugged. \"She seems happy. It's no big deal. It's cool.\" She paused awkwardly, cleared her throat, and added, \"Anyways, let's go.\"\n\nWithout another word, she strode over to her mom's old Corolla, yanked open the door, and got inside. She slammed the door behind her and didn't look at me. It took me a second to snap myself out of it and join her. As soon as my door was shut, she started the car, threw it into reverse, and backed out of the driveway. She switched quickly to drive, cut the wheel sharply, and peeled out from the curb, like she couldn't get away from her house fast enough.\n\n# chapter 9\n\nThe party was in full swing by the time we got there. I followed Jennica and Brian toward the house, feeling more nervous than I usually did. Even though I'd been to parties before with Jennica, I knew I didn't belong. I didn't drink. I didn't have a boyfriend. I didn't make out with random guys. And I didn't really care whether people thought I was cool or not.\n\nAs we walked through the front door, we were blasted immediately by a wave of thumping bass turned up as loud as it could go. An old Kanye West song was throbbing from the speakers, and more people than should ever be crammed into any space were jostling and gyrating all over the Newells' perfect living room.\n\nMost of the girls were dressed skimpily and were laughing too loudly and swaying a little bit on their stiletto heels. The boys were talking in unnaturally booming voices, slapping one another on the back and shamelessly ogling the girls. And everyone was carrying big red plastic cups filled with what I guessed was beer. In fact, I saw several people sloshing it onto the carpet as they talked.\n\nJennica turned to me with a big smile. \"Isn't this _awesome?\"_ she asked, her eyes sparkling with enthusiasm.\n\n\"Um...,\" I responded.\n\n\"Let's go get some beer!\" she said loudly, close to my ear so that I could hear her over the music.\n\n\"I don't really want any!\" I said back.\n\n\"What?\" she shouted. I repeated myself, but she shook her head again. The music was too loud. I shrugged and followed her and Brian through the living room, out the French doors in the back. There was a line of about a dozen people waiting for beer while Scott Moore, who was in my English class, cheerfully pumped the keg handle. There was a couple kissing on the hanging swing near the house, and a stressed-out-looking senior girl, whose name I thought was Trish, was furiously texting on her phone while she chewed on her lower lip.\n\nJennica, Brian, and I got in line.\n\n\"Wassup?\" Scott said as we got close to the keg. He grinned and handed us empty red cups. \"Who's first?\"\n\nJennica filled up her cup. \"Your turn!\" Scott told me as she stepped away from the keg and took a sip of her beer.\n\nI hesitated. I'd always been so against drinking. But wouldn't it be nerdy to say no with a keg right in front of me?\n\nJust then, I saw Sam come out of the house, scanning the yard. My jaw dropped. What was he doing here? At the same time, he caught sight of me, smiled, and waved. I ducked my head, immediately feeling guilty, like I'd been caught doing something wrong.\n\n\"Lacey?\" Scott prompted, glancing at the growing line behind me. I snapped to attention and looked from him to the beer keg and back.\n\n\"Um, no thanks,\" I mumbled.\n\n\"You sure?\" he asked.\n\n\"Yeah,\" I said. \"I'm sure.\" Brian filled up and then slipped an arm around Jennica's waist.\n\n\"It's freezing out here,\" she said. I couldn't help noticing that she wrinkled her nose a little bit every time she sipped, like the beer tasted bad. Why would you drink something you didn't even like? \"Can we go inside?\" she asked.\n\nI followed her and Brian back into the hot, loud, crowded living room. It felt like a sauna. A tall guy I didn't recognize splashed beer on me as he walked by.\n\n\"C'mon, Lacey!\" Jennica shouted over the music. \"Dance with us!\" She took another big sip of her beer.\n\nI shook my head and glanced around the room. I never should have come.\n\nJust then, Logan and Sydney walked by, both of them clutching beer cups. From the looks of it, they'd been here for a while. One side of Logan's shirt was untucked, and his hair was a little messed up. I wondered how much he'd been drinking.\n\n\"Hey,\" he said when he saw me. \"What's up?\"\n\nI could smell the beer on his breath. I shrugged. \"Nothing.\" I glanced pointedly at the cup in his hand. Logan shifted it to his other hand.\n\n\"What are you doing here?\" he asked.\n\n\"I'm with Jennica and Brian,\" I said.\n\nLogan's eyes landed on my cup. \"You're drinking?\" he asked incredulously.\n\nI realized it must look like I was holding a beer I'd finished, rather than one I'd never started. \"So what if I am?\" I asked.\n\n\"You don't drink,\" he said flatly.\n\nI rolled my eyes. \"I didn't think _you_ drank either,\" I said.\n\n\"Yeah, well.\" He paused. \"Maybe you don't know everything about me.\"\n\n\"Yeah, well,\" I said. \"I guess I don't.\"\n\nAfter I walked away from my brother and Sydney, I looked for Jennica and Brian, but I didn't see them anywhere. Amy Tan, from my trig class, told me she'd spotted them walking upstairs.\n\n\"To make out,\" she added unnecessarily. \"Lots of people make out up there.\"\n\n\"Thanks,\" I said. \"I've got it.\"\n\nI felt more out of place than ever. I walked through the backyard, past the beer keg, past the handful of couples making out near the deck. The backyard was larger than I would have thought, and there was a small lake at the end of the lawn. I made my way down to the old wooden dock, pulling Jennica's cardigan more tightly around me as the wind whipped in stronger now that I wasn't shielded by the trees in the backyard anymore. I shivered, but I liked the feel of the breeze against my face.\n\nI sat down on the edge of the dock, took off my strappy heels, and dangled my feet over.\n\nThe night was cold around me, and I was surprised at just how far away the sounds of the party seemed. It was quiet enough that I could hear crickets chirping and the occasional splash of a fish or a bird in the water. Across the lake, the darkness was punctuated by porch lights of houses, which looked much farther away than they did during the day\n\nI was so tuned in to the sounds of the water that I didn't realize I wasn't alone until I heard a voice just behind me.\n\n\"I've been looking everywhere for you.\"\n\nI jumped about a mile in the air and whipped my head around, my heart pounding double time.\n\nIt was Sam, standing there, looking down at me. He was backlit by the lights from the Newell house far behind us, and he seemed to almost glow in the shadows. I blinked a few times and tried to slow my racing heart. By the time my eyes adjusted, I noticed he had two cups, one of which he was holding out to me. \"No thanks,\" I mumbled. \"I don't drink.\"\n\nSam looked amused. \"Me neither,\" he said. \"All you need to do is take a walk through the party back there, and you realize how stupid it makes people act.\"\n\nI looked at the cup again and raised an eyebrow.\n\nHe laughed. \"It's not beer. It's Coke. I had a few cans in my Jeep.\"\n\nI didn't know what to say. I took the plastic cup from his hand. \"Oh. Thank you.\"\n\nSam sat down beside me, close enough that our thighs were almost touching. I could feel the heat from him. It made me shiver.\n\n\"So what are you doing down here?\" he asked.\n\nI shrugged and looked out at the water. \"I don't know. I just wanted to be alone, I guess.\"\n\nHe seemed to consider this for a second. \"Do you want me to leave?\"\n\n\"No,\" I said, surprising myself with how quickly the word came out of my mouth. \"I mean, that's okay. I don't care what you do.\"\n\n\"Why are you mad at me?\" he asked.\n\n\"I'm not mad,\" I said.\n\n\"Was it something I said the other day?\" he persisted. \"When I drove you home?\"\n\n\"Don't worry about it.\"\n\n\"Well, I _am_ worried about it. You've been avoiding me since then. And I don't know what I did.\"\n\nI squinted, wishing I didn't have to explain it to him. He'd never understand. \"It's nothing personal. I just don't need another friend like you,\" I said.\n\nHe stared. \"What do you mean?\"\n\nI gazed out at the lake without answering. After a moment, I felt his hand close over mine. It was big and warm and reminded me a little bit of the way my father's hand had fit around mine when I was little. I could feel my heart thudding in my chest.\n\n\"Please,\" he said. \"Tell me what I did wrong.\"\n\nI hesitated. His hand didn't move. And strangely, I realized I didn't want it to. \"Look, I know I'm being dumb,\" I said. \"But I didn't want to talk about my dad with you. I'm sick of having to explain it to people who have no idea what it feels like. Okay? Can you just drop it?\"\n\nHe looked surprised and withdrew his hand. \"I'm sorry.\"\n\n\"I'm just tired of people feeling sorry for me,\" I added.\n\n\"I don't feel sorry for you,\" Sam said.\n\n\"Whatever,\" I muttered. I paused. \"And I hate it when people say they know how I feel. Okay? Because you don't know how I feel.\"\n\n\"Fair enough,\" he said. \"I'm sorry. You're right. I don't know.\" He paused. \"But I do understand, Lacey. Better than you think.\"\n\nOur eyes met in the darkness, and he held my gaze. I blinked a few times. I didn't want to talk about this anymore. \"So how come you don't drink?\"\n\nHe gave me a half-smile. \"First of all, I hate the taste of beer. Why would I drink something I don't like?\"\n\n\"True,\" I said. I'd never had it, but it smelled terrible.\n\n\"It tastes like socks,\" Sam said, reading my mind. \"Dirty socks.\"\n\nI giggled.\n\n\"Plus, it makes people act like idiots.\" I laughed. \"True again.\"\n\n\"But the biggest reason, I guess, is that it's dangerous,\" he said. \"Think about how many people in that house are going to drive home tonight. What if they get into an accident and get hurt or cause an accident that hurts someone else?\"\n\nI felt cold, the way I did whenever I thought of car accidents. Suddenly, I couldn't fathom ever wanting to drink anything in my entire life, if it could lead to something like that.\n\n\"I'd never drink and drive,\" Sam added.\n\n\"Me neither,\" I agreed. \"No way.\"\n\nI looked up at the sky. It was clear out tonight, with just a few wispy clouds drifting across the nearly full moon like pieces of gauzy silk suspended in space. I searched for the brightest star and recited the familiar words in my head: _Star light, star bright, first star I see tonight, I wish I may, I wish I might, have this wish, I wish tonight_. Then, without even thinking about it, I silently wished that Sam Stone would kiss me.\n\nImmediately, I regretted it. I didn't necessarily believe in wishes coming true or anything like that, but what if they did? Shouldn't I have wished for my dad to be safe in heaven? Or for my mom to stop crying in her room at night? Or for Tanner to come out of his shell? Or for Jennica's mom to snap out of her weird teenager phase? What if I'd just wasted a wish? And why, of all the things I could wish for, would I wish for Sam to kiss me?\n\n\"So, I think I'm going to go,\" Sam said after a minute. \"My mom worries when I'm out too late.\"\n\n\"My mom doesn't worry about anything anymore,\" I said before I could think about it.\n\nSam looked at me closely. \"I bet she worries more than you realize.\"\n\nI wanted to tell him that he had no idea what it was like in my family, and he had no idea what my mom was thinking. But there was something in his eyes that stopped me from speaking.\n\n\"Are you okay getting home?\" he asked.\n\nI hesitated. \"I'm actually spending the night at Jennica's,\" I said.\n\n\"She's driving?\"\n\nI nodded.\n\n\"But she's drinking,\" Sam said. \"I saw her.\"\n\nI shrugged. \"I'll figure it out,\" I said. \"Don't worry.\" But I was worried. I didn't have my license, so I couldn't get us home, and there was no way I was climbing in a car with someone who'd had a few beers. I figured we'd have to call Jennica's mom, which I knew Jennica would argue with me about.\n\n\"How about I drive you home?\" he asked. \"You and Jennica and her boyfriend, I mean.\"\n\n\"You don't have to\u2014\" I started to say.\n\nBut he cut me off. \"I'm not leaving you in a situation like that,\" Sam said firmly. He stood and pulled me up. \"Let's go get her and tell her it's time to leave.\"\n\nSam didn't let go of my hand as he led me into the party and upstairs to find Jennica and Brian. Ten minutes later, his fingers were still intertwined with mine as the four of us walked out to the street to pile into Sam's Jeep. I realized I didn't want to let go.\n\n# chapter 10\n\nThe next morning, back to thinking about the conversation I'd had with Kelsi, I Googled \"grief counseling for teens\" and \"starting a group for people whose parents have died.\" I read through all the entries, taking notes as I went, although there wasn't much I didn't know already. Most of the tips I found were pretty obvious, like letting everyone have a chance to talk and not pressuring anyone to open up.\n\nBesides, I reminded myself, my goal wasn't to start some kind of grief group. I intended to make sure it was casual and not at all like the stupid counseling sessions Mom made us go to with Dr. Schiff. I was sure we'd all had enough of well-intentioned adults who didn't have a clue, who wanted to believe we were little kids they could fix with simple words from textbooks on grief.\n\nI found a group in Atlanta called Kate's Club that sounded a lot like what I wanted to do. Kate was a woman in her thirties whose mom had died when she was twelve, and now she ran a group for more than a hundred kids. According to the group's Web site, they hung out together once a week, and once a month they did something fun, like go to a baseball game or to the aquarium. I imagined that one day I'd be like Kate. _Lacey's Club_ , I thought.\n\nBut I was getting ahead of myself again.\n\nI started an e-mail.\n\nHi, guys. Lacey Mann here. As you probably heard, Kelsi Hamilton's mom died last week, and Kelsi's back in school. I've been trying to figure out how to help her feel better, and then I realized that all of us could pitch in to make things easier on her. It might even help us, too. I was thinking that we could get together once in a while to hang out. We don't have to talk about anything if we don't want to. It's just a chance for us to feel like ourselves again and to hang out once in a while with people who get us. What do you think? Can you meet at the McDonald's on Samoset Street on Tuesday after school?\n\nI thought about it for a moment. Then, I deleted _McDonald's_ and typed in _Plymouth Diner_. It was only fitting that the place we'd meet for the first time would be the restaurant I thought of as belonging to me, my brothers, and Dad, the place we went for Saturday-morning pancakes. I hadn't been back there since the accident.\n\nI sent the e-mail to Cody, Mindy, and Logan. Then I sent a different e-mail to Kelsi, telling her the plan.\n\nAfter feeling so helpless at home, it felt good to finally be in control of something that had a real chance of helping people.\n\n\u2022 \u2022 \u2022\n\nBy Sunday night, there was still no word from any of the people I'd e-mailed. So I decided to call them.\n\n\"Hey,\" Cody said gruffly after his little sister handed the phone off to him.\n\n\"Hi, Cody. It's Lacey Mann. Did you get my e-mail?\"\n\n\"Yeah.\"\n\n\"So? What do you think?\" I asked. Cody and I didn't have classes together this year because I was in honors courses and he was in regular, but we'd gone to the same elementary school and junior high, and we knew each other well, even if we hadn't hung out in ages.\n\n\"I think it sounds kind of dumb,\" he said. \"You want to get together just because we have dead parents? I mean, get over it, Lacey.\"\n\nI took a deep breath. \"I _am_ over it, Cody. This is about Kelsi.\"\n\n\"So? What does that have to do with me?\"\n\n\"Look,\" I said. \"Let's just try this. Once. And if it feels stupid, you don't have to come again. But I just think it will be good for Kelsi to be around us now. Remember how weird it feels to have everyone treating you like you're some kind of alien?\"\n\nI could hear him breathing. \"Yeah,\" he said in a low voice.\n\n\"I just think it would help if we could show her that there are people who know how she feels.\"\n\n\"So, what, are we supposed to talk about grief and stuff?\" he asked. \"I already had enough of that crap with the military psychologist my mom made us go to. It was stupid.\"\n\n\"No,\" I said. \"No grief talk. Unless someone wants to.\"\n\nThere was another long silence. In the background, I could hear a television.\n\n\"Fine,\" Cody said finally. \"But if it's stupid, I'm leaving.\"\n\n\"Okay,\" I agreed. We hung up, and as I placed the phone back in the receiver, I felt a little bubble of hope float up inside me.\n\nI didn't know Mindy's number, so I called Kelsi next, and after a brief conversation about school stuff, I asked her if she was planning to come Tuesday.\n\n\"I guess so,\" she said. \"Being in my house is depressing.\"\n\n\"I know the feeling,\" I said.\n\n\"My dad just cries all the time,\" she said. \"Does your mom do that too?\" I hesitated. \"No.\"\n\n\"I wish I could forget about it,\" Kelsi murmured.\n\n\"Yeah, me too,\" I said. Silence crackled over the line. \"So how are you doing?\" I asked. \"I mean, really? Are you okay?\"\n\n\"I guess,\" she said. \"It's hard.\"\n\n\"Yeah,\" I said. \"I know.\"\n\n\"So I'll see you at school?\" she said.\n\nI agreed, and we said our goodbyes. I mentally ticked Cody and Kelsi off my list. Two down. One to go.\n\nA moment later, I was knocking on the door to Logan's bedroom.\n\n\"What?\" His voice was muffled.\n\n\"I need to talk to you,\" I shouted.\n\n\"About what?\"\n\n\"Can you just let me in?\" I asked.\n\nI heard a rustling, and then Logan pulled open the door, looking irritated. His room was dark, save for the light emanating from the monitor of his computer. An IM window was open. I figured he was probably talking to Sydney. Apparently, the world would end if they went more than a few hours without contact.\n\n\"What do you want?\" Logan demanded, blocking the doorway.\n\n\"Can I come in?\" I asked.\n\n\"Why? To snoop?\" He didn't move.\n\n\"I just want to talk to you about Tuesday.\"\n\n\"Your stupid meeting thing?\" Logan asked. I noticed that his eyes were bloodshot, which startled me. Had he been crying? The last time I'd seen his tears was Christmas morning, nearly ten months ago, when he'd come into the kitchen first thing in the morning and found me sitting alone there, staring at the wall, my hands wrapped around a mug of the Twinings Christmas tea that our dad used to drink all December. Logan had murmured, \"He's really gone, isn't he?\" before sinking into the chair across from me and starting to sob. He had cried, while I sat there, feeling uncomfortable, wondering why my own tears wouldn't come. From that day on, he had avoided looking me in the eye.\n\n\"It's not stupid,\" I said.\n\n\"Whatever,\" Logan muttered. \"I don't see why we have to hang out with some girl I don't even know.\"\n\n\"Because it'll help her. So what's a couple of hours one afternoon if it makes her feel better?\"\n\n\"Why do you have to save everyone, Lacey?\" Logan asked. He raked his hand through his hair and shook his head. \"I don't get you.\"\n\n\"I'm just trying to help.\"\n\n\"Yeah, well, you can do it without me,\" he said. \"Some of us have better things to do.\" He slammed the door without another word.\n\n# chapter 11\n\nSam and I were different in class now. The time we'd spent together at the party had changed us. Or maybe it had just changed me by teaching me to relax a little and not judge him so harshly. In any case, we chatted easily before first period, and in sixth period, we worked together on an assignment, and he even showed me the picture he'd been doodling while Mr. Henchey droned on during the first ten minutes of class. In the time it had taken our teacher to explain our assignment, Sam had sketched him in pencil, only he had given him a Colonial soldier's uniform instead of normal clothes. I couldn't believe how good the drawing was.\n\n\"What, this?\" Sam asked dismissively. He crumpled it up and looked embarrassed. \"This is nothing. I draw a million of these a day.\"\n\nBy lunchtime on Tuesday, I was practically bubbling over with excited nerves about the meeting after school. Cody had nodded at me as we passed each other in the hall, and Kelsi had shot me a small smile.\n\n\"So can you help me with trig after school today?\" Jennica asked as she and Brian plopped down across from me in the cafeteria.\n\n\"I can't,\" I said. \"I've got that meeting after school today. Remember?\" I'd told her about it on Saturday night when we went to the movies. I couldn't believe she'd forgotten.\n\nJennica looked at me blankly. \"What?\" she asked. \"Oh, that death-group thing you're doing?\"\n\n\"It's not a _death group,\"_ I said. \"It's just some people getting together to support each other. And Kelsi.\"\n\nJennica nodded, and I could tell she was trying to look interested. \"Yeah, sounds great,\" she said.\n\nI tried not to let her forced enthusiasm bother me.\n\n\"It _is_ going to be great,\" I said firmly.\n\n\"So who's going?\"\n\nI ticked off the short list.\n\n\"Pretty sad, huh?\" she asked. \"That there are that many kids whose parents have died?\"\n\n\"Actually, in a school this size, I would have thought it would be more, you know?\" I said.\n\nBrian looped his arm around Jennica's shoulder and pulled her close. He whispered something in her ear and she giggled. It was like they'd both forgotten I was there.\n\nAs I dumped my tray and made my way alone toward the doors of the cafeteria, I looked up and saw Sam midway across the room, eating lunch with a small group of popular seniors. Summer was gazing at him from two seats away. But his attention wasn't on her. He was watching me.\n\nStartled, I stopped for an instant longer than I should have. He raised his hand in a wave and smiled. Summer and a few of the others looked to see who he was waving at, then, apparently satisfied that it was no one important, they returned to their conversation.\n\n\u2022 \u2022 \u2022\n\nAs I walked down the street after school to the Plymouth Diner\u2014about a half mile away\u2014my heart was thudding so loudly that I was afraid everyone passing by would be able to hear it.\n\nThe restaurant was mostly empty, save for an elderly couple who were sitting on the same side of a booth, sharing an order of spaghetti and meatballs. I stood in the doorway for a moment, memories washing over me.\n\nThere was the booth in the back where we used to sit almost every Saturday; the waitresses knew to reserve it for us. I blinked a few times, images playing like a movie across the backs of my eyelids. Dad making airplane sounds and flying a spoon of oatmeal toward Tanner when he was little. Logan and Dad laughing and flinging whipped cream at each other from their strawberry pancakes, until a giant glob of white landed right on the tip of Dad's nose. Dad cutting Tanner's fried eggs into bite-sized pieces. Dad putting his arm around me and giving me an affectionate noogie with his other hand while I complained, pretending to hate it, even though I couldn't hide my grin.\n\n\"Can I help you?\" The hostess had appeared out of nowhere, someone I didn't recognize. But it had been almost a year since we'd last been here. I didn't know why I'd expected that the diner would be frozen in time, the way the memory of my dad was.\n\nI asked for a table for five\u2014just in case Logan and Mindy decided to show\u2014and then waited nervously at the table.\n\nThe next seven minutes felt like an eternity. Finally a tiny girl with a mass of jet-black curls walked through the door and looked around, her eyes wide and unblinking. I recognized her immediately from her Facebook profile.\n\n\"Hi!\" I exclaimed, hopping up. \"Mindy?\"\n\n\"Yeah,\" she said.\n\n\"Hey,\" I said. A tidal wave of relief washed over me as she stepped closer. \"I'm Lacey Mann. I'm the one who organized this.\" I felt proud to say those words.\n\n\"Where is everyone?\"\n\n\"You're the first one here.\"\n\n\"Oh.\" She hesitated.\n\n\"Here, sit down,\" I said before she could change her mind and bolt for the door. I couldn't think of anything to say. Not without everyone else here. I didn't want to get into anyone's stories without the whole group present.\n\n\"So you're a freshman, right?\" I asked finally. The seconds ticked by.\n\n\"Yeah,\" Mindy said.\n\n\"You like Plymouth East so far?\"\n\nShe shrugged. \"I guess.\"\n\nJust then, the door opened and Kelsi strode in. \"Hey,\" she said, joining us. She sat down hard, throwing her book-laden backpack on the floor, where it landed with a loud thump.\n\nBefore I had the chance to say anything, the door opened again, and Cody came in, looking annoyed. \"I'm here,\" he said. He was tall and a little stoop-shouldered with long, dark hair that flopped over his piercing dark eyes. He pushed a shaggy shock of hair behind his ears and ducked his head.\n\n\"Hi,\" I said. I felt immensely relieved; I realized I'd been expecting him not to show.\n\n\"This better not be dumb,\" he muttered. I felt a tight feeling in my chest. In my head, this had all gone so well; everyone would be glad to be here, we'd laugh together and cry together and feel better at the end. But now I was beginning to wonder just how dumb that was.\n\n\"So I guess we can get started,\" I said, suddenly unsure of how to begin.\n\n\"Whatever,\" Cody said. \"Can't we order or something, though?\"\n\nWe ordered Cokes and a few orders of fries to share. Just as we handed the waitress our menus, the door of the restaurant opened again, and Logan appeared in the slice of sunshine from outside, followed closely by Sydney. My jaw dropped.\n\n\"What's up, man?\" he said to Cody as he strode over to our table. He nodded at Kelsi and Mindy, throwing a \"What's up\" their way, too. Since there was only one chair left, Logan grabbed one from another table and wedged it beside the empty one so that Sydney could sit next to him. She was eyeing me warily, a little smile on her face.\n\nI could feel my blood boiling. \"Logan, can I talk to you for a minute?\" I asked, trying to keep my voice level.\n\nHe shrugged. \"Whatever.\" But he allowed himself to be led away, back toward the entrance.\n\n\"I'm glad you came,\" I said evenly. \"But what's Sydney doing here?\"\n\n\"She's with me,\" he replied.\n\n\"I know _that,\"_ I said. \"But this is a meeting for people whose parents have died. Sydney's mom and dad are fine!\"\n\nLogan shrugged. He knew he was bugging me. \"Yeah, well,\" he said noncommittally.\n\n\"Can you maybe get rid of her for like an hour?\" I asked. \"And meet her after?\"\n\nLogan shook his head firmly, but I couldn't help thinking he looked a tiny bit guilty. He glanced toward the table, where Sydney was standing, hands on her hips, lips pursed, watching us with narrowed eyes.\n\n\"She wanted to come with me,\" he said. \"And she's my _girlfriend.\"_\n\n\"Yes, I'm aware of that.\"\n\nLogan glanced at Sydney again and then back at me. He lowered his voice. \"Seriously, Lacey, can you loosen up a little?\"\n\n\"Whatever,\" I muttered. I didn't have the energy to fight.\n\nWe made our way back to the table, and Logan whispered something in Sydney's ear. She giggled and the two of them sat down. Everyone looked at me expectantly. The waitress arrived with our Cokes, and Sydney and Logan ordered. I tried not to roll my eyes as Sydney asked for a sparkling water and a salad with low-cal dressing.\n\nI took a long sip of my soda. \"Hi,\" I said. \"I know we all know each other, but I thought maybe we could start today by going around and introducing ourselves briefly and saying why we're here.\"\n\nCody snorted. \"I thought you said this wasn't going to be like therapy.\"\n\n\"I already went to grief counseling,\" Mindy mumbled.\n\n\"I hated it.\"\n\n\"Is that what this is?\" Cody demanded. \"Because if it is, I'm leaving.\"\n\n\"No,\" I said quickly. \"It's not like I expect us to sit around and talk about death, you know?\" I glanced at Kelsi, and she looked away. \"But for today, I thought it would be a good idea if we at least all know each other's stories.\"\n\nThe smirk slipped from Cody's face. He looked down at his lap.\n\nThe rest of the group watched me in silence. I didn't think it was my imagination that Sydney looked uncomfortable.\n\n\"Fine, I'll go,\" I said finally. I took a deep breath. \"I'm Lacey Mann. I have two brothers, Logan and Tanner. My dad died in a car accident last November. We were all with him in the car. All of us except for my mom, I mean.\"\n\nI said the words matter-of-factly. I didn't expect them to make me feel weird, because it wasn't like they were anything new. These were all facts I had accepted. But there was a lump in my throat when I finished, and my eyes stung a little bit. \"Logan?\" I said. \"Do you want to go next?\"\n\n\"What do you want me to say?\" he asked. \"I have the same story as you.\"\n\n\"Duh,\" Sydney said under her breath.\n\n\"I just thought...,\" I said. I stopped, because I wasn't sure _what_ I'd thought.\n\n\"I'll go,\" Cody said. \"I'm Cody. My dad died in Iraq when I was in eighth grade.\"\n\nHe paused, and I thought he was done. I was about to open my mouth to thank him when he spoke again.\n\n\"He was with his battalion,\" Cody continued. \"It was just a normal day. They were driving along a road. And then all of a sudden, a bomb went off in the road in front of them. They had driven over some wire and tripped it. The bomb totally ripped apart the convoy. A few other soldiers were hurt. But my dad died. Right there.\"\n\nHe took a deep breath and then looked down at his lap.\n\nIt was Mindy who finally spoke. \"That must have been really hard on you,\" she said. \"To have him so far away. And not be able to say goodbye.\"\n\n\"Yeah, it sucked,\" Cody said. He paused. \"What happened to your mom?\"\n\nSomething inside me lurched. It was working. The people around the table were talking.\n\n\"She died,\" Mindy said simply. \"Last year, when I was in eighth grade. She had been sick when I was younger. But the cancer went away, and we thought she was done with it. After a while, she stopped going to the doctor as often as she should have. And then, when they found it again, it was too late. It had already spread. She died really fast. I mean, in a couple of months. They tried chemo, but it didn't work. My little sister and I were with her. We had to move in with my dad after. He and my mom were divorced.\"\n\n\"Was he sad?\" Kelsi asked in a soft voice.\n\n\"My dad?\" Mindy asked, turning to her. Kelsi nodded. Mindy considered this for a minute. \"I don't know. I never saw him cry or anything. He told me and my sister he was sad. But he's remarried. He has a new wife and a little baby now. I think it's weird for him that we live with him.\"\n\n\"Do you like your stepmom?\" The question came from Logan. It surprised me that he was participating instead of mocking.\n\nMindy shook her head. \"Not really. She's really young. She doesn't like us. Me and my sister, I mean.\"\n\n\"That must be hard,\" Cody said.\n\nMindy glanced at him. \"Yeah,\" she said. \"It is. We don't talk about my mom very often anymore. My dad gets uncomfortable when we bring her up.\"\n\nI caught her eye. \"You can talk about her here,\" I said.\n\n\"With us.\"\n\nMindy smiled at me, a little sadly. \"Yeah. I know.\"\n\n\"My mom died,\" Kelsi said in a tumble of words. \"But you all know that. _Everyone_ knows that. Don't they?\"\n\nThere was a brief silence, then Cody laughed. It sounded out of place after her somber declaration. I looked at him, startled.\n\n\"Yeah, we're pretty much all famous,\" he said. To my surprise, Kelsi laughed too.\n\n\"Everyone knows you,\" Cody went on, \"but no one knows what the heck to say to you.\"\n\n\"Sure they do,\" Kelsi said. She batted her eyelashes and adopted a high-pitched voice. \"We're _so_ sorry!\"\n\nWe all laughed. I hadn't expected this. I was feeling better about this meeting idea every moment.\n\n\"Yeah,\" Mindy chimed in. \"And then they just stare at you. And avoid you. Because they don't know what to say after that.\"\n\nEveryone laughed except Logan and Sydney. \"That's not true,\" Sydney interjected. The laughter died down, and everyone looked at her. \"People don't do that.\"\n\nCody narrowed his eyes. \"Yeah they do.\"\n\n\"You're just being paranoid,\" Sydney retorted.\n\n\"Really?\" Cody shot back. \"And what makes _you_ the expert?\"\n\nSydney's face was turning red. \"I'm just saying that I think you're all blowing things out of proportion,\" she said, her voice rising. She looked to Logan for support, but he was looking at his feet. \"Besides, it's not like anyone means badly by it.\"\n\nI hated to make things smoother for Sydney, especially when she didn't belong here. But I also hated to have us fighting at the first meeting. So before Cody could reply, I cut in. \"Sydney, I think Cody just means that people don't know how to act around us,\" I said. \"Because they don't know what to say.\"\n\n\"Well, what are we supposed to do?\" Sydney said. \"Act like you're some kind of royalty or something? Just because you had one bad thing happen to you?\"\n\nI stared at her. \"None of us expect to be treated like _royalty_. We just want to be treated normally. And it's not like having your parent die is just some random 'bad thing,' you know. It's a huge deal.\"\n\n\"Or maybe you're just _making_ it a big deal,\" Sydney said. \"Honestly, Lacey. I think this whole thing is a little silly. Don't you?\"\n\nShe looked around the table, smirking, as if it were full of people who would agree with her. I was a bit heartened to see that Kelsi, Cody, and Mindy were staring stonily back at her. I opened my mouth to reply, but before I could, there was a deep voice from the direction of the doorway.\n\n\"I don't think it's silly at all.\"\n\nWe all turned to see who had come in unannounced. I practically fell out of my chair. Sam was standing by the hostess stand, his Red Sox cap pulled low over his forehead.\n\n\"Hey,\" he said, looking directly at me. My heart was pounding, and my cheeks felt like they were on fire. \"Um, thanks for saying that. But, um, what are you doing here? This is a group for people who have lost a parent.\"\n\nSam nodded slowly. \"I know,\" he said. \"That's why I'm here.\"\n\nI was confused. I stared at him for a minute, uncomprehending.\n\n\"My dad,\" Sam said. He cleared his throat. \"I lost my dad.\"\n\n# chapter 12\n\nI couldn't believe it.\n\n\"Oh,\" I said. My cheeks grew even warmer. \"I'm sorry. I didn't know.\" Suddenly, the conversation in the car came flooding back to me. Sam telling me he knew how I felt. Me getting defensive and mad. I felt a little sick.\n\nSam glanced at Sydney. \"At my old school, everyone was weird to me. After they found out about my dad. I didn't want to have to deal with it with a whole new group of people when I moved here, you know?\"\n\nI knew exactly what he meant.\n\n\"I was trying to tell you,\" he said, looking straight at me. \"That's what I was trying to say to you that day in the car.\"\n\n\"Oh.\" I swallowed hard.\n\n\"What happened, man?\" Cody asked. \"If it's cool for me to ask.\"\n\n\"A stroke,\" Sam said. \"He had a stroke.\"\n\nSydney seemed to have been shamed into silence. The rest of us mumbled words of apology.\n\n\"Was it recent?\" I asked. \"With your dad?\"\n\n\"Yeah,\" he said in a barely audible voice. \"It was a few months ago. He just...\" Sam paused, like he wasn't quite sure what to say next. He took a deep breath. \"He was fine, you know? And then all of a sudden he wasn't. It was like something just went wrong in his face, like something short-circuited, you know, like a light that flickers all weird or something.\"\n\n\"You were with him?\" Cody asked.\n\nSam nodded. \"Yeah,\" he said. \"I mean, I kept asking him what he was doing. I thought for a minute maybe he was joking, you know. But then I knew he wasn't. And I called nine-one-one.\"\n\nSilence settled over us again.\n\n\"So, um, do you want to sit down?\" I asked, clearing my throat.\n\n\"Yeah,\" Sam said. \"I do.\"\n\nLogan glanced at Sydney again and then back at me.\n\n\u2022 \u2022 \u2022\n\nSydney left about five minutes after Sam joined the group. To my surprise, although he remained largely unresponsive, Logan stayed.\n\nIn the next hour, with me sort of leading the group, we talked a bit about our parents who had died, a little about what it was like with a whole new family dynamic, and what it was like when everyone at school treated you like a weirdo. But mostly, we just talked, awkwardly at first but then more like friends.\n\nI learned all sorts of things I didn't know about people. Kelsi wanted to try out for softball this spring; Mindy had done gymnastics until her mom got sick and had even competed twice at the state level. Cody had just gotten a job at the local movie theater, tearing tickets, and he was thinking about signing up for the army next year, despite what had happened to his dad.\n\nThere were a million things I wanted to ask Sam, like when his dad had died and why his family had moved to Plymouth or how he seemed so much better adjusted. But unlike the rest of the group, he didn't seem to be volunteering any information. And I didn't want to make him uncomfortable. So I didn't say anything.\n\nA few minutes later, after we had complained a little more about therapists and other adults who thought they knew exactly how we were supposed to feel, Cody looked at his watch and stood up. \"I gotta go,\" he said. \"My shift at the movie theater starts at four-thirty.\"\n\nI checked my watch too. It was almost four. I couldn't believe we'd been talking for that long. It felt like just minutes ago that Sam had made his surprise appearance.\n\n\"Yeah, I guess we should get home,\" I said, glancing at Logan. I took a deep breath. \"I am so glad all of you came today. I wasn't really sure how this would go. But I wanted, I don't know, a place for us to feel normal, you know?\"\n\n\"A place for weirdos like us,\" Cody said. I thought for a split second that he was making fun of me until he winked and smiled.\n\n\"Yeah, weirdos like us,\" Mindy echoed. \"I like that.\"\n\nWe all laughed.\n\n\"So, should we do this again?\" I ventured after a moment. \"Next week maybe?\" I held my breath.\n\nKelsi and Mindy exchanged glances. Cody shrugged. Logan didn't reply. But Sam was nodding enthusiastically.\n\n\"Yeah,\" he said. \"I like that idea. Don't you?\"\n\n\"Yeah,\" I said, glancing around.\n\n\"Yeah,\" Kelsi said. \"That'd be cool, I guess.\"\n\n\"Okay,\" Mindy said.\n\n\"Whatever,\" Cody said. We all turned to Logan.\n\n\"I guess,\" he mumbled, looking down.\n\nI couldn't stop the smile from spreading across my face. This was really going to work.\n\n\"Can I make a suggestion, though?\" Sam asked. \"What if we met somewhere else?\"\n\n\"Like where?\" Kelsi asked.\n\nSam smiled. \"What if we went bowling?\"\n\n\"Bowling?\" Logan repeated.\n\n\"Yeah,\" Sam said. \"Why not? My aunt Donna owns Lucky Strikes Lanes over off Main. I bet she'll give us a big discount. Or maybe she'll even let us bowl free.\"\n\n\"That sounds cool,\" Cody said.\n\nI looked at the girls. I was worried that bowling would sound dorky and they wouldn't want to go. But they both nodded.\n\n\"Okay,\" Kelsi said.\n\nI looked at Logan. He seemed annoyed, but he shrugged. \"Yeah, whatever,\" he muttered.\n\nI turned back to Sam and smiled. \"That sounds like a good idea. So next Tuesday, then? A week from today?\"\n\nEveryone nodded.\n\n\"If anyone needs a ride, maybe we can just meet in the parking lot after school,\" Sam said. \"I drive a Cherokee. I can fit a bunch of people.\"\n\n\"Okay, next Tuesday it is,\" I said. \"And guys?\"\n\nEveryone looked at me, expectant. I paused.\n\n\"Thanks,\" I said finally. \"Really. Thanks.\"\n\nNo one said anything for a minute. Then Mindy said softly, \"Well, thanks for setting this up. It's nice to be someplace where you don't feel like a weirdo. Where you can feel like you did...\"\n\nHer voice trailed off. I knew exactly what she meant. But it was Kelsi who put it into words.\n\n\"Before,\" she filled in, her voice soft. \"Where you can feel like you did before everything changed.\"\n\nI beamed. This felt like the most important thing I had ever done. I was helping people.\n\n\"Thanks for coming,\" I said quietly.\n\nAnd then, with a bunch of mumbled goodbyes, everyone went their separate ways. Sam glanced back and smiled at me as he walked out the door, but he didn't wait or ask if I needed a ride. A wall had gone up between us, and I'd been the one to put it there, all because I'd assumed that he was just like everyone else.\n\n\u2022 \u2022 \u2022\n\nThat night, Mom tried to get us to talk about the meeting, and I told her a little bit about it. Logan was strangely quiet, muttering only yes or no to Mom's questions. Tanner, as usual, pushed his food around on his plate and was silent. I felt a knot starting to form in my stomach as I looked around the table at my silent little brother, my sad-eyed mother, and grumpy Logan. For the millionth time, I missed Dad so much I could feel the pain in my chest.\n\nAfter dinner, everyone shut themselves away in their rooms, even Mom. It made it feel like we were living in four separate little universes.\n\nI did my trig homework at the dining room table, puzzling over one particularly complicated cosine problem. Then, closing my books, I walked upstairs and knocked on Logan's door.\n\n\"What?\" he barked.\n\n\"It's me,\" I said. \"Can I come in?\"\n\nThere was a moment of silence. \"Whatever.\"\n\nI hadn't been in Logan's room in a while, and I was struck by how unfamiliar it felt. He had the same blue and green bedspread, of course, and the same white blinds that were a little bent on the lower right side. But he had taken down the surfing posters he used to have on his walls. In their place, he had a big collage made out of pictures of him and Sydney, with little hearts drawn all over it. Sydney had made it, of course, but I couldn't believe he had actually put it up.\n\nHe was sitting at his desk, shoulders slumped, staring at the bright screen of his computer. He had his history textbook spread in front of him and a few IM windows open.\n\n\"I, um, just wanted to say thanks for coming today,\" I said. I stood awkwardly in the doorway for a minute, then I crossed the room and sat on his bed. Logan sighed, typed a few things into the IM windows, and then turned around to look at me.\n\n\"Thanks,\" I continued after a pause. \"For staying. After Sydney left, I mean.\"\n\n\"Yeah, well, now she's pissed at me,\" Logan said.\n\n\"Oh,\" I said. I didn't want to say that I was sorry, because I wasn't. \"Well, maybe she shouldn't have been there in the first place.\"\n\nEvidently, this was the wrong thing to say.\n\n\"Who are you to tell my girlfriend where she can and can't go?\" Logan exploded.\n\n\"I'm not trying to do that,\" I said defensively.\n\n\"Whatever,\" Logan said bitterly. \"You made her feel so uncomfortable. And now she's mad at _me.\"_\n\n\"Logan, I didn't do anything to make her feel uncomfortable,\" I said. \"She got all defensive. Remember?\"\n\n\"Yeah, well,\" Logan said. But he didn't continue.\n\nWe sat in silence. Then all of a sudden, Logan blurted out, \"What's the point, anyways?\"\n\nI was startled. \"The point of what?\"\n\n\"Of your stupid club?\" Logan asked. \"What, like it's supposed to make us feel better?\"\n\nI shrugged. \"I don't know. I just thought it might help. I thought today went well.\"\n\n\"Yeah, for you, maybe,\" he said.\n\nI stared at him.\n\n\"You know, you say you hate that we feel different from everyone else,\" he said. \"But then you start some group that makes us feel even _more_ different.\"\n\n\"It's not supposed to make us feel like that,\" I protested. \"It's supposed to give us a place to just feel normal.\"\n\n\"It's all about you, isn't it?\" he said, an edge of bitterness creeping into his voice.\n\nI couldn't understand why he'd say something like that. Everything I did these days was for other people. I worried about Mom. I tried to get Tanner to talk. I put up with Logan's stupid girlfriend just to keep the peace. \"What are you talking about?\" I asked.\n\nLogan rolled his eyes. \"I know, I know, you've been Saint Lacey since Dad died,\" he said. \"But don't you ever get sick of being good? I mean, don't you just want to get pissed off at the world sometimes?\"\n\n\"No,\" I said. How would that help?\n\nLogan made a face. \"Yeah, well, _I_ don't always want to be perfect, you know? And Sydney doesn't want me to be.\"\n\nHe gazed at me triumphantly, like the fact that he had a \"supportive\" girlfriend was the answer to everything.\n\nI stared at him for a minute. \"How does Sydney even _know_ what she wants, anyhow? She's so joined at the hip with you that I think you two are sharing a brain.\"\n\n\"Shut up, Lacey,\" he said. \"You don't know everything.\"\n\nI stood up. \"Sometimes I don't think you know anything at all.\"\n\n\"You can't bring him back, you know,\" Logan said. \"You can't bring anyone's parents back or make things like they were before. And it's stupid to try.\"\n\nI stormed out of his room, slamming the door behind me. I went into my room, slamming that door too, and collapsed on my bed.\n\nI waited for a minute, figuring that Mom would come to see what the problem was. After all, I was sure that the slamming doors could probably be heard down the block, especially since our house was so silent these days.\n\nBut she never came. And Logan didn't come to apologize. Instead, the loneliness settled down on me like a fog, and I lay slowly back on my bed, soaking in the silence.\n\n# chapter 13\n\nAfter our Saturday-afternoon appointment with Dr. Schiff, Mom, Logan, and Tanner had once again shut themselves away in their rooms. Feeling lonely and bored, I called Jennica.\n\n\"Want to go to the mall or something?\" I asked. Silence. Then, \"I'm busy, Lacey.\"\n\n\"With Brian?\" I ventured.\n\n\"Not exactly,\" she replied. More awkward silence. Then she said, \"Look. I found out on Thursday that my dad's getting remarried, okay? And things are just a little weird around here. I don't really feel like going to the mall.\"\n\nI was stunned. \"Your dad's getting remarried? To Leanne?\"\n\n\"Yeah.\"\n\n\"I didn't know it was that serious,\" I said.\n\n\"Yeah, well,\" Jennica said. I could hear her sigh on the other end of the line. \"There's a lot you don't know, Lacey.\"\n\nI wondered what she meant. \"But... why didn't you tell me?\"\n\nJennica was silent for a minute. \"I guess I didn't really expect you to understand.\"\n\n\"What?\" Jennica and I talked about everything. Or at least we used to.\n\n\"Well, it's not like he's dead or anything,\" Jennica said. \"I mean, you're always going on and on about how your life is so different because your dad died.\"\n\n\"I never talk about it,\" I interjected, surprised. I really didn't.\n\n\"Yeah, well,\" Jennica said. \"I guess I just didn't expect you to take my problem that seriously.\"\n\n\"You're my best friend,\" I said. \"Of course I'd take your problem seriously.\"\n\n\"Be honest,\" she said. \"You think my thing is so much less important than yours, don't you?\"\n\nI hesitated. Part of me wanted to say yes, of course. No matter how sad she was, at least her dad was still alive. She still got to see him sometimes. Her whole world hadn't been shattered. Not the way mine had been. But I knew she didn't see it that way. And I knew that admitting that would be the wrong thing to say. \"Um,\" I said instead.\n\nShe made a muffled sound. \"Like I said. Don't worry about it, Lacey.\"\n\nAnd then, for the first time in our friendship, Jennica hung up without saying goodbye.\n\nI sat down at the kitchen table and put my head in my hands. Jennica was mad at me. Logan barely talked to me. My mom was trying to put on a happy face, but she avoided the house and her kids as much as she could. And then there was Tanner.\n\nI walked upstairs and knocked lightly on Tanner's door. He didn't reply, so I knocked again. \"Tanner?\" I called out. \"Can I come in?\"\n\nI waited a minute, and hearing no reply, I pushed open the door.\n\nThe shades were drawn and the room was dark, even in the middle of the afternoon. The lamp beside Tanner's bed was on, but he was crouched in the shadows next to McGee's cage.\n\n\"Hey, buddy,\" I said. I crossed the room and knelt beside him. \"How's it going?\"\n\nTanner was staring into the cage like his life depended on it, his concentration entirely fixed. I glanced into the cage to see what McGee was doing.\n\nExcept McGee wasn't there. I bent my head to look inside his little plastic cave. No McGee. Nor was he on the hamster wheel. And the cage was small, only a few feet long and a few feet tall.\n\n\"Tanner?\" I asked, starting to feel alarmed. \"Where's McGee?\"\n\nWithout looking at me, he raised his right arm and pointed toward the window.\n\n\"He's over by the window?\" I asked. Tanner shook his head.\n\nI struggled to figure out what he meant. \"He's outside?\" That didn't make sense. \"You let him outside?\" But Tanner shook his head again. And then I noticed a tear roll down his right cheek. He blinked quickly and wiped it away as he went on staring at the empty cage.\n\nSuddenly, I got it. \"Tanner?\" I asked. \"Did McGee die?\"\n\nTanner nodded once, still without looking at me. \"Oh, Tanner,\" I breathed, blinking back tears. \"I'm so sorry. Why didn't you tell anyone?\" Tanner kept staring at the cage.\n\n\"Tanner, where is he?\" I glanced toward the window. \"Did you bury him out back?\" Tanner nodded again.\n\nI swallowed hard. \"Well, come on,\" I said resolutely. \"McGee needs a proper funeral.\"\n\nTanner finally looked up at me, surprise playing across his face. \"A funeral?\"\n\nAn hour later, I had helped Tanner make a little cross-shaped headstone out of Popsicle sticks and glue. With a thin Sharpie he wrote \"Good Bye McGee\" on the horizontal sticks and drew a little picture of the hamster. While he drew, I downloaded \"Amazing Grace\"\u2014the song that had played for much of our dad's funeral\u2014on my iPod and grabbed my portable speakers from my room. Then, I got Mom and Logan and told them we needed to do something in the backyard.\n\nMom was mystified at first, but her face crumpled when I told her what had happened. She excused herself, and I could hear muffled sobs coming from her bathroom. Logan, on the other hand, just rolled his eyes.\n\n\"You're making me come outside for a _hamster's_ funeral?\" he demanded.\n\nI glared at him. \"No, I'm making you come outside to be supportive of our brother.\"\n\nLooking annoyed, he got up and followed me downstairs, grumbling under his breath.\n\nA few minutes later, we all stood under the old, arching oak tree in the left corner of the backyard, where Tanner had buried McGee. With a solemn look on his face, Tanner carefully stuck his Popsicle-stick cross in the ground and secured it with a pile of little pebbles. Then he stood up and pushed play on my iPod. The strains of \"Amazing Grace\" drifted through the yard, and as we all stood in silence, clustered around the tiny grave, the song and the solemnity of the moment reminded me uncomfortably of Dad's funeral. I gulped.\n\n\"Do you want to say a few words in McGee's honor?\" I asked my little brother.\n\n\"Lacey,\" my mom said, \"you know he doesn't like to talk. Don't push him.\"\n\nBut Tanner surprised us all by turning to face us and clearing his throat. \"McGee was my friend,\" he began. I turned the iPod down a little. \"He always understood me. He didn't try to make me talk. But he listened if I wanted to talk.\"\n\nWe stared at him. He hadn't spoken this much at a stretch since last November.\n\n\"He was just there for me,\" Tanner went on. He looked at the ground. \"He was fun to play with. And I never had to talk about Dad or about being sad with him.\" He paused. \"Thank you for coming to the funeral.\" Then before any of us could respond, he walked quickly away, toward the house. We stood and watched him in shocked silence until he disappeared into the house, pushing the door closed behind him.\n\n\u2022 \u2022 \u2022\n\nAfter Tanner disappeared into the house, Mom went back to cleaning the kitchen, as if all her meticulous scrubbing and organizing could restore order to our lives, too. Sydney came and picked up Logan, who left without a word to any of us. And as our house fell silent again, I knew I had to get out.\n\nI changed into running shorts, a sports bra, and a long-sleeved T-shirt and laced up the running shoes I hadn't put on in nearly a year. I used to love running, but I hadn't gone out once since the accident. At first, it was because my leg had been broken. But then, after it healed and after the doctors told me I should try to ease back into my normal routine, I couldn't bring myself to do it. Running made my leg ache, a dull, throbbing pain in the two places where the bone had been crushed. And the last thing I needed was a physical reminder of the accident.\n\nBut today, I wanted to feel it. I wanted to hurt. I wanted to feel _something_. And so I pulled my hair back in a ponytail, plugged earbuds into my iPod, and left the house without saying goodbye.\n\nEvening was approaching, and with it, cooler temperatures. I shivered as I stretched in the driveway, but I knew that I'd warm up as I ran. I took off down the street, no particular route in mind. I pulled up Star Beck's latest album, the one she'd written herself, on my iPod, and let myself slip into the music as my feet pounded the pavement.\n\nMy leg ached, as I knew it would, every time my left foot hit the ground. I tried to imagine the exact places my femur had broken, tried to imagine the bone shattering as our car crumpled around us. It seemed unfair that my leg would be able to heal almost entirely, while my dad's injuries had stolen him in a matter of seconds. In a way, it was comforting that my leg still hurt, and I found myself wishing that it would ache more, as if hanging on to the pain of that day would give me a do-over.\n\nI avoided, as I always did, the intersection where the accident had happened. It used to be part of my jogging route, but now I went the other way, winding deeper into our subdivision. I ran back toward the cranberry bogs, which were awash in red, ripe fruit. It was harvesting season, and even as the sunlight waned, I could see a few men in hip boots in what appeared to be a brick-colored sea, raking floating cranberries into containers. My dad had harvested cranberries as a side job when he was putting himself through college. I tried to imagine him out there with the other men, but I couldn't fix the image in my head. I used to be able to close my eyes and see the outline of his face so clearly, but now he had all but disappeared.\n\nI turned away from the bogs. I ran along the main road for a little while, then dipped into the next neighborhood. Jennica lived here, and I ran by her house, not sure what I was intending to do or say. But the lights were all off, and her mom's car wasn't in the driveway. Perhaps she and her mother and sister had gone out to dinner, like a normal family.\n\nI ran on. My leg still ached, but the pain felt like a companion now instead of a burden. I was running with it, not against it. I turned down a street I hadn't been on before and noticed, way off at the end, a guy in a long-sleeved gray tee, a baseball cap, and running shorts mowing the lawn of a big house that sat a little way up a hill. As I ran toward it, I thought about what an insurmountable task it seemed like with the push mower he was moving around the enormous yard. My feet took me closer, and just as I was about to pass by the house and loop down another street, the guy mowing the lawn turned, and I realized with a start that I knew him.\n\nIt was Sam.\n\nI stopped in my tracks without meaning to, and our eyes met. He stared for a moment and then shut off the mower.\n\n\"Lacey?\" he yelled down the lawn a little uncertainly. \"Um, hi,\" I said. I took my earbuds out and glanced around, unsure of what to do. I was suddenly conscious of how I must look. I was drenched in sweat, my hair was frizzing out of my ponytail, and I didn't have any makeup on, which meant that the two pimples on my chin were probably staring right at Sam, in all their angry red glory.\n\nAs Sam made his way down the lawn, I was surprised to see a tattoo on his left calf. I couldn't help staring. It was a Celtic claddagh, a pair of hands clasping a heart with a crown on top. My dad had the exact same one. I knew it meant love, friendship, and loyalty. My mother's wedding ring had the same design on it too, and my dad had once explained to me that it meant he had married his best friend, the woman he loved most in the world, and someone he'd be loyal to forever.\n\n\"You have a tattoo,\" I said.\n\n\"What?\" He looked surprised and glanced down at his leg. \"Oh. Yeah. I got it after my dad...\" His voice trailed off. He looked down, then he smiled at me. \"I thought my mom was going to kill me when I came home with it. The guy at the tattoo place thought I was eighteen.\"\n\nI smiled. \"My dad got a claddagh tattoo too. On his arm. He got it when he and my mom got married.\"\n\n\"Oh yeah?\" Sam said. \"That's cool.\"\n\nWe stood there awkwardly for a minute. \"So,\" Sam finally said. \"What are you doing here?\"\n\nI could feel the color rise to my cheeks. I probably looked like I was stalking him. \"I was just going for a run,\" I said, and added hastily, \"I had no idea you lived down here.\"\n\nSam glanced back at the house. \"I'd invite you in, but my mom's sort of freaking out right now. My little brother just gave her his report card, and he failed English. They're screaming at each other. That's why I came out to mow the lawn.\"\n\n\"You have a brother?\"\n\n\"Yeah,\" Sam said. \"Joey. He's eight.\" He paused. \"Is it just you and Logan?\"\n\n\"I have another brother too,\" I said. \"Tanner.\" I paused and added, \"He's eleven. He doesn't talk very much anymore. Since the accident.\"\n\n\"It's crazy how much things change, isn't it?\" Sam said. \"You know, after.\"\n\n\"Yeah,\" I agreed. I suddenly wanted to change the subject. I glanced up at the lawn. \"So you mow this whole thing by yourself?\"\n\nSam laughed. \"Yeah, it's crazy,\" he said. \"Our old house had a much smaller yard, so it was a lot easier. But you know, I don't really mind. It's kind of nice to have a reason to be outside.\"\n\n\"I know,\" I said. Silence settled over us.\n\n\"So the other day was really cool,\" Sam said after a minute. \"I mean, I think it was a really good idea.\"\n\nI smiled. \"Thanks.\"\n\nSam took off his cap raked a hand through his hair, getting a few tufts of grass stuck in his thick, dark strands. \"Hey, could I run with you for a little while?\"\n\n\"You want to run with me?\"\n\nHe shrugged. \"If that's cool,\" he said. \"I used to run track at my old school. My dad was the coach, actually.\" A shadow flickered over his face.\n\n\"Sure,\" I said. \"I haven't run in a while, though.\" I paused. \"Not since the accident, actually. So I'm not very fast.\"\n\n\"Good,\" Sam said. \"Then you'll be easy to beat when I race you.\"\n\nI laughed. He pushed the mower back up to the house and then jogged back down the driveway.\n\n\"You're not going to change clothes?\" I asked.\n\nHe glanced down at his grass-stained sneakers, his faded running shorts, and his sweaty shirt. \"Nah,\" he said.\n\nWe set off at a slow jog, and until we reached the end of the block, neither of us said a word. I was conscious of the silence between us and of my pounding heart, which was pumping blood so loudly that I feared Sam could hear it too. It wasn't until we were at the end of the street that Sam spoke.\n\n\"So do you have a boyfriend?\" he asked.\n\nStartled, I looked up at him. \"Um, no,\" I said. I cleared my throat and focused on my pace. \"Do you have a girlfriend?\"\n\n\"Nah,\" he said. He paused and added, \"I had one at my old school. But that was a while ago.\"\n\nWe jogged in silence again, and then Sam blurted out, \"I think we should go out. You and me, I mean.\"\n\n\"What?\" It sounded ruder than I'd meant it to.\n\n\"It's what I was trying to ask you that day in the car. Before you got mad. I think we should go out. Like, together.\"\n\nI could feel myself blushing. \"Really?\" I asked. \"Why?\"\n\n\"You don't want to?\" Sam asked. I noticed he wasn't looking at me, but his face seemed redder than it should have, considering that we weren't jogging that hard.\n\n\"No, no, I do,\" I said quickly. \"I'm just not used to...\" I didn't know what to say. What, that I wasn't used to guys liking me? That I wasn't used to being asked out? \"You don't even know me,\" I finally concluded.\n\n\"What are you talking about?\" he asked. \"I sit next to you in two classes, and we've talked pretty much every day for the last month and a half.\"\n\n\"I guess,\" I said. I didn't know why I was being so reluctant. I was completely attracted to him; how could I not be? And I knew he wasn't asking me out just because he felt sorry for me or wanted to gossip about how great he was for taking out the poor little fatherless girl.\n\n\"Besides,\" Sam continued, \"how are you going to get to know me if you don't let me take you out to dinner?\"\n\n\"When?\" I asked.\n\n\"Tomorrow night?\" he said.\n\nI thought for a minute. \"Yeah,\" I said. \"I could do that.\"\n\nWe both fell into silence again, and as I ran, my mind swirled, thinking about the fact that at this time tomorrow, I'd actually be out with Sam Stone. Who was hot and sweaty\u2014and really, really gorgeous\u2014as he jogged next to me right now.\n\n\"So tell me about your little brother,\" Sam said as we turned out of his neighborhood onto Long Pond Road.\n\nI hesitated, then began to tell him about how much Tanner liked animals and video games and how he used to love searching for the prize in the bottom of the Cracker Jack boxes at ball games. And before I knew it, I found myself telling Sam about Tanner's almost constant silence and how much it worried me. He told me that he was really scared to see his brother withdraw from everything he used to love. And I was surprised to realize that our mothers seemed to have reacted to losing our fathers the same way: by throwing themselves into their work and social lives instead of spending time with us.\n\n\"It's like she thinks that if she just works hard enough, she can forget,\" Sam said, glancing down at me.\n\n\"That's exactly how my mom is too,\" I said. Somehow, it helped to know that my family wasn't the only one crumbling, the only one where the remaining adult had retreated rather than dealing.\n\nIt was the best conversation I'd had since the accident.\n\nAs we jogged and talked, our feet eventually carried us to my house, like that's where we'd been going all along. By the time we got there, we'd covered everything from Jennica to Sam's best friend Chris at his old school who didn't call anymore to how hard it was to come into a close-knit community like this one and make friends, when everyone had known one another since preschool.\n\nWe stopped in my driveway, and as we stood catching our breath, I asked, \"Do you want to come in and get some water or something?\"\n\nSam looked at his watch. \"Nah,\" he said. \"My mom's probably wondering where I am. I'd better get home.\"\n\n\"Want me to get my mom to drive you?\" I asked. Then I added apologetically, \"She won't let me drive, even though I'm sixteen. It has to do with the accident.\"\n\nSam nodded, like he understood and wasn't going to judge me. \"Nah, I'm good,\" he said. \"I should be able to make it home in twenty minutes if I pick up the pace a little.\"\n\n\"You saying I'm slow?\" I teased.\n\nHe laughed. \"No. I'm saying I enjoyed our conversation too much to put any thought into actually running.\"\n\nThen, before I realized what was happening, he leaned down and gave me a quick peck on the lips. He pulled back, looking embarrassed, before I could get my mouth to unfreeze long enough to reply.\n\n\"Pick you up at six,\" he said.\n\nHe was already disappearing down the street before I raised my hand in a silent goodbye. My lips were still tingling as he vanished around the corner.\n\n# chapter 14\n\nI practically floated up to my bedroom and booted up my computer. After a quick shower to rinse my hair and wash the run away, I went online. Jennica's screen name popped up in a little IM window, accompanied by her AIM tone, which was the sound of a kiss.\n\nJENNICAJENNICA: Hey Lacey.\n\nJENNICAJENNICA: I'm sorry.\n\nI gulped. I paused and typed back.\n\nLACEYLOO321: it's ok\n\nJENNICAJENNICA: i was a jerk.\n\nLACEYLOO321: u weren't a jerk.\n\nJENNICAJENNICA: i was.\n\nLACEYLOO321: weren't\n\nJENNICAJENNICA: was\n\nLACEYLOO321: agree to disagree?\n\nJENNICAJENNICA: only if u accept my apology.\n\nLACEYLOO321: deal.\n\nLACEYLOO321: :-)\n\nLACEYLOO321: hey, i'm sorry u r upset about ur dad.\n\nJENNICAJENNICA: :-(\n\nJENNICAJENNICA: it's no big deal.\n\nLACEYLOO321: yeah it is. i shouldn't act like it's not. i'm sorry if i do that.\n\nJENNICAJENNICA: it's ok.\n\nJENNICAJENNICA: besides. it's not a big deal. Not like ur dad. i know that, ok?\n\nLACEYLOO321: i don't want to talk about that.\n\nJENNICAJENNICA: u never do.\n\nJENNICAJENNICA: is that what you talked about at ur group? that group for kelsi?\n\nLACEYLOO321: not really. we just kinda hung out.\n\nJENNICAJENNICA: why? they're not even ur friends.\n\nLACEYLOO321: i dunno. it's just nice. to have people who understand you.\n\nJENNICAJENNICA: i understand u.\n\nLACEYLOO321: i know.\n\nJENNICAJENNICA: but u don't talk to me.\n\nLACEYLOO321: it's different w\/ people who have lost a parent 2.\n\nJENNICAJENNICA: but i try to understand.\n\nLACEYLOO321: i know.\n\nLACEYLOO321:...\n\nLACEYLOO321: maybe i don't give you enough credit for that.\n\nJENNICAJENNICA: so anyway.\n\nJENNICAJENNICA: my dad's stupid wedding is in 2 months.\n\nLACEYLOO321: what????!!!! 2 MONTHS???? but he just got engaged!!!!!!!!!!!!\n\nJENNICAJENNICA: ya\n\nLACEYLOO321: that's CRAZY.\n\nJENNICAJENNICA: ya\n\nJENNICAJENNICA: i hate his stupid girlfriend.\n\nLACEYLOO321: she's like our age.\n\nJENNICAJENNICA: almost. she's like 23 or something.\n\nLACEYLOO321: what does brian think?\n\nThere was a long pause. I thought maybe Jennica had signed off without seeing my last question. I was just about to close the IM window when she wrote back.\n\nJENNICAJENNICA: he thinks i'm being dumb\n\nLACEYLOO321: what????\n\nJENNICAJENNICA: i dunno\n\nLACEYLOO321: are u 2 fighting?\n\nJENNICAJENNICA: not exactly. kinda.\n\nLACEYLOO321: about what???\n\nJENNICAJENNICA: he doesn't get it\n\nJENNICAJENNICA: his dad is on wife #3. and his mom just got married last year\n\nLACEYLOO321: so?\n\nJENNICAJENNICA: he just doesn't think it's a big deal. doesn't get why i'm upset\n\nLACEYLOO321: that's crazy\n\nJENNICAJENNICA: yeah. well.\n\nLACEYLOO321: i'm sorry.\n\nJENNICAJENNICA: ya. thanx.\n\nJENNICAJENNICA: gotta go. my mom's yelling at me.\n\nLACEYLOO321: hey, i've got something to tell you.\n\nJENNICAJENNICA: can we talk tmrow? seriously, mom's pissed.\n\nLACEYLOO321: yeah. u ok?\n\nJENNICAJENNICA: ya. see ya. can u come over tmrw? like lunchtime? we can go to the mall.\n\nLACEYLOO321: yeah. c u at noon?\n\nJENNICAJENNICA: c u. bye!\n\n\u2022 \u2022 \u2022\n\n\"We're not invited to the wedding,\" Jennica told me as she opened her door the next day.\n\n\"What?\" I asked, my heart aching for her. \"You're not?\" Jennica nodded. \"My dad's _fianc\u00e9e_ \"\u2014she spat the word out like it tasted terrible\u2014\"is apparently afraid Anne and I will make a _scene_.\"\n\nAs much as Jennica disliked her dad's bride-to-be, a spandex-wearing, yoga-practicing blond waif who was the polar opposite of Jennica's dark-haired, pleasantly plump Cuban American mom, she and Anne had been raised to be polite in every situation. I knew as well as her father did that Jennica would never, in a million years, make a scene at someone's wedding\u2014even if she hated the person.\n\n\"But what about your dad?\" I asked. \"Isn't he insisting you come? I mean, he's your _dad!\"_\n\nJennica's eyes filled with tears, which she wiped away angrily, as if furious that they were even there in the first place. \"No.\" The word sliced out of her mouth.\n\n\"No?\"\n\n\"No,\" she repeated. \"He says it's Leanne's day. And he wants it to be perfect for her.\"\n\n\"But it's his day too,\" I protested. \"And you're his kids.\"\n\n\"Yeah, well.\" Jennica shrugged. \"I guess that doesn't matter.\"\n\n\"He still loves you, Jennica,\" I said. I knew the words were weak. I didn't know what else to say.\n\n\"Well, it doesn't feel like it.\"\n\nI thought about my family and about how far apart we'd all drifted. I thought about my dad, and how he wasn't here for us when we needed him most. And weird as it was, I thought about Sam and the fact that in just a few minutes of talking to him, he'd made me feel more understood than I'd felt in the last ten months.\n\n\"My mom's out and Anne's at her friend's house,\" Jennica said, changing the subject. \"Is it cool with you if we just go to the mall now and eat there? I promised I'd have the car back by three.\"\n\n\"Sure,\" I said.\n\nIt wasn't until we were walking into Macy's that I blurted out, \"So I'm going out with someone tonight.\"\n\nJennica stopped. \"Who?\" she practically shrieked.\n\n\"Sam Stone.\"\n\n\"What? Since when? Why didn't you tell me?\"\n\n\"I'm telling you now.\" I quickly recounted the story of my jog yesterday and of running into him as he mowed his lawn.\n\n\"He's, like, completely gorgeous,\" Jennica said. \"I can't believe you're going out with him.\"\n\n\"I know,\" I said.\n\n\"Omigod,\" Jennica said.\n\n\"I know,\" I repeated with a smile.\n\n\"Well, we totally have to get you a new outfit, shoes, a top, earrings\u2014\"\n\n\"It's not really that big of a deal,\" I protested. I felt silly. \"I mean, it's a Sunday night. We're just going to dinner.\"\n\n\"And you need to look hot.\"\n\nFor the next hour, Jennica seemed to forget entirely about her dad's upcoming remarriage as we raced around the mall. She was on a mission as she rifled through sale racks, throwing dresses, skirts, and cute tops at me. And she was chattering a mile a minute.\n\n\"So you have to ask a lot of questions, but not too many, because you want to seem interested, but not annoying,\" she rambled. \"And you want to make eye contact, but you can't, like, stare, because that comes off as creepy, you know? And you should order a real meal, not just a salad, because guys don't like girls who don't eat, but you shouldn't finish it all, because you don't want to look like a pig. And you should remember to cross your legs, bat your eyes, and sometimes lick your lips, because it's been proven that guys find that attractive.\"\n\n\"Jennica,\" I said after we got our lunch and found seats in the crowded food court, \"I appreciate all the advice. But I think I'm just going to be myself.\"\n\n\"Be yourself?\" Jennica repeated. She looked horrified.\n\n\"I'm not a total loser or anything,\" I said, taking a bite of my hot dog.\n\n\"No... but you aren't exactly used to going out with cute guys.\"\n\nI gave her a look. \"I think I can manage.\"\n\nJennica took a sip of her soda. \"You are going to have so much fun. I'm actually kind of jealous.\"\n\n\"What, of me going out with Sam?\" I asked, surprised.\n\n\"No,\" she said. \"Of how excited you are.\"\n\n\"Don't you feel like this with Brian?\" I asked.\n\nJennica paused. \"No,\" she said. \"Not anymore.\"\n\n# chapter 15\n\nWhen the doorbell rang, I took one last look in the mirror, then raced downstairs. I'd bought a new black top with a deep pink rose stitched up the side, and I'd paired it with my favorite jeans, black boots, and gold hoop earrings. I looked good.\n\n\"Hey,\" Logan said, beating me to the door. He stared at Sam, who looked ridiculously hot in dark jeans, a white button-down shirt, and his leather jacket.\n\nSam looked past Logan and saw me coming down the stairs. \"Hey.\" His smiled widened.\n\n\"Tell Mom I'll be home by ten,\" I told my wide-eyed brother.\n\n\u2022 \u2022 \u2022\n\nWe ate at a place called Saltwinds, which looked out over Plymouth Bay. And luckily I didn't need Jennica's rules\u2014I didn't even feel nervous. It was like talking to a friend who happened to be super cute.\n\n\"I want to show you something,\" Sam told me after dinner. \"Do you have another couple of hours?\"\n\nI doubted Mom would even notice if I was late.\n\n\"Yeah,\" I said.\n\nSam turned east on Route 44, but I didn't figure out where we were going until we were well outside the city limits.\n\n\"Are we going to where you used to live?\" I asked.\n\n\"Near there,\" Sam said mysteriously.\n\nJust after we passed a WELCOME TO TAUNTON sign, Sam took a left and then another left onto a dirt road. We drove until I could see a collapsed bridge across a river ahead of us.\n\n\"You're not going to try to cross that or something, are you?\" I asked, realizing as I said it that I was being silly.\n\n\"No,\" Sam laughed. \"Just trust me, okay?\" He pulled off the dirt road and parked several yards from the water. He shut off the ignition and crossed around to the passenger side to help me out. \"Come on,\" he said. \"It's not muddy.\"\n\nI hopped out of the Jeep. He took my hand, lacing his fingers through mine the way he had at the party. We walked in silence under the bridge. I looked at the water, half expecting to see a boat or something down there. But when we reached the bank, Sam gently put his hands on my shoulders and turned me around so that I was facing the underside of the bridge.\n\nI stared without speaking. I didn't know what to say; it was like nothing I'd ever seen.\n\nThe entire underside of the bridge abutment was painted with a mural in all the colors of the rainbow. Every available inch of space was filled with individual scenes. Kids played together; families sat around the dinner table; stars sparkled in the sky.\n\n\"Wow,\" I said. \"This is amazing.\"\n\n\"It's where I used to come before we moved to Plymouth. You know, when I needed to get away,\" Sam said.\n\n\"How did you find this mural?\" I asked.\n\nSam laughed. \"Find it? I painted it.\"\n\nI looked at him in disbelief. \"What?\"\n\nHe looked a little embarrassed. \"I mean, it's not that big of a deal. It's just how I get stuff out, you know? Like this.\" He walked over and pointed to a scene of three people standing together, their backs to us. The man in the middle was pointing upward. The boys to either side of him, one younger than the other, were looking in the direction of his finger. \"This is the first thing I painted. It's my dad and my brother and me when we were little. He used to take us to the air show in Chicopee. For some reason, after his stroke, this was the image I couldn't get out of my head.\"\n\n\"It's beautiful,\" I said. I reached out and touched the paint, feeling its texture beneath my fingers.\n\n\"And this\"\u2014he walked me a few feet to the right, where a man and a woman knelt laughing under a Christmas tree\u2014\"was the Christmas that my dad gave my mom a pair of socks and then surprised her by pulling a necklace out of his bathrobe pocket while she was trying to pretend she liked the socks.\"\n\nHe took me down the mural, pointing out scenes here and there. I couldn't get over the level of skill; some of the figures looked real enough to reach out and touch. As Sam talked me through some of the pictures, I couldn't help feeling like I was reading the CliffsNotes to his life. I liked it.\n\nAfter a while, he led me over to a cement block near the river, and we sat down.\n\n\"So, you never told me what happened to your dad,\" he said after a minute. He reached for my hand and squeezed. \"You don't have to talk about it. But if you want to, I'd like to hear.\"\n\nI looked down at the water. \"I'm sure you've heard the story,\" I said. \"Everyone at school talks about it.\"\n\n\"I don't listen to rumors,\" Sam said.\n\nI took a deep breath. \"Okay.\" I looked up and cleared my throat.\n\n\"It happened last November,\" I began. Slowly, I told him about that crisp, bright autumn morning eleven months ago, when everything in the world had seemed so perfect. The words poured out, as they'd never done before. No one had ever asked for my story; everyone assumed they already knew.\n\n\"The worst part about it is...\" I couldn't finish the sentence. Sam put his hand on my shoulder, and I knew he was trying to comfort me. I couldn't meet his eyes. \"I think it was my fault,\" I said, so softly that I wasn't sure Sam could even hear me.\n\n\"It _wasn't_ your fault,\" he said right away.\n\n\"Yeah, it was.\" I still couldn't look at him. \"Sam, if I hadn't spent that extra time in the bathroom, if I hadn't been so stupid and shallow\"\u2014I took a deep breath\u2014\"my dad would still be here.\"\n\n\"Lacey,\" Sam said firmly.\n\nI continued to look down. I swallowed the lump in my throat.\n\n\"Lacey,\" Sam repeated. \"Look at me.\" His face was inches from mine. I could feel his warm breath. He was looking at me intensely.\n\n\"What?\" I whispered\n\n\"It wasn't your fault,\" he said.\n\n\"But it was,\" I said. \"If I had just taken less time\u2013\"\n\n\"No.\"\n\n\"But if I had screamed or something when I saw the other car\u2013\"\n\n\"No, Lacey.\"\n\n\"If I hadn't been dragging my feet to annoy Logan\u2013\"\n\n\"No,\" Sam cut me off, his voice leaving no room for argument. \"It was _not_ your fault. Just like it wasn't my fault with my dad. I beat myself up about it for a while, Lacey. Even after the doctors said there wasn't anything I could have done. But it wasn't me. And it wasn't you. And as unfair as this is, and as hard as it is to understand, it was just their time for something to happen.\"\n\nI swallowed hard. I didn't believe that. How could it have been my dad's time? He was thirty-eight. Just the other day, we'd read in history class about a man in Puerto Rico who had lived to be 115. How was that possible? He had lived three of my father's lifetimes.\n\nI stared out into the blackness for a while and tried to process what Sam had said. He had felt like it was his fault too when his dad died. But what could he have done to stop a stroke?\n\nIt was different for me. There were a thousand things I could have done to change the outcome that day. I could have gotten up earlier that morning. I could have taken less time in the bathroom. I could have chosen not to deliberately annoy Logan. I could have looked up a second sooner in the car and seen the SUV barreling toward us. I could have warned my dad before it was too late.\n\nIf I'd done any of those things, my dad would still be alive.\n\nBut I knew Sam wouldn't understand that or would try to talk me out of it, the way Dr. Schiff did whenever we touched on the topic. So instead, as I did with her, I changed the subject. \"The anniversary is in three weeks,\" I said. I looked out in the blackness of the night and tried to focus on one of the porch lights across the river. Sometimes, if I stared into the darkness long enough, I could see the shape of my dad's face in the shadows, his familiar form coming out of the blackness. But not tonight.\n\n\"The anniversary of the accident?\" Sam asked.\n\nI nodded. \"November fifteenth,\" I said. \"It's weird thinking it's been a whole year.\"\n\nSam slipped his arm around my shoulder and scooted a little closer so that the sides of our bodies were pressed together. I should have felt nervous, or at least that tingly, anticipatory feeling of being with someone I really liked. But instead, all I could think about was my dad.\n\n\"You must miss him,\" Sam said, his breath tickling my ear.\n\nI nodded and he gave my shoulder a long squeeze, pulling me closer. \"So much has changed,\" I said. \"I miss him more than I could even say. But I miss _us_ , too. I miss my family. I miss being normal. I miss the sound of my little brother's voice. I miss seeing my mom smile. I miss being able to feel happy, even for an instant, without feeling guilty.\"\n\nI paused, embarrassed, and looked at Sam. \"I'm glad you're here,\" I said. \"I've never been able to talk to someone who understands before. I mean, I know other people whose parents have been sick or have had cancer or who have gotten divorced. And that's really sad. But it's not the same thing. Talking to you just makes me feel safe.\"\n\nSam shifted, and I thought he was going to say something, but he didn't. I settled back against his shoulder and gazed out at the river.\n\n\"Sometimes, I miss my dad so much it literally hurts,\" I whispered. I wasn't even sure I'd said the words aloud until I felt Sam's arm tighten around me.\n\n\"I know,\" he said. \"Can I show you something?\"\n\nHe led me back to the mural, to the far right side, and pointed up, above the heads of the paintings of himself and his mom and brother watching a baseball game. \"See that rainbow?\"\n\nSam had painted a sunny sky with only a few wisps of clouds. But in the middle of it, so faint that you had to strain to see it, there was the lightest wash of red, orange, yellow, green, blue, and purple, all in an arching ribbon of translucent color.\n\n\"My uncle Joe died when I was ten. Cancer,\" Sam continued. \"We were all really close, so it was really tough on me.\"\n\n\"I'm sorry,\" I said.\n\nSam shook his head. \"No, it's okay. But my point in telling you is that my dad used to say that anytime we could see a rainbow in the sky, that was Uncle Joe telling us he was all right.\"\n\n\"A rainbow?\" I asked.\n\nSam shrugged, embarrassed. \"I know it sounds dumb.\"\n\n\"No, it doesn't,\" I said gently. \"But do you believe that? I mean, really believe it?\"\n\n\"I didn't at first. But you know, I started noticing that there were rainbows in the sky at the weirdest times. Like the afternoon my dad had his stroke. It wasn't even rainy that day. But I swear, when we got to the hospital with the paramedics, I looked up, and there was this really faint rainbow in the sky.\"\n\n\"Really?\"\n\n\"Maybe it's not as crazy as it sounds. I mean, if you believe in heaven and all.\"\n\n\"I do,\" I said simply. You had to believe in heaven when your dad died. The alternative, that your father's soul simply vanished, was too awful to even consider.\n\n\"So what I meant was, I think maybe your dad's been here all this time,\" Sam said. \"Maybe he does see you. You just haven't known where to look for him.\"\n\nI nodded quickly. I was trying to fight a strange feeling welling up inside me. It almost felt like I was going to cry, but I hadn't done that in almost a year. Not since that day in the cafeteria with Tali and Tatiana. It wasn't that I didn't want to. But every time I felt like the tears should come, they didn't. This was the closest I'd felt. My insides swam uncomfortably. I fought the feeling. I didn't want to cry; I couldn't afford to crack now, for so many reasons.\n\n\"Thanks,\" I said finally.\n\nI looked back at the mural and realized that the figures of Sam, his mother, and his brother weren't watching the baseball game after all. They were looking up at the rainbow, which seemed to rise out of the horizon, behind Fenway Park's Green Monster. I smiled at Sam. He was already staring at me, and for a moment, we just held each other's gaze.\n\nI knew it was going to happen a second before it did. Our breath grew short. The space between us grew smaller. And then, Sam's hand was on my cheek, brushing it gently. When he finally leaned in and touched his lips to mine, my eyes were already closed, and I was already leaning toward him.\n\nSam tasted a little like Coke, but sweeter. As our lips touched, it was like someone had cranked up all my senses. I could smell something burning in the distance and the leaves turning to fall and the almost imperceptibly salty smell wafting in from the river. I could hear the chirping of the crickets and the splashes of the water and a train whistle in the distance. And Sam's hand touching my cheek ignited every cell in my skin.\n\nWe kissed for a long time without saying anything.\n\nFinally, Sam pulled back a little.\n\n\"Wow,\" he said, his nose still just a couple of inches from mine.\n\nI smiled. \"Yeah,\" I agreed.\n\nSam kissed my forehead. \"Mind if we just sit here for a while?\"\n\n\"Nope,\" I said.\n\nTogether, we turned and looked out into the darkness.\n\n# chapter 16\n\nForty minutes later, we were pulling into my driveway. I didn't want the evening to end.\n\n\"I had a great time with you,\" I said.\n\n\"Me too,\" Sam said. He paused. \"Listen, Lacey, I need to tell you something.\"\n\nI turned to him. He looked worried. And all of a sudden, I realized that whatever it was, I didn't want to hear it tonight. I didn't want anything to ruin our perfect night.\n\n\"Tell me tomorrow,\" I said.\n\n\"But\u2014\"\n\n\"Is it that you have some girlfriend back in Taunton or something?\" I asked, trying to sound like I was making a joke. But I meant it, actually. It would be just my luck to have truly connected with the perfect guy and to find out he was keeping something huge hidden from me.\n\nSam laughed. \"No.\"\n\n\"Then can you tell me tomorrow?\"\n\nHe nodded. I looked out the window at my front lawn, which was covered in a blanket of autumn leaves in varying shades of orange, red, and yellow, illuminated under the shallow glow cast by the streetlights. They reminded me of the first three colors of Sam's rainbow.\n\n\"My dad used to rake the leaves and make a big pile for all of us to jump in every fall,\" I said. \"Even when we were too old for it. Even Logan and I would jump in.\"\n\n\"That sounds really nice,\" Sam said.\n\n\"Is it weird that I miss things like the leaf pile, instead of just missing my dad?\" I asked.\n\n\"It's not weird at all,\" Sam said. He leaned across the center console and touched his lips lightly to mine. We lingered there for a minute, our lips just barely touching. Finally, he pulled back and looked me right in the eye again.\n\n\"Lacey Mann,\" he said, \"you're pretty amazing.\"\n\nI smiled. \"You're not so bad yourself, Sam Stone.\"\n\nHe grinned. \"See you tomorrow, then?\"\n\n\"See you tomorrow.\"\n\nSam walked me to my front door and after one last, quick kiss, I said goodbye and turned the key quietly in the lock. I smiled once more at Sam before shutting the door behind me. As I tiptoed up the stairs, avoiding the steps that creaked, I thought about how one day could make such a difference in your life. A year ago, it had been losing Dad\u2014and my family\u2014in a morning, in the blink of an eye. But today, it had been finding something new with Sam that I had the feeling would last for a long time.\n\n\u2022 \u2022 \u2022\n\nAt school now, everything was different. Even though we'd never said the words, never officially made some sort of declaration of togetherness, Sam and I were a couple.\n\nIt was strange. I'd never had a boyfriend before. And I certainly wasn't used to people staring at me in the halls with jealousy in their eyes instead of pity. It was kind of nice to be the center of a different kind of attention.\n\nI figured at first that Logan and Sydney must have gossiped about me and Sam, but as the day went on I realized that we were creating our own waves. And although we had agreed on the disgustingness of Jennica-style PDA, Sam didn't seem embarrassed in the slightest to greet me in trig class with a peck on the cheek, like we'd been dating forever, or to walk me to my locker after class with my bookbag slung over his shoulder. He ate lunch with me, Jennica, and Brian and seemed completely oblivious to the stares from other tables.\n\n\"So is Sam, like, your boyfriend now?\" Jennica whispered as we went to throw out our trash. Brian and Sam were several paces behind us, talking about the Patriots game this weekend.\n\n\"I don't know,\" I said. \"I guess so. It's weird.\"\n\n\"That happened fast.\"\n\nThe words unsettled me, because I wasn't exactly sure what she meant. \"What do you mean?\"\n\nShe shrugged. \"Sometimes relationships that develop so quickly aren't really based on anything real.\"\n\nI wondered whether she was talking about me or about her dad and Leanne.\n\n\"But he seems to really like you,\" she added hastily, as if she'd just realized what she'd said and how it had sounded. \"I'm sure things are fine.\"\n\n\u2022 \u2022 \u2022\n\nThe next afternoon was supposed to be our second group meeting. I'd been so caught up with thinking about Sam over the weekend that I hadn't even sent out a reminder e-mail. I spent the whole day feeling guilty that I had dropped the ball. It was so unlike me.\n\nI waited in the parking lot after school, near the fence around the football stadium, wondering who would show up for a ride. Sam had offered to drive everyone last week, and he had reminded me in sixth period today. Logan hadn't said one way or the other whether he was coming, but I figured that if he did show up, he'd get a ride from Sydney. That left Mindy, Kelsi, and Cody.\n\nFive minutes after the final bell, I saw Sam striding out, his Red Sox cap and leather jacket on, as usual. His bag was slung over his shoulder, and he was grinning as he approached.\n\n\"Hey you,\" he said as he reached me. \"You ready to go?\"\n\nI fell into step beside him. \"We should probably wait and see if anyone else needs a ride.\"\n\n\"Sure thing,\" he said cheerfully as we reached his Jeep.\n\nSam started the engine, fiddled with the heat for a minute, and then pushed Play on his CD player. A song I recognized from my dad's CD collection started playing.\n\n\"You like Jimmy Buffett?\" I asked, surprised. I didn't know anyone else our age who did. I'd always liked \"Cheeseburger in Paradise\" and some of his other songs. My dad used to make goofy faces when he sang along.\n\nSam seemed equally surprised. \"Yeah,\" he said. \"You know Buffett?\"\n\nI nodded. \"My dad really liked him.\"\n\n\"Yeah?\" he said. \"My dad too.\" He smiled a little. \"He was actually a Parrothead. Official member of the Jimmy Buffett fan club.\"\n\nI laughed. \"Same with my dad!\"\n\n\"Did he go last time Jimmy played at Gillette Stadium?\"\n\n\"Yep.\"\n\n\"So did my dad,\" Sam said. \"Isn't it weird to think that they sat in the same stadium at the same show? And we hadn't even moved here yet?\"\n\n\"Yeah,\" I said. Actually, I thought, it seemed stranger to me that there existed a time, in the not-so-distant past, that my dad and Sam's dad had been out enjoying a rock concert, maybe just rows away from each other, with no idea that their days were numbered. It made me feel so suddenly sad that my throat closed up. I glanced at Sam, and the smile had fallen from his face too. I wondered if he was thinking the same thing.\n\nAfter a few minutes of waiting, the crowd of students flowing out from the school had slowed to a trickle, and the parking lot was nearly empty. Sam checked his watch. \"Think everyone found a ride?\"\n\nI nodded and took a deep breath, which I exhaled in a nervous laugh. \"Actually, I'm really worried that no one will show up at all.\"\n\n\"Why?\" Sam asked.\n\nI shrugged. \"I don't know. Maybe everyone thought last week was really stupid. I mean, maybe they thought about it later and realized they didn't want to hang out again with a bunch of sad people.\"\n\nSam seemed to think about this for a minute. \"No,\" he said firmly. \"I know it helped. And I know people felt good about it.\"\n\n\"Are you sure?\" I asked in a small voice.\n\n\"Yes,\" Sam said. We waited another few minutes, the silence hanging over us, then Sam shifted into drive. \"I guess we should go.\"\n\nI nodded, feeling discouraged. What if it _was_ just me and Sam? I'd feel like such a failure. And I'd look like an idiot.\n\nAs we drove, I glanced at him a few times out of the corner of my eye when I knew he was paying attention to the road. I liked how angular his face was from the side. Sharp nose, sharp chin. But he didn't seem sharp-featured when you looked straight at him. It was funny how different people could appear when you simply looked at them from different angles.\n\nWhen Sam and I pulled into the Lucky Strikes parking lot it was almost totally empty. There was a beat-up, dusty pickup truck I didn't recognize and a Cadillac with a dented front end. But I didn't see Kelsi's car. Or Sydney's. Even her snob-mobile would have been a relief at this point.\n\n\"No one's here,\" I murmured.\n\nSam glanced over at me as he shifted his Jeep into park and cut the motor. \"Lacey, it's still early,\" he said. \"Don't worry yet.\"\n\n\"What if no one shows?\" I asked.\n\n\"Then you and I will have a great time bowling together in really ugly shoes.\"\n\nWe got out of the Jeep and walked into the bowling alley. I had never been there before. We'd never been big on bowling in my family. Come to think of it, we weren't big on much of anything anymore.\n\nThe entrance area was dimly lit, while bright fluorescent lights illuminated the wood-paneled lanes. There were only three people bowling: a man and a woman together at a lane toward the far end of the room, and, midway down, a man in a FedEx uniform.\n\n\"I bet he's on his lunch break,\" Sam whispered, elbowing me gently in the side. I giggled as the FedEx guy bowled a strike and jumped up and down a few times in apparent glee.\n\nThe deep ping of the balls hitting pins punctuated the background music piped from various old-looking speakers around the room. There was a counter near the door with lots of bowling shoes lined up behind it, and another counter farther down with a couple of beer and soda taps, a popcorn machine, and a little warmer rotating some decidedly stale-looking hot dogs. I decided I wasn't hungry.\n\n\"Let me introduce you to my aunt,\" Sam said, reaching for my hand. A dark-haired woman was walking out from a door near the concession area. \"Donna!\" Sam called. She squinted toward the doorway and grinned.\n\n\"Hey, kiddo!\" she said. It sounded funny to hear Sam called kiddo. His aunt's enthusiasm was electric, though, and I could feel myself smiling at her even before she reached us. She was about five feet eight with cropped hair, a few freckles across the bridge of her nose, and clear green eyes that matched Sam's.\n\nShe reached us quickly and gave Sam a hug. Then she extended a hand to me. \"You must be Lacey,\" she said. \"Sam's told me about you.\"\n\nI blushed, wondering what he'd said\u2014and when. \"Nice to meet you. Thanks for letting us use your bowling alley.\"\n\n\"Of course!\" she said. She glanced around, then looked back at me. \"Where is everybody?\"\n\nSam answered before I could. \"They should be here in a few minutes. And if they don't show up, Lacey and I will bowl.\"\n\nDonna smiled at us again. \"Sounds fun! Help yourselves to bowling shoes. Sam, you know the drill,\" she said. \"Can I get you anything from the concession stand? A hot dog, maybe?\"\n\n\"No thanks,\" Sam and I both chorused immediately. We exchanged glances and tried not to laugh. Donna looked bewildered.\n\n\"Okay, then,\" she said. \"Have fun! I'll be back to check on you guys in a bit. Sam, you know where everything is when your other friends show up.\" She kissed him on the cheek. \"Nice to meet you, Lacey,\" she said before walking away.\n\n\"She seems nice,\" I said to Sam as we turned toward the wall of bowling shoes.\n\n\"She's the best. She and my uncle are a ton of fun. Our family Trivial Pursuit matches are pretty fierce.\"\n\nI suddenly wished that I had an aunt like that. Or another family member\u2014any family member\u2014who wasn't full of sympathetic looks. My uncle Paul and his wife, Sherry, came around from time to time, but Aunt Sherry was always casting sad glances my way, and Uncle Paul didn't seem to know how to talk to any of us anymore.\n\nSam led me over to the shoes and asked me for my size. A moment later, he pulled out a pair of pink and white shoes that were slightly scuffed at the toes. They were pretty silly-looking.\n\n\"Trust me,\" Sam said, reading my expression, \"no one looks good in bowling shoes.\"\n\nJust then, the front door opened, pouring a large sliver of bright sunshine into the bowling alley. I was relieved to see Kelsi and Mindy standing there, blinking into the darkness.\n\nAs Sam was helping Kelsi and Mindy pick out bowling shoes, Cody arrived, eyes downcast. As the five of us were heading toward the lanes, the door opened again, and Logan walked in, tailed a few steps behind by a sullen-looking Sydney.\n\nRelief washed over me again, along with an unfamiliar sense of gratitude for my brother, who didn't look at me as he walked quickly toward the shoe counter.\n\n\"Hey, man,\" Sam said to my brother as he walked over to the shoe counter. \"Hey, Syd.\"\n\nThey both nodded, but neither of them said anything. Typical. Too cool to talk to anyone. But, I had to give Logan credit for being here. And, I supposed, I had to grudgingly give Sydney a little credit too, even if she didn't belong here. But clearly, she and Logan were a package deal.\n\nAfter everyone was fitted, we headed to the lanes and picked two adjoining ones on the right. Donna came over and asked us if we wanted anything from the snack bar. Logan ordered two hot dogs. Sam and I shared amused looks, but neither of us said anything. I liked that we were on the inside of a private joke. The hot dogs arrived a few minutes later, along with a hot pretzel for Cody and three Cokes for the girls. I tried not to giggle as my brother gazed in horror at the shriveled-looking meat like it was something from another planet. It was even funnier when, unaware that he was being watched, he shrugged and bit into a hot dog anyway.\n\nSam, Kelsi, and I were in one lane, and Mindy, Cody, Sydney, and Logan took the lane beside us.\n\nNeither Kelsi nor I had bowled much before, so Sam took a few minutes to patiently explain technique. He showed us how to grip the ball, how to take steps forward to support our weight, and how to aim for the center pin by angling the ball in slightly from the side. He demonstrated three different styles for us and got strikes each time.\n\nOn Kelsi's first turn, she knocked over three pins with the first ball and another three pins with the second. Sam grinned and told her she was an excellent student. She blushed and sat down. I glanced at the lane next to us, where Logan and Sydney had each knocked over eight pins, and I wondered how my brother was so good at this. Maybe I would be too.\n\nI stood up, took a deep breath, and put my fingers into the purple, glittery ball I had picked out. I tested its weight and hesitantly carried it toward the lane. I took a deep breath, moved my arm back like Sam had shown me, and then took a step forward as I moved my arm forward and released the ball.\n\nIt dropped into the gutter almost immediately.\n\nI groaned as I watched it make its way down the edge of the lane.\n\n\"Nice gutter ball, Lacey,\" Sydney piped up, laughing. Sam rolled his eyes. \"It took me _forever_ to learn to bowl,\" he told me. \"Here.\" He got up and touched my arm lightly. \"I'll show you.\"\n\nI picked up my ball and we walked to the lane. \"Put your arm back,\" Sam said.\n\nI grasped the ball tightly and did what he said, stepping back with my left foot, like Sam had showed me. I was just about to ask him what to do next when I felt his warmth right behind me.\n\n\"Like this,\" he said softly, his breath grazing my ear as my cheeks flamed. He was so close that I could feel him, yet not close enough to be touching my back. Every single hair on my body was standing on end. He placed his right hand over mine, and I was so electrified by his touch that I nearly dropped the ball. Thankfully, I didn't.\n\n\"Okay, now,\" he said. \"Let's do this slowly once to practice. Don't let go.\"\n\nI knew he was talking about the bowling ball, but for a second, I wanted to tell him not to worry, that I would never let go. I glanced around and realized that everyone was watching us.\n\nSlowly, with his right hand resting on mine and his left hand gently clasping my left hand, Sam guided my right arm forward, urging me, \"Step forward with your left foot, like I showed you.\" He took the step with me, his much longer left leg shadowing my leg and his arm guiding mine forward.\n\n\"Just like that,\" he said in my ear. Then he cleared his throat loudly, glanced around, and stepped away, as if he'd just noticed that everyone was staring. \"Um, nice job, Lacey,\" he said. \"You're a natural. Now let's try it for real.\"\n\nI knew my face was on fire. I glanced around and quickly returned my attention to the ball and the lane in front of me. I closed my eyes for a minute and tried to center myself. Repeating Sam's words in my head, I stepped back, then stepped forward, slowly swinging my right arm in one fluid motion.\n\nI watched as the ball rolled slowly down the lane. It seemed to take forever, but this time, it didn't roll toward the gutter. In fact, it went straight down the middle. At the last second, it veered to the right and hit near the center. It knocked over seven pins!\n\nI leapt up, ecstatic. I knew it wasn't exactly a strike, but it felt pretty good to me.\n\nI wanted to see if everyone had noticed, but before I even had a chance to look around, Sam put his arms around me. And then, in front of everyone, he kissed me like it was the most normal thing in the world.\n\n\u2022 \u2022 \u2022\n\nTwo hours later, we had bowled two games, and I had gotten a little bit better. I even knocked over all ten pins on two tries. Still, I was by far the worst bowler in the group. I didn't care. What I cared about was that we _were_ a group. All of us, who had nothing in common except for the biggest thing, had talked and laughed and had fun.\n\nAfter we had changed back into our street shoes, we all thanked Donna and walked outside. The sun had gone down, and there were just a few remaining streaks of light in the sky to the west, the last remnants of the day. With the sunshine gone, a chill had set in, and none of us were dressed warmly enough. I shivered as we stood in the parking lot, looking at one another.\n\n\"That was fun,\" Cody said.\n\n\"Yeah,\" Mindy agreed.\n\n\"It _was_ fun,\" Kelsi said after a minute. \"But we didn't really talk about anything.\"\n\nShe was right. It just hadn't seemed like the time or the place to bring up sad stuff when, for once, we were having fun without thinking about it.\n\n\"Maybe we didn't need to,\" Cody said. \"Maybe it's cool to hang out with each other sometimes and not have to talk about it.\"\n\n\"Yeah,\" Kelsi said. \"All you guys want me to do is be myself. It's kind of nice.\"\n\nAfter exchanging goodbyes and saying we'd see one another in school tomorrow, we started drifting toward our cars.\n\n\"Need a ride?\" Sam asked, putting his hand lightly on my shoulder.\n\nSydney, seeing this and apparently deciding that she couldn't bear for someone else to be happy for a millisecond, jumped in. \"I'll drive her home. I'm going there anyhow. Obviously.\"\n\nSam and I exchanged glances, and I shrugged.\n\n\"Okay,\" Sam said. I'd been kind of hoping he'd insist on driving me. He kissed my cheek and walked over to his Jeep. I watched him go, feeling a lot warmer than I should have in the evening air.\n\n\"Let's go,\" Sydney said, clapping her hands together. \"It's cold out here.\"\n\nI followed her and Logan to her car and climbed into the back. Sydney started the engine, and as we sat there for a minute to let it warm up, she and Logan whispered something to each other. I ignored them. Finally, Sydney pulled out of the lot.\n\n\"So,\" she said, glancing at me in the rearview mirror, \"what's up with you and Sam?\"\n\nI knew it was coming. I took a deep breath. \"We're going out, I guess,\" I said. The words tasted sweet in my mouth. \"But then again, you knew that,\" I heard myself add. \"What are you talking about?\"\n\n\"Haven't you been gossiping about it all day?\" I asked her.\n\nSydney huffed indignantly. \"I don't gossip,\" she said. \"How dare you accuse me of that?\"\n\nI laughed. \"Right.\"\n\nSydney nudged my brother. \"Lacey, I don't want to burst your bubble or anything, but it can't last, you know.\"\n\n\"What?\" I looked at her face in the rearview mirror. I had no idea what she was talking about.\n\n\"I mean, it's just not a logical match,\" she said.\n\n\"What on earth do you mean?\" I asked.\n\n\"Well,\" Sydney said slowly, like she was talking to a child, \"you have to admit, it's not like you have anything in common. Other than your dead fathers.\"\n\nThe way Sydney said the words sliced into me.\n\n\"You barely know him,\" I said after a minute. \"How could you possibly say that?\"\n\n\"Think about it,\" she continued. \"You're brainy. He's hot. And you're up against Summer Andrews. I mean, do you really think he's going to choose you over her in the long term?\"\n\nI had just opened my mouth to reply when my brother spoke up. \"Syd, leave her alone.\"\n\nDead silence. I was as taken aback as Sydney was. Logan never came to my defense. Not anymore. And certainly not against Sydney.\n\nSydney sputtered for a second. I knew she was flailing for a retort.\n\nLogan sighed again. \"We'll talk about it later, Sydney,\" he said with more finality than I'd ever heard in his voice.\n\n\"But\u2014\"\n\n\"Not now,\" Logan said. Then he turned and looked out the window, effectively ending the conversation.\n\nI turned and looked out the window too, biting my lip and trying not to smile. I didn't know what had just happened, but somehow, Logan seemed to be back on my side, even if only a little bit.\n\n\u2022 \u2022 \u2022\n\nEven though I knew she was just being a jerk, I couldn't shake Sydney's words. They had wormed their way into my brain, making me wonder if I was being na\u00efve to believe that what Sam and I had was real. Maybe Sydney was right. I didn't want to think that the only reason Sam liked me was because he saw a reflection of his own pain. But maybe that was all it was.\n\nMom actually made it home for dinner that night, so the four of us sat down to a meal that she had \"cooked\"\u2014spaghetti with sauce out of a jar and a bagged salad.\n\n\"This is nice,\" Mom said as we chewed in silence. Tanner slurped a noodle noisily and looked up, the faintest trace of a smile on his face. \"You know, we hardly ever eat together anymore.\"\n\n\"Maybe it's because you're never home,\" Logan said.\n\nMom sighed heavily. \"Logan, someone has to support this family. You know I'd love to spend more time with you. It just isn't able to work out that way right now.\"\n\nLogan was silent for so long that I thought he was going to let it go. And for a moment, I was very relieved. Dinners together were so rare that I didn't want this one to be spoiled by a fight. But then Logan said slowly, \"That's bullshit.\"\n\nMom flinched, like she'd been struck. \"What did you say?\"\n\n\"I said it's bullshit,\" Logan said.\n\nTanner and I exchanged glances. \"Logan,\" I said as my mom gaped at him. \"I really don't think\u2013\"\n\n\"Shut up, Lacey,\" he cut me off. \"I'm so sick and tired of everyone tiptoeing around the truth.\"\n\n\"Young man,\" my mother began. But her voice was shaky and lacked conviction.\n\n\"Don't 'young man' me,\" my brother snapped. \"You don't have the right anymore.\"\n\n\"I'm your mother,\" she said.\n\nLogan shook his head. \"My mother disappeared last November.\"\n\nI hated the way he was hurting my mom, but despite myself, I agreed with him. I wanted to defend her, but I couldn't. I held my breath.\n\n\"You know, you can't run from it, Mom,\" he said. \"Dad's dead, okay? Dead, dead, dead.\"\n\n\"Logan!\" she exclaimed. There were tears welling in her eyes.\n\n\"You keep acting like if you just work enough, if you just avoid your family, if you go out and have fun and play tennis and keep your hair perfect and your clothes ironed, it will all go away,\" he went on. \"But it won't. You're just lying to yourself. You can't make everything perfect, because it's not. Dad is dead, Mom. He _died_. I watched him, okay? I watched him die. You can't pretend.\"\n\nMy mom was crying and I felt a tightening in my own chest. There were tears in Tanner's eyes too, and he made a little choked sound before setting down his fork.\n\nLogan wasn't done. \"You don't even act like a mom anymore,\" he continued. \"You remember what it used to be like? Huh? Do you? You used to ask us about school. You used to joke around. You used to be fun. You used to care. But now all you care about is forgetting. You think that's how Dad would want you to be?\"\n\nMy mother was sobbing full force now, but Logan didn't seem to notice. His face was red, and his hands were clenched into fists, like he was waiting to defend himself against some unexpected attack.\n\n\"Logan,\" I began.\n\n\"And you!\" he exclaimed, turning on me. \"You think that by being Little Miss Perfect, you can fix everything,\" he accused. \"Well, you're as stupid as she is!\" He nodded in Mom's direction. \"You don't even have a clue. You think you're so much better than me just because you make straight As and you take care of everyone and you never cry. But you know what? That's really screwed up.\"\n\n\"What?\" I choked out.\n\n\"You're such a phony,\" he spat.\n\n\"Shut up,\" I whispered. \"You don't know what you're talking about.\"\n\n\"Yeah, well, what are you going to tell me?\" he mocked. \"I mean, you seem to have all the answers, right?\"\n\n\"I never said I had all the answers.\"\n\nLogan snorted. \"You didn't have to.\" He pulled his napkin off his lap, balled it up, and tossed it onto the table. \"Thanks for the great dinner. This has been fun.\"\n\nHe got up and strode away without another word. We all watched him go, shocked into silence. Then we all looked at one another.\n\nTanner was the first to move. As he stood up from the table, he knocked his milk glass over with his elbow. It crashed onto the floor, shattering into a hundred little pieces. His eyes filled. \"Sorry,\" he whispered.\n\n\"It's okay, honey,\" my mom said, her voice pinched. \"I'll take care of it.\" Tanner darted out of the room. I could hear his footsteps on the stairs, then the slamming of his bedroom door.\n\nSilence settled over us. My mother and I looked at each other, then down at the floor, where the shards of shattered glass reflected the light.\n\n\"We'd better clean that up,\" my mother said. But she didn't move. She just kept staring at the glass, like she was wondering whether it would really be possible to ever pick up all the pieces.\n\n# chapter 17\n\nI told Sam about the fight the next day, and he said that sometimes people don't think before they speak, and that Logan probably hadn't meant the things he said.\n\n\"But he did mean it,\" I said as we sat across from each other at McDonald's after school, sharing a large chocolate milk shake in alternate slurps. \"And the thing was, he was right.\"\n\n\"About what?\" Sam asked.\n\n\"About everything,\" I admitted. \"I mean, all the things he said about my mom were the things I've been thinking. Maybe he was right about me, too.\"\n\n\"Or maybe Logan was just telling you the way he sees things,\" Sam said, \"which doesn't necessarily make it right.\"\n\nOn Friday night, he and I went out with Brian and Jennica to the movies, and as we sat in the darkened theater, with our fingers intertwined, I thought how nice it was not to feel like a third wheel for once. I hated that I needed another person to make me feel like I belonged. But if I had to have someone at my side to help me fit, I was glad it was Sam.\n\nOn Saturday, Sam had practice for a soccer league he'd joined in town. He asked me if I wanted to come sit in the park with a few of the other girlfriends while he kicked the ball around with the guys, and I agreed instantly. It wasn't that I wanted to spend every waking second with him or anything. It was that I was avoiding my house. It was even more silent than usual, which was weird, because Mom was actually home. Logan's words had evidently penetrated; she had come home every night before seven, and she canceled her Saturday tennis plans to catch up on some housework. That was a first.\n\nSunday, Jennica and I went to the mall and then got sundaes at Brigham's. We talked about guys, and for the first time in ages, I had something to contribute. I told her what Sydney had said earlier in the week, and she reassured me that Sydney was just jealous and mean. I knew this, but even with Jennica's words of comfort, I still couldn't shake the feeling that something was wrong.\n\nSam didn't call on Sunday night, which was weird, because he had started calling me every night so that we could wish each other sweet dreams. But I tried not to read into it; he was probably just busy.\n\nOn Monday morning, though, he wasn't in first period. As the final bell rang, and his seat remained ominously empty, a funny feeling settled over me.\n\n\"Where is he?\" Jennica mouthed as she glanced at Sam's seat, then at my confused face. Mrs. Bost had already started class and was babbling something about vectors, but I couldn't seem to tune in.\n\n\"I don't know,\" I mouthed back.\n\nMy phone vibrated a moment later, and I snuck a look. Jennica had texted im sure theres an explanation. I nodded and looked away, trying to focus on Mrs. Bost. But I had a feeling that something was wrong.\n\nAt lunchtime, I snuck outside to call Sam, but his voice mail picked up on the first ring. We weren't supposed to talk on our cells at school, but I left him a quick message asking him to call me when he could.\n\nHe wasn't in sixth period either, and he hadn't called back. I tried him again as I was getting my books out of my locker after school, but his phone still went straight to voice mail. I was so busy agonizing over the reasons behind his absence that I didn't even notice Cody approach until he was right in front of me.\n\n\"Hey,\" he said. His hands were jammed in his pockets, and he looked nervous.\n\nI stopped at my locker and looked at him. \"Hey.\"\n\n\"So, um,\" he began. He coughed and looked down. \"Are we on for tomorrow?\"\n\n\"Tomorrow?\" I asked.\n\n\"Well, it's Tuesday, isn't it?\" he said. \"Are we still having a meeting?\"\n\n\"Yeah,\" I said, feeling good for the first time today. \"What do you want to do?\"\n\nCody shrugged. \"I dunno. I'm supposed to watch my sister, Sarah, tomorrow afternoon. She's ten. Think we could go somewhere where I could bring her?\"\n\nI nodded. \"Sure. I'll talk to everybody.\"\n\nBy that evening, I had gotten in touch with Kelsi and Mindy and we had agreed to meet at the ice rink at Plymouth Center.\n\nI told Logan, who reluctantly agreed to try to come, even though we were barely talking to each other, and I asked my mom if I could take Tanner, too, since Cody's sister was about his age. I thought it might be good for my little brother.\n\n\"Whatever you want,\" my mom said with a shrug. I wasn't even one hundred percent sure she heard me.\n\nThat evening, I called Sam once more and left another message. And then, because I didn't know whether I should be hurt or worried, I went onto our local newspaper's Web site to check for traffic accidents. There hadn't been anything serious enough to be covered. Sam wasn't lying by the side of the road somewhere. Just to be sure, I searched the site for his name, but nothing came up.\n\nI swallowed my pride and sent Sam an e-mail, telling him that the group was meeting the next day at four at the ice rink if he wanted to come.\n\nThe next morning, I checked my e-mail as soon as I got up, but there was no reply from Sam. He wasn't in school all day either. I couldn't figure out what was wrong or why he couldn't call me back. Why was he avoiding me?\n\nAfter school, Sydney drove me and Logan home, and we picked up Tanner to take him to the ice rink. Cody, Kelsi, and Mindy were already waiting for us when we got there. Cody's little sister Sarah turned out to be a tiny girl with long, frizzy hair. She talked a mile a minute and sounded like a miniature adult.\n\n\"Hi, you must be Lacey,\" she began rapid-fire, without taking a breath. \"I'm Sarah and my brother told me about you and I love ice skating, so he thought I'd want to come along, and my dad died, but it's not like we have to just talk about that, because there are lots of other interesting things we can talk about too, like ice skating or school or sports or something, and is that your brother over there?\"\n\nShe finally paused for breath and cocked her head inquisitively.\n\nI followed her eyes to Tanner, who was hanging back from the crowd. He had brought his own knee pads and helmet and didn't seem the slightest bit worried about appearing dorky. The only thing I'd been able to get out of him when I asked why he'd come prepared was, \"Dad always told me better safe than sorry.\"\n\nI nodded at Sarah. \"Yeah, his name's Tanner,\" I said. \"He doesn't talk much, though.\"\n\nShe nodded wisely. \"Some kids don't talk much after their parents have died,\" she said. \"But I've always talked, and when my dad died, I just started talking some more, and now I talk all the time, and I think it drives my brother and my mom crazy, but I can't really help it, you know, and maybe if I talk enough to your brother, maybe he'll talk back to me, and we can be friends, even though he's a year older than me, but we go to the same school. I see him on the playground at lunch and he's usually by himself, even though kids like him, but he's really quiet, and maybe we can hang out sometime.\"\n\nI blinked at Sarah a few times, trying to keep up with the words pouring out of her mouth. \"Um, yeah,\" I said. \"That sounds good.\" I glanced over at Tanner, who was carefully pulling on his bright blue knee pads. I felt sad for him. I looked back at Sarah. \"I think he could use a friend.\"\n\nI expected another torrent of words, but instead she just said, \"Me too.\"\n\nI watched as Sarah went over to Tanner and said something to him. He looked at her blankly, then nodded. She launched into another long-winded sentence, which I couldn't hear, and when she finally paused for breath, I watched as Tanner searched her face for what felt like an eternity and then finally broke into a hesitant grin. I was startled; I hadn't seen him smile in a while.\n\nFeeling relieved, I went to pick up skates for Tanner and me at the counter.\n\nTen minutes later, all of us were out on the ice.\n\n\"Where's Sam?\" Kelsi asked as we inched along, trying to get our balance.\n\n\"I don't know,\" I said.\n\n\"You told him about the meeting?\" Mindy asked in a soft voice.\n\n\"Yeah,\" I said. \"I mean, I left him a message. He hasn't called back.\"\n\nI looked up, expecting to see judgment or pity on their faces. After all, they knew Sam and I were going out, and now he wasn't even replying to me. But they only looked concerned. \"Well, it's not really the same without all of us here,\" Kelsi said. \"When you talk to him, tell him we missed him today, okay?\"\n\nKelsi and Mindy partnered up after a few minutes, and, holding each other's hands and giggling, they picked their way around the rink. I stopped and just watched them. Before I'd put this group together, they'd hardly known each other. And here they were, laughing on a Tuesday afternoon, just weeks after Kelsi's mom had died, when she might otherwise have been at home, wallowing in grief.\n\nLogan and Sydney made their way a little more quickly. It looked like Sydney was leading Logan, who was a bit slower, dragging him by the hand and chiding him when he couldn't keep up. Still, he appeared content.\n\nCody was off in his own world, whizzing across the ice like he was on the Olympic speed skating team. Each time he passed, his cheeks were flushed, and his eyes were focused straight ahead. I wondered if he was doing more than exercising; it looked like he might have been getting something out of his system.\n\nI skated alone, and it gave me time to think about Sam\u2014and about my dad. The anniversary of the accident was fast approaching, and it seemed like I should be in a different place. I knew that what I had done with setting up this club was good; it seemed to be helping. And I knew that was something I had to do: help other people come to terms with a parent's death, like I had. But still, the emptiness loomed inside of me, big and cold. I'd never felt so lonely.\n\nWhile the rink was cleared temporarily for the ice to be Zambonied, I sat down and closed my eyes. No one was paying attention to me. Kelsi and Mindy were talking about some sophomore guy Mindy liked. Logan and Sydney were cooing at each other in the corner\u2014I was afraid they were going to start making out any minute. Cody had disappeared to the other side of the rink, where he apparently knew three girls in skating outfits.\n\nAnd for the first time in ages, I heard Tanner's voice loud and clear for more than a few words at a stretch as he talked to Sarah.\n\n\"What happened?\" I heard him ask. I strained to hear, feeling a little bad that I was eavesdropping.\n\n\"My dad was in the military, in Iraq, you know, which is really far away, and we couldn't see him very often because his job was dangerous and he had to be gone for a long time,\" Sarah was saying, speaking at the speed of light. I glanced over and was surprised to see my little brother staring at her with rapt interest. \"I was always scared that something would happen to my dad, because I heard about Army guys getting hurt, and he always told me not to worry because he'd be here forever, so I tried not to worry. But he was supposed to come home on March sixteenth, and it was March ninth, and I was really excited and I was making him a big picture of our house so he could see what everything looked like while he was gone, and I was outside doing the drawing, and two military men pulled up in the driveway.\"\n\n\"Military men?\" Tanner repeated.\n\n\"Yeah, they were wearing really fancy uniforms with lots of ribbons and stuff, and I think they were really important,\" Sarah babbled on. \"And they called me 'little girl,' even though I'm not that little, and they looked really serious, and they asked if my mom was home, and I said yeah, and I asked what they wanted, and they wouldn't tell me, but I had a bad feeling about it, so I ran and got my mom, and they whispered something to her and then she just started crying. I didn't know what to do because I'd never seen her cry before, and she fell down on the driveway, and they didn't know what to do either; they just stood there looking down at her and saying it would be okay, and they could help her.\"\n\n\"Oh,\" Tanner said softly.\n\nSarah went on. \"I went and got Cody, I don't even remember what I said, but I think I was screaming really loud, and he came out of the house and bent over and hugged my mom, and he asked the military men what was wrong, and they told him that my dad had died, I heard them tell him that, and then he started crying too, and I started screaming again, because I didn't know what else to do, and I wanted to ask them if my dad was in heaven, and I wanted to ask my mom that too, but she was crying, and the military men looked mean, and I didn't know what they'd say, and besides, they were treating me like a baby.\"\n\nTanner was quiet for a minute, and my heart sank for Sarah. She had opened up to him the way I had to Sam, and my brother wasn't going to answer her, simply because he couldn't.\n\nBut then, my brother spoke, which surprised me so much that I nearly fell over. \"I believe in heaven,\" he said quietly. \"I know your dad has to be there. Because he was doing the right thing when he died.\"\n\nI heard Sarah sniffle a little. \"You think?\" she ventured.\n\nI was afraid Tanner wouldn't answer. Then he said,\n\n\"Yeah.\"\n\nThey sat there in silence for a moment, and when I was pretty sure that the conversation was finally over, I peeked my head around a chair to take a look at them. Perhaps Tanner, having spoken his entire word quota for the past three months, had gotten up and left, or spontaneously combusted from the mental exertion.\n\nInstead, they were sitting side by side in companionable silence, staring out at the Zamboni making its slow loops around the rink, smoothing the surface of the roughed-up ice. I waited for Sarah to say something more\u2014affter all, she seemed to be overflowing with words\u2014but she didn't seem to have anything else to say. After a minute, she put her head on my brother's shoulder. He paused and then put an arm around her shoulders. From the back, they looked like miniature adults. I could hardly believe it was my little brother, the one who hid in his room, watching TV and obsessing over animals.\n\nMaybe, I thought, he was better off than I'd given him credit for. Maybe he'd get better with or without my help and concern. Maybe I was wasting my time fearing for his mental health.\n\nMaybe he didn't need me at all.\n\n# chapter 18\n\nBy Friday, Sam still hadn't shown up at school. He wasn't calling me back either, and I was really worried. There was still a part of me that was scared it had to do with me, but I reassured myself that no one in his right mind would skip school for five days because of a girl. I'd e-mailed him twice more, but I'd gotten no reply. I was starting to feel like a stalker.\n\nAfter school, I caught a ride with Jennica and asked if she'd mind dropping me off at Lucky Strikes. \"Sure, but I can't stay and drive you home,\" she said. \"Anne has dance practice, and I have to take her. My mom's having a spa day, so I'm stuck babysitting.\" She rolled her eyes for emphasis. \"Why are you going there anyhow?\"\n\n\"Just meeting the group,\" I lied. I knew I should tell her that I was looking for Sam, but I didn't want to feel any more pathetic than I already did.\n\nFive minutes later, I was standing in front of Lucky Strikes, staring at the door and wondering if this was stupid. I took a deep breath and walked in.\n\nIt took a moment for my eyes to adjust to the darkness inside the alley, but when they did, I spotted Donna seated behind the cash register, reading a paperback James Patterson novel. She was so absorbed in the book that she didn't even look up until I was standing right in front of her. I had to clear my throat to get her attention.\n\n\"Oh, Lacey!\" she exclaimed. She peered at me. \"Hi! How are you?\"\n\nI shrugged, suddenly feeling embarrassed to be there.\n\n\"I'm okay,\" I said. \"Um, I was just wondering if Sam is around.\"\n\nI felt stupid the moment the words were out of my mouth.\n\nDonna looked confused. \"Sam?\" she said. \"No, Lacey. He's at the hospital.\"\n\nMy heart caught in my throat. \"The hospital?\" I croaked. \"What happened? Is he okay?\"\n\nGuilt flooded through me.\n\nDonna was looking at me more closely now. \"You don't know?\" she asked.\n\n\"Know what?\" I demanded. I felt like I was on the verge of panicking.\n\nShe studied me while my heart pounded double time. It looked like she was trying to decide whether to tell me or not.\n\n\"Please, just tell me if Sam's okay,\" I pleaded. I didn't think I could handle it if something happened to someone else I cared about. In an instant, all the awful things that could have happened to Sam flashed through my head. And for some reason, my mind got immediately stuck on Sam in a car crash. A cold chill ran through me.\n\n\"Sam's fine,\" she said.\n\nRelief flooded through me, followed quickly by conffusion. \"Why's he at the hospital, then?\"\n\nShe put down her book. \"It's his dad, Lacey.\"\n\n\"His dad?\" I repeated. What was she talking about? Hadn't he died months ago?\n\n\"He woke up,\" Donna said softly.\n\nMy jaw dropped. \"Woke up? But... he's dead.\"\n\nNow it was Donna's turn to look confused. \"Dead?\" she repeated. \"Where did you get that idea?\"\n\nA feeling began to creep through my veins like ice. Every conversation I'd ever had with Sam began to replay itself in my mind. Is this what he had wanted to tell me? But even if it was, how could he let me go on believing something so huge when it had been a lie all along?\n\n\"He's not dead?\" I whispered. Donna shook her head. \"But Sam said he had a stroke.\"\n\nDonna nodded. \"He did. He's been in a coma since July. They moved him to Plymouth Regional Hospital in September. That's why Sam and his mom moved here.\"\n\nI stared at her in disbelief. She must have thought I was totally crazy, but I couldn't help repeating, \"You're telling me he's alive?\"\n\n\"Yes. And he woke up on Sunday night. The doctors are calling it a miracle. Sam and Joey and their mom have been at his bedside since then.\"\n\nI stared at her. I couldn't form words. I couldn't think of anything to say.\n\n\"Lacey?\" she asked. Her face radiated concern. \"Are you all right, honey?\"\n\n\"Um\" was all I could manage. I shook my head. \"I'm sorry. Thank you.\"\n\nI felt like the walls were closing in on me. I slowly backed away from her and out of the bowling alley. It wasn't until I was outside, in the crisp fall afternoon air, that I realized I didn't have a ride home. Numbly, feeling like the wind had been knocked out of me and I couldn't quite catch my breath, I began walking toward my house.\n\n\u2022 \u2022 \u2022\n\nBy the time I walked up my own driveway I didn't feel any better. I knew I should be happy for Sam that he'd gotten his father back. Wouldn't I have given anything in the world to hear the same kind of news about my dad? But the fact was, I never would hear that news; my father was gone for good.\n\nI'd believed the same about Sam's dad. Sam had _made_ me believe the same about his dad. And that's why I'd trusted him with my feelings, my secrets. That's why I'd believed, in the very depth of my soul, that he understood me. But the truth was, he didn't know any more about how I felt than Jennica or Dr. Schiff or any of the kids at school who lived in their perfect homes with their perfectly complete families.\n\nI began to replay in my head every conversation I'd ever had with Sam. He'd never directly lied, I couldn't actually remember the words \"My dad died\" coming out of his mouth. But from the day he showed up at our first meeting, saying that he'd lost his dad, I had trusted him and had assumed that he'd meant his dad was no longer alive. Why would I think anything different? But just because he hadn't blatantly lied didn't make the betrayal any less serious. He knew what he'd led us to believe. He knew what he'd led _me_ to believe. And it hadn't mattered.\n\nWhy had he done it? Had he been that desperate to fit in with us? Sam didn't seem to care about being popular, and it wasn't even like we were a popular group. Besides, who in their right minds would fake a parental death to become part of a clique? No, it went deeper than that. I had no doubt that losing your dad to a coma was really hard. And I was sure that to some extent, Sam _had_ understood us and identified with us. But the fundamental difference was that _his dad had woken up_. Mindy, Kelsi, Cody, Logan, and I would never have that experience. We couldn't. And for Sam to think it was okay to trick us in this way made me feel sick.\n\nThe sky darkened as I walked home, and as I reached my front door, the first fat raindrops of an approaching storm began to fall, splashing on the driveway and pinging off the roof of the house. I put my key in the lock and closed my eyes before turning it, steadying myself.\n\nI would never let anyone in again. I couldn't trust anyone. The world around me had crumbled, and once again, it was still me, only me, standing there on my own.\n\nI should have known better.\n\n\u2022 \u2022 \u2022\n\nMom was home from work early again, standing in the kitchen and absentmindedly beating something in a big bowl, when I walked in.\n\nShe smiled. \"Hi, honey.\"\n\nI raised a hand to wave without a word. I didn't feel like talking to her. Or anyone else. Sam's betrayal had been the final straw.\n\n\"Sam called,\" Mom said. She wiped her hands on her apron and crossed over to the notepad that we kept by the kitchen phone. \"He left a number and an extension and asked that you call him back as soon as possible. He said he tried your cell, but it went straight to voice mail.\"\n\nI gazed at her in disbelief. _Now_ he had called? After I'd been trying to reach him all week? Donna had probably called him and told him what had happened.\n\n\"Lacey?\" Mom asked. \"Don't you want the number?\"\n\nI glanced at the pad of paper and then back at her. \"No.\"\n\nShe shrugged. \"Well. I'm just about to put a souffl\u00e9 in the oven,\" she said, turning away from me and returning to the hand beater. \"I thought I'd try something new for dinner.\"\n\n\"Really?\" I asked, surprised. Mom used to love to cook\u2014she subscribed to _Bon App\u00e9tit_ and _Food & Wine_ and a few other cooking magazines\u2014and before the accident, she would try something new and exotic at least once a week.\n\n\"Yeah,\" she said. \"I think I need to stop.\"\n\n\"Stop what?\" I asked.\n\nShe looked down at the bowl. \"Stop wallowing. Logan was right the other day.\"\n\nThe rain had started to fall harder now, and the fat droplets had given way to an insistent waterfall that made it look fuzzy and almost dreamlike outside.\n\n\"I've been awful,\" she added, gazing out the kitchen window. \"I've really failed you kids.\"\n\n\"No, you haven't,\" I said. It felt like the right thing to say, but I realized, after the words were out of my mouth, that I meant it.\n\n\"Yes, I have,\" she said. She took a deep breath and let it out slowly. \"It's been almost a year, Lacey. A year of our lives that I've lost. Your father wouldn't have wanted it this way.\"\n\n\"I think he would have understood, Mom.\"\n\n\"Understood what?\"\n\n\"Understood that we all needed to figure out things in our own time.\"\n\nMom blinked a few times. \"Maybe it's time to start living again.\"\n\nAs I walked slowly out of the kitchen, I thought about the last thing she had said. At least we _had_ the luxury to start living again. Logan and Tanner and Mom and I, Sam and his mom, even Sam's dad, could start over at any time. It made me even sadder to think about it in those terms. Because it seemed unfair, like a betrayal of Dad, to be able to just reinvent ourselves any day, didn't it? We'd all have a thousand\u2014a million\u2014second chances. Dad wouldn't even have one.\n\n\u2022 \u2022 \u2022\n\nWhen I checked my e-mail before going to bed that night, there was one new message waiting for me in my in-box. It was from Sam.\n\nLacey,\n\nI don't even know how to begin. I know you feel like I lied to you. And I don't blame you. But I didn't do it on purpose. I heard about the group you were starting, and I didn't know until the end of the first meeting that it was only for people whose parents had actually died. But by that time, I felt so much better just being there. I know you might not understand, but it felt like my dad had died, just like yours. He wasn't supposed to wake up, ever, and in a way, I felt sometimes like it would have been better if he did just die, because then we could at least have a funeral and say a real goodbye and everything.\n\nI was with my dad when he had the stroke. The doctors said he would never regain consciousness. And then, Sunday night, we got a call from the hospital. The nurse on duty had noticed the call light from his room was on. She went in to check on him, figuring it was a mistake, and he was sitting up in bed, looking confused. He didn't know where he was. They called the doc and then they called us. My mom hasn't wanted to leave his side since then. We've been sleeping at the hospital. She keeps saying it's our second chance.\n\nI know you're mad at me. I tried to tell you, but I guess I didn't try hard enough. I was scared about how you'd react. I thought you wouldn't believe anymore that I knew how you felt. But I do. I'm sorry. I can't even tell you how sorry I am. But I never meant to hurt you. And it doesn't take away the fact that I do understand you. Please call me.\n\nSam\n\nI read the e-mail three times before closing the screen. My finger hovered over the Delete key, but finally, I hit Save instead.\n\nI understood what he was saying. But that didn't make his actions easier to understand. Or to forgive.\n\n# chapter 19\n\nI tried to talk to Dr. Schiff about Sam on Saturday during my half-hour session with her. She told me I needed to stop holding other people to an unrealistic standard. I'd asked her what was so unrealistic about expecting someone to be honest. I called Jennica and filled her in on everything, and she was totally sympathetic. \"I'm beginning to think that all guys are more trouble than they're worth,\" she told me. I wasn't sure I agreed with her, though. No matter how mad I was at Sam.\n\nTanner had Cody's sister, Sarah, over on Sunday. They watched TV and played video games, and I could hear them laughing. I felt a strange blend of relief, pride, and jealousy. Relief, because it meant there was hope for Tanner. Pride, because if I hadn't started the group that included Cody, Tanner wouldn't have met Sarah. And, embarrassingly, jealousy, because Tanner was learning how to cope while I seemed to be getting more and more lost by the day.\n\nOn Monday morning, I walked into trig class to find Sam waiting by my seat.\n\n\"Hey, Lacey,\" he said, like we were the only two people in the room.\n\n\"Hey,\" I mumbled, both wanting and not wanting to see him.\n\n\"Lacey,\" Sam said, putting his hand on mine. I bit my lower lip and moved my hand away. \"I don't want to talk to you.\"\n\n\"Look, I'm sorry,\" he said. \"I really am. Did you get my e-mail?\"\n\n\"Yeah,\" I said. I paused. \"And I'm glad your dad woke up.\" I really was, and I wanted him to know that.\n\n\"Thanks,\" he said. \"And Lacey, for what it's worth, I'm sorry.\"\n\n\"You lied to us,\" I whispered. \"You lied to me.\"\n\n\"I never lied,\" he said, shaking his head. \"I just\u2014I just didn't correct the misunderstanding.\"\n\n\u2022 \u2022 \u2022\n\nAfter school I walked home by myself. I didn't know where Logan was, and Jennica had to stay after school to work on a history group project.\n\nThe sun was low in the sky. The days were getting shorter and the nights longer, but that was okay; I liked the darkness. It was only four in the afternoon, but the first colors of sunset were starting to gather on the horizon.\n\nI was so lost in thought five minutes later that it barely registered when a vehicle slowed beside me.\n\n\"Lacey?\"\n\nIt was Sam in his Cherokee, his window rolled down. \"Lacey, get in,\" he said. \"It's cold out.\"\n\nI shook my head, not stopping. \"I'm fine.\" But Sam kept inching his Jeep along.\n\n\"I'll follow you the whole way home if you want,\" Sam said. \"But wouldn't it be easier to just get in? It's not getting any warmer.\"\n\nI snorted and quickened my pace. \"I like to walk.\"\n\nBut Sam was right. I only had a denim jacket on, and the cold was starting to seep into my bones. It was another fifteen minutes home. I'd be fine, but the heated interior of a car was admittedly tempting.\n\n\"Please, Lacey? Just give me a chance to talk to you for five minutes.\"\n\nI hesitated, watching my warm breath crystallize into little white clouds. Finally, I got in.\n\n\"Thanks,\" Sam said. He glanced in the rearview mirror as I buckled my seat belt. Then he pulled slowly away from the curb.\n\nWe didn't say anything for a little while. Then Sam said, \"Look, Lacey. I'm sorry.\"\n\nI shrugged and looked out the window. The oranges and pinks to the west were inching farther up the sky as the horizon began to tug the curtain down on the day.\n\n\"I'm glad for you,\" I said. \"I'm glad your dad is fine.\"\n\n\"No you're not,\" Sam said. His words sliced into me, and I turned to look at him.\n\n\"I am,\" I said. \"Really. I would give anything in the world to have my dad back. And I'm glad that's happening to you. But the thing is, you tricked me. You made me feel like you understood me.\"\n\n\"I _do_ understand.\"\n\nMy breath felt heavy, and the air around me seemed suddenly in short supply. I gazed at the sky again and thought about what Sam had said about rainbows. It had all been just words. \"You _can't_ understand!\" I said. My eyes felt dry, and I blinked a few times, trying to get the burning sensation to go away. \"Your dad is alive, Sam! You have another chance with him. You can talk to him and tell him about your day and tell him you love him. Even when he was in a coma, you could say all those things to him, and there was a chance he could hear you.\"\n\n\"Lacey, don't you think your dad can hear you too?\" he asked.\n\nI rolled my eyes. His words that night about rainbows and my dad looking over us just sounded ridiculous now. \"No,\" I said. \"And I think you're pretty much the last person who should be saying something like that to me.\"\n\nWe had pulled into my neighborhood. I was silent as Sam parked his Jeep alongside the curb in front of my house. I glanced at him and was surprised to see how wounded he looked. I suddenly felt a little bad.\n\n\"Is he doing okay?\" I asked. \"Your dad, I mean?\"\n\nSam nodded. \"It's hard to watch him,\" he said. \"He can't move the right side of his face. He talks funny, and he can't remember a lot of words.\"\n\n\"But he's alive,\" I couldn't help but add.\n\n\"Yeah.\"\n\nThen, before he had a chance to say anything else, I climbed out of the Jeep and slammed the door behind me. I could feel Sam watching me the whole way to the house. I had to stop myself from looking back at the street when I let myself in the front door.\n\n\u2022 \u2022 \u2022\n\nThat night, Sam sent an e-mail to everyone in the group.\n\nWhen I came to the first meeting, I didn't realize right away that it was supposed to only be for people whose parents had died. By the time I realized, I didn't know how to tell you guys. I felt really good around you; it feels weird to have a parent in a coma too, and we didn't think he was going to wake up, so I felt like I'd lost my dad too. I didn't mean to trick anyone, and I'm really sorry if anyone feels that way. You guys really helped me, and I would love to keep spending time with you if you'll have me. Cody wrote back an e-mail, copied to the rest of us:\n\nGlad your dad's okay. You don't have to apologize to us.\n\nNo one else responded\u2014or if they did, they didn't CC everyone. I wondered how Cody could act so forgiving. Did Mindy and Kelsi feel the same way I did? Or was I the only one who was upset?\n\nBut the thing was, I was the one who had opened myself up.\n\nI was the one who got hurt.\n\n# chapter 20\n\nThe next two weeks passed quickly. Sam was absent from school pretty often, and when he was there, I avoided him. Soon he stopped trying so hard to talk to me or to get me to forgive him. I think he knew it wasn't going to happen.\n\nIn English class, where he and I usually partnered up, he began working on projects with Matt Alexander, and I started working with Gillian Zucker. We had two Tuesday meetings of our group\u2014one at McDonald's (where we all got Happy Meals and giggled our way through playing with the toys like little kids) and one at the ice rink again\u2014and I don't think I was the only one who felt Sam's absence.\n\nSunday, November fifteenth dawned gray and bleak, which seemed fitting. It was officially the anniversary. It had been an entire year. Today we'd begin a whole new year of days my father would never get to live, things he'd never get to see. But saying it, admitting it had been nearly a year already, was more difficult than it should have been.\n\nIt had been fifty-two Saturdays since I'd taken my sweet time in the bathroom and cheerfully headed out the door for the five-minute car ride that would change our lives. I felt tears prickle at the backs of my eyes as I lay in bed.\n\nDespite myself, I went to the window to look for a rainbow, and I almost wanted to kick myself for believing there was even a chance one would be there. Not only did I not believe in stuff like that, but it would have been scientifically impossible, given the overcast skies. You needed sunshine for a rainbow, and I had the feeling there wouldn't be any today.\n\nI looked at the sky anyhow, hoping that there would be some kind of sign that my dad was up there, watching. But still, nothing.\n\nThen, something made me look down. My window overlooked the front yard and the street, and as I glanced at the grass, I noticed the strangest thing.\n\nThe lawn, which had been covered for the past few weeks in a growing blanket of orange, red, and yellow leaves, had been raked, and there was a big pile of leaves in the corner, almost exactly where my dad used to put the leaf pile.\n\nFor a fleeting instant, I was sure my dad had done it, that it was the sort of sign Sam had talked about, except that instead of painting a rainbow in the sky, my dad had done something much more personal.\n\nThen I remembered. I had told Sam about the leaves, hadn't I? But he couldn't have done this. With as coldly as I'd been treating him, it was hard to believe that he would show up with a rake in the wee, cold hours of the morning and do something so incredibly touching.\n\nI stared down from the window for a long time at the leaf pile. And while I looked, a little bit of the ice melted from around the outside of my heart.\n\n\u2022 \u2022 \u2022\n\nMom surprised us all by making light, fluffy blueberry pancakes for breakfast.\n\n\"I thought it would be a start to a tough day that your dad would appreciate,\" she said as she brought the platter to the table. Logan shuffled over to the fridge to grab the maple and blueberry syrups, and Tanner poured juice for all of us, sloshing a little over the side of Mom's glass.\n\n\"Sorry,\" he said.\n\nShe smiled at him. \"No problem.\"\n\nIt was like we were in a time warp and had gone back to normal. Well, almost. Logan didn't look at all like himself; his eyes were bloodshot, his hair was a mess, and I could swear I could smell alcohol on him, although Mom seemed oblivious. Mom still looked vacant, but I knew she was trying. And Tanner, of course, was being his usual quiet self.\n\nOr so I thought. After we'd downed our pancakes and Mom had stood up to start clearing the table, he suddenly said, \"Knock, knock.\"\n\nWe all looked at him. Logan and I exchanged glances. Mom stopped in her tracks.\n\n\"What?\" I asked, sure that I must have heard him wrong.\n\n\"Knock, knock,\" Tanner repeated. We hadn't heard a joke come out of Tanner's mouth in a year. \"Um, who's there?\" I asked. \"Little old lady,\" he said.\n\n\"Little old lady who?\" my mom asked, coming back to the table.\n\nTanner smiled at her and then at Logan and me. \"I didn't know you could yodel, Mom.\"\n\nIt was a stupid joke, really, the kind that we only would have laughed at a year ago to be polite. But hearing Tanner tell it today, after a year of barely hearing his voice, never mind his humor, unleashed something in all of us.\n\nMom started laughing first, in high, tinkling tones that I hadn't heard in so long I had almost forgotten what they sounded like. Logan joined in next with an amused chuckle. Before I knew it, I was laughing too.\n\n\"I've been saving that one for today,\" Tanner said. \"I think Dad would have liked it.\"\n\nThe words brought the laughter to a halt. Finally, Mom broke the silence. \"Yes, Tanner,\" she said. \"I know he would have.\"\n\nAnd in that moment, sitting around the kitchen table with my mom and two brothers I felt like maybe, just maybe, our dad was with us after all.\n\n\u2022 \u2022 \u2022\n\nLogan disappeared after breakfast with promises that he'd meet us back at the house by two to go to the cemetery, a trip I was dreading. I'd managed to avoid it for an entire year, but I knew I had to go. I had to do it. For my mom, for Tanner, and, I guess, for myself. After I took a shower and got dressed, I knocked on Tanner's door.\n\n\"Want to go out and jump in the leaf pile in the yard?\" I asked.\n\nHe followed me outside. We spent the next hour jumping around together, like we used to when we were younger. We threw handfuls of leaves at each other, made leaf angels in the yard by lying on our backs and spreading our arms, and dove into the pile again and again, breathing in the familiar, slightly musty smell of autumn all around us.\n\nWe laughed like we used to when our dad would dive in with us, and as I grabbed my little brother for a tickle attack, like Dad used to do to me, I looked up at the gray sky once again, foolishly half expecting a rainbow. Instead there were just low, dense clouds and the promise of rain. The leaves, I knew, would get wet and soggy and would disperse around the yard again when the sky opened up. But for now, they were perfect, and when I closed my eyes, I could almost believe that it was like before, a crisp fall day when everything in the world was right.\n\n\u2022 \u2022 \u2022\n\nLogan didn't come home.\n\nAs we waited for him at the kitchen table, my mom got more and more mad.\n\n\"Maybe he's just running late,\" she said at 2:10.\n\n\"He must be on his way,\" she said at 2:20 when she called his cell phone and it went straight to voice mail.\n\n\"What could they be doing?\" she demanded at 2:30 when she called Sydney's phone and got her voice mail too.\n\n\"Fine, he can meet us there,\" she huffed at 2:45 when Logan still hadn't appeared.\n\nSo Mom, Tanner, and I climbed into the car and headed to the cemetery.\n\nAfter we parked, Mom led us up the little hill to Dad's grave, as easily as if she had a map of the place imprinted on her mind. I supposed maybe she did.\n\nDad's gravestone was a thick slab of dark gray marble, and as we walked up to it, the words imprinted on it burned into me.\n\n**P ETER MANN  \nBELOVED FATHER, HUSBAND, AND SON**\n\nA single ray of sunshine poked through the gloomy mass of clouds as we stood in silence, looking at Dad's grave. I had no idea how to act. Was I supposed to kneel and say a prayer? Or look up at the sky and try to talk to him? Was I supposed to touch the gravestone or the flowers that seemed to have no right to be alive while my father lay dead?\n\nMy mom started crying. Tanner stood beside her, holding her hand, his head leaning against her arm.\n\n\"Lacey,\" she said, turning toward me.\n\nI swallowed hard and wondered what was wrong with me that I wasn't crying too. I joined them, putting my arm around Mom. She pulled me into a hug, and the three of us stood there for what felt like a small eternity, blanketed in a silence that was only punctuated by the occasional sounds of Mom's sniffles.\n\nAfter a few minutes, Tanner pulled away and announced that he was going to go look for a squirrel he'd just seen run by.\n\n\"I have some peanuts in my pocket,\" he said solemnly. \"And maybe he's hungry.\"\n\nMom nodded, and we watched Tanner head off. After a few paces, he broke into a run.\n\nAfter a moment, Mom began crying again. I didn't know what to do. It felt awkward to be around a grieving person, especially my mother.\n\n\"It'll be all right, Mom,\" I mumbled.\n\n\"I've been a terrible mother,\" she whispered.\n\n\"No, Mom,\" I said, shaking my head. \"It's okay.\"\n\n\"I'm the mom,\" she said, pulling a tissue from her pocket and blowing her nose. \"I'm supposed to be the one who holds it all together. For all of you. And I haven't been able to do even that.\"\n\n\"You've done your best. I've done my best. We've all done our best. And it's going to get better.\"\n\n\"But your dad would have\u2014\"\n\n\"Dad would have understood,\" I said, \"that you can't be perfect.\"\n\nThe words settled around us, and as they did, I realized that maybe I needed to take them into account too.\n\n\"I'm going to go sit in the car,\" Mom said with a sigh, turning away from Dad's grave.\n\nTen minutes later, I found Tanner sitting under a tree, gazing at a pair of squirrels, and together, we returned to the car. Mom already had the engine running and the heater going.\n\n\"Ready?\" she asked.\n\nWe both nodded.\n\nIt wasn't until we'd pulled out of the parking lot that I realized I'd been so busy comforting my mom, I hadn't had a chance to say anything to my dad. I still wasn't sure that he could even hear me. I wasn't sure what I believed. But once again, I'd failed him.\n\n# chapter 21\n\nThere was a message on the machine from Logan when we got home.\n\n\"Sorry I missed the cemetery,\" he said, his voice sounding slurred. \"I'm still out with Sydney. See ya later.\"\n\nMom hung up her coat and began sorting through a stack of mail.\n\n\"Mom?\" I said, biting my lip. I'd always kept up the unspoken sibling rule of honor by not telling my mother if I saw Logan drinking or smoking at a party, but the fact that he sounded drunk at four on a Sunday afternoon worried me. \"Doesn't Logan sound kind of... funny?\"\n\n\"It's been an emotional day for all of us, Lacey,\" she said, sighing. \"I'm sure he's shed a few tears of his own.\"\n\nThat wasn't what I meant, but there was no point arguing with her.\n\nLater, after dinner, I decided to go for a run. I needed to get out. Logan still wasn't home, and Tanner and Mom were watching some show about pandas on Animal Planet.\n\nThe night had turned cold. The rain that had started just after we got home from the cemetery\u2014and had dried up another hour after that\u2014had brought with it a chill in the air that hadn't been there before.\n\nAs I set out at a slow jog, I wasn't quite sure where I was going at first. I just knew that I needed to be alone.\n\nAs I ran, my feet carrying me farther from home, I thought about my dad, I mean _really_ thought about him, for the first time in a very long while. It was easier not to think about him most of the time. I'd stopped letting the memories in. I'd stopped talking to him in my head, pretending he could hear me. I'd stopped looking obsessively at his pictures. A little part of me _wanted_ to forget his face, his warmth, his deep voice, his lopsided smile, because it would be easier that way, wouldn't it?\n\nAnd now he was back. Seeing his gravestone for the first time since the funeral had brought it all home. No matter how fast I ran, I couldn't escape the reality that he was gone.\n\nMy feet carried me the two miles to the cemetery. I didn't even stop to consider that I shouldn't be coming this far by myself after dark. It was like I was numb to everything: good judgment, logic, even the bitter cold that was seeping in through my double-layered sweatshirts. I patted the pocket of my sweatpants and felt the familiar shape of my cell phone.\n\nSlowly, I made my way up the shallow hill until I could see my father's headstone emerge from the darkness. A moment later, I stood in front of it, gazing down for the second time today at his name, the year of his birth, the year of his death. My knees suddenly felt weak, and I reached for the headstone to steady myself.\n\n\"Hi,\" I said softly. \"I'm sorry.\" My voice didn't sound like my own. In fact, it took me several seconds to register that the voice was mine, that I had spoken the words aloud instead of just thinking them. I took a deep breath and repeated the words a little louder. \"I'm sorry,\" I began again, \"for not always being a very good daughter. I'm sorry for all the fights we had. I'm sorry for the times you told me I was being a brat and you were right. I'm sorry for the times I yelled at you that I hated you. I never meant it. Not once. I wish I could take them all back.\"\n\nMy knees were growing weaker; my legs felt like jelly. My hand still on the headstone, I eased myself down on the dead grass. The rain had left the ground damp, and I could feel it seep through my sweatpants almost immediately. But I didn't care.\n\n\"I'm sorry for that morning,\" I went on. \"I'm sorry I took so long getting ready, just to bug Logan. I'm sorry I took my time coming downstairs. I'm sorry I thought that was funny. I'm sorry I thought it wouldn't matter.\"\n\nMy heart was pounding quickly now, and that familiar icy feeling was back. But still no tears. \"I'm sorry that I didn't look up sooner. I'm sorry that I saw the SUV but didn't say anything. I didn't have time, but I should have. I should have thought more quickly. I'm sorry I blacked out. I'm sorry I couldn't hold your hand. I'm sorry I couldn't save you.\"\n\nI felt short of breath. The words were coming faster, piling out on top of each other. \"I'm sorry it was you and not me.\" I heard myself say the words, and they surprised me even as they came out of my mouth. I hadn't known I'd felt that way until that very moment. I hadn't let myself think about it. But if I'd been just a second faster, if I'd snapped my seat belt right away instead of giving Logan a hard time, if I'd spent one less stupid second in the bathroom making sure my lipstick was just right, then we would have been several inches farther along the road, and the car would have missed Dad and plowed into me in the backseat instead.\n\nMaybe that was the way it was supposed to happen. Maybe I had cheated fate.\n\n\"I'm sorry I haven't done better,\" I went on. The more things I apologized for, the more miserable I felt. \"I'm sorry I haven't done a better job of taking care of everyone. I don't know how, sometimes, Daddy. It's really hard. But I know it's what I have to do. I know I have to do that for you. And I'm sorry I haven't done better. I promise to try harder.\"\n\nI sat there, staring at his headstone. I wasn't sure what I was waiting for. But there was only silence.\n\n\"I'm sorry,\" I said again. I leaned forward and felt the cold marble of the headstone on my forehead. The cold was cutting into me now, but I didn't care. I fervently hoped that somewhere, my dad could hear me. \"I'm sorry. I'm sorry. I'm sorry.\"\n\nI repeated the words, again and again, until the pain in my chest was so great that I couldn't take it anymore. I couldn't feel my dad's presence. Not at all. I realized I was talking to myself.\n\nI stood up, cleared my throat, and touched the gravestone once more. I dusted what dirt I could off my sweats and, with one last, long glance, turned away.\n\n\u2022 \u2022 \u2022\n\nI walked toward the parking lot and saw a vehicle parked in the far corner of the lot, in the shadows. Who would be here this late? My heart hammered and I reached for the phone in my pocket. I shouldn't have come here. What if I had gotten myself into a dangerous situation?\n\nAnd then, as I tentatively walked closer, I suddenly recognized it. And the person leaning against it, watching me approach.\n\nSam straightened up and began walking toward me at the exact instant I realized it was him.\n\n\"Hi,\" he said as we approached each other.\n\n\"Hi,\" I said, staring up at him as the distance between us closed. We were standing face to face, under a dim puddle of light from a flickering streetlight. \"What are you doing here?\"\n\n\"I went to your house, and your mom said you'd gone for a run,\" he said.\n\n\"But how did you know I'd come here?\"\n\n\"I couldn't think of anywhere else you'd go. Not today, anyhow.\"\n\n\"Oh\" was all I could manage. There was something about realizing how well he knew me that made my stomach flip. We stared at each other for a moment. Then I asked, \"Did you rake the leaves in my yard this morning?\"\n\n\"Yeah,\" he said.\n\n\"Why?\"\n\nHe looked a little embarrassed. \"It was important to you. It was a memory you had with your dad.\"\n\n\"You can't bring him back, you know.\" My voice sounded angry, and I wasn't sure why I was directing any of that toward Sam. But my stomach was all tied up in knots. \"Just by raking leaves. He's gone.\"\n\n\"I know.\"\n\nI looked away. \"It's not fair.\"\n\n\"What's not fair?\"\n\nI swallowed hard. \"Your dad loved you enough to stay. My dad... didn't. And sometimes I hate him for it.\"\n\nThere. I had finally shown Sam the last of the cards I had kept so close to my chest, the cards I hadn't even known were there. How could I hate my father, even a little bit? Surely it made me the worst person in the world. And now I'd shown Sam just what a despicable human being I really was.\n\nHe stepped forward and pulled me into his arms.\n\nI was startled, but I finally let myself relax into the embrace. I tentatively wrapped my arms around him and returned the hug. He responded by holding me tighter, like he would never let go.\n\n\"It's going to be all right, you know,\" he whispered, ruffling my hair with his breath.\n\nI opened my mouth to tell him he was wrong, but before I could even get a syllable out, he had put his hand gently over my mouth.\n\n\"Stop, Lacey,\" he said. \"Stop always having to be so tough. Just have some faith.\"\n\n\"Sam,\" I said after a minute, \"I still haven't seen a rainbow.\" I paused and added, \"I've looked.\"\n\nSam stroked my hair. \"Maybe you haven't really needed your dad yet,\" he said. \"You know, it's okay to hate him a little. He _did_ leave you, even if he never would have wanted to, Lacey. But it made life hard for you. Life is _still_ hard for you. He'd understand.\"\n\n\"How can I feel like that and still love him so much?\" I asked in a small voice.\n\nSam was silent. \"I think,\" he said, \"that's exactly what love is.\"\n\nSam's words, and the fact that he was finally absolving me of everything while he held me tight, made something inside me snap. I didn't even know it had happened until I felt the first tear roll down my right cheek, followed soon after by a single tear from the other eye. And then, they were coming like a deluge, one after another, tears falling from eyes that had been dry for a year.\n\n\"You're crying,\" Sam said, leaning back. He looked concerned. He reached in to gently wipe a tear away.\n\n\"I know,\" I said. I reached up and touched my cheek. \"I know.\" And for the first time that day, I smiled.\n\nWe stood in the middle of the cemetery parking lot for a long time, under the glow of the flickering light, enveloped in a dark silence. But I'd never felt so safe in all my life. I didn't want to move, didn't want to go back to reality.\n\nAnd then, my cell phone rang, a sharp jangle that invited reality back in.\n\nThe spell was broken. I looked at Sam as I pulled away. I looked at the caller ID. _Mom's Cell_. I didn't know why she'd be calling from her cell instead of the home phone, but I knew she was probably wondering where I was.\n\nI snapped my phone open. \"Hello?\"\n\n\"Lacey?\" Her voice sounded frantic. I felt immediately bad.\n\n\"Mom, don't worry; I'm fine,\" I said quickly. \"Sam's here with me, and\u2014\"\n\nShe cut me off. \"It's Logan. There's been an accident. He's at the hospital. I need you to come right away.\"\n\n# chapter 22\n\nWe got to Plymouth Regional Hospital's emergency room in record time. Sam dropped me off near the ambulance bay and promised to be inside as soon as he parked. I dashed inside and wildly scanned the waiting room for my family.\n\nI spotted them immediately. Mom was standing in the corner, looking disheveled, and Tanner was sitting in a chair, his head down, mumbling to himself. Sydney was standing several yards away, her face tear-streaked.\n\n\"What happened? Is he okay?\" I demanded, running up to them. All three of them looked up. \"Please!\" I snapped. \"Is he okay? Tell me!\"\n\n\"I don't know,\" Mom said. She appeared exhausted. \"He's in surgery now. The doctors will be out to talk to us as soon as he's done.\"\n\nI stared at her. My whole body felt cold. _Surgery. Doctors. An accident_. It was all so familiar.\n\n\"What happened?\" Just then, Sam came through the doors of the waiting room and jogged over to me. I introduced him to my mom, who nodded vaguely. I glanced down at Tanner and realized that he had reverted to sucking his thumb, something I hadn't seen him do since those dark weeks after Dad died. \"What happened, Mom?\"\n\n\"Apparently, Logan was drinking,\" she said in a tight voice. \"With Sydney.\" She glared at Sydney, who seemed to shrink under her gaze. \"He took the keys to Sydney's car,\" Mom continued through gritted teeth, \"and went out to drive around the neighborhood. To find the spot of your accident with Dad.\"\n\nI felt tears in my eyes.\n\n\"He didn't come back for a while,\" Sydney cut in, glancing nervously back and forth between me and my mom. \"So finally, I got worried and took my dad's car out to look for him. I found him on Old Port Road. You know, the one by the harbor that curves? I guess he took the turn too fast and hit a telephone pole.\"\n\nMom made a muffled sound and turned away. I sucked in a deep breath. Tanner curled up on the seat and closed his eyes, sucking his thumb more furiously now. Sam wrapped both of his arms tightly around me and squeezed.\n\n\"The police were already there,\" Sydney continued. \"And the ambulances. They were just taking him away on a stretcher. That's when I called your mom.\"\n\n\"Did you see him?\" I demanded. \"As they were taking him away?\"\n\nI wanted to ask her if he had been conscious, if there had been blood, how he had looked. But she just shook her head. \"They were already shutting the doors to the ambulance. I only knew it was him 'cause of the car. It's totally ruined. My parents are going to _kill_ me.\"\n\nIn this moment, with my brother lying somewhere behind closed doors and possibly dying, _she was worrying about her car?_ I wanted to wring her perky little neck with my bare hands. But Sam held on to me and murmured in my ear, \"It's not worth it.\"\n\nHe was right. But I'd never hated someone quite as much as I hated Sydney right then.\n\nAn hour passed without any word. My mom paced for a while, then sat down, chewing so hard on her lower lip that it started to bleed. She didn't even seem to notice. Tanner's eyes glazed over as he continued rocking back and forth, sucking his thumb. Sydney sat several seats away from us, alternately staring at the wall and texting on her phone. The whole time Sam sat next to me, rubbing my back gently and occasionally whispering things like, \"It's going to be okay, Lacey.\"\n\nHis words weren't much comfort. But his being there was. At least a little bit.\n\nFinally, a doctor in pale blue scrubs came out of the swinging doors leading to the operating room. \"Mrs. Mann?\" he asked, scanning the waiting room.\n\nMy mom jumped up immediately. \"Yes, that's me,\" she said. \"I'm here. How is he?\"\n\nI was on my feet before I knew it, standing at Mom's side. Sam appeared behind me a second later. Tanner stood up and grabbed my hand. Sydney just sat there, staring nervously.\n\nThe doctor glanced around at our little group. \"Logan's a very lucky young man,\" he said. \"He's going to be fine.\"\n\nI didn't think I'd ever felt so relieved in my entire life. My knees buckled a little, but Sam was there to catch me.\n\n\"He is?\" Mom demanded, almost as if she didn't believe it. \"Are you sure?\"\n\nThe doctor still looked concerned. \"Yes,\" he said slowly. \"He suffered a concussion and several broken bones, but it appears his internal injuries are minimal, aside from the trauma to his liver. He should make a full recovery.\"\n\n\"Oh, thank God,\" Mom breathed. I could see tears glistening in her eyes as she turned to me and smiled. I could feel the tears in my own eyes too.\n\nBut the doctor didn't look as happy as we did. \"Mrs. Mann,\" he said slowly. \"It seems to me that we have a difficult situation here.\"\n\n\"What?\" my mom asked, sniffling a little.\n\nHe cleared his throat. \"Your son's blood alcohol level is quite high. I asked him some questions, as did the police, and it seems that this isn't the first time he has gone overboard with drinking.\"\n\n\"What?\" My mom looked at the doctor blankly.\n\nThe doctor cleared his throat again. \"I suspect he will have to deal with the legal ramifications of this incident. I need to strongly recommend that you get him into some sort of rehab program.\"\n\n\"Rehab?\" my mother whispered.\n\n\"He's a minor, Mrs. Mann. This is extremely serious. He's very lucky that no one besides himself was hurt.\"\n\nShe looked down. \"Today's the anniversary of his father's death,\" she said. \"I don't know if he told you that.\"\n\n\"Ah,\" the doctor said. For the first time, he looked a bit sympathetic instead of judgmental. \"I see. I'm sorry to hear that. Was it long ago?\"\n\n\"A year ago today,\" my mother whispered.\n\n\"I'm sorry,\" the doctor said. \"But this is a wake-up call, Mrs. Mann. Your son needs help.\"\n\nI could feel my face flaming. I'd known my brother drank. I hadn't done anything to stop it. And he had almost gotten himself killed.\n\nAs if reading my mind, Sam leaned down and whispered softly in my ear, \"Don't you dare go blaming yourself, Lacey Mann. You are _not_ responsible for Logan. He did this on his own.\"\n\n\"But\u2014\" I started to whisper back.\n\n_\"Not your fault,\"_ Sam said in a tone that left no room for argument.\n\nThe doctor was saying something to my mom about how Logan was under anesthesia and was a little groggy but could talk to her if she wanted to go in. The rest of us would have to wait until visiting hours tomorrow.\n\n\"After you see him,\" the doctor said, \"the police will want to interview you. And I'd like to recommend a few rehab centers to you before Logan is released.\"\n\n\"Okay,\" she said in a small voice.\n\nWhen the doctor disappeared, my mother crumpled to the floor. It was as if all her bones suddenly turned to jelly. \"My God, my God, my God,\" she was murmuring to herself. I bent down and wrapped my arms around her.\n\n\"I'm sorry,\" I murmured. \"I'm sorry I didn't do anything to stop him.\"\n\n\"My God, Lacey,\" she said. \"It's not your responsibility. When did it start being your responsibility?\"\n\n\"But\u2014\"\n\n\"Lacey, you're sixteen,\" she said. \"You're not in charge of your brother's actions. It's my fault. I should have known.\"\n\nI tried to reassure her that it wasn't her fault. But the words fell on deaf ears. Tanner hopped up from his chair and joined me and Mom on the floor. He put his arms around both of us, and the three of us sat there in a messy, crying heap.\n\n\"It's not anyone's fault,\" he said. My mom and I both looked at him. Mom sniffled. \"You can only do your best. And you can either get upset about the past, or just plan on doing things differently in the future. That's what the Crocodile Hunter said, anyhow. In a show I used to watch.\"\n\n\"Thanks, Tanner,\" my mom said.\n\nHe shrugged and put his hands in his pocket. \"Whatever.\"\n\n\u2022 \u2022 \u2022\n\nSydney's parents came to pick her up a little while later. To their credit, they didn't say one word about the car. Mom decided to stay at the hospital overnight. Sam offered to drive me and Tanner home, and my mom gratefully accepted.\n\nSam walked us to the door, and after I unlocked it and watched Tanner disappear into the house, Sam pulled me into a long embrace on the doorstep.\n\n\"Lacey, I don't know if this is the right time to say this,\" he said, \"but I'd really like it if maybe you'd come meet my dad sometime.\"\n\nI didn't know what to say.\n\n\"I mean, he's not himself,\" Sam continued. \"He can't move one whole side of his body. And sometimes I feel like he doesn't even remember me. But he's still my dad.\"\n\nI swallowed hard. I thought of all the things I'd said to Sam, all the selfish, warring emotions I'd felt over his father coming out of his coma. I thought about how I'd never see my dad again and about how lucky I was to not have lost my brother, too. I thought about what Tanner had said about how you couldn't live in the past and how you had to do things differently in the future.\n\nFinally, I smiled. \"I'd like that,\" I said.\n\n\"Good,\" Sam said, smiling back at me. Then he kissed me goodnight.\n\n\u2022 \u2022 \u2022\n\nMy cell phone rang early the next morning, jolting me awake. I glanced at the clock as I dove for the phone: 6:55. My blood ran cold. Was it my mom, calling with bad news about Logan?\n\n\"Hello?\" I answered breathlessly.\n\n\"Lacey?\" It was Sam, and he sounded concerned.\n\nI let out a huge sigh of relief. \"I was afraid it was my mom and something was wrong with Logan.\"\n\n\"Oh jeez, I'm sorry,\" he said. \"I just wanted to make sure you're okay.\"\n\nI smiled. \"I'm fine.\"\n\n\"Can you go to the window?\"\n\nI sat up in bed. \"What?\"\n\n\"I just want you to look outside.\"\n\nA warm feeling spread through me. I wondered if he'd raked the leaves again. I got out of bed, pulled open the curtain, and looked down. But the leaf pile had dispersed in yesterday's rain, and no one had put it back together again. Early-morning sunlight beamed down on a front lawn that looked absolutely ordinary.\n\n\"I don't see anything,\" I said to Sam.\n\n\"Are you looking down?\"\n\n\"Yes,\" I said, puzzled.\n\n\"Try looking up,\" he said mysteriously.\n\nI did as he said, and right away, I saw why he'd called. I gasped.\n\nStretching across the sky and dipping down again in the distance was the prettiest, brightest rainbow I'd ever seen. It was just like the one in Sam's painting under the bridge.\n\n\"Oh my God,\" I breathed. I blinked a few times. I couldn't believe my eyes.\n\n\"Lacey,\" Sam said, \"it's not even raining. Look. It's all sunshine.\"\n\nI looked around. He was right. A few wispy white clouds floated by, but there wasn't a rain cloud, nor a drop of rain, in sight. There was no logical reason for there to be a rainbow.\n\n\"You can't tell me you don't believe now,\" Sam said. \"Your dad's up there, Lacey.\"\n\nI gazed at the rainbow. Then I craned my neck as far as it would go and strained to look up, my chin pointing heavenward. I smiled at my dad.\n\n\"Thank you,\" I whispered to Sam. \"Can I call you later?\"\n\nWe hung up and I stared at the rainbow for a long time. \"Thanks, Dad,\" I said.\n\nThen I sent Jennica a text. No way could I call her this early.\n\nLACEYLOO321: call me when u wake up. miss u.\n\nThen I dialed Mom's cell number.\n\nShe answered on the first ring. \"Hi, honey. Is everything all right?\"\n\n\"Yeah,\" I said. I took a deep breath, realized that it was the first time in a year I'd meant it. \"In fact, I'm pretty sure that everything's going to be okay from now on.\"\n\n# epilogue\n\n**ONE YEAR LATER**\n\nThe second anniversary of Dad's accident fell on a Monday, so we couldn't go to the cemetery until the evening, when we were through with school and Mom was home from work.\n\nSam had bowed out of the visit. He had gone with me to the cemetery over the past year, but today, he said, was for my family. He didn't want to intrude. And that was just one of the many reasons I loved him. He was always thinking about things like that.\n\nHis dad was doing a lot better. He liked to play board games, so Sam and I would get out Monopoly or Battleship and sit with him for hours. It sounded crazy, but it was one of my favorite things to do now. Tanner even visited sometimes, and he entertained all of us with his new jokes\u2014he'd decided he might want to be a stand-up comic. He and Mr. Stone really liked each other.\n\nLogan's car accident had been the wake-up call he needed. Because it was a first offense, he wasn't sent to jail, but he had to enroll in a program for teen alcohol abusers, which met twice a week. He had stopped partying, and he had started hanging out with his old friends Josh and Will again. He and Sydney broke up, and she had a brand-new BMW and a brand-new boyfriend.\n\nMom was finally closer to being her old self again. I still heard her sobbing at night sometimes. But those nights were a lot fewer and farther between. And her smiles at the dinner table were real.\n\nAs for Kelsi, Mindy, Cody, Logan, Sam, and I, we'd become even closer. A freshman named Amber had joined our group a few months earlier; her dad had died when she was five. And Jennica, who had broken up with Brian in January, sometimes came too. The group had decided that it would be okay, from time to time, if kids whose parents were getting divorced joined us.\n\nToday my family met at the cemetery, just after the sun had gone down. The last remnants of the sunset\u2014a few streaks of orange and fuchsia across a deep indigo sky\u2014hung above us, lighting our way. I had a car now, an old Toyota, and I had driven Tanner and Logan. Mom came straight from work, trading her high heels for sneakers in the parking lot.\n\nI came here more often now to ponder things. In fact, I'd come here just last week when I needed to think about a big college decision. I had asked Dad's advice. And in the silence, with the sunshine dappling through the trees around us and the wind stirring the leaves on the ground ever so slightly, I think I'd gotten it.\n\nWe gathered around his headstone and I swallowed hard. _Two years ago_. It was hard to imagine that it had been two whole years since my dad had smiled at me or hugged me or said my name.\n\nMy mom laid down a bouquet of roses and murmured something under her breath. They weren't words for us. They were for Dad. Tanner told a few jokes. He came to Dad's grave with me sometimes, and he told a few each time. And, he'd told me, he was pretty sure that Dad could hear his jokes wherever he was and was proud of him. I'd had to blink several times to stop myself from crying when he said that.\n\n\"I miss you, Dad,\" Logan said in a deep voice that was growing deeper by the day. He bent down on one knee and closed his eyes, and when he stood, there was a tear running down his cheek. He didn't bother to wipe it away.\n\nI took a deep breath. \"I have some news,\" I said. \"I got a letter from Boston University yesterday.\" I paused and grinned. \"I got in. I got accepted. And I think there's a pretty good chance I'll get that scholarship.\"\n\nI had applied for a scholarship for children whose parents had died, sponsored by Kate's Club in Atlanta\u2014the club that had inspired our group. Every year, the founder, Kate Atwood, chose a few kids to send through college. You just had to write an essay about how your life had changed since your parent's death and what your plans were for the future. I had sat down to write an essay. Instead, I wrote twenty-two chapters. I couldn't stop writing. And Ms. Atwood had called to say that my story had moved her to tears, and she thought that with some editing, it could maybe even be turned into a book.\n\nMom was the first to hug me. \"I'm so proud of you. And I know Daddy would be too.\"\n\nI hugged her back and imagined Dad's arms also wrapped around me. I imagined what his face would look like, so full of pride and joy for me. And for a moment, I felt like he was there with us.\n\nI'd wanted to tell my family first, but I could hardly wait to tell Sam later tonight. He'd been accepted at Northeastern. I knew we were young, and who knew what would happen in the future? But at least this meant we were going to be in the same city and we wouldn't have to deal with the whole long-distance thing. If we were meant to work out, we would.\n\nLogan cleared his throat. \"Well, I haven't heard back yet, but I applied to Suffolk,\" he said, naming a small university in the center of Boston. He'd taken a year off after graduation. \"And I think my grades and SAT scores will get me in. So I guess we'll both be in the city.\"\n\nLogan and I hugged. He drove me crazy sometimes, but we'd become a lot closer in the past year, and I couldn't imagine being far away from him. Plus, he'd probably need to hit me up for rides home to visit Mom.\n\nTanner was grinning. \"This is perfect!\" he announced. \"There's a comedy club in Boston that me and Sarah read about. And on Monday nights, they have amateur night for comics under eighteen. We're gonna work on our act. We should be ready by next fall. And you guys can come watch us and bring all your friends!\"\n\nI grinned back at my little brother. \"You bet! You're going to have the biggest BU cheering section any comedian has ever had.\"\n\n\"Not to mention the biggest Suffolk cheering section,\" Logan added.\n\n_\"And_ probably the biggest Northeastern cheering section too,\" I said, thinking of Sam. \"Actually, it just sounds like you're going to have the biggest cheering section ever.\"\n\nTanner smiled from ear to ear. \"Cool,\" he said.\n\nMom was looking at all of us, her eyes glistening. \"Dad would be really proud of you,\" she said. _\"All_ of you.\"\n\nAs we walked away from Dad's grave that night, Mom held hands with my brothers, and I held Tanner's right hand, my own right hand outstretched. I was reaching for Dad. I knew he was right there with us, as much a part of our family as he had ever been. Just because we couldn't see him didn't mean he wasn't there.\n**Kristin Harmel** is a longtime contributor to _People_ magazine. It was while working on a _People_ story that she got to meet Kate Atwood, who runs Kate's Club, an organization for grieving kids in Atlanta, which inspired this novel. Kristin lives in Orlando, Florida, and admits that she spends far too much time at Walt Disney World, which is just fifteen minutes from her house.\n\nTo learn more about Kristin, visit her Web site at www.KristinHarmel.com.\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.\n\nCopyright \u00a9 2010 by Kristin Harmel\n\nAll rights reserved. Published in the United States by Delacorte Press, an imprint of Random House Children's Books, a division of Random House, Inc., New York.\n\nDelacorte Press is a registered trademark and the colophon is a trademark of  \nRandom House, Inc.\n\nVisit us on the Web\n\nEducators and librarians, for a variety of teaching tools, visit us at www.randomhouse.com\/teachers\n\n**_Library of Congress Cataloging-in-Publication Data_**  \nHarmel, Kristin.  \nAfter \/ Kristin Harmel. \u2014 1st ed.  \np. cm.  \nSummary: When her father is killed in a car accident, Lacey feels responsible, so when she is given a chance to make a difference in the lives of some of her fellow students, she jumps at the chance.  \neISBN: 978-0-375-89488-6  \n[1. Grief\u2014Fiction. 2. Guilt\u2014Fiction. 3. Family life\u2014Fiction. 4. High schools\u2014Fiction.   \n5. Schools\u2014Fiction.] I. Title.  \nPZ7.H2116Af 2010  \n[Fic]\u2014dc22   \n2009001367\n\nRandom House Children's Books supports the First Amendment and celebrates the right to read.\n\nv3.0\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nCopyright \u00a9 2016 by Stephen Coonts\n\nAll rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means electronic or mechanical, including photocopy, recording, or any information storage and retrieval system now known or to be invented, without permission in writing from the publisher, except by a reviewer who wishes to quote brief passages in connection with a review written for inclusion in a magazine, newspaper, website, or broadcast.\n\nRegnery\u00ae is a registered trademark of Salem Communications Holding Corporation\n\nThis is a work of fiction. Names, characters, businesses, places, events, and incidents are either the products of the author's imagination or used in a fictitious manner.\n\nFirst e-book edition 2016: ISBN 978-1-62157-529-0\n\nOriginally published in hardcover, 2016\n\nCataloging-in-Publication Data on file with the Library of Congress\n\nPublished in the United States by\n\nRegnery Publishing\n\nA Division of Salem Media Group\n\n300 New Jersey Ave NW\n\nWashington, DC 20001\n\nwww.Regnery.com\n\nManufactured in the United States of America\n\n10 9 8 7 6 5 4 3 2 1\n\nBooks are available in quantity for promotional or premium use. For information on discounts and terms, please visit our website: www.Regnery.com.\n\nDistributed to the trade by\n\nPerseus Distribution\n\n250 West 57th Street\n\nNew York, NY 10107\n**ALSO BY STEPHEN COONTS**\n\n_The Art of War_\n\n_Saucer: Savage Planet_\n\n_Saucer: The Conquest_\n\n_Saucer_\n\n_Pirate Alley_\n\n_The Disciple_\n\n_The Assassin_\n\n_The Traitor_\n\n_Liars & Thieves_\n\n_Liberty_\n\n_America_\n\n_Hong Kong_\n\n_Cuba_\n\n_Fortunes of War_\n\n_The Intruders_\n\n_The Red Horseman_\n\n_Under Siege_\n\n_The Minotaur_\n\n_Final Flight_\n\n_Flight of the Intruder_\n\n**WITH WILLIAM H. KEITH**\n\n_Deep Black: Death Wave_\n\n_Deep Black: Sea of Terror_\n\n_Deep Black: Arctic Gold_\n\n**WITH JIM DEFELICE**\n\n_Deep Black: Conspiracy_\n\n_Deep Black: Jihad_\n\n_Deep Black: Payback_\n\n_Deep Black: Dark Zone_\n\n_Deep Black: Biowar_\n\n_Deep Black_\n\n**NONFICTION**\n\n_The Cannibal Queen_\n\n**ANTHOLOGIES**\n\n_The Sea Witch_\n\n_On Glorious Wings_\n\n_Victory_\n\n_Combat_\n\n_War in the Air_\n\n**WRITING AS EVE ADAMS**\n\n_The Garden of Eden_\nTo all those persons, wherever they are, who believe in Liberty. \nThe oath to be taken by the president on first entering office is specified in Article II, Section 1, of the United States Constitution.\n\n\"I do solemnly swear (or affirm) that I will faithfully execute the office of President of the United States, and will to the best of my ability, preserve, protect and defend the Constitution of the United States.\" \nCONTENTS\n\nPrologue\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nChapter Fifteen\n\nChapter Sixteen\n\nChapter Seventeen\n\nChapter Eighteen\n\nChapter Nineteen\n\nChapter Twenty\n\nChapter Twenty-One\n\nChapter Twenty-Two\n\nChapter Twenty-Three\n\nChapter Twenty-Four\n\nChapter Twenty-Five\n\nChapter Twenty-Six\n\nChapter Twenty-Seven\n\nChapter Twenty-Eight\n\nChapter Twenty-Nine\n\nChapter Thirty\n\nChapter Thirty-One\n\nChapter Thirty-Two\n\nChapter Thirty-Three\n\nChapter Thirty-Four\n\nChapter Thirty-Five\n\nChapter Thirty-Six\n\nRussia Account - Demo chapter 2\nPROLOGUE\n\nOn that third Saturday in August, four separate events came together and snowballed into an avalanche that forever changed life in the United States.\n\nThe first occurred on a ranch in west Texas, a few minutes after one in the morning. There was no moon, so the night was dark, enlivened only by a million stars in the clear sky. The ranch belonged to Joseph Robert Hays, Joe Bob to his friends. For many years Joe Bob had made a modest living raising cattle on his twenty-two-thousand-acre spread, but drought and economics finally forced him out of that business. Like the very first Texans, he had no intention of giving up his land, so he decided to try something else.\n\nToday the ranch raised African game animals, a dozen varieties of antelope, which rich sportsmen paid Joe Bob serious money to hunt. Why go to Africa to hunt, Africa with its desperate poverty and brutal Islamic terrorists? Hunt right here in Texas, in the beating heart of the good ol' US of A. That was what his brochures said that he mailed to anyone who inquired about his ranch. His youngest son was a schoolteacher and had cleaned up the message so it read smoother in the brochures, but that is the way Joe Bob wrote it.\n\nJoe Bob also picked up a little money by hosting scout camps on weekends over the winter and making sure every camper got to see and photograph some of the exotic species.\n\nHis ranch adjoined the Rio Grande, the river that formed the boundary between the United States and Mexico, with its poverty, caste system, and systemic corruption. So the poor Mexicans migrated. Over thirteen million of them, over a fifth of the Mexican population, had crossed that border illegally in the last fifty years and were grubbing for work in the United States, usually for minimum wage, or living on welfare and food stamps. Illiterate, unskilled, and usually unable to speak English, they flooded the schools with their children, kept blue-collar wages low, and formed an underclass that resisted assimilation and required huge amounts of public assistance dollars.\n\nAmerican politicians had done little through the years to stem the flood. Hispanic voters wanted their kinsmen to be able to enter the United States regardless of their ability to contribute to the economy or pay their own bills, yet this wasn't the decisive factor. Farmers and small-business men wanted a source of cheap labor, and were content to pass the true costs, the social costs, on to the taxpayers. Generous public welfare programs also drew millions of Mexicans, more than small business or agriculture could possibly use. Even draining off an eighth of the population didn't really help Mexico, which found itself racked by turf wars between vicious criminal gangs that smuggled drugs into the United States to supply the richest narcotics market in the world.\n\nJoe Bob's ranch had six miles of riverfront, and unfortunately sat astride an ancient trail up from old Mexico, one that had been used for millennia. The tread of thousands of feet for thousands of years had left their mark on the land. The trail began somewhere in the Mexican state of Coahuila, hundreds of miles to the south, but it could be accessed from a dirt road that crossed it two miles south of the river. From there it descended into an arroyo, avoiding the sandstone escarpments that the river had left in the tens of millions of years it had been eroding the land. The escarpments, cliffs of hard, dense rock from eight to twelve feet high, were vertical and formed walls that spread out from the arroyo in a fan pattern. On the north side of the river, the trail, about six feet wide and packed hard, climbed another arroyo into the scrub brush of the Hays ranch. The trail was the easiest and most direct way to get from the dirt road south of the river to the hard road on the north side of the ranch. Drug smugglers sent the mules\u2014men carrying drugs in backpacks\u2014from the road on short summer nights after dark. They would wade the river, cross the Hays ranch on the north side, throw the drugs over the fence there to men waiting with a van, then walk back and be south of the river, safe in old Mexico, by dawn.\n\nWhen he ran cattle, Joe Bob Hays had used a three-strand barbed wire fence across the trail about three hundred yards north of the river to keep his cattle in. Illegal immigrants and drug smugglers had to merely lift the top wire and press one down to crawl through. When he got into the hunting business, Joe Bob had to build a much better fence to hold the exotics, an eight-foot-high chain-link affair topped with a strand of barbed wire. The fence was more expensive than the animals. He borrowed money from the bank at the county seat to finance both. In addition to keeping the antelope in, the fence kept the Mexicans out, so they cut it, allowing the various species of expensive antelope to escape the ranch.\n\nJoe Bob was nothing if not determined. After he had repaired holes in the fence a half-dozen times, he decided he had had enough. He complained to the Border Patrol, the DEA, and the county sheriff, and he wrote letters to his congressman and senators and members of the Texas legislature. All to no avail. The DEA, mysterious as always, didn't reply to his letters. Those who replied said they were sorry, but nothing could be done. Neither the Border Patrol nor the sheriff's department had the manpower to guard his fence.\n\nThe politicians pointed their fingers at the president, who, for political reasons, was in a squabble with Congress about immigration and refused to compromise. Of course, he was merely the latest president, and this was the latest Congress, to do little or nothing about the unarmed invasion from Mexico. Someday, someway, all those illegals would become American voters, and when it happened in that distant, hazy someday, both political parties would want their votes, but none more so than the Democrats, who had bet their political future on the bedrock of welfare and food stamps for the uneducated, the unskilled, the addicted, and the shiftless unable or unwilling to find work in an American economy increasingly fueled by science, technology, and government employment.\n\nIt never occurred to Joe Bob to complain to the Mexican government, which actively encouraged its citizens to migrate illegally to the United States and was infamously corrupted by criminals in the drug business.\n\nSo the last time he repaired his fence, Joe Bob put tin cans with small rocks in them on the top strand of barbed wire. The cans tinkled when the wind moved the wire, and they should tinkle when Mexicans operated on the chain links with wire cutters.\n\nTonight Joe Bob sat under some scrub brush on the bank of the arroyo on his side of the fence. Across his knees was an old Marlin lever action in .30-30, with a nightscope mounted on it that he had ordered from a Cabela's catalog.\n\nHe had been here for two nights, had seen and heard no one, and was tired. Yet this evening before twilight he had seen dust to the south, so he thought some Mexicans might come tonight. If they were drug smugglers, they wouldn't cut the fence by the hard road. Illegal immigrants would cut the northern fence, however, to squeeze through.\n\nDamn them all, anyhow.\n\nJoe Bob opened his snuff can and put a pinch in his mouth. He really wanted a cigarette, but they might see the glow or smell the smoke. He wanted to surprise them, throw some shots around, run them back across the river. The sons of bitches could find another place to cross, and no doubt would. But he was sick and tired of working on his goddamn fence.\n\nHe was thinking about a drink of water when he heard the cans rattle down in the arroyo. Someone, man or animal, was fooling with the fence.\n\nJoe Bob lifted his rifle and began scanning with the scope, looking for people.\n\nWhat he didn't know was that two Mexican gunmen on the other side of the fence were also looking for him with nightscopes, better ones than Joe Bob could afford. They had been hired to escort eight mules to the paved highway on the northern side of Joe Bob's ranch, where a vehicle would meet them to take the packages of cocaine on to Los Angeles.\n\nThe lead mule rattled the fence while the gunmen searched. One of the shooters, Jesus Morales, spotted Joe Bob Hays seated under a bush and settled the crosshairs of his scope on him. He squeezed the trigger.\n\nThe bullet smacked Joe Bob in the chest, a mortal wound, and he went over backward.\n\nNothing else moved on the ranch side of the fence, so after a twenty-minute wait to be sure, the fence was cut and the mules moved through the opening up the ancient trail. Morales climbed the bank of the arroyo to where Joe Bob Hays lay bleeding out. He found him with the nightscope.\n\nTo Morales' amazement, the rancher was still alive. Morales pointed his rifle at the dying man's head and pulled the trigger. His head exploded.\n\nThe Mexicans moved on, walking north with their loads. The wheels of commerce were turning, as they had to turn, for that was the way of the world.\n\nAt eleven o'clock that Saturday morning four clean-shaven, skinny young men bought tickets for the Amtrak Express to New York at the BWI Airport station between Washington and Baltimore. They had arrived in a stolen car that they parked on the upper level of the garage adjacent to the station. Carrying backpacks, they took the stairs down and into the train station and stood in line to buy tickets. When their turns came, they each paid cash for a ticket to New York, then went out onto the platform to wait for the train. There were no metal detectors to pass through; no one inspected their backpacks.\n\nTen minutes later the train arrived right on time. They climbed aboard, each entering a different car.\n\nThey found seats. The train was crowded, as usual. The young men looked around and were pleased to see that there were no uniformed police, no armed guards of any type, not that they expected any. This was America, the most under-policed nation on earth.\n\nThe train pulled out right on time, at twenty-two minutes after the hour. There was no clanking and jerking. Powered by electric locomotives, the train merely glided into motion.\n\nThe traveler who had boarded the last car, Salah al Semn, found that the only empty seat was in the middle of the car, facing two fit young men, one white, one black, clean-shaven, with military haircuts, wearing jeans and pull-over short-sleeve shirts. He had seen that type before in Iraq, and suspected, rightly, that they were in the American military. He ignored them. Beside him was a young person with unkempt long hair wearing ear buds and apparently listening to an iPod.\n\nWith their backpacks on their laps or in the overhead bins, all four of the men who boarded at BWI sat back in their seats, avoided eye contact with their fellow passengers, and checked their watches. They had some time to wait, so they watched the countryside pass outside the windows and thought private thoughts as the train ran along through suburbs and into downtown Baltimore.\n\nIn the Chicago suburb of Arlington Heights, a van pulled up outside a parochial school. There were three men in it, brown, clean-shaven skinny men in jeans. They sat watching as families parked their cars and took children into the school. Today was registration day for a new school year that was to begin Monday. Nuns ran the school and taught some of the classes. In the office, nuns supervised the registration process and shook hands with the parents and greeted the students, most of whom were returning for another year. The school was for children in grades one through six. It had been in operation for over a hundred years, and many of the parents were graduates.\n\nThe name of the school, Our Sisters of Mercy, was emblazoned above the main entrance, but the men in the van couldn't read the words. Not only did they not know how to read and write English, they were illiterate in all the world's languages, including their own, which was Farsi. The only education any of the three had ever received was in an Islamic school, where the sole item in the curriculum was the memorization of the Koran. The Prophet's message, their teachers knew, was all the boys really needed to know to wend their way through this vale of tears and earn their way into Paradise.\n\nThe men in the van checked their watches. As the two in the front seat scrutinized every vehicle and watched traffic on the street, the man in back began opening bags and extracting semiautomatic AR-15 assault rifles, into which he inserted magazines.\n\nAt Yankee Stadium in the Bronx the players were on the field warming up, tossing balls around, taking batting practice, and signing autographs for the kids and fans who hung over the rails. The Yanks were not having a good year; they were third in the American League East standings, ten games off the pace, so management expected that only half the seats in the stadium would have bodies in them when the game against the Detroit Tigers started at precisely one o'clock.\n\nThe jihadist, Nuri Said, sat in the top tier of seats watching the activities on the field as fans wandered in. He had attended two games in the past few weeks and had a rudimentary understanding of the game, which he thought boring. Mainly he watched the uniformed police who stood here and there at the portals that people had to pass through to get to and from the stands to the vast galleries where there were restaurants, fast food stands, and restrooms.\n\nNuri had chosen Yankee Stadium for his jihad strike because of the television cameras that would make him and his three mates famous and immortal. The police were a necessary evil, he thought, and would kill all four of them, but not until after the cameras had captured the naked power of Islam for all the world's infidels to see and ponder. Nuri Said and his three fellow believers would please Allah, he knew, which was more than most men accomplished in this life. That would be enough.\n\nSalah al Semn found the waiting hard. He fidgeted. He tried to avoid eye contact with his fellow passengers on the train to New York, but found that he was watching them, sizing them up, wondering who they were as they got on and off at the depots in Baltimore and Wilmington. He repeatedly checked his watch. He was acutely aware that the two young men opposite him were watching him. Every time he glanced their way their eyes were upon him, and they didn't look away.\n\nHe would kill them first, he thought. Infidel dogs.\n\nThe train was sliding into the station in Philadelphia when Salah al Semn checked his watch for the last time, picked up the backpack, which he had placed on the floor under his legs, and made his way toward the restroom.\n\nMarine Sergeant Mike Ivy and Lance Corporal Scott Weidmann were from Brooklyn. They were on their way home for a week's leave before they shipped out for tours in South Korea.\n\n\"He's got a gun in that bag,\" Weidmann whispered to Ivy.\n\n\"Something hard, with angles,\" Ivy agreed. \"Ain't his underwear.\"\n\nAs al Semn opened the door to the restroom and went inside, Ivy and Weidmann got up and went to the restroom door. Ivy put his ear to the door. The train coasted to a stop in the Philadelphia station.\n\nThe two Marines had to make way for people getting on and off the train, but in a moment the rush was over. Ivy leaned nonchalantly against the restroom door and listened while Weidmann watched the passengers in the car to see if anyone was paying attention. They weren't, he decided. Everyone was getting settled for the ride on to Newark, then Pennsylvania Station in New York.\n\nIvy said to Weidmann, \"Bastard's putting his weapon together. Ain't nothing else sounds like that.\"\n\n\"What do you want to do?\" Weidmann asked. He automatically deferred to the senior man.\n\n\"I figure it's a rifle or something. He'll come out of there with the thing pointing up so he can make the turn. Not much room. You slam the door on him and I'll take it away from him.\"\n\nThey took their positions and waited.\n\nIn Arlington Heights, the three men in the van inspected their weapons. Each made sure he had two extra magazines in his pocket, and pulled a ski mask down over his head. They doubted that they would survive this strike so it didn't matter if their faces were seen: they wore the masks to create terror in the heart of everyone who saw them. Terrorized people don't think or fight back, so they are easy to slaughter. Not that any of the three thought the nuns and children and suburban parents would fight back. These people were Christians, who routinely defamed and ridiculed the Prophet, may he rest in peace. They deserved what was coming.\n\nIn Yankee Stadium Nuri Said met his fellow terrorists at a trash can near a service door. One of them, from Iraq, had worked at the stadium for two weeks and had smuggled in weapons and ammunition, which were hidden in the can. As the last minutes ticked by and the national anthem played on the loudspeakers throughout the stadium, Nuri and his three jihadists reached into the can, dug out the trash that covered the weapons, and removed them. Checked that they were loaded. Pocketed spare magazines. And pulled black ski masks over their heads.\n\nThen they walked toward the nearest portal to the stands. There was a woman policeman there, and Nuri saw her before she saw him. He shot her. Even though she was wearing a bulletproof vest, she went down from the impact. The report of the weapon seemed magnified inside that concrete gallery, like a thunderclap. It triggered screams. Or perhaps the sight of the ski masks and weapons triggered them.\n\nPeople panicked and tried to run. One of the terrorists stood there methodically firing single shots as fast as he could aim his weapon. His three colleagues ran out the portal into the grandstands.\n\nSalah al Semn stood in the tiny restroom aboard the express train with his AR-15 at port arms, loaded, with the safety off, and looked at his watch again. One minute to go. The train was accelerating out of the station. He could see the concrete and roofs moving through the little window and feel the motion of the car on the uneven rails.\n\nHe knew precisely what he had to do. Exit the restroom and start shooting people in this car, the nearest first.\n\nWhen he had shot everyone in this car, he was to proceed forward to the other cars, where three other shooters were working. When everyone in all four cars was dead, he and any surviving shooters were to proceed all the way forward, executing people until they reached the engine.\n\nSalah al Semn knew he would see Paradise soon, and he was ready. He would go with the blood of infidels on his hands, one of the holiest martyrs. The Prophet would be proud!\n\nHe took a deep breath and opened the door.\n\nAs it opened, he saw one of the American soldiers standing there, the black man, within a foot. He grabbed for Salah's weapon and jerked it toward him. Salah grabbed for the trigger, and the door slammed into him with terrific force. He lost control of the rifle.\n\nSergeant Mike Ivy didn't hesitate. He merely pulled the rifle toward him, then drove the butt at al Semn's Adam's apple with all the force he could muster. The blow pushed the Syrian back into the restroom. The commode caught the back of his legs, and he lost his balance and fell.\n\nMike Ivy was already examining the AR-15. It was loaded, with a round chambered. Ivy and Weidmann both heard muffled shots from the passenger car ahead of this one. Ivy glanced at Weidmann, nodded to the restroom, and Weidmann said, \"Go.\"\n\nMike Ivy began running forward as people screamed and tried to cower behind their seats.\n\nLance Corporal Scott Weidmann jerked the door open and reached down for Salah al Semn, jerking him upright. The Syrian decided to fight, which was a fatal error. Weidmann's first blow was aimed at his solar plexus, which took the air out of the Syrian and doubled him up. His second blow, an elbow to the man's left ear, was delivered with so much force that the man's neck snapped. Dead on his feet, Salah al Semn collapsed. . .and started his journey to Paradise. Or Hell, depending on your faith.\n\nScott Weidmann left the Syrian sprawled half in, half out of the restroom and ran after Sergeant Ivy, toward the sound of shots.\n\nHe jerked open the door to the car ahead just in time to see Ivy shoot a terrorist and drop him in the aisle. People were sobbing and shouting; an unknown number had been shot. Ivy reached the body of the hooded man first, grabbed his weapon, and tossed it back to Weidmann, who fielded it in the air. As Ivy turned to go forward, Weidmann stomped on the terrorist's larynx, crushing it. Then the two Marines ran on, toward the next car.\n\nThe three shooters walked into the parochial school and the first person they saw was a nun, so they shot her. One of them, a Yemeni named Hassan, stopped to cut her throat with a knife as the other two shot down several families standing there, men, women, and kids.\n\nVinnie Latucca was in the principal's office with his granddaughter talking to Sister Mary Catherine, who had been one of his teachers when he was a pupil at Our Sisters of Mercy forty-some years ago. He heard the sound of gunfire and reached into his pocket for his .38 Smith & Wesson with a four-inch barrel. Vinnie never went anywhere without it.\n\nTelling Sister Mary Catherine and his granddaughter to stay where they were, he opened the door a crack. One of the gunmen entered the office area with his weapon up. As the gunman fired at the ten or twelve people in the room, Vinnie Latucca cocked his revolver, steadied it on the door jam, and fired. One shot. The masked man with a rifle went down.\n\nGunfire continued to sound. A woman was bent over her limp child, cradling him, sobbing softly as Vinnie Latucca shot the gunman again, this time in the head, and then helped himself to the AR. He eased the outer office door open so he could see down the hallway.\n\nNo shooter in sight, so he pocketed his revolver and stepped out.\n\nHe walked toward the sound of gunfire and found the next shooter in a classroom. The fool had his back to the door and was shooting kids. Vinnie shot him twice in the back with the AR, then rolled him over and jerked the ski mask from his head. He put the barrel of the rifle in the man's mouth and pulled the trigger, exploding his head. The man might have been dead by then, but Vinnie hoped not.\n\nThe third man must have wondered why there was no more gunfire, because when Vinnie Latucca stepped out of the classroom into the corridor he fired a shot at him. Vinnie was quicker. Three fast, aimed shots dropped the man. He didn't even twitch. If he had, Vinnie would have blown his head off too. He ripped off the ski mask, saw the fixed eyes, and stood listening for shots.\n\nWhat he heard was the sound of a siren. For the first time in his life, the sound filled Vinnie Latucca with relief.\n\nDetective Victor Goldman, NYPD, was in the middle of his seating section when the gunmen who exited the portal into the grandstand area opened fire. He heard the shots and saw two of them. He didn't know there were three.\n\nHe had a .380 automatic strapped to his ankle, so he pulled it out and tried to get a shot. People were sobbing, shouting, diving for cover so he couldn't get a clear shot. And he was too far away. At least thirty feet, with a pistol with a three-inch barrel and a million people behind the gunmen, so if he missed he would hit a civilian or two.\n\nHe had to get closer. He made his two boys get down under the seats, then he started trying to crawl over people to get closer to the shooters, who were blazing away.\n\nHis chance came when the nearest gunman realized his magazine was empty and bent down to pop it out of his weapon and insert another. Vic Goldman had closed to ten feet. He took careful aim, using both hands on the hideout pistol, and shot the gunman in the chest. He half-turned and Vic shot him again.\n\nThat was when the gunman Vic hadn't seen shot him high in the back. Vic went down on his face, fatally wounded. He was dead when police shooting from the portals killed all the gunmen still standing, and still dead an hour later when a paramedic team found him with his two sons, ages seven and nine, holding his hands.\n\nSomeone pulled the emergency cord on the Amtrak train, so the brakes locked on every car and it screeched to a stop. Mike Ivy and Scott Weidmann had killed the third shooter by then.\n\nAfter the train stopped, the fourth gunman leaped from the train onto the gravel beside the tracks. He was on a dead run heading for Newark when Sergeant Mike Ivy dropped him from a distance of one hundred yards with one shot between the shoulder blades.\n\nAs Ivy and Weidmann stood in front of the locomotive looking down at the terrorist, Weidmann said, \"Nice shot, Sarge.\"\n\nIvy pointed the rifle at the dead man's crotch and fired a shot.\n\n\"Bastard won't be able to fuck his virgins in Paradise,\" he explained.\n\n\"You believe that shit?\"\n\n\"Hell, no, but they do. Send them cock-less.\"\n\nIt was late afternoon in Arlington Heights when Assistant District Attorney Ronald Farrington walked into the room where Vinnie Latucca sat with two uniformed police officers and motioned to them. They stood and left, closing the door behind them.\n\nThe lawyer laid Vinnie's .38 on the table and nodded to it. \"If we get a bullet for comparison, are we going to find any bullets from old open cases that match it?\"\n\n\"Of course not,\" Vinnie said disgustedly. \"That's a clean gun.\"\n\n\"Or you wouldn't have been carrying it.\"\n\nVinnie nodded and lit a cigarette.\n\n\"The nuns don't allow smoking in the building.\"\n\n\"I don't think they'll mind this evening,\" Vinnie replied, and blew smoke around.\n\nFarrington sighed. \"How many guys have you hit, anyway? Off the record.\"\n\nVinnie smoked in silence.\n\n\"We have you on a weapons charge if the DA decides to prosecute. I doubt if he will. You did good today. Saved a lot of lives.\"\n\nVinnie didn't say anything.\n\n\"Put your gun in your pocket and go home,\" Farrington said.\n\nVinnie pocketed the piece and stood.\n\nFarrington held out his hand. \"I'd like to shake your hand,\" the lawyer said.\n\nVinnie grinned, shook hands, and walked out. His daughter and granddaughter were waiting for him on the school lawn.\nONE\n\nOccasionally people ask me, What were you doing that day? You know\u2014that day, that Saturday the terrorists hit the United States hard? Again. Fifteen years after 9\/11 had dropped the World Trade Center, more American blood had been spilled on the altar of global jihad.\n\nMy name is Tommy Carmellini, and the people who ask that question know that back then I worked for Jake Grafton. At the time he was the director of the CIA, the Central Intelligence Agency. Perhaps I should tell you a bit about Jake Grafton, a retired two-star navy admiral, a former attack pilot, a genuinely nice guy, and the worst enemy you could imagine in an alcohol-soaked nightmare. He was a pretty good spook too. So-so shuffling paper. He had an uncanny ability to connect the dots, not just the ones you and I could see, but the ones that only a savant could have suspected might be there.\n\nYet Jake Grafton was pretty closemouthed. He never talked about his boss. He took orders and gave orders and you never knew what the man who lived behind those gray eyes was really thinking. Until the shooting started. Then. . .well, then you found out that Jake Grafton was the perfect attack pilot. Away up there in the blue going fast, out of sight of the people on the ground, he could roll in, draw a bead with his bomb, and turn it loose. To kill you. Then he pulled out and dodged the flak and pointed his ass at the blast and left the vicinity to get on with his life. While your doom was falling from the sky, toward you. That was Jake Grafton.\n\nSo. . .what was I doing that day, the day the old world came to an end? Well, I was in Colorado watching the windup to a Federal Emergency Management Agency (FEMA) exercise, Jade Helm 16.\n\nWhen I got back to my hotel, the television in the lobby said over a hundred people were dead and another hundred injured, some seriously, not expected to live, after the three terror strikes. At least three of the terrorists had been Syrian refugees, and several of the others were here illegally.\n\nAround the world, the news was all bad, but especially in the Middle East, where it looked like the Sunnis and Shiites were well on their way to a Hundred Years' War, each sect trying to exterminate the other, and any Christians who happened to be available. There were rumors of stray nuclear weapons, and there were definitely floods of refugees\u2014and who knew, maybe terrorists among then\u2014pouring into Turkey and Europe.\n\nBack in the good old USA, we were already getting started on a presidential election campaign, and it was ugly. Both sides assured the voters that if the other side won, it was the end of civilization as we know it. And then there was the Soetoro government, getting ready for a civil war.\n\nOn Sunday I flew back to Washington. The airports looked like armed camps. Armed soldiers in full battle dress were everywhere, and there weren't many people volunteering to be victims of an airliner bombing. My plane was less than half full.\n\nOn Monday I finished my report on the FEMA exercise at my cubbyhole office at the CIA facility in the Langley, Virginia, neighborhood. When I ran out of words I decided to print out my opus and proof it. I stamped the report secret using my desk inkpad, stapled it together, and read it through. Signed it.\n\nI had spent the two weeks of the exercise in Colorado at exercise headquarters, the buildings that the Federal Emergency Management Agency occupied on the federal reservation on West Sixth Avenue in Lakewood, a suburb of Denver. The head dog was a Homeland Security career civil servant who had obviously impressed his political bosses with his zeal and commitment to the cause of federal supremacy against all domestic foes.\n\nWhen my report was ready for prime time on Monday morning, I walked it and the classified summary down the hall to the director's office. Admiral Grafton was in, the receptionist said.\n\nI just had time to pour myself a cup of coffee before the receptionist sent me in. Grafton was sitting there behind his desk looking sour, and Sal Molina, the president's man Friday, was sitting across from him. Molina looked sour too. I guess the view from the White House wasn't much better than it was from my apartment.\n\nGrafton motioned me to a seat. I handed him my report, with the classified summary attached, and he flipped through it. He was a tad over six feet tall, lean and ropy, with thinning, graying hair combed straight back. No one would ever call him handsome, not with a nose that was a size too large. When you looked straight at him, you forgot about the nose. It was those cold gray eyes that captured you.\n\nMolina, on the other hand, was a middle-sized guy with a twenty-pound spare tire and a shiny dome. He looked as if he were about ten years younger than Grafton, in his mid to late fifties.\n\nThe admiral tossed the report at Molina and said to me, \"Tell us about it.\"\n\n\"Jade Helm is an exercise about how the government will put down a right-wing uprising, or rebellion, and arrest everyone they think might be sympathetic with the rebels. They'll use these paramilitary police they have tucked into every government alphabet agency as storm troopers and SS troops\u2014\"\n\nThat was as far as I got. Molina exploded. \"Comparing the federal government to Nazis is unacceptable. I am not going to sit here listening to that kind of shit, Carmellini.\"\n\nI didn't say anything. Sal Molina couldn't fire me, and if Grafton did, I was ready to be on my way. Truth was, I had been in the belly of the beast for far too long.\n\n\"Go on, Tommy,\" Grafton prompted, ignoring Molina.\n\n\"They'll arrest every prominent Republican they can find and hold them in guarded camps, mainly at military bases. They have computer-generated lists. Gun owners, people who run their mouths on Facebook and Twitter, radio talk-show hosts, editors and publishers of Republican newspapers. . .you know, dangerous enemies of society.\"\n\n\"Who ran the exercise?\"\n\n\"A senior Homeland Security dude named Zag Lambert. Wore a uniform shirt and a belt with a holstered pistol. Honest to God, all he needed was a Hitler mustache. That guy should be kept in a padded room.\"\n\nGrafton sighed. Molina threw the report back onto the desk. Grafton picked it up and said to me, \"I'll read this. Thanks, Tommy.\"\n\nI got up and beat it.\n\nOutside I rescued my cup, decided the coffee was still warm enough to be drinkable, punched the door code, and strolled into the executive assistants' office. I worked with and liked both of them: Max Hurley, a skinny long-distance runner, and Anastasia Roberts, a black woman with a PhD whose IQ was probably up in the stratosphere.\n\n\"Hey.\"\n\n\"Tommy,\" Hurley acknowledged. \"You were just in the pit\u2014how is it going with Molina?\"\n\nI shrugged. \"Tense.\"\n\n\"They've been arguing for a week,\" Roberts said. \"These agency police forces and huge ammo buys. The White House wants the CIA to establish our own paramilitary force, and Grafton has said no. He's defying the White House.\"\n\nThey stared at me and I stared back. That meant Grafton was on the way out, and we probably were too. The new man, or woman, would bring his or her own management team.\n\n\"They don't trust us,\" Anastasia Roberts remarked, quite unnecessarily. I knew whom she meant. The brain trust at the White House, hunkered down on Pennsylvania Avenue ever since the Democrats lost control of the Senate in the last off-year election, two years ago. The Republicans already had the House. This was August. The presidential election was in November, and no matter which way it went, the current president, Barry Soetoro, was leaving on January 20. The Constitution limited the president to two terms, so the end of his eight-year occupation of the White House was in sight at the end of a long, dark tunnel. Only 151 days of Soetoro left to endure, according to the countdown counter on Fox News that one of the hosts opened his show with every day.\n\n\"You know I was out in Denver last week at the Jade Helm 16 exercise,\" I remarked. \"The National Oceanic and Atmospheric Administration, NOAA, has their own private army, and some of the troopers were at the exercise. A couple dozen of them came down from Boulder, decked out in camo clothes and helmets and armed to the teeth. They bonded with the storm troopers from other agencies. In my opinion, if the water and air gurus need paramilitary police, this agency certainly does.\"\n\n\"Boulder is a hotbed of sedition,\" Max Hurley observed. \"Washington is a hotbed of sheep.\"\n\n\"The revolution will start there, no question,\" I agreed. \"The faculty of the University of Colorado is packed with dangerous right-wing fanatics who will lead their students in a wild charge against the Bureau of Standards, burn it down, then attack NOAA.\"\n\n\"If they fire Grafton, will you stay with the agency?\" Roberts asked me.\n\nNeedless to say, I hadn't thought about that possibility. I had an apartment just up the road, my car was paid for, I was single, my mom was doing okay out in California. When I didn't answer quickly enough, Roberts added, \"I'm resigning. I've been offered a faculty position at the University of Chicago. If the job is still open, I can start when the new semester begins.\"\n\nI grunted. The University of Chicago was notoriously left-wing, very politically correct, and Roberts was a level-headed, pragmatic genius who had worked for Republicans on the Hill early in her career. On the other hand, she was a she, and black, and consequently could get away with a lot that would sink a white male faculty member.\n\nHurley admitted he was on the fence. He loved the game of analyzing raw intelligence. He said so now, and expressed the hope that he could return to the Middle East Desk.\n\n\"Nothing but bad news there,\" I said, trying not to sound too downbeat.\n\n\"I think I can take it for a while longer,\" he said. The cockeyed optimist.\n\n\"Negativity is the problem with this agency,\" Anastasia Roberts declared. \"Eventually it overwhelms you and your shit bucket overflows.\"\n\n\"I wouldn't express that opinion quite that bluntly in the faculty lounge in Chicago, if I were you,\" I told her. \"Clean it up for the civilians.\"\n\nWe chuckled, locked up, and went to the cafeteria for lunch, where we discussed the weekend terror attacks.\n\nI was working on a chicken salad sandwich with mustard and a slice of pickle on the side, plus a little bag of barbeque potato chips, when the televisions mounted high in the corner of the cafeteria broke away from their coverage of the investigation of the terrorist incidents to televise a live news conference with the president, Barry Soetoro. He had complete faith in the professionalism and competence of the FBI and Homeland Security Department. They were investigating. The terrorists were obviously criminals, he said, but they certainly didn't represent the vast bulk of American Muslims or the refugees who had been admitted to the United States. He and his security team were reviewing the information the crime scene investigators were producing, and when more was known, they would be taking any steps called for.\n\n\"Does that mean you will reconsider your decision to admit Muslims to America?\"\n\n\"We can't classify people by their religion.\"\n\n\"Obviously refugee screening was inadequate. What will the administration do to find the jihadists and keep them out?\"\n\n\"We are looking at that.\"\n\n\"A lot of people in Congress are saying your policies on illegal immigration and the admission of Middle Eastern refugees are abject failures, as proved by the events of the weekend. Would you comment on that?\"\n\n\"My political enemies say a lot of things, every day. I haven't the time or inclination to listen and comment.\"\n\nThere was more, a lot more. The public was frightened and angry, and Barry Soetoro was defiant.\n\nWhen the press conference was over, the cafeteria was quiet.\n\n\"It's a miracle someone hasn't shot at him before now,\" Max Hurley observed, leaning forward at the waist and speaking softly.\n\nBut why shoot him? The Democratic nominee was Cynthia Hinton, who, according to the polls, was going to be the victim of a landslide. The Republican nominee was Jerry Duchene, the Wisconsin governor, and if the polls could be believed, he was going to be elected by a landslide. And the Congress would get a veto-proof Republican majority. The country had had more than enough of Barry Soetoro, his left-wing agenda, and his political allies, and was waiting, more or less patiently, for his final day in office. Yet the terror strikes had stirred the pot.\n\nBack upstairs after lunch things began to pop. Were there any indications among the intelligence bits trapped in the intestines of our intelligence systems that some evil foreign power or narco-criminals or terrorist groups had plotted with or funded the Saturday monsters?\n\nWe three EAs were told to contact every department head and find out.\n\nWe spent the afternoon talking to people throughout the agency who were trying their best to find a hint, a clue, a sniff. They failed. While it is theoretically impossible to prove a negative, you can often get close enough for government work. And we did. Nothing. Nada. Zilch. Of course, on television every terror organization in the Middle East was claiming credit.\n\nI was elected to tell Grafton, and did so a bit after five p.m. He just nodded. He had spent the afternoon on the phone, presumably talking to other heads of agencies and political big shots all over town.\n\n\"Are you going home soon?\" he asked. After all, five o'clock is traditionally quitting time, although not in the CIA.\n\n\"Not if you need me, sir.\"\n\n\"Hang around. Sal Molina is coming over again later. I may need a witness.\"\n\nOh boy. I wandered out past the receptionist and walked the halls a while with my hands in my pockets. Was Grafton going to resign? Or get fired?\n\nPeople were standing in knots here and there, chewing the rag over the terrorist attacks. The news shows, they told me, said that Cynthia Hinton had scheduled a news conference for prime time this evening.\n\nI was sitting in the director's reception area when the vice director, Harley Merritt, strode by on his way to the inner sanctum. He ignored me. He had an EA with him, and she ignored me too. It was that kind of day.\n\nThey were in there about a half hour and came marching out. Grafton stood in the doorway as they crossed the reception room. He motioned to me. I went in and he closed the door.\n\n\"Molina is on his way. Sit down.\"\n\n\"Is he going to ask for your resignation?\" I asked. Why beat around the bush?\n\n\"I don't know,\" Grafton said crossly.\n\nI also suspected he didn't give a damn, but I kept my mouth shut and seated myself on the couch. Laid my notebook on my lap, so I'd be ready to scribble down orders or telephone numbers or order flowers for funerals.\n\nGrafton picked up something from his in-basket, glanced at it, tossed it back, then rose from his chair and stretched. He reminded me of a caged lion. Waiting. In a darkened office with the lights off. Behind him the day was slowly coming to an end.\n\n\"Nations don't just happen,\" he remarked, as if he were talking to himself, or perhaps composing an essay. \"They are put together by groups who are convinced that the people who live within a certain area will be better off as one political entity, this thing called a nation. Nations are fragile. Homogenous nations seem to have done best through written history. Ours is anything but homogenous, a grand experiment with many people from diverse racial groups, cultures, and religious heritages, all mixed together willy-nilly and bound together politically.\"\n\nLooking back, I think at that moment Jake Grafton had a glimpse of the future, a future that disturbed him profoundly.\n\nHe sat in silence for a while, then remarked, \"A government that loses, or forfeits, the consent of the governed is doomed. Invariably. Inevitably. Irreversibly.\"\n\nHe was sitting in silence with the light from the window behind him throwing his face in shadow when the squawk box buzzed. \"Mr. Molina.\"\n\n\"Send him in.\"\n\nI went to open the door and close it behind Molina. He sat in the chair across the desk from Grafton and glanced at me. \"You won't need him,\" he said to Grafton.\n\n\"He stays. Say what you want to say.\"\n\n\"You need a witness?\"\n\n\"I won't know until I hear it.\"\n\n\"The president is declaring martial law tomorrow. He wants you standing behind him tomorrow at ten o'clock in the press room when he announces it.\"\n\nJake Grafton didn't look surprised. I was flabbergasted, but since I was sitting on the couch against the wall Sal Molina couldn't see the stunned look on my face unless he turned his head, and he didn't.\n\n\"Why?\" said Grafton.\n\n\"These terrorist conspiracies need to be rooted out. We must make sure the American people are safe, and feel safe.\"\n\n\"Horseshit,\" Grafton roared, and smacked the desk with both fists. \"Pure fucking horseshit! Oh, a million or two jihadists would love to murder Americans, including Soetoro, if they could get here, but if they were a credible threat we'd have heard about it. This is just an excuse for Soetoro to suspend the Constitution and declare himself dictator.\"\n\n\"The American people must be protected, Admiral. The president is taking no chances. No one wants to be the next victim of Islamic terrorists.\"\n\n\"So he is going to rule by decree.\"\n\n\"We face a national emergency.\"\n\n\"And he is going to postpone or cancel the election in November. Isn't that the real reason for martial law?\"\n\n\"I'm not going to debate it, Grafton. Tomorrow at ten at the White House. Be there an hour early and we'll have a decree signed by the president detailing the actions that he wants from this agency.\"\n\n\"His staff can e-mail me a copy,\" Grafton said softly. \"I am not going to be a prop in a presidential power grab. Not now, not ever.\"\n\nMolina ran his hands over his face. \"Jake, you don't have a choice,\" he said reasonably. \"You'll either be there or your name will go on the list as an enemy of the president. They'll lock you up. Soetoro is playing for keeps. You can kiss your pension good-bye. Do you want to spend the rest of your life in prison?\"\n\nMolina stood, put both fists on the desk, and leaned forward. His voice dropped. \"You think _I_ want to be a part of this? I have a wife and two kids. I don't have a choice. By God, you don't either.\"\n\nGrafton was silent, looking at nothing for a moment or two. Finally, he said, \"Soetoro has been waiting for a terror strike so he could declare martial law, become a dictator, and fix all the things he doesn't like about America.\"\n\n\"You don't know that.\"\n\n\"I'll bet any sum you want to name he is going to call off the election and remain in office.\"\n\nMolina straightened and made a gesture of irritation. He glanced around and saw me, which obviously startled him. Apparently he had forgotten I was in the room.\n\nHe took a step in my direction. \"One word from you outside this room will put you in a cell, Carmellini.\" I'd had confrontations with Molina before. I wasn't stupid enough to open my mouth this time.\n\nMolina swung back to Grafton.\n\n\"Be there tomorrow morning. If you aren't, I can't help you.\"\n\n\" _Me_? You can't help _me_?\" Grafton was standing too, and he was beyond fury. He had a scar on his temple that was throbbing red. \"That bastard is going to rip this country apart, and you worry about your family and pension? You think there's a lifeboat handy that will keep you and yours comfortably afloat in this sea of shit while the ship sinks? What the hell kind of man are you, Molina? He doesn't need _you_ and he doesn't need _me_. Get a grip, fool.\"\n\nMolina was holding on to the desk, as if he were trying to stay erect. \"Jake. . .\"\n\n\"You get out of my office and don't ever come back.\"\n\n\"It won't be your office long. That's what I'm trying to tell you.\"\n\n\"I don't ever want to see your face again, Molina. Get the fuck out.\"\n\nMolina turned and walked from the room. Neither fast nor slow. He merely walked. The door closed behind him.\n\nI was too stunned to open my mouth or move.\n\nGrafton looked at me and gestured toward the door. \"You too, Tommy. Out.\"\n\nI got my muscles working and went.\n\nIn west Texas, Joe Bob Hays' hired man stood in the yard of the ranch house and watched the helicopter approach. It came from the east and slowed as it descended. It touched down in a cloud of dust and, after the sound of the engine subsided, the rotors slowly wound down.\n\nA man in a suit but without a tie climbed out. A state trooper got out with him. They came walking over.\n\n\"I found him this morning, Governor, down by the arroyo trail. They killed him and cut the fence early Saturday, it looks like.\"\n\nGovernor Jack Hays was Joe Bob's nephew. He had grown up on the ranch back in the cattle days, and had gone on to law school, then into politics.\n\n\"The sheriff and his men are down there taking pictures and whatnot. I think the body is still there.\"\n\n\"Let's go. I want to see him.\"\n\n\"They shot him in the head, Governor. Executed him. Blew the top half of his head clean off.\"\n\n\"I want to see him. Let's go.\"\n\nThey went by jeep. In the late afternoon sun, the blood and bits of brain had turned black. Ants had gathered, and bugs. . .\n\nThe county sheriff was there, Manuel Tejada, and he shook hands with the governor. \"I'm sorry, sir,\" the sheriff said. \"You know about this trail. He complained for years, and I did what I could, but I only got so many men and this is a damn big county. . .\"\n\n\"I know.\"\n\n\"They came up the trail, at least ten of them. Judging by their tracks, at least eight of them were carrying a heavy load going north, but not when all ten of them went back south. One man came up the hill here and executed Joe Bob. He would probably have died anyway from that bleeding hole in his chest, but. . .shit!\"\n\n\"Yeah.\"\n\n\"The first bullet was fired from the other side\u2014\" the lawman pointed \"\u2014over there. We found a spent .223 cartridge. Probably one of them ARs. Tracks. The tracks went through the hole and up here to where Joe Bob is, and here's the second cartridge.\"\n\nHe opened his hand for the governor's inspection. Jack Hays merely glanced at the open hand, then said, \"He's lain out here long enough. You got your photos?\"\n\n\"Yes, sir.\"\n\n\"Get him out of here. Take him to the funeral home in Sanderson.\"\n\n\"Yes, sir,\" Sheriff Tejada said to the governor's back, for he was walking away, trying not to look again at his uncle's remains.\n\nBack at the ranch there was a trim, fit man in his early forties waiting beside a large pickup. His name was Joseph Robert Hays Junior, but everyone called him JR.\n\n\"They're bringing him out of there now, JR,\" the governor said, after he hugged the younger man. \"Better stay here. You don't want to see him like that. He wouldn't have wanted you to.\"\n\nJR nodded. His eyes were dry. He had seen his share of bodies in Iraq and Afghanistan and had not the slightest desire to see his father's remains.\n\nThe governor continued, telling his cousin what he knew. JR had just retired after twenty years in the army, retired as a lieutenant colonel, and was working as a consultant for a military contractor in El Paso, one supplying state-of-the-art night-vision equipment to the army. After he got the news, JR threw some things in his pickup and drove east.\n\n\"He was trying to protect his fence,\" Jack Hays said. \"They killed him and cut it.\"\n\n\"I told him to put a gate in that damn fence,\" JR said, \"but he wouldn't.\"\n\n\"No. . .\" the governor said thoughtfully. \"That wasn't him. There was no backup in him.\" He eyed his cousin. He suspected there was no backup in Joe Bob's son, either.\n\n\"They'll be back,\" JR said matter-of-factly.\n\n\"You going to wait for them?\"\n\n\"Hunting assholes in the desert was my business for a lot of years. I suspect I know more about it than Sheriff Tejada and his deputies do.\"\n\nJack Hays didn't try to talk him out of it. All he could hope for was that JR didn't get shot or caught. But JR was JR, and Joe Bob was his dad. And this was Texas. If JR shot some Mexican drug smugglers who had killed his dad, no Texas jury was going to convict him of anything.\n\n\"Fred coming down?\" Fred was the younger brother, teaching school somewhere in the Dallas area.\n\n\"For the funeral. He and his wife can't get off just now.\"\n\n\"Call me when you get the funeral scheduled,\" the governor said. \"Nadine and I will want to be there.\"\n\n\"I will, Jack.\"\n\nJack Hays hugged JR again, then went to the helicopter and climbed in. \"Let's go,\" he told the pilot.\nTWO\n\nMartial law! Rule by decree from the White House! Barry Soetoro, emperor of the United States. People had been whispering for years about the possibility, but like most folks, I dismissed the whisperers as alarmist crackpots. Now, according to Sal Molina, the president's longtime guru, the crackpots were oracles.\n\nI sat at my desk in my cubbyhole and thought about things. I wondered if there was any truth to Grafton's crack that Soetoro and company had been waiting for a terrorist incident so they could declare martial law. Well, why not? The nation was fed up with the Democrats. Seniors and the white middle class had deserted the party by the millions. Cynthia Hinton didn't have a chance. The Republicans were going to take over the government in November if there was an election.\n\nI felt hot all over. Suddenly the room was stifling. It looked as if the nation I had grown up in, the crazy, diverse republic of three hundred million people all trying to make a living and raise the next generation, was going on the rocks. And all the king's horses and all the king's men weren't going to be able to put it back together again. That must have been the thrust of Grafton's remark before Molina arrived.\n\nI felt as if I were on the edge of the abyss, like Dante's hero, staring down into the fiery pit. What next?\n\nGrafton would be gone. Like tomorrow. The agency would become another arm of Soetoro's Gestapo. Molina had implied that much.\n\nI opened the locked drawer where I kept my stuff. I had a shoulder holster and a little Walther in .380 ACP in there. Since I did bodyguard duty for Grafton, I had a permit for it signed by the director, who was Grafton. I took off my jacket, put on the shoulder holster, checked the pistol, and made sure I had a round in the chamber and the safety engaged. Put the pistol in the holster and put my coat back on.\n\nI stood there looking around. There was nothing else in my office I wanted. Not the CIA coffee cup, the free pens, the photo of me and the guys on a big campout in Africa that hung on the wall. . .none of it. I locked the drawer and cabinets, left the room and made sure the door locked behind me, then headed for the parking lot.\n\nDriving out of the lot was surreal. There were still some cars there, and people trickling out, just as there were every evening. The streetlights were on; traffic went up and down the streets obeying the traffic laws; news, music, sports, and talk emanated from my car radio. . . _and it was all coming to an end_.\n\nAs I drove I took mental inventory of my arsenal. If you live in America, you gotta have some guns, so when the political contract falls apart. . .yeah!\n\nI drove over to a gun store I had had prior dealings with. A few people in the store, about as usual. I bought two more boxes of Number Four buckshot for the shotgun, another box of .380 ACP for my Walther, and four boxes of .45 ACP for my Kimber 1911, which was in my apartment. Three boxes of .30-30s for my old Model 94 Winchester.\n\n\"Expecting a war?\" the clerk asked.\n\n\"Comes the revolution, I want to be ready,\" I replied.\n\nI used a credit card to pay for the stuff. If the future went down the way I suspected, in a few days no one would be able to buy guns or ammo for love or money. Soetoro would shut down the gun stores. Screw the Second Amendment.\n\nThen I drove over to Maryland to visit the lock shop I owned with my partner, Willie \"the Wire\" Varner. He was a black man about twenty years older than me, and had been up the river twice. Now reformed, he was my very best friend. Don't ask me why a two-time loser should be the only guy in the world I really trust\u2014besides Jake Grafton\u2014but he is. Maybe because he's so much like me. As I unlocked the front door and went into the shop, I realized that I couldn't tell him about the bomb Molina dropped, but I did have news.\n\nWillie was in the back room of the shop wiring up the motherboard of an alarm system for installation in an old house. The final innings of an Orioles game were on the radio. \"Hey,\" he said.\n\n\"Hey. Stopped by to tell you, I quit the agency this evening.\"\n\nHe stared. \"No shit?\"\n\n\"Honest injun. I am not going back.\"\n\n\"They give you any severance?\"\n\n\"Uh, no.\"\n\nHe turned back to the alarm system. \"They goin' to be lookin' for you, Carmellini?\"\n\n\"Naw. It'll be days before they figure out that I'm gone. Maybe weeks.\"\n\n\"Want to tell me about it?\"\n\n\"Just did. All I can.\"\n\nHe straightened up and gave me another look. \"And I thought I had a monopoly on fuckin' up my life. If you ain't gonna tell me nothin', just why the hell did you drive over here tonight?\"\n\nI was at a loss for words. Why did I? I knew the answer, of course\u2014because I needed some company\u2014but I wasn't going to tell him that.\n\n\"Don't think you're gonna start workin' here on salary,\" Willie declared. \"We ain't got barely enough work for me. We divide it up and neither one of us will be eatin'.\"\n\nI nodded. Stood looking around. Maybe I should just give Willie a bill of sale for my half of the place and be done with it. He would never leave the metro area, and I wasn't staying. I didn't know where I was going, but I did know I wasn't staying in Washington.\n\nI decided that was a problem for another day. Said good night and left.\n\nI wasn't ready for my apartment. Hell, I had nothing better to do, so I headed for Jake Grafton's condo in Rosslyn. I had certainly been there often enough these last few years, so I knew the way. I was going to try to find a parking place on the street, but instead decided to cruise by the building and see who was sitting outside in cars. Sure enough, a half block from the entrance there was a parked car with two men in it. They were of a type. FBI. After a while you get a feel for them. Trim, reasonably fit, wearing sports coats to hide a concealed carry, maybe a tie. Who, besides middle-level government employees, dresses like that at ten o'clock at night?\n\nI decided I didn't give a damn if they saw and photographed me. There were no parking places on the street, so I steered the Benz into the parking garage and found a spot on the third deck. Took the stairs down, crossed the street, and went into Grafton's building.\n\nGrafton buzzed the door open and I went up. Knocked and he opened the door. Callie was sitting in the kitchen. The admiral led me there and asked, \"Want a drink?\"\n\n\"Sure. Anything with alcohol.\"\n\nCallie Grafton was a tough lady, but she looked about the way I felt. Bad. \"Tommy,\" she said, trying to smile.\n\nI realized then that coming over to Grafton's was a really bad idea. But I couldn't just walk out. The admiral opened the fridge and handed me a bottle of beer. I unscrewed the top and sipped it. \"Car out front with two men in it. Maybe FBI.\"\n\n\"A dirty gray sedan? They followed me home,\" he said.\n\n\"So are you going in tomorrow?\" I asked.\n\n\"Of course,\" he said, scrutinizing my face.\n\n\"Not me. I'm done. Gonna hit the road tomorrow. I think the time has come for Mrs. Carmellini's boy Tommy to go on to greener pastures.\"\n\nThe admiral didn't say anything to that. Mrs. Grafton hid her face behind her tea cup.\n\nOn the way over here I wondered if Grafton had told his wife about the conversation with Sal Molina. From the silence and the way she sat looking at the dark window, I knew that he had.\n\n\"I shouldn't have come,\" I said. \"I'll take this road pop with me to remember you by, Admiral. Good-bye.\" I stuck out my hand. He shook it.\n\n\"Mrs. Grafton.\" She rose from the table and hugged me. Fiercely.\n\nThen I left. Pulled the door shut until the lock clicked. I took the elevator down, put the half-empty beer bottle in my side pocket, crossed the street, and climbed the stairs.\n\nThe next morning, Tuesday, August 23, I was wide awake at five in the morning. The sky was starting to get pink in the east. I hopped out of bed, showered, shaved, put on jeans and a golf shirt, and got busy packing. Everything had to go in my car, which was a 1975 Mercedes. Guns and ammo, of course, plus some of my clothes. No kitchen utensils, pots, pans, dishes, or coffee pot. No television or radio. I did decide to take my laptop and charger, but I left the printer.\n\nWhen I had made my selections and the stuff was stacked in the middle of the little living area, I began shuttling stuff down to the car in the elevator.\n\nWhen I got the car loaded, I stood in the middle of my apartment and took stock. Nothing else here I wanted.\n\nI wrote a short letter to the landlord and enclosed my key and building pass. He could have everything left in the apartment. The stuff in the refrigerator I emptied into a garbage bag and carried down with me.\n\nIn light of what happened subsequently, perhaps I should have been worried about the country and martial law and what was to come, and perhaps I was on a subconscious level. I must have suspected the future might be grim or I wouldn't have worried about the guns and ammo. Still, after I packed the car, I was thinking about what I was going to do with the rest of my life.\n\nIt was a nice problem. I had daydreamed about _afterward_ for years, after the CIA, but that eventuality was always somewhere ahead in a distant, hazy future. Now, boom, the future was unexpectedly here, and it wasn't hazy.\n\nOf course I didn't have to plot my next fifty or sixty years today. I decided that this day would be a good one to head west, following the sun. A few weeks of backpacking in Idaho or Montana would suit me right down to the ground.\n\nAlready I was late for work\u2014at Langley\u2014as if I were ditching school. Feeling rather bucked with life, I drove to a breakfast place in a shopping mall and ordered an omelet and coffee. I scanned a newspaper while I waited for my omelet. The journalists had dug up a lot more on the dead terrorists. They were from Syria, Yemen, and Iraq. The experts were speculating on where and how they acquired their weapons, all of which were legally for sale in many states in America. Two more of the Saturday gunshot victims had died, bringing the grand total of deaths to 173.\n\nAt 9:45 I was standing in line in the lobby of the suburban Virginia bank where I had my accounts. When I reached the window, I wrote a check for the amount in my checking account, leaving only a thousand bucks in the account to cover outstanding checks.\n\n\"And how would you like this, Mr. Carmellini?\" The teller was a cute lady wearing an engagement and wedding ring. The best ones are always snagged early.\n\n\"Cash, please. Half fifties and half hundreds.\"\n\nShe tittered. \"Oh, good heavens. Since it's over ten thousand, we must fill out a form. Are you sure you don't want a cashier's check?\"\n\nTitterers set my teeth on edge. On the other hand, she wasn't still swimming around in the gene pool looking for a man. I silently wished her husband luck. \"Pretty sure,\" I replied. \"Cash, please. And while you are at it, I want to close out my savings account. I'll take that in cash too.\"\n\nShe had to go get more cash from the vault, then the paperwork took another few minutes. When I had my money, a little over twenty-two thousand monetary units\u2014they gave me a little cloth envelope with the bank's name printed on it to carry it in\u2014I opened my safe deposit box with the help of one of the ladies who didn't titter.\n\nBack in my younger days, when I thought the day might come when I wanted to leave town in a hurry\u2014like today, for instance\u2014I had stashed thirty grand in cash in the box, along with a couple of false driver's licenses in various names, credit cards, and a genuine false passport. Getting that paper had taken time and money years ago, but I did it and kept the stuff. Of course, the credit cards had long expired, but they added heft to my wallet and looked good to anyone who happened to glance into my wallet while I had it open. Some people think that people with credit cards are more trustworthy than those without.\n\nUnder the money at the bottom of the drawer was another 1911 .45, an old Ithaca made during World War II with brown plastic handles and most of the bluing gone from the slide, plus two extra magazines and a box of cartridges. The pistol was marked \"United States Property M 1911 A1 US Army.\" It had either been liberated from the army's clutches many years ago or sold as surplus. It was serviceable, although it didn't have the good sights and fancy grips of my Kimber.\n\nIf there is a possibility that you might get shot at, you should at least be prepared to shoot back. In this brave new world that Emperor Soetoro envisioned, I thought the odds of getting shot at would be increased for a great many people, me included. I emptied the metal box into my briefcase, then with the help of the vault lady, who had discreetly faded while I plundered my treasure box, put the box back into its slot where it would rest undisturbed, safer than a pharaoh's sarcophagus, for all eternity, or until my annual box rent was due and I wasn't around to pay it, whichever came first.\n\nAs I was leaving the lobby with my now-bulging briefcase, Barry Soetoro was on the television high in the corner, reading from a teleprompter. That was, I had long ago concluded, his one skill set. The audio on the TV was off, so I was spared his mellifluous tones. There were people standing behind him, but since I knew Jake Grafton wasn't among them, I didn't bother to check out the crowd of toadies. I walked out of the bank with my money\u2014earned, not stolen, with taxes paid on every dime. I kinda wished I had stolen it, then I would have felt better about this whole deal. I was just too goddamn conventional.\n\nTo hell with all of it! I walked out of the bank into the rest of my life.\n\nBarry Soetoro's declaration of martial law stunned the nation. His reason\u2014the need to protect the nation from terrorism\u2014met with widespread skepticism. After all, at least three of the Saturday jihadists had entered with Soetoro's blessing, over the objections of many politicians and the outraged cries of all those little people out there in the heartland, all those potential victims no one really gave a damn about.\n\nHis suspension of the writ of habeas corpus went over the heads of most of the millions of people in his audience, since they didn't know what the writ was or signified. He didn't stop there. He adjourned Congress until he called it back into session, and announced an indefinite stay on all cases before the courts in which the government was a defendant. His announcement of press and media censorship \"until the crisis is past\" met with outrage, especially among the talking heads on television, who went ballistic. Within thirty minutes, the listening audience found out what the suspension of the writ of habeas corpus meant: FBI agents arrested select television personalities, including some who were literally on camera, and took them away. Fox News went off the air. Most of the other networks contented themselves with running the tape of Soetoro behind the podium making his announcement, over and over, without comment.\n\nDuring the day FBI agents arrested dozens of prominent conservative commentators and administration critics across the nation, including Rush Limbaugh, Mark Levin, Michelle Malkin, George Will, Ann Coulter, Bill O'Reilly, Glenn Beck, Ralph Peters, Judge Jeanine Pirro, Matt Drudge, Thomas Sowell, Howard Stern, and Charles Krauthammer, among others. They weren't given a chance to remain silent in the future, but were arrested and taken away to be held in an unknown location until Soetoro decided to release them.\n\nSenators and congressmen, from both sides of the aisle, were told in no uncertain terms that they too would be arrested if they publicly questioned the administration's methods and motives.\n\nPlainly, life in America had just been stood on its ear. All the usual suspects who had supported Barry Soetoro for seven and a half years, no matter what, through thick, thin, and transparent, rushed to find a reporter with a camera so that they could say wonderful things on television about their hero, the self-proclaimed messiah who had said when he was first elected that he would lower the level of the sea and allow the earth to heal.\n\nWhile all this was going on, Jake Grafton was fired as director of the CIA. Two White House aides arrived in Langley with FBI agents in tow and delivered a letter from the president. Grafton was summarily relieved and the assistant director, Harley Merritt, was named acting director.\n\nAs Grafton departed with the FBI agents, the two White House aides remained for a talk with Merritt about what was expected of him.\n\nThe FBI took Grafton to a federal detention center that had been set up at Camp Dawson, a National Guard facility near Kingwood, West Virginia. Grafton should have been surprised to find that the holding facility had concertina wire, kitchens, latrines, and a field full of erect army tents containing a dozen cots each, but he wasn't. Obviously someone had done the staff work to have facilities ready and waiting, with only the date that they were to be used remaining to be selected.\n\nGrafton arrived in time to shuffle through the lunch line, which contained about forty people. Most were men in their twenties and thirties, with here and there a few women salted in. The women huddled together. Everyone was in civilian clothes. He recognized several of the other detainees, or prisoners: two army four-star generals and a couple of former cabinet members. He picked up an aluminum tray from the stack, and a soldier in uniform spooned out mashed potatoes, mystery meat, and corn. At the end of the food line, he could select paper napkins and plastic tableware. No one trusted the detainees with real knives or forks.\n\nAfterward Jake was given a plastic Walmart bag for his stay, one containing a disposable razor, soap, a towel, a toothbrush, and toothpaste. The tube of toothpaste was small, TSA size, and he hoped that was an indicator of how long he would be here. He suspected it wasn't.\n\nHe still had his cell phone, but he had no charger, so he turned it off in the car on the way here. He had managed a call to Callie before he left the Langley facilities, so she knew he wasn't coming home this evening, even if she didn't know where he was.\n\nHe sat on the side of the cot he had chosen in his assigned tent. He was the only occupant of the tent, so far, but he expected plenty of company. Finally he unrolled his sleeping bag and stretched out on it.\n\nBarry Soetoro had just decapitated the American government in a coup d'\u00e9tat. Furthermore, Soetoro and his aides knew that Grafton was politically unreliable. How long they would hold him, if indeed he would ever be released, was unknowable.\n\nJake Grafton was a political prisoner.\n\nThe suspension of the writ of habeas corpus and declaration of martial law in the United States stunned the world. Abraham Lincoln did both during the American Civil War in the 1860s, so there was precedent. The Constitution itself, Article 1, Section 9, stated: \"The privilege of the Writ of Habeas Corpus shall not be suspended, unless in Cases of Rebellion or Invasion the public Safety may require it.\" Clearly, this past week there had been no rebellion, as there had been during the Civil War. What there was, Soetoro declared, was an \"invasion by terrorists,\" and in Soetoro's opinion, \"public safety did indeed require martial law.\" During the Civil War Lincoln had also declared martial law, claiming he had a right to do so to preserve the Constitution; his actions were quickly ratified by Congress and the Supreme Court. Army officers arrested several politicians, including one prominent one, Ohioan Clement Vallandigham, and closed down several newspapers. Lincoln's generals caused him more trouble than the people they arrested; the newspaper editors were quickly freed, and Vallandigham, a copperhead Democrat, was taken south and handed over to the Confederates, who didn't want him either. He wound up in Canada, slipped back across the border, and ran for governor of Ohio. Lincoln ignored him and told his generals to do likewise. Vallandigham lost the Ohio governor's race of 1864.\n\nThe Constitution was silent on Soetoro's two other declarations: the adjournment of Congress until he recalled it and suspension of all federal cases in which the government was the defendant. There was absolutely no precedent for either action, which hadn't been attempted in the history of the republic, which spanned a civil war and two world wars. Critics immediately claimed that Soetoro had unconstitutionally attempted to seize power, subordinating the legislative and judicial power to that of the executive. Strident voices compared him to Hitler and Napoleon, both of whom took over the government and made themselves dictators. Soetoro's supporters\u2014including ardent white leftists and more than ninety percent of black Americans, who had backed everything he had done in office since his first election and damned his critics as virulent racists\u2014loudly supported him now. Amazingly, those who cheered his actions were given space in newspapers and time on television, while critics weren't. Those editors and producers who were not inclined to fall in line, and most of them were, were threatened with arrest. If that didn't make them behave, they were hustled away to detention camps.\n\nSocial media websites also received government attention and were told if they allowed \"criticism of the government\" on their websites, they would be shut down. Since they had no way to stop the wired-in public from posting anything they wanted, these websites were soon shut down by their corporate owners. Pirate social media websites quickly sprang up, but unhappy people could make little noise on them in the near future. Mouse squeaks, someone said.\n\nThe result of all this in much of America was an ominous silence that afternoon.\n\nThe news that Soetoro had declared martial law and suspended the holy writ arrived like an incoming missile in Austin, Texas. Legislators crowded the governor's office and all wanted to talk to the governor, Jack Hays. And they all wanted to talk at once.\n\nState Senator Benny \"Ben\" Steiner copped a seat in a corner and listened. The consensus was that Barry Soetoro had declared himself dictator.\n\n\"Anybody have any idea of when America will get its Constitution back?\" Charlie Swim asked. He was the most prominent black politician in the state, a former Dallas Cowboys star. He was, arguably, also one of the smartest and most articulate politicians in Texas.\n\nThe hubbub subsided somewhat. Everyone wanted to know what Charlie Swim thought. \"The problem here is that Washington politicians haven't had the guts to impeach Soetoro. And I'll tell you why. He's black. They're afraid of being called racists. If Soetoro had been white, he'd have been thrown out of office years ago. Rewriting the immigration laws; refusing to enforce the drug laws; siccing the IRS on conservatives; having his spokespeople lie to the press, lie to Congress, lie to the UN; rewriting the healthcare law all by himself; thumbing his nose at the courts; having the EPA dump on industry regardless of the costs; admitting hordes of Middle Eastern Muslims without a clue who they were. . . . Race in America\u2014it's a toxic poison that prevents any real discussion of the issues. It's the monkey wrench Soetoro and his disciples have thrown into the gears that make the republic's wheel turn. And now this! Already the liberals are screaming that if you are against martial law, you're a racist; if anyone calls _me_ a racist, he's going to be spitting teeth.\"\n\nCharlie Swim wasn't finished, and his voice was rising. \"The black people in America were doing all right, working their way up the ladder, until drugs came along. Then welfare, and payments to single mothers\u2014when you pay poor people not to work and not to marry they are going to take the money. Barry Soetoro had a real chance to do something about what's taken black America down\u2014drugs, welfare rather than work, kids without wedlock\u2014but he didn't bother.\" Swim's voice became sarcastic. \"Climate change is his cause, and discrimination against Muslims. And expensive golf vacations.\" His voice rose to a roar. \"I'm sick of this self-proclaimed black messiah!\"\n\n\"That won't do any good, Charlie,\" Jack Hays said conversationally. He was standing behind his chair and now addressed the crowd. \"I have no doubt we'll hear from Washington soon, and in great detail, and when we do I'll pass it on. You'll know what I know just about as fast as I get it.\"\n\n\"What are _you_ going to do about this mess?\" someone demanded.\n\n\"What am I going to do if it rains?\" Hays said. \"What am I going to do if it doesn't? You people go back to your chambers and make speeches, hold press conferences, tell the people of Texas what you think. That's all we can do right now. Tomorrow is another day. Now git!\"\n\nAnd they did. All except Ben Steiner. A lawyer from Abilene, he had tried civil and criminal cases all over Texas for forty years. Politics was his hobby. Now he closed the door behind the last of his colleagues and seated himself in one of the chairs across the desk from Hays.\n\n\"You are avoiding the issue, Jack, and you know it.\"\n\n\"I know a lot of things I don't talk about in public,\" Jack Hays replied curtly.\n\n\"Barry Soetoro is ripping up the Constitution and declaring himself dictator. All he needs is a crown. That's indisputable. This crap about terrorism\u2014the FBI can find terrorists, and they don't have to go any farther than the nearest mosque. What's really happening here is Barry Soetoro taking out his political enemies. What are we Texans going to do about this? Are we going to knuckle under?\"\n\nHays moved around in his chair, trying to get comfortable. He rearranged his scrotum. \"You're working up to something, Ben. What?\"\n\n\"We need to secede from the Union. Declare the Republic of Texas, again.\"\n\nHays made a face. \"This isn't 1836. There are forty-nine other states and the U.S. Army, Navy, Air Force, and Marine Corps. The last time Texas got uppity, back in 1861, the roof caved in. It would again.\"\n\n\"Really?\" Ben Steiner leaned forward and lowered his voice. \"The roof has already caved in. Give me a better idea, Jack. Tell me what we are going to do if Soetoro calls off the election. If he declares himself president for life.\"\n\n\"He hasn't done that,\" Hays shot back.\n\n\"Not yet,\" Steiner admitted. \"What he has done is declare martial law, adjourn Congress, shut down the courts, muzzle the press, and arrest his critics. How are we going to preserve our way of life, preserve our liberty, preserve our democracy with a dictator in the White House?\"\n\n\"I don't know,\" Jack Hays admitted. \"I need to think on it.\"\n\n\"Better not think too long,\" Ben Steiner said as he got out of his chair. \"There's a lot of people in Texas who won't think long at all. They hate that son of a bitch and they won't take this lying down. While you're thinking, think about how to head them off if they get out of hand. If you don't, or won't, or can't, we're talking anarchy. No man's life or property will be safe. Think about that. Also think about what you're going to do if Soetoro sends some federal agents to drag you out of this office and throw you into a prison somewhere. Until such time, if ever, that he decides it's safe to let you out. Think about that too.\"\n\nBen Steiner walked out of the governor's office and closed the door behind him.\n\nJack Hays put his hands on his face and tried to force himself to relax. Various right-wing groups in Texas had argued for independence for years. They were the lunatic fringe, the village idiots. Hays had kept his distance. Now Ben Steiner had taken his turn at the independence podium, and he was no crackpot.\n\nThe way people lived in early-twenty-first-century Texas depended on the American monetary system, Social Security, military retirement, banks stuffed full of U.S. Treasury bonds as their capital, the national telephone grid, the power grid, all of that. Companies here paid wages to Texans to manufacture goods and sold them all over the United States\u2014all over the world\u2014and the stores in Texas that supplied the stuff of life were filled with goods manufactured all over the world; Texans used their paychecks to pay for what they needed. Independence, he thought, would take a civil war, and that would destroy the very fabric of life for a great many Texans. Cutting Texas out of the United States would be like trying to cut Mona Lisa's face out of her portrait and arguing that the operation wouldn't harm it.\n\nJack Hays didn't believe it could be done. In this interdependent world, Texas had to be part of the United States, a state in the Union.\n\nOr did it?\n\nHe was thinking about his deceased uncle, Joe Bob Hays, and the drug smugglers who killed him when the phone on his desk summoned him to duty.\nTHREE\n\nThere were five people in Grafton's tent, all males, when he went in after sunset. Everyone introduced himself: three civil servants, one broadcaster, and one congressman.\n\n\"Where are the women?\" Grafton asked.\n\n\"They have their own tents,\" he was told. \"Politically incorrect, but those are army regulations.\"\n\n\"If Elizabeth Warren only knew.\"\n\nThe tentmates had just arrived, and were still outraged that they had been arrested. Being taken in handcuffs from their homes or work, with family or colleagues watching, and physically transported to Camp Dawson, a three-hour ride from Washington, had filled them with adrenaline that had to be burned off. They had been frightened, humiliated, and shamed, and now they were very angry. They told each other their stories and talked long into the night while Jake Grafton slept.\n\nOn his second evening in Camp Dawson, Jake Grafton ran into _Washington Post_ columnist Jack Yocke in the chow line. Yocke was in his late thirties, lean and ropy, with shoulder-length hair and a fashionably grizzled face, the lumberjack look. His name was pronounced Yockkey.\n\n\"When did you get here, Admiral?\" a plainly surprised Yocke asked.\n\n\"Yesterday at noon.\"\n\n\"Seems to be a lot of people here,\" Yocke said, looking around.\n\n\"Welcome to the American gulag archipelago. I think I was one of the first, but there were a bunch of people already here. Spies, I think. Stool pigeons. I would be careful what I said and who heard it, if I were you.\"\n\nThey ate together in silence, put their leftovers in a large garbage can, and stacked their trays, then went to sit under a shade tree near the wire, where they could talk privately.\n\nGrafton managed to get the first question in, always a feat with Yocke. \"Did you piss on the establishment or did they dump you here on general principles?\"\n\n\"I'm an unreliable bastard. I wrote a column that was uncomplimentary to the administration, and a political apparatchik in the editor's office called the troopers. Needless to say, I don't think my column will be in tomorrow's paper.\"\n\n\"Brave editors.\"\n\n\"They were threatened with arrest, their families were also going to be arrested, their bank accounts and property seized, and the IRS would prosecute them. Not audit them, but prosecute them. The only thing they weren't threatened with was execution.\"\n\n\"Why did you flout them?\"\n\n\"Stupid, I guess. And you?\"\n\n\"The same.\"\n\n\"There's a lot of that around. Soetoro is going to be surprised.\"\n\n\"They've made their preparations. The administration didn't decide this after they got a look at Saturday's terror strikes. They've been getting ready for this for years.\"\n\n\"When this is over,\" Yocke mused, \"someday, the only heroes will be the people who stood up to them and went to prison.\"\n\n\"Martyrs,\" Grafton murmured.\n\n\"Christians versus the lions.\"\n\n\"Martyrs don't win wars,\" Grafton stated. \"That's a law, like gravity. So what's happening out there beyond the fence?\"\n\n\"The country's falling apart. Inner-city riots: Chicago, Detroit, Saint Louis, LA. Just getting worked up, getting the car fires set. Agitators and race-baiters screaming about overturning white America once and for all. What they are going to do is loot Walmarts and Safeways and burn down the inner cities, then starve. We've got martial law, but there's no National Guard, no soldiers, no police stopping the rioters, there's no fire departments putting out the fires, and there's apparently no Border Patrol at the border. Go figure.\"\n\nGrafton didn't say anything.\n\n\"The cops have got the message. Let it burn, baby.\"\n\nYocke got out his cell phone and checked his messages.\n\n\"You have a charger for that?\" Jake asked.\n\n\"Yep. All I need is a place to plug it in. If cell phones go flat, civilization as we know it will be stone cold dead. Teenagers, millennials, reporters, and real estate agents will go through seismic withdrawal and drop dead left and right.\"\n\n\"The camp authorities will pass out chargers when they can lay hands on some,\" Jake said.\n\n\"Why?\"\n\n\"The NSA can listen to every cell phone and telephone transmission in America. They've been working on it for over a year. Soetoro's orders. It used to be all they got was your number and the number you dialed. Now they can record the conversations digitally and mine them for key words or names. They _want_ you to talk on your cell phone. That's why they didn't confiscate the things.\"\n\nJack Yocke sat with his cell phone in hand watching the shadows lengthen. Finally he put the device on the ground, took off his shoe, and pounded on it with the heel until the glass screen broke. Then he threw it over the fence.\n\nAfter a while Yocke calmed down. \"So when do you think we'll get out of here?\"\n\nGrafton snorted. \"They didn't let me pack my crystal ball.\"\n\n\"A few days, months, years?\"\n\nWhen Grafton remained silent, Yocke decided to answer his own question. If you are going to make your living writing newspaper columns, you must have opinions, on everything. Yocke did. Almost every living human had opinions, but no one wanted to hear them. People paid to read Yocke's because his were better thought out and expressed. \"People are upset and angry right now, but few if any are willing to risk everything they own, everything they have, even their lives, to oppose Soetoro and the federal government. That will change over time. Government oppression in the short run pisses people off. In the long run it transforms them into revolutionaries.\"\n\n\"Conquer or die,\" Grafton mused. \"Too bad you weren't there at the White House when the aides discussed how to keep Soetoro in office for life.\"\n\nYocke wanted to talk. Like most writers, his head buzzed with words. Sooner or later he had to spew them out so that he could have room to think about something else. \"Being a revolutionary is very romantic,\" he said. \"It isn't for everyone. The hours are brutal, you can get seriously hurt or dead, even if you win you'll be a pauper, and you'll probably wind up unhappy with whoever emerges from the chaos as the head dog. Sooner or later the optimistic revolutionary becomes the disillusioned veteran. If he is still above ground.\"\n\n\"Was this your column that won't get printed?\"\n\n\"Yeah. Good solid stuff.\"\n\n\"So, Jack, are you willing to kiss your pension, 401(k), Mazda sports car, and Washington condo good-bye and sign on for the voyage? Are you ready to pledge your life, your fortune, and your sacred honor?\"\n\n\"Not yet, Admiral. I'm working up to it. Soetoro is dragging me to it by the hair. He's dragging a whole lot of people there. If Soetoro doesn't stop this shit pretty soon, there is going to be a major explosion.\"\n\n\"He thinks not.\"\n\n\"Barry Soetoro is a damn fool. President of the United States, and he doesn't know Americans.\"\n\nOn Thursday, the twenty-fifth of August, Jack Hays and his wife, Nadine, rode a helicopter from Austin to Sanderson, Texas, where a funeral home had Joe Bob Hays laid out. JR and his brother, Fred, and Fred's wife and eldest son were there. The grandson was only four. JR had been divorced for the past ten years. His ex-wife had custody of their children. The wife had had an affair while her husband was in Afghanistan, and divorce followed. She didn't remarry. The kids were teenagers now and knew everything about everything. JR wrote them a note about their grandfather and mailed it, and that would have to do.\n\nThe sheriff, Manuel Tejada, was there with some of his deputies in uniform. One of them, a man with bright, garish yellow and green tattoos that started at both wrists and ran up his forearms, took the time to shake JR's hand and tell him how sorry he was. \"Knew your dad,\" he said. \"Good man.\" His name was Romero, according to the silver name tag he wore over his left shirt pocket.\n\nThe sheriff, his deputies, the mayor and county commissioners lined up to shake hands with Governor Jack Hays. Funerals aren't normally places to talk politics, but they were very worried about terrorism and martial law and asked Hays what it meant.\n\n\"Washington hasn't said much. We'll know more soon,\" was his stock answer. Actually, he was lying. Washington had sent him a directive that ran over a hundred pages. He had scanned it and turned it over to the attorney general for comment. His aides had run off some copies. He gave a copy to Ben Steiner and one to Charlie Swim, and told them to keep their mouths shut. He had taken another copy home and he and Nadine had read it.\n\nAs he stood listening to the preacher drone on, he was thinking of some of the major points in the directive. In effect, Soetoro and his administration were deputizing the state government to enforce their orders in Texas. That was Nadine's verdict as she read the thing. She was an archaeology professor at the University of Texas and considered herself middle of the road politically. In Texas, that put her a little left of center, but not much. At the university, that made her a conservative oddity among the faculty, most of whom didn't think much of her husband either.\n\nHays glanced around. Against the back wall stood two Texas state troopers in uniform who had flown out to Sanderson in the helicopter with him. They were now his official bodyguards. This morning he asked them point-blank: \"What will you do if federal agents try to arrest me?\"\n\n\"They better come a-shootin',\" the little one said. He was the senior man. The other man merely nodded.\n\n\"I doubt if it will come to that,\" Jack Hays told them, \"but it might.\"\n\n\"You're our elected governor. Ain't nobody in Washington gonna drag you outta the state house. Period.\"\n\n\"Thanks.\"\n\n\"Them guys and gals at the FBI office in Austin, some of them are Texans too. If they get orders to come and get you, they'll call us first. They promised.\"\n\nAfter the service, Jack and Nadine stood on the lawn and watched the funeral home personnel load Joe Bob's coffin in a hearse. JR and Fred and his wife were going to follow the hearse to the ranch, where Joe Bob would be interred beside his wife, who had died of cancer ten or eleven years ago. No, Jack Hays thought. Twelve years ago. Damn, but time slides right along.\n\nBefore they closed the rear door of the hearse, he went over to the coffin and touched it. \"Good-bye, Uncle Joe Bob.\" He started to say more but choked up. \"Good-bye,\" he whispered and walked away.\n\n\"Drug smugglers,\" Nadine said as they walked to the helicopter, which was a block away in the courthouse square. Texas flags hung everywhere, from windows and poles mounted on buildings. \"They killed him,\" she said, \"and now their poison is ready for consumption all over.\"\n\n\"Ready to supply the addicts and recreational users who don't give a damn about violating the law or who gets killed,\" Jack Hays muttered, \"as long as they are having a good time.\"\n\n\"Why haven't we sealed that border?\" Nadine asked.\n\n\"We tried,\" he shot back. Nadine knew that. He had tried and the federal government sued and the judges said only the feds could control the border. We have to leave it open so the illegals can get in, Jack Hays told himself. Can't take a chance on pissing off the Latino voters. And all those illegals who Soetoro wants to turn into voters. Hays was in a foul mood. Drug smugglers, now Soetoro and his martial law. It's a hell of a world we live in.\n\nHis cell phone rang. He looked at the number. His aide.\n\nThe engine on the helicopter began to make noise.\n\n\"Yes.\"\n\n\"The Houston police got troubles. A riot broke out several hours ago in the projects. They are burning cars and building barricades. Doing some looting. Some black congresswoman is shouting into microphones about the racist right-wing conspiracy trying to keep people of color down.\"\n\nHe was tempted to order her arrested for inciting a riot, but that would only pour gasoline on a fire. \"I'll be back in Austin as soon as I can,\" he told the aide. \"Get out the riot plan and act on it.\"\n\n\"Yes, sir.\"\n\nHe got in the back of the helicopter with Nadine; the two troopers climbed aboard after them.\n\nJack saw JR watching as the machine lifted off.\n\nAfter the interment, Fred Hays and his wife and son shook hands with JR, got into their car, and started driving back to Dallas. Fred and JR had just inherited a twenty-two-thousand-acre ranch in a very dry corner of Texas, and now didn't seem to be the time to discuss what they were going to do with it. Fred and his wife were schoolteachers, had two kids, and needed every dollar they could get. Neither wanted to live along the Rio Grande miles from civilization\u2014if Pumpville, Texas, was civilization. Fred had grown up on that ranch and that was precisely the place he wanted away from when he went to college. He had never come back except for brief visits. And his parents' funerals.\n\nJR, on the other hand, had spent too many years in Iraq and Afghanistan to look at desert chaparral with affection. The ranch was a big, windy, dry place, and in August hot as the doorstep of Hell with the fire doors open. His grandfather had settled here way back when because the land was cheap. It wasn't worth much now, either. His father had stayed because he loved it, and he had gone broke there. Oilmen had drilled some exploratory wells yet never found anything. Probably never would. The place was mortgaged for the fence and exotic animals. If JR and Fred didn't sell it, they'd need to find a way to make money to pay the bank. Hosting hunters was probably the only way.\n\nMaybe, JR thought sardonically, he should sell it to the dope smugglers.\n\nJR gave his father's sole employee a check worth two weeks' pay. \"Take some time off. Visit your family. I'll call you when we need you. We'll probably sell the place, and while it's for sale we'll need someone to look after it and keep the fences repaired, so we don't lose the animals.\"\n\n\"I don't want to get shot by them dopers,\" the hired man said.\n\n\"I don't blame you. Just stay the hell out of their way and fix the fences in the daytime. You could do that, couldn't you?\"\n\n\"I reckon.\"\n\n\"Two weeks. See you then.\"\n\nJR went in the house and called his boss in El Paso. Quit his job on the phone. \"I won't be coming back. Dad's dead and there's the ranch.\"\n\n\"I understand.\"\n\n\"I brought some of the company's stuff with me, and I'll bring it all back in a couple of weeks.\"\n\n\"Sure. Sorry about your dad.\"\n\nJR inventoried the grub in the house and his father's meager collection of weapons. The sheriff had returned the Marlin, with its nightscope. JR looked it over. It seemed intact and should be workable if he charged the battery, but he had a much better one under the rear seat of the pickup. There was a twelve-gauge pump shotgun and an old thirty-eight revolver, a double-action Colt that his father had used to execute pigs years ago, when he kept pigs and cured his own hams and ate his own bacon.\n\nThen JR got in his pickup and headed for Del Rio, eighty miles away. He drove fast, so he got there before the stores closed. Went into the first gun store he saw. The man behind the counter was sitting on a stool watching television. He wore a holstered pistol on his belt.\n\n\"I need to buy a couple of guns,\" JR said. \"And some ammo.\"\n\nThe proprietor gestured toward the television. \"Barry Soetoro says he is shutting down all the gun stores nationwide. We ain't supposed to sell guns and ammo anymore to anybody but law enforcement. _Fuck_ that raghead commie son of a bitch. I ain't seen nothin' in writing from the ATF, and until I do I'm still open. Sell you ever'thing in the whole goddamn store if you got room on your credit card.\"\n\n\"Not that much. I'll limit myself to a small fraction of your inventory.\"\n\n\"Help yourself. I'm gonna sit here and watch the riots. Put what you want on the counter and we'll dicker. I'm easy, long as you're not a convict or illegal chili-picker and you've stopped beatin' your wife. Comrade Barry is gonna put me out of business pretty damn quick and I'll need some money until the welfare checks start arrivin' in the mailbox.\"\n\n\"You have any black powder?\"\n\n\"Six or eight cans of the stuff. It's on a shelf in the back. Help yourself.\"\n\n\"I have a cannon. Need some fuses for it, too.\"\n\n\"Same place. I supply the local pyro club, you know, the nutcases that make their own fireworks. Got all the stuff to make their rockets go up and pop. Take all you want. That asshole Soetoro will probably shut them down too and I can't return that stuff or eat it. I'll probably end up piling it up and setting it afire in my backyard.\"\n\n\"You have a big backyard?\"\n\n\"Couple hundred acres. That's where the pyro club does their thing.\"\n\nAn hour later when JR Hays paid for his purchases, the proprietor tossed in a couple of NRA bumper stickers into one bag and two that said \"Fuck Soetoro.\"\n\n\"Classy,\" JR said.\n\n\"Yeah. Kinda to the point. I'm all outta the ones that say 'Soetoro Sucks.'\"\n\nJR Hays loaded his purchases into his pickup and visited the local hardware store. While there he purchased four five-gallon cans for gasoline, among other things. At the supermarket he stocked up on canned goods, dry beans, two cured hams, bacon, and coffee. He hit the liquor store for two big bottles of bourbon and a case of beer. As people in this sparsely populated country normally did, he stopped at the filling station on the edge of town, topped off the truck, and filled his gas cans.\n\nHe took his time getting back to the ranch. It was after eleven o'clock when he closed and locked the gate behind him, drove the half mile over the rutted dirt road to the ranch house, a low single-story with two bedrooms and a bath, with a telephone but no TV, and got busy carrying his purchases inside. After he had his food and hardware put away, he opened one of the bottles of bourbon and poured himself a drink, neat, just the way Joe Bob used to drink it. He turned out the lights and went out on the ramada to escape the heat of the house. Sitting there sipping whiskey, he could hear the whisper of the wind in the brush. Somewhere a coyote howled.\n\nAbove him, the obsidian sky was full of stars.\n\nWhen Jack Hays got home from the state capitol, a Texas flag was stirring on the flagpole in the yard. It was always there, but tonight he paused to look at it. Inside, Nadine was watching television. He flopped on the couch and watched a little in silence. The \"ghetto rats,\" as he called them when reporters weren't around, were burning and looting in Houston, St. Louis, Chicago, Detroit, Los Angeles, and Philadelphia. Screaming about the right-wing white conspiracy.\n\n\"How do they know it's whites?\" he asked Nadine.\n\n\"All right-wingers are white Republicans. Ninety-eight percent of blacks are Soetoro Democrats. You know it, I know it, everybody knows it. Soetoro lit the fuse and it's burning.\"\n\nWhen the television people began a commercial, Nadine killed the savage beast. In the silence that followed, he told her about more of the federal government's demands. And about his talk with Ben Steiner.\n\nNadine listened in silence and sipped Chardonnay. When he ran out of words, he went to the bar and poured himself a drink, vodka over ice. God knows, he needed it. What a hell of a day!\n\nSeated again near Nadine, he sipped the liquor. \"I feel like I'm chained to a railroad track with locomotives coming fast from both directions. Soetoro is ripping up the American Constitution and there are a large number of people in Texas who would rather fight than submit. Lincoln must have had similar feelings when he watched the Southern states pass secession resolutions. We're headed for a smash and I haven't a clue what to do about it.\"\n\n\"Maybe Ben Steiner is right. Texas should become its own country.\"\n\nJack Hays snorted. \"Texas will become a nation over Barry Soetoro's dead body. If he lets Texas go, a lot of other states will follow. Why should people who work for a living pay taxes to provide welfare to all those rats in the center cities? Explain that one to me.\"\n\n\"Extortion?\"\n\n\"Pay or we'll burn it down and live in the ashes. The only people who worry about that kind of logic are politicians.\"\n\n\"Texas could make it as an independent nation,\" Nadine said, eyeing her husband.\n\n\"Horseshit. American dollars are our currency\u2014\"\n\n\"Issue your own currency, backed by the state's full faith and credit. That's easy enough.\"\n\n\"Hundreds of thousands of people rely on Social Security and federal and military retirement. We can't abandon them. Without those pensions\u2014\"\n\n\"Texas can assume those obligations.\"\n\nHe stared at her.\n\nNadine took another sip of Chardonnay, then said, \"If people paid income and Social Security taxes to Texas instead of the federal government, and if Texas didn't have the federal debt to service, I suspect that the finances would be pretty close to a wash. Dollar for dollar, in and out. Texas could guarantee U.S. government bonds held by Texas banks and pension funds. If you made welfare recipients who are able-bodied work for their check or forfeit it, that would help a bundle. And make welfare recipients take a drug test. You know, straight out of Charlie Swim's platform. No more money for single women to have kids.\"\n\nShe leaned forward, pleading her case. \"Texas has energy to sell to the world, a great banking system, world-class hospitals, automobile factories, cutting-edge high-tech industries, a solid agricultural base, and we're on the Gulf Coast so we can import and export. Texas has an annual GDP of 1.6 trillion dollars. That is a larger economy than the state of New York, just a little less than California. Texas generates roughly ten percent of the economic activity in the United States. Our Texas economy is a third larger than Mexico's, just ten percent behind the United Kingdom's. If Texas were an independent nation, ours would be the twelfth-largest economy on earth, a smidgen less than Canada, but more than Australia, Spain, or Switzerland. And you think Texas couldn't go it alone?\"\n\nJack Hays eyed his wife coldly. \"I didn't know you were an independence crackpot.\"\n\n\"I'm not. But the people of Texas will not live in a dictatorship. _Will not_.\"\n\n\"The United States won't let us go without a fight.\"\n\n\"We're heading for a fight regardless,\" Nadine said flatly. \"Even if independence isn't your end game, it might give you leverage to demand a return to constitutional government on the federal level. Texas has a hell of a lot better hand than you think.\"\n\nJack Hays took a swig from his drink and sat staring at his wife. \"We could seal the border,\" he suggested. \"Demand the Mexican government stop allowing drug smugglers and illegals to cross. We could seal the border so tight a bat couldn't get across.\"\n\nNadine put her hand on his arm. \"Sure you could, but you'd need to make it clear that no one is against immigration _per se_ , from Mexico or anywhere else. The problem is _illegal_ immigrants; they're swarming in faster than we can absorb them in the schools or in the labor force or with social services. When illegal, unskilled laborers flood the market, it's our own low-skilled citizens\u2014black, white, brown\u2014who pay the price. People understand that, and they understand that it's high time someone stood up for _them_. So sure, seal the border, cut off all trade to Mexico if necessary, and force Mexico to patrol its own borders and crush the drug trade that does even more harm to Mexico than it does to us. Make Mexico an offer it can't refuse. Not a single dollar, truck, railroad car, or immigrant, legal or illegal, crosses the border until Mexico cleans up its own house.\"\n\n\"That might precipitate a revolution in Mexico,\" Jack Hays said. \"Or Mexico might declare war on us.\"\n\n\"Another Mexican revolution or another Mexican-Texas war, let it come,\" Nadine shot back. She had steel in her.\n\nJack Hays wasn't sure he bought all that. And yet, \"We've been Mexico's safety valve for a long, long time,\" he admitted.\n\n\"Think about it,\" Nadine said, and finished her wine. \"I'm sorry about your uncle. His was a needless, useless death. And as long as we leave that border open to drug gangs and criminals, we'll have more needless, useless deaths. I'm going to bed. I've had all I can take today.\"\n\nJack Hays was tired too, but he sat in the silent house thinking. About his uncle. About the possibilities of a free Texas, out from under the yoke of Barry Soetoro, Washington bureaucrats, and a feckless Congress stymied by cries of racism. Was freedom worth all the blood it would take to reap the benefits? No one but God knew how much blood freedom would cost.\n\nHis thoughts drifted back to the American Revolution. The revolutionaries then knew the British would fight. Great Britain had the finest navy in the world and a solid, although small, professional army. All the colonists had were farmers and a dream.\n\nAnd yet, who today would argue that the lives of American patriots killed during the revolution had been squandered? Or the lives of the Texans who fought at the Alamo and at San Jacinto? Sometimes those half-seen, fog-shrouded dreams of national destiny and the unpredictable future must be anointed with blood to make them reality. Not someone else's blood, but yours. Or your son's or daughter's. Your blood is your gift to future generations.\n\nHe was thinking of the Texans at the Alamo who knew they were doomed but fought to the last man. Then the telephone rang. He looked at the number before he answered it. A 301 area code. The Maryland suburbs of Washington, D.C. There was no one in Washington he wanted to talk to at that hour of the night. After ten rings the phone fell silent.\n\nPerhaps he should think of independence as a maneuver, not an end in itself. The name of the game is bettering the lot of your constituents. Nadine was right: in politics there is always an end game. You say you want A, the opposition offers E or F, and all of you settle for C. Or B or D. Something in the middle. The real problem, as Jack Hays saw it from Austin, was that Barry Soetoro refused to settle for anything less than the whole enchilada, everything he wanted, which in a democratic republic is simply impossible. He was the prophet, the messiah, and he was driving a stake through the heart of the Great Republic.\n\nJack Hays was a good working politician. All he wanted was to move the needle in his direction. He well knew that every political question is not black or white, but some shade of gray. He still believed that most Americans were well-meaning people, not ideological crazies, and that compromise was possible.\n\nThe telephone rang again. He looked at the number and saw that it was the state director of the Department of Public Safety, Colonel Frank Tenney. The man wanted to talk about the riot in Houston. Hays listened carefully, grunted twice, said yes three times, then hung up.\n\nAfter he finished his drink, he turned off the lights in the living room and stretched out on the couch.\n\nJake Grafton was wide awake at four in the morning. The camp was lit only by floodlights on the perimeter fences, yet there was just enough illumination leaking through the front flap of the tent to see by. All the cots were occupied. It was August and hot and the cicadas outside, and the farting, snoring, deep-breathing sleepers inside made it anything but silent.\n\nGrafton had spent the evening talking to his fellow detainees. They were almost all white and perhaps forty years old or older. Some had been arrested at home and allowed to bring their medications; others had been arrested at their places of business or in restaurants or golf clubs or bars. The police or federal agents knew whom they wanted, and they came and cuffed them and led them away without much fuss or bother. Several said they were pretty liquored up and loudly denouncing Soetoro and the feds, but the cops treated them decently anyway. Maybe the fact that they were spouting anti-government sentiments when arrested made a deeper impression on the witnesses.\n\nThe detainees were small-business men, middle or senior managers or officers in major enterprises, civil servants, state or county politicians, a few preachers, a lot of military and civil retirees. A couple of sheriffs. Basically, the feds had taken a large sample of white America. Apparently federal officers had taken a similar sample from the female population; the women were housed in other tents at Camp Dawson, and males and females mingled inside the compound until lights out was called. The detainees were a talkative bunch, gathering in ever-shifting groups, talking, talking, talking. They also gabbled endlessly on cell phones to the folks at home.\n\nA lot of these people needed medications, and they didn't have them. Grafton thought this meant the detention was intended to be only for a short period, or whoever had planned it had planned it poorly. After many years spent in large bureaucracies, he suspected the latter was the case.\n\nGrafton got up from his cot and headed for the latrine. Once outside the tent, he pulled the cell phone from his pocket and turned it on. In a moment the device locked onto the network. Still had a charge.\n\nHe pushed the buttons and held it to his ear. He could hear the ring signal.\n\n\"Jake, is that you?\" Callie's voice.\n\n\"Yes. I\u2014\"\n\n\"Where are you?\"\n\n\"Camp Dawson. It's a detention facility in West Virginia.\"\n\n\"Are you okay?\"\n\n\"Oh, sure, Hon. Got a cot in a tent and they feed us three times a day, all the food anyone wants.\"\n\n\"Jake, your name was in the paper this morning. The government said you are being investigated to see if you were a member of the conspiracy that planned to assassinate the president.\"\n\n\"Who said that?\"\n\n\"Some spokesperson for the FBI.\"\n\nSo Sal Molina was correct. Jake changed the subject. \"Are you doing okay?\"\n\n\"Oh, sure. Missing you and worried stiff. Why didn't you call sooner?\"\n\n\"They are monitoring and recording all telephone calls. All of them.\"\n\n\"Oh,\" Callie said, and fell silent.\n\n\"Talk to me,\" Jake said. \"I need to hear your voice. Talk about Amy and the grandbaby.\"\n\nHe leaned against the cinderblock latrine, closed his eyes, and listened to Callie's voice. She had been his rock for so many years. He was damned lucky to have had her to share his life with, and he knew it.\n\nWhen they finally broke the connection, Jake Grafton stood looking at the ten-foot chain-link fence topped by three strands of barbed wire, with guard towers at the corners. This thing wasn't built overnight. Fence, latrines, sewage and water lines, showers, kitchens with natural gas stoves, electric refrigerators, concrete pads for the tents. . .construction must have taken months. The phone in his hand rang. He looked at the number. Tommy Carmellini.\n\n\"Hey, Tommy.\"\n\n\"I heard you are now famous, Admiral. Saw the news on television last night when I was eating dinner. Been trying to call you.\"\n\n\"My fifteen minutes.\"\n\n\"Where are you?\"\n\n\"Camp Dawson, West Virginia.\"\n\n\"You got a charger for that phone?\"\n\n\"I can get one. Why?\"\n\n\"Keep it charged and on. I may want some investment advice. The stock market has the giggling shits, and you know how I am about bargains.\"\n\n\"Sure.\"\n\n\"Don't bend over to pick up the soap.\" And Tommy was gone.\n\nJake snorted, smiled, and put the phone in his pocket. Tommy Carmellini was one of the good guys he had known through the years. Amazing that there had been so many.\nFOUR\n\nI turned the iPhone off and looked at the ceiling in the motel room. Since I heard that news broadcast while munching a burger at the bar of a TGI Friday's at a little town in Ohio, I had tried Grafton's phone eight times before midnight, and two times since. Then, voila!, he answered.\n\nNot that he had anything to say. I remembered that classified file that crossed his desk about the NSA going to comprehensive monitoring of all American telephone conversations. And I well knew how good they were at triangulating cell phone signals. They could put you within a few meters, whether you were using the phone or not, just as long as it was logged into a network. I was on teams that used that technique to find wanted terrorists in Pakistan and Syria and Yemen.\n\nThe way to defeat that was to wrap your phone in tinfoil. So I wrapped mine back up and put it in my pocket.\n\nThe thing that bothered me was the announcement by the FBI that former CIA director Jake Grafton\u2014note that \"former\"\u2014was being detained and investigated for a possible role in the right-wing conspiracy to assassinate the president. They could have just locked him up and thrown away the key, but no, they decided to create a conspiracy to help justify martial law. I had no doubt when the trolls in the White House were finished writing this fiction the guilty bastards would make quite a list. I might even be on one of them. Along with the many enemies of the administration who didn't believe in global warming or Soetorocare or his give-a-pass-to-terror treaty with the death-to-America regime in  Iran. Soetoro's enemies would be in deep and serious shit that no doubt would ruin them for life. Maybe they would get a show trial before a military commission. And afterward, be put against a wall in front of a firing squad, or permanently locked in a cell somewhere to figure out where they went wrong. Barry Soetoro had that in him. He was the savior of the planet, after all.\n\nSo the question became, what was Mrs. Carmellini's little boy Tommy going to do about it?\n\nWell, at least I knew where Grafton was. Tonight. I suspected they would not keep him long at Camp Dawson. They would want him to sign a confession they were busy writing now, so I suspected they would move him soon and go to work on him with torture and drugs.\n\nPersonally, I didn't give a damn what he signed. I had to get to him before they killed him.\n\nI crawled out of bed, took a shower, and shaved because I had no idea when I would get another chance, then loaded my stuff into my car. I paused for a good look at the Benz. What an impractical car. I needed a pickup. Tomorrow, maybe.\n\nI filled the car at an all-night station, got a cup of coffee, and pointed the front bumper east. There wasn't much traffic. The sky lightened up and the tires hummed on the pavement and I passed some trucks. I left the radio off.\n\nNormally I don't think much about politics. I am like most people, I suppose. I get wrapped up in the business of earning a living, giving pleasure to select members of the opposite sex, spending time with friends, and following the fortunes of my favorite sports teams. I vote for people to represent me at every little meeting from city council to Congress and the White House; they can worry about the public's business, about filling the potholes in the streets, the state of the sewage treatment plants, and how much, if any, foreign aid we should give to Egypt: I vote for them because I don't want to do that stuff, and they say they do.\n\nAnd yet, they need to stay within certain boundaries. I don't want them messing with me any more than they absolutely must. I am choosing my path through life: I want to be responsible for my choices and the results.\n\nJust like most people.\n\nI sat there driving through America wondering about Barry Soetoro and his disciples. I have never trusted people who think they know how everyone else should live, and demand those other people obey. I am not a good follower.\n\nAaugh!\n\nThe highway spun along toward the horizon and the sky got lighter. Another day in America!\n\nWhen Jack Hays woke up on his couch that Friday morning, Nadine was leaning over, brushing her lips on his. She liked to wake him with a kiss.\n\n\"The coffee is on,\" she said, and went back toward the kitchen, where the cook reigned. Jack padded along behind and found the cook wasn't in yet.\n\nWith both of them sipping coffee, Nadine said, \"You are going to have a hell of a day.\"\n\nHe nodded. \"I think it'll come to a boil today, or tonight.\"\n\n\"What are you going to do, Jack?\"\n\n\"Ask God for the wisdom to make the right decision and for the courage to see it through.\"\n\nShe rested her head on his shoulder and they stood holding each other, feeling the warmth of each other's bodies.\n\nJR put his Beretta 9-mm in his belt and went for a tour of the ranch in the pickup. He wanted to see the terrain again, to refresh his memory, to see how it had changed through the years. Joe Bob had built some shooting stands here and there, boxes for hunters to stand in fifteen or twenty feet above the ground. The sports would climb up there with their rifles, hunker down, drink beer, and wait for something wonderful to wander into range, where they would assassinate it.\n\nJR climbed up into several of the stands just to look at the terrain. Shooting at people from one of these things, with people shooting back, would be suicidal.\n\nSo what were the possibilities? Ambush the bad guys as they exited their vans in Mexico, or on the trail to the river, or as they crossed the river, or cutting the Hays fence, or somewhere on the Hays land, or out near the highway as they threw the backpacks over the fence, or anywhere along the return journey.\n\nHe saw no people during his tour, but he did spot two kudu. Gorgeous creatures.\n\nAny ambush site would have to allow him to shoot, move, and survive. The shooting would be easier with his state-of-the-art night-vision equipment.\n\nWhat if he got two or three of them? Or five or six? Those who escaped would tell their bosses back in Mexico, and next time he would be facing a company of hired killers, perhaps as many as fifteen or twenty heavily armed gunmen with automatic weapons.\n\nLate in the afternoon, JR got out his new AR-15, cleaned it thoroughly, and mounted a scope on it, a regular 3 by 9 variable. He suspected the battle might drag on into the morning, and he should be well armed if it did.\n\nAfter fifty shots he was sure of the scope's zero and comfortable with the trigger. He took the rifle into the house and opened all the windows to let the breeze air out some of the heat. He cleaned his rifle thoroughly again. Then he got busy fixing dinner. Poured some bourbon and drank it as he ate out on the ramada with the sun setting.\n\nWhile JR was scouting the ranch, Jack Hays was under political siege in Austin. The Texas independence crowd was getting really worked up, especially after they saw copies of the directives\u2014there were four directives, so far\u2014about life in an America ruled by martial law under Barry Soetoro. The press was to be censored; television shows preapproved; news would be government press releases, which would be read without comment; and military courts would replace civilian ones. Gun sales were forbidden, and all guns would be turned in to military arsenals that would be designated in a few weeks.\n\nThe directives said nothing about the upcoming November election, but the feds obviously were planning a long spell of martial law, so pessimists could read between the lines, and did.\n\nMeanwhile, inner-city riots around the country were getting worse, as the civil authorities let crowds burn and loot. Any persons in the riot zones were fair game for the mobs. The military that now were under federal control, the U.S. Army and National Guard, did nothing. Government spokesmen on television blamed the right-wing conspiracy, evil men who didn't believe in progressive goals and wanted to use low-wage earners as slaves in the capitalist economy. Translated, that meant evil whites who wanted to exploit semiliterate, unskilled minorities for the minimum wage.\n\nJack Hays spoke to the National Guard brigadier in charge of Houston, James Conrad, three times that day. The first call went like this: \"What's happening?\"\n\n\"I need orders from Washington, Governor. I was told to await written orders. Until I get them, I can't do anything.\"\n\n\"Washington knows that people are getting murdered in Houston and having their homes and businesses destroyed, right?\"\n\n\"Sir, I have sent in reports every hour. I don't know what else to do. If I go into the riot zone on my own hook in disobedience of orders, I'll be relieved and court-martialed and they will put someone else in my place, someone who will obey orders.\"\n\n\"Are you going to keep the mob inside the riot zone?\"\n\n\"No one has said anything to me about that. Governor Hays, I'm just a soldier. I obey orders and I give orders. Right now, I am awaiting orders from the national command authority.\"\n\n\"That's Soetoro, right?\"\n\n\"Yes, sir. The president.\"\n\n\"Call me when you hear something,\" Hays said, and General Conrad promised he would.\n\nJack Hays called in Colonel Frank Tenney, director of the Texas Department of Public Safety (TxDPS), who commanded the state police. Hays told him about the call with Brigadier General Conrad of the National Guard. \"We can't let those rioters burn down the city and murder people. I want you to get as many of your men as you can and encircle the area. Let the National Guard do its thing, but don't let those rioters out of the zone they are in right now. And evacuate anyone willing to leave. You have a copy of the riot plan?\"\n\n\"Yes, sir.\"\n\n\"Then use it.\"\n\n\"I would, but FEMA's Texas chief told me I have no authority, except as _he_ gives it to me in obedience to the president.\"\n\nJack Hays had pretty much had all he was willing to take. Without really thinking through the possible ramifications, he said, \"You go get that bastard and take him with you. I want him right up front when I give the order to go in there.\"\n\n\"You know there will be trouble. FEMA has their own private army, armed to the teeth.\"\n\n\"And they aren't doing anything about this riot. Go get the bastard. Disarm and arrest anybody that gives you trouble. That office is in Texas, and in Texas we run the show. _Texas is ours_.\"\n\n\"You're goddamn right it is, Governor.\"\n\n\"Then get ready to go into that riot zone and arrest those thugs when I give the order. Get the Houston police to help. Call me when you are ready to do it. Got that?\"\n\n\"Yes, sir.\"\n\nWhen Colonel Tenney left his office, Hays sensed he had crossed the line. He asked the Texas Ranger outside the door to come in and explained the situation. \"I need your boss as soon as he can get here. We are coming to a crisis.\"\n\n\"Yes, sir.\" The ranger was on his cell phone as he walked from the room. Primarily criminal investigators, the Texas Rangers\u2014there were only about 140 of them\u2014were a division of the TxDPS.\n\nThe Constitution of the State of Texas required the governor to maintain public order and enforce the laws\u2014and Jack Hays meant to do that. Under state law, he could assume command of the TxDPS during a public disaster, riot, or insurrection, \"or to perform his constitutional duty to enforce the law.\" As Jack Hays saw it, Barry Soetoro could not relieve him of this responsibility or void the statutes or Constitution of the State of Texas for any reason whatsoever. Jack Hays had sworn to uphold the law and, by God, he was going to do it or die trying.\n\nHis decision made, he called in the leaders of the legislature to brief them.\n\nIt was three o'clock that Friday afternoon when Jake Grafton was led into an office in the admin building of Camp Dawson. He wasn't wearing handcuffs. The room looked like what it used to be, a crowded office for low-level bureaucrats and staff officers of the West Virginia National Guard. Now it appeared to be full of FBI agents.\n\n\"We want to ask you some questions,\" the man behind the desk said. He was a White House aide, maybe in Soetoro's inner circle, or only one level away. His name was Harlan Sweatt, known to the world as Sluggo. He was balding, with a double chin and a serious spare tire that was hidden behind the desk. Jake recognized him, although the two had never met.\n\nGrafton dropped into the chair across from Sweatt. Scanned the other agents in the room, four men and one woman. All looked as if they hadn't had much sleep, and no wonder, busy as they must have been rousing citizens from offices, golf clubs, bars and beds, and transporting them here to this mountain concentration camp.\n\n\"Ask away,\" Grafton said.\n\n\"I am not going to read you your rights,\" Sluggo said, \"because your rights have been suspended by the declaration of martial law.\"\n\n\"I didn't know that the president had the power to suspend the rules of criminal procedure or the presumption of innocence or the right to be represented by counsel.\"\n\n\"Are you a lawyer?\"\n\n\"No.\"\n\n\"He has been advised by good lawyers, including the attorney general. He is fulfilling his constitutional duty to protect the nation.\"\n\n\"If you say so.\"\n\n\"We want to ask you about your role in the conspiracy to remove the president from office.\"\n\nJake sat silently, watching the man drone on. He had suspected this might be coming since Callie told him of the FBI's announcement to the press.\n\nWhen Sluggo Sweatt paused for air, Grafton said, \"I deny any involvement whatsoever.\"\n\n\"Four people have confessed, so far. They swear you knew about the planning for a coup d'\u00e9tat.\"\n\n\"Who?\"\n\nHe named names. Two names Jake thought he recognized from the CIA, low-level staffers. The other two he didn't.\n\n\"I don't care what they signed. I deny any involvement whatsoever, nor did I know of any plot.\"\n\n\"You had better rethink that, Admiral. You have a daughter, a sonin-law, and a grandson. Your wife lives on your pension. You have money in the bank and property. With a stroke of a pen, all that can be taken away from you.\"\n\nGrafton said nothing.\n\n\"I don't think you realize how serious the crime is that you are accused of,\" Sweatt explained, as if Grafton had a 75 IQ and his wife had to help him put on his pants in the morning. \"The penalties are catastrophic, for you and your family. We have drafted a confession for your signature.\" He opened a drawer and removed the confession, tossed it on the desk. \"As you will see, you are charged with nothing but failing to report treasonous activity. There is no suggestion that you committed any overt act. I suggest you read it, please.\"\n\nGrafton didn't even pick it up. \"Sluggo, I am not going to put my fingerprints on that. I have no doubt you can forge my signature, if you want it, and no doubt whatsoever that you have sold your soul to the devil. Currently there is nothing I can do about this situation, or you, but I'll remember you. Not fondly.\"\n\n\"I won't try to persuade you,\" Sluggo Sweatt said coolly. \"But I want you to consider the fact that the world has turned, and you are in serious danger of being roadkill. There won't be another day in your life when you can do anything about it, about me, or about your situation. Not a day, not an hour, not a minute. You can only save yourself and your loved ones a great deal of grief by signing that document.\"\n\n\"Is that why you sold out? Saving yourself grief?\" Grafton replied.\n\nThe man shrugged. \"Unlike you, I have some common sense,\" he said, and gestured to the agents against the wall.\n\n\"I am delighted to hear that, Sweatt,\" Grafton shot back. \"Common sense is almost as rare as hen's teeth, and equally hard to find.\"\n\nThe agents led Grafton back to the compound.\n\nThe members of the Texas legislature that packed into the governor's office were a mixed lot. Some were demanding that the legislature pass a declaration of independence and declare Texas a free republic. Others looked damned worried.\n\n\"Are you people out of your minds?\" It was Smokey Bryan from Hall County. \"I fought for the United States in the army. I am a citizen of the United States. My family have all been American citizens, and my great-great-grandparents who came to Texas when it was Comanche country and got scalped\u2014they were Americans. I'll be goddamned if I am gonna commit treason and try to take Texas out of the Union. Again. The last time we tried that they shot a lot of Texans but didn't hang anybody. This time they might. Barry Soetoro is, no question, a would-be tin-pot dictator, but he _is_ the president of the United States. And let's call a spade a spade\u2014no pun intended\u2014he's black. Most black people will stick to him even if he declares he is the risen Christ.\"\n\nLuwanda Harris, a black woman representing a district in Houston, said, \"Gangs of terrorists are running around killing people. People are plotting a coup. I don't know who, but it's probably Republicans. They hate him. You are damn fools to sit here discussing treason when the FBI hasn't finished its investigation.\"\n\nSomeone shouted from the back. \"You don't seem very worried about your constituents who are caught in the middle of a riot.\"\n\n\"Fuck you,\" she shot over her shoulder. She was looking straight at Bryan when she said, \"And you too, Smokey, you Nazi bigot. Black people have been shit on for centuries, ever since they were dragged to Texas as slaves. You people have segregated them, won't educate them, won't give them a leg up. You won't even increase the minimum wage. Let the niggers rot. That's\u2014\"\n\n\"You racist bitch!\" Senator Bryan roared. \"I have had\u2014\"\n\n\"Quiet,\" the governor shouted. \"If you people are going to cuss at each other, go outside on the lawn to do it. You can use your fists, shout, pull hair, act like children, get your names and photos in the papers. Go on. Get the hell outta my office.\" Silence descended.\n\nJack Hays lowered his voice. \"Ms. Harris, Mr. Bryan, you two seem to have lost sight of the fact you are on the same side. You are both against Texas independence. Yet we all share a common concern, I hope. We all care deeply about the people of Texas, all of them, and what is best for them.\"\n\n\"I'm concerned about what is best for _black_ Americans,\" Ms. Harris shot back. \"All you white people can worry about your own damned selves. We black people are going to stick together.\"\n\n\"You speak for yourself, woman,\" interjected Charlie Swim. \"You don't represent me, and when the fires finally go out, don't come begging the legislature for money to rebuild the projects. You won't get it. You helped them burn.\"\n\nThat caused another frenzy of shouting.\n\n\"Shut up,\" Jack Hays roared. \"The question is, How are we going to stop the riot? If the feds interfere, what are we going to do?\"\n\n\"You're goin' to Houston and shoot a bunch of black people,\" Luwanda Harris said. \"I know it, they know it, and the White House knows it.\"\n\n\"We're going to arrest rioters and hold them responsible for their crimes,\" the governor said in a normal voice. \"Murder, rape, looting\u2014nobody gets a free pass. Nobody. I have sworn to uphold the law and I will, whether you are white, black, brown, yellow, or green. If you want to do your community a service, Ms. Harris, you will get yourself to Houston and help stop the riot.\"\n\n\"Who do you think I am?\" Luwanda Harris demanded. \"You think I own them?\"\n\n\"Anybody else?\" the governor said.\n\nA delegate from the Dallas suburbs wanted to discuss threats. Her name was Melissa McKinley. She didn't know whether Soetoro was right about a right-wing conspiracy, but her constituents were worried about security. Terrorist threats, insane people, drug violence, the list went on. \"My constituents want to be free from fear, free to raise their children in a safe environment. Guns scare them, enraged homicidal maniacs that shoot kids in schools and theaters scare them, terrorists and assassins scare them. The specter of a civil war would horrify them. They don't want to live in Baghdad or Beirut or Syria. They want their children to have a chance to reach adulthood free from fear.\"\n\n\"How much freedom are they willing to trade for their security?\" Ben Steiner asked.\n\n\"They don't want to bury their kids, Ben.\"\n\n\"So they would be happy in Hitler's Germany or Stalin's Russia in a comfortable little cell, just as long as their blood didn't flow?\"\n\n\"I doubt it, but freedom doesn't do you a lot of good if you're dead.\"\n\n\"Amen to that,\" several of the legislators muttered.\n\nThey wanted to mention the grievances of their constituents, introduce them into the discussion, things such as EPA regulations designed to save the climate at the expense of the working men and women of Texas, even though there was no scientific evidence that the changes demanded would have any impact on the problem as defined by the EPA. And the EPA's demands to shut down coal-fired power plants, which would raise electric bills dramatically. Several wanted to talk about the financial and social burden of illegal aliens on the school districts and the education of American children, whose parents were paying the taxes to fund the schools. Others wanted to talk about federally mandated school curriculums and school lunches. Many were sick and tired of being dictated to by Washington bureaucrats who thought they knew more than the people ruled by their edicts.\n\nAnother just wanted to talk about a federal government many of her constituents perceived as an out-of-control, fire-belching, meat-eating monster that could not be tamed, controlled, or killed, a monster that increasingly stuck its nose into every facet of American life and propagandized their children every minute of the school day. A minister denounced a government that he believed was not just neutral on religion, but actively antireligious.\n\nCharlie Swim broke in. \"The bottom line is we need to stop these riots. You want to help black people?\" He scowled at Luwanda Harris. \"The people getting crippled and maimed and killed are black. The people doing it are black. A lot of the businessmen getting looted and burned out are black. If the federal government won't stop it, the state government must: it's that simple. A government that fails to protect its citizens from violence has forfeited its claim to legitimacy. And if bucking Soetoro and the feds leads to a confrontation, it's time for Texas to face the issue head-on and declare its independence.\"\n\nCharlie Swim stood on a chair and looked around the room. \"I tell you now,\" he continued, \"I'm for independence. The people of Texas would be better off without the other forty-nine states, all the Texans, white, black, and brown, for all the reasons that have been mentioned here this morning. We would be better off without those fools in Washington.\n\n\"Luwanda, you, the Republicans, and everyone in the country with a brain know that Cynthia Hinton doesn't have a chance to win the November election. She knows it too. She has plenty of her own ghosts, but carrying the Soetoro record on your back would have defeated anybody. All Hinton is doing is jacking off the faithful.\n\n\"And as for Soetoro and his gang. You know what their motto is: Never let a crisis go to waste. I don't trust them or believe anything they say.\n\n\"I think the time has come for us to start our own country. When you don't trust your spouse, or your boss, or your government, it is time to say good-bye and go on down the road.\"\n\nWhen JR Hays considered the tactical possibilities, he decided the only answer was booby traps, or mines. One man shooting wasn't going to get it done. Oh, he might get a few of the drug smugglers, but he wouldn't get them all, and if he didn't get them all, every last one, he would be signing his own death warrant.\n\nNot that JR thought he was going to live forever, because he doubted that he would.\n\nThe problem with booby traps was that they kill anyone who trips them\u2014illegal pregnant women trying to get across the river to have their babies in Texas, men looking for work, as well as any drug smugglers and professional killers who happened by. Anyone and anything, including kudus, elands, oryx, springbok, nyalas, impalas, whitetail deer, and coyotes.\n\nUnless he wanted to bury a lot of relatively innocent people and very innocent animals, he needed mines he could detonate at the proper moment.\n\nHe unlocked the toolbox in the bed of his pickup. Using the truck's tailgate as a table, he laid out all the devices he had borrowed from his former employer, the defense contractor, and looked them over carefully. Nothing there was explosive. What he had was sensors, miniature control boxes, radio controllers, batteries, and the other bits and pieces of high-tech booby traps. With the black powder and fuses, he should be able to construct some seriously lethal homemade Claymore mines.\nFIVE\n\nAfter the crowd filed out of Jack Hays' office, Ben Steiner stayed behind and closed the door. He dropped into a chair and lit a foul little cigar. Jack Hays sat in his executive chair, which his wife had bought from Office Depot and he had assembled in his garage.\n\n\"Looks like you've crossed the Rubicon, Jack. Ain't no going back from here.\" Steiner blew smoke around, then looked for an ashtray. There wasn't one. \"You're sort of in the position of the fellow that found himself astride a fence when the ladder gave way and he came down with one leg on either side.\"\n\n\"If you introduce a declaration of independence in the legislature,\" Hays asked, \"will it pass?\"\n\n\"That's the question, isn't it?\" Ben Steiner said, puffing lazily. \"And damn, I don't know. It might. Just might.\"\n\n\"Or it might not,\" Jack Hays said disgustedly. \"Don't you think you ought to start counting noses? If it's DOA, I'd like to know it before I manage to piss off every federal employee from the postman to Soetoro.\"\n\n\"I'm all for it,\" Steiner declared, \"but it's a big step. Soetoro is arresting everybody in Texas he can get his hands on\u2014whoever intimated, hinted, or told his wife that he didn't like Soetoro. FEMA has a camp for them up in Hall County. They got a list and are rounding 'em up.\"\n\n\"How come you aren't on it?\"\n\n\"Oh, I am, but my wife told them I was in Argentina fishing for a couple of weeks.\"\n\n\"Ben, it would be silly to introduce such a resolution, or bill, unless we knew it was going to pass.\"\n\n\"By how much?\"\n\n\"Simple majority.\"\n\n\"That isn't much.\"\n\n\"We'll be lucky to get that,\" Jack Hays said. \"We must have something to paper our ass with. Unlike Soetoro, I want to hear the people's representatives speak. One way or the other. Yea or nay.\"\n\n\"It's that 'lives, fortunes, and sacred honor' thing that has them worried.\"\n\nThe governor took his time answering. \"I think everyone would like to wake up and find this is just a nightmare. But it's real. None of us are going to be able to bury our head in the sand and hope the wolves don't bite our asses. The revolution has started. Soetoro has suspended the United States Constitution and the Bill of Rights. Lincoln did it under his war powers. Unfortunately for Soetoro, we aren't in a war. A rebellion, or revolution, will change the life of _everyone_ in America. Indeed, perhaps everyone on the planet. We can't start it\u2014and the Texas legislature can't\u2014because Barry Soetoro already did.\"\n\n\"That wasn't what you told me yesterday.\"\n\n\"I've changed my mind.\"\n\nBen Steiner took a deep drag on his cigar and let the smoke out slowly. \"Our people need a little time,\" he said. \"They gotta work up to being brave. They gotta examine all the options before they can screw up their courage for this one.\"\n\n\"How much time? The Soetoro administration has been planning martial law for years.\"\n\n\"Tomorrow or the next day.\"\n\n\"We better not have the vote if we aren't going to win. Barry Soetoro is too much of an egotist to ignore an independence vote, win or lose.\"\n\n\"We'll win,\" Steiner said grandly. In his fifties, with a booming voice, he knew how to sway people, persuade them. Jack Hays was a more difficult sell than the average juror, however.\n\n\"When you're sure you know how the vote will go, after you've talked to every member, come back and see me.\"\n\nBen Steiner leaned forward. \"Jack, as we sit here Luwanda Harris and some of her friends are burning up the wires to Washington. If you don't want the capitol surrounded by tanks and army troopers from all over, you had better start talking to people, tell them what's at stake. We must get this done, and soon. If you don't, my best guess is the government of Texas is going to get arrested en masse and accused of treason. In the interim, let's cut off access to Washington.\"\n\n\"Can we take down the telephone system and the internet?\"\n\n\"Of course. The only question is how fast.\"\n\n\"Let's do it,\" Jack Hays said. \"Who do we call?\"\n\n\"The state director of disaster response, Billy Rob Smith.\"\n\nThe governor picked up the phone and made the call.\n\nBilly Rob Smith heard the governor out, then asked, \"Are you nuts? Every business in America bigger than a lemonade stand relies on telephones, landline and cell, and the internet. Millions of people use the system to send or get business information and to buy and sell securities. Medical records are transmitted via fax or over the internet. The feds have been working like beavers to digitize every medical record in the nation\u2014shutting off the internet may mean people can't get proper medical care. And the telephone system\u2014you can't shut one system down without turning off the other. In a lot of places, voice and digital use the same wires. In some places the telephone system is completely digital. Turning off cellular and landline telephones will drop us right smack dab back into the nineteenth century. Shutting those systems down is insanity.\"\n\n\"I didn't ask for your opinion\u2014I am giving you an order.\"\n\n\"And I'm telling you that you're crazy. Hell, I don't even know that you are the governor. You sound like an idiot jabbering on the telephone.\"\n\nWhat ended the argument and decided the matter was an announcement at precisely that moment that was carried on television networks nationwide: The president had directed the military to work with civilian law enforcement agencies to confiscate all the guns in America in private hands. In the future, only the military and law enforcement officers would have guns.\n\nBilly Rob Smith had a television in his office airing a twenty-four-hour news channel, which was limiting itself to government press releases these days, and he paused his conversation with the governor while an aide told him the news as rapidly as possible and pointed at the television set.\n\nSmith was not stupid. \"Did you hear that?\" he demanded of Jack Hays.\n\n\"Yes.\"\n\n\"Holy damn. It's like the British marching to Lexington and Concord. This tears it. Americans won't stand for it. Hell, the people of Texas won't stand for it.\"\n\nJack Hays took a deep breath. He had other things to attend to. \"Smith, I want you to shut off the telephone and internet systems in east Texas. Start right here in Austin, right now. Then Houston. Get busy.\"\n\nA very subdued Billy Rob Smith said, \"Yes, sir,\" and hung up.\n\nJack Hays repeated the news to Ben Steiner, who was taking a last puff of his little cigar. Steiner stared, slack-jawed. Finally he said, \"Soetoro isn't just temporarily suspending the Constitution, he's tearing it up.\"\n\nJack Hays rubbed his forehead.\n\nSteiner said, \"Luwanda Harris will never change her mind, but this will get us Smokey Bryan and a whole lot of others who were on the fence. Of course, a lot of liberals will have a spontaneous orgasm when they hear Soetoro has repealed the Second Amendment, people like Melissa McKinley, but they weren't going to vote for independence no how, no way. They don't mind a dictator repealing the Second Amendment as long as they think he's on the side of social justice and the planet, like they are.\"\n\n\"Ben, if you are going to introduce a declaration of independence, and I don't mean an ordnance of secession, hadn't you better write one? After you count noses.\"\n\nBen Steiner rushed from the room, taking his cigar butt with him.\n\nTrust Jack Yocke to know when something was going on, Jake Grafton thought. He was standing under a tree watching it rain from a low overcast sky when the _Washington Post_ columnist found him.\n\n\"I saw them take you into the admin building, Admiral,\" Yocke said. \"Rumor has it you are now part of the conspiracy that planned a coup d'\u00e9tat.\"\n\n\"Where did you hear that?\"\n\n\"It's being whispered around.\"\n\n\"They wanted me to sign a confession.\"\n\n\"Did you?\"\n\n\"I am not going to confess to anything I didn't do. Ever. Once you start that, there's no end to it.\"\n\n\"No matter how bad you think the Soetoro White House gang is,\" Yocke said, \"you're wrong. They're worse.\"\n\n\"They certainly think they are on the side of righteousness and history.\"\n\n\"Hitler and Stalin were sure of it too\u2014didn't work out so well for them.\"\n\n\"Now I feel better.\"\n\nJake Grafton had his hands in his pockets. He looked around. No place to sit that wasn't wet. He leaned against the tree trunk, which wasn't wet yet. The rain was falling in greater volume.\n\n\"So what are your politics, Admiral? In all the years I've known you, I never got an inkling.\"\n\nGrafton snorted. \"Long ago, when I was very young, I learned that all political points of view were valid for the people who held them, except for the fanatics on the fringes who are usually incapable of rational thought. Think about the blind men and the elephant. Honorable people can hold very different opinions because they have very different life experiences. Liberals, conservatives, middle-of-the-roaders, big-government types, libertarians, old, young, middle-aged, highly educated or average or uneducated, skilled or unskilled, stupid, average smarts, or genius, they all see a little bit of how the world works and process it into a worldview, and they are all correct. The genius of representative democracy is that it takes all these viewpoints and grinds them up and arrives at some kind of resolution, most of the time. Look at the federal tax code: government policy has tried to accommodate all major and many minor concerns and still raise revenue. Any dictator with half a brain could put a tax code together that is simpler and more efficient and raises more revenue. But the United States still has one of the highest, if not the highest, rate of voluntary tax compliance of any country in the world. So something must be working right.\"\n\n\"Democracy can't handle every problem; you have to admit that.\"\n\n\"Slavery was too big for representative government,\" Grafton acknowledged. \"The story of this century is the haves versus the have nots, and illegal immigration is one aspect of that. Drugs are another piece of that problem. The disintegration of the black family is a piece. The desire of Barry Soetoro to drastically increase the number of non-white voters in America as quickly as possible to enhance the political power of blacks and Hispanics and Muslims and dilute the power of the whites is another. Representative democracy hasn't figured these problems out and may not be able to do so. Still, no other form of government has a better chance.\"\n\nLightning flashed, then two seconds later came the clap of thunder. The wind picked up.\n\n\"So how will the story turn out?\" Yocke asked.\n\n\"I don't know, Jack. I really don't.\"\n\n\"I'm getting wet,\" the _Post_ 's man complained, and brushed wind-driven raindrops from his hair.\n\n\"See you later,\" Grafton said.\n\n\"Good luck, Admiral.\"\n\n\"Thanks. You too.\" Grafton moved a few degrees around the tree and stood watching the rain.\n\nI parked in front of the lock shop and went in to see Willie Varner, my partner. He knew more about locks than I ever hoped to know, and much of that knowledge was acquired in prison. They say prison will broaden a man; I couldn't testify to that, but the experience seemed to have stretched Willie's mind somewhat, even if it didn't do anything for his morals or ethics.\n\n\"Damn, Carmellini,\" he said, \"I thought you was gone out west somewhere on the lone prairie learnin' to rope and ride and sing to the dogies, whatever they are.\"\n\n\"I've only been gone three days, Willie.\"\n\n\"Come back to reenlist in the CIA, have you?\"\n\n\"Nope. Come back to break Jake Grafton out of prison.\"\n\n\"I saw the _Post_. And heard about him on TV. He's famous now. Arrested and all for tryin' to kick Barry Soetoro outta the White House and get him started on his way to Hell. You ain't serious about bustin' him out, are you?\"\n\n\"I am.\"\n\nHe made a rude noise. \"You are a real damn fool, Tommy. I've known some real losers in my day, people so damn stupid they needed help to pee, but you take the prize. Where they got 'im?\"\n\n\"Camp Dawson.\"\n\n\"Never heard of it.\"\n\n\"It's a National Guard camp over in West Virginia.\"\n\n\"Ahh, the beatin' heart of civilization. I should of heard of it, cultured as I am. And after you get him outta there, where pray tell are you two gonna go? Yemen? You can share a goat herder's hut with some holy warriors. I heard the summers are kinda warm there. Maybe you can summer up at the North Pole in an igloo.\"\n\nThat was Willie, always asking the tough questions. \"I don't know. Haven't thought that far ahead.\"\n\n\"Better get that figured out before you cross the line, Tommy. Send me your address in a year or two when you're settled so I can send you birthday cards.\"\n\n\"How do you like living in a dictatorship? Transition going okay?\"\n\n\"So far so good. There's a kid down the street teachin' me the Sieg Heil salute. Want a beer?\"\n\n\"Why not?\"\n\nWe settled down with longnecks in the back room of the shop. That was where I broke the news that I needed some help.\n\n\"Oh, no!\" Willie roared. \"Forget that! Wash out your filthy mouth, Carmellini. I ain't ever goin' back to the joint, and how I know that is because I ain't ever goin' to do anythin' that would get me sent back there. Livin' in the joint with a bunch of losers who would as soon kill you as look at you, eatin' mac and cheese, no liquor or beer or women, jackin' off under the sheets. . .nope. Ain't gonna do it again, Tommy, so you just forget whatever shit is in your twisted head.\"\n\n\"I know you're a patriot.\"\n\n\"The hell I am! Who told you that? You go wave the fuckin' flag somewheres else.\"\n\n\"One of the sons of liberty.\"\n\nHe said a crude word that is illegal to say on the television or radio. Maybe even on the telephone. I knew I could talk him around, so we each had another beer and talked about Barry Soetoro and martial law and all that.\n\nThat evening I stopped in to see if Mrs. Grafton was home. I buzzed the door in the front lobby, told her who I was, and she let me in. Rode the elevator up.\n\nCallie Grafton looked tired and out of sorts. She offered me something to drink and I chose bourbon. She poured me a healthy drink over ice.\n\nShe knew all about what the government spokesmen were saying to the press about her husband. \"None of it is true. He has devoted his life to serving America. I can't believe that anyone could say these things about him with a straight face. Tonight on television they named two other men they said were coconspirators. I've never even heard their names before.\"\n\n\"They're sacrificial goats,\" I said, and watched her face.\n\nShe reached for my drink and took a sip. \"I think so too.\"\n\n\"I'm thinking of busting him out of Camp Dawson, or wherever they move him. It can be done, but afterward he'll be a fugitive.\"\n\n\"So will you. And anyone who helps you.\"\n\n\"Can you go visit him? Like tomorrow?\"\n\n\"I don't know. I can call him and ask.\"\n\n\"Please do so. Right now. Don't tell him that I'm here.\"\n\nShe went into the master bedroom, I suppose, and I sat at the little kitchen dining nook working on my drink and looking at the lights of Washington. Lots of lights, all the way to the horizon. Thought about being a fugitive in Barry Soetoro's America.\n\nI also thought about the possibility that the Grafton condo was bugged. It was a very slim chance, I thought. There hadn't been enough time, and why Grafton? Sure, they were setting him up as a human sacrifice, but why would they care what Callie Grafton said? There was nothing she could do about it.\n\nFinally Callie returned. \"I can see him tomorrow afternoon. They are still allowing visitors.\"\n\n\"Good,\" I said. \"I doubt if they'll have the visitor's rooms wired up already, but they might.\" I handed her a watch. \"Put this on and wear it. Pushing the stem in turns on a very high pitched sound, too high for human ears, but it will overpower any listening device and mask a conversation.\"\n\n\"Where did you get this?\" she asked.\n\n\"Liberated it from the CIA. I thought that someday I might need it more than they did, and darn if that day didn't come. When your conversation is over, don't forget to push the stem again to turn the squealer off.\"\n\n\"How will I know it's working?\"\n\n\"The second hand will cease to move when the squealer is on, and resume when it's off.\"\n\n\"Okay.\"\n\n\"You need to ask him if he wants out. That's the only question, and it's yes or no. He'll understand about being a fugitive if we get him out. Maybe they've been threatening him, maybe they haven't, but Jake Grafton will know the score. Yes or no. Can you do it?\"\n\n\"Of course.\" She acted as if that were a silly question.\n\n\"On your way home, please call me. I'll give you my cell number. If his answer is yes, he wants out, you will tell me that he looks good. If the answer is no, tell me he looks tired.\"\n\n\"He said they were listening to telephone calls.\"\n\n\"It's worse than that,\" I admitted, and decided to share some classified information. \"NSA is recording and data mining every telephone call in America. All of them. Have been for at least six months. Never say anything on any telephone that you don't want the government to hear.\"\n\nShe sniffed. \"Handling that much information, they couldn't be very efficient.\"\n\n\"Computers are marvelous things. Never bet on bureaucratic sloth and incompetence. Just pray for it.\"\n\nShe stared straight into my eyes. \"Tommy, how are you going to get him out?\"\n\n\"I don't know just yet,\" I said. \"I'll get some help and we'll put our heads together and see what is possible.\"\n\nShe started to say something, thought better of it, and examined her hands.\n\nI hoped Jake Grafton would say yes, and I told Mrs. Grafton that.\n\n\"Why?\" she said.\n\nShe was a tough broad, so I looked her straight in the eyes and explained. \"Cynic that I am, I suspect if we don't spring him the Admiral is bound for a maximum security prison. Or a graveyard. Accused, convicted, and executed, he wouldn't be around to embarrass the crowd that needs him as a scapegoat.\"\n\nShe kept her eyes on mine. \"You may be right,\" she said softly.\n\n\"Mrs. Grafton, if the White House didn't need some scapegoats, why did they accuse your husband of something ridiculous?\"\n\nShe took a deep breath and exhaled slowly. \"I'll call you tomorrow on the way home, Tommy. Thank you for coming.\"\n\n\"He looks good or he is tired.\"\n\nI finished the drink, punched my cell number into her phone, said good-bye, and left. In the elevator down I thought about the fact that Callie Grafton didn't once mention herself, ask what she would do if her husband escaped custody. In her own way she was as tough as Jake Grafton. If I were Barry Soetoro, I wouldn't want to share an elevator with her.\n\nWhen I was out of the parking garage and tooling through the city toward the lock shop, I got back onto the problem of how my helpers\u2014they didn't yet know they were going to be my helpers\u2014were going to snatch Grafton from the arms of the law. I had an idea or two about how we might evade afterward, for a little while at least, but first things first.\n\nI decided to call my girlfriend, Sarah Houston. She used to be a dataminer at the NSA, with the world's biggest and best computer system to play with. It helped that she was also a genius and the most gifted hacker alive on this side of the Pacific. Hacking and selling secrets had gotten her into serious trouble a few years ago and she went to the joint, but Jake Grafton had sprung her to help him. That worked out, so her name was changed and she was given a new identity. Grafton had gotten her transferred to the CIA, and she had an office two floors below mine. I didn't know what she was doing at the agency, and I hadn't asked. Not that she would have told me anyway. If there was ever a woman who thrived on secret shit, Sarah Houston was her name.\n\nShe and I had an up-and-down relationship. Just now we were down. It was an old, old story: she wanted to get married and I didn't.\n\nShe answered the phone on the third ring. \"What is it, Carmellini, you jerk?\" I am not a fan of caller ID, and this is why.\n\n\"Hey, gorgeous. I was thinking of dropping by in about a half hour to run something\u2014\"\n\n\"No.\" She hung up.\n\nWe Carmellini men are made of stern stuff, so I went anyway. I buzzed her apartment from the lobby. No answer. Maybe she had a guy up there tonight, but I didn't think so. Men who could handle that edgy personality were rare indeed. I was one, sort of, but there is only one Tommy Carmellini.\n\nI pushed the buzzers on three or four apartments, and was rewarded with a click. I was elevated to the fourth floor and marched purposefully to her door and rapped politely.\n\nShe must have looked through the security eye. \"Get out of here, Carmellini, before I call the police.\"\n\n\"I'm here to talk about Jake Grafton.\"\n\nTen seconds. . .then she opened the door and stood there. She was wearing a robe and slippers.\n\n\"Well?\"\n\n\"It would probably be better if we talked inside your place.\"\n\nShe pulled the door open and headed for her living room. I came in and closed the door.\n\n\"Well?\" she said again.\n\n\"You have probably been reading about Jake Grafton being accused of conspiring to do a coup d'\u00e9tat. I need your help with a jail break.\"\n\nShe sat down and ran her hand through her hair. \"Damn you, Tommy.\"\n\n\"I had nothing to do with it, and you know damn well Jake Grafton didn't. You know Jake Grafton. But Soetoro and his staff are going to frame him and either lock him up forever or execute him. If he doesn't get hanged in his cell while he is awaiting trial.\"\n\n\"They wouldn't do that,\" she whispered.\n\n\"You think?\"\n\nShe put her face in her hands. Finally she whispered, \"Okay. They would.\"\n\n\"Right now he's being held in a detention center at Camp Dawson in West Virginia. They'll move him soon to the federal holding center in Washington. We need to know when they plan to do that, and how many agents will transport him. I assume they will be FBI agents, but I don't know that for a fact. You could help with that.\"\n\nShe studied the carpet. After a bit she said, \"You know if they catch me getting out of line they'll send me right back to the women's prison at Alderson. A knock on the door, handcuffs, and I'm gone for the rest of my life.\"\n\n\"If they catch me and Willie and the guys, we're going up the river too. If we're still alive.\"\n\nShe went into the kitchen and I heard her knocking around. In a few minutes she was back with two drinks. I sipped mine. Gin. I don't think much of gin, but I sipped along as if I drank it every day. She sipped hers too.\n\n\"So you get him out. Then what?\"\n\nI told her my idea.\n\n\"That won't work for long.\"\n\n\"Soetoro is lighting a fuse on a rebellion. We just need to be out of the blast zone until it blows up in his face, then he will have a great many more pressing problems than you, me, Grafton, and Willie the Wire.\"\n\n\"And if you are wrong?\"\n\n\"If I'm wrong, I'll be dead. The rest of us too, maybe.\"\n\n\"You would take that chance for Jake Grafton?\"\n\n\"Yes.\"\n\nShe took her drink and went to the window. Pulled back the drapes so she could see out. Stood there a while, taking an occasional sip of her drink. Finally she said, without turning around, \"You would.\"\n\n\"I need your help to pull this off,\" I told her.\n\nBack in his hotel that night, Ben Steiner went to the business center and used his computer to call up a copy of the U.S. Declaration of Independence of 1776 and the Texas Declaration of Independence of 1836. He printed them out, then logged off and went up to his room.\n\nHe read both documents carefully. The authors of the Texas declaration had obviously used the U.S declaration as a format. First there was a statement of their authority, then a list of grievances that justified what was to come, then the declaration itself, which severed the political ties with the mother country. The language of both was stirring, defiant, a political act that could not be undone except by military defeat by the mother country. Both were written for a wide audience, all the people in the nascent new nation and the citizens of the mother country, England and Mexico, respectively, and everyone in the world. The drafters of both knew they were writing a historical political document. They wrote for the people who would fight the battle and for all the generations yet to come.\n\nWriting such a document would require the best that was in him.\n\nBen Steiner turned on his computer and began.\nSIX\n\nJR Hays dug his hide at noon. Before he turned over the first shovelful of earth he rigged up a listening device with an eighteen-inch parabolic dish. The dish picked up sound that was too faint for the human ear to detect, magnified it, and delivered it to the operator via earphones or on a speaker.\n\nJR laid the dish antenna on the ground so he was listening to the sky. He heard jets come and go and birds flapping their wings. He began digging. The hide was on the side of the arroyo in hardpan. He had to use the pick to break it up enough to shovel. The dirt had to go in a wheelbarrow. He dumped the wheelbarrow fifty yards back where the dirt was fairly well concealed.\n\nWith the hide finished and bottles of water and weapons put inside, he installed a night-vision periscope. Twice he thought he heard piston engine sounds from the sky, so he quickly covered the hide with a green tarp. Then he lay on the ground and used night-vision goggles set for infrared. He saw the drone going northwest up the Rio Grande. When it was gone, he removed the tarp and got busy. An hour later the drone came back, so he did it all again.\n\nWhen he had the hide finished, JR installed his homemade mines on both sides of the arroyo. He got them in by ten in the evening. He could clean up the sites in the morning, but they were concealed well enough to not be seen at night.\n\nHe went to his hide, checked that the parabolic dish antenna for the audio device was well concealed thirty feet away in a bush and aimed right where he wanted it, then climbed in and pulled the tarp over the hole. He donned the headset and scanned with the night-vision periscope he had borrowed from his employer.\n\nHe was tired. He ate a few energy bars, drank water, and waited.\n\nHe doubted they would come tonight. Or tomorrow night. But eventually they would. And he had an unlimited quantity of time. The rest of his life, actually.\n\nAmbushes aren't for everyone. Few people have the patience to wait, and wait, and wait some more on the off chance that the opportunity you prepared for will actually happen. Snipers have that kind of patience, but most people don't. Most people want to attack right now. Or sooner. Yesterday would be preferable. Do it and get it over with.\n\nRevenge isn't that way. The juice in revenge, JR knew, is in the anticipation. The longer you wait the sweeter it will be.\n\nWhen he had repaired the fence that morning, he had attached a thin bare wire to it and run it to the hide. Now he twisted the end of the wire around one finger. Maybe he would feel it. His dad, he knew, had tried tin cans with rocks on the fence, but when the smugglers saw them, they knew he was nearby. JR doubted that they would see the wire. From his position near the fence, he should be able to count how many came through\u2014and he could make sure that none got back.\n\nSo what could go wrong? Well, despite his precautions a drone might have spotted him digging the hide or planting the mines. Federal agents might be on their way here right now.\n\nThere was nothing he could do about that contingency, so he dismissed it. Never worry about things you cannot control. That was one of the hard lessons he had learned in the army. He had taken all the precautions he could, and that would have to do.\n\nAs he sat in the hide with the periscope, listening to the audio on the earphones, he reviewed the timetable again. If they didn't come by two hours before dawn, they weren't coming. They needed at least an hour to hike to the paved road on the north side of the ranch and an hour to get back here. He thought they would want to be back across the river in Mexico by dawn. Maybe.\n\nBut if they didn't come or he missed them, he could get them some other night. They would keep coming as long as this delivery route worked. As the hours passed he consoled himself with the thought that the smugglers were dead men walking.\n\nBy midnight he was having a devil of a time staying awake. Ten hours of hard manual labor in the heat of the west Texas summer had about done him in. That's what you get for not staying in shape, he thought, for letting yourself get soft.\n\nHe dozed off finally, wearing the earphones. Awoke with a start. Thought about giving up on tonight and heading back to the ranch house. But if he did that, they would come tonight. That was the way God rolled the dice.\n\nJR checked his watch. Almost one in the morning. He decided he would give himself one more hour, and if they didn't come, go home to sleep. That decision made, he scanned with the periscope, saw nothing, and waited.\n\nAnd dozed. When he awoke again with a start, he found that it was almost two. Something woke him up.\n\nWhat?\n\nNow he felt it again. A tug on the wire wrapped around his finger. Something was brushing against the fence. An animal? He unwrapped the wire and let it dangle.\n\nHe listened on the parabolic dish, adjusted the volume in the earphones. Looked through the periscope and saw three men operating with wire cutters on the fence.\n\nThey were here!\n\nThe internet and telephone service in the Austin area went down at ten that evening. Ben Steiner knew the system was dead because he saw legislators fiddling with their cell phones and pocketing them in disgust.\n\nThe legislature was in joint session, considering a declaration of independence for Texas. The balconies were packed, standing room only.\n\nSteiner thought the declaration would pass, but figured it would take all night. Everyone, pro or con, had something to say.\n\nThose for independence were outraged at the president's announcement that he was stopping all gun sales and confiscating firearms from Americans nationwide. Was he afraid of armed, law-biding American citizens? Hell yes. And what further destruction of the American way of life was in the works? Freedom of speech was already gone. Freedom from arbitrary arrest was gone. Was freedom of religion next? Federal officers were arresting people and incarcerating them for no crime other than the fact that they had been political opponents of the administration. That was deeply troubling. Even worse was the fact that no one had a clue when martial law would be over, when the country could get back to normal, or if it ever would.\n\nThe delegates and senators opposed to the declaration were equally passionate. A Texas declaration of independence was a declaration of war. It was a bold step into the unknown. War. With all the power and might of the federal government against them. Several delegates argued that the threat from terrorism justified martial law, and others pointed out that it was Soetoro himself who demanded that some of the terrorists be admitted as refugees. \"He manufactured a bloody crisis and now he's using it to take the country where he wants it to go,\" a senator shouted acidly.\n\n\"Are you ready to lay down your life in opposition to the federal government?\" one representative demanded. \"Are you ready to lose everything, your family, your home, your savings, your means of making a living? Make no mistake; all those things are on the table. Are you ready to watch your children be killed in the violence? What will you say when your sons and daughters lie dead at your feet? Are you ready to turn your back on the American flag, the flag so many Texans have given their lives to defend? What the hell kind of people are you?\"\n\nAnother representative wanted to argue about the process. \"This question is so important that it should be voted on by the people of Texas, not passed here by majority vote. This isn't a convention of delegates elected to consider independence and draft a declaration\u2014it's the state legislature, for God's sake.\"\n\n\"Texas voters will get their chance,\" someone shouted. \"We're here to ensure that they do.\"\n\n\"Freedom isn't free,\" another speaker pointed out. \"Freedom in America has been bought with blood. And that freedom purchased at such a precious price has been taken from us, ripped from our hands. The feds didn't declare martial law after Lincoln, Garfield, or JFK were assassinated. Are our institutions so flawed that a dictator can destroy them before our eyes, yet we lack the moral and physical courage to fight for our heritage? Mr. Speaker, if we won't fight to preserve our freedom, we don't deserve it. And Barry Soetoro will take it from us. He's trying to do that as we sit here this evening. There is only one thing for an American patriot to do, and that is vote to remove Texas from the tyranny of Barry Soetoro and the federal government.\" A roar went up from the audience.\n\nBen Steiner went into the governor's office and found him conferring with several senior National Guard officers. A glance out the window showed troops in the yard, a lot of them. Two tanks were visible, and three armored personnel carriers. Jack Hays had called out the Guard.\n\nFinally Hays came over to Steiner and whispered, \"How is it going over there?\" He meant on the other side of the capitol building, in the House chamber.\n\n\"They're debating.\"\n\n\"Will we win?\"\n\n\"I think so, but I guarantee nothing. Think of them as a large jury. Soetoro is on trial.\"\n\n\"They'd better get it done tonight. Federal agents are out there with some regular army troops, and they sent word in that everyone in this building is under arrest.\"\n\n\"Will the Guard hold?\"\n\n\"I don't know, Ben.\" The skin of Jack Hays' face was drawn tightly over his cheekbones and his eyes seemed to have sunk back into his skull. \"I suspect that if the legislature decides to surrender, the guardsmen will go back to their armory, turn in their weapons, and go home. What else is there for them to do?\"\n\n\"I'll go tell the legislature,\" Steiner said.\n\nHays stopped him with a tug at his sleeve. \"Make damned sure every person in that chamber understands that if they declare independence, their necks are on the line.\"\n\n\"I think they know that.\"\n\n\"If we can't win our independence, we're all dead, including you and me. Once they vote for independence, we've crossed the river of fire and burned our boats.\"\n\n\"Jesus carried his cross,\" Ben Steiner said gently. \"We have to stand for something or the gift of life was wasted on us.\" He walked out the door and along the hallway through lines of state troopers.\n\nThe peons laden with backpacks full of narcotics trudged along in the darkness about six feet apart. There was starlight and a sliver of moon, but the old Indian trail up from Mexico would have been easy to follow regardless.\n\nWith the periscope, JR saw the lead man with a backpack and started counting. One. . .two. . .he quit at eight. Eight mules. No doubt there were armed guards, perhaps even the same ones who had killed his father, but they weren't on the trail. One was probably behind him, paralleling the trail.\n\nJR glanced at the luminescent hands of his watch when the last man went by. At the speed the peons were walking, he thought it would take about a minute and a half for all of them to get into the kill zone. He had walked it himself that morning, timing it.\n\nCarefully, ever so carefully, he rotated the periscope. If he hadn't already passed the hide, the man or men on this side of the arroyo guarding the column must be close. JR had to get them first.\n\nThe second hand of his watch was swinging, past forty-five seconds. Come on, man, where are you?\n\nAh, there, moving slowly and carefully. JR zoomed in on his head, which was partially obscured by brush. But for an instant he got a good look. Yep, he was wearing a night-vision headset. But there was only the one man. A quick sweep revealed no others.\n\nJR lifted the edge of the tarp an inch or so, located the man. He was about forty feet away, moving right along so as to keep up with the mules. He was relying on the goggles, so he wasn't situationally alert. JR poked his AR-15 with the night-vision scope out under the tarp. He flicked off the safety, aimed it, and squeezed the trigger. The man went down.\n\nAbandoning the rifle for a moment, JR located his lighter and the detonator cord by feel. Applied the flame. That cord burned at several thousand feet a second. It seemed to explode, dissolve into ashes. Then he heard the explosions, just one big roar. At least two screams, of men in mortal agony. The blast was followed by a patter on the ground and brush, like rain. JR knew what it was: he had used ten pounds of screws and nails in the mines.\n\nNow for the shooter or shooters on the other side of the arroyo. JR hadn't seen any, but he knew someone was there. These guys didn't take chances.\n\nHe came out of the hide on his belly, wearing the night-vision goggles, with the AR cradled in his arms. He crawled as he scanned around. Black powder smoke oozed through the brush and acted like fog, reducing visibility. Still, the other men might have caught the muzzle flash of the AR or seen the flash of the burning det cord.\n\nHe caught a glimpse of a man, then saw the muzzle flash and heard the bullet strike brush near his head.\n\nJR shot back, three shots as fast as he could squeeze the trigger, then he rolled sideways away from the spot where he had been.\n\nLay in the brush on his face, waiting.\n\nSilence.\n\nHow much patience would these shooters have? They weren't trained soldiers and they had no idea how many opponents they faced.\n\nRaising his head, JR scanned again with the goggles. There was a lot of brush, so he could be sure of nothing, except he didn't see anyone.\n\nIt occurred to him that the man behind him might be only wounded. So he crawled that way to check on him. The little .223-caliber slug had hit him square in the chest and killed him almost instantly.\n\nNow for the other man. JR thought anyone on the other side of the arroyo would make for the hole in the fence as quickly as they could get there. They had heard explosions, screams, and shots from two different weapons, and had certainly gotten a good whiff of the stench of that black powder smoke. They knew they had walked into an ambush; they didn't know how many people they faced; they'd get out of there as fast as they could.\n\nJR crawled to an old juniper, which screened him from the west side of the gully and allowed him to see where the fence crossed the arroyo. He waited, lying absolutely still.\n\nTwo minutes, and then he saw a man break from the brush and run toward the hole in the fence. JR shot him in the back. Down he went on his face, the rifle falling ten feet away. JR took careful aim and shot the prone man again.\n\nHe waited, listened, scanned with the goggles, felt his heart pounding in his chest.\n\nHe consciously willed his heart to slow, which was ridiculous, but it did, finally. Ten minutes passed. . .eleven. Now he heard a man. Sounded as if he were in the arroyo, moaning softly, dragging himself along.\n\nJR tried to become one with the earth. Put his head down and listened.\n\nYes, the man was dragging himself along, moaning, \"Madre de Dios. . .\"\n\nHe was just to JR's right, down in the arroyo, crawling for the fence. He couldn't have been more than twenty-five feet away from where JR lay, but JR didn't move. Didn't even twitch. There might be another shooter out there, one with steel nerves, and if there were, to move was to die.\n\nFinally the man made the gap in the fence and JR saw him with the goggles. Shot him with the rifle, twice. Now the man lay absolutely still, the stillness of death.\n\nJR leaped to his feet and ran away from the fence, out of the area at an angle.\n\nHe loped along, turned north, and went around the kill zone and finally joined the trail. Jogging along with his rifle at port arms, wearing the goggles; he could do this for hours. Or used to be able to, anyway. Tonight, with his nostrils full of the black powder smell and his ears still ringing from the gunshots, he fell into a rhythm. Only two miles to go, two miles, run, run, run.\n\nHe wanted to see that van, get the license number.\n\nHe got to the fence, ran eastward along it fifty feet, and lay down. The road was empty. Checked his watch. Forty-five minutes had passed since he detonated the mines.\n\nHis breathing returned to normal and he waited.\n\nSeems like he had spent the major portion of his life waiting. He tried not to think, just became one with the night. The van would come, if the driver wasn't waiting for a cell phone call to summon him. Waiting just up the road, around the bend.\n\nHe saw the glare of the headlights in the goggles before he heard the engine.\n\nIt came on, slowing. It wasn't a van; it was a car. Down to a creep as it approached the spot where the trail and fence met. No doubt the driver was looking for a signal. Didn't see it, so he began to accelerate on by.\n\nJR got a good look as the car passed the fence. It had a bank of emergency lights on the roof and on the side it said \"Sheriff of Upshur County,\" and under that, \"To serve and protect.\"\n\nFive minutes later it came slowly back. JR was tempted. Taking out the driver would be an easy shot, but then what? He let it go by. Big man driving. Maybe the sheriff himself, ol' Manuel Tejada.\n\nFive or six minutes later the sheriff's car returned heading east. Five or six minutes after that, it passed again, westbound, and as the taillights went on along the road, JR heard the engine wind up to highway speed and saw the dimly glowing taillights fade into the darkness of the rolling plains.\n\nIn the House, Ben Steiner signaled to the speaker that he would like the floor. The speaker recognized him. The chamber was silent as he approached the podium.\n\n\"My fellow Texans,\" he said. \"This building is surrounded by federal agents and regular army troops, who have sent in word that everyone in this building is under arrest. Defending us are Texans from our National Guard. There has been no shooting yet, but there might be, at any moment. Armed Texans are trying to defend this building, this seat of Texas government, and defend you, the elected representatives of the people of Texas. And some in this chamber worry that blood might be shed, so they advocate our surrender to tyranny.\"\n\nSteiner paused and surveyed his audience on the chamber floor and in the balconies. \"I _do not_ believe\u2014I _cannot_ believe\u2014that such sentiments are representative of the sentiments of the people of Texas, the physical and spiritual descendants of the defenders of the Alamo, those patriots who laid down their lives rather than surrender to the tyranny of the Mexican government. I say to you, Remember the Alamo! Remember those thirteen days of glory. Remember those brave men who laid down their lives so that Texans might be free.\"\n\nThe applause rose like thunder in the chamber. Ben Steiner mopped his brow with his handkerchief. He was on a roll now, and he knew the jury was with him. He waited until the noise died somewhat and said, \"Hard, cold, and cruel will be the road ahead. Many difficult decisions will have to be made. Many will suffer, some will die. Yet I say to you, Americans everywhere will judge us by what we do here tonight. We can so conduct ourselves that future generations will glorify our deeds and honor our lives, and remember our deaths if need be. . .or we can surrender and throw ourselves on the mercy of a tyrant. Is life so precious that you would shame yourself to keep it? As for me, I want to repeat\u2014and I hope someday they engrave these words upon my tombstone\u2014the immortal words of Colonel William Barret Travis at the Alamo: 'Victory or Death.'\"\n\nThe applause and cheering rose to a staggering volume. Ben Steiner turned around, leaned toward the speaker, and shouted to be heard. \"Mr. Speaker, I move the question.\"\n\nThe Senate passed the declaration by two-thirds vote, and the majority was almost as large in the House.\n\nBen Steiner went back to the podium. \"My fellow Texans, we are making history tonight, history that Texans will talk about as long as there are people in Texas and men yearn to be free. We cannot tell our children and our children's children that we passed this by a mere majority vote. I move that the vote be made unanimous.\"\n\nThe speaker called for a voice vote. The yeas had it.\n\nSteiner was so relieved he had to hang on to the podium to stay erect as the legislators cheered wildly.\n\nThe leaders of both chambers signed the document and took it to the governor to be signed, which he did. He handed the signed document to the colonel in charge of the National Guard troops, one with the unfortunate name of Buster Bean, and said, \"Get a loudspeaker and read this on the steps of the capitol.\"\n\nWhen the crowd in the governor's office had thinned somewhat because many of them wanted to be outside to hear the declaration read, Jack Hays asked Ben Steiner, \"What did you say to them?\"\n\n\"I paraphrased Winston Churchill and Colonel Travis and appealed to their honor.\"\n\n\"I guess you convinced them.\"\n\n\"No. They knew the right thing to do. They just needed to hear someone say it.\"\n\nThe floodlights of several television stations almost blinded Colonel Bean, but at least their illumination helped him read the document.\n\n\"The unanimous Declaration of Independence made by the elected representatives of the people of Texas in General Convention in the City of Austin on the twenty-third day of August, 2016.\n\n\"When a government has ceased to protect the lives, liberty, and property of the people from whom its legitimate powers are derived, and for whose happiness it was instituted, and ceases to be a guarantor of those inalienable rights which are granted to every human by God Almighty, and becomes an instrument in the hands of evil rulers for their oppression:\n\n\"When the federal Constitution of their country, which they have sworn to support, has been declared a nullity by the leader of their country and the whole nature of their government has been forcibly changed without their consent from a limited federal republic into a military dictatorship:\n\n\"When, after the spirit of representative, constitutional government has been forcibly usurped, when the semblance of freedom has been removed and the sole power in the land is the whims of a dictator, the first law of nature, the right of self-preservation, the inherent and inalienable rights of the people to preserve their liberty, rights, and property by taking the political power into their own hands becomes a sacred obligation to their posterity to abolish such a government and create another in its stead, one calculated to rescue them from impending dangers and secure their future welfare and happiness.\"\n\nInside the governor's office the amplified voice outside was quite clear. Jack Hays said to Ben Steiner, \"Good stuff, but I've read much of that before.\"\n\n\"I cribbed it. I couldn't do better.\"\n\nColonel Bean read a list of grievances, including Barry Soetoro's declaration of martial law, the arrest of political opponents, and the de facto repeal of the First Amendment.\n\nHe ended with this paragraph:\n\n\"It has been demanded that we deliver up our arms, which are essential to our defense, the rightful property of free men, and formidable only to tyrannical governments.\"\n\n\"The necessity of self-preservation therefore now demands our separation from the United States of America. We, therefore, the duly elected representatives of the people of Texas, in solemn convention assembled, do hereby resolve and declare that the political connection with the United States of America has forever ended, and the people of Texas do now constitute a free, sovereign, and independent republic, and are fully vested with all the rights and attributes that properly belong to independent nations; and conscious of the righteousness of our intentions, we fearlessly and confidently commit the issue to the decision of the Supreme arbiter of the destiny of nations and mankind.\"\n\nColonel Bean stepped away from the podium as applause and wild cheering broke out. Beyond the National Guard troops, many of the U.S. Army soldiers began leaving in twos and threes. Here and there sergeants and officers tried to stop them, but many went anyway. The regular army officer in charge, a colonel, knew when to fight and when to regroup. He ordered his soldiers to return to base. In less than fifteen minutes, only National Guard troops remained on the capitol lawn, facing a sea of cheering civilians. Thousands of them. People poured from the side streets as the news swiftly spread and soon packed the area as far as the eye could see. Texas flags were waved defiantly and proudly.\n\nTexas was once again an independent nation. If the Texans could make it stick.\nSEVEN\n\nIn Washington, Thurman Truax, the senior U.S. senator from Texas, was appalled at the spectacle on television that morning. He had been in politics since he was twenty-seven years old, which was thirty-five years ago, and he kept his ear close to the ground in Texas to find out what people were thinking, so close to the ground that the people said he had dirt in it. He had been worried for years about this independence movement and had talked about it at length with the governor, Jack Hays, who he thought was against it too. Apparently Jack Hays had changed his mind or found he was caught in a tide he couldn't resist.\n\nTruax had suspected something of this sort might happen when Soetoro announced martial law, and had called the White House to tell the president so. He wound up speaking to some junior aide. The president had made his decision, Truax was told. He also shared his misgivings with the other senator from Texas and the members of the Texas congressional delegation, some of whom shared his concern, and the leadership in the Senate.\n\nThe television was still showing video of people cheering and celebrating independence in front of the capitol in Austin when Truax called his chief of staff. He had tried five times to call the governor and had sent him three e-mails during the broadcast, but had been unable to get through. Nor could he reach any of his political or social friends in Austin. Texas seemed to have dropped right out of the United States.\n\nNot that he blamed Texas. Truax had fought the good fight against admitting Muslim refugees from the Middle East to America, many of whom, he suspected, were jihadists. Of course, despite Soetoro's and the secretary of state's bromides about security checks and vetting them, the reality was that the refugees had no identification whatsoever, a fact the president and his administration chose to ignore. And jihad had come to pass. Murder in a parochial school, on a train, in Yankee Stadium. . .sometimes Truax thought that the administration actually wanted some terrorist incidents. So now Texas had rebelled.\n\nHis chief of staff had watched the broadcast too. And she also had tried repeatedly to call people in Austin and had been unable to get through. Truax didn't wait to hear her take on the whole mess, but told her to make airline reservations to get the senator back to Texas as soon as she could this morning.\n\nHe heard pounding on his door. When he answered it, a television reporter and cameraman were standing there, wanting an interview.\n\n\"As you can see, I'm still in my pajamas. My office will have a statement for the press later this morning.\"\n\n\"Did you know this Declaration of Independence was going to happen, Senator?\"\n\n\"No comment.\" He closed the door on the reporter, a woman with NBC, locked it, and went upstairs to dress.\n\nThe truth was, he was appalled. Those fools in Austin had smashed Pandora's box. Barry Soetoro would be outraged, and he was the commander in chief of the armed forces. No telling what that damned fool would do. The United States was tearing itself apart, and the senator felt powerless to prevent it. No one in Washington wanted to listen to reason. Truax well knew that every decision government made had consequences, intended and unintended. Barry Soetoro and Jack Hays were on a collision course.\n\nAfter he was dressed, the senator went to the kitchen for coffee and a boiled egg. He ate his meager breakfast in front of the television watching national coverage of the news, whatever Soetoro's censors would permit to be aired, which was universal condemnation of the Texas political system and everyone in it. Terrorism seemed to have dropped off the news radar. Texas treason, one talking head said. Another speculated that since the president had declared martial law, the governor of Texas and members of the legislature could be tried by court-martial, and probably would be.\n\nTruax had had his fill and turned the television off when he heard another knock on the door. He looked out the security peephole. It wasn't a reporter. He opened the door and found four FBI agents, who had orders to arrest him. As it turned out, the White House had ordered that Senator Truax and every member of the Texas delegation were to be arrested and held in a Washington prison for treason. An FBI agent accompanied him upstairs to get his medications.\n\nAs he rode away in the back of a car in handcuffs, Truax pondered on the reaction in Texas when this news got out.\n\nOne of the people who heard the Declaration of Independence read aloud heard it over the radio. As it happened, he was the captain of a tugboat in Galveston Harbor. He was always up early, planning the morning's work on the boat before it had to get under way for the day's tows or pushes. He took his cup of coffee out and climbed the ladder to his bridge.\n\nAcross the harbor he could see an attack submarine berthed, USS _Texas_ , a _Virginia_ -class boat, only a few years old, moored port side to a pier. She had come in yesterday for a three-day port call to show the flag, entertain visitors, and let the good people of Texas see where the navy's share of their federal taxes was being spent.\n\nHow long would she be here now? he wondered. Bet they'll get under way as soon as they hear the news.\n\nHe set his cup down and ran down the ladder to his crew berthing, where his engineer and first officer were sound asleep. Those two were all the crew he had right now. The seamen who fixed things and handled lines wouldn't come aboard until half past seven.\n\n\"Wake up,\" he urged as he shook them. \"We're going to move the tug.\"\n\nHe gave hurried explanations as they pulled on jeans and tugged on shoes.\n\nTen minutes later, the tug, _Mabel Hardaway_ , named after his wife, got under way. Captain Hardaway took it over to where the sub was berthed and maneuvered to anchor immediately behind it. To ensure the tug didn't swing on her anchor and damage the sub's screws, he dropped an anchor from the stern as he came up slowly, then a bow anchor. He backed down and killed the engines, then went down the outside ladder to the deck to help the first mate secure the anchors.\n\nThat sub isn't leaving until I say so, he thought, vastly pleased with himself.\n\nHe got on the radio to another tug, managed to wake up the skipper, and asked it to come anchor immediately beside the submarine. \"As soon as you can get here,\" Captain Hardaway added for emphasis.\n\nAboard _Texas_ , the watch officer awakened the captain, Commander Mike Rodriquez, who had spent the previous evening at a dinner in his honor in a hotel in Galveston, one attended by the mayor, most of the city councilmen, and everyone who was anyone in the Chamber of Commerce. He had probably had one or two too many glasses of wine, but toasts were offered right and left and he had to do it, he told himself then.\n\nHis head was a little thick as he listened to the watch officer. \"We have a tugboat anchored immediately behind us.\"\n\n\"In the prohibited zone?\"\n\n\"Yes, sir. I've notified the harbor police.\"\n\n\"They can probably handle it,\" the captain said. \"Are our guards on the pier?\"\n\n\"Yes, sir. And armed.\"\n\nIn the age of terror one can't be too careful, the captain well knew. Local jihadists would secure undying fame in Paradise if they could damage a U.S. nuclear submarine. The FBI had assured him they were keeping a close eye on the local Muslims, of whom there were only a few. Still. . .\n\nThe captain quickly donned his uniform, khakis because he wore camos only when under way and he hated them. He went to the control room, satisfied himself that everything was as it should be, then climbed the tiny conning tower to the miniscule bridge.\n\nYep, there was the tug, _Mabel Hardaway_. What in the world was that thing doing there? He picked up a loud-hailer and pointed it at the tug's bridge.\n\n\"You are in a prohibited zone. Get under way and move your boat immediately.\"\n\n\"Sorry,\" came the shouted reply, quite audible in the pre-dawn stillness.\n\n\"You will be arrested if you don't move that boat.\"\n\nNo reply.\n\n\"Sir,\" the watch officer said. \"One of the sentries is running toward us. There are some civilians up there at the head of the pier.\" He was using his binoculars. \"Looks as if some of them are carrying rifles.\" He handed the binoculars to the CO, who was staring through them as the sentry came halfway across the gangway and shouted, \"Sir, those civilians say they have closed the pier. They say they won't let our liberty party back aboard.\"\n\n\"Why?\" the officer of the deck asked loudly enough to be heard.\n\n\"They say Texas declared its independence an hour ago.\"\n\nThe captain rubbed his head. Jesus Christ, he thought. Of all the time for a port visit! He glanced back at the tug. Well, he couldn't back out of here, even if he got all the lines off the boat.\n\nHe looked to his starboard side. If he could swing the stern, perhaps he could back and forth using the rudder until he could go behind the tug, like a car getting out of a parallel parking place. He used the binoculars in the half-light and saw the line running off the stern of _Mabel Hardaway_ at an angle, out into the open area he would have to use. He knew he was looking at a chain with an anchor on the end. Backing the naked screws of his boat into the chain would disable _Texas_.\n\n\"Here comes another tug, sir,\" the OOD said, and pointed.\n\nSure enough, there it was, maybe a mile away down the harbor, coming slowly. It would certainly be here before he could get _Texas_ free of the pier and maneuver her out of this slip. And even if he did get _Texas_ out of the slip, the tugs could ram her and make sure she didn't get out of the harbor.\n\nDamn!\n\n\"Go below,\" Rodriquez told the OOD, \"and get off a flash message to SUBLANT. Tell them we are blocked in by tugboats, with armed civilians on the pier. Go.\"\n\n\"Aye-aye, sir.\"\n\nGulls wheeled above him as he stood alone weighing his options. Half his crew was on liberty in Galveston. While he could operate the boat with the duty section, he had nowhere to go with tugs in the way. His sentries could keep the civilians off the pier, for a while, anyway, unless they started shooting.\n\nHe could scuttle the boat, sink her here in this slip. But the navy brass would have his balls if he did that and the independence news was some kind of misinformation or a political ploy to embarrass the Soetoro administration, something that could be cleared up or would go away in a few hours or days. He certainly didn't know. All he knew was what the sentry had told him. If he scuttled _Texas_ , she could be raised of course, and eventually returned to seaworthy condition, after she had spent a year or so in the Electric Boat shipyard in Connecticut where she was built.\n\nHe decided to wait and see what SUBLANT said to do. He wanted someone in a much higher pay grade to point to if recriminations started. Let the admiral earn his pay, he thought as he watched the other tug ease into the slip and tie up starboard side to the pier abeam _Texas_. Now he was blocked in.\n\nWhen the OOD came back up the ladder he said, \"Message sent, Captain.\"\n\nThe skipper pointed at the tug on the starboard side. \"Go send another one. Tell them we are corked good.\"\n\nThe OOD took a quick look and disappeared back down the ladder.\n\nThe skipper looked at the people milling on the pier. At least thirty of them, only a couple of sailors in uniform, and a police car. Maybe he should go up there and talk to the cop.\n\nWhen the OOD came back, the captain gave his instructions, went below for his ball cap, then went to the forward torpedo room and climbed the ladder through the open hatch to the main deck. He paused on the gangway and saluted the flag flying on its portable flagpole on the stern, then went ashore.\n\nAfter Colonel Curt Wriston, commander of the Texas National Guard in Abilene, saw the declaration read on TV, he tried to call his headquarters in Austin, with no success. The telephone didn't even ring.\n\nWriston dressed, skipped his morning coffee, and got into his car. He picked up his deputy commander. They discussed the situation and were in agreement: the Soetoro administration would use force against Texas, just as quickly as they could.\n\nWriston drove the county roads to a spot near the perimeter fence of Dyess Air Force Base. From there, they could see the two runways: the main runway, 13,500 feet long, and a short parallel runway, 3,500 feet long. Also visible in this flat country in the clear air of early dawn were the big hangars and flight line between the two runways. Wriston looked around. Not a cloud in the sky.\n\nNo doubt the commander of the base, Brigadier General l'Angistino, was on the wires right now with bomber headquarters in Nebraska and the air force brass in Washington, asking for instructions. Everyone in the chain of command would bump the decisions up the ladder, Wriston thought. He knew how the military bureaucracy worked these days. Initiative had been ruthlessly and remorselessly squeezed out of the system. Obey orders was the mantra, and, whatever you do, don't make your bosses look bad. General l'Angistino was a good man, but he would undoubtedly have to wait awhile for orders, which would have to come from the very top, perhaps even the White House, which would have a ton of other red-hot problems to deal with today.\n\nThe deputy commander said it first. \"We need to block those runways, make sure the air force doesn't fly those bombers and Hercules transports out of there. Texas will need them.\"\n\n\"They'll probably sabotage them,\" Wriston said thoughtfully, \"if they can't fly them out.\"\n\n\"Either way, they can't use them to transport troops or bomb us.\"\n\n\"We could use tanks, just go through the fence,\" Wriston mused.\n\n\"We only have four tanks, and one of them has the fire control system disassembled for upgrade.\"\n\n\"It'll move.\"\n\nThe deputy said, \"That big runway is about three hundred feet wide, as I recall. Take a serious amount of iron to block it. And the Hercs can use the short runway.\"\n\n\"We can get some construction equipment, road graders, and bulldozers,\" Wriston suggested. \"They can follow the tanks. We'll block the long runway, and if we have any equipment left, leave it on the small one. We'll have to block the long one in at least two places. Three would be better.\" He used a small set of binoculars he kept in the car for looking at birds to examine the distant buildings, which looked like toy blocks sitting out there on the horizon.\n\nWriston added, \"They've got cranes and such to handle crashed airplanes. If they can't start the engines and drive our stuff off, they'll drag it off.\"\n\n\"We can disable everything.\"\n\n\"Only delay them for a day, maybe two.\"\n\n\"That might be enough. Let's do it.\"\n\nWriston started his car and they drove away planning where to get the yellow equipment, people to drive it, and how to summon their tankers.\n\nAt the head of the pier in Galveston where _Texas_ was moored, Commander Mike Rodriquez found out that the Declaration of Independence news the sentry had given him was as real as a heart attack. Thirty or so civilians carrying rifles, some of them civilian versions of the M16, were standing there watching him. The sheriff had him sit in the right seat of his patrol car, which had its front windows down, then got behind the wheel. When he was comfortably settled, he gave the naval officer the news about the declaration.\n\n\"Texas is now a free republic,\" the sheriff said in summary. The captain scrutinized the lawman's face to see if he was kidding. He didn't appear to be. The fucking idiot! Secession in this day and age!\n\nOne of the civilians came over and leaned on the car to hear what was being said inside. The sheriff ran him off.\n\n\"Now, Captain, this is the way I see it,\" the sheriff continued. He had a serious pot gut that lapped over the buckle of his gun belt. His shirt needed pressing and he needed a shave. \"I haven't talked to anybody in Austin 'cause the phones are out and, anyway, they're probably drunk and asleep, which I ought to be. When they wake up they're gonna be mighty busy. In any event this declaration thing sorta upset the applecart. Did you watch it on TV a while ago?\"\n\nNo.\n\n\"County commissioners are asleep too, and even when they get up this morning, they're goin' to tell me what they always say, which is use my own judgment. That way if people start squallin' I have to take the heat and not them. Being an elected official and all, I suppose it comes with the territory. But you probably ain't interested in my problems, since you got a big one your own self.\"\n\nGet on with it, you oaf, Rodriquez thought.\n\n\"Your problem is that these voters here aren't going to let your sailors get on your submarine. And it looks to me like those tugboat captains ain't goin' to move their boats to allow you to get goin', even if you had all your sailors. That's kinda it in a nutshell.\"\n\n\"And you aren't going to clear the pier and tell the tugboat captains to get out of the prohibited area?\"\n\n\"That's about the size of it.\"\n\nRodriquez thought of a common dirty word but didn't say it. He pulled at the door handle.\n\nThe sheriff laid a hand on his arm. \"You stay right here. I think this whole situation will go better if you sit right here with me. Keep the crowd calmed down. If these people start shootin' your sailors, we'll both have more problems than we do now.\"\n\n\"If they shot my sailors, you'd arrest them, wouldn't you? A crime committed in your presence.\"\n\n\"Well, I don't know. I haven't got my thinkin' that far down the road. Been my experience that problems are best headed off, if possible, rather than tackled afterward. That's what I'm tryin' to do. Now are you gonna just sit here like I told you, or do I have to handcuff you and lock you in the back?\"\n\nRodriquez couldn't contain himself. \"You son of a bitch!\"\n\n\"Be that as it may, I need a yes or no.\"\n\n\"I'll sit.\"\n\n\"Fine. I'll radio for one of my deputies to stop by McDonald's and get us some McMuffins and coffee. Or do you want something else?\"\n\n\"That'll do, thanks.\"\n\nThe sheriff picked up the dashboard mike and started talking.\n\nForty-five minutes later, after they had eaten, the sheriff had the deputy, with the crowd's help, disarm Rodriquez' sentries and take them to jail. \"Just to hold for a little while,\" the sheriff told Rodriquez, \"until somebody with more brains than me can figure out what we oughta do.\"\n\nFive minutes after the sentries had departed with the deputy, a sailor from the sub came looking for his captain. He had a wad of paper in his hand. He spotted Rodriquez in the patrol car and came over to the open window. \"Messages, Captain,\" he said and offered them.\n\n\"I'll take those, son,\" the sheriff said, holding out his right hand. When he had them, he told the sailor, \"You're under arrest. Now you get in the back of the car here.\"\n\nThe sailor looked beseechingly at his commanding officer.\n\n\"Do as he says,\" Rodriquez said listlessly. Shit, he thought, I should have stayed on the boat. What a fool I was! There goes my naval career!\n\nBy ten o'clock the crowd had swelled to at least fifty people, most of them carrying rifles. They were having a high old time. Some of them had brought beer, which they shared. Sailors who were ashore and wanted back aboard their submarine were arrested and taken away.\n\n\"Crowd's gettin' a little rowdy, don't you think?\" the sheriff asked Commander Rodriquez.\n\n\"Yes,\" he agreed.\n\n\"I kinda think it's time we put an end to this and let these folks go home or to work or to a bar someplace to tank up. Let's you and me walk down the pier and you get all your people out of that thing and bring them along. I'll get a bus to take them to a hotel.\"\n\nRodriquez felt like a cornered rat. Aboard the boat he could overpower the sheriff, scram the reactor, and order her scuttled. But should he scuttle her? If this political thing blew over. . . . He looked longingly at the classified messages the sheriff had read and tucked into a pocket in the driver's door. It wasn't as if these civilians knew how to operate a nuclear submarine, for Christ's sake. USS _Texas_ wasn't going anywhere. And the U.S. Navy could destroy her with a Tomahawk cruise missile or two anytime they got around to it.\n\n\"Let me read those messages,\" he said.\n\n\"Nope. It's my way or I send you off to join your sailors in jail. Then I'll go down there with these voters and arrest all of them aboard. Your only choice is to go with me or go to jail.\"\n\n\"I'll go with you.\"\n\nThe sheriff got out of the car. He stopped the captain and pulled out handcuffs. \"We'll put these on you,\" he said, \"in case you get any big ideas. Just to protect myself, you understand.\"\n\nHe cuffed the captain's hands in front of him, then pulled his pistol. He waved it at the crowd. \"You people back off and give me room. Don't want anyone comin' down the pier. Don't want anyone doin' anything we'll all regret. Come on, Captain.\"\n\nAt the gangway, the sheriff could see an officer or sailor on the bridge. He told the captain, \"Tell them to turn off the reactor and come out. All of 'em.\"\n\n\"Aren't we going aboard?\"\n\n\"No. They ain't goin' no place in that boat and you ain't neither. So get them out here.\"\n\nWhen the remainder of the submarine's crew were on the pier, about two dozen men, the sheriff thought, although he didn't bother to count them, he asked one of the chiefs, \"Did you turn that reactor off?\"\n\n\"Yeah.\"\n\n\"Got everybody out of there?\"\n\n\"Yeah.\"\n\n\"That's good, 'cause after you're gone, I'm goin' aboard and look around, and if I find anybody they might scare me and I'll probably have to shoot 'em. Hate to do it, but if I fear for my personal safety, and I will, there ain't nothin' else I can do. Promised my wife when I first run for sheriff that I wouldn't endanger myself.\"\n\n\"There's no one else aboard,\" the chief said sullenly.\n\nThe sheriff looked down the pier and saw that a blue bus marked SHERIFF had arrived on the quay. \"There's your ride, Captain. Lead them down the pier and climb aboard.\"\n\nThat was how the brand spanking new Republic of Texas acquired its first warship.\nEIGHT\n\nIn the dim light of dawn JR inspected the bodies. Some of the nails and screws in the mines had ripped open the backpacks and blasted white powder everywhere. JR didn't know if it was cocaine or heroin, and he didn't care. Only two of the mules had obviously tried to crawl away and bled to death; the others died almost instantly, perforated by the metal from the mines\u2014four with nails and screws that had ripped into their brains, another eviscerated.\n\nThe two at the fence were well and truly dead, too. The man who had dragged himself had his gut torn open and intestines were trailing behind him. The bullet that killed him had been a mercy.\n\nJR got his first surprise when he looked at the first man he shot, the man near his hide. The man had yellow and green tattoos that started at his wrists and ran up his forearms.\n\nThat deputy sheriff\u2014he had tattoos like that, very distinctive. What was his name? Morales? He seemed to recall that was it.\n\nJR didn't recognize the other man wearing night-vision goggles. JR pulled the goggles off. He looked like he might have a lot of Mexican in him, but with his face contorted in death, it was difficult to say.\n\nHays walked back to the ranch house and poured himself a stiff tot of bourbon. Sat on the porch with the AR across his lap sipping the whiskey as the sun poked over the horizon and sunlight began illuminating the high places in the brush. Cloudless blue sky. Another scorching hot day in the works. Those bodies were going to get ripe pretty quick.\n\nThe syndicate that sent those drugs across the border would send more men, probably pretty soon. JR had no idea how much money the drugs represented, but he knew it was a lot. Enough to buy the deaths of a thousand peons and a whole lot of Americans. Enough to buy half the sheriffs in Texas.\n\nJR took out his cell phone and called his cousin the governor. Nothing. No ringtone on the thing. He looked at how many bars he had. Two. Well, that should be enough. But the cell didn't work. He went inside and tried the landline. No luck there either.\n\nHe was exhausted and needed sleep. Yet Manuel Tejada would be along in a little while to find out what had happened to his deputy and all the drugs he was supposed to pick up. He wouldn't phrase it quite that way, but that would be what he wanted. Mainly, however, he would want the drugs. If Tejada could show the syndicate the drugs he might get out of this with a whole skin. If he couldn't, he was going to be in trouble, although how much JR didn't know. Maybe he could dig Tejada's pit deeper.\n\nJR placed the guns in the floor of the backseat of his pickup and drove down to the arroyo, as close as he could get. He retrieved the gear from the hide, including the periscope and parabolic antenna, stored all this stuff in the tool chest in the bed of the truck. Went to the bodies of the mules and removed the backpacks. Two were so torn up the white powder spilled all over the ground. JR thought each backpack had contained twenty-five pounds or so of the stuff.\n\nThe syndicate was going to be pissed.\n\nJR put the six reasonably intact backpacks in the chest, locked it, and drove off. When he got to the main gate, he stopped and opened the gate, then got back in the truck.\n\nHad the sheriff been in on it? Apparently. But JR wanted to be sure. He pulled out his cell phone and let it log on the network. Two bars. He called 911, got the sheriff's office number, then dialed it.\n\n\"Sheriff Tejada.\"\n\n\"JR Hays, Sheriff, out here at the Hays ranch.\"\n\nA pause, then, \"What can I do for you, JR?\"\n\n\"Hell of a shootout last night here at the ranch, Sheriff, a little after three. Woke me up. A real firefight. Kinda scared me. I went down this morning for a look, and bodies are lying all over the place. Looks like a drug gang ambush. The dead men had about two hundred pounds of some kind of drug on them.\"\n\nHe paused, but the sheriff said nothing.\n\n\"It's pretty bloody, Sheriff. Goddamn mess is exactly what it is. Might have been some of the bastards who killed my dad.\"\n\n\"The drugs are still there?\"\n\n\"Yeah.\"\n\n\"Huh! What kind of drugs?\"\n\n\"Damn if I know. It isn't marijuana, that's for sure. Some kind of white powder. Anyway, I'm going to call the staties and DEA, but I wanted to give you a courtesy heads-up first.\"\n\n\"Appreciate that, JR. Much obliged. But before you call those other agencies, let me run out there for a look. I'll bring the county coroner and we'll see about the bodies.\"\n\n\"When can you get here?\"\n\n\"Couple of hours.\"\n\n\"I'm pretty worried. God only knows what all that powder shit is worth. I kinda suspect somebody might come back to get it.''\n\n\"This is my county, JR.\" Like the slob owned it.\n\n\"Yes. Yes, it is.\" He paused as if he hated to wait. \"Okay, Sheriff, you come out and look around and call them. These guys aren't going anywhere. Gonna get hot again today and they'll get real ripe fast. Better bring some body bags.\"\n\n\"Two hours. I'm on my way.\"\n\n\"Sure.\"\n\nHe broke the connection. The sheriff hadn't even asked how many dead men there were.\n\nHe pulled the truck through the gate, got out, and shut it carefully. As he walked back to the truck a buzzard in the cloudless sky caught his eye, circling over the old trail. Two of them; no, three. Little dots up there riding the thermals. There would be more buzzards soon.\n\nHe remembered the bumper stickers. Got one out of the truck, peeled the paper off the back, and stuck it on the gate. Stood back and admired it. FUCK SOETORO. He liked it so much he put the other one on the truck's rear bumper.\n\nJR got into his pickup and headed southeast toward Del Rio. He decided he owed himself a treat, so he reached across to the glove box and pulled out a pack of unfiltered Camels. Opened it and lit one.\n\nThe raw smoke tasted delicious. JR adjusted the bill of his ball cap to keep the rising sun out of his eyes and smoked in silence.\n\nThe television clip of Colonel Bean reading the Texas Declaration of Independence on the steps of the capitol in Austin, and the shots of the delirious crowd, went to television stations nationwide. Networks worldwide rebroadcast the scenes over and over. In the United States, many station managers had qualms, and at some stations federal officers demanded that the feed not be aired. Some stations caved, but most didn't. Managers argued that other stations would show it, and while they were arguing with federal censors, many staffs flipped switches and put it on the air. The scenes ran over and over again. Usually the scenes were aired without comment because the people in the stations were leery of the gun-toting bureaucratic squads who occasionally walked their halls, but the scenes spoke for themselves.\n\nThe spectacular act of defiance by the Texas legislature had immediate consequences. Here and there groups of armed citizens waylaid federal officers hauling away political prisoners, disarmed them, and released the prisoners. Several of these federal officers chose to fight it out and were shot dead. Others were taken to a county jail.\n\nThe armed federal police forces from bureaucracies nationwide became nervous. The mood of the public was turning ugly. Some of the agents stayed home and locked their doors.\n\nBarry Soetoro nationalized the National Guard nationwide. Less than half the guardsmen reported to their armories to be inducted into federal service. Officers resigned on the spot. In two cities, small groups of guardsmen called local television stations, which sent crews to watch the guardsmen take off their uniforms in public, put them in a pile, and burn them.\n\nIn Oklahoma City a half-dozen armed officers from the FAA trying to arrest a local newspaper columnist, a conservative, panicked and opened fire on a crowd of vociferous unarmed citizens. Four people were killed and seven wounded, four of them severely. The payback came within an hour. A mob of armed civilians arrived at the FAA's basement office where the armed enforcers hung out and put it under siege. When the officers came out four hours later with their hands up, the crowd opened fire. The last one ran a block and took refuge in someone's basement; he was dragged out and executed with a shot in the head. No one knew if the four murdered officers were the ones who shot the unarmed civilians, nor did anyone really care. Civil wars are messy.\n\nUp and down the plains, in the Rockies and the Midwest, people gathered in spontaneous groups to cheer Texas and wave homemade Texas flags.\n\nIn Austin, Jack Hays saw snatches of this activity on television before he, Charlie Swim, Luwanda Harris, and Colonel Tenney of the Department of Public Safety boarded a helicopter for a flight to Houston. They were met by the National Guard commander there, Brigadier General James Conrad, the mayor of Houston, and the chief of police.\n\nUnfortunately they were downwind of some tire fires, and stinking, heavy smoke was almost overpowering.\n\n\"Have you got the riot area surrounded?\" Hays asked.\n\n\"Yes, sir,\" Conrad said. The senior law officers nodded.\n\n\"Where's that FEMA dude,\" Jack Hays asked, \"the one I wanted at the pointy end of this expedition?\"\n\n\"He got cold feet and split.\"\n\nHays frowned.\n\n\"Would have had to handcuff him and put a gun in his back, Governor, to walk him into that riot.\"\n\n\"Obviously he didn't think his liberal credentials would protect him,\" Charlie Swim said, and Hays chuckled.\n\nHays explained to the politicians, \"We want to capture the rioters and not let them rampage through the rest of the city.\" Luwanda Harris and Charlie Swim looked grim.\n\n\"Is everyone ready?\" Hays asked the police chief, Colonel Tenney, and General Conrad.\n\nReceiving affirmatives all around, Hays said, \"Start 'em moving.\" Conrad spoke into his handheld radio. Hays turned back to the politicians. \"Ms. Harris, Mr. Swim, will you accompany me?\"\n\n\"You got a grandstand seat picked out?\" Luwanda Harris asked sourly.\n\n\"Indeed I do. I am going to walk ahead of the troops and talk to anyone I meet. I would like you both to accompany me.\"\n\n\"They may shoot us,\" Charlie Swim pointed out. Even as he said this, several random gunshots could be heard.\n\n\"They might,\" Jack Hays agreed, grabbed two elbows, and started off with Swim on the left and Harris on the right. The troops in riot gear followed, then the state police carrying shotguns and wearing helmets.\n\nDown the street, right into the middle of the riot zone.\n\nWhen they were seen, young men threw some rocks, then turned and ran. Hays kept advancing. The three of them passed burning cars, looted stores, and melting asphalt. On they went.\n\nSomeone fired a shot at them from an upstairs window. Guardsmen fired back, and two soldiers charged into the building to find the shooter and arrest him, or kill him if need be.\n\nJack Hays pretended he didn't notice the shooting.\n\nIt took thirty minutes, but an ever-tightening cordon of law enforcement and guardsmen had brought the rioters, mostly young men, into the middle of a large intersection. Surrounded, and scared, they threw down guns, chains, tire irons, and knives.\n\nJack Hays was handed a loudspeaker. He climbed up on the hood of a fire truck that had followed the skirmish line and turned on the speaker.\n\n\"Folks, the party is over. Texas in an independent nation, and as governor I am going to enforce the law. You and the folks who live around here will be questioned. If anyone here is guilty of murder, he will stand trial. For the rest of you, I am here to tell you nothing will happen to you if you obey the law from this minute on. No more looting, no more stealing, no more fires, none of that.\"\n\nHays paused and silence reigned except for the moan of a siren a long way off.\n\n\"I know, Charlie Swim knows, and Luwanda Harris knows that you and your families have many grievances, from failing schools to horrific unemployment rates, to police harassment for the crime of being black.\n\n\"But the time has come for a new beginning for Texas and for its citizens. I swear to you that the Texas legislature and I are going to take action.\n\n\"We are going to have every complaint about police brutality investigated by the staff of a legislative committee, and both these folks standing beside me, Charlie Swim and Luwanda Harris, are going to be on that committee. If you think they will sweep harassment and brutality under the rug, you don't know them.\n\n\"We're going to set up a private-public partnership so that people in your community, people who study, can qualify for the thirty thousand new high-tech, high-paying jobs that are projected to grow in Houston in the next few years. . .and you are the people who are going to fill them. Industry will pay part of the cost of your training and the Republic of Texas will pay part. All you have to do to qualify is put your butt in a chair and study hard.\n\n\"Texas needs you right now. We are going to be invaded by United States forces in the near future. The Texas Guard needs recruits. You can do yourself and Texas a favor by enlisting. I am not going to pretend it will be easy or without danger. You may get wounded, maimed, or killed. But Texas needs your help. Make your life mean something. Fight for Texas.\n\n\"Folks, the riot is really over. Stay and talk to the guardsmen or go home. No more rioting. This is your city and your nation.\"\n\nJack Hays got down off the fire truck. \"Charlie, get on in there and talk to them. We need all the soldiers you can get. These guys like to fight\u2014let's point them in the right direction and give them some discipline and leadership. Hell, let's give them a country to fight _for_.\"\n\nHays looked at Luwanda Harris and added, \"You tell them I'm sincere\u2014because I am.\" Then he turned and walked alone the mile and a half back to the helicopter.\n\nIn Abilene, Colonel Wriston had his column of tanks and construction vehicles ready to go by ten that morning. It had been hectic. He had received an unexpected assist from the president, who had announced via television that he was nationalizing the National Guard, so many of the soldiers had reported to the armory without waiting to be summoned.\n\nWriston and his officers explained that since Texas had declared its independence the Texas Guard was going to defend Texas and take its orders from the governor. All but four of the guardsmen\u2014who were sent home\u2014agreed to defend an independent Texas, and Colonel Wriston quickly had them organized into units and loaded them aboard trucks and buses pressed into service. With tanks in the lead, the column got rolling at ten o'clock.\n\nOne of the soldiers who was sent home instead drove straight to the main gate at Dyess and told the sergeant of the guard he wanted to see the commanding general. A call was made, and the sergeant climbed into his air force SUV and led the way to the headquarters building.\n\nWithin a minute the guardsman was standing in front of the commanding general, Brigadier General Lou l'Angistino, explaining what the Texas Guard was up to. \"They're going to block your runway, General.\"\n\n\"When?\"\n\n\"About as fast as they can get there, sir, I reckon.\"\n\n\"Do you know where they intend to breach our perimeter?\"\n\n\"No, sir. I didn't hear anyone say.\"\n\nThe general thanked the man and watched him leave the office. He nodded to his chief of staff, who closed the door. They had been poring over a stream of classified messages that flowed into the office just as fast as the message center could get them decoded and printed.\n\nGlobal Strike Command, GSC, headquarters had ordered him to get his airplanes ready to fly. They might be sent on bombing missions. . .or they might be sent to Offutt Air Force Base in Nebraska. . .or. . . . In the next message, GSC headquarters hedged. Belay the first message: Stand by for further orders. Let no civilians onto the base. Consult with local authorities and advise of the political situation in Abilene ASAP. Were the people loyal to the federal government or to the Texas rebels?\n\nOn it went. Action messages were interspersed with messages from Washington, from the Joint Chiefs, and every command all over. The army needed his C-130s in Colorado and Alabama. Send them immediately. No, wait. Get them ready to fly and when higher authority had sorted out the priorities, mission orders would be issued.\n\nGeneral l'Angistino shoved the whole pile to a corner of his desk. \"Get the base security officer in here. Roust every air policeman on the base and get them suited up.\"\n\n\"Yes, sir.\"\n\nL'Angistino had seen the morning news footage of the declaration. He had been horrified; his comfortable peacetime command had just been transformed.\n\nHe looked out his office window at the runway. Rows and rows of B-1 Lancer bombers and C-130 Hercules aircraft were parked on the ramps. He had thirty-six B-1s assigned, the only B-1 wing on active duty in the air force. Twenty-eight C-130s were assigned here, but five were flying, doing overnight training missions or hauling troops and supplies from one military installation to another, the usual peacetime flight schedule. Now this.\n\nHe had already issued orders to get all the airplanes serviced, fueled, and ready to fly. He didn't tell anyone to bring the bombs for the B-1s from the magazines, and wouldn't until they had missions assigned. Parking weapons on the ramp when they weren't needed violated air force safety regulations.\n\nBlock the runway. Was the guardsman telling the truth, or was this only a rumor? Or was he a plant to spread disinformation?\n\nThe general was consulting a map of the base with his security officer, a major, when Colonel Wriston stopped his column at the place he and his deputy commander had been that morning.\n\nA lowboy behind the colonel's Humvee off-loaded a bulldozer, which scraped dirt to fill in the ditch between the road and the perimeter fence. The job was done in less than two minutes. More bulldozers off-loaded from lowboys. They quickly tore out fifty yards of eight-foot-high, chain-link, barbed-wire-topped fence and shoved it to one side. A tank covered with soldiers went through the gap.\n\nWriston watched his tanks, bulldozers, road graders, earthmovers, trucks, and buses full of guardsmen as they rolled through the gap and disappeared in a cloud of dust. Many of Wriston's men were heavy-equipment operators in civilian life, which had made commandeering so much equipment relatively easy; it was theirs, and by parking it on the Dyess runway, they'd be putting themselves out of work. Well, he suspected Jack Hays would want full-time soldiers.\n\nThe drivers of the tanks soon spread out until all four were running abreast raising dust clouds. This had the unintended consequence of blinding the drivers of the vehicles behind them, who were also trying to spread out to avoid colliding with everyone in front. Watching his column disintegrate, Colonel Wriston was reminded of his experience in tanks in the deserts of Iraq. He hated deserts. He climbed into his Humvee and headed off behind them to supervise this operation. If they didn't get the runways blocked, this whole adventure was for naught.\n\nIn the leftmost tank, the tank commander saw an air force SUV charging across the prairie toward him. There were four people in it, apparently. It came to a stop fifty yards in front of him, and the driver jumped out, holding up his right hand in the universal signal to stop.\n\n\"Shoot out his radiator,\" the tank commander told his machine gunner.\n\nThe burst of the .30-caliber apparently holed the SUV's radiator, because a cloud of steam shot forth from under the hood. The other three occupants of the vehicle jumped out with their hands in the air. Two guardsmen dropped off the tank, which speeded back up. The guardsmen disarmed the air police, pointed them at the hangar complex two miles away, and told them to start hiking.\n\n\"You can't do this,\" the air force sergeant protested.\n\n\"We already did,\" a soldier answered. \"Git.\" They grabbed the guns in the SUV and on the ground, then ran to the left, the west, to get away from oncoming vehicles. Colonel Wriston saw them as he came up and stopped to give them a ride.\n\nA small group, one bulldozer and three earthmovers, peeled off to block the short runway. The main column clanked up to the long runway, an awesome sight with more than two miles of thick concrete stretching before them, three hundred feet wide.\n\nWriston sped past the lead tanks and went a third of the way down the runway. He knew how far he was because he stopped just past the big \"8\" sign, marking eight thousand feet remaining to the end. He gestured to two of the tanks, and they came to a stop, then turned sideways. Other equipment would also park there.\n\nWriston got back in the Humvee and rolled on to the \"4,000 feet remaining\" sign. He stopped there and awaited his vehicles.\n\nThe operation went well, he thought. As some of the soldiers stood guard, the two tanks and four pieces of construction equipment were parked. Mechanics worked on the treads of the tanks, then the tanks ran off the treads. Bulldozers were similarly disabled. The tires of the earthmovers were shredded with automatic weapons fire.\n\nWhile this was going on, four air police vehicles came rushing toward them, two on the runway and two on the adjacent taxiway. Machine-gun fire and automatic weapons fire over the top of the vehicles convinced the drivers to turn around and retreat.\n\nHand grenades were placed in engine bays, and guardsmen ran from the explosions. It was over in less than eight minutes. When all his guardsmen were on buses and trucks going back toward the hole in the fence, Colonel Wriston surveyed the blockade and followed along in his Humvee. The men and women of the Guard couldn't have done it any better if they had practiced it every day for a week, he thought proudly. Then he followed his retreating vehicles.\n\nGeneral l'Angistino had watched the dust cloud and activity on the runway from his office with binoculars. When the guardsmen had departed, he rode out to the mess of abandoned equipment and surveyed it with his fists on his hips.\n\nHis chief of staff rolled up in an air police sedan. \"You know what to do,\" he said to the colonel. \"Get busy and get this stuff off the runway. As quickly as possible.\"\n\nAir force crash crews were still moving equipment at dark, when General l'Angistino went home. He had of course notified GSC and Washington of the runway obstructions, but other than a terse message to report when the runway was open again, nothing else was said.\nNINE\n\nThere was an old sleeping bag in the workroom of the lock shop, and I spent the night in it. Willie had the television on when I woke up.\n\nThe news this Sunday morning was that Texas had declared its independence during the wee hours of the morning. I listened while I helped myself to a cup of coffee.\n\n\"The world is movin' right along, Tommy,\" Willie said. \"Texas declared itself free of the US of A, and Barry Soetoro is havin' a shit fit. He says that the right-wing conspiracy was more virulent than he and his advisors suspected. This should silence any critics of martial law. And so on.\"\n\nThe coffee was hot and black, and strong enough to take the enamel off your teeth, but a man can't have everything. Idly, I thought about Sarah's coffee\u2014hers was several times better than this stuff. Maybe I should have tried to wheedle her into letting me sleep on her couch last night. Or in her bed.\n\n\"Guess we're back to forty-nine states,\" Willie said philosophically, \"if Texas can make Soetoro eat it. Kinda doubt that they can, but who knows. He was on the tube a minute ago, and was he ever pissed! Babbled about treason. Treachery. Betrayal. The malignant tumors in high offices.\"\n\n\"Good help is hard to find these days.\"\n\n\"Want an egg?\"\n\n\"Yeah, that would be good.\"\n\n\"Well, we ain't got any, this bein' a lock shop. No pancakes or bacon or ham or toast. I brought in a half-dozen doughnuts for me this mornin'; if you want one I can spare it.\"\n\n\"That's mighty white of you.\"\n\n\"Don't get racial, dude.\"\n\n\"What else don't we have?\"\n\n\"Lots of stuff. Got toilet paper, though. White toilet paper.\"\n\n\"That's the best kind.\"\n\n\"Can't believe that they let us black folk wipe our asses with it.\"\n\nI took my coffee to the restroom and settled on the throne. I reflected that Willie Varner had reminded me once again why no female on the planet had succumbed to his charms and leaped into matrimony.\n\nThe Texas revolt was good news, I thought. The dung beetles at the White House now had something to think about besides forcing confessions from people like Grafton. Perhaps. Maybe they would decide that Grafton was partly responsible for the bad attitude in Austin.\n\nI dressed, drank another cup of Willie's coffee-colored enamel-eater, and then headed over to McDonald's for a sausage and egg biscuit and a cup of decent coffee. I made some phone calls. Called some of the covert warriors I knew, guys I had served with in various third world shitholes. Two were at home. I asked if I could come by. They said yes.\n\nWillis Coffee lived in Bethesda. His wife answered the door and told me he was around back in the garden. It looked more like a flower bed with vegetables, a few onions and some scraggly lettuce. He was hoeing.\n\n\"What the hell you doing in Washington? I thought you quit your job,\" he said.\n\n\"I did quit.\"\n\n\"Hell, so did I. When they arrested Jake Grafton, I turned in my building pass and drove out of there.\"\n\n\"I still have the building pass. Maybe I ought to mail it in. They might come looking for it.\"\n\nWillis snorted and leaned on his hoe. \"I doubt if anyone at Langley has the time. Ol' Harley Merritt is on the bridge now and Soetoro is cracking the whip. Merritt is looking for traitors within the agency. Maybe he'll find one, but I doubt it.\" He spotted a weed and attacked it.\n\n\"Maybe they'll invent some, like they did with Grafton.\"\n\nWillis leaned on his hoe again. \"Yeah,\" he said.\n\n\"I need your help, Willis, to rescue Grafton. You probably heard they accused him of conspiring to depose Soetoro. Blow him up. Try to turn America into a democracy.\"\n\n\"I heard.\" He stood there awhile, surveying the weeds in his agricultural project. Then he threw down the hoe. He dragged over a chair and sat in it. \"I got a wife and two kids. The kids were on a sleepover last night. I need another job, one that will pay the mortgage and grocery bill. I can't afford to go tilting at windmills.\"\n\nI played my ace. \"If you were in some ISIS dungeon waiting for your appointment with the knife, you _know_ that Jake Grafton would move heaven and earth to get you out. Whatever it took. Whether State gave its okay or not.\"\n\nHe jerked as if I had stuck the knife in him right there. He refused to meet my eyes.\n\nAfter a bit he said, \"Tell me about it.\"\n\nThe chairman of the Joint Chiefs, General Martin L. Wynette, was summoned to the White House that morning. Wynette was a sycophant, a paper-pushing soldier who had never seen combat but had kissed a thousand asses on his way up the ladder. He was known throughout the military for his role in destroying the career of a lieutenant colonel teaching a course at the Joint Forces Staff College about Islamic extremism and jihad, a course suggested and approved by the college. Some Muslims got wind of it and wrote a hot letter to Barry Soetoro, who ordered the offending officer disciplined and the course dropped. Wynette did the dirty work without protest. Of course Wynette knew that Soetoro's father was a Muslim, his chief political advisor was a Muslim, and a sizable chunk of the American population thought he was too, but after all, the American people had voted Soetoro into the White House, twice, so Wynette was certainly willing to let the prevailing wind flap his flag.\n\nThis morning the general was escorted into the presence of the anointed one, who was beyond fury. He was outraged and shaking, at times almost incoherent. Texas had to be punished, he told General Wynette. \"Texas must be taught a lesson that the people there will never forget. What are you people in the Pentagon going to do to smash them?\"\n\nThe truth was that the military had no contingency plans to attack Texas, or New York City or Honolulu or Des Moines or anywhere else in the United States. But General Wynette told the president, \"Staff is working on it, sir. I assume you want boots on the ground.\"\n\n\"Boots on the ground, bombs on target, and the heads of every one of those sons of bitches in the legislature. And that governor\u2014I want him alive. You go get them, General. Go to Texas and kick ass. Go as soon as you can get there.\"\n\n\"Yes, sir.\"\n\nDismissed, he marched out to the limo waiting to take him back to the Pentagon wondering if the president really meant for Martin L. Wynette to personally go to Texas to direct the invasion. Certainly not. He must have meant that figuratively, General Wynette decided. He settled back into the comfy leather seat of the limo.\n\nOther than that, Wynette thought, the president had been specific enough. Bomb the hell out of those rebels, then invade. Air force fighter-bombers blasting refineries, factories, power plants, and oil fields would get those fools' attention. A naval blockade would stopper their ports and screw them down hard. Then the U.S. Army would go charging through Texas like Wynette's hero, George Patton, went through Germany. As Georgie used to say, \"Like crap through a goose.\"\n\nWhen he drove through the little town of Langtry, JR Hays saw Texas flags flying in front of every house and building, every business. Must have been a hundred of them. Was this Texas Independence Day? No, that was in March. He shrugged and kept driving.\n\nDel Rio also looked as if it were having a flag festival. Texas flags were everywhere, flying, hanging, tacked to buildings, strung across the street. He pulled into a filling station and went in for a piss and a Coke.\n\nThe people inside greeted him like a long-lost cousin. \"Happy Independence Day.\"\n\n\"I thought that was in March.\"\n\n\"That was then, this is now. Today. Early this morning Texas declared its independence. Haven't you heard?\"\n\n\"No.\"\n\nThe young man behind the counter with rings in his ears and one in his nose pointed to a newspaper. \"Special edition,\" he said. The headline took up all the space above the fold: \"Texas Free, Again.\" The kid was wearing a pistol in a belt holster.\n\n\"How about that,\" JR said.\n\n\"Already we got some troubles,\" the kid said. \"Mexicans tried to force the bridge from Ciudad Acuna this mornin', tryin' to get across. Must have figured that without the feds we'd be runnin' around with our thumbs up our asses. Some of the guys went down there with their rifles and put a stop to that shit. Shot some of 'em. They're layin' out there on the bridge bleedin' all over. The Mexicans won't expose themselves to drag them away and our guys ain't goin' to go to their rescue. They can crawl back to Mexico or lay there and die.\"\n\n\"Hmm.\"\n\n\"State trooper was in here a little bit ago and tol' me all about it. Eight or ten got across before the shootin' started. Folks are roundin' 'em up and gonna make 'em walk back over the bridge. Some other guys are goin' through the city right now roundin' up illegals to take the walk. Feds ain't protectin' them anymore.\"\n\n\"Where are the feds?\"\n\n\"Home, I reckon. They're all Texans. They don't like that asshole Soetoro either, but it was do it his way or get fired.\"\n\nAfter he went to the men's room, JR picked a Coke from the cooler and took it to the counter.\n\n\"American money still good?\"\n\n\"The boss ain't tol' me not to take it. Reckon the politicians will have to figure all that stuff out.\"\n\n\"I guess so,\" JR said. \"Happy Independence Day.\" He paid and walked out.\n\nOutside, he looked around. There wasn't an American flag in sight. Just lots of Lone Star flags.\n\nBeing human, he wondered about his pension. Twenty years in the army and now no pension. He felt like one of those Mexicans bleeding to death down on the bridge over the river.\n\nHe started the truck and screwed the plastic lid off the plastic bottle and took a sip. Then the implications of independence hit him. Texas was going to need an army. Maybe he could join. Hell, soldiering was what he knew how to do; it was the only thing he knew how to do. He would ask Jack about that.\n\nAt a stoplight he lit another Camel.\n\nDamn! _The Republic of Texas_. How about _that_!\n\nTravis Clay had been home from the Middle East for only two days and had next week off. He was just getting out of bed when I knocked on the door of his apartment. He was in his underwear when he opened the door and motioned me in. I could hear television audio from the bedroom.\n\nHe went to the kitchen and put coffee in the basket and added water. As he did the work, I leaned on the door jam and asked, \"How was Syria?\"\n\n\"The fires of hell have leaked through the crust there. Never trust a man who wipes his ass with his bare hand. We thought we knew where that British dude was who likes to lop off heads with a knife, but he was gone when we hit the place. The Brits were royally pissed. Words cannot express how badly they want that murderous prick. They would sell a prince and maybe a princess or two to lay hands on him for just an hour.\"\n\n\"That's what they get for letting every raghead who can get there into the country.\"\n\n\"Don't say that aloud to them. They don't have warm fuzzies about the politicians. And Soetoro is doing it too. Welcome to diversity.\"\n\n\"So where's your significant other?\"\n\n\"Rachel? She hit the road. I don't know the straight of it, but I think she got tired of waiting for me to come home and started picking up men in bars. Anyway, she left a note. Want to read it?\"\n\n\"No.\"\n\n\"That's good, because I tore it up.\"\n\nThe coffee was dripping through, and he poured me a cup. He had to wait a minute for the pot to deliver enough for another cup.\n\nWhen we were sitting in his little living room, I said, \"I suppose you heard about Jake Grafton getting arrested.\"\n\n\"Yeah. And parochial school murders and martial law and Texas declaring independence and all of that. The whole damned country is going to hell in a wheelbarrow. I'm thinking about pulling the plug and going to Montana. You know I grew up there?\"\n\n\"I didn't know that.\"\n\n\"Yeah. My folks are outfitters, fishing trips during the spring and summer and hunters in the fall. My dad told me last night I've got a job there if I want it. I'm sorta thinking I do. I don't want to go back to Syria. They're all pedophiles and wife-beaters. Sunnis and Shiites will be fighting each other for a century or two, and the truth is, I don't think it matters a single teeny tiny little goddamn who wins.\"\n\n\"Probably not,\" I murmured.\n\n\"The only thing I am absolutely convinced of, I don't want to die in that shithole.\"\n\n\"Montana would be good.\"\n\n\"I'm thinking about it.\"\n\n\"Before you run off, I need some help.\" I told him about Jake Grafton and my project to rescue him.\n\nTravis Clay took it like a man and didn't cry. What he said was, \"Fuck you, Carmellini.\"\n\n\"You aren't cute enough.\"\n\nWe batted it back and forth awhile, and I told him Willis Coffee was on board.\n\n\"Oh, hell,\" he finally said. \"Why not?\"\n\nHalf an hour later, after we had gone through my plan from end to end, he said, \"If you have to shoot an FBI agent, can you do it?\"\n\nI answered honestly. \"I don't know.\"\n\n\"Better get that figured out before we saddle up. I guarantee you they will shoot you and me and Willis Coffee in a heartbeat if we stop that car. That's what they train them to do at Quantico. They won't even think about it\u2014they'll just throw lead.\"\n\n\"I suppose.\"\n\n\"What you need, Tommy, is a serious diversion. Think about that for a while. The feds will pull out all the stops if we snatch Jake Grafton, whether we shoot an agent or two or not. Barry Soetoro will turn purple. We must give Soetoro and the rest of them something else to think about, something with a higher priority.\"\n\nI was in a McDonald's munching a Big Mac when the phone rang. It was Callie Grafton.\n\n\"I saw him,\" she said. \"He looks good.\"\n\n\"Great. Maybe I'll stop by this evening for a beer.\"\n\n\"Sure,\" she said.\n\nWe hung up.\n\nSo it was a go.\n\nJR Hays rolled into Austin late that Sunday afternoon. He was fighting to stay awake, but he parked by the state capitol and walked across the lawn. Upstairs, he told the governor's receptionist who he was and took a seat in the waiting room. Legislators came and went, striding purposefully, almost trotting. He gathered that the legislature was in session on the other side of the building, arguing about and passing the legislation needed to convert Texas from a state in the United States to an independent republic.\n\nAn hour passed. JR dozed in the chair. The governor shook him awake. \"Come into my office, JR. I apologize for the wait. We're making history and trying to give every Texan a decent place to live.\"\n\nHe went into the office, and Jack Hays closed the door behind them. \"Talk to me,\" the governor said, and sat down behind the desk.\n\nJR dropped into a chair and told it. \"There are ten dead men at the ranch. I ambushed them last night. They were carrying about two hundred pounds of some kind of drug, and I have about a hundred fifty pounds of it in the truck. Two of the backpacks the mules carried were too full of holes to hold the stuff. One of the guards was that deputy sheriff we met before the funeral, Morales I think his name was. There couldn't be two men in west Texas tattooed like that. After the ambush, I hot-footed it out to the highway, and who should be driving up and down but Sheriff Manuel Tejada.\"\n\n\"Was he in on it, you think?\"\n\n\"I called him this morning, told him there had been a shootout between two drug gangs, and the stuff was lying all over. Told him I wanted to call the state police and DEA. He begged me to wait until he had come out to look the scene over. He would have probably tried to shoot me, so I boogied.\"\n\nJack Hays was a quick study. \"How do you want to handle this?\" he asked his cousin.\n\n\"We have to fix it so the drug syndicate guys don't come to the ranch with enough firepower to conquer Israel and whack little old me. Plugging Tejada would have felt mighty good, but it wouldn't have solved that problem. I want to take these backpacks over to DPS headquarters, and the colonel needs to have a press conference. Show the drugs to the press. He needs to thank Sheriff Manuel Tejada for his cooperation, which was an essential element in the investigation that allowed the Texas DPS to break up this gang of smugglers.\"\n\nJack Hays smiled. \"The phones here are down. I'll take you over there. Let's go.\"\n\nThe cousins drove to the state police headquarters in JR's truck. They went in to see Colonel Frank Tenney. Fifteen minutes later two state troopers armed with the key to JR's toolbox in the bed of his truck carried the backpacks full of dope up to Tenney's office.\n\nTenney looked the governor in the eye. \"There was a warrant issued for JR over in Upshur County today. He's wanted for murder and drug trafficking. It's signed by a justice of the peace. They radioed the news in.\"\n\n\"Squash it,\" Jack Hays said, waving the warrant away as if shooing a fly. \"He was working as an undercover agent for the Department of Public Safety. I want you to hold a press conference, for the evening news if possible, and have the department take full credit for recovering a hundred and fifty pounds\u2014or whatever it is\u2014of narcotics and smashing a smuggling gang. And I want you to tell the world that it wouldn't have happened without the active help of the sheriff of Upshur County, Manuel Tejada, who gave you the intelligence necessary to break up this gang. It is unfortunate that the smugglers chose to fight rather than submit to arrest and trial by jury, but that was their choice. I want you to make the point that the Republic of Texas will seek out and actively hunt down narco-criminals. Tell the world that Governor Jack Hays has personally assured you the Department of Public Safety will get the funding and manpower needed to finally do the job right.\"\n\nThe lab did a quick check and established the drug was pure, uncut cocaine, and the cops weighed the stuff. The street value they came up with was $1,360,000 at twenty grand a kilo.\n\nDriving back to the capitol, Jack Hays told JR, \"Come on over to my place for dinner tonight. We need to talk. Washington is gearing up for a war against Texas.\"\n\n\"Breakfast tomorrow,\" JR said. \"I've been up over thirty-six hours and am going to a hotel to crash.\"\n\n\"Breakfast at my house,\" Jack Hays said, shook his cousin's hand, and walked into the capitol.\n\nJR did indeed crash, but not until after he had a shower and watched Colonel Tenney on the evening news. The camera lingered on the pile of cocaine on Tenney's desk. \"Breaking this gang would not have happened without the intelligence provided by and the active cooperation of Sheriff Manuel Tejada of Upshur County,\" Colonel Tenney intoned, staring into the camera. \"He was instrumental in helping us smash a major narcotics smuggling operation. All of Texas thanks you, Sheriff Tejada.\"\n\nJR hit the bed and slept for ten hours.\n\nThe aftermath was not slow in coming. Two mornings later Mrs. Tejada found her husband wired to a tree in their backyard. He was dead, strangled with bailing wire. She was pretty broken up about it, until she found over a quarter of a million dollars in an old chest in the guest bedroom, wrapped in a quilt her mother made over a half century ago. Since no one knew where the money had come from, she kept it and lit a candle for her husband in the local church.\n\nWhen the state police finally got around to visiting the Hays ranch, they found the bodies, which had been worked on by buzzards, coyotes, and feral pigs, one of which was lying dead with the mules. It had apparently ingested enough of the cocaine scattered around to kill it, so presumably it went to pig heaven happy. The only positive identification the cops made was the body of Deputy Sheriff Jesus Morales, identified by his fingerprints and distinctive tattoos, but his boss had been dead almost a week by then, so it was decided to not make a fuss and embarrass the Morales family, who were third-generation Americans, and by all accounts good people. The other dead men were apparently Mexican nationals, so their fingerprints were passed to the Mexican DEA, which didn't bother to acknowledge the receipt of them.\n\nWhen the Hays' hired man returned from his two-week vacation, he repaired the ranch fence.\n\nBut all that was aftermath, and the lives of Jack and JR Hays had moved on by then.\nTEN\n\nOn Monday morning, August 29, JR got his pickup from the hotel valet and drove to the governor's mansion. There he discovered that the governor had a maid, who admitted him and led him to the dining room, where Jack and Nadine were buried in the _Austin Statesman_. The events of the previous day and the full text of the Declaration of Independence filled the front page. Inside were interviews with legislators and quick man-in-the street quotations from celebrating citizens of the new republic. The _Statesman_ , a liberal newspaper, editorialized that the governor and legislators who voted for independence were irresponsible radicals whose actions bordered on insanity.\n\nAfter the trio had discussed the events of the previous day, JR remarked about the maid. Jack said he did a lot of official entertaining so the legislature paid the salaries of a maid and a cook.\n\n\"He's in the kitchen now whipping something up. You ready?\"\n\n\"Sure,\" JR said.\n\n\"Jack, read that editorial aloud,\" Nadine urged. So he did.\n\nJack said dryly, \"If the _Statesman_ had editorialized that we had done the right thing, I would have been really worried.\"\n\nSoon the maid served eggs Sardou with crumbled bacon, unbuttered toast, and white wine.\n\nJack said to his cousin, \"If it's too early for you for wine, we have coffee and the juices.\"\n\nJR glanced at his watch. \"I have an ironclad rule that I never drink before seven in the morning,\" he said, \"and it's ten after. I'll do the wine.\"\n\nNadine took coffee with cream.\n\nJack and Nadine expressed the hope that the Houston rioting was at last at an end. As they ate they discussed the new status of Texas.\n\n\"Tell us what you think,\" Nadine said to JR.\n\nJR thought about his response, then said, \"I'm a natural-born Texan, and I've had it with Soetoro. A few terrorist incidents don't seem to be a good reason to declare martial law. The FBI and local police can sort that stuff out. I sorta suspect Soetoro thought all that was a good enough excuse to become a dictator, but I don't know. I just caught snatches of the news, here and there.\"\n\nNadine zeroed in. \"Are you happy with independence?\"\n\n\"Anyone who isn't can hit the road,\" JR said. \"I'm staying. But I hope you folks know that you have bought a ton of trouble. I doubt if all the U.S. soldiers and sailors and airmen will stick to Barry, but a lot of them will, and that'll be plenty. They can cause you lots of grief. Air strikes against concentrations of troops or civilians, against industry, refineries, armories, storage tanks, power generation facilities, everything you can think of, plus armored columns and infantry going through the towns and cities to take them house by house and block by block, seeking out and killing or defeating the rebels. . .it could get damned rough. The feds will ultimately lose, of course, but they will give it the old college try and kill a lot of Texans before they throw in the towel.\"\n\nNadine jumped right on it. \"Why will they lose?\"\n\n\"Because control of the cities is strategically worthless. Whoever controls the countryside always wins in the end, if they keep their nerve and are willing to take the casualties. People in cities have to eat, and the food comes from the countryside. Not to mention electrical power, gasoline, and every other commodity known to man. It all has to be produced in the country or transported through the country, which means it is militarily vulnerable.\"\n\n\"You think?\"\n\n\"I know. The American Revolution, the French, Russian, Chinese, and Cuban Revolutions, Vietnam, Afghanistan, you name it. Control of the countryside was the essential element every time. And successful revolutions or rebellions are not the victory of a pissed-off majority, but the triumph of a dedicated minority who won't quit. It doesn't take many men. But the revolutionaries must be willing to suffer and be quite ruthless with the enemy.\"\n\n\"There will be casualties.\"\n\n\"A lot of them,\" JR agreed, and sipped his wine. \"Bloodless revolutions are usually military coups\u2014the generals win because no one else has weapons. The people of Texas are armed. Everyone has guns, and a lot of people know how to use them. More important, some of them are willing to do so. Just having a gun isn't enough. Successful rebels must be willing to fight, to kill, and if necessary, be killed. But you know all that and declared independence anyway, so I assume a lot of legislators have some guts. Or their constituents do, which is better. Whether a dedicated minority has enough guts and determination remains to be seen. Time will tell.\"\n\n\"We are hearing very little from Washington,\" Jack Hays said. \"They aren't going to make idle threats. When the blow falls, it will be heavy.\"\n\n\"Don't wait for it,\" JR advised. \"You must take it to them. Seize the initiative and put them on the defensive. That's the only way. The Confederates in the American Civil War were strategically hampered by the politicians' desire to take the defensive. In war the defense always loses. If you try to defend everything, you are spread so thin you end up defending nothing. If you try to defend just a few key places or installations, the attackers will bleed you to death someplace else.\"\n\nNadine had abandoned her breakfast. \"So how do we prevail and make our independence stick?\"\n\n\"Attack. As U. S. Grant said, find out where they are, hit 'em as soon as you can, as hard as you can, and keep moving on.\"\n\n\"The best defense is a good offense,\" Jack Hays said thoughtfully.\n\n\"Amen to that. In the military we call it seizing the initiative, forcing your enemy to react to your moves rather than you reacting to his.\"\n\n\"It's the same way in politics.\"\n\nNadine looked at her watch and said she had to leave for the university. Her breakfast was only half-eaten. She rose, got a kiss on the cheek from both men, grabbed her purse, and hurried for the garage.\n\nJack Hays leaned back in his chair. The maid came in with the coffee pot and poured.\n\nWhen she had left again, Jack Hays asked JR, \"If you were running the military show, how would you go about it?\"\n\n\"That's a big _if_.\"\n\n\"A hypothetical.\"\n\n\"The commander must figure out what he has to fight with. That's Job One. What we have in the way of people, weapons, ammo, and transport defines our options. Our hypothetical commander must start there.\"\n\nJack Hays nodded, sipped his coffee, and nodded again. \"If I offered to make you a general,\" he said, \"and put you in charge of the Texas Armed Forces, which is the National Guard, Air National Guard, and every military unit within Texas, all you can grab, you'd be in charge of defending Texas. Would you take the job?\"\n\nJR grinned. \"I came here this morning,\" he said, \"to ask for a job in the Texas Army. Any job. Soldiering is all I know. I suspect that you are going to need an army very badly, very soon.\"\n\n\"General Twilley has wanted to retire for the last year, and I have been asking him to put it off and hang in there. I want you to take his place. The air guard guy is Major General Elvin Gentry. He'll answer to you.\"\n\n\"Okay,\" JR said.\n\n\"Before you go, I have to add the Texas Navy to your list of responsibilities. We got a nuke attack sub yesterday morning. USS _Texas_. She's sitting at a pier in Galveston, and we've got to do something with her quick before the U.S. Navy sinks her or steals her back.\"\n\n\"Is she undamaged?\"\n\n\"The sheriff down there thinks she is, but his nautical experience is limited to bass boats.\"\n\n\"I've got an old army friend who got fed up with grunts and transferred to the navy,\" JR said slowly. \"He's retired now. As I recall, he was in attack subs. Smart as a tack. Went to nuke power school and did well. He's a law student now at UT. I can send him down to evaluate the boat. If we can't move and hide her, Jack, we probably ought to scuttle her right where she is so the SEALs can't steal her out from under our noses.\"\n\n\"They could do that?\"\n\n\"You can bet they're noodling on how to do it right now.\"\n\nJack Hays scooted his chair back and rose. \"Sounds like you need to get busy.\"\n\n\"Yes, sir,\" JR Hays said. \"I'll do my best.\"\n\n\"I'd like to introduce you to the press and the Guard brass hats this morning, but I've got to go to Houston again. We'll do the paperwork when I get back. I'll scribble a note to General Twilley. You run out to Camp Mabry, give it to him, and take command\u2014and have him muster you up a major general's uniform.\"\n\n\"Where is Camp Mabry?\"\n\nHis cousin stared at him a moment before he answered, \"West Thirty-Fifth Street, west of Highway One.\" Then he grinned. If you don't know, ask. JR would do nicely.\n\nThe governor wrote the note in longhand, they shook hands, and JR headed for the front door and his pickup.\n\nJR drove to the University of Texas Law School and went in. He found his friend, a muscular black man named Loren Snyder, standing in a hallway outside a classroom talking to two fellow students.\n\n\"Lorrie.\" JR smacked him on the shoulder.\n\n\"JR Hays, folks. Long time no see, JR. What are you doing here?\"\n\n\"Thinking of getting a law degree and wanted to talk to you about that.\"\n\n\"Well,\" Loren glanced at his watch. \"I have ten minutes.\"\n\n\"Terrific.\"\n\nJR led Loren away from the other students to a quiet corner. \"Weren't you in attack submarines?\"\n\n\"Yep. Sixteen years of it after the army, which my wife said was plenty long enough. Now I'm going for the gold. Going to be a personal injury lawyer and screw those insurance companies down hard.\"\n\n\"Before you get to that, I need some help. I'm now a major general commanding all the military forces of the new Republic of Texas.\"\n\n\"You're _what_?\"\n\n\"You heard me. We acquired a nuke attack sub yesterday morning down at Galveston, and I need a quick evaluation of the boat by someone who knows what they are talking about.\"\n\n\"I saw in the paper _Texas_ was making a port visit there.\"\n\n\"Will you go to Galveston right now and evaluate the condition of the boat? Then answer some questions for me. Specifically, can we get enough ex-sailors to move her, can we hide her, or should we just sink her at the pier so the SEALs can't snatch her back?\"\n\n\"You haven't even asked me if I'm a loyal Texan.\"\n\n\"Are you?\"\n\n\"Well, I don't know. Haven't thought much about it.\"\n\n\"You do this, I'll give you a medal to frame and hang in your law office, when you get that office.\"\n\n\"You want me to go now, miss today's classes?\"\n\n\"An hour ago would have been better.\"\n\n\"How do I get hold of you?\"\n\n\"Any National Guard armory. They can radio me a message.\"\n\nLoren gave him a sheet of paper from a notebook and JR wrote upon it, \"Please allow Loren Snyder to inspect USS _Texas_ ,\" and signed it JR Hays, Major General, Commanding, dated it, and gave it to Loren.\n\n\"Well, _that_ looks official,\" said Loren.\n\n\"I gotta run,\" JR said. \"And you do too. Saddle up.\"\n\nBrigadier General Lou l'Angistino was fifty years old, from Nebraska, an ROTC graduate who had worked his way up the ladder, flying and performing staff jobs. He flew F-4 Phantoms and F-16s, and considered it ironic that he now commanded a bomber wing. The Global Strike Command dudes must have been very unhappy when they heard. Of course, he had served a tour on the GSC commander's staff, and maybe that was why he was selected.\n\nL'Angistino habitually went to bed at nine o'clock in the evening unless he had an official function to attend, and rose between four thirty and five o'clock a.m. He usually put a leash on his black Lab and then ran five miles, rain or shine.\n\nThe events of the previous week troubled him deeply. He knew all about Jade Helm, the plan of the Federal Emergency Management Agency to put Americans in concentration camps, and he also knew that liberals, minorities, and Democrats weren't the intended detainees. He had been appalled when Soetoro announced martial law and invoked Jade Helm.\n\nThe National Guard's blockade of the runway yesterday was only the first shot in the war, he told himself. Texas has a lot more bullets. Last night his staff thought that the crash crews would have the runway cleared by late this morning. Then he would fly the planes out, if he could find enough flight crews. He suspected that might be a problem. But he would worry about all that when he got to the office.\n\nNormally the general ran on base, but this morning he put the lab in the car and headed for the main gate. He drove past the thirty aircraft that lined the boulevard to the highway, aircraft dating from World War II right up through the present day. One was a retired B-1 and another a retired C-130.\n\nHe drove the seven miles into town, marveled at the display of Texas flags, and, as the sun rose, was jogging in a park with his dog.\n\nTwo miles along he saw a man in a baseball cap sitting on a bench with a rifle across his knees. He had a golden retriever on a leash. As l'Angistino got closer, he saw the man was probably Latino and well past retirement age. He was also wearing a gun belt with a pistol in a holster.\n\nThe general stopped to talk. As the dogs sniffed each other and got acquainted, the man said, \"Was you in the air force?\" L'Angistino was wearing a faded air force T-shirt and red shorts. He nodded.\n\n\"I was too,\" the man said. The hands that caressed the worn old lever-action Winchester were the hands of a working man. \"Wound up in Thailand turning wrenches on F-105s. Now them was airplanes!\"\n\n\"Why the rifle?\" the general asked.\n\n\"Oh, a bunch of us are going out to the base this mornin'. Going to talk to those people out there. We're gonna meet at nine o'clock. Can't sleep very well anymore, so came out here to the park to sit. Gonna get hot today\"\u2014it was already pushing 80\u2014\"but with the clear sky and still breeze, it's mighty nice right here right now. At my age, you enjoy ever' day because you don't know how many more you got.\"\n\n\"Think there'll be trouble at the base?\"\n\n\"Hope not, but you never know about the blue suits. They's good 'uns and bad 'uns, just like ever'where. But if there's any shootin', I fully intend to shoot back until they get me.\"\n\n\"I see.\"\n\n\"My folks was in Texas before the white and black people ever showed up. One of my great-great-great-grandpappies died at the Alamo with Travis and them. His nephew rode with Terry's Texas Rangers during the Civil War and lost a leg at Shiloh. Yankee doctors cut it off for him. I've had granddaddies and uncles and men kin fight in ever' war this country ever fought. The world wars, Korea, and me in Vietnam. We're Texans.\"\n\n\"What kind of pistol is that in your holster?\"\n\n\"It was my daddy's Colt Police Positive. He was a policeman in San Antone until he retired and moved here to Abilene to be near his daughters. I got it when he died.\"\n\n\"So what do you think of independence?\"\n\n\"Some more of us are gonna have to fight for Texas again.\"\n\n\"When did you get out of the air force?\"\n\n\"Seventy-five. Came back here and opened a garage. It was a close squeak at times, but we have six bays now. My two sons run it, and I sit and watch baseball on TV and drink beer.\"\n\nGeneral l'Angistino glanced at his watch. He needed to get going, but. . .\n\n\"Texans don't seem to like illegals. What is your opinion?\"\n\n\"I'm like ever'body else. They flood in here and take jobs away from poor Texans because they'll work for the minimum wage or less. Down in Mexico the Church won't let 'em use contraceptives. Lots of kids guarantees they'll never get ahead and will always be poor. I'm Catholic, but believe me, after the two boys arrived I used rubbers back when the old lady could still get knocked up. I tol' the priest about it, and he said I had to do what God tol' me to do. I tol' him that I was gonna do what my wife tol' me to do, and if that got me sent to Hell, at least I'd know a lot of the people there.\" The old man chuckled. Apparently he had told this story many times before and still liked it.\n\n\"What does your wife think about you going out to the base this morning carrying a pistol and rifle?\"\n\n\"She tol' me to be careful and never forget my family or Texas.\"\n\n\"Good luck to you,\" Brigadier General l'Angistino said. As he jogged back to his car, dog in tow, he thought, I'm the one who's going to need the good luck.\n\nNewspapers all over Texas carried the news about independence in headlines in the largest type they had. The _Dallas Morning News_ devoted its entire front section to the declaration and interviews with lawmakers, including a short one with the governor. Of the paper's editorials and op-ed pieces, all but one favored independence in order to preserve the freedom of the people of Texas. The lone dissenter was the paper's token liberal, whose column most _Morning News_ subscribers read only for aggravation.\n\nThe publisher defied federal edicts when he published the paper. He got away with it because the federal censors spent Sunday at FEMA headquarters getting briefed on the latest orders from Washington, which was in a dither, apparently, unsure how to handle those goddamn Texans.\n\nAt eight that morning five FBI agents, two women and three men, plus a FEMA representative, showed up at the publisher's house in one of Dallas' toniest neighborhoods to arrest him.\n\nThey were met by several dozen armed civilians. In the shootout that followed, one civilian was killed and another wounded, but all six of the federal officers died on the scene. Two minutes after the shooting stopped, there was one more shot, which may have been a coup de gr\u00e2ce, but afterward none of the participants could recall hearing it.\n\nLeaving the agents and their weapons where they lay, the victors of this encounter took their dead comrade to a funeral home and the wounded man to a hospital. Then they went to Dallas FBI headquarters and arrested everyone they could find, even the office help. The sheriff incarcerated all the prisoners in the Dallas County jail. He had to release some drunks and potheads to make room.\n\nWhen the crowd, which had swelled to more than two hundred armed men and women, arrived at the Dallas FEMA building, they found it empty. The FEMA employees had fled: that was probably a good thing since the crowd was in an ugly mood.\n\nLater that morning a television reporter on _Good Morning Texas_ questioned the sheriff, Milo Makepeace. Milo claimed he was a direct descendant of Comanche war chief Quanah Parker, and he had enough Indian blood in him to make that plausible, even if newspaper reporters had been unable to ever prove or disprove the relationship. Not that it mattered. With his dark skin tint and Indian features, Makepeace was Texas \"to the bone.\" The interview ran on _Good Morning Texas_ , a popular morning staple for many in the Dallas area. The show ran fifteen seconds of footage of ambulance crews in front of the publisher's house loading the bodies of the FBI and FEMA agents. Then the station aired the interview. The reporter asked about the slayings of the FBI and FEMA agents.\n\n\"I don't know a solitary thing about it,\" Sheriff Makepeace said, \"other than the fact they're dead. I also was told that they had federal credentials and weapons on them. Maybe they got in a shootout and killed each other, or maybe they tangled with persons unknown. If they had packed up and gotten out of Texas yesterday, that incident wouldn't have happened. It's very sad that they didn't.\"\n\n\"I understand you jailed some FBI agents this morning.\"\n\nThe sheriff nodded. \"As of yesterday morning federal employees got no authority whatsoever in Texas. The FBI people had a lot of concealed weapons on them that they didn't have permits for, which is a violation of the laws of Texas and Dallas County. Texans are big on self-help, and the folks in the crowd that brought them in looked like voters to me. I'm holding them until Jack Hays or a Texas judge tells me what to do with them.\"\n\n\"How about a federal judge?\"\n\n\"Federal judges have no authority in Texas. I just explained that. All their summonses, orders, warrants, and such don't mean diddly-squat. If they want to keep drawing federal checks, they'd better get themselves back to Soetoro-land. If they want to stay here, they need to get a real job. That goes for all federal employees, from the janitor at the federal courthouse to the people at FEMA, ICE, the DEA, the FAA, the EPA, and the Federal Reserve Bank. All of 'em. Get out of Texas or get a real job.\"\n\nMajor General Twilley read Governor Jack Hays' note and came around his desk to shake JR's hand. \"I can't tell you how relieved I am,\" he said. \"I have a son in the U.S. Army Special Forces and a daughter in the U.S. Air Force in Germany. I couldn't fight against them under any circumstances, and you know as well as I do that Barry Soetoro won't let Texas go without a fight. I was going to write Jack Hays a letter and ask for immediate retirement. He saved me the trouble.\"\n\nHe called in his staff, introduced JR, and read the governor's letter aloud. \"I have been relieved by Major General JR Hays.\" He and his staff saluted JR. JR returned the salute.\n\nThen Twilley turned to a colonel. \"Major General Hays will need a uniform. Until he can get some greens, get him some camos. I'll give him my stars.\" And he took them off right there and pinned them on JR's collar. Meanwhile, Major General Gentry, the officer in command of the Texas Air Guard, came into the room and was introduced. He read Governor Hays' order and saluted. JR saluted him back.\n\nTwilley took a few moments to shake the hands of every officer on the staff, then he put a photo of his wife, son, and daughter that sat on his desk under his arm and walked out of the room.\n\n\"Let's go somewhere that we can sit down,\" JR said. \"Do y'all have a conference room?\"\n\n\"Sure do, sir. Follow me.\"\n\nWhen everyone was sitting down with pads of paper and pens handy, JR got to it. \"Ladies and gentlemen, you know all about the Declaration of Independence. The people of Texas, acting through their elected representatives, have declared themselves a free, independent republic. Our job is to build a military that can and will defend the Republic of Texas against all enemies. The governor has asked me to lead the military effort. Yes, I'm Jack Hays' cousin. I grew up in west Texas, graduated from West Point, and spent twenty years as an infantry officer in the United States Army. I fought in Kosovo, the Gulf War, Iraq, and Afghanistan before I retired from that army. I'm proud of my service, and I am very proud Jack asked me to lead Texas' military in a fight for freedom.\n\n\"I understand the emotional muddle many of you find yourselves in. Many of my closest friends still wear United States uniforms. They will do their duty as they see it, as I will mine.\n\n\"Still, I want you to understand the depth of my commitment. I am absolutely committed to the Texas cause. One of my kinsmen, Captain Jack Hays, was the very first captain of the Texas Rangers. Hays men have fought, bled, and sometimes died fighting for Texas, for the Confederacy, and for the United States in world wars and police actions. As a soldier, I was fully prepared to give my life for my country in every place I fought, just as I am now fully prepared to give my life for the Republic of Texas if God demands it of me. I expect no less from every one of you.\"\n\nHe surveyed the audience, tried to gauge their mood. He concluded most of them were with him, which was more than he hoped for.\n\n\"Every one of us in this room swore an oath to defend freedom. Every one of us swore to defend the United States Constitution. But our Constitution, and the United States as we knew it, no longer exist. They've been hijacked by a power-mad tyrant bent on transforming America into a socialist dictatorship. The people of Texas have chosen not to be a party to the destruction of their liberties. _And since freedom is never free, we are going to have to pay for ours_.\"\n\nA murmur of approval swept through the room.\n\nHeartened, JR said, \"Our first job is figuring out how many soldiers we have. I want the National Guard troops individually polled today. I intend to muster everyone in the Guard into full-time Texas service. I know every guardsman has agreed to that in his enlistment papers or officer's commission, but we need to be realistic. We're going to have a civil war. If you can't in good conscience defend Texas against other Americans in Barry Soetoro's army, please excuse yourself right now and no questions will be asked. If you can't in good conscience take the risk because of your family obligations\u2014again, fine, leave now and no questions will be asked. The rest of us need to get ready.\"\n\nOne captain in the rear of the room got up and left. The door closed behind him.\n\n\"Adjutant, take this down. Everyone who stays will be sworn into Texas service with this oath: 'I swear to support and defend the Constitution of the Republic of Texas against all enemies, foreign and domestic, and to obey the orders of the officers appointed over me, so help me God.' Make a lot of copies. Administer the oath before a Texas flag, with the oath-taker standing at attention with his right hand raised. Afterward, the oath-taker will sign a hard copy of the oath that will remain in his service record. Anyone accused of violating the oath will be tried by court-martial, and if found guilty, will be imprisoned or shot. Any questions?\"\n\nThere were none.\n\n\"Okay, let's get at it. Chief of staff, have a list prepared of every United States military installation in the state, all of 'em. We'll divide up the list and take as many troops with us as we can find, go to the CO of every installation this afternoon or as soon as we can get there, and ask for the formal surrender of the base with all its weapons, ammo, and equipment. Prepare a short paragraph for the COs to sign.\n\n\"Our policy is this: Every person in Barry Soetoro's federal service who wishes to leave Texas will be allowed to take his family and personal possessions, no weapons or military gear of any kind, and depart Texas expeditiously. East, west, or north, no questions asked. Any man or woman in federal service, officer or enlisted, who wishes to serve the Republic of Texas will be mustered in by taking the oath and signing it. Enlisted will serve for four years, or until sooner discharged or our legislature decides otherwise. Officers serve at the pleasure of the governor. Time in service and pay grade will transfer directly. Any questions?\"\n\n\"These folks who want to join us\u2014the feds will probably list them as deserters.\"\n\n\"That is not a question, but I'll comment on that point anyway. The feds are going to do whatever they want, and that's sort of a fact of nature. Anyone in federal service wishing to join the Texas Guard by taking the oath will be allowed to do so; in fact, they will be encouraged to do so. We need all of the soldiers we can get. Civilian volunteers will be enlisted after a physical and an abbreviated background check. No crazy people, felons, dope addicts, or congenital idiots.\"\n\n\"We have our share of idiots now,\" someone remarked. \"We don't need any more.\"\n\nJR laughed and the tension was broken. He clapped his hands once and said, \"Break out the sidearms and ammunition. I want every officer armed.\" After a few more housekeeping details, he said, \"We don't know how much time Barry Soetoro will give us, so let's get at it, people.\"\n\nAs the staff dribbled out, JR and Elvin Gentry moved chairs together at the end of one table. \"Call me JR,\" the newest general said. \"Will the air guard stick?\"\n\n\"Most of them,\" Gentry replied.\n\n\"What do you have in the way of airplanes, and where are they?\"\n\nGentry told him. A reconnaissance wing equipped with Predator drones, an airlift wing flying C-130 Hercules planes, and a fighter wing flying F-16s stationed at Kelley Field at the joint base in San Antonio, Lackland. There was an air reserve C-5 outfit there too.\n\n\"We're going to need those fighters PDQ. How about you sending someone down there this morning to make sure we keep them?\"\n\n\"Yes, sir. The air reserve also has a wing of F-16s at the Joint Base in Fort Worth, the old NAS Dallas.\"\n\n\"And doesn't the air force have a wing of B-1Bs over at Dyess in Abilene?\"\n\n\"They do. Plus some more Hercs.\"\n\n\"We need them too. Fact is, we need every military asset we can lay hands on. We've got to grab everything we can reach before it is sabotaged or flown or driven out of state. We have to turn it over to loyal people.\"\n\nGentry nodded his understanding.\n\n\"Send the best people you can find on to Fort Worth and San Antone. The critical assets, however, are the B-1s. They compose our only real transcontinental offensive capability. I suggest you go to Abilene as fast as you can get there.\"\n\n\"I'm on my way, sir.\"\n\n\"We also need someone to invade the air traffic control facilities and shut them down. It would be nice to ground every commercial flight in Texas and scoff up all the planes.\"\n\n\"I'll do my best, sir.\"\n\n\"Thanks, Elvin.\"\nELEVEN\n\nAt Dyess Air Force Base in Abilene that morning General l'Angistino was trying to digest a message from Washington directing the Dyess B-1 wing and a B-52 outfit in Louisiana to prepare strikes against the heart of Austin. Before the Pentagon used this blunt weapon, however, an armored division from Fort Hood was ordered to surround the city to isolate it and, after the bombing, capture every politician they could find still alive.\n\nAn armored column cannot be organized and set in motion instantly, and the air force general knew that. He didn't know how long the army would need to comply with the directive, but he thought he had a couple of days before anyone would demand that Dyess bombers smite Austin.\n\nAnd it was going to take a couple of days to get ready. The runway was now clear, but a hundred armed civilians were blockading the main gate and dozens of others blocked the other gates.\n\nL'Angistino was rapidly running out of air policemen. Last night he had directed that machine-gun emplacements be dug on the edges of the ramp area.\n\nHe certainly didn't have the personnel to patrol the entire base perimeter. The base comprised more than six thousand acres, and it was surrounded only by the fence, which, as Colonel Wriston had proved, could be easily breached. L'Angistino did the best he could. He ordered the digging of three machine-gun emplacements to deter an attack from the front gate and had his air police patrol the base in six armored cars with mounted machine guns, the same kind of armored cars FEMA was distributing to police departments nationwide.\n\nL'Angistino picked up the file he had on Colonel Wriston, the National Guard commander who opposed him. He was a warrior. A tanker who had done four tours in Iraq and Afghanistan, he left active duty after fifteen years and had taken a commission in the Texas National Guard. He was married, with three teenage daughters\u2014undoubtedly the reason why he'd transferred to the Guard. Wriston knew all there was to know about bulldozer blades burying machine gunners alive, and had mounted them on tanks. Now he had bulldozers, if he could find some more, and l'Angistino thought he probably could. Wriston wasn't done, not by a long sight. The only question was what he would do next.\n\nThe major in charge of base security, Timothy Toone, had already had a confrontation with the people out front, who were standing around the county sheriff's car.\n\nAs the major reported it to l'Angistino, he told the Taylor County sheriff, \"You need to get these people out of here.\"\n\n\"I ain't movin' nobody who's not on federal property. They've got ever' right to be here.\"\n\n\"They have no right to blockade our gates. Interference with U.S. military operations is a federal crime.\"\n\n\"Call the FBI and report it,\" the sheriff said calmly. \"I'm sure they'll come roaring right out here and arrest everybody.\"\n\n\"These people are armed.\"\n\nThe sheriff looked around, acting as if he hadn't noticed the guns before. Then he told the major, \"People have a right to openly carry firearms in Texas, except in places where it's prohibited, like courthouses. This isn't a courthouse, but a public road. Fact is, these streets and roads belong to the City of Abilene, Taylor County, or the Republic of Texas. These folks don't have to leave unless I tell them to.\"\n\nThe sheriff grinned, the major told l'Angistino, while he waited for the major to ask him to do just that, a request that he would cheerfully refuse in front of an audience of his constituents. So the major had kept his mouth shut and returned to headquarters. Now what did the general want him to do?\n\nMore than half the officers and airmen assigned to Dyess lived outside the gates, mostly senior people. Many of the pilots did too.\n\n\"Major, I want hourly reports on conditions at all seven gates of the base; I want you to double the base guards and ensure they're armed. If armed civilians are foolish enough to try to force their way onto the base, I want the guards to respond with lethal force. Do you understand?\"\n\n\"Yes, sir.\"\n\n\"If this blockade continues, we won't have enough pilots, crew chiefs, and ordnance specialists to accomplish our missions. We have to break it.\"\n\n\"Yes, sir.\"\n\nAt noon, Major Toone estimated that the crowd on the streets had swelled to more than ten thousand civilians\u2014including women and children. He estimated that half the men were armed. If they rushed the base, his troops would be in a hell of a fix if ordered to shoot. He wanted written orders from General l'Angistino.\n\nNothing in the brigadier's military education or experience readied him to meet this situation. Shooting unarmed women and children would be an atrocity, a war crime . . . and, he thought, a sin. His wife would never forgive him. He wondered if the air force would.\n\nHis operations officer entered with a mission assignment. As many B-1s as l'Angistino could get airborne were ordered to bomb Austin tonight. They were to use JDAMs, which were precision-guided munitions. A detailed target list would follow.\n\n\"But there is no fighter protection laid on,\" the ops officer said. \"The Texas Air Guard has a squadron of F-16s at the joint base at Lackland. If they sortie to intercept the B-1s, the Bones will be toast. They have to have fighter protection, General. We could lose them all on the way to the target, over it, or on the way home. It's only seventy or eighty miles from San Antonio to Austin. Sending those guys without fighter protection is ridiculous. Foolhardy.\"\n\nA knock on the door, and his aide appeared. \"General, there are two squadron commanders and nine pilots waiting to see you.\"\n\nThe ops officer and the general exchanged glances. Did they know about the lack of fighter protection? Already?\n\n\"Send them in,\" he said. Then he turned to Major Toone and added, \"Major, let's talk later.\" The two colonels, squadron commanders, passed Major Toone in the doorway. \"We have a problem, General. Some of our pilots want to talk to you.\"\n\n\"Send them in.\"\n\nThe pilots were wearing flight suits. The first man in line stood at attention in front of the general's desk, saluted, and laid his silver wings insignia on the desk. \"Sir, I wish to turn in my wings and be removed from flight status, immediately.\"\n\nL'Angistino stared at the captain, who met his gaze. In the American military pilots and flight crewmen were all volunteers. No one could order an officer to be a pilot.\n\n\"Do you want to give me an explanation, Captain?\"\n\n\"Sir, I find that in good conscience I cannot fight other Americans. I may be obligated to remain in the air force, but I am not going to fly again.\"\n\nThe next man laid his wings on the table and saluted. He repeated the formula, \"I wish to turn in my wings and be removed from flight status, sir.\"\n\n\"Why?\"\n\n\"My wife and I are from Texas. Born and raised here. I'm not going to take a chance that you want me to bomb Texas, maybe kill some of our relatives or some friends I grew up with or went to school with. Or their kids. I can live with a court-martial, but I couldn't live with that.\"\n\nWhen the last man left, nine silver wings lay on the general's desk. The squadron commanders stood at parade rest.\n\n\"Where's the other squadron commander, Colonel Hurley?\"\n\n\"Somewhere off base, sir, we think.\"\n\n\"How many pilots do we have available to fly the Bones?\"\n\n\"Twelve, sir, including us. Nine command pilots and three copilots. Using some of the command pilots as copilots, we can launch six planes.\"\n\nThe general sank into his chair. A hundred twenty pilots in the wing, and he could muster just a dozen?\n\n\"A lot of them are trapped off base, sir. To get them in, we'd either have to run the civilians off or slip one of our own over the fence to find our guys and organize a mass break-in.\"\n\n\"That would take all night.\"\n\n\"Or longer. And we have enlisted manpower problems. Our muster list shows about thirty percent of our personnel are present for duty.\"\n\n\"I saw the morning muster rolls.\"\n\n\"It's a bad situation, sir.\"\n\nThe general dismissed the colonels and sat thinking. The bomb wing wasn't ready for combat. With only six flight crews and thirty percent of its enlisted personnel, it wasn't ready for anything. Not even morning colors.\n\nThe ops officer left, but he soon came back. \"There is a turboprop inbound, sir. National Guard.\"\n\n\"Send them up here when they land.\"\n\nHe watched the turboprop taxi to base ops and shut down. One or two people in uniform got out, climbed into a waiting sedan. Ten minutes later they were in his office.\n\nHe recognized the Air National Guard general, Elvin Gentry, whom he saluted since Gentry was a two-star. The man with Gentry was a colonel. \"Please be seated, gentlemen,\" l'Angistino said.\n\n\"This independence thing,\" Gentry said, \"it's turned the world upside down. I've come to ask for you to surrender the base, its personnel, and all the military property on it.\"\n\n\"Are you kidding?\"\n\n\"Lou, I wish I was. But I'm deadly serious. My boss, Major General JR Hays\u2014do you know him?\"\n\n\"Not that I can recall.\"\n\n\"He was army. Anyway, he sent me up here to get your surrender. If you refuse, he'll have to launch an F-16 strike on the base to take out the planes. Texas wants them or wants them in ashes.\"\n\n\"I'll fly them out of here,\" l'Angistino said stoutly. \"The first ones leave in less than an hour.\"\n\n\"Lou, you couldn't have enough pilots to fly more than a handful of Bones and Hercs out of here, and I doubt you have the enlisted mechs and specialists it takes to even launch that many. We know the situation here. I've been on the radio with Colonel Wriston. Fact is, you could tell me to go to hell and sabotage all those planes, shoot holes in the spars, whatever, but we're going to take this base before very long, and General Hays is going to be royally pissed with you people if those airplanes are harmed. I don't know exactly what the Geneva Convention says about the treatment of prisoners of war, but this isn't a war we've declared. You are now trespassers on property owned by the Republic of Texas. Lawyers love tangles like this, but the local Texans won't. If you don't surrender they will be even more pissed than JR Hays. You know those people aren't under our control.\"\n\nLou l'Angistino's thoughts tumbled around.\n\n\"Lou, for God's sake. I am not trying to threaten you, although maybe it came out that way, and if so, I apologize. Most of your people don't want to fight other Americans, many are Texans who won't fight other Texans even at the point of a gun. You can't sit here on a base protected by nothing but a wire fence and defy the whole population of Texas! There is no realistic chance for victory. None. For God's sake, do the right thing and save some lives.\"\n\nGentry pointed to the little pile of wings on l'Angistino's desk. \"Even your pilots are trying to tell you something. Your air force is disintegrating.\"\n\nGeneral l'Angistino picked up the op order directing a strike on Austin, glanced at it in disgust, then dropped it on the table. \"What are your terms?\"\n\nAn hour later, after the surrender document was signed and sent to be posted in barracks, ready rooms, and maintenance shops, Colonel Wriston of the National Guard was escorted into the office. He was wearing jeans and a faded Texas A&M T-shirt. Lou l'Angistino reached for his hand and perfunctorily shook it.\n\n\"Lou, here's the man responsible for the ten or so thousand people standing outside on the street,\" Gentry told the air force general. \"Wriston and his men spent the night recruiting their friends and neighbors.\"\n\n\"You mean some of that crowd were National Guardsmen in civvies?\"\n\n\"They were. Wriston did what he could to block your runways, but he didn't go home afterward to watch television. He knew the vast majority of the civilian community was behind him, so he used his men to mobilize them.\"\n\nA loud voice interrupted them, to l'Angistino's relief. Colonel Wriston went to meet the man, who was standing in the reception area.\n\n\"Wriston, you bastard. I got back from Dallas this morning and nobody's workin' my job site. My foreman says the equipment operators stole ever'thin' that would move on your orders. Where the hell is my construction equipment?\"\n\n\"Out beside the runways, Carroll. We used it yesterday to block the runways here. Did you watch the declaration read night before last?\"\n\n\"Sure did! All I can say is, it's about damn time.\"\n\n\"I couldn't call and ask to borrow your stuff, but I thought since Carroll is a good man, he won't mind.\"\n\n\"By the runways, you say?\"\n\n\"It's damaged and tore up some, but it kept all the planes from taking off. You should be able to repair some of it. Anyway, your project is going to go slow until you get some more equipment operators. Most of yours are in the Guard and they are now on active duty and won't be back for a while.\"\n\nCarroll took a deep breath. \"The yellow iron is insured, but the insurance company will lawyer up and refuse to pay unless I sue 'em, then offer ten cents on the dollar. You know that.\"\n\n\"Tell you what,\" Wriston said, and put his arm around the construction man. \"If you eat the repair costs, we'll give you an airplane. Any one of those along the road into the base, your pick. You can put it in your front yard. When things calm down, we'll move it for you.\"\n\nCarroll's eyes lit up. \"Got pecan trees in the front yard, but I could put it in the horse pasture out back. Damn, I'd like that.\"\n\nThey shook hands on it.\n\nIt was noon when JR Hays, wearing a camo uniform and a pistol on a web belt, arrived at the front gate at Fort Hood, sixty miles north of Austin in Killeen. He was in the right seat of a sedan with Texas flags flying from the corners of the front bumper. Two guardsmen, a male captain and a female major, were with him. An enlisted woman was driving.\n\nThe soldier at the gate wanted to see ID, but the sergeant was right there immediately and said, \"Sir, you can enter the base, but the carrying of firearms around the administrative and living areas is forbidden.\"\n\n\"Who is the commanding general?\" JR asked the sergeant.\n\n\"Lieutenant General Gil Ellensberger, sir.\"\n\n\"Call him. Tell him Major General JR Hays of the Texas Army is sitting at his main gate and wants in to see him. You may tell him we are wearing sidearms, if you wish.\"\n\nThe sergeant did as he was told. When he hung up the phone, he came out and explained to the driver of the sedan how to get to the headquarters building. Then he saluted. JR returned it.\n\nThe commanding general was in a staff meeting. The receptionist had a television in her office, and JR stood in front of it a minute watching. Armed citizens were taking over federal office buildings statewide. The FBI agents in Waco had been arrested en masse, disarmed, and jailed. DEA and ICE headquarters had been occupied, the agents disarmed and sent home.\n\nEllensberger came striding in. He was a tall, lanky man. He didn't look happy, but he said, \"Good lord, JR Hays, as I live and breathe. I haven't seen you since Afghanistan. Come on into my office.\" Ellensberger led the way and closed the door.\n\nJR thought commanding generals' offices all looked alike: big desk, carpet, U.S. flags, mementos of the current occupant scattered around. Unbidden, he dropped into a chair.\n\n\"I retired from the army last year, General, and my cousin, Governor Jack Hays, just this morning put me in charge of the Texas Guard. Raw nepotism.\"\n\nEllensberger let that one go by. \"All our off-base telephones are down, as well as the internet. Did you have anything to do with that?\"\n\n\"Jack Hays did, not me. I am here today to accept your surrender of the base and all of its personnel and military equipment to the Republic of Texas.\"\n\nEllensberger snorted. \"You know I can't do that. You can't just march in here and take over a United States military installation!\"\n\n\"Gil, you don't have a choice. Texas is now an independent republic, and Fort Hood is right smack in the middle of it.\"\n\nEllensberger waved that away. \"Texas is a state in the United States that has tried to secede from the Union. We settled all that back in the 1860s. Surely you read about that. It didn't work then and it isn't going to work now.\"\n\n\"We're not lawyers and I can't read tea leaves. We're soldiers, and you have an impossible military problem. How many of your troops reported for duty this morning?\"\n\nFrom the look on General Ellensberger's face, JR knew he had scored a hit.\n\n\"Half? Was it fifty percent?\"\n\nEllensberger didn't reply.\n\n\"Last I heard, you had over forty-five thousand soldiers assigned here. If we blockade the base, how are you going to feed them? And for how long?\"\n\nStill no reply.\n\n\"Are you going to deploy your troops around your perimeter\u2014how many miles of it do you have, anyway?\u2014and defend it? How many U.S. Army troopers do we have to kill before you will surrender? Or are you going to defend this dirt to the last man and go down like they did at the Alamo? Tell me now so I can brief my staff and get at it.\"\n\n\"Pfui. All the good ol' boys in Texas aren't going to whip an armored division.\"\n\nJR Hays rubbed his nose.\n\nEllensberger pushed the intercom button. \"Bring in this morning's classified message traffic.\"\n\nIn a moment a soldier came in and handed Ellensberger a clipboard. He automatically said thank you, and the soldier left.\n\nThe commanding general flipped through the messages, then handed the clipboard to JR.\n\n\"It's the one on top. Op Immediate from the chairman of the JCS, Wynette. He has ordered me to take an armored column from the First Cavalry down the interstate to Austin and surround the city. The air force is going to bomb it. We will go in after the bombers are finished and capture every politician left alive.\"\n\nJR took his time with the messages. He read the first one, then saw that Ellensberger was an info addee on a message to the B-1 bomber wing at Dyess and a B-52 outfit in Louisiana ordering them to prepare a strike on the Texas capitol in Austin. They were to wait to launch until First Cavalry had the city surrounded.\n\n\"This is insanity,\" JR said, gesturing with the clipboard. \"They are going to indiscriminately slaughter everyone in central Austin.\"\n\n\"They're not thinking very straight,\" Ellensberger admitted.\n\n\"But you are willing to be a part of this? Murdering civilians from the air? Americans?\"\n\nEllensberger sighed. After a bit he said, \"If I surrender to you, Wynette will just order the bombers to obliterate Austin ASAP. A dozen B-52s should be able to convert the heart of the city to rubble and kill a whole bunch of civilians.\"\n\nJR carefully placed the clipboard back on Ellensberger's desk. \"Which side are you on, Gil?\"\n\nEllensberger took his time answering. \"As I see it, the governor of Texas and the president of the United States are locked in a hell of a political dispute. I wish they would settle it between themselves without dragging the American flag through the dirt and asking American soldiers to kill other Americans. Honest to God.\"\n\n\"It isn't the governor. It's the legislature and the people of Texas who are locked in a dispute with Soetoro. Haven't you been watching television?\"\n\nEllensberger didn't reply to that remark, either.\n\n\"But politics isn't my business,\" JR murmured. \"I'm a soldier.\"\n\n\"Soldiering _is_ politics. You know that!\"\n\n\"Yes or no, Gil. I have responsibilities too.\"\n\nEllensberger took in a bushel of air and sighed deeply. \"What are your terms?\"\n\nIt took a half hour for the surrender document to be typed and signed. Meanwhile JR sent the captain to the flight line to take a helicopter to National Guard headquarters at Camp Mabry in Austin with a message to Major General Gentry. Bombers were coming sooner or later to flatten Austin, and he'd better get fighters ready to fly with pilots willing to fight for Texas.\n\nLieutenant General Ellensberger signed the surrender. Then he went to the U.S. flag in the corner and carefully removed it from its display pole. He folded it reverently and put it into his briefcase.\n\n\"Two mornings ago,\" he said conversationally to JR, \"when I heard the legislature had passed the declaration and the governor had signed it, I knew this moment was coming. And I didn't know what to do. I could have asked Washington, but all I would have gotten was bullshit. I wanted some time to see what my staff thought, what the troops thought\u2014you can't fight if the troops aren't with you body and soul. The moment of decision just came sooner than I thought it would.\n\n\"JR, I am sick to death. You and I were both at West Point, served our country\u2014all of it. Then along came Soetoro. A progressive fascist, if there is such a thing. I figured the country could stand eight years of even the devil's rule, but I was wrong. Race was the wild card. Everyone is scared to death of being labeled a racist. If Soetoro were white he would have been impeached years ago. . . . Do you mind if my wife and I stay in our quarters for a few days? I need to figure out what to do next.\"\n\n\"Whatever you need,\" JR replied. Ellensberger took a deep breath and looked around the office one more time. \"I fear for my country,\" he said softly. \"The United States may not survive this mess.\"\n\n\"Texas will,\" JR said with more confidence than he felt. He saluted, Ellensberger returned it, and then JR walked out to address the office staff.\n\n\"You folks in the United States Army who wish to leave can go. You folks who want to enlist in the Texas Guard can stay. You civilians have a job right here if you want it.\"\n\nThe civilians all stayed. Most of the soldiers asked permission, which was granted, to go home and discuss it with their wives or just to think about things.\n\nJR's next problem was easily solved. The aide he had brought with him, Major Judy Saar, asked, \"What are you going to do about Major Nasruli?\"\n\nNasruli was an American-born jihadist who had murdered thirteen Fort Hood soldiers several years before and wounded thirty-two more.\n\n\"Is he still here?\"\n\n\"So they tell me. In the jail or detention facility or whatever they call it.\"\n\n\"I thought he was convicted by a court-martial and sentenced to death.\"\n\n\"Yes, sir. But he's very much alive.\"\n\n\"We are not going to waste people running a jail or spend a dime of taxpayers' money feeding him,\" JR Hays said. \"It's high time he was dead, anyway. Dictate an execution order addressed to yourself. Put in Nasruli's rank, full name, service number, and a place for my signature. Reference the death sentence. Then get a half-dozen volunteers, get them some M4s, and put him up against a wall. Make sure he's real dead. Then come back here and dictate a press release. The army and the civilians in Washington have screwed around and screwed around, and now he's history.\"\n\n\"And the body, sir?\"\n\n\"Burn it.\"\n\n\"Yes, sir,\" Judy Saar said, came to attention, and saluted. Apparently she too thought Major Nasruli had lived long enough. In seven minutes she was back with a one-paragraph order she had apparently typed herself. JR Hays read it, signed it, handed it back to her, and went on to the next problem, which was the armored division at Fort Bliss, in El Paso.\n\nHe doubted that the commanding general there would surrender quite as quickly as Major General Ellensberger had. JR knew Major General Lee Parker, knew him to be a perfect bureaucrat who wouldn't want to buck the system. JR thought Parker personified everything wrong with the army: bureaucratic inertia, lack of initiative, a craven capitulation to political correctness, and a pathological fear of casualties. The media's fondness for trumpeting casualties meant that a career officer on the way up wanted as few as absolutely possible, so he took as few risks as possible, and accomplished very little. He also kicked difficult decisions up the line, so that he wouldn't be blamed if anything went wrong. JR thought that before he surrendered, Parker would want the blessing of higher authority, which he was unlikely to get.\n\nGiven some time, JR thought Parker could be conned into thinking his military bosses wanted him to surrender, but time was a diminishing asset for JR. He needed that armored division in his pocket right now. He was going to have to convince Parker that he was facing a mountain of casualties in a losing cause.\n\nMajor Judy Saar drove a staff car and parked at the first barracks she saw. Inside she found groups of male soldiers loafing in the lounge, loudly discussing Texas independence and the takeover of the base. She said, \"Attention please.\"\n\nSome of the soldiers looked around. \"I am here to ask for volunteers for a firing squad.\"\n\nStunned silence greeted her. One black sergeant said, \"Who do you want to shoot, Major?\" His name tag read HILL.\n\n\"Major Nasruli. I have an execution order here in my hand.\"\n\nEvery man in the room raised his hand, including the black staff sergeant, short and wiry and buff, with close-cropped, prematurely gray hair. \"One of the men he shot was my brother, who is paralyzed from the waist down.\"\n\n\"I need six people,\" she said. \"Sergeant Hill, will you select five other men and follow me to the base armory?\"\n\n\"Yes, ma'am.\"\n\nAt the armory she requisitioned six M4s and a cartridge for each of them. She passed the carbines to her volunteers and pocketed the cartridges.\n\n\"Turn these weapons in here afterward,\" she told them. \"Now the detention facility.\"\n\nShe parked in front of the building and waited for the other vehicles, three private cars, to arrive. She felt as if she were watching herself outside of her body.\n\nHer husband, a private physician, would not approve. But then he didn't approve of her service in the National Guard. He wanted her to stay home with the two children, who were now in junior high and didn't need her sitting at home. She wanted to make a larger contribution.\n\nThe cars drove up and the soldiers got out with their weapons.\n\nMajor Saar led them inside, showed the officer at the desk the execution order.\n\n\"You can't do this,\" he said. \"The death sentence has to be approved by the president.\"\n\n\"You have heard that Texas has declared its independence and Lieutenant General Ellensberger has surrendered Fort Hood to the Republic of Texas, have you not?\"\n\n\"Yes, but\u2014\"\n\n\"The president of the United States has no authority here. Would you care to call base headquarters and verify the order with Major General Hays?\"\n\nHe would. He did so. After a moment of listening, he said, \"Yes, sir,\" and hung up and looked askance at Judy Saar.\n\n\"Do you have an exercise area?\" Major Saar asked.\n\n\"Yes, ma'am.\"\n\n\"Bring him out there. In handcuffs.\"\n\nShe had the sergeant arrange the squad in a line and handed a cartridge to each of them. Major Nasruli protested as the guards led him out. Apparently he had been told what was about to happen, because when he saw her he shouted, \"I have written to President Soetoro demanding clemency. Allah protects the faithful. Allah has\u2014\"\n\n\"The post that holds up the basketball backboard,\" Major Saar told the guards. \"Cuff his hands behind the post.\"\n\nNasruli continued to shout, to rant. Sergeant Hill asked, \"Do you want him blindfolded?\"\n\n\"He can take this with his eyes open,\" she said.\n\nNasruli refused to stop shouting. He was still shouting when Major Saar told the marksmen to aim at the center of the chest and gave the commands: Ready, aim, fire. The shots came as one report and Nasruli went down, held semi-erect by the pole. She heard the spent shells tinkling on the concrete. She walked over to the body. Blood stained his shirt. His eyes were open, staring at nothing.\n\nLike an automaton, she drew her pistol, looked to ensure the safety was off, and, using both hands to steady and aim the pistol, shot Nasruli in the head from a distance of three feet. Brains and bloody tissue flew out the back of his head.\n\nShe engaged the safety of her Beretta, holstered it, and turned to the officer commanding the detention facility, who was staring slack-jawed at the remains of Major Nasruli. \"Pour gasoline on the body and set it afire, Captain.\"\n\nThe sergeant called the firing squad to attention, turned them, and marched them back into the detention facility.\n\nIt took twenty minutes for the detention facility staff to come up with a five-gallon can of gasoline. _They are probably robbing a civilian on a lawnmower_ , Judy Saar thought. She stood and looked at the sky, at the windows of the detention facility, at the body against the pole. She thought she was going to be sick, but she choked it down. _Later_ , she whispered. A bird skittered along the top of the wall. A mockingbird, she noted.\n\nAfter they put the body against an exterior stone wall, drenched it with gasoline, and set it ablaze, she marched back through the detention facility and vomited by her car. Then she drove back to headquarters.\n\nThe staff sergeant and the five other men from the firing squad were waiting for her in front of the building. They had apparently turned in the carbines to the base armory. All of them saluted and she returned their salute. \"Major, we'd like to enlist in the Texas Guard,\" Sergeant Hill said.\n\nShe nodded and motioned for them to follow her inside.\n\nThere was a handwritten letter waiting for Major Judy Saar in the commanding general's office.\n\n\"You are now the CO of the base and the 1st Cavalry Division. Get as many soldiers enlisted as possible, and get the 1st Cavalry ready to fight. I am on my way to Fort Bliss to grab the 1st Armored, Old Ironsides. We'll need them too. You are a good soldier. I'll back you in every decision you make. Texas needs you.\" It was signed by JR Hays, Major General.\nTWELVE\n\nOn the flight line at the base airfield, JR Hays went into a ready room full of helicopter pilots. They were gathered around a television, watching the feed from Washington. Someone saw JR enter the room and called everyone to attention. JR walked to a spot in front of the television, turned it off, and told everyone, \"Please be seated.\"\n\nHe surveyed the faces. Most army pilots are warrant officers. He was looking at a bunch of them, with a few commissioned officers scattered among them.\n\n\"I'm JR Hays of the Texas Guard. As you know, Major General Ellensberger surrendered to the Texas Guard just an hour or so ago. You've been watching television, so you know the current political situation. Barry Soetoro declared martial law and ripped up the Constitution, and consequently Texas declared its independence. General Ellensberger surrendered Fort Hood because it is indefensible. Circling the wagons in a lost cause struck him as ridiculous unless he was prepared to cut his way out of Texas, and he wasn't.\n\n\"Which gets me down to you. Every one of you has a decision to make: you can go home, pack your family, and leave Texas, or you can join Texas in our attempt to build a free nation dedicated to the principles that the Founding Fathers laid down when they wrote the U.S. Constitution. I suspect Barry Soetoro's army will not be pleased if you choose to join Texas in its fight, and it will be a fight, a second American Civil War. Barry Soetoro is going to use the armed forces of the United States to try to conquer Texas, so if you sign on, you will be fighting U.S. forces. Americans against Americans, as if it were 1861 all over again.\n\n\"Finally, if you choose to join the Texas Guard and fight with us, you can't change your mind later. It's sort of like getting baptized down at the creek: as the preacher would say, once you're in, you're all in, and you can't wash it off.\n\n\"Any questions or comments?\"\n\nOne of the warrant officers stood up and said, \"Sir, Chief Warrant Officer Three Buck Johannson.\"\n\nJR nodded and Johannson said, \"My dad is a state representative in Wisconsin. His politics are right of center and he's loud. The feds arrested him yesterday and put him in a camp because they don't want other people to hear the opinions of a free man. Far as I'm concerned, Texas is on the side of freedom. I'd like to join the Texas Guard.\"\n\n\"Fine,\" JR said. \"Anyone else?\"\n\nAnother warrant said, \"I think Soetoro wants to be a dictator. I don't want my kids to grow up in that kind of country. I'm from Georgia, but from now on I'm a Texan.\"\n\n\"Welcome to the Alamo,\" JR said, which drew a chuckle from his listeners.\n\nAbout half the pilots volunteered to serve with Texas. JR dismissed the others, told them to go home and pack. \"If, while you're doing that you decide to join us, you know where the headquarters building is.\"\n\nWhen only his volunteers remained, JR said, \"Our first priority is the First Armored in Fort Bliss. I want to go over there and capture the whole outfit. We need the tanks, helicopters, ammo, and all the rest of it. I'll need three Apaches and a Blackhawk armed to the teeth. We are going to do some violence, enough to make the CG there, Major General Lee Parker, surrender. Who wants to go?\"\n\nSpecialist Fourth Class James B. Cassel, a name that he and his kin had always pronounced Castle, spoke for thousands of his fellow soldiers when he got home to the tiny apartment he shared with his wife, Linda Sue, and their infant daughter. Jimmy Cassel was from a tiny town in the coalfields of southern West Virginia. He told Linda Sue, who was from Killeen and had married James just a year ago, about the surrender of Fort Hood to Texas forces.\n\n\"They say I can enlist in the Texas Guard, or we can pack up and get outta Texas,\" he said as he took off his uniform and put on his jeans and tennis shoes. \"Get packed up. We're leavin'.\"\n\n\"I was born and raised here,\" Linda Sue protested. \"I'm Texan clear through to my backbone. I'm not turning my back on my family.\"\n\n\"I joined the army to get the hell out of the coalfields,\" Jimmy explained as he pulled on a T-shirt that advertised the local Harley dealership, although he didn't own a motorcycle because he couldn't afford one, not even a used one. \"I didn't join the army to shoot Americans. If I was willin' to do that when push come to shove, I'd have joined the police. I got no love for that son of a bitch Soetoro, but America's my country from coast to coast. I ain't goin' to shoot Texans or Hoosiers or Californians or anybody else from America. We're leavin'.\"\n\n\"I'm not,\" Linda Sue declared. \"And the baby is stayin' with me. You just load your stuff in the car, Jimmy, and get the hell out. Go ahead, run off! If you won't fight to defend _us_ , I don't want _you_.\"\n\n\"Now, hold on! You married me and I'm the man of the family. My dad was in the army and fought in Kuwait. My granddad fought in Vietnam and got shot for his troubles. Us Cassels been fightin' _for_ this country since before it was a country. _I ain_ ' _t_ _turnin_ ' _traitor_.\"\n\n\"Jimmy Cassel, I am not turnin' traitor neither. I want to hear exactly nothin' about your daddy and granddaddy. The baby and I are your family now. And if you won't fight for your family, then you just hit the road. I'm takin' the baby and walkin' down to Mom's place.\"\n\nAn hour later, sitting alone in his apartment, Jimmy Cassel started to cry.\n\nSergeant Claude Zeist handed beers to three of his sergeant friends at his house on base. The television was on: scenes of federal agents making arrests alternated with scenes of riots in Baltimore, St. Louis, LA, and Chicago.\n\n\"The Texans have bit off a big chunk, and I doubt if they can chew it,\" Zeist said. \"But that's neither here nor there. Fact is, I took an oath to defend the United States of America, and when this is all over I want my kids and grandkids to know that I did my duty. Did what I swore I would. And there is no way in hell I am going into combat against my fellow American soldiers.\"\n\n\"It'll be over soon,\" his friend Benny Straight said. \"Thing I can't figure is why everybody is so damned upset. Barry Soetoro will be gone in January. He can't run again. The next president can set things right.\"\n\n\"What if he doesn't?\"\n\n\"That's tomorrow's problem. You don't burn the house down just because the sewer is backed up.\"\n\n\"So what are you going to do, Claude?\"\n\n\"I'm going to pack up the wife and kids and get outta Texas and find an army base somewhere so I can be an American soldier again. That's what I always wanted to be, and if we have to kick ass again like we did during the Civil War in the 1860s, so be it. That damn General Ellensberger hasn't got enough guts to make a sausage.\"\n\n\"Generals get paid to decide when to fight and when not to,\" Benny remarked.\n\n\"One good fight and Texas will crack like a rotten egg,\" Claude Zeist insisted, and drained his beer. Then he reached for another. \"We should have had it today. Never put off until tomorrow kicking ass today.\"\n\nNo one smiled; they were worried.\n\nBenny Straight put into words a thought that all of them had and none of them had yet voiced. \"After Texas folds, the U.S. Army is going to court-martial any United States soldier who did the turncoat trick. They'll be called traitors, and you know it.\"\n\n\" _If_ Texas folds,\" Jeff Hanifan said.\n\n\"Oh, it will,\" Benny Straight scoffed. \"For God's sake, one state against forty-nine? Texas against the United States Army, Navy, Air Force, and Marine Corps?\"\n\n\"Well, something good came out of this shit storm, anyway,\" Claude Zeist said. \"The Texans put Nasruli up against a wall and shot him. I would have bet my left nut that Soetoro was going to wait until his last day in office and commute the sentence to life in prison.\"\n\n\"This is Texas,\" Jeff Hanifan said, as if that explained everything. His comrades, all career soldiers, nodded knowingly and drank more beer.\n\nLoren Snyder went down the open hatch in front of the small sail of USS _Texas_ and found himself in the torpedo room. He looked around with his flashlight. The reactor was scrammed of course, and the boat was dead iron. The torpedoes in their cradles looked sleek and fat and ominous.\n\nHe wandered along, inspecting everything. The sailors hadn't even been able to take their personal gear. It seemed they would return any moment, but he knew they wouldn't.\n\nThe flashlight's beam in that dark ship was sorta spooky. The gentle, barely perceptible motion of the ship riding the little waves of the harbor made it even more so.\n\nIn the control room, the realization hit him that he was standing right dead center in a cruise missile target. Tomahawks could be climbing for their final dive right this instant. Each breath he took could be his last. He felt perspiration break out on his forehead and forced himself to concentrate on what he could see with the flashlight's beam.\n\nIn the reactor spaces, he examined everything and could find nothing amiss. The crew had simply secured the reactor and the batteries, then trooped up the forward ladder out of the ship.\n\nAssuming he could get the reactor started again, how many men would he need to move this boat? Lorrie Snyder thought hard. No more than five, he thought.\n\nMove her where? Satellites could see her submerged in shallow water, even if the water were muddy, using infrared. Where could he put a submarine so that the U.S. Navy couldn't find her?\n\nEven if he could find such a place, did he really want to do it? JR Hays had asked the question point-blank: Was he willing to fight for Texas? Well, was he?\n\nIf he planned on living and practicing law in Texas, Loren Snyder thought he had better get that figured out. Along with everything else.\n\nThe easy way out of this personal nightmare would be to just scuttle the submarine right here at the pier. Then the U.S. Navy wouldn't need to sink her or send SEALs to steal her. JR Hays would tell him he had done his best, and thank him. Loren Snyder thought about that too.\n\nThat Sunday afternoon chairman of the JCS General Martin L. Wynette was back at the White House. He hadn't had a day this bad since he was a plebe at West Point, way back when. President Soetoro, Vice President John Rhodes, and their aides surrounded him at a conference table and wanted to know how and when the armed forces of the United States were going to crush the Texas rebellion. The general had two aides with him, a Major General Stone and a brigadier, but the questions were directed at him, and the politicians weren't happy. They wanted action now.\n\n\"Willy-nilly bombing and invasion without a plan will get us nowhere,\" the general explained. \"We are working around the clock to formulate a coherent plan that will accomplish a military objective, which is the occupation of an enemy state.\"\n\n\"That's not it,\" Soetoro said, thumping the table. \"The military objective is to destroy the political opposition in Texas.\"\n\n\"Your political opposition.\"\n\n\"You're damned right. Those who oppose the progressive policies of this administration, earth-friendly policies that will benefit all future generations, policies designed to take care of those today who are unable to participate in our high-tech economy, whether from institutional racism or white privilege or the circumstances of birth, are indeed _my_ political opposition. They oppose America! Your job is to kill or capture them. Now\u2014how are you going to get it done?\"\n\n\"The navy will launch two Tomahawk cruise missiles at power-generating facilities in Houston later tonight, after dark in Texas. Your staff told me they want bombs falling immediately, so I gave the order. We are planning more strikes on the power plants\u2014\"\n\n\"Planning?\"\n\n\"Scattering cruise missiles around like grass seed isn't going to kill or capture your political opposition, Mr. President. These strikes must be in coordination with armed invasion, or we are simply wasting missiles.\" Wynette felt his irritation leaking through. _He_ was the military expert. None of these political types had ever spent a single day in uniform, unless they did a stint at scout camp once upon a time. Hell, they didn't even _like_ soldiers, whom they often referred to as neolithics.\n\n\"So when is the invasion?\"\n\n\"Sir, as I said, we are working around the clock to produce a plan. Going in half-cocked and getting our asses shot off isn't going to get us any closer to your objective. When we go in, we want to win.\"\n\n\"So when? Tomorrow? This week? Next week? Next month? Next year? _When_?\"\n\n\"I would say next week. We must move soldiers and equipment from all over the country, figure out the logistics\u2014\"\n\n\"Bullshit,\" Soetoro's senior political advisor, Sulana Schanck, said acidly. \"This isn't the invasion of Germany or Iraq! Your opposition is a mob of crackers with deer rifles who will shit their pants when the shooting starts and run like rabbits.\" She obviously was a believer in direct speech.\n\nThe thought shot through Wynette's head that British General Thomas Gage had that same opinion when he marched his troops from Boston to seize the arms and powder at Lexington and Concord, but he had the sense not to air it. He did, however, screw up the courage to say, \"The Texans did a number on General Santa Ana, as I recall.\"\n\n\"Damn it, General,\" Soetoro roared. \"I don't want a history lesson! I want you to take the United States armed forces down there and kick butt. If you can't do it, we'll find someone who can.\"\n\nWynette automatically dropped into his ass-kissing mode. \"We'll get it done, sir.\"\n\n\"And how come the brass in charge of these military facilities in Texas are busy seeing how fast they can surrender? Are they a bunch of traitors?\"\n\n\"Sir, I have ordered investigations. The commanders will be held responsible for their actions.\"\n\n\"Firing squads will stiffen some backbones. The sooner the better.\"\n\n\"Yes, sir.\"\n\n\"When the invasion starts, I want you down there in the lead tank, General. Do you understand? If you fuck this up, don't come back alive.\"\n\n\"Yes, sir,\" General Wynette said.\n\nLate Sunday afternoon as JR Hays settled into one of the passenger seats of a U.S. Army executive jet, normally used to ferry flag officers around, he took stock of all the things he needed to get done and hadn't been able to attend to today. Everything needed to be done immediately. He hoped that Air National Guard Major General Elvin Gentry had hit the ground running. Air traffic facilities and their radars, in addition to fighter planes, bombers, helicopters, had to be seized by the Guard. Without ground control sites pinpointing incoming enemy planes, fighters were handicapped severely. Gentry also needed to ground civilian air traffic and confiscate every jetliner he could lay hands on so they could be used to ferry troops.\n\nJust thinking of all the critical tasks and decisions that had to be attended to made JR's head throb and gave him a sense of anxiety that he was having trouble shaking off. The fact that the feds were equally inundated didn't help much to ease his frame of mind. If the feds got licked, they had forty-nine other states to play in. If Texas got licked, a whole lot of Texans were going to die in the aftermath.\n\nThe three Apaches and one Blackhawk that he had launched from Fort Hood were going to make a pit stop for fuel in Pecos. Unfortunately the weather was rotten around El Paso. Thunderstorms full of rain and lightning were drifting in from the west and southwest, bringing low ceilings and visibility in addition to their usual goodies.\n\nHe had no plan for forcing the 1st Armored to surrender. He had learned long ago the truth to the old maxim that no plan survives contact with the enemy, so he made none. First he had to learn what the situation was in El Paso and at Fort Bliss, then he could plan. The Apaches and the Blackhawk were arrows in his quiver. The National Guard commander, Wiley Fehrenbach, scion of the Hill Country Fehrenbachs, and his old civilian contractor boss, Pete Taylor, would know, so he would land at the civilian airport and seek them out.\n\nHe had been lucky to get into West Point, and soon hated the place. He decided to stick with it and do his required service afterward, then bail. Before he could get out, along came Kosovo. The experience left him with a profound respect for the men and women who served in the army. In Afghanistan, then Iraq, and two more combat tours in Afghanistan, their valor had left him humbled and awed. Leading troops had been the great experience of his life. The military bureaucracy\u2014full of paper-pushers and desk soldiers angling for promotion\u2014had defeated him when the Holy Warriors could not. He knew he had to get out when he hit twenty years, and he did.\n\nNow he was on the cusp of a horrible dilemma, which he had helped bring on. It looked as if killing some American soldiers might well be on the menu. The hard fact was that if Texas was going to win its freedom, mortal combat was inevitable. If not here, then other places. And the sooner it was done, the sooner the bloodletting would be over.\n\nCombat had taught JR to find strength in God when he doubted if he had enough, and so he prayed a little as the jet lifted off. He had long ago come to grips with his own mortality. He had once written a quotation from Stonewall Jackson in the front of his Bible: \"God has fixed the time for my death. I do not concern myself about that, but to be always ready, no matter when it may overtake me. That is the way all men should live, and then all would be equally brave.\"\n\nBe always ready. Go meet your maker with a clear conscience.\n\nAnother lesson he had learned in combat was to sleep whenever possible. He had done all he could, and could make no plans until he knew the situation he faced, so he reclined his seat as far as it would go, leaned his head back, and went to sleep.\nTHIRTEEN\n\nTravis Clay, Willis Coffee, Willie Varner, and I sat in the work area behind the lock shop display room drinking beer and watching television. The front door was locked.\n\nBarry Soetoro was on the tube breathing fire and damnation. Beside him stood General Martin L. Wynette, USA, looking every inch a soldier, with enough ribbons on his chest to decorate the Light Brigade. I had heard that Wynette had actually never heard a shot fired in anger, except for some outgoing artillery barrages fired several miles away, yet he looked fierce and determined, ready to chew nails. I thought it was his square jaw and steely eyes that created that impression, which had taken him far.\n\nApparently Willie the Wire was also impressed by the general, because he remarked, \"He oughta be in movies. Central Casting must have sent him over to the White House.\"\n\nSoetoro was reading from a teleprompter, as usual. I wondered who wrote his stuff: \". . .are going to crush the rebellion in Texas. The traitors who survive will be tried for treason. I appeal again to the sane people in Texas to put a stop to the foolishness of the legislature and the governor. They are the ones who will suffer, who will pay for the stupidity of their state officials. The price will be high. . .\"\n\nHe went on, telling about the Texas press release reporting the execution of Major Nasruli, the convicted Fort Hood jihadist. To hear Soetoro tell it, the execution was a personal insult to him. \"True, Major Nasruli was awaiting execution, but the timing and manner of that execution, if I allowed it to go forward at all, was at _my_ discretion. Many and diverse interests were at stake, including our relationship with many Muslim nations, and my judgment on this matter was rendered a nullity by a Texas National Guard officer who violated federal law. . . .\"\n\nHe talked some more about the heavy burdens of the presidency, then got back to the sins of Texas. \"I have ordered General Wynette to prepare a military response to Texas' blatantly illegal and violent act of secession. We will use the entire might of the federal government to stamp it out, to crush it. We owe the loyal citizens of the nation nothing less. One hundred fifty years ago Texas and other states tore this Union apart in a futile attempt to defend indefensible slavery. Now Texas is tearing this Union apart in order to defend an indefensible, reactionary vision of America that the rest of the country rejects. I can assure you that as president of the United States and commander in chief of the armed forces, I will do my duty as Abraham Lincoln did his, I will not let this stand. I will preserve the Union.\"\n\nWynette nodded several times during this rant, almost as if he were whispering amens.\n\nSoetoro took no questions from the gathered reporters, but stepped aside to give Wynette the podium. \"You may have heard rumors,\" Wynette said, \"that the commanding generals of a few of the United States military installations in Texas surrendered today. Actually, the facilities were delivered to the enemy by treachery. We are investigating. I promise you that the Benedict Arnolds responsible will be court-martialed for treason. If they are found guilty and given the sentence that the law prescribes for that crime, they will be executed. You may have also heard that some of our soldiers and airmen have joined the enemy's ranks to fight against United States forces. I cannot comment on the truth of that rumor, but I will state that any American soldier, sailor, airman, or Marine who does indeed join the enemy's ranks will be charged with desertion. I remind any member of the American military listening to this broadcast to remember where their loyalty lies.\n\n\"We will soon begin military operations against the rebels in Texas. We cannot be responsible for the loss of innocent lives; that responsibility rests with those who have rebelled against the lawful government of the United States and taken up arms against it.\"\n\nWynette ducked questions too. He followed Soetoro and Vice President Rhodes back into the bowels of the White House.\n\n\"Lots of treachery down in Texas,\" I remarked.\n\nThe network went back to showing footage of the rioting in Baltimore.\n\n\"Those scenes were shot at the riot last year,\" Travis said. \"I've seen those shots a dozen times. The TV people get around the censor by showing old footage.\"\n\nI went over and snapped off the television. I would have used the remote but Willie had laid it somewhere, lost it I suppose.\n\n\"They're going to start killing people,\" Willis Coffee said bitterly. \"He isn't even going to negotiate.\"\n\n\"I doubt if Texas would negotiate with him,\" I remarked. \"If you were them, would you negotiate with that megalomaniac?\"\n\n\"No,\" Willis admitted.\n\n\"I think those folks down in Texas are going to need a lot more killing than Barry Soetoro thinks they will,\" Travis said softly. \"It's that Alamo thing. They get it with their mother's milk. Texas, Texas, Texas, like it's the promised land that God gave them.\"\n\n\"Maybe he did,\" Willie the Wire muttered. \"For sho', he didn't give us anythin' to brag about here in Washington. I wouldn't risk a fingernail for the whole damn district.\"\n\nWe were batting things around when someone knocked on the front door of the lock shop. Willie went to see who it was, and came back with Sarah Houston. She looked particularly delicious that evening in her going-to-work outfit, a nice, knee-length dress with a belt that emphasized her figure. She was shod in a set of black pumps and had her purse over her shoulder.\n\nI introduced her to Willis and Travis. She looked us over and said, \"All the usual suspects.\"\n\n\"Want a beer?\" Willie asked, ol' Mr. Hospitality.\n\n\"No,\" she said, and looked around for something to sit on. Willie took a box of junk off a chair and arranged it for her. She seated herself, arranged her legs in the required position for female television journalists, and tugged her dress down a millimeter. She placed her purse on the floor beside her. Royalty come to call on the peasants.\n\n\"So when are they going to move Jake Grafton?\" I asked.\n\n\"I don't know. I doubt that it will be any time soon. They are frying other fish. They have a long list of people to arrest and incarcerate. They are working on a list of people who have shot their mouths off on Facebook and other social media.\"\n\n\"So there is no hurry,\" Travis remarked.\n\n\"I wouldn't say that,\" she said. \"The White House classified net is full of e-mails about this right-wing conspiracy, and Jake Grafton is near the head of the list. They're manufacturing evidence, trying to decide the best way to spin it for the public. They're going to try a dozen or so people to justify Soetoro's decision to invoke martial law.\"\n\n\"What about terrorism? All those jihadists Soetoro let in?\"\n\n\"The FBI is having some difficulty finding a sufficient number. They have their hands on a lot of Soetoro's domestic enemies, and Grafton, so. . .\"\n\nNone of us had anything to say to that. If they got Grafton into a federal prison, not just a concentration camp, there was no way we could get him out without an army.\n\nShe let that soak into our beetle brains, and then said, \"A snatch on the highway isn't going to work. That was Tommy's idea, I think.\"\n\nI nodded.\n\n\"You were also talking about a diversion, Tommy, and I decided the best one was probably to kill the power grid in the northeastern United States.\"\n\nWillis Coffee's eyes bulged. Travis whistled. I wasn't surprised, knowing as I did how Sarah's mind worked. This was a woman who arranged for a gang of Russian ex-sailors to steal an American attack submarine a few years ago. When Sarah Houston set out to do something, she didn't believe in half measures.\n\n\"Holy damn,\" Willie the Wire said.\n\n\"So how in the world are we going to do that?\" Willis Coffee demanded.\n\n\"We don't have any explosives, and we can't easily lay hands on any,\" Travis Clay pointed out. \"Even if we had a truckload, we can't run around the countryside blowing up a hundred substations.\"\n\n\"Maybe drop a hair dryer in the bathtub,\" Willie the Wire suggested.\n\nSarah Houston went on as if she hadn't heard them. \"The power grid is stretched to the max in August in the Northeast. It operates at one hundred percent of capacity much of the time running air conditioners and the like. The power companies use computer programs to automatically feed power around problem spots to prevent taking down the net. Computers are cheaper than new power-generation plants. They have hardened that computer system somewhat over the last few years in response to the perceived terrorist threat, but it is still vulnerable. I can put some code into the programs that will make the system default into the problems, not away from them, and that will quickly overload the system and take it down. All over the Northeast. From Cleveland to Maine and down to Cincinnati and Richmond.\"\n\nWe sat in silence digesting that. Finally Willis asked the obvious question. \"How are we going to create problems?\"\n\n\"We are going to have to knock out some key transformers and sub stations. I have compiled a short list of the most critical ones.\"\n\nHe was a sucker. \"So how are we going to do that?\"\n\n\"With explosives,\" Sarah said matter-of-factly. \"Shouldn't take a whole lot, but it will take some. As you may know, most of the federal agencies are stockpiling ammunition at warehouses in secret locations to use against the right-wing conspiracy, or if the locals get rowdy. Also in those warehouses are modest stocks of C-4 and enough tear gas to gas everyone east of the Mississippi. I made a list of the four closest warehouses. One of them is in Leesburg, a huge facility FEMA leased from Walmart.\"\n\n\"So you want us to start by breaking into a warehouse?\"\n\n\"If you want to give Barry Soetoro a crisis to worry about besides chasing you and Jake Grafton, you are going to have to make it something that really gets his attention.\"\n\n\"Texas might be enough,\" Travis Clay opined.\n\n\"You think?\"\n\n\"Uh, no.\"\n\nWillis Coffee said, \"Maybe killing the power grid is overkill. Modern cities can't work without electricity. Windows won't open, water pumps won't work, commodes won't flush, elevators won't work, lights won't work, medical equipment won't work, refrigerators won't work, microwaves won't work. Depending on how long the power stays off, some people could starve or die of heat exhaustion or dehydration.\"\n\nThat's when I got into the conversation. \"Barry Soetoro has torn up the Constitution. He's going to try a dozen innocent men for a crime he's invented. He's declared war on America. Texas has taken up the gauntlet. Now we must decide if we are willing to fight for America and let the chips fall where they may, or whether we would rather just pull our heads down, tuck our tails between our legs, and let Soetoro and Martin Wynette kill anyone they want. They are going to whack Texas hard. They are going to whack Jake Grafton. And believe me, given half a chance they'll whack us.\"\n\nThey sat staring at each other.\n\n\"I was listening to the president on the radio while I drove down here,\" Sarah Houston said. \"I would rather crawl into a hole out of the line of fire, but the fact is we have reached the point in America when it is time to choose a side.\"\n\n\"Jeez,\" Travis said softly. \"So we have to burgle a government warehouse, blow up some power substations, and then break into Camp Dawson and snatch Grafton from under the noses of God knows how many troops and feds. You and your little projects, Tommy.\"\n\n\"Yep,\" I said heartily. \"Gotta choose sides and smell armpits, guys. What say we all go to dinner and think this over before the power goes out. I'm buying.\"\n\nWillie Varner nearly broke his leg hopping off his stool.\n\nWe went to a white-tablecloth restaurant, even though the only one of our group dressed for it was Sarah. She led the way inside and favored the ma\u00eetre d' with a smile, so we were seated in a corner.\n\n\"Sorta like the last supper,\" Willie opined, then asked the waiter, \"What's the most expensive Scotch you have on your shelf?\"\n\nIt was something I'd never heard of.\n\n\"I'll take a double of that, neat,\" my lock shop partner told the waiter, and smiled at me. Sarah ordered a bottle of eighty-four-dollar wine, and my two covert warriors ordered draft beers. I ordered a bourbon on the rocks.\n\nAll of us had the sense not to even whisper about our planned operation to spring Grafton, or any of the other mayhem we had planned. We talked about riots and politics and whether Texas could win.\n\nAfter they had sipped their drinks and studied the menu, Willie ordered the most expensive steak, and Willis Coffee and Travis Clay did the same. I shrugged and ordered one too. Sarah took her time and ordered a piece of bare salmon with some lemon wedges and a small salad.\n\nThe bill was going to be a whopper, but I wasn't worried. I planned to use my CIA credit card to pay for it. I figured it would be a week or so until the clerks at Langley got around to turning the card off, and anyway, they could just deduct the amount from my severance pay, which I doubted I would ever get.\n\nI was so tense the liquor hit my stomach hard. I began to feel the glow down there instantly. I sat back in my chair, smiled vacuously, and tried to relax. Some of us were almost certainly going to be dead soon. I wondered if one of them would be me.\n\nWhen Willie Varner's steak came, it was still bloody. Travis pointed to it and said, \"A good vet could have saved that cow.\"\n\n\"Thank God he didn't,\" Willie said, and stuffed a piece in his mouth.\n\nThe copilot woke JR somewhere over west Texas. \"General, ATC is off the air. No one on any of their freqs or on El Paso Approach.\"\n\n\"Can you get into the civilian field?\"\n\n\"We're in solid goo. If the ILS is on the air, no problem. If we have to shoot a GPS approach, we may have to go below minimums, we think.\"\n\n\"Get me on the ground. However you have to do it.\"\n\n\"Yes, sir.\" The copilot went back to the cockpit.\n\nAfter another twenty minutes, the plane was maneuvering, answering the controls and responding to throttle input. They came out of the clouds perhaps three hundred feet above the ground, JR estimated.\n\n\"Good job,\" he told the pilots as he was getting out of the plane.\n\nThey saluted.\n\nThe ramp of the El Paso Fixed Base Operator's executive terminal was packed with planes, most of them jets or turboprops, yet the terminal was almost empty. The place reeked of luxury, with leather-covered sofas, ornate glass coffee tables, big flat-screen televisions, and subdued lighting\u2014perfect for important business executives or people who wanted to think they were important. JR approached the woman standing at the desk, the only human in sight, a cute twenty-something brunette wearing stiletto heels and a little black dress that ended well above her knees.\n\n\"What's happening?\"\n\n\"The airspace is closed to civilian traffic, General. These planes are stranded.\"\n\n\"I need a car.\"\n\n\"All our courtesy and rental cars are gone, sir. The passengers and crews of the planes outside took every one.\"\n\n\"Do you have a mechanic's van?\"\n\n\"Yes, but\u2014\"\n\n\"I'll take it. Send for whoever has the keys.\"\n\n\"General\u2014\"\n\n\"Now.\"\n\nHe wanted to see his old civilian contractor boss Pete Taylor and then look up the local National Guard commander, Wiley Fehrenbach, who was probably at the National Guard armory.\n\nOn his drive to Taylor's house, a helicopter flew past. It was an Apache scooting along at perhaps two hundred feet.\n\nHe knocked on the door of the house, which was a modest rancher in a modest neighborhood, and Zoe Taylor answered it.\n\n\"Oh, JR. Come in.\"\n\n\"No time. Is Pete here?\"\n\n\"No. The army came for him this afternoon. Arrested him.\"\n\n\"What for?\"\n\n\"They had a list.\"\n\n\"I see.\" Lee Parker was following the Jade Helm plan, no doubt on orders from Washington. \"Thank you, Zoe.\"\n\n\"Can you talk to them, JR, get him out? People have been talking for over a year about these Jade Helm things, saying it looks as if Soetoro was planning martial law.\" Tears leaked down her face. \"I can't believe this is really happening. It's like a nightmare. Is this still the United States of America?\"\n\n\"I understand,\" JR said, and against his better judgment, he added, \"I'll do what I can, Zoe.\"\n\nTears burst forth and she closed the door.\n\nJR got back in the van and headed for the armory.\n\nThe armory was a hive of activity. Bulldozers, generators, trucks, and construction equipment were swarmed by soldiers painting the Texas flag on every flat surface they could find. Plainly, these Texans were willing to fight, but they didn't have a lot of stuff to fight with: this was an engineering battalion. JR parked his mechanic's van in a handicapped spot and went inside.\n\nWiley Fehrenbach was delighted to see him. He wrung JR's hand and touched the stars on his camos. The pistol belt didn't escape his notice. He was wearing one too.\n\n\"I'm in command of the Texas Guard now, Wiley.\"\n\n\"Thank God.\"\n\n\"I need to know what's happening in town and at the base. Everything you know.\"\n\n\"When the news came out about the declaration, the town went wild. They've had it with the federal government. Martial law really ticked them off, then the gun thing. This morning civilian patrols started rounding up illegals and pushing them to the border crossings. The ICE people there tried to stop them, but they were surrounded and disarmed and told to disappear. Civilians shut down the border crossings. Only Mexican nationals can cross going south. All the trucks waiting to cross are lined up\u2014someone said the line is two miles long already.\"\n\n\"Okay.\"\n\n\"Our people came here as fast as they could this morning. I issued weapons, and it's a good thing I did. Some colonel and ten army troopers with weapons showed up at ten this morning and wanted to secure all the weapons and send everyone home. I refused, and since they were outnumbered twenty to one, they climbed into their car and left. They'll be back, and it's going to be bloody. My troops won't surrender. Right now, though, I think the army is out arresting civilians. They want all those Republic of Texas people who have been shouting for independence for years. They've arrested all of them they could find, plus newspaper people, the television and radio people, the sheriff, anyone who is anybody. It's all rumors, but everyone heard something and they're buzzing. Looks like they've opened the Jade Helm playbook and are going down the checklist.\"\n\n\"Where are they taking the prisoners?\"\n\n\"They have some railroad cars equipped with shackles on base. The army got them ready during the last Jade Helm exercise.\" JR already knew about the railcars with shackles, which had been hot news and stoked the rumors about martial law being planned. \"No one knows for certain,\" Wiley Fehrenbach said, \"but probably there.\"\n\n\"Are you sure your troops will fight?\"\n\n\"'I talked to them this morning. Told anyone that couldn't in good conscience fight for Texas to turn in his weapons and leave. Less than ten percent did. We're Texans and that's that.\"\n\n\"What's the situation out at Fort Bliss?\"\n\n\"It's on lockdown. Only U.S. Army soldiers admitted. I've had people out watching the gates, and as near as we can figure, a lot of the soldiers living in town haven't gone in. Maybe a hundred went in since we started watching, all told. You know there are maybe ten thousand soldiers living in town, so that's good.\"\n\n\"Yes,\" JR agreed.\n\n\"Parker ordered the television and radio stations shut down this morning, and all the phones and the internet are off. Electricity and water are still on, but who knows for how long.\"\n\n\"You need to get some troopers out to the water plant as soon as possible.\"\n\n\"Already sent a squad.\"\n\n\"Good man.\"\n\n\"It looks as if Parker has troopers patrolling the fences around the main part of the base, but you know how big Bliss is. I doubt anyone is out on the fence in the boonies. I don't know what Parker has planned, but no one has been back to get our weapons, so maybe he has some loyalty troubles. A lot of soldiers may have refused to fire on fellow Americans.\"\n\nJR Hays rubbed his head and tried to concentrate upon the problem. As he looked out the window, he realized the day was almost gone. It was twilight outside, under a gloomy sky. He heard another helicopter shoot overhead. With night-vision goggles, they could see everything that moved on the streets below.\n\nWiley Fehrenbach read his thoughts. \"Supposed to get some thunderstorms in here soon. How soon, I don't know. Maybe that will ground the choppers. I didn't think it wise to deploy my people until they were grounded or had left.\"\n\n\"I have four helos coming in from Fort Hood. They're supposed to land at the civilian airfield. Send some armed troops to meet them. Do you have some handheld radios? Our pilots will need them.\" The truth was, he thought wryly, he should have thought of that before they left Fort Hood. Maybe he was too tired, or maybe he wasn't thinking clearly.\n\n\"Sure.\" They discussed frequencies and JR made notes. A female sergeant appeared, and he handed her a note that contained a freq, told her about the helos, and sent her off with five enlisted soldiers carrying M4s and four radios with fully charged batteries.\n\nAfter they left, JR said, \"Wiley, our long-term objective is to take that base. We need all the military equipment they have and all the people who will fight for Texas.\" He tried to visualize General Lee Parker's situation. A lot of his soldiers had stayed home. The base, with base housing running right up to the perimeter fence, was basically indefensible. If Parker had any sense, he would arrange his tanks and loyal troopers into a strong defensive position where the tanks could cover each other and his artillery could provide support. Parker's helicopters were already patrolling, searching for threats.\n\nParker must be very worried, JR thought, wondering if his troops would fight. No doubt he was sending messages as quickly as he could dictate them to Washington, requesting instructions. These messages would go out over the army communications net, which was radio. JR doubted that Parker would do anything without orders from Washington. Then he would move slowly, carefully.\n\nHe and Wiley Fehrenbach discussed the situation as night fell. JR didn't want a battle, but he suspected he was going to get one before long. Eight hundred or so National Guardsmen in this armory were the only organized military unit in the area, so Washington would eventually tell Parker to take the armory. Parker outnumbered the guardsmen at least ten to one and had enough armor and artillery to invade Mexico and take Mexico City.\n\n\"Will U.S. soldiers fight Texans?\" JR whispered to the gods, who didn't answer back.\n\n\"Food?\" Wiley asked.\n\nJR hadn't eaten since breakfast, which seemed like years ago. \"Hell, yes.\"\n\nHe was soon handed a paper plate with three hot dogs in buns smothered in chili, along with a plastic knife and spoon and a bottle of water. JR found he was ravenous.\n\nHe had just started on the first hot dog when the radio on the desk came to life. It was \"Milestone One Six,\" the senior army aviator, who was flying a Blackhawk\u2014CWO-4 Erik Sabiston, Sabby to his friends.\n\n\"JR, we're fueling at the FBO at El Paso International. Weather is turning to crap. We flew at a hundred feet to get in here.\"\n\nJR answered, \"Fort Bliss has Apaches on patrol. Be careful, but I want you to do a recon over the base. I need to know what they're up to. Can you do that?\"\n\n\"Yes, sir. As soon as we finish fueling. Maybe fifteen minutes.\"\n\n\"I'd like to know if there are any units from Bliss out on the street. Your primary mission, though, is to shoot up everything on the flight line at Fort Bliss. Here are the coordinates. Ready to copy?\"\n\n\"Go.\"\n\nJR read them off, and Sabiston read them back. \"We know the base,\" he said. \"Trained there many times.\"\n\nJR ended with an admonition. \"Shoot and get out, Sabby. Hit them as hard as you can but don't be a hero. We need to deny them the sky.\"\n\n\"Yes, sir.\"\n\nJR attacked the food on his plate and said to Wiley Fehrenbach, \"They're going to come looking for you people sooner or later. You are going to have to abandon the armory. What do you have in the way of munitions?\"\n\n\"Dynamite, of course. Locked in vaults out back. And a couple hundred AT4s. Maybe a dozen .30-caliber machine guns. Ammo and grenades.\"\n\nJR felt a bit better. AT4s were handheld, single-shot anti-tank weapons. They came with the rocket pre-installed and could not be reloaded, so they were discarded after use. They weighed about five and a half pounds each and fired a rocket with a 1.6 pound HEAT warhead, HEAT standing for high explosive anti-tank. The rocket was marginal against an Abrams, which had the finest tank armor in the world, unless the rocket took a tread off or was fired into the rear, where the armor was thinnest. It was better against Bradley Infantry Fighting Vehicles and whatever other version of the armored personnel carrier 1st Armor had. It was hell on unarmored vehicles, such as trucks, or buildings.\n\nJR said, \"Get the explosives out of here. Ammo, weapons, radios, whatever you need, let's get it gone as soon as possible.\"\n\n\"Yes, sir.\"\n\n\"Get some people with AT4s out to El Paso International. Sooner or later Parker will send a tank column to occupy it. Tonight it is our airport. Let's get cracking.\"\nFOURTEEN\n\nGovernor Jack Hays was in uncharted political territory. He had to deal with threats from the federal government, demands for interviews from newspaper and television reporters, and the myriad of details that had to be addressed and resolved to turn Texas from a state into a nation. He had the leaders of the legislature in his office all morning while he sought consensus on a wide range of issues: the republic's assumption of U.S. debt held by Texas banks and financial institutions; the issuance of currency by the new republic; collection of federal taxes by the republic; payment of federal pensions and closing the Mexican border; and organizing a system of civil defense that had been pretty much dormant since the end of the 1960s since the feds were threatening military action against targets in Texas. No one knew if that would entail mass bombing of cities, but it certainly might.\n\nOne other thing happened that afternoon that would have far-reaching consequences, not only in Texas and throughout America, but around the world. Barry Soetoro announced that legislation would be introduced in Congress to phase in a completely electronic currency and retire all paper money. The implications were unstated but obvious: the federal government could control or confiscate anyone's wealth, whether it was corporate, individual, or nonprofit. A more effective way of whipping people into line probably could not be devised. Instead of locking up people, the federal government could simply take their money. Part of it or all of the loot could be used to fund the federal deficit, recapitalize banks, pay off political friends, or all of the above. Passage of the legislation was a foregone conclusion because the president's bitterest political enemies were already incarcerated, which helped cow the rest.\n\nWithin seconds of the announcement, precious metals prices on the world's commodity exchanges took off like a rocket. Within a minute, trading limits had been reached and trading was suspended. Hours later, the government announced that all trading in precious metals was suspended indefinitely.\n\nTexas was already committed to moving from U.S. currency as quickly as possible, but now the urgency became stark. It also hardened the resolve of those legislators who were still unsure they had done the right thing by declaring independence.\n\nThe legislators demanded that the governor make a televised speech to the legislature at midnight tonight, and Jack Hays agreed. When he was going to sort out his ideas on what he might say he didn't know. He assumed he was going to have to speak impromptu, which might be disastrous if he came across as tired, harassed, scared, or uncertain of the course of the new nation. He asked his speechwriters to consult with Ben Steiner and draft some talking points.\n\nIn the meantime, Jack Hays had an interview with the Mexican consul, Fernando Ferrante. They had a good working relationship, but Ferrante was not inclined to listen politely to protests of Mexican government policy, allegations of corruption, or complaints about illegal immigration and drug smuggling. His job, Ferrante said, was to smooth the flow of trade, not to advise the Mexican leadership on how to run the government.\n\n\"As you know, Se\u00f1or, we are embarking on a war with the United States to win our freedom,\" Jack Hays began. \"Unfortunately, we cannot guarantee the safety of Mexican nationals, nor the protection of civil commerce. Consequently Texas must temporarily close the border between Texas and Mexico.\"\n\nFerrante was sitting up straight. More than $90 billion in Mexican imports passed through Texas every year. A lesser amount, an estimated $60 to $70 billion, passed through Texas on the way to Mexico. In addition, Mexicans in the United States legally and illegally sent home hundreds of millions of dollars a year\u2014for some families, it marked the difference between poverty and starvation.\n\nJack Hays lowered the boom. \"It is very unfortunate, but for the moment we have no choice but to shut down all financial transactions transferring money into, out of, or through the new nation of Texas.\"\n\nFerrante protested. Hays cut him off. \"I know this will be a severe hardship to people south of the border. It will be an even greater hardship to Texans as we sever our commercial and financial relationships with the people and businesses of the other forty-nine American states. I wouldn't even suggest such a course were it not absolutely necessary.\"\n\nThe Mexican consul tugged thoughtfully at his lip. \"May I smoke?\" he asked.\n\n\"Of course,\" Hays said, and produced an ashtray from a desk drawer.\n\nWhen Ferrante had a cigarette alight, Hays continued. \"Since we cannot guarantee anyone's safety, we're asking Mexican nationals to leave Texas as soon as possible, and I'm asking you to let Texans in Mexico return to Texas.\"\n\n\"What about the citizens of other American states?\"\n\n\"If they cannot prove Texas residency, they will be refused entry.\"\n\nFerrante was shocked. He took a moment to organize his thoughts, then said, \"Factories producing goods for export are the economic bedrock of the Mexican economy. Shutting them down for any significant period, more than a weekend, gives the drug cartels more recruits. People _must_ feed their families.\"\n\n\"Mexico is in a hell of a hole,\" the Texan acknowledged, \"that you folks dug for yourselves. Mexico has dumped its problems on us for a great many years.\"\n\n\"Mexico is a democracy,\" the Mexican diplomat shot back, \"and elected politicians cannot ignore the will of our proud, poor people. It is in Texas' best interest that Mexico remain a democracy governed by the rule of law. A fascist dictatorship on your southern border will create many more problems in Texas than it will solve. You have a phrase: don't throw us under the bus. While you and your government are making policy, do not forget that the United States is the world's largest, richest market for recreational drugs of all kinds. Your 'War on Drugs' has been an abject failure. We are in the unfortunate position of being next-door neighbors to this hedonistic hell of addicts and abusers with too much money and not a shred of honor.\"\n\n\"I know, and I agree that a great many federal programs, including the 'War on Drugs' and the 'War on Poverty,' to name just two, were ill-conceived or abject failures,\" Jack Hays replied. \"But we're going to change that. The Republic of Texas is no longer going to be a pawn for feckless politicians in Washington who play to the mobs elsewhere and ignore the real problems we face here. We hope to be a better neighbor to the Republic of Mexico, but both our nations need to get our houses in order.\"\n\n\"When will Texas cease to isolate itself and resume free trade with my country?\"\n\nJack Hays engaged in a diplomatic lie. He planned on using trade as a weapon to force the Mexicans to stop illegal immigration, or at least to choke it down on their side of the border, and to crack down on the drug cartels and corrupt officials. He thought Mexico needed to clean the sty with a fire hose. Without Mexican help, the problems of the border would never be solved. Trade was the only issue that would force Mexico to change, Hays thought. At least he hoped it would, because it was the only big lever he had. He didn't voice this opinion, however, but said, \"As soon as our position with the other American states stabilizes. I cannot foretell the future. A week, a month, a year. . .\"\n\n\"Would Texas consider lifting this trade embargo if Mexico recognizes the new Republic of Texas?\"\n\n\"That would certainly help,\" Jack Hays said warmly. \"In fact, it would be a precondition.\"\n\nThe governor's answer committed him to nothing, a fact that did not escape the consul, who merely said, \"Our conversation will be passed along to my government, of course. When I receive their instructions, I will call you to arrange an appointment to discuss matters.\"\n\nHays stood, signifying the interview was over. He escorted Ferrante out of the office and reception area, which was packed with people all wanting a few minutes of his time.\n\nOne of the people was Charlie Swim.\n\nSwim was an ally that Jack Hays absolutely had to have, so he lightly grasped his elbow, escorted him into the office, and closed the door.\n\n\"Sit down, Charlie, please.\"\n\nCharlie Swim did so and took a folded sheet of paper from an inside pocket of his jacket. \"Governor, we've got a marvelous opportunity to finally do something positive for poor people in Texas.\" He tapped the paper and then passed it across the desk.\n\nAs the governor scanned it, Swim explained. \"Liberal progressive policies for the last fifty years or so have devastated the poor people of America. Welfare; aid to dependent families; food stamps; essentially free medical care; schools that try to prepare everyone for a four-year college degree, when only a fraction of the poor people will ever want or get one; lack of technical training; the breakdown of the black family\u2014all those things have led us to where we are.\n\n\"When Lyndon Johnson was lobbying Congress to pass his Great Society programs, he reportedly said, 'If we pass this the niggers will all vote Democratic for the next two centuries.' I don't know if he said that, but that has been the consequence. People do whatever it takes to get free money, because without an education and job opportunity they can't make it in America. We have to change that or we won't want to live in the poor socialist empire that will result.\"\n\nJack Hays sighed and pointed out, \"Luwanda Harris and her Democratic allies will be outraged, accuse you and me of abandoning the poor people to exploitation and starvation, or worse.\"\n\n\"I know that. Medicine often tastes bad, but until we fix the government policies that breed poverty, we have condemned the poor, black, white, and brown to a life of economic slavery. Goddamn, Jack, a hundred fifty years after Lincoln and the Union Army freed the slaves, we're still enslaved! Enslaved to the government! If there is to be a new life, a better life, for the poor people of Texas it has to start here and now. We can't waste another hour.\"\n\nJack Hays read the note, which was Swim's political wish list, a libertarian charter for abolishing everything from public employee unions to welfare to the minimum wage.\n\n\"Why do you want to repeal the minimum wage?\" Jack Hays asked.\n\n\"Without trade and technical training our supply of unskilled workers is limitless,\" Charlie Swim explained. \"We are awash in illegals. Every economist I have talked to tells me that the minimum wage really means that unskilled labor cannot be hired and trained unless they can immediately contribute to their employer the minimum wage and the value of their benefits, plus an amount sufficient to pay for supervision and the expenses of doing the paperwork they require, such as payroll, deductions, and the rest of it. All that, plus a profit. The higher the minimum wage, the greater economic incentive for employers to automate or move jobs out of the country. We are _never_ going to get wages up unless we let the free market determine the value of labor. Stopping the flow of illegals into Texas and getting some of them to leave will help. But as long as our schools turn out nothing but an endless supply of hamburger flippers and nail techs, industry goes begging for skilled labor and the free market can't work.\"\n\nJack Hays kept Charlie talking for another fifteen minutes, looked at his watch, and knew he had to come to a decision.\n\nThe governor looked Charlie Swim in the eye. \"The legislature will never pass most of these things, and right now you and I lack the political capital to even push them hard. My suggestion is that you pick the most important thing on the list and push just that. For example, education reform. We need a public education system that trains people for the jobs we have and are going to see in the foreseeable future. That we can sell, maybe.\"\n\n\"We need that and a lot more.\"\n\n\"We can't change the world in a week, a month, or even a year. We have to convince the voters we are advocating needed change. If you draft education reform as a war measure and tell every delegate and senator I'm for it, and shepherd it through, I'll sign it if they don't committee it to death or amend it beyond recognition. Tell them Texas can't afford to waste valuable education dollars. Right now we need every able-bodied Texan without a job to enlist in the National Guard. But when our future is secure, we need an educational system in place that will prepare people for good jobs, veterans, high school kids, everyone.\"\n\nSwim jumped out of his chair and shook the governor's hand. \"Thanks, Jack.\"\n\n\"Thank me after I sign it. Now go get at it.\"\n\nAt ten that night the war began in earnest. Two cruise missiles smashed into one of the main power plants in the Houston area, leaving a section of the city without electrical power. No doubt similar strikes would soon be forthcoming for power generation facilities all over Texas. Hospitals and key public institutions had to have emergency generating facilities up and running as soon as possible and be prepared to handle mass casualties. The director of emergency preparedness, Billy Rob Smith, left the governor's office on the run in company with Lieutenant Governor Bullet G. Fitzroy. Jack Hays had already loaded Fitzroy with more tasks than the man could conceivably handle, but Fitzroy had a background as an executive at a large conglomerate and knew how to prioritize, delegate, and supervise.\n\nBen Steiner remarked to Jack Hays that they would soon find out what Texans were made of.\n\nSluggo Sweatt, the president's man, sent for Jake Grafton, and within a few minutes he was escorted into the office where he had been interrogated. Grafton, like all the prisoners, was now clad in a red one-piece jumpsuit. That morning all the prisoners had been lined up, required to take off their civilian clothes, and issued jumpsuits. It wasn't that the authorities thought any of them could escape; the jumpsuits were designed to lower their morale and emphasize their status as prisoners.\n\nSweatt addressed him. \"Mr. Grafton\u2014you notice that I don't call you Director Grafton or Admiral Grafton, because you are no longer entitled to either honorific\u2014are you ready to talk sense and sign a confession?\"\n\n\"No,\" Jake Grafton said and dropped into a chair.\n\n\"Stand up when I talk to you,\" Sweatt said sharply. Jake did so.\n\n\"Your wife, Callie, and daughter, Amy\u2014have you heard from them?\"\n\n\"Yes.\"\n\n\"Your cell phone, please.\" Sluggo held out his hand.\n\nJake removed it from the pocket of his jumpsuit and passed it across. Sluggo played with it a moment. He called up the numbers and jotted them down.\n\nAllowing detainees, or prisoners, to retain their cell phones was counterintuitive, but Sluggo and his friends knew precisely what they were doing. Prisoners could make and receive calls from their friends, or anyone else. The prisoners would tell their sad tales and fear would spread like a hothouse fungus. Friends on the outside would soon cease to reach out to the prisoners, who would quickly become psychologically isolated.\n\nFinally Sluggo slid the phone back across the desk. Jake didn't reach for it.\n\n\"Three more people have confessed their roles in the plot to kill the president and take over the government. They implicated you. Swore that you knew, that they had discussed key items of the plan with you on several occasions.\"\n\nThe assassination of the president was a new wrinkle on the coup, Grafton noted sourly. When he said nothing, Sweatt added, \"The prosecutors are thinking of asking for the death penalty for you.\"\n\nStill no response.\n\nSluggo Sweatt sighed. \"Well, I've done all I can for you. I've told you the situation. You need to go back to your tent and think about your future. A confession would keep you alive.\"\n\nJake stood totally relaxed.\n\n\"Take the phone.\"\n\nJake pocketed it, and Sweatt nodded to the man behind Grafton, who took his arm and led him out.\n\nHe thought that the next time they brought him in the rough stuff would start, physical abuse, and threats against his family.\n\nJake Grafton knew that most men can be broken if the captors have the time to create enough pain. He didn't know if he was one of those rare men who could summon the inner strength to resist to the death, but he hoped\u2014make that prayed\u2014he was. Many years ago when he flew combat missions over enemy country in constant danger of being shot down, he had made up his mind to never surrender. Ever. Sluggo might make him prove it.\n\nAs he walked through the compound, he wondered what Sluggo Sweatt knew about the shenanigans at the White House.\n\nThe compound was crowded now. Jake estimated there were about two thousand people milling around. He recognized at least three congressmen and two senators. And then he saw someone whose face he knew well: Sal Molina, the president's right-hand political op. Now, apparently, his former political op. Wearing a red jumpsuit.\n\n\"Well, well, well,\" Grafton said as Molina recognized him. \"Fancy meeting you here.\"\n\nMolina turned his back on Grafton, who grabbed an arm and spun him around. That was when he realized tears were leaking from Molina's eyes.\n\n\"Did the hard-liners throw you out of the inner sanctum?\" Grafton asked roughly. \"Or did you just decide you needed a summer vacation courtesy of the taxpayers?\"\n\nMolina's Adam's apple went up and down a few times. \"Texas insulted Soetoro with their Declaration of Independence. He took it real personal. Since I'm from Texas and have relatives there, he decided he didn't want me around.\"\n\n\"Can't say that I blame him.\"\n\n\"I tried to warn you, Jake.\"\n\n\"So you did.\"\n\nAfter they had eaten dinner, some kind of stew with a little hamburger in it, Jake Grafton, Sal Molina, and Jack Yocke, the _Washington Post_ 's erstwhile columnist, settled under Grafton's favorite tree. The ground was damp from a morning shower, but they could talk in semi-privacy here, something they couldn't do elsewhere, not even in the latrine, which consisted of rows of commodes with no stalls.\n\nYocke rattled off the latest news, gleaned from his cell phone; Grafton and Molina made few comments. Then Yocke asked Sal Molina point-blank, \"So what's the big plan over at the White House? I'll bet they almost creamed their pants when the Saturday terrorists went hog wild.\"\n\n\"I don't know,\" Molina replied.\n\n\"You lying bastard. They've been planning martial law for years. Some people have even suggested Soetoro's boys gave the terrorists the weapons.\"\n\n\"That isn't true.\"\n\n\"But Schanck and Al Grantham jumped all over it, didn't they?\"\n\n\"Is this off the record?\"\n\n\"Oh, lighten up, dude. Like I'm going over the fence tonight and this interview will be in the _Post_ tomorrow.\"\n\n\"They might eventually let you out.\"\n\n\"Might?\"\n\n\" _Might_. Maybe after Soetoro drops dead of old age or cancer or something.\"\n\n\"Answer the question, Sal,\" Grafton prompted.\n\nMolina took a deep breath and looked around for eavesdroppers. Finally he said in a low voice, \"Yes. They told the president he had to do it. It would be unpopular, but martial law was the only way to save the progressive revolution. Soetoro loved it. This was his chance to change the course of history, to save the planet. The bastard has a messianic complex.\"\n\n\"He's got a lot of complexes,\" Jake Grafton muttered.\n\n\"More than you can imagine. For example, Barry and Mickey do S and M. She's a dominatrix. I guess he needs it, although don't ask me why. They didn't talk about that kind of stuff in psych class when I went to college.\"\n\n\"Hell, that's old news,\" Yocke scoffed. \"For seven years I've heard rumors that Soetoro is gay. People have even accused him of being a gay prostitute when he was younger, servicing old queens for drugs.\"\n\nGrafton asked Yocke, \"So how come your fine newspaper hasn't investigated these rumors about Soetoro?\"\n\n\"The editors don't think that crap is news,\" the _Post_ 's man explained. \"They're liberals. Some of them are gay, and for all I know some of them are swingers or dig S and M. Soetoro is liberal and black. He gets a pass. Now if he were some white Republican presidential candidate, they'd have had reporters investigate every day of his life from the moment his mom popped him out. You'd be reading about spitwads he threw in second grade and how many hours of detention hall he got in junior high.\" Yocke made a gesture dismissing the whole subject.\n\nAfter a pause he asked Molina, \"So why does Soetoro want to frame Grafton for plotting an assassination?\"\n\n\"Spymasters are good villains,\" Molina explained. \"They do a lot of secret shit they can never tell about, so people will believe almost any accusation. And the president doesn't like Grafton. And, of course, right-wing plots give the public something to talk about instead of terrorism and jihad in America. And S and M. Matt Drudge got the story from some Secret Service guy and was trying to get confirmation when a White House maid ratted him out. Still, Drudge might have broken the story anyway, so Soetoro had that hanging over his head when the terrorists did their thing. That helped push Soetoro to martial law now.\"\n\n\"He doesn't like a lot of people,\" Yocke replied. \"Is he going to frame them all?\"\n\n\"Oh no. He's just going to lock them up in concentration camps. Hitler and Stalin wrote the playbook.\"\n\n\"I suppose they grabbed Matt Drudge.\"\n\n\"He was locked up before the declaration. He's in solitary someplace. Drudge isn't the _Washington Post_ ; he would have run the story.\"\n\n\"So you're telling me that we're sitting in a concentration camp and the United States is about to bomb Texas because Soetoro is a pervert?\"\n\n\"That's about the size of it. As my old Marxist professors used to say, 'The personal is the political.'\"\n\nWe were leaving the restaurant when Sarah Houston said to me, \"Are you going to sleep at the lock shop tonight?\"\n\n\"Yes, unless I get a better offer.\"\n\n\"I feel the need for your manly presence to reassure me,\" she said.\n\n\"That's a better offer.\"\n\nIn the parking lot we agreed to meet at the lock shop tomorrow morning at eight. \"This is it, guys,\" I told them. \"Bring whatever you need for the op. I have no idea when we'll be back.\"\n\n\"After Barry Soetoro is dead,\" Travis Clay said gloomily.\n\n\"Christmas, maybe,\" Willis Coffee offered.\n\n\"The Fourth of July,\" Willie the Wire chimed in. \"Bring an extra set of underwear.\"\n\nOn that note we parted.\n\nBack at Sarah's place, she fixed drinks, Grand Marnier this time. \"I didn't know you kept this stuff around,\" I remarked.\n\n\"For the road,\" she told me, and lifted her glass.\n\nIn bed she whispered, \"You know we will probably all soon be dead.\"\n\n\"No one lives forever.\" That was a stupid remark. I sounded brave, which was a lie. Bravery is not on my short list of virtues. I'm anything but.\n\n\"I want more of this,\" she said.\n\n\"Me too,\" I agreed. The hell with it. Live today. . .\n\nWiley Fehrenbach and JR Hays decided to welcome any contingent that came to take the El Paso National Guard armory with a little ambush, then the ambushers would evade. Washington was probably lighting a fire under Lee Parker, so it was just a matter of time before he sent troops to the armory. This time it wouldn't be ten troopers and a colonel. This time he'd send the first team, some tanks, and maybe an infantry company, all with orders to shoot to kill.\n\nArmy Apache helicopters were already circling the area. Armed with Hellfire missiles and rockets, they could incinerate any vehicle, and their Gatling guns were hell on exposed troops.\n\nThe Apaches were the reason the Guard hadn't moved from the compound all day. Let the army open the ball, Wiley Fehrenbach and JR Hays reasoned, while they waited for the Fort Hood helicopters that were the equalizers. Every minute brought them closer.\n\nJR was in Fehrenbach's office. He heard thunder and watched lightning from the window. A soldier rushed in; three colonels followed.\n\n\"Sir,\" the soldier blurted. \"Four tanks and four Bradleys are coming out the main gate of Fort Bliss.\"\n\nWiley Fehrenbach looked at his colonels and said, \"You know what to do.\"\n\nThe colonels saluted, \"Yes, sir!\"\n\n\"Wait!\" JR roared. He got on a handheld radio. \"Milestone One Six, this is JR.\"\n\n\"One Six, go ahead.\"\n\nJR could hear the engine; the Blackhawk was airborne.\n\n\"At least four tanks and four Bradleys are coming from Fort Bliss, probably headed toward the armory. We need you to take out any airborne Apaches you can find, over.\"\n\n\"Can do.\"\n\n\"Leave the stuff on the ground to us. Over.\"\n\n\"One Six copies. Out.\"\n\n\"Some of those Apaches are ours, along with a Blackhawk,\" JR told the colonels. \"Don't let your men shoot at a helicopter unless they are absolutely sure it's the enemy. Now go.\"\n\nFor the first time that day, JR felt optimistic. Lee Parker had dithered too long.\n\nHe got more news when a trooper announced, \"We're destroying the decryption gear, sir, so the army doesn't get it.\" JR nodded, and the trooper handed him a batch of messages from Camp Mabry.\n\nThe first was from Loren Snyder: \"She can be moved. I'm searching for men.\"\n\nAnother, from Elvin Gentry: \"Dyess surrendered. Airplanes, weapons depot, and fuel facilities not sabotaged. Am recruiting crews. Awaiting further orders.\"\nFIFTEEN\n\nLightning was flashing from the clouds and gusts of rain and wind were pounding on the Blackhawk as it ran at a hundred feet above the housetops toward the El Paso National Guard armory, the coordinates of which the crew had punched into their GPS systems. The Blackhawk rocked and rolled in the turbulence. Fortunately the myriad lights of the city were still on, houses alight, street lights, traffic on the boulevards, so they had a good ground reference. Two Apaches were behind the Blackhawk, one on the right, one on the left.\n\nIf the city had been blacked out, Sabiston would have kept his crews on the ground. Still, they could go into inadvertent IFR conditions at any moment if some of this cloud dripped toward the ground, or if they hit a column of rain, or if the ground rose up into a cloud. If they flew under a thunderstorm, with its river of cold air descending out the bottom, all bets would be off: It would be all the pilots could do to keep their machines from being driven into the ground. Or a house. Or a school. Or a telephone pole. Of course, the same held true for the army pilots in their Apaches. Sabiston was listening on the Fort Bliss air traffic frequencies, trying to discover how many of their Apaches were airborne.\n\nIt sounded like only one base Apache was still airborne, and the pilot was bitching about the weather. \"I gotta get on the ground,\" he told the tower.\n\nSabiston keyed the intercom to talk to his copilot. \"Good news. Only one enemy Apache in the air, and he wants to come down. So what do ya think?\"\n\nThe copilot, who was from Albany, New York, keyed his mike and replied, \"We are fucking crazy. Once more into the suck. Will Texas pay our widows death benefits?\"\n\nOne of the Apaches behind him keyed the radio. \"Sabby, I got him on infrared. Clear to the left.\"\n\nThe copilot initiated a turn. They were almost on the housetops. Flying a helicopter was an unforgiving art, and in filthy weather this close to the ground, it attained the level of black magic.\n\nThe Apache behind them came abreast, accelerating. The Apache was an attack helicopter, manned by a crew of two seated in tandem. The pilot sits in the rear seat, the copilot\/weapons operator, or gunner, sits in the front. Both were usually rated pilots and both had controls to fly the machine, but in combat the front-seater operated the sensors and aimed and fired the weapons, which included a chain gun under the fuselage and whatever rockets or missiles were loaded for the mission. It was designed to provide close air support to infantry, armor, and artillery, and it did it well.\n\nThe Apache gunner had his target in sight; the chain gun sent a finger of fire shooting across the gloom.\n\nThe target absorbed two seconds' worth of 30-mm, then, with its tail rotor gone, lost control and tilted sideways, rotating viciously, then went into the ground and exploded.\n\nErik Sabiston saw the flash of the explosion in his night-vision goggles.\n\n\"The base,\" he told the copilot. \"Turn toward it.\"\n\nThey turned right. The base was lit up with streetlights, house lights, lights in parking lots. Tanks and artillery were bunched up, parked in a large grass area behind the exchange, facing the main gate.\n\n\"Go down the flight line,\" Sabby said.\n\nThey got lost once, flying just over the tops of the buildings, then miraculously they saw the field dead ahead: Blackhawks, Apaches, and a few old Chinooks were lined up in rows illuminated by floodlights on poles. They should have at least turned the lights off.\n\n\"I have the controls,\" Sabby said. He turned the Blackhawk and pulled the nose up, bleeding off airspeed dramatically. When he was down to fifty knots, he straightened out, about fifty feet from the ground, and flew between the two rows closest to the hangar. He spoke on the intercom to the door gunners. \"Shoot 'em up, guys.\"\n\nThe gunners fired one-second bursts at each target. One helicopter caught fire. _Brap_ , _brap_ , _brap_ , the gunners worked methodically; the noise bursts were out of sync. Another Apache in the line caught fire.\n\n\"Some ground fire from the hangars,\" the copilot said, and within seconds a hole appeared in the right front quarter of their windshield. It was a strange feeling, being fired on intentionally by Americans.\n\nWhen they finished the line, Sabiston accelerated and turned to fly back to El Paso International. No warning lights on the panel. All systems looked normal. \"Any damage in back?\" he asked his crew chief.\n\n\"Don't think so. I'll inspect.\" He turned the controls over to the copilot, then flipped freqs and got on the radio to JR Hays, who needed to know about the disposition of the base armor and artillery.\n\nThe Apache flown by Harvey Williston was following the Blackhawk down the line. \"I have the target,\" his gunner said. Dustin Bonner, from Tupelo, Mississippi, was the gunner. Earlier, Dustin was wondering if he had made the right decision signing on with the Texas Guard. There was going to be a lot of flying, a lot of shooting, and a lot of dying done before this thing was over. Maybe, he thought, he should have sneaked back to Mississippi and got back to playing blues guitar and working on his uncle's catfish farm. One thing was sure, there was a future in catfish. Being a gunner on an Apache in the middle of a shooting war, not so much.\n\nCertainly not when you were flying in a helicopter in shitty weather like this. Even if the bad guys didn't whack you, Mother Nature might. He fired rockets at the first few helicopters in the third row, which look undamaged. Three of them were obscured by the warhead's blast. Locked up a TOW wire-guided missile and launched it. Another. Then he was aiming the 30-mm M230 chain gun mounted on the fuselage between the landing gear. He pulled the trigger, moving from parked chopper to chopper.\n\nThe Apache flown by Mike Berk from Bemidji, Minnesota, followed along behind, with Mike's gunner doing the dirty work. Despite soldiers sheltered behind hangar doors taking pot shots, there was no opposition. First Armored had not yet got it into their collective heads that they were in a war. They'd figure it out pretty soon, though, so the next trip down the flight line wasn't going to be as pretty. Ahead of him he saw Williston turn left. \"Follow me, Mike. Hellfires into the hangars. You have the one on the right, I'll take the left.\"\n\nThe two attack helicopters made a sweeping 270-degree turn as lightning flashed and rain came in waves, under that low ceiling, until they were lined up. The ramp lights were off by then\u2014someone had gotten to the switches. It didn't matter to the Apaches, which had night-vision and infrared sensors that allowed the crew to fly and employ their weapons as if it were high noon on a cloudless day. The gunners fired the Hellfire missiles through the open hangar doors, and the explosions caused at least one fire that they could see.\n\nThen the two Apaches swept away southward toward El Paso International.\n\nWiley Fehrenbach worked feverishly with his officers and NCOs to get their stuff loaded and out of the Guard's compound. When the trucks were rolling, men jumped in their cars and left as fast as they could get out of the garage. The last of the cars were still pouring from the parking lot when the tanks rolled into view.\n\nThe tanks stopped, then the Bradleys behind them. Only when the parking lot was empty did the tanks move forward again, carefully.\n\nFrom his vantage down the street three blocks, JR Hays and two volunteer troopers watched the tanks and Bradleys go by. JR had an AT4 under his right arm and an M4 carbine on a sling across his back. One of the troopers was also carrying an AT4, an extra, just in case.\n\nBefore they left the armory, JR asked the young guardsmen, \"Have either of you ever actually fired an AT4?\"\n\n\"No, sir,\" each of them said.\n\n\"Then you get to watch me miss tonight. Your job is to act as lookouts, to ensure we don't get jumped by scouts.\"\n\nBut to JR's amazement, there were no scouts. This was _America_ , for Christ's sake, not Baghdad or Mosul or some other squalid Arab town. Well, the soldiers would learn. And fast. The next time the Guard tried this, it wouldn't be so easy.\n\nJR decided he would try for a Bradley when the troopers had re-embarked and were headed back to base. Patrols looking for guerillas or hidden troops took manpower. A constant low-level threat also took a toll on morale. JR knew because he had done his tours in the Middle East.\n\nJR found a basement stairwell to hide in, and took the extra antitank rocket.\n\n\"If you see a scout, open up, force him to take cover, then scatter to the rendezvous point. Tonight's goal is to ratchet up their fear a notch. You got that?\"\n\n\"Yes, sir.\"\n\nBoth these young troopers looked to be about twenty years old, but they were game. Given time, they would be good soldiers. Time. That's what JR had to buy them by arranging some serious air attacks on the 1st Armored. Fuel storage tanks were probably the top priority if he could get some planes in the air carrying bombs. Without fuel, 1st Armored was going nowhere.\n\nWith just the top of his camo cap showing, JR watched the troops set up a perimeter around the armory, with tanks on the four corners. Bradleys each carried six troopers, so that meant there were twenty-four troopers out there afoot, searching and guarding and looking to shoot the first man they saw with a gun.\n\nTime passed. Perhaps a half hour. The idling tanks were surprisingly quiet. The thunderstorm drifted off to the east and the wind was just a zephyr.\n\nFinally JR realized they had fired the armory. Probably by pouring gasoline around. Some of the windows must have been broken or shattered on their own, because soon smoke was oozing around the lights on poles around the place and the lights illuminating the parking lot. He hoped the fire department had the sense to stay in the station tonight.\n\nHe checked his sentries, who were out of sight. Waited.\n\nWaiting was the hardest part, he thought. You never get used to it. You wait for everything in the army, literally everything. Take a number, soldier. Or get in line. Then in combat, you wait some more. Wait to shoot and wait to die.\n\nFinally, with visible fire coming from three of the armory windows, the Abrams tanks started snorting and moving. Two of them led off up the street.\n\nJR Hays ignored them and watched the troopers return to the Bradleys. The Bradleys lined up; two tanks guarded the rear of the column.\n\nDarn.\n\nPicking up his AT4 and the spare, JR scuttled out of his hidey hole\u2014he didn't want to be there if the tank or Bradleys cut loose. The Abrams main battle tank was a formidable foe. Equipped with a 120-mm gun, a .50-caliber machine gun, and two .30-caliber machine guns, it was a rolling sixty-ton fortress protected by massive armor. Quite simply, the M1A1 Abrams was the finest tank on planet earth.\n\nThe Bradley was also armored, more lightly than a tank, but for protection it did have a nice 25-mm gun that fired up to two hundred rounds a minute. Twenty-five millimeters meant the shells were about an inch in diameter. Throwing three of those monsters every second, the gun could shred buildings, vehicles, and people very nicely, thank you, at terrific ranges.\n\nJR took up a new position, partially hidden by a corner of a building. He laid his spare AT4 on the ground against the building. The lead pair of tanks clattered past JR at perhaps eight to ten miles per hour. He turned on the battery in the AT4. Now the Bradleys came, in formation, at the same speed. Kneeling, JR glanced at the trailing tank, then sprinted forward to get a square shot at the rear of the last Bradley. He kneeled, pushed the safety button forward, quickly made sure he had the crosshairs where he wanted them, and pushed the fire button. The job took no more than four seconds. Just a tiny delay and the rocket shot out of the tube, leaving an enormous blast of glowing hot exhaust gases pouring from the rear of the launch tube. . .and almost instantly the rocket hit the end of the Bradley, punched through, and exploded. A jet of fire shot back out the entry hole.\n\nJR had already dropped the empty tube and was running for the corner of the sheltering building when he heard the chatter of a machine gun. That was from the tank behind him, he thought. He tore around the side of the building, out of sight of the tanks, ran right by the extra loaded tube lying by the building, and ran hard. Troopers from the other Bradleys would be after him in seconds.\n\nHe quickly found himself in an old neighborhood of mature trees and lawns and iron fences. Vaulted a fence and ran as if the hounds of hell were behind him, which they were, then got into an alley and ran on the gravel.\n\nFrom somewhere behind him he heard a shot. Not too loud. One of his kids, he hoped, slowing down the pursuit. He checked street signs and kept moving, now jogging.\n\nThe carbine on his back was slapping him at every step, slowing him, so he pulled it off and carried it in his hands. His pistol belt was also rubbing him with every step. Damn, he was going to be sore. He must have run three miles or more before he came to the parking lot of a Walmart. He found Wiley Fehrenbach sitting behind the wheel of his SUV; his two guardsmen were already seated in the back.\n\n\"I'm getting too old for this shit,\" he told Wiley as he motioned him to drive and put on his seatbelt. Then he tried to ease the pistol on his raw, aching hip.\n\nFehrenbach headed downtown.\n\nJR thought about the troopers in the Bradley he'd shot. No doubt they were all dead, or wished they were. They had been American soldiers, and perhaps he had even served with them somewhere in the last twenty years. When he recovered his breath, he turned to the two soldiers in the backseat.\n\n\"I'm a soldier,\" he offered in way of explanation, \"which is an ancient, honorable profession. I had absolutely nothing to do with independence. I wasn't even asked my opinion before the legislature did it. They did it because they thought their constituents wanted it desperately and without independence, Texas didn't have a chance. I don't know if they were right or wrong, yet I'm a Texan, and I'm all in. Do you understand?\"\n\n\"Yes,\" the two young men murmured. They were Texans too. JR wasn't sure they did fully understand, so he continued: \"Soldiers fight for their country. Ours is Texas. Freedom isn't free, and if we're going to get it, we're going to have to fight for it. We're going to have to hurt them worse than they hurt us, and we can't ever give up. You see that?\"\n\nOne of the soldiers, his name tag said he was Murray, replied, \"My dad is locked in a railroad car at the base. He's the president of the El Paso Rotary. Wrote some stuff for one of those independence movements. Fight for Texas? Hell yes.\"\n\nThe other soldier, his name tag said Tyler, nodded his head. At the wheel Wiley Fehrenbach was nodding too.\n\n\"Some of our enemies have to die and some of us will too,\" JR Hays said. \"Blood is the fertilizer of freedom. Maybe yours and mine.\"\n\nHe fell silent and watched the street with old, careful eyes. Fehrenbach pulled into a McDonald's parking lot. Cars full of National Guard soldiers were waiting. \"Murray, Tyler, run on over there and tell them to follow us to the airport. We have some work to do tonight.\" The young guardsmen trotted off, carrying their weapons.\n\nOn the way to the airport, JR said to Wiley, \"Our objective is to isolate First Armored, make sure it can't be reinforced or resupplied and can't run. I want you to pull all those executive jets onto the runways and taxiways and then shoot out their tires so they can't be moved easily. We may not be able to hold the airport, but at least no airplane will land on it until the army takes it back.\"\n\n\"And the airport on base?\"\n\n\"We'll take care of that in a day or two,\" JR said. \"After you do the international airport, I want you to get busy blowing up railroad trestles, as far out of town as you can. No trains in or out. Then bridges on the highways.\"\n\n\"We can do that. We're engineers.\"\n\n\"Do some ambushes, one or two, after you blow a trestle or bridge and they come to look. Try to hit a patrol in town occasionally. Shoot, then skedaddle. Don't get in any stand-up fights when you're outnumbered and outgunned. Just worry them.\"\n\n\"Hit and run.\"\n\n\"Precisely. The playbook is so old the pages are crumbling, but the tactics still work.\"\n\nAfter a moment he added, \"The army will soon be trying to ambush your men and doing searches house to house looking for weapons and uniforms. You'll be amazed at how fast the army's combat veterans will catch on, even anticipate your tactics. They're pros, not twenty-year-old amateurs like the two with me tonight.\"\n\n\"I understand.\"\n\n\"You have to watch out for your boys, Wiley, or soon we won't have any soldiers to fight with, just a bunch of bodies.\"\n\nJR thought about his comment to the soldiers that he had had nothing to do with independence. Perhaps Joe Bob's death at the hands of smugglers had pushed Jack toward independence. Certainly, he thought, his father's death had convinced him, when he heard about independence, that _he_ was going to fight.\n\nNot being an introspective man, he left it there and began thinking about how to win the war of independence. When Wiley Fehrenbach climbed out of the car and went inside the terminal to wait for his soldiers to assemble, JR found a notebook in the car and wrote an order to Major General Elvin Gentry.\n\n\"It is essential that we take the offensive and give Washington something to think about besides pounding Texas into submission. Have your B-1 people study up on railroad trestles and bridges out of the Powder River Basin in northeastern Wyoming and southeastern Montana. Send as many planes as possible as soon as possible to hit those trestles and bridges. I want to stop all the trains into and out of the Powder River Basin. The coal-fired power plants they service will soon run out of coal and shut down. The second-priority targets are pumping stations on natural gas delivery lines to the Upper Midwest and Northeast. If we can shut some of those gas lines down, many of the power plants there will have to shut down too.\"\n\nHe signed it JR Hays, Major General Commanding, Texas Guard. Then he went into the executive terminal, found the pilots of the executive jet that he had flown in on, gave them the note, and told them to fly to Dyess right away, before the runway was blocked. They were to deliver the message to Elvin Gentry.\n\nFehrenbach posted guards armed with rifles and AT4s on the access roads to the airport. He set the rest of his men to towing planes onto the runways with the little tractors and tow bars the FBO had parked on the ramp. \"Park the crash truck out there too,\" he said.\n\nWiley Fehrenbach and JR Hays were called to the lobby television by the desk lady, who apparently had no idea that the jets on her ramp were being moved. She pointed to the television. Jack Hays was giving a speech.\n\n_President_ Jack Hays\u2014the legislature had awarded him that new title along with declaring itself the Congress of Texas\u2014was escorted by the leaders of the Texas House and Senate. They walked past television cameras from local stations whose feed was beamed to satellites that were broadcasting across the world. Soetoro's censors might prevent it from being aired outside of Texas, but it would circle the earth and eventually reach every person upon it.\n\nAfter shaking dozens of hands on his way to the podium, Jack Hays at last took his place behind it. His writers and Ben Steiner had given him a speech, but to Steiner's dismay, he left the speech in his pocket. He was going to wing it.\n\nIn the packed gallery he saw his wife, Nadine.\n\n\"My fellow Texans,\" he began. Then he changed that, \"My fellow Texans and American patriots everywhere. I speak to you tonight after a tumultuous few days, a historic period that marks the beginning of our fight for freedom, a fight that we hope patriots everywhere in America will join and stand shoulder to shoulder with us against tyranny.\"\n\nHe detailed President Soetoro's transgressions, laying special emphasis on his imposition of martial law and the jailing of political opponents. \"Who would have thought that what is being done now was possible in the United States: that we live in fear of the midnight knock on the door; that many of our leading citizens are in concentration camps, where at any moment we might join them as prisoners. Let us be frank. America is now being ruled by a tyrant who has shredded the Constitution of the United States. In the last week, one man has seized all power unto himself, and the rights of no man or woman in America are safe.\n\n\"He has chosen to rip up the Bill of Rights, destroying the right of free speech, which is absolutely essential in a democracy. He has destroyed the right to bear arms, which is a free people's only defense against tyranny. He suspended the Writ of Habeas Corpus, an ancient writ created hundreds of years ago in England and brought to America by our first colonists to ensure the rule of law and protect the populace from government lawlessness. He has chosen to eliminate the currency. He has chosen to rule by fiat, dismissing Congress and flouting the courts. By his actions, he defines the word tyrant. In response to the dictates of a tyrant, we here in Texas have chosen to exercise our God-given right to self-government, our right to choose our own destiny and our own leaders, our right as a free people to resist tyranny and create a government worthy of a free people. In a sublime act of courage, the elected representatives of the people of Texas have done so. Yesterday morning in the very early hours they declared our independence. Today they established the Republic of Texas.\"\n\nHe paused in response to loud, sustained applause.\n\n\"We face difficult days ahead. The federal government has already fired the first shots, which were cruise missiles launched from a navy ship at sea off our coast. Today the navy has declared a blockade of our ports in an attempt to deny us freedom of the seas.\n\n\"The road ahead will not be easy. No doubt the federal government will escalate its pressure upon us. Still, precious as it is, freedom is worthless unless it is defended, and I fear blood will be required. How much, no man can say. At least a dozen people died and two dozen were wounded when a power plant in the Houston area was struck by those cruise missiles. Those Texans, _who wore no uniform_ , were our first casualties. I am reminded of the words of that great American patriot Thomas Paine: 'If there must be trouble, let it be in my day, that my child may have peace.'\"\n\nThe applause was thundering.\n\nWhen the noise had at last subsided, he said: \"Tonight we ask lovers of freedom all over America, indeed, lovers of freedom all over the world, to join us in our struggle. Let us here in Texas resolve to fight, no matter the price that may be required, for all that we loved about our country, for all that we treasured and hoped to pass on to our children, and their children, and the generations yet unborn. Let us here dedicate ourselves to enshrining freedom, justice, and the rule of law in the Republic of Texas, for ourselves and our posterity. _So help us God_.\"\n\nThe applause and shouting died after a while, because the hour was late and the day had been long for everyone. Still standing at the podium, Jack Hays shouted, \"Ben Steiner, you wrote our Declaration of Independence, what is your favorite song?\"\n\nTexans argued for years afterward whether Steiner knew that question was coming, but his answer was quick and his voice carried throughout the chamber. \"'The Eyes of Texas.'\"\n\nOne of the television producers was about to send the program back to the studio for commentary by instant experts, but he now waited, sensing that the best might still be ahead.\n\nJack Hays started singing. He had a nice baritone. Everyone in the chamber was still on their feet, including the spectators in the gallery. Nadine's eyes were locked upon her husband as he sang, \"The eyes of Texas are upon you, all the live long day. The eyes of Texas are upon you, and you cannot get away . . .\"\n\nWhen the roar died, Hays looked and gestured at the Speaker, who shouted, \"'The Yellow Rose of Texas.'\"\n\n\"There's a yellow rose in Texas, that I am going to see. Nobody else could miss her, not half as much as me . . . She's the sweetest little rosebud that Texas ever knew. Her eyes are bright as diamonds, they sparkle like the dew. You may talk about your Clementine and sing of Rosalee, but _the yellow rose of Texas_ is the only girl for me. . . .\"\n\nAll over Texas, people were sitting in front of their televisions or radios, many singing at the top of their lungs, as Jack Hays thought they would.\n\nBarry Soetoro watched the speech and singing on television in the family quarters of the White House. \"That's a dangerous man,\" he remarked to Mickey. \"He's firing up every half-wit cracker in the country.\"\n\n\"You'd better have someone shoot him quick,\" she said. \"You knew those Texas bastards were going to give you trouble.\"\n\nThe president nodded. He knew good advice when he heard it.\n\nJack Hays said, \"My favorite now, 'Deep In the Heart of Texas,'\" and he led off.\n\n\"The stars at night . . . are big and bright\"\u2014clap, clap, clap\u2014\" _deep in the heart of Texas_. The prairie bloom . . . is like perfume\"\u2014clap, clap, clap\u2014\" _deep in the heart of Texas_. Reminds me of the one I love\"\u2014clap, clap, clap\u2014\" _deep in the heart of Texas_. . . .\"\n\nThe last notes had barely died when Jack said, \"Let's end with the anthem of Texas, 'Texas, Our Texas.'\"\n\nThe voices rose loudly, if not melodiously. \"Texas, our Texas! All hail the mighty state! Texas, our Texas! So wonderful, so great. . . .\"\n\nThe last stanza was the best, and although many of the legislators didn't know the words, Jack Hays did. He sang it with every ounce of fervor that was in him. \"Texas, dear Texas! From tyrant's grip now free, shines forth in splendor, our star of destiny! Mother of heroes, we come your children true, proclaiming our allegiance, our faith, our love for you. . . . God bless you, Texas! And keep you brave and strong, that you may grow in power and worth, throughout the ages long. . . .\"\n\nLong after the singing had died and they turned off their televisions and radios, in cities, towns, and hamlets and at isolated homes and ranches, from the islands and low flatlands near the gulf and the thickets and pine forests of east Texas, to the prairies, plains, and semi-deserts of west Texas and the windswept tableland of the Panhandle, people hummed the tunes and thought about Jack Hays' words. In truck stops, cafes, and big rigs rolling along lonesome highways, people thought and pondered, about America and Texas and the dreams men carry for a someday that may or may not ever come.\n\nAs Jack Hays once remarked to Nadine, \"Texas isn't a place; it's a religion.\"\n\nAt Fort Bliss, Major General Lee Parker had a nightmare on his hands. His flight line had been shot to hell, an Apache had been shot down, he had lost a Bradley and every soldier in it, and three helicopters had shot up his flight line. As Jack Hays spoke on the television, Parker's air officer was trying to get a count on how many helicopters were flyable.\n\nIn addition to these problems, the Pentagon was bombarding him with messages directing him to attack in all directions, disarm all civilians, and arrest every male Texan he could find. \"What about the women?\" he asked his chief of staff, who had no answers. Parker had served in Texas long enough to know that many Texas women were, if anything, made of even sterner stuff than the men. Given sufficient reason, they could and occasionally did shoot a man as dead as a man can get. On the other hand, arresting women, some of them mothers of young children, would not play well in Washington. And he had no decent facilities to hold them in.\n\nSo how was he going to do all this attacking and arresting? He huddled with his ops officer, the brigade commanders, and their ops officers trying to figure out what his objectives and priorities should be. Staff officers flitted around like moths around a flame. Given enough time, something might have come out of the blizzard of orders from headquarters and all this staff work, but time ran out for Lee Parker at about three a.m. He was whipped. He hadn't slept in eighteen hours, and was keeping himself running on strong black coffee.\n\n\"Sir,\" one of his aides whispered to him. \"There is a delegation of NCOs in your outer office. They want to talk to you.\"\n\n\"A delegation?\"\n\n\"Yes, sir. That's what they said.\"\n\n\"We don't do delegations in the army. Tell them to return to duty or their quarters.\"\n\n\"General, they insist on seeing you.\"\n\nParker stormed out of the conference room and down the hallway to the reception area outside his office, fully intending to blister some soldiers. A delegation! Just who did these sergeants think they were, anyway?\n\nHe faced a group of command sergeant majors. \"What the hell do you want that can't go through the chain of command?\"\n\n\"We wanted to bring this to your attention right now, General. The troops in the barracks are packing their duffle bags, getting in their cars, and driving out the main gate. Over in base housing, officers and men are loading their families and leaving.\"\n\n\"This base is on lockdown. No one in or out. You know that!\"\n\n\"Yes, sir, but the gate guards have left too. Our gates are wide open and unmanned.\"\n\n\"Get them manned immediately. Anyone leaving this fort in violation of orders will be arrested and court-martialed.\"\n\n\"Sir, the soldiers we have left refuse to man the gate. The main road outside the gate is lined with armed civilians, and more are coming every minute. The sheriff's deputies are out there trying to keep them from flooding onto the base.\"\n\nLee Parker stared, his jaw agape. In all his years in the army he had never even heard of mass disobedience. \"This is mutiny,\" he said to the top sergeants.\n\n\"Yes, sir, it is that. But we can't stop our soldiers short of shooting them, and they won't do the army any good if they are shot. What it boils down to is that less than ten percent of the troops will stick. That's just an estimate. More like a guess, maybe. The rest are scattering like leaves in the wind. Some say they are going to fight for Texas, others are going home, wherever that is for them. Bottom line, General, is _we have no one to fight with_.\"\n\nOn the way to headquarters, the sergeants agreed that having soldiers arrest local civilians and incarcerate them had been a major mistake. They didn't think it was Parker's fault; he was just following orders. Ill-considered orders. The men and women of the 1st Armored were soldiers, damn good ones too, not KGB or Gestapo or Brown Shirts. Or FEMA or Homeland thugs. \"We're soldiers, sir,\" the division sergeant major told the general now by way of explanation, although without context the general thought that comment inane.\n\n\"Our troops aren't acting like soldiers,\" Parker shot back heatedly. \" _Mutiny_! By God, when this is over I'm going to send a whole lot of people to Leavenworth. Just watch.\"\n\nThe command sergeant major, Alfredo Mendez, five feet, six inches of professional soldier from McAllen, Texas, said, \"General, I don't think you understand the situation. Perhaps we weren't clear enough. Your troops are leaving. They will not fight Texans. They refuse to serve in Barry Soetoro's army. Your choice right now is to get in a plane as quickly as possible and fly out of here, or stay and surrender to the National Guard. When Wiley Fehrenbach figures out the situation here at Bliss, which will be sooner rather than later, he and his troops will be coming, armed, and our people will not fight.\"\n\nThe general went into his office, slammed the door, and tried to get control of himself. Never in his wildest nightmares had he ever imagined this. _Mutiny_!\n\nAfter five or six deep breaths, he walked out, past the waiting NCOs, and headed for the staff conference room. He broke the news to his staff and his generals in four sentences.\n\nOne of the brigadiers exploded. \"We'll get the loyal soldiers and kick the snot out of those guardsmen and civilians. Let's get at it.\"\n\n\"So you want to go out like Custer, eh?\" another brigadier shot back. \"This isn't Iraq. These civilians will shoot first, just like the Sioux did. The people of Texas are fighting for their freedom from what they believe is a tyrannical government that has suspended the United States Constitution. So far we have fifteen dead and thirty-two wounded and all we've accomplished is burning down an empty National Guard Armory. What do you plan to do, fight house to house to get the hell out of El Paso? Make a last stand at a Walmart or on some lonesome, windswept hill in the middle of a cow pasture, if you get that far?\"\n\n\"We could get our loyal troops and some of the equipment into New Mexico, and the Texans wouldn't follow us across the border.\"\n\n\"You think this is chess?\" another officer retorted. \"If I were making the decisions for them, I would follow you all the way to Hell to force you to surrender. And we're just not ready to move. It would take a couple of days to get ready, and we don't have two days.\"\n\nLee Parker made up his mind. The brass would court-martial him if he ran, and, in truth, he didn't have running in him. Nor did he want to fight for Barry Soetoro. He had been doing what he had done for the past thirty-two years: obeying orders because he was in the United States Army, serving under the Stars and Stripes. Now he lacked the means to fight. \"We'll surrender,\" he said. He glanced at the chief of staff and told him to draft a message to all the higher headquarters telling them of his decision.\n\n\"Sir, shouldn't we disable the tanks, artillery, Bradleys?\"\n\n\"If we had the people to accomplish that, we wouldn't be surrendering,\" Lee Parker said bitterly. \"This command has just disintegrated. I didn't see it coming, and I doubt if anyone else in this room did either. If you did have an inkling, you certainly didn't do your country any favors by keeping your mouth shut.\" Yet, after all, in a vast bureaucracy, one didn't get ahead by pointing out statistically remote disastrous possibilities that had never occurred in the past. A mutiny! For heaven's sake, this _is_ the United States Army, and 1st Armored was a hell of a good outfit!\n\nLee Parker went back to the NCOs who still stood in the reception area.\n\n\"Sergeant Major Mendez, will you please go to the main gate and tell the sheriff or his deputy to send for General Fehrenbach? I'll surrender Fort Bliss to him. Have the sheriff bring him here.\"\n\n\"Yes, sir,\" Mendez said, saluted, and marched from the room.\n\nThe thunderstorms were gone and it was drizzling rain when JR Hays and Wiley Fehrenbach were ushered into the commanding general's office at Fort Bliss. Seeing that JR was wearing major general's stars, Lee Parker, standing at attention beside his desk, saluted and said, \"Gentlemen, my troops have mutinied and I am unable to defend the base or the military equipment here. In order not to squander lives uselessly, I wish to surrender the base and its personnel to the Texas forces.\"\n\nJR and Wiley returned the salute. JR told Wiley, \"You accept the surrender. Write it out on a computer.\" He dictated the terms: All military equipment would be surrendered along with the troops. Those soldiers who wished to leave Texas were welcome to do so, and those who wished to enlist in the Texas Guard would be encouraged to do so after they took a loyalty oath and signed it. Anyone caught sabotaging surrendered military equipment would be dealt with summarily.\n\n\"If you or your staff or senior officers wish to leave, General Parker, I suggest you get in one of your C-130s or executive transports and leave immediately. We are going to block the runway with tanks as soon as you depart.\"\n\n\"I'll stay,\" Lee Parker said. \"My officers can make their own decisions.\"\n\n\"I understand you have some civilians locked up.\"\n\n\"Orders from Washington,\" Parker replied curtly. \"FEMA has lists.\"\n\n\"Let them out, Wiley, and get them rides home. And haul down the American flags on base. Find some Texas flags and run them up.\"\n\n\"Yes, sir.\"\n\nWiley Fehrenbach unbuttoned his shirt and produced a Lone Star flag. He grinned at JR and handed it to the nearest soldier. \"You heard him. Run it up the pole outside and get one for the pole at the main gate.\"\n\nJR glanced at the leather couch, and asked the two generals to conduct their business in the outer office. When the door was closed behind them, he sacked out on the couch. He glanced at his watch. The Republic of Texas was just a bit less than forty-eight hours old. The window was open and the breeze felt good. He was asleep ten seconds later.\nSIXTEEN\n\nThe riots continued in inner cities around the country. Baltimore was probably the worst: it had been racked by riots the previous year, and this time the mob at the core expanded across downtown and into the suburbs.\n\nPolice and National Guardsmen had disappeared. Much of their leadership had already been imprisoned by the feds. Many of those left on duty went home to protect or move their families. Others just threw up their hands. Why try to bring a mob under control when the physical risks were high and the politicians were frightened that they might lose some votes, so none of the political elite or police brass would back the men and women in uniform on the streets? Police and guardsmen went into bars, had a few, then found their cars and went home.\n\nIn the suburbs, people were getting into a state of near panic. Rumors were rampant. In subdivisions and neighborhoods, mothers and fathers surrounded by children met in front yards and culs-de-sac, exchanging rumors and fears. People talked about blocking off streets as they faced the prospect of having to defend their homes against marauders. It seemed as if much of America now had two ravenous domestic enemies\u2014rioting, looting mobs, and the federal government. Many of the suburbanites had an old lever-action Winchester or Marlin, or a bolt-action Winchester, Remington, or Ruger in the closet, and a couple boxes of ammunition for it. They decided what they were going to do if the mobs invaded their neighborhoods to rob, loot, rape, and burn.\n\nIn Detroit, Chicago, Philadelphia, St. Louis, and Los Angeles, the mobs were still in the ghetto, but as in Baltimore, those who lived in the riot-torn area and were not a part of it were trying to flee. People left on foot and in cars, streams of refugees, some with the contents of looted stores on their backs, but all convinced they had had enough.\n\nLocal and network television were showing some of this, where censors would allow it, and radio stations were on the spot with breathless reporting. Social media filled in with some truth, rumor, and wild speculation. As usual on social media, budding writers of sardonic fiction posted absurd tales they thought only fools would believe; of course the fools did believe, but so did many frightened people who were definitely not foolish.\n\nEveryone had someone they needed to talk to desperately: Telephone networks were at maximum capacity. Calls, e-mails, and text messages inundated city and state officials high and low, all those remaining after the FBI, FEMA, Homeland Security, and cooperating county sheriffs had carried off the disloyal for incarceration. Some of the less cooperative sheriffs and police chiefs had also been arrested, decapitating their law enforcement departments. The only thing observers could agree on was that the situation was getting worse. In the White House and congressional offices, staffers stopped answering telephones and e-mail servers crashed. Monday night, August 29, was another wild one in America.\n\nThey came for Jake Grafton at Camp Dawson at three in the morning, Tuesday, August 30. Four of them, in green coveralls with FEMA badges on the right shoulder. They woke him up by dragging him from his cot, slamming him to the floor, and kicking him.\n\nThen they cuffed his hands behind his back and dragged him from the tent, across the common area, by the mess tents, to the building Sluggo Sweatt used as headquarters. Up the stairs into Sluggo's lair. He was up, with a light on, waiting. The four thugs lifted Grafton bodily from the floor and threw him into a chair. Another man came in and dropped Grafton's watch and cell phone on Sluggo's desk.\n\n\"Good morning, Grafton,\" Sluggo said pleasantly. \"I decided it was time to take the gloves off and confront you with the reality of your situation.\"\n\nGrafton tried to ease himself in the chair. It felt as if one of his ribs on the left side was broken. Sharp pain with every breath.\n\n\"My conscience requires me to tell you in advance that the road ahead for you is filled with pain. I need you to sign a confession of complicity in the attempted assassination of President Soetoro. Of course, there will be television cameras. You will need to speak slowly and coherently about your crimes.\"\n\nJake Grafton looked around the room, the same one he had visited twice before.\n\nOne of the men on his right used a fist into his side. He gasped at the blow and almost fell from the chair.\n\n\"Be polite and pay attention,\" Sluggo said. \"I told my colleagues that you would undoubtedly need a lot of persuading, and they thought it would be fun to do it. There isn't much to do to pass the time here in the boonies.\" With that, Sluggo nodded.\n\nThe thugs dragged him from the chair and took him along a hallway to a jail cell, complete with bars and a cot and a honey bucket. There they started pounding on his ribs. One of them stomped on his scrotum. At some point he passed out.\n\nWhen he came to, the lights were on, but he had no idea whether it was day or night or how long he had been unconscious.\n\nTelevision. That was why they hadn't touched his face.\n\nThe good news was that he was still alive. The bad news was that Sluggo's men were going to beat him to death by inches.\n\nLoren Snyder had been busy. He used the Houston telephone book to find the address of a former naval officer, Julie Aranado, also known as Jugs. Apparently the Aranado men of prior generations had favored big-bosomed women, so Julie was awesomely endowed. Lots of exercise kept the rest of her figure slim and trim, showing off the trophies. She had acquired her nickname at the Naval Academy and, although it reeked of political incorrectness and sexism, she liked it, so it stuck. \"If you got 'em, be proud,\" she had been heard to remark when questioned about the appellation.\n\nAfter eleven years of active duty, she decided the GI Bill's offer of a free advanced education beat the navy's retention bonuses. So she quit the navy and was earning a PhD in physics at the University of Houston. She returned to her apartment on Sunday evening, after watching Jack Hays' speech at a girlfriend's house, and found Loren Snyder sitting on the front stoop waiting for her.\n\n\"Hey, Jugs. You're looking good.\"\n\n\"Mr. Snyder! I haven't seen you in what, two or three years?\"\n\n\"About that. And it's Loren. Hey, I need some help and you were the first person I thought of.\"\n\n\"I heard you were in law school at UT.\"\n\n\"Yep.\"\n\n\"What kind of help?\" she asked as she unlocked the door. Snyder was at least ten years her senior, and she had served with him aboard an attack sub. Romance hadn't been on the agenda then, and she knew it wasn't now. The Loren Snyder she had known was all business.\n\n\"The Republic of Texas is now the proud owner of a _Virginia_ -class sub, USS _Texas_. She's lying in Galveston. I'm the new skipper and you are now my XO.\"\n\nShe snorted. \"Don't bullshit me, Snyder. School starts again next week and I need to study. What do you want?\"\n\nHe told it as he had gotten it, then added, \"I went aboard her yesterday evening. The crew scrammed the reactor, secured the batteries, and left, arrested by the county sheriff, who doesn't know jack about ships, boats, or submarines. I inspected everything I could see and couldn't find any sign of sabotage. All _Texas_ needs is a crew.\"\n\nJugs snorted. \"Where, pray tell, are you going to find sixty people to man her?\"\n\n\"I'm not. I figure with five people who know what they are doing, I can get her under way. We can't leave her lying at the pier. I figure there is probably one chance in five the navy will destroy her with Tomahawks, and four chances in five the navy will send a SEAL team to take her.\"\n\n\"SEALs couldn't get her under way,\" Jugs objected with a frown. \"They don't have that kind of training.\"\n\n\"They could if they brought five or six certified people with them. And you know they can do that.\" Both these former naval officers had a very healthy respect for the navy's special operations warriors, arguably the best in the world. If anyone could steal a submarine, they could.\n\n\"They're probably planning a mission right now,\" she said thoughtfully.\n\n\"If we are going to save that boat for Texas, we have to get in gear. Are you for independence?\"\n\n\"Hell, yes. I'm from San Antone. I've had more than enough of Soetoro pissing on the Constitution. It's high time we went our own way.\" Although Aranado didn't say it, like many Mexican American Catholics, she was socially conservative. Same-sex marriages, she believed, were an insult to the sanctity of that institution. Abortion horrified her\u2014especially late-term abortions, doctors sucking the brains from viable infants\u2014and Soetoro's and his party's fervid support of the practice had cost them her vote years ago. In fact, she had sworn in church at the altar of God she would never vote for one of those baby-butchering sons of bitches as long as she lived.\n\nJugs always was blunt, Snyder reflected. \"I need three more qualified people,\" he said. \"Who do you know that we can get?\" Then he added, \"In Texas?\"\n\nAnother group, five young men in their late twenties or early thirties, was also busy that Monday night. They were unemployed coal miners in southern West Virginia. They had been following Soetoro's declaration of martial law and Texas' reaction to it on television, in bits and pieces. They were nonpolitical high school grads who had become certified underground miners and worked in the mines since their early twenties. Their mines had laid them off some months back when demand for coal forced mines to lay off shifts. Their fathers had been miners, and their fathers before them. Underground mines were the last remaining sources of good jobs in southern West Virginia since NAFTA had sent factory jobs to Mexico twenty years before. They believed Barry Soetoro's EPA was killing coal, and with it, their way of life, and they were bitter. They still had fishing, hunting, riding their ATVs, and chasing girls, but without a decent paycheck, their futures looked bleak. None wanted to leave the hills to look for work elsewhere. Here was where they had spent their lives, here was where their friends were, here was where their relatives had been buried for over two centuries in the little graveyards surrounding the one-room white churches that dotted the hills. _This_ was their place.\n\nNow evil politicians, rich environmentalists, and Washington bureaucrats had robbed them of it, they believed. They had never thought of themselves as terrorists, but for months now they had been talking about getting even with those distant bastards who had taken everything they had. These young men despised Barry Soetoro and everything he stood for and admired the Texans. Unlike the miners in West Virginia, those Texans hadn't just hunkered down and let the big shits fuck them. They were fighting back.\n\nHarlan Greathouse was the natural leader of this little group, and the biggest talker. Sunday, while they were fishing the eddies in a quiet little river shaded with verdant sycamores and drinking beer, Greathouse prodded them into action.\n\nOne of them still had a key to the explosives locker at the mine where he used to work. The padlocks on the locker were supposed to be changed periodically, but who knew when the mine foreman would get around to it. The key still worked, and for that they were grateful. The locker was a grounded steel building as far away from structures and dwellings as was practical. Sunday night they used that key, opened the locker, and helped themselves to three cases of dynamite, blasting caps, a roll of wire, and three detonators that passed their battery checks. The roll contained about a thousand feet of wire. They really needed three rolls, so they could plant three charges, but they decided to make do with one.\n\nHarlan Greathouse led in his pickup, and his friends in two more pickups followed him to the interstate. They stopped at a convenience store on a freeway exit, gassed up, and bought more 3.2 percent beer, the so-called non-alcoholic beer, then got back onto the highway. As they finished each can of beer, they crushed it and with a practiced flip of the wrist, tossed it into the beds of their pickups. They drove into the great valley of Virginia and across the Blue Ridge to the rolling countryside cut by old rivers that ran into the Chesapeake.\n\nOn a two-lane asphalt road that ran through bucolic countryside they found a pumping station on one of the natural-gas trunk lines that ran from Louisiana northeastward all the way to Boston. Anyone could see it was a pipeline right-of-way because the tree-less terrain covered in low weeds ran from one horizon to another and was about a hundred feet wide. This line serviced a myriad of smaller feeder lines that supplied natural gas to factories, cities, towns, and gas-fed power plants.\n\nNone of the miners had the slightest idea how big the explosion would be when they blew the pumping station. Big, they figured, big enough to perhaps ignite this stand of dry pines that stood on either side of the right-of-way. They saw in the moonlight\u2014it was four in the morning\u2014that each stand consisted of about five acres of trees. A quick reconnaissance revealed that these two stands were surrounded by pastures and meadows as far as the eye could see, with here and there a modest house and its associated barn. Cattle grazed in the pastures. The nearest house was perhaps five hundred yards beyond the edge of the trees, so they figured no one there would be injured by the blast.\n\nHarlan thought this a good place. They could set one case of dynamite, unroll perhaps four hundred feet of wire off the roll, cut it, and rig it to a detonator. The loss of line pressure after the explosion would cause emergency shutoff valves farther up and down the line to secure the flow of gas. Those power plants to the northeast that depended on this line would be down until gas from other, interconnecting lines, could be routed to them. The explosion would no doubt obliterate this pumping station, and it would eventually need to be rebuilt.\n\n\"They should have stayed with coal,\" one of the miners said, chuckling, just loud enough to be heard.\n\nThe pumping station, about a half-acre in size, was surrounded by a ten-foot-high chain-link fence topped with three strands of barbed wire and was lit by floodlights on poles. There was a gate, of course, and it was padlocked.\n\nThe gate wasn't a problem. The miners hooked a tow chain around one of the fence posts, hooked the other end to a tow-hitch, and pulled it down.\n\nThey all knew how to handle dynamite. In less than five minutes they had divided a case of dynamite into three charges, one of which was set on the main inlet line\u2014about three feet in diameter\u2014another on the line out, and one on the main pump itself. Between the pump and the charges on the lines were the safety cutoff valves, which were going to be destroyed too. One car went by without slowing while they worked. They inserted the blasting caps, wired up a harness that they mated to the caps, then unrolled an estimated four hundred feet of wire, cut it, and turned the pickups around.\n\nHarlan Greathouse thought he should be the one to trigger the blast. The other two pickups went on west a half mile or so to the crest of a low hill as he wired up the detonator. He took cover behind his pickup and lifted the safety lever. Took a deep breath and pushed the button.\n\nThe resulting explosion wasn't really that bad. But it was followed by a hurricane of noise as natural gas under pressure hissed from the ruptured line. That lasted just long enough to register on Harlan's ears, then the gas was ignited by molten hotspots in the steel. A giant explosion resulted. Trees were flattened to the east and west. The stupendous fireball from the blast rose in a monstrous flaming mushroom cloud.\n\nThe pickup truck absorbed the peak pressure of the shockwave from the concussion of the gas explosion, thereby saving Harlan from being crushed. However, even with the dubious shelter of his shattered truck, he perished within a second or so as the pulse of superheated air scorched and fried him to blackened gristle. The heat pulse also set the ten acres of now-flattened pines instantly aflame.\n\nWithin a minute the gas flowing from the ruptured lines slowed as pressure bled off. Air rushing back into the blast area and escaping gas fed a blowtorch flame that rose at least three hundred feet in the air. The initial fireball, now expanding into a mushroom cloud and turning from yellow to red and orange, rose and rose into the sky, lighting the countryside as bright as day.\n\nHarlan Greathouse's friends came driving madly back, but one look in the light of the burning gas told the story. They turned their pickups around in the road and roared away to the west toward the distant mountains.\n\nAs dawn was breaking Tuesday in Galveston, Snyder, Aranado, and three men, all of whom Jugs knew from her naval reserve weekends, were aboard _Texas_ checking her out. Speedy Gonzales was a nuclear engineer, Mouse Moore was a first-class petty officer with twelve years in attack subs, and Junior Smith was a third-class who had served aboard Polaris boats. All Texans, all foursquare for independence, they had volunteered immediately.\n\nUsing flashlights, they inspected everything they could see, opened panels and examined wiring and fittings, checked the galley for provisions, and all came to the same conclusion. _Texas_ was ready for sea. The former crew's personal effects were still aboard, uniforms, underwear, hygiene items, letters from wives and girlfriends. The batteries had a good charge on them. It was as if the crew had mustered on the pier and marched off, leaving everything. Although Snyder and his crew didn't know it, that was pretty much what had happened.\n\nAll five gathered in the control room and discussed their inspections. \"She's ready to go, I believe,\" Speedy said. \"A full load of Tomahawks and torpedoes, plenty of food and water, more than ample for five people. The batteries seem okay, the checklists are in place and apparently complete.\" He spoke like a judge, weighing every word before he uttered it because it would appear on the court reporter's transcript.\n\n\"Mouse?\" Loren asked.\n\n\"She's ready to go, Mr. Snyder.\" Snyder was an officer, and under no conceivable circumstances would Mouse Moore address him familiarly. He had spent too many years in uniform. In his bunkroom he might tell his shipmates his opinion of Loren or Jugs, but he would never address either of them that way to their faces. It was a mark in his favor: Mouse was a good sailor who would always obey orders.\n\nJunior Smith was cut from a slightly different pattern. He had been doused in naval tradition and most of it had washed off. He was a civilian at heart, and so he said, \"Loren, I'm willing to go to sea in her.\"\n\n\"Just precisely what _do_ you plan, Mr. Snyder?\" Jugs asked, preferring to address Loren formally.\n\n\"I want to get the reactor cooking again, check that every system is working properly, run some drills to ensure we don't entomb ourselves, and if we're all cool, cast off and get the hell out of Dodge before the SEALs show up. They can't get at us if we're submerged.\"\n\n\"We have no secure way to communicate with JR Hays,\" Jugs objected.\n\n\"After a while we can poke up the mast, listen to the radio, and learn what's happening. Right now, I think it imperative we get gone before the SEALs come, and you all know they will.\"\n\n\"Sure as God made little green apples,\" Junior agreed.\n\n\"So let's check all the circuit breakers and emergency alarms, then fire off the tea kettle. Stations everyone.\"\n\n\"Your first command,\" Speedy said with a grin.\n\n\"And probably my last,\" Loren Snyder admitted. \"Miz Aranado, you and Speedy bring the batteries online and let's do it.\"\n\nFour minutes later the batteries brought the boat to life. Lights came on, air began circulating, computer displays came to standby. Back aft Speedy Gonzales checked the emergency alarms one by one. Loren Snyder snapped off his flashlight and smiled. It was as if he had returned to something he had loved and missed. He thought for three seconds about law school, and snorted. Someday, maybe.\n\nGeneral Martin L. Wynette, the Joint Chiefs, and their staff were having a terrible morning. The news of the surrender of Fort Bliss, after a mutiny, cast a pall on their planning to invade Texas. Large numbers of troops that refused to obey orders, or refused to fight, or went AWOL was a nightmare that the U.S. armed forces had never before dealt with. It raised the question of whether any troops ordered to attack Texas could actually be relied upon to do so. It seemed to the planners that the answer to that question would determine what could be done, and when. Of course, the White House staff was outraged and said the military was dragging its feet in the face of treason. That comment was grossly unfair, and even Martin Wynette was severely irritated by it. Everyone in the E-Ring offices of the Pentagon knew that imprudent action would lead to even more severe condemnation of the military.\n\nThe loss of USS _Texas_ gave the navy serious heartburn. Some advocated launching Tomahawk cruise missiles at the attack submarine while she lay at the Galveston pier, but the chief of naval operations, the CNO, Admiral Cart McKiernan, was having none of it. \"We spent 2.6 billion dollars for that boat that we had to squeeze out of Congress like it was blood,\" he roared to the Joint Staff. \"I'll be damned if I'm going to order her destroyed until we've tried every other option. We may desperately need her if Iran and China get feisty. Those rodeo cowboys in Galveston are going nowhere in that boat; the very idea is ludicrous. Now you people get a SEAL team saddled up to go down there and get her. Have them take some submariners with them. I don't give a damn who the SEALs have to kill or how they do it, but I want that submarine back in one piece. Understand?\"\n\nThat was yesterday. In the wee hours of this morning it looked as if the SEAL team needed at least another twenty-four hours to get ready. People and equipment had to be moved into position and it all took time, a fact that infuriated the White House staffers sitting in on the pre-dawn meeting, who knew absolutely nothing about logistics. While they ranted, the lights and computers in the Pentagon flickered and went out for a few seconds until the building's massive emergency power system automatically came online.\n\nThe sabotage of the natural gas trunk line from Louisiana had forced several natural gas power plants in the area to shut down until gas could be rerouted over the network. The shutdowns of the power plants blacked out cities in northern Virginia and Maryland. Then the problems began to cascade. The computer system that controlled the electrical grid, automatically rerouting electrical power to restore it to deprived areas, began to do precisely the opposite. It demanded power from the stricken plants, and when there was none to be had, began shutting down the grid across the northeastern United States. In seconds, the power was off from Chicago to Boston and south all the way to Richmond. Air conditioners quit, elevators jammed, computers died, the telephone system went down, water and sewage pumps failed.\n\nI found out about the power failure about seven that morning when I sneaked from Sarah Houston's bed and padded into her kitchen to make coffee. The kitchen lights wouldn't illuminate. Suspecting the worst, I opened the door of a very quiet refrigerator. No light inside. Oh boy. I jabbed the remote to turn on the television, just in case, but no soap. I thought maybe it was the circuit breakers, but I didn't know where her panel was. I tried my cell phone: no service. So it wasn't the circuit breakers.\n\nI went back to the bedroom, woke Sarah, and told her the news.\n\n\"Perhaps my little program worked,\" she chirped, pleased with herself.\n\n\"Maybe the juice is only off in this neighborhood.\"\n\n\"You are always so cheerful, Tommy. And at this hour of the morning.\"\n\n\"I'm a natural-born optimist,\" I objected. \"In fact, I'm so optimistic that I think we should throw on some clothes and hot foot it over to the lock shop. If the outage is regional, we don't have to wait until tonight to hit that warehouse. We can do it as soon as we can get there, and should.\"\n\n\"But I'm not packed.\"\n\nI was already dressing and didn't reply. Sure enough, forty-five minutes later we were in my car on our way. Sarah's a trooper.\n\nAnd the power was off everywhere. Traffic was light. Why go to work if nothing at the office or factory will function, if the malls, grocery and convenience stores, and gas stations are closed?\n\nThe guys were waiting at the lock shop. \"How'd you do it, Sarah? How did you kill the power?\"\n\n\"I waved a wand,\" she said.\n\nIn addition to the Wire, Willis Coffee, and Travis Clay, there was one other guy there, a big black guy, really buff, who hadn't had a haircut or shaved in months. His name was Armanti Hall, and I knew him, although not very well, because he and I had done some training together a few years back. He was in a sour mood, didn't say a word.\n\n\"Armanti was waiting for me last night at my place,\" Travis said. \"He wants to go with us, and he has a pickup with a bed cover.\"\n\n\"Did you brief him?\"\n\n\"No. He doesn't give a damn what we're up to. I'll tell you about it later.\"\n\nWe unloaded the lock shop stuff from the van and began packing it with stuff we thought we might need in our war on FEMA and Barry Soetoro. Took some propane bottles and a torch, a box of tools, two crowbars, and some other things. I took my bag of cash and my weapons and ammo from the car and packed them in the van. The other guys had some small duffle bags of personal items, so we threw them in too.\n\nArmanti and Willis muttered to each other while we loaded up. They decided to ride in Armanti's pickup together. We locked up the shop and my car and saddled up. Willie Varner and Travis rode in the back of the van and I drove, with Sarah Houston in the right seat.\n\nAfter we were off the Beltway headed for Leesburg, I asked Travis what the story was on Armanti.\n\n\"He just got back from Syria a couple days ago. He thinks the agency will be looking for him soon, maybe to turn him over to civilian prosecutors.\"\n\n\"Lovely. Want to tell us about it?\"\n\n\"They had him working with the Brits, trying to find the executioner. Last week sometime he went into a building to drag out a guy they wanted to question, guy who they thought was a big dog in ISIS. Hall is an expert in unarmed combat and he thought he could put him down quick, minimum fuss, minimum time, and carry him out.\"\n\nTravis glanced at Sarah and stopped talking. I prompted him.\n\n\"Anyway, he got in okay and started searching the house. Couldn't find his guy. He went up the stairs to the third floor and walked in on the guy. The shit was trying to get his dick into a six-year-old girl. You know those guys are pedophiles, child-fuckers?\"\n\n\"Yeah. I know.\"\n\n\"The kid was sobbing and had been hit a couple of times. Naked from the waist down. Armanti didn't hesitate, just came up behind the guy, grabbed him, and broke his neck. Crack. So with the guy there dead and the kid sobbing, Armanti castrated the corpse and stuffed his genitals into his mouth. That took just seconds.\n\n\"He had the kid under his left arm and was on his way out when a woman walked in. She took one look at the corpse and started to scream. He hit her once in the chest as hard as he could. Maybe he wasn't trying to kill her, just wind her good so she couldn't scream, but. . .anyway, the way he told it to me, her heart stopped dead. Probably burst like a balloon. He's a strong man and was all pumped on adrenaline. . .\"\n\nTravis took a few seconds, then continued, \"Met a man coming up the stairs as he and the kid went down. The guy decided to shoot Armanti, but he was a hair slow. I think Armanti actually stuck his pistol in the guy's mouth and blew his head off.\"\n\nWe all thought about that for a moment.\n\nTravis went on. \"The Brits took the kid and said they would send her to a British charity that is trying to get orphans out of Syria and into the UK. Of course he had to tell the Brits why they didn't have a prisoner to sweat. They said to forget it, but you know how these things are. Someone will whisper about it, and when the agency gets wind of it, killing the mother and kidnapping the girl, the shit will hit the fan. Armanti just wants to be gone.\"\n\n\"How does he know the woman was the child's mother?\" Sarah asked.\n\n\"She was. He was briefed before he went in. But when she walked into that room, she didn't care about the child\u2014she was screaming about the holy warrior who was going to do a Muhammad on the kid. So he killed her. Instant justice, I guess.\"\n\n\"Can he be trusted?\" I asked.\n\n\"You've trained with him, Tommy. I'd trust him with my life, but I don't do kids.\"\n\nWe left it there.\n\nAs we approached Leesburg I glanced to my left and saw a strip mall with one store all lit up. It was a drugstore. I wheeled the van into the parking lot. We locked it and went inside.\n\n\"How come you're open?\" I asked the guy behind the counter.\n\n\"We have an emergency generator. We're open twenty-four\/seven, all year around, rain, snow, or power outages. People sometimes need medications in the middle of the night. That's our edge.\"\n\nWe stocked up on bandages, antiseptics, needles and thread, surgical tape, aspirin, and a box of surgical gloves. \"Be prepared,\" my scoutmaster always said.\n\nThe warehouse district is on the south side of Leesburg, in an industrial district that looked as if it contained only warehouses and light industry. Without power, there were only a few vehicles there today.\n\nSarah pointed out the warehouse we wanted. It was a big steel building and the sign said \"Walmart. Always low prices. Always.\" It was locked up tight, with a steel personnel door and a code pad.\n\nI parked the van so people down the street couldn't see what we were doing. Armanti parked a block away in the other direction.\n\nWe put on surgical gloves, and then used a propane torch and a crowbar. Took about ten minutes but we got that door open. No alarm sounded. The place was dark as King Tut's tomb. We used flashlights and right in front of us was a deuce and a half and four pickups with FEMA markings, plus a gaggle of big forklifts. I left the Wire outside to warn us if anyone came along, then, using flashlights, the rest of us explored.\n\nThe place looked like the hold of a ship heading for D-Day in Normandy. More pickups, trucks, Humvees, electrical generators on trailers, mobile kitchens, tanks for water and fuel, even some weird looking things that Travis said were microwave radar for crowd control, plus mobile radio setups and com units mounted on the backs of trucks. The stuff was painted a dark green and had a white star stenciled on each side. It wasn't marked U.S. Army. This was FEMA stuff, for Barry Soetoro's army.\n\nThat was one side of the warehouse. On the other side, arranged so there was room for forklifts to go between the stacks, were pallets of ammo, several tractor-trailer loads; more pallets with boxes full of one-piece green coveralls emblazoned with a FEMA badge on the right shoulder and an American flag on the left; tractor-trailer loads of MREs, meals ready to eat; mountains of weapons; crates of M4s, AT4s, heavy belt-fed .30-caliber machine guns and M279 light machine guns, hand grenades, belted ammo, and pistols; and even some small wooden boxes containing two sniper rifles each. There were some industrial-sized coffee pots, a truckload of first aid supplies, including anti-coagulant pads, and medical emergency kits for corpsmen. Basically, it looked to me like enough military supplies to outfit an infantry brigade for a trek across Africa even if they had to fight every step of the way.\n\n\"When the revolution comes, these folks planned to come out on the winning side,\" Willis Coffee remarked. The rest of us just looked around, stunned.\n\n\"Did you know all this was here, Tommy?\" Armanti asked.\n\n\"Nope. But I was hopeful we'd find some weapons. Our pistols aren't going to be enough to pry Jake Grafton out of Camp Dawson.\"\n\n\"So that's what's going down.\"\n\n\"Yeah. You still want in?\"\n\n\"Why not.\"\n\n\"Okay, people,\" I said. \"Let's get at it. We'll load two of their pickups, the van, and Armanti's ride. Use that forklift over there to load up some pallets of MREs. Take four of those ten-gallon jerry cans full of fuel. We want a crate of AT4s, a couple of machine guns with boxes of belted 7.62 for them, a couple of light machine guns, a couple M4s for each of us, lots of ammo, and anything that looks interesting, like those boxes of hand grenades and the medical supplies. I don't want to die for lack of a Band-Aid. I'd also like a sniper rifle for my personal collection in case I decide to take up groundhog hunting. But what I'd really like to find in here is some C-4, timers, and detonators. Chop chop.\"\n\nThe good news was that Willis, Travis, Armanti, and I knew how to use all these weapons and keep them in good working order. Sarah didn't, of course, and neither did Willie the Wire. On one trip to the van with a crate of MREs, I asked Willie, \"You want a rifle or pistol for a souvenir?\"\n\n\"I'm a two-time loser, man, and you know it. If I got a pistol in my pocket when they arrest me for jaywalking while black, it's mandatory life. Thanks, but no thanks.\"\n\nHe was going to bet his life on our ability to rescue Grafton, but wanted to do it disarmed. Explain that logic if you can.\n\nSince it was already ninety degrees outside, we threw our jeans and shirts in the van before we stepped into the new duds. Everyone but Willie strapped a web belt and pistol holster on, including Sarah. Beretta nines were the flavor of the day. \"You know how to use that shooter?\" I asked her.\n\n\"No, but it's the fashion accessory of the season, so I want one.\"\n\nThere were boxes of army combat boots in the warehouse, so we each took a pair. Sarah, of course, said, \"I'm not wearing those.\"\n\n\"Find a pair that fits, try them on to make sure, then throw them in the van, just in case we have to wade a swamp.\"\n\nShe nodded and did it.\n\nWe spent fifteen minutes opening the overhead door so we could get the pickups out. Using the forklift, they raised me as high as possible and I unlatched it from the opening mechanism, then we used one of the door cables to pull it open. The forklift pulled and up it went. Willis and Travis climbed into the cabs of the pickups. The keys were in them, lying on the dashes.\n\n\"Look around and get all the people out of this area. You're FEMA guys, tough dudes. Government orders. Don't take any backtalk.\"\n\n\"You aren't going to blow this warehouse, Tommy,\" Willis Coffee said.\n\n\"I thought I would.\"\n\nWillis lowered his head onto the steering wheel for a moment. When he raised his head, he said, \"And I thought we were just going to burgle and run.\"\n\n\"Hey, Walmart's lawyers undoubtedly got FEMA to agree to indemnify them. The surrounding owners can sue in the sweet by and by, if the courts ever get back up and running.\"\n\n\"I don't care about that lawyer shit. I would prefer not to be chased. Not anytime soon, anyway.\"\n\n\"An opportunity like this comes along only once in a lifetime, if that,\" I told him.\n\nSo they drove through the open door and I walked over to the C-4 pile and got busy. I figured the C-4 would ignite all the ammo in the warehouse, so there would be a pretty good pop. Even if it didn't, the blast should wreck all this stuff, turn it into junk. Just to make sure, I poured a jerry can of gasoline on the ammo pile and opened three or four others. I gave us twenty minutes on the timer, checked my watch and saw it was two minutes after one o'clock, and pushed the button. The countdown began.\n\nI used the forklift to lower the overhead door, then walked out of the warehouse through the buckled personnel door and pushed it shut. The three or four civilian vehicles that had been in front of other warehouses were now gone. I climbed into the van with Sarah and drove away. The pickups were waiting by the front gate. We headed west.\n\nI was glancing at my watch when the whole thing went off. I saw the top of the mushroom cloud in my rearview mirror.\n\nSarah saw me looking, twisted her right side mirror, and took a squint.\n\n\"Tommy, what if some civilian was killed in that explosion?\"\n\n\"We all have to die sometime. I'll pray for 'em.\" I wouldn't, though, if I heard they were Soetoro voters.\n\nIt took a little under half a minute for the sound of the blast to reach us. The concussion probably broke windows in Leesburg.\nSEVENTEEN\n\nThe mushroom cloud was still hanging over Leesburg when General Martin L. Wynette and two staff officers, both generals, arrived at the Executive Office Building across from the White House. President Soetoro and thirty or so of his staff were waiting in a large conference room. The emergency generators were apparently running sweetly: the building was well lit and the air conditioners were pumping cool air.\n\n\"So what is your plan to crush Texas?\" the president asked the chairman of the JCS.\n\n\"The briefer has some maps. He'll run through it and we'll answer questions.\"\n\nThe briefer, Major General Strong, stood in front of a huge computer screen, upon which a PowerPoint presentation was projected. \"Our first problem is manpower. Given desertions, we're estimating our combat effectives are fifty percent of what they should be.\"\n\nThe president's chief of staff, Al Grantham, blew up. By reputation, he was one of the most aggressive leftists on the president's staff, and, although he was white, was of the opinion that white America would have to be conquered. He thought most whites were racists and Nazis. \"You mean to tell me that in the armed forces only the people who want to fight have to fight?\"\n\nWynette said flatly, \"We have a volunteer army. It's hard to make someone fight if they refuse to do so.\"\n\nGrantham glared. \"What the hell have we been paying them for?\"\n\n\"We have been paying them to defend the United States. Not to put too fine a point on it, a lot of our personnel don't think shooting their fellow Americans meets those criteria.\"\n\n\"Court-martial the bastards.\"\n\n\"Oh, we can do that, if the president orders us to do so. We can convict them of cowardice, give them bad discharges, maybe some jail time, but that still doesn't put people in ranks willing to fight.\"\n\nThe president gestured at the briefer to continue.\n\nThe major general nodded and said, \"We will take two divisions, one armored, one infantry, from Georgia and Alabama; put them on trains, trucks, and air force transports; and assemble at Barksdale Air Force Base in Louisiana. From there we will proceed to Austin and take it, engaging any Texas military units or guerilla bands we encounter along the way. Meanwhile we will have the Fourth Infantry division at Fort Carson in Colorado proceed by road to Amarillo, and from there to Austin. So we will have three divisions in a two-pronged assault. Operating on two fronts\u2014\"\n\n\"How will they get across the rivers and all that?\" Grantham interrupted, glowering.\n\n\"I was about to cover that, sir,\" the briefer said patiently. \"We will drop paratroops to seize the key bridges and hold until relieved. Then\u2014\"\n\n\"So how are you going to get there? Across cow pastures and rice paddies?\"\n\n\"We will use the interstates and other roads where possible. A division cannot move on only one road. It must move on a wide front, yet not so wide that one brigade cannot reinforce the other. Where we must cross rivers without a bridge available, we will use pontoons. We will have close air support from attack helicopters and air force fighters every foot of the way. We'll use satellite reconnaissance, aerial reconnaissance, and drones to keep us apprised of the enemy's movements.\"\n\n\"Those crackers are going to shoot at you,\" Al Grantham said. \"Probably a lot. Every one of those racists has a gun, or two or three or four.\"\n\n\"No doubt,\" Martin Wynette replied. \"We'll take casualties, yet we'll annihilate all opposition and proceed forward as fast as possible to our objective.\"\n\nThe president smiled at that comment. He apparently liked to think of his opposition being annihilated. Then the smile faded. \"When?\" he asked.\n\n\"It will take at least two days to get people sorted out and transferred to fill up our three assault divisions. Another four days to get our people and equipment to Barksdale, and another two days to get them under way. The Fort Carson division commander says he can get his division under way in two days, after he gets his personnel sorted out and is reinforced by willing fighters. That shuffling will take at least two days, maybe three. Then it will take another three days to get them to the Texas line. Seven days total. If anything slips, eight or nine.\"\n\n\"What will the rebels be doing while we are getting our show ready to go on the road?\"\n\n\"Making a nuisance of themselves and getting ready to block our moves.\"\n\n\"How will they know what we intend?\"\n\n\"Texas' commander is JR Hays, Jack Hays' cousin, and he was a career army officer, although retired now. He could probably write our op order. If he hadn't burned out in the Middle East and retired, he would have become a general. I've seen his service record. JR Hays is a soldier from head to toe, and he doesn't shrink from combat. He's seen more than his share and knows precisely how to fight. And how to win. The Taliban had a price on his head: ten thousand American dollars. No one in the Middle East was able to earn it.\"\n\n\"Can you whip him?\"\n\n\"The United States Army can.\"\n\n\"Eight days to get combat troops into Texas,\" Al Grantham stated. \"Or nine. Or ten. Or eleven. That's too long. Can't we use fewer troops and go sooner?\"\n\n\"Even if we cut our invasion force to only one division, we will only save one day,\" Wynette said flatly. \"So the tradeoff is one division that can possibly be surrounded and cut off, or a two-pronged assault that will force the Texans to divide their forces to fight them both. In my military judgment, and the judgment of the Joint Chiefs, if we are going to hit Texas with a hammer, it should be a really big hammer, as big as we can put together in a reasonable amount of time. Given a month, we could hit them with every American soldier willing to fight.\"\n\nThe president nodded his agreement.\n\nGrantham asked, \"And how many is that?\"\n\n\"I don't know yet,\" Wynette said. He continued, \"There are around twelve thousand fighting soldiers in a division. We'd like the divisions at full strength, if possible. In addition to their weapons, artillery, air support, food, and ammo, we must also move all the support equipment and manpower required to keep the warriors eating, sleeping, and fighting, the planes and choppers flying, the artillery supplied with ammo, and enough extra stuff to provide humanitarian relief. We are doing all we can to get this organized and moving, which is everything humanly possible.\"\n\n\"So we sit on our asses for eight days and wait,\" Al Grantham summed up.\n\nWynette gestured to the briefer, who went on with his presentation. JCS envisioned beginning air operations against Texas tomorrow. The targets would be all the surrendered military equipment at the military bases. Missions would be flown by B-52s escorted by F-16s during the day and B-1s at night targeting Fort Bliss, the Texas Guard armories, and other military targets. The navy can bring an aircraft carrier around Florida and begin air operations in two days against the military bases around San Antonio and Killeen. \"Our goal,\" the briefer summed up, \"is to attrite their armor and air assets by seventy-five percent by D-Day, which is the day we plan to cross the Texas border.\"\n\n\"Why not hammer their industry, their refineries, and factories and power plants?\"\n\n\"Those are legitimate strategic military targets,\" Major General Strong said, \"but the primary goal of the air campaign must be the destruction of the enemy's combat power\u2014the opposition we'll face when we put boots on the ground. After we knock out their combat power and render it impotent, then we can bomb strategic targets.\"\n\n\"But before we do that,\" interjected Wynette, \"you must decide how much of an economy you want standing after we take over. If everyone is destitute and starving, the assets to feed them and rebuild Texas must come from the rest of the United States.\"\n\n\"Just the military targets,\" the president said. Then he added, \"Unless this invasion gets bogged down. If push comes to shove, we are going to win if we have to flatten every building and kill every cow in Texas.\"\n\n\"Yes, sir,\" General Wynette said.\n\nBarry Soetoro leaned forward in his chair and looked straight into Wynette's eyes. \"I expect you to crush the rebels, General. If you don't, don't come back alive.\"\n\nIt was the second time that the president had told him that, and though Wynette had kissed ass for a lot of years, he was fed up with Barry Soetoro. \"Mr. President, if you don't think I can win, fire me and get a general you think can. The army has plenty of experienced combat leaders for you to pick from.\"\n\n\"You're the man I want,\" Soetoro shot back. \"I _know_ you'll obey orders.\"\n\n\"And you think these others might not? What kind of orders wouldn't they obey?\"\n\nSoetoro's eyes were locked on Martin Wynette. \"We'll cross that bridge when we get to it,\" he said.\n\nWynette was the first to look away.\n\nBack in the air-conditioned Pentagon, Wynette had another bad moment. The staff had framed the loyalty question to the troops as \"Are you willing to fight for the United States of America to stamp out a rebellion?\" _Yes_ or _No_.\n\nLast night on television he saw commentators talking about \"Barry Soetoro's army\" versus Texas. Wynette thought\u2014and he knew many of his troops thought\u2014there was a huge difference between \"Barry Soetoro's army\" and \"the United States Army,\" and the more commentators talked like that, the more desertions he would have.\n\nThe Joint Chiefs assembled in his office. They wanted to know their role in putting down the riots that were raging in the big cities.\n\n\"Forget about that for now. That doesn't seem to be the president's priority,\" Wynette replied. \"He seems to think that if he squashes Texas, all his other problems will go away. However, in fifteen minutes Grantham may call and want us to invade Detroit.\"\n\nWhat he didn't say, although he thought it, was that the president and his staff were fixated on the wrong problem. In his years of service he had served on joint staffs on numerous occasions and knew it was the job of a commander to define the priorities and keep his staff focused on them. Wynette thought Barry Soetoro didn't understand what his problems were or couldn't prioritize; if either was the case, he was incompetent. As the general saw it, the primary problem in America just now was that civil authority in much of the nation was about to collapse. It wasn't just Texas that the president might lose, it was America.\n\nWhen JR Hays arrived in Austin that afternoon, he headed straight for the capitol and was ushered into the governor's (now the president's) office. He waited in a corner while some politicians briefed Jack.\n\nSeveral thousand people a day were pouring into Texas from other states. Many of these people said their extended families, neighbors, and coworkers were only a day or two behind them. More people were coming, a lot more, and they would need housing and jobs. After the politicians had spent ten minutes discussing how the flood of refugees might be accommodated temporarily, Hays shooed them out and locked the door. He and JR sat in chairs facing each other.\n\n\"We've had some good luck,\" JR said, \"because a lot of the people in the army and air force refused to fight for Barry Soetoro. Any commander in that position would have had to surrender. Still, those services are going to find people who _will_ fight for Soetoro, and then the shooting will begin in earnest.\"\n\n\"So what's your plan?\"\n\n\"We can't sit here waiting for Soetoro to hammer us. I would bet my soul they are plotting to do that right now in Washington. If Soetoro lets us get away with leaving the Union and setting up as an independent nation, other states will do it too, one by one, and eventually he won't own anything but the federal district in Washington. He can't let that happen. He has to whip us, and he has to do it as soon as he can assemble the forces to do it with. Every day he doesn't win is a victory for us. If we can pile up enough little victories, we can win the war.\"\n\nJack Hays nodded. He was delighted JR was thinking about all this, because he hadn't had time. Politics was his business, not the military.\n\n\"We need to seize the initiative and force Soetoro's military forces to react to us. We need to put them on the defensive, derail _their_ plans. And, if possible, we need to move the fight out of Texas; we want the front line to be somewhere else, not here.\"\n\n\"So how do you propose to do that?\"\n\nJR Hays began explaining his plans.\n\nWhen he had finished, Jack nodded.\n\n\"We're going to have a lot of civilian problems to deal with,\" JR added. \"Soon the enemy will target our power plants. Houston and Dallas and the high-rises all over Texas will instantly become uninhabitable. We need to get organized now to take care of what will become a huge humanitarian crisis.\"\n\n\"Houston and Dallas are using school buses to evacuate all the people stranded at the airports,\" the governor mused, \"so we have a start, anyway. I'll get our emergency people involved.\"\n\n\"In my view, Jack, your number one job is to buck up the people of Texas with your courage and determination to see this through to the end.\"\n\n\"My courage? How much do you think I have?\"\n\n\"Enough, or we're doomed. Leaders must lead. But the more effective you are, the more likely Soetoro will send assassins or commandos to take you out. If they kill you, the backbones of a lot of people will soften. Get some bodyguards, and good ones. Use them.\"\n\n\"The Texas Rangers,\" Jack Hays decided.\n\n\"Your call. But they must keep you alive. That said, you and the Congress need to get out of this building and set up at an undisclosed location. I suggest somewhere underground, like a parking garage under a hotel. Right now a handful of cruise missiles into the capitol would decapitate the new republic. No doubt Soetoro is thinking about that right now, trying to figure out what the repercussions of a mass assassination would be on his political base up north.\"\n\n\"We'll be out by five o'clock.\"\n\nJack Hays nodded, stood, and shook hands. JR left. He had a ton of things to do, all of which needed to be done at once. Or yesterday.\n\nHis jailers came in midafternoon and tossed a plastic water bottle on the cot. Jake Grafton was still on the floor. One of the jailers came in and kicked Grafton in the ribs, repeatedly.\n\nThe man who delivered the water stopped the kicker. \"Don't kill him. Sluggo wants him alive.\"\n\nGrafton was still conscious. His ribs were on fire. If one of the broken edges penetrated a lung, he would die quickly. If he started coughing blood, he would know.\n\nSteeling himself, he moved. The pain was searing. He managed to reach the water bottle.\n\nThe plastic cap almost defeated him. He had to open it with his teeth. After he drained it, he lay back on the floor. And passed out.\n\nAs it happened, the White House political staff was indeed trying to estimate the damage decapitating the Republic of Texas would cause in the president's political base. Would it fuel insurrection elsewhere? There were no easy answers, so the staff was having a wonderful time wrestling with these imponderables, preparing a list of options for the chosen one.\n\nWhile staff was staffing, Barry Soetoro signed an order temporarily closing all stock and commodity exchanges. Since the power was off in New York and Chicago, this order wouldn't create much of a sensation today, but it would when the power came back on. Tomorrow, perhaps. Or the next day. No one knew when the juice would again flow. The power company execs were doing all they could, they told his staff. One of them darkly hinted at sabotage, but Soetoro wasn't buying that excuse. One natural-gas trunk line in Virginia had been severed, stopping the gas flow through that line, yet that amount was only a drop in the bucket. Sheer damned incompetence, he thought angrily. One of these days he was going to have to nationalize all the public utilities, replace the executives with reliable people.\n\nHis thoughts turned back to Texas. Using cruise missiles or JDAMs on all the power plants in Texas was certainly an option. In August and September it was a lot hotter in Texas than it was in most of the Northeast. Texas was closer to Hell.\n\nHad Soetoro been a fly on the wall in Jack Hays' office that afternoon, he would have been delighted at what he heard. A delegation of oil and gas and refinery executives had called in a body upon the new president. Many rich men and women value predictability and stability above all else, so this crowd had been cold to the idea of independence from the start, hinting strongly, as it did, of civil war. There is a lot of money to be made in war, but people with multibillion dollar capital investments in the line of fire wouldn't be making much of it. If anything, they stood to have their investments wiped out.\n\nJack Hays listened patiently to the executives.\n\n\"Mr. President, the fact is that any damned fool with a machine gun or a few sticks of dynamite could take down the refineries on the southeast coast of Texas.\" Even if that didn't happen, the feds could destroy the refineries and oil storage tanks by air attack. And the U.S. Navy could prevent tankers carrying foreign oil from discharging their cargoes, restricting supply to what Texas could produce, which in fact was a hell of a lot, since Texas was the biggest producer of hydrocarbons among the fifty states. Still, guerillas or federal forces could stop the flow of natural gas and gasoline out of Texas, destroying their markets. All together, the group spokesman said, the picture was \"bleak.\"\n\n\"What assurances, Mr. President, can you give us that the armed forces of the Republic can protect our facilities?\"\n\n\"Very few, gentlemen,\" Jack Hays said. \"In fact, I was thinking of asking you to stop pumping oil and gas to the northeastern United States and California. That would bring a significant amount of political pressure to bear on the Soetoro administration.\"\n\nThe executives were horrified. Such an action would cut off their cash flow, and it would mean that many of their facilities would have to shut down, oil and gas wells would be shut in, people would be laid off, and money would cease to percolate through the economy.\n\n\"Do you realize, sir, how many people make their livings directly from the petroleum industry in Texas? And twice that many indirectly. You are talking about a depression, _millions_ jobless.\"\n\nAnother said, \"The people of Texas didn't sign up for _that_!\"\n\nJack Hays refused to be riled. \"I think the people of Texas knew that they would have to fight for their independence when the idea was first discussed. In a war one stands to lose not only his livelihood, his home, and everything he owns, but also his life and the lives of his family. Texans aren't stupid; they knew that. They were for independence anyway, if that meant they could preserve the benefits of a free society with a representative democratic government that they had enjoyed all their lives, benefits they hoped to again enjoy, benefits that would be their legacy to their children and the generations of Texans still to come. They were willing to pay the price. Or most of them were, anyway.\"\n\nEveryone tried to talk at once, but Jack Hays silenced them with a gesture.\n\n\"We have taken a political step that cannot be reversed,\" he said.\n\n\"Of course it can be reversed,\" a big oil executive said loudly, to drown out other voices. \"It's time to make peace with Soetoro. You've made your political points, Hays. Now let's settle this mess and get on with business.\"\n\n\"Talk loud, then surrender. Is that your advice?\"\n\n\"Now see here. That isn't what I said. Oil and gas are the heart of Texas industry. Hell, of American industry.\"\n\n\"Gentlemen, thank you for your time today,\" Jack Hays said. \"I will carefully consider all your points. For my part, I'm glad that Colonel Travis had men with him at the Alamo who had more backbone than you have.\"\n\nOne of the executives\u2014dressed in an Armani suit, hand-tooled alligator boots, and a two-hundred-dollar silk tie\u2014snarled: \"Travis didn't own a goddamn thing but his horse and some worthless scrub land. If he had owned something he'd have been a bit more careful.\"\n\nAnd you'd be speaking Spanish and working for Petromex, Jack Hays thought savagely. He didn't say that, of course. What he did say was, \"I will meditate upon that insight. Now if you ladies and gentlemen will excuse me. . .\"\n\n\"How much time do you think we have before the electrical wizards figure out what you did to their computer, and fix it?\" I asked Sarah as we rolled into the West Virginia panhandle.\n\n\"I have no idea,\" she said distractedly. She was looking at the huge towers carrying their high-voltage transmission wires across the countryside.\n\nWhen I realized what she was looking at, I said, \"Are you thinking what I'm thinking?\"\n\n\"There's no power on those wires right now. If we could lay down some towers, they wouldn't know we did it until the grid came back up. People who see it fall can't even call in.\"\n\n\"You are a natural-born terrorist,\" I acknowledged.\n\nI pulled over to the side of the road and the pickups pulled up behind me. I got out, and we all huddled over a roadmap. \"Here is where we're going, Camp Dawson, near Kingwood, West Virginia, in Preston County. I thought we would stay off the interstates and do the back roads. But along the way, I'd like to take down some of these transmission towers. Two or three on each right of way, to put the wires on the ground. Use C-4, set the timers to the max on the dial.\"\n\n\"That's an hour,\" Armanti Hall said.\n\n\"You guys can drop off, do a couple of towers, then catch back up. Try to make them fall in the woods or streams, not on the road. We'll meet here.\" I jabbed my finger at a crossroads, near Kingwood.\n\n\"Okay by me,\" Travis said. The others nodded their heads.\n\nI went back to the van and climbed in.\n\n\"I could use a bathroom,\" Sarah said.\n\n\"The side of the road is brushy,\" I pointed out. \"No one will see you. Climb on down there.\"\n\n\"I don't have any toilet paper.\"\n\nI reached around the seat to my duffle bag, extracted a roll, and passed it to her. \"I stole a roll of yours this morning when we were leaving.\"\n\nShe scanned the roadside weeds, then observed, \"There might be poison ivy or snakes.\"\n\nI started the engine and got the van rolling. \"Maybe we'll find an open filling station with clean restrooms,\" I said brightly, \"or even a McDonald's.\"\n\n\"Jerk.\"\n\nShe ended up using a port-a-potty on a road bridge rebuild project. The construction crew wasn't around. After she finished, I used it too. It smelled like every port-a-potty I'd ever been in, but it was like the facilities at the Ritz compared to the places I had pooped in the Middle East, often merely a hole in the floor you squatted over. Or a patch of desert. The miracle of toilet paper has not yet been revealed to most of the sons of Islam. Muhammad never said a word about it. If you don't believe me, read your Koran.\n\nWhen we were back rolling again, I told her, \"There may come a day when you dream longingly of that port-a-potty.\"\n\n\"Did you see the graffiti in there?\"\n\n\"Yes.\"\n\n\"Men are such pigs.\"\n\nI let that one go by without comment.\n\nA little while later Sarah began to laugh.\n\n\"What's so funny?\"\n\n\"Oh, I was just thinking about the irony of it all. Jake Grafton and I have been listening to the goings-on at the White House for about six months. He knew all about Soetoro's plan to declare martial law and tear up the Constitution.\"\n\nI stared at her, trying to decide if she was telling the truth. And almost ran off the road.\n\nShe chuckled. \"He refused to do anything about it, of course. Said there was nothing he could do. And maybe he was right. If he told people about Soetoro's plan, they would have thought him crazy. It would have gotten back to the White House, and they would have arrested him and locked him up. So he decided to do nothing and he got blamed for a fake coup and assassination plot and he's locked up anyway. Life is crazy.\"\n\n\"Tell me more.\"\n\n\"Only Grafton and I know about it. When the Iran treaty was being negotiated in Switzerland, he asked me if we could bug the hotels where the delegates were staying. He wanted to know what the Iranians were talking about, what their negotiating strategy was. The problem was that the hotels were going to be swept repeatedly, and any bugs with transmitters would be quickly discovered. So I ginned up a program to use all the hotels' computers and security systems as listening devices and have the feed sent to me over the internet.\n\n\"But when I got into their systems, I found that the Israelis had been there first. They had a surveillance system in place using the computers and security cameras and even the personal computers that everyone brought with them and that used the hotels' Wi-Fi systems to connect with the internet. You may have heard about it last year. The Russians had the same idea, and they announced the Israelis' espionage.\"\n\n\"I did hear about that.\"\n\n\"The Israeli system was better than mine. So I got all their computer code and we just listened in.\"\n\n\"Jesus,\" I said, trying to think as I steered the vehicle. That Grafton!\n\n\"Then about six months ago, he asked if I could use the Israeli system on the White House.\"\n\n\"Jesus!\"\n\n\"He and I have been listening for six months. All the plotting, all the plans, all the bullshit. But he wouldn't do anything about it.\"\n\n\"You are saying he knew about the coming of martial law?\"\n\n\"Oh yes. He and I knew. They were merely waiting for an excuse. They thought the excuse would be a domestic terror incident, but if that hadn't happened, it would have been something else. Martial law was going to happen. We were the only ones who knew outside the inner circle at the White House. I demanded the admiral do something, but he just gave me those cold gray eyes and asked, 'What?'\"\n\nIndeed, I thought, what? Whom do you tell? Who will believe?\n\n\"So here we go, riding to the rescue,\" she said sourly, \"and he knew all along.\"\n\n\"So did you.\"\n\n\"Yeah. I had to agree with Grafton. What do you do when the president is plotting to become a dictator?\"\n\n\"Assassinate him,\" I suggested.\n\n\"Who? Me? Grafton? Or should Grafton have sent you to do the dirty deed?\"\n\nShe had a point.\nEIGHTEEN\n\nThe Blackhawk helicopter settled onto the tarmac at the Longview, Texas, airport, shut down, and JR Hays went forward to speak to the pilots. CWO4 Erik Sabiston was in the right seat.\n\n\"Wait for me,\" JR said. \"Be back late this evening. Fuel the chopper and get something to eat.\"\n\n\"Yes, sir.\"\n\nJR climbed out and walked across the tarmac into the FBO. \"I need a car,\" he said to the lady on the desk.\n\n\"We have a courtesy car, sir. It's kinda old and wrinkled, like me, but it'll probably get you there and back again. Always has so far, anyway.\"\n\nShe handed him the keys and he made a pit stop, then went outside and climbed in. It was an old Ford with sun-scorched paint and more than a hundred fifty thousand miles on the odometer. It started on the first crank.\n\nHe had gotten the address from the Texas Department of Motor Vehicles. It took him a while to find it that afternoon. There were high cirrus clouds up there, making the afternoon light gauzy. It didn't do much to soften the heat, though.\n\nJR found his address in a newer subdivision, parked on the street, and walked up the driveway. Inside he heard a dog barking, a little one from the sound of it. Rang the doorbell.\n\nIn a moment a man in shorts and an old army T-shirt opened the door, a man in his mid-fifties.\n\n\"JR Hays! As I live and breathe!\"\n\n\"Hello, Nate. May I come in?\"\n\n\"Of course.\" The man threw the door wide, then closed it behind JR. His name was Nathaniel Danaher, and he was a retired army colonel with thirty years service. JR had served under him on his last combat tour in Afghanistan. Danaher was from Connecticut originally, but he hadn't lived there since he went away to VMI for college. He hadn't been able to score a West Point appointment so he joined the VMI corps of cadets, got a reserve commission, which, after a few years of outstanding service, the army transformed into a regular commission.\n\n\"I like the gleam of those stars on your blouse, JR. Somehow they look exactly right on you. Want a beer?\"\n\n\"Sure.\"\n\nWith beers in hand, they sat on the covered porch in the backyard, a _ramada_ as the old Texans called it. It kept the sun off and allowed the people sheltered under it to savor any breeze. The dog, some kind of terrier, was friendly enough. He did some exploratory sniffing and then found a shady spot to lie down.\n\nDanaher was still lean and fit. He looked, JR thought, exactly as he had when he was in Afghanistan, only a little older and grayer. JR remarked on it.\n\n\"Still get up at five o'clock every morning and run five miles,\" Danaher said. \"Might as well; can't sleep past five anyway. Heard your cousin put you in charge. He couldn't have found a better man.\"\n\n\"That remains to be seen. Where do you stand on independence?\"\n\n\"Well, when I first heard about it, I thought, there goes my fucking pension and health benefits unless I get the hell out of Texas. That was pretty small of me, I suppose, but then I heard on TV that Texas is taking over all the federal government's obligations to military and Social Security retirees, so that was a relief. I've got some money saved up but nowhere near enough without a pension. I despise that son of a bitch Soetoro and everything he stands for. It's a big club so I have lots of friends. Independence is great if you folks can make it stick, because the country that elected that bastard twice is going somewhere most people in Texas don't want to go.\"\n\n\"I need some help,\" JR said. \"I need some civilian duds, and then if you are willing, let's the two of us drive over to Louisiana and take a look around.\"\n\n\"You mean it?\"\n\n\"I do.\"\n\n\"My wife is playing bridge this afternoon. Went over after lunch. I'll leave her a note. We'll be back tonight?\"\n\n\"I hope.\"\n\n\"I think I may have some clothes that will fit you. If you haven't had lunch, mine the refrigerator while I root around. Make yourself a sandwich or something. Last night's meat loaf was pretty good.\"\n\nJR was halfway through a cold meat loaf sandwich when Nate returned with a pair of baggy shorts, an ancient VMI T-shirt, and a set of worn tennis shoes. He also handed JR a pistol, an old double-action revolver, small and trim. \"If you're going to Louisiana you better take this, stick it in your pocket, just in case. It's loaded.\"\n\nJR checked the cylinder, snapped it back in place. The gun was an old Smith & Wesson in .38 Special with about half its bluing remaining. \"That thing's about ninety or so years old,\" Danaher said. \"Used to carry it in my pocket when a service pistol wouldn't do. Louisiana is enemy territory for you.\"\n\nNate Danaher's car was a late model sedan. \"Where are we going?\"\n\n\"Barksdale Air Force Base, east of Shreveport and Bossier City.\"\n\n\"I know where it is. Take Gina to the doctor there on a regular basis. She's got lymphoma. It's under control now, we think, but . . .\" he shrugged, \"it's in God's hands. I shop at the PX while she's getting examined.\"\n\n\"Stay off the interstate tonight. Take the back roads. We don't need to run into a roadblock.\"\n\n\"Sure.\"\n\n\"Got that postcard from you a while back,\" JR explained. \"So I knew you were in Texas. Why here?\"\n\n\"Our daughter is here. Her husband is an engineer in the oil business. Gina wanted to be near the grandson, Little Nate, who just turned seven. He's a pistol.\"\n\n\"I seem to recall you had a son, too.\"\n\n\"Yep. Got on drugs in high school and dropped out. Pot at first, then crack, then heroin and meth. We put him in rehab twice, but it didn't take. Haven't seen him in. . .well, it's been twelve years now. A few years ago someone said they saw him in New Orleans, living on the street. For all I know he may be dead now. All those drugs\u2014it figures he won't last too long.\"\n\nJR changed the subject. \"So how is Longview taking to independence?\"\n\n\"Was out at Walmart today. The place was packed. People on welfare were cashing their last checks, loading up their cars, and getting out of Texas. They heard Texas isn't paying welfare anymore, so a lot of them are heading for greener pastures. Everyone else is stocking up. Everything they can get, food, toilet paper, everything. People in line said the liquor stores were mobbed. I wanted to buy a little generator\u2014figured I could wire it into the house circuits some way\u2014but Walmart was out of them. None in the hardware stores. People sense that times are going to get hard.\"\n\n\"Yeah,\" JR said dryly.\n\nSarah and I drove the van along the road by Camp Dawson and sure enough, there was the compound that held the detainees, though we didn't see any. The compound, surrounded by a chain-link fence topped by barbed wire, with guard towers about ten feet above the ground on all four corners, was about a hundred yards on each side. It was lit up in the late afternoon like Macy's on Christmas Eve, so obviously they had generators going. All the comforts. . .\n\nThe gate was manned by four guys in FEMA dark-green coveralls carrying carbines and wearing green caps. They weren't soldiers, lounging around like that, smoking, laughing, and grab-assing. And, I suspected, they were not well disciplined. No army sergeant I ever met would allow his troops to goof off on guard duty. They were armed thugs.\n\nI got all this on one slow drive-by. The gate guards paid no attention to us. The guy on the last guard tower was leaning on the rails of his perch, smoking a cigarette, looking into the compound.\n\nWhich made me suspect that they weren't worried about people breaking in, but their prisoners breaking out. The thought that someone might assault them with intent to kill apparently had not entered their hard little heads. When the shooting started in earnest, many would probably boogie. No one wants to be dead any time soon, which can happen when people shoot at you.\n\nAcross the road from the compound was an up-sloping pasture, maybe fifty yards wide, with what looked to me like yearling steers in there munching grass. Maybe dreaming of the girlfriends they would never have. Perhaps those were the virgins the jihadists would find in Paradise. Beyond the pasture and higher was a strip of forest on a low ridge. Over the top of the ridge I got a glimpse of a green mountain.\n\nI kept on driving, thinking about how we could pop Jake Grafton out of that compound. Since we had no idea where in there he was, we were going to have to ask someone. That would be my job. I am pretty good at getting answers in a hurry from people who initially thought they didn't want to be bothered.\n\nThe designated rendezvous was a crossroads about eight more miles along. I pulled over to the side of the road and turned off the engine. The sun was just setting, so we had at least another half hour of evening, then maybe another fifteen minutes of twilight.\n\nAll I needed were my troops.\n\n\"You know how to use that pistol?\" I asked Sarah.\n\n\"Never fired one in my life.\"\n\nI showed her how the Beretta worked, popped out the magazine, jacked out the shell from the chamber, made her dry fire it, and put everything in and reloaded. \"Just disengage the safety, point, and pull the trigger. It will fire thirteen shots, one with every squeeze of the trigger. The gun will kick in your hand, so use both hands. Don't use it unless the bad guy is very close, and keep shooting until he's dead on the ground. Not wounded on the ground, but obviously dead, so he can't hurt you.\"\n\n\"Okay,\" she said, hefting the weapon.\n\n\"Never point a gun at a man unless you are willing to shoot, and never shoot unless you are willing to kill. This isn't Hollywood.\"\n\n\"Okay,\" she repeated, and holstered the weapon.\n\nI felt better. She seemed to be getting into this warrior gig. If I could just keep finding her bathrooms or port-a-potties.\n\nI rooted in my duffel and came up with my Kimber 1911 in a holster. I added it to my web belt and put it on the right side. On the left I put my Marine Corps fighting knife with the eight-inch blade.\n\nThe Beretta was a 9-mm: it shot a .357-caliber, 125-grain full-metal-jacket bullet since it was a weapon of war\u2014Geneva Convention and all that\u2014and would make nice holes in people. Magazine capacity was thirteen rounds. The .45 shot a 230-grain bullet, and I used hollow points. Under fifty feet, one of those to the body would kill King Kong. It held eight cartridges, but if eight wasn't enough, I was probably gonna soon be dead anyway.\n\nI made sure my shooter was cocked and locked, then sat there wondering where my troops were. Civilian cars and pickups came by from time to time, and after a glimpse of my FEMA green, ignored us. Apparently the boys in Soetoro's army were not yet winning the hearts and minds of the locals. I glanced at my watch from time to time.\n\n\"Stop fidgeting,\" Sarah said.\n\nI loaded up some M4s, passed one to Sarah, and laid a couple behind the passenger seat where I could reach them. Broke out some grenades and put one in each shirt pocket.\n\nFinally I got a couple of boxes of MREs and dug through them. Sarah took a fruit cup, and I munched a cardboard cookie that had come out of the oven during the first Bush administration. We certainly weren't in danger of gaining weight on this adventure.\n\nBefore they went onto the base, JR Hays and Nate Danaher stopped at a beer joint, which was packed, every stool and booth full, with people standing and drinking beer. The conversations were loud. A television was on up in the corner, showing the devastating effects of the power outage in the northeastern United States. Philadelphia and Baltimore were rioting as usual.\n\nJR kept an eye on the television as he waited for Danaher to work his way to the bar and order beers. There was a short segment about rioting in Watts in LA, then a parade of Soetoro administration officials being interviewed. JR couldn't hear the audio, but he thought he knew what the officials were saying. Everything was under control. The administration was taking steps, and so on.\n\nThen he heard a snatch of a conversation between two men at the bar. \"This place is going to be packed when those soldiers get here. . . . Yeah, I heard the day after tomorrow. . . . Someone said the Fourth Brigade. . . . Gonna come in dribs and drabs, I suppose. . . . Thirty-five hundred men and equipment is a lot to move. . . .\"\n\nThe Fourth Brigade Combat Team of the 10th Mountain Division. JR knew about them. The fact that they were being deployed from Fort Polk, the massive joint training base further south in Louisiana, to Barksdale was certainly news. There was also a brigade of airborne troops at Fort Polk, JR thought, and he listened intently to see if the garrulous bar buddies knew about them. A brigade of paratroopers dropping into Texas, or Barksdale, could cause massive problems.\n\nHe wandered on, listening. Most of the men and women in the bar were talking about the run on grocery stores and Walmarts. The lines were horrendous. One woman said she waited over an hour in line to check out. One gasoline station was completely out of gas and the clerk said they didn't know when they could get more.\n\nAfter they drank some of their beer, JR and Danaher left the mugs on the bar and went outside. Their retired military ID cards got them onto the base. They drove over by the flight line and looked at the rows of B-52s parked there. Barksdale was home to the 2nd Bomb Wing, the only outfit in the air force that still had B-52 Stratofortresses.\n\nHuge hangars, flood-lit ramps, here and there a security vehicle. Half-full parking lots. Activity at the barracks.\n\nThe parking lots at the commissary and PX were packed, with almost every space occupied. A long line waited to get to the fuel pumps at the base filling station.\n\nJR told Danaher about the conversation he had overheard.\n\n\"That's no surprise,\" Nate replied.\n\n\"I want you to lead an assault team in here tomorrow morning. We need to take this base and be prepared to hold it. If we can't, we need to destroy those B-52s. Can an assault team arriving on C-130s pull it off?\"\n\n\"Let's go back to the flight line and take a look,\" Danaher said.\n\n\"If it can't, we can do an air attack tomorrow,\" JR explained. \"Strafe the flight lines, drop some JDAMs on the hangars and fuel farm, make a royal mess.\"\n\n\"Hold that thought. I have a small set of binoculars in the glove box. Let's trade places, and I'll look while you drive.\"\n\nThey did so. The only plane in the traffic pattern was a B-52 shooting landings, apparently on a training mission. They could hear the engines roar every time it lifted off and watch it in the pattern, a big dark-green metal cloud.\n\n\"They're not bombing up the BUFFs,\" Danaher said after a while. \"No missile batteries or missile-control radars or AAA in sight.\" AAA was anti-aircraft artillery. Five more minutes of looking, then Danaher said, \"Let's go home. We've seen all that there is to see.\"\n\nSluggo Sweatt had Jake Grafton brought to his office that Tuesday evening. Grafton couldn't walk, so the jailers dragged him. They didn't bother putting him in a chair. Sluggo came around his desk and rested a hip on the edge of it and looked down at Grafton lying on the floor. Sluggo had a smile on his face.\n\n\"How are your ribs?\"\n\nGrafton tried to focus. Being dragged here had made him want to scream, so he had bitten his tongue. Now blood was leaking out his lips. He could feel it, warm and slick.\n\n\"I think we'll take you back to your cell and let you sleep through the night. If tomorrow you don't sign the confession in front of a television camera and read the little script we have prepared\u2014it's only about a hundred words\u2014we'll beat you to death tomorrow night. The other prisoners will hear your screams. I'll be honest, Grafton, I don't like you. Still, I urge you to be tough. Don't give us an inch. Then I will have the pleasure of helping the boys work on you.\"\n\nSluggo Sweatt smiled at Grafton. He picked up a sheet of paper from his desk, fluttered it, then handed it to one of the thugs and made a gesture. They dragged Grafton back to the cell. There they threw the sheet of paper on his chest and left him lying on the floor, after one of them had kicked him in the balls.\n\nThe overhead lights were on. Although Jake Grafton didn't know it, the power for the camp was being supplied by several large emergency generators since the grid was down. With the generators snoring away, grid problems didn't really matter to Sluggo Sweatt. He was the king of his own little empire, and he liked the feeling.\n\nEvery breath Grafton drew was agony. When the fierce pain in his testicles finally subsided to a dull ache, exhaustion overcame him and he went to sleep. He dreamed of Callie.\n\nArmanti Hall and Willie the Wire showed up first. I got out of the van and Willie started motormouthing. \"Damn, Tommy, did we have fun! You should have seen those towers come down. Man, if someone would pay me for doing this, I'd give up the locksmith business in a heartbeat.\"\n\nI didn't have the heart to tell the fool that he was probably permanently out of the locksmith business unless a meteor hit the White House and all its inhabitants were instantly obliterated.\n\n\"How'd it go?\" I asked Armanti Hall.\n\n\"We dumped towers on two different transmission lines. Just walked up to them, rigged the charges, and went on to the next one. We watched one stretch of them go down. Some of the lines broke, and the others went on the ground.\"\n\n\"You two get some MREs, take a whiz, and when the other guys get here I'll brief everyone.\"\n\nTen minutes later Travis Clay rolled in, and five minutes after that Willis Coffee. They had each found a transmission line and put three towers on the ground. Travis, however, had done more. He came across a substation and used an AT4 to put it out of business. \"That box blew apart into a thousand pieces, Tommy. It was kinda fun.\"\n\n\"I'll bet. You didn't leave the tube there, did you?\"\n\n\"Oh, no. It's in the back of the truck.\"\n\n\"Good man.\"\n\nI addressed my lock shop partner. \"Tell me, Willie, now that you are back in the felony business, are you willing to pull a trigger or not?\"\n\n\"Well. . .\"\n\n\"One life sentence, two, three, what does it matter?\"\n\n\"You're suckin' me into a life of crime, Carmellini. I'm not ready to give up pussy. I got a few good years left, dude, me and Viagra, and a couple of women who are countin' on me to help them find a little joy in this colorless life.\"\n\n\"Sarah, would you put a first aid box in each truck while I brief these guys?\"\n\nWe gathered around the hood of one of the FEMA pickups. I spread out the map. \"Here is where we are going to rendezvous, this bridge over the Greenbrier at Bartow. Then we'll go to the CIA's safe house near Greenbank. I want each of you to go to Bartow by a different route.\" We traced routes with fingers in the twilight.\n\nThen I explained the setup at Camp Dawson, how the internment compound was laid out, where the four guard towers were.\n\n\"Now, Sarah and I are going to drive in the main gate of the internment compound in a FEMA pickup. We'll want to find out where Grafton is being held. We'll ask to see the commandant of the camp. Meanwhile Armanti and Willie Varner, you will go through the main gate of the National Guard base and come around behind the compound. That gate was open when I went by and the Guard looked like it had moved out. Set up an M279 machine gun out back. There is undoubtedly a rear gate through the compound wire, and maybe a barracks where these FEMA dudes are bunking.\n\n\"When the shooting starts up front, the guards in the rear towers are going to be trying to see what's happening, and from the way the camp is laid out, I don't think they can see. They might get interested in you. If they do, open fire. If the FEMA guys stream out the back gate after the shooting starts, let them all get out. Wait until they are out, then kill them quick and fast, including anyone left in the rear towers. If the fleeing guards go into a barracks, use an AT4 on it. If they get into vehicles, use the machine gun. It is imperative that no one follow us.\" I looked at Armanti and asked, \"Can you do that?\"\n\n\"These people aren't soldiers?\"\n\n\"Some of them might have some military experience, but now they're civilians. FEMA paramilitary thugs, Barry Soetoro's army. What we have going for us is surprise. We want them dead before they can figure out that they oughta shoot back. They aren't holy warriors: being a martyr for Barry Soetoro isn't on their bucket list.\"\n\n\"You're asking an awful lot of one man with one gun.\"\n\n\"Willie will help.\"\n\nArmanti looked at Willie Varner, who for once kept his mouth shut.\n\nI explained, \"I don't want the guards in the compound taking hostages, and I don't want them following us. If we can't take them down quick and fast, we're going to have to clean that camp building by building.\"\n\n\"Okay,\" Hall said, and shrugged. FEMA's reputation was going downhill fast.\n\n\"Willis and Travis, you guys are the front shooters. You are to wait one minute after Sarah and I go through the gate, exactly sixty seconds, then shoot the guys in the guard towers beside the road. They may have a machine gun in each tower, although I doubt it. But they might. Shoot each of them and toss a grenade up into the tower, then do the guys at the front gate.\"\n\n\"I'll take the south tower,\" Willis said, and Travis nodded.\n\n\"Then come into the compound. Drive through the compound and kill anyone in FEMA green. Try not to shoot any of the detainees. My idea is to let the guards get out of the compound through the back gate before we lower the boom. When the shooting starts out back, go help with the rear towers and anyone in FEMA green still standing. No FEMA people are to be left alive.\"\n\n\"Got it, Tommy.\"\n\n\"Wish I had a better plan,\" I admitted, \"and I wish we had a few days to sniff this out, but we don't have any more time. It's tonight or never. Any questions?\"\n\nWe cleaned up a few details, then mounted up.\n\nAnother half-assed plan with insufficient reconnaissance. That was a prescription to get my guys killed, as all of us knew, but it couldn't be helped. We didn't have days to set this up.\n\nSarah and I rolled up to the main gate of the compound in our brand-new stolen FEMA truck and I leaned out the window, which was down. I had my Kimber in my left hand, out of sight behind the door.\n\nThree guys were lounging around, two sucking cigarettes and one arranging a pinch of Skoal in his mouth. One of the smokers looked inquisitive.\n\n\"The guy who runs this place?\"\n\n\"Sluggo Sweatt.\" He pointed. \"That building on the left.\"\n\n\"Thanks.\"\n\nI rolled on over and parked in front. I holstered the Kimber.\n\n\"Sluggo Sweatt is on the White House staff,\" Sarah said.\n\n\"I've heard the name. Are you ready?\"\n\n\"Let's go in.\"\n\nWe turned off the engine, left the keys in the ignition, walked up the three steps to the porch and went inside. The receptionist's desk was empty, but the next room had a window and a desk with Sweatt seated behind it in an executive chair that he had apparently liberated from Office Depot. Sarah and I pulled our pistols and pointed them at him.\n\n\"See who else is in here,\" I told Sarah. As she went down the hallway looking in offices I scanned the room.\n\n\"You have precisely ten seconds to tell me where Jake Grafton is, or I'm going to shoot you.\" The words were no more out of my mouth than I heard M4s begin to fire bursts.\n\nSweatt looked startled. His eyes went to the windows. I fired a shot into his computer, and the bits of glass flew out. \"Pay attention,\" I said.\n\nI heard a shot from down the hallway. Then another.\n\nHis eyes were frozen on the pistol in my hand now. One of the interesting things about a .45 is how big the muzzle looks when it is pointed right at your eyes. Only a half inch in diameter, the hole in the barrel looks like a howitzer at close range. I lined up the sights and shot his right ear off.\n\nHe jerked and blood flew all over the wall behind him as a fusillade of M4 fire behind me filled the room with noise. Then a hand grenade went off. And another.\n\nSluggo got the message. \"He's in a cell, down the hallway.\"\n\nSarah came trotting back. I gave her the news.\n\n\"The keys?\"\n\nThey were on Sluggo's desk. Sarah grabbed them and ran. \"If he isn't there,\" I told Sweatt, \"I'm going to start shooting parts off.\"\n\nMore M4 bursts, a cacophony. Blood ran down Sluggo's neck and his face looked pasty.\n\nIn a moment Sarah was back. \"He's in terrible shape. A lot of broken ribs.\"\n\n\"You keep Mr. Sweatt occupied. If he twitches, empty your pistol into him.\"\n\nShe stood precisely in front of the desk and used both hands to steady the gun on his chest.\n\nI ran outside, grabbed a medic's pack from the bed of the truck, glanced at the gate and saw all three guards sprawled there. I ran back inside. If anyone shot at me they missed. Still some shooting going on. It would have been nice to know how many FEMA dudes we had strapped on, but we hadn't had time for an extended recon.\n\nI found Grafton lying on the floor in a cell, the door of which was standing open.\n\n\"Tommy,\" he whispered. \"Lots of broken ribs on both sides, I think.\"\n\nI cut his shirt off with my fighting knife. His sides were black and blue. Digging into the medic pack, I got out several rolls of gauze. \"I gotta sit you up, sir.\"\n\n\"Do it.\"\n\nI took his arms, which were bruised badly where he'd tried to cover up, and pulled him into a sitting position. He groaned. Working as quickly as possible, I wrapped him in gauze from his armpits down to his belly button. Needed three rolls to do it. Then I began wrapping him with surgical tape, as tightly as I could.\n\nA few more shots. I was listening for the sound of a machine gun, but I hadn't heard it yet. \"Who did this?\" I asked.\n\n\"Sweatt had it done. Wanted a confession. Said if I didn't sign, he was going to personally help beat me to death tomorrow.\"\n\n\"So we're right in the nick. You lucky dog.\"\n\nNow I heard the stutters of a machine gun.\n\nArmanti Hall had set up the M279 beside a small wooden building with a good view of the guard towers and the barracks. The fact that the only lights were in the compound and the towers were backlit probably helped. The guards, one in each tower, were looking into the light, watching the people in the compound and smoking. Armanti got the belt arranged in the gun and chambered a cartridge. When he had that attended to, he gave Willie Varner four hand grenades.\n\n\"I want you to go around on the other side of this building,\" Armanti said, \"where you can see the front of the barracks. Then put all four of your hand grenades on the ground. Wait until I fire, then pick up one grenade. See this pin on each one\u2014hold the lever, pull the pin, then wind up and throw it in from the outfield. Pick up another, pull the pin, and throw it. Do it until you have thrown all four. Then lay down, right where you are, and don't move a muscle until you hear me call your name. I don't want you running around out here in the dark. I'll be shooting at everything that moves. If anyone comes up on you, play dead.\"\n\n\"Okay, man.\"\n\n\"Can you do it?\"\n\n\"I guess.\" Willie Varner took a deep breath and exhaled explosively.\n\nFive minutes later the shooting started, and to Armanti's amazement, the man in the north tower climbed down and ran for the barracks. The man in the south tower wasn't far behind. Thirty seconds later, as gunfire popped in the front of the compound, guards in FEMA green came running through the compound toward the back gate, jerked it open\u2014apparently it wasn't locked\u2014and ran for their cars or the barracks.\n\nArmanti waited until no one wearing green wanted out of the compound, then opened fire.\n\nI heard the M279 open up, followed by grenade blasts. I hoped that was Armanti Hall behind the compound gunning every FEMA guard who had came out the back gate and jumped in a car or pickup. Or anyone who wanted out of the barracks to join in the fray, if there was a barracks back there.\n\nWhen I finished with the tape, Grafton said, \"Cut this jumpsuit off. I shit in it.\"\n\nI knew that by the smell, but was too polite to mention it. After I used my knife and he was naked except for the tape, I got a look at his swollen balls. They were bruised almost black. I helped Grafton to his feet. \"You're going to have to walk, Admiral.\"\n\n\"Give me a shoulder to hang on to.\" I put the medic bag over my shoulder, put my left arm around Grafton, and took an experimental step. He wasn't going to go down; that was one tough man. I drew the Kimber and led him down the hall.\n\nSilence had descended on the compound. Sweatt was still in his chair, holding his ear. Blood was oozing through his fingers and running down his neck, staining his collared shirt.\n\nGrafton paused in front of the desk and picked up a watch, put it on. Then he reached for a cell phone and handed it to me. He put a hand on my Kimber and I gave it to him.\n\n\"Sluggo, you were born eighty years too late,\" Jake Grafton said as he looked down to check the safety on the .45. \"You should have been an SS colonel in charge of Auschwitz or Dachau.\"\n\nHe pointed the pistol and shot Sluggo in the center of his forehead. The back of the man's head exploded onto the wall and his body rocked back in the chair. The corpse stayed in the chair, its arms dangling, its eyes pointed at the ceiling.\n\nGrafton handed me back my gun.\n\n\"Let's go, Tommy. Sarah.\"\n\nWe both helped him down the steps and into the right seat of the pickup. Then Sarah ran around and entered through the driver's door and scooted over.\n\nA knot of civilians was standing there. Willis and Travis were policing up weapons and tossing them into a stack in the yard.\n\nJack Yocke and Sal Molina came over to the right-side window, which was down. \"We want to go with you, Admiral.\"\n\n\"Get in the back.\"\n\nI addressed the crowd while Yocke and Molina climbed over the tailgate. \"Folks, your guards have skedaddled or died, I am not sure which. Help yourselves to the weapons. You must decide if you wish to remain here or take your chances outside. We can't stay, and you know they'll be back, sooner or later, when they figure out what went down here. All I can tell you is, good luck.\"\n\nI got in the pickup, carefully backed up, then put it in drive and steered toward the gate. I ran over a body of a FEMA warrior sprawled there because I was in no mood to get out and move the corpse or wait for someone else to do it.\n\n\"Who'd you shoot?\" I asked Sarah.\n\n\"A couple of men who thought I wouldn't.\"\n\n\"Good.\"\n\n\"This pistol doesn't kick as much as I thought it would.\" Oh, man! I glanced at her, but she was looking straight ahead at the road.\n\nThe breeze coming in the open windows felt good.\n\n\"Where are we going, Tommy?\" Grafton asked.\n\n\"A place I know. You need a vacation and Sarah needs access to a real bathroom.\"\n\n\"Where?\" he said. That was Jake Grafton. No nonsense at all.\n\n\"The CIA safe farm near Greenbank.\"\n\nHe grunted. Then his head tilted back onto the headrest and he was asleep, or maybe passed out. He had had a really bad time.\nNINETEEN\n\nCongressman Jerry Marquart was one of the civilians who watched Tommy Carmellini and the gunmen depart through the gate and down the road into the night. He recognized Jake Grafton, former CIA director, and Sal Molina, who was presumably no longer employed at the White House. The fashionably grizzled younger man who climbed into the back of the pickup with Molina he didn't know.\n\nJerry was in his late thirties. He was an ROTC grad, had spent six years in the Marines, had done the Afghanistan gig twice, and then had gotten out and gone into politics in Iowa. He was in his second term in the House of Representatives when FBI agents arrested him and brought him here. He didn't even bother to ask why. He was no friend of the Soetoro administration and denounced their policies at every opportunity. He actually had a lot of opportunities, because he was one of the very few members of congress with recent military experience. Or any military experience, for that matter.\n\nHe looked at the pile of carbines the attackers had left behind, walked over, and picked one up. Worked the action, checked the magazine, then went over to one of the bodies and helped himself to several full magazines.\n\nAnother man came over and asked him, \"You know anything about guns?\"\n\n\"A little.\"\n\n\"I'm from New Jersey, and I don't know shit about guns.\" He was about twenty-five pounds overweight, had saggy jowls, and combed his hair over his bald spot. He picked up a carbine and hefted it. \"But I don't think I want to stay here.\"\n\n\"Don't take one unless you're willing to use it.\"\n\n\"I'm getting there. Name's Evan Bjerki.\"\n\n\"Help yourself to some ammo,\" Marquart advised. \"The price is right.\"\n\nJerry Marquart went into the admin building and spent two seconds looking at the remains of Sluggo Sweatt. He had seen a lot of corpses so Sluggo's didn't affect him one way or the other. Nor did the two dead men sprawled on the floor of a room with cots and porn mags scattered around. He helped himself to a pistol belt that he had to pull off one of them, strapped it around his middle. He checked the pistol, a Beretta, made sure it was loaded, then moved on. The cell gave him pause. He smelled the feces, saw the jump suit on the floor, connected it to the naked Grafton, and walked back through the building and out into the compound. Knots of people, maybe a hundred by now, were talking earnestly and loudly to each other and gesturing. Bjerki trailed along behind Marquart.\n\nMarquart went back through the camp, taking his time. There might be some guards still around, and they would undoubtedly be in a pissy mood.\n\nThe back gate of the compound was standing open. More bodies lying round. He surmised this was from the machine-gun fire he had heard. Six more bodies lay on the porch and dirt in front of the guards' barracks. One of the men wasn't dead; he was groaning and his legs worked back and forth in the dirt. Marquart didn't get near him.\n\nThe wooden sides of the building had been raked by machine-gun fire. Maybe there were more dead or wounded in there, but Marquart wasn't curious enough to go inside to find out.\n\nHe examined the vehicles. One car with a body lying beside it seemed undamaged. As he checked the pockets of the corpse, which hadn't bled much, he noted the man had taken four rounds in the chest, any one of which would probably have been fatal. He found a set of keys. They fit in the ignition. He started the engine, which seemed to run okay. Half a tank of gas. Bjerki stood by the driver's door. Marquart ran the window down. \"I'm leaving,\" he said. \"You want to come, get in.\"\n\nBjerki walked around the front of the car and climbed into the passenger's seat. He held his M4 between his knees. \"Where are you going?\"\n\n\"To the revolution.\"\n\n\"Be a shame if they had one without us,\" New Jersey Bjerki said.\n\n\"Put on your seatbelt.\"\n\nMarquart pulled the lever to get the car into drive, and they rolled.\n\nOn their way back to Longview, Nate Danaher said to JR Hays, \"You understand that if we attack Barksdale, the gloves will be off.\"\n\n\"Sure.\"\n\n\"You've talked this over with your cousin?\"\n\n\"Yes.\"\n\n\"He understands that this is not a declaration of independence; it's a declaration of war?\"\n\n\"Nate, you and I know Barry Soetoro isn't going to let Texas go without a fight. For us, the only decision to be made is whether we let Soetoro strike the first blow. Politically, it would be wise to let him be the aggressor. Militarily, not so wise. If Texas is going to win its independence, it must seize the military initiative and _never let it go_.\"\n\nDanaher nodded.\n\n\"If we let Soetoro pick and choose his points of attack, we will ultimately lose our organized military forces and be reduced to years of guerilla warfare. In the long run, I think we could win a guerilla war, but it will destroy Texas and ultimately cost more lives than an offensive that takes the fight out of Texas and into Soetoro's territory. Jack thought that a Texas offensive would, in the long run, cause Soetoro to lose political control of the country. Soetoro must show his supporters he can win the battles, or else he will lose the war. He's already on record as saying that he will crush Texas. I don't think he thought that statement through very well, because Jack can say we are responding to an imminent threat, and everyone south of Canada will believe him. Barry Soetoro doesn't want to negotiate: he wants war. We must give it to him in spades.\"\n\n\"An assault on the base really ought to happen at night. Tonight would have been ideal. Tomorrow night would be the next choice.\"\n\n\"We can't wait. By tomorrow night they may have flown those B-52s out of here or arranged AAA and SAMs, plus a reception committee on the ground. In addition to air police, they can fly some troops up from Fort Polk. By tomorrow night they might be ready to kick our butts. So we must go as soon as we can get ready. The C-130s are already at Hood, and the troops, all volunteers, are getting ready. We just need you to brief them, set it up, and go. Tomorrow morning at perhaps nine o'clock is about the earliest possible time. In my judgment, we dare not wait. _We cannot wait_.\"\n\n\"What are you going to do about that brigade combat team from Fort Polk? And those paratroops? They could push us right off Barksdale and back into Texas.\"\n\n\"I'm going to bomb them while you are taking Barksdale.\"\n\nDanaher thought for a few minutes as the miles rolled by. Finally he said, \"Okay, I'll do it. Gina can stay with our daughter. Let's saddle up.\"\n\n\"Welcome to the Texas Guard.\"\n\n\"Welcome to the war, you mean.\"\n\n\"Yeah, that too.\"\n\n\"I don't know if I have another war in me, but I guess we'll all find out,\" Nate Danaher said softly.\n\nThe CIA safe house was in the woods of a large farm that the locals thought belonged to an eccentric novelist. That was the agency's cover story, anyway. It was midnight when we entered by a gravel driveway, passing by signs that announced \"Private Property, No Trespassing\" and \"Trespassers Will Be Persecuted and Prosecuted, This Means You.\" The one-lane road led across a large meadow, passing a wooden hangar and a barn, and crossed a grass runway and then a bridge across a creek. Security cameras were mounted unobtrusively on trees and under the eaves of the hangar and barn. I led the way.\n\nThe safe house was used for interrogating defectors, Russians and Eastern Europeans back in the day, and now Islamic jihadists. I doubted if there was anyone there just now due to the current state of national affairs, but I was ready in case we met anyone. We didn't. No one was at the guard's cottage, and the gate was locked. Willie the Wire worked on it awhile and couldn't get it open, so we used a tow chain to pull the gate down and off the road. Willie's one skill in life is opening any lock without a key, yet he had just had his first taste of combat so he was a little shook up.\n\nThere was no one at the main house. After an incident a couple years ago when some bent FBI agents and former cops burned the house down, the place had been rebuilt. I was involved in that fracas, and hadn't been here since.\n\nWillie opened the front door for us, partially redeeming himself. While the guys fired up a gasoline generator out back, I explored the layout and found that the new building had a small medical room. It contained an X-ray machine and one that I thought was probably an EKG machine. Some other equipment that I couldn't identify. I had the guys take Jake Grafton in there and put him on the gurney.\n\nGrafton was conscious and obviously hurting. \"He needs a doctor,\" Sarah said with a frown.\n\n\"I'll go get one.\"\n\nI drove back to the hard road and went into Greenbank, and found a small white cinderblock building that said \"Clinic\" on the sign. It was closed of course, but a sign by the door gave a number to call in case of medical emergencies.\n\nBack in the FEMA truck, I fired up the GPS, played with the options, and found one labeled \"phone number.\" I clicked on it and a prompt appeared. I put in the area code, which was 304, and the number. In about two seconds a red pin appeared. Five more seconds, and the computer filled in a map with directions from my present position to the pin.\n\nIt was eight miles away. I rolled.\n\nThe doctor's house was on a secondary road at the top of the grade, in a saddle where there was a nice view. I went up his drive and, late as it was, found a man and woman sitting beside an outdoor fireplace with drinks in their hands. I got out and went over.\n\n\"Doctor?\"\n\n\"Yes. Nathan Proudfoot.\" He was about six feet, thin, perhaps sixty years old, with cropped hair and a mustache.\n\n\"My name is Tommy Carmellini. I'm with FEMA. We have a medical emergency down the road a little ways and could certainly use your services. Could you come with me?\"\n\nTo his credit, he didn't hesitate. \"I'll get my bag.\" He charged into the house. There was a lighted kerosene lamp on the porch and apparently at least one in the house.\n\n\"Sorry to ruin your evening, ma'am,\" I told the lady.\n\n\"Goes with the territory,\" she said. \"What happened?\"\n\n\"Car wreck. One hurt.\"\n\nDr. Proudfoot came trotting out with his black medical bag. He got into the passenger seat of the truck, and we headed back for the safe house. I told him about the fictitious wreck.\n\n\"How did you find me?\"\n\nI gestured to the GPS. \"FEMA can find anyone,\" I said, which was true.\n\n\"How are you making out without electricity?\" I asked.\n\n\"Fine,\" he said confidently. \"Rural nets occasionally go down when there are thunderstorms or someone knocks down a pole with a car, but only for a few hours or overnight. That's just a nuisance. Still, a few years ago we had a blizzard that took a lot of lines down and left us without power for eight days. That was a real pain, so I'm set up now. Even have a little generator that keeps the refrigerator and water pump running. We'll be fine.\"\n\nAs we drove up the road I told him about the patient. \"He's a little over sixty-five, I think, six feet, not obese, in fairly good health as far as I know, but he has a bunch of cracked or broken ribs on each side. I taped him up as best I could; he's in a lot of pain and needs a doctor.\"\n\n\"He got busted ribs in a car wreck?\"\n\n\"I confess, I lied to your wife. Some men beat him badly with fists and shoes. Kicked him in the balls too.\"\n\n\"FEMA sounds like tough duty to me,\" he said acidly. I didn't argue.\n\nIf he didn't know about the safe house in the woods, he didn't show surprise. I guess in his practice he gave up surprise some years back.\n\nDr. Proudfoot glanced at Grafton, looked around at the equipment in the room, then went to work. He cut off the tape I put on his ribs, X-rayed the admiral, asked him about his general health and how he was feeling, checked his heart and vitals. After a careful exam and a study of the X-rays on a computer screen, he taped him again, a much better job than I did. He also gave Grafton a shot to make him sleep. \"Six ribs are cracked on the right side, five on the left,\" he told Sarah and me, \"but none are severed, as far as I can determine. I think he'll heal okay, but he should be in a hospital where he can be observed.\"\n\n\"We'll try to get him there as soon as possible,\" Sarah assured him.\n\n\"Used to be I'd give him some pain pills, but the government is so tight on pain pills now I don't carry any. The good news is that the damned pill-billies aren't tempted to rob me. It's a hell of a world.\"\n\n\"Isn't it though,\" I remarked.\n\n\"If he's hurting when he wakes up, he can have a shot of whiskey. No aspirin. Keep him as inactive as possible. Now, I need all his information so I can get paid for this house call.\"\n\n\"I'll give you cash. Is two hundred dollars enough?\"\n\n\"That's more than the government would pay me.\"\n\nI paid him on the spot.\n\nWhen I was taking him home, the doctor asked, \"Is that a government facility?\"\n\n\"Doctor Proudfoot, you appear to be a good man, and I'd like to answer your question, or questions, because I know you have more than one. But I cannot.\" I smiled at him benignly. \"I don't know where you stand on our current national difficulties, nor do I care. What I can say is this: I want you not to tell anyone about the facility you just visited or the patient you saw there. Or me. Or the other men there.\"\n\n\"It's a government secret, huh?\"\n\n\"Indeed it is.\" We were on the secondary road by then, about a mile from his house. I stopped the truck in the middle of the road and turned in the driver's seat to face him. The panel lights made his face quite plain. \"If we get visitors of any kind, sheriff, locals, FEMA people, FBI, state police, Homeland Security, anyone at all, I'll know you told someone the secret. You won't be prosecuted because you'll be dead. I'll find you like I did tonight and kill you. Do you understand?\"\n\nHe stared at me with fear in his eyes.\n\n\"I don't want to kill you, but I will if you tell anyone at all. Even your wife. Tell me that you understand.\"\n\nHe nodded.\n\nI took my foot off the brake and drove him the rest of the way home. As he got out of the truck, I said, \"I told your wife it was a car wreck. Make her believe it. Good night.\"\n\nI felt dirty and ashamed of myself, but I had to put the fear in him. I hoped for our sakes I scared him enough.\n\nBack at the ranch, I sent Willis and Travis to spend the night in the guard cottage by the gate. Told them to drag the gate back across the road.\n\nI put loaded weapons around the house, with a couple of grenades at each window, just in case, checked on Grafton, who was asleep, and Sarah, who was asleep in a bedroom upstairs. Armanti and the Wire, Jack Yocke and Sal Molina were sharing bedrooms. I took off my boots and flaked out on the couch downstairs.\n\nEarly that Wednesday morning, while most Americans were in bed, the Oklahoma legislature passed a declaration of independence and the governor signed it. The news had been out all day Tuesday that the legislature had been called into special session to consider the measure. Washington had instructed the FBI and FEMA to arrest the governor and the entire legislature to ensure the declaration wasn't even debated. The commander at Fort Sill was instructed to send a thousand troops to assist the federal agents in maintaining order in Oklahoma City.\n\nThe general at Fort Sill was willing, but as the evening progressed he found he didn't have a thousand troops willing to go. He had, at the most, about a hundred, so finally he sent them, armed and wearing battle dress. They went in trucks that convoyed up I-44 from Lawton. They were rolling through the open prairie south of Chickasha when the front tires of the lead truck were shot out. As the truck rolled to the side of the road, more heavy reports were heard and the tires of several following trucks went flat. The final truck had its dual rear wheels shot out while it was almost stopped.\n\nThe soldiers piled out and took up formation around the trucks, but there were no more shots. An hour later soldiers searching the prairie found where someone had apparently fired from a low hill three hundred yards from the highway toward the convoy. Not only was dirt scraped away and grass pulled to provide a decent field of fire, a single spent .50 Browning machine-gun cartridge was found in the grass. A little more searching located another firing position about equidistant from the highway on the other side of the interstate, but there were no more cartridges. Nor, apparently, were there any shooters remaining around. Whoever the marksmen were, they had retreated into the darkness with their weapons, undoubtedly bolt-action .50-caliber rifles set up for long-distance target competition.\n\nThe officer in charge of the column had already informed his commander of his predicament by radio, so the troops sat alongside the interstate smoking and munching whatever snacks they had in their packs as civilian cars and trucks rolled by. It looked like it was going to be a long evening.\n\nTwo hours later four replacement trucks from Fort Sill were fired upon from an overpass. Each truck was hit once in the radiator. The drivers didn't even walk up onto the overpass to look around. They reported the incident on their radios and settled in to spend the night sleeping in their cabs.\n\nThe FBI agents and FEMA troops found an estimated eight hundred armed National Guardsmen in battle dress surrounding the state capitol. The federal officers were disarmed and told to go home or they would be arrested. They went home.\n\nDuring the course of the night, as debate raged on in the legislative chambers, civilians crowded onto the capitol grounds. They passed through the guardsmen's lines carrying lawn chairs and picnic baskets, and many had small children asleep in strollers. The floodlights around the capitol gave the warm evening a festive air. A local band set up amplifiers and microphones and got busy jamming to entertain the crowd.\n\nInside the building, every member of the statehouse and senate got his or her turn at the microphone. The current national situation, and Barry Soetoro's proclamations, were discussed and dissected. Oklahoma was one of only two states in the Union where Soetoro had failed to carry a single county in the 2012 election. His popularity had continued to sink since then, and it was soon clear that he had few friends in the legislature.\n\nOne woman delegate from Norman, a university town and the state's liberal bastion, argued that Soetoro would be out of office on January 20, 2017, a mere five months away, so there was no need for drastic action. \"He's not only a lame duck, he's a dead duck. Why shoot ourselves in the head when he's going to be gone in five months, regardless of what we or Texas or any other state does?\"\n\nThe following speaker took issue with her. \"You are the wildest optimist in the history of representative government in Oklahoma. What makes you think there will be an election? Soetoro's party will lose if there is one, so he has manufactured this crisis to give himself a plausible excuse for calling off the election. He wants to be president for life. Or maybe king. Or emperor. Emperor Barry. We need to stand up for representative government here and now, regardless of the cost. We owe it to ourselves, for our own self-respect, and we owe it to our children and grandchildren. Five years from now, how will you explain to your grandchildren what happened to Oklahoma after you refused to do what you knew to be right? And we all know the _right_ thing to do. But the right thing is _hard_. Let us do it now, and someday we can all stand proudly, shoulder to shoulder, in heaven before the ruler of the universe.\"\n\nThere was more, lots more. One of the low points was a plea by a delegate from one of the districts encompassing the poorer section of Oklahoma City. \"Nothing we can do here tonight will alter the course of our nation's history. We here in Oklahoma are a sideshow. We are a thinly populated state, with only three million nine hundred thousand people. Do you really think we can realistically defy the federal government? The decisions that matter will all be made in Washington. I urge you to not compound the president's problems by being defiant. Let us not beard the lion to see if, indeed, he will bite.\"\n\nSeveral of the following speakers heaped scorn on her position. One speaker summed it up: \"Submit, submit, submit. Don't anger the tyrant. I never thought I would hear such words from a free American.\"\n\nThe criticism of the Soetoro administration kept rolling, mixing with a broad criticism of liberalism and federal judges. \"I am sick of federal judges deciding that the United States Constitution requires abortion and same-sex marriage,\" a state senator from Enid said. \"I challenge you to read that document from end to end, and if you can find the word 'abortion' in it I will kiss your ass tomorrow at high noon on the capitol steps. Ditto gay marriage. What's next? Plural marriages? Legalizing infanticide? We're practically there now. I say it's time we seized control of our own lives here in Oklahoma. Anyone wanting an abortion or to marry a homosexual partner can move to California or New York. We shouldn't be forced to put up with it, and my constituents don't want to. The real problem here is federal judges who enshrine their liberal philosophies in federal decisions instead of letting individual states vote their consciences in open, fair elections. Abortions, gay marriage, legalized pot, all of that should be decided by the states. Whatever happened to the governmental powers reserved to the states? Let's declare ourselves independent, give the people of Oklahoma the right to decide which laws they want to live under, and tell Barry Soetoro where to go and what to do to himself when he gets there.\"\n\nAnother delegate in the House had this to say: \"Oklahomans are tired of being ruled by federal bureaucrats and judges, none of them elected. They decide everything from what can be taught in the public schools to what can be served to kids for lunch and whether the kids can have a prayer. They decree that welfare recipients are entitled to a color television and cell phone, all paid for by the working families of Oklahoma, some of whom can afford neither. They claim they have the right to regulate every creek, farm pond, mudhole, and wet spot in America, including here in Oklahoma. We have to pay for their crackpot regulations based on crackpot science, or no science at all. We have to pay the salaries of the bureaucrats and put up with the endless delays and mountainous paperwork. It's high time to put a stop to bureaucrats and judges running our lives. Let's take back control. _Independence today_ , _tomorrow_ , _and forever_.\"\n\nThe Oklahoma Senate and House passed the declaration by overwhelming majorities and made the vote unanimous by voice vote, and the governor signed it. As in Texas, the declaration, which was almost word for word identical to Texas', was read before television cameras on the statehouse steps to a wildly cheering crowd that commentators estimated at more than ten thousand people.\n\nIn New Mexico the legislature also met that evening, but decided to defer any action until Soetoro had made a definitive announcement about whether the presidential election would proceed in November. If it was canceled altogether, the New Mexico legislature agreed to revisit the issue. The governor of Arizona called the legislature to meet the following evening. The governors of Kansas, Nebraska, Arkansas, South Dakota, North Dakota, Wyoming, and Utah scheduled special sessions two days hence. The governors of Montana and Iowa called for a special session of the legislature in three days time, to give lawmakers a chance to canvass their communities. Other states, too, were mulling their options.\n\nAlthough the legislatures had yet to be called, in Alaska and Hawaii the question of independence was also being weighed and debated, for different reasons. The previous year Soetoro had announced his intention to ignore the U.S. statutes and declare a huge chunk of northern Alaska off-limits to oil exploration. Many of the people of that sparsely settled state were outraged; oil development created good-paying jobs, of which Alaska had far too few, and severance taxes funded state and local governments and generated a check every year for every Alaskan. Oil development had never been the ecological disaster the save-the-earth crowd swore it would be. Soetoro's announcement would slowly upend the Alaskan economy and affect every man, woman, and child who lived there. The devil of it was that the only people who visited the undeveloped Arctic were Alaskans who went to hunt and fish; the limousine liberals in Soetoro's audience rarely if ever trekked the frozen north dribbling dollars as they went. Still, Soetoro would be gone in five months, they hoped, and his extralegal imperial declarations would then be history.\n\nIn Hawaii, independence talk had been around for years, especially among native Hawaiians, many of whom were still on the bottom rung of the economic ladder. There was also a large number of people of all races that felt the Hawaiians had gotten a raw deal in 1893 when white American businessmen played a large role in toppling Hawaii's last monarch, Queen Lili'uokalani, an overthrow that even then-president Grover Cleveland thought an illegal act of war. The current political crisis on the mainland looked to many native Hawaiians like a rare, once-in-a-lifetime opportunity: perhaps the U.S. government would be too busy chasing Texas traitors to worry about the islands in the sea's middle. On the other hand, the economic ties to the mainland were the bedrock of the economy. Could trade and tourism from Japan and China replace lost American dollars? Would the people of the islands be better or worse off as an independent nation?\n\nGeneral Martin L. Wynette read the news summaries of all this \"grandstanding,\" as he called it, at seven o'clock on Wednesday morning when he got to the Pentagon, and thought if this news didn't wake up the fools in the White House, nothing short of nuclear war would. Those people in flyover land were pissed off and feisty.\n\nOne of his aides had brought him a copy of the Minerva Research Initiative, which the president had directed the armed forces to draft and study after he was elected in 2008. Minerva, the Roman goddess of wisdom and war. Idly, Wynette wondered about the subtle mind that had dreamed up that title. The Minerva Research Initiative was a military plan to put down a civil insurrection in the United States.\n\nWynette scanned it and tossed it aside. The plan assumed that the members of the armed forces would willingly participate in armed action against angry citizens. That was a forlorn and foolish assumption, Wynette now realized. He also had on his desk a flash message from the commanding general at Fort Sill in Lawton, Oklahoma, telling him that he had scoured his command for men and women willing to fight Oklahomans. They were willing to go to Iraq, Afghanistan, Syria, and if necessary Iran to fight for America, but only a few were willing to fight Oklahomans.\n\nHe was getting briefings on the result of other army commanders' attempts to muster soldiers who would fight for the Soetoro administration against domestic enemies when he was summoned to the White House. Wynette stuffed the messages in his briefcase along with a copy of the Minerva Research Initiative and called for his aide and his driver.\n\nIn Colorado a group of FBI agents and a sheriff's deputy searching houses to confiscate guns got into a shooting scrape with a homeowner and his son. The homeowner and son were killed, but not before they shot an FBI agent and the sheriff's deputy to death. Another agent was in the hospital. Social media was aflame, with citizens promising the agents and local law officers who cooperated with them in confiscating guns more of the same.\n\nAn FBI office in Seattle was attacked, one agent wounded: perpetrators unknown. In Idaho a county sheriff who agreed to help search the homes of citizens of his county to find and confiscate guns was ambushed, stripped naked, dipped in tar and feathers, and carried to his office on a fence rail. He was now hospitalized with burns over sixty percent of his body. A county in Utah with a significant percentage of Mormon fundamentalists declared its independence from the United States and the State of Utah. Polygamy there was now legal. Finally, a dispatch from Mexico City: the Mexican government was considering diplomatic recognition of the Republic of Texas.\n\nIn Baltimore, a suburban sporting goods warehouse had been looted overnight. The gun counters were stripped clean and the looters helped themselves to every box of cartridges on the premises, then amused themselves by shooting at stuffed animal heads displayed high on the walls. The good news was that due to the federal government's massive orders for ammunition over the last two years, and the president's oft-repeated remarks about his desire for gun control that had induced civilians to buy and hoard ammo, the sporting goods store had only a small supply of cartridges, most in unpopular hunting calibers. The bad news went unspoken: the inner-city rioters were now armed.\n\nIn other riot-plagued big cities around the country, the police and National Guard contented themselves with trying to prevent the destruction from spreading. It was a losing fight. The centers of many of America's largest cities now resembled the core of German cities after World War II.\n\nPeople living in the suburbs nationwide were armed and organizing. They were also emptying the grocery and hardware stores, buying everything in sight, to the limits of their credit cards. Canned and dry food items were almost completely gone in some stores. Hardware stores sold out of emergency generators, charcoal, and gasoline cans. Gasoline stations found that many of their customers were filling up as many as ten five-gallon cans with fuel. Sporting goods stores were selling every gun on the shelf and all the ammunition in stock. In Howard County, Maryland, a bedroom suburb of Washington and Baltimore populated with a large percentage of federal civil service employees of all races, the county police and Homeland Security officers tried to search homes for guns, only to be met at four houses by armed householders who threatened to shoot to kill.\n\nThe chief of the Howard County police announced that henceforth his officers would concentrate on arresting criminals, answering domestic violence calls, and helping motorists involved in traffic accidents. The chief was quoted by a reporter as saying, \"If Barry Soetoro wants to confiscate guns, he can figure out how to do it. The people here are frightened by what's going on in Baltimore and elsewhere and want to be able to protect themselves. I can't say I blame them.\" After the story was published, two black Maryland legislators called the police chief, who was also black, a racist.\nTWENTY\n\nIn Galveston that morning, after the sun came up, the sheriff drove his car down the pier and parked adjacent to the gangway of USS _Texas_. He walked across the gangway and shouted down into the open hatch, \"Anybody home?\"\n\nIn less than a minute, a man appeared below and looked up at him. \"Yep, we're home.\"\n\n\"Mind if I come down and visit?\"\n\n\"Please do.\"\n\nSpeedy Gonzales escorted the sheriff to a small wardroom, where he found Loren Snyder studying several large bound volumes and sipping a cup of coffee.\n\n\"Coffee, Sheriff?\"\n\n\"Don't mind if I do.\"\n\n\"Best coffee in the world,\" Loren Snyder said.\n\nThe sheriff sipped at his, which he took black. Almost as good as Dunkin' Donuts coffee, he thought, but he didn't say it. Instead, he got straight to the point. \"When are y'all going to nuke yourselves out of here?\"\n\nLoren laughed. \"Well, we're working on that right now. Before we go, I want my crew, all five of us, to run through every emergency procedure in the book and figure out how we're going to handle it. We don't have sixty people, just five. We don't want to die in this boat.\"\n\nThe sheriff looked around and nodded. \"I sure understand that.\" Just sitting here in this steel cigar gave the sheriff a mild case of claustrophobia. What it would be like being submerged he didn't want to think about.\n\n\"How long can you guys stay submerged, anyway?\" the sheriff asked.\n\n\"Until we run out of toilet paper.\"\n\nThe sheriff chuckled at that, thinking Loren Snyder was being facetious. He wasn't. With only five people aboard eating the stores, _Texas_ could stay submerged for a long, long time.\n\n\"We're going to spend today running emergency drills,\" Snyder said, \"making sure everyone knows what is expected of him and we are all on the same page. I hope by tonight we'll be ready to leave this pier.\"\n\n\"What about the U.S. Navy? I'll bet they're kinda unhappy that they lost this thing.\"\n\n\"They'll probably send SEALs to take it back,\" Lorrie admitted.\n\n\"You mean like those guys who whacked bin Laden?\"\n\n\"Yep. Naval Special Warfare commandos.\"\n\n\"Maybe y'all oughta get outta here and do your drills someplace else.\"\n\n\"Sheriff, I agree one hundred percent. As soon as we feel we can safely move this submarine, we will. In the interim, it would help if you would station some officers with radios out there around the harbor to keep a lookout. I suspect the SEALs will come at night. Probably tonight. We hope to be gone when they get here, but just in case, if your lookouts see anything suspicious\u2014anything\u2014I would appreciate a heads-up so we can cast off and get going. Once we close the hatches, the SEALs can't get inside the boat.\"\n\nThe sheriff nodded reluctantly. \"Today and this evening?\"\n\n\"Yes, sir.\"\n\n\"You'll try like the devil to get this stuff done and get out of Galveston?\"\n\n\"Cross my heart.\"\n\n\"Okay, Captain. But I ain't asking my deputies to get in a shootout with SEALs. No way. They're law enforcement officers, not soldiers.\"\n\nThey discussed radio frequencies for a moment, then Loren Snyder said, \"Thanks for stopping by, Sheriff.\"\n\nThe sheriff had one last gulp of coffee, then said, \"Good luck to y'all out there, Captain.\" After he and Loren shook hands, he followed Speedy to the forward torpedo room and the ladder topside.\n\n_Captain_. Loren Snyder liked the sound of that.\n\nSecret Service sniper Tobe Baha drove slowly around Austin looking things over. He had had a private interview with President Soetoro's chief of staff, Al Grantham, then went home and packed for a trip. He put his rifle in its aluminum airline case in the toolbox behind the cab of his pickup. He carefully locked the toolbox with the best padlocks money could buy.\n\nThe rifle wasn't his service rifle. This was his personal rifle, a Remington Model 700 in .308, or as it was known in the service, 7.62\u00d751 NATO. It certainly wasn't the best cartridge for extreme long-range shooting, but Tobe had used it extensively while in the military and knew the ballistics cold, so he was very comfortable with it. And ammo for it was available everywhere, if need be. Tobe had loaded his own with match bullets and had two boxes in the airline case.\n\nUnder his rifle was another airline case stuffed with a quarter of a million U.S. dollars and fifty thousand dollars' worth of gold. That was his down payment on the assassination of Jack Hays.\n\nThe problem was that Tobe Baha wasn't an assassin. He was a sniper, pure and simple, so he didn't even bother trying to come up with a second method of taking out the president of Texas if setting up a snipe proved difficult. Actually, he couldn't conceive of a set of circumstances that would cause him to miss a rifle shot, if and when he got one. And he would get one, sooner or later. Everyone was vulnerable to a sniper, unless they lived in a prison, and politicians especially. They had to make public appearances, they got into and out of limos and helicopters on a routine basis, and most of them, including Jack Hays, had families.\n\nPatience was the sniper's golden asset, and Tobe Baha had more than his share. He could and would wait until he was presented with a shot he knew he could make during one of Jack Hays' inevitable public appearances. After that, with a cool million in his jeans, he would disappear.\n\nOf course, he worried a little about the possibility that the Soetoro administration might eventually want him permanently removed from the land of the living. If they just had him arrested, he might talk. So arrest wasn't the risk.\n\nTobe Baha had thought it over when approached for this shoot, and decided he could handle the risk of treachery by his employers. After all, three or four of the Secret Service people knew of the plot.\n\nHe had said as much on his last interview with Al Grantham. \"If you don't pay me the money you owe or if you send people after me, I'll come after you,\" he told Grantham, \"and I won't miss.\"\n\nAustin certainly had possibilities, Tobe concluded as he drove around. The capitol was surrounded by buildings, although they were several hundred yards from the capitol itself, which sat on a small knoll surrounded by scattered large trees and lots of grass. The governor's mansion also had buildings within range of a .308. The real question was whether Jack Hays' bodyguards included snipers. Protecting a public figure from bombs and maniacs with pistols and knives was what the Secret Service did best. Snipers, however, were the worst threat, which was why Tobe Baha had been recruited by the service. It takes a sniper to kill a sniper.\n\nIf the Texas crowd didn't have snipers protecting Jack Hays, Tobe Baha's mission would be a whole lot easier. So his first task was to determine if they did.\n\nTobe Baha smiled. This was going to be a good hunt.\n\nMajor General JR Hays launched his first offensive that morning, the thirty-first of August. He watched Texas guardsmen file aboard six C-130 Hercules transports, four-engine turboprops, at Fort Hood, sixty-four combat-equipped soldiers to each plane. Two other C-130s were being loaded with howitzers, ammunition, rations, water, and a portable field hospital.\n\n\"I'm banking on surprise,\" JR told Colonel Nathaniel Danaher, who was leading the attacking force. \"I think you can get on the ground and establish a perimeter before the people on the ground figure out that something is going down. I want you to clear the planes and let them take off immediately for another load. Ideally, I'd like to get a brigade on the ground over there with some artillery to give it teeth. F-16s will provide close air support and top cover. But it's up to you to stop our assault if you find you are in way over your head. You must remain in radio contact with the planes in the air at all times, keep them advised of how things are going.\"\n\nNate Danaher looked ten years younger than he did last night. The challenge of leading men in combat had always energized him.\n\nThe six transports bearing soldiers took off first, escorted by a high top cover of F-16s from Lackland. The attacking force would fly east of Barksdale, turn and approach the base from that direction, calling the control tower for landing clearance. While the panicked air controllers sorted through messages trying to find one about incoming Hercs, the Hercs would land, discharge their troops, and take off again. The C-130s bearing howitzers and ammo would land an hour later, after the soldiers of the first wave had secured the flight line.\n\nWould they achieve surprise? JR Hays asked himself that question, but he didn't know the answer. If the bad guys had gotten wind of the invasion of Louisiana, he would be among the first to hear about it.\n\nMaybe yes, maybe no, he decided.\n\nPerhaps he should have given his major general stars to Nate Danaher and commissioned himself a colonel, then led the troops invading Barksdale. Jack Hays would have said okay, if that was the way he wanted it. But would Nate Danaher have laid on this attack if he had been the general in charge? That hypothetical had no possible answer, because JR had made the decision. Nate had saluted and marched off to give every ounce he had in him. That quality, JR thought, was the salvation of the professional soldier. Regardless of whether the professional thought the order wise or foolish, he said, \"I will do my best, sir,\" and the rest of the sentence was unspoken: \"Even if it kills me.\" So generals ordered men into combat, knowing that some of them, an unknown number, would die. Generals hoped and prayed that the objective would be worth the sacrifice, and, in the end, only they and God would know how the scales balanced.\n\nJR thought ruefully about the old observation that doctors buried their mistakes. Truly, so did generals.\n\nAnd yet, even if he lost every soldier and airplane he sent this morning, JR Hays would win a strategic victory simply by attacking. He knew that in the depths of his military soul. Soetoro would stop worrying about invading Texas and wreaking havoc and start worrying about protecting what he had. People the world over expect their government to protect them, and when it doesn't, or can't, they begin to worry.\n\nAnd if Danaher was victorious and captured a fleet of intact B-52s, Barry Soetoro would start fretting about where they might be used against him. Would they bomb Washington? New York? Los Angeles? A squadron of B-52s carpet-bombing with unguided weapons could destroy a city, just as they did Hanoi. Fighters would be detailed to guard the skies over cities and military bases. Soetoro _must_ commit his air force to protecting those places, and if he did, those air assets would be unavailable to attack Dallas, Houston, San Antonio, Austin, or the military bases Hays had captured.\n\nJR walked across the tarmac when the troop-laden transports were out of sight and went into the base's air traffic control facilities. \"Are the Lancers from Dyess airborne?\"\n\n\"Yes, sir. Target time is less than an hour away.\"\n\nThe B-1s were targeted against the military equipment at Fort Polk. Many of the soldiers at Hood had trained at Polk, and they helped annotate maps. The Lancer crews knew precisely where they were going, and they had air cover, F-16s from Lackland. In and out fast like a rabbit was their credo. Leaving smoldering wreckage.\n\nJR got a cup of coffee from the pot and sat down in front of a temporary theater map taped to the wall. He had launched his strikes; now there was nothing to do but wait.\n\nWait, wait, wait.\n\nI found Jake Grafton alert that morning when I took a cup of coffee into the dispensary. He was still on the gurney.\n\n\"Tommy, you've got to get me off this thing and help me to the restroom.\"\n\nI did that, and then I put him in a large easy chair in the main room of the facility, or lodge, or whatever they called it, with a blanket wrapped around him.\n\n\"Thanks for rescuing me, Tommy,\" Grafton said with coffee in hand. He sniffed it, savoring the smell before he took the first sip.\n\n\"Any old time, Admiral. The guys and I had nothing to do since you got kicked out of the agency. So we thought, let's go spring the admiral and take a nice vacation.\"\n\n\"And Sarah Houston?\"\n\n\"She's got the hots for me something terrible. I think that's affected her brain. Whatever, she came along.\"\n\nAbout that time all the folks upstairs came down, so I got busy fixing breakfast. Needless to say, we didn't have eggs or milk or any of that, but we had beans and MREs and a lot of canned meats and veggies. I made a stew. Tasted it and added some salt and a generous dollop of Cholula sauce.\n\nWhen I brought it into the main room and put it on a table, Sal Molina and Jack Yocke were in earnest conversation with Grafton. I ladled some of the stew out for the admiral, gave it to him with a spoon, and told everyone else to help themselves.\n\nSarah was eating tiny little bites. \"The first person who complains gets to do the cooking,\" I said with no-nonsense authority.\n\nWillie Varner made a face. \"Tastes like shit, Tommy, but good.\"\n\nWhen the chuckles died, he started telling about the fare in the prisons he had resided in. According to the Wire, prisons were good feeders. He was lying, again. After he got out the second time, he told me he never wanted to see a macaroni or spaghetti noodle again as long as he lived.\n\nI sent Willie and Armanti down to the guard shack to relieve Travis and Willis. \"We're going to have visitors, probably sooner rather than later.\" I told them about the doctor and my threat. \"I doubt if he believed me. I don't have a face that will scare anybody.\"\n\n\"He'll blab for sure,\" Willie said, nodding.\n\nWhen Willis Coffee got there, he went upstairs and got an extra shirt and jeans for the admiral. He was about the same size. Travis Clay loaned Grafton his tennis shoes; he said the boots were fine for him. I left the guys to clean up, took a carbine, checked to see that the magazine was full and there was a round in the chamber. Strapped my pistol belt around my middle. It had been a few years since I was here, and I wanted to refresh my memory about how the land laid. Grafton, Molina, and Yocke were busy solving the world's problems as I left.\n\nI walked up behind the lodge, stood for a moment listening to the muffled generator, then hiked straight up the hill to the ridge. At first the hill behind the lodge was steep, then the grade lessened and it was just a walk in the forest, which was beautiful. The chain-link fence on the ridge ran north-south, surveyor straight. The trees and brush had been cleared for ten feet on either side, and this late in the summer, the open space was full of knee-high weeds. I walked the fence for about a half mile north, going downhill when the ridge turned west. I crossed a little creek that didn't have any water in it and then followed the fence back steeply uphill.\n\nI kept track of the security cameras mounted unobtrusively in trees on our side of the fence. The cameras were battery-operated and broadcast to a receiver in the security shack. I could just discern a trail agency people had walked through the years changing the camera batteries, and no doubt replacing cameras that broke or got water in them or someone in the National Forest on the other side of the fence shot for the hell of it.\n\nWhen I had had enough I turned eastward, downhill in the general direction of the guard cabin. Ended up climbing another ridge. This ground was cut up by meandering little creeks and steep slopes, all heavily wooded.\n\nMainly by accident I finally found the access road and followed it to the guard cabin. I could hear the generator running a hundred feet away.\n\nI walked in without knocking and startled Willie Varner and Armanti Hall, who were listening to a radio\u2014police calls, or maybe FEMA calls. The digital feed from the security cameras was on a monitor beside the radio, but they weren't watching it.\n\n\"With that generator going, you dudes won't even hear them coming,\" I remarked.\n\n\"Sit down, Tommy,\" Armanti said. \"You should hear some of this. People are shooting at federal officers. I don't know if they are FEMA or Homeland, and I don't guess it matters.\"\n\n\"Where?\"\n\n\"Well, I don't recognize any of the place names, but I kinda think up in Maryland or Pennsylvania someplace. One guy was talking about getting more vehicles and agents out of Harrisburg.\"\n\nWillie chimed in, \"Two federal guys shot and need evacuation. They claim they killed three of the locals. Civilians. Ambushers, they called them.\"\n\n\"We thought we should keep an ear open for transmissions around here,\" Armanti explained, reading the expression on my face.\n\n\"You guys start watching the barn and hangar security cameras on the monitor. The feds won't sneak down through the woods. Someone will drive up that road sooner or later and they won't give you a heads-up call on the radio. You'll see them on the barn and hangar cameras.\"\n\n\"We can't stay here,\" Willie declared. He wasn't Einstein but he got there eventually.\n\nI tromped out and headed up the hill to the lodge. If we didn't leave we were running the risk of being trapped. I should have stuck my pistol in that doctor's mouth and scowled until he crapped his pants. Grafton was asleep again. He was certainly in no condition to be moved, so we had to stay.\n\nI got Willis Coffee and Travis Clay to dig a nest for two heavy machine guns across the road from the parking area where they could engage any vehicles that drove up. They were pros: they knew how to set up a machine-gun nest. They took some AT4s along, just in case, and got busy moving the guns and ammo.\n\nJack Yocke and Sal Molina were not thrilled when I told them they were now soldiers in the Army of the Rebellion. \"I'm a reporter,\" Yocke said stiffly.\n\n\"I just drafted you,\" I replied. \"When this is over, you'll probably have enough material to write a couple of books and eat on the rubber chicken circuit until you die of constipation. Right now, however, your problem is staying alive. I'm about to do you a big favor and show you how you can do that, and help the rest of us stay alive too.\"\n\n\"And if I say no?\"\n\n\"You walk down to the hard road and hitch a ride anywhere you want to go.\"\n\n\"I'll stay.\"\n\n\"I'm so thrilled.\"\n\nMolina said, \"I'm fat, out of shape, and never touched a weapon in my life.\"\n\n\"When this is over, you'll want to join the NRA.\"\n\n\"What about me?\" Sarah asked.\n\n\"You are my inside surprise. You can toss grenades and shoot if they come through the door in front or back. I suggest that you pick a few spots to watch the back of the building. If we get visitors with something nasty on their minds, they will drop someone off to come through the woods behind us. Your job is to guard the rear.\"\n\nI showed the three of them how to operate the carbines, grenades, and AT4s. \"Don't fire one of these AT4s in the house. The back blast will burn the building down.\"\n\nWhen I thought they had the basics, I gave them a little heart-to-heart about combat. \"You are going to be very scared when the shooting starts. Concentrate on making your weapon function and keep firing it at the bad guys. It's really easy to shoot the wrong people, which will not help you nor the rest of us. The main thing is to stay in the fight.\"\n\n\"What about prisoners?\"\n\n\"What about 'em?\"\n\n\"Well, what if they throw down their weapons and surrender?\"\n\n\"Anybody who gets into a shooting scrape with us wants our weapons, vehicles, and food. If you surrender, they'll kill you. I suggest you do the same to them.\"\n\n\"I can't do that,\" Molina said frankly. Yocke nodded his agreement.\n\n\"Don't worry. Someone will do it for you,\" I said. \"Just don't let 'em run off.\"\n\n\"Could you shoot a man with his hands up?\" Yocke asked Sarah.\n\nShe looked at him as if he had asked if she were still a virgin. Women are usually tougher and more realistic than men.\n\nOne of the troopers in the back of the first C-130 in the string flying just above Louisiana was Specialist Jimmy Schaffran from Minnesota. His story was unique, as was the story of every man in the plane, but perhaps similar to many. He had been a chubby nerd in high school, addicted to video games, partly because he wished to find some way to escape a bad home situation and partly because he was unattractive to girls. He had no idea what to say to them. Certainly he wasn't a jock or rocket scientist. There was no money in the family to send him to college when he graduated from high school, a fact he didn't fret because he didn't know what he wanted to do with his life and doubted that he was smart enough for college, anyway. He got a job delivering sandwiches in his father's old work car, then pizza because the tips were better, and finally decided to join the army.\n\nRecruit training nearly killed him. Pushed mercilessly by the sergeants, the pounds began melting off and his stamina increased dramatically. After thirty pounds of fat were gone, he began gaining muscle.\n\nJimmy Schaffran found a home in the army. He had some buddies and they went into town together. He met a girl, a cute waitress in Killeen with a little tattoo over her heart, which happened to put it on the top of the swell of her left breast; she loved to neck in his car, the first one he had ever owned, cherry red, only three years old, with a loud aftermarket muffler.\n\nWhen this Texas thing went down, a Guard officer asked him if he wanted to go back to the U.S. Army or fight for Texas. Jimmy hadn't hesitated. \"I'm from Minnesota,\" he said, \"but now I'm a Texan.\" His buddies, from California, Michigan, and South Carolina, also decided they were Texans. \"Be a shame to break up a good team,\" one of them said. So all four were in this assault on Barksdale, two on this plane and two on another.\n\n\"Hell, it's all an adventure,\" Jimmy Schaffran told himself, wished his stomach would stop doing flips, and squeezed his weapon a little tighter.\n\nNathaniel Danaher sat behind the pilots in the cockpit of the first C-130 to approach Barksdale. The planes, strung out in trail about a mile apart, had flown the entire distance from Fort Hood at a hundred feet above the ground. They had managed to avoid several radio towers, which would have made flying at this altitude suicidal at night.\n\nAs briefed, the pilot called Barksdale Approach, gave his position from the field, and asked for clearance to land. \"I'm leading a flight of six. My playmates are in trail and would like to land behind me.\"\n\nThere was a long silence, then, \"We don't have a flight plan on you. Where did you take off?\"\n\n\"Fort Rucker.\"\n\nAnother pause, then, \"Make a modified straight-in to Runway Three-Three, Altimeter two niner niner six, wind three one zero at seven. Switch to Tower and report five miles.\"\n\n\"Wilco.\"\n\nThe copilot flipped the radio freq and made the call, trying to keep his voice airline-pilot, ah-shucks cool.\n\n\"Flight of six, cleared to land.\"\n\nThe copilot turned to Danaher. \"They'll get on the phone to Rucker, sir.\"\n\n\"Regardless of what they say, land. Taxi right over in front of base ops and drop the ramp.\"\n\nDanaher went into the back and got his troops ready. They had been carefully briefed, and knew they were to go off running as soon as the loadmaster lowered the ramp.\n\nIn Barksdale Approach Control, confusion reigned. The only planes scheduled to arrive at noon were a flight of four F-22s. If Ops had received messages about arriving Hercs, no one had seen them, but that didn't mean they didn't exist somewhere. And there was something else. Approach Control radar showed blips without transponder codes, up high and approaching from the south. What were these airplanes? The duty ops officer called his boss, a colonel, who confessed his ignorance. Flipping madly through the messages on the message board, and calls to the message center, didn't help. Nor would calling Center do any good: Center was off the air and no one was answering the telephones.\n\nThe first Herc touched down and, ignoring orders from Ground Control, taxied to a stop in front of the Ops building; armed, helmeted troops in battle dress piled out of the plane.\n\nAn enlisted controller in the tower remarked, \"Rucker must have sent an advance party to augment base security.\"\n\nVery shortly, everyone in the tower was disabused of that notion and jerked headlong into the reality of war. Troopers entered the tower, pointed their guns, and waved the air force controllers away from the scopes and microphones. An NCO growled, \"You people get on the floor, hands in your laps, and no one will get hurt!\" Troopers bound the air controllers' wrists with plastic ties. Cell phones were confiscated. Another trooper sat at a microphone to guide approaching aircraft.\n\nSimilar scenes were enacted at the base ops center, where Colonel Danaher established his command post, and at the message center. It all happened so quickly that no message of the attack was transmitted. As far as the Pentagon knew, Barksdale was still owned by the United States Air Force.\n\nDanaher couldn't believe his good fortune. Lady Luck had just given him a gift of a few hours.\n\nThe second C-130 taxied to the B-52 parking mat. As the troops disembarked, an air police SUV came roaring up and two armed men jumped out. When a couple of the troopers fired bursts over their heads, the air policemen jumped back into the SUV and started off, but now someone shot the tires out. It kept going anyway. Another burst into the rear of it brought it to a stop. One of the air policemen was slightly wounded. They were disarmed and led away across the mat to a holding area as the troops fanned out and the C-130 began taxiing for takeoff. There were more troops at Fort Hood that needed transport.\n\nTwo minutes after the sixth and last transport off-loaded its men, Colonel Danaher could look at the base's mechanics, officers, and pilots seated in rows, hands fastened with plastic ties, and under guard. It was a quick victory for Texas. Hearing the reports over handheld radio, Colonel Danaher breathed a sigh of relief. For the first time in his life, he understood the ennui that engulfed the military personnel in Pearl Harbor in the weeks before the Japanese attack on December 7, 1941. It is devilishly difficult to instantly transition from peace to war. Danaher knew he wasn't up to speed yet, but thought maybe he better get that way fast. No doubt all the air force personnel on the base were waking up mighty quick.\n\nThe B-1 Lancer surprise attack on the war materiel stockpiled at Fort Polk was a complete success. Not a SAM or artillery shell rose to meet them. Using JDAMs, the six bombers hit the large tank and artillery depots. Then the F-16s flying top cover came down and used rockets and cannons on armored vehicles and artillery pieces that appeared undamaged. Several JDAMs went into the fuel storage facilities. Post-strike photos snapped by the F-16 strike leader suggested that perhaps forty percent of the tanks and artillery were no longer serviceable. The black column of smoke rising from the fuel storage areas was visible in the sky from a distance of ninety miles.\n\nWhile that strike was going on, General Martin L. Wynette was in his limo on his way to the Executive Office Building. When he received a call from the JCS duty officer informing him of the attack on Fort Polk, Wynette hung up the phone with a frown. The president and his disciples were going to eat him alive. He briefed his general officer aides, a male and a female, so they would know what was coming.\n\nAt the Executive Office Building, Wynette and his two aides were ushered to a conference room where Soetoro, his national security advisor, and a dozen top political aides were waiting, including Sulana Schanck, the Muslim. She had always intimidated Wynette. Those eyes, glaring at everyone who didn't share her vision of a Muslim America. Wynette thought her the most evil woman he had ever met. He thought that one of these days she might snap and start cutting off heads with a butcher knife. He hoped she would begin with Al Grantham.\n\nWynette opened his briefcase as the men and women in the room debated the implications of Oklahoma's rebellion and the scheduled independence votes in other plains states. Soetoro seemed to have himself under control this morning, Wynette thought, as he listened to machine-gun bursts of terrible news.\n\nWynette dropped into a chair and tried to keep his face deadpan. His aides sat down beside him. No one mentioned the attacks on Fort Polk in Louisiana. Maybe they don't know yet, he thought.\n\nFinally the president addressed a question to the general, his first acknowledgment of the officer's presence. \"What can the military do to put a stop to this treason?\"\n\n\"Nothing,\" Wynette said, \"except maybe bomb the statehouses involved. And I'm not sure what that would achieve.\"\n\nAl Grantham let out a roar. \"Goddamnit, General, it would kill some traitors.\"\n\n\"You folks have a red-hot political crisis on your hands and the U.S. armed forces are melting away. A couple more days of this and we won't have enough people to turn the lights on and off at the Pentagon.\"\n\nSilence descended upon the room. Wynette thought about all the ways the president had disrespected the men and women in uniform during his administration, including refusing to make appearances and public statements during Armed Forces Day, and refusing to salute the flag. His contempt of the people in uniform was now being returned in spades.\n\n\"We are going to have to recruit an army of progressives who are willing to fight for America,\" Barry Soetoro said.\n\nGood luck with that, Wynette thought. What he said aloud was, \"By the time you get your army recruited and equipped, with enough training to teach them which end of the rifle the bullet comes out of, you are going to be out of office.\"\n\nThe political aides merely stared ahead silently, Schanck included. Soetoro didn't say a word. Even Grantham managed to control himself. All of which proved to the chairman of the Joint Chiefs that the White House knew there was not going to be an election in November. That was still their little secret.\n\nFinally Grantham said, \"Maybe you should start shooting some of your reluctant warriors. That would inspire the rest to do their sworn duty.\"\n\n\"I don't have the authority to hold drumhead courts-martial and execute soldiers.\"\n\n\"The president can give you that authority.\"\n\n\"I don't want it. If you like, I'll tender my resignation right now and you can dig down through the officer corps until you find someone willing to shoot American soldiers. There must be one or two ambitious assholes in uniform that would shoot their own mothers for a big promotion. I've never met any, but they say there are rotten apples in every barrel.\"\n\nGrantham snarled, \"Why don't you start saying yes, sir, and no, sir, and stop this damned insubordination?\"\n\n\"I thought you wanted me here for professional advice. I just gave you some.\"\n\n\"Enough,\" Soetoro said. He rubbed his face with both hands. \"We have a political crisis that is fed by social media and the press pouring gasoline on hot embers. What we need to do is shut down the power grid nationwide to stop all the bitching, plotting, and conspiracies.\"\n\nMartin Wynette lost control of his face. He stared slack-jawed at the president. That had to be the most idiotic suggestion he had ever heard.\n\n\"We must do something, and that might have a good effect,\" Al Grantham opined.\n\nIronically, Martin Wynette thought that comment proof that Grantham was a total, complete flaming fool, and a world-class ass-kisser to boot! Had his senior aide only known the general's thoughts, he would have probably laughed aloud. Wynette managed to close his mouth and put on his poker face again.\n\nThe civilians around the table discussed it. Indeed, they thought that something had to be done to douse the political fires, and this was something. If those rebels were sitting in the dark without air conditioning or the internet or telephones, at least they wouldn't be damning the administration and fomenting treason before a national audience, the members of which would have their own problems to deal with. And it was the president's own idea, which was nice. No one there had to take the risk of offering a suggestion that might be rejected. It never hurts to say yes to the boss.\n\nWhat wasn't addressed, Wynette noted grimly, was how cutting the juice was going to stop the social collapse that he thought almost inevitable. In fact, Wynette thought that leaving people nationwide without power to stay cool and preserve and prepare food in the dead heat of August was likely to accelerate the process, not impede it. Not to mention the havoc it would play on nursing home residents and the elderly who lacked emergency generators. Police and firefighters could not be summoned in an emergency. This callous decision would kill American citizens, whether they were progressives or conservatives, loyal or disloyal, whether they worshipped the ground Barry Soetoro walked upon or urged God every night to take the bastard quick. It would also stop the American economy dead in its tracks. Factories would be left without not only electricity but natural gas, because electricity powered the compressors needed to move it through pipelines. Without pumps, water and sewage would cease to flow. And every filling station in America would be unable to pump gasoline or diesel fuel. Truck deliveries would stop. If the power outage went on long enough, urban Americans would begin to starve or die of thirst. Cutting power might be justified as a military necessity, Wynette thought, but certainly not as a political expedient to silence dissent. He almost said aloud that JR Hays would turn off America's juice if he could, but being Martin Wynette, he kept his mouth shut.\n\nSoetoro made the decision, as his inner circle of committed progressives knew he would. \"Do it,\" he said, and gestured toward the door.\n\nSome moron asked, \"How?\"\n\nGrantham fielded that one. \"Call the heads of the various power companies and tell them to shut off the juice, and if they don't, send the FBI around to arrest them and every officer in the company. Crack the damned whip.\" When you have dictatorial powers, you can iron out all the little difficulties.\n\n\"Yes, sir,\" they said, and scattered.\n\n\"You stay,\" the president said to the general and his aides.\n\nWhen the room was empty, the president said, \"Tell me about that attack in Louisiana.\"\n\nSo he had heard after all. \"I got a telephone call in the car on the way over here,\" Wynette said, \"so all I know are the basics. Apparently B-1 Lancers. They probably came from Dyess Air Force Base in Abilene.\"\n\n\"What can we do about those Texas traitors?\"\n\n\"Sir, we are putting together an invasion, as you directed. JR Hays just made the invasion a little more difficult, but he can't stop it.\"\n\n\"What will he do next?\"\n\n\"We need to destroy those B-1s on the ground at Dyess. I was thinking of using the B-2s at Whiteman Air Force Base in Missouri to do that as soon as possible.\"\n\n\"Fine,\" Barry Soetoro said. \"We should have retired those old B-1s years ago. Instead we wasted mountains of money on them that could have been better spent elsewhere.\"\n\nWynette didn't argue that point.\n\n\"I also want you to turn off the lights in Texas, General. I don't think calling the president of the power company will do it. Do it any way you can. As soon as you can. Texas started all this trouble.\"\n\n\"Yes, sir.\"\n\nBarry Soetoro would have been furious if he had known that JR Hays was already one jump ahead of him. Another half-dozen B-1 Lancers were already in the air on their way to Missouri to bomb Whiteman Air Force Base. An hour later, as the carcasses of the B-2s at Whiteman were still burning, he found out.\n\nIn the limo with his general officer aides, Martin Wynette said, \"He knew about that Louisiana attack when he ordered the power turned off nationwide.\"\n\nHis generals both nodded.\n\n\"And he knew about the state legislatures giving him the finger.\"\n\nYes.\n\n\"Did he do it to punish the American people?\" Wynette asked aloud.\n\n\"Ten to one that he blames the Texans for the loss of power,\" the female two-star said.\n\n\"No bet,\" her male colleague said.\n\n\"A hundred to one,\" she offered.\n\n\"No bet.\"\n\nBut with the power off, only a few will hear him, Wynette thought. And who will care? The one fact every American will understand is that the federal government can't keep electricity flowing through the wires.\n\nAt Barksdale Air Force Base four F-22s broke over the runway and swung into trail on the downwind. They slowed, dropped their landing gear and flaps, and the controller in the tower cleared them to land. Once down, Ground Control directed them to park on one end of the B-52 ramp.\n\nEverything appeared normal to the pilots as they followed the directions of linesmen, parked in a row, and one by one shut down. Number Four was the last to shut down, of course, and the pilot was the last to exit his cockpit onto a boarding ladder that had been pushed to the side of his plane.\n\nHe was standing with one foot in the cockpit and one foot on the ladder when he looked around and realized that the other pilots had their hands in the air and soldiers in battle dress were pointing weapons at them.\n\nHe drew his pistol from a holster under his left armpit and began shooting into the instrument panel, which was composed of complex multifunction displays.\n\nThe air force officer had fired three shots when Specialist Jimmy Schaffran triggered a three-shot burst from his M4 carbine from a distance of eighteen feet. The pilot tumbled backward without even trying to grab the ladder and fell to the concrete.\n\nJimmy Schaffran, late of Minnesota and now of Texas, walked over to the body. The man's head was at an odd angle. Obviously a broken neck. If the carbine bullets didn't kill him, the fall to the concrete did.\n\nSchaffran was still staring at the corpse when his buddy from South Carolina came running over.\n\nOne look at the dead man was enough. Carolina threw an arm over Schaffran's shoulders. He turned him away from the body and said, \"You had to do it, Jimmy. We may need these planes.\"\n\n\"Fuckin' shit,\" said Jimmy Schaffran.\n\n\"Hey, man. We chose our side of the fence and he chose his. Not much any of us can do about it now. God will have to figure it out.\"\nTWENTY-ONE\n\nIn Galveston, Loren Snyder had a visitor. The man shouted down the open hatch, got no answer, then climbed down and wandered aft. He found Loren in the control room.\n\n\"Hi. I'm George Ranta. The sheriff sent me to see you.\"\n\n\"Oh.\" Loren was more than a little surprised. The sheriff was supposed to be guarding the pier and preventing the locals from meandering over for a look at a real submarine.\n\n\"I used to serve in attack boats. In fact, I used to be the head sonarman on this one.\"\n\n\"On this boat?\"\n\n\"Yes, sir. Could you guys use some help? I'd kinda like to volunteer, if you could use me.\"\n\n\"Volunteer for what?\"\n\n\"For whatever you have in mind, Captain.\"\n\nThat captain thing did it for Loren. This guy could be a SEAL in civvies, he reflected, here to kung fu the whole crew, all five. On the other hand, that captain thing sounded automatic, and he didn't look like a muscle man who spent four hours a day in the gym. Maybe he was on the level. \"Prove it,\" Loren said.\n\nRanta sat down at the main sonar console and began flipping switches. In less than a minute the sonar was running through built-in tests. Yep, he knew what he was doing.\n\n\"We're going to sea in a few hours. If you've served in these boats, you know what we're up against. The navy won't like us out cruising around in an armed attack submarine.\"\n\n\"You have torpedoes in the tubes and Tomahawks in the wells?\"\n\n\"Yep.\"\n\n\"Going to use them?\"\n\n\"We might.\"\n\n\"To free Texas?\"\n\n\"Yep.\"\n\n\"I'll go if you'll have me.\"\n\n\"Got any stuff?\"\n\n\"It's on the other side of the gangway.\"\n\n\"Go get it, and find yourself a bunk.\"\n\nTwo hours later, another person showed up, a woman. Loren heard her call and went to meet her as she came out of the torpedo room.\n\n\"I heard you guys were getting ready to go to sea, so I talked to the sheriff and he let me come down here to talk to you.\"\n\n\"So talk.\"\n\n\"Got out last year after three years aboard _Colorado_.\"\n\n\"Why'd you get out?\"\n\n\"Oh, the usual. I had a boyfriend and he wanted me home to fuck him every night. So\u2014\"\n\n\"The navy will try to sink this boat. You understand?\"\n\n\"Sure.\"\n\n\"And you still want to go?\"\n\n\"I was born and raised in Texas.\" She stopped, thought about that answer, and decided it was adequate. She was of medium height, trim, with a firm mouth and thin lips. Her hair was in a ponytail. The T-shirt she was wearing had a Texas flag on the front and back.\n\n\"What was your rate?\"\n\n\"Quartermaster.\"\n\n\"Can you handle the helm?\"\n\n\"Yes, sir!\"\n\n\"Get your stuff and find a bunk.\"\n\n\"I already dropped my bag through the hatch.\"\n\n\"Welcome aboard.\"\n\nShe stuck out her hand. \"My name is Ada Fuentes.\"\n\n\"Loren Snyder.\" He grabbed her hand and pumped it.\n\nFifteen minutes later Jugs met Ada and shook her hand. She sent Ada aft to meet the rest of the crew, who were running engine room drills.\n\nWhen they were alone, Jugs said, \"Lorrie, we gotta get outta here.\"\n\n\"As soon as the engine room drills are complete.\"\n\n\"No, Loren. Now.\"\n\n\"Are you getting worried?\"\n\n\"You are goddamn right I am. What if those SEALs come before we submerge and shoot holes in the outer casing? Or shoot out the photonics masts? Or throw a chain around the screw?\"\n\n\"Well. . . .\"\n\n\"For God's sake, Lorrie. We can't do Texas any good if they disable us right here at the pier.\"\n\nLoren Snyder ran his hand through his short hair. He had been so worried about his ability to handle this ship, perhaps losing her at sea and killing these volunteers, that he had not sufficiently considered the risks of sitting here at the pier. At the pier, _Texas_ was only a harmless steel sculpture. At sea submerged, she was a powerful warship.\n\n\"You're right, Jugs,\" he acknowledged. \"Let's get two guys topside to dump the gangway and cast off lines, you take the conn from the bridge. I'll do the control room, and we'll get the hell out of Dodge.\"\n\nThat was the way it worked. Julie Aranado gave the orders from the tiny bridge, and using her rudder and screw in reverse, _Texas_ backed out of the slip in which she was moored and began forward motion toward the mouth of Galveston Harbor. Julie had her at five knots when she saw the speedboats with machine guns on the forward deck come through the harbor entrance at high speed and turn toward the submarine.\n\n\"The SEALs are here,\" she shouted into her voice-activated microphone on her headset. \"Give me more turns.\"\n\nShe felt the screw of the sub biting. Behind her a rooster tail was forming. The screw was partially out of the water and was much less efficient than it would be when fully submerged.\n\nAs the three speedboats rounded the far pier, a ragged fusillade rang out. Julie didn't hear it, but she saw the faint traces of smoke and flashes from the rifles on the shore. The sheriff must have stationed sharpshooters on the piers, she thought.\n\nOne of the boats lost way. The other two turned hard to fall in formation with the sub. Julie asked for more turns on the screw.\n\n\"We're going to have to submerge the hull,\" she told Loren in the control room.\n\n\"For God's sake, stay in the channel,\" he replied.\n\nShe looked for the buoys. Fortunately this harbor was dredged regularly for cruise ships and freighters. The wind was playing with her hair as she scanned with the binoculars.\n\nJugs heard the snapping of bullets passing close by. A glance aft. The machine guns on the speedboats were flashing. And the hull was settling under the surface and the submarine was accelerating. Still, the bullets from the machine guns could damage the small conning tower and the photonics masts, all that remained of the submarine above water. Without those masts, _Texas_ was blind at periscope depth. The photonics masts had replaced periscopes. They contained low-light, natural-light, and infrared cameras, and their video was displayed on monitors in the control room.\n\nShe timed the turn to the outbound channel and got it right. The boat answered the rudder nicely and the bow swung, and now they were going southeast into the rollers toward the ocean.\n\nAnother glance aft. One of the speedboats was dropping back, but one was staying with _Texas_ , now doing at least twenty knots.\n\nThe speedboat might have managed to come alongside in calm waters, but now that they were out of the harbor the vessels hit the swells of the sea. Except for a slight pitching motion, _Texas_ was unaffected, but the speedboat began to buck, rising and falling with every down thrust raising a cloud of spray.\n\n\"Give me all you've got,\" Julie said to Loren on the sound-powered phone.\n\nIncredibly, the bow wave that the tower was making became larger. She could hear and see the curl of water against the tower and feel the drops of spray. She held out her tongue and collected a few drops. They tasted salty. Riding the bridge as the sub ran on the surface was a sublime sensatory experience, just as she remembered it from her submarining days, a sensual experience that would stay with her all the days of her life.\n\n\"Twenty-two knots,\" Loren reported.\n\nJulie was watching the buoys. She wanted the safety of the deepest part of the channel. She was in it now, and she needed every foot. The coastal Gulf of Mexico was a shallow sea, unsuitable for submarine operations, the seabed dropping slowly away from the land.\n\nFinally the swells were too much for the last speedboat. A few more bursts, the spang of bullets smacking the steel conning tower, then the boat slowed. The submarine ran on into the empty ocean, past a coaster that may have been the SEALs' mother ship, into the afternoon.\n\nFinally, an hour later, with two hundred feet of water beneath the keel, Julie Aranado said into her sound-powered mike, \"Dive, dive, dive.\" She unplugged the headset and dropped through the hatch, then pulled the hatch down behind her. Perched on the ladder, she spun the crank to dog it down. Then she went down the ladder and lowered herself through the opening in the pressure hull. She dogged that hatch behind her too, sealing the hull.\n\nAt the helm, Ada Fuentes didn't use the planes to help drive _Texas_ under because the water was so shallow. The attack submarine sank slowly as her ballast tanks filled. When the conning tower disappeared under the surface in a boil of white water, the surface of the sea became a slick as the water continued to roil. While gulls soared above the place where _Texas_ submerged looking for small marine life lifted by the swirling water, _Texas_ ran southeastward, toward deeper water. She was in her element now, a powerful warship hidden under the surface, in the great wide sea.\n\nOn Thursday morning, the first day of September, the power came back on in the Baltimore area. One power company, Potomac Electric Power, had figured out that the master computer that controlled the northeast grid had been sabotaged with bad code, so it began manually restoring power in portions of their service area. Still, restoring power to their entire service area would take a while, and restoring service to the entire northeastern United States would take days.\n\nOne of the suburban residents, Lincoln B. Greenwood, a senior executive service employee of the Department of Health and Human Services, had not gone to work that day because without power nothing could be done at the office. He was delighted when his television came back on and lights illuminated in his house. He could hear the toilet tanks filling as water once again surged through the pipes. He grabbed his car keys and opened his garage door, which rose majestically.\n\nGreenwood was worried about the uncertainties the future held and had concluded that he and his wife didn't have sufficient food in the house that would not spoil without refrigeration. And his daughter, with the four-month-old, undoubtedly needed baby food, formula, and diapers. He called her on his cell phone, and she affirmed his shopping list. She and her husband also needed more staples, she said.\n\nThe lot at the mall in Clarksville was packed with cars when Greenwood arrived, which surprised him. All of these stores closed when the power went out because their registers and computer systems were nonfunctional. Greenwood glanced at his watch; the power had only come back on twenty minutes ago. All of these people must have been here waiting, probably for hours, hoping and praying the power would be restored.\n\nThe queue to get into the supermarket, which also had a large pharmacy department, was four deep and extended around the corner of the store into the two-acre mall lot. Lincoln Greenwood got in line, resigned himself to a long wait, and began fretting that the store shelves would be empty when he got inside. The checkout lines would fill every aisle, blocking shoppers' access to the shelves. What a nightmare!\n\nThe man in front of Greenwood said he had parked on the grass across from the main entrance, and the store was not yet open. The clerks were just coming to work, he thought.\n\nAround the corner, out of Greenwood's sight, the manager of the store stood in front of the locked doors and spoke to the crowd. \"Folks, we are going to open the doors in a few minutes and admit ten shoppers a minute from the front of the line. When two hundred are inside, we will admit one additional shopper when one customer comes out. We have to comply with the fire codes, and besides, our checkout clerks can only work so fast. Due to the number of people waiting, we are limiting each shopper to the contents of one grocery cart, so there will be items on the shelves for everyone. Thank you for your cooperation and your patience.\"\n\nThen, five minutes later, as he unlocked the doors, the crowd, many of them white-collar workers from the vast bureaucracies of the federal government, scientists from the nearby Johns Hopkins Applied Physics Laboratory, or mathematicians from the National Security Agency at Fort Meade, ten miles away, rushed the door. The surge was unplanned and unstoppable. The manager was swept out of the way. The exit door, on the other end of the store, shattered, apparently broken by someone in the crowd. People surged in through that door too.\n\nBehind the people in front shoving to get through the doors, the queue disintegrated and became a mob as people ran, shoved, pushed, and forced their way forward. Lincoln Greenwood gave way to panic. His daughter _needed_ the baby supplies. He and his wife _needed_ food and bottled water, and so did his daughter and her husband. Without it, _they might starve if the power went off and water once again stopped coming from the tap_.\n\nSo Lincoln B. Greenwood fought his way forward. He threw several women to the ground and stepped on another who had already fallen. As he came around the corner of the building he could see the huge supermarket doorway, now standing wide open. A man took a swing at him but Greenwood parried the blow and continued his odyssey through the human sea.\n\nHis shirt was torn and his face was bleeding from a woman's fingernails when he made it through the door. People were already pushing shopping carts containing whatever they could grab, pushing them not toward the checkout counters, but toward the doors where people were trying to get in. People coming in began looting the carts. This milling, pushing, shouting, screaming swarm of humanity was no longer a group of civilized beings who attended church, obeyed the traffic laws, and were courteous to strangers; they were a primal force, much like a herd of charging elephants, driven only by their survival instincts.\n\nThe store manager who had unlocked the doors and been swept aside ran into the parking lot and used his cell phone to call 911. Within two minutes a Howard County police car rolled to a stop with lights flashing and siren wailing. The officer killed the siren and met the manager, who ran toward him. Seemingly oblivious to the presence of the officer, the crowd surrounding the doors continued to push, shove, and fight.\n\nThe police officer stood silently, watching the melee in disbelief as the manager shouted to be heard, \"You have got to stop this madness. They'll kill each other in there.\"\n\nIndeed, the officer could see several people sprawled on the sidewalks and in the loading lane, apparently trampled or injured. They were being ignored by the surging mob. The officer tried to estimate how many people he could see, and concluded there were more than a thousand people outside the building.\n\n\"What the hell do you think I can do?\" the officer asked the manager without taking his eyes from the panicked mob.\n\n\"Tear-gas them. My God, people are going to be _killed_ in there! Can't you see that?\"\n\n\"Tear gas isn't going to stop them,\" the cop said, and began talking to his dispatcher through the radio transmitter pinned to his lapel. He got into the patrol car and locked the door so he could hear better. The manager tried to jerk the door open, then pounded on the window with his fist.\n\nWhat the dispatcher knew and the officer didn't was that this scene was being played out in supermarkets all over the county. Smaller mobs, but equally frightened, were looting hardware stores and stealing gasoline at service stations as quickly as it could be pumped.\n\nAt the police station, the chief listened to the calls describing the looting and shook his head. Nothing could be done.\n\nThroughout the Pepco service area, similar scenes were being enacted. What the violent looting would have looked like if the crowds had known that just hours before Barry Soetoro had ordered electrical power shut off nationwide is something that defies speculation.\n\nInside the Clarksville supermarket, Lincoln B. Greenwood managed to fill his pockets with little jars of baby food. He grabbed one box of six-quart cartons of Similac Infant Formula from a shopping cart and made for the door. He had to fight his way out, just as he had fought his way in. Now he had to keep both hands on the box of infant formula to keep it from being torn from his grasp, hug it into his belly, and use his elbows to create a pathway. When he finally reached his car, he still had the Similac, but two of the glass jars in his pockets were broken. He was bleeding from the nose where he had been punched and his shirt was in tatters.\n\nHe got into the car, started the engine, and tried to get out of the parking lot, only to find that people trying to get in had abandoned their cars in the entranceways and ran for the store. He began bumping cars, trying to shove them out of the way. And succeeded. He got to a median, jumped it with his car, and drove away quickly. He was an animal fighting to survive, and he suspected he wasn't going to make it.\n\nWhen Greenwood did get home that evening, the power was off again. Officers from the Department of Homeland Security had visited Pepco headquarters and demanded at the point of a gun that power be shut off throughout Pepco's multi-county service area. When the lights again went out across the Pepco area, they handcuffed every executive they could find and led them away. Everyone else was told to leave the building immediately. The last officer out of the building seized the keys from a terrified janitor and locked the doors behind her.\n\nOblivious to the panic that had seized suburban Maryland and was spreading like an internet virus across America, on Friday morning, the second day of September, Barry Soetoro went before the cameras in his best gray suit and blue tie, a combination that his makeup artist had once assured him was flattering.\n\nIt would take hours, probably at least twenty-four, before the power went off all across the lower forty-eight states, or forty-six since Oklahoma and Texas had tried to go their own ways, so the president and his advisors thought he should use the time to build political support for the battles yet to come. \"Comfort your friends and afflict your enemies,\" Al Grantham advised; Soetoro thought that nugget summed up his mission. He had his best speechwriter prepare the remarks, and they were on the teleprompter, so he could look the unseen audience straight in the eyes as he delivered his truth.\n\n\"My fellow Americans. As I address you today, many of our fellow Americans sit in the dark, sweltering in the heat, with food rotting, without any access to electricity because of the violent acts of desperate and dangerous men. Our nation is at war\u2014at war with ideological fanatics who take the slave-owning Confederate States of America as their model. They want to destroy not only our nation's electrical power grid, they want to destroy this country in pursuit of an extreme ideological vision that would deny women, minorities, and everyday Americans their basic rights. Already they have attacked United States military installations and killed brave servicemen and women who were defending freedom.\n\n\"As you know, I have said repeatedly through the years that the two greatest threats to our nation are right-wing constitutionalists and climate change. I have been ridiculed in the conservative press for those statements, but as I foretold, the threat from the Right has become a deadly peril to our national life.\n\n\"Tonight I ask all loyal Americans for their support, patience, and understanding as we fight to preserve the Union. One hundred fifty years ago, one of my predecessors had to fight the same battle against an enemy that would have kept half our nation as a haven for slavery. Today we battle a similar enemy, an embittered minority who cannot break with the past, whose political beliefs are grounded in ignorance, hate, and bigotry, and who are now in open rebellion against the United States. We face trying days ahead. But I pledge, as President Abraham Lincoln did before me, to preserve our Union and ensure that this nation shall have a new birth of freedom, and that government of the people, by the people, and for the people shall not perish from the earth. Have faith, and I will lead us through the fire to the promised land. Thank you.\"\n\nWhere electricity still flowed through the wires, people nationwide sat staring at their television screens as picked liberal commentators talked about the president's resolve, his vision. His forceful delivery struck just the right note, one woman said. Another commentator, a tenured university professor infamous for urging all white people to commit suicide so the nonwhites of the earth could flourish, pounded the racial drum. Only through Barry Soetoro could the promise of racial justice and equal rights be realized, and white privilege once and for all be defeated and banished from the land.\n\nWhere it was seen, the presidential speech had the opposite effect from the one he presumably intended. Panicked people quite beyond rational thought got in their cars and joined mobs looting stores.\n\nGeneral Martin L. Wynette watched the speech on television in his Pentagon E-Ring office and shrugged sadly. Climate change!\n\nHe asked himself, Were chaos and anarchy the president's real goals, so he could build his socialist dictatorship upon the rubble? Or had the damned fool miscalculated once again? Was he a sublime evil genius, or simply a bumbling, incompetent believer in his own bullshit that fate and poisonous racial politics had raised to a very high place? Not that it mattered\u2014the result was the same in either case. The apocalypse had finally arrived.\n\nA few offices down the E-Ring of the Pentagon from the office of the chairman of the JCS, the chief of naval operations, Admiral Cart McKiernan, was staring at a hard copy of the president's order for the destruction of the power plants in Texas. The best way to do that was with Tomahawk Land Attack Missiles, the admiral thought, but he had no idea how many power plants there were in Texas.\n\nHe had a much better grasp on how many Tomahawks the navy had, which was a little less than 3,400. The Soetoro administration had ended production in Fiscal Year 2015. The missiles cost $1.4 million each and the manufacturer, Raytheon, had stated that restarting the factory and suppliers' production would take two years and increase costs. The next-generation missile was not scheduled into the fleet for ten more years.\n\nA Tomahawk was a subsonic cruise missile that carried a one-thousand-pound conventional warhead. To put a power plant out of action, the missile would need to score a direct hit. To do that, one needed to program the precise GPS coordinates, the latitude and longitude, of each target into the missiles. On a big power plant with large generators\u2014as many as twenty, mounted on thick, reinforced concrete\u2014direct hits by multiple missiles would be required to do significant damage. Perhaps five missiles for each target, because inevitably, as with all complex state-of-the art weapons, Tomahawk reliability was not one hundred percent. More like ninety percent, assuming they were properly and meticulously programmed before firing.\n\nThe missile depended on an accurate satellite survey of the terrain it would fly over to ensure it didn't hit an obstacle, a system called Terrain Contour Matching. This feature allowed the missile to fly as close to the earth as possible, thereby making it difficult for defenders to acquire on radar and shoot down. GPS was used to guide it over the water to its preprogrammed coast-in point and in its terminal guidance phase. So precisely where were the power plants that Soetoro wanted destroyed? It would require several days of staff work to come up with that information from existing satellite databases and then pass it on in a targeting order to the ships selected to launch the missiles.\n\nCart McKiernan wasn't thrilled about using Tomahawks in this manner. Blasting the hell out of Texas could deplete the navy's inventory of Tomahawks, which might hurt America down the road, assuming that down the road there still was an America and a United States Navy that needed the weapon. The Sunnis and Shiites were fighting each other in the Middle East, North Korea's dictator was strutting as usual, China was bullying its neighbors, and Iran was once again giving the world the finger over its nuclear ambitions. Israel was worried about ISIS and Iranian attacks. And what if next week Soetoro decided to punish Oklahoma, Louisiana, or Florida?\n\nThe alternative to Tomahawks was strikes against the power plants using carrier aircraft. USS _Texas_ had just escaped from Galveston, so she was at sea somewhere in the Gulf of Mexico. Giving her an aircraft carrier for a torpedo target didn't appeal to the CNO's military mind. Losing an aircraft carrier or two off the Texas coast would be a poor trade for some power plants, many of which were probably scheduled to be retired in a few years anyway and replaced with more efficient ones.\n\nOf course, McKiernan could pass the request on to the air force and ask if they wanted a piece of this action, but that didn't strike him as a good idea, either. Funding for the next generation of Tomahawk was the stake on the table, and if the navy couldn't complete assigned missions with the missiles it had, perhaps it didn't really need those new, more expensive missiles after all. And no doubt the air force already had a full plate.\n\nMcKiernan attached a memo to the order authorizing the use of one hundred Tomahawks against Texas power plants, and he directed that the plants with the largest generating capacity be attacked first using five missiles per plant. Losing the generating capacity from twenty big power plants would play hob with the Texas grid and leave millions of treasonous Texans sweltering in the dark, which should satisfy even Barry Soetoro, Cart McKiernan thought.\n\nWith CNO's proviso, the presidential order went off to the strike planners.\n\nColonel Nathaniel Danaher spent the morning and afternoon in the B-52 hangar spaces talking to the pilots, crewmen, and ground personnel attached to the squadrons. He wanted to know if any of them would fight for Texas and Oklahoma. A few people from those two states volunteered, but the vast majority didn't want to fight for anybody. He was about to give it up as a bad job when his handheld squawked. \"Major General Hays is here, sir. He came on the last C-130.\"\n\nThe officer on the other end was a major whom Danaher liked because he was competent and could think on his feet. \"You know where to put the troops?\"\n\n\"Yes, sir. Augment our people at the ammo depot and fuel farm.\"\n\n\"Tell General Hays I'll meet him at base operations.\"\n\nNate Danaher got into his staff car and rode across the parking mat the two miles to base ops.\n\nJR Hays was standing there in his camos. Danaher saluted, and it was returned. It felt a little strange saluting JR, who was ten years younger than he was and had been a newly minted major when he served with him, but he did it proudly, with a grin.\n\n\"It went well, sir,\" he said. \"Total surprise. We even got into the message center before they notified the Pentagon, which bought us a few hours, anyway.\" Of course, with cell phones, everyone in Bossier City and Shreveport knew the base had been taken.\n\nThey walked into base ops and headed for the planning room as Danaher reported. \"The commanding general was very unhappy when we stormed into his office and captured him.\"\n\n\"I'll bet he was,\" JR said with a smile.\n\nThey stood in front of a large wall-mounted map and the two career soldiers examined it with practiced eyes.\n\n\"As you suspected,\" Nate Danaher said, \"most of the people here don't want to fight anybody, but we have enough volunteers with the right skills to make up a couple of crews.\"\n\n\"Fine.\"\n\nThe primary reason JR had wanted Barksdale was to prevent B-52s from bombing Texas cities or military bases. Taking as many of the bombers as possible to Texas was not in the cards since the infrastructure and equipment to maintain and fly the planes, not to mention their weapons, was here. It would take weeks, if not months, to move all that to a new base.\n\nThen there was the fact that B-52s, and B-1s for that matter, were essentially defenseless against modern jet fighters equipped with air-to-air missiles with ranges up to a hundred miles. They were dinosaurs and could only be used when one had absolute air supremacy. The B-1s had managed strikes yesterday on railroad bridges in the Powder River Basin and today on Fort Polk and Whiteman Air Force Base in Missouri because there was no air opposition. In the future, there would be. Meanwhile the U.S. Air Force would be getting its act together, and strikes against Barksdale, as long as it was occupied, and Dyess and the other air bases in Texas would soon be forthcoming.\n\nJR Hays and Nate Danaher knew that their window of opportunity would close as soon as Soetoro's brain trust could slam it shut, so they intended to use the bombers while they still could.\n\n\"We hammered Whiteman,\" JR said, jabbing at the map with a finger, \"but of course we didn't get all the B-2s. Expect a few to visit tonight.\"\n\n\"We have four F-22s,\" Danaher said. JR had already seen them as his ride taxied in. \"But no one to fly them. One of the pilots shot up the instrument panel of his before he got off the boarding ladder. The other three aren't interested in joining Texas.\"\n\nJR merely nodded. A competent F-22 pilot\u2014if he had one, which he didn't\u2014might have been able to find B-2s in the night sky, but F-16 pilots certainly couldn't. \"At least those are four F-22s that can't be used against us,\" he said to Danaher.\n\nJR went back to the map. \"We are loading an armored brigade at Fort Hood onto a train. Tanks and troopers and artillery. They'll be rolling tonight. At first they said it couldn't be done. Anyway, they'll be coming through here tomorrow morning. By tomorrow night I want them here.\" He pointed to a position between Barksdale and Fort Polk.\n\n\"A flight of four F-16s will be along in\u2014\" he consulted his watch\u2014\"about an hour. Since we don't have aerial tankers, we'll have to refuel them, top them off. As soon as we can get some B-52s ready, assuming they aren't destroyed by B-2s from Whiteman, launch them and their fighter escorts at the bridges.\"\n\nJR jabbed at the map, which only showed rivers, towns, and interstates. \"They know their targets. I want every highway and railroad bridge across the Mississippi from Baton Rouge to above Memphis in the river by morning. Elvin Gentry says it can be done, and he swore he could do it.\"\n\n\"How many bridges is that?\" Danaher asked.\n\n\"I don't know, but Elvin does. All he has to do is drop at least a span of each one into the river. He says JDAMs will do it. Any intact bridges left standing tomorrow will be attacked with F-16s, or any B-52s or B-1s we have left.\" JDAM was an acronym that stood for Joint Direct Attack Munition. It was a guidance package that screwed into a dumb\u2014freefall\u2014bomb, enabling it to make a direct hit on a preprogrammed target.\n\nJR took a deep breath and let the air out slowly as he surveyed the map. His strategy was simple. He didn't want to fight in Texas, but Louisiana would do fine. If Soetoro's army could get across the Mississippi River to fight. An opposed crossing of a big river was the most difficult maneuver an army could undertake, the equivalent of an amphibious assault against a dug-in enemy.\n\nThey had discussed this objective before, but now that they were on the cusp of trying it, they looked at it again, discussed logistics, roads, what the enemy might do.\n\n\"I wish we could get more B-52 crews,\" Nate Danaher said, a tad wistfully JR thought.\n\n\"If you think we have problems getting people to fight, Soetoro's forces have them worse,\" JR assured him. \"I suspect the U.S. Army and Air Force are on the verge of falling apart, and will unless Soetoro starts putting people against a wall and shooting them. Still, mutiny and mass desertions will certainly slow them down. Our edge is that our people are fighting _for_ something, for a free and independent Republic of Texas. Soetoro is fighting to become an absolute dictator, and the people in uniform aren't stupid. They'll figure out the difference, if they don't know it already.\"\n\n\"You put a lot of faith in average, run-of-the mill people,\" Danaher murmured.\n\n\"Average, run-of-the-mill people won their independence from Great Britain,\" JR shot back, \"and have fought in every war this country ever had. They were at Valley Forge and the Alamo, at Shiloh, Gettysburg, and the Wilderness. Not to mention Belleau Wood, Normandy, Iwo Jima, Vietnam, and Afghanistan. You and I spent our military careers leading them. They'll fight for freedom, all right, to the last drop of blood. Barry Soetoro is on the wrong damned side.\"\nTWENTY-TWO\n\nWith the power out again in suburban Maryland on Friday morning, Lincoln B. Greenwood was a changed man. His adventures the previous day in the supermarket had shaken him to the core. To be in the midst of a mob of people savagely fighting for basic necessities\u2014and fighting just as hard as anyone else\u2014had given him a glimpse of the monster in all of us.\n\nEat or starve. Move or die. Kill or be killed.\n\nThose monsters were waiting out there in the darkness now. Evil people, unrestrained by the bonds of civilization or religion. People willing to do _anything_ to survive.\n\n\"We gotta get outta here,\" he said to his wife, Anne.\n\n\"Where will we go?\" she asked reasonably as she placed candles around the house for the coming evening.\n\nHe gestured vaguely. He hadn't the foggiest idea, but here there was chaos, so instinct told him to leave. To run. To escape.\n\n\"What about Suzanne and her family?\"\n\nThe daughter, the son-in-law, and the baby; Lincoln B. Greenwood hadn't thought about them all morning. He glanced guiltily at the box of Similac powder and the baby food jars still resting where he had put them on the kitchen counter.\n\n\"She married that moron; they are going to have to take care of themselves.\"\n\nHis wife glowered at him, but Lincoln didn't notice. He walked around the living room looking out the windows at the darkness. He could see faint light in a neighbor's window across the cul-de-sac. Candles, he figured. The other houses on the cul-de-sac appeared dark. Maybe the neighbors had already left. Maybe that was the smart thing to do. Get in the car and go. Somewhere. Escape.\n\nHe felt the urge to run, to flee. Adrenaline. He broke into a sweat.\n\n\"Get packed up,\" he said to his wife. \"Your meds, some clothes. Some food. Nothing else. We're leaving.\"\n\n\"But _where_ are we going?\" she demanded.\n\n\"I don't know. We can't stay here. They've been rioting in Baltimore all week. They rioted at the supermarket yesterday. Power is off, phones are off, internet is off. When the inner-city thugs come to the suburbs to loot and burn and rape, we had better be gone.\"\n\n\"I don't want to leave.\"\n\n\"Dear wife, we don't even have a gun, because you wouldn't have one in _your_ house.\" That's when Lincoln B. Greenwood lost it. \" _I don_ ' _t give a shit what you want_!\" he roared to his shocked wife. \"I am not going to sit here waiting to be murdered or die of starvation. Now get upstairs and pack what you want to take.\"\n\nGreenwood ran upstairs and threw three pairs of jeans and some shirts into a bag. Some underwear and socks. He added his blood pressure medicine and his prostate pills to the bag, his toothbrush and toothpaste, his razor and shaving cream, plus some laxatives and a bottle of aspirin.\n\nThen he went to a safe in his closet, opened it, and got out the strips of gold he had invested in when the economy was going to hell in 2008 and 2009. A few Krugerrands. It was damn little, but paper dollars weren't going to be the coin of the realm and credit cards were worthless. Not that it mattered. He had maybe fifty dollars in his wallet and, since the power was out, no prospect of getting more from his bank, even if the ATMs worked or the bank was open and willing to convert every dollar in his savings and checking accounts to cash, which they wouldn't be.\n\nHe stuffed the gold into his pocket and zipped up his bag. Carried it downstairs. Anne was still upstairs packing.\n\nA car pulled up in the driveway and he went to the window. His daughter, Suzanne. He opened the door for her. \"We're leaving, Dad. Going to Gerald's parents' place in Front Royal. We're going to ride it out there.\"\n\n\"Good idea. We're getting ready to leave too. I got some Similac and baby food for you. I'll put it in a bag while you go upstairs and say goodbye to Mom.\"\n\nWhen Suzanne left, Lincoln Greenwood went upstairs to check on his wife. She was sitting on a stool in her bathroom crying.\n\n\"Are you packed?\"\n\n\"Oh, Lincoln. I feel as if I am saying good-bye to my life. What is to become of us?\"\n\n\"If you don't get a move on, woman, we're going to be dead.\" He could feel the evil out there in the night. \"Pack your meds and a few clothes and let's get in the car and go while there is still time.\"\n\nShe sobbed, trying to pull herself together. And nodded. \"You're right. Another few minutes.\"\n\nSo he went downstairs and put his bag in the car, which was in the garage. He would pull the handle that disconnected the door and raise it to get the car out. But not until they were ready to go.\n\nFive long minutes later, as he threw all the dry and canned food they had in garbage bags and stuffed them in the car, he heard engine noises.\n\nHe ran to the living room window and looked out. A police car and a late-model pickup were examining the houses in the cul-de-sac. Lincoln Greenwood went back to the kitchen and helped himself to a carving knife from the block on the counter. He put it up his left sleeve, leaving only a bit of the handle sticking out.\n\nThen he went back to the window. Four young black men were coming up the walk, and all four had pistols in their hands.\n\nOne of them pounded on the door. \"Open up in there or we'll kill all of you and burn this goddamn thing down around your bodies.\"\n\nGreenwood unlocked the door and they rushed in. One of them pointed a pistol in his face. \"Hello, asshole. Who else is here?\"\n\n\"My wife is upstairs.\"\n\nHe jerked his head at his compatriots and they went charging up the stairs.\n\n\"You and me are goin' to the kitchen, motha-fuck. We want the food. All of it. And anything else you got.\"\n\nGreenwood led the way.\n\nThe man immediately began opening cupboards and rooting through the pantry. He turned on Greenwood and pointed the pistol in his face. \"Where is the grub, honkey? Don't tell me you people ain't got no grub in the house. Cause if you do, I'll just shoot you now and be done with it.\"\n\n\"In the car in the garage. We were just about to leave.\"\n\n\"So we got here just in the nick of time. Ain't that sweet? You lead. Get it out.\"\n\nHe went into the garage and began emptying the garbage bags of spaghetti noodles and cans onto the floor.\n\n\"Pick it up. Take it to the front door.\"\n\nGreenwood hoisted a bag in each hand and led off. The thug picked up another and followed him, gun in hand.\n\nWhen the bags were at the front door, the man said, \"Let's go get the rest of it. Seems like you oughta be carryin',\" and he laughed.\n\nAnother trip cleaned out the car. The men who went upstairs were rooting around and shouting to each other, as if they were on an Easter egg hunt.\n\nIn the kitchen, the punk with Greenwood said, \"You got any guns?\"\n\n\"No.\"\n\n\"You better not be lying, 'cause we're gonna look. If I find you lied, I'll just shoot you like a dog and that will be that.\"\n\n\"I'm not lying.\" Lincoln Greenwood was scared and his voice was an octave high and quavered.\n\n\"Pills. We want all the pills you got, motha-fuck. And your grass and powder and smack.\"\n\n\"Pills are upstairs.\" That was a mistake, Greenwood realized. There was nothing in the medicine cabinet in his bathroom, and if the man looked, there would be hell to pay. \"We don't have any dope,\" he added.\n\n\"Like shit! You lyin' asshole. All you white motha-fucks got shit to get high on. You buy it in Baltimore from the guys in the 'hood. Us niggers ain't got the money for nothin' but pot. It's white trash like you that buy the high-dollar shit and then convict the poor dudes sellin' it who ain't got no other way to make a livin'.\"\n\nThe man, who was perhaps twenty or twenty-one, looked around, surveying the crystal and kick-knacks in the kitchen. He pointed his pistol at the counter television that Anne watched every morning when she made breakfast and pulled the trigger. The shot sounded like a cannon. The front of the television showered glass on the counter.\n\nThen the gunman turned his back on Lincoln B. Greenwood. Greenwood pulled the knife from his left sleeve and rammed it between the man's ribs on his right side up to the hilt. Gave it a savage twist and jerked the knife out. Blood squirted out, under pressure.\n\nThe young gunman turned with a funny look on his face, tried to bring the pistol around. Greenwood pushed his arm up and rammed the knife into his solar plexus, then jerked it loose. The gunman collapsed on the floor, bleeding copiously.\n\n\"Hey, Joey!\" A shout from upstairs. \"You havin' fun, man?\"\n\nLincoln B. Greenwood removed the pistol from his victim's grasp and went to the hallway, with the stairs on his left. He crouched against the wall so anyone coming down the stairs wouldn't see him. He waited. When they came down each had an armload of stuff. After the first two got down the stairs and went through the front door, he shot the third one in the back from a distance of three feet. At that range he couldn't miss.\n\nThe man fell the rest of the way down the stairs and piled up on the floor. Greenwood shot him again.\n\nHe ran to the door of the house and tried to align the sights of the pistol, a black thing without a cylinder. Greenwood had just fired the first two shots of his life, and now the problem of hitting anything or anyone who wasn't five feet away became a bit much. He pulled the trigger and the gun kicked and to his amazement the closest man fell flat on his face.\n\nHe aimed as well as he could in the darkness and began firing. Missing. The pistol bucked with every shot and the muzzle flash blinded him. He kept squeezing the trigger anyway.\n\nThe fourth man jumped in the right seat of the pickup and roared off as Greenwood emptied the pistol in that general direction. The truck rocketed out of the cul-de-sac and down the street with its engine howling.\n\nGreenwood walked over to the man lying face-down on the lawn. He had a red spot dead center in his lower back, just visible in the dim evening light. Sheer dumb luck, Greenwood thought, and helped himself to the man's pistol, which lay on the grass by his outstretched hand, along with Anne's jewelry box. Without thinking, he began scooping up the baubles and dumping them back in the box. Most of it was junk, but she had a few nice pieces.\n\n\"I can't move my legs,\" the man whispered.\n\n\"Tough shit,\" Lincoln B. Greenwood said, and began going through the man's pockets. He found an extra magazine for his pistol. A roll of bills. A pack of Marlboros with one cigarette missing and a lighter. Some more jewelry, whether Anne's or someone else's, he didn't know. He put the money and jewelry in his pocket. He almost left the cigs and lighter on the grass, and changed his mind. Someone might trade him something he needed for them.\n\n\"Don't leave me like this,\" the man pleaded. \"Please.\"\n\n\"Die slow, black mother-fucker,\" said Lincoln B. Greenwood, lately of the U.S. Department of Health and Human Services.\n\nUpstairs, he found someone had smacked Anne across the face with a pistol. She was half out of it, with a terrific welt, but apparently otherwise uninjured.\n\nHe threw the rest of her meds in the suitcase and looked at her stuff. Everything neatly folded, dresses and sandals like she was packing for Paris. He shoved some underwear and slacks into the suitcase and closed it. Took it downstairs, walked around the man he had knifed and the man he had shot, and loaded it into the car. Then he began the chore of reloading all the food bags. That took three minutes. He tracked in the blood on the kitchen floor, now a small lake, and began leaving footprints.\n\nThe man he had knifed was apparently dead, his eyes focused on infinity, his face a grimace. Greenwood went through his pockets and found two magazines for the pistol, a wad of bills, and a cellophane baggy that apparently contained marijuana. A lighter, keys, a pack of cigarette papers, some change.\n\nHe took the pistol and a spare magazine from the man he shot coming down the stairs and dragged him into the living room, leaving a bloody streak on the carpet. The guy was still alive, apparently, because he was still bleeding, but Greenwood didn't check. Or care.\n\nGreenwood went back upstairs and used a wet towel to bring Anne around. Helped her downstairs and through the kitchen, trying to avoid the puddles of blood. In the garage he put her in the passenger seat and belted her in.\n\nAfter he got the garage door raised manually, he backed out, put the car in park, and went over to the police car and looked in. Piles of electronic gear, some silverware, and bags of food. He pulled out two bags of canned goods and left the rest. Stowed it in his car and drove off. He didn't even look to see if the man sprawled on the lawn was still alive.\n\nAs he went through Clarksville on Route 32, Greenwood turned off the highway and threaded his way past darkened fast food joints and a closed filling station into the parking lot at the mall. Three cars sat in the huge lot.\n\nGreenwood got out of the car, taking a pistol, car keys, and a flashlight from the glove box with him. He passed a darkened wine store with its windows smashed out. An AT&T store had received similar treatment. He adjusted the pistol in his belt as he walked around to the front of the supermarket. The doors were open, the glass smashed out, and there were no lights.\n\nHe went inside, using the flashlight. The place had been ransacked. Not a crumb was left on the shelves, not even in the candy section. No cereal boxes, bags of flour, cans, none of that. The freezers were empty and the doors standing open. The pharmacy windows were shattered and the door that led behind the counter was wide open. A glance with the flashlight was enough. The pharmacy shelves were completely empty.\n\nNear the back of the store he found a body lying in the aisle. It was a man, in his sixties, perhaps, balding, a modest spare tire. His eyes were open, staring at nothing, and a dried trickle of blood showed on one corner of his mouth. He looked to Greenwood as if he had been trampled.\n\nGreenwood started to turn away when he realized he recognized the man. He couldn't remember his name, but he saw him occasionally in church and they nodded to each other.\n\nWe're all going to end up like this, Greenwood thought, and used his flashlight to leave the store and walk to his car.\n\nAnne was fully conscious. \"Where were you?\"\n\n\"In the supermarket. They cleaned it out.\" He didn't tell her about the body.\n\nHe used his flashlight to inspect the pistols. The empty one was a Glock with a fat handle. There didn't appear to be a safety. He managed to get the empty magazine out and a full one in. Pulled the slide back and let it go. He guessed it was ready to go, but he would have to try to shoot it to find out.\n\nThe other pistol was an old army .45. He tried to pull the slide back, but it wouldn't move. He found the safety. Clicked it off and now the slide came back, showing a gleam of brass. The hammer was all the way back. He carefully put the safety back on. The third pistol was similar, and also loaded.\n\nLincoln Greenwood started the engine of the car and steered through the empty parking lot and out onto the road that led to the highway, Route 32. Turned west and fed gas.\n\nIn Arizona that Thursday night, a crowd of four thousand people carrying candles marched on a Homeland Security detention facility. The facility, on an unused corner of Luke Air Force Base, was off-limits to the public, which tore down the fence with chains and trucks so the crowd could walk through.\n\nThe crowd stood in the darkness with their candles singing hymns for almost an hour. Then they walked up to the gate and went through it, even though the Homeland employees tried to stop them by threatening to arrest the whole crowd.\n\nThe officer in charge gave orders for his employees to fire upon the crowd, yet not a single shot was heard. The prisoners were released and accompanied the crowd, as did many of the Homeland Officers.\n\nIn Pittsburgh a similar crowd of peaceful protesters intent on storming a detention facility were fired upon by several guards. Two people died and three were injured. The crowd pressed in relentlessly, and when it left with the prisoners, two of the guards were dangling from light poles with barbed wire twisted around their necks.\n\nIn Michigan two people were trampled and three shot to death by guards when a crowd attempted to storm a detention facility. The crowd didn't get the prisoners, but all involved knew there would be a next time, and when it came the crowd would be armed.\n\nThe widespread power outage never became total, and neither did censorship at local radio and television stations and newspapers where federal censors had been driven out. It was small towns served by small power plants that informed the larger public about what was going on, and that became the equivalent of the colonists' committees of correspondence before the Revolutionary War.\n\nMore radio and television stations said whatever they pleased on the air. They were becoming more strident over Barry Soetoro's attempts to muzzle them or force them to report only government propaganda as contained in press releases. Of course, for every rebel radio or television station, there were three or four that obeyed the government's edicts, either because ownership or management were progressive liberals who believed wholeheartedly in Barry Soetoro or the censors had them buffaloed: it was impossible to tell which was the case by listening or watching the broadcasts.\n\nRadio audiences were almost exclusively in automobiles and pickup trucks. People at home who had solar power or an emergency generator watched television. The satellites were on the air, and a set of rabbit ears could pull in a local television station if there was one. Some of the rabbit ears were made out of coat hangers. Television audiences tended to be large: family and neighbors gathered in a living room that had service.\n\nAnd in some rural communities served by small local power plants, the electricity stayed on. Either the managers of the plants ignored federal orders or intimidated the Homeland Security or FEMA Gestapo. As long as the natural gas continued to flow through the pipe or the stockpile of coal lasted, the power plants were still in business, supplying hospitals, nursing homes, residences, and everyone else who used power, which was everyone, within their service area. In a blacked-out nation, a few islands of light continued to defy the darkness.\n\nDinner on Thursday evening at our hideout was another culinary masterpiece of MREs, hot sauce, and canned beans. I sat down beside Sarah with my plate. Everyone else was talking about the political situation, damning Soetoro, wondering what the tidbits meant that Willie and Armanti had gleaned from the short-wave.\n\nTimes were tough and getting tougher in Soetoro land. Power seemed to be off in all directions\u2014and the guys weren't hearing any utility repair crews chattering back and forth.\n\nWhile the others gabbed, Sarah whispered, \"What is going to become of us, Tommy?\"\n\nSarah Houston never needs an arm to lean on, but still she made the gesture, and I was touched. \"Hey, babe, I wish I knew.\"\n\n\"When do you think Admiral Grafton will be in good enough shape to travel?\"\n\nI thought about that. I'm not a doctor or trained medic, so I didn't want to move Grafton until it became absolutely necessary. And we had no better place to go. We were in a tactical trap with only one road in and out, yet being on the dead-end of a road to nowhere meant we would have to entertain few tourists. I didn't think the feds were looking for us; I suspected they had a lot of bigger problems to keep them busy. I explained this to Sarah.\n\n\"They could find us from the air,\" she pointed out.\n\n\"If they are looking. In the right place, that is. They probably aren't looking for us at all.\"\n\nCounting on an enemy's incompetence struck me as foolish, yet expecting efficiency from a bureaucracy was the definition of insanity.\n\nGrafton was definitely in less pain this evening. If he had to, he could walk to the restroom. Every other minute was spent sitting or lying down, and talking. Just now he was in the corner of the living room with Jack Yocke on one side and Sal Molina on the other. They were discussing all things Soetoro.\n\nIt seemed that Grafton's adventures with Sluggo Sweatt and his friends had loosened his tongue a good deal. In my on-and-off association with Jake Grafton in the past, I never heard him express a political opinion, which was proper for a serving officer. Don't criticize your superiors in front of the troops. Aye-aye, sir, and all that. However, after his boss fired him and tried to frame him for a murder plot and coup, he probably felt he owed his former superior nothing\u2014not deference, not respect, not silence, not the benefit of the doubt.\n\nI suspected that deep down Grafton thought he owed Barry Soetoro a bullet, the same debt he had paid to Sluggo Sweatt.\n\n\"So explain what is happening to America,\" Jake Grafton asked Sal Molina, the career White House insider.\n\nMolina took a moment to gather his thoughts. \"What we are seeing,\" he said, \"is a classic political reaction to a threatened loss of power. Politics as usual meant that the progressive liberals, who have captured the Democratic Party body and soul, were going to be voted out of office and would probably be out for decades, if they ever got back in. The world is changing quickly, which has profound implications for the Democrats' power-base, which rests solidly on the uneducated and unskilled in the center cities who are being increasingly marginalized in a world economy that is going to grow like a mushroom on steroids in the years ahead.\"\n\nJack Yocke, _Washington Post_ columnist, made a noise with his lips that sounded a bit like a Bronx cheer.\n\nSal Molina ignored the columnist and continued: \"You remember Moore's Law and what happened to computing power in the past fifty years. Gordon Moore was a tech executive who made a prediction in nineteen sixty-five that computing power would double every two years. It was a prediction for exponential growth, and those kinds of predictions rarely come true, and if they do, the growth doesn't last long. But the growth Moore predicted has lasted for fifty years, and the end of exponential growth is not in sight. Intel's latest microprocessor is thirty-five hundred times faster and ninety thousand times more efficient than its first one, the Intel 4004, which came out in nineteen seventy-one.\n\n\"Moore's Law applies to _all_ technological applications, although no other technologies grow at such a multiple of efficiency. The one that will change our world is hydraulic fracking of shale formations. Drilling a well two miles deep and running horizontal lines out as far as fifteen thousand feet in undulating formations is becoming more efficient, more technologically advanced, and cheaper. The cost for these wells keeps dropping. The ocean of oil and gas being produced drives the cost of these commodities down. Shale wells produce over half their output over their lives in the first year, so that makes the frackers the marginal producers; when the market can absorb it, they can supply vast quantities of oil and gas at lower and lower prices.\"\n\n\"I think I see it,\" Jake Grafton said. \"Traditional oil-producing nations will find they get less and less for their oil and their economies will stagnate.\"\n\n\"Ah,\" Molina replied, \"but as the price of oil drops, the world benefits in countless ways. Industries can develop, billions of poor people will get better-paying jobs, prosperity will lift a great many boats. America will prosper. Natural gas is so cheap and abundant that industries that need lots of feed stock are coming back onshore. Low prices for gasoline and natural gas will stimulate every industry in America.\"\n\nYocke shook his head slowly. \"All that may be happening, but who can see it coming? Only fortune-tellers or readers of tea leaves.\"\n\n\"Barry Soetoro and the people on his staff see it coming,\" Sal Molina said bitterly. \"Why do you think he continually says climate change is one of the worst problems facing America and the world, when in fact there is no scientific proof whatsoever that man's activities on this planet have any statistically significant effect on the climate? Because the world of cheap oil and natural gas, with frackers here and in shale formations worldwide providing more production any time it makes economic sense to do so, is a direct threat to the Democratic Party power base. Good-paying new jobs at home mean the unions lose power, which means less money for Democratic candidates. The oil and gas industry's demand for skilled workers will require the companies involved to demand the school systems be reformed to teach the skills required, or they will teach the workers themselves. That threatens the teachers' unions, who are one of the main fund-raisers for Democrats and a huge source of votes, and they indoctrinate the young. So Soetoro has been trying to slow the oil and gas tidal wave with cries of climate change, which polls say eighty percent of the public think is a hoax, and by refusing to approve pipelines or allowing the bureaucracies to issue permits, and causing the bureaucracies to issue reams of regulations that drive up the cost of production. Still, as the cost of drilling and fracking goes down, more oil and natural gas can be produced at cheaper and cheaper prices.\n\n\"In our lifetimes\u2014indeed, in the remainder of the century\u2014oil and natural gas, like coal, will never be scarce; these commodities will become progressively cheaper, like computing power. And as they become cheaper, the economic and technical hurdles for renewable energy, such as solar and wind, become higher and higher with every passing day. In this brave new world we live in, once you get behind the technological curve, you can never catch up. Never, because the state of the art is progressing at an exponential pace! That's a corollary of Moore's Law.\"\n\n\"All this will drive the leftists bonkers,\" Grafton said.\n\n\"Indeed,\" Molina agreed. \"And they fund the Democratic Party.\"\n\nYocke jumped in again. \"So you are saying that Soetoro understands all this and has bet everything on his ability to turn the country into a socialist dictatorship?\"\n\nMolina frowned. \"I don't know that he understands what is happening. He is not a brilliant man. Average intelligence, perhaps. But he understands the political pressures he is getting from unions, from big-city Democrats, from environmentalists, and he can read polls. He hears from OPEC nations worried that their domination of the world oil industry is coming to an end, and with it their prosperity, of which, by the way, only a little trickled down. Islamic fundamentalism is on the rise, and as prosperity in the Arab world drops, it will become more virulent. Barry Soetoro understands _that_!\n\n\"The future of socialism is on display in Venezuela, which will collapse one of these days, done to death by cheap oil. Socialism depends on a huge percentage of the population being unable to survive in a changing world without government help. Entrepreneurship and technical progress promise a world with abundant cheap energy that will raise prosperity for everyone who has the education to participate. Two centuries of cheap energy have made America the most prosperous nation on earth.\n\n\"At heart Barry Soetoro is a socialist, and he loves power. Soetoro understands that in this evolving world of cheap energy, the Democratic Party as it exists will become an anachronism. So he is trying to change the game and come out on top. He and his allies are screaming about climate change and proposing regulations and taxes on energy as a way to increase the cost of energy. Regulations and taxes have devastating consequences on the poor because all those costs must be passed on. In effect, the climate changers have declared war on the poor people of the earth, and they blame the carnage on evil capitalists, banks, hedge funds, and the like: those rich bastards are the enemy.\"\n\n\"All this was discussed in your presence at the White House?\" Jack Yocke asked.\n\n\"In and out of my presence.\"\n\n\"And you fought Soetoro's political vision?\"\n\n\"Why do you think he threw me in a concentration camp?\"\n\n\"So why did Texas secede, or declare independence, whatever you want to call it?\"\n\n\"Texas is going to do well in the cheap-energy future,\" Sal Molina said. \"The people there understand that. The legislature didn't vote for poverty. They voted for a new, better, more prosperous future for everyone in Texas that felt threatened by Barry Soetoro's vision of a socialist utopia, with himself at the helm. Socialism drives taxes up\u2014to fund social justice, the socialists say\u2014and that makes everyone poor. That is socialism's fatal flaw. It has others, but that one always destroys socialism eventually.\"\n\n\"You are implying everyone is an economist,\" Yocke scoffed. \"They aren't.\"\n\nMolina made a gesture of impatience. \"Politics is about macro forces. Texas and the plains states are responding to macro forces that people feel. All thinking people do that, even the uninformed. When you fill up your car, you don't need a PhD in economics to understand that something profound is happening to the price of gasoline, and that something has huge, sublime implications.\n\n\"And you don't have to be a computer scientist to see and understand how computer technology has changed the lives of everyone on earth, except perhaps some pygmies in darkest Africa or headhunters in the Amazon. Cell phones are bringing the internet to places without electricity or running water. People in central Asia are selling goods worldwide on eBay. Computers are revolutionizing life on earth, and that revolution has just begun. Changes are going to happen faster and faster\u2014that's Moore's Law\u2014and change threatens politicians who are invested in the status quo.\"\n\n\"So Texas' actions after the declaration of martial law was the monkey wrench in Soetoro's plan,\" Jake Grafton said thoughtfully. \"That they didn't expect.\"\n\n\"They didn't,\" Molina acknowledged. \"They also thought the paramilitary police they installed in every federal bureaucracy would be able to control the population. And they thought the military would be loyal; they have been purging independent thinkers from the top ranks for years, people in whom they had political doubts.\"\n\n\"Civil war,\" Jack Yocke mused.\n\n\"Like Crackerjacks,\" Jake Grafton said. \"Remember those, with a surprise in every box?\"\nTWENTY-THREE\n\nAfter dinner Travis Clay and Willis Coffee went down to the guard cabin and in a little bit Willie Varner and Armanti Hall walked into the house. They were full of radio news, which they passed to Grafton, Yocke, and Molina.\n\nWe settled in for another night. Before we did, I took off Grafton's tape and bandages and rewrapped them. His bruises were turning yellow and green. That was good, I thought. There were no hematomas that I could see, and no bulges from busted ribs pushing against his skin. He really needed to be in a hospital, but he would never agree to that, even if there were a hospital we could get him into, which there wasn't.\n\n\"Thanks for getting me out of that camp,\" he said. \"If it weren't for you, I'd be dead by now.\"\n\n\"Forget it,\" I replied. \"But I must say, you have a real talent for getting yourself in messes.\"\n\nHe just grunted. I figured he must be doing some serious thinking about where we were going to go and how we were going to survive the next few days, or weeks, or years, when Yocke and Molina weren't bending his ear.\n\n\"We only have so much gasoline for the generators,\" I told him, \"and we need to save what we have for the one in the guard shack so we can monitor the security cameras. I'm going to turn off the one here in the house. There are candles and some kerosene, and we'll cook on the outdoor fireplace. Pour water from the creek into the commodes.\"\n\n\"Oh boy,\" Jake Grafton said.\n\n\"If it's yellow, let it mellow; if it's brown, flush it down.\" Tomorrow, I decided, I'd dream up something to keep Yocke and Molina busy. I told him that.\n\n\"Good,\" he said. \"Neither of them can handle being alone with their thoughts for very long. They've had no practice.\"\n\n\"I'll probably shoot a deer and let them butcher it. Fresh meat would be a treat.\"\n\nThen, out of nowhere, Grafton said, \"Molina is a cynical bastard. He's an economist, so maybe it's his training. He thinks all political behavior, or most of it, can be predicted based upon where the money is going. He's right to some extent, but life is a lot more complicated than that. He's sat over at the White House for years preaching that welfare, Social Security, disability, food stamps, and cell phones would win the hearts and minds of the low-skilled and unemployed. He knows that poor people are easily bought. It's everyone else he doesn't understand.\"\n\n\"How so?\" I ventured.\n\n\"People are motivated by a myriad of things. Religion, tradition, a sense of service, loyalty, curiosity, challenge, accomplishment, praise, patriotism, sometimes a kick in the ass, a sense of rightness. . .and greed, the most basic of human emotions. Greed has built civilization; greed is the reason entrepreneurs start businesses, inventors invent, businessmen try to earn profits. Greed is the reason we aren't still living in caves. _Most_ people want to earn more money so they can have a better life. Yet we could make a long list of human motivations and still not get every one on it.\n\n\"The people at the White House, including Barry Soetoro, don't understand America. None of them has ever been in the military, so they don't understand the men and women in uniform. They aren't religious, so they don't understand the deep antipathy so many feel toward abortion or gay marriage. They never worked manual labor jobs, so they don't understand those who do. They think marriage and traditional morality are old fashioned, so yesterday, so they don't understand those who believe in them. Most of them have never worked in private industry, so they think business is crooked and contemptible. Their political base is in the inner cities, yet they advocate policies that will keep people poor and fight policies that would give the poor a leg up. They are perfect hypocrites, con artists, traitors to the people who believe in them. They willingly tell lies to advance their political agenda, and are amazed when that outrages people.\n\n\"They think they can ram things down people's throats, and maybe they can, to some extent. Remember Willie Varner's comment the other night: 'Tastes like shit, but good'? No matter why you put up with something that tastes like shit, you can't get the taste out of your mouth. Shit is shit.\"\n\nHe paused, so I said, \"Soetoro picked staffers who thought like he did.\"\n\n\"Indeed. Yes-men. And of course women. That may be good for one's ego, but it's a lousy way to ensure you get good advice. Only a man who never ran anything would surround himself with staff that has only one point of view. Barry Soetoro is a lousy manager and a lousy politician; we're all paying for that. And he has another fatal flaw: he doesn't want to hear anything that conflicts with his opinions, or prejudices. He refuses to listen to intelligence that might make him revise an opinion or consider other options.\"\n\n\"There's a lot of that going around these days, especially in the universities.\"\n\nJake Grafton nodded. \"People with closed minds are always the ones who get the worst surprises,\" he said.\n\n\"One thing is for sure,\" I said. \"Soetoro's managed to change the political landscape in the United States, and I doubt if he likes the changes.\"\n\nI wanted to ask the admiral what he had learned from eavesdropping on the White House for the last six months, but decided not to. Sarah shouldn't have told me about it, and if I mentioned it to Grafton he would know I got it from Sarah. So I kept my mouth shut. The thought occurred to me that he had just told me his conclusions.\n\nBut I wondered. If I had listened to the conniving and plotting at the White House for six whole months, what would I have done? Whom would I have told? Who would believe me when I accused the president of the United States of plotting to subvert the Constitution, the Constitution that he was sworn to uphold, and declare himself a dictator? Who would have believed me if I accused him of waiting for a terrorist incident so he could declare martial law?\n\nThe answer of course was no one. Not a solitary soul on planet Earth. That was undoubtedly the conclusion that Jake Grafton reached.\n\nI finished my doctoring and told the admiral he was good to go.\n\nThe attack submarine _Texas_ , now the flagship of the Republic of Texas' Navy, ran just below periscope depth in the Gulf of Mexico. Loren Snyder called an all-hands conference in the control room. He would rather have convened his little congregation of seven in the wardroom, but he wanted to keep a person on the helm at all times. The water was only three hundred feet deep here, so if the sub rammed into the bottom, she might never come up again. Fortunately the floor of the gulf fell away as one proceeded away from the coast, becoming well over a mile deep in places.\n\nSnyder checked the depth, 240 feet; the heading, 130 degrees; the boat's speed on the inertial readout, eight knots.\n\nHe surveyed the faces of his crew. Submarine duty attracted smart, technically savvy people who were interesting to be around, which was why smart, technically savvy people enjoyed it. The challenge was constant and boredom rare.\n\nAda Fuentes was on the helm, Jugs Aranado was sipping coffee, George Ranta, Speedy Gonzales, Mouse Moore, and Junior Smith were drinking water or eating toast from a loaf Mouse made in the galley last night.\n\n\"Okay, folks,\" Loren said. \"We made it to sea. That was the first hurdle, and we got over it, and I thank you. I thought our chances of getting out of Galveston about fifty-fifty. In any event, we are out.\n\n\"A few words on how this Texas Navy sub is going to be run. I am the captain, and I will make all decisions and expect my orders to be obeyed. That said, I want and need advice from each and every one of you on how to run the boat and use it as a military weapon. I hope you will give me honest opinions, and I will use them to make the best decision I can. But once I have decided, that is the way it is going to be. No more debate.\"\n\nHe got nods from everyone standing around the plotting table in the center of the room.\n\n\"Our first problem for discussion is this: What are we going to do with this boat? Are we going to find someplace to hide and wait out the war, making the U.S. Navy worry about where we are and what we might be doing every minute of every day? Or are we going to use her as an attack boat? If we are, what are our targets? Where and how can we do the new Republic of Texas the most good? Your thoughts, please.\"\n\n\"If we don't do anything, the navy will stop worrying before long,\" Speedy Gonzales said. \"They'll assume we managed to submerge forever.\"\n\n\"Someone put two or three Tomahawks into power plants around Houston the other night,\" Jugs said. \"I assume they were launched from a surface ship. At least, I hope they were. If there is another attack boat out here we have major problems. They are fully manned and we aren't.\" She shrugged. \"Anyway, I suggest we put a fish into that surface combatant, then get out of this pond and into the Atlantic, preferably the Gulf Stream, where we can go deep.\"\n\n\"Ranta, you've been on the sonar. Any idea where that destroyer or frigate might be?\"\n\n\"No, sir.\"\n\n\"I've been looking at the chart,\" Jugs said. \"If I were the skipper of that ship, I'd be in the middle of the deep water rigs off Louisiana and Texas. If I were him or her, I'd be worrying about this submarine.\"\n\n\"Tough operating around those rigs,\" Ranta said. \"Sonar will be crap.\"\n\n\"Our main problem is another attack boat out here. It'll be just as tough for them as it will be for us.\" Snyder's audience liked the idea that someone might be worried about what they would do.\n\nSnyder studied the chart. Deep down, he thought the best and safest course of action was to get out of the Gulf of Mexico and look for a warship in the Atlantic. The drawback was that choice would cede the gulf to the United States Navy.\n\n\"Can we operate among those platforms without ramming a platform leg?\" he mused aloud.\n\nJunior Smith said, \"We have to threaten Soetoro's navy some way, and keeping them away from the shipping channels to Houston seems worthwhile to me. Let's make 'em sweat.\"\n\n\"What about torpedoing a Louisiana production platform?\" Mouse Moore asked. \"Or a tanker loaded with Arabian oil? Soetoro's navy has to protect those tankers and platforms or the people of Louisiana are going to get huffy. Not to mention what will happen to insurance rates if one of those crude haulers gets torpedoed.\"\n\n\"Let me think about this,\" Loren Snyder said. \"We certainly can't go under a rig, but we can thread our way around them using the photonics mast. We'll have to get GPS fixes as often as possible, but let's not update the inertial until we are absolutely sure the feds haven't tinkered with the GPS satellites.\" He used a parallel ruler to plot a new course and gave the course to Ada Fuentes at the helm. She brought the boat around to the heading.\n\n\"And slow the boat. Five knots, I think. Ranta, we need you on the sonar for as long as you can stand it. Then I'll relieve you. I was the sonar officer on my first boat, and I think I remember most of it.\"\n\n\"Aye-aye, sir,\" they said.\n\n\"Thank you for your input,\" Snyder told his crew, who went off to the reactor and engineering spaces and, if they were off duty, to try to nap in a bunk. Sleep was precious.\n\nNow on a more easterly course, _Texas_ ghosted along through the heart of the sea.\n\nLoren Snyder busied himself in the control room, checking the computers and torpedo data computer, the TDC. He and Jugs were going to have to run all this stuff. As he worked he thought about his first submarine skipper, who drilled his crew mercilessly and ended up convincing himself and everyone aboard that the crew was the best in the fleet. Incidentally, they passed their Operational Readiness Inspection (ORI) with flying colors and won the Battle Efficiency E.\n\nSnyder picked up the intercom mike and keyed it: \"This is a drill, this is a drill. Runaway torpedo in Tube Two. This is a drill.\"\n\nHe hung the mike in its bracket and heard a loud \"Oh, shit!\" and then the sound of running feet.\n\nThat first night in September, F-16 Falcons from Lackland landed at Barksdale Air Force Base in Louisiana. An hour after midnight, the F-16s were gone again, fanning out to defend the B-52 Stratofortresses, which were beginning their start rituals. They were loaded with JDAMs, two-thousand-pound dumb bombs with a GPS seeker and steering that would guide them to their targets.\n\nJR Hays knew the GPS system was controlled by the United States government, which had the capacity to induce errors into the system, or shut it down altogether, but such an action would affect air navigation all over the earth, no doubt causing a few airliners to crash, and he doubted that the Soetoro administration was ready for the inevitable international political backlash that would cause. Not yet, anyway.\n\nThe B-52s came to life\u2014three of them, because another crew volunteered\u2014and slowly taxied to the takeoff end of the duty runway. The wind was still out of the northwest, so the runway was 33.\n\nElvin Gentry was in the lead bomber. He had flown in at dusk and had a hurried conference with JR and Nate Danaher, then went to the crew briefing.\n\nNo doubt Soetoro loyalists all over the area would have liked to alert Washington when the fighters arrived and took off, and burn up the lines when the B-52s serenaded the city on their climb-outs, but the local power company had obeyed Soetoro's orders and the electricity was off in the greater Shreveport area.\n\nB-52s were old airplanes. The first one flew in 1952. Between 1952 and 1962, when the production line was closed, the air force bought 744 of them at a cost of a couple of million dollars each. Informally and affectionately known by their crews as BUFFs, which stood for Big Ugly Fat Fuckers, they carried up to seventy thousand pounds of bombs at high subsonic speeds and were relatively cheap to operate. The design intended to replace them, the B-70 Valkyrie, was too expensive. The variable-geometry B-1 Lancer and the stealth B-2 Spirit, both of which actually made it into service, were also too expensive to acquire in large numbers, and had high operating costs. Despite the air force's institutional predilection for faster, sexier, and newer, economics reared its ugly head; the air force continually upgraded the B-52s and planned to keep them in service until 2045, over ninety years after the first one had flown. The only version still flying was the B-52H. The air force had invested an estimated $100 million into each one, so far, mere peanuts compared with the cost of newer warplanes. Twenty B-2 stealth bombers had cost Uncle Sam $2 billion _each_.\n\nThe B-52 crews planned on delivering two JDAMs on a support for each targeted bridge. The hope was the two bombs would drop one span in the water, or at least do enough damage that the bridges would no longer support sixty-three-ton M1A2 Abrams tanks.\n\nAnd if tomorrow some bridges were still standing, Gentry planned on launching F-16s carrying two one-thousand-pound JDAMs each. He was willing to trade planes for the bridges.\n\nGentry would have loved to have an airborne early-warning airplane in the sky tonight, but he didn't have one. His F-16s would have to make do by listening to the freqs GCI sites used to control the U.S. fighters. Gentry worried about F-22s, stealth fighters, which could detect and shoot down fighters and bombers of ranges as far as a hundred nautical miles. What he didn't know was that the F-22 wing had sent all the pilots who were willing to fight for Barry Soetoro, all four, to Barksdale. So there were not going to be F-22s in the air tonight. Had he known, he would have been much less apprehensive than he was, and he might even have stayed on the ground tonight. As it was, he thought the risks were so high that he was unwilling to send his aircrews into combat unless he shared the risks with them.\n\nJR Hays, no man to evade risks himself, reluctantly agreed. He didn't want to lose Elvin Gentry, but he had to trust Gentry's judgment and leadership abilities or get someone else. Barry Soetoro would have never understood.\n\nGentry had never before ridden in a Stratofortress, so the pilot's exercise of the Crosswind Crab Control while they taxied felt spooky. The wheels continued to track the centerline of the taxiway, but the airplane turned to point up twenty degrees to the left, then swung back to point twenty degrees to right.\n\nOn the runway, the wind from the right demanded a crab in that direction, so the centerline of the runway was visible out the left side of the pilot's windshield. The BUFF accelerated with all eight engines pulling, they reached decision speed right on time for the load they were carrying, and began rotating five to ten knots before liftoff speed. It wasn't much of a rotation, a bit over five degrees. Then the giant green bomber parted company with the earth.\n\nOn climb out the pilot turned to the general, just to see how he was doing. His face was lit by the glow of the red instrument lights. Gentry was struck by his youth. Captain Rogers, flying a bomber from the 1950s, was all of twenty-seven years old. Gentry felt like a fossil.\n\nJDAMs were units that screwed into freefall bombs. They were comprised of a GPS receiver, a small computer, and canards that steered the bomb to its target, which was a preprogrammed bulls-eye defined by GPS coordinates. Accuracy was only as good as the GPS coordinates programmed in, so satellite maps of the earth had to be consulted.\n\nThe delivery crew, in this case in a B-52, had to use the onboard weapons system to drop the bomb into an invisible cone with its tip resting on the target and the large open end up in the sky. If the bomb were placed within the cone, it could steer itself to a bulls-eye. If it were released outside the cone, the canards would not be able to get the bomb back into the cone, so it would miss. This nebulous cone was defined by the capability of the canards that steered the bomb, by the prevailing wind, and by the angular velocity imparted to the weapon by the airplane that released it.\n\nGuided weapons were the future of aerial warfare, Elvin Gentry believed. The days of dropping huge numbers of dumb bombs in the hope that one or two would hit the target you wanted destroyed were history.\n\nGPS-guided bombs were a technological leap into the future from laser-guided bombs, which steered themselves to a dot of laser light projecting upon the target, projected by the bombing aircraft or a spotter aircraft, occasionally a person on the ground. Unlike laser-guided systems that were useless in bad weather, GPS-guided bombs hit their bulls-eyes all the time, whether they were falling through clear air, clouds, rain, snow, blowing dust, or smoke\u2014as long as you had the correct coordinates for your target: type in a wrong digit somewhere and you missed.\n\nThe cockpit of the B-52 was cramped, almost like a two-seat tactical jet. Gentry sat in the jump seat aft of the pilots, and he didn't have an ejection seat. After everyone else ejected, he was supposed to go to the lower level, or deck, and jump through the hole in the fuselage left by the recently departed navigator or bombardier. It sounded iffy, but if worse came to worst. . . .\n\nThe F-16s were out there somewhere ahead on a fighter sweep, looking for bad guys, protecting the bombers from beyond the range of fighter missiles. That was the theory, which was only as good as the fighter pilots. Elvin Gentry consoled himself with the thought that we all have to die sometime. At least, he reflected, he wasn't in a B-17 on the way to Berlin, harassed every mile by flak and German fighter pilots who knew their business. Those B-17 guys had balls, he thought. This little jaunt tonight was a piece of cake.\n\nHe keyed the intercom and told the crew, \"A piece of cake.\"\n\n\"Yeah,\" the copilot said. \"Sir.\"\n\nIn minutes, as they were still climbing for altitude, the B-52s split up, each headed for its initial fix, to begin a series of bomb runs on bridges. The bombardiers had been plotting their courses and run-ins to their targets, and were now checking their ordnance panels.\n\nGentry heard the cryptic transmissions on the intercom of his BUFF, heard the pilot and copilot running through checklists, and heard the countdown begin to the first bomb release, on the highway bridge on I-20 at Vicksburg. And on the adjacent railroad bridge. The tops of the cones overlapped, so the BUFF would drop four one-ton weapons on this run. He saw the light on the instrument panel as the bomb bay doors came open, he heard the countdown, then the bombs released and he felt the airplane give a jump upward as it became four tons lighter in a fraction of a second. Felt the plane bank into a turn. The next targets were the bridges at Natchez.\n\nSo far, so good, Gentry thought. Then he realized he had been holding his breath. He exhaled and forced himself to breathe deeply.\n\nWalter Ohnigian was a career F-16 pilot. Flying fighters was all he had ever wanted to do since he watched the Thunderbirds perform at an air show when he was twelve. He had attended the Air Force Academy, worked like a slave to get into flight school, and once in gave it everything he had to get fighters. He had fought in Iraq and Afghanistan, had graduated from two courses at USAF Weapons School, and had served a tour as an instructor on F-16s. Along the way he found time to serve a tour in a Navy F\/A-18 squadron, which meant a nine-month cruise aboard an aircraft carrier. He had planned to stay in the air force until they forced him to retire.\n\nThe Texas Declaration of Independence changed his mind. Now he was a Texas fighter pilot. His decision had been easy; born and raised in Brady, Texas, he loathed Barry Soetoro and all he stood for.\n\nSusie Ohnigian, from Colorado Springs, was a tougher sell. She had met Walt when he was a cadet and knew the blood, sweat, and tears he had put in to succeed at his chosen profession. Basically nonpolitical, Susie loved her husband. She knew military aviation has its risks, even in peacetime, and she consoled herself with the indisputable truth that God was in charge of our lives, and He would take Walt when it suited His purpose. He hadn't yet, and she prayed that He wouldn't until they were both old and full of years. She took her marriage vows before the altar of God, and thought it her duty to stand by her husband for as long as they both lived, so with some misgivings, she concurred with his choice.\n\nTonight he was over southeastern Mississippi, listening to the published approach and departure frequencies for Eglin Air Force Base in the Florida panhandle. Ohnigian thought that by the time the air force figured out that bombers were attacking the Mississippi bridges, it would be too late to launch and catch the bombers, which would be several hundred miles west. On the other hand, if Eglin had fighters on a combat air patrol, they could intercept the BUFFs. Or intercept the Texas F-16s.\n\nSo he listened on all the frequencies they might use, and he used the radar in his fighter to sweep the skies for airplanes. Targets. Bad guys. Fighters that might attack the friends in the BUFFs. Fortunately civilian traffic was prohibited by the Soetoro regime. Any targets Ohnigian and Free saw tonight on their radars were enemy airplanes. Or outlaw airplanes whose pilots had decided to roll the dice and take their chances.\n\nThe F-16s flown by Walter Ohnigian and his wingman Drew Free had two AMRAAMS (advanced medium-range air-to-air missiles) and two Sidewinders each, an internal M-61A Vulcan 20-mm cannon, and a two-thousand-pound external fuel tank. No doubt if there were Eglin F-16 fighters aloft, they were similarly armed.\n\nThe AIM-120C AMRAAM was seven inches in diameter and twelve feet long, flew at Mach four, had an active radar homing seeker, carried a forty-pound high-explosive warhead, and had a maximum range of fifty-seven miles. The AIM-9 Sidewinder was a short-range (up to twenty-two miles) missile with infrared homing; in other words, a heat seeker. It was five inches in diameter and nine feet long and carried a twenty-pound warhead. The latest versions could turn over ninety degrees to chase their targets at speeds up to 2.7 Mach, and could even lock on a target up to ninety degrees off the airplane's boresight. Sidewinder was the perfect dogfight weapon: when it locked on your quarry's tailpipe signature, the hunter squeezed it off and the Sidewinder did the rest. Sidewinder even had a limited head-on capability.\n\nTonight Walter Ohnigian hoped and prayed that there were no F-22 Raptors aloft. If there were, he would never see them on radar. His first indications of an F-22 would be a Raptor radar locked on him, so he kept his radar warning indicator in his instrument scan. Nothing so far.\n\nHe checked that he was on Eglin Air Force Base tower frequency. Yes, two fighters were taxiing. A flight of two. The lead had a laconic, gravelly voice.\n\nHe headed that way and eased his fighter into a climb. He wanted to be as high as possible so he would have an energy advantage. His wingman to his right and aft stepped up several hundred feet.\n\nNow the Eglin fighters were airborne and switching to Departure Control. He pushed the button on the radio for the new frequency.\n\nAnd he heard that voice again. Jesus, it sounded like Johnny O'Day! Of all people, Johnny O'Day, his roommate at the Air Force Academy, way back when.\n\nAnother transmission to Departure. Hell yes, it _was_ Johnny O'Day, and he flew F-16s. Headed for the B-52s over the Mississippi.\n\nThe bombs from Gentry's BUFF smashed into the bridges at Vicksburg. They were falling supersonic, so no one on the ground had a clue except for the faint, distant rumble of jet engines way up there in the night. The explosions on each bridge were so close together they sounded like one big bang, which rolled through Vicksburg and woke up several thousand folks.\n\nSlowly, ponderously, the weight of the now unsupported bridge spans carried them down into the dark water of the big river. There were only two trucks on the highway bridge, since traffic on the interstates these days was down to a trickle. One driver on the highway bridge managed to stop his truck; the other rode the span into the river and drowned in his cab.\n\nThe railroad bridge actually had a train on it, rumbling along at eight miles per hour. The bombs went through a railcar, penetrated the track and ballast, and detonated against the targeted abutment. The spans on either side of the abutment began sagging, dragging the train along, down, down into the river.\n\nThe scene would be repeated tonight up and down the river. America was being cut in half with surgical precision.\n\nVictory in a modern dogfight usually goes to the pilot in the most technologically advanced fighter, who will usually detect his enemy first and shoot first. Once missiles are launched, the rest is up to the missiles, those marvels of modern weaponry, which, if fired within their operating envelope, are quite deadly.\n\nTonight Walter Ohnigian fired two AMRAAMs at the Eglin fighters at a distance of fifty miles, head on. They raced off downhill at their targets and had soon accelerated to four times the speed of sound, the active radar in the nose of the missiles probing the night for their targets.\n\n\"Fox Three,\" Walter Ohnigian said over the radio, a transmission he knew Johnny O'Day would hear. He held the transmit button on the stick down and continued, \"Johnny, this is Oboe. You better eject.\" Johnny was married to an operating room nurse and they had two kids. Ohnigian owed him the warning.\n\nIn his fighter, climbing through ten thousand feet, Johnny O'Day's eyes automatically scanned the sky for the pinpoint exhausts of the rocket engines in missiles. Oboe\u2014Ohnigian! After wasting several seconds, he looked at his radar screen.\n\nAnd saw the tiny dots streaking toward his aircraft and that of his wingman.\n\nHe pumped off chaff and tried to turn a square corner. He was pulling eight Gs when the first missile went off just below the belly of his fighter and showered it with shrapnel that penetrated into the delicate internal organs of his steed. One second later the fighter exploded.\n\nThe second AMRAAM exploded as it went through the expanding cloud of pieces.\n\nO'Day's wingman had also turned violently to avoid the oncoming missiles, so after he was sure they had missed him, he had to turn back into the threat to acquire a firing solution on the bogeys on his radar screen. He was turning hard when the first AMRAAM from Ohnigian's wingman actually struck his machine and exploded. Like Johnny O'Day, he died in the fireball.\nTWENTY-FOUR\n\nTexas Ranger Parker Konczyk went to see Colonel Tenney of the TxDPS. \"We think there's a sniper casing the roofs of buildings around the capitol,\" he said. \"He's dressed in a jumpsuit that bears the logo of an air conditioning company. We spotted him with a drone.\"\n\n\"What air conditioning company?\"\n\nKonczyk told him. \"We talked to the owner. He had the van for sale and an Anglo came along, paid him ten grand for it. He wanted fifteen, but the most the guy would pay was ten, cash, and the owner was way behind on his child support, so he took it. He signed the title and never even got the guy's name.\"\n\nKonczyk used an iPad to show Colonel Tenney video from the drone. The man in a jumpsuit on the roof of a bank three hundred yards from the capitol didn't even bother looking at the rooftop-mounted HVAC units, but inspected the roof and lased the capitol and some other buildings, including the hotel with the underground parking garage that was being used by the Texas government as a bomb-proof bunker. \"That location hasn't been published, but half the people in Austin know the government is down there.\"\n\n\"A rangefinder?\"\n\n\"It looks like a laser rangefinder, a small unit that he holds in both hands up to his eye.\"\n\nThe picture on the iPad went to another building and apparently the same man scouted out that roof. Finally, pictures from the drone of the van parked by the curb.\n\n\"So what is your recommendation?\"\n\n\"Right now all we have this guy for is not registering the van in his own name, and a few trespass charges. If we arrest him he'll be out on bail in an hour. And he might not be a sniper; he might be a scout.\"\n\n\"Go on.\"\n\n\"Or we can wait until someone appears on the roof with a rifle.\"\n\nThey discussed it, and decided that the best course was to keep the van under constant surveillance, and the best way to do that and not spook the suspect was to use drones. Konczyk only had access to one.\n\n\"Get a couple more from the National Guard,\" Colonel Tenney said. \"Let's just watch this guy for a while, find out where he is staying and who he sees, and try to figure out how big this conspiracy really is, if there is one.\"\n\nChairman of the JCS General Martin L. Wynette was working late at his office in the Pentagon. The problem he faced was the disintegration of the United States armed forces, all of them, Army, Navy, Air Force, and Marines. The reports from commanders all over the nation were appalling: huge numbers of troops were not available for duty. In some major commands the AWOL rate approached forty percent. Another thirty or forty percent refused to bear arms against Americans, or as they phrased it, to fight for that son of a bitch Soetoro. Sailors on navy ships were refusing to go to sea. Commandos and paratroopers were refusing to go to Texas, Oklahoma, or Alabama, which had just declared its independence. Pilots were refusing to fly, which made it impossible to get fighters aloft to protect military targets or to attack targets in Texas. The most powerful military force on the planet was shattering like old crystal right before his eyes.\n\nMaybe Soetoro was right, Wynette mused. Maybe it was time to start standing some people against the wall and shooting them to inspire the rest.\n\nWynette and several senior members of the JCS staff were trying to figure out just how many willing fighters Barry Soetoro actually had and how to get the willing to where they could fight when the news came in that the interstate and railroad bridges over the Mississippi at Vicksburg had been bombed and were impassable. Even as he tried to digest this information, he learned that bridges were being bombed from Baton Rouge to well above Memphis. Four bridges in Memphis had gone into the river. It was thought that the bombers were B-52s from Barksdale, but of course that was merely speculation.\n\nOn top of all of this were the plights of cities such as Washington, Baltimore, Philadelphia, metropolitan New York, and Boston. And Pittsburgh, Cleveland, Chicago, Milwaukee, Detroit, St. Louis, and Los Angeles. For seventy-five years architects had created urban buildings that were sealed units and uninhabitable without electrical power. Millions of city dwellers were abandoning the cities for the supposedly better life in the countryside, where some planned to throw themselves on the mercy of the rustics while others planned to rob, steal, and kill their way to a better life.\n\nWynette wondered what the heck was going to come of all this. According to radio reports, they were partying in Montgomery tonight. The governor had made a speech, a \"rant\" according to the reporter on the radio, in which he told Barry Soetoro to go to hell and do something anatomically impossible to himself when he got there. The lights were back on in most of Alabama, and the governor vowed they were going to stay on even if the Alabama National Guard had to defend the plants against Soetoro's troops and thugs. He also vowed that a copy of the Ten Commandments were going up in every courtroom and classroom in Alabama; if the justices of the United States Supreme Court didn't like it, he said, they could come to Alabama and take them down, if they could.\n\nIt was obvious to Martin Wynette that Soetoro's propaganda campaign to blame the electrical outages on Texans and right-wing fanatics hadn't moved the needle. Barry Soetoro and his minions were taking the blame.\n\nWynette was trying to put this mess into perspective when the assistant chairman, a four-star admiral, knocked on the sill of the open door and, when Wynette glanced up, strolled into his office and closed the door behind him. He was the only officer in the navy that outranked the chief of naval operations, Admiral Cart McKiernan.\n\nHis name was Hiram Gregory Ray. He was a feisty little cuss, a fighter pilot, and somewhere along the line he had acquired the nickname of Sugar. He was anything but sweet, but the people who worked for him regarded him in awe. Brilliant, technically savvy, aggressive, and competent, he could fire up a room full of sailors and he could kiss a congressman's ass so subtly and perfectly that the bastard would fart red, white, and blue for months.\n\nSugar Ray knew Wynette's peccadillos and usually tried not to fret the boss unnecessarily. After a day spent watching the United States and the armed forces come apart at the seams, he was in no mood tonight to stroke the chairman.\n\n\"I think we can wave good-bye to America,\" he said, \"unless that damned fool in the White House turns the juice back on. New York, Chicago, and LA are in meltdown. Soldiers, sailors, and Marines deserting in droves, refusing to enforce Jade Helm mandates, refusing to fight, refusing to back up the police. . . . Why in the name of God did that idiot turn off the power?\"\n\n\"He blamed it on the Texans,\" Wynette said sourly. \"He's a disciple of Joseph Goebbels. The truth will never catch up to a lie. 'If you like your doctor, you can keep him. If you like your health insurance, you can keep it.' He's that kind of guy.\"\n\nSugar Ray tossed a message on the desk. \"Here's a tidbit that will make your evening. Soldiers at Fort Benning are deserting and taking their weapons with them. They are driving out of the base in trucks. The CG there says all order and discipline are lost. If he tries to arrest people, he is afraid that the MPs will refuse to obey, and if they do obey, he's afraid the people he wants to arrest will shoot back. He asked the chief of staff for guidance.\"\n\nWynette picked up the message and read it. \"A complete breakdown of order and discipline,\" he muttered.\n\n\"I think it's high time we arrest Soetoro and take over the government.\"\n\nMartin L. Wynette stared at Sugar Ray for several seconds, took a deep breath, and said, \"I'll pretend you didn't say that.\"\n\n\"Oh, shove it, Marty! Soetoro is attempting to become a dictator, and he has got to be stopped. We should arrest him or shoot him. Personally, I'd like to shoot him, and I volunteer to pull the trigger, but I'll settle for arrest and solitary confinement.\"\n\nWynette shook the message at Ray. \"And just who the hell do you think we're going to lead over to Pennsylvania Avenue to do all this arresting? Or will it be just you and me with a couple of pistols and any beggars with signs that we can pick up on street corners along the way?\"\n\nSugar Ray cocked his head as he looked at his boss. \"Have you sent any of these numbers\u2014\" he gestured at the messages on Wynette's desk \"\u2014over to the White House?\"\n\n\"Not yet. Tomorrow morning is soon enough.\"\n\n\"What do you think the reaction will be?\"\n\n\"By God, I don't\u2014\"\n\nSugar Ray interrupted and finished the sentence for him. \"You don't know. Civil society in this country is coming apart in the large cities. Old people and babies are dying like flies in un-air-conditioned apartments and tenements; people are fighting for food, looting grocery stores, banks, liquor, and jewelry stores; breaking into ATMs; shooting at police at every opportunity. . .and the military is collapsing. Man, we went back to the stone age in less than ten days! I hope you appreciate the delicious irony of the fact that Soetoro fucked the very people who voted for him.\"\n\nWynette grunted. He thought political loyalty was an oxymoron.\n\nSugar Ray wasn't done. He said to the general, \"Tomorrow morning Soetoro will probably want some heads, and yours is first on the block.\"\n\nWynette didn't reply to that comment.\n\n\"But that's in the short term,\" the admiral said, dismissed that little problem with a flip of his hand. \"Eventually Soetoro is going down hard, and anyone who saluted and said, 'Yes, sir,' may go on the gallows with him. Hitler's and Mussolini's generals didn't fare so well.\"\n\nAdmiral Ray stood and leaned toward Wynette, braced himself with his fingertips on the general's desk, and said, \"My assessment is that this situation is completely out of Soetoro's control. If we lock up Soetoro and everyone else in the White House we can lay hands on, maybe we can stop a humanitarian disaster and save millions of lives. Maybe we can even save our miserable country and some of those morons who voted for Soetoro. . . _twice_.\"\n\nWynette looked at Sugar Ray for a long moment, then asked softly, \"Who have you talked to about this?\"\n\nRay straightened up and took a deep breath. \"All the other chiefs. I was hoping it would be unanimous, but it isn't. The commandant and army chief are with me, but CNO and the air force want to think about things.\"\n\n\"Well,\" Wynette said dryly, \"treason _is_ a big step.\"\n\n\"Yeah\u2014and Barry Soetoro is striding out. How long are we going to wait, Marty, before we call him on it? In a better day to come, Americans are going to ask that question of us.\"\n\nWynette sat stolidly, eyes focused on infinity.\n\nSugar Ray shrugged, then headed for the door. \"I'm going home and getting some sleep,\" he tossed over his shoulder, and pulled the door shut behind him.\n\nWalter Ohnigian actually flew two flights that night, and landed as dawn streaked the eastern sky. The B-52s were safely back on the ground at Barksdale and the Mississippi bridges all had at least one span in the river, from Baton Rouge to Memphis.\n\nOhnigian was numb. He let his wingman do the debrief while he stretched out on a couch in the ops building.\n\nSo Johnny O'Day was dead and he had killed him. Holy mother. . .\n\nWhat was he going to say to Johnny's wife, Ruby? Two little kids. . .\n\nHow was he going to tell Susie, his wife, about this? She and Ruby had double-dated the roommates. The marriages were just a year apart.\n\nStaring at the ceiling, he decided that Ruby and Susie might forgive him, someday. The real problem was how he was going to forgive himself.\n\nAt Fort Carson in Colorado Springs, Major General Douglas Seuss was trying to figure out how to comply with Pentagon demands that he send an armored column from the 4th Infantry Division to fight the rebels in Texas. Most of his soldiers were refusing to fight fellow Americans, and Washington was demanding court-martials. That didn't strike Doug Seuss as a productive idea. He needed soldiers who would fight, not people looking for an opportunity to desert to avoid a combat they thought morally wrong.\n\nSeuss had been trading messages with the Pentagon. West Texas was the finest terrain on this side of the Atlantic for tank operations, but he was unwilling to commit his tanks without air protection. There was no place on the naked plains for tanks to hide if they were attacked from the air. Seuss told the generals in the Pentagon that he was unwilling to sacrifice his troops needlessly to make political points. \"You must guarantee me air cover for my tanks or they will not be committed,\" he said flatly. His worry was that he would get the promise of air cover, commit the tanks, and friendly fighters would never appear while Texas fighters would. That, he thought, was the way the wind was blowing.\n\nSifting through the readiness reports and the results of interviews with his soldiers, he found a company of the 10th Special Forces Group had sixty percent of their troops willing to fight. He called the colonel in command of the group, Colonel Kevin Crislip, into his office. After a heart-to-heart talk, he decided to send the colonel and his volunteers to Texas to blow up some highway and railroad bridges.\n\n\"We've got to do something,\" Seuss said. They looked at maps and decided to blow some bridges on U.S. Route 287 north of Amarillo and several bridges on the nearby railroad. Route 287 was a major truck route between the Pacific Northwest via Denver and Dallas and east Texas. The railroad carried a lot of freight. Bridges were good targets for tactical air, yet the Pentagon was demanding action from the Carson troops, so the ball was in Seuss' court.\n\nCrislip wanted to use CH-47 Chinooks to insert and extract his men, and Seuss agreed. In at dusk, out at dawn was a tactic that would minimize the chance of air attack while the commandos were on the ground. Both officers thought the chances of the Special Forces troopers running into Texas ground forces were slim or none at all, but just to be sure Predators would be launched tomorrow at dawn and reconnoiter. Tomorrow the Green Berets would ride Chinooks to the Army's Pinon Canyon Maneuver Site on the Purgatoire River, and launch from there for Texas at dusk.\n\nAs Colonel Crislip was leaving, Seuss said, \"And colonel\u2014I never dreamed I'd have to say this\u2014make sure the men you take are politically reliable.\" That was the jargon of the latest Pentagon directive. General Seuss found that phrase offensive in the extreme, smacking as it did of the old soviet military and their political commissars, but what could one do?\n\nGeneral Martin L. Wynette was a worried man when he rode to the White House that Friday morning, the second day of September, in his limousine. Arizona had declared its independence, the fourth state to do so, along with Texas, Oklahoma, and Alabama. Other states were meeting this afternoon and tonight and no doubt some of them would pass declarations of independence.\n\nThe people of the big cities from coast to coast were about out of endurance. Without electricity, there was no way to escape the summer heat, no running water, no way to flush toilets, no way to preserve food. Soon there would be no food to preserve, since trucks couldn't deliver without fuel, and even if they could, they wouldn't deliver to looted stores. Calls to police, fire departments, and paramedics went unanswered. Houses burned down with no one there to rescue the kids or fight the blaze. People died from heart attacks because they couldn't get to the hospital. People ran out of medications and couldn't get more. Given time, people would learn to cope, those who survived, but in the interim a lot of people were going to die.\n\nIn Chicago, the Black Panthers had attacked a police station. It looked as if a race war was about to explode in the city. The mayor was begging the governor for National Guard troops.\n\nWynette's aide, Major General Stout, wisely kept his mouth firmly shut that morning as the limo carried them through the streets of the nation's capital.\n\nInside the executive mansion, the soldiers found the president flanked by his chief of staff, Al Grantham, and his senior political advisor, Sulana Schanck.\n\n\"The Texans bombed the bridges across the lower Mississippi last night, Mr. President,\" Wynette said. \"The reports we received at the Pentagon said all the highway and rail bridges were down from Baton Rouge to above Memphis. It's going to take at least a month, perhaps six weeks, before we are ready to mount an invasion.\"\n\n\"Why not drop paratroopers?\"\n\nWynette explained that lightly armed paratroopers didn't have the combat power to hold out long without relief. They were shock troops and not equipped to invade and conquer enemy territory.\n\n\"And then there is the problem of numbers. We are having extreme difficulty keeping people who will fight. About half the army and air force is AWOL. The navy's numbers are better only because they have ships at sea. There are dozens of ships on the east coast that are unable to get under way because the crews have abandoned the ships.\"\n\nSulana Schanck's eyes narrowed and her voice was hard. \"It is time you shot some people, General. I think perhaps ten people from every unit, while the rest of them watch.\"\n\nAl Grantham seemed inspired. \"You've got to teach those damned kids that they have no choice. They are in the United States armed forces, and by God, they'll fight or die.\"\n\n\"As I've said before, I don't have the authority to issue such an order,\" Wynette objected. Indeed he didn't. The Uniform Code of Military Justice didn't provide for summary executions. Islamic militaries might do them, Wynette knew, but those of civilized nations didn't.\n\n\"You do now,\" Grantham said. \"The president has declared martial law and he is the commander in chief.\"\n\nWynette recognized that he was being made the fall guy. \"I'll need a direct order signed by the president,\" he insisted.\n\n\"No, by God, you won't,\" Grantham roared. \"You are going to take the responsibility, General. _You_! _You_ will write the order and sign it. _You_ will have it transmitted to every major command and ship. _You_ will demand that the commanding generals or officers or whoever is in charge find ten people who refuse to fight and have them executed. The names will be reported to _you_. Have I made myself clear?\"\n\n\"Write it out, Grantham.\"\n\n\"No,\" Barry Soetoro said in his coldest voice. \"You'll do it, General. That's a direct order from me. And summary justice for those who disobey orders applies to the Pentagon too, to the E-Ring.\"\n\nSo there it was. Shoot people or we shoot you.\n\nWhile Martin Wynette was swallowing that, Sulana Schanck started in. \"We hear that there is some talk in the E-Ring about a coup. What have you heard about that, General?\"\n\nWynette's first impulse was to deny he had heard anything, but under the stares of Soetoro, Grantham, and the bitch Schanck, he decided that answer might get him shot. There was no telling what they had heard, who had whispered, if anyone had. Schanck was probably just shooting in the dark. Perhaps. Or perhaps not.\n\nSoetoro smacked the table with his open palm. \"Answer, damn it. Don't sit there thinking up a lie.\"\n\n\"The assistant chairman, Admiral Sugar Ray, told me that he, the army chief of staff, and the air force chief of staff did discuss a coup. That is all I know.\"\n\n\"Did you put him under arrest?\"\n\n\"No.\"\n\n\"Ray discusses treason with you and you do not arrest him? Whose side are you on, General?\"\n\nFor the first time in many years, Martin Wynette felt the cold hand of fear grip him with paralyzing fierceness. He had a powerful urge to urinate and somehow managed to hold it in. But he lost control of his face, and knew it.\n\nSoetoro took obvious pleasure at Wynette's discomfort. He glanced at Schanck and made a little motion with his head. She got up and left the room.\n\n\"Did you order the Tomahawks launched?\" Al Grantham demanded.\n\n\"Yes. We should have waited for night, but the missiles will be on their way momentarily, as soon as they can be programmed. Two destroyers will shoot fifty each. They will take out the twenty largest power plants in Texas.\"\n\nGrantham nodded. Once.\n\nWynette said, \"All of the missiles won't get through. In the daytime fighters can find cruise missiles and shoot them down.\"\n\n\"They might get a few,\" Barry Soetoro said, \"and the people of Texas will hear and see them flying over, on their way to cause havoc.\" He smiled. \"The missiles will deliver an unmistakable message that we are in charge and that disobedience has its price.\"\n\nMartin Wynette was enough of a soldier to know that using military weapons to deliver political messages was a good way to lose a war, but he held his tongue. Hitler had tried delivering messages with V-1 and V-2 rockets and that hadn't worked so well. Lyndon Johnson tried to send explosive messages to the North Vietnamese and failed rather dismally. Truly, Wynette thought, Soetoro was a fool.\n\nArmanti Hall and I were exploring the roads in his pickup truck when we saw a little house fifty yards or so from the road, a strip of twelve-foot-wide asphalt that wound around over the hills following the contours. It was a nice enough little clapboard house, but the reason it attracted my attention was the large garden beside it. The flora it contained was big and tall.\n\nWe parked and strolled over. We didn't get very far before we realized that lying near the garden gate was a body.\n\nAs we walked up I could see it was a man. He had that totally collapsed look of the dead that are in the process of returning to the earth. From ten feet away, I could see the dark mottled color of his skin and the bloating of his abdominal cavity, so I guessed he had been dead at least a day, and perhaps two.\n\n\"Don't go any closer to that gate,\" a woman's voice said.\n\nWe turned to face the house. A small old woman with iron-gray hair was sitting in a rocking chair under a roof on a flagstone patio that was just inches above ground level. Across her knees was a lever-action rifle. Her right hand rested on the stock above the trigger.\n\n\"Looks like this gent expired suddenly,\" I said conversationally.\n\n\"It come on him quick,\" she acknowledged. \"I gave him fair warnin' and he decided he needed my tomatoes and beans more than I did. Didn't think I would shoot, I suspect.\"\n\n\"Did you know him?\"\n\n\"Not by name. Seems like I seen him across the mountain at Walmart from time to time, but only to nod to. He was one of the hollow rats, I'm thinkin', used to sittin' on his porch drinkin' beer, waitin' for the welfare check. That and huntin' and fishin' all year 'round. Doubt if he had a garden or much food laid by.\"\n\n\"So he wanted yours.\"\n\n\"So he did. He don't need it now.\"\n\n\"Where's his ride? How did he get here?\"\n\n\"The people he was with drove off after the shot like the hounds of hell were chasin' them. I thought they'd go get the sheriff, but I ain't seen hide nor hair of a lawman, unless you're lawmen.\"\n\n\"We aren't. My name is Tommy Carmellini. My friend is Armanti Hall.\"\n\n\"My name is Angelica Price,\" she said. \"I see you're wearing pistols. Are you with the gover'ment?\"\n\n\"No,\" I said. \"The pistols are just fashion statements in these troubled times, strictly for social purposes. I'm a peaceful man, myself.\"\n\n\"We don't see many black folks up this way. Wasn't ever ver' many in these mountains, and after the Civil War most of those few traipsed off for the big cities and bright lights.\" She said that as if she could remember it. \"Mr. Hall, you're the first black man I've seen in years.\"\n\n\"I don't know whether that's good or bad,\" Armanti told her. \"If I stay I'll have to find me a white girl, I suppose.\"\n\nAngelica Price supposed so too.\n\nThe garden didn't have a weed in it. Rich dark earth was heaped up along several rows of plants that I thought were probably potatoes. There were several rows of tall plants tied up with strips of rag loaded with green tomatoes, and row after row of beans climbing poles, with cucumbers growing among them. A fence surrounded the whole thing, which was perhaps forty yards wide and sixty or seventy yards long. Above the fence were a couple strands of wire that raised the fence too high for a deer to jump. Just to make sure, strands of wire ran across the top of the garden festooned with strips of cloth that flapped in the breeze. Over it all was fishing line strung from pole to pole to discourage birds.\n\nBeyond the garden was a pasture. Up higher on the hill, right on top, I could see a few headstones sticking up inside a wooden fence, which apparently had been erected to keep cattle away from the stones. Three black cattle grazed on the hillside. To the right, almost behind the house, was a three-sided pole shed containing piles of loose hay inside a fence with an open gate. Chickens and a rooster or two wandered around inside and outside the fence.\n\n\"If you'd like, Mrs. Price, we'll tuck this gent under the sod for you. Say. . .up there on the hill in that cemetery.\"\n\nShe turned that offer over, then said, \"No. I think we'll leave him lay right there as a warnin' to any other fool that happens by. He's already startin' to get ripe and I figure he'll get riper, but I can put up with it. And I don't want him up there on the hill with my folks and my man.\"\n\n\"He _is_ getting a little strong,\" Armanti remarked, and headed back down the hill toward his truck.\n\nI liked the old lady, who looked to be in her mid-seventies. She was spry and lean, so it wouldn't have surprised me to learn she was ten years older. It's a wonder some lonely man didn't try to marry her years ago. Maybe some did and were refused.\n\n\"After he gets rotted down some, I'll put him on the compost pile,\" Angelica Price said.\n\nIt took me a moment to get my head around that. Then I asked, \"So how are you getting along without electricity?\"\n\n\"Just fine. Only used it for lights. Got candles and a kerosene lantern, a wood stove and an outhouse, so life is goin' just the way it has for years, twenty-two since my man died. I got ever'thin' I need right here, young man. I was born in this house and hope to live out the rest of my days here, on this piece of earth. This is a good place.\"\n\nI had to agree. Across the valley I could see clouds building. The breeze, smelling of the land as summer came to an end, was rippling the leaves of the distant trees, making the forest look like the surface of the sea. And it was quiet; the only sound was the whisper of the wind.\n\n\"Good luck to you, Mrs. Price,\" I said, and walked down the hill to where Armanti was waiting in the pickup.\n\nAs we drove off, I told Armanti about Mrs. Price's remark about the compost pile.\n\nAll he said was, \"I saw plenty of 'em in Afghanistan and Syria that I would have enjoyed tossing on a compost pile. Killed a few of 'em myself. God bless her.\"\nTWENTY-FIVE\n\nA board the flagship of the Texas Navy, George Ranta, sitting at the sonar console, removed his headset. The boat was at periscope depth amid a large area of drilling and production rigs. \"It's like listening to a mechanical orchestra warm up,\" he told Loren Snyder. \"Machinery noises transferred into the water, drill strings going up and down, hammering, clanking, sucking, gurgling. . .\"\n\nThe photonics mast was out of the water and the video was on the scope. Loren rotated it slowly around the horizon, stopping every few seconds to make a note on the chart he used to back up the computer plots. Paper didn't crash and forget things. It was a decent day up there above the ocean, with a high thin overcast and enough breeze to give the water a bit of chop.\n\nWhat Loren was looking for was a destroyer or frigate, a gray warship. He wanted to torpedo it, then leave the gulf and head around Florida for the Atlantic. First, he thought, put the fear in them. No, first you must find a target. The good news was that any submarine or surface warship amid the rigs was as acoustically deaf as he was.\n\n_Always look on the bright side_ , Loren told himself. _Be optimistic_. _That's one of the rules for successful people_. And we are highly successful people, looking for a place to get a little more of it.\n\nHe gave orders to Ada Fuentes on the helm. He wished he knew more about drilling rigs: he wondered if they were stabilized with underwater cables that fanned out from the surface to the seabed. Stay between them, he told himself. Don't get near one.\n\nHe looked again at the chart. _Texas_ was off Louisiana's southwestern coast and proceeding into deeper water on a course just a bit east of south. Over a thousand feet of water below the keel. If he didn't find a surface warship by the time he reached the southern tip of the area, he thought he should swing more westerly to get into the main channel to Houston and Galveston.\n\nHe went back to the monitor. He was looking southwest, almost on the right beam, when something airborne passed quickly from left to right. He tried to focus the image, pan out, and catch it. If it was a patrol plane, they were in trouble. But it had been so small. A chopper servicing rigs?\n\nWhatever it was, it was gone now. Off to the northwest.\n\n\"What was that?\" Jugs Aranado asked. She was behind him, watching over his shoulder.\n\n\"I don't know.\"\n\n\"Play it back and freeze-frame it.\"\n\n\"You do it. You're better at this.\"\n\nHe got out of the chair, and she sat and began manipulating the controls. In fifteen seconds she had it on the screen.\n\n\"Tomahawk.\"\n\nLoren Snyder looked at the chart and gave Fuentes a new course to steer, one twenty degrees to the right of her current course. \"Let's kick it up to about twelve knots, see if we can close on this guy. I'll keep you away from the rigs. George, those rigs should stand out like sore thumbs on the sonar.\"\n\n\"They do, but there is so much noise in the water. . .\"\n\n\"We're very shallow for twelve knots,\" Fuentes objected. She was worried that an aircraft or satellite scanning the surface with radar or in optical wavelengths might detect the wake.\n\nThe problem, Snyder knew, was that the surface ship, if that was what shot the Tomahawk, could simply run away from a sub cruising slowly. Snyder wanted to put a fish into a destroyer or frigate, and to do it he was going to have to take some chances. On the other hand, if a sub had fired the missile, going in there at twelve knots was asking to be smacked, although Snyder doubted an attack sub would be shooting missiles in water this noisy.\n\n\"Twelve knots,\" he said.\n\nFive minutes later Snyder saw another Tomahawk fly past, just a little to the right, or west, of the sub, on a reciprocal course. It was low, no more than a hundred feet above the ocean. This one seemed to come from almost dead ahead.\n\nHe picked up the ICS mike and keyed it. \"Folks, I think we should all take our general quarters stations. We have a ship ahead, surface or subsurface, that is punching off Tomahawks heading for Texas. I intend to try and torpedo it.\"\n\nLoren lowered the photonics mast and told Ranta to listen carefully. To give the hydrophones a little better angle, he turned another twenty degrees to the west. A half hour later, he had Fuentes go a little deeper and slow to ten knots.\n\nNow Ranta heard the destroyer, or thought he did. It was just a low, deep throb amid the cacophony, one of the echoes bouncing off the bottom. There it was again! Three-three-five degrees, relative. Twenty-five degrees left of the bow.\n\n\"There are two destroyers out there,\" George Ranta announced. \"Both at slow speed, probably launching missiles.\"\n\n\"Retract the photonics mast,\" Loren Snyder said, and to Ada Fuentes on the helm he added, \"Take us down to two hundred fifty feet. Maybe the acoustics are better down there.\"\n\nThe first Tomahawks from the navy's salvo slammed into power plants in the Houston area and knocked out the grid.\n\nJR Hays and Elvin Gentry thought this moment might come, so they had some planes on alert, with the pilots sitting in cockpits. Four planes on alert at Lackland in San Antonio were scrambled and fanned out to the east to look for cruise missiles inbound. They stayed relatively low so their radar would be more effective against the missiles with tiny radar cross-sections, a choice that gave them a high fuel burn.\n\nThe fighter pilots were forbidden to cross the coastline. Neither general wished to risk those precious airplanes in attacks on destroyers, which were very capable of defending themselves.\n\nThere wasn't much else they could do. Except give a heads-up to Jack Hays, who had spent a long half hour with Billy Rob Smith, the Texas emergency coordinator. Billy Rob had been busy borrowing National Guard emergency generators and wiring them into nursing homes and hospitals that didn't have their own. He had even signed a contract with a machine shop in Bryan, Texas, that normally made custom oil-field equipment. Now the fifty machinists employed there were busy manufacturing emergency generators. It would be a week or two before the first ones were ready to be installed, but as Billy Rob told Jack Hays, it was the best he could do. Every generator he could buy, borrow, or steal was being positioned and wired up.\n\nJack Hays gave him a slap on the back and told him, \"Good work!\"\n\nThe acoustics were indeed better at two hundred fifty feet. Ranta found a cluster of rigs ten degrees to port, and found both destroyers. One was dead ahead, the other ten degrees starboard. They were heading northwest, toward Galveston.\n\nTo get the range to the target, Ada Fuentes turned the boat, which was trimmed up and doing about ten knots. After a few minutes, plotting the bearing change, the range was resolved at ten miles to the port target, ten and a half to the starboard one. The targets were moving from left to right, but this would be a fairly simple shot for the Mk-48 torpedoes. They had active sonar seekers and trailed a fiber optic cable behind them, which would allow the submarine crew to ensure they were heading toward the proper targets. If the cables didn't break. If they did, the pump-jet torpedoes would continue on course at fifty-five knots searching for their targets on passive sonar based on the internal logic of their onboard computers, which were programmed by Jugs Aranado. At the very last moment the torpedoes' sensors would go active, ping. Nineteen feet long, twenty-one inches in diameter, the weapon would run under the target and a proximity fuse would trigger its 650-pound warhead. The explosion would break the target ship's back. Time to cover the ten nautical miles to target\u2014eleven minutes.\n\n\"Flood Tubes One and Two,\" Snyder ordered. Jugs Aranado was on the torpedo control console, programming each torpedo. She worked her way through the checklist quickly.\n\n\"Torpedoes ready, Captain,\" Jugs sang out.\n\n\"Fire One,\" Snyder said, and Jugs checked the panel, saw that all lights were green, and pushed the fire button on Tube One. The boat bobbed slightly as the torpedo was ejected by compressed air. On the sonar, Ranta said, \"It's running.\"\n\n\"Fire Two.\" Another little bob as the boat reacted to the loss of the weight of the torpedo and the rush of incoming water to replace it.\n\n\"Close outer doors,\" Snyder ordered.\n\nNow the data from the torpedoes began coming in. Number One was running almost straight, so the chances of the fiber optic cable breaking were small. Number Two turned to a course to intercept the second destroyer. Both were soon up to fifty-five knots, rising from the depths to just under the surface. Both were now armed, but they weren't pinging from their seeker heads.\n\nJugs Aranado was watching their track, waiting. She didn't want to activate their seeker heads until the last possible moment, because the active pinging would be plainly audible to the destroyers' sonar operators. Amazingly, the propulsion system, a pump jet, was very quiet, and so the targets of the torpedoes might not hear them until they were very close. Too close. Especially in these noisy waters.\n\nAboard USS _Harlan Jones_ the cry \"Torpedo incoming!\" from the sonarman in the Combat Control Center galvanized the watch. They knew _Texas_ might be in the area, but had relied upon the noise from the drilling rigs to shield them from attack. Obviously they had been detected and fired upon. The sonar operator had picked up the telltale sound of the pump-jet engines in the torpedoes. He didn't know how close the torpedo was. Actually, it was less than a mile away, approaching at fifty-five knots.\n\nThe tactical action officer, the TAO, a lieutenant commander, ordered decoys fired and used the squawk box to notify the bridge. \"Torpedo inbound starboard side.\"\n\nOn the bridge, the captain didn't waste a second. He shouted, \"All ahead flank. Full right rudder. Give me a ninety-degree turn to starboard.\"\n\nA destroyer is a large ship, and accelerating it takes time, time the captain didn't have. He was only making eight knots to give the Tomahawk missiles a stable platform to launch from. Now, even with full right rudder, it would take time to turn the ship, and time was what he didn't have. Still, he could feel the four turbines answer the engine telegraph. The ship seemed to squat as the twin screws bit deep into the water and her bow slowly began to swing.\n\nAboard _Texas_ , the sound of the destroyer's screws was a signal to George Ranta. \"Port target is accelerating.\"\n\n\"Take her down to a thousand feet,\" Loren Snyder ordered. Ada Fuentes repeated the order and pushed the control yoke forward to use the planes to help drive her down as Jugs was busy on the panel flooding tanks. \"Give me twenty knots.\" Fuentes pushed the throttle forward.\n\n\"Twenty knots, aye.\"\n\n\"Launch the decoys,\" Snyder ordered. Jugs pushed the buttons and the sound of the decoys being launched could be faintly heard; the panel showed four were launched. Decoys were noise- and bubble-makers, which hopefully would attract any ASROC missiles the destroyer might launch. ASROC, an antisubmarine rocket-propelled torpedo, was launched from a vertical tube. The rocket engine carried the Mk-46 torpedo well away from the ship, where it would plunge into the sea and begin searching for a submarine. The noise of the decoys would attract an acoustic seeker, and the bubbles would create a return for an active, pinging sonar.\n\n\"The fiber optic wires are going to break,\" Snyder told Jugs. \"Go active on the torpedoes.\"\n\nIn _Harlan Jones_ ' Combat Control Center, the TAO plotted the probable launching position of the submarine and instructed the man on the ASROC panel where to put the missiles. The TAO decided to launch four. One would hit six miles up the bearing of the torpedo, another at eight, another at ten, and the last one at twelve. Once they were in the water, they would circle and search with active sonar for the submarine.\n\nThen the TAO remembered the oil-production platforms. There was a cluster of them, six or seven, ten degrees right of the bearing line. Would they attract the ASROCs? She shrugged the possibility away and ordered four missiles fired.\n\nBut time was up! The torpedo ran under the hull of _Harlan Jones_ in front of the bridge and exploded. Water being essentially incompressible, the explosion blew a large hole in the bottom of the ship, breaking the keel, and water began rushing in. The ship shook from the hammer blow.\n\n\"All stop,\" the captain ordered, which was merely a term that meant the adjustable-pitch screws were to be brought to fine pitch so the ship wouldn't drive headlong into the ocean and increase the possibility of bulkheads giving way. She began drifting to a stop, which would take a while.\n\nMeanwhile the ASROC launchers spit out four missiles, which roared along the bearing the torpedo had followed.\n\nThe crew of _Harlan Jones_ began trying to save their ship. Fifteen _Harlan Jones_ crewmen were dead. Others would probably die if the watertight bulkheads inside the ship weren't shored up against the sea fighting to invade the vessel. _Harlan Jones_ had fired thirty-three of the fifty Tomahawks she had been ordered to launch.\n\nThe second destroyer, USS _O_ ' _Hare_ , also heard the pinging of the incoming Mk-48 torpedo, and like her sister ship, turned into it so as to present as narrow a profile as possible. She fired her ASROCs up the bearing line of the incoming ship-killer. She had fired off two when the Mk-48 from _Texas_ went under her bow and exploded. The explosion literally cut the ship in half, severed the bow from the ship twenty feet aft of the sonar dome. She wasn't going at flank speed, or the sea would have blown out every internal bulkhead and doomed her. As it was, she lost speed as several watertight bulkheads buckled under the pressure and she began going down at the head. The captain let her drift to a halt.\n\nBoth destroyers had been at General Quarters when torpedoed, with all watertight hatches dogged down, so the damage was not as extensive as it could have been. Aboard _O_ ' _Hare_ , as in _Jones_ , the fight to save the ship began immediately. _O_ ' _Hare_ had launched thirty of the fifty Tomahawks she had been ordered to launch.\n\nAboard _Texas_ , George Ranta and the control room crew heard the whumps of the torpedoes exploding. Snyder had the sound on the loudspeaker. A moment later, they heard the splashes of the antisubmarine torpedoes launched from _O_ ' _Hare_ , followed by the sound of the ASROCs fired by _Harlan Jones_ hitting the water.\n\nLoren Snyder looked at the computerized plot. The cluster of oil platforms were to his port side, perhaps two miles away, so he told Ada Fuentes to turn in that direction. Perhaps the sound of the platforms, at least one of which was drilling a well, would attract the torpedoes searching for his boat.\n\nHe glanced at the depth meter, which read seven hundred feet. They were still going down.\n\nHe had hoped the torpedoes he had fired would catch the destroyers flat-footed, but apparently the crews were well-trained and alert, just in case. Snyder and his small band of fools might well have run flat out of luck.\n\nRose-Marie McGarrity's F-16 was over Galveston when her radar showed a low target running fast to the northwest; it had to be a Tomahawk.\n\nShe rolled her fighter and plunged down, pulling Gs and getting her nose well in front of the missile on a course to intercept. Down through a layer of clouds, down into the gray day underneath, closing on the blip that had to be a cruise missile. It was doing about five hundred knots. Due to the angle at which she was intercepting, she didn't need her afterburner. Yet. She flipped switches, arming the Sidewinders. If she could get a lock-on. . . .\n\nIntercepting at a forty-five-degree angle, still diving into the hot, humid turbulent summer air, Rose-Marie McGarrity found that visibility underneath the goo was no more than four or five miles. She doubted that she would see the missile. This air was like thin soup and she was bouncing in turbulence. She checked to ensure her radar altimeter was set at two hundred feet: it would give her an audible warning if she got within two hundred feet of the surface of the earth.\n\nThen she heard a tone from the Sidewinder, indicating it was locked on a heat source. She was down to five hundred feet above the ground, doing about Mach .9. The target was dead ahead, crossing slowly from right to left.\n\nWith the tone in her ears, she punched off a Sidewinder, a heat-seeker.\n\nIt left the rail with a blast of fire and shot forward into the haze almost too fast for the eye to follow.\n\nMcGarrity was looking through the heads-up display, the HUD, at the target symbol, when she saw the flash. The Sidewinder had scored a kill.\n\nInstantly she was off the juice and soaring upward and right, to point her radar out to sea, just in case.\n\nAnd, by golly, here came another one. Four or five hundred feet above the earth, scorching along to the northwest. McGarrity got that one with a Sidewinder too. Elation flooded her. This fighter pilot gig was hot shit! Again she soared up and turned southeast, toward the sea.\n\nTwo minutes later, she found a third Tomahawk on radar, this one going almost north. Catching it meant a chase, so she engaged the burner and let her fighter accelerate as it again went down toward Mother Earth. She didn't see the Tomahawk until she was about four miles from it\u2014it was a little thing, only visible because the radar told her exactly where to look. She kept the juice on, coming in from an angle, nose well in front to bounce it by sliding up behind it. Gun selected. She kept the missile just below the visible horizon, because to dip below it was to risk flying into the ground. Flying this fast this close to the planet was sublime, a sensory overload.\n\nShe was only a mile from it, flying at just above two hundred feet on a course to intercept, closing at Mach 1.2, when she saw something out of the left corner of her eye. Even as the object registered as a radio tower, she hit one of the supporting cables with her left wing.\n\nAt that speed, about 1,300 feet per second, the steel cable sliced halfway through the wing as if it were cheese; the spar in the left wing broke and the wing separated from the racing F-16.\n\nThere was just no time to react. In a tiny fraction of a second the F-16 rolled hard left, the nose dropped, and the fighter smacked into the ground inverted. The fireball rolled along the land for a thousand yards, dribbling pieces of airplane and Rose-Marie McGarrity. Two houses and one barn caught fire. Smoke mixed with the thick, humid haze.\n\nNo one spoke in the control room of _Texas_. They knew that passive antisubmarine torpedoes were hunting them. And the pundits said the age of robots was still in the future!\n\n\"Put out some more decoys,\" Loren Snyder said. Jugs Aranado went to the control panel and launched four.\n\n\"Where's the bottom?\" Loren asked.\n\n\"Two thousand feet down.\"\n\n\"Take us to fifteen hundred,\" he said to Ada Fuentes.\n\nThe sub continued its descent as water poured into the ballast tanks. Snyder was worried. _Virginia_ -class submarines were the quietest ever made, and the antisubmarine torpedoes weren't designed to find subs this quiet. But. . .\n\nThe tension mounted. They could be dead in a moment. Each breath could be their last, each heartbeat.\n\n\"Do you hear the torpedoes?\" Loren asked George Ranta.\n\n\"Too much noise,\" he whispered. \"I hear pinging but I can't get a direction.\"\n\nBoom. The explosion rocked the boat. One of the torpedoes had found a decoy.\n\nAnd another boom.\n\n\"More pinging,\" Ranta said.\n\nWhere had the other torpedoes gone?\n\n\"We've got to turn,\" Ada said. \"That production platform is dead ahead.\"\n\n\"Right ninety degrees.\" The boat was still going down. Fourteen hundred feet and sinking.\n\nBut they were still alive.\n\nThey heard two more explosions. Well away.\n\n\"The torpedoes went for a platform,\" Ranta said.\n\nA wave of relief swept over the little crew of _Texas_.\n\n\"There are more of them out there,\" Ranta said. \"I can hear at least one. Maybe circling.\" They turned the boat toward the noise and waited. Finally the noise from the torpedo's engine faded.\n\nSnyder said to Fuentes and Aranado, \"Go back up, so we can use the photonics mast.\" To Ranta he said, \"You must keep us clear of those platforms.\"\n\n\"I can hear them.\"\n\nSo they rose slowly from the depths. When the photonics mast was raised, it revealed the injured destroyers lying dead in the water at least six miles to the west. The damaged production platform still stood, but no doubt the crew on it was on their radio reporting the torpedoed destroyers and the torpedoes that struck the platform. And trying desperately to prevent a major oil spill.\n\nLoren Snyder was exhausted. He'd slept six hours in three days. \"Let's get the hell out of the gulf,\" he said. \"Jugs, lay a course for the Straits of Florida. When we are clear of these platforms, take us back down to a thousand feet so the P-3s can't find us. I'm going to sleep.\"\n\nHe staggered along to the tiny captain's cabin and collapsed into the bunk.\n\nFifty-five of the sixty-three Tomahawk cruise missiles launched by USS _O_ ' _Hare_ and _Harlan Jones_ actually impacted Texas power plants. The resulting explosions took seventeen power plants off the grid instantly. Subsequent inspections revealed that at least nine of them could be repaired, and they began producing electricity, at least at a reduced level, within a week or two. The remaining eight were damaged beyond salvage.\n\nThe Texas government kept the amount of damage a closely held secret, although within a day or two satellite reconnaissance would allow analysts in Washington to make reasonably accurate assessments.\n\nNo doubt more Tomahawks were in Texas' future.\n\nA few minutes before three that Friday afternoon, six Secret Service agents climbed from an SUV in front of the main entrance of the Pentagon and went inside. They were escorted to the E-Ring, where they arrested Admiral Sugar Ray, the army chief of staff, and the air force chief of staff. They put all three men in handcuffs and took them to the ground level of the building and into the interior courtyard. The sun was shining and the temperature was already in the low nineties.\n\nEach man was handcuffed to a small tree with his hands behind his back. Admiral Ray knew what was coming. He cursed himself for waiting so long. _We should have done it yesterday_ , he thought bitterly.\n\nThe senior agent drew his weapon and shot each of them in the head. Sugar Ray just happened to be last. \"Rot in Hell,\" Ray told the agent, who then pulled the trigger.\n\nThe agents left the bodies handcuffed to the trees, walked back through the Pentagon, past those horrified officers and enlisted who had actually managed to get to work today, and out the main entrance to their waiting car.\n\nAl Grantham was worried. He had visions of squads of armed troops coming into the White House and arresting the president and everyone around him, taking them to some dungeon and chaining them to the wall. Shooting three senior officers at the Pentagon was an in-your-face insult the armed services couldn't ignore.\n\nHe broached the subject to the president, who sneered. \"They'll do nothing,\" he said. \"They are bureaucrats, paper-pushers, and they achieved their high ranks by not making waves.\" The president lit a cigarette and puffed on it contentedly. \"We have nothing to fear from the generals. They have taken orders since their first day in uniform. Nothing in their experience has prepared them for the day when their superiors might use violence to make them behave.\"\n\n\"They aren't cowards.\"\n\n\"Oh,\" said Soetoro with a hint of derision in his voice, \"but they are. They believe in nothing but the holy flag, keeping the boss happy, and collecting their pensions in the good by and by. The man who believes in something and will use any means to get it will leave them at a loss.\"\n\nGrantham's face reflected his doubt.\n\n\"Relax,\" Barry Soetoro said. \"Whatever they are, they are not gamblers. When have you ever known one of them to take a risk?\"\nTWENTY-SIX\n\nA couple of days after our first visit, Armanti Hall and I decided to call on Angelica Price to deliver a deer haunch. A little fresh meat always goes well, we thought, and maybe we could trade for some fresh potatoes and beans.\n\nWe were in civvies and wearing our web belts that morning, and each of us had an M4 beside us in the cab of Armanti's pickup. There weren't many vehicles on the roads, but the pickups we passed were piled high with firewood and one was hauling a steer. I wondered if it was stolen.\n\nWe followed the little ribbon of asphalt into the hills. When Angelica Price's house came into view, we saw three cars parked nearby. One looked as if it were about eight years old, the other two were show-room new. The new cars didn't have license plates.\n\nWe coasted on by for about fifty yards, then Armanti stopped and I got out with my M4. \"I'll go look the cars over. How about you snuggling up against the bank there and give me cover if I need it.\"\n\n\"Notice that there are only two cows in the pasture now?\"\n\nI hadn't, but a quick scan showed he was correct.\n\nI strolled back with the M4 under my arm, just in case. The new cars weren't locked. One had 170 miles on it, the other 180. The older car, a gray Toyota, wore a Maryland license plate.\n\n\"Hey, man!\" A black guy with a rifle was walking toward me from the house. At a glance, the rifle looked like Angelica Price's old lever action.\n\n\"Get away from them cars!\"\n\nI was partially shielded by the front end of the old one, and I retreated one step to get a little more metal between us. I snicked the safety off the M4.\n\n\"Where's Mrs. Price?\"\n\n\"Never heard of her.\"\n\n\"This is her house.\" I was scanning on both sides. I could see someone at the window of the house watching, and the window was open. If there were anyone to the right or left in the pasture or garden, I didn't see him.\n\n\"You mean that old white woman? She's out in the chicken coop, man. Gave us some shit, she did.\"\n\n\"She dead?\"\n\n\"Not yet. If you don't get the fuck outta here, you\u2014\"\n\nThat's when I swung the M4 up and fired a burst at his legs. He went down hard and lost the rifle.\n\nSomeone fired from the house. I heard the bullet smack into the car. The report sounded like a pistol to me. The distance was about sixty yards, and whoever fired wasn't a good pistol shot.\n\nI couched down, used the car hood for a rest, and put a burst into the window. Silence followed.\n\nOn my right, I could see Armanti removing two AT4s from the back of his pickup. Apparently sneaking up on the house and taking a chance on getting shot didn't appeal to him, either. I hoped the thug lying in the yard had told the truth about Mrs. Price.\n\nArmanti ran up the road, using the embankment of a drainage ditch as cover.\n\nTo keep their heads down, I fired another burst through the window.\n\nThe guy lying in the yard was moaning, holding on to his left thigh. I could see blood at this distance, about twenty yards. Looked like a bullet had clipped an artery.\n\nI moved aft along my mobile fortress, with just the top of my head showing. Armanti was about a hundred yards away now, looking back at me. I gave him a thumbs-up.\n\nHe stood. He had one of the AT4 tubes on his right shoulder. Five seconds, six, then the exhaust blast behind him raised a cloud of crap from the road.\n\nHe had fired at the base of the chimney of the house, which was probably the only thing hard enough to trigger the detonator of the armor-piercing missile warhead.\n\nThe windows blew out, flame gushed forth, and the roof rose a few feet, then crashed down. In seconds the house was on fire.\n\nI began a bent-over trot toward the house. Looked at the guy lying in the yard with blood pumping between his fingers.\n\n\"Help me, man,\" he pleaded.\n\nI grabbed the pistol in his waistband and left him there.\n\nThe house was blazing nicely. No one in the yard or garden. One of the exterior walls of the house was tilting out, falling slowly. I glanced through the open door into the fire. Anyone in there was too far gone to save, even if I wanted to be a hero, which I didn't. Near the garden was a hole with a fire smoldering. Looked like a barbecue pit. Pieces of cowhide and half a carcass were lying near it.\n\nI went on around to the chicken coop, the M4 ready to go. Only one chicken was in sight.\n\nMrs. Price was lying on the hay in the shed. She had been smacked in the side of the head with a pistol several times; one of the blows had laid open her scalp. Now her gray hair was matted with blood.\n\nBeside her were a dead white man and an unconscious white woman. Sparks from the house were causing the hay to smoke. I stepped on the hot spots, and pulled the two women and the dead man out of the shed.\n\n\"Mrs. Price. Mrs. Price, it's Tommy Carmellini. We were by to see you a couple of days ago. Remember?\"\n\nArmanti walked up, looking grim. \"The one in the front yard is still alive.\"\n\n\"Find out who these people were,\" I said. He trotted off.\n\nI went through the dead man's pockets. His driver's license in his wallet, which was empty of cash, said his name was Lincoln B. Greenwood, of Clarksville, Maryland.\n\nMrs. Price was stirring. She was a tough one.\n\n\"They killed him for the fun of it,\" she said. \"He refused to beg. That's his wife, Anne.\" Only her left eye tracked. \"They got here an hour before the others showed up.\"\n\n\"Mrs. Price, I'm going to carry you to the pickup. Then I'll come back for Mrs. Greenwood. We've got to get you two ladies to a doctor.\"\n\nShe couldn't have weighed more than ninety pounds. After I deposited her in the truck, I stopped by where Armanti was squatting beside the wounded man.\n\n\"He says they're from Baltimore,\" Armanti told me. \"Four guys and a whore. Stole the new cars from a dealer and hit the road. Nothing to eat in Baltimore. Stopped here because they were about out of gas.\"\n\nBlood was still pumping from that hole in the guy's leg. He had three or four other holes in his legs, and the right leg was obviously broken, but the one high in his left thigh was a real bleeder. His jeans were sodden. He was lying back on the grass and had relaxed his hold on his thigh.\n\n\"Let me have the keys to your truck. Got to get two women to a doctor. I'll be back for you after a while.\"\n\nArmanti handed me the keys from his pocket. \"This one's gonna be gone soon.\"\n\n\"They pistol-whipped the women and killed the man driving the gray sedan,\" I told him. \"Don't forget Mrs. Price's rifle.\"\n\nI went on to the chicken coop, picked up Anne Greenwood, who had been struck at least twice recently. She also had an old welt across her face. I carried her to the pickup. The wreckage of the house was completely aflame when I drove off.\n\nDr. Proudfoot was in at the clinic in Greenbank. I carried Anne Greenwood in first. The doctor was attending to his nurse, who had been whacked on the head.\n\n\"Got two women for you, Doctor. They've both been pistol-whipped. This is Mrs. Greenwood.\"\n\n\"Just like my nurse. An hour ago. We were held up at gunpoint by a gang of pill-billies looking for drugs. We didn't have any painkillers, but they took every drug I had.\"\n\nI carried Mrs. Greenwood into the examining room and put her on a gurney. Went back to the truck and brought Angelica Price in. I put her on a gurney in the second examining room.\n\n\"My God,\" the doctor said. \"I know Mrs. Price. Why on earth?\"\n\n\"Baltimore thugs. They were after her food. Have you called the law?\"\n\n\"No phone. They wouldn't have come, anyway. Everyone is busy getting robbed or robbing the neighbors. It's anarchy. Maybe the lawmen are home taking care of their families. I would be if I were one of them.\"\n\nHe finished bandaging the nurse and sent her home. Then he spoke to Mrs. Price. \"Can you hear me, Angelica?\"\n\n\"Yes, Doctor.\"\n\n\"I want you to just lie here quietly and let me look at Mrs. Greenwood. Will you do that?\"\n\n\"Yes.\"\n\nI sat holding Mrs. Price's hand while Proudfoot worked on Mrs. Greenwood.\n\n\"Pills,\" she said bitterly. \"The hollow trash is on meth and OxyContin. Surprised they aren't robbing drugstores.\"\n\n\"No doubt they are,\" I said. \"This little clinic looked too good to pass up, I suspect.\"\n\n\"Thanks for coming by, Tommy,\" she said.\n\n\"We've got to stop meeting like this.\"\n\nTwenty minutes passed before the doctor returned. \"Mrs. Greenwood is in a deep coma. She needs to be in a hospital, but the one nearest here is closed.\"\n\n\"What can you do for her?\"\n\n\"Pray.\"\n\nHe began examining Mrs. Price. \"You have a concussion too, Angelica. I'm going to clean up that cut on your scalp and stitch it up, but that's about all I can do. You should be in a hospital too, but since there isn't one around, you need to stay in bed. You're going to have a terrible headache. We'll pray for you too.\"\n\n\"I don't think much of prayer,\" Angelica Price told him. \"I've been praying every night that Barry Soetoro would wake up dead, but apparently God hasn't taken him yet.\"\n\nI went out to the pickup while Dr. Proudfoot worked. Clouds were building over the mountains to the west.\n\nI changed magazines in the M4 and examined the pistol I had taken off the bleeder. A 9-mm, and the magazine was full. God only knows where the bastard got it, but I would have bet a thousand to one he didn't buy it. I decided to give it to the doctor. We were getting a nice collection of weapons, but no matter how hard you try, you can only shoot one at a time.\n\nAnother guy pulled up in an old truck. His son was in the right seat, shot once above the heart. I helped him carry the boy inside. He was maybe fifteen. People were stealing the cows, the man said, and the boy put up a fight.\n\n\"It's like trying to stop an avalanche,\" the old man said. Tears were running down his weathered cheeks.\n\nI emptied my wallet for the doctor, who tried to wave the money away. \"Got nothing to spend it on,\" he said.\n\n\"It won't always be like this,\" I said, with more conviction than I felt. I told him I would be back tomorrow to check on Mrs. Greenwood, and carried Angelica Price out to the truck. I got the deer haunch from the pickup that I had intended to give Mrs. Price and gave it to the doctor instead.\n\nI took her to the CIA safe house and made the introductions.\n\nAfter Sarah had Mrs. Price in bed, I made sure Yocke and Molina were standing by the machine guns in the pits and drove down to the guard shack where Travis Clay and Willie the Wire were playing gin. \"Big-city punks are out, hillbillies are hunting drugs, and scared people are looking for food. All of them are armed. You guys better cowboy up and be ready.\"\n\nWillie was appalled. He wanted to argue, but I told him, \"It's us or them, Willie. If you want to keep on living, you'd better be willing to shoot.\"\n\nMrs. Price's house was down to smoking boards when I got back. The bleeder was dead, and Armanti Hall had dragged him around the house and put him beside Lincoln Greenwood.\n\nThere were four corpses in the remains of the house, burned beyond recognition. The boards were still hot and smoking, and we didn't have body bags, so we left them there.\n\nWe buried Greenwood and the bleeder up on the hill in the Price family plot. Before we tossed the bleeder in the hole, I checked his pockets. He had a nice roll of bills on him.\n\n\"You can't take it with you when you go,\" Armanti Hall said with a sigh.\n\nSome of the bills were blood-soaked. I peeled them off and tossed them in the hole. \"He can take these,\" I said, and handed the rest to Armanti. \"Grab his feet.\"\n\nWe tossed him in, then went down the hill to the garden gate for the man lying there. He stunk to high heaven. Each of us grabbed a foot; we dragged him up the hill and dumped him in on top of the bleeder.\n\nI heaved my cookies before we got the holes filled up.\n\nAs we walked down the hill for the last time, Armanti said, \"I don't want to live in Barry Soetoro's new empire. I'm thinking of going to Texas.\"\n\nThe taste of vomit was strong in my mouth and the smell of death in my nose. \"Maybe I'll go with you,\" I said.\n\nHe had picked some potatoes and green beans from Mrs. Price's garden while he waited for me, and had them in five-gallon buckets. We loaded the buckets into the truck and headed up the road to find a place to turn around.\n\nOn the way back by Mrs. Price's, before we got there, a pickup pulled up below the three cars in the parking area and three white males got out. The oldest one had a rifle. He aimed it at one of the cows in the pasture and pulled the trigger. The cow staggered a few feet, then went down. As the younger males, apparently teenage boys, climbed the fence, the guy with the rifle turned to face our stopped pickup. He held the rifle in both hands and looked at us defiantly.\n\n\"Protecting his kill,\" Armanti muttered. \"I could drop the bastard before he gets a shot off.\"\n\n\"To what purpose?\" I asked, and put the truck in motion.\n\nWe drove on by. The shooter never took his eyes off us.\n\n\"This place is like fucking Syria,\" Armanti remarked.\n\nI didn't argue.\n\nThe evening after the Tomahawk strikes, Jack Hays held a press conference at an \"undisclosed location,\" which was the bottom floor of an underground parking garage in Austin, which fortunately was still on the electrical grid. Three print reporters were there, and two local television reporters, whose cameramen were set up with lights and sound and all the bits and pieces, including a set with a podium for the president of Texas and folding chairs the reporters.\n\nJack Hays started by reading a statement about the progress of the government in converting a state in the United States to a standalone nation. Much had been accomplished by the legislature, which was in session twelve hours a day, seven days a week. A new currency had been approved and a Texas Border Patrol and Customs Service established. The tax department was expanded and statutes passed adopting federal tax rates for the new nation.\n\n\"Everything has to be done at once,\" Jack Hays said, \"and we are up to our elbows in it. Inevitably there will be glitches, but we will try in good faith to correct any mistakes and injustices, if everyone will help us find them.\"\n\nThe first question was, \"Mr. President, what can you tell us about last night's missile strikes in the Houston area?\"\n\n\"The missiles were launched from at least two United States Navy surface ships, both of which were subsequently damaged by an attack from a Texas naval vessel. We know that much because crews on nearby oil-production platforms radioed what they had seen to their companies, who passed it to news media. We are doing our best to get power restored in the Houston area. We understand that at this time of year, loss of electrical power in that area is a humanitarian crisis.\"\n\nAfter a half hour of answering questions about the measures the legislature had passed and was considering, Hays said, \"One more question and we'll call it an evening.\" Three hands went up and he pointed to a reporter from the _Wall Street Journal_.\n\nShe asked, \"Under what circumstances would Texas consider rejoining the United States?\"\n\n\"Under the old Constitution?\" Jack Hays asked.\n\n\"Yes.\"\n\n\"I can't think of any,\" Hays said curtly. He had learned long ago that the best tactic for a politician was to just answer the question asked or evade it. In the silence that followed that short sentence, he reconsidered his answer. Texans deserved to know his thinking, and if they didn't like it, they could say so.\n\nAs he tried to decide what to say, the reporter followed up with the question, \"What if it was no questions asked, all forgiven?\"\n\n\"I'm certainly not going to engage in international diplomacy via your newspaper,\" he said tartly.\n\n\"Even if President Soetoro were removed from office?\"\n\n\"My answer stands.\"\n\n\"You mean, sir, there is no peaceful way to restore the Union?\"\n\nJack Hays weighed his answer as the cameras scrutinized his face and the reporters watched.\n\n\"The old nation was seriously divided,\" he said, \"with political power split between large urban populations and the people in the heartland. Even Texas has some of that. Some of the policies that the elected politicians in Houston wish to follow have been resoundingly rejected by the rest of the state's residents. In a free nation there will always be the push and pull of conflicting views, conflicting desires, conflicting interests. Yet in my judgment, in the old nation the system had broken down, irreparably, and that is why Barry Soetoro chose to become a dictator, to force his political vision on people who rejected it repeatedly at the polls and in the Congress.\n\n\"Be that as it may, the reality is that if the people of Texas wish to continue to enjoy the rights granted by the old Constitution, such as free speech, freedom of religion, freedom of the press, the right to own a gun, the sovereign right to control our borders, the right to be ruled by elected representatives and not be dictated to by the executive or the courts or bureaucracies. . .if Texans want those things, they need to be an independent nation.\"\n\nJack Hays paused, gathering his thoughts. \"Our parting from the United States has not been amicable. Barry Soetoro is raining Tomahawk cruise missiles on the people of Texas. If he wants Texas back in the Union, I would tell him what the citizens of Gonzales, Texas, told Mexican army Colonel Ugartechea in 1835 when he demanded return of a cannon. If you want it, 'come and take it.'\"\nTWENTY-SEVEN\n\nThe interview with the Texas president Jack Hays was broadcast via satellite to those stations and networks still broadcasting in an America with limited electrical assets. It also was soon on the internet. Yet it was on clandestine radio stations that it was picked up by the refugees hidden in the CIA safe farm in the Allegheny Mountains.\n\nI was there when it was played on the recorder that Friday night to the assembled audience in the cabin on the mountainside. I had spent the evening worrying about what would happen when we were discovered, which was bound to happen in the near future. I inspected the machine-gun pits, strategically located around a kill zone in front of the house where any vehicles would have to come to a stop, and inspected each and every rifle and pistol and AT4. I was a worried man, and tired of waiting.\n\nSarah Houston watched me fret and said nothing. Perhaps she was becoming fatalistic. It would be a miracle if any of us got out of this mess alive. I wondered if she was resigned to the inevitable.\n\nYet she was at my side when the tape played, and Jack Hays' clear, confident baritone voice spoke of the problems of the United States and the future of Texas. I watched Jake Grafton's face\u2014the man should have been a poker master in Vegas\u2014and the much more expressive faces of Sal Molina and Jack Yocke. And, I confess, cynic that I was, I wondered how all this squared with the White House plotting that Grafton had overheard. I had quizzed Sarah about that\u2014she said she had listened to little of it. Grafton kept her too busy with other things. But, she said, Jake Grafton had listened. By the hour. Night after night. He knew!\n\nHe knew what?\n\nWhen the tape was over, Sal Molina spoke first. \"When Puerto Rico and Illinois melt down, America has two choices. We can let those two go bankrupt and default on their bonds, or the federal government can take over their debts. If the latter, the states as we know them are doomed: They will cease to exist as sovereign entities. The federal government\u2014actually the executive\u2014will be the ruler of America, able to dictate the smallest decisions, the minutiae of American life, dictate how it will be for his allies and his enemies, of whom he has a great many.\"\n\nYocke snorted. \"It will never happen,\" he declared.\n\nMolina merely gave him a derisive glance, stood, and went up the stairs to bed. Yocke piddled and diddled, looked out the window a bit, then followed Molina upstairs.\n\nGrafton and I were the only two left in the room. I decided to brace him. \"How long are we going to hide here?\"\n\nHe looked at me with two raised eyebrows. \"Are you getting impatient?\"\n\n\"Yes, sir.\"\n\nHe nodded, readjusted his fanny without wincing, and sipped at a cup of cold coffee that rested on the stand beside him. After all his years in the navy, it seemed that he was impervious to caffeine.\n\n\"The whole country is going to hell,\" I said, \"and I feel like a tit on a boar sitting around here. I'm ready to shoot somebody.\"\n\n\"I thought you did that earlier today.\"\n\n\"It wasn't enough. I want to shoot some of those Soetoro sons of bitches, the assholes who decided to rule America and everyone in it. I want to kill those bastards for what they did to my country.\"\n\nHe grunted.\n\n\"We can't just sit here! What about your wife? Your daughter and her husband? What about _America_?\"\n\nHe smiled at me, which drove my blood pressure up another ten points. \"Tommy, there is a time for everything. This pot has to simmer before the country is ready to throw Soetoro out. We're almost there, I suspect, but not quite. Another day or two, then we'll hit the road. We'll have lots of help.\"\n\n\"Oh,\" I said, less than enthusiastically. \"And where the hell are we going?\" I wanted to be sure the old fart had a plan.\n\n\"Why, to Washington of course.\"\n\n\"And this help? Like who?\"\n\n\"We'll pick them up on the way.\"\n\n\"You hope!\"\n\nGrafton looked at me askance. \"You don't really believe in the American people, do you?\"\n\n\"I've killed too many of 'em.\" He didn't say anything, so I added, \"They voted for Soetoro twice. They've sat on their collective thumbs watching the bastard pervert the Constitution, lie like a dog, and poison race relations, and they haven't done anything about it other than elect some gutless Republicans who refuse to stand up to Soetoro. The American people don't seem to give a damn about their country or the future that their kids are going to have to live in. Americans just _don't care_ anymore. Naw, I don't think much of the American people. I wish I'd gotten out years ago.\"\n\nWhen I wound down he cocked his head and looked me in the eyes as he said, \"These are the descendants of the people who hacked out homes in the wilderness. They fought Indians, the British, the Mexicans, and each other. Over a half million Americans died in the Civil War. They peopled a continent and built a nation. They helped win two wars in Europe and defeated Japan. They fought in Vietnam to help a poor people resist communism. They've done their best to fight terrorism and help people in the third world get a leg up. You grossly underestimate them.\n\n\"True, they voted for Soetoro, and a lot of them did it because they naively thought Soetoro would be _good_ for race relations in America, and they thought that was a larger good. This race thing. . .,\" he shook his head, \". . .people want America to include everybody. Martin Luther King left a huge legacy, and America wants his vision, wants an American to be judged by his character rather than the color of his skin. _That_ is the society we want to live in, but we're not there yet. Our first black president got into office not because of his character or his politics, but because he's half-black\u2014or in the parlance of today, black. He gets away with pissing on the Constitution because he's black. He gets away with lying because he's black. He gets away with poisoning race relations because he's black. Even the liberals on the Supreme Court have given him pass after pass.\"\n\nGrafton sighed. \"His time has run out. The American people have gotten a good look at Soetoro this past week, and I don't think they like what they saw. I thank my stars that I'm not Barry Soetoro. He won't like his future.\"\n\nI wanted to believe him, but I didn't. For once I did the smart thing: I kept my mouth shut.\n\n\"Help me to bed,\" Jake Grafton said.\n\nAs I hoisted him, my resolve melted. I asked, \"Do you really think Joe Six-Pack and the missus will shoot at Soetoro's thugs?\"\n\n\"This republic is their heritage,\" he said. \"If they don't value it enough to fight for it, a great many men have wasted their lives fighting for them.\"\n\nThe next morning, Saturday, the first day of the three-day Labor Day weekend, the radio gave us the news that seven more states\u2014Kansas, Nebraska, South and North Dakota, Wyoming, Idaho, and New Mexico\u2014had declared their independence. Georgia had tried to, but federal paramilitary police broke up the legislature and arrested half the politicians. In South Carolina, a gun battle had broken out in the statehouse and at least ten people had died.\n\nThe governor of New Mexico read a statement to the press after the Declaration of Independence was read. \"The proud citizens of New Mexico will never escape poverty unless the flood of illegal immigrants from Mexico and Central America is drastically curtailed. New Mexicans are being robbed of the American dream, the dream that by hard work and thrift they can improve their lot in life and provide a better life for their children. We have taken a stand here tonight. Let history be our judge.\"\n\n\"The liberals are going down hard,\" Jake Grafton remarked.\n\n\"You knew they wouldn't go easy,\" Sal Molina shot back.\n\n\"Yes. I did know that,\" Grafton replied, glancing at Molina's face. I was watching him. No doubt that is why he kept his mouth so firmly shut about Soetoro's plans, which he had overheard on Sarah Houston's White House bugging operation. I wondered what Molina's reaction would be when he learned\u2014if he ever did\u2014that Grafton had been listening to all the White House bullshit and plotting for the last six months, including Molina's.\n\nThat Saturday was the day the Mexican army invaded Southern California. Maybe the Mexicans thought they could carve off a chunk for themselves, or maybe the troops were funded by the drug cartels that wanted their own country.\n\nAs the day wore on, we heard that the Marines at Camp Pendleton were fighting back. All up and down the west coast, U.S. military units raced south to engage. Two carriers left San Diego and began launching strikes against the invading troops and fighting to maintain air superiority.\n\nWhen I had had all of the news I could stand, I went out onto the porch, carrying my M4. Sarah joined me and we climbed the hill and sat under a tree. A breeze whispered in the pines, and we sat for so long and so quietly that a doe and her two fawns eventually wandered by.\n\nWhen they were out of sight, she whispered, \"Life goes on.\"\n\n\"With or without us,\" I said.\n\nTen or so minutes later an airplane broke the silence, flying low, just above the trees. A piston-engine plane. Then I got a glimpse of it through the forest canopy. A tail-dragger. A little Cessna by the look of it. It circled the safe house twice, and the pilot probably got a look at the trucks, even though they were parked under the trees.\n\nI was up and running, searching for a hole in the canopy so I could track the plane, which was still humming pleasantly. The sound was fading though. Then and I saw it in the distance, to the south, apparently circling to land on the grass runway in the valley.\n\n\"Come on,\" I shouted at Sarah. We trotted down the hill, jumped into a pickup, and raced down the road toward the valley.\n\nThe little plane was sitting by the hangar when we arrived. It looked like a Cessna 170, all polished aluminum. I took the carbine as I got out of the truck. A man was helping a woman and two small boys. I didn't see any weapons on them.\n\n\"Hey,\" I said as I walked up.\n\n\"Hello. Is this your place?\"\n\n\"It's private property, but I don't own it.\"\n\n\"The thugs from Philly are looting and burning houses in our neighborhood. We got to the airport and I got my plane. I didn't know where to go, and when I saw this runway, I said, 'Guess we'll try our luck here.'\" He had been eyeing my carbine and the pistol on my web belt. Then his eyes shifted to Sarah, who walked by us over to the woman.\n\n\"My name's Johnson. That's my wife,\" he told me. \"We had to get out. I think thugs killed the woman next door and left her body in the house when they burned it down.\"\n\nWe opened the hangar and shoved his plane in tail-first, chocked it, and closed the doors. I loaded everyone in the truck and took them to the safe house.\n\nJake Grafton was sitting in an easy chair in the main room. He perked right up when I told him about the plane. He skipped the social pleasantries with Johnson. \"How much fuel is in it?\"\n\n\"Both tanks are about half full.\"\n\n\"Tommy, go back to the hangar and see if there is any avgas there.\"\n\nAs I left, Grafton was asking Johnson about bridges and roadblocks he might have seen from the air. _Maybe this will galvanize Grafton_ , I thought. _Get him moving_. God, I was tired of sitting doing nothing while America went back to the stone age.\n\nA plane would be a good thing to have if we could keep it fueled. Our own air force. I opened a panel of the sliding hangar door and went inside. And Lady Luck smiled. I found a fifty-five-gallon plastic drum full of fuel in the hangar. The drum had a hand-crank pump mounted on top and a hose. I was maneuvering the drum under the left wing when I heard a pickup truck drive up. I figured it was Armanti and I needed him to crank the pump while I stood on the ladder with the hose.\n\nI turned. Two scraggly faced locals in filthy jeans and T-shirts stood at the door of the hangar and had me covered with scoped deer rifles. Both were grinning at me with yellow teeth.\n\n\"Well, well, well! By God, we heard it and here it is,\" said one of them.\n\n\"Just shuck that pistol, asshole, and maybe we won't shoot you,\" said the other.\n\nI pulled out the Kimber and tossed it in the dirt.\n\n\"Look the plane over, Benny. You, get over here against the wall.\" He waggled the barrel of his rifle and I went.\n\nThe one called Benny picked up my shooter, examined it, and tucked it into his pants. The other kept his rifle pointed at my belt buckle while Benny opened the door to the plane and looked around inside.\n\n\"Jearl, that kid is gettin' away!\" A call from outside. So there were more of them out there.\n\nJearl must have been the stalwart guarding me, because he forgot me and ran back to the open panel in the door. \"Hey!\" Jearl went dashing out of sight, shouting, \"Get off your asses and catch her!\"\n\nI grabbed a heavy wrench off the shelf and stuck it up my sleeve. Benny strolled over from the plane, pulling my Kimber from his waistband. He had a big wad of snuff under his lower lip. \"You're a big one, ain't you?\"\n\n\"Your mom know you boys are out causing trouble?\" I asked.\n\n\"Man, the country has gone to hell. We can be just as bad as we wanna be and ain't nobody to say we can't.\"\n\n\"And how bad is that?\"\n\nI heard the sound of another truck. So did Benny, and he turned his head to his right toward the door. I let the wrench slide down into my hand; as he turned back toward me I hit him in the jaw with it with everything I had, right on top of his snuff wad. The blow put him down hard and I was all over him. Got my pistol and his rifle. He was only partially conscious. His jaw was obviously broken. Blood, saliva, and brown tobacco juice dribbled from his open mouth.\n\nThe rifle was some cheap piece of Walmart crap with a plastic stock, but it had brass in the chamber when I pulled the bolt back for a peek.\n\nI stepped to the left edge of the hangar door and looked around. Jearl was on the runway, about fifty yards from me, pulling a girl about nine or ten years old along by the arm. There were two men in the back of their pickup, and they had rifles pointed at Armanti, who was stepping from his truck with his hands up.\n\nI braced the rifle against the door and shot the man on the right in the bed of the truck. Worked the bolt. The other one was quick as a cat. He spun toward me, leveled his rifle, and fired. Something burned my neck and my shot went wild. I worked the bolt again and got on him, but he was already going down. Armanti had shot him in the back.\n\nJearl, the guy in the meadow, held the girl against him with his left hand and pointed his rifle toward me with his right. I didn't figure he could even hit the hangar with that rifle shooting one-handed from the waist. I rested the rifle against the edge of the hangar door again and looked through the scope. Steadied the crosshairs on Jearl's head and squeezed one off. He went over backward.\n\nI walked out for a look. The bullet had taken his head clean off. Above his neck only his lower jaw remained.\n\nThe girl was sobbing. I picked her up and walked back to the hangar. Armanti was standing, pistol in hand, over the guy I had tamed with a wrench. The guy was coming around.\n\n\"You want me to finish him?\" Armanti asked me.\n\n\"Be as bad as you wanna be,\" I told him flippantly.\n\n\"Who is this kid?\" Armanti asked Benny, who was now moaning and writhing in the dirt.\n\nBenny mumbled something, holding his mouth. Armanti kicked him, and he squirmed and moaned louder.\n\n\"I asked you a question, Jack,\" Armanti said, \"and if you don't tell it to me straight, things could get really iffy for you. Hold your jaw together and answer me! Who is she?\"\n\nWith a supreme effort, holding his jaw with both hands, Benny said, \"Some kid we picked up. Jearl was porkin' her.\"\n\n\"Where's her folks?\"\n\n\"Jearl killed 'em.\"\n\nI didn't even see it coming. Bang. The pistol in Armanti's hand went off, and the guy lying in the dirt was instantly dead with a 9-mm bullet through his head.\n\nArmanti Hall holstered his pistol and came over to me, looked at the girl's face streaked with dirt and tears. \"It's gonna be all right,\" he said softly.\n\n\"Take her up to the house,\" I said, \"then come back and help me fuel this plane.\"\n\nHe carried the child out to his truck, and I got busy tossing bodies into the back of the junky pickup they had arrived in. The corpses had almost stopped oozing blood, but I got some blood and brains on my shirt anyway. I figured the stuff would wash off. The key was still in the ignition of the truck, so I didn't have to go through their pockets.\n\nMy neck burned like fire and I could feel blood trickling down into my shirt. Another fucking scar! Welcome to the revolution.\nTWENTY-EIGHT\n\nThe CH-47s dropped Colonel Kevin Crislip and his troops of the 10th Special Forces Group at six bridges across the Canadian River in the Texas panhandle, five highway bridges and one railroad bridge. The Canadian was not much of a river, merely a wet, sandy depression in that cap-rock country, but knocking the railroad bridge down would prevent any trains from using the railroad until it was replaced. The destruction of the five highway bridges across the Canadian would severely inconvenience truckers, who would have to go east to the main body of Oklahoma or west to New Mexico to find an alternate route south.\n\nColonel Crislip thought this whole mission a bad joke, political revenge on the Texas politicians who had embarrassed Barry Soetoro, but General Seuss and his staff had been trading messages with the Pentagon, so here the Green Berets were, blowing up bridges in the panhandle, each demolition team delivered by helicopter. Crislip consoled himself with the thought that these demolition jobs were good training, if nothing else.\n\nEach bridge had one demolition team assigned and it was delivered by a Chinook, which moved safely away from the bridge after off-loading the team, their explosives, and a few guards. Colonel Crislip accompanied the team blowing the bridge north of Borger. He stood in the warm Texas night listening to crickets and inhaling the faint aroma of cow manure drifting on the breeze while the team worked. Crislip sent the guards up the highway on either side of the bridge to stop traffic. There wasn't much. A semi came from the north fifteen minutes after they arrived and was waved on through. Five minutes later a pickup full of Mexicans who had been drinking came from the direction of Borger. They were going back to the ranch, they said, so the guard waved them across the bridge. They went by Crislip saluting and shouting and laughing. Although the Mexicans could see the helo parked in a nearby pasture, they couldn't see the soldiers working under the bridge, so they certainly couldn't warn anyone that the bridge was soon to be destroyed.\n\nThe colonel had never actually demolished a real bridge before; he went down the riverbank and stood underneath, looking up, ten feet, with a flashlight to see where his troops put the charges. They seemed to know what to do and how to do it.\n\nThey were planting C-4 charges, which the experts at Fort Carson had assured the colonel were quite enough to put the bridge in the sand of the Canadian River, if, the experts said, they were placed properly.\n\nAlways the big _if_ , Crislip fumed. So if any bridge remained standing after its charge was detonated, his troops would take the blame. Wonderful!\n\nHe climbed back up the bank and was standing beside the highway listening to the crickets and savoring that stockyard smell when a battered old pickup coming from the north was stopped by the guard. Crislip walked over, just in time to hear his soldier tell the driver to turn around and go home. There were two other people in the truck's cab, Crislip saw, two women.\n\n\"Let him across the bridge if he wants to go,\" the colonel told the guard as he walked up.\n\nThe driver, who looked to be in his fifties and was wearing a ratty ball cap, asked, \"Who is the head man here?\"\n\n\"I am,\" Crislip said. \"Colonel Kevin Crislip, United States Army.\"\n\n\"I live just a little west of here, and we saw you people come in on that helicopter after dark and we been watching you. What the hell is going on?\"\n\nThe dashboard lights let Crislip see the other passengers, one a woman about the driver's age and the other a teenage girl. \"That's none of your business, sir. What's your name, anyway?\"\n\n\"Zeke Lipscomb, buddy. And telling me to mind my own business ain't the way we do things here in Texas.\"\n\n\"Mr. Lipscomb, this is army business. Cross the bridge or go home.\"\n\n\"I'll cross.\" He put the truck in motion, drove it a hundred yards and stopped right in the middle of the bridge. He killed the headlights, parked the truck, and he and the two women got out.\n\nCrislip strode toward them. The guard was going to accompany him, but Crislip growled for him to stay put.\n\n\"I told you to drive across,\" he said to Mr. Lipscomb, who had a female on each side of him.\n\n\"Well, I didn't. And I ain't a gonna. We kinda think you soldiers are up to no good, and we're not going to let you get away with it.\"\n\nCrislip sighed.\n\nThe older woman, presumably Mrs. Lipscomb, spoke up. \"You federal troops got no damn business in Texas, Colonel, and you know it. We done declared ourselves a separate nation.\"\n\nCrislip looked back at the guard. There was just a sliver of moon and enough starlight to see him clearly, standing there in the road looking this way, no doubt wondering what the colonel was going to do about this stubborn rancher.\n\n\"Mr. Lipscomb and Mrs. Lipscomb\u2014\" he looked at the girl. \"What's your name?\"\n\n\"Ruby.\"\n\n\"And Ms. Ruby Lipscomb. I am here obeying the orders of my superior officers, and the men with me are obeying my orders. We are doing our duty. Now I am asking you nicely to please get in your truck and drive on into Borger or return to your home.\"\n\n\"You're gonna blow up this bridge, ain't ya?\" Lipscomb said, scrutinizing Crislip's face.\n\n\"Yes, we are.\"\n\n\"Well, we ain't goin' anywhere. We use this bridge to get back and forth to town, and so do our neighbors. Our tax dollars built this bridge, and we ain't gonna let a bunch of Soetoro's soldier boys blow it up. You people get in your helicopter and get the hell outta here.\"\n\n\"There are ten of us, Mr. Lipscomb, and we're all armed.\"\n\n\"I ain't packin'. My wife and daughter ain't packin'. But if we have to go home and get our rifles and start shootin', we will. You people ain't blowin' up this bridge without a fight . . . and that's my final word.\"\n\nCrislip walked over to the guardrail on the edge of the bridge and looked down. The soldiers in the riverbed had finished placing the charges under the bridge and were unrolling det cord.\n\nHe turned around and found Lipscomb beside him.\n\n\"You people must be idiots,\" Lipscomb said. \"Blowin' a bridge in the middle of the Texas panhandle ain't no way to win friends. You think that'll make us submit?\" He spat onto the pavement. \"When they hear about this glorious military raid in Austin, no doubt they'll decide to drag Texas back to Soetoro's slimy embrace, kiss his shitty ass, and beg for forgiveness.\"\n\nCrislip tried to decide what to do.\n\n\"Meanwhile the folks who live around here ain't got no bridge, thanks to the United States Army and Barry Soetoro.\"\n\nThe colonel examined his options. He could have his soldiers drag these three people off this bridge and blow it. Or he could tell the Lipscombs to go get their rifles and blow it while they were gone. Or. . .\n\nHe took a deep breath of that foul stink of cow shit. \"How the hell do you stand the smell?\" he asked Lipscomb.\n\nLipscomb sniffed the air. \"Oh, the cows. You get used to it.\"\n\nKevin Crislip grew up in Des Moines, son of a lawyer. His mother's father had been a farmer, growing corn on three sections of land every summer. Kevin had loved his visits to his grandparents' farm. There he learned to drive a tractor, shoot a rifle\u2014learned what hard work was.\n\nAfter four years at West Point and twenty-three years in the army, four deployments in two wars, here he was standing in the darkness on a bridge in the middle of nowhere breathing that pure Texas smell, arguing with a rancher who really didn't deserve to lose his bridge to make Barry Soetoro happy.\n\nThe colonel made his decision. He leaned over the guardrail of the bridge. \"Lieutenant,\" he called.\n\n\"Yes, sir.\"\n\n\"There's been a change of plans. Remove the charges from under the bridge, and let's go back to Colorado.\"\n\n\"Ahh . . .\"\n\n\"Do it,\" the colonel said.\n\n\"Yes, sir.\"\n\nAnd that is what they did. The three Lipscombs were still standing in the middle of the bridge when the twin-rotor helicopter lifted off with all the soldiers aboard.\n\nWith the electricity off in much of east Texas, the prison, its power provided by emergency generators, seemed an oasis of light that Saturday evening. Although it was only six and darkness was several hours away, the institution's floodlights were all lit. That was an irony that didn't escape the seven armed men in National Guard camo uniforms who pulled up to the main gate in two Humvees with fresh Lone Star flag insignia painted on the doors. Behind them was a National Guard bus that contained another ten soldiers, also sporting newly painted Lone Star flags.\n\n\"It's after visiting hours,\" the bored gate guard said. The officer in charge, a colonel, displayed a letter. He passed it through the window to the guard, who picked up a telephone on his desk and made a call.\n\nFrom his right front seat in the Humvee, the colonel had a good view of the star-shaped building, the tiny barred windows, the guard towers, and the double-chain-link fence topped by concertina wire that encircled the entire facility. Popular legend had it that there had never been an escape from the prison, and the colonel could see why.\n\nAfter a few minutes, the guard said, \"I'm going to open the gate and you drive in and park by the stairs. Someone will be down shortly to escort you.\"\n\nThat is how it went. Ten minutes later the colonel, whose name was embroidered on his left chest, and a captain were sitting in the warden's office. The warden was eating from a heaping plate on his desk, apparently his supper.\n\nThe colonel passed the warden the letter, which was on the stationery of the governor, now president, of Texas. The warden dropped his eyes to the signature. Jack Hays.\n\nThe warden, Arlen Kirkpatrick, was forty or so pounds overweight, was balding, and had prominent jowls. Kirkpatrick picked up a bite of fried chicken with his fingers as he started to read. He read in silence. In the document, President Hays summarily relieved him, thanked him for his past service, and appointed Colonel Ezekiel Holly in his place. Warden Kirkpatrick was told to report to the Bureau of Corrections as soon as possible to be reassigned or, if he wished, placed on the retired list.\n\nHe read the letter quickly, abandoned his dinner, then read it again much slower.\n\nHe dropped the two sheets of paper on the desk and looked at the colonel. \"What did I do to earn this honor?\"\n\n\"Obviously, the president is putting the military in charge of the prisons for the time being. He said he intends to see that you are reassigned to another prison when the crisis is past.\"\n\nKirkpatrick shook his head in amazement. \"Colonel Holly, someone has lost their senses. Soldiers aren't trained to run prisons. Our inmates are some of the worst in the system. Only a fool would send you here.\"\n\n\"You are entitled to your opinion.\"\n\nKirkpatrick picked up the letter and read aloud, \"The Republic can no longer afford the past level of outlay on prisons. . . . Having full faith and confidence in Colonel Ezekiel Holly, I have ordered him to assess the prison population at your facility and recommend which prisoners should be released early.\"\n\nThe warden stared at Holly. \"Does this mean . . .\"\n\n\"Indeed, there may be some early releases,\" Colonel Holly said with a curt nod. \"Texas is fighting for its life and must save dollars wherever it can. The president thought extraordinary measures were necessary.\"\n\n\"I must verify this with the president,\" the warden said.\n\n\"Certainly.\"\n\n\"But I cannot. The electricity is out and the telephones are dead.\"\n\nColonel Holly's face was impassive.\n\n\"I will not admit you to the prison proper until I can verify this with President Hays.\"\n\n\"Just how do you propose to do that?\" Holly asked softly.\n\n\"Well, wait until power is restored, I suppose.\"\n\n\"Mr. Kirkpatrick. I have sixteen armed soldiers with me. Do I have to bring these troops in here and forcibly remove you from this office?\"\n\n\"Now see here\u2014\"\n\n\"If that is what I need to do, please excuse me.\" Colonel Holly stood. \"I must be about it. I have my orders.\"\n\n\"Sit, Colonel, sit. Please.\" Arlen Kirkpatrick knew when he was beaten. He pushed his unfinished dinner out of the way. \"What can I do to help?\"\n\n\"Bring your senior staffers in, tell them you have been relieved, and go home.\"\n\n\"Only the night guards are here. We are finishing the dinner hour, then the prisoners will be locked down for the night.\"\n\n\"That will do.\"\n\n\"What do you intend to do, Colonel?\"\n\n\"That isn't your problem. As I said, I have my orders. I suggest you make a copy of that letter, keep the original, and let me have the copy.\"\n\nArlen Kirkpatrick rose from behind his desk, made the copy on a machine in the outer office, called in the senior people on the night shift, and introduced Holly. The warden shook hands all around, the guards wished him well, and then he departed, leaving his half-eaten dinner on the desk.\n\nHolly called for the records. His armed staff found seats in the outer office while the night shift, mostly guards, carried in the records in alphabetical order.\n\nHolly read for several hours as darkness fell and made notes. He sent the captain and the senior NCO, a staff sergeant, to ensure the prisoners were indeed locked in their cells. Then he sat in the warden's office and watched the security monitor high on the wall shift automatically around the security doors and corridors. About midnight, the guards were called in. \"Gentlemen, we are sending all of you home for the evening.\"\n\n\"You can't do that,\" one of the guards said curtly. \"Regulations require\u2014\"\n\n\"The military is now in charge of this facility. With the prisoners locked up, I have enough men to see that they remain behind bars through the night. Report tomorrow at your usual shift time.\"\n\nThe guards didn't want to go, but Ezekiel Holly looked stern and every inch a senior military officer used to being obeyed. They went by the armory, turned in their weapons, which were locked up, and filed to the courtyard in front of the prison for their cars. One of the soldiers closed the gate behind them. Soldiers replaced guards at key checkpoints throughout the prison.\n\nThe colonel nodded at the security monitor. \"Get all the tapes, or if the feed goes on a computer, the hard drive.\"\n\nWhen that was done, the colonel led a half-dozen soldiers, all that remained after the guard positions within the prison and at the gate were manned, to the security checkpoint outside Cell Block A. When they got there, the colonel consulted a list he had made from examining the files.\n\n\"James Abbott,\" the colonel said. \"Bring him here.\" Two soldiers left their weapons on the desk and went through the checkpoint. Another manned the panel that opened the cell doors in the block.\n\nIn a few minutes, Abbott appeared. He was a pasty-faced man of medium height with a prominent spare tire. His hands were cuffed into a wide leather belt that encircled his waist, and he had cuffs on his ankles that were held together with about fifteen inches of chain. He had lively eyes and a semipermanent smile upon his lips. One of the Texas Guard soldiers that had accompanied Holly to the prison stood behind him.\n\n\"Mr. Abbott, according to your file, you were convicted of raping and murdering four girls. The Texas Rangers believed you raped and murdered at least six other girls over a period of nine years, but you refused to admit the crimes or tell where the bodies were buried.\"\n\nAbbott said nothing, merely looked from face to face with nervous eyes, wearing that smirk.\n\n\"You were sentenced to life in prison without parole.\"\n\nThe smirk didn't change.\n\n\"Do you want to tell us now how many other young women you murdered?\"\n\n\"You're shitting me, right?\"\n\nIn the silence that followed, Ezekiel Holly looked at his list. When he looked up, Abbott had said nothing and was still wearing that semipermanent smirk.\n\nHolly nodded at two of the guards who were still wearing sidearms.\n\nThe soldiers grasped Abbott, one on each arm, and started leading him to the corridor that led to the courtyard one story below.\n\n\"Hey,\" Abbott said, trying to resist. \"Where are you taking me?\" That is when he really looked at the face of the soldier on his left side. \"I know you,\" he shrieked. \"You are the brother of\u2014\"\n\nHe refused to walk, so the soldiers dragged him along, supporting his weight.\n\nA minute later a young man was brought in, also wearing shackles and manacles.\n\n\"Jason Brodski. Apparently you opened fire with an assault rifle in a movie theater and killed a dozen people and wounded thirty-three more. Your attorneys argued that you were insane, and the jury rejected that defense. They convicted you but couldn't agree on the death penalty, so you were sentenced to life without parole. Is that correct?\"\n\nA sneer crossed Brodski's lips. He was a slightly built white man with a mop of unruly black hair and pimples. \"Yeah,\" he said.\n\n\"Mr. Brodski, the world has turned. The Republic of Texas is not going to force taxpayers to pay for your maintenance and medical care, nor for the guards to watch you. You will be executed tonight.\"\n\n\"What the fuck! You can't do that! Goddamn, I know my rights. I want my lawyer. I\u2014\"\n\nHolly nodded to the two armed soldiers near Brodski, who grabbed his upper arms and removed him through the open security door along the corridor. The smell of feces was in the air. Holly glanced down the corridor and saw a dark stain spreading on the seat of Brodski's pants.\n\nThe next prisoner was standing in front of Holly when the muffled sound of a shot could be heard through the window overlooking the interior basketball court.\n\n\"What was that?\" the prisoner, a Latino, asked nervously. \"What the fuck is going on here?\" He had a thick accent, glowered, and shifted from foot to foot.\n\n\"Alfredo Mendez, citizen of Mexico. Apparently you were an assassin for a Mexican drug syndicate, and you were convicted of murdering six men with an automatic weapon as they sat in a Del Rio beer joint.\"\n\nMendez merely glared. \"What the hell is this, anyway?\"\n\nAnother muffled shot could be heard from the basketball court.\n\nAlfredo Mendez looked around wildly as the first two soldiers returned carrying the empty shackles and manacles. They handed them to the unarmed soldiers and grabbed Mendez.\n\n\" _Madre de Dios_! No! I can pay. My _patron_ swore\u2014\"\n\nThe soldiers took Mendez down the corridor, still swearing and shouting.\n\nThe next man was a hulking black with scars on his face and tattoos on his knuckles and forehead. He had apparently been spending a lot of time in the weight room, because he was heavily bulked up.\n\n\"James Elvin Dallas,\" Colonel Holly said. He looked Dallas straight in the eyes as he recited, \"You were convicted of raping three women. Then, while in prison, you beat a man to death, apparently because he refused to be your butt-boy. It is thought you killed another with a homemade shiv, but you were never charged due to lack of evidence.\"\n\n\"So?\"\n\n\"Did you ever wonder what became of your victims?\" Holly's eyes scrutinized Dallas' face.\n\nDallas' eyes were roaming, measuring the men in the room.\n\nAnother shot was heard from the courtyard.\n\nJames Elvin Dallas went nuts. He lunged sideways and tried for the rifle on the table. Four of the soldiers tried to subdue him. That task was only accomplished when one of the soldiers struck him repeatedly on the head with a rifle butt. As Dallas lay immobilized upon the floor, Holly pulled his service pistol and, from a distance of one foot, shot him between the eyes. Brains and blood splattered across the concrete floor.\n\n\"Take him to the courtyard,\" Holly ordered, \"and shoot him again.\"\n\nThe next prisoner was large and sloppy, with greasy, curly black hair springing from his head and his chest. He had a full beard too\u2014something that had been banned in Texas until last year. \"Muzzaffan Mehsud. You were convicted of throwing acid in your wife's face because she went shopping without your permission. You were sentenced to twenty years.\"\n\nThe man spat at Holly, who merely nodded to the soldiers. They took Mehsud away as he shouted, over and over, \" _Allahu Akbar_.\"\n\nAfter three more men were removed from Cell Block A, the colonel led his soldiers to Cell Block B.\n\n\"Francisco Colon, you are a serial rapist. At least six girls, none older than fourteen.\"\n\n\"You fuck! I know my rights. You can't revisit a sentence.\"\n\n\"Did you ever wonder what happened to the girls you raped?\"\n\n\"Everyone heard the shots from the courtyard. You can't get away with this.\"\n\n\"One of the girls, Judy Martinez, committed suicide six months ago. She had been in psychiatric care for four years. Apparently she could never come to grips with the fact that animals like you roam the streets. Her father paid ten thousand dollars to hear that you were dead.\"\n\n\"Fuck you!\" Colon lowered his head and launched himself at Holly. He didn't get there. The men on either side dropped him on his face on the concrete floor, smashing his nose and releasing a torrent of blood. Semiconscious from the impact, he was carried to the courtyard.\n\nThe next man was a white man, medium-sized, with a full head of hair. He could even be called handsome. He was calm. \"We heard shots. Are you executing people?\"\n\n\"Robert Winston Carrington. You were convicted of running a Ponzi scheme that took in over twelve million dollars, most of which you squandered to pay for an extravagant lifestyle.\"\n\nCarrington glanced at the bloodstain on the floor from Colon's nose, then his eyes came back to Holly. \"I didn't kill anybody,\" he said.\n\n\"Did it ever occur to you,\" Ezekiel Holly said conversationally, \"that prisons exist for two reasons? The first of course is to keep the guilty in, and the second is to keep the victims out.\"\n\n\"They were all greedy bastards and got what they deserved.\"\n\n\"As we all shall, rest assured. Two of your victims committed suicide. Many were reduced to penury after a lifetime of work because they believed in you, trusted you. We are here tonight as surrogates for your victims.\"\n\nHolly nodded at the soldiers, and they took Robert Winston Carrington away. He walked with his head high. Maybe, thought Colonel Holly, he doesn't believe he will really be executed. Or, perhaps, he doesn't care.\n\nThree minutes later another shot was heard.\n\nWhen Colonel Holly and his soldiers left the prison at three that morning, thirty-two corpses were laid out side by side on the prison basketball court, where they were found by the day shift.\n\nWarden Arlen Kirkpatrick was summoned, and he sent a man to Austin. When the man returned two days later, he reported that no one at the Bureau of Prisons, in the governor's office, or at Texas Guard headquarters had ever heard of Ezekiel Holly. The governor's signature on the letter was a forgery.\n\nPerhaps fingerprints might have identified Colonel Holly, but all the other soldiers wore tactical gloves. When the Texas Rangers finally sent a man around to hunt for prints, more than a week had passed and the task was hopeless.\nTWENTY-NINE\n\n\"We leave tomorrow,\" Jake Grafton said on Sunday morning.\n\nBoy, that was good news to me!\n\nSarah Houston was carrying the little girl around, everywhere, and gave me The Look every time she passed me, as if it were my fault the kid got raped. She didn't even say anything about my neck wound. Mrs. Johnson was a nurse and bandaged me up after she had smeared some sort of antiseptic on it. My neck was so sore I couldn't turn my head.\n\nWillie Varner said, \"Goddamn, Tommy. You keep lettin' these sons of bitches shoot at you. It's just a matter of time, dude.\"\n\n\"Hey, Willie, I\u2014\"\n\n\"Don't want to hear it. I done tol' ya. Just a matter of time. Ain't goin' to cry at your funeral, Tommy. Sarah might, or Mizz Grafton, but I ain't a gonna. See you in Hell, dude, and we'll catch up then.\"\n\n\"I can hardly wait. Thanks, asshole.\"\n\n\"You're welcome.\"\n\nI was fed up to here. I broke out the two sniper rifles from FEMA's Walmart stash and took them down to the meadow. Put a target out at two hundred yards\u2014measured with one of the laser rangefinders the military had thoughtfully included in the box\u2014and laid down by the hangar. Used a box of MREs as a rest and commenced shooting. The rifles were .308 caliber, actual designation 7.62\u00d751 NATO, and we had  plenty of ammo. I played with them a while and got them zeroed. Just in case.\n\nI am not a sniper: I am not good enough with a rifle, and I don't have the patience for it. However, the concept of whacking bad guys from beyond the effective range of their weapons strongly appeals to me. I have no sporting instinct whatsoever and am a disciple of W. C. Fields: Never give a sucker an even break, and its corollary, do it unto others before they do it unto you.\n\nWhen I got back to the safe house, the sun was down. In the twilight everyone was sitting around outside eating venison that I had shot, Molina and Yocke had butchered, and Jake Grafton had cooked on the outdoor fire. Burned on the outside, pink in the middle. Everyone but me complimented him on his outdoor culinary skills. To accompany the venison we also had Mrs. Price's green beans and baked potatoes with margarine and ketchup, for those so inclined. With the smell of wood smoke in the delightful evening air and plenty of good, wholesome food, some of the folks around the fire looked like they were dumping some stress. The meal was filling and a nice change from MREs, but I wasn't ready to sing Kumbaya.\n\nI figured that there was a lot of shooting and dying coming up in the days ahead. Going to Washington to clean up the government wasn't on my bucket list.\n\nBut what the hell! A man can only die once. That's a good thing, by the way. We've all gotta go sometime, and, truth be told, the sooner you check out, the more shit you miss. That's the gospel according to Reverend Carmellini. Amen.\n\nThe girl spent the evening sitting by Sarah Houston. Armanti was sitting with Mrs. Price. The Johnsons were huddled together, the parents taking care of the kids. Yocke and Molina sat engaged in earnest conversation, solving the nation's problems, probably. The warriors kept by themselves, although they had included Willie Varner in their little group. They liked Willie's brand of pessimism, I suspected: I certainly did.\n\nAfter the fire died down to glowing coals, Sarah Houston picked up the kid and carried her into the house. I waited a moment, then tagged along. I found them upstairs in the bedroom we had been using, and the door was open a crack. I eavesdropped. It's one of my failings. But, to paraphrase that great American philosopher Yogi Berra, you can learn a lot by listening.\n\n\"My name is Sarah too,\" the girl said.\n\n\"We have a lot in common,\" Sarah Houston said warmly.\n\n\"I saw that man shoot my parents. He was really mean. He hurt me terrible down there. Then Mister Tommy shot him and his whole head came off. After what he did to me, I was glad.\" I knew that Mrs. Johnson, the nurse, gave the kid a vaginal exam and had to do some stitches, after she had numbed her.\n\n\"I suppose so,\" Sarah Houston said. \"Tommy is a good man. Was your father a good man?\"\n\n\"Oh yes. He wasn't tall, and he was sort of heavy, not a bit like Mister Tommy. But he loved me very much. So did Mommy.\"\n\nAfter a bit I heard Sarah Houston say, \"I am sure you will miss them very much.\"\n\n\"Mommy and Daddy loved me.\"\n\n\"I am sure they did.\"\n\n\"I like Mister Armanti too. He's a real nice man. Sort of like a bear.\"\n\nThe two sat in silence for a while, then the big Sarah said, \"You and I are going to sleep right here. If you have a nightmare, you wake me up. Will you do that?\"\n\n\"Daddy always read me a book before bed.\"\n\n\"I don't have any books. Maybe I can tell you a story, after you get in bed.\"\n\nFive minutes later Sarah began, \"Once upon a time . . .\"\n\nA half hour later, Sarah came out. I was sitting on the top of the stairs. She sat down beside me.\n\n\"That kid has been through a hell of a lot.\"\n\n\"I guess.\"\n\n\"She's in denial right now. Sooner or later the implications of the murder of her parents, rape, all that is going to hit her hard. She is only eight years old and she saw all that mayhem.\"\n\n\"God help her,\" I murmured.\n\n\"You can sleep on the couch downstairs.\"\n\n\"Okay.\"\n\n\"Oh, Tommy. What a disaster . . . for all of us.\"\n\n\"Yes.\"\n\nI put my arm around her. After a while she said, \"I'm staying here when you leave tomorrow. Someone has to take care of this child.\"\n\n\"Okay.\"\n\n\"Will you come back? Afterward?\"\n\n\"You can bet your life on it.\"\n\nSunday evening two Muslim male refugees from Syria, ages nineteen and twenty, raped a thirteen-year-old black girl in St. Louis. She screamed and they beat her. Despite the perilous state of law enforcement in St. Louis after a week of rioting in the black neighborhoods, the police apprehended the pair. They were taken to a police lockup.\n\nThat night a crowd of almost eight hundred people, mostly black, surrounded the police station. These were not ghetto dwellers, but middle-class suburbanites, and many were armed. They held the police at gunpoint and removed the two rapists from the cells. The two were taken outside and hanged by their necks from a nearby tree with ropes some members of the crowd had thoughtfully brought along.\n\nThen the crowd, now containing about 1,500 people, walked in a body to the downtown mosque that the imam had made infamous by preaching jihad from the pulpit; the mosque, incidentally, where the two rapists had worshiped. The crowd found the cleric cowering in a closet in a nearby house, dragged him outside, and hanged him too. The mosque was set on fire.\n\nWhile the imam dangled and strangled, a few people in the crowd fired some shots in the air and shouted catcalls, but mainly the crowd was quiet. Some police officers sat on the hoods of their cruisers, watching and smoking. An intrepid television cameraman filmed the holy man swinging in the wind for broadcast whenever. An hour or so later, the crowd began to dissipate and trudge away into the night.\n\nAmazingly, the energy seemed to go out of the rioters in other sections of town, many of whom actually went home. For the first time since Barry Soetoro declared martial law, the hours from midnight to dawn on Labor Day were quiet in St. Louis.\n\nAt nine o'clock that Labor Day morning, a convoy of two companies of Marines from Quantico arrived at the Pentagon. A colonel was there to meet the company commanders, both captains. After a short conversation, the troops set up machine guns inside sandbagged positions at the entrances to the Pentagon, other Marines were sent to guard the Metro station downstairs (even though it wasn't running) and to guard the entrances to the parking lots. They set up a bivouac on an empty section of the vast parking lot on the western side of the massive building, a lot that looked relatively empty because, despite the crisis engulfing the nation, many of the civilians had Labor Day off.\n\nThe chairman of the Joint Chiefs, General Martin Wynette, knew nothing about the Marines' arrival. He was upstairs in his office on the E-Ring going over readiness reports from the U.S. armed forces around the world, with special attention to those units in the United States. The United States armed forces were in full mutiny, he said to his staff after a quick perusal of the reports. People in uniform willing to fight for Barry Soetoro against Americans were a rare commodity. The only bright spot was the Marines in Southern California, who had strapped on the Mexican military as though they were God's gift to starving men. At last, an enemy to shoot at. The crews of the navy's two carriers now cruising off the coast of San Diego were apparently happy as pigs in slop launching strikes at the Mexican invaders. They had achieved complete control of the air, left Mexican armor burned-out wrecks, destroyed Mexican staging areas on the American side of the border, and flown support missions for the Marines. It was a proverbial turkey shoot.\n\nThe rioting Mexicans in the slums of LA weren't the military's problem. What the civil authorities were going to do about them was up to Barry Soetoro and the politicians in LA and Sacramento who wanted those Hispanic votes more than they wanted salvation. If they wanted salvation, which was doubtful.\n\nMartin Wynette was trying to figure out what he was going to tell the president and his disciples when he went over to the White House to brief them at eleven o'clock when a group of flag officers led by CNO Admiral Cart McKiernan came into his office unannounced and closed the door. The commandant of the Marine Corps was there, as well as the deputy chiefs of staff of the army and air force. The four officers stood in front of the desk looking down on Wynette.\n\n\"Marty,\" said the commandant, Morton Runyon, \"tell us why you threw Sugar Ray, Jack Williams (the army chief of staff), and Harry Miller (the air force chief) to the wolves.\"\n\nWynette stood up. \"You don't know what you are talking about.\"\n\n\"We've talked to Major General Stout, who was there at the White House with you. Remember?\"\n\n\"Now, listen, people. Someone told Soetoro that a coup was being planned over here in the Pentagon. He already knew. What could I say?\"\n\n\"He didn't know shit, Marty. Schanck tried a shot in the dark and you spilled your guts. You pulled the trigger on Sugar, Jack, and Harry.\"\n\n\"Well, Jesus, they _were_ planning a coup! Talking about it, anyway. For Christ's sake, he's the _commander-in-chief_. He's the _president_!\"\n\n\"And you took an oath _to support and defend the Constitution of the United States_. Soetoro has become a dictator. He's ripped up the Constitution.\"\n\n\"These are perilous times,\" Wynette explained. \"The president has a right to do whatever is required to maintain the government. You know that.\"\n\n\"He doesn't have the right to convert the country into a dictatorship,\" Cart McKiernan said, and made an angry gesture. \"But we aren't here to debate politics. This has gone too damned far. Three senior officers were executed without a trial in the courtyard downstairs. This isn't Nazi Germany or Soviet Russia. Get your head out of your ass, Marty.\"\n\nWynette sank into his chair and his gaze went from face to face.\n\n\"What do you want of me?\" he said softly.\n\nThe CNO, who was in short-sleeve summer whites, nodded to the commandant, who was in greens. He lifted his blouse and pulled out a pistol. \"This is yours, Marty. I stopped by your quarters and your wife let me in. I got this from the desk in your study.\"\n\nMartin Wynette stared at the faces. \"I'm not going to shoot myself, if that is what you are implying.\"\n\n\"We'll call Mrs. Ray. What is her name? Naomi, I think. Maybe she'll do it for you. Or Barry Soetoro can send his goons over to do you in the courtyard.\"\n\nWynette said nothing. He was sweating and licking his lips.\n\nMorton Runyon walked around the desk and fooled around with the pistol. Then, quick as a flash, as Wynette looked at the other officers, he put it to the right side of Wynette's head and pulled the trigger. Blood, tissue, and little pieces of skull spurted out the other side. Wynette slumped in the chair.\n\nRunyon picked up Wynette's dead hand, put it around the pistol, got fingerprints all over it, then dropped the gun on the floor.\n\n\"Damn,\" said Cart McKiernan. \"I think he shot himself. Get the staff in here for the bad news.\"\n\nJake Grafton sat everyone down after breakfast and announced that Mr. and Mrs. Johnson and their children were staying at the safe house, along with Mrs. Price and the young Sarah. \"Armanti, would you be willing to stay here with them and keep an eye on things?\"\n\n\"Yes, sir,\" Armanti Hall said.\n\n\"I'm staying too,\" Sarah Houston announced.\n\n\"No, you're not,\" Jake Grafton said, eyeing her. \"Too much is at stake.\"\n\nSarah looked at me, then shrugged.\n\n\"Uh, Admiral,\" said Willie Varner. \"Maybe I could stay too. I ain't much of a shooter and all, and\u2014\"\n\n\"We may need your lock skills,\" Grafton said crisply. \"Tommy, load the trucks. Leave what you can for the people who are staying, and let Armanti keep whatever weapons he might need. Pistols for all the adults who want one.\"\n\n\"What about the plane?\" I asked.\n\n\"I'll fly it. Mr. Johnson has given me his permission and the ignition key. Take me down to the hangar and let's get it out.\"\n\nWe pulled the plane from the hangar, spun it around, and I helped Grafton in. He wasn't spry and obviously had some discomfort, but he seemed able to move without pain.\n\nHe looked over a sectional chart that Johnson had used to get here, and said, \"You get everything loaded up. I'll be back in about an hour.\"\n\nHe started the engine, taxied down to the far end of the runway, swung the plane around, and ran the engine up to a pleasant hum. I looked at the sky: clear above, hazy, only a couple of knots from the north, just enough to stir the wind sock a little. The morning dew hadn't yet burned off in the sunny places, so people and the plane left tracks in the grass. If you didn't know any better you'd think it was just another late summer day in paradise.\n\nAfter a moment the Cessna accelerated down the runway with its tail-wheel off the ground and got airborne. It flew away to the north, climbing slowly. The plane got smaller and smaller and more indistinct, then it merged into the haze and the sound of the engine faded completely away.\n\nThe guys and I loaded the trucks. Willie the Wire did some bitching. \"Hell, he don't need me to open locks when he's got you.\"\n\n\"The only place you know how to rustle grub is in a grocery store,\" I said. \"You eat too much to leave you here.\"\n\n\"We get back to Washington, dude, there ain't gonna be no grocery stores. Not ones with anythin' in them to eat, anyway. Did you think of that?\"\n\n\"No liquor stores or beer joints either,\" Armanti offered. \"Gonna be like Baghdad or Damascus. Nice of you to share the pain, Willie.\"\n\nWillie Varner said a crude phrase.\n\n\"Look on the bright side,\" I suggested, just to buck him up. \"It couldn't be as bad as the sewers of Cairo. Did I ever tell you about the month I\u2014\"\n\n\"You too, Carmellini.\"\n\nSarah Houston and I got to spend a few minutes with young Sarah before we left. The girl was sobbing, finally letting her emotions out, which was a good thing. The Sarahs put their foreheads together and hugged. Finally Sarah kissed the kid and said, \"I'll be back.\"\n\nOn the way down the hill, I told her, \"You're optimistic.\"\n\n\"Live every day until you die,\" she retorted. Then she touched the pistol butt in its holster on her web belt. I doubt if she even realized she did it.\n\n\"You're taking your computer along, I see.\"\n\n\"Yes.\"\n\n\"When do you suppose you'll get a chance to use it?\"\n\n\"You never know.\"\n\nWhen Grafton landed, he said the roads were clear to the airport in Elkins, which looked deserted. \"Before we go, run me down to that clinic.\"\n\n\"Okay.\"\n\nIn Greenbank Dr. Proudfoot didn't seem surprised when Grafton and I walked through his door. He and Grafton shook hands.\n\n\"Could we have a little talk in your office?\" Grafton asked him.\n\nWhile they were talking, I went into the room where Mrs. Greenwood was. She was still in a coma. The nurse and I chatted.\n\nAfter about fifteen minutes, Grafton and the doctor came back. \"Dr. Proudfoot is going with us. He needs to run up to his house for some things, and he'll join us at the hangar.\"\n\nBack at the hangar, Grafton spread out the sectional chart and our one roadmap on the hood of my pickup, and Travis, Willis Coffee, and I studied them.\n\n\"If there are rebels around,\" Grafton said, \"I suspect we will find them at Camp Dawson, where FEMA had their concentration camp. I want to fly up to Elkins, wait for you there, and then we'll fuel the plane if we can and I'll fly up to Dawson and look around. If it's safe, we can all go.\"\n\n\"Why Dawson?\" Travis asked.\n\n\"A National Guard base figures to have an armory. People with deer rifles are guerillas. To turn them into an army you need machine guns, mortars, and artillery, if you can find some.\"\n\nRight then I began to suspect that Grafton wasn't leveling with us. Maybe everyone else thought he was, but I was no virgin. I had worked with him too many times in the past. I kicked myself for not cornering him several days ago and getting the lowdown. If anything happened to Jake Grafton, Sarah and I and all these other fools were going to be up the proverbial creek without a paddle. Too late to brace him now, though.\n\nGrafton got back in the plane and we climbed into the trucks. The doc and Sarah rode with me. As we rolled along he wanted to talk about Jake Grafton. He was obviously star-struck and called him \"Admiral\" in every sentence, finishing with, \"You didn't tell me he was a retired admiral.\"\n\n\"I didn't think it mattered.\"\n\n\"Or director of the CIA. Why didn't you say so?\"\n\n\"Because I didn't want you telling anyone anything.\" I turned my head and locked my eyes on him.\n\nDr. Proudfoot got uncomfortable and shifted his eyes to the road ahead. \"Well, I'm glad he asked me to go along. It's a great honor.\"\n\nI thought he should talk to Willie Varner about that, but I kept my mouth shut and drove. Sarah just sat looking out the window.\n\nThe JCS staff was shocked by the suicide of General Martin Wynette. Still, everyone knew he had been under tremendous pressure from the White House, and after the deputy chairman and the army and air force chiefs of staff were summarily executed by Secret Service personnel, his personal choice to end his own life was understandable, if tragic. While his remains were being carried away to a freezer in the cafeteria, to wait for a better day for his funeral and burial, the four surviving heads of their services met in a conference room behind locked doors.\n\n\"Gentlemen,\" Cart McKiernan said, \"we have some critical decisions to make, and not much time to make them. We must announce Wynette's demise, and no doubt the White House will have a serious reaction to the news. Either Soetoro will send people over here to take over the Pentagon, or he will think this is the start of a putsch. Your thoughts, please.\"\n\nThe army deputy chief Franklin Rodriquez said, \"I think it would be a terrible precedent if the armed forces were involved in decapitating a president or in assisting a popular uprising to overthrow him. Or in assisting in keeping a hated president in office in the face of a revolution. In my opinion, the best thing for America is for the armed forces to remain neutral.\"\n\n\"As if we could,\" Morton Runyon scoffed. \"We're already up to our necks in this.\"\n\nThe acting air force chief of staff, Erhard \"Bud\" Weiss, said, \"We can't win, gentlemen. If we fight for or against Soetoro the people will never trust us again. We must let the American people sort this out.\"\n\nRodriquez tapped his chest. \"This isn't the uniform of Barry Soetoro's army; it's the uniform of the United States Army. There's a big difference. And this afternoon the order is going out to every army commander: we're not arresting civilians anymore, and we're releasing the political prisoners from every army-run camp.\"\n\nThe Marine commandant's gaze went from face to face. \"Well, that's a start, but I think we should go over to the White House, drag Soetoro and his staff out into the Rose Garden, and execute them. That prick is a traitor! He violated his oath to uphold the Constitution. He ordered officers murdered without trial. He deserves a bullet. I volunteer to take a company of Marines across the river and personally deliver one between that bastard's eyes.\"\n\n\"You're wrong, Mort,\" Bud Weiss said. \"The military _must_ remain neutral. We must publicly announce it. Confine all our forces to base. Defend ourselves, yes. But not take sides. California is a different story. Southern California has been invaded by the Mexican Army. It's our job to defend America and shove them back.\"\n\n\"What about defending America from Soetoro?\" grumbled the Marine Corps commandant.\n\nCart McKiernan took his time before he spoke. \"Mort, you know damn well we can't lead a revolution. But that said, I'm going to start carrying a pistol, and if I ever come face to face with Soetoro, I'm going to exercise my rights as a free American and shoot him dead. Now let's get the staff in here and get orders drafted. All offensive operations against Texas and other states are to stop immediately. All forces in the United States are confined to base except in Southern California. Bud, you are going to have to use the air force to supply our forces in SoCal. The navy will cooperate fully. Are we in agreement?\"\n\n\"You understand that if we wash our hands of the Soetoro administration, Barry Soetoro is doomed,\" Franklin Rodriquez remarked.\n\n\"That's up to the American people,\" McKiernan shot back. \"Our problem is to preserve the American armed forces to defend future generations of Americans from foreign threats. I repeat, are we agreed?\"\n\nThey were. They opened the door and the staff trooped in for orders.\n\nThe news that General Martin L. Wynette, chairman of the Joint Chiefs, had committed suicide in his office was merely a footnote to the press release issued by the Pentagon. Henceforth, the release announced, United States armed forces would take no part in or play any role in the political problems the country was enduring. All offensive operations were canceled, all troops confined to base, all ships ordered into port, and all airplanes grounded. Except, however, in Southern California, where United States forces were actively engaged in armed combat with invading forces from Mexico. The statement went further: \"Unless the Republic of Mexico desires a wider war with the United States, it will recall its troops from United States soil immediately. If all Mexican forces are not back across the international border within twenty-four hours, United States forces will attack Mexican forces wherever they can be found.\"\n\n\"Those Pentagon bastards just revolted against the government and issued an ultimatum to the government of Mexico!\" Al Grantham roared as he read the press release. \"What in hell is going on over there?\"\n\nHe found out within two minutes. An icy Cart McKiernan told Grantham on the scrambled telephone, \"You people at the White House are on your own, Grantham. We won't obey your orders and we won't fight rebel forces. We will defend the Pentagon and armed forces bases worldwide, and kick the shit out of Mexico if they don't wise up fast.\"\n\n\"This is mutiny, McKiernan. Treason. You know the penalty for treason.\"\n\n\"Label it anything you like.\"\n\n\"Are you demanding that President Soetoro resign?\"\n\n\"I don't think anyone on this side of the Potomac gives a flying fuck what Barry Soetoro does or doesn't do. Please tell him I said so.\" And Admiral Cart McKiernan hung up on Al Grantham.\nTHIRTY\n\nWe were sitting in our pickups in the parking lot of the little one-story brick office building at the Elkins airport when Jake Grafton landed in the Cessna tail-dragger and taxied up. He shut down, got out of the plane, and came strolling over. It looked to me as if his ribs weren't hurting him too badly; his stride was almost normal.\n\nWillis Coffee and Travis Clay had gone up the road to the main entrance of the airport and were settled in there behind trees, just in case.\n\nExcept for the two on guard duty, we gathered around Grafton. \"Okay,\" he said. \"The road to Dawson is open. I'll take Yocke with me in the plane. The rest of you drive on up there. There is a roadblock about five miles from the southern entrance, but they know you're coming and will let you through. Any questions?\"\n\n\"Yeah,\" Jack Yocke said angrily. \"Just what the hell is going on?\"\n\n\"We're joining the revolutionary army,\" Jake Grafton said calmly, as if that were as plain as the nose on his face.\n\n\"Did you land there?\"\n\n\"No. I talked to them via radio. Tommy,\" he said, \"keep yours handy. Call me on one-twenty-two point nine if you have any trouble. I'll be listening on that freq.\"\n\n\"But who's there?\" Yocke asked, his puzzlement evident.\n\n\"We'll find out when we get there.\"\n\nHe walked back to his plane with Yocke trailing along. The admiral climbed in, and in less than twenty seconds the prop began turning, a little cloud of black smoke puffed from the exhaust, then the prop spun up to a blur as the engine settled into a nice idle. He swung the tail of the bird with a little blast of power and began taxiing for takeoff.\n\nSarah and I were sitting in the front seats of the truck, Dr. Proudfoot in the back, when the Cessna lifted off and turned northward.\n\n\"Well,\" Sarah said with a sigh, \"let's go to the war.\"\n\n\"You knew all about this, didn't you?\" I growled.\n\nShe glanced at me and smiled. \" _He is Jake Grafton_ , Tommy. You, of all people, should have known that he'd be a mile ahead of Barry Soetoro on the best day Soetoro ever had.\"\n\nI couldn't think of a thing to say. We picked up Willis and Travis at the airport entrance and headed up the asphalt ribbon through the mountains for Camp Dawson.\n\nOn the way I fiddled with the radio. Got a station with a seductive female on the mike who said her name was Dixie Cotton. She read the latest news releases from Washington, including one from the Pentagon that said they would no longer fight Americans, on whichever side of the political spectrum, and the ultimatum to Mexico. I wondered if that threat would frighten the Mexicans.\n\nI found myself rubbing my sore neck and, to take my mind off it, kept playing with the radio. I finally got a station that identified itself as being in Kingwood, West Virginia, which I knew was just a mile or two up the road from Camp Dawson. \"Guess the folks up there have their power back,\" I said brightly.\n\nSarah just grunted.\n\n\"Hey, electricity means commodes flush. Don't knock it.\"\n\nThe announcer was telling people in the Kingwood area which stores were open, where they could buy food and fuel. The senior center was open, she said, and would feed anyone who was hungry.\n\nMaybe America was starting to get back to normal. I rubbed my sore neck some more.\n\nA reporter came crashing into the governor's office in the parking garage under the Austin hotel with the Pentagon press release, which his newspaper had downloaded off the satellite. An aide took it into Jack Hays, who was in a meeting with bankers, college professors, and Dallas Federal Reserve officials. The subject of the meeting was the new Texas currency. As one of the Fed's bankers, now working for the Republic of Texas, held forth on the value of money, Hays read the press release.\n\nHays held up his hand, which silenced the moneymen. He read the press release aloud, all of it, including the ultimatum to Mexico.\n\nThe bankers cheered. \"We've won!\"\n\n\"If this is true,\" Jack Hays muttered, too softly for anyone to hear.\n\nHe sent an aide to find his cousin JR.\n\nThe bankers were leaving when JR came in, so they all had to tell him the news and shake hands and congratulate him. \"Best general since Sam Houston,\" one of them told JR, who looked a little stunned.\n\nWith the door closed, JR read the press release. \"Is this true?\" he asked Jack.\n\n\"I don't know,\" Jack Hays said, shrugging. \"But the implications are vast. Either Soetoro wants a political peace, or the U.S. armed forces have mutinied against him.\"\n\n\"Hadn't we better find out which it is?\"\n\n\"We'll find out soon enough. If Soetoro wants a settlement, we'll be hearing from them. In the meantime, let's stop all offensive military operations until we know more.\"\n\n\"What about that attack boat, _Texas_?\" JR asked.\n\n\"You know where she is, what she's doing?\"\n\n\"Hell, no. Loren Snyder and a handful of volunteers took her to sea. Apparently they torpedoed two destroyers busy squirting off Tomahawks at our power plants, but there have been no more ships torpedoed nor has the U.S. Navy shot any more Tomahawks.\"\n\n\"Do we have any way to contact them?\" Jack Hays asked.\n\n\"I certainly don't. I think all we can do is rely on Loren Snyder's good sense and hope for the best.\"\n\n\"Could you call the Pentagon and talk about this?\"\n\n\"I can send them a message through the National Guard message system.\"\n\n\"If they try to locate and attack that submarine, your Mr. Snyder might well fight back.\"\n\n\"Might? Of course he will. But let's not get our knickers in a twist. Loren can take care of himself. I'll pass along a heads-up to the Pentagon, however.\"\n\nPresident Hays leaned forward. \"I've got another project for you too. Texas has roughly a billion dollars in gold on deposit in a vault in a New York bank, the Bank of Manhattan. I want you to make a withdrawal and bring that gold back to Texas. All the bankers say that backing the new Texas dollar with gold will give it instant credibility. That's their prescription for the next few years. After that, they want the Texas dollar to float so the money supply isn't linked to the price of gold, which is nothing but a commodity. We'll see how that goes, but a gold standard does sound like a place to start.\"\n\n\"How much does a billion dollars' worth of gold weigh?\" JR asked.\n\nJack Hays said, \"The treasurer's office said a pound or two less than forty tons. Actually, it's probably worth a lot more than a billion dollars since Soetoro shut down the markets and gave investors worldwide the jitters. In the past, state government sort of ignored the gold. It's like oil under the southeast forty or Grandma's diamond ring\u2014no one gave it a thought. They simply talk about a billion dollars' worth of gold. However, it's there and it's ours, and now we need it.\"\n\nJR whistled. \"New York City,\" he said thoughtfully. \"Forty tons.\"\n\n\"Ingots, I guess,\" Jack said. \"I hope it isn't in wafers in boxes or some such thing.\" He grinned. \"I've got a feeling that the sooner we get that gold to Texas, the better. If Soetoro knew we were coming for it, we wouldn't get it without some kind of political settlement. And there isn't going to be a political settlement between Texas and the United States with Barry Soetoro in the White House.\"\n\n\"I'll need some kind of paper from the treasurer, and maybe a letter from you. You know how bankers are. They'll want to paper their ass.\"\n\nJack Hays frowned. \"They aren't going to turn loose of that gold without Washington's say-so even if you have a letter from Jesus. Take some guns.\"\n\n\"Oh, we will,\" JR said with a disarming smile. \"Rest assured, we won't forget to pack those. But give me a letter from the treasurer and one from you.\"\n\nJack called in a secretary and dictated one. While she typed it, he called the treasurer and told him what was wanted. \"He'll send it over to the Guard this afternoon,\" he told JR.\n\nWhen he left with Jack's letter in hand, JR went to the Texas State Library and Archives building on Brazos Street. He asked to see the head librarian and told him what he wanted. Twenty minutes later he left with a copy of a letter from the White House in his hand, one congratulating some scout for achieving the rank of eagle. That kid's parents knew somebody, he thought.\n\nHis next stop was a printer at the Texas Department of Public Safety. He asked to see the head printer, and after introducing himself, presented him with the copy of the White House letter to the Eagle Scout. \"I need at least four sheets of a nice white bond with the president of the United States' seal on it. Exactly like this, identical.\"\n\n\"Whoa. On whose say-so?\"\n\n\"Mine. Or we can get the president's, or Colonel Tenney's. Whom do you want?\"\n\n\"This isn't going to be used for a forgery, is it?\"\n\n\"Perish the thought.\"\n\nThe man sighed and asked, \"When do you want it?\"\n\n\"Well, I have to go see Colonel Tenney upstairs. That should take no more than an hour. I'll pick it up when I finish.\"\n\n\"An absolute rush job will take about a week. Every office in government wants new stationery now that we're a republic again. We have\u2014\"\n\n\"One hour, or I go get Colonel Tenney and bring him here to talk to you.\"\n\nThe man frowned at the letter with distaste, as if JR had been using it for toilet paper, then at JR Hays. \"You're serious?\"\n\n\"As a heart attack.\"\n\n\"Who the hell are you?\"\n\n\"I told you. Major General Hays, commanding the Texas Guard.\"\n\n\"Never heard of you. When the Guard wants something, they always send a sergeant, a gal named Dooley.\" He waved the letter. \"But okay. One hour.\" Then he wheeled and marched away from the counter.\n\nJR went upstairs to see Colonel Frank Tenney and explained the problem. \"Maybe the vault will be open when we get there, but probably not. Maybe they'll open it for me, but maybe not. More than likely I'll have to help myself. I need some expert advice about how to get into a gold bullion vault.\"\n\n\"Forget it,\" Tenney said. \"It can't be done. There is no way in the world to get into either of the big gold vaults in New York. Texas' gold is in the Bank of Manhattan, which has a helluva vault. But the biggest gold repository on earth, bigger than Fort Knox or the Bank of England gold vaults, is under the New York Federal Reserve Bank. They have at least five thousand tons of gold there, and it's guarded by a private army.\"\n\n\"Explosives?\"\n\n\"Two Mosler vaults were in banks in Hiroshima when we dropped the bomb there. The banks were obliterated, but the vaults were intact. In 1957 the air force set off a thirty-seven-kiloton bomb near a vault, and it remained intact. JR, you ain't going in one of those things unless they let you in. And the chances of them doing that are essentially microscopic. Go back and tell your cousin Jack that it can't be done.\"\n\n\"That's probably good advice,\" JR acknowledged. He said good-bye and went to the Texas Treasury Department, where he had a private interview with the treasurer of the Republic. The man was prematurely bald and wore a suit and tie even though the building wasn't air-conditioned. All the juice from the emergency generator was being used to run lights and computers.\n\nThey discussed Texas' gold reserve, how many ounces and so forth.\n\n\"So where is Texas' gold?\" JR Hays asked.\n\n\"Bank of Manhattan.\"\n\n\"Have you ever visited the facility, looked at the gold?\"\n\n\"Oh yes. Impressive vault. The bank installed it when people started speculating in bullion ten or fifteen years ago. They didn't want to store the stuff at home or in a suburban safe deposit box, so the bank got into the business of storing gold for a fee. Modern facility, great vault, as secure as any vault on earth. We put about half a billion dollars of the state's funds into gold, and they stored it for us. Our gold has essentially doubled in value, so it's worth about a billion, or was until the current political difficulty arose. Probably worth twice that now.\"\n\n\"Good investment.\"\n\nThe treasurer nodded and looked pleased.\n\n\"What about the New York Federal Reserve's vault?\"\n\n\"I got a tour once,\" the bureaucrat acknowledged. \"Didn't get into the vault, of course, since they never let humans inside. The gold is moved on trolleys by remote control. Robots stack the ingots and load and offload the trolleys.\"\n\n\"I've heard they have a private army guarding the vault.\"\n\nThe treasurer nodded. \"Yes, indeed. Most of what I know about the vault I picked up in casual conversation from the assistant treasurer, who used to work at the Bank of Manhattan. He wanted to get back to Texas so I hired him. Guy named Chuy Medina.\"\n\n\"May I talk to him?\"\n\n\"Sure. Great guy. You'll like Chuy. I talked to the governor about the gold, but why are you interested in it?\"\n\n\"Oh, that gold has to come back to Texas someday. We thought we should ask some questions.\"\n\n\"Sure.\"\n\nChuy Medina was of medium height, about fifty years of age, from McAllen, Texas, and had spent fifteen years at the Bank of Manhattan. Left two years ago when he scored a job at the Texas treasurer's office.\n\n\"Tell me about the Bank of Manhattan,\" JR prompted. \"They have about forty tons of Texas gold, and I have been ordered to make a withdrawal.\"\n\nMedina laughed. \"That's a joke, right?\"\n\n\"Perhaps.\"\n\n\"This is like some weird plot from _Mission: Impossible_. There ain't no way, man. No way at all.\"\n\n\"Talk to me,\" JR Hays said with a smile. \"Convince me.\"\n\nThe FEMA concentration camp guard towers on the edge of Camp Dawson were empty when we rolled by and went between the guards at the main gate. Several of the guards were wearing old army shirts, but most were in jeans and T-shirts. They were armed to the teeth and looked to me like they knew precisely what they were doing. This might be amateur hour, but there was some military discipline and brains guiding the amateurs. There wasn't a FEMA uniform in sight.\n\nThe place was as crowded as a state fair, but without the animals. I estimated I could see over a thousand people, all adults, most in civilian clothes, all armed and doing army stuff, like working on weapons, loading trucks, and doing calisthenics. Cars were parked in rows, men wearing pistols directed us to a parking place, and a girl who looked as if she had ditched her classes in high school that afternoon escorted us toward the headquarters building, not the one in the concentration camp, but the main National Guard building. I could hear rifles popping, no doubt over at the shooting range. And a buzzing overhead. I looked up and saw a Predator drone taking off with a Hellfire under each wing.\n\nI glanced over my shoulder and got a good gander at Sal Molina's face. The man was stunned. Almost stupefied. Obviously Grafton hadn't been whispering to him, either. If he had been doing any whispering, I supposed it was to Sarah Houston, who looked as if she were trooping up to the director's office to be given another twenty-hour-a-day assignment.\n\nWillie Varner was looking around wide-eyed. He had been clueless too. Willis and Travis were almost as surprised as the Wire.\n\nI confess, I was a bit pissed at Grafton. I would have bet the ranch that _he_ wasn't surprised, that he well knew what we would find here. Why hadn't the spook bastard confided in me? Need-to-know and all that spy shit, I suppose.\n\nThey confiscated all cell phones as we came through the front door, and put a sticky on each one with the owner's name. Then they patted us down.\n\nWe ended up in the back of a conference room standing against the wall, all of us, including Dr. Proudfoot. Grafton was sitting at a table right up front, and that _Washington Post_ weenie Jack Yocke was sitting beside him as if he were number two in the chain of command. Three big bananas, all in their fifties, were standing in front of a map that covered a blackboard, I suppose, taking turns briefing Grafton. They had started a few minutes ago, and they didn't bother starting over for us. Another dozen or so people, perhaps half of them women, all wearing pistols, were in the chairs behind Grafton and Yocke and in front of us. One was a congressman I recognized from television, Jerry Marquart.\n\n\"So our plan is to have First Corps . . .\" Yep, I thought, these are army dudes. \". . . proceed east on I-Sixty-Eight to Cumberland and Hagerstown. Second Corps will go east on U.S. Route Fifty to Winchester and then to Leesburg and into the District along that route. All this is subject to change if we hit opposition or find bridges have been blown. We'll be close enough together on parallel routes that we can mass if necessary. Keep the drones up and looking, use our Special Forces veterans as scouts, and take whatever comes.\"\n\nGrafton had a few questions, then asked to see the Pentagon's press release again. He read it carefully, then laid it on the table in front of him and said, \"This is too good to be true.\"\n\n\"It could be disinformation, deception,\" the head dog agreed. \"We don't have their crypto codes, but from all the plain-language traffic we are hearing, perhaps there is some truth in it.\"\n\n\"What plain-language traffic?\"\n\n\"FEMA and Homeland. They are complaining bitterly that Soetoro has betrayed them.\"\n\n\"Even if we get into a firefight with that crowd, that doesn't mean the Pentagon's press release is inaccurate. It may only mean that the paramilitary boys are taking orders directly from the White House. If we see army troops, however, we'll know this is a pretty little lie.\"\n\n\"Yes, sir.\"\n\nThey chewed the rag about trucks, ammo, food, weapons, and all of that for another half hour, then I ducked out to find a restroom. There was toilet paper in there and the commode flushed. Life was looking up.\n\nWhen I got back, the conference had broken up, the rebel officers were leaving, and only our little crowd remained. Everyone had taken seats around the conference table so they could talk to Admiral Grafton, who looked at Willie and said, \"Please escort Dr. Proudfoot to the hospital. They may need his services. Is that all right with you, Doctor?\"\n\nIt was, and the two of them left.\n\nJack Yocke jumped right in before the door shut behind them. \"This rebel enclave didn't just happen, Grafton. Someone made it happen and you knew all about it.\"\n\n\"I made it happen,\" Grafton said, looking around and taking in faces. \"Sarah and I knew several months before Soetoro declared martial law that he was going to do it. We knew he was waiting for an incident that would justify martial law. The terrorists obliged. I have spent my adult life in the military and intelligence business. I talked to people I knew I could trust, told them Soetoro's intentions, and asked for their help.\"\n\n\"How did you know Soetoro was going to seize power? Did Molina tell you?\"\n\n\"Sal, do you want to answer Yocke?\"\n\n\"No,\" Molina said. He had to force the word out, and it came out unnaturally loud.\n\n\"But you knew Soetoro's plans,\" Jack Yocke persisted, staring at the president's man.\n\n\"I'm not going to\u2014\"\n\nGrafton spoke, which cut off Molina. \"Sarah.\"\n\nShe was seated at the end of the table. She had her computer out of its case and was fiddling with the keyboard. \"I bugged the White House,\" she said, \"at Admiral Grafton's order. We used every electronic device in the White House as a listening device, including computers and cell phones.\"\n\nMolina turned ashen.\n\n\"Including yours, Mr. Molina, and President Soetoro's.\"\n\nMolina gaped at her. The way she said it, matter-of-factly, as if she were making a report to her boss, made it impossible to disbelieve her. Then Sarah pushed a button.\n\nThe president's voice came from the speaker, quite plain. \"Martial law will give us the opportunity to remake America the way it should be, take charge of industries and banks, tax the rich, redistribute income, give full citizenship to illegals, take power from the states, and rule from Washington. We'll make America into a progressive socialist country that all of us will be proud to live in, and, incidentally, we'll make a good start on saving the planet.\"\n\nMolina's voice: \"It won't work, Mr. President. The majority of Americans will never approve. Revolutions from the top down never work. You can't take the American people where they don't want to go.\"\n\nSarah pushed a key and the sound stopped. She hit a few more and closed the computer.\n\nIn the silence that followed, Molina turned his attention to Jake Grafton, who had his eyes on him.\n\nJack Yocke broke the silence with a question aimed at Sarah. \"What have you done to that file?\"\n\n\"The background noises have been digitally suppressed so the speakers' voices are clearer. That's it.\"\n\nHe grunted and faced Jake Grafton. \"You knew that they were waiting. For a terrorist incident? Did they arrange those incidents?\"\n\nGrafton turned those gray eyes on the reporter. \"They let those people into the country, lied about the vetting they would receive. They played for a terrorist incident, or incidents, and they got them. Considering who they were letting into the country, it would have been a miracle if there weren't any terror strikes.\"\n\n\"You could have stopped it. Hundreds of innocent people were killed. Obviously you didn't stop it.\"\n\n\"And just how do you think I should have accomplished that feat?\"\n\n\"You sacrificed those people.\"\n\nGrafton's face didn't even twitch.\n\n\"You are a ruthless man, Admiral,\" Yocke said softly.\n\n\"I think this has gone quite far enough,\" the admiral said. \"Jack, go find someone to interview. You might start with Congressman Jerry Marquart. I am sure he has quite a story to tell.\" His eyes moved to Molina. \"You stay,\" he said.\n\nYocke stomped out with little grace. That's the free press for you. When the door to the room was once again closed, Grafton said, \"I think it is time for a confession from you, Sal. Not one in the hearing of the _Washington Post_ , but here before me and Sarah and these men who risked their lives to drag us out of that concentration camp a few hundred yards away.\"\n\nMolina seemed to have shriveled and aged ten years. He tried to compose himself, but it was a lost effort.\n\n\"Let me start your confession for you,\" Jake Grafton said. \"You were never Barry Soetoro's advisor\u2014you were his controller. Your boss is Anton Hunt, the billionaire left-wing financier. He created Barry Soetoro, and you were there to tell him what to do, to make him obey Anton Hunt, so he could make more billions and create the kind of world he thought we all should live in.\"\n\nMolina licked his lips. \"I\u2014\"\n\nGrafton smacked the table a healthy lick with his palm. It sounded like a pistol shot. \"I'll do the talking. You even suggested that Soetoro arrest me as one of the conspirators in the fake plot to take over the government. You argued that spies are easy to blame, and people would automatically give credence to any story of nefarious activities at the CIA. When you reported Soetoro's plans to Anton Hunt, he was horrified. He hadn't signed on to a communist dictatorship.\n\n\"He thought Soetoro was a black man of modest intelligence with a good gift of gab who would be grateful for all Hunt had done to lift him to the highest place in America and make him the most powerful man in the world. He thought he could control Barry Soetoro because he had written evidence of all he had done for him: a fake birth certificate, passport applications removed from the State Department, bribes to get him into school, bribes to conceal his academic records, all of it. He thought the evidence would ruin Soetoro if it ever came out, but the evidence was a two-edged sword. Soetoro knew the evidence would also take down Anton Hunt, so Hunt didn't dare to ever reveal it.\"\n\nMolina licked his lips and wiped a sheen of perspiration from his forehead.\n\n\"But somewhere along the line,\" Grafton continued, \"Hunt began to realize that he had no control over Soetoro, but the reverse was true. Soetoro controlled _him_. Perhaps the revelation occurred when Soetoro demanded Hunt fund demonstrators to protest racial injustice, demonstrations designed to drive a wedge between white and black America. Or perhaps the light dawned for Hunt when Soetoro sacrificed an ambassador and several Marines to the Taliban. Perhaps you can tell us, Sal. When did Hunt see the evil in Soetoro?\"\n\nSal Molina was staring at the tabletop.\n\n\"Certainly both of you were in no doubt when Soetoro plotted martial law and suspension of the Constitution. You knew then, didn't you, Sal?\"\n\nSilence.\n\n\" _Answer me_!\" Grafton roared.\n\n\"Yes.\"\n\n\"One of the most amazing things I heard on Sarah's eavesdropping program was Soetoro telling you that Hunt thought he had a nigger slave in the White House, and the nigger had made a slave of him. And he made a slave of you, the slave driver. Do you remember that? Remember his laughter?\"\n\n\"He's a monster,\" Molina whispered. \"He loathes white people. He wants to rule a nonwhite America. He's willing to ignite a race war, burn America, and rule in the ashes.\"\n\n\"And you didn't think it would work.\" That wasn't a question, but a statement.\n\n\"I didn't,\" Molina said.\n\n\"You argued, unsuccessfully, and only managed to convince him you were disloyal and a danger, so he sent you to the gulag.\"\n\nGrafton leaned back in his seat, his eyes fixed on Molina. \"You were lucky that sadist Sluggo Sweatt decided to have his fun with me before he got to you, because Soetoro wanted you dead. But Soetoro gave Sweatt his priorities. First the scapegoat, then the traitor.\"\n\n\"You don't know that,\" Sal Molina whispered.\n\n\"I deduce it. I thought it was a stroke of luck that FEMA brought me to the concentration camp here at Dawson, because that is where we\u2014my friends and I\u2014agreed to rendezvous two weeks after Soetoro declared martial law. Then Sweatt began his program of forcing a confession. The irony is, I was and am guilty of a conspiracy to remove the president of the United States from power, which was Sweatt's accusation. I thought it likely he would beat me to death.\n\n\"Not that my death would have made any difference. If I weren't here, the others still would be. There are two thousand five hundred men and women here at Camp Dawson, and they are committed to the hilt. It's victory or death for them. If they don't kill Soetoro, he will kill them. They understand that.\"\n\nGrafton smacked the table again. \"Yocke accused me of being ruthless. I _am_. The life of the United States is at stake. If I had thought it could be done, I would have shot Soetoro myself.\" He pointed his finger at Molina. \"If I thought your death would move Soetoro one inch away from the White House, make an iota of difference, I would shoot you myself, here and now. Do you understand?\"\n\nMolina bit his lip.\n\nGrafton smacked the table again, and the map fell off the blackboard. \"Answer me!\" he roared.\n\n\"Yes.\"\n\n\"Consider yourself a prisoner. If you try to escape, you will be shot.\" He turned to Travis. \"Lock him in one of those cells in the concentration camp. See that he is guarded twenty-four hours a day and arrange to have him fed.\"\n\n\"Yes, sir.\" Travis Clay grabbed Molina's arm, hoisting him from the chair in which he sat. I pulled out my .45.\n\n\"Get rid of the web belt,\" I told Molina. \"Take it off.\" He was wearing a pistol.\n\nHe reached down, released the buckle, and let the belt fall on the floor.\n\n\"Your leather belt too,\" Grafton said. \"We'll save you for a firing squad.\"\n\nThe belt came off and went onto the floor.\n\nMolina could barely walk, so Travis almost dragged him.\nTHIRTY-ONE\n\n\"You could have confided in me,\" I told Grafton.\n\nHe looked surprised. \"I told Sarah to tell you about the eavesdropping scheme. Did she tell you?\"\n\n\"Well, yes, but not about all this other stuff.\"\n\n\"Tommy, you have a good brain between those ears and you had better start using it.\"\n\nYou would think that after all these years around Grafton I would know how to keep my mouth shut. One of these days I am going to get that trick down.\n\n\"The local radio station is back on the air,\" Sarah told the boss. That female could read minds. \"I don't know if the power is on or if they are using a generator.\"\n\n\"Okay,\" Grafton said. \"Tommy, take Sarah over there. She is going to put some of that stuff from the White House on the air. She has winnowed it down to about sixty hours. Convince the radio staff to do it, and then set up an ambush around the station and transmission tower. Use Travis and Willis Coffee. Take whatever weapons you need. If the military is still in the game, they'll take the tower out with a Hellfire or commandos. If it's FEMA or Homeland, expect a few truckloads of thugs. Don't take any prisoners\u2014we don't have anywhere to keep them. The beds in the concentration camp are being used as barracks.\"\n\n\"Yes, sir.\"\n\n\"Sarah, you know what to do.\"\n\n\"This will set off an explosion in the White House,\" she said flatly.\n\n\"I hope. Infuriated, frightened men don't think very well. Go.\"\n\nSarah repacked her computer and we left, with Willis Coffee trailing along behind. We picked up Travis ten minutes later and took my stolen FEMA pickup truck.\n\nDowntown Kingwood was a typical American small town, in my opinion. A Walmart on the edge of town had pretty much turned the old downtown into a wasteland of vacant stores interspersed with insurance agencies, lawyers' offices, gift and craft shops. All of them looked closed, and there were no parked cars.\n\nThe radio station's offices were in one of the old storefronts on the east side of the street in the middle of the block. The transmission tower was obviously offsite, probably on a nearby hill. I parked right in front, and Sarah and I strolled in while Willis and Travis, each with an M4 in their hands, walked to the adjacent corners.\n\nThe lady at the front desk was still on the right side of forty and had a cute hairdo and a ready smile. She even had on a plastic name tag: \"Sue.\"\n\n\"Good afternoon,\" she said brightly. The studio was right behind her, visible through a double-pane window. A woman was in there talking into a boom mike, and a young guy in a ponytail was fielding telephone calls. We could hear the station feed over a hidden loudspeaker system, background noise. Above the window was a large clock with a sweep second hand.\n\n\"Are you with the government?\"\n\n\"Not anymore. We were federal employees and left under a cloud.\" I smiled.\n\n\"Really!\" she said, her eyes widening.\n\nI confided in a low voice, \"I stole our truck.\" Then I introduced Sarah and myself.\n\nThe desk lady stared. I continued smoothly, as if stealing a government vehicle needs no explanation. \"How long has the power been back on?\"\n\n\"Since yesterday morning. We got back on the air as quickly as we could.\"\n\n\"Don't you have an emergency generator?\"\n\n\"We ran out of gas for it. The station manager is down waiting in line at the filling station to fill some cans now.\" With only a little prompting from us, she chattered on. The station was licensed at one thousand watts, sunrise to sunset. The transmitter was outside town on Mount Morgan, named after a local farmer who leased the site to the station. \"He's such a nice gentleman,\" she added.\n\n\"We should probably wait for the manager,\" Sarah said, glancing at me. \"When do you expect him back?\"\n\n\"In a little while, certainly, unless the line is too long or the filling station runs out of gasoline. We close the office here at five.\" It was ten till. \"And go off the air at . . .\" she glanced at her calendar. \". . . seven thirty-two.\"\n\nThere was a hallway that looked as if it went all the way through the building, and a door at the end of it. The door opened and a portly man of medium height with a fringe of gray hair around a white pate came bustling through it. He opened the door to the studio and went in. In less than a minute, he came out. He addressed Sue. \"I got the last of the gas at Plunkett's. I just told Jan. She'll put it on the air immediately.\"\n\n\"These folks want to talk to you,\" Sue told him. She said his name, Howard Shinaberry. He glanced at us, at our web belts and holsters, and waited.\n\n\"Sarah Houston,\" I said, nodding at my companion, \"and I'm Tommy Carmellini. Sarah wants to talk to you in your office.\"\n\nHe shrugged and led the way down the hall to another door. Sarah followed with her computer case.\n\nI smiled at Sue, then walked down the hallway and went out back. There was an alley and a parking lot. Three cars and an old Chevy pickup were parked there. I surveyed the alley. All I could see was a cat wandering around and a bunch of garbage cans. The gas cans were in the back of the truck, so I unloaded them and put them in the hallway.\n\nI closed the alley door and waited by the front desk with Sue. \"Does Mr. Shinaberry own the station?\"\n\n\"Oh, no. He's just the manager. Three doctors own it.\"\n\n\"Local doctors?\"\n\n\"They live in Maryland, Bethesda I think.\"\n\nSue chattered on. She was a local and had worked at the station for five years come November. She liked it. She met such interesting people. \"Do you have an ad you want us to air?\"\n\n\"Something like that,\" I replied.\n\nShe got busy locking the cash register and putting things away. Five o'clock came and went.\n\n\"If you want to go home, that's all right,\" I said. \"I'll tell Mr. Shinaberry.\"\n\n\"I'll just wait, in case he has something else for me to do.\" She was obviously getting nervous. I didn't blame her. I gave her my best innocent smile that had melted a thousand hearts.\n\nAt nine after five, Mr. Shinaberry and Sarah came from the office out to the front desk. She paused beside me and said, \"He doesn't want to do it.\"\n\n\"Our license is up for renewal in three months,\" Shinaberry explained. \"That stuff on that computer is dynamite. The FCC\u2014\"\n\nI went out the front door to the sidewalk and gave Travis Clay the Hi sign. He came walking back, his M4 cradled in his arms. We went back into the radio station together.\n\nShinaberry was explaining to Sarah why the owners would fire him if the file on the computer were put out on the air. \"I know the president's voice, and it certainly sounds like him, but if the file is fake, airing it would be libel, and if it's real I can't imagine how that recording was obtained legally\u2014\"\n\n\"You know about the rebels down at Camp Dawson?\" I asked as I rubbed my sore neck. I realized I was doing it and stopped.\n\n\"The general in charge\u2014at least he said he was a general\u2014was in here and asked us not to mention all the people there over the air. And we haven't. Haven't said a peep about Camp Dawson. I gave our staff strict instructions.\"\n\n\"This gentleman is Travis Clay. Travis, take Mr. Shinaberry over to Dawson and let him talk to Jake Grafton. Use Mr. Shinaberry's truck. It's out back.\"\n\n\"Now, see here\u2014\" Shinaberry protested.\n\nTravis put his hand on the guy's shoulder and smiled. \"Don't be difficult,\" he said. \"You can drive.\"\n\nAfter they left, I suggested to Sue that it was time for her to go home. \"We'll lock up when we leave, after Mr. Shinaberry gets back.\"\n\nShe was obviously relieved. She grabbed her purse without saying good-bye, trotted down the hallway, and closed the alley door behind her.\n\n\"It's all yours,\" I said to Sarah. \"Send Jan out and get that guy in the ponytail to show you how the equipment works.\"\n\nSarah took her computer and went into the studio. After ten minutes the announcer lady came out, frowned at me, and left via the alley door too. Ponytail was busy with a thumb drive Sarah had given him. Then Sarah got on the mike.\n\n\"We are going to air segments of a recording that was made at the White House over the previous six months. Not all of it, but segments. The voices you will hear are those of President Soetoro, his staff, and other public officials. There are about sixty hours of recorded material, a small fraction of the whole, and this station will be on the air day and night until the entire sixty hours has played, then we will run it again. Josh, let it rip.\"\n\nAnd he did.\n\nBarry Soetoro's voice came over the loudspeaker. Three minutes later the telephone rang. I answered it with the station's call letters.\n\nA man's voice: \"Where in the hell did you guys get that tape?\"\n\n\"How does it sound?\" I asked.\n\n\"Holy shit! President Soetoro said that?\"\n\n\"He did.\"\n\n\"Jesus Christ!\"\n\n\"He didn't have anything to do with it,\" I told him and put the phone back in its cradle. It rang again. I figured that we were going to get a lot of calls, so I unplugged the phone. I looked into the studio and saw that Sarah was doing the same thing to the phone in there. I could hear the phone ringing in the manager's office, so I walked down and unplugged that one too.\n\nTobe Baha, the Secret Service sniper, was having dinner that evening at his hotel on Congress Avenue in Austin. It was a nice hotel, perfect for expense-account executives and rich oilmen bringing their wives or girlfriends to see the bright lights of the big city. Tobe thought his odd hours would bring less notice here and he would have to answer fewer well-meaning questions than he would have at some cheap motel on the interstate where guests rarely stayed more than a night or two.\n\nSo he was studying the menu and contemplating ordering a steak when three men entered the dining room, looked around, and seeing him, walked purposefully toward his table. They were in civilian clothes wearing sports coats, and from the slight bulges he could see that they were packing pistols in their armpits. After years in the Secret Service, he could spot an armed man at fifty yards.\n\nThe man in front seated himself on Tobe's left and put an iPad on the table. The other two took the remaining chairs.\n\n\"Good evening, Mr. Baha,\" the man on his left said. He was the older of the three, in his mid-fifties, with salt-and-pepper hair getting thin on top. \"I'm Colonel Frank Tenney. I'm the head of the Texas Department of Public Safety. These gentlemen are colleagues of mine.\"\n\nTobe tried to hide his surprise, and did fairly well, he thought. He was registered at this hotel under a false name, so the use of his real name put him on notice.\n\n\"Are you carrying this evening?\" Tenney asked, just making conversation.\n\nTobe tried to look surprised. \"Of course not.\"\n\nTenney just nodded. The waitress came over, delivered Tobe's Scotch on the rocks, passed out menus to the new arrivals, and inquired about drinks. The lawmen all ordered iced tea.\n\n\"I have some video on my iPad I'd like to show you,\" Tenney said, then picked up the tablet and began playing with it. In a few seconds, he placed it so Tobe could see it.\n\nThe screen began showing aerial shots. Tobe Baha instantly knew what he was looking at: drone surveillance video. And there he was, in the van, parking it, getting out, looking around, strolling the street. Then there were shots of Tobe up on roofs, using the laser rangefinder, back on the street, driving through the city, going into stores and public places. . .\n\nAfter three or so minutes, Tenney picked up the iPad and shut it off. He put it on his left.\n\nTenney smiled at Baha. The waitress came back with the drinks. Tenney told her that they would not be staying for dinner. She looked at Tobe, who told her, \"Later.\"\n\nWhen she had moved off, Tenney said, \"We were surprised when you showed up in Austin, since Texas is no longer a part of the United States and Barry Soetoro isn't planning a visit, at least to the best of my knowledge.\"\n\nTobe picked up the Scotch and sipped it. His hand was steady, and he hoped that the colonel noticed that. If he did, he gave no sign.\n\n\"We thought that perhaps you were here to use your sniper skills on someone in Austin. Of course, we haven't yet seen you with your rifle. No doubt it is somewhere in Austin, and if necessary we could arrest you and search and find it. It will probably have your fingerprints on it and so forth. But President Hays thought that an arrest and trial would not be good for future relations between Texas and the United States.\"\n\nColonel Tenney leaned toward Tobe Baha. He was speaking softly, and his eyes were impossible to avoid. \"I also thought about disappearing you. That would solve any diplomatic problems, and the justice system wouldn't have the expense of fooling with you. Do you understand?\"\n\nThose eyes boring into his made evasion impossible. \"Yes,\" Tobe said.\n\n\"That's good. We're tying up a lot of people flying these drones and keeping tabs on you, and enough is enough. So I stopped by this evening to let you know. If a sniper fires a shot anywhere in Austin and you're still around, we'll come for you. You will be killed resisting arrest and be buried somewhere in west Texas in an unmarked grave. Have I made myself clear?\"\n\n\"Yes.\"\n\n\"Barry Soetoro or your Secret Service colleagues may decide that you have lived long enough, so you may want to rethink your return to the States. Be that as it may, you may reside in Texas as long as you never again show your face in Austin. If someone fires a rifle in Austin and you are around after this evening, you are a walking dead man.\"\n\nTenney stood and picked up his iPad. \"Just a friendly warning. You can pay for our tea.\"\n\nHe and his colleagues walked out of the restaurant.\n\nTobe Baha drained his Scotch. He glanced at the menu, decided he wasn't hungry, and ordered another drink.\n\nAbout ten after six, Travis Clay came through the alley door of the radio station with four buffed-up guys. \"Grafton sent Mr. Shinaberry home. The sheriff and city police chief were there and we won't have any trouble with them.\"\n\n\"Patriots are they?\"\n\n\"With twenty-five hundred armed people at Dawson, they saw the light, whether they are patriots or Soetoro loyalists.\" He gestured to the other men. \"Grafton thought we could use more help.\"\n\n\"Get Willis in here.\"\n\nThe ex-soldiers, for that is what they were, stood listening to the radio feed on the loudspeakers, shaking their heads. One of them muttered, \"That son of a bitch.\"\n\nI briefed the troops. Two of them at each end of the alley. I sent Willis across the street and asked him to put an M240 machine gun on the roof of the old bank building on the corner; the false brick front would give him a little protection. Travis was to be on the roof on the other corner with another M240. These were belt-fed guns that fired the 7.62\u00d751 NATO cartridges. I would have our third machine gun, an M249 that was fed by a belt of 5.56x45 NATO cartridges, inside here on the counter. \"Lots of grenades and AT4s. We'll make the street in front our killing zone.\"\n\nEveryone trooped out to the FEMA truck, where Willis passed out weapons and ammo. We carried some MREs into the station, and I drove the truck around back and backed in up to the alley door. We carried stuff in. I brought in two boxes of ammo for my machine gun, an M4 carbine, a dozen grenades, and a couple of AT4s.\n\nI was feeding a belt of ammo into the M249 when Josh came out. He looked at the weapons and ammo and at me. \"Where did you people get that recording?\"\n\n\"What did Sarah tell you?\"\n\n\"That a little bird gave it to her.\"\n\n\"There you are.\"\n\n\"I'm getting the hell outta here,\" he said, and marched for the alley door. I heard his old ride fire up. Josh needed new mufflers. Then it went away down the alley.\n\nAfter a while Sarah came out. \"It's all automatic,\" she said. \"I don't need to sit there and watch it.\"\n\n\"Want some dinner?\"\n\nShe gave me The Look.\n\n\"I put some MREs in the break room. There's a microwave. I'd like meatloaf, some potatoes, and corn.\"\n\n\"Yes, General,\" she said, and marched away.\n\nAs I dug into my gourmet repast\u2014Sarah could do MREs, let me tell you\u2014individual cars and pickups, each full of people, kept creeping down the street and looking into the radio station. Finally I wised up and turned off the lights in the office.\n\nIt was after nine o'clock and Soetoro was plotting with Al Grantham and Sulana Schanck on how they would turn off the power and blame it on the right-wing constitutionalists, when a van pulling an army generator drifted to a stop at the curb outside. The van had a big, flexible aerial mounted on the rear bumper.\n\nI cradled the M4 and waited. A woman walked around the van, tried the door to the station, found it unlocked, and came in. I could see a guy still in the van.\n\nThe woman looked at the machine gun, the grenades on the counter, and me. \"I'm Dixie Cotton,\" she said. She couldn't have been a day over thirty, with a sexy bedroom voice and a figure to match. She was wearing tight jeans and a T-shirt that revealed everything she had, which was a lot.\n\n\"Tommy Carmellini.\"\n\nSarah came from the hallway. I introduced the two.\n\n\"I've heard of you,\" Sarah said. \"Aren't you 'The Mouth of the South'?\"\n\n\"It's been said,\" she admitted modestly. It sure had. She had a talk show on an Atlanta radio station and thrived on controversy, which she created by trashing everyone who disagreed with her, which was practically everybody.\n\n\"I thought Soetoro had FEMA lock you up as a dangerous subversive. How did you get out?\"\n\n\"A doctor certified that I was crazy and some of my friends paid a few bribes, so they turned me loose.\"\n\n\"Could I get a certification like that?\" I asked hopefully.\n\n\"So where did you people get that recording?\"\n\n\"You know the old story: if I told you I'd have to kill you,\" I said deadpan.\n\n\"Bullshit,\" she said dismissively. \"Is it real?\"\n\n\"Of course.\"\n\n\"I run a mobile pirate radio station these days, during the current difficulty, while my station in Atlanta is up to its armpits in federal censors. I'd like a copy of that recording. I'll cruise Washington and broadcast it.\"\n\n\"They'll kill you if they catch you,\" I told her, \"tits, mouth, and all.\"\n\n\"Not the way I work it, they won't. They've been trying for four days and haven't caught me yet.\"\n\n\"You're living on borrowed time.\"\n\n\"That's my lookout.\"\n\n\"Sarah, what do you think?\"\n\nShe shrugged. \"Can you use thumb drives?\"\n\n\"Sure.\"\n\n\"Let's make you some.\" And Sarah led her into the studio.\n\nCars and pickups crept by at random intervals all evening. The locals were getting an earful and they were curious.\n\nAbout midnight, an army truck pulled up outside and soldiers piled out of the back. Jake Grafton climbed down from the cab, carefully, and led the soldiers, six of them, inside. The soldiers were in full combat gear, with helmets, weapons, and body armor. Grafton was wearing a camo shirt and trousers. Willie Varner was the only one in civilian duds, and he was carrying an M4.\n\nGrafton introduced the soldiers. Two army officers and four senior sergeants, all with combat experience. \"They came to Dawson with General Netherton,\" he explained. \"Where do you want them?\"\n\n\"On the roofs on this side of the street,\" I said to them. \"The street is the kill zone. Don't let any of the bad guys get into this office.\"\n\nWe talked about frequencies, because they all had handheld radios, and they trooped out.\n\n\"FEMA and Homeland have at least a dozen people on the way,\" he said. \"They're on the clear-voice radio. Soetoro is raving mad.\"\n\nI told him about Dixie Cotton. \"She's nuts,\" I added. \"Literally and figuratively. Certified even. They'll execute her.\"\n\n\"That's her problem,\" Jake Grafton said. He looked around. \"Break out those windows. You want the glass on the sidewalk, not flying around in here.\"\n\n\"I'm worried about the radio tower, which is out on some knoll called Mount Morgan.\"\n\n\"We have it covered,\" he said. He glanced at the machine gun on the counter. \"Is that where you want it?\"\n\n\"I doubt if they'll be stupid enough to drive up in front of the joint, but if they do . . .\"\n\n\"A man can always hope,\" he said.\n\n\"You could ambush these dudes on the way into town,\" I pointed out.\n\n\"It'll take most of the night to get ambushes set up. We'll whack the second wave. You deal with the first bunch.\"\n\n\"If they get a bullet into the equipment in the studio we are well and truly fucked,\" I remarked.\n\n\"Make sure they don't.\"\n\nI almost said something I would probably have regretted later, but I managed to stifle myself.\n\n\"What are you doing here?\" I asked Willie.\n\n\"I'm your bodyguard.\"\n\nAs if I needed something else to worry about.\n\n\"You had dinner?\"\n\n\"Oh, yeah. They're good feeders over at that camp.\"\n\n\"You should have joined the army when you were a kid.\"\n\n\"Maybe you're right.\"\n\nGrafton, Sarah, and I chatted for a bit, the admiral shook Sarah's hand and mine, then went back out and climbed into the army truck, which got under way in a cloud of diesel exhaust.\n\n\"There goes the next president of the United States,\" Sarah said.\n\n\"Not after Jack Yocke gets through with him,\" I replied.\n\n\"Screw Jack Yocke,\" Sarah said.\n\nSarah went into the break room, which had a cot, and sacked out. I broke out the office windows, as Grafton had suggested.\n\nWillie was in a talkative mood. He carefully laid his M4 on the counter. \"Nice shooter,\" he said with feeling.\n\n\"You know which end the bullet comes out of?\"\n\n\"The little tiny round end with the asshole. I shot that thing this evenin' at the range and the guy in charge said I was a natural-born marksman.\"\n\n\"Was coming over here your idea?\"\n\n\"Yeah. I was sittin' beside Grafton participatin' in a high strategy session when a radio dude came runnin' in and told him all about these Soetoro dudes coming to shut this radio station down. I volunteered to come help. Knowin' you, I figured you'd need all the help you could get.\"\n\nThat must have been the first time in his life Willie ever volunteered for anything but beer. \"It's good to see you, shipmate. You can stay in here with me, but why don't you lay down in the corner and try to catch some Zs.\"\n\nHe did so, after bitching about how hard the floor was and having to use his jacket for a pillow. \"Turn off that damn radio noise out here,\" he said. \"I've had enuffa Soetoro to last me a lifetime.\"\n\n\"I thought you were a Soetoro voter.\"\n\n\"Don't remind me.\"\n\nI cranked the volume of the speaker to zero and settled down to wait. Willie and Sarah were sound asleep when I went into the break room at one a.m. and made a pot of coffee. While it dripped through, I went in to the studio and put on the earphones. The prez was talking about his enemies. I put the earphones down and went back to the break room for a cup. Nothing but that white powdered stuff for creamer, so I silently cussed the Maryland doctors and drank it black.\n\nWaiting was hard. I went out and surveyed the street. Two or three truckloads of them\u2014we would kill them right there.\n\nWaiting has never been my long suit. I must have been at the head of the line for good looks and natural charm; when I got to patience there wasn't much left\u2014I only got a teaspoon full, if that.\n\nI found myself rubbing my sore neck again. The doctors at Camp Dawson had put more antiseptic on it and a sticky bandage. The muscles were still stiff.\n\nI wondered about Willie, why he was here. A warrior he wasn't. Growing up in the Washington ghetto and a couple of stretches in the pen had taught him to stay out of the line of fire and keep his head down. Willie was a survivor. That was one of the reasons I liked him. When I had had my fill of agency operators full of bullshit and testosterone, I could visit Willie at the lock shop and come back down to earth.\n\nMusing along those lines, my handheld squawked. The voice was Travis Clay's. \"We have a truck two blocks north, and someone standing beside it looking the situation over with binoculars.\"\n\n\"Okay.\"\n\nI nudged Willie with my foot. He came right awake.\n\n\"Uh-oh,\" Willis Coffee said. \"I hear helicopters. . . . Coming this way. Getting louder.\"\n\nDamn!\n\nIt was beginning to look like the bleeding wasn't going to be one-sided at our little party.\n\nI walked to the busted window and listened. I could hear the choppers now, definitely coming this way. If these were paramilitary thugs, from FEMA or Homeland or the IRS or wherever, they were catching on fast. If they were military, oh boy.\n\n\"Trucks are moving, at least three. Guys walking along beside them. All armed. Looks like FEMA uniforms.\"\n\nThe choppers were above us somewhere.\n\n\" _They stopped in the wrong block_! They're in the next block north.\"\n\n\"Choppers overhead. Two Blackhawks. Guys rappelling down onto the roofs on the east side of the street. But they're in the wrong block too.\"\n\nI keyed the mike on my hand-held. \"Machine gunners, take out the choppers. Everyone else, hit 'em.\"\n\nAnd the world split apart. The hammering of heavy machine guns rolled up and down the street. I grabbed an AT4, fired it up, and stepped right through the empty window onto the sidewalk. The lead truck was in the middle of the next block. Perfect. I didn't waste time and got the round off within three seconds. It went right into the engine compartment and exploded. Pieces of the truck went flying everywhere.\n\nBullets were whanging off the concrete sidewalk and brick facade, so I dived right back through the window socket with the empty tube in my hands.\n\nThe sound of combat rose to a roar.\n\nThose soldiers\u2014I saw uniforms and helmets\u2014would quickly figure out there was no radio station in that block and be heading this way if the guys on our roofs didn't manage to keep them pinned.\n\nThen I heard a chopper crash. The explosion was tremendous. The other one was trying to get away, it sounded like.\n\nI grabbed two grenades, pulled the pins, and went over to the window. Risked a quick squint. Guys coming down both sides of the street, shooting up at the roofs. I threw one as far as I could across the street at an angle, then leaned out and tossed the other left-handed up the street.\n\nWillie was hunkered down in the corner, trying to see up the street through the empty window socket.\n\n\"Shoot low,\" I shouted. \"Ricochet the bullets off the walls over there.\"\n\nHe began squirting bursts.\n\n\"More, more,\" I urged.\n\nI became aware that Sarah was behind me, and she handed me a couple more grenades. I sent them down the street, and the explosions were gratifying.\n\nThis went on for what seemed like an hour, but couldn't have been more than a couple of minutes, if that. Willie changed magazines twice.\n\n\"Keep your goddamn head down,\" I told him when he kept bobbing up to squirt off a burst.\n\nI glimpsed a grenade flying into the street in front of our position. \"Down! Grenade!\"\n\nIt went off and showered the office with shrapnel. I looked at the studio window, which was grazed but intact.\n\nThen I realized the shooting was tapering off. Another burst or two, and a deafening silence descended.\n\n\"Willis? Travis?\"\n\n\"The survivors are running for the trucks,\" Willis shouted into his radio. \"Don't let 'em get away!\"\n\nAbout that time the alley door crashed open. Willie Varner spun on his knee, a very athletic move, and fired a burst from the hip. Then another burst that emptied his weapon.\n\nI was there with my M4, waiting, so I cranked my head to see. Two soldiers in uniform down.\n\nWith the carbine at the ready, I went down the hallway. One was still alive, a black kid. The other was seriously dead. From the streetlight in the alley I could see the patches on their shoulders. New Jersey National Guard.\n\nWillie was there, kneeling, checking on the wounded man. The guy looked at Willie, gurgled something, then his eyes froze and he stopped breathing.\n\nWillie dropped his weapon and put his hands over his face.\n\n\"Hey, man,\" I said. \"It was them or us.\"\n\nSarah put her hand on his shoulder.\n\n\"If you had waited another half second to shoot,\" I told Willie, \"you'd be the one lying dead.\"\n\nWillie straightened up, left his weapon right where it lay, and walked out the alley door and turned right, away from the fight.\n\n\"Let him go, Tommy,\" Sarah said.\n\n\"I just hope there are no more bad guys out there.\"\n\n\"Let's check on the broadcasting equipment.\"\n\nThe radio came to life. It was Willis Coffee. \"There was a fire fight over west of town, about where that radio tower should be. Maybe they tried to take it too.\"\n\nOne of our guys was dead and three more wounded. The soldiers who lay on the floor in the hallway had apparently come south down the alley and gunned the two good guys on guard at the north entrance, then kicked in our door.\n\nAmong the attackers on the ground there were nineteen bodies and eight wounded. The rest had gone north running or riding the surviving trucks.\n\n\"If they had stopped in the right block, we'd have gotten them all,\" Travis Clay said. And they would have destroyed the radio broadcast equipment, I thought, but I managed to bite it off before it came out. \"And we have one prisoner, a FEMA guy who surrendered. His name tag says his name is Lambert. What do you want me to do with the wounded and this guy?\"\n\n\"Put all the wounded on trucks and take them out to the camp. Maybe the doctors can save them.\"\n\n\"Our guys already left. Grafton said no prisoners.\"\n\n\"I'm giving the damned orders. Take all the wounded out to the base. And bring that prisoner over here. I want to look at him.\"\n\nThree minutes later Travis had him standing in the radio studio with a plastic tie around his wrists. Yep, it was Zag Lambert, whom I had met in Colorado a lifetime or two ago. He was even porkier than he had been in Colorado, with a truly awesome gut jutting out above his belt. I doubted if he had seen his dick in the last ten years unless he used a mirror. It was a wonder he could even reach it. He didn't look as feisty now as he had in Colorado.\n\n\"Take him to Grafton,\" I told Travis. \"After they interrogate him, lock him up with Sal Molina. Don't feed him for a few days. Maybe a week. Water only. He needs to lose some weight. His wife will thank us.\"\n\n\"Yo. Come on, fatso.\" And he led Lambert away.\n\n\"New Jersey National Guard,\" I told Grafton when he called on the radio a few minutes later. \"FEMA guys in trucks and two Jersey guard helicopters with grunts who rappelled down. Travis is bringing you a prisoner to interrogate, Zag Lambert, the guy who ran Jade Helm 16.\"\n\n\"Good work, Tommy,\" he said. \"We'll send some people to relieve you when the sun comes up, and you, Sarah, and Willie can get some sleep.\"\n\n\"Yeah.\" I didn't mention that Willie had bugged out. I figured that I would run into him at Dawson in the chow hall. At least, I hoped so.\n\nOne of the choppers had crashed on a baseball diamond, and the other went into a block of old houses a quarter mile away. There were no survivors from the Blackhawks. Someone said six or eight civilians were killed in the crash into the houses; no one knew for sure. The smoke was still rising from the fire at dawn.\n\nThus ended the battle of Kingwood. Maybe someday they'll put up a commemorative plaque.\n\nI just hoped that somewhere people were listening to the radio.\nTHIRTY-TWO\n\nJR Hays had four C-17s lined up, fueled, and ready to go. Aboard them were twelve trucks, three apiece. For now, the trucks were loaded with ammo, welding torches, and C-4 explosive. On the trip back, they'd be loaded with gold. He had selected and briefed his men\u2014all one hundred of them. They were dressed in U.S. Army combat gear that would have passed the inspection of any sergeant major. The men had been briefed to shoot only in self-defense. He meant this to be a bloodless adventure.\n\nJR had confirmed, in three satellite calls with the Pentagon, that the United States armed forces were in a state of armed truce and officially neutral in the war between the United States and Texas, and he had letters in his pockets, all forgeries on good paper with appropriate letterheads affirming that he was Lieutenant General Robert Been, United States Army, with written orders from the president of the United States, Barry Soetoro, and the secretary of the Treasury to transport the gold in the Bank of Manhattan to the New York Federal Reserve Bank for safekeeping until the current political crisis had passed. To further his ruse, he had five Texas Rangers, three men and two women, in civvies carrying FBI pistols and credentials, which Colonel Tenney had confiscated from agents in Austin. Chuy Medina had told him the bank had at least a hundred tons of gold on deposit. JR hoped to take every ounce.\n\nSarah and I went to the big head honchos' meeting in the conference room of the headquarters building on Tuesday night after dinner. The place was packed, standing room only.\n\nThere were four generals: Jose Martinez, an active-duty two-star who either took leave or deserted (he wasn't telling); Mort Considine, a retired brigadier; Lee Netherton, a retired three-star; and Jerry Marquart, a congressman if Congress ever got back in session. Jake Grafton was the general commanding, by the unanimous vote of the four, and he presided.\n\nThe big news was that radio stations along the East Coast had received duplicate thumb drives of Sarah's recordings from Dixie Cotton; and Dixie herself was making a splash as she flitted through Washington, Baltimore, Philadelphia, and New York, broadcasting on her mobile radio. FEMA and Homeland were after her, but I figured they would drop the chase soon enough\u2014news of the recordings had already gone nationwide, and the rumor was that even FEMA and Homeland were now having doubts about Soetoro.\n\nWithin twenty-four hours of the first Kingwood broadcast, more than a thousand people joined our little army\u2014veterans, truck drivers, steel workers, mechanics, carpenters, dentists, students, housewives, eccentrics, whackos, and no doubt some true psychopaths, all angry about Soetoro's violation of their \"rights\" and the \"Constitution.\" Many brought their own firearms.\n\nThe generals fretted about the willingness and ability of civilian volunteers to follow orders. As usual, Grafton cut to the chase. \"We've got to keep control of our troops or we are nothing but a mob. Let's agree right here, right now, that anyone caught robbing, stealing, raping, or murdering noncombatants will be summarily executed on the spot. Anyone accused of these crimes but not caught in the act will be court-martialed as soon as possible with the accuser and any witnesses testifying. If found guilty, he or she will be executed immediately. That will be General Order Number One.\"\n\nFurther orders followed swiftly. Jose Martinez, with Mort Considine as his deputy commander, would take the units designated as the First Army, or our northern army, to Washington via I-68. Lee Netherton, with Jerry Marquart as his deputy, would lead the units organized into the Second Army, or our southern army, to Washington via Leesburg. Grafton would fly the Cessna, our only observation plane, and keep in touch with the columns via radio. Predators would scan the ground for bad guys and ambushes.\n\nThen they got into logistics. The generals told their staff officers to stay but ordered the rest of us to get busy.\n\nThinking that good advice, I wandered out with Sarah and asked, \"Wanta get laid?\"\n\nShe stopped and did a double take, then said, \"Why, Mr. Romantic, I thought you would never ask. You must be overwhelmed by my feminine charms.\" She held up a palm. \"Don't explain. I would rather keep my illusions.\"\n\n\"Wise woman,\" I acknowledged.\n\n\"Where do you plan to conduct our tryst? The barracks is full of people playing poker, shooting craps, and listening to Barry Soetoro on the radio, and I'm not doing it in a pickup truck, period.\"\n\n\"I was thinking of walking a little way up into the woods and finding a leafy glade that we could remember fondly all our days.\"\n\n\"You animal! Lead on.\" She placed her hand in mine.\n\nApparently some other couples had similar ideas, so we had to go a bit further uphill into the woods than I wanted. It was so dark we tripped over tree roots twice.\n\nWhen we thought we had a private spot free from brush and snakes, we sank to the ground. \"Ooh,\" she said as she ran her hand around, \"moss covered with sticks and stones and spiders. I've always dreamed of getting laid on a bed of moss, our very own private bower of carnality.\"\n\n\"I'll bet,\" I said, and got busy brushing the debris off the moss.\n\nHours later gently pattering raindrops woke us. The night was as black as the inside of a coal mine but a lot noisier, what with drops loudly whacking leaves, which were beginning to drip on us. Sarah and I hurriedly put on our clothes and threaded our way through the trees downhill toward the barely visible lights of the camp.\n\nWhen we got back to our barracks we were a little damp, so we hung our trousers and shirts and web belts on the posts at the end of the bunk and both of us crawled under my blanket. When I woke up, it was dawn and Sarah was still sound asleep in my arms.\n\nOther people were stirring, but they studiously ignored us.\n\nJake Grafton came thumping in. I pretended to be asleep. He shook my shoulder anyway and said, \"Come on, Tommy. See you at the plane in fifteen minutes.\"\n\n\"Yessir.\"\n\nGrafton was adding a quart of oil to the Cessna's engine when I came walking up. I put my M4 carbine and a little bag of extra loaded magazines and a dozen grenades in the plane. The sun was trying to come up under a high overcast. The earth smelled of late summer, pungent, fertile, and hinting of fall. The temperature was in the fifties so my sweatshirt felt good. Truly, we had a marvelous piece of the planet.\n\nI stood there inhaling it all and watching the sun fire the tops of the trees as Grafton finished his preflight. I could hear the PA system squawking, wakening the troops. I had been hesitant to wake Sarah so I didn't kiss her good-bye; now I wished I had.\n\n\"You ready?\" he asked.\n\n\"I suppose.\"\n\nWe got aboard and put on seat belts and headsets, and he fired up the engine. It caught on the first crank, and the prop spun into a blur with a nice little roar, blasting the morning dew from the windscreen. I checked the fuel gauges on the butts of both wings: we were full.\n\nThere was no wind, so after waiting a moment for the engine to warm and doing a short run-up and mag check, we were rolling down the runway. The tail came up and in less time than it takes to tell, we were airborne. Out over the camp and the trees, climbing into that morning sky between the low green mountains, then turning eastward into the morning sun.\n\nHe gave me a brief on the ICS. \"We'll check the roads the two columns are going to take, then we're going to Washington.\"\n\n\"I thought we were the eyes of the army?\"\n\n\"For a little while. Then we have places to go, things to do, people to see.\"\n\n\"Right.\"\n\n\"It's a great morning to fly,\" he said. That was Jake Grafton. He was wearing a little smile.\n\nAfter an hour in the air, he reported on his handheld to the generals. No ambushes were evident. We did find a couple of campsites in the woods, but apparently the people there were refugees from the cities. Fires were giving off smoke, and we saw no evidence of heavy weapons. Our scouts would see the smoke and be forewarned.\n\nThen Grafton set a course to the east. No low clouds, excellent visibility, so he climbed to four thousand feet. Soon Leesburg came into view, and a few moments later the long runways at Dulles airport.\n\nThe C-17 Globemasters landed one after the other at LaGuardia airport in Brooklyn. There were no flight plans, of course, since the FAA was out of action because the power was out, but these were air force planes on official business, so they landed and that was that.\n\nThe ground controller parked the four giant cargo planes on the cargo ramp, appropriately enough, and the loadmasters and their soldier passengers got busy off-loading the trucks. Also on the trucks were little cargo donkeys driven by gasoline engines, just in case.\n\nThe caravan got itself arranged, the soldiers got their weapons and got into cabs and on the backs of the trucks, two armed guards were posted at each plane, and the fliers stayed with their steeds. The rest of the hardy band of adventurers set off through the wilderness of Brooklyn toward Manhattan.\n\nThe place reminded JR Hays of Baghdad. Trash was everywhere, windows were broken out, and knots of idle young men congregated on corners, looking like packs of feral dogs. Few women could be seen, and those that were, were always walking with several men. Carcasses of burned-out cars sat pushed to the side of the street. Other cars had been stripped of wheels and even doors.\n\nNone of the stoplights worked, which didn't matter because there was little traffic, probably because there was little or no fuel available, so the caravan rolled steadily at twenty-five miles an hour onto the main thoroughfares that led to the bridge into Manhattan.\n\nHe looked at his watch. At least this Wednesday, the seventh of September, there weren't a couple million commuters and an endless stream of over-the-road tractor-trailers and local trucks fighting to get into Manhattan. The roads were essentially empty, with pieces of cars strewn randomly that the army trucks had to drive around. Wrecks were abandoned against the median barriers. It looked to JR as if Barry Soetoro had finally managed to choke America, and it was dying.\n\nIt was ten minutes before nine. His convoy would arrive at the bank a few minutes after the hour. He straightened his uniform, the dress uniform of a lieutenant general in the United States Army. He had given himself a promotion. He had used his own ribbons on his left breast, which was a dazzling collection for a twenty-year light colonel, but rather sparse for a three-star. JR doubted that the bankers had met many three-stars in full regalia.\n\nAs the truck rumbled along, he sat in the right seat on the lead truck praying that the Bank of Manhattan was going to be open today. If it weren't, this trip would be nothing but two airplane rides and a short jaunt on abandoned highways and streets.\n\nFrom the right seat of the Cessna buzzing over suburban Virginia, I didn't see any airliners in the air, but I saw a bunch parked around the terminals at Dulles. Every gate was full and the ramps were crowded.\n\nWe flew on. The Washington Monument rose like a finger ahead of us. Grafton flew toward it. What a view that was, with the Potomac winding into town, the Lincoln Memorial, the Capitol Building, the White House, the Jefferson Memorial, and the long slash of the Mall.\n\nI was nervous. I figured someone might decide to take a pot shot at us with anti-aircraft artillery or a surface-to-air missile, but apparently not. The streets looked almost deserted, yet the Mall and area around the White House certainly weren't. People everywhere, a sea of people.\n\nGrafton swung the airplane to fly around the White House counterclockwise, with the good view on his side. But I could see plenty. Uniformed police and cops in riot gear were arranged outside the fence that encircled the executive mansion. They faced a sea of people, ten, perhaps twenty thousand flooding toward the White House. It was the damnedest sight I ever saw.\n\nBooks have been written about what was going on in the White House that morning of the seventh day of September, about how the president and his advisors and staunchest legislative allies weighed options and tried to figure out what to do next. At the risk of stating what you already know, I will summarize by telling you that Barry Soetoro was in denial, according to later accounts, and so were Al Grantham and Sulana Schanck. They raved about the treason of the military, demanded summary executions.\n\nThe vice president thought the mob outside could be handled by the Secret Service and police riot squads, augmented if necessary by fire trucks with high-pressure nozzles. He urged calm and assured everyone who would listen that America's progressives and people of color would ignore the crap spewing over the radio (and now some television stations), and support their president with their lives, if necessary. According to an account written by a senator, a delegation from Capitol Hill tried to warn the president that the fury of the American people was real and widespread, and had been dismissed as traitors for their pains.\n\nOf all that drama Grafton and I were blissfully ignorant. After Grafton had made two complete circles, he leveled the wings and aimed the plane across the river toward the Pentagon, that massive stone structure between the Potomac and Reagan National Airport.\n\nGrafton circled the Pentagon, eyeing the vast parking lot. On his second circuit, I had a good view of armed soldiers, machine-gun nests, military vehicles, and tents. We were only a few hundred feet above the parking lot by then, but no one started shooting.\n\nGrafton swung out and began a straight-in to the parking lot. There were light poles here and there, but most of it was empty. He pulled up on the bar between the seats, which put in half flaps, then pulled again for full flaps and we were on final doing about seventy miles per hour. He plunked that thing in a three-point landing within twenty feet of the edge, just clearing some power wires, avoided all the light poles, and slowed to a taxi. Then he braked to a stop and pulled the mixture knob out. The prop swung to a stop as a Humvee came rushing up.\n\nGrafton killed the mags and master switch and we got out. Two soldiers jumped from the vehicle with guns in hand. Optimist that I am, I left my M4 and bag of grenades in the plane.\n\n\"My name is Jake Grafton. I want to see the CNO or army chief of staff, if they are around.\"\n\n\"Sir, you aren't supposed to land here.\"\n\n\"Right. Now get on the radio and find out if Admiral McKiernan has the time to see Jake Grafton.\"\n\nFifteen minutes later we were in some kind of situation room still wearing our sidearms. At least they weren't going to arrest us on the spot, I thought, which was a relief.\n\nGrafton shook hands all around\u2014the room was full of admirals and generals\u2014enough brass to make a few dozen monkeys. He was even courteous enough to introduce me, although all I got from the heavies were nods, then they ignored me. They all knew him and were obviously happy to see him. The commandant whacked him on his back so hard I worried about his ribs, but Grafton didn't wince.\n\n\"Was that you we saw flying around the White House a few minutes ago?\" someone asked, and Grafton admitted it was.\n\n\"The FAA will mail you a flight violation.\"\n\nOn a console were three large screens showing the mob surrounding the White House. It only took me a moment to figure out that these pictures were the datalink video from drones. A large map of downtown Washington covered one wall. It was held there with masking tape, so it hadn't been there long.\n\nI watched the video while Grafton chatted and the brass nodded at the screens and shook their heads. \"He's going down before long,\" one general said.\n\nEveryone seated themselves in chairs and Grafton got right to it. \"Is it true that the military is no longer taking sides in this civil war?\"\n\n\"That's right,\" Bud Weiss, the air force general, said. \"We're America's armed forces, not Barry Soetoro's.\"\n\nJake Grafton nodded. \"I had hoped that you would see it that way.\"\n\nCart McKiernan explained, \"Marty Wynette committed suicide in his office two days ago. This war against Texas and Soetoro's enemies had gone far enough, so we decided the best course for the military was to remain neutral.\"\n\nFifteen minutes later I thought I had the picture. The military was devoting its efforts to pushing Mexican forces out of California. A very unhappy Barry Soetoro was hunkered down at the White House fulminating and making big noises, but so far he had left the Pentagon, and the Marines surrounding it, alone\u2014probably because he had nothing to bother them with.\n\n\"What does Jack Hays down in Austin say about all of this?\"\n\n\"I talked to him earlier today on the radio,\" the army general, Frank Rodriquez said. \"He says if we leave Texas alone, Texas forces will leave our troops and military installations alone. I guess you could call it a truce.\"\n\nGrafton gratefully accepted a cup of coffee from an aide. He sipped it and told the brass, \"There are a bunch of folks, about three thousand, but the number is growing by the hour, heading this way from Camp Dawson in West Virginia. They'll probably be here tomorrow.\"\n\n\"Who is in charge of this group?\"\n\n\"I guess I am,\" Grafton said with a smile. \"We intend to enter the White House and arrest Soetoro, if we can get there before that mob beats us to it. His days are almost over.\"\n\n\"Then what?\" some general asked.\n\n\"We need to get the United States up and running again. Get the power turned on nationwide, get water flowing through the pipes, and restore public order.\n\nThey weren't yet ready to talk about tomorrow. \"What do you know about this White House recording that is all over the radio dial? We think three or four stations are broadcasting it.\" General Weiss said that and he gestured at the video screens. \"That is what has them stirred up. The big problem is that that mob is made up of people who hate Soetoro and people who think he is the risen Christ and is being viciously slandered. We have people down there reporting on what's happening. That thing may turn into a battle royal between the two groups right there in Lafayette Park, a bloody riot.\"\n\nGrafton replied, \"I authorized secret electronic monitoring of the White House about six months ago. We used an Israeli program to turn all their cell phones, computers, and surveillance equipment into listening devices. The signals were gathered by the White House Wi-Fi system, encrypted, and sent to us. My tech staff\" (that was only Sarah Houston, by the way\u2014she was going to smile when I related this remark to her) \"waded through hundreds of hours of conversation, but edited our take down to the pithiest sixty hours. That is what the radio stations are broadcasting.\"\n\nRodriquez whistled. \"That stuff is dynamite.\" He jumped right to the key point. \"So you knew Soetoro was planning to declare martial law for weeks before he did it?\"\n\nGrafton merely nodded.\n\n\"How many weeks?\"\n\n\"Two months,\" Grafton said.\n\nAs they digested that revelation, General Runyon said, \"You should have told us.\"\n\nGrafton made a face. \"There is always the question of whether clandestine recordings are genuine, and that cannot be answered to a certainty by listening to them. Even if you concluded that I was as honest as Diogenes, what would you have done after you listened? The American people needed to _see_ the reality of a dictator in the White House, not listen to him scheming. _Now_ they have seen and believe and most are ready to listen. The die-hards, a minority, are convinced the recordings are a plot to slander the saint; nothing on God's green earth will make them change their minds.\"\n\nThe military brass sat and looked at each other. \"He's right, you know,\" Cart McKiernan said. No one wanted to argue. All eyes went to Grafton.\n\nGrafton took another sip of coffee. \"You made the right decision when you pulled your troops to the sidelines. The American people need to solve this problem. And I think they're about to.\"\n\nThat was the moment when I knew my country had a future. Jake Grafton talked about the rebuilding mission ahead, and the Pentagon generals and admirals listened carefully to every word.\n\nI slipped out of the office, closed the door behind me, and asked the aides in the reception area how to get to the men's room. A major escorted me, and when I had lightened the load, I asked if there was food available. There was. The major and I had a delightful late breakfast of scrambled eggs, sausage, fried potatoes, and toast with real butter.\n\nI was in an expansive mood. The major wanted to talk about the splash the Soetoro White House conversations were making. I wasn't about to tell him that was a Sarah Houston\/Jake Grafton production, so I just listened. When he had expressed his and his colleagues' stupefied amazement, he segued to the subject of the rebels coming to town. I told him what I knew, which wasn't much.\n\n\"Who is leading the rebels?\"\n\n\"Admiral Grafton, the officer who flew me here. I think you lead a rebel army by moral suasion. That's Jake Grafton. I used to work for him but I quit. Now I do what he asks or tells me to do because he's Jake Grafton and I'm me.\"\n\n\"What's going to happen to Soetoro?\"\n\n\"I haven't the faintest idea,\" I said. \"If Grafton has an idea, he hasn't shared it with me. I doubt if he does. He's sorta playing the melody by ear. May I have another cup of coffee?\"\n\nWe both went and filled our cups. Seated again, the major said confidentially, \"The betting in my shop is that Soetoro will fly to Iran and ask for asylum.\"\n\n\"Maybe the ayatollahs will put him to work in a bomb factory,\" I suggested.\n\n\"I don't think he's going to get rich making speeches,\" the major declared.\n\n\"Probably not,\" I agreed and finished the coffee.\nTHIRTY-THREE\n\nJR Hays' convoy arrived at the Bank of Manhattan. He walked across the plaza, accompanied by his five fake FBI agents and two officers armed with M4 carbines. He pushed on the revolving door.\n\nTo the vast relief of JR Hays, the door wasn't locked. In seconds they were inside and crossing the lobby, which actually had a good crowd of civilians lined up facing only three tellers. JR strode over to the receptionist and announced he was here to see the president of the bank.\n\n\"Mr. Gottlieb?\"\n\n\"If you please.\"\n\n\"I don't know if he is available just now. I'll check.\"\n\nShe made a call and read his rank and name tag into the telephone. With the instrument in her hand, she asked, \"May I tell him what this matter is about?\"\n\n\"Government business,\" JR said curtly and directed his gaze around, as if he were a bit peeved to be kept waiting. The two soldiers in combat gear, Colonel Adam Holt and Lieutenant Colonel Charley Grayson, adjusted their helmets and fingered their carbines, which looked black and ominous and very out of place in this marble temple to capitalism. People eyed the soldiers, who didn't seem at all self-conscious.\n\n\"If you will follow me, gentlemen. . . .\" The receptionist opened a short door and admitted them behind the counter, then led the way to a bank of elevators. They were lifted up, up, up.\n\nThe president's office was in the executive suite. They were shown to a conference room, one with a long, polished mahogany desk and portraits on the walls of past bankers who had presumably gone on to an honorable retirement and whatever was awaiting them after that.\n\nJR and his men cooled their heels for four minutes by JR's watch when the door opened and a man in his fifties bustled in. He was wearing a rumpled shirt and slacks and carrying his shoes in his hand.\n\nHe proved he was a top-notch executive by going straight for JR, whose silver stars gleamed on each shoulder. \"I apologize, General,\" he said, \"but since the power has been off I have been sleeping at the office.\"\n\nJR looked the president up and down and gave a quick, tight smile. He stuck out his hand. \"Lieutenant General Been, sir.\"\n\n\"I'm Abraham Gottlieb.\"\n\nJR introduced the two soldiers in combat gear and the FBI agents, who whipped out their credentials.\n\n\"The army and the FBI,\" Gottlieb said, merely glancing at the credentials. The agents put them away and JR tried not to relax. None of the photos on the credentials matched the faces of the people holding them. That was one of the little hurdles he had to clear, and he was over.\n\n\"Let's sit down,\" JR said to Gottlieb. He reached into his tunic and pulled out two letters and handed them to the banker. One was a letter on White House stationery to Lieutenant General Robert Been, United States Army, ordering him to proceed with whatever troops he thought appropriate to the Bank of Manhattan and transport the gold in the bank's vault to the New York Federal Reserve Bank for safekeeping. The other letter was on Treasury Department stationery and was addressed to Mr. Gottlieb. The secretary of the Treasury regretted the necessity of moving the bank's gold, but threats from mobs and various unnamed rebel forces required that the gold in bank vaults in New York be moved to one central location where it could be guarded by the army.\n\nGottlieb seemed to shrink. He wiped his forehead and read both letters again while the lieutenant general reached into his tunic and brought out another sheet of paper. \"Your copy of the president's letter, sir. I need to keep the original. If you don't mind.\"\n\nThe banker surrendered the document without a murmur. \"I never thought it would come to this,\" he said, and swabbed his brow again. \"I'll have to verify these letters of course, and if they are genuine, you may have the gold. Unfortunately most of our staff aren't in the bank today, although the vault is open so our customers can withdraw gold.\"\n\n\"Just how do you propose to verify these letters?\" JR snapped.\n\n\"Well. . .\" Gottlieb tried to compose himself. \"The telephone system, internet, and telex are down, so I suppose I'll have to send a bank officer to the New York Fed to see the chairman there. He should have received similar documents.\"\n\nThis was the make or break moment. JR looked at the banker, overweight, with fleshy features, measuring him. \"Mr. Gottlieb, as you know, the president has declared martial law. The army is running America now, subject to the president's orders. As far as you are concerned, _I am the army. I own New York and everyone in it_.\"\n\n\"Yes, sir, but we have our procedures, which the SEC and banking authorities require us to\u2014\"\n\n\" _Mr_. Gottlieb, my troops are now surrounding the Fed\u2014your messenger would not get through. You have just read the president's and secretary of the Treasury's orders, and I am obeying them. If I run into any difficulties, these agents of the FBI are authorized to arrest you and your staff.\" JR simply stared at the banker, daring him to open his mouth. As briefed, the senior woman removed a set of handcuffs from her purse, and Gottlieb's eyes went to them. Obviously he knew that people on Barry Soetoro's shit list were being hustled off to concentration camps.\n\n\"We are not going to be delayed by disloyal people, Mr. Gottlieb,\" JR intoned, as if he were talking to a buck private who had his shoes on the wrong feet. He stood, signaling he was through talking. \"Now call down to your lobby and tell the head cashier to stop passing out gold. All of it is going to the Fed. When the army has it in our trucks, I'll give you a receipt for every ounce.\"\n\nWithout waiting for a response from the banker, JR turned to the colonels and said, \"Gentlemen, let's get at it.\"\n\nHe turned back to Gottlieb. \"If you wish to put on your shoes, sir, you may come to the vault and help supervise my troops, and ensure the receipt is properly prepared.\"\n\nThe banker slammed his feet into his shoes.\n\nJR spoke to his FBI agents. \"It looks as if your services aren't needed today.\"\n\n\"We'll stay, just in case,\" the senior woman said and slid her handcuffs back into her purse.\n\nJR Hays made sure his missive from Barry Soetoro was safely in his pocket. The letter from the Treasury secretary lay on the polished mahogany where Gottlieb had left it. Maybe, JR thought, he should frame the president's letter and a copy of the one from Treasury. They were great pieces of work, signed by the best forger in the Texas prison system.\n\nHe strode out of the conference room, looking every inch a man in complete command, the general from central casting.\n\nWhen I got back upstairs after brunch, my escort found that the brass had moved from a situation room to an office on the E-Ring. They were huddling behind closed doors. I asked one of the outer-office types if Grafton was in there, and informed he was, headed for the door.\n\n\"You can't go in there unless they send for you,\" I was told.\n\nI smiled to show I could forgive a social faux pas. \"I'm Grafton's official biographer. He wants me there unless Mother Nature shrieks for my attention. It's just one of his peccadilloes.\" I opened the door, slipped in and closed the door behind me.\n\nGrafton glanced my way but continued talking. I dropped into an empty chair near the door.\n\nGrafton said, \"As I see it, our first priority must be getting people out of Soetoro's concentration camps. Then, in no particular order, we must get electrical power, telephone, and internet service restored nationwide; get police and firefighters back on the streets and highways; and tackle the humanitarian problems this mess has caused. I would bet there are forgotten and abandoned elderly, sick, and addicts tucked away in odd corners dying of malnutrition and dehydration. In other words, we must get the nation moving again.\"\n\n\"What about the states that declared their independence?\" the bluesuiter, Bud Weiss, asked.\n\n\"It wasn't just Barry Soetoro who caused this mess,\" Grafton replied, \"it was a vast overreach by the federal government, by which I mean the executive, judiciary, congress, and bureaucracies.\"\n\n\"That's not our business.\"\n\n\"It's our business if we're rebuilding this country. Frankly, gentlemen, if we're going to restore the United States of America, we need a constitutional convention to decide if we really want the federal government to rule America, or if we even want a federal system. I don't know the answer, but I know that without a political settlement to resolve lots of festering issues, this nation will fracture into several nations.\"\n\n\"You're saying we need a new constitution,\" Cart McKiernan murmured with his chin down, looking at Grafton over the top of his glasses.\n\n\"The states are going to have to figure that out,\" Grafton said with a gesture of irritation. \"The military needs to stabilize the country and get it running again so the politicians can ruminate and negotiate without the house burning down around them.\"\n\nGrafton stood up and started shaking hands. \"Gentlemen, I want to thank you for your time this morning. This is our country. Soetoro won't be here long. The sooner he's gone, the better.\"\n\nGeneral Rodriquez said, \"Still think we should call the White House and offer to fly him out of the country?\"\n\nWhen I heard that my eyebrows went up toward my hairline.\n\n\"Yes,\" Jake Grafton said. \"Tell him the military won't protect him. In my opinion America will be better off going forward if people don't have his blood on their hands, but\u2014\" He raised his hands in a shrug. Then he said his good-byes. I opened the door and followed him out.\n\nFifteen minutes later, when we were in the Cessna and he was taxiing around the parking lot to find a lane for takeoff, I asked Grafton why he recommended flying Soetoro into exile.\n\n\"None of the leaders at Dawson can control our little army, and that's only one of at least eight or ten armed mobs marching on Washington. They'll kill Soetoro if they get their hands on him. If they do, his supporters will try to make him a martyr. A lot of people still think he's the black messiah, beset by evil enemies on all sides.\"\n\n\"Think he'll go?\"\n\n\"No, but it's worth a try.\"\n\nA minute later we were airborne and climbing over the Potomac for the White House. Maybe it was my imagination, but the crowd outside seemed larger. As we crossed the Mall, we could see people walking toward the mansion, like an incoming tide.\n\nI looked away from the scene below. There was a house fire somewhere up to the northeast, and the plume was rising and drifting on the wind. I wondered if the fire department was on the job. Grafton finally leveled his wings heading west and added power to climb.\n\nThe flagship of the Texas Navy, the attack submarine _Texas_ , was fifty miles east of Cape May, New Jersey, running at three knots when Loren Snyder poked the telescoping photonics masts\u2014 _Texas_ had two of them\u2014above the surface. In less than a minute, the video from the mast confirmed what the sonar was telling the crew, that there were no surface ships of any kind within their visible horizon.\n\nFive days had passed since the combat with the destroyers among the oil rigs offshore of Louisiana. _Texas_ had transited the Florida Straits and headed north. Snyder had it in his mind that if he torpedoed a couple of container ships in the approaches to New York and Newark, he could probably shut down those ports for a while. Days of cruising deep and listening via sonar for ships and submarines had been unproductive. The ocean seemed extraordinarily empty.\n\n\"Maybe the war is over,\" Jugs Aranado suggested.\n\n\"We should hope,\" Snyder said, but just in case, he decided to listen to East Coast radio stations to see what he could learn.\n\nThe AM band was remarkably quiet, but there were a few stations on the air. He channel surfed, looking for a news show. What he found was a station broadcasting the White House eavesdropping show. Barry Soetoro's voice sounded in his ears. The fidelity was quite good, and he could readily understand the conversations. They were talking about declaring martial law and arresting subversives.\n\nAs he listened on a headset, Snyder wondered what he was listening to. Gradually the idea dawned that someone had recorded a White House conversation weeks ago, perhaps months.\n\nThirty minutes later he was sure. They were talking about the upcoming Republican nominating convention. _This had to be recorded in late July or early August_!\n\nHe flipped a switch to put the audio on the loudspeaker in the control room.\n\nJugs was there, and Ada Fuentes was on the helm.\n\nThe two women sat, startled at first, then mesmerized.\n\n\"How did this get on the radio?\" Fuentes asked, dumbfounded. What she was hearing just didn't compute.\n\nWhen the scene was over, an announcer came on. \"You were listening to President Soetoro and his advisors, Al Grantham and Sulana Schanck. Now for the next scene.\"\n\nSnyder reached for the dial and turned it. He found a news station. The announcer was interviewing a Long Island congressman. \"We have fifteen hundred people assembling at the Meadowlands parking lot. Tomorrow we will begin our march on Washington. Food has been donated from the local food bank and some local farmers. Anyone who wishes to join us should do so today. Bring your own weapon and ammunition and whatever camping gear you think you will need. Bring whatever food you can.\"\n\n\"What will you and your 'army' do in Washington?\"\n\n\"We are going to drag Soetoro from the White House and hang him.\"\n\n\"And you are sure the military won't interfere?\"\n\n\"They said they wouldn't\u2014you have been reading their press release every hour on the hour. They're fighting Mexico, not Americans. We're taking them at their word. If they want a fight, however, we'll give it to them.\"\n\nSnyder looked at Jugs. \"What do you think?\"\n\n\"Holy Christ!\"\n\nHe lowered the photonics masts and told Ada to speed up to arrive off the Narrows at dusk. He suggested she descend to two hundred feet, and as Jugs flooded tanks, she did.\n\nStabilized at that depth, they discussed what they had heard.\n\n\"Let's go home,\" Ada suggested.\n\nJugs didn't say anything, merely scrutinized Loren Snyder's face.\n\nHe had to make a decision, so he did. \"We'll take a look at New York Harbor and listen again this evening, with everyone not on duty in the control room. We'll let everyone have his say, and I'll make a decision then.\"\n\nThings began to go wrong pretty quickly when JR Hays saw the Bank of Manhattan's vault. It had a massive circular door that weighed about twenty-five tons, Gottlieb said proudly. The ingots were stacked in the vault and almost filled it. Around the walls on shelves and in drawers were the packs of small wafers, small bars called kilobars, Krugerrands, and other gold holdings, all labeled with owner's names. The sight of all that gold was awe-inspiring, the wealth of nations.\n\nThe bank had precisely two electric forklifts and four dollies to move the gold ingots, but they weren't set up for speed. Each bar had a serial number, and two men were busy writing down the number on each bar. JR put a stop to that. \"The gold is going to the Fed,\" he said. \"You already have the numbers.\"\n\n\"Can you guarantee that all the bars will arrive in good condition?\" Gottlieb demanded.\n\n\"Sure as shootin',\" JR said, and told him to watch and simply count bars.\n\nColonel Holt took him aside and said, \"I figure it will take two days to empty this vault, if nothing breaks.\"\n\n\"How long to load up fifty tons? Can we get that accomplished today?\"\n\n\"Maybe.\"\n\n\"Make it happen. Get soldiers down here to help carry the ingots out by hand. Fifty tons is our goal. I'll tell Gottlieb we'll be back for the rest tomorrow.\"\n\n\"Yes, sir.\"\n\nWith the help of a dozen soldiers who were soon sweating profusely, the work speeded up. The soldiers grabbed an ingot in each hand and carried it to the elevator while Gottlieb and another bank officer counted them. For the receipt, of course.\n\nJR tried to stay calm. Fifty tons would be a good haul. _Be satisfied with that_ , he told himself.\n\nHe had been there an hour when one of the officers came to the vault holding a handheld radio. \"Just talked to the airport,\" he said. \"Some FAA guy came around. He stayed ten minutes and left.\"\n\n\"Keep me advised,\" JR said, and watched the gold being loaded onto dollies. He was learning a lot about gold ingots. The standard bars in the vault were Good Delivery bars, each with a serial number. They weighed 12.4 kilograms each, contained 400 troy ounces, or 438.9 ounces. That translated to about 27.5 pounds each.\n\nHe did some figuring in his head. Fifty tons was one hundred thousand pounds, which equals about 3,600 bars, more or less.\n\nThey weren't going to get it done using dollies or a dozen soldiers. He had Colonel Holt assemble a conga line of thirty soldiers, and they passed gold bars from hand to hand into the elevator, and when it held a couple of hundred, sent it up to be off-loaded into a truck while more were stacked at the entrance. The work went faster.\n\nThe men ate MREs in shifts at midday and took a five-minute potty break in shifts. The pile of gold shrank slowly.\n\nWe should have brought some dollies, JR thought. Well, one can't think of everything. He found himself glancing at his watch every few minutes. The minute hand crawled.\n\nOur flight back to Dawson was uneventful. The weather was benign, a typical September day in the eastern United States. We watched for ambushes and found none. We did see our columns snaking along. They had made almost a hundred miles since we saw them in the early morning. We saw another convoy of trucks and cars heading west from Baltimore, approaching Frederick, about fifty miles behind our northern army column. This convoy wasn't ours.\n\nGrafton circled the convoy, low enough that we could see flags with Soetoro's image on them (a sort of Che Guevara T-shirt look) fluttering from car and truck aerials. Then we headed for Dawson.\n\nI decided to deliver myself of an opinion. \"The problem with democracy,\" I told Grafton on the ICS, \"is that fools elect fools.\"\n\nHe snorted. \"And the problem with hereditary kings is that too often you get the pampered, coddled village idiot running the country.\"\n\n\"Life is tough,\" I told him.\n\nThe National Guard camp near Kingwood was almost deserted. We taxied up and shut down in the precise spot where we had manned the plane hours before.\n\nI chocked the plane and tied it down after Grafton went off to find whoever was manning the radios. I got busy fueling the plane. An army portable generator supplied the power to pump the avgas.\n\nAs I finished I noticed some metal blossomed out on the left wing. I climbed off the ladder and took a look. There was a bullet hole in the wing, about six feet in from the port wingtip. A bullet had gone in the bottom of the wing and out the top. A .30 caliber, from the look of it. And neither Grafton nor I had felt a thing. Someone we flew over today was unhappy with us, with life, maybe with the world.\n\nAfter I got the fueling hose put away and the generator secured, I looked the airplane over carefully for any more punctures, didn't find any, and then strolled away carrying my M4. I found Sarah with Grafton in the headquarters building by the radios.\n\nHe was on the horn to General Martinez. \"I'll meet you at first light at the Hagerstown airport. I suspect you are going to meet that bunch coming from Baltimore tomorrow mid-morning between Hagerstown and Frederick. I want to be with you.\"\n\n\"Roger. I'll meet you there.\"\n\nSarah looked at me and I looked at her. She didn't say anything, and I couldn't think of anything except, \"Want to go see if we can find something to eat?\"\n\nShe did, so with a nod to Grafton, we left. He got busy talking to the southern column.\n\n\"Seen the Wire around?\" I asked.\n\n\"Not hide nor hair.\"\n\nI wondered where in the world that fool was. Giving him a weapon the other night in Kingwood was a bad mistake. He wasn't any part of a warrior, which was why I liked him. I was worried. It was a tough world out there these days, and he wasn't a tough man.\n\nThe Bank of Manhattan's president, Abe Gottlieb, wanted to know if the Fed was waiting for the gold. \"Of course,\" JR said. \"I have troops there. They'll stay open until we arrive and if necessary will work all night getting it into their vault.\"\n\n\"Ah, the army!\"\n\n\"Indeed.\"\n\n\"How do you know that a few soldiers won't steal some bars?\"\n\nThat sally drew a frosty stare from the general. Gottlieb said he needed something to eat, and wandered away.\n\nJR was upstairs in the bank when the power came back on in Manhattan. He knew it was on because the outside telephone lines began ringing. The receptionist smiled broadly, shouted to the other tellers, and answered the phone.\n\nUh-oh.\n\nJR looked around for Gottlieb. Two FBI agents were with him, and he signaled to one of them, who jogged over.\n\n\"Power's on, phones are up,\" said JR. \"We've got to move. Close the doors, round up the staff\u2014all of them including Gottlieb\u2014and put them in a conference room upstairs. Confiscate all cell phones and remove the regular phones from the room. Get cracking.\"\n\n\"Yes, sir.\"\n\nJR went back to the vault. Soldiers were passing bars along at a good clip. \"How many ingots have you loaded?\" he asked Holt.\n\n\"By my count, about two thousand.\" JR looked at his watch. It was almost one o'clock.\n\n\"Sixteen hundred to go. Get the bank employees out of here and give them to the FBI agents. Get more troops down here.\"\n\n\"Yes, sir.\"\n\nJR went along encouraging his soldiers. \"Come on, men. You can do it. We're over half way there.\"\n\nHe went up on the street and checked the trucks. As soon as a truck had four hundred bars in it, a new truck was pulled into position. Armed soldiers were stationed around the trucks, and they kept the curious moving on. Not all of them, of course, since the sight of all that gold stopped people in their tracks.\n\nThree policemen had been enlisted to help keep the crowd moving along the sidewalks. As JR watched, another police car pulled up and a captain in uniform came over. He had scrambled eggs on his hat. He saluted JR, who returned it.\n\n\"We got no notice of this move.\"\n\n\"We can handle it. We figured you had enough troubles as it was.\"\n\nThe cop took off his hat and wiped the sweat from his hair, then put it back on. \"We sure do, General. We sure do. But with the electricity back on, maybe things will start returning to normal.\"\n\n\"We can only hope.\"\n\nHe pointed to JR's combat infantryman's badge on his chest. \"I got one of those,\" the captain said. \"The Gulf War, Desert Storm.\"\n\n\"Thank you for your service,\" JR replied.\n\n\"Yeah. Got out and joined the police. Probably should have stayed in the army. It was a great experience, but I wanted to come home to New York. You know, you're the first general I ever talked to.\"\n\n\"Well, you're my first police captain. I hope we never meet professionally.\"\n\nThe cop grinned. \"You're pretty young too.\"\n\n\"Good whiskey,\" JR confided. \"Never drink the cheap stuff.\"\n\nThe captain held up his hand and adjusted the earpiece in his ear. He rogered the transmission, then said to JR, \"Gotta go. Got some dead people in an apartment house. Someone just found them. Been dead a few days.\"\n\nThey shook hands, and the captain trotted over to his cruiser and jumped in. The driver hit the lights and siren, and away the cruiser went up the street, howling madly.\n\n_Everyone has problems_ , JR thought, and got back to attending to his.\n\n_Texas_ poked her photonics masts up as she approached the narrows. Loren could see the Verrazano Bridge across the narrows, and he saw ships. Lots of ships, none of them going anywhere.\n\n\"Water is pretty shallow, Captain,\" Jugs said.\n\n\"Surface,\" Loren said. \"I'll go up to the bridge. I want to see what's in the harbor. Listen to the radio and brief me over the sound-powered phone.\"\n\nSo _Texas_ rose from the depths and her sail broke water. Loren opened hatches and was soon standing on the small bridge. He plugged in a sound-powered headset and talked to Jugs.\n\nGiving heading commands, he went around ships that were anchored and under the bridge. Not much traffic on it, he noted, and he used his binoculars to examine the freighters and tankers anchored in the lower harbor, waiting for pier space.\n\nHe saw no navy ships. Not a one. Maybe there was a submarine outside the narrows, but maybe not. Maybe peace had broken out all over. The sky was empty of airplanes, even helicopters.\n\nStaying at ten knots, Loren took the boat around Liberty Island. \"Jugs, come up here.\"\n\nIn about a minute she was standing beside him, gazing at Lady Liberty, the ships, and the Manhattan skyline.\n\nAfter a bit she said, \"It's time to go home, Lorrie.\"\n\n\"I think so too,\" he said, and put his elbows on the rail in front of him and breathed deep of the tangy, salty air. \"Why don't you go below and send the others up here for a look, one by one.\"\n\n\"Aye-aye, sir,\" she said, and went down the ladder.\n\nLoren used the sound-powered phone to order a turn back toward the narrows. George Ranta came topside, looked and laughed and pounded Loren on the back, then went below and sent up Mouse.\n\nTwo hours later, safely back through the narrows and with good water under the keel, _Texas_ slipped beneath the waves. In the control room, Jugs briefed him on what she had heard on the radio. The power was back on in New York. The Pentagon was adamant that the military was not taking sides in the Soetoro administration's squabble with the states. People were being released from concentration camps in droves. Politicians were lining up at radio and television stations to be interviewed.\n\nMaybe America\u2014and Texas\u2014will make it after all, Loren thought, and gave orders for the voyage back to Galveston.\n\nIt was four in the afternoon when the 3,600th gold ingot was laid in a truck and the tailgate closed. The bank's personnel were locked upstairs in conference rooms, and the fake FBI agents had put the fear of God in them.\n\nFive minutes later the last of the soldiers and Texas Rangers were aboard the trucks and they were rolling. The traffic signals were working again, although the streets were still almost devoid of traffic. The trucks didn't stop for lights\u2014they simply drove on through.\n\nAt LaGuardia the planes were sitting with their ramps down when the trucks rolled up, and the loadmasters used hand signals to guide the drivers into the cavernous bays of the C-17s. Every soldier helped with the tie-down chains, then the loadmasters checked everything as the ramps came up and the planes started taxiing.\n\nWhen they were airborne, a sergeant passed out bottles of water and MREs to the troops. JR went up to the front of the plane and came back with an open packing case. He walked down the line of soldiers sitting beside the trucks passing out bottles of champagne. \"You guys have to share. We only brought a case for each plane.\"\n\nCorks popped and happy smiles broke out.\n\nJR went up to the flight deck and sat down in the jump seat. _By God_ , _we did it_ , he thought. _Fifty tons of gold_!\nTHIRTY-FOUR\n\nAriot in the streets in front of the White House and in Lafayette Park broke out between supporters and opponents of Barry Soetoro that evening. The melee quickly got out of control, so the police called for fire trucks with water cannons, which were waiting a half-mile away. And they fired tear gas grenades.\n\nThe mob wavered under the gas, but it was the fire trucks that finally dispersed the crowd. A dozen people were dead, either beaten to death or trampled, and several hundred injured.\n\nWhile the tear gas wafted into the White House, the survivors of the battle surged through the streets smashing out store windows, looting, and overturning cars and setting them on fire.\n\nIn the White House, loyalists gathered around Barry Soetoro and urged him to accept the Pentagon's offer of a plane to take him into exile.\n\n\"Their price is a letter of resignation,\" Soetoro said, \"and I am not going to resign this office. It would be a betrayal of all those people who believe in me.\" His chin quivered. \"I am America's hope, the hope of all people everywhere to build a just society and save the planet. That is my destiny.\"\n\nSulana Schanck believed. \"You are the hope of the _world_! And the world will come to your rescue. These racist pigs _will not prevail_!\"\n\nAmid the coughing and fervid pledges of loyalty, the realization sank in that they couldn't stay in the White House. The mob would return. And when it did . . .\n\nThey took the tunnel to the Executive Office Building across the street, and from there went to the basement, where their staff had a fleet of cars waiting. Not everyone got into the cars, of course. Most of the senators and representatives decided not to go. One said later that he knew when Soetoro's car pulled away that he would never see Barry Soetoro again.\n\nThe standoff between the crowds and the police and Secret Service guarding the executive mansion ended at about midnight. A crowd of almost two hundred people, mostly men, came walking out of a side street on the west side of the grounds. With them was a large tow truck, one used to rescue tractor-trailer rigs. Leading them was a black man in the uniform of a captain of the D.C. police. They came straight to the west gate, where four D.C. police in riot gear stood guard. Behind the gate, which was closed, were a half-dozen federal police, also in riot gear. Accompanying the crowd was a television reporter and her cameraman.\n\nThe police captain, who was unarmed, walked up to the cops, who knew him. \"Guys, we are going to open that gate and go through it. You have two choices: you can shoot me or get out of the way while we pull the gate down.\"\n\n\"What the hell do you think you're doing, Captain?\"\n\n\"I've joined the rebels. It is time to stop the bloodshed over Barry Soetoro. We're going in.\"\n\nOne of the cops fingered his radio. While he was doing that, the captain gestured to the tow truck, which moved up to within six feet of the gate. Men carrying chains went around the cops and ran the chains around the gate and hooked them to the massive bumper hooks of the truck. Then the helpers got out of the way and the truck backed up with its audible warning beeping madly.\n\nThe federal cops backed away from the gate with their weapons at the ready. One of them was already on his radio.\n\nThat was when the senior cop on duty, a sergeant, staring at the captain whom he had served under for more than a dozen years, gestured to his mates. \"Get out of the way, fellows. The captain is pulling it down.\"\n\nThe captain nodded once, and the tow truck engine revved and the driver popped the clutch. The slack came out of the chains and the gate came off its hinges and went skidding as the truck backed across the street, blocking it.\n\nThe captain strode up the now-open drive and said to the federal police. \"Shoot me or get out of the way.\"\n\nThey looked at the crowd surging forward, the television camera catching it all, and moved aside. The crowd surged onto the lawn and made for the White House. The television reporter and police captain walked, but many of the men in the crowd\u2014it was almost exclusively male\u2014ran ahead.\n\nIn fifteen minutes the police captain and television reporter learned to their satisfaction that the president was not in the mansion. Only a few servants remained. Not a single staffer or aide or politician could be found.\n\nAs the crowd surged through the first family's quarters and the Oval Office grabbing souvenirs and vandalizing furniture, the reporter and cameraman went trotting out the way they had entered. They had footage that they needed to get on the air fast. The reporter could smell a Pulitzer.\n\nGrafton and I were off the ground Thursday morning when the sky was black as coal and the morning star was just ooching up over the horizon. He climbed to 4,500 feet and headed straight for Hagerstown. The little plane didn't have a nav aid or GPS, so Grafton took a squint at the sectional chart, decided on a course, and hi-de-ho, here we go. As we flew along, I communed with Venus. Like most people, I rarely visit with the morning star. Praying that we wouldn't make this a habit, I gazed with wonder at the sprite. The night faded, and almost as if God had taken a hand, at the proper time the Hagerstown airport appeared in the dawn haze.\n\nThe northern army was camped on the airport grass. It was a sea of military vehicles; a few APCs; several howitzers; lots of trucks, generators, tents, portable kitchens; and several thousand people, about half in uniform. Pickup trucks and cars were parked in rows.\n\n\"Wow,\" I said.\n\n\"That's only about half the troops,\" Grafton said. \"The rest are camped at the fair ground, and a lot of the veterans are on picket duty. Martinez thinks he has about five thousand people now.\"\n\nWe landed and parked near the control tower. General Martinez was there to meet Grafton. They went over to Martinez' ride, a pickup, and conferred while I chocked the Cessna and tied it down. I looked to see if we had collected any more bullet holes. Not yet today.\n\nI faced into the dawn, surveyed the encampment, and took a leak. I gave thanks that I hadn't chosen the military as a career; the hours are terrible. Zipped up and yawned. Okay, I was ready.\n\nI strolled over to the meeting of the general staff at the pickup truck.\n\n\"General Martinez says Soetoro isn't in the White House. Civilians got in last night and found he had skedaddled.\"\n\n\"Terrific,\" I said, yawning again. \"If the Pentagon didn't fly him to some third world paradise, this will be like looking for Elvis.\"\n\n\"Oh,\" Jake Grafton said with a gleam in his eye, \"I have a feeling he's close. Like up at Camp David.\" He pointed to the east. \"Just twenty miles that way, on the other side of that low mountain.\"\n\nI turned and looked east at the mountain bulging against the dawn sky. Actually, it sort of figured that Barry Soetoro might run to earth in that rustic presidential getaway, which was designed for defense by Secret Service and federal police. Local crackers couldn't get within five miles of the place without alarms going off. If I were going to hide out for a while and had the federal government to pay the help, chefs included, Camp David would be high on my list.\n\n\"Maybe so,\" I said to Grafton.\n\n\"Indeed,\" he said, \"maybe so.\"\n\nHe turned back to General Martinez, so I walked around the pickup truck to see if it had any dings. It looked clean. After this mess was over, maybe I could make an offer on one that FEMA didn't need anymore. I had decided that I needed a truck. My old Benz convertible was cool, but a truck had more possibilities for a man of my m\u00e9tier.\n\nGrafton and Martinez gabbled on their handhelds a while, then Grafton motioned toward the Cessna. He shook hands with Martinez and conferred some more while I untied the plane and stowed the chocks. I climbed into the right seat and put on my belt and headset. Arranged my little bag of grenades behind me so I could reach them easily and made sure my M4 on the backseat was loaded and handy. I wished I had a flak vest to sit on, but I didn't.\n\nFinally Grafton strode over, jumped into the left seat, and cranked the engine. With it at idle he put on his seat belt and headset. \"Martinez will get the Predators up. They are flying them out of Dawson, so until they get here we are the eyes of the army.\"\n\n\"Roger eyes.\"\n\n\"We need to find out what happened to that column of people coming from Baltimore along the interstate and see what's happening at Camp David.\"\n\n\"The feds will likely shoot at us if we go swanning over in this crate.\"\n\n\"Then we'll know, won't we?\"\n\nThe asshole! It was on the tip of my tongue to tell him that if he had an ounce of sense he'd send a Predator over David, but not-a-minute-to-waste Grafton had made his decision and he wouldn't change it. How come I always get stuck with the heroes?\n\nBoth our side windows opened on hinges at the top to a limit of about three inches. I checked mine. It was a bit too small for me to push a grenade through the opening. Not to worry, I could always open the door against the slipstream and drop them like eagle shit on the multitudes below. Maybe they would be inspired to keep their heads down. I reached behind me and got a couple, which I put in my lap.\n\nTwenty minutes later we realized that the interstate east all the way to Frederick was essentially empty. That column of Soetoro volunteers had to be somewhere, but where?\n\nGrafton turned toward Camp David. He was only about a thousand feet above the trees. Plumes of smoke rose from the forest, formed a thin cloud in the still air, and pointed the way to Camp David. Lots of fires down there, so there were probably lots of people.\n\nAnd sure enough, we found them. Grafton got looks through the trees at people camping, then he dropped lower and we saw vehicles by the dozens, mainly trucks. Saw the presidential buildings surrounded by lawns and stately mature trees, and many people on those lawns. Most of the people I saw had rifles. Then a few of them pointed their weapons skyward and I saw flashes against the dark of the forest floor.\n\n\"They're shooting at us,\" I told Grafton.\n\n\"We're leaving,\" he said, and headed west over the low mountain.\n\nWhen we were clear, he got on the radio to Martinez. \"Many people around Camp David. I think you need to check it out. The man may be there.\"\n\n\"Wilco.\"\n\nWe landed at Hagerstown and I tied the plane down after inspecting it again for bullet holes. The shooters all missed. Maybe this was going to be a lucky day for me. Sarah Houston drove up in our stolen FEMA pickup, the one that had my money in it, along with spare weapons, AT4s, and my sniper rifles. I was ready for a real war.\n\nShe was wearing jeans, a green army T-shirt, and a web belt with her pistol holster attached. Her hair was pulled back in a ponytail. \"You're looking great this morning, lady,\" I told her.\n\n\"Have you heard that Soetoro left the White House sometime yesterday?\"\n\n\"I have.\"\n\n\"The Pentagon said he refused an offer of a flight into exile.\"\n\n\"Probably no one would accept him. He'd want to take Mickey with him, and that's a deal killer.\"\n\nSarah sighed and looked at the sky and army and mountains. \"I'll be glad when this is over,\" she said. She flipped a hand at the ad hoc army, now getting ready to move. \"The officers say that the former soldiers and guard troopers and veterans follow orders. The civilians are here on a toot. They don't do what they're told unless they feel like it. They were up drinking and partying all night. Some of them didn't get an hour's sleep.\"\n\n\"My prediction is they're going to get shot at today,\" I said. \"Some of them will run like rabbits. Don't get caught behind them or you'll get run over.\"\n\nI jumped in the driver's seat of the truck, Sarah climbed in beside me, and we went looking for Grafton, who would be at headquarters if we could find it.\n\nTurned out HQ was in the airport office building. Outside, I ran into Willis Coffee. \"How goes the war?\" I asked.\n\nHe looked disgusted. \"Two accidental shootings last night. Civilians! One dead, one injured. Amateur hour.\"\n\n\"We'll see if Soetoro's army can whittle them down today. They're at Camp David, just over that little mountain to the east. The man himself may be there, so Grafton will probably have us humping hard to surround the place so he can't sneak out.\"\n\n\"Fine with me,\" Willis said. \"Let's pop him and get on with the program.\"\n\n\"You'd shoot him?\"\n\n\"That son? In a New York minute. _Vaya con Dios_ , asshole, and bang!\"\n\n\"Where's Travis?\"\n\nHe gestured vaguely. \"Scouting somewhere. Martinez sent him out before dawn.\"\n\n\"Good luck today,\" I said, and Sarah and I went inside the building.\n\nGrafton was conferring with Generals Martinez and Considine. I listened in and gathered that they wanted to surround Camp David as quickly as possible. Trucks full of troops and the APCs would get on the highway and go around to the east as fast as they could. Another load of troops and APCs would go around to the north. The civilians would be pointed east and told to hike over the mountain, with some professionals along to ensure they didn't get lost in the woods.\n\nWhen the meeting broke up, Grafton said he was riding with me. \"Which column are we going with?\"\n\n\"The civilians, through the woods.\"\n\nMy face must have fallen, because he said, \"There're a couple of dirt roads. We'll take the pickup. If we do this right, the people at Camp David will think the mob coming through the woods is the main assault and leave the front door open for the pros.\"\n\nI wondered if he was having a senior moment. \"If they aren't stupid,\" I suggested tactfully, \"they might think the main assault is coming through the front gate.\"\n\n\"Didn't you see them when we flew over this morning, Tommy? The pros are dug in to defend the front gate and perimeter fence. They're well dug in, with at least two machine-gun nests and a couple of artillery pieces that I saw. Our troops out front will set up ambushes a couple of miles from the front gate, and the defenders won't even see them or know that they are there. With a little bit of luck, if the civilian volunteers coming through the woods can make enough noise, Soetoro will flush and boogey out the front and we'll bag him.\"\n\nSo he intended to capture the president of the United States. \"What are you going to do with him when you have him?\" I asked.\n\n\"Lock him up and let the new government worry about him. A significant percentage of Americans still think he's God's other son. We have got to bring people together, not drive them apart. The next government can have a trial, send him to Switzerland or Kenya, whatever floats their boat. And we can start putting America back together again.\"\n\n\"What about all these civilian volunteers? They're undisciplined, don't know tactics, are poorly armed, won't obey orders\u2014they don't know shit about combat. They'll panic and get shot in droves.\"\n\n\"We're rebuilding a nation here, Tommy. It takes blood to create legends and myths. These people want to fight for their country. We'll let 'em.\"\n\nThat was the Jake Grafton I knew, one hard man. God help all these civilians.\n\nThere must have been three or four thousand of them, armed with everything from shotguns and deer rifles to black civilian versions of the M16. Lots of pistols. It seemed a quarter of them carried pistols and nothing else. I was appalled. If you were within pistol range of the enemy, you were too damned close.\n\nTrucks passed out water bottles, and cases of water were tossed in the beds of our pickups. For all those people, it was not enough. A lot of them were going to get seriously thirsty, even though the temperature was only seventy degrees. I suspected many would pass out from heat exhaustion, especially those who were overweight. Today they had a mountain to climb and a fight on the other side ahead of them. It was at least fifteen miles, I suspected, to the Camp David perimeter fence, and most of it uphill. The crest of the mountain was about a thousand feet in elevation above us.\n\nLooking them over, I thought the average age might be around forty. Everyone who claimed he was a U.S. Army or Marine veteran or retiree had already been winnowed out, given a uniform and a military rifle, and those folks were in trucks and APCs, going to fight the Secret Service and Federal Security police on the other side of the mountain. These were people who claimed no military experience, which meant they knew nothing of tactics or how to handle military weapons and hardware. They probably had minimum experience obeying orders, our modern world being what it is.\n\nAnd yet . . . it was the men over forty who interested me. Many were apparently construction workers or farmers, wearing bib overalls or work trousers and leather boots. Lean and tanned, they carried their rifles like they knew how to use them and had a rucksack or backpack over their shoulders with water, rations, and ammo. Lots of ball caps; some of them were my very favorite, John Deere. I had no doubt most of these guys could walk me into the ground.\n\nThen there were the outdoor types, men and women, also lean, wearing walking shoes with shorts and logoed T-shirts. They all had backpacks, some of them with the logos of purveyors of outdoor gear. Many wore floppy sun hats with strings that hung under their chins. A few even had bicyclists' water bags over their shoulders. They carried their rifles or shotguns as if they were unsure how to do it.\n\nAnd then there was everyone else. A few were teenagers, but many looked to me like they were professionals or middle managers, some pudgy, some downright overweight, wearing jeans and everything else you could imagine. Their T-shirts were from colleges, high schools, and state parks. At least a third of these folks looked as if a walk across a large parking lot would wear them out. I would have bet some of the women were soccer moms.\n\nBlack, white, brown, Asians, with ancestors from all over the globe, they looked like America to me.\n\nAt least three thousand of these volunteers gathered around the spot where the first dirt road left the pavement. They had walked over two miles through suburban Hagerstown to get there. It was getting on toward ten o'clock.\n\nWith General Considine beside him, Grafton stood in the bed of the truck and shouted for them to gather around. They did. He raised his voice, and I swear, I think everyone in that mob heard him. Grafton in full cry was a primal force.\n\n\"We're going up this road to the top of that mountain and will hit the Camp David perimeter fence on the other side. It's a good hike up there, and you need to keep up. Don't fire your weapons until we make contact with the enemy. Obey your officers and stay together. No straggling. When you tell your grandkids about this someday, you'll want to be able to say you were there at the finish, there when the dictator was captured and a new America was born. Keep your head down and shoot low. Let's go.\" And he waved his arm up the road.\n\nThere was a fork a mile or so up the road, and he had stationed guardsmen there to divide the civilians, sending half on one road, half on the other. Travis Clay had reconnoitered both, he told Sarah and me, and both roads led to a bald spot on the mountain crest; the Camp David perimeter fence was just beyond that. \"Considine will take the north fork and I'll take the south. We expect to meet most of Soetoro's volunteers at the bald crest,\" Grafton told us as we watched our crowd trudge up the road. \"That's the fight that will flush Soetoro, I hope, and Martinez will bag him on the other side of the mountain.\"\n\n\"If he's there,\" I said. \"For all we know he may be in Hawaii playing golf.\"\n\n\"If he is, he swam over,\" Grafton shot back. \"Tommy, you drive. Follow that howitzer. The guardsmen with their mortars will follow you.\"\nTHIRTY-FIVE\n\nThe trek up the mountain was the most frustrating experience I have ever had. We averaged two miles every hour. I would move the truck ahead a couple of hundred yards and shut off the engine to save fuel.\n\nThe western side of that mountain, the crest of which ran generally north and south, was a mix of pastures and woodlots with farm houses and ramshackle barns thrown in, and here and there a mobile home surrounded by the owner's junk collection. Rotting tractors, curious cows staring at us over fences, abandoned pickups manufactured during the Truman administration, stray dogs, yards full of weeds, fences covered with poison ivy, it was rural America in late summer in all its glory.\n\nThe fat people had it worst. They began dropping out, just sitting down. Some of the skinny people put their weapons in the truck and on the army trucks behind us carrying mortars, MREs, and water, just to lighten the load. People trudged and trudged up the edge of our road, raising clouds of dust.\n\nGrafton sat in the rear seat and was on the handheld radio constantly. He gave Sarah and me updates on the southern army. They were through Leesburg and had collected another two or more thousand civilians, who were walking and driving cars and pickups and vans. Everyone seemed to want to go to Washington. Our ambushers, Martinez' bunch, were in position blocking the roads into and out of Camp David.\n\nThe power was back on in eastern Virginia and Maryland, and television and radio reporters were giving their audiences the blow by blow. Dixie Cotton was with the army marching through Leesburg, heading for the eastern Virginia suburbs, and she was on the air and on fire, urging all loyal Americans to join with the army of volunteers on its way to liberate Washington.\n\nIt was nearly one o'clock when I saw Travis Clay standing beside the road. I stopped beside him.\n\n\"This is like herding cats,\" he said. \"Got any water?\"\n\n\"In the bed. Help yourself.\"\n\nWhen he had guzzled a bottle and had another bottle in his hand, he came back to the driver's door. \"You going to sit there riding along in your limo, or are you going to help?\"\n\n\"I'm an officer. Rank has its privileges.\"\n\n\"I'm going to write a letter to your mother. 'Tommy doesn't play well with other children.'\"\n\nI told Sarah to drive the truck and got out with my M4.\n\nI helped Travis and Willis herd the troops up the road. Every little bit a shot would echo around. The wannabe warriors got bored and shot into a tree or a deer or whatever. I saw a guy with a shotgun drop a crow that was flying over.\n\n\"Save your ammunition,\" I admonished the trekkers. \"You're going to need every damn bullet before the day is over. And for God's sake, don't shoot the cows: they don't vote, don't have guns, and can't shoot back, so it isn't sporting.\" Some listened, some didn't.\n\nWe came upon a farm where the lady of the house had gone all out. Apparently she knew the column was hiking up the road, so she had a folding table set up by the gate and she and her daughters, both early teens, were pouring good well water for anyone who wanted a drink. And serving homemade cookies.\n\n\"Thank you, ma'am,\" I said as I helped myself to an oatmeal raisin cookie and filled up my water bottle. \"How'd you know this mob was coming?\"\n\n\"Your scouts came up the hill at dawn this morning, and I met them coming back. They said a lot of people would be along.\"\n\nSo I sipped water and munched my cookie as the troops did the same, then we moved along while other people crowded the table. Everyone had a good word to say to the lady and her daughters, and she had a good word for everyone. America walking by your door, on a dirt road that leads nowhere in particular. It was a strange experience.\n\nTwo miles farther up the road, I found a woman sitting with her shoes and socks off, looking at broken blisters, now leaking blood. A double-barrel shotgun lay beside her. \"Are you going to be able to keep going?\" I asked.\n\nShe looked to me to be in her fifties. She cocked her head to eye me, squinting against the sun. \"I'll make it, Jack,\" she said.\n\n\"My name's Tommy Carmellini.\"\n\n\"Betty Connelly.\"\n\nShe took a pair of dry socks from her backpack. \"My daughter died in that parochial school in Arlington Heights a couple of weeks ago. She was a teacher. One of those jihadists Soetoro let into the country shot her in the face. I'll get up this mountain if I have to crawl it.\"\n\nWhile she put her shoes and socks back on, I inspected her shotgun, an elegant old side-by-side. I opened the breech and extracted one of the shells. Number six birdshot, perfect for pheasants. I put the shell back in, snapped the breech closed, checked the safety, and put her on the tailgate of our truck. Gave her a bottle of water and her shotgun. \"You ride there until we get on top,\" I told her.\n\nShe nodded and brushed the hair back out of her eyes. \"Thanks,\" she said. I just hoped she didn't get shot.\n\nAfter two hours, I got back in the truck. Although the temp was only seventy-five degrees, according to the truck's thermometer, I was hot and sweaty, and so was everyone hiking up that low mountain to get to whatever fate awaited us. I guess I was a little nervous, right along with everyone else.\n\nSomehow, someway, we made it up the grade. The dirt road got worse and worse the higher we went, until it was just a rutted road full of dried-up mud-holes. No farms up here, just woods. I glanced at the truck's odometer. It had driven fourteen miles to cover the twelve miles direct distance to the edge of the bald.\n\nGrafton had received radio messages from the Predator crew long before. Soetoro's army was on the crest of the mountain, and at least three hundred yards of cow pasture lay between the forest on the top of the western slope and the naked crest.\n\nIt was four o'clock by my watch when I first sighted the bald. Sarah was at the wheel of the truck, so I got out and started directing our tired volunteers into the woods. I estimated we had lost at least half through straggling and heat exhaustion, but that was just a guess.\n\n\"Get the troops spread out,\" Grafton told me. \"Link up with the people on the other road and stay in the woods. Have the mortarmen take their weapons out there a ways for max coverage.\"\n\nAlready the people on the crest were popping away at us. The bullets pattered on the trees and leaves like rain, but if they hit anyone, I didn't see him or her go down. With all the dust and engine noise and gunfire all afternoon from our crowd as they climbed the mountain, there was no possibility of surprise. Not that Grafton wanted surprise.\n\nOur troops retrieved their weapons from the vehicles and went scurrying out through the woods as the distant firing and pattering of bullets encouraged them on. The howitzer was turned and set up in the road. The truck pulling it had already run over the cattle gate, flattening it. A three-strand barbed-wire fence on ancient, half-rotted posts ran away on both sides of the gate. The artillery officer, a captain, came over to confer with Grafton. \"Not yet,\" the admiral said.\n\nI went into the woods, trying to show the civilians how to take advantage of cover, advising them not to fire their weapons, but to wait. Some of the fools huddled down behind a bush or sapling that wouldn't stop a BB, so I moved them to rocks and behind big trees. Inevitably a few of them began banging away at the distant crest, wasting ammo; they probably had no idea how far their bullets would drop at that distance. Some were shooting into the air at a thirty-degree angle; maybe they were trying to hit Camp David.\n\nOne guy was walking around like it was Sunday afternoon in the park, shouting to his fellow warriors, \"Hang tough. We'll kick the shit out of those stupid sons of bitches.\"\n\n\"Get down, you idiot,\" I told him.\n\nHe looked at me with distain and struck a pose. \"At this distance, they can't hit\u2014\"\n\nWhap! There is no sound on earth like that of a bullet striking a living body.\n\nI heard the sound and saw the hole appear in the side of his head. Blood began leaking out. He swayed like an old oak in a storm, his eyes fixed on infinity, dead on his feet. He fell beside me.\n\nHe had a nice rifle, an old 1903 Springfield with a four-power scope. I laid it across his chest and moved on, shouting, \"You morons get behind something solid and stay down! Save your ammo!\"\n\nAfter twenty minutes of that, when I had positioned the men and women who had made the climb on the left side of the road, I went back to the pickup.\n\n\"Get out your sniper rifle, Tommy, and look at the people on the crest,\" Grafton said. \"When the action starts, shoot anyone who looks as if he is directing troops.\" That was always the advice to snipers: kill the officers.\n\n\"Yo,\" I said and got out the best rifle, deployed the bipod, filled my pockets with cartridges, and set up using a pile of dirt that some snow scraper had deposited there in past years.\n\nI lased the crest. Three hundred fifteen yards, give or take.\n\n\"Start shooting, Tommy,\" Jake Grafton said.\n\nI picked out some fool who was standing up looking this way with binoculars and let him have it. After the recoil, I didn't see him. I scared or hit him.\n\nI had fired ten shots when Grafton said, \"Do you have a machine gun in the truck?\"\n\n\"Yes.\"\n\n\"Put it up there and get ready.\"\n\nI had no more than gotten the bipod deployed and the belt in it when the howitzer began firing at a high angle. I saw the shells popping on the crest. Then the mortars opened up, dropping their shells along the crest too.\n\nThis is it, I thought. They'll break for the woods behind them and we'll charge up there to take the crest.\n\nGrafton was running to the left, telling everyone who would listen that we were going to charge the crest, but to stop there. In all likelihood, the people on the crest would retreat to the woods on the other side and be waiting for our bunch to charge them.\n\nBut. . .\n\nI was astounded when the enemy on the crest stood up and began running downhill toward us. They charged, at least two thousand of them, screaming at the top of their lungs and firing wildly. They were dedicated Soetoro fanatics, not professionals.\n\nI hunkered down over the M249 and began firing bursts. They went down in handfuls. To my right and left the woods came alive as the civilian volunteers let loose with everything they had, shotguns, rifles, and pistols.\n\nThe charge broke halfway to the trees. The ground was carpeted with people when, suddenly, the survivors began running back up the hill en masse, some of them carrying and dragging wounded people.\n\nI shot the whole belt at them as the howitzer banged away to my left and the mortarmen dropped their shells among the survivors. Then the artillery shells that had been popping viciously moved their aim point and I no longer saw the shells land. They were obviously shooting to land their shells on the back side of the ridge.\n\nAll along our line a shout went up and people who thought they didn't have another erg of energy left in them left the trees in a trot, charging up that hill. That's when my admiration of the American volunteer went through the roof. By God, they had guts.\n\nThey swarmed up that hill.\n\nSarah motioned to me, so I grabbed the machine gun and belt and got in the back. She put the truck in motion and I hung on. I wanted to change the belt in the machine gun but with the uneven ground tossing the truck around, there was no way. I grabbed my M4 and squirted a burst at any of the enemy who paused in flight to shoot at people charging up the hill.\n\nWhen we made the crest, it was empty. The enemy was running down the other side. Sarah stopped the truck. I dumped the carbine and grabbed a belt of ammo and slapped it in the M249 as bullets snapped around the truck. People running, guns blazing: it was the damnedest battle I have been in yet, like something from an American Civil War movie, blues versus grays. I dismounted, set up the gun, and shot at the retreating people dashing into the trees on the east side of the bald.\n\n\"Hose the tree line,\" Grafton shouted. He was outside the truck, crouching, watching everything. \"They may have an ambush there.\" Out of the corner of my eye, I saw him run over to the howitzer crew and point. In seconds the artillery shells began falling just back of the treeline: explosions, clouds of dirt, trees falling. The mortarmen came up to the crest in the pickups that they had used to transport their tubes, recoil plates, and ammo. After taking a moment to get set up again, they began lofting shells into the woods below.\n\nTo my amazement, our guys who had scaled the crest stopped for only a moment to get their breath, then set off running downhill for the trees.\n\nI finished the belt and got another into the gun, which was getting damned hot.\n\nGrafton jumped into the truck and Sarah raced it downhill. I sprayed lead, then grabbed the gun and followed them.\n\nShe stopped forty feet from the edge of the trees. I threw my machine gun in the bed and picked up my M4.\n\nWillis Coffee came running up. Grafton shouted at him, \"Get some AT4s and shoot them into the trees.\"\n\n\"We've got about a dozen.\"\n\n\"Get them running for Camp David.\"\n\nWillis did as he was told. Stood in the bed and launched the rockets as fast as he could.\n\nWhen our troops were no longer in sight, Willis got down. Our guys and gals had gone into the trees. They had literally jerked the old fence posts out of the ground rather than climb over or through the barbed wire.\n\nGrafton, Sarah, and Willis each got an M4, and we trotted toward the trees. We hadn't taken five steps when Willis grunted and fell. I stopped and went back to check on him. He had taken a bullet in the chest. He looked at me and said, \"Tell my wife. . .\"\n\n\"What?\" I demanded. \"Tell her what?\"\n\nBut he was dead. I realized then that I really didn't know Willis Coffee very well. And I would never know him better. \"God bless you,\" I whispered, and ran on toward the trees.\n\nDead and wounded lay everywhere. We disarmed the wounded and kept going. Our troops were in front of us, driving the enemy toward the perimeter fence somewhere in the woods ahead.\n\nWhen we hit the fence, it was down. Who tore it down I never learned. It was down when we got there and that was the reality of it. We kept going.\n\nSomehow in the woods amid the smoke and bodies, we lost Grafton. He must have run on ahead. I was too old a dog for loping through the woods when people could be hiding behind any tree praying for a good shot at their pursuers. Ahead I could hear the cacophony of gunfire. Bodies lay every which way, a lot of them shot in the back. The wounded were groaning. The rocky forest floor looked like hell's half acre.\n\nA moment later I saw the first body that had been scalped. The head was a bloody mess and the hair was gone. At the time I thought, maybe shrapnel did that.\n\nI kept going, and soon found another. Scalped.\n\nA hundred yards later, I met an unarmed man wandering amid the shattered trees and rocks. He had long hair, at least on the fringes; his scalp had been cut and torn off. The top of his head and his face were masses of blood.\n\nI stopped him, forced him to lie down. \"Whoa. What the hell happened?\"\n\n\"A shell hit near me. I was out for a bit, and when I woke up some guy was ripping the top of my head off. He had a big knife. He left me there.\"\n\n\"Lie still. The medics will be along after a bit.\"\n\n\"Help me, mister! For God's sake!\" He clutched at me but I drew back and scanned the woods.\n\n\"Lie still,\" I repeated. \"Your war is over.\"\n\nI picked up the pace. I had covered maybe two hundred yards when I came upon a big tattooed guy with a long knife and a black rifle. He had a bag on a strap over his shoulder. I could see hair protruding from it. He was bending over a figure on the ground, a woman in shorts with long blond hair, and he had his knife out. She had an arm up, trying to fend him off. \"For God's sake,\" she screamed. He grabbed a handful of hair, lifted her head a little, and jabbed the knife into her scalp.\n\n\"Stop,\" I roared. He turned toward me and I shot him.\n\nI ran toward him as he went down. The woman on the ground looked at me stunned, then she was dead, as if someone had turned a switch. He had her scalp half off.\n\nHe was still alive. He looked at me with the strangest expression. I kicked his rifle away.\n\nThen a shot rang out. He took the bullet in the head. I turned and saw Sarah Houston standing there with her carbine at her shoulder.\n\nShe shot into him three or four more times, turned, and began walking downhill, east toward Camp David and the rolling racket of gunfire.\n\nThe woman on the ground was wearing a Penn T-shirt\u2014University of Pennsylvania\u2014now soaked with blood from a mortal wound caused by a large shard of a tree that was still sticking two feet out of her chest. She had bled a lot before the scalper got to her. Blood, almost black, was everywhere. She and the scalper lay in it.\n\nThe sun was already behind the bald crest above me, leaving the woods in dark shadow. Below on the slope, Sarah threaded her way through trees still standing and those blasted by shellfire, around downed trees, limbs, and rocky outcrops, and disappeared from view. I got myself in motion, following along.\n\nGrafton must have passed these wounded people on his trot down the hill, sore ribs and all, trying to get to Camp David before the mob killed Soetoro. He was a man on a mission.\n\nI wasn't. I didn't give a damn what happened to Barry Soetoro.\n\nIt got dark as I went through the woods. There was just enough moon and starlight to allow me to see trees and rocks except under dense foliage, when I had to literally feel my way along. I wished I had some night-vision goggles, but I didn't. And of course, neither did anyone else. I only tripped and fell four times.\n\nAfter a while I got glimpses of fires burning around the presidential enclave. I moved carefully, the M4 at the ready. I came out of the trees and walked along a graveled path toward the biggest of the fires. People were everywhere, and all of them were armed. I figured they were our guys, and was sure when I saw fifty or sixty people sitting on the ground wearing white plastic ties around their wrists. There must have been a thousand people in the lawn and flower beds, most of them shouting like fiends.\n\nNear the front door of what I took to be the main building or lodge, I saw Grafton and some of the people from the camp this morning confronting a knot of men and women in business attire. They had to be Secret Service. Barry and Mickey Soetoro were not in sight. I went around the corner of the house away from the group. The house, or lodge, was a two story. Looking around and concluding I was unobserved, I leaped for the bottom of a balcony. Got my hands on the concrete floor of the thing and pulled myself up with every muscle screaming about all the exercise I hadn't been getting.\n\nChecking over my shoulder, I decided I still didn't have an audience, so went up like I was climbing a rope. Hooked an ankle over the top of the rail and voila, I was in. The door, unlocked, led to a bedroom. The lights were on inside and it was empty.\n\nI closed the balcony door and stood listening with my pistol in my hand as I scanned the room. Actually, it was the sitting room of a suite. The crowd noise outside was now only a murmur. First I checked the bedroom, which was dark and empty. So was the bathroom.\n\nThe interior door of the sitting room opened into a hallway. I could hear voices from my left. That was the way I wanted to go, but only after I checked these other suites, for there appeared to be four of them off this hallway. When I went toward the voices, I wanted to know that there was no one behind me. The second suite I checked was empty of people, but the bed and bathroom had obviously been used.\n\nIn the third suite I found the body. It was lying beside the wet bar, as if it had fallen off a bar stool. The remnants of several drinks were on the bar. His throat was cut and he had done a lot of bleeding. I tried not to step in the blood, but to get a look at the face to see if I could recognize it. Yep. Al Grantham, the chief of staff.\n\nWhoever cut his throat knew exactly how to do it. It looked like just one vicious swipe had severed the carotid arteries and his windpipe. Apparently done from behind. Unconsciousness had followed within a second or two as blood pressure in the victim's brain dropped toward zero.\n\nI reached and touched his hand. It was still supple, although just beginning to cool off. He hadn't been dead long, not more than a few minutes. The blood was red and sticky.\n\nI found that the palm of my hand on my pistol was sweaty. I dried it on my jeans and checked to make sure the suite was indeed empty of living people. A surprise by a knife fighter of that caliber was something to be avoided.\n\nThe hallway still empty, I tried the door of the fourth suite. Sucked it up and went in fast with the pistol ready. No one there.\n\nBack down the hallway, gliding along beside the wall, listening intently. The voices got louder as I moved.\n\nI could see that the wall I was against turned into a railing, and the hallway became a balcony leading to a stairway down into a great room. I got down on the floor, and after crawling, inched the top of my head around the edge of the wall and peeked between it and the first balcony upright.\n\nThere in the main room below, no more than fifteen feet from me, were Barry and Mickey Soetoro. . .and Sulana Schanck and a male aide I didn't recognize, talking to a couple of Secret Service types carrying M4s. Vice President Rhodes was there, the veep from central casting, with the superbly barbered white hair and square chin, in a gray suit that fit perfectly. Two other people were facing the agents: I couldn't see their faces and didn't know who they were. Rhodes' aides or politicians, no doubt, and true believers to the core.\n\n\". . .There are at least a thousand of them, Mr. President. Perhaps twice that. They have the buildings surrounded and have complete control. We have six people left. The rebels can come into this building anytime they decide to walk over us and do it.\"\n\n\"Have you called for reinforcements? Assistance? Whatever you call it?\"\n\n\"Yes. No one answers our radio transmissions, and no one is picking up the scrambled landlines.\"\n\n\"You're going to have to talk to Grafton,\" the veep said to the prez.\n\n\"I am not going to surrender,\" Soetoro declared. I thought I could detect a slight tremor in his voice, but it may have been only the acoustics. \"Where are our supporters? Where are the liberal armies that were going to preserve order and support the federal government against the reactionaries? _Where are they_?\"\n\nI thought that his loyal supporters lying dead or maimed on the mountainside or sitting outside with their hands shackled by plastic ties were beyond caring how much they had disappointed ol' Barry.\n\nWhich of these people killed Al Grantham with a knife, and why? If you were going to do it, why not years ago? Truthfully, his mother should have done it way back when she realized what a twisted, diseased monster she had foisted upon the world, but that was water under the bridge, until today.\n\nOf course, the knife artist could be somewhere else in the building, not down below. I glanced back down the hallway, a bit nervously, I suppose, to ensure that it was still empty. I certainly didn't want that dude within twenty yards of me.\n\nMeanwhile they were jabbering away just below me. Everyone talking at once. Just beyond the door was a seriously unhappy crowd, or if you were inside looking out, an angry armed mob. These people in the lodge had no idea what fate awaited them. Jake Grafton didn't know either. Not only did I not know, I didn't give a damn.\n\nI became aware that Sulana Schanck was having a serious private conversation with Barry Soetoro, just a few steps away from the others. No one else was apparently paying attention to what was being said, and they were talking too low for me to eavesdrop, even though my hearing is excellent. I tried to read lips and body language. She was adamant and he was resisting.\n\nWhatever fate awaited these two, it would probably be worse for Soetoro. Schanck was merely a bit player. Or so I thought.\n\nThen, in a twinkling of an eye, I found out how wrong I was. Sulana Schanck pulled a large knife from her sleeve and with one vicious backhand sliced Soetoro's throat from ear to ear. Blood geysered forth, showering Schanck, as the president sank toward the floor.\n\nI scrambled to my knees and pointed my pistol, but I was too late. She spun like a ballet dancer, took one bound, and used the knife on the veep's neck, with similar results. John Rhodes went down in a welter of blood.\n\nOne of the Secret Service agents beat me to the trigger. He put a burst in Sulana Schanck's chest, hammering her to the floor.\n\n\"Drop it,\" I shouted. I had the Kimber .45 at arm's length pointed right at his head. If he tried to swing that carbine in my direction he was going to die.\n\n\"Drop the weapons,\" I roared again. Both carbines hit the floor.\n\nThe outside door swung open and a man appeared there with a pistol in his hand. I shouted, \"You in the door. Get Admiral Grafton and send him in here _now_!\"\n\nDown below, Mickey had freaked. The aides and pols were fluttering around uselessly, staring horrified at the corpses of Barry Soetoro and his vice president. There was nothing anyone on earth could do for them. Sulana Schanck hadn't twitched since she hit the floor. Maybe she was in Paradise now or shaking hands with Muhammad in Hell.\n\nTo my eternal relief, Jake Grafton and General Considine walked into the room accompanied by four guys carrying weapons.\n\nI sat down on the floor and holstered my shooter.\n\nAbout two hours later the bodies of the president, vice president, chief of staff, and chief political advisor were carried out of the house and placed on a stack of firewood in the middle of a grassy area. The crowd had raided the presidential woodpile. They piled the bodies on that rick of wood, poured a couple of gallons of gasoline on them, and set them afire.\n\nThe National Guard had arrived by then and the volunteers had stopped shooting their guns into the air. The prisoners were loaded on trucks and driven away. I didn't ask where they were being taken.\n\nA huge silent crowd encircled the fire. As I watched, the woman from the hike up the mountain, Betty Connelly, stepped from the crowd, leveled her shotgun into the fire, and fired twice.\n\nThen she turned and walked away.\n\nGrafton and Considine came over to where I was standing.\n\n\"Tell me what happened in there, Tommy.\"\n\nSo I told it, from climbing the balcony, to finding Grantham's corpse, to watching the Soetoro party trying to decide what to do. . .to Schanck's unexpected knife work.\n\n\"So you didn't hear what she and the president said?\"\n\n\"No, sir. It looked like she was urging him to do something that he didn't want to do. Maybe she wouldn't take no for an answer.\"\n\n\"Workplace violence,\" General Considine remarked flippantly.\n\nThey had a few more questions, but I had no more answers.\n\n\"ISIS or Al Qaeda will claim they got him,\" Grafton said gloomily.\n\n\"Soetoro is the one who chose Sulana Schanck to sit beside him and whisper in his ear,\" Considine remarked. \"The true believers are going to have to swallow that, Jake, whether they want to or not.\"\n\n\" _Et tu, Brute_ ,\" Grafton muttered.\n\nI scored a flashlight off a soldier on the water truck and went looking for Sarah. Meanwhile she found Grafton. The funeral pyre was burning steadily now. The admiral had a handheld radio up to his ear, so I gave him the Hi sign and he acknowledged. With the fire illuminating a thousand faces, Sarah and I turned our backs to it and plunged into the darkness.\n\nIt was a five-mile hike through the woods, all uphill, and we came out on the bald about a half-mile north of the pickup. A sliver moon was hanging in the sky and the stars were out. This old earth just keeps on turning. Walking toward the truck, I asked her, \"How are you feeling?\"\n\nShe didn't reply.\n\n\"If that truck isn't hors de combat, I thought we might head west.\"\n\nShe didn't say anything.\n\n\"You got the keys to the truck?\" I asked.\n\n\"I left them in the ignition.\"\n\nOh boy.\n\nThat half-mile hike through the grass in the moonlight, with corpses lying on the ground in a random pattern, was one of the memories I will carry with me all my days. There were at least two army trucks out there, lights ablaze, looking for wounded. The whole scene was surreal. The dead didn't even whisper.\n\nWe passed a young woman wandering along, trying in the moon and starlight to see the faces of the dead. She didn't have a weapon. Maybe she never did, or threw hers away or lost it. She didn't speak to us, so we passed her and kept hiking. I wondered which side of the fight she had been on, then decided that really didn't matter.\n\nIt was a little after midnight when we got to the truck. The keys were dangling from their slot. Is this a great country or what? All four tires had air. The windshield had taken at least three bullets and was in bad shape. One of the bullets had gone through the windshield and out the rear window. Fortunately Sarah had been lying on the seat at the time, protected by the motor and lots of metal, so she wasn't tagged. One of the truck's headlights was shot out. Some of the sheet metal had holes or gouge marks from bullets, and the radio aerial was missing, shot off. I opened the hood and examined the radiator and hoses with the flashlight. No visible leaks. Maybe the antifreeze all ran out. I looked at the ground under the engine, which was dry. We were good to go.\n\nAbout a hundred yards to the south was an army truck with every light on. I walked over and saw a white cross painted on the side. Dr. Proudfoot was there, and he said the medics were out looking for wounded.\n\n\"We found some guy who had been scalped,\" he said. \"Hell of a wound. He's a professor from some little college in New England. I sedated him.\"\n\n\"Is he going to make it?\"\n\n\"Probably, if infections don't kill him.\"\n\nI shook Proudfoot's hand and walked back to my stolen FEMA truck. Sarah was already in the passenger seat, buckled up.\n\n\"Idaho,\" Sarah said.\n\n\"Idaho,\" I agreed.\n\nI fired up the motor. The lone headlight bravely stabbed the darkness.\nTHIRTY-SIX\n\nWe spent what was left of the night at Camp Dawson, which was manned by a skeleton crew of guardsmen. I gave them the machine gun and extra ammo and three AT4s that Willis hadn't managed to shoot. After lunch, we hit the road.\n\nIn a little town in Ohio I found a glass repair shop that was open. They replaced the windshield, rear window, and headlight. The head man wanted to talk, so I told him about the battle for Camp David.\n\nWhen I finished he said, \"I have been really worried about America for years, and martial law was my worst nightmare come true. I think the socialists and left-wing radicals want to change America into a nation my kids won't want to live in. It seems like they don't know the basics of economics, don't believe in work, don't believe that a person should earn and keep the fruits of their labor. They'll run America into the ground, then what?\"\n\n\"Maybe now the future will be better,\" Sarah said.\n\n\"Then there is terrorism, all those Muslims admitted willy nilly,\" he said. \"I can only hope and pray.\"\n\nThe power was back on in Ohio and Indiana, so we spent a night in a chain motel that was open. We ate a free breakfast at the bar off the lobby, which consisted of cornflakes and milk. I asked about the milk, and was told cows keep giving it regardless.\n\nFilling stations were open again, and before the tank in the truck was empty, we found one with fuel to pump. Life was looking up.\n\nIn Illinois a state trooper took offense because I was driving at eighty miles an hour when the speed limit was sixty-five. He pulled us over.\n\n\"I told you to slow down,\" Sarah said primly as the trooper walked up.\n\n\"You with the government?\" he asked, looking us over. The pickup had federal government plates, although it lacked logos on the doors. Sarah and I were still wearing our web belts and pistols. The trooper was a big black man with hair going gray at the tips. For a man who spent most of his working life sitting behind a wheel, he was reasonably trim and fit.\n\n\"Ah, no,\" I admitted. \"We quit. We were with the CIA.\"\n\n\"Spies, huh?\"\n\n\"I stole the truck,\" I said brightly, \"from FEMA.\"\n\n\"Those assholes? No shit! You got ID?\"\n\nI dug out my wallet and passed him my CIA Langley pass.\n\nHe looked it over and passed it back. \"What you got in the cooler in the bed?\"\n\n\"A six-pack. Filling station back in Indiana had some. Want one?\"\n\n\"Man, I haven't had a beer since Soetoro declared martial law. Yeah, I'd like one.\"\n\nWe got out and opened the cooler, and all three of us took a beer.\n\n\"If you have a camera in your cruiser, they might get unhappy seeing you with a beer,\" I said.\n\n\"Camera's broken. Piss on 'em.\" He popped the top on his can and took a swig. \"Ahh! Tell me about the bullet holes in your ride.\"\n\nSo we sat on the tailgate of the truck and sipped beer while I told him about the attack on Camp David. As I talked and he asked questions of Sarah and me, he visibly relaxed. He believed us. If he only knew how good a liar I was, he would have been more suspicious, but ignorance is bliss, so they say. And for a change I stuck strictly to the truth.\n\nWhen he finished asking questions about the death of Barry Soetoro, the trooper, whose name was Davis, waxed philosophical. \"Soetoro made life a living hell for us cops, made us targets, turned people against us, and stirred up racial hatred we sure as hell didn't need. Sure, there are a few bad cops, the same as there are bad dentists, doctors, CEOs, and plumbers, but all these body cameras and shit, and the constant second-guessing of cops who put their lives on the line\u2014that's bullshit. That bastard Soetoro killed a lot of people by making criminals feel free, taking their side, and giving carte blanche to illegal aliens with criminal records. He destroyed a lot of trust, especially with law enforcement. And you know, without the rule of law, we don't have a civilization. It's that simple.\"\n\nI'd seen enough to know that.\n\nHe stood and dusted off his trouser seat. \"You two slow it down, huh?\"\n\n\"Yes, sir.\"\n\nDavis got into his cruiser and drove away. We put all three empties in the cooler and got our chariot under way, heading west.\n\nI didn't want to read newspapers or watch television or listen to radio. I had had enough of the world's troubles. Sarah and I chatted and watched the countryside pass by and the road unwind endlessly before us. Although traffic was light, things were getting back to normal. We saw tanker trucks sitting in filling stations, food trucks rolling the highways, trucks hauling cattle and hay, and farmers in the fields running combines. Trains went by on tracks that paralleled the highway. Here and there construction crews were back at work on road and bridge projects. Jets were flying again, so contrails streaked the blue sky.\n\nYet even political hermits like Sarah and me found the political crisis impossible to avoid. Every diner or bar we went into had televisions going full blast. The generals in the Pentagon had asked Jake Grafton to get an interim civilian government up and running and to hold elections in every state that wanted to remain in the old Union. Texas was independent and intended to stay that way, President Jack Hays said. The commentators were still aghast, and delighted, at the effrontery of the Texas military in stealing\u2014or \"replevying,\" Jack Hays' word\u2014fifty tons of gold. At the quoted market price that morning\u2014$2,132 an ounce\u2014the metal was worth $3.4 billion. Jack Hays assured an interviewer that Texas would return any excess after Texas' claims against the federal government were settled by negotiation.\n\nIn California, the Mexican Army had been driven out, but Mexican gangs and their radical supporters were now engaged in a civil war against everyone else. They had supported the Mexican Army, and now were fighting for an independent Mexican Southern California they planned to call Aztlan. They were being crushed, but Southern California, and Los Angeles in particular, would never be the same again. Television cameras lingered lovingly on columns of smoke rising over the LA basin.\n\nIn Mexico, another civil war had broken out. The reasons seemed to be manifold: the flood of illegals back to Mexico, Texas closing the border, the failed invasion of California (some said at the behest of the drug lords), and massive unemployment. The good news was that without the United States as a safety valve, Mexico was finally going to have to come to grips with poverty, monopoly, corruption, and lack of opportunity for most of the people who lived within its borders.\n\nThe violent death of Barry Soetoro had, as Jake Grafton feared, transformed him into a cultural icon among certain groups. His sins were forgotten in the pathos of his demise. Bogus eyewitness accounts aired between newscasts. Mickey Soetoro publicly and loudly blamed \"white people.\" A waitress at a truck stop told us that Oprah was in tears for her entire show. All this despite the fact that the conversations Sarah captured in the White House in which Soetoro plotted to become a dictator were still airing on some radio stations.\n\nWe had been on the road for four days when we rolled into Idaho. We examined the brochures at a visitor's center and signed up for a float trip down the Salmon, the River of No Return. That took six wonderful days under a September sky. The nights were spent camping on a beach, and the days riding the river with a guide who paddled occasionally while Sarah and I fished the riffles and rapids for steelhead going upriver to spawn. We actually caught several good ones, which we immediately released back into the river.\n\nThe whole experience was magical. The canyon was wild and glorious, the eternal river flowing through rapids and down long, languid stretches, then through more rapids. We saw mule deer and coveys of chukar. Eventually we ended up on the Snake and spent a day drifting with the current to the pullout. People along the banks of the Snake on farms and in yards waved to us.\n\nSarah and I were laughing and smiling when the experience was over. America was still here, still glorious.\n\nAfter another week of driving through the mountains, we ended up in Idaho Falls. That evening we finally turned on the television to a news channel and began catching up.\n\nA constitutional convention had been announced. Jake Grafton was on television with the leaders of the House and Senate asking the governors of states both in and out of the Union to send delegates. He finished with this statement: \"I think a great many people feel that the constitutional mandate for separation of powers between the three branches of government, and between the states and the federal government, got badly warped through the years. We hope a convention can fix that, especially by putting more teeth into the Tenth Amendment.\"\n\nGrafton continued, \"The judges decided the interstate commerce, due process, and some other clauses were loopholes big enough to swallow the states and give the federal government control of every aspect of American life. That control was not exercised by Congress, an institution totally inadequate for the task, but by unelected, unaccountable bureaucrats, sometimes controlled by the executive but often controlled by no one at all. That has to change. I don't know what devices the convention delegates will come up with to harness the Cheshire Cat, but they can try or fail or surrender, as they choose.\n\n\"The delegates may also choose to revise our democratic institutions to make them more efficient and responsive to the electorate.\n\n\"What is not on the table are the basic civil rights we Americans as a free people enjoy. We are seeking new ways to preserve those rights, not diminish them.\n\n\"If the delegations do their jobs well, we will have added safeguards to preserve liberty, the rights of the states, and the freedom of the people. It is my hope that the states that have declared their independence will return to the family of states that we call the United States, a family that has provided shelter and livelihoods for a free people for over two centuries, and I believe, with tweaking, can shelter us and our descendants for many more.\n\n\"May God bless a restored and reunited America.\"\n\nAfter the speech a commentator appeared on camera. I stared. Yes, it was Jack Yocke, clean-shaven, with a haircut, wearing a suit and tie. He was now the network's expert on all things Grafton.\n\nJack Hays was next.\n\n\"Texas is getting its act together,\" he said. \"We are in talks with Oklahoma, New Mexico, and Arizona to form some kind of federation. How that will work out, I don't know, but I am encouraged. The illegals who don't speak English and have no job skills are going back to Mexico; we have about two thousand families a day moving to Texas to find jobs, families that do speak English and have trades and job skills to support themselves and make positive contributions to the economy and tax base. We are reforming the education system, training Americans, and putting them to work. Texas has a bright future.\"\n\nWhen we turned off the television in the wee hours of the morning, Sarah asked, \"So what are we going to do with our lives?\"\n\n\"I don't know,\" I replied, truthfully.\n\n\"We can't keep doing nothing.\"\n\n\"I know.\"\n\n\"I want to go home,\" she said.\n\nThe following day we pointed the truck east. The highways were more crowded, almost back to normal, I thought, and every filling station and truck stop had gas and lots of customers.\n\nFour days later we rolled into West Virginia and stopped by the safe house near Greenbank. Dr. Proudfoot was there making a house call. Mrs. Price sat on the porch with a jacket around her shoulders and a blanket over her legs enjoying the fall colors, which I thought were near their peak. Little Sarah threw herself at Big Sarah, and Armanti Hall shook my hand until I had to jerk my appendage out to save it.\n\n\"I thought you were boogying off to Texas,\" I said, flexing my fingers.\n\n\"Gonna stay here and rebuild Mrs. Price's house. Then the three of us are going to live in it.\"\n\n\"Got enough money for lumber, toilets, and pipes?\"\n\n\"I have a little saved up,\" he said, looking down his nose at me. \"Need a loan?\"\n\n\"Ah, right now, no. But if in the uncertain and unpredictable future I unexpectedly find myself in a fiscal hole, I know where to find you.\"\n\n\"Right up the road. We should be in by spring.\"\n\n\"It's great to have friends.\"\n\n\"So they say.\"\n\nWe drove on to Washington and stopped in front of the lock shop. We went in, and there sat Willie the Wire Varner.\n\n\"Where the hell you been?\" he demanded. \"I thought you two were dead.\"\n\n\"Still kicking,\" I said. \"What happened to you after the battle of Kingwood?\"\n\nHe said he had hitchhiked back to Washington. \"I'm no warrior,\" he declared defensively. \"Ain't got it in me.\"\n\nThis was the Willie Varner I knew and liked.\n\nWe were catching up, telling him of our adventures and listening to him describe his odyssey back to Washington, when my cell phone rang.\n\nI looked at the caller ID\u2014Jake Grafton\u2014and answered it. \"Hey.\"\n\n\"Tommy, where are you?\"\n\n\"In Washington.\"\n\n\"Good. Come see me tomorrow. I need you.\"\n\n\"See you where?\"\n\n\"Callie and I are bunking at the White House temporarily.\"\n\n\"Okay.\"\n\n\"I want you to go to Europe. Some of the Middle East refugees flooding in there turned out to be jihadists, which seemed to surprise the Europeans. Maybe you can help keep us advised of what's going on.\"\n\n\"See you tomorrow.\"\n\nI hung up.\n\nSarah looked at me and raised her eyebrows.\n\n\"Jake Grafton,\" I said. \"He wants me to go to Europe.\"\n\n\"It's about time,\" she said, and smiled.\n\n**Chapter Two**\n\n**I** used the secure satellite communication system in the SCIF at the embassy to call Sarah Houston in Virginia. She answered warmly. After a few delightful boy-girl moments on Uncle Sam's dime, I got down to it.\n\nI gave her the information on the passport and from the documents Armanti Hall had stolen from the watcher. \"Anything you can tell me.\"\n\n\"When do you need this?\"\n\n\"We've been up all night doing that branch bank. I'm going to bed. By the way, are you getting anything from their server?\"\n\n\"Oh, yes. We've been working it for a couple of hours. A river of money flows through that bank, mostly from Russia. It doesn't stay long. Then it flows all over the world. About half goes to South America and a big chunk comes to the States. Another nice chunk goes to the UK. We'll be trying to figure out where it comes from and where it goes.\"\n\nI grunted, trying to fit these revelations into what I knew of the world of finance and money laundering. We said our good-byes and I walked the old streets to my hotel.\n\nI kept thinking about the kidnapped little girl, wondering where she was, why she was snatched. Since there had been no ransom note, I wasn't optimistic.\n\nI examined the pistol Armanti had taken from the guy outside the bank. A Walther in 9 mm, loaded. I took out the magazine, thumbed out the cartridges, cocked the pistol and dry fired it, then wiped off each cartridge with a hanky and reloaded the magazine. Kidnapping, loaded guns, a billion dollars a week through a branch bank in Estonia... I pocketed the piece and decided it was time for food and bed.\n\nThe city was alive, bustling, a tourist magnet. My hotel was on the edge of the Old Town. I ate a continental breakfast while watching my fellow diners. I noted two men, sitting apart but both in their thirties, dressed in similar business attire. I committed their features to memory, ensuring I would recognize them if I saw them again. Then I went upstairs and crashed.\n\nTwo hours later, I was wide awake. Got to thinking about the kid, Audra, and about the interview with the father, Frank Rogers, and how he had suppressed his emotion. He obviously knew something.\n\nBut why had there been no ransom note? That had me stumped. Only perverts grabbed kids without ransom notes. Kidnapping is a business. It's done for money. Or revenge. Or something.\n\nI kept coming back to the bank. A river of money. There was the cash, right there, Tommy, you twit.\n\nThat interview with Frank Rogers. Had he been lying? Fact is, I am a professional liar. I know all the tells. Recalling Frank Rogers' face, I suspected that the emotion had covered up some of the tells. On purpose. Perhaps.\n\nMaybe I should find out.\n\nI gave the watcher's documents, wallet, and cell phone to the station chief at the embassy to put in the diplomatic bag for delivery to the company in Langley. I kept the pistol. Dulcie Del Rio wanted a debrief, so I told her how it went down.\n\nShe said, \"Isn't it an interesting coincidence that they tried to get into the bank the same night you did?\"\n\nI didn't believe in coincidences. Oh sure, random chance rules the world, but always bet on cause and effect. A little paranoia will take you a long way... and keep you alive.\n\nThe Rogers lived in a two-bedroom flat on the second story of a five-story building in a new section of town. At least the plumbing was modern, and they had electricity. The place had no doorman and a selfservice elevator. No security cameras. With Armanti Hall and Joe Kitty standing guard front and back, I went in during the afternoon and took a look around.\n\nThe first things I found were wireless electronic bugs. The tiny microphones were transmitting, according to my hand-held receiver. Uh-oh.\n\nI went out twice as fast as I came in and started searching for the booster. There had to be one someplace nearby, but I didn't see it.\n\nThe three of us held a huddle by the embassy car and I told Joe and Armanti about the bugs.\n\nArmanti whistled softly. \"Oh, man.\"\n\n\"It isn't just the snatched kid,\" Joe Kitty said. \"These people are in trouble to their eyes.\"\n\n\"And they've been lying to State and to me,\" I said. \"Let's get the dad first when he gets out of school, and intercept the mom coming home from work. Then we can have a quiet little prayer meeting, just the five of us.\"\n\n\"Not together,\" Joe Kitty said.\n\n\"Oh, no,\" I agreed. \"One at a time.\"\n\nWe went back to the embassy to check out another car, and while we were there I again visited with Dulcie Del Rio. \"I need three guns and shoulder holsters.\"\n\n\"Who is the opposition?\" she asked flatly, just like that.\n\n\"Don't know yet, but I kinda doubt it's Estonians.\"\n\n\"Check their identity papers before you shoot them.\"\n\n\"Of course. And we need a safe house.\"\n\n\"For Christ's sake, this is _Estonia_.\"\n\n\"So I'm told, but we need a secure place to have some interviews.\"\n\nShe didn't ask who we planned to interview, which was a credit to her. She dug in her purse and gave me the key to her flat. \"Check for bugs, don't mess the place up, stay out of the liquor, and don't let the cat out. Her name is Oreo.\"\n\n\"We may need it all evening.\"\n\n\"I'll spend the night with a friend. Feed the cat.\"\n\n\"Thanks,\" I said.\n\nWe ended up with old Baretta Nines. The U.S. government bought them by the millions. I gave Dulcie my liberated Walther for her collection. We loaded the Barettas, put them in holsters and went to get the other car.\n\nI sent Bill Leitz to check Dulcie's apartment for bugs. As a tech support guy, this was his area of expertise. \"We'll meet you there,\" I told him, and repeated Dulcie's instructions about the liquor and the cat. \"Make sure you aren't followed,\" I added, \"and that there aren't any eyes on the place.\"\n\n\"Got it,\" he said, took the keys, and went back into the embassy to get his gear.\n\nI was waiting beside one of the cars when Frank Rogers came walking along. He was going home. This was probably near the place where someone snatched his daughter.\n\nI was friendly. \"Mr. Rogers, Jim Wilson. We met the other day.\"\n\nHe looked me over, obviously was not glad to see me. He edged around as if he wanted to walk on.\n\n\"I was hoping that you and I could spend another few minutes together.\" I opened the car door and held it.\n\n\"I don't think\u2014\" he began.\n\n\"You haven't been thinking\u2014that's the problem. Now if you want to ever see Audra again, get in the car.\"\n\nHe got in the passenger seat. I went around and settled myself behind the wheel, snapped my seatbelt. \"Put on your seat belt,\" I told him as I inserted the key in the ignition and fired up the tiny motor.\n\n\"What\u2014\"\n\n\"Save it.\"\n\nI put the car in motion and pulled into traffic. I was paying attention to the mirrors and saw another car pull out as we passed. I made the second turn and it followed. Terrific.\n\nTwo turns later and I was sure: we had a follower. I stopped at a light and checked the rearview mirror. He had hung back, but I could see that there were two guys in the car and one was on his cell phone. When the light changed I headed for Old Town, with its traffic and narrow streets. We crossed a few bridges, then we were there. I didn't know the street pattern, but apparently neither did they, because they followed me into a narrow street where only one of us could turn around\u2014and that was me\u2014before getting wedged in by traffic.\n\nI worked the car around as the guy behind me laid on his horn, then drove back the other way. As I went by our pursuers, I looked them over; they didn't look at me. The passenger was glued to his phone and the driver stared straight ahead. But there was a car in front of him and a delivery truck behind, and it was going to take him a while to get turned around. Tough luck for him.\n\nOut on the boulevard, I headed out of the district. Four turns later, I was sure I was clean. I used the map feature on my phone to direct me to Dulcie Del Rio's apartment.\n\nFrank Rogers started talking as we left Old Town. He was chattering as we crossed the bridge and chattering as we entered Dulcie's neighborhood.\n\n\"I've told you everything I know. Told the people at the embassy. We're Americans... for God's sake, you can't treat us like this.\" There was more, a lot more, about his wife knowing a senator and the fact they had written their Congresswoman.\n\n\"Can it,\" I said and concentrated on my cell phone.\n\nAt one stoplight, he popped the buckle on his seat belt and reached for the door handle. I backhanded him gently across the chops. \"Put the damn belt back on,\" I said harshly. If I had had any doubts that he had lied in our previous interview, they would have evaporated then. But I didn't. He was frightened. Scared. Whatever he and his wife had gotten themselves into, he knew the stakes were blood. His, his wife's, and his daughter's. He knew it and now I did.\n\nLeitz let us into Dulcie's pad. He handed me the keys, glanced at Rogers, and walked out, pulling the door shut behind him.\n\nI took Rogers by the elbow and guided him into the little living room. \"Sit.\" He did, in a little couch just big enough for two bottoms.\n\nI checked out the apartment. Two bedrooms, one bath, a kitchen with a dining area that doubled for a home office. As if Dulcie Del Rio had stuff she could work on at home. She had a little laptop, however, probably to send emails to the family back in the States. Maybe check on how the investments in her IRA were doing. I hoped she was getting rich.\n\nI opened the refrigerator. Dulcie had beer.\n\n\"You want a beer?\" I asked Frank Rogers. He shook his head.\n\nI sat down opposite him in a stuffed chair and made myself comfortable. Dulcie's cat climbed up on the couch beside Frank and presented itself to be petted. He stroked it once, mechanically, and eventually it wandered away.\n\n\"Frank, we have a problem. You've been lying to the folks at the embassy, and perhaps the local police. You haven't given us a damned thing that will get us to Audra. How are you going to feel if Audra shows up dead? If the local cops find her corpse in some alley? Knowing that you told a bunch of lies when the truth might have saved her, knowing you didn't tell all you know? How are you going to feel?\"\n\nBig tears leaked from both eyes. He began sobbing and finally buried his face in his hands. I felt like a jerk. Still, if he ever wanted to see his kid again, we were his main chance. In fact, probably his only chance.\n\n\"I've never had to deal with anything like this. I don't know what to do.\"\n\n\"You _have_ heard from the kidnappers, haven't you? What did they say?\"\n\n\"That if we made any more noise about that bank they'd kill her. Told us to shut up and Penny was to keep doing her job.\"\n\n\"Un-huh.\"\n\n\"It was those letters to the big honchos in Stockholm that Penny wrote. The bank is making a lot of money from the deposits and withdrawals flowing though the bank. The fees are a fraction of one percent, but the amounts are so large...\"\n\n\"How large?\" I instantly regretted that question. His wife knew, but Frank would only know what Penny told him. Hearsay. And Sarah Houston was going to have it chapter and verse very soon.\n\n\"Some days it's more, some less. Never less than a hundred million U.S. dollars. Sometimes three or four times that.\"\n\n\"A day?\" I was incredulous. It was as if the bank were a depository of the Treasury Department, collecting tax checks after the 15th of April.\n\nHe nodded. After a bit he stopped sobbing. I went to find Dulcie's liquor cabinet. She liked vodka and bourbon. I poured some vodka on the rocks and brought him a tumbler full.\n\n\"Let's go back to after Audra was snatched. Did they call you?\"\n\n\"They called my wife. Penny told me what they said.\"\n\n\"Tell me.\"\n\nHe did. Going over it took half an hour. The threat was the kid would get killed or maimed if Penny Rogers didn't shut up and behave. If she did, the kid would come home alive and well. If she didn't, they would start mailing her parts.\n\nHe had finished the vodka and I thought I knew most of it when I heard a knock on the door. I went and checked with my gun in my hand. It was Joe Kitty and Armanti Hall, and they had Penny Rogers with them.\n\n\"We were followed,\" Armanti said. \"We managed to ditch them.\"\n\nPenny Rogers was a sight. She was not a large woman, never pretty, and the last few days, or perhaps the ride with Armanti and Joe as they ditched her tail, had wrung her out.\n\n\"She didn't say a word,\" Joe Kitty said. He went outside to keep an eye on the cars.\n\nI nodded towards Frank and whispered to Armanti, \"Take him into the bedroom and keep him quiet.\"\n\nPenny watched them go in silence, then sat on the little couch.\n\n\"A glass of wine, or perhaps water?\" I suggested.\n\nShe thought wine, so I poured her a glass from a bottle of red that Dulcie had corked in a cabinet, then sat down across from her.\n\n\"Who are you men?\" She asked.\n\nI showed her my fake passport and State ID. She scrutinized the documents, then handed them back.\n\n\"Frank said you talked to him at school.\"\n\n\"I did. And again this afternoon, right here.\"\n\n\"Whose apartment is this?\"\n\n\"A friend's.\"\n\n\"Why here? Why not the embassy or our place?\"\n\n\"Your apartment is bugged and I didn't want anyone to see you entering the embassy.\"\n\nShe stared at me. \"How do you know our apartment is bugged.\"\n\n\"We went in and looked.\"\n\n\"The door was locked.\"\n\n\"We picked the lock, Mrs. Rogers. Just like the men did who bugged the place.\"\n\nShe sipped at the wine, remained silent, and looked around. The cat sniffed her feet, then wandered off. It was black, with one splotch of white on its face. Oreo. Looked like a nice cat.\n\n\"They'll kill Audra if we talk.\"\n\n\"Not if we get to her first. And there is no way on earth for that to happen unless you tell us the truth.\"\n\n\"She doesn't deserve this.\"\n\nShe lost control of her face. The refrigerator hummed and somewhere in the building a door slammed. I could faintly hear traffic down on the street below the windows. She wiped at her eyes, and finally her eyes dried up.\n\nPenny Rogers began talking. All that money flowing through that tiny little branch bank aroused her curiosity. Her employees, all Estonians who had worked there for years, told her the branch had always been profitable. More than ninety percent of the accounts were Russian, and they were always opened over the internet. The only people who came into the branch were locals. Their accounts were small and legitimate, with paychecks deposited into savings or checking accounts. But the Russians...\n\nBig deposits, in the tens of millions. Day in and day out. The money didn't stay long, but was wire transferred on to America, the UK, South America, southern Europe. All over the world.\n\n\"How much money?\"\n\n\"It varies. The deposits have averaged a billion a week during the last year.\"\n\n\"Fifty billion dollars in the past year?\"\n\n\"Yes.\"\n\nShe finished her glass of wine and put it on the little table to her right.\n\n\"They're laundering money,\" I suggested.\n\n\"Obviously,\" Penny Rogers said, as if I needed a dozen more IQ points. \"All the businesses in Russia don't generate fifty billion dollars of profit in a year.\"\n\n\"So what did you do?\"\n\nShe began to talk. As she spoke the words poured out, faster and faster. She tried to check on the accounts where the money was being transferred. Used the internet to track down the accounts, which were all corporations. She finally realized they were all shell corporations, without assets. She wrote emails and memos to her superiors in Stockholm. She was told to remember the bank's business was the bank's business, and not to discuss it with anyone. Her emails became more strident. Then Audra was kidnapped.\n\n\"And someone came to talk with you...\" I suggested.\n\nShe nodded. \"At the bank. It was a man I didn't know. He was blunt. Told me to keep my mouth shut and run the branch. If I discussed the bank's business with anyone we would never see Audra again. In one piece. He said that. ' _In one piece_.' I told him we had already reported the kidnapping to the police. He sneered at that.\"\n\n\"Tell me about this man.\"\n\n\"In his forties, I thought. Spoke with a heavy accent. Not Swedish. Russian, perhaps. Terrible teeth, stained yellow from cigarettes.\" She made a face. \"A horrible man. Evil. Pure, unadulterated evil.\"\n\n\"Did he demand money, anything like that?\"\n\n\"No.\"\n\n\"Did you ask for proof that they had Audra, that she was still alive?\"\n\nShe nodded and sniffled. Dug in her purse. Produced a photo. A snapshot. It was Audra all right, against a white wall, with a welt across her face. No glasses.\n\n\"May I keep this?\"\n\nShe said yes, so I put it in my inside jacket pocket.\n\n\"She's all we have,\" Penny said, so softly I almost missed the comment.\n\nWhat do you say to that? I couldn't think of anything.\n\n\"How are you going to get her back?\"\n\nThat was an excellent question. \"Who did you write to at the bank in Stockholm?\"\n\n\"I sent emails to Arne Soderman, who supervises the branches, and then finally a letter to the president, Isak Dahlberg. Three letters, actually. A scandal like this could wreck the bank.\"\n\n\"We'll talk to them,\" I said.\n\nI went to the bedroom door and told Armanti and Frank to come out. I said to Armanti, \"You and Joe take the cars back to the embassy. Get some new wheels, rental cars. Then come back here.\"\n\nArmanti left. The Rogers sat beside each other on the couch and held hands.\n\nJake Grafton came by invitation to Sarah Houston's office in Langley. When he was seated with the door shut, she turned a computer monitor so he could see it. \"The branch bank server gave us the encryption codes. This is what we have so far from that branch. Money in, money out. Euros, dollars, pounds, yen, Swiss francs, you name it. It'll take a few more days to get into the main servers in Stockholm.\"\n\nAs Grafton scrolled through the transactions, Sarah said, \"I didn't believe it, but the size of these transactions is awe-inspiring. It could easily go a billion dollars' worth a week. It's money-laundering on a massive scale. They must be washing money for Iran, North Korea, Syria, and every drug syndicate on the planet.\"\n\n\"Get all the information you can as fast as you can. I'll have a department head meeting and get you some help. Someone at the bank knows there is a leak or Audra Rogers wouldn't have been kidnapped. This river of money is going to stop flowing soon. We want to know where it comes from before it goes into the bank and where it goes from there. Let's find out before the river stops flowing.\"\n\nThey discussed the logistics of the operation. \"If I can get into the bank's main computers, the sheer volume of data could take man-years to unravel,\" Sarah noted. \"And we are just seeing a tiny piece of the operation.\"\n\nGrafton didn't look up from the screen. \"We need to see the entire watershed.\"\n\nSarah said, almost as an afterthought, \"We happen to have a full plate already. I don't think you understand how labor intensive it will be to dig into records at other banks, assuming we can get in by hook or crook or even\u2014God forbid\u2014invitation. It will be like trying to sort plates of spaghetti. The whole reason washers transfer money hither and yon is to make it difficult to trace. Sometimes tracing becomes impossible.\"\n\n\"This is priority One,\" Jake Grafton shot back. \"We'll get the people and equipment you need as soon as we can. We don't have man-years.\"\n\n\"Okay.\"\n\n\"I need your help to figure this out, Sarah.\"\n\nAfter Grafton left, Sarah made a face at the door, then began drafting a memo for him to sign.\n\nBack in his office, Jake Grafton ignored the pile of paper in his in-basket. He looked at the paintings, at the flags, thought about the river of money. And he thought about Audra Rogers.\n\nHe was sitting at his desk doodling on a pad when the phone rang. The receptionist. \"Tommy Camellini on the secure link from Estonia.\"\n\n\"I'll take that.\"\n\nTwo button pushes later he heard Tommy's voice.\n\nAfter Tommy had brought the admiral up to date, he continued, \"Seems to me, boss, that the first priority is Audra Rogers. Truth is, she may be dead, and the trail is what, thirteen days old now? To get to her, we're going to have to go to Stockholm and sweat the people at the bank. One or more of them is dirty. They won't want to talk to us and we don't have the horsepower to put serious pressure on unless you're willing to create an international incident with an American friend.\"\n\nJake Grafton took a deep breath and exhaled slowly. He too thought Audra was probably dead. Why would the kidnappers keep her alive? \"You need to get that branch server. Go in and get it before the bad guys do.\"\n\n\"I've got a guy watching the bank to make sure it doesn't walk out the front door,\" Tommy said. \"We'll snatch it ASAP.\"\n\n\"What time is it there?\"\n\n\"Ten p.m.\"\n\nGrafton looked at the clock on the wall. Seven hours difference. \"Get it tonight.\"\n\n\"Yes, sir.\"\n\nGrafton hung up and pushed the intercom, summoning his executive assistants.\n\n\"I want to go to Stockholm as fast as possible,\" he told them. \"Get me a plane, an executive jet out of Andrews. And I need more people.\" He named them. \"They can come with me. Make it happen.\"\n\nTable of Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Dedication\n  5. Contents\n  6. Prologue\n  7. Chapter One\n  8. Chapter Two\n  9. Chapter Three\n  10. Chapter Four\n  11. Chapter Five\n  12. Chapter Six\n  13. Chapter Seven\n  14. Chapter Eight\n  15. Chapter Nine\n  16. Chapter Ten\n  17. Chapter Eleven\n  18. Chapter Twelve\n  19. Chapter Thirteen\n  20. Chapter Fourteen\n  21. Chapter Fifteen\n  22. Chapter Sixteen\n  23. Chapter Seventeen\n  24. Chapter Eighteen\n  25. Chapter Nineteen\n  26. Chapter Twenty\n  27. Chapter Twenty-One\n  28. Chapter Twenty-Two\n  29. Chapter Twenty-Three\n  30. Chapter Twenty-Four\n  31. Chapter Twenty-Five\n  32. Chapter Twenty-Six\n  33. Chapter Twenty-Seven\n  34. Chapter Twenty-Eight\n  35. Chapter Twenty-Nine\n  36. Chapter Thirty\n  37. Chapter Thirty-One\n  38. Chapter Thirty-Two\n  39. Chapter Thirty-Three\n  40. Chapter Thirty-Four\n  41. Chapter Thirty-Five\n  42. Chapter Thirty-Six\n  43. Russia Account - Demo chapter 2\n\n# Guide\n\n  1. Cover\n  2. Contents\n  3. Title Page\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. v\n  6. vi\n  7. vii\n  8. ix\n  9. x\n  10. xi\n  11. xii\n  12. xiii\n  13. xiv\n  14. xv\n  15. xvi\n  16. xvii\n  17. xviii\n  18. xix\n  19. xx\n  20. xxi\n  21. xxii\n  22. xxiii\n  23. xxiv\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  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328. \n  329. \n  330. \n  331. \n  332. \n  333. \n  334. \n  335. \n  336. \n  337. \n  338. \n  339. \n  340. \n  341. \n  342. \n  343. \n  344. \n  345. \n  346. \n  347. \n  348. \n  349. \n  350. \n  351. \n  352. \n  353. \n  354. \n  355. \n  356. \n  357. \n  358. \n  359. \n  360. \n  361. \n  362. \n  363. \n  364. \n  365. \n  366. \n  367. \n  368. \n  369. \n  370. \n  371. \n  372. \n  373. \n  374. \n  375. \n  376. \n  377. \n  378. \n  379. \n  380. \n  381. \n  382. \n  383. \n  384. \n  385. \n  386. \n  387. \n  388. \n  389. \n  390. \n  391. \n  392. \n  393. \n  394. \n  395. \n  396. \n  397. \n  398. \n  399. \n  400. \n  401. \n  402. \n  403. \n  404. \n  405. \n  406. \n  407. \n  408. \n  409. \n  410. \n  411. \n  412. \n  413. \n  414. \n  415. \n  416. \n  417. \n  418. \n  419. \n  420. \n  421. \n  422. \n  423. \n  424. \n  425. \n  426. \n  427. \n  428. \n  429. \n  430. \n  431. \n  432. \n  433. \n  434. \n  435. \n  436. \n  437. \n  438. \n  439. \n  440. \n  441. \n  442. \n  443. \n  444. \n  445. \n  446. \n  447. \n  448. \n  449. \n  450. \n  451. \n  452. \n  453. \n  454. \n  455. \n  456. \n  457. \n  458. \n  459. \n  460. \n  461. \n  462. \n  463. \n  464. \n  465. \n  466. \n  467. \n  468. \n  469. \n  470. \n  471. \n  472. \n  473. \n  474. \n  475. \n  476. \n  477. \n  478. \n  479. \n  480. \n  481. \n  482. \n  483. \n  484. \n  485. \n  486. \n  487. \n  488. \n  489. \n  490. \n  491. \n  492. \n  493. \n  494. \n  495. \n  496. \n  497. \n  498. \n  499. \n  500. \n  501. \n  502. \n  503. \n  504. \n  505. \n  506. \n  507. \n  508. \n  509. \n  510. \n  511. \n  512. \n  513. \n  514. \n  515. \n  516. \n  517. \n  518. \n  519. \n  520. \n  521. \n  522. \n  523. \n  524. \n  525. \n  526. \n  527. \n  528. \n  529. \n  530. \n  531. \n  532. \n  533. \n  534. \n  535. \n  536. \n  537. \n  538. \n  539. \n  540. \n  541. \n  542. \n  543. \n  544. \n  545. \n  546. \n  547.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**zest **\n\nCatherine Saxelby & Jennene Plummer\n\n**More than 120 recipes for vitality and good health**\n\nPublished in 2007  \nby Hardie Grant Books  \n85 High Street  \nPrahran,Victoria 3181, Australia  \nwww.hardiegrant.com.au\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers and copyright holders.\n\nThe moral rights of the authors have been asserted\n\nCopyright Introduction \u00a9 Catherine Saxelby  \nCopyright Recipes \u00a9 Jennene Plummer  \nCopyright photography \u00a9 Ian Hofstetter\n\nNational Library of Australia Cataloguing-in-Publication Data: Saxelby, Catherine.   \nZest: the Nutrition for Life cookbook \u2013 more than 120 recipes for vitality and good health.\n\n1st ed.  \nISBN 9781740664790 (pbk.).\n\nISBN 1 74066 479 5 (pbk.).\n\n1. Cookery. 2. Functional foods. 3. Nutrition.   \nI. Plummer, Jennene. II. Title.\n\n641.563\n\nConsultant editor: Philippa Sandall  \nEditor: Lucy Malouf  \nStylist: Jane Collins  \nFood preparation: Mandy Sinclair  \nCover and text photography: Ian Hofstetter  \nCover design, text design and typesetting: saso content & design pty ltd  \nDiagram: Clare Forte  \nPrinted and bound in China by SNP Leefung\n\n10 9 8 7 6 5 4 3 2 1\n\n**www.foodwatch.com.au**\nFor our children Guy and Georgia, Marcus and Emma, Richard and Emma \u2013 the next generation, for a lifetime of good health \nAcknowledgements\n\nThere are always many people who make a book come together and this book is no exception.\n\nA big thank-you to Julie Pinkham, Fran Berry, Mary Small and Catherine Cradwick from Hardie Grant Books for their vision and enthusiasm in bringing _Zest_ to life.\n\nOur consultant editor Philippa Sandall has been a wonderful guiding force and helped us through the critical planning and editing stages.\n\nCatherine would like to thank dietitian and colleague Karen Kingham, for her invaluable work on analysing the recipes, and dietitian Anne Gregory, who helped with research and checking.\n\nJennene would like to thank Mandy Sinclair and Sharon Reeve for their support and contribution in testing some of the recipes in this book. She would also like to thank her wonderful husband and family for allowing her the space to work on the recipes, and the fabulous team of Ian Hofstetter (photographer), Jane Collins (stylist) and Mandy Sinclair (food preparation) for their commitment to producing the beautiful food shots featured in this book.\n\nOur thanks also go to all those who have generously supplied props and equipment for use in the photography: Honey Bee Homewares, Clay & Flax, Mud Australia, Alfresco Emporium, The Fine Food Store and Gina Cucina.\n\nContents\n\nPreface\n\n**Nourish your whole self the whole time**\n\nThe Nutrition for Life healthy eating guidelines\n\nWhat's to drink?\n\nPortion sizes \u2013 downsize, don't supersize\n\nNutrition for Life superfoods\n\nCooking \u2013 the quicker the better\n\nCooking lighter and healthier\n\nMaking the most of your freezer\n\nModifying your recipes for health and wellbeing\n\n**Seven-day summer and winter meal plans**\n\nSummer meal plan\n\nWinter meal plan\n\n**Recipes with zest**\n\nBreakfast and brunch\n\nLight meals and snacks\n\nMain meals\n\nAccompaniments\n\nDesserts and sweet treats\n\nBasics\n\nShopping with zest\n\nConversion tables\n\nPreface\n\n**_'It's good food and not fine words that keeps me alive!'_**\n\nMoli\u00e8re (1622\u20131673)\n\nDespite great gains in our knowledge about food and nutrition, healthy eating hasn't become easier. In fact for many of us it has become harder because of our busy, 'no time to cook' lifestyles. But what we eat can make a huge difference to how we feel and how healthy we are. Along with exercise and stress management, it's the third of that vital 'trio' of things we can do for ourselves and our families to improve long-term health and wellbeing.\n\nIn fact, today's hectic lifestyle makes the need to eat right all the more important. That's why we are firm believers in the value of cooking for yourself and your family. Studies show that people who prepare food at home eat better \u2013 with less saturated fat, less salt, more vegetables, more vitamins and antioxidants \u2013 than those who eat out a lot. And there's so much fun and enjoyment to be had when you prepare and share meals with family and friends.\n\nThe book _Nutrition for Life_ talks about how a healthy diet gives us vitality and energy, helps us stay at a weight that's right for us, boosts our immune system, delays the effects of ageing and builds strong, dense bones.\n\nSince then, many readers have begged for recipes so that they can put 'nutrition for life' into practice in their own kitchens and create healthy meals in their own homes. So we have teamed up to 'put it all on the plate' in _Zest_ , the _Nutrition for Life_ cookbook.\n\n_Zest_ is packed with recipes that are easy to prepare as well as being easy on the waistline and good for heart health.You'll also find they taste every bit as good as they look \u2013 a key part of our food philosophy. These well-balanced recipes are rich in slow carbohydrates, lean protein, the good fats and lots of fibre. They are not designed to be ultra-low-fat nor extra-high-protein, but if you are following _The Low GI Diet_ , _The_ _CSIRO Total Wellbeing Diet_ or a heart-healthy diet, you will find many of them suitable.\n\nWe hope that you will enjoy cooking from our great range of meals. In the recipe section of this book you'll find lazy brunches, snacks and meals to rustle up in minutes, dinners for families or for entertaining, scrumptious desserts and tasty baked goods. We have certainly enjoyed creating (and eating) them. Well, someone had to test them!\n\nGood health and good eating!\n\n_Catherine Saxelby_ and _Jennene Plummer_\n\n[Nourish your whole self  \nthe whole time](Plum_9781742734859_epub_c5_r1.html#d7e1895)\n\nWhat you eat can make a huge difference to how you feel and how healthy you are. In fact, it can really boost your zest for life. And that's what this book is about. You probably already know that you need seven essential nutrients for health and growth: carbohydrates, fibre, protein, fat, vitamins, minerals and water. The body can't do without these nutrients. They provide fuel for the body in the form of kilojoules (calories); they are the raw material for building new tissues, bones and teeth; they regulate the speed of metabolic reactions and they release energy from food. All of which keeps you fit, well and functioning at top speed.\n\nThe healthy eating guidelines in this first section get down to the nutrient nitty-gritty with practical tips on how to nourish your whole self the whole time, how to maintain a healthy weight, have more energy and delay the effects of ageing. We also show you how to moderate your intake of sugar and sweet foods and why what you and your family drink really matters. We give tips on how to avoid portion creep on your plate when eating out or dining at home and ways to make the most of nutrient-dense superfoods.\n\nThere are also plenty of ideas to help you cook the light and easy way, and at the back of the book our shopping guide will ensure that you have those healthy foods on hand in your cupboards, fridge and freezer to create deliciously nourishing meals in minutes, whatever the occasion.\n[The Nutrition for Life   \nhealthy eating guidelines](Plum_9781742734859_epub_c5_r1.html#d7e1955)\n\nForget dieting \u2013 eat for life!\n\nMake it your goal to eat well for vitality, good health and to look after your body \u2013 to nourish your whole self the whole time and boost your zest for life.\n\nLove those vegetables and snack on fruit\n\nVegetables, salads and fruit carry an abundance of vitamins, minerals and natural antioxidants (also called phytochemicals), all for very few kilojoules. They are what nutritionists call 'nutrient dense'. The more you eat them, the more you'll like them. And there are so many ways to enjoy them, as you'll discover in our recipes.\n\nKeep hydrated\n\nWater is the best thirst quencher, yet most of us don't drink enough. The body needs at least 2 litres (8 glasses) of water each day to keep it hydrated, to maintain vital biochemical reactions and to keep the kidneys flushed. Ideally about half of this should be plain water, while the rest can come from other healthy beverages such as juices, clear soups, herbal infusions and weak tea. Carry a bottle of water with you in the car and have some on your desk at work. Don't wait until you feel thirsty to drink \u2013 sip regularly throughout the day.\n\nFocus on the good fats\n\nOpt for monounsaturated and polyunsaturated fats that are derived from oils, nuts, seeds and avocado. Use them to replace any saturated fats you now consume. The omega-3 fats from fish help stabilise the heartbeat and lower inflammation. There's also emerging research showing the omega-3s may influence brain function and help to alleviate depression, schizophrenia and learning difficulties such as ADHD (Attention Deficit Hyperactivity Disorder).\n\nEat less of the bad fats\n\nHigh levels of saturated fat and trans fats raise the 'bad' LDL (low-density lipoprotein) cholesterol and put you at risk of heart disease, stroke and diabetes. We over-consume this type of fat instead of keeping it low. So limit your intake of butter, cream, sausages and fatty meats, deli meats and heavy sauces. And think twice about pies, pastries, confectionery and cakes.\n\nEat smart carbohydrates\n\nChoose foods like wholemeal and multigrain breads, brown rice, oats, barley, fibre-enriched or wholegrain cereals and legumes that offer higher concentrations of fibre, vitamins and minerals. These generally have a lower glycaemic index (GI). This means they are absorbed more slowly \u2013 your body is doing all the work not the food factory. Low-GI carbohydrates can help with weight control and managing your blood glucose levels, vital if you have diabetes or pre-diabetes.\n\nEat less refined carbohydrates and kilojoule-dense foods \u2013 save them for when you eat out or as an occasional treat. Limit how much white bread, white rice and pre-sweetened cereals you eat, along with cakes and biscuits.\n\nWatch the sugar\n\nFeeding that sweet tooth? Too much sugar adds unwanted kilojoules and supplies no fibre, vitamins or minerals, so it makes no contribution to your nutrition intake. A spread of jam on your toast or sugar in yoghurt or flavoured milk is okay, but limit your consumption of sugary soft drinks, juices, lollies, chocolate, cakes, pastries and ice creams.\n\nWatch the salt\n\nWe eat twice as much salt as we should. Stop sprinkling salt over your food and start buying reduced-salt products when you shop. The ones to focus on are salt-reduced stock, soy sauce and cheese as these can make a big difference quickly without your noticing the drop in salt. Don't forget to boost the flavour of meals with fresh herbs, garlic, chilli and citrus zest \u2013 we show you in our recipes how these salt-free ingredients can make all the difference to the final taste.\n\nDon't overdo the alcohol\n\nModerate-to-heavy intake of alcohol is associated with cirrhosis, high blood pressure and cancers of the digestive system. Too much alcohol can also put weight on easily. Gram for gram, alcohol has almost twice the kilojoules of either carbohydrate or protein.\n\n**_Glycaemic Index (GI)_**\n\nThe Glycaemic Index (or GI) is a ranking of foods from 0 to 100 that tells us whether a carbohydrate food will raise blood sugar (glucose) levels dramatically (high GI), moderately (medium GI) or just a little (low GI).\n\nHigh GI = 70 or more\n\nMedium GI = 56\u201369\n\nLow GI = 55 or less\n\n_**Healthy eating in a nutshell**_\n\n\u2022 Avoid saturated fat.\n\n\u2022 Choose fewer refined carbohydrates and more slow carbohydrates (low-GI), wholegrains and high-fibre foods.\n\n\u2022 Protein is essential, whether from animal or plant sources \u2013 just keep it lean.\n\n\u2022 Bone up on calcium-rich foods.\n\n\u2022 Make sure you get your two serves of fruit and five serves of vegetables every day.\n\n\u2022 Moderate your consumption of sugar and sweet foods.\n\n\u2022 Cut back on salt.\n\n\u2022 Limit alcohol.\n\n\u2022 Drink plenty of water.\n\nOn the positive side, a modest amount of alcohol is good for your heart and red wine with its grape antioxidants can keep your arteries from becoming clogged. But it really is a case of less is best!\n\nAim for balance\n\nEating well involves getting the balance right. If 90 per cent of all your foods are nutritious, then the remaining 10 per cent can be a treat or indulgence.\n\nVariety\n\nVariety doesn't mean having ten different cereal packs in your cupboard, but rather eating a variety of botanically different foods. Pasta, bread, puffed wheat and couscous all look and taste different but are all derived from the one basic but versatile grain (wheat). So they all provide similar nutrients.\n\nSubstituting other grains like oats, barley, corn or rye for some wheat adds diversity to your diet and ensures a variety of different vitamins and antioxidants, each of which has a different function in the body. Eating a wider range of foods ensures that the nutrients you miss from one food you can gain from another.\n\nSo be adventurous. Try to introduce new foods or experiment with a new dish occasionally \u2013 it will broaden your horizons, both gastronomically and nutritionally.\n\nBe positive\n\nRather than longing for foods you shouldn't eat, try to focus on all the delicious enjoyable foods you can eat. After all, what can beat a ripe luscious mango, a crisp green salad or a perfectly grilled fillet of fish?\n\n**_Remember, what you eat today walks and talks tomorrow!_**\nWhat's to drink?\n\nOne of our key guidelines is to moderate your intake of sugar and sweet foods. That includes what you drink. These days, beverages contribute a lot of kilojoules (between 15 and 20 per cent) of your average day's intake.\n\nResearch is now linking what we drink to the obesity epidemic. It seems that drinks are less filling than solid food and don't register with our brain's appetite control centre. So it's easy to gulp down large amounts of juice or soft drink, which together supply over half of all the sugar we consume. Here are our guidelines for what's good and what's not-so-good to drink.\n\nThe best drinks\n\nWater\n\nWater has no kilojoules and no extra sugar or sodium. An inexpensive water filter jug removes the taints and off-flavours without taking out the 'good for your teeth' fluoride \u2013 it really does make water taste better.\n\nChill your water or serve it with a slice of lime or lemon.\n\n_**Which water?**_|   \n---|---  \n_Tap water_|  Okay taste in most places. Check whether fluoride is present. Tank rainwater won't contain fluoride; bore water may.   \n_Filtered tap water_|  Pleasant taste. A filter removes taints and that strong chlorine smell. Charcoal filters are inexpensive and retain fluoride in the water. Undersink reverse osmosis filters remove both fluoride and harmful bacteria like _cryptosporidium._  \n _Bottled water_|  Clean neutral taste. Lacks fluoride so don't allow your children to drink this all the time.   \n_Still mineral water_|  Pleasant taste. Certain spring waters can be high in minerals (sodium carbonate, sodium chloride and salts of calcium, magnesium, iron and sometimes hydrogen sulphide) depending on the source.   \n_Sparkling mineral water_|  Fizzy so more thirst-quenching than still water. Some are naturally aerated with carbon dioxide. Taste varies from brand to brand.   \n_Flavoured mineral water_|  Sounds healthy, but has only a little less sugar content than soft drink. Think of it as another sweetened drink with kilojoules to count.\n\nWeak green or black tea (or iced tea)\n\nTea is a major source of flavonoid antioxidants, natural compounds that can keep your heart healthy and may even slow the ageing process. It also contains L-theanine, an amino acid unique to tea that keeps you alert yet relaxed. It's a pick-me-up that's different from caffeine. Tea does contain caffeine but only at around half the level found in coffee \u2013 so it's a healthier choice. Enjoy two or three cups a day, with milk if you prefer.\n\nHerb and floral teas\n\nThese infusions add zero kilojoules (unless you sweeten them). Some have medicinal properties \u2013 camomile before bed to induce sleepiness, peppermint or ginger as a natural 'stimulant', and so on.\n\nFruit juice\n\nLimit juice consumption to one small glass of unsweetened juice a day \u2013 it's healthy but it's fruit in concentrated form without the fibre. We recommend you dilute juice 50:50 with water or ice-blocks. This way, it's still refreshing but lower in kilojoule density. Check the label \u2013 don't buy 'fruit juice drinks' or 'fruit drinks', which contain a lot of sugar and only a small percentage of real fruit juice.\n\nJuice bars have made juice trendy \u2013 you can get vitamin-packed orange and carrot juice with a shot of wheat grass, or apple, celery and ginger in minutes. But the serve sizes are supersized. Even a standard foam cup of juice packs in 1040 kilojoules (250 calories). Order the smallest size or share with a friend. And remember you still need to get two serves of whole fruit each day as well for fibre.\n\nLow-fat milks\n\nMost of these milks are enriched with extra calcium so you get half the fat yet 30 per cent more calcium per glass. They're good value. You don't need to use skimmed (very low-fat or no-fat) versions unless you're being super-strict on fat intake, or love drinking milk but are on a weight-loss diet. Low-fat milks suit families with kids over the age of two. Children need four serves of dairy foods a day, which can come from milk as well as yoghurt and cheese.\n\nLow-fat calcium-enriched soy drinks\n\nMake sure the soy drink is fortified with calcium (most are) otherwise you'll short-change yourself on calcium.\n\nFruity frapp\u00e9s and smoothies\n\nSmoothies made with low-fat milk, yoghurt or ice cream (or soy alternatives) plus fruit and perhaps some wheatgerm, provide an excellent source of bone-building calcium. They'll also provide long-lasting energy, as they are usually low-GI. But they're supersized, so regard them as a complete meal, not just a snack or thirst quencher.\n\nThings to drink occasionally\n\nSports drinks\n\nSports drinks are useful if you're an endurance athlete exercising hard for more than an hour straight. The rest of us weekend warriors will do fine on plain water. However if you need to replenish lost sweat quickly, sports drinks do a good job with their formulation of lower sugar content (at 5\u20136 per cent which is half that of soft drinks) plus added sodium and potassium.\n\nSports waters\n\nAt only 1\u20132 per cent sugar, sports waters are like drinking water with a splash of sugar and flavouring. You'll find these more refreshing than plain water, yet they are lightly sweetened. Forget the claims about added B vitamins \u2013 often what's added is just enough to make an enticing claim on the label, rather than a difference to your life.\n\nIced tea drinks\n\nLightly sweetened (4\u20135 per cent) and low on carbonation, commercial iced teas are great thirst quenchers. Make your own using cold brewed black tea, a little sugar, lemon slices and mint leaves.\n\nNot-so-good things to drink\n\nSoft drinks\n\nOnce reserved for parties and special occasions, fizzy sugary drinks are now everyday staples in super sizes. The small 200 ml Coke 'waist' bottle of the 1950s has been superseded by 375 ml cans, large 600 ml Coke 'buddies' or even 900 ml fast food cartons to go.\n\nYet soft drinks give us no nutrients apart from water and load us up with kilojoules \u2013 a 375 ml can will hit you with around 10 teaspoons or 41 grams of sugar and 655 kilojoules (155 calories).\n\nDiet or zero-sugar versions that are sweetened with aspartame, acesulphame K or sucralose are handy if you want to cut back on your sugar intake, but they too can erode enamel and dentine. And you don't want to be on a high intake of sweeteners of any sort \u2013 despite their safety record to date. Stick to one or two diet drinks a day.\n\nEnergy drinks\n\nThese sound as if they're doing you good, crammed with healthful sounding ingredients like B vitamins, taurine, amino acids and guarana. But the truth is they're really just a fizzy drink with added caffeine (guarana is actually another plant that contains caffeine). The combination of caffeine with alcohol (say when you sip a Vodka Red Bull) is not a good one \u2013 one picks you up while the other settles you down! Steer clear of these.\n\nCoffee\n\nCoffee-to-go is a real trap. Gourmet coffee chains such as Starbucks, Jamaica Blue and Gloria Jeans offer the tempting combination of supersize sofas, supersize serves and killer kilojoule syrups. A jumbo latte with caramel syrup plus whipped cream packs in a hefty 1000 kilojoules (240 calories) and 10 grams of fat, not forgetting almost 8 teaspoons of sugar (31 grams). It's rather like drinking the equivalent of a regular size Mars Bar.\n\nIf you are a regular milky coffee drinker, it's a good idea to opt for a 'skinny' latte or cappuccino. A full-cream version of these has 4 grams of sugar, 2 grams of fat and 190 kilojoules (45 calories), whereas a skinny latte has the same sugar content but almost no fat and only 115 kilojoules (27 calories).\n\n**_Not teeth friendly_**\n\nOver the past 20 years, soft drinks have gradually replaced milk as the main drink of teenagers. Apart from the fact that soft drinks lack calcium, they are not bone-friendly for another reason \u2013 it seems the phosphoric acid they contain interferes with calcium absorption. And this also makes them overly acidic which, together with their high sugar level, is a harmful combination for teeth.\n\nCoffee is fine in moderation \u2013 there are even early reports that caffeine may help reduce the risk of Parkinson's disease \u2013 but too much can leave you sleepless, jittery and with an upset stomach. Limit yourself to no more than two real coffees (this means a capuccino or short black made from espresso) or four instant coffees each day.\n\nHot chocolate\n\nIf you choose to drink a hot chocolate with whipped cream you get a whopping 10 teaspoons of sugar (41 grams) and 24 grams of fat. This is not a mere beverage \u2013 this is equivalent to a mini-meal!\n\nWe recommend you drink\n\nWith meals\n\n\u2022 water \u2013 tap or bottled\n\n\u2022 milk \u2013 for children with meals (full-fat up to the age of two and then reduced-fat or light after that)\n\n\u2022 juice, unsweetened and diluted with water \u2013 a small glass\n\n\u2022 wine \u2013 1 small glass (100 ml).\n\nIn between meals\n\n\u2022 tea or herbal infusions\n\n\u2022 milk \u2013 reduced-fat, plain or flavoured.\n\n_**How much to drink over the day?**_|   \n---|---  \n_Water_|  4\u20138 glasses   \n_Milk_|  1 glass   \n_Teas_|  2\u20133 cups   \n_Unsweetened juice, diluted_|  1 small glass (optional)   \n_Other_|  1 glass\/cup (optional)   \n_Wine_|  1 small glass (optional)\n\nSugar in drinks\n\nBecause drinks come in a range of sizes, we have listed the sugar content as a percentage so you can compare. All hot beverages below were made with full-fat milk. The sugar content won't vary much if made with low-fat milk \u2013 only the fat content drops. If you like your tea or coffee with a heaped teaspoon of sugar, add another 8 grams of sugar to the figures below. We have indicated the drinks that will also be high in saturated fat with an asterisk *.\n\n**_Beverage_**|  _ **Sugar % or g per 100g**_  \n---|---  \nBlack tea, unsweetened | 0   \nEspresso, unsweetened | 0   \nTea with milk, unsweetened | 1   \nTomato juice | 2   \nFlat white coffee, unsweetened | 2   \nCappuccino, unsweetened | 4   \nStarbucks Caf\u00e9 Latte | 4   \nSports drink (Gatorade\/Powerade) | 6   \nMilkshake | 6   \nLipton Iced Tea, Green | 7   \nGloria Jeans Chai Tea | 7   \nSmoothie | 7   \n**_Beverage cont._**|  _ **Sugar % or g per 100g**_  \n---|---  \nGloria Jeans Caf\u00e9 Latte |  8   \nOrange juice, no added sugar |  8   \nStarbucks Hot Chocolate (with whipped cream) | 8\u20139 *   \nFlavoured milk, strawberry |  9   \nStarbucks Frappuccino |  9 *   \nGloria Jeans Creamy Hot Cocoa |  10 *   \nCordial, made up |  10   \nStarbucks Iced Coffee |  11   \nApple juice, no added sugar |  11   \nCola drink |  11   \nBoost Juice Dairy Banana Buzz Smoothie |  12 * \n[Portion sizes \u2013 downsize,   \ndon't supersize](Plum_9781742734859_epub_c5_r1.html#d7e2105)\n\nPortion sizes have been getting bigger and bigger over the past 20 years. It's now clear that they've been a major contributor to the obesity problem. There's even a name for it \u2013 'portion creep'. The problem is that the bigger the portion in front of you, the more you tend to eat.\n\nUS researchers have tracked this upward trend in serve sizes. They report that in the past ten years the size of juices increased by 30 per cent, wine and soft drink by 50 per cent, while beer rose a whopping 200 per cent.\n\nWhat used to be considered a family-sized block of chocolate is now the standard size. Fizzy drinks come in huge 1.5 litre bottles (25 per cent more for free; but you also get to keep the sugar and the extra kilojoules). Movie popcorn is sold in buckets, muffins balloon out of their paper cases and fresh juice comes in a 650 ml cup \u2013 equal to 6 or 7 pieces of whole fruit.\n\nFast food operators have led the way with upsizing. Supersized serves and 'two-for-one' meal deals (for a fraction more money) may be great value \u2013 but it's a bargain that our waistlines don't need.\n\nTips for preventing portion creep\n\nEating out\n\n\u2022 Only eat what you need. Listen to your stomach and stop when it says 'I'm full'.\n\n\u2022 If there's a choice, opt for the smaller size.\n\n\u2022 Share large portions with a friend.\n\n\u2022 Ask for a 'doggie bag' to take leftovers home for later.\n\n\u2022 If you do buy two-for-one offers, don't eat it all at one sitting.\n\n\u2022 When it comes to treats, be satisfied with less. A small indulgence of the real thing is often enough to satisfy.\n\n_**How much protein, carbohydrate and vegies should you put on each plate?**_\n\nHere's an easy way to think about balancing your meal: fill half the plate with salad or non-starchy green vegetables such as zucchini, green beans or broccoli; then fill quarter of the plate with meat or fish (protein); and the final quarter with pasta, rice or potatoes (starchy carbohydrates). These proportions will create a balanced meal that's filling and nutritious.\n\nAt home\n\n\u2022 Watch how much you serve up \u2013 you can always save leftovers for the next day.\n\n\u2022 Keep your portions moderate.\n\n\u2022 Don't go back for seconds.\n\n\u2022 Women generally need smaller portions than men. If you eat together as a couple, don't eat the same sized serves.\n\n\u2022 Serve meals on smaller plates in the kitchen rather than helping yourself at the table.\n\n\u2022 You don't have to clean your plate \u2013 stop when you feel full.\n\n\u2022 With main meals, make sure that half the plate is green or salad vegetables (this does not include starchy ones like potatoes).\n\nDanger zone\n\nDrinks\n\n_**How big should those glasses really be?**_|   \n---|---  \nWine | 1 small glass (100 ml)*   \nBeer | 1 glass (250 ml)*   \nBourbon, vodka or other spirits | 1 nip (30 ml)*   \nSoft drink or juice | 1 glass (200 ml)\n\n* The drinks marked with an asterisk provide 10 grams of alcohol, which is used as the standard measure of alcoholic drinks.\n\nSnacks and confectionery\n\nMost popular snacks are more than a quick bite \u2013 they're really a mini-meal.\n\nTake doughnuts for instance: at 20 grams of fat and more than 2000 kilojoules (475 calories), two cinnamon doughnuts pile on one-third of the day's recommended intake of fat and kilojoules for a sedentary woman. Most of this fat is saturated, the type that clogs arteries and thickens waistlines. Ditto for pastries, a bucket of hot chips or a large slice of banana bread. So watch those portions and think small!\nWhat is a standard serve size?\n\nVegetables\n\nOne serve is:\n\n\u2022 \u00bd cup cooked vegetables (such as broccoli, beans, peas)\n\n\u2022 1 tomato\n\n\u2022 1 cup salad leaves\n\nFruit\n\nOne serve is:\n\n\u2022 1 medium apple, banana, orange or pear\n\n\u2022 2 plums, apricots or kiwi fruit\n\n\u2022 1 cup fruit salad or canned fruit\n\n\u2022 2 tablespoons sultanas\n\n\u2022 4 dried apricot halves\n\nLegumes\n\nOne serve is:\n\n\u2022 \u00bd cup (75 g) cooked or canned beans or lentils\n\n\u2022 1 small can (100 g) baked beans\n\nLean meat, fish, chicken or eggs\n\nOne serve is:\n\n\u2022 125 g meat (cooked) \u2013 2 slices roast meat, 2 medium chops, 1 small steak, \u00be cup mince\n\n\u2022 150 g fish or seafood (cooked) \u2013 1 large fish fillet or 120 g can tuna or salmon\n\n\u2022 125 g chicken (cooked) \u2013 1 small chicken breast, 2\u20133 drumsticks\n\n\u2022 2 eggs\n\nNote: 150 grams raw weight becomes around 125 grams when cooked. 175 grams raw weight trims down to 150 grams.\n\nDairy\n\nOne serve is:\n\n\u2022 1 cup (250 ml) low-fat milk\n\n\u2022 1 tub (200 g) low-fat yoghurt\n\n\u2022 2 slices (40 g) reduced-fat cheese\n\nNuts and seeds\n\nOne serve is:\n\n\u2022 a small handful (30 g) almonds, walnuts, cashews, macadamias, pecans, peanuts\n\n\u2022 2 tablespoons (50 g) peanut butter or tahini\n\nWhole grains and cereals\n\nOne serve is:\n\n\u2022 1 thick slice bread\n\n\u2022 \u00bd bread roll\n\n\u2022 \u00bc cup (90 g) cooked rice\n\n\u2022 \u00bd cup (90 g) cooked pasta or noodles\n\n\u2022 \u00be cup (30\u201340 g) breakfast cereal\n\n\u2022 \u00bd cup (140 g) cooked porridge\n\n\u2022 \u00bc cup (30 g) muesli\n\nFats\n\nOne serve is:\n\n\u2022 1 tablespoon (20 g) of any oil\n\n\u2022 2 tablespoons (40 g) light margarine (also called low-fat spread)\n\n\u2022 \u00bd avocado\n\n**_Your daily nutrition goals: How much to eat each day \u2013 the minimum for good health_**\n\n_Vegetables_|  5+ serves   \n---|---  \n_Whole grains_|  4+ serves   \n_Dairy, low-fat_|  4 serves   \n_Fruit_|  2+ serves   \n_Legumes_|  2+ serves a week   \n_Lean meat (includes fish, chicken and eggs)_|  1\u20132 serves   \n_Fats_|  2\u20133 serves   \n_Nuts and seeds_|  1 serve (a small handful, about 30 g)   \n_Salt_|  Shake the habit   \n_Sugar_|  Go easy\n\nHow much should you be eating?\n\nHow many kilojoules you should eat each day will depend on your age, level of physical activity, body size (larger bodies require more), sex and stage of life. Children and teens need more kilojoules, due to the demands of growth, as do women who are pregnant or breastfeeding. The following levels are only approximate but serve as a general guide.\n\n**_Women_**  \n---  \n _Moderately active adult_|  9000 kJ   \n_for weight maintenance_|  (2100 cal)   \n_Sedentary adult for_|  7000 kJ   \n_weight maintenance_|  (1700 cal)   \n_Fat loss_|  6000 kJ   \n| (1400 cal)   \n_**Men**_  \n---  \n _Moderately active adult_|  10,000 kJ   \n_for weight maintenance_|  (2400 cal)   \n_Sedentary adult for_|  9000 kJ   \n_weight maintenance_|  (2100 cal)   \n_Fat loss_|  7000 kJ   \n|  (1700 cal)\n\n_**Daily intakes**_\n\nUse these suggested daily kilojoule intakes to give you an idea of what you should be eating over a whole day.\n\nThese daily intake figures are based on a standard diet ratio of:\n\n\u2022 50 per cent of kilojoules from carbohydrate (10 per cent from sugars)\n\n\u2022 30 per cent of kilojoules from fat (10 per cent from saturated fat)\n\n\u2022 20 per cent of kilojoules from protein \n[Nutrition for Life  \nsuperfoods](Plum_9781742734859_epub_c5_r1.html#d7e2155)\n\nFoods are not created equal. Some are packed with more nutrients than others or have therapeutic effects beyond nutrition. Take fruit, for instance. While all fruit is nutritious, different types vary enormously in the nutrients they offer.\n\nPut simply, an orange is not equivalent to an apple. An orange has ten times more vitamin C and beta-carotene, four times more thiamin and a huge 40 times more folate (a B vitamin that prevents birth defects).\n\nBroccoli is nutritionally superior to beans or zucchini (and so are its relatives cauliflower, cabbage, kale and Brussels sprouts). Garlic lords it over leeks, onions, eschalots and chives, even though they are cousins in the plant kingdom.\n\nSo overwhelming is this difference that US researchers have gone so far as to call them 'powerhouse' foods and call for them to replace the others we now consume. For instance, when you shop for fruit and vegetables, the most popular choices are apples, bananas, iceberg lettuce, potatoes and corn. The researchers say these should be replaced with these powerhouse ones.\n\nSuperfoods help you make every kilojoule count\n\nAt a time of global obesity, it pays to make each kilojoule count. Use this concept of nutrient density to get the maximum vitamins, minerals and protein without overloading your system with kilojoules.\n\nHere's our pick of the superstar foods to include in your diet.\n\nFruit\n\nAll fruit is a nutritionist's delight, but here are the leaders:\n\n\u2022 citrus (orange, grapefruit, mandarin, lemon, lime, tangelo) \u2013 a nutrition all-rounder for vitamin C, minerals, as well as B1 and folate. Even the peel contains nutritional 'goodies' that lower cholesterol and ward off cancer\n\n\u2022 kiwi fruit \u2013 low in kilojoules and high in vitamin C, folate and fibre; rich in lutein, a phytochemical that's good for the eyes\n\n\u2022 berries (blueberries, strawberries, cranberries, blackberries, raspberries) \u2013 bursting with vitamin C, potassium, fibre and folate; their blue-red pigments are powerful antioxidants that can neutralise 'bad' bacteria and fight off cancer.\n\nVegetables\n\nVegetables are a nutritionist's delight too, but these are the superstars:\n\n\u2022 spinach, silverbeet, Asian greens \u2013 packed with vitamin C, folate, antioxidants and fibre for almost no kilojoules\n\n\u2022 dark-green lettuces (mignonette, rocket, baby spinach leaves) \u2013 a nutrient-rich addition to any meal, big on volume but small on kilojoules\n\n\u2022 cruciferous (cabbage, cauliflower, broccoli, Brussels sprouts, kale, rape, turnips) \u2013 contain cancer-fighting sulphur compounds, lots of fibre, beta-carotene, a host of minerals, a little iron and calcium\n\n\u2022 avocados \u2013 for the good fats that lower your LDL-cholesterol\n\n\u2022 tomatoes \u2013 high in lycopene, vitamin C and fibre.\n\n**_Superstar spinach_**\n\nTry to eat spinach or silverbeet three times a week, if not in salads, then add a handful of leaves to a stir-fry, risotto or a curry. They soften in the heat of the finished dish.\n\n_**Superstar salad dressing**_\n\nIf you want to make your salads and vegetables interesting (and have everyone eat them), drizzle over this dressing. It's also doing your heart a big favour. In a screwtop jar, place 3\/4 cup extra-virgin olive oil, the juice of one lemon (around \u00bc cup), 2\u20133 teaspoons Dijon mustard, 1\u20132 cloves crushed garlic, and a little ground black pepper. Shake well to combine. Makes about 1 cup of dressing.\n\n**_Superstar green salad_**\n\nMix together a 200 g bag of baby spinach leaves with a bunch of rocket (discard the stalks). You can also throw in 100 grams of lightly blanched sugar snap peas for more crunch.\n\nLegumes\n\nAll legumes such as beans, split peas, chickpeas and lentils are good for fibre, protein for vegetarians, B vitamins and low GI. The star-players are:\n\n\u2022 soy beans \u2013 for phyto-oestrogens, fibre, excellent protein quality, good quality oil\n\n\u2022 chickpeas \u2013 a good dose of B vitamins, protein and significant fibre content\n\n\u2022 lentils \u2013 quick and easy to cook, no pre-soaking; for protein, a little iron, zinc, potassium and fibre.\n\nProteins\n\nFor maximum protein, iron, vitamin B12 , folate, potassium and omega-3s. Winning proteins are:\n\n\u2022 eggs \u2013 a compact package of nutrition, giving you every vitamin except vitamin C, plenty of protein and a host of essential minerals including vitamin B12\n\n\u2022 pink salmon \u2013 omega-3s, heart-healthy, protein, minerals from the ocean such as magnesium and potassium\n\n\u2022 lean lamb \u2013 high-quality protein; minerals iron, zinc and potassium; and a range of B vitamins including thiamin, niacin, vitamins B6 and B12\n\n\u2022 liver \u2013 packed with iron, protein, B vitamins and vitamin A\n\n\u2022 tofu \u2013 low in fat and a key source of protein and B vitamins in many Asian diets.\n\nDairy\n\nYou'll get lots of calcium, protein and riboflavin here:\n\n\u2022 low-fat milk \u2013 lots of calcium, protein and riboflavin\n\n\u2022 low-fat yoghurt (choose a probiotic one with friendly bacteria) \u2013 calcium, protein, B vitamins, probiotic.\n\nHerbs and spices\n\nAll green leafy herbs are nutrient-rich for almost no kilojoules but here's our pick:\n\n\u2022 basil, rosemary, oregano \u2013 rich in vitamin C and antioxidants\n\n\u2022 chillies \u2013 packed with antioxidants and beta-carotene\n\n\u2022 cinnamon \u2013 may help lower blood sugar in diabetes\n\n\u2022 garlic \u2013 good for the heart, anti-bacterial\n\n\u2022 turmeric \u2013 its vibrant yellow pigment curcumin can inhibit the formation of cancer and reduce inflammation.\n\nNuts and seeds\n\nAll nuts are full of 'healthy fats' that keep your heart in top shape. But you'll also max out on vitamin E, fibre and many minerals. Choose:\n\n\u2022 almonds \u2013 important for vitamin E and arginine for a healthy heart, monounsaturated fats that can lower the bad LDL-cholesterol; high in fibre and minerals including a little calcium\n\n\u2022 walnuts \u2013 a source of fibre and vitamin E as well as the minerals potassium, magnesium, zinc, copper and selenium\n\n\u2022 linseeds (flaxseeds) \u2013 a storehouse of plant omega-3s, healthy fats and lignans (a type of plant oestrogen).\n\n_**Superstar sprinkle**_\n\nTop cereal or yoghurt with a couple of tablespoons of this easy sprinkle \u2013 it's a great way to take in the good fats from nuts and seeds. Place 1 cup of almonds (whole or pieces) in the bowl of a food processor along with \u00bd cup of walnuts and \u00bd cup of linseeds (flaxseeds). Process until finely ground and then store in a jar in the refrigerator to keep fresh. Use within a month. You can also grind these up in a coffee grinder but you'll have to do it in batches. Makes about 1 cup.\n\nGrains\n\nThink wholegrain or high-fibre. You get the most B vitamins, vitamin E, lignan antioxidant and fibre. Choose:\n\n\u2022 brown rice, wild rice \u2013 gluten-free grains packed with B vitamins and fibre\n\n\u2022 wholegrain bread \u2013 B vitamins, nutrient-rich\n\n\u2022 oats \u2013 lower LDL-cholesterol, soluble beta-glucan fibre, low-GI\n\n\u2022 wholegrain breakfast cereals \u2013 convenient and nutritious; buy these in preference to refined cereals\n\n\u2022 barley \u2013 good for soluble fibre, low-GI\n\n\u2022 wheatgerm \u2013 chock-full of B vitamins and good fats; sprinkle it over your usual cereal\n\n\u2022 gluten-free grains \u2013 rice, maize\/corn, buckwheat, amaranth, millet, quinoa.\n\n**_Superstar mega-muesli_**\n\nStart with any natural muesli of your choice from the supermarket. Tip it into a large bowl and toss in \u00bd or 1 cup of linseeds (flaxseeds), slivered almonds or walnuts, pumpkin seeds, wheatgerm or lecithin. If you need to watch your cholesterol, finish off with \u00bd cup psyllium husks. If you need help to stay regular, add in 1 cup of bran breakfast cereal as well. Makes it crunchy and slows down your rate of eating. It's also low-GI so will keep you powering along full speed until lunchtime.\n\nFats and oils\n\nChoose the good fats:\n\n\u2022 extra-virgin olive oil \u2013 healthy monounsaturated fats, squalene, polyphenol antioxidants\n\n\u2022 any other oil, cold-pressed if possible.\n\n_**Recipes using rice**_\n\nTo help you reduce the GI of your overall diet, we've used a low-GI rice such as Doongara or Moolgiri in all our recipes that call for white rice. Some recipes use brown rice, which has a medium GI but gives you the extra nutrients from a whole grain.\n\nDrinks\n\nIt's vital to drink plenty of water to stay hydrated. Here are some healthy benefits of other drinks:\n\n\u2022 tea \u2013 antioxidants, good for the heart\n\n\u2022 red wine \u2013 high in polyphenols, good for keeping the blood thin and free-flowing.\n\n**_Herbal tea recharger_**\n\nIn a tea mug, dangle a green tea bag and add a 2 cm chunk of ginger (no need to peel) plus a slice of lemon. Pour boiling water over and leave to steep for 2\u20133 minutes. Remove tea bag, ginger and lemon. Add a little honey if you like. This really gives you a lift!\n[Cooking \u2013 the quicker  \nthe better](Plum_9781742734859_epub_c5_r1.html#d7e2205)\n\nHealthy cooking doesn't mean that you have to become a gourmet chef or invest in expensive cookware. Here we show you 12 basic cooking methods for preparing foods in healthy ways without adding excessive amounts of fat or salt.\n\nCooking for good health has three main aims. It aims to:\n\n\u2022 retain the most vitamins and minerals\n\n\u2022 remove saturated fat from meats and chicken\n\n\u2022 make recipes taste good without adding salt or sugar.\n\nNo matter how careful you are, all forms of cooking deplete nutrients to some extent \u2013 heat inactivates three heat-sensitive vitamins (vitamin C, thiamin or vitamin B1 , and folate, another B vitamin) while water leaches out minerals. The trick is to minimise the loss, and some cooking methods are better at this than others.\n\nAs a rule of thumb, the quicker the cooking time and the less water used, the better your nutrition will be. On the other hand, cooking improves the digestibility of fibre and proteins. It also increases the availability of lycopene and other fat-soluble nutrients. Mineral levels such as zinc or magnesium are not affected by cooking.\n\n_**Tips to preserve the goodness of your vegetables**_\n\n\u2022 Dice or slice vegetables about the same size so they all cook evenly.\n\n\u2022 Cook vegetables in a small quantity of water for as short a time as possible. Cook until just tender but still crunchy.\n\n\u2022 Don't leave cooked vegetables standing for long periods.\n\n\u2022 Try not to peel vegetables thickly as the nutrients are generally concentrated near the skin.\n\n\u2022 Avoid buying vegetables that look 'tired' or wilted. Try to use fresh produce soon after buying.\n\n12 ways to cook for your health\n\nSteaming\n\nWhether you opt for a double-boiler or a foldable metal basket that fits into a pot, steaming ranks at the top of the cooking methods. It minimises loss of vitamins, cooks quickly and needs no fat. It's the ideal way to cook vegetables rather than boiling, where nutrients are leached out into the cooking water. Steam ovens rate highly too. They inject steam into a cooking chamber and cook food quickly without discoloration or softening, while retaining good flavour.\n\nMicrowaving\n\nMicrowaving cooks fastest and needs little or no water and no salt. Forget the rumours about microwaves destroying the goodness in food. Like radio waves, microwaves are a form of electromagnectic energy. Microwave cooking works by vibrating the water molecules within the food so they heat up. It is as safe as cooking on a conventional stovetop. It is simply a way of heating.\n\nIt's best to use containers that have been specially designed for microwave heating \u2013 or stick to glass, ceramic or paper. Don't use plastic containers which are high in polyvinyl chloride (PVC), such as margarine or ice-cream containers. The plastic molecules may pass into the food.\n\nPoaching\n\nTo poach foods, gently simmer ingredients in water or a liquid such as broth, wine or juice until they're cooked through and tender. The food retains its shape during cooking. For stovetop poaching, choose a covered pan that best fits the size and shape of the food. You want to just cover the food with the poaching liquid.\n\nGrilling\n\nGrilling on a slotted tray or barbecue plate allows any fat to drip away from food and also creates its own unique flavour. An electric health grill is a great idea for quick steaks or melts when space is limited. Some grills cook from the top and bottom at the same time, so your meat cooks faster. You can use them for toasted sandwiches too.\n\nFast boiling\n\nShort rapid boiling is better than long simmering for vegetables. The more water and the longer the cooking time, the more nutrients you lose. Ideally bring the water to the boil first, then add your vegetables and cover the pan (this speeds up the cooking time). Bring back to the boil and cook until just tender. Drain. Use the cooking water in stocks, sauces and soups.\n\nStir-frying\n\nStir-frying delivers a lot more heat than a conventional pan so your food cooks faster and doesn't 'stew' in its own juices. Go for a nonstick surface to minimise sticking so you don't have to use a lot of oil \u2013 you can even stir-fry with just a spray of oil or a little stock or water. Remember to dice or slice your ingredients to a uniform size so they cook at the same rate. Cook food in order of cooking time \u2013 the foods that take the longest to cook go in first.\n\nSaut\u00e9ing\n\nFrying in oil piles on the fat \u2013 in fact, half the kilojoules in a fried chicken breast come from the oil it is cooked in. Additionally, the high frying temperatures are quite destructive of nutrients. But saut\u00e9ing \u2013 cooking quickly over high heat in a little oil and tossing frequently \u2013 is fine. Use a little oil, or brush or spray the pan first. You shouldn't have any oil left over, so use just enough to do the job.\n\nRoasting\n\nRoasting uses dry heat at high temperature to cook the food. For poultry and meat, make sure you place a rack inside the roasting pan so that the fat can drip away during cooking.\n\nOven bags\n\nThese work like a mini-slow cooker so you 'steam' your meat in an individually sealed bag in the oven. Best of all, you simply throw the bag out once you've finished.\n\nBarbecuing\n\nThis high-heat method needs to be controlled so you minimise how much you char (burn) the food. It's not just meat you need to be careful with, this applies to poultry and char-grilled vegetables too. When fat drips onto hot coals (or other source of heat), potentially dangerous compounds are then deposited on the food from the rising smoke and flames. It tastes delicious but it does set the scene for cancer possibly later in life. So to minimise the risk, our recipes suggest you:\n\n\u2022 buy lean meats or trim away all fat to reduce fat-flares\n\n\u2022 marinate meats before barbecuing\n\n\u2022 don't cook directly over the coals\n\n\u2022 wait for the fire to die down a little before you cook so it's not smoking\n\n\u2022 cut away any charred bits.\n\nIf you use the barbecue hotplate, watch the amount of oil you add or you'll end up frying the food in fat, not barbecuing it.\n\nSlow cookers\n\nWhen the weather gets cold, it's time to bring out the electric slow cooker or a heavy-based casserole that you leave on low in the oven or over a gentle heat. The direct heat from the pot, lengthy cooking and steam created within the tightly covered container combine to destroy bacteria and make the slow cooker a safe process for cooking. Always defrost meat or poultry completely before putting it into a slow cooker.\n\nWrap and cook routines\n\nWe're big fans of wrapping fish fillets or single chicken breasts in baking paper \u2013 or even bamboo, banana leaves, fig leaves, grape leaves, corn husks, spinach leaves or cabbage leaves \u2013 along with lots of fresh herbs and baking in the oven or on the barbecue. Although you don't get any browning with this method, it seals in the flavour and works wonders with lean, fat-free cuts.\n\n**_Cooking with wine_**\n\nWine, vermouth, sherry, madeira and port make wonderful marinades or sauces. They impart a mellow flavour and zing and then are evaporated during cooking. Just how much is removed will depends on cooking time \u2013 the longer you cook it and the higher the temperature, the greater the evaporation and the less alcohol is retained in the final dish. If you add it towards the end of your cooking, the resulting dish will have a much higher alcohol content (in some cases as much as 85 per cent of the alcohol can remain) than if you add the alcohol at the beginning and heat it for an hour or so.\n\nWhat to use instead of alcohol:\n\n\u2022 savoury dishes \u2013 use stock with garlic and chopped fresh herbs, soy sauce, or Worcestershire sauce\n\n\u2022 sweet dishes \u2013 use orange juice with grated orange zest, dark grape juice, apple juice or a light sugar syrup with a dash of bitters and good lime cordial.\n[Cooking lighter  \nand healthier](Plum_9781742734859_epub_c5_r1.html#d7e2255)\n\nUse our food preparation tips and tricks to make your cooking lighter and healthier. They will save you time and money too.\n\nInvest in a non-stick fryingpan and simply brush or spray with a little oil for browning. Often an inexpensive pan works as well as a more costly one and you can replace it when it gets too 'used' on the surface.\n\nGrill, roast on a rack, steam, barbecue or cook in a microwave. Don't fry meat or chicken in oil, margarine, butter or ghee. Try wrapping in baking paper before baking or wrapping in foil for the barbecue.\n\nTrim off any visible fat from meat. At the supermarket or butchers, look for lean meat with the least amount of fat or marbling.\n\nTrim the fat and remove the skin from chicken pieces before cooking. The skin carries most of the chicken fat.\n\nBrush filo pastry with a little oil or water to achieve crisp golden results. If you love pastry, go for a one-crust pie at the top or bottom. Use sheets of frozen puff pastry.\n\nDrizzle a little oil into the wok when you cook a stir-fry \u2013 just enough to stop the food sticking. Or you can brush or spray a film of oil over the bottom of the wok. Alternatively stir-fry in stock for a change.\n\nReduce the fat of soups, casseroles and curries by cooking one day ahead and then refrigerating. Any fat will rise to the surface and can be easily skimmed off once it solidifies.\n\nMarinate lean cuts of meat and chicken in wine, fat-free sauces, garlic, mustard or even salad dressings. This tenderises lean meats and imparts a richer flavour.\n\nToss your peeled and diced vegetables in a plastic bag with a tablespoon of oil. This makes it really easy to get a nice even coating of oil on each piece. Or you can brush your vegetables with a little oil before roasting.\n[Making the most   \nof your freezer](Plum_9781742734859_epub_c5_r1.html#d7e2315)\n\nFrozen food, if stored correctly (at \u201318\u00baC\/0\u00baF for no more than 6 months), is the most nutritious and efficient way to preserve food. Make sure you label and date any foods or meals you freeze yourself.\n\nCompared with canned or dried food, frozen food retains a lot more of the important nutrients (particularly vitamin C, thiamin and folate which are heat-sensitive) and has a better texture. Fibre and minerals such as potassium and calcium are not affected by freezing. And there's little additional salt, a big drawback with canned vegetables.\n\nA freezer can help you eat more healthily \u2013 a frozen dinner thawed and heated with the addition of your own vegetables or fresh herbs makes for a more balanced meal than just cheese on toast or re-heating food that's been hanging around in the fridge for days.\n\nA freezer saves you time too. You can make up a large pot of soup at the weekend and freeze portions to heat during the week when you're pressed for time, adding a handful of fresh parsley or baby spinach leaves before you serve. Curries, casseroles and 'wet' dishes are ideal to freeze. And you can pack diced lamb, chicken breasts, steaks and salmon cutlets in family-friendly portions.\n\nHow long to freeze for?\n\n**_At-a-glance freezing times_**\n\n_Meat_|  Sausages | 1 month   \n---|---|---  \nOther raw cuts | 4\u20136 months |   \n_Poultry_|  Raw cuts | 6 months   \n_Seafood_|  Raw fatty fish (tuna, salmon) | 3 months   \n| Lean fish (whiting, snapper) | 4\u20136 months   \n| Cooked and shelled prawns | 3 months   \n_Fruit and vegetables_| |  3\u20136 months   \n_Cakes, biscuits, muffins_|  Sponge cake | 6 months   \n| Muffins and biscuits | 8 months   \n| Pancakes | 2 months\n\n_**Don't re-freeze**_\n\nDon't re-freeze meals once they've thawed out (it's okay to put them back if they're still frozen and firm to touch). If you're using the microwave, make sure you heat them thoroughly. If liquid, they must come to the boil.\n[Modifying your recipes  \nfor health and wellbeing](Plum_9781742734859_epub_c5_r1.html#d7e2365)\n\nMost of your existing recipes can be given a healthy makeover \u2013 you can cut the butter or oil, use plenty of garlic, ginger, lemon zest and fresh herbs instead of so much salt, trim the fat from meat or add chickpeas or soy beans for more fibre.\n\nMost (but not all) recipes work well with less butter, margarine or oil. Hang on to those family treasures and bake them on special occasions. For the rest, see how we give these recipes a makeover so they're better for you.\n\n**_Where your recipe has:_**|  _ **Try the following options instead:**_  \n---|---  \n _Cream_|  Use canned evaporated low-fat milk mixed with cornflour   \n_Sour cream_|  Try yoghurt mixed with a little cornflour or arrowroot   \n_Oil_|  \u2022 Halve the quantity   \n| \u2022 Cook in a non-stick wok or pan   \n| \u2022 Use cooking spray or brush on oil sparingly rather than pouring it in   \n**_Melted cheese topping_**|  \u2022 Halve the quantity of cheese and mix with breadcrumbs, rolled oats or crushed cornflakes   \n| \u2022 Use less of a strongly flavoured cheese like parmesan   \n_Pastry such as shortcrust or puff_|  Substitute filo, and spray lightly with oil between every second sheet (brushing with orange juice, skim milk or water instead of oil also works well)   \n_Lots of meat in casseroles or curries_|  \u2022 Halve meat quantity, replace with drained canned beans   \n| \u2022 Serve smaller portions of pasta, noodles, couscous or basmati rice   \n_Salt_|  \u2022 Omit, especially if the dish already contains salty ingredients like soy sauce, stock powder or bacon   \n| \u2022 Use fresh herbs, chilli, garlic, lemon zest, ginger and curry powder to add flavour   \n_Bread_|  Use low-GI varieties, such as soy and linseed or mixed grain   \n_White rice or potato_|  Use brown rice, pasta, bulghur wheat for low GI or use whole grain types   \n_Salad_ _dressings_|  Use vinegar or lemon juice (their acidity helps to slow down the digestion of carbohydrate) mixed with extra-virgin olive oil \nRecipe makeovers\n\nHow our recipes will boost your health and energy\n\nOur _Zest_ recipes will give you a healthy balance of the essential nutrients you need without overloading you with excess kilojoules. They are suitable for people with diabetes or high cholesterol or anyone who needs to lose a few kilos. You can check how each recipe stacks up by checking the nutrition analysis we have calculated. You'll see the quantity of kilojoules, total fat, saturated fat, fibre and sodium. In addition, we've highlighted those recipes that are especially low in fat, high in fibre, low in salt, or are low GI or gluten-free.\n\nThe recipe analyses are based on the listed ingredients in each recipe. We tell you how much of the rice or bread accompaniments are included in the analysis. Where a range of serve sizes is given (such as serves 4\u20136), the analysis has been done on the midpoint (that is, 5 serves).\n\n**_Recipe symbols_**\n\nThe recipes in this book use five special symbols to show at a glance which recipes are especially:\n\nKilojoules\n\nEach recipe shows its kilojoule (kJ) count. As most of us need to lose or maintain weight, try to eat foods that give you a low kilojoule-density. Plenty of salads, soups, vegetables, fruits, low-fat dairy and water will help you with that.\n\nIf you still think in calories, divide the kilojoule figure by 4.2.\n\nFat\n\nNot all fat is bad! Our bodies need a small amount of fat for good health and vitality. But it's got to be the good fats from foods such as oils, spreads, nuts, avocados and seeds. These foods will also boost your unsaturated fats and add valuable heart protectors.\n\nDon't drop below a minimum of 40 grams of fat a day. Around 50\u201380 grams of fat a day is advisable for women, 70\u2013100 grams for men.\n\nWe have classified recipes as low-fat if they have less than 15 grams of fat per serve for main meals, and less than 10 grams of fat per serve for light meals and desserts.\n\n_**Gluten-free recipes**_\n\nWe have highlighted those recipes that we are confident will contain no gluten. Many more recipes however can easily be made gluten-free by simply substituting gluten-free accompaniments (gluten-free bread, gluten-free pasta, rice noodles, rice, mash) for those listed. In addition, if you shop carefully, checking the label of sauces, pastes, stocks, mustard, yoghurt, ice cream and custard, you can make almost all our recipes suitable. Recipes with flour and pastry need special flour blends \u2013 you're better off buying gluten-free baking mixes and using the recipes they provide.\n\nSaturated fat\n\nLess than one-third of the total fat should be derived from saturated fat which translates to somewhere between 15 grams and 25 grams of saturated fat a day, depending on your total fat intake.\n\nAll the recipes concentrate on keeping saturated fat low, generally under 4 grams a serve, by minimising butter, cream, coconut and cheese. However it is impossible to avoid saturated fat entirely as it is found in all foods even those classified as 'monounsaturated' or 'polyunsaturated'.\n\nFibre\n\nAt least 30 grams of fibre a day is a good idea for adults. But as most people barely manage to take in just over 20 grams of fibre, this means that all of us would do well to increase our fibre intake from vegetables, legumes, whole grains, wholemeal bread, fruits and nuts. Our recipes make it easy to take in enough fibre, which will help keep your digestive tract working smoothly, lower your cholesterol and make weight control easier (high-fibre foods are usually filling and satisfying).\n\nFor children, a handy rule of thumb to working out how much fibre they should have is their age plus 5. So a 10-year-old should aim for 15 grams a day, being 10 + 5.\n\nWe have classified recipes as high fibre if they have 3 or more grams of fibre per serve.\n\nSodium\n\nA maximum of 1600 milligrams of sodium a day is recommended but most of us consume over twice this amount. We don't add salt to the recipes, but some salt will come from ready-made sauces, stock, mustard, cheese, bread and ham. Asian-based recipes with their reliance on soy, fish or oyster sauces tend to be high in salt too. We've used the salt-reduced versions, but go easy with these, as they are still quite high in salt. Wherever possible, we suggest you buy salt-reduced or no-added-salt versions. And remember, too, that it's easy to make your own salt-free stock.\n\nIf you are on a low-salt diet, you can consider main meal recipes with under 400 mg per serve as being low sodium. Breakfasts, light meals and desserts need to come in under 300 milligrams per serve.\n\nProtein\n\nA higher protein intake \u2013 without going to the extreme of an Atkins' diet \u2013 is a good idea if you're trying to shed weight or have high triglycerides or the metabolic syndrome. Protein is satiating (filling) and may boost your metabolism.\n\nThere's no need to count protein so we haven't listed any figures in the recipe analyses. It's more important to keep an eye on how much fat, saturated fat, fibre and kilojoules you eat. Most of us get enough protein without having to work at it.\n\nCarbohydrate\n\nWe've incorporated the 'good' carbohydrates into our recipes, which are those that are a whole grain (like brown rice) or low-GI (like chickpeas, kidney beans, traditional oats or grainy bread) and high in fibre (like wholemeal bread or sultanas). Like protein, there's no need to count carbohydrates unless you're on a strict Atkins' diet \u2013 which we don't recommend. So no carbohydrate figures appear in the recipe analyses. It's better to choose the 'good' carbohydrates than worry over how much of them you take in.\n\nSugar is a carbohydrate too! We haven't cut out all sugar, but in line with recommendations we use it moderately to enhance the flavour of a low-fat dessert or high-fibre loaf. You won't be consuming more than 10 per cent of your intake as added sugar this way, which we feel is still quite compatible with good health. A little sugar makes the good things go down better!\n\nWe also don't see the value of sugar-free cakes and desserts. Some sugar is needed for the flour and fat to work together and produce a nicely browned, well-risen baked product. When we make muffins or slices, we want them to work. But we don't want anyone to eat a lot of these!\n\n[Seven-day summer and  \nwinter meal plans](Plum_9781742734859_epub_c5_r1.html#d7e2445)\n\nTime to get started with the Nutrition for Life way of eating with plenty of healthy protein foods, all the right carbs, the right fats and an abundance of fruit and vegies to keep you and your family functioning at top speed \u2013 whatever your respective ages and stages of life!\n\nIn this section we have created two special weekly meal plans, one for easy summer eating and one for winter's cooler days when you'll feel more like hearty and warm 'comfort' foods.\n\nFor each day of the week we've put together meal ideas for breakfast, lunch and dinner, and snacks in between. You'll find plenty of suggestions about how to achieve the right balance at mealtimes \u2013 and how to get those two servings of fruit and five of vegetables every day.\n\nKeep in mind that our meal plans are a guide only to help you get started \u2013 they can be as flexible as you are creative. Simply substitute your favourite recipes from the book if you wish. Don't wait for Monday to get started \u2013 join the plan any day of the week that suits. We think that Monday is a good day for new beginnings, however, because you can take a little time over the previous weekend to stock the pantry.\n\nWhen you reach the end of your seven-day set meal plan, use the ideas we've provided to devise your own meal plans \u2013 or simply repeat the program if you enjoyed it.\nSummer\n\nMonday\n\nThe first healthy eating tip for your new way of eating is a simple reminder that it's all about balance.\n\nIt's important to eat a wide variety of foods. Make sure each week that you have a couple of meals with legumes, a couple of meals based on fish, and the rest based around lean meat, chicken, eggs or tofu.\n\nBreakfast\n\n  2\u20133 wholegrain breakfast biscuits such as Weet-Bix or Vita Brits with low-fat milk, topped with a sliced banana and a handful of sultanas\n\nSnack\n\n  Handful (30 g) of unsalted almonds or walnuts or a nut bar low in grains such as Be Natural\n\nLunch\n\n  Chicken, tabbouleh and salad wrap: use a thin mountain bread wholemeal wrap (or \u00bd a large wholemeal pitta pocket) and fill with sliced chicken breast, lots of tabbouleh, lettuce, tomato slices, onion rings and a smear of hummus. (Most delis and sandwich shops have tabbouleh, but if you want to make your own, see our recipe)\n\nSnack\n\n  Glass of chilled low-fat milk with a spoon of Milo or low-fat iced coffee\n\n  2 digestive biscuits such as Shredded Wheatmeal\n\nDinner\n\n  Fillet Steak with Pawpaw Salsa (opposite), served with steamed potatoes and a large mixed salad (made with mesclun, cucumber, capsicum and tiny tomatoes) tossed in an olive oil and lemon juice vinaigrette\n\n  Low-fat berry yoghurt topped with fresh blueberries and strawberries\n\n**_Energy tip: Put on your walking shoes..._**\n\nJust ten minutes of brisk walking can boost your mood and energy for 1\u20132 hours. So put on your walking shoes and stride out each day. Set a brisk pace \u2013 fast enough for you to be puffing slightly but still able to have a conversation. Try to include a few hills to improve your cardiac output.\n**Fillet steak with pawpaw salsa**\n\nServes 4\n\nPreparation time   15 minutes\n\nMarinating time   20 minutes\n\nCooking time   8 minutes\n\n4 x 100 g eye fillet steaks (or scotch fillet)\n\n\u00bd cup red wine\n\n1 tablespoon Dijon mustard\n\n1 tablespoon honey\n\n4 garlic cloves, crushed\n\nPawpaw salsa\n\n\u00bd pawpaw (papaya), chopped\n\n1 small red onion, finely chopped\n\n1 tablespoon chopped mint\n\n1 tablespoon chopped coriander\n\n1 teaspoon chopped chilli\n\njuice of \u00bd lime\n\nPlace the steaks in a shallow dish. Combine the wine, mustard, honey and garlic in a small bowl, mix well then pour over the steaks, coating thoroughly. Marinate for at least 20 minutes \u2013 in the refrigerator if the weather is very warm.\n\nTo make the pawpaw salsa, combine the chopped pawpaw, onion, mint, coriander and chilli in a bowl with the lime juice. Mix well, cover and set aside.\n\nPreheat the barbecue plate or char-grill on high. Cook the steaks for 3\u20134 minutes each side until cooked to taste.\n\nServe with the salsa, steamed potatoes and a large mixed salad.\n\nNUTRITION ANALYSIS 1 serve (including 3 potatoes and a generous serve of salad) = 1430 kJ, 9 g fat (includes 2 g saturated fat), 6 g fibre, 125 mg sodium\n\n**_Super ingredient: Pawpaw\/papaya_**\n\nPawpaw packs in two key vitamins \u2013 vitamin A (thanks to generous levels of beta-carotene which gives pawpaw its lovely golden-orange colour) and vitamin C \u2013 present at a concentration as high as in oranges. There's fibre as well as potassium and magnesium. One-quarter of a medium pawpaw supplies only 160 kJ (40 cal).\nTuesday\n\nStress, tension, rushing and eating on the run all take their toll on your digestion and health. When you sit down to eat, take a long deep breath and take the time to appreciate the food in front of you. Try to avoid mindless eating with the television on or the day's newspaper grabbing your attention! Focusing on your food will increase your enjoyment, help your digestion, and let you learn to stop eating when your stomach registers 'comfortably full' \u2013 a technique recommended as a way to help people lose weight.\n\nBreakfast\n\n  Large bowl of fresh fruit salad topped with low-fat vanilla yoghurt and toasted flaked almonds\n\n  Blackcurrant tea\n\nSnack\n\n  Carrot and celery sticks with hummus or tomato salsa\n\nLunch\n\n  Make up a thick wholegrain sandwich filled with thin slices of cold meat (rare roast beef is delicious), ripe tomato and fresh basil leaves. Add a small tub of coleslaw for extra fibre if you're a big eater\n\n  Sparkling mineral water\n\nSnack\n\n  Bunch of grapes with reduced-fat cheese and 3\u20134 wholegrain crackers\n\nDinner\n\n  Sang Choy Bow (opposite, crisp lettuce cups filled with a spicy mixture of chicken mince, sprouts, onions, coriander leaves, ginger and chilli), served with steamed low-GI rice\n\n  Jasmine tea or iced tea\n\n  Wedge of rockmelon topped with the pulp of one passionfruit. Add a scoop of low-fat vanilla ice cream if you like\n\n_**Energy tip: Work out with a friend**_\n\nIf you commit to meeting up with a friend for a walk or a work-out on a regular basis, you get to keep in touch while keeping fit. Find a friend or neighbour and make them your 'walking buddy'. Just one walk or swim a week with your buddy can make a difference.\n**Sang choy bow**\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   15 minutes\n\n1 tablespoon peanut oil\n\n1 red onion, chopped\n\n1 garlic clove, crushed\n\n1 teaspoon grated ginger\n\n500 g chicken mince\n\n\u00bc cup sweet chilli sauce\n\n1 tablespoon oyster sauce\n\n1 tablespoon salt-reduced soy sauce\n\n1 cup bean sprouts, plus extra for serving\n\n4 green onions (shallots), sliced\n\n2 tablespoons coriander leaves\n\n1 small red chilli, chopped\n\n8 lettuce cups (preferably iceberg or butter lettuce)\n\nHeat the oil in a wok or large frying pan on high. Stir-fry the onion, garlic and ginger for 1\u20132 minutes until the onion is tender.\n\nAdd the mince and stir-fry for 3\u20134 minutes, browning well and breaking the meat up as it cooks.\n\nCombine the sweet chilli, oyster and soy sauces and blend in. Lower the heat and simmer, stirring occasionally, for 4\u20135 minutes.\n\nJust before serving, stir in the sprouts, green onions, coriander and chilli. Toss over the heat for 1\u20132 minutes to quickly warm through.\n\nDivide the mixture evenly between the lettuce cups. Top with extra bean sprouts and serve immediately, accompanied by steamed low-GI rice.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup rice) = 1600 kJ, 16 g fat (includes 4 g saturated fat), 3 g fibre, 860 mg sodium\n\n**_Super ingredient: Chillies_**\n\nChillies pack a punch. They are concentrated in vitamin C \u2013 around 2\u20133 times greater than citrus fruit, weight for weight \u2013 and are high in fibre, minerals and the B vitamins, particularly riboflavin and niacin. Chillies can raise your metabolic rate, and thus give dieters the edge, as a body with a 'super-speed engine' burns fuel faster.\nWednesday\n\nBeing organised will help you avoid the temptation to grab takeaway food, or to skip a meal altogether. It may seem like a chore to start with, but when you think of the return on your investment, then it's a small price to pay. Keep a shopping list. Make sure you always have the basics on hand in the kitchen \u2013 like pasta, rice, tuna, herbs, frozen vegetables, frozen meat, eggs and grated cheese \u2013 which make it easy to whip up last-minute meals. Don't buy chocolate, crisps and other junk food if you know you can't resist them.\n\nBreakfast\n\n  Spoon some low-fat ricotta or cottage cheese into a bowl and top with whatever fresh fruit you have on hand \u2013 diced rockmelon or sliced banana with a handful of blueberries are both nice. Add a drizzle of honey if you like. Serve with a thick slice of good quality toasted fruit loaf\n\nSnack\n\n  Peach or nectarine\n\nLunch\n\n  Open sandwich of ham, mustard and salad: smear 2 thick slices of chewy wholegrain bread with a little spread and Dijon mustard, then add shaved ham and top with lettuce, cucumber, tomato and artichoke halves\n\nSnack\n\n  Low-fat honey yoghurt \u2013 or top a low-fat natural yoghurt with a dollop of honey if you prefer\n\n  A handful of mixed nuts\n\nDinner\n\n  Fusilli with Salmon and Baby Spinach (opposite), served with a large tossed leaf salad with an oil\u2013vinegar dressing (or try our No-oil Vinaigrette)\n\n  Scoop of lemon gelato\n\n_**Energy tip: Watch your alcohol**_\n\nAlcohol drains your energy. While a drink or two can help you relax and unwind, any more than that acts as a depressant. Worst of all, alcohol uses up the body's stores of B vitamins, especially thiamin (vitamin B1) and folate. We need B vitamins to release energy from carbohydrates. So moderate your drinking.\n**Fusilli with salmon and baby spinach**\n\nServes 4\n\nPreparation time   5 minutes\n\nCooking time   10 minutes\n\n500 g fusilli, or another pasta of choice\n\nspray oil\n\n250 g punnet cherry tomatoes, halved\n\n2 garlic cloves, sliced\n\n210 g can red or pink salmon, drained and flaked\n\n\u00bd cup extra-light cream or light evaporated milk\n\njuice 1 lemon\n\n60 g baby spinach leaves\n\nchopped chives and grated parmesan cheese to serve\n\nCook the pasta in plenty of boiling water, following packet instructions. Drain the pasta well and set aside, keeping warm.\n\nHeat a large frying pan on high. Spray with oil. Saut\u00e9 the tomatoes and garlic for a minute then stir in the salmon, cream and lemon juice. Lower the heat and simmer gently for 2\u20133 minutes, stirring from time to time, until the mixture has thickened slightly.\n\nToss the sauce through the hot pasta with the spinach leaves. Serve topped with chives and a little parmesan. Accompany with salad.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 2550 kJ, 13 g fat (includes 4 g saturated fat), 8 g fibre, 295 mg sodium\n\n**_Super ingredient: Canned salmon_**\n\nLow in saturated fat, rich in zinc and a great source of omega-3s \u2013 there are plenty of reasons to eat canned salmon. It is also high in protein and full of iodine, potassium and zinc. Make sure you eat the small edible bones, one 100 g can provides 200\u2013230 mg of calcium \u2013 20 per cent of the recommended daily intake.\nThursday\n\nMake your fridge friendly. Don't fill it up with tempting high-kilojoule food that other family members can eat but you can't. You won't be able to stay that strong and committed every time you open the door! Instead, fill your fridge with food to make healthy eating easy \u2013 low-fat milk, low-fat yoghurt, vegetables, salad ingredients, tomatoes, chilled water, tomato juice, cold meats, hard-boiled eggs. These all add bulk for few kilojoules. Store leftover cooked vegies in small containers \u2013 they make nice nibbles.\n\nBreakfast\n\n  Banana smoothie made with low-fat milk, banana and honey\n\n  Or try our Banana and Berry Breakfast Smoothie\n\nSnack\n\n  1 Rhubarb Muffin\n\nLunch\n\n  Toss together a large pasta salad: start with cooked penne or macaroni and add hard-boiled eggs, diced celery and eschalots and half an avocado. Stir in a light creamy dressing (don't drown it \u2013 just enough to coat everything) or try our No-oil Creamy Dressing, and finish with a sprinkle of toasted pine nuts\n\nSnack\n\n  Fresh strawberries with low-fat yoghurt, sprinkled with flaked almonds\n\nDinner\n\n  Stir-fry Prawn Salad (opposite) on a bed of rocket, served with a slice of sourdough bread\n\n  Half a mango topped with the pulp of 1 passionfruit\n\n_**Energy tip: Fit exercise in around your day**_\n\nCan't get the closest spot in the car park? No hassles. View the extra walk as something healthy that can invigorate you. Or take the stairs (two at a time if you like) instead of the escalator; walk to work or the shops; stretch as you wait for your emails to download; jog on the spot while you wait for the microwave.\n**Stir-fry prawn salad**\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   5 minutes\n\n500 g green prawns, peeled, de-veined and tails intact\n\n\u00bc cup lemon juice\n\n1 tablespoon thinly sliced lemongrass\n\n1 tablespoon sweet chilli sauce\n\n2 teaspoons salt-reduced soy sauce\n\n2 teaspoons fish sauce\n\n2 teaspoons peanut oil\n\n1 Lebanese cucumber, seeded and sliced\n\n60 g snow peas, trimmed and halved diagonally\n\n1 red onion, finely sliced\n\n\u00bc cup chopped mint\n\n\u00bc cup coriander leaves\n\n3 green onions (shallots), sliced baby rocket leaves\n\nCombine the prawns, lemon juice, lemongrass and sweet chilli, soy and fish sauces with the oil in a large bowl.\n\nHeat a wok or a large frying pan on high. Add the prawns with the marinade, tossing well. Stir-fry for 2\u20133 minutes until the prawns change colour.\n\nTip the prawns and sauce back into the same cleaned bowl. Add the cucumber, snow peas, onion, herbs and green onion, tossing to combine. Serve immediately on a bed of baby rocket with a slice of sourdough bread.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 955 kJ, 5 g fat (includes 1 g saturated fat), 5 g fibre, 935 mg sodium\n\n**_Super ingredient: Prawns_**\n\nPrawns are rich in protein, a number of B vitamins and many essential minerals like zinc, iron and potassium. They have little fat \u2013 and what fat there is gives you a bonus of omega-3 fatty acids. Their cholesterol once took prawns off the okay list, but now dietitians recognise that their saturated fat is so low, that this overrides the cholesterol.\nFriday\n\nPlan and shop ahead so you have ingredients on hand in your freezer or your cupboard to make healthy dinners when you're tired at night. That way, you avoid the temptation to grab takeaway meals, which are almost always higher in fat, salt and sugar than a healthy home-cooked meal. Cook double the quantity and freeze portions for later.\n\nBreakfast\n\n  Bowl of natural muesli with low-fat milk topped with low-fat yoghurt and fresh or canned apricots\n\n  If you like to make your own muesli, try our recipe for Superstar Mega-Muesli or Home-made Muesli\n\nSnack\n\n  Small pack of pretzels\n\nLunch\n\n  For a quick-and-easy portable office lunch, grab a can of tuna in spring water and a can of 3-bean mix. Drain both and mix together, leaving the tuna chunky. Make or buy a takeaway mixed salad. Top the salad with the tuna-bean mix and enjoy with 3\u20134 rye crispbreads\n\nSnack\n\n  Toasted wholegrain English muffin with low-fat cottage cheese and tomato slices\n\nDinner\n\n  Spinach and Cheese Filo Pie (opposite) with a Greek salad (made with chopped tomatoes, sliced cucumber, cubed or crumbled fetta cheese, sliced red onion and black olives. Drizzle with extra-virgin olive oil and red wine vinegar)\n\n  A nectarine or slices of fresh pineapple\n\n_**Energy tip: Sleep well, recharge your batteries**_\n\nHere's how to beat insomnia.\n\n\u2022 Forget caffeine (coffee or hot chocolate) after 4 pm.\n\n\u2022 Sip warm low-fat milk and honey before bedtime. Milk contains the amino acid tryptophan, which increases levels of serotonin in the brain. This induces a more restful state.\n\n\u2022 No spice. Chilli and pepper raise the body's temperature and metabolic rate, which can have a stimulating effect.\n**Spinach and cheese filo pie**\n\nServes 4\u20136\n\nPreparation time   20 minutes\n\nCooking time   30 minutes\n\n2 tablespoons olive oil\n\n1 bunch fresh silverbeet, trimmed and chopped, or 500 g frozen spinach, thawed and well drained\n\n6 green onions (shallots), sliced\n\n1 onion, chopped\n\n4 eggs, lightly beaten\n\n125 g fetta cheese, crumbled\n\n125 g ricotta cheese\n\n\u00bc cup grated parmesan\n\n\u00bc cup chopped parsley\n\n\u00bc teaspoon ground nutmeg\n\n12 sheets filo pastry\n\nspray oil\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Lightly grease a 20 cm springform pan, or a round or square casserole dish.\n\nHeat the oil in a frying pan on high. Saut\u00e9 the spinach and onions for 4\u20135 minutes until the spinach has wilted. Press the mixture into a strainer to remove as much excess liquid as possible and allow it to cool slightly.\n\nTip the spinach mixture into a large bowl and blend in the beaten eggs, cheeses, parsley and nutmeg. Mix well.\n\nFold 6 sheets of filo into halves. Spray the surface of each folded piece with oil and layer them in the base of the prepared pan. Spoon in the filling and spread evenly. Fold the remaining sheets of filo in half and spray with oil.\n\nLayer on top of filling to cover.\n\nBake for 20\u201325 minutes until crisp, set and golden. Cut into portions and serve hot or warm with a Greek salad.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 1740 kJ, 24 g fat (includes 7 g saturated fat), 7 g fibre, 735 mg sodium\n\n**_Super ingredient: Spinach_**\n\nSpinach is an excellent source of vitamin C, folate, beta-carotene as well as vitamin E. It also contains a number of antioxidants, particularly lutein and zeaxanthin, which help keep macular degeneration (a leading cause of blindness in older people) at bay. Try to include it (cooked or as a salad of baby spinach leaves) in your meals at least twice a week.\nSaturday\n\nThis is the weekend \u2013 a chance to relax and fit in some serious exercise. It's also an opportunity to cook something special for family or friends. When dinner is served, take your time. Before beginning to eat, try the 'one-minute pause'. Relax, breathe out deeply and concentrate on yourself and your inner feelings of hunger. Check mentally just how hungry you really are and eat to your appetite.\n\nBreakfast\n\n  2 eggs (poached or scrambled) on wholegrain toast with grilled tomato and mushrooms\n\n  Or try our Big Healthy Breakfast with Baked Beans\n\nSnack\n\n  3\u20134 rice cakes or crispbreads topped with reduced-fat cheese and tomato slices\n\nLunch\n\n  Lunch pack of small sushi rolls or 2\u20133 California rolls\n\nSnack\n\n  Low-fat passionfruit yoghurt\n\n  Slice of Banana and Nut Loaf\n\nDinner\n\n  Barbecued Greek Lamb with Minted Couscous (opposite), served with a large mixed salad (made with mesclun, cucumber, capsicum and tiny tomatoes) tossed in an olive oil and lemon juice vinaigrette (or try our No-oil Vinaigrette). This meal is ideal for entertaining. Serve with a slice of crusty wholemeal or multigrain bread\n\n  Never-fail Berry Fool\n\n_**Energy tip: Exercise gives you so much**_\n\nExercise burns off fat, helps raise the 'good' HDL-cholesterol, keeps your bones strong (if it's weight bearing) and leaves you feeling good about life. If you are desk-bound, exercise will clear your mind and improve your mood. Aim for 30 minutes of vigorous exercise a couple of times a week \u2013 vigorous enough that you are a little out of breath but not exhausted.\n**Barbecued Greek lamb with minted couscous**\n\nServes 6\u20138\n\nPreparation time   10 minutes\n\nCooking time   20 minutes\n\nMarinating time   20 minutes\n\n1.5 kg boned lamb leg, all visible fat trimmed\n\n\u00bc cup olive oil\n\ngrated zest and juice of 2 lemons\n\n4 garlic cloves, crushed\n\n2 tablespoons chopped parsley\n\n\u00bc cup honey (optional)\n\nlow-fat natural yoghurt to serve\n\nMinted couscous\n\n1 cup boiling water\n\n\u00be cup couscous\n\n\u00bc cup sliced pitted black olives\n\n\u00bc cup chopped sun-dried tomatoes\n\n\u00bc cup chopped mint\n\n1 green onion (shallot), chopped\n\nPlace the lamb in a shallow dish. Combine the olive oil, lemon zest and juice, garlic and parsley in a small bowl. Mix well then pour over the lamb, turning it over to make sure it is well coated in the marinade. Cover and set aside in the refrigerator to marinate for at least 20 minutes.\n\nPreheat a barbecue plate to high. Barbecue the lamb, basting it occasionally, for 8\u201310 minutes each side or until cooked to your liking. Drizzle the honey over the lamb if desired.\n\nTo make the minted couscous, combine the water and couscous in a bowl, cover and leave to stand for 10 minutes until all liquid has been absorbed. Fluff the couscous with a fork then stir in the olives, sun-dried tomatoes, mint and green onion.\n\nSlice the lamb thinly and serve with the minted couscous and yoghurt, and a mixed salad and crusty wholemeal or multigrain bread.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup couscous, a dollop of yoghurt, 1 slice bread and a generous serve of salad) = 2810 kJ, 26 g fat (includes 8 g saturated fat), 5 g fibre, 420 mg sodium\n\n**_Super ingredient: Lamb_**\n\nLean lamb offers high-quality protein plus iron, zinc and potassium and a range of B vitamins. Like all red meat, it's a good source of haem iron, the type most easily absorbed by the body. Most women don't eat nearly enough iron-rich foods to meet their body's daily needs, but recipes like this provide a delicious way to get that vital 18 milligrams a day!\n\nSunday\n\nFor Sunday night we have opted for a fast and friendly meal. Burgers are one of the best fast-food choices when eating out. They're easy to make at home too \u2013 and the kids will love them! The combination of meat pattie with bread and salad makes a hamburger a healthy choice. It's lightly pan fried, so there's less fat, less kilojoules and less 'bad' saturated fat. While little kids usually tuck into them plain with tomato sauce, teens and parents can pile on beetroot, onion, tomato, cheese or chilli sauce \u2013 whatever takes your fancy.\n\nBreakfast\n\n  Thick slice of a good quality fruit loaf, toasted and topped with ricotta, slices of pear and chopped walnuts\n\nSnack\n\n  Slices of fresh plum or peach mixed into low-fat fruit yoghurt\n\nLunch\n\n  1\u20132 burritos with kidney beans, lettuce, tomato and diced avocado\n\nSnack\n\n  Handful of dried apricots and unsalted walnuts\n\nDinner\n\n  Homeburger with the Lot (opposite), served with baked potato wedges\n\n  Or try making burgers using Tofu and Cannellini Bean Patties\n\n  Scoop of low-fat ice cream topped with fresh fruit salad\n\n_**Energy tip: Don't skip breakfast**_\n\nPower through your day on a proper breakfast. It:\n\n\u2022 fuels your body and lifts its metabolic rate, helping you burn more kilojoules\n\n\u2022 boosts your mental performance (memory, concentration) for the day\n\n\u2022 means you're less likely to be tempted by fattening snacks later on.\n**Homeburger with the lot**\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   10 minutes\n\n400 g trim beef or lamb mince\n\n1 onion, finely chopped\n\n2 tablespoons chopped parsley\n\n1 tablespoon tomato sauce\n\nspray oil\n\n2 slices prosciutto, halved\n\n2 onions, sliced\n\n2 slices reduced-fat tasty cheese, halved\n\n4 eggs (optional)\n\n4 wholegrain rolls, split\n\n60 g mixed lettuce leaves\n\n2 tomatoes, sliced\n\n4 slices beetroot, drained\n\ntomato sauce and mustard pickles to serve\n\nCombine the mince, onion, parsley and tomato sauce in a large bowl until well mixed and form into 4 even-sized, flattened patties.\n\nSpray a large non-stick frying pan with oil and heat on high. Cook the patties for 4\u20135 minutes each side until well browned and cooked through. Add the prosciutto to same pan and cook for 1\u20132 minutes until crisp.\n\nMeanwhile, spray a separate non-stick pan with oil and saut\u00e9 the onion on medium heat for 5\u20138 minutes until soft and golden. Keep warm.\n\nTop each pattie with a slice of cheese and continue cooking until the cheese has melted. If you are having the eggs, now's the time to break the eggs into the pan and cook to your liking.\n\nWhen ready to serve, toast the rolls and arrange them on each plate. Layer lettuce, tomato, beetroot and a homeburger pattie on one half of each roll. Top with the onion, egg and prosciutto, sauce and pickles and finish with the top half of the roll. Serve immediately with baked potato wedges.\n\nNUTRITION ANALYSIS 1 serve (including 4 wedges of potato) = 2565 kJ, 21 g fat (includes 6 g saturated fat), 11 g fibre, 780 mg sodium\n\n**_Super ingredient: Parsley_**\n\nParsley is incredibly rich in vitamin C, folate, beta-carotene and fibre, with generous quantities of potassium and magnesium. It's also a top source of antioxidants which help prevent damage to our body's cells and so lessen the risk of cancer and eye damage as we age. So be generous with the parsley and make these flavoursome greens a regular part of your cooking.\nWinter\n\nMonday\n\nWatch those 'low-fat snacks'. Most snackfoods labelled 'low-fat' or '97% fat-free' should be eaten only occasionally. You can't eat twice as much just because they're labelled 'fat-free'. Most of these foods have similar kilojoule counts to their full-fat cousins, because the fat is replaced with enough sugar or starch to make up for the missing fat.\n\nFor instance, a regular, medium-size banana muffin contains 8 grams of fat and 710 kilojoules (170 calories). In its low-fat form, it has a tiny 1 gram of fat but still hits 675 kilojoules \u2013 hardly a huge saving for your waistline or hips. Keep in mind as well, that many 'lite' biscuits, muesli bars and cereals are also sold in smaller serve sizes (20 grams for a low-fat biscuit compared to 40 grams for the regular version) so the kilojoule savings appear sizeable simply because you're eating less food.\n\nBreakfast\n\n  Baked beans on a thick slice of toasted wholegrain English muffin\n\nSnack\n\n  A nut or seed bar (look for one with mostly nuts or seeds and few grains)\n\nLunch\n\n  Toasted wholegrain sandwich filled with roasted Italian vegetables and a dollop of pesto (try our delicious pesto recipes)\n\nSnack\n\n  Hot chocolate made with low-fat milk\n\n  2 date slices or Full-o-Fruit biscuits\n\nDinner\n\n  Veal Goulash with Fettucine (opposite), served with a spoonful of low-fat natural yoghurt and lightly steamed broccolini or a side salad\n\n  Low-fat ice cream topped with canned plums or peaches\n\n_**Energy tip: Do it in the morning**_\n\nIf you can, slot in your exercise first thing in the morning. This keeps your metabolism 'revving' high over the day. And there's evidence suggesting a morning jog or walk is more effective than an afternoon or evening one. Your body stores of glycogen are at their lowest after the night's sleep, so you tend to turn burn fat more readily.\n**Veal goulash with fettucine**\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   1\u00be hours\n\n1 tablespoon olive oil\n\n750 g diced veal\n\n1 onion, chopped\n\n1 carrot, diced\n\n2 garlic cloves, crushed\n\n1 tablespoon sweet paprika\n\n2 tablespoons tomato paste\n\n400 g can diced tomatoes\n\n1 cup salt-reduced beef stock or water\n\n300 g egg fettucine\n\nlow-fat natural yoghurt to serve\n\nHeat half the oil in a large, heavy-based saucepan on medium. Brown the veal in two batches for 3\u20134 minutes then transfer to a plate.\n\nReduce the heat and add the remaining oil to the pan. Saut\u00e9 the onion, carrot and garlic for 5 minutes until soft and lightly golden.\n\nBlend in the paprika and cook, stirring, for 1 minute. Stir in the tomato paste followed by the tomatoes, stock and browned veal. Simmer, covered, over a low heat for 1\u00bd hours, until the meat is very tender. Uncover for last 15 minutes to allow the sauce to thicken slightly.\n\nCook the fettucine according to the packet instructions. Drain well. Serve the goulash with fettucine and a dollop of yoghurt and accompany with lightly steamed broccolini spears or a side salad.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup fettuccine and a generous serve of broccolini) = 2545 kJ, 11 g fat (includes 2 g saturated fat), 7 g fibre, 205 mg sodium\n\n**_Super ingredient: Veal_**\n\nVeal is a dieter's best friend. It is one of the leanest meats with a mere 1 per cent fat and a low kilojoule count. It's light yet filling. And you'll still get a good serve of protein and the B group vitamins. It falls midway between red and white meat, so you get iron and zinc, although not as much as from red meat.\nTuesday\n\nDon't forget to keep up your fluid intake. The human body is largely made up of water and every day you need to drink enough fluid to prevent dehydration. If you are dehydrated, you may experience symptoms such as headache, fatigue, heat intolerance, loss of appetite, flushed skin and muscle cramps. Over the years, not drinking enough fluid has been linked with a greater chance of kidney stones, urinary tract infections and bladder cancer.\n\nNutritionists recommend that women drink at least 8 glasses and men 12 glasses of fluid a day \u2013 more in hot weather or if you are sweating through work or exercise. Water is the best way to quench your thirst but other fluids such as fruit juices, mineral waters, tea, herbal teas and clear soups can count towards your total fluid intake.\n\nBreakfast\n\n  Porridge made with low-fat milk and topped with sliced banana and raisins\n\nSnack\n\n  1 piece of fruit toast topped with low-fat ricotta and drizzled with a little honey\n\nLunch\n\n  Noodle stir-fry with chicken and vegetables bought from an Asian food bar\n\nSnack\n\n  3\u20134 wholegrain crackers with reduced-fat cheese slices\n\nDinner\n\n  Potato, Cauliflower and Chickpea Curry (opposite), served with steamed low-GI rice and pappadams (these only take 1 minute each in the microwave)\n\n  Poached pears with low-fat vanilla yoghurt or custard\n\n_**Energy tip: Keep up the exercise**_\n\nWhen it's too cold or rainy to go walking, bring your work-out indoors. Invest in a few pieces of home gym equipment like a stationary bike, a mini-trampoline or just a skipping rope and some weights. Or buy one of the many home-exercise videos available and follow the routines every day or every other day.\n**Potato, cauliflower and chickpea curry**\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   25 minutes\n\nspray oil\n\n1 onion, roughly chopped\n\n2 teaspoons grated fresh ginger\n\n1 garlic clove, crushed\n\n2 tablespoons Madras curry powder\n\n1 teaspoon ground coriander\n\n1 teaspoon ground cumin\n\n2 potatoes, peeled and cubed\n\n\u00bc cauliflower, cut into florets\n\n400 g can chickpeas, drained\n\n4 roma tomatoes, chopped (or 400g can diced tomatoes)\n\n1 cup water\n\n200 g green beans, trimmed and chopped into 4 cm lengths\n\n\u00bd cup frozen peas\n\n\u00bd cup roughly chopped toasted cashews\n\n\u00bc cup chopped coriander\n\nHeat a large heavy-based saucepan on medium. Spray with oil and saut\u00e9 the onion, ginger and garlic for 2 minutes until the onion is soft.\n\nStir in the curry powder, coriander and cumin. Cook, stirring, for 2 minutes until it is aromatic.\n\nAdd the potatoes, cauliflower, chickpeas, tomatoes and water. Simmer, covered, for 20 minutes.\n\nStir through the beans and peas and simmer for a further 5 minutes until the potatoes are tender.\n\nAdd cashews and coriander and serve with steamed low-GI rice and pappadams.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup rice and 2 pappadams) = 1885 kJ, 12 g fat (includes 2 g saturated fat), 15 g fibre, 660 mg sodium\n\n**_Super ingredient: Chickpeas_**\n\nWith their low GI, chickpeas also offer substantial amounts of protein \u2013 which makes them very useful to vegetarians \u2013 as well as a good dose of B vitamins and significant fibre content. Their fibre is of the soluble type, which researchers have found helps remove cholesterol from the body (in a similar way to oat bran).\nWednesday\n\nSit down to eat, even if you're only having a cup of tea or an apple. This forces you to think twice about stopping to eat (are you really that hungry?) and you may find you cannot be bothered to stop. It also helps you to register that you're actually eating. Most overeating is thoughtless: snacking while you walk, picking at food while cooking, nibbling in front of television. Concentrate on the meal in front of you and enjoy every mouthful.\n\nBreakfast\n\n  Scrambled eggs with fresh herbs on wholegrain toast\n\nSnack\n\n  Handful of unsalted mixed nuts\n\nLunch\n\n  Bowl of vegetable soup and a wholegrain roll\n\nSnack\n\n  Low-fat blueberry muffin\n\nDinner\n\n  Slow-roasted Salmon with Teriyaki Sauce (opposite), served on a bed of steamed spinach with buckwheat (soba) noodles\n\n  Low-fat vanilla yoghurt and slices of kiwi fruit\n\n_**Energy tip: Incidental exercise**_\n\nIt's vital that you move more in your day-to-day life. This incidental exercise (climbing the stairs, walking to the bank or post office) is a key to weight control. Researchers have shown that people who maintain their weight over the years \u2013 even if they aren't doing a formal sport or going to the gym \u2013 are the ones who move more in the course of their day.\n**Slow-roasted salmon with teriyaki sauce**\n\nServes 4\n\nPreparation time   5 minutes\n\nCooking time   30 minutes\n\nspray oil\n\n2 tablespoons teriyaki sauce, plus extra to serve\n\n1 tablespoon rice vinegar\n\ngrated zest and juice of \u00bd lime\n\n1 teaspoon sesame oil\n\n1 teaspoon grated fresh ginger\n\n1 garlic clove, chopped\n\n4 salmon fillets, bones removed\n\n1 bunch spinach, trimmed and lightly steamed\n\nPreheat the oven to very slow, 120\u00baC (250\u00baF). Lightly spray a baking pan with oil.\n\nIn a small bowl, combine the teriyaki sauce, vinegar, lime zest and juice, oil, ginger and garlic.\n\nPlace the salmon in the prepared baking pan and brush liberally with the marinade.\n\nBake the fish for 25\u201330 minutes, basting occasionally, until the flesh flakes easily when tested with a fork (it will still be pink).\n\nServe the fish on a bed of steamed spinach drizzled with extra teriyaki sauce and accompany with buckwheat (soba) noodles.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup noodles and a generous serve of spinach) = 1670 kJ, 13 g fat (includes 3 g saturated fat), 7 g fibre, 210 mg sodium\n\n**_Super ingredient: Limes_**\n\nLimes like other citrus fruit are nutritional all-rounders, packed with vitamin C, which enhances iron absorption, speeds up wound healing and reduces the risk of a heart attack. They also provide small amounts of minerals and some B vitamins. Don't just juice them. There are plenty of nutritional goodies in the skin too. So bring out the zester.\nThursday\n\nBe fat-smart, not fat-obsessed. A small amount of fat is crucial for good health, vitality, clear skin and shiny hair. Your body needs essential fatty acids to absorb certain vitamins and antioxidants like beta-carotene and lycopene. The trick is to choose the 'good' monounsaturated and polyunsaturated types, and cut back on the 'bad' saturated fats found in butter and cream, deli meats, fatty meats, sausages, pies, pastries, and cakes.\n\nBreakfast\n\n  Cinnamon, Pear and Date Porridge\n\nSnack\n\n  3\u20134 rye crispbreads topped with avocado\n\nLunch\n\n  Takeaway burger with lettuce, tomato and onion. Ask for a wholegrain bun or bread\n\nSnack\n\n  Raisin and walnut snack-pack\n\nDinner\n\n  Tofu with Mixed Vegetables (opposite), served with steamed basmati rice or Asian (such as hokkein) noodles\n\n  Rice Custard\n\n_**Energy tip: Get your spine flat**_\n\nLie down flat on your back for 10 minutes each day, especially in the afternoon when the mid-afternoon energy slump sets in. This is a good thing to do if you're working at the computer all day. Lying down takes the pressure off the spine and allows the discs to decompress. So it helps you stay alert and overcome drowsiness.\n**Tofu with mixed vegetables**\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   8 minutes\n\n300 g firm tofu\n\n\u00bc cup seasoned plain flour\n\n1 tablespoon oil\n\n1 red capsicum, seeded and sliced\n\n1 onion, sliced\n\n1 teaspoon chopped fresh ginger\n\n1 garlic clove, crushed\n\n1 bunch broccolini, trimmed\n\n1 bunch baby bok choy, quartered\n\n1 bunch choy sum, sliced\n\n\u00bc cup salt-reduced vegetable stock or water\n\n2 tablespoons salt-reduced soy sauce\n\n1 tablespoon sweet chilli sauce\n\n1 teaspoon sesame oil\n\nCut the tofu into slices. Place the flour and tofu in a plastic bag and shake to coat well. Shake off any excess flour and reserve the tofu.\n\nHeat the oil in a wok or large frying pan on high. Stir-fry the tofu in batches for 2\u20133 minutes until golden, then drain on paper towels. Add the capsicum, onion, ginger and garlic to the wok and stir-fry for 1 minute. Add the broccolini, bok choy and choy sum and stir-fry for 2 minutes.\n\nCombine the stock, soy sauce, sweet chilli sauce and sesame oil in a jug. Pour the mixture over the vegetables and stir-fry for 1 minute. Return the tofu to the wok and toss gently until heated through. Serve immediately with basmati rice or hokkein noodles.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup rice or noodles) = 1305 kJ, 8 g fat (includes 1 g saturated fat), 7 g fibre, 470 mg sodium\n\n**_Super ingredient: Tofu_**\n\nTofu is a key source of protein and B vitamins and also contains a number of minerals. Tofu has less than 5 per cent fat, no cholesterol and is virtually free of sodium (salt). If the setting agent used is calcium sulphate (sometimes it's nigari, which is potassium sulphate), it will be high in calcium, making it a well-rounded superfood.\nFriday\n\nDid your mother or grandmother insist you eat three different-coloured vegetables at every meal? Nutritionists now confirm that 'eating by the rainbow' makes good nutrition sense because it increases the variety of vegetables you're consuming and ensures you eat the ones richest in the beneficial antioxidants. For instance, yellow-orange vegetables give us beta-carotene, which is converted into vitamin A. Red vegetables like tomatoes are rich in lycopene, an antioxidant that can help protect the prostate. Green vegetables have lots of chlorophyll, which is high in magnesium.\n\nBreakfast\n\n  2 slices of raisin toast with light margarine\n\nSnack\n\n  Low-fat strawberry yoghurt\n\nLunch\n\n  Bowl of penne pasta with green peas, ricotta and crispy pancetta\n\nSnack\n\n  2 mandarins or 1 orange\n\nDinner\n\n  Hearty Chowder (opposite), served with crusty wholegrain bread\n\n  Baked apple with a dollop of low-fat yoghurt and a sprinkle of chopped walnuts\n\n  Or try Chunky Apple Bake\n\n_**Energy tip: Beat the cold with exercise and sun**_\n\nWhatever exercise you choose, aim to spend at least 10 minutes a day outdoors during winter months. The pineal gland, located at the base of the brain, is 'switched off' by lack of light. It appears to control the mechanisms for weight gain, depression, lack of energy and poor sleep during these grey months, so it makes sense to keep it functioning.\n**Hearty chowder**\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\nCooking time   25 minutes\n\n1 rasher shortcut bacon, finely chopped\n\n2 stalks celery, finely chopped\n\n1 onion, finely chopped\n\n1 large potato, peeled and diced\n\n2 cups water\n\n2 tablespoons plain flour\n\n2 cups low-fat milk\n\n300 g can corn kernels, drained\n\n200 g smoked cod, skinned and chopped\n\n\u00bd cup frozen peas\n\n1 tablespoon chopped celery leaves\n\nHeat a large saucepan on high. When the pan is warm, add the bacon, celery and onion. Saut\u00e9 for 2\u20133 minutes until the onion is tender.\n\nAdd the potato and cook, stirring, for 1 minute. Stir in the water and bring to the boil, then reduce the heat and simmer, covered, for 10 minutes.\n\nBlend the flour with a little of the milk to form a smooth paste. Add to the pan with the remaining milk, and blend in well. Stir constantly until the chowder comes to a gentle boil and thickens a little.\n\nStir in the corn, smoked cod and peas. Simmer for 5 minutes until heated through. Serve sprinkled with celery leaves. Serve with crusty wholegrain bread.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 1155 kJ, 4 g fat (includes 2 g saturated fat), 5 g fibre, 750 mg sodium\n\n**_Super ingredient: Corn_**\n\nCorn has the virtues of both a vegetable and a grain. It gives you some vitamin C and folate but also offers those key grain nutrients such as the B group vitamins, fibre, a little potassium and iron. It has a low GI making it ideal to serve as a snack, light meal or side dish to help you keep those blood glucose levels low. Just hold the salt and butter.\nSaturday\n\nToday's main meal was inspired by Mediterranean cuisine. Just count the long list of nutritional blessings from Italian, Greek or Moroccan cuisines, the best-known examples of Mediterranean cuisines, all of which have been thoroughly studied.\n\nHere's what you get:\n\n\u2022 vegetables, salads and herbs, which are rich in antioxidants, fibre and folate\n\n\u2022 seafood (calamari, sardines, prawns and fish), which is a top source of omega-3 fatty acids\n\n\u2022 tomatoes, noted for their lycopene, a powerful antioxidant and prostate-protector\n\n\u2022 nuts, which supply arginine, a blood vessel 'relaxant', and vitamin E, another antioxidant\n\n\u2022 garlic and onions, which both contribute sulphur compounds, good for the blood\n\n\u2022 pasta and bread, which are rich in B vitamins, of which niacin and folate actively help the heart\n\n\u2022 red wine, which adds polyphenol antioxidants.\n\nBreakfast\n\n  Muesli with low-fat milk topped with low-fat yoghurt and canned apricots\n\nSnack\n\n  A pear or an apple\n\nLunch\n\n  Reduced-fat cheese, ham and tomato melt on wholegrain bread\n\nSnack\n\n  3\u20134 rice cakes spread with peanut butter\n\nDinner\n\n  Chicken Tagine with Olives and Couscous (opposite), served with a green vegetable such as lightly steamed green beans, asparagus or spinach\n\n  Dried Fruit Compote\n\n_**Energy tip: Time away to recharge**_\n\nTreat yourself to a short break or weekend away. A change of scene (and time off) gives you a new perspective on things and 'recharges your batteries'. It's a good idea to occasionally 'escape' from home, with its constant reminders of things waiting to be done.\n**Chicken tagine with olives and couscous**\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   50 minutes\n\n1 tablespoon olive oil\n\nsize 18 chicken (1.8 kg), segmented, skin and fat removed\n\n1 onion, finely sliced\n\n3 garlic cloves, crushed\n\n1 teaspoon ground cumin\n\n1 teaspoon ground paprika\n\n\u00bd teaspoon ground chilli\n\n\u00bd teaspoon ground ginger\n\n1 cup salt-reduced chicken stock\n\n\u00bd cup water\n\npinch saffron threads\n\n\u00bc cup green stuffed olives\n\n  cup chopped coriander leaves\n\nCouscous\n\n1\u00bc cups water\n\n1 cup couscous\n\n2 tablespoons chopped dates (optional)\n\n\u00bc cup toasted flaked almonds\n\nHeat the oil in a large heavy-based pan on high. Cook the chicken pieces in 2 batches for 3 minutes on each side until golden brown. Transfer to a plate.\n\nSaut\u00e9 the onion and garlic in the pan for 4\u20135 minutes, until tender. Add the cumin, paprika, chilli and ginger and stir for 1 minute until aromatic.\n\nReturn the chicken pieces to the pan with the stock, water and saffron and bring to a gentle boil. Reduce the heat and simmer, covered, for 30 minutes. Stir in the olives and simmer, uncovered, for 15 minutes until the liquid has reduced slightly.\n\nMeanwhile, prepare the couscous by bringing the water to the boil in a small saucepan. Add the couscous and stir. Remove from the heat and stand, covered, for 5 minutes. Fluff with a fork, and use your fingers to rub out any small lumps. Add the chopped dates, if you're using them, and top with the toasted almonds.\n\nStir the coriander leaves through the chicken tagine. Serve with couscous and a green vegetable such as steamed green beans, asparagus or spinach.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup couscous and a generous serve of green beans) = 2415 kJ, 23 g fat (includes 6 g saturated fat), 6 g fibre, 505 mg sodium\n\n**_Super ingredient: Coriander_**\n\nCoriander is packed with minerals like potassium and magnesium, with smaller amounts of iron and calcium. It's not short on vitamins either. Its seeds and roots have been used medicinally for centuries in Asia to treat digestive upsets and ease stiff joints.\nSunday\n\nDon't rush. Eat slowly, enjoying each mouthful. Chew each mouthful well and aim to be the last person to finish, not the first. Try to stretch out your meal to 15 minutes so that your stomach can send the signal to your brain's appetite centre that it's full (known as the 'eye\u2013mouth gap'). This will help ensure you leave the table feeling satisfied but not bloated.\n\nBreakfast\n\n  Poached eggs on wholegrain toast with grilled tomato and mushrooms\n\nSnack\n\n  Handful of dried apricots and almonds\n\nLunch\n\n  Bowl of pea and ham soup and a thick slice of rye toast\n\nSnack\n\n  Toasted wholegrain English muffin with tomato and reduced-fat cheese\n\nDinner\n\n  Slow-cooked Lamb Shanks (opposite), served with mashed potato and green vegetables or a green salad of your choice\n\n  Almond wafer biscuits with fresh pears\n\n_**Energy tip: Laughter the best medicine**_\n\nLaughter is a great stress buster too! A good laugh improves your outlook and wellbeing. Research shows a really serious belly-laugh can trigger the release of endorphins, the body's natural pain-killers, which also produce a feeling of exhilaration or wellbeing. So enjoy a laugh more often and you'll feel brighter.\n**Slow-cooked lamb shanks**\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   2 hours 40 minutes\n\n6 lamb shanks, all fat and sinews scraped from bones\n\n2 tablespoons flour\n\n1 tablespoon olive oil\n\n2 onions, finely chopped\n\n2 carrots, sliced\n\n1 stalk celery, sliced\n\n2 garlic cloves, crushed\n\n400 g can diced tomatoes\n\n1 cup red wine, beef stock or water\n\n1 tablespoon chopped rosemary\n\nPreheat the oven to moderately slow, 160\u00baC (325\u00baF). Dust the shanks lightly with flour, shaking off excess.\n\nHeat the oil in a large heavy-based frying pan on high. Cook the shanks for 3\u20134 minutes, turning, until browned all over. Transfer to an ovenproof casserole dish.\n\nAdd the onion, carrot and celery to the frying pan and saut\u00e9 for 5 minutes, until the onion is golden and tender. Add the garlic and cook for 1 minute more. Blend with the tomatoes, wine and rosemary and transfer to a casserole dish.\n\nBake, covered, for 2\u20132\u00bd hours, until the meat is very tender and virtually falling off the bone. Serve with mashed potato and vegetables or salad of your choice.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup mash and a generous serve of green vegetables or salad) = 2690 kJ, 25 g fat (includes 10 g saturated fat), 8 g fibre, 260 mg sodium\n\n**_Super ingredient: Rosemary_**\n\nRosemary is one of the most powerful of culinary herbs. Just a tablespoon or two of chopped fresh rosemary can significantly boost your intake of polyphenol antioxidants, which work to neutralise damaging free-radicals and cut the risk of heart disease. It's said to aid memory and is being trialled in a treatment for patients with Alzheimer's disease.\n\nRecipes with zest\n\nWhat you eat can really boost your zest for life. Our recipes for busy morning breakfasts, lazy brunches, meals or snacks in minutes, dinners for families or for entertaining, scrumptious desserts and tasty treats are rich in slow carbohydrates, lean protein, the good fats and lots of fibre.\n\nSo get cooking and boost your zest for life!\nBreakfast and brunch\n\nYou've heard it many times before, but it's worth repeating. Breakfast really is the most important meal of the day.\n\nBreakfast literally 'breaks the fast' after the night's sleep. It refuels the brain and body for the day ahead, so you can concentrate and have better problem-solving abilities, it 'switches on' your metabolism to help you burn kilojoules faster and it stops those mid-morning hunger pangs that entice you to grab a doughnut or pastry.\n\nChildren who skip breakfast are more likely to suffer fatigue or irritability and don't fare as well at morning lessons. As a minimum, a child should eat a banana (or another piece of fresh fruit) and drink a glass of milk before setting off to school.\n\nUnless you have a physically demanding job or play a lot of sport, a light cereal-and-toast style of breakfast is perfectly adequate and will meet your morning nutrition needs. Here's a checklist of the best foods to eat:\n\n\u2022 fruit, eaten whole or cut up over cereal. Fruit juice is a quick alternative with the same valuable vitamin C, but it has virtually no fibre\n\n\u2022 wholegrain cereal, muesli, high-fibre bran cereals (especially for anyone who needs extra fibre), or porridge sweetened with a little sugar or honey and moistened with skim milk or juice\n\n\u2022 milk or yoghurt, or milk and yoghurt whipped up with fruit into a simple smoothie\n\n\u2022 wholemeal or mixed grain bread, toast or muffins topped with Vegemite, peanut butter, ricotta and honey, cottage cheese with slices of pear and chopped walnuts, baked beans, mushrooms etc.\n\n\u2022 egg (boiled, poached or microwaved) with wholemeal or wholegrain toast and baked beans or grilled tomatoes (optional)\n\n\u2022 milk, juice, tea or coffee.\n\nNone of us has much time to spare in the morning. That's why our suggestions in this chapter are very quick and easy to prepare \u2013 even a delicious cooked brekkie can be on the table in 30 minutes.\n**Dried fruit compote**\n\n_If you are short of time, microwave the tea and fruit mixture on high (100%) power for 5 minutes to speed up the re-hydration process._\n\nServes 4\n\nPreparation time   5 minutes\n\nStanding time   5 minutes and overnight\n\n\u00bd cup dried apricots\n\n\u00bd cup pitted prunes\n\n\u00bd cup dried peaches\n\n\u00bd cup dried pears\n\n\u00bd cup dried apple slices\n\n6 dried figs\n\n6 cloves\n\n2 cinnamon sticks\n\n4 strips orange peel\n\n4 tea bags, your choice of flavour\n\n\u00bc cup caster sugar\n\n3 cups boiling water\n\nno-fat vanilla yoghurt and chopped pistachio nuts to serve\n\nIn a large bowl, combine the apricots, prunes, peaches, pears, apple slices and figs with the cloves, cinnamon sticks and orange peel.\n\nPlace the tea bags and caster sugar in a heat-proof bowl and pour on the boiling water. Stir to combine and dissolve the sugar then allow the mixture to infuse for 5 minutes.\n\nDiscard the tea bags and pour the liquid over the dried fruit. Mix everything together well, cover with plastic wrap and place in the refrigerator to chill overnight.\n\nServe the compote topped with a dollop of yoghurt, and sprinkled with pistachio nuts. If you wish, add some fresh apple slices, orange segments or berries before serving.\n\nNUTRITION ANALYSIS 1 serve (including a dollop of yoghurt and pistachio nuts) = 1725 kJ, 6 g fat (includes 1 g saturated fat), 13 g fibre, 40 mg sodium\n\n**_Super ingredient: Tea_**\n\nBoth green tea and regular (black) tea are rich in powerful antioxidants known as catechins, which can lower your risk of heart disease and help to keep cancer at bay. Best of all, if you take your tea black, with no milk or sugar, it has no kilojoules and has the ability to both relax and revive you.\n**Home-made muesli**\n\n_This crunchy muesli will sustain you right through the morning until lunchtime. Stored in an airtight container, the mix will keep well for 2\u20133 weeks._\n\nServes 4\n\nPreparation time   5 minutes\n\n1\u00bc cups rolled oats\n\n1 cup oat bran\n\n1 cup dried fruit medley\n\n\u00bd cup sultanas\n\n\u00bc cup shredded coconut\n\n\u00bc cup slivered almonds\n\n1 tablespoon sunflower seeds\n\n1 tablespoon psyllium\n\nnatural low-fat yoghurt and maple syrup or honey to serve\n\nCombine all the ingredients in a large bowl and toss well.\n\nServe \u00bd cup muesli per person, topped with yoghurt and drizzled with maple syrup. Add fresh fruit and low-fat milk if you like.\n\n**VARIATIONS**\n\n  If you prefer toasted muesli, spread the mixture onto a baking tray and bake at 180\u00baC (350\u00baF) for 10\u201315 minutes until golden. There's no need to add any extra oil.\n\n  Add your favourite dried fruit, nuts and seeds, as well as (or instead of) the ones we suggest.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup yoghurt and a tablespoon of honey or maple syrup) = 1385 kJ, 7 g fat (includes 2 g saturated fat), 6 g fibre, 95 mg sodium\n\n**_Super ingredient: Oats_**\n\nThe humble oat is a winner among grains. Traditional rolled oats oats have a low GI, which means that the carbohydrate is slowly absorbed into your system, keeping you satisfied for hours after eating. Oats are rich in the soluble type of fibre that can lower your cholesterol. They contain small amounts of 'good' fats \u2013 more than either wheat or rice. And they are nutritious, giving you B vitamins and vitamin E as well as protein and minerals.\n**Cinnamon, pear and date porridge**\n\n_Perfect porridge is about getting the right consistency for you and your family. For a 'runnier' texture, just add extra milk or water during cooking as required._\n\nServes 4\n\nPreparation time   5 minutes\n\nCooking time   15 minutes\n\n1 litre low-fat milk, plus extra to serve\n\n4 pears, cored, unpeeled and sliced, or canned pears, drained and chopped\n\n1\u00bd cups rolled oats\n\n1 cup chopped pitted dates (or prunes)\n\n1 tablespoon cinnamon sugar\n\nmaple syrup to serve (optional)\n\nIn a saucepan, combine the milk, pears, oats and dates and bring to the boil over a low heat, stirring constantly.\n\nSimmer the porridge for 10\u201315 minutes, stirring occasionally, until the pears are tender. Remove from the heat.\n\nSpoon the porridge into 4 serving bowls, sprinkle with cinnamon sugar and serve with extra milk and maple syrup, if desired. A sprinkle of chopped walnuts before serving adds a lovely crunch.\n\nNUTRITION ANALYSIS 1 serve = 2105 kJ, 7 g fat (includes 3 g saturated fat), 11 g fibre, 155 mg sodium\n\n**_Super ingredient: Cinnamon_**\n\nIf you only have room for one spice in your kitchen, cinnamon is it! Research has revealed that small amounts of cinnamon, (around half a teaspoon or 3 grams), taken each day may lower blood sugar levels in diabetes. So be generous with the cinnamon; it goes beautifully in many desserts, such as creamy rice, baked custard, apple crumble, apple strudel and stewed pears.\n\n**Banana and berry breakfast smoothie**\n\n_This is the ideal 'whip up and serve in an instant' breakfast. For a thicker smoothie, double the amount of oat bran or breakfast biscuit. Feel free to experiment with other favourite fruits as well. Frozen mixed berries work particularly well in this smoothie._\n\nServes 4\n\nPreparation time   3 minutes\n\n1 litre vanilla soy milk, chilled\n\n2 ripe bananas, peeled and cut into chunks\n\n250 g strawberries (or mixed berries of choice)\n\n1 breakfast biscuit such as Weet-Bix or\n\nVita Brits, crumbled, or 1\/4 cup oat bran\n\nice to serve (optional)\n\nPlace all the ingredients in a food processor or blender and process until smooth.\n\nServe immediately, over ice if you like.\n\nNUTRITION ANALYSIS 1 serve = 1075 kJ, 8 g fat (includes 1 g saturated fat), 4 g fibre, 185 mg sodium\n\n**_Super ingredient: Soy milk_**\n\nFoods made from soy beans are nutritional 'all-rounders'. Soy has twice as much protein as other beans \u2013 and it's the best quality protein the plant kingdom has to offer, including the 8 essential amino acids normally found in animal protein. Soy milk is enriched with calcium, to the same level as in milk. And it's an easy way to obtain soy's phyto-oestrogens, natural plant hormones known as isoflavones, that may help women through the menopause. Additionally, soy milk contains polyunsaturated oil, some omega-3 essential fatty acids and a number of B vitamins.\n**Buckwheat pancakes with mixed summer berries**\n\n_Here's a tip for cooking perfect pancakes if you haven't made them before. After you've poured the batter into the pan, wait for the bubbles that form on the surface to begin to break before turning them._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   15 minutes\n\nStanding time   30 minutes\n\n\u00bd cup self-raising flour\n\n\u00bd cup buckwheat flour\n\n1 tablespoon caster sugar\n\n1\u00bc cups buttermilk\n\n1 egg\n\n250 g strawberries, hulled and quartered\n\n150 g raspberries\n\n150 g blueberries\n\n1 tablespoon icing sugar\n\nspray oil\n\nlow-fat vanilla yoghurt to serve\n\nSift the self-raising flour into a bowl, then stir in the buckwheat flour, followed by the sugar, and make a well in the centre.\n\nWhisk the buttermilk and egg together in a jug. Pour into the dry ingredients, then mix slowly, drawing in the flour to make a smooth batter. Cover and allow to stand for 30 minutes.\n\nIn another bowl, combine the berries and icing sugar and set aside.\n\nHeat a non-stick frying pan (large enough to cook 2 pancakes at a time), on medium. Spray with a little oil.\n\nPour \u00bc cup of batter into the pan for each pancake, making sure they are well spaced. Cook for 1\u20132 minutes until bubbles form on the surface. Turn the pancakes and cook the other side for a further 1 minute until just golden. Transfer to a plate, cover with a tea towel and keep warm. Repeat with the remaining batter to make 8 pancakes.\n\nTo serve, place 2 warm pancakes on each plate. If they go cold, you can reheat them in the microwave on high (100%) power for 20 seconds. Top with a spoonful of berries and a dollop of yoghurt.\n\n**VARIATION**\n\nBlend together 200 g low-fat ricotta, \u00bd cup low-fat vanilla yoghurt and \u00bc teaspoon ground cinnamon. Use as a topping to dollop on top of the berries. Drizzle with honey or maple syrup.\n\nNUTRITION ANALYSIS 1 serve (including a dollop of yoghurt) = 1130 kJ, 2 g fat (includes 1 g saturated fat), 7 g fibre, 210 mg sodium\n\n**_Super ingredient: Buckwheat_**\n\nBuckwheat is a good source of protein, two of the B group vitamins (thiamin and niacin) as well as magnesium and iron. It's a handy flour to use if you want to reduce your reliance on wheat. Buckwheat is derived from the seed of a grass and is not technically a cereal grain, but it's used in similar ways to bulgur wheat or wheat flour. It's free of gluten and has a low GI. It's also high in soluble fibre and contains several interesting phyto-chemicals such as rutin.\n\n**Rhubarb muffins**\n\n_Ever wondered how to get muffins to rise evenly? Simply drop the batter into the muffin pan in one go and don't top up with extra batter later. Topping up is what causes uneven rising. Don't worry about a few lumps of flour in the batter \u2013 they help create that characteristic crumbly appearance._\n\nMakes 12\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\nspray oil\n\n2\u00bd cups self-raising flour\n\n1 teaspoon mixed spice\n\n1 cup brown sugar\n\n1 cup buttermilk\n\n125 g light margarine, melted\n\n1 egg\n\n1 teaspoon vanilla extract\n\n2 cups chopped fresh rhubarb (use frozen if fresh is unavailable)\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Spray a 12-hole muffin pan lightly with oil, or line with paper patty cases.\n\nSift the flour and spice together into a large bowl. Stir in the sugar.\n\nIn a jug, whisk together the buttermilk, melted margarine, egg and vanilla.\n\nMake a well in the centre of the dry ingredients. Pour in the buttermilk mixture all at once and stir until just combined \u2013 about 16 strokes is all you need.\n\nCarefully fold in the rhubarb. Spoon the mixture in even quantities into the muffin pan, until each mould is two-thirds full.\n\nBake for 25\u201330 minutes then transfer to a wire rack to cool. Serve warm.\n\n**VARIATION**\n\nTry making a pear and streusel muffin for a delicious change. Replace the rhubarb with 2 ripe pears (peeled, cored and chopped), and \u00bc cup chopped glac\u00e9 ginger. To make the streusel topping, rub 30 g light margarine into   cup brown sugar, then add 2 tablespoons plain flour and \u00bc teaspoon ground ginger and rub to a fine crumb consistency. Sprinkle over each muffin before baking.\n\nNUTRITION ANALYSIS 1 serve = 925 kJ, 7 g fat (includes 1 g saturated fat), 2 g fibre, 265 mg sodium\n\n**_Super ingredient: Rhubarb_**\n\nRhubarb has little fat, little sodium and is low in kilojoules, even with the addition of a little sugar for sweetness. It contributes fibre, about the same as celery, another stringy vegetable \u2013 remember technically rhubarb is a vegetable, but we eat it as a fruit! Its strong flavour is good to include if you're on a diet \u2013 it is surprisingly satisfying without overloading you with sugar or kilojoules.\n**Banana and nut loaf**\n\n_Buttermilk is great for baking. It not only adds flavour, but its acid content reacts with raising agents and can give baked products a lighter texture. If you don't have any to hand in your fridge, use the same amount of orange juice or low-fat milk instead._\n\nMakes 1 cake\n\nPreparation time   15 minutes\n\nCooking time   1 hour\n\n125 g light margarine\n\n1 cup brown sugar\n\n3 eggs\n\n1\u00bd cups wholemeal self-raising flour\n\n1 cup mashed bananas (about 3 bananas)\n\n\u00bd cup desiccated coconut\n\n\u00bd cup buttermilk\n\n  cup chopped mixed nuts of choice\n\n  cup chopped dried apricots\n\n\u00bd teaspoon mixed spice\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Lightly grease a 10 x 20 cm loaf pan.\n\nPlace the margarine and sugar into the bowl of an electric mixer and beat until the mixture is pale brown and creamy. Add the eggs, one at a time, beating well after each addition. If you don't have an electric mixer, cream the ingredients together with a wooden spoon.\n\nLightly fold in the flour, bananas, coconut, buttermilk, nuts, apricots and spice, until well combined. Spoon the mixture into the prepared pan.\n\nBake for 50\u201360 minutes until cooked when tested with a skewer. Do this by inserting the skewer into the centre of the loaf. If it comes out clean and dry the loaf is cooked. Cool in the pan for 10 minutes, then turn out onto a wire rack to cool completely. Serve sliced plain or toasted.\n\nNUTRITION ANALYSIS 1 serve = 1115 kJ, 12 g fat (includes 3 g saturated fat), 4 g fibre, 200 mg sodium\n\n**_Super ingredient: Buttermilk_**\n\nButtermilk with its thick, slightly acidic flavour is similar in food value to skim milk. It's low in fat, a good source of protein and high in the minerals calcium and phosphorus. Traditionally it was the liquid drained from the churn as butter was being made. Today it is made by culturing skim milk and is more akin to yoghurt \u2013 and it's pre-digested so it easier to digest for those with digestive troubles.\n**Baked eggs with tomato and spinach**\n\n_If you don't have prosciutto, you can make these baked eggs with slices of light ham._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\n\u00bd cup baby spinach leaves\n\nspray oil\n\n1 small red onion, finely chopped\n\n2 slices prosciutto, finely chopped\n\n125 g cherry tomatoes, halved\n\n8 eggs\n\n  cup grated reduced-fat tasty cheese\n\n2 tablespoons finely chopped chives\n\nwholegrain toast fingers to serve\n\nPreheat the oven to moderately slow, 160\u00baC (325\u00baF). Lightly grease four 1\u00bd cup ramekins and arrange them on a baking tray.\n\nDivide the spinach leaves evenly between the ramekin dishes.\n\nHeat a medium frying pan on high. Spray with oil. Saut\u00e9 the onion and prosciutto for 3\u20134 minutes until the onion is tender. Add the tomatoes and cook for another minute. Remove the pan from the heat and leave to cool.\n\nTransfer the onion mixture to a bowl. Whisk with half the eggs, half the cheese and all the chives. Spoon the mixture evenly into the prepared ramekins.\n\nBreak the remaining eggs, one at a time, carefully into each ramekin. Sprinkle each with an even amount of the remaining cheese. Bake for 20\u201325 minutes until the egg yolks are just set. Serve with wholegrain toast fingers.\n\n**VARIATION**\n\nTo reduce the quantity of eggs by half, don't add them to the saut\u00e9ed onion mixture. Just use 1 egg per ramekin \u2013 spoon it onto the top of the onion mixture as described above.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 1215 kJ, 16 g fat (including 6 g saturated fat), 3 g fibre, 585 mg sodium\n\n**_Super ingredient: Eggs_**\n\nAn egg is a compact package of nutrition. For a very modest 355 kilojoules (85 calories), it gives you every vitamin except vitamin C, plenty of protein and a host of essential minerals. Particularly worth mentioning is vitamin B12, which is hard to obtain on vegetarian diets, and folate, a B vitamin which can help minimise birth defects. Eggs are a surprising source of two carotenoids, lutein and zeaxanthin, natural compounds related to the beta-carotene in carrots and usually found only in vegetables and fruits. These two anti-oxidants are now under study for their role in preventing macular degeneration, a common cause of blindness as we age.\n**Big healthy breakfast with baked beans**\n\n_To help the eggs 'set' faster, splash some hot water over them during cooking. And if you like your eggs 'hollandaise style', top the eggs with a little low-fat whole egg mayonnaise._\n\nServes 4\n\nPreparation time   5 minutes\n\nCooking time   15 minutes\n\n4 flat mushrooms, trimmed\n\nspray oil\n\n4 slices prosciutto\n\n1 bunch asparagus, trimmed\n\n1 tablespoon water\n\n1 tablespoon vinegar\n\n4 eggs\n\n420 g can baked beans\n\n4 wholemeal English muffins, halved and toasted\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF).\n\nArrange the mushrooms in a baking tray. Spray with oil and bake for 10\u201315 minutes until tender.\n\nPlace the prosciutto on another baking tray lined with baking paper. Bake for 3\u20135 minutes until crisp.\n\nPlace the asparagus in a microwave-safe dish with the water. Cover with plastic wrap and cook, on high (100%) power for 1\u20131\u00bd minutes. Stand for 3 minutes.\n\nMeanwhile, bring a frying pan of water to a gentle simmer and add the vinegar (this helps the eggs to set). Poach the eggs one at a time by breaking each egg onto a saucer, and sliding it gently into the water. Bring the water back to a simmer and poach until it is cooked to your liking.\n\nWhile the eggs are poaching, heat the baked beans in a saucepan or in the microwave, according to the instructions on the can.\n\nTo serve, arrange the toasted muffins on serving plates. Top each with a spoonful of baked beans, an egg, and accompany with a mushroom, a few asparagus spears and crispy prosciutto.\n\nNUTRITION ANALYSIS 1 serve = 1350 kJ, 8 g fat (includes 2 g saturated fat), 9 g fibre, 820 mg sodium\n\n**_Super ingredient: Baked beans_**\n\nThe humble can of baked beans is a nutritional package of protein, fibre, slowly digested carbohydrate and a host of vitamins and minerals. They are a much better choice than canned spaghetti. If you're trying to cut back on salt, look for salt-reduced varieties at your supermarket. Keep a couple of cans in your cupboard to have on multigrain toast for a satisfying light meal when you don't feel like cooking.\n\n**Celeriac rosti**\n\n_If you haven't prepared celeriac before, here's a tip. Squeeze the grated celeriac very thoroughly (and the potato too), to remove any excess liquid before combining them with the other ingredients. Rosti make a delicious accompaniment to many meals._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   20 minutes\n\n500 g celeriac, peeled and grated\n\n500 g potato, peeled and grated\n\n2 eggs\n\n2 teaspoons wholegrain mustard\n\n1 teaspoon chilli flakes\n\n1 tablespoon olive oil (or spray oil)\n\ncrispy prosciutto and baby spinach to serve\n\nIn a large bowl, combine the celeriac, potato, eggs, mustard and chilli flakes.\n\nHeat the oil in a large non-stick frying pan on medium.\n\nDivide the rosti mixture into 12 patties. Add to the frying pan in batches of 6, pressing down firmly with a spatula to form an even disc.\n\nCook for 4\u20135 minutes on each side until golden brown. Spray the pan with oil between batches, as required.\n\nDrain the rosti on a paper towel and keep warm in a preheated oven until ready to serve. Serve with prosciutto and baby spinach.\n\nNUTRITION ANALYSIS 1 serve = 790 kJ, 7 g fat (includes 1 g saturated fat), 4 g fibre, 220 mg sodium\n\n**_Superfood: Celeriac_**\n\nThis large round root, with its celery-like flavour, makes a nice change from the blander potato in winter months. You can serve it boiled, mashed or roasted and it's a wonderful addition to soup. The French even grate it and add to salad with a little vinaigrette dressing or mayonnaise. It adds lots of fibre and potassium plus a little vitamin C if you eat it raw. It has little starch and is low in kilojoules.\n**Corn fritters**\n\n_If you are using fresh corn, simply remove the kernels by slicing along the cob with a sharp knife. You can also replace the wholemeal flour with the same amount of plain white, buckwheat or rice flour._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   25 minutes\n\n1 cup wholemeal plain flour\n\n1 teaspoon baking powder\n\n\u00bd cup reduced-fat grated cheddar cheese\n\n2 tablespoons chopped chives\n\n2 tablespoons chopped coriander\n\n1 cup low-fat milk\n\n2 eggs\n\n420 g can corn kernels, drained spray oil\n\nsweet chilli sauce, smoked salmon or crisp prosciutto, and chives to serve\n\nSift the flour and baking powder together into a large bowl, returning the husks left in the sieve to the bowl. Stir in the cheese, chives and coriander. Mix well.\n\nIn a jug, whisk the milk and eggs lightly to combine. Add to the dry ingredients with the corn, and beat the mixture with a wooden spoon to thoroughly combine.\n\nHeat a large non-stick frying pan on medium. Spray with a little oil. Drop 2 generous tablespoons of mixture into the pan for each fritter and cook for 2\u20133 minutes on each side until golden brown. Drain on a paper towel and keep warm. Repeat with the remaining batter (this mixture makes about 12 fritters).\n\nServe with sweet chilli sauce, smoked salmon or crisp prosciutto. Fritters are best eaten immediately, but you can reheat them in the microwave for a few seconds if necessary.\n\nNUTRITION ANALYSIS 1 serve (including a slice of smoked salmon or prosciutto and a dollop of sweet chilli sauce) = 1150 kJ, 6 g fat (includes 2 g saturated fat), 6 g fibre, 945 mg sodium\n\n**Mushrooms and tomatoes on toast**\n\n_To prepare the mushrooms, simply trim the stems and wipe the mushrooms with a damp piece of paper towel to remove any dirt._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   40 minutes\n\n8 roma tomatoes, halved (or use round tomatoes, quartered)\n\n1 tablespoon olive oil\n\nfreshly ground black pepper\n\n30 g light margarine\n\n400 g button mushrooms, quartered\n\n1 tablespoon finely chopped parsley, plus leaves to garnish\n\n8 slices seeded bread, toasted\n\nPreheat the oven to hot, 200\u00baC (400\u00baF). Arrange the tomatoes in a single layer in a large baking tray, cut-side up. Drizzle with oil and top with a good grinding of black pepper.\n\nBake for 25\u201330 minutes until the tomatoes begin to collapse.\n\nMeanwhile, melt the margarine in a large frying pan over a medium heat. Saut\u00e9 the mushrooms for 5\u20136 minutes, until golden brown. Stir in the chopped parsley.\n\nServe hot toast topped with the tomatoes and mushrooms. Sprinkle with extra parsley leaves.\n\n**VARIATIONS**\n\n  Instead of roma tomatoes, try sweet cherry tomatoes for a change. You'll only need to bake them for 15\u201320 minutes.\n\n  For a great taste of summer, try sprinkling the tomatoes with finely shredded basil before baking.\n\nNUTRITION ANALYSIS 1 serve (including 2 slices bread) = 1415 kJ, 12 g fat (includes 2 g saturated fat), 8 g fibre, 480 mg sodium\n\n**_Super ingredient: Mushrooms_**\n\nMushrooms impart loads of flavour with very little fat or kilojoules. They are often under-rated in nutritional value \u2013 they are actually an excellent source of many of the B group vitamins. For example, 100 grams of mushrooms provides over 50 per cent of our daily requirement of niacin, which is equal to a small lean steak. Mushrooms are also a source of vitamin D.\n**Cheese and herb omelette**\n\n_To cook the omelette evenly (and faster), tilt the pan a little during cooking. This allows the uncooked mixture to come into contact with the hot surface of the pan. Add saut\u00e9ed sliced mushrooms to the omelette as well, if you like._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   15 minutes\n\n8 eggs\n\n  cup water\n\n\u00bc cup chopped mixed herbs (parsley, basil, chives)\n\nspray oil\n\n  cup reduced-fat grated tasty cheese\n\n2 tomatoes, chopped\n\n\u00bd cup chopped light ham\n\nwholegrain toast to serve\n\nIn a jug, whisk together the eggs, water and herbs until well combined.\n\nHeat an omelette pan on medium. Spray with oil. Pour in quarter of the egg mixture \u2013 it should begin to set around the edge of the pan almost immediately.\n\nUse a spatula to pull the edge of the omelette in from the side of the pan, allowing the uncooked mixture to heat and cook.\n\nWhen the egg mixture no longer runs freely and the surface looks creamy, sprinkle on a quarter of the cheese, tomatoes and ham.\n\nFold the omelette in half to cover the filling and gently slide onto a plate and serve immediately with hot wholegrain toast. Repeat with the remaining egg mixture and filling to make 4 omelettes.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 1350 kJ, 13 g fat (includes 4 g saturated fat), 3 g fibre, 785 mg sodium\n\n_**Super ingredient: Green herbs**_\n\nThe culinary herbs that grace our meals can make a sizeable contribution to our nutrition intake \u2013 if we eat enough of them.\n\n\u2022 Basil, parsley and mint are high in mono-terpenes, which are thought to have cancer-delaying properties, especially with mammary tumours.\n\n\u2022 Rosemary, thyme and oregano have been found to be high in polyphenols, a class of antioxidant that may cut the risk of heart disease.\n\n\u2022 Parsley is also high in coumarins, noted for their anti-coagulant and anti-bacterial effects, while extracts of rosemary are being tested to see if they can be used as a natural food-grade preservative.\n\nMost fresh green herbs are also a rich source of potassium and magnesium, with smaller amounts of iron and calcium \u2013 as well as being high in vitamin C, folate, vitamin B1 and vitamin K.\nLight meals and snacks\n\nHow often have you wanted something light to eat but not felt like a huge meal? We've gathered together our favourite sandwiches, soups and salads for you, and we've created some healthier, lighter versions of other favourites, such as quiche, p\u00e2t\u00e9, risotto and pizza. There's even a light laksa to try!\n\nMany of these light dishes can be prepared for lunch, which these days, sadly, is almost in danger of disappearing. Everyone knows that it's important to eat a good breakfast to provide fuel for the morning ahead, and dinner is a nice way to wind down at the end of a hectic day. But lunch is often missed out, either because we get too busy to stop or because we're grazing on other snacks throughout the day to keep us going. We think this is a bad idea!\n\nLunch helps you maintain momentum during the day and avoid that mid-afternoon slump called the '3 pm low'. Generally speaking, this slump occurs either because you've eaten an inadequate lunch, or because you've skipped it altogether. As a result, you end up feeling drained of energy, unable to concentrate and you are much less efficient.\n\nDesktop dining is not conducive to working well. If you can, take a 10-minute break and change location to eat a healthy lunch, such as a sandwich made with wholegrain bread and lots of salad filling. And if you have access to a microwave at work, bring in leftovers from dinner to reheat for lunch the next day. Or bring frozen soup, which can thaw during the morning and be reheated in the microwave. Both of these options make a great change from salad and sandwiches, and are especially good in the winter months.\n\nIf you eat out at lunchtime, try not to order a huge meal that will put you to sleep. Heavy meals prime you for a nap rather than a productive afternoon's work.\n\nMaking interesting school lunches for kids is always a challenge, particularly if refrigeration is not possible and the food needs to be appealing several hours after it is made. The trick is to keep it simple, especially for younger kids. Peanut butter and cheese make perfectly adequate sandwich fillings \u2013 even if you get sick of them! Save fillings such as ham or chicken for when the weather is cooler and there is little potential risk of food poisoning.\n**Spicy salmon p\u00e2t\u00e9**\n\n_This is delicious as part of a colourful antipasto platter. Serve it with crudit\u00e9s, dips, lean ham slices, artichoke hearts, reduced-fat cheeses, gherkins and olives \u2013 black and green._\n\nServes 4\u20136\n\nPreparation time   5 minutes\n\n200 g can pink or red salmon, drained and flaked\n\n100 g low-fat ricotta (fresh from the deli, or a supermarket tub)\n\n2 green onions (shallots), finely chopped\n\n1 tablespoon chopped parsley\n\n2 teaspoons sweet chilli sauce\n\njuice \u00bd lemon\n\ncrispbread, wholegrain toast or rolls to serve\n\nPlace the salmon, ricotta, green onion, parsley, chilli sauce and lemon juice in a medium-sized bowl. Blend until smooth.\n\nServe as a dip, or spread onto crispbread, wholegrain toast or rolls.\n\n**VARIATION**\n\nIf you like, stir through half a chopped avocado, but this will increase the fat content of the p\u00e2t\u00e9.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 755 kJ, 6 g fat (includes 2 g saturated fat), 2 g fibre, 310 mg sodium\n\n**_Super ingredient: Ricotta_**\n\nRicotta \u2013 the low-fat type \u2013 is one of the most useful ingredients in your kitchen. It has less than 5 per cent fat, very little cholesterol, no added salt and adds protein and B vitamins. Made from whey it has around half the calcium of cheddar cheese, but is still a significant source. Research has found that whey protein can help muscles recover after a bout of exercise. It is quickly digested by the body and provides an ideal mix of high-quality amino acids. Best of all, ricotta is a great substitute for high-fat cream cheese or sour cream in dips and p\u00e2t\u00e9s, as in this recipe.\n**Cucumber dip**\n\n_Make this dip up to 2 hours before serving (just cover and keep chilled) and serve with crudit\u00e9s \u2013 sliced mixed raw mushrooms, carrots, celery, capsicum, radishes, green beans and cherry tomatoes are all delicious accompaniments._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\n200 g tub low-fat natural yoghurt\n\n1 Lebanese cucumber, finely chopped\n\n2 green onions (shallots), thinly sliced\n\n2 tablespoons chopped mixed herbs (flat-leaf parsley, oregano, basil)\n\n1 tablespoon lemon juice\n\n1 garlic clove, crushed\n\n\u00bc teaspoon cayenne pepper (optional)\n\nmixed raw vegetables (crudit\u00e9s) to serve\n\nPlace all ingredients in a mixing bowl. Stir well to combine, then spoon into a serving bowl.\n\nServe with mixed raw vegetables (crudit\u00e9s).\n\n**VARIATIONS**\n\n  Char-grill or roast assorted vegetables to use for dipping.\n\n  Toast Turkish bread fingers, sliced pitta or sliced lavosh until golden and crisp to use for dipping.\n\n  Pan-fry, char-grill or barbecue chicken, lamb, prawns or fish and serve topped with the dip and steamed vegetables for a complete meal.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup crudit\u00e9s) = 170 kJ, negligible fat, negligible saturated fat, 2 g fibre, 55 mg sodium\n\n**_Super ingredient: Yoghurt_**\n\nPacked with calcium, protein and B vitamins (especially riboflavin, which is needed for healthy skin and eyes), yoghurt offers the nutrients of milk but in a more concentrated form. It's more easily digested than milk, and suitable for people with lactose intolerance. Studies show the protein, fat and lactose it contains are better absorbed and the calcium is more available. Additionally, it is low GI \u2013 the perfect snack for people with diabetes.\n\n**Beef and bean burritos**\n\n_To keep the remaining avocado half to use another day, just brush the flesh with a little lemon juice, wrap it in plastic wrap and store in the fridge. If possible keep the half with the stone still attached._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\nCooking time   12 minutes spray oil\n\n400 g rump steak, trimmed and sliced thinly\n\n1 red onion, halved and sliced\n\n420 g can of four bean mix, rinsed and drained\n\n1 cup water\n\n50 g packet reduced-salt burrito seasoning\n\n8 burrito tortillas\n\n\u00bd ripe avocado, lightly mashed\n\n8 lettuce leaves, shredded\n\n\u00bd cup grated reduced-fat mozzarella cheese\n\nHeat a large frying pan on high. Spray with oil. Cook the beef and onion in batches for 4\u20135 minutes until well browned.\n\nStir in the beans, water and burrito seasoning. Bring to the boil, stirring continuously. Then lower the heat and simmer for 2 minutes, until the sauce has thickened.\n\nMeanwhile, warm the tortillas according to the packet instructions. Spread each with a little mashed avocado.\n\nTop each burrito with lettuce and a spoonful of the beef and bean mixture.\n\nSprinkle with cheese, roll up securely and serve straight away.\n\n**VARIATIONS**\n\n  You can also use chicken strips or chicken or beef mince for these, or add some diced tomato.\n\n  For a vegetarian version, replace the beef with an additional can of beans of your choice.\n\nNUTRITION ANALYSIS 1 serve = 1800 kJ, 18 g fat (includes 7 g saturated fat), 6 g fibre, 995 mg sodium\n\n**_Super ingredient: Four bean mix_**\n\nBeans \u2013 whether kidney, soy, haricot, cannellini, broad beans or chickpeas \u2013 are one of our best friends. They have almost no fat and are packed with fibre, protein (important for vegetarians), B vitamins and minerals. Their GI of around 30, is one of the lowest of any carbohydrate \u2013 so they fill you up and keep hunger pangs away.\n**Steak sandwich**\n\n_For a great look and an extra flavour boost for a sandwich with a difference, cook the steak and the bread on a char-grill._\n\nServes 4\n\nPreparation time   5 minutes\n\nCooking time   15 minutes\n\n1 tablespoon olive oil\n\n2 onions, sliced\n\n\u00bc cup vinegar\n\n2 tablespoons brown sugar\n\n4 beef minute steaks or veal steaks\n\n4 wholegrain or rye rolls, split\n\n1 tablespoon Dijon mustard\n\n1 bunch rocket, trimmed\n\nHeat oil in a large frying pan on medium. Add the onions to the pan and cook for 5 minutes, stirring occasionally, until softened. Blend in the vinegar and sugar and simmer, stirring occasionally, for another 5 minutes.\n\nSpoon the onions into a mixing bowl and return the frying pan to a high heat. Add the steaks and cook for 1\u20132 minutes on each side.\n\nToast the rolls and spread each one with mustard. Serve topped with a generous mound of rocket, a piece of steak and a spoonful of onions. Add the top half of the roll and serve straight away.\n\n**VARIATIONS**\n\n  If preferred, use thick slices of wholemeal bread instead of rolls.\n\n  Roasted tomatoes and sliced beetroot make delicious additions.\n\n  For a change, try using thin slices of chicken breast instead of steak.\n\nNUTRITION ANALYSIS 1 serve = 1495 kJ, 11 g fat (includes 2 g saturated fat), 5 g fibre, 425 mg sodium\n\n**_Super ingredient: Rocket_**\n\nRocket (also known as arugula) rates highly among the salad leaves. It is exceedingly rich in many vitamins (such as beta-carotene and vitamin C), minerals and antioxidants. It has a unique peppery, mustardy flavour, showing that it belongs to the cabbage family that are such nutrition stars. Best of all, it's low in kilojoules (calories) with virtually no fat or sodium. Eat up!\n**Our favourite wrap**\n\n_We made our wrap in a sandwich press, if you don't have one, cook the wrap in a lightly oiled frying pan over medium heat and turn once or twice._\n\nServes 1\n\nPreparation time   5 minutes\n\nCooking time   2 minutes\n\n1 wholemeal pitta bread, mountain bread or lavosh\n\n1 tablespoon light mayonnaise\n\n\u00bd cup sliced poached chicken breast, or skinless barbecued chicken\n\n\u00bc avocado, thinly sliced\n\n\u00bc bunch rocket\n\nPreheat a sandwich press.\n\nSpread your choice of bread with the mayonnaise and top with chicken, avocado and rocket.\n\nRoll up and toast lightly in sandwich press for 1\u20132 minutes. Serve immediately.\n\n**VARIATION**\n\nTry low-fat ham, tomato, bocconcini, mustard and spinach as an alternative filling.\n\nNUTRITION ANALYSIS 1 serve = 2240 kJ, 24 g fat (includes 5 g saturated fat), 7 g fibre, 640 mg sodium\n\n**_Super ingredient: Avocados_**\n\nAvocados are in a class of their own. At 23 per cent fat, they are rich and filling \u2013 but it is largely monounsaturated, one of the healthiest fats around. Avocados are also rich in niacin, vitamin E and potassium. Contrary to what was long believed, avocados contain no cholesterol and Australian research reveals that an avocado-enriched diet is actually quite effective at lowering blood cholesterol \u2013 more effective, in fact, than a standard low-fat, high-carbohydrate diet.\n**Emma's sandwich**\n\n_Sandwiches make a great snack or light meal in minutes. But they do taste better made with fresh bread. Keep a loaf in the freezer and simply defrost as many slices as you need each time. This salad combo is a great favourite with Jennene's daughter._\n\nServes 1\n\nPreparation time   5 minutes\n\n2 slices wholegrain bread\n\n4 slices tomato\n\n4 slices cucumber\n\n2 slices beetroot\n\n\u00bc cup alfalfa sprouts\n\n\u00bc cup grated carrot\n\n1 slice reduced-fat fetta cheese\n\n1 tablespoon chopped black olives\n\nTop one slice of bread with toppings. Finish with second slice of bread. Halve and serve.\n\n**VARIATION**\n\nThese are numerous, of course, but we also particularly like using reduced-fat tasty cheese and lettuce or rocket in place of alfalfa.\n\nNUTRITION ANALYSIS 1 serve = 1215 kJ, 7 g fat (includes 3 g saturated fat), 6 g fibre, 945 mg sodium\n\n**_Super ingredient: Alfalfa sprouts_**\n\nIn food value, sprouts rate midway between a dry seed and a green vegetable. Compared to the dry seed from which they grow, sprouts have 3\u20135 times more vitamin C. However, because sprouts have 8 times more water than a dry seed, their other nutrients are diluted, making them less concentrated in B vitamins. Crunchy, light and tasty, they are nutritious \u2013 but not as impressively as is often claimed.\n**Tuna salad ni\u00e7oise**\n\nFor deliciously crisp, al dente vegies, blanch them by plunging into boiling water for 1 minute and then refreshing in cold water to stop the cooking process. This salad is also delicious when all the ingredients are warm.\n\nServes 4\n\nPreparation time   20 minutes\n\nCooking time   10 minutes\n\n350 g chat potatoes, quartered\n\n4 eggs\n\n1 oak leaf lettuce, leaves separated, washed and dried\n\n425 g can tuna in olive oil, drained and flaked\n\n250 g cherry tomatoes, halved\n\n125 g green beans, trimmed and blanched\n\n\u00bc cup pitted kalamata olives\n\ncrusty sourdough bread to serve\n\nDressing\n\n2 tablespoons extra-virgin olive oil\n\n1 tablespoon white wine vinegar\n\n1 tablespoon lemon juice\n\n1 tablespoon Dijon mustard\n\nCook the potatoes in a saucepan of boiling water for 8\u201310 minutes until tender. Drain well and set aside to cool.\n\nMeanwhile, place the eggs in a saucepan of cold water and bring it to the boil. Lower the heat and simmer for 3\u20134 minutes. Cool under cold running water. When cool enough to handle, peel the eggs and cut them into quarters.\n\nIn a large bowl, combine the potatoes with the lettuce leaves, tuna, tomatoes and beans.\n\nTo make the dressing, combine all the ingredients in a jug and whisk them together well.\n\nPour the dressing over the salad and toss well. Serve topped with the quartered eggs and olives, and with crusty bread.\n\n**VARIATIONS**\n\n  For a more traditional ni\u00e7oise salad, add some chopped anchovies.\n\n  And for a delicious change, try using fresh tuna, char-grilled and broken into rough flakes.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 1905 kJ, 24 g fat (includes 4 g saturated fat), 6 g fibre, 635 mg sodium\n\n**Prawn laksa**\n\n_Overcooked seafood becomes tough. Prawns are cooked when they are opaque and orangey pink \u2013 they cook very quickly \u2013 2\u20133 minutes for small prawns and about 5 for larger ones._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   5 minutes\n\n\u00bc cup Thai red curry paste (or laksa paste)\n\n2 x 375 ml creamy coconut flavoured light evaporated milk\n\n2 cups water\n\n1 cup low-salt chicken stock\n\n500 g green prawns, peeled and de-veined, tails intact\n\n250 g rice stick noodles\n\n1 cup bean sprouts\n\n2 green onions (shallots), sliced\n\nsliced chilli, coriander leaves and lime wedges to serve\n\nHeat a large saucepan on high. Cook the curry paste, stirring, for 1 minute until fragrant. Stir in the coconut milk, water and stock. Bring to the boil, then lower the heat and simmer for 2 minutes.\n\nAdd the prawns and noodles and simmer gently for 2\u20133 minutes, stirring occasionally.\n\nDivide the noodles, sprouts and prawns between 4 bowls. Pour on the soup.\n\nServe topped with the green onion, sliced chilli and coriander leaves and a lime wedge on the side.\n\n**VARIATIONS**\n\n  Add other Asian greens of your choice.\n\n  This laksa can also be made with sliced chicken, mixed seafood or tofu.\n\nNUTRITION ANALYSIS 1 serve = 1725 kJ, 8 g fat (includes 2 g saturated fat), 3 g fibre, 865 mg sodium\n\n**_What's in a name?_**\n\nOften confused, here are descriptions to help you work out which onion is which:   \n**Green onion:** an immature onion with an unformed bulb, milder in flavour than a spring onion. Used in soups, salads and stir-fries. These onions are sometimes referred to as 'shallots'.   \n**Spring onion:** an immature onion with a semi-developed bulb (about the size of a small pickling onion). Can be used as the green onion, but is stronger in flavour.   \n**Eschalot:** this is a mild onion, about the size of a large garlic bulb, with brown skin and purplish flesh. It has a refined flavour and is used in both French and Asian cooking in particular to infuse sauces such as b\u00e9arnaise.\n\n**Pumpkin and red lentil soup**\n\n_Croutons always seem to make soup extra-special. It's easy to make simple ones by just cutting thick slices of toast into cubes. Otherwise cut thick slices of wholegrain bread into cubes and bake them in a moderate oven (180\u00b0C \/350\u00b0F) for 5\u201310 minutes, or until golden brown._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\n1 tablespoon olive oil\n\n2 onions, chopped, or 1 leek, thinly sliced\n\n1 rasher rindless bacon, chopped (optional)\n\n2 garlic cloves, crushed\n\n2 teaspoons curry powder\n\n700 g pumpkin, peeled, seeded and chopped\n\n1 carrot, cut into chunks\n\n1 potato, cut into chunks\n\n\u00bd cup red lentils\n\n6 cups water or low-salt vegetable stock\n\n2 cups low-fat milk\n\nchopped parsley, croutons and crusty bread to serve\n\nHeat oil in a large saucepan on high. Saut\u00e9 the onion, bacon, garlic and curry powder for 2\u20133 minutes until the onion has softened.\n\nAdd the pumpkin, carrot, potato and lentils to the pan and cook, stirring, for 1 minute. Pour on the water or stock and bring to the boil.\n\nLower the heat and simmer, covered, for 25\u201330 minutes until the vegetables are very tender.\n\nPur\u00e9e the mixture using a hand blender or food processor. Stir in the milk.\n\nWhen ready to serve, reheat the soup gently and serve sprinkled with parsley and a few croutons. Accompany with crusty bread.\n\n**VARIATION**\n\nThis soup is also delicious made with sweet potato. Replace the potato and carrot with 4 sweet potatoes and add a little brown sugar to the pan as they are saut\u00e9ing.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 1325 kJ, 8 g fat (includes 2 g saturated fat), 7 g fibre, 345 mg sodium\n\n**_Super ingredient: Pumpkin_**\n\nA versatile and nutritious vegetable, the pumpkin is a rich source of beta-carotene, which is converted into vitamin A in the body. Beta-carotene is needed for eyesight and is currently being studied for its role as an antioxidant. It helps fight off dangerous free radicals that would otherwise damage cell membranes and DNA genetic material.\n**Chicken and sweetcorn soup**\n\n_Chicken and sweetcorn soup is often served with 'egg flowers'. Whisk an egg lightly and stir it quickly through the soup just before serving._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   10 minutes\n\n2 cups low-salt chicken stock\n\n2 cups water\n\n\u00bc cup sherry\n\n1 tablespoon grated ginger\n\n3 garlic cloves, crushed\n\n1 chicken breast fillet, thinly sliced\n\n400 g creamed corn\n\n50 g baby corn, halved lengthwise\n\n\u00bd bunch choy sum, sliced\n\n6 green onions (shallots), sliced\n\nsalt-reduced soy sauce to serve (optional)\n\ncrusty bread to serve\n\nIn a large saucepan combine the stock, water, sherry, ginger and garlic. Bring to the boil then lower the heat and simmer for 3 minutes.\n\nStir in the chicken and corn and simmer for 5 minutes. Add the choy sum and green onions and simmer for another minute.\n\nLadle the soup into bowls and serve immediately with a drizzle of soy sauce if desired. Accompany with crusty bread.\n\n**VARIATIONS**\n\n  125 g of thinly sliced button mushrooms can be added to the soup.\n\n  Add some sliced chilli for added bite.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 1215 kJ, 6 g fat (includes 1 g saturated fat), 6 g fibre, 815 mg sodium\n\n**_Super ingredient: Choy sum_**\n\nChoy sum, like other Asian greens, is rich in beta-carotene, fibre, vitamin C and folate, as well as being an important source of calcium and iron. Because of their lower levels of oxalic acid (a compound that interferes with mineral absorption), the iron and calcium in Asian greens is more readily absorbed than from the traditional Western leafy greens such as spinach and silver beet.\n\n**Roasted tomato and capsicum soup with pesto**\n\n_For a lovely vibrant pesto, make it just before serving to prevent discoloration. This soup can also be made using just tomatoes or just capsicum._\n\nServes 6\u20138\n\nPreparation time   20 minutes\n\nCooking time   1 hour 10 minutes\n\n8 large ripe tomatoes, halved\n\n12 garlic cloves, skin on\n\n2 tablespoons olive oil\n\n4 large red capsicums, seeded and quartered\n\n4 cups low-salt vegetable stock or water\n\ncrusty sourdough bread to serve\n\nPesto\n\n  cup basil leaves\n\n\u00bc cup olive oil\n\n2 tablespoons toasted pine nuts\n\n2 tablespoons grated parmesan\n\nPreheat oven to moderate, 180\u00baC (350\u00baF).\n\nPlace the tomato halves and garlic cloves on a large baking tray and drizzle with olive oil. Place the capsicums on another baking tray, skin-side up.\n\nRoast both trays of vegetables for 50\u201360 minutes until the tomatoes collapse and the capsicum blisters and blackens. Cover the capsicum with plastic wrap and leave it to steam for 5 minutes. Carefully peel away the skin and any membranes.\n\nWhen the garlic is cool enough to handle, squeeze out the flesh and discard the skins.\n\nPlace the capsicums, tomatoes, garlic and half the stock into a food processor or blender. Process until smooth.\n\nTransfer the mixture to a large saucepan and stir in the remaining stock. Heat on medium, stirring occasionally, for 3\u20134 minutes, until the soup is hot. If it is very thick, you can thin it with a little more water.\n\nTo make the pesto, place all the ingredients in a food processor or blender and process. Add a little stock if you like a smoother-textured pesto.\n\nServe the soup with a big dollop of pesto and some good crusty bread.\n\nNUTRITION ANALYSIS 1 serve (including a dollop of pesto and 1 slice bread) = 1300 kJ, 19 g fat (includes 3 g saturated fat), 6 g fibre, 935 mg sodium\n\n**Mango and chicken salad**\n\n_Although fresh is best, mango is not always available. Out of season, you can substitute canned or thawed frozen mango for fresh in this recipe. You can also use \u00bd sliced pawpaw instead of the mango \u2013 or use some of each._\n\nServes 4\n\nPreparation time   20 minutes\n\n1 small, hot barbecued chicken\n\n100 g mixed salad leaves\n\n50 g watercress\n\n1 mango, seed removed, peeled and cubed\n\n1 red onion, thinly sliced\n\n60 g reduced-fat fetta cheese, crumbled\n\n8 black olives\n\ncrusty sourdough bread to serve\n\nDressing\n\n\u00bd cup orange juice\n\n2 tablespoons red wine vinegar\n\n2 teaspoons grain mustard\n\n2 teaspoons olive oil\n\nRemove all the skin and bones from the chicken. Break the meat into smallish pieces and place in a shallow dish.\n\nTo make the dressing, combine all the ingredients in a jug and whisk together well.\n\nPour over the chicken and leave to marinate for at least 15 minutes, basting occasionally.\n\nIn a large serving bowl, toss together the mixed leaves, watercress, mango, onion, fetta and olives.\n\nJust before serving, add the chicken with dressing to the salad and toss everything together well. Serve with crusty sourdough bread.\n\n**VARIATIONS**\n\n  You can also add \u00bd sliced avocado to the salad, but remember it will add to the fat content of the dish.\n\n  Instead of chicken, use cooked prawns, smoked salmon or trout (you won't need to marinate them).\n\n  A few sliced strawberries make a lovely summery addition to this salad.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 1600 kJ, 15 g fat (includes 4 g saturated fat), 4 g fibre, 555 mg sodium\n\n**_Super ingredient: Mango_**\n\nMango is blessed with a nutritional bounty. High in beta-carotene and related carotenoids, this fruit is loaded with protective antioxidants. Mangoes are also a great source of vitamin C, the B vitamin folate and essential minerals such as potassium.\n\n**Chickpea salad with roasted vegetables**\n\n_We used olive oil to roast the vegetables, but if you would like to make a lighter version, you can simply spray the vegetables with oil before baking._\n\nServes 4\n\nPreparation time   20 minutes\n\nCooking time   25 minutes\n\n500 g pumpkin, peeled, seeded and cubed\n\n2 zucchini, sliced\n\n1 red capsicum, seeded and sliced\n\n1 tablespoon olive oil\n\n420 g cans chickpeas, drained\n\n50 g baby spinach leaves\n\n\u00bc cup roughly chopped semi-dried tomatoes\n\nwholemeal pitta bread to serve (optional)\n\nDressing\n\n2 tablespoons extra-virgin olive oil\n\n1 tablespoon red wine vinegar\n\n2 teaspoons chopped tarragon\n\n1 teaspoon Dijon mustard\n\n1 garlic clove, crushed\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF).\n\nScatter the pumpkin, zucchini and capsicum onto a baking dish and toss with the olive oil so they are evenly coated. Bake for 20\u201325 minutes until golden and tender.\n\nTo make the dressing, combine all the ingredients in a jug and whisk them together well.\n\nTransfer the roasted vegetables to a large serving bowl and add the chickpeas, spinach and tomatoes. Toss with the dressing and serve warm with pitta bread, if you like.\n\n**VARIATION**\n\nFor a slightly lighter salad, replace 1 can of beans with some fluffy couscous. Pour 1 cup of couscous into a large bowl with 1 cup of boiling stock, water or white wine. Stir briefly, then leave to stand until all the liquid has been absorbed. Fluff with a fork to separate grains and toss through the salad.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd round of pitta) = 1565 kJ, 17 g fat (includes 3 g saturated fat), 8 g fibre, 440 mg sodium\n\n**Spanish omelette with spinach and pancetta**\n\n_This omelette is easily adapted to a delicious vegetarian version by omitting the pancetta._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\n1 tablespoon olive oil\n\n3 potatoes, peeled and thinly sliced, or sweet potato\n\n1 onion, thinly sliced\n\n4 slices pancetta, chopped\n\n6 eggs, lightly beaten\n\n30 g baby spinach, chopped plus extra leaves to serve\n\nspray oil\n\nextra baby spinach and wholegrain toast to serve\n\nHeat oil in a large non-stick frying pan on medium. Add the potatoes. Cook, covered, for 10\u201315 minutes, turning occasionally, until the potatoes are tender but not browned.\n\nAdd the onion and pancetta to the pan and cook, covered, for 4\u20135 minutes, until the onion is tender. Spoon the mixture into a large bowl. Wipe out the frying pan with a paper towel.\n\nAdd the eggs and chopped spinach to the potato mixture in the bowl and mix everything together well. Preheat the grill to high.\n\nHeat the frying pan on medium and spray with oil. Pour the omelette mixture into the pan and cook for 5\u20136 minutes until lightly browned on the bottom and almost set.\n\nPlace the frying pan under the grill and cook the top surface for 2\u20133 minutes until lightly browned and firm.\n\nCut the omelette into wedges and serve with extra baby spinach leaves and hot toast.\n\n**VARIATIONS**\n\n  Why not add some roasted tomatoes to the mix? Place quartered roma tomatoes on a baking tray, drizzle with 1 teaspoon oil and bake at 180\u00baC (350\u00baF) for 15\u201320 minutes until they start to collapse. Once cool, chop them coarsely.\n\n  Vary the mix with roasted pumpkin, zucchini and fetta cheese, peas and mint or capsicum, mushroom and caramelised onion.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread) = 1425 kJ, 15 g fat (includes 4 g saturated fat), 4 g fibre, 435 mg sodium\n\n**Fish cakes**\n\n_Instead of baking the fish cakes as we have done, you can cook them in a non-stick frying pan with 1 tablespoon of oil. They'll need about 3\u20134 minutes on each side for the perfect golden-brown crust._\n\nMakes about 10\n\nPreparation time   15 minutes\n\nCooking time   25 minutes\n\nspray oil\n\n500 g boneless fish fillets (red fish, ling), chopped\n\n4 green onions (shallots), chopped\n\n1 tablespoon sweet chilli sauce\n\n1 tablespoon red curry paste\n\ngrated zest and juice of \u00bd lime\n\n1 teaspoon fish sauce (optional)\n\n\u00bd cup finely chopped snake beans (or green beans)\n\n2 tablespoons chopped coriander leaves\n\nsweet chilli sauce and lime wedges to serve\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Line a baking tray with foil and spray with oil.\n\nPlace the fish, green onions, chilli sauce, curry paste, lime zest and juice and fish sauce into a food processer or blender. Process to a uniform, fine paste.\n\nTransfer to a large mixing bowl and fold through beans and coriander leaves. Form into 10 even-sized patties and arrange on the prepared baking tray.\n\nSpray the patties lightly with oil and bake for 20\u201325 minutes or until golden brown. Serve with sweet chilli sauce and lime wedges.\n\n**VARIATION**\n\nUse the mixture to make Asian-inspired dumplings. Spoon into wonton wrappers, form into dumplings and steam until cooked through.\n\nNUTRITION ANALYSIS 1 serve = 670 kJ, 5 g fat (includes 1 g saturated fat), 1 g fibre, 480 mg sodium\n\n**Light salmon quiche**\n\n_Here's a simple tip when using filo pastry: cover the sheets with a clean tea towel while you work, to help prevent them drying out. You can also wrap any leftover sheets and freeze them to use later \u2013 do not refreeze more than once as they will break up._\n\nMakes 8\n\nPreparation time   15 minutes\n\nCooking time   25 minutes\n\nspray oil\n\n8 sheets filo pastry\n\n200 g can red or pink salmon, drained and flaked\n\n1 cup reduced-fat grated tasty cheese\n\n30 g reduced-fat brie or camembert, chopped\n\n\u00bd cup roughly chopped baby spinach\n\n4 green onions (shallots), finely sliced\n\n1 tablespoon pine nuts\n\n1\u00bd cups low-fat milk\n\n5 eggs\n\nsalad to serve\n\nPreheat oven to moderate, 180\u00baC (350 \u00baF). Spray eight 10 cm quiche pans with a little oil.\n\nSpray half the sheets of filo pastry with oil and layer them together. Cut into 4 even-sized squares. Press into 4 of the tins. Repeat with the remaining sheets of pastry, so you have a total of 8 quiche pans lined with filo.\n\nDivide the salmon, cheeses, spinach, onions and pine nuts evenly between the pans. Arrange on a baking tray.\n\nIn a bowl, whisk together the milk and eggs until combined. Pour evenly into each pan and bake for 20\u201325 minutes until golden and firm.\n\nServe hot with salad.\n\n**VARIATIONS**\n\n  Saut\u00e9ed mushrooms and leeks or caramelised onions make a delicious alternative filling.\n\n  Instead of filo, make your own pastry if you have the time. Try our crisp, flaky wholemeal pastry.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 950 kJ, 12 g fat (includes 4 g saturated fat), 2 g fibre, 320 mg sodium\n\n**_Super ingredient: Pine nuts_**\n\nPine nuts are incredibly nutritious. They contain around 54 per cent fat, a nice mix of both polyunsaturated and monounsaturated types, as well as thiamin, niacin, vitamin E and a little iron. Surprisingly they have one of the highest protein counts of all nuts.\n\n**Smoked trout pizza**\n\n_Kids love pizza and if you use a good-quality purchased pizza base it makes a quick-and-easy dinner. Pizzas also provide a great opportunity to snack on 'healthy' ingredients like vegies (see our variation below)._\n\nServes 4\n\nPreparation time   5 minutes\n\nCooking time   15 minutes\n\nspray oil\n\n1 good-quality plain pizza base\n\n\u00bd cup low-fat ricotta cheese\n\n2 teaspoons horseradish\n\n1 teaspoon chopped capers\n\nfinely grated zest \u00bd lemon\n\n100 g smoked trout slices\n\n50 g watercress\n\ngreen salad to serve\n\nDressing\n\n1 teaspoon lemon juice\n\n1 teaspoon extra-virgin olive oil\n\n\u00bd teaspoon Dijon mustard\n\nfreshly ground black pepper\n\nPreheat the oven to very hot, 220\u00baC (430\u00baF). Spray a pizza tray with oil.\n\nPlace the pizza base onto the prepared tray and bake for 10\u201315 minutes until golden.\n\nMeanwhile, place the ricotta, horseradish, capers and zest in a mixing bowl and blend well. Spread over the hot pizza base. Top with slices of smoked trout.\n\nTo make the dressing, whisk all ingredients together in a jug. Toss the watercress with the dressing in a bowl. Pile on top of the pizza to serve.\n\nServe with salad.\n\n**VARIATIONS**\n\n  Make your own pizza base \u2013 it's a great way to get the whole family involved in some hands-on cooking.\n\n  Replace the smoked trout with a selection of sliced vegetables: zucchini, capsicum, mushrooms, onions, tomatoes, eggplant \u2013 or whatever you have on hand. Top with bocconcini or shavings of parmesan, scatter with fresh herbs and drizzle on a little olive oil.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 1045 kJ, 10 g fat (includes 3 g saturated fat), 3 g fibre, 290 mg sodium\n\n**Salmon and chive risotto**\n\n_For that wonderfully creamy, tender, slightly al dente risotto texture, it is essential to use hot stock and to stir constantly. Add some halved cherry tomatoes with the salmon, chives and capers for extra flavour._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   25 minutes\n\n2 cups salt-reduced fish stock\n\n2 cups water\n\nfinely grated zest and juice of 1 lemon\n\n2 teaspoons olive oil\n\n1 onion, finely chopped\n\n2 cups arborio rice\n\n\u00bd cup white wine\n\n200 g can pink or red salmon, drained and flaked, or canned tuna\n\n2 tablespoons finely chopped chives\n\n1 tablespoon capers\n\ngrated parmesan cheese and green salad to serve\n\nCombine the stock, lemon zest and juice in a large sauce pan. Bring to the boil, then lower the heat and keep at a low simmer.\n\nHeat the oil in another large saucepan on medium. Add the onion and saut\u00e9 for 1\u20132 minutes. Stir in the rice and cook for 1 minute.\n\nStir in the white wine, stirring frequently for 2 minutes until all the liquid has been absorbed. Continue adding hot stock, 1 cup at a time, until all of the liquid has been used and the rice is tender and creamy (about 15\u201320 minutes). You'll need to keep stirring the risotto frequently, as this helps break down the starch to create that lovely creamy consistency.\n\nTowards the end of the cooking time, stir in the salmon, chives and capers. Serve topped with parmesan, and accompanied by a green salad.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 2260 kJ, 11 g fat (includes 2 g saturated fat), 4 g fibre, 650 mg sodium\n\nMain meals\n\nFor most of us, dinner means time to wind down. The day is coming to a close and we can relax and enjoy the evening meal. Whether it's a barbecue outdoors or a spicy stir-fry in the wok, the evening meal is the perfect time for protein foods with vegetables or salad plus a carbohydrate.\n\nHere you'll find some of our favourite recipes. These are the ones that we cook for our families, and we'd love to share them with you.You'll find a healthy mix of recipes for fish and seafood, lean red meat, chicken, pork and veal. If you're vegetarian, we have recipes for you too. And we also give suggestions for how to substitute tofu, lentils or chickpeas \u2013 those key vegetarian protein foods \u2013 for meat.\n\nSitting around the table and unloading after a long day is good for the stomach and the soul. If you have kids, try to have a sit-down dinner with them as often as you can, or at least three times a week. A shared family meal is the best way to strengthen family ties and keep track of your children's lives. So turn off the television, make conversation, exchange ideas and ask them about their day.\n\nResearch shows that frequent family meals can actually lead to better nutritional intake \u2013 more fruits and vegetables and fewer snack foods \u2013 compared with that of children who eat alone or in front of the TV. Now isn't that what your grandmother would have told you?\n\nDon't forget that your kids watch and mimic your behaviour. If they see you eating and enjoying a balanced meal, they will want to copy that too \u2013 even if it's a few years down the track. Here are a few things to remember:\n\n\u2022 Serve sensible portion sizes.\n\n\u2022 Don't always salt your food.\n\n\u2022 Try new foods and new recipes occasionally, and offer the kids a taste as you cook.\n\n\u2022 Encourage your kids to help prepare meals, set the table and help with dishes.\n\nFinally, everyone loves to eat out, but the truth is that when you cook at home you have more control over the nutritional value of that meal. You can serve a larger portion of beans, carrots and other vegetables (you often have to pay extra for vegetables at restaurants). You can go easy on the butter, cream or oil when you cook.You can hold back on the salt if you need to. All things you have no control over when a chef cooks for you.\n\nSo here's to enjoyable, delicious dinners!\n**Mexican rice stuffed capsicums**\n\n_This classic Mexican 'beans 'n' rice' combination is also delicious stuffed into golden nugget pumpkins. Remove the tops, scrape out the seeds and partially cook each pumpkin in the microwave oven on high (100%) power for 5 minutes. You will also need to increase the total cooking time to 30 minutes._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   20 minutes\n\n2 cups cooked low-GI rice, such as Doongara or Moolgiri, or \u00be cup raw rice, cooked\n\n400 g can mixed beans or Mexican\n\nbeans, drained and rinsed\n\n2 zucchini, chopped\n\n1 tomato, chopped\n\n1 corn cob, kernels removed\n\n\u00bc cup sweet chilli sauce\n\n4 red capsicums\n\n\u00bd cup reduced-fat grated tasty cheese\n\n2 tablespoons toasted pine nuts\n\n2 tablespoons chopped parsley or coriander, plus extra to serve\n\nsalad and salsa of choice to serve\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Line a baking tray with baking paper.\n\nCombine the rice, beans, zucchini, tomato, corn and chilli sauce in a large mixing bowl and toss together well.\n\nSlice the top off each capsicum and remove all seeds and white membrane. Arrange the capsicums on the prepared tray and spoon the rice filling into each one. Sprinkle with the cheese, nuts and parsley.\n\nBake for 15\u201320 minutes until the capsicums are tender. Serve topped with parsley or coriander and with salad and a spicy salsa of your choice.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 1715 kJ, 11 g fat (includes 2 g saturated fat), 12 g fibre, 535 mg sodium\n\n**Polenta wedges with mushroom sauce**\n\n_We love to serve this with roasted tomatoes. Cut them into quarters and toss in 2 teaspoons of olive oil. Sprinkle on a few twists of freshly ground black pepper and a little fresh oregano and bake in a preheated oven (200\u02daC\/400\u02daF) for 5\u201310 minutes._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   20 minutes\n\n3 cups water\n\n\u00bd cup polenta\n\n\u00bc cup semolina\n\n\u00bc cup grated parmesan\n\nspray oil\n\n4 roma tomatoes, quartered and roasted to serve\n\ngreen salad to serve\n\nMushroom sauce\n\n30 g light margarine\n\n2 green onions (shallots), sliced\n\n1 garlic clove, crushed\n\n500 g mushrooms, quartered\n\n1 tablespoon lemon juice\n\n1 tablespoon chopped basil\n\n30 g ricotta, crumbled\n\nfreshly ground black pepper\n\nPour the water into a large saucepan and bring to the boil. Mix the polenta and semolina together and add them gradually to the pan, whisking vigorously.\n\nReduce the heat and cook, stirring constantly, for 15\u201320 minutes until thick and smooth. Stir in the parmesan and pour the polenta into a 20 cm square baking tray. Flatten the top. Chill until firm.\n\nTo make the mushroom sauce, melt the light margarine in a large frying pan over a medium heat. Saut\u00e9 the onions and garlic for 1 minute then add the mushrooms and cook, stirring, for 5\u20138 minutes until tender. Stir in the lemon juice, basil and a good grinding of pepper. Keep warm.\n\nPreheat a char-grill or barbecue on high. Cut the cold polenta into 8 wedges, spray with oil and cook for 2\u20133 minutes on each side until golden.\n\nServe the hot polenta wedges with the mushroom sauce sprinkled with ricotta and roast tomatoes on the side. Accompany with a green salad.\n\n**VARIATIONS**\n\n  Add chopped fresh herbs of your choice to polenta to vary the flavours.\n\n  Serve polenta wedges with a salsa of choice.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 1130 kJ, 12 g fat (includes 3 g saturated fat), 7 g fibre, 160 mg sodium\n\n**Tofu and cannellini bean patties**\n\n_We love leftovers \u2013 they are great to take to the office for lunch. Wrap any leftover patties well in plastic wrap and freeze for the future or store in the fridge for a day or two._\n\nMakes 8\n\nPreparation time   15 minutes\n\nCooking time   6 minutes\n\n300 g can cannellini beans, rinsed and drained\n\n300 g firm tofu\n\n\u00be cup fresh wholemeal breadcrumbs\n\n\u00bd cup dry breadcrumbs\n\n  cup plain flour\n\n3 green onions (shallots), sliced\n\n1 egg, beaten\n\n1 tablespoon chopped mint\n\n1 tablespoon chopped basil\n\n1 tablespoon chopped coriander\n\n1 tablespoon sweet chilli sauce\n\n1 tablespoon lime juice\n\n1 tablespoon olive oil\n\nsweet chilli or teriyaki sauce and salad or steamed vegetables to serve\n\nPlace all the ingredients except the oil in a food processor or blender and process to a smooth, even consistency. Refrigerate until firm enough to handle.\n\nShape into 8 even-sized, flattened patties. Heat a non-stick frying pan on high and brush with a little oil.\n\nCook the patties in 2 batches for 2\u20133 minutes each side until golden and crisp. Drain on paper towels. Serve with sweet chilli or teriyaki sauce and salad or steamed vegetables.\n\n**VARIATION**\n\nFor a delicious wrap, try serving these patties stuffed into wholemeal pitta bread with snow pea sprouts and bean sprouts.\n\nNUTRITION ANALYSIS 1 serve = 475 kJ, 4 g fat (includes 1 g saturated fat), 3 g fibre, 80 mg sodium\n\n_**Super ingredient: Cannellini beans**_\n\nPopular in Italian cuisine, the creamy white cannellini bean is fairly large \u2013 about the same size as a kidney bean. Like other legumes, it has little fat and is high in fibre and vitamin B1. Its carbohydrate is the 'slow- digesting' type with a low GI, so it's filling and satisfying. Because it maintains its shape well when cooked and has a mellow flavour, the cannellini bean is excellent in many braised dishes, and can be used interchangeably with other white beans in many recipes.\n**Spiced barbecued prawns**\n\n_Bamboo skewers need to be soaked in warm water for 30 minutes before use otherwise they will char and burn on your barbecue. You might also like to wear disposable gloves when handling the marinated prawns as turmeric's vivid yellow can stain your hands._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\nCooking time   5 minutes\n\n1 small onion, finely chopped\n\n1 tablespoon rice bran oil\n\n1 tablespoon lemon juice\n\n1 tablespoon chopped basil\n\n1 tablespoon chopped coriander\n\n2 teaspoons ground cumin\n\n\u00bd teaspoon turmeric\n\n\u00bd teaspoon paprika\n\n1 garlic clove, crushed\n\n1 teaspoon grated ginger\n\n1 kg green prawns, peeled and de-veined, tails intact\n\nspray oil\n\nmixed salad and steamed low-GI rice to serve\n\nCombine the chopped onion, oil, lemon juice, fresh herbs, spices, garlic and ginger in a large mixing bowl.\n\nAdd the prawns, tossing to coat well. Thread about 5 prawns onto each of 6 pre-soaked bamboo skewers.\n\nPreheat your barbecue or char-grill to medium. Spray with a little oil and cook the skewers for 1\u20132 minutes on each side until cooked through.\n\nServe straight from the barbecue with a mixed salad and steamed rice.\n\n**VARIATION**\n\nYou can also turn this dish into a simple stir-fry. Cook the herbs, spices and aromatics in a wok until aromatic, then add the prawns, a splash of stock and some Asian greens of your choice. Serve with steamed rice.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup rice and a generous serve of salad) =1240 kJ, 8 g fat (includes 1 g saturated fat), 3 g fibre, 375 mg sodium\n\n**Broad bean and prawn pilau**\n\n_We leave the skin on the tomatoes for this dish, but if you prefer removing them, it's easy to do. Pierce the skin in a few places with a sharp knife or skewer, cover in boiling water and leave to stand for 30\u201360 seconds. Drain, refresh in cold water and peel. The skins will slip off easily._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   25 minutes\n\n1 tablespoon olive oil\n\n1 onion, chopped\n\n2 garlic cloves, crushed\n\n1 teaspoon ground cumin\n\n\u00bc teaspoon turmeric\n\n1 cup low-GI rice, such as Doongara or Moolgiri\n\n2 cups low-salt vegetable or fish stock\n\n\u00be cup water\n\n250 g fresh or frozen broad beans, blanched and peeled\n\n250 g green prawns, peeled, or cubes of firm white fish\n\n2 roma tomatoes, chopped\n\njuice \u00bd lemon\n\n2 tablespoons chopped coriander\n\nfreshly ground black pepper\n\ngreen salad to serve\n\nHeat oil in a large saucepan on high. Saut\u00e9 the onion, garlic, cumin and turmeric for 2\u20133 minutes until the onion is tender and mixture is aromatic.\n\nAdd the rice and cook, stirring, for 1 minute. Stir in the stock and water and bring to the boil. Reduce the heat and simmer, covered, for 12\u201315 minutes, until the rice is just tender, adding more stock if required.\n\nStir in the broad beans, prawns, tomatoes and lemon juice. Cook, uncovered, for 3\u20135 minutes until all the liquid has evaporated. Stir in the coriander and a good grinding of pepper just before serving and accompany with a green salad.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 1470 kJ, 9 g fat (includes 1 g saturated fat), 4 g fibre, 475 mg sodium\n\n**_Super ingredient: Turmeric_**\n\nRelated to ginger, turmeric has an active ingredient, curcumin, whose pigment gives curries the familiar vivid yellow. It is also an approved natural food colour (additive No. 100). Many studies have discovered that curcumin has a number of health benefits. It can destroy viruses and possibly cancer cells, reduce inflammation, heal wounds, and inhibit the oxidation of the 'bad' LDL-cholesterol. It acts as an antioxidant as well. Definitely a spice to use liberally!\n\n**Curried mussels**\n\n_We have also made this dish with prawns (shelled, de-veined and tails left intact), and with a mixed seafood combination. Remember, only cook seafood until it turns opaque._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   8 minutes\n\n2 teaspoons olive oil\n\n6 eschalots, sliced\n\n2 garlic cloves, crushed\n\n1 teaspoon curry paste\n\n1 kg mussels, scrubbed and beards removed\n\n1 cup light evaporated milk\n\n\u00bd cup white wine\n\n2 small red chillies, sliced\n\ncoriander leaves, lemon wedges, wholegrain toast and mixed salad to serve\n\nHeat oil in a large heavy-based saucepan on high. Saut\u00e9 the eschalots, garlic and curry paste for 1\u20132 minutes until the eschalots are tender.\n\nAdd the mussels, evaporated milk, wine and chilli to the pan and bring to the boil. Cover the pan and cook for 4\u20135 minutes, shaking occasionally, until the mussels open. Discard any unopened mussels.\n\nServe in deep bowls, sprinkled with coriander. Accompany with lemon wedges, toast and salad.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread and a generous serve of salad) = 1200 kJ, 8 g fat (includes 1 g saturated fat), 4 g fibre, 760 mg sodium\n\n**Seafood stew**\n\n_There's a simple trick to remember when buying mussels in the shell. Tap any opened ones and discard them if they do not close. To remove the beard, pinch it firmly between your forefinger and thumb and pull it away from the mussel shell. You can add a little chopped chilli to the stew if you like your food spicy._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   25 minutes\n\n1 tablespoon olive oil\n\n1 leek, finely sliced, or 1 fennel bulb, finely sliced\n\n3 garlic cloves, crushed\n\n400 g can diced tomatoes\n\n1 cup water\n\ngrated zest 1 lemon\n\n300 g firm white fish fillets (ling,\n\nblue eye), cut into chunks\n\n1 cleaned squid hood, cut into rings\n\n8 medium green prawns, peeled and de-veined, tails intact\n\n8 mussels, scrubbed and beards removed\n\n  cup flat-leaf parsley, roughly chopped\n\ncrusty sourdough bread and green salad to serve\n\nHeat oil in a large heavy-based saucepan on high. Saut\u00e9 the leek and garlic for 4\u20135 minutes, until tender.\n\nStir in the tomatoes, water and lemon zest and bring to the boil. Reduce the heat and simmer, partially covered, for 10\u201315 minutes.\n\nAdd the seafood to the pan and simmer for 4\u20135 minutes. Discard any unopened mussels. Sprinkle with parsley, and serve with a crusty sourdough and salad.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread and a generous serve of salad) = 1490 kJ, 12 g fat (includes 2 g saturated fat), 5 g fibre, 685 mg sodium\n\n**Fish, vegetable and peanut stir-fry**\n\n_Use any firm-fleshed white fish for this stir-fry \u2013 blue eye, ling or snapper all work well. With a mild chilli buzz and crunchy vegetables and peanuts, this is sure to become a family favourite._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   11 minutes\n\n2 teaspoons peanut oil\n\n500 g boneless fish fillets, sliced\n\n1 bunch choy sum, roughly chopped\n\n150 g sugar snap peas, trimmed\n\n4 green onions (shallots), sliced\n\n250 g hokkein noodles (or noodles of choice)\n\n\u00bd cup bean sprouts\n\n2 tablespoons chilli sauce\n\n2 tablespoons hoi sin sauce\n\n1 tablespoon salt-reduced soy sauce\n\nroughly chopped toasted peanuts and chopped coriander to serve\n\nHeat oil in a wok or large frying pan on high. Stir-fry the fish in 2 batches for 2\u20133 minutes or until just cooked through. Remove to a plate.\n\nAdd the choy sum, sugar snap peas and green onions to the wok. Stir-fry for 2\u20133 minutes until just beginning to wilt.\n\nCook the noodles according to the packet instructions. Drain well. Add the noodles to the wok with the bean sprouts and sauces. Toss well and stir-fry for 1\u20132 minutes to heat through.\n\nServe immediately sprinkled with peanuts and coriander.\n\nNUTRITION ANALYSIS 1 serve = 1575 kJ, 10 g fat (includes 2 g saturated fat), 6 g fibre, 690 mg sodium\n\n**_Super ingredient: Bean sprouts_**\n\nSprouts are germinated seeds that have sent out a baby shoot and root. As they absorb water and grow, they convert some of the stored carbohydrate and protein in the seed to energy, new growth and vitamins. The bean sprouts available from fresh produce stores are either mung, lentil or pea seeds. They are more a 'texture' than a vegetable, adding crunch to a sandwich and interest to a stir-fry. They are tasty, light and nutritious.\n**Fish, leek and potato pies**\n\n_Everyone loves pie \u2013 kids especially \u2013 so try something different with this recipe and make it with chicken sometimes. You'll need to replace the fish stock with chicken stock._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\nspray oil\n\n3 potatoes, peeled and cut into 2 cm cubes\n\n2 leeks, trimmed, washed thoroughly and sliced\n\n  cup plain flour\n\n1 cup low-fat milk\n\n\u00be cup fish stock\n\n750 g firm white fish fillets (blue eye, ling etc), cubed\n\n\u00bc cup chopped parsley\n\n4 sheets filo pastry\n\nmixed salad to serve\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF).\n\nHeat a heavy-based frying pan on high. Spray with oil and saut\u00e9 the potato and leek for 3\u20134 minutes until the leek is tender. Stir in flour and cook for 1 minute.\n\nRemove the frying pan from the heat. Stir in the milk and stock then return to the heat and cook, stirring, until the sauce boils and thickens. Reduce the heat and simmer for 3 minutes. Stir in the pieces of fish and the parsley.\n\nSpoon the filling into four 1 cup ramekin dishes.\n\nSpray each sheet of filo with a little oil. Scrunch up roughly and use to top the pies.\n\nBake for 15\u201320 minutes until the pastry is crisp and golden. Serve with salad.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 1850 kJ, 10 g fat (includes 3 g saturated fat), 6 g fibre, 470 mg sodium\n\n**Fish parcels**\n\n_Fish wrapped in parcels like this should cook in a preheated oven in 15\u201320 minutes, but it will depend on the thickness of the fish. Check after 15 minutes \u2013 the fish is cooked when it flakes easily when tested with a fork._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   20 minutes\n\n4 boneless fish fillets (snapper, blue eye)\n\n1 red capsicum, seeded and finely sliced\n\n1 red onion, finely sliced\n\n1 lemon, thinly sliced\n\n\u00bd cup dill sprigs\n\nfreshly ground black pepper\n\nsteamed vegetables or salad and extra dill to serve\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Cut 4 large squares of baking paper.\n\nPlace a fish fillet in the centre of each piece of baking paper. Top each fillet with an even amount of the capsicum, onion, lemon and dill. Sprinkle with a good grinding of black pepper.\n\nWrap up, folding in the ends to form a parcel. Place seam-side up (tuck ends under) in a baking tray. Bake for 15\u201320 minutes until cooked when tested.\n\nUnwrap the parcel and serve the fish with extra dill and salad or steamed vegetables of choice.\n\n**VARIATION**\n\nWhiting fillets, with their delicate texture and flavour are also delicious cooked in this way \u2013 but you will need two of these per serve as they are quite small.\n\nNUTRITION ANALYSIS 1 serve (including 1 potato and a generous serve of green beans) = 1315 kJ, 4 g fat (includes 1 g saturated fat), 7 g fibre, 150 mg sodium\n\n**_Super ingredient: Dill_**\n\nDill's soft ferny leaves and slight aniseed flavour are used to flavour many fish and potato dishes and smoked salmon. The seeds are also used to flavour pickles. Dill must be used fresh, as it rapidly loses its flavour once dried. Like many other fresh green herbs, dill is a source of potassium and magnesium, with smaller amounts of iron and calcium \u2013 as well as being high in vitamin C, folate, vitamin B1 and vitamin K.\n**Country chicken and vegetables**\n\n_We used a whole chicken for this all-in-one-dish roast, but you may prefer to buy your favourite chicken pieces so there's no fighting over who gets the drumsticks. Cooking time will be a little shorter with chicken pieces._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   1 hour\n\n500 g chat potatoes, halved\n\n300 g sweet potato, peeled and chopped\n\n2 leeks, trimmed, thoroughly washed and sliced\n\n1 knob garlic, separated and unpeeled\n\n1 bunch thyme\n\n1.8 kg chicken, cleaned, dried, quartered and skin removed\n\n1 cup white wine or water\n\nfreshly ground black pepper\n\n4 cup mushrooms, trimmed\n\n2 roma tomatoes, quartered\n\n2 zucchini, quartered\n\n2 baby eggplants, quartered\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF).\n\nArrange the potatoes, leek, garlic and thyme in a large baking tray.\n\nPlace the chicken portions on top of the vegetables and pour over the wine or water. Sprinkle with a good grinding of black pepper. Bake for 30 minutes.\n\nAdd the mushrooms, tomatoes, zucchini and eggplant to the baking tray. Bake for a further 30 minutes until the chicken is cooked through.\n\nNUTRITION ANALYSIS 1 serve = 2185 kJ, 22 g fat (includes 7 g saturated fat), 9 g fibre, 150 mg sodium\n\n**_Super ingredient: Thyme_**\n\nThanks to its aromatic oils, thyme imparts a wonderful savoury flavour to your cooking and goes well with any meat, soup or slow-cooked casserole. Like rosemary and oregano, it is high in polyphenols, a class of antioxidant that may cut the risk of heart disease. Thyme is easy to grow and is wonderful to have on hand so you can add a sprig of fresh thyme when you need it.\n\n**Chicken chilli con carne**\n\n**_This makes a good change from the usual beef chilli. It is lower in fat and has a lighter flavour. If you prefer to be traditional, simply replace the chicken with 500 g of beef mince._**\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   1 hour\n\n1 tablespoon canola oil\n\n500 g chicken mince\n\n1 large onion, chopped\n\n2 garlic cloves, crushed\n\n2 teaspoons plain flour\n\n2 teaspoons ground cumin\n\n1 teaspoon ground coriander\n\n1 teaspoon ground chilli\n\n1 cup salt-reduced beef stock\n\n400 g can diced tomatoes\n\n1 teaspoon dried oregano\n\n400 g can red kidney beans, rinsed and drained\n\nchopped coriander, steamed low-GI rice and green salad to serve\n\nHeat oil in a large heavy-based saucepan on high. Brown the mince for 3\u20134 minutes. Transfer to a plate.\n\nReduce the heat to medium and saut\u00e9 the onion and garlic for 3\u20134 minutes until tender. Add the flour, cumin, coriander and chilli and cook, stirring, for 1 minute.\n\nMix in the stock gradually, stirring to scrape up the sediment from the base of pan. Return the browned mince to the pan with the tomatoes and oregano and stir well.\n\nBring to the boil then reduce the heat and simmer, covered for 30 minutes. Uncover and simmer for a further 15 minutes until the sauce has thickened slightly.\n\nStir in the kidney beans and heat gently for 2\u20133 minutes. Sprinkle with coriander and serve on a bed of rice with salad.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup rice and a generous serve of salad) = 2680 kJ, 18 g fat (includes 5 g saturated fat), 10 g fibre, 575 mg sodium\n\n_**Super ingredient: Kidney beans**_\n\nLike all legumes, kidney beans fill you up and rank among the lowest foods for GI \u2013 kidney beans have a GI of 28. This means they take a long time to be digested, and the glucose enters your bloodstream more slowly than almost any other carbohydrate \u2013 even multigrain bread or porridge. If you can swap some of your potatoes or rice for beans, you're way ahead in the filling-power stakes. Kidney beans are an excellent source of protein for vegetarians. And for meat eaters they 'extend' the protein value of a meal \u2013 when you add a can of kidney beans to a recipe, you save on cost yet still have a nutritious high-protein dinner.\n**Glazed pork fillet**\n\n_We cooked our pork and potato slices on a char-grill for that lovely smoky flavour and attractive griddle markings, but if you prefer, simply arrange the potatoes in a single layer on a baking tray and bake them at 180\u02daC (350\u02daF) for 10\u201315 minutes. As an alternative to grilled potatoes, you can serve mashed or baked jacket potatoes._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   25 minutes\n\n500 g pork fillets, trimmed\n\n2 teaspoons olive oil\n\nfreshly ground black pepper\n\n4 potatoes, peeled, thickly sliced\n\nspray oil\n\nradicchio and green salad with low-fat dressing to serve\n\nDressing\n\n\u00bc cup balsamic vinegar\n\n\u00bc cup maple syrup\n\n1 tablespoon roughly chopped oregano\n\n1 garlic clove, crushed\n\nPreheat a char-grill pan to medium. Brush the pork fillets with oil and sprinkle with freshly ground black pepper.\n\nChar-grill the pork for 8\u201310 minutes on each side until cooked through. Rest, covered, for 5 minutes.\n\nSpray the potato slices with a little oil. Cook on the char-grill for 4\u20135 minutes each side until golden and crisp.\n\nTo make the dressing combine the vinegar, maple syrup, oregano and garlic in a small jug and whisk together well.\n\nTo serve, slice the pork fillets thickly and arrange on plates with the grilled potatoes and salad. Drizzle over the dressing and eat straight away.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 1505 kJ, 10 g fat (includes 2 g saturated fat), 4 g fibre, 100 mg sodium\n\n**_Super ingredient: Radicchio_**\n\nRadicchio, a red-leafed Italian chicory, has a bitter taste that works well with strong flavours like duck or venison. Its blue-purple colour comes from anthocyanins, natural plant antioxidants that can kill off harmful bacteria. Anthocyanins are also found in red grapes, blueberries, cranberries and beetroot. Like all lettuces, you'll get plenty of vitamin C, folate and vitamin B1, three heat-sensitive vitamins, which are hard to get.\n\n**Pork and Asian greens stir-fry**\n\n_The secret to stir-frying is the heat. You need a really high heat to cook the ingredients quickly and prevent them stewing in their own juices. That way your stir-fries will always be fresh and crisp, not limp and soggy._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   11 minutes\n\n1 tablespoon rice bran oil\n\n500 g pork fillet or skinless chicken breast, trimmed and sliced\n\n1 onion, sliced\n\n1 garlic clove, crushed\n\n1 teaspoon grated ginger\n\n2 bunches baby bok choy, quartered\n\n1 bunch broccolini\n\n100 g snow peas, trimmed\n\n\u00bc cup sweet chilli sauce\n\n\u00bc cup lime or lemon juice\n\n2 tablespoons salt-reduced soy sauce\n\ncoriander leaves and steamed rice or noodles to serve\n\nHeat oil in a wok or large frying pan on high. Stir-fry the pork in 2 batches for 2\u20133 minutes. Remove to a plate.\n\nAdd the onion, garlic and ginger to the wok and stir-fry for 1\u20132 minutes until just tender.\n\nReturn the pork to the wok with the baby bok choy, broccolini, peas, sweet chilli sauce, citrus juice and soy sauce. Stir-fry for 2\u20133 minutes until greens are just beginning to wilt.\n\nSprinkle with coriander and serve immediately with steamed rice or noodles.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup rice or noodles) = 1385 kJ, 7 g fat (includes 2 g saturated fat), 5 g fibre, 555 mg sodium\n\n**_Super ingredient: Asian greens_**\n\nAsian leafy greens such as bok choy, choy sum, on choy and gai lum are tops for nutrition, being rich in vitamin C, beta-carotene, fibre and many B vitamins. Surprisingly they can also contribute a lot of calcium and iron. Best of all, they are low in kilojoules with almost no fat. All good reasons to tuck into them!\n**Honey-rosemary rack of lamb**\n\n_Lamb cutlets can be high in saturated fat. That's why we like to buy a 'frenched' rack of lamb \u2013 all the fat has been trimmed off and the bones scraped clean for a neat presentation._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   40 minutes\n\n500 g chat potatoes, halved\n\n2 x 6 racks lamb, frenched\n\n2 teaspoons olive oil\n\nrosemary sprigs\n\n4 field mushrooms, stalks trimmed\n\n2 roma tomatoes, halved\n\n\u00bc cup honey\n\n1 bunch asparagus, trimmed\n\ngreen salad to serve (optional)\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF).\n\nArrange the potatoes in a baking dish. Rub the lamb racks with olive oil and sprinkle with rosemary. Place them on top of the potatoes and roast for 20 minutes.\n\nAdd the mushrooms and tomatoes to the baking dish and continue roasting for 5\u201310 minutes until the lamb is cooked to your liking.\n\nRemove the lamb racks from the oven, drizzle with honey and rest, covered with foil, for 10 minutes. Return the vegetables to the oven until ready to serve.\n\nArrange the asparagus in a microwave-safe dish with a sprinkling of water. Cover with plastic wrap and cook on high (100%) power for 1\u20132 minutes until just tender.\n\nTo serve, place a roast mushroom in the centre of each plate. Cut each lamb rack in half and arrange on top of the mushrooms. Serve with the roast potatoes and asparagus. If you like you can also serve with a green salad.\n\n**VARIATION**\n\nSweet, smoky maple syrup makes a great change from honey if you have any in the fridge.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of salad) = 1945 kJ, 18 g fat (includes 6 g saturated fat), 6 g fibre, 135 mg sodium\n\n**_Super ingredient: Honey_**\n\nHoney's anti-bacterial properties have been well known for over 80 years and it is considered effective in the treatment for wounds, ulcers and burns. Being 'natural', honey is often considered a healthy sweetener. In fact, honey has fractionally less sugar than processed sugars, so use judiciously. It has tiny quantities of B vitamins and minerals like potassium and magnesium, and some phytochemicals and antioxidants.\n**Easy biryani**\n\n_Coconut-flavoured light evaporated milk is a wonderful product \u2013 it gives you all the flavour and creaminess you love, without the high saturated fat of coconut milk. Choose a variety of rice with a low GI for this carb-rich dish._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\n1 tablespoon olive oil\n\n2 onions, sliced\n\n1 garlic clove, crushed\n\n1 teaspoon grated ginger\n\n1 teaspoon ground coriander\n\n\u00bc teaspoon paprika\n\n\u00bc teaspoon ground cumin\n\n\u00bc teaspoon ground turmeric\n\n500 g lamb backstraps or fillet, cubed\n\n1 cup low-GI rice, such as Doongara or Moolgiri\n\n2 cups low-salt chicken stock or water\n\n\u00bd cup coconut flavoured light evaporated milk\n\n1 tablespoon chopped coriander\n\n1 teaspoon caraway seeds\n\nsteamed green vegetables to serve\n\nHeat oil in a large frying pan on medium. Saut\u00e9 the onions, garlic and ginger for 2\u20133 minutes until the onion is tender.\n\nStir in the spices and cook for 1 minute or until aromatic. Add the lamb and cook, stirring, for 4\u20135 minutes until browned.\n\nAdd the rice and stock or water and stir well. Bring to the boil, then lower the heat and simmer, covered, for 15\u201320 minutes until the rice is tender, adding more liquid if required.\n\nStir in the coconut-flavoured milk and heat gently. Just before serving blend in the coriander and caraway seeds. Accompany with steamed green vegetables.\n\n**VARIATIONS**\n\n  If you like, add some sultanas and garnish with a dollop of natural yoghurt and cashews.\n\n  Use cubed pork fillet or chicken as a change from lamb.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of green vegetables) = 1895 kJ, 10 g fat (includes 3 g saturated fat), 4 g fibre, 435 mg sodium\n\n**_Super ingredient: Caraway seeds_**\n\nThese tiny crescent-shaped brown seeds add a wonderful aromatic flavour to any dish. Since Roman times, they have been taken to aid the flow of digestive juices (such as saliva and bile) and to prevent flatulence. Their characteristic flavour comes from two terpenes called carvone and limonene. These are natural plant chemicals now being studied for their role in blocking the development of cancer tumours in animals. Terpenes are also found in lemon, ginger and cumin seeds.\n**Dead-simple Mediterranean veal**\n\n_Here's our favourite quick-and-easy recipe. It's made in minutes and will have family and friends asking for it time and again. You can vary this dish by adding thin slices of eggplant or replacing the grated mozzarella with slices of baby bocconcini._\n\nServes 4\n\nPreparation time   5 minutes\n\nCooking time   10 minutes\n\nspray oil\n\n4 veal schnitzels (or veal steaks), pounded lightly\n\n  cup passata sauce\n\n  cup grated reduced-fat mozzarella cheese\n\n\u00bd cup basil (or baby spinach) leaves\n\nsalad and vegetables of choice to serve\n\nHeat a non-stick frying pan on high. Spray with oil and cook the veal for 1\u20132 minutes on each side. Transfer to a foil-lined baking tray.\n\nSpread each piece of veal with about a tablespoon of sauce. Sprinkle each evenly with cheese.\n\nPreheat the grill to high. Grill the veal for 3\u20135 minutes until the cheese melts and the topping is bubbly. Top with basil leaves.\n\nServe with salad or vegetables of choice.\n\nNUTRITION ANALYSIS 1 serve (including 1 potato and a generous serve of salad) = 1235 kJ, 8 g fat (includes 2 g saturated fat), 5 g fibre, 205 mg sodium\n\n**Osso bucco**\n\n_The secret ingredient in a successful osso bucco is the marrow inside the bone. That's why it is important when making this dish to ensure that the marrow is exposed on both sides._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   1 hour 40 minutes\n\n1 tablespoon olive oil\n\n6\u20138 pieces veal osso bucco (sliced veal shank)\n\n2 onions, sliced\n\n1 garlic clove, crushed\n\n2 new potatoes, cut into chunks\n\n2 carrots, sliced\n\n2 stalks celery, chopped\n\n400 g can crushed tomatoes\n\n1 cup white wine\n\n1 cup water\n\n1 bay leaf\n\nsweet potato mash, crusty sourdough bread and steamed green vegetables to serve\n\nHeat oil in a large saucepan on high. Brown the meat well on all sides for 5\u20136 minutes and transfer to a plate.\n\nAdd the onions and garlic to the same pan and saut\u00e9 until the onion is tender.\n\nStir in the potatoes, carrots and celery and saut\u00e9 for 2 minutes more. Blend in the tomatoes, wine, water and bay leaf and stir well.\n\nBring to the boil and return the meat to the pan. Reduce the heat and simmer gently, uncovered, for 1\u00bc \u20131\u00bd hours until the meat is very tender. Remove the bay leaf. Serve immediately with sweet potato mash, crusty sourdough bread and steamed green vegetables.\n\n**VARIATION**\n\nServe this dish sprinkled with the traditional gremolata. Mix together \u00bc cup chopped parsley, 2 finely chopped garlic cloves and the grated zest of 1 lemon.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup mash, 1 slice bread and a generous serve of green vegetables) = 2115 kJ, 10 g fat (includes 2 g saturated fat), 10 g fibre, 345 mg sodium\n\n**_Super ingredient: Onions_**\n\nA close cousin of garlic, onions are high in the phytochemical quercetin, an antioxidant that helps to protect us from cancer-causing agents as well as keeping our arteries unclogged and free-flowing. Onions are also a good source of vitamin C.\n**Fillet steak with chilli jam**\n\n_We like our steak on the pinkish side. If you prefer it more well done, turn the steak and cook for a further 2\u20134 minutes each side._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   40 minutes\n\n4 tomatoes, halved\n\n2 zucchini, cut into chunks\n\n2 red capsicums, seeded and quartered\n\n2 red onions, halved\n\n1 fennel bulb, thickly sliced\n\n1 sweet potato, peeled and cut into chunks\n\nspray oil\n\n4 fillet steaks, trimmed\n\ngreen salad to serve\n\nChilli jam\n\n400 g can diced tomatoes\n\n\u00bd cup white wine vinegar\n\n\u00bd cup brown sugar\n\n1 onion, chopped\n\n1 garlic clove, crushed\n\n2 teaspoons chopped chilli\n\n1 teaspoon chopped ginger\n\nPreheat the oven to hot, 200\u00baC (400\u00baF).\n\nArrange the vegetables in a baking tray. Spray with oil and bake for 20 minutes. Toss well and bake for a further 20 minutes.\n\nWhile the vegetables are roasting, make the chilli jam. Place all the ingredients into a heavy-based saucepan and bring to the boil. Reduce the heat and simmer for 20\u201325 minutes until thick and jam-like. Set aside.\n\nHeat a non-stick frying pan on high. Spray with oil and cook the steaks for 3\u20135 minutes each side for medium-rare, or until cooked to taste.\n\nServe the steaks with a spoonful of chilli jam, the mixed roast vegetables and a green salad of choice.\n\n**VARIATION**\n\nThe roast vegetables and chilli jam work equally well with chicken, lamb or pork.\n\nNUTRITION ANALYSIS 1 serve (including a generous serve of green salad) = 1640 kJ, 5 g fat (includes 2 g saturated fat), 11 g fibre, 110 mg sodium\n\n**_Super ingredient: Red meat_**\n\nLean red meat is a vital source of B vitamins, zinc and easily-absorbed iron. Iron is important for healthy blood, the development of our brain and for improved work performance. Zinc aids the functioning of our immune system. The CSIRO Diet showed how lean meat can be a great diet food. It's filling and satisfying so you don't look for a snack between meals. Remember to trim all visible fat from meat. When shopping for meat, look for cuts with the least fat and marbling.\n**Beef in red wine**\n\n_When making sauces, always blend a cold liquid into a hot roux off the heat to prevent lumps from forming. For a perfectly smooth sauce, stir constantly over a medium heat until the sauce comes to the boil and thickens._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   1 hour 10 minutes\n\n2 tablespoons olive oil\n\n30 g light margarine\n\n2 rashers shortcut bacon, sliced\n\n2 garlic cloves, crushed\n\n750 g round steak, trimmed and cubed\n\n12 pickling onions, peeled\n\n250 g button mushrooms, trimmed\n\n\u00bc cup plain flour\n\n\u00bc cup tomato paste\n\n2 cups water\n\n1 cup red wine\n\nmashed potatoes and steamed green vegetables to serve\n\nHeat oil and light margarine in a large frying pan on medium-high. Add the bacon and garlic and saut\u00e9 for 1\u20132 minutes. Transfer to a plate.\n\nAdd the steak to the same pan in 2 batches, browning well for 3\u20134 minutes. Transfer to a plate. Repeat with onions and mushrooms, transferring each to a plate once they're browned.\n\nAdd the flour to the pan and cook, stirring vigorously, to scrape up any sediment. Cook until golden brown. Add the tomato paste, and stir in well.\n\nRemove the frying pan from the heat. Gradually blend in the water and wine until smooth. Return to the heat and cook, stirring continuously, until the sauce boils and thickens.\n\nLower the heat and return the bacon, steak, onions and mushrooms to the pan. Simmer for 50\u201360 minutes until the meat is tender, adding more water if required. Serve with mash and vegetables.\n\nNUTRITION ANALYSIS 1 serve (including 1 potato and a generous serve of green vegetables) = 2190 kJ, 19 g fat (includes 5 g saturated fat), 8 g fibre, 430 mg sodium\n\n**_Super ingredient: Red wine_**\n\nRed wine has been suggested as the answer to the so-called 'French paradox' \u2013 why the French have the second lowest rate of heart disease in the world (after Japan), despite their love of rich cholesterol-laden food. In modest amounts, red wine is good for your heart. Red wine (and grapes) contain more than 50 phenolic compounds known as flavonoids; they act as antioxidants, reduce thickening of the arteries and keep the blood 'thin' and smooth flowing. Red wine has 9\u201310 times more of these natural chemicals than white wine.\n**Sweet soy beef with cabbage**\n\n_For a vibrant green dish, add a handful of spinach leaves and some sliced Asian greens to the pan with the shredded cabbage._\n\nServes 4\u20136\n\nPreparation time   10 minutes (plus marinating time)\n\nCooking time   10 minutes\n\n\u00bc cup sake\n\n2 tablespoons light soy sauce\n\n1 tablespoon sugar\n\n1 tablespoon grated ginger\n\n1 teaspoon sesame oil\n\n500 g beef, thinly sliced\n\n1 tablespoon rice bran oil\n\n\u00bc Chinese cabbage, thickly sliced\n\n1 tablespoon sesame seeds\n\nudon noodles (cooked according to packet instructions) to serve\n\nCombine the sake, soy, sugar, ginger and sesame oil in a large mixing bowl. Add the beef and toss well. Leave to marinate for 15 minutes.\n\nHeat oil in a wok or large frying pan on high. Remove the beef from the marinade and reserve the marinade.\n\nStir-fry the beef for 4\u20135 minutes, then add the cabbage, reserved marinade and sesame seeds. Stir-fry for 2\u20133 minutes until the cabbage begins to wilt. Serve straight away with udon noodles.\n\n**VARIATION**\n\nUse minced steak in place of beef strips. Omit the marinating step and simply stir-fry the mince in all the marinade ingredients before adding cabbage and sesame seeds.\n\nNUTRITION ANALYSIS 1 serve (including \u00bd cup noodles) = 1430 kJ, 11 g fat (includes 3 g saturated fat), 2 g fibre, 365 mg sodium\n\n**_Super ingredient: Cabbage_**\n\nKids turn their noses up at it, but studies show that those of us who eat large helpings of cabbage (and its fellow brassicas, cauliflower, broccoli and Brussels sprouts) show a reduced risk of several cancers. Their active ingredients \u2013 sulphur compounds known as indoles and isothiocyanates \u2013 activate cancer-fighting enzymes. As well, they rate tops for fibre (hence their reputation for flatulence), beta-carotene, vitamin C and folate \u2013 and have very few kilojoules.\n\n**Tagliatelle with chilli meatballs**\n\n_We used tagliatelle \u2013 long strips of egg pasta \u2013 for this dish, but you could use any egg pasta such as fettuccine or the wider pappardelle \u2013 or even spaghetti if that's the family favourite. Or for added health benefits use wholemeal pasta._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   35 minutes\n\n500 g lamb or beef mince\n\n\u00bd cup fresh breadcrumbs\n\n1 egg, lightly beaten\n\n1 onion, finely chopped\n\n1 tablespoon pine nuts\n\n2 teaspoons chopped oregano\n\n1 long red chilli, chopped\n\n1 teaspoon grated lemon zest\n\n1 garlic clove, crushed\n\n1 tablespoon olive oil\n\n375 g tagliatelle or other pasta of choice, cooked and drained\n\noregano leaves, grated fresh parmesan, crusty bread and green salad to serve\n\nRich tomato sauce\n\n1 tablespoon olive oil\n\n1 onion, chopped\n\n1 garlic clove, crushed\n\n400 g can diced tomatoes\n\n\u00bd cup white wine\n\n\u00bd cup salt-reduced chicken stock\n\n\u00bc cup semi-dried tomatoes\n\n1 tablespoon tomato paste\n\n1 teaspoon brown sugar\n\nPlace all the ingredients except the oil and pasta into a large mixing bowl. Mix thoroughly and form into walnut-sized balls. You should make around 24 meatballs from the mixture.\n\nHeat a large non-stick frying pan on high. Add the oil and cook the meatballs in 2 batches for 4\u20135 minutes, turning until evenly browned. Drain on a paper towel.\n\nTo make the sauce, heat oil in a heavy-based saucepan on high. Saut\u00e9 the onion and garlic until tender. Stir in the tomatoes, wine, stock, semi-dried tomatoes, paste and sugar. Reduce the heat and simmer for 15 minutes.\n\nAdd the meatballs to the sauce and simmer for 10\u201315 minutes until cooked through (add a little more liquid to the sauce if it becomes too thick).\n\nToss the sauce through the hot pasta and serve immediately, sprinkled with a few oregano leaves and a little grated parmesan. Serve with crusty bread and green salad.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread and a generous serve of salad) = 2900 kJ, 23 g fat (includes 5 g saturated fat), 8 g fibre, 380 mg sodium\n\n**Pasta with tuna and capers**\n\n_To prevent pasta from sticking together when cooking, add a splash of olive oil to the boiling water, then stir for a few minutes after you add the pasta to the water._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   10 minutes\n\n400 g shell pasta or other pasta of choice\n\n400 g can tuna in brine, drained\n\n125 g reduced-fat fetta cheese, crumbled\n\n\u00bc cup chopped parsley\n\n2 tablespoons baby capers\n\nfinely grated zest and juice 1 lemon\n\n2 garlic cloves, crushed\n\n2 teaspoons olive oil\n\nsalad and crusty bread to serve\n\nCook the pasta in plenty of boiling water following the packet instructions for timing. Drain well and transfer to a large mixing bowl.\n\nToss all the remaining ingredients through the hot pasta. Serve with salad and crusty bread.\n\n**VARIATION**\n\nToss through some sliced char-grilled capsicum and some blanched peas or sugar snaps for a heartier sauce.\n\nNUTRITION ANALYSIS 1 serve (including 1 slice bread and a generous serve of salad) = 2570 kJ, 13 g fat (includes 4 g saturated fat), 7 g fibre, 600 mg sodium\n\n**_Super ingredient: Pasta_**\n\nOf all the starchy carbohydrates, pasta is the one nutritionists love. It's low in fat, has little salt and is low GI. Compared with potato and many varieties of rice, the carbohydrate in al dente pasta is slowly digested and absorbed. This makes it a great advantage for people with diabetes and for endurance athletes. Pasta is also blessed with protein, B vitamins (notably vitamin B1 needed to release energy from food) plus a little iron and calcium. Use wholemeal pasta for added health benefits.\n**Pad Thai**\n\n_The trick with stir-fries is to make sure you have everything ready before you start to cook. Making preparation a team effort saves you a little time and can be a lot of fun._\n\nServes 4\n\nPreparation time   20 minutes\n\nCooking time   20 minutes\n\n250 g rice stick noodles\n\n2 tablespoons canola oil\n\n125 g firm tofu, cubed\n\n1 red onion, sliced\n\n2 garlic cloves, crushed\n\n1 chicken breast fillet, sliced\n\n8 green prawns, shelled and de-veined, tails intact\n\n2 eggs, beaten\n\n\u00bc cup lime or lemon juice\n\n\u00bc cup basil leaves\n\n2 tablespoons coriander leaves, plus extra to serve\n\n2 tablespoons palm sugar or brown sugar\n\n2 tablespoons sweet chilli sauce\n\n1 tablespoon fish sauce\n\n2 small red chillies, chopped\n\ncrushed roasted peanuts to serve\n\nSoak and cook the rice stick noodles following the packet instructions. Drain well and set aside.\n\nHeat oil in a wok or large frying pan on high. Add the tofu in batches for 1\u20132 minutes. Drain on a paper towel.\n\nAdd the onion and garlic and stir-fry for 1 minute. Toss through the chicken and prawns and stir-fry for a further 3\u20134 minutes.\n\nPush the mixture to one side of the pan and pour in the beaten egg. Stir briefly in the pan until scrambled and mix through the chicken mixture.\n\nAdd the citrus juice, herbs, sugar, sauces and chillies with the noodles and tofu, tossing well. Stir-fry for 2 more minutes then serve immediately topped with crushed peanuts and some extra coriander leaves.\n\nNUTRITION ANALYSIS 1 serve = 2560 kJ, 22 g fat (includes 4 g saturated fat), 2 g fibre, 840 mg sodium \nAccompaniments\n\nVegetables and salads can turn a basic meal into a balanced dinner. Meat and potatoes alone are not enough! For energy and glowing good health we need to include vegetables or a salad to supply those all-important nutrients.\n\nVegetables\n\nUse our wonderful recipe ideas to pile your plate high with vegetables. Aim for half your plate to be filled with vegetables, with the remaining half comprising lean protein and starchy carbohydrates (this includes starchy vegetables such as potato, and rice, lentils or pasta) (see our diagram). Nutritionists suggest you eat five serves of vegetables every day and we hope our recipes will inspire you to achieve this. We team them with grains and legumes so some of them are a complete nutritious package. For added zest, we also include plenty of fresh herbs, chilli and a splash of good oil.\n\nNutritionists confirm that 'eating by the rainbow' makes good nutrition sense. So let colour be your guide. The more colour you put on your plate, the better your intake of antioxidants from colourful vegetables and fruit. Think of carrots, beetroot, spinach, red carrots, purple onions, yellow capsicum, mint leaves \u2013 they are all brightly coloured and all so good for you.\n\nSalads\n\nWe like to say that 'a salad a day keeps the doctor away'. Not only are salads healthy in their own right, they are also an easy way to make any meal \u2013 even pizza or fish and chips \u2013 more balanced.\n\nEating vegetables raw means that you maximise your intake of heat-sensitive vitamins such as vitamin C, B1 and folate that often get destroyed during cooking.\n\nSalads are also a good source of phyto-chemicals, natural plant compounds that act as antioxidants to protect the body and slow the ageing process. For the highest antioxidant value, choose lettuces with darker leaves (such as dark oak leaf, mignonette, radicchio or coral), baby spinach leaves or rocket \u2013 the darker the leaves, the better.\n\nDon't say 'No' to a little dressing either. Research shows the oil from a dressing or mayonnaise improves the absorption of the fat-soluble vitamins and antioxidants in the salad itself \u2013 and also means you enjoy it more!\n\n**Parmesan and olive damper**\n\n_This damper is delicious with Roasted Tomato and Capsicum Soup. It freezes very well \u2013 wrap in foil, thaw at room temperature and re-heat at 180\u02daC (350\u02daF) for 5\u201310 minutes._\n\nServes 6\u20138\n\nPreparation time   10 minutes\n\nCooking time   50 minutes\n\nspray oil\n\n3 cups self-raising flour\n\n45 g light margarine\n\n\u00bd cup grated parmesan\n\n\u00bc cup sliced pitted olives\n\n\u00bc cup snipped chives\n\n1\u00bc cups low-fat milk\n\n2 teaspoons sesame or sunflower seeds\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Spray a small baking tray with oil.\n\nSift the flour into a large bowl. Rub in the light margarine gently, using your fingertips. Stir in the parmesan, olives and chives.\n\nMake a well in the centre of the flour mixture. Pour in milk all at once. Using a bread and butter knife, mix quickly to a soft, sticky dough. Do not over-mix.\n\nTurn the dough out onto a lightly floured surface. Knead gently and form into a round. Place on the prepared tray. Brush with milk and sprinkle with sesame seeds.\n\nBake for 45\u201350 minutes until the loaf is golden, firm and sounds hollow when tapped. Cool on a wire rack. Serve warm.\n\n**VARIATIONS**\n\n  You can increase the fibre in this loaf by using wholemeal flour and adding some oat bran. You may need to increase the liquid slightly.\n\n  Experiment with other flavour combinations such as garlic and pine nuts, grated zucchini and tasty cheese, sun-dried tomato and basil.\n\nNUTRITION ANALYSIS 1 serve = 1260 kJ, 8 g fat (includes 2 g saturated fat), 3 g fibre, 605 mg sodium\n\n**Vietnamese salad**\n\nThis salad is a fresh and zesty accompaniment to Tofu and Cannellini Bean Patties.\n\nServes 6\u20138\n\nPreparation time   15 minutes\n\n2 cups finely shredded Chinese cabbage\n\n2 Lebanese cucumbers, cut into matchsticks\n\n1 small turnip, peeled and cut into matchsticks\n\n1 carrot, peeled and cut into matchsticks\n\n1 bunch mint leaves, picked\n\n1 bunch chives, cut into 2 cm lengths\n\n1 small red chilli, finely sliced\n\n1 tablespoon chopped peanuts\n\nDressing\n\n\u00bc cup white vinegar\n\n2 tablespoons sugar\n\n2 teaspoons fish sauce\n\nPlace all the ingredients, except the peanuts, into a large mixing bowl.\n\nTo make the dressing, whisk together all ingredients until the sugar dissolves.\n\nJust before serving, pour on the dressing and toss well. Serve sprinkled with chopped peanuts.\n\nNUTRITION ANALYSIS 1 serve = 215 kJ, 1 g fat (includes negligible saturated fat), 3 g fibre, 150 mg sodium\n\n**_Super ingredient: Mint_**\n\nPeppermint contains menthol, a powerful therapeutic ingredient, that can give relief from indigestion and increase the flow of digestive juices. Clinical trials have shown peppermint oil can relieve symptoms of irritable bowel syndrome (IBS). Peppermint tea is one of the most popular herbal infusions as it's naturally reviving. Try it instead of coffee when you need a quick lift. Make fresh peppermint tea by pouring boiled water over fresh mint leaves. Steep for 2 minutes, strain and add a teaspoon of honey, to taste.\n\n**Asparagus and bamboo shoot salad**\n\n_This Asian-inspired salad is full of wonderful crunchy ingredients. Dress it at the last minute so it doesn't go soggy._\n\nServes 6\n\nPreparation time   10 minutes\n\n2 bunches asparagus, trimmed, blanched and halved\n\n1 cup canned shredded bamboo shoots, drained\n\n1 cup bean sprouts, trimmed\n\n1 cup thinly sliced radish\n\n1 cup coriander leaves\n\n3 green onions (shallots), sliced diagonally\n\nzest of 2 limes\n\nDressing\n\njuice of 1 lime\n\n1 tablespoon honey\n\n1 teaspoon sesame oil\n\n\u00bd teaspoon chilli flakes\n\nCombine all salad ingredients in a bowl.\n\nTo make the dressing, whisk all the ingredients together. Just before serving pour the dressing over the salad and toss well.\n\nNUTRITION ANALYSIS 1 serve = 185 kJ, 1 g fat (includes negligible saturated fat), 2 g fibre, 10 mg sodium\n\n**Bean and spinach salad**\n\n_Turn this salad into a meal by adding some smoked trout or canned tuna, a few black olives, cherry tomatoes and some shaved parmesan._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   2 minutes\n\n200 g baby spinach leaves\n\n200 g green beans, trimmed, blanched\n\n100 g sugar snap peas, blanched\n\n2 hard-boiled eggs, quartered\n\n1 tablespoon pine nuts, toasted\n\nLemon-mustard dressing\n\n2 tablespoons flaxseed oil\n\n1 tablespoon lemon juice\n\n1 teaspoon grated lemon zest\n\n1 teaspoon Dijon mustard\n\nCombine the spinach, beans and peas in a large bowl.\n\nMake the dressing by whisking all the ingredients together.\n\nWhen ready to serve, pour on the dressing and toss gently. Top with the hard-boiled eggs and toasted pine nuts.\n\n**COOK'S TIP**\n\nTo blanch beans and peas place them in a saucepan of boiling water. Simmer for 30\u201360 seconds, drain well, refresh under cold water and pat dry with paper towels.\n\nNUTRITION ANALYSIS 1 serve = 740 kJ, 14 g fat (includes 2 g saturated fat), 4 g fibre, 55 mg sodium\n\n**Carrot salad**\n\n_Try this yummy salad wrapped in lavosh with sliced cold meat or chicken, or use it to accompany your favourite roast. For a change, try this with roasted pepitas._\n\nServes 4\n\nPreparation time   15 minutes\n\n500 g carrots, cut into matchsticks and blanched\n\n\u00bc cup chopped coriander leaves\n\n2 tablespoons currants or chopped raisins\n\n2 garlic cloves, crushed\n\n1 teaspoon roasted cumin seeds, crushed\n\n2 tablespoons extra-virgin olive oil\n\n1 tablespoon lemon or orange juice\n\n1 tablespoon pepitas (peeled pumpkin seeds)\n\nCombine the carrot, coriander, currants, garlic and cumin in a large mixing bowl.\n\nPour on the oil and lemon juice and toss together well. Serve topped with pepitas.\n\nNUTRITION ANALYSIS 1 serve = 655 kJ, 11 g fat (includes 2 g saturated fat), 5 g fibre, 60 mg sodium\n\n**Fattoush**\n\nThis traditional Arabic salad is often made more substantial by adding torn pieces of toasted pitta bread through the salad with the dressing.\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\n4 roma tomatoes, cut into chunks\n\n2 Lebanese cucumbers, quartered and thickly sliced\n\n2 stalks celery, thinly sliced on diagonal\n\n1 small red onion, sliced\n\n\u00bc cup roughly chopped parsley\n\n\u00bc cup roughly chopped mint\n\ntoasted pitta bread to serve\n\nDressing\n\nfinely grated zest and juice of 1 lemon\n\n1 tablespoon extra-virgin olive oil\n\n2 garlic cloves, crushed\n\nCombine the tomatoes, cucumber, celery, onion and herbs in a large mixing bowl.\n\nTo make the dressing, whisk the lemon zest and juice with the oil and garlic. Toss through the salad. Serve with toasted Lebanese bread on the side or tossed through the salad.\n\nNUTRITION ANALYSIS 1 serve = 230 kJ, 4 g fat (includes 1 g saturated fat), 2 g fibre, 30 mg sodium\n\n**Couscous and chickpea salad**\n\n_This salad is great with our Fish Parcels (page 145) or Country Chicken (page 146) \u2013 simply roast the chicken without the potatoes. If you like, add some sliced roasted capsicum or sliced black olives._\n\nServes 4\n\nPreparation time   10 minutes (plus standing time)\n\nCooking time   10 minutes\n\n1 cup couscous\n\nfinely grated zest and juice 1 lemon\n\n1 cup boiling water or low-salt stock\n\n400 g can chickpeas, drained and rinsed\n\n1 Lebanese cucumber, sliced\n\n\u00bd cup shredded mint\n\n1 tablespoon olive oil\n\nMix the couscous and lemon zest in a mixing bowl. Pour over boiling water and stir briefly. Cover and leave to stand for 5 minutes, or until all the liquid has been absorbed.\n\nFluff the couscous with a fork and stir through the chickpeas, cucumber, mint, lemon juice and oil.\n\nNUTRITION ANALYSIS 1 serve = 1250 kJ, 6 g fat (includes 1 g saturated fat), 4 g fibre, 160 mg sodium\n\n**Tabbouleh**\n\n_A fabulous accompaniment to any grilled, barbecued or roasted meat, chicken or fish. Or try with Our Favourite Wrap (page 107) or on our Steak Sandwich (page 106)._\n\nServes 6\u20138\n\nPreparation time   15 minutes (plus standing time)\n\n\u00bd cup burghul (cracked wheat)\n\nboiling water\n\n1 cup roughly chopped parsley\n\n1 cup roughly chopped mint\n\n2 roma tomatoes, diced\n\n3 green onions (shallots), finely sliced diagonally\n\n\u00bc cup lemon juice\n\n2 tablespoons olive oil\n\nfinely grated zest 1 lemon\n\n2 garlic cloves, crushed\n\nPlace the burghul in a large bowl and add enough boiling water to cover. Allow to stand for 15 minutes.\n\nDrain the burghul if necessary and return it to the bowl. Stir through the remaining ingredients, tossing well.\n\nChill, covered, until required.\n\nNUTRITION ANALYSIS 1 serve = 395 kJ, 6 g fat (includes 1 g saturated fat), 3 g fibre, 10 mg sodium\n\n**Fennel and lima bean salad**\n\nServes 4\n\nPreparation time   15 minutes\n\n1 fennel bulb, trimmed, thinly sliced\n\n400 g can lima beans, drained and rinsed\n\n100 g snow peas, trimmed and blanched\n\n1 small red onion, thinly sliced\n\n\u00bd red capsicum, seeded and thinly sliced\n\n\u00bd cup basil leaves\n\n\u00bc cup capers\n\n1 teaspoon celery seeds\n\nDressing\n\n2 tablespoons red wine vinegar\n\n1 tablespoon extra-virgin olive oil\n\n1 garlic clove, crushed\n\n1 teaspoon Dijon mustard\n\nCombine all the ingredients, except the celery seeds, in a large mixing bowl.\n\nTo make the dressing, whisk all the ingredients together.\n\nJust before serving pour on the dressing and toss the salad well. Sprinkle with celery seeds.\n\nNUTRITION ANALYSIS 1 serve = 555 kJ, 5 g fat (includes 1 g saturated fat), 5 g fibre, 350 mg sodium\n\n**Mixed bean salad with haloumi cheese**\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   5 minutes\n\n400 g can three bean mix, drained and rinsed\n\n50 g green beans, trimmed, blanched\n\n1 small red onion, sliced\n\n\u00bc cup semi-dried tomatoes\n\n2 tablespoons sliced pitted green olives\n\n1 tablespoon chopped basil\n\n50 g haloumi cheese, sliced thinly\n\n2 tablespoons seasoned flour\n\nspray oil\n\nDressing\n\n1 tablespoon olive oil\n\njuice \u00bd lemon\n\nCombine the bean mix, green beans, onion, tomatoes, olives and basil in a large bowl.\n\nTo make the dressing, whisk together the oil and juice and toss through the bean mixture.\n\nPat the cheese slices dry with paper towels and dust in flour, shaking off any excess.\n\nHeat a non-stick frying pan on medium. Spray with oil and cook the cheese slices for 1\u20132 minutes on each side until golden. Drain briefly on paper towels and scatter into the salad.\n\nNUTRITION ANALYSIS 1 serve = 740 kJ, 9 g fat (includes 3 g saturated fat), 5 g fibre, 480 mg sodium\n\n**Brown rice salad**\n\n_Wonderful as an accompaniment to Honey-rosemary Rack of Lamb \u2013 just make the lamb without the potatoes. We've used brown rice for this salad, which has a lovely nutty flavour and is so good for you. It does take a bit longer to cook though, and if you prefer you can use a low-GI rice, such as Doongara or Moolgiri._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\nCooking time   35 minutes\n\n1  cups long-grain brown rice\n\n2 sweetcorn cobs\n\nspray oil\n\n2 finger eggplants, sliced\n\n1 Lebanese cucumber, finely chopped\n\n1 red onion, sliced\n\n2 tablespoons finely chopped parsley\n\nDressing\n\n2 tablespoons olive oil\n\n1 tablespoon lemon juice or balsamic vinegar\n\npinch sugar\n\nCook the rice in a large saucepan of boiling water for 25\u201330 minutes, or until tender. Drain well and set aside to cool.\n\nMeanwhile, microwave or boil the sweetcorn cobs for 4 minutes until tender. Set aside to cool. Use a sharp knife to cut all the corn from the cobs.\n\nHeat a non-stick frying pan on medium. Spray with oil and cook the eggplant for 1\u20132 minutes on each side.\n\nCombine the cooled rice, sweetcorn, eggplant, cucumber, onion and parsley in a large mixing bowl.\n\nMake the dressing by whisking all the ingredients together. Pour over the salad and toss well.\n\nNUTRITION ANALYSIS 1 serve = 1410 kJ, 9 g fat (includes 1 g saturated fat), 5 g fibre, 10 mg sodium\n\n**_Super ingredient: Brown rice_**\n\nBrown rice is a starchy carbohydrate and a surprisingly good source of high-quality protein. It has three times as much fibre as white rice and gives you plenty of B vitamins and minerals. Like all grains, it has no cholesterol, only 2\u20133 per cent fat and very little sodium. At 350 kilojoules (or 85 calories) per half-cup serve, it has a place on any weight-loss diet. Being gluten-free and relatively non-allergenic it is also useful for anyone battling allergies or food intolerance. Rice is often used as the basis of a 'rice diet' to lower blood pressure.\n**Roasted sweet potato, eschalot and asparagus salad**\n\n_This salad can make a wonderful light meal on its own. For a really colourful salad, roast baby beetroot and fennel and add these too. Roasted pistachio nuts also make a delicious addition, but if you do add them, remember the fat content of the salad will increase._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\n500 g sweet potato, peeled and cut into chunks\n\n8 eschalots, peeled and halved\n\n\u00bc cup roughly chopped oregano\n\n1 tablespoon olive oil\n\n2 bunches asparagus, cut into 5 cm lengths\n\n50 g haloumi cheese, cubed\n\n1 bunch watercress, sprigs picked\n\nDressing\n\n1 tablespoon extra-virgin olive oil\n\n1 tablespoon honey\n\n1 tablespoon white wine vinegar\n\nPreheat the oven to hot, 200\u00baC (400\u00baF).\n\nArrange the sweet potato and eschalots in a baking dish. Toss through the oregano and olive oil. Bake for 20 minutes then add the asparagus and cheese and bake for a further 10 minutes. Remove from the oven and leave to cool slightly.\n\nTo make the dressing, whisk all the ingredients together.\n\nWhen ready to serve, arrange the roasted vegetables and watercress in a large serving bowl, pour on the dressing and toss gently.\n\nNUTRITION ANALYSIS 1 serve = 1115 kJ, 13 g fat (includes 3 g saturated fat), 5 g fibre, 225 mg sodium\n\n**_Super ingredient: Extra-virgin olive oil_**\n\nExtra-virgin olive oil is one of the healthiest oils for us. It has a high content of natural squalene and plant sterols, which lower cholesterol levels. Polyphenols, the compounds responsible for its deep greenish colour and strong flavour, also help thin the blood and keep it free-flowing. As extra-virgin oil is cold-pressed, it doesn't lose heat-sensitive vitamins or antioxidants. It aids the absorption of fat-soluble nutrients, and best of all, it makes vegetables and salads taste delicious.\n**Mixed vegetable couscous**\n\n_Try serving this salad with a roast in place of the traditional roast vegetables._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   30 minutes\n\nspray oil\n\n2 cups cubed pumpkin or sweet potato (about 500 g)\n\n250 g punnet grape tomatoes\n\n2 zucchini, chopped\n\n1 capsicum, seeded and chopped\n\n1 red onion, chopped\n\n1 cup couscous\n\n1 cup boiling water or low-salt stock\n\njuice \u00bd lemon\n\n\u00bc cup low-fat natural yoghurt (optional)\n\nPreheat the oven to hot, 200\u00baC (400\u00baF). Spray a baking dish with oil. Arrange the vegetables in the dish and spray with a little more oil. Bake for 25\u201330 minutes until tender and golden then remove from the oven and keep warm.\n\nMeanwhile, measure the couscous into a mixing bowl. Pour over the boiling water and stir briefly. Cover and leave to stand for 5 minutes, or until all the liquid has been absorbed.\n\nFluff with a fork and stir through the roasted vegetables. Top with a dollop of yoghurt.\n\nNUTRITION ANALYSIS 1 serve = 1095 kJ, 2 g fat (includes negligible saturated fat), 4 g fibre, 25 mg sodium\n\n**Barley risotto with chickpeas**\n\n_This has a nuttier, chewier texture than traditional risotto made with rice, and makes a delicious accompaniment, a light lunch or supper dish with a green salad on the side \u2014 or try with a little grated parmesan and a rocket salad._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   45 minutes\n\n2 teaspoons olive oil\n\n2 red onions, finely chopped\n\n1 cup pearl barley\n\n3 cups hot salt-reduced chicken stock\n\n2 cups hot water\n\n400 g can chickpeas, drained and rinsed\n\n2 red capsicums, char-grilled, peeled, seeded and finely sliced\n\n2 zucchini, finely chopped\n\nHeat oil in a heavy-based saucepan on medium. Saut\u00e9 the onion for 2\u20133 minutes until it starts to colour. Add the barley and cook for 2 minutes until toasted.\n\nAdd the combined hot stock and water, one ladle at a time, stirring after each addition, until absorbed. Repeat until all stock has been used and the barley is cooked until just tender (al dente), about 40 minutes. Add the capsicum and zucchini with the last ladle.\n\nUse a fork to separate the grains, then stir through the chickpeas and cook for a further 2 minutes, stirring continually.\n\nNUTRITION ANALYSIS 1 serve = 1220 kJ, 5 g fat (includes 1 g saturated fat), 10 g fibre, 630 mg sodium\n\n**_Super ingredient: Barley_**\n\nBarley is an ancient grain with three great virtues over wheat and rice. It is rich in beta-glucan fibre, which can 'sweep' cholesterol out of the body. It contains tocotrienols, vitamin E-related compounds, and has a low GI of 25, so its carbohydrate is slowly absorbed, making it useful for staving off hunger pangs and steadying blood sugar levels. Like flax seeds, it contains lignans which are plant oestrogens that can benefit menopausal women. Barley is low in fat and packed with energy-giving carbohydrates.\n\n**Spinach and split pea dahl**\n\n_This is a perfect accompaniment to Spiced Barbecued Prawns._\n\nServes 4\n\nSoaking time   1 hour\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\n2 cups water\n\n\u00bd cup split peas\n\n\u00bc teaspoon turmeric\n\n1 bunch spinach, chopped\n\n1 tablespoon olive oil\n\n1 tablespoon ginger\n\n3 garlic cloves, crushed\n\n1 teaspoon garam masala\n\n\u00bd teaspoon cayenne\n\njuice of \u00bd lemon\n\nCombine the water and split peas in a large saucepan and stir well. Leave to soak for at least 1 hour.\n\nStir in the turmeric and bring to the boil over a medium heat. Reduce the heat and simmer gently for 10 minutes, stirring occasionally. Add the spinach and continue to simmer for 10\u201312 minutes until almost all liquid has been absorbed and the peas are tender.\n\nMeanwhile, heat oil in a non-stick frying pan on high. Saut\u00e9 the ginger, garlic, garam masala and cayenne for 2\u20133 minutes until aromatic. Stir into dahl with lemon juice.\n\nNUTRITION ANALYSIS 1 serve = 605 kJ, 5 g fat (includes 1 g saturated fat), 5 g fibre, 20 mg sodium\n\n**Warm lentil salad**\n\n_This is a fabulous accompaniment to Easy Biryani._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   2 minutes\n\n1 tablespoon olive oil\n\n400 g can brown lentils, drained\n\n2 medium tomatoes, diced\n\n50 g baby spinach leaves, shredded\n\njuice of 1 lemon\n\nHeat oil in a frying pan on medium. Add the lentils and cook, stirring, for 1\u20132 minutes to warm through.\n\nStir in the tomatoes, spinach and lemon juice, and cook for 1 minute until the spinach has wilted.\n\nNUTRITION ANALYSIS 1 serve = 390 kJ, 5 g fat (includes 1 g saturated fat), 3 g fibre, 180 mg sodium\n\n**_Super ingredient: Lentils_**\n\nOne of the oldest legumes and integral to the cuisines of the Middle East, India and Eastern Europe, lentils are important for their protein, iron, zinc, potassium and fibre. Lentils are a source of phyto-oestrogens \u2013 natural plant chemicals which help protect against cancer and heart disease and can minimise the unpleasant side effects of menopause. Lentils are often combined with rice to make a complete and satisfying meal.\n**Lentil dahl**\n\n_This dahl is delicious as a meal or try it with Curried Mussels._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   25 minutes\n\n3\u00bd cups water\n\n400 g can diced tomatoes (or 4 fresh tomatoes, skinned and chopped)\n\n1 cup red lentils (or mixed dahl)\n\n2 potatoes, peeled and cubed\n\n1 teaspoon ground cumin\n\n1 teaspoon ground coriander\n\n\u00bd teaspoon turmeric\n\n\u00bd teaspoon chilli powder\n\n1 tablespoon canola oil\n\n1 onion, finely chopped\n\n4 garlic cloves, crushed\n\n1 teaspoon mustard seeds\n\n\u00bd teaspoon cumin seeds\n\n1 teaspoon garam masala\n\nCombine the water, tomatoes, lentils, potatoes, cumin, coriander, turmeric and chilli in a large saucepan.\n\nBring to the boil over a high heat, stirring. Reduce the heat and simmer for 15\u201320 minutes until the potatoes are tender.\n\nMeanwhile, heat oil in a frying pan on high. Saut\u00e9 the onion, garlic and seeds for 3\u20134 minutes until the onion is tender and golden. Stir in the garam masala.\n\nBlend the onion mixture into the lentils and cook, stirring, for 1 minute.\n\nNUTRITION ANALYSIS 1 serve = 850 kJ, 5 g fat (includes negligible saturated fat), 7 g fibre, 10 mg sodium\n\n**_Super ingredient: Garlic_**\n\nOften called 'nature's penicillin', fresh garlic can slow the growth of bacteria and fungi and has a long medicinal history. It was used by the ancient Egyptians, Vikings and Chinese to ward off illness. Use with abundance in your cooking! Fresh garlic is always the best choice for anti-bacterial or anti-viral qualities. Both fresh and dried garlic have been shown to lower LDL-cholesterol, lower high blood pressure and help dissolve blood clots, although not all studies agree. The dose required is quite large for most of us \u2013 10 to 20 grams of fresh garlic (2 to 4 cloves) a day or 600\u2013900 milligrams of powder garlic. Garlic's pungent odour comes from its active agents, allicin and other sulphur compounds. Odourless garlic tablets are not as effective.\n**Saut\u00e9ed red cabbage**\n\n_Serve this cabbage hot, warm or even cold, with roast pork or grilled meats._\n\nServes 4\n\nPreparation time   5 minutes\n\nCooking time   6 minutes\n\n2 teaspoons olive oil\n\n2 teaspoons fennel seeds\n\n\u00bc red cabbage, finely shredded\n\n2 tablespoons chilli jam (or redcurrant or cranberry jelly)\n\n1 tablespoon currants\n\n1 tablespoon red wine vinegar\n\nHeat oil in a frying pan on medium. Saut\u00e9 the fennel seeds for 30 seconds.\n\nStir in the cabbage and saut\u00e9 for 3\u20134 minutes until it begins to wilt.\n\nAdd the chilli jam, currants and vinegar and cook, stirring, for 1 minute.\n\nNUTRITION ANALYSIS 1 serve = 265 kJ, 2 g fat (includes negligible saturated fat), 2 g fibre, 10 mg sodium\n\n**_Super ingredient: Red cabbage_**\n\nRed cabbage is a wonderful winter vegetable. When cooked briefly it retains much of its vitamins \u2013 especially vitamin C of which it's a rich source. There's also vitamin B1, vitamin K, folate, potassium and a good dose of fibre. Its colour comes from anthocyanins, flavonoids that function as antioxidants and keep you looking youthful and in top health.\n**Herbed crisp potatoes**\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   40 minutes\n\n4 potatoes, unpeeled, quartered\n\n8 whole unpeeled garlic cloves\n\nspray oil\n\n\u00bc cup roughly chopped thyme\n\n\u00bc cup roughly chopped rosemary\n\n\u00bc cup roughly chopped oregano\n\nfreshly ground black pepper\n\nPreheat the oven to hot, 200\u00baC (400\u00baF).\n\nPlace potatoes in a baking pan, skin-side down. Scatter on the garlic cloves and spray with oil.\n\nSprinkle with the herbs and pepper. Bake for 35\u201340 minutes until the potatoes are cooked through, golden brown and crisp.\n\nNUTRITION ANALYSIS 1 serve = 505 kJ, 1 g fat (includes negligible saturated fat), 4 g fibre, 5 mg sodium\n\n**Roasted fennel, tomato and garlic**\n\n_Try this dish as an accompaniment to Fish, Leek and Potato Pies (page 143) or Glazed Pork Fillet (page 149)._\n\nServes 4\n\nPreparation time   15 minutes\n\nCooking time   35 minutes\n\nspray oil\n\n2 fennel bulbs, trimmed, sliced (or cut into wedges)\n\n1 leek, trimmed, washed, sliced\n\n2 roma tomatoes, chopped\n\n2 garlic cloves, crushed\n\n\u00bd cup white wine or salt-reduced chicken stock\n\n1 cup fresh wholegrain breadcrumbs\n\n\u00bd cup grated parmesan\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Spray a 2 litre casserole dish with oil.\n\nArrange the fennel, leek, tomatoes and garlic in the prepared dish, tossing to combine.\n\nPour the wine over the vegetables. Mix breadcrumbs with the parmesan and sprinkle over the top.\n\nBake for 30\u201335 minutes until tender and golden.\n\nNUTRITION ANALYSIS 1 serve = 630 kJ, 4 g fat (includes 2 g saturated fat), 5 g fibre, 295 mg sodium\n\n**Celeriac gratin**\n\n_This dish is a great alternative to potato and makes a delicious accompaniment to many meals._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   45 minutes\n\nspray oil\n\n500 g celeriac, peeled and thinly sliced\n\n500 g potatoes, peeled and thinly sliced\n\n1\u00bd cups grated reduced-fat tasty cheese\n\n1 cup salt-reduced vegetable stock\n\n2 slices prosciutto, chopped\n\n1 tablespoon chopped rosemary leaves\n\nPreheat the oven to moderately hot, 190\u00baC (375\u00baF). Spray a 3 litre casserole dish with oil. Layer in the celeriac, potato and \u00be of the cheese.\n\nPour on the stock and sprinkle with the remaining cheese, prosciutto and rosemary.\n\nBake, uncovered, for 40\u201345 minutes until tender and golden.\n\nNUTRITION ANALYSIS 1 serve = 850 kJ, 4 g fat (includes 2 g saturated fat), 3 g fibre, 760 mg sodium\n\n**Cherry tomato and zucchini bake**\n\n_This delicious combination of flavours is perfect to accompany many meals, but we love it with Honey-rosemary Rack of Lamb \u2013 simply make the lamb without the tomatoes. If you don't have cherry tomatoes, use 500 g of roughly chopped tomatoes. Sliced leek makes a delicious addition to this dish._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   45 minutes\n\n2 x 250 g punnets cherry tomatoes, halved\n\n2 zucchini, thinly sliced\n\n1 tablespoon oil\n\n1 cup fresh wholegrain breadcrumbs\n\n\u00bd cup grated reduced-fat tasty cheese\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Lightly grease a 1\u00bc litre ovenproof dish.\n\nPlace the tomatoes and zucchini in a mixing bowl and toss with the oil. Spoon into the ovenproof dish.\n\nMix the breadcrumbs with the cheese and sprinkle over the vegetables.\n\nBake for 40\u201345 minutes, until golden and bubbly.\n\nNUTRITION ANALYSIS 1 serve = 540 kJ, 6 g fat (includes 1 g saturated fat), 3 g fibre, 175 mg sodium\n\nDesserts and sweet treats\n\nIf you have a sweet tooth, you'll enjoy the light and healthy sweet endings we have created here. They'll help to satisfy that desire for sweetness without blowing your total kilojoule intake or burdening you with unwanted fat and sugar.\n\nAs you'd expect, many of our desserts start with fruit. You'll find berry strudels, pears poached in wine, spiced peaches, banana br\u00fbl\u00e9e, date and butterscotch pudding, easy mango crumble \u2013 and of course the ultimate fruit salad. Fruit is naturally sweet, refreshes and cleanses the mouth, is full of vitamins, minerals and antioxidants, and is low in fat. And if you don't have time to prepare a dessert, simply finish off with fresh fruit in season.\n\nIn our recipes, we haven't cut out all the sugar, but we've added only the minimum to give a balance of flavour and function. Sugar has indispensable properties in baking and browning which we've used to our advantage. Similarly with fat: we've included some 'good' fat when it's needed, but not in huge amounts.\n\nAnd what to use as topping? We believe you don't need thickened cream or high-fat premium ice cream. Instead, we've suggested different and equally delicious ideas, such as thick natural yoghurt with a sprinkle of cinnamon, low-fat custard, low-fat ricotta mixed with a little honey and grated lemon zest or low-fat vanilla ice cream.\n\nAnd remember, you can always add a sponge finger biscuit, thin almond wafer or a biscotti for crunch with minimum fat.\n\nWhen you're dining out and feel that it's impossible to resist that delicious dessert, keep these tips in mind.\n\n\u2022 Ask for a small portion \u2013 just enough to satisfy, without overloading yourself.\n\n\u2022 Share a dessert between two.\n\n\u2022 Eat just a few mouthfuls, then push your plate away.\n\n\u2022 If you order a tart or slice, eat the filling only, leaving the crumb base or pie crust.\n\n\u2022 And ask the waiter to hold the cream!\n**Pears poached in red wine**\n\n_To make sure that the pears are tender, try gently inserting a skewer \u2013 it will be easily removed if the fruit is ready. Use the same technique for testing other poached fruit such as peaches, nectarines, plums etc._\n\nServes 4\n\nPreparation time   10 minutes\n\nCooking time   45 minutes\n\n2 cups water\n\n2 cups sugar\n\n1 cup red wine\n\n2 strips orange peel\n\n1 cinnamon stick\n\n1 vanilla pod, split and scraped\n\n4 pears, peeled and cored, but left whole\n\nlow-fat custard or Fr\u00fbche to serve (optional)\n\nCombine the water, sugar, wine, orange peel, cinnamon, vanilla pod and seeds in a large saucepan. Stir over a low heat for 2\u20133 minutes until the sugar dissolves. Increase the heat until the syrup comes to a boil \u2013 do not stir.\n\nAdd the pears to the saucepan and return to the boil. Lower the heat and simmer for 15\u201320 minutes, until the pears are just tender.\n\nRemove the pears from the syrup with a slotted spoon and set aside. Remove the aromatics from the syrup and bring it back to the boil. Simmer for 15\u201320 minutes until reduced by half. Serve the pears drizzled with the syrup and top with custard or Fr\u00fbche.\n\nNUTRITION ANALYSIS 1 serve (including a dollop of custard or Fr\u00fbche) = 2280 kJ, negligible fat, negligible saturated fat, 3 g fibre, 25 mg sodium\n\n**_Super ingredient: Pears_**\n\nPears have one of the highest fibre contents of all the fruits, supply lots of potassium and magnesium, have few kilojoules and no fat. If you have any type of food intolerance, you will find them indispensable, as they are the least allergenic of all the fruits. They also make an ideal first food for babies. No wonder they are one of the world's most loved fruits. As is usually the case with fruit, fresh is best. If buying canned pears, look for the ones in natural juice, rather than syrup.\n\n**Walnut and spice baked peach halves**\n\n_If you can, buy slip-stone (sometimes called free-stone) peaches. With these varieties it is very easy to remove the stone without bruising the peach. Of course if fresh peaches aren't available, you can use drained, canned peach halves. And pecans can be used instead of walnuts or use a combination of your favourite nuts._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\nCooking time   20 minutes\n\nspray oil\n\n2 tablespoons wholemeal plain flour\n\n\u00bd teaspoon ground cinnamon\n\n\u00bd teaspoon ground nutmeg\n\n20 g light margarine\n\n  cup coarsely chopped walnuts\n\n2 tablespoons rolled oats\n\n2 tablespoons brown sugar\n\n6 peaches, halved and stone removed\n\nlow-fat vanilla ice cream and honey to serve\n\nPreheat the oven to hot, 200\u00baC (400\u00baF). Spray a deep muffin tray with oil.\n\nIn a small bowl combine the flour and spices. Rub in the margarine using your fingertips, then stir in the walnuts, oats and sugar.\n\nPlace the peach halves, cut-side up into the muffin tray. Divide the nut mixture evenly among the peaches. Bake for 15\u201320 minutes until golden and tender.\n\nServe the peaches warm with ice cream and drizzled with honey.\n\nNUTRITION ANALYSIS 1 serve (including 1 scoop ice cream and a tablespoon of honey) = 965 kJ, 9 g fat (includes 1 g saturated fat), 4 g fibre, 35 mg sodium\n\n**_Super ingredient: Walnuts_**\n\nA source of omega-3 polyunsaturated fats, associated with reduced blood cholesterol and blood pressure \u2013 walnuts are a must for a healthy heart. Like all nuts, they are also a source of fibre and vitamin E as well as the minerals potassium, magnesium, zinc, copper and selenium. A serving is a tiny handful \u2013 about 30 grams.\n**Chunky apple bake**\n\n_To make this chunky, crunchy bake look really professional, try to keep the bread and apple cubes the same size. If you want to be creative and use a combination of apple and pears for a change, the same applies._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   25 minutes\n\n3 green apples, peeled, cored and cut into chunks\n\n2 slices wholegrain bread, crusts removed and cubed\n\n\u00bd cup low-fat cottage cheese\n\n2 tablespoons brown sugar, plus extra to sprinkle (optional)\n\n200 ml low-fat milk\n\n1 tablespoon lecithin\n\nlow-fat custard or ice cream to serve\n\nPreheat the oven to very hot, 220\u00baC (430\u00baF).\n\nCombine the apples, bread, cottage cheese and sugar in a large mixing bowl and toss them together well.\n\nStir in the milk then spoon the mixture into a shallow ovenproof dish. Sprinkle with lecithin and some extra brown sugar if you want an extra-crunchy topping.\n\nBake for 20\u201325 minutes until bubbling and golden. Serve hot from the oven with low-fat custard or ice cream.\n\nNUTRITION ANALYSIS 1 serve (including a dollop of custard) = 725 kJ, 3 g fat (includes 1 g saturated fat), 2 g fibre, 125 mg sodium\n\n**_Super ingredient: Apples_**\n\nRich in flavonoid antioxidants and the soluble fibre pectin (both renowned for their heart protective capabilities), is it any wonder that an apple a day keeps the doctor away? Fat-free and low in kilojoules, an average apple has only a modest 315 kilojoules (75 calories), making it a good between-meal snack or a crisp, sweet finish to a meal. Their carbohydrate (mostly fructose) is low GI and is digested slowly, keeping blood sugar levels steady. Apples are packed with essential minerals like potassium and magnesium.\n\n**Baked fruit medley**\n\n_In this baked medley you can use the softer type of dried figs, called 'dessert figs', or even fresh figs if you prefer. If you do opt for fresh figs in season, add them during the last 10 minutes of cooking time._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\n\u00bc cup pure maple syrup\n\n30 g light margarine\n\n1 tablespoon brown sugar\n\n1 teaspoon ground cinnamon\n\n3 green apples, quartered (retain skin and core)\n\n3 pears, quartered (retain skin and core)\n\n200 g dried figs\n\n3 stalks rhubarb, trimmed and sliced\n\nlow-fat ice cream or custard to serve\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF).\n\nCombine the maple syrup, margarine, sugar and cinnamon in a small saucepan. Heat gently, stirring, until melted and well combined.\n\nArrange the apples, pears and figs in a baking dish. Pour in the syrup and toss gently so the fruit is evenly coated.\n\nBake for 15 minutes. Add the rhubarb to the dish, stirring in gently so it is coated with syrup.\n\nBake for a further 10\u201315 minutes until the fruit is tender. Serve warm with low-fat ice cream or custard.\n\n**VARIATION**\n\nTry making this with fresh stone fruit: top a selection of peaches, nectarines, apricots or plums with flaked almonds and bake for 15\u201320 minutes.\n\nNUTRITION ANALYSIS 1 serve (including 1 scoop ice cream) = 1325 kJ, 5 g fat (includes 1 g saturated fat), 11 g fibre, 65 mg sodium\n\n**_Super ingredient: Figs_**\n\nOne of the oldest known fruits, figs add a marvellous flavour and texture to any dessert. Like all fruit, they have almost no fat or sodium but their tiny seeds are what sets them apart. These are a great source of protein as well as minerals such as potassium, calcium, magnesium, phosphorus and iron. They also have mild laxative properties (handy if you're troubled by constipation). Dried figs are concentrated with around half their weight being fruit sugars.\n**Fruit salad with cinnamon yoghurt**\n\n_This is one of those recipes where you can be as creative as you like. Try any colourful combination of ripe fruit that are in season. For a simpler dessert, serve chunks of chilled honeydew or rockmelon with a dollop of low-fat Greek-style natural yoghurt, a sprinkling of sugar or sweetener and a scattering of chopped pistachios._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\n150 g (1 punnet) blueberries\n\n\u00bd small pineapple, peeled, cored and sliced\n\n\u00bc rockmelon, seeds removed and sliced\n\n2 tablespoons shredded mint\n\n1 tablespoon chopped glac\u00e9 ginger\n\nCinnamon yoghurt\n\n200 g tub low-fat natural yoghurt\n\n1 tablespoon honey\n\n\u00bc teaspoon cinnamon\n\nCombine all the ingredients in a large mixing bowl and toss gently to combine. Chill until required.\n\nTo make the cinnamon yoghurt, combine all the ingredients in a small bowl and mix together well.\n\nServe the fruit salad with a generous dollop of cinnamon yoghurt.\n\nNUTRITION ANALYSIS 1 serve (including cinnamon yoghurt) = 395 kJ, negligible fat, negligible saturated fat, 2 g fibre, 35 mg sodium\n\n**_Super ingredient: Rockmelons_**\n\nRefreshingly sweet and juicy, rockmelons have no fat and a mere 140 kilojoules (33 calories) a serve. They are equally delicious as a light summer dessert, a fresh-tasting breakfast, or a quick, healthy snack. Half a small rockmelon (around 1 cup of diced melon) will provide one-and-a-half times your daily requirement of vitamin C and up to 20 times more beta-carotene than either honeydew or watermelon. It also gives you around 1.5 grams of fibre, as much as a slice of wholemeal bread.\n\n**Mixed berry and ricotta strudels**\n\n_This recipe can readily be adapted for other fruits including sliced plums, pears, apples or rhubarb, on their own or combined. Apple and rhubarb makes a delicious combo._\n\nServes 6\n\nPreparation time   10 minutes\n\nCooking time   15 minutes\n\nspray oil\n\n8 sheets filo pastry\n\n300 g fresh ricotta\n\n2 tablespoons brown sugar\n\n150 g punnet blueberries\n\n125 g (\u00bd punnet) strawberries, hulled and chopped\n\n120 g (1 punnet) raspberries (or frozen)\n\n1 teaspoon caster sugar\n\nicing sugar and low-fat ice cream to serve\n\nPreheat oven to very hot, 220\u00baC (430\u00baF). Spray 2 baking trays with oil and line them with baking paper.\n\nWorking with one sheet of filo pastry at time, place it flat on a clean, dry surface. Spray lightly with oil. Top with a second sheet of filo and spray with oil. Repeat this process until you have 4 layers.\n\nMake a second stack with the remaining 4 sheets of filo pastry, following the same procedure. Cut each stack crosswise into 3 even strips.\n\nIn a small bowl, combine the ricotta with the brown sugar, blending well. Place an equal amount onto the centre of each filo strip, leaving a 2 cm border. Distribute the berries evenly between the filo strips, mounding them onto the ricotta.\n\nFold the long sides in, then roll up to enclose the filling, forming a parcel shape. Arrange seam-side down on prepared trays.\n\nSpray with oil and sprinkle with the caster sugar. Bake for 10\u201315 minutes until golden brown. Serve immediately dusted with icing sugar and accompany with ice cream.\n\nNUTRITION ANALYSIS 1 serve (including 1 scoop ice cream) = 830 kJ, 6 g fat (includes 3 g saturated fat), 2 g fibre, 245 mg sodium\n\n**_Super ingredient: Strawberries_**\n\nStrawberries are packed with vitamin C, potassium, fibre and folate. They boast a number of phyto-chemicals, naturally occurring plant compounds that are now being researched for their medicinal properties. One of these is ellagic acid, a substance that appears to 'neutralise' carcinogens in the intestines and so reduces damage to DNA and other genetic material. Best of all, strawberries have a low GI and a punnet of strawberries (125 grams) supplies a mere 100 kilojoules (24 calories).\n**Vanilla pannacotta with strawberry salsa**\n\n_If you don't use gelatine very often in your cooking, remember to let it cool to the same temperature as the yoghurt mix. This minimises the risk of lumps forming and ensures you achieve that desirable creamy smoothness._\n\nServes 6\n\nPreparation time   10 minutes\n\nCooking time   5 minutes (plus chilling time)\n\n2 x 200 g tubs no-fat vanilla yoghurt\n\n2 tablespoons honey\n\n\u00bd teaspoon vanilla paste\n\n2 teaspoons gelatine\n\n\u00bc cup just boiled water\n\nStrawberry salsa\n\n250 g (1 punnet) strawberries, hulled and chopped\n\n1 tablespoon icing sugar\n\n1 tablespoon Grand Marnier\n\nChill six 60 ml moulds. Combine the yoghurt, honey and vanilla in a large mixing bowl.\n\nDissolve the gelatine by whisking vigorously in hot water in a small jug. Allow to cool slightly.\n\nBeat a little of the yoghurt mixture into the gelatine to equalise the temperature, then whisk this back into the yoghurt mixture until well combined.\n\nPour into the prepared moulds and chill until almost set. Cover with plastic wrap and chill overnight.\n\nTo make the strawberry salsa, combine the strawberries, icing sugar and Grand Marnier in a mixing bowl and toss together gently. Cover and chill until required.\n\nTo unmould the pannacottas, carefully run a blunt knife around the rim of each mould and then dip them into hot water for a few seconds. Invert onto serving plates and shake firmly. Carefully lift away the moulds.\n\nServe the pannacottas with a spoonful of strawberry salsa.\n\nNUTRITION ANALYSIS 1 serve = 435 kJ, negligible fat, negligible saturated fat, 1 g fibre, 60 mg sodium\n\n**Never-fail berry fool**\n\n_This is such a simple dessert you'll want to whip it up time and again. It's also an easy one to vary \u2013 instead of berries you can use chopped mangoes, bananas or passionfruit._\n\nServes 4\n\nPreparation time   5 minutes\n\n3 egg whites\n\n\u00bc cup caster sugar\n\n2 cups low-fat pouring custard\n\n2 cups mixed berries of choice\n\ntoasted slivered almonds to serve\n\nIn a clean, dry bowl, beat the egg whites with an electric beater until soft peaks form. Take care not to over-beat them, though, or they will collapse and cannot be used.\n\nGradually add the sugar, beating constantly, until thick and glossy.\n\nGently fold in the custard and berries and spoon into serving glasses. Chill until required. Serve topped with almonds.\n\nNUTRITION ANALYSIS 1 serve = 810 kJ, 3 g fat (includes 1 g saturated fat), 2 g fibre, 100 mg sodium\n\n**_Super ingredient: Berries_**\n\nNutritionally, berries are in a class of their own. They are all good-to-excellent sources of vitamin C and dietary fibre and supply lesser amounts of many essential minerals, like potassium, calcium, iron, magnesium and phosphorus. The purple and red pigments (anthocyanins) in berries function as antioxidants, minimising the effects of ageing. Eaten fresh, they're a top supplier of two key B vitamins including folate (essential for younger women as it helps prevent birth defects in babies) and niacin (which releases energy from food). And they won't pile on the weight either \u2013 half a punnet of strawberries (125 grams) supplies a mere 100 kilojoules (24 calories).\n**Fruity ricotta cakes**\n\n_Although fresh and tub ricotta are often interchangeable, for this recipe you really need to use fresh ricotta; the tub version is a little bit too soft. You can use other dried fruit of your choice for variety._\n\nMakes 4\n\nPreparation time   10 minutes\n\nCooking time   20 minutes\n\nspray oil\n\n250 g fresh low-fat ricotta cheese\n\n1 teaspoon vanilla paste\n\n  cup sultanas\n\n\u00bc cup chopped dried apricots\n\n2 tablespoons honey\n\nfinely grated zest of 1 orange\n\nlow-fat ice cream, custard or yoghurt to serve\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Spray four 120 ml ramekin dishes with oil.\n\nCombine all the ingredients in a large mixing bowl and mix until well combined. Spoon into the prepared ramekin dishes and arrange on a baking tray.\n\nBake for 15\u201320 minutes until firm. Serve warm or cold with low-fat ice cream, custard or yoghurt.\n\nNUTRITION ANALYSIS 1 serve (including 1 scoop ice cream) = 885 kJ, 6 g fat (includes 4 g saturated fat), 1 g fibre, 140 mg sodium\n\n**_Super ingredient: Dried apricots_**\n\nDried apricots pack more of a punch than fresh as they are more concentrated. They boast 5 times as much beta-carotene (an antioxidant that's converted to vitamin A), which is great for healthy skin and eyes. They also have around twice the fibre and 4 times the amount of iron as fresh apricots. But you only need a few \u2013 5\u20136 halves is the equivalent of a serving.\n**Yoghurt and caramel ice cream**\n\n_If the caramel is a little bit too gooey, you can thin it down slightly by adding a little warm water. If you find caramel a bit sweet, try chocolate topping instead. For something a little fruitier, add a cup of your favourite berry pur\u00e9e._\n\nServes 6\u20138\n\nPreparation time   20 minutes\n\nCooking time   15 minutes\n\nFreezing time   overnight\n\n2 x 500 g tubs low-fat vanilla yoghurt\n\n2 teaspoons vanilla paste\n\n2 eggs\n\n\u00be cup icing sugar, sifted\n\n\u00bc cup pistachio nuts\n\n\u00bd cup caramel topping, beaten until runny\n\nCombine the yoghurt and vanilla paste in a large mixing bowl and whisk together until smooth.\n\nWhisk the eggs and sugar in a heat-resistant bowl. Place over a saucepan of gently simmering water and continue whisking until pale and mousse-like (it will take 10\u201315 minutes). Fold into the yoghurt.\n\nPour into an ice cream machine and churn according to the manufacturer's instructions. When the ice cream is semi-frozen add the nuts and allow it to churn for a few more moments. If you don't have an ice-cream machine, freeze the mixture until semi-frozen then beat vigorously or pulse in a food processor for a few seconds before folding in the nuts.\n\nTip half the ice cream into a 10 x 20 cm loaf pan and swirl through half the caramel. Repeat with another layer and freeze overnight.\n\nNUTRITION ANALYSIS 1 serve = 1090 kJ, 4 g fat (includes 1 g saturated fat), 1 g fibre, 140 mg sodium\n\n**Watermelon sorbet**\n\n_For that delectably smooth sorbet texture, you need to add egg white \u2013 it helps to prevent ice crystals forming._\n\nServes 4\u20136\n\nPreparation time   15 minutes\n\nCooking time   10 minutes\n\nFreezing time   overnight\n\n4 cups chopped seedless watermelon flesh (around 900 g)\n\njuice 1 lemon\n\n1\u00bd cups water\n\n1 cup sugar\n\n1 egg white\n\nfresh fruit to serve (optional)\n\nPlace the watermelon and lemon juice in a food processor or blender and process until smooth. Push through a strainer.\n\nCombine the water and sugar in a saucepan and bring to the boil over a medium heat, stirring, until the sugar has dissolved. Lower the heat and simmer for 5 minutes without stirring.\n\nRemove from the heat and allow to cool before stirring in the watermelon pur\u00e9e. Pour into an ice cream machine and churn according to the manufacturer's instructions until semi-frozen. If you don't have an ice cream machine, freeze the mixture until semi-frozen then beat vigorously or pulse in a food processor for a few seconds.\n\nWhile the sorbet is churning place the egg white into a clean bowl and whisk until frothy. Fold through the semi-frozen sorbet then pour into an 18 x 28 cm tray. Freeze overnight.\n\nServe in scoops on its own or with fresh fruit.\n\n**VARIATION**\n\nThere is no limit to the variety of delicious fruit sorbets you can try: replace the watermelon with 4 cups of pur\u00e9ed stone fruit, or 8 kiwi fruit, 1 cup of lemon juice and the finely grated zest of 2 lemons, or the pur\u00e9ed flesh of 4 mangoes.\n\nNUTRITION ANALYSIS 1 serve = 825 kJ, negligible fat, negligible saturated fat, 1 g fibre, 15 mg sodium\n\n**_Super ingredient: Watermelon_**\n\nWith its high water content, watermelon is a refreshing fruit for the hot summer months. A good source of vitamin C, watermelon is also a source of lycopene so it can top up your levels of cancer-protective antioxidants. And all for a tiny 274 kilojoules (65 calories) for a cool, sweet thick slice! Buy the seedless varieties for a fuss-free thirst-quencher.\n\n**Banana br\u00fbl\u00e9e**\n\n_A combination of two favourite desserts: br\u00fbl\u00e9e and banana custard! We love the layer of caramelised fruit on top of this br\u00fbl\u00e9e, and it works just as well with strawberries, mangoes or other fruits in season. Or be traditional, and omit the fruit altogether._\n\nServes 4\n\nPreparation time   10 minutes (plus chilling time)\n\nCooking time   8 minutes\n\n2 cups low-fat milk\n\n\u00bc cup skimmed milk powder\n\n\u00bc cup caster sugar\n\n2 tablespoons cornflour\n\n2 eggs, beaten\n\n1 banana, sliced\n\nsponge fingers or biscotti to serve\n\nCombine the milk, milk powder and 2 tablespoons of the sugar in a heavy-based saucepan. Blend a little of the liquid into the cornflour, mixing until smooth. Tip this back into the saucepan and add the beaten eggs.\n\nBring to the boil over a medium heat, stirring constantly. Reduce the heat and simmer for 3 minutes to cook the flour. Spoon into 4 ramekin dishes and chill overnight.\n\nPreheat the grill to its highest setting. Just before serving, top each ramekin with an even layer of banana slices. Sprinkle with the remaining sugar, coating the bananas evenly. Grill for 2\u20134 minutes until the sugar caramelises.\n\nServe with sponge fingers or biscotti.\n\nNUTRITION ANALYSIS 1 serve (including 1 sponge finger biscuit) = 1045 kJ, 5 g fat (includes 2 g saturated fat), 1 g fibre, 160 mg sodium\n\n**Rice custard**\n\n_A baked custard is such a wonderful dessert, and so different from the kind from the packet or dairy cabinet in the supermarket. As with cakes, you need to test to make sure it's done. Simply insert the point of a knife into the centre of the custard. If it comes out clean it is cooked._\n\nServes 4\u20136\n\nPreparation time   5 minutes\n\nCooking time   35 minutes\n\n1 cup cooked rice (use a low-GI rice, such as Doongara or Moolgiri)\n\n\u00bd cup sultanas, raisins or dried fruit medley\n\n4 eggs\n\n  cup sugar\n\n600 ml low-fat milk\n\n\u00bd teaspoon ground cinnamon\n\n\u00bd teaspoon ground nutmeg\n\nlow-fat ice cream to serve\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF).\n\nSpread the rice over the base of a 1\u00bd litre casserole dish. Sprinkle over the sultanas.\n\nWhisk the eggs with the sugar until well combined. Beat in the milk then pour the custard mixture over the rice and fruit. Dust with cinnamon and nutmeg.\n\nPlace the casserole dish into a large baking dish and pour in enough warm water to come halfway up the sides of the casserole dish.\n\nBake for 30\u201335 minutes until cooked through. Serve with low-fat ice cream.\n\nNUTRITION ANALYSIS 1 serve (including 1 scoop ice cream) = 1220 kJ, 7 g fat (includes 3 g saturated fat), 1 g fibre, 145 mg sodium\n\n**Easy mango crumble**\n\n_Don't think you have to wait for summer to enjoy this crumble. It works just as well with canned and frozen mangoes. In fact it works with all sorts of fruit and nuts, such as peaches and pears topped with macadamias or pistachios._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\nCooking time   25 minutes\n\n100 g almond bread or biscotti\n\n\u00bd cup rolled oats\n\n2 tablespoons brown sugar\n\n1 tablespoon wheatgerm\n\n2 x 400 g cans mango slices in syrup, drained (reserve 2 tablespoons of the syrup) or 8 frozen or fresh mango cheeks\n\n1 tablespoon chopped pistachio nuts\n\n200 g natural yoghurt\n\n2 tablespoons maple syrup\n\nlow-fat vanilla yoghurt or Fr\u00fbche and maple syrup to serve\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF).\n\nPlace the almond bread or biscotti in a sealed plastic bag and crush it with a rolling pin to coarse crumbs. Alternatively, use a food processor, but be careful not to process too finely \u2013 you need the pieces to be quite chunky.\n\nPour the crushed almond bread into a large mixing bowl with the oats, sugar, wheatgerm and 2 tablespoons of reserved syrup. (If using frozen or fresh mango, use orange juice in place of the syrup.)\n\nCut the mangoes into chunks and arrange in a shallow ovenproof dish. Scatter on the crumble topping and bake for 20\u201325 minutes until crisp and golden.\n\nServe with yoghurt and a drizzle of maple syrup.\n\nNUTRITION ANALYSIS 1 serve (including yoghurt and maple syrup) = 1110 kJ, 4 g fat (includes 1 g saturated fat), 2 g fibre, 25 mg sodium\n\n**_Super ingredient: Wheatgerm_**\n\nIf you feel tired and run down, wheatgerm is the natural way to top up your tank. It'll boost your B vitamins, protein, 'good' fats, vitamin E and fibre, and give you a healthy dose of the antioxidants and minerals that too much alcohol and refined carbohydrates drain out of the body. Wheatgerm is especially rich in the B vitamins that your body needs to release energy from food, particularly carbohydrates.\n**Butterscotch and date pudding**\n\n_Don't feel you have to rush out and buy vanilla paste \u2013 although it's a lovely ingredient to have on hand. For this recipe vanilla extract or even vanilla essence will do just as well._\n\nServes 6\u20138\n\nPreparation time   15 minutes (plus cooling time)\n\nCooking time   45 minutes\n\nspray oil\n\n1\u00bc cups chopped dates\n\n1 teaspoon bicarbonate of soda\n\n1\u00bd cups boiling water\n\n90 g light margarine\n\n\u00bd cup firmly packed brown sugar\n\n2 eggs\n\n1 teaspoon vanilla paste\n\n1\u00bd cups self-raising flour, sifted\n\nLight vanilla custard\n\n3 egg yolks\n\n\u00bc cup caster sugar\n\n1\u00bd cups low-fat milk, scalded\n\n1\u00bd teaspoon vanilla paste\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Lightly spray a 20 cm square cake pan or casserole dish with oil.\n\nCombine the dates and bicarbonate of soda in a small saucepan. Add the water and bring to the boil, stirring well. Remove from heat and leave to cool for 10 minutes.\n\nCombine the margarine and sugar in the bowl of an electric mixer and beat until creamy. Add the eggs one at a time, beating well after each addition. Mix in the vanilla.\n\nFold half the flour into the creamed mixture with half the dates and liquid. Repeat and mix lightly until combined. Pour the batter into the prepared pan and bake for 40\u201345 minutes until cooked. Test by inserting a skewer into the centre; if it comes out clean, it's cooked.\n\nTo make the custard, beat the egg yolks and sugar in a bowl until pale and creamy. Gradually whisk in the warm milk and vanilla paste. Transfer to a saucepan. Cook over a low heat, stirring constantly, for 3\u20134 minutes, until the custard thickens and coats the back of a wooden spoon. Serve the custard warm or chilled with the warm pudding.\n\nWrap any leftovers well and keep frozen or in the fridge. The custard will keep for 2 days in the fridge and can be gently warmed through before serving, if desired.\n\n**VARIATIONS**\n\n  For extra crunch, add \u00bc cup chopped pecans to the pudding mixture.\n\n  To make an even lower-fat custard, make it with custard powder and low-fat milk, following the packet instructions.\n\nNUTRITION ANALYSIS 1 serve (including custard) = 1795 kJ, 12 g fat (includes 3 g saturated fat), 4g fibre, 320 mg sodium\n\n**Panforte**\n\n_Here's a tip for mess-free chocolate melting. Place the chocolate in a microwave-safe bowl and microwave on medium power for 1 minute bursts. Stir after each minute until the chocolate has melted evenly. Don't use a wooden spoon for stirring \u2013 it can retain moisture that could make the chocolate seize and become unusable._\n\nMakes about 16 slices\n\nPreparation time   15 minutes\n\nCooking time   45 minutes\n\nspray oil\n\n\u00be cup macadamia nuts\n\n\u00be cup hazelnuts, almonds or pecans\n\n\u00bd cup dried fruit medley\n\n\u00bc cup mixed glac\u00e9 fruit\n\n\u00bc cup mixed peel\n\n  cup plain flour\n\n1 tablespoon cocoa\n\n\u00bd cup golden syrup or honey\n\n  cup caster sugar\n\n60 g dark chocolate, melted\n\nicing sugar\n\nPreheat the oven to moderate,180\u00baC (350\u00baF). Spray a 20 cm square or round cake pan with oil and line with baking paper.\n\nSpread all the nuts onto a baking tray. Bake for 4\u20135 minutes until lightly toasted. Chop them roughly and transfer to a large mixing bowl.\n\nReduce the heat to moderately slow, 160\u00baC (325\u00baF). Add the fruit and peel to the chopped nuts then sift in the flour and cocoa. Mix together well.\n\nCombine the syrup and sugar in a small saucepan. Heat on low, stirring, until the sugar dissolves. Bring mixture to a simmer, then cook, without stirring, for 4\u20135 minutes until a small amount dropped into a glass of water forms a ball.\n\nPour the syrup mixture and the melted chocolate into the dry ingredients. Mix well.\n\nPour the panforte mixture into the prepared pan, pressing it in well. Bake for 30\u201335 minutes until the panforte is firm to the touch. Cool completely in the pan.\n\nWhen cold, remove from the pan, wrap in foil and leave overnight. To serve, cut into squares or wedges and dust with icing sugar.\n\n**VARIATIONS**\n\n  Use all glac\u00e9 fruit instead of dried fruit if preferred.\n\n  Sprinkle with a few sesame seeds before baking.\n\nNUTRITION ANALYSIS 1 serve = 835 kJ, 11 g fat (includes 2 g saturated fat), 2 g fibre, 20 mg sodium\n\n**Mixed berry friands**\n\n_When making friands, there's a technique for beating the egg whites so they are easy to incorporate into the mixture. Don't beat them to a thick foam, just whisk them until frothy bubbles begin to appear._\n\nMakes 12\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\nspray oil\n\n1\u00bd cups icing sugar, sifted\n\n1 cup dry breadcrumbs\n\n\u00bd cup plain flour\n\n90 g light margarine, melted\n\n100 ml low-fat milk\n\n6 egg whites\n\n1 cup frozen mixed berries\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Spray a 12-hole muffin or friand pan with oil.\n\nCombine the icing sugar, breadcrumbs and flour in a large mixing bowl. Blend in the melted margarine and milk, stirring gently until just combined.\n\nIn a clean bowl, lightly whisk the egg whites until frothy. Using a metal spoon, lightly fold the egg whites through the breadcrumb mixture. Gently fold in the berries.\n\nSpoon the mixture evenly into the friand pan. Bake for 25\u201330 minutes until cooked when a skewer inserted comes out clean. Cool in the pan for 5 minutes before turning out onto a wire rack and leaving to cool completely.\n\n**VARIATIONS**\n\n  If you prefer, use wholemeal breadcrumbs. It's easy to make your own using a few slices of stale wholemeal bread \u2013 simply process in a food processor or blender to the right fine consistency.\n\n  Replace the mixed berries with the same amount of chopped apple, pear, dried fruit, fruit and nuts \u2013 or whatever takes your fancy.\n\n  For an interesting flavour variation, replace the milk with a flavoured tea.\n\nNUTRITION ANALYSIS 1 serve (including 1 scoop ice cream) = 890 kJ, 6 g fat (includes 1 g saturated fat), 1 g fibre, 150 mg sodium\n\n**Date, prune and craisin brownie slice**\n\n_The dried fruit makes this brownie slice lusciously sticky and moist, and the craisins (dried cranberries) add a tang that cuts through the richness. Remember to let it cool after baking: just-cooked brownies often have a soft gooey centre which firms on cooling._\n\nMakes 16\n\nPreparation time   15 minutes\n\nCooking time   25 minutes\n\n\u00bd cup chopped pitted dates\n\n1 cup self-raising flour\n\n  cup cocoa\n\n\u00bd teaspoon bicarbonate of soda\n\n\u00bd cup brown sugar\n\n\u00bd cup chopped pitted prunes\n\n\u00bd cup craisins or raisins\n\n\u00bd cup chopped pecans (or mixture macadamias and pecans)\n\n1 cup low-fat milk\n\n1 teaspoon vanilla extract\n\n3 egg whites\n\nIcing\n\n1 cup icing sugar\n\n1 tablespoon cocoa\n\n1 teaspoon light margarine\n\n1\u20132 tablespoons hot water\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Lightly grease a 20 cm square cake pan and line the base with baking paper.\n\nPlace the dates in a bowl, cover with boiling water and leave to soak for 15 minutes. Drain well.\n\nSift the flour, cocoa and bicarbonate of soda together into a large bowl. Stir in the sugar, prunes, craisins and nuts.\n\nMake a well in the centre of the dry ingredients. Pour in the milk and vanilla and stir in well.\n\nIn a clean bowl, using an electric beater, whisk the egg whites until soft peaks form. Fold into the chocolate mixture.\n\nPour the mixture into the prepared pan. Bake for 20\u201325 minutes until cooked when tested. Turn onto a wire rack to cool. Ice the brownies when completely cold.\n\nTo make the icing, combine the icing sugar, cocoa and margarine, and enough water to make a spreadable consistency. Drizzle over the brownies. Cut into squares to serve. Store in an airtight container.\n\nNUTRITION ANALYSIS 1 serve = 725 kJ, 4 g fat (includes 1 g saturated fat), 2 g fibre, 90 mg sodium\n\n**Carrot and zucchini cake**\n\n_Cakes freeze well. Allow the cake to cool completely before wrapping firmly, whole or in serving portions, in plastic wrap or place in a freezer bag. Seal (making sure all the air is expelled), label and freeze. Your cake will keep in the freezer for up to 3 months._\n\nServes about 12\n\nPreparation time   20 minutes\n\nCooking time   50 minutes\n\nspray oil\n\n1\u00bc cups self-raising flour\n\n1\u00bc cups wholemeal self-raising flour\n\n2 teaspoons mixed spice\n\n\u00bd teaspoon bicarbonate of soda\n\n1 carrot, grated (around \u00bd cup)\n\n1 zucchini, grated (around \u00bd cup)\n\n\u00bd cup sultanas\n\n\u00bc cup chopped pecans\n\n1 cup low-fat milk\n\n\u00bd cup orange juice\n\n2 eggs\n\nlow-fat vanilla ice cream to serve (optional)\n\nOrange icing\n\n1\u00bd cups icing sugar, sifted\n\n2 tablespoons orange juice\n\nzest of 1 orange\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Spray a 10 x 20 cm loaf pan with oil and dust with flour, shaking out any excess.\n\nSift the flours, spice and bicarbonate of soda into a large mixing bowl, returning any husks into the mix. Add the grated carrot and zucchini, the sultanas and pecans, and stir until well combined.\n\nWhisk together the milk, orange juice and eggs and gently fold them into the dry ingredients. Spoon the mixture into the prepared pan.\n\nBake for 45\u201350 minutes until cooked when tested. Leave to cool in the pan for 5 minutes then turn out onto a wire rack and leave to cool completely.\n\nTo make the orange icing, combine the icing sugar and orange juice in a large mixing bowl and stir well until smooth. Spread over the cooled cake. Use a zester or very sharp knife to cut a long ribbon of orange peel, taking care not to include any of the bitter white pith. Cut into fine shreds and use to garnish the cake.\n\nServe on its own or with ice cream. You can store this cake in an airtight container after cooling \u2013 it also freezes well.\n\n**VARIATIONS**\n\n  Use lemon instead of orange for a delicious lemon cake.\n\n  This cake could also become a berry and nut cake by replacing the zucchini and carrot with 1 cup of frozen mixed berries.\n\nNUTRITION ANALYSIS 1 serve (including 1 scoop ice cream) = 1115 kJ, 4 g fat (includes 1 g saturated fat), 3 g fibre, 245 mg sodium\n\n**Peach and banana clafoutis**\n\n_This delicious dessert is like a fruit flan and is traditionally made with black cherries. You can also try making it with rhubarb, plums or berries. It puffs on cooking and will sink slightly as it cools \u2013 when set it should still wobble slightly._\n\nServes 4\u20136\n\nPreparation time   10 minutes\n\nCooking time   30 minutes\n\nspray oil\n\n425 g can sliced peaches, drained\n\n1 large banana, peeled and sliced\n\n2 eggs\n\n2 tablespoons caster sugar\n\n\u00bc cup plain flour\n\n2 cups low-fat milk\n\n1 teaspoon vanilla paste\n\n1 tablespoon brown sugar\n\nicing sugar and low-fat custard to serve\n\nPreheat the oven to moderately hot, 190\u00baC (375\u00baF). Lightly spray a 20 cm round casserole dish with oil.\n\nArrange the peaches and banana over the base of the prepared dish.\n\nIn a bowl, whisk together the eggs and sugar until pale. Gradually blend in the flour, then the milk and vanilla until smooth.\n\nPour this mixture over the fruit and sprinkle with brown sugar. Bake for 25\u201330 minutes until set. Serve dusted with icing sugar and drizzled with custard.\n\nNUTRITION ANALYSIS 1 serve (including a dollop of custard) = 900 kJ, 4g fat (includes 2g saturated fat), 2g fibre, 105mg sodium\n\nBasics\n\nWe have put together a selection of basic recipes to make your life healthier and easier! Healthier, because our dressings and sauces really add zing to salad or vegetables without overloading you with unwanted fat, salt or kilojoules; and easier, because they're also a very simple way to add extra flavour and colour to your meals.\n\nSome of the recipes here are modified versions of old favourites; others we have created especially to fit in with the Nutrition for Life way of eating. All of them are versatile and you'll soon be using them freely with all your favourite meals.\n\nDon't think of dressings as just for salads, for instance! Use them, as well as the sauces, to liven up everything from a plain grilled fish fillet to a bowl of broad beans. In traditional Mediterranean cooking, vegetables always have something added \u2013 a splash of good virgin olive oil or a dollop of fresh basil pesto. We think it's the secret to making vegetables more attractive \u2013 and your kids will want to tuck into them with relish. And do please try our collection of simple flavoursome pestos, which do double duty as either dips or as toppings for pasta or fish.\n\nIf you're following a special diet, these recipes are definitely for you. You can make up a batch of salad dressing without any fat, or simmer a pot of stock without adding salt and use your home-made versions in other recipes. When you have to cut out things like salt, sugar or fat (or you have a food allergy or intolerance), it's always easier if you cook from scratch. That way, you know exactly what's in your food. And you don't have to worry about additive numbers or hidden salt or fat.\n**Basil and pine nut pesto**\n\nMakes about 1 cup\n\nPreparation time   10 minutes\n\n2 cups basil leaves\n\n  cup grated parmesan or pecorino\n\n\u00bc cup toasted pine nuts\n\n2 garlic cloves, crushed\n\n\u00bc cup salt-reduced vegetable stock\n\n1 tablespoon olive oil\n\nCombine the basil, parmesan, pine nuts and garlic in a food processor or blender and process until finely chopped.\n\nWith the motor running, gradually drizzle in the stock and oil until a paste-like consistency is achieved.\n\nStore, chilled, in an airtight container for about 1 week.\n\n**VARIATIONS**\n\n  If you like, make the pesto with 1 cup of basil leaves and 1 cup of flat-leaf parsley leaves.\n\n  For a change of flavour, replace the basil with coriander or parsley \u2013 or combine all three.\n\n  All the pestos can be stored in an airtight jar in the refrigerator where they will keep happily for up to a week. The ingredients may separate into layers, but you simply give them a stir before using.\n**Spinach and cashew pesto**\n\nMakes about 1 cup\n\nPreparation time   10 minutes\n\n2 cups baby spinach leaves\n\n  cup grated parmesan\n\n\u00bc cup toasted unsalted cashew nuts\n\n1 tablespoon sunflower seeds\n\n2 garlic cloves, crushed\n\n  cup salt-reduced vegetable stock\n\n1 tablespoon olive oil\n\nCombine the spinach, parmesan, cashews, sunflower seeds and garlic in a food processor or blender and process until finely chopped.\n\nWith the motor running, gradually drizzle in the stock and oil until a paste-like consistency is achieved.\n\nThis pesto is best served as soon as it's made \u2013 try tossing it through hot pasta. Although it will keep, chilled, in an airtight container for about 1 week.\n\n**VARIATION**\n\nYou can also replace the cashew nuts with walnuts or pecans.\n\nClockwise from top: Fetta, Garlic, Basil and Lemon Pesto; Spinach and Cashew Pesto ; and Char-grilled Capsicum and Ricotta Pesto.\n\n**Semi-dried tomatoes, basil and pecan pesto**\n\nMakes about \u00be cup\n\nPreparation time   10 minutes\n\n\u00bd cup low-fat, semi-dried tomatoes, halved\n\n\u00bd cup basil leaves\n\n\u00bc cup grated parmesan or pecorino cheese\n\n2 tablespoons chopped pecans\n\n2 garlic cloves, crushed\n\n\u00bd teaspoon chopped chilli\n\n  cup salt-reduced vegetable stock\n\n1 tablespoon olive oil\n\nCombine the tomatoes, basil, parmesan, pecans, garlic and chilli in a food processor or blender and process until finely chopped.\n\nWith the motor running, gradually drizzle in the stock and oil until a paste-like consistency is achieved.\n\nStore, chilled, in airtight container for about 1 week.\n\n**VARIATION**\n\nFor a change in flavour, try using flat-leaf parsley leaves instead of basil and char-grilled capsicum instead of tomatoes.\n**Fetta, garlic, basil and lemon pesto**\n\nMakes about   cup\n\nPreparation time   10 minutes\n\nCooking time   20 minutes\n\n1 head garlic\n\n125 g creamy fetta cheese, Greek or Danish are both good\n\n\u00bc cup basil leaves\n\n\u00bc cup toasted pine nuts\n\n1 tablespoon salt-reduced\n\nvegetable stock\n\n1 tablespoon olive oil\n\njuice \u00bd lemon\n\nfew drops Tabasco sauce\n\nPreheat the oven to moderate, 180\u00baC (350\u00baF). Bake the garlic for 20 minutes.\n\nCombine the remaining ingredients in a food processor or blender. Cut the end off the head of garlic and squeeze the softened garlic into mixture.\n\nProcess until a paste-like consistency is achieved. Use to toss through hot pasta for a delicious creamy sauce.\n\nStore, chilled, in an airtight container for about 1 week.\n\n**VARIATION**\n\nFor an extra zing, add some chopped chilli, to taste.\n**Asian-style pesto**\n\nMakes about   cup\n\nPreparation time   10 minutes\n\n\u00bd cup basil leaves\n\n\u00bd cup coriander leaves\n\n\u00bc cup unsalted peanuts\n\n2 tablespoons tamari (or light soy)\n\n1 tablespoon grated fresh ginger\n\n1 tablespoon red wine vinegar\n\n2 garlic cloves, crushed\n\n1 long red chilli, seeded and chopped\n\n1 teaspoon water\n\nCombine all the ingredients in a food processor or blender and process until a paste-like consistency is achieved.\n\nThis pesto is particularly good tossed through hot Asian noodles, instead of through Italian pasta.\n\nStore, chilled, in an airtight container for about 1 week.\n\n**VARIATION**\n\nUse as a flavour base for Asian stir-fries \u2013 it is equally delicious with chicken, fish, meat and vegetables.\n**Char-grilled capsicum and ricotta pesto**\n\nMakes about 1 cup\n\nPreparation time   10 minutes\n\n1 red capsicum\n\n250 g low-fat ricotta cheese\n\n\u00bd cup sliced basil leaves\n\n2 tablespoons toasted pine nuts\n\n1 tablespoon grated parmesan\n\n2 garlic cloves, crushed\n\nChar-grill the capsicum in a moderate oven, 180\u00baC (350\u00baF), or place it directly over the gas flame on your stove. It needs to be black and blistered all over. Place in a plastic bag, seal and allow to steam for a few minutes before carefully peeling off the skin and removing the membrane and seeds. Leave to cool then chop.\n\nPlace the capsicum into a large mixing bowl with all the remaining ingredients and mix everything together thoroughly.\n\nTo serve, toss through hot pasta, steamed vegetables or use as a dip. It can be stored, chilled, in an airtight container for about 1 week.\n\n**VARIATIONS**\n\n  Try using toasted, chopped flaked almonds in place of the pine nuts.\n\n  \u00bd cup chopped semi-dried tomatoes can be used in place of the capsicum.\n\n**Celeriac remoulade**\n\nMakes about 2 cups\n\nPreparation time   10 minutes (plus standing time)\n\n500 g celeriac, peeled and finely grated\n\njuice \u00bd lemon\n\n  cup low-fat whole egg mayonnaise\n\n2 teaspoons Dijon mustard\n\nCombine the celeriac and lemon juice in a bowl. Mix thoroughly so that the celeriac is well coated with the lemon juice and leave to stand for 20 minutes. Drain well, squeezing out any excess liquid.\n\nReturn the celeriac to the same bowl and add the remaining ingredients. Toss together well and serve with fish or meat.\n\n**VARIATION**\n\nAdd some freshly cracked black pepper, a spoonful of capers, some chopped dill pickles and fresh dill to make a tangier, chunkier remoulade.\n**Rouille**\n\nMakes about 1\u00bd cups\n\nPreparation time   10 minutes\n\n2 char-grilled red capsicum, chopped (for method)\n\n\u00bd cup fresh wholegrain breadcrumbs\n\n4 garlic cloves, crushed\n\n1 red chilli, seeds removed, chopped\n\n2 tablespoons olive oil\n\njuice \u00bd lemon\n\nCombine the capsicum, breadcrumbs, garlic and chilli in a food processor or blender and process until smooth.\n\nWith the motor running, gradually add the oil and lemon juice.\n\nRouille is traditionally served with seafood dishes, such as bouillabaisse. Spread it onto toasted bread or croutons or dollop straight into soup as you serve it.\n\nStore in an airtight container and refrigerate for up to 1 month.\n\n**VARIATION**\n\nFor a simpler version, blend the garlic, chilli and lemon juice directly into \u00bd cup of low-fat, whole egg mayonnaise.\n\nClockwise from top: No-oil Creamy Dressing; Rouille; No-oil Vinaigrette.\n**No-oil creamy dressing**\n\nMakes about 1 cup\n\nPreparation time   10 minutes\n\n200 g tub no-fat natural yoghurt\n\n1 tablespoon chopped dill\n\n2 teaspoons lemon juice\n\n1 teaspoon horseradish\n\nCombine all the ingredients in a mixing bowl and whisk together. Serve with your choice of salads. It is particularly good with potato salads and coleslaw.\n\nStore in an airtight container in the fridge, where it will keep well for up to 1 month.\n\n**VARIATIONS**\n\n  If you find the dressing a bit acidic, add a pinch of sugar.\n\n  Stir in a spoonful of grainy mustard for an extra zing.\n**No-oil vinaigrette**\n\nMakes about \u00bc cup\n\nPreparation time   5 minutes\n\n2 tablespoons lemon juice\n\n1 tablespoon finely chopped mixed herbs (parsley and thyme)\n\n1 teaspoon red wine vinegar\n\n1 teaspoon seeded mustard\n\npinch of sugar\n\nWhisk all the ingredients together in a small jug. Serve with any salads of your choice.\n\nIt's worth making larger quantities of this vinaigrette and storing it in a screw-top jar in the refrigerator. It will keep well for up to 1 month.\n\n**VARIATIONS**\n\n  Use your own choice of fresh herbs: parsley, basil, mint and dill are all delicious.\n\n  For a dressing with more depth, try adding a drizzle of balsamic vinegar to the dressing with the red wine vinegar. Or try one of the caramelised versions for a sweeter finish.\n**Low-fat white sauce**\n\nMakes 1 cup\n\nPreparation time   5 minutes\n\nCooking time   5 minutes\n\n30 g light margarine\n\n1 tablespoon plain flour\n\n1 cup low-fat milk\n\nPlace the margarine in a small saucepan and heat gently until it melts. Blend in the flour and cook over a medium heat, stirring, for 1 minute.\n\nRemove the pan from the heat and gradually blend in the milk until smooth. Return to the heat and cook, stirring, until the sauce boils and thickens. Lower the heat and simmer for 3 minutes to cook the flour thoroughly.\n\n**VARIATIONS**\n\n  Add reduced-fat cheese, chopped herbs, saut\u00e9ed mushrooms or green onions.\n\n  Use as the basis for a creamy fresh tomato soup.\n**Salsa verde**\n\nMakes about 1 cup\n\nPreparation time   10 minutes\n\n2 cups parsley leaves, finely chopped\n\n\u00bc cup roughly chopped dill pickles or gherkins\n\n2 tablespoons baby capers\n\n1 tablespoon olive oil\n\n1 tablespoon lemon juice\n\nfinely grated zest 1 lemon\n\n1 garlic clove, crushed\n\nCombine all the ingredients in a mixing bowl and toss together thoroughly.\n\nServe this tangy salsa with fish or lamb. It is also delicious with chicken, tossed through steamed vegetables, spread on bruschetta or stirred into risotto.\n\nStore in an airtight container in the fridge for 2\u20133 days.\n\n**VARIATION**\n\nFor a slightly sweeter version, replace the lemon juice and zest with orange.\n**Healthy home-made vegetable stock**\n\n_Making vegetable stock is easy, and it's a great way to clear out the fridge and to use up all sorts of bits and pieces of vegies that might be past their best._\n\nMakes about 2 litres\n\nPreparation time   15 minutes\n\nCooking time   1 hour\n\n2\u20133 carrots, chopped\n\n3\u20134 stalks celery\n\n1 large onion, chopped\n\nother vegetables of your choice, such as zucchini, mushrooms, peas, corn, green beans and capsicum\n\n6\u20138 garlic cloves\n\na few stalks of parsley\n\nother fresh herbs of your choice\n\n1\u20132 bay leaves\n\n1 tablespoon whole black peppercorns\n\nabout 2 litres water\n\nAssemble your selection of vegetables and fresh herbs. There is no need to peel the vegetables, just chop them roughly and place them in a large stock pot. Throw in the black peppercorns and a bay leaf or two for added flavour.\n\nCover your ingredients with water (a good rule of thumb is to have about half vegetables to water) and bring to the boil.\n\nReduce the heat and simmer, covered, for about an hour. Cool and strain.\n\nThat's all there is to it. You've just made vegetable stock. Store in containers in the freezer.\n**Healthy home-made chicken stock**\n\nMakes about 2 litres\n\nPreparation time   15 minutes\n\nCooking time   2 hours\n\n2 kg chicken or turkey carcass or wings\n\n2\u20133 carrots, chopped\n\n3\u20134 stalks celery\n\n1 large onion, chopped\n\n6\u20138 garlic cloves\n\n1 tablespoon whole black peppercorns\n\na few sprigs of parsley\n\nabout 2 litres water\n\nPlace the chicken in a large stock pot with the carrots, celery, onion, garlic, peppercorns and parsley. Cover with water and bring to the boil.\n\nReduce the heat and simmer, covered, for about 2 hours. Periodically skim off the foam as it rises to the surface.\n\nStrain the stock and chill for a few hours. Any fat will rise to the surface and congeal where it can be easily skimmed off. Your stock is now ready for use or for the freezer.\n**Healthy home-made beef stock**\n\nMakes about 2 litres\n\nPreparation time   15 minutes\n\nCooking time   3 hours\n\n3\u20134 kg lean beef and bones\n\n4 large carrots, chopped\n\n4 stalks celery\n\n2 large tomatoes, chopped\n\n2 large onions, chopped\n\n1 garlic head, peeled and chopped\n\n4 bay leaves\n\n1 tablespoon whole black peppercorns\n\na few stalks of parsley\n\nabout 2 litres water\n\nFor a richer flavour, first roast the beef and bones in a hot oven (200\u00baC\/ 400\u00baF) for about 45 minutes.\n\nPlace the beef in a stock pot with the carrots, celery, tomatoes, onions, garlic, bay leaves, black peppercorns and parsley. Cover with water.\n\nBring to the boil then reduce the heat and simmer, covered, for 3 hours.\n\nStrain the stock and chill for a few hours. Skim off any fat that rises to the surface. Your stock is now ready for use or for the freezer.\n**Pizza dough**\n\nMakes enough for 1 large pizza\n\nPreparation time   20 minutes (plus rising time)\n\n1 cup warm water\n\n8 g sachet dried yeast\n\n1 teaspoon caster sugar\n\n2\u00bd cups plain flour (wholegrain if liked)\n\n1 teaspoon salt\n\nCombine the water, yeast and sugar in a small jug. Stir well and leave to stand in a warm place for 10 minutes until the mixture is frothy.\n\nSift the flour and salt into a large bowl. Stir in the yeast mixture to form a soft, sticky dough. Turn the dough out onto a lightly floured work surface and knead for 5\u201310 minutes until smooth and elastic.\n\nPlace the dough in a large oiled bowl. Cover with plastic wrap and leave to rise in a warm place for around an hour until the dough doubles in size. Knock down the dough to remove the air. Knead briefly into a smooth ball. Proceed with the toppings of your choice.\n**Wholemeal pastry**\n\nMakes enough for 23 cm tart pan\n\nPreparation time   15 minutes (plus resting time)\n\nCooking time   15 minutes\n\n2 cups wholemeal plain flour\n\n\u00bd cup wheatgerm\n\n125 g light margarine\n\n2 egg yolks\n\n2\u20133 tablespoons water\n\nWholemeal is a healthier option than plain flour being rich in B vitamins and fibre. However, some people find that wholemeal pastry is heavier and crumbly. This recipe will give you a smooth, light and crisp-textured pastry that you can use for both sweet and savoury dishes.\n\nSift the flour into a large bowl, returning the husks to the bowl. Stir through the wheatgerm then add the margarine and use fingertips to rub it in thoroughly.\n\nMix in the egg yolks and enough water to make a firm dough. Wrap in plastic wrap and rest in the refrigerator for 15 minutes.\n\nRoll out the pastry between 2 sheets of baking paper. Use to line a 23 cm tart pan then rest in the refrigerator for a further 30 minutes.\n\nWhen ready to blind bake the pastry shell, preheat the oven to hot, 200\u00baC (400\u00baF). Line the pastry with baking paper and fill with dried beans or rice and bake for 10 minutes. Remove the paper and beans and bake for a further 5 minutes. If filling and cooking immediately, lower the oven temperature to moderate, 180\u00baC (350\u00baF), and proceed as directed.\n\nShopping with zest\n\nThere are many staple ingredients that we love and use all the time and recommend that you keep in your kitchen. They'll help you rustle up quick meals at the last minute and give you the basics for any recipe \u2013 you'll only have to 'top up' on your way home with some fresh herbs, fresh salad ingredients, fish or meat.\n\nSome tips to remember while you are shopping:\n\n\u2022 Always keep a shopping list in the kitchen to jot down items as you run out.\n\n\u2022 Group your shopping list into categories that mirror the aisles in the supermarket (such as dairy, freezer, meats, fresh produce, bakery, deli and general groceries) so you streamline your shopping and don't have to backtrack.\n\n\u2022 Skip that aisle! If you don't need food from a particular aisle in the supermarket (say the confectionery, snacks and soft drinks aisles), don't visit it. Don't be tempted into purchases that are 'nutritional extras'.\n\n\u2022 Buy your fruit and vegetables when it is in season. That way you'll get them at their lowest price, in peak quality and you can afford to buy up lots of different produce for better variety and nutrition.\nDairy and deli\n\nMilk (low-fat or no-fat, skim)\n\nButtermilk\n\nSoy alternatives (low-fat, calcium-enriched soy milk)\n\nLow-fat drinking yoghurt\n\nYoghurt (low-fat plain, low-fat fruit or vanilla)\n\nLow-fat dairy desserts (low-fat custard, Fr\u00fbche)\n\nReduced-fat cheeses (reduced-fat grated or sliced cheese, low-fat ricotta and cottage cheese, parmesan cheese, haloumi, reduced-fat fetta)\n\nMargarine (light olive, canola spread)\n\nTofu\n\nDips (hummus)\n\nPesto (you can make your own fresh pesto using our recipes)\n\nSemi-dried or sun-dried tomatoes\n\nOlives\n\nFreezer items\n\nFrozen vegetables (peas, corn cobs, beans, spinach, broad beans, mixed vegetables, cauliflower, stir-fry mix)\n\nOven-bake chips (8% fat or less)\n\nFish fillets\n\nVegetarian dinner options (soy burger, tofu)\n\nFrozen yoghurt or gelato\n\nIce cream (low-fat vanilla)\n\nFilo pastry\n\nFresh fruit, vegetables and herbs\n\nBasics\n\nChillies\n\nGarlic\n\nGinger\n\nLemons or limes\n\nOnions\n\nPotatoes\n\nFresh seasonal vegetables\n\nAsian greens (bok choy, on choy, Chinese broccoli)\n\nAsparagus\n\nBeans\n\nBroccoli or broccolini\n\nBrussels sprouts\n\nCabbage or Chinese cabbage\n\nCarrots\n\nCauliflower\n\nCeleriac\n\nCorn cobs\n\nEggplant\n\nFennel\n\nLeeks\n\nMushrooms\n\nPeas\n\nPumpkin\n\nSnowpeas\n\nSpinach or silverbeet\n\nSweet potato\n\nZucchini or squash\n\nSalad ingredients\n\nAvocado\n\nCapsicum\n\nCelery\n\nCucumber\n\nGreen onions and spring onions\n\nRed (Spanish) onions\n\nSalad leaves (lettuce varieties, baby spinach, rocket leaves, salad mix)\n\nSprouts (mung bean, snowpea, alfalfa, mixed)\n\nTomatoes\n\nFresh herbs\n\nBasil\n\nChives\n\nCoriander\n\nDill\n\nLemongrass\n\nMint\n\nParsley\n\nFresh fruit\n\nApples\n\nBananas\n\nBerries\n\nGrapes\n\nKiwi fruit\n\nMelon (rockmelon, watermelon)\n\nOranges, mandarins or grapefruit\n\nPeaches, nectarines or apricots\n\nPears\n\nPineapple, pawpaw, mango\n\nRhubarb\n\nMeat\n\nLamb (chops, backstraps, fillets, cutlets, mince)\n\nLean beef (steaks, mince, diced or in strips)\n\nChicken (skinless breast fillets, mince, strips)\n\nPork (strips, chops, fillets, mince)\n\nHam, pancetta or prosciutto\n\nFish\n\nFish fillets (cutlets or whole)\n\nPrawns\n\nMussels\n\nBread and bakery items\n\nBread or rolls (grainy, sourdough, rye)\n\nEnglish muffins\n\nPitta bread or wraps\n\nRaisin bread\n\nGrocery items\n\nEggs\n\nDried legumes (lentils, chickpeas, split peas, beans \u2013 all types are healthy)\n\nPasta\n\nRice (low-GI rice such as Doongara and Moolgiri; basmati or brown rice)\n\nNoodles (hokkien, rice sticks)\n\nBurghul (cracked wheat)\n\nPearl barley\n\nCouscous\n\nPolenta\n\nSemolina\n\nWholemeal breadcrumbs\n\nFlour\n\nBuckwheat flour\n\nLecithin\n\nWheatgerm\n\nOat bran\n\nOils (see box below)\n\nSpray oil\n\nVinegar (red and white wine, balsamic, cider, rice)\n\nStock (salt-reduced, or make your own using our recipes)\n\nNuts and seeds (unsalted varieties \u2013 almonds, walnuts, pine nuts, pumpkin seeds \u2013 all types are healthy)\n\nDried fruit (sultanas, apricots, raisins, prunes, apples, figs, dates)\n\nSpreads (honey, jams, marmalade, peanut butter)\n\nMaple syrup (100% pure)\n\nBeverages (tea leaves and bags, milk flavourings such as Ovaltine, Horlicks, Milo)\n\nJuices (no added sugar \u2013 orange, apple, pineapple, grapefruit, cranberry, breakfast blends. Dilute with water before drinking)\n\n**_Which oil is best?_**\n\nThere is no single perfect oil for all recipes or for good nutrition. All oils are low in saturated fats, which raise blood cholesterol levels. Oils can be high in either monounsaturated or polyunsaturated fat and are therefore a healthier choice than saturated fats such as butter and lard.\n\nChoose the oil that complements the dish you're cooking \u2013 olive is the essence of Italian and Greek cuisines; peanut and sesame add oomph to Asian dishes; rice bran, canola and grape seed are more neutral in flavour.\n\nRice bran and grape seed oils are light with a relatively high smoking point, which means they are suitable for high-heat pan-frying (the temperature at which they start to break down and burn is higher than some oils). They have little flavour so are useful for cooking subtly flavoured foods such as fish.\n\nFlaxseed oil is useful for anyone on a vegetarian diet as it is a good source of omega-3.\n\nOlive oil is one of our favourite oils and we use it often. Use 'pure' olive oil for cooking and reserve the more expensive virgin olive oil for splashing over vegetables or making into a salad dressing. We use sesame oil and peanut oil when we want a more pungent, spicy overtone.\n\nRemember that all oils are high in fat and therefore high in kilojoules, so make sure you use the smallest quantity you need for the job.\n\n_**Types of oil**_\n\n**_Monounsaturated oils_**\n\nOlive\n\nCanola\n\nPeanut\n\nSunola\/monosun\n\n_**Polyunsaturated oils**_\n\nSunflower\n\nGrape seed\n\nSoy bean\n\nMaize (corn)\n\nCottonseed\n\nWalnut\n\nSafflower\n\nFlaxseed\n\n**_A mix of poly- and monounsaturated_**\n\nSesame\n\nRice bran\n\nCanned goods\n\nBaked beans (salt-reduced)\n\nThree bean mix or chick peas (for salads)\n\nKidney or soy beans (to stretch out casseroles or meat dishes)\n\nLentils\n\nCorn kernels\n\nCreamed corn\n\nTomatoes, whole or diced\n\nTuna (in freshwater or in sachets, plain or flavoured)\n\nSalmon (pink or red)\n\nSoups (low-fat and salt-reduced)\n\nLight evaporated milk (plain and coconut-flavoured)\n\nSauces and dressings\n\nTomato pasta sauces\n\nTomato paste (no added salt)\n\nCurry paste, curry powder\n\nBottled pasta sauces\/dinner sauces (handy for quick meals \u2013 tomato-based ones are low fat and look for less than 5 grams of fat per 100 grams for others)\n\nAsian sauces (salt-reduced soy, fish, oyster, teriyaki, hoi sin \u2013 these are high in salt even if salt-reduced so use sparingly)\n\nSauces\/marinades (sweet chilli sauce, Tabasco, Worcestershire \u2013 can be high in salt but add flavour and make fat-trimmed meats and vegetables taste delicious!)\n\nSalad dressings (regular, no-oil, or make your own vinaigrette using our recipe)\n\nMayonnaise or creamy dressing (fat-free, low-fat, or make your own using our recipe)\n\nRelishes, condiments and spices\n\nRelishes and condiments (mustard (Dijon or grainy), fruit chutney, tomato relish, horseradish \u2013 add flavour to lean meats, fish and sandwiches)\n\nSpices and dried herbs (oregano, mixed herbs, Italian herbs, cumin, five spice, lemongrass, turmeric, tarragon, thyme, dill seeds, sage, fennel, bay leaves, paprika, cayenne, garam marsala, saffron, cinnamon, cloves, nutmeg, vanilla pod, vanilla paste, mixed spice; bottles of minced ginger, chilli, coriander; tubes of herb paste \u2013 can be high in salt so use sparingly) \nConversion tables\n\n_**Liquid measures**_|   \n---|---  \n**metric**|  **imperial**  \n30 ml | 1 fl oz   \n60 ml | 2 fl oz   \n100 ml | 3 fl oz   \n125 ml | 4 fl oz   \n150 ml | 5 fl oz (\u00bc pint)   \n190 ml | 6 fl oz   \n250 ml | 8 fl oz   \n300 ml | 10 fl oz (\u00bd pint)   \n500 ml | 16 fl oz   \n600 ml | 20 fl oz (1 pint)   \n1000 ml (1 litre) | 35 fl oz (1\u00be pints)   \n**_Dry measures_**|   \n---|---  \n**metric**|  **imperial**  \n15 g | \u00bd oz   \n30 g | 1 oz   \n60 g | 2 oz   \n90 g | 3 oz   \n125 g | 4 oz (\u00bc lb)   \n155 g | 5 oz   \n185 g | 6 oz   \n220 g | 7 oz   \n250 g | 8 oz   \n280 g | 9 oz   \n315 g | 10 oz   \n345 g | 11 oz   \n375 g | 12 oz (\u00be lb)   \n410 g | 13 oz   \n440 g | 14 oz   \n470 g | 15 oz   \n500 g | 16 oz (1 lb)   \n750 g | 24 oz (1\u00bd lb)   \n1 kg | 32 oz (2 lb)   \n_**Teaspoons, tablespoons and cups**_  \n---  \n1 Australian metric teaspoon = 5 ml   \n1 Australian metric tablespoon = 20 ml   \n1 Australian metric cup = 250 ml   \n**_Oven temperatures_**|  |   \n---|---|---  \n| **\u00b0Celsius**|  **\u00b0Fahrenheit**  \nvery slow | 120 | 250   \nslow | 150 | 300   \nmoderately slow | 160 | 325   \nmoderate | 180 | 350   \nmoderately hot | 190 | 375   \nhot | 200 | 400   \nvery hot | 220\u2013250 | 450\u2013500   \n_**Length**_|   \n---|---  \n**metric**|  **imperial**  \n5 mm | \u00bc in   \n1 cm | \u00bd in   \n2 cm | \u00be in   \n2.5 cm | 1 in   \n5 cm | 2 in   \n7.5 cm | 3 in   \n10 cm | 4 in   \n15 cm | 6 in   \n20 cm | 8 in   \n30 cm | 12 in \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n\n\nProduced by John Bickers and Dagny\n\n\n\n\n\nPIERRE GRASSOU\n\n\nBy Honore De Balzac\n\n\n\nTranslated by Katharine Prescott Wormeley\n\n\n\nDedication\n\nTo The Lieutenant-Colonel of Artillery, Periollas, As a Testimony of the\nAffectionate Esteem of the Author,\n\nDe Balzac\n\n\n\n\n\nPIERRE GRASSOU\n\n\nWhenever you have gone to take a serious look at the exhibition of works\nof sculpture and painting, such as it has been since the revolution\nof 1830, have you not been seized by a sense of uneasiness, weariness,\nsadness, at the sight of those long and over-crowded galleries? Since\n1830, the true Salon no longer exists. The Louvre has again been taken\nby assault,--this time by a populace of artists who have maintained\nthemselves in it.\n\nIn other days, when the Salon presented only the choicest works of art,\nit conferred the highest honor on the creations there exhibited. Among\nthe two hundred selected paintings, the public could still choose: a\ncrown was awarded to the masterpiece by hands unseen. Eager, impassioned\ndiscussions arose about some picture. The abuse showered on Delacroix,\non Ingres, contributed no less to their fame than the praises and\nfanaticism of their adherents. To-day, neither the crowd nor the\ncriticism grows impassioned about the products of that bazaar. Forced to\nmake the selection for itself, which in former days the examining\njury made for it, the attention of the public is soon wearied and the\nexhibition closes. Before the year 1817 the pictures admitted never went\nbeyond the first two columns of the long gallery of the old masters; but\nin that year, to the great astonishment of the public, they filled the\nwhole space. Historical, high-art, genre paintings, easel pictures,\nlandscapes, flowers, animals, and water-colors,--these eight specialties\ncould surely not offer more than twenty pictures in one year worthy of\nthe eyes of the public, which, indeed, cannot give its attention to a\ngreater number of such works. The more the number of artists increases,\nthe more careful and exacting the jury of admission ought to be.\n\nThe true character of the Salon was lost as soon as it spread along\nthe galleries. The Salon should have remained within fixed limits of\ninflexible proportions, where each distinct specialty could show its\nmasterpieces only. An experience of ten years has shown the excellence\nof the former institution. Now, instead of a tournament, we have a mob;\ninstead of a noble exhibition, we have a tumultuous bazaar; instead of\na choice selection we have a chaotic mass. What is the result? A great\nartist is swamped. Decamps' \"Turkish Cafe,\" \"Children at a Fountain,\"\n\"Joseph,\" and \"The Torture,\" would have redounded far more to his credit\nif the four pictures had been exhibited in the great Salon with the\nhundred good pictures of that year, than his twenty pictures could,\namong three thousand others, jumbled together in six galleries.\n\nBy some strange contradiction, ever since the doors are open to every\none there has been much talk of unknown and unrecognized genius. When,\ntwelve years earlier, Ingres' \"Courtesan,\" and that of Sigalon, the\n\"Medusa\" of Gericault, the \"Massacre of Scio\" by Delacroix, the \"Baptism\nof Henri IV.\" by Eugene Deveria, admitted by celebrated artists accused\nof jealousy, showed the world, in spite of the denials of criticism,\nthat young and vigorous palettes existed, no such complaint was made.\nNow, when the veriest dauber of canvas can send in his work, the whole\ntalk is of genius neglected! Where judgment no longer exists, there is\nno longer anything judged. But whatever artists may be doing now, they\nwill come back in time to the examination and selection which presents\ntheir works to the admiration of the crowd for whom they work. Without\nselection by the Academy there will be no Salon, and without the Salon\nart may perish.\n\nEver since the catalogue has grown into a book, many names have appeared\nin it which still remain in their native obscurity, in spite of the ten\nor a dozen pictures attached to them. Among these names perhaps the most\nunknown to fame is that of an artist named Pierre Grassou, coming from\nFougeres, and called simply \"Fougeres\" among his brother-artists, who,\nat the present moment holds a place, as the saying is, \"in the sun,\" and\nwho suggested the rather bitter reflections by which this sketch of\nhis life is introduced,--reflections that are applicable to many other\nindividuals of the tribe of artists.\n\nIn 1832, Fougeres lived in the rue de Navarin, on the fourth floor of\none of those tall, narrow houses which resemble the obelisk of Luxor,\nand possess an alley, a dark little stairway with dangerous turnings,\nthree windows only on each floor, and, within the building, a courtyard,\nor, to speak more correctly, a square pit or well. Above the three or\nfour rooms occupied by Grassou of Fougeres was his studio, looking over\nto Montmartre. This studio was painted in brick-color, for a background;\nthe floor was tinted brown and well frotted; each chair was furnished\nwith a bit of carpet bound round the edges; the sofa, simple enough, was\nclean as that in the bedroom of some worthy bourgeoise. All these things\ndenoted the tidy ways of a small mind and the thrift of a poor man. A\nbureau was there, in which to put away the studio implements, a table\nfor breakfast, a sideboard, a secretary; in short, all the articles\nnecessary to a painter, neatly arranged and very clean. The stove\nparticipated in this Dutch cleanliness, which was all the more visible\nbecause the pure and little changing light from the north flooded with\nits cold clear beams the vast apartment. Fougeres, being merely a genre\npainter, does not need the immense machinery and outfit which ruin\nhistorical painters; he has never recognized within himself sufficient\nfaculty to attempt high-art, and he therefore clings to easel painting.\n\nAt the beginning of the month of December of that year, a season at\nwhich the bourgeois of Paris conceive, periodically, the burlesque idea\nof perpetuating their forms and figures already too bulky in themselves,\nPierre Grassou, who had risen early, prepared his palette, and lighted\nhis stove, was eating a roll steeped in milk, and waiting till the frost\non his windows had melted sufficiently to let the full light in. The\nweather was fine and dry. At this moment the artist, who ate his bread\nwith that patient, resigned air that tells so much, heard and recognized\nthe step of a man who had upon his life the influence such men have\non the lives of nearly all artists,--the step of Elie Magus, a\npicture-dealer, a usurer in canvas. The next moment Elie Magus entered\nand found the painter in the act of beginning his work in the tidy\nstudio.\n\n\"How are you, old rascal?\" said the painter.\n\nFougeres had the cross of the Legion of honor, and Elie Magus bought his\npictures at two and three hundred francs apiece, so he gave himself the\nairs of a fine artist.\n\n\"Business is very bad,\" replied Elie. \"You artists have such\npretensions! You talk of two hundred francs when you haven't put six\nsous' worth of color on a canvas. However, you are a good fellow, I'll\nsay that. You are steady; and I've come to put a good bit of business in\nyour way.\"\n\n\"Timeo Danaos et dona ferentes,\" said Fougeres. \"Do you know Latin?\"\n\n\"No.\"\n\n\"Well, it means that the Greeks never proposed a good bit of business\nto the Trojans without getting their fair share of it. In the olden time\nthey used to say, 'Take my horse.' Now we say, 'Take my bear.' Well,\nwhat do you want, Ulysses-Lagingeole-Elie Magus?\"\n\nThese words will give an idea of the mildness and wit with which\nFougeres employed what painters call studio fun.\n\n\"Well, I don't deny that you are to paint me two pictures for nothing.\"\n\n\"Oh! oh!\"\n\n\"I'll leave you to do it, or not; I don't ask it. But you're an honest\nman.\"\n\n\"Come, out with it!\"\n\n\"Well, I'm prepared to bring you a father, mother, and only daughter.\"\n\n\"All for me?\"\n\n\"Yes--they want their portraits taken. These bourgeois--they are crazy\nabout art--have never dared to enter a studio. The girl has a 'dot' of a\nhundred thousand francs. You can paint all three,--perhaps they'll turn\nout family portraits.\"\n\nAnd with that the old Dutch log of wood who passed for a man and who was\ncalled Elie Magus, interrupted himself to laugh an uncanny laugh which\nfrightened the painter. He fancied he heard Mephistopheles talking\nmarriage.\n\n\"Portraits bring five hundred francs apiece,\" went on Elie; \"so you can\nvery well afford to paint me three pictures.\"\n\n\"True for you!\" cried Fougeres, gleefully.\n\n\"And if you marry the girl, you won't forget me.\"\n\n\"Marry! I?\" cried Pierre Grassou,--\"I, who have a habit of sleeping\nalone; and get up at cock-crow, and all my life arranged--\"\n\n\"One hundred thousand francs,\" said Magus, \"and a quiet girl, full of\ngolden tones, as you call 'em, like a Titian.\"\n\n\"What class of people are they?\"\n\n\"Retired merchants; just now in love with art; have a country-house at\nVille d'Avray, and ten or twelve thousand francs a year.\"\n\n\"What business did they do?\"\n\n\"Bottles.\"\n\n\"Now don't say that word; it makes me think of corks and sets my teeth\non edge.\"\n\n\"Am I to bring them?\"\n\n\"Three portraits--I could put them in the Salon; I might go in for\nportrait-painting. Well, yes!\"\n\nOld Elie descended the staircase to go in search of the Vervelle family.\nTo know to what extend this proposition would act upon the painter, and\nwhat effect would be produced upon him by the Sieur and Dame Vervelle,\nadorned by their only daughter, it is necessary to cast an eye on the\nanterior life of Pierre Grassou of Fougeres.\n\nWhen a pupil, Fougeres had studied drawing with Servin, who was\nthought a great draughtsman in academic circles. After that he went to\nSchinner's, to learn the secrets of the powerful and magnificent color\nwhich distinguishes that master. Master and scholars were all discreet;\nat any rate Pierre discovered none of their secrets. From there he went\nto Sommervieux' atelier, to acquire that portion of the art of painting\nwhich is called composition, but composition was shy and distant to him.\nThen he tried to snatch from Decamps and Granet the mystery of their\ninterior effects. The two masters were not robbed. Finally Fougeres\nended his education with Duval-Lecamus. During these studied and\nthese different transformations Fougeres' habits and ways of life were\ntranquil and moral to a degree that furnished matter of jesting to the\nvarious ateliers where he sojourned; but everywhere he disarmed his\ncomrades by his modesty and by the patience and gentleness of a lamblike\nnature. The masters, however, had no sympathy for the good lad; masters\nprefer bright fellows, eccentric spirits, droll or fiery, or else gloomy\nand deeply reflective, which argue future talent. Everything about\nPierre Grassou smacked of mediocrity. His nickname \"Fougeres\" (that\nof the painter in the play of \"The Eglantine\") was the source of much\nteasing; but, by force of circumstances, he accepted the name of the\ntown in which he had first seen light.\n\nGrassou of Fougeres resembled his name. Plump and of medium height, he\nhad a dull complexion, brown eyes, black hair, a turned-up nose, rather\nwide mouth, and long ears. His gentle, passive, and resigned air gave a\ncertain relief to these leading features of a physiognomy that was full\nof health, but wanting in action. This young man, born to be a virtuous\nbourgeois, having left his native place and come to Paris to be clerk\nwith a color-merchant (formerly of Mayenne and a distant connection of\nthe Orgemonts) made himself a painter simply by the fact of an obstinacy\nwhich constitutes the Breton character. What he suffered, the manner in\nwhich he lived during those years of study, God only knows. He suffered\nas much as great men suffer when they are hounded by poverty and hunted\nlike wild beasts by the pack of commonplace minds and by troops of\nvanities athirst for vengeance.\n\nAs soon as he thought himself able to fly on his own wings, Fougeres\ntook a studio in the upper part of the rue des Martyrs, where he began\nto delve his way. He made his first appearance in 1819. The first\npicture he presented to the jury of the Exhibition at the Louvre\nrepresented a village wedding rather laboriously copied from Greuze's\npicture. It was rejected. When Fougeres heard of the fatal decision,\nhe did not fall into one of those fits of epileptic self-love to which\nstrong natures give themselves up, and which sometimes end in challenges\nsent to the director or the secretary of the Museum, or even by threats\nof assassination. Fougeres quietly fetched his canvas, wrapped it in\na handkerchief, and brought it home, vowing in his heart that he would\nstill make himself a great painter. He placed his picture on the easel,\nand went to one of his former masters, a man of immense talent,--to\nSchinner, a kind and patient artist, whose triumph at that year's Salon\nwas complete. Fougeres asked him to come and criticise the rejected\nwork. The great painter left everything and went at once. When poor\nFougeres had placed the work before him Schinner, after a glance,\npressed Fougeres' hand.\n\n\"You are a fine fellow,\" he said; \"you've a heart of gold, and I must\nnot deceive you. Listen; you are fulfilling all the promises you made in\nthe studios. When you find such things as that at the tip of your brush,\nmy good Fougeres, you had better leave colors with Brullon, and not take\nthe canvas of others. Go home early, put on your cotton night-cap, and\nbe in bed by nine o'clock. The next morning early go to some government\noffice, ask for a place, and give up art.\"\n\n\"My dear friend,\" said Fougeres, \"my picture is already condemned; it is\nnot a verdict that I want of you, but the cause of that verdict.\"\n\n\"Well--you paint gray and sombre; you see nature being a crape veil;\nyour drawing is heavy, pasty; your composition is a medley of Greuze,\nwho only redeemed his defects by the qualities which you lack.\"\n\nWhile detailing these faults of the picture Schinner saw on Fougeres'\nface so deep an expression of sadness that he carried him off to dinner\nand tried to console him. The next morning at seven o'clock Fougeres was\nat his easel working over the rejected picture; he warmed the colors; he\nmade the corrections suggested by Schinner, he touched up his figures.\nThen, disgusted with such patching, he carried the picture to Elie\nMagus. Elie Magus, a sort of Dutch-Flemish-Belgian, had three reasons\nfor being what he became,--rich and avaricious. Coming last from\nBordeaux, he was just starting in Paris, selling old pictures and living\non the boulevard Bonne-Nouvelle. Fougeres, who relied on his palette\nto go to the baker's, bravely ate bread and nuts, or bread and milk, or\nbread and cherries, or bread and cheese, according to the seasons. Elie\nMagus, to whom Pierre offered his first picture, eyed it for some time\nand then gave him fifteen francs.\n\n\"With fifteen francs a year coming in, and a thousand francs for\nexpenses,\" said Fougeres, smiling, \"a man will go fast and far.\"\n\nElie Magus made a gesture; he bit his thumbs, thinking that he might\nhave had that picture for five francs.\n\nFor several days Pierre walked down from the rue des Martyrs and\nstationed himself at the corner of the boulevard opposite to Elie's\nshop, whence his eye could rest upon his picture, which did not obtain\nany notice from the eyes of the passers along the street. At the end of\na week the picture disappeared; Fougeres walked slowly up and approached\nthe dealer's shop in a lounging manner. The Jew was at his door.\n\n\"Well, I see you have sold my picture.\"\n\n\"No, here it is,\" said Magus; \"I've framed it, to show it to some one\nwho fancies he knows about painting.\"\n\nFougeres had not the heart to return to the boulevard. He set about\nanother picture, and spent two months upon it,--eating mouse's meals and\nworking like a galley-slave.\n\nOne evening he went to the boulevard, his feet leading him fatefully to\nthe dealer's shop. His picture was not to be seen.\n\n\"I've sold your picture,\" said Elie Magus, seeing him.\n\n\"For how much?\"\n\n\"I got back what I gave and a small interest. Make me some Flemish\ninteriors, a lesson of anatomy, landscapes, and such like, and I'll buy\nthem of you,\" said Elie.\n\nFougeres would fain have taken old Magus in his arms; he regarded him as\na father. He went home with joy in his heart; the great painter Schinner\nwas mistaken after all! In that immense city of Paris there were some\nhearts that beat in unison with Pierre's; his talent was understood and\nappreciated. The poor fellow of twenty-seven had the innocence of a lad\nof sixteen. Another man, one of those distrustful, surly artists, would\nhave noticed the diabolical look on Elie's face and seen the twitching\nof the hairs of his beard, the irony of his moustache, and the movement\nof his shoulders which betrayed the satisfaction of Walter Scott's Jew\nin swindling a Christian.\n\nFougeres marched along the boulevard in a state of joy which gave to his\nhonest face an expression of pride. He was like a schoolboy protecting\na woman. He met Joseph Bridau, one of his comrades, and one of those\neccentric geniuses destined to fame and sorrow. Joseph Bridau, who had,\nto use his own expression, a few sous in his pocket, took Fougeres to\nthe Opera. But Fougeres didn't see the ballet, didn't hear the music; he\nwas imagining pictures, he was painting. He left Joseph in the middle\nof the evening, and ran home to make sketches by lamp-light. He invented\nthirty pictures, all reminiscence, and felt himself a man of genius. The\nnext day he bought colors, and canvases of various dimensions; he piled\nup bread and cheese on his table, he filled a water-pot with water,\nhe laid in a provision of wood for his stove; then, to use a studio\nexpression, he dug at his pictures. He hired several models and Magus\nlent him stuffs.\n\nAfter two months' seclusion the Breton had finished four pictures. Again\nhe asked counsel of Schinner, this time adding Bridau to the invitation.\nThe two painters saw in three of these pictures a servile imitation\nof Dutch landscapes and interiors by Metzu, in the fourth a copy of\nRembrandt's \"Lesson of Anatomy.\"\n\n\"Still imitating!\" said Schinner. \"Ah! Fougeres can't manage to be\noriginal.\"\n\n\"You ought to do something else than painting,\" said Bridau.\n\n\"What?\" asked Fougeres.\n\n\"Fling yourself into literature.\"\n\nFougeres lowered his head like a sheep when it rains. Then he asked and\nobtained certain useful advice, and retouched his pictures before taking\nthem to Elie Magus. Elie paid him twenty-five francs apiece. At that\nprice of course Fougeres earned nothing; neither did he lose, thanks to\nhis sober living. He made a few excursions to the boulevard to see what\nbecame of his pictures, and there he underwent a singular hallucination.\nHis neat, clean paintings, hard as tin and shiny as porcelain, were\ncovered with a sort of mist; they looked like old daubs. Magus was out,\nand Pierre could obtain no information on this phenomenon. He fancied\nsomething was wrong with his eyes.\n\nThe painter went back to his studio and made more pictures. After seven\nyears of continued toil Fougeres managed to compose and execute quite\npassable work. He did as well as any artist of the second class.\nElie bought and sold all the paintings of the poor Breton, who earned\nlaboriously about two thousand francs a year while he spent but twelve\nhundred.\n\nAt the Exhibition of 1829, Leon de Lora, Schinner, and Bridau, who all\nthree occupied a great position and were, in fact, at the head of the\nart movement, were filled with pity for the perseverance and the poverty\nof their old friend; and they caused to be admitted into the grand salon\nof the Exhibition, a picture by Fougeres. This picture, powerful in\ninterest but derived from Vigneron as to sentiment and from Dubufe's\nfirst manner as to execution, represented a young man in prison, whose\nhair was being cut around the nape of the neck. On one side was\na priest, on the other two women, one old, one young, in tears. A\nsheriff's clerk was reading aloud a document. On a wretched table was a\nmeal, untouched. The light came in through the bars of a window near\nthe ceiling. It was a picture fit to make the bourgeois shudder, and\nthe bourgeois shuddered. Fougeres had simply been inspired by the\nmasterpiece of Gerard Douw; he had turned the group of the \"Dropsical\nWoman\" toward the window, instead of presenting it full front. The\ncondemned man was substituted for the dying woman--same pallor, same\nglance, same appeal to God. Instead of the Dutch doctor, he had painted\nthe cold, official figure of the sheriff's clerk attired in black; but\nhe had added an old woman to the young one of Gerard Douw. The cruelly\nsimple and good-humored face of the executioner completed and dominated\nthe group. This plagiarism, very cleverly disguised, was not discovered.\nThe catalogue contained the following:--\n\n     510. Grassou de Fougeres (Pierre), rue de Navarin, 2.\n     Death-toilet of a Chouan, condemned to execution in 1809.\n\nThough wholly second-rate, the picture had immense success, for it\nrecalled the affair of the \"chauffeurs,\" of Mortagne. A crowd collected\nevery day before the now fashionable canvas; even Charles X. paused to\nlook at it. \"Madame,\" being told of the patient life of the poor Breton,\nbecame enthusiastic over him. The Duc d'Orleans asked the price of\nthe picture. The clergy told Madame la Dauphine that the subject was\nsuggestive of good thoughts; and there was, in truth, a most satisfying\nreligious tone about it. Monseigneur the Dauphin admired the dust on\nthe stone-floor,--a huge blunder, by the way, for Fougeres had painted\ngreenish tones suggestive of mildew along the base of the walls.\n\"Madame\" finally bought the picture for a thousand francs, and the\nDauphin ordered another like it. Charles X. gave the cross of the Legion\nof honor to this son of a peasant who had fought for the royal cause\nin 1799. (Joseph Bridau, the great painter, was not yet decorated.) The\nminister of the Interior ordered two church pictures of Fougeres.\n\nThis Salon of 1829 was to Pierre Grassou his whole fortune, fame,\nfuture, and life. Be original, invent, and you die by inches; copy,\nimitate, and you'll live. After this discovery of a gold mine, Grassou\nde Fougeres obtained his benefit of the fatal principle to which society\nowes the wretched mediocrities to whom are intrusted in these days the\nelection of leaders in all social classes; who proceed, naturally, to\nelect themselves and who wage a bitter war against all true talent. The\nprinciple of election applied indiscriminately is false, and France will\nsome day abandon it.\n\nNevertheless the modesty, simplicity, and genuine surprise of the good\nand gentle Fougeres silenced all envy and all recriminations. Besides,\nhe had on his side all of his clan who had succeeded, and all who\nexpected to succeed. Some persons, touched by the persistent energy of a\nman whom nothing had discouraged, talked of Domenichino and said:--\n\n\"Perseverance in the arts should be rewarded. Grassou hasn't stolen his\nsuccesses; he has delved for ten years, the poor dear man!\"\n\nThat exclamation of \"poor dear man!\" counted for half in the support\nand the congratulations which the painter received. Pity sets up\nmediocrities as envy pulls down great talents, and in equal numbers.\nThe newspapers, it is true, did not spare criticism, but the chevalier\nFougeres digested them as he had digested the counsel of his friends,\nwith angelic patience.\n\nPossessing, by this time, fifteen thousand francs, laboriously earned,\nhe furnished an apartment and studio in the rue de Navarin, and painted\nthe picture ordered by Monseigneur the Dauphin, also the two church\npictures, and delivered them at the time agreed on, with a punctuality\nthat was very discomforting to the exchequer of the ministry, accustomed\nto a different course of action. But--admire the good fortune of men who\nare methodical--if Grassou, belated with his work, had been caught by\nthe revolution of July he would not have got his money.\n\nBy the time he was thirty-seven Fougeres had manufactured for Elie Magus\nsome two hundred pictures, all of them utterly unknown, by the help of\nwhich he had attained to that satisfying manner, that point of execution\nbefore which the true artist shrugs his shoulders and the bourgeoisie\nworships. Fougeres was dear to friends for rectitude of ideas, for\nsteadiness of sentiment, absolute kindliness, and great loyalty; though\nthey had no esteem for his palette, they loved the man who held it.\n\n\"What a misfortune it is that Fougeres has the vice of painting!\" said\nhis comrades.\n\nBut for all this, Grassou gave excellent counsel, like those\nfeuilletonists incapable of writing a book who know very well where a\nbook is wanting. There was this difference, however, between literary\ncritics and Fougeres; he was eminently sensitive to beauties; he felt\nthem, he acknowledged them, and his advice was instinct with a spirit\nof justice that made the justness of his remarks acceptable. After\nthe revolution of July, Fougeres sent about ten pictures a year to the\nSalon, of which the jury admitted four or five. He lived with the most\nrigid economy, his household being managed solely by an old charwoman.\nFor all amusement he visited his friends, he went to see works of art,\nhe allowed himself a few little trips about France, and he planned to go\nto Switzerland in search of inspiration. This detestable artist was an\nexcellent citizen; he mounted guard duly, went to reviews, and paid his\nrent and provision-bills with bourgeois punctuality.\n\nHaving lived all his life in toil and poverty, he had never had the time\nto love. Poor and a bachelor, until now he did not desire to complicate\nhis simple life. Incapable of devising any means of increasing his\nlittle fortune, he carried, every three months, to his notary, Cardot,\nhis quarterly earnings and economies. When the notary had received\nabout three thousand francs he invested them in some first mortgage, the\ninterest of which he drew himself and added to the quarterly payments\nmade to him by Fougeres. The painter was awaiting the fortunate moment\nwhen his property thus laid by would give him the imposing income of two\nthousand francs, to allow himself the otium cum dignitate of the\nartist and paint pictures; but oh! what pictures! true pictures! each a\nfinished picture! chouette, Koxnoff, chocnosoff! His future, his dreams\nof happiness, the superlative of his hopes--do you know what it was?\nTo enter the Institute and obtain the grade of officer of the Legion\nof honor; to side down beside Schinner and Leon de Lora, to reach the\nAcademy before Bridau, to wear a rosette in his buttonhole! What a\ndream! It is only commonplace men who think of everything.\n\nHearing the sound of several steps on the staircase, Fougeres rubbed up\nhis hair, buttoned his jacket of bottle-green velveteen, and was not a\nlittle amazed to see, entering his doorway, a simpleton face vulgarly\ncalled in studio slang a \"melon.\" This fruit surmounted a pumpkin,\nclothed in blue cloth adorned with a bunch of tintinnabulating baubles.\nThe melon puffed like a walrus; the pumpkin advanced on turnips,\nimproperly called legs. A true painter would have turned the little\nbottle-vendor off at once, assuring him that he didn't paint vegetables.\nThis painter looked at his client without a smile, for Monsieur Vervelle\nwore a three-thousand-franc diamond in the bosom of his shirt.\n\nFougeres glanced at Magus and said: \"There's fat in it!\" using a slang\nterm then much in vogue in the studios.\n\nHearing those words Monsieur Vervelle frowned. The worthy bourgeois drew\nafter him another complication of vegetables in the persons of his wife\nand daughter. The wife had a fine veneer of mahogany on her face, and\nin figure she resembled a cocoa-nut, surmounted by a head and tied in\naround the waist. She pivoted on her legs, which were tap-rooted,\nand her gown was yellow with black stripes. She proudly exhibited\nunutterable mittens on a puffy pair of hands; the plumes of a\nfirst-class funeral floated on an over-flowing bonnet; laces adorned\nher shoulders, as round behind as they were before; consequently, the\nspherical form of the cocoa-nut was perfect. Her feet, of a kind that\npainters call abatis, rose above the varnished leather of the shoes in a\nswelling that was some inches high. How the feet were ever got into the\nshoes, no one knows.\n\nFollowing these vegetable parents was a young asparagus, who presented\na tiny head with smoothly banded hair of the yellow-carroty tone that a\nRoman adores, long, stringy arms, a fairly white skin with reddish spots\nupon it, large innocent eyes, and white lashes, scarcely any brows, a\nleghorn bonnet bound with white satin and adorned with two honest bows\nof the same satin, hands virtuously red, and the feet of her mother. The\nfaces of these three beings wore, as they looked round the studio, an\nair of happiness which bespoke in them a respectable enthusiasm for Art.\n\n\"So it is you, monsieur, who are going to take our likenesses?\" said the\nfather, assuming a jaunty air.\n\n\"Yes, monsieur,\" replied Grassou.\n\n\"Vervelle, he has the cross!\" whispered the wife to the husband while\nthe painter's back was turned.\n\n\"Should I be likely to have our portraits painted by an artist who\nwasn't decorated?\" returned the former bottle-dealer.\n\nElie Magus here bowed to the Vervelle family and went away. Grassou\naccompanied him to the landing.\n\n\"There's no one but you who would fish up such whales.\"\n\n\"One hundred thousand francs of 'dot'!\"\n\n\"Yes, but what a family!\"\n\n\"Three hundred thousand francs of expectations, a house in the rue\nBoucherat, and a country-house at Ville d'Avray!\"\n\n\"Bottles and corks! bottles and corks!\" said the painter; \"they set my\nteeth on edge.\"\n\n\"Safe from want for the rest of your days,\" said Elie Magus as he\ndeparted.\n\nThat idea entered the head of Pierre Grassou as the daylight had burst\ninto his garret that morning.\n\nWhile he posed the father of the young person, he thought the\nbottle-dealer had a good countenance, and he admired the face full\nof violent tones. The mother and daughter hovered about the easel,\nmarvelling at all his preparations; they evidently thought him a\ndemigod. This visible admiration pleased Fougeres. The golden calf threw\nupon the family its fantastic reflections.\n\n\"You must earn lots of money; but of course you don't spend it as you\nget it,\" said the mother.\n\n\"No, madame,\" replied the painter; \"I don't spend it; I have not the\nmeans to amuse myself. My notary invests my money; he knows what I have;\nas soon as I have taken him the money I never think of it again.\"\n\n\"I've always been told,\" cried old Vervelle, \"that artists were baskets\nwith holes in them.\"\n\n\"Who is your notary--if it is not indiscreet to ask?\" said Madame\nVervelle.\n\n\"A good fellow, all round,\" replied Grassou. \"His name is Cardot.\"\n\n\"Well, well! if that isn't a joke!\" exclaimed Vervelle. \"Cardot is our\nnotary too.\"\n\n\"Take care! don't move,\" said the painter.\n\n\"Do pray hold still, Antenor,\" said the wife. \"If you move about you'll\nmake monsieur miss; you should just see him working, and then you'd\nunderstand.\"\n\n\"Oh! why didn't you have me taught the arts?\" said Mademoiselle Vervelle\nto her parents.\n\n\"Virginie,\" said her mother, \"a young person ought not to learn certain\nthings. When you are married--well, till then, keep quiet.\"\n\nDuring this first sitting the Vervelle family became almost intimate\nwith the worthy artist. They were to come again two days later. As they\nwent away the father told Virginie to walk in front; but in spite of\nthis separation, she overheard the following words, which naturally\nawakened her curiosity.\n\n\"Decorated--thirty-seven years old--an artist who gets orders--puts his\nmoney with our notary. We'll consult Cardot. Hein! Madame de Fougeres!\nnot a bad name--doesn't look like a bad man either! One might prefer a\nmerchant; but before a merchant retires from business one can never know\nwhat one's daughter may come to; whereas an economical artist--and then\nyou know we love Art--Well, we'll see!\"\n\nWhile the Vervelle family discussed Pierre Grassou, Pierre Grassou\ndiscussed in his own mind the Vervelle family. He found it impossible to\nstay peacefully in his studio, so he took a walk on the boulevard, and\nlooked at all the red-haired women who passed him. He made a series of\nthe oddest reasonings to himself: gold was the handsomest of metals; a\ntawny yellow represented gold; the Romans were fond of red-haired women,\nand he turned Roman, etc. After two years of marriage what man would\never care about the color of his wife's hair? Beauty fades,--but\nugliness remains! Money is one-half of all happiness. That night when he\nwent to bed the painter had come to think Virginie Vervelle charming.\n\nWhen the three Vervelles arrived on the day of the second sitting the\nartist received them with smiles. The rascal had shaved and put on clean\nlinen; he had also arranged his hair in a pleasing manner, and chosen\na very becoming pair of trousers and red leather slippers with pointed\ntoes. The family replied with smiles as flattering as those of the\nartist. Virginie became the color of her hair, lowered her eyes, and\nturned aside her head to look at the sketches. Pierre Grassou thought\nthese little affectations charming, Virginie had such grace; happily she\ndidn't look like her father or her mother; but whom did she look like?\n\nDuring this sitting there were little skirmishes between the family\nand the painter, who had the audacity to call pere Vervelle witty. This\nflattery brought the family on the double-quick to the heart of the\nartist; he gave a drawing to the daughter, and a sketch to the mother.\n\n\"What! for nothing?\" they said.\n\nPierre Grassou could not help smiling.\n\n\"You shouldn't give away your pictures in that way; they are money,\"\nsaid old Vervelle.\n\nAt the third sitting pere Vervelle mentioned a fine gallery of pictures\nwhich he had in his country-house at Ville d'Avray--Rubens, Gerard Douw,\nMieris, Terburg, Rembrandt, Titian, Paul Potter, etc.\n\n\"Monsieur Vervelle has been very extravagant,\" said Madame Vervelle,\nostentatiously. \"He has over one hundred thousand francs' worth of\npictures.\"\n\n\"I love Art,\" said the former bottle-dealer.\n\nWhen Madame Vervelle's portrait was begun that of her husband was nearly\nfinished, and the enthusiasm of the family knew no bounds. The notary\nhad spoken in the highest praise of the painter. Pierre Grassou was, he\nsaid, one of the most honest fellows on earth; he had laid by thirty-six\nthousand francs; his days of poverty were over; he now saved about ten\nthousand francs a year and capitalized the interest; in short, he was\nincapable of making a woman unhappy. This last remark had enormous\nweight in the scales. Vervelle's friends now heard of nothing but the\ncelebrated painter Fougeres.\n\nThe day on which Fougeres began the portrait of Mademoiselle Virginie,\nhe was virtually son-in-law to the Vervelle family. The three Vervelles\nbloomed out in this studio, which they were now accustomed to consider\nas one of their residences; there was to them an inexplicable attraction\nin this clean, neat, pretty, and artistic abode. Abyssus abyssum, the\ncommonplace attracts the commonplace. Toward the end of the sitting the\nstairway shook, the door was violently thrust open by Joseph Bridau; he\ncame like a whirlwind, his hair flying. He showed his grand haggard face\nas he looked about him, casting everywhere the lightning of his glance;\nthen he walked round the whole studio, and returned abruptly to Grassou,\npulling his coat together over the gastric region, and endeavouring, but\nin vain, to button it, the button mould having escaped from its capsule\nof cloth.\n\n\"Wood is dear,\" he said to Grassou.\n\n\"Ah!\"\n\n\"The British are after me\" (slang term for creditors) \"Gracious! do you\npaint such things as that?\"\n\n\"Hold your tongue!\"\n\n\"Ah! to be sure, yes.\"\n\nThe Vervelle family, extremely shocked by this extraordinary apparition,\npassed from its ordinary red to a cherry-red, two shades deeper.\n\n\"Brings in, hey?\" continued Joseph. \"Any shot in your locker?\"\n\n\"How much do you want?\"\n\n\"Five hundred. I've got one of those bull-dog dealers after me, and if\nthe fellow once gets his teeth in he won't let go while there's a bit of\nme left. What a crew!\"\n\n\"I'll write you a line for my notary.\"\n\n\"Have you got a notary?\"\n\n\"Yes.\"\n\n\"That explains to me why you still make cheeks with pink tones like a\nperfumer's sign.\"\n\nGrassou could not help coloring, for Virginie was sitting.\n\n\"Take Nature as you find her,\" said the great painter, going on with his\nlecture. \"Mademoiselle is red-haired. Well, is that a sin? All things\nare magnificent in painting. Put some vermillion on your palette, and\nwarm up those cheeks; touch in those little brown spots; come, butter it\nwell in. Do you pretend to have more sense than Nature?\"\n\n\"Look here,\" said Fougeres, \"take my place while I go and write that\nnote.\"\n\nVervelle rolled to the table and whispered in Grassou's ear:--\n\n\"Won't that country lout spoilt it?\"\n\n\"If he would only paint the portrait of your Virginie it would be worth\na thousand times more than mine,\" replied Fougeres, vehemently.\n\nHearing that reply the bourgeois beat a quiet retreat to his wife, who\nwas stupefied by the invasion of this ferocious animal, and very uneasy\nat his co-operation in her daughter's portrait.\n\n\"Here, follow these indications,\" said Bridau, returning the palette,\nand taking the note. \"I won't thank you. I can go back now to d'Arthez'\nchateau, where I am doing a dining-room, and Leon de Lora the tops of\nthe doors--masterpieces! Come and see us.\"\n\nAnd off he went without taking leave, having had enough of looking at\nVirginie.\n\n\"Who is that man?\" asked Madame Vervelle.\n\n\"A great artist,\" answered Grassou.\n\nThere was silence for a moment.\n\n\"Are you quite sure,\" said Virginie, \"that he has done no harm to my\nportrait? He frightened me.\"\n\n\"He has only done it good,\" replied Grassou.\n\n\"Well, if he is a great artist, I prefer a great artist like you,\" said\nMadame Vervelle.\n\nThe ways of genius had ruffled up these orderly bourgeois.\n\nThe phase of autumn so pleasantly named \"Saint Martin's summer\" was\njust beginning. With the timidity of a neophyte in presence of a man of\ngenius, Vervelle risked giving Fougeres an invitation to come out to\nhis country-house on the following Sunday. He knew, he said, how little\nattraction a plain bourgeois family could offer to an artist.\n\n\"You artists,\" he continued, \"want emotions, great scenes, and witty\ntalk; but you'll find good wines, and I rely on my collection of\npictures to compensate an artist like you for the bore of dining with\nmere merchants.\"\n\nThis form of idolatry, which stroked his innocent self-love, was\ncharming to our poor Pierre Grassou, so little accustomed to such\ncompliments. The honest artist, that atrocious mediocrity, that heart\nof gold, that loyal soul, that stupid draughtsman, that worthy fellow,\ndecorated by royalty itself with the Legion of honor, put himself under\narms to go out to Ville d'Avray and enjoy the last fine days of the\nyear. The painter went modestly by public conveyance, and he could not\nbut admire the beautiful villa of the bottle-dealer, standing in a park\nof five acres at the summit of Ville d'Avray, commanding a noble view\nof the landscape. Marry Virginie, and have that beautiful villa some day\nfor his own!\n\nHe was received by the Vervelles with an enthusiasm, a joy, a\nkindliness, a frank bourgeois absurdity which confounded him. It was\nindeed a day of triumph. The prospective son-in-law was marched about\nthe grounds on the nankeen-colored paths, all raked as they should be\nfor the steps of so great a man. The trees themselves looked brushed and\ncombed, and the lawns had just been mown. The pure country air wafted\nto the nostrils a most enticing smell of cooking. All things about the\nmansion seemed to say:\n\n\"We have a great artist among us.\"\n\nLittle old Vervelle himself rolled like an apple through his park, the\ndaughter meandered like an eel, the mother followed with dignified step.\nThese three beings never let go for one moment of Pierre Grassou\nfor seven hours. After dinner, the length of which equalled its\nmagnificence, Monsieur and Madame Vervelle reached the moment of their\ngrand theatrical effect,--the opening of the picture gallery illuminated\nby lamps, the reflections of which were managed with the utmost care.\nThree neighbours, also retired merchants, an old uncle (from whom were\nexpectations), an elderly Demoiselle Vervelle, and a number of other\nguests invited to be present at this ovation to a great artist followed\nGrassou into the picture gallery, all curious to hear his opinion of the\nfamous collection of pere Vervelle, who was fond of oppressing them with\nthe fabulous value of his paintings. The bottle-merchant seemed to have\nthe idea of competing with King Louis-Philippe and the galleries of\nVersailles.\n\nThe pictures, magnificently framed, each bore labels on which was read\nin black letters on a gold ground:\n\n               Rubens\n     Dance of fauns and nymphs\n\n                         Rembrandt\n     Interior of a dissecting room. The physician van Tromp\n             instructing his pupils.\n\nIn all, there were one hundred and fifty pictures, varnished and dusted.\nSome were covered with green baize curtains which were not undrawn in\npresence of young ladies.\n\nPierre Grassou stood with arms pendent, gaping mouth, and no word upon\nhis lips as he recognized half his own pictures in these works of art.\nHe was Rubens, he was Rembrandt, Mieris, Metzu, Paul Potter, Gerard\nDouw! He was twenty great masters all by himself.\n\n\"What is the matter? You've turned pale!\"\n\n\"Daughter, a glass of water! quick!\" cried Madame Vervelle. The painter\ntook pere Vervelle by the button of his coat and led him to a corner on\npretence of looking at a Murillo. Spanish pictures were then the rage.\n\n\"You bought your pictures from Elie Magus?\"\n\n\"Yes, all originals.\"\n\n\"Between ourselves, tell me what he made you pay for those I shall point\nout to you.\"\n\nTogether they walked round the gallery. The guests were amazed at the\ngravity in which the artist proceeded, in company with the host, to\nexamine each picture.\n\n\"Three thousand francs,\" said Vervelle in a whisper, as they reached the\nlast, \"but I tell everybody forty thousand.\"\n\n\"Forty thousand for a Titian!\" said the artist, aloud. \"Why, it is\nnothing at all!\"\n\n\"Didn't I tell you,\" said Vervelle, \"that I had three hundred thousand\nfrancs' worth of pictures?\"\n\n\"I painted those pictures,\" said Pierre Grassou in Vervelle's ear, \"and\nI sold them one by one to Elie Magus for less than ten thousand francs\nthe whole lot.\"\n\n\"Prove it to me,\" said the bottle-dealer, \"and I double my daughter's\n'dot,' for if it is so, you are Rubens, Rembrandt, Titian, Gerard Douw!\"\n\n\"And Magus is a famous picture-dealer!\" said the painter, who now saw\nthe meaning of the misty and aged look imparted to his pictures in\nElie's shop, and the utility of the subjects the picture-dealer had\nrequired of him.\n\nFar from losing the esteem of his admiring bottle-merchant, Monsieur\nde Fougeres (for so the family persisted in calling Pierre Grassou)\nadvanced so much that when the portraits were finished he presented them\ngratuitously to his father-in-law, his mother-in-law and his wife.\n\nAt the present day, Pierre Grassou, who never misses exhibiting at the\nSalon, passes in bourgeois regions for a fine portrait-painter. He earns\nsome twenty thousand francs a year and spoils a thousand francs' worth\nof canvas. His wife has six thousand francs a year in dowry, and he\nlives with his father-in-law. The Vervelles and the Grassous, who agree\ndelightfully, keep a carriage, and are the happiest people on earth.\nPierre Grassou never emerges from the bourgeois circle, in which he\nis considered one of the greatest artists of the period. Not a family\nportrait is painted between the barrier du Trone and the rue du Temple\nthat is not done by this great painter; none of them costs less than\nfive hundred francs. The great reason which the bourgeois families have\nfor employing him is this:--\n\n\"Say what you will of him, he lays by twenty thousand francs a year with\nhis notary.\"\n\nAs Grassou took a creditable part on the occasion of the riots of May\n12th he was appointed an officer of the Legion of honor. He is a major\nin the National Guard. The Museum of Versailles felt it incumbent to\norder a battle-piece of so excellent a citizen, who thereupon walked\nabout Paris to meet his old comrades and have the happiness of saying to\nthem:--\n\n\"The King has given me an order for the Museum of Versailles.\"\n\nMadame de Fougeres adores her husband, to whom she has presented two\nchildren. This painter, a good father and a good husband, is unable to\neradicate from his heart a fatal thought, namely, that artists laugh at\nhis work; that his name is a term of contempt in the studios; and that\nthe feuilletons take no notice of his pictures. But he still works on;\nhe aims for the Academy, where, undoubtedly, he will enter. And--oh!\nvengeance which dilates his heart!--he buys the pictures of celebrated\nartists who are pinched for means, and he substitutes these true works\nof arts that are not his own for the wretched daubs in the collection at\nVille d'Avray.\n\nThere are many mediocrities more aggressive and more mischievous than\nthat of Pierre Grassou, who is, moreover, anonymously benevolent and\ntruly obliging.\n\n\n\n\nADDENDUM\n\nThe following personages appear in other stories of the Human Comedy.\n\n     Bridau, Joseph\n       The Purse\n       A Bachelor's Establishment\n       A Distinguished Provincial at Paris\n       A Start in Life\n       Modeste Mignon\n       Another Study of Woman\n       Letters of Two Brides\n       Cousin Betty\n       The Member for Arcis\n\n     Cardot (Parisian notary)\n       The Muse of the Department\n       A Man of Business\n       Jealousies of a Country Town\n       The Middle Classes\n       Cousin Pons\n\n     Grassou, Pierre\n       A Bachelor's Establishment\n       Cousin Betty\n       The Middle Classes\n       Cousin Pons\n\n     Lora, Leon de\n       The Unconscious Humorists\n       A Bachelor's Establishment\n       A Start in Life\n       Honorine\n       Cousin Betty\n       Beatrix\n\n     Magus, Elie\n       The Vendetta\n       A Marriage Settlement\n       A Bachelor's Establishment\n       Cousin Pons\n\n     Schinner, Hippolyte\n       The Purse\n       A Bachelor's Establishment\n       A Start in Life\n       Albert Savarus\n       The Government Clerks\n       Modeste Mignon\n       The Imaginary Mistress\n       The Unconscious Humorists\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Pierre Grassou, by Honore de Balzac\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**EARLY BIRD BOOKS**\n\n**FRESH EBOOK DEALS, DELIVERED DAILY**\n\nLOVE TO READ?\n\nLOVE GREAT SALES?\n\nGET FANTASTIC DEALS ON BESTSELLING EBOOKS\n\nDELIVERED TO YOUR INBOX EVERY DAY!\n\nThe Galaxy's Greatest Newsletter  \nDelivered to Your Inbox\n\nGet awesome tales of fantasy and science fiction once a week.\n\nVisit us at www.theportalist.com\n\n# The Dark Imbalance\n\n## Evergence: Book 3\n\nSean Williams & Shane Dix\n\n__\n\n__\n\n_For Richard Curtis and Ginjer Buchanan, without whom this project would have remained forever incomplete._\n\"One knows what a war is about only when it is over.\"\n\nH. N. Brailsford\n\n__\n\n_\"Unser Leben geht hin mit Verwandlung.\"_\n\n(Our life passes in transformation.)\n\nRainer Maria Rilke\n**PART ONE:**\n\n****\n\n**SOL SYSTEM**\n\n****\n\n****\n****\n\n****\n\n**PROLOGUE**\n\nThe former COE Intelligence Head of Strategy didn't need to study her stolen fighter's instruments to know that something strange was going on in Sol System. Something strange and very unsettling.\n\nPage De Bruyn swung her fighter down into the plane of the ecliptic, braving a navigational nightmare as she went. The reopening of the Sol anchor point behind her had allowed\u2014and continued to allow\u2014a flood of vessels into the system. In the first few minutes, she catalogued fifty vessels whose design matched none in her records, and logged markings of fifteen new nations. None of them was the one she sought\u2014and she had barely touched the surface. According to the fighter's instruments, the total number of ships, stations, and launchers present in the system might well be on the order of several hundred thousand. Given that she hadn't properly surveyed the innermost and outermost extremes, she wouldn't be surprised if that figure doubled by the end of the day.\n\nPossibly a million ships, then, representing maybe tens of thousands of nations, near and far. She had heard of larger gatherings, but never in a solar gravity well. Even the combined fleet that had assembled in this very place to destroy the Sol Apotheosis Movement two thousand years earlier had, according to records, numbered barely ten thousand ships. Whether or not that record was accurate, she was now unsure, but the point remained: nothing like this had occurred in or near the Commonwealth of Empires before. And it would make finding her quarry that much more difficult.\n\nAs she skimmed the morass, she was scanned and hailed twice but not challenged. There didn't seem to be a central authority operating anywhere. The system was a mess. But the longer she looked at it, the more she realized that this might not be a bad thing after all. It might even work to her advantage. She could travel freely through it, confident that no one would notice a single fighter among the other ships. That was indeed a good thing, for the journey to Sol System had been long and exhausting, and she was going to need rest to prepare for the days ahead.\n\nShe had to work out what was going on, and how it related to an unaspiring orphan whom she appeared to have completely underestimated. And to do that, she needed to be closer to those who had spurned her.\n\nShe instructed the fighter to hunt for COE signals among the babble of transmissions filling the spectra around her. It wasn't a sophisticated craft, but it would do that for her. Once registered as TBC-14, she had renamed it _Kindling_ upon stealing it from Intelligence HQ. Although she was, theoretically, a fugitive from justice, in reality she had enough friends remaining in high places to divert attention from her, provided she didn't ruffle anyone's feathers too soon.\n\nThe time would eventually come, though, when she wouldn't care whom she offended or how she offended them. The question of why she had been so abruptly dismissed from her post in COE Intelligence was proving a vexing one, and one that became increasingly far-reaching the more she probed. She refused to let it go unasked.\n\nObtaining an answer was all that mattered to her, now. That, and revenge...\n\nSix hours after she had arrived in Sol System, _Kindling_ detected signals from a vanguard of the COE Advance Fleet. De Bruyn ordered the fighter to approach, carefully. She didn't know quite what to expect\u2014although, given the COE's proximity to Sol System, it was only natural to suppose that it would have a role to play in the emerging power base in the system, however small. That there would be such a power base before long she didn't doubt, for it was the nature of Humans to coalesce into groups. Maybe not one single group, but something larger than isolated clusters. Looking for such an emerging group in the obvious Pristine camp was something she was sure others would be doing also.\n\nWhether this focus of attention on the Advance Fleet would work to its advantage or detriment was difficult to tell. De Bruyn wasn't convinced the. COE Armada commanders had the ability to exploit such a situation properly. It needed someone with a flair for intrigue, someone prepared to be ruthless, someone who knew an opportunity when she saw it.\n\nShe smirked in the dim light of _Kindling'_ s cockpit. It would be the COE's loss, disposing of her the way they had. She would show them that she wasn't someone to be trifled with, to be used up and tossed away. She would pursue the mystery of her dismissal no matter where it led. And if it brought down the Eupatrid himself, then so be it. She would allow nothing and no one to come between herself and the answer...\n\nAnd Roche.\n\nThe thought of that name made her fists clench, as it always did. _Damn_ that woman! Roche had disobeyed her superior officers, jeopardized her mission as an Intelligence Field Agent, even caused a diplomatic incident over the theft of the _Ana Vereine\u2014_ and yet she had been allowed to walk away\u2014 _free._ And the sole person who seemed to care about righting this wrong was penalized for being \"unduly enthusiastic.\"\n\nDe Bruyn would give Burne Absenger\u2014chief liaison officer with the COE Armada\u2014 _unduly enthusiastic._ That she promised herself. She would expose the truth: a truth so large even _he_ would choke on it; a truth she sensed hiding deep in the data, deep in the mystery that was Morgan Roche.\n\nAll she needed was information. All she wanted was _proof._ No matter how long it took, she was dedicated to finding it.\n\nShe sent a coded message to a drone on the edge of the Advance Fleet. It relayed her message to a nexus deeper within the COE camp. There, her message triggered a coded response from a communications AI, which sent another message higher still in the command structure. From there, it was out of her hands\u2014but she was sure one of her contacts would see the message and work out what it meant. It was just a matter of tracing her message to its source. To her.\n\nIn the middle of the second largest fleet ever assembled by Humanity, she settled back to wait.\n\nAnd when, finally, _Kindling_ told her that it had recognized the distinctive camouflage signature of the _Ana Vereine_ as it entered the system, she clasped her hands together with something approaching eagerness. This was precisely what she had been hoping for. If Roche thought she could just walk in and throw everything into a spin to suit her own ends, whatever they were, she was about to be disappointed.\n\nDe Bruyn sent a brief, coded message to a Dato warship she had found lurking nearby, notifying it that the stolen property of its Ethnarch had arrived in the system.\n\nThen she settled back to see what happened next.\n****\n\n**1**\n\n****\n\n****\n\nCOEA Lucence-2\n\n955.1.29\n\n1860\n\nThe feet of Morgan Roche's suit came away sticky as she stepped across the bridge of the _Lucence-2_ toward the commander's chair. She stopped a meter from it, staring with a mix of apprehension and disgust at the fist-sized object lying on the brown-spattered cushion. She didn't need to touch it to know that it was organic.\n\n<It's a heart,> said the Box through her implants.\n\nShe nodded mutely as her gaze panned around the bridge, the light from her suit's helmet cutting through the dark to reveal the carnage: here, a dismembered body, there, walls splashed with swaths of blood. She couldn't smell the blood through the triple-thickness armor of her powered Dato suit, but she could imagine its stench.\n\n\"Commander Roche?\" The voice of the Basigo first officer crackled loudly in her ears, his accent as thick as that of a Hum peasant, and not dissimilar.\n\nShe didn't respond for almost thirty seconds; it took that long for her to find her voice\u2014and even then all she could manage was a grunt of acknowledgment.\n\n\"Commander?\" the first officer repeated.\n\n\"Forget the 'Commander,' \" she said. \"I'd prefer that you just call me by my name.\"\n\n\"Whatever,\" the voice shot back impatiently. \"Have you found what you were looking for?\"\n\nHer helmet light once again caught the organ in the commander's chair, and she winced. \"Yes and no,\" she said, turning from the disturbing sight. \"You say you intercepted this vessel on your last orbit?\"\n\n\"We were in close to the primary when it intersected our orbit. We hailed it, but it didn't respond. We thought it was a derelict, so we boarded it.\"\n\n_Looking for bounty,_ she didn't doubt.\n\n\"That's when we saw your name.\"\n\nShe nodded. She had seen it too, painted in blood on the wall in front of the main airlock, where no one could miss it. The fact that it was painted in letters six feet high made certain of that.\n\n\"And its orbit was highly elliptical?\" she said.\n\n\"Aye, that it was,\" he said. \"Would've swung past us and headed way out-system if we hadn't slowed it down a touch during docking.\"\n\n_Headed right for us,_ she concluded, privately. The Box had superimposed trajectories before she had come aboard. Barely had they arrived at Sol System's anchor point when the ship they were chasing had been hurled at the _Ana Vereine_ like an insult, filled with the blood of its crew.\n\nBut even if the Basigo scout hadn't intercepted it, Kajic would have seen the ship approaching long before it became a serious threat, and avoided it with ease. Such a crude tactic would never have worked. Roche knew that it was never intended to.\n\n\"Repeating herself,\" Ameidio Haid had said upon the discovery. Jelena Heidik, the clone warrior who had hijacked the _Lucence-2,_ had committed the same atrocity in Palasian System within days of her first awakening, that time to the crew of the _Daybreak._ \"Honing her skills,\" he added somberly.\n\nHeidik had gone on to single-handedly kill more than five hundred thousand people in Palasian System before escaping. Roche shuddered to imagine what she could accomplish here, in Sol System.\n\n\"It might be a trap,\" said Uri Kajic from the _Ana Vereine,_ on a channel the Basigo weren't listening to.\n\n<I sense no life.> Maii's words came from the same source but by utterly different means. The reave's voice sounded like a whisper in Roche's skull, as though the very cells of her brain were listening. It came with an image of a bone picked clean by the elements. <But that means little. Cane is closed to me now\u2014even his senses\u2014and Heidik could be doing the same. If she were standing next to you, I might not know.>\n\nRoche nodded, waiting to see if Cane himself would say anything, but he didn't. The clone warrior she had once been happy to call _ally_ \u2014who was at least distantly related to the woman Jelena Heidik\u2014had been reticent since his awakening from the coma in which he'd been imprisoned by Linegar Rufo. Under the circumstances, she wasn't sure she blamed him. Nevertheless, it still made her uneasy....\n\n\"We've lost her, haven't we?\" said Haid from elsewhere in the ship.\n\nRoche glanced at the pools of blood around the bridge. \"I think so,\" she said, unsure whether to feel relieved or piqued. The clone warrior presumably had more important things to worry about now that she was in Sol System. And Roche would have no chance of finding her unless Heidik chose to attack\u2014a notion she didn't particularly care to entertain.\n\nSwitching back to the Basigo channel, Roche came to a decision. \"We're going to disable all the drives except for attitude adjustment and program a warning beacon. It shouldn't be disturbed any more than it already has been. Do you agree to that?\"\n\n\"It's not my place to decide,\" said the first officer with some relief. \"They're your bodies, not mine.\"\n\n\" _My_... ?\" Roche started, a sick feeling rising in her stomach.\n\n\"Hey, they were addressed to you,\" he said. \"And that's good enough for me.\"\n\n* * *\n\nBy the time Roche and Haid returned to the _Ana Vereine,_ the Basigo ship had already gone, powering in-system on a torch of blue energy as though its crew was keen to put as much distance between it and the death-ship as possible. Roche could at least empathize with this. Behind her, the _Lucence-2_ had been scuttled with cold efficiency, its navigation AIs wiped. Its only remaining sign of life was the beacon, warning people away.\n\n\"Heidik knew we were following her,\" Roche said aloud as she stepped out of the back of her suit and down onto the rubberized floor of the changing room. The moment the suit was empty, it walked itself to an empty niche in the wall for recharging.\n\nHaid watched her from a bench in one corner, his dark skin and biomesh glistening with sweat. \"It couldn't just be a lucky guess?\"\n\n\"She wrote my name in six-foot letters on the bridge of that ship, Ameidio, using the blood of the people she'd murdered.\" Roche ran a hand across her stubbled scalp. \"Trust me, she knew we were coming after her, and exactly when we would arrive, too.\"\n\n\"She could have destroyed us if she'd really wanted to,\" Haid mused.\n\n\"But she didn't,\" said Roche. \"My name was written there for someone to find, and that wouldn't have happened if the ship had been destroyed.\" She slipped a loose top over her head. \"No, the _Lucence-2_ was only intended as a parting shot\u2014a spit in the eye.\"\n\n\"That's one hell of a spit,\" said Haid humorlessly.\n\nShe shrugged wearily, as though settling a burden on her back. \"Our options now are limited. We keep looking for her\u2014although just how we're going to do that, I don't know. Or we warn whoever's in charge to keep an eye out.\"\n\n\"You really think someone is in charge, here?\"\n\n\"Not yet. But that won't stop someone trying.\"\n\nHaid paused before saying: \"There's something I still don't understand, though, Morgan.\" He didn't wait for her to respond before continuing: \"How _did_ we know where she was going?\"\n\nRoche avoided meeting his eye. \"I told you, the Box talked about the gathering here before we left Palasian System. Before it was destroyed.\"\n\n\"Yeah, but how did _it_ know?\" said Haid. \"We could have been heading into a trap.\"\n\nRoche snorted. \"Didn't we just do that?\"\n\n\"You know what I mean,\" said Haid. \"The Box could have been sending us\u2014\"\n\nKajic's voice over the intercom interrupted him: \"Morgan, you're receiving another hail.\"\n\n\"Me specifically?\"\n\n\"Yes.\"\n\n\"I don't suppose the Basigo simply forgot something?\"\n\n\"No,\" said Kajic. \"It's a representative of the Eckandar Trade Axis in what looks like a Commerce Artel ship. They're radiating an impartial sigil, anyway.\"\n\n\"What do they want?\"\n\n\"They haven't said. I can open a line if you like.\"\n\n\"Give me a minute to get to the bridge.\" Roche indicated for Haid to come with her. He tossed the towel aside and followed her from the changing room, along a stretch of corridor and to an internal transit tube. Two harnesses awaited them there, ready to whisk them across the ship.\n\nNot that their physical presence was actually required on the bridge. The _Ana Vereine_ was as advanced as anything the Dato Bloc could build; in some areas it was even slightly ahead of the Commonwealth of Empires. Roche could run the ship in every respect from any point within it\u2014or beyond its hull, if necessary. But being at the heart of the ship helped her concentrate, she had found, and it was as good a place as any for everyone to gather.\n\nMaii was there when they arrived. So was Cane. The dark-skinned clone warrior watched impassively from where he stood off-center in the large room, facing the main screen. On it was an image of a ship: flat, petal-shaped, with a sheen to it like that of polished bone. There were no visible markings, although on ultraviolet a repeating pattern of symbols raced around the undulating rim. Artel sigils, as Kajic had already noted.\n\nThere was no obvious means of propulsion to the ship, but it advanced steadily toward them.\n\nThinking of Heidik, Roche said: \"Be careful, Uri. It could be a trap.\"\n\n\"I am battle-ready,\" said Kajic.\n\n\"I would not attack like this,\" said Cane, facing Roche. \"They are foolishly exposed. Until it is clear who are your enemies and who are your allies, it would be best to wait.\"\n\n\"Then what is it they want?\" asked Haid.\n\n\"Let's find out.\" Roche indicated for Kajic to open a line to the Artel ship. \"This is Morgan Roche of the vessel _Ana Vereine._ What is\u2014?\"\n\n\"Ah, Roche.\" The long, gray face of an Eckandi in middle age appeared on the screen. \"My name is Alwen Ustinik. I am sorry to trouble you, but, having been advised of your arrival, I thought it prudent to contact you as soon as possible.\"\n\n\"Advised? By whom?\"\n\n\"An associate. I do not speak for myself, of course. I am merely the representative of a number of interested parties. The Commerce Artel has many such representatives scattered throughout this system, as I'm sure you would expect. Even at a time such as this, the possibilities of trade are enormous. So many new contacts to make and avenues to explore...\"\n\n_She's trying to distract me,_ Roche realized. \"Get on with it, Ustinik.\"\n\nThere was a pause, then a smile. \"Naturally,\" Ustinik said. \"The people I represent have an interest in seeing justice served, as I'm sure you do too, Roche. When people are hurt, they desire recompense\u2014or, at the very least, a sense that some attempt at retribution has been made. How one dispenses punishment depends on one's society, of course, but there tends to be more overlap than dissent, I have found. The majority decides, and, where the justice system fails, it is often up to the Artel to facilitate corrective dialogue.\"\n\nRoche sighed. \"Can we get to the point here? I have no idea what it is you're talking about.\"\n\n\"I am talking about war, Roche,\" the Eckandi said evenly. \"The ultimate destabilization an economy can experience. Yes, it may have its short-term benefits, but in the long term it leads to nothing but hardship. The legacy of death and heartbreak is enduring; everyone pays in the end.\"\n\nRoche thought of the clone warriors, spreading dissent throughout the galaxy, and guessed that Ustinik had been sent to get her hands on Cane. Why? For a show-trial, perhaps, to suggest that her \"associates\" knew what they were doing. Or in a last-minute, desperate attempt to obtain information...\n\n\"I'm not turning him over,\" she said, despite her own misgivings about having him around.\n\n\"Please reconsider. I speak on behalf of those who have had the misfortune in the past to be on the receiving end of his business dealings. He is a mercenary and a terrorist who has not fully atoned for his crimes\u2014\"\n\n\"Wait a second.\" Roche gestured the other woman to silence. \"Are you talking about _Haid_?\"\n\nThe Eckandi frowned. \"Yes, of course.\"\n\nRoche frowned also. \"But what the hell would you want with him?'\n\n\"I am here to ensure his return to a corrective institution,\" said Ustinik, \"where the remainder of his sentence can be carried out.\"\n\nRoche was momentarily taken aback. \"His sentence was repealed by the High Equity Court\u2014\"\n\n\"Not formally\u2014and under some duress, if the information I have at my disposal is correct. I am told that, quite apart from the crimes committed before his capture, he was also the leader of a resistance movement on Sciacca's World, and that this movement overthrew the legally appointed warden of the planet.\"\n\n\"The warden was corrupt, and colluding with the Dato Bloc\u2014\"\n\n\"The Artel doesn't get involved in regional disputes, Roche.\" Ustinik's tone was calm but commanding; not once did her pitch rise, nor her face display any annoyance or anger. \"There is still such a thing as due process. My clients are dissatisfied with a pardon extracted at gunpoint. If they do not make an example of his flagrant disregard for the law, where will it end?\"\n\n\"It wasn't like that. If you'll let me explain\u2014\"\n\n\"No explanations are necessary,\" Ustinik cut in again. \"Or desired. To resist would only implicate yourself further, Roche.\"\n\n\"Are you threatening me?\"\n\n\"My clients' words, not mine.\" The woman's smile was economical and short-lived. \"I am a mediator, nothing more.\"\n\nRoche's fists clenched. \"And I have more important things to worry about.\"\n\n\"Regardless, the facts remain: you helped Ameidio Haid evade justice, and you continue to shield him from those who wish to see that justice served in full. I doubt they will smile on your venture, no matter how important you think it is. Turn him over to my custody, and you will have nothing further to worry about.\"\n\nAnger flared, but Roche kept it in tight check. \"Give me ten minutes to think about it.\"\n\n\"You have five.\" Ustinik killed the line without any change in facial expression.\n\n\"You should've asked her who she was representing,\" said Haid after a few moments.\n\n\"I was hoping you might be able to answer that one,\" said Roche.\n\n\"Well, there are a number of people it could be.\" The ex-mercenary shrugged. \"Maybe all of them. I was busy for a long time, Morgan.\"\n\n\"Great.\" Roche sighed. A representative of the Commerce Artel would be easy to ignore if the woman was on her own; but if some of her clients showed up to back her claim...\n\n<She represents several factions from the M'taio and Reshima Systems,> said Maii, <plus a venge-seeker from Imi.>\n\n\"You can read her?\" Roche asked.\n\n<Her shield is strong, but not leakproof.> The blind Surin smiled from her place in one corner of the bridge, black lips pulling back to crease her ginger-haired cheeks. <It's nice to be useful again.>\n\nRoche smiled also; she had missed Maii's input in Palasian System, where the reave's abilities had been dampened. \"How serious does she think her clients are? Are they prepared to use force if we don't give them what they want?\"\n\n<She believes so. She has been authorized to threaten us with it.>\n\nHaid hissed between his teeth. \"I should have known that i-Hurn thing was going to cost me one day.\"\n\n\"We're not handing you over,\" Roche said. \"It's not even an option. There must be some way to convince her to see reason.\"\n\n\"Will her side of the conversation be monitored?\" asked Cane.\n\n\"Probably,\" said Roche. \"Uri, can you detect any signals leaving her ship?\"\n\n\"None,\" said Kajic. \"But given the strong possibility that she would use a tightbeam, and the large amount of noise in this system, I doubt that I could detect anything at all.\"\n\n\"Then we'll have to assume that she's being monitored,\" Roche concluded. \"Which means we can't just blow her out of the sky.\"\n\n\"You'd really do that?\" asked Haid.\n\nRoche shrugged, and grinned. \"No, but it _is_ tempting.\"\n\nThey discussed a number of more or less fanciful options for several minutes, until Kajic interrupted with the news that he was receiving another hail.\n\n\"Our friend Ustinik again, I presume, telling us that time is up?'\n\n\"No, Morgan. It's coming from elsewhere.\"\n\n\"What?\"\n\n\"From a Surin _imaret_ closing in on our position, to be exact.\"\n\n\"I don't believe this,\" said Roche. \"We've been in-system just over half a day and we've already had one attempt made on our lives, one threat, and now...\" She shook her head. \"Put them through.\"\n\n\"Morgan Roche.\" The face of a large male Surin adult appeared on the main screen? \"I am Fighter-For-Peace Jancin Xumai. You have one of our citizens aboard your ship, and we request that she be returned to us.\"\n\nRoche was confused. \"Returned? Why?\"\n\n\"So that she may be reunited with her mother.\"\n\n_< No!>_\n\nRoche called out in pain as a bolt of anger and fear slammed into her mind. Clutching her head, her vision swimming with intense secondhand anxiety, she turned to face Maii. Through the discomfort she saw Cane move over quickly to the girl's side and take her shoulders in his large hands. A second later, as he eased her back into her seat, the debilitating emotions ebbed and died.\n\n<I'm sorry, Morgan. I\u2014I just...> The mental equivalent of tears soaked the girl's words, diluting her emotions.\n\n\"It's okay, Maii. I understand. It's all right. We're not going to let them take you. Did you hear that, Jancin?\"\n\n\"I advise against that course of action.\" The Surin's unnerved expression belied the threat in his words. Roche supposed that he had felt a backlash of the girl's epsense projection. \"The Surin Caste has a strong military presence in Sol System. Should you not comply with the wishes of the ruling Agora, I am instructed to call for backup.\"\n\n\"Then you'd better do just that,\" said Roche bluntly. \"Because we won't be surrendering her to you\u2014certainly not against her wishes.\"\n\n\"Her wishes are irrelevant,\" said Jancin. It is the mother's wishes, and that of the Agora, which are important here.\"\n\n<I have no mother,> said Maii. Her words were edged with bitterness, and Roche could feel the anger inside the Surin girl wanting to break free. <She sold me for medical experiments! These people are more family to me than she or the Agora could ever be!>\n\nIgnoring the girl's outburst, Jancin addressed Roche once more: \"I urge you to consider the implications of going against the Agora. They only want the girl; they do not wish you or your crew any harm.\"\n\n\"No,\" said Roche. \"No one ever does, yet everyone keeps threatening us.\"\n\nShe killed the line before Jancin could speak again, then turned to Cane.\n\n\"Thanks,\" she said to him. The clone warrior nodded a brief acknowledgment.\n\n\"We can't take on the Surin as well,\" said Kajic, his hologram appearing on the bridge.\n\n\"And we can't give them what they want, either.\" Roche tapped the arm of her chair. \"Maybe this is what it's all about. Uri, have any other ships changed course to intercept us?\"\n\n\"It's hard to say, Morgan.\" Kajic called up a display of the portion of the system surrounding the _Ana Vereine._ Even in that small bubble of space, there were over fifty ships following a wide variety of vectors and ranging in size from small, anonymous fighters to bulky cruisers. The display was awash with energy and particulate wakes. As Roche watched, a new cluster of six medium-sized attack craft appeared, following a high-energy elliptical orbit around the system's sun; who they were, Roche didn't know, nor did she care. All that mattered was that they weren't homing in on her ship.\n\nKajic ringed three craft. \"There is a Dato pursuit vehicle that seemed to react to our appearance an hour ago, but so far has not displayed any hostile intentions. This ship, here, which I have not been able to identify, is almost certainly following us. And this one\"\u2014the third ship was a stationary speck in the center of the swirl of orbits\u2014\"has done nothing at all.\"\n\n\"Trying to remain inconspicuous?\" suggested Haid.\n\n\"Trying a little _too_ hard,\" said Cane.\n\n\"Exactly my feeling.\" Roche turned to the young epsense adept. \"Maii? Anything?\"\n\n<This Jancin Xumai wants me because the Agora has heard about the experiments that gave me full epsense ability.> The girl's voice still had a thick edge to it. <They want to know how it worked.>\n\n\"Is that what you're reading from him?\"\n\nThe girl hesitated. <No. I'm assuming, and nothing he's giving me contradicts it.>\n\nRoche could understand her suspicions, but wanted hard facts, not suppositions. \"What _is_ he giving you?\"\n\n<He knows that we've been in Sol System for three hours, and he knew which region of the system to stake out. His orders came from his superiors, though, so he doesn't know where that information came from originally.>\n\nRoche nodded. \"What else?\"\n\n<There's a lot of overlap in this system; it's hard to isolate individual thoughts without knowing precisely who to focus on....> Roche appreciated the girl's difficulties, reaching out across space, clutching at any thought that seemed important out of the millions flung her way. <But I do sense a growing awareness of you, Morgan. Word is spreading.>\n\n\"What do you mean?\"\n\n\"Don't be modest, Morgan,\" Haid put in lightly. \"You've made a lot of enemies in the last few weeks. It's only natural they're going to be talking about you.\"\n\n<Wait... There's something else...> Maii frowned and fell silent.\n\nAnd into the silence came a new voice, a voice that resounded through their minds with discomforting familiarity:\n\n<You have given my people everything we needed.> The speaker was a strong but faceless epsense presence. <For that, we will permit your existence.>\n\nOn the main screen, the stationary dot suddenly moved to a new course, away from them.\n\n\" _Now_ what?\" asked Roche, increasingly bewildered.\n\nMaii's voice was hushed. <Grayboot.>\n\nHaid stiffened over the weapons board. \" _Olmahoi_? Here?\"\n\n<He's leaving us now.> The girl's relief was touched with an underlying fear. <He read me without me even knowing! He knows what happened in Palasian System; he knows about the _irikeii.... >_\n\n\"Great,\" said Roche dryly, rubbing at her forehead. The _irikeii_ \u2014linchpin of the epsense-dependent Olmahoi Caste\u2014had been killed by a representative of the Kesh. If the grayboot had suspected that they were involved\u2014and why else would he have tracked them down so quickly?\u2014they were lucky to have escaped some sort of automatic reprisal. The Olmahoi retribution squads weren't known for their patience.\n\nStill, Roche thought, having her brain instantly fried might just solve her problems right now....\n\n\"Ustinik is hailing us again,\" said Kajic. \"As is the Surin.\"\n\n\"Okay.\" Roche sat forward. \"Uri, take us somewhere else\u2014somewhere a _long_ way from here, and as fast as possible.\"\n\n\"In-system?\"\n\n\"Yes, but make it hard for someone to follow, without being too obvious about it. Use camouflage if you think it will help. Ustinik might be bluffing, and so might the Surin. Either way, I don't like being an open target.\"\n\nRoche felt a gentle thrum through her fingertips and thighs as the ship broke orbit.\n\nShe waited a moment, then checked the main screen. Kajic's words only confirmed what she saw.\n\n\"Ustinik is changing course, at a discreet distance, and continuing to hail us. The Surin _imaret_ has broken off communications and is heading away. That Dato ship I mentioned is still keeping quiet, but looks like it's going to follow too. There is another ship...\" Kajic ringed a newcomer to the screen. \"It's a COE fighter we passed before. Might be tagging along for the ride as well.\"\n\nRoche used her controls to expand the view and scan the regions ahead of them. There were ships everywhere\u2014all moving in wildly varying directions with dangerously different velocities, all orbiting the yellow star at the heart of the system. She was glad it was Kajic, and not her, piloting the ship.\n\n\"No sign of the Kesh?\" she asked.\n\n\"None yet.\"\n\n\"Good.\" That was one less thing to worry about. If the Olmahoi were annoyed at the Kesh for killing the _irikeii,_ she was sure the Kesh would be just as annoyed with her for having destroyed one of their prized ships.\n\n<Box? Any thoughts?>\n\n<I think you correctly summarized the situation a moment ago, Morgan.> The voice of the AI whispered solely through her implants. Now that she was becoming used to the idea that it was actually part of her, living in her cells, she found its voice less discomfiting. It was almost like hearing another part of herself think.\n\n<Someone is attempting to draw attention to you,> it said. <Any hopes you might have had of slipping unnoticed into Sol System and quietly going about your investigation have been effectively dashed. You are exposed, compromised, threatened. If, somehow, you were to\u2014>\n\n\"Another hail,\" Kajic interrupted the voice only Roche could hear. \"Another new one, I mean.\"\n\nShe shook her head. \"Who now?\"\n\n\"Assistant Vice Primate Rey Nemeth of the Second Ju Mandate, according to his ID.\"\n\n\"I don't recognize the name.\" She glanced about the bridge; no one volunteered anything. \"I suppose he's following us, too?\"\n\n\"No. He's coming in on a relay.\"\n\n\"Ignore him, then. Now\u2014\" She stopped herself in time and subvocalized: <Now, Box, what were you saying?>\n\n<Nothing of any great importance. For now I suggest you simply continue your avoidance tactics. We will talk later.>\n\nShe took that as a sign that, at least in the Box's eyes, she wasn't doing anything outrageously wrong. That made a nice change.\n\n\"Uri, ignore further hails, unless you think it's something particularly important. We've got better things to do than listen to other peoples' grievances.\"\n\nHaid grinned wryly. \"You figure we have so many enemies already that making a few more won't make much difference?\"\n\n\"That, and I'm loath to believe _anyone_ at the moment. If, as we think, the clone warriors are interested in infiltrating and stirring up dissent, then they could be anywhere. Who's to say which complaint is legitimate and which a trap? I'd prefer not to take the risk either way. And anyway, it'll be easier for us to keep dodging than it will be for someone to catch us, no matter how many of them there are.\"\n\nCane nodded. \"True.\"\n\nRoche turned to face him. \"And while Uri, Maii, and Ameidio see to that, maybe you and I should take the opportunity to have a private talk.\"\n\nCane shrugged. \"Whatever you say, Morgan.\"\n\n\"Good.\" Roche stood. \"I like the sound of that.\"\n\n* * *\n\nIn the small room at the rear of the bridge, Roche sat in a chair opposite the large hologram emplacement where Uri Kajic had once projected his image. On a display she studied a detailed image of Sol System composited from old map records and incoming data. She had lost count of the number of ships they'd passed since leaving the anchor point, but the Box estimated that around seven hundred Castes were represented in various forms\u2014from the fringe-lovers out where a comet cloud might once have once been to the hot-bloods in close. The sun had seen better days; there was evidence of large-scale waste-dumping in its outer atmosphere\u2014unsurprising, she thought; it had to go _somewhere\u2014_ but thankfully no one had tried any tricks such as the Kesh had in Palasian System. One system utterly destroyed in a month was more than enough for the region.\n\nNot that there was much to lose. Discounting the ships, the system was mostly empty. There was a faint but well-defined ring around the sun, approximately half a million kilometers in width and less than a thousand thick, just straddling the regions that might have been mundane-habitable had there been a planet to live on. Apart from the ring and the ships, the system contained nothing but vacuum. Anything larger than a pebble had been stripped back to molecules long ago, leaving behind only a wisp of smoke around the system's star.\n\nIf the system _had_ ever been inhabited\u2014let alone the birthplace of Humanity, as a few scholars had once suggested\u2014nothing remained to show it.\n\nRoche watched the endlessly chaotic dance of ships for a long moment, wondering who was in them and what they wanted. Then she turned to Cane.\n\nHe sat opposite her, his expression unreadable. The overhead light reflecting off his scalp made it look as if he had a third eye.\n\n_Appropriate,_ she thought.\n\n\"You wanted to talk to me,\" he prompted.\n\nShe paused, wondering, then asked: \"Are you reading my mind?\"\n\n\"Why do you ask that?\"\n\n\"Just answer me, Cane.\"\n\n\"No,\" he said. \"I'm not reading your mind.\"\n\n\"Could you, if you wanted to?\"\n\nHe frowned. \"Morgan, why are you asking me these questions?\"\n\nShe held his gaze for a moment, then let it wander back to the screen. \"On the way here, I talked to Maii. She told me in detail everything she'd picked up from the _irikeii_ before he died. She says...\" Roche sought the words, not sure she herself understood everything the young reave had told her. \"She says that the _irikeii_ was like a pit, sucking in thoughts. For him, minds were lights, or suns, and he was the black hole dragging them in. He experienced the universe through the minds around him, like a reave but with less selectivity; he experienced everything at once, all at once\u2014which was why the Kesh and Linegar Rufo had him kidnapped. Once Palasian System had been enclosed he was able to search it thoroughly. And nothing could hide from him.\"\n\n\"Not even a clone warrior,\" said Cane.\n\nRoche nodded. \"In theory.\"\n\n\"It makes sense,\" Cane went on. \"Had it worked, the advantage might have outweighed the inevitable backlash.\"\n\n\"It _did_ work. To the _irikeii,_ Jelena Heidik and you stood out like supernovae, by far the brightest things he had ever seen. He called you 'The Shining Ones.' \"\n\n\"We radiate thought,\" mused Cane. \"Is that what he meant?\"\n\nShe studied him closely; he was still frowning, although now apparently at the puzzle posed by the _irikeii,_ not at her. \"Possibly,\" she said. \"But we have no evidence to back it up.\"\n\n\"So...?\"\n\n\"So there's more to it than that.\" Roche leaned forward slightly in her seat. \"Maii says that one of the _irikeii's_ last impressions was of your mind while under the influence of Xarodine. He was aware of a dark space behind the glare\u2014a dark space similar to the one inside his own mind. He thought you and he might have had a lot in common.\"\n\n\"I don't see how that follows.\"\n\n\"Obviously the metaphor is strained.\" She couldn't tell if Cane was prevaricating. \"As far as I can understand it, he thought that you too could absorb thoughts from the people around you. You're a sponge, soaking everything up. And the glare he described\u2014\"\n\n\"Was just a form of camouflage?\" Cane finished. \"Something to hide our epsense ability?\"\n\nRoche nodded slowly. \"Something like that, yes.\"\n\n\"I am unaware of any such ability, Morgan,\" Cane said evenly.\n\n\"But how can I be sure you're telling the _truth_? How do I know you're not reading my mind right now?\"\n\n\"Because I give you my personal assurance, Morgan.\"\n\nShe studied him for a few moments. He was perfectly still, hands folded in his lap, eyes not leaving hers for an instant. Even at rest, the air of strength remained with him. She had seen how fast he could move; she knew what he was capable of. And having witnessed what his siblings could do if they turned against the people around them, she was reluctant to trust him without reservation. She needed reassurance.\n\n\"That's all well and good,\" she said, \"but I still can't help wondering. Heidik knew we were coming here; she even knew when. I can't believe it was just a good guess\u2014so who told her? The COE squadron we left behind at Palasian System might have sent word to expect us, but how would she have got hold of that information? We were less than a day behind her. That's not long enough to infiltrate the COE presence here. We haven't even _found_ them yet.\n\n\"And I keep thinking of that dark speck\u2014and Maii. She's proof that epsense ability can be bioengineered. If you _were_ made to blend in and to fight, what better way could there be to gather intelligence than to act as an _irikeii_ \u2014passively absorbing data from the minds of the people around you? Even if you couldn't actually read minds, you could at least see and hear through them\u2014and maybe even communicate with others like yourself. If these black specks linked up somehow, you could share information, talk, plan, whatever you needed without anyone knowing.\"\n\n\"Yes\u2014 _if_ ,\" said Cane. \"But ask yourself this, Morgan: if I _was_ in touch with Heidik or any of the other clone warriors, why would I be here? My siblings were clearly made for a purpose; they have spread across the galaxy seeding dissent and destruction wherever they go. But I have not. So _why_ would I bother with you if I shared their goals? Why would I not be with _them_?\"\n\n\"Evidence of absence is not absence of evidence.\"\n\n\"Guilty until proven innocent?\" Cane smiled slightly. \"I'm surprised at you, Morgan.\"\n\n\"There's too much at stake to take chances, Cane.\"\n\n\"The only way to be sure is to take no chances at all.\" His smile disappeared and he relaxed back into his seat. \"Space me, or imprison me\u2014get me out of the way entirely. Better to do that than to be perpetually in doubt. The chance that I might betray you will never leave your mind until I am gone.\"\n\nShe nodded. That had occurred to her. For all the times he had saved her on Sciacca's World, the casual cruelty and treachery of Jelena Heidik had tainted him in her eyes. She would never be free of it.\n\nAnd if he ever _did_ betray her, she stood to lose everything.\n\nNot just her life, but the lives of her companions and every other Pristine in the galaxy.\n\nShe wasn't sure she had the right to take that chance.\n\n<What do you think, Box?>\n\n<I know that if he could read minds, he would know that I continue to exist,> said the AI smoothly.\n\n<What about using the command language Rufo was using? Could we force the truth out of him?>\n\n<If we knew how to speak the language properly, we might\u2014 assuming, of course, that he isn't already telling the truth. But the exact use of the command language eludes even me; it seems a fairly crude device. I suspect it may relate to the capsules in which the clone warriors were born rather than the finished product. The best we could hope for is to nudge him to do our will.>\n\nRoche fought a twinge of annoyance. Why did things have to be so uncertain all the time? She would welcome a single, uncomplicated fact with open arms.\n\n<Besides,> added the Box, <if we use the command language, correctly or not, he will suspect that I gave you the knowledge to use it.>\n\n\"Morgan?\"\n\nShe looked over at Cane to see him watching her suspiciously.\n\n\"Are you all right?\" he said.\n\nShe brushed his suspicions aside by ignoring the question altogether: \"If you _were_ reading my mind, you'd know that spacing you isn't an option. And as there's no way you can prove conclusively that you're not in communication with the other clones, then all I can do is follow my gut instincts.\"\n\nCane nodded. \"But again, I assure you that I am telling you the truth, Morgan.\"\n\nShe remembered the words he had tapped in code shortly after he wakened from his coma: _I'm as Human as you are_. That was patently untrue in the details\u2014after all, she was not a genetically engineered combat soldier designed to blend in with the Pristine Humans and kill them\u2014but in essence it might not be far from the truth.\n\n\"Okay.\" She exhaled slowly. \"I'll believe you. Commander Gent must have sent word to his superiors after we left Palasian System. It's possible that Heidik already had a contact in the COE Armada, through which she found out when we were due to arrive. That's the only alternative I can think of.\"\n\n\"It is certainly a less speculative hypothesis.\" There was a glint of humor in Cane's eyes. \"But if the rest of my siblings _do_ communicate by epsense, I would be keen to find out why I have been excluded from the conversations.\"\n\nThat was a point Roche had not missed. \"Maybe something went wrong with you: your capsule was damaged, or corrupted. There might even be others out there like you\u2014others who could help us \"\n\nBefore Roche could pursue the thought, Kajic's image appeared in the empty corner of the suite.\n\n\"My apologies again, Morgan.\"\n\nShe swiveled to face him. \"Problems?\"\n\nHis broad, pleasant face was concerned. \"I have been ignoring hails as you instructed, although their numbers are increasing\u2014as is the number of ships following us. There are eight currently matching our course, and I have detected emissions from another five suggesting that they might also attempt to do so. No one has actually made a move against us; although it is difficult to project the precise makeup of the regions ahead of us, I am doing my best to keep us out of any regions where forces are massing. But we can't keep this up indefinitely; sooner or later, we _will_ miscalculate.\"\n\nRoche could see what Kajic was saying: they ran the risk of running headlong into a trap. \"So you suggest we stop running?\"\n\n\"No, Morgan. There's something else.\" Kajic changed the view in the screen to a recent telemetry display. The eight ships tailing them were marked clearly in red; a handful in yellow were the ones he suspected were about to join the convoy. As Roche watched, one green dot darted into view from off-screen, angling down and toward her to match velocities with the white _Ana Vereine_ at the center.\n\n\"Is that real-time?\" she asked.\n\n\"Yes. This happened only a minute ago. I should point out that I am currently accelerating at seventy percent of my design tolerance.\"\n\nThe green dot braked effortlessly to a relative halt a hundred kilometers away. \"What the hell is it?\" Roche asked.\n\n\"A large drone or singleship. I'm not familiar with the design or its markings. But we clearly can't outrun it.\"\n\n\"Has it tried to contact us?\"\n\n\"Not yet.\"\n\n\"Are we camouflaged?\"\n\n\"Mildly, only in order to give the appearance of trying. Our position has been well-known since we arrived and we are currently too well-observed to successfully drop out of sight.\"\n\nShe nodded. \"Drop the pretense, then. Hail that drone, or whatever it is. I want to talk to it.\"\n\nKajic's hologram abruptly dissolved.\n\nRoche stood, and Cane followed her out of the small office. \"As you said, Morgan: absence of evidence is not evidence of absence. Just because they haven't threatened you yet doesn't mean that they won't.\"\n\n\"At the moment, that's good enough for me.\" She assumed her usual station at the first officer's post. \"Maii? Anything?\"\n\n<The craft is either unoccupied or its occupants are extremely well-shielded,> said the reave.\n\n\"I have a lock on it,\" said Haid. \"Its E-shields and hypershields are down.\"\n\n\"I doubt it's defenseless,\" Roche said, watching a close-up of the craft on the main screen. It resembled a mushroom in shape: flat, circular cap with a trailing stem five meters long and two meters wide. There were no visible drive outlets or weapons ports. \"Whoever it belongs to, they're more advanced than us.\"\n\n\"I have a reply,\" said Kajic. \"The drone is a relay.\"\n\n\"Open a direct line. Let its source talk to me.\"\n\nSeconds later, a voice issued from the bridge's speakers:\n\n\"Welcome to Sol System, Morgan Roche.\" The female voice was precise and clipped, and unfamiliar. \"The Interim Executive Pristine Council has been expecting you.\"\n\n\"You're not the only ones, it seems.\"\n\n\"Your arrival has created something of a disturbance. As the news spreads, we expect the situation to worsen.\"\n\n\"Meaning?\" Roche wished for an image to give her something to focus on.\n\n\"In case you failed to notice, the atmosphere in this system is somewhat tense. There have been many skirmishes in the last few days\u2014even several attempts at outright war. As we speak, Olmahoi forces are preparing to engage the Kesh\u2014acting on information you brought with you. You are a catalyst, Roche, a destabilizing influence. The council would ask you to restrict your activities before you cause more damage.\"\n\n\"Is that a request or an instruction?\"\n\nThe woman's voice sounded amused. \"It is an appeal,\" she said, \"to your better judgment.\"\n\nRoche was silent for a moment. \"Perhaps you should tell me who you are and what exactly the council is.\"\n\n\"It might be easier to demonstrate,\" she returned. \"Turn your instruments to the following coordinates...\"\n\nKajic swung the view on the main screen accordingly, but only starlight dusted the empty space.\n\n\"There's nothing there,\" said Roche irritably.\n\n\"Give the light a chance to reach you,\" said the woman.\n\nEven as she spoke, something appeared on the screen. It looked like a ship, but the perspective was all wrong. Where a dot might represent other craft, there glowed a tiny arrowhead.\n\n\"Whatever it is,\" said Kajic, magnifying the view, \"it's millions of kilometers away.\"\n\nThe display was suddenly taken up with a huge vessel, and Roche found herself gasping at its immensity. It was shaped like a long cone flattened on one side, hollow at tip and base and bristling with instruments and weapons emplacements\u2014some as large as the _Ana Vereine_ itself. It had to be at least a thousand kilometers long and as much as one hundred and fifty wide; it made COE Intelligence HQ look like a drone.\n\n\"You're seeing the _Phlegethon_ ,\" said the woman. \"It's a consistory vessel of the Skehan Heterodox. You have been invited to approach.\"\n\nRoche stared at the screen a moment longer. The name meant nothing to her. \"Why?\" she said eventually.\n\n\"To discuss the situation,\" the woman replied. \"For the duration of those discussions, at least, we can offer you our protection.\"\n\n\"Again: why?\"\n\nThe woman hesitated slightly, as though Roche's suspicion annoyed her. \"The IEPC exists to assess the threat presented by the clone warriors you seek. To do that, we must gather as much information as possible. Contacting you is an important part of that process. Understand, Roche, we are not asking you to join forces; we are not asking you to surrender control to us. We ask merely to exchange information, in return for which we will get your pursuers off your back.\"\n\nRoche hesitated, thinking of the Surin backup Jancin Xumai had threatened her with, and the Kesh, and the Commerce Artel, and Jelena Heidik\n\n<Morgan,> said the Box, <I urge you to comply.>\n\n<I thought you might>\n\n<They are exactly what we need: a significant presence in this system with the resources to investigate it properly. You will note that she is communicating with the drone by faster-than-light methods. Access to such technology will make our quest that much easier.>\n\n<And what exactly _is_ our quest, Box? I'm still a little uncertain about that.>\n\n<All will become clear soon enough, Morgan. I suggest you take the opportunity given to us and see what happens.>\n\nThe AI was making sense. Any group with a ship that big would be a fair contender for the role of central authority in the system\u2014and she had to take on allies sooner or later. She couldn't do it on her own.\n\n\"Roche? Are you still there?\" The woman's voice sounded more amused than concerned.\n\n\"I was just considering your offer.\" Roche glanced at Haid, who shrugged: _Your decision_... \"Very well. We agree to talk, at least.\"\n\n\"Good. I will instruct the Heresiarch to give you an approach vector to match orbits. We will contact you again when everything is in order.\"\n\nBefore Roche could reply, the woman had cut the line. A moment later, the drone accelerated impossibly fast, outward, away from the sun and away from them. Within seconds, it was gone.\n****\n\n**2**\n\n****\n\n****\n\nAVS-44\n\n955.1.30\n\n0640\n\nSomething stirred inside Roche as they approached the _Phlegethon._\n\nFor the most part, the uneven surface of the giant ship's hull was bleak and lifeless, with only the occasional beacon sporadically flaring in the darkness. But as they moved along the length of the _Phlegethon_ 's vast exterior, a patch of quivering energy some fifty kilometers wide followed, lighting up the ship's black, moist-looking skin. There were no windows of any description to be seen, yet Roche couldn't help but feel she was being watched. Not by the ship's instruments perhaps, but by the ship itself. It seemed... _alive_ to her. And the _Phlegethon_ 's only identifying mark, a mural of a giant, half-lidded eye on the flattened underbelly of the beast, only enhanced that feeling.\n\nShe suppressed a shudder as the bulk of it passed between her and the distant, yellow sun. This close to the monster craft, she felt intimidated, ineffectual. Worse, she felt vulnerable.\n\n\"The ships following us have broken off,\" said Kajic via the scutter's intercom. \"Whatever the IEPC said to them, it's had the desired effect.\"\n\nShe could tell by the tone of his voice that he wasn't saying what was foremost on his mind. \"You think I'm doing the wrong thing, don't you?\"\n\nHe didn't reply immediately. When he did, he sounded almost relieved. \"Yes, Morgan.\"\n\nHe didn't need to say anything more than that. Roche could see his reasoning: after Linegar Rufo had captured half her crew by luring them into his station in Palasian System, Kajic had every right to point out that she might be making the same mistake twice\u2014and for similar reasons, too. But those reasons were sound, and they outweighed any risks to her personally.\n\nShe needed information, first and foremost, and she had information others might find useful. She had to take the chance that this Interim Executive Pristine Council\u2014whatever _that_ was\u2014 was in the system to help, not hinder.\n\nAnd she had Maii with her, as well as the Box, hidden away in her flesh. Haid hadn't liked being left behind, but he could see that Roche needed the sort of help only a reave might be able to provide. If it was a trap, then she was never going to be able to fight her way out of the _Phlegethon_ by force alone.\n\n\"Trim,\" said the traffic controller, guiding her in a perfunctory, almost disinterested manner.\n\nShe concentrated on flying the scutter. It had drifted slightly off course. She corrected easily, following the trajectory she had been given to three decimal places.\n\n\"We don't have a better option at the moment,\" she told Kajic.\n\n\"I know,\" he replied. \"That's the main reason I've kept silent.\"\n\nThe scutter arced gracefully toward an open dock two thirds from the hollow tip of the _Phlegethon_ to its base. A line of docks encircled the ship, one every fifty meters. Roche performed the arithmetic in her head: assuming the ring went right around the ship, that made almost a thousand docks in that band alone, and she could see several more bands in either direction along the hull. She could only wonder why they needed so many. Fighter launchers, perhaps?\n\nThere was no denying the sophistication of the vessel. How far it had come was still unknown, but she had no doubt it belonged to an empire of similarly spectacular proportions.\n\n\"You getting anything from the crew, Maii?\" Roche turned unnecessarily to the girl. \"Any clues as to where they're from?\"\n\n<Not much,> said the girl. She sat next to Roche in the copilot's station, wearing an undersized hazard suit that brought her up to Roche's height and twice her thickness. Inside she would be safe from Xarodine or any other physical anti-epsense attacks. Roche wore a simple environment suit in Dato colors with a bare minimum of ceramic armor and an energy pistol at her side. <What I am getting suggests that the ship _is_ their home. If there was a place prior to this, it didn't leave much of an impression on them. Most are simply focused on their jobs or their daily lives.> She paused for a few seconds. <And there's a lot of prayer going on, too\u2014perhaps unsurprising given there are several million people on board.>\n\n\"Prayers? To whom?\"\n\n<No one I've ever heard of,> said Maii. <Whoever it is, they don't seem to be answering.>\n\nRoche smiled. \"What about the Interim Executive Pristine Council? Anything there?\"\n\n<There are a number of chambers warded by high-grade reaves toward the top of the cone. I haven't been able to worm my way in yet. My bet is the council work is done there, where it's most secure.>\n\n\"Well, keep trying,\" said Roche. \"And let me know if you learn anything important.\"\n\n<I will, Morgan.>\n\nRoche eased the scutter into the large dock, bringing it to a halt in exactly the spot indicated. There followed a series of clangs and small bumps; then the traffic controller spoke again.\n\n\"You're docked,\" he said. \"Praise Weryn, and welcome aboard the _Phlegethon.\"_\n\n\"Thanks.\" Roche unclipped her harness and stepped from the couch.\n\n\"Air outside is normal,\" said Kajic. \"And so far our transmissions aren't being interfered with.\" He still sounded concerned.\n\n\"Good. I think we're going to be okay, Uri.\"\n\n\"You'd better hope so, Morgan,\" Haid put in over the open line. \"Because if something _does_ go wrong, I don't fancy our chances of getting you out of there.\"\n\n\"Personally, I don't give you any chance at all. Not against this thing.\" Roche forced herself to sound casual. \"But let's hope it doesn't come to that.\"\n\n<You could say a prayer,> said Maii. <That's what most people here would do.>\n\n<Mine would just be another voice among the millions, then,> said Roche. <A sigh in a storm.> She let the hint of a smile carry with her reply. <But you feel free to try, Maii, if you think it will make a difference...>\n\n* * *\n\nA tall woman with a solid build met them outside the airlock bay. She was dressed in a sky-blue uniform that seemed part robe, part jumpsuit. It was hard to tell where the folds of fabric stopped or started. Her face was long and strong-boned, her chin curved and slightly protruding.\n\n\"My name is Hue Vischilglin,\" she said, taking both of Roche's hands in hers and pressing them to her forehead. She repeated the ritual with Maii, when Roche introduced her. The young reave, made awkward by the hazard suit, bowed slightly in return. \"Be welcome here.\"\n\n\"Thank you,\" said Roche distractedly, glancing along the empty, curving corridor that connected all the various docks on the inside of the ship. It was so long that the air blurred the details in the distance, and so wide that, with gravity pointing down away from the center of the ship, it almost appeared flat. She shook her head. \"I never expected... _this_!\"\n\n\"Few do.\" Vischilglin smiled warmly and gestured for them to follow her across the plain toward a distant pillar. There was no one else in sight. \"The Heterodoxies have come from the Far Reaches on the other side of the galaxy. They've known about the problem longer than most, and have possibly suffered its worst effects. This ship is all that's left of one of their fleets. Its Heresiarch\u2014its 'captain'\u2014rebelled when he was ordered to destroy a civilian outpost inhabited by several billion people. It would seem his superiors had been infiltrated by the enemy. He managed to escape reprisal and kept on running. Eventually he was contacted by others in similar situations and directed here.\n\n\"Like some of the other outermost Castes, their greater lead-in time has given him more chance to prepare for being here. On the other hand, his crew is exhausted from having come so far. That's probably why they're being so open-minded about the council running the show.\" She smiled widely. \"Although I suspect they were as glad to get their hands on our ftl relays and advanced camouflage as much as we were glad to get our hands on such a figurehead. What a beast, eh? And to think this was just one ship from one of the Heterodox fleets!\"\n\n\"So you're not one of them?\"\n\n\"Oh, no,\" she said, surprised by Roche's misunderstanding. \"I'm from the Rond-Spellor Outlook, myself.\" Catching Roche's reaction, she went on with even more surprise: \"You've heard of us! That makes us practically family around here.\"\n\nThey reached the pillar, which turned out to be much thicker than Roche had first imagined; the lack of perspective was playing tricks on her eyes. Vischilglin waved a hand across a black panel and it slid silently open, revealing an elevator cab.\n\nRoche hesitated outside. \"Where are you taking us?\"\n\n\"For debriefing,\" said Vischilglin. \"Don't worry; you won't come to any harm.\"\n\n\"Sentiments I have had expressed to me in the past,\" said Roche cynically, then added: \"No offense.\"\n\n\"None taken, I assure you,\" said Vischilglin.\n\n\"I just want my crew to know, that's all.\"\n\nVischilglin nodded. \"We're aware that you're in contact with them; we wouldn't have it any other way.\" Vischilglin stood on the threshold. \"Is there anything we can do to put your mind at ease?'\n\nRoche shook her head slowly. \"I'm just habitually nervous these days, that's all.\"\n\n\"As you should be. I'm taking you to the secure areas on level 391. Your reave would have noted them already, I'm sure. We keep them shielded as best we can to keep word getting out. Maybe it's effective; maybe it's not. Either way, we have to try. But we're not keeping secrets from our allies. That would be counterproductive. We're just trying to maintain security against our common foe.\"\n\n\"And do you know who _they_ are?\"\n\nVischilglin grimaced. \"If you mean do we know their origins or the identities of the individuals, then no, I'm afraid not. But we are hoping you might be able to help us.\" She indicated the interior of the elevator. \"Won't you?\" she said. \"They're waiting.\"\n\nRoche forced herself to ignore the nagging uncertainty and stepped into the cab. Besides, what choice did she really have? If they wanted to spring a trap, then her position was already so compromised that she wouldn't be able to do anything about it, anyway.\n\nMaii followed her in. As the doors closed, Vischilglin turned to the girl with an amused expression.\n\n\"You know, you're free to remove that suit any time you like,\" she said. When Maii didn't respond she added: \"I hate those things. Too confining, constricting\u2014and they _chafe._ We have more suitable clothing if you're uncomfortable.\"\n\n<Are there any of my people where we're going?> Maii asked. Vischilglin looked uneasy for a moment, and Roche suspected the girl had known the answer before she'd asked. \"There is one, yes,\" Vischilglin replied. <Then I am quite comfortable as I am, thank you.> Roche felt the slight tickle in her mind that meant the Box wanted to talk to her. <What is it?>\n\n<I thought you might like to know that the Rond-Spellor Outlook has been in a state of civil war for some weeks, now.>\n\n<You think they've been infected by the clone warriors?>\n\n<Most likely. Or at the very least, _af_ fected.>\n\n<Could she be one of them?>\n\n<I have no reason to believe so. Neither her name nor her appearance match any in my database, and one must assume that any organization devoted to the investigation of the clone warriors would take precautions against such an infiltration. Nevertheless...>\n\nThe Box left the sentence unfinished, but the sentiment was clear.\n\n<Consider me warned,> she said.\n\n<I have every confidence in your abilities, Morgan.>\n\n<Let's hope I can live up to your expectations,> she said. <For both our sakes.>\n\nThe elevator didn't seem to have moved, but when the doors opened a second or two later, an entirely different vista was spread out before them. Water from gentle waterfalls washed down numerous curved walls into undulating ground between them, collecting in valley floors to form small, slow-moving streams which curled and divided in unpredictable directions, some emptying into numerous ponds scattered about the area. The air was moist and sweet\u2014scented, Roche suspected, by the various plants growing in the waters.\n\nThe banks of the waterways, however, were gray and sterile\u2014a striking contrast to the exotic flowers and reeds. And high above it all hung featureless white clouds. The vista gave Roche the impression of an attempt at terraforming by a clerical AI.\n\nShe moved out of the elevator. \"Is this the right level?\" she asked.\n\n\"Incredible, isn't it?\" Vischilglin stepped up beside Roche. \"The waterways erode giant, mazelike circuits around the ship. Given enough time, the Heterodoxies believe they will one day spell out the name of God. Or something like that.\" Vischilglin shrugged helplessly. \"It all sounds like nonsense to me. Yet I can't help admiring it whenever I see it.\"\n\nShe led them through the strange landscape, across modest but elegant bridges and along the narrow valleys. As they climbed over each rise, Roche could clearly discern the curve of the floor beneath them; they were obviously higher along the cone than they had been before.\n\nShe realized then that nowhere on their journey had they seen another person.\n\n<Are we blocked here, Maii?>\n\nThe head of Maii's suit rose when she spoke via epsense, uncannily as though she were looking at Roche. The visor was black, however, and the girl had no eyes to see with behind the white bandage she wore across her face; she was using Roche's eyes to guide herself.\n\n<We have passed through several barriers,> she said, <and are now... Well, the best analogy I can think of is that we have reached the eye of a storm. I am not being interfered with, although I am finding it difficult to reach the outside world.>\n\n<Can you pick up anyone else near here?>\n\n<There is a large group up ahead,> Maii said. <They seem to be waiting for us.>\n\n<Is it the council she was telling us about?>\n\n<I think so. Look...> Roche received a mental impression of many minds congregated in one place, focusing intently on one thing. She couldn't make out any individuals in that crowd, but she sensed their combined will. <I suspect this might be an illusion,> the girl went on. <A powerful one. I can't break through it.>\n\n<Interesting that they've made the effort,> said Roche. They reached a flight of stone steps that twisted and turned around a sharp rise in the landscape like a thread around a screw.\n\n\"The place we're going is known as the _fane_ ,\" Vischilglin said, pausing at the base of the steps. \"You and I would probably call it the ship's bridge, but that doesn't do it justice.\" She hesitated for a moment, then went on: \"The Heterodox are great believers in ritual. There is some protocol you'll need to observe. When you reach the nave, in the center of the fane, bow to the Heresiarch\u2014you'll see me do it ahead of you, so you'll know who he is. When you're asked to speak, always address at least part of your reply to him. He may not speak directly to you, but if he does, look him right in the eye. Should you hear bells at any point, be prepared for everything to stop. That means the ship requires his attention.\"\n\nRoche nodded her understanding, and Vischilglin began their ascent up the broad and shallow steps. After a while, cloud obscured not only their destination above, but also the area around them. It was composed of thick and surprisingly dry mist that smelled of ozone and left no residue as they passed through it.\n\nRoche followed Maii, allowing the girl to use her eyes to navigate her way up the stairs. With each step the girl took, the suit struck sparks from the stones, but she expressed no discomfort to Roche.\n\n\"Not far now,\" said Vischilglin.\n\n<Kajic, are you tracking us?> Roche asked via her implants. <Yes,> he replied, <and monitoring via Maii's suit.>\n\n<No problems at your end?>\n\n<We're still holding in the position they gave us. There have been no other attempts to approach us. I suspect we are being camouflaged; we've fired a couple of base-line probes away from our position, but so far none have reported back. They seem to keep losing our location.>\n\n<At least you're safe,> said Roche.\n\n<If a little frustrated,> he replied. <I envy Ameidio: he can sleep at times like these.>\n\nRoche smiled. <Years of practice, I guess.>\n\n<It wouldn't be so bad if Cane was better company.>\n\nHer smile slipped a notch. <Listen, make sure\u2014>\n\n<I know, Morgan,> he cut in. < I'm keeping an eye on him for you. He hasn't done anything suspicious, and if he did, I would notify you immediately. But I don't think he will. He knows he's being watched.>\n\n<He also knows you're going to have to sleep _some_ time. You're not the Box.>\n\n<Don't worry about me, Morgan. Just take care of yourself, okay?>\n\nHer smile returned. Under the concern in his voice she heard a genuine warmth. If they had become friends in the weeks since she'd taken control of his ship, then that was all to the better. It took some of the edge off the uncertainty she felt about her situation.\n\nRoche's first feeling as she emerged from the cloud a few minutes later and looked out over the vast bridge\u2014the _fane,_ she reminded herself\u2014was relief that it had been the Dato Bloc she'd fought on Sciacca's World and not the Skehan Heterodox.\n\nShe was standing in the middle of a wide, concave space carved out of what looked like dark gray stone. This space was one of many\u2014like the petals of a flower\u2014abutting a central bowl almost two hundred meters across. The bowl was stepped in the fashion of an ancient amphitheatre, but with no sharp edges; everything was rounded, molded\u2014smooth, perhaps, from the generations of people that had sat on those seats and worn them down. A few were occupied now, as were spaces in the petals, where people stood rather than sat and observed what was happening in the bowl. At the bowl's center was a rough-hewn font filled with water.\n\nRoche looked up. If symmetry was anything to go by, local gravity had taken a turn through ninety degrees in the clouds. Far above, hanging from the central point of a convex roof was a slender spike, pointing downward like a stiletto poised to strike. Its tip burned white, with enough light to cast a shadow from everything it illuminated below. Roche guessed that the spike and the font at the center of the bowl delineated the long axis of the ship.\n\nVischilglin led her along a short walkway through the petal, and down, toward the central bowl. When they stepped across its lip, the woman stopped and turned to face a man dressed in gold, who stood on the far side.\n\nShe bowed. Assuming this man to be the Heresiarch they'd been told to watch for, Roche bowed also. Beside her, Maii did the same.\n\n\"Morgan Roche wishes an audience with the Heresiarch.\" Vischilglin, speaking in a voice only slightly louder than normal, gestured toward Roche.\n\n\"Bring her down.\"\n\nRoche couldn't tell who had spoken, yet the voice was as clear as if it came from someone standing directly beside her. The Heresiarch didn't appear to have moved.\n\nThey descended step by step into the heart of the central bowl\u2014the nave, Vischilglin had called it. When they reached the lowest circle, they stopped and waited. Even at the edge of the nave, the font was still some distance away.\n\nOnly when they came to a halt did the voice speak again: \"Do you know who we are?\" Roche was still uncertain as to who had spoken, but she knew it was directed at her.\n\nShe looked around. Apart from the Heresiarch in his gold attire, nobody else stood out. Most wore white robes or shipsuits; only a few, like Vischilglin, wore blue. All were watching Roche, waiting on her reply. She didn't dare presume that the Heresiarch was the one who had spoken, so when she did reply it was to the space in general: \"No.\"\n\nIt was a few moments before the speaker continued, and when he did, the words still seemed to issue from everywhere at once: \"Five hundred thousand years ago, more or less, Humanity diversified to the point where its origins were forgotten.\" The man spoke slowly and with a crisp, nasal tone. \"Only the dimensions and attributes of the Pristine form remained known. In order to ensure that the cause of the Pristine would never be lost among those of the other mundane Castes, the framework for a council was established\u2014a council that would surface from obscurity _only_ when it was needed. All Pristine governors of all Pristine governances know how to summon the council into being, and all know that to do so improperly would have its... consequences.\" The word was chosen carefully. \"Only the gravest of circumstances can justify such a summoning\u2014as, for example, when the genetic code of our distant ancestors becomes threatened.\"\n\n\"But this is not such an occasion, is it?\" said Roche. The silence which followed was filled with unspoken disapproval for her interruption.\n\n\"This council,\" continued the voice shortly, \"was called forty-six months ago, and is now in full session.\"\n\n\"Forty-six _months_?\" Roche exclaimed, not caring whose sensibilities she offended. She wanted answers, not speeches.\n\nMovement to her right caught her eye as a figure in blue took a step toward her. She interpreted it as a warning against further interruptions, and ground her teeth together.\n\n\"We have been aware of this threat for that long. Only recently, however, did we learn about Sol System. Our data showed an apparent convergence upon this region, although not enough on its own to fix the location precisely. An attack on a nearby system helped us triangulate traffic among the civilizations we've been keeping an eye on, suspecting them to be corrupted. We were among the first to arrive here, barely a week ago.\"\n\nThe figure to Roche's right shifted once again.\n\n\"The speed with which word has spread is phenomenal,\" the speaker continued. \"Ships continue to arrive at the rate of over one hundred every hour. We have reopened several secondary anchor points on the fringes of the system, to act as exits should congestion worsen. If that is not enough, we might have to close the main anchor point altogether. That way, only the most determined will be able to come here.\"\n\nThe figure in blue took several more steps forward, close enough now so that Roche could make out the face of a man, the blue-white light from the spike above casting deep shadows in the lines of his aging features. He was the one talking, not the Heresiarch.\n\n\"The situation here is approaching a watershed,\" he said. \"The council senses a change coming, but does not know what form it will take, or to what purpose it comes. Some of us suspect that you might lie at the heart of it, Morgan Roche, and believe that you can help us with an answer to this question. Will you do so?\"\n\n\"Of course,\" she said without hesitation. Looking at the Heresiarch, she added: \"After all, That's why I'm here.\"\n\nShe saw Vischilglin nod approvingly as she turned back to the speaker.\n\n\"I am Esko Murnane,\" he continued. \"My superiors in Pompili sent me as their plenipotent envoy to the council, and the council in turn has declared me chairperson for this hearing. You have already met Hue Vischilglin, co-adjutant to the leaders of the Rond-Spellor Outlook. Although a minimum of thirty Pristine nations are required to allow the full and proper council to sit, at present we number four hundred and seven. All have representatives here today, although few, if any, will be known to you. We will, therefore, forgo introductions for the time being. Should you be asked to join our cause, the identities of your questioners will become known to you then.\"\n\nAgain, Roche nodded. \"I understand.\"\n\n\"Good. You stand before the council as a witness to the aftermath of the atrocity that recently occurred in Palasian System, and as someone who appears to have a deeper association with the enemy than most of us here.\" The slow steadiness of his speech combined with what he was saying lent Murnane an air of deep, long-standing authority. \"All of us have been touched by the enemy, in one way or another, to our detriment and lasting regret. So we are keen now to hear all that you have learned.\"\n\nHe paused and looked around the enormous chamber, his eyes eventually finding their way back to Roche. When he spoke, they remained upon her, but his words were directed to everyone present.\n\n\"Who will begin?\" he said.\n\n\"I will.\" The voice came from the far side of the chamber. Another male, but younger, and fair complexioned. \"Each of the many nations in the council was drawn here under a different pretext, none seemingly more convincing than any other. We hope to find one that predominates, for that one might contain a shred of truth. By what name do you refer to the enemy, Morgan Roche?\"\n\n\"At first,\" she said, speaking slowly and clearly, addressing her reply equally between her questioner, Murnane, and the Heresiarch, \"we thought they were Wunderkind created by the Sol Apotheosis Movement. They had a base in this system, a couple of thousand years ago\u2014\"\n\n\"We are familiar with their history,\" the speaker interrupted. \"So, have you ascertained another name for them now?\"\n\n\"No,\" said Roche. \"I'm afraid not.\"\n\n\"We are told that you have one of the enemy aboard your ship.\"\n\n\"Yes, we do.\"\n\n\"And what does he have to say on the matter?\"\n\nRoche shook her head. \"Nothing.\"\n\nAnother voice spoke, this time a woman to Roche's left: \"But he _does_ have a name?\"\n\n\"Yes,\" said Roche. \"His name is Adoni Cane.\"\n\n\"A name of your choosing?\" said the woman.\n\n\"No, it's what he called himself when we first met. I've never had cause to doubt him. Later it produced a match in Dato Bloc's historical records, confirming a link to the Sol Apotheosis Movement.\"\n\n\"Which later turned out to be spurious?\"\n\nRoche nodded.\n\n\"How do you account for that?\"\n\n<Answer that question carefully,> the Box cautioned.\n\nShe frowned, fighting her automatic urge to answer with the truth. The AT had faked the historical data in order to mislead the COE and other neighboring governments\u2014and also to throw any of the \"enemy\" off the trail. If the enemy knew how close the High Humans behind the Crescend and the Box were getting\u2014even if it wasn't very close at all\u2014it might work to their advantage.\n\nThe fact that it still might, in the midst of the Interim Emergency Pristine Council, gave her cause to reconsider.\n\n\"Would you like the question repeated?\" said Murnane.\n\n\"No, that's okay,\" she said. \"I guess I can't account for the discrepancy. Maybe the data was deliberately corrupted by the enemy in order to throw us off the trail.\"\n\n<Well done, Morgan,> enthused the Box.\n\n\"That is certainly a possibility,\" said Murnane, coming forward. \"There is a risk of infiltration and perversion at every level. I fear we have not yet seen the full extent of the enemy's abilities or motivations. Until we do, we must assume the worst\u2014 even of ourselves.\"\n\n\"Has Adoni Cane ever revealed any detail regarding his origins?\" The speaker, another woman, was very close and directly behind Roche.\n\nShe turned toward the voice, but was unsure which of the many faces looking back at her had asked the question. \"He seems to have no knowledge of his origins,\" she said, addressing them all. \"He doesn't know where he came from or why he's here.\"\n\n\"You're saying he has no memory?\" This time Roche saw who had spoken: a young girl, tall and thin, with flaxen hair brushing the shoulders of her blue robe.\n\n\"Everything since his awakening is clear,\" said Roche. \"But nothing before then.\"\n\n\"And you are convinced he is telling the truth?\"\n\nShe hesitated, remembering her most recent conversation with Cane. \"I trust him as much as I can,\" she said. \"Under the circumstances.\"\n\n\"Because he claims to be one of the enemy?\"\n\n\"Yes. That is, he talks about them as if they are his siblings; he shares certain characteristics with them.\"\n\n\"What characteristics, precisely?\"\n\n\"Well, his genetic profile is profoundly abnormal,\" she said.\n\n\"And his body is patently modified in order to make him a good soldier. I haven't seen hard data on others like him, but I do know that if he set his mind to it, he'd be more than capable of the same destructive force that they have displayed. And when in Palasian System he did respond to a command language understood by the other clone warriors\u2014\"\n\nMurnane held up a hand. \"We will return to Palasian System in a moment,\" he said. \"First we'd like to hear how you met up with this Adoni Cane, and what you have observed about his behavior to date.\"\n\nShe took a moment to organize her thoughts, then began to talk\u2014describing succinctly how she and Cane had met on the _Midnight,_ how they had escaped and crash-landed on the surface of Sciacca's World, and their pursuit and eventual escape from the penal colony.\n\n\"He helped you escape?\" The question was from another council member whose thick accent was unfamiliar to Roche; she had to concentrate to understand what he was saying. \"From prison wardens corrupted by a rival government? Do you know _why_ he did this?\"\n\n\"No,\" she said, with a shake of her head. \"And I have to admit that it's puzzled me.\"\n\n\"Can you explain why his behavior is so different from the others?\"\n\nShe shrugged lightly. \"The best explanation I can come up with is that he's a freak,\" she said. \"A mistake.\"\n\n\"You mentioned genetic data, earlier,\" said one of the previous speakers, the man with the fair complexion. \"Will you give us access to this data?\"\n\n\"Gladly,\" she said. \"If I may contact my ship...\"\n\n\"Your lines of communication are not being interfered with in any way,\" said Murnane.\n\n<Uri?> she said, checking to see if this was true.\n\n<We're still receiving you loud and clear,> Kajic replied. <I have all the info the Box compiled before it was destroyed. Do you want me to send it?>\n\n<Yes, but I don't know where, though.>\n\n<They've allocated me a buffer,> he said. <Leave it to me, Morgan.>\n\nShe was about to turn back to Murnane when she remembered Maii's suit standing immobile beside her. <You okay, Maii?> she asked via epsense.\n\n<I'm fine,> came the reply. <I'm having fun poking at shields. Did you know that there are five high-grade reaves within the nave alone? They're the hooded ones around the edge. I've never heard of so many being in one place at the same time before!>\n\n<What are they doing? Probing me without my knowledge?>\n\n<I wouldn't let them do that, Morgan,> the girl reassured her. <I think they're just trying to make sure I don't get up to anything.>\n\nMurnane cleared his throat. \"Thank you,\" he said. \"We have received the data and will examine it later.\" He folded his arms and took a couple of thoughtful paces around the font. \"But I am curious. At the time Cane was examined on the _Midnight,_ news had not yet reached your corner of the galaxy that there even was a problem he might be part of, otherwise his capsule would have been instantly identified. And on Sciacca's World, your rebel friends had access to even more limited information about the outside world. Yet our sources in the Commonwealth of Empires reveal that in a very short space of time you determined precisely what was going on\u2014bearing in mind the Sol Apotheosis Movement fallacy\u2014and confronted your superiors with that knowledge. When was it that you managed to piece it all together?\"\n\nRoche opened her mouth to speak\u2014then shut it again. If they had sources in the COE, chances were they already knew the answers to every question they had asked so far. So why go through the motions?\n\nThen she reminded herself: trust no one. They could no more believe their sources than they could believe her\u2014even if one corroborated the other.\n\nShe didn't envy them their position.\n\n\"It wasn't me so much who put it all together.\" She half expected a nagging voice in her ear telling her to be careful what she said. \"It was the Box.\"\n\n\"What is this 'Box'?\" It was asked in the same thick, unfamiliar accent as before, except that this time the questioner was female. \"I take it you are referring to some sort of intelligence- gathering device?\"\n\n\"An AI, yes.\" Roche nodded. \"I was carrying it to Intelligence HQ when I was intercepted by the Dato Bloc. That's how I ended up on Sciacca's World in the first place.\"\n\n\"This device reasoned that Adoni Cane was one of the enemy?\"\n\n\"Yes,\" she said. \"And everything afterward seemed to confirm it.\"\n\n\"How was this device able to do something you yourself were unable to do?\"\n\n\"The Box was no ordinary device,\" Roche said, remembering to use the past tense. \"It was a truly remarkable piece of engineering. It suspected from the very start who Cane was. It even faked the distress call that led to the capsule's discovery.\"\n\n\"So it had access to information which you did not?\"\n\n\"Yes, like the command language. But it wasn't just that. It actually thought better than I did.\"\n\n\"Impossible. No AI has yet surpassed a Human intelligence.\"\n\nRoche shrugged. \"I told you it was remarkable.\"\n\n\"And who built this amazing device?\"\n\n\"It was manufactured on Trinity,\" said Roche. \"They specialize in AIs there.\"\n\nThere was a muted whisper. Then Murnane spoke. \"We have no record of such a place.\"\n\n\"No?\" She looked around and out of the corner of her eye caught the gold robes of the Heresiarch. She had forgotten he was there. \"Go ask your sources,\" she said. \"They'll confirm it exists.\"\n\nMurnane stirred. \"What say you, Trezise?\"\n\nStartled first by the familiar name, Roche almost jumped as a familiar voice followed: \"We know the place. It's administered by a High Human we have had some dealings with\u2014an entity calling himself the Crescend.\" The man's voice was flat, emotionless, almost dead. \"The AI Roche refers to did indeed come from this place, but as to its other abilities...\"\n\nSalton Trezise, Roche remembered\u2014senior aide to Auberon Chase, head of COE Intelligence. She should've guessed someone like him would be here.\n\n\"You are not aware of any facility capable of making Human- superior AIs in COE jurisdiction?\" Murnane pressed.\n\nTrezise's tone didn't change as he said: \"I'd sooner believe in aliens.\"\n\nMurnane turned back to Roche. \"You will understand if we hesitate to accept this aspect of your story without any hard evidence to back it up,\" he said. \"Unless you could produce this AI for us to examine, perhaps?\"\n\nShe didn't need the tiny prod the Box gave her. \"I'm afraid it was destroyed along with Palasian System.\"\n\n\"I see.\" A sigh carried his words. \"Well, the exact manner of your discovery of the enemy is not the issue here. What is important is the fact that you learned of their existence and went seeking more data. What can you tell us about Adoni Cane that we have not already covered?\"\n\n\"The Box thought we should check the introns of Cane's genetic code,\" she said. \"But I don't know what for.\"\n\nMurnane nodded as though the suggestion was trivial. \"And your young charge here.\" He pointed to Maii. \"Does she have nothing to contribute to this discussion?\"\n\n<Maii?>\n\n<Ask him how I'm supposed to talk with everyone's minds walled up,> the girl shot back.\n\nMurnane raised a hand before Roche could pass the message on. \"Simply speak to me,\" he said, \"as you would to Roche, and a relay will announce the message for all to hear.\"\n\n<Oh. Only\u2014>\n\n\"\u2014words?\"\n\nRoche heard the girl's voice directly through her own senses and a split second later through the relay, aloud. The relay stood on the far side of Roche; it was disconcerting to hear the girl's voice coming from two directions almost simultaneously.\n\n\"And appropriate images, where necessary.\" Murnane inclined his head in welcome. \"Please feel free to share with us any impressions you received regarding the mind of Adoni Cane and any other member of the enemy's number you have encountered.\"\n\nMaii did so, conveying as best she could a number of conflicting visions. Cane possessed a mental shield that was difficult to penetrate, but did allow him to communicate with her by epsense and occasionally offered strange glimpses of what lay beyond. Sometimes, Cane's mind seemed to spin like a top; at other times it was as still and clear as a lake, or a mirror. The _irikeii_ had imagined him as a glowing light-source with a speck of black at its heart, and also as a snake coiling and uncoiling around itself.\n\n\"What sense do you make of these impressions?\" she was asked.\n\n\"None of them are necessarily true representations of his mind,\" she said. \"They're like the different reflections you get off the facets of a diamond, or the different meanings one collection of sounds has in different languages. I'm not seeing the underlying reality, just the secondary effects.\"\n\nShe shrugged, and the heavy shoulders of the suit magnified the gesture. She sent an image, via the relay, of a crystal turned inside out: smooth and spherical outside, facets crossing and tangling inside.\n\n\"It's hard to find words for this,\" she said.\n\n\"Evidently,\" said Murnane. \"But if you had to choose just one word to describe him... ?\"\n\n\"I'm not sure. 'Complex' isn't enough. 'Incipient,' perhaps? 'Numinous' has too many spiritual overtones, and I don't believe 'unknowable' applies to anything. There's a great potential within him. I don't know what for, but it's there.\"\n\nMurnane waited a moment, to see if she would add anything else\u2014or perhaps to confer mentally with the reaves surrounding them. After a moment he said: \"And what of the _irikeii_? What did he think of you?\"\n\nMaii was silent so long, Roche thought she wouldn't answer. Finally, she said: \"He disapproved of me.\"\n\n\"We thought as much,\" said Murnane, nodding. \"The Olmahoi Caste petitioned strongly for your capture prior to your arrival\u2014as did your own government. Somehow the word of your existence has spread, although exactly _how_ has yet to be determined. We decided not to become involved, for very good reasons; there are enough inter-Caste tensions as it is without the council seeming to take sides\u2014and what happens in non-Pristine Castes is, ultimately, none of our concern.\" Murnane stopped and took a deep breath. \"Still, it is clear that the events that occurred within Palasian System have had far-reaching repercussions\u2014many, perhaps, still to be felt. Morgan Roche, would you care to explain to us what happened there?\"\n\nRoche did so, outlining the exploration of the system after it had been ransacked by the clone warrior, her disastrous attempts to cooperate with Linegar Rufo, and her clash with the Kesh. Later, she hoped, she would be able to discuss things in more detail, but for the time being she contented herself with an overview.\n\n\"You say that the name of the enemy in this case was Jelena Heidik?\" someone asked when she reached the aftermath of the destruction of Palasian System.\n\n\"Yes. It's one of a list of names we... found in an old archive. The others included Vani Wehr, Sadoc Lleshi, Ralf Dreher\u2014\"\n\n\"Do you know who they refer to?\" Murnane interrupted. \"Was there any other information in that archive, apart from the names?\"\n\n<Box?>\n\n<No. But I can give you a full list of names, if you require it.>\n\n\"No,\" she said. She would give them the rest of the names later.\n\n\"And where is this Jelena Heidik now?\"\n\n\"I don't know,\" Roche admitted. \"We came here looking for her, but she's managed to get away.\"\n\n\"But you do think she's still somewhere in Sol System?\"\n\n\"Yes.\"\n\n\"Why do you believe that?\"\n\n\"Well, this seems to be where it's all coming to a head. She would hardly leave so soon.\"\n\nMurnane leaned forward, his hands on each side of the font supporting him. \"But _why_ Sol System? Are we here following the enemy, or has the enemy followed us? We see patterns of movement across the galaxy, leading here, but we still cannot be one hundred percent certain that we are not fulfilling our own prophecy.\" He shrugged. \"That is always a risk, I suppose, in any war of espionage; words and hints and suppositions carry little weight compared to maps and soldiers and bullets. So little is certain.\"\n\n\"We heard that Sol System was the location of an ancient battle,\" said Roche.\n\n\"It is the location of many things, if you believe the records; few stand up to strict examination. Which battle do you refer to?\"\n\n\"I'm not sure,\" she said. \"We've begun to suspect that the clone warriors\u2014Cane and Heidik and the others\u2014are seeking revenge for a war lost a long time ago. A war won by the Pristines.\"\n\n\"Do you know when?\"\n\nShe shrugged. \"As far back as we can remember. Half a million years or more. Back when there were only Pristines; the other Castes didn't exist yet.\"\n\n\"Do you have records to support this?\"\n\n\"Nothing concrete\u2014but surely that indirectly supports this theory? If there _were_ records, someone would have found them by now. The fact that we haven't implies that they no longer exist\u2014that the events we're looking for lie back in the earliest times.\"\n\n\"Perhaps.\" Murnane's expression remained impassive. \"Remember, though, that many millions of civilizations have risen and fallen since then. That is an awful lot of data to sift through; if the records indeed are lost, not hidden, then we might never know. And without knowing when this battle you refer to took place\u2014and who it was that lost\u2014we have little to go on.\"\n\nRoche conceded the point. \"That's partly why we came here,\" she said. \"We were following Heidik, yes, but we were also interested in seeing what happened. _If_ the clone warriors attacked, then who they attacked first\u2014and last\u2014could reveal who their allies are, or who is related to their creators.\"\n\n\"Tell me, Roche,\" Murnane said. \"Did you have any idea how complex the situation here would be before you came?\"\n\n\"The Box had mentioned a gathering of sorts, and the COE commander I spoke to confirmed it, but that's all. I expected nothing like this.\"\n\n\"Did this Box of yours also happen to say anything about the composition of this system?\" asked another voice. \"There are several anomalies we have not yet fathomed, and I fear they may become hazards to navigation. More of these we do not need.\"\n\nIt took Roche a second to realize that it was the Heresiarch himself speaking. When she replied, she made certain she followed Vischilglin's advice and looked him directly in the eye\u2014 or at least in the direction of where he stood.\n\n\"I'm sorry, I don't know anything about that.\"\n\n\"The behavior of the solar wind is quite peculiar, and its effect on the gaseous volatiles of the planetary ring even more anomalous. If your AI had _anything_ to say about that, I would've been grateful.\"\n\n\"Like I said,\" Roche replied evenly, keeping her attention fixed firmly on the Heresiarch. \"It never mentioned a thing. I'm sorry.\"\n\nShe thought she saw him shrug, but he was too for away to tell for certain.\n\n\"There is no need for apology if one speaks the truth,\" he said, with wry humor to his tone.\n\n\"We have asked the High Humans for this information, too,\" said Murnane into the silence that followed. \"They haven't told us anything that might conceivably help, on that or any other subject. I for one find their silence unnerving. Do you know why this might be the case?\"\n\n\"No,\" said Roche.\n\n\"Given that your Box came from this Trinity, which had connections to this High Human called the Crescend, do you think its destruction would be of some concern to him? Would he respond to a call for more information, perhaps?\"\n\n\"I really don't know.\" Roche hoped he would not respond; if the Crescend revealed to the council that she had lied about the Box's destruction, that certainly wouldn't count in her favor.\n\n\"The Crescend never contacted you while the Box was in your presence?\"\n\n\"No, never.\"\n\n\"Do you expect him to?\"\n\nShe resisted the urge to ask where this line of questioning was going. \"Look, I went to Trinity to collect the Box, but met no one while I was there. I was rendered unconscious in orbit, and when I woke up the Box was... in my possession. That's all. You're obviously hoping that I can act as some sort of link between yourselves and the High Humans, but I don't see that as being an option. I've never communicated with them, and I doubt I ever will. Why should they bother with me? I'm just someone who happens to be caught in the middle of all this.\"\n\n<I hope you know what you're doing, Box,> she whispered via her implants to the AI in her body.\n\n<Trust me,> it said.\n\n\"We are _all_ caught in this,\" Murnane said. \"But outside of the enemy, few individuals have had such a catalytic effect as yourself.\" He paused. \"Is there anything else you would like to tell us, while this council is in session?\"\n\n<Box?>\n\n<No>\n\n\"No,\" she repeated.\n\n\"Will you submit to a probe by one of our reaves to verify the answers you have given us?\"\n\nThe question surprised her. \"Why do you need that? The hard data speaks for itself, and I've no reason to deceive you.\"\n\n\"Nevertheless\u2014will you?\"\n\nIf she said yes, they would know that she was lying about the Box. Although she knew it would look suspicious, she had no choice but to say: \"No. I'm sorry.\"\n\n\"Will you allow us, then, to examine you and, if necessary, take a genetic sample?\"\n\nShe squirmed. <Box, could they find you that way?>\n\n<The Dato missed me when you were captured,> the Box replied, <but that was partly because I had infiltrated their systems. I cannot rule out the possibility that the council will see something the Dato missed, and that I will be unable to suppress the information once it emerges.>\n\n<Haven't you got into their systems?>\n\n<Of course I have. But the Skehan Heterodox places great faith in the role of people in or alongside all its systems. At least one Human doctor would be present during your examination, and I see no easy way to suppress any knowledge that might emerge at that time.>\n\nAgain, she had no choice. \"I'd prefer not to,\" she said. \"I'm sorry.\"\n\nMurnane studied her for a long moment. \"As are we,\" he said. \"But we cannot force you to submit to either examination\u2014nor would we wish to.\" He gestured helplessly. \"This meeting is now concluded. We would ask you to return to your ship, Roche, and\u2014\"\n\n_\"What!\"_ Roche snapped. \"Aren't you even going to discuss what I've told you?\"\n\n\"There is no need,\" he said. \"We've been conferring by epsense the entire time.\"\n\n\"But you can't just dismiss me!\"\n\n\"Can't we?\" He took a step toward her. \"Roche, we had hoped that you would provide us with information that is both new and verifiable. We had hoped that this might show us a way to combat the enemy we fear has infiltrated every group we deal with and perverts everything we attempt to do to stop them. Now it seems certain that you yourself have fallen into the same trap\u2014either willingly or by accident.\"\n\nRoche felt herself straighten, her tired back and stomach muscles tensing as though ready for attack. \"Meaning?\"\n\n\"You have told us _nothing,_ Roche. You claim that Adoni Cane is one of the enemy, yet you can offer no explanation for this surety nor a reason for his atypical behavior. Of what value is his genetic data under those circumstances? You offer us names that you assure us are relevant, but do not give us a context in which to place them or access to the records you say they came from. On what grounds can we possibly use them as means to uncover the enemy among us? You cannot tell us why Sol System has become the focus of so much concern\u2014you can't even tell us why _you_ came here without resorting to vague explanations involving this mysterious AI of yours! And as to _that,_ well, I hardly need to state how the council feels. _If_ it existed at all, its tenuous connections to the High Humans might have been exploitable, but as it stands\u2014\"\n\n\"I'm telling you the _truth_ ,\" she broke in angrily.\n\n\"Are you?\" Murnane moved closer again, his own anger evident in his face. \"There is much to suggest that what you are doing is far from innocent. Ameidio Haid is a convicted criminal who, as the Commerce Artel points out, has not served his full term; who is to say you don't have criminal intent in mind as well? Add to that the fact that both your young friend here and the pilot of your vessel are the subjects of biological experiments; if Adoni Cane's genetic data and physiognomy turn out to be peculiar, could he not also be an experimental subject, and not the enemy you claim he is?\n\n\"Then there are the credibility gaps in your story. How did you come to the conclusion that Adoni Cane was one of the enemy? How did you survive Palasian System when even the Kesh destroyer sent to monitor the situation did not? Why did you come here? And why has your arrival caused such a furor among all those who have known you: the COE, the Dato Bloc, the Commerce Artel, the Surin, the Kesh, the Olmahoi... ?\n\n\"Even if what you are telling us is the truth, and Adoni Cane _is_ one of the enemy, then how can we trust someone who openly admits to having one aboard her ship\u2014as part of her _crew_?\"\n\nMurnane shook his head. \"It may seem like we pre-judged you, but we have done nothing of the sort. We simply considered all possible conclusions prior to your arrival and allowed you to show us the one that best fitted the circumstances. Because you seem not to be dealing honestly with us, we are forced to conclude that Adoni Cane is a fake, or a misdiagnosis, or an enemy plant. We are unsure of _your_ motives, but we are sure that we will no longer allow our precious time to be wasted examining your spurious claims and false offers. We have work to do, Roche, and a distraction such as this, even if not maliciously intended, does the enemy's work for them.\"\n\nTight-lipped, Roche forced herself to speak calmly. \"If I could just say\u2014\"\n\n\"There is nothing more to be said,\" Murnane cut in. \"Hue Vischilglin will escort you and your companion to your vessel. Once you're on board, the protection offered by the _Phlegethon_ will be withdrawn.\"\n\nVischilglin appeared, expressionless, at Roche's side as Murnane turned his back and moved away without another word. The Heresiarch made no move at all. Roche let herself be taken by the arm and led away, furious but impotent, as a growing murmur filled the fane.\n\n****\n\n****\n****\n\n****\n\n**3**\n\nSHCV __ Phlegethon\n\n955.1.30\n\n0900\n\n__\n\n_< That_ went well,> Roche muttered to the Box as Vischilglin directed her through the council and back into the petal from which they had first emerged. <'Talk to them,' you said. 'Exactly what we need,' you said. Whatever happened to getting access to their resources and getting on with the job?>\n\n<You seem upset, Morgan,> the Box responded smoothly.\n\n<Of course I'm upset! The questions you wouldn't let me answer were the ones that convinced them to send us away!>\n\n<Be patient, Morgan. Time will tell if this trip has been wasted or not.>\n\n<Yeah, right.> Feeling humiliated and frustrated, Roche avoided the eyes of everyone around her as she walked by. They thought she was a fool\u2014or, worse, some sort of collaborator. <Sometimes I wonder why I don't just shut you up like De Bruyn did back in HQ.>\n\n<The 'Silence between thoughts' override phrase would allow you to do it any time you wished,> the Box intoned casually. <I wasn't faking that period of disconnection.>\n\n<I never thought that\u2014> She stopped midsentence as its admission sank in. <Why _did_ the Crescend give you an override phrase that actually worked? It doesn't make sense\u2014especially in the light of recent knowledge.>\n\n<Given the degree of invasion that I am upon your person, it seemed reasonable that you be provided with a means to turn me off, should I become too much of an inconvenience.>\n\n<In other words, the Crescend gave me an out.>\n\n<Exactly.>\n\n<Without _telling_ me about it?>\n\n<You learned in your own time.>\n\n<Only by accident\u2014the same way I found out that you're inside me!>\n\n<I can only assure you that, had you learned the latter before the former, you would have been given the override.>\n\n<Oh, naturally.> She couldn't help the sarcasm. The modified ulnar nerve along her arm was throbbing with remembered sensation. Data had burned along that path to the valise she had once believed the Box to inhabit. She wondered which pathways it used, now that it was completely inside of her....\n\n\"Well, that could have gone better.\"\n\nRoche recognized the man's voice as one from the interrogation. She looked up to see a fair-haired, diminutive figure waiting for them at the top of the stairs, an almost condescending smile beaming from his small, triangular face.\n\nVischilglin didn't give Roche a chance to reply. \"Stand aside, Junior Primate Nemeth,\" she said, pushing past the man and heading back down into the thick cloud they had climbed through to get to the council.\n\n\"That's _Assistant Vice_ Primate to you, Co-adjutant Vischilglin,\" he objected, following them down the steps.\n\nOnly then did Roche realize something odd about the council\u2014or rather the people who comprised the council. They were all plenipotentiary envoys, co-adjutants, assistant vice primates, senior aides\u2014underlings with fancy titles. None of them were the real operators. Perhaps, she thought, the situation in Sol System was too risky for the superiors to come, so their assistants had been sent instead.\n\nThen she realized another thing: she had heard the name of Assistant Vice Primate Nemeth somewhere before. She stopped and turned to the man. He stopped also, a couple of meters away, behind Maii. Mist from the cloud created a slight haze between them, but not enough to obscure the man's crooked grin.\n\n\"It was you who hailed us before we came here,\" she said. \"Before that drone intercepted us.\"\n\nNemeth executed a slight bow of the head. \"I'm flattered you remembered me.\"\n\n\"What do you want?\" Roche was in no mood for small talk.\n\nHe gesticulated expansively. \"Perhaps it would be more appropriate to ask what it is _you_ want\u2014from _me_?>\n\nShe studied him for a moment, then turned and continued after Vischilglin down the steps. \"I haven't got time for these games,\" she muttered irritably.\n\n<Can you read him, Maii?>\n\n<Yes,> replied the girl, maintaining a steady plodding pace behind her. <He is devious, but not impenetrable.>\n\n<So what's he after?>\n\n<To strike some sort of deal with you,> said the girl. <Although I don't know exactly what kind of deal.>\n\n<What do we know for sure about him?> said Roche. <What is most prominent in his mind?>\n\n<It seems the government he represents has been proven to be compromised\u2014therefore his influence within the council is on the wane. He spoke first during the session in an attempt to remind people that he was still there.> She paused for a few seconds before adding: <And he doesn't seem to like Esko Murnane.>\n\n<That's the best thing you've said about him so far.> Roche could hear the footsteps of the man following her down. Vischilglin strode ahead of her, a tall broad shape plowing through the mist. Far from the chatty, affable guide she'd been when they first met, she had hardly said a word since the hearing.\n\n<What's on _her_ mind?> Roche asked Maii.\n\n<She's embarrassed. She thought you would be more help to the council\u2014to the people she represents. She feels... betrayed, perhaps.>\n\n<By me?>\n\n<No. By the people who encouraged her to think that you might offer a solution.>\n\n<And who are they?>\n\n<Some members of the council and various lower-echelon officials.>\n\nRoche sighed. <I only came here looking for answers. I mean, what were they expecting to find in me? A savior or something?>\n\n<To some, that's exactly what you are,> said Maii. <To others you are the heart of the enemy itself. But most people are just curious. They're listening to the rumors, but they're waiting to see what happens.>\n\n<But how can so many rumors about me be started in such a short period of time?>\n\n_< Chen chen, fe,>_ said Maii.\n\nRoche frowned. <What...?>\n\n<Something Veden once said to me,> the girl replied. <Words attract words. 'Morgan Roche' are two words that have been uttered many times in the last few weeks. It is inevitable that other words would attach themselves to them.>\n\nRoche smiled now. <Well, whatever the rumors are, I think there are going to be a lot of disappointed people,> she said. <Because without the help of this council, I'm not going to be doing much.>\n\nThey stepped out of the clouds and back into the landscape of rolling valleys and trickling waters. Roche groaned inwardly when she remembered the distance they'd come to get this far. Her legs and back were sore from standing for so long.\n\nAs though someone had read her mind, an air-car resembling a large silver spoon hummed into view. There were seats for four people in the bowl, all empty.\n\n\"I thought you might be weary,\" said Nemeth from behind diem. \"As attractive as the scenery is, there's no need to view it on foot twice.\"\n\nRoche glanced at Vischilglin, who was frowning. \"You arranged this?\" Roche asked, suspecting an ulterior motive.\n\n\"It is not the council's will,\" said Vischilglin, scowling.\n\nNemeth shrugged expansively. \"Since when did the council start dictating courtesy? I'm offering you all a lift\u2014including yourself, Co-adjutant Vischilglin.\" He smiled. \"Well, are you coming or not?\"\n\n* * *\n\nThe air-car sped quietly across the uneven terrain, leaving the steps they had just descended far behind. Although they didn't move alarmingly fast, there were a couple of moments when the car slued to avoid a jutting ridge, making Roche feel a little uneasy.\n\nNemeth didn't appear to be troubled by the craft's sudden movements. He sat beside her, looking out at the rolling landscape sweeping beneath them, his face split by a seemingly perpetual smile.\n\nAs if sensing her staring at him, Nemeth turned to face Roche, and his smile widened.\n\n\"Now isn't this so much easier?\" he said. \"Perhaps we could even take a more interesting route back to the docks.\" Over his shoulder to where Maii and Vischilglin sat, he said: \"Do you think the council would approve, Vischilglin?\"\n\nThe woman grunted an affirmation. She really had little choice now, Roche thought. Nemeth laughed and turned back to look at the scenery.\n\n'Tell me,\" said Roche. \"What exactly _is_ it you think I want from you, Nemeth?\"\n\n\"All, now, that's the question, isn't it?\" he said. \"Make no mistake: I can do any number of things for you, Roche.\" He glanced over at her. \"If I were so inclined, of course.\" When she didn't react, he went on: \"You come at a peculiar time, Roche\u2014when the council is desperate for answers that none of us have. It feels constrained by the very precepts that allowed it to come into being so quickly. It is... _limited_ by its nature.\"\n\n\"You mean it's for Pristines only,\" said Roche.\n\nHe nodded. \"But some of us fear that 'Pristines only' may not be enough to combat this threat.\" He watched the view silently for a few moments; when he spoke again, the smile had faded. \"I lost my family back home, you know,\" he said. \"They were caught in an insurrection while I was serving in a completely different system. A local terrorist branch whipped up enough anti-government action\u2014in the form of riots and infrastructure sabotage\u2014to warrant calling in the army. Thousands of innocent people died in the ensuing repression, including my family, and it achieved nothing for either side. It turned out that the enemy was responsible for the whole thing. The terrorists were just a tool\u2014the means to an end. And that end was to cause as much destruction and misery as possible.\"\n\nNemeth looked at Roche, who sat watching him carefully. He remained outwardly relaxed, except for his hands: his knuckles were white where they gripped the armrest. When he realized this, he quickly loosened his grip and his smile returned.\n\n\"So, do you have any family, Roche?\" Nemeth asked.\n\nRoche felt a stab of pain. Never knowing her parents had been a constant regret throughout her childhood. As an adult, she had aspired to COE Intelligence in order to track them down. Upon reaching that goal, however, she had forgotten about her parents entirely, too busy with her own life to worry about the one she might have had.\n\n\"No,\" she said. Another part of her was glad that she could forestall his obvious gambit. While she could feel compassion for his loss, he would have to engage her intellect, not her emotions, in order to get what he wanted. Whatever that was.\n\nIf he was disappointed by her reply, he made no sign. He simply nodded and changed the subject.\n\n\"In a second we'll be entering one of the main longitudinal ducts that run down the hull from minaret to crypt,\" he said.\n\nVischilglin leaned forward in the cab. \"That's fore to aft to us,\" she said.\n\n\"Even at the speeds we will be going,\" Nemeth continued, \"it will take us ten minutes or so. But please don't be concerned by that,\" he added in response to a look of alarm in Roche's eyes: to travel any significant length along the giant ship so quickly would demand speeds greater than one or two thousand kilometers per hour. \"We'll be perfectly safe.\"\n\nThey raced toward what at first appeared to be nothing more than a wall, but as they flew closer, Roche saw it for what it actually was: a giant tube lying on its side across their path, suspended by invisible forces ten meters or more above the rolling hills. It was so thick that its top was obscured by the cloud cover, and for a moment Roche wondered how they were going to get past it\u2014or into it, if this was in fact one of the ducts Nemeth had mentioned.\n\nA moment later the craft swept beneath the massive cylinder and into its shadow. Their speed eased slightly as the air-car rose toward an enormous portal on the underbelly of the tube, easily thirty meters across and hanging open like a slack and lipless mouth. From it issued a cold breeze; not strong, but enough to make Roche shiver.\n\n\"An air duct?\" she said, hearing a faint susurrus coming from within. \"Seems a bit primitive on a ship like this.\"\n\n\"Believe me,\" said Nemeth, \"it's purely for aesthetics.\"\n\nThen they were inside\u2014and caught by a tremendous, rushing wind. The air-car lurched violently as it began to accelerate along the tube. Roche gripped her armrests as she was pressed back into her seat and knocked from side to side with every buffeting motion. Beside her, Nemeth laughed at her obvious alarm.\n\nAnother air-car\u2014this one a single-passenger model shaped more like an egg with two limp, trailing spines\u2014swept past them, barely missing by a meter. Startled, Roche looked around properly for the first time. Inside, the tube was easily wide enough to hold a hundred air-cars. Lines of lights trickled along the walls; every now and again, larger, brighter patches would rush by, too quick to take in. Other air-cars continued to pass theirs, less quickly than before, but thankfully none came as close as the first one.\n\n\"Aesthetics, huh?\" she said to Nemeth over the sound of the wind; some sort of field-effect was keeping the worst of the turbulence at bay; otherwise he would never have been able to hear her.\n\nHe laughed out loud again. But this time it was with an almost childlike delight: he was enjoying the ride.\n\n\"You would've loved Palasian System,\" she said. <Are you okay back there, Maii?>\n\n<Yes, Morgan. The view is spectacular. If you know when to look, you can see out the duct entrances and glimpse the levels as we go past.>\n\nRoche looked around her, concentrating for the first time on the bright patches as they went past. Indeed, now that she looked, she could make out brief impressions of the levels as they flashed by: here, deep purple and icy, there, soft pastels. One of the portals was much larger than the others, and through it she glimpsed angular structures in the distance, across flat, metallic plains; levels devoted to the ship's working, she supposed.\n\nClearly Maii had lifted this method of looking at the levels from the way Nemeth moved his eyes. <What is our guide thinking, Maii?>\n\n<He's getting a huge thrill,> said the girl. <I can't read much beyond that. I suspect he's taken us this way because it gives him a natural shield to hide behind.>\n\nRoche turned back to Nemeth. \"Scenery is all very well,\" she said, \"but when are we going to _talk_?\"\n\n\"We can talk now, if you like.\" He swiveled in his seat to look at Vischilglin, who regarded him stonily. \"Do you think the council would object to us having a little privacy?\"\n\nBefore the woman could answer, the field-effect protecting the passengers from turbulence clove in two, leaving Roche and Nemeth in a bubble of their own. Absolute silence suddenly pressed against her ears.\n\nThe air-car had settled itself into a gentle, rocking motion, and swept along the tube with the other air-cars as though on any conventional road. If Nemeth had brought her along the duct in order to unnerve her, Roche refused to let it.\n\n\"I want to talk to you in a frank and open manner,\" said Nemeth after a few seconds.\n\n\"You've displayed little intention of that so far,\" said Roche.\n\n\"Games. I know.\" He dismissed her accusation with the wave of a hand. \"The council is a bureaucracy; whether one is working within it or despite it, one is necessarily limited in one's options by this very fact.\"\n\nRoche sighed. \"Open and frank, remember?\" she said, making no attempt to conceal her annoyance. \"Can we just get to the point?\"\n\nHe sighed, too, and looked away for a moment. Behind him, a white landscape flashed by. \"Things are not going well for us here in Sol System,\" he said. \"In that much, at least, Murnane and I agree. The enemy were here before we even arrived, and have made their presence felt in a thousand ways\u2014sometimes subtle, other times not so subtle. Although there has been something of a lull in the last few days, every hour dozens more ships arrive, and with each ship the chances are high that more of the enemy are coming too. And we are not finding _any_ of them.\"\n\nHe looked at her, then. \"I am being completely and utterly frank about this, Morgan. I hope you realize that. Not even Vischilglin knows the depths of our failure. For all the council's collective experience and wisdom, for all the technology of groups like the Skehan Heterodox, for all that we have been studying the enemy for four and a half years, we are not even close to solving the problem here. Can you understand how galling that is?\"\n\nShe didn't have to think hard about that. She had been banging her head against the problem for less than three months.\n\n\"So why not take a chance on me?\" she asked. \"If you're so desperate, what have you got to lose?\"\n\n\"That's an interesting question, isn't it?\"\n\n\"Do you have an answer?\"\n\n\"A kind of answer,\" he said. \"But it starts with a question.\"\n\nHe paused. \"There are more than just Pristines in this system. Do you know what the Exotics are doing here, along with us?\"\n\n\"Following the flow, I guess,\" she said. \"Maybe coming to settle old scores. Naturally they'd be swept up in any regional conflict that might have started among the Pristines. I can see how they would be dragged here along with everyone else.\"\n\nHe nodded. \"It's certainly a valid assumption. According to your theory, the enemy comes from a time in which the Exotic strands of Humanity did not exist, or at least may not have been so prevalent. Indeed, maybe they are a weapon created by an ancient alliance of _all_ Exotics, in response to the age-old grudge that Pristines have it better than the others simply and unjustly because they are more like the original\u2014although why this alliance would wait so long to wreak its vengeance is somewhat of a mystery. And why would a weapon created by Exotics allow the descendants of its masters to be dragged into such a dispute?\"\n\n\"There may be only one way to find out,\" she said.\n\n\"Precisely. Here we come to your plan to wait until the fighting starts and see who doesn't end up dead at the end of it all, apart from the enemy. If anyone _is_ left standing, they must be guilty. Simple.\" He raised a hand as Roche started to protest. \"I'm sorry for seeming disrespectful. Your plan is ruthless and, perhaps because of that, likely to be more effective than most of the others bandied about. I simply fear that we will find out the truth only when it's too late.\"\n\n\"So what do _you_ suggest?\"\n\nHe shrugged, palms raised\u2014and for the first time Roche noticed that the little finger on each of his hands was missing. \"I told you: your plan is better than any of the others I've heard\u2014including my own.\" He grimaced. \"It's a hard thing to admit. If ever you doubt my sincerity, please recall this conversation\u2014although I'd be happier if you kept it to yourself, otherwise.\"\n\nShe allowed herself a half-smile. \"I don't know,\" she said. \"Blackmail has a certain appeal.\"\n\n\"A kindred spirit.\" His own smile was wide and natural. \"Perhaps we can come to terms, after all.\"\n\nAn air-car going the opposite way rushed past them; Roche gripped her seat until the rocking of their own car settled. When it had, Nemeth went on.\n\n\"We thought the lull recently might have something to do with you,\" he said. \"Your ex-superiors in COE Intelligence have kept us up to date with your movements. Ever since we heard about Cane's existence, we've been quite curious to see what would happen next. Many of us expected the COE to start falling apart as a result. In fact many of us felt that the Commonwealth's proximity to Sol System, the very focus of everything, would put it under much more pressure than other nations farther out. But apart from that brief fracas with the Dato Bloc, nothing much seems to have happened. It's almost disappointing.\" He flashed his grin at her again before adding: \"For some, that is.\"\n\n\"What does all this have to do with _me_?\" she said, conscious that the ride would be coming to an end soon and wanting some answers before it did.\n\n\"You're an anomaly, Roche. An outlier. You claim to have survived two verifiable encounters with two self-declared clone warriors. For that alone you're worth observing. And\u2014\" He hesitated slightly. \"And worth having on our side.\"\n\nRoche shook her head. \"Why? Because you think I'm _lucky_ or something?\" She was desperately trying to make some sense of what he was saying.\n\n\"No,\" he said. \"Nothing to do with luck.\" Again the smile, but this time forced and uneasy. \"But there is something about you. Something that doesn't quite add up. And, unlike Murnane, I don't think it's wise to turn you away without knowing what that thing is.\"\n\n\"But Murnane _has_ turned me away,\" she said bitterly. \"The council has already made its decision.\"\n\n\"It made _a_ decision,\" Nemeth corrected her. \"It wasn't necessarily the right one, and it certainly wasn't unanimous. It needn't necessarily be the _only_ one it makes. I happen to know that there is enough support to back up the offer I'm about to make you\u2014 if only because in hindsight it may prove wise for the council to be seen as having made the other decision it couldn't officially make, where everyone could see it. By that I mean that the council has to cover every base it sees open, even though here and now it can't acknowledge even to itself what it is doing. For posterity's sake\u2014for the sake of the future itself\u2014every chance must be taken.\"\n\nRoche was just managing to keep up. \"You're talking about some covert group within the council?\"\n\n\"One with its own agenda,\" he said, nodding. \"Does it surprise you that such a thing might exist?\"\n\nRoche shrugged heavily. \"Every bureaucracy supports such groups,\" she said. \"I guess I just didn't expect one here, that's all. I mean, we all have the one common enemy, right? We have the same _aim_.\"\n\n'True,\" he said. \"But we all work differently to achieve those aims. The council has become concerned with method, whereas the Ulterior concentrates on intention.\"\n\nRoche laughed at this. The _Ulterior_... \"And every such group has to have a catchy name, right?\"\n\nNemeth ignored the gibe. \"We have no firmly entrenched protocol,\" he said. \"If we see an opportunity, or even the potential for an opportunity, we will take it. We are less... scrupulous, perhaps, than many of our colleagues. And for that reason, we must remain as our name suggests: behind the scenes.\"\n\nRoche regarded him steadily. \"And you and your friends in this 'Ulterior' regard me as some sort of 'opportunity'? Is that what you're trying to tell me?\"\n\n\"Isn't that what you wanted the council to believe?\" he said.\n\n\"Yes,\" she said. \"I guess it was.\"\n\n\"Working for us, you would obtain that goal, Roche. Indirectly. If you fail, of course, the council has no knowledge of you, having turned you away from the one and only official hearing it was obliged to give you.\"\n\n\"Of course,\" said Roche dryly.\n\n\"But if it looks as if you might succeed, then you will have the full support of the Ulterior\u2014and ultimately the council itself.\"\n\n\"And why should I believe anything you're saying?\" she asked him. \"How do I know you're telling me the truth?\"\n\nHe dismissed the objection with a shrug. \"You don't,\" he said. \"But you don't have many other options right now. And we need each other.\"\n\nRoche sighed and, despite the apprehension she was feeling, said: \"So what exactly are you offering?\"\n\n\"A deal,\" he answered quickly, and with sudden enthusiasm. \"We can't give you any formal protection or recognition, obviously, but we can give you information. This information has to flow both ways\u2014unconditionally. If you learn anything new, we want to know about it. And if you find anything you think might work, we want to know about that most of all.\"\n\nShe didn't need the Box to tell her that she should take the deal. If she couldn't get the full approval of the council, this might be the next best thing. But she still had her doubts...\n\n\"It can't be that simple,\" she said.\n\n\"Well, there is something else we would like you to do for us,\" he admitted. \"But I can't see how it doesn't fit in with your plans, anyway.\"\n\n_Here we go_ , she thought. \"Meaning what exactly?\"\n\n\"That you're probably going to want to go buzzing around the system, looking for the enemy, right? Poking your nose in here, seeing what turns up there; waiting for the fight to start so you can see who kills who. Well, that's exactly what we want you to do, too. Specifically, we want you to see what the Exotics are up to. That's the one area this damned Pristine council of ours can't see into properly\u2014and any blind spots in situations like this are dangerous.\"\n\nShe nodded: that much at least was true.\n\n\"Do you have any other agents working in this area?\" she asked.\n\n\"A few,\" he told her. \"But nowhere near enough. Right now there are seven hundred and fifty-eight known Castes in Sol System, Morgan, not counting Pristine. Some are wildly Exotic; some are down the other end of the scale from the Skehan Heterodox\u2014almost Low Castes.\"\n\n\"And High Humans?\" she said.\n\nHe shook his head briefly. \"None that we are aware of,\" he said. \"But if you find anything that suggests there are, we'd be keen to hear about them too.\"\n\nShe was keen on the Box's behalf to avoid that subject. \"So basically,\" she said, \"if I find something, you take the credit. If I don't, or if I get into trouble, you disown me, right?\"\n\n\"Obviously we will do everything in our power to help you,\" Nemeth said, \"but our power is not unlimited. Unless you give us a reason to come forward, I'm afraid the Ulterior must remain just that.\"\n\nShe nodded slowly. \"And will I have to pledge allegiance to the Ulterior? Swear a secret oath? Sign my name in blood, perhaps?\"\n\nHe grinned. \"Your word will be fine,\" he said.\n\n<Is he telling the truth, Maii?>\n\n<He thinks he is, for what that's worth. He has the mind of a man who could convince himself of anything, though.>\n\n<That doesn't surprise me.>\n\n\"Okay,\" Roche said after a deep breath. \"For lack of a better option at this time, we have a deal.\"\n\n\"Good,\" he said, smiling and extending a hand. She just looked at it. \"If you're still worried about that genetic sample,\" he said, \"you should know that I'm more likely to get a decent one from the armrest you've been leaning on than from shaking your hand.\"\n\nShe relented and took his hand.\n\n\"And not a moment too soon,\" he said.\n\nThe air-car had begun to decelerate and drift toward the wall. The bright patches passed more slowly than before, and Roche caught glimpses of endless docks like the one through which she'd arrived: row after row of airlock inner doors, ramps, and floating cargo-lifters. All empty. For all the traffic she had seen, the ship might have been completely sealed.\n\nAnd maybe it was, Roche thought. That might have been the only option open to the Heresiarch and the council in order to prevent contagion.\n\n\"Oh,\" said Nemeth as they approached an opening and braked still farther, \"there is one more thing.\"\n\n\"There always is,\" she said.\n\n\"We'd like you to take one of us with you.\"\n\n\"What? _You?_ Forget it.\"\n\nHe managed to affect a hurt expression. \"No,\" he said. \"Not me. And not on board your ship, either. He'll have his own. But we'd like him there as backup, an observer\u2014or a bodyguard, if you like.\"\n\nThey slid smoothly out of the duct and into the docks.\n\n\"As insurance?\" she said.\n\n\"The only true necessity in all the universe,\" he said. \"Or so I've been led to believe.\"\n\nBefore she could say anything, the air-car reached a safe travel speed and the partition between the front and back seats evaporated along with the rest of the cushioning bubble. They decelerated still further, heading for the dock where the scutter was waiting.\n\nUnable to talk in privacy, Roche could only stare in alarm at the atypically enormous Surin warrior standing in full battle-dress at the inner door of their dock.\n\n\"You can't be serious,\" she said.\n\n<He is,> said Maii, her mental voice sharp with dismay.\n\n\"It was the only way to get the Surin off our backs,\" Nemeth said. \"Officially they want to make sure your young ward here is treated well; unofficially, they want in on the action.\" His eyes were hard. \"You should be glad it's not an Olmahoi grayboot as well.\"\n\n\"Someone with a little more subtlety would've been better.\"\n\n\"I think you'll find our friend here quite suited to your task.\"\n\nShe grunted dubiously. \"Any other surprises I should know about?\"\n\n\"No,\" Nemeth said as the air-car slid to a halt. \"At least, none that _I'm_ aware of...\"\n****\n\n**4**\n\n****\n\n****\n\nHHAB Dark Stressor\n\n955.1.30\n\n1155\n\nFinding the sort of people she wanted was almost ridiculously easy. Finding the right _person,_ however, was proving to be a little more difficult.\n\n\"I don't give a damn what you think, De Bruyn,\" said the obese Exotic on the far side of the partition, his voice a deep and guttural drawl. He had a tic on the left side of his body that seemed to move of its own accord: first his eyelid would twitch, then one finger, then a muscle in his neck, then something under the table would thump as his foot kicked out at nothing.\n\n\"You don't, huh?\" She leaned forward and slid the partition aside, not caring anymore about his Caste's preference to avoid close personal contact.\n\n\"No, I don't,\" he repeated, backing away uneasily. His entire left side twitched\u2014eye, finger, neck muscle, et al.\u2014simultaneously. \"What do we need someone like you for, anyway?\"\n\n\"I told you,\" she said. \"I have contacts; I can make things _easier_ for you.\"\n\nHe snorted. \"I don't see how getting dragged into this personal grudge of yours will make life easier,\" he said. \"Grudges are bad for business. They can get messy.\"\n\nShe feigned indignation. \"Now, who said anything about a grudge, Ken'an?\"\n\n\"It's in your eyes,\" he said. \"It's in the way you bargain. You're after something real bad\u2014so bad you're practically drooling. People don't salivate for money, in my experience. The stomach rumbles for betrayal, revenge, hatred, jealousy...\"\n\nShe retreated slightly. Maybe he wasn't so stupid after all. Still, she'd hoped for better. \"So much for mercenaries,\" she said dismissively.\n\n\"If it's mercenaries you want, talk to Uyeno Lenz. He'll do anything for a quick credit.\"\n\n\"Yeah, including knife me the moment my back is turned.\"\n\n\"A distinct possibility,\" he said. \"But I can't help you, De Bruyn. Like I said, getting involved in your personal grudge would be bad for business.\" He slid his seat back and shrugged. \"I'm sorry, but I have standards.\"\n\n\"Yeah,\" she muttered to herself, watching him waddle away from the table. \"Just not very high ones.\"\n\nHe was right, of course. She had no interest in the petty power-squabbles boiling in the vacuum of Sol System. She didn't care who came out on top in whose transplanted regional politics. All she wanted was someone to help her keep an eye on Morgan Roche\u2014and more, if necessary.\n\nDe Bruyn shouldered her way back to the bar, where an orange-clad Exotic refilled her glass. She wasn't drinking anything alcoholic; she wanted to keep her head clear.\n\n\"I'm looking for Uyeno Lenz,\" she said.\n\nThe bartender gave her a noncommittal shrug as he took some empty glasses away.\n\n\"You don't want to do business with him,\" said a deep voice at her side.\n\nShe turned. Another Exotic leaned against the bar, green-skinned, a mug of clear liquid clasped in his large, oil-stained hand. His eyes were deep-set and red; two thick strands of black hair ran down his head from forehead to nape. He flashed her an amused expression which seemed strangely out of place on his otherwise hard features. He didn't have to say another word. She knew he was the mercenary called Lenz. Only a hack would try a line like that on someone.\n\n\"You're right,\" she said, walking back to her booth. \"I don't.\"\n\nAn alarm went off in her implants before she sat down. Being fired from COE Intelligence hadn't meant the loss of equipment standard for upper-echelon agents. Her eyes and ears were artificial; much of her nervous system had been enhanced to run faster under stress, as well as to act as conduits for many different types of data; her skeletal strength had been increased by the addition of materials far stronger than Human bone. Although she could fight as well as most COE Intelligence operatives, she had not been trained for that; instead, she was wired to receive and transmit data\u2014like a Human antenna, complete with two-way listening and viewing devices.\n\nShe recognized the alarm instantly; indeed, she had been expecting a call from this source for the last hour or two. Putting the drink carefully in front of her, she activated scramblers and ciphers and opened a link to her ship.\n\n_Kindling_ was stationed just inside the protective bubble of the _Phlegethon,_ hidden by the big ship's camouflage and given clearance by her contact in the council. She had sent it there to act as a relay after catching a tug to the _Dark Stressor_ compact habitat to look for allies. She'd been hoping for a little more time, though; if Roche was already on the move, she would have to hurry to keep up.\n\nWhen the connection was made and secured, she spoke via her implants directly to her contact.\n\n<What's happened, Trezise?>\n\n<Nothing unexpected.> Via tightbeam, the lack of emotion in the man's voice was only magnified; she'd never decided whether it was an affectation or a genuine condition. <The worm is hooked and wriggling. Now all we have to do is wait for the fish....>\n\nThe words came with an image of Roche's scutter leaving its dock and heading for the _Ana Vereine,_ closely preceded by another ship\u2014a long-range fighter of some kind, angular and harsh. The design was unfamiliar.\n\n<Who's the escort?>\n\n<That I am still trying to ascertain.>\n\n<What about the other data I asked you for?>\n\n<I have some.> An icon winked in the corner of her field of vision, indicating an attachment to the transmission. <The rest is on its way.>\n\n<How long?>\n\n<I can't be exact, Page,> he said. <We're a long way from home, you know.>\n\n<I _know_ that.> She hated it when people used her first name\u2014a fact that wasn't lost on Trezise, she was sure. She forced herself not to rise to the bait, glancing instead at the data and searching for any of a handful of details she was hoping to find. One was there, as obvious as a nova now that she knew what to look for, and she smiled to herself.\n\nShe wasn't going to share her small victory with Trezise, though. <What about one of those ftl links? We'll need one if we're to keep in touch>\n\n<Can't help you there, > he said. <The best I can do is swing one by periodically, and give you the codes to call for one should you have the need. There are thousands all over the system. The longest delay you'll likely suffer is about ten minutes or so.>\n\n<Then that'll have to do.> She looked up to see the green-faced mercenary still watching her. She caught his image and sent it to Trezise. <What Caste is this?>\n\n<None I've seen before, but that's not surprising around here,> he said. <Where the hell are you, anyway? I thought you were in your ship.>\n\nShe smiled to herself. <If I wanted you to know where I was, I would've already told you.>\n\n<Suspicious to the end,> he said. <Auberon always said that the feeling was mutual.>\n\n<Auberon was a fool.> Her smile became a snarl. Auberon Chase, his boss and once hers, _was_ a fool, but he was still head of COE Intelligence and safe in HQ, while she was out hunting among the predators. <What did _you_ do wrong, Salton, to get such a lousy assignment?>\n\n<Nothing,> he said. <I volunteered.>\n\n<I don't believe you. Only a fool would want to come here.>\n\n<Only a fool would fail to see the opportunity,> he said. <When I think of all that your friend Morgan Roche is doing to squander her unearned leverage, I can't help but want to kill her myself.>\n\nAgain, De Bruyn refused to rise to the bait. <And where is our little worm headed?>\n\n<Who?>\n\nShe fought to contain her annoyance at his games. <Roche, of course.>\n\n<Oh, did you think I was referring to Roche?> His voice was smooth and amused. <I thought you knew me better than that. I was referring to a _much_ bigger target than her.>\n\n<What the hell are you talking about, Trezise?>\n\n<My dear Page,> he said. <I'm talking about the council itself, of course.>\n\nShe broke the line abruptly when she saw the mercenary approaching.\n\n\"I heard you talking to Ken'an, before.\" The words rolled from somewhere deep in the back of his throat, sounding as though they were having to fight their way through food to get out.\n\n\"You have a problem with that?\"\n\nHe sat down opposite her. \"Not at all,\" he said. \"But you should listen to him. Grudges are dangerous.\"\n\n\"I don't recall asking either you or Ken'an for your opinion.\"\n\n\"Well, make the most of it anyway,\" he said. \"Advice is about the only thing you'll get for free around here.\"\n\n\"And what's the price of a little peace and quiet?\"\n\n\"Quiet I can give you.\" He activated some sort of device in his jacket and a bubble of silence enfolded the booth. \"Peace, however, will be more difficult.\"\n\nDe Bruyn's implants buzzed, warning her of the field-effect he was using to give them privacy. She ignored the alarm, doubting the bubble was anything more dangerous than a toy. Still, her right hand slipped to her thigh-holster and disengaged the safety on her pistol.\n\nShe smiled. \"Okay,\" she said. \"I'm looking for someone to watch my back while I go about my business.\"\n\n\"What sort of business?\"\n\n\" _My_ business,\" she repeated firmly. \"For now, at least.\"\n\n\"In Sol System?\" The words continued to rattle in his throat.\n\n\"I wouldn't be here otherwise.\"\n\n\"For how long?\"\n\n\"Until the job is done.\" De Bruyn kept her stare firmly on his gold-flecked irises. \"It may require a bit of muscle.\"\n\n\"And how would you pay for this... muscle?\"\n\n\"I have influence in the Interim Emergency Pristine Council. What I can't provide in credit, I can make up in IEPC clearance and access. The breadth of your clientele will increase overnight.\"\n\n\" _If_ we survive.\" His lips tightened. \"Perhaps Ken'an was right: maybe you are a bomb just waiting to go off. Who's to say you won't take us with you?\"\n\n\"There are ways to avoid that,\" she said. \"And the right person working with me would find out how. But I'll need more than a handful of people to see this through.\"\n\n\"Promises and plans are easy to make,\" he said, his voice a low rasp. \"So who is the target, anyway?\"\n\nShe hesitated a second. Ken'an hadn't asked that, nor had any of the others. She'd been glad to assume it wasn't relevant,\n\n\"Morgan Roche\u2014\"\n\nShe was cut short by a hand under her chin, jerking her head back. She clutched at her pistol, but another hand gripped her wrist and yanked it away. She kicked, flexed, strained\u2014then relaxed when she realized it was futile to resist. The hands were just too strong.\n\nShe cursed silently. The privacy field had kept her from hearing her assailant creep up behind her. But she wasn't at a complete disadvantage yet...\n\n\"Call him off, Lenz,\" she hissed. \"Or I swear I'll blow this place apart.\"\n\nThe mercenary smiled calmly at her. \"And how do you intend to do that?\"\n\n\"With the nugget of turcite I slipped under the bar,\" she said. \"One word, and it'll detonate.\"\n\n\"Blowing yourself up in the process,\" he said with a slight, forced laugh.\n\n\"A risk I'm prepared to take,\" she said. \"But chances are this thug of yours will offer me some protection from the blast. As for you...\"\n\nThe mercenary looked nervous and cast a glance at the person holding her. The grip about her neck tightened.\n\n\"Tell me why we should help you with this Roche person.\" This came from the man squeezing her neck.\n\n\"What\u2014?\" She attempted to turn around but was barely able to move at all.\n\n\"If I'm going to be doing business with you,\" he said, \"then I want to know what's so important about her.\"\n\n_\"You're\u2014?\"_\n\nAgain the grip tightened. \"Lenz,\" he said. \"That's right.\" He released her throat and arm and pushed her facedown onto the table. She reached for her pistol, but he beat her to it and snatched it away, slamming it down in front of her. \"Now, no more games; no more threats. You talk.\"\n\nHe moved a few paces from behind her to where she could see him. He looked much like the mercenary sitting opposite her, but broader, older, and without the hair.\n\n\"What do you know about Morgan Roche?\" De Bruyn asked, sitting up and rubbing at her neck.\n\n\"Only what we've heard,\" he said. \"There's a lot of stories going around about her. Her name keeps cropping up. Not many of the details match, though. The general impression is she's somehow relevant to everything going on here. Someone who might be dangerous.\"\n\n\"Yes, she is\u2014but to whom? Us or the enemy?\"\n\nHe frowned. \"Meaning?\"\n\n\"All those stories you've heard,\" she said. \"They're all lies. Every one of them. The purpose of the stories is to hide the truth, and to keep attention focused on her\u2014so that when she's ready, she can act.\"\n\nHis skeptical look didn't change. \"And what is the truth?\"\n\n\"I'm not sure,\" De Bruyn said thoughtfully. \"But I think I can find out. All I need is a little more time, and\"\u2014she hesitated significantly\u2014\"some help.\"\n\nHe studied her for a long time. She looked patiently back.\n\n\"We have a ship,\" he said eventually. \"It doesn't look much, but that's the idea.\"\n\n\"It's not your ship I'm interested in,\" she said. \"What's your crew like?\"\n\n\"Hand-picked.\"\n\n\"How many?\"\n\n\"Eight.\"\n\n\"And you trust them?\"\n\n\"With my life.\" He smiled. \"But not my money.\"\n\nShe leaned back into her seat and returned the smile. \"Okay, then. Let's talk business.\"\n\nLenz relaxed and moved around the table. His buddy slid over to make room. \"You should know that we don't come cheap,\" he said. \"For what you're asking\u2014\"\n\nAs soon as his hand came off her pistol, De Bruyn grabbed it and shot him through the chest. She shot his buddy too, before he had a chance to register what had happened. Screams erupted around her before the bodies had even hit the floor.\n\nDe Bruyn took the lights out with her next two shots, then slipped through the panicked crowd and out of the bar before anyone realized that she had gone. At the first sign of pursuit, she triggered the nugget of turcite with a quick burst from her implants. The explosion tore through pressure-walls and bulkheads, the shock wave hurling her and her pursuers through a locked door and into a storage room full of cartons. She sustained only minor bruising and temporary hearing loss, and was back on her feet in time to ensure that none of her pursuers would ever wake again.\n\nThe authorities believed her story about a clash between rival mercenaries. Using her EEPC pass, she was on the tug within the hour, and back on _Kindling_ an hour after that.\n\n* * *\n\n<Let me guess,> said Trezise when she had reopened communications with him. _ <Dark Stressor,_ was it?>\n\n<How did you find out?> she said with studied indifference. <It wasn't difficult. There was a disturbance there, a few hours ago, involving a Caste much like the fellow you showed me. I looked into it, of course, to see if you'd come to any harm. The habitat's surveillance cameras caught a couple of good shots of you. Black suits you, my dear\u2014much more so than a Commonwealth uniform ever did.> He paused for a moment. <Did everything go according to plan?>\n\nShe shrugged noncommittally. He knew damn well it hadn't, she guessed, and that ate at her. She was no better off than she had been the day before. But it was only a matter of time before she found someone suitable for her needs. There were many other places to look, and she would have plenty of other opportunities to do so while she followed Roche across the system.\n\n_Kindling's_ engines hummed softly through the walls of its cramped cockpit. In a way, she was glad to be on her own. Relying on other people was dangerous, albeit a necessary danger at times. It was much better, she'd always thought, to have them rely on you....\n\n<I hope you know what you're doing, Page,> said Trezise across the expanding distance between the two ships\u2014one as large as a good-sized moon, the other barely a speck. There was still no emotion in the man's voice. <You seem to be expending a lot of energy on something that will ultimately get you nowhere.>\n\n<Is that what you think I'm doing?>\n\n<I'm hypothesizing,> he said. <That's all.>\n\n<And does your hypothesis explain why Morgan Roche's COE birth records have been erased?>\n\nThere was a slight pause. <I'm not sure what you mean.>\n\n<The records you gave me,> she said. <Look at them yourself. Try to find her date of birth.>\n\n<Why?>\n\n<Just _do_ it,> she snapped. <But you won't find it, Trezise. You won't find it because it's not there\u2014along with a lot of other information that should be there too.>\n\n<A woman without a past,> he said, with the barest hint of dryness in his voice. <Whatever will an intelligence operative come up with next?>\n\n<Listen, you idiot. _We_ didn't wipe those records. Someone else did. Someone's covering up. The same someone who got in the way when I tried to stop her. The same someone who sacked me.>\n\nTrezise sighed heavily. <Listen to yourself, Page. Listen to what you're saying. This is ridiculous!>\n\n<You think I'm crazy?>\n\n<I think you're paranoid,> he said. <I think you _believe_ you're making sense.>\n\n<You're entitled to your opinion, I guess,> said De Bruyn.\n\nHe acknowledged this with a nod of the head. <And you yours, Page.>\n\n<But only one of us can be right.>\n\n<True,> he said. <And both of us could be wrong.>\n\nShe shook her head. Trezise enjoyed arguing for the sake of it; she shouldn't let him get her worked up so easily. <Do as I say. Look it up. You'll have to at least admit that it looks peculiar.>\n\n<This whole thing is peculiar,> he said. <This entire _system_ is. Did you know that the Heresiarch has advised all council vessels to avoid traveling in or near the ring after a probe found evidence of nanoware in the dust?>\n\n<No, but\u2014>\n\n<And were you aware that rumors suggesting a cult worshipping the enemy arising among the more destabilized elements have been less ambiguously verified by no less than three council agents? And that\u2014?>\n\n<Enough, Trezise.> She felt weary just listening to him. <If you think you have more important work to do, then just go. I'm not exactly here on holiday either.>\n\n<No, you're not. If you do find anything out\u2014something of note\u2014please let me know.>\n\n<Sure. And when the rest of that data comes through\u2014>\n\n<I will pass it on to you,> he said, cutting her short. <I know how much it means to you.> He almost seemed to smile as he added: <But be careful of the ring, Page. I wouldn't lie about something as important as that.>\n\nThen he was gone, leaving De Bruyn half-smiling to herself. Trezise annoyed her, but he played a good game. She'd rather have one single adversary like him than ten allies of Uyeno Lenz's ilk. Not that Trezise _was_ an adversary, she hoped.\n\nFollowing the _Ana Vereine's_ trace at a discreet distance, she drilled deeper into the nugget of data Trezise had given her. What she found did little to put her mind at ease, and what she _didn't_ find only added to her frustration. If only, she thought, she could get at the data directly and not worry about elements of corruption along the way. Or better yet, get her hands on _Roche_ , and extract the data in a way that would leave no doubt at all....\n\n****\n\n****\n****\n\n****\n\n**PART TWO:**\n\n****\n\n**PERDUE**\n\n****\n\n****\n****\n\n****\n\n**5**\n\nAVS-44 955.1.30\n\n1100\n\nRoche let Kajic pilot the scutter while she watched their new companion break dock. Defender-of-Harmony Vri flew a compact rapid fighter that looked like a cross between a throwing-star and a dagger. Roche didn't recognize it as a Surin military ship. Their designs were normally more hospitable. Only when the craft were attacked did they sprout numerous means of retaliation, suddenly taking on a more aggressive look.\n\nBack on the _Phlegethon,_ the warrior had spoken barely a dozen words to Roche before turning and moving off to where his ship was docked.\n\n\"Does he have a first name?\" Roche had asked Nemeth, staring at the back of the receding warrior. On the back of his lightly furred skull was a triangle of darker hair, pointed upward like an arrowhead. Whether it was natural or dyed, Roche couldn't tell; and she wasn't about to ask him in a hurry, either. His wide-spaced, dark eyes had discouraged any personal questions.\n\n\"Not that I'm aware of,\" Nemeth had replied. \"Or Vri might be it. Like your friend here, he doesn't seem to have a family name.\"\n\n\"Which would make him a renegade, right?\"\n\n<Soldiers who renounce their family line,> said Maii, <usually do so only to demonstrate that they are willing to die for the principle they embrace. In Vri's case, it is harmony.>\n\nRoche caught an image from the girl of something that looked anything _but_ harmonious. \"So he's a fanatic?\"\n\n<No> said Maii. <Just extremely dedicated.>\n\nRoche couldn't argue with that. When introduced, Vri had nodded to Roche and Maii in turn and said: \"I will defer to your instructions unless they conflict with the directives given to me by the Agora.\"\n\nShe could tell he would be a force to be reckoned with. Even from a distance he looked intimidating, with his sheer size\u2014strange for his Caste\u2014and the strange orange and yellow overlapping garments the Surin called ceremonial armor. It looked more like some sort of thick fungus.\n\n\"Is he any relation to Fighter-for-Peace Jancin Xumai?\" she asked, thinking of the Surin who had threatened them earlier.\n\n\"Maybe,\" Nemeth said with a shrug. \"We don't know exactly how many Surin there are in the system,\" he had said. \"There could be numerous factions. You'd be more familiar than we are with how they operate.\"\n\nThe air-car had waited patiently for him while they talked. Vischilglin was watching silently and suspiciously on the sidelines. Roche had half expected Nemeth to say something more, but he obviously felt constrained by the woman's presence. He had bowed at Roche and Maii in turn, then climbed back into his seat.\n\n\"Perhaps we will meet again,\" he said.\n\n\"Perhaps.\" Roche didn't return his wave as the air-car sped off along the curving floor.\n\n<He says to contact him when you're on the move,> said Maii.\n\n<How?>\n\n<Vri will tell you.>\n\n\"I must apologize for him,\" said Vischilglin. \"If his behavior offended you\u2014\"\n\n\"No, it's all right.\" Roche suddenly felt sorry for the woman. If her hopes had been as high as Maii had said, then acting as Roche's guide must have been something of an honor. To see that hope dashed, then have that honor usurped by someone else, must have been disappointing.\n\nTaking Vischilglin's hands in her own, Roche pressed them to her forehead, in the same way Vischilglin had done when they first met. \"Thank you for your hospitality,\" she said. \"I will do my best to prove that your faith in me was warranted.\"\n\nVischilglin looked in turn confused and embarrassed, then relieved. Then she smiled warmly. \"Thank you, Morgan Roche. And you.\" She bowed to Maii. \"My thoughts go with you.\"\n\nShe turned and walked away, leaving Roche and Maii to make their own way through the airlock doors and into the scutter. The same terse traffic controller as before guided them out of the dock in the same perfunctory manner, adding almost as an afterthought once they were clear: \"Weryn guide you and keep you safe.\"\n\nVri's ship rapidly overtook the scutter, darting through space on jets of blue energy.\n\n\"How do you feel about him, Maii?\"\n\n<He means me no harm,> the girl said, her voice less strained than before. <But his definition of 'harm' is open to interpretation. If he feels that I am being mistreated by you, then he will try to take me back to my family.>\n\nRoche shook her head. \"But they _sold_ you,\" she said. \"Surely that's just an excuse to get a look inside your head.\"\n\n<Possibly,> said Maii. <But _he_ believes only what he's been told. He is very traditional and holds the Agora in high regard, as one would expect of anyone in the military. But how he came to be here, working with the Ulterior, I don't know. His thoughts do not reveal whether he thinks it an honor or a punishment.>\n\n\"Maybe it's neither,\" said Roche. \"Maybe it's simply a chance to prove himself, an opportunity for advancement.\"\n\n<Or an early death in defense of his principle.>\n\n\"That could hardly be regarded as harmonious.\"\n\n<Perhaps not for us, Morgan,> said Maii. <But he is different.>\n\nRoche couldn't argue with that....\n\n* * *\n\nBy the time the scutter docked with the _Ana Vereine,_ Vri had placed the _Esperance_ in formation nearby. Roche and Maii went straight to the bridge to debrief the others, and to open communications with the Surin warrior.\n\n\"I don't like it,\" said Haid. \"He's potentially dangerous.\"\n\n\"His ship is no match for ours,\" she said. \"Would you agree, Uri?\"\n\n\"Without question. If he tried to attack, he would be disabled or destroyed with little effort.\"\n\n\"But if he catches us off guard\u2014\" Haid began.\n\n\"He won't.\" Kajic's voice was firm. \"His every move is being monitored.\"\n\n\"But\u2014\"\n\n\"Enough,\" said Roche. \"There's no point arguing about this. We can't do anything about it right now, so let's just accept that we're stuck with him and get on with it.\"\n\nA signal came from the angular craft and Kajic put it through to the main screen.\n\n\"Commander Roche.\" Vri's elongated face was fuzzy with tightbeam static and hair. \"I am instructed to accompany you on your journey and to lend assistance where I see fit. In order to do this, I will need notice of your destinations _and_ intentions. I trust this will not be a contentious issue.\"\n\n\"Of course not,\" she said. \"But as far as my 'destinations and intentions' go, I haven't thought that far ahead.\"\n\n\"Assistant Vice Primate Nemeth instructed me to advise you of the communication channels used by the Ulterior, and to ask you to call him immediately. A description of how to contact him accompanies this message.\"\n\nRoche looked up at Kajic's hologram. \"Got it,\" he said after a momentary pause.\n\n\"Thanks,\" she said, turning back to Vri. \"We'll call him now.\"\n\nThe Surin warrior nodded slightly. \"When you have decided what to do next, I may be contacted on this frequency.\"\n\nHe disconnected the line before Roche had a chance to say anything else. She shrugged and addressed the hologram once again. \"Kajic, open a line to the Ulterior.\"\n\n\"Doing so now, Morgan,\" he said.\n\n\"I presume this will be a secure line?\" she said.\n\n\"The content of the transmissions is encrypted, yes, but the transmissions themselves are not hidden. Signals in both directions travel in the same way the _Phlegethon_ communicates with its ftl drones.\"\n\n\"As though we have nothing to hide, eh?\" Roche could see the reasoning, but she didn't feel entirely comfortable with it. \"I hope they're right...\"\n\n\"Ah, Roche.\" Nemeth's voice and image simultaneously burst from the main screen. \"Glad to see you received my message in good faith.\"\n\nShe was in no mood for pleasantries. \"Are you sure this is the best way to talk to each other?\"\n\n\"No method of communication would be a hundred percent safe from prying ears,\" he said. \"But this is certainly the safest option at our disposal right now. Just as I cannot possibly hope to safeguard against every security breach, so too are they\u2014\"\n\n\"Okay, okay,\" said Roche. \"Just tell me what it is you've got to say. I presume you're going to tell me what you want me to do.\"\n\n\"More or less,\" he said, smiling at her impatience. \"Rather than wandering all over the system in the hope of stumbling across something useful, we feel you would be better served having your own area to investigate. A file will follow this conversation; it maps out that area for you. Obviously it will change as ships arrive and leave, but it's a starting point. I have listed all major known Castes and alliances, and marked key congregation points. Infiltrate them and see what you can find, then move on to the next site. Report back as you are able.\"\n\n\"Anything else?\" she asked, not even attempting to disguise her irritability. She listened to Nemeth with a growing sense of unease. She was becoming a lackey again, a pawn in someone else's game\u2014something she thought she'd left behind with Intelligence HQ.\n\n\"You know the score, Roche,\" he said, suddenly serious. \"Our main objective is to deal with the enemy, but first we have to know how to _find_ them. There has to be a way of determining who they are\u2014a test of some kind that can apply to _all_ of them, collectively rather than just as individuals. Your reave might be able to help with that: if the enemy does possess a unique n-body signature, that might be a way we can distinguish between them and us. Our reaves have had no success at seeing what you've reported, but that doesn't mean you might not have better luck.\"\n\nRoche nodded. \"And once we have found a means of doing this, what then? How do you propose we deal with them?\"\n\nHis shrug was both heavy and helpless. \"That's a completely different issue,\" he replied. \"And one we will address at a later date. But the matter of where they come from might assist us in this, just as why they're here might help us work out where to find them. They _must_ be communicating somehow, so if you can work that out too, that would be excellent.\" There was a slight pause as the signal broke up momentarily. \"Oh, and see if you can find out why so much energy is being wasted talking about you, too. The _Lucence-2_ gives us a direct link between one of the enemy and the propagation of your name; it would be foolish to ignore the possible ramifications of this link. Clearly your Box wouldn't have kept you alive for so long if it too didn't have some sort of plans for you.\"\n\nThe mention of the Box threw her for a second. \"I thought you didn't think the Box was important.\"\n\n\"Me? I said nothing of the sort. Even the council isn't so stupid as to ignore what it knows to be true\u2014although it denies it in public. We've had some dealings with High Humans in the past months and years, but not as many as we would like. Two in particular\u2014Aquareii and the Catiph\u2014were quite frank until they suddenly stopped communicating with us.\" He paused again, as if in thought. \"It's common knowledge in some circles that High Humans limit or actively suppress AI technology\u2014except when it suits them, of course. That the Crescend runs a factory in Commonwealth space where you obtained this mysterious Box only supports your theory that it is somehow important.\"\n\n\" _Was_ important\" she corrected.\n\nHe shrugged again. \"Anyway, one can only wonder what it would have thought of your situation now.\"\n\nPrivately she agreed, and promised to deal with that question as soon as possible.\n\n\"Is there anything else you'd like of me?\" she said.\n\nHe either missed the sarcasm or didn't care. \"That will be enough for now, I think. We can call each other another time, should something dramatic occur or some important need arise; otherwise we'll just get on separately with our work. Agreed?\"\n\nAgain, she had little choice. \"Agreed.\"\n\n\"Good. Until then...\" With a curt nod he was gone.\n\nRoche turned to face Haid and Cane, watching from the sidelines.\n\n\"I'm liking this situation even less, now,\" said Haid.\n\n\"It's better than nothing,\" she said. \"We need _some_ sort of contact with the council. At least this way we have a chance of making headway.\"\n\n\"Covert organizations can operate more effectively than their parent bodies,\" said Cane. \"They can respond to changes more rapidly, and can work in areas prohibited to officials. I believe that this is a good sign, Morgan. Working for the council, you would have been just one agent among many; your voice could have been lost. Now you have a greater chance of gaining the attention of the entire Ulterior, and in time the council itself. Now, I believe we will start to make progress.\"\n\nShe couldn't remember the last time he had shown such enthusiastic support for one of her decisions. Her satisfaction was tempered only by the part of her that wondered if he was telling the truth.\n\n\"Well, **** I hope you're right, Cane, and Ameidio is **** wrong. No offense.\" She smiled at the ex-mercenary, who shrugged affably back. \"Maybe we should look at the file Nemeth gave us, to see where we're supposed to go.\"\n\nKajic displayed a map on the main screen. It showed the entirety of Sol System, as mapped by the _Phlegethon'_ s network of ftl drones over the previous days, with particular attention to a region beyond the planetary ring, one hundred and fifty million kilometers out from Sol. There, a moderately large collection of ships and habitats had gathered, including\u2014if Roche wasn't mistaken\u2014no less than five outrigger spines. Someone must have piggybacked them into the system, since they didn't use hyper-space technology. But why they were here at all, Roche didn't know; the system itself didn't even have an asteroid belt. She wondered if Nemeth had given her this region because she had worked with outriggers before. Certainly, there didn't seem to be any other reason.\n\n\"What does the file say about the people here?\" she asked.\n\n\"There's a wide mix,\" said Kajic. \"Some extremely Exotic Castes and some Pristines, with numerous variations in between. Some Castes segregate except to negotiate; others mix freely. There are three mobile habitats around which most of the activity takes place; the largest of these is called Perdue. There has been word of fighting from its vicinity, and remote observations of weapon-use. This is to be our first destination.\"\n\n\"How long will it take us to get there?\"\n\n\"Twenty hours,\" Kajic replied.\n\n\"And does the file tell us what we're supposed to do when we get there?\"\n\n\"No,\" said Kajic. \"Nor in the other destinations we've been given, either.\"\n\n\"So I guess we'll just have to wing it,\" said Roche. \"And no doubt stir up trouble in the process.\"\n\n\"I bet that's what Nemeth is hoping for,\" said Haid. Shaking his head, he added: \"Look, if you really think this is the right thing to do, Morgan, I'll go along with it\u2014but...\"\n\n\"I know.\" She stood. \"Uri, advise Defender-of-Harmony Vri of our destination, and let him know the course you set. Get us on our way as soon as possible. And keep us camouflaged. The less attention we draw to ourselves, the better.\"\n\nShe looked at the faces of the people in her charge. Maii, out of the hazard suit, seemed older, thinner, and paler beneath her hair than when they had first met. Haid's dark black skin and biomesh looked out of place against the warm browns of the bridge, lending him an air of discomfort. Neither Kajic nor Cane had changed at all\u2014the former's image artificially generated and never looking as tired as he felt, the latter seemingly untouchable. The one and only time she had seen Cane at a loss had been when he was thawing from the coma Linegar Rufo had used to keep him contained on Galine Four. And even then, she had sensed dangerous aura around him\u2014like a bomb that could explode at any time.\n\n\"Okay,\" she said after a moment. \"We need to be fresh when we arrive at Perdue Habitat. Unfortunately we no longer have the luxury of the Box to keep an eye on things, so we're going to have to take shifts keeping watch.\"\n\n\"I am alert,\" said Vri, his face appearing on the main screen in response to Kajic's hail. \"I'd be more than willing to keep watch.\"\n\n\"I appreciate it,\" said Roche, \"but I'd like one of my own crew awake too. And don't you volunteer either, Uri; you can only run for so long on stimulants. _I'll_ take the first watch. If something comes up that Vri and I can't handle, I'll sound the alarm. But until then, I want everyone to get some rest. That goes for you too, Cane.\"\n\n\"If you insist, Morgan,\" said Cane.\n\nRoche had expected some objection from him, but was thankful it didn't come. Whether or not his obedience was offered in response to her earlier suspicions or for completely innocent reasons, she didn't know. Nor did she care. She was simply grateful not to be getting into an argument right now. She was just too tired.\n\nShe watched as Cane stood with his easy, smooth grace, and strode from the room without another word. Haid was close behind, with Maii in step beside him. They stopped at the doorway and Haid turned to face her.\n\n\"You _will_ call, right?\" he said.\n\nShe smiled. \"You know I can't handle this ship without you or Uri.\"\n\nThe ex-mercenary returned the smile and then, with Maii using his eyes for guidance, left the bridge.\n\n\"Now you, Uri,\" she said to the hologram standing in the center of the bridge.\n\n\"I won't deny that I am tired, Morgan,\" he said. \"But I am concerned that you are, too.\"\n\n\"Don't worry about me,\" she said. \"I'll be okay. _And_ I'll be watching your systems to make sure you're doing as you're told.\"\n\n\"Very well. I will rest for four hours, the most I need at this time. When I wake, it'll be your turn.\"\n\nShe raised her hand in mock salute. \"Sweet dreams.\"\n\nHis image flickered out, and she was left with Vri's face on the big screen. \"We'll speak if something happens,\" she said. \"Otherwise, stay alert.\"\n\n\"I will,\" said the warrior, and closed the link.\n\nEven then, she wasn't alone.\n\n<Box, put a forward view on the main screen, along with a navigation chart. And dim the lights.>\n\nAlmost instantaneously a chart was displayed before her, showing numerous ships in a wide variety of orbits, none with any likelihood of crossing their path. Several were traveling in directions similar to the _Ana Vereine,_ but that didn't necessarily mean anything; there was so much traffic in the system the chances were high that at any given time there would be such a coincidence.\n\n<Any sign of the ftl drones?> she asked.\n\nTwo green circles winked around objects in the display. Neither was following them.\n\n<I'm assuming they work by sending information through hyperspace to and from the _Phlegethon. >_\n\n<That would be a reasonable assumption,> the Box said.\n\n<You've seen technology like this before?>\n\n<No, Morgan, but I am aware that it exists. There are many ways to communicate.>\n\n<Does the Crescend use technology like this?>\n\n<No.>\n\nThe simple response held a wealth of meaning. _No,_ because the Crescend had far surpassed such simple beginnings. _No,_ the Box wouldn't tell her any more if she asked. She tried to imagine what sort of communications a being thousands of years old and comprised of many millions of mundane Human minds would use. Instantaneous? She wasn't prepared to rule anything out....\n\n<Does the Crescend know you're still alive?> she asked.\n\n<Yes.>\n\n<Have you communicated with him?>\n\n<After a fashion. We do not exchange information in the way mundane Humans do. It is more complicated than that.>\n\n<Could you communicate with him now if you needed to? Could you ask him something?>\n\n<Yes.>\n\n<So you have access to the High Human's communications systems?>\n\n<Some. For the most part, I am restricted to the means of communication you have at your disposal\u2014namely those of the _Aria Vereine. >_\n\nShe thought for a few seconds before responding. <Would you be offended if I chose not to believe you?>\n\n<Not at all,> the AI replied evenly. <Although I would stress that I have no reason to lie.>\n\nShe could think of plenty of reasons why the Box might not want that information freely disseminated. <Would you ask the Crescend a question for me?>\n\n<If you wished it.>\n\n<Would he answer?>\n\n<I am in no position to speak for him, Morgan.>\n\n<But you wouldn't rule it out?>\n\n<No. He might.>\n\nShe nodded. <What you're saying, then, is that we could have contact with the High Humans at any time we want\u2014or one of them, at least.>\n\n<Potentially, yes.>\n\nThe admission didn't make her smile. <In other words, you made me lie to the council.>\n\n<You didn't lie, Morgan, because you didn't know the truth.>\n\n<That's splitting hairs, Box, and you know it. Why don't you want the council to talk to the Crescend?>\n\n<Because the Crescend has no desire to talk to them.>\n\n<But _why_? _>_\n\n<Because he doesn't believe that the council is relevant to the solution of this situation.>\n\n<But they're the biggest force in the system. If _they_ can't...> She stopped, feeling cold at the thought. If the council, with the united forces of the four hundred plus Pristine nations behind it, couldn't fight back\u2014who could?\n\nThe answer came to her almost immediately. <The High Humans,> she said. <The High Humans could wipe out the enemy any time they wanted, couldn't they?>\n\n<It's not that simple, Morgan.>\n\n<No? They have so much more experience, and their technology is better; there must be thousands, maybe _millions_ of them\u2014>\n\n_< It is not that simple, Morgan.>_ the Box stressed. There was a harsher tone to the Box's voice, but that didn't stop her.\n\n<Why not?>\n\n<You are extrapolating wildly. The High Humans are not super-Human. They too need to know _who_ they are attacking. When the enemy as an individual looks like any other Pristine Human and as a group has become enmeshed in many Pristine governments, will any amount of higher thought or technology prevail?>\n\nShe wasn't satisfied with this answer. <So why can't they at least _talk_ to the council?>\n\n<Because the risks outweigh the benefits, Morgan. If the council has no contact with High Humans, the enemy will not know for certain that the High Humans are involved. Uncertainty may lead to indecision; indecision may lead to tactical error.>\n\n<But how _are_ the High Humans involved? As far as I know, you're the only evidence that they're showing the slightest interest, and even then, you're just a go-between.>\n\n<Trust me, Morgan,> said the Box earnestly. <I am much more than that.>\n\n<Then what are you? Why are you here, and who sent you? Is it just the Crescend, or are there more of them? And if they ever work out how to find the enemy, will we be able to rely on them to help?>\n\nThe Box hesitated, then said: <I cannot answer these questions, Morgan.>\n\n<You're saying you don't trust me?>\n\n<I am saying that you are Human,> said the Box. <Nothing more.>\n\n_< You_ are saying that, or the Crescend?>\n\nThere was another slight pause. <I am in communication with my maker, yes.>\n\n<He's listening right now?>\n\n<In a manner of speaking, yes.>\n\nThat stopped her. The High Human was eavesdropping on them; the being that had grafted the Box to her very cells and sent her headlong through the galaxy was actually paying attention to what she said! The thought was unnerving. Nevertheless, she had the ear of someone a million times more evolved than she was; she knew she should use the opportunity while she had it.\n\nOnly one question concerned her at that moment.\n\n<Does he actually care, Box?>\n\n<What do you mean, Morgan?>\n\n<Does he care what happens to us lesser Humans?>\n\n<I would not be here if he didn't,> said the Box. They had come full circle: the Box was proof that at least one High Human was interested in what happened on mundane levels\u2014was, perhaps, even concerned\u2014but beyond that refused to say anything at all. He had access to technology undreamed of, but wouldn't allow them to use it. He could step in at any time and be of great help in the struggle to understand and repel the enemy, but he did not. He preferred lurking in the shadows....\n\nRoche saw no point in pursuing the matter for the moment. She had more immediate things to worry about. Things she could actually do something about\u2014or at least feel like she was doing something.\n\nOnce again, as the _Ana Vereine_ powered its way across the solar system, Roche suspected that they were being followed. Not overtly; two ships hung back a long way and changed their trajectory several times, presumably in an attempt to allay suspicion by diverting attention away from their true activities. But their signature always reappeared on the navigation chart, and there was no doubt in Roche's mind why: they were in pursuit of the _Ana Vereine._\n\nThey could have been Ulterior drones or ships making sure she was doing the right thing; they could have been completely unrelated to her situation in the system\u2014security probes or freebooter scouts, establishing the ship's status as either threat or opportunity. Regardless, Roche's first thought was to shake them, but the difficulty of doing so outweighed the benefits; evasive maneuvers were less effective at high velocities, and any change in course at all would mean recalculating their orbit around the sun. No large feat, but it would mean waking Kajic.\n\nHer best chance of losing them would come when they reached a relative halt at Perdue Habitat. That was just under a day's travel. Until then she would simply have to try to ignore them, and take action only if either ship made a hostile move.\n\n<If I fall asleep,> she said to the Box, make sure you keep an eye on things, okay?>\n\n<Of course,> the AI said. <And I will doctor the surveillance records to show that you were awake. Kajic will never know.>\n\n<And Cane?>\n\n<He is in another section of the ship, exercising.>\n\nRoche frowned. <Have you ever seen him sleep?>\n\n<He has infrequently and briefly entered states that are sleep- _like,_ but that is all.> The Box's voice was soft in her mind, soothing. <I will ensure he does not intrude. If he were to find you asleep\u2014>\n\n<He'd suspect something is up. I know.> She commanded her first officer's chair to unfold, allowing her to recline more comfortably. Remembering everything Nemeth had said about the importance of finding the enemy, she asked one last question:\n\n<You recognized Cane, and you knew Jelena Heidik's name. Would you be capable of recognizing another clone warrior if you came across one?>\n\n<It is possible,> the Box said. <Either by name or face, if neither had changed. However, I suspect that given time and opportunity, both would change to reflect the environment the clone warrior is trying to infiltrate. Remember, Morgan, that both Heidik and Cane were recent awakeners; and remember also just how effectively Heidik disguised herself as an outrigger.>\n\nRoche nodded. <There has to be another way, then.>\n\n<You may be right,> said the Box without conviction.\n\n<You don't sound too confident, Box,> she said.\n\n<I'm not, Morgan.>\n\nThe blunt and frank response surprised Roche. She had grown accustomed to the AI's self-assurance, and despite feeling a certain trepidation at times, had come to take comfort in the idea that she could rely on the Box. To hear its uncertainty now was somewhat unsettling.\n\n<Hey, who knows ?> she said lightly, trying to reassure herself as much as the Box. <Maybe things won't be as bad as we imagine.>\n\n<Maybe,> agreed the Box. <But I fear they will get much worse before they get better.>\n****\n\n**6**\n\n****\n\n****\n\nIND Ana Vereine\n\n955.1.31\n\n0050\n\nWhether the Box's prediction had been specific to their journey or not, it turned out to be correct. Three hours after falling dreamlessly asleep, Roche woke to the sound of alarms: a Kesh interceptor was moving in to attack. The alarms brought the rest of the crew to the bridge, where Roche, still shaking off sleep, coordinated their response.\n\n\"How the hell did they find out who we were?\" she muttered to no one in particular.\n\nThe interceptor\u2014not one of the two ships she'd had her eye on earlier\u2014was determined. Its relentless assault ended only when Defender-of-Harmony Vri dispatched it with a sustained blast from his A-P cannon. Before they could even begin to work out what to do next, an entire Kesh squadron slow-jumped to their location and opened fire.\n\n\"I have no idea,\" said Haid, operating the weapons systems with Cane as fast as he could. \"But they want us ** __**_real_ bad.\"\n\nRoche glanced up from where she and Kajic were plotting evasion tactics and escape routes. \"It's only one squadron,\" she said encouragingly.\n\n\"One could be enough,\" said Haid. \"And to jump like that, at a moment's notice\u2014they must've been waiting for the word. This didn't happen on a whim, Morgan.\"\n\nRoche returned to the task at hand without agreeing or disagreeing. The sound of incoming weapons-fire was distraction enough without trying to have a conversation at the same time.\n\n<They want revenge,> said Maii, lifting the information from the minds of their attackers. <Their war with the Olmahoi is going badly, and they blame us for getting them into it.>\n\n\"I can see why they'd miss the _Sebettu_ at a time like this,\" Haid said. \"Which they probably blame us for too.\"\n\n\"Forget the small talk, Ameidio,\" said Roche. \"Stay focused! Maii, how did they find us so _easily_?\"\n\n<They weren't looking for us,> the reave explained. <Since we left the _Phlegethon_ , they have been looking for Vri's vessel. He wasn't camouflaged. They simply deduced that any ship like his with a companion vessel was likely to be the target they were seeking.>\n\nRoche cursed Nemeth's insistence that they take an escort\u2014and herself for allowing this chink in their armor. \"How's Vri doing out there, anyway?\"\n\n\"Exceptionally well, actually,\" admitted Haid. \"There are only six ships left. If I were them, I would've called off the attack long ago.\"\n\n\"They won't do that,\" said Cane. \"Theirs is a suicide mission: it's a matter of win or die.\"\n\nRoche didn't need him to tell her that. The Kesh pilots were fighting for their lives in the truest sense of the expression. Failure was not an option.\n\n\"Well, I hope they've made their peace with Asha,\" said Haid. \"Because the way Vri's going out there, they'll be meeting her pretty soon.\"\n\n\"Kajic,\" said Roche. \"Tell Vri to dock when this is over. I want him in _close_ from now on, under our camouflage. He's going to be one hell of an inconvenience if he keeps on giving us away like this.\"\n\n\"Yes, Morgan,\" said Kajic.\n\n\"And Maii,\" Roche went on, \"can you determine _who's_ been leaking information to them?\"\n\n<Their orders came from superior officers. Beyond that, they know nothing.>\n\nRoche cursed again, although the news wasn't all bad. Someone had set them up, yes\u2014but that someone had only known who they'd be traveling with, not _where_ they were headed. This at least put to rest her fears of an ambush at Perdue.\n\nNevertheless, it was frustrating. Word about her was obviously continuing to spread. The only time she'd been left alone since arriving at the system was while under the protection of the council. She wondered if the superior camouflage technology of the Skehan Heterodox alone was sufficient to explain that brief lull.\n\nCane exploited weaknesses in the engineering of three of the Kesh fighters, to cripple rather than destroy them. The remaining three were taken out by Haid and Vri with less compassion, or less skill. Roche plotted a high-energy course away from the area, which Kajic set off upon the moment Vri's ship was safely enclosed within the _Ana Vereine_ 's camouflage field. Disguised as an innocuous freighter, they accelerated rapidly toward the sun.\n\n\"Our route takes us through or near several densely occupied regions\u2014regions we know next to nothing about,\" Kajic warned. \"We're battered but by no means unable to fight. However, I am going to require some time to do repairs.\"\n\n\"I understand,\" said Roche. \"Vri, do you know anything about where we're going?\"\n\n\"No.\" The Surin's stolid mien was unchanged by the battle. Despite being docked to the _Ana Vereine,_ he remained locked in his ship, ready for anything. \"I suspect that no matter where we go we will enter regions in which the risk of conflict is high. Such is the nature of this environment.\"\n\n\"All we can do, then, is keep our guard up.\" The two ships that had followed them from the _Phlegethon_ seemed to have wandered off, but that didn't reassure her. If someone was still watching the _Ana Vereine,_ their new attempt at camouflage wouldn't fool them, and neither would the change of course. And chances were that the Kesh probably weren't the only people who had known about the Surin escort.\n\nDespite that, when they ran afoul of a minefield an hour later, then triggered a security alert two hours after that, the occurrences seemed unconnected to their mission. They were random incidents exacerbated by the tension and uncertainty in the system. As they traveled closer to the sun, then past it, the density of ships, and therefore the possibility of conflict, increased. Their sheer velocity was considered by some a serious threat, especially with so much debris already filling the system. They spotted two hulks in close orbit to the sun\u2014strange spindly things that looked as though they'd been tied in knots. Roche couldn't tell how they'd been scuttled; she couldn't even imagine how they'd looked before being damaged.\n\nThey passed beyond the innermost regions and reached the domain of the ring. Ships seemed to avoid the dust-filled area, choosing orbits that arced out of the ecliptic or never crossed its aegis. Apart from the ablative effect of the dust on shields and hulls, Roche could see no good reason to take such dramatic steps, yet she did the same. There may have been a reason of which she was not yet aware.\n\nThe ring itself didn't look like much by visible light. Viewed in artificial colors revealing frequencies in the infrared and ultraviolet, and shown in rapid motion so that all the observations Kajic had made since their arrival in the system roughly two days before lasted only a fleeting minute, strange patterns swirled through the dust like standing waves in a torus made of water. What this meant, if anything, Roche didn't know, but it did give her something to look at apart from the endless parade of other vessels. Compared to Palasian System, Sol had very little to offer in the way of natural spectacles.\n\nBeyond the ring, their velocity decreased. The number of ships in the region surrounding them decreased also, until they reached a distance from the sun similar to that maintained by the _Phlegethon._ Roche recalled how crowded it had seemed when they arrived; now it felt like a vacuum.\n\nAs a result, she was forced to concede the possibility that their close pass by the sun might have shaken off any pursuit. Fifteen hours into their voyage, and feeling the effects of another long, stress-filled stint on the bridge, she decided it was safe enough to call another break. Kajic, not knowing that she'd had little sleep while the Box kept watch, insisted that she retire to her cabin\u2014or at least get something to eat.\n\nThe latter she couldn't argue with. Leaving the ship in Kajic's capable hands, she went to the mess and ate as much of a standard meal as she could stomach. Then, anxious about what lay ahead, she went to her cabin.\n\n* * *\n\nThe whirring of thousands of electric scalpels disturbed her rest. Tiny machines, ranging in size from a pinhead to her thumbnail, were drilling somewhere nearby. They burrowed. They buzzed. The noise was maddening.\n\nIt seemed to be coming from inside her mattress, or possibly from under the bed. She got up and turned on the light to look, but there was nothing there. Nevertheless, the sound continued\u2014 but behind her now. She turned. The room was empty. Still the noise persisted, growing louder\u2014whining, sawing, grating.\n\nThen something tickled her ear. She flicked it away in irritation: a black speck, like a bug. Another ran down the back of her neck. She flicked it away too, and felt more. She shook her head violently as a sense of unease rushed through her.\n\nThe noise became louder. It was coming from behind her head.\n\nIn the mirror, she saw dozens of minuscule machines crawling through her stubbled hair, the area blurred and hazy from the frenetic movements of their razor-sharp mandibles. She brushed them away in fright, but others quickly took their place. She couldn't get rid of all of them; there were just too many.\n\nWith a growing sense of horror, she turned her head to one side to see the hole in the back of her skull, where hair, skin, fat, and bone had been carefully cut away, allowing the tide of machines egress from where they lived _inside_ her....\n\nShe woke with a start to the buzzing of her alarm.\n\nSitting up, she ran a hand across her scalp and tried to gather her thoughts. Her first concern was for the ship. A quick check of her implants showed that she had been asleep for almost five hours\u2014the longest she could recall sleeping for ages. Presumably nothing dramatic had happened, or else she would have been awakened, but she'd be surprised if nothing had happened at all.\n\n\"Box?\" She swung her legs out of bed and thought about standing. She needed a shower and a change of clothes. All she could smell was the sweat the nightmare had left on her skin.\n\n\"Box?\" she said again. \"Why the hell aren't you talking to me?\"\n\n<Because I was destroyed in Palasian System,> it said into her mind. <Remember?>\n\n<Damn!> She cursed her stupidity. It was just fortunate that she hadn't made the mistake of speaking out loud to the Box with the others present. <I totally forgot.>\n\n<You have to be careful, Morgan.>\n\nShe ignored the reprimand and headed off for the showers. <Any dramas while I was out?> she asked.\n\n<Nothing of any consequence,> the Box replied. <We will arrive at Perdue Habitat shortly. Population eleven thousand, plus or minus a few hundred passers-through. Our IEPC clearance has taken us this far, but Haid suspects we'll be challenged once we're in range of their cannon.>\n\n<Cannon?> A wave of hot water hit her skin. <What sort of habitat is it?>\n\n<Ex-military. It's seen some service in the past, by the look of it, although it's not what you'd call serviceable now>\n\n<Who runs Perdue?>\n\n<That has yet to be determined,> the AI said. <Kajic has spoken to three different people so far, each claiming to be the representative of the local powers-that-be. They don't seem to be exchanging information. I suspect Perdue may be in conflict with one of its neighboring habitats. The transmissions we are receiving could be coming from more than one source in the command chain.>\n\nRoche exhaled heavily, and breathed in steam. <I assume Nemeth gave us _some_ information on who to contact. A name, at least?>\n\n<He did give us a name, yes: Atul Ansourian, the self-styled _eminence grise_ to the Administer of this habitat and, by default, of the region. The Ulterior had come to some sort of arrangement with him, I gather, in which he traded resources for assistance when or if it was required. He was the one who was to have provided access visas and such support as we would require to carry out our mission.>\n\n_< Was_ to have provided?> said Roche. <Why past tense?>\n\n<Because he is dead,> said the Box. <He was murdered yesterday.>\n\n<About the same time we left the _Phlegethon, >_ she mused.\n\n<It would seem so,> it said.\n\n<Do you think it's a coincidence?>\n\n<I couldn't say, Morgan.>\n\nShe savored the last few moments of the shower. _ <Eminence grise,_ you said.>\n\n<Yes. His rank was only adviser, but he wielded the real power. From isolation, apparently; according to the files he was something of a recluse.>\n\n<It's a shame the two of you never got the chance to meet,> she said, stepping from the cubicle. <You would have gotten along well.>\n\n<Do I detect a hint of sarcasm, Morgan?>\n\nIgnoring the remark, she began to dry herself. <So, let me guess,> she went on. <The murderer is still at large, identity unknown?>\n\n<Far from it. She is in custody in Perdue Habitat and has already pleaded guilty to the crime.>\n\n<Really?> said Roche. <Things aren't normally so cut and dried.>\n\n<Indeed, Morgan,> said the Box. <There are several legal and ethical problems dogging the case. Although having said that, no one really expects her to escape the death penalty in the end.>\n\n<What sort of ethical problems do you mean?>\n\n<It appears that the girl is Ansourian's daughter.>\n\nRoche wondered if any of this would prove relevant, but noted it anyway. <How do you know all this, Box?>\n\n<I took the liberty of infiltrating the habitat's news services when we were first contacted. Once I had the correct frequencies, it wasn't difficult.>\n\nRoche nodded thoughtfully as she finished toweling herself down. <Tell me about this Perdue Habitat, then,> she said, slipping into a simple, unadorned uniform. <Is it segregated or open?>\n\n<The dominant Caste is called the Vax, but they allow free movement from the other two main habitats. Tocharia 13 is occupied solely by members of the Zissis Caste; Random Valence is an open space designed for trade. I would opt for the latter if I had the choice of destination, but there's no denying that Perdue Habitat has the influence over the region, and is therefore the main congregating point.>\n\n<How long until we're ready to dock?>\n\n<One hour. Cannon will come to bear, if Haid is right, in ten minutes.>\n\n<Guess I'd better get moving, then.> She took one last chance to be still, standing in the middle of the room and breathing deeply three times. Then, rubbing vaguely at the back of her head, she set off for the bridge.\n\n* * *\n\n\"As I said, I don't _care_ who you say you are,\" said the figure on the main screen. \"You have _no_ papers we recognize, _no_ jurisdiction over us, and, as far as I can tell, no reason to even _be_ here. Therefore, we have no reason to let you dock. So unless you change your orbit and move away, I will assume your intentions to be hostile and be forced to take appropriate action.\"\n\n\"And I've told _you_ ,\" Roche said. \"We had private business with Atul Ansourian. He was supposed to meet us here!\"\n\n\"I'm not stupid, Roche,\" said the official, his bald, yellowish scalp crinkling as he spoke. \"Your ship is camouflaged, and you won't tell us what your business with Atul was. Yes, we've heard of you, but not through him. He never mentioned you at all.\"\n\n\"There has to be someone else there we can talk to, surely?\" snapped Roche.\n\nThe man sighed tiredly. \"I can pass your query through to the administer if I really have to, but I don't think it'll do you any good.\"\n\n\"I don't care what you think,\" Roche said. \"Just get her on the line! I'd rather talk to her than waste my time with you.\"\n\nThe line closed without another word from the man. Roche vented her frustration by thumping the station in front of her.\n\n\"Maybe we should just try bribing him,\" said Haid.\n\n\"On an open line?\" She shook her head. \"That'd just give them another excuse to turn us away.\"\n\n\"And if they turn us away, anyway?\"\n\nShe looked over at Haid and forced a smile. \" _Then_ we might give it a try,\" she said.\n\nFive minutes later, the line opened again to reveal another yellow-skinned, bald male. Except that his face was rounder and his eyes more deeply set than the previous official, Roche would have had trouble distinguishing between them.\n\n\"I am Dockmaster Rench,\" he said, his voice smooth. \"I apologize for the misunderstanding. Dock 14-B will be cleared for your approach\u2014with the proviso that you drop your camouflage and declare your crew. Should you fail to comply with these conditions, access to this habitat _will_ be denied.\"\n\n\"Agreed.\" Roche's response was immediate; she had little choice. She instructed Kajic to reveal the _Ana Vereine_ and Vri's ship to the habitat; then she named each of her companions in turn. \"Is that sufficient, Rench?\"\n\nThe dockmaster studied something off-screen. \"I don't recognize your configuration. Somewhere local?\"\n\n\"Dato Bloc, a Commonwealth of Empires splinter government.\" She figured it didn't hurt to be open about some things.\n\nHe nodded. \"Looking a bit rough around the edges for something clearly so new.\"\n\n\"We've seen some action,\" she admitted.\n\n\"Who hasn't?\" He half smiled. \"Prepare to dock, Roche. I'll have someone meet you down there.\"\n\nKajic followed navigation buoys into the crowded docks. Numerous ships of various types occupied most of the available gantries; some seemed to be undergoing repairs while others were idle, perhaps loading or unloading cargo and passengers. Most of them were support craft for the various military forces massing in the system. Roche recognized a COE Armada cruiser among them, although the name painted on its side\u2014 _Paraselene\u2014_ didn't ring a bell.\n\nWith a clang, the _Ana Vereine_ docked with the massive structure. Shaped like a mutated sea anemone, the former military station had sprouted numerous access tubes and containers, crossing and recrossing, branching and rebranching away from a barely glimpsed central section. Its asymmetry reminded Roche of a coral, yet its angular edges and corners made her think of crystal deposits.\n\nOutfitted with side arms and hazard suits, Roche and Maii stepped from their ship into the dock's grease-smelling antechamber. The entire area rang to the sound of metal striking metal, over the rumble of a thousand voices speaking at once. There was a striking contrast between the habitat and the vast empty spaces of the _Phlegethon._ It seemed to be full of crates, machines, and people of all shapes and sizes. None of it looked familiar to Roche, used to the homogeneity spread by the Eckandi Trade Axis.\n\nFrom among the bustle, a woman stepped forward to greet them. Short and muscular, wearing a purple uniform with black trim and a close-fitting cap, she had the same yellowish tinge to her skin as the other two officials Roche had spoken to. She assumed that they were all members of the Caste the Box had mentioned to her earlier: the Vax.\n\n\"Hello,\" said the woman. Her voice was brisk but not unfriendly, and raised slightly to be heard above the clamor of the other voices around them. \"I am Overseer Pacecca. Dockmaster Rench sent me to welcome you.\"\n\nRoche introduced herself and Maii. Pacecca eyed the girl's blank visor for a second, then asked: \"Your friend is blind?\"\n\n\"Yes.\" That seemed the simplest answer. \"Her suit's navigation systems are linked to mine; she won't get in the way.\"\n\n\"Very well.\" Pacecca looked around her, as though realizing for the first time just how busy it was. \"Perhaps we should go elsewhere to discuss why you're here.\"\n\n\"I'd prefer to talk to the administer,\" said Roche.\n\n\"There isn't much chance of that, I'm afraid,\" said Pacecca. \"She has taken the loss of Atul Ansourian very badly. You probably won't get to see her for a while, when things settle down.\"\n\nThe implication that the habitat failed to run without Atul Ansourian around backed up everything the Box had said. \"Nevertheless, I'd like to try.\"\n\nPacecca looked at her evenly, patiently. \"Very well. I shall see what I can do. My assistant\u2014\" The overseer looked around irritably. \"Quare!\" she barked.\n\nA man stepped forward from the crowd, dressed in a uniform similar to Pacecca's, but green with gray trim. He looked like any number of faceless, middle-management lackeys Roche had seen over the years\u2014slightly overweight, balding and stooped, yet with eyes that watched everything, keen to find an advantage.\n\n\"Yes, Overseer?\" he said softly.\n\n\"This is Quare,\" Pacecca said to Roche. \"He will take you somewhere quieter.\" She paused thoughtfully, as if considering her options.\n\n\"Perhaps Stateroom B?\" the little man suggested.\n\nShe scowled at him. \"Remember your station, Quare,\" she warned disdainfully. \"However,\" she continued, turning her back to him, \"Stateroom B _will_ be fine.\"\n\n\"Yes, Overseer,\" said Quare, his head lowered.\n\nPacecca nodded, then faced Roche once more. \"I'm sorry if we're not more hospitable,\" she said, distracted by something happening on the other side of the dock. \"But what with the murder and the trouble with Guidon...\" She shrugged helplessly. \"Things have just been falling apart around here, I'm afraid. So, if you'll excuse me, I'll have to talk to you later.\"\n\nRoche barely had time to nod before the woman was off. She didn't doubt that \"later\" meant \" _much_ later\"....\n\n<What was that she said they were having trouble with?> she asked the Box.\n\n<Guidon,> said the Box. <A sibling-habitat that appears to have been recently destroyed. It seems that most of these people around you are refugees and survivors.>\n\nQuare stepped forward. \"This way, please.\" Natural caution made Roche double-check: \"Where did you say you were taking us?\"\n\n\"Somewhere to wait,\" he said. \"Away from all of this.\" He gestured at the chaos around them. His expression remained blandly pleasant, with a hint of indifference. \"The overseer will report to the dockmaster, who will in turn report to his superior. Your request to speak to the administer will be forwarded to her in due course. I'm sure it won't take too long.\"\n\n\"How long, exactly?\"\n\n\"No more than a couple of days, I'm sure,\" he said.\n\n\"A couple of _days_!\"\n\nHe nodded. \"Perhaps a little longer,\" he said. \"If you'd care to follow me\u2014\"\n\n\"We don't have the time to sit around doing nothing while your precious administer decides whether or not to see us!\" Roche was finding it difficult to keep her annoyance in check. \"And even if we did, I'd do it on my own ship!\"\n\n\"That is your decision, of course,\" he said. \"We would not expect you to...\" He stopped, suddenly turning his attention to Maii. \"Why is your reave attempting to probe my mind?\"\n\n<Maii? What are you doing?>\n\n<Trying to get in. He's blocking me>\n\n<How?>\n\n<He has a very powerful, learned shield. I can't... quite ... get through it\u2014but I'm sure he can't keep it up forever!>\n\n<Was Pacecca the same way?>\n\n<Couldn't have been more different. She was as clear as glass.>\n\n<Then he's hiding something.> Roche considered her options for a moment, then said: <Stop, for now, and we'll go with him. Try again later, but be more subtle if you can. I'd like to know why he can do it but Pacecca couldn't.>\n\nSomething in Quare's face relaxed. \"Thank you,\" he said, to both of them. Then to Maii in particular: \"Please do not try that again. It is considered by my people to be highly impolite.\" And to Roche: \"Now, do you wish to return to your ship?\"\n\n\"No, we'll come with you,\" she said. \"For now, at least.\"\n\n\"Very well,\" said Quare, then turned and led them through the chaotic activity on the dock.\n\n<Looks like we're wasting our time,> said Haid via her implants.\n\n<Maybe.> Roche relayed what Maii had told her. <He might lead us somewhere.>\n\n<And he might not. Maii can't even read him, Morgan. For all you know, he could be one of the enemy!>\n\n<Now you're being overly suspicious,> Roche said, hoping Haid was wrong. <Why would the enemy clone someone who looks like him?>\n\n<To fit in, of course,> he shot back. <I don't know. It's just frustrating to sit here and watch. Even if you do get past him, it doesn't sound like this administer is going to be much help.>\n\n<But without her, we've no way of getting anything done.>\n\nQuare took them through two large hangar doors, then along a corridor lined with a silvery metal. A hairpin bend brought them to another chamber, where it was at least quieter if still crowded. He waved them through some sort of security checkpoint, then took them deeper into the habitat.\n\n<What exactly _do_ you hope to get done?> Haid asked.\n\n<Whatever Nemeth sent us here to do: spy, I guess. Look for the enemy. To do that, we need access to all areas and as much data as we can lay our hands on.> The Box would take charge of the latter, but she needed to put up a front for its behavior before she started producing conclusions based upon it. <With Ansourian gone, we don't really have anyone else to go to but this administer.>\n\n<What about Ansourian's daughter?> said Haid. <She might know if he'd had anything planned for the Ulterior, should one of their agents come to town.>\n\n<Yes, but I gather she's locked up somewhere. I doubt we'd get near her.>\n\nHaid grunted his dissatisfaction with the situation. <One other thing I want to know is: what's the administer's name? The way they talk about her, she sounds like she's handling things pretty badly.>\n\n<Her name is Inderdeep Jans,> said the Box to Roche. <It seems she inherited her position when her father, the previous administer of the habitat, died eight years ago.>\n\nRoche couldn't pass that on without explaining where the information had come from. <Try hacking into the news channels she said. <Make yourself useful.>\n\nHaid grunted again. <I'll see if I can find a map, for starters.\n\n<I have arranged the information so that it will be easy to access,> the Box whispered to Roche.\n\n<Excellent.> To Haid she replied: <Let me know if you find anything unusual, Ameidio.>\n\nShe returned her attention to where they were going. The journey seemed to be taking a while, and had brought them to a relatively clean and quiet section of the habitat. White walls and ceiling and a gray floor made the area seem sterile, although the air smelled vaguely of Human sweat.\n\n<I'm still not getting through,> said Maii. <But at least he's not sensing me probing at the moment.>\n\nRoche remembered what Nemeth had said about using epsense to find a way past the enemy's natural camouflage\u2014and Haid's half-serious suggestion that Quare might be a clone warrior. <Does his mind feel at all like Cane's?>\n\nThe girl sent a mental shrug. <There _is_ something odd about him. I can use his senses, like I can with Cane when he lets me. But he doesn't _feel_ like Cane. Then again, apart from Cane, I've never really studied another clone warrior, so who's to say they'll all be exactly like him?>\n\n<What about the business of the black speck?>\n\n<That was just the _irikeii_ 's impression. He saw much deeper into things than I can. I can't see that speck in Cane at all.>\n\nRoche mulled this over. <So although you don't think Quare is a clone warrior, you can't be certain, right?>\n\n<That's right, Morgan.>\n\n<Not very reassuring, Maii.>\n\n<None of this is.>\n\nQuare stopped at a door midway along the curving corridor they were following. He produced an old-fashioned key from his pocket and inserted it into a lock in the center of the door. It clicked open, and he gently pushed the door inward. It retreated a foot, then swung smoothly to one side, reminiscent of how some airlocks operated.\n\nHe took two steps inside, then gestured ahead of him. \"Stateroom B,\" he said. \"You will be comfortable here.\"\n\n\"Not if I have to wait two days, I won't be.\"\n\nHe didn't smile. \"We shall see,\" he said, then urged them inside: \"Please...\"\n\nRoche hesitated.\n\n\"I can stay with you if it will put your mind at ease,\" he said, seeing her apprehension.\n\n\"That's okay,\" said Roche. \"Just leave us the key, and we should be fine.\"\n\n\"I'm afraid I can't do that,\" he said. \"Besides which, it is ineffective from the inside anyway.\"\n\n\"In that case,\" said Roche, \"after _you_.\"\n\nHe shrugged easily and stepped all the way into Stateroom B, which consisted of three connected rooms. From the comfortably furnished antechamber, Roche could see a conference room, with what looked like a small kitchen or toilet facility beyond that. There was a stale smell about the place, as though the air vents hadn't been cleaned for a while.\n\n<No one here but us,> said Maii. <That much I can tell.>\n\nDeciding that they could deal with him if he tried anything, Roche stepped into the antechamber with Maii right behind her.\n\n\"Would you like refreshments while you wait?\" Quare asked. \"A drink, perhaps?\"\n\n\"No, I'm fine.\" Maii also declined.\n\n\"Then perhaps you would like to rest your feet.\"\n\nRoche glanced around at the soft-cushioned chairs in the room, the legs so slender and graceful they looked as though they couldn't take so much as the weight of Maii's undersized hazard suit.\n\nShe laughed. \"No, I really don't think\u2014\"\n\nMovement out the corner of Roche's eye startled her: the door was sliding shut.\n\n\"Security,\" said Quare, catching her alarm. \"We could not guarantee your safety if just anyone could get in.\"\n\n\"Nor the habitat's if we were to get out, right?\" said Roche cynically.\n\nThe little man smiled briefly, but it didn't touch his eyes. The door clicked shut. \"Now, about that seat...\"\n\n\"It's not necessary,\" said Roche stepping over to the door to check it.\n\n\"We have something more practical through here,\" Quare said, waving them farther into the suite. \"Come with me, please.\"\n\n<Morgan!> Maii's sudden interjection was loud in Roche's mind. <The ship has lost contact with our telemetry feeds!>\n\n<What?> Roche slid her helmet closed and studied its instruments. Sure enough, it hadn't received a return signal from the ship for almost half a minute. But Maii wouldn't have seen that: she must have learned from someone else.\n\n<Uri?> she called. <Ameidio?>\n\nThere was no answer.\n\n<Tm telling them we're okay,> Maii said, <but they don't know what to do.>\n\nAngry, Roche drew her side arm and followed Quare into the conference room, where she grabbed him roughly by the shoulder and spun him around.\n\n\"What the hell is going on?\" she demanded. \"Why have we been cut off from our ship?\"\n\nHe stared at her helmeted visage, visibly startled. \"I don't understand\u2014\"\n\nShe wasn't in the mood for denials. \"Just open that damned door now,\" she said. \"We're leaving.\" When he hesitated, she snapped, _\"Now!\"_\n\nHe drew himself up in her gauntleted grasp. \"No.\"\n\nShe pushed the pistol into his cheek: _\"Yes.\"_\n\nHe flinched but didn't relent.\n\n<Haid wants to know if you need reinforcements,> said Maii.\n\n<Tell him...> Roche thought for a second. <Tell him to be ready just in case.>\n\n\"You won't be harmed,\" Quare was saying. \"I promise you. This _isn't_ a trap.\"\n\n\"You've locked us in here!\" Roche said, her voice rising with her anger. \"You've severed our communications with my ship! What would _you_ call it?\"\n\n\"An opportunity,\" he said, wincing as the pistol dug deeper into his cheek. \"An opportunity to talk.\"\n\n\"I've got nothing to talk to you about. Let us out of here.\"\n\n\"Look, you can see I'm unarmed. Can't you at least put your weapon down? Please?\"\n\n\"How do I know there aren't troops waiting just outside?\"\n\n\"You don't,\" he said. \"But I assure you there aren't.\"\n\nRoche snorted derisively. \"What the hell do I care about your assurances?\"\n\n\"Don't be stupid, Roche,\" the man snapped. \"Think about it! The administer wouldn't waste her time on a stunt like that.\"\n\n\"But _you_ might,\" said Roche.\n\n\"I might consider it, yes,\" he said. \"If I was truly desperate. But I'm not. Not yet, anyway. So again I ask you, please _hear me out._ If you've been cut off from your ship, then that only proves that I've done the right thing by bringing you here.\"\n\nThis took Roche aback. \"What? Why?\"\n\n\"This is a secure area,\" he said. \"Electronically speaking, no one can get in or out. Once the door is shut, we're sealed in.\"\n\n\"And why is _that_ so important?\"\n\nHe stared at her then with a look that could not possibly be misinterpreted: it was desperation.\n\n\"Because my real name is Atul Ansourian,\" he said. \"I need your help. Without it, my daughter\u2014and maybe everyone else on this habitat\u2014will die.\"\n****\n\n**7**\n\n****\n\nPerdue Habitat\n\n955.1.32\n\n0150\n\nRoche held on to the little man for a while longer, searching his eyes for some sign of a lie. When she failed to find it, she let him go, saying, feebly: \"But you're _dead_!\"\n\n\"A necessary ruse, I'm afraid,\" he apologized. \"I needed to disappear in order to survive. If I hadn't done that, the chances are I really would be dead right now.\"\n\n\"But Pacecca\u2014\"\n\n\"Doesn't know anything,\" he said, cutting her short. \"To her I'm just another faceless drone to boss around. And that's what I want her to think. Her mind is weak. I couldn't trust the likes of her with the truth; she'd be too easily read.\"\n\nRoche remembered how Maii had described the woman's transparency. Quare\u2014no, _Ansourian,_ if he was to be believed\u2014was making sense in this respect, at least.\n\n\"How do I know you're telling me the truth?\" she said.\n\n\"I'm not asking you to trust me,\" he said. \"All I ask is that you hear me out.\"\n\n\"Why?\" said Roche.\n\n\"Because I think we can help each other,\" he said. \"At most I've only got another day or so before the truth comes out. And once that happens, there is every chance that _both_ my daughter and I will wind up dead.\"\n\nRoche was curious despite herself. \"But if your daughter's going to die for killing you, why not confess to the truth so she'll be set free?\"\n\n\"It's not that simple.\" Ansourian stepped over to the conference table and sat in one of the chairs. \"Please,\" he said, gesturing to the chairs opposite him.\n\nRoche glanced at Maii. The girl was still, concentrating.\n\n<Maii?>\n\n<I'm still not getting anything from him,> she said. <But there's definitely nobody waiting for us outside, Morgan. It seems that no one is too concerned about us being here.>\n\n<Okay. Tell the others to sit tight. I want to find out what the hell is going on here.>\n\nRoche took a seat at the wide wooden table opposite Ansourian. The roomy, low-backed chair creaked beneath her weight, but held. Maii positioned herself a couple of seats down.\n\n\"You've no doubt heard the official story,\" said Ansourian.\n\n\"That your daughter killed you a couple of nights ago and then turned herself in?\" said Roche. \"Yes, we had heard something.\n\nAnsourian nodded, his expression earnest. \"It's an open and shut case,\" he said. \"Security has a body and a killer, with no evidence to suggest anything out of the ordinary. But for the fact that my daughter will almost certainly be charged with patricide if I maintain the fiction of my death, I would be content to let the situation rest. But obviously I cannot do this. In the next day or so the deception will be exposed, and my daughter will be forced to reveal the truth.\"\n\nRoche was still wondering what she had to do with this. \"And then what?\"\n\nAnsourian shrugged. \"There is no legal precedent for this situation,\" he said. \"Understand that we follow reproductive customs that are regarded as unusual by many Castes. The Vax do not have two parents as most do; we have just the one, who creates a child by combining his or her own genetic code with another's, sometimes chosen at random. The child, always the opposite gender of the parent, is gestated artificially, then released to its parent\u2014and that parent is the sole caregiver for that child. But just as we have only one parent, so do we have only the one child. Perhaps you can appreciate that the bond between father and daughter or mother and son is _very_ strong.\"\n\n\"So the murder of one by the other,\" said Roche, \"would be considered one of the worst crimes imaginable.\"\n\n\"The most heinous of crimes,\" he said. \"Punishable by death. It doesn't matter if the child is murdered or the parent, the consequences are the same: _two_ lives are ultimately lost\u2014and along with them is lost a long line of descent.\"\n\nRoche could understand what he was saying, but she still didn't see the relevance of it all to herself.\n\n\"You say security has a body,\" Roche broke in. \"Did you clone yourself and kill the clone?\"\n\nThe look of surprise and disgust was genuine. \"No, of course not!\" he said. Then, seeing Roche's confusion, Ansourian took a deep breath and continued slowly. \"Please understand that this is very difficult for me. Under normal circumstances, I am very much a recluse; I am uncomfortable with face-to-face contact. Only one person is allowed into my chamber and knows my face\u2014and that is my daughter, Alta. Until two days ago, she shared my apartment in a high-security wing of the habitat not far from where the administer herself lives.\"\n\n\"Alta lived with you?\"\n\n\"Yes, and would have until I died, with her son\u2014should she have chosen to bear one, of course. But she is not as antisocial as I. Although she respects the lifestyle I have chosen, she does not feel the same need to remain isolated from the rest of the community. She works\u2014or worked, I should say\u2014in the Logistics Department, supervising the distribution of resources that pass through here to those who need it the most. Perhaps she was reckless in believing that the situation was not as dangerous as indicators suggested\u2014and it does seem that my opinion on that score has been vindicated. But the fact remains that had she not gone out and returned when she did she would have died with me, or I would have died alone.\"\n\nRoche listened closely. Again, the subtleties of Vax relationships escaped her. Did they take lovers from outside the family line, or was incest the norm? The question was irrelevant, yet it nagged at her just the same.\n\n\"Two nights ago,\" he went on, \"Alta returned home late. When she came in to say good night, she found me asleep and another person in the room with me. This person, she says, was in the process of giving me a dose of poison that would have killed me in seconds and left no trace whatsoever.\" Ansourian stopped for a moment before going on. \"Alta is a proficient fighter, Roche. Perhaps too proficient. She killed the assailant with little effort, but she did so before we could determine _who_ he worked for.\"\n\n\"But did you at least find out who _he_ was?\"\n\nAnsourian shook his head. \"He carried no papers,\" he said, \"nor did he have any DNA files in the habitat records. And I was not in a position to call security to find out, either. Disregarding the fact that I had already explored all the avenues they have open to them, my would-be killer had not broken into my rooms by force; he simply walked through my extensive security system as though it hadn't even existed. Someone _must_ have shown him how to do that, and only a handful of people have access to that information.\"\n\n\"They're all high-up in the security chain, no doubt,\" Roche put in.\n\nAnsourian leaned forward on the table, nodding. \"I couldn't risk reporting the incident for fear of alerting whoever was responsible that they had failed.\"\n\n\"So you had to find a way to make the problem disappear, in other words,\" said Roche, wanting him to get to her relevance in this scenario.\n\nHe nodded again. \"Smuggling the body out of the habitat was not an option, either,\" he said. \"The moment I stepped out of my room, my enemy would have known something had gone wrong and would make sure security was watching every dock. And I couldn't keep the body in my rooms for any length of time for similar reasons. There seemed to be no avoiding the fact that I had survived by mistake; no matter which way I turned, that mistake looked likely to be rectified soon.\n\n\"The only way I could hope to find out what happened was to doctor habitat records to indicate that the body was mine, and convince whoever was responsible that I was dead. Under the cover of an alias I could watch to see what happened next: who would be looking to take over my position; who would advocate a speedy trial to see the matter closed quickly\u2014\"\n\n\"Basically,\" said Roche, \"who would benefit the most from your death once the dust had settled on the whole unpleasant affair.\"\n\n\"Exactly,\" said Ansourian. \"The most difficult problem to get around was the fact that Alta had left genetic evidence all over the body. But there was no avoiding that. We figured in the end that it would be best if she turned herself in and thereby forestalled a thorough inquiry. Her story wouldn't stand up under a detailed forensic examination, and no doubt my enemy is puzzled as to why my assassination didn't go quite as planned\u2014it must have startled him to see Alta accused of the crime, especially if the assassin was supposed to report in, and has not\u2014but I hope his acceptance of the situation will continue a little longer. While Alta is imprisoned, the thought that he might soon discover that the body is actually that of his assassin, and not mine, concerns me greatly.\"\n\n\"But surely he would be aware of that already?\" Roche found this aspect of the story difficult to swallow. \"I mean, didn't habitat files reveal a mismatch between your genetic profile and that of the body?\"\n\nAnsourian shook his head. \"I had my own records removed a long time ago. It seemed a sensible precaution to take, especially for someone in my position. As far as preferring anonymity goes, doesn't it seem reasonable that the person who wielded the true power in this habitat should not seek recognition of any kind? The temptation to use it for personal gain would always be there. And the fact that I could walk the length of every corridor in this habitat and not be recognized by anyone but my daughter actually pleased me. As long as I could continue making the right decisions for Inderdeep to follow, _that_ was the main thing.\"\n\n\"What about the administer?\" Roche asked. \"Doesn't she even know what you look like?\"\n\n\"I couldn't take the chance.\" He shrugged. \"I know this may seem paranoid to you, Roche, but if I _hadn't_ taken such precautions, I might have died a long time ago. Everything I have feared appears to have come to pass. And now, I must find out who tried to kill me, and save my daughter.\"\n\nRoche nodded her understanding. \"This is where I come in, right?\"\n\n\"I can't do this alone,\" he said soberly. \"I need your help.\"\n\n\"Why do you think I should help you?\" she asked. \"It's not my brief to become involved in domestic politics.\"\n\n\"But you are,\" he insisted. \"You are one of the Ulterior's agents, and the Ulterior is dealing with a much larger enemy. Our goals may overlap.\"\n\n\"How?\"\n\n\"A week ago, Guidon, one of Perdue's sibling-habitats, was destroyed.\"\n\n\"I heard about that on the way here,\" Roche half-lied. The Box had found the information en route, but hadn't told her until she arrived.\n\n\"No doubt,\" he said. \"But what you wouldn't have heard is that Guidon Habitat was destroyed from within using security codes known only to a handful of people. Exactly _how_ they were obtained remains a mystery, but I suspect that the enemy\u2014 _your_ enemy\u2014was involved. Perhaps he is my enemy too.\"\n\n\"Why?\"\n\n\"Inderdeep tends toward a policy of indifference and nonintervention regarding the problems we left at home. It took a lot of convincing just to get her here. Ultimately, though, I wonder how much good we can do here, particularly now with Guidon destroyed\u2014but I have always felt that it is important to at least try.\"\n\n\"So it was you who persuaded Inderdeep to come here?\"\n\n\"Yes,\" he said. \"And I've maintained a steady influence over her not to change her mind and return home. Maybe someone took offense at that, finally\u2014this faceless man, pulling her strings so freely. Maybe that someone decided the war effort could do without me helping it along. But if that is the case, then this habitat has already been infiltrated, and we may all be close to the same fate that awaited those on Guidon.\"\n\n\"Let me get this straight,\" Roche said. \"Someone in Guidon, working for the enemy, somehow managed to get the codes that led to its destruction, and now you believe this same person has turned up here on Perdue?\"\n\n\"And is attempting to do the same thing again, yes,\" Ansourian said. \"It would be easy to sneak on board right now. We're still collecting life-capsules from the wreckage of Guidon. In fact, we have inadvertently picked up a couple belonging to the enemy, but we disposed of them before they could open.\"\n\n\"You think someone in an ordinary capsule could have sneaked in unnoticed?\"\n\n\"If they were carrying the right papers,\" said Ansourian, \"there would be no cause to suspect anything. And once in, they could go about working their way up the chain of command. From there it would be a simple matter of working on Inderdeep to change her mind and go home. It would be an efficient way to get rid of the Vax.\"\n\n\"Efficient, yes, but that's not normally how they work,\" Roche said. \"The more destruction and loss of life, the better for them\u2014at least in my experience.\"\n\n\"Perhaps that is a generalization deserving examination,\" he countered. \"The most destructive actions are the ones we see most clearly, and remember. There may be more subtle plots going on around us all the time.\"\n\n<And remember Cane,> said Maii, privately.\n\n\"Perhaps,\" she said in response to both of them. \"But I still don't know _how_ you expect us to help. Your daughter's locked up somewhere. What do you want us to do? Break her out using brute force?\"\n\n\"I'm not naive enough to think that would work\u2014or that you would agree to such an action.\"\n\n\"What, then?\"\n\n\"Your name precedes you, Morgan Roche. If my adversary hears that you have spoken to Inderdeep Jans, he may become anxious. If she can be reminded that the enemy may be whispering to her even now, she might take his advice less to heart. I may yet be able to come out of hiding in a way that will not place me or Alta in any more danger than we already are. I hope to use you, in other words, as a catalyst to change Inderdeep's mind.\"\n\nRoche stared at him for a long time. He wanted to use her in much the same way the Ulterior and the Crescend both did: as a pawn in a personal power game. She wasn't sure she liked this role at all\u2014but neither did she want to rule out the possibility that she could use it to her own advantage.\n\n\"You can get me to the administer?\" she asked.\n\n\"I believe so, yes,\" he said. \"I should be able to get you into her chambers without anyone knowing. You will have as much time alone with her as you need\u2014as long as you can convince her to let you stay. You see, her chambers aren't monitored. Not even by me.\"\n\n\"And what's to stop her simply throwing us out?\"\n\n\"She won't. She has heard of you, and I know her well enough to say that she will be curious.\"\n\n\"If that's so, then why would it take so long to get an _official_ meeting with her?\"\n\n\"Because the chances are she is unaware of your presence right now,\" said Ansourian. \"Whoever is behind all of this is more than likely protecting her from you, making sure your request to meet her goes through official channels\u2014which would ensure a delay of a couple of days, at least.\" Roche opened her mouth to object, but before she could speak, Ansourian jumped in with: \"Believe me, Roche, this _is_ the best option available to us at the moment.\"\n\nRoche carefully considered what he was saying. \"Okay, but once I've talked to her, then what?\"\n\n\"That depends on how it turns out. If it goes as well as I hope it to go, there's a good chance I will reveal myself there and then in order to press home my case. If it goes badly, I will make other plans. I know of various flaws in security's prisoner-holding bays. I may still be able to set Alta free and find a way off the habitat.\"\n\nRoche was under no illusions as to where she might fit into the latter part of such a plan. There was no way, though, that she intended to commit herself to anything but the most basic level of support for Ansourian\u2014who was still, after all, a complete stranger whom she had little reason to trust.\n\n<What do you think, Maii?>\n\n<I think you should follow your instincts, Morgan,> the girl said. <I have little to go on, and would hate to mislead you.>\n\nThat was fair enough, Roche thought. She couldn't ask any member of her crew to give her advice when they didn't have enough information to decide; that was her job, after all.\n\nNot that she thought of Maii as merely a crew member; she had become more than that in the previous weeks, especially after her capture and imprisonment by Linegar Rufo.\n\nRoche had felt bad enough over that; she could only imagine what Ansourian had been feeling since his daughter's arrest.\n\n\"Okay,\" she said. \"Let me talk to the administer and we'll see what happens. I can't guarantee you anything, but it's worth a try.\"\n\n\"Thank you.\" He smiled then. Surprisingly, it looked genuine. \"Inderdeep will not be in her quarters for a couple of hours yet; I will endeavor to find out precisely how long. Also, Overseer Pacecca will be expecting Quare back at some point and I don't want to needlessly arouse suspicion.\"\n\nRoche nodded. \"Can you give me some way to communicate with you?\"\n\n\"I think it's best if you remain completely isolated in here,\" he said. \"Even from your own ship.\" Seeing concern on her face, he added: \"It really _is_ the only way to be certain that you won't be discovered before time.\"\n\nHe seemed sincere, and his reasoning was sound, if a little overcautious. And she did have Maii, after all.\n\n<Would you be able to find him if he leaves us here too long?> Roche asked the girl.\n\n<Yes,> Maii said. <Or I could find someone else nearby and make them let us out.>\n\nRoche nodded. The reave's power to influence those around her, not just read them, hadn't been necessary so far in Sol System. She hoped it wouldn't be necessary at all. At the very worst, though, Maii could force the administer to give them what they wanted.\n\n\"Okay,\" she told Ansourian. \"I'll give you two hours. If we don't hear from you by then, the deal is off.\"\n\nHe nodded. \"Don't worry,\" he said. \"I'll be back before then.\"\n\nThey all stood, and he left the room. The door leading out of the suite hissed open, then clicked shut. There was no handle and no keyhole on the inside.\n\n\"I've got a bad feeling about this,\" Roche muttered to herself.\n\nMaii slid her helmet back and sniffed the air. <Haid says that the inner defenses are fairly poor, now that we're past the perimeter cannon emplacements. He says he could probably blast his way in and get us out of here in under ten minutes.>\n\n<Tell him he's being optimistic. It would take him at least fifteen.>\n\n<He wants to know if that's a bet.>\n\n<No, it's not.>\n\n<He says: Then don't blame me if we arrive too late to save you.>\n\nRoche smiled. <He would say that. He's probably just jealous he's not here himself.>\n\nMaii didn't disagree.\n\n* * *\n\nTime passed slowly. Roche hadn't realized how dependent she was on data from the _Ana Vereine_ 's datapool to keep her occupied. The hazard suit's capabilities didn't include much in the way of sophisticated software. Even though the Box had access to vast amounts of data, she still didn't entirely trust the AI to give her what she wanted. There were too many ways it could exploit her ignorance.\n\n<Do you believe what Ansourian is telling us?> she did ask it at one point.\n\n<His story does match the data I have obtained through the news services\u2014but that doesn't mean it is true. On the face of it, it is somewhat implausible. I do note, however, that Ansourian's security records have been skillfully tampered with, and that his relationship with his daughter has previously shown every sign of being exceedingly positive. If this is _not_ Ansourian, then that leaves us with the original mystery: why did Alta Ansourian kill her father in such a brutal and apparently unprovoked way?>\n\n<Not that it's any business of ours.>\n\n<It is in the sense that it provides us with another reason to entertain this hypothesis a while longer.>\n\n<Is that what all this is to you, Box: a hypothesis?>\n\n<Naturally, Morgan. At present, all I _can_ do is hypothesize.>\n\nShe gave up on that line of conversation. Talking to the Box for too long when it was bored could give anyone a headache. But she needed to do something, too, to stave off her own boredom. Sleep wasn't an option, and neither was eating; her stomach was too tense to make an easy meal of the concentrates stored in the hazard suit's compartments.\n\n<Can you show me what's happening outside here?> she asked Maii.\n\n<Sure,> said the girl. Her mind touched Roche's gently once, then again with more pressure. <Don't be alarmed, Morgan. I'm not going to take you over. I'm just going to give you a glimpse of how I see things.>\n\nRoche forced herself to relax. After all, she had already done this a couple of times before\u2014on Sciacca's World, before Maii had agreed not to go digging around in her mind. As on those occasions, when Maii touched a true sensory experience in someone else's head, that experience conveyed itself to Roche with the same vividness as if it had been her own. She could easily see how the girl survived on the senses of the people around her.\n\nFor a second, she seemed to see an echo of Maii, as she saw directly through her own eyes and _through her own eyes via Maii_ simultaneously. But the effect was fleeting. Her own vision seemed to fold in on itself as Maii moved to another viewpoint.\n\nThey belonged to a woman who was performing repairs on an air filter somewhere along the corridor just outside the quarters they were in. Barely had Roche determined this when Maii skipped to another pair of eyes\u2014these belonging to a courier on his way to deliver a package. A quick succession of viewpoints from various people followed as they moved ever deeper into the habitat, catching glimpses of people Roche didn't know doing things that didn't concern her. Maii never lingered for more than a few moments at a time; no sooner had they found an open mind than they were moving off in search of another. And none of the people seemed aware they had been touched by a reave, for Maii's mind was gentle and fleeting. But Roche knew that if provoked, the girl's butterfly touch could just as quickly become the sting of a wasp.\n\nFor a while, Roche forgot about Ansourian and their situation. As she and Maii danced across the minds of the habitat's populace, she became aware of another level situated beyond the sensory experiences she was receiving\u2014or beneath it; it was difficult finding words to describe how she was feeling. Having never before gone along as the reave's willing passenger, she hadn't had the chance to appreciate the subtleties of what Maii did.\n\nEach mind was separated by a moment of subtle dislocation, as old sights and sounds were replaced by new ones. In between, Roche felt Maii's mind searching, and for that split second she caught a glimpse of n-space\u2014the theoretical realm in which the reave operated. It was like looking into the mind of a creature that used sound to echo-locate rather than sight to see. Maii was at the center of her universe, and the minds of everyone around her stood out like bumps on a flat plain\u2014but in three dimensions. Some minds jutted out like peaks; others were no more than slight swellings on the surface. Roche understood intuitively that this impression bore no relation to the quality of the minds in the \"real\" world; they were no more or less intelligent, or epsense-adept, or Human for having odd-Shape n-space contours. They were just different, in the same way that people's physical characteristics were different. Roche couldn't be sure from the brief glimpses, but every one seemed unique in its own way, like a signature or a fingerprint.\n\nAs they jumped from mind to mind, like someone circling an island on stepping stones, Roche became more and more intrigued by what she saw between the jumps. Eventually, she asked Maii to stop jumping entirely and show her the reave's world without any sensory input whatsoever.\n\nIt was wildly disorienting.\n\n<I'm amazed you can see it at all,> Maii said. <In all the years Veden rode my mind, he never saw anything. And you've never shown any talent at epsense before.>\n\n<Is this actually talent, though?> Roche wondered, not letting herself get her hopes up. Like most children in the COE, she had dreamed of epsense powers blossoming at puberty. The life of a trained reave was much better than average, orphan or not. To be in demand, to travel to different systems, to delve into minds for government or private business... Roche had dreamed but, also like most children, had never shown any promise.\n\n<I guess not,> Maii admitted. <But I've never known anyone to be able to visualize n-space who wasn't able to access it too.>\n\n<So I'm a freak?>\n\n<A fluke, perhaps,> said Maii.\n\n<Sounds better.>\n\n<Where's Ansourian?> Roche asked, curious to see a mind with a shield.\n\nMaii guided her to the spot where the man's mind should have been. All Roche could see was a steep, circular lip, like the edge around a very deep crater. No matter how Maii tried, she couldn't get inside or even look over the wall.\n\n<It's like a fortress,> Roche said. <This is fascinating, Maii. Keep going, please.>\n\nMaii took her on a whirlwind tour of the habitat, showing her shielded and unshielded minds, minds with epsense powers and no epsense at all, minds that had been damaged by epsense attacks and minds that possessed strange outgrowths into n-space that the reave couldn't explain, except to say that she had seen their like before and that they didn't seem to serve any purpose. Roche tagged along for the ride, an eager student delighted to have discovered a new skill.\n\n<Could this have anything to do with you?> she asked the Box, explaining briefly what she was experiencing.\n\n<Impossible, Morgan. AIs, no matter how sophisticated, have never been able to master epsense to any degree.>\n\n<Could the Crescend have knocked something out of whack when he was installing you?>\n\n<Unlikely. I'm sure he knew what he was doing.>\n\n<But\u2014> Roche stopped, feeling a new pressure on her mind.\n\n<Who are you talking to?> Maii asked. Roche went cold. She hadn't thought that the link with the girl might expose the existence of the Box inside her.\n\nShe was unable to think of anything even remotely convincing. <No one,> she said lamely.\n\nRoche felt a short, sharp probe penetrating deep inside her mind\u2014then abruptly the girl was gone.\n\nRoche rocked back into her chair, stunned by the girl's absence. The real world flooded her senses, dispelling the gray clarity of n-space.\n\n<I'm sorry,> Maii said, reaching out and taking her hand, gripping it tightly through two layers of hazard suit glove. <I shouldn't have looked. But when I felt you talking to it...> A profound sense of remorse came with the words. <Oh, Morgan. I won't tell anyone\u2014I promise!>\n\nRoche didn't know what to do. Although the girl hadn't actually said it, there was no doubting that she now knew about the Box. That went against everything Roche and the Box had arranged; it could even jeopardize the Box's mission for the Crescend.\n\nBut it didn't _have_ to be a problem. If Maii told no one, the secret stopped there\u2014and unless Roche told the Box, it would never know either.\n\nThe simplest thing, she thought, might be to trust the girl.\n\n<It's okay, Maii,> she said. <Keeping the Box a secret is only a precaution. As long as you don't tell anyone about it, it'll stay a secret until we're ready.>\n\nThe girl nodded. <But it's _inside_ you!>\n\n<Yes.> Roche couldn't put the girl's mind at ease on that score, <It's been inside me all along, Maii. I just didn't know. The valise was a backup, and a decoy.>\n\n<You mean you dragged that thing from one end of the COE to the other for no reason?>\n\nMaii's thoughts were tinged with an annoyance Roche could relate to. But that was all history now; the present had given her a whole new set of problems to deal with.\n\n<It doesn't matter anymore,> said Roche. There was a long stretch of silence which she ended with: <Are you okay about all of this, Maii?>\n\n<Yes,> said Maii eventually. <Just... surprised, I guess.>\n\n<I know,> said Roche. <But for now I don't want the others to know, okay? Even Ameidio.>\n\n<I understand, Morgan,> said the girl. <And I am sorry for looking without\u2014>\n\n<There's no need to apologize,> Roche cut in quickly. Her attitude surprised even herself. A month ago when she had first met the girl, she would have been furious with what Maii had just done. But her time with the young Surin had tempered her hostility toward reaves\u2014or at least toward this one.\n\n<Shall we continue, Maii?> she said.\n\nMaii hesitated for a second, then diffused once more into Roche's mind. This time they headed in a different direction, outward and away from their current location. As they traveled, the number of minds they passed slowly increased, then abruptly fell away to virtually nothing, until they were left facing just four anomalies in the n-space plain.\n\nThree of them she wouldn't have recognized, but the fourth one she knew immediately. It was a sudden hole in n-space, as though someone had dropped a ball bearing made of neutronium onto a rubber sheet.\n\n<That's Cane, isn't it?> she said.\n\nMaii seemed momentarily taken aback. <How did you know that?>\n\n<Just a hunch,> Roche said. <He looks so different from everyone else.>\n\n<Different? How?>\n\nAgain Roche struggled for words. <He's not the way you describe him. I see him as empty and very deep\u2014perhaps even bottomless.>\n\n<Maybe you're seeing the hole the _irikeii_ saw.>\n\n<Or maybe I'm just imagining things.>\n\n<Either way, it's strange.> Maii thought for a moment, then said: <From what you've said, I gather we don't see things the same way in n-space. You know how I saw Cane: like a complicated machine moving too quickly for me to grasp; to you, he's a hole. It's the same with Ansourian: to me he's a slippery rubber egg, while you see him more like a cup. This isn't unusual; all people view n-space differently. And we do see _some_ things the same way. Those odd outgrowths, for instance; I knew exactly what you were talking about when you mentioned them. And n-space itself; I also view it as a gray, smooth plain. So I wonder why the extreme examples should be so different?>\n\n<It's what comes from being a freak, I guess,> said Roche. <Sorry, 'fluke.'>\n\n<The _irikeii_ called you an enigma,> Maii said thoughtfully. <It was curious about you. Maybe that means something.>\n\nRoche nodded to herself, thinking: _An enigma.._. <Do you think I could learn to do this on my own?> she asked.\n\n<I doubt it. Your talent is reactive only. You can't see n-space on your own; a reave will always have to show it to you secondhand. Just like when we converse by epsense; it's not so much you talking as me hearing.>\n\n<But why would I be able to see it at all?> Roche wondered, thinking aloud. <It seems a pointless talent to have.>\n\n<Perhaps.> Maii drifted away for a second as she concentrated on something else. <Unless... Morgan, maybe now's not the right time to let you talk to Haid and Kajic. I'm taking you back to the habitat instead to have a proper look around. There are a lot of minds here, and you're new at this, so you're going to find this hard. But I'm going to sweep across as many as possible as quickly as possible. We won't be looking at individual minds, but rather at the landscape they create when combined\u2014a bit like ignoring the trees in order to admire the forest. Let yourself drift with the flow, and let me know if you see anything at all that stands out.>\n\nRoche did as the girl said, guessing immediately what she was after. If Roche could see one clone warrior so clearly, why not another? Ansourian had seemed so certain that there was another one on board the habitat....\n\nThey swept rapidly over the population of the habitat. Minds blurred and merged into a strange landscape that dipped and fell around Roche. As she became accustomed to it, she started to find a sort of coherence to what she saw: there were few sudden dips or highs, as though minds that were alike tended to congregate even without being aware of it, or else individual minds were influenced by those around them. Only a few stood out, and then only because they were so tall among the others. She didn't know what that meant; possibly nothing. It wasn't what she was looking for, anyway.\n\nShe didn't know how much time passed before she saw something. Time seemed meaningless. Likewise, she had no idea where her mind might have been in the real world....\n\n<Wait!> she suddenly shouted. <I see one!>\n\nMaii brought them to an immediate halt. <Where?>\n\n<There.> Roche, unable to move of her own will, could only describe what she was seeing as best she could. <See that group of minds over there? A lot like Ansourian's all in a bunch? There are three in a line to one side. Next to them\u2014yes, that one. It looks just like Cane.>\n\nAnd it did. The same abrupt drop in n-space to a depth she could neither see nor imagine.\n\n<This one?>\n\n<Yes. And... Wait, pull back a bit\u2014no, more...> Mail's mind roamed across the gray vista. <That knot of people. In there.>\n\n<Another one?>\n\n<Yes.> Roche's stomach fell at the sudden realization: there were two clone warriors on the habitat!\n\nMaii seemed to be having trouble deciding which mind to look at. <I can't pin them down as well as you can, Morgan. Wait a second.> They headed back to the location of the first one. <No, I still can't see it. Here?>\n\n<Close. Across a bit\u2014yes! There.>\n\n<And it's the same as Cane?>\n\n<Exactly the same.>\n\n<I can't imagine what else that could mean,> said Maii, <except that this person is a clone warrior.>\n\nRoche's concern slipped back a notch as another realization hit home: <This is it, Maii! We can _find_ them!>\n\n<Maybe,> Maii said, jumping back to the place where Roche had spotted the second clone warrior. It had moved, and Roche had to keep giving Maii directions so they could keep up. <I can't enter this mind, Morgan, so we can't tell who it belongs to that way. Can _you_ tell where this person is, precisely?>\n\n<I, uh...> Roche realized then that it wouldn't be so easy. N-space bore little relation to the real universe, except in the broadest terms. She could tell that the clone warrior was one of many in a group of people, and that that group of people was a subset of the larger group that comprised the population of the habitat. But beyond that...\n\n<Hang on.> Maii let go of her for a moment. When she returned a second or so later, she explained that she'd been exploring the scene more closely on her own. <Dragging you around can be a strain if I move too quickly.>\n\n<Sorry, Maii. I didn't realize.>\n\n<No. This is interesting\u2014and maybe important.> The girl sounded alive in a way that Roche hadn't heard before; maybe the thrill of guiding someone around on her own turf for a change accounted for that. <It's hard to tell for sure,> she said, <but the first mind you pointed out seems to be in an audience of some kind. There are lots of important people nearby, most of them with shields like Ansourian's. I touched a couple of ancillary staff members who weren't as well protected. The audience _seems_ to be the one the administer is holding but I can't be absolutely sure.>\n\n<That would make sense,> Roche said. <That's where the clone warrior would want to be\u2014as close as possible to the center of power. Can you determine which one of them is the clone warrior?>\n\n<No.> There was both excitement and frustration in the reave's voice. <To be honest, I can't even tell myself which one it is. When you pointed it out, I could tell it looked different, but not so remarkably different that I couldn't lose him in a crowd.>\n\n<But I could spot him,> Roche said, confident in this newfound sense. Cane's mind had been as different from the others as a crater was to a mountain.\n\n<I don't doubt that, Morgan,> Maii said. <But the other ones you spotted _could_ be unrelated. We need to get you face to face with another clone warrior, alone, so there won't be any doubt at all...> She sent a mental shrug. <Until then I guess we won't know for certain if it really will work on the rest.>\n\nRoche accepted that. The hope she had felt a moment before was tempered by the thought that her gift might prove too unwieldy to rely upon. But it was a step in the right direction. If the difference was a real one, and she could detect it, there was always a chance that there were others like her who also had this ability. Out of all the high-power reaves on board the _Phlegethon_ there had to be at least one who would replicate it.\n\nThe only trick would be proving it, and that meant coming into contact with another clone warrior.\n\nAs she and Maii lingered around the impenetrable mind in the administer's audience, Roche couldn't help but feel a little apprehensive about that.\n****\n\n**8**\n\n****\n\n****\n\nPerdue Habitat\n\n955.1.32\n\n0330\n\nRoche took the chance to speak to Haid and Kajic once she and Maii had tested her vague ability to its limits.\n\n<Part of me still thinks you should tell the Ulterior to go to hell,> Haid said when she had finished bringing them completely up to date on the situation.\n\n<Part of me agrees,> Roche replied. <But this is about more than the Ulterior. What's happening here could have a bearing on every habitat, every station, every ship throughout Sol System\u2014throughout the _galaxy._ If we don't understand why and how it's happening here, we'll never be able to stop it elsewhere. Slowly, but surely. In that sense, I think Nemeth _has_ done the right thing in sending us here.> She added: <Even if it does put us in the line of fire.>\n\n<And I guess we're always going to be in the line of fire until this is over,> Haid said. <Or we leave the system.>\n\n<Exactly.>\n\nWhen she asked Kajic about the status of the ship, he was more relaxed. <Everything here is fine, Morgan. The repairs are complete, and we are completely restocked.>\n\n<What's Vri up to?> Roche asked.\n\n<Both he and Cane are keeping a very low profile,> said Haid. <Cane just watches the habitat. Vri just thinks. I suspect he's wishing he was with you, so he could keep an eye on Maii. I have to admit, though, he's a lot better at this waiting game than me. He stays in his ship at all times, always alert, always... poised, I guess.>\n\nRoche pondered Haid's choice of word, and found it apt. He might exude calm and patience, but she knew Vri was wound like a spring. She wondered what it would take to make him snap.\n\n<I'll see if I can convince Maii to send him some words of reassurance,> she said, knowing Maii would overhear. <I don't want him rushing in with guns blazing because he thinks I'm up to no good.>\n\n<Good idea,> Haid said. <I imagine your radio silence isn't helping any.>\n\nRoche dropped back into the real world while Maii conversed with the Surin warrior. They didn't talk long, and Maii's expression was sour when they were finished.\n\n<He doesn't like my attitude,> she said. <He thinks I should be more responsible until I come of age.>\n\n<How many years would that be?>\n\n<Three. It's irrelevant, anyway. As far as I'm concerned, I became an adult the day my mother sold me for medical experiments.>\n\nRoche didn't want to intrude upon that pain. She and Maii sat in silence for a long while, thinking private thoughts. It was odd for Roche after such mental intimacy to be alone again in her skull.\n\nOr nearly so...\n\n<Have you found a way to identify the enemy?> the Box asked.\n\n<Not really,> she said. <I could probably pick them out face to face, with Maii's help, but that doesn't help us very much here.>\n\n<It's something, though,> the Box reassured her, echoing her own thoughts.\n\n<I guess so.> She was glad to be reminded that epsense gave her a way to converse with the others without the Box overhearing\u2014especially at times like these, when she didn't feel like talking to the AI. She was afraid of letting slip that Maii knew about the Box's survival. The Box probably wouldn't approve, and she couldn't help but wonder whether it would take any steps to ensure the girl's silence.\n\nVery little time had passed when Maii lifted her head and said: <Ansourian is on his way back>\n\n<Alone?>\n\n<Yes>\n\nRoche stood and shut the helmet to her suit. Maii did likewise. A minute later, the door clicked and opened.\n\n\"I'm sorry I took so long,\" he said as he entered, his voice loud as it filled the quiet of the room. \"Pacecca's kept me busy, and Inderdeep is running behind schedule.\"\n\n\"Are we going now?\" Roche asked.\n\n\"Yes.\" He started to lead the way, then turned back. \"I do understand your need for security,\" he said, \"but it would attract a lot less attention and suspicion if your visors were open.\"\n\nRoche did as he suggested. The risk of physical assault was small, and the hazard suits wouldn't be as effective as combat armor anyway. If they were recognized, so be it; as it was, she had made no attempt to hide her identity when they arrived at the station. Someone looking for her would have found her regardless of an alias, or a visor covering her face.\n\nMaii did likewise, and when Ansourian caught his first glimpse of her face, he smiled amicably.\n\n\"It's nice to see you properly,\" he said. \"You have a very strong mind.\"\n\n<You too. Where did you learn to shield like that?>\n\n\"The Vax were taught the technique by a senior adept passing through from Guo.\" Seeing no sign of recognition, he explained: \"The Guo Sodality is dedicated to epsense training and study in the Middle Reaches; its senior adepts are renowned throughout our region for their strength and subtlety. The technique they taught us has been handed down along certain family lines in order to secure their places in the hierarchy of our culture.\"\n\n\"We noticed a lot of shielded minds in one particular spot in the habitat,\" Roche said.\n\nHe nodded. \"That would be the audience chamber,\" he said. \"That is where they would gather at this time.\" He half smiled. \"Sometimes I wonder if there are any actual thoughts going on behind those shields.\" He shrugged. \"But that's politics for you.\"\n\nRoche debated whether to tell him about the two clone warriors, but decided to wait until she was sure. \"Shall we go?\"\n\n\"Yes, of course,\" he said, leading them out the door and into the corridor.\n\nIt was refreshing to be able to move again, and Roche relished the sensation of walking, even though it was in the cumbersome hazard suit. There seemed to be fewer people around than there had been earlier. Maybe that was because the habitat was between shifts, or the conservative Vax still maintained a consensus \"night.\" Either way, Roche was glad for the relative anonymity.\n\nThey came to a major branching-point, where numerous corridors met at a wide variety of angles. The artificial gravity maintained in the habitat decreased slightly to accommodate the sudden shifts in orientation. Ansourian took them around a curved wall, then down into an undulating tube barely tall enough to accommodate Roche and her suit. This way was completely deserted; they passed no one, nor any doors or windows. But for her reopened link with the _Ana Vereine,_ Roche would have had no idea where she was.\n\n<You're heading deeper into the structure,> said Kajic, throwing a 3-D map of the habitat into her left eye. They were a red dot inching through a twisted tube that led, as Kajic indicated, to the heart of the habitat. <There's a junction up ahead. If he's taking the direct route, he'll turn left. But it doesn't seem likely.>\n\nAnsourian turned right, taking them along a corridor that curved smoothly upward, then abruptly dropped 90 degrees in only a few meters. Roche negotiated the incline with care, trying not to let the wildly shifting gravity throw her off balance. The corridors from that point became decidedly cramped, with odd protrusions and corners and, overall, a makeshift air, as though they had been assembled from spare parts over many decades with little or no forethought as to their final function.\n\n\"Why don't you use transit tubes?\" Roche asked.\n\n\"There are only a handful for freight,\" Ansourian explained. \"Otherwise we don't care for them. There was a terrible accident a few years ago, and the previous administer discouraged their use.\"\n\n\"Where exactly are you taking us now?\"\n\n\"Into the maintenance infrastructure. Security is relatively lax there, and we'll most likely pass as workers. The area is rarely monitored firsthand; only a basic AI checks for movement.\"\n\n\"We'll register, won't we?\"\n\n\"Yes, but my Quare persona has clearance.\"\n\n<Box, can you make sure of that?> Roche asked privately.\n\n<I can disable the security systems in your vicinity, if you like.>\n\n<Yes, do it.> She thought for a second. <I don't suppose you could get us to the administer more easily, could you?>\n\n<Very possibly,> the Box replied. <But I feel you will have a greater chance of talking to Inderdeep Jans with Ansourian's guidance.>\n\n<Do you think she'll listen?>\n\n<That depends on what you have to say, I imagine.>\n\nRoche had thought about that. She didn't have much to offer the administer apart from a vague hint about the enemy among her number and reassurances that the council was doing everything in its power (and more besides, in the form of the Ulterior) to rectify the problem. She was really only there to ask questions of her own, and if the administer was feeling uncooperative, then it was unlikely those questions would be answered.\n\n\"Is there a proper way to address the administer?\" she asked.\n\nAnsourian glanced over his shoulder. \"I have no suggestions on how to get her to do what you want, if that's what you're asking.\" He shrugged and returned his attention to the way ahead. \"Inderdeep is unpredictable at best, and can be willfully destructive at worst. Her father left me to keep an eye on things when he died. He never expected me to have to run things the way I have been. But if Inderdeep was left to act as she wished, the habitat would fall apart within months.\"\n\n\"What will happen if you're not going to be there?\"\n\nAgain, he shrugged. \"Maybe things will go well when you talk to her and I'll be able to reveal myself,\" he said. \"If not, there are a couple of options still available. Oren Quare may prove to be eminently suitable for an advisory post closer to the administer's office. I know enough to work my way back in; given time, I could regain lost ground. But time is something I do not have, I'm afraid.\"\n\nRoche knew what he was referring to. \"How long do you think Alta has?\" she said.\n\n\"Not long,\" he replied. \"The evidence alone would have been enough for a guilty verdict. Her confession will hasten the legal proceedings. Only the fact that I was so close to Inderdeep is keeping her alive right now. Who knows? Sentence may already have been passed. The matter was bound to come up in the current round of audiences, so Inderdeep may have already signed the execution order.\"\n\n\"She has power of life and death in the habitat?\" Roche said, shocked that so much authority could reside in one person\u2014especially one such as this Inderdeep Jans seemed to be.\n\n\"Indirectly she has _some_ power,\" Ansourian explained. \"She ratifies the decisions of the judicial system. Without their approval she can't impose the death penalty, but she can overturn one at will. I am hoping she will do this in Alta's case.\"\n\n\"Why should she do that?\"\n\nHe faced Roche again. \"Because you are going to ask her to,\" he said. \"Tell her you came here because Alta and I called you. Tell her you're here to help, but you don't know how. Only Alta knows, now that I'm dead. The habitat may be riddled with the enemy for all anyone can tell, but with Alta's help you might be able to ferret them out.\"\n\n\"That's ridiculous,\" Roche scoffed. \"She won't believe that.\"\n\n\"She might,\" he said. \"Besides, it's not so far from the truth. Someone did try to kill me, after all.\"\n\nIgnoring the obvious\u2014that someone might've killed Ansourian simply because they disagreed with him\u2014Roche said, \"Is there anything else I'm supposed to be asking her? Anything else I should know?\"\n\nHe thought for a long while as he continued to lead them through the habitat's maintenance labyrinth. \"Just don't take her for an idiot,\" he eventually replied. \"She's not stupid. She's just... wayward.\"\n\nRoche absorbed the comment as they walked. Since Ansourian had worked with the administer so closely for so long, Roche had to assume that he would know her better than anyone else would. If he said she wasn't an idiot, then Roche had to accept that she wasn't\u2014even though it was difficult to believe, given everything she had heard.\n\n\"How much farther?\"\n\n\"We're practically there,\" he said, negotiating a narrow pass between two large ducts that intruded on the passage. \"Just around this corner.\"\n\n\"Good, because I'm getting claustrophobic.\" That wasn't exactly true; she was just tired of squeezing through the tiny spaces. Her suit scraped the ducts even when she turned sideways to slip through.\n\n<Could you get us out of here if we lost Ansourian?> she asked Kajic.\n\n<Absolutely,> came the confident reply, echoed a split second later by the Box, privately. <Why, do you think that's likely?> Kajic went on.\n\n<Anything is possible,> she said.\n\n\"Here,\" said Ansourian, taking them up a short corridor that ended in a cul de sac and bringing them to a halt. \"I prepared this entrance in secret when Ehud Jans, the last administer, was still alive. It was intended as an escape route only, but it can work both ways, of course.\" He produced some silver tape from the pocket of his uniform, along with what looked like a small battery. \"This will only work once,\" he said, affixing a length of tape to the wall at head-height and another down by his feet. The tape slowly changed in color from silver to red. \"It may look like an ordinary section of wall, but it's not. It's barely solid at all: enough to fool a rapping knuckle, or even a gentle punch, but barely more than that. When I run a current through it, the alignment of its molecules will change, and it will dissolve completely. You'll be able to walk through without any trouble at all.\"\n\n\"And then?\" said Roche uncertainly.\n\n\"Inderdeep will be on the other side,\" he said, meeting her gaze squarely. \"I'll wait here and listen. The less she knows about me, the better\u2014as her old friend Atul, or Oren Quare.\"\n\nRoche nodded, and drew her side arm in readiness.\n\n\"That won't be necessary,\" he said.\n\n\"No?\" She didn't put the pistol back into its holster. \"You told me not to trust you before. Why should I start now?\"\n\nHe smiled. \"Just don't shoot _her,_ whatever you do.\"\n\nShe smiled in return, but there was no humor to it. \"I'm not stupid either, Ansourian.\"\n\nHe turned back to the wall.\n\n<Can you sense her?> Roche asked Maii.\n\n<There's someone there, yes> said the reave, <and it matches the echo I perceive in the minds of those who know her. Those I can read, at least. She's shielded too, though.>\n\n<Naturally,> said Roche dryly. So much for her backup plan of forcing the administer to do what she wanted.\n\nAnsourian reached up to affix the battery to the top strip of tape.\n\n\"Get ready,\" he said.\n\nRoche tensed and watched expectantly. Little happened at first, then the red plastic seemed to soften and ran. As though it was composed of grains of sand slipping through a person's fingers, the wall simply fell away. After barely ten seconds, all that was left was a spreading accumulation of dust on the floor, and a smell like ozone.\n\n\"Right,\" Ansourian whispered. \"In you go. Good luck.\"\n\nRoche nodded and, with Maii following, slipped through.\n\n* * *\n\nRoche's first impression of Inderdeep Jans was that she looked older than she'd expected. Her skin was paler than that of the other members of the Caste whom Roche had met, and she actually had hair: a long ponytail of perfect white that hung from the back of her skull and was bound in three places with bronze clasps. She wore a simple yellow robe adorned with a stylized sun\u2014a motif echoed throughout the room.\n\nShe was seated on a wide couch, drinking deeply from a glass containing a pink liquid. She stopped drinking the moment Roche stepped from behind the wall hanging that hid the secret entrance, and turned coolly to face her.\n\n\"Who are you?\" she demanded, seemingly unsurprised by the sudden intrusion. She put down her glass calmly but didn't rise from her seat. \"How did you get in here?\"\n\nRoche kept her distance, not wanting to alarm the woman with any gesture that might be construed as hostile. \"I apologize for the intrusion, Administer, but I\u2014\"\n\n\"What are you _doing_ here?\" Jans said with a hint of irritability when she saw Maii emerge from behind the wall hanging, also.\n\n\"We mean you no harm,\" said Roche. \"I assure you.\"\n\nThe administer snorted. \"Why should I believe, you?\"\n\nRoche hefted her side arm. \"We could have killed you already, if that was what we really intended.\"\n\nA sly look passed across the woman's face. \"Then what _do_ you want?\"\n\nBefore Roche could answer, the administer raised a hand and said: \"Wait. I know you, don't I?\" Roche opened her mouth to speak again, but again never got the chance. \"Roche!\" she said, clicking her fingers and nodding her head triumphantly. \"I was told about you barely an hour ago. They showed me footage of your arrival and said you wanted to talk to me.\"\n\nThis surprised Roche, given what Ansourian had said about how her request for a meeting with the administer would probably be deliberately delayed.\n\n\"Yes, Administer,\" said Roche. \"And I apologize for the manner we went about it, but\u2014\"\n\n\"This must be your blind companion.\" The woman stood now, and Roche realized with a shock that Jans was almost as tall as she was, hazard suit included. The administer took a step closer, scrutinizing with some fascination the bandages about Maii's eyes.\n\n\"This is Maii,\" Roche said. The girl nodded in greeting.\n\n\"I don't recognize her type.\" She looked at Roche. \"Local, I assume?\"\n\n\"The Surin are neighbors of the Commonwealth of Empires.\"\n\n\"Ah, yes, I've heard of them.\" The woman nodded with private satisfaction. Then, as if remembering something, she said: \"You were meant to be in Stateroom B, waiting to be granted an audience.\"\n\n\"We were, but we were told it could take up to two days for us to see you...\"\n\n\"Very likely,\" Jans said. \"I am a busy person, you know.\" She glanced away for a second, her expression sad. \"A dear friend will be consigned to the sun tomorrow. And _that_ is more important to me than anything you or anyone else might have to say at this time.\"\n\nThe sorrow on the woman's face seemed completely genuine. Roche would have liked to reassure her on that score, but knew she couldn't do that just yet.\n\n\"I understand that, Administer,\" she said. \"But please hear me out. My mission is of the gravest importance, and I need your assistance to complete it. In two days, it might be too late.\"\n\n\"Too late? For what?\"\n\n\"For me to make a difference.\" Roche was loath for the moment to stoop to the story Ansourian had suggested. She had to at least see if something closer to the truth would work first. \"I've been sent here by the Interim Emergency Pristine Council in response to claims that your habitat has been infiltrated by agents working for the enemy. With Atul Ansourian's help, I had hoped to investigate these rumors and, if they proved to have some foundation, determine precisely who among your staff could no longer be trusted.\"\n\n\"Atul knew you were coming?\" Jans turned easily and returned to her seat.\n\n\"Yes, he did.\"\n\n\"And he was going to help you?\" she said, leaning back and looking over at Roche with some suspicion.\n\n\"Yes.\"\n\n\"How?\"\n\n\"I'm not sure exactly. All I know is that he was to be my contact here.\"\n\nThe administer's expression became one of distaste and annoyance. \"But Atul is dead now,\" she said. \"Killed by his own daughter.\"\n\nRoche nodded. \"We were told,\" she said. \"I'm sorry. Perhaps...\" Roche vacillated regarding how much to say. \"Perhaps there is more to it than meets the eye.\"\n\n\"What are you saying?\" Jans studied Roche. \"Are you implying she might _not_ have been responsible?\"\n\n\"It is a possibility, Administer.\"\n\n\"But why would she lie?\" The woman looked confused.\n\n\"She could be covering for the _real_ assassin,\" suggested Roche.\n\nThe administer's confusion deepened. \"The person you came here to warn me about?\" she said. \"Why would she do _that_?\"\n\n\"I'm not sure,\" said Roche. \"But if you would allow me to talk with her, perhaps we could find out just how much she knows.\"\n\n\"And what makes you think she would tell you anything?\"\n\n\"Maii, here, is a reave,\" said Roche. \"She could read her mind.\"\n\n\"Really?\" Jans turned to face the girl. \"Can she read mine?\"\n\nMaii shook her head.\n\nThe woman looked smug. \"So why assume she could read Alta's? She was Atul's daughter through and through, and _his_ shield was perfect. I should know. I once hired a reave to crack it, just to see if she could. She failed.\"\n\n\"Be that as it may, Administer, I do feel it is worth a try.\"\n\nThe woman shrugged, and with it Roche knew the possibility had been dismissed. \"It doesn't really matter anyway,\" said Jans. \"Alta is guilty of _something._ I signed her execution order barely an hour ago. Whether she is interrogated or not, she will be dead this time tomorrow.\"\n\nRoche took a deep breath. That closed off that line of inquiry, for the time being. \"Even so, that doesn't change what I have come here to tell you.\"\n\n\"No? Without Atul, what can you do?\"\n\n\"I can still _try_.\"\n\n\"How?\"\n\n\"With your help.\"\n\n_\"Mine?\"_\n\nRoche tried to contain her impatience. \"Administer, I am not exaggerating when I try to impress upon you the urgency of my mission. The council needs the help of the Vax, and in return I will try to help you. Atul Ansourian freely offered us his assistance. It continues to be my hope that you will decide to offer us the same.\"\n\nThe administer looked bored. \"Why should I care about your Pristine Council? The Vax can take care of themselves.\"\n\nRoche recalled the number of ships in the habitat's docks, at least one of them\u2014the COE's _Paraselene_ \u2014a Pristine vessel. \"You already lend support to the IEPC's campaign. And I note, without meaning to offend, that you yourself are of different stock from the other Vax I have met. With such diversity\u2014\"\n\n\"Mind your words, Roche,\" Inderdeep Jans stood abruptly, taking one menacing step forward. Even with her side arm, Roche felt threatened and instinctively stepped back. \"How _dare_ you suggest that\u2014\"\n\n\"That's not what I meant, Administer,\" said Roche quickly. \"I was merely trying to reinforce the fact that different Castes _can_ work together for a common good\u2014be they Vax, Pristine, or any other. If you\u2014\"\n\n\"If _you_ hadn't come here, maybe Atul would still be alive.\"\n\nThe sudden shift in topic caught Roche off guard. \"What? That's ridiculous! There is no evidence to suggest that\u2014\"\n\n\"Really? Atul calls for your help, and within days he is dead. The coincidence seems striking, does it not?\"\n\n\"Then surely you must see that _your_ life is at risk also?\"\n\n\"Why should _that_ be?\"\n\n\"Because if your chain of command has been compromised by the enemy, then their ultimate aim will be to dispose of you too.\"\n\n\"I don't see why they'd want to do that,\" Jans said, gesturing dismissively with one hand. \"I don't even want to be here. It was Atul who talked me into it, and see what it cost him! Besides\u2014\" She took another couple of steps forward, her relaxed expression belying her words, \"the only person who has even remotely threatened me to date, Roche, is _you_.\"\n\n\"That's not true, Administer,\" said Roche defensively. \"I pose no threat to you whatsoever!\"\n\n\"No? You break into my private chambers and exhort me to assist you in your mission\u2014a mission, I might add, that requires turning my staff upside down to search for a hypothetical spy\u2014while muttering vague suggestions that if I don't, my life will be forfeit. That sounds like a threat to me, Roche.\"\n\n\"Everything I have said is true,\" Roche stated patiently, although she could feel her patience crumbling. \"Your life _is_ in danger, and inaction on your part only increases that danger. But I'm not the one threatening you. It is the enemy\u2014our _common_ enemy. It's this person, and many more besides, that we should be fighting\u2014 _not_ each other.\"\n\n\"So you say,\" Jans remarked dubiously.\n\n\"Because it is true,\" insisted Roche.\n\n\"Then show me the proof.\"\n\n\"It's all around you! The death, the destruction, the distrust, the disorganization...\" She struggled to remember the name she'd heard just hours before. \"And what about Guidon?\"\n\n\"The cause of that accident has yet to be verified.\" The administer looked uncomfortable, but was unwilling to take the point. \"None of this is proof. Your words are empty, Roche. Why should I believe you over one of my own advisers?\"\n\n\"Because one of your advisers may well belong to the enemy!\"\n\nThe administer smiled wryly. \"As might _you_ ,\" she said. \"Your reputation precedes you, Roche. It is said that wherever you go, trouble follows. If I were to give you the help you request, how could I be sure that the Vax won't become your next victims?\"\n\n\"Nothing could be further from the truth!\"\n\n\"So you say.\" The administer held a palm outstretched, silencing Roche, who had opened her mouth to object again. \"I have no desire to put my people at risk on such flimsy evidence! A few disputes and the threat of war, an accident, and a failed assassination attempt closer to home\u2014it will take more than _this_ to convince me, Roche. And if you _cannot_ convince me, you might as well leave.\"\n\nThe administer's words sent a chill down Roche's spine. She knew, then, that there was no chance of convincing the woman to change her mind\u2014not on that score, and especially not on the matter of Alta Ansourian.\n\n\"Very well, then,\" Roche said, backing away toward the secret exit. \"We'll trouble you no further, Administer.\"\n\nThe woman watched them leave, a wary expression on her face. \"Wait,\" she said, just as Roche pulled the sun-motif wall hanging aside. \"Who told you about this entrance?\"\n\nRoche thought fast. She really had only two options, given that she didn't want the administer to know the truth. She could refuse to answer, or she could lie. How to do the latter convincingly was the trick.\n\n<Box?>\n\n<Tell her it was Gurion Egarr, one of her senior councilors. She will want to believe that. The news reports them as being old antagonists>\n\n\"It was Councilor Egarr,\" said Roche a second later. \"And I tell you that now only to demonstrate my openness and honesty with you. His intentions were good, I assure you.\"\n\nThe administer's broad smile was cut with cynicism. \"I bet.\" Then she nodded. \"Go now,\" she said. \"I shall allow you five minutes grace before I send my guards after you. And I do this now to demonstrate _my_ magnanimity.\"\n\nHer smile dissolved as she glared at Roche.\n\nWhether she was serious or not, Roche could not tell. But she couldn't afford to take the chance. With Maii ahead of her, she slipped behind the wall-hanging and back into the cramped confines of the maintenance infrastructure.\n\n****\n\n****\n****\n\n****\n\n**9**\n\n****\n\n****\n\nPerdue Habitat\n\n955.1.32\n\n0600\n\nAnsourian led them quickly through the tunnels, taking them by a different route from the one they had followed on the way in. Maii was directly behind him with Roche close on her tail, glancing back uneasily now and then to see whether they were being pursued.\n\n<Please stop doing that, Morgan,> said Maii. <It's hard for me to see where I'm going when you keep looking over your shoulder.>\n\n<I'm sorry, Maii,> Roche said, <It's just...>\n\n<Trust me, there's no one following us,> said the girl. <The administer seems to be keeping to her word so far.>\n\nAnsourian stopped suddenly, ushering both Maii and Roche through a hatchway; he closed the heavy bulkhead behind them.\n\n\"That'll stop them,\" he said, belligerently punching some commands into a keypad. \"I've let the air out of the tunnels we just came through.\"\n\n\"I don't think it'll stop them,\" said Roche. \"At best it will only slow them down.\"\n\nThe small man shrugged but said nothing. Roche could see the hurt in his eyes.\n\n\"You heard what she said, didn't you?\" she asked. He didn't need to reply; his expression spoke volumes. _Failed assassination attempt_ , the administer had said. Roche could believe that she had guessed it was an assassination rather than simple patricide\u2014but how could she have known it had failed if she hadn't been a party to the attack in the first place? At the very least, she had known about it.\n\n\"I'm sorry,\" Roche said.\n\n\"Don't be,\" he said. \"It has made my decision easier. Escape seems the surest course, now.\"\n\n\"What about Alta?\"\n\n\"As I said earlier, there are blind spots in the security system. I will try to get her away first.\"\n\n\"More secret doors you put in place?\"\n\n\"No, just exploitable flaws,\" he said. \"Even the best security system has its weaknesses\u2014I simply have the advantage of _knowing_ what those weaknesses are. I am confident that I can get into the holding cells. Getting out will be more difficult, but not impossible.\"\n\nRoche followed him along the corridors while conducting a conversation via her implants with her crew back on the _Ana Vereine._ <Ameidio, Uri\u2014have you been following this?>\n\n<Of course,> said Haid. <Are you thinking of helping him?>\n\n<We'll need him if we're to have any chance of fixing the situation here. The administer is going to be worse than no use to us, since the enemy has obviously gotten to her and turned her against him; they're aware that he runs the shop, and that they couldn't influence him. With him out of the way, or with his daughter as leverage, they figure they have control.>\n\n<But why should they need leverage if they think he's dead?>\n\n<They're not stupid; they might at least suspect that he switched bodies, especially when their assassin hasn't turned up since. And when the administer tells them about our little visit, that'll probably just confirm their suspicions.>\n\n<How can we help him, though?>\n\n<The old fashioned way, I guess,> she said.\n\n<A show of strength?> asked Haid.\n\n<We haven't got time for subtlety here, Ameidio,> Roche replied. <You and Cane\u2014or Vri, even\u2014can act as a distraction while we make a move on the holding cells. Do you think you could organize something big enough and fast enough to attract everyone's attention?>\n\n<Undoubtedly.>\n\n<Feel like some action?>\n\nHaid laughed. <I can't deny that I could use the change,> he said. <And I'm sure Vri would be up for it. One mention of you putting Maii on the firing line and he'd move for certain.>\n\n<Okay. You and Vri it is. We'll leave Cane to mind the ship with Uri. Are they both listening?>\n\n<We're here, Morgan,> said Kajic.\n\n<Do either of you have any reservations about this?>\n\n<It's your decision,> said Kajic. <I am confident we can pull something like this off successfully. But there are no guarantees.>\n\n<There never are,> said Roche.\n\n<I agree with Uri,> said Cane. <However, I feel that I might be more use to you alongside Vri in the diversionary attacks.>\n\nRoche paused before replying. <I appreciate what you're saying, but\u2014>\n\n<But you need to have someone there that you can trust,> he said with no hint of indignation. <I understand this, Morgan. I felt I should voice my feelings just the same.>\n\nRoche smiled to herself. <Thanks,> she said. <Okay, I'll talk with Ansourian and see what he thinks. Then we move\u2014as soon as possible. The longer we take, the more chance we give the enemy to be ready.>\n\n<And we don't want to do that,> said Haid. <They have enough advantages as it is....>\n\n* * *\n\nAnsourian looked surprised when she offered to help.\n\n\"Why?\" was the first question he asked.\n\n\"Because if I just let you rescue Alta and leave, it's tantamount to handing the enemy this station along with everyone in it. The same if I let you try and you fail. You'll be dead, and that doesn't serve anyone. If we're to stand any chance of fixing this, we're going to need you alive.\"\n\n\"What difference does it make to you, either way?\"\n\nThat wasn't so easy to answer. \"It makes a difference to how I feel about myself,\" she said. \"It's a matter of pride. This mission is a test, if you like; maybe metaphorically rather than literally, but a test all the same.\" She shrugged. \"I don't want to fail.\"\n\nHe nodded slowly. They had stopped to rest in an unlit stores cubicle; the only light came from the necks of Roche's and Maii's hazard suits, lending their heads a surreal, disembodied look.\n\n\"So your crew will create a diversion while we get Alta,\" he said. \"Then what?\"\n\n\"Then you help us get back to the _Ana Vereine_ ,\" she said. \"You know your way around this place; I'm going to need a less obvious way to get off the habitat than the main docks. The ship will have to cast off once things heat up, but it can send a scutter to get us when we're ready. Just name the place and we'll head for it.\"\n\n\"Okay,\" he said. \"There _is_ a way, but it will be tricky. And I'm going to need pressure suits for myself and Alta. There are\u2014\"\n\n<Morgan.> Kajic's voice broke in sharply; Roche raised her hand to silence Ansourian. <There's been a development.>\n\n<I'm listening.>\n\n<I've been monitoring traffic to and from the docks. Exactly twenty minutes ago, every commercial and military vessel from the Random Valence habitat disengaged and retreated to a safe distance. Now a contingent of ships has arrived from the same habitat and is demanding to dock.>\n\n<How many ships?>\n\n<Fifteen, all armed.>\n\n<A takeover force ?>\n\n<It seems likely, striking while the chain of command here is uncertain.>\n\nRoche relayed the information to Ansourian, who simply nodded. \"It doesn't surprise me. Frane Yugen has been itching to move on Inderdeep since she rejected his offer to form a partnership against Tocharia 13. With me out of the way, it's a perfect opportunity.\"\n\n\"Can she hold him off?\" Roche asked.\n\n\"That depends. I've kept the defenses well-stocked over the years, and made sure the staff know what to do. If she doesn't interfere, they'll manage well enough.\"\n\nThat sounded ominous. \"Can you guess how she'll respond?\"\n\n\"Again, it depends on what the enemy are telling her. If they want to increase conflict, they might feed her inappropriate advice.\"\n\n\"Then we can't assume things will go well,\" said Roche. She switched to her implants. <Uri, where are Haid and Vri?>\n\n<In the habitat,> he said. <Preparing for the distraction.>\n\n<It doesn't look like we'll need them, after all,> she said. <Recall them and disengage. I don't want you caught in the crossfire.>\n\n<Understood.>\n\n\"How far is it to where they're keeping Alta?\" she asked Ansourian.\n\n\"Fifteen minutes or so, going the back ways.\"\n\n\"Then let's get moving before things start heating up.\"\n\n\"Not without the pressure suits,\" he reminded her. Roche nodded and indicated that he should lead the way. They left the cubicle and headed off through the labyrinth.\n\n<He's worried,> said Maii.\n\n<You can tell that? Good. That means his shield must be slipping. Can you pick up anything else?>\n\n<No, nothing. Whoever those people from the Guo Sodality are, they taught the Vax a good technique.>\n\n<What about the administer? Did you get anything at all from her?>\n\n<A few stray thoughts; most likely just planted to fool a casual glance. The core of her mind is impenetrable, like Ansourian's.>\n\n<You said you thought you might be able to break into his, given time.>\n\n<I'm sure of it,> said Maii. <But this is hardly the time to try. At the very best, I would distract him; at the worst, I might seriously injure him. The Guo shields are deep-rooted; cracking them would probably have consequences for the mind beneath.>\n\n<Maybe I should set you loose on the administer anyway. It's not as if we have much to lose.>\n\n<That wouldn't be ethical, Morgan.>\n\n<Maybe not, but it is an option.>\n\nA siren sounded in the distance, echoing along the winding corridors like the baying of an enormous beast. \"What's that?\" Roche asked Ansourian.\n\n\"Security alert, level 5. Ambient gravity will drop by twenty percent to conserve power.\"\n\nEven as he said it, a wave of dizziness rushed over Roche, leaving her feeling somewhat lighter after it had passed. \"Does that mean the habitat is under attack?\"\n\n\"No, not yet. You'll know if that happens. Perdue is designed to absorb the energy of an impact and spread it across its structure.\"\n\n\"Meaning we'll feel it regardless of where we are?\"\n\n\"Yes, but it shouldn't be too bad. Habitats like these tend to absorb almost anything up to a point, and fall apart completely only if you cross that point. That's what happened to Guidon; it was pushed too far. Since a ruined habitat will be of no use to Yugen, he'll play it fairly safe.\"\n\n\"And if he _does_ want a ruined habitat?\"\n\nAnsourian looked sharply at her. \"You think the enemy might have got to him, too?\"\n\n\"It's a possibility we can't ignore.\"\n\nThey stopped at a locker, from which Ansourian produced two transparent pressure suits. \"We call them OSFA suits,\" he explained, slipping one over his uniform. \"One Size Fits All. They're designed to maximize survival through a wide range of conditions\u2014heat, cold, vacuum, pressure, etcetera; they'll even stop a measure of coherent light\u2014but they won't last long in combat.\"\n\n\"Do you have access to weapons?\"\n\n\"They'll be in the armory. We won't get in there at a time like this.\"\n\nRoche touched the pistol at her side. An identical weapon rested on Maii's hip; unlike most reaves, the girl had proven herself more than capable of killing on Sciacca's World. By looking through her victim's eyes and aiming along the barrel from the other end, she made quite an effective fighter.\n\n\"We'll have to make do, then,\" Roche said, hoping the distraction would be enough.\n\nAnsourian finished sealing the suit, leaving only the hood open, then continued to lead them along.\n\n<Are Vri and Haid back on board yet?> Roche asked Kajic.\n\n<Vri is unwilling to leave until Maii has returned,> Kajic replied.\n\n<But it's unlikely we'll be able to exit that way.>\n\n<He's aware of that.>\n\n<Then what does he think he's going to do?>\n\n<Find you and take her himself>\n\n<Damned fool!> Roche snapped irritably.\n\n<As Maii pointed out, he's quite dedicated.>\n\nA deep vibration rippled through the tube surrounding them. Ansourian placed a hand against one wall.\n\n\"It has begun,\" he said. \"That one hit the shields. Maybe nothing more than a warning volley. If there are more, then we can assume Inderdeep isn't going to give in without a fight.\"\n\n<Dockmaster Rench has ordered all docks sealed,> said Kajic. <Haid and Vri are still not aboard.>\n\n<Okay,> she said. <Disengage. Tell them to sit tight. We'll just have to work something else out.>\n\nThey headed off along the corridor. The floor still moved slightly as the energy of the attack ran through the entire habitat and dissipated, ultimately, as radiant heat. In theory, Roche could see how such a passive defense might pay off; she could also see, however, how disastrous it might be. One sustained attack could ruin everything, decisively.\n\n<Disengaging now,> said Kajic. <Cane will fly the scutter when the time comes to pick you up.>\n\n<Fine.> She didn't have much choice, really. With Vri and Haid stuck on the habitat with them, and Kajic potentially dodging fire, there _was_ no one else. Even with Ansourian's help, the chances of finding a ship and breaking the dockmaster's embargo had to be almost zero.\n\nShe explained what had happened to Ansourian. \"We'll need a rendezvous point. Can you give us one away from the docks?\"\n\n\"Yes, but it's too complicated to explain how your friends should get there. They can access a map using Quare's security code.\" He rattled off a string of letters and numbers, which Roche memorized. \"They want to reach the maintenance airlock at the end of corridor 14 in Sector Green-D. It's not far from the holding cells in Sector Blue-J. Tell them to wait for us there. We shouldn't be more than an hour.\"\n\nRoche relayed the information through Maii, using the girl as a medium the way she had earlier.\n\n<You be careful,> Haid said, via Maii.\n\n<You too.>\n\nShe could clearly taste the worry in his mind, and the tangle of plans and counterplans as he mentally prepared for any contingency. <I'll call if we need help. What equipment do you have?>\n\nUsing mental shorthand, he sent her a list that included shaped charges, compact percussion rifles, pressure mines, and flash- bombs\u2014anything that would fit into the relatively low-key armor the two men had donned in order to enter the habitat. Dock security hadn't been especially tight, not since they'd been given approval to disembark, but cannon and full combat suits would have attracted attention.\n\n<I wish we had some of that,> Roche said, <but there's no helping it, I guess. Make sure you're at the rendezvous when we get there.>\n\nHaid mentally nodded and went back to studying the map Vri had accessed via a wall terminal. That left Roche with half a mind on where Ansourian was leading them, and the other half in n-space. Again she experienced a moment's disorientation as she simultaneously saw through her own eyes and those of Maii, who was also looking through her eyes. <Could the Box help us?> Maii asked.\n\n<It could probably open the docking bay doors, but that would be too obvious. All it's good for at the moment is gathering information and keeping automatic security off our tails.>\n\n<It seems to be doing a good enough job at that, anyway.> Another vibration rippled through the corridor, followed quickly by another. The wailing alarm changed pitch, becoming more shrill and urgent. <It's a shame it can't do anything about the Random Valence ships....>\n\n\" _That_ was no warning shot,\" said Ansourian, steadying himself against a wall.\n\nRoche forced herself to keep walking even as the floor moved beneath her feet. <Box, are they firing back?>\n\n<All A-P cannon emplacements have been ordered to retaliate. Nine free vessels have sided with Perdue and are joining the fight. However, more vessels are on their way from Random Valence.>\n\n<Who has the advantage?>\n\n<Perdue should hold. Its structure is more than capable of bearing the brunt of such an attack, and given the extra ships, I doubt the attackers will be able to board at any point.>\n\n<What about our scutters? Could one of them make it through the defense?>\n\n<If it declares for Perdue's side, then it will most likely be allowed into the fray.>\n\n<Okay. I'll get Cane out there now, to save time later.> She got Kajic on the line and instructed Cane to take the scutter as the Box had suggested. Kajic didn't ask how she had found out about the battle; she hoped that he simply assumed that she had learned about it via Maii.\n\nStill linked via n-space, she noted that the reave was occasionally glancing away from where they were headed in the tunnels to the people surrounding them in the habitat. Even in the maintenance infrastructure, they weren't completely alone. The ones that weren't shielded were mostly thinking about what was going on: curious, concerned, frightened, angry... Only a handful were hoping the attackers would win, and none of them had any specific thoughts about the enemy. If there was a connection between the attack and the clone warriors, Roche had yet to find proof of it.\n\n<Any sign of the enemy?> she asked Maii.\n\n<I still can't pinpoint them the way you can. Want to try?>\n\n<It couldn't hurt.>\n\nRoche tried to concentrate on walking as well as the mental landscape of the habitat. The rapid progression of minds was as disorienting in its own way as the incessant but irregular grumbling of the walls and floor. Mind after mind rolled by, until\u2014\n\n<There's one! Back up. _There. >_ The pitlike mind was in a group of three or four others, all stationary. <Where is it?>\n\n<Hard to tell,> said Maii. <I think it's the same place we were looking before.>\n\n<The administer's audience chamber?> Roche asked.\n\n<Possibly. Or maybe even her private rooms.>\n\nAnsourian brought them to a halt when they came to a closed door.\n\n\"On the other side of this bulkhead is the corridor leading to the holding cells,\" he said. \"I don't know what's waiting for us there. It _could_ be messy.\"\n\n<Box?>\n\n<It's empty. Two guards are at the entrance to the holding cells, though not in line of sight.>\n\n<Good. Keep me posted as we get closer.>\n\n<There is one thing, Morgan,> the Box said, before she could turn her attention back to Ansourian. <I have just intercepted a coded security bulletin. Inderdeep Jans is dead. She was just found in her chambers with her neck broken.\n\n<Who killed her?>\n\n<That is the curious thing. Obviously the crime scene is awaiting a thorough forensic examination, and no firm conclusions should be drawn without\u2014>\n\n<Come _on,_ Box!>\n\n<It would seem that you are the main suspect, Morgan.>\n\n< _Me?_ >\n\n<All available security officers have been advised that you and other members of your crew are presently on the loose and, according to the security bulletin, quote, required to assist the investigation, unquote. They have been instructed to use force to bring you in.>\n\n_< Lethal_ force?>\n\n<If necessary.>\n\n<Damn!> Until now, it had just been necessary to keep a low profile. Now they were suspects and would be hunted through every corridor of the habitat until caught. Or killed.\n\nHer face must have shown something. When she focused once again on Ansourian, he was staring at her.\n\n\"What's happened?\"\n\n\"The administer...\" Roche began, then hesitated, debating for a moment whether or not to tell him. The news had the potential to distract him, a distraction she couldn't afford right now. Nevertheless, he did have the right to know. \"The administer is dead,\" she finally said. \"Assassinated, it would seem.\"\n\nHis expression flickered, and for the briefest moment she glimpsed a grief that surprised her. Even knowing that Jans was involved in the attack on him, Ansourian still felt compassion for the woman.\n\nCatching Roche's surprise, he said: \"For all her failings, she didn't deserve _this_.\"\n\nRoche nodded. \"I understand,\" she said. She allowed a moment's pause before gesturing to the door. \"I'm sorry, Ansourian, but we have to keep moving.\"\n\nAll trace of sadness vanished from his expression as quickly as it had appeared. \"Do you want me to go first?\" he said, jerking a thumb at the door in front of them.\n\nRoche flipped her helmet closed. \"No, I'll go. The suit will hold for a second or two, if it has to. You just open it.\"\n\nHe nodded and stepped back. The lock glowed green, and the door slid aside.\n\nRoche stepped into the corridor, confident that the Box knew what it was talking about but wary of any new developments it may have overlooked. As promised, though, there were no guards in sight.\n\n\"Clear,\" she whispered.\n\nAnsourian stepped out of the tunnel with Maii. The girl's visor was also closed, and she kept one hand close to her pistol at all times.\n\nAnsourian indicated the passage to their right, and they walked that way. Two guards came into view thirty seconds later, one on either side of a door two meters wide. They seemed alert but not especially concerned at seeing Ansourian and his companions as they approached.\n\n<What's on the other side of the door?> she asked the Box. <A rectangular grid thirty-five meters across, with ten corridors in two groups of five crossing at right angles. There are normally twenty guards, one at each end of each corridor, but half have been reassigned during the attack. There are one hundred cells in all. Alta Ansourian is in number 77.>\n\n<Are the guards shielded, Maii?>\n\n<Neither of the ones ahead,> said Maii. <But there are two on the other side of the door that are.>\n\nRoche didn't know what Ansourian had planned, so to forestall anything precipitous, she said to Maii: <Tell Ansourian his daughter is in Cell 77, in case he doesn't know. We'll take care of these guards.>\n\n<Okay, Morgan.> There was a brief pause as Maii relayed the message. <He sends his thanks,> she told Roche.\n\nRoche nodded. <Now, what about these guards, Maii?>\n\n<Already taken care of,> she said.\n\n<Okay, then let's move.>\n\nInfluenced by Maii's powers, the guards stepped aside as they approached and waved them through; then, when the heavy door had closed behind them again, they promptly forgot they had ever seen the intruders.\n\nAnsourian automatically took off along the nearest of the black-gray corridors. He seemed unconcerned by the strong jolts that occasionally rattled and shuddered through the walls. Roche followed him, keeping a close eye out for the two shielded guards. They turned right, then left, passing sturdy-looking doors with numeric keypads instead of locks. There was no way to determine whether the cells were empty or not, however, as the window in each door was shuttered closed.\n\nWhen they reached Alta's cell, Roche noticed a guard standing motionless at the far end of the corridor. She assumed he was unshielded and controlled by Maii, since he didn't seem too concerned about the presence of visitors to the cell block.\n\nAnsourian quickly tapped the appropriate code into the keypad of his daughter's cell. Nothing happened.\n\n\"They've changed the codes,\" he hissed. \"We'll have to blast our way in. Is your pistol up to it?\"\n\n\"It'll have to be.\" She removed it from her holster and took careful aim. <Box? Help me out here, okay?>\n\n<The door is already unlocked,> it said.\n\n<Thanks.> She fired, and the door sprang open. Roche looked up. The guard hadn't reacted to the noise, but she could hear the inquiries of someone who had.\n\n\"Inside!\" She pushed Ansourian and Maii into the cell, followed them in and shut the door.\n\nAlta was sitting on a low bunk. She looked up, surprised, when she saw them enter.\n\n\"Father!\" A smaller, female version of Atul Ansourian, with dark eyes and a small tattoo on her throat, sprang to her feet to embrace him.\n\n\"Shhh!\" Roche waved for silence. Footsteps were coming toward the cell.\n\n<One guard is approaching your location,> said the Box.\n\n<I hear him> She waited, pistol at the ready. The smell of energy-fire would linger for a while. If it was strong enough, the guard might realize immediately where they were. If it wasn't, he might just keep walking, which would at least give them an extra minute or two. As far as Roche was concerned, every second was valuable.\n\nThe footsteps stopped directly outside the cell.\n\nRoche didn't hesitate: she opened the door and fired her pistol. The bolt of energy caught the guard on the shoulder of his armor, spinning him heavily into the wall. While he was distracted, Maii penetrated his shield and rendered him unconscious.\n\nHe fell to the ground with a thud, his weapon sent clattering along the floor. But Roche didn't dare hope that he would be the last of their problems. There was one more unshielded guard in the complex, and she had now fired two shots.\n\n\"Quickly!\" said Roche back over her shoulder. \"Get her into that suit!\"\n\nWithout a word, Alta slipped into the OSFA suit her father had brought.\n\nRoche edged back out of the corridor, using the Box to scan the corridor of the holding cell grid. Aside from the unconscious guard outside Alta's cell, she counted eight others: five along one wall and three along the other, all subdued by Maii. One was missing.\n\n<Any guesses, Maii?>\n\n<He must've slipped out to get help.>\n\nThe Box thought otherwise. <The checkpoint door has not opened since we came through it.>\n\n<So check the cells, then,> said Roche. <He must be in one of them.>\n\n<The cells do not have surveillance cameras,> it said.\n\n<You didn't _see_ where he went?>\n\n<I was distracted by the guard approaching you,> the Box confessed. <The resources I have available here are hardly what I would call optimal, Morgan.>\n\nShe shrugged off its excuses. They would just have to hope the guard planned to lie low while they made their exit.\n\nAlta had finished donning her suit and stood, slightly stunned, with one hand on her father's shoulder.\n\n\"We're leaving here?\" she asked.\n\n\"Yes,\" Roche said. \"The sooner the better.\"\n\nThey filed out of the cell and into the corridor, Roche leading, keeping alert for any sign of the missing guard. They made it around the first corner, then the next. The checkpoint appeared ahead of them, its doors invitingly open.\n\nRoche had barely passed the last cell door when it burst open. The guard managed to get off three shots before Roche brought him down with two shots from her own weapon\u2014the first bolt hitting his side and spinning him away, the second taking him in the back and throwing him forward onto the cell's bunk, which collapsed noisily beneath him.\n\nRoche ignored him and went to check on Alta and Ansourian instead. The woman, clutching an injured arm, was scrambling her way to her father who had taken two direct hits, one to the chest and one to the stomach.\n\nRoche checked Ansourian for broken ribs, but couldn't feel anything through her glove and the blackened material of his OSFA suit. Even unconscious, he winced with pain as she probed.\n\n\"I can carry him,\" she said to Alta, \"if you want to take that risk.\"\n\nThe woman nodded, all expression gone. \"He needs medical attention,\" she said, \"and he's not going to get it here.\"\n\n\"He'll be all right if we can just get him back to the _Ana Vereine._ But before that,\" she said, remembering Ansourian's instructions to Haid and Vri, \"we need to find corridor 14 in Sector Green-D. Do you know where that is?\"\n\n\"Yes, of course,\" she said. \"But why do we need to go there?\"\n\n\"Because that's our escape route,\" Roche explained hastily. \"I'm arranging a pickup at the maintenance airlock there.\"\n\nAlta frowned. \" 'Maintenance airlock'\u2014is that what Father said?\"\n\n\"Yes, why?\"\n\n\"It's not a maintenance airlock,\" she said. \"It's an old refuse dumper.\"\n\nRoche stared at the woman for a second. Dumpers were little more than chutes designed to fire pellets of unreclaimable material at a suitable disposal site\u2014be it the atmosphere of a dead world, the nearest sun, or anywhere the waste would do little harm. They would be lucky if it had even the most primitive airlock facilities, let alone somewhere to dock.\n\n\"This just gets better and better,\" she said, shaking her head. \"But it doesn't matter. If it's the only way out, we're going to take it.\"\n\nShe rose slowly to her feet, bringing Ansourian with her. The suit gave her the strength to carry him, and the fall in ambient gravity helped, but he was still going to be a burden\u2014especially if they encountered any more guards and she had to use her pistol.\n\n<I'm going to need as much warning as I can get if there are any signs of guards approaching\u2014okay, Box?> she said. <I don't want to have to drop him like a dead weight at the last moment.>\n\n<You've got it, Morgan,> said the Box.\n\nShe staggered as the floor shifted under her; then, regaining her balance, she headed for the checkpoint. They passed through unhindered, and headed off along the corridor.\n\n<I presume you can't read her,> Roche said to Maii.\n\n<Not at all.>\n\n<Trusting bunch, aren't they?> She shifted Ansourian's body to a more comfortable position. Alta took them along a series of wide, curving corridors, many of them ringing to the sound of sirens, but each as deserted as the last. The vibrations seemed to lessen when they stuck to the main routes, and Roche was grateful to he able to forget about not banging her head on the maintenance tunnels' low ceilings. She did feel exposed, though, and would be glad when they reached the exit point\u2014even if it was just a refuse dumper.\n\n<Is Cane going to be able to find us, Box?>\n\n<He and Kajic are monitoring our progress through the habitat, and the egress point is clearly visible from the outside. That is probably why Ansourian chose it. Once we get close, there won't be a problem.>\n\n<How far away are Ameidio and Vri?>\n\n<They are a little farther away than you.>\n\n<And any word on the situation with the administer?>\n\n<No change, Morgan. You are still wanted, and I am still blocking security's attempts to trace you electronically.>\n\n<Well, that's something at least.>\n\nA further violent judder swept through the habitat just as they turned down another corridor, this time forcing Roche to one knee in order to keep herself from falling. \"What was _that_?\"\n\nAlta looked frightened as she helped Roche to her feet. \"We've been holed!\" she said as a sudden shift in air pressure made everyone's ears block. The sirens took on a new note of alarm. \"Pressure-doors will be up all over the station.\"\n\n\"We have suits,\" said Roche. \"Do you know the override commands?\"\n\nAlta shook her head. \"My father would have.\" Roche felt trapped. Keeping the Box's existence a secret was becoming harder and harder. <Box, give me the codes!>\n\nThe Box complied and she quickly relayed the codes to Maii. <Give them to Ameidio. Tell him Ansourian told us what they were. Hopefully they don't know he's down.> Via implants again, she said: <Box, give us some interference on the line to the _Ana Vereine_ and open those doors for us when we reach them. We'll just have to bluff our way through this one.>\n\n<Yes, Morgan.>\n\nRoche nudged Alta on. \"Come on, keep going. We're going to make it, okay?\"\n\nThey continued through the vibrating corridors of the habitat, though now with more pace and urgency. There was still no one around, and Roche assumed the nonessential Vax personnel were holed up in their pressurized compartments. If the majority of the security personnel were down at the docks or elsewhere in the habitat where the defense was concentrated, that at least decreased the chances of their being found. But it wasn't enough to help Roche relax. Not until she was standing on the bridge of the _Ana Vereine,_ with the habitat far behind her, would she allow herself that luxury....\n\nThey passed through two open pressure-doors, then a third. Roche could see the puzzlement on Alta's face at the blatant lapse in safety precautions, but there was no time to explain. The only time they stopped was when a particularly violent attack shook the habitat so badly that Roche was flung into one of the walls, and Maii and Alta were thrown onto the shuddering floor.\n\nSteadying herself, Roche asked: <How far now, Box?>\n\n<Two more junctions. The way is clear. Haid and Vri are two corridors over, barely a minute behind.>\n\n<Uri? Tell Cane to start moving in. We're almost there.>\n\nFrom the _Ana Vereine,_ Kajic said: <We have the location. I'll let you know when he's there.>\n\nThey passed through the first junction, taking a little-used passage that obviously connected to the maintenance tunnels they had left earlier. The second junction took them to a wide, straight corridor that seemed to stretch for miles. In the distance, loping toward them, were two figures, one dressed in black, the other in gold.\n\nAlta started when she saw them.\n\n\"It's okay,\" Roche assured her. \"They're with us.\"\n\nYet another violent shudder ran through the walls and floor.\n\n\"Quickly,\" said Roche, stepping over to the dumper's enormous steel hatch. \"Get that thing open!\"\n\n\"We haven't used the dumpers for years,\" explained Alta, tapping at a keypad. \"Father kept this one in commission to use as an emergency exit, if he ever needed it. It was used occasionally, but not regularly. But the codes should still work.\"\n\nThinking of the dissolving door in the administer's chambers, Roche said: \"If we do get out of this in one piece, we'll have your father's devious mind to thank for it.\"\n\nHaid and Vri arrived in a clatter of armor and heavy footsteps.\n\n\"Any problems?\" Roche asked.\n\n\"None,\" Haid said. \"Which only makes me more nervous.\"\n\n\"It would,\" she said. Alta glanced up but there wasn't time for introductions. \"Cane should be waiting for us out the other side of the dumper. If they're designed like others I've seen, we should be able to crawl into the induction tube and out into vacuum easily enough. Then we'll have to jump to the scutter.\"\n\n\"And then...?\" Haid prompted.\n\n\"Then we wait to see what happens here,\" said Roche. \"The administer has been killed, so it'll be easier to deal with things here if we can reinstate Ansourian somehow. It might be worth sticking around to see if the habitat survives the attack.\"\n\nAlta had looked up again when Roche mentioned that Inderdeep Jans was dead, but returned quickly to her task. Seconds later she had finished tinkering with the keypad and tried to open the hatch. The hinges were stiff; with Haid's assistance the hatch finally came open a little.\n\n<Morgan,> asked Kajic via her implants, <is everything okay?>\n\n<As much as it can be. Why?>\n\n<Cane is reporting something unusual on the hull near the dumper outlet.>\n\n<'Unusual'? In what way?>\n\n<An object of some kind. He can't make it out yet, but he's concerned you might be about to walk into a trap.>\n\n_A trap._ Roche watched as Haid and Alta continued to pull at the dumper's hatch. Haid might have been right to be nervous, after all.\n\n\"Just blow it, Ameidio,\" she said. Then to Maii: <Maii, link up! I want to see what's ahead.>\n\n<I can sense nothing, Morgan.>\n\n<Let's do it, anyway,> said Roche. <I want to make sure.>\n\nShe stepped back from the recalcitrant hatch as Haid positioned the charges to blow it open. She closed her eyes as she lowered Ansourian to the ground, allowing the gray vistas of n-space to unfold around her.\n\n<Cane is in position,> said Kajic. <But the object he saw has gone.>\n\n<Wait...> Roche's mind drifted with Maii's, leaving the cluster of minds that were Haid, Vri, Alta and Ansourian. She had trouble seeing herself, and Maii.\n\nNot far away from them, though, she clearly saw the distinctive dip of a clone warrior.\n\n<Who's that?> said Roche. <Is it Cane?>\n\n<I _think_ so,> said Maii, uncertainty stressing the word. <He's closer now than he was.>\n\n<That would make sense. Can you see anything else?>\n\n<No...>\n\nThe corridor rocked beneath them as the charges blew open the hatch. Roche opened her eyes on a cloud of smoke obscuring everything in the corridor. Haid stepped to the hole in the wall where the hatch had been.\n\n<No, wait!> said Maii, her mental voice urgent. <There are two of them!>\n\nHaid hesitated by the entrance to the refuse dumper, startled by the mental shout. The gray of n-space overwhelmed Roche's ordinary vision\u2014and she too saw the second hole in the field that meant a clone warrior. At first she didn't realize how Maii had known there were two, but then it all fell into place. The second one was farther away and moving gradually closer. It had all the hallmarks of Cane's mind.\n\nThe first one they had seen was _inside_ the refuse dumper. Before anyone could move away, there was a second explosion, this time from the other side of the open hatchway. A hurricane of air roared past them as the atmosphere inside the corridor was sucked out the open outlet, into space. Roche braced herself, and grabbed Ansourian's body as it slipped toward the hole. Pressure doors slammed closed along the corridor; alarms screamed, fading gradually as the air pressure dropped.\n\n<Check your seals!> she shouted via her implants. <Make sure your suits will hold!>\n\nThen, out of the hatchway, moving easily against the whirlpool currents of air, her white ponytail whipping in the wind, stepped Inderdeep Jans.\n\n****\n\n****\n****\n\n****\n\n**10**\n\n****\n\n****\n\nPerdue Habitat\n\n955.1.32\n\n0810\n\nJans was wearing an OSFA suit not dissimilar to the ones Ansourian and his daughter had on, except her face mask wasn't in place. That she had blown the dumper outlet without sealing her suit first surprised Roche, but she didn't have time to dwell on the matter\u2014the clone warrior was advancing toward her. Jans didn't appear to be armed, but that didn't lessen the potential threat. There was a spark of malice in her eyes, and her attention was fixed firmly upon Roche.\n\nBefore Roche could react, Haid grabbed Jans's shoulder from behind and spun her around. Jans used her momentum to kick out and up, throwing him with ease across the corridor. Then, with a single, fluid motion, she was at Vri's side, disarming him of the rifle he had leveled at her only a second earlier. One shot blew a hole in the far wall as the weapon was wrenched from his grasp and turned on him. In that same instant, Roche managed to snap off a shot from her own pistol, knocking the rifle out of Jans's hands. The warrior didn't seem the slightest bit fazed; instead, her free hand now struck out twice at Vri's head. The third time she lashed out at him she grabbed the Surin's arm and spun him around, hurling him at the nearest wall. He bounced off the gray-black surface heavily and fell back to the floor, his body prostrate and writhing.\n\nThe warrior now turned to face Roche. For the briefest moment their eyes met.\n\nThen the clone warrior pulled her mask closed and took two running steps toward Roche\u2014unbelievably fast. Roche fired just as her assailant jumped. Jans's suit flared mirror-bright for an instant; then she was on top of Roche, forcing her down and knocking the pistol out of her hands. Roche struggled, but two incredibly strong hands lifted her from the ground and threw her across the floor. She slid into a wall; even through the hazard suit, the impact knocked the breath out of her. She raised her hands to ward off another blow, but none came. It wasn't _her_ Jans was after.\n\nPlacing her feet on either side of Atul Ansourian's prone body, Inderdeep Jans thrust down with her left fist, penetrating his suit, rib cage, and heart with one, smooth punch.\n\nEven in her dazed state, Roche knew what the clone warrior was doing: she was finishing the job. Now there would be no easy way that the situation in Perdue Habitat could be fixed.\n\nInderdeep Jans rose to her full height now and looked around, her hand dripping red. The drop in air pressure cast an eerie half-silence across the scene. Roche could hear her own breathing loudest of all, over the calling of voices in her implants. Vri's roar of defiance from behind the clone warrior went unheard entirely, as did Alta Ansourian's cry of horror.\n\nRoche felt a wave of giddiness wash over her as Maii leveled a spear of mental force at the clone warrior, and thrust with all her strength. Jans didn't even react. She just stood there for a couple of seconds, as though contemplating what to do next. Then, having made a decision, she stepped away from Ansourian's body and over to Roche. Without wasting time, she steadied Roche with one hand to the chest and raised her bloodied fist to strike. There was no emotion in the woman's eyes, just cold detachment. And in the seconds that Roche looked into those eyes, part of her felt a fear deeper than any she had known before. But another part of her was just tired, and the breath she released at that moment was almost a sigh of relief that it was finally over....\n\nThen the clone warrior's fist fell.\n\nA flash of white, and the fist missed her helmet and embedded itself in the corridor wall.\n\nRoche sagged, her legs turned to water. Only Jans' hand on her chest held her upright as the clone warrior turned, eyes flashing at Cane standing in the open dumper hatch, a pistol steadied in both hands, still aiming the shot that had knocked Jans' punch aside.\n\n<Morgan, get the others out of here while you still can!> Cane's voice traveled via his pressure suit through the _Ana Vereine,_ then to Roche's implants. <The scutter's holding by the outlet. I'll take care of this.>\n\nJans's face became an angry sneer and the hand on Roche's chest tightened.\n\nCane shot her again. The suit's reflective powers had been compromised by the two shots it had already received, and this time the reflective flash was more purple than white. Jans recoiled from Roche, then sprang forward so fast that her figure became a blur.\n\nCane didn't waste time firing again. The pistol had hardly begun to fall when her lunge met his defensive crouch. Their limbs moved too rapidly for Roche to make out anything clearly. One would strike, the other would defend and counterstrike\u2014the exchange almost too quick to follow. There was a dizzying effect to the battle as each whirled, ducked, and thrust, with no indication as to who was gaining the upper hand.\n\nAfter a few moments the two of them came apart, as if thrown from each other by a small and silent explosion. Cane bounced back into a wall as Jans skidded along the corridor. Quicker than it would have taken Roche to blink, the two were poised again, ready to attack.\n\n_< Move,_ Morgan!> The inside of Cane's pressure mask was spattered with fresh blood.\n\nHis call brought her to life and, as the two clone warriors clashed again\u2014with Cane leaping forward to prevent Jans getting any closer to the hatch\u2014Roche reached for Alta, who was crouched over the body of her father.\n\n\"Come on!\" she shouted. But Alta ignored her, whimpering as she pulled ineffectually at her lifeless father, as if trying to drag him to his feet. \"We have to _go!\"_\n\nRoche dragged her to the hatch. Maii followed, clutching the back of Roche's suit as though it were a lifeline. Once there, Roche took Alta's face in her hands and forced the woman to look at her.\n\n\"I need you to help Maii through the hatch. Do you understand?\"\n\nAlta blinked back tears, sniffed, nodded.\n\n\"Okay, _in_ ,\" said Roche.\n\nThe floor bucked beneath them as first Alta, then Maii, climbed through.\n\n\"Help Maii into the scutter at the end of the outlet,\" she said to Alta while their suits were still touching. \"I'll follow in a second.\"\n\nVri was standing over Haid, his gold armor dusted with frozen water vapor. Haid was out cold, his armor heavy and inert. Roche helped Vri drag him toward the hatch, each taking one leg. She briefly considered using one of the weapons in his suit pouches to shoot Jans again. But when she glanced up the other end of the corridor, she knew there was no way she would get a clear shot.\n\nThe two clone warriors were a blur of motion with only occasional, unpredictable pauses. There was a strange beauty and grace to their movements, almost balletic. Each was as superb as the other at combat, each forced to rely on subtleties and surprise rather than brute strength to gain the edge. Roche had never seen anything like it, and knew that in such a conflict she would have barely lasted a second against either of them. And in realizing that, she also realized just how lucky she was to be alive right then. Had Cane reached her a second later...\n\n\"Lift,\" Vri grunted, returning her attention to her injured friend. She bent her knees to slide Haid's body into the hatch. The floor shifted beneath them again, and Roche distinctly felt the ambient gravity dip. That was a bad sign, even if it did make Haid easier to carry. Somewhere, something was going terribly wrong for the station.\n\n<Uri, what's happening?>\n\n<Reinforcements have arrived from Random Valence,> Kajic replied. <They're concentrating fire in several areas, including power generators and life support. The habitat is taking a beating in the process.>\n\n<I can feel it,> said Roche. <Do you think it'll hold?>\n\n<Unless their tactics change, I have no doubt.>\n\n<I bet that's what they thought on Guidon, too.>\n\nWith Haid through the hatch, Vri also went through, and lent Roche a hand. Together they maneuvered Haid along the chute and to the open end. The hatch there was blackened and bent open from the outside. Seeing it, Roche realized that _Cane_ had blown the outer door on the outlet, not Jans, thereby forcing the other clone warrior to attack before she was ready. Had he not done this, it was unlikely they would have survived so long.\n\nThe scutter hung tethered two meters from the outlet. Vri jumped the gap, easily negotiating the change from gravity to zero-g. A quick glimpse of the local region of space revealed several ships accelerating brightly across the starfield, some of them under fire from cannon out of sight behind the tangled bulk of the habitat. Gas clouds, glowing and expanding, hung in numerous places between the ships, although Roche couldn't tell whether they were the remains of destroyed fighters or missiles that had missed their targets. Another shudder rocked the section Of the habitat she clung to while leaning out to pass Haid's body to Vri. Looking back along the length of the outlet chute, she could actually see the structure flexing.\n\nWhen Haid was safely aboard the scutter, Vri reached out for Roche's hand.\n\nShe shook her head. <Give me a rifle,> she said. <I'm going back for Cane.>\n\nHe turned away, returned a second later with two weapons pulled from Haid's armor, then jumped across the gap. Handing one to her, he said: <I'm coming with you.>\n\n<You don't understands she protested, waving the weapon under his nose. <This isn't to help him; it's for self-defense!>\n\n<I know, but I'm still coming.>\n\nShe shrugged, turned, and began crawling back along the chute. She didn't have time for arguments.\n\nNeither Cane nor Jans had followed them, so they had to be still inside\u2014unless one or the other had forced their way through the pressure doors at either end of the section of the corridor.\n\n<Be careful,> she said as Vri approached the open hatch.\n\nHe nodded, shifting the rifle into the optimal position and taking one side. Roche took the other.\n\nPeering out of the hatch, to the right along the corridor, she could see nothing. Vri, on the other hand, nodded, and she shifted over to his side.\n\nCane and Jans were still fighting furiously, although both had sustained injuries now. During one standoff, Roche noticed that one of Jans's fingers stuck out at an odd angle, and that she seemed to be favoring the hand in combat too. But the blood on Cane's face mask was thicker and darker, and it had to be obscuring his vision.\n\nJans lunged again; Cane parried her blows with familiar grace. But this was different from the fights he'd had before. Then he had been fighting almost for the fun of it; now he was fighting for his life.\n\n<Box, is there nothing we can do?>\n\n<No. We can only watch.>\n\n_And learn,_ she thought to herself, hoping it was worth risking a companion's life in exchange for such knowledge.\n\nThe two combatants separated and maneuvered for position. Jans caught sight of Vri and Roche in the open hatchway and kicked off for them immediately. Taken by surprise by the move, Cane was a split second slow in jumping after her. Roche recoiled as Jans's hands reached out for them, but the clone warrior rolled at the last minute and kicked off the wall. Cane was caught mid-leap and flung across the room.\n\nJans was on him before he could recover, rolling him over so he was between her and Roche, with an arm firmly around his neck. Cane's spine bent backward, and he twisted to look into Jans's eyes. As Roche aimed her rifle on the slight chance Jans miscalculated and gave her a clear shot, Roche saw their eyes lock.\n\nThen, with a sickening lurch, gravity failed entirely. Using the sudden shift to his advantage, Cane pushed away from the floor and kicked Jans into the ceiling. As she rebounded, he was ready with a second kick that sent her spinning into a wall. He followed, catching her around the throat and wrenching her backward. It was her turn to be pinned from behind.\n\nJans writhed, managing to twist her hands behind her and pull Cane's mask off. He hung on even tighter, provoking a grimace of pain from the woman. Roche could see air puffing out of his suit, around his cheeks and jaw, but he seemed oblivious. Jans screamed silently into the vacuum, and for a moment Cane's grasp seemed to relax. Only slightly, but enough to allow Jans to twist her own head around to look at him. He leaned into his sibling, then suddenly tightened his grip once more. With one savage twist he snapped her neck.\n\nHer body went limp, but he waited a further minute before letting her go. Then he collected the face mask dangling about his throat and carefully replaced it. The exposure to vacuum didn't seem to have harmed him at all.\n\nRoche climbed out of the hatch, followed by Vri. Cane looked up. His face was black with frozen blood.\n\n<Are you okay?> she asked, kicking closer to him through free-fall.\n\n<Are _you_? _>_ His voice, even via implants, sounded weary.\n\n<Yes, but we should get out of here.>\n\nHe nodded and let Vri take him by the arm.\n\n<The body,> said the Box. <You must take the body with you.>\n\nRoche eyed the corpse of Inderdeep Jans with distaste; it was one of two in the corridor. <I presume you mean hers, not Ansourian's.>\n\n<It will contain important data,> said the Box.\n\n<Yeah, I know.> She approached it cautiously, afraid that even now Jans might be dangerous. The clone warrior's skin was much paler than it had been only seconds ago, and mottled with hundreds of odd-shaped bruises.\n\n<Guards are coming,> the Box nudged her.\n\nShe grabbed the body with both hands and pushed it toward the hatch as Vri and Cane passed through. With considerably less difficulty than she'd had with Haid\u2014for Jans was smaller and unarmored\u2014Roche got her through the hatch and dumper chute and into the scutter. She sealed the airlock behind her, glad to be in familiar territory once again.\n\nMaii and Alta were already braced and ready in the cockpit. Alta looked up as Roche brought Jans's body inside, then glanced away, her eyes watering.\n\nVri helped Roche stow the body in the rear hold, where the Box could keep an eye on it. Then she went back to the pilot's station, which Cane had left vacant, and took control of the scutter from Kajic.\n\n<Where are you, Uri?> she asked.\n\nA chart appeared on the console, with major concentrations of fire and wreckage marked. <I've marked the optimal course.\"\n\n<Thanks.> She glanced around the cockpit. Everyone looked shell-shocked. She supposed she looked the same. Surprisingly, Cane looked worst of all; his skin was pale and his cheeks were hollow. He looked sick, something Roche would never have thought possible. And it wasn't just exhaustion, either. It was something much more.\n\nWith a simple series of commands directed through her implants, she fired the scutter's thrusters and moved them away from Perdue Habitat.\n\n* * *\n\nThe scutter was attacked the moment it disengaged. Fighters and cannon converged on it, urged on by reports that they were responsible for the death of the administer\u2014which in essence was true. Roche guessed that the remaining clone warrior was behind it, continuing the ploy that had begun with the report that she had assassinated Jans\u2014only this time it was for real.\n\nSurprisingly, the Random Valence ships joined in, local rivalries forgotten for the moment\u2014or perhaps directed by another clone warrior among their own numbers. Roche wondered what lies they might be being fed, but it was impossible for the Box to access their command network.\n\nBarely had they managed to evade pursuit long enough to dock with the waiting _Ana Vereine_ when something exploded in the habitat behind them. They couldn't tell what it was\u2014maybe a major power plant, or a weapons store. Roche could only watch with a sinking feeling as the great tangle of corridors began to disintegrate, unraveling like a knot and breaking into chunks as local stress points flexed too far and snapped.\n\nPursuit abruptly fell off as all the ships in the area retreated to search for survivors. Roche wanted to assist also, but Haid\u2014awake but concussed\u2014talked her out of it.\n\n\"Even if they don't shoot us out of the sky,\" he said, \"they probably wouldn't let us get close enough to do any good. Remember: one of them is still in there, and they're _not_ going to make it easy for you.\"\n\nBy \"them\" he meant the clone warriors. Roche was beginning to think that \"they\" were everywhere.\n\nAs the _Ana Vereine_ retreated from the vicinity, injuries were attended to while everyone discussed what had happened. Haid was still recovering. Alta Ansourian was under sedation; Roche didn't know if she would ever be united with the habitat survivors, or indeed any of her people, but she would do her best for the woman. Vri was back in his ship. Maii was resting on a bench on the bridge, where Roche and Kajic went over the small amount of data they had managed to gather.\n\nCane was asleep in the medical unit.\n\n<He's genuinely unconscious,> the Box informed Roche.\n\n<How can you be sure?> she said.\n\n<I coopted the diagnostic AI to my service,> it replied.\n\n<So his injuries were severe, then?>\n\n<I don't believe his need to rest has anything to do with his injuries or his being tired, Morgan. At least not a physical tiredness.>\n\nShe knew what the Box was implying. She herself had done little else but think about it since leaving the habitat, and even now the significance of what had happened was still difficult to fully appreciate.\n\n<He killed one of his own kind,> she said. <He killed her in order to save us\u2014at no small risk to his own life, either.>\n\n<Yes.> Even the Box seemed to be caught on that simple fact. Why would Cane be prepared to do something like that if he wasn't worthy of their trust? She should have been glad now to finally have proof that he was in fact on their side. So why was she still suspicious of him? Because he had hesitated at the end? She wasn't sure she could blame him for that....\n\n<What about the dead warrior?> she asked needing to think about something else. <Anything there?>\n\n<Some developments I've not seen before. The cellular composition of her skin is changing at a rapid rate, returning to a decidedly more Pristine appearance rather than that of the Caste she was impersonating. The chameleonic abilities of these creatures is literal, it seems, to a degree.>\n\nRoche added that to the list of things they'd learned. <So you've studied their bodies before?>\n\n<Others have, and I have access to some of their data. Most bodies are found in capsules or ships destroyed by energy fire. Very few intact remains in other words. Never has one been studied so soon after death.>\n\n<Well, that's something,? she said. <At least we know how she managed to look so much like the Vax, even if it wasn't perfect. What I don't understand is, how could she have fooled Ehud Jans into believing that she was his daughter?>\n\n<She didn't have to.>\n\n<What do you mean?>\n\nThe Box hesitated for a second. When it spoke again, its voice was contrite. <It is easy to see in hindsight, Morgan. The data was in the habitat records all the time. Despite their strict laws controlling breeding, the Vax do have adoption policies. A parent may adopt a new child if their first has been killed by accident, or died of a disease. In such circumstances, they may take a parentless child as their own, in order to continue both lines. In the case of Inderdeep Jans, she is recorded as an orphan recovered from the wreckage of a mining vessel that drifted into Vax space. She was taken in and became a close friend of Ehud's real daughter, Eir. Not long after, Eir Jans was killed in an accident, and Ehud adopted Inderdeep as his own.>\n\nRoche remembered something Ansourian had said about the previous administer discouraging the use of transit tubes after \"a terrible accident,\" years ago. <All arranged, I imagine.>\n\n<It seems likely,> said the Box. <Since then, she simply bided her time until she could do the most damage, doing her best in small ways to impede progress against the enemy.>\n\nRoche could see that, too. Ansourian had also mentioned that Inderdeep Jans wouldn't have agreed to come to Sol System at all if he hadn't pressured her into it. Certainly her odd moods had kept the efficiency of the habitat at a low ebb. And ultimately, with Roche on her way and Ansourian a genuine thorn in her side, she had been forced to make her move.\n\nRoche couldn't help but wonder what might have happened had they realized the truth earlier. Maybe she could have saved the lives of Ansourian and everyone else on the station, or maybe it would've turned out as it had. There was no way of knowing.\n\n<And what do we do now?> Roche mused. <Turn around and go back? Proceed to the next place on Nemeth's list? Return to the _Phlegethon,_ perhaps? We could even just head off on our own.> She sighed tiredly. Despite the options, she still felt trapped. She wanted a clear goal\u2014or even better, a decisive way to strike at the enemy. The ambiguity was driving her crazy. <Or maybe we could just do nothing at all,> she finished.\n\n<It is up to you now.>\n\n<Yes, I know,> she said irritably. <But what do _you_ think?>\n\n<I believe proceeding to the next place on the list would be the best course of action right now.>\n\n<There's no guarantee it'll be any better than where we've just been.>\n\n<Of course not.>\n\n<No, and that's the point, isn't it?> Perdue Habitat had been ready to explode, and Roche had been the lit fuse Nemeth had thrown at it. The chances were high that the other places on the list would be equally volatile.\n\n<Now that we know that the enemy is able to pose as members of marginally Exotic Castes,> said the Box, <we can at least be more careful in future. And we've also learned that they can be patient for years before cutting loose.>\n\n<That only makes them harder to find, though,> said Roche.\n\n<Not for you,> returned the Box.\n\n<But that doesn't do me much good when I have to stand practically face to face with one to be certain!> They hadn't spent much time examining her newly found talent. When Cane was awake, perhaps, they would look at it more closely. When they could find a way to fix the precise locations of people in the real universe from the impressions they left in n-space, maybe there would be a chance. Maybe.\n\nShe sagged back in her seat. If Kajic wondered why she was no longer showing much interest in the data they were supposedly examining, he said nothing at all. Maybe he just assumed she was tired like everyone else\u2014which she was. It was made worse by the fact that she knew she wouldn't be getting the chance to rest any time soon.\n\n<What _is_ next on the list?> she asked.\n\n<We have a choice of two. There's either an out-station maintained by the Drys or a loose collective known as the Katajalin Serai. Both are slightly out-system from our present position. We can be at either place in approximately fifteen hours.>\n\nRoche's weariness intensified at the thought. In just fifteen hours they could be going through it all again.\n\n<First, I suppose I'd better put together some sort of report for Nemeth,> she said, feeling no enthusiasm at all for the task.\n\n<I can do that for you, if you like,> said the Box. <At least the first draft. You can look at it when I'm done, and send or amend it as you wish.>\n\n<Thanks, Box.> As always, she was unsure how far she could trust the AI, but the offer was a welcome one.\n\nSpeaking aloud, she said: \"Uri, I'm sorry. I'm really not up to this at the moment. I'm feeling a little tired and distracted. Let me get a couple of hours' sleep and we'll finish up then, okay?\"\n\n\"Of course, Morgan,\" Kajic said. \"Is there anything you'd like me to get on with while you rest?\"\n\nRoche thought for a second. \"Yes. Plot a course for the Katajalin Serai and get us under way. Its location is in the file Nemeth gave us. Send a drone to check things out first, though; this time I want to know exactly what we're getting ourselves into.\"\n\n\"Understood.\" As his holographic image began to dissolve, he added: \"Sleep well, Morgan.\"\n\n\"I will,\" she said to the suddenly empty space before her.\n\n<I can make sure of that,> said Maii softly into her mind.\n\nFor once, Roche was tempted. All the other times the girl had made that offer, she had turned it down out of fear that the reave might tamper with her mind. She'd had plenty of opportunities now to see that Maii's intentions were innocent. If Roche could trust her with the knowledge that the Box had survived the destruction of the _Sebettu,_ then she could probably feel safe in her hands for a couple of hours.\n\nBut the thought still nagged at her that she would be defenseless. It would take longer still for her to get over that feeling.\n\n<Okay,> she said. <Just get me to sleep, though, and I'll do the rest.>\n\nThe girl radiated warmth. <Of course, Morgan. Gladly.>\n\nLater, though, Roche would regret that she hadn't given herself wholly over to the girl's care. Her sleep was disturbed by dreams that left her frightened, bewildered, confused, and sad. Most unnerving was a recurring image in which she was floating gently over a flat gray landscape in which there suddenly appeared a circular hole. First a pinprick, then a gaping mouth, then a yawning chasm, it sucked her into its lightless maw\u2014and she fell, much as she had in the dream that had haunted her through COE Armada Military College and before her arrival in Palasian System. Only this time there was no bottom. This time her fall went on forever.\n\n****\n\n****\n****\n\n****\n\n**PART THREE:**\n\n****\n\n**PHLEGETHON**\n\n****\n\n****\n****\n\n****\n\n**11**\n\n****\n\n****\n\nTBC-14 (a.k.a. IND Kindling)\n\n955.1.33\n\n0390\n\n\"Who is the enemy, Page?\" Saltan Trezise asked via the ftl link with the _Phlegethon._\n\nDe Bruyn didn't answer immediately. She was too busy watching the recording of Perdue Habitat disintegrate. It was the third time she had sat through it; this time she was paying special attention to one particular section of the structure.\n\nSigns still remained of where Roche's tame clone warrior had blown his way in through the outer hatch of the refuse dumper, and where the ex-Dato scutter had left distinctive exhaust marks upon its departure. Unfortunately, that portion of the habitat had been one of the most severely damaged, and De Bruyn's recording was the only remaining proof of what had taken place there.\n\n\"Someone got rid of the evidence,\" she said, watching again the puff of energy that tore apart the section of the habitat, atomizing anything that might have remained within and slagging the dumper itself. \"There's nothing left at all.\"\n\n\"Are you listening to me, Page?\" he said. \"I asked\u2014\"\n\n\"I heard you,\" she said. And indeed she had: she had switched to an ordinary audio link, tired of his voice insinuating through her implants. After a session with him, she felt as if her brain had been soaked in oil. \"I'm busy.\"\n\n\"Busy doing what?\"\n\n\"The work you and your ineffectual council should be doing.\"\n\n\"We have enough on our hands without wasting time pursuing grudges.\"\n\n\"Is that really what you think this is?\" She made no attempt to hide her anger. She'd had enough of that sort of talk from the mercenaries on _Dark Stressor._\n\n\"You haven't given me sufficient evidence to think otherwise, I must say.\"\n\n\" _Must_ you?\"\n\nHis sigh was audible over the link. \"Page, you're acting like a child.\"\n\n\"I'm investigating every possibility,\" she said. \" _That's_ what I'm doing.\"\n\n\"No you're not,\" he countered quickly. \"You're investigating just _one_ possibility from every possible angle.\"\n\n\"And what's wrong with that? It's still a possibility.\"\n\n\"A remote one, at best,\" he said. \"Really, Page, what have you to go on? A suspicion; a scrap or two of barely circumstantial evidence\u2014\"\n\n\" _More_ than a scrap! We know Roche was in the habitat. One of your own agents lifted her image from security. But no one saw her leave.\" She indicated the image before her, speaking as much to herself as to Trezise, mulling over the problem again. \"This whole area was blacked out at the time. We have no hard evidence of anything!\"\n\n\"We _do_ know she escaped, Page. She sent a report to Rey Nemeth ten hours later.\"\n\n\" _Someone_ did. We can't be sure it was her.\"\n\nHis laugh held more exasperation than amusement. \"You think she stayed behind?\"\n\n\"Maybe. Or maybe the scutter acted as a distraction while she went elsewhere.\"\n\n\"Where?\"\n\n\"I don't know.\"\n\n\"Why?\"\n\n\"I don't know.\"\n\n\"With whom?\"\n\n\"I don't _know_.\" Her fists clenched.\n\n\"So in the absence of any alternative theory, do you _really_ believe it wasn't her?\"\n\n\"All I'm saying is that it pays to be careful. She's slippery, and she's not stupid. If she thinks she's being followed, who knows what lengths she would go to to avoid me?\"\n\n\"Possibly,\" he muttered. \"But you have to admit that these lengths would be ridiculous.\"\n\nShe opened her mouth to protest, but decided against it. The evidence overwhelmingly suggested that Roche _had_ been on the scutter when it left the habitat, and that she had therefore been aboard the _Ana Vereine_ when it left the vicinity of the disaster. If De Bruyn could allow herself to feel satisfied on that point, she could move on. There were other, more immediate things to worry about.\n\nShe killed the recording, deciding that she did in fact agree with Trezise. She would not, however, give him the satisfaction of knowing that she had given in.\n\n\"Have _you_ learned anything of note?\" she challenged him as she checked local space. Still no sign of the ship she was waiting for.\n\n\"Things are moving apace,\" he said. \"The Kesh-Olmahoi conflict has ended as expected, with the Kesh decisively routed. Surprisingly, no evidence has been found of enemy involvement. But the council is lending its support to those attempting to clean up the mess.\"\n\n_No evidence_ , De Bruyn repeated to herself, disappointed that he should be so blind.\n\n\"We've been investigating the cult I mentioned earlier, the one purported to worship the enemy,\" he went on. \"They have caused a moderately large amount of damage and their influence appears to be growing. It seems there are several splinter sects. Their devotion is quite genuine, albeit misplaced and lacking any foundation. We have interrogated a number of devotees and found that they possess no more knowledge about the enemy than we do ourselves. If the enemy _does_ have any involvement with them at all, then it is purely symbolic. They share no information with their worshippers, and no plans, and seem to bear them no sympathy at all when they are purged. Five pockets of the religion have been flushed out of as many fleets, and not once has the enemy raised a hand to stop us.\n\n\"The interesting thing, though, is that the cults believe, as do some of us, that the enemy has been sent to rid the galaxy of Pristines. While this does conflict with some of the evidence we have gathered, it _is_ suggestive. There might yet be a grain of truth to that belief, after all.\"\n\nDe Bruyn let him ramble on. She didn't know whether he was talking for her benefit, or his own, but she doubted he would be telling her anything simply because he thought it would help her. Maybe he thought she knew something about the cults and might let it slip if he encouraged her. She did, but she wasn't going to let him know.\n\nThe near-space screen was still empty.\n\n\"There have been a few setbacks,\" he continued. \"One of the council's major allies, the Espire-Mavrodis Coalition, was struck from within just hours ago. A group of military officers attempted to capitalize on discontent in the lower ranks, and mounted a coup that might very well have been successful, had not one of their own had a change of heart at the last moment and turned against them. The debacle cost the life of nearly every person we had come to rely on in that quarter, and has left the Coalition in complete disarray. They'll be no use to us in the short term, I'm afraid\u2014and maybe never, as there is talk of them pulling out to regroup. Unfortunately, the defecting officer has not been found.\"\n\n\"Has the Commonwealth of Empires been affected?\" De Bruyn asked.\n\n\"No. That continues to puzzle me. It is such an easy target, and so close to the focus of things. I would've thought it would be riddled with the enemy.\"\n\nDe Bruyn nodded to herself, agreeing with him about the COE's vulnerability but not really surprised that it seemed to have been spared.\n\n\"And I hope you've taken my warning about the system's ring,\" he said. \"We've lost five ships in there, that we know of. Four others are incommunicado. With the little information we've had to go on, it's hard to say why this is happening. Given the conflict within the system at the moment, the likelihood of a ship being attacked simply because it is not recognized by another Caste has risen dramatically. It is possible that the ships were attacked out of paranoia. It's hard to know for sure.\" He paused. \"It may not have anything to do with the ring itself at all. Still, it doesn't hurt to be cautious, and until we _do_ know for certain what's going on, I advise you to stay away from it.\"\n\nDe Bruyn had been doing just that, not because of Trezise's warning, but because Roche herself had stuck to more distant regions of the system.\n\n\"Could the enemy be hiding within the ring?\" she asked.\n\n\"No, it's too thin\u2014thinner than even the most meager of atmospheres. Add it all together and you get enough for a medium-sized gas giant or two, but spread out so far it amounts to practically nothing at all. The Heresiarch still hasn't worked out where it came from, but that's a low priority at this point. He's only concerned because he's obsessed by navigational anomalies. With a ship that big to look after, even the slightest snag could be disastrous. You know, I think he spends all his time\u2014\"\n\n\"Look, Salton,\" she interrupted, \"was there an actual _reason_ for this call?\"\n\n\"I always have a reason, Page. You know that.\"\n\n\"Would it have anything to do with the information I requested?\"\n\n\"No,\" he said. \"But I do happen to have that as well.\"\n\nShe gritted her teeth. \"Can I have it?\"\n\n\"It should be coming through with this transmission.\" She could hear the smile in his voice, but kept her annoyance in check.\n\n\"Thank you,\" she forced herself to say, but not without a hint of sarcasm, and not before checking that he was telling the truth. He was. As soon as the data transfer was complete, she would get rid of him.\n\n\"But before you go,\" he said, irritating her still further by seeming to read her thoughts, \"can I ask you a small favor in return?\"\n\n_Here we go,_ she thought \"Which is?\"\n\n\"That idiot Murnane is thinking of letting Exotics onto the council. Obviously that would be a dangerous idea, even if they were allowed only an advisory or associate status. Things are complicated enough without adding more interests into the mix. If we can't guarantee our own safety\u2014as the downfall of the Espire-Mavrodis Coalition demonstrates all too well\u2014why should we make ourselves responsible for anyone else's?\"\n\n\"Unless they can help you more in return.\"\n\n\"Which is exactly what I think Murnane is hoping for. But I for one am skeptical.\"\n\nShe refrained from commenting on his motives. \"What does any of this have to do with me?\"\n\n\"Well, I'm presuming that you are following Morgan Roche as she gallivants across this forsaken system. That seems to be your mission in life, after all.\"\n\n\"You could presume that,\" she said, thinking: _But you'd be wrong._\n\n\"And I happen to know that her mission is to reconnoiter the Exotic Castes to see how they are faring against the enemy. But we both know this is a waste of time. _Everyone_ is performing badly against the enemy\u2014be they Exotic or Pristine.\"\n\nThe data had arrived. \"What's your point, Salton?\"\n\n\"I need a reason to keep the Exotics off the council, Page. You can help me find that reason.\"\n\n\"I can?\"\n\n\"You're out there, among it all. You're seeing it firsthand. You can give me the flip side to Roche's reports. She makes it sound like the Exotics are suffering as much as we are\u2014which may be true, but I'd prefer if it wasn't quite so obvious.\"\n\n\"Why not, if it's the truth?\"\n\n\"That's just the point I'm trying to make. We should ignore it _because_ it is true. It _has_ to be deliberate. What's the biggest problem in Sol System at the moment? It's not the enemy\u2014although I don't doubt their presence is being felt in a thousand small ways: disrupting communications, sabotaging procedures, corrupting information, and more. Even without the enemy, even if everything ran smoothly, there'd still be chaos: there are simply _too many people_! And when these people start fighting among themselves, the situation inevitably worsens. We're balancing constantly on the precipice of all-out mayhem; only the most super- Human effort stops us from doing the enemy's job for them.\"\n\n\"So you figure that if there are less people, things might improve?\"\n\n\"Of course! Reorganize and integrate our resources, limit the number of governments to a manageable number, reduce the council's active concerns to a feasible few, and _maybe_ we'll prevail. Given that the majority of people here are Pristines, and that apart from a few exceptions the enemy seems to target Pristines, wouldn't it be better if everyone else left and let the Pristines sort it out?\"\n\n\"With you in charge, no doubt.\"\n\nHe laughed lightly. \"I knew I could count on you to see my reasoning,\" he said. \"And if my proposal goes through the council, hopefully others will too.\"\n\n\"So you want me to lie about what's happening to the Exotics?\"\n\n\"No; I just want you to balance the scales.\"\n\n\"It amounts to the same thing,\" she said.\n\n\"It's all about _how_ you put it, Page. How else do you think politics works? Give me the right words and I could move entire solar systems.\"\n\n\"Not if there isn't anyone listening to you.\" She found it hard to imagine greasy Salton Trezise, known for so long as Auberon Chase's lackey, being a force in his own right in something like the IEPC. \" _Is_ anyone going to be listening to you, Salton?\"\n\n\"I have the ear of several councilors,\" he said. \"Enough to make a difference.\"\n\n\"Why do you need me, then? Why not use the council's own agents?\"\n\n\"Because _your_ reports will help discredit Roche's. That's a cause to which I know you will apply yourself.\" He chuckled softly to himself for a second. \"Look, even discounting personal grudges, Page, this will benefit us both. Think of the enemy\u2014the _common_ enemy. _They're_ the ones we're ultimately after\u2014not Roche, not Murnane. If we can do what we want to do while at the same time achieving what we need to achieve, then we approach success. Ultimately the ends do justify the means, and, as we're in this together, we might as well use each other to the best of our abilities. I'm not naive, Page; I _know_ you've used me to suit your needs.\"\n\n\"I would never call you 'naive,' Salton,\" she said soberly.\n\n\"Good,\" he returned. \"Then let's start working together, because together we _can_ do some damage\u2014to the enemy, and to those who oppose us.\"\n\n_And to each other_ , she thought. But all she said was: \"Okay, Salton. I'll see what I can do.\"\n\nHe was silent long enough for her to suspect he might have gone, even though the line was still open. That suited her. A beacon had begun to flash on the fighter's main display, indicating that a ship had just entered her region of the system. She didn't know if Trezise could tell where she was by tracing which ftl drone had detected her replies to him; maybe the broadcasts traveled from drone to drone in a complicated chain, not just via a single one direct to the receiving stations on the _Phlegethon._ Either way, she was glad he seemed to think she was still hot on Roche's heels. If he knew where she actually was, he might reconsider his deal.\n\nThe Hum ship drifted smoothly toward _Kindling,_ looking like a cross between a sailing ship and processional barge, its bulky, bullet-shaped body almost completely obscured by tapering instrument spines, crisscrossing antennae, and curved flanges that seemed to serve no actual purpose. This far from the sun, out where an O\u00f6rt cloud might once have existed, there was very little light to view anything effectively. In artificial color, the vessel looked like it was painted electric blue, with highlights of bright orange and green. Its vividness was unsettling enough without its contents to consider.\n\nIt decelerated to a relative halt and hung there, waiting.\n\n\"I have to go,\" she said to Trezise.\n\n\"Understood,\" he replied without any appreciable lag. \"We'll speak again soon, Page.\"\n\nFor once, despite herself, she hoped that this would be true.\n\n* * *\n\nThe circular bridge of the _Apostle_ was dimly lit and smelled of steel. De Bruyn sat at the lowest point of the cavernous space, surrounded by ranks of instruments like steps in an amphitheatre. Tall figures moved among those instruments, but the light was too faint for her to make them out. She half-glimpsed robes and cowls; very occasionally, eyes glinted at her.\n\n\"We know who she is,\" said the shadowy figure sitting opposite her. His face was completely obscured by the hood of a black combat suit that had been modified to give it a more ceremonial air. \"But do you know who _we_ are?\"\n\n\"You are the Disciples of Evergence,\" she said.\n\n\"And what does that name mean to you?\"\n\nShe hesitated. \"I'm not sure,\" she said finally. \"The word 'Evergence' doesn't appear in any language I have access to. It's not a name, it's not a code\u2014\"\n\n\"It has no meaning of itself,\" the hooded figure interrupted her. \"It could be said to be a confluence of many essences: of convergence, forever, emerging, divergent, evolution, emergency, and even vengeance. But 'Evergence' is none of these things. It is a word for something that has, until now, needed no words. It has existed in silence, and will return to silence when the need for words is gone.\" The ominous figure inclined its head as the echoes of his voice faded. \"But I feel you do not understand.\"\n\n\"No, I'm afraid you've lost me,\" she admitted with a shrug.\n\n\"It doesn't matter. Your comprehension is neither essential nor desired. We simply wish to ensure that you bring with you no misconceptions about our purpose here in Sol System.\"\n\n\"You worship the enemy,\" she said softly. The words carried much more significance when said in context.\n\n\"So it has been said,\" the figure observed. \"And if that is what you choose to believe, so be it.\"\n\n\"You're telling me you _don't_ worship them?\"\n\n\"We do not _worship_ ,\" the figure intoned. \"That is all I am telling you.\"\n\n\"But you are on their side?\"\n\n\"Yes.\"\n\n\"And Morgan Roche is your enemy.\" She didn't need to phrase that as a question.\n\n\"Yes, she is.\"\n\n\"Then I believe I can help you,\" De Bruyn said, feeling a catch in her voice that surprised her.\n\n\"Really?\" Was there a hint of mockery in his voice? She couldn't be sure.\n\n\"Yes,\" she said, the word emerging as little more than a croak. What was wrong with her? Finally she had found people that suited her needs, and she seemed to be having second thoughts! But she wasn't about to back out. Not now. She _couldn't._\n\n\"I _can_ help you,\" she said more assertively, adding: \"If you help _me_.\"\n\n\"Ah, I see.\" The figure nodded thoughtfully. \"And what is it exactly that you want from us?\"\n\n\"A deal,\" she said. \"We work together and we _both_ get what we want.\"\n\n\"You would serve us?\"\n\n\"No,\" she said quickly. \"I would work with you. For a time.\"\n\n\"How?\"\n\n\"I am alone, here. My resources are meager. But I know what I'm doing, and I have access to information at the heart of the IEPC. Give me one of your cells to command, and I will do your job for you.\"\n\n\"Which job do you think that would be?\"\n\nShe leaned forward. \"With my knowledge and your Disciples, we can trap Roche. I _know_ we can. All we have to do is cooperate, and she will be out of your hair forever. It's that simple.\"\n\n\"Nothing is ever that simple,\" he said darkly.\n\n\"Look, all I need is to get my hands on her,\" De Bruyn said. \"I don't even care if she's alive, just as long as I have her body.\"\n\n\"Why would you want her body so badly?\" he asked.\n\n\"Because it contains the truth,\" she said. \"The truth of who she is.\"\n\n\"I've told you once that we already know who she is,\" he said. \"Why not just ask?\"\n\nDe Bruyn didn't state the obvious: how could he know anything more than _she_ did? The rumors she'd heard\u2014and which, she was sure, comprised the bulk of the Disciples' knowledge\u2014 were wild and contradictory. Some proclaimed Roche as a savior, others as a traitor. De Bruyn lent none of them credence, just as she wouldn't waste her time listening to the views of a religious fanatic.\n\n\"Because I want to see the truth with my own eyes,\" she said.\n\nThe robed figure pondered this for a few moments, then asked, \"And what _exactly_ do we receive in turn?\"\n\n\"Apart from Roche out of the way?\" She leaned back into the seat and shrugged. \"What do you want?\"\n\n\"You say that you have access to the Interim Emergency Pristine Council.\"\n\n\"I have a contact\u2014\"\n\n\"Will you give us information?\"\n\nShe hesitated. \"What sort of information?\"\n\n\"The information we require.\"\n\nShe waited for him to elaborate, but he said nothing more\u2014and he obviously wasn't going to until she had agreed or disagreed. An icy silence filled the air of the bridge, and again her doubts returned.\n\nShe shook herself free of them by reminding herself why she was here. This _wasn't_ a grudge-match. This was about justice. Everything she had unearthed suggested that she was doing the right thing. If she was doing this for herself, why was she going to such lengths? Any sane person would have given up weeks ago\u2014and there could be no question of her sanity.\n\nShe was _so close_ to the truth....\n\nAs she waited for docking instructions to arrive from the _Apostle,_ she'd used the time to scan the data Trezise had given her. A cursory glance had been all she needed. The records weren't complete, but they did fill in portions of a bigger picture. They had come from the second moon around Bodh Gaya, the former capital system of the Commonwealth of Empires, where the Armada housed its Military College. Morgan Roche had served there during her training for COE Intelligence, years ago. Trezise had managed to get his hands on various reports, assessments, essays, and test scores that demonstrated how average a student Roche had been. Only in one area had she excelled, and that had been the handling of AIs. She had preferred to grapple with artificial minds rather than those of the people around her.\n\nPerhaps, De Bruyn thought, that might explain why she'd had so few friends. It certainly explained how she had been selected as a courier for JW111101000, the Box that had ultimately helped her escape from Sciacca's World and the clutches of the Dato Bloc. That was one AI De Bruyn was glad to see the end of.\n\nBut that information was not specifically what De Bruyn was after. The information she sought hadn't been there at all. And in some ways, the gaps were more telling. All of Roche's physical records were unavailable. Not missing or deleted: _unavailable._ When Trezise had asked why, he had been told that access to those records was restricted by special order VSD5278.\n\nDe Bruyn recognized that order number. It was department shorthand for the COE's previous Eupatrid, Enver Buk. Eupatrid Buk himself had specifically ordered those records kept secret, no matter who asked for them.\n\nThat in itself might not have been significant. Trezise had dug back a little further, not just trying to find a birth date but _any_ physical record at all, prior to Roche's enrollment at Military College. Her name was recorded at an orphanage on Ascensio, but there was little else of note: no medical records, no education reports, no informal recollections. Her application scores were on file in Ascensio's COE Armada recruit database, along with the form she must have filled out to apply for the test, but the results of her medical exam were missing. Special order VSD5278 had cast a cloak over them, too, it seemed.\n\nIt had taken De Bruyn only a minute or two to confirm what she already suspected: that the details of Roche's early life were being kept secret by the government of the COE, and that this secret had been ratified by the Eupatrids past and present who had issued the special orders required to ensure that no one ever found out the truth.\n\nWhat that truth was, though, she wasn't yet entirely sure. But she had suspicions. She had been chasing those suspicions, along with Roche, across Sol System and half of the COE in the hope that they might be verified. If Roche's early life was being kept secret, it was entirely possible that the details Trezise had uncovered were completely fictitious. Where had Roche come from before Military College? Out of thin air, it seemed\u2014which made her suspiciously like the clone warriors the IEPC were trying to fight.\n\nAs soon as the idea occurred to her, De Bruyn had been caught by its ramifications. If it was true, Roche _must_ have been planted by the enemy to seed chaos and disorder the way that they had in so many other systems. The fact that she had not, until recently, shown any destructive or even subversive tendencies did not necessarily invalidate this theory.\n\nDe Bruyn noted that at about the time Roche enrolled in the Military College the High Human known as the Crescend had joined the COE in a partnership designed to foster trade and joint industry between the Caste echelons. Maybe the Crescend knew about Roche and had decided to see if she could be contained rather than destroyed. Maybe he hoped to bend her to his will, or at least make her an ally rather than an enemy; that might explain why he was so keen to keep her existence a secret, to the point of penalizing those who came even close to the truth, like De Bruyn. Maybe he didn't care what happened to the COE at all, and was only interested in seeing what happened at firsthand when the time bomb called Morgan Roche finally went off.\n\nMaybe he wasn't involved at all, and the giant shadows De Bruyn saw on the wall before her were cast by shapes much tinier. Either way, she had to be sure. Something odd was going on, and she had been caught up in it. Now that she was close to finding proof\u2014of _anything at all_ \u2014it was finally time to act. But she couldn't do it on her own; she was going to need powerful friends if she was to see justice served.\n\nShe needed to fight fire with fire.\n\n\"I'll give you information,\" she said to the man in black, \"if you give me Morgan Roche.\"\n\nHis posture didn't change, but in the shadows that hid his face she thought she had seen him smile.\n\n\"Good,\" he said. \"In that case, _God's Monkey_ will meet with you in precisely seven hours. Its pilot and contingent of Disciples will obey your commands\u2014unless, of course, those commands are contrary to the goals of our movement. You may use them as you will until such time as our mutual obligations have been discharged. Does that suit you?\"\n\nShe swallowed with relief. \"Thank you, yes,\" she said. \"But where exactly will I meet them?\"\n\nHe gave her the coordinates. \"They will know you as 'Reverence,' \" he said. \"Use them well, Page De Bruyn.\"\n\nHe stood. She stood too, although her muscles felt weak. The gray-clad Disciples who had led her to the bridge stepped out of the shadows to stand by her side.\n\n\"Wait,\" she said to the figure in black. \"You know my name?\"\n\nHe turned to face her, but said nothing.\n\n\"Couldn't I at least know yours?\"\n\n\"Mine is not relevant,\" he said.\n\n\"If you're worried that I will tell the council\u2014\"\n\n\"The thought would never cross our minds,\" he said, taking a step toward her. \"Nor should it cross yours again.\"\n\nDe Bruyn swallowed. \"I assure you,\" she said nervously, \"I wouldn't tell anyone\u2014\"\n\n\"Oh, I know you won't,\" he interrupted her. \"Not just because I tell you that should you betray us by divulging anything that has passed between us this day, we would hunt you down, Page De Bruyn, and we would kill you. Of that _you_ can be certain.\"\n\nHe took another step forward, into the light, and smiled as she recoiled a pace in alarm.\n\nEven with the hood up, there was no mistaking the face of Adoni Cane.\n\n\"I know,\" he said, \"because we have a deal...\"\n\n****\n\n****\n****\n\n****\n\n**12**\n\n****\n\n****\n\nAVS-38\n\n955.2.12\n\n1770\n\nDefender-of-Harmony Vri carried the injured girl over the threshold of the airlock, ignoring the gunfire that insistently peppered the back of his combat suit. Once he had safely passed the girl to the armored figure waiting for him, he turned and fired four shots in rapid succession in the direction he had come.\n\nThere was an explosion. Immediately, the gunfire stopped.\n\nThe scutter shuddered noisily as the airlock closed with a hiss.\n\nVri steadied himself. The scutter lurched away from the station and weathered a battering on its way back to the _Ana Vereine._ He walked the short distance to where the girl had been strapped to a stretcher and attached to a portable autosurgeon. The shot had taken her in the shoulder, piercing her hazard suit and making a mess of the flesh beneath, then exploding messily out the back. Only the suit's small first-aid facility had kept her alive while Vri and Haid fought their way back to the scutter through near-endless waves of Fathehi custodians.\n\nThe autosurgeon's display was blinking red: the girl needed the full version on the main ship before she would begin to recover.\n\nVri stood. His faceplate clicked open, swinging up and back to reveal not just his face but most of his head too. Even through the light hair that covered every inch of his features, his anger was obvious.\n\n\"It was too close,\" he said. His voice was deep, and every word perfectly enunciated.\n\nHaid too had shucked the helmet of his combat suit. \"We were unlucky,\" he said, wiping sweat from his hairless black forehead with the palm of his glove. \"Even when they sprang the trap, we thought Maii had it covered. But all it took was one lucky shot...\" He looked down at the girl, rocking in her stretcher as the scutter endured another battering. \"Maybe we weren't so unlucky. At least we managed to get out.\"\n\n\"It was too close,\" Vri repeated with the same, slow precision to his words.\n\nHaid looked at him. \"So you keep telling us.\"\n\n\"It is not _you_ I am telling.\" His intense eyes were as golden as his armor, and focused on the back of the person piloting the scutter.\n\n\"I hear you,\" said Roche. She didn't need to turn to know he was referring to her. Nor did she particularly care what the Surin thought at this moment; there was too much happening to worry about that. Besides, she had been watching via the suit monitors and the Box's patch into the consulate's security channel; she knew better than either of them just how narrow their escape had been. Another ten seconds in the dock and a full squadron of custodians would have pinned them in a crossfire from which none of them would have emerged alive.\n\n\"I'm taking her back,\" the Surin warrior said. His firm tone conveyed more than the words themselves.\n\nRoche did turn, at that. \"You're not taking her to Erojen.\" Her voice was as hard as his.\n\n\"No, of course not,\" he said. \"I meant to the _Phlegethon.\"_\n\nRoche was silent for a long moment; then her expression softened slightly. \"Okay,\" she said. \"The _Phlegethon_ it is. But if you don't let me fly this thing home first, we won't be going anywhere at all....\"\n\nThe Surin nodded, and Roche turned back to the controls. He took the seat closest to Maii's injured form.\n\nSensing Haid's eyes on him still, he turned to face the ex-mercenary. They stared at each other for a few seconds.\n\n\"You fight well,\" Vri said finally, adding: \"Despite your handicap.\"\n\nHaid's eyes flashed. \"As do you, despite yours.\"\n\nVri frowned a question, and Haid indicated the girl.\n\n\"I do not consider my protecting this child to be a handicap,\" said Vri indignantly.\n\nHaid raised one arm, indicating where an energy bolt had passed clear through his arm and out the other side.\n\n\"Nor do I consider this to be one, either,\" he said.\n\nVri pondered this for a moment, then turned away and was silent for the remainder of the trip back to the _Ana Vereine._\n\n* * *\n\nEarlier, they had argued about the mission to the Fathehi Consulate.\n\n\"It is too dangerous,\" Vri had insisted.\n\n\"Dangerous, yes; _too_ dangerous, no,\" Roche had shot back. \"The junior consul herself has assured us that we will have free passage through the station.\"\n\n\"And you _believe_ her?\"\n\n\"Why shouldn't I?\"\n\n\"After all that has happened, I find your naivete disturbing.\"\n\nRoche felt her face turn red. \"We are more than capable of handling anything they might throw at us.\"\n\n\"We are a handful of people against an entire station!\"\n\n\"Maii can\u2014\"\n\n\"Yes, she can. And she _has._ You rely on her too much. You are _using_ her! You are using her as I would use a percussion rifle\u2014to be tossed aside when its usefulness has expired.\"\n\n\"That's not true!\" Roche was finding it difficult containing her emotions, and a blast of anger from the reave in response to Vri's accusation only enhanced her irritation. \"We rely on her help, but that's _not_ the same as using her.\"\n\n\"She is a child that is\u2014\"\n\n\"That is still capable of making up her own mind!\" snapped Roche. \"If you want to talk about people being used, then take a good look at yourself!\"\n\nThis caught the Surin warrior by surprise, and he frowned in confusion.\n\n\"You're just a stooge of the Surin Agora,\" Roche went on. \"They don't care about the truth, and they don't care about Maii! They just want her back so they can take her apart and see how she works. And you're helping them!\"\n\nVri had reared back in the screen: \"Be careful what you say, Morgan Roche.\"\n\n\"If you're threatening me,\" Roche spat, \"I swear I'll have you shipped back to the council faster than you can say 'mindless pawn'!\"\n\n\"Easy, you two,\" said Haid, putting a hand on Roche's shoulder. \"This is getting us nowhere.\"\n\n<I agree,> put in Maii herself, her emotions back under control. <I don't require either of you to argue on my behalf.>\n\nRoche took a deep breath and looked down at her feet. Maii was right; she was as guilty as Vri of assuming that she knew what was right for the girl.\n\n<I will go into the consulate,> Maii said. <And if you so desire, Vri, you may accompany me as my bodyguard. Would that be acceptable?>\n\nFor a moment, Roche thought Vri would argue with her, but instead he simply nodded. \"It is a reasonable compromise,\" he said.\n\n<Good. And Haid will go with us while Morgan stays with the scutter.>\n\n\"But\u2014\" Roche began.\n\n<Morgan, you need a rest from the front line. Let someone else take the risks for a change.>\n\n\"She's right, Morgan,\" said Haid, smiling. \"It's my turn to be the hero.\"\n\nRoche knew he was only joking, but she couldn't help feeling slightly stung by his words. She was reluctant to lose control because she was desperate for something to go right\u2014just once! The fact that nothing had gone right for anyone in Sol System didn't change things. She still felt like it was _her_ that was somehow getting it wrong.\n\nAnd Maii did have a point. She _was_ tired of facing hostile envoys and suspicious security forces. This could be a good way for Vri to save some face, and to relieve the restlessness eating at Haid. Besides, she and Maii had swept for clone warriors upon their arrival and the place seemed clear. That made it safer than any of the other ports they had visited.\n\nOr so it had seemed. Roche had become so used to looking for clone warriors that she had blinded herself to base Human treachery. When the junior consul decided that Roche had extended herself far enough into her station, she ordered her custodians to open fire\u2014on the boarding party and on the _Ana Vereine_ itself, forcing the ship to retreat to a safe distance and leaving the others to scramble for their lives through the station. Had the scutter not already docked, and had the Box not been available behind the scenes to keep the custodians at bay, the situation could have been a lot worse than it was.\n\nEven without knowing what lay ahead, Roche had approached the mission with apprehension. Before leaving, she had gone to see Cane.\n\nHe had taken up residence in an observation blister on one of the _Ana Vereine's_ seven nacelles. The curved window allowed him an unobstructed view of the space around the ship. Not that there was much to see. The only object visible to the naked eye was the crossed rings of the Fathehi Consulate, tumbling slowly against the starry backdrop.\n\nThe Box had observed him there on numerous occasions over the previous two weeks, since the events on Perdue Habitat. When Roche didn't ask for him specifically, that was where he went. She suspected that he was avoiding her.\n\n\"Will you tell me why you're doing this?\" she had asked.\n\nAt first he hadn't answered, his brown skin soaking up the light from distant stars.\n\n\"Cane?\"\n\n\"I am thinking.\"\n\n\"What about?\"\n\n\"About what it is like to be alone.\"\n\nRoche had glanced at the stars, at the galaxy around them. All those systems, all those worlds, all the Humanity filling them up: High and Low, Pristine and Exotic, old and young\u2014and almost a thousand of those Castes were now crammed into Sol System. She didn't feel alone anymore. Not at all.\n\nAnd that didn't even take into account the AI sharing her body.\n\n\"Why?\" she had asked.\n\n\"I killed one of my own kind,\" he said. He turned to face her.\n\n\"You still have us.\" She had attempted a smile, then regretted it. \"I'm sorry. I didn't realize it was bothering you this much.\"\n\nHe had shaken his head and returned, expressionlessly, to watching the stars.\n\n\"Nor did I,\" he said quietly.\n\n\"You mightn't be the only one in this position, you know.\" She didn't want to take his black mood with her to the consulate; if there was anything she could do to crack his reserve, she would try it\u2014even if it meant inviting him to join the landing party to the consulate in Haid's place. \"Your siblings don't seem terribly indiscriminate. We only know of one time when they cooperated, and that was on Perdue, Maybe _that_ was the exception. Maybe they kill each other as easily as ordinary Humans do.\"\n\nBut he hadn't responded.\n\nEventually, she had left him alone and gone to see Alta Ansourian. Another mistake. The woman was still grieving for her father and for all the friends destroyed with her habitat. Since coming on board she had barely emerged from the stateroom Roche had given her\u2014not even to see what was going on when the _Ana Vereine_ had come under attack.\n\n\"Do you think I'll ever get back to Vacishnou?\" she had asked Roche.\n\n\"I'll do my best,\" Roche told her, hoping the words didn't sound as empty as they felt. The homeworld of the Vax was on the other side of the galaxy and seemed far removed from their current situation. \"That's the most I can promise, I'm afraid.\"\n\n\"I understand.\" But if she truly understood, it hadn't given her any comfort. And that more than anything made Roche wonder if Alta Ansourian was wiser than she looked.\n\n* * *\n\n<Are you sure this is the right thing to do?> the Box asked Roche as she rested in her cabin, the disastrous sortie to the Fathehi Consulate fresh in her mind.\n\n<What do you think?> She scratched absently at the back of her head. <Despite recent history, running away really isn't my thing.>\n\n<You know that it's not what you will be doing.>\n\n<No, but it feels like it,> she said. <And I don't like it.>\n\n<That much is obvious. But ask yourself, Morgan, who you are most angry at\u2014and why.>\n\nShe thought seriously about the question. Was she angry at Rey Nemeth for giving her a mission she couldn't finish? Or at Vri for forcing her to face the inevitable? Or at Maii for being shot?\n\nThe last was ridiculous. If anyone, she should resent the person who had shot the girl. That in itself was tempting, but not as tempting a target as the junior consul who had ordered the attack. Or the lieutenant who had turned them away from the LaGoc barracks that had been their previous port of call, as had leaders of the three previous habitats they had tried to contact. Then there were the Noske saboteurs who had planted a bomb on their usual scutter, nearly killing everyone on board; the clone warrior in the Katajalin Serai, responsible for triggering a mass riot that had torn the normally tight collection of vessels apart within a day of Roche's arrival; and finally Inderdeep Jans and her unknown enemy cohort who, together or apart, had brought about the appalling failure that had been the very first of Roche's missions for the Ulterior.\n\nShe could hate the council for not helping her, but that was pushing the boundary too far. She might as well hate the COE, or the Crescend, or the galaxy itself.\n\nExcept the Box hadn't asked about _hate,_ had it? It had asked her about her anger, and once she separated the two, the answer came to her.\n\n<I'm angry at myself,> she said in the end.\n\n<And why is that, Morgan?> he pressed.\n\n<Because I'm losing control, Box. I'm losing control of it all.> She paused as a dream image momentarily flashed across her thoughts. <I'm falling.>\n\n<Morgan, you never _had_ control.>\n\n<But I had a direction! At least that was _something_.> She rolled onto her side, into a fetal position. <Now...>\n\n<Now you just have momentum.>\n\nShe couldn't tell if the Box was being facetious or not. <What good will _that_ do me when it comes to the crunch?>\n\n<Perhaps a lot of good, depending on what you hit.>\n\n<If it's the council, they're not likely to give.> Vri had been pestering her ever since they left the volume of space the Fathehi Consulate controlled, but she hadn't yet confirmed the order to return to the _Phlegethon._ <Come on, Box, what do you think? I want your honest opinion.>\n\n<I would give you nothing less, Morgan.> The AI seemed pleased that she had asked its advice. <I think the _Phlegethon_ is as good a place as any.>\n\n<For what, though? What do I do once I get there?>\n\n<You can see what happens. Maybe the Ulterior will not regard your mission as a failure at all. You have, after all, learned some things.>\n\n<And got my name blacklisted from every station this side of the system.>\n\n<That's something, too. One would have to assume that was deliberate.>\n\n<I considered this. Why else would everything fall apart just when I arrived?> The destruction of Perdue had taken less than two days; the Katajalin Serai had dissolved in half that time. By the end of the week, no one in their right mind would let Roche on board their stations\u2014except the Fathehi Consulate who, it turned out, had planned a trap. <It's not me doing it, so it must be someone else hoping to make it _look_ like me. Unfortunately, it seems to have worked.>\n\n<No doubt the Ulterior will guide your return to the _Phlegethon_.>\n\n<I hadn't even thought of that! What happens if they don't let us in? What if\u2014?>\n\n<Morgan, you don't have to worry about that. They will let you return; you have something they do not.>\n\nShe rolled onto her back again. <Such as?>\n\n<Two things, actually. The most obvious being the experience you have gained.>\n\n<Half as much as their usual agents, I'm sure.>\n\n<Possibly\u2014but none of their agents have discovered a means of identifying the clone warriors, have they?>\n\nShe was about to protest that the talent was so vague as to be almost useless, except to sweep a large number of people to make sure they were clean. But then she stopped herself. The council didn't need to know about her talent's limitations\u2014not initially, anyway. The very fact that her ability existed would at least guarantee her a hearing.\n\n<So I become their test subject while people like Nemeth get all the credit,> she said cynically. <Sounds like a lot of fun, Box.>\n\n<But at least you'll have some direction again,> the Box responded.\n\nShe thought about that for a few seconds.\n\n<That's true,> she admitted. <And somewhere to channel that momentum.> She forced herself to let her eyelids close and her muscles relax. The last thing she wanted right now was to move _anywhere._\n\n<How many more stops were there on Nemeth's list?> she asked.\n\n<Two\u2014outposts run by the Saa-hurod and Yemena Castes.>\n\nNeither of the names rang a bell. <Does either look likely to tell us anything new?>\n\n<Impossible to say, from here.>\n\n<Does Nemeth highlight either as being especially important?>\n\n<No.>\n\n<Then we can safely ignore them, I think. And without Maii to help me sweep for clone warriors, I wouldn't feel comfortable in approaching them.>\n\n<Her prognosis is good, Morgan. She will be conscious within twenty hours.>\n\n<That's not the point,> Roche said, recalling the accusation leveled at her by Vri about using the Surin girl. <I don't want to push her unnecessarily. She needs rest.>\n\nRoche was silent for a while, on the verge of making the decision but still balking. It would only take a call to Kajic to put them on their way. She wanted to make absolutely certain\u2014as certain as she could be, anyway\u2014that she was doing the right thing.\n\n<Have you heard from the Crescend?> she asked.\n\n<If you are asking me whether he would approve of your return to the _Phlegethon_ , I can tell you that he would.>\n\nShe hadn't dared ask the question so blatantly, so to hear the reply threw her off balance for a moment. She didn't want to push her luck by asking _why_ he approved.\n\n<He's told you that?>\n\n<Yes.>\n\n<When?>\n\n<While we were just talking.>\n\nThe ramifications of that information were profound. <So you and he are communicating by ftl, like the council does?> she said. <Why didn't you tell me?>\n\n<It didn't seem necessary. Should I have?>\n\n<Well, no. It doesn't make that much difference, I guess. I just assumed...> She wondered how much else she had assumed about the Box's relationship with the Crescend that would one day turn out to be wrong. That was the problem in dealing with the AI: without asking exactly the right question, she couldn't be entirely sure she was learning what she needed to know.\n\nAnd yet it had offered the Crescend's advice on the matter of the _Phlegethon._ That was out of character. Either it had a hidden agenda\u2014which was all too possible\u2014or the High Human had finally decided to become involved.\n\nShe wasn't sure if the latter would be a good thing or not.\n\nYawning, she rolled back onto her stomach. <Box, open a line to Vri's ship. Audio only.>\n\nThe AI did as it was told, making sure to maintain the illusion that Roche herself had placed the call. When Vri responded, she said:\n\n\"Vri, it's Roche.\" The Surin didn't respond, so she kept talking. \"I need to ask you something. About Maii.\"\n\n\"Which is?\"\n\n\"Do you know the name of Maii's mother? Or even her family name?\"\n\nThere was a pause. \"No. Why?\"\n\n\"Would the Agora know?\" she said, ignoring his question.\n\n\"I would assume so.\"\n\n\"And would you think it unreasonable of me to ask for that information before I even consider handing Maii over?\"\n\nHe paused again, then answered, \"No, not at all.\"\n\n\"Good, because that's as far as I'm prepared to compromise at the moment. Unless you can convince me that the Agora knows what it's doing\u2014and is doing it for the right reasons\u2014then you will never complete your mission. If, however, you _can_ convince me, then your chances of convincing Maii will improve. Ultimately, _she_ is the one you have to deal with. Unless she agrees, you'll go home empty-handed.\"\n\nThere was a third pause as he seemed to consider this. \"I understand,\" he said finally.\n\n\" _Do_ you, Vri?\"\n\n\"I am not stupid, Roche. There is a bond between you and the child; that is indisputable. I may question the nature of the bond on your side in the same way that you question the sincerity of my superiors\u2014but, as you have pointed out, neither of us can do anything without Maii's consent. We are not monsters, you and I.\"\n\nRemembering the concern with which he had brought the injured girl back to the scutter, Roche could only agree.\n\n\"Then let's leave it at that,\" she said. \"We'll go back to the _Phlegethon._ She'll be safer there. You can contact the Agora, or their representatives in the system, and we'll talk about it. Openly, and _with_ Maii. She'll be conscious by the time we get back, and she'll be able to tell if anyone's lying. When we've all talked it through, she can decide what she wants to do, and we'll abide by her decision. Can we agree on that?\"\n\n\"We can.\" Through his gruff reticence, she sensed a certain satisfaction. \"I will contact my superiors as soon as we arrive and advise them of our decision.\"\n\n\"Good.\" She went to close the line, but stopped at the last second. \"And Vri? Thanks for everything you did for her today. You saved her life over there.\"\n\n\"Your thanks are not necessary. I was doing my duty.\"\n\n\"I know that,\" she said. \"But to you, duty is everything, isn't it?\"\n\nHe didn't reply. A second later, the line closed.\n\nRoche smiled to herself. <Box? Get me Uri. I want us under way as soon as possible. And draft a report for Rey Nemeth describing what happened with the Fathehi.>\n\n<What else would you like me to tell him?>\n\n<Just that. Let him assume we're going on to the Yemena Caste. I think I'd like to surprise him by turning up unannounced.>\n\n<Yes, Morgan.>\n\n<And...>\n\n<Yes, Morgan?>\n\n<The next time the Crescend thinks I should do something, ask him to tell me himself. I think it's about time we dispensed with the go-between.>\n\n<I hope you mean that metaphorically, Morgan.>\n\nShe smiled again as she closed her eyes. <Of course, Box,> she said. <Of course.>\n****\n\n**13**\n\n****\n\n****\n\nSHCV Phlegethon\n\n955.2.13\n\n1975\n\nThe conference room was conical in shape, its walls tapering smoothly up from the circular floor to a point far above, from which shone a single, bright light. A round table filled most of the floor space, cut, like the walls, from heavy gray stone. Apart from the table, the light, eight chairs, and a single door, the room was featureless. Roche had been assured that it was completely secure: no information could get in or out by any means, including epsense, without the knowledge of the room's inhabitants.\n\n\"We hardly expected you back so soon,\" said Esko Murnane. Roche had been surprised to see him at the meeting of the Ulterior, but not as surprised, it seemed, as Rey Nemeth, whose customary charm\u2014superficial though it might have been\u2014was still clouded by a scowl. There were four others at the table Roche hadn't been introduced to.\n\n\"I thought it best to return, for a number of reasons,\" she said, feeling extremely uncomfortable. Even though she was dressed in full combat armor and armed to the point where even the hospitable Skehan Heterodox had thought twice about letting her aboard their ship, she still felt vulnerable. \"There's something you need to know.\"\n\n\"Is it related to your mission?\"\n\n\"Perhaps indirectly,\" she said. \"I might not have found out about it if I hadn't gone.\"\n\n\"I have her reports,\" said Nemeth, attempting to reclaim some power in the assembly.\n\n\"I'll read them later,\" said Murnane dismissively.\n\n\"You won't find anything in the reports about this,\" Roche said. \"It's not information I'd like to go public with just yet.\"\n\n\"What is it, then?\"\n\nShe glanced at Haid, who had accompanied her this trip. Maii was still in the _Ana Vereine_ 's medical center, conscious but weak. Haid caught her look and shrugged.\n\n\"You have a problem,\" she said slowly. \"There are five of the enemy on board the _Phlegethon_.\"\n\nStartled mutterings broke out among those gathered, but it was Nemeth's voice which rose above them all: \"Are you sure?\"\n\n\"I have no doubts whatsoever,\" she said. \"In fact, there may be even more. You have too large a crew to scan all of them effectively and quickly. But I can tell you that three are amidships, one is down in the crypt and the other is up near the minaret.\"\n\n\" _Where_ exactly?\" pressed Nemeth.\n\n\"I can't tell,\" she said with a slight shrug. \"I just know they're there.\"\n\nThe mutterings continued among the other members of the Ulterior. Except for Murnane, who hardly reacted at all.\n\n\"How could you possibly know this?\" he asked calmly.\n\n\"It's hard to explain,\" said Roche.\n\n\"Nevertheless,\" said Murnane smoothly, \"you're going to have to try. We have no intention of taking you at your word.\"\n\n\"I understand that.\"\n\n\"Is there _any_ way to get a precise fix on them?\" said Nemeth.\n\n\"Only by looking for them in person,\" she said. \"By coming face to face with them.\"\n\n\"At grave personal risk, no doubt,\" said Murnane, leaning back in his seat.\n\nShe met and held Murnane's icy blue eyes. \"Yes,\" she said.\n\n\"Do you expect the council to sanction such an undertaking?\" There was a hint of a smile at the corners of his mouth.\n\n\"She hasn't come to the council,\" said Nemeth.\n\n\"The person I _should_ speak to is the Heresiarch,\" Roche said. \"It's his ship, after all.\"\n\nMurnane dismissed this with a wave of his hand. \"How long have you known?\"\n\n\"Little more than an hour. We swept the ship on the way in to dock.\"\n\n\"And how do you know that one of _us_ isn't a clone warrior?\" asked a woman to Roche's left.\n\n\"I checked before you sealed the room.\" Although Maii was on the _Ana Vereine_ and physically resting, her mind was still strong. The n-space link was harder to maintain over a distance, but still viable.\n\n\"That's not to say, of course, that one of us couldn't be in league with them,\" said Murnane, glancing around the table. \"The Ulterior is an organization designed for covert dealings, after all.\" Then, returning to Roche, he said: \"Perhaps it's time you told us exactly how you came by this knowledge.\"\n\nShe agreed and proceeded to describe how her chance link with Maii had brought to their attention Roche's ability to detect the minds of the enemy in the way they distorted the fabric of n-space. She still couldn't explain what those distortions meant or why it seemed she alone possessed this ability, and was as open about this as she was about her inability to pin down a clone warrior's precise location.\n\n\"Despite this limitation,\" interrupted someone from the end of the table, \"you are certain you _can_ identify them?\"\n\nRoche nodded. \"It enabled me to identify Inderdeep Jans as one of the two clone warriors on Perdue Habitat,\" she said. \"And of the seven locations we visited, I was able to scan them prior to boarding and determine which of the locations had been compromised by the enemy and which of them had not been.\"\n\n\"How many hadn't been?\"\n\n\"Just one.\"\n\nThis provoked another round of muttering, until Murnane broke in.\n\n\"Would it be possible to replicate this procedure with another reave?\" he asked.\n\n\"I don't know,\" she admitted. \"But I'm willing to try.\"\n\n\"Good.\" He nodded. \"We've been looking for a way to use epsense to reveal the enemy. This might just be it.\"\n\n<You are taking a risk doing this,> said the Box into her mind.\n\n<Less of a risk than a full probe.> Then, heading off an argument, she said: <It's a risk we have to take, Box.>\n\nThe conference room was unsealed for a moment, to summon one of the high-grade reaves warding the room\u2014one of the many Maii had observed in the fane during their first visit. She was a short Pristine woman dressed in white robes and a ceremonial headdress wrapped about her eyes, ears and mouth.\n\n<You are allied to the Surin Abomination,> she said to Roche. It wasn't a question.\n\nRoche colored slightly. Answering in kind, via epsense, she said: <The Olmahoi have permitted her existence.>\n\n<They _suffer_ her existence,> the woman said, taking a seat vacated by one of the Ulterior. <I am Stryki of Taborca. You are shielded. I cannot help if you will not open yourself to me.>\n\nRoche felt the faintest suggestion of Maii at the back of her mind; before Roche had left the _Ana Vereine_ , the reave had installed a shield guarding the knowledge that the Box had survived Palasian System. She forced herself to think of something else.\n\nThe woman entered her mind like a sheet of silk sliding into water. There was no sensation of invasion or penetration; she was suddenly _there,_ among Roche's thoughts, as though she always had been.\n\n<How does the Abomination access n-space ?> the woman asked. <Like this?>\n\nRoche felt her mind swept up by the woman's and tugged into another place\u2014a place where there were no walls, no boundaries, just the faintest suggestion of lines all around them, some intersecting, others stretching out to infinity. Where they met, they glowed white.\n\n<No,> Roche said. <It wasn't like that.>\n\n<This, then?>\n\nRoche was suddenly pulled in a thousand directions at once\u2014as though her skin had been plucked by fishhooks and stretched like the fabric of a balloon to the breaking point\u2014\n\n<No!> she managed.\n\nThe reave made no apology for the obvious discomfort she had caused, but the sensation vanished and Roche found herself floating over the familiar gray field of n-space.\n\n<That's it,> she said, unable to keep the relief from her voice.\n\n<Ah,> said the reave. <A simple two-dimensional Nyberg representation. I should have anticipated something unsophisticated like this. It's often the way with the self-taught.>\n\nIgnoring Stryki's disdain, Roche forced herself to look around with the woman.\n\n<What do you see?>\n\nShe described the congregation of shielded bumps that she guessed was the meeting. A steep ridge surrounded the gathering\u2014the shield, she presumed, that kept outside observers at bay.\n\n<Can you see yourself?>\n\n<Not clearly, no.>\n\n<And me? Can you see me?>\n\n<The same. You're indistinct, kind of fuzzy around the edges.>\n\n<That's because our minds are linked,> explained the reave. <Were we to disconnect and you to look at the view on your own, or with the help of another mind, I would spring back into focus.>\n\n<Can you look at yourself?>\n\n<Of course,> Stryki said. <All reaves are taught to do this before they learn to examine another's mind. But we do not see ourselves as you would see us.> The woman paused. <Your mind has an unusual flavor, Roche. One I have not tasted before.>\n\nRoche recalled what Maii had said about the _irikeii_ calling her an \"enigma.\" The possibility that it might be Maii's block confusing the issue didn't occur to her until the reave went on: <I can sense the Abomination's hand in your mind. You are aware of this?>\n\n<Yes.> Wary that the reave might penetrate the block, she automatically withdrew.\n\n<I would not violate your privacy,> the reave assured her. <I am not here to probe you, and I would not do so without your permission. That would not be ethical.>\n\nAlthough leery of the woman's intent\u2014and tired of her incessant insinuations regarding Maii\u2014Roche did allow her back into her mind. She had little choice but to trust the reave if she was to get anywhere.\n\n<Okay,> said Roche. The woman's gaze began to wander, through a short-lived gap in the barrier around the meeting and out into the _Phlegethon._\n\n<Do you know where we are?>\n\n<No.> Within seconds, Roche had become completely lost. <Do you?>\n\n<More or less. Fixing spatial coordinates to n-space equivalents is one of the hardest things a reave must learn to do,> the woman explained. <It is only complicated when the real-space environment is unfamiliar or inconstant. Fortunately, I am both trained and familiar with this environment.>\n\n<So pinpointing the clones shouldn't be too difficult, then?>\n\n<Theoretically.>\n\nRoche allowed Stryki to whisk her through the concatenated minds of the crew of the _Phlegethon_ for a minute or two before asking: <Have you studied clone warriors before?>\n\nThe reave hesitated a second. <No, but I do have a great deal of experience with Exotic and near-Transcendent Humanity.>\n\nRoche would have been interested to hear more about the latter. She was, however, forced to concentrate on the task before her. The sea of thoughts was rushing by much faster than it had with Maii. It was all she could do to keep up.\n\nThen she saw it: a dip representing a clone warrior.\n\n<Stop! Go back!>\n\nThe reave retraced the way they had come, more slowly. Now Roche could tell that the concentration of people was less dense than it had been before.\n\n<Where are we?> she asked.\n\n<Find the one you are looking for, and I will tell you.> Roche swept invisibly among the crew of the _Phlegethon_ until the dip reappeared.\n\n<There. No, there! Can you see it?>\n\n<I see nothing out of the ordinary.>\n\n<Look at that group of five minds,> Roche said. <The bottom right-hand individual\u2014that's the one!>\n\n<Are you sure?>\n\n<I'm positive,> said Roche.\n\nThe reave was silent for a moment. <I can't read the person you have identified. That in itself is not unusual, but it does lend some credence to your story. Other reaves have reported difficulty reading the enemy's minds. They seem to be totally impenetrable, if they so desire. They may, at times, allow a reave to access sensory data, but little else....> She paused again. <The people speaking to the person you have indicated know her as Advocate Janil Coriett. They are in compartment 43 on Deck 25B of the minaret.>\n\nThe view of n-space disappeared and Roche found herself back in the meeting, facing the hooded reave.\n\n\"Are you okay?\" asked Haid, his hand resting on her armored shoulder.\n\nShe blinked and looked around, dazed. \"Fine, I think.\" She turned to Murnane. \"We found one.\"\n\n\"So I am hearing.\" The councilor listened to the reave's mental voice a moment longer.\n\n\"We need to handle this very delicately,\" said Nemeth, leaning forward.\n\n\"Obviously,\" said Murnane. \"What do we know about this person?\"\n\n\"Advocate Coriett has been with us for six months,\" said the woman who had spoken before. Her eyes were out of focus; she was obviously studying information through her implants. \"She came from Ceyle's Hub and, as well as taking a position in Environment Control, helped us negotiate a settlement with the H'si F'ta.\"\n\n\"The Hub was destroyed not long before she arrived here,\" said Nemeth. \"The H'si F'ta didn't last much longer before they were decimated by a rival Caste. The settlement encouraged them to drop their defenses.\" He shrugged. \"At the time, we thought it was just a bad call.\"\n\n\"That makes her a prime candidate, then,\" said Roche. She faced Murnane. \"Do you believe me now?\"\n\nMurnane turned from Roche to the others. \"Until we have definitive proof\u2014\" he began.\n\n\"What sort of proof do you need?\" said Roche sharply. \"Her body on a slab?\"\n\nMurnane ignored her.\n\n\"We'll _have_ to move in,\" said Nemeth.\n\n\"No,\" said Murnane firmly. \"That would only alert the others, and that could be disastrous. If one can bring down a civilization, imagine what _four_ could do to this ship!\"\n\n\"Then perhaps we should we tell the Heresiarch,\" Roche said.\n\n\"I disagree,\" said Nemeth. \"The fewer who know about this the better\u2014at least until we've dealt with the situation.\"\n\n\"And how do you propose we do that?\" asked Roche.\n\n\"Well, first, you're going to have to find the others,\" said Murnane. \"We can keep track of Coriett. Once we have all five, _then_ we can act.\"\n\n\"And I repeat, _how_ do you propose to do this?\"\n\nMurnane glanced away for a second, then returned. This time, the fear in his eyes was obvious. Roche didn't need to be a reave to know what he was thinking. It was all very well to contemplate a nebulous, almost unreal enemy, but to actually come directly in contact with them was another matter altogether....\n\n\"We'll deal with that when the time comes,\" he said. But the uncertainty in his tone did not inspire confidence.\n\n* * *\n\nIt wasn't difficult to find the other four, but it was time-consuming. By the time Roche had helped the reave pinpoint the location of each of the five clone warriors on the _Phlegethon,_ she felt weary right down to her bones. Her brain ached in ways she had never imagined before. When she was released from n-space for the final time, she sagged back into her chair with a groan, and only Haid's hands under her armpits stopped her from slipping to the floor.\n\nWith a whine of servo-assisted joints, he helped her upright.\n\n\"Morgan, this is crazy.\"\n\n\"No, it's done now.\" She turned to Nemeth for confirmation.\n\nHe nodded. \"We have the identity of the fifth. He's a fusion technician, of all things.\"\n\n\"In a prime location to sabotage the power core,\" said Murnane. \"Roche, if you're right about these people, you have undoubtedly saved the lives of everyone on this ship.\"\n\n\"They're not safe yet,\" said Roche.\n\n\"Quite,\" said Nemeth. \"I suggest we move on all five simultaneously\u2014send multiple containment teams to pin them down; then, if they resist\u2014or if there's even the slightest chance they'll get away\u2014neutralize them permanently.\"\n\nMurnane nodded slowly. \"That at least gives them a chance to prove their innocence,\" he said. \"A blood test would be enough to reveal the truth.\"\n\n\"They've obviously managed to avoid blood tests before now,\" objected Nemeth.\n\n\"And if they do come quietly and are guilty?\" Roche asked. \"What then?\"\n\nNemeth glanced at Murnane. \"Execution.\"\n\n\"No,\" said the older man. \"We're not barbarians.\"\n\n\"They would kill _us_ out of hand!\"\n\n\"But we are not like them, Nemeth,\" said Murnane severely.\n\n\"And I have no desire to become like them, either. There _must_ be some way to subdue them.\"\n\nRoche described the crystalline cocoon Linegar Rufo had used to neutralize Adoni Cane. \"He was in a coma,\" she said. \"There was no way he could escape.\"\n\n\"We could easily implement something like that,\" said Murnane thoughtfully. \"Later, higher authorities could decide what to do with them.\"\n\nNemeth still didn't look happy, but dropped the argument. \"What happens after this?\" he said. \"Can we replicate these results? Can we use this process to keep the _Phlegethon_ clean?\"\n\nHe was addressing the white-robed reave, whose posture stiffened as she replied.\n\n<I do not understand how Morgan Roche is able to identify these individuals\u2014assuming they are what she says they are. To me, they stand out from a crowd no more than anyone else. In the same way that mandatory blood tests have not isolated them, detailed scans of the crew might not turn up more of those we seek. We need more people with Roche's ability.>\n\n\"And to find them,\" interrupted Nemeth, \"we need to know _how_ she is doing it in the first place. We're back where we started.\"\n\n\"Not quite,\" said Murnane. \"We have a test. Bring in those five, or attempt to. Once we have ascertained whether or not we're on the right track, _then_ we can work out what to do next.\" He turned to the other members at the meeting. \"I want those containment teams in place in half an hour. Prepare for any contingency, no matter how unlikely. Advise the Heresiarch by epsense that an exercise is about to take place, but that he is under no circumstances to alert the crew. Everything proceeds as usual until the operation commences. No one leaves here until it is concluded. Is that understood?\"\n\nThere was a chorus of assent. Nemeth offered his last of all, still clearly displeased at the way Murnane had taken over his operation.\n\n\"Can I talk to my crew?\" Roche asked.\n\n\"A brief message relayed via epsense only,\" Murnane conceded. \"When everything is concluded here, you may converse freely.\"\n\nRoche and the reave put together a short message for Maii explaining that they would continue to be incommunicado for an hour or so. After that, they would know if it was safe to bring Maii aboard the _Phlegethon,_ as Vri insisted they do. Should they receive any unusual communications from the _Phlegethon_ at all, Kajic was to move to a safe distance and wait for news.\n\nThe reave sent the message and, a moment later, confirmed that it had been received.\n\nAs they waited, the mood around the conference table became increasingly restless. How arrangements were being made, Roche couldn't tell exactly, but she noted the far-off expressions of those using implants. She worried that word might somehow get to the clone warriors, warning them of what was to happen, but as she could see no way to organize things without taking that risk, she said nothing.\n\nHaid tapped her on the shoulder and indicated for her to close her faceplate. Roche nodded, and instructed her suit to seal. The people around the table looked up as her visor and Haid's hissed shut, but when she and Haid made no other move they returned to their work.\n\n\"What do you think?\" asked Haid over the private link between their suits. \"Is this actually going to work?\"\n\n\"It had better,\" she replied. \"Because if it doesn't, we're really in trouble. If we can't fight them individually, even when we know who they are, then there's no point even trying. We either give up, or we advocate extreme solutions like completely destroying habitats and stations that we know have been infiltrated. And even then, we could never be certain that we've wiped out the last of them. There could always be one capsule left in deep space, or one survivor hiding out in the Far Reaches. And where there's one...\"\n\nHaid grunted. \"No one's mentioned the alternative to killing them.\"\n\n\"Which is?\"\n\n\"Conversion.\"\n\n\"Like Cane?\"\n\n\"Maybe.\"\n\nShe frowned. \"It would be hard to trust them after what they've done\u2014or what their kind has done.\"\n\n\"I know. But not impossible, surely?\"\n\nShe shrugged the suit's heavy shoulders, thinking of Cane and her own uncertainty. \"Only time will tell, I guess.\"\n\n\"Speaking of trust,\" Haid went on: \"Why's Murnane being so chummy all of a sudden? You were definitely persona non grata last time you two met.\"\n\n\"I don't know. Maybe something's happened to change his mind. I'm sure he doesn't trust me too far. At the most, I'd say he's decided I'm useful.\"\n\nRoche examined the councilor while she talked. His face seemed more deeply lined than ever, and the faint wisp of hair at the back of his skull had faded almost to invisibility. He looked older, more tired, and less inclined than ever to tolerate fools.\n\n\"How desperate do you think he is?\" Haid asked.\n\n_Very,_ she thought, but said nothing. After a moment, she unsealed her suit and leaned back into her chair with her eyes shut, thinking.\n\nThey waited another twenty-five minutes. Then, finally, everything was ready. Security in the conference room was eased slightly to allow data to flow in and out as the five containment teams moved into position. The teams, consisting of fifteen security officers each, had strict instructions to use lethal force at the slightest sign of resistance. Protecting the lives of bystanders was considered a lesser priority than ensuring the death or capture of the targets.\n\nRoche watched via her implants as the containment teams closed around the areas where the clone warriors were situated. It was a complicated display, showing all five teams simultaneously. There was no way she could follow all of them at once, so she focused on the team advancing on Coriett.\n\nThe woman was in one of the uppermost levels of the giant ship, sitting in a room with four other people. To all appearances she seemed an ordinary Pristine dressed in a plain shipsuit, discussing the day's activities with colleagues. They chatted amiably, laughing now and then, and sipping occasionally from their mugs. It all seemed so innocuous to Roche, which paradoxically lent the scene a sinister air\u2014because she _knew_ what this woman was truly capable of.\n\n\"Pull out two of those people,\" she heard Murnane say.\n\n\"Why not pull them all out?\" said Nemeth. \"It would make it easier to deal with her.\"\n\n\"Because we risk arousing her suspicions,\" said Murnane.\n\nThrough her implants as well as the ship's internal intercom system, Roche heard the names of two of the people being summoned. They exited the room a minute or so later with smiles and polite bows of the head, leaving just Coriett and the two others behind.\n\nA joke was made by one of the remaining colleagues; Coriett smiled politely and sipped from her cup. As she did so, she glanced up at the room's monitor.\n\nRoche went cold: for the second it lasted it seemed as though the clone warrior was looking directly at her.\n\n\"She suspects something,\" she said. \"I'm sure of it.\"\n\n\"What?\" It was Nemeth. \"That's impossible...\"\n\n\"Move your team in now,\" insisted Roche.\n\nThe security guards entered the room just as Coriett made to stand. All three guards took position around the woman, leveling their weapons carefully at her. The two other people present in the room leapt up from their seats in alarm, spilling their drinks.\n\n\"Advocate Janil Coriett,\" said one of the guards loudly, firmly, \"by order of the Interim Emergency Pristine Council, I am placing you under arrest. Please put your hands together behind your head and step away from the desk.\"\n\n\"What is the meaning of this?\" said one of the other women angrily. \"This is an outrage!\"\n\nInstead of answering, one of the guards indicated that she should step away from Coriett, out of the line of fire. With a weapon leveled directly at her, the woman's indignant protest became alarm.\n\nCoriett, meanwhile, had responded to the orders, but not before coolly appraising the three guards.\n\n\"Don't I at least deserve some sort of explanation?\" she asked.\n\nThe guard didn't respond, and her two companions were ushered from the room by one of the other guards. When the room was cleared, another guard produced a set of handcuffs and approached the clone warrior.\n\n\"Surely I have rights?\" she said, holding her hands to be cuffed. But it wasn't to the guards she was speaking. She was looking again at the monitor. Even from this safe distance, Roche couldn't help but feel unnerved by the woman's unflinching confidence.\n\nAt the same moment, the sound of secondhand gunfire came from inside Haid's helmet. Clearly, one of the other missions wasn't going as well as this one. Roche was about to switch viewpoints when a subtle change in Coriett's expression caught her eye. It was as though she too had somehow heard the gunfire, and knew what it meant: that one of her siblings was in danger\u2014and that this was therefore much more than an administrative error or a mere suspicion she could talk her way out of.\n\nCoriett pulled back from the cuffs and elbowed the guard in the face. As he went down, she grabbed him by one arm and swung him in front of her. The two other guards in the room contemplated shooting through him to get to her, but in the second they hesitated, the warrior had found the stock of the rifle and brought the weapon up firing.\n\nThe rifle was set on rapid repeat; all she had to do was swing the barrel to cut down the guards and the two other people in the room. One of them was half out the door when the shots took him in the back, throwing him forward into the hall with enough force to disrupt the formation of guards waiting there. Coriett followed a second later, capitalizing on the surprise. Roche watched with mounting alarm as the guards recoiled in confusion, only a few of them managing to get off even a single shot before Coriett targeted them herself.\n\nHer shots never missed.\n\nBut killing the guards wasn't her main priority. Escape was more important, and there would always be more guards if she stayed in one spot too long. Roche could only watch anxiously as the warrior paused to evaluate her position: she was in the middle of a long corridor with an open elevator at one end and a junction at the other. If she could reach the elevator, she could go anywhere on the ship. Security would be hard pressed to catch her. And once she had slipped through the net, she would have the entire ship to hide in.\n\nRoche wanted to cry a warning to the remaining guards as Coriett strafed them with one wild wave of her gun, then sprinted for the elevator. Behind her, amid the tangle of wounded and dead, only one guard had the forethought to guess what she was doing. He pushed a limp body aside, raised his rifle, and fired.\n\nAt first, Roche thought the woman wasn't going to stop. Round after round struck her in the back, propelling her onward through a mist of blood. She was still running even when she hit the back wall of the elevator cab, her speed unchecked. With a sound like the crack of bone, she rebounded and fell to the ground.\n\nAs she fell, her hand struck the access panel on the inside of the cab, and the doors slid shut.\n\nRoche quickly changed her view to the inside of the cab. Surely there was no way Coriett could have survived so many shots at such close range. Hitting the switch _must_ have been an accident.\n\nThe cab slid silently away from the carnage, taking the immobile and bloody body of Coriett with it. The gun was on the floor where she had dropped it, a crimson puddle quickly pooling around the barrel.\n\nA map in the side of the channel tracked the cab as it traveled out toward the hull of the ship and down to the docks. Another team was being sent to meet the cab at its destination.\n\nRoche watched. Coriett didn't move.\n\nSatisfied that the clone warrior wasn't going anywhere, Roche skipped to the other channels. Only one was still active. Another clone warrior\u2014a male with close-cropped red hair, the fusion technician\u2014had managed to get his hands on a weapon and taken a hostage to use as a shield. His containment team had hesitated long enough to allow the clone warrior to regain the initiative. He shot five, and the hostage, while they were making up their minds, then slipped away before they could regroup.\n\nThe point of view of the channel followed him easily, however\u2014jumping crazily from camera to camera as he ran headlong through what looked like a cargo section of the ship. Guards converged on the area from all directions; blast doors slammed shut in his path. But he seemed to know what he was doing. The area was riddled with access ways and maintenance shafts. Where he couldn't run, he crawled; where he couldn't climb, he jumped. Roche didn't know the big ship well enough to guess where he was headed, but she didn't doubt he had _somewhere_ in mind.\n\nIn the end, though, his luck ran out, and a stuck hatch forced him into another team's path. When he realized his mistake, he tried to double back on his pursuers and take them by surprise. They realized just in time, and the concentrated fire from three security guards finally brought him down.\n\nRoche took a deep breath. That made four confirmed kills. As Coriett's cab came to a halt, she felt sure that it would soon be five.\n\nThe team were ready for anything as the doors slid open. From the viewpoint of one of the guards, Roche watched as they inched forward, weapons at the ready, until they were within meters of the woman.\n\nShe didn't move.\n\nOne guard reached gingerly forward to slide the still-smoking gun out of arm's reach.\n\nStill she didn't move.\n\nMore confidently, another approached to test for vital signs while the others kept their weapons trained on her. If she was conscious, or even alive at all, she gave no indication as his fingers sought for a pulse in her throat.\n\nThe diagnosis wasn't one Roche expected to hear.\n\n\"She's alive,\" the guard called.\n\nThe containment team had begun to relax in the face of her lack of response. Now they tightened formation again and began to inch away nervously.\n\n\"What do we do with her?\" said one of the team anxiously.\n\n\"Restrain her,\" came the response. \"Bring her in for examination and interrogation if you can. But shoot her if she so much as moves. Whatever you do, _don't_ let her get away from you. If she\u2014\"\n\nA siren began wailing through the ship. Distracted from the view through her implants, Roche looked up. Murnane's eyes were flickering rapidly, intent on an internal feed.\n\n\"What's happening?\" she asked.\n\n\"Overheated life-support module,\" said Nemeth.\n\nThe siren grew louder and more strident.\n\n\"Is that all?\" she said.\n\nHe stared at her. \"You still haven't grasped how big this ship is, have you? Each of those modules is bigger than a small moon, and there are _five_ of them. If one blows, it could start a chain reaction through the ducts that'll tear the whole place apart.\"\n\n\"What's causing it?\"\n\n\"The Heresiarch is trying to find out. Something is interfering with the module's normal operation. A virus of some sort, perhaps, triggered from the outside.\"\n\n\"Sabotage?\"\n\n\"Could be. We don't know yet. But it's going to blow in thirty seconds if we can't get control of it, so hope for both of us someone works out quickly what the hell is going on.\"\n\nRoche's attention was drawn back to the channel in which Coriett was being dragged out of the cab. The woman was limp, unprotesting, to all appearances completely unconscious. And as she watched, a sudden realization brought with it a sense of terrible panic.\n\n\"Tell your team to shoot her!\"\n\nNemeth's eyebrows shot up. _\"What?\"_\n\n\"It's _her_ ,\" Roche found herself shouting. __ \" _She's_ doing it.\"\n\n\"Don't be ridiculous,\" said Nemeth, but the uncertainty was evident in his tone.\n\n\"Didn't someone say she worked for Environment Control? She would have been in a perfect position to set something like this up. If she has implants, and if she's faking unconsciousness...\"\n\nRoche didn't need to go any further. The alarm blossoming in Nemeth's eyes matched her own.\n\nHe turned away to rattle off orders to the containment team and to the councilors around him. The siren wailed on as the guards readying Coriett for mechanical restraint looked up at their superior and listened to the new orders coming through their armor's communication links. They let go of her and backed away.\n\nBut even as they did this, Roche felt it was too late.\n\nThe warrior's eyes snapped open, and in an instant she had rolled toward the nearest guard and grabbed him by the legs, blood squirting from her injured back in a high-pressure jet. Confused, surprised, frightened, the guard didn't have time to react, and fell heavily to the floor. His head hit the ground with a sickening thud.\n\nCoriett seized the fallen guard's rifle and raised herself up on one knee, leveling it effortlessly at the others. But three members of the containment team had managed to raise their own weapons first and had already targeted the injured woman.\n\nA volley of shots flung the clone warrior back into the cab, and they kept firing until her body stopped moving altogether...\n\nNemeth acknowledged the woman's death only in passing. He didn't relax until word came that the interference with the life-support module had ceased. Its operations were being normalized, and the threat to ship integrity would soon pass. He sagged visibly as the siren decreased in volume and then fell silent.\n\n\"That's it,\" he said. \"I think we can call this operation a success.\"\n\n\"Bring her body,\" said Murnane. Outwardly he seemed unaffected by the events. \"I want _all_ of the bodies in for postmortem examination. Then we'll know for certain.\"\n\n\"We already know,\" said Nemeth.\n\n\"We can take nothing for granted,\" said Murnane evenly.\n\n\"How could you even doubt it? Look at them! They ran and they fought\u2014they fought even when there was no chance they could win! If they weren't the enemy, then what were they?\"\n\nRoche looked, and knew what he was feeling. Five of the clone warriors lay dead on the _Phlegethon_ 's decks\u2014killed by the Pristine Humans they impersonated. _That_ was progress.\n\n\"Now we're getting somewhere!\" Nemeth gloated.\n\nBut as the surviving security guards picked themselves up and saw to their injured colleagues, and as the casualty reports came in listing thirty dead guards and fifteen dead civilians, Murnane's face grew grim.\n\n\"At what cost?\" he asked, perhaps of himself.\n\nRoche looked down at her trembling hands aid wondered the same thing.\n****\n\n**14**\n\n****\n\n****\n\nSHCV Phlegethon\n\n955.2.14\n\n0560\n\nWithin the hour, chaos had erupted all around the system. Border skirmishes broke out and became firelights; grudges became battles; enemies forgot diplomacy, along with the greater good, and exchanged open, sometimes devastating attacks. Within three hours, virtually every Caste present in Sol System was engaged in some form of dispute.\n\n\"I don't understand.\" Nemeth watched the screens with a growing confusion. Perhaps, Roche thought, he was watching his newfound position of dominance in the council dissolving before his eyes. She hoped he had more Humanity than that; she herself saw nothing but lives wasted, nothing but more death and destruction\u2014with the potential for it to become worse than anything she had witnessed in Palasian System. Worse, even, than anything she had ever imagined.\n\n\"I don't _understand_!\" Nemeth said again, hitting a console with the palm of one hand. Roche could appreciate his frustration, but his anger was serving no purpose. A feeling obviously shared with the Heresiarch, who turned to Nemeth and said: \"If _you_ don't understand, then who does?\" The question was clearly rhetorical, for the Heresiarch didn't wait to hear what Nemeth would say next. Instead he returned to the business of running the ship, turning his attention to the influx of data coming at him from all the monitors about the room.\n\nHe was standing in the center of the small room adjoining the fane to which Roche and Haid had been moved following the **** Ulterior's apparent success with the five clone warriors. The room was fifteen meters across, at most, with glowing blue walls that looked as though they had been fashioned from crystal. Set into the walls and floor, and even the ceiling, were consoles and stations for dozens of crew members. The air was full of whispered instructions, burbling data, and an impression that everything was running perfectly to order. Busy yet not chaotic, the adytum hummed to its own driving rhythm.\n\nIn this space, the Heresiarch did the real work involved with the running of the _Phlegethon,_ rather than the ceremonial. Roche knew that being permitted here, among the highest officers and critical decision-makers, watching the information pouring in live from tens of thousands of ftl drones, was something of a privilege. Also a high honor, if Nemeth's expression upon arriving was anything to go by.\n\n\"We killed those five easily enough,\" persisted Nemeth, turning now to Murnane with his concerns. \"They had no chance to warn the others. All this...\" He gestured at the mess on the screens. \"It _has_ to be a coincidence.\" He searched the room now, looking for support but finding none. \"It's always been at flashpoint,\" he insisted. \"The whole system was unstable from the moment we arrived\u2014and it's been getting worse every day! There have been skirmishes, conflicts, even small wars, before. This is just more of the same. Only worse.\"\n\n\"Much worse,\" the Heresiarch said dryly, leaning against the steel rail surrounding his station. He seemed to be completely hairless; his eyes were a deep brown, like his skin.\n\n\"It's a chain reaction, that's what it is.\" Nemeth began pacing and gesticulating agitatedly. \"Civilization A attacks civilization B, who calls in C as an ally. That would be fine, except D has been waiting for the chance to move on C and ropes in E and F to stack the odds. G is caught in the crossfire, and H and I come to its rescue. And so on. Perfectly sane and comprehensible.\" His words trailed off as he stopped and looked up at the screens. \"What we did has nothing to do with this,\" he finished more calmly. \"It _can't_ have.\"\n\n\"I fear it did,\" said Murnane.\n\nThe elder councilor didn't waste energy posturing as he talked. He simply stood, composed, on the other side of the Heresiarch. \"The timing is too precise. Every new conflict was initiated within moments of the deaths of those five. Medical tests confirm that they were the enemy, so their exposure and attempted capture _have_ to be connected. The others, the rest of them\"\u2014his eyes flickered for a second to the screens\u2014\"the ones that are still alive are fighting now because they know that we have learned how to find them. They feel vulnerable\u2014perhaps even afraid. We were able to kill those five because we managed to take them by surprise; the others are not going to allow the same thing to happen again.\"\n\n\"But _how_ did they know?\" Nemeth's frustration was palpable.\n\nMurnane shrugged. \"I'm not sure,\" he said. \"But the information _must_ have been transmitted by either an epsense or hyperspace signal. There's no other way it could've spread so quickly.\"\n\nRoche remembered the black speck at the heart of Cane and Jelena Heidik's minds, as viewed by the _irikeii,_ and the look on Janil Coriett's face as firing broke out on the far side of the _Phlegethon._\n\n\"It was epsense,\" she said.\n\nNemeth rounded on her. \"How can you know that? None of our reaves have ever detected anything.\"\n\n\"Neither have we,\" she said. \"But look at the recordings: Coriett suspected she'd been discovered, but she didn't know for sure until someone opened fire on one of the others. She wasn't anywhere near any sort of hyperspace communicator; there's no way we know of to hide one inside a Human body\u2014and we would've found it if one had been there. So it must have been epsense.\"\n\n\"That's not proof,\" Nemeth said.\n\n\"It's all we have to go on,\" said Murnane. \"We need to recall our field agents and warn anyone who might not have realized what's going on. Without extensive ftl communications or epsense on _our_ side, word might take time to spread that the disturbances aren't local. We have to save as many people as we can, starting with our own.\"\n\n\"And then what?\" Nemeth asked.\n\n\"Then we wait and see what happens.\" The elderly councilor suddenly looked very tired. \"This might blow over; it might just be a warning. We might receive some sort of communication, at last. Who knows? If they are that concerned that we have the ability to find them now, we might even be able to negotiate a settlement.\"\n\n\"What _have_ we got to negotiate with?\" The short bleat of a laugh from Nemeth was cynical and derisive. \"If we don't find someone else who can do this, it'll take us months to sweep the entire system. Maybe forever, if Roche is killed!\"\n\n\"The enemy doesn't _know_ that!\"\n\n\"How long do you think it'll take them to figure it out?\" said Nemeth. \"They're not stupid, you know.\"\n\n\"Exactly\u2014they're not. They know they can't afford to take too many chances while they're so outnumbered.\"\n\n\"But what if they can?\" continued Nemeth. \"What if they _don't_ negotiate?\"\n\nMurnane shrugged. \"Then we save what we can,\" he said. \"That's all we can do.\"\n\n\"There is another alternative.\" A new voice entered the discussion, this one hauntingly familiar to Roche.\n\nMurnane looked up. \"Yes, Trezise?\"\n\nThe senior aide to Auberon Chase stepped into the center of the adytum. Roche hadn't noticed him there before, but recognized his narrow, almost equine features immediately.\n\n\"Roche claims to have among her crew one of the enemy.\"\n\n\"Yes; the council decided Adoni Cane was a fake,\" Murnane said.\n\n\"What if he isn't? What if he _is_ as tame as Roche suggests? Surely his opinion would be worth seeking at this juncture.\"\n\n\"Bring another one of the enemy onto the ship?\" objected Nemeth. \"We've just killed a lot of people clearing out the first lot! If he is a clone warrior, it would be insane to allow him on board.\"\n\n\"What other means do we have to decipher the enemy's intentions?\" Trezise opened his hands in supplication. \"If an epsense link _does_ exist, he might be able to tap into it.\"\n\n\" _If_ he is genuine,\" Murnane said.\n\nTrezise nodded, agreeing calmly. \"If he is genuine, yes.\"\n\nMurnane turned to face Roche. \"What do _you_ think?\"\n\nShe wanted to say that this was what she'd wanted to do two and a half weeks earlier, that if they'd listened to her the first time, then maybe everything that had happened since then might have been avoided.\n\nBut she didn't. She was too conscious of the fact that this could be her last opportunity to speak to the council. She couldn't afford to miss that chance.\n\n\"On two conditions,\" she said instead. \"One: you will grant any member of my crew asylum on this ship should they seek it. And two: you will make me a participating member of the council, effective immediately.\"\n\nMurnane chuckled softly. \" _I_ have no objection to either condition,\" he said. \"Obviously, though, the granting of asylum would need to be ratified by the Heresiarch.\" He nodded to where the Heresiarch stood at his station, preoccupied with the running of the ship. \"I'm not sure he would extend the honor to Adoni Cane, enemy or not. And your membership in the council would have to be on a _pro tem_ basis, to be ratified by a formal sitting\u2014\"\n\nShe raised a hand to silence him. \"Okay, okay, you've made your point,\" she said, and sighed. \"Just give me your assurance that you'll _listen_ to me, at least.\"\n\nHe nodded once. \"Make arrangements to transfer Adoni Cane across and we'll prepare for the interview.\" He swept his gaze across the room. Nemeth looked relieved, but said nothing. \"If there are no further urgent issues to be raised, we will adjourn to await further developments. The Heresiarch has work to do, and our being here can only be a distraction.\"\n\nEven as he spoke, a bell chimed loudly, sending the assembled officers to their stations. A raiding party from a small but aggressive government had stumbled into the _Phlegethon'_ s camouflage field and opened fire on the middle decks. Roche turned to the monitors, watching as hundreds of pod-shaped fighters swarmed out of the giant ship's many docks to repel the intruder.\n\nThe council moved out to the fane. Hue Vischilglin took Roche's arm, nodding recognition but saying nothing. As she and Haid were led away, Roche caught a glimpse of Salton Trezise, who could barely contain his look of triumph.\n\n* * *\n\nFrom the safety of another conference room, Roche made arrangements for Cane and Maii to cross by scutter to the _Phlegethon._ She also instructed Kajic to assume a close parking position under the big ship's shadow. While the raid had in no way threatened the _Ana Vereine,_ it did highlight the potential for conflict nearby. If things did get too hot, Kajic had permission to seek protection in the _Phlegethon_ 's larger docks.\n\nOnce that was organized, Roche had nothing else to do but observe. Even as only a _pro tem_ member of the council, the information she now had access to was overwhelming. What had once been chaos had now become a slaughter. Traditional alliances dissolved; defensive agreements were torn apart; like fought like as intra-Caste conflicts expanded to consume entire fleets. Conservative estimates put the number of ships and stations lost in the first three hours at twenty thousand. That only amounted to barely two percent of the million or so ships known to have entered the system, but the sheer loss of life could not be ignored.\n\n<I hope you're watching this, Box.>\n\n<Indeed I am, Morgan. This is an unexpected development. The ability of the enemy to mobilize such a response so quickly is unprecedented. Few could have anticipated this sort of capability\u2014or this sort of generalized conflict so early in the endgame.>\n\nIts choice of words made her frown. <I hope this is more than just a game to you, Box.>\n\n<Of course it is, Morgan. Of course.>\n\nShe couldn't decide whether the AI was being insincere or patronizing. Perhaps it was both. <Do you see any biases toward any particular Castes?>\n\n<None at this stage. Taking regional peculiarities into account, the targets are randomly distributed. There doesn't even seem to be any broad trend, either\u2014such as Pristine versus Exotic, for example.>\n\n<Does that blow your theory?>\n\n<No. I may need to see more data before a trend emerges.>\n\n< _If_ a trend emerges,> she said, thinking of the twenty thousand ships destroyed already and wondering how many more it would take to give the Box enough data.\n\n<Naturally, Morgan.>\n\nShe went back to watching the screens. Sitting beside her, heavy and brooding in his black combat suit, Haid was as silent as she was. He kept his emotions carefully hidden, only a slight tightness to his jaw revealing anything of the tension he must have been feeling.\n\nAfter a few moments, she addressed the Box again. <Box, someone mentioned being able to identify the enemy by blood test. Do you know how they do this?>\n\n<It's a genetic analysis,> the AI replied. <As Rufo and I previously noted, the enemy has an unusual genetic structure. Where normal Humans possess long stretches of 'junk' genes called introns, Cane and the others have sequences that _appear_ to serve a purpose, yet result in no known proteins. Whatever else they might do remains a mystery, but these stretches can be used to distinguish a clone warrior from any other ordinary Human.>\n\n<That's why Rufo wanted us to look at Cane's introns more closely?>\n\n<Yes. He suspected that an important clue to the origins and\/or purpose of the enemy would lie within.>\n\n<Do you agree?>\n\n<I agree that it's a possibility.>\n\n<What about the council? They obviously knew about the introns when I mentioned them at the first meeting. Have they learned anything new in the time they've had?>\n\n<Well, if they have, it hasn't explained anything about the enemy we didn't already know.>\n\nRoche absorbed this. <What about the Crescend, then? What does _he_ know?>\n\n<If he knows anything conclusive about this, he hasn't told me.>\n\nTired of repeatedly butting the same brick wall, Roche fell silent. A few minutes later, Kajic reported that Maii and Cane were ready to leave. Roche switched her implants to a view of the scutter's cramped interior. Cane sat in the pilot's seat, his face expressionless. Maii sat beside him, still somewhat pale, but looking better than she had been earlier; the medicinal pack covering her wound was less bulky than it had been, indicating that its healing work was almost done. Behind her...\n\n<What's _he_ doing there, Uri?> she asked Kajic.\n\n<Defender-of-Harmony Vri insisted on coming,> he replied.\n\n<But we're getting Maii to safety,> she said. <That's what he wanted, wasn't it?>\n\n<He seems determined to keep an eye on her in person.> Kajic sounded almost amused. <It wasn't my place to stop him.>\n\n<No, of course not. Nor mine.> Roche found the Surin warrior's persistence admirable, if a little obsessive. <But are you going to be okay there by yourself?>\n\n<Everything will be fine, Morgan. Now that normal channels are open again, I'll be able to watch what you're doing. Besides,> he added, <I'm not completely alone. Alta is still here. And there's still plenty of work to do shipside. I think everything must be perfectly in order if we're to get out of this one.>\n\nRoche agreed. Part of her was still nervous about giving the ex-Dato captain the chance to escape with the ship, but the rational side of her knew that this was simply unjustifiable paranoia. Had he wanted to, he could have killed or lost them dozens of times already.\n\n<Vri will attempt to make contact with the Surin delegates in the system,> Kajic went on. <That is one thing you haven't been able to do as yet.>\n\n<It's not as if I've had the time!>\n\n<He realizes that. He just wants to resolve this issue once and for all.>\n\nRoche nodded to herself. She couldn't do everything\u2014especially when this particular job held little appeal. <How long until they arrive?>\n\n<They should be there in about half an hour.>\n\n<Okay. Thanks, Uri.>\n\nThe scutter disengaged from the _Ana Vereine_ and arced smoothly toward the larger ship. Cane flew the small craft with competence and ease. He was a natural at everything he turned his hand to, even a complicated task such as flying a space vessel. Somewhere in his lost memory, Roche supposed, was the knowledge he needed, accessible at will. How it had got there in the first place, though, was the question\u2014one question among many. She could only hope that some of them would be answered when he came under the council's spotlight.\n\nThere had been no mention of the Ulterior outside the sealed conference room where Roche had revealed her knowledge concerning the five clone warriors. She assumed that it was still considered at best to be an informal group by most of its members\u2014although Murnane's presence at that meeting was a strong indication that its activities were partially sanctioned by its parent, or would be gratefully absorbed into the greater body of work if things went well.\n\nHow long her partial acceptance by the council would last she didn't know, but while she _was_ a member, she resolved to take full advantage of it. She couldn't just sit by and watch while everything was potentially falling apart around her. Even if the council ended up dismissing her again, then at least she could say that she'd tried.\n\n\"Ameidio?\"\n\nHaid turned to face her.\n\n\"If you were the enemy, and this was your doing\"\u2014she indicated the images of destruction displayed on the monitor which was built into one wall of the conference room\u2014\" _why_ would you be doing it?\"\n\nHe faced the monitor and contemplated the question for a few moments. \"To reduce the resources of the enemy,\" he said at last.\n\nShe shook her head. \"No,\" she said, \"They're far too outnumbered. Even a ninety-nine percent reduction in our capability would leave them way behind.\"\n\n\"To disorganize the enemy, then?\"\n\nShe considered this for a short while before offering another shake of the head. \"That's a hell of a lot of effort for so little gain.\"\n\n\"Depends on how you look at it.\"\n\n\"Not really,\" she said. \"Wouldn't any sensible campaign concentrate its energies here, on the _Phlegethon_? That's where the potential for organization exists. Even if it's the only surviving ship, it'd stand a chance of victory against a small enough enemy force.\"\n\nHe shrugged. \"This could just be a smoke screen, then, and they are already working on us. We just haven't realized it.\"\n\n\"We got rid of their agents; they don't have anyone else to work through.\"\n\n\"You heard what Murnane said: the council is recalling its field agents. How many of the enemy do you think will slip in with that lot?\"\n\nShe nodded. \"I've considered that,\" she said. \"And who's to say the enemy _has_ to be a clone warrior at all? There are bound to be collaborators we'll never detect, small-time operators who might slip past even high-grade reaves because they aren't aware that what they're doing is even wrong.\"\n\nHis dark eyes watched her closely. \"You could be right about the agents,\" he said. \"But there's something else on your mind, isn't there?\"\n\nShe half-smiled, then sobered. \"This mass killing,\" she said slowly. \"It's a message of some sort.\"\n\n\"A message?\" Haid frowned. \"Saying what?\"\n\n\"I'm not sure,\" she said. \"I'm not even sure for whom it was intended.\"\n\n\"It would have to be for the council, surely,\" said Haid. \"Who else could it be for?\"\n\nShe didn't answer that, because were she to voice her suspicions, she was sure that Haid would think her totally paranoid. Nevertheless, she couldn't shake the feeling that the message was aimed at _her._ It was as though they somehow knew that _she_ was responsible for having located the five clone warriors, and that now they were making her pay the price. If so, this was retaliation on the largest possible scale; they were warning her not to do it again....\n\nBut that all presupposed the existence of the epsense link Cane had already denied knowing anything about. She doubted the council could decide in a single sitting whether he was telling the truth or not, but she was looking forward to seeing them try.\n\nOn the screen, a habitat shaped like a spinning top broke up under centrifugal forces, spraying fragments into the small flotilla arrayed against it. Roche couldn't even begin to imagine the scale; the habitat could have been home for dozens or thousands of people, and the ships may have been fighters or cruisers. There was no way to tell the scale from the display on the screen alone.\n\nShe stood abruptly, turning from the destruction to face Haid. \"Come on,\" she said. \"I'm going to get Vischilglin to take us to the docks to meet Cane and Maii.\"\n\n\"You don't trust the council to do it for you?\" said Haid.\n\n\"That's not the problem,\" she said. \"The last time I sent those two somewhere on their own, I very nearly didn't get them back.\"\n\n\"What about Vri?\" Haid said, getting to his feet.\n\nShe had forgotten the Surin. Vri had been with them for twenty-four days, but had never integrated into the group. He kept apart, following his own agenda, only working with them when their goals meshed. The moment their goals came into conflict, she had no doubt whose orders he would follow. On the _Phlegethon_ , he would be close to getting what he and the Surin Agora wanted. She didn't entirely trust his ability to compromise if his superiors didn't follow suit.\n\n\"All the more reason to go down there.\" She turned to leave.\n\n\"Morgan?\" Haid said suddenly. She stopped and faced him again. \"Do you think you could handle Vri?\"\n\nThe question startled her.\n\n\"If you had to,\" he went on quickly. \"One on one.\"\n\n\"I've no idea,\" she said. \"Probably not. I've never even thought about it, to tell the truth.\" The soldier had performed very well in the Fathehi Consulate. Not as well as Cane, but better than anything Roche could ever hope to perform. \"Why do you ask?\"\n\nHe shrugged. \"Just curious.\"\n\n\"And what about you?\" she said.\n\n\"Me? I wouldn't stand a chance.\" His smile was disarming. \"But both of us at once...? Well, that would be a different story.\"\n\nShe smiled slightly and patted the ex-mercenary's shoulder. \"Let's just hope it doesn't come to that.\"\n\n* * *\n\nThe scutter was delayed slightly by an unscheduled course-change undertaken by the _Phlegethon_ in order to avoid a cloud of debris too extensive to tackle head on. Even with the enormous ship's shields, the chance of a large fragment slipping through was too great to risk. Changing the ship's orbit gave a certain tactical advantage too: camouflaged or not, the more it moved, the less chance someone hostile had of tracking it from its last known location.\n\nApart from a containment team already in position when they arrived, the civilian docks were virtually empty. Of all the airlocks Roche could see, the one through which Cane and Maii disembarked seemed to be the only one in use.\n\n\"You don't get many visitors here, I take it?\" she said to Vischilglin, who waited with them by the inner door.\n\nThe tall woman shook her head. \"Security is very tight at all times, and especially so now. I am told that docks like these are usually a hive of activity when consistory vessels return to their home system. With such a large crew, the reunions can go on for weeks. The Heterodoxies are renowned for their devotion to family as well as to faith.\"\n\nThe woman spoke quickly and with animation, but never meeting Roche's eyes. It seemed to Roche that she was nervous, anxious\u2014trying, perhaps, to suppress an uncomfortable thought.\n\n\"Something's bothering you?\" Roche asked.\n\nVischilglin kept her gaze on the dock's inner door. \"I've lost contact with my superiors back home,\" she said softly. \"Signals stopped arriving three days ago.\"\n\nRoche nodded slightly, but didn't know what to say. She was saved from having to by the inner door hissing open. The suits of the containment team whirred as they stood at the ready.\n\nCane stepped out first, followed closely by Maii. She wore a new hazard suit with additional armor provided by Vri that lent the normally gray exterior an air of gilt decoration. Cane wore nothing but a typical brown Dato shipsuit. There was a tension in his posture which only heightened when he saw the containment team.\n\nMaii looked tired; her lips were thin, her pale features drawn. She stepped over to Roche and lightly touched her arm.\n\n<I'm not going to seek asylum,> she said, her mental whisper directed at Roche alone. She sent a picture of herself standing on the top of thick battlements.\n\nThe image might have been meant to make the girl look strong. To Roche, it made her look very small and alone.\n\n<I'm glad to hear it, Maii,> Roche said, quashing her impression. <But I don't think Vri will be.>\n\nThe Surin girl touched her mind with a mental shrug.\n\nVri was the last to step from the airlock. His visor was in place and his eyes were hidden, but Roche could tell from the way the helmet moved that he had scanned the containment team, Vischilglin, Haid, and herself with one appraising glance. He knew what had happened on Galine Four and was obviously prepared for anything.\n\nHe stopped just behind Maii and waited silently.\n\n\"The council will convene in fifteen minutes,\" said Vischilglin, stepping forward. \"Transport has been arranged.\"\n\nAs she spoke, a large, flat vehicle slid quietly to a halt nearby. The containment team reorganized itself to create a clear space leading from the airlock to the transport.\n\n\"Do we all go together?\" Roche asked her.\n\n\"That would be simplest.\"\n\nRoche nodded, but instead of heading for the transport, she moved to face Cane.\n\n\"Do you know why I've brought you here?\" she asked.\n\n\"To testify before the council,\" he said.\n\n\"Are you ready for that?\"\n\nHe returned her stare evenly. \"Are you?\"\n\nThe question was a challenge, although she didn't know why it should be. \"I have nothing to hide,\" she said. \"And I'm assuming that you don't, either\u2014that you've been telling the truth from the start.\"\n\n\"Why would I do otherwise?\"\n\n\"Because...\" She faltered in mid-sentence. There were no words to frame the suspicion she still felt, deep in her gut. \"Because you _could_.\"\n\n_Because I stand to lose everything if you haven't been telling the truth...._\n\n\"Having the potential to do something is not the same as intending to use it, Morgan. You of all people should know that.\"\n\n\"What do you mean?\"\n\n\"I mean that five of my people died today as a direct result of your involvement in this conflict. Uri showed me the footage. Now that you know for certain that your ability works, you have the potential to track down and hunt every one of my kind and bring them all to their deaths. Whether I agree with them or not, whether you think I might be one of them or not, whether I am lying to your or not\u2014it's all irrelevant. Ultimately, all that matters is your _intent_ , isn't it?\"\n\nShe took a step back from his intense gaze. \"Killing all of your kind would be genocide.\"\n\n\"Exactly. And since I know that this is what you would call it, you have nothing to fear from me.\" He reached out and put a hand on her shoulder. \"Morgan, any intelligent being can only do what it thinks is right. Have faith in my ability to do that, and everything will _be_ all right.\"\n\nHis eyes never once left hers. His hand gripped her shoulder tightly. She felt he was pleading with her, trying to make her understand something important\u2014but she couldn't quite grasp it. Genocide? Was he talking about the destruction of the Caste that had made him and his kind over half a million years ago? Or was he just using that possibility to illustrate a more general point?\n\n\"If you have any surprises in store for me,\" she said, \"you'd better let me know now.\"\n\n\"I'm not the one with the surprises, Morgan,\" he said.\n\nTurning on his heel, he stepped through the containment team and onto the transport.\n****\n\n**15**\n\n****\n\n****\n\nSHCV Phlegethon\n\n955.2.14\n\n1000\n\nThe fane was full. By the time Roche and her party took their places in the front row, with a clear view of the font and the small group of people around it, she had lost count of the number of unfamiliar Castes mixed in with the thousands of Pristines: short ones, tall ones, broad ones, thin ones, Castes that required filters to weed out atmospheric irritants, and Castes that, judging by the thickness of their skins and protective coverings over their eyes, could have survived just as comfortably in a vacuum. She recognized only three types: a Surin not far from where they stood, an Eckandi toward the rear, and a robed Hum looming to one side. Where they had come from, she couldn't guess.\n\n<Some people aren't happy about this,> Maii's voice whispered to her.\n\n<Anyone in particular?> asked Roche, continuing to scan the crowd.\n\n<No,> said the girl. <It's just a general feeling, underlying the tension.>\n\nThe \"tension\" sprang in part from the news that the Heresiarch had placed the ship on red alert. The conflict in the system showed no signs of abating; if anything, it seemed to be spreading. All peripheral civilian ships, including the _Ana Vereine,_ were to dock, and a protective sphere of fighters would patrol the space between the camouflage and the body of the _Phlegethon_ itself. If anything got through, it would be instantly dealt with.\n\nWhen the time came, Esko Murnane stepped forward and bowed respectfully to the Heresiarch. He raised his hands and, gradually, silence fell about the fane.\n\n\"We have taken the unprecedented step,\" he began, without preamble, \"of inviting our non-Pristine guests to join us today. We intend this as a sign of solidarity in these difficult times, when _all_ of Humanity seems endangered, not just the Pristine Caste. For many of us, this meeting could well be the first time we come face to face, knowingly, with the enemy.\"\n\nAlthough Murnane had made no gesture nor mentioned any names, heads began to turn toward Roche and her party.\n\n\"We have among us again a person you all know at least by reputation\u2014a person who was initially rejected by this council but who has, despite that, worked for us in an unofficial capacity for the last two and a half weeks. The information distributed immediately prior to this meeting explains how Morgan Roche has stumbled across a means of identifying the enemy. If this ability is unique to her, it may be of little long-term benefit in our fight with the enemy. But if it is not, if there are others among us who share this ability, then we have a very real chance of victory. To overcome the enemy we must explore _every_ possible avenue\u2014and at this moment in time, this is the best option available to us.\"\n\nRoche couldn't tell from Murnane's expression whether he believed what he was saying or not, but the triumphant glint was back in Nemeth's eye. It probably wouldn't matter from his point of view if Murnane believed it at all, as long as the council gave _him_ the credit.\n\n\"The sudden and unexpected escalation in conflict we've seen around us,\" Murnane went on, \"may be connected to the death of five clone warriors earlier today on board this very ship. Immediately following their deaths, fighting broke out around the system, and it has not stopped or even eased since. The speed with which the news spread suggests that some sort of epsense link might be involved, and so we have asked Morgan Roche, here, if she will help us in determining whether or not this might be the case.\"\n\nThe whispers around the fane became slightly louder, and in the general murmur Roche heard her own name being mentioned over and over again. She wished that she could shut the helmet of her combat suit to block the voices out; the attention focused on her from those present in the fane was almost suffocating.\n\nMurnane stepped back to let Nemeth take the floor.\n\n\"I have been declared chairperson for this meeting,\" said the younger man. \"As someone who has recently worked with Morgan Roche, I am in a unique position to guide the council to the conclusions it _must_ reach. As my colleague has just told you, this development may prove crucial to the success of our defense of the galaxy against our enemy. Indeed, it may prove critical to our very survival.\"\n\nThe murmurings rose in pitch again, threatening to become a clamor of alarm.\n\n\"Please!\" Nemeth raised both his hands, gesturing for calm. \"There is no need for panic!\" he called out over the noise. \"We mustn't be unnerved by what the enemy has done this day! Don't allow yourselves to think that they have the measure of us. What you are seeing is merely the winnowing of the weak\u2014of those corrupted and influenced by the enemy! Those seduced by evil have died by evil's hand! But the same fate does not await us. We are equal to the task ahead. _We_ are strong; we _will_ prevail!\"\n\nHe lowered his hands and cast his gaze across the crowd. If he expected cheers, he didn't get them, but he did get the crowd's full attention. After a while, relative quiet returned to the enormous room.\n\nRoche wondered how many people had allies, friends, or family among those already killed in the chaos. She didn't think that any of the Castes she had encountered deserved to be labeled \"weak\" or \"evil\" simply because they had been destroyed by the enemy before the others. After all, only chance might have spared the _Phlegethon_ itself from the five clone warriors that had infiltrated it.\n\nBut this was politics, not reasoned debate, and the reminder was a timely one. In order to get what she wanted, she would have to score points, not make them.\n\nWhen he had finished scanning the crowd, Nemeth faced Roche's party. \"Morgan Roche and Adoni Cane, please step up to the font.\"\n\nCane waited for Roche to move before stepping out of the crowd. Together they walked the twenty-odd steps to the heart of the fane, where Nemeth and Murnane and a dozen other people waited for them. The gaze of the council was almost unbearable now: as heavy as a planet and no less impersonal. A subtle prompt from Maii buoyed Roche slightly, made her feel that she could actually face them successfully.\n\n<You've tackled worse than this,> said the girl. <Many of them are more scared than you are, Morgan.>\n\n<Yes,> she sent back, <but it seems they're more scared _of_ me than _with_ me.>\n\n<That's only natural, Morgan.>\n\n<I'm not sure it helps me much, Maii...>\n\nNemeth's nod to each of them was formal and perfunctory. His only interest was in beginning the interrogation.\n\n\"Morgan Roche, why are you here?\"\n\n\"To determine the origins of the enemy,\" she said briskly; she was tired of answering the same old questions over and over. \"And, if possible, to find a way to stop them.\"\n\n\"Who sent you?\"\n\n\"No one sent me,\" she replied. \"I used to work for the Commonwealth of Empires, but I am now independent.\"\n\n\"Is it not true that your mission has been sanctioned by at least one of the High Humans?\"\n\n\"I am unable to answer that question,\" she said after some consideration, \"because I'm not sure myself of the truth.\" That much, at least, was honest. \"There have been times when I was convinced of High Human intervention, but I've never had the evidence to prove or disprove this.\" That, also, was true; she only had the Box's word that the Crescend was involved. \"The fact that I once had in my possession a fully conscious artificial intelligence\u2014something far beyond the capabilities of mundane science\u2014was all I really had to suggest that I was being helped by someone in the High Human ranks.\"\n\n\"And this AI is now destroyed,\" said Nemeth. \"Is that correct?\"\n\n\"Yes.\" Having been said so many times now, the lie came easily.\n\n\"Do you claim that your companion here is one of the enemy?\" He looked at Cane as he said this.\n\n\"Yes.\"\n\n\"We'll come back to this in a moment,\" he said, returning his attention to Roche. But first I'd like to address something else.\" He paused, posturing loftily. \"The last time you came before us,\" he said, \"you refused to submit to a genetic test. Why was that?\"\n\n\"At the time I was unaware of your reasons for wanting me to,\" she said. \"The thought simply hadn't occurred to me that you wanted to determine whether or not I myself was one of the clone warriors. Having said that, however, I should point out that I will _still_ resist such a test, because I believe that I have clearly demonstrated my allegiances in this last week. Even if I _was_ a clone warrior, I have given the council information on the whereabouts of five others. Why would I allow my own kind to be killed if I wasn't on your side?\"\n\nNemeth nodded\u2014approvingly, she thought. He knew what she was doing. She was setting up her argument for the acceptance of Cane. If she could convince the council that the matter of her genetic origins was irrelevant given that she was clearly working for them, not against them, then it would be easier to convince them about Cane.\n\n\"Neither would you submit to an in-depth epsense probe, though,\" he went on.\n\n\"Because I believe such probes are invasive and unnecessary,\" she responded calmly. __ \" _And_ they are open to misinterpretation. My actions should be taken into consideration, not what takes place in the privacy of my own thoughts.\"\n\nHe nodded again. \"And do you speak for Adoni Cane, here?\"\n\nShe glanced to her left, to where Cane stood patiently, awaiting his turn to speak.\n\n\"In what sense?\"\n\n\"Would _he_ allow himself to be genetically examined or probed by an epsense adept?\"\n\nShe shrugged. \"That's up to him,\" she said. \"But I've already given you his genetic data. As for probing him, I don't believe it's possible. The reave in my crew finds his mind impenetrable\u2014\"\n\n<In both senses of the word,> interrupted Maii. <His barriers are naturally strong; he himself might not be able to dismantle them. Even the surface levels of his mind are too complicated for me to interpret; the impressions I receive are in no way typical of any minds I have known.>\n\nNemeth turned to face the girl, whose thoughts had easily filled the fane, relayed by the other reaves around them. \"Your testimony is not called for at this time, child,\" he said, \"The council will address you if and when it is required.\"\n\n< _Squt,_ > the girl sent to Roche alone. Roche suppressed a quick smile, recognizing the Surin word for a closed-minded fool.\n\n\"Your reave is young and inexperienced,\" said Nemeth to Roche, \"although I am told she does possess a formidable raw talent. It's possible she may be right, but I would prefer to trust the judgments of the high-grade epsense adepts the council normally employs.\" He turned to Cane. \"So I ask you now, _would_ you allow such an examination?\"\n\n\"Your reaves have been attempting to read my mind ever since I arrived on the ship,\" Cane said. \"They have not been able to.\"\n\nA flicker of a smile crossed Nemeth's face. \"Then will you at least drop your barriers for them?\"\n\n\"I am not able to do that,\" Cane said. \"The barriers I have around my mind are not artificial. They are part of me. I am as unable to remove them as you are unable to remove your skin.\"\n\n\"You realize that this will make it difficult for us to trust you? After all, we have nothing but your word that this is the case.\"\n\n\"I understand that,\" said Cane implacably, as though daring the entire council to change his mind.\n\nNemeth shrugged helplessly. \"Then all we can do is proceed,\" he said. \"Do we at least have your permission to take a genetic sample, to confirm the data Morgan Roche gave us earlier?\"\n\nHe didn't hesitate: \"Yes.\"\n\nNemeth waved forward two of the people standing by the font. Cane held out his hand as one produced a small device designed to take a blood sample from his thumb. There was a small _click,_ and the two women stepped away.\n\nA few moments later, the results were displayed for all the council to study. In a giant hologram hanging above her head, Roche could see a stylized representation of Cane's genetic code alongside the data she had given the council before. She recognized the scientific shorthand standardized by the Commerce Artel across the galaxy: chunks of code common to all Humans, no matter how divergent their Castes, lay scattered through Cane's genes like islands in an otherwise unfamiliar sea. For the first time, she saw the vast stretches of introns laid bare, incomprehensible patterns of base pairs lined up like words in a language she completely failed to understand.\n\n\"They are the same,\" observed one of the women who had taken the sample. One of the two patterns disappeared, allowing the remaining to be seen in more detail.\n\n\"He possesses the features we have come to associate with the enemy?\" Nemeth asked.\n\n\"There can be no doubt.\" Several of the unknown sections were highlighted in red.\n\n\"You are convinced that this man is one of the enemy, then.\"\n\n\"Genetically speaking, yes.\" The woman stared balefully at Cane. \"I am convinced.\"\n\nNemeth turned away from her, but Roche cut him off before he could speak.\n\n\"Wait,\" she said, addressing the woman. \"What can you tell me about these features?\" She indicated the sections highlighted in red.\n\n\"Nothing, I'm afraid.\" The woman seemed unsettled by the question. \"They don't correlate to any known Human code.\"\n\nRoche raised her eyebrows. \"What does that mean?\"\n\n\"Just what I said.\"\n\n\"That he's not Human?\"\n\n\"No... no, of course not.\" She frowned at the question. \"What else _could_ he be? I just meant that the features we find in his introns are not seen in any other Caste.\"\n\n\"But why is that so unlikely? Every Caste is different. Surely there must be some that stand apart from the rest?\"\n\n\"No.\" The woman was emphatic. \"There has been much genetic intermingling between the Castes since the Primordial strains speciated, five hundred thousand years ago. One always shares _some_ common features with another, no matter how different they might appear in the flesh.\"\n\n\"Then what happened to Cane's introns? Where are the sequences that should be there, and where have the new ones come from?\"\n\n\"The only way the common features could be missing was if they were somehow removed and replaced with new, maybe random, sequences. But I can't see why anyone would want to do that. The introns are ignored, for the most part, since they serve little or no function.\"\n\n\"But if someone _did_ have the capability to do this, might they want to do it to conceal the origins of a new Caste?\"\n\n\"They might.\" The woman shrugged. \"But, again, I can't see why. Only the High Humans have this sort of technology\u2014and why would they create a new Caste just to kill us? There must be many more certain ways to do that.\"\n\nRoche nodded. The woman had raised an interesting point, and allowed Roche to assert her presence in the meeting. Satisfied that the council knew that she was not going to sit back and let Nemeth railroad her to whatever conclusion he was hoping for, she indicated for him to continue.\n\nHe nodded with exaggerated politeness. \"Thank you,\" he said. \"Now, having ascertained that Adoni Cane is in fact one of the enemy, several questions arise that cannot be easily answered. Why he chose to ally himself with Morgan Roche at all is one such issue; why he chose to risk his own life to save hers and that of her companions would be another. These were key sticking points at Roche's last appearance before this council, and they have yet to be resolved. The possibility also remains that he is in fact still working for the enemy\u2014a possibility which cannot be completely discounted, and _must_ be the context within which his replies to our questions are considered.\n\n\"Do you understand what I am saying, Adoni Cane?\"\n\n\"Of course I understand you.\" Cane's expression didn't change, but Roche noted the contempt in his tone.\n\nIf Nemeth heard it, he ignored it.\n\n\"Very well,\" he said. \"We'll proceed. Tell me, Adoni Cane, _do_ you possess epsense abilities?\"\n\n\"In the sense that I can make myself heard to a reave? Yes, I do. But if you are asking whether I can actually read minds or stop people's hearts\u2014then no, I don't.\"\n\n\"And you are certain of this?\"\n\n\"I would hardly be unaware of such an ability,\" said Cane.\n\n\"Morgan?\" Nemeth turned to her, asking her to corroborate Cane's statement.\n\n\"Obviously I don't know what goes on inside his head,\" she said, \"but I haven't seen anything to suggest that he's a reave of any kind.\"\n\n\"His shield?\" Nemeth suggested.\n\n\"It _could_ be innate. I've never seen him hurt anyone that way, or even been addressed by him that way. He has never tried to influence my decisions\u2014\"\n\n\"Are you sure of that?\" Nemeth was quick to jump on this.\n\n\"Positive.\" She was certain Maii would have alerted her to any mental tampering, had it occurred.\n\n\"Then what makes you think an epsense link could be responsible for the chaos that has broken out around us? Either the link exists, and therefore Cane has it, or it doesn't exist and he is as mute as he appears to be.\"\n\nRoche remembered the conversations she'd had with Maii and Cane immediately prior to coming to the _Phlegethon_ the first time. \"Like his shields, the link could be innate. Before his death, the Olmahoi _irikeii_ expressed the opinion that the clone warriors were like him: absorbers of thought\u2014 _all_ thought, from all around them. This would include each other's thoughts, of course, assuming they can penetrate each other's shields. That would turn an innate ability to absorb thought into a means of communicating with each other.\"\n\nNemeth frowned. \"Wouldn't this make them some kind of collective mind?\"\n\nRoche shrugged. \"I raise it merely as an hypothesis to be tested.\"\n\n\"But how could we possibly test it?\"\n\n\"Maybe we already have, inadvertently,\" she said. \"By alerting one clone warrior to the knowledge that we can now find them, we may be alerting the others and\u2014\"\n\n\"You cannot produce the phenomenon you are attempting to explain as evidence to support your hypothesis,\" said a voice from the crowd.\n\nRoche looked around and saw Salton Trezise stepping forward to confront her.\n\n\"How do you plan to prove your argument?\" he continued. \"We need more data. How do you propose we go about getting it?\"\n\nRoche glanced at Nemeth, who looked furious at the interruption but didn't himself in turn interrupt. An intrigued sussurrus spread through the fane.\n\n\"I have no specific experiment in mind,\" Roche admitted. \"That's why I'm here, to talk to the council.\"\n\n\"Well, maybe it's time the council started asking the right questions, instead of skirting the issue. Tell me, Roche, have you personally ever seen anything in the time that you have known Adoni Cane to suggest that he shares a connection with the other clone warriors?\"\n\nRoche thought about this for a second. \"Only once,\" she said. \"When we arrived in Sol System. Jelena Heidik, the clone warrior we were following, knew exactly when we would arrive and where we were headed. That information could have been transmitted through such a link.\"\n\n\"But is it possible that this information could have been obtained through other means?\"\n\n\"Yes, it's possible\u2014\"\n\n\"Then such evidence is circumstantial, _not_ conclusive, and comes from a source one might describe as unreliable: _you_.\"\n\nHe smiled broadly, but Roche didn't respond. Nemeth stepped forward to regain control, but Trezise refused to stop.\n\n\"And what about you?\" Trezise asked, turning to Cane. \"Do you share a connection with the others?\"\n\n\"No,\" said Cane bluntly.\n\n\"You're not aware of any such a connection? Or are you saying that such a connection does not exist?\"\n\n\"It doesn't exist.\"\n\n\"Good, because I'd hate to think the enemy was listening in on us.\" He turned to Roche again. \"Did you think about that when you brought him here? That if such a link _did_ exist, he could broadcast every word we said to the enemy in this system?\"\n\n\" _You_ suggested it,\" she said.\n\n\"Yes, I did\u2014and not because I believed your crazy theory, but rather to clear this matter up once and for all. It's time this nonsense was laid to rest and we returned to serious business.\"\n\nA dissatisfied mutter from the crowd echoed his words. Rey Nemeth took advantage of the slight pause to break in:\n\n\"What are you suggesting, Trezise?\"\n\n\"I am suggesting that we are wasting our time here!\" he said loudly. \"That _Morgan Roche_ is wasting our time, and that you, Nemeth, are letting her!\" He turned to the crowd. \"It is not any mysterious epsense link which will allow the enemy to win; it is meetings such as this! While we stand around here listening to _her_ outrageous claims and _his_ pontificating, we are doing the enemy's work for them!\"\n\nNemeth drew himself up. \"What exactly are you accusing me of, Trezise? Collaboration with the enemy?\"\n\n\"I accuse you of nothing more than incompetence, Councilor. Morgan Roche came to us with vague hints and rumors and she was rightly rejected. _You,_ however, took it upon yourself to pursue her cause in another forum. Perhaps at the time the gamble seemed justified, but her reports now reveal how disastrous that course of action was.\"\n\nRoche felt Maii's anger boiling over, but she forced herself to remain calm as Trezise ranted on.\n\n\"Then she returns, spouting even more wild allegations. They serve no purpose. Worse\u2014they actively impede any progress we might make toward ascertaining the truth! Word about today's 'exercise' could just as easily have spread by means of ordinary Human spies and hyperspace communications. There is no need to hypothesize beyond that. All we have to do is look for those spies and the problem will be solved. But no, instead we're off in search of phantoms, while the very real enemy continues to work among us!\"\n\n\"Not any more,\" said Nemeth. \"The five she helped us locate _were_ clone warriors. There is no doubt of that.\"\n\n\"I don't dispute this,\" said Trezise. \"But how many _more_ might there be, that she _hasn't_ told us about? We have only her word that the ship is now clean.\"\n\n\"There has been no overt move against us\u2014\"\n\n_\"Yet.\"_ Trezise turned to Esko Murnane. \"And _you_! Bad enough that Assistant Vice Primate Nemeth should already have wasted so much of the council's time\u2014but you had to give him more. You encouraged this 'exercise' which has brought the entire Sol System to war. Do you call this progress? Thousands are dying every minute!\"\n\nTrezise turned to address the council as a whole. \"I call for a vote of no-confidence in the leadership and guidance of Esko Murnane and Rey Nemeth!\"\n\nRoche understood, then, why Trezise had asked for her to appear in front of the council.\n\nShe glanced at the Heresiarch, who was watching the proceedings with a frown. The crowd was unsettled; she heard confusion and anger in the mingled voices surrounding her. How a no-confidence vote would go she couldn't guess, but the fact that it had been called was bad enough. Even if it failed to get rid of Murnane or Nemeth, it had placed Trezise firmly in the minds of the councilors, and it would disrupt normal proceedings for some time.\n\nThis was his chance to seize power, and he wasn't going to waste it. He wasn't interested in her testimony one way or the other. She was just a tool to help him get what he wanted.\n\nShe was being used yet again\u2014and the worst thing was that she had to go along with it. Regardless of who ran the council, it was still her best chance to do any good in the system. She knew the ship was clear of the enemy, and she also knew that whether Cane was himself telling the truth, he was still their best hope of learning anything new about the enemy.\n\n<Box, can you do anything to stop this?>\n\n<If you wish, Morgan.>\n\n<Nothing too drastic. I just want it postponed until after we've dealt with the real problem.>\n\n<I understand. Give me a minute.>\n\nRoche returned her attention to Trezise. His expression was guarded, outwardly restrained, but she could see the delight behind his eyes. He was pleased with his work, was relishing the growing dissent about the room. The council wasn't entirely on his side\u2014but he had upset the balance; he'd had a direct effect on its mood. Where Nemeth had been simply power-hungry, Trezise looked like he was enjoying the disruption purely for its own sake.\n\nMurnane tried to quiet the crowd, but to no avail. There was too much tension in the air now for it to be so easily quelled. Even when Nemeth added his voice to the call for calm, the racket continued. Trezise took a step back and smiled openly at the chaos.\n\nThen a single, clear chime cut through the noise. A Heterodox officer ran through the crowd to talk to the Heresiarch, whose face instantly became grim.\n\nThe bell chimed a second time. As the Heresiarch headed for the adytum, the officer came to the font and spoke to Murnane. Roche wasn't close enough to hear what was being said, even though the noise of the crowd was finally ebbing.\n\nAfter the third chiming of the bell, Murnane stepped forward to address the council.\n\n\"We are under attack,\" he said simply. \"The Heresiarch has been called to attend to the vessel. This meeting is therefore adjourned until the emergency is past.\"\n\nThe crowd erupted once again, thousands of voices shouting out in a mix of fear and anger. Robed officials stepped into the fane and moved among them, trying to get the people to head toward the exits. Roche saw scuffles break out in a number of places.\n\n<Thanks, Box,> she said. <A little dramatic, perhaps, but effective nonetheless.>\n\n<This has nothing to do with me, Morgan,> the AI said. <We really are under attack. A large fleet previously stationed several million kilometers from here just slow-jumped to the edge of the camouflage field and launched base-line probes. No shots have been exchanged yet, but that is only a matter of time. Fighters have been launched by both sides.>\n\nShe tried to call Kajic, but a precautionary scrambling system was in place. <What about the _Ana Vereine_?> she asked, concerned. <Is it okay?>\n\n<It is docked and safe.>\n\n<Good. I\u2014>\n\nA hand came down on her shoulder. \"We should leave,\" said Cane.\n\nShe looked around. Haid, Maii, and Vri were being herded toward an exit on the far side of the fane, and the containment team was closing in around the font.\n\n<We'll meet you back at the ship,> Maii called, her voice faint through jamming of a mental sort. The council's reaves, it seemed, weren't taking any chances, either.\n\n\"You go back to the ship too,\" she told Cane. \"You're not a prisoner. They'll take you there and let you go.\" She directed the words at the leader of the containment team, who nodded. \"Wait for the others. I won't be long.\"\n\nCane hesitated for a moment, then nodded and was led away by the squad of soldiers.\n\nRoche approached Murnane, who stood, looking stunned and confused, with one hand on the font supporting him. Trezise was arguing loudly with him.\n\n\"This is exactly what I said would happen if we allowed Exotics into the council meetings! We've become caught up in someone else's dispute!\"\n\n\"Our ftl drones are being destroyed across the system,\" Murnane said. \"This is a coordinated assault, not a random skirmish.\"\n\n\"All the more reason to resolve this issue _now\u2014\"_\n\n_\"No!\"_ Then, more calmly, meeting Roche's eye, Murnane said: \"I don't think talking will resolve anything anymore.\"\n\nHe turned and walked away. Trezise glared at her, then followed.\n\nRoche was at a loss for a moment. She had hoped to find out how she could help, but the fane was rapidly emptying. Nemeth had gone with the others. The only ones remaining were a handful of Exotics trying to get closer to her, and a ring of guards around the central area keeping them at bay.\n\n<What now, Box?>\n\n<We watch,> it said. <This is an interesting development, Morgan. Before, the attacks were random and local. Now they have a target. The enemy is moving against the council in a manner that cannot be disguised as anything but purposeful and destructive. At last, we have confirmation of their intent.>\n\n<But what can I _do_?>\n\n<Nothing for the time being. You are safer here than you would be anywhere else in this system. Even if the _Phlegethon is_ overcome, you will have ample warning, and therefore time to escape.>\n\n<Has the firing started yet?>\n\n<No. The Skehan Heterodox will not initiate the conflict. They would rather avoid it if possible, since it is likely that such an exchange would result in the deaths of many innocent people.>\n\n\"Roche!\" Vischilglin's voice echoed in the emptying space. \"What are you still doing here?\"\n\nRoche turned to face the tall woman who had breached the ring of guards and now stood on the far side of the font.\n\n\"I don't know,\" said Roche. \"I feel like I'm missing out on something important.\"\n\nVischilglin came closer, until she reached the font Then she did as Murnane had done during the first council meeting Roche had attended: she dipped her hand into the water and sipped it.\n\n\"The Heterodoxies say it brings clarity of thought,\" she said, wiping her hand lightly on her robe. \"Something we could all use at the moment.\"\n\nRoche nodded, willing to accept the superstition but not to indulge it. \"There's nothing for me here,\" she said. \"I should get back to my ship.\"\n\n\"I'll take you,\" said Vischilglin.\n\n\"No, that's all right. I can find it.\"\n\n\"Please,\" she insisted. \"I have little else to do while the warriors blunt their swords on each other.\"\n\nRoche acquiesced, and was led out of the fane via the same exit Murnane and the other senior councilors had used. It opened onto a series of featureless white corridors that could have come from any center of bureaucratic power anywhere in the galaxy\u2014a far cry from the streams and valleys she had witnessed on her first trip to the fane.\n\nThankfully, Vischilglin seemed to know where she was going. She said nothing as she guided Roche through the warren. The only sounds were the soft pad of her footfalls, almost entirely drowned out by the heavy footfalls of Roche's combat suit.\n\n<The battle outside has commenced,> the Box announced matter-of-factly. <The first shot was fired forty seconds ago. Another fleet has moved into position alongside the first, and the Heresiarch has ordered a course-change.>\n\nRoche didn't respond. The news wasn't good. With two fleets now engaging the _Phlegethon,_ the possibility that more might join in was very real. How long the Skehan Heterodox could last against a sustained assault she didn't know\u2014and she didn't want to have to find out the hard way, either. She just wished there was something constructive she could do to ease the situation.\n\nInstead, she was stuck in a warren, led by a woman whose silence was starting to make Roche nervous.\n\n<Do you know where she's taking me?> she asked the Box.\n\n<You are on an administration level directly beneath the fane.>\n\n<That's not what I asked,> Roche said.\n\nThey turned a corner. Ahead was a row of doors that suggested elevators or some other intraship conveyance like the one they had used on her first visit. Vischilglin took her to the nearest and pushed a button. The door opened with a hiss and they stepped into the small capsule. Vischilglin selected a destination and the doors hissed shut again.\n\n<She is taking you to the docks,> the Box confirmed. <As expected.>\n\nRoche didn't know why Vischilglin's behavior was bothering her. All she knew was that there was something odd about her, something not quite right....\n\nAlthough she hadn't felt the capsule begin its journey, she did feel it decelerate. Before it could come to a halt, Vischilglin tapped something into the pad by the door, and the capsule coasted a second before recommencing its braking.\n\n<Something's wrong,> said the Box. <She's overshot the level you need.>\n\nRoche didn't give herself time to think. Her combat suit was sealed and a weapon in her hand just as the capsule slid to a halt.\n\n\"Any sudden moves and I won't hesitate to pull the trigger,\" she warned Vischilglin, her voice booming via the helmet's speakers into the confined space.\n\nThe woman's eyes widened. \"How...?\"\n\nThen the doors opened, and Roche saw the welcoming party intended for her: five tall figures dressed in a mixture of spiky Hum armor and robes, all with weapons raised and aimed directly at her.\n\n\"Put the weapon down, Roche,\" said one. \"You can't possibly hope to fight us all.\"\n\nShe hesitated, ready to fire. They all had heavy-duty rifles, and she didn't dare doubt that they were all equipped with armor-piercing ammunition. If she so much as raised a hand, they would cut her down where she stood.\n\n<Box?>\n\nThe lights went out. In the same instant Roche dropped to the floor and switched to infrared. Her welcoming committee was slow to respond, giving her the few precious split-seconds she needed to get out of their sights. She took one robed figure in the throat and another in the hip before any of them returned fire. When they did, the elevator exploded with light. Vischilglin's scream was short-lived.\n\nRoche used the suit's attitude jets to propel herself along the floor. Sparks flew from her stomach-plating as she fired at another of her attackers. The first two she had shot were down but still moving. The armor of the remaining three was tougher; the third one she hit barely flinched.\n\nTheir heat-images were turning to follow her. She scrambled to where one of the fallen figures lay and wrenched the rifle out of its grasp. Rolling, she fired at the other three. The recoil of the rifle took her by surprise, even through her suit. One of her attackers flew backward into a wall. The two others split up and darted away.\n\nShe took the opportunity to look around her. In infrared, the scene was confusing. Airlocks glowed red with flashing lights above them; floors, walls, and ceilings were lukewarm gray; energy from the shots splashed the area around the elevator with bright swaths of white-yellow. Her attackers were green-blue on either side of her, trying to pin her between them.\n\nShe turned and ran as fast as the suit would allow her.\n\n<Box, I need a way out!> she said quickly. <An elevator, an open lock\u2014 _anything_!>\n\nSomething red flashed in her implants to her left: another elevator. She headed toward it. Energy flashed past her and blossomed on a far wall: her attackers were firing at her. She crouched to decrease her profile, dodging as much as she could without lessening her speed.\n\nShe switched to visible light for a second to judge the distance. The elevator doors hung invitingly open, barely fifteen meters ahead. Yellow light shone from between them. Gunfire flashed past her again, and she realized that she was silhouetted against that light, giving her assailants a perfect target.\n\n<Box, kill the light and get ready to close the doors! Warn the others!>\n\n<I've already told Kajio The Box's calm voice was in almost surreal contrast to her mad dash for safety. <He says\u2014>\n\nSomething smashed into her from behind, throwing her forward, sprawling. Pain exploded in her right shoulder and back. She skidded helplessly along the floor, moving fast enough to reach the elevator but missing the doors by a meter and crashing heavily into the wall. She tried to move, to stand, but her suit only whined ineffectually at her. She could smell ozone and smoke and burning blood.\n\nLots of blood.\n\n<Box?> Through the pain, she managed to tip the dead weight of her suit onto its back.\n\nSomeone was running toward her with a rifle trained on her stomach. She tried to raise her own weapon, but her hands wouldn't respond. Her attacker came closer, slowing to a cautious walk. The weapon's aim didn't waver for a second.\n\n<Box! Help me!>\n\nOne of the other suited figures appeared, asking, \"Did we get her?\"\n\n\"Don't ask stupid questions,\" said the first. \"Call the others. We're going to need help getting her on board\u2014and make sure the surgeon is ready!\"\n\nThe other nodded and turned away. The first suited figure approached closer still, until it was an arm's length away. Reaching out with a boot, the figure tapped Roche on the chest. She could do nothing but grit her teeth on the pain.\n\nThe light spilling out of the elevator seemed to be fading.\n\nSomewhere in the distance\u2014or perhaps from deep inside her\u2014she thought she heard a voice calling her. A girl. She knew she should respond, but she didn't have the strength.\n\nIn the fading light, the first figure crouched on one knee beside her. \"Morgan Roche.\" It was a woman's voice. \"At last.\"\n\nRoche had barely a second of consciousness to realize that she knew that voice.\n\nThen a wave of darkness broke over her and took her with it.\n\n****\n\n****\n****\n\n****\n\n**PART FOUR:**\n\n****\n\n**THE CRESCEND**\n\n****\n\n****\n****\n\n****\n\n**16**\n\n****\n\n****\n\nHIC God's Monkey\n\n955.2.14\n\n1380\n\nPage De Bruyn watched closely as three Disciples carried Morgan Roche into the Hum cruiser. Roche's face was red-lit through the blood-spattered visor of her damaged suit, painted oddly by warning lights and alarm signals from within. She was very pale beneath the blood. De Bruyn caught herself thinking that Roche was lucky to be alive\u2014although from Roche's point of view, \"lucky\" was hardly the right word.\n\nThey hauled the injured woman through the cramped, convoluted crawlspaces of the ship and placed her on the autosurgeon's table. Cutting devices flared as they stepped away. Something in De Bruyn's stomach dropped as the cruiser disengaged from the _Phlegethon_ and accelerated into the battlefield, broadcasting clearance codes to ensure their safe passage. De Bruyn waited anxiously for any sign of attack, but none came. The besieging fleets ignored them as the Disciples had assured her they would.\n\nBit by bit, Roche's suit fell apart down her right side, exposing the woman within. De Bruyn was surprised at how small she was, but supposed that was only in contrast to the sheer bulk of the suit. They were approximately the same height, and De Bruyn was taller than most men she knew. Or maybe it was just Roche's vulnerability that made her seem so small\u2014lying there now, finally, helpless and alone. Without her crew of freaks around her, she wasn't as impressive as the rumors would suggest.\n\nRoche's body was covered with gore. The shot had taken her low in the right shoulder and gone straight through her, leaving a hole easily a hand's-breadth wide. Shattered bone, torn muscle, and liquefied organs filled the hole. Blood still pulsed weakly from it, even through the cauterized ends of veins and arteries. De Bruyn could have pushed her hand through the mess and out the other side had she wanted to.\n\nBut the torment could wait. The important thing for now was keeping the woman alive. It was inconceivable that Roche could have survived such an injury. She should have died on the spot.\n\nHissing and licking sounds emanated from the autosurgeon as it went to work on Roche. De Bruyn faced the Disciple who had fired the wounding shot.\n\n\"You're very fortunate,\" she said quietly. \"Had she died, I would have killed you myself. As it is, you'll just be disciplined.\"\n\nThe Disciple paled, but bowed in deference and backed out of the room. The others followed, sensing De Bruyn's mood. She didn't bother to hide the fact that she was displeased, even though the mission had, in almost every respect, been a success. But the Disciples didn't respond as well to reward as they did to punishment.\n\nWhen they were gone, De Bruyn unsealed her own suit and slipped out of the helmet. While she watched the autosurgeon stabilize its patient, she patched into the command network via her implants and summoned the pilot of the vessel.\n\n<Wamel, how long until we're out of here?>\n\n<A few minutes. The Rebuli have given us a mine-free vector taking us up the long axis of the camouflage shield. That will mean a slight delay, but less risk of\u2014>\n\nThe ship lurched. De Bruyn grabbed for support as the deck fell out from underneath her and the lights flickered.\n\n<What the hell was that?> she asked.\n\n<I don't know.> She could hear a racket in the background as the pilot fought for control of the ship. <We're changing course !>\n\n< _What?_ >\n\n<The instruments\u2014> There was another lurch, more violent than the first. Voices shouted at each other over the command network. <I'm shutting down the main drives and going to auxiliary. Hold on!>\n\nFree-fall came suddenly, and just as abruptly ended. De Bruyn's feet lifted off the ground for a second, then slammed back down with twice her normal weight. She slipped and fell, skidding across the floor as acceleration sent the ship into a tight turn. The lights flickered again, and didn't return to their full strength. Red emergency lighting came on, and stayed.\n\n<I don't understand it!> the pilot shouted. <The instruments are giving me readings that don't make any sense, and the controls aren't responding\u2014>\n\n<Don't be a fool,> De Bruyn snapped, gripping the lip of the autosurgeon's operating table and scrambling to her knees.\n\n<We're being dragged back to the _Phlegethon_! _>_\n\n<Call the Rebuli!> she ordered.\n\n<I can't! We're broadcasting something but I can't tell what it is. It's like the ship's been taken over!>\n\nA chill ran the length of De Bruyn's spine. \"This can't be happening,\" she muttered. \"Not again...\"\n\nThe last time she'd had Roche in her grasp, something much like this had occurred. The AI that Roche had babysat too well had somehow taken over a Dato Marauder and COE Intelligence HQ, bending them to its will as easily as De Bruyn used the Disciples. But a recurrence was not possible. That particular AI had been destroyed back in Palasian System. Or so she had thought.\n\nDe Bruyn clambered to her feet, leaning over the operating table, studying its patient intently. Despite all the power fluctuations, the autosurgeon's work on Roche continued unabated.\n\nRoche's lips were moving. It was hard for De Bruyn to hear over the racket in her implants, but Roche was definitely trying to say something. De Bruyn leaned in closer still, and in doing so heard one word being repeated over and over again. It was faint, but unmistakable: \"Box... Box ...\"\n\nDe Bruyn stood upright, aghast. _How_ it was possible, she didn't know, but she couldn't afford to have any doubts. Not now, when she was so close.\n\n<Shut down the navigation and communication systems!> she ordered. <Hit the manual override! Shut down _everything_ \u2014even life-support! Do it now, before it's too late!>\n\nThe ship lurched beneath her again as the pilot obeyed.\n\n<Control system not responding\u2014trying module overrides. Drive module not responding. Navigation down. Communication not responding. Life-support down.>\n\nThe emergency lights went out completely for a second. De Bruyn could hear noises from the bridge that sounded like panels being opened. <Attempting physical disconnection.> There was a pause and then: <Drive module out.> Gravity disappeared completely. <Communications\u2014>\n\nThe line died, and everything went quiet.\n\nDe Bruyn anchored herself on Roche's table. Her suit had closed automatically, and she had just enough light to see by. Roche's face was in shadow, but parts of her body were visible under the autosurgeon's lasers. It was still operating, using its internal emergency power. Roche's lips had stopped moving.\n\nDe Bruyn grabbed a cutter and began to slice away the remaining fragments of Roche's suit. The autosurgeon resisted, especially as she cut at the glove encasing Roche's right hand\u2014where Roche's standard COE Intelligence implants provided her with an external data link. But the autosurgeon had nothing strong enough to cut living armor, and as the glove came free, its resistance ceased.\n\nDe Bruyn heard someone moving toward her, through the crawlspaces.\n\n\"Reverence?\" called a voice. \"Reverence!\"\n\n\"Here,\" she replied, turning from Roche.\n\n\"The interference has ceased,\" said the pilot, climbing into the room. His robes fluttered like the wings of a giant moth. \"But we are drifting blind and vulnerable!\"\n\nShe heard reproach in his voice, and didn't rise to it. \"Bring the systems up slowly,\" she said. \"One by one. Keep automation to a minimum. If that means doing without communications and life-support for the time being, then that's what we do. Navigation, too. All we really need is a working drive to get us away from here. Once we're out of range, everything will operate properly again, I'm sure.\"\n\n\"Out of range?\" The pilot frowned. \"Of what?\"\n\n_Of the damnable Box_ , she wanted to tell him, but couldn't bring herself to say it. She hardly believed it herself.\n\n\"The _Phlegethon_ ,\" she said instead. \"They must be interfering with us somehow.\"\n\nIt was only a half-lie. If the Box still existed, then it had to be broadcasting from the big ship. Roche's suit was in pieces, now, and it wasn't anywhere to be found on her, so it _had_ to be somehow communicating via her implants. If they could just get away from its influence, they would be able to continue their work. With the only possible link between Roche and the Box severed, now that she was entirely free of the combat suit, it would have no way of communicating with her when she awoke. Or so De Bruyn hoped. Her only alternative was to try the \"Silence between thoughts\" shutdown code again\u2014although Roche had ordered the machine to ignore De Bruyn if she said it, and there was no guarantee it would listen to any of their transmissions anyway.\n\nThe pilot looked doubtful. \"Reverence, I\u2014\"\n\n\"Do as I tell you, Wamel.\" Her tone was smooth and cold; argument would not be tolerated. \"I want those drives working even if you have to stoke them with coal. Take us away from the _Phlegethon_ as quickly as possible. We can discuss what happens later. Just get us moving before someone decides to do it for us.\"\n\n\"Yes, Reverence.\" He bowed and left the room.\n\nDe Bruyn returned her attention to Roche. The sight of her lying there in the dark, so near to death, filled De Bruyn with a sense of satisfaction. Finally, Roche was in her hands. Finally, she would know the truth. And _nothing_ was going to keep her from that.\n\nThe lights flickered weakly. Gravity came and went. Deciding that the Disciples needed all the help they could get, she left the autosurgeon to its work\u2014confident in the knowledge that, at least for the moment, she and the machine were on the same side....\n\n__\n\n* * *\n\n__\n\n_God's Monkey_ limped through the battle zone and out of the _Phlegethon_ 's camouflage screen on the tip of a fluttering, poorly tuned fusion flame. An hour later, when the need for accurate navigation overrode De Bruyn's sense of caution, she allowed the pilot to risk switching on some of the ship's higher functions. Gradually, when it became apparent that nothing untoward was going to happen, all of the systems were reconnected. When the ship was fully operational again, she sent it along an orbit that would take them close to the sun, then out to the system's dark fringes, where they would linger in the lesser-populated regions until they had to return.\n\nWithin another hour, the embattled _Phlegethon_ was far behind them, along with the council, the Rebuli, and Siriote fleets, beyond even Salton Trezise and his devious little schemes. Originally, his price for letting her and the Disciples into the _Phlegethon_ had been a disturbance that would justify his push to get the Exotics off the council. But events turned out to be a little more dramatic than anyone had anticipated, what with the Hum kidnapping _and_ the attack of the Rebuli at once. Nevertheless, from De Bruyn's point of view, the outcome had been more than satisfactory.\n\nSeparating Roche from her friends had been ridiculously easy, and Trezise had happily turned Hue Vischilglin to his will, filling her head with the notion that Roche was consciously working for the enemy and convincing her to set Roche up. Whether it was true or not, De Bruyn neither knew nor cared. She had what she wanted, and that was all that mattered.\n\nWhen she was certain they weren't being pursued, she returned to the operating room to see how her captive was doing.\n\nHum autosurgeons were notoriously simple-minded in their relationships with Pristine Humans, and this one was no exception. It took her much longer to access Roche's medical data than it should have, and even then it didn't make much sense.\n\nRoche was stable. Her wound had been cleaned and sealed, and tissue regeneration had begun. It would be days before she was able to move again, and it was still a mystery how she had survived such enormous blood-loss and trauma, but at least she was out of immediate danger.\n\nTrezise had given De Bruyn the council's information on the enemy, and she ran Roche's genetic code past it, to see if there was a match. She was half surprised to receive a negative response: Roche was _not_ a clone warrior. But she wasn't normal, either. Roche's code was riddled with irregularities that neither De Bruyn nor the autosurgeon could explain.\n\nShe patched into the command network. <Wamel? Send Lemmas down here.>\n\n<Yes, Reverence.>\n\nWhen the reave arrived, De Bruyn was busy programming the autosurgeon to remove Roche's implants.\n\n<You summoned me, Reverence?> The man's voice in her mind was like a smooth dark fluid, yet conversely sharp and penetrating at the same time. The first time his mind had touched hers had been disturbing, but she had quickly accustomed herself to this epsense adept's \"tone.\"\n\n\"Yes. Wait a moment.\"\n\nLemmas waited patiently behind her, his arms at his sides in the folds of his black robes. No ordinary reave, he was unskilled at long-distance communication or remote sensing but frighteningly precise at close range. His specialty was the extraction of information from unwilling subjects, and his methods were notoriously effective.\n\nThe autosurgeon whirred and set to work, prepping several places on Roche's body for surgery. De Bruyn turned to face Lemmas, folding her arms across her chest. In doing so, she felt a stickiness there and looked down; some of Roche's gore had made it onto her, perhaps during the brief free-fall when the ship had been drifting.\n\nNot that it mattered. Undoubtedly there would be more in the hours to come.\n\n\"Lemmas,\" she said, absently wiping Roche's blood off her uniform. \"I have some work for you to do.\"\n\nThe man nodded slowly, his hairless face, like most Hums, finely boned and long. He wore his ritual mutilation openly: ears removed, eyes sewn shut, tongue gone. His skin was bluish in the harsh light; through it, De Bruyn imagined that she could see not just his veins but his bones as well\u2014yellow and decayed, like his teeth.\n\n<I am yours to command, Reverence,> he said.\n\n\"I want you to take her apart,\" she said. \"Slowly. I don't want you to kill her. Just break her open so I can look inside.\"\n\n<What is it you are looking for?>\n\nShe looked over to Roche on the table and shook her head. \"I'm not entirely sure.\"\n\n<That will make it difficult.> There was an unhealthy relish to his voice.\n\n\"Just do what you have to do.\"\n\nThe reave inclined his head. <You will be required to observe,> he said.\n\n\"Naturally. You need my eyes.\"\n\n<I will also need you to tell me when I have found the information you seek.>\n\n\"I can assist you in other ways, if you like.\"\n\n<That won't be necessary.> He paused. <You are not squeamish, then?>\n\nThe thought of Roche being tortured didn't bother her at all. Not that she could have hidden it from the reave even if it did. \"That's not something you have to worry about,\" she said.\n\nHis smile was an open wound between his cheeks.\n\n<When shall I begin?>\n\n\"By epsense, immediately,\" she said. \"You will have full access to her body once the autosurgeon has finished. In theory, you will have as much time as you need. In practice, however, I think you should proceed as quickly as you can. There's always a possibility that we'll be traced.\" She was still nervous about the Box. However it had survived, and whatever it was doing on the _Phlegethon,_ the fact that it was out there at all made her anxious. The one thing she couldn't take into account in her plans was a rogue, hyperintelligent machine.\n\n<I have her shielded,> said the reave. <Unless someone knows where she is, she will not be found.>\n\n\"Be that as it may, I'd still like you to hurry.\"\n\nLemmas moved closer to the table and rolled up the sleeves of his robe. His hands were as slender as the rest of his body; his right hand possessed six fingers. He had no fingernails, and below each knuckle were tattoos like rings. He stood for a moment with his head bowed over the operating table, uncannily as though gazing at Roche's face.\n\nThe autosurgeon whirred as it unwound artificial nerves from Roche's arm.\n\nDe Bruyn wondered when and how Lemmas would start.\n\n<I have already started,> he said.\n\nHe reached out with one hand to stroke Morgan Roche's face and, even though she was unconscious, she flinched from his touch.\n\n* * *\n\nIt was less crude than De Bruyn had anticipated. Barely minutes after the autosurgeon had finished\u2014leaving Roche with several wounds across her body, one hand crippled and an empty eye socket\u2014Lemmas began in earnest. All he did was touch her. De Bruyn couldn't tell whether his mind had powerful psychosomatic effects, or if his nail-less fingertips held hidden tools, but his slightest touch pierced skin, parted fat, and slit through muscle with disturbing ease.\n\nRoche remained unconscious throughout the procedure. De Bruyn didn't ask if that was Lemmas's decision. The autosurgeon might have been keeping her sedated while she recovered from its ministrations. A couple of times De Bruyn had to override its attempts to intervene in Lemmas's work, but she resisted turning it off completely; she didn't want Roche dying from shock before she had learned everything there was to learn.\n\n<Ask.> Lemmas held one hand over Roche's mouth as though he were trying to keep her silent. A tiny line of blood trickled down her cheek and onto the table.\n\n\"Where does she come from?\"\n\n<You already have that information.>\n\n\"That's not the point. I want to know what _she_ thinks.\"\n\n<Ascensio,> he said, with the faintest hint of irritation.\n\n\"When was she born?\"\n\n<Day thirty-six of the eighth month, '933.>\n\nDe Bruyn nodded. That accorded with COE Intelligence and Armada records, but still had not been verified independently.\n\n\"Who were her parents?\"\n\n<She doesn't know.>\n\n\"There are no deep memories at all?\"\n\nHe paused for barely a couple of seconds; Roche's body stiffened. <There are vague recollections,> he said. <But they could just as easily be memories of her early caregivers at the orphanage or her host family. It is impossible to say.>\n\n\"So she does remember her childhood?\"\n\n<Yes,> he said, as if at that very moment his mind was caressing those particular memories from Roche's past. <Vividly, at times.>\n\n\"Give me an example.\"\n\n<She had a nightmare when she was four,> he said. <It was raining so hard that she couldn't see through it. She was trying to get home but was unable to find the back door to her house. Something large and hairy brushed her\u2014>\n\n\"Not dreams,\" snapped De Bruyn. \"Are there any _real_ memories?\"\n\nLemmas didn't hesitate: <She had a favorite toy in the orphanage. It was a talking book called Paz. It was yellow, with a round display, and specialized in M35 children's stories. It\u2014>\n\n\"Enough of that,\" she said. \"Tell me what she was afraid of.\"\n\n<She was afraid of being alone,> he said. <Of having no one to talk to.>\n\n\"Was that why she put her name down for the Armada intake?\" De Bruyn asked.\n\n<Partly,> said Lemmas. <But she also disliked being poor and felt that her options on Ascensio were limited. She believed joining the COE Armada would give her more opportunities. One of which was to be able to look for her parents.>\n\n\"Did she ever find them?\" De Bruyn was suddenly very interested in this line of questioning.\n\n<No. She never looked.>\n\nDe Bruyn nodded thoughtfully to herself. \"Was she ever sick before joining the Armada?\"\n\nThere was another slight pause. <She contracted a severe viral infection at the age of ten,> he said. <And she broke her leg at fifteen.>\n\n\"She was treated on Ascensio?\"\n\n<Yes.>\n\nDe Bruyn noted that treatment of neither condition appeared in Roche's official records. \"Go back to the orphanage,\" she said. \"Does she remember any of the caregivers' names from there?\"\n\n<Yes.> Lemmas rattled off five names, two of which De Bruyn recognized from her research.\n\n\"And did she have friends in the orphanage, or outside?\"\n\n<Both.> More names followed. De Bruyn consigned them to her implants; she would check them later.\n\n\"What about emotional or physical intimacy?\"\n\n<What do you mean, exactly?>\n\nDe Bruyn couldn't help a slight sneer. \"Was she ever in love?\"\n\nLemmas didn't reply immediately. <She did not seek such relationships.>\n\n\"That doesn't mean there weren't any,\" said De Bruyn. \"I want someone who will remember her\u2014somebody who couldn't possibly forget her. Caregivers can forget, and even friends might with time\u2014but a lover never forgets.\"\n\nLemmas recounted several instances that, on the surface at least, suggested a willingness to open up to friends and colleagues\u2014a willingness that De Bruyn knew Roche had not shown in Military College nor any time after graduating. She had always been considered aloof by those who came to know her\u2014emotionally distant and efficient, very much like the machines she had once regarded as friends. Yet what Lemmas recounted now of Roche's past portrayed a woman who at least had dabbled with the idea of sharing life with someone else, but who had ultimately rejected it\u2014maybe because it made her feel vulnerable; maybe because her sexual needs simply weren't that great; maybe because she was self-sufficient within herself. For whatever reason, there were only a handful of people, male and female, who featured in Roche's memories as ones who might have been regarded as \"lovers.\"\n\nDe Bruyn had hoped for more, but she was content with anything at all. She at least had more knowledge, now, of Roche's life on Ascensio, and that knowledge could be verified in time. All she needed was one person to say that they recalled Roche, and De Bruyn would have the proof that the official information had been covered up.\n\nShe was still missing the _why,_ though. _That_ would be much harder to find, she was sure.\n\nAt her instigation, Lemmas dug deeper. A life as unremarkable as that of any other orphan from an out-of-the-way world presented itself: her hopes, her fears; her delights, her disappointments; her ambitions, her failures; her dreams, and her everyday anxieties. The COE was full of people like her.\n\nSo why, then, De Bruyn wondered, had she been chosen? And, more importantly, for _what_?\n\nAfter four hours, they took a break. De Bruyn was tired and, although he displayed nothing but cool aloofness, she suspected that Lemmas was also feeling the strain. Roche's condition was a concern, too. De Bruyn couldn't tell exactly what the autosurgeon's data meant, but the patient _was_ showing signs of extreme stress. That was the idea, of course, but it was possible to push too far too soon.\n\n<Do you know now what you are looking for?> Lemmas asked.\n\n\"No.\" She didn't feel inclined to discuss her quest with the reave; the fact that he could reach into her mind and pluck out the information himself only made his asking all the more insincere. \"How deep can you dig?\"\n\n<As deep as required,> he said with unfaltering confidence.\n\n\"Is it possible to hide information from you?\"\n\n<Conceivably, but not permanently,> he admitted. <She is already doing it.>\n\n\"How?\"\n\n<There is a block in her. I touch it whenever we talk about her mission for Intelligence.>\n\n\"Which aspect of the mission?\"\n\n<The AI.>\n\n\"What about the AI?\"\n\n<I don't know. That's what the block is for.>\n\nShe ignored his sarcasm. \"Can you break through it?\"\n\n<It is crude but effective. I suspect the girl, Maii, put it in place at Roche's request. Given time, it will unravel.>\n\n\"Could it be that she doesn't want anyone to know that the Box still exists?\"\n\n<Perhaps.>\n\nShe studied Roche's face in silence for a moment. Bruised, missing one eye, encrusted with blood, the woman was barely recognizable. Fleetingly De Bruyn wondered if she might be wrong\u2014if Roche wasn't as important as she had first thought. What would she do if all this had been for nothing?\n\nBut there was no getting past the enemy's fixation on her: the way they had disseminated her name and interfered with her work among the Vax, the Fathehi, and the Noske. And what of Adoni Cane? It all had to fit together somehow. If she wasn't herself a clone warrior, then there had to be another explanation.\n\nDe Bruyn glanced again at Roche's genetic code. The unidentifiable sections remained just as mysterious as they had been before, different from those of the clone warriors _and_ any known Caste. Random mutations? She didn't know. But at least now she had that data.\n\n<Reverence?> The voice came from the command network, not the reave. <We've spotted a ship that looks like it might be matching our orbit,> said the pilot.\n\nShe felt a tiny shot of adrenaline. <Are you sure?>\n\n<Not one hundred percent. It could be chance...>\n\n<Unlikely,> she said. <Any stray emissions?>\n\n<A few. It might be the _Apostle,_ but we haven't been signaled yet.>\n\nA slight apprehension tightened her gut. The idea of the Disciples' leader arriving made her uneasy. <They wouldn't announce themselves too soon for fear of detection,> she said. <Keep an eye on the ship and let me know if there's any change.>\n\n<Yes, Reverence.> The pilot went back to his work with no mention of Roche. That side of their mission was not relevant to him.\n\nBut cracking Roche _was_ relevant to De Bruyn, and she was conscious now of time running out.\n\n\"Let's continue,\" she said, approaching the table.\n\nThe reave inclined his head. <As you command, Reverence.> Earlier, he had removed the pack covering the great wound through Roche's chest. Smoke came from where his index finger now brushed the stump of her shattered clavicle. <What is it you wish to know now?>\n\nShe only had to think for a second; there were so many questions to choose from. \"Find out if she knew anything about the enemy prior to her meeting with Adoni Cane.\"\n\nHe probed Roche's mind at the same time as he sent her nerves jangling with pain. <No.>\n\n\"Then did she know anything unusual about the Box prior to commencing her mission on the _Midnight_?\"\n\n<No, nothing at all.>\n\n\"Has she ever had any contact with Eupatrid Gastel or his predecessor?\"\n\n<No.>\n\n\"Does she know why I was sacked?\"\n\n<No. She doesn't even know you've _been_ sacked.>\n\nDe Bruyn sighed. She hadn't really believed it would be so easy\u2014but it would have been nice.\n\nShe tried another tack: Did Roche know how the clone warriors communicated among themselves? Did she know why Cane was helping her? Did she know who made him? Did she know why she seemed to be the only one who could find them?\n\nThe answers came as rapidly as De Bruyn fired the questions, and each time the response was the same: _No._\n\nHer questioning became bolder, and Lemmas's probing blunter: Was Roche aware of any plan to the engagements in Sol System? Was the fact that they were in Sol System in the first place significant, or was that just chance? To her knowledge, was the planetary ring as dire a navigation hazard as the Heresiarch feared\u2014and if so, why?\n\nBut again, Roche had no knowledge of these things.\n\nDe Bruyn moved down to details. Had Proctor Klose, captain of the _Midnight,_ known anything about Cane? What about Uri Kajic, ex-captain of the _Ana Vereine_? Why did she think Cane's introns were so important? Did she know where Jelena Heidik was hiding, or how many of the enemy were still at large in the system? Did she know anything _at all_ about the movements of the enemy?\n\nWithin fifteen minutes De Bruyn guessed that Roche in fact didn't know anything about the big picture; two hours more and she was convinced of it. Nevertheless, she persisted, digging for what she suspected might remain behind a veil she hadn't pulled back yet, working through her own fatigue and the continuing fluctuation of Roche's condition. If the reave's finer efforts weren't successful, maybe sheer persistence would win the day.\n\nThe trouble was, she was running out of questions. Since the only area she had taken steps to avoid was that of the Box, it was there that De Bruyn finally turned. She didn't know why it was important, but Roche clearly thought so, and that was enough for her.\n\n\"What can we do about that block?\"\n\n<We can try two things: break it down, or circumvent it.> The reave was weary but still compliant.\n\n\"How difficult is the latter?\"\n\n<Less difficult than complicated. No knowledge can be completely hidden; even when removed from conscious thought it leaves echoes. Locating those echoes, those secondary and tertiary iterations, is a time-consuming task. But it can be rewarding.>\n\n\"Give me an example.\"\n\n<Well, I can't tell you whether Roche thinks this Box is destroyed or not, but I can tell you that she thinks that she has talked with it in the recent past.>\n\n\"What about?\"\n\n<That I cannot tell you.>\n\n\"Then where is it?\" De Bruyn said, then added: \"Or where does she _think_ it is?\"\n\n<That I cannot tell you either. But I do know that the valise that once contained it was a decoy, and that _was_ destroyed in Palasian System.>\n\n\"What has the Box been doing since?\"\n\n<Specifics are hidden in regard to this.>\n\n\"Does anyone else know about it?\"\n\n<Her reave friend, presumably: Maii. No other name is mentioned, except that of the High Human you refer to as the Crescend.>\n\n\"Really? How is he involved?\"\n\n<Again, I have no access to information regarding that.> De Bruyn felt her frustration growing. This was worse than getting nothing at all. However, she was beginning to see how it worked. Direct questions would get her nowhere; she had to go around the issue and tackle it from behind. \"Can you tell me why the Box is important?\"\n\n<No.>\n\n\"Is it connected to the Crescend somehow?\"\n\n<Roche seems to think so.>\n\n\"How?\"\n\n<Not connected so much, perhaps, but _part of._ Her understanding of the situation here is particularly vague. It has troubled her\u2014hence the leakage around the block.>\n\nDe Bruyn pondered this with interest. The Box and the Crescend _were_ connected, it seemed\u2014perhaps more intimately than anyone could have guessed. \"Where is the Crescend?\"\n\n<She doesn't know.>\n\n\"What is he doing?\"\n\n<She doesn't know\n\n\"Is she working for him?\"\n\n<If she is, she isn't aware of it.>\n\n\"Has she at least spoken to him?\"\n\n<Through the Box, perhaps, although that also is vague.>\n\n\"Why is he so important?\"\n\n<She doesn't know.>\n\n\"Why is _she_ so important?\" Frustration gave De Bruyn's voice a bitter edge, one she instantly regretted. Even though Lemmas couldn't actually hear it, he would certainly read the emotion behind it.\n\nThe reave stiffened slightly, but didn't reply. \"What? Have you found something?\"\n\n<No. I was simply listening\u2014>\n\n<Reverence?> broke in the pilot.\n\n<What, Wamel?>\n\n<We have received orders to dock with the _Apostle._ It has assumed an approach vector and will be in position in ten minutes.>\n\nShe closed her eyes, annoyed. The timing couldn't have been worse. <What do they want?>\n\n<They have not said, Reverence.>\n\n<Are they waiting for a reply?>\n\n<No. Confirmation is not necessary.>\n\nThat was a polite way of saying that nothing would stop him from obeying that order to dock\u2014not even De Bruyn.\n\n<Give me an airlock and I'll meet you there.>\n\n<Deck 4, Reverence.>\n\nShe killed the line.\n\n\"We're done here for now,\" she told Lemmas. \"Maybe it's not a bad thing. She'll have a chance to heal.\" She glanced down at Roche's broken and bloodied body on the table and nodded to herself. \"We'll resume later.\"\n\n* * *\n\nWamel was at Deck 4 ahead of her, straightening his robes. They watched together through the command network as a singleship detached from the many-spined _Apostle._ Again, their environment was dark, so the images were either gloomily portentous or painted in surreal colors. De Bruyn remembered the first time she had encountered the cruiser, and felt a similar dread.\n\nThe singleship approached with bright flares from its thrusters, moving confidently across the gap between the two ships. Its approach took barely two minutes, during which time De Bruyn did her best to maintain her composure. She didn't have any doubt who would be in the singleship.\n\nIt docked with a bump heard clearly through the rigid bulkheads of _God's Monkey._ Via the command network she saw it anchor firmly in place not far from her own fighter, _Kindling._ There was silence for a moment, then came the sound of the outer door of the airlock opening. There was another pause, followed by a hiss as the door shut again. Then the inner door was open and Cane's twin walked through.\n\n<Master.> Wamel bowed at the sight of the black-robed figure.\n\nThe clone warrior didn't acknowledge him. His face was exposed, and his eyes sought De Bruyn. \"You have her?\"\n\n\"Yes.\" As uneasy as he made her feel, she refused to defer to him the way the Disciples did. \"How goes the campaign against the _Phlegethon_?\"\n\nHe smiled. \"The arrangement has been profitable for both of us.\"\n\nShe thought of Trezise for a moment, and wondered if her old colleague had any idea what had happened\u2014if he even suspected just how thoroughly she had betrayed him.\n\n\"What are you doing here?\" she asked.\n\n\"I want to see her. Is she still alive?\"\n\n\"Of course.\"\n\n\"Good,\" said the clone warrior.\n\nDe Bruyn frowned. \"Why? What does it mean to you?\"\n\n\"We are curious.\"\n\n\"About what?\"\n\n\"That is not your concern.\"\n\n\"I disagree,\" said De Bruyn. \"Everything to do with her is my concern.\"\n\n\"I think the truth would appall you.\"\n\n\"But _the truth_ is exactly what I'm looking for.\"\n\n\"Very well,\" he said. His smile widened. \"Morgan Roche may yet be the key to our defeat. In her lies the potential for our destruction. She alone could be the undoing of all we have worked for.\"\n\nDe Bruyn paused to digest this. \"Do you expect me to believe that?\"\n\n\"I expect nothing of you.\"\n\n\"Could you explain\u2014\"\n\n\"No. I will not,\" he said. \"We appreciate your efforts in neutralizing this threat to us. With Roche out of the way, our work can continue unchecked.\"\n\nDe Bruyn felt suddenly cold. Whether he was merely posturing or not, the thought was sobering. \"And what _is_ your work, exactly?\"\n\n\"What do you see?\"\n\nHer laugh was humorless. \"I see nothing but destruction.\"\n\n\"What you fail to see is _justice_.\" There was no mistaking the passion and anger in his voice. \"There is no higher aspiration, De Bruyn. You should know that.\"\n\nHis smile was gone. And something in his expression warned her not to push any further.\n\n\"Take me to her,\" he said. \"I want to see her with my own eyes.\"\n\nShe turned and led the way to where Roche was being held. Cane's twin followed her silently, with Wamel bringing up the rear. Along the way, De Bruyn turned over in her mind everything the clone warrior had said. What if he was telling the truth? What if she had somehow ruined any hope at all of defeating the enemy?\n\nThe autosurgeon had tended to Roche's minor injuries by the time they reached her. Her skin was a patchwork of healing strips and salves, and the pack was back on her shoulder wound. Her readings had stabilized slightly, although they still seemed odd to De Bruyn.\n\nThe clone warrior stepped up to the table and looked down upon the patient. \"What have you learned so far?\" he said, his dark eyes studying Roche's embattled body with dispassionate interest.\n\n\"Very little,\" De Bruyn confessed. \"Perhaps if we knew what to look for\u2014\"\n\n\"You still wouldn't find it.\" Cane's twin faced her. \"Because you're looking in the wrong place.\"\n\n\"But... _you_ said she was the key.\"\n\n\" _May yet_ be the key, is what I said. Even if she is, the key itself holds nothing. What the key is _for_ is the important thing, and to know that you must have everything else: the lock, the door, the room beyond...\" He seemed to be enjoying her discomfort.\n\n\"You're talking about Sol System, aren't you?\" she asked, riding a hunch. \"That's where I should be looking?\"\n\n\"Yes.\"\n\n\"And that's why you've come here: you're looking for something too.\"\n\n\"In a sense,\" he said. \"If Roche _is_ the key, and Sol System is the lock, then the room beyond contains the justice we seek.\"\n\nRemembering his metaphor, she asked: \"And what is the door?\"\n\nHis smile returned. \"That is the one remaining issue we must deal with,\" he said. \"Then our business in the galaxy can truly begin.\"\n\nHis expression was relaxed enough, but De Bruyn sensed a terrible energy radiating from him. She felt like a moth flying into a furnace, only slowly realizing just how dangerous her environment was becoming.\n\nIf the enemy's work in the galaxy hadn't even begun, where would it stop? When every Pristine Human was dead? Every mundane, including the Exotics? When all that remained was High and Low Humans?\n\nIt was all very well to think in abstract terms about the enemy and their apparent desire to destroy Humanity, but De Bruyn found it disturbing to be confronted with the possibility that it might actually come true.\n\nOr was he just bluffing? She clutched at this thought. Maybe he was trying to put her off balance. And if that were so, then she would have to find some way to regain control of the situation\n\n\"We did find something unexpected,\" she said.\n\nHis stare was cold and penetrating, his silence demanding she continue.\n\n\"The Box,\" she said. \"It still exists. Roche was lying when she said it had been destroyed.\"\n\nHis eyes narrowed. \"How do you know this?\"\n\n\"It interfered with our escape from the _Phlegethon_ ,\" she said. \"It infiltrated the ship's systems and tried to take us over. We very nearly didn't get away. If I hadn't guessed what was happening and shut everything down in time, we wouldn't have made it a thousand kilometers.\"\n\n\"Are you sure you're not mistaken?\" said the clone warrior, suddenly alert and interested.\n\nShe bristled at the question. \"Of course I'm sure,\" she said. \"I've seen it in action before.\"\n\nHis gaze drifted back to Roche. \"But it's not possible,\" he mused aloud. \"It couldn't be...\"\n\nHe stepped back to the table, leaning slightly over Roche's helpless form. In a loud and clear voice, he said: \" _Silence between thoughts_.\"\n\nThe readings on the autosurgeon instantly changed, going haywire for a moment, then settling down into a new pattern. Roche shuddered; her mouth opened, gasping for air; her one good hand clutched at nothing\n\nThen she relaxed. The readings changed again, returning to how they had been previously, as though nothing at all had happened.\n\nDe Bruyn watched in amazement, her jaw hanging.\n\n\"It's inside her!\" The clone warrior's hands gripped the edge of the operating table, knuckles slowly whitening. \"There's no other way to explain it! The Box's shutdown code affected her physically, and that could only happen if it was interfering with her in some way. And given there are no signals passing between her and any part of this ship\u2014\"\n\n\"That's why she didn't die,\" De Bruyn muttered incredulously. \"It's been keeping her alive!\"\n\n\"Perhaps. But did you note how her readings returned to normal so quickly? Something or someone must have countered the shutdown code, and the only person that could have done that is Roche herself. She must be aware, on some level at least, of what is happening around her.\" His gaze was fixed on Roche's face, as though daring her to wake and contradict him.\n\n\"But...\" De Bruyn shook her head. Another realization had come to her while he talked. _How could he have known about the shutdown code?_\n\n<Lemmas?> She broadcast her mental summons as loudly as she dared. <Lemmas!>\n\nThe reave answered her from the far side of the ship. <Yes, Reverence?>\n\n<Have you detected any interference from the _Apostle_ since it arrived?>\n\n<No, Reverence.>\n\n<Were there any unusual events before then?>\n\n<None>\n\n<Is there anything going on now?>\n\n<I will check, Reverence.>\n\nCane's twin had glanced up, and was looking at her closely. \"Something is troubling you?\" he asked.\n\n\"Nothing.\" She shook her head.\n\n<Quickly !> she urged the reave.\n\n<I detect no interference.>\n\n<Nothing from the ship at all?>\n\n<No emissions of any kind, Reverence. Why do you ask?>\n\nShe didn't have time to answer. The clone warrior had returned his attention to Roche and the autosurgeon.\n\n\"I will be returning to my ship immediately,\" he said. \"I will, of course, be taking Roche with me.\"\n\n\"No, wait\u2014you can't!\"\n\n\"Do not defy us, De Bruyn,\" he said, gesturing for Wamel to disconnect Roche from the autosurgeon. The pilot moved obediently forward to help his master.\n\nDe Bruyn backed away a step. She had to decide, and fast. Roche had theorized about some sort of connection between the clone warriors; that would have explained how Jelena Heidik had known when she was arriving in the system. Perhaps it explained now how this clone warrior had known about the shutdown code for the Box\u2014codes known only by a handful in the COE, but which Cane had heard used before the Box had taken over Intelligence HQ. Maybe she was overreacting.\n\nOr maybe there was some deeper treachery at work\n\nThe clone warrior watched as Wamel disconnected Roche from the autosurgeon. As the last of the contacts fell away, he indicated that the pilot should swing her around so the two of them could lift her. They seemed to have forgotten all about De Bruyn, or perhaps their ignorance was deliberate. She had played her role. She was no longer important. She had become irrelevant.\n\nShe sent a command to _Kindling,_ instructing it to prepare for launch, and drew a pistol from her suit's thigh compartment.\n\n\"Put her down,\" she said, aiming the pistol at the clone warrior.\n\nWamel stopped, looking to his master for guidance. Cane's twin simply stared at De Bruyn, totally expressionless.\n\n\"Put her down,\" she repeated, lowering her aim. \"Or I'll shoot _her_.\"\n\n\"Don't be ridiculous, De Bruyn,\" said the clone warrior. \"Think what you'll be throwing away. Think of how hard you've worked for this.\"\n\n\"I _am_ thinking of how hard I've worked for this,\" she spat vehemently. \"That's why I can't let you take her.\"\n\n\"But to kill her would mean never learning the truth,\" he said.\n\n\"I've learned nothing anyway!\"\n\nHe shook his head. \"But you would have found out eventually,\" he said. \"This way, you'll never know.\"\n\n\"Just put her _back_ on the table!\" De Bruyn waved the gun nervously toward the bloodied tabletop.\n\nWamel let go of Roche's legs and went for his weapon. De Bruyn had a split second to think that he would have been on her side\u2014if only she'd had time to explain\u2014before she shot him. He fell back onto the operating table, smoke sizzling from the hole in his chest, and slid to the floor.\n\nAn alarm rang. De Bruyn guessed that Wamel must have sent a warning through the command network before he died. Already she could feel Lemmas batting at her mind, trying to find out what was going on. Once the Disciples learned that she had threatened their leader and killed one of their own, it would be as good as over.\n\n\"I want to know the _truth_ ,\" she hissed. \"Or I'll kill her now!\"\n\nThe clone warrior raised an eyebrow. \"You wouldn't believe me if I told you.\"\n\n\"Try me!\"\n\nHis eyes shone from his dark complexion. \"If my people do not receive the justice we deserve, we will eliminate every Human in the galaxy.\"\n\nDe Bruyn was taken aback for a moment. Although she had considered the possibility before, stated so boldly it sounded almost ridiculous.\n\n\"That's\u2014\"\n\n\"Inhuman?\" he offered.\n\n_\"Insane,\"_ she said.\n\nSomething moved in the doorway behind him\u2014the other Disciples had arrived\u2014and De Bruyn was out of time. She had nothing else to bargain with.\n\n\"Let me go,\" she said, trying anyway.\n\nThe clone warrior shook his head. \"No,\" he said. \"I couldn't possibly allow that.\"\n\nShe fired: the shot took Roche in the hip and spun her off the table.\n\nThen, moving impossibly fast, the clone warrior was upon De Bruyn, pushing her off balance onto the floor with her arms pinned beneath her suit. His face was close to hers; she could feel his breath as she vainly attempted to break free. A hand in her hair twisted her backward, making her gasp. The air was full of the sound of footsteps as Disciples rushed into the room\u2014but all she had ears for were the words he spoke to her as her neck twisted\u2014\n\n\"You would have been right the first time,\" he whispered. \"And _that's_ the truth.\"\n\nWith her last strength, she instructed _Kindling_ to blow its antimatter fuel reserves.\n\nShe never felt the explosion.\n****\n\n**17**\n\n****\n\n****\n\nJW111101000\n\n955.2.14\n\n1380\n\nMorgan Roche was dreaming.\n\nShe had never considered herself a terribly imaginative person. Through most of her life, her dreams had consisted of everyday things and simple imagery, easily interpreted. They reflected the logical and rational person she was, and demonstrated a lack of creativity\u2014something COE Intelligence appreciated in their agents. They wanted them to be reliable and thorough, not innovative.\n\nBut in accordance with the dramatic change to her life in recent weeks, her dreams had become much more disturbing and vivid, the symbolism darker and more profound. It had gotten to a point where she almost became reluctant to close her eyes for fear of what images she might meet in her sleep.\n\nMore often than not the images fragmented and disappeared soon after she awoke, leaving her with just a vague impression of the emotions that the dream evoked\u2014and even this tended to dissipate as the day progressed. But now and then a dream would be too powerful, too provocative, to ignore and would stay with her long into the waking hours.\n\nTwo dreams she'd had in Palasian System alone would stay with her forever: the lizard she had been trying to trap, which had in turn caught her, and the meeting with the twins on the deck of the stone boat. There had been other dreams that had left impressions, but none like these. These were dreams she would simply never forget.\n\nAs with the dream she was having now. It hadn't even finished yet, but she knew it would be a dream she would not be rid of in a hurry. It felt so real, and the fact that she was unable to wake herself up from it disturbed her terribly.\n\nShe was standing inside a hollow sphere barely ten meters wide. Gravity pointed outward from the center of the sphere, with no odd tidal effects arising from the height difference between her head and her feet. No matter where she walked, the sphere was the same: white and featureless. Light seemed to emanate from all around her; there was no obvious source.\n\nBut something was wrong. She could feel it. Something terrible was happening outside the sphere. Something wanted to get in to where she was. No matter how hard she tried, she couldn't see beyond the sphere to make out what was trying to get her, and that just scared her all the more. She didn't know even vaguely what would happen if _it_ penetrated those walls, if it did get inside, but she knew it would be horrible.\n\nThe unspecified threat made her cold, and she wrapped her arms about herself. She didn't know _anything._ All she remembered was being shot in the back and falling, calling for the Box, hearing Maii calling her... and the suited figure that had approached her. There had been something about that figure's voice. Something familiar. She had heard it before. But where?\n\nShe couldn't remember. It was hard to concentrate while trapped in the bubble, cut off from the rest of the universe with no way out, and something terrible lurking just outside wanting to get in.\n\n\"Hello?\" she called, for what felt like the thousandth time. Her voice echoed dully in the chamber.\n\nShe bent down and touched the floor. It was warm and yielded slightly, like rubber. Underneath it, though, was something firm. She reached into the pockets of her shipsuit in the hope of finding tools of some description, something that might have helped her dig her way out. But her pockets were as empty as the sphere itself. Nothing in them but _her._\n\n<Box? Maii?>\n\nAs when she had called their names before, there was no reply.\n\nShe resumed pacing. There was nothing else for her to do. Eventually something would happen. The bubble would burst and the thing outside would break in, or she would simply wake up.\n\nShe wanted the latter more than anything.\n\nWas it possible to sleep within a dream? Possible or not, she woke with a start. Her entire body spasmed, recalling the shot she had taken in the shoulder. And the face leaning over her, the voice speaking to her...\n\n\"This isn't over yet.\"\n\nNo. Those weren't the words\u2014but it _was_ the same voice.\n\nThe words rolled around in her thoughts. She reached for her shoulder, feeling for the wound, but there was nothing there now. She was undamaged; there was no blood whatsoever. All she had were the memories of pain, a pain worse than any she had experienced before; a pain too huge to comprehend. And the tide of darkness which had followed, pulling her into its depths.\n\nA shudder passed through her. She had dreamed during her brief sleep, and it came to her now with a viciousness that stung. She had been lying on a slab somewhere, in a dark space, and someone had been looming over her, hurting her. There was pain all through her body\u2014her left arm, hand, and eye; her right shoulder; her face and throat. And in her mind. Someone was cutting into her, slicing her thoughts open, piercing the inner depths of her psyche\u2014and behind that someone, behind that pain, standing in the shadows, was _Page De Bruyn...._\n\n* * *\n\nA door seemed to close in her thoughts; images faded. She found herself on her side in the sphere, mouth slack, nose running. She sat up and wiped her face. Her hands were trembling.\n\n_\"This isn't over yet.\"_ Page De Bruyn, of course: the woman who had betrayed her on Sciacca's World; the words De Bruyn said to her the last time they'd seen each other...\n\nOutside, the terrible thing was still trying to get in.\n\nShe wasn't so sure she was dreaming anymore.\n\n<Box?> Her voice shook slightly. <Box, answer me! I _order_ you to talk to me!>\n\nThe sphere seemed to tremble beneath her.\n\n<I can't talk right now, Morgan,> came the reply.\n\nRelief washed through her. Although it was unable to disobey her when she issued a direct order, she'd feared the worst. \"Box! I thought you'd abandoned me!\"\n\n<I can assure you that I have not, Morgan,> it said. <I am doing everything I can to help you. Please be patient for a little longer. I will return to you soon.>\n\nThen it was silent again, and she was alone with the echoes. She listened to them uneasily. The Box had sounded weary, strained. She had never heard it like that before.\n\nThe sphere was solid as it had ever been. A stab of pain in her left eye reminded her of her dream's dream, and of De Bruyn....\n\nShe wrapped her arms around her legs and waited for the Box to speak again.\n\n* * *\n\nA long time passed before the sphere trembled beneath her again. She woke immediately and looked around.\n\nThe light had changed. She had a shadow pooling beneath her now. Above her, in the exact center of the sphere, was a point of light too intense to look at directly. She glanced away, blinking.\n\n<Box? Is that you?> she said, rubbing at her stinging eyes.\n\n<Not as you would understand it, Morgan,> it said. <But yes, it is me. Or at least the part of me that believes itself to be conscious. The part that _lives. >_\n\nShe looked around again at the sphere, avoiding the light. <So where am I?> she asked. <And why am I here?>\n\n<You are inside me; inside my mind.>\n\n_< Inside_ you? That's not possible!>\n\n<It is difficult to explain,> it said. <Your mind is attempting to interpret sensory inputs of types it has never encountered before, and the only way it can do that is by analogy. The truth is that our minds are cohabiting cognitive spaces I normally reserve for myself and my creator. Or to use another analogy, we are thinking with the same mind in the same way that two programs might run simultaneously on the same processor. Just as I have infiltrated your cells in order to exist within you, your mind is now operating within me. Does that make things any clearer?>\n\nShe fought an image of two Klein bottles constantly filling and emptying each other. It wasn't helping.\n\n<I guess,> she said, sitting up. <This place isn't real, but it isn't a dream, either?>\n\n<It is _like_ a dream in that it is an unreal construct of your subconscious mind comprised of real images. Its details might be wrong, but it does contain some truth. It would be wrong to dismiss it as a complete fabulation.>\n\n<So you really are talking to me right now?>\n\n<Yes, Morgan. I am.>\n\n<Then you can tell me why I'm here. What's going on\u2014and how is Page De Bruyn involved?>\n\nThe Box paused for a long moment. When it spoke, its voice was softer than before, almost tentative.\n\n<I must confess to having miscalculated, Morgan,> it said. <Page De Bruyn was somehow able to infiltrate the _Phlegethon_ and ambush you on the way to the _Ana Vereine._ You were shot in the process.>\n\n_< That_ much I remember.>\n\n<You should have died. Shock very nearly killed you instantly. Only by my sacrificing part of myself were you able to survive long enough to make it to an autosurgeon.>\n\n<You 'sacrificed' yourself?> she echoed. <What does that mean, exactly?>\n\n<The function of many of your cells has been subverted in order to maintain my existence. By reducing the number of such cells in critical places and thereby allowing them to return to normal duties, or by enhancing their activity in others, I was able to stabilize your condition long enough for you to receive treatment.>\n\nShe didn't know what to say at first. <And now that I've been treated...?>\n\n<The capacity I sacrificed is beyond immediate recall. Too many cells have died. And there is a complicating factor.>\n\n<That being?>\n\n<You are being tortured, Morgan. The damage to your body is continuing at a rate too great for even the autosurgeon to contain. Far from reclaiming my lost components, I am forced to sacrifice more to ensure your continued survival. I am only able to talk with you now because they have temporarily ceased with the torture.>\n\nRoche felt something much like sadness welling in her. But she wasn't sure who it was for: herself or the Box. <How long can you keep this up?> she asked.\n\n<A while yet, Morgan. Once this crisis is past\u2014or once it has overtaken my capacity to maintain this refuge\u2014you will be returned to your normal mode of being. If you are allowed to recover, I will eventually regain my former stature. All we need is time\u2014time to _heal. >_\n\nShe nodded. <But that's not all, is it? That doesn't explain why I'm here. If your capacity is reduced\u2014and running _me_ must be taking a large chunk of what's left\u2014why are you going to such trouble? It's not just to keep me alive, surely?>\n\n<No.> Again the Box hesitated. <You are here because your torturer is a reave.>\n\nRealization dawned. <And reaves can't read AIs,> she said. <If my mind is running on your components, then I'm safe from him, right?>\n\n<That is approximately correct, Morgan. In fact, minds as complicated as mine _can_ be partially read by a reave, but he simply does not know where to look. That makes all the difference.>\n\nRoche remembered her fuzzy self-image in n-space. <So are _you_ a reave?>\n\n<No\u2014although I can, at times, read your surface thoughts. That is not my purpose. My purpose is\u2014and has always been\u2014 to protect you.>\n\nThe Box had said something very much like this before. She still wasn't sure she believed it, even in the current circumstances. <So if I'm in here,> she said, <who is the reave torturing?>\n\n<You, still. Although the essence of you is in here, everything else remains outside. To him you appear unconscious, and he has free access to those memories of yours not blocked by Maii. Eventually, though, these blocks will fall, but for the moment everything you do and say in here will be safe.>\n\n<If the block fails, they'll know you weren't destroyed!>\n\n<They already know that. I was able to take direct action only once you were stable in the autosurgeon; even so, I was too late to prevent our kidnap. Since then, I have had no chance to subvert De Bruyn's command. She ordered your implants removed; I no longer have access to the outside world, except by the passive observation of your sensory input.>\n\nHer thoughts were reeling, and she found herself wishing this really had been a dream. <Okay,> she said. <So what are we going to do?>\n\n<Before De Bruyn realized what was going on, I managed to broadcast a brief call for help. I also programmed the ship to broadcast a tightbeam beacon back to the _Phlegethon_ every hour. If someone has followed us here, there may be a chance of rescue.>\n\n<And if not?>\n\n<Morgan, there is much I must say to you, and I fear we don't have a lot of time. When your torturer resumes, I must return to the maintenance of your body and will not have the capacity to hold a conversation. Do you understand?>\n\nShe nodded. <You have to talk now, while you can\u2014and I don't have a problem with that. It's about time you told me what's going on.>\n\n<I can hardly blame you for feeling this way, Morgan, but do try to see why the truth was withheld from you. It was not that you couldn't be trusted, but that it could have been got at. Had you known prior to your capture what you are about to learn, your torturer would have already extracted it. For that reason, also, I have brought you here, with me. Were we to discuss this in your normal state, your torturer would access those memories instantly.>\n\n<So let's get on with it.> She lay back on the curved floor of the sphere, closing her eyes against the Box's glare. She was tired, apprehensive, even scared. But her curiosity overrode all of these. <Before you're called away.>\n\n<Very well,> the AI said. <The first thing you should know is that you were partially correct in regard to what you said the other day about the High Humans. They _could_ destroy the enemy at any time; they possess the technology to do so with ease. What they lack is the inclination.>\n\nAs vindicated as Roche felt to learn that she had seen through at least that part of the conspiracy, the conclusion she had avoided disturbed her deeply.\n\n<You mean they'd rather let us all _die_ than help us?>\n\n<Morgan, you have to understand that High Humans like the Crescend are rare. Most of them follow a strict anti-Interventionist path\u2014not out of any profound principle, but because they have little interest in the affairs of mundanes. They are as different from you as you are from an insect. Would you save one species of insect over another invading one? Maybe not, unless your interest happened to be myrmecology.>\n\nRoche could tell where it was heading. <So the Crescend is a myrmecologist?> she said dryly. <It studies ants?>\n\n<Essentially, yes. He is concerned enough about mundane Humanity to become directly involved in this situation. The Crescend\u2014perhaps alone of all the High Humans\u2014has considered ridding the galaxy of the threat you face.>\n\n<So why doesn't he?>\n\n<It's not so simple, Morgan. As we discussed earlier, there are many issues he must face before coming to such a decision.>\n\n<Like finding the enemy,> she remembered.\n\n<That, and reaching consensus with the rest of the High Caste. At present, there are four thousand and seventeen High Humans active in this galaxy. Approximately thirty are Interventionists and willing to support the Crescend. The majority has ruled that it is not the Interventionists' place to decide whether or not to destroy the enemy. They will not allow a massacre of mundanes to occur with the help of the High Caste\u2014even though inaction might well allow a greater massacre to take place. For the time being, the Interventionists can only observe and assist in the decisionmaking process.>\n\nRoche fought the urge to argue that the anti-Interventionists' stance neither made sense nor was fair. <That's where you come into it, I take it\u2014on the information-gathering side?>\n\nThe Box didn't sound as smug as she might have expected. <Suffice it to say that the Interventionists' role is _mainly_ passive, for the moment. They have access to the little information available regarding the enemy, and although they are not, strictly speaking, allowed to spread that information, they can ensure that the right people come across it at the right time.>\n\n<Is that what you're doing now?>\n\n<Yes. The Interventionists are on the brink of finalizing their plans. The decision whether or not to assist in the eradication of the enemy is about to be made\u2014and the High Caste majority _will_ support it, if it is made in the agreed fashion.>\n\nRoche thought of the thousands of people dying every minute in Sol System, and wondered what was happening outside the system. <Is there any way we can hurry them along?>\n\n<Not at the moment, I'm afraid. Our options are decidedly limited.>\n\n<Well, what _can_ we do, then?>\n\n<In a moment, I will give you access to much of the information gathered in recent weeks about the enemy. It will help you decide whether you want to be a part of this process.>\n\n<Why wouldn't I want to be part of it? If I don't kill the enemy, they'll kill me!>\n\n<I feel obliged to point out, Morgan, that your life is at present in more danger from one of your own kind than from the enemy.>\n\nShe conceded the point. <Well, all right, Box. I'll look at this information, whatever it is. But then what? Where do we go from there?>\n\n<That remains to be seen, Morgan, and it depends to a large part on what happens to you\u2014your body outside of this shell. If no one rescues you and you die at the hands of De Bruyn's torturer, then all our plans are undone.>\n\n<And if I fall into the enemy's hands?>\n\nThere was a lengthy pause. <Then there is no telling what might happen.>\n\nThe Box's tone made her nervous again. <But _why,_ Box? Why am I so important?>\n\n<Ask yourself one question, Morgan, and think about the answer while I am gone: why can you detect the enemy when so many who have tried before have failed?>\n\n<How the hell am I supposed to know the answer to that, Box?>\n\n<Just think about.>\n\n<But I _have_ thought about it!>\n\n<I'm sorry, Morgan, I can't explain. You must work it out for yourself. Your torture has resumed, and I must attend to your physical well-being. I will return as soon as I can, even if it comes to the worst. Be assured that I am\u2014>\n\nThe light above her suddenly went out, and the Box fell silent. Roche sat bolt upright, looking in alarm to where the Box's light had been. It was suddenly very quiet, and the sense of threat from beyond the sphere returned.\n\nHer skin tingled all over as a patch of air one meter in front of her clouded over, as though a self-contained mist had suddenly formed out of nowhere. It swirled around itself for a moment, becoming thicker and darker, then faded to reveal a three-dimensional tank not dissimilar to the instrument displays on the _Ana Vereine._ Inside was a single, flashing icon, shaped like a gold key.\n\nThere being no other visible way to interface with the display, she reached in and touched the key.\n\nIt turned into an embedded document containing numerous chapters and headings. The glossary was full of references to things she had never heard of before. There were links to diagrams and charts, statistics and formulae. There were texts from the fields of biology, sociology, anthropology, and archaeology. There were maps of regions long since distorted by millennia of stellar movements, and others so up to date that they included the destruction of Palasian System. There was even a mention of her, although when she touched the link, the display returned a message saying: \"Access Denied.\"\n\nShe sat back with her legs crossed, the display following her every move. Then when she was relatively relaxed, she began to browse....\n\n* * *\n\nThe second name she recognized was that of Adoni Cane. There were several Canes listed, and some with aliases; one was the Cane she knew, his activities extensively chronicled thanks to the Box's proximity. Other Adoni Canes had appeared in diverse parts of the galaxy, always to sow chaos, then to disappear. One had left a swath of disorder from the core to the Middle Reaches, his path pointing directly to Sol System. Where they were now was not listed, although the anonymous authors of the text speculated that at least some of them had made it to Sol System already.\n\nShe followed two links from that article. One led to the original Adoni Cane. The other explored the history of Humanity, as near to its origins as the High Humans could get. She was amazed to learn that even they didn't know for certain where their progenitors came from. She had always assumed that there was nothing they didn't know\u2014or couldn't find out, if they wanted to. But clearly that wasn't the case.\n\nHumanity had diverged from the original, Pristine genetic strain somewhere between five and six hundred thousand years ago. Its dispersal throughout the galaxy could be plotted by studying the aging of certain anchor points known to have been constructed at that time. Anchor points didn't decay like matter; over hundreds of thousands of years, they dissipated back into the universe's natural background vacuum fluctuations in gradual, known ways. The remnants of the network that had first allowed Humanity to spread outward into the galaxy later gave archaeologists a rough guide to how that expansion had taken place.\n\nBy following it backward, a vague approximation could be made as to where it had all started.\n\nThe study of the propagation of the four known Primordial Castes suggested that the original Human homeworld had once been located near the space currently occupied by the Commonwealth of Empires. This region itself was now totally empty, with any ruins that might have existed long since removed or destroyed. No hard evidence remained to isolate a single system out of the many possibilities, but around twenty had been singled out as likely possibilities.\n\nSol System was one of them, despite its emptiness. The proponents of this theory raised the history of the system as their main evidence. Time and time again, it had become the focus for fringe groups or obsessive cults as though a subconscious collective memory guided them there. The Sol Apotheosis Movement was just one of many that had used the empty system as a home base, free from observation and interference. The system's name had accrued a certain notoriety among the High Caste observers, and the current convergence only added to that.\n\nFrom Roche's point of view, the difficulty lay in knowing whether the convergence occurred because of the system's history, or regardless of the fact. The Box had admitted that the Crescend sowed rumors of the enemy's origins in order to draw people to the system, but the reasons for his doing this were unclear. Roche wasn't sure whether the rumors had been started _because_ the enemy was already converging there, or whether the enemy had been lured there by the rumors, along with everyone else.\n\nThe history of the system itself, though, did intrigue her. It had once possessed a number of planets\u2014at least eight, if the records were accurate, plus a large number of dark bodies, an asteroid field, and a cometary halo. Their fate was a mystery, although one observer grimly hypothesized that the composition and mass of the ring suggested that the entire system had somehow been ground to dust and put in orbit around the primary. Why anyone would want to do this remained unknown.\n\nAmong the ancient records that did remain from the older days of Humanity were scraps pertaining to the present situation. The name Adoni Cane was among those scraps, as were the other names the Box had mentioned in Palasian System. They had once been real people.\n\nOn a list of military honors, Field Admiral Adoni Cane of the Old Earth Advance Guard had received a Military Star for extraordinary acts of valor against the enemy. General Jelena Heidik distinguished herself against the same enemy in a place called Alpha Aurigae and received a Mars St. Selwyn Medal for her trouble. Vani Wehr was a civilian whose quick thinking on the Clarke Cylinder thwarted an enemy incursion and earned him an Honorable Mention. Captain Sadoc Lleshi was one of many Ground Corps officers posthumously recognized for excellence in battle after the long and bitter campaign had ended with the enemy's defeat. And so on.\n\nAlthough there was no explanation for the names of the medals awarded or places mentioned, and nothing placing the battles in any context, Roche recognized the pattern immediately. The names used by the present enemy were all taken from those distinguished in the battle against them in the distant past. No doubt it was intended as an insult or a grand irony. That lent credence to the theory that the \"enemy\" referred to but never actually named in the old records was indeed the source of Adoni Cane and his siblings\u2014but it didn't really tell her anything new about the enemy, past or present. There were still no recognizable names or locations, no descriptions, no clues at all as to where they came from or what they had looked like.\n\nThere were some tantalizing snippets, however. One concerned the command language Linegar Rufo had used in his attempts to communicate with the clone warrior in Palasian System. It appeared to be an actual language, not specifically restricted to military applications\u2014although, again, its origins were clouded. Whether the Box had lied when it denied recognizing the language upon first hearing it, or whether this was new information added since then, Roche couldn't tell. Either way, its unique syntax and dissimilarity to any tongue currently in use marked it as enigmatic. Why it remained when so little else did was not explained, and Roche had a feeling that if she pursued the matter, she would run up against another Access Denied warning.\n\nWhen she hunted for a genetic reference to the ancient enemy, she also found no data available. That didn't surprise her as such\u2014if the records didn't contain even a name, then a DNA record was too much to hope for\u2014but it did disappoint her. Hard evidence of a connection between the ancient enemy and the new would have been good. It would have silenced the doubt that nagged at her even now, asking her how it was possible for a connection to exist across such a gulf of time.\n\nBut then she remembered that to people like Adoni Cane, no time at all had really passed. The capsules that had created them had been drifting through the galaxy for over half a million years, their contents frozen, waiting for the moment to loose a new clone warrior. Their creators had programmed them and set them loose, then been destroyed forever. The legacy of their clone warriors was all that remained.\n\nAs such, their own genetic code was of particular interest to the High Humans. Were the unique intron passages somehow responsible for the unusual structures in Cane's brain that had baffled Sylvester Teh on Sciacca's World? These in turn might have been related to their odd n-space impression. But how? Minds greater than hers had grappled with these problems and had come to no firm conclusions. All were convinced that the introns of the enemy contained important information or played a critical role, but no one knew exactly how.\n\nAfter what felt like an eternity browsing through the file, Roche closed her eyes and leaned back on the yielding floor of the sphere. She really wasn't learning terribly much. Yes, there had been a war in the distant past, whose losers had seeded this peculiar revenge. And yes, Adoni Cane was one of them. But she still didn't know who the enemy was, and she still didn't know how she fit into it all.\n\nThe Box had asked her to think about why she alone could detect the enemy. She was no closer to the answer than when she had started, and she suspected that no amount of random browsing would find it, either. But if she knew that, then the Box knew it too. It obviously hadn't meant that she would find the answer there.\n\nBut where, then?\n\nShe got up again and began to pace. The misty screen followed her for a while, then collapsed to a fuzzy point and fell behind. There was nothing else in the sphere. It was as featureless as ever, its air perfectly breathable and temperature perfectly comfortable. Her only distraction was the occasional urge to sleep, which she resisted. Even if such urges meant that the Box was having problems running her on its components, she didn't care. Its components were part of _her._ She had every right to use them, too...\n\nShe stopped in mid-pace, struck by an idea.\n\nWas _that_ what the Box had meant? Could it be so simple?\n\nThe galaxy she knew was about to be destroyed by a relatively small number of superior warriors partly because Humanity lacked the ability to tell these warriors from their own. If the High Humans did in fact possess the ability to wipe out the enemy, then presumably they also knew how to find them. But if the Crescend wasn't allowed to intervene directly, he also couldn't stand back and let Humanity be slaughtered. He therefore had to find another way to help.\n\nOne way would be to provide Humanity with a means of detecting the enemy. Since mundane Humanity already had access to epsense abilities, a slight enhancement of those abilities could be enough to give them an edge. If it could be done subtly, without obviously interfering, all the better. In short, the ability Roche had could be a \"gift\" from the Crescend. It might have been implanted within her along with the Box.\n\nIf it was true, she had been tinkered with yet again.\n\nAnd now she was a _tool._\n\nShe began to pace again, angrily. It all made perfect sense. The Surin had learned how to engineer for epsense abilities, and the High Humans surely had superior abilities. Why not give her the ability to perform this feat and allow her to discover it by accident? No one could accuse the Crescend of creating a weapon designed explicitly for retaliation: after all, she was unable to access n-space without the help of another, and her ignorance of the ability meant that it might never have been found. From the outside looking in, it could even be mistaken for a fluke of genetics.\n\nBut why _her_?\n\nShe cursed aloud and strode on, working her anger out. She hadn't asked for this! What was she supposed to do? Devote what little of her life remained to the hunting down and destruction of the enemy? She didn't even know how many there were in the galaxy; there might be millions! High Executioner wasn't a role she relished playing alone, and without respite\u2014and, ultimately, with little chance of success. It was too much for one person.\n\nUnless, she thought, there were _more_ like her....\n\nBut there was little she could do except stew over it until the Box returned, and she had no idea how long that might be. She walked around the sphere to where the reduced display wavered in the air, and passed a hand through it. It returned instantly to its full size, displaying the key once again. She sat down on her haunches and searched every link she thought might be even remotely promising. Anything to distract her.\n\nShe learned some things she hadn't known before. The Crescend wasn't the most powerful Interventionist. One called Aquareii\u2014whom Rey Nemeth had once mentioned in passing\u2014 coordinated that faction in the High Caste. The Crescend's value, it seemed, lay in his close proximity to the convergence\u2014to Sol System\u2014although his precise location was never specified. Roche didn't know whether members of the High Caste retained a physical component when they Transcended; for all she knew, they might have written their minds on the fabric of space itself, never to be erased. But if they did have components that could be damaged or even destroyed, she could understand why they kept their locations a secret, even from each other. When one's potential for life was equal to millions of mundane lives combined, death was a tragedy only comprehensible in the same terms.\n\nThere were other details, too, that she couldn't see connected in any particular way to the matter of the enemy. One struck her as being so far afield that it couldn't possibly be right: the discovery in a distant part of the galaxy of several anchor point remnants that appeared to be older than Humanity itself. Either the dating of their decay was wrong, or Humanity was simply older than first thought\u2014\n\nThe sphere suddenly and violently vibrated, flexing as though it had been struck by a giant hammer.\n\nThe display dissolved as an inrush of sensory data flooded through Roche\u2014pain, fear, nausea, paralysis...\n\nAlmost buried beneath it, she heard two words:\n\n_\"... between thoughts.\"_\n\nShe knew instantly what had happened. Someone had used the Box's shutdown codes! She hung on desperately as the sphere threatened to unravel beneath her. Clutching for the appropriate response before she lost herself totally to the overwhelming sensations, she called out as loudly as she could: _\"The game begins! The game begins!\"_\n\nAs the rush ebbed slightly, she fell back with a gasp. The sphere was still unstable, but at least the pain had relaxed its grip on her.\n\nA flash of light above her heralded the return of the AI.\n\n<Box! What's happening?>\n\n<I don't have time to explains it said, its voice thin and strained, its light weak. <And I won't be able to protect you for much longer, Morgan.>\n\n<But\u2014> she began.\n\n<There isn't _time_ , _>_ it snapped. <Tell me now: do you _know_ what you are?>\n\n<The Crescend made me like this, didn't he?>\n\n<Yes, but why?>\n\nShe paused, reluctant to say what she knew to be true. <I'm a weapon,> she said finally.\n\n<That's right, Morgan.> The Box sounded almost relieved. <You are. And it is important that you contact the Crescend as soon as you are able.>\n\n<What?> Roche felt confused. __ < _How?_ >\n\n<With the code phrase 'Dawn comes,'> it said, its voice growing softer. <The High Caste has agreed to abide by your judgment on the matter of the enemy. When you have weighed up the information you have learned here, contact the Crescend and let him know your decision. He will act immediately.>\n\nThe sphere shuddered around her. <What are you talking about?> she said, fighting down panic.\n\n<On its own, the High Caste will not sanction the eradication of a weaker enemy, but it _will_ if the decision is made by an ordinary Human. That is the way they have resolved to break the moral dilemma. The decision is yours, Morgan, and yours alone.>\n\nFor a moment she couldn't speak. This was far more than she had guessed. The Crescend was putting the lives of Cane and all his siblings in her hands!\n\nThe light of the AI flickered, then returned at a reduced intensity. <You must decide, Morgan,> it repeated.\n\nStunned, Roche closed her eyes. This wasn't what she wanted to hear. <I don't believe you, Box,> she said.\n\nThe sphere seemed to be unraveling again beneath her, and the Box's voice grew fainter every second. <Yes you do, Morgan; I feel it. You _have_ to believe. It is why you are here!>\n\n<I...> She shook her head. Her thoughts were becoming fuzzy, as though whipped by a rising wind.\n\n_< Please,_ Morgan!>\n\nSomething caved inside her. She had never heard the Box so anxious, so desperate.\n\n<But I don't even know where he is!> she said, her panic rising steadily as the Box's voice gradually faded.\n\n<Just aim for Sol, Morgan. Use the words I gave you. Use the\u2014>\n\nBefore it could finish, the sphere was torn apart by forces beyond her comprehension and the Box's light faded completely. Pain exploded through her. Her skin was afire and every cell of her being cried out in agony. She dimly heard voices\u2014someone shouting her name\u2014and felt hands roughly on her shoulders.\n\nShe opened her eyes to a darkness broken by the faint flicker of light.\n\n\"Box?\" she said weakly.\n\nBut all she saw, looming from the shadows, was Cane's face.\n\n****\n\n****\n****\n\n****\n\n**18**\n\n****\n\n****\n\nHIC God's Monkey\n\n955.2.15\n\n1210\n\nAfter the solitude of the sphere, his voice struck her like a whip.\n\n\"She's alive!\"\n\nRoche reached for him with hands bent into claws. \"Help...\"\n\n\"Don't move,\" he said, putting his arms beneath her to lift her up. He placed her down again on something hard and cold.\n\n\"The Box...\" The world grayed for a moment, and she clutched at consciousness with the last of her strength. \"Don't let me die!\"\n\n\"Trust me,\" Cane said. \"I have no intention of allowing that to happen.\"\n\nShe felt an incredible pain surge through her as he stretched her out. Her gut heaved and she tasted blood\u2014just as something exploded nearby and she was flung back onto the floor. Someone called out in pain; she didn't recognize the sound of her own voice.\n\n\"That idiot blew her ship!\" Cane said loudly. \"We'll have to manage as best we can.\"\n\nWas he talking to her? Roche couldn't tell. But her mouth moved feebly in response anyway.\n\nRobed figures suddenly loomed over her, trying to pick her up. She recoiled from them, confused. Was she still back on the _Phlegethon,_ trapped by Page De Bruyn? Was all that had gone before merely a dream, and the nightmare proper only just beginning?\n\nWanting to cry out, she let her body go limp. She was simply too weak to resist.\n\nHer head lolled back over her shoulder, and she glimpsed a body dressed in a black uniform lying in one corner, its head twisted at an impossible angle. The face had once belonged to Page De Bruyn. It didn't seem to belong to anyone now.\n\nShe heard Cane's voice as though from a great distance, ordering the robed people to move faster. She thought he sounded different somehow, but was unable to be sure with the wailing of alarms and the pounding of machines booming through the bulkheads. He sounded colder, more efficient perhaps. He sounded _dangerous._\n\nHer body spasmed as the terrible realization spread like burning ice through her mind: _it wasn't Cane_!\n\n\"Where\"\u2014Her mouth was full of blood. She tried her best to spit it out\u2014\"are you taking me?\" she managed.\n\nOne of the robed figures turned to face her. Beneath the cowl, the woman's skin was pale-blue and waxy. Her eyes were red.\n\n\"To Hell,\" she said matter-of-factly.\n\nRoche closed her eyes; despair threatened to overwhelm her. She could feel it gathering like the black clouds of a dust storm on the horizon. If she let it in, it might never leave. She had to fight it.\n\n<Box? Box\u2014can you hear me?>\n\nNothing.\n\n< _Box!_ >\n\nSilence.\n\n_< The game begins,>_ she tried lamely, but even as she spoke the words she knew it was pointless.\n\nHer bearers slowed and she heard the hissing of an airlock.\n\n\"Through here,\" she heard Cane say. \"On the acceleration couch. Careful!\"\n\nShe was brought forward and laid gingerly on a reclined, cushioned seat. The sound of alarms faded slightly. She tried to look around, but her vision was blurred and hazy. Her left eye was completely blind.\n\nThe hands that had held her fell away, and a series of footsteps led out of the room. Then there was a voice:\n\n\"Master?\"\n\n\"What is it?\" Cane snapped.\n\nRoche could hear the speaker's obsequious tone; she imagined him bowing, but couldn't see to be sure. \"Master, I would accompany you to safety.\"\n\n\"That is not necessary. The _Apostle_ is only minutes away\u2014\"\n\n\"Allow me to serve you, Master.\"\n\n\"You have served me,\" he said. \"But now you must return to the others and tend to repairs.\"\n\n\"But our pilot\u2014\"\n\n\"Another ship shall be summoned,\" he said, his patience wearing thin. \"You will be rescued.\"\n\n\"Master\u2014\"\n\n\"I _command_ you to wait.\" The frost in Adoni Cane's voice could have cooled stars. \"Leave me now, or invoke my displeasure!\"\n\n\"Yes, Master.\" The owner of the voice didn't believe he would be rescued; that much was clear. Yet he obeyed. His footsteps slowly shuffled away, then were cut off by the closing of the airlock.\n\nThe baying of alarms ceased, and for a second all was silent.\n\n\"Fools.\" Cane's voice so close to her made her jump.\n\n\"Where... ?\" she tried, then: \"Why...?\"\n\n\"Don't talk.\" His strong hands strapped a harness around her broken body, tying her hands together in the process. Then a medical pack was pressed against her hip. \"I only have one of these, I'm afraid. I didn't think you'd be this bad.\"\n\nThe pack attached itself with a slight sting.\n\n_Why do you want me alive_? It was nothing more than a thought. She was unable to control her voice enough to do anything other than moan.\n\n\"That's it,\" he said, his tone almost encouraging. \"Keep fighting, Morgan, and you might even make it.\"\n\nShe shuddered, feeling a strange coldness in her mind. She wanted to succumb to the physical and mental exhaustion, wanted to sleep. But that was a luxury she couldn't afford just yet. For now she only had grief to distance her from the pain....\n\n* * *\n\nGradually, as the medical pack took effect, the pain began to ease. The sharp edges in the world softened. Blinking, she could make out flashing lights around her, blurred as if she were looking at them through rain over a pane of glass. Cane sat not far away, his back to her.\n\nInstruments chattered briefly; she felt a gentle nudge of acceleration. Then something clanged, and the acceleration became more insistent. She clutched the sides of her couch as the pressure mounted. It might have lasted only a minute or so, but seemed like an hour.\n\nAs the minutes ticked by, she found her vision clearing even more. Not very much at first, and only in her right eye, but she appreciated any improvement.\n\nShe was in an ordinary-looking cockpit, with Cane operating the pilot's station. All she could see was his scalp and the lights reflecting from it like multicolored stars in a chocolate sky.\n\n\"Where are you taking me?\" she asked eventually.\n\nHis chair swiveled to face her. He pointed to a display. In it she could make out, vaguely, a large Hum ship against a starry backdrop.\n\nDespair rippled through her. \"Am I going to die?\"\n\n\"Not yet,\" he said, returning to his console. \"That would be counterproductive.\"\n\n\"If you think I'm going to help you\u2014\"\n\n\"Conserve your strength, Morgan. You're going to need it.\"\n\nSomething in his voice made her look at him again. _Was_ he the Adoni Cane she knew or not? He looked and sounded exactly like him, apart from the coldness in his tone. But that was the whole point: the enemy was composed of clone warriors, many of them identical. He could very well be one of Cane's siblings with the same face, but with the killing instinct intact.\n\nRegardless of who he was, he was right about conserving her strength. She felt weak right down to her core, and the coldness was still in her mind. The pain was manageable now, thanks to the ministrations of the medical pack, but that meant she could look down and see how badly she had been injured. When she did, she instantly wished she hadn't.\n\nThe Hum ship grew larger in the display. Cane had mentioned something about \"the _Apostle_ \" being only minutes away. Presumably they were one and the same. Although she had never seen this particular ship before, the connection between it and the black-robed figures they had left behind seemed clear. Some sort of organization staffed and supplied by Hum backers had obviously assisted Page De Bruyn in hunting her down. Why, she didn't know, but the presence of a clone warrior high on the command chain seemed ominous. If this _was_ her Cane, how could he have infiltrated such a group so quickly?\n\nHer wrists chafed where he had tied her hands together. Her left hand in particular ached as though stiff from a half-healed wound. The back of her head felt like someone had hammered a nail into it, and the vision still hadn't cleared in her left eye. When she blinked, the socket itself even felt odd, unnatural\u2014\n\n_Empty._ The Box had said that her implants had been removed. The harsh reality of that fact was only now sinking in. Without them, she felt hollow, incomplete.\n\n\"Why did you rescue me?\" she asked.\n\nHe turned again to face her. \"If you can't answer that question, then perhaps I have wasted my time.\"\n\n_Him and the Box,_ she thought to herself. \"Maybe you have.\"\n\nHe shrugged. \"It might change nothing.\"\n\nShe winced as another wave of pain swept through her.\n\nHe came over to check her medical pack. \"I haven't come all this way to watch you die,\" he said dispassionately. \"You'll be treated properly when we arrive.\"\n\nHer words came through clenched teeth. \"How much longer?\"\n\n\"Not long. We're almost in range.\" He turned back to the display. \"It'll be over soon.\"\n\nHe adjusted something on the pack and warmth rushed through her. At first she resisted it, wanting to remain alert. Maybe all hope was not quite lost; if a chance came to escape, she had to be ready.\n\nBut then she remembered what she had seen when she had looked down at her body. There was nothing she could do. She closed her eyes and let the warmth caress her pain, blunt the icy coldness inside her.\n\nA moment later, it disappeared completely. In its absence, she felt strangely light, as though it had been tying her down. In its wake, she felt almost free....\n\nThat was crazy, she thought. She was half-dead, the captive of an unknown organization with links to the enemy. Not only had she no way of escaping, but she wouldn't even live much longer if they chose not to help her.\n\nCane cocked his head as though listening to something.\n\n\"That's close enough,\" he said, turning back to her. \"Morgan, I have someone who wishes to speak to you.\"\n\nRoche steeled herself for another grim surprise, glancing around the cabin to see if anyone else had entered.\n\nThen she heard the voice\u2014loud and clear in her thoughts.\n\n<It's so good to feel your mind again, Morgan,> said the reave. <We thought we'd lost you for good!>\n\n<Maii?> Roche tried to sit, but pain forced her back. <Maii, is that you?>\n\n\"Look at the screen,\" said Cane.\n\nThrough her one remaining eye, Roche watched as they passed through the fringes of a camouflage screen and the Hum ship became the _Ana Vereine._\n\n\"It really _is_ you?\" There was both uncertainty and relief in her whispered words.\n\nHe smiled. \"Does that surprise you?\"\n\n<We couldn't let you know until you passed out of range of Lemmas, their reave,> Maii said. Her voice hinted at dark truths Roche didn't want to explore. <I'm sorry we kept you hanging.>\n\n\"But how...?\"\n\nShe felt Cane's hand on her left shoulder, pressing gently but firmly. \"This can be discussed later, Morgan. Right now I want to dock and get us out of here before anyone back there suspects what has happened\u2014before their _real_ contact shows up.\"\n\nShe nodded weakly. Maii filled her mind with a radiant warmth. She felt as though she had been dipped in a bath of light, and the cold, dead touch of the Hum reave faded like ice in the sun. For the first time, Roche allowed herself the luxury of really _believing_ that she might live long enough to see her friends again.\n\nAnything beyond that could wait.\n\n* * *\n\nHaid and Vri met the scutter with a fully equipped stretcher. Barely had she been placed in its embrace than her treatment began. The autosurgeon dictated the list of her injuries all the way to the medical center: beginning with her shattered hip and pelvis, her punctured lung and blood loss, and working its way down to relatively minor muscle damage and gashes. It was still droning on when Haid cut off its output in order to let her rest.\n\nAt the same time, Kajic sent the ship accelerating back in-system, away from Roche's captors. She was conscious just long enough to learn that the _Phlegethon_ was still under intense attack, so was not considered a safe port. Kajic had plotted a relatively innocuous orbit instead, bypassing the major concentrations of fighting in the system and skimming close to the outer edge of the ring where traffic was light. The ship would travel under heavy camouflage and in a constant state of alert. If they _were_ spotted, they would be ready to defend themselves.\n\nHaid was sitting with her when the autosurgeon put her under, his black skin and artificial eyes gleaming in the medical center's bright lights.\n\n\"Don't worry, Morgan,\" he said, touching her arm lightly. \"We'll still be here when you come back.\"\n\n\"How long?\" The anesthetic was already beginning to work; her voice sounded like it was coming from kilometers away.\n\n\"As long as it takes, I guess.\"\n\n\"Two hours,\" she said. \"There's something... something I have to do.\"\n\nHaid glanced at the autosurgeon's holographic display. \"It'll take at least six to clean you up, not to mention fitting the new eye.\"\n\n\"Forget the eye.\" She could barely keep her remaining one open. \"Make it three, or so help me I'll\u2014I'll\u2014\"\n\n_\u2014send you back to Sciacca's World._\n\nShe never found out whether she finished the sentence.\n\n* * *\n\nWhen she woke, the pain was gone. That more than anything else convinced her that survival had been worthwhile.\n\nShe couldn't move, though. The autosurgeon had her carefully encased in a body cast that allowed the use of her right arm only. When she tried to sit up, it correctly interpreted her feeble movements and tilted the entire bed instead.\n\n\"It won't let you out of its clutches just yet,\" said Haid. He was sitting with his feet up on one of the other operating tables with his back to the holographic \"cybercorpses\" rotating slowly in one wall.\n\n\"You're still here?\" she asked. \"Haven't you anything better to do?\"\n\n\"It's not as if I've been sitting around idly waiting for you to wake up.\" He smiled at her warmly. \"You said three hours, and it's been exactly that. I just had to be here on time.\"\n\nShe smiled also, envying him his mobility and fitness\u2014even with his cybernetic mesh and patchwork limbs. \"How am I?\"\n\nHe swung his feet off the table, but didn't stand. \"Much better. Not one hundred percent by any means, but at least you look\"\u2014he shrugged\u2014\" _better._ \"\n\n\"Is there a mirror in here?\"\n\n\"No, but I'm sure Maii can arrange something.\"\n\nRoche felt the girl's featherlight touch in her mind, and full stereoscopic vision poured through her, from Haid's eyes. She saw a white-wrapped corpse half in and half out of a gleaming sarcophagus. One eye was covered with a patch. Her mouth was swollen; yellowing bruises spread down one cheek to her jaw. Her head had been shaved and half-covered with bandages.\n\n<It's a vast improvement,> said Maii.\n\n<Is it? Show me.>\n\n\"That's not a good idea,\" Haid warned.\n\n<It's therapeutic,> Roche insisted. <If this is better, how bad was I?>\n\nA flash of red passed before her secondhand eyes, but it didn't really register. The naked woman curled up in pain, the one arm nearly severed and vertebrae visible through wounds at the back of her neck, the messy crater on her right hip, the blood... surely this couldn't have been her?\n\n\"Enough,\" she said, swallowing. If the Box really was dead this time, at least she knew why. Nothing else could have kept her alive through such mistreatment. She tried shaking the image from her mind by changing the subject altogether.\n\n\"Where's Cane?\"\n\n\"Up in the observation blister,\" said Haid. \"He's been there since we got you back.\"\n\n\"I want to talk to him later.\" She couldn't help the tiredness in her voice. She was alive, yes, but there was still so much to do. \"First, tell me _how_ you got me back. How did you know where I was?\"\n\nHaid stood, frowning, and stepped up to her. \"I'm not sure I understand all of it myself, Morgan. We knew something had gone wrong almost immediately, when you didn't arrive at the _Ana Vereine_ and Maii couldn't find you. There'd been a disturbance in the docks below us, and security arrived just minutes too late. Automatic monitoring in the area had been shut down somehow during the ambush, so we never did get a good look at what was going on, and the ship they had you on had detached and hot-launched before anyone could work out it was involved. Things were pretty messy in the area because of the attack. It wasn't until we received a tightbeam squirt from the ship that we guessed.\"\n\n\"What did the message say?\"\n\n\"It was fairly short, telling us basically that you were aboard and injured and that a pulse would be sent every hour telling us where the ship was, but it didn't tell us who it was from. The Heresiarch picked it up and passed it on. I wanted to follow straightaway but Cane was adamant we shouldn't. Quite apart from getting through the siege around __ the _Phlegethon_ , he felt there was also the matter of the people who captured you to take into consideration. We couldn't afford to take the chance that they might kill you if we came in with guns blazing. So we kept track of the ship and thought of another way.\"\n\n\"By masquerading as one of them.\"\n\n\"Basically, yes.\"\n\n\"And who _are_ they?\"\n\nHe looked uncomfortable. \"To be honest, I don't know.\"\n\n\"So how did you know what to do?\"\n\n\"I didn't. It was all Cane's idea. He got us through the blockade and gave us the specifications of the ship we were to impersonate. When we caught up with the ship you were on, he gave us the codes to broadcast to convince them that we were who we said we were. And when we were in range he insisted that he should go aboard alone. He didn't tell us what he was going to do, just told us to trust him. I didn't know whether I should, but couldn't think of anything better to do. He seemed to know what he was doing, and if it got you back...\" Haid shrugged. \"It worked out in the end, I guess.\"\n\nShe was silent for a while, remembering Cane's tone, remembering how he had dealt with the groveling Hum. And she thought of the epsense link that possibly connected the clone warriors....\n\n\"He was different back there,\" she said. \"For a while there, it was almost as if he _was_ the enemy, you know? I think he was close to becoming one of them.\"\n\n\"That's what I was afraid of,\" Haid said, concern etched deeply in his face. \"I couldn't help think that if he went too far, he wouldn't come back to us.\" He shrugged again. \"I didn't want to lose him as well.\"\n\n\"I don't believe he was ever ours to begin with.\"\n\n\"You know what I mean,\" he said. \"We need him here.\"\n\n_Perhaps a little too much_ , she thought, but said nothing.\n\n\"Alta Ansourian is still with us, by the way,\" he went on. \"She refused to disembark when she had the chance. She's still in her quarters.\"\n\n\"Doing what?\" Roche asked.\n\n\"Staring at the wall as far as I can tell,\" he said. \"Cane has tried talking to her a couple of times, but to no avail. She just won't snap out of it.\"\n\n\"Give her time, Ameidio,\" she said. \"She just witnessed her father being murdered. It's going to take more than a few days to snap out of that.\"\n\nHe nodded wearily. \"Who knows?\" he said. \"Maybe she has the right idea. At least she doesn't have to worry about... everything.\"\n\nHe pulled his gaze away from hers; Roche realized he was embarrassed.\n\nShe reached out and took his hand lightly in hers. \"If it's any consolation, Ameidio,\" she said, \"I think this will all be over soon.\"\n\nHis hand squeezed hers back. \"Not soon enough for my liking.\" He forced a smile.\n\n\"Have we heard anything from the council?'\n\n\"Nothing yet. There's an ftl drone following us, though. We can call them when you're ready. If the fighting's done at their end, they might be willing to reconvene.\"\n\nIt felt like weeks had passed since the last meeting. \"How long was I gone?\"\n\n\"Just over thirty hours,\" he said. \"You still haven't told us what happened to you.\"\n\n\"I'm not sure I'm ready to.\" Her scalp itched, and although she wanted to scratch it, doing so would mean letting go of his hand. She wasn't ready to do that, either, even though she'd already held it longer than she'd intended to.\n\nAs though through a fog she saw Page De Bruyn's face as it had looked, lifeless, on the deck of the Hum ship. She still had no idea what her former superior had been doing to her, and why. If the Box had been around, she could have asked it, but this time it seemed to be irrevocably gone. Having lost it once before, she found it hard to believe that it wouldn't come back to her again\u2014but she could _feel_ its absence all through her body. It was gone forever.\n\n\"Hey,\" Haid said, letting go of her hand and wiping her cheek. \"I'll go and let you get some rest.\"\n\nShe took a deep breath. \"How about _you_ get some rest? I'll bet you haven't slept for two days. Besides, I want to talk to Uri. Then Cane. I need to sort this out now, before I convince myself it was all a bad dream.\"\n\n\"The surgeon says\u2014\"\n\n\"I don't care what it says, Ameidio,\" she cut in. \"It's keeping me comfortable enough in here, and I'm not planning on going anywhere for a while.\"\n\nHe nodded reluctantly. \"Okay, but you call if you need anything, all right?\"\n\nShe assured him she would, and watched as he turned and strode from the room.\n\nWhen he was gone, she turned her eyes to the ceiling and asked: \"Okay, Uri, what _does_ the autosurgeon say?\"\n\n\"That you are responding unexpectedly well to treatment.\" The voice of the ex-captain of the ship came from one side of the room, not all around as she'd expected. She glanced around to find that three \"cybercorpses\" had disappeared. In their place, Kajic's hologram reclined comfortably in a standard bridge chair, affecting a warm and slightly amused expression.\n\nHer bed rotated to face him.\n\n\"Your fractures have already knit,\" he went on, \"and all tissue grafts are proceeding ahead of schedule. Although the autosurgeon doesn't anticipate your returning to full mobility for at least two days, I wouldn't be surprised if you were out of the cast in eight hours or so and walking within the day.\"\n\n\"That seems unreasonably fast,\" she said.\n\n\"As I said, you are recovering quicker than expected. I've had a quick look to ascertain why and found some evidence of nanotech tampering here and there. It looks like you were being helped along. Not so much now, but certainly when you were first brought here.\"\n\nShe nodded slowly, not wanting to say anything in case it made him suspicious. \"I guess I was lucky.\"\n\nHe smiled then. \"It's okay, Morgan. I guessed the Box was still around after Perdue Habitat. You had too many lucky escapes that could not have occurred any other way. And since it wasn't anywhere on the ship, it had to be on you\u2014or inside you. It helped you escape from the destruction of the habitat, it sent the message when you were kidnapped, and it somehow kept you alive long enough to reach here. Am I right?\"\n\n\"Yes,\" she said. \"But it's dead, now.\"\n\n\"Are you sure?\"\n\n\"You said the evidence of nanotech had faded. That's the only way you would have picked it up\u2014and that's why it didn't want me examined back on the _Phlegethon._ A thorough search would've found signs of it for sure. Since you can't find it now, it must be gone.\"\n\n\"I'm sorry, Morgan,\" he said.\n\nShe brushed aside his sympathy, genuine or not. \"Don't be, Uri,\" she said. \"It lied and it manipulated me and I'm still not entirely sure what its hidden agenda was. Maybe in the long run I'm better off without it.\"\n\n\"Maybe.\" He paused for a moment, the light from his hologram flickering minutely. \"Was this what you wanted to talk to me about?\"\n\nRoche sighed. \"Uri, I need to make a decision,\" she said. \"One that could affect millions, maybe even trillions of people.\"\n\n\"Regarding the enemy?\"\n\n\"Yes.\" She cast about for a way to phrase her question, but in the end decided to be blunt. The chances were he would take it for a metaphor, anyway. Not even she could take the idea seriously yet.\n\n\"If you found a way to wipe them all out,\" she said slowly, \"would you do it?\"\n\n\"That depends,\" he said.\n\n\"On what?\"\n\n\"On why I was doing it, of course.\"\n\n\"Because if you didn't do it, there is every chance that Pristine Humanity could wind up extinct!\" She blurted it out, and, having done so, realized how ridiculous it sounded. She sighed again, this time in annoyance. \"There's only a few of them, Uri, but their method of turning us against each other might actually work.\"\n\n\"But why are they doing it, Morgan? Ask yourself that. They might have good reasons\u2014or think they have, anyway. Whoever created them may have felt justified in unleashing them against us.\"\n\n\"Justified half a million years ago, maybe\u2014but _now_? So much time has passed; Humanity has moved a long way since then. Surely we shouldn't be held responsible for the crimes of our ancestors? There must be another way for them to achieve retribution\u2014or whatever the hell it is they want.\"\n\n\"I agree. But if they're programmed to attack\u2014\"\n\n\"Exactly: they're _programmed._ There _is_ no other way, for them. But does that make it _right_?\"\n\n\"There is no right and wrong in war, Morgan. There is only expediency, efficiency, and capability\u2014all untainted by emotions or morals. Nearly all wars are won or lost without regard for Human values. As a result, the right side loses as often as the wrong. Only when the odds are stacked highly in favor of one side can such qualities be called into play. Mercy, after all, relies on the certainty that one party can kill another any time they wish. Without that certainty, mercy is meaningless. Only the most powerful can afford the luxury of forgiveness.\"\n\nShe half-smiled. \"Once again, you sound like my old Tactics lecturer.\"\n\nHe returned the smile, briefly. \"Ultimately, though, Morgan, all the theory in the world will only get you so far. In the end you reach a point where you have to decide for yourself. When you have to _act._ War is as much about instinct as it is about higher thought. Indeed, one could argue that if we thought _enough,_ there would be no war at all.\"\n\n\"Now what are you trying to say?\"\n\n\"That it's your choice, and I don't feel qualified to advise you. If what I think you're saying is true, and you do somehow have this capability, then I don't envy your position. I don't think I could make a decision like that. I'm too narrowly defined.\"\n\nShe frowned. \"I don't know what you mean.\"\n\n\"I mean that in some ways I'm like the enemy. I'm programmed to obey a small set of rules, inasmuch as a Human can be.\" His image shrugged. \"I don't remember my previous life. Maybe I was no different from who I am now, the person I became after the experiment. But all that I am, now, is here within this hull. All I really care about is the ship and the people who travel within it.\"\n\n\"Well,\" she said, \"it's nice to know we're in good hands.\"\n\nHe disregarded the compliment, his image staring over at her with a sober expression. \"Morgan, I would be just as happy to leave this system and never come back, since we would all be safer that way. But I know we can't do that, and never will be able to until the business with the enemy is sorted out I wouldn't be surprised if fighting has already started escalating outside Sol System. Soon, perhaps, if we don't do anything about it here and now\u2014nowhere will be safe.\"\n\n\"If I could only be certain that it did in fact boil down to a 'them or us' decision,\" she said. \"That would make it simpler. Or if there was some way we could negotiate, find some other solution, or...\"\n\nShe ran her hand across her face. Her skin was clammy, and she felt tired, but she didn't want to rest anymore. She wanted to push this through to the finish.\n\n\"Have you told anyone about the Box?\" she asked.\n\n\"No, of course not.\"\n\n\"Don't, then. Not that it matters anymore, I suppose.\"\n\nShe rested her head back on the bed, and the autosurgeon misinterpreted it as a request to lie flat. She didn't stop it lowering the bed, though. She just closed her eyes for a moment and put her forearm over them, to block out the glare from the ceiling light. Her mind felt full, heavy. There was too much to think about, too much to _do,_ and simply not enough of her to go around....\n\n* * *\n\nWhen she woke an hour later, it was on the crest of a soothing dream. She was a plant, absorbing nutrients and turning them into cells one by one, growing and stretching at a patient, steady rate. She existed; she was. Stripped of all fears, all concerns, she delighted in the simplicity of just _being...._\n\nThen the memory of the decision she had to make came rushing back, and she realized at once what was going on.\n\n<Maii?>\n\n<Yes, Morgan. It's me,> replied the girl, without the smallest trace of guilt. <You needed it.>\n\n<Did I?> she said, stretching her one free arm and raising her head. She could at least feel her other limbs now, under the cast <Actually, I _do_ feel better.>\n\n<That's the idea. Epsense therapy assists the biofeedback process better than any other known technique.>\n\n<Did you have anything to do with my healing before?>\n\n<No. This was the first time. It must have been the Box, as Uri said.>\n\nRoche rested her head back on the bed's cushioned support. <Can I ask you something, Maii?>\n\n<Of course, Morgan.>\n\n<When I was kidnapped, why couldn't you find me?>\n\n<I don't know.> The girl's reply was instantaneous and frank. <Even when I knew where to look, all I could pick up was a faint shadow. It was like you had retreated inward, hidden yourself somehow. I couldn't try too hard or else Lemmas would have noticed me, but I still should have detected _something_.>\n\nRoche thought of the white sphere that had enclosed her from the torturer-reave. <How's my image now?>\n\n<Rock-solid. More so than ever, in fact.>\n\n<I thought it might be. The Box told me that it was complex enough for you to read, if you'd known how to look. Maybe it was making my n-space signature a little odd. Interfering with it.>\n\n<Probably, I'd say, given the fact that you're almost back to normal now.>\n\n<Then that could be what the _irikeii_ meant when he referred to me as an 'enigma.'>\n\n<It's hard to say,> Maii responded <I never understood what he meant by that. But I do find it hard to believe that an AI could ever have an n-space image....>\n\nRoche was too tired to explain\u2014not that she was entirely sure about it herself. Part of her wished she could return to the peace of mind her dream had offered, even though she knew it had been false.\n\n<Seduced at last, eh?> Maii asked with a smile in her voice.\n\n<Don't get too cocky,> Roche returned, <It's not every day I need it that bad.>\n\n<True. But the first time is always the hardest...>\n\nRoche resisted the girl's soothing touch. <Sorry, Maii, but I need some time on my own to think right now. I'll call if I need you, okay?>\n\n<I'll be here, Morgan.>\n\nHer mind felt cold when the reave had gone, but she was glad for the privacy. There were things she needed to consider that the girl had probably already read from her mind but which she didn't want to dwell on with an audience watching. The Box had left her in a difficult situation which, if everything it had said was true, required a clarity of thought she found difficult to achieve even at the best of times.\n\nShe went to fold her hands across her chest but was stopped by the white plastic shell of the surgical cocoon enclosing her left arm. She rested her right hand on her neck instead, finding comfort and reassurance in the beating of her own pulse. She was still on the board.\n\n_You must decide, Morgan..._\n\nThe Box's words turned idly in her thoughts. It had said that the High Humans were reluctant to act against an inferior enemy, even if there was a possibility that the enemy might actually win against the mundanes they chose to attack. It had said that the Crescend, a noted Interventionist, was concerned about mundane affairs and would act against the enemy if allowed to by the rest of the High Caste. It had concluded by saying that the High Humans had agreed to abide by the decision of one mundane Human, thereby taking the moral dilemma out of their own hands. How they would do that, exactly, it hadn't said. Maybe the Crescend would modify others as he had modified her, to enable mundanes to locate the enemy within; or maybe he would act directly, using weapons more superior than she could imagine.\n\nHowever they did it, she felt safe in assuming that the enemy would be destroyed. The High Caste rarely acted, but when it did, it always got what it wanted.\n\nSo the decision was hers: to wipe out the enemy or not.\n\nShe was amazed by how difficult it was. On the one hand, she could end all the squabbling in Sol System and throughout the galaxy, at the cost of a Caste which might have a genuine grievance. On the other hand, she could let the conflict run its course. If the enemy won, so be it. She had no idea just how great a cost that would be to the trillions of other mundane Humans inhabiting the galaxy.\n\nIn terms of lives, it was relatively easy to judge. She didn't know exactly how many clone warriors there were, but they seemed to seed one or two per organization they were trying to infect. On the _Phlegethon_ there had been only five. There might be millions scattered across the galaxy\u2014but even those sorts of figures paled if the chaos they had caused in Sol System could be extrapolated everywhere. Millions of lives versus trillions: on any scale, the test was simple.\n\nBut of course it _wasn't_ that simple. If it was, she wouldn't have been Human.\n\nRestless, she tried to roll over, but of course she couldn't. The autosurgeon simply rolled the bed onto a disconcerting angle, then swung it back when she clutched at it in alarm.\n\n\"Uri! How long do I have to stay in this damned thing?\"\n\n\"An hour or two,\" came Kajic's voice. A second later, his form appeared in the holographic tank. \"Nerve reconstruction and bone marrow grafts have yet to be finalized in your injured pelvis. I'm told complete immobility is advisable.\"\n\nShe grunted. \"I'm not hungry or thirsty, either. Is that normal?\"\n\n\"Completely. The autosurgeon is taking all your bodily needs into account. Except for boredom, I suppose.\"\n\n\"I'm not bored.\" She sighed. \"I do need some input, though. It's hard to keep up without my implants. Can you give me a screen or something down here? I'd like to see what's happening in the system,\"\n\n\"Of course.\" Kajic's image dissolved, leaving a complicated display in its wake. \"I'll leave you with full access to the data we have on the current state of play. Some of it is coming from the IEPC drones; the rest I'm extrapolating as we go. I think it'll be enough to give you an overview, anyway. It is voice-activated.\"\n\n\"Thanks, Uri.\"\n\nShe settled back to browse through the charts. The system was a mess of conflicts, the largest concentrated around the blue triangle of the _Phlegethon._ As far as she could tell, five whole fleets had declared war against the massive vessel; it looked as if it was holding its own, but she couldn't tell how much longer that would last.\n\nElsewhere the situation was more difficult to analyze. Red patches, indicating conflict in one form or another, had spread to cover entire sections in the system. Most were concentrated around the plane of the ecliptic, but there were some hot spots farther out. One concentration of fire was high above the sun's north pole, where two swarms of comet-chasers appeared to be fighting over a third party's observation complex. Another was very close to the sun, whipping up strange currents in the chromosphere.\n\nMines were marked with yellow stars and tended to form drifting sheets like two-dimensional shoals around protected fleets. Regions of weakened space were marked with purple cross-hatching and avoided by all. Areas suspected to be under enemy control were delineated by sharp black lines. There were odd white patches in the rings that Roche couldn't identify but guessed were installations of some sort designed to interfere with passing ships. The gray crosses indicating derelict vessels were everywhere\u2014whether largely intact, smashed to fragments, or completely gaseous.\n\nShe could discern no pattern to the conflict, and the Box wasn't around to tell whether any one Caste was being consistently spared the fighting. All she could do was go by its earlier statement that no such bias was being shown. If the enemy did have an agenda, then she didn't know what it was.\n\nThat didn't _necessarily_ mean that the enemy's goal was to wipe every other mundane from the face of the galaxy. It was possible that the events in Sol System did not represent how things would go elsewhere. Yes, she had heard of civilizations that had fallen under the influence of the enemy\u2014Rey Nemeth's and Hue Vischilglin's were two\u2014but they were only a handful out of millions. It was barely conceivable that all would fall. Once the element of surprise was lost, as it surely would be after Sol System, the enemy would face a stiffer, more organized resistance.\n\nUnless that resistance was undermined from within. She had to assume that there was nowhere the enemy couldn't penetrate if it wanted to. The only way to keep them out was to conduct rigorous genetic testing\u2014which would never be rigorous enough, as the _Phlegethon_ had demonstrated\u2014or to rely on others like her to find the clone warriors before they could do more damage.\n\nIf that was what the Crescend had in mind, she might agree to support it. But she had no way of knowing until she actually spoke to him.\n\n\"Uri?\"\n\n\"Yes, Morgan?\"\n\n\"This is going to seem a little strange, but I need you to give me an open communications channel.\"\n\n\"Communicating with whom?\"\n\n\"I want to broadcast a message aimed at the sun. I don't want you to wait for any reply protocol or anything. If the only way you can do that is by radio or laser, that'll have to do.\"\n\n\"I can arrange that for you at any time, Morgan.\" Kajic didn't ask why she would want to, but the question was present in his tone. \"A tightbeam would be less likely to give away our location.\"\n\n\"Okay. Give me the open line; audio only. And...\" She paused slightly. \"I'd appreciate it if you didn't let anyone listen in.\"\n\n\"Of course, Morgan. Opening the line now.\"\n\nRoche didn't say anything at first. If the Crescend responded, that reinforced everything the Box had said; she still wasn't sure she wanted that. But if he didn't respond, that was just as bad. She would have no idea at all what was going on, then.\n\nThere was only one way to find out.\n\n\"Dawn comes,\" she said.\n\nA familiar voice responded immediately from behind her right ear:\n\n\"\u2014to bring an end to the dark imbalance.\"\n\nThe table spun beneath her as she twisted by reflex. \"Box? If that's really you, I'm going to\u2014\"\n\n\"I am not the Box, Morgan,\" interrupted the voice, this time coming from the other side of the room. \"You know who I am.\"\n\nHer skin goose-bumped. Who else could it be? For a few moments she didn't know what to say. \"But your voice...\" was all she could manage.\n\n\"The Box was a part of me in the same way that your eye is a part of you\u2014or your finger, or your anterior cingulate cortex. It was not me, and I am not it.\"\n\nRoche looked around her. The voice changed position constantly. She couldn't tell if that was an effect of the way it was being broadcast into the room, or whether it was something more significant. Maybe it was just trying to keep her _off_ balance.\n\n_He,_ she corrected herself. The Crescend wasn't an _it._\n\n\"Where are you?\" she asked.\n\n\"It has been a long time since last we met,\" he said with some amusement, deliberately avoiding her question.\n\n\"It hasn't been that long since Trinity.\"\n\n\"I'm not talking about Trinity.\"\n\nShe frowned. \"Then I don't know what you're talking about.\"\n\n\"Don't you? I thought you might have guessed by now.\"\n\n\"Guessed _what_?\"\n\nThe Crescend was silent for a long time, long enough for her to wonder if the line had been broken.\n\n\"Uri,\" she called, \"is this line still open?\"\n\n\"He can't hear you,\" said the Box's voice, this time coming from somewhere behind the autosurgeon. \"No one can hear us talk. It's just you and me. And your decision.\"\n\nShe shifted nervously in the cocoon. \"You're not seriously going to leave it up to me, are you?\"\n\n\"Why not?\"\n\n\"Because it's crazy, that's why not! I'm just one person. I'm not in any position to judge\u2014\"\n\n\"You're better qualified than you realize.\"\n\n\"Well, I don't _feel_ qualified.\"\n\n\"You can locate the enemy, for a start.\"\n\n\"How does that help my decision?\"\n\n\"It's proof that I can back up my offer to assist you. Without such proof, my offer could seem empty.\"\n\n\"And that's all you're offering? A way to find the enemy so we can finish them off?\"\n\n\"Is that the help you would like the High Caste to provide?\"\n\nShe opened her mouth, then shut it. \"You're not going to trick me into a decision like that.\"\n\n\"It's no trick. I'm genuinely interested in your answer: what do you want us to do, Morgan?\"\n\n\"I don't know,\" she said, feeling trapped. \"I don't feel I have enough information to decide.\"\n\n\"You have as much as you need.\"\n\n\"But I don't know why the enemy are here. I don't know where they come from, even. I don't know if they mean to kill everyone or just some of us. I don't know why the war began in the first place. I just don't _know_!\"\n\n\"Some of these questions have no answers,\" said the Crescend. \"At least for the moment. But I can answer that last one for you.\"\n\n\"Then please do,\" she said, annoyed that he had made her ask.\n\n\"The original war was fought over territory, as all wars are.\"\n\nShe waited, but nothing more was forthcoming. \"What territory?\"\n\nThe Crescend chuckled. \"Now you are trying to trick _me._ You are hoping that I will specify a location which will help you identify the enemy's original Caste. Unfortunately, I cannot do that, Morgan. And it's not because I won't, but because I _can't._ I don't know where the enemy originated\u2014not for certain. All I know is that the original war engulfed the entire inhabited galaxy.\"\n\n\"Ending when the enemy were ultimately defeated.\"\n\n\"They weren't just defeated,\" he said. \"They were completely erased. All trace of them vanished until forty years ago, when the first confirmed capsule sighting was made.\"\n\n\"Do you know the enemy's original name?\"\n\n\"Unfortunately, that information has also been lost.\"\n\nShe shook her head. \"You're not giving me much to go on. I mean, how can I judge what's right now when I don't even know if the enemy are justified in what they're doing? Maybe they're in the right, and I shouldn't interfere.\"\n\n\"Have you spoken to Adoni Cane about this?\"\n\n\"More or less.\" She had given him the _chance_ to speak, anyway.\n\n\"And hasn't the enemy had many chances in the past to reveal the truth behind their motives?\"\n\n\"Yes, I suppose\u2014\"\n\n\"So why should it fall on me to justify their actions when they themselves feel no need to do so?\"\n\nShe accepted his point. \"But _do_ you know why they're doing this?\"\n\n\"Facts are hard to come by in this matter. We suspect, that is all. If it were more clear-cut, the High Caste would find it easier to reach consensus. There was guilt on both sides, perhaps.\"\n\n\"So why are you so keen for me to let you destroy them?\"\n\nThe Crescend's voice was shocked. \"Do you really think that this is what I want, Morgan?\"\n\n\"Isn't it?\"\n\n\"Nothing could be further from the truth. I am not asking you to do anything but decide. I am an advocate of neither position.\"\n\n\"But you're an Interventionist,\" she said. \"That means you want to help us defeat the enemy.\"\n\n\"No. It means that I am _willing_ to help you, should you decide that way. There is a great difference. Should you decide not to accept my help, I will retire into the background once again. My role will be complete, and future events will play themselves out without my interference.\"\n\n\"Even if the enemy destroys us completely?\"\n\n\"Even so.\"\n\n\"That's a little harsh.\"\n\n\"Unfortunately, it is the nature of things. The nature of _nature,_ if you like. I cannot say if your annihilation at the hands of the enemy will or will not come to pass, or will necessarily be a bad thing if it does happen. Not because I am reluctant to tell you, or because it is too close to call, but because it is not my role to judge such issues. I am a facilitator, not an instigator. The people of my Caste who _do_ instigate have decided to pass their role to you. Perhaps, if the projected outcome _was_ more clear-cut, the High Caste would have divined its role differently. Perhaps there would be no situation such as this at all: the enemy would have been eradicated before they even left their capsules. You might take hope, if you like, in the fact that this didn't happen\u2014for it may mean that you have a chance. You might not. I am in no position to argue or suggest either way. All I want is for you to make a decision.\"\n\nHe paused now at length, then asked: \"Do you have one for me, Morgan Roche?\"\n\nRoche didn't know what to say. \"I'm not sure. I want to be _certain_ \u2014\"\n\n\"As do we all. There is no certainty to be had here, except on one thing: that if you decide to accept my help, you will prevail over the enemy, and they _will_ be destroyed.\"\n\n\"All of them?\" she asked.\n\n\" _All_ of them.\"\n\n\"Including the Cane I know?\"\n\n\"He is one of them, isn't he?\" the Crescend's voice chided her.\n\n\"Yes, but\u2014\"\n\n\"Would I be true to my word if I spared him?\" he said. \"One spared here might mean another elsewhere\u2014and another, and another. This compact must be sealed in the sure knowledge that your victory will be total, and the enemy's defeat complete.\"\n\nShe grimaced.\n\n\"You don't like that?\" he asked.\n\nShe was flustered for a moment by the fact that he could obviously see her. \"It disturbs me.\"\n\n\"Then turn down my offer and end this phase of the war,\" was all he said. \"I will abide by your decision.\"\n\n\"That's my only other option?\"\n\n\"Would you care to suggest another?\"\n\nShe took a deep breath. He was trying to trick her again. She wanted to scream that it wasn't fair, that she didn't deserve this, that she hadn't asked for it, that she didn't want it\u2014but all she said was:\n\n\"Why me? You still haven't explained that to me. Why not the council?\"\n\n\"The Interim Emergency Pristine Council is too large and unwieldy, and too exposed, for this purpose. It will play a role in organizing resistance or mopping up the damage\u2014depending on which way you decide, of course\u2014but it is not suited to the task before you. Like me, it is a facilitator, not an instigator.\"\n\n\"But why _me_ and not someone else?\"\n\n\"Because you alone are the one who must decide.\"\n\n\"But _why_ am I the one?\"\n\n\"Because that is your purpose.\"\n\n\"Stop avoiding the question! Tell me _why_ it has to be me!\"\n\nHe paused again. \"It is not something you will want to hear, Morgan.\"\n\nThe Crescend's voice was full of sympathy but she didn't accept it for a second. She didn't believe a creature that advanced would use language to communicate; any emotion that appeared in his voice therefore had to be artificially generated. Either way, she wasn't going to give him any leeway.\n\n\"How can you know that?\"\n\n\"I know you well enough to be certain of your feelings on this matter,\" he said. \"Besides, it would undoubtedly influence your decision. Since I have gone to great lengths to ensure your freedom, strange as that may seem, and to keep you as impartial as possible, I would not have those efforts wasted. Page De Bruyn came close enough to doing that already.\"\n\nRoche froze. \"What do you know about her?\"\n\n\"I know that she resented the fact that I was in cahoots with your Eupatrid regarding your mission with the Box, and your subsequent freedom. When she got in the way, she was dismissed and told to keep her nose out of the affair. I would not have anyone interfering in the process of your education. She misinterpreted what happened, believing that you were part of a conspiracy to undermine her power base, and embarked upon a personal campaign of revenge.\"\n\nRoche absorbed the news in silence for a moment; the Crescend _had_ been behind everything, after all. \"So that's why she was after me, then.\"\n\n\"She unwittingly put the entire project at risk. Your purpose was to learn about the enemy and about your own abilities in your own time. The Box was to guide you until you were informed enough to decide. That was its prime directive, beside studying Adoni Cane. Losing the Box has forced us to move sooner than many had anticipated\u2014but perhaps we would have reached this point now, anyway. Either way, we are here.\"\n\nShe nodded slowly, feeling oddly sorry for her old boss. De Bruyn had betrayed her, but had herself been the victim of forces beyond her control. She had been caught in the middle of the High Caste's convoluted plot. Like Roche herself.\n\n\"You know all you need to know,\" the Crescend said. \"Does it truly matter who created the enemy? The fact that they are here is all that should concern you. Does it matter if they plan to kill all of you or just half? All death is tragic. Does it matter if you are the only who can find them or one of millions? _This_ decision is yours alone. Decide, Morgan, and be done with it. I have waited forty years for this moment.\"\n\nRoche didn't say anything. Why the war had started, half a million years ago, _did_ indeed seem irrelevant. The same with who had made the enemy in the first place. It was too long ago. She had to concentrate on the situation before her. On the problem as she saw it.\n\nThat was what they wanted, she supposed.\n\n\"You're going to have to wait a little longer, I'm afraid,\" she said. \"I need to think on it some more.\"\n\nThe voice chuckled. \"You mean that you need to talk to Adoni Cane.\"\n\n\"He has a right to know what's going on,\" she said.\n\n\"And will you believe him, no matter what he says?\"\n\n\"I think I owe it to him to listen.\"\n\nThere was a sound much like a sigh. \"Very well, Morgan. I will continue to wait. Just remember one thing, though: your will is paramount in this instance. You must not let yourself be coerced by anyone\u2014not even me. _Especially_ me. You must decide as objectively as you can. That is all we ask of you.\"\n\nShe couldn't help a bitter laugh.\n\n\"Understand what I am trying to say, Morgan. Adoni Cane is one of the enemy. I would not lose all now to false sympathy or wishful thinking.\"\n\n\"You won't,\" she said. \"Because that's what this is all about, really.\"\n\n\"Yes, it is.\" The Crescend's voice softened slightly. \"Listen and think well, Morgan Roche.\"\n\nThe Crescend said nothing more, and she knew she was alone.\n****\n\n**19**\n\n****\n\n****\n\nIND Ana Vereine\n\n955.2.15\n\n1960\n\nRoche settled back onto the bed and ran her hand across her face. \"Uri?\" she called. \"Can you hear me?\"\n\n\"Of course, Morgan.\"\n\nShe was relieved to hear his voice. \"That's good,\" she said. \"I was afraid you wouldn't be able to.\"\n\n\"Why not?\" he said. \"I've been monitoring your room since your awakening.\"\n\nRoche frowned. \"But he said no one would hear us talk.\"\n\n\"Who said this?\"\n\nShe hesitated for a moment, confused. \"Didn't you hear me transmit my message?\"\n\n\"Yes, but there was no reply. I closed the line when it became apparent that you weren't going to say anything else.\"\n\n_No reply._ Part of her wasn't surprised. This was the Crescend she was dealing with, after all. Interfering with mundane technology\u2014and mundane minds\u2014probably came as easily as toying with an insect.\n\n\"Is everything all right, Morgan?\"\n\n\"Everything's fine, Uri,\" she said. \"Just a little tired, I guess. Listen, get hold of Cane for me, can you? I need to see him.\"\n\n\"Yes, Morgan.\"\n\n\"And make sure you monitor _this_ conversation,\" she added hastily. \"I want you to be ready if anything goes wrong.\"\n\n\"Understood.\"\n\nRoche went back to watching the display of Sol System while she waited. Nothing much had changed, except the patterns of white on the ring. Now that the ship was closer, she could see them in more detail. They shifted like oil on water, sweeping in swirls with ponderous grace across vast sections of dust and gas. Indicating them, Roche asked Kajic what they meant.\n\n\"Electrical activity,\" he said. \"Source unknown. This data comes direct from the Heresiarch's general navigation service. He believes them to be a hazard to shipping.\"\n\n\"A trap of some sort?\"\n\n\"Conceivably.\"\n\nShe wondered if it had anything to do with the convergence on the system. The clone warriors might have rigged some sort of trap involving the ring, although she couldn't imagine what kind of trap it could possibly be. A solar-system-sized laser would be better based in the sun itself, and there was no sign of the exotic types of matter usually associated with hyperspace weapons. It _might_ have been a natural phenomenon, but she couldn't afford to make such assumptions at this stage. It was too risky, and could end up being costly....\n\n\"Cane is on his way,\" Kajic announced.\n\n\"Okay, Uri.\" She breathed deeply and slowly for a full minute, composing herself for what was to come. She had no idea what she would say, and even less of an idea how he would respond to what she had to say. _I have the power to kill you and all of your people unless you give me a reason not to._ If she was lucky, he would think she was crazy.\n\nIf she was unlucky...\n\n\"Hello, Morgan.\" Cane stepped into the room.\n\n\"Thanks for coming, Cane.\" Her voice was edgy; she hadn't heard the door open. \"I wanted to\u2014\"\n\nShe stopped.\n\nHe stood before her with his arms folded, to all appearances completely at ease, and waited for her to continue.\n\n\"I wanted to thank you for rescuing me,\" she said, realizing only as she said it that it was the truth. \"You took a great risk, and it paid off. Thank you.\"\n\n\"I did what I had to do,\" he said. His eyes revealed nothing about his thoughts.\n\n\"How _did_ you know what to do, by the way?\" she asked.\n\n\"I didn't. I just took a chance,\" he said. \"The ship that took you passed through the blockade of the _Phlegethon_ without resistance, so it seemed likely that it had an allegiance with the clone warriors. Using fragments of the command language we detected in Palasian System, I was able to convince the leaders of the blockade to let us through too. Once they believed that I was in fact one of the enemy, the rest was simple.\"\n\nHe seemed to be telling the truth, but she still couldn't read him. \"What about the camouflage you ordered around the _Ana Vereine?\"_\n\nHe shrugged. \"You were taken by Hums, so I assumed that they would respect the authority of a superior Hum vessel. It wasn't difficult to retrieve the design of such a ship from the datapool.\"\n\n\"And the name of the ship? You called it the _Apostle_.\"\n\n\"I overheard the name when I arrived.\"\n\n\"What else did you overhear? Anything useful\u2014like who they were, for instance?\"\n\n\"They call themselves the Disciples of the Evergence,\" he said. He didn't wait for her to ask: \"I don't know what it means.\"\n\n\"But they are in league with the enemy.\"\n\n\"It seems so. They accepted me readily enough.\"\n\nShe nodded. His explanations made sense, even if they were a little glib. She suspected that no matter what she threw at him he would be able to explain it away.\n\n\"I'd wondered,\" she said, \"whether this might be proof that an epsense link of some sort does exist between you and your siblings. That way it would have been easy to know just what to say and who to imitate in order for my captors to be convinced.\"\n\n\"I'm sorry to disappoint you, Morgan,\" he said. \"But that wasn't the case.\"\n\nShe studied him closely for a few seconds \"Are you lying to me?\" she asked, as she had when they had first reached Sol System.\n\n\"Why would I lie?\"\n\n\"Why does anyone lie?\" she said. \"To conceal the truth, obviously.\"\n\n\"And if I were trying to conceal the truth, what would it gain me to admit that _now_?\"\n\n\" _Are_ you lying to me?\" she pressed him.\n\n\"What do you want to hear, Morgan? That I'm in communication with my siblings? That I'm in league with them?\"\n\n\"I just want you to answer the question! Why is this so difficult for everyone?\"\n\n\"If I am lying now, then I must have been lying all along!\"\n\nThe passion in his voice surprised her, but it didn't sway her from her own anger.\n\n\"Just answer the damned question, Cane!\" she snapped. __ \" _Are_ you lying to me?\"\n\nThere was a long pause in which he breathed deeply a few times, almost as if trying to calm himself. Seeing him so agitated was not a common experience; it was only the second time she had really seen him angry. It was a side of himself he kept carefully hidden.\n\n\"I'm not lying, Morgan,\" he said eventually, his even voice cutting across her thoughts. \"Everything I have told you is the truth.\"\n\nShe sighed wearily. She wanted his assurances to take away her doubts, but they were still there, lingering, continuing to eat away at her.\n\n\"I want to believe you,\" she said. \"But it's hard\u2014\"\n\n\"Why, Morgan?\" he broke in. \"Why is it so hard? How many times have I saved your life now? How many times more must I do it before you will believe me? How many more of my own must I kill?\"\n\nRoche was speechless for a second. For an instant, she remembered how he looked when he had rescued her\u2014cold and dangerous\u2014but that wasn't what she saw on his face now. He looked... _hurt._\n\n_I can't afford to trust you_ , she wanted to say, _because I don't have the courage to risk so much. Because if I'm wrong there's no limit to what I might lose._\n\nBut she didn't say that. She couldn't. It exposed her vulnerability, it cut too deeply to the core of her uncertainty. That's what it all boiled down to, after all: balancing the uncertainty of his trustworthiness against the damage he could do. If he was lying, if he was leading her into some sort of trap, if he was really one of the enemy and had been faking it all along\u2014and if she alone could destroy the enemy...\n\nInstead she said: \"You told me once that we shouldn't trust you completely because you yourself don't know what you might do. Without knowing who made you, and why, we can't guess how you will respond to every situation. There's a chance you might be compelled, some day, to act in a way that goes against or subverts your conscious intentions. Do you still feel that way?\"\n\nHis tension eased slightly. \"That is a fair point,\" he said. \"I guess I would still consider it were I in your shoes.\"\n\nShe decided to take a chance and ask him the question she really wanted an answer to: 'Tell me, Cane. If you _were_ in my shoes and you had the means to destroy the enemy at no cost to any other Human Caste, would you do it?\"\n\nHe looked thoughtful for a moment. \"It would be an attractive option.\"\n\n\"Even at the cost of millions of lives\u2014of the lives of the enemy?\"\n\n\"Yes, of course,\" he said. \"Because so many more would stand to be saved.\"\n\n\"Is that how you justify genocide?\"\n\n\"I'm not justifying anything, Morgan. I am merely answering your question.\"\n\nWas he? she wondered. \"But what if _one_ of the enemy overcame its makers' wishes and demonstrated that it could achieve redemption? Would you kill that one, too, to make absolutely certain that there would be no recurrence of war?\"\n\nAgain he hesitated. \"For total victory?\" he asked.\n\n\"For total victory.\"\n\nThis time he didn't hesitate: \"I would let that one fall with the others, yes.\"\n\nThe answer chilled her. Not because of its ruthlessness\u2014for she had guessed that this would be his answer\u2014but because of what it said about him and how he regarded himself.\n\nHe knew what she was hinting at. He wasn't stupid. He was one of the most frightening individuals she had ever met, simply because he was so much better at everything than she. He could outrun, outfight, and probably outthink her to degrees she didn't dare imagine\u2014and he had been watching her and the Box the whole time they had known each other. If he hadn't already guessed at the Box's connection to the Crescend, then she had surely given him enough clues to work it out.\n\nHe knew that she was talking about him, and he was telling her that if she had the power to do so, that she should use it and eliminate them all, including himself.\n\nOr was he outthinking her even now? Was he gambling that this display of selflessness would in fact convince her _not_ to wipe out the enemy?\n\nHis eyes stared calmly into hers, as if they were discussing ordinary politics rather than genocide. But what _would_ he do were she to choose to destroy him with the others?\n\nShe couldn't allow that question to influence her. The decision had to be made on its own merits. What happened afterward was an entirely different matter.\n\nIf he could turn against his programming, maybe others could too. She couldn't justify the extermination of an entire Caste if that possibility existed. No matter how superior they seemed to her, they wouldn't stand a chance against the High Humans. And if they _were_ superior to her, maybe they deserved a chance to prove it.\n\nThe galaxy had never been a peaceful place, and it probably never would be. If she sanctioned the destruction of the enemy, would that guarantee any sort of peace? Maybe for a while, as long as alliances lasted. But the enemy's influence would still be felt. Old grudges wouldn't go away, even if they had been inflamed to further a third party's ends. There would still be conflict and injustice.\n\nAnd letting the enemy survive didn't necessarily mean that everyone else would die. She clutched at that thought, even as she made her decision. It wasn't a case of millions versus trillions. It was a case of an end to the present hostilities versus its continuation. Who would be victor was not clear. The enemy was too greatly outnumbered to take its success for granted.\n\nOr so she hoped. She didn't like to think what it would mean if she was wrong.\n\n\"Morgan?\" Cane was still standing in front of her, waiting patiently for her to say something.\n\n\"I'm sorry,\" she said. \"I was distracted.\"\n\nHe nodded stiffly. \"Is there anything else you want to ask me?\"\n\n\"No, that's all,\" she said. \"Thanks.\"\n\nHe looked as though he was about to say something, then turned and left the room.\n\nWhen he had gone, she said: \"Uri? Open that communications channel again.\"\n\n\"Opening now, Morgan.\"\n\n\"Dawn comes,\" she said.\n\n\"Does this mean you've decided?\" asked the Crescend, his voice emanating instantly from somewhere near the door.\n\n\"Yes.\" But she was reluctant to say the words.\n\n\"And what _is_ your decision, Morgan?\"\n\nShe paused, not for effect, but to give herself one last chance to change her mind. She didn't.\n\n\"I can't allow you to destroy them,\" she said slowly, carefully. \"So I guess I decide to let them live. We'll fight them ourselves, and either win or lose on our own merits.\"\n\n\"You know what this means?\"\n\n\"Yes.\" She hoped so. \"War\u2014at best.\"\n\nHe seemed to pause a long while before speaking again. \"May I ask why you have chosen this?\"\n\n\"Because Cane doesn't deserve to die,\" she said. \"He's helped me too much. He's proven\u2014to me, at least\u2014that the clone warriors _can_ rise above their programming.\"\n\n\"And what if I said that Cane could be spared?\"\n\n\"That's not the point. Who's to say there isn't another like him out there who doesn't deserve to be killed? You?\"\n\n\"I could not make such assurances.\"\n\n\"Exactly. It wouldn't be right to take that chance.\"\n\n\"So you make this decision by weighing the certain harm to an individual against potential harm to the masses?\"\n\n\"I guess so.\" She took a deep breath. __ \" _Would_ you spare Cane if I asked you to?\"\n\n\"No, of course not,\" he said. \"Nor do I believe Cane would wish it.\"\n\nShe thought about this and nodded to herself. \"What happens now, then? Is there anything else I need to do?\"\n\n\"Your role is played out,\" he said. \"For the time being, anyway. I will leave you to communicate your decision to the rest of my Caste. All the information at our disposal will be disseminated simultaneously to concerned parties throughout the galaxy, including the IEPC. We will no longer try to hide this knowledge from the enemy. It will be crucial in the times to come.\"\n\n\"And what happens if I need to talk to you again?\"\n\n\"Why? Do you feel you might change your mind?\"\n\n\"No, I don't think so,\" she said, even though, now that she had made her decision, she was filled with a terrible sense of doubt.\n\n\"If you did, there is no guarantee that the High Caste would accept your change of heart,\" he said, his voice almost scolding.\n\n\"I understand.\" She wasn't sure if that option would have made her feel better, anyway. If things went badly with the enemy, she didn't want that decision hanging over her, to be made again and again.\n\nShe had given the enemy the freedom to fight; she had given her own kind the freedom to lose. When gods interfered in the affairs of mundane Humans, she doubted any decisions were easy. But that didn't make her feel any better.\n\nThe Crescend waited for a moment, as though to see if she would speak, then said: \"Perhaps we will talk again some day.\"\n\nThe thought didn't fill her with pleasure. \"Perhaps.\"\n\n\"Goodbye, Morgan Roche.\"\n\n* * *\n\nHow much time passed after the Crescend left, she didn't know. She wasn't really thinking at all. Making the decision had drained her, leaving her feeling strangely empty.\n\n\"You have a visitor, Morgan.\"\n\nRoche turned to see Kajic's hologram coalescing in the wall display.\n\n\"Who is it?\" she said wearily.\n\n\"Defender-of-Harmony Vri.\"\n\n\"What does he want?\"\n\n\"He hasn't said.\"\n\n<Can't you guess?> asked Maii.\n\nRoche sighed. <Can't you stall him?>\n\n<I\u2014> The girl stopped in mid-thought. <I'm sorry, Morgan. I didn't realize you were this tired. Is everything all right?>\n\n<I'm not sure.> Maii didn't seem to know what had happened. She too must have been unable to read what had happened with the Crescend.\n\n_< What_ happened?> asked the girl. <Did I miss something?>\n\n<I'll tell you later, Maii.>\n\nRoche tilted the bed closer to upright and braced herself. She wasn't doing anyone much good moping around.\n\n\"It's all right,\" she said aloud to Kajic. \"I'll have to deal with him eventually, I guess. Send him in, Uri.\"\n\nThe doors to the medical center opened and the tall soldier stepped through. His armor shone as always, golden and feathered like some Humanoid phoenix. He took four precise paces to the end of Roche's bed, where he stopped and bowed slightly from the neck.\n\n\"Thank you for seeing me,\" he said.\n\nThe bed tilted farther so they were closer to eye level. \"How can I help you, Vri?\"\n\n\"I wish to suggest another compromise.\"\n\nShe suppressed a sigh. \"What now?\"\n\n\"First, I want to say that I understand why it was not possible for negotiations to take place between you and the Agora on the _Phlegethon._ The circumstances at the time were not conducive to such discussions. So I do not blame you for this additional delay.\"\n\nShe was relieved about that. She'd half expected him to storm in, making demands and claiming she was deliberately stalling.\n\n\"Secondly?\" she prompted.\n\n\"I have become aware that your work here may ultimately benefit the Surin Caste as a whole,\" he went on. \"While I do not feel that Maii should play any role in this, I accept that she has made her decision\u2014and since that decision can be interpreted as one serving the interests of the Agora, I also accept that it is not my role to intervene.\"\n\nRoche watched him as he spoke. His broad, furred features were composed and thoughtful. He had obviously considered these words in great depth before coming to her with them.\n\n\"I'm glad about that, Vri,\" she said, a touch cautiously, wary of the sting in the tail. \"But what's your point?\"\n\n\"It is my hope,\" he said, \"that you will also see my superiors' point of view, which is that Maii has been separated from the culture that might arguably be best for her development. Before her childhood is over, it may be beneficial to expose her to aspects of our Caste that give us pride. There is no denying that she has been hurt by members of our Caste, but all the Agora desires is the opportunity to right that wrong.\n\n\"I ask, therefore, that when the matter of the enemy is resolved you will see fit to return with me to Essai, to discuss the matter of Maii's custodianship with the Agora, and that I might be allowed to serve formally as her bodyguard until then.\"\n\nHe bowed again, and stepped back a pace.\n\nFor a moment, Roche didn't quite know what to say. It looked like he was handing her a simple way to put the problem aside for the time being, but she couldn't help but wonder if there was more to it.\n\n<He means it,> said Maii. <He's trying to find a balance between his orders, which he must obey, and the situation he finds himself in. Although he knows our work is dangerous, he also knows, now, that I am an essential part of that work. Without access to a reave, you cannot find the enemy, and at the moment I'm the only reave you have. Although the Agora might prefer him to take me home by force, he feels that doing so might actually cause more harm. If you agree to this proposal, all his needs will be **** satisfied.>\n\n<What about you, though, Maii? How do you feel about it?>\n\n<That's kind of irrelevant at the moment, Morgan. I don't know if we'll _ever_ make it to Essai. But if the chance does arise, I might take it. I suppose I should at least see what the Agora has to offer. Maybe they really do have my best interests at heart.>\n\nRoche could accept that. <Naturally, I wouldn't try to stop you.>\n\n<I know you wouldn't, Morgan. And I'm glad that you would be coming with me\u2014if you agree, of course. I'd feel uncomfortable facing them alone.>\n\nVri shifted his feet. Roche snapped out of the mental conversation.\n\n\"I'm sorry,\" she said. \"I was just talking to Maii about it. We think the compromise is fair, so I'll agree to meet the Agora when the time is right. Until then, you can travel with us and help keep Maii safe. After all, that is something we are _both_ concerned about.\"\n\nThe Surin soldier bowed a third time, this time more formally. \"Thank you,\" he said. \"I believed that this was the decision you would reach. You have not disappointed me.\"\n\nHe turned and strode heavily out of the room. Roche smiled as he went. _You have not disappointed me._ That was probably as close to a compliment as she would ever get from him.\n\n<I think you're right,> said Maii. <He's as stiff as they come.>\n\n<May he never unbend,> said Roche.\n\n<Are you going to tell me now what happened before?>\n\n<I'm not sure I fully understand it myself, Maii.>\n\n<The fragments I detect in your mind paint a strange picture. The events themselves are hidden, but I can read where you have thought about them\u2014as though the knowledge is leaking out from behind a shield of some kind. But I've never seen one quite like this before. You were actually talking to the Crescend?>\n\n<Yes.>\n\n<I guess it shouldn't be a surprise that High Humans have access to epsense as well as everything else.>\n\n<Maybe what they have makes epsense look primitive.>\n\n<Maybe. It's something to think about, anyway....>\n\nRoche felt the reave withdraw slightly, and she tilted the bed back. Under the cocoon, she wriggled the fingers of her left hand and felt them clench. Whatever the autosurgeon had done to her in the last few hours, it had left her feeling almost Human again. Almost.\n\n\"How much longer now, Uri?\"\n\n\"Three hours or so, Morgan.\"\n\n\"Can I at least have my other hand back now? I can feel it moving under this thing, so I know it's working.\"\n\n\"I'll check.\" Kajic conversed silently with the autosurgeon, then said: \"On the condition that you don't exert yourself, it will allow free movement to be restored to that limb.\"\n\nEven before he had finished speaking, the cocoon slid back down her left side, retracting like a fluid to reveal her left shoulder, arm and upper chest. She appeared to be wrapped in close- sticking bandages made from a white, tissue-thin material she didn't recognize. Through it, she could make out the healing red wounds through which her implants had been removed. Her left palm and wrist were still stiff.\n\nAt some point, she would have to be refitted. Like Haid, she had grown accustomed to working intimately with machines. _Too_ intimately at times\u2014an issue she still hoped to take up with the Crescend, one day. When the war eased enough for her to take some time off\u2014the sooner the better\u2014she would commit herself to the care of the medical center again and get the upgrade she had once hoped to obtain from COE Intelligence.\n\nExactly when that would be, she didn't know. First she had to get back in touch with the council and work out strategies for the coming war; they also had to find out if her talent could be replicated, naturally or otherwise. That was assuming, of course, that the _Phlegethon_ survived its assault by the fleets of enemy-infected nations. Then there was working out what to do about Sol System itself: did the enemy actually have a reason for gathering here, or was it safe to go back home? If the latter, could the council coordinate the battle across the galaxy as a whole, or would its efforts need to be restricted to those areas considered the most important to save? And how would the governments of the Far Reaches\u2014often overlooked by core-based interests, and the source of an ages-old resentment\u2014react to that decision?\n\nThe future was full of uncertainties, as the Crescend had promised it would be if she chose this particular path. Roche didn't regret her decision, but she did wish it could have been otherwise.\n\n_Your role is played out,_ the Crescend had said.\n\nAs far as she could tell, her job was only just beginning.\n****\n\n**20**\n\n****\n\n****\n\nIND Ana Vereine\n\n955.2.16\n\n0290\n\n\"Morgan?\"\n\nShe snapped out of her thoughts. \"What is it, Uri?\"\n\n\"I'm picking up a ship on an approach vector,\" he said. \"Its configuration matches that of the Hum ship we impersonated to rescue you.\"\n\n\"Is it hailing us?\"\n\n\"No.\"\n\n\"And we're still camouflaged?\"\n\n\"Yes.\"\n\n\"Then how does it know who we are?\"\n\n\"Maybe it doesn't,\" Kajic said. \"Maybe its similarity to the other ship is only a coincidence. We may have just blundered into Hum territory.\"\n\n\"And you believe that?\"\n\n\"Not for a moment,\" he said.\n\nShe vacillated briefly. \"Okay, broadcast an anonymous query for ID. Ask Cane and Ameidio to be on the bridge, if they aren't already there. And give me that screen again. I want to see what's going on.\"\n\nHer bed turned back to the holographic display. In it, she saw the relatively empty space around the _Ana Vereine_ , plus the colorful overviews of the system. The approaching ship appeared as a red triangle, swooping closer along a gently curving trajectory.\n\n\"I have broadcast the query,\" said Kajic. \"If they're going to reply, the earliest we'd hear is in a minute or two.\"\n\nRoche rubbed her chin with the fingers of her newly healed left hand. \"If we had to fight, how long could we last?\"\n\n\"Without the Box? Not long at all,\" he said. \"I'd prefer to avoid conflict entirely.\"\n\n\"I agree, but if we _don't_ have a choice, I'd like to know what our options are.\"\n\n\"It depends if they have independent fighters or not. Against more than three or four, I will be hard-pressed to maintain much of a defense.\"\n\n\"A minute?\"\n\n\"Maybe two,\" he said. \"At most.\"\n\n\"They may not even use fighters,\" Roche mused. \"Depends what they want, I guess.\"\n\n\"True.\" Kajic's voice was cautious. \"No response as yet.\"\n\n\"How long until they're within combat range?\"\n\n\"One hour.\"\n\n\"Give them ten minutes to respond, then put us on alert. At fifteen we'll broadcast a warning. If we still haven't heard back by thirty minutes, we'll assume they're hostile and change course. _If_ they follow, we'll take further evasive action. And if _that_ doesn't deter them...\" She shrugged. \"Then we fight, I guess.\"\n\nShe watched the screen. The ship didn't seem to be changing course of its own accord, no matter how much she might wish it would.\n\n\"Where's Vri?\" she asked.\n\n\"Back in his fighter. I've advised him of the situation. He's battle-ready, should we require him.\"\n\n\"And Maii?\"\n\n<I'm in my quarters, Morgan.>\n\n\"Perhaps you'd better head to the bridge too.\" Not that it made a great deal of difference where the reave actually was, Roche reminded herself; just as long as their minds were in contact. Still, she felt better knowing where the girl was. \"Is everyone else on deck?\"\n\n\"Cane has arrived,\" said Kajic, \"but Haid is\u2014\"\n\n\"Right here,\" said the ex-mercenary as the doors hissed open. He was dressed in a mirror-finished Dato combat suit and trailing another that echoed his movements perfectly. \"I figured you'd probably prefer being where the action was.\"\n\nRoche smiled. \"You figured right.\"\n\nReturning the smile, he tapped a series of codes into the autosurgeon's manual console. It protested with a series of alarms and warnings, but capitulated under the weight of Haid's overrides. The cocoon enclosing Roche's half-healed body clicked, then hissed, then began to recede back into the bed.\n\nRoche looked down in amazement as the rest of her body appeared, blotched red in places and wrapped in white like a barely formed chrysalis. Various sensors and drips retracted into the bed like worms diving away from sunlight. Her legs still felt numb. She moved one tentatively. It responded, but she wasn't confident of its holding her weight.\n\n\"Are you sure this will be okay?\" she asked.\n\n\"Uri assures me you'll manage well enough, once you're in the suit.\"\n\n\"But I don't have any implants.\"\n\n\"We've thought of that. What the suit can't work out, I'll do for you.\" Haid swung her legs off the bed, then manually walked the spare armor into position with its back facing her. The ceramic shell split and cracked open at the touch of a pressure pad. Its interior was black and moist-looking.\n\nShe couldn't stand on her own. A twinge of pain shot up her left side as soon as she tried to take her full weight. Haid instructed the suit to go down on its knees; then he picked her up and put her inside. The suit's padding cradled her, allowing her to rest as though she were sitting, with the merest twitch of her legs magnified to become steps.\n\nShe paced the room, enjoying the newly found freedom of movement. Control was limited to a primitive electrode net draped across her shoulders. Luckily the autosurgeon's repairs of her damaged nervous system had progressed far enough for the device to work. It was years since she had trained to use one, and manual control wasn't an option.\n\n\"Ameidio, you're a genius!\"\n\n\"Realize that I'm only doing this out of self-interest,\" he said lightly. \"Trying to coordinate things with you stuck down here would have been just too damned awkward. So, if you will...\"\n\nHe indicated the door.\n\nShe didn't move. \"Since when are you giving the orders around here?\"\n\nAgain, he smiled. \"It's good to have you back, Morgan.\"\n\nTogether they clanked heavily from the room and headed for the bridge.\n\n* * *\n\nThe ship's complement was complete by the time they arrived. As the ten-minute deadline came and went, a siren echoed through the ship, announcing full alert.\n\n\"What was that for?\" asked Haid, assuming his position by Cane at the weapons board.\n\n\"Don't forget our passenger,\" Kajic chided him. \"Alta's still down below.\"\n\nRoche settled into the spot where Kajic's second-in-command had once sat. \"Any change in the situation?\"\n\n\"The ship is proceeding as per its expected course, decelerating with a constant delta-v. It has neither responded to my hail nor issued one of its own.\"\n\n\"Does it look like it's going to attack?\"\n\n\"Apart from heading our way, it doesn't seem to be doing anything at all.\"\n\n\"Maii?\"\n\n<I'm getting nothing, Morgan. Their shields are tight.>\n\nShe turned. \"Any thoughts, Cane?\"\n\n\"If what we suspect is true,\" he said, \"then I think we can expect at least one of my siblings to be aboard this ship. And they will be much more difficult to deceive than the Disciples who captured you.\"\n\n\"Do you think they'd respond if we hailed them in your command language?\"\n\n\"They might, but that would only confirm their suspicions that we are the people they seek,\" he said. \"We may yet be able to bluff our way out of this, though.\"\n\n\"I'd just like to know how they found us,\" said Haid. \"Something's not right about all of this. I can feel it.\"\n\n\"I agree.\" Roche looked at the main screen, glad to have access to its greater area and clarity. The one in the medical center had been barely adequate. \"Any idea where this ship came from, Uri?\"\n\n\"It appears to have altered its course from this orbit here.\" A red ellipse circled the sun. \"How long it followed that orbit, however, I can't tell.\"\n\nThe orbit didn't seem to intersect any hot-spots or suspicious-looking regions.\n\n\"The Disciples must have contacted them somehow,\" said Roche. \"They must have traced our course.\"\n\n\"Sounds reasonable,\" said Cane.\n\n\"So what do we do?\" asked Haid.\n\n\"We wait them out,\" said Roche. \"They might be bluffing. If they intended to destroy us, they would've come in faster or slow-jumped right on top of us. They must want something else.\" She studied the creeping dot on the screen. \"How long now, Uri?\"\n\n\"Forty minutes until we change course.\"\n\n\"That gives us a little breathing space, anyway. In the meantime, I want to get in touch with the council, if we can. Uri, is that drone still following us?\"\n\n\"Yes.\"\n\n\"Good. Send a tightbeam message requesting a conference with whoever's in charge at the moment. Tell them I have something they need to know.\"\n\n\"I'll try.\" Kajic's hologram faded into static.\n\n\"Why are we wasting our time talking to them?\" Haid asked. \"How many times do they have to knock us back before you take the hint, Morgan?\"\n\n\"They still need us,\" she said, adding: \"And unfortunately we still need them. Besides, unless they've found someone else who can locate the enemy like I can, I'm pretty sure they'll be prepared to talk.\"\n\n\"Yes,\" said Haid, \"but will they listen?\"\n\nShe didn't have long to wait. Barely five minutes passed before Kajic announced the receipt of a reply from the council.\n\n\"That was fast,\" she said. \"Do we have a live feed?\"\n\n\"Connecting as we speak. The signal is heavily encrypted and therefore low on detail but at least delay-free.\"\n\n\"Put it through.\"\n\nA window opened on the main screen, revealing a grainy black-and-white image.\n\n\"Well, if it isn't Morgan Roche,\" said Salton Trezise. \"How nice of you to get in touch. Rumor had it you were dead.\"\n\n\"As much as I'd love to exchange pleasantries with you, Trezise, I don't have the time. I need to speak with Murnane.\"\n\n\"He's no longer on the council, I'm afraid.\" Trezise's expression was smug. \"Both he and Nemeth have been censured, along with their Exotic friends. The council doesn't take kindly\u2014\"\n\nShe cut him short. \"So who's in charge?\"\n\n\"Me, for the time being.\" He smiled broadly.\n\nHer first instinct was to defend Murnane. Nemeth, she was sure, had been involved in all manner of underhand deals, but she doubted the older councilor had ever acted improperly. He had the change-resistant, inflexible air of someone who was not easily diverted from the straight and narrow.\n\nBut this wasn't the time or the place to get involved with petty politics. There was too much else at stake right now.\n\n\"If you're really in charge,\" she said, \"then there's something you need to know.\"\n\n\"I hope it involves the imminent arrival of reinforcements,\" he said. \"I've just been told that another fleet is jockeying for firing rights\u2014\"\n\n\"I'm sorry, Trezise, but there won't be _any_ reinforcements.\" She didn't like to say it. He looked genuinely harried under the self-satisfied exterior. \"I've been in contact with the Crescend. You can forget about the High Caste. They're not going to get involved.\"\n\nHis face dropped. \"Why not?\"\n\nShe hesitated momentarily, not wanting to tell Trezise that the reason the Crescend wasn't getting involved was because she had specifically asked him not to.\n\n\"Because it's not their fight. Would we intervene in a squabble between two Low Caste tribes on a mud planet somewhere? That's how this looks to them. Yes, one tribe might get wiped out, and things will certainly be messy for a while afterward, but that's just the way it goes. It's no big deal.\"\n\n\"Tell _that_ to the tribe being wiped out,\" he said.\n\nShe was trying to defend the Crescend, but found it difficult to keep the bitterness out of her voice. \"They will provide us with all the data they've collected over the years, but that's all you can rely on.\"\n\n\"Well, it's something, I guess.\" He looked thoughtful for a moment. \"Tell me, how exactly did you get in contact with the Crescend?\"\n\n\"I haven't got time for explanations right now,\" she said, certain the Crescend wouldn't talk to Trezise even if she told him how. \"I have a situation here I need to deal with. When things have settled down at both your end and mine, I'll be in touch again to work out what to do next.\"\n\n\"This situation\u2014it wouldn't have anything to do with the ring, would it?\"\n\n\"No. Why?\"\n\n\"The drone receiving your broadcast is on its outer fringes, and activity is rising in that area. I just thought there might be a connection.\"\n\n\"Not that I'm aware of,\" she said. She was about to close the line when she added on impulse: \"Oh, and by the way, Trezise. I ran into one of our former colleagues the other day.\"\n\n\"Really?\"\n\nRoche nodded. \"Page De Bruyn.\"\n\nHe raised both eyebrows. \"What's she doing here?\"\n\n\"Not much anymore,\" said Roche. \"She's dead.\"\n\nSomething passed behind the man's eyes, but otherwise he didn't react. \"That is unfortunate,\" he said. \"Did she happen to\u2014?\"\n\nThe screen went black.\n\nRoche waited a moment, but the image didn't return.\n\n\"Uri, what happened?\"\n\nThere was no reply.\n\n\"Uri?\"\n\nHaid looked up from his station. \"I'm getting a damage report from section gold-two.\"\n\n\"Which is what?\"\n\n\"Uri's maintenance support and information management,\" he said. \"Seems a cable has malfunctioned.\"\n\n\"What sort of cable?\"\n\n\"One linking his higher functions to the rest of the ship, from what I can tell.\"\n\n\"He's cut off?\"\n\n\"For the moment. Repair agents are moving in now.\"\n\nShe forced herself to quell a twinge of alarm. \"Good, because we're going to need him soon to coordinate the course change.\"\n\nHaid was about to respond when the floor shook slightly and a low rumble passed through the bulkheads. The lights flickered.\n\nRoche was instantly on her feet. \" _Now_ what's going on?\"\n\n\"I have red lights all through sections gold-one and gold-two,\" Haid said, tapping frantically at his board. \"Security is down across that level.\" He looked over to Roche, the concern evident in his expression. \"I think there's been an explosion, Morgan.\"\n\nRoche met his gaze silently for a few seconds until she was able to ask the question: \"What about Uri?\"\n\n\"I can't tell.\"\n\nShe checked with the reave \"Maii?\"\n\n<I'm not picking him up at all,> the girl replied.\n\nRoche's sense of alarm became one of rising dread. \"I'm going down there.\"\n\nHaid also rose to his feet. \"I'm coming with you.\"\n\n\"If anything's happened to him...\" She hurried for the exit to the bridge. \"You have the bridge, Cane.\"\n\nHaid followed with heavy, urgent footsteps. \"It _can't_ be an attack,\" he said. \"Our disrupters haven't registered the use of hyperspace weapons. There's no way they could've got something in here without us knowing.\"\n\n\"That's as may be,\" she said. \"But I have a gut feeling that this was no accident.\"\n\nShe forced herself to hurry, ignoring the pain in her hip every time her left leg took a loping stride. Section gold-one lay midway between the officers' decks and the warren, in the middle of the ship's main inhabited nacelle. It wasn't far from the bridge, but it seemed to take forever.\n\n\"You don't think it might have been Vri?\" Haid asked.\n\nShe was trying not to think at all, so she didn't answer him.\n\nPressure-doors had come down around sections gold-one and gold-two. Roche and Haid sealed their suits and keyed in the appropriate overrides. The heavy door before them slid aside, setting free a cloud of smoke.\n\nRoche stepped cautiously through it. The walls bulged inward in places, reminding her that this area was close to the ship's main life-support vats. For a moment, she wondered if the attack might have been directed there, and not directly at Kajic.\n\nBut then they rounded a corner and found what remained of the four rooms that had contained his body and the equipment required to support it.\n\nThe external door was off its hinges. The walls were blackened and twisted. Roche passed through the outer chambers, crunching over piles of twisted wreckage to the control room. Nothing had survived intact. The window over the main console had shattered into a million pieces. Through the hole in the wall where it had once hung, Roche saw the smashed air-conditioning units, the melted fiber-optic cables, the cracked tank...\n\nThe tank itself was empty of anything but ash. Its organic contents had either boiled or burned away. \"Look at this,\" said Haid from behind her. She turned mutely.\n\nHe was at the entrance to the rooms, holding up a section of the outer door. One gloved finger traced the smooth, curved edge of the fragment.\n\n\"It's been cut through,\" he said.\n\nRoche had seen enough. Not trusting her voice, she called Maii by epsense.\n\n<Maii? Is Cane still with you?>\n\n<Why? What's happening, Morgan?>\n\n<Uri's dead,> she replied bluntly. <Murdered.>\n\nShe felt the Surin girl's grief immediately but didn't indulge it.\n\n<Maii, _is_ Cane still with you?>\n\n<Yes,> she said. <He's trying to patch into the security systems.>\n\n\"It wasn't him,\" said Haid, overhearing their thoughts. \"Someone had to cut through the door while Uri's defenses were down, when the cable was severed.\"\n\n<Where's Vri?> Roche asked Maii.\n\n<He's still in his ship.>\n\nRoche shook her head in frustration. Then, like an energy bolt, realization hit her.\n\n\"Alta!\" she exclaimed.\n\n\"What? That's not possible,\" said Haid. \"She doesn't know the ship well enough.\"\n\n\"Who else _could_ it be?\" snapped Roche, <I've got her, Morgan,> said Maii. <She's heading few the bridge.>\n\n\"We're on our way,\" said Roche, turning to run back the way she had come. \"Can you find out for sure if it _was_ her?\"\n\n<I'm trying now,> said Maii. <Her shields are tight\u2014>\n\n\"Don't be gentle.\" Roche turned grief and apprehension into powerful nervous impulses that sent the suit hurrying through the ship. \"Tell Cane!\"\n\n<He knows,> said Maii. <She's close; he can hear her.>\n\nRoche urged her suit faster, even though at fall stretch she knew they wouldn't arrive in time. \"Just be careful!\"\n\n<I'm going to try to knock her out.> A wave of secondhand mental force rolled through Roche. <Her shields are\u2014no, _wait_!>\n\nA terrifying flash of fear caught Roche in mid-step, making her stumble. The suit fell heavily as a second explosion tore through the ship. This time it was much closer. The floor bucked as she landed on it, sending her flying into the air. For a giddy moment she was in free-fall; then she hit a wall with a solid crunch and skidded to a halt.\n\nHer vision grayed for a moment.\n\n\"Ameidio... ?\"\n\nA silver-gray figure loomed out of the suddenly billowing smoke, hand extended. Haid helped her upright. \"That came from up ahead,\" he said.\n\n\"This can't be happening!\" She was moving even as she spoke.\n\nBlast doors had come down across the entrance to the bridge. Roche keyed an override and stepped back as a wave of oily, black smoke exploded out of the entrance.\n\n<Maii?>\n\nThe reave was silent. She had said nothing at all since that last, terrible thought. As Roche stepped cautiously into the slowly fading smoke, all she could think of was the fear she had felt in the girl's mind.\n\nThe air around them, as measured by the suit's sensors, was hot\u2014much too hot to sustain life, even if the oxygen in the room hadn't been consumed by the fire. The crew stations were half- melted and spattered with fire-retardant foam. The walls had buckled, the main screen imploded.\n\n\"Pressure-mine,\" said Haid dully. \"The same used on Kajic, but not as effective in an open space.\"\n\n\"Effective enough,\" Roche muttered numbly.\n\nHaid stared helplessly at the blackened wreckage.\n\n\"Help me look,\" Roche said, refusing to give up hope.\n\n\"Morgan, they couldn't have survived...\"\n\nRoche ignored him, using the powerful limbs of the suit to sweep debris aside. The explosion had blown piles into corners or burned the tougher fixtures where they stood. Haid stepped over to her as she attempted to wrench a large sheet away from where it had stuck to the wall.\n\n\"I just want to make sure,\" she said distractedly. \"She might have been taken hostage\u2014or Cane got her out in time\u2014or\u2014\"\n\nShe fell silent with a large piece of wreckage raised in one hand, staring at what she'd uncovered\u2014at the evidence that dashed her last, desperate hope.\n\nHaid stepped up beside her. \"Is that... _her_?\"\n\n\"Yes.\" The face and head of the body was scorched down to the bone where it showed above the neck of the blackened suit, but there could be no doubt. The body was so small, curled in a fetal position with one arm outstretched.\n\nBlackness rose up in Roche like bile. She thought she could smell burning, even though she knew that was impossible through the suit. Her gaze fixed upon the open hand of Maii's outstretched arm, as if it were reaching for help.\n\nAn overwhelming guilt washed over Roche for not having been there...\n\n\"She's going to pay for this,\" Roche whispered.\n\n\"Maybe Cane's here too\u2014\" Haid began.\n\n\"I don't care.\" Roche heard her own voice as though listening to another person from a great distance. \"I just want Alta Ansourian.\"\n\n\"Morgan...\" Haid's hand had come to rest on her shoulder.\n\n\"We have to find her,\" said Roche without looking up, resisting his efforts to turn her away from the sight of Maii's body.\n\n\"But how? With Uri down and the bridge wrecked we've no way of tracing her.\"\n\nShe looked up and faced him. \"The suit beacons.\"\n\n\"You think she might be wearing one?\"\n\n\"If she's using Dato mines, why not a suit as well?\"\n\nThe shoulders of his own suit jerked. \"It's possible, I guess.\"\n\nShe clenched her left fist so tight, she imagined the ceramic finger joints buckling. \"I don't have any implants,\" she said, keeping her anger in check. \"So I can't operate my suit properly. You'll have to do it for me.\"\n\nHe slaved her instruments to his and searched for beacons anywhere in the ship. There was one, and it was moving toward the scutter docks.\n\n\"She could be trying to escape,\" said Haid.\n\nRoche nodded. \"But we can catch her if we hurry.\" She reached for a weapon, and realized only then that she was unarmed. Haid was in a similar position.\n\nHe raised his arms in a helpless gesture. \"I wasn't anticipating combat within the ship,\" he explained.\n\n\"And we don't have time to go to the armory,\" Roche said. \"We'll just have to manage without.\"\n\n\"Maybe we should split up,\" Haid suggested. 'Tackle her from two directions at once\u2014\"\n\n\"No,\" Roche said. \"We stay together. But we won't use the suit intercoms; otherwise she'll be able to hear us. Just stick to the suit speakers.\"\n\nHaid nodded in agreement.\n\n\"What about Cane?\" he said, the audio system of his suit clearly audible through hers. \"Do you think he's dead also?\"\n\n\"Not a chance,\" said Roche. Turning her back completely on Maii's body, she stepped past Haid.\n\n\"Morgan?\" he called after her. \"What happens when we find her?\"\n\n\"She dies,\" she said, leading the way from the bridge. \"Beyond that, I don't really care.\"\n\n* * *\n\nThe internal transit tubes were inactive, killed along with Kajic, so they were forced to run again. The light brown corridors seemed too bright as they headed in a different direction this time, down through connecting corridors and access tubes toward the ship's fat central drive section. There the scutters and other smaller support vehicles were docked or stored between outings.\n\nHalfway there, a growing suspicion about what the woman had in mind became a certainty as the first of a new wave of explosions rocked the ship.\n\n\"What the hell is she doing?\" Haid gasped. The data available through his instruments confirmed that at least one of the five remaining scutters had been damaged.\n\nRoche watched\u2014beyond horror, beyond surprise\u2014as a chain of detonations ripped the docks to pieces. \"This didn't just happen overnight,\" she said. \"Those mines must have been placed and armed in advance.\"\n\n\"We gave her the freedom to roam anywhere in the ship,\" Haid said. \"We let her do whatever she wanted. I didn't _suspect_ \u2014\"\n\n\"You had no reason to,\" Roche said. \"How could we have known she'd do something like this?\"\n\nThe explosions continued, damaging more of the scutters. Roche couldn't tell how many craft had been crippled or destroyed, but she knew the docks were ruined. The chain reaction would have left the place a raging inferno. If Alta herself hadn't got out in time, she wouldn't last a minute.\n\nRoche found it difficult to muster any sympathy for the woman\u2014especially given that Vri had been down there, too.\n\nBut the transponder kept moving. Roche tried to think one step ahead of the woman. No matter how hard they ran, the suits were identical; they would never catch her by dogging her heels. They had to try to cut her off.\n\nThe docks were in an outer layer of the drive section. The next obvious place to hit was two levels down.\n\n\"She's headed for the drive chamber,\" Roche said. \"She's probably going to try and blow that, too.\"\n\n\"Can she do that?\" asked Haid.\n\n\"She obviously thinks she can, otherwise she wouldn't be heading there.\" Roche considered her options while trying to maintain a steady pace, but the sheer size of the _Ana Vereine\u2014_ previously an asset\u2014was now proving to be a disadvantage. \"I doubt we'll arrive in time to stop her,\" she said, \"so there's no point trying. But somehow I don't think she's going to be willing to blow herself up along with the ship.\"\n\n\"She'll be looking to escape,\" said Haid.\n\n\"And fast.\"\n\n\"But how? She's blown up all the EVA\u2014\" He stopped, realizing the only real option that Alta had. \"The Hum ship!\"\n\n\"It's the only possibility I can think of.\"\n\n\"You think she's a clone warrior?\"\n\n\"Maybe not,\" Roche said, \"but certainly on their side. It's the only explanation I can think of for why she's doing this.\"\n\n\"So she'll rig the drive to blow, then jump ship?\"\n\n\"That would be the simplest solution.\"\n\n\"Then she's going to need access to the hull.\"\n\n\"Exactly,\" Roche said. \"See if you can pull up any plans on airlocks close to the central drive chamber.\"\n\nWithin seconds, Haid had the plans before them, via their suit's displays.\n\n\"Which do you think she'll go for?\" Roche asked, studying the map.\n\n\"There's one likely candidate, and a couple of close seconds.\"\n\n\"Take us to the most likely, and we'll just have to wait for her there.\"\n\n\"Listen, Morgan,\" said Haid, \"I know you want to keep us together, but wouldn't it make sense to split up now? If we risk all on one airlock, and she picks another, that puts us in a bad situation. After all, we've only got one shot at this.\"\n\nRoche pondered this. As much as she didn't want to let Haid out of her sight right now, his suggestion _did_ make sense. If Alta had worked out how to locate them via her own suit, they could end up walking into a trap.\n\n\"Okay,\" she relented. \"Ameidio, you take a secondary airlock and I'll wait for her at the main one.\" Haid was about to protest. \"I have just as much chance against her as you, Ameidio. And besides,\" she added, \"I'd like to be the one that takes her down.\"\n\n\"Okay, Morgan,\" he said. \"But just take care, all right?\"\n\nShe nodded. \"Keep an eye on my beacon so you know where I am, but _don't_ communicate unless you have to. The less she knows about us, the better.\"\n\n\"Okay. And if all goes well, I'll meet you out on the hull.\"\n\nAt the next intersection, he turned left while Roche kept going. She watched the dot of his suit angle away from hers with a feeling of apprehension. The pain in her side was a constant ache punctuated by spikes of agony. A rumble of distressed machinery came to her through her feet rather than her eardrums. She ignored everything, and concentrated instead on the task ahead.\n\nEach stride was like the tick of a clock, taking her ever closer to some unknown fate. She hurried through the _Ana Vereine's_ extensive warren, through tunnels she hadn't visited since first occupying the ship. It was here that Cane, Haid, and Maii had hidden while Kajic left Sciacca's World and the Hutton-Luu System with the intention of handing Roche and the Box over to his superiors in the Ethnarch's Military Presidium.\n\nOn the tracking screen, Alta had reached the main drive chamber. Within minutes, a siren began to sound.\n\n\"That'll be the drives,\" Roche mumbled to herself. The lack of response from either Maii or the Box only intensified her sense of isolation and the ever-growing emptiness inside her.\n\nShe had hoped it would take the woman longer, but a warning through her suit only confirmed her worst suspicions: the drives were going to overload in about fifty minutes. She didn't know exactly what Alta had done, but it had to be something slow-acting, something she had prepared in advance. Had she freed the antimatter reserve, they would already be dead. But whatever it was, it was unlikely it could be stopped. It could take them hours just to figure out what she had done, let alone begin reversing whatever process Alta had set in motion....\n\nRoche studied the tracking screen. According to the display, Alta was on the move again. She was heading for the airlock that Haid suspected she would go for, but Roche wasn't going to be able to beat her there.\n\nShe considered calling Haid, but then thought better of it. Hopefully he would see what was going on. He was making good time toward his airlock; even if they didn't make it in time, there was a chance he would be able to intercept Alta out on the hull.\n\n_Fifty minutes,_ Roche thought to herself. In half an hour, the entire ship would be gone. Unless they could find a way off the ship, or someone to rescue them, they would die with it.\n\nPart of her didn't mind that at all. The Crescend himself had stated it quite bluntly the last time they spoke: _Your role is played out._ As long as she took Alta Ansourian with her, she might be happy with that....\n\nThe woman was also making good time, unfortunately. Hoping to distract her, Roche activated the open channel connecting the suits.\n\n\"Alta? Can you hear me?\"\n\n\"I hear you.\" The reply was immediate. \"And I can see you, too, Roche! But you can't stop me! You're too late! There's no turning back now.\"\n\nThe woman's voice was feverish, breathless. There was a raw edge to it. Roche couldn't tell if it was fear or determination\u2014 or perhaps both.\n\n\"What have you done to the engines, Alta?\"\n\n\"The matter\/antimatter mix is in disequilibrium,\" the woman replied. \"The mix will become more and more unstable until it spirals out of control, and then the ship will blow. Trust me\u2014you can't stop it; there isn't enough time.\"\n\n\"But there is enough time to get away?\"\n\n\"You can get out of the ship, Roche, but you'll never escape!\" The mockery in her voice angered Roche, and the image of Maii's helpless form with its outstretched arm returned.\n\n\" _Why_ did you kill them, Alta?\"\n\n\"To stop them from interfering, of course.\"\n\n\"But Maii was just a _child._ She never did you any harm!\"\n\nThere was a pause before Alta spoke again, and when she did her voice was steeped in uncertainty.\n\n\"Those were my orders,\" she said. \"I couldn't disobey.\"\n\n\" _Whose_ orders, Alta?\" Roche pressed, anger giving way to curiosity. \"Why are you doing this?\"\n\nMore assured now, Alta replied: \"Because if it hadn't been for you, he wouldn't be dead now!\"\n\n\"Who? Your _father_?\"\n\n\"You killed him, Roche!\" The hatred in her voice was intense and frightening. \"You took him away from me!\"\n\n\"What are you talking about! You _saw_ Jans kill your father! You were there\u2014\"\n\n\"Lies that you planted!\" Alta's voice was raised to a shout now. \"But I know the truth! He showed me. He showed me your _lies._ Justice will prevail!\"\n\nThe edge to her words unsettled Roche. There was a sense of desperation to them that suggested Alta was beyond reasoning with. She believed what she was saying; she was convinced Roche was responsible for her father's death. Why, though, Roche didn't know. Maybe it was simply because she was the nearest available target for reprisal...\n\nAlta was much closer now. The air in the warren became thick with smoke as Roche neared the airlock. She passed emergency doors that had failed to shut. Some hung loose, half open, as though they had changed their mind in the act of closing. Without Kajic to run the ship, even the most basic systems were gradually running down. The rumble of complaining engines became steadily louder as she ran.\n\n\"Who showed you, Alta?\" said Roche.\n\nThe woman said nothing.\n\n\"Alta?\"\n\nA loud explosion knocked Roche off her feet and sent air whipping along the corridor.\n\n\"The airlock,\" she gasped in the sudden hurricane, thinking: _We're open to vacuum_!\n\nShe scrambled to her feet, hurried along the corridor and around a corner to find the source of the hurricane: a blackened breach where the outer door of an emergency airlock opened onto space. The inner door hung invitingly open. Roche approached it cautiously, wary of a trap.\n\nThen, she glimpsed a movement: a suit much like hers had just stepped out onto the hull.\n\nRoche didn't hesitate. Using every joule at her suit's disposal, she jumped through the gap and after the fleeing woman. Even if Alta was armed, Roche could take a direct hit or two before her suit failed. And if she could catch her and somehow overpower her before she jumped ship....\n\nUpon exiting the hatch, Alta dropped and rolled, bringing her rifle up to cover anyone following her. Roche burst out of the airlock, only a thin veil of artificial gravity preventing her from rebounding off into space. Alta's first shot missed. Her second caught Roche full on the chest as she threw herself forward. The recoil of the shot knocked her to one side with her ears ringing, but she didn't hesitate. She surged forward again.\n\nAlta kicked back to avoid the charge and scrambled to her feet, her gun coming up again. The third shot took Roche in the shoulder, spinning her around. Alarms in her suit flashed red, and the acrid smell of smoke burned at her nostrils.\n\n\" _Die!_ \" The voice came over the open line as Alta swung the rifle's sight up to her eye, training it on Roche.\n\nRoche knew she had only one more chance. Ignoring the pain in her hip, she dropped to a crouch, forcing Alta to readjust her aim; then, taking advantage of the split second she had gained, she hurled herself forward\u2014\n\nShe heard Alta's gasp at the same instant the energy bolt grazed her helmet. Unable to stop herself, Roche lunged awkwardly forward, colliding with Haid as he took Alta from behind, swinging her around and down onto the hull. The impact cost him his grip on the woman.\n\nAs both he and Roche tumbled backward, Roche saw Alta regain her footing and collect the rifle that Haid had knocked from her hands.\n\n\"Quickly!\" Roche called.\n\nThey were barely on their feet when the rifle discharged. She felt the edge of the energy bolt that took Haid in the back, forcing him forward into Roche and sending them sprawling across the hull again.\n\nHaid came to a halt, totally inert, a meter or so away from where Roche had fallen, a massive smoking scar stretching across his suit's backplate.\n\n\"Ameidio!\" She heard the panic in her voice. The lack of response from the ex-mercenary filled her with dread. The bolt had taken out the suit's servomotor control; she wasn't even sure whether his immobility resulted from death or a simple inability to move.\n\nShe looked over to Alta again, saw the woman lifting the rifle once more to aim at her.\n\n\"I'm going to kill you, Roche.\" There was no longer any hysteria in Alta's voice, just a quiet determination.\n\nRoche wanted to feel angry, but everything seemed to have been knocked out of her. All she could feel as she sat there, her friend lying prostrate on the hull before her, was a terrible exhaustion. All the fight went out of her; only acceptance remained as she stared down the barrel of Alta's rifle, and waited....\n\nA single rifle shot flickered through the vacuum, taking Alta at the weakest point of her suit between chest and neck plates. The woman jerked upright and staggered backward, trying to stanch the flow of air out the broken seal. Roche caught a glimpse of her terrified expression before the water vapor sucked out of her lungs plastered ice across the inside of her faceplate. She kicked spasmodically, staggering backward.\n\nRoche heard the woman's cries over the suit's intercom, __ \" _Master_!\" The words were carried on molecules of freezing air. \"I have failed you!\"\n\nRoche turned to look at the figure who had fired, just in time to see him fire again at Alta Ansourian. The woman's suit stiffened, then went limp. It fell to the hull with a thud.\n\nThe sudden silence in Roche's helmet was broken only by the sounds of her own breathing.\n\n* * *\n\nIt was a few moments before Cane spoke. He seemed to be waiting to make sure Alta wasn't going to get up again.\n\n\"Morgan?\"\n\nShe watched him striding over to where she still sat; then his hands were taking hers and helping her to her feet.\n\n\"Morgan?\" he repeated. \"Are you all right?\"\n\n\"I thought...\" she began, before realizing she didn't know _what_ she had thought. She hadn't believed he was dead, but she had been so taken up with pursuing Alta that she hadn't stopped to consider just what _had_ happened to him.\n\n\"The explosion threw me through the blast doors before they could close,\" he explained quickly. \"I realized it must have been Alta and so went after her.\"\n\n\"But the suit,\" said Roche. \"It didn't register on the tracking display.\"\n\n\"I disabled the beacon,\" he said. \"It was the only way I could get the advantage of surprise over her.\"\n\nRoche was still very much shaken by everything that had happened, but she saw a flaw in Cane's explanation. \"That would have taken time. How\u2014?\"\n\n\"Time is something we _don't_ have right now, Morgan. Look.\"\n\nHe pointed above her. Only then did Roche look up and see the Hum ship. All spines and strange glassy towers, it looked like a nightmarish sea anemone magnified a thousand times. It was enormous, ominous. Behind it, a faint glow marked where the ring surrounding Sol System's primary obscured the stars. Strange, pale sheets of lightning flickered in slow motion through the glow, lending the ship a surreal backdrop.\n\n\"How...?\"\n\n\"It must have slow-jumped,\" he said. A rumble came through the _Ana Vereine_ 's hull. \"Quick, Morgan. Use your thrusters and jump over to D nacelle.\" He pointed. \"That one.\"\n\nShe faced him, puzzled.\n\n\"Just _do_ it,\" he insisted.\n\n\"I'm not leaving Ameidio.\" She nodded toward her friend's lifeless form. His suit was immobile, but she still didn't know whether or not he was dead.\n\n\"Understood,\" he said. \"I'll slave his suit to mine and follow you over. Now go!\"\n\nHis urgent tone overrode her confusion and she did as he asked, though minus the usual grace with which she might have piloted the suit. Limited by her lack of depth through just one eye, as well as the lack of implants, she took the suit awkwardly up and out of the drive section's artificial gravity well and to the one enclosing the nacelle Cane had indicated. She resisted the disorienting change in gravity at first, then relaxed and let it hold her.\n\nWhen she had adjusted, she was standing on a patch of hull apparently slightly above where she had been before. From this new vantage point, she could see a reddish glow clinging to the drive ports, the sole visual sign of the ship's impending destruction.\n\nShe looked around, expecting Cane to be following, but saw him land instead on another nacelle some distance away, Haid's suit following like a ball on a string.\n\n\"Cane?'\n\n\"I'm sorry, Morgan.\" There was something different in his tone\u2014something of the Cane that had rescued her from the Disciples. \"But this is where we part, I'm afraid.\"\n\nRoche felt a wave of nausea wash over her.\n\n\"What are you talking about?\"\n\n\"Really, Morgan\u2014think.\"\n\nShe did, and it hurt. \"It was you Alta was talking about,\" she said softly. __ \" _You're_ her master.\"\n\n\"You sound surprised,\" he said.\n\n\"No,\" she said. She was beyond surprise; she felt utterly dead inside. \"Disappointed.\"\n\n\"I'm only doing as my nature dictates,\" he said.\n\nShe didn't say anything for a moment. She was remembering what Haid had said about Alta being unresponsive, and that Cane had spent a lot of time talking to her. That was when it must have happened. With a veil of suspicion still hanging over him, he would not have been able to move as she had. He had bent her will to his; he had made her his puppet.\n\nThe silence on the open link between them seemed to hum with repressed energy.\n\n\"It was all lies,\" she said, more to herself than to him. \"Your amnesia, saving me on Sciacca's World\u2014everything. It was all just lies.\"\n\n\"Not just lies, Morgan. But lies in a game\u2014a game between the Box and myself, between my kind and the Crescend.\"\n\n\"What about Inderdeep Jans?' said Roche. \"Why\u2014?\"\n\n\"All strategic games require sacrifices, Morgan,\" he said. \"But don't think I didn't feel her pain. It touched me deeply, as did the deaths of the five you killed on the _Phlegethon._ For a group mind, the loss of even one component is painful. But you couldn't appreciate that, could you? Coming as you do from such a hapless and disparate race...\"\n\n\"A group mind?'\n\n\"The _irikeii_ glimpsed it in the black speck at the heart of me and Jelena Heidik. You saw it too, when you looked at us through a reave. There's more to us than meets the mental eye.\"\n\n\"So you _are_ linked?\" she said.\n\n\"Of course,\" he said. \"Everything I have seen and experienced was shared with my siblings.\"\n\n\"Another lie.\"\n\n\"All of them were necessary,\" he said. \"It was part of the game.\"\n\nRoche felt herself reel. \"But why didn't you just kill me? Why have you kept me alive? Why all of _this_?\"\n\n\"Because we want you to know the reason for it all.\" She could almost feel the coldness in his voice. \"We want you to be aware of the consequences of your _decision_.\" He emphasized the word deliberately, almost viciously. \"We suspected from the beginning that the High Caste would become involved. Coming into contact with the Box when I awoke confirmed it. I watched it as closely as it watched me. I didn't guess that it was inside you, but I knew you were connected, that you were involved. You were the key, and the Crescend was the lock. When you asked me about killing me, and when you told Trezise that High Humans wouldn't help, I knew that the lock had been turned.\"\n\nShe wasn't entirely following him, but that didn't seem to matter at this time.\n\n\"I let you live,\" she said weakly. \"I gave your race another chance.\"\n\nHe raised the assault rifle.\n\n\"That was a big mistake, I'm afraid, Morgan,\" he said, and fired.\n\nShe took a step back by reflex, but the bolt didn't come anywhere near her. Instead it struck the nacelle's support girders. She couldn't believe he had missed; it had to have been deliberate. He was tormenting her, perhaps, before finally killing her.\n\nShe took a step forward again and stood her ground. She would defy him, she decided; she would not give him the satisfaction of seeing her afraid to die.\n\n\"Do you really believe you can purge the galaxy?\" she said.\n\nAnother shot, again striking the support girders. This time she didn't move at all.\n\n\"You're going to wipe out everything that's not a part of your precious group mind?\" she continued, trying to control the tremor in her voice.\n\nA third shot, and she felt the impact vibrate through her suit.\n\n\"It's called justice, Morgan,\" he said.\n\n\"It's called genocide, Cane!\"\n\nShe felt the hull shift beneath her as a fourth, fifth and sixth shot struck the nacelle girders in rapid, angry succession.\n\n\"You dare stand there and talk to us about _genocide_!\" he flung at her vehemently. \"We're here to avenge a crime no worse!\"\n\n\"But that was half a million years ago, Cane! You can't blame everyone alive today for something that happened so long ago!\"\n\nWhen next he spoke a few moments later, the icy calm had returned to his voice. \"Don't impose your values on us, Morgan,\" he said.\n\n\"What does that mean?\"\n\n\"That you shouldn't expect mercy from us simply for mercy's sake.\"\n\nHe fired another shot, this one causing her to momentarily lose her balance as the nacelle rocked under her.\n\n\"Charge is running low,\" he said. \"But you should tear free in a moment or two and drift away.\"\n\n\"What good will that do?\" she said, regaining her footing. \"Why not just kill me now and be done with it?\"\n\n\"Like I said, I want you to have time to think. I want you to think about everything that has happened. I want you to consider what your people have done, and I want you to know why it would have been utterly wrong for the Crescend to have intervened.\" He paused before adding, with almost a hint of amusement: \"Besides, it seems only fitting, given that you spared me.\"\n\n\"I spared _all_ of you!\" she snapped angrily.\n\n\"Yes,\" he said. \"And I want you to think about that, too, as you slowly die, Morgan.\"\n\nThe nacelle rocked again.\n\n\"It doesn't have to be like this, Cane,\" she said desperately, hoping to reach that part of his nature she was sure existed, the part she had come to know over the weeks since they had first met.\n\n\"Yes it does, Morgan. We are programmed to exact revenge and we shall do so to the best of our capabilities. And we _will_ win.\"\n\n\"But this is crazy! There is no grudge left to pursue! Surely we can get along?\"\n\n\"History would disagree with you.\"\n\n\"No\u2014history _agrees_ with me. Humanity has diversified so much since your creators were around. We come in all shapes and sizes now, and we live in all sorts of places. But underneath it all, we're still all Human. Some may think they're better, but in the end we're all the same. And regardless of what has happened in the past, or what might happen in the future, eventually you will fit in too. It's inevitable.\"\n\nThere was a mocking laugh which filled her helmet. \"Assimilation? How typically arrogant of the victors.\"\n\n\"There _was_ no victor,\" she stressed, feeling the need for urgency as the nacelle continued to shift uneasily beneath her. \"That's what I'm trying to tell you, Cane. There's just _Humanity.\"_\n\n\"Exactly,\" he said. \"Humanity... the victor.\"\n\nHis answer caught her with her mouth open to respond, but the words stuck in her throat.\n\n_Humanity... the victor._\n\n\"No,\" she said. \"That can't be, Cane. You're Human. You _have_ to be.\"\n\n\"You're wrong, Morgan,\" he said.\n\n\"But... it's not possible!\"\n\n\"It's more than possible,\" he said. \"It's the essence of this entire conflict. You deny our existence, so we repeal yours.\"\n\n\"But...\" The concept was so difficult for her to grasp, she found the words almost impossible to say. \"You're _aliens_?\"\n\n\"We are the indigens,\" he said. \" _You_ are the invader.\"\n\n\"Now I know you're lying.\"\n\n\"No, this time I _am_ telling the truth,\" he said. \"It's simply a matter of perspective. My creators were exploring the galaxy before Humanity's ancestors had even reached its moon. We had created the beginnings of an anchor-point network before you learned how to use hyperspace. By the time our expanding empires came into contact, you were catching up fast but you were still our inferiors, and would have been for some time. You were hasty, impetuous, prone to sudden advances followed by long periods of decay\u2014the brutal disequilibrium of individuality. We on the other hand were patient, persistent, and compassionate. Ours was the steady growth of unity.\n\n\"Neither of our civilizations had encountered alien life before, so we were unprepared to deal with such differences; we had no warning of what would come. We didn't expect you to learn from us as quickly as you did. We didn't expect your expansion to be so rapid, and so inconsiderate of our own. You stole our worlds, Morgan; you appropriated our anchor-point network and you encroached upon the society we had accreted so carefully over thousands of years. And you hurt us! Maybe you didn't mean to. Maybe you didn't even realize what you were doing. But you did it nonetheless. And we, like any injured organism, struck back.\"\n\nShe shook her head, dizzy with the concept of what she was hearing. \"There's no evidence for any of this. How do I know you're telling me the truth?\"\n\n\"What reason do I have to lie to you now?\"\n\nShe didn't say anything; she couldn't. There didn't seem to be anything _to_ say. The reality upon which her entire life was founded now seemed as shaky as the nacelle that rocked and moaned beneath her.\n\nShe thought of the anchor points supposedly older than Humanity itself, and found herself believing him. As incredible as it seemed, maybe she _had_ finally learned the truth.\n\n\"You won't find evidence of our existence,\" he went on, \"because it was either suppressed or destroyed in the process of your expansion. And in time, it was forgotten entirely. Even the war that engulfed our two species was forgotten. The origin of my makers was one of the first details to go\u2014for although wiping out an entire losing Caste was not unheard of, destroying the only other intelligent life you had found in the universe _was._ Even those military leaders who sanctioned the genocide ultimately realized this. Perhaps we _could_ have learned to live together. Perhaps you were hasty in assuming that, where more than one of my kind existed, the threat remains that the group mind can emerge again, as strong as before\u2014or even stronger. Perhaps you had been wrong in pursuing the war to its ultimate, deadly conclusion.\"\n\n\"And _were_ we wrong?\" she asked. \"Could we have lived together?\"\n\n\"No,\" he said bluntly. \"You had proven too many times that peaceful coexistence\u2014in the long-term\u2014simply wasn't a possibility. There had been too many broken treaties, smashed peace accords, violated cease-fires... We had grown tired of tolerating you, and resolved to rid our galaxy of you.\"\n\n\"But you lost.\"\n\n\"No, Morgan. We didn't,\" he said. \"We are tenacious. I am proof of that.\"\n\n\"But what _are_ you? You look like one of us; your anatomy is at least based on ours; you could almost _be_ one of us, if you tried. What's happened to your precious origins now?\"\n\n\"It exists in our minds, and will exist again when the data encoded in our introns is released. We carry within us the knowledge to re-create our race in its original form. Given time, we can restore the galaxy to the way it should be\u2014the way it was _meant_ to be\u2014and we will reclaim our planets and systems. We will travel _your_ anchor-point network. We will take from you all those things that should have been ours, and _make_ them ours. The balance of justice shall be restored.\"\n\nThe nacelle suddenly lurched beneath Roche, throwing her off balance. There was a crunching, tearing sound, and artificial gravity failed. She used her thrusters to keep her against the hull while the suit anchored itself firmly into place. When next she looked up, everything was silent\u2014even the rumble of the main drive was gone\u2014and the nacelle was drifting away from the _Ana Vereine._\n\n\"I want to thank you, Morgan,\" he said. The growing distance between them and the unsteadiness of the nacelle made it difficult for her to focus on him. \"You have enabled us to continue the work we were created to do. Without the High Caste to interfere, the battlefield will be even. We can fight openly, if we choose to. We will prevail if we can, or we will die. History does not allow us a third option.\"\n\n\"There are always options,\" she muttered, thinking of the many times she could have turned away from the path that had led her here. \"We make our own destinies.\"\n\n\"But sometimes they are made for us.\"\n\n\"Then they can be unmade.\"\n\n\"If you truly believe that, Morgan Roche, then you're more of a fool than I thought.\"\n\nLight flashed from his thrusters as he leapt off the _Ana Vereine_ and arced toward the ship waiting for him. She was too far away to make out whether or not he had taken Haid with him. Not that it mattered anymore. In fact, she thought it would be better for Haid to be left behind. If he wasn't already dead, she felt it would be better for him to die there rather than at the hands of Cane and his siblings later....\n\nShe clutched the hull and waited for Cane to say something more, but nothing came. The flare of his thrusters disappeared into a wide-mouthed airlock leading into the Hum ship. A minute passed in silence, then space became crowded in the vicinity of the _Ana Vereine._ Temporal echoes converged on the slow-jumping _Apostle,_ creating a halo of flickering, short-lived ghosts around the spiny craft. As the echoes converged, the warp in space reached peak flexure until, with a flash of light so bright it left Roche blind in her sole eye for almost thirty seconds, it disappeared.\n\n****\n\n****\n****\n\n****\n\n**21**\n\n****\n\n****\n\nAna Vereine Fragment (Nacelle D)\n\n955.2.16\n\n0430\n\nSeventeen minutes later, the _Ana Vereine_ exploded.\n\nRoche didn't see it; she was inside the nacelle, trying to find anything that might be of use. But she felt the expanding wave-front as it hit. It shook her like a die in a cup. When it passed, she disentangled herself from the remains of a solar antenna and went back out onto the hull to see. There was nothing else for her to do.\n\nCane had chosen the nacelle well. It contained little but packed storerooms full of raw materials for repairs and hyperspace disrupters that were useless without power to run them. There was no communicator, no long-term life-support and no means of turning the nacelle into a powered vehicle of any kind. If she could plug the leaks, she could survive a day or two, but that was all. Had she found a way to contact the council drone that had shadowed the ship, she might have got away a distress call to Trezise; she might have had a chance of being rescued. But even if Cane's friends hadn't destroyed the drone to prevent anyone's overseeing their actions, she had no way of finding it, let alone sending a message. She was trapped.\n\nSo, when the _Ana Vereine_ exploded, she went up on the hull to witness the pyre of her former friends and allies, and the place she had called home for the previous nine weeks.\n\nAnd she saw something else, too\u2014something completely unexpected.\n\nAn elongated star flew unsteadily out of the cloud of debris that was all that remained of the Dato Marauder. Bobbing and weaving like a drunken bird, it angled in the general direction of the tumbling nacelle. Where it wasn't burned black, its hull plating shone bright gold.\n\nPerched on the truncated end of an access corridor that had once joined the nacelle to the main drive section of the ship, Roche watched it come. Despite the damage, she immediately recognized the ship. When it was closer, she moved out onto an exposed section and waved for attention. It changed course to rendezvous with her.\n\n\"Your suit beacon was moving,\" said Vri over the radio. \"I followed it on the off chance you were still alive.\"\n\nHer relief at hearing the Surin soldier's voice could not be measured. \"I'm glad to see you, Vri.\"\n\n\"And I you,\" he said as he brought his fighter down.\n\nWhat relief she had felt on finding the Surin alive was quickly tempered when she climbed aboard his ship and realized his condition.\n\nHis golden armor looked worse than the ship. He had taken two powerful shots: one to the right thigh and one in a line across the back of his head. The back of his skull was a mess of blood, bone, and fur. The tiny cockpit\u2014with barely enough room for the two of them\u2014stank of his blood.\n\nRoche didn't say anything because there didn't seem to be anything to say.\n\n\"I was coming to assist you,\" he said, his eyes going in and out of focus as he talked. \"I shouldn't have left Maii. She needed me, and I wasn't there. But I\u2014\" He coughed, winced, and tried again. \"Alta took me by surprise,\" he said feebly. \"Shot me and left me for dead.\" He smiled despite the pain. \"When I woke up, the docks were in flames. _Esperance_ was undamaged, though. It had sealed automatically and could weather worse than a mine or two, if it had to\u2014and it did, when the drive went up. I couldn't move before then in case I was exposed. I wouldn't have stood a chance against that other ship. I _had_ to wait...\"\n\n\"You did the right thing,\" she reassured him.\n\n\" _Esperance_ ...\" He stopped, breathing rapidly, and shut his eyes. He seemed to be running out of energy. \"I give it to you, now.\"\n\nRoche unsealed one of her suit gloves and touched his forehead. His skin was extremely cold, but she had no idea whether or not this was a bad sign for a Surin. The same ignorance struck her when she examined the instrument board in front of her. Although she had flown Surin vessels before, this was nothing like what she was used to. It had been designed to suit him, she suspected, with every screen and switch tailored for his needs, his mind. She suspected that she would be able to open the airlock if she ever had the need to, but she doubted she would be able to pilot the ship.\n\nShe wasn't about to tell him that, though.\n\n\"Thank you, Vri.\"\n\nHe opened his eyes briefly, but didn't see her.\n\n\"I fear,\" he said faintly, \"that I have failed.\"\n\nHe died before she had a chance to say anything more.\n\n* * *\n\nThe first thing she did was seal Vri's suit and get his body out of the pilot's seat. He had set _Esperance_ on a course to nowhere; the engines were firing at a constant, gentle rate, with no apparent intention of stopping and no clear destination other than away from the nacelle. Maybe he had intended to change the settings before he died, or perhaps he had assumed that she would be able to do so herself.\n\nDistantly, she felt embarrassed at being so helpless. With the Box she might have had a chance. Even without implants, it could have told her what to do. At the very least she would have had someone to talk to.\n\nShe played with the controls at random and managed to find the attitude jets. All that did was send the fighter into a slow tumble around its long axis and made its course even more chaotic. Fearing she would touch the wrong control completely and blow the drive core, she sat back and tried to think. The life-support seemed to be working fine, and there was no shortage of power. She could survive in the fighter longer than she would have on the nacelle, so that was an improvement. Given time, she could work out how to operate the drive or the communicator, which would improve her situation even more...\n\nThe one thing she _could_ understand was the navigation display. The small fighter's projected course\u2014as near as she could tell\u2014took it through the outer edge of the ring system; she would strike the fringes of it within a day. The electrical anomalies Kajic had noted seemed to be fading, but she was still wary of them. When she wasn't puzzling over the controls, she kept her eye on the screen. What she would do if it _was_ a trap, she didn't know.\n\nEven as fatigue wore her down into a feverish creature talking to itself and weeping in fits and starts, she resisted sleeping. She was afraid of her dreams, of what she would see on closing her eyes. She didn't want to see Maii alive _or_ dead: that wound was too fresh, the grief too keen. The same with Haid, and Kajic, and the Box\u2014even Alta Ansourian and Cane. _Especially_ Cane.\n\nShe had trusted him, and he had betrayed her. Put boldly, in black and white, she could accept it. As soon as she looked beneath that pronouncement, however, at the emotional consequences and potential ramifications, she saw the pain. She didn't want that. She would have the rest of her life to dwell upon it, however long that might be. If she could forget about it just for a moment, now, she would be grateful.\n\nIn the end she did sleep. Not even she could put it off forever. But there were no dreams\u2014and for that she was grateful.\n\n* * *\n\nFourteen hours after she left the nacelle, the glowing haze of the ring had expanded to fill most of her forward view. It looked like smoke: yellowish and acrid, with denser wisps almost blocking out the stars behind them. Strange, glowing sheets rippled through it at odd moments, but with little more definition than a predawn glow. She could no longer make out the plane of the ecliptic, or tell by sight alone at what angle she was approaching it. It was just there, before her, something she could neither avoid nor find any pleasure in viewing.\n\nOnce again, she regretted Sol System's lack of grandeur. If she had been about to crash into the Soul around Sciacca's World, or be electrocuted by the ion bridge snapping across the gap between Kukumat and Murukan, Palasian System's double-jovian, that would have at least been something to marvel at. But here, all she was heading for was a cloud of dust. She didn't even know what it would do to her. It might not do anything at all, if it was thin enough. Even if it was as dense as the average planetary ring, the most the _Esperance_ would suffer as it passed through the cloud was some wear on its ablative shield.\n\nThat was exactly what it _was_ suffering from when she began to hear voices.\n\nAt first she thought she was dreaming. A snatch of phrase caught her ear, apparently from somewhere behind her, followed quickly by another. Then another, and another, until the sounds came in a continuous, muddled stream. The whispers belonged to people of all ages and many different accents; most spoke galactic standard\u2014although she could discern maybe one word in three\u2014and some spoke in tongues she had never heard before. There were too many of them, too many fragments, for her to follow. All she could pick up were the individual flavors as they rushed by her, like ghosts in an echo park, or a half-remembered dream. They were happy, sad, angry, hurt, proud, joyous\u2014all colors of the emotional spectrum.\n\nWhen her suit's life-support kicked in, she knew she wasn't asleep. The temperature in the fighter had risen to an uncomfortable point without her noticing, so intrigued had she become with the whispering. She must have underestimated the velocity the fighter had accrued during its hours of constant thrust. Even as she studied the data, annoyed with herself for not paying attention, she could feel a slight vibration through the bulkheads. Having no knowledge of how to access the ship's diagnostic systems, she couldn't tell if the hull was being damaged by the high-speed rain of dust. All she could do was worry about it. Eventually, the ship would pass through the ring and out the other side. Or it wouldn't. The situation was as simple as that.\n\n_\"Oh, for an axe...\"_\n\nThe whispered phrase caught her ear. There was a pause, filled with other whispers, then the fragment continued. The words were buried under others, but she could make them out. She knew them. She had heard them before.\n\n_\"\u2014nuclear strike from one hundred meters. I know, I know, but if it wasn't for you, I wouldn't be in this mess. Can you understand how frustrating it is to be cooped up in here with nothing to do?\"_\n\nRoche listened, stunned. That was _her_ voice. She remembered the conversation, but not the context.\n\n_\"The sooner we're back in HQ, Box, the better...\"_\n\nAnd with these words, she placed the memory: in her quarters on the _Midnight._ She had been bemoaning her lot\u2014bored by her mission, resentful of her attachment to the Box and frustrated by Captain Klose's refusal to let her examine the mysterious life-capsule the ship had picked up in deep space. This was where it had all started.\n\nBut why was she hearing it now?\n\nThere was only one answer: _this wasn't just dust she was falling through..._\n\n\"Hello?\" she called.\n\nShe didn't use her suit's radio. She simply spoke aloud.\n\n\"Hello, Morgan,\" replied a familiar voice. \"I wondered when you would guess.\"\n\nShe wasn't as surprised as she thought she should be. \"These are your voices?\" she said. \"Your memories?\"\n\n\"The boundaries of this identity known as 'the Crescend' are difficult to define at the best of times,\" he said. \"In my long life I have been many individuals, have spawned many components. Like the Box, for example: it came from me, and its memories now form part of me, but it was not _me_ \u2014at least not in the way you would understand the concept. All the voices you hear around you, they are memories to which I have access, and yet none of them are truly mine.\"\n\nShe imagined the Crescend as a spider sitting in the middle of an enormous web, reeling in experiences along silken threads, capturing and absorbing entire minds full of information....\n\n\"A colorful analogy,\" he said. \"And perhaps not totally inaccurate.\"\n\n\"You're reading my mind?\"\n\n\"I am aware of what you are thinking,\" he said.\n\n\"Why can't you just give me a straight answer?\"\n\n\"Because the questions you ask do not allow it.\"\n\nThe voice of the High Human came clearly over the babble, but the endless whispering\u2014combined with the sound of the ship slowly being battered\u2014was making it difficult to concentrate.\n\nShe glanced at the forward view screen, at the dust particles of the ring that were taking up the entire display now, as impenetrable as a dust storm.\n\n\"It's you, isn't it?\" she said. \"The ring is you. Or the other way around.\"\n\n\"Both, and neither. The ring is a physical construct upon which this identity is presently generated.\"\n\n\"A computer?\" said Roche, lifting her voice unnecessarily to be heard above the growing noise.\n\n\"The term is grossly inadequate,\" he said with no hint of condescension. \"It is composed of the mass of an entire solar system liberated and allowed to interact as computational components: every atom of an ever-changing matrix circling\u2014and powered by\u2014the system's sun. A sun which, half a million years ago, birthed the species from which we sprang.\"\n\nIt was hard not to be impressed by the sheer scale of what the Crescend was describing. As camouflage, it was perfect: of everyone she had met, only the Heresiarch had any serious idea that the ring might be more than it seemed. The unsuspected truth explained the odd electrical impulses, and the navigational hazard the ring occasionally posed: even something as nebulous and innocent-looking as a cloud of dust could be disturbed by passing ships and would possess the means to defend itself.\n\nBut this wasn't all he was telling her. _Birthed,_ he had said.\n\n\"That's why we're here,\" she said. \"You, and everyone. Sol System is where it all began.\"\n\n\"It may not be much to look at anymore,\" he said, \"but yes, this is where it all began. For those who know, the system is something of a symbol. Not a shrine; one's origins are not to be worshipped. Sol System just _is_ , and that is enough.\"\n\n\"But what happened to the planets?\" One of them, she assumed, must have been Humanity's homeworld.\n\n\"That is a long story, Morgan,\" he said. \"Too long for now.\"\n\n\"You won't tell me?\"\n\n\"There isn't time,\" he said. \"You haven't that much left.\"\n\n\"So I am going to die?\"\n\n\"Do you _want_ to die?\"\n\nAnother evasive answer. Before she could respond, however, the ship lurched violently, tossing her from side to side in her seat.\n\nWhen the ship settled again, she said: \"Can you get me out of here? Can you fix the fighter so I can use it and get back to the _Phlegethon_? If it hasn't already been destroyed, that is.\"\n\n\"Far from it, Morgan. The council has experienced good fortune since the enemy became aware of your decision. No longer required to concentrate their efforts on one location\u2014in order to draw you out, to force you to make a decision, and to influence what that decision would be\u2014many of their number have retreated from the system and begun the long journey home. To their _hosts'_ homes, I should say. As the war in Sol System winds down, preparations for the war throughout the galaxy are heating up. We sit on the brink of a new age, Morgan: today, the peaceful domination of Humanity, founded on near-genocide; tomorrow, the battlefield of justice, in whose days lives will be lost and civilizations will fall, and which might, ultimately, lead to a more balanced future.\"\n\n\"Is that what you want, then?\" Roche felt the same confusion about his motives that she had the first time she and the High Human had talked. \"I still don't know whether you believe I made the right decision or not.\"\n\n\"Does my opinion matter?\"\n\n\"Of course it does!\"\n\n\"If I tell you that you made the right decision, I will be accused of wanting war. If I tell you that you made the wrong decision, I will be accused of wanting to commit genocide upon the enemy.\"\n\nRoche fell quiet for a while.\n\n\"I just wish there had been another alternative.\" Her words were soft and low, barely audible above the whispers that filled the space around her.\n\nIf the Crescend had heard, he didn't answer.\n\nThe temperature continued to rise, along with the turbulence, and the voices were louder, harder to think through. A golden haze tinged the air around her; the walls of the ship themselves seemed to glow.\n\n\"Tell me about your final conversation with Adoni Cane,\" the Crescend said.\n\n\"What? _Why?_ \" She was irritated that the Crescend seemed to be trying to distract her rather than doing anything to actually help her.\n\n\"Did he reveal anything to you that you didn't already know about him?\"\n\n\"Like the fact that he's an alien, perhaps?\"\n\nThere was a sound like a sigh. \"He said this?\"\n\n\"He said Humanity wiped out his creators and took over the galaxy. Did you know about this?\"\n\n\"We suspected,\" the Crescend said. \"Few records exist from that time, and only the oldest memories of the most inward-seeking of my Caste speak of such events, but we have always known of another race that preceded ours\u2014which may have even co-existed with us for a time. It seemed likely that it was destroyed in the war about which we had also heard rumors. The possibility also existed that these two suspicions were linked to the emergence of the enemy and their convergence on Sol System. Linking all three was the most elegant solution.\"\n\n\"That explains why we couldn't find his parent Caste,\" Roche said. \"He never had one, did he? It was a waste of time looking.\"\n\n\"Absence of evidence is not evidence of absence, Morgan\u2014as you yourself recently said. We needed confirmation before we could be sure.\"\n\nShe suspected he was still trying to distract her, but she didn't care. She needed the distraction to take her mind from the sounds of the ship being bombarded by the dust outside.\n\n\"So who were they?\" she said loudly. \"Where did they come from?\"\n\n\"They are referred to only by euphemism.\" The Crescend's voice never seemed to rise in pitch despite the ever-increasing noise within the cockpit. \"And then most frequently as 'the Concinnity.' Where they originated, however, remains a mystery.\"\n\n\"Cane said that they're a group mind, and that they plan to resurrect the species from the data in their introns.\"\n\n\"It was always thought unlikely that revenge was the only thing on their agenda,\" the Crescend said. \"Resurrection of the species was always considered a possibility.\"\n\n\"What are you going to do about it if they do?\"\n\n\"What we have always done: observe.\"\n\n\"You'll let it happen?\"\n\n\"It is in accordance with your decision, Morgan.\"\n\n\"But my decision was made with the understanding that they were _Human_! That if they wiped us out, there would still be Humans left, even if it was _them_!\"\n\n\"There will always be Humans, left,\" the Crescend assured her. \"They will never destroy the High Caste. At least not until they have themselves evolved to our level, in their own way. And numerous Low Castes will survive too. The worst-case scenario would be that the mundanes will be wiped out, and then only for a time. In the enemy's eyes, it would even the score; in our eyes, it changes little.\"\n\nRoche pondered this as best she could through the racket. It was true. As important as it seemed to her, the activities of the mundanes didn't amount to much in the big scheme of things. The High Humans were doing the real work, whatever it was, on a galactic scale. The mundanes just filled in the gaps, gave their superiors something to watch in their spare time....\n\nThe cockpit's life-support suddenly failed, sending a blast of hot air into her face. She made sure her suit was completely sealed, then shut her eyes. Clutching the arms of the crash-couch, she rode out the turbulence, not knowing how much longer it would last and, irrationally, afraid that it might never stop.\n\n\" _Would_ you change your mind?\" The Crescend's voice was clear and calm in her helmet. \"Knowing what you now know, do you think you made the wrong decision?\"\n\nShe kept her eyes closed and fought down the fear by focusing on his question. \"I don't think so,\" she said. \"Cane and his race deserved at least a fighting chance. I just wish I hadn't been so stupid.\"\n\n\"In what sense?\"\n\n\"Cane told me he was Human, and I believed him!\"\n\n\"He never said that, Morgan.\"\n\n\"Yes he did,\" she said. \"After Palasian System, when we woke him from that coma Linegar Rufo put him in. He tapped out a message in code\u2014!\"\n\n\"Yes, but that's not what he said,\" said the Crescend. \"His exact words were: 'I am as Human as you are.'\"\n\nHer eyes opened, as if upon a realization she had been blinded to.\n\n\"What are you saying?\" she asked. \"That I'm not Human?\"\n\n\"You are as Human as I am, Morgan. As Human as Cane. Even as Human as the Box, if you like,\" he said. \"That's the way you were made.\"\n\nShe wanted to recoil from what the Crescend was saying, but she was trapped in her suit, in a disintegrating fighter. She had nowhere to hide from the words, no way to avoid them. All she could do was listen to him.\n\n\"The High Caste needed someone to make a decision it was not capable of making\u2014or was not prepared to make. But we could hardly trust such a judgment to someone lacking the necessary attributes. Since mundanes are inherently unreliable, and since the person we required simply did not exist, we decided to make one. We made _you,_ Morgan.\"\n\nShe shook her head. __ \" _Why?_ \"\n\n\"You are determined and not easily swayed. You see all sides of a dispute and try to be fair. You have a keen sense of duty, on many levels. You are honorable, and will not shirk from the truth. You may not see yourself as such, Morgan, for we also gave you a sense of humility, but you are a good person. A good _Human._ If the fate of the mundanes was to rest in your hands, it was important for you to be so.\n\n\"On the other hand, you needed access to information and capabilities beyond the access of a normal mundane\u2014especially once the time came to bring you face to face with the enemy, in the form of Adoni Cane\u2014so my relationship with the Commonwealth of Empires was exploited to allow the Box to fall into your hands.\n\n\"The only thing that set you apart from the mundanes around you was your ability to detect the enemy, and even that was limited. You were, to all intents and purposes, an ordinary person, but one fashioned in such a way that you would not break under extraordinary circumstances. That was our gift to you, Morgan\u2014one which has served you in good stead these last few weeks.\"\n\n\"How much of me...?\" She couldn't finish the sentence. Her mind was full of conflicting images, thoughts, and feelings. Everything seemed to be shaking, falling apart around her.\n\n\"I can assure you that you are as real as anyone.\"\n\n\"Ascensio\u2014the orphanage\u2014?\"\n\n\"Real memories,\" he said. \"Taken from someone else.\"\n\nShe closed her eyes. \"Bodh Gaya?\"\n\n\"Your own experiences. Everything from your arrival at the Military College was you. But that makes those memories no more 'real' to you than the others. They are all yours, Morgan. They all contribute to who _you_ are.\"\n\nShe thought of the parents she had hoped to find one day, and whom she had forgotten upon joining COE Intelligence. She remembered her friends in the orphanage, and the conditions that had led her to flee her home planet. She saw again, as clearly as though it happened only yesterday, the flash of the COEI _Gegenschein's_ engines as it broke orbit and headed for her new home, her new future.\n\nAll hers.\n\nAll faked.\n\nA shrieking of tortured metal rose around her, as though the ship were tearing up.\n\n\"I've never had any choice, have I?\" She raised her voice to be heard over the noise, even though she knew the Crescend could read her mind just as easily as it ever had.\n\n\"Of course you have, Morgan. That was the whole point.\"\n\n\"But you made me in order to do something. There was no way I could avoid that. There's no way you would've let me!\"\n\n\"Perhaps not, but\u2014\"\n\n\"And could I have avoided all of _this_?\" She saw Maii's body, the _Ana Vereine_ 's pyre, the golden glow of the cockpit around her. \"Was I always intended to end up _here_?\"\n\n\"That question is irrelevant,\" said the Crescend. \"You are here now, and the 'now' is all that matters.\"\n\nA siren wailed in her ears.\n\n\"Why are you bothering to talk to me at all?\" she said angrily. \"Why ask me about Cane? Why not just lift the information from my memories? What is it you are after? You want me to absolve you for what you've done? Is that it?\"\n\n\"I have no need of absolution, Morgan. I have no ulterior motives, either. Your role in this phase of the war is truly finished.\"\n\n\"So now I am being thrown out with the trash?\" she shouted. \"Is that it?\"\n\n\"You are not a robot, Morgan,\" the Crescend said.\n\n\"But I'm not _real_!\"\n\n\"You may find it difficult to accept, but you are as genuine a being as anyone else you have met. You have mind, you have will, and you have character. Where your body actually came from is irrelevant.\"\n\n\"Do you expect me to accept that?\"\n\n\"In time, I think you will.\"\n\n\"But I don't _have_ time,\" she said. \"The fighter's burning up!\"\n\n\"Yes,\" the Crescend said with no suggestion of remorse. \"It is. In fact, you have less than a minute before it disintegrates completely.\"\n\nShe fought down a surge of panic, resisted the tears pressing at the backs of her eyes.\n\n\"I'm frightened,\" she said, the words both a whisper and a sigh.\n\nThe Crescend said nothing.\n\nShe closed her eyes again, bracing herself as the fighter began to shake violently. The sound of voices was drowned out by the rattling and creaking of the ship. She thought she might be screaming, but she could hear nothing at all over the noise. She was sound: sound and movement: movement and pain: pain and\u2014\n\nWith a burst of heat, everything went silent.\n****\n\n**Epilogue**\n\nThere was no pain; there was no grief. There was only the darkness drawing her in, consuming her. She didn't resist the warm sensation; she allowed herself to be taken.\n\n<That's it, Morgan. Just accept.>\n\nThe familiar voice of the Crescend filled her with a strange relief. But he sounded different somehow. Closer\u2014almost as though the words were emanating from herself.\n\n<The fighter...> she said, as if remembering something from long ago. <It...burned up?>\n\n<Yes, Morgan. It did.>\n\nShe hesitated for a moment. <I'm dead?> she said.\n\n<How could that be possible if you are speaking to me now?>\n\n<So I'm alive, then?>\n\nThe Crescend didn't respond, and an interminable silence followed. She felt something approximating panic wash over her, soaking the empty dark around her.\n\n<Tell me!> Her voice resounded through the void. <Am I _alive_?>\n\nAnother silence followed before the Crescend spoke again.\n\n<Define 'alive,' Morgan.>\n****\n\n**APPENDIX**\n\n****\n\n**THE ORIGINS OF HUMANITY**\n\n**AN OPEN-ENDED QUESTION.**\n\n(by Provost Rejuben Tade, extracted from his welcoming address to the Guild of Xenoarchaeologists' 13,333rd Decannual Intake Expo.)\n\nIt is said that unless you know where you started, it is difficult to tell where you are heading. You can plot your course with as much precision as you like; you can map vectors, measure velocity and distance to the nth degree, but without those vital initial coordinates, you might as well be flying blind.\n\nThe authors of this axiom were, of course, referring to navigation on land or sea, or even in space. But why should it not be equally applicable to Humanity as a whole?\n\nAnyone with an education would know that the origins of our species are clouded in mystery, buried under the obfuscating weight of five hundred millennia. Half a million years: that's an awful lot of dust. And if we look closely at this dust, we can make out lumps and bumps along the surface which suggest things that _might_ be buried there. But unless we actually brush away these layers of dust, we would never know exactly what lies beneath. When we do, sometimes we find what we imagined we would; other times we find nothing at all. Most of the time, though, we simply reveal new landscapes of dust which seem to bear little relation to the ones above and which might, too, reveal nothing about what remains hidden beneath.\n\nThe mystery of the origins of Humanity is one known to all, although appreciated by few. Any individual fortunate enough to resolve this mystery would not so much earn himself or herself a footnote in some dusty xenoarchaeological journal, as guarantee themselves a place among the greats of science. For that person will not only return to us the sense of place, of identity, that has been denied us these long centuries, but will also thereby enable us\u2014to return to our original metaphor\u2014to know where we, as a species, are going. Not in the sense that evolution, social or physical, has a \"destination\" or a \"purpose\" in \"mind,\" but in the sense that changes we do see occurring could finally be measured against a single fixed reference point\u2014the elusive \"Alpha Point\" (as some scholars refer to it). Without this point, it is inevitable that any observations we make will be corrupted by our own subjective viewpoints, and any objectives we aspire toward will be difficult to achieve.\n\nSome argue that we aren't flying blind at all, that the question of Humanity's origins has already been answered. Such people usually, in my experience, possess barely enough knowledge on the subject to have formed an opinion but a profound insufficiency to prove that opinion to anyone's satisfaction but their own. Exponents of the Out-of-Sol theory spring immediately to mind, along with their archrivals, the Multiple-Genesis-ites. Where they all fall down is in the assumption that we _can_ know such things, that the evidence exists and has been misinterpreted or deliberately suppressed. The truth is in fact that information, once set loose in the massive information flows of the galaxy, is very difficult to contain\u2014especially if it is of such revolutionary nature, and even more especially if it is completely verifiable. Were such evidence to exist, more people would know about it in an hour than will ever hear this speech in my lifetime.\n\nIn short, conclusive evidence simply does not exist.\n\nSo let's look at what we _do_ know...\n\nRoughly five hundred thousand years ago, probably slightly longer, at least four Primordial Castes colonized a large number of systems in a migration we would today call an \"outsweep.\" This region of space contains several hundred stars, including Sol, and is commonly referred to in old records as the Exordium Worlds; we suspect all were visited around the same time, making it difficult to isolate one as a definite home system. Of course, this difficulty might reflect the limitations inherent in our only available method of dating this expansion. In the absence of actual ruins of any kind, only the remnants of the anchor-point network established at this time gives us any kind of date at all, and even that is uncertain after so long.\n\nTo explain why this is uncertain, I always fall back on an old fishing metaphor. I imagine myself casting a line into a pond. On the end of my line is a sinker. As the sinker falls into the water, it creates a disturbance. Ripples spread out from the disturbance with decreasing magnitude until all trace of the disturbance is gone. But the line remains, and it too may create disturbances. My hand may vibrate, or I might tug the rod to attract a fish.\n\nNow, if the line is the crack in space that allows us to break through to hyperspace, and the sinker is the shock that created the hole in the first place, and the surface of the pond is space itself, then the ripples are the echoes not only of the anchor point's creation but of its continued use. Although these ripples in space do not propagate the same way as ripples in water\u2014tending to radiate in the temporal dimensions rather than those of space, forming localized distortions often and misleadingly referred to as standing waves\u2014they are frequently used as navigational aides, or to find an anchor that has disappeared from charts. Xenoarchaeologists can use these ripples too, since their amplitude decreases at a known rate. One can tell at a glance whether an anchor point is a thousand years old, or ten thousand, or if it was created yesterday. That much is very simple.\n\nThe difficulty arises when an anchor point is more than three hundred thousand years old, or has not been used regularly for half that time. The amplitude of the ripples may decrease to the point where they are indistinguishable from the background fluctuations of the universe. While we can still detect ripples from the ancient anchors among the Exordium Worlds, we are unable to tell whether the decrease has come about because of age or disuse. If the former, they might be eight hundred thousand years or more old; if the latter, they might be as young as four hundred and fifty thousand. All we can say with any certainty is that each and every one of the anchors in this area was created around the same time\u2014suggesting that hyperspace technology was only developed _after_ the region was colonized.\n\nWe do not know where this technology came from or who developed it, but we know roughly when it occurred. Four hundred and twenty thousand years ago, Humanity suddenly boiled out of the Exordium Worlds in an outsweep known as the Second Expansion. This surge is much easier to account for. Lines of datable anchor points expand radially from the region, riding on the back of faster-than-light technology and forming the skeletal remains of vast trade routes that literally spanned the galaxy. Humanity, initially in the form of the four known Primordial Castes, spread like ink through water from planet to planet, star to star, jumping across gulfs previously unimaginable and daring even to send probes out of the galaxy itself\u2014probes that have not as yet reached any of their destinations. What they will see, you and I will probably never know. Only the High Caste\u2014the first members of which Transcended during this time\u2014have that possibility open to them.\n\nSome records paint this Second Expansion as a time of great conflict for Humanity. Some researchers suggest that there might have been one single, mighty war, or there might have been innumerable smaller conflicts. Certainly it was a time of tremendous change, during which Humanity began the speciation that has led to such diversity and richness today. Legends were founded and, almost as quickly, forgotten. We will never know exactly what happened in those times, and for that very reason we will never tire of asking the questions.\n\nInevitably, as the Second Expansion slowed, the centers of power shifted away from the Exordium Worlds to the core. The ancient shipping routes shifted too, until they settled into the familiar pattern we now call the Great Lanes. This almost certainly happened around three hundred thousand years ago\u2014although, oddly enough, we cannot independently verify precisely when. The Lanes have been used almost continuously since their creation, and have been frequently re-created, so their ripples show few signs of natural aging. That hasn't stopped people trying to date them, of course, but the results are inconsistent and anomalous. One research group actually dated a pivotal Middle Reach anchor point to be in excess of nine hundred thousand years old\u2014a conclusion which is patently not tenable, inasmuch as it is far older than Humanity itself.\n\nBut this is a small mystery, usually raised to trigger the what-if instinct in all of us. The fact remains that the vast proportion of evidence is in favor of the story as I have told it: that Humanity expanded outward from a single system only slowly at first, then much more rapidly when it discovered anchor point technology\u2014changing as it went. And so we continue to change today, even though expansion halted long ago, with the colonization of the entire galaxy. The only new territories we can dream about are those across the intergalactic gulfs or within our own minds. Many observers note that since the most advanced of Humans always seem to choose the latter path, perhaps that says something about the long-term possibilities for physical expansion. Others point out that High Humans may indeed have found a way to cross the gulfs, but have either not yet returned or choose simply to keep their discoveries a secret.\n\nWhichever way one looks at it, the question remains: Humanity has most likely not reached the end-point of its evolution, and where that end-point _might_ be depends very much on its beginning. Has Humanity always been so changeable, or so insular at its higher reaches? Is the present ratio of High Humans to mundanes, which has been constant for hundreds of thousands of years, one we can assume indefinitely? Or are we just going through a phase\u2014one that might change with little or no warning, plunging the galaxy into chaos once more?\n\nCertainly, attempting to plot trends in the behavior of Humans throughout the last four hundred thousand years has been a thankless task. Castes tend to develop in isolation, usually from a Low form that has itself devolved some time in the past, occasionally with the help of a benefactor's biotechnology. Newly vitalized, the Caste then undergoes a period of expansion, sometimes fragmenting as it goes, leaving pockets of itself behind that might in turn one day also expand, depending on the Caste's ambition or its Batelin Limit. At the same time, many other Castes are behaving similarly, and these expansive types may meet and overlap, or meet and clash, or meet and rebound, depending on their compatibility. The possibilities for trade and conflict are endless, as attested to by the prevalence of the Commerce Artel throughout all reaches of the galaxy.\n\nOther Castes are no longer expansive, having reached the peak of their development, and preparing\u2014whether they know it or not\u2014to change into something new. Lots of Castes advance, devolve, then rise again thinking they're the first to do so; legends and ancient folk tales tell of angels and the like, all metaphors for former glory days that goad them on, upward again. Some Castes disappear, of course, destroyed by war or technological suicide or absorbed by neighbors. Others never devolve, just go on to greatness, Transcending at the peak of their rise to become immortals of a type we can barely comprehend\u2014secretive and elusive, and capable of understanding beyond our wildest imaginings.\n\nThose who don't devolve, disappear, or Transcend, achieve homeostasis in the mundane and remain that way indefinitely. Only the names of their empires change, rising and falling like the vibrations in a cosmic string. The Pristines are most notable among these types. The ancient remnants of Primordial Humanity are reluctant to change, but tolerate it in others\u2014for perhaps that is the way it must be, since to grow, one must change. But to change is to risk devolution, and that risk is a great one.\n\nJust one Caste in ten thousand Transcends. The rest devolve or disappear. One school of thought says that the handful of Castes that achieve homeostasis, apart from the Pristine, might only be delaying inevitable decline. But we cannot be certain of this without greater knowledge of our origins. Perhaps Humanity has always been like this, and will never change.\n\nHere we return to our original question: how can we know where we will end up if we don't know where we began? For an example of how the answer to this might have very real ramifications for all of us, one has only to ask: why _has_ the ratio of High to mundane Humans been so constant? It might very well be a natural state for our race, as most people assume\u2014but it might _not_ be just as easily. And if not, it can tip either way, in favor of either the High Caste or the mundanes. We know that the ratio changed from no High Humans at all to a relatively fixed proportion in the early days of the galaxy, suggesting that Human nature did favor the High Caste at one point. If that trend had continued, we would not exist today: the galaxy would be populated only by the members of the High Caste, everyone else having devolved or Transcended at a rate too great for mundane stocks to replenish themselves. Clearly this has not happened\u2014but _why_ not? What shifted the balance away from the High Caste?\n\nThe most obvious possibility is that there is a natural rate of accretion of which we were not previously aware. The effects of a High Caste death-rate would only become visible after the initial members began to age, and there are indeed High Caste deaths on record. But these are exceedingly rare, and it is generally doubted that they would even brake the initial expansion of the High Caste, let alone halt it entirely. So what else is going on? Perhaps High Humans don't need to die before they need to be replaced: perhaps being old and insular is enough. After all, we are only aware of _active_ High Humans; there may be many more who choose not to communicate with anyone, or who have entered a state of prolonged hibernation, or have undergone transformation to another plane of being we cannot imagine.\n\nWhatever the truth, this issue raises a disturbing possibility: that the High Caste maintains the ratio artificially, by either limiting its numbers somehow or maintaining an artificially high rate of mundane replenishment. The latter, of course, might simply be to restock its own numbers\u2014for if High Caste expansion continued unchecked and there were no mundanes left, where would future members come from? Or it might be to give them something to watch, just as some mundane and Low Castes keep inferior species as pets.\n\nIs this our ultimate fate, then, to amuse, or to act as breeding stock for new High Humans? We will never know until we learn the truth about our origins. And to do _that,_ we need more data.\n\nThis is where you come in. As alumni from institutions all over the galaxy taking the bold leap of faith into the rarefied air of xenoarchaeology, your job is to probe deep into these questions and to expose the truth. Or if not the truth, then a fragment of it. Or if not even a fragment of the truth, then another question for someone else to answer. This process is ongoing, and will outlast me just as it has outlasted two hundred thirty-seven Provosts before me. It will probably outlast you, too, and the ones who follow you. Perhaps future xenoarchaeologists will look back on our work with an indulgent smile for our ignorance\u2014or perhaps they will regard our work as cornerstones in the great edifice of understanding under constant construction. I cannot say which will be the case, just as my predecessors did not know. All I _can_ say is that these questions are worth asking, even if we can never answer them. Not knowing where we came from does not stop us from moving on\u2014and that is perhaps the most important thing about our race that has brought us to where we are today, wherever that is. We are not inclined to stay still.\n\nOnce, millions of years in the past, a small, barely bipedal creature rose up on its hind legs and squinted at the stars above. Well, we own those stars now, and we're still moving. Only time will tell where we will ultimately end up....\n****\n\n**GLOSSARY**\n\n**A-14 Higher Collaboration Network:** an amalgamation of core-based High Caste members whose intentions include attempting to establish an objective frame of reference with respect to Humanity's occupation of the galaxy. _The Objective Reference Calendar_ is one result of this work.\n\n**A-P cannon:** a weapon that fires accelerated particles of various types. Common on spacefaring warships.\n\n**Absenger, Burne:** chief liaison officer, COE Armada.\n\n**adytum:** Skehan Heterodox term for the control room of a consistory vessel.\n\n**Alpha Aurigae:** an ancient system, the precise location of which is presently unknown.\n\n**Alpha Point:** a name for the single point in time and space from which some xenoarchaeologists surmise Humanity evolved.\n\n**Ana Vereine, IND:** the first of a new class of warships\u2014the Marauder\u2014manufactured by the Dato Bloc as part of the Andermahr Experiment. Its design incorporates a captain surgically interfaced with the ship. Once part of the Ethnarch's Military Presidium, it is now an independent vessel registered to Morgan Roche.\n\n**anchor drive:** the usual means of crossing interstellar space, but by no means the only one (see **slow-jump).** Indeed, the anchor method has undergone several radical redesigns over time; current technology is rated at 49th-generation.\n\n**anchor points:** regions of \"weakened\" space from which translation to and from hyperspace is both easier and less energy-expensive; jumps from anchor points are therefore of a greater range than from \"normal\" space and usually terminate in another anchor point. They are typically located near inhabited systems (but far enough away to avoid distortion by background gravitational effects) or in locations in deep space which are considered strategically important. There are approximately ten thousand million anchor points currently in existence\u2014one for roughly every ten stars, scattered across the galaxy.\n\n**Andermahr Experiment:** a covert project specializing in cybernetic interfaces designed to allow mind and machine to merge. Founded by Ataman Ana Vereine, who desired captains that were as much a part of their ships as was the anchor drive\u2014an integral, reliable system rather than a flesh and blood afterthought. Continued in secret until the Ataman Theocracy emerged from the COE as the Dato Bloc. Culminated in the DBMP _Ana Vereine,_ the first Marauder-class warship, with Uri Kajic its captain.\n\n**Ansourian, Alta:** only child of Atul Ansourian.\n\n**Ansourian, Atul:** adviser and self-styled _eminence grise_ to the administer of Perdue Habitat, Inderdeep Jans.\n\n**Apostle:** a Hum vessel in the service of the Disciples of Evergence.\n\n**Aquareii:** a High Human renowned for being the most powerful member of the Interventionist faction.\n\n**Armada:** see **COE Armada.**\n\n**Ascensio:** the homeworld of Morgan Roche.\n\n**Asha:** the single, warlike deity of the main Kesh religion.\n\n**Ataman Theocracy:** a tightly knit empire that existed as an independent entity until its absorption into the COE after the Second Ataman War in '442 EN. After several centuries, it eventually seceded as the Dato Bloc ('837 EN).\n\n**AVS-38 & AVS-44:** two of the _Ana Vereine'_ s large contingent of scutters.\n\n****\n\n**Basigo:** a Caste not native to the COE region.\n\n**Batelin Limit:** the ceiling above which the complexity of a nation exceeds the biological capabilities of the individuals inhabiting it. In the case of the Pristine Caste, the value of the Batelin Limit is approximately three and a half thousand systems.\n\n**Black Box:** the generic term for an AI. Usually abbreviated to \"Box.\"\n\n**Bodh Gaya:** former capital system of the COE and the Dominion. Its second moon houses the Military College of the COE Armada.\n\n**Box, the:** an AI commissioned by COE Intelligence. Its binary identification number (JW111101000) is one digit longer than normal, indicating its unique status. Created by the High Human known as the Crescend, the Box is designed to infiltrate and subvert all available systems, thereby increasing its own processing powers until, at its most powerful, it resembles its creator. Once thought to be contained within a small black valise, it is now known to inhabit the cells of Morgan Roche.\n\n**Buk, Enver:** the COE Eupatrid prior to Felix Gastel.\n\n****\n\n**Calendar:** the galactic standard timekeeping method consists of: 100 seconds per minute, 100 minutes per hour, 20 hours per day, 10 days per week, 4 weeks (40 days) per month, 10 months (400 days) per year. All dates are expressed in the form of year (usually abbreviated to the last three digits, i.e. '397), month, and day from the _Ex Nihilo_ reference point. See also **Objective Reference Calendar.**\n\n**Cane, Adoni:** the occupant of an unidentified life-support capsule recovered by the COEA _Midnight_ near Ivy Green Station anchor point while en route to Sciacca's World. A genetically modified combat clone designed to mimic a Pristine Human, his origins are known not to lie with the Sol Apotheosis Movement, although who did create him, and many others like him, remains a mystery.\n\n**Cane, Adoni:** field admiral, Old Earth Advance Guard; received a military Star for extraordinary acts of valor against the enemy.\n\n**Castes:** following the speciation of the Human race, numerous Castes have proliferated across the galaxy. These Castes are too numerous to list, but they can be classified into three broad groups: High, Low, and mundane (which includes Pristine and Exotic). There are seven predominant Exotic Castes to be found in the region surrounding the COE: Eckandar, Hum, Kesh, Mbata, Olmahoi, and Surin.\n\n**Catiph:** a High Human.\n\n**Ceyle's Hub:** former home of Advocate Janil Coriett.\n\n**Chase, Auberon:** head of COE Intelligence.\n\n**Clarke Cylinder:** an ancient habitat.\n\n**COE:** see **Commonwealth of Empires.**\n\n**COE Armada, the:** the combined armed forces of the COE, responsible for external security. Active soldiers are referred to as Marines.\n\n**COE Enforcement:** the policing body responsible for security and information gathering within the COE. Field agents are referred to as Enforcers.\n\n**COE High Equity Court:** the department responsible for intersystem justice within the COE. Its usual purpose is to settle territorial disputes.\n\n**COE Intelligence:** the body responsible for information gathering outside the COE. Originally and still nominally a subdepartment of the Armada, but an independent body in practice.\n\n**COE Intelligence HQ:** the command center of COE Intelligence, a large, independent station located in deep space near the heart of the Commonwealth.\n\n**COE Military College:** the main training institution of COE Armada personnel; situated on the second moon of Bodh Gaya.\n\n**COEA:** COE Armada vessel identification prefix.\n\n**COEI:** COE Intelligence vessel identification prefix.\n\n**Commerce Artel:** a galaxy-wide organization devoted to initiating and coordinating trade between Castes and governments that might otherwise have no contact. It prides itself on remaining aloof from political conflict yet has some strict behavioral standards to which it expects its customers to adhere (such as the Warfare Protocol). Structurally, it is divided into chapters managed by indigenous Caste-members with only loose control from above. It has strong links, locally, with the Eckandar Trade Axis.\n\n**Commonwealth of Empires:** often abbreviated to COE or Commonwealth. A relatively ancient Pristine nation currently in its 40th millennium of nominal existence\u2014\"nominal\" in that the membership of the COE is fluid by nature, with provinces joining and seceding on a regular basis. It has had many different capitals and its borders have changed radically over the centuries. Indeed, it has drifted with time, and now occupies territories quite remote from its original location. One thousand inhabited systems currently fall under its aegis, and another three thousand uninhabited systems have been annexed. It is ruled by a democratically elected Eupatrid and a council of representatives who, when united, wield supreme executive power. Its security departments include Intelligence, Armada, and Enforcement.\n\n**Coriett, Janil:** an advocate formerly of Ceyle's Hub, currently employed in Environment Control of the SHCV _Phlegethon._\n\n**Crescend, the:** a High Human of some note and great history. His time of Transcendence is not recorded. Little is known about him, beyond the facts that he is the founder and overseer of Trinity, an ally of the COE, and a key supporter of the Interventionist movement. He is assumed to be a singular entity simply because the first person singular is his pronoun of choice.\n\n**crypt:** the Skehan Heterodox term for the engineering rooms of a consistory vessel.\n\n****\n\n**Dark Stressor, HHAB:** a compact habitat of ill-repute, known to be a gathering-point for mercenaries.\n\n**Dato Bloc:** an independent nation founded on the ruins of the Ataman Theocracy that recently broke free of the COE. Although not hierocratic in nature, the Ethnarch exerts a strict rule. Its security departments include the Ethnarch's Military Presidium and the Espionage Corps.\n\n**Daybreak, COEI:** a courier vessel belonging to COE Intelligence.\n\n**DBMP:** vessel identification prefix for the Ethnarch's Military Presidium.\n\n**De Bruyn, Page:** former head of Strategy, COE Intelligence.\n\n**Disciples of Evergence:** a covert organization with Hum connections devoted to serving the enemy.\n\n**disrupters:** see **hyperspace disrupters.**\n\n**Dreher, Ralf:** a name from ancient records connected to the enemy.\n\n**Drys:** a caste not indigenous to the COE region.\n\n****\n\n**E-shield:** an electromagnetic barrier designed to ward off particle and energy weapons. Used mainly by medium-to-large spacefaring vessels.\n\n**Eckandar:** ( **Eckandi** , adj & sing, n): a Caste flourishing in the regions surrounding the COE. Its members are typified by their slight size, gray skin, bald scalps, and unusual eyes. They are a gregarious Caste, preferring trade and communication over conquest. They are also well-advanced in genetic science. Their past stretches back beyond that of the COE, although they lack the continuity of history that strong nationhood often provides. Their sole uniting body is the Eckandar Trade Axis.\n\n**Eckandar Trade Axis:** the main society of the Eckandi Caste, devoted, much like the Commerce Artel (with which it has close ties), to facilitating free and nondiscriminatory trade with and between the COE and its neighbors.\n\n**Egarr, Gurion:** senior councilor to Inderdeep Jans.\n\n**EN:** see ** __ Ex Nihilo.**\n\n**enemy:** Interim Emergency Pristine Council shorthand for the clone warriors.\n\n**Enforcement:** see **COE Enforcement.**\n\n**Enforcer:** see **COE Enforcement.**\n\n**epsense:** an ability encompassing telepathy and empathy. The ritual training of epsense adepts generally takes decades and incorporates elements of sensory deprivation. Note: telekinesis and precognition are not covered by epsense and are assumed to be nonexistent. Skilled utilizers of epsense are referred to as **epsense adepts** or **reaves.**\n\n**Erojen:** a town on an outpost far from the heart of the Surin domain.\n\n**Esperance, SAS:** the fighter piloted by Defender-of-Harmony Vri.\n\n**Espionage Corps:** see **Dato Bloc.**\n\n**Espire-Mavrodis Coalition:** one of the IEPC's major allies.\n\n**Essai:** home of the Surin Agora.\n\n**Ethnarch:** the title of the leader of the Dato Bloc.\n\n**Ethnarch's Military Presidium:** see **Dato Bloc.**\n\n**Eupatrid:** the title of the chief executive officer of the COE.\n\n**Ex Nihilo:** refers to the date upon which the COE is believed to have been founded. Evidence exists to cast doubt upon the accuracy or relevance of this date\u2014notably the fact the Commonwealth as a single body did not exist at all between the 13th and 15th millennia\u2014but the date remains as a reference point. Usually abbreviated to EN.\n\n**Exordium Worlds:** the region first colonized by the Primordial Castes.\n\n**Exotic:** any mundane Caste that differs physiologically from the Pristine. There are a vast number of Exotic Castes, and, although no one type of Exotic comes close to outnumbering Pristine Humans, the Exotics as a whole mass far greater than Pristines alone.\n\n****\n\n**fane:** the Skehan Heterodox term for a ship's bridge.\n\n**Far Reaches:** the name of the outermost fringes of the Outer Arms.\n\n**Fathehi Consulate:** a habitat sent to Sol System by a Caste originating on the far side of the galaxy.\n\n**flicker-bombs:** devices used in space warfare to attack an enemy vessel. Employing the fact that small masses (under a few kilograms) can slow-jump a small distance within a gravity well, these missiles skip in and out of space on their way to their target, which, it is hoped, they will materialize within, causing massive amounts of damage. They are easily deflected by hypershields, however, which form a barrier in hyperspace that no such weapon can cross.\n\n**40th millennium:** the current millennium in the history of the COE. See ** _Ex Nihilo._**\n\n****\n\n**Gastel, Felix:** the current Eupatrid of the COE.\n\n**Gegenschein, COEI:** a cruiser owned by COE Intelligence.\n\n**God's Monkey, HIC:** a small vessel operated by the Disciples of Evergence.\n\n**grayboots:** see **Olmahoi retribution squad.**\n\n**Great Lanes:** the network of shipping routes delineated by anchor points across the galaxy.\n\n**Ground Corps:** a military body referred to in ancient records.\n\n**Guild of Xenoarchaeologists: (GOX)** an ancient and highly regarded professional organization of xenoarchaeologists.\n\n**Guo Sodality:** an organization from the Middle Reaches dedicated to the study and training of epsense adepts.\n\n**Guidon Habitat:** sibling-habitat to Perdue Habitat.\n\n****\n\n**H'si F'ta:** a Caste known to the IEPC.\n\n**Haid, Ameidio:** former transportee, Sciacca Penal Colony, and ex-mercenary.\n\n**Heidik, Jelena:** the name by which the clone warrior responsible for the destruction of Palasian System identified herself.\n\n**Heidik, Jelena:** in ancient times, a general and recipient of a Mars St. Selwyn Medal for valor against the enemy in Alpha Aurigae.\n\n**Hek'm:** the Olmahoi Caste homeworld.\n\n**Heresiarch:** Skehan Heterodox name for the commanding officer of a consistory vessel.\n\n**High Humans** (or **High Castes** ): Superior intelligences that have evolved (Transcended) from the mundane. Enormously long-lived and farseeing, they concentrate on issues quite removed from the rest of the galaxy; indeed, due to their enormous scale, they are the only beings capable of comprehending the galaxy in its entirety. They generally leave mundanes alone, to let them progress (and, ultimately, to Transcend) in their own time. See **Castes** and **Transcendence.**\n\n**High Equity Court:** see **COE High Equity Court.**\n\n**Hurn:** a Caste typified by ritual and complexity. In appearance they are lean and muscular, averaging greater than Pristine height. They are predisposed toward music and mathematics. Socially they prefer oligarchies with a baroque middle class.\n\n**Hutton-Luu System:** a much-disputed system of the COE near its border with the Dato Bloc. See **Sciacca's World.**\n\n**hypershield:** a barrier erected in hyperspace to deflect or inhibit the passage of anything traveling by that medium. Commonly used as a prophylactic against hyperspace weapons. Hypershields operate under a maximum volume constraint: i.e. they will only operate as intended under two thousand cubic kilometers.\n\n**hyperspace disrupters:** a form of hypershield which actively combats incoming hyperspace weapons, such as flicker-bombs. Unlike anchor points, which \"weaken\" space, disrupters do the opposite, making it more difficult for anything nearby to emerge from hyperspace.\n\n****\n\n**i-Hurn Uprising:** a civil dispute which broke out between two rival factions of the Hurn Caste.\n\n**IEPC: see Interim Emergency Pristine Council.**\n\n**imaret:** a Surin single-passenger fighter.\n\n**Imi:** a system in which Ameidio Haid once worked.\n\n**IND:** independent vessel identification prefix.\n\n**Intelligence:** see COE **Intelligence.**\n\n**Interim Emergency Pristine Council:** (IEPC) an organization that exists only in times of greatest duress for the Pristine Caste. With no fewer than thirty nations required to call it into being, it exists only so long as the threat remains.\n\n**Interventionism:** a movement among High Humans\u2014and some mundanes\u2014that advocates closer links between High and mundane Castes. See **The Crescend.**\n\n**irikeii;** one of very few Olmahoi \"sound-thoughts\" which can be equated with audible words; often translated as _unnamed_ or _unnamable._\n\n****\n\n**Jans, Ehud:** Inderdeep Jans's father and the previous administer of Perdue Habitat.\n\n**Jans, Eir:** biological daughter of Ehud Jans.\n\n**Jans, Inderdeep:** administer of Perdue Habitat.\n\n**Ju Mandate, Second:** a government represented by Assistant Vice Primate Rey Nemeth.\n\n**JW111101000:** see **Box, the.**\n\n****\n\n**Kajic, Uri:** former captain, **** DBMP _Ana Vereine,_ physically bonded to his ship.\n\n**Katajalin Serai:** a loose collection of trade and security vessels.\n\n**Ken'an:** a mercenary operating from the habitat _Dark Stressor._\n\n**Kesh:** the most primal of the local Castes. The Kesh are typically warlike and predisposed toward violence. In appearance, they tend to be larger than the Pristine average and have mottled, multicolored skin. Their social structure is heavily ritualized, with a strong tribal or family base. They are known for being highly racist.\n\n**Kindling, IND:** the name given by Page De Bruyn to her stolen fighter, formerly registered as TBC-14 and owned by COE Intelligence.\n\n**Kukumat:** one of a gas giant pair occupying the sixth planetary orbit around Palasian System; the pair shares a single moon.\n\n****\n\n**LaGoc:** a Caste that contributed a mobile barracks to the crisis in Sol System.\n\n**Lemmas:** A Hum reave employed by the Disciples of Evergence.\n\n**Lenz, Uyeno:** a mercenary frequently to be found on _Dark Stressor._\n\n**Lleshi, Sadoc:** captain, Ground Corps; recognized for excellence in battle against the enemy in ancient times.\n\n**Low Castes:** devolved mundane Humans. These animal-like creatures come in many forms and occupy many niches across the galaxy. Some evolve back up to mundane status, given time and isolation, while others become extinct as a result of the forces that led to their devolution in the first place.\n\n**Lucence-2, COEA:** COE Armada Escort & Assault Craft.\n\n****\n\n**M'taio System:** a system notable for its Caste Wars in recent times.\n\n**Maii:** Surin epsense adept.\n\n**Marauder:** an experimental class of warship developed by the Dato Bloc. See **DBMP __ Ana Vereine.**\n\n**Marines:** see **COE Armada.**\n\n**Mars St. Selwyn Medal:** an honorific bestowed in ancient times.\n\n**Mbata:** ( **Mbatan,** adj & sing, n): a well-regarded Caste known for its peace-loving and familial ways. In appearance they resemble the ursine species, larger and stronger than the Pristine. Their culture is egalitarian and open to trade.\n\n**Middle Reaches:** the region of medium stellar density between the Outer Arms and the galactic core.\n\n**Midnight, COEA:** COE Retriever-class frigate.\n\n**Military Star:** an honorific bestowed in ancient times.\n\n**Military Presidium:** see **Dato Bloc.**\n\n**minaret:** Skehan Heterodox term for the prow of a consistory vessel.\n\n**Multiple Genesis Theory:** a theory that the various Castes of Humanity evolved\u2014or at least emerged into the greater galaxy\u2014from several systems simultaneously.\n\n**mundane Castes:** Castes of Humanity that are essentially similar to the Pristine in terms of size, mental capacity, worldview, etc. Naturally there is a spectrum of types across the mundane Caste\u2014from the highly evolved (some might say near-Transcendent) Olmahoi, through the socially complex Surin and Hurn Castes, to the Eckandar and Pristine Castes with their societies based on trade and empire-building, and beyond, via the earthy Mbata, to the relatively primal Kesh. Mundanes are typically short-lived (a century or so, when allowed to age naturally) and build empires up to four or five thousand systems in size. There is a ceiling of complexity above which mundanes rarely go without Transcending. See **High Humans** and **Batelin Limit.**\n\n**Murnane, Esko:** plenipotentiary envoy sent by the citizens of Pompili to stand on the IEPC.\n\n**Murukan:** one of a gas giant pair occupying the sixth planetary orbit around Palasian System (see **Kukumat** ).\n\n****\n\n**n-body:** the epsense \"counterpart\" to the physical body.\n\n**n-space:** a word used by epsense theorists to describe an environment completely empty of thought.\n\n**Nemeth, Rey:** Assistant Vice Primate sent by the Second Ju Mandate to stand on the IEPC.\n\n**Noske:** a Caste not normally found in the COE region.\n\n**Nyberg:** the founder of school of epsense training regarded as second-rate by superior adepts.\n\n****\n\n**Objective Reference Calendar:** a system of date-keeping established by the A-14 Higher Collaboration Network.\n\n**Old Earth Advance Guard:** a military body mentioned in ancient records.\n\n**Olmahoi:** an exotic Caste that communicates entirely by epsense. Physically they are of similar size to Pristines, but are much stronger; their skin is black and they possess little in the way of distinguishing features, apart from the epsense organ which dangles like a tentacle from the back of the skull. Their social structure is too complex to explore in detail here. They are renowned fighters, capable of feats of great skill, yet also possess a capacity for peace far in excess of any other local Caste.\n\n**Olmahoi retribution squad:** renowned fighters able to combine perfectly their physical and epsense abilities. Also known as \"grayboots.\"\n\n**Outer Arms:** the low stellar-density regions of the galaxy between the Middle and Far Reaches.\n\n**Out-of-Sol theory:** a theory suggesting that all the Castes of Humanity originated on one world.\n\n**outrigger:** a unique Caste of miner\/explorer found in sparsely populated systems; living within all-suits that double as mobile homes, outriggers typically scout uncharted dark body halos and asteroid belts, looking for viable mineral sources, which they then either mine or report to a centralized authority (if any) for a modest fee. Outriggers are notoriously self-sufficient, avoiding even other outriggers as much as possible, and have been known to exist for years out of contact with another being. Spending much of their lives drifting in hibernation between dark and cometary bodies, some live longer than three centuries. Few outriggers have family names, coming as they do from such small communities that single given names usually suffice.\n\n**outsweep migrations:** brief, outward surges by expansionist empires. These usually occur in the crowded environment of the core or Middle Reaches, in the direction of the Outer Arms.\n\n****\n\n**Pacecca:** an overseer of the docks of Perdue Habitat.\n\n**Palasian System:** a system of the COE quarantined by the COE Armada as a result of an enemy outbreak, then destroyed by an ancient Kesh device (see **Solar Envelope** ).\n\n**Paraselene, COEA:** a COE cruiser stationed near Perdue Habitat.\n\n**Paz:** an educative device once owned by Morgan Roche.\n\n**Perdue Habitat:** a habitat populated by approximately **** 11,000 **** people, predominantly Vax Caste, stationed to Sol System during the crisis. Its administer, Inderdeep Jans, is known to have been reluctant to commit her resources to solving the crisis.\n\n**Phlegethon, SHCV:** a Skehan Heterodox consistory vessel from the far side of the galaxy.\n\n**Pompili:** a founding nation of the IEPC.\n\n**Primordial Castes:** precede the earliest confirmed records, half a million years ago. Little is known about them, except that they exist; ruins of several unique types are to be found throughout the galaxy. They are called Castes A, B, C, and D, for even their names are unknown. (See Appendix.)\n\n**Pristine Caste:** the form of Humanity which most closely resembles the original race that evolved an unknown time ago on an unknown planet somewhere in the galaxy. The Pristine Human Genome, handed down from antiquity and regarded with near-veneration, is stored in innumerable places among the civilized worlds. Pristines themselves, however, are accorded no special status.\n\n****\n\n**Quare, Oren:** assistant to Overseer Pacecca, Perdue Habitat.\n\n****\n\n**Random Valence:** a habitat stationed close to Perdue Habitat.\n\n**reave:** see **epsense.**\n\n**Rebuli:** a Caste antagonistic to the cause of the IEPC.\n\n**Rench:** dockmaster of Perdue Habitat.\n\n**Reshima System:** a system in which Ameidio Haid once worked.\n\n**Roche, Morgan:** former commander, COE Intelligence.\n\n**Rond-Spellor Outlook:** home of Hue Vischilglin.\n\n**Rufo, Linegar:** renowned xenoarchaeologist\n\n****\n\n**Saa-hurod:** a Caste maintaining an outpost in Sol System during the crisis.\n\n**Sciacca's World:** the only habitable world of the Hutton-Luu System; once an agricultural planet of the Dominion, now a desert penal colony of the COE **** ( **Sciacca Penal Colony** ). Its ring of moonlets\u2014the Soul\u2014is owned and mined by DAOC Inc.\n\n**scutter:** a small, swift spacegoing vessel with many uses, both military and civilian; also known as a singleship.\n\n**Sebettu, SRF:** Kesh destroyer.\n\n**Second Expansion:** an ancient outsweep known to have led from the Exordium Worlds into the greater galaxy.\n\n**Siriote:** a government antagonistic to the IEPC.\n\n**Skehan Heterodox:** a tolerant theocracy on the far side of the galaxy, significantly more advanced than the COE.\n\n**slow-jump:** a common alternative to the anchor drive that utilizes similar technology. Most ships with an anchor drive can slow-jump if necessary. It is essentially a jump through hyperspace from any point in real space. A certain degree of kinetic energy is required before translation can be achieved, so ships must accelerate for some time beforehand. Even then, the hyperspace jump is short-lived, and the vessel emerges soon after (typically less than a light-year away from its departure point) with significantly less kinetic energy. The process must be repeated from scratch if another slow-jump is required. As a means of crossing interstellar space, it is inefficient and time-consuming, hence its name. Slow-jumping becomes increasingly nonviable closer to a gravity-well, but more efficient as mass (of the traveling object) decreases.\n\n**Sol Apotheosis Movement:** a quasi-religious organization devoted to the pursuit of Transcendence via genetic manipulation and biomodification that reached its peak and was destroyed in the 37th millennium. Its fanatical followers were a source of unrest for decades, until an alliance was formed among their neighbors dedicated to putting a stop to them. In '577 EN, at the climax of the Scion War, a flotilla of allied forces encircled their base, which the Movement destroyed in order to prevent its capture. The resulting explosion annihilated them as well, of course, but also decimated the flotilla. Of the four stations involved in the battle, only one survived, and that was severely damaged. So embarrassed was the alliance that the leaders of the day ordered the event stricken from history. They even closed the anchor point leading to the system to prevent anyone learning what occurred there. Nothing survived of the base, and the rest of the system is an unsalvageable ruin.\n\n**Sol System:** an uninhabited system in a nonaligned region near the Dato Bloc, one known for its antiquity and a possible contender for birthplace of the Human race. Former home of the Sol Apotheosis Movement and many other such sects. (See Appendix.)\n\n**Sol Wunderkind:** genetically modified clone warriors designed and bred by the Sol Apotheosis Movement.\n\n**Solar Envelope:** a device intended to provide a jump shield large enough to enclose an entire solar system. Two prototypes were built by an early Kesh Government. \"Asha's Gauntlet\" was used on one system, with disastrous results: the system's primary sun, modified to power the Envelope, was exhausted within two months; the entire system collapsed shortly thereafter. Useless as a defensive weapon, and forbidden by the Convention on Extraordinary Weapons, the second Gauntlet remained in the hands of the Kesh until it was used in Palasian System.\n\n**Soul, the:** the local name for the orbiting ring of mineral-rich moonlets girdling Sciacca's World.\n\n**spine:** the collective noun used to describe a loosely linked group of outriggers; from their means of traveling between systems, on the back of a naked real-space drive known as \"the spine.\"\n\n**squt:** a Surin word for someone who is both foolish and closed-minded.\n\n**Stryki:** an epsense adept from Taborca.\n\n**Surin:** a relatively minor Caste found in the regions surrounding the COE. They exist in isolated clumps overseen by a governing body that guides rather than rules. They are social beings yet fond of isolation, giving them a reputation for occasional aloofness. They are technically accomplished, especially in the biological sciences. In stature, they tend to be slight and have hair covering much of their bodies. It is occasionally speculated that they have re-evolved from Low Caste status.\n\n**Surin Agora:** the ruling body of the loosely knit Surin nation.\n\n****\n\n**Taborca:** a region in the Middle Reaches.\n\n**Tade, Rejuben:** 238th Provost of the Guild of Xenoarchaeologists (see Appendix).\n\n**TBC-14:** see ** __ Kindling.**\n\n**Teh, Sylvester:** transportee, Sciacca Penal Colony.\n\n**Tocharia 13:** a habitat maintained by members **** of the **Zissis Caste.**\n\n**Transcend:** to break free of the constraints of mundane Humanity. A being or Caste that has Transcended typically has an extremely long life span and spreads its consciousness across a number of primary containers\u2014such as neural nets, quantum data vats, and the like. Transcended entities, singular or collective, are referred to as High Human and accorded the highest status.\n\n**Transcendence:** the state of being Transcended. Usually achieved when consciousness research and computer technology overlap, allowing an organic mind to be downloaded into an electronic vessel, thereby gaining the potential for unlimited growth.\n\n**Trezise, Salton:** senior aide to Auberon Chase and envoy to the IEPC.\n\n**Trinity:** the world on which AIs are made in the region dominated by the COE. The AI factory was founded and is overseen by the High Human known as the Crescend.\n\n**turcite:** a type of high explosive.\n\n****\n\n**Ulterior:** a covert organization operating within\u2014but not without the official disapproval of\u2014the IEPC.\n\n**Ustinik, Alwen:** a representative of the Commerce Artel.\n\n****\n\n**Vacishnou:** homeworld of the Vax Caste.\n\n**Vax:** a Caste renowned for its peculiar breeding customs.\n\n**Veden, Makil:** an Eckandar Trade Axis citizen and Commerce Artel ex-delegate. Deceased.\n\n**Vischilglin, Hue:** co-adjutant sent by the Rond-Spellor Outlook to assist the IEPC.\n\n**Vri:** a Surin soldier, who has forgone his family name and taken the tide \"Defender-of-Harmony.\"\n\n**Wamel:** a Hurn pilot employed by the Disciples of Evergence.\n\n**Warfare Protocol:** the code by which war is conducted within and between those nations that trade with the Commerce Artel.\n\n**Wehr, Vani:** earned an Honorable Mention in ancient times for quick thinking on the Clarke Cylinder.\n\n**Weryn:** deity of the Skehan Heterodox.\n\n****\n\n**Xarodine:** an epsense-inhibiting drug.\n\n**Xumai, Jancin:** a Surin soldier who has taken the title \"Fighter-For-Peace.\"\n\n****\n\n**Yemena:** a Caste not native to the COE region.\n\n**Yugen, Frane:** leader of Random Valence.\n\n****\n\n**Zissis:** a Caste not indigenous to the COE region.\n\nAll rights reserved, including without limitation the right to reproduce this ebook or any portion thereof in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.\n\nThis is a work of fiction. Names, characters, places, events, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.\n\nCopyright \u00a9 2001 by Sean Williams and Shane Dix\n\nCover design by Open Road Integrated Media\n\nISBN 978-1-4976-1153-5\n\nThis edition published in 2014 by Open Road Integrated Media, Inc.  \n345 Hudson Street  \nNew York, NY 10014  \nwww.openroadmedia.com\n\n**EVERGENCE**\n\nFROM OPEN ROAD MEDIA\n\nFind a full list of our authors and\n\ntitles at www.openroadmedia.com\n\nFOLLOW US\n\n@OpenRoadMedia\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Larry B. Harrison and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n\n\n\n\n  His Last Week\n\n  THE STORY OF THE PASSION AND\n  RESURRECTION OF JESUS\n\n  IN THE WORDS OF THE FOUR GOSPELS\n\n  PREPARED BY\n  WILLIAM E. BARTON, THEODORE G. SOARES\n  SYDNEY STRONG\n\n  HOPE PUBLISHING COMPANY\n  CHICAGO\n\n\n\n\nPREFACE.\n\n\nONE HUNDREDTH THOUSAND.\n\nThe evangelists have devoted one-third of the Gospel record to our Lord's\nPassion and Resurrection. A comparison of the four narratives clearly\nindicates the order of events upon the several days of the Holy Week. The\ndevotional reading of the story is a most natural and helpful observance of\nthe Easter season. As an aid to such observance this booklet has been\nprepared. It is the story, day by day, of the last week in our Lord's\nearthly life in the words of the four evangelists, containing all that they\nrecord, but without repetition. Messrs. Thomas Nelson and Sons have\ngenerously co-operated in permitting the use of the best translation.\n\nOriginally planned for the churches of all denominations in a single\ncommunity, the booklet has proved a blessing to many thousands of\nChristians. May this new edition help in the fulfillment of the great\npurpose which the Gospel epilogue expresses.\n\n    ------------------------------------------------------\n    | COPYRIGHT, 1905, BY THE OAK PARK PASTORS' UNION.   |\n    | THE TEXT OF THE AMERICAN STANDARD REVISED BIBLE,   |\n    | COPYRIGHT 1901, BY THOMAS NELSON AND SONS, IS USED |\n    | BY SPECIAL ARRANGEMENT AND WITH THEIR PERMISSION.  |\n    ------------------------------------------------------\n\n\n\n\nHIS LAST WEEK\n\n\nGOING UP TO JERUSALEM.\n\nAnd it came to pass when the days were well nigh come that Jesus should\nbe received up, he stedfastly set his face to go to Jerusalem. And he\ndeparted from Galilee, and passed through the borders of Samaria and\nGalilee, and came into the borders of Judaea beyond the Jordan. And great\nmultitudes followed him, and he healed them there.\n\nAnd they were on the way, going up to Jerusalem; and Jesus was going\nbefore them: and they were amazed; and they that followed were afraid.\n\nAnd he took again the twelve, and began to tell them the things that\nwere to happen unto them, saying, \"Behold, we go up to Jerusalem; and\nthe Son of man shall be delivered unto the chief priests and the\nscribes; and they shall condemn him to death, and shall deliver him unto\nthe Gentiles; and they shall mock him, and shall spit upon him, and shall\nscourge him, and shall kill him; and after three days he shall rise again.\"\n\nThomas therefore, who is called Didymus, said unto his fellow-disciples,\n\"Let us also go, that we may die with him.\"\n\nAnd he entered and passed through Jericho and went on before, going up\nto Jerusalem.\n\nNow the passover of the Jews was at hand: and many went up to Jerusalem\nout of the country before the passover, to purify themselves. They\nsought therefore for Jesus, and spake one with another, as they stood in\nthe temple, \"What think ye? That he will not come to the feast?\"\n\nNow the chief priests and the Pharisees had given commandment, that, if\nany man knew where he was, he should show it, that they might take him.\n\n\nTHE FEAST AT BETHANY.\n\nJesus therefore six days before the passover came to Bethany, where Lazarus\nwas, whom Jesus raised from the dead. So they made him a supper there in\nthe house of Simon the leper; and Martha served: but Lazarus was one of\nthem that sat at meat with him. Mary therefore took a pound of ointment of\npure nard, very precious, and anointed the feet of Jesus, and wiped his\nfeet with her hair; and the house was filled with the odor of the ointment.\n\nBut Judas Iscariot, one of his disciples, that should betray him, saith,\n\"Why was not this ointment sold for three hundred shillings, and given\nto the poor?\"\n\nNow this he said, not because he cared for the poor; but because he was\na thief, and having the bag took away what was put therein.\n\nJesus therefore said, \"Suffer her to keep it against the day of my\nburying. For the poor ye have always with you; but me ye have not\nalways. She hath done what she could; she hath anointed my body\nbeforehand for the burying. And verily I say unto you, Wheresoever the\ngospel shall be preached throughout the whole world, that also which\nthis woman hath done shall be spoken of for a memorial of her.\"\n\nThe common people therefore of the Jews learned that he was there: and\nthey came, not for Jesus' sake only, but that they might see Lazarus\nalso, whom he had raised from the dead. But the chief priests took\ncounsel that they might put Lazarus also to death; because that by\nreason of him many of the Jews went away, and believed on Jesus.\n\n\n\n\nPALM SUNDAY--THE DAY OF TRIUMPH.\n\n\nTHE TRIUMPHAL ENTRY.\n\nOn the morrow when they drew nigh unto Jerusalem, unto Bethphage and\nBethany, at the mount of Olives, he sendeth two of his disciples, and saith\nunto them, \"Go your way into the village that is over against you: and\nstraightway as ye enter into it, ye shall find a colt tied, whereon no man\never yet sat; loose him, and bring him. And if any one say unto you, 'Why\ndo ye this?' say ye, 'The Lord hath need of him; and straightway he will\nsend him back hither.'\"\n\nNow this is come to pass, that it might be fulfilled, which was spoken\nthrough the prophet, saying,\n\n        \"Tell ye the daughter of Zion,\n        Behold, thy King cometh unto thee,\n        Meek, and riding upon an ass,\n        And upon a colt the foal of an ass.\"\n\nAnd they went away, and found a colt tied at the door without in the\nopen street: and they loose him. And certain of them that stood there\nsaid unto them, \"What do ye, loosing the colt?\" And they said unto them\neven as Jesus had said: and they let them go. And they bring the colt\nunto Jesus, and cast on him their garments; and he sat upon him.\n\nAnd the most part of the multitude spread their garments upon the way;\nand others branches, which they had cut from the fields. And as he was\ndrawing nigh, even at the descent of the Mount of Olives, the whole\nmultitude of the disciples began to rejoice and praise God with a loud\nvoice for all the mighty works which they had seen. And they that went\nbefore, and they that followed, cried, \"Hosanna to the Son of David;\nBlessed is he that cometh in the name of the Lord; Blessed is the kingdom\nthat cometh, the kingdom of our father David: Hosanna in the highest.\"\n\nThese things understood not his disciples at the first: but when Jesus\nwas glorified, then remembered they that these things were written of\nhim, and that they had done these things unto him.\n\nThe multitude, therefore, that was with him when he called Lazarus out\nof the tomb, and raised him from the dead, bare witness. For this\ncause also the multitude went and met him, for that they heard that he\nhad done this sign.\n\nAnd some of the Pharisees from the multitude said unto him, \"Teacher,\nrebuke thy disciples.\"\n\nAnd he answered and said, \"I tell you that, if these shall hold their\npeace, the stones will cry out.\"\n\nAnd when he drew nigh, he saw the city and wept over it, saying, \"If thou\nhadst known in this day, even thou, the things which belong unto peace! but\nnow they are hid from thine eyes. For the days shall come upon thee, when\nthine enemies shall cast up a bank about thee, and compass thee round, and\nkeep thee in on every side, and shall dash thee to the ground, and thy\nchildren within thee; and they shall not leave in thee one stone upon\nanother; because thou knewest not the time of thy visitation.\"\n\nAnd when he was come into Jerusalem, all the city was stirred, saying,\n\"Who is this?\"\n\nAnd the multitude said, \"This is the prophet, Jesus, from Nazareth of\nGalilee.\"\n\nThe Pharisees therefore said among themselves, \"Behold, how ye prevail\nnothing; lo, the world is gone after him.\"\n\nAnd he entered into Jerusalem, into the temple; and when he had looked\nround about upon all things, it being now eventide, he went out unto\nBethany with the twelve.\n\n\n\n\nMONDAY--THE DAY OF AUTHORITY.\n\n\nTHE CURSING OF THE FIG TREE.\n\nAnd on the morrow, when they were come out from Bethany, he hungered.\nAnd seeing a fig tree afar off having leaves, he came, if haply he might\nfind anything thereon: and when he came to it, he found nothing but\nleaves; for it was not the season of figs. And he answered and said unto\nit, \"No man eat fruit from thee henceforward for ever.\"\n\nAnd his disciples heard it.\n\n\nTHE CLEANSING OF THE TEMPLE.\n\nAnd they come to Jerusalem: and he entered into the temple, and began to\ncast out them that sold and them that bought in the temple, and\noverthrew the tables of the money-changers, and the seats of them that\nsold the doves: and he would not suffer that any man should carry a\nvessel through the temple. And he taught, and said unto them, \"Is it not\nwritten, 'My house shall be called a house of prayer for all the\nnations'? but ye have made it a den of robbers.\"\n\nAnd the blind and the lame came to him in the temple; and he healed\nthem. But when the chief priests and the scribes saw the wonderful\nthings that he did, and the children that were crying in the temple and\nsaying, \"Hosanna to the son of David\": they were moved with indignation,\nand said unto him, \"Hearest thou what these are saying?\"\n\nAnd Jesus saith unto them, \"Yea: did ye never read, 'Out of the mouth of\nbabes and sucklings thou hast perfected praise'?\"\n\nAnd the chief priests and the scribes and the principal men of the\npeople sought to destroy him: and they could not find what they might\ndo; for the people all hung upon him, listening.\n\nAnd he left them, and went forth out of the city to Bethany, and lodged\nthere.\n\n\n\n\nTUESDAY--THE DAY OF CONTROVERSY.\n\n\nTHE LESSON FROM THE WITHERED FIG TREE.\n\nAnd as they passed by in the morning, they saw the fig tree withered\naway from the roots. And Peter calling to remembrance saith unto him,\n\"Rabbi, behold the fig tree which thou cursedst is withered away.\"\n\nAnd Jesus answering saith unto them, \"Have faith in God. Verily I say\nunto you, Whosoever shall say unto this mountain, 'Be thou taken up\nand cast into the sea'; and shall not doubt in his heart, but shall\nbelieve that what he saith cometh to pass; he shall have it. Therefore\nI say unto you, All things whatsoever ye pray and ask for, believe\nthat ye receive them, and ye shall have them. And whensoever ye stand\npraying, forgive, if ye have aught against any one; that your Father\nalso who is in heaven may forgive you your trespasses.\"\n\n\nTHE CHALLENGE OF CHRIST'S AUTHORITY.\n\nAnd they came again to Jerusalem. And all the people came early in the\nmorning to him in the temple to hear him. And as he was teaching the\npeople in the temple, and preaching the gospel, there came upon him the\nchief priests and the scribes with the elders; and they spake, saying\nunto him, \"Tell us: By what authority doest thou these things? or who is\nhe that gave thee this authority?\"\n\nAnd Jesus answered, and said unto them, \"I also will ask you one question,\nwhich if ye tell me, I likewise will tell you by what authority I do these\nthings. The baptism of John, whence was it? from heaven or from men?\"\n\nAnd they reasoned with themselves, saying, \"If we shall say, 'From\nheaven'; he will say unto us, 'Why did ye not believe him?' But if we\nshall say, 'From men'; all the people will stone us: for they are\npersuaded that John was a prophet.\"\n\nAnd they answered Jesus, and said, \"We know not.\"\n\nAnd Jesus said unto them, \"Neither tell I you by what authority I do\nthese things.\"\n\n\nTHE TWO SONS.\n\n\"But what think ye? A man had two sons; and he came to the first, and\nsaid, 'Son, go work to-day in the vineyard.' And he answered and said,\n'I will not': but afterward he repented himself, and went. And he came\nto the second, and said likewise. And he answered and said, 'I go, sir':\nand went not. Which of the two did the will of his father?\"\n\nThey say, \"The first.\"\n\nJesus saith unto them, \"Verily I say unto you, that the publicans and\nthe harlots go into the kingdom of God before you. For John came unto\nyou in the way of righteousness, and ye believed him not; but the\npublicans and the harlots believed him; and ye, when ye saw it, did not\neven repent yourselves afterward that ye might believe him.\"\n\n\nTHE WICKED HUSBANDMEN.\n\n\"Hear another parable: There was a man who was a householder, who\nplanted a vineyard, and set a hedge about it, and digged a winepress\nin it, and built a tower, and let it out to husbandmen, and went into\nanother country. And when the season of the fruits drew near, he sent\nhis servants to the husbandmen to receive his fruits. And the\nhusbandmen took his servants, and beat one, and killed another, and\nstoned another. Again, he sent other servants more than the first; and\nthey did unto them in like manner. But afterward he sent unto them his\nson, saying, 'They will reverence my son.' But the husbandmen, when\nthey saw the son, said among themselves, 'This is the heir; come, let\nus kill him, and take his inheritance.' And they took him, and cast\nhim forth out of the vineyard, and killed him. When therefore the lord\nof the vineyard shall come, what will he do unto those husbandmen?\"\n\nThey say unto him, \"He will miserably destroy those miserable men, and\nwill let out the vineyard unto other husbandmen, who shall render him\nthe fruits in their seasons.\"\n\nJesus saith unto them, \"Did ye never read in the scriptures,\n\n        'The stone which the builders rejected,\n        The same was made the head of the corner;\n        This was from the Lord,\n        And it is marvellous in our eyes'?\n\nTherefore say I unto you, The kingdom of God shall be taken away from\nyou, and shall be given to a nation bringing forth the fruits thereof.\nAnd he that falleth on this stone shall be broken to pieces; but on\nwhomsoever it shall fall, it will scatter him as dust.\"\n\nAnd when the chief priests and the Pharisees heard his parables, they\nperceived that he spake of them. And when they sought to lay hold on\nhim, they feared the multitudes, because they took him for a prophet.\n\n\nTHE MARRIAGE OF THE KING'S SON.\n\nAnd Jesus answered and spake again in parables unto them, saying, \"The\nkingdom of heaven is likened unto a certain king, who made a marriage feast\nfor his son, and sent forth his servants to call them that were bidden to\nthe marriage feast: and they would not come. Again he sent forth other\nservants, saying, Tell them that are bidden, 'Behold, I have made ready my\ndinner; my oxen and my fatlings are killed, and all things are ready; come\nto the marriage feast.' But they made light of it, and went their ways, one\nto his own farm, another to his merchandise; and the rest laid hold on his\nservants, and treated them shamefully, and killed them. But the king was\nwroth; and he sent his armies, and destroyed those murderers, and burned\ntheir city. Then saith he to his servants, 'The wedding is ready, but they\nthat were bidden were not worthy. Go ye therefore unto the partings of the\nhighways, and as many as ye shall find, bid to the marriage feast.' And\nthose servants went out into the highways, and gathered together all as\nmany as they found, both bad and good; and the wedding was filled with\nguests. But when the king came in to behold the guests, he saw there a man\nwho had not on a wedding-garment: and he saith unto him, 'Friend, how\ncamest thou in hither not having a wedding-garment?' And he was speechless.\nThen the king said to the servants, 'Bind him hand and foot and cast him\nout into the outer darkness'; there shall be the weeping and the gnashing\nof teeth. For many are called, but few chosen.\"\n\n\nTRIBUTE TO CAESAR.\n\nThen went the Pharisees, and took counsel how they might ensnare him\nin his talk so as to deliver him up to the rule and to the authority\nof the governor. And they send to him their disciples, with the\nHerodians, saying, \"Teacher, we know that thou art true, and teachest\nthe way of God in truth, and carest not for any one: for thou\nregardest not the person of men. Tell us therefore, What thinkest\nthou? Is it lawful to give tribute unto Caesar, or not?\"\n\nBut Jesus perceived their craftiness, and said, \"Why make ye trial of\nme, ye hypocrites? Show me the tribute money.\"\n\nAnd they brought unto him a denarius. And he saith unto them, \"Whose is\nthis image and superscription?\"\n\nThey say unto him, \"Caesar's.\"\n\nThen he saith unto them, \"Render therefore unto Caesar the things that\nare Caesar's; and unto God the things that are God's.\"\n\nAnd when they heard it, they marvelled, and left him, and went away.\n\n\nTHE QUESTION OF THE RESURRECTION.\n\nAnd there came to him certain of the Sadducees, they that say that\nthere is no resurrection; and they asked him, saying, \"Teacher, Moses\nwrote unto us, that if a man's brother die, having a wife, and he be\nchildless, his brother should take the wife, and raise up seed unto\nhis brother. There were therefore seven brethren: and the first took a\nwife, and died childless; and the second; and the third took her; and\nlikewise the seven also left no children, and died. Afterward the\nwoman also died. In the resurrection therefore whose wife of them\nshall she be? for the seven had her to wife.\"\n\nAnd Jesus said unto them, \"Ye do err, not knowing the scriptures, nor the\npower of God. The sons of this world marry, and are given in marriage; but\nthey that are accounted worthy to attain to that world, and the\nresurrection from the dead, neither marry, nor are given in marriage: for\nneither can they die any more: for they are equal unto the angels; and are\nsons of God, being sons of the resurrection. But that the dead are raised,\neven Moses showed, in the place concerning the Bush, when he calleth the\nLord the God of Abraham, and the God of Isaac, and the God of Jacob. Now he\nis not the God of the dead, but of the living: for all live unto him.\"\n\nAnd when the multitudes heard it, they were astonished at his teaching.\n\n\nTHE GREATEST COMMANDMENT.\n\nAnd one of the scribes came, and heard them questioning together, and\nknowing that he had answered them well, asked him, \"What commandment is\nthe first of all?\"\n\nJesus answered, \"The first is, 'Hear, O Israel: The Lord our God, the\nLord is one: and thou shalt love the Lord thy God with all thy heart,\nand with all thy soul, and with all thy mind, and with all thy\nstrength.' The second is this, 'Thou shalt love thy neighbor as\nthyself.' There is none other commandment greater than these.\"\n\nAnd the scribe said unto him, \"Of a truth, Teacher, thou hast well said\nthat he is one: and there is none other but he: and to love him with all\nthe heart, and with all the understanding and with all the strength, and\nto love his neighbor as himself, is much more than all whole\nburnt-offerings and sacrifices.\"\n\nAnd when Jesus saw that he answered discreetly, he said unto him, \"Thou\nart not far from the kingdom of God.\"\n\n\nTHE UNANSWERABLE QUESTION OF JESUS.\n\nNow while the Pharisees were gathered together Jesus asked them a\nquestion, saying, \"What think ye of the Christ? whose son is he?\"\n\nThey say unto him, \"The son of David.\"\n\nHe saith unto them, \"How then doth David in the Spirit call him Lord,\nsaying,\n\n        'The Lord said unto my Lord,\n        Sit thou on my right hand,\n        Till I put thine enemies underneath thy feet?'\n\nIf David then calleth him Lord, how is he his son?\"\n\nAnd no one was able to answer him a word, neither durst any man from\nthat day forth ask him any more questions.\n\nAnd the common people heard him gladly.\n\n\nDISCOURSE OF JESUS AGAINST THE SCRIBES AND PHARISEES.\n\nThen spake Jesus to the multitudes and to his disciples, saying, \"The\nscribes and the Pharisees sit on Moses' seat: all things therefore\nwhatsoever they bid you, these do and observe: but do not ye after their\nworks; for they say, and do not. Yea, they bind heavy burdens and grievous\nto be borne, and lay them on men's shoulders; but they themselves will not\nmove them with their finger. But all their works they do to be seen of men:\nfor they make broad their phylacteries, and enlarge the borders of their\ngarments, and love the chief place at feasts, and the chief seats in the\nsynagogues, and the salutations in the market-places, and to be called of\nmen, 'Rabbi.' But be not ye called 'Rabbi,' for one is your teacher, and\nall ye are brethren. And call no man your father on the earth: for one is\nyour Father, even he who is in heaven. Neither be ye called masters: for\none is your master, even the Christ. But he that is greatest among you\nshall be your servant. And whosoever shall exalt himself shall be humbled:\nand whosoever shall humble himself shall be exalted.\n\n\"But woe unto you, scribes and Pharisees, hypocrites! because ye shut\nthe kingdom of heaven against men: for ye enter not in yourselves,\nneither suffer ye them that are entering in to enter.\n\n\"Woe unto you, scribes and Pharisees, hypocrites! for ye compass sea and\nland to make one proselyte; and when he is become so, ye make him\ntwofold more a son of hell than yourselves.\n\n\"Woe unto you, ye blind guides, that say, 'Whosoever shall swear by the\ntemple, it is nothing; but whosoever shall swear by the gold of the\ntemple, he is a debtor.' Ye fools and blind: for which is greater, the\ngold, or the temple that hath sanctified the gold? And, 'Whosoever\nshall swear by the altar, it is nothing; but whosoever shall swear by\nthe gift that is upon it, he is a debtor.' Ye blind: for which is\ngreater, the gift, or the altar that sanctifieth the gift? He therefore\nthat sweareth by the altar, sweareth by it, and by all things thereon.\nAnd he that sweareth by the temple, sweareth by it, and by him that\ndwelleth therein. And he that sweareth by the heaven, sweareth by the\nthrone of God, and by him that sitteth thereon.\n\n\"Woe unto you, scribes and Pharisees, hypocrites! for ye tithe mint and\nanise and cummin, and have left undone the weightier matters of the law,\njustice, and mercy, and faith: but these ye ought to have done, and not\nto have left the other undone. Ye blind guides, that strain out the\ngnat, and swallow the camel!\n\n\"Woe unto you, scribes and Pharisees, hypocrites! for ye cleanse the\noutside of the cup and of the platter, but within they are full from\nextortion and excess. Thou blind Pharisee, cleanse first the inside of\nthe cup and of the platter, that the outside thereof may become clean also.\n\n\"Woe unto you, scribes and Pharisees, hypocrites! for ye are like unto\nwhited sepulchres, which outwardly appear beautiful, but inwardly are\nfull of dead men's bones, and of all uncleanness. Even so ye also\noutwardly appear righteous unto men, but inwardly ye are full of\nhypocrisy and iniquity.\n\n\"Woe unto you, scribes and Pharisees, hypocrites! for ye build the\nsepulchres of the prophets, and garnish the tombs of the righteous,\nand say, 'If we had been in the days of our fathers, we should not\nhave been partakers with them in the blood of the prophets.' Wherefore\nye witness to yourselves, that ye are sons of them that slew the\nprophets. Fill ye up then the measure of your fathers. Ye serpents, ye\noffspring of vipers, how shall ye escape the judgment of hell?\nTherefore, behold, I send unto you prophets, and wise men, and\nscribes: some of them shall ye kill and crucify; and some of them\nshall ye scourge in your synagogues, and persecute from city to city:\nthat upon you may come all the righteous blood shed on the earth, from\nthe blood of Abel the righteous unto the blood of Zachariah son of\nBarachiah, whom ye slew between the sanctuary and the altar. Verily I\nsay unto you, All these things shall come upon this generation.\n\n\"O Jerusalem, Jerusalem, that killeth the prophets, and stoneth them\nthat are sent unto her! how often would I have gathered thy children\ntogether, even as a hen gathereth her chickens under her wings, and ye\nwould not! Behold, your house is left unto you desolate. For I say unto\nyou, Ye shall not see me henceforth, till ye shall say, 'Blessed is he\nthat cometh in the name of the Lord.'\"\n\n\nTHE WIDOW'S TWO MITES.\n\nAnd he sat down over against the treasury, and beheld how the multitude\ncast money into the treasury: and many that were rich cast in much. And\nthere came a poor widow, and she cast in two mites, which make a farthing.\nAnd he called unto him his disciples, and said unto them, \"Verily, I say\nunto you, This poor widow cast in more than all they that are casting into\nthe treasury: for they all did cast in of their superfluity; but she of her\nwant did cast in all that she had, even all her living.\"\n\n\nTHE GENTILES SEEK JESUS.\n\nNow there were certain Greeks among those that went up to worship at the\nfeast: these therefore came to Philip, who was of Bethsaida of Galilee,\nand asked him, saying, \"Sir, we would see Jesus.\"\n\nPhilip cometh and telleth Andrew: Andrew cometh, and Philip, and they\ntell Jesus.\n\nAnd Jesus answereth them, saying, \"The hour is come, that the Son of man\nshould be glorified. Verily, verily, I say unto you, Except a grain of\nwheat fall into the earth and die, it abideth by itself alone; but if it\ndie, it beareth much fruit. He that loveth his life loseth it: and he\nthat hateth his life in this world shall keep it unto life eternal. If\nany man serve me, let him follow me: and where I am, there shall also my\nservant be: if any man serve me, him will the Father honor. Now is my\nsoul troubled; and what shall I say? Father, save me from this hour. But\nfor this cause came I unto this hour. Father, glorify thy name.\"\n\nThere came therefore a voice out of heaven, saying, \"I have both\nglorified it, and will glorify it again.\"\n\nThe multitude, therefore, that stood by, and heard it, said that it had\nthundered: others said, \"An angel hath spoken to him.\"\n\nJesus answered and said, \"This voice hath not come for my sake, but for\nyour sakes. Now is the judgment of this world: now shall the prince of\nthis world be cast out. And I, if I be lifted up from the earth, will\ndraw all men unto myself.\"\n\nBut this he said, signifying by what manner of death he should die.\n\nThe multitude therefore answered him, \"We have heard out of the law that\nthe Christ abideth forever: and how sayest thou, 'The Son of man must be\nlifted up'? who is this Son of man?\"\n\nJesus therefore said unto them, \"Yet a little while is the light among\nyou. Walk while ye have the light that darkness overtake you not: and he\nthat walketh in the darkness knoweth not whither he goeth. While ye\nhave the light, believe on the light, that ye may become sons of light.\"\n\nThese things spake Jesus, and he departed and hid himself from them.\n\n\nTHE JEWS REJECT JESUS.\n\nBut though he had done so many signs before them, yet they believed not on\nhim: that the word of Isaiah the prophet might be fulfilled, which he\nspake,\n\n        \"Lord, who hath believed our report?\n        And to whom hath the arm of the Lord been revealed?\"\n\nFor this cause they could not believe, for that Isaiah said again,\n\n        \"He hath blinded their eyes, and he hardened their heart;\n        Lest they should see with their eyes, and perceive with their\n              heart,\n        And should turn,\n        And I should heal them.\"\n\nThese things said Isaiah, because he saw his glory; and he spake of him.\nNevertheless even of the rulers many believed on him; but because of the\nPharisees they did not confess it, lest they should be put out of the\nsynagogue: for they loved the glory that is of men more than the glory that\nis of God.\n\nAnd Jesus cried and said, \"He that believeth on me, believeth not on me,\nbut on him that sent me. And he that beholdeth me beholdeth him that\nsent me. I am come a light into the world, that whosoever believeth on\nme may not abide in the darkness. And if any man hear my sayings, and\nkeep them not, I judge him not: for I came not to judge the world, but\nto save the world. He that rejecteth me, and receiveth not my sayings,\nhath one that judgeth him: the word that I spake, the same shall judge\nhim in the last day. For I spake not from myself; but the Father that\nsent me, he hath given me a commandment, what I should say, and what I\nshould speak. And I know that his commandment is life eternal; the things\ntherefore which I speak, even as the Father hath said unto me, so I speak.\"\n\n\nDISCOURSE CONCERNING THE FUTURE.\n\nAnd Jesus went out from the temple, and was going on his way; and his\ndisciples came to him to show him the buildings of the temple.\n\nBut he answered and said unto them, \"See ye not all these things? Verily\nI say unto you, There shall not be left here one stone upon another,\nthat shall not be thrown down.\"\n\nAnd as he sat on the mount of Olives over against the temple, Peter and\nJames and John and Andrew asked him privately, \"Tell us, when shall\nthese things be? and what shall be the sign when these things are all\nabout to be accomplished?\"\n\nAnd Jesus began to say unto them, \"Take heed that no man lead you\nastray. Many shall come in my name, saying, 'I am he,' and shall lead\nmany astray. And when ye shall hear of wars and rumors of wars, be not\ntroubled: these things must needs come to pass; but the end is not yet.\nFor nation shall rise against nation, and kingdom against kingdom; there\nshall be earthquakes in divers places; there shall be famines: these\nthings are the beginning of the travail.\n\n\"But take ye heed to yourselves: for they shall deliver you up to\ncouncils; and in synagogues shall ye be beaten; and before governors\nand kings shall ye stand for my sake, for a testimony unto them. And\nthe gospel must first be preached unto all the nations. And when they\nlead you to judgment, and deliver you up, be not anxious beforehand\nwhat ye shall speak; but whatsoever shall be given you in that hour,\nthat speak ye; for it is not ye that speak, but the Holy Spirit. But\nye shall be delivered up even by parents, and brethren, and kinsfolk,\nand friends: and some of you shall they cause to be put to death. And\nye shall be hated of all men for my name's sake.\n\n\"And then shall many stumble, and shall deliver up one another, and shall\nhate one another. And many false prophets shall arise, and shall lead many\nastray. And because iniquity shall be multiplied, the love of the many\nshall wax cold. But he that endureth to the end, the same shall be saved.\n\n\"But when ye see Jerusalem compassed with armies, then know that her\ndesolation is at hand. Then let them that are in Judaea flee unto the\nmountains; let him that is on the housetop not go down to take out the\nthings that are in his house; and let him that is in the field not\nreturn back to take his cloak. For these are days of vengeance, that all\nthings which are written may be fulfilled.\n\n\"But woe unto them that are with child and to them that give suck in\nthose days! And pray ye that your flight be not in the winter, neither\non a Sabbath: for then shall be great tribulation, such as hath not been\nfrom the beginning of the world until now, no, nor ever shall be. And\nexcept those days had been shortened, no flesh would have been saved:\nbut for the elect's sake those days shall be shortened. Then if any man\nshall say unto you, 'Lo, here is the Christ,' or, 'Here,' believe it\nnot. For there shall arise false Christs, and false prophets, and shall\nshow great signs and wonders; so as to lead astray, if possible, even\nthe elect. But take ye heed: behold, I have told you all things\nbeforehand. If, therefore, they shall say unto you, 'Behold, he is in\nthe wilderness,' go not forth: 'Behold, he is in the inner chambers,'\nbelieve it not. For as the lightning cometh forth from the east and is\nseen even unto the west, so shall be the coming of the Son of man.\nWheresoever the carcase is, there will the eagles be gathered together.\n\n\"But immediately after the tribulation of those days the sun shall be\ndarkened, and the moon shall not give her light, and the stars shall\nfall from heaven, and the powers of the heavens shall be shaken: and\nthen shall appear the sign of the Son of man in heaven: and then shall\nall the tribes of the earth mourn, and they shall see the Son of man\ncoming on the clouds of heaven with power and great glory. And he\nshall send forth his angels with a great sound of a trumpet, and shall\ngather together his elect from the four winds, from the uttermost part\nof the earth to the uttermost part of heaven.\n\n\"Now from the fig tree learn her parable: when her branch is now become\ntender, and putteth forth its leaves, ye know that the summer is nigh; even\nso ye also, when ye see all these things, know ye that he is nigh, even at\nthe doors. Verily I say unto you, This generation shall not pass away till\nall these things be accomplished. Heaven and earth shall pass away, but my\nwords shall not pass away. But of that day and hour knoweth no one, not\neven the angels of heaven, neither the Son, but the Father only.\n\n\"But take heed to yourselves, lest haply your hearts be overcharged with\nsurfeiting, and drunkenness, and cares of this life, and that day come\non you suddenly as a snare; for so shall it come upon all them that\ndwell on the face of all the earth. But watch ye at every season, making\nsupplication, that ye may prevail to escape all these things that shall\ncome to pass, and to stand before the Son of man.\n\n\"And as were the days of Noah, so shall be the coming of the Son of man.\nFor as in those days which were before the flood they were eating and\ndrinking, marrying and giving in marriage, until the day that Noah entered\ninto the ark, and they knew not until the flood came, and took them all\naway; so shall be the coming of the Son of man. Then shall two men be in\nthe field; one is taken, and one is left; two women shall be grinding at\nthe mill; one is taken, and one is left. Watch therefore: for ye know not\non what day your Lord cometh.\n\n\"But know this, that if the master of the house had known in what watch\nthe thief was coming, he would have watched, and would not have suffered\nhis house to be broken through. Therefore be ye also ready; for in an\nhour that ye think not the Son of man cometh.\n\n\"Take ye heed, watch and pray: for ye know not when the time is. It is as\nwhen a man, sojourning in another country, having left his house, and given\nauthority to his servants, to each one his work, commanded also the porter\nto watch. Watch therefore: for ye know not when the lord of the house\ncometh, whether at even, or at midnight, or at cockcrowing, or in the\nmorning; lest coming suddenly he find you sleeping. And what I say unto you\nI say unto all, Watch.\n\n\"Who then is the faithful and wise servant, whom the lord hath set over his\nhousehold, to give them their food in due season? Blessed is that servant,\nwhom his lord when he cometh shall find so doing. Verily I say unto you,\nthat he will set him over all that he hath. But if that evil servant shall\nsay in his heart, 'My lord tarrieth'; and shall begin to beat his\nfellow-servants, and shall eat and drink with the drunken; the lord of that\nservant shall come in a day when he expecteth not, and in an hour when he\nknoweth not, and shall cut him asunder, and appoint his portion with the\nhypocrites: there shall be the weeping and the gnashing of teeth.\"\n\n\nTHE PARABLE OF THE TEN VIRGINS.\n\n\"Then shall the kingdom of heaven be likened unto ten virgins, who\ntook their lamps and went forth to meet the bridegroom. And five of\nthem were foolish, and five were wise. For the foolish, when they took\ntheir lamps, took no oil with them: but the wise took oil in their\nvessels with their lamps. Now while the bridegroom tarried, they all\nslumbered and slept. But at midnight there is a cry, 'Behold, the\nbridegroom! Come ye forth to meet him.' Then all those virgins arose,\nand trimmed their lamps. And the foolish said unto the wise, 'Give us\nof your oil; for our lamps are going out.' But the wise answered,\nsaying, 'Peradventure there will not be enough for us and you: go ye\nrather to them that sell, and buy for yourselves.'\n\n\"And while they went away to buy, the bridegroom came; and they that\nwere ready went in with him to the marriage feast: and the door was\nshut. Afterward came also the other virgins, saying, 'Lord, Lord, open\nto us.' But he answered and said, 'Verily I say unto you, I know you not.'\n\n\"Watch therefore, for ye know not the day nor the hour.\"\n\n\nTHE PARABLE OF THE TALENTS.\n\n\"For it is as when a man, going into another country, called his own\nservants, and delivered unto them his goods. And unto one he gave five\ntalents, to another two, to another one; to each according to his several\nability; and he went on his journey. Straightway he that received the five\ntalents went and traded with them, and made other five talents. In like\nmanner he also that received the two gained other two. But he that received\nthe one went away and digged in the earth, and hid his lord's money.\n\n\"Now after a long time the lord of these servants cometh, and maketh a\nreckoning with them. And he that received the five talents came and brought\nother five talents, saying, 'Lord, thou deliveredst unto me five talents:\nlo, I have gained other five talents.' His lord said unto him, 'Well done,\ngood and faithful servant: thou hast been faithful over a few things, I\nwill set thee over many things; enter thou into the joy of thy lord.'\n\n\"And he also that received the two talents came and said, 'Lord, thou\ndeliveredst unto me two talents: lo, I have gained other two talents.'\n\n\"His lord said unto him, 'Well done, good and faithful servant: thou\nhast been faithful over a few things, I will set thee over many things;\nenter thou into the joy of thy lord.'\n\n\"And he also that had received the one talent came and said, 'Lord, I\nknew thee that thou art a hard man, reaping where thou didst not sow,\nand gathering where thou didst not scatter; and I was afraid, and went\naway and hid thy talent in the earth: lo, thou hast thine own.'\n\n\"But his lord answered and said unto him, 'Thou wicked and slothful\nservant, thou knewest that I reap where I sowed not, and gather where I did\nnot scatter; thou oughtest therefore to have put my money to the bankers,\nand at my coming I should have received back mine own with interest. Take\nye away therefore the talent from him, and give it unto him that hath the\nten talents. For unto every one that hath shall be given, and he shall have\nabundance; but from him that hath not, even that which he hath shall be\ntaken away. And cast ye out the unprofitable servant into the outer\ndarkness: there shall be the weeping and the gnashing of teeth.'\"\n\n\nTHE JUDGMENT SCENE.\n\n\"But when the Son of man shall come in his glory, and all the angels with\nhim, then shall he sit on the throne of his glory: and before him shall be\ngathered all the nations: and he shall separate them one from another, as\nthe shepherd separateth the sheep from the goats; and he shall set the\nsheep on his right hand, but the goats on the left. Then shall the King say\nunto them on his right hand, 'Come, ye blessed of my Father, inherit the\nkingdom prepared for you from the foundation of the world: for I was\nhungry, and ye gave me to eat; I was thirsty, and ye gave me drink; I was a\nstranger, and ye took me in; naked, and ye clothed me; I was sick, and ye\nvisited me; I was in prison, and ye came unto me.'\n\n\"Then shall the righteous answer him, saying, 'Lord, when saw we thee\nhungry, and fed thee? or athirst, and gave thee drink? And when saw we\nthee a stranger, and took thee in? or naked, and clothed thee? And when\nsaw we thee sick, or in prison, and came unto thee?' And the King shall\nanswer and say unto them, 'Verily I say unto you, Inasmuch as ye did it\nunto one of these my children, even these least, ye did it unto me.'\n\n\"Then shall he say also unto them on the left hand, 'Depart from me, ye\ncursed, into the eternal fire which is prepared for the devil and his\nangels: for I was hungry, and ye did not give me to eat; I was thirsty, and\nye gave me no drink; I was a stranger, and ye took me not in; naked, and ye\nclothed me not; sick, and in prison, and ye visited me not.' Then shall\nthey also answer, saying, 'Lord, when saw we thee hungry, or athirst, or a\nstranger, or naked, or sick, or in prison, and did not minister unto thee?'\nThen shall he answer them, saying, 'Verily I say unto you, Inasmuch as ye\ndid it not unto one of these least, ye did it not unto me.' And these shall\ngo away into eternal punishment: but the righteous into eternal life.\"\n\n\nTHE CONSPIRACY AGAINST JESUS.\n\nAnd it came to pass, when Jesus had finished all these words, he said\nunto his disciples, \"Ye know that after two days the passover cometh,\nand the Son of man is delivered up to be crucified.\"\n\nThen were gathered together the chief priests, the elders of the people,\nunto the court of the high priest, who was called Caiaphas; and they took\ncounsel together that they might take Jesus by subtlety, and kill him. But\nthey said, \"Not during the feast, lest a tumult arise among the people.\"\n\nAnd Satan entered into Judas, who was called Iscariot, being of the number\nof the twelve. And he went away and communed with the chief priests and\ncaptains, how he might deliver him unto them. And they were glad, and they\nweighed unto him thirty pieces of silver. And from that time he sought\nopportunity to deliver him unto them in the absence of the multitude.\n\n\n\n\nWEDNESDAY--THE DAY OF RETIREMENT.\n\n\n[There is no record of the events of this day. Jesus spent it in\nretirement, almost certainly in the home of his friends at Bethany.]\n\n\n\n\nTHURSDAY--THE DAY OF FELLOWSHIP.\n\n\nPREPARATION FOR THE PASSOVER.\n\nAnd on the first day of unleavened bread, when they sacrificed the\npassover, his disciples say unto him, \"Where wilt thou that we go and\nmake ready that thou mayest eat the passover?\"\n\nAnd he sendeth two of his disciples, and saith unto them, \"Go into the\ncity, and there shall meet you a man bearing a pitcher of water; follow\nhim; and wheresoever he shall enter in, say to the master of the house,\n'The Teacher saith, My time is at hand. Where is my guest-chamber, where\nI shall eat the passover with my disciples?' And he will himself show\nyou a large upper room furnished and ready: and there make ready for us.\"\n\nAnd the disciples went forth, and came into the city, and found as he\nhad said unto them: and they made ready the passover.\n\n\nSTRIFE AMONG THE DISCIPLES.\n\nAnd when it was evening he cometh with the twelve. And there arose also\na contention among them, which of them was accounted to be greatest. And\nhe said unto them, \"The kings of the Gentiles have lordship over them;\nand they that have authority over them are called Benefactors. But ye\nshall not be so: but he that is the greater among you, let him become as\nthe younger: and he that is chief, as he that doth serve. For which is\ngreater, he that sitteth at meat, or he that serveth? Is not he that\nsitteth at meat? But I am in the midst of you as he that serveth. But ye\nare they that have continued with me in my temptations; and I appoint\nunto you a kingdom, even as my Father appointed unto me, that ye may\neat and drink at my table in my kingdom; and ye shall sit on thrones\njudging the twelve tribes of Israel.\"\n\n\nJESUS WASHING THE DISCIPLES' FEET.\n\nNow before the feast of the passover, Jesus knowing that his hour was\ncome that he should depart out of this world unto the Father, having\nloved his own that were in the world, he loved them unto the end.\n\nAnd during supper, the devil having already put into the heart of Judas\nIscariot, Simon's son, to betray him, Jesus, knowing that the Father had\ngiven all things into his hands, and that he came forth from God, and\ngoeth unto God, riseth from supper, and layeth aside his garments; and\nhe took a towel, and girded himself. Then he poureth water into the\nbasin, and began to wash the disciples' feet, and to wipe them with the\ntowel wherewith he was girded.\n\nSo he cometh to Simon Peter. He saith unto him, \"Lord, dost thou wash my\nfeet?\"\n\nJesus answered and said unto him, \"What I do thou knowest not now; but\nthou shalt understand hereafter.\"\n\nPeter saith unto him, \"Thou shalt never wash my feet.\"\n\nJesus answered him, \"If I wash thee not, thou hast no part with me.\"\n\nSimon Peter saith unto him, \"Lord, not my feet only, but also my hands\nand my head.\"\n\nJesus saith to him, \"He that is bathed needeth not save to wash his\nfeet, but is clean every whit: and ye are clean, but not all.\" For he\nknew him that should betray him; therefore said he, \"Ye are not all clean.\"\n\nSo when he had washed their feet, and taken his garments, and sat down\nagain, he said unto them, \"Know ye what I have done to you? Ye call me\nTeacher, and Lord: and ye say well; for so I am. If I then, the Lord\nand the Teacher, have washed your feet, ye also ought to wash one\nanother's feet. For I have given you an example, that ye also should do\nas I have done to you. Verily, verily, I say unto you, A servant is not\ngreater than his lord; neither one that is sent greater than he that\nsent him. If ye know these things, blessed are ye if ye do them.\n\n\"I speak not of you all: I know whom I have chosen: but that the scripture\nmay be fulfilled, He that eateth my bread lifted up his heel against me.\nFrom henceforth I tell you before it come to pass, that, when it is come to\npass, ye may believe that I am he. Verily, verily, I say unto you, He that\nreceiveth whomsoever I send receiveth me; and he that receiveth me\nreceiveth him that sent me.\"\n\n\nTHE BETRAYER POINTED OUT.\n\nWhen Jesus had thus said, he was troubled in the spirit, and testified, and\nsaid, \"Verily, verily, I say unto you, that one of you shall betray me.\"\n\nThe disciples looked one on another, doubting of whom he spake. And\nthey were exceeding sorrowful, and began to say unto him every one,\n\"Is it I, Lord?\"\n\nAnd he answered and said, \"He that dipped his hand with me in the dish,\nthe same shall betray me. The Son of man goeth, even as it is written of\nhim: but woe unto that man through whom the Son of man is betrayed! Good\nwere it for that man if he had not been born.\"\n\nAnd Judas, who betrayed him, answered and said, \"Is it I, Rabbi?\"\n\nHe saith unto him, \"Thou hast said.\"\n\nThere was at the table reclining in Jesus' bosom one of his disciples,\nwhom Jesus loved. Simon Peter therefore beckoneth to him, and saith unto\nhim, \"Tell us who it is of whom he speaketh.\"\n\nHe leaning back, as he was, on Jesus' breast, saith unto him, \"Lord, who\nis it?\"\n\nJesus therefore answereth, \"He it is, for whom I shall dip the sop, and\ngive it him.\"\n\nSo when he had dipped the sop, he taketh and giveth it to Judas, the son\nof Simon Iscariot. And after the sop, then entered Satan into him.\n\nJesus therefore saith unto him, \"What thou doest, do quickly.\"\n\nNow no man at the table knew for what intent he spake this unto him. For\nsome thought because Judas had the bag, that Jesus said unto him, \"Buy\nwhat things we have need of for the feast,\" or that he should give\nsomething to the poor. He then having received the sop went out\nstraightway: and it was night.\n\nWhen therefore he was gone out, Jesus saith, \"Now is the Son of man\nglorified, and God is glorified in him; and God shall glorify him in\nhimself, and straightway shall he glorify him.\"\n\n\nTHE LORD'S SUPPER.\n\nAnd he said unto them, \"With desire I have desired to eat this passover\nwith you before I suffer: for I say unto you, I shall not eat it until\nit be fulfilled in the Kingdom of God.\"\n\nAnd he took bread, and when he had given thanks, he brake it, and gave\nto them, saying, \"This is my body; which is given for you; this do in\nremembrance of me.\"\n\nAnd he took a cup, in like manner after supper, and gave thanks, and gave\nto them, saying, \"Drink ye all of it; for this is my blood of the new\ncovenant, which is poured out for you, for many, unto remission of sins.\nTake this and divide it among yourselves; for I say unto you, I shall not\ndrink from henceforth of the fruit of the vine, until the Kingdom of God\nshall come.\"\n\n\nTHE FAREWELL CONVERSATION.\n\n\"Little children, yet a little while I am with you. Ye shall seek me: and\nas I said unto the Jews, 'Whither I go, ye cannot come,' so now I say unto\nyou. A new commandment I give unto you, that ye love one another; even as I\nhave loved you, that ye also love one another. By this shall all men know\nthat ye are my disciples, if ye have love one to another.\"\n\nSimon Peter saith unto him, \"Lord, whither goest thou?\"\n\nJesus answered, \"Whither I go, thou canst not follow me now; but thou\nshalt follow afterwards.\"\n\nAnd Jesus saith unto them, \"All ye shall be offended: for it is written,\nI will smite the shepherd, and the sheep shall be scattered abroad.\nHowbeit, after I am raised up, I will go before you into Galilee.\"\n\nBut Peter said unto him, \"Although all shall be offended, yet will not I.\"\n\nAnd Jesus saith unto him, \"Verily I say unto thee, that thou to-day,\neven this night, before the cock crow twice, shalt deny me thrice.\nSimon, Simon, behold Satan asked to have you, that he might sift you as\nwheat: but I make supplication for thee, that thy faith fail not: and do\nthou, when once thou hast turned again, establish thy brethren.\"\n\nBut he spake vehemently, \"If I must die with thee, I will not deny\nthee.\" And in like manner also said they all.\n\n       *       *       *       *       *\n\nAnd he said unto them, \"When I sent you forth without purse, and wallet,\nand shoes, lacked ye anything?\"\n\nAnd they said, \"Nothing.\"\n\nAnd he said unto them, \"But now, he that hath a purse, let him take it,\nand likewise a wallet; and he that hath none, let him sell his cloak,\nand buy a sword. For I say unto you, that this which is written must be\nfulfilled in me, 'And he was reckoned with transgressors': for that\nwhich concerneth me hath fulfillment.\"\n\nAnd they said, \"Lord, behold, here are two swords.\"\n\nAnd he said unto them, \"It is enough.\"\n\n       *       *       *       *       *\n\n\"Let not your heart be troubled; believe in God, believe also in me. In\nmy Father's house are many mansions; if it were not so, I would have\ntold you; for I go to prepare a place for you. And if I go and prepare a\nplace for you, I come again, and will receive you unto myself; that\nwhere I am, there ye may be also. And whither I go, ye know the way.\"\n\nThomas saith unto him, \"Lord, we know not whither thou goest; how know\nwe the way?\"\n\nJesus saith unto him, \"I am the way, and the truth, and the life: no one\ncometh unto the Father, but by me. If ye had known me, ye would have\nknown my Father also: from henceforth ye know him, and have seen him.\"\n\nPhilip saith unto him, \"Lord, show us the Father, and it sufficeth us.\"\n\nJesus saith unto him, \"Have I been so long time with you, and dost thou\nnot know me, Philip? He that hath seen me hath seen the Father; how\nsayest thou, 'Show us the Father'? Believest thou not that I am in the\nFather, and the Father in me? The words that I say unto you I speak not\nfrom myself: but the Father abiding in me doeth his works. Believe me\nthat I am in the Father, and the Father in me: or else believe me for\nthe very works' sake. Verily, verily, I say unto you, He that believeth\non me, the works that I do shall he do also: and greater works than\nthese shall he do; because I go unto the Father. And whatsoever ye shall\nask in my name, that will I do, that the Father may be glorified in the\nSon. If ye shall ask anything in my name, that will I do. If ye love\nme, ye will keep my commandments. And I will pray the Father, and he\nshall give you another Comforter, that he may be with you for ever, even\nthe Spirit of truth: whom the world cannot receive; for it beholdeth him\nnot, neither knoweth him: ye know him, for he abideth with you, and\nshall be in you. I will not leave you desolate: I come unto you.\n\n\"Yet a little while, and the world beholdeth me no more; but ye behold me:\nbecause I live, ye shall live also. In that day ye shall know that I am in\nmy Father, and ye in me, and I in you. He that hath my commandments, and\nkeepeth them, he it is that loveth me: and he that loveth me shall be loved\nof my Father, and I will love him, and will manifest myself unto him.\"\n\nJudas (not Iscariot) saith unto him, \"Lord, what is come to pass that\nthou wilt manifest thyself unto us, and not unto the world?\"\n\nJesus answered and said unto him, \"If a man love me, he will keep my\nword: and my Father will love him, and we will come unto him, and make\nour abode with him. He that loveth me not keepeth not my words: and the\nword which ye hear is not mine, but the Father's who sent me.\n\n\"These things have I spoken unto you, while yet abiding with you. But\nthe Comforter, even the Holy Spirit, whom the Father will send in my\nname, he shall teach you all things, and bring to your remembrance all\nthat I said unto you. Peace I leave with you; my peace I give unto you:\nnot as the world giveth, give I unto you. Let not your heart be\ntroubled, neither let it be fearful. Ye heard how I said to you, I go\naway, and I come unto you. If ye loved me, ye would have rejoiced,\nbecause I go unto the Father: for the Father is greater than I.\n\n\"And now I have told you before it come to pass, that, when it is come\nto pass, ye may believe. I will no more speak much with you, for the\nprince of the world cometh: and he hath nothing in me; but that the\nworld may know that I love the Father, and as the Father gave me\ncommandment, even so I do.\"\n\n       *       *       *       *       *\n\n\"I am the true vine, and my Father is the husbandman. Every branch in me\nthat beareth not fruit, he taketh it away: and every branch that beareth\nfruit, he cleanseth it, that it may bear more fruit. Already ye are clean\nbecause of the word which I have spoken unto you. Abide in me, and I in\nyou. As the branch cannot bear fruit of itself, except it abide in the\nvine; so neither can ye, except ye abide in me. I am the vine, ye are the\nbranches: He that abideth in me, and I in him, the same beareth much fruit:\nfor apart from me ye can do nothing. If a man abide not in me, he is cast\nforth as a branch, and is withered; and they gather them, and cast them\ninto the fire, and they are burned. If ye abide in me, and my words abide\nin you, ask whatsoever ye will, and it shall be done unto you. Herein is my\nFather glorified, that ye bear much fruit: and so shall ye be my disciples.\nEven as the Father hath loved me, I also have loved you: abide ye in my\nlove. If ye keep my commandments, ye shall abide in my love; even as I have\nkept my Father's commandments, and abide in his love.\n\n\"These things have I spoken unto you, that my joy may be in you, and\nthat your joy may be made full. This is my commandment, that ye love\none another, even as I have loved you. Greater love hath no man than\nthis, that a man lay down his life for his friends. Ye are my friends,\nif ye do the things which I command you. No longer do I call you\nservants; for the servant knoweth not what his lord doeth: but I have\ncalled you friends; for all things that I heard from my Father I have\nmade known unto you. Ye did not choose me, but I chose you, and\nappointed you, that ye should go and bear fruit, and that your fruit\nshould abide: that whatsoever ye shall ask of the Father in my name,\nhe may give it you. These things I command you, that ye may love one\nanother. If the world hate you, ye know that it hath hated me before\nit hated you. If ye were of the world, the world would love its own:\nbut because ye are not of the world, but I chose you out of the world,\ntherefore the world hateth you. Remember the word that I said unto\nyou, A servant is not greater than his lord. If they persecuted me,\nthey will also persecute you; if they kept my word, they will keep\nyours also. But all these things will they do unto you for my name's\nsake, because they know not him that sent me. If I had not come and\nspoken unto them, they had not had sin: but now they have no excuse\nfor their sin. He that hateth me hateth my Father also. If I had not\ndone among them the works which none other did, they had not had sin:\nbut now have they both seen and hated me and my Father. But this\ncometh to pass, that the word may be fulfilled that is written in\ntheir law, 'They hated me without a cause.' But when the Comforter is\ncome, whom I will send unto you from the Father, even the Spirit of\ntruth, which proceedeth from the Father, he shall bear witness of me:\nand ye also bear witness, because ye have been with me from the beginning.\n\n\"These things have I spoken unto you, that ye should not be caused to\nstumble. They shall put you out of the synagogues: yea, the hour\ncometh, that whosoever killeth you shall think that he offereth\nservice unto God. And these things will they do, because they have not\nknown the Father, nor me. But these things have I spoken unto you,\nthat when their hour is come, ye may remember them, how that I told\nyou. And these things I said not unto you from the beginning, because\nI was with you. But now I go unto him that sent me; and none of you\nasketh me, 'Whither goest thou?' But because I have spoken these\nthings unto you, sorrow hath filled your heart. Nevertheless I tell\nyou the truth: It is expedient for you that I go away; for if I go not\naway, the Comforter will not come unto you; but if I go, I will send\nhim unto you. And he, when he is come, will convict the world in\nrespect of sin, and of righteousness, and of judgment: of sin, because\nthey believe not on me; of righteousness, because I go to the Father,\nand ye behold me no more; of judgment, because the prince of this\nworld hath been judged. I have yet many things to say unto you, but ye\ncannot bear them now. Howbeit when he, the Spirit of truth, is come,\nhe shall guide you into all the truth: for he shall not speak from\nhimself; but what things soever he shall hear, these shall he speak:\nand he shall declare unto you the things that are to come. He shall\nglorify me: for he shall take of mine, and shall declare it unto you.\nAll things whatsoever the Father hath are mine: therefore said I, that\nhe taketh of mine, and shall declare it unto you. A little while, and\nye behold me no more; and again a little while, and ye shall see me.\"\n\nSome of his disciples therefore said one to another, \"What is this that\nhe saith unto us, 'A little while, and ye behold me not; and again a\nlittle while, and ye shall see me': and 'Because I go to the Father'?\"\n\nThey said therefore, \"What is this that he saith, 'A little while'? We\nknow not what he saith.\"\n\nJesus perceived that they were desirous to ask him, and he said unto them,\n\"Do ye inquire among yourselves concerning this, that I said, 'A little\nwhile, and ye behold me not, and again a little while, and ye shall see\nme'? Verily, verily, I say unto you, that ye shall weep and lament, but the\nworld shall rejoice: ye shall be sorrowful, but your sorrow shall be turned\ninto joy. A woman when she is in travail hath sorrow, because her hour is\ncome: but when she is delivered of the child she remembereth no more the\nanguish, for the joy that a man is born into the world. And ye therefore\nnow have sorrow: but I will see you again, and your heart shall rejoice,\nand your joy no one taketh away from you. And in that day ye shall ask me\nno question. Verily, verily, I say unto you, If ye shall ask anything of\nthe Father, he will give it you in my name. Hitherto have ye asked nothing\nin my name: ask, and ye shall receive, that your joy may be made full.\n\n\"These things have I spoken unto you in dark sayings: the hour cometh,\nwhen I shall no more speak unto you in dark sayings, but shall tell you\nplainly of the Father. In that day ye shall ask in my name: and I say\nnot unto you, that I will pray the Father for you; for the Father\nhimself loveth you, because ye have loved me, and have believed that I\ncame forth from the Father. I came out from the Father, and am come into\nthe world; again, I leave the world, and go unto the Father.\"\n\nHis disciples say, \"Lo, now speakest thou plainly, and speakest no dark\nsaying. Now know we that thou knowest all things, and needest not that any\nman should ask thee: by this we believe that thou camest forth from God.\"\n\nJesus answered them, \"Do ye now believe? Behold, the hour cometh, yea, is\ncome, that ye shall be scattered, every man to his own, and shall leave me\nalone: and yet I am not alone, because the Father is with me. These things\nhave I spoken unto you, that in me ye may have peace. In the world ye have\ntribulation: but be of good cheer: I have overcome the world.\"\n\n\nTHE INTERCESSORY PRAYER.\n\nThese things spake Jesus; and lifting up his eyes to heaven, he said,\n\"Father, the hour is come; glorify thy Son, that the Son may glorify thee:\neven as thou gavest him authority over all flesh, that to all whom thou\nhast given him, he should give eternal life. And this is life eternal, that\nthey should know thee the only true God, and him whom thou didst send, even\nJesus Christ. I glorified thee on the earth, having accomplished the work\nwhich thou hast given me to do. And now, Father, glorify thou me with thine\nown self with the glory which I had with thee before the world was. I\nmanifested thy name unto the men whom thou gavest me out of the world:\nthine they were, and thou gavest them to me; and they have kept thy word.\nNow they know that all things whatsoever thou hast given me are from thee:\nfor the words which thou gavest me I have given unto them; and they\nreceived them, and knew of a truth that I came forth from thee, and they\nbelieved that thou didst send me. I pray for them: I pray not for the\nworld, but for those whom thou hast given me; for they are thine: and all\nthings that are mine are thine, and thine are mine: and I am glorified in\nthem. And I am no more in the world, and these are in the world, and I come\nto thee. Holy Father, keep them in thy name which thou hast given me, that\nthey may be one, even as we are. While I was with them, I kept them in thy\nname which thou hast given me; and I guarded them, and not one of them\nperished, but the son of perdition; that the scripture might be fulfilled.\nBut now I come to thee; and these things I speak in the world, that they\nmay have my joy made full in themselves. I have given them thy word, and\nthe world hated them, because they are not of the world, even as I am not\nof the world. I pray not that thou shouldest take them from the world, but\nthat thou shouldest keep them from the evil one. They are not of the world,\neven as I am not of the world. Sanctify them in the truth: thy word is\ntruth. As thou didst send me into the world, even so sent I them into the\nworld. And for their sakes I sanctify myself, that they themselves also may\nbe sanctified in truth. Neither for these only do I pray, but for them also\nthat believe on me through their word; that they may all be one; even as\nthou, Father, art in me, and I in thee, that they also may be in us: that\nthe world may believe that thou didst send me. And the glory which thou\nhast given me I have given unto them; that they may be one, even as we are\none; I in them, and thou in me, that they may be perfected into one; that\nthe world may know that thou didst send me, and lovedst them, even as thou\nlovedst me. Father, I desire that they also whom thou hast given me be with\nme where I am, that they may behold my glory, which thou hast given me: for\nthou lovedst me before the foundation of the world. O righteous Father, the\nworld knew thee not, but I knew thee; and these knew that thou didst send\nme; and I made known unto them thy name, and will make it known; that the\nlove wherewith thou lovedst me may be in them, and I in them.\"\n\nAnd when they had sung a hymn, they went out into the mount of Olives.\n\n\n\n\nFRIDAY--THE DAY OF SUFFERING.\n\n\nTHE AGONY IN GETHSEMANE.\n\nAnd they come unto a place which was named Gethsemane: and he saith unto\nhis disciples, \"Sit ye here, while I pray.\"\n\nAnd he taketh with him Peter and James and John, and began to be greatly\namazed, and sore troubled. And he saith unto them, \"My soul is exceeding\nsorrowful even unto death: abide ye here, and watch.\"\n\nAnd he went forward a little, and fell on the ground, and prayed that,\nif it were possible, the hour might pass away from him.\n\nAnd he said, \"Abba, Father, all things are possible unto thee; remove\nthis cup from me: howbeit not what I will, but what thou wilt.\"\n\nAnd there appeared unto him an angel from heaven, strengthening him.\n\nAnd being in an agony, he prayed more earnestly; and his sweat became as\nit were great drops of blood falling down upon the ground.\n\nAnd when he rose up from his prayer, he came unto the disciples, and\nfound them sleeping for sorrow, and said unto Peter, \"Simon, sleepest\nthou? Couldest thou not watch one hour? Watch and pray, that ye enter\nnot into temptation: the spirit indeed is willing, but the flesh is weak.\"\n\nAgain a second time he went away, and prayed, saying, \"My Father, if\nthis cannot pass away, except I drink it, thy will be done.\"\n\nAnd he came again and found them sleeping, for their eyes were heavy.\nAnd he left them again, and went away, and prayed a third time, saying\nthe same words.\n\nThen cometh he to the disciples, and saith unto them, \"Sleep on now,\nand take your rest: behold, the hour is at hand, and the Son of man is\nbetrayed into the hands of sinners.\n\n\"Arise, let us be going: behold, he is at hand that betrayeth me.\"\n\n\nTHE BETRAYAL AND ARREST.\n\nAnd straightway, while he yet spake, cometh Judas, one of the twelve,\nand with him a multitude with swords and staves, from the chief priests\nand the scribes and the elders.\n\nNow he that betrayed him had given them a token, saying, \"Whomsoever I\nshall kiss, that is he; take him, and lead him away safely.\" And when he\nwas come, straightway he came to him, and saith, \"Rabbi,\" and kissed him.\n\nBut Jesus said unto him, \"Judas, betrayest thou the Son of man with a\nkiss?\"\n\nJesus, therefore, knowing all the things that were coming upon him, went\nforth, and saith unto them, \"Whom seek ye?\"\n\nThey answered him, \"Jesus of Nazareth.\"\n\nJesus saith unto them, \"I am he.\"\n\nAnd Judas also, who betrayed him, was standing with them. When therefore\nhe said unto them, \"I am he,\" they went backward, and fell to the ground.\n\nAgain therefore he asked them, \"Whom seek ye?\"\n\nAnd they said, \"Jesus of Nazareth.\"\n\nJesus answered, \"I told you that I am he; if therefore ye seek me, let\nthese go their way\": that the word might be fulfilled which he spake,\n\"Of those whom thou hast given me I lost not one.\"\n\nAnd when they that were about him saw what would follow, they said,\n\"Lord, shall we smite with the sword?\"\n\nSimon Peter therefore having a sword drew it, and struck the high priest's\nservant, and cut off his right ear. Now the servant's name was Malchus.\n\nBut Jesus answered and said, \"Suffer ye them thus far.\" And he touched\nhis ear, and healed him.\n\nThen saith Jesus unto Peter, \"Put up again thy sword into its place:\nfor all they that take the sword shall perish with the sword. Or\nthinkest thou that I cannot beseech my Father and he shall even now\nsend me more than twelve legions of angels? How then should the\nscriptures be fulfilled, that thus it must be? The cup which the\nFather hath given me, shall I not drink it?\"\n\nAnd Jesus said unto the chief priests and captains of the temple, and\nelders, that were come against him, \"Are ye come out as against a\nrobber, with swords and staves? When I was daily with you in the\ntemple, ye stretched not forth your hands against me: but this is your\nhour, and the power of darkness.\"\n\nThen all the disciples left him, and fled.\n\nAnd a certain young man followed with him, having a linen cloth cast\nabout him, over his naked body: and they lay hold on him; but he left\nthe linen cloth, and fled naked.\n\n\nTHE TRIAL BEFORE THE JEWISH AUTHORITIES.\n\nSo the band and the chief captain, and the officers of the Jews,\nseized Jesus and bound him, and led him to Annas first; for he was\nfather in law to Caiaphas, who was high priest that year. Now Caiaphas\nwas he that gave counsel to the Jews, that it was expedient that one\nman should die for the people.\n\nAnd Simon Peter followed Jesus, and so did another disciple. Now that\ndisciple was known unto the high priest, and entered in with Jesus into\nthe court of the high priest; but Peter was standing at the door\nwithout. So the other disciple, who was known unto the high priest,\nwent out and spake unto her that kept the door, and brought in Peter.\n\nThe maid therefore that kept the door saith unto Peter, \"Art thou also\none of this man's disciples?\"\n\nHe saith, \"I am not.\"\n\nNow the servants and the officers were standing there, having made a\nfire of coals; for it was cold; and they were warming themselves; and\nPeter also was with them standing and warming himself.\n\nThe high priest therefore asked Jesus of his disciples, and of his\nteaching. Jesus answered him, \"I have spoken openly to the world; I even\ntaught in synagogues, and in the temple, where all the Jews come together;\nand in secret spake I nothing. Why askest thou me? Ask them that have heard\nme, what I spake unto them: behold, these know the things which I said.\"\n\nAnd when he had said this, one of the officers standing by struck Jesus\nwith his hand, saying, \"Answerest thou the high priest so?\"\n\nJesus answered him, \"If I have spoken evil, bear witness of the evil;\nbut if well, why smitest thou me?\"\n\nAnnas therefore sent him bound unto Caiaphas the high priest.\n\nNow the chief priests and the whole council sought witness against Jesus to\nput him to death; and found it not. For many bare false witness against\nhim, and their witness agreed not together. And there stood up certain, and\nbare false witness against him, saying, \"We heard him say, I will destroy\nthis temple that is made with hands, and in three days I will build another\nmade without hands.\" And not even so did their witness agree together.\n\nAnd the high priest stood up in the midst, and asked Jesus, saying,\n\"Answerest thou nothing? What is it which these witness against thee?\"\n\nBut he held his peace, and answered nothing.\n\nAnd the high priest said unto him, \"I adjure thee by the living God,\nthat thou tell us whether thou art the Christ, the Son of God.\"\n\nAnd Jesus said, \"I am: and ye shall see the Son of man sitting at the\nright hand of Power, and coming with the clouds of heaven.\"\n\nAnd the high priest rent his clothes, and saith, \"What further need have\nwe of witnesses? Ye have heard the blasphemy: what think ye?\"\n\nAnd they all condemned him to be worthy of death.\n\nThen did they spit in his face and buffet him. And they blindfolded him\nand smote him with the palms of their hands, saying, \"Prophesy unto us,\nthou Christ: who is he that struck thee?\"\n\n\nTHE DENIAL OF PETER.\n\nAnd as Peter was beneath in the court, there cometh one of the maids of\nthe high priest; and seeing Peter warming himself, she looked upon him,\nand saith, \"Thou also wast with the Nazarene, even Jesus.\"\n\nBut he denied, saying, \"I neither know nor understand what thou sayest,\"\nand he went out into the porch; and the cock crew.\n\nAnd after a little while they that stood by came and said to Peter, \"Of\na truth thou also art one of them; for thy speech maketh thee known.\"\n\nThen began he to curse and to swear, \"I know not the man.\" And\nstraightway the cock crew.\n\nAnd the Lord turned, and looked upon Peter. And Peter remembered the\nword of the Lord, how that he said unto him, \"Before the cock crow twice\nthou shalt deny me thrice.\"\n\nAnd he went out, and wept bitterly.\n\nAnd straightway in the morning the chief priests with the elders and\nscribes, and the whole council, held a consultation, and bound Jesus,\nand carried him away, and delivered him up to Pilate, the governor.\n\n\nTHE REMORSE OF JUDAS.\n\nThen Judas, who betrayed him, when he saw that he was condemned, repented\nhimself, and brought back the thirty pieces of silver to the chief priests\nand elders, saying, \"I have sinned in that I betrayed innocent blood.\"\n\nBut they said, \"What is that to us? See thou to it.\"\n\nAnd he cast down the pieces of silver into the sanctuary, and departed;\nand he went away and hanged himself.\n\nAnd the chief priests took the pieces of silver, and said, \"It is not\nlawful to put them into the treasury, since it is the price of blood.\"\nAnd they took counsel, and bought with them the potter's field, to\nbury strangers in. Wherefore that field was called, \"The field of\nblood,\" unto this day.\n\nThen was fulfilled that which was spoken through Jeremiah the prophet,\nsaying, \"And they took the thirty pieces of silver, the price of him\nthat was priced, whom certain of the children of Israel did price; and\nthey gave them for the potter's field, as the Lord appointed me.\"\n\n\nTHE TRIAL BEFORE PILATE.\n\nThey led Jesus therefore from Caiaphas into the Praetorium: and it was\nearly; and they themselves entered not into the Praetorium, that they\nmight not be defiled, but might eat the passover. Pilate therefore went\nout unto them, and saith, \"What accusation bring ye against this man?\"\n\nThey answered and said unto him, \"If this man were not an evil-doer, we\nshould not have delivered him up unto thee.\"\n\nPilate therefore said unto them, \"Take him yourselves, and judge him\naccording to your law.\"\n\nThe Jews said unto him, \"It is not lawful for us to put any man to\ndeath\": that the word of Jesus might be fulfilled, which he spake,\nsignifying by what manner of death he should die.\n\nAnd they began to accuse him, saying, \"We found this man perverting our\nnation, and forbidding to give tribute to Caesar, and saying that he\nhimself is Christ a king.\"\n\nAnd when he was accused by the chief priests and elders, he answered\nnothing. Then saith Pilate unto him, \"Hearest thou not how many things\nthey witness against thee?\" And he gave him no answer, not even to one\nword: insomuch that the governor marvelled greatly.\n\nPilate therefore entered again into the Praetorium, and called Jesus, and\nsaid unto him, \"Art thou the King of the Jews?\"\n\nJesus answered, \"Sayest thou this of thyself, or did others tell it thee\nconcerning me?\"\n\nPilate answered, \"Am I a Jew? Thine own nation and the chief priests\ndelivered thee unto me: what hast thou done?\"\n\nJesus answered, \"My kingdom is not of this world: if my kingdom were of\nthis world, then would my servants fight, that I should not be delivered\nto the Jews; but now is my kingdom not from hence.\"\n\nPilate therefore said unto him, \"Art thou a king then?\"\n\nJesus answered, \"Thou sayest that I am a king. To this end have I been\nborn, and to this end am I come into the world, that I should bear\nwitness unto the truth. Every one that is of the truth heareth my voice.\"\n\nPilate saith unto him, \"What is truth?\"\n\nAnd when he had said this, he went out again unto the Jews, and saith\nunto them, \"I find no crime in him.\"\n\nBut they were the more urgent, saying, \"He stirreth up the people, teaching\nthroughout all Judaea, and beginning from Galilee, even unto this place.\"\n\nBut when Pilate heard it, he asked whether the man were a Galilaean. And\nwhen he knew that he was of Herod's jurisdiction, he sent him unto\nHerod, who himself also was at Jerusalem in these days.\n\n\nJESUS BEFORE HEROD.\n\nNow when Herod saw Jesus, he was exceeding glad; for he was of a long time\ndesirous to see him, because he had heard concerning him; and he hoped to\nsee some miracle done by him. And he questioned him in many words; but he\nanswered him nothing. And the chief priests and the scribes stood,\nvehemently accusing him. And Herod with his soldiers set him at nought, and\nmocked him, and arraying him in gorgeous apparel sent him back to Pilate.\n\nAnd Herod and Pilate became friends with each other that very day: for\nbefore they were at enmity between themselves.\n\n\nTHE TRIAL BEFORE PILATE RESUMED.\n\nAnd Pilate called together the chief priests and the rulers and the people,\nand said unto them, \"Ye brought unto me this man, as one that perverteth\nthe people: and behold, I, having examined him before you, found no fault\nin this man touching those things whereof ye accuse him: no, nor yet Herod:\nfor he sent him back unto us; and behold, nothing worthy of death hath been\ndone by him. I will therefore chastise him, and release him.\"\n\nNow at the feast the governor was wont to release unto the multitude one\nprisoner, whom they would. And they had then a notable prisoner, called\nBarabbas, lying bound with them that had made insurrection, men who in\nthe insurrection had committed murder. And the multitude went up and\nbegan to ask him to do as he was wont to do unto them.\n\nAnd Pilate answered them, saying, \"Will ye that I release unto you the\nKing of the Jews?\" For he perceived that for envy the chief priests had\ndelivered him up.\n\nNow the chief priests and the elders persuaded the multitudes that they\nshould ask for Barabbas, and destroy Jesus.\n\nBut the governor answered and said unto them, \"Which of the two will ye\nthat I release unto you?\"\n\nAnd they said, \"Barabbas.\"\n\nPilate saith unto them, \"What then shall I do unto Jesus who is called\nChrist?\"\n\nThey all say, \"Let him be crucified.\"\n\nAnd he said unto them a third time, \"Why, what evil hath this man done? I\nhave found no cause of death in him: I will therefore chastise and release\nhim.\"\n\nThen Pilate therefore took Jesus, and scourged him.\n\nAnd the soldiers led him away within the court, which is the Praetorium;\nand they call together the whole band.\n\nAnd they stripped him, and arrayed him in a purple garment. And they\nplatted a crown of thorns and put it upon his head, and a reed in his\nright hand; and they kneeled down before him, and mocked him, saying:\n\"Hail, King of the Jews!\" and they struck him with their hands. And they\nspat upon him, and took the reed and smote him upon the head.\n\nAnd Pilate went out again, and saith unto them, \"Behold, I bring him out\nto you, that ye may know that I find no crime in him.\"\n\nJesus therefore came out, wearing the crown of thorns and the purple\ngarment. And Pilate saith unto them, \"Behold, the man!\"\n\nWhen therefore the chief priests and the officers saw him, they cried\nout, saying, \"Crucify him, crucify him!\"\n\nPilate saith unto them, \"Take him yourselves, and crucify him: for I\nfind no crime in him.\"\n\nThe Jews answered him, \"We have a law, and by that law he ought to die,\nbecause he made himself the Son of God.\"\n\nWhen Pilate therefore heard this saying, he was the more afraid; and he\nentered into the Praetorium again, and saith unto Jesus, \"Whence art thou?\"\n\nBut Jesus gave him no answer.\n\nPilate therefore saith unto him, \"Speakest thou not unto me? Knowest thou\nnot that I have power to release thee, and have power to crucify thee?\"\n\nJesus answered him, \"Thou wouldest have no power against me, except it\nwere given thee from above: therefore he that delivered me unto thee\nhath greater sin.\"\n\nUpon this Pilate sought to release him: but the Jews cried out, saying,\n\"If thou release this man, thou art not Caesar's friend: every one that\nmaketh himself a king speaketh against Caesar.\"\n\nWhen Pilate therefore heard these words, he brought Jesus out, and sat\ndown on the judgment-seat at a place called The Pavement, but in\nHebrew, Gabbatha.\n\nAnd while he was sitting on the judgment-seat, his wife sent unto him,\nsaying, \"Have thou nothing to do with that righteous man; for I have\nsuffered many things this day in a dream because of him.\"\n\nNow it was the Preparation of the passover: it was about the sixth hour.\nAnd he saith unto the Jews, \"Behold, your King.\"\n\nThey therefore cried out, \"Away with him, away with him, crucify him!\"\n\nPilate saith unto them, \"Shall I crucify your King?\"\n\nThe chief priests answered, \"We have no king but Caesar.\"\n\nSo when Pilate saw that he prevailed nothing, but rather that a tumult\nwas arising, he took water, and washed his hands before the multitude,\nsaying, \"I am innocent of the blood of this righteous man; see ye to it.\"\n\nAnd all the people answered and said, \"His blood be on us, and on our\nchildren.\"\n\nAnd they were urgent with loud voices asking that he might be crucified.\nAnd their voices prevailed.\n\nAnd Pilate, wishing to content the multitude, gave sentence that what\nthey asked for should be done. And he released unto them Barabbas, him\nthat for insurrection and murder had been cast into prison, whom they\nasked for; but Jesus he delivered up to their will.\n\nAnd when they had mocked him, they took off from him the robe, and put\non him his garments, and led him away to crucify him.\n\n\nTHE SORROWFUL WAY.\n\nThey took Jesus therefore: and he went out, bearing the cross for himself.\n\nAnd as they came out, they laid hold upon one Simon of Cyrene, the\nfather of Alexander and Rufus, who was passing by, coming from the\ncountry; him they compelled to go with them, and laid on him the cross,\nthat he might bear it after Jesus.\n\nAnd there followed him a great multitude of the people, and of women who\nbewailed and lamented him.\n\nBut Jesus turning unto them said, \"Daughters of Jerusalem, weep not\nfor me, but weep for yourselves, and for your children. For behold,\nthe days are coming, in which they shall say, 'Blessed are the barren,\nand the wombs that never bare, and the breasts that never gave suck.'\nThen shall they begin to say to the mountains, 'Fall on us'; and to\nthe hills, 'Cover us.' For if they do these things in the green tree,\nwhat shall be done in the dry?\"\n\nAnd there were also two others, malefactors, led with him to be put to\ndeath.\n\n\nTHE CRUCIFIXION.\n\nAnd when they were come unto a place called Golgotha, that is to say,\nThe place of a skull, they gave him wine to drink mingled with gall: and\nwhen he had tasted it, he would not drink.\n\nThere they crucified him, and the malefactors, one on the right hand and\nthe other on the left.\n\nAnd Jesus said, \"Father, forgive them; for they know not what they do.\"\n\nAnd Pilate wrote a title also, and put it on the cross. And there was\nwritten:\n\n    --------------------------------------------\n    | JESUS OF NAZARETH, THE KING OF THE JEWS. |\n    --------------------------------------------\n\nThis title therefore read many of the Jews, for the place where Jesus\nwas crucified was nigh to the city; and it was written in Hebrew, and in\nLatin, and in Greek.\n\nThe chief priests of the Jews therefore said to Pilate, \"Write not, 'The\nKing of the Jews,' but that he said, 'I am King of the Jews.'\"\n\nPilate answered, \"What I have written I have written.\"\n\nThe soldiers therefore, when they had crucified Jesus, took his garments\nand made four parts, to every soldier a part; and also the coat: now the\ncoat was without seam, woven from the top throughout. They said\ntherefore one to another, \"Let us not rend it, but cast lots for it,\nwhose it shall be\": that the scripture might be fulfilled, which saith,\n\n        \"They parted my garments among them,\n        And upon my vesture did they cast lots.\"\n\nThese things therefore the soldiers did; and they sat and watched him\nthere.\n\nAnd the people stood beholding.\n\nAnd they that passed by railed on him, wagging their heads, and saying,\n\"Thou that destroyest the temple, and buildest it in three days, save\nthyself: if thou art the Son of God, come down from the cross.\"\n\nIn like manner also, the chief priests mocking him, with the scribes and\nelders, said, \"He saved others; himself he cannot save. Let the Christ,\nthe King of Israel, now come down from the cross, that we may see and\nbelieve. He trusteth on God; let him deliver him now, if he desireth\nhim: for he said, I am the Son of God.\"\n\nAnd one of the malefactors that were hanged railed on him, saying, \"Art\nnot thou the Christ? Save thyself and us.\"\n\nBut the other answered, and rebuking him said, \"Dost thou not even fear\nGod, seeing thou art in the same condemnation? And we indeed justly; for\nwe receive the due reward of our deeds: but this man hath done nothing\namiss.\" And he said, \"Jesus, remember me when thou comest in thy kingdom.\"\n\nAnd he said unto him, \"Verily I say unto thee, To-day shalt thou be with\nme in Paradise.\"\n\nBut there were standing by the cross of Jesus his mother, and his\nmother's sister, Mary the wife of Cleopas, and Mary Magdalene. When\nJesus therefore saw his mother, and the disciple standing by whom he\nloved, he saith unto his mother, \"Woman, behold thy son!\"\n\nThen saith he to the disciple, \"Behold thy mother!\"\n\nAnd from that hour the disciple took her unto his owns home.\n\nAnd when the sixth hour was come, there was darkness over the whole land\nuntil the ninth hour. And at the ninth hour Jesus cried with a loud\nvoice, \"Eloi, Eloi, lama sabachthani?\" which is, being interpreted, \"My\nGod, my God, why hast thou forsaken me?\"\n\nAnd some of them that stood by, when they heard it, said, \"Behold, he\ncalleth Elijah.\"\n\nAfter this, Jesus, knowing that all things are now finished, that the\nscripture might be accomplished, saith, \"I thirst.\"\n\nThere was set there a vessel full of vinegar: so they put a sponge full\nof the vinegar upon hyssop, and brought it to his mouth. When Jesus\ntherefore had received the vinegar, he said, \"It is finished.\"\n\nAnd Jesus, crying with a loud voice, said, \"Father, into thy hands I\ncommend my spirit,\" and having said this, he gave up the ghost.\n\nAnd behold, the veil of the temple was rent in two from the top to the\nbottom; and the earth did quake; and the rocks were rent; and the tombs\nwere opened; and many bodies of the saints that had fallen asleep were\nraised; and coming forth out of the tombs after his resurrection they\nentered into the holy city and appeared unto many.\n\nNow the centurion, and they that were with him watching Jesus, when they\nsaw the earthquake, and the things that were done, feared exceedingly,\nsaying, \"Truly this was the Son of God.\"\n\nAnd all the multitudes that came together to this sight, when they\nbeheld the things that were done, returned smiting their breasts. And\nmany women were there beholding from afar, who had followed Jesus from\nGalilee, ministering unto him; among whom was Mary Magdalene, and Mary\nthe mother of James and Joses, and the mother of the sons of Zebedee.\n\nThe Jews therefore, because it was the Preparation, that the bodies\nshould not remain on the cross upon the sabbath (for the day of that\nsabbath was a high day), asked of Pilate that their legs might be\nbroken, and that they might be taken away.\n\nThe soldiers therefore came, and brake the legs of the first, and of\nthe other that was crucified with him: but when they came to Jesus, and\nsaw that he was dead already, they brake not his legs; howbeit one of\nthe soldiers with a spear pierced his side, and straightway there came\nout blood and water. And he that hath seen hath borne witness, and his\nwitness is true: and he knoweth that he saith true, that ye also may\nbelieve. For these things came to pass, that the scripture might be\nfulfilled, \"A bone of him shall not be broken.\" And again another\nscripture saith, \"They shall look on him whom they pierced.\"\n\n\nTHE BURIAL.\n\nAnd after these things, when even was come, there came a rich man from\nArimathaea, named Joseph, a councillor of honorable estate, a disciple of\nJesus, but secretly for fear of the Jews; and he boldly went in unto Pilate\nand asked for the body of Jesus. And Pilate marvelled if he were already\ndead: and calling unto him the centurion, he asked him whether he had been\nany while dead. And when he learned it of the centurion, he granted the\ncorpse to Joseph.\n\nHe came therefore, and took away his body. And there came also Nicodemus,\nhe who at the first came to him by night, bringing a mixture of myrrh and\naloes, about a hundred pounds. So they took the body of Jesus, and bound it\nin linen cloths with the spices, as the custom of the Jews is to bury.\n\nNow in the place where he was crucified there was a garden: and in the\ngarden a new tomb wherein was never man yet laid. There then because of\nthe Jews' Preparation (for the tomb was nigh at hand), they laid Jesus;\nand rolled a stone against the door of the tomb.\n\nAnd Mary Magdalene and Mary the mother of Joses beheld the tomb, and how\nhis body was laid. And they returned, and prepared spices and ointments.\n\n\n\n\nSATURDAY--THE DAY OF SILENCE AND SORROW.\n\n\nTHE WATCH AT THE TOMB.\n\nNow on the morrow, which is the day after the Preparation, the chief\npriests and the Pharisees were gathered together unto Pilate, saying, \"Sir,\nwe remember that that deceiver said while he was yet alive, 'After three\ndays I rise again.' Command therefore that the sepulchre be made sure until\nthe third day, lest haply his disciples come and steal him away, and say\nunto the people, 'He is risen from the dead,' and the last error will be\nworse than the first.\"\n\nPilate said unto them, \"Ye have a guard: go, make it as sure as ye can.\"\n\nSo they went, and made the sepulchre sure, sealing the stone, the guard\nbeing with them.\n\n\n\n\nSUNDAY--THE DAY OF RESURRECTION.\n\n\nTHE EARTHQUAKE.\n\nAnd behold, there was a great earthquake; for an angel of the Lord\ndescended from heaven, and came and rolled away the stone, and sat upon\nit. His appearance was as lightning, and his raiment white as snow: and\nfor fear of him the watchers did quake, and became as dead men.\n\n\nTHE EMPTY TOMB.\n\nNow on the first day of the week cometh Mary Magdalene early, while it\nwas yet dark, unto the tomb, and seeth the stone taken away from the\ntomb. She runneth therefore, and cometh to Simon Peter, and to the other\ndisciple whom Jesus loved, and saith unto them, \"They have taken away\nthe Lord out of the tomb, and we know not where they have laid him.\"\n\nPeter therefore went forth, and the other disciple, and they went toward\nthe tomb. And they ran both together: and the other disciple outran\nPeter, and came first to the tomb; and stooping and looking in, he seeth\nthe linen cloths lying; yet entered he not in.\n\nSimon Peter therefore also cometh, following him, and entered into the\ntomb: and he beholdeth the linen cloths lying, and the napkin, that was\nupon his head, not lying with the linen cloths, but rolled up in a place by\nitself. Then entered in therefore the other disciple also, who came first\nto the tomb, and he saw, and believed. For as yet they knew not the\nscripture, that he must rise again from the dead. So the disciples went\naway again unto their own home.\n\n\nTHE APPEARANCE TO MARY.\n\nBut Mary was standing without at the tomb weeping: so, as she wept, she\nstooped and looked into the tomb; and she beholdeth two angels in white\nsitting, one at the head, and one at the feet, where the body of Jesus\nhad lain. And they say unto her, \"Woman, why weepest thou?\"\n\nShe saith unto them, \"Because they have taken away my Lord, and I know\nnot where they have laid him.\"\n\nWhen she had thus said, she turned herself back, and beholdeth Jesus\nstanding, and knew not that it was Jesus.\n\nJesus saith unto her, \"Woman, why weepest thou? Whom seekest thou?\"\n\nShe, supposing him to be the gardener, saith unto him, \"Sir, if thou hast\nborne him hence, tell me where thou hast laid him, and I will take him\naway.\"\n\nJesus saith unto her, \"Mary.\"\n\nShe turneth herself, and saith unto him in Hebrew, \"Rabboni\"; which is\nto say, \"Teacher.\"\n\nJesus saith to her, \"Touch me not; for I am not yet ascended unto the\nFather; but go unto my brethren, and say to them, 'I ascend unto my\nFather, and your Father, and my God and your God.'\"\n\nMary Magdalene cometh and telleth the disciples, \"I have seen the Lord\";\nand that he had said these things unto her.\n\n\nTHE APPEARANCE TO THE WOMEN.\n\nAnd the women which had come with him out of Galilee came unto the tomb,\nbringing the spices which they had prepared. And they were saying among\nthemselves, \"Who shall roll us away the stone from the door of the tomb?\"\nand looking up, they see that the stone is rolled back: for it was\nexceeding great. And entering into the tomb, they saw a young man sitting\non the right side, arrayed in a white robe; and they were amazed. And he\nsaith unto them, \"Be not amazed: ye seek Jesus, the Nazarene, who hath been\ncrucified; he is risen; he is not here: behold, the place where they laid\nhim! But go, tell his disciples and Peter, 'He goeth before you into\nGalilee: there shall ye see him, as he said unto you.'\"\n\nAnd they departed quickly from the tomb with fear and great joy, and ran\nto bring his disciples word.\n\nAnd behold, Jesus met them, saying, \"All hail.\" And they came and took\nhold of his feet, and worshipped him.\n\nThen saith Jesus unto them, \"Fear not: go tell my brethren that they\ndepart into Galilee, and there shall they see me.\"\n\n\nREPORT OF THE WATCH.\n\nNow while they were going, behold, some of the guard came into the city,\nand told unto the chief priests all the things that were come to pass.\nAnd when they were assembled with the elders, and had taken counsel,\nthey gave much money unto the soldiers, saying, \"Say ye, 'His disciples\ncame by night, and stole him away while we slept.' And if this come to\nthe governor's ears, we will persuade him, and rid you of care.\"\n\nSo they took the money and did as they were taught: and this saying was\nspread abroad among the Jews, and continueth until this day.\n\n\nTHE APPEARANCE AT EMMAUS.\n\nAnd behold, two of them were going that very day to a village named\nEmmaus, which was three-score furlongs from Jerusalem. And they communed\nwith each other of all these things which had happened.\n\nAnd it came to pass, while they communed and questioned together, that\nJesus himself drew near, and went with them. But their eyes were holden\nthat they should not know him.\n\nAnd he said unto them, \"What communications are these that ye have one\nwith another, as ye walk?\"\n\nAnd they stood still, looking sad. And one of them, named Cleopas,\nanswering, said unto him, \"Dost thou alone sojourn in Jerusalem and not\nknow the things which are come to pass there in these days?\"\n\nAnd he said unto them, \"What things?\"\n\nAnd they said unto him, \"The things concerning Jesus the Nazarene, who\nwas a prophet mighty in deed and word before God, and all the people:\nand how the chief priests and our rulers delivered him up to be\ncondemned to death, and crucified him. But we hoped that it was he who\nshould redeem Israel. Yea, and besides all this, it is now the third\nday since these things came to pass. Moreover, certain women of our\ncompany amazed us, having been early at the tomb; and when they found\nnot his body, they came, saying, that they had also seen a vision of\nangels, who said that he was alive. And certain of them that were with\nus went to the tomb, and found it even so as the women had said: but\nhim they saw not.\"\n\nAnd he said unto them, \"O foolish men, and slow of heart to believe in\nall that the prophets have spoken! Behooved it not the Christ to suffer\nthese things, and to enter into his glory?\"\n\nAnd beginning from Moses and from all the prophets, he interpreted to\nthem in all the scriptures the things concerning himself.\n\nAnd they drew nigh unto the village, whither they were going: and he\nmade as though he would go further. And they constrained him, saying,\n\"Abide with us; for it is toward evening, and the day is now far spent.\"\n\nAnd he went in to abide with them. And it came to pass, when he had sat\ndown with them to meat, he took the bread and blessed; and breaking it,\nhe gave to them. And their eyes were opened, and they knew him; and he\nvanished out of their sight.\n\nAnd they said one to another, \"Was not our heart burning within us,\nwhile he spake to us in the way, while he opened to us the scriptures?\"\n\nAnd they rose up that very hour, and returned to Jerusalem, and found\nthe eleven gathered together, and them that were with them, saying, \"The\nLord is risen indeed, and hath appeared to Simon.\" And they rehearsed\nthe things that happened in the way, and how he was known of them in the\nbreaking of the bread.\n\n\nTHE APPEARANCE TO THE DISCIPLES.\n\nWhen therefore it was evening, on that day, the first day of the week, and\nwhen the doors were shut where the disciples were, for fear of the Jews,\nJesus came and stood in the midst and saith unto them, \"Peace be unto you.\"\n\nBut they were terrified and affrighted, and supposed that they beheld a\nspirit. And he said unto them, \"Why are ye troubled? and wherefore do\nquestionings arise in your heart? See my hands and my feet, that it is I\nmyself: handle me, and see: for a spirit hath not flesh and bones, as ye\nbehold me having.\"\n\nAnd when he had said this, he showed them his hands and his feet.\n\nAnd while they still disbelieved for joy, and wondered, he said unto\nthem, \"Have ye here anything to eat?\"\n\nAnd they gave him a piece of broiled fish. And he took it, and ate\nbefore them.\n\nJesus therefore said to them again, \"Peace be unto you: as the Father hath\nsent me, even so send I you.\" And when he had said this, he breathed on\nthem, and saith unto them, \"Receive ye the Holy Spirit: whose soever sins\nye forgive, they are forgiven unto them; whose soever sins ye retain, they\nare retained.\"\n\n\n\n\nAFTER THE RESURRECTION DAY.\n\n\nTHE APPEARANCE TO THE DISCIPLES AND TO THOMAS.\n\nBut Thomas, one of the twelve, called Didymus, was not with them when Jesus\ncame. The other disciples therefore said unto him, \"We have seen the Lord.\"\n\nBut he said unto them, \"Except I shall see in his hands the print of the\nnails, and put my finger into the print of the nails, and put my hand\ninto his side, I will not believe.\"\n\nAnd after eight days again his disciples were within, and Thomas with\nthem. Jesus cometh, the doors being shut, and stood in the midst, and\nsaid, \"Peace be unto you.\"\n\nThen saith he to Thomas, \"Reach hither thy finger, and see my hands; and\nreach hither thy hand, and put it into my side: and be not faithless,\nbut believing.\"\n\nThomas answered and said unto him, \"My Lord and my God.\"\n\nJesus saith unto him, \"Because thou hast seen me, thou hast believed:\nblessed are they that have not seen, and yet have believed.\"\n\n\nTHE APPEARANCE TO THE SEVEN BY THE SEA.\n\nAfter these things Jesus manifested himself again to the disciples at\nthe sea of Tiberias; and he manifested himself on this wise. There were\ntogether Simon Peter, and Thomas called Didymus and Nathanael of Cana in\nGalilee, and the sons of Zebedee, and two other of his disciples.\n\nSimon Peter saith unto them, \"I go a fishing.\"\n\nThey say unto him, \"We also come with thee.\"\n\nThey went forth, and entered into the boat; and that night they took\nnothing. But when day was now breaking, Jesus stood on the beach: yet\nthe disciples knew not that it was Jesus.\n\nJesus therefore saith unto them, \"Children, have ye aught to eat?\"\n\nThey answered him, \"No.\"\n\nAnd he said unto them, \"Cast the net on the right side of the boat, and\nye shall find.\"\n\nThey cast therefore, and now they were not able to draw it for the\nmultitude of fishes.\n\nThat disciple therefore whom Jesus loved saith unto Peter, \"It is the\nLord.\" So when Simon Peter heard that it was the Lord, he girt his coat\nabout him (for he was naked), and cast himself into the sea.\n\nBut the other disciples came in the little boat (for they were not far from\nthe land, but about two hundred cubits off), dragging the net full of\nfishes.\n\nSo when they got out upon the land, they see a fire of coals there, and\nfish laid thereon, and bread. Jesus saith unto them, \"Bring of the fish\nwhich ye have now taken.\"\n\nSimon Peter therefore went up, and drew the net to land, full of great\nfishes, a hundred and fifty and three; and for all there were so many,\nthe net was not rent.\n\nJesus saith unto them, \"Come and break your fast.\"\n\nAnd none of the disciples durst inquire of him, \"Who art thou?\" knowing\nthat it was the Lord.\n\nJesus cometh, and taketh the bread, and giveth them, and the fish likewise.\n\nThis is now the third time that Jesus was manifested to the disciples,\nafter that he was risen from the dead.\n\nSo when they had broken their fast, Jesus saith to Simon Peter, \"Simon,\nson of John, lovest thou me more than these?\"\n\nHe saith unto him, \"Yea, Lord; thou knowest that I love thee.\"\n\nHe saith unto him, \"Feed my lambs.\"\n\nHe saith unto him again a second time, \"Simon, son of John, lovest thou\nme?\"\n\nHe saith unto him, \"Yea, Lord, thou knowest that I love thee.\"\n\nHe saith unto him, \"Tend my sheep.\"\n\nHe saith unto him the third time, \"Simon, son of John, lovest thou me?\"\n\nPeter was grieved because he said unto him the third time, \"Lovest thou\nme?\" And he said unto him, \"Lord, thou knowest all things; thou knowest\nthat I love thee.\"\n\nJesus saith unto him, \"Feed my sheep. Verily, verily, I say unto thee, When\nthou wast young, thou girdedst thyself, and walkedst whither thou wouldest;\nbut when thou shalt be old, thou shalt stretch forth thy hands, and another\nshall gird thee, and carry thee whither thou wouldest not.\"\n\nNow this he spake, signifying by what manner of death he should glorify\nGod. And when he had spoken this, he saith unto him, \"Follow me.\"\n\nPeter, turning about, seeth the disciple whom Jesus loved following; who\nalso leaned back on his breast at the supper, and said, \"Lord, who is he\nthat betrayeth thee?\" Peter therefore seeing him saith to Jesus, \"Lord,\nand what shall this man do?\"\n\nJesus saith unto him, \"If I will that he tarry till I come, what is that\nto thee? Follow thou me.\"\n\nThis saying therefore went forth among the brethren, that that disciple\nshould not die: yet Jesus said not unto him, that he should not die,\nbut, \"If I will that he tarry till I come, what is that to thee?\"\n\n\nTHE APPEARANCE TO THE ELEVEN ON THE MOUNTAIN.\n\nThe eleven disciples went into Galilee, unto the mountain where Jesus\nhad appointed them. And when they saw him, they worshipped him; but some\ndoubted. And Jesus came to them and spake unto them, saying, \"All\nauthority hath been given unto me in heaven and on earth. Go ye\ntherefore, and make disciples of all the nations, baptizing them in the\nname of the Father and of the Son and of the Holy Spirit: teaching them\nto observe all things whatsoever I commanded you: and lo, I am with you\nalways, even unto the end of the world.\"\n\n\nTHE LAST APPEARANCE AND ASCENSION.\n\nAnd he said unto them, \"These are my words which I spake unto you, while I\nwas yet with you, that all things must needs be fulfilled, which are\nwritten in the law of Moses, and the prophets, and the psalms, concerning\nme.\"\n\nThen opened he their mind, that they might understand the scriptures;\nand he said unto them, \"Thus it is written, that the Christ should\nsuffer, and rise again from the dead the third day; and that repentance\nand remission of sins should be preached in his name unto all the\nnations, beginning from Jerusalem. Ye are witnesses of these things. And\nbehold, I send forth the promise of my Father upon you: but tarry ye in\nthe city, until ye be clothed with power from on high.\"\n\nAnd he led them out until they were over against Bethany: and he lifted\nup his hands, and blessed them.\n\nAnd it came to pass, while he blessed them, he parted from them, and was\ncarried up into heaven. And they worshipped him, and returned to\nJerusalem with great joy: and were continually in the temple, blessing God.\n\n       *       *       *       *       *\n\nMANY OTHER SIGNS THEREFORE DID JESUS IN THE PRESENCE OF THE DISCIPLES,\nWHICH ARE NOT WRITTEN IN THIS BOOK: BUT THESE ARE WRITTEN, THAT YE MAY\nBELIEVE THAT JESUS IS THE CHRIST, THE SON OF GOD; AND THAT BELIEVING YE\nMAY HAVE LIFE IN HIS NAME.\n\n\n\n\n[Transcriber's Note:\n\n\n* p. 11: Replaced the word \"Caesar\" with \"Caesar\" located in the phrase\n\"Tribute to Caesar\" to be consistent with other similar spellings.\n\n* p. 14: Corrected spelling of word \"cribes\" to \"scribes\" located in the\nphrase \"But woe unto you, cribes\".\n\n* p. 23: Added missing closing quotation after the word \"hour\" located\nin the phrase \"not the day nor the hour\".\n\n* p. 24: Added missing closing quotation after the word \"teeth\" located\nin the phrase \"and the gnashing of teeth\".\n\n* p. 34: Corrected spelling of word \"m\" to \"me\" located in the phrase\n\"If a man abide not in m\".\n\n* p. 43: Corrected spelling of word \"ever\" to \"even\" located in the\nphrase \"I ever taught in synagogues\".]\n\n\n\n\n\nEnd of the Project Gutenberg EBook of His Last Week, by\nWilliam E. Barton and Theodore G. Soares and Sydney Strong\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Annie R. McGuire\n\n\n\n\n\n\n\n\n[Illustration: HARPER'S YOUNG PEOPLE]\n\n       *       *       *       *       *\n\nVOL. III.--NO. 121. PUBLISHED BY HARPER & BROTHERS, NEW YORK. PRICE FOUR\nCENTS.\n\nTuesday, February 21, 1882. Copyright, 1882, by HARPER & BROTHERS. $1.50\nper Year, in Advance.\n\n       *       *       *       *       *\n\n\n\n\n[Illustration: JESSE'S VALENTINE.--DRAWN BY W. ST. JOHN HARPER.]\n\nMISS HOLSOVER'S \"TREASURE.\"\n\nA Story of St. Valentine's Day.\n\nII.\n\n\nThe voices went on, but Jesse's power of listening seemed gone. He had\nto clutch the ladder for support. What could he do?\n\nIt was clear to his mind that Bill had come back for some evil deed.\nWhat was it he had come to steal? Jesse shook with terror, but tried to\nremember just what Bill had said, and just where this hidden treasure\nlay. _The cupboard to the right of the fire-place, under the floor._ It\nwas always locked, he knew that. Oh, if only he dared move!\n\nThe wind and rain seemed to make the most unearthly noises as poor\nlittle Jesse crept to the door and sped across the yard. There was no\nlight in the old kitchen but that of the fire. At first he was glad of\nthis; it seemed to protect him from Bill and his wicked companion. But\non second thoughts he remembered that if they saw the house in darkness,\nthey might conclude it safe to come in for their evil purpose; and so he\nlighted two candles, placing them on the table near the window. This\ndone, he went to the cupboard door, and looking at it in perplexity and\nterror, wondered what he ought to do next.\n\nHis one distinct idea was that he must in some way save Miss Holsover's\ntreasure, whatever it might be. He did not care how or with what he\nwrenched the cupboard door open; but when it fell back there was a new\ndifficulty, and every sound outside seemed to increase his nervousness\nand dread.\n\nNothing of importance was on the shelves; but Bill had said _under the\nfloor_. How could he move the flooring? Finally, taking the screw-driver\nhe had used in forcing open the door, he put the edge of the instrument\ninto the crack, and to his delight found that the old plank yielded.\n\nHe searched narrowly, and there, protected by a wooden box, lay a\nsmaller one of tin, and a package tied with faded ribbon.\n\nJesse did not stop to look at what he had found. Blowing out his candle,\nhe gathered them up as quickly as his trembling fingers would allow, and\nthen, with one hasty thought what to do, he rushed to the front door,\nand sped away in the darkness.\n\nFor the first few moments he had no definite idea of where he meant to\ngo. His main thought was to get away from Bill, and save Miss Holsover's\ntreasure.\n\nSuddenly he thought of Mr. North, and turning down the lane by the\ncross-roads, he walked along toward his house as swiftly as possible.\nThat there was money in the box he held, Jesse never doubted; but what\nthe package contained was more mysterious. There were papers, and he\ncould feel a frame and glass. But he did not even think what it might\nbe. He was too worn out by excitement to have any clear ideas. The road\nahead of him seemed endless, and so dark that it seemed as if he could\nnever find his way.\n\nBut happily his long walks on Miss Holsover's errands stood him in good\nstead. He knew all the country for miles around, and had grown fearless\nfrom habit.\n\nAs he went along he thought with a pang of his preparations to leave the\nfarm, and that perhaps no chance would come again when he would feel\ncourageous enough to go.\n\nThen another thought came. Suppose Miss Holsover refused to believe that\nBill had meant to rob her cupboard, and asserted that the real thief was\nJesse himself? This was more than likely, and the child shuddered,\nthinking of the result. Well, he knew he was doing his duty.\n\nPoor, tired little Jesse! He looked up at the dark sky, and knew that\nbeyond it there was One who always _knew_. Comfort came with the\nthought. He pushed on bravely, holding the box and package with a firmer\ngrasp.\n\n       *       *       *       *       *\n\nMr. North had had his supper, and was sitting at his comfortable\nfireside smoking his pipe, while Mrs. North, in the little inside\nkitchen, was making a cheerful clatter with the supper dishes.\n\nSuddenly their attention was arrested as by something falling against\nthe outer door.\n\n\"What's that noise?\" exclaimed the old lady.\n\n\"I don't know,\" Mr. North said, jumping up and putting down his pipe. He\ndisappeared from the kitchen, and Mrs. North, accustomed to her son's\nalways doing the right thing, quietly waited.\n\nHe was only gone two minutes, but he returned with a slower step. In his\narms he bore Jesse's apparently lifeless figure. His poor garments hung\nwet about him--his face against Mr. North's rough coat was like some\nrain-washed, beaten-down white flower; but his thin little hands still\nclasped the \"treasure.\"\n\n\"Why, Peter North!\" exclaimed the old lady.\n\nMr. North slowly shook his head.\n\n\"Ef that thar old fiend has killed him,\" he whispered, solemnly, \"I vow\nI'll hev her hung.\"\n\nVery tenderly he deposited the child on the big soft lounge which stood\nnot far from the fire-place, and it was not long before he and his\nmother had devised restoratives.\n\nMrs. North knew just what to do. She ran up stairs, and in a few moments\na bright wood fire was crackling in the spare room, while she and her\nlittle servant Jenny aired sheets, and made a comfortable bed. Jesse was\nsoon brought up stairs, and snugly tucked in.\n\nThen after administering a hot drink, and bidding him go to sleep, the\ngood lady went down to her son again.\n\nHe was standing before the fire, holding Miss Holsover's treasure in his\nhands. Jesse had contrived to make them understand what had happened at\nthe farm, but something else was absorbing the good-hearted man.\n\n\"Mother,\" he said, \"jest look here. These things are that child's. Look\nwhat's written on the package.\" And he handed it to Mrs. North. The old\nlady scanned it curiously. In a faded but delicate and refined hand was\nwritten the following:\n\n\"For my child Jesse. To be sent to my brother, Paul Martin, ---- Beacon\nStreet, Boston, July 6, 1865.\"\n\n\"I see through it all,\" said Mr. North. \"That old skinflint has kep' 'em\nfrom him. Seems to me we ought to look to it that he never goes back.\"\n\nMrs. North looked at her son with moistened eyes.\n\n\"It's the Lord as sent him,\" she said, softly, and Mr. North bowed his\nhead.\n\n       *       *       *       *       *\n\nThe sun was shining brightly St. Valentine's morning when Mr. North,\nleaving Jesse on the lounge near the fire, started for Boston.\n\nOnce in that city, he proceeded directly to the Beacon Street address.\nIt was a quiet old-fashioned house on a corner, and on Mr. North's\ninquiry of the butler for Mr. Martin, he was told that that gentleman\nhad been dead five years, but his widow lived there. So Mr. North was\nushered into a beautiful room, full of books and pictures and quiet\ncolors, to wait for the lady of the house.\n\nIn a few moments an elegantly dressed, gentle-looking lady came into the\nroom, and good Mr. North turned to her, stammering out his story.\n\n\"How remarkable!\" exclaimed Mrs. Martin. \"I am sure it is my husband's\nnephew Jesse. Oh, if he had but lived to see this day! His sister Helen\nran away from home to marry a young sailor, and she was never heard from\nagain. This Miss Holsover must have known her. But why should she not\nhave brought the child to us, when poor Helen so evidently desired it?\"\n\nMrs. Martin opened the package eagerly. Within were several letters from\nPaul Martin to his sister, and one long one from Helen to her brother,\nwhich had evidently been written during her last illness. In it she\nspoke of having found board with Miss Holsover, to whom she had\nintrusted her valuable jewels and all her money and her poor little\nchild, Miss Holsover promising to take all to Boston when she was dead.\n\n\"There is the secret,\" exclaimed Mrs. Martin, looking up from the letter\nwith streaming eyes. \"The wretched woman kept the jewels and money, and\nmade the child her drudge. Oh, my poor little Jesse, if I can only find\nyou!\"\n\nMr. North had a sumptuous dinner before he and Mrs. Martin started out\nfor B----. Then some time was devoted to making purchases for Jesse's\ncomfort and convenience. Mr. North looked on in mute admiration while\nMrs. Martin purchased a warm suit, a stylish little overcoat trimmed\nwith handsome fur, and a warm seal cap, not to speak of shoes and\nstockings and all sorts of underwear.\n\nThen they were on board the train, and whirling on toward B---- and\nJesse.\n\n       *       *       *       *       *\n\nIt had not been a long day to the poor boy. He lay on the sofa, glad of\nthe repose. As the twilight deepened, night and a snow-storm set in.\nJesse watched the window, and listened eagerly for every sound of\nwheels; but he fell asleep before Mrs. Martin and Mr. North arrived, and\nawoke to find himself in his aunt's arms.\n\nIt was a blissful evening. Jesse was too weak to move; but he lay still,\nholding his new aunt's hands, and half crying with joy. Mr. North had\ngone over to the farm, and on his return brought the word that Miss\nHolsover refused to make any explanation, and declared herself well rid\nof a troublesome charge. A little threatening, however, induced her to\ngive up the key of the tin box. In it were found the money and jewels\nwhich the miserly old woman had evidently been hoarding.\n\nWhat became of Bill and his companion they never knew, and Mrs. Martin\ndeclared herself so happy to have Jesse that they could afford to forget\nthe Holsovers forever.\n\n       *       *       *       *       *\n\nIt was about three years later, when Mr. and Mrs. North were spending\nthe 14th of February at Mrs. Martin's house, that Jesse, a very\ndifferent-looking person from the poor little waif of former times, put\nhis hand on his first friend's shoulder, and, smiling, said,\n\n\"Mr. North, do you remember the day you gave me that lift?\"\n\n\"Of course I do, Master Jesse,\" he answered, looking at the boy's\nbright, happy face as he stood near him. Then he glanced around the\nbeautiful room, to which the handsomely dressed figure seemed so well\nadapted.\n\n\"Well,\" said Jesse, \"you remember how you told me what valentines\nmeant?\"\n\n\"Now you speak of it, I believe I do.\"\n\n\"Don't you think I got mine?\" the boy continued, gently. \"And it was you\nwho brought it to me, after all.\"\n\nThey did not speak for a moment; then the good-hearted man said,\nquietly: \"No, my boy; it was you doing your duty. Instead of seeking to\nbe revenged on that poor woman, who's in her grave now, you did what was\nright, and God sent you your valentine.\"\n\nAnd Jesse, happy in his aunt's home, seeing his old friends often,\ncheering many lives, and being grateful for his blessings, brings new\nhappiness and comfort to others with every year. And the sunshine of the\npresent has put far into the past the night he left the lonely\nfarm-house unconsciously carrying his first valentine.\n\n\n\n\nTHE GAVOTTE.\n\nBY MRS. JOHN LILLIE.\n\n\nA dear old friend of mine once showed me a fragment of manuscript music\nby John Sebastian Bach.\n\nThis bit of music was part of a _gavotte_; and as in this paper I mean\nto tell you Bach's story, and his special association with that quaint\ndance-music, I think we had better first see what the _gavotte_ and the\n_chaconne_, the _passacaille_ and the _sarabande_, mean. All of this\nmusic was popular in Bach's day.\n\nIn a French gallery there is a picture of splendidly dressed ladies and\ngentlemen dancing the gavotte. They wear the costume of the latter part\nof the seventeenth century. They have smiling faces, nourish large fans,\nand wear high-heeled slippers, which they lift gracefully; for the\ngavotte was a very brilliant dance in its movement.\n\nThe name came from a people in Dauphiny, known as Gavots. They danced it\nmore wildly than the stately people of Louis's court; but the music of\nevery gavotte seems to me to be best suited to them. One can fancy them\non their village green clattering away to the quaint gay music, flinging\ntheir arms about, or beating time with their hands. But when the gavotte\nwas introduced into the upper classes, and with it various other dances\nof the people, it became more refined, dignified, even more serious.\n\nBach wrote many gavottes, some singly, some in what are called _suites_,\nor sets of short pieces of music. Just now all who can play well at all\nare interested in this dance-music of the eighteenth century. It holds\nits own perhaps as much from the fact that its form is very generally\nclassic as from its charming melodies.\n\nIt is always well, even for beginners, to understand the _principle_ on\nwhich any kind of music is written. You will find your practicing much\nmore interesting if you look deeper than the mere sounds. Suppose we\ntake some simple gavotte, and examine into the way it is written. Here,\nfor instance, is the first strain in one of Bach's most popular\ngavottes:\n\n[Illustration]\n\nNow let us see what the few rules created for its composition are. They\nare these:\n\nIt must be in _common time_, which really means _equal time_, two or\nfour beats to the bar, although the term is generally applied to that of\nfour crotchets to the bar, marked by the Italian C.\n\nThe movement is rather quick, and it is generally in two parts. These\nparts are, in accordance with a custom peculiar to old dances, repeated.\n\nOriginally the gavotte consisted of four bars in the first part and\neight bars in the second; but if the gavotte is only one of various\nparts of a suite, no fixed number of bars is given. Now, as a general\nrule, the gavotte begins on the third beat of the bar, so that you will\nsee, if you calculate, that each part must finish with a half-bar\ncontaining a minim, not two crotchets.\n\nI know that to many of my young readers this may sound very dull and\nuseless; but if you will only give a little careful study to a few rules\nwhich apply to your first \"pieces,\" lessons in real harmony and\nthorough-bass will seem much more interesting to you later on. The\nchaconne and the passacaille, the passaglia and the sarabande, are all\ndances of about the same period as the gavotte, and have certain\ngoverning principles. The chaconne is slow, and is usually written in\nthe major key. This is always a semitone greater than the _minor_.\n\nThe passacaille is written in the minor key. What is called the\n_theme_[1] in the chaconne is invariably in the bass; in the passacaille\nit may be in any part. The passacaille has a very curious kind of\ninterest, since in the last century composers made use of it to show\ntheir skill--what is known as contrapuntal skill. It must consist of a\nshort theme of two, four, or eight bars. Bach, Frescobaldi, and Handel\nall wrote famous passacailles.\n\n[1] The _theme_, or, as it is sometimes called, _tema_, is the _subject_\nof the work; that is, the principal idea of melody--the \"plot,\" as it\nwere, of the whole thing.\n\nThe sarabande is more stately in its movement. It was a popular dance in\nthe sixteenth century, and some say it was introduced then by a famous\ndancer called Varatanda. I think that it might often have formed part of\nvery picturesque scenes in the sixteenth and seventeenth centuries, when\npeople were full of a certain kind of poetry, and enjoyed whatever was\nsplendid and stately. Sometimes dancers were hired to perform it;\nsometimes ladies and gentlemen of quality danced it for their sovereign.\n\nOld songs are full of references to sarabandes as being danced at times\nwhen sadness or even deep regret filled the minds of the performers; so\nthat we may picture it as a slow, pathetic movement, with melancholy and\nsweetness in its train.\n\nAnd from these dances of olden time we come to a great name--to the\nstory of a great man who wrote, amidst hundreds of finer things, the\nmost exquisite gavottes, chaconnes, and passacailles that we know. I\nmean John Sebastian Bach.\n\nIn a certain part of Germany lived the Bach family, famous for\ngenerations for their musical ability. Finally we hear of one of them\nliving in the quaint town of Eisenach, and having a little son named\nJohn Sebastian, who from his tiniest years glowed with musical fire. All\nthe genius of that family seemed to have taken root in him, yet when his\nfather died, his elder brother, with whom the boy lived, seemed to think\nit better for him not to study.\n\nWell, little John Sebastian craved music. I fancy the house in Eisenach\nin the year 1695 was very dull. German children then as now were kept\nvery strictly, and when the elder brother forbade music to the little\nboy, he did not know the pain he was inflicting.\n\n[Illustration: YOUNG BACH COPYING MUSIC BY MOONLIGHT.]\n\nIn the old house John Sebastian well knew that there was a rare old book\nof manuscript music, and this he longed to copy. It was hidden, locked\naway in a cupboard; but the door had a lattice-work panel in it, and so\nthe child watched his opportunity, climbed up, and pulled the book\nthrough the lattice. But even then it was hard to know how to copy the\nmusic, since candles or lights were refused him. So he waited for\nmoonlight nights, and on every one worked hard in his window, finally\nsucceeding in copying the entire book. I have often thought of the\npicture of the dear little German boy working away in his old-fashioned\nroom, the moonlight tenderly bathing his head and eager fingers, and the\nmanuscript page on which he worked. How little he knew that two hundred\nyears later all music-loving nations would reverence his name.\n\nHow he worked on with many trials, the usual ups and downs of an earnest\nlife, I can only tell you briefly, but employment and leisure in which\nto compose came to him quite young in life, and he had the happiest kind\nof a domestic circle.\n\nBach was twice married, and his children inherited enough of his musical\nability to make it a pleasure for him to instruct them. The young Prince\nof Anhalt-Cothen loved him so dearly that he could scarcely bear to be\nseparated from him, and later he was given an important position at\nLeipsic.\n\nOne grief clouded his later years. From overwork in copying and writing\nmusic he became quite blind--a calamity which all his dear friends felt\ndeeply. In 1750 he died suddenly.\n\nNow in Bach's lifetime great progress had been made in piano-forte\nmusic. Among other things, he invented a new style of fingering. In\nBach's day the thumb and little finger were rarely used. This he changed\nentirely, and our present ideas are due to his influence.\n\nWhen I spoke of him as the composer of famous gavottes it was only\nbecause that form of composition is one which is peculiarly adapted to\nyoung students, which is now newly popular, and which he wrote with\nwonderful grace. But Bach, as you may hear later, filled the world with\nmany grander sounds. They come in solemn majesty in his famous _Passion\nMusic_. They come, filling the air with sweetness, in his many preludes\nand symphonies. They are in concertos and sonatas. Bach worked with his\nheart full of exaltation. About him was the court of Frederick the\nGreat, of George II. in England. Piano-making was beginning to be a\nstudied question. People saw the need of good music at home as well as\nin public. Bach passed through those years when piano-forte music was,\nyou might say, on trial, well knowing what it might one day be.\n\nPeace and gentleness were always about him. He was a kindly, keen, busy\nman, full of generosity and goodness, and he lived and died believing in\nthe future of his art.\n\n\n\n\nTHE TALKING LEAVES.[2]\n\n[2] Begun in No. 101, HARPER'S YOUNG PEOPLE.\n\nAn Indian Story.\n\nBY WILLIAM O. STODDARD.\n\nCHAPTER XX.\n\n\nAs Steve walked away with Red Wolf, Many Bears at once turned his\nattention to Murray, and the great affairs to be decided by the chiefs\nand councillors.\n\nThe chiefs and warriors whose fame and rank entitled them to such a\nprivilege soon gathered for the expected \"talk.\"\n\nMurray found himself regarded as an honored guest. Not only were his\nhosts indebted to him for past favors, but they were anxiously expecting\nmore.\n\nAt first he merely listened as brave after brave replied to the mention\nof his name. He saw that only the very gray-headed men had anything to\nsay in favor of peaceful action, and a prompt \"getting away.\"\n\nHe was even surprised at the ardor with which many of the warriors\ndeclared their eagerness for a blow at the Lipans, and the good reasons\nthey were able to give.\n\nThe presence of the band of Two Knives was a sort of invasion of the\nApache hunting grounds.\n\nThe Lipans had no business this side of the mountains. They had come to\nstrike the Apaches, and if they should be allowed to get away unhurt,\nthey would surely come again.\n\nSend Warning had already told how many there were of them. If there were\nno more than that, none of them ought to be allowed to get away.\n\nMurray could but think that a party of Apaches in the Lipan country\nwould probably be talked about and dealt with very much in the same way,\nbut it seemed to require a special effort for him to think at all.\n\nHis head had been in a sort of whirl for some minutes before the time\nwhen Many Bears turned suddenly upon him with the question:\n\n\"What Send Warning say? His head is very white.\"\n\nMurray was muttering to himself at the moment, while Dolores handed her\nhusband a stick with a piece of corn bread on the point of it. \"She is\nnot an Apache. She is a full-blooded Mexican. Yes, I've seen that woman\nbefore.\"\n\nBut the chief's inquiry startled him out of that train of recollection.\nHe could not have answered instantly to save his life, but it was\naccording to Indian notions that he should not speak too quickly, so he\nhad time to recover himself.\n\n\"More enemies besides Lipans,\" he said at length. \"Apaches better not\nforget pale-face miners.\"\n\n\"Ugh!\"\n\nThe exclamation went all around the circle, for that was the very thing\nnone of them had thought of.\n\n\"Pale-faces fight Lipans,\" remarked Many Bears.\n\n\"Is the great chief sure of that?\" asked Murray. \"Suppose they come all\ntogether. Apaches need more braves then. Suppose they fight each other\nfirst, then Apaches eat up all that are left. Great chief better find\nout.\"\n\n\"Ugh!\"\n\nIt was a very loud grunt indeed to come from the throat of Many Bears,\nand the chiefs and braves looked at one another in a way that spoke a\ngood deal for the value they set on the advice of their white friend.\n\nWhipping sixty Lipans was one thing, attacking them with a strong force\nof pale-face riflemen to help them was quite another.\n\n\"What Send Warning say do?\"\n\n\"Do?\" almost sharply exclaimed Murray, with his eyes upon the retreating\nform of Mother Dolores. \"I'll tell you. Send your whole camp across the\nriver. They can surround it here. Then send out your best braves to\nwatch for the Lipans. They'll attack you before morning. That's what\nthey came for. They won't fight the miners.\"\n\nHe was partly right and partly wrong, but Many Bears and his chiefs rose\nto their feet as one man.\n\n\"The words of Send Warning are wise. He is very old, and he is a chief.\nNo use talk any more. All braves go and eat a heap. Tell squaws bring up\nall ponies. Get ready to cross river. No lose time.\"\n\nMurray was not a \"general,\" and he had never studied war, but he knew it\nwould be a good thing to have deep water between that camp and any\nassailants, instead of behind it. Many Bears was a chief of great\nexperience, but it had never occurred to him that it would cost him all\nhis horses if he should be beaten in a fight with a river behind him.\nThe blunder was to be remedied now with a rapidity which astonished even\nMurray, for he had not known how good a ford there was right there.\n\n\"Hope the Lipans won't find that out,\" he said to himself. \"They'll\nthink twice before they try to swim their horses. I've given these\nfellows good advice. May prevent a battle. But if one should come, how\ncould I fight the Lipans? What am I doing in an Apache camp anyhow?\nSteve and I must make haste out of this.\" And then a puzzled, pained,\nanxious look came over his wrinkled face, and he seemed to be looking\naround him very wistfully indeed, as if he wanted to see somebody. \"Not\nto-night, perhaps; but I'll see her again in the morning. Steve and I\nmust get away to-morrow. It'll be easy enough to give him his\ndirections, and I can find Two Knives and his braves in a few hours.\"\n\nAs the deepening gloom of the evening settled slowly down, he stood\nbeside Many Bears on the bank of the river, and watched the young braves\ndrive in the last squads of ponies from their pasturage, and urge them\nacross the ford. He had no idea how much quiet fun Steve and his friend\nRed Wolf had already enjoyed. The squaws had insisted upon making all\nthe boys and girls who were big enough swim instead of going over on\npony-back, and the youngsters, in their turn, had revenged themselves by\nall the mischievous pranks they knew.\n\nIf talk could have raised the river, the chatter of nearly two hundred\nsquaws of all ages, added to the scolding of Too Many Toes, would have\nmade a torrent of it. And yet a number of the squaws, wives and\ndaughters of men of character and station, attended to the business of\nfording the stream with the silence and gravity of the most dignified\nwhite matrons. Dolores would have scorned putting herself on a level\nwith such a squaw as Too Many Toes even in the use of her tongue, and as\nfor Ni-ha-be and Rita, they never forgot to whose family they belonged.\n\n\"Rita,\" said Ni-ha-be, as they rode down to the river, \"your blanket is\nloose. Red Wolf and Knotted Cord are watching us.\"\n\n\"Send Warning is not there.\"\n\n\"No, of course not. He is with the chiefs. Don't let them see we are\nlooking at them.\"\n\n[Illustration: \"IN AN INSTANT SHE WAS FLOUNDERING IN THE RIVER.\"]\n\nNi-ha-be had better have been attending to the feet of her own pretty\nmustang. The ford was not very wide just there, and the two girls were\ncompelled to get a little out of the way of two mules loaded with lodge\npoles. Alas for the vanity of the chief's self-confident daughter! Her\nhorse's fore-feet went over the ledge, and in an instant more she was\nfloundering in the river, while every squaw and young Indian who could\nsee her broke out into merry laughter. It was well, perhaps, that she\nslipped from the ford on the up-stream side; but she did not need a bit\nof help from anybody. No Apache girl of her age ever needed to be taught\nto swim. In a moment she had caught her mustang by the head, turned it\nto the ledge, and found her own footing on the rock, from which position\nshe encouraged the unlucky quadruped to follow.\n\nThus, although the water was at her shoulders, she managed, all dripping\nas she was, to clamber into the saddle again. It was so dreadfully\nprovoking, though, and she had certainly heard Red Wolf laugh.\n\nIt had been the chief's order that the lodges should be set up on the\nsafe side of the ford, and so there was work enough before the squaws.\nEven some of the younger braves were called upon to lend a hand, and in\nless than an hour's time there was a very respectable Indian village.\nLodges, ponies, fires, dogs, everything belonging to an Apache hunting\ncamp was there, and between them and any probable danger the river was\nrolling now, and the Lipans did not know where to look for the ford.\n\n\"Ni-ha-be,\" exclaimed Dolores, sharply, a little later, \"go into lodge.\nToo late for young squaw. What will the great chief say?\"\n\n\"It is early yet.\"\n\n\"Go in. Lipans come and carry you off. Old pale-face see you, and say\nfoolish young squaw. Not know enough to keep dry. Fall off pony. Ugh!\"\n\nThat was a sharp hit, and Ni-ha-be obeyed Dolores rather than stay for\nanother reminder of her ducking, but Rita followed her very slowly. \"If\nI could see him again,\" she murmured, \"I feel sure he would speak to me.\nI don't care what they say. Dolores may scold as much as she pleases. I\nwill ask Send Warning about those words, and all about those pictures.\"\n\n[TO BE CONTINUED.]\n\n\n\n\nPETIT JEAN.\n\nAT THE BATTLE OF THE PYRAMIDS, JULY 21, 1798.\n\nBY MARY A. BARR.\n\n\n  Up rose the sun o'er Egypt's tents,\n    O'er Egypt's pyramids and sands,\n  O'er fierce and fiery Mamelukes,\n    And o'er Napoleon's veteran bands;\n  The palms stood still in the hot air,\n    The sad and silent Sphinx looked on,\n  While over all the Afric sun\n   In burning, blinding splendor shone.\n\n  The Mamelukes fretted on their steeds,\n    Their cimeters all bright and bare;\n  The French stood grimly watching them,\n    Napoleon in the centre square.\n  He pointed to the Pyramids:\n    \"Comrades, from those grand heights, I say,\n  The brave of forty centuries\n    Will watch you draw your swords to-day!\"\n\n  They answered him with ringing shouts,\n    And ere the echoes died away\n  The van, like a tornado, charged,\n    Led by the brave and bold Desaix.\n  Then while the trusty \"Forty-third\"\n    Stood waiting for the word to charge,\n  They saw their little drummer-boy\n    Come from the column of Dufarge.\n\n  With tottering steps and bleeding breast,\n    But bravely beating still his drum,\n  He said, with sad and tearful face,\n    \"Oh, Forty-third, to you I've come;\n  I've come to you, my Regiment,\n    For nothing but a child am I;\n  I've come to you, my comrades brave,\n    That you may teach me how to die!\n\n  \"I'll never shame you, Forty-third;\n    I want to be as brave and true;\n  I want to die as brave men die;\n    So tell a poor child what to do.\"\n  Then Regnier gnawed his long gray beard,\n    And Joubert turned his head away:\n  The lad had been the pet of all,\n    And now they knew not what to say,\n\n  Till Regnier kissed the boy, and spoke:\n    \"Our Petit Jean, I see 'tis plain\n  Your place is with the Forty-third;\n    So beat us now the 'charge' again,\n  Then follow, and we'll show you how\n    Death comes unto the soldier brave.\n  Comrades, salute the nine-year-old\n    Who'll bravely fill a soldier's grave!\"\n\n  The men's hearts glowed like living coals,\n    And Regnier cried, \"Why do we stay?\"\n  And to the roll of the little drum\n    They rode upon their vengeful way;\n  But each one as he passed the child\n    His sword with earnest purpose drew,\n  And cried in brave or tender tones,\n    \"Mon Petit Jean, adieu! adieu!\"\n\n  \"I come, my Regiment, I come!\"\n    But never Petit Jean again\n  His drum beat for the Forty-third:\n    They found him lying with the slain.\n  They put the medal on his breast,\n    Together clasped his childish hands,\n  And dug, with many a bitter tear,\n    A grave for him in Egypt's sands.\n\n  'Tis near a century ago,\n    But still his memory is green;\n  The Regiment has not a name\n    So dear as that of Petit Jean;\n  And many a weary soldier has\n    To brave and noble deeds been stirred\n  By the tale of the little nine-year-old\n    Who died among the Forty-third.\n\n\n\n\nTHE MILKMAIDS OF DORT.\n\n\nGirls often declare that boys have all the fun. Well, they certainly do\nseem to get the larger share of it in a good many ways. Then, when they\ngrow up, they are very apt, too, to carry off all the honors, the\nliterary fame, the military glory, the professional success, while the\ngirls are left at home to do worsted-work.\n\nNow and then, however, the girls come to the front in art, in\nliterature, in science, and even in war. You all know how Joan of Arc\nled the armies of France to victory, and how Moll Pitcher stood at the\nmouth of her cannon, pouring confusion into the British ranks.\n\nNot so great as these women of martial fame were the \"Milkmaids of\nDort,\" but still they have their place in history. If any of you ever go\nto Holland, the land of wooden dikes and windmills, it is quite possible\nthat you may find yourselves some day in the ancient town of Dort, or\nDordrecht. It is a grand old city. Here among these antiquated\nbuildings, with their queer gables and great iron cranes, many an\ninteresting historical event has taken place.\n\nIn the centre of the great market-place of Dort stands a fountain, and\nif you will look close you will see upon the tall pyramid a _relievo_\nrepresenting a cow, and underneath, in sitting posture, a milkmaid. They\nare there to commemorate the following historical fact:\n\nWhen the provinces of the United Netherlands were struggling for their\nliberty, two beautiful daughters of a rich farmer, on their way to the\ntown with milk, observed not far from their path several Spanish\nsoldiers concealed behind some hedges. The patriotic maidens pretended\nnot to have seen anything, pursued their journey, and as soon as they\narrived in the city, insisted upon an admission to the burgomaster, who\nhad not yet left his bed. They were admitted, and related what they had\ndiscovered. The news was spread about. Not a moment was lost. The\nCouncil was assembled; measures were immediately taken; the sluices were\nopened, and a number of the enemy lost their lives in the water. Thus\nthe inhabitants were saved from an awful doom.\n\nThe magistrates in a body honored the farmer with a visit, where they\nthanked his daughters for the act of patriotism which saved the town.\nThey afterward indemnified him fully for the loss he sustained from the\ninundation, and the most distinguished young citizens vied with each\nother who should be honored with the hands of the milkmaids. Then, as\nthe years went by, the fountain was erected, and the story commemorated\nin stone.\n\n\n\n\nMAX RANDER'S YOUNG NOBLEMEN.\n\nBY MATTHEW WHITE, JUN.\n\n\nIt was only a day or so after my bicycle ride that I received a letter\nfrom father, telling me that he and mother were detained in London, and\nasking if Thad and I could do without them for a week longer.\n\nSo in view of our \"to-be-continued\" lonely condition, the landlady\nkindly offered to introduce us to \"three young gentlemen\" who were\nattending school in the town, and who boarded with her sister.\n\nThey all came of noble families, she informed me, adding that it would\nbe a great honor for me to know them.\n\nThus my expectations were raised to the highest pitch, and the morning\nof the day they were to call I spent in polishing my watch chain to its\nbrightest, and deciding whether a blue or a red bow would be most\nappropriate to wear.\n\nI know now that I must have seen scores of little dukes and lords\nwalking about the streets of London just like other boys, but I didn't\nknow it then, and always had an idea that they all wore red velvet\ncloaks and cream-colored tights, carried swords dangling at their sides,\nand never went out except in a gilded chariot, preceded by two men on\nhorseback blowing brass horns to clear the way.\n\nTherefore I was considerably astonished when I saw three boys of about\nmy own age turn in at the gate, all dressed in short black coats and\ntall black hats, exactly like any common English school-boy.\n\n\"Now, Thad,\" I said, as I turned away from the window, \"be sure and make\na low bow when you are introduced, because--\" But the Landlady's knock\nwas on the door, and in she came, followed in close order by Malcolm\nHeppingham, the son of a Duke, Douglas Galton, whose father was an Earl,\nand Ralph Maisley, who was an \"Honorable.\"\n\nI was presented to these high and mighty individuals with great\nceremony, and then the landlady went out and left us to ourselves.\n\nFor a moment there was dead silence, while Thad stood staring at our\nvisitors as if afraid to sit down in their presence. The three English\nchaps stared at us too, and then all at once began to ask questions\nabout America.\n\nAnd such questions! Why, they made me think of it as some island away\noff in the Arctic Ocean, which people never heard from except every two\nor three years, when a ship managed to break its way through the ice,\nand carried back the news that a new barn had been built or half the\ninhabitants killed off by the natives.\n\nNevertheless, I was deeply impressed by the grand airs the young\nnoblemen gave themselves, and the queer way they had of pronouncing\ntheir words.\n\nTheir call was a very formal one, for they only staid about ten minutes,\nand then filed out in solemn procession, after making me promise to come\nand see them the very next afternoon. Nothing was said about Thad, so I\nconcluded they thought him too young for company.\n\nWell, of course I felt very much honored by the pressing invitation to\nvisit them so soon, and promptly at the hour named presented myself at\ntheir boarding-house, where I was immediately pounced upon by all three\nof my titled friends, who in a very lively manner, and all talking at\nonce, informed me that I had arrived just in the nick of time to see the\n\"jolliest sight\"--no less than the ascent of the dining-room chimney by\na boy sweep.\n\nThen they took me into the long dining-hall, with its great fire-place,\nwhich, though big enough to hold two boys, was anything but a pleasant\nspot to poke one's head into.\n\nThe sweep quickly and quietly made his preparations, and with our four\npair of eyes riveted upon his sooty form, speedily clambered out of\nsight.\n\n\"I say,\" cried Duke Malcolm, when there was nothing more to be seen,\n\"while he's up there, come on out and take a look at my dog-kennel.\"\n\n\"And my rabbit-burrows,\" put in the Earl.\n\n\"And you must see my pony too, you know,\" added the Honorable Ralph.\n\nThis last inducement was not to be resisted, so I hurried out into the\nback yard after my hosts, wondering why the chaps with the biggest\ntitles should have only dogs and rabbits to show, while the fellow who\nwas merely an Honorable rejoiced in the possession of a pony.\n\nThe kennel and burrows were duly inspected and admired, and we were\ncrossing the yard to visit the stables, when the Earl suddenly stopped,\nstruck one hand against his forehead, and with the other pointed to the\nchimney over the dining-room, _out of which smoke was ascending_.\n\n\"The sweep!\" cried the Duke, staggering back against the barn door,\nwhich banged to with a crash.\n\n\"He'll be roasted alive! Quick!\" exclaimed the Honorable, and then they\nall three started on a run for the house.\n\nI followed as fast as I could, but not knowing the ins and outs of the\nyard as well, I completely lost sight of them before I reached the back\ndoor. However, I remembered where the dining-room was, and dashing in,\nfound nobody there but the maid, who was blowing the already fiercely\ncrackling fire into a brighter blaze.\n\nNot stopping to wonder why the others were not there, I rushed up to\nher, and snatched the bellows out of her hands so suddenly that she\nimmediately set up a cry of \"Murder!\" while I began to shout \"Fire!\" at\nthe top of my voice, expecting every instant to see the charred body of\nthe sweep come tumbling out of the chimney at my feet.\n\nBefore I could explain to the girl the awfulness of the deed she had\ndone, the landlady, the butler, the rest of the servants, and all the\nlady boarders had crowded into the room, and as the maid was too\nfrightened to say anything except that one word, they all began to stare\nat me as if I were a burglar.\n\nBut all my thoughts were with that poor sweep roasting alive in the\nchimney, and rushing up to the landlady, I entreated her to have the\nfire put out, and-- But at that moment I saw looking in at the window\nthe faces of my three noble friends, all distorted with suppressed\nlaughter, while directly behind them stood the sweep himself, grinning\nfrom ear to ear.\n\n\"I say, Rander,\" called out the Duke, \"why don't you cable to New York\nfor one of your American fire-engines?\"\n\n\"Perhaps we can induce the Queen to present you with a leather medal in\nhonor of your mighty efforts at life-saving,\" added the Earl.\n\nThen all three vaulted into the room to explain how they had caught a\nglimpse of the sweep walking off just as they reached the house, and\nforgetting me entirely, had run after him to see if he was scorched.\n\nHowever, I consoled myself with the reflection that not one of those\nchaps with handles to their names can ever be King of England, while I\nhave the chance of becoming President of the United States.\n\n\n\n\n[Illustration: THE GAME OF TCHUNGKEE.]\n\nTHE GAME OF TCHUNGKEE.\n\n\nThis is the favorite game of the Indian tribes of the Upper Missouri,\nwith the exception of the Sioux. It is played by sides, each choosing a\nchampion, and they in turn selecting their players alternately,\naccording to their merits. The ground on which they play in summer is\nnear the village; and is a clay-covered space which from constant use\nhas become hard like pavement.\n\nThe scene of the accompanying sketch is laid in winter, when the game is\nplayed on the ice or frozen snow. It is begun by two of the players, one\nfrom each side, running abreast of one another, and rolling in advance a\nring, which is cut out of stone, and is from two to four inches in\ndiameter. The players follow it up with their tchungkees, spears or\nsticks about six feet long, with little projections of leather an inch\nand a half in length placed at intervals on the sides. These implements\nthey throw before them as they run, sliding them along the ground beside\nthe ring, and endeavoring to place the stick in such a position that the\nring will fall upon it when it stops rolling, receiving one of the\nleather projections through it. They count for game one, two, or three,\netc., from the point, according to the number on which it has fallen.\nThe winner rolls in the next run, and the loser is counted out, while\nanother from his side takes his place.\n\nThus the game proceeds until the sides have had their innings, the\nlargest count being the game. They sometimes become so excited that they\nbet away everything they possess, and have even been known to sell\nthemselves into slavery for a given period.\n\nThe game of tchungkee originated with the Mandans, a race now almost\nextinct. Prince Maximilian, in his travels among the North American\nIndians, called them the \"gentlemen of the plains,\" from their courteous\nmanners and gallantry to strangers.\n\n\n\n\n[Illustration: MR. BARNUM'S GREAT SHOW IN WINTER-QUARTERS.]\n\nMR. BARNUM'S GREAT SHOW IN WINTER-QUARTERS.\n\n\nTwo-thirds of all the boys who read this article have without doubt been\nto the circus. But who has seen a show in winter-quarters? Not more than\nhalf a dozen of you, I fancy. And if you were to apply to the great gate\nof the mysterious inclosure at Bridgeport, you would not be let in, for\nthere are very strict regulations, and the public are \"not admitted.\"\nSomehow or other our artist found an \"open, sesame,\" and he has given us\na page of sketches showing some of the characteristic features of a\ngreat show _not_ on exhibition.\n\nTo any one entering at the gate of the grounds two secretaries will\nusually present themselves. One of these is Mr. Barnum's private\nsecretary, the other a pet bird of like appellation, to which is\npermitted a dignified freedom. There are also several pelicans strolling\nabout, and a coach-dog, a great favorite with the elephants. On the\nright is the building in which the painting, carpentering,\nharness-making, and general tinkering for the show are going on. Here\nare a score of chariots in different stages of construction,\norchest-melochors with the exasperating tune part taken out, broken\ncages and wagons. There are forges in different parts of the building.\nOn the left is the car shed, a building over three hundred and fifty\nfeet long, and four or five tracks wide, in which the cars for\ntransporting the show are kept. It is full to overflowing, a number of\nflat cars being outside.\n\nUnder the wide eaves of the car shed are ranged the gorgeous and\nluxurious show wagons in which the animals are exhibited. The three\nlarge buildings form three sides of a quadrangle. Behind them is the\nten-acre lot where the Bridgeport show is held at the beginning of every\nseason.\n\nHow many elephants do you suppose Mr. Barnum has now? Can you fancy\nit?--there are twenty-two, big and little, young and old. Just think of\nthe noise they can make! At a signal from the keeper they will begin to\ntrumpet at once, and then the noise is like several thunder-storms with\nthe rain and lightning left out.\n\nAs a matter of course you have all heard of the baby elephant recently\nborn in the show. It came on the 2d of February. The mother is the\nelephant Queen. The father is named Chief. He is the largest in Mr.\nBarnum's herd of twenty-two. Those who have seen the elephant pyramid\nact will recollect him posed with uplifted trunk at the top of the heap.\nHe is of wayward disposition, and has of late exhibited some savage\ntraits of character. When the little one was born she weighed one\nhundred and forty-five pounds; her trunk was seven inches long, and she\nwas about as big as a full-grown Newfoundland dog. Mr. Barnum has named\nthe new-comer Bridgeport. He is going to exhibit her with Chief and\nQueen, as \"The Elephant Family,\" and of course we are all going to see\nthem.\n\nOne of the most interesting things about these elephants is the\nintelligence they show, and the attachment they form for their keepers,\nand even for the other animals about them. An amusing incident was daily\nwitnessed last season in the greenroom of the show. Donald Melville, a\nlittle child two and a half years old, son of a trainer, formed a\nstrange attachment for Gypsy, one of the largest elephants in the herd.\nGypsy was equally fond of the child, and would follow it anywhere.\n\nDonald could scarcely talk, but he would pull and tug at Gypsy's trunk\nuntil the intelligent animal would comprehend what was wanted, and\ncarefully lay its huge carcass upon the ground, when the child would\nclimb upon its back. To play hide-and-seek between Gypsy's legs was a\nfavorite pastime with Donald.\n\nThere is a mammoth sloth bear in the menagerie which attracts a great\ndeal of attention from such visitors as are admitted. He weighs upward\nof four hundred pounds, and when standing erect is nearly six feet tall.\nAt the word of command the bear stands upright upon his hind-feet,\ncloses both eyes, opens his mouth, and makes a guttural sound that the\nmen call preaching. His calm, dignified attitude is ridiculous in the\nextreme, and has earned for him his clerical nick-name of \"the\nPreacher.\"\n\nAmong the leopards there is a full-grown animal called Pet. It is so\ntame that the keeper will enter its cage, take it in his arms, and\nhandle it as familiarly as if it were a house cat. Yet this is dangerous\nbusiness. The men whose profession is the training of wild animals had\nsad stories to tell of their own and others' experience. One of Mr.\nBarnum's tamers carries a great scar on his forehead. He was in the cage\nwith some lions one day when one of them took advantage of his looking\naway, and sprang at him. Its paw struck him in the face, and a claw made\nthe dent in his skull. He did not lose his presence of mind, but seeing\nthat the lion was temporarily frightened at having hit him, he got out\nof the cage as fast as possible. The claw just failed to reach the\nbrain. Another time he tried to tame some hyenas, and one of his fingers\nwas bitten short off. \"The hyenas,\" he remarked to our artist, \"are\namong the most cowardly and treacherous of wild beasts.\"\n\nA rhinoceros is not supposed to be a playful animal, but there is one in\nthe show that has a decided taste for playing with a ball. He will roll\nit up and down the wall of the cage with his absurdly shaped nose, and\napparently finds great enjoyment in the sport. He expresses his delight\nin what may be called deep grunts of satisfaction.\n\nThe circus business demands a small army of performers and attendants.\nMr. Barnum in the coming season will have over six hundred persons in\nhis pay. The \"master of the sails\" will have a force of one hundred and\ntwenty-five tent-setters; the head groom, sixty grooms: the\nloading-master, one hundred \"razor-backs\"; the menagerie, sixty\nattendants; and the advertising department, seventy men always ahead of\nthe show. There will be performers of all sorts, caterers, side-show\nmen, etc., at least two hundred more in number. The master of the sails\nsaid that in eighteen minutes his force of one hundred and twenty-five\nmen had taken down and packed in the wagons fifteen tents containing two\nhundred and sixty thousand yards of canvas, to say nothing of the ropes,\npoles, and other appointments.\n\nA novel feature of the show the next time it starts out in a procession\nis to be \"Mother Hubbard's Shoe.\" A shoe of gigantic size, mounted on\nwheels, and filled by the convenient children of the company, will take\nits place among the chariots and cars. This new attraction will\nunquestionably be a source of delight to all small people.\n\n\n\n\nTONY'S BIRTHDAY, AND GEORGE WASHINGTON'S.\n\nBY AGNES REPPLIER.\n\n\nIt was the great misfortune of Tony Butler's life to have been born on\nthe Twenty-second of February.\n\nThere was no comfort in reflecting that there were doubtless plenty of\nother boys in the country who labored under the same disadvantage. The\nother boys might perhaps be better fitted for the honor, but for poor\nTony the distinction was a crushing one.\n\nIn the first place, he had an older brother, and that older brother's\nname was George. Now it is generally conceded that one of a name is\nenough for any family; but when Tony was born on the Twenty-second of\nFebruary, how was poor Mrs. Butler to act?\n\nNot to have called him after the Father of his Country would have been,\nin that good woman's opinion, a positive slight to the illustrious dead.\nAs long as her boy was fortunate enough to have the same birthday as our\ngreat President, it became her plain duty to give him one other point\nof resemblance, and then trust to time to complete the likeness.\n\nIt was a pity that they had a George already, but that difficulty could\nbe done away with by calling her second son Washington. Washington\nButler sounded well, and seemed all that was desirable; only there was\njust a little too much of it for every-day use. Sometimes the boy was\ncalled Washie, and sometimes Wash, and sometimes Wall, and sometimes\nTony, until, as he grew older, and able to talk, he evinced a decided\npreference for the last title, and would answer to no other.\n\nBut although this lessened his troubles, it by no means ended them; for\nwhen a child has so many nicknames to choose from, everybody is apt to\nselect a different one; and to confess the truth, he was not at all the\nright sort of a boy to be called George Washington.\n\nThere was nothing of the soldier, nothing of the patriot, nothing at all\nremarkable, about poor Tony in any way. He was a shy, homely little boy,\nwho would have passed well enough as plain Sam, which, being his\nfather's name, would also have been his had it not been for his\nunfortunate birthday. But as a George Washington, even his doting mother\nwas forced to realize he was not a complete success.\n\nThe first day he went to school the master sonorously read out his name\nas Antony Butler, whereat his brother giggled, and Tony, blushing fiery\nred, stammered out that he was not an Antony at all.\n\n\"Not Antony?\" said the teacher, in natural surprise. \"Why, then, are you\ncalled Tony?\"\n\n\"Because my name is George Washington, and we had a George already,\" was\nthe embarrassed answer.\n\nAfter this the boys with one accord dubbed him Washing Tony, as if he\nwere a Chinese laundryman, and Washing Tony he continued to be called.\n\nUnder these circumstances, perhaps he was excusable in wishing he had\nbeen born on some less illustrious day, and when the Twenty-second came\nduly around, it required all the delights of a new pair of skates and a\nfur cap to reconcile him entirely to his fate.\n\nIt being a general holiday, all the boys proposed spending it on the\nice, and Tony could skate a great deal better than he could write or\ncipher; although even here he was never what boys consider brave, and\nwhat their parents are apt to more accurately define as fool-hardy.\n\nThe truth is, there was not in the child a spice of that boyish daring\nwhich seems so attractive in its possessor, and which is in reality so\nwanton and useless.\n\nTony never wanted to climb high trees, or jump from steep places, or pat\na restive horse, or throw an apple at a cross old farmer. All these\nthings, which were clear to the hearts of his companions, were totally\nunattractive to him. He could never be dared to any deed that had a\ntouch of danger in it, and the contrast between his prudent conduct and\nhis illustrious title was, in the eyes of all the other boys, the\ncrowning absurdity of the case.\n\nOn this particular birthday the weather, though clear, was mild for the\nseason, and some apprehension had been felt as to the complete soundness\nof the ice. A careful investigation, however, showed it to be all firm\nand solid except in one corner, where the lake was deepest, and where\nthe ice, though unbroken, looked thin and semi-transparent, with the\nrestless water underneath. Around this uncertain quarter a line was\ndrawn, and soon some thirty or forty boys were skimming rapidly over the\nfrozen surface.\n\nFred Hazlit and Eddy Barrows were the champion skaters of the district,\nand their evolutions were regarded with wonder and delight by a host of\nsmaller boys, who vainly tried to rival their achievements.\n\nNot so Tony. Although perfectly at home on the ice, he seemed to have no\nmore desire to excel here than elsewhere, but skated gravely up and\ndown, enjoying himself in his sober fashion, his cap drawn over his\neyes, his little red hands thrust in his overcoat pockets.\n\nGeorge, who did not think this at all amusing, was off with the older\nboys, trying to write his name on the ice, and going over and over it\nwith a patient persistency that, practiced at school, would have made\nhim the first writer in his class.\n\nGradually the forbidden ground began to be encroached on, some of the\nolder boys skimming lightly over it, and finding it quite hard enough to\nbear their weight. Soon the line was obliterated by a dozen pairs of\nskates, and the children, never heeding it, spread themselves over every\ninch of ice on the lake.\n\nAll but Tony. With characteristic prudence he had marked the dangerous\ncorner well, and never once ventured upon it. As he stopped to tighten\nhis skates, four of the younger boys, hand in hand, came bearing down\nupon him.\n\n\"Catch hold,\" shouted Willie Marston, \"and we'll make a line. Hurrah!\nHere goes!\" and Tony with the rest shot across the smooth sheet of ice\nuntil they came to the inclosed quarter. The others were keeping right\non, but Tony stopped short.\n\n\"It is not safe,\" he said, \"and I am not going on it.\"\n\n\"Nonsense!\" cried Dick Treves. \"What a coward you are, Tony! We have\nbeen over it a dozen time already this morning, and it is just as safe\nas the rest.\"\n\n\"Of course it is,\" said Willie. \"Come ahead.\"\n\nBut Tony did not go ahead. Neither did he discuss the matter, for\nargument of any kind was not at all in his way. He merely stopped and\nlet go of Willie's hand. \"It isn't safe,\" he persisted. \"You can do as\nyou like, but I am not going on it.\"\n\n\"Well, stay there,\" said Ned Marston, giving him a little shove--\"stay\nwhere you are, General Washington, and cross the Delaware on dry land if\nyou can.\"\n\n\"Three cheers for General Washington!\" shouted Dick, derisively. \"Hurrah\nfor the bravest of the brave!\" and then the three boys skated on,\nleaving Tony standing there upon the ice.\n\nHis face flushed crimson with shame, but he never stirred. He hated to\nbe laughed at and called a coward, but he was afraid to venture, and no\namount of ridicule could urge him on.\n\nSlowly he turned to go, when at that instant an ominous sound struck his\near. The treacherous ice was cracking in all directions, a dozen jagged\nseams spreading like magic over the smooth surface. There was a sharp\nsnap, a cry of terror, a splash, and _three_ boys, white with fright,\nstarted back from the yawning hole, barely in time to save themselves\nfrom falling.\n\nIn the excitement and fear of that moment no one of them thought of his\ncompanion; but Tony, who stood beside, had seen poor Willie's despairing\nblue eyes fixed on him with a mute appeal for help as he staggered and\nfell into the dark water.\n\nSomehow all his habitual caution, which was so falsely termed cowardice,\nhad disappeared; he never even thought of being afraid, with that\npitiful glance still before his eyes, but, urged on by some great\nimpulse, cleared the space between them in an instant, and plunged down\nafter his drowning friend.\n\nAnother minute and both boys re-appeared, Willie clutching fiercely at\nhis preserver, and Tony holding him off as well as he could with one arm\nwhile he struck out bravely with the other.\n\nIt was but the work of a moment before help reached them, but that\nmoment had saved poor Willie's life, and changed forever the opinions of\nthe school.\n\nThey had learned what true courage was. Tony Butler might be timid and\ninsignificant, but he had proved himself beyond a doubt worthy of his\nillustrious name, and a fit hero for the Twenty-second.\n\n\n\n\n[Illustration]\n\nTHE SAD FATE OF SEVENTEEN CUPIDS.\n\n\nPUZZLED EDITOR.\n\n  How did we come by this marvellous flower?\n    Say, can you tell us what is it?\n  Did it spring from the earth in some wonderful hour,\n    Dainty, and rare, and exquisite?\n\nBRIGHT YOUNG CONTRIBUTOR.\n\n  Why, seventeen Cupids went wild with despair\n    When Valentine's Day left each solus,\n  So they picked off the flowers and hung themselves there,\n    On the stalk of a tall gladiolus.\n\n\n\n\nFUN AND FACTS FOR LITTLE FOLKS.\n\n\n\"How does a hardware dealer differ from a bootmaker?\" asked a bright boy\nof one of his playmates. The latter, somewhat puzzled, gave it up.\n\"Why,\" said the other, \"because the one sold the nails, and the other\nnailed the soles.\"\n\n       *       *       *       *       *\n\nLittle Lucy fell and hurt her knee badly, which her mother, when she\nwent to bed in the dark, tried to bandage. Soon the little one was heard\ncalling. \"Mamma,\" she said, \"this bandage is not in the right place. I\nfell down higher up.\"\n\n       *       *       *       *       *\n\nA grandfather, coming to read his paper, found that he had mislaid his\nspectacles, and thereupon declared, \"I have lost my glasses somewhere,\nand can't read the paper.\" A little three-and-a-half-year-old girl,\ndesiring to assist him, answered; \"G'an'pa, you go outside and look froo\nze window, and I'll hold ze paper up so you can read it.\"\n\n       *       *       *       *       *\n\nA \"Menagerie Race\" was recently the source of great amusement to a party\nof army officers in India. Each competitor had an animal to enter, which\nhe drove as straight as he could. There was a frog, a goose, a young\npig, a cock, a cat, a dog, a turkey, a kid, a duck, a young monkey, and\na pelican. The latter got away from his string and flew up into a high\ntree just as the race was going to begin. The animals had ribbons round\ntheir necks. The goose won the race, as he was the only one who went\nstraight; the dog made for the pig, and a battle-royal ensued; the\nmonkey and the cat laid down and would not move a step. It was a very\namusing scene, so say the spectators, and the curious antics of the\nastonished animals caused a vast amount of laughter.\n\n       *       *       *       *       *\n\nWe had a parrot once which knew how to talk so well that it seemed as if\nhe must certainly be able to think and reason as well as any of us. Two\ninstances may be given to show what Polly could do in the way of\nconversation: One fine summer's morning, a young woman bringing a\nmessage to the house was asked into the kitchen, and while, as she\nsupposed, quite alone there, a rather gruff voice remarked that it was\n\"a very hot day,\" which it certainly was. As she did not know the parrot\nwas there, she was considerably startled, and would scarcely believe it\nwas the bird which had spoken to her. Another day Polly's cage was hung\nup on a tree near the poultry-yard, where a fight for supremacy was\ngoing on between two cocks, and the gardener, who was at work hard by,\ndistinctly heard Polly say, \"You idiots! Bran\" (calling to the big dog\nwhich lay asleep in his kennel)--\"Bran, bite them! bite them!\"\n\n       *       *       *       *       *\n\nWho could have believed that even among the famous riders of Hungaria\nwould be found one who could perform the following feat? While a noble\nstag of ten was being hotly chased by the K\u00e0posz\u00e1tsmegyerer hounds--a\nsubscription pack--one Karl P\u00f6r\u00f6s, a discharged hussar, managed to bring\nthe terrified animal to a stand-still in some close cover through which\nit was forcing its way, and, by an almost superhuman effort of strength\nand agility, to vault upon its back. After several desperate but\nunsuccessful attempts to dislodge its rider from his seat, the stag,\nstimulated anew to flight by the cry of the fast-approaching hounds,\nresumed its course, but it soon broke down under the weight of its\nunaccustomed burden, and died from sheer exhaustion and terror.\nP\u00f6r\u00f6s--at least so the story goes--was found by the huntsmen sitting on\nthe unwounded carcass of the stag, which he had literally ridden to\ndeath and resolutely claimed as the just reward of an achievement,\nunprecedented in the annals of the chase.\n\n\n\n\nA NOVEL CHAIR-SLEIGH.\n\n\n[Illustration: FIG. 1.]\n\n[Illustration: FIG. 2]\n\nHere is a design of a chair-sleigh, which can easily be constructed, and\nwhich will enable some of the ladies who do not skate to have a very\npleasant time on the ice. The runners (see Fig. 1) are thirty-nine\ninches long, and are shod with iron for their whole length. They are\nabout thirteen inches apart, and are braced in three places, namely, at\nthe back, at the front, and in the middle (directly under the front legs\nof the seat). The foot-boards are two inches wide by five-eighths of an\ninch thick. The seat is about fifteen inches square, and strengthened\nunderneath before and behind. The legs of the seat are fifteen inches\nhigh, and are fastened to the runners and the seat by hinges, which\nallow the seat to fall forward over the foot-board, as shown in Fig. 2.\nIn like manner the long arms, instead of being firmly fixed, work on a\nround iron rod, about a quarter of an inch in diameter, connecting the\nrunners in front. The arms are fifty inches long, and are mortised into\nthe back piece, which is twenty-five inches long by two inches and\nseven-eighths wide, thus leaving a few inches at each end for handles.\nThe arms are connected by a cross-piece in such a position that when the\nseat is perpendicular, the cross-piece shall rest upon little brackets\nplaced on the front legs. This cross-piece has a screw-hole in it, by\nwhich it is screwed to the front brace of the seat, thus firmly securing\nthe whole.\n\nTo fold the chair for carrying to and from the pond, draw the screw and\nraise the arms, and the seat will fall forward between the arms and\nunder the cross-piece, as shown in Fig. 2. The advantages of this\nchair-sleigh over an ordinary chair on runners are that it is perfectly\nsafe, does not interfere with the action of the skater who is pushing\nit, and can be easily carried in the hand or in a carriage.\n\n\n\n\nPINKETY WINK.\n\n\n  Miss Pinkety Wink she had eyes that would blink,\n  And small feet that turned out at the toes;\n  She had neat little hands, hair braided in bands,\n  And her cheeks they were red as a rose,\n  And she always was dressed in fine clothes.\n\n  Miss Pinkety Wink, I am sure you will think,\n  Was a very nice Dolly indeed;\n  She was pretty and still, and she never got ill,\n  For the doctor there never was need;\n  To her mistress she always gave heed.\n\n  But Pinkety Wink (I'm afraid you will sink\n  When I tell you how sad was her fate),\n  For an airing gone out, and while riding about\n  In her fine little carriage in state,\n  It turned over, oh, sad to relate!\n\n  Poor Pinkety Wink, with her eyes that would blink,\n  From her shoulders had dropped off her head!\n  Little Carrie, she cried, while vainly she tried\n  Her dear Dolly to mend; then she said,\n  \"_I'm afraid my sweet Pinkey is dead!_\"\n\n  Now what do you think? Well, Pinkety Wink\n  She carried right straight to mamma,\n  Who melted some glue, and made her quite new;\n  And now Carrie is laughing, \"Ha! ha!\n  Was there ever so nice a mamma?\"\n\n\n\n\n[Illustration: HOMELESS.]\n\n\n\n\n[Illustration: OUR POST-OFFICE BOX.]\n\n\nLast week we had Valentine's Day, and this week brings us a holiday of a\ndifferent kind. Americans celebrate Washington's Birthday because that\ngreat and good man was our leader in the war for our separate existence\nas a nation. Many of you study the history of your country at school,\nand so you have learned that George Washington was born February 22,\n1732. His father died when he was only eleven, but he had a good mother,\nand he early learned to honor her in all things. We have not room to\ntell you much about Washington's boyhood, but one thing we will say,\nbecause you little folks who are going to school may imitate George in\nthis particular: his copy-books were always as neat and clean as\npossible, and are models of beautiful writing. You see, he was\nconscientious in little things. Now if you want to refresh your memories\nabout the French and Indian War, the Revolution, and the early days of\nour Independence, you can not do better than to begin reading those\nportions of our history in the week that brings Washington's Birthday.\n\n       *       *       *       *       *\n\n  BROOKLYN, NEW YORK.\n\n     I have three pet cats, but Mr. Tibbs is the wisest. One morning it\n     was so cold there was ice on the window so that you could not see\n     out, and Tibbie is very fond of looking out of the window; so, as\n     he could not see, he took his tongue and licked a place big enough\n     for him to look out. It was so cunning. He is a very high jumper.\n     Every meal-time all the cats come around Mr. Tibbs, and then he\n     cries for all of them. He is very large, and weighs sixteen pounds.\n\n      I go to school, and learn music, and I am very fond of it. When I\n      grow older, I am going to take singing lessons and lessons on the\n      organ. I also have my dolls. I have a large French doll, which has\n      everything like a little girl. Then my baby is very sweet, with\n      big brown eyes and golden curls. My French doll has blue eyes, and\n      dark brown hair, like myself; I also have brown eyes. I have been\n      a constant reader of your paper ever since it was published. I\n      have a dear little turtle which is as large as a quarter of a\n      dollar, but he sleeps all the time now.\n\n  NETTIE M. T.\n\n       *       *       *       *       *\n\n  WASHINGTON, D. C.\n\n     I am a little girl just nine years old, and I am the only girl my\n     papa and mamma have got, and the only grand-girl my two grandpapas\n     and one grandmamma has, too, but I tell you he has lots of\n     grand-boys, though.\n\n      I have four little brothers, all younger than I am, and once we\n      had some cows, and Harvie--that's one of my brothers, who is\n      eight--came running into the house, and told our nurse, Buty, he\n      was \"afraid to go out into the yard where the married cow was\"--he\n      meant the one that had the calf. Everybody laughed so much when\n      mamma told them, and I have read some things not any funnier than\n      that, so I thought it would be a nice thing to put in my letter to\n      you. Harvie says I must put how old he was then, because you all\n      might think he did it when he was eight. He was nearly four when\n      he was frightened so badly. My aunts in Virginia take HARPER'S\n      YOUNG PEOPLE for us, and Harvie, Lewis, and I like it ever so\n      much. Lewis is just five, and wears pants, too. Maurice sits up\n      and looks as though he knew what mamma was reading; but he don't,\n      though. Isn't this a long one? I did not write it all the same\n      day. If you can, I wish you would please copy this in the paper\n      you send Harvie, Lewis, and me. I don't see many letters from\n      Washington. Harvie says he just bets you won't.\n\n  SERENA HELEN S.\n\nOh, Harvie boy, you shouldn't bet, especially when you bet so foolishly.\nRena's letter is \"copied\" right here by the types, and now we shall be\nexpecting, one of these days, another letter from Rena's brother.\n\n       *       *       *       *       *\n\n  KEOKUK, IOWA.\n\n     I am a little girl almost six years and a half old. I think YOUNG\n     PEOPLE is the nicest paper published. Papa and auntie have read\n     every number to me since the first. My dear mamma is in heaven, and\n     I live at grandpa's with my auntie. Papa comes to see me three\n     times a day, and every Friday he stays all night with me. YOUNG\n     PEOPLE comes that night, and papa reads it to me. I think Jimmy\n     Brown's stories are so funny. I liked \"Lady Rags\" so much! Then I\n     thought I would try and help the poor, too. The Christmas number\n     was very nice! After I read my paper I send it out to my grandma in\n     the country; they have no little children there, but they like it\n     ever so much. I have the nicest play-house; it is built out in the\n     yard, is furnished like a big house, and will hold six or eight\n     grown people. I have so many dolls that I can not tell you all\n     about them. Yesterday I felt so badly! I took my newest doll out to\n     walk; it was a gentleman, and I called it Frank, after my papa. I\n     let it fall, and broke its head all to pieces. But papa got me\n     another right away, just like the first. I have a kitty named after\n     Toby Tyler. I do not go to school, but I study at home. I can spell\n     very well, and print, but can not write yet, so my auntie is\n     writing this for me. We read all the letters in YOUNG PEOPLE, and\n     like them very much. I sometimes see the names of those who send\n     money for the Young People's Cot; if I knew how to send it, I would\n     send some also. I am getting a collection of picture cards. I have\n     over four hundred now. Will you please publish this, for I have\n     never seen a letter from Keokuk.\n\n  LULU W.\n\nWe are glad you have a kind auntie and a dear loving father and grandma,\nsince God has taken your mamma to live in the home above. About sending\nmoney to Young People's Cot, it is very simple. You need merely write a\nlittle letter, inclose your money, and address it to Miss E. Augusta\nFanshawe, 43 New Street, New York city. This lady is Treasurer of the\nCot Fund, and she will duly acknowledge all the money received by her,\nwhether the amount be large or small.\n\nHow nice it was that your papa was able to buy a new head for his\nnamesake so soon! Now if we should fall and break our heads, we could\nnot get new ones so easily.\n\n       *       *       *       *       *\n\n  CHARLOTTE, NORTH CAROLINA.\n\n     I have a black-and-tan dog; his name is Bismarck--for short, Bis. I\n     think he is beautiful--short, stubby, bandy-legged, and very fat,\n     and wobbles when he runs. Papa got him for a pure-blooded rat\n     terrier, but I do not think him very pure. Will B. and Willie S.\n     caught a mouse and put it in a dry-goods box, and put Bis in with\n     the mouse. The mouse ran all over Bis, and Bis would not touch it.\n     So one held Bis's mouth open, and the other caught the mouse with\n     the tongs and put it into Bis's mouth. Bis caught the mouse in his\n     teeth, but was careful not to hurt it. It is dinner-time now, and I\n     will close my letter.\n\n  NED L.\n\n       *       *       *       *       *\n\n  CUSTER CITY, PENNSYLVANIA.\n\n     I am a little boy seven years old. Papa reads the stories to us. We\n     want the man to write about Toby Tyler some more; and Jimmy Brown\n     is the \"boss\" boy.\n\n      There are lots of wells here, with deep holes and high derricks.\n      They get oil out of the holes, and send it to the 'finery, and\n      they make it into 'finery oil for the stores, and they sell it to\n      the people to burn in lamps. They get thick par'fine out of the\n      wells too, and the men come with pails and kegs, and sell it to\n      the factory, and they make candy and chewing-gum out of it for\n      stores to sell to boys and girls. I am tired. That's all.\n\n  JOE A. V.\n\n     P.S.--I can't spell good, 'cause I've been sick 'bout four weeks.\n\n  JOE.\n\n       *       *       *       *       *\n\n  NEW YORK CITY.\n\n     Papa has given my brother and me HARPER'S YOUNG PEOPLE, both years,\n     bound, but this year we are taking it by the week, because we\n     thought we would have more time to read it, and we could make out\n     the Wiggles. I have four brothers, all older than myself. We have a\n     very large black Newfoundland dog called Carlo, and a white setter\n     called Bob. We have a gray cat, and a cunning little black kitten,\n     which papa called Janauschek, after the actress. This is the first\n     letter I have ever written to any paper, and I hope you will print\n     it.\n\n  AGNES R. L.\n\n       *       *       *       *       *\n\n  BOONEVILLE, CALIFORNIA.\n\n     I am eleven years old, and live in the mountains in California,\n     among the big redwood trees. I have a doll that shuts her eyes and\n     opens them again. We have twenty-six geese and a lot of sheep--but\n     they are dying now--and a pet lamb. To-morrow father and brother\n     Simon go out in the mountains to get two wild hogs that Bob Bawles\n     killed for us to-day. I have five sisters and three brothers.\n\n  SARAH A. R.\n\n       *       *       *       *       *\n\n  SAN ANTONIO, TEXAS.\n\n     DEAR \"YOUNG PEOPLE,\"--Mamma gave us a year's subscription to your\n     dear self for a birthday present, and when that expired, renewed it\n     for a Christmas present. There are four of us boys, and our great\n     favorite of all the papers we read, or get mamma to read to us, is\n     YOUNG PEOPLE. We love Mr. Otis, and feel well acquainted with him\n     and the dear Postmistress, and our little friends' letters are such\n     pleasure to us! We copy them very often. We send you a small\n     bouquet of violets in a box, and hope you will get them; they will\n     be mailed with this letter. We have violets, pinks, roses, and\n     tuberoses almost all winter. We would have sent some long ago, but\n     mamma was too ill for us to think of anything pleasant. Our climate\n     is so very mild that flowers, even geraniums, begonias, ice-plant,\n     etc., grow out of doors all winter. Sometimes a frost comes and\n     kills a few, but not often. Dear YOUNG PEOPLE, we all love you so!\n     and when we go to New York we will call to see you. Grandpapa has\n     been a subscriber to HARPER'S MONTHLY ever since it was first\n     published. Mr. Nic Tengg, the bookseller here, ordered YOUNG PEOPLE\n     for mamma, as we get all our papers through him. We will soon send\n     something for Young People's Cot.\n\n      We can all swim, even four-year-old Edward, and ride our pony, and\n      three of us can drive mamma's buggy all over town. We take lessons\n      on the piano, and can sing several songs.\n\n      Your loving little admirers,\n\n  GEORGE C. F.\n  JOSEPH F. F.\n  STERLING I. F.\n  EDWARD F.\n\n\"What can be in this little box, so soft, yet so bulky, and oh! so\nsweet?\" we said, when, the other day, a box came to our office. Opening\nit, dear boys, there were your violets, wrapped so nicely in the wet\ncotton that they, were still blue and beautiful. It took us quite a\nlittle while to discover who were the kind donors, for your letter was\none of a great many which came to our Post-office Box that day. But when\nwe did find out, we stopped writing and reading long enough to waft a\nkiss all the way to San Antonio. How nice it is for you to be able to\nride, drive, and swim, as well as to sing and play! Each of these\naccomplishments will have its use, and give you pleasure all your lives.\nWe are glad your dear mother is better, and if you come to New York, we\nwill be much pleased to see you.\n\n       *       *       *       *       *\n\n  STAMFORD, CONNECTICUT.\n\n     We have two little guinea-pigs, an old cat named Tomas, and four\n     little bantams. I drive the old rooster all about the yard,\n     harnessed to a little wagon. We have a nice horse named Saxon,\n     which eats sugar. The little guinea-pigs drink coffee and eat brown\n     paper. I am a little boy nine years old.\n\n  WILLIE R. D.\n\nIt must be fun to drive a rooster in harness, and the little wagon must\nlook \"cute.\" Did you begin when he was a little chick? Are you not\nafraid the guinea-pigs will have \"nerves\" if you give them coffee?\n\n       *       *       *       *       *\n\n     I haven't any pets to tell about, but I have six dolls. I live in\n     Philadelphia all winter, but in the spring I go to Mayville, on\n     Chautauqua Lake, near the Sunday-school Assembly Grounds. I go\n     there a great many times, and I think it is very nice.\n\n      I have a horse of my own in Mayville. Her name is Daisy. I ride\n      horseback on her. Once she threw me, and nearly broke my arm. It\n      was bruised so I couldn't use it for a week.\n\n      I am beginning to get together a cabinet of curiosities. I will\n      exchange a 25 and 50 cent stamp from Germany, a 2 and 5 cent from\n      Finland, and 15 and 25 cent from France, and some silk cocoons and\n      spun silk, for ore; shells, minerals, or anything fit for a\n      cabinet.\n\n  LODIE TOURGEE,\n  303 S. 11th St., Philadelphia, Penn.\n\n       *       *       *       *       *\n\n     DEER MISTER HARPER,--my Pa tok yor paper for me. i be much plaes\n     wid it. i tink \"Torkin laves\" is very inrestin. i got a cro an I\n     named um Toosulom out ob de book. i cotched a Rabet an de way he\n     did run! you ought to se um blink whin I put um in de kage. Put dis\n     note in de paper kos I want to se how it looks.\n\n  JACOP WASHINGTON J.\n\nDid you mean to be as funny as you could, young gentleman? We fancy it\ntook not a little trouble to get up that bad spelling, and to put\ninstead of capital I's those modest little i's. After all, it does not\nlook so nicely as the letter you are going to write us next week, with\nevery word spelled correctly, and all the capitals and stops in the\nright places, as prim as young ladies at a party. When you send that\nletter, please remember to write your address plainly in the upper\nright-hand corner.\n\n       *       *       *       *       *\n\n  MEADVILLE, PENNSYLVANIA.\n\n     I have been a sick boy. But as I am now much better, I hope to be\n     well again soon.\n\n      I have asked my mamma to write for me. Mamma reads a great deal to\n      us, and we have HARPER'S YOUNG PEOPLE and _St. Nicholas_.\n\n      We have them bound at the end of the year, and we think the two\n      volumes of HARPER'S YOUNG PEOPLE the very nicest books we have. I\n      have two sisters and one brother, and we have several pets--a\n      white pony, a canary-bird, a black cat, and some _mice_.\n\n      Mamma read to us about having mice for pets in one of HARPER'S\n      YOUNG PEOPLE, and I thought I would try to have some. I made a\n      cage out of a starch box; on one side I fastened a large\n      baking-powder can, and I put some cotton in that. I put wire\n      netting over the front, and then I tried to catch the mice.\n\n      When I caught one, I put it in the cage, and after a few days he\n      looked lonely, and seemed to have grown thin, so I caught another\n      and put him in, and now they are fat, and seem very happy.\n\n      They keep pretty still all day in the dark, but at night come out\n      in the box, and eat and nibble just enough to keep their teeth\n      sharp, I guess.\n\n      They are cunning little creatures, and I think very nice for pets,\n      and perhaps if you should publish this letter some other little\n      folks would like to try the same plan.\n\n      We always enjoy the stories in YOUNG PEOPLE, but we liked \"Toby\n      Tyler\" best of all.\n\n  LOUIS DE V. M.\n\n       *       *       *       *       *\n\n  OMAHA, NEBRASKA.\n\n     My little sister and I take HARPER'S YOUNG PEOPLE. She is nine, and\n     I am twelve. My brother has a cat nine years old. He is gray, and\n     striped with black. My aunt has a Maltese cat twenty-seven years\n     old. I go to the Central School, and I am in the sixth class. There\n     is a parlor game called capping rhymes. One person gives two or\n     three lines of some piece of poetry, and some other person takes\n     two or three lines of another piece, the first word beginning with\n     the first letter of the last word of the preceding person's verse.\n\n  BLANCHE A.\n\nAn easier way to play at rhymes is this: I begin, for instance, by\nsaying:\n\n  \"The moon came up in the clear night sky.\"\n\nSarah follows, making a line to rhyme with mine, thus,\n\n  \"As round and full as an apple-pie.\"\n\nTom's turn is next, and he says,\n\n  \"Some people think it is made of cheese.\"\n\nAnd then Theodore takes it up with,\n\n  \"But that is silly, as any one sees.\"\n\n       *       *       *       *       *\n\n  BARTLE, INDIANA.\n\n     I am a little boy ten years old. We live in the country, on the\n     Knobs. It is very high land; my pa says it is a thousand feet above\n     the level of the sea. We have lots of fruit trees, mostly peaches.\n     I have lots of peaches to eat in summer. I know a word longer than\n     the one sent by L. L. H., of Orange, New Jersey. It is\n     A-ber-con-way-co-pen-ha-gen-nic-o-de-mus-an-a-bap-tist-dal-a-mu-tha-\n     o-ba-di-ah. My pa learned this word when he went to school.\n\n  FRANK C. P.\n\n       *       *       *       *       *\n\nExchangers will please address Alice C. Little, formerly of Institution\nfor the Blind, Janesville, Wisconsin, at Oberlin, Ohio.\n\n       *       *       *       *       *\n\nA WORD TO EXCHANGERS.--Do not write with pale ink, green ink, blue ink,\nyellow ink, red ink, or brown ink. Do not write with a lead-pencil.\nWrite very plainly, please, with black ink, on white paper or on a\npostal card. Before beginning to write, think over what you have to\noffer, and state it as briefly as possible. We can not make room for any\nexchange which covers four pages of note-paper. Observe our standing\nnotice, and always communicate by mail with the boys and girls with whom\nyou desire to exchange before sending away your own treasures. Remember\nthat letters which are not fully prepaid will probably go to the\nDead-letter Office, and in sending heavy articles, take pains to have\nthem weighed, and put on a sufficient number of postage stamps.\n\n       *       *       *       *       *\n\nC. Y. P. R. U.\n\nLucy wants to know something about the agate. She has a very pretty one\nset as a breastpin, and her brother prizes agates as among the most\nvaluable things which he collects and exchanges. They have an\ninteresting history.\n\nAgates are composed of layers of quartz, generally of different\nvarieties, intimately joined together, and are found of all colors,\nsometimes exceedingly vivid. In modern mineralogy this stone is an\nimpure variety of chalcedony, which derives its name from Chalcedon,\nthat once famous city of Bithynia, in Asia Minor. The rocks near this\nplace, which is not far from the present Scutari, contain this stone in\nconsiderable quantities. Chalcedony consists of silica and alumina, and\ncomprises besides agate, heliotrope, onyx, plasma, and sard, differently\ncolored by metallic oxides. It is found in grape-like masses, but more\nfrequently in rolled pebbles. The finest Oriental chalcedony presents in\nits interior a fine mottled appearance.\n\nThe first engraved gem that Pliny mentions is an agate that belonged to\nPyrrhus, King of Epirus. This was in the first half of the third century\nbefore Christ. The same monarch is said to have had in his possession an\nagate on which were figured the nine Muses, and Apollo holding a lyre;\nthe work not of an engraver, but of Nature herself! The veins in the\nstone were said to be arranged so naturally that each of the Muses had\nher particular attribute.\n\nAt a late period, and even in the Middle Ages, it was a popular belief\nthat the engraved gems found in digging the ground of ancient sites were\nnatural objects, and that the representations on them were a mere\nnatural indication of the special power or quality each possessed.\n\nBusts and heads in full and bass relief were executed by the Romans on\nchalcedony in the grandest style; the finest specimens of these that we\npossess are the Marlborough \"Medusa,\" and the bust of \"Matidia,\"\nsupported on a peacock, and three inches high. The chalcedony was\nsupposed to cure lunatics, and make them \"amiable and merry.\"\n\nThe agate was an object of the most fanciful delusions to the ancients.\nOrpheus says, \"If thou wearest a piece of the tree-agate on thy hand,\nthe immortal gods shall be pleased with thee; if the same be tied to the\nhorns of thy oxen when ploughing, or round the ploughman's sturdy arm,\nwheat-crowned Ceres shall descend from heaven with full lap upon thy\nfurrows.\" He adds that every kind is an antidote to the asp's bite, if\ntaken in wine.\n\nBy burning the agate it was believed that storms would be averted, the\nproof of their efficacy being that if thrown into a caldron of boiling\nwater they immediately cooled it; but in order to do good, they must be\nstrung on the hair of a lion's mane. The stone, colored like a hyena's\nskin, was believed to be the cause of domestic strife, and was viewed\nwith horror.\n\n       *       *       *       *       *\n\nWhat do you think of the wardrobe of Queen Elizabeth, girls? It must\nhave been about the most varied and extensive ever recorded in royal\nannals, to judge from a list of her wearing apparel recently gathered\nfrom the State papers. When the Maiden Queen was sixty-eight, and might\ntherefore have been supposed to have outlived some of her youthful\nvanity, she possessed 99 complete official costumes, 102 French gowns,\n100 robes with trains and 67 without, 126 antique dresses, 136 bodices,\nand 125 tunics, not to mention such trifles as 96 mantles, 85\ndressing-gowns, and 27 fans. With all these dresses, however, it is\ncurious to note that Queen Bess owned only nine pairs of shoes. When she\ndied, in 1603, three thousand articles of apparel were found in her\nwardrobes, duly catalogued.\n\n       *       *       *       *       *\n\n\"A GERMANTOWN GIRL.\"--Thanks for your cordial indorsement of the article\non cigarette smoking in No. 117. We hope you will ask the boys of your\nacquaintance to read it, and we are sure that you and your girl friends\nwill do much to put an end to the bad habit in your set if your young\ngentlemen friends know that you disapprove of it.\n\n       *       *       *       *       *\n\nAmong the thousands of young people who read this paper there must be\nsome who suffer from timidity or bashfulness. The Postmistress wishes\nall such to read the following quotation from an address made to\nchildren at a recent service in Westminster Abbey by Dr. Bradley, the\nsuccessor of Dean Stanley, of whom you all have heard:\n\n     \"Dean Stanley was, as I said, a man whom not only children, but all\n     persons, rejoiced to meet, because, as an American whose name is\n     known all over Europe said, at a great meeting held close by the\n     other day, 'he was so pleasant'; simple words, but very true. Yet,\n     strange to say, he was in his childhood and boyhood so shy and\n     timid that his mother and his father--who, besides being a very\n     good man, could tell stories all day about birds and insects and\n     country life, and could draw children after him as a magnet draws\n     the needle--were alarmed about him. They could not teach him to\n     enjoy games like other children; and though he was not unhappy at\n     school, they were much distressed because they could not get him to\n     speak a word or seem happy in company. He seemed not to like--some\n     of you, perhaps, know the feeling--to notice anything, or to be\n     noticed when others were by him that he did not know quite well. He\n     cured himself of this so completely that if it were not for the\n     many letters describing him as he was then, no one would now\n     believe this about him. How did he do so? By always trying to\n     instruct himself about other things, and to interest himself about\n     other people.\n\n      \"It was not merely because he was so clever, and wrote such good\n      poetry, that boys christened the corner where his little study was\n      'Poet's Corner,' that the Rugby boys were so fond of him, but\n      because he was so affectionate, pure-minded, warm-hearted, and\n      kind; and years afterward we young men, as we were at Oxford,\n      learned to love him as our tutor, not because he was becoming\n      famous in the land, but because he cared so much about us, and was\n      always ready with a kind word and act, and we thought him so good.\n      If any of the children to whom I am speaking suffer very much now,\n      as boys and girls often do, in a way their elders never know of,\n      from this same shyness and timidity, let them learn from the story\n      of Dean Stanley the best way to cure themselves, and be of good\n      cheer.\"\n\n       *       *       *       *       *\n\nThe members of the C. Y. P. R. U. will find in this number an\ninteresting article, by Mrs. John Lillie, entitled \"The Gavotte,\" giving\na description of this picturesque old dance, together with a sketch of\nthe boy life of the famous musician John Sebastian Bach. Our boy readers\nwill be interested in the description of \"Mr. Barnum's Show in\nWinter-Quarters,\" some of them may be inclined to try the Indian game of\n\"Tchungkee,\" and others may like to busy themselves in the construction\nof \"A Novel Chair-Sleigh.\" The girls will be interested in the story of\n\"The Milkmaids of Dort.\"\n\n       *       *       *       *       *\n\nPUZZLES FROM YOUNG CONTRIBUTORS.\n\nNo. 1.\n\nDIAMOND.\n\n1. A letter. 2. To descry. 3. A vulgar woman. 4. One of the United\nStates. 5. A place where watches are manufactured. 6. A loud noise. 7. A\nletter.\n\n  BOB.\n\n       *       *       *       *       *\n\nNo. 2.\n\nFOUR EASY WORD SQUARES.\n\n1.--1. Low, wet ground. 2. Part of a circus. 3. Kingly. 4. An animal. 5.\nParts of a building.\n\n  JIMMY BROWN.\n\n2.--1. A present. 2. Unemployed. 3. To run away. 4. A bird.\n\n3.--1. Pulverized grain. 2. A name. 3. An exclamation. 4. Final.\n\n  NELSON M.\n\n4.--1. A number. 2. A metal. 3. Empty. 4. Extremes.\n\n  CAPTAIN KIDD.\n\n       *       *       *       *       *\n\nNo. 3.\n\nENIGMA.\n\n  My first in heart as well as hand.\n  My second figures in the land.\n  My third appears in every plain.\n  My fourth is found in every train.\n  My fifth in hope, but not in fear.\n  My sixth in sound, but not in hear.\n  My seventh rides in every drive.\n  My eighth's in all, and in alive.\n  My ninth's in crown, but not in ring.\n  My tenth's a favorite with the king.\n  My eleventh may be seen in sail.\n  Also in storm, though not in gale.\n  My whole's a well-known mountain chain\n  That's girdled round by grove and plain.\n  Poets have added to my fame,\n  And Cooper often wrote my name.\n\n  DOLLIE.\n\n       *       *       *       *       *\n\nANSWERS TO PUZZLES IN No. 118.\n\nNo. 1.\n\n  P           P\n  I n v a d e R\n  L           O\n  G           G\n  R e f i n e R\n  I           E\n  M           S\n  S p o n g e S\n  B           B\n  Y           U\n  J a v e l i N\n  O           Y\n  H           A\n  N a n k e e N\n\n_Pilgrim's Progress_, by John Bunyan.\n\nNo. 2.\n\n  L ute  J ars  G ate\n  A nne  A lps  R oof\n  D ime  N oon  E lms\n  Y ore  E ase  Y ear\n\nLady Jane Grey.\n\nNo. 3.\n\n  M O T H    O G R E\n  O L I O    G R E W\n  T I P S    R E D E\n  H O S E    E W E R\n\nNo. 4.\n\nSnow-storm. Whittier.\n\nNo. 5.\n\nTongue-tied. Cap-a-pie. Blackboard.\n\nNo. 6.\n\n  W H E A T    T R A S H\n    H E A T      R A S H\n      E A T        A S H\n        A T          S H\n          T            H\n\n       *       *       *       *       *\n\nAnswer to Enigma on page 240--Courtship.\n\n       *       *       *       *       *\n\nCorrect answers to puzzles have been received from \"Helen of Troy,\"\nLudwdg L. Dessoir, \"Commodore Nutt,\" Ella Price, \"Fanchon,\" Gustave D.\nMott, Georgie Wardell, Edith Abbey, Mary Hathaway, M. F. Tomes, H. L.\nPruyn.\n\n       *       *       *       *       *\n\n[_For Exchanges, see 2d and 3d pages of cover._]\n\n\n\n\n[Illustration: A QUEER AQUARIUM.]\n\n\n\n\nPARALLELS.\n\nARRANGED BY G. B. BARTLETT.\n\n\nThere is no better amusement among the many in which old proverbs are\nbrought to mind than the one just introduced under the above title,\nwhich exercises at the same time the inventive and guessing powers.\n\nAny number can join in this game, which is begun by one of the players,\nwho relates some real or fancied experience, or tells a story in which\nsome proverb which he has in mind is expressed. The person who guesses\nthe proverb thus indicated has to tell another story, which must be\ncontinued until it has also been guessed.\n\nWhen played by a large company, it is well to divide into equal sides,\narranged opposite each other in two lines. In this case the first player\nat the right upper side begins to tell the story, which must be guessed\nonly on the opposite side. The guesser tells his story next, which must\nbe guessed by some one on the right side of the room. The sides before\nbeginning to play choose a time-keeper, who decides the duration of each\ncontest, which should be from half an hour to an hour, according to the\nnumber of players. He keeps his watch in his hand, and calls out \"Stop!\"\nthe instant the time has expired, and then declares the side to have won\non which the unguessed story was being told at that moment. This keeps\nall the players on the alert, as each one is eager to guess while he is\ninventing his own story, so there may be no delay if he succeeds in\nguessing.\n\nThis effort to do two things at once is very good for the mental powers,\nas well as very amusing, as the players often become nervous and\nconfused. The best stories are those which suggest several proverbs, as\nthey may be guessed incorrectly at first, and thus give more time to the\nside on which the narrator is. To make this game clear to the youngest\nreaders, who often play as well as their elders, a specimen of a simple\nstory is given, as follows:\n\n\"An old farmer had lived very frugally on his farm for many years, until\nhe had acquired a small competence. His old gray mare had worked as hard\nas he, and now, grown old in the service, was seldom driven fast, but\nwent slowly from door to door, dragging the milk-cart, which stopped at\nevery farm-house, in the morning for the full cans, and on the return\ntrip from the station left the empty ones to be refilled. The old animal\nhad become very lazy from this habit, and as she slowly jogged home, the\nold man would be able to read his daily paper, or to count over the\ncoppers with which the sale of milk had filled the old leather bag which\nhe always carried. One fine day, as he left the station on his homeward\nway, a telegram was put into his hands containing the startling news\nthat his barn was on fire. Eager to save his stock, he plied the whip on\nhis poor beast, whose hide was so thick that very little effect was\nproduced. The farmer continued his blows until the whip-lash was worn\nout, and then tried the whip-stock, which soon broke also, and the old\nman was in despair, as the mare only jumped up and down, without\nincreasing her forward progress.\" At this point \"More haste the worse\nspeed,\" is incorrectly guessed by one of the players, and the narrator\nproceeds: \"The mare shook her head angrily, and leered at her master,\nshowing the whites of her eyes as if in scorn, and this action reminded\nthe farmer that she was very sensitive about the head, being always\nannoyed when her bridle was put on. 'Ah! old Betsey,' said he, 'I know\nhow to make you go now;' and taking a handful of coppers from his bag,\nhe threw them with all his might at the mare's head. Surprised at this\nnovel attack, old Betsey darted off with the speed of an unbroken colt,\nand brought the old man home quicker than he had ever before gone over\nthe road.\" \"Money makes the mare go,\" guesses a player on the right, and\nbegins the second story.\n\nThis example will show the method of playing, as the stories are as\nvaried as the tastes of their authors, and old and young alike enjoy\nthis game, which is adapted for the summer picnic as well as for the\nwinter fireside.\n\n       *       *       *       *       *\n\nThe answer to the Monogram Puzzle on page 237 of No. 119 as follows:\n\n  _Swift_ ran the deer in mortal fright,\n    Over _Hawthorne_ bushes leaping;\n  Passing the _Hamlet_ with footsteps light,\n    Near where the _Cooper_ his shop was keeping.\n\nSwift (Dean), Hawthorne, Hamlet, Cooper.\n\n\n\n\n[Illustration: \"Hi! here's fun. Daddy thinks I've been in School this\nhalf-hour.\"]\n\n[Illustration: Does he, though?]\n\n[Illustration: GRAND FINALE!]\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Harper's Young People, February 21,\n1882, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n  1. Title Page\n  2. Acknowledgements\n  3. List of Figures\n  4. List of Abbreviated Citations\n  5. Opening\n  6. Relationality and the Work of Art\n    1. Mattering, Particularity and the Work of Art\n    2. Othering Relationality\n    3. On Hands\n  7. Engaging Philosophy's Art\n    1. Matter and Movement's Presence\n    2. Art's Work\n    3. Colouring Philosophy\n    4. And Colour?\n    5. Endless Touching\n  8. Bibliography\n  9. Index\n\nArt's Philosophical Work\n\nArt's Philosophical Work\n\nAndrew Benjamin\n\nLondon \u2022 New York\n\nPublished by Rowman & Littlefield International, Ltd.\n\nUnit A, Whitacre Mews, 26-34 Stannary Street, London SE11 4AB\n\nwww.rowmaninternational.com\n\nRowman & Littlefield International, Ltd. is an affiliate of Rowman & Littlefield\n\n4501 Forbes Boulevard, Suite 200, Lanham, Maryland 20706, USA\n\nWith additional offices in Boulder, New York, Toronto (Canada), and London (UK)\n\nwww.rowman.com\n\nCopyright \u00a9 2015 by Andrew Benjamin\n\n_All rights reserved._ No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without written permission from the publisher, except by a reviewer who may quote passages in a review.\n\n**British Library Cataloguing in Publication Information Available**\n\nA catalogue record for this book is available from the British Library\n\nISBN: HB 978-1-7834-8289-4\n\nISBN: PB 978-1-7834-8290-0\n\n**Library of Congress Cataloging-in-Publication Data**\n\nBenjamin, Andrew E.\n\nArt's philosophical work \/ Andrew Benjamin.\n\npages cm\n\nIncludes bibliographical references and index.\n\nISBN 978-1-78348-289-4 (cloth : alk. paper) \u2014ISBN 978-1-78348-290-0 (pbk. : alk. paper)\u2014 ISBN 978-1-78348-291-7 (electronic) 1. Art\u2014Philosophy. I. Title.\n\nN66.B45 2015\n\n701\u2014dc23\n\n2015007800\n\n TM The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences Permanence of Paper for Printed Library Materials, ANSI\/NISO Z39.48-1992.\n\nPrinted in the United States of America\n\n# Index\n\nA\n\nactative,  ,  ,  ,  ,  , 6.1-6.2 ,  ,  , 9.1-9.2 ,  ,  ,  ,  ,  ,\n\nAdorno, Theodor,\n\nafter-effect,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , 14.1-14.2 ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,\n\nAlberti,\n\nanoriginal,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , 19.1-19.2 ,  ,  ,  ,\n\nAppel, Karel,  ,  ,  ,  ,\n\nArendt, Hannah,\n\nAristotle, 1.1-1.2 ,  ,  ,  , 5.1-5.2 ,  ,\n\nArtaud, Antonin,  , 2.1-2.2 ,  , 4.1-4.2 ,\n\nart's self-staging,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , 24.1-24.2 ,  ,\n\nAtlan, Henri,  , 2.1-2.2 ,  , 4.1-4.2\n\nat-the-same-timeness,  ,  ,  ,  ,  ,  ,  ,  ,  ,\n\nB\n\nBarker, Stephen,\n\nBaselitz, George,\n\nBellori,\n\nBenjamin, Walter,  ,  , 3.1-3.2 ,  ,  ,  , 7.1-7.2 ,  , 9.1-9.2 ,  ,  ,  ,  , 14.1-14.2 ,  ,  ,  ,  , 19.1-19.2\n\nBernini,  ,  ,  ,  ,  ,  ,  ,  ,\n\nBilderverbot,\n\nBlanchot, Maurice,\n\nBoccaccio, Giovanni,\n\nBorromeo, Federico,\n\nBotticelli,\n\nBronzino, Angelo,\n\nL\n\nle Brun, Charles,\n\nB\n\nBruno, Giordano,\n\nBulwer, John,\n\nBurgkmair, Hans,\n\nC\n\nCamerarius, Joachim,\n\nCanisius, Petrus,  ,  ,\n\nCaravaggio, Poloforo,  ,\n\nCastiglione,\n\nCaygill, Howard,\n\nCenini, Cennino,\n\nCicero,  ,\n\ncoming-into-relation,  ,  ,  ,  ,  ,  ,  ,  , 9.1-9.2 ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , 24.1-24.2 ,\n\nD\n\nDerrida, Jacques,  ,  , 3.1-3.2\n\nDescartes, Ren\u00e9,  ,  , 3.1-3.2 ,  ,  ,\n\nDiderot,  ,  , 3.1-3.2 ,  ,  ,\n\nDidi-Huberman, Georges,\n\nD\u00fcrer, Albrecht,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , 11.1-11.2 ,  ,  ,  ,  ,  ,  ,  ,\n\nE\n\nekphrastic impulse,  ,  ,  ,  , 5.1-5.2\n\nF\n\nFocillon, Henri, 1.1-1.2\n\nD\n\ndella Francesca, Piero,  ,  , 3.1-3.2 ,\n\nG\n\nGorgias, 1.1-1.2\n\nGossaert, Jan,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,\n\nGoya,  ,\n\nGrien, Hans Baldung,  ,  ,\n\nGuarino of Verona,\n\nH\n\nHamlet,  ,  ,\n\nhand,  ,  ,  ,  ,  ,  , 7.1-7.2 ,  ,  , 10.1-10.2 ,  ,  ,  ,\n\nhand\/instrument relation,  ,  ,  ,  ,  ,  ,  ,  , 9.1-9.2\n\nHegel, G. W. F.,  , 2.1-2.2 ,  ,  ,  , 6.1-6.2 ,  ,  ,  ,  ,\n\nHeidegger, Martin,  ,  , 3.1-3.2\n\nHerder, Johann Gottfried, 1.1-1.2\n\nhyponia,\n\nI\n\nideational,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,\n\nincomplete,  ,  ,  ,  ,  ,  ,  ,  ,\n\nK\n\nKant, Immanuel,  ,  ,\n\nL\n\nLavin, Marilyn Aronberg, 1.1-1.2\n\nLessing, G. E.,  ,  ,\n\nLevinas, Emmanuel,\n\nE\n\nEl Lissitzky,  ,  ,  ,  ,\n\nL\n\nLotto, Lorenzo,\n\nLucian,\n\nLuther, Martin, 1.1-1.2\n\nLyotard, Jean-Francois,  , 2.1-2.2\n\nM\n\nMachiavelli,\n\nMant, \u00c9dourad,\n\nMarmion, Simon,  ,  ,  ,  ,  ,  ,  ,\n\nMary,  ,  ,  ,  ,  ,\n\nmattering,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , 22.1-22.2 ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,\n\nMeilon, Walter S.,\n\nMelanchthon, Philip, 1.1-1.2\n\nMichaud, Philippe-Alain,\n\nMichelangelo,  ,  ,  ,  ,\n\nMonet, Claude,  ,  ,  ,  ,\n\nMosca, Francesco di Simone,  ,  ,  ,  ,  ,  ,  ,\n\nmotif,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,\n\nN\n\nNagel, Alexander,\n\nNancy, Jean-Luc,\n\nNiobe,  , 2.1-2.2 ,\n\nO\n\nOlkowski, Dorothea,\n\nOvid,  ,  ,  ,  ,  ,\n\nP\n\nPanofsky, Erwin,  , 2.1-2.2 ,  ,  ,\n\nParmigianino,  ,  ,\n\nPascal, Blaise,\n\npathosformel,  ,  ,\n\nPericolo, Lorenzo,\n\nPhilostratus, 1.1-1.2 ,  , 3.1-3.2\n\nD\n\nde Piles, Roger,\n\nP\n\nPlato,  ,  ,  ,  ,  ,\n\nPliny the Elder, 1.1-1.2 , 2.1-2.2 ,\n\npotentiality,  ,  ,  ,  ,  ,  ,  ,  , 9.1-9.2 , 10.1-10.2 ,  ,  ,  ,  ,  ,\n\nPoussin, Nicolas, 1.1-1.2 ,\n\nD\n\nDu Pr\u00e9au, Garbiel,\n\nQ\n\nQuintilian,  ,\n\nR\n\nRaphael,  ,  , 3.1-3.2\n\nrelational ontology,  ,  ,  ,  ,\n\nRipa, Cesare,\n\nD\n\nde Ronsard, Pierre,\n\nS\n\nSadeler, Jan,  ,\n\nSallis, John,  , 2.1-2.2\n\nD\n\ndel Sarto, Andrea,  , 2.1-2.2 ,  ,\n\nS\n\nSchelling,  ,  ,  ,  ,  ,  ,  ,\n\nShakespeare, William,  ,\n\nT\n\nTitian,  ,  ,  ,  ,  ,  ,\n\nV\n\nVasari,\n\nVeronica,  ,  ,  ,\n\nvirtue in being,\n\nW\n\nWarburg, Aby,  ,  ,  ,  , 5.1-5.2 , 6.1-6.2 ,  ,  , 9.1-9.2 ,  ,\n\nWeigel, Sigrid,\n\nWinckelmann, J.,  ,\n\nWind, Edgar, \n\n# Acknowledgements\n\nThis book could not have been written without the support and encouragement of friends and colleagues. Its strengths\u2014 _s'il y'en a_ \u2014are derived from the constant questioning by art historians and philosophers. All the limitations are mine. Institutions have also been important. I have had the opportunity to present elements of it in a number of academic contexts. Chapter 1 was first presented as a seminar in the Department of Art History at the University of Warwick. Karen Lang and Paul Smith were wonderful hosts and interlocutors. Karen Lang's wisdom and generosity of spirit have been extremely helpful in allowing me to understand in what way certain arguments needed clarification. Chapters 2 and 3 formed part of a graduate seminar I taught in 2014 in the Centre for Research in Modern European Philosophy at Kingston University. I want to thank the students for their sustained engagement with the work of Warburg, Benjamin and D\u00fcrer. In addition, I want to thank the centre's Director, Peter Osborne, for his support. A version of chapter 2 was given as a lecture at the Warburg Institute. It was a profound honour to speak in such a prestigious environment. I want to thank the Bilderfahrzeuge: Aby Warburg and the Future of Iconology research group for their kind invitation to present my work. I also gave a lecture on Warburg and Benjamin in the Department of Art History at the University of Leeds. I want to thank Griselda Pollock for her kind invitation to give the lecture. Her engagement with some of my arguments was equally decisive in their reformulation. Additionally, I would like to thank the following for the kindness with which they discussed my work with me; in every instance I learnt a great deal: Christopher D. Johnson, Joanne Woodall, Paul Hills, Howard Caygill, Philip Ekardt, Christopher Wood, Helen Hills, Jas Elsner and Teresa Stoppani. I also want to thank James Kent for his help in the preparation of the manuscript.\n\nThe second part of this book consists of already published papers that form part of the overall project. I wish to thank the publishers, Blackwell, Northwestern University Press and _Research in Phenomenology_ (Brill), for their permission to reprint them. \n\n# List of Figures\n# List of Abbreviated Citations\n\nBenjamin, AP | Walter Benjamin, _The Arcades Project_ , ed. Rolf Tiedemann, trans. Howard Eiland and Kevin McLaughlin (Cambridge: Harvard University Press, 1999).\n\n---|---\n\nBenjamin, GS | Walter Benjamin, _Gesammelte Schriften_ , eds. Rolf Tiedemann and Herman Schweppeng\u00e4user (Frankfurt: Suhrkamp Verlag, 1980).\n\nBenjamin, OGTD | Walter Benjamin, _The Origin of German Tragic Drama_ , trans. John Osborne (London: New Left Books, 1977).\n\nBenjamin, SW | Walter Benjamin, _Selected Writings_ , 4 vols., eds. Marcus Bullock, Michael W. Jennings, Howard Eiland and Gary Smith (Cambridge, MA: Harvard University Press, 1996\u20132003).\n\nDescartes, AT | Ren\u00e9 Descartes, _Oeuvres de Descartes_ , ed. Charles Adam and Paul Tannery (Paris: Libraire Philosophique J. Vrin, 1996).\n\nHegel, LFA | G. W. F. Hegel, _Hegel's Aesthetics: Lectures on Fine Art_ , vol. 2, trans. T. M. Knox (Oxford: Oxford University Press, 1998).\n\nPanofsky, AHAT | Erwin Panofsky, 'On the Relationship of Art History and Art Theory: Towards the Possibility of a Fundamental System of Concepts for a Science of Art', trans. Katharina Lorena and Jas Elsner, _Critical Inquiry_ 35 (Autumn 2008): 43\u201371.\n\nPanofsky, AV | Erwin Panofsky, 'The Concept of Artistic Volition', trans. Kenneth J. Northcott and Joel Snyder, _Critical Inquiry_ 8, no. 1 (Autumn 1981): 29.\n\nWarburg, TRPA | Aby Warburg, _The Renewal of Pagan Antiquity_ , trans. David Britt (Los Angeles: Getty Research Institute, 1999).\n\nWarburg, WEB | Aby Warburg, _Werke in Einem Band_ (Frankfurt: Suhrkamp, 2010). \n\n# Opening\n\nRepresented here is a project the aim of which is to locate a philosophy of art within the parameters of a relational ontology.[1] The project has two interconnected elements. The first is a philosophical engagement with art's history. The second involves indicating the ways that the particularity and the materiality of the work of art are to be understood. Philosophy usually engages with art as though the latter did not have a history.[2] As a result, not only has the history of art been excluded from any real philosophical consideration, but equally what is left unaddressed is the more general question of how that history\u2014art's history\u2014is to be taken up philosophically. Any attempt to rectify this situation needs to locate a thinking of art history within a more general thinking of the philosophical. In other words, while the specificity of 'art history' has to be maintained, it remains the case that art's history does not fall beyond the province of philosophical thought. As already indicated, the position taken here, an approach whose justification can only occur within the analyses themselves, stems from a broader philosophical project that locates relationality at its core. There is a general organizing philosophical proposition within the development of a relational ontology that is at work within this project. Indeed, it orientates the argumentation of each of the subsequent chapters. The proposition is the following: A singularity is always the after-effect of a relation. The force of this claim, as it is presented here, is that it results in undoing the hold of the naturalization of objects and events while also working to undo their incorporation into the naturalization of time as chronology. There is a twofold process of naturalization. Undoing these processes gives rise to the denaturing of nature. Nature was, after all, a construct. The question then is, how are presentations of that which occurs in the wake of this process to be presented? As what prompts these moves is recovering the particularity of art's work, the approach to be taken in this context entails letting the centrality of relationality appear as the basis of a response to a thinking of art within philosophy.\n\nThe three chapters that comprise part I are concerned with developing some of the elements central to any philosophical response to the history of art (accepting the delimitation given to the philosophical through the attribution of centrality to relationality). As this book will continue to argue, this means the creation of a twofold philosophical engagement. First, there has to be an engagement with the particularity and the materiality of works of art. The contention here is that the question of art's presence within the philosophical will have been delimited from the start by having to take up how a given work of art works as art. The latter is a position that continues to return in terms of a concern with the work of the work. Holding to the centrality of the work's work, and thus to the question of how a work works as art, both yields and demands the primacy of particularity. Second, responding philosophically to the history of art can only occur adequately when a central component of the project is the temporality of history. From within the purview of the philosophical, history cannot have the status of the given. Its temporality has to be thought (hence the inevitability of divergences within and as thought). Both of these elements, which here are taken as fundamental to a philosophical engagement with the interplay of the work of art on the one hand and the ineliminable historicity of art on the other, are necessary if art's history is to be maintained as an object of thought.\n\nWhile philosophy has engaged and continues to engage works of art, the identification of the limitations of those encounters is itself instructive, since what is often effaced within them are the differing ways in which the materiality of art's work is central to its presence as the work of art. Meaning, and this is the argument of chapter 1, is always the after-effect of the way materials work. As a result of arguing for the centrality of the material and thus locating its presence as forming an integral part of a materialist philosophy of art, part II consists of a series of connected essays that attempt to establish the limits that occur in the way fundamental components of the philosophical have addressed the work of art hitherto. What these analyses bring out are the differing ways philosophy has continued to resist art's inherent materiality. Identifying the ways art's material presence has been resisted or effaced creates openings.\n\nAfter a general consideration of what is involved in responding philosophically to the history of art, an undertaking which orientates the interrelated trajectories of chapter 1, the implications of what is at work within that question and its response will then be pursued via the writings of Walter Benjamin and Aby Warburg.[3] (The latter is the task of chapter 2.) In terms of its detail, chapter 1 involves the development of specific terminology and a mode of thought that sustains the project as a whole. The ontology of art's work has to be connected to a rethinking of the temporality of history that insists both on relationality and on the process of works _coming-into-relation_. This is a difficult and demanding undertaking. As has been suggested, part of the challenge resides in letting 'work'\u2014a term which is traditionally understood as substantive, naming a static object, an object with both a title and a determined location within chronological time\u2014cede its place to 'work' as a verb and therefore to that which has a dynamic and active quality. (The move is to the _actative_. The actative underscores the primacy of activity. Activity\u2014thus work\u2014is a quality intrinsic to a work.) As will be presented in much greater detail, it is only by allowing for the move from the static to the dynamic that it then becomes possible to take up what is meant by the work of the work and thus to approach specific artworks within the terms set by the question of how a particular work works as art. Activity remains the defining characteristic of work.\n\nChapter 2 is a development and clarification of these earlier concerns. Nonetheless, it has a specific orientation. What it seeks to establish is the nature of the difference between Warburg and Benjamin, both of whom are concerned, inter alia, with the way a reconsideration of the temporality of art history opens up another thinking of the image. The approach to their work is given a specific location. In order to contextualize their writings, points of intersection, divergence and difference will be identified as they emerge from their respective interpretations of Albrecht D\u00fcrer's _Melancholia I_. What connects their projects is the recognition that thinking art cannot be separated from questions of historical time (and therefore how the latter is itself thought). While the connection between Warburg and Benjamin has a founding importance, insofar as the affinity that can be identified in their writings is initially productive, in the end what is central are the differences between them.\n\nDeveloping these differences, which here involves extending Benjamin's thought beyond his actual writings on D\u00fcrer, turns around the repetition of a particular motif. The motif in question is the hand\/instrument relation. A motif, in regards to the way this term is to be used within this project, is an aspect or component of the work of art whose existence is defined as much by its ease of repetition as it is by its having been positioned beyond any determination that would attribute an essential quality to it. Its form allows it to be informed differently. The premise is that it does not follow from the presence of the same motif\u2014here the recurrence of the hand\/instrument relation\u2014that each instance is informed in the same way. Drawing this conclusion arises from allowing different instances of the motif to be connected to each other. It results from the process of coming-into-relation; the latter is the process that is central to the rethinking of particularity. A consequence of the way coming-into-relation figures is that the repetition of that which has the same form could, while appearing the 'same', also be the site of genuine differences. Once difference takes over from sameness, then the processes of coming-into-relation could then be deployed as a counter to the way Warburg understands the relation and repetition of the 'same' images. While Warburg's _Atlas of Images_ ( _Der Bilderatlas_ ), known as the _Mnemosyne_ project, consists of the reiterations of sets of images in which what is repeated is the same, their repetition\u2014and thus the way relationality is understood\u2014is premised on the claim that the presence of the same image entails that what is presented is itself the same in each instance.[4] (Remembering that sameness is not identity.) If there is a counter to this possibility, then, as intimated above, it can be located in the proposition that the repetition or relation of that which appears to be the same could in fact be the relation and reiteration of genuine differences. This is a counter based on the propositions that (1) difference need not be equated with mere variety, (2) difference need not be reduced to variation within a prevailing unity and (3) difference creates spaces, marked by their own sense of tension, which, in general, are defined in terms of irreducibility.[5] What this then means is that the identity of a given motif would then stem from the relations that its repetition establishes. Identity is neither fixed nor determined; it is always an after-effect.\n\nWalter Benjamin uses the term 'motif' throughout his writings in order to identify a particular topos or locus of activity that has an abstract quality and whose meaning is created by the way that quality provides a locus of engagement. Hence, in 'Central Park', writing of Baudelaire, he argues that 'Baudelaire's imagination is occupied by stereotyped images (stereotype Bilder). He seems to have been subject to a very general compulsion to return at least once to each of his motifs' ( _einmal zu jedem seiner Motiven zur\u00fcckzukehren_ ).[6]\n\nEqually in the same text, Jugendstil is characterized by 'motifs'. Furthermore, the work of the Romantics is described as having defining epistemological and philosophical 'motifs'.[7] The repetition of a motif is it be understood here as a coming-into-relation. The motif does not have an essential quality. It _is_ \u2014is what is\u2014in its repetition, that is, within its being repeated. Once the motif predominates, what then becomes possible is a form of repetition that need not be defined in terms of sameness. Indeed, the force of the term 'motif' is that it will allow the repetition of the same to be the site of genuine differences. As mentioned, the motif in question\u2014as will be seen, a motif central to any understanding of the presence of the figure of Melancholia in D\u00fcrer's engraving\u2014is the hand\/instrument relation. More is at work here than the presence of this motif in _Melancholia I_. In fact the argument is that a motif's repetition, and it is essential to the argument that it is a motif and thus not a symbol or figure with an already determined and singular designation, provides the limit condition of Warburg's thought while questioning at the same time any identification of a symbol or a figure with a single original determination, be that determination conceptual or just affective (as though there could ever be the purely affective). Hence, the further contention is that even an affective state that is thought to have had a single determination will always have had a doubled and thus an originally irreducible quality. Original irreducibility, understood as an ontological set up and not a semantic one (i.e., the referent is to ontology and not to semantics) will continue to be identified here by the term _anoriginal_.[8]\n\nThe development of the hand\/instrument motif beyond its role in D\u00fcrer and the works to which _Melancholia I_ can be related forms the project of chapter 3. Moreover, that motif has a history, the presentation of which continues to question Warburg's project while at the same time furthering it. While this appears to be a paradox, what it identifies is the presence of a fundamental limit within Warburg's work with images. The limit involves a mode of repetition within his project\u2014thus what could be seen as his thinking of the motif\u2014that is structured by sameness rather than a conception of difference defined by the interplay of continuity and discontinuity. The limitation depends upon how the repetition of motifs is to be understood. What is retained, and here is the momentary affinity with Benjamin, is the refusal to define the temporality of history in terms of chronology. That refusal also allows automatic assumptions concerning art history's assumed periodization as well as its geographical specificity to be questioned.[9] The response to Warburg, a response that is intended to have an affinity with fundamental aspects of Walter Benjamin's thought, is to argue that it is discontinuity, rather than continuity, which is central. In other words, at work here is a conception of difference that maintains forms of irreducibility at its core. However, rather than develop this position or introduce the terminology through which the project takes place at this point, it will continue to be developed throughout the arguments to come.\n\nNonetheless, if there were a formulation that organizes this project as a whole, then it can be located in one of Walter Benjamin's attempts, this time in _The Arcades Project_ , to delimit how the relation between the past and the 'present instant' ( _Aktualit\u00e4t_ ) is to be conceived. 'In order for a part of the past ( _ein St\u00fcck Vergangenheit_ ) to be touched by the present instant ( _Aktualit\u00e4t_ ) there must be no continuity between them.'[10]\n\nWhile a passage of this complexity warrants detailed analysis, there are two aspects that need to be noted at the outset. The first is that the constitution of an event as historical occurs as a result of the concerns of the present being brought to bear on that which would otherwise have been nothing other than what Benjamin identifies elsewhere as 'fore-history'. The present here is not the present moment as defined by the instant within chronology. The present\u2014hence Benjamin's use of the term ' _Aktualit\u00e4t_ '\u2014is the informing, thus the forming of that instant by what can best be described as the concerns of the present. Those concerns yield a task. In this way, once the present is no longer identified with the instant, or with the 'now' defined chronologically, then the time of the present yields tasks. (The determination of these concerns and related tasks is itself the site of an already present, thus anoriginal, conflict. Hence the inevitability of what can be called a 'politics of time'.[11]) The second aspect is the demanding claim that what occurs is a relation formed by discontinuity rather than continuity. Discontinuity is a mode of relationality, and so what this passage underscores is the centrality within Benjamin's thought of a specific mode of relationality. Moreover, it is discontinuity as a mode of relationality that creates the 'force field' sustaining meaning. The historical event, or more exactly the constitution of the occurrence as an historical event, is the after-effect of a coming-into-relation. The relation in question refuses any form of synthesis or reconciliation. Any possible equilibrium is eschewed in the name of tensions that will always have to be as much affective as they are ideational. Benjamin's use of Henri Focillon is instructive here. On one level Benjamin's attraction to his work had to do with Focillon's insistence on tensions, the bodily and the affective. Indeed, Benjamin's use of bodily language is a fundamental aspect of his own undertakings. There is, however, a dimension of Focillon's work that departs from a complete identification with the bodily and thus with the interplay of tensions, feelings and the affective. Part of the accompanying material to his _On Some Motifs in Baudelaire_ contained citations from Focillon. One of the most instructive concerns a passage from the latter's _Vie des forms_. The passage is the following: 'Focillon on the work of art: \"The instant at which it is born, it is a phenomenon of rupture ( _elle est phenomene de rupture_ ). A current expression provides a lively sense of it: 'make the day an event' ( _faire date_ ). It is not to intervene passively in chronologies it is to break the moment\" ( _c'est brusquer le moment_ ).'[12]\n\nBenjamin's conception of 'the work of art', as well as Benjamin's own insistence of the creative potential of 'destruction', accords with Focillon. Chronology is broken; however, it is chronology as 'semblance' ( _Schein_ ). The problem emerges because, as argued by Benjamin elsewhere, 'the persistence of this semblance of persistence provides it with continuity' [ _stiftet eben die Best\u00e4ndigkeit dieses Scheins der Best\u00e4ndigkeit die Kontinuit\u00e4t in ihr_ ].[13] It is the recognition of semblance as semblance that is the destructive act. It occurs, to shift register slightly, in 'the moment of recognizability' ( _im Augenblick seiner Erkennbarkeit_ ).[14] It constitutes that moment as a moment. It is the denaturing of nature. Revealed in the process is semblance as semblance and continuity as a ruse which has come to be naturalized. Benjamin's project, to deploy Focillon's formulation, is 'to break the moment' ( _c'est brusquer le moment_ ). On the other hand, Warburg's project, the argument will be, is to find within the 'moment' forms of equilibrium and harmony the possibility of which is predicated on Warburg's failure to see the connection between 'semblance' and 'continuity'. In other words, while Warburg is concerned to undo continuity of time, hence the critique of the identification of historical time with chronology, there is the concomitant failure to recognize that this demands that destructive moment in which semblance is undone and another conception of the image emerges within and as a result of that undoing. On one level the argument has to be that the image is to be made. It is not already there to be repeated.\n\nFinally, allowing for the insistence of particulars, locating meaning as an after-effect of the way in which materials are at work and demanding that relationality provide the way into a rethinking of the history of art's history define a set of concerns that continue to be worked through in the chapters to come. Works of art, located beyond any logic of exemplarity, continue to provide the loci through which arguments are advanced and interpretive claims made. Art figures. It figures, however, as the work of art. The move is from works taken as ends in themselves, to work reconfigured as an activity in which a particular emerges as the after-effect of the network of relations that constitute its particularity. Constructed here, held in place at the outset by the work Benjamin and Warburg, are the elements central to a philosophical engagement with the work of art.\n\nPart II, 'Engaging Philosophy's Art', consists of a series of chapters, all of which are central to the project. While there has been an engagement with art and in certain cases with specific topics that are directly germane to a philosophical thinking of art in which matter and thus art's material presence need to be taken up, and this is particularly the case once a concern with colour orientates the approach taken, an engagement with both the specificity of art and art's materiality rarely occurs. Hence, the strategy of chapters 4, 5, 6 and 7 is to show in what way, and this despite their enormous value, the writings of Heidegger, Derrida, Lyotard and Sallis advance a philosophical thinking of art that remains indifferent to the work of materials. Part of what arises as a result is the impossibility of thinking about the particularity of art. In other words, they remain indifferent to what will be developed throughout this book in terms of mattering. While there are limits within the way philosophy has engaged the work of art, chapter 8 argues that there is a great deal to be recovered from Herder's writings on sculpture for the development of a materialist philosophy of art. In Herder, and in the way that Herder deploys Diderot, there are genuine intimations of a concern with materials and the way that materialism should not be reduced to empiricism. Herder's concern with 'touch' allows his philosophical project to be positioned beyond the opposition, between idealism and empiricism, a concern in affinity with a number of the arguments presented in the chapters in part I.\n\n## Notes\n\n1.\n\nHence this book needs to be seen as an attempt to further the project that has its clearest expression in my _Towards a Relational Ontology: Philosophy's Other Possibilities_ (Albany: SUNY Press, 2015). The latter is itself the development and refinement of a project first identified in my _The Plural Event_ (London: Routledge, 1993). A further extension of the development of such a conception of the ontological, this time in regard to a concern with the interconnection of ethical and the ontological, is forthcoming in my _Virtue in Being_ (Albany: SUNY Press, 2016).\n\n2.\n\nGeneralizations bring with them attendant risks. There is a vast range of work written within the broad domain that could be identified with the philosophy of art. Clearly there will be points of overlap between this project and other bodies of writing. Equally there will radical points of separation. The second part of this book involves an engagement with the aspects of the work of John Sallis, Jean-Fran\u00e7ois Lyotard, Jacques Derrida and Martin Heidegger. In every instance the attempt has been both to uncover limits while at the same time deploying the nature of those limitations productively (i.e., to deploy them in the furtherance of the development of a materialist philosophy of art). Clearly what is sidelined in the process is a more exhaustive engagement with the work of other philosophers. Thinkers such as Noel Carroll, John Hyman, Peter Osborne, Christine Buci-Glucksman and Christine Batteresby have all produced work of sustained importance. However, both the limitations of space and the necessity to develop a project that is not defined simply in terms of identifying limits has necessitated inevitable restrictions.\n\n3.\n\nAll references to Walter Benjamin are to the German and English editions: _Gesammelte Schriften_ , eds. Rolf Tiedemann and Herman Schweppeng\u00e4user (Frankfurt: Suhrkamp Verlag, 1980); and _Selected Writings_ , 4 vols, eds. Marcus Bullock, Michael W. Jennings, Howard Eiland and Gary Smith (Cambridge, MA: Harvard University Press, 1996\u20132003). Henceforth the reference will be GS and SW. Reference to _The Arcades Project_ is to Walter Benjamin, _The Arcades Project_ , ed. Rolf Tiedemann, trans. Howard Eiland and Kevin McLaughlin (Cambridge, MA: Harvard University Press, 1999)\u2014henceforth AP.\n\n4.\n\nAby Warburg, _Der Bilderatlas Mnemosyne_ (Berlin: Akademie Verlag, 2003).\n\n5.\n\nWhile its detail cannot be pursued here, the claims concerning original difference and thus a founding irreducibility have to be seen as pitted against the conception of the ontological that inheres in the work of Deleuze. Deleuze argues the following: 'To say that being is univocal signifies not that being is said in only one way, but that it is said in one and the same way in all its individuating differences.' ( _Dire_ _de l'\u00eatre qu'il est univoque signifie non pas que l'\u00eatre se dise en un seul sens, mais qu'il se dise, en un seul et m\u00eame sens, de toutes ses diff\u00e9rences individuantes_.) Gilles Deleuze, _Diff\u00e9rence et R\u00e9p\u00e9tition_ (Paris: PUF, 2000), 119. The counter here involves the argument that that understanding time in terms of continuity and then in terms of interruption and discontinuity marks the presences of an irreconcilable divide that demands to be thought ontologically. Hence, _pace_ Deleuze, anoriginal ontological difference demands the presence of a relational ontology.\n\n6.\n\nBenjamin, GS I, 669\/SW I, 172.\n\n7.\n\nBenjamin, GS I, 20\/SW I, 122.\n\n8.\n\nFor a more detailed discussion of this term, see _Towards a Relational Ontology._\n\n9.\n\nWhile it is neither avowedly Warburgian or Benjaminian in orientation, what is clear from the work of Clare Farrago, for example, are the productive possibilities opened up once geographical determinations are resisted and questions of relationality occasioned. See her 'Reframing the Renaissance Problem Today: Developing a Pluralistic Historical Vision', in _Crossing Cultures: Conflict\/Migration\/Convergence_ , 32nd Congress of the International Committee of the History of Art, ed. Jaynie Anderson (Melbourne: University of Melbourne Press, 2009), 227\u2013232; and 'Thinking beyond the Baroque', in _Reframing the Baroque_ , ed. Helen Hills (Aldershot: Ashgate Press, 2011), 99\u2013122.\n\n10.\n\nWalter Benjamin, _Das Passagen-Werk,_ [N7, 7].\n\n11.\n\nThe formulation first occurs in the work of Peter Osborne. His book _The Politics of Time_ (London: Verso, 2011) endures as the most significant engagement with the ways divisions within an understanding of historical time are always overdetermined insofar as they involve differing and inevitably conflictual ethical and political commitments. While this use of the term here and elsewhere\u2014for example in my _Working with Walter Benjamin: Recovering a Political Philosophy_ (Edinburgh: Edinburgh University Press, 2013)\u2014refers to but differentiates itself from Osborne's use, it remains a term that is essential to an understanding of what Walter Benjamin means by the 'present instant' and thus to any understanding of modernity.\n\n12.\n\nHenri Focillon, _Vie des forms_ (Paris, 1934), 94, cited in GS, I, 1230. Benjamin's relation to Focillon is of course ambivalent. There are important critical comments in _The Arcades Project_. See, for example, AP, 488.\n\n13.\n\nBenjamin, GS VI, 609\/AP, 486.\n\n14.\n\nBenjamin, SW IV, 390.\nI\n\n# Relationality and the   \nWork of Art\n\nChapter 1\n\n# Mattering, Particularity and the Work of Art\n\nThe work of art exists as a historical document. It yields and defines a determined context. Equally, the work of art can come to acquire a value. Values attributed to works often involve a hierarchy (one that can be either affirmed or denied). Equally, artworks can be attributed a significant place within estimations of a culture's strength. All of these different though related settings are attended by possible arguments that the work of art _matters_. And yet, within such settings, 'work' designates, more often than not, a completed entity. And, as it is complete, its completion would then be the precondition for the work's _mattering_. While, for example, a given work's iconography can be contextualized and the work's location within a given historical domain can be adumbrated such that a work's possibility then flows from either a certain conception of patronage or a specifically determined milieu that the work is then taken to evidence, such possibilities define in advance a specific conception of work. In all such instances what continues to insist is the work's already completed nature. Within this set of parameters, the work of art documents. As a result, works take on the quality of documents. That there is a work is the central point around which such understandings of the work of art would then be orientated. The aim of this chapter is the development of an argument whose intention is to show that such an approach, while important in its own right, fails to satisfy the requirements for a philosophy of art in which art is thought of in relation to history. Moreover, while works of art document, the complete identification of art's work with the process of documentation involves a set of arguments that are predicated on stemming art's inherent capacity to work beyond the hold of any determined context. The identification of these limitations needs to be understood as a philosophical claim about the nature of the art object as well as about art's history. The latter is the locus of philosophical thought and is determined by the relationship between the temporality proper to a philosophical thinking of art's history and the related conception of object and thus work of art. Indeed the entire project of this and the subsequent chapters of this book could be understood as the development of a philosophical thinking of art's history and the related conception of object and work of art.\n\nThe movement beyond the complete determination of work by context identifies the presence of a potentiality that is integral to art's work. Moreover, it is a potentiality whose release depends upon the processes of relationality. Here the hold of context is not being denied. In fact, forms of contextualism are integral to any understanding of work. The point, rather, is that once context does not determine meaning, then this demands an account first of how a work is able, continually, to mean beyond any context, and then, second, the further consequences that can be drawn from this state of affairs. Accounting for this potentiality both to address context while not being held by it necessitates a redefinition of what is entailed in any particular work of art being the work of art.[1] The presence of a potentiality of this nature, and the subsequent question of its release, identifies the presence of an opening within which the work of art remains incomplete. Here, an important qualification needs to be introduced since a further and fundamental part of the argument being developed here is that works of art are not incomplete in the sense that they await completion. It is, rather, that the 'incomplete' describes as much a state as it does an activity. Works of art work to incomplete. Incompletion therefore marks the impossibility of work being present as a self-completing finality where finality would be present as a form of reconciliation. With any work, what is seen or what will come to be seen is always more than that which would have been given at any specific point in time. Excess here, understood as an inherent quality of work, has the consequence of relativizing any sense of a 'period eye'.[2] The impossibility of completion therefore repositions finite determination as that which completes by incompleting. The claim concerning the overcoming of any context and thus any one work's refusal to be determined as a singularity by a singular relation to a singular context is not one made about a work's semantic content, even if the consequences pertain to meaning. It is a claim that concerns the ontology of art's work and thus what can be described as the work of art as the site of work. The term 'ontology' and its subsequent use in arguments to come identify a work being art, thus opening up as a genuine area of philosophical investigation the question of what is entailed by an object being a work of art. What a description of art that emphasizes work entails is that a work's iterability is a quality of a work. Iterability forms an integral component of the work of the work. While this has an impact on the attribution of meaning and thus on the practice of interpretation, it has to be noted that both meaning and interpretation are a result of work's capacity to sustain interpretive conflict within and as a result of the work's own iteration (a state of affairs that depends upon its already present iterability). Iterability here names potentiality. And yet, it should not be thought that such a proposition is without important precedents. A different but similar version of this position can be located, for example, in Derrida's arguments concerning the impossibility of context to delimit particularity, which, as a consequence, means that particularity has to be thought in terms of iterability.[3] Derrida will argue that a context does not 'saturate' meaning.\n\nEven though Derrida's own arguments are of fundamental importance, they are essentially limited since, in the end, they privilege meaning. There is a systematic failure within them to locate iterability within what is proper to a work (object) being what it is. In other words, and consistent with Derrida's overall mode of argumentation, there is an effacing of the ontological in favour of meaning.[4] (It is as though 'ontology' resists any form of reiteration that would have freed it from the determinations of traditional metaphysics. Ontology, within such a characterized position, metaphysics as caricature, must always be the same as itself.) Moving on from this restriction means that there needs to be a return to work and thus a return to a conception of the object as a site of work. Precisely because as authors they stage the issue with such clarity, a way into how this position is to be understood can be found in Alexander Nagel's and Lorenzo Pericolo's definition of another art-historical project that is linked to what they term the 'aporetic'. They argue the 'goal' of an 'aporetic methodology' is to understand within a 'work' 'how its contradictions and non-resolutions participate in its identity even as they render a fixed identity questionable'.[5] This approach accords in important ways with the opening position outlined here. Nagel and Pericolo underscore the centrality of the argument in which claims that are made in relation to meaning and restricted to meaning need to be rethought in terms of claims about the way works work. And yet, there is still a need to go further. What remains a problem is the place of any one work's relation to other works. Part of what endures as a question is not the presence of such relations. They can be assumed. Rather, what continues to be a problem is different. It has two specific forms. First, it pertains to the way constitutive elements _within_ a work's work are to be understood in terms of relationality such that relation becomes a question rather than an assumption. Second, the question of relationality and thus the constituting power of relations are integral to any understanding of the connection _between_ works. In sum, the question of relationality involves both internal and external concerns. As a result, their interconnection is central.\n\nFrom the position sketched above concerning both the limits of context and the emergence of relationality, it follows that the identification of the work of art with a document is only one aspect of a more general impediment that stands in the way of developing a philosophical thinking of art in which art's history also has to be thought about philosophically. Philosophical writing on art often occurs as though art does not have a history, as though there is no need for philosophy to engage with art's history. Rather than accept this limit, there needs to be an extension of the philosophical that opens up the space for an inscription of this element. For this to occur, art's history needs to be understood as a question for philosophy rather than as automatically incorporable into the philosophical.[6] The result of the absence of any consideration of either history or materiality, where both are thought philosophically, is that artworks are often only present as examples. This is clearly the case in Heidegger's use of Vincent van Gogh in _The Origin of the Work of Art_. There are two elements involved in the way that Van Gogh figures in that text. Heidegger writes in the first instance, 'We choose as example ( _Wie w\u00e4hlen_ _als Beispeil_ ) a common sort of equipment\u2014a pair of peasant shoes.' The second is, 'We shall choose ( _Wir w\u00e4hlen_ ) a well known painting by Van Gogh.'[7] Ensuing from the presence of art as an 'example' is that the materiality and the work's specificity then lose any possible centrality. It works as an 'example' not as art's work. (This limitation also characterizes Derrida's approach to art, a position taken up in considerable detail in chapter 5.)\n\nOnce the work of art comes to evidence a historical period where this is taken as an end in itself, or it becomes evidence of the presence of a specific affective register or, finally, it evidences the philosophical, though at the cost of art's own material presence, then there are two related negative consequences. The first is the systematic failure to think of the particular work as the site of work. The second is the inability to think of its relation to 'history', which, instead of being simply assumed, needs to become a topos demanding philosophical engagement. The response to these limitations arising from the misidentification of the locus of art's work or its reduction to one single register, both possibilities that resist art's work, is not to be found in a simple historicization of philosophy's encounter with the work of art. The equation of the work with either a historical document or an affective domain, or with its presence as an example that is extra-philosophical, succumbs to the same type of limitation. For the most part, as suggested at the outset, those limitations centre on the assumption of completion and then on the conception of singularity or particularity that figures within this assumption. Contrary then to the insistence on the interplay of the document, the example, the affective domain and the already complete, the emphasis here will be significantly different. Work, in being recast, needs to be understood as an activity. Once work designates the activity of the work of art, then the organizing question concerns the way in which a work of art works as art. Allowing for work reopens as much the question of philosophy's relation to the necessity of art's history as it puts into question the assumed meaning of the constituent terms 'art' and 'history'. Furthermore, what then has to return is how the affective is understood when it is incorporated into work. There cannot be mere affect.\n\nMoving a philosophy of art away from a concern delimited by exemplarity, the giveness of the affective and the epistemological towards one defined by activity and thus by the work's work, involves a reconsideration of the nature of the art object and how the truth content of an object is to be recovered. Truth here does not refer to content, as though a concern with meaning could be extracted from modes of presentation. Nor does truth refer to the possibility that artworks present arguments and that these arguments could be reformulated propositionally. Again, the realization of such an approach would be premised on the necessary effacing of art's materiality as that through which the 'propositional' was enacted. As was already suggested, such a possibility would deny the workful nature of the work insofar as it would deny that the artwork's self-realization as art was, at the very minimum, inextricably bound up with its material presence. Truth here pertains to what it means for an object to be the work of art. Truth therefore has an ontological rather than a narrowly epistemological register. Work occurs within and as the work's material presence. Consistent with centrality of activity is the claim that the work of art, rather than having a complete and already determined form, is in a continual state of self-realization.[8] Hence, it becomes possible to attribute to the work, precisely because it is the site of work eschewing the possibility of completion, an economy. Being is always already at work. The economic maintains the work's incompletion. Incompletion becomes a form of finitude. Indeed, it is only in being incomplete, by incompleting, that a work is able to continue to incomplete. Incompletion, as will be suggested, is linked to the continuity established by the process of _coming-into-relation_.\n\nThe initial question to be addressed therefore concerns the relationship between the presence of an economy and the work of art. It is simply not sufficient to argue that the work is always economic since works are active and that, as a result, an account of art involves a dynamic formalism rather than the static presence of the iconographic or mere colour. What has to be brought into consideration is the presence of that work\u2014and thus any one work\u2014as the work of art and thus as _art's work_. To reiterate the point that has already been made, the term 'work' is central to the argumentation of this book. While it is taken up in more detail, fundamental to 'work' is its twofold nature. First, it is present as a substantive. That is, it is possible to identify a work of art. Second, the 'work' has a verbal quality. As a result, it is possible to ask how a work works as art. This doubled quality will continue to return in a variety of formulations. At every stage what is being identified, in relation to any one particular work of art, is the question of how it works as art: hence, the concern will always be with the work's work.\n\nThere is therefore a need to maintain the particularity of art while allowing the question of the nature of the particular instance and the insistence of art's work to continue. While this brings the question of particularity into play, a question to which it will be essential to return, the project at hand concerns a philosophical thinking of the object that opens up the possibility of thinking the particular work in terms of activity on the one hand and its being as art on the other. If there is a methodological imperative that orientates the approach to the work of art that is being thought through here, then it can be located in the question, how does the work\u2014any one work\u2014work as art? The question has a precise determination since what is being asked is the question of how a given work, thus a particular work, works as art. The question therefore frames the insistent presence of singularity or particularity.\n\nAs a consequence of holding to the centrality of the question of how the work works as art, a different task emerges. What needs to be developed is an account of work in which art's material presence is the site of work. Here the work of art, as a site of work, refers therefore both to itself and, in a complex way, beyond itself, or more accurately it is referred to beyond itself. There is a doubled sense of movement. It is the 'beyond', the ineliminability of this beyond, that has to be taken up. Such a project entails the introduction of a sense of relationality. Conceding the presence of a 'beyond' means that what then becomes important is the way work is understood in terms of a singular entity whose singularity is constituted by both internal and external relations. At work therefore is a singular object whose presence as a singularity is always as an after-effect of a network of relations. It is this network that locates and defines differing loci and modalities of incompletion. Relationality therefore, while addressing internality, also pertains to the way a work's work comes to the fore as a result of a relation to other works whose externality allows for the content of any particular work to attain and continue to attain new and different understandings. This is the discontinuous continuity of incompletion. It arises with relationality, and it is within the context created by the centrality of differing forms of relationality that it becomes possible to deepen a materialist approach to art's work.\n\nMaterialism provides the _countermeasure_ to empiricism and idealism insofar as it cannot be recuperated by either and yet does not deny that which is viable in each.[9] (As will emerge, the countermeasure to idealism, within the structure of this argument, is the 'immanent'.) An account of materialism, which in this context means, as a beginning, the work of materials, is integral to the sustained development of a materialist philosophy of art. Materials are at work. Empiricism merely notes their presence but cannot think the activity of materials. Hence, the claim is that materials occur within and as the activity that is art's work. (Moreover, once a quality that attributes more to the work's empirical presence than mere inert matter, what is then introduced is a realm of non-necessity insofar is concerns as the attribution of meaning to that work.) An insistence on materialism, an insistence that privileges the actative over the locative, gives rise to a concern with what will be described as the work's _mattering_. _Mattering_ , as the term is both deployed and worked out here and in the chapters to come, concerns the co-presence of the ideational and the material.\n\nThe term 'ideational' identifies the _content_ within what would otherwise be the empirical registration of the material. There are only ever instances of the ideational. It may be the symbolic dimension of colour; it may be the presence of animals that stages the complex distinction between the tamed and wild; it may be that the painting of a set of eyes in one work occurs in a way that, while it conveys a sense of seeing, that sense is not replicated in the painted presentation of another set of eyes in the same work. Seeing while present resists any generalizable account. (It is precisely possibility that is central to the works by Nicolas Poussin discussed at a later stage.) These differing accounts of seeing are ideational. Different understandings of what seeing is are at work. Possibilities proliferate. There is an inexhaustibility that accompanies the inexhaustibility of possibilities within arts' continual presentation. The ideational is an essential part of the immaterial content that materials stage, and in staging that content, the material is positioned beyond any possible reduction to the merely empirical. Matter is always informed. There can never be just form. Hence the material and the ideational do not comprise an opposition. The contrary is the case. There is an already present interarticulation of the ideational and the material. As indicated, this is a set-up that marks the presence of an economy at work, a complex economy that defines and delimits the work's work. As a result, meaning is only ever the after-effect of a work's mattering. Moreover, meaning may itself be the site of a set irreconcilable tensions, the viewing of which is the occasion of the work's being seen. This gives rise to a related point whose very possibility repositions the connection between time, iteration and meaning, namely that what can be seen 'now' could always be that which could not have been seen at the time of the work's creation. This possibility opens up the need to account for the release of a potentiality. This release will always be a result of the unpredictable nature of what has already been identified as coming-into-relation. This point will return later as a fundamental component within a brief engagement with one of the frescoes comprising Piero della Francesca's cycle _The Legend of the True Cross_ in Arezzo.\n\nThe next stage of the argument involves a return to 'art' now that it has been repositioned within the formulation 'the work of art'. The point of departure is that art, in the different ways the presentation of art takes place, has always evinced a concern with its own material presence. The latter is an integral part of the work's presence as art. Moreover, that concern is legible in artworks, as well as in the writings that accompany art's production. What is legible not only has a history, but that history consists of radically distinct forms of presentation and thus radically different conceptions of art's material presence. The latter\u2014art's material presence, art's work, its mattering\u2014has differing dimensions. One, which is often fundamental to a work's work, is what will be presented here in terms of _art's self-staging_.[10] As a beginning it can be suggested that self-staging has two particular loci of concern. The first is the way art stages itself and draws attention to itself as art, the way a work is present\u2014and presents itself\u2014as art. This may be no more than a moment within a work: the drawing held by the Venetian woman in Lorenzo Lotto's _Venetian Woman in the Guise of Lucretia_ (1553) inscribes work within and as part of a work. Equally, the unresolved hand in \u00c9douard Manet's 1878 _Self-Portrait_ opens up the question of resolution. Incompletion is literally staged and thus calls attention to the work as part of its work and as integral to its presence as art.[11]\n\nThe second sense of self-staging concerns the way that staging is present in a particular work and then the way it is essential to the relationality that constitutes a work's particularity. This second is the more complex. It works with the already noted assumption that a particular is an after-effect of the network of relations that constitute it. The particular can be understood therefore as a knot that ties together threads that constitute the relations, the effect of which is the particular. Those relations change and develop; some threads drawn into the work disappear while new ones are added. Fundamental to the argument here is that threads will include institutions, writings and other works of art. (The latter holds open the possibility that the force of the work emerges as the result of a decontextualizing juxtaposition with other works. This is the constitution of identity as a result of relationality. As such, it is another dimension of the claim that a singularity is always an after-effect.[12]) However, the substantive point at this stage is that the work's work cannot be equated in a definitive way with any one thread (or preclude the emergence of new threads). There is no one relation\u2014understanding the posited determining finality of context as a form of relation\u2014that fixes the work. Moreover, what has to be allowed, precisely as a condition of a work being the work of art, is that other possible relations are always present _in potentia_. What this means is that this second aspect can also involve writings that accompany art's self-staging to the extent that they also comprise threads that can be drawn into the work and thus are, in certain instances, integral to effecting the particular's construction (and of course potential reconstruction) as a particular. Finally, self-staging is a term that needs to be given a type of generality insofar as it refers as much to differing forms that the work of art may have, as it does to the presentation of art within specific discursive practices. As has been suggested, self-staging concerns attempts within art's own production to stage specific conceptions of self-presentation; in addition, writings on art that are or can be made contemporaneous with art's work cannot be separated from questions of presentation as well as their effect on the role of materials within (and as) that presentation.\n\nPosed at a general philosophical level, what is at work here is the question of the particular. Once it can be argued that any one particular is an after-effect of the relations that constitute it, then what constitutes particularity cannot have been determined in advance. Relationality is both there and there to be established. It is never merely given. As already noted, relationality is both internal to any one particular\u2014for example, in regards to the work of art the relation between its material elements, in sum its materiality\u2014and equally connected to external relations that also bear on the particular's constitution as a particular. What can never be excluded are those additional elements the after-effect of which is the specific particular that is there within a new work (as its after-effect) and thus subject continually to the reconstitution of its particularity. As intimated above, the external elements may include relations to other artworks, both past and future (hence undoing the identification of historical time with chronology), as well as works that while not being works of art can nonetheless be deployed effectively as that which is integral\u2014in any one instance\u2014to what constitutes a given particular's particularity. In sum, within a relational ontology, art's self-presentation is a particular within a network of relations in which, as a more general claim, the identity of any one relation is always indeterminate insofar as it is always there to be established (again the necessity of incompletion.)\n\nIn broader terms, it is essential to note that what is attested to by the presence of differing conceptions of art's presentation is the presence of three interrelated elements whose occurrence devolves into questions. In the first instance, it pertains to work as the site of continual interpretive conflict. The second is that what cannot be separated from the presence of work as a site of interpretive conflict is what has already been identified as self-staging. There is an important additional point that needs to be made here. Despite the presence of self-staging, it neither provides the ground for the resolution of conflict nor makes conflict's identification otiose. There is no one account of the presentation of conflict. Unanimity is absent. The third element is the devolution of the other two just noted into a guiding question. It concerns how that absence is itself to be thought. In other words, the additional question, and thus the one addressed continually throughout these considerations, is how the registration of the ineliminability of a complex economy and a conception of the differing modes of art's presentation as that site, a complex economy that obtains _ab initio_ , is itself to be understood. At work here are concerns that are directly philosophical, though equally they also obtain within the practice of interpretation.\n\nHow is the presence of a complex economy to be thought? Once this is allowed as a question, it means that terms such as 'mimesis', 'presentation', 'appearance', 'figuration' and the like, all understood as accounting for work's presentation, cannot be differentiated from this overall site, a site to be understood as an economy defined by difference and conflicts rather than the assumption of unanimity (be it ontological or semantic). (Indeed, the site forms these terms.) That there is appearance, in other words, that art appears and _is_ in its appearing, cannot be separated either from the history of appearance nor from the inscription of conflict and irreducibility at the centre of that history. The results of the presence of a complex economy pertains as much to interpretation as it does to differing senses of propriety that are brought to the object. The third element is the nature of art's presence. What are present, what appears, are materials at work. The contention guiding the chapters to come is that an engagement with the interplay of history and original irreducibility, whose continual presentation is the work of materials, is integral to any project that seeks to set the measure for a philosophical encounter with art. Consistent therefore with the self-definition of a project in terms of the centrality of particularity is the claim that self-staging can only ever be present in different ways. (That there is no one presentation adequate to self-staging does not mean the questions of what self-staging is or can be endure as unanswered. It is rather that the proliferation of possibilities refers to the idea of self-staging without having to be thought of as its representation.) There are only particulars. Reference within the chapters to come will be made to a range of artworks, including works by the following: Poussin, Piero della Francesca, Karel Appel, Georges Baselitz, Jean-Michel Atlan, Anotnin Artaud, Andrea del Sarto, Raphael, Francesco Moscha, Gian Lorenzo Bernini. The movement is between particulars and thus the way networks are created and sustained. Beginnings therefore must always be particular\u2014 beginning with particulars.\n\nIn Poussin's _Cephalus and Aurora_ (1627\u20131630)\u2014a painting whose narrative content is to be found, in part, in Ovid's _Metamorphoses_ Book VII, though equally, it has been argued, in Pierre de Ronsard's _Le ravissement de Cephale_ \u2014Aurora's attempt to seduce Cephalus is undone by the latter's continual recall of his wife, Procris.[13] (See figure 1.1.) The textual sources have their own life. Their own internal preoccupations are neither clear nor obscure. As texts, _Metamorphoses_ and _Le ravissement de Cephale_ make their own interpretive demands. They make them, however, as texts. (How could they not?) Yet more is at work in the work of painting than mere narrative. This is not the painting of a literary narrative regardless of the source. Equally, from within the domain of painting, there are stylistic references to Titian, among others, insofar as the positioning of figures is concerned. In other words, there are clear modes of appropriation and the formal presence of influence. Poussin did not work alone. Nor does his work. The painting engages with the presentation of memory within and as art's work. While that seeing occurs within the recall of myths, what is staged is, nonetheless, a relation between painting's presentation of myth and the way the same work stages differing and in the end irreducible forms of seeing. Memory itself, insofar as this work's work is concerned, becomes a type of seeing. With the painting, as its work, there are different determinations of seeing.[14]\n\nNicholas Poussin, _Cephalus and Aurora_ (1627\u20131630). National Gallery, London.\n\nA painting of Procris is held before Cephalus\u2014a painting within a painting. Within the painting as a whole, what is seen is that he sees Procris without seeing Aurora. Viewing this works means that what is seen is a process of a seeing that is equally and at the same time a non-seeing. Self-staging is not just the presence of the painting within the painting. It is there in the paintings own self-engagement with seeing; thus as part of its work. Cephalus' all-encompassing seeing is the seeing that is undone by the way the painting works as a site to be seen. The key point is that the viewer cannot occupy the space of Cephalus. The viewer sees the limitation of seeing and thus the link between seeing and the (im)possibility of desire.[15] The viewer knows that Cephalus is held by a moment that demands both a remembering and a forgetting (the latter will always be present within the remembered necessity to forget). As a result, Poussin's painting stages, firstly, the inherently incomplete nature of seeing, incomplete in the precise sense that the work stages different incompatible modes of seeing whose formal coherence reinforces the way the work works to incomplete, and then secondly the impossibility of separating, in the context of this work, seeing from touch.[16]\n\nAurora not only holds Cephalus, his left leg is between hers and she is pressing against him. Pulling him towards her while pressing her groin against his thigh. The viewer confronts an eroticism in which what is seen has force precisely because it could only have been felt and thus not seen. Cephalus continues to see, even though his arm and hand adopt a position that is inevitably associated with the proposition _noli me tangere_ , a proposition that has no one presentation. Forbidding here takes on a demanding quality; insisting and asking. The link with Christ's refusal is present through denial since the forbidding of touch is occurring at the centre of touching. Within the tradition of such paintings Christ, while refusing touch does so with a hand gesture that Poussin's painting recalls, looks at Mary. (A position staged by a range of different works, though emphatically present in Angelo Bronzino's _Noli me Tangere_ [1561].) In Poussin's painting Cephalus looks while refusing touch. Looking at her he refuses her touch. Here the complexity of the tradition of _noli me tangere_ paintings is recalled in a way that reworks the defining bodily elements within it. There can be no radical separation of touch and not touching. Forbidding and enticing, deny, allowing and forcing all become differing and irreconcilable modalities of touch. What also has to be noted is Cephalus' hand. It is as though while being touched the hand evokes at the limit of (im)possibility a form the _noli me tangere_ that can be taken as a defining element in the history of touch. The resistance to touch which demands of course touch's literalization is undone by the touch that is noticed while being refused\n\nLooking at the frame(s), seeing Cephalus seeing is to see the impossibility of a singular view within what is ostensibly a singular view (i.e., the latter is that which suggests that the painting could be viewed, were its complexity to be stilled, as an overall singularity). Two incompatible modalities of seeing are already in play. There is a division within the ideational that obtains from the start. Her left leg and his right leg, the white and yellow folds of clothing, clothing that is only ever falling away from their bodies, gradation within colour achieving this effect, have created a meld in which a separation, while possible, would only be a moment gestured at within a set of processes. Legs and the folds of drapery lap and overlap each other. They are present as the work's own check to the space\u2014also present and created within the frame\u2014that has been opened between Cephalus's eyes and the canvas held by the putto. A spacing that opens at the focal point is closed within the folds of legs and draped clothing. These elements are integral both to the way in which the work matters and to what has been identified as self-staging. The differing forms of seeing and touching, both of which are viewed without the viewer being able to reconcile them, when taken together, identify the component elements of a complex economy. The impossibility of reconciliation, an impossibility that retains, nonetheless, formal coherence, within conciliation's impossibility forms a defining part of the work's work.\n\nThe work, understood as internal sets of relations, therefore continues to incomplete. Incompletion cannot be described. Were it to be, then the work would have become a simple empirical manifestation and as a result would have lost its work-like nature. As suggested at the outset, works work to incomplete. Incompletion here is sustained by the ideational, the latter's already fractured presence. Different and non-synthesisable conceptions of seeing continue to resist reconciliation despite the presence of what has been described as formal conciliation. In fact, it can be argued that since formal conciliation and sameness relate to the size and proportion of bodies\u2014and thus the presence of the same bodies in the same space, at the same time, and so on\u2014both conciliation and the irreconcilable pertain at the same time. The work becomes a site whose complex internal relationality is held in place by the temporality of _at-the-same-timeness_. All positions are the work of colour and of painting. All are materially present. Complexity here has to be understood then as the mark of a founding irreconcilability and can be understood as a play of forces that resists an overall synthesis. In other words, at work here is an economy rendering impossible either narrative completion or any form of simple unity other than one which would be given by an act of description in which all the described elements would have been given, or would have acquired, a similar status. Such a description, while accurate on one level, would have failed to capture the relationship between simple appearances (the work as described within an empiricist as opposed to an materialist account of art) and the role of a complex economy within the frame. Poussin's painting therefore allows for indications of what is at play when, rather than understanding the painting as an already determined entity that has to be described\u2014a project in which writing on art succumbs to the ekphrastic impulse\u2014cedes its place to an approach that is orientated around work and in which the original presence of a complex economy constructs the work as a site or a form of irreducibility that is already present (a semantic effect dependent on the ontology of the work).[17] As such, the work is present as a plural event, the latter as the particularization of a complexity that is anoriginal and which is defined continually in terms of at-the-same-timeness.[18] The further significant point here is that temporal simultaneity; that is, at-the-same-timeness does not entail ideational simultaneity.\n\nWhile it would always be possible to pursue further specific paintings (or other works of art) and continue a project that insists upon particularity with the continual encounter with given particulars, what has to be addressed are the questions that are already at work within a concern with the presence of particular works when they are understood as the setting in which materials are at work. As has already been intimated, the work of materials is another way of identifying elements essential to art's self-staging. Beginning with painting, what is of genuine interest is how the material presence of painting informs accounts of art's presentation. And yet, a far more preliminary question is necessary, a question the response to which cannot be just assumed (and this despite the already present encounter with Poussin noted above; indeed, it might be added that this encounter prepares the way for this now preliminary question, if only because questions still remain concerning how self-staging occurs). The question is the following: What counts as the material presence of painting? There is an initial answer, one to which allusion has already been made and which haunts attempts to develop a workful sense of material presence. It is the answer that involves that form of reduction in which art's material presence becomes its empirical presence.\n\nWere this to have been accepted as the point of departure, it would have meant that a description of materials and the way they are used to create certain effects would provide an account of art's work since the material would have been equated with the empirical. If this were an exhaustive account, then art's work would be no more than its materials, though where, as noted, the latter had become no more than an empirical presence. Materials, as material elements within ontology and an epistemology structured by empiricism, have a history, and thus the use of clearly identified materials at a certain conjuncture evidences that history. And yet, there is an attendant problem that both limits and delimits what is meant by the material in such a context. The problem arises due to the possibility that what informs material presence is itself not material. It is of course precisely this possibility that has already been suggested by a move to an insistence on work understood as an activity and therefore on artworks as involving the original presence of a complex economy, complexity as always already present and thus anoriginal. If this is the case, what then has to be taken up is how the mattering of materials is interarticulated with an immaterial force; hence, the mattering of art becomes the way in which immateriality as the presence of the ideational, and relations involving force cannot be differentiated from matter's work nor from the process of mattering. Precisely because they pertain at the same time, even though they are irreducible, one to the other, they are apart from each other while being a part of each other. What can be described as the _apart\/a part_ pertains at the same time. As has already emerged in the encounter with Poussin's painting, the temporality of at-the-same-timeness provides the basis of irreducibility. In addition, at-the-same-timeness contains a spatial dimension; there is an inscribed spacing at the centre of the temporality of at-the-same-timeness. Part of the argument to be developed here is that it is precisely this possibility that concerned, among others, Erwin Panofsky, Alois Riegl and Edgar Wind.[19] The development of a concern with what comes to be named as _Kunstwollen_ , _Stylfrage_ or in the case of Wind the 'autonomous formal impulse' can all be traced to a refusal to identify the material with the empirical. The dynamic processes in questions mean that mattering and the material (the conception of 'material' within a materialist philosophy of art) become the complex interarticulation of the material and the immaterial. Indeed, it is possible to go further and argue that the inherent complexity of their interrelation is precisely what allows for irreconcilability and thus the possibility that works always work to incomplete and that incompletion is a continuous process.\n\nPrior to any engagement with texts by Panofsky or Riegl, even though it is predominantly the work of the former that is central here, a number of introductory points need to be made. Clearly, what is involved in the framing of materials in terms of mattering and the consequent refusal to identify the material with the empirical, since the latter precludes the possibility that what is at work within art's work and as a part of any one work's mattering is the immaterial, is what might be described as the problem of expression or the problem of appearance, problems captured in the following questions: What is it that appears? What comes to be expressed?[20] The contention here is that there is a double movement within which what has to be given a form of priority are the questions of _how_ the expression occurs and thus what _allows_ for appearance. The 'how' and processes that 'allow', when taken together, bring art's material presence to the fore. They are that through which expression and appearance occur. What appears cannot be reduced to it, and yet what appears cannot be thought of other than that always already interarticulated with it. Once, however, the interrelated presence of what appears and the necessary materiality of its appearing are both central, then the work's work has to incorporate the way in which materials are themselves at work. The interconnection of the material and the immaterial in both forming and informing the work of art, and so the conception of what has already been identified as any one art work's complex economy, introduce the question of how this dynamic process is itself to be understood. The significance of the immaterial is that it captures the force of the ideational. Understanding mattering as the interarticulated presence of the material and the immaterial is to argue that matter is informed and the 'informing' cannot be separated from what is captured by the term 'ideational'. The point suggested above is that there are already present intimations of this array of possibilities in the thought of Panofsky and Riegl.\n\nWhile Panofsky's concern with Kunstwollen comes to be articulated in terms of iconography as his writings develop, the initial interest that the term identified can be located in an attempt to reposition art history such that it was no longer the expression, in the first place, of a simple idealism such that matter, and thus what mattered, was the presence of the 'soul' or 'spirit' within the material. The latter is a position whose most exacting formulation occurs in the words of Hegel and Schelling. Equally, his concerns were to distance any attempt to conflate the material with the empirical.[21] Indeed, what will be suggested is that in his papers of 1920 and 1925\u2014'The Concept of Artistic Volition' ( _Der Begriff des Kunstwollen_ ) and 'On the Relationship of Art History to Art Theory' ( _\u00dcber das Verh\u00e4ltnis der Kunstgeschichte zur Kunsttheorie_ )\u2014there is a genuine attempt to formulate what might be called, in order to be consistent with the argumentation presented above, a materialist account of art's presentation (materialist in its opposition to both empiricism and idealism).[22] The repeated point made by Panofsky in both texts is that what is central, and it is a position that he also attributes to Riegl, to the project of art history is the 'aim' of 'revealing meaning immanent ( _immanenten_ ) in artistic phenomena'.[23] There is a strong point of connection between the immanent and what is also identified in Panofsky's texts as the a priori. Hence, one of the inescapable questions that occurs here concerns what is meant by the 'immanent' and how to account for the presence of the immanent. The necessity to pose this question arises since the term 'immanent' brings both an account of presence and an account of content into play. Immanence allows them to be thought of together. Moreover, there is the additional point, which may steer these concerns on from Panofsky, that the immaterial, once an account of its presence becomes necessary, can only ever be immanent.\n\nWhat defines the a priori quality of art works that Panofsky privileges has to do, for example, with the way the relationship between form and volume works within a given work and the way the formal possibility of the resolution of their necessary relation involves time and space. Hence for Panofsky, if the antithesis between form and volume is the a priori precondition for the existence of artistic problems, then the interaction between time and space is the a priori precondition for their potential solution.[24] The force of this formulation is that what it opens up is the retention of a quality that takes intentionality away from the domain of the artist, thus pitting it against an assumed automatic pyschologization of artistic production and equally away, at least initially, from any straightforward essentialization of a historical period. Kunstwollen has the potential therefore to become a conception of the economic and to form the basis of an account of the effective presence of the immaterial within an artwork, precisely because the resolution of artistic problems, problems that are resolved by the work of Kunstwollen, are both realized and resolved materially. (In this way, a gesture is made towards a conception of the economic which is itself explicable in terms of the ontology of the work of art and therefore in terms of art's mattering.) In other words, the economic would become, as a consequence, a quality integral to any account of the ontology of the work of art. It is thus that in relation to what has been identified as the economic, which would be the presence of the immaterial, that Panofsky is then able to suggest, in relation to the concepts that form the basis of artistic problems (i.e., 'optic\/haptic', 'depth\/surface', 'fusing\/splitting') that they have an indeterminate presence within a given artwork. Hence, for Panofsky, they 'refer not to contrasts actually encountered within artistic reality but to contrasts out of which artistic reality generates _conciliation_ of some kind' (emphasis added).[25]\n\nIn regards to specific 'artistic problems' what this means is that the relationship between figures within a work\u2014and thus of relationality as a question within painting\u2014involves the optical which deals with connection, over the haptic which is concerned with individuation. What becomes of interest is the way that relationality operates as a resolution to the question of what the distinction between separation and connection (the optical and the haptic) makes possible. This determines the relationship to materials. As a consequence, Panofsky then goes on to argue that what he identifies as 'artistic productions' is not, and his own words here are decisive, 'the expression of subjects but the informing of materials, not the given events but the results'.[26] The formulation 'the informing of materials' ( _Formungen von Stoffen_ ) will re-emerge as central. It also needs to be noted that it could also be the case, though this is not Panofsky's own supposition, that what materials and the ideational in their interarticulation could stage is the necessary impossibility of any form of conciliation. In other words, the informing of matter may open up a lack of reconciliation other than the merely formal. What these other possibilities mean is that the intrusion of the ideational into the dynamic, thus forming part of the way in which Kunstwollen is present, indicates that 'conciliation' pertains to a formal arrangement of elements. A formal presence cannot preclude the possibility of the irreconcilable, precisely because the latter is as much mere presentation as it is ideational. Indeed, what Poussin's painting instantiates is the co-presence of formal reconcilability, namely the maintenance of sameness and the irreconcilable, insofar as the latter pertains to senses of seeing that refused any form of overall synthetic unity. They are held together, staged as the work's mattering. While interpretation could always emphasize the overcoming of impossibilities and tensions in certain cases, such a possibility always encounters its own impossibility. What this entails is that interpretation, once sustained by the works' work, in which work stages the impossibility of a founding ideational synthesis, becomes as a result a practice whose own possibility cannot be separated from the work's material presence\u2014mattering as the interarticulation of the material, and the immaterial becomes this possibility.\n\nIt is essential to take a particular site of engagement in order to pursue the informing of matter, leading to formed matter and thus opening the way to mattering, where the latter, as already suggested, is understood as the interarticulated presence of the ideational and the material. That site is provided here by colour. Indeed, a concern with colour is fundamental to the project of this book as a whole. If colour could exist independently of the ideational, then not only is empiricism possible, a possibility located in the separation of the empirical and the material, but a philosophy of art that concerned itself with nothing other than the affective dimension of colour would have also become plausible. The impossibility of the former and the implausibility of the latter demand that arguments be adduced, arguments to which the general conclusion would be that, despite the link between colour and affect, colour is only ever part of the work's mattering. There can never be just 'colouring'. Colour is always already informed. Hence, colour is not just there; there is no 'there is' of colour. On the contrary, colour matters.\n\nDespite the difference between Panofsky and Hegel, there is a productive affinity between the two. It can be established between the presence of colour in the development of Panofsky's argumentation, which would be the link between colour and Kunstwollen, and then the way colour also appears in Hegel's _Lectures on Fine Art._ Points of overlap and differentiation are, it will be argued, pertinent to the position played by colour in the development of mattering as a productive concept within a materialist philosophy of art. The differentiation between Panofsky and Hegel takes place on at least at two levels. The first occurs in relation to materials and thus the way both the work and the presence of colour are to be understood. The second relates to the presence of the immaterial and thus the question of how the immaterial is to be understood and its presence registered. Recalled here is of course what has already been identified as the immanence of the immaterial. In an exact formulation of the position of colour as that which is always secondary in relation to the resolution of 'artistic problems', Panofsky writes the following: 'Even concepts characterizing colour style describe nothing other than different solutions to the very same artistic problems, which (insofar as they are fundamental problems) are located wholly beyond the differentiation between colour and noncolour qualities but which receive their solution in the choice of colour as much as in bodily or spatial composition.'[27]\n\nAt work here is a more nuanced positioning of colour. Colour is not abstract. Rather, colour cannot be separated from its location within what Panofsky refers to as 'artistic problems'. Having made this point, Panofsky then goes on to locate, to take an example from his investigation, colour as already present within 'colourism' and 'shadedness', both of which, when conceived of as modes of presence, are integrated into, and thus only possible as part of, the resolution of 'artistic problems'. There is, however, a fundamental additional point, namely: 'Purely polychrome colour design is the necessary correlate to an isolating, surface-orientated and crystalline space and body composition insofar as it forms a solution to the fundamental problems, while coloristic specifically shaded, colour design is necessarily linked to unifying depth-orientated, and pictorial space and body composition.'[28]\n\nWhat is significant about this point is that it reinforces the position that colour is neither present in its own terms nor original. There is never just colour. Colour is positioned in relation to the force of Kunstwollen and thus as always there\u2014indeed, only there\u2014in relation to that which is 'immanent' in the work. Kunstwollen is, after all, to deploy Panofsky's own formulation, the presence of 'formal style that is immanent'. Hence, colour is at work in the formal elements that comprise the actual presentation of an artistic problem. Panofsky identifies the latter in terms of a work's 'sensible qualities'.[29] What has to be undertaken here is an approach to the sensible, to use Panofsky's term, that allows for its transformation.\n\nThe sensible needs to be held apart from its incorporation into empiricism. Empiricism reduces the work either to those sensible qualities or an affective response that assumes their giveness. Thus they figure as a complete presence (present as complete), such that an analysis of artwork, art's work, would then become no more than the provision of differing forms of description. What is described is the object as a completed and thus self-completing entity on the one hand, or, on the other, the affective (aesthetic) response to such an entity. Moreover, if this set-up then moves into the conceptual, it only does so by abstracting from the sensible and so generating concepts that assume, hold in place and finally are derived from the primacy of what Panofsky refers to as the 'sensible'. If there is to be an opening in which what occurs is a move from empiricism to materialism, then such a move retains the sensible. Its retention occurs insofar as the sensible would no longer be defined within the framework that would have taken it to be an end in itself. The further and indispensible point is that there would be a transformation of the sensible beyond the opposition between the sensible and the intelligible, a move already suggested in the refusal to identify the empirical with the material. Consequently, such a transformation is consistent with the move from the empirical as the simply given to the material as a locus of force. Implicit therefore in the way Panofsky formulates his position is a disjunctive relation between the sensible as the simply given (what would amount to the sensible within empiricism) and what the sensible actually is, which is the incorporation, thus the transformation of the sensible, within the actative; that is, the sensible then becomes the site of work, mattering. In other words, the sensible is then present in terms of its interarticulation with the ideational as the material. What this means is that mattering generates a complex domain that cannot be understood as engendering let alone enjoining any form of unity that would then be a necessary correlate to its presence. In sum, mattering eschews its inscription within the problematic of representation. Hence, what is crucial here and moreover what provides the decisive limit to Panofsky's approach is the way the relationship between the sensible and work is understood.\n\nWithin the context of the papers under consideration, art for Panofsky works through the 'organization of sensibility'. What enacts this 'organization' is of course the presence of what he identifies as 'artistic problems'. And yet, these 'problems' have a twofold determination. In the first instance they are abstract and thus external to the work of the frame even though they are worked out within it. As a consequence, their presence remains purely formal. Second, the way these problems are presented insists, within Panofsky's argumentation, on the unicity of the framed, and the presence of problems whose preconditions are present a priori cannot, by definition, involve either an economy marked by original complexity, and thus the possibility of finality as the incomplete, nor the necessity of a work's end as a formal irresolution. Both of these latter possibilities would remain open to the continuity of incompletion (hence the already problematic presence of the insistence on 'conciliation', especially when the latter is thought in terms of abstract formal conditions within the frame). Formal irresolution, an irresolution whose force is sustained, in part, because of what was refereed to earlier as at-the-same-timeness, is the precise state of affairs staged in Poussin's _Cephalus and Aurora_. The problematic status of this twofold determination\u2014abstraction and the necessity of unity and completion\u2014becomes clearer when they are positioned in relation to a further claim made by Panofsky in terms of an expansion of what is intended by the use of the term Kunstwollen. To recall the detail of the passage already noted, Panofsky is arguing that:\n\n> we must be able to characterize artistic volition or artistic intention ( _Kunstwollen_ ) . . . as a concept which can be described immediately from every artistic phenomenon, no matter how limited, whether it be the total creation of a period, a people, or a particular region, the oeuvre of a particularor finally any work of art at all. Such a concept defines the phenomenal characteristics of the work in question not generally as discovered by abstraction, but fundamentally, revealing the work's immanent meaning and laying bear the actual root of its being.[30]\n\nThere are two aspects of this identification of Kunstwollen that are fundamental here. The first is the reiteration of the identification of 'immanent meaning'. This sense of the 'immanent' works to recall what Panofsky has already identified in terms of the a priori. From this it is clear that for Panofsky the circumvention of both empiricism and idealism means that he understands the presence of an immaterial force and an economy within a given work of art in terms of a posited distinction between the a priori and the a posteriori (the latter is the 'sensible' present as that which has been organized). This occurred, however, without there being any concomitant attempt to think through the necessity of the way that the a priori and the a posteriori pertain at the same time. This is what is at stake in any use of the term 'immanent'. The immanent, to the extent that it is thought in relation to appearance where what appears is at the very minimum the predominating quality of the sensible, brings a specific temporal register into play. What is immanent is always there at the same time as that which is sensible. And yet, neither one can be reduced to the other. Allowing both their distinction and for the relation to be effective is the temporality of at-the-same-timeness _._ This is a conception of time that has an inherently spatial dimension. Spatiality here, though this is a position that needs sustained demonstration, is part of what sustains the complexity that a certain interpretation of art's always already inherent formalism tries to exclude.\n\nThe temporality of at-the-same-timeness is demanded by the simultaneity and yet the irreducibility of the immanent and material. Moreover, it is this at-the-same-timeness that begins to define the work that is crucial to understanding the way in which mattering twists free from any reduction of work to the giveness of sensibility and consequently it is precisely this form of temporality that is implicit in the construction of any one work as a plural event. However, it needs to be noted what this also accounts for is the presence of the risk that is inherent in any interpretive practice. In sum, that risk is the ever-present possibility of defining the work in terms of formal and hence a possible semantic conciliation. It should be added to this that within such a setting, thus as a result of the realization of the risk, art would then be understood in terms that were premised on the abnegation of its workful nature.\n\nThe next aspect in the explanation of the function of Kunstwollen in this context that has to be taken up concerns the identification of Kunstwollen as internal to a work, though equally it has a possible, perhaps necessary, externalization. In regards to the latter it is clear that Panofsky is using the term Kunstwollen in a way that attempts to escape the conception of idealism in which Spirit ( _Geist_ , _Seele_ ) allows the particular work to attain the identity that any one given work may have. Nonetheless, he wants to argue for the possibility that Kunstwollen could equate to a broader sense of the historical such that a singular Kunstwollen and a singular style could identify a given art-historical period. Hence, Kunstwollen thus construed would have organized the frame in advance. In addition, it would assume a form of unanimity within any historical period, which would then show itself within the frame. This is, after all, the sense of Kunstwollen that is also at work in Riegl.[31]\n\nThe response that needs to be made to the above, a response that will reiterate part of what has already emerged, is to argue that Panofsky is correct in his refusal to identify a work's sensible qualities with the totality of the work. There is always a fundamental aspect of work that exceeds the 'sensible'. As he argues in 'On the Relationship of Art History to Art Theory', an inherent restriction emerges were the sensible to be taken as delimiting the locus of interpretation: 'Taken on its own a classification that only captures the sensible quality of that at work has no right whatever to draw any fundamental difference between those lines that were produced by the artist's hands and those lines that appear because of the cracks in the varnish.'[32]\n\nThere is an important rigour at work here. Not only has Panofsky pointed to the impossibility within description, were it to be taken as an end in itself, he has opened up the space in which there is the necessity to think of the ineliminable presence of the immaterial as integral to any account of the way art works. And yet, his resolution to the impossibility of empiricism involves the identification of a priori principles as operative within a work and whose realization is only ever formal. Indeed, he goes so far as to argue that Kunstwollen 'can only be grasped by an interpretation of phenomena which proceeds from a priori categories'.[33] (In a sense this is Panofsky's Kantianism.[34]) These categories have particularity in that they are at work in the creation of a specific 'artistic problem'; however, for Panofsky, they also have a form of generality. The difficulty here is that in identifying the immaterial in terms of 'a priori categories', what is then precluded is the possibility that a particular work works in a way that refuses a form of resolution, and moreover the work would then be the affirmation of that refusal. In other words, what becomes impossible, with the insistence on the a priori organization of the sensible, is that a particular work is the belonging together of irreconcilable possibilities that comprise the work's work. The primacy of the irreconcilable creates interpretive tensions. The attribution of tension is not a claim about meaning as though meaning were simply that which could be attributed to a work independently of the way in which the work's material presence was itself at work (and thus was, in fact, the work). Hence, claims about the impossibility of reconciliation still hold to the proposition that meaning and mattering cannot be disassociated. The impossibility of an effective disassociation entails that claims about meaning are always after-effects of mattering. (As already suggested, this would be a basic tenet within a materialist philosophy of art.)\n\nIn sum, what is at work in Panofsky's approach to Kunstwollen is the attempt to identify what has been described as an immaterial force within the work of art. (This is its central importance.) The difficulty is that Kunstwollen (as the name that force is given in the context both of Panofsky's writings and in the work of others who deployed it and similar concepts in the same period) is attributed a general quality which, while it may be unique to a given work of art, also has a type of generality such that it then has to account both for a historical period as well as the presence of specific works. Hence, Riegel argues in _The Leading Characteristics of Late Roman Kunstwollen_ that 'to obtain an understanding of the nature of late antique art (that is the art of the middle and late Roman period) we may study individual works or the surviving literary sources. In either case, we obtain an insight of the same proposition: that there was in general at that time only one direction for the _Kunstwollen_ to take.'[35]\n\nHowever, Kunstwollen, as the term is used directly in Panofsky's project, is as a consequence not to be understood as a theoretical account of the presence of a particular within relationality. The difference between works would be accounted for in terms of the different ways they resolved the same 'artistic problems'. The 'artistic problems', however, are accounted for in terms of a presence that is 'a priori'. Hence, while the language of 'immanence' is correct insofar as the presence of the ideational and the force of its presence, and thus the economic as an account of work is only ever immanent in a particular work, the difficulty here is that what is immanent is simply a generalizable abstraction such that a set of abstract relations (e.g., haptic\/optic) are at work within a work leading to the completed resolution of the 'problems' staged. Such limitations are not limitations imposed on Kunstwollen understood as a condition of the particular work and thus as an immaterial presence within the frame. On the contrary, the limitation is the explication of Kunstwollen in terms of a singular economy which, precisely because of the way formal relations are construed with it, eschews the possibility of anoriginal complexity and thus effaces the sense of spacing inherent in the temporality of at-the-same-timeness. What is shut down therefore is the space in which mattering occurs.\n\nIn order to understand the way Kunstwollen was understood as operative within the frame emphasis was given to Panofsky's comments on colour. The important conclusion is that colour cannot be disassociated from its place in the way a given work works as art.[36] While there may be disagreements as to the way work is understood, again the significant contribution of Panofsky in this regard is the identification of colour as part of a work's work. The significance of this identification is that colour cannot as a result be taken as a generalized concern on the one hand nor, on the other, can the work in question be understood merely in regards to its ideational content as though its material presence\u2014here colour\u2014were incidental to its work as art. The value of Panofsky's approach to colour needs to be interpreted in relation to the limitations inherent in the way Kunstwollen is understood. Both need to be retained. The real contrast to the formulation of colour's presence provided by Panofsky is found in Hegel, specifically in his _Lectures on Fine Art_.[37] The general point at issue concerns the question of what the material presence of artwork actually is. Pursuing the contrast between Panofsky and Hegel will aid significantly in helping to clarify what is to be understood by the terms 'self-staging' and 'mattering'.\n\nHegel's _Lectures on Fine Art_ mark a moment of fundamental importance within the history of philosophical writing on art. With the _Lectures_ , the concerns of aesthetics\u2014which were defined by feelings held by a subject in relation to an object though where the object (in terms of the quality of the object's existence) did not figure philosophically and thus the object as the work of art was doubly distanced from the concerns of the philosophical\u2014were no longer primary. Indeed, it is possible to go further and argue that Hegel's _Lectures_ mark one of the decisive moments at which aesthetics finally cedes its place to a philosophy of art. This occurs primarily because of the insistent presence of the object within the latter. The object can be both thought and evaluated. Colour, shading and the presence of shadows, and landscape as opposed to portraiture, are all allowed to figure as topics within the philosophical. Moreover, Hegel discusses named and identifiable works of art. His travels to Dresden registered philosophically. In sum, Hegel's work marks the end of aesthetics and, drawing on a tradition that defined the philosophical in relation to the history of art rather than aesthetics (Baumgarten and Kant), continued the project of securing a philosophy of art through an incorporation of the object within a structure of expression in which expression was to have been regulated in advance. Hegel's interests were located in ways specific artworks worked as art. In addition, though this is only implicit in the project of the _Lectures_ , he is able to discriminate between particular works. The locus of the discrimination is the works themselves. Discrimination between individual works can occur for Hegel because discrimination involves a form of evaluation linked to the recognition of art having, in relation to content, a founding sense of propriety. Hegel's critique of Jan van Eyck's _Ghent Altar Piece_ , for example, is located in Hegel's belief that the presentation of God in his separation from the other member of the Trinity or from the Holy Family presented a systematic misunderstanding of God. The work of art should be constrained by the content of the ideational. Precisely because it is possible to misunderstand the nature of God, it is equally possible that such a misunderstanding would find expression in a specific work of art. Discrimination between works involves the works' mattering, since evaluation assumes the effective presence of the ideational.\n\nHegel was not, however, the first to note a sense of propriety. Even if the argumentation is importantly different, Gotthold E. Lessing in his _Laoco\u00f6n_ writes, in his attempt to differentiate himself from Johann Joachim Winckelmann's interpretation of the statue, that 'only the ill-informed observer would judge that the artist had fallen short of nature and had not attained the true pathos of suffering' ( _das wahre Pathetische des Schmerzes_ ).[38]\n\nHegel, too, will refer to the eponymous statue. While Winckelmann, Lessing and Hegel take importantly different positions in regards to how the question of universality and the latter's presentation within a particular work is to be understood, what is significant is that all three take the grief\u2014as an instance of pathos\u2014as that which can be both properly and improperly expressed. What the statue presents was for Lessing, as noted, 'the true pathos of suffering' ( _das wahre Pathetische des Schmerzes_ ). Hegel is in part continuing this tradition. While it is to adopt a language that is not Hegel's, it can still be argued, nonetheless, that in _Lectures_ not only is there a philosophy of art that is linked to art works but it is also the case that thinking philosophically about works of art is sustained by a sense of propriety. (Propriety is the possibility that inheres in the relation between the material and a very specific conception of the material's relation to the immaterial.) It is thus that Hegel's philosophical position occasions the development of modes of argumentation giving rise to the presence of evaluative distinctions between works. Both a form of judgment and discernment have become an actual possibility. The emergence of such a possibility will be taken up here as part of the following engagement with a number of Hegel's decisive references to colour, as they appear in the _Lectures._\n\nHegel charts a complex path in relation to the presentation of colour in the _Lectures on Fine Art_. The point of departure is the identification of painting in terms of a form of 'particularization'. Note that with this delimitation of painting there is an already present reciprocity between the presence of colour and what is described by Hegel as a work's 'inner life'. Thus far colour eschews abstraction. To evoke a term such as 'inner life', to be able to define that life, is already to have named the ideational. Hegel writes of colour that it is the particularization of the appearance in the picture, and as such demands equally a particularization of the 'inner life'.[39] The 'inner life' is the place of the 'soul', though it is also redescribed as the 'heart's feeling'.[40] All are immanent in the work. Two questions emerge here: First, could colour ever be extracted from this reciprocity? In other words, could matter be systematically differentiated from the immanent presence of the ideational, the immaterial in, and as, the work? While the detail of the answer needs to be worked out, it is essential to note in advance that such a position would only be possible if the conventions that attend mattering were undone such that the material was the equated with\u2014thus reduced to\u2014the empirical. The second is more obvious and yet it is a far harder question to answer. What is meant by colour? What is colour such that it can be thought of philosophically\u2014though not as an end in itself but rather as part of art's work? Is colour taken in general and then in terms of specific colours, or, is it simply the presence of certain hues within the palette, which in their application to a surface then colour it?\n\nThe way to approach both these questions, and in the end both are decisive, is to proceed with the already identified reciprocity between appearance and what it is that appears. In other words, once posed in relation to art's work, the emergence of colour has to be considered within the general ambit of the problem of expression (or appearance and in that way of how what appears comes to appear). Once this latter point is allowed, then there cannot be just colour. Nor is there merely a formal or abstract relationship. Assuming it is possible to sustain the argument that there is a relationship between appearance and the 'inner life', as Hegel claims\u2014a relationship in which what appears may be 'inner life', though equally what appears may be mere appearance, or an illusion, and as a result the undoing through the non-appearance of that 'inner life'\u2014then, to follow the path that has already been noted in Lessing's use of the word 'true', there has to be that which regulates such a relationship and therefore regulates appearance. In sum, not only does there need to be an already determined sense of what counts as the 'inner life', it is also the case that what counts as appropriate for its expression needs to be established and maintained. Evaluation has to be possible because it is necessary. With this framework, both expressions' failures, and equally the expression of that which is untrue, are continual possibilities haunting expression. What this entails is that there cannot be pure and unregulated colour (leaving open the question of the exact signification colour would have in such a context). This has to be the case precisely because for Hegel everything that is 'distinct' and particular within painting is the work of colour: 'Shape, distance, boundaries, contours, in short all the spatial relations and differences of objects appearing in space, are produced in painting only by colour.'[41]\n\nHence the remarkable claim that the entire difference between two specific human beings, in fact 'two men', who are described by Hegel as being 'altogether different', is the work of colour. What this means is that difference is the effect of the work of colour. As Hegel states, 'In a painting this entire difference is reduced to a difference of colours.'[42] Hence the question concerning colour, the second question noted above\u2014What is colour?\u2014now has an even more insistent quality since it defines the possibility of difference. And yet, the formulation of this difference is accompanied by a number of attendant problems. The argument here is that this conception of the work of colour, in which difference is given by colour, and by colour alone, which is Hegel's own position, is undone by the way that Hegel accounts, not just for the nature of the difference between Niobe and Mary but on the necessity that there be a difference between them. Niobe and Mary are not just two women.\n\nThe necessity of their difference and that their difference be expressed reveals an instability inherent to the way colour is deployed in staging the interarticulated presence of the ideational and the material within the overall formulation of Hegel's position. The argument has to be that in order for colour to be effective in sustaining differences\u2014which in this instance has to occur since Niobe and Mary are, in terms of the way they are understood, thus ideationally, inherently different\u2014then there cannot be just colour. Were there to be just colour, and the suggestion implicit in the argumentation to come, drawing on what has already been noted is that this is an impossible state of affairs, then it would depend upon the identification of colour with the sensible and not the material. Once it can be argued that materiality is marked in its inseparability from the immaterial and that their co-presence is thought of in terms of at-the-same-timeness, as the temporal quality of a founding site of irreducibility and difference at the origin, then the proposition that there is just colour becomes simply untenable. This is the argument that has to be developed.\n\nIn _Lectures_ , Hegel argues that the presence of colour, and therefore the process of 'colouring', is precisely that which 'makes a painter a painter'.[43] However, it is the way that this insistence on colour occurs, plus the conception of colour at work within it, that introduces the potential instability noted above within his conception of both sculpture and painting. The instability pertains to abstraction. If difference is carried by colour, then colour is informed from the start. The informing of colour precludes the possibility that colour could ever be pure form; that is, it precludes the possibility that colour could ever be abstract. And yet, it is this exact possibility that arises. The instability noted above cannot be avoided, as it results from an essential element of Hegel's overall argumentation insofar as it is precisely this instability that allows painting to pass into music. That passage within the formal sense of movement demanded by _Lectures_ takes place as a process of abstraction. Abstraction is that which allows colour to becomes 'colouring'. Moreover, the instability in question can be linked to this specific version of abstraction since it is abstraction as a projected overcoming of mattering. It is abstraction as the posited overcoming of the ideational, an overcoming in which form\u2014now colour\u2014was no longer informed.[44] The argument is of course that these possibilities are premised on a refusal of mattering.\n\nAbstraction results from what Hegel describes as 'the magical effect of colouring'. To this end Hegel writes that 'the magic consists in so handling all the colours that what is produced is an inherently objectless play of pure appearance which forms the extreme soaring pinnacle of colouring, a fusion of colours, a shining of reflections upon one another which becomes so fine, so fleeting, so expressive of the soul that they begin to pass over into the sphere of music.'[45]\n\nAbstraction then is that moment in which colour appears as an end in itself. While this is one possibility in and for painting, a potential already acknowledged in the possibility that appearance may be that which attracts, while content may have become irrelevant\u2014a position that reiterates, for example, the debate occurring in the Renaissance concerning the role of _disegno_ in the response to Michelangelo's _The Last Judgment_ as well the argument emerging in both the Northern and Southern Renaissance concerning the paintings of Mary and the necessity that the artist's 'hand' be regulated in advance (in part the subject matter of chapter 2).[46] Nonetheless, for Hegel, to the extent that painting has to present the 'inner life', it also needs to be the case that a work\u2014and it is the work that is also the work of colours\u2014needs to exist such that 'appearance is animated and ensouled from within'.[47]\n\nThe problem is clear from the start. There are two different demands. The question is therefore, How could it be true that, first, colour is all and that, second, what appears be 'ensouled from within'? That process, the 'ensouling', is not colour _qua_ colour. Yet it could not be present\u2014and this presence has both negative as well as positive determinations\u2014were it not for colour. While there is always an 'inner' element within all paintings, significant differences between works still endure. This becomes clear in the contrast of Niobe and Mary. There are sculptures, figurines, vase paintings and paintings of Niobe and Niobids. She is present as herself as well as metonymically via her children's death (i.e., in regards to the Niobids, there are numerous vase painting, for example, in which only the Niobids appear). Moreover, there is an extraordinary figurine of Niobe (dating from the third century CE) in which she resembles the Pieta, _avant la letter_ , insofar as the dead male child, who is the size of an adult, lies cross her lap.[48] (See figure 1.2.) In regards to painting Abraham Bloemaert's (1564\u20131651) _Niobe and Her Children_ also presents Niobe as a woman and mother whose anguish locates her within a complex relation to the work of fate. (See figure 1.3.) Moreover, both the figurine and the famous set of sculptues, now in the Uffizi, in which she vainly attempts to shield a child with her cloak and which recall her ore Ovid's dramatic lines, 'the last one remained, whom her mother with all her form ( _toto corpore_ ), with all her robes concealed', all evince nothing other than the most emphatic form of love.[49] (See figure 1.4.) Moreover, there is the possible reiteration of the motif in the martyrdom of the children of Saint Felicitas. And yet for Hegel, all works that included Niobe\u2014directly or indirectly\u2014would be limited. However, the limitation has nothing to do with the work's appearance. Equally, the array of colours that allows for her presentation could be identical to those occasioning Mary's presentation. Precisely because of this possibility, the difference has to reside elsewhere. Niobe is presented as one characterized as having a life that, for Hegel, is no more than the 'empty endurance of fate'.[50] What a painting or sculpture of Niobe has to encounter, and this would have to be the case despite the appearance of colour, is the fact that for Hegel her 'inner life, her heart, has lost the whole burden of its love and its soul'.[51] Again, the significance of this is found in the earlier position in which the project of art's work and its relation to colour for Hegel is not just defined in terms of particularization, but more emphatically 'colour as the particularization of the appearance in the picture, demands also a particularization of the inner life'.[52] The 'inner life' does not just bring the 'soul' into consideration; rather, its effect is far stronger. The soul and the 'inner life', when taken together, delimit what needs to be understood as the propriety of painting. In other words, there cannot be just colour precisely because there has to be a relationship between 'inner life' and colour. The latter's project is the presentation of the former. In other wards, colour in order to realize the project entrusted to it has to bear the ideational.\n\nFigurine of Niobe (dating from the third century CE).   \n_Altesmuseum_ , Berlin.\n\nAbraham Bloemaert, _Niobe and Her Children_ (1591).   \nUffizi Gallery Museum.\n\nUffizi Gallery, _Sala della Niobe_ (Niobe Room).\n\nConsequently, while Niobe has no 'inner life', hence the non-presentation of the presentation of that 'inner life', it is that which is allowed for by the work of colour. However, the locus of the work's evaluation and judgment has to do with 'inner life' and not just with colour. Hence, while it may be the case that 'painting gives to the life of the soul its really living external appearance' through colour, which is Hegel's actual argument, it is also true that colour would be that which presented the soul as having turned to 'stone'.[53] These are terms in which Niobe is present in Hegel's text. Colour is able to do both. Both would be coloured. Thus the body, and here it would be Niobe's body, could not be present as that which the 'inner shines through', this despite the work of colour, precisely because she has no inner life.[54] The painting of Niobe would not be inappropriate; it would verge on the impossible. Though it is, of course, an impossibility grounded on the ineliminability of its actual possibility; hence, it is there verging on the impossible. Any painting would therefore be an impossible possibility. On the other hand, Mary, whose 'whole inner life is love', has to figure in a radically different way.[55] If it can be claimed that a work's 'spiritual content' is such that, as Hegel argues, it 'cannot be separated from the mode of presentation', then this has the effect of destabilizing the claims he has made about the centrality of colour.[56]\n\nTherefore, while it may be possible to argue that the difference between two given individuals\u2014the already identified 'two men' identified in the _Lectures_ \u2014may be no more than the work of colour, that is only true if the 'men' in question are present and thus appear as universal abstractions. The difference between Niobe and Mary cannot be explained in the same way. And this is the crucial point. Colour on its own cannot sustain the division, the difference, that such works demand. There is an important gesture here to the argument that has already been advanced in which there was a twofold claim that there can never be colour on its own, and as a consequence the empirical as the given will have always been informed in advance. That informing\u2014as the impossibility of the uninformed\u2014is of course a fundamental part of the move from the empirical to the material, a move that would highlight the fact that empiricism is self-contradictory in nature. 'Inner life' in Hegel's philosophical thought as a locus of different possibilities, which as already noted have both positive and negative determinations and are thus linked to an external sense of propriety, have been undone from the start. Hegel's conception of colour allows for that which can have 'magical effects' and so exists in a move towards abstraction. Yet it is precisely this sense of colour that cannot sustain the ineliminable difference between Mary and Niobe. Colour, while allowing for abstraction, is also essential to the presentation of their real differences. Colour presents those differences because once colour is at work in painting, thus forming part of art's work, a work's mattering, it is always already informed. Such a set-up precludes the possibility that colour could ever be abstract and be there as simply given within the process of its becoming abstract. Were abstraction to pertain, obviating the demands of the ideational, then Niobe and Mary, as was the case in the already noted 'two men', would be the same. They cannot be. The latter is an ideational claim that demands that art\u2014thus colour\u2014be regulated in advance by the ideational and just as significantly that art's work be the always already present meld of the ideational and matter, which when thought together, as has been argued, appears as the work's mattering.\n\nFor Panofsky 'artistic problems' are abstract and singular in determination. Colour becomes no more than that through which the work of singular economies are at work within the formal resolution of these problems. What can be found in Hegel is the inverse position. Namely that while colour is that through which the work is presented\u2014 on one level colour is all\u2014it is equally the case that the divisions within the radically differing conceptions of 'inner life', here taken as an instance of the ideational, means that in the end colour, while allowing for abstraction, has equally to refuse that possibility. Colour within Hegel's _Lectures_ , though it has to be noted that this occurs _pace_ Hegel, marks as a result the presence of both a complexity and a plurality that will be integral to any account of the art's work.\n\nIf there is a way of summing up the preceding argument, then it resides in the claim that neither Panofsky's attempt to deploy the concept of Kunstwollen nor Hegel's attempt to engage the relationship between what here is being called the 'ideational' and the 'material' within art were successful. And yet, both exerted a specific demand. What remains is the necessity to think the presence of an immaterial force. The value of Panofsky's in this instance is that he allows, if only as suggestion, to recognize that Kunstwollen, rather than being an externally organizing system, and thus as providing a synthetic account of a historical period, could be a quality of a particular work. What has to be taken up in the guise of a conclusion is this repositioning of Kunstwollen in terms of mattering.\n\nHaving begun to detail some of the concerns that are, in the first instance, inherent to mattering and self-staging and, in the second, already at work in what has been identified as the effective presence of a complex economy, it is essential that what is taken up at this stage are the preconditions for identifying and delimiting how they are at work within particular works and thus how they are to be employed in an account of art's work. Distancing mattering from formalism and equally from abstraction is to allow for the presence of modalities of complexity within the frame. Complexity that includes mattering and which, despite the apparent presence of differing formal resolutions (e.g., the arrangement of the constitutive elements within the frame), holds open the possibility that the work's presence continues to resist any form of direct or deliberate synthesis. The work remains at work, present as a site of potential or actual irreconcilability, working with the insistent presence of the aporetic demands that this is the way the work works. A specific task arises here. What has to be taken up are the ways in which that state of affairs, complexity as an anoriginal presence, is to be recovered. Again this is to allow the actative predominance over the static (and moreover to position the presence of any static configuration as providing a locus of meaning as a contingent after-effect of the actative).\n\nA more general way of asking the question of the status of the particularity of a single work is to concentrate on the way a given particular's particularity is itself constituted. Particularity emerges then as a question rather than having an assumed presence. The force of particularity as a question is that it excludes, first, the possibility that a work's particularity can be posited and, second, that a particular is the particularization of either a context or a universal. In both instances the particular is no more than the particular of a universal. The conjecture is that as such its particularity remains unthought. The countermeasure to the positing of context and the conception of particularity entailed thereby is to allow for acts of constitution. Such acts involve relations. They hold open, as a continual possibility, the reconstitution, at any moment, of the particular as a particular. Identified here is the presence of a particular within relationality. Relationality despite the fact that any one instance can be attributed a specific determination is always incomplete. Coming-into-relation is a process that yields particularities. Hence, any specific determination retains the quality of the continuity of a coming-into-relation such that relationality is always there and to come, thus precluding the possibility that there could be a single determination that had the quality of a self-completing finality. And yet, of course a single finite determination, despite its being inherently incomplete and therefore as sanctioning forms of contestation, becomes as a result of these qualities a locus of responsibility. Responsibility is the result of judgments staged in relation to the necessity of the incomplete. Taken as a set of determinations, the relationship between the incomplete, finitude and responsibility differs fundamentally from processes of contextualization and the sense of epistemological exactitude that it demands. Moreover, particularization occurs as a consequence of a relation. Particularity is at its most exacting when a relation yields the particular as its after-effect. This is the point at which a possible continuity of concerns could be established, be they already thematic or a theme's creation, and where the establishing of forms of continuity would allow particularity to have a greater exigency and a new determination. Indeed, the possibility of allowing for the continual determination of any one particular as a particular and thus the possibility of a particular acquiring new determinations is allowed for by the continuity of its coming-into-relation and the interpretive responsibility that this continuity entails. The content of relationality, once the possibility of the continuity of coming-into-relation cannot be excluded, means that the determination of any one particular's particularity can be neither predicted nor controlled. Relations, once established, call for judgement precisely because they operate outside the realm of that which is determined in advance.\n\nRelationality, as the term is being used in this context, has a number of determinations. One is that relationality involves forms of juxtaposition: in this instance, relationality defined by externality. Juxtaposition is a coming-into-relation (a process that demands judgment rather than as having assumed efficacy). The encounter between particulars allows the content of one particular\u2014an instance of art's work (i.e., the work of art)\u2014to emerge in a way that is a distinct and occurs as a result of its juxtaposition with another specific work. In other words, it occurs as a result of the process of coming-into-relation where the latter is defined in terms of relations between externalities. What is occasioned is a sense of movement in which the truth of a work comes to acquire a different quality and a different content as a result of its juxtaposition. In one of the 'theses' comprising his _On the Philosophy of History_ , Walter Benjamin wrote the following: 'There is no document of civilization which is not at the same time a document of barbarism.' ( _Es ist niemals ein Dokument der Kultur, ohne zugleich ein solches der Barbarei zu sein._ )[57]\n\nIt is essential to note that Benjamin inscribes a fundamental temporal marker into this claim. A document is both of 'civilization' and 'barbarism'. While the presence of these qualities is paramount, their importance is that this double designation holds 'at the same time' ( _zugleich_ ). And yet, of course, while this temporal dimension, which is a version of at-the-same-timeness and thus a further sense of anoriginal complexity, may be true insofar as both 'civilization' and 'barbarism' pertain at the same time, a question remains: How is this quality to appear? In fact, this may not be the right question. It is better to ask how this doubled quality can be made to appear. The force of this as a question can be located in what might be described as the non-appearance at any one time of one of these elements, given that they are, as determinations, present 'at the same time'. In practical terms it would usually be the case that the presence of 'civilization' would be taken to have muted or effaced the possibility of 'barbarism'. Art is usually thought to console.\n\nIn the cycle of frescoes in Arezzo painted by Piero della Francesca between 1452 and 1456 now titled _The Legend of the True Cross_ , one specific component of the overall cycle and which is placed in the middle register of the choir is known as _The Torture of the Jew_. (See figure 1.5.) Within the overall narrative structure of the cycle, what is occurring within this fresco takes place as a result of Helena, the mother of Constantine, having arrived in Jerusalem. Suspecting that the whereabouts of the Cross were indeed known, she had a Jew, Judas, whom she thought knew where the Cross was buried, tortured in order to find out. This involved placing him in a well and providing neither food nor water. After six days, and as a result of his pleading, he was let out. The fresco depicts him being pulled from the well. He is being pulled by a rope around his body. He is also being pulled out of the well by his hair. Taken together, they underscore the presence of violence within the scene.\n\nThere are at least two ways of approaching this fresco. In the first instance, a path could be traced between the faces. This would be an approach in which internal relationality was predominant. The beauty of each face would prevail: Judas and 'torturers'. It would even be possible to contextualize these faces by linking them to other faces by Piero della Francesca, thereby opening these faces to classification as instances of early Renaissance portraiture. The point that would continue to return is that the face of Judas cannot be directly differentiated from the face of the one pulling his hair and the others actively engaged in the torture. Here, the interplay of faces renders the scene of the torture less emphatic. Moreover, the universality both of the faces within the frame, and then other faces that could be chosen, holds open the possibility of the Jew's conversion. After all, his face is for all intents and purposes the same. Recalled here is Blaise Pascal's description of Jews. In Fragment 289 of the _Pens\u00e9es_ he wrote, 'The Jews are of two sorts. Those who have only pagan feelings, the others that have Christian feelings.' ( _Les uns n'avaient que les affections pa\u00efennes, les autres avaient les affections chr\u00e9tiennes_.)[58]\n\nIt is as though the face of Judas holds open the spurious hope of conversion since his face cannot be distinguished from the Christian face. What this means is that with this locus of torture, his alterity can be mastered. His face indicates the presence of that alterity already having been effaced. There is an invitation to see sameness. That face, however, gazes at his tormentor. The seeing, his being seen, the imploring look and the response it receives, invite another seeing. With the latter, what would be seen is the violence of the tormentor. Judas's hair is wound around the latter's fingers. Violence and beauty coincide with the possible distancing of one in order that another form of occurrence prevails. The argument has to be that to see the violence as violence demands the presence of a coming-into-relation that undoes both contextualization and the fresco's own internal relations. What is continually forgotten is the torture. It is forgotten because it would not have been seen at the time of its creation as staging a set-up that invites repugnance nor its reconfiguration as an injustice. That this is 'a document' ( _ein Dokument_ ) that is equally one of 'civilization' as well as 'barbarism' becomes clear when it is juxtaposed with the aftermath of the Jew's actual torture. The latter is present in the image of 'a sick Polish survivor in the Hanover-Ahlem concentration camp'. (See figure 1.6.) It is only when the Jew's torture as normalized is seen within a setting that traces the aftermath of the process of normalization that the incipient violence in the fresco becomes clear. The clarity is now there in the image. The relation between the nurse and the one who is no longer the 'prisoner' but the 'survivor' marks the presence of a relation that refuses violence but incorporates violence's presence into the image in terms of its aftermath. It is precisely this set-up that allows relations internal to the images to be contrasted (and thus brought into a relation). In regards to figure 1.6, it can be further suggested that it is an image in which violence resisted any aesthetic hold. It would therefore be an image whose presentation of violence is so muted that the violence that attends it becomes even more insistent.\n\nPiero della Francesca, _The Torture of the Jew_ , a fresco within the larger cycle titled _The Legend of the True Cross_ (1452\u20131456). By concession of the Ministry of Cultural Heritage and Activities and Tourism \u2013 Superintendence for the Architectural, Landscape, Historic, Artistic and Ethnoanthropolical Heritage of Arezzo\u2013foto: Alessandro Benci\/\n\nBy concession of the Ministry of Cultural Heritage and Activities and Tourism\u2014Superintendence for the Architectural, Landscape, Historic, Artistic and Ethnoanthropolical Heritage of Arezzo\u2014foto: Alessandro Benci.\n\nA sick Polish survivor in the Hanover-Ahlem concentration camp receiving medicine from a Red Cross worker. United States Holocaust Memorial Museum.\n\nWhat is at work in both images is barbarism. In the case of Piero della Francesca's fresco, this would not have been known. And yet it has become a state of affairs that is now known. To be clear: not only was it not known at the time\u2014except of course by Jews\u2014it was not, nor could it have been seen as such at the time. After the event, its presence as barbarism would come to be seen. While there is a sense of obviousness about the claim (i.e., that an image can retrospectively be seen to form part of history or acquire a meaning that was unavailable to it at the time), that possibility means that in its juxtaposition with other images, it acquires another possibility. In other words, that the truth of the image is given in its capacity to stage another possibility that at the same time reveals a content that may not have existed at the time the work was constructed or even in its immediate afterlife. If, as Marilyn Aronberg Lavin argues, the cycle provides to 'the world of the Renaissance, its full-fledged Christian epic', the question to be asked is about the role or the necessity of the 'torture of the Jew' as part of that epic.[59] The faces in Piero della Francesca's fresco can be conflated. One is like the other. One becomes the other. There is a moment in which this becomes impossible. The possibility of universal humanity that is thought to be have been staged by the faces and realized by simultaneity of expression and style\u2014combining as component parts of mattering\u2014is undone by the recognition of violence. What has then to be argued is that the face of Judas and his torturers cannot be reconciled and that this is the case despite appearances. It is the aesthetic hold of the entwined fingers that works to preclude such a possibility. That also matters. And this will be the case despite the ease with which they can be reconciled. They are _apart_ from each other precisely because they are a _part_ of each other. Reconciliation and the irreconcilable hold and held within a temporality of at-the-same-timeness. Reconciliation and the irreconcilable are allowed because the presence of paint, its form and forming that are the fresco's construction, are informed and can continue to be informed. Repetition is the discontinuous continuity of informing. Constructed here is a possibility that occurs to the extent that a coming-into-relation is driven by differing instances of externality.\n\nIn more general terms, however, the argument is that what was identified earlier as 'iterability' now reappears\u2014no longer to be thought of as the simple continuity of interpretation, even though it is a sense of continuity that took the form of the continuously discontinuous. Rather, what is emerging is the necessity to recognize that iterability is not about the repetition of a singularity. Rather, iterability is located, at its most productive, within relationality. Hence, iterability names the continuity of any one work's coming-into-relation; a state of affairs defined in terms of a potentiality actualized through relationality. While the process of coming-into-relation may have a certain generality, one may object that all that has been presented thus far are 'examples'. There is a rejoinder. Neither the work by Poussin nor Piero della Francesca is an example. Of what would they have been examples? Instead, reference to any one work of art is a reference to a particular work, and thus if there is a generalizing move it is to particularity. Understanding the work of any one particular involves responding to the question of how it works as art. If there is a methodological question, then it concerns the place of a particular within a network of relations of which it\u2014the particular\u2014is the after-effect. This is the formulation that will continue to return in the ensuing chapters. While the full force of what is intended by terms such as 'relationality' and 'mattering' will come to be worked out in the chapters that follow, if there is a summary position that can be presented at this stage, then it is the already noted general claim concerning particularity. The particular work, its singularity, or particularity, is always the after-effect of a set of relations. Those relations are, however, not given. Hence what always remains, remaining as an open possibility that defines a work, is the possibility of a coming-into-relation. It names an ineliminable potentiality. This process involves as much internal relations as it does both external relations as well as the effect of the external on internal relationality. Constitution through relationality does not entail any form of synthesis where that synthesis would result in a type of unity whose coherence was both formal and ideational.\n\nWhat follows from this opening attempt to situate that development of a philosophy of art that insists equally on particularity and work\u2014mattering as work\u2014is that it demands that works, themes, topoi all emerge as providing the settings within which to think through and thus to work out that development. As already noted, after tracing the presence of different modes of relationality in the work of Walter Benjamin and Aby Warburg, another beginning will be made\u2014beginning, though it will involve a form of continuity, with the hand\/instrument relation. The hand becomes a site. The hand continues to appear. It is painted into art; it paints; it has its own history within both artworks and the writings with which they are directly concerned; the hand is equally the locus of art's self-staging. All these modalities of the hand's presence involve relations. What the hand's anoriginal mediation opens up is a set of possibilities. The second part of the book comprises chapters that are concerned with tracing the limits of various philosophical approaches to writing on art. The philosophers whose work is discussed are Heidegger, Derrida, Lyotard and Sallis. While in each case the way philosophy's relation to art is posed is different, and while in each case the artworks to which references is made are different, nonetheless what emerges is the way that philosophy in different and differing ways attempts to avoid encounters with what has been described as mattering. In the case of Heidegger it is the specific materiality of both sculpture and painting that escapes his analysis. In regards to Lyotard and Sallis the limit lies in the ways colour, the work of colour, is understood. While in the case of Derrida, the reduction of painting to text, a consequence of having yielded to what has already been described as the ekphrastic impulse entails, though this time for other reasons, that the presence of art as the site of work thought in terms of mattering resists his approach at every turn.\n\nDiffering terminological formulations\u2014mattering, self-staging, at-the-same-timeness, coming-into-relation\u2014are all intended to mark the presence of the language deployed to develop a materialist philosophy of art. They are terms that underscore movement, acts of constitution and the presence of the actative over the static. They are intended to progress the concept of Kunstwollen, perhaps progressing it by leaving it behind once the work of art is understood as that which is at work within a network of relations that, in being open to constitution and reconstitution, continues to yield a work as that which occurs as an after-effect within a discontinuous continuity. There is, however, one final conceptual or terminological issue that has to be addressed. Addressing it, however, is not to resolve it. This project has been described as the development of a philosophy of art that works by insisting on a certain version of materialism and the interarticulation of the ideational and the material engendering a set of processes occurring under the rubric of mattering. Furthermore, a specific claim has been made about 'art'. Formulations such as art's self-staging are intended to capture a concern with the particularity of the work of art. Art is the predominate and predominating term. However, once a move is made to the work of Walter Benjamin and Aby Warburg, while the term 'art' has some currency, it is for the most part replaced by the term 'image' ( _Bild_ ). Moreover, it is in terms of different formulations of the 'image' that both develop their own engagement with that specific history of the image that is most readily connected to writing on the history of art. Part of the force of the 'term' image is that it allows different objects to play a role within a more general analysis without the attendant concern to establish their presence as 'art'. Hence, figurines, coins and paintings can be brought together into an interpretive constellation without having to make the attendant argument that they are or are not art. While there is an importance that can be attributed to the move to the image in that it obviates the need to engage the question of art, there is, nonetheless, an attendant naivety. What has to endure is the recognition that even with recourse to the history of the image, or even in allowing for a specific discursive domain of ' _Bildwissenschaft_ ', it remains the case that a concern with art is already taking place with it. While there is no one original moment, it is already clear that art as specific form of production within, for example, Greek thought, has a structuring effect on the way art itself then comes to be understood. A set of interrelations has already been established. In the _Ecomium of Helen_ , to take one possible text, Gorgias, in an attempt to explain Helen's action, indeed in order to defend her, identifies the creations of 'painters' and the effect of these creations as playing a fundamental role.[60] Objects and forms are created that 'please the sight' (\u03c4\u03b5 \u03c1\u03c0\u03bf\u03b8\u03c3\u03b9 \u03c4\u03b7\u03bd \u03bf\u03c8\u03b9\u03bd). Certain objects have a specific effect. While that may be the case that art is bound up with a mode of persuasion, the results of which can be regretted, it is equally true that such an evaluation identifies a domain of activity\u2014in which there are both agents and objects\u2014that can be differentiated from other agents and other objects. In other words, there is an already present discourse of art which is delimited by a set of motifs, practices and modes of argumentation. That discourse cannot be equated with the concerns of the image as though there were not an already existing identification of art. Consequently, while the language of the image will come to play a more significant role\u2014precisely because it is the language deployed by Warburg and Benjamin\u2014what has to be retained is the recognition that specificity of art will complicate recourse to an exclusive language of the image.\n\n## Notes\n\n1.\n\nThrough a range of writings on art, I have attempted to ground a philosophy of art in the interconnection between particularity and 'materiality'. While the argument is developing continually, the project here continues earlier work to develop a materialist philosophy of art. In this regard, see books such as my _Object Painting_ (London: Academy Editions, 1994); _Disclosing Spaces: On Painting_ (Manchester: Clinamen Press, 2004); _Style and Time: Essays on the Politics of Appearance_ (Evanston, IL: Northwestern University Press, 2006).\n\n2.\n\nThe term 'period eye' is associated with the work of Michael Baxandall. While it is not defined in a rigorous way, what is at stake within it is clear from the following description of the fifteenth-century 'beholder' in Michael Baxandall, _Painting and Experience in Fifteenth Century Italy_ (Oxford: Oxford University Press, 1988):\n\n> The beholder must use on the painting such visual skills as he has, very few of which are normally special to painting, and he is likely to use those skills his society esteems highly. The painter responds to this; his public's visual capacity must be his medium: Whatever his own specialized professional skills, he is himself a member of the society he works for and shares its visual experience and habit. (40)\n\nWhile what is claimed above cannot be disputed and it is clearly important to recover what was seen and to understand the conditions in terms of which it could have been seen, it is also the case that what marks out any period\u2014any document within it\u2014is that which is there within it but could not have been seen. To go further, it can be suggested that with many works what was not seen can only be seen after the event. As such, the event of seeing takes on a different quality, one defined by that which would always occur 'after'. For detailed engagement with the concept of the 'period eye', see Allan Langdale, 'Aspects of the Critical Reception and Intellectual History of Baxandall's Concept of the Period Eye', _Art History_ 21, no. 4 (December 1998); and Michael Podro, 'Literalism and Truthfulness in Painting', _British Journal of Aesthetics_ 50, no. 4 (October 2010). In addition, while it moves in a different direction, another investigation of not seeing can be located in Rebecca Zorach, 'Love, Truth, Orthodoxy, Reticence; or, What Edgar Wind Didn't See in Botticelli's Primavera', _Critical Inquiry_ 34 (Autumn 2007).\n\n3.\n\nThe terms 'singularity' and 'particularity' will be used interchangeably throughout this and subsequent chapters.\n\n4.\n\nJacques Derrida, 'Signature, \u00e9v\u00e9nement, context', in _Marg\u00e8s de la philosophie_ (Paris: Les \u00c9ditions de Minuit, 1978). This is a paper whose importance cannot be overstated. Even though limits have been suggested here, the significance of Derrida's project in this regard is the redefinition of what can be described as either a singularity or particularity in terms of 'iterability' rather than contextualization.\n\n5.\n\nAlexander Nagel and Lorenzo Pericolo, _Subject as Aporia in Early Modern Art_ (Farnham: Ashgate, 2010), 10. While the argument is different, there are other possible formulations of the aporetic that define it uniquely in terms of semantic undecidability. See as an instance of the latter Wojciech Ba\u0142us, 'D\u00fcrer's 'Melencolia I': Melancholy and the Undecidable', _Artibus et Historiae_ , 15, no. 30 (1994).\n\n6.\n\nThere have recently been a number of investigations from within the discipline of art history regarding questions of how the temporality of art history is to be understood. A work that needs to be noted, in part because of its affinity with this project, is Alexander Nagel and Christopher S. Wood, _Anachronic Renaissance_ (New York: Zone Books, 2010). The 'anachronic', when pursued rigorously, demands a rethinking of the nature of art object. It is in terms of the 'object' that the differences between these projects can be located.\n\n7.\n\nMartin Heidegger, 'The Origin of the Work of Art', in _Martin Heidegger: Basic Writings_ , ed. and trans. David Farrell Krell (San Francisco: Harper Collins, 1993), 199. Originally published as 'Der Ursprung des Kunstwerkes', in _Holzwege_ , vol. 5 of _Gesamtausgabe,_ (Frankfurt: Vittorio Klostermann, 1977), 158\/15. For an important attempt to rescue Heidegger's work on art from Heidegger's privileging of the written over the plastic and thus poetry over painting, see Gregory Schufreider, 'The Art of Truth', _Research in Phenomenology_ 40 (2010).\n\n8.\n\nWhile the argument advanced here is not Romantic in any direct sense, there is a real affinity with the conception of artwork that can be found in Schlegel, and in particular in _Fragments_. This overall position can be developed in relation to _Athen\u00e4ums-Fragmente_ 116:\n\n> Romantic poetry is still in the process of becoming; that is its true essence, that it is only ever becoming, and can never be completed. ( _Die romantische Dichtart ist noch im Werden; ja das ist ihr eigentliches Wesen, da\u00df sie ewig nur werden, nie vollendet sein kann_.) (Friedrich Schlegel, _Studienausgabe_ , vol. 2 [Paderborn: Ferdinard Sch\u00f6ningh, 1988], 115)\n\n9.\n\nThe term 'countermeasure' is intended to be a mode of critique that marks a limit. The limit of idealism is reached by the impossibility of sustaining the truth that it carries. Its truth is its irreducibility to the empirical. Its failure is its location as external to the work of material. Hence, what is created is idealism and empiricism as a type of either\/or. The transformation of the ideal\/idea into the immanent allows for the retention of truth but also the location of the immaterial as an inherent part of the work of materiality; hence, there is a countermeasure to the trap created by having to oscillate between idealism and empiricism.\n\n10.\n\nWhile the argument is different, there is an affinity between what will be developed in the course of this book as _art's self-staging_ and what Victor I. Stoichita has developed in terms of the 'self-aware image'. The distinction resides in the claim that art's self-staging pertains to the way a work works as art. It is not quality that is delimited by a historical period, nor is it defined in terms of either self-reflection or as delimiting a 'meta' quality in relation to other images. See Victor I. Stoichita, _The Self-Aware Image: An Insight into Early Modern Meta-Paintings_ (Cambridge: Cambridge University Press, 1977).\n\n11.\n\nI have tried to make a more extensive argument in connection to the overall significance of Manet for this project in my 'The Matter of a Materialist Philosophy of Art: Bataille's Manet', in _Style and Time: Essays on the Politics of Appearance_ (Evan-  \nston, IL: Northwestern University Press, 2006), 167\u2013178.\n\n12.\n\nAn engagement with the work of art as having a form of force is not limited to claims about decontextualization. Indeed, the argument here involves arguing for the position in which the interrelationship between force and work gives rise to the presence of an economy. For other developments of the concept of force as integral to an understanding of the work of art, see Krzysztof Ziarek, _The Force of Art_ (Stanford: Stanford University Press, 2004); and Christoph Menke, _Die Kraft der Kunst_ (Frankfurt: Suhrkamp, 2013).\n\n13.\n\nFor the argument that pertains to the possibility of source in Ronsard, see Philip L. Sohm, 'Ronsard's Odes as a Source for Poussin's _Aurora and Cephalus_ ', _Journal of the Warburg and Courtauld Institutes_ 49 (1986). There is a range of differing arguments concerning the 'sources' for paintings by Poussin. See in addition Malcolm Bull, 'Poussin and Nonnos', _Burlington Magazine_ 140, no. 1148 (November 1998).\n\n14.\n\nPaul Barolsky, 'Poussin's Ovidian Stoicism', _Arion_ , Third Series, 6, no. 2 (Fall\u2013Winter, 1998).\n\n15.\n\nHere I agree with Richard Neer's important claim concerning Poussin: 'Poussin's pictures do not illustrate or even allegorize moral philosophy: rather they put ethics into practice.' The question here is what the practice of the ethical would involve. See Richard Neer, 'Poussin and the Ethics of Imitation', _Memoirs of the American Academy in Rome_ 51, no. 52 (2006\/ 2007). For the question of desire in Poussin see: David Carrier, 'Blindness and the Representation of Desire in Poussin's Paintings', _RES: Anthropology and Aesthetics_ , no. 19\/20 (1990\/1991).\n\n16.\n\nOn the wider context of touch and the positioning of the proposition _noli me tanger_ as a philosophical trope, see Jean-Luc Nancy, _Noli me tangere: Essai sur la lev\u00e9e du corps_ **(** Paris: Bayard, 2013).\n\n17.\n\nThe 'ekphrastic impulse' sits behind the equation of the project of responding to the work of art by incorporating its 'narrative' within a secondary or written narrative and where the story recounted by the artwork is taken to have occurred almost indifferently to its material presence. There is a school of thought that defines art history in terms of ekphrasis. I have engaged with this topic in a systematic way in chapter 5.\n\n18.\n\nThe term 'anoriginal' is used here to signal the presence of an original form of irreducibility. It is this quality, its already being there, which allows the original to be recast as the anoriginal. The plural event is therefore the specific setting of this generalized state of affairs. For another important use of the term 'plural' in the context of art history, see Christopher Wood, 'Painting and Plurality', _Yearbook of Comparative Literature_ 56 (2013).\n\n19.\n\nWhile Panofsky, Riegl and Wind all approach the question differently, the important constellation here is that all three are involved in trying to understand the sense in which either a historical period or a work of art has a dynamic and therefore active quality. Wind argues for the development of this possibility in his _Art and Anarchy_ (London: Duckworth, 1985). In addition, his own overall evaluation of Warburg is important for understanding the way in which Wind identifies the particularity of artwork. See Edgar Wind, 'Warburg's Concept of Kulturwissenschaft and Its Meaning for Aesthetics', in _The Eloquence of Symbols. Studies in Humanist Art_ (Oxford: Clarendon, 1993), 21\u201336.\n\n20.\n\nWhen Aby Warburg uses the formulation ' _energetische Ausduckswerte_ ', what he is alluding to is the presence of universal psychological states that find expression within a range of artworks. The problem of expression is one way of locating art's relation to the problem of the universal and the particular.\n\n21.\n\nFor an important discussion of Riegel and Panofsky's writings that provides an account from which it becomes possible to draw such a conclusion, see Jas Elsner, 'From Empirical Evidence to the Big Picture: Some Reflections on Riegl's Concept of _Kunstwollen_ ', _Critical Inquiry_ 32 (Summer 2006).\n\n22.\n\nErwin Panofsky, 'The Concept of Artistic Volition', trans. Kenneth J. Northcott and Joel Snyder, _Critical Inquiry_ 8, no. 1 (Autumn 1981): 29\u2014henceforth AV. See also Erwin Panofsky, 'On the Relationship of Art History and Art Theory: Towards the Possibility of a Fundamental System of Concepts for a Science of Art', trans. Katharina Lorena and Jas Elsner, _Critical Inquiry_ 35 (Autumn 2008)\u2014henceforth AHAT.\n\n23.\n\nSee in this regard when Panofsky writes that Kunstwollen pertains to 'an immanent sense within an artistic phenomenon', AHAT, 44. Erwin Panofsky, _Aufs\u00e4tze zu Grundfragen der Kunstwissenschaft_ (Berlin: Wissenschaftsverlag Volker Speiss, 1985), 37. The word 'immanent' poses a range of complex problems. Once the term is employed and attributed an explanatory function, then the inescapable question concerns how the relationship between presence and immanence is to be understood. Part of the difficulty is formulating the question accurately concerns not just the status of the 'immanent' but its location. Is the 'immanent' simply 'in' the work, immanent but not present; or its presence to be defined in terms of that work staging an externality that is immanent where that means external? Instances of such a conception of immanence would be Hegel's conception of _Geist_ or Schelling's conception of _Seele_. It is in terms of the latter that Schelling is able to write of the soul's 'ensouled' presence in sculpture. This position is taken up in a brief discussion of Schelling's _Philosophy of Art_ in chapter 4.\n\n24.\n\nPanofsky, AHAT, 46.\n\n25.\n\nIbid., 49.\n\n26.\n\nPanofsky, AV, 18. See also Erwin Panofsky, _Aufs\u00e4tze zu Grundfragen der Kunstwissenschaft_ (Berlin: Wissenschaftsverlag Volker Speiss, 1985), 29.\n\n27.\n\nPanofsky, AHAT, 49.\n\n28.\n\nIbid., 51.\n\n29.\n\nIbid., 57.\n\n30.\n\nPanofsky, AV, 26.\n\n31.\n\nSee in this regard, Benjamin Binstock, 'Alo\u00efs Riegl in the Presence of \"The Nightwatch\"', _October_ 74 (Autumn, 1995). See also Erwin Panofsky, 'The Concept of Artistic', trans. Kenneth J. Northcott and Joel Snyder, _Critical Inquiry_ 8, no. 1 (Autumn, 1981).\n\n32.\n\nPanofsky, AHAT, 58.\n\n33.\n\nPanofsky, AV, 31.\n\n34.\n\nPanofsky's relation to Kant is an important topic in its own right. In part it is a relationship that is mediated by the presence of Ernst Cassirer. For an important historical account of the relationship between Warburg, Cassirer and Panofsky, see Emily J. Levine, _Dreamland of Humanists: Warburg, Cassirer, Panofsky, and the Hamburg School_ (Chicago: University of Chicago Press, 2013). On the direct use of Kant's conception of the category, see Maurizio Ghelardi, 'La violence de l'interpr\u00e8te', in _Relire Panofsky_ (Paris: Les \u00e9ditions Beaux-arts de Paris, 2008), 96\u201397.\n\n35.\n\nAlois Riegel, 'The Leading Characteristics of Late Roman Kunstwollen', in Donald Preziosi, _The Art of Art_ _History: A Critical Anthology_ (Oxford: Oxford University Press, 2009), 155\u2013161.\n\n36.\n\nIn the context of this book, this is a position that will be argued in much greater detail as it is the problematic presence of colour that established the limits of the work of both Jean-Fran\u00e7ois Lyotard and John Sallis. (See chapters 6 and 7, respectively.)\n\n37.\n\nG. W. F. Hegel, _Hegel's Aesthetics: Lectures on Fine Art_ **,** vol. 2, trans. T. M. Knox (Oxford: Oxford University Press, 1998)\u2014henceforth LFA.\n\n38.\n\nGotthold. E. Lessing, _Laoc\u00f6on_ , trans. Edward Allen McCormick (Baltimore: The John Hopkins University Press, 1984), chapter 1.\n\n39.\n\nHegel, LFA _,_ 802\n\n40.\n\nIbid., 803.\n\n41.\n\nIbid., 810.\n\n42.\n\nIbid., 810.\n\n43.\n\nIbid., 838.\n\n44.\n\nI have taken up the question of abstraction\u2014both in relation to its presence as a form of artistic practice in painting and as a mode of philosophical thought in Hegel\u2014in my 'On Abstraction: Notes on Mondrian and Hegel', in _Paradoxes of Appearing: Essays on Art,_ _Architecture and Philosophy_ , ed. Michael Asgaard Andersen and Henrik Oxvig (Baden: Lars M\u00fcller, 2009).\n\n45.\n\nHegel, LFA, 848. A return is made to this passage in chapter 7 insofar as it plays a fundamental role in John Sallis's interpretation of Hegel. This attempt to underscore the impossibility of pure 'colouring' can be read as a prelude to the engagement with the work of Sallis in that chapter 7.\n\n46.\n\nSee in this regard Melinda Schlitt, 'Painting, Criticism, and Michelangelo's _Last Judgment_ in the Age of the Counter Reformation', in _Michelangelo's 'Last Judgment',_ ed. Marcia Hall (Cambridge: Cambridge University Press, 2005).\n\n47.\n\nHegel, LFA, 846.\n\n48.\n\nFigurine of Niobe (dating from the third century CE), _Altesmuseum_ , Berlin. I hope to return to this figurine in a different context.\n\n49.\n\nOvid, _Metamorphoses_ , book 6, trans. Z. Philip Ambrose (Peterborough: Focus, 2004), 297\u2013299. On Ovid's use of sources, see Andrew Feldherr, 'Flaying the Other', _Arethusa_ 37, no. 1 (2004).\n\n50.\n\nHegel, LFA, 817.\n\n51.\n\nIbid., 826. What would be interesting in this regard is what Hegel would have made of the musical presentation of Niobe, namely Agostino Steffani's _Niobe_ _, regina di Tebe_.\n\n52.\n\nHegel, LFA, 802.\n\n53.\n\nIbid., 839.\n\n54.\n\nIbid., 846.\n\n55.\n\nIbid., 826.\n\n56.\n\nIbid., 800.\n\n57.\n\nBenjamin, GS 1, 696\/SW 4, 123.\n\n58.\n\nBlaise Pascal, _Pens\u00e9es_ (Paris: Hachette), 289. I have analysed this fragment in considerable detail in the context of a general discussion of the figure of the Jew in the _Pens\u00e9es_. See my _Of Jews and Animals_ (Edinburgh: Edinburgh University Press, 2010), 130\u2013150.\n\n59.\n\nMarilyn Aronberg Lavin, 'Piero the Storyteller: Tradition and Innovation in the \"Legend of the True Cross\"', in _Piero della Francesca: The Legend of the True Cross in the Church of San Francesco in Arezzo_ , ed. Anna Maria Maetzke and Carlo Bertelli (Milan: Skira, 2001), 36.\n\n60.\n\nGorgias, 'Ecomium of Helen', _Greek and Roman Aesthetics_ (Cambridge: Cambridge University Press, 2010). There is genuine danger in taking examples from the history of philosophy. However, the point in this instance is to argue that within the history of philosophy there is a thinking of art such that were there to be the move from art to the image, there would still need to be the identification within the later of those 'images' that were contextually, and thus in their decontextualization, defined in relation to 'art'. For a detailed discussion of Gorgias as a philosopher of art, see James I. Porter, _The Origins of Aesthetic Thought in Ancient Greece: Matter, Sensation and Experience_ (Cambridge: Cambridge University Press, 2010), 275\u2013307.\n\nChapter 2\n\n# _Othering_ Relationality\n\nWarburg, Benjamin and the Presence of D\u00fcrer\n\nComing back to a conception of art that emphasizes work and recasts the question of history in terms of what has already been presented as a coming-into-relation, an opening whose justification resides in the continuity of the project that allows for a philosophical engagement with the complex set of connections between the terms 'art' and 'history', demands that the concept of relationality be given greater extension. This needs to occur, because relations within works and between works continue to constitute singularities as after-effects. The project that this philosophical setting discloses needs a further setting. Relationality needs to become, once again, a specific locus of engagement. The generality within the initial formulation of coming-into-relation necessitates it. The project of giving the concept of relationality greater extension contains a series of elements whose concerns are partially reflected in each other. The first is straightforward since it incorporates the two bodies of work that remain central to the strategy that attributes priority to relationality (whether the relation is within a work or between works). The writings central to the undertaking that seeks to further a thinking of relationality are associated here with the proper names Aby Warburg and Walter Benjamin.[1] And yet, while there are ways of drawing these names into a concert of concerns, it is equally as important to devise strategies occasioning forms of differentiation (i.e., to establish those moments at which affinity and differentiation encounter each other). Comparisons structured by the search for affinities often fail to discover the particularity of that which is being compared. The friction arising from the ways in which bodies of thought differ can often be far more illuminating than the simple plotting of points of comparison. Hence, the first part of this chapter involves an engagement with the question of how the differences between Benjamin and Warburg are to be understood. The image allowing it to be staged is D\u00fcrer's _Melancholia I_. (See figure 2.1.) However, what the comparison between them opens up is not just the limits that obtain within Warburg's interpretation of _Melancholia I_ ; what will also emerge are the possibilities that remain\u2014still remain\u2014in Benjamin's _The Origin of German Tragic Drama_.[2] Taken together, this delimiting process allows for the development of a further, though nonetheless connected, project (i.e., allowing relationality a determinate form). As a result, the final part of the chapter seeks to take up what will be identified as an essential component of _Melancholia I_ , namely the relation between hand and instrument. The task is to show the presence of an inherent complexity within that relation once it is present as a singularity\u2014a presence that will have emerged as the result of a coming-into-relation in which the relation figures as a founding irreducibility and thus as a plural event. This will involve the presence of other images. Procedurally, the argument for giving priority to what will become a specific set of relations will continue to appear as the chapter develops. However, the immediate justification for this development can be found in the compass in Melancholia's hand. The holding and releasing of objects and instruments yields a relation as the point of departure. However, rather than assuming that this relation is either the expression of a universal condition having a singular determination which is then repeated, what endures as central is the continual refusal of any form of synthetic universality.[3] Indeed, relationality understood as the site of anoriginal complexity fractures such a possibility in advance. The positing of either the universal or the syncretic is refused. Moreover, that refusal and thus the opening(s) enjoined are the demand of the work's work (and therefore by the way in which images figure). Hence, works are there at (and in) the service of another philosophical thinking. Here is the setting in which she holds the compass. Melancholia holds what Warburg describes as 'the compass of genius'. The hand\/instrument relation endures as a motif.\n\nAlbrecht D\u00fcrer, _Melancholia_ (1514). National Gallery of Victoria.\n\nThere is a further element that needs to be added here. While this point will be developed, it is nonetheless clear that for Warburg the return of the antiquity is often registered in the presence of 'expressive gestures'. In this regard he argues in connection to the unearthing of the Laoco\u00f6n, for example, that it was the discovery of what had been 'sought' ( _gesucht_ ) from the ancient world, namely 'extremes of gestural and physiognomic expression, stylized in tragic sublimity'.[4] Equally there is a concern with 'extremes of gesture' such that what then becomes important is that certain Renaissance works, D\u00fcrer in particular, owed their 'dramatic force of expression to the surviving\u2014inherently Greek\u2014 _Pathosformeln_ that had reached him by way of Northern Italy.'[5] What Warburg identifies as 'agitated mobility' ( _bewegung_ ) is a fundamental element insofar as it is taken as what survives, such that its survival becomes that which is then generalized across images establishing relations between them. While the gesture may be a site of the reconciliation of a play of disjunctive forces (e.g., Apollo and Dionysus), its capacity for repetition is itself dependent upon the attributions of a form of sameness to each gesture and thus to each image. Philippe-Alain Michaud expresses a similar point when he argues that an 'analysis of expressive gestures opens up an unusual, intuitive path to the figures of the past and allows one to identify their recurrence in contemporary imagery'.[6] Again, while the formulation will be different, it is clear, nonetheless, from entries in his _Tagebuche_ that Warburg approached his study of Giordano Bruno in terms of a recovery of that form of universality in which universality\u2014even if a locus of tensions\u2014could nonetheless be displayed. He formulated the project as an 'attempt at a psychology of the human orientation as the basis of a universal image history' ( _versuch einer Psychologie der menschlichen Orientierung auf universell bildgeschichtlicher Grundlage_ ).[7] As is clear from this formulation, not only is there an assumption of universality\u2014where universality becomes a version of sameness\u2014it is equally the case that the presentation is unproblematic.\n\nSigrid Weigel writes of the ' _pathosformel_ ' that it is his 'most prominent concept' and moreover that it 'pinpoints the expressive gestures depicted in paintings and other visual media, which he considered to be memory symbols of fears and excitements that purportedly had been overcome'.[8] Expression and gesture therefore have the status of symbols in which, to further the argument that has already been advanced, what continues to be present is variation within the same. Sameness is held apart from identity and thus _path\u00e9_ , as the plural form of pathos, necessitates, though only in this instance, what has to be understood as plurality within sameness. The challenge posed by the hand\/instrument relation is that whilst not being a symbol in the Warburgian sense, it becomes a relation whose reiteration within and across works has the effect, or this will be the argument, of undoing a conception of difference that is ground in sameness. It will be in these terms that the already noted distinction between the symbol and the motif reappears in a way that could be attributed the quality of calling into question the ease with which the symbol is taken to operate. The undoing of sameness holds open another possibility of repetition. And equally it brings into question the viability of what may be described as Warburg's acritical acceptance of Vischer's conception of the symbol.[9] For the latter, the symbol had a twofold presence: one described in terms of a domain of clear or direct meaning; the other structured by the 'sacramental', the religious or the superstitious. It is the latter that is taken to have opened up a link to the affective or the bodily. Even if the symbol could be understood in terms of having a singular determination\u2014and this despite the symbol being understood as that which arrested a continuity of movement between the possibilities of what Karen Lang describes as 'the perpetual rhythm . . . of opposing forces'\u2014it is still the case that the presence of differing symbols within an image, their presence and hence their relation, could still have the effect of introducing another and this time an originally discordant sense of polarity.[10] In other words, the presence of 'symbols' is able to open beyond their incorporation into Warburg's project. Moreover, the element taken to be significant need not be equated with Vischer's definition of the symbol.[11] Part of this process would be the recasting of symbols in terms of motifs to the extent that such a process overcomes the imposition of sameness that the retention of the symbol would seem to have involved.\n\nIn a minor text\u2014a review of an exhibition that took place at the Volksheim in 1907\u2014Warburg engaged with the question of what was involved in the presentation of works by Rembrandt. In his brief deliberations he referred directly to the exhibition that he had organized in 1905 on D\u00fcrer. Warburg wrote a defence of his own use of the 'colour reproductions of D\u00fcrer drawings'. That reproductions were used within this exhibition, and not the originals, is a curatorial and thus conceptual move that signalled the inescapability of reproduction. The substitution of the reproduction for the original should not pass unnoticed insofar as it signals Warburg's awareness that reproductions allow for the possibility of images coming-into-relation. In other words, developments within art history that resist chronology are themselves the result of the forms of technological innovation that made them possible. Warburg claims that if these 'reproductions' had 'failed to have their effect, this was because no interpreter was constantly present to help the visitor look beyond the tough and unfamiliar historical shell of the work to its universal humanity' ( _der die_ _Aufmerksamkeit von der fremdartig harten historischen Schale ablenkend, das allgemein Menschliche den Zuschauern f\u00fchlbar machte_ ).[12]\n\nWhile no more than a comment made in the course of an engagement with an already staged exhibition, it could be argued, nonetheless, that this line amounts to a generalized claim both about 'humanity' and its relation to that which is present within the image. The identification of 'universal humanity' ( _das allgemein Menschliche_ ) becomes the promulgation of a specific philosophical anthropology the evidence for which, according to Warburg, is present within the image. Moreover, it can be argued that it is precisely this anthropological claim which, despite later concessions concerning divisions and thus polarity, exerts a regulating force on the image. The positioning of an external constraint\u2014here 'universal humanity' perhaps as the discovery of a conception of _decorum_ that allows for conflict\u2014that exerts a regulating force on interpretation is repeated elsewhere in Warburg's work. It occurs when he writes that processes of exchange between 'past and present and North and South . . . afford a clearer understanding of the early Renaissance as a complete area of European civilization' ( _die Fr\u00fchrenaissance als Gesamtgebiet europ\u00e4ischer Kulturgeschichte_ ).[13] Warburg's position concerns the anthropological, which is understood here as given by the interplay, for example, of 'universal humanity' and the position of the early Renaissance as a _Gesamtgebiet_ (area, territory) marked by forms of neutrality rather than a locus of dispute and conflict, such that what then occurs is a form of regulation insofar as what the work is taken to present is now the stage on which that interplay occurs. The key is that it is there taking place rather than having to be uncovered and thus 'remembered'. Created, as a result, is a set of interconnected elements in which both a sense of propriety and externality prevail. This is the precise setting against which Benjamin writes. As such it is also the position allowing for the genuine differences between Warburg and Benjamin to be discerned. However, rather than argue in purely general terms, the staging of that difference will be developed here in connection to the role that D\u00fcrer's _Melancholia I_ plays in their differing projects.\n\nIn part, the difference between their approaches resides in how the nature of the polarity or tension that an image yields is understood. For Warburg its presence is formulated in terms of what he describes as 'a kind of polar functioning of the empathetic _memory-image_ ' ( _eine Art polarer Funktion des einf\u00fcheln den Bildged\u00e4chtnisses_ ).[14] Here memory works beyond the hold of intentionality. Memory attains a form of actuality that marks what Benjamin terms in _The Origin of German Tragic Drama_ the 'death of intention'.[15] The presence of that 'death' in Warburg is secured and an interpretive possibility opened in its wake by the retention of empathy; the 'memory-image' is 'empathetic'. Warburg's recourse to the language of empathy is an essential part of his project. Benjamin, however, eschews empathy as a ground of understanding. After all, he writes in _On the Concept of History_ that there 'is no better way of characterizing the method with which historical materialism has broken. It is a procedure of empathy' ( _Es ist ein Verfahren der Einf\u00fchlung_ ).[16] In addition, Warburg's formulation is posed at a distance from both the temporality of the image and the way in which the content of the image is there to be recovered in Benjamin's work.[17] Here the distance between them is fundamental. It is not a distance simply in terms of emphasis. On the contrary it concerns two importantly distinct senses of the way an image is caused to present.\n\nBenjamin describes the presence of the image (and it should be noted that he writes 'the true image' and in so doing refuses the possibility of the mere giveness of the image) as that which appears in 'the moment of its recognizability' ( _im Augenblick seiner Erkennbarkeit_ ). The full passage is the following: 'The true image of the past _rushes past_. Only as an image, which flashes its final farewell in the moment of its recognizability, is the past to be held fast.' ( _Das wahre Bild der Vergangenheit huscht vorbei. Nur als Bild, das auf Nimmerwiedersehen im Augenblick seiner Erkennbarkeit eben aufblitzt, ist die Vergangenheit festzuhalten_.)[18]\n\nThe complexities here are clear. This is a demanding passage. The claim does not concern the recognition of the image. Rather what is involved is predicated on a distinction between the 'true image' and the image as semblance. The 'true image' arises at that moment in which it is recognizable as an image. Hence, the claim made above that what is at work here is a conception of the image which refuses the possibility that the image could be identified with what appears, as though the image qua image was simply given to perception. In its emphasis on the centrality of 'recognizability', the importance of the concept of potentiality in Benjamin's work is reinforced. Semblance becomes the locus of the 'true'. It always had that potential. As a result, the question that attends Benjamin's formulation need not insist on the actual moment\u2014the ' _augenblick_ '\u2014as though it were simply an already determined experience. At work here is more than a simple phenomenology. The real question to which the passage gives rise hinges on what 'recognizability' means, since 'recognizability' cannot be separated from the supposition that there is a moment in which what is there, in being constituted as such, comes to be recognized. There is a capacity. With recognition, with its taking place, a potentiality qua potentiality attains actuality.\n\nWhile not all of the detail can be pursued in this instance, it is vital to note, nonetheless, that there is an important link here between this 'moment' and the workings of what Benjamin identifies in his _A Little History of Photography_ as the 'optical unconscious'.[19] It is, after all, the 'optical unconscious' that allows for the presence of that aspect of an image in which 'the future rests so eloquently that we, looking back, may rediscover it'.[20] Fundamental to this is what Benjamin refers to elsewhere as 'devotion to the insignificant' ( _Andacht zum Unbedeutenden_ ).[21] Opened up here is the possibility of being able to claim that an image remembers its future. Indeed, it is possible to suggest that this is precisely what occurred in the juxtaposition of images that allowed the violence already there in Pierro della Francesca's fresco to be remembered (the argument presented in chapter 1). It was the relation between images, thus relationality as a methodological prompt, that allowed violence to cut through what would otherwise have been mere beauty. Recall that the beauty of the faces in the fresco\u2014Christian faces and the face of the Jew\u2014would have been able to elide the differences that the violence was enacting. There had to have been the suspension of that beauty. Mere beauty is, of course, 'beautiful semblance' attempting to function as an end itself. The future's remembrance, occurring in a work that is caused to remember its own future, would not have been available in the context in which the image was produced and thus in which its initial occurrence as art's work took place. The image's capacity to remember, the image's own future as an already present condition of the image, is integral to that understanding of the ontology of the image in which its capacity to have an afterlife endures as a constitutive part. The distancing of 'beautiful semblance' ( _der sch\u00f6ne Schein_ ) as a mode of appearing is recast such that what it is that appears is replaced by 'the inconspicuous' (i.e., by _das Unscheinbare_ ). This is a move, a shift in register, that is of fundamental importance. As will continue to be argued throughout this chapter, what is essential to _das Unscheinbare_ , present here as a concept that allows for another presentation of the image, is that it be understood as a mode of appearance. Hence, Benjamin writes in a formulation that links the latter to the work's truth, 'It is the inconspicuous\u2014or also the offensive aspect (the two together are not a contradiction)\u2014which survives in\u2014true\u2014works and which constitutes the point where the content reaches the breaking point for an authentic researcher.' ( _Sondern es ist das Unscheinbare oder auch Anst\u00f6\u00dfige [beides ist kein Widerspruch], das in den wahren Werken \u00fcberdauert und der Punkt ist, an welchem der Gehalt f\u00fcr einen echten Forscher zum Durchbruch kommt._ )[22] The relation between the 'insignificant', the 'breaking point' ( _Durchbruch_ ) and the presence of truth as that which pertains to the object (hence the ontological having precedence of the epistemological) will return as fundamental to Benjamin's conception of both the temporalization and the force of the image. Benjamin's term here will be 'image'.\n\nOnce it is taken more generally, 'recognizability' stages its own set of demands. First, there is the question, what is it that is recognized? Second, how is non-recognition to be understood? (The 'moment' could never be mere experience; rather, there will be the more demanding temporality of _having-had-experienced_. The latter occurs in 'the moment' of the image's 'recognizability'.) And finally there is the question as to why there cannot be mere or pure expression or gestures. Moreover, precisely because both recognition and non-recognition are connected to memory, memory does not only open in one direction. In Warburg, as indicated in the passage cited above, memory and polarity are connected. Benjamin is equally a thinker of polarity. In _The Arcades Project_ , polarity, as a named presence, appears in the following way: 'The fore- and after history of a historical phenomenon shows up in the phenomenon itself on the strength of its dialectical presentation. What is more, every dialectically presented historical circumstance polarizes itself and becomes a force-field ( _einem Kratfeld_ ) in which the confrontation between fore-history and after-history is played out. It becomes such a field insofar as the present instant ( _Aktualit\u00e4t_ ) interpenetrates it.'[23]\n\nAgain the question that returns does not concern the mere presence of the same or even similar terms in Warburg and Benjamin\u2014here the reiterated presence of 'polarity'\u2014but rather how their difference is to be understood. What always matters is the thinking of difference that comes into play once there is a form of fracturing within what may have been assumed to comprise a singular conception of polarity.[24] If what has to be taken up is the fracturing of 'polarity, and it should be noted that even the possibility of this fracturing is a transformation of what is meant by polarity insofar as the term now loses any sense of ever having had an original singular determination, then D\u00fcrer's _Melancholia I_ is the specific instance within which that possibility can be worked out. In other words, _Melancholia I_ becomes a site within which to begin to understand the status of the 'image' within Warburg's and Benjamin's respective projects and respond to the questions raised by Benjamin's use of the term 'recognizability'.[25] It will be within this setting that the connection between the image and the motif occurs.\n\nThe differences between Warburg and Benjamin have little to do with the knowledge of historical art in any direct sense. Benjamin's work was a specific type of engagement. Moreover, the use and the meaning of the term 'image' ( _Bild_ ) in their differing projects is itself sufficiently far from settled for any straightforward comparison to be made. Taken generally, the contention to be argued here is that the difference between their projects lies in how 'life' is to be understood and how the image's relation to life is to be staged. Life here is not just human life. It is equally the life of images. The life of an image is its afterlife. The afterlife of antiquity\u2014what has already been designated by Warburg as 'the renewal of pagan antiquity'\u2014concerns its life. Here, however, caution is necessary. The invocation of life, no matter from which direction it comes, cannot be taken as an end itself. Life cannot emerge as a philosophical topos with any real sense of productive efficacy were it to have been separated from the world. There can never be just life. On the contrary, life occurs in the world such that life is always worldly. Life is also lived\u2014lived by humans and, to deploy a term used by Benjamin and Warburg, though in importantly different ways, also lived by 'creatures'. Warburg will write in relation to D\u00fcrer, in a passage that will return in the course of this chapter, that within _Melancholia I_ , he (D\u00fcrer) 'showed the spirit of Saturn made harmless ( _unsch\u00e4dlich gemacht_ ) by the individual mental efforts of the thinking creature ( _Kreatur_ ) against whom its rays are directed'.[26]\n\nAnd it is vital to note that captured in these lines is the claim that the hindrances and restrictions that had been imposed by Saturn, while they may still be present, have already dissipated. What had been a threat has been 'made harmless' ( _unsch\u00e4dlich gemacht_ ). Polarization, as operative, involves in this context a stilling, perhaps even forms of reconciliation within the frame. The difficulty is always going to be how to maintain a sense of polarity thought in terms of productive tensions, and not to let it give rise to the calm repose that defines the artwork in philosophers such as Schelling. Moreover, the challenge in so doing, as suggested earlier, recalls the interplay of the aesthetic and the moral at work in the use of decorum present in Cicero and Quintilian and the tradition of Roman aesthetics.[27] The poles are given together at the same time and recognized as such.\n\nBenjamin's 'creature' undoes the possibility of either repose or decorum by introducing a radically different conception of life whose effect would be to render both terms not redundant but precipitate. It would be too early for either repose or decorum. In a complex formulation, Benjamin argues that within the world created by _Trauerspiel_ , 'once human life has sunk into the merely creaturely, even the life of apparently dead object secures life over it' ( _Denn \u00fcber das Menschenleben, ist es einmal in den Verband des blo\u00dfen kreat\u00fcrlichen gesunken, gewinnt auch das der scheinbar toten Dinge Macht_ ).[28]\n\nThe terms 'creature' and 'creaturely' are central.[29] They will continue to provide points of orientation. They are the already present intersection of 'time' and a thinking of life. In other words, what these two formulations of the 'creature' underscore at this stage is the necessary connection between creaturliness and differing modalities of life. The latter is the case (i.e., the insistent presence of anoriginal difference within a thinking of 'life' obtains); even though life is present in each case, since both creatures are bound up with life, life does not yield an essential quality. In other words, life is present in ways that resist any form of equation that would allow them to be simple variations within life. Life is a locus of contestation.\n\nTo the extent that it is possible to distinguish between worldly life, human life and creaturely life, it must also be possible to plot points of intersection and overlap. However, it does not follow from the presence of types of imbrication that what overlaps exists within the same identical time frame and thus can be given in a single view. Hence the distinction between perceptions of 'renewal' and the presence of polarities in Warburg, on the one hand, and, on the other, what would be demanded of cognition from claims concerning the 'moment of recognizability' in Benjamin. It is precisely this distinction that complicates conceptions of polarity. Once it is remembered that the creature names the intersection of time and life, then Agamben's question\u2014How can an image charge itself with time?\u2014is the right question to be asking at the beginning.[30] However, answering it has to maintain the link between the image and life. The image cannot be taken as an end in itself. Images are charged with life: timed life and the lives of time. Neither time nor life starts as a singularity. Hence, what matters is the way different and contesting senses of life figures and are timed in their difference. Within this setting the fracturing of a single view involves coming to an understanding that 'creaturely life' is that modality of human life in which life, expressed now in terms of the 'worldliness' of the world, has had 'worldliness' taken from it. Life then is 'mere life'. Worldliness is an integral part of that which begins to define the being of being human. As will become clear, 'creaturely life', for Benjamin, is fated life; therefore, it is life equated with the continuity of guilty indebtedness.[31] And yet it is still a life. It is a life, however, that cannot be defined simply in terms of its actuality. It is not present to itself as life. Such a response would be the pragmatic response that equates the normative with the given. Pragmatism is nothing other than the naturalization of the creaturely. The creaturely is that modality of life that is situated, as much in relation to the potentialities inherent in it, as it is by its current actualization. Once recognized for what it is, creaturely life, rather than being taken as fated and thus subject to own its continual return, is also characterized by the possibilities that attend it. Attending is a sign of potentiality. The possibility of the actualization of that potentiality distances the hold of the continuity of debt. (Or this is, at least, the risk to be taken.) The recognition of this polarity is central.\n\nThe most sustained engagement with D\u00fcrer's _Melancholia I_ occurs in Benjamin's _The Origin of German Tragic Drama_. It is situated as a moment within the continuity of an engagement with the topics of 'allegory' and 'melancholy'. While it is always possible to sum up Benjamin's engagement, rather than proceed with summary, Benjamin's one methodological imperative will be followed: _Method ist Umweg_.\n\nBenjamin quotes Hamlet. He quotes him in relation to the sketch of a Baroque anthropology. In the context of the play the lines that appear in Benjamin's text formed part of a soliloquy. Hamlet speaks:\n\n> What is a man\n> \n> If his chief good and market of his time\n> \n> Be but to sleep and feed? A beast, no more.\n> \n> Sure, he that made us with such large discourse,\n> \n> Looking before and after, gave us not\n> \n> That capability and godlike reason\n> \n> To fust in us unused. (Act 4, Scene 4)[32]\n\nThe 'beast' is the creature, not the animal but 'mere life'. In contradistinction to the 'beast' there is both 'capability' and 'godlike reason'. Both are there. More significantly, they are there from the start, there as potentialities awaiting their own actualization. They signal the possibility of another life. However, they are not there 'to fust in us unused'. (Present in Benjamin as _Um ungebraucht in uns zu schimmeln_ ). 'To fust' is both to rot and have the smell of decay. To write as Shakespeare does that 'capability and godlike reason', while they may lie dormant, may remain yet to be actualized has a doubled meaning. In the first instance, their force remains hidden, and thus, second, they do not form part of any 'now'. What this means is, of course, that the 'now', as _Aktualit\u00e4t_ , provides the point of departure. The break-up of semblance, here the naturalization of life as mere life, the creaturely, the 'beast', occurs within and creates the now as 'actuality'. Hamlet's formulation brings two questions into play.\n\nThe first is the question of their nonappearance: What holds 'capability and godlike reason' at bay? Second, given their non-appearance, what then is at stake in their actualization? What the second question opens up is, of course, their use within the present. In other words, what would be involved were they to become actual and given rise to specific tasks? The latter of these is the move from presence to present, and this is the actualization of an immanent possibility in the world, which cannot be understood other than as occurring in relation to that world which, were its actualization to occur, would then have become integrated into the world's transformation. This might be described as the _othering_ of the world.[33] And this is the possibility that continues to attend. Moreover, the second question is not an ethical one, as though the ethical could be adduced. Rather, it is the moment at which the presence of what will be called _virtue in being_ is acknowledged insofar as what would then be affirmed is the anoriginal presence of the ethical within the ontological.[34] Evoking the world therefore cannot be thought as independent both from a sense of value and the possibility that value is an intrinsic quality of worldly being. The values that pertain to worldliness are precisely those that are effaced by the subjection of human beings to fate; namely the province of the creaturely. This is a claim about life\u2014both worldly life and by extension the life of images. Here there is a momentary affinity with Warburg. It is in this context that Geroges Didi-Huberman is right to argue the following: 'Time conceived as a succession of direct relationships ('influences') or conceived in the positivist way as a succession of facts had no appeal for Warburg.'[35]\n\nThis refusal of a specific sense of time, and with that refusal the return to the time in which the afterlife of images is to be thought, is to work against this very tendency\u2014namely working against a tendency which has to be understood as prevailing and which not only maintains time _as_ chronology but also identifies chronology as commensurate with time itself. Fate appears here as the unrelenting 'succession of facts' that could not be otherwise. 'Succession' does not pertain to content per se; rather, 'succession' here is the articulation of that content within a specific structure of time. In other words, what this means is that positivism cannot be separated from processes involving the naturalization of continuity. There is, however, a limit here. Taken on its own, it is simply not sufficient to argue against 'succession' as though that claim were simply discursive. There has to be that form of interruption\u2014the countermeasure\u2014that repositions 'succession' as having been created. This is, in part, what Benjamin means by the formulation 'moment of recognizability'. And, as stated at the outset, it is the interruptive or destructive element that is absent from Warburg. The justification for this destructive act\u2014and it is destructive in the precise sense that it generates an image, it does not presuppose one\u2014resides in the potentiality for another life. There are therefore two specific, albeit interrelated components at work. The denaturing of nature, the undoing of 'succession', has to be charged with an occasioning or an allowing. And yet, precisely because what occurs in that opening has no determined direction, since not only is there the absence of an already present image to be repeated, it is also the case that the 'worst' is always a possibility, there needs to be that which continues to maintain a concern with life.[36] Theodor Adorno's claim that there cannot be a 'correct' ( _richtiges_ ) sense of life within a prevailing sense of the 'false' necessitates that the falseness of the world be seen as what it is (i.e., as life condemned to be lived _im falschen_ ).[37] Hence, there is a countervailing language of truth. What is therefore opened up is a possibility that hinges on the development of such an understanding. As a consequence there is a further coalescence with the image.\n\nIn _The Origin of German Tragic Drama_ , Benjamin's opening evaluation is of the world in which D\u00fcrer's _Melancholia I_ appears. The opening is there almost as a result of the demand for that form of recognition that inheres in the interplay between world, life and potentiality. _Melancholia I_ occurs within the world that it shows, and it is important that the anthropological determination with which Benjamin is concerned be understood as concerned with the world. The anthropological here is a determining relation between world and subjectification. In this regard, Benjamin's claim is both diagnostic and anthropological. He writes: 'The religious man of the Baroque era clings so tightly to the world because of the feeling that he is being driven along to a cataract with it. There is no Baroque eschatology.' ( _Der religi\u00f6se Mensch des Barock h\u00e4lt an der Welt so fest, weil er mit ihr sich einem Katarakt entgegentreiben f\u00fchlt. Es gibt keine barocke Eschatologie_.)[38]\n\n'The world' ( _die Welt_ ) is already given. It is a world that holds itself in place and reciprocally, for this will be the policing function of sovereignty, is held in place without the eschatological. Benjamin's claim that 'there is no Baroque eschatology' ( _Es gibt keine barocke Eschatologie_ ) is decisive. The latter\u2014the eschatological\u2014names that which when thought provides the interruption of the continuity established by religion: the interruption of the naturalization of religion is the project of a political theology. In other words, the absence of the eschatological\u2014and its absence is Benjamin's contention\u2014means that for a certain subject position, named in the passage cited above as the 'religious man' ( _Der religi\u00f6se Mensch_ ), there was no possible recognition of a day in which the world could be other than that which it was always going to be. The world remains held within its earthly eternal return. As has already been noted, this is the position that is understood by Benjamin as 'creaturely life'. If there is a further element in Benjamin's engagement with the Baroque and thus with D\u00fcrer, it is the recovering of the potentiality for the eschatological within a set-up in which it could not be thought. In other words, the project is the recovery of hope in an 'empty world' ( _leere Welt_ ). Hope's absences and the naturalization of the 'empty world' as the world accounts for what was unavailable within the Baroque. It is precisely the possibility of avoiding the repetition of that possibility that makes Benjamin's _The Origin of German Tragic Drama_ equally a work concerned with modernity.\n\nFate and guilt are for Benjamin as much modalities of time as they are the creation of subject positions. What is at work in both is therefore a particular modality of subjectification. Within their hold, life becomes 'guilty life'. As noted above, it is within this life (i.e., this anthropological setting) that the eschatological, understood as the mark of a political theology, would become an interruption in time. Not only is there a projected countermeasure in which the interruption takes on the quality of a catastrophe\u2014which would be the interruption, though as a caesura of allowing\u2014it is equally and simultaneously the creation of another possibility for subjectivity.[39] Within this guilty life, Benjamin notes that life is subjected to law's naturalization. Were such a setting to be accepted, it would mean that there could be nothing other than nature's continuity. Responding has to be nature's undoing, which is the recognition that the natural is itself an effect. There needs to be a presentation of life which would be an expression of life in which it was not 'an expression of the guilty life falling under the hold of the law of natural life' ( _ein Ausdruck der Verfallenheit des verschuldeten Lebens an das Gesetz des nat\u00fcrlichen_ ).[40] Fundamental to this formulation is a sense of 'fallenness' ( _der Verfallenheit_ ). This is of course the world to which Hamlet responds. It is a world in which, in the apparent absence of value, the 'empty world' ( _leere Welt_ ) of _Trauerspiel_ , there arises for Benjamin another sense of _Verfallenheit_. Subjects in this world have in Benjamin's terminology 'a grim belief in having been given over to fate' ( _an die Schicksalsverfallenheit_ ). There is a belief of having fallen under the hold of fate ( _an die Schicksalsverfallenheit_ ).[41] And yet, were the creation of images to have no other task or form of appearance than the one bequeathed by _Schicksalsverfallenheit_ , by this having fallen, then image creation would be formed by the continual reiteration of that 'grim belief'. As a result, the creation of images would have had a singular constraint. The argument will be that D\u00fcrer's engraving is situated at the heart of the problem of a possible making present that undoes the hold of this world and its assumed sense of subject creation. The latter is nothing other than the naturalization of subject positions as subjugated to the temporality of fate. What awaits in D\u00fcrer is the undoing of that temporal schema.\n\nIn a world in which actions have been robbed of authenticity\u2014and Benjamin writes of 'inauthentic actions' ( _unechter Handlungen_ )\u2014what stands against their repetition and the continuity of the 'empty world' is life. In a line that is both prescient and charged with the possibilities marking his own project, he writes that, 'life itself struck back against this'. ( _Dagegen schlug das Leben selbst aus._ )[42] In this extraordinary line there is the emphatic suggestion that life, perhaps life itself, is charged with possibilities that enable it to resist. The body as the locus of affect has to undo its having been deadened. Life striking back, a movement that is not merely physiological but is the emergence of the countermeasure is present in this context in terms of 'genius'. What defines the problematic presence of the countermeasure is its incorporation into a structuring polarity that holds it in place. Hence, for Benjamin, the dog in D\u00fcrer's work is both 'vapid' and a 'tireless investigator', and yet there is always the possibility that the dog's dreams remain uncreative. With melancholy, wisdom is positioned by the creaturely. Nonetheless, there is an element that continues out of the creature's reach. What endures unheard within it is for Benjamin 'the voice of revelation'.[43] In the realm of thought there are uncreative possibilities\u2014possibilities that are held in place by the languor, ' _acedia_ ', the influence of Pseudo-Aristotle, endure a set-up that will reappear later on in Benjamin's writings as 'boredom',[44] namely an affective state charged with possibilities. However, as with 'boredom', the dulled body continues to confront itself. Thought becomes agitated. Allegory as a creative possibility is already there for Benjamin. It is located in the 'stone' in D\u00fcrer's work. It is an instance of the trap which when confronted does not give rise to any form of renunciation. Rather, the tension is located. It 'lies ready to burst into bloom' ( _zu sprengender Entfaltung bereit liegt_ ). This interruptive _Entfaltung_ marks the moment of possibility, what has been referred to already as the countermeasure. The latter does not just occur; it takes place in the name of life.\n\nThe German _Trauerspiel_ was never able to inspire itself to a new life: 'It was never able to awaken within itself the clear light of self-awareness ( _den Silberblick der Selbstbesinnung in seinem Inneren nie zu erwecken vermocht_ ). It remained astonishingly obscure to itself and was able to display the melancholic only in the crude and washed out colours of the mediaeval complexion books. . . . The images and figures presented in the German _Trauerspiel_ are dedicated to D\u00fcrer's genius of winged melancholy. The intense life ( _Leben_ ) of its crude theatre begins in the presence of this genius.'[45]\n\nIf what took place within _Trauerspiel_ were to encounter a limit, then it would be a form of self-limitation in which what was both possible and impossible would then have occurred. It becomes the project of genius. While it will be necessary to return to the presence of the 'genius', and therefore to the figure of genius as staging a relation within time to time\u2014genius hints at the countermeasure\u2014the clue to understanding what is at stake has already been located in the claim that life itself will always have had the possibility of striking back against it own subjugation. 'Genius' here names that exact possibility. In _Fate and Character_ Benjamin expresses the presence of this destructive eventuality in the following terms: 'It was not in law rather in tragedy ( _Nicht das Recht, sondern die Trag\u00f6die was es_ ) that the head of genius lifted itself ( _sich . . . erhob_ ) for the first time from the mist of guilt, for in tragedy demonic fate was breached ( _das d\u00e4monische Schicksal durchbrochen_ ).'[46]\n\nHer head is yet to have lifted. _Trauerspiel_ is not tragedy. The compass in her hand as allowing for that other measuring of the world, the one in which she would have taken the world's measure, remains unused. She awaits.[47] Melancholy becomes for Benjamin 'an overwhelming indeterminate mood'.[48] The question to be addressed concerns the extent to which this 'indetermination' has another possibility that awaits being remembered. Perhaps it is even a possibility which, while implicit in Benjamin's argument, is yet to be articulated.\n\nAnd Warburg's D\u00fcrer, his _Melancholia I_ , what awaits her? Again what matters has a twofold quality. In the first instance, it is the identification of those elements of his engagement with the work that are germane to this project and then, second, what is implicit within them. Precisely because what is at stake is the presence and thus registration of antiquity, what is important is the way Warburg traces its registration. A fundamental element of the overall project consists of different responses to astrology (hence the fundamental importance of his 1920 paper _Pagan-Antique Prophecy in Words and Images in the Age of Luther_.)[49] Central to the argumentation of this paper is Warburg's engagement with Philip Melanchthon's letter to Johann Carion written 17 August 1531. The letter, discovered by Warburg himself, concerns, initially, the response to the sighting of a comet. The letter sets in play that particular presentation of astrology that reveals, for Warburg, that astrology is not a singular occurrence. Moreover, it is not antithetical to humanism. Indeed, what Warburg uncovers in the letter is the way humanism and astrology can be combined. In Warburg's terms, Melanchthon 'adopted the astrology of the ancient world as an intellectual defence against a cosmically predetermined earthly fate' ( _gegen das kosmisch bedingte irdische Fatum_ )[50] (hence the difficulty of a simple interpretation of images such as Hans Burgkmair's _The_ _Young Maximilian_ _Instructed in_ _Magic_ [1514\u20131516], an image that figured in _Der Wei\u00df Kuning_ in which there is an overlap of kingship, statecraft and magic). Thus it is deployed in the service of that which occurred as one removed from magic, or statecraft.[51] However, the language of fate in Warburg's formulation cannot go unremarked. Indeed, part of the argument to be developed in what follows is that the limitation of Warburg is not found in the presence of the 'pagan' or in its 'renewal' but rather in the way fate registers both in terms of its self-image and equally in terms of the conception of time that it demands. The Benjaminian response to 'fate' and to 'myth' had not registered.\n\nLuther did not accept astrology. Contra Melanchthon, or at least contra the complex ambivalence in his relation to both astrology and magic, Luther's stand is unequivocal. His words, quoted by Warburg, are explicit: ' _Es ist ein dreck mit irer kunst_ '.[52] Moreover, Luther mocked Saturn, equating the impact of having been born _sub Saturno_ to the same restrictive effects on his life as the lateness of friends and family. As Warburg shows, what Melanchthon's letter reveals, once contextualized, is the presence of divergent views of astrology. It then needs to be added, since this is also part of Warburg's project, that astrology as a prophetic model, and thus as deploying a specific structure of fate, was open to forms of manipulation. Warburg recounts the attempt to control the political dimension of Luther's impact by altering the date of his birthday such that his 'fate' was repositioned as a result. The astrological as evidence of the immutability that underwrites astrology and allows it to be prophetic is rendered mutable in the name of the political. Indeed, it might be added here that the evidence afforded by both Machiavelli and Shakespeare is that what is occurring is a fracturing of a world in which immediate recourse to the temporality of fate, which is the temporality in which astrology and prophecy work\u2014remembering equally that this is taking place within what Benjamin aptly describes as an 'empty world' in which objects are endowed with powers such that they orchestrate events, orchestrating them beyond the hold of direct intervention (even Iago is in the end subject to his own machinations)\u2014is becoming mutable in the name of the political. Or rather, the political is emerging as an opening provided by refusing to allow fate to have the final determining effect on sovereignty. What, after all, is the rebellion of the Percys and Harry's defence of his father in Shakespeare's _Henry IV_ parts I and II other than the clash of two modalities of sovereignty, which is in fact the clash of two temporal orders. It will be essential to return to consequence of a type of fracturing enjoining forms of mutability since what is always at stake is the recognition of those temporal orders as co-present in their irreducibility.\n\nIn the case of Warburg, what is occurring in D\u00fcrer's _Melencolia I_ is the opening in which Saturn, while present, is no longer to be feared. As a work, _Melencolia I_ forms a fundamental part of the 'struggle to cast off pagan cosmological fatalism' ( _heidnisch-kosmologischen Fatalismus_ ).[53] There is a type of optimism here. Moreover, it is an optimism that is linked to success.[54] Warburg began by noting that magic had destroyed the space between the human realm and the realm of objects by allowing that space to be filled by superstition. This is a position that is also at work in the manuscript on Giordano Bruno in which there is the clear suggestion that what is significant about Bruno is 'belief in the utility of rapturous thought' ( _den Glauben an d. Zweckma\u00dfigkeit des hingerissenen Denkens_ ).[55] The fundamental point is the link between 'utility' ( _Zweckma\u00dfigkeit_ ) and 'rapturous thought' ( _des hingerissenen Denkens_ ), since what is staged is the relationship between utility and a mode of thought that, even though it cannot be directly equated with the rational utility, remains.\n\nWhat is to be seen in _Melancholia I_ is a different relation between humans and objects. This is taking place for Warburg in the age of Faust. What is occurring is that the scientist, in Warburg's terms, is 'trying to insert the _thought-space_ of rationality ( _Denkraum der Besonnenheit_ ) between himself and the object'.[56] In other words, the image itself stages or presents the movement from the dominance of magic, astrology and thus fate towards the opening up of a 'thought-space' ( _Denkraum_ ). Moreover, it is a conception of thought that takes on the hue of objective rationality. It would be as though the 'thought-space' ( _Denkraum_ ) just opened up. What this means of course is that Benjamin's sense of both the 'dialectical image' and the already noted 'moment of recognizability' would have become otiose. The opening would not have depended upon a form of destruction. Rather, it would have emerged as part of a move towards 'enlightenment'. The difficulty is not the presence of rationality. On the contrary, the difficulty is understanding the temporal framework within which it appears and then Warburg's inability to recognize that even the rational and the enlightened will have a doubled existence.\n\nAt the end of his long and detailed analysis in which not only all relevant literature is reviewed, important additional qualifications are made in relation to the interpretation of the contents of the image, Warburg is able to write the following:\n\n> The truly creative act\u2014that which gives D\u00fcrer's _Melencolia I_ its humanistic comforting presence ( _zum humanistischen Trostblatt_ ) from the fear of Saturn\u2014can be understood only if we recognize that the artist has taken a magical and mythical logic and made it spiritual and intellectual. The malignant child-devouring planetary God, whose cosmic contest with another planetary ruler seals the fate of the sunlit creature ( _das Schicksal der beschienenen Kreatur_ ) who becomes the sculptured embodiment of a thinking working human being as a result of D\u00fcrer's humanizing metamorphosis ( _wird bei D\u00fcrer durch humanisierende Metamorphose die plastische Verk\u00f6perung des denkenden Arbeitmenschen_ ).[57]\n\nWhat is significant is the language of a type of contentment and reconciliation. What counts is that the forces, the polarities within the frame have become becalmed. While it is true that the strength of Warburg's methodology means that another return, another interruption, or another refusal of continuity, would allow that setting to become displaced, it remains the case that the 'compass of genius' held by Melancholia is the result, again in Warburg's own terms, of D\u00fcrer having 'showed the spirit of Saturn made harmless ( _unsch\u00e4dlich gemacht_ ) by the individual mental efforts of the thinking creature ( _Kreatur_ ) against whom its rays are directed.'[58] For Warburg, there is still a limit. D\u00fcrer's Melancholia is held fast by it. That she still wears the _teukrion_ reinscribes her back into the world from which she cannot escape. She is subject both to ancient remedies for melancholia as well as to Marsilio Ficino's advice concerning how to hold off the effect of Saturn by the use of a 'numerological magic square'.\n\nThere is a further and fundamental point that needs to be made here. To the extent that Saturn is 'made harmless', then the threat of fated life may be thought to be over. This creature\u2014'the thinking creature' ( _Kreatur_ )\u2014is projected as having won out over the threat of what for Benjamin would be 'the merely creaturely'. While this may be a possibility that Warburg would have come to know as having been too hasty, this does not diminish the remarkable set of discoveries that his text presents. The detail is not the issue. And yet, it has to be added that it is important to recognize that in regards to what appears within the image, its force as an image, is for Warburg the undoing of the creaturely. As a claim, however, this is simply untrue. The threat of what Benjamin identifies as 'the merely creaturely' endures. It is there, for example, as the threat within totalitarianism in which, to deploy Arendt's terminology, the subject becomes 'superfluous' and thus as nothing other than 'the merely creaturely'. It endures to the extent that poverty, slavery, sexism, racism and the like deprive subjects of worldliness. What the endurance of the creaturely indicates is that not only is a more complex understanding of the image needed, it is also necessary to see the subduing effect of the 'the thinking creature' as premature.[59] Again what endures is a sense of the precipitate. If the image really were of 'a humanistic comforting presence' ( _zum humanistischen Trostblatt_ ), then not only would the image become yet another instance of 'beautiful semblance', it would be an end-in-itself. The limitation of the Warburgian project does not lie in the details, nor in the undoing of chronology and then the latter's naturalization. Indeed, these are its genuine strengths. Rather, the limitation of the overall argument is the way in which it dramatizes the intellectual paucity of humanism, that impoverishment is not located in its aspirations but in the thought of its accomplishment, of its having been achieved. A return needs to be made to a sense of polarity in which instead of an increasingly benign oscillation between the poles, the tension and thus the incompleteness can be regained and thus retained. The question that insists, therefore, is how this incomplete state is to be understood. The answer is that it is allegory's other side. It is the position announced by Benjamin in _Central Park_ thus: 'Majesty of the allegorical intention: to destroy the organic and the living\u2014to eradicate semblance' ( _Zerst\u00f6rung des Organischen und Lebendigen\u2014Ausl\u00f6schung des Scheins_ ).[60]\n\nThe interruption of the temporality of what is named here as 'semblance' is equally the destruction of that 'semblance'. Destruction is not nihilism. Destruction is more destructive. It involves the identification of appearance with 'semblance' and thus demands not another appearance but that which appears be understood differently and what appears be transformed in the process. In the context of the chapter, this sets up a return to D\u00fcrer's _Melancholia_ or, more exactly, a return to the hands holding 'the compasses of genius' (i.e., a return to anoriginal relationality in which the relation is present as a motif).\n\nThere have been many returns to Warburg, attempts to find a moment of summation in which fundamental aspects of the project were both captured and presented. Writing of Warburg, Edgar Wind formulated one such position; moreover, it is a formulation that allows elements of the analysis outlined above to be carried a stage further. Wind wrote, 'In analyzing early Renaissance men, could he penetrate through to that level at which the most violent conflicts are reconciled and develop a psychology of equilibrium ( _Ausgleichpsychologie_ ) which assigns to opposing impulses different \"loci\", poles between which the subject oscillates, whose distance from each other is the measure of the oscillation.'[61]\n\nEqually, even though he gives greater emphasis to the presence of reconciliation and what he called 'equilibrium', Howard Caygill suggests that in Warburg's engagement with D\u00fcrer, the latter's work 'represents a moment in which the drives for knowledge and for beauty find themselves in rare equilibrium.'[62] Responding to both these summations of a project and the presence of D\u00fcrer within it has to begin with what is staged by terms such as 'reconciliation' and 'equilibrium'. To the extent that this formulation is retained, then what exists within a state of harmony is a play of forces that is present both inside and outside the work. Works of art\u2014images\u2014then become differing loci of reconciliation. Art's project, as has been intimated, will then be equated with a form of reconciliation leading to forms of consolation. The consolation is neither aesthetic nor religious. Rather, it is the consolation of a harmony in which rationality prevails. The response to this position, the response to Warburg, perhaps despite the imperatives that may have been driven by biography were it to be located in Walter Benjamin, is to be found in the proposition that, despite the significance of a project that resists conceptions of temporality that are structured by positivism and, more importantly, one that allows for the interruptive force of memory, it nonetheless remains a version of cultural history that 'lacks the destructive element' ( _das destruktiv Moment_ ). Destruction is linked, inseparably, to the 'moment of recognition'.\n\nThat it is possible to keep turning Benjamin back on Warburg, and that their differing undertakings be noted, means that the question that has to be addressed, once again, is, what is at stake in this difference? How is it to be thought? What has to be created is what Benjamin identifies elsewhere as the 'offensive part' ( _anst\u00f6\u00dfig_ ) of 'the work', the moment where content reaches what he describes as the 'breaking point'. The identification of the affective, or the muscular, of path\u00e9 at their most discordant, is as much the identification of the bodily and atmospheric as it is the identification of the ideational. Indeed, it cannot be one without the other. Form and appearance are always informed. The illusion of appearance is that it is either just appearance or the merely abstract, and therefore only ever forms itself. Despite the limitations that may have emerged, or rather in making use of them as limitations, this project must be taken a stage further. Warburg's lead needs to be followed. The project does not concern a single work, a single image. At stake here is what occurs once a complex sense of relationality is allowed to set the measure. Motifs present within the process are already named as coming-into-relation.\n\nHenceforth what matters are as much relations within singular works as relations between works. Both allow for another understanding of singularity. Rather than looking for similarities and an array of images that allow for a coalescence in which a sense of sameness endures, even though the affinities established would still distance both the idea and the attempt to discern identity by having established a causal relation between the idea and the particular, what has to be sought, following Benjamin, is the 'offensive'. It has to be located within what would otherwise have been understood as the reiteration of sameness. 'Reconciliation', both as a project and as that which had an image, would encounter both its limit and its destruction if what were secured is the 'breaking point'. The 'breaking point', arising as it does through an emphasis on the 'insignificant'\u2014again as a mode of appearing\u2014is the moment of reconciliation's impossibility. It is the moment in which that impossibility is recognized as such. Warburg's sense of polarity will have appeared as the attempt to efface the 'offensive'. (This 'breaking point' can be further understood as the moment in which a singularity appears with such intensity that it cannot be subdued by incorporation into any sense of universality of which it would then be an instance.)\n\nAnother beginning. Returning to D\u00fcrer's _Melancholia I_ is to start again. Starting now with 'genius', this other beginning therefore involves starting with a more complex conception of production, since the presence of production is named as 'genius'. This has become possible insofar as, within Benjamin's terminology, 'genius' names a form of agency that marks the depositioning of fate. As a result, 'genius' twists free from both a conventional Kantian determination, as well as the latter's reworking within Romanticism. Genius becomes a destructive _ingenium_.63] Even for Warburg, for whom the thinking of destruction is necessarily foreign, 'genius' is still linked to that which stills the effect of astrology. Melancholia, as the figure of genius, is therefore imbued with different potentialities. While her head is yet to lift, it is nonetheless ready. She awaits. The question that she brings into play therefore is how this potentiality is to be understood. What is awaiting? What is awaiting as a potentiality? These questions insist precisely because the necessity enjoined by the presence of a potentiality is always accompanied by a sense of contingency in relation to any action resulting from the presence of a potentiality. In sum, the necessity of possibility cannot be separated from the contingency of its actuality. This is Aristotle's key insight in regards to potentiality; the latter, potentiality, is named in the following as _dynamis_ (hence the inevitable interplay in English between 'potentiality' and 'capacity'). The formulation of _dynamis_ that is central for the argumentation being presented here occurs in the _Metaphysics_ (1046a30). In the passage in question, Aristotle argues that every 'capacity' ( _dynamis_ ) is linked to a contrary 'incapacity' ( _adynamis_ ). Aristotle's formulation is the following: 'Incapacity and the incapable ([\u1f21 \u1f00\u03b4\u03c5\u03bd\u03b1\u03bc\u03af\u03b1 \u03ba\u03b1\u1f76dunami\/a) \u03c4\u1f78 \u1f00\u03b4\u03cd\u03bd\u03b1\u03c4\u03bf\u03bd) is the privation contrary to capacity (\u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9) in this sense; so that every 'capacity' has a contrary incapacity (\u03c4\u1f78 \u03b1\u1f50\u03c4\u1f78 \u03c0\u1fb6\u03c3\u03b1to%5C) \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2 \u1f00\u03b4\u03c5\u03bd\u03b1\u03bc\u03af\u1fb3) for producing the same result in respect of the same subject.'[64]\n\nAristotle's argument is that what defines all and every 'capacity' is the simultaneous co-presence of an 'incapacity'. Potentiality is therefore defined in terms of relation between necessity and contingency (in which, it needs to be added, the contingency of result takes the necessity of existence\u2014the existence of a potentiality\u2014as its condition of possibility). Once awaiting can be thought as a form of potentiality, then awaiting\u2014now present as a philosophical concept\u2014has two defining elements. The first is that which charges any moment is the potentiality that awaits. The second is that because of the contingency of actualization, the question that pertains with equal force is how actualization is to be realized. Potentiality brings the question of actualization, a question determined by actualization's non-necessity, into consideration. Melancholia, the figure in D\u00fcrer's work, stages the complex relation between necessity and contingency that defines, following Aristotle, potentiality.\n\nMelancholia is there. There within the image. Her left hand is at rest. It awaits. Potentiality ( _dynamis_ ) is present as accompanied by im-potentiality or incapacity ( _adynamis_ ) only insofar as actualization is contingent, since after all, 'There is no Baroque eschatology' ( _Es gibt keine barocke Eschatologie_ ), or at least there was not an eschatological impulse that was perceived as such at the time, so the image has to be made to remember such a possibility. If this is the case, then the compass in _Melancholia_ 's hand\u2014that which has already been described as 'the compasses of genius'\u2014is not the key element if either the compass or the hand were taken on its own. Rather, what is at stake is the relation between the hand and the compass. Neither one nor the other; on the contrary, there is a relation. Internal to this work\u2014 _Melancholia I_ \u2014is the relation between the hand and the instrument. And yet, it has to be added immediately, and this point is decisive, this relation even with the frame of one work, cannot have a singular determination. Moreover, it is only by insisting on the relation between the hand and instrument that what then occurs is the possibility of emphasizing the constancy of the oscillation between production and non-production (i.e. between _dynamis_ and _adynamis_ ). Melancholia is literally unproductive. The argument has to be that the recognition of that limitation as a limitation is precisely that which overcomes the limit. The moment of recognition is the moment in which the potentiality attains a form of actuality. Potentiality presses on the present. And, since what is at stake is the relation between the compass and the hand in which the practice of geometry has ceased all that can be said is that she awaits.[65] It is only by taking this relation as central that there is a productive affinity such that relationality emerges as a locus of divergent possibilities.[66] Such an eventuality points to the anoriginal complexity of relationality. The hand is not being adduced to D\u00fcrer. As is well known in his _Elementa Rhetorica_ (1541) as part of a more general _ekphrasis_ of _Melancholia I_ , Joachim Camerarius wrote of D\u00fcrer as the artist 'from whose divine hand ( _diuine manus_ ) many immortal works still exist'.[67] This evocation of the hand is not an arbitrary motif. On the contrary it establishes the link that allows for the possibility of the history of the hand\/instrument relation, a history moreover that is the connection between St Luke, Michelangelo and, as will emerge, George Baselitz. (That set of relations will be taken up in greater detail in chapter 3.)\n\nIf there is a way into _Melancholia I_ that brings out the tension inherent, both in the work itself and then in the way it figures within the writings of Benjamin and Warburg, it inheres in the relation between the hands and literal implements of production (assuming measurement and thus knowledge can be characterized in this way). The relationship between the hand and questions of production and creation occur in a complex set of ways. Melancholia's estrangement from production, an estrangement that remains, nonetheless, a mode of relationality, needs to be seen as opposed to other moments in which the hand is inherently productive or where the possibility of the hand being productive no longer pertains. It is the link between inherent production and what can be described as the immediacy of production\u2014the latter a formulation that identifies a specific temporality of production\u2014that is important here. Again, it is the nature of production that is fundamental when it is presented in terms of a polarity defined by an either\/or. This either\/or, either the hand is productive or it is not, stands at an important distance from awaiting. Indeed it can be argued further that 'awaiting' works to deconstruct the opposition that this either\/or seeks to secure. This is the point that has to be developed.\n\nOther relations are both possible and necessary. Note the play of hands in Hans Baldung Grien's _Bewitched Groom_ (1544) (figure 2.2).[68] The groom has been undone.[69] His hands have been robbed of their utility. Utility, however, is defined in terms of the literal utility and then the symbolic intensity of the tools that he no longer holds. At the window there is a witch. What of her hand and its relation? The torch with its flame and which she holds in her hand is at work. Its symbolic dimension with its evocation of lust and female power is on display. Even though there is a literal threat imposed by the flame, it is also the case that a malevolent power works through the torch. Her hand\/torch relation\u2014and it is essential to underscore that what is at work here is a relation\u2014is implicated in an undoing; there is a doing and undoing. What have been undone are the groom's hands. As he lies unconscious, his hands have released their hold. Whatever utility they had has ended. They do not await. They have been separated from the implements that they had held. The undoing of utility needs to be held apart from the potentiality inherent in awaiting. There are many other possible hands that can be brought into play at this precise point.\n\nHans Baldung Grien, _Bewitched Groom_ (1564). British Museum, London.\n\nIn a woodcut illustration to Petrus Canisius's _De Maria Virgine_ (1583), Gabriel's hand mediates the stream of light that is God's power at work (figure 2.3). The hand is not causally related to the Annunciation. And yet, as Walter S. Meilon has pointed out, 'by showing Gabriel's sceptre immersed in the ray of divine light extended from the holy name of God to the virgin, illustrates the notion that he salutes her _luculenter_ , revealing the connection between what transpires within her and its majestic source on high'.[70] It should be added here that the 1577 version of the text, while not featuring the same set of images, maintains in its images the use of the Latin INRI ( _I\u0113sus Nazar\u0113nus, R\u0113x I\u016bdae\u014drum_ ) to identify Jesus. Here the Hebrew is deployed to name God. And yet, of course, what is named is precisely that which can be neither said nor represented. The sun is adorned with 'the name' ( _haShem_ ). In other words, present here is that name that endures as the necessarily unpresentable. Understanding the use of the Hebrew raises an interesting problem. It could have been deployed, as indeed it was elsewhere, as an integral component of the iconoclasm of the Reformation. Here, however, in a work that is central to sixteenth-century Marianism, it signals the triumph over the Jews. The Hebrew accords with the broken step in the image itself. Gabriel's sceptre is therefore productive. It marks the production of Christianity. What it marks out, at the same time, is that which is integral to that production. The latter is the undoing of Judaism, its having become unproductive. One cannot be separated from the other. Production and the rendering unproductive are implicated in the work enacted by the hand\/instrument relation.\n\nIf there is a contrast to this productive hand (i.e., this conception and direction of a productive relation), then an emphatic instance can be linked to the figure of Prometheus. Of interest here is Parmigianino's _Prometheus Animating Man_ (1524\u20131527) (figure 2.4). There is the possibility that Parmigianino drew on a tradition, exemplified in a text to which he would have had access, namely Giovanni Boccaccio's _Genealogia deorum gentilium_ in which there is doubling of the figure of Prometheus and in which 'the first is the true omnipotent God' ( _Primus autem deus verus et omnipotence est_ ).[71] And yet, despite this link and the obvious history of the Christianization of the 'pagan', here in Parmigianino it is equally as possible that what is being staged is creation itself\u2014a position part of whose justification would lie in the image's avoidance of any obvious Trinitarian register. Indeed, it might even be argued that Boccaccio's second Prometheus\u2014described thus, _id est doctus homo_ \u2014means that the human as educator becomes the mark both of the secular and of production.[72] Prometheus names the possibility of forms of production that have the potential to remain indifferent to the presence of the God of Christianity. Prometheus thus understood inscribes a form of ambivalence into questions of production.\n\nPetrus Canisus, _De Maria Virgine_ (1583). British Library, London.\n\nParmigianino, _Promethus Animating Man_ (1524\u20131527). Morgan Library and Museum, New York.\n\nPanofsky argued that within _Melanchoia I_ there was a division between forms of production: one defined in relation to 'art' ( _Kunst_ ), the other to mechanical forms of use (i.e., _Brauch_ ). Within the setting of this interpretation, Melancholia is identified with art and the putto with _Brauch_ (use).[73] While there may be a strategic decision to identify two different modes of production within the frame, a decision sustaining Panofsky's overall project, it is also possible to allow for production to have a greater sense of generality. It should be noted that the putto writes, while Melancholia sits. Her hands do not move. The compass is inoperative. However, if Benjamin's use of Hamlet is recalled, then her hands, once understood as given within the hand\/instrument relation, remain _ungebraucht_ (unused). The question, however, is not, what occurs when art is put to use? What arises here is importantly different. There are two questions. The first concerns the already defined relation between hands and instruments in terms of modalities of use. The use of the plural here is fundamental. The other question, the question that attends a setting that is not caught in an endless oscillation between _ungebraucht_ (unused) and _brauch_ (use) is the following: What occasions use? Yet there are modalities of the hand\/instrument relation that are useful in advance. They are already at work. In Parmigianino's _Prometheus Animating Man_ , that relation oscillated between a possible reiteration of a secularized version of the creative power of God, or the recognition that human beings are in a relation of non-relation to the divine in every act of creation. In creating, humans have never been further from God, and this despite the fact that each act of creation is an imitation of God. Hence, it can be conjectured that the _imitatio Dei_ has nothing to do with images but rather with activity and production. Form is displaced by the necessity to find form. There can be no reconciliation between these possibilities. There is no harmony. Hence, following this line of argumentation, what has to be noted in the context of Parmigianino's drawing is that the human produces itself, producing itself as itself and undoing the identification of human beings with the image of God. The human becomes an image of itself. _Imitatio Dei_ becomes _imitatio homini_.\n\nIn the engraving in Petrus Canisius's _De Maria Virgine_ (1583), Gabriel's hand (again there is a hand\/instrument relation) works to mediate by calling attention to the Annunciation while at the same time reinforcing the Pauline position that the effect of the Annunciation signals the rendering inoperative of the Jews.[74] There is a double move defined by the temporality of at-the-same-timeness. The hand\/instrument relation is implicated in a doubled sense of production. The hands in Hans Baldung Grien's _Bewitched Groom_ are divided between doing and having been undone. Tensions abound. The hands\u2014and once relationality is emphasized and images are present as the effect of a process of coming-into-relation in which the hand\/instrument relation has continual primacy\u2014then their work, precisely because it is an after-effect of that process, cannot be quickly assimilated. No one determination of the motif can predominate. What this means is that pathos is always path\u00e9. The projects in which they are implicated remain importantly different. And yet, in every instance, what is not in question is the possibility of use. The relation between the hand and the instrument is the immediacy of the already mediated in the precise sense that utility is uppermost, even if that utility cannot be given a singular orientation (and can if certain instances are presented negatively).\n\nIf there is a history of hands where the latter are defined continually in terms of modalities of relationality\u2014the hand\/instrument relation as a motif\u2014then what is significant in the case of _Melancholia I_ is if the defeat of Saturn, which needs to be understood as the suspension of fate, is destructive, following Benjamin, then what is involved is the presence of a setting for creativity's possible actualization. What emerges is a caesura that allows. The 'expressionless' opens the way for expression. That possibility cannot be incorporated within an oscillation between use and its opposite. (The latter would be the groom's hands in Hans Baldung Grien's _Bewitched Groom_.) If there is a genuine polarity, then it is between the hand\/instrument relation in this work and the 'same' relation in _Melancholia I_. Moreover, their own relation is itself productive. Hence, there is another possibility. Melancholia holds the compass. There is an already present relation even if it cannot be assimilated to the opposition use\/non-use. If the hand\/instrument relation in this instance (i.e., in _Melancholia I_ ) has force, then it is acquired as an after-effect of its relation to other images of the hand\/instrument relation. There is a productive constellation. However, one image does not repeat another. Despite the temptations exerted by their repetition, these images\u2014motifs\u2014rework each other precisely because difference is not a form of variety. Potentiality cannot be assimilated to either use or non-use. Here is the force of the hand\/instrument relation as a motif in _Melancholia I_. 'The moment of recognizability' occurs as a result. What cannot be brought to all these images\u2014and this despite a proliferation of productive\/unproductive hands\u2014is unanimity of project, even one that is structured by a polarity that would both have a singular determination and be located within an oscillation, the subduing of which would create a form of rational equanimity. There are different possibilities of use. Standing opposed to all is that sense of potentiality that holds fate at bay. Potentiality, however, cannot be thought outside a relation to life. And Melancholia, the 'thinking creature'\u2014has she not now become, precisely, in awaiting, in attending, the very signature of hope?\n\nHaving come this far, there is a need to return. Hope both forms and informs the present. As such it is charged with possibilities. In other words, the present is defined in terms of potentiality. Hence, there is the need to maintain Benjamin's definition of creaturliness. Creaturliness is the other possibility that attends life: the 'worst' remains as a possibility. Integral to that definition is that it marks the presence of a possibility. As a possibility, it attends worldliness, such that what would be lost once creaturliness prevailed, and it needs to be recalled that it prevails with the context of _Trauerspiel_ in an 'empty world', is not the world per se. Within the Baroque, as presented by Benjamin, life endures. What was not available, thus what can now be seen as lost, is the possibility for it to have been other. And yet, even within the dense realm of disparate objects, present with varying degrees of power, in the differing countenances with which individuals face their predicament, what can be recovered, still, are intimations of the possibility for it to be other. This occurs not just through a hermeneutic exercise that seeks to recover meaning but as resulting from the constant interplay between processes of relationality and mattering. It is within this context that what is to be brought to the fore is the possibility that images remember. Hence, the methodological question is, what will cause remembrance on the level of the image?\n\nAll of the images dealt with in this chapter are engravings. In addition, they were produced within a similar historical time frame. Nonetheless, relations were not established as the result of thematic continuity or mere historical contiguity. Rather, what established relations, and thus the nature of the continuity involved, was the hand\/instrument relation. It was a motif. The continuity entailed and continues to entail discontinuity. The continuity allowed for the recognition of a 'breaking point' in which utility, in both its positive and negative definition, can be opposed to that presentation of the hand\/instrument relation that awaits and in awaiting underscores the centrality of potentiality. In broader philosophical terms, this reinforces the claim that potentiality\u2014following Aristotle\u2014always involves a relation between capacity and incapacity. Actualization is always, and of necessity, contingent. The discontinuity between the images, which is itself a form of relationality, once set within the context of what is at stake ideationally within these works, brings into play a twofold sense of dislocation. In the first instance, it occurs on the level of ideas; on the second, it pertains to the way life, time and the status of the image intersect.\n\nIt needs to be noted in conclusion that caution should be exercised here. This is in part because the majority of works in question have similar mode of production, and as a result there is a type of visual solicitation driven by forms of similitude that needs to be resisted\u2014a resistance that will prove productive. To be precise, what has to be resisted is the possibility that a continuity between the images take precedence over their differences. It is essential to recognize that the ground of any such continuity would be a prevailing interpretive continuity that would have emerged as the direct result of having operated simply on the level of that equated the visual with semblance. Such a move would have excised the ideational and as such would have remained indifferent to the ways in which mattering is itself reconfigured by the process of coming-into-relation. What are underscored in all these instances are different senses of utility. If there is a final point that needs to be made in order to underscore that what are staged are different relations to production in which potentiality is exercised, then it stems from the recognition that description of potentiality, _Melancholia 1_ as the signature of hope, can only occur and thus can only be explicated adequately in relation to the sustained centrality of mattering.\n\n## Notes\n\n1.\n\nFor the details of Warburg's life set in the context of his thought, see Marie-Anne Lescourret, _Aby Warburg ou la tentation du regard_ (Paris: Hazan, 2013). Of the many valuable introductions to the thought of Aby Warburg, see Georges Didi-Hubermann, _L'image subrvivante: Histoire de l'art et temps des fant\u00f4mes selon Aby Warburg_ (Paris: Les \u00c9ditions de Minuit, 2008); and Christopher D. Johnson, _Memory, Metaphor and Aby Warburg's Atlas of Images_ (Ithaca, NY: Cornell University Press, 2012). For an important identification of the significance of Warburg with contemporary art history, see Griselda Pollock, 'Whither Art History?' _Art Bulletin_ 96, no. 1 (March 2014). In addition, the Warburgian legacy entails that what continues is the question of the status of the 'image'. See in this regard Karen Lang, _Chaos and Cosmos: On the Image in Aesthetics and Art History_ (Ithaca, NY: Cornell University Press, 2006). References to Warburg will for the most part come from Aby Warburg, _The Renewal of Pagan Antiquity_ , trans. David Britt (Los Angeles: Getty Research Institute, 1999); and Aby Warburg, _Werke in Einem Band_ (Frankfurt: Suhrkamp, 2010). They will be referred to as TRPA and WEB, respectively. Occasionally translations have been modified.\n\n2.\n\nReferences to the text will be to the German edition followed by the English translation: _The Origin of German Tragic Drama_ , trans. John Osborne (London: New Left Books, 1977)\u2014henceforth OGTD.\n\n3.\n\nFor the counterargument, see Jason Gaiger, 'The Idea of a Universal Bildwissenschaft', _Estetika: The Central European Journal of Aesthetics_ 51, no. 2 (2014).\n\n4.\n\nWarburg, TRPA, 558\/WEB, 151.\n\n5.\n\nIbid., 563\/374.\n\n6.\n\nPhilippe-Alain Michaud, _Aby Warburg and the Image in Motion_ , trans. Sophie Hawkes (New York: Zone Books, 2004), 270\u2013271.\n\n7.\n\nAby Warburg, _Tagebuch der Kulturwissenschadtlichen Bibliothek Warburg_ (Berlin: Akademie Verlag, 2001), 547.\n\n8.\n\nSigrid Weigel, 'Aby Warburg's Schlangenritual: Reading Culture and Reading', _New German Critique_ , no. 65, Cultural History\/Cultural Studies (Spring\u2013Summer, 1995): 135\u2013153.\n\n9.\n\nFor a more detailed investigation of Warburg's conception of the symbol, see Charlotte Schoell-Glass, 'Aby Warburg's Late Comments on Symbol and Ritual', _Science in Context_ 12, no. 4 (December 1999), 621\u2013642. See in addition Michael Neumann, '\"Wilde Sc\u00f6nheit\": Die D\u00e4monen der Natur und die Magie des Begehrens bei Friedrich Theodor Vischer und Aby Warburg', in _Friedrich Theodor Vischer Leben\u2014Werk\u2014Wirkung_ , ed. Barbara Potthast and Alexander Reck (Heidelberg: Universit\u00e4tsverlag, 2011), 277\u2013298. Finally, Matthew Rampley, 'From Symbol to Allegory: Aby Warburg's Theory of Art', _Art Bulletin_ 79, no. 1 (March 1997), offers an interpretation of Warburg that sees far greater connection with Benjamin than the one argued for here. One of the most crucial issues, as Rampley rightly indicates, concerns how the relationship between symbolism and empathy is understood.\n\n10.\n\nLang, _Chaos and Cosmos_ , 252.\n\n11.\n\nFor Vischer's text 'Das Symbol', see Friedrich Theodor Vischer, _Ausgew\u00e4hlte_ _Werke_ , vol. 8 (Anstall: Deutsche Verlags, 1918), 312\u2013347. For a discussion of the links between Warburg, Cassirer and this aspect of Vischer's work, see Edward Skidelsky, _Ernst Cassirer: The Last Philosopher of Culture_ **(** Princeton, NJ: Princeton University Press **,** 2011), 91\u201395.\n\n12.\n\nAby Warburg, 'Die Bilder ausstellungen des Volksheims', in _Gesammelte Schriften_ , vol. 2 (Leipzig: B.G. Teuuber, 1932), 591.\n\n13.\n\nWarburg, TRPA, 558\/WEB, 182.\n\n14.\n\nWhile English does not contain the word 'memory-image', the current translation of 'pictorial memory', in failing to retain the centrality of _Bild_ fails to capture the force of what Warburg is doing. In English 'memory-image' would have the same status as ' _Bildgedachtnisses_ ' in German (i.e., an invented compound noun that forms part of an attempt to develop a science of the image).\n\n15.\n\nBenjamin, OGTD, 35\u201336. For a discussion of this formulation in the broader context of Benjamin's writings see: Rodolphe Gasch\u00e9, 'Saturnine Vision and the Question of Difference: Reflections on Walter Benjamin's Theory of Language', _Studies in Twentieth Century Literature_ 1, no. 1 (1986), 69\u201390.\n\n16.\n\nBenjamin, GS I, 696.\n\n17.\n\nThere is an extensive range of discussions of Benjamin's theory of the image. Three writers whose work touches on central components of what is at stake with his theory are Rebecca Comay, Eli Friedlander and Gerhard Richter. See Rebecca Comay, 'Materialist Mutation of the Bilderverbot', in _Walter Benjamin and Art_ , ed. Andrew Benjamin (London: Continuum, 2005), 32\u201359. Her reexamination of the _Auseinandersetzung_ between Adorno and Benjamin is of fundamental importance in locating the specificity of what Benjamin intends by the dialectical image. Gerhard Richter, _Afterness: Figures of Following in Modern Thought and Aesthetics_ (New York: Columbia University Press, 2011), 139\u201341. Richter locates Benjamin's theory of the image in relation to a range of contemporary philosophers. Most productively in relation to Heidegger, Richter emphasizes the complex link that Benjamin's conception of the image had to withdrawal and thus to its presence in the 'moment'. Eli Friedlander's work accords with aspects of the argument advanced in this chapter. He rightly underscores the link between presence and the 'expressionless'. This is Benjamin's link to H\u00f6lderlin. Eli Friedlander, _Walter Benjamin: A Philosophical Portrait_ (Cambridge, MA: Harvard University Press, 2012), 37\u201339.\n\n18.\n\nBenjamin, GS I, 695.\n\n19.\n\nSee in this regard the suggestion made by Griselda Pollock in the context of the photographs of the Pueblo Indians: 'Through the photographs immediacy, something profoundly significant in the clash of temporalities (Indian and Euro-American cultural time), in space (made possible by railways) becomes visible because the optical unconscious may disclose to analysis what is not spontaneously grasped in seeing. The image becomes ostensive rather than indexical' ('Aby Warburg and Mnemosyne: Photography as Aide-m\u00e9moire, Optical Unconscious and Philosophy', in _Photo Archives and the Photographic Memory of Art History_ , ed. Constanza Caraffa, ed. [Berlin: Deutscher Kunstverlag, 2011], 87).\n\n20.\n\nBenjamin, SW II, 1505\/GS II, 1, 371.\n\n21.\n\nBenjamin, GS III, 365.\n\n22.\n\nIbid., 366.\n\n23.\n\nBenjamin, AP N, 7a.\n\n24.\n\nWhile there is a growing literature on the relationship between Warburg and Benjamin, by far the most sustained engagement is still the one presented by Beatrice Hanssen. See her 'Portrait of Melancholy (Benjamin, Warburg, Panofsky)', _MLN_ 114, no. 5 (December 1999). See in addition Jane O. Newman, 'Luther's Birthday: Aby Warburg, Albrecht D\u00fcrer and Early Modern Media in the Age of Modern War', _Daphnis\u2014Zeitschrift f\u00fcr Mittlere Deutsche Literatur_ 37, nos. 1\u20132 (2008): 32. In regards to D\u00fcrer, see Lynne J. Miles-Morillo, 'Composing a Self: Translation and Transformation in D\u00fcrer's Humanism', _Prose Studies: History, Theory, Criticism_ 32, no. 2 (2010). For a more general but nonetheless essential discussion of melancholy in Benjamin's work, see Ilit Ferber, _Philosophy and Melancholy: Benjamin's Early Reflections on Theatre and Language_ (Stanford, CA: Stanford University Press, 2013).\n\n25.\n\nThe move to a reconfiguration of art history in terms of the image as this pertains to Waburg's work has been analyzed in detail by Horst Bredekamp. See his 'A Neglected Tradition? Art History as Bildwissenschaft', _Critical Inquiry_ 29, no. 3 (Spring 2003).\n\n26.\n\nWarburg, TRPA, 601.\n\n27.\n\nF. W. J. Schelling: 'Serenity ( _Die Still_ ) is the condition unique to beauty, just as repose ( _die Ruhe_ ) is that of the undisturbed sea. Only in repose can the human figure as such, and human countenance in particular, be the mirror of the idea ( _der Spiegel der Idee_ ). Here, too, beauty points to unity and indifference as its true essence (154\/387).' _The Philosophy of Art_ , ed. and trans. Douglas W. Scott (Minneapolis: University of Minnesota Press, 1989); originally published as \"Philosophie der Kunst,\" in his Ausgew\u00e4hlte Schriften, bd. 2 (Frankfurt: Surhkamp, 1985), 154\/387. (Translation modified on occasion.) Reference will be to the English pagination followed by the German.In regards to Roman aesthetics, the argument is not that Warburg is in any sense intending to deploy terms such as 'decorum' to present his project. Rather, it is not as though the problem of a play of forces between antithetical emotions was foreign to either a Roman anthropology\u2014especially Cicero\u2014or Roman aesthetic theory. What was important was their reconciliation. For an extended discussion of decorum, see Ellen Perry, _The Aesthetics of Emulation in the Visual Arts of Ancient Rome_ (Cambridge: Cambridge University Press, 2005), 28\u201350.\n\n28.\n\nBenjamin, OGTD, 132\/GS I, 311. This insight of Benjamin's could be understood as the _Othello moment_ insofar as there could not be a more accurate formulation of the role of the handkerchief in the play. In regards to the handkerchief, Andrew Sofer argues, rightly: 'In a startling reversal of a prop's usual function, Othello's handkerchief uses and discards its victims rather than vice versa. Instead of merely symbolizing its human couriers, the handkerchief absorbs and literally inscribes then ads absences within its ghostly palimpsest' ('Felt Absences: The Stage Properties of _Othello_ 's Handkerchief', _Comparative Drama_ 31, no. 3 [Fall 1997]: 388). However, what is missing from the claim is that for the handkerchief to function in this way, there must be a corresponding quality that allows it to occur. This is Benjamin's great insight. For a discussion of the purely symbolic register of the handkerchief, see Lynda E. Boose, 'Othello's Handkerchief: \"The Recognizance and Pledge of Love\"', _English Literary Renaissance_ 5, no. 3 (September 1975).\n\n29.\n\nIt should not be thought that the terms 'creature' and 'creaturely' have an unequivocal presence in contemporary thought. For another interpretation of the 'creaturely', see Eric Santner, _On Creaturely Life: Rilke, Benjamin, Sebald_ (Chicago: University of Chicago Press, 2006).\n\n30.\n\nGiorgio Agamben, 'Nymphs', in _Releasing the Image_ , ed. Jacques Khalip and Robert Mitchell (Stanford, CA: Stanford University Press, 2011), 62.\n\n31.\n\nAnother version of this position can be located in John Donne. See his _Devotions upon Emergent Occasions_ , 'Meditation 1', (1624).\n\n> VARIABLE, and therefore miserable condition of Man; this minute I was well, and am ill, this minute. I am surpriz'd with a sodaine change, and alteration to worse, and can impute it to no cause, nor call it by any name. We study _Health_ , and we deliberate upon our _meats_ , and _drink_ , and _ayre_ , and _exercises_ , and we hew, and we polish every stone, that goes to that building; and so our _Health_ is a long and regular work; But in a minute a Canon batters all, overthrowes all, demolishes all; a _Sicknes_ unprevented for all our diligence, unsuspected for all our curiositie; nay, undeserved, if we consider only _disorder_ , summons us, seizes us, possesses us, destroyes us in an instant. O miserable condition of Man, which was not imprinted by _God_ , who as he is _immortall_ himselfe, had put a _coale_ , a _beame_ of _Immortalitie_ into us, which we might have blowen into a _flame_ , it out, by our first sinne.\n\nThe originally 'miserable condition of Man' would occur in what Benjamin describes as an 'empty world'.\n\n32.\n\nThe edition to which Benjamin refers is Hamlet, _Dramatische Werke_ , Nach der Ubers. Von August Wilhelm Schlegel u. Ludwig Tieck, sorgf\u00e4ltig revidirt u. theilweise neu bearbeitet, mit Einleitungen u. Noten versechen, unter Redaction von Hermann Ulrici, durch die Deutsche Shakespeare Gesellschaft, VI (Berlin: G. Reimer, 1877).\n\n33.\n\nI have developed this term as playing a pivotal role in the work of Walter Benjamin. See my _Working with Walter Benjamin_ _: Recovering a Political Philosophy_ (Edinburgh: Edinburgh University Press, 2013).\n\n34.\n\nThis formulation\u2014virtue in being\u2014is intended to capture the setting in which there is what can best be described as the anoriginal interarticulation of the ethical and the ontological. A detailed discussion of this setting is forthcoming in my _Virtue in Being_ (Albany: SUNY Press, 2016).\n\n35.\n\nGeorges Didi-Huberman, 'Artistic Survival: Panofsky vs. Warburg and the Exorcism of Impure Time', _Common Knowledge_ 9, no. 2 (Spring 2003): 274.\n\n36.\n\nThe 'worst' is a term that has an important contemporary currency. The worst, as the term is being used here, indicates that what occurs with a form of interruption is that it leads as much direction that is viewed as positive as it can lead in the other. That other direction is the worst. Destruction opens up as a problem of direction. In Leonard Lawlor's work, the 'worst' has a slightly different though equally important register. In _This Is Not Sufficient_ the 'worst' is explicated in terms of 'a form of totalisation'. The worst violence is the violence acted out against what can be called following Derrida 'undecidability'. See Leonard Lawlor, _This Is Not Sufficient: An Essay on Animality and Human Nature in Derrida_ (New York: Columbia University Press, 2007), 38.\n\n37.\n\nTheodor W. Adorno, _Minima Moralia_ (Frankfurt: Gesammelte Schriften, Suhrkasmp, 1997), 43.\n\n38.\n\nBenjamin, OGTD, 66\/GS I, 1, 246.\n\n39.\n\nThis is a formulation developed to account for the role of the 'caesura' in Benjamin's thought. The key element is that it has to be understood as a mode of destruction which, while itself 'expressionless', presages. In other words, it is the condition of a 'true image's' creation without itself having an image.\n\n40.\n\nBenjamin, OGTD, 310\/GS I, 131.\n\n41.\n\nIbid., 138.\n\n42.\n\nIbid., 139.\n\n43.\n\nIbid., 152.\n\n44.\n\nI have taken up the question of boredom as a determining motif in Benjamin's work in my, 'Boredom and Distraction: The Moods of Modernity', in _Walter Benjamin and History_ , ed. Andrew Benjamin (London: Continuum, 2005).\n\n45.\n\nBenjamin, OGTD, 158.\n\n46.\n\nBenjamin, SW I, 203\/GS II, 1, 174.\n\n47.\n\nEven though its detail cannot be pursued, the following engagement with 'awaiting' is prompted by Maurice Blanchot's, _L'Attente l'oubli_ (Paris: Gallimard, 1981).\n\n48.\n\nHanssen, 'Portrait of Melancholy', 1003.\n\n49.\n\nThis paper appears in English in Warburg, TRPA, 597\u2013698.\n\n50.\n\nWarburg, TRPA, 603\/WEB, 435. For a general discussion of astrology in the period, see Claudia Brosseder, 'The Writing in the Wittenberg Sky: Astrology in Sixteenth-Century Germany', _Journal of the History of Ideas_ 66, no. 4 (October 2005).\n\n51.\n\n_Der Wei\u00df Kuning_ is an allegorical tale written by Marx Treitzaurwein. Its significance as a sixteenth-century work is that its illustrations as well as the text itself highlight a number of the complex relations between the 'pagan' and the 'modern' within the court of Maximillian I. Of particular note are the illustrations by Hans Burgkmair.\n\n52.\n\nWarburg, TRPA, 607\/WEB, 438.\n\n53.\n\nIbid., 645\/478.\n\n54.\n\nEqually of course the success in question\u2014optimism's successful outcome\u2014can have more negative consequences. As Maurizio Ghelardi argues, 'This gradual liberation from the violence of primitive mythical forces has a cost: language increasingly becomes a mere sign' ('Das Klopfen auf der andern Seite des Tunnels', _Cassirer Studies_ 1 [2008]: 165).\n\n55.\n\nAby Warburg, 'Giordano Bruno', ed. Maurizio Ghelardi and Giovanna Targia, _Cassirer Studies_ 1 (2008): 50. See in addition Donald Phillip Verene's introduction to the text.\n\n56.\n\nWarburg, TRPA, 650\/WEB, 480.\n\n57.\n\nIbid., 644\/476. Christopher D. Johnson also cites this passage and in a conclusion that is similar to the one presented here goes on to make the additional point that the very attempt to stem the hold of melancholia\u2014both personal and historical\u2014was equally impossible. See his _Memory, Metaphor and Aby Warburg's Atlas of Images_ , 228\u2013229.\n\n58.\n\nWarburg, TRPA, 645\/WEB, 477.\n\n59.\n\nHannah Arendt, _The Origins of Totalitarianism_ (New York: Harcourt, Brace, Jovanovich, 1973), 459.\n\n60.\n\nBenjamin, SW IV, 172\/GS I.2, 669\u2013670. There is a similar argument advanced elsewhere that allegory is the 'antidote' ( _das Antidoton_ ) to myth'. See Benjamin, SW IV, 1979\/GS, I.2, 677. Allegory ruptures the time of a projected unending continuity.\n\n61.\n\nEdgar Wind, 'Warburg's Conception of Kulturwissenschaft and Its Meaning for Aesthetics', in _The Eloquence of Symbols_ (Oxford: Clarendon, 1993), 25. Wind's position finds further confirmation for Michael Diers' argument that 'Warburg defines culture as the historical sum of all efforts made by man to overcome his fear, which tries to hold him captive again and again. The act of liberation from irrationality, pagan stupidity, and the rule of desire, is carried out in a slow process of humanization, a process which is nevertheless subject to setbacks at any given moment.' See his 'Warburg and the Warburgian Tradition of Cultural History', _New German Critique_ , no. 65, Cultural History\/Cultural Studies (Spring\u2013Summer, 1995): 68.\n\n62.\n\nHoward Caygill, 'Walter Benjamin's Concept of Cultural History', in _Cambridge Companion to Walter Benjamin_ , ed. David Ferris (Cambridge: Cambridge University Press, 2004), 87.\n\n63.\n\nFor another possibility underscoring the link between _ingenium_ and creation in this context, see Christopher D. Johnson, _Hyperboles: The Rhetoric of Excess in Baroque Literature and Thought_ (Cambridge, MA: Harvard University Press, 2010), 360.\n\n64.\n\nThe edition of Aristotle used is Aristotle, _Metaphysics_ , trans. Hugh Tredennick (Cambridge, MA: Harvard University Press, 1933).\n\n65.\n\nThe argument concerning the compass as a sign of potentiality is intended to call into question as emphatically as possible the claim that with Melanciolia, D\u00fcrer was recalling the notion of God as _Geometre_. This is the position that emerges in the claim made by Philip L. Sohm, who suggests that 'Melancholy can be seen as an imitator of God and hence a perfect illustration of Agrippa's _fiuror divinus_ ' ('D\u00fcrer's _Melencolia I_ : The Limits of Knowledge', _Studies in the History of Art_ 9 [1980]: 25).\n\n66.\n\nSee in addition Wojciech Balus, 'D\u00fcrer's _Melencolia I_ : Melancholy and the Undecidable', _Artibus et Historiae_ 15, no. 30 (1994). Balus's position is summed up in the following: 'On the contrary, the world of _Melencolia I_ is the world of human activity. The problem is that this activity, expressed in an allegorical language, has been subjected to diffusion imposed by the rules of composition. The creative element has thereby been stopped in its tracks by dynamism in immobility and by heterotopia.' The position argued here is there is an affinity in that both hold to the centrality of the stopping of creative activity.\n\n67.\n\nJoachim Camerarius, _Elementa Rhetorica_ , 1564. The reference to D\u00fcrer is available in Latin and English translations in Frank Baron, _Joachim Camerarius (1500\u20131574): Beitr. zur Geschichte d. Humanismus im Zeitalter d. Reformation_ (Munich: Fink, 1978). For a detailed investigation of 'ekphrasis' in the Renaissance, see Claire Preston, 'Ekphrasis: Painting in Words in Sylvia Adamson', in _Renaissance Figures of Speech_ , ed. Gavin Alexander and Katrin Ettenhuber (Cambridge: Cambridge University Press, 2011).\n\n68.\n\nMargaret A. Sullivan describes her in the following terms: 'Holding aloft her burning brand the old woman in Baldung's _Bewitched Groom_ is fitting as a Fury, a witch-like creature of the night, and her fierce expression is appropriate for the vengeful Infamia, the spirit of punishment who disturbs the mind and causes insanity.'See Margaret A. Sullivan, 'The Witches of D\u00fcrer and Hans Baldung Grien', _Renaissance Quarterly_ 53, no. 2 (Summer 2000). See in addition the discussion of what are taken to be the complex and perhaps unresolvable interpretive questions raised by this work in Bodo Brinkmann, 'Witches' Lust and the Fall of Man', in _Witches' Lust and the Fall of Man: The Strange Fantasies of Hans Baldung Grien_ (Frankfurt: Michel Imhof Verlag, 2007), 191\u2013196.\n\n69.\n\nSee to this end Dale Hoak, 'Art, Culture, and Mentality in Renaissance Society: The Meaning of Hans Baldung Grien's Bewitched Groom (1544)', _Renaissance Quarterly_ 38, no. 3 (Autumn 1985).\n\n70.\n\nWalter S. Melion, ' _Quae lecta Canisius offert et spectata diu_ : The Pictorial Images in Petrus Canisius's _De Maria Virgine_ of 1577\/1583', in _Early Modern Eyes_ , ed. Walter S. Melion and Lee Palmer Wandel (Leiden: Brill, 2010), 255. Melion's work has been an indispensible guide both to this engraving and to Canisius's work as a whole.\n\n71.\n\nReference is made here to the edition edited by Jon Solomon and published as _Genealogy of the Pagan Gods_ , I Tatti Renaissance Library (Cambridge, MA: Harvard University Press, 2001), book 4, chapter 44, 8.\n\n72.\n\nBoccaccio, _Genealogy_ , book 4, chapter 44, 12.\n\n73.\n\nSee Erwin Panofsky, _The Life and Art of Albrecht D\u00fcrer_ (Princeton, NJ: Princeton University Press, 2005), 164.\n\n74.\n\nI have discussed this process in relation to Giorgio Agamben's interpretation of Paul in my 'The Inoperative Jew: Agamben's Paul', in _Place Commonality and Judgment_ (London: Continuum, 2012), 139\u2013158.\n\nChapter 3\n\n# On Hands\n\nAgain, handing on. Coming-into-relation names forms of possible movement. If the preceding chapter can be summarized, then the argument was that the hand\/instrument relation in D\u00fcrer's _Melancholia 1_ introduced the possibility of differing approaches to questions of use. Within that context, it was by allowing for the possibility that the hand\/instrument relation could be configured as a locus of awaiting, and thus not defined in terms of either use or non-use, that had the effect of removing such a configuration of relationality from an economy defined in terms of strict utility. If taken on its own, however, such a position becomes no more than an empty abstraction. Hence there was the additional argument that a concern with the suspension of use had to be understood in terms of the enactment of a critical distancing from the hold of fate. This distancing was brought about by the identification of potentiality and hope. The figure of Melancholia in D\u00fcrer's work was therefore reconfigured in terms of hope, precisely because the instruments awaited use. Once juxtaposed with Jan Sadeler's engraving of _Geometria_ \u2014(after Maarten de Vos, _Geometry,_ from the series _The Seven Liberal Arts_ (1575) (figure 3.1)\u2014in which the compass is measuring the world, the force of awaiting becomes clear. Images of _Geometria_ present the world as a given. Acting in the world, measuring and remeasuring do no more than confirm the given. What is not there within such an image is the potentiality for the world to be other. Othering the world is the possibility that inheres in life once othering is understood as the depositioning of fate. It would not be a world beyond measure. Rather, it would be a conception of the world that was not reducible to the already measured. The potentiality for the world to be other, which when taken back to the Baroque would have been the claim that there was indeed a Baroque eschatology, is a possibility that cannot have a predetermined image. On the contrary, it is an image to be made. Enduring as a continual possibility, potentiality and hope are then bound together (the latter emerging from the recasting of the hand\/instrument relation in terms of a motif).\n\nJan Sadeler, _Geometria_ , engraving (1575), after Maarten de Vos, _Geometry_ , from the series _The Seven Liberal Arts_. Bristish Museum, London.\n\nThis distinction, as presented thus far, is between potentiality on the one side and use\/non-use on the other. However, the development of the hand\/instrument relation in these terms is, by definition, only part of a more general sketch of the hand\/instrument relation. If there is another dimension that has to be introduced, and this has to occur prior to the presentation of any conclusions which could only be drawn at the chapter's end, then it concerns the recognition of relationality itself. In other words, while there are instruments and while it is the case that the hand is already implicated in forms of production, there is, for the most part, a failure within the accompanying literature that engages the hand as a locus of production to account for what has already been described as the anoriginality of the hand\/instrument relation it terms of the capacities of the instrument. Retaining and recalling the instrument is to inscribe the centrality of production into a concern with art. It is as though the hand worked without any form of mediation. In other words, it is as though the relation between hand and instrument was either secondary or, as significantly, it could be ignored. Again, it is essential to note that the hand is mediated in advance, and thus the hand\/instrument relation has anoriginal force. The claim will always be that relationality precedes singularity. This is the position that the centrality of a relational ontology demands.\n\nAgain, there is the hand. In Walter Benjamin's engagement with Eduard Fuchs, an essential component of Benjamin's celebration of the latter's 'collecting', was that as an activity it incorporated Fuchs's avowed emphasis on what Benjamin describes as 'techniques of production'. This was one of the aspects of Fuchs's project that, in Benjamin's formulation, 'could only prove destructive to traditional conceptions of art'.[1] Benjamin goes on to argue that Fuchs's 'greatest achievement maybe that he cleared the way for an art history to be freed from the fetish of the masters' signature'.[2] Here the hand is to be understood as a locus of original or obtained mastery such that its reiteration is its presence as a 'signature', allowing thereby the term 'signature' greater extension, while at the same time noting the interplay of mastery and completion. Again, an opening emerges as a result of a process of delimitation.\n\nThe insistence on reproduction and the destruction of what can only be described as the master's hand opens up at least two possible paths. The first path necessitates that reproduction is taken literally. As a result, all that would then be of interest are works whose point of origination is mass production. While this is a genuine possibility and one that could be supported by aspects of Benjamin's own argumentation, it does involve a literalization of his project. And, indeed, misses the larger point of the argument advanced in _The Work of Art in the Age of its Technical Reproducibility_ , namely that the advent of 'technical reproducibility' not only renders certain categories or concepts of interpretation redundant (e.g., 'genius', 'beauty'), but the direct correlate is that this break demands both a reinvention of the concepts and categories of interpretation as well as the removal of any limit to their field of application. This is the demand staged by an insistence on production. Allowing for such an opening means that it no longer follows from a reconfiguration of the basis of interpretation that it would then only have projective force. If this projective force were taken on its own, it would amount to the introduction of a different form of historicism. Moreover, it is precisely the possibility of the retroactive, named in part by the process of coming-into-relation, that opens up the other prompt given by an insistence on both reproduction and reproducibility. Moreover, retroactive force is precisely what Benjamin's conception of the 'dialectical image' necessitates. This other possibility, and the one to be pursued here, involves the argument that what the insistence on reproduction brings into play is not a fetishization of the present and the future (which again are terms given within a historicist framework)\n\nbut the necessity to recognize that works are produced and therefore that they are the after-effect of the technical. Understanding their presence as produced means noting the possible inscription of the presence of production as part of the work's work. The technical has to be thought. Production becomes an essential element in accounts of art's self-staging. The immediate consequence, as Benjamin indicates in connection to Fuchs, is that terms such as 'beautiful semblance' ( _Schone Schein_ ), 'harmony' and 'unity' are no longer appropriate. An emphasis on production undoes the 'master's signature' by demanding, in this context, another interpretation of the presence of the hand\u2014present within both relationality and the processes named coming-into-relation. As is clear, this may mean the literal presence of the hand in painting, photography and the accompanying writing integral to the hand's presence and art's production is always overdetermined. The literal hand always stages more than its literal presence.\n\nThe presence of the hand that would have been lost comes to acquire actuality through the recognition that would be afforded to it by an insistence on reproduction; its presence within the field opened by reproduction and reproducibility. Benjamin's interests are specific insofar as his writings on reproduction and reproducibility are concerned. And yet, once reproduction opens up the need to take up the question of production, then what is brought out is a connection between production and mattering. This connection repositions work. Within this chapter hands are present as a result of their location within differing processes of relationality; these processes are modalities of coming-into-relation. The hand is implicated in advance. Precisely because an opening occurs, it becomes possible to begin, albeit cautiously. Integral to that caution is the recall of Warburg. What endures is the possibility that instead of the repetition of motifs being a repetition defined in terms of sameness\u2014and here the motif in question is the hand\/instrument relation\u2014repetition becomes the staging of irreducible differences. This would be the way in which the formal possibilities within the 'dialectical image', possibilities that tied meaning to the emergence of those points of connection that were equally points of disconnection, could be distanced from the link between meaning and iteration that underscores the _pathosformel_. Clearly the hand\/instrument relation is not an 'expressive gesture' (hence the argument that what are involved here are formal possibilities). The hand\/instrument relation is an interplay of the ideational, the material and the expressive. Indeed, the overall contention is that one cannot be separated from the other. Moreover, there is an additional point: namely that such a description would be true of the components of any set of motifs or gestures. So once the ideational is built in, then what becomes difficult to sustain is any argument for reconciliation as opposed to irreducibility. The position has to be that in regard to the relation between the hand and the instrument when it is taken as a motif\u2014and thus it would be an open question as to whether it could be classed as a Warburgian symbol\u2014what is seen is a differing set of operations. With this reorientation the hand\/instrument relation resists the possibility of the immediacy of expression. Indeed, there cannot be immediate expression, because there is no such thing as 'pure affect' let alone the singularity of expression. (There will be an important corollary here with the impossibility of pure unmediated colour. Colour as form is always informed.) This is the case even if what were thought to be possible is the affect's singular presence within a set defined in terms of polarity. As a result, there would be the need for what Warburg terms an 'energetic theory of human expression' ( _energetische Lehre des menschlichen Ausdrucks_ ) that was constrained by the identification of expression with the realm of the purely affective.[3] A return will need to be made to this point.\n\nTo begin, from the start the evocation of what is at hand, hence the supposition that there is an element of a work or an event that is at hand, would seem to bring with it a relation to a form of immediacy. At hand, what is at hand, if the idiom ' _at hand_ ' is followed, then it is there immediately\u2014there, as the formulation indicates 'at hand'. And yet, the already present status of the given cannot be understood as merely _at hand_. Within such a claim the given would be naturalized. It would be as though there were mere objects. (The naivety of realism would prevail.) Rather, once it can be assumed that modalities of relationality pertain from the start, it would have to follow that the history of hands\u2014the history of being at hand, simply at hand, even in hand\u2014cannot be understood in terms of the immediacy of the given. Presentation is only ever presence as mediation; im-mediacy, in other words, is only ever mediacy. Coming to be at hand, a state of affairs, which, once taken up, perhaps taken in hand, therefore is what brings the primacy of mediation into play. If there is a dominant trope at work here, then it concerns what has already been identified as the anoriginality of relationality. Relationality and mediation have to be thought of together. The history of the hand, the history of the hand's presentation (which already has folded into it the history of presentation) is as much the history of the image as it is the history of the way art stages its relation to questions of its presentation which, in this context, is the hand\/instrument relation\u2014a relation that can play a decisive role in the presence of art's self-staging. As has been argued from the start, there is a doubling. There is both the image as a locus of investigation and the image as that which evinces more than its presence as a document. Drawing on the argument that has already been advanced, if the former is the undoing of context, or at the very least allows for an undoing of the primacy of context, then the latter is the way the 'work' as the result of insisting on the primacy of relationality both refers to a context while at the same time refusing that context a unified and thus singular designation. (And this will be as true of context as the locus of history as it is context as the work itself.) If the former can be understood, within the terms set by Walter Benjamin's reference to quotation in his writing on Epic Theatre (namely, 'Quoting a text implies interrupting its context'), then quotation and interruption can be taken here as naming the process by which an image can be presented, or re-presented, via an act of its coming-into-relation.[4] The latter, the image's own position within the history of presentation, suggests that even a form of contextualization is itself the interruption of context in the precise sense that it shows the ways that any context is able to resist the attribution to it of original singularity. Contexts, and thus the process of contextualization, involve establishing relations of indetermination, rather than strict determination. Moreover, the claim about the limit of context is not a claim that concerns meaning as though the resistance to context gave rise to forms of semantic relativism. On the contrary, as has been suggested from the start, it is a claim about the ontology of the work of art, a dynamic and relational ontology in which work is an activity.\n\nThe development of the overall argument of this chapter involves a number of interrelated moments. As should be clear from what has been argued thus far, they are not moments within a generalized history of the hand, as though they would then need to be supplemented by others in order to complete the project, nor is there a simple genealogy of the hand in which its origin would emerge as having been given by a sleight of hand. Rather, what hands hold open, an opening that is equally a form of allowing, is a way of drawing together elements that are central to a philosophical engagement with the work of art and its history (where the latter is thought philosophically). And yet, what is drawn together resists both a generalized form of abstraction, though equally the history of hands serves, as has already been indicated, as a disruption of the project that seeks to sustain a connection between works that is merely chronological (historicism), abstractly thematic (idealism) or finally forms of decontextualization that are driven by repetition defined in terms of sameness. Hands continue to show themselves. The place of hands is equally present, for example, in a more generalized history of gestures. A history that incorporates the history of rhetoric\u2014evident, for example, in writings by both Cicero and Quintilian and also, to further the range of examples, in seventeenth-century writing on both iconology and emotional expression in painting\u2014attempts to interarticulate theories of langue with the hand. The possibilities can be clearly discerned in, for example, the presentation of hands in Cesare Ripa's _Iconologia overo Descrittione Dell'imagini Universali cavate dall'Antichit\u00e0 et da altri luoghi_ (1645) or Charles le Brun's _Conf\u00e9rence sur l'expression g\u00e9n\u00e9rale et particuli\u00e8re_ (1688), while the latter can be located in John Bulwer's _Chirologia: Or the Natural Language of the Hand_ (1644).[5] More generally therefore, while not included directly, though which would need to be there as part of any concern with the hand, is that which can be understood as taking place under the general heading of _manus loquens_ : this artful hand; the hand that speaks is an integral part of both the history of rhetoric as it is the history of gesture.[6]\n\nHere, however, the hand has another slightly altered presence. There are other hands at work. Each of these moments has a specific project, one that is as much internal to the works in question as it is projective. Nonetheless, prior to taking up the actual presence of works, there is a preceding moment; its development necessitates working through, once again, some of the earlier arguments that allow for the presence of work to take on a dynamic and thus active quality. What has to be allowed to insist is the work and thus its activity. Once work is recast in terms of the interrelationship between ideational and the material, then what endures as central is not an insistence on the content as opposed to work but working henceforth with the supposition that both are structured by a sense of process and activity.\n\nThe question that can be identified as there prior to any engagement with work\u2014with the work of specific works\u2014concerns the truth of the image. Once the question of truth is moved from the narrowly epistemological, such that it addresses the being proper to the work of art and is thus posed in relation to work understood as a locus of activity, then two interrelated domains of investigation are opened up. The first is given by the necessity to engage with content. The second pertains to the ontology of work: a positioning which concerns nothing more nor less than what it means for an image to be an image and thus for a work to be at work. And yet, posed in this way, these concerns are disarmingly simple. Part of the argument that forms a continual locus of investigation within this project, and which has already emerged, is that the workful nature of the work is already more complex than would have been signalled by terms such as 'content' or 'depiction'. Complexity, however, is not the identification of the complex with the difficult. On the contrary, the complex is a description of that within which the work is located. Even when posed in this way, the formulation still lacks the necessary precision. An insistence on location and thus on defining the image in terms of the locative, as has already been suggested, needs to give way to the presence of differing conceptions of activity. In sum, and to repeat the position that has already been argued, the locative must continue to cede its place to the actative.\n\nWhile allowing for the centrality of content and the static to identify part of what it means for an image to be an image, namely that specific part of the image that would be in play if the image could be, first, reduced to depiction\u2014captured in the claim, albeit one whose force is merely putative, that the image _is_ in its depicting\u2014or, second, when the image is equated with the object to be described such that it _is_ in its being described, precisely because both moves are reductive neither captures what is true of the image. Depiction and description only address the equation of the image with its having a determined presence\u2014a presence that can be dated and which would figure within any attempt to write the history of the image's reception (again, as though the image _is_ what it is in its being received). What is not captured by terms such as 'depiction', 'description' or 'reception' is the possibility that the image's appearance, the work\u2014its finitude\u2014is an after-effect of the work of an economy which itself demands that attention be given to the constitutive power of relationality and within which meanings are only ever the after-effect. Meaning as an after-effect of what has already emerged in terms of mattering. This economy cannot be separated from the work's material presence. However, it is not reducible to that which is given within description and analysed in terms of iconography or interpreted solely in relation to a 'beholder's previous visualizing activity.'[7] More is at stake, if only because any image allows itself not just to be seen again, but through its always being the effect, after-effect, of relations, the continuity of its coming-into-relation means that it is continually constituted and reconstituted. Acts of constitution are the result of the continual actualization of processes of relationality.\n\nIn order to take these opening concerns a stage further, an opening that is of course a type of continuing, two hands will be brought into play\u2014two moments in which the hand\/instrument relation has a centrality even if the nature and consequence of that centrality is yet to be determined. Two versions of St Luke painting the Virgin, one by Jan Gossaert and the other by Simon Marmion, provide the points of orientation. While employed by Philip of Burgundy (a Catholic) and thus while the problems posed by iconoclasm were never far away, between 1520 and 1526 Jan Gossaert produced one of the most remarkable engagements with the complex presentation of art's self-staging, namely his _St Luke Painting the Virgin_ (figure 3.2).[8] The thematic content of the painting has its own history, a history that continues to turn around the question of art's capacity to present the truth. Marmion's work is an illustration from a fifteenth-century _Book of Hours_ (figure 3.3). Taken together, and they are of course only two possible instances, the question that emerges concerns the making of content. Both are paintings of St Luke painting the Virgin. A topos has been identified. As such, a motif has been named. While it is not a simple either\/or, there are questions that arise here. Are these works paintings of the same motif? Were the answer to be in the affirmative, it would be as though there were an external element that the works addressed and which provided them with their content. If there is another answer, then it does not hinge on the denial of content. On the contrary, it depends upon the rethinking of content such that the claim then becomes the following: The motif of St Luke painting the Virgin is nothing other than the continuity of the motif's repetition. It would then follow from that claim that while there would be a sense of sameness, what cannot be precluded is the possibility that some instances of the motif's reiteration could not be equated with each other. Moreover, were they to come-into-relation, then the inherent fracturing of the motif, where what defined a motif was the necessity of its always already being fractured, would be demonstrated. The latter is a point of central importance. Its possibility is integral to a reconsideration of the philosophy of art. It results from holding to the centrality of mattering understood as the interarticulation of the ideational and the work of materials.\n\nJan Gossaert, _St Luke Painting the Virgin_ (1520\u20131526). Kunsthistorisches Museum, Vienna.\n\nSimon Marmion, _St Luke Painting the Virgin_ , from a _Book of Hours_ (1460\u20131470). British Library, London.\n\nPart of the argument to come involves linking the motif of St Luke painting the Virgin to the possibility of immediate presentation. This builds on the quality of 'Mary' that has already emerged and which locates that possibility within the larger framework of the immediate presentation of content. Immediacy endures as a question, even within the positing of its assumed possibility (and its subsequent impossibility). One of the most emphatic forms in which art stages its own concerns as given by immediacy occurs in the Veronica or the Mandylion.[9] Within differing traditions, both were presentations of the face of Christ. Both allowed for the possibility that while Christ himself could not be present, the image was the 'actual' face of Christ. As 'actual', it provided Christ with a form of immediate presence. While that presence, where presence is understood as pure immediacy and as such should be self-sustaining, it is nonetheless marked by an inherent instability. In the first instance this occurs because Christ's actual and complete presence within an image is an impossible possibility. This is true even if the Veronica were indeed the 'true icon'. The bodily presence of Christ will always be bound by the moment, and this will be the case precisely because the presentation has a twofold quality. It is firstly the presentation of that which is taken to be 'Eternal' within the image. Hence, there is a form of immediacy at work (albeit one in which immediacy is simply posited). Second, immediacy founders, because the image is itself always the presentation of the true icon, as opposed to its being the true icon itself. In other words, an initial immediacy always encounters an ineliminable and necessary mediation. That mediation is the presentation of immediacy. Immediacy's presentation, precisely because there will always be the means and the process of presentation, is that in which immediacy is undone. Again this is the consequence of an insistence on production.\n\nArt's own undoing of immediacy (i.e., the retention through an undoing and exposure of the original's impossibility) of the Veronica has a number of importantly different forms. It is there, for example, in the presentation of St Bartholomew holding the now undone face, a face there in its undoing, in Michelangelo's _The Last Judgment_. The undone face is the possibility that will always have haunted the Veronica. It exposes the face as a form of construction, one that can be undone. Immediacy is mediated from the start by the possibility of the impossibility of its own continuity. The undoing of the face is the depositioning of fate. This is a deconstruction that continues to appear once other instances are brought into consideration. The relation to St Bartholomew's undone face cannot be refused. It is always there as a possibility; hence the force of coming-into-relation. What this means is, of course, that there is no true Veronica, there are just a series of interrelated instances, all of which attest to the impossible possibility of immediacy. If further evidence of instability of immediacy were necessary, then it could be found in outlines or preliminary sketches of the Veronica\u2014not in their detail but rather in their actual existence. After all, would not an outline of the true image be a form of deconstruction of immediacy in advance? This is what occurs in Polidoro di Caravaggio's sketches for what would become the altar piece _The Way to Calvary_ (1534) (figure 3.4). The work is an oil sketch for a larger work. Hence the question of whether or not there can be a preparatory sketch of the Veronica. Or rather, given that there are such sketches, the question then is: What effect would the possibility of such a sketch have on the status of the image once the image is supposed to be the presentation of an immediate relation? In other words, the presence of the sketch announces the presence of the Veronica as a moment within art's self-staging and as such underscores the inherently mediated nature of immediacy.\n\nPolidoro di Caravaggio, _The Way to Calvary_ (1534). National Gallery, London.\n\nThe immediacy of presentation sets a specific measure. Its impossibility is significant. Nonetheless, as a project governing art's self-staging it provides the way into an engagement with the content of paintings of St Luke. And it needs to be noted that these are often paintings of St Luke painting. As such, it opens up the possibility that what is also at work is the possibility of self-presentation and thus a modality of the self-portrait in which the artist is present as St Luke. One of the most famous is Vasari's painting of 1565 in which he has become St Luke.[10] In other words, Vasari paints himself as St Luke painting. This doubling of presentation continues to take on different forms. Even though it may be a continual point of reference, it is worth recalling the description of Mary in the Christian Bible that is taken to be central. The passage is from St Luke's Gospel 2:19. It presents Mary in terms of an already present relation between an inside and outside. As such, Mary cannot be separated from the already present problem of expression: 'But Mary kept all these things, and pondered [them] in her heart' (\u1f21 \u03b4\u1f72 \u039c\u03b1\u03c1\u03af\u03b1 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c3\u03c5\u03bd\u03b5\u03c4\u03ae\u03c1\u03b5\u03b9 \u03c4\u1f70 \u1fe5\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03c3\u03c5\u03bd\u03b2\u03ac\u03bb\u03bb\u03bf\u03c5\u03c3\u03b1 \u1f10\u03bd \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 \u03b1\u1f50\u03c4\u1fc6\u03c2).\n\nWhile Mary is represented in paintings, she is also the outward appearance, thus the presentation, of the totality of knowledge which she 'ponders' (\u03c3\u03c5\u03bd\u03b2\u03ac\u03bb\u03bb\u03bf\u03c5\u03c3\u03b1) 'in her heart' concerning what is known of Christ, her representation, as noted in the preceding chapter, becomes a fundamental moment within a more generalized history of the image and thus of art's self-staging.[11] Mary is both an image to be imitated and a self-completing finality, precisely because the continuity of her reflection upon the totality of all that she knows; she is complete as a sign. It is this completion that has to be given in her presentation. There is the complete identification of the signifier and the signified. While there is no one precise word to describe Mary's presentation, it is clear that with Mary there is a coincidence of representation, imitation and emulation. Mary is therefore as much a figure within the history of theology as she is that which causes theological positions, perhaps stripped of their religious force, to form integral moments within the history of art's own self-conception. Hence, in broader terms, what this means is that representations of Mary when taken in relation to the Veronica should be seen as the inversion of Plato's arguments concerning the impossibility of a complete and self-completing totality. Platonism, the Veronica and then Mary, become defining moments in fundamentally separable moments within the history of the image. Their separation\u2014their connection is defined in these terms\u2014concerns the interplay of the possibility and impossibility of both representation and completion. As will be noted, the presence of Moses within Jan Gossaert's painting _St Luke Painting the Virgin_ is equally part of the same regime of signification. Present here is therefore a regime that resists any form of synthetic or syncretic summation or expression. When taken together, all the modalities of presentation that the names Plato, Mary and the Veronica underscore is the presence of a form of regulation in which the force of the regulative is located in the nature or quality of the content that has to be represented. Thus, it matters that the 'shadows' in book 7 of _The Republic_ are not the presentation of what is real and that there is a distinction between that which is self-identical, and which can be located in relation to the Platonic conception of the Idea, and that which in its presentation is always inadequate in regards to the idea's presentation. The latter is the presence of the particular. In relation to both the image of Christ and of Mary, it is precisely the determination of content that means that their representation, which means their presentation, must always be adequate to that content. Platonism, if interpreted in terms of the necessary impossibility of the Idea's self-presentation, founders at the point at which it is possible for Mary and Christ to be present 'as' themselves.\n\nMary brings the question of the truth of (re)presentation into play in an emphatic manner. Present here is a conception of truth that does not just attend the Virgin, it must attend any attempt to paint or present her in an image. The concerns of Jan Gossaert's _St Luke Painting the Virgin_ cannot be divorced from the work of hands. There are three hands that are central: St Luke's hand, the hand of the Archangel and Moses's hand. These embody the different forms taken by the hand\/instrument relation (reaffirming thereby its presence as a motif). What is significant about these hands is the way each is bound up with what has been referred to as art's self-staging. Each point must be taken in turn in order that the effect of the network of relations is then established. Here the painting works as a result of already present forms of relationality. In the first instance, Moses's hand points to the Law, the relation to the Law (and the 'Law' is what it is as given within that relation.) The Law is objectified. It is present and determined by a relation. Part of what is indicated\u2014pointed out, pointing as the hand's work\u2014is the presence of the law forbidding the creating of images. While the Commandments (Exodus 20:4) refer to the representation of the contents of the 'heavens', elsewhere there is a more direct interdiction in terms of the representation of God (e.g., Leviticus 19:4). Part of what is indicated is therefore both the Law that forbids images; this is Moses's hand, its work, and at the same time the overcoming of that interdiction. As a result, the Law becomes a law. At-the-same-timeness grounds the relation of non-relation that produces the movement from Law to law. An overturning occurs, though it has to be noted that what takes place here is a very specific interpretation of the _Bilderverbot_ , since the paintings\u2014Gossaert's and St Luke's\u2014present the image of that which would have been forbidden. The latter is there in the figure of Mary, present as that which is imitated and then present in the painting of her presence. It is doubled as it appears as a result of St Luke's painting and is of course the content of Gossaert's painting. In both instances, the law is literalized, and thus made into a commandment which, in being shown only having pertained to literal images, is, as a result, also shown to have been overcome. In other words, Moses is not just present as a figure, rather a figure whose hand's relation to the law\u2014the pointing hand almost as though it were a textual manicule\u2014when viewed in relation to the hands of both St Luke and Gabriel, is implicated in the creating of a setting, one that is an after-effect, in which the presence of Jewish law has been undone and thus rendered inoperative. Even though what is present is only the figure of the Jew, the relation between the hands\u2014and it is hands as productive and thus as mediated\u2014not only has an effect, that effect is itself only possible because neither the hands nor the gestures can be assimilated to each other.[12] Moses and St Luke are both situated in a world whose disconnection is enacted by the inscription of Moses into the frame (rather than his exclusion from it). There is a form of synecdoche here insofar as while Moses is there as a sculptured presence, he is equally, therefore at the same time, the figure of the Jew. That figure is encapsulated in his hand. Moses becomes his hand. It is only the irreducibility of the hands of Moses and St Luke\u2014of one to the other\u2014thus their relation, which is a relation of non-relation, that is integral to the production of the work's meaning. In other words, while there is a form of repetition, it is guided by the proposition that here are hands, the connection between which signals that they have never been further apart. While Moses has horns, thereby underscoring the anti-Semitic nature of the image, if what were wanted is the overcoming of Judaism, that addition is an embellishment.[13] The real work of the identification and then the exclusion of the Jews occurs as a result of the two hands coming-into-relation. Once again the painting stages the double move that has already been noted in the woodcut illustration in Petrus Canisius's _De Maria Virgine_ (figure 3.5) in which Mary announces both the coming of Christ and the overcoming of Jewish law. It is not just that one cannot be thought without the other. The more significant point is that both are given by the work of hands such that their separation is equally their co-implication.\n\nPetrus Canisius, _De Maria Virgine_ (1583). British Library, London.\n\nThe Virgin, in Gossaert's painting, appears in a divine revelation, evidence for which can be found in the clouds surrounding her. She is in her appearing. The clouds signify immediacy and in this instance the presentation of that which is divine. However, her presence within St Luke's painting is only possible if that particular presentation is controlled from the start; what this means of course is that immediacy is always being mediated by that which has a form of externality. (This is the condition of presence.) In other words, the hand that produces the Virgin might betray her. However, it must not. Recalled here is the question of art's fidelity to an outside. While what occurs here has an ostensibly theological setting, what the painting stages is of course those moments in which externality provided the work's possibility as well as its ground of judgment. Within this work those external concerns have been staged by the painting itself. In a sense the theological here could be viewed as subservient to larger questions that have always informed writing on art. As the opening of Philostratus's _Imagines_ makes clear, that painting, imitation and thus the necessity that there be a fidelity between what is presented, and its presentation, is a setting that, while governing Christian theology's concen with fidelity, also predates it.[14] Moreover, the determining effect of externality as a dominant position within the history of philosophy has an important register both in Plato and in the Platonic tradition.[15] What marks out this position is a relationship between image and content and a capacity for showing or presenting (mimesis) which, even though it has different determinations, continues to be repeated within the history of philosophy. Presenting, thus understood, would always have a content that is essentially indifferent to art (where the latter is understood as a specific form of making present) in the precise sense that what is taken to be the content has a force that is independent of what is presented. There is an inevitable space between presentation and content.[16] Thus, in the case of Plato, the modes of presentation fail to capture that which defines what is proper to externality. Art, understood as a form of presentation, mimesis as presentation, betrays the propriety of that which is to be presented. The determination by externality finds its negative instance. However, that instance depends upon a relation of indifference between the 'Idea' and its presentation within art. Indifference has a precise sense here: indifference means that externality has a non-necessary relation to its presentation _within_ and _as_ art. Nonetheless, within these repetitions, what is maintained is the position in which ideational content is attributed a quality that can be, on the one hand indifferent to modes of presentation, while, on the other, presentations can be indifferent to that content (indifferent to a retained sense of propriety). Thus, when identifying why it is that the rhapsodes were 'stupid' (\u03b7\u03bb\u03b9\u03b8\u03bf\u03c4\u03b7\u03c9), Xenophon in his _Symposium_ has Socrates specify the reason with great precision: 'The reason is clear, they do not know the inner meaning of the poem (\u03c4\u03b1\u03c9 \u03c5\u03c0\u03bf\u03bd\u03bf\u03b9\u03b1\u03c9 \u03bf\u03c5\u03ba \u03b5\u03c0\u03b9\u03c3\u03c4\u03b1\u03bd\u03c4\u03b1\u03b9).'[17]\n\nThe ' _hyponia_ ' is indifferent to art (where the latter is understood as a mode of presentation and thus as appearance).[18] And it is the very structure of indifference that allows for the nature of the distinction between the concerns of the rhapsodes as opposed to the concerns of philosophy. Indifference within the determination by the conception of externality at work within the philosophical tradition is that which holds externality and materiality apart. There is a further point that needs to be made in relation to the conception of externality that plays a founding role in the determination by that form of externality that is constitutive of art's work. Externality, even though it marks the presence of forms of propriety, does not have a necessarily determined quality. (Externality may itself be the site of forms of interpretive conflict in which differing forms of the proper continue to encounter each other.)\n\nThese concerns already bear on St Luke. They are borne out by the hand\/instrument relation. In Gossaert's painting the next instance in which this relation is being worked out concerns the way St Luke's hand is controlled. His hand is controlled from the start; the control is neither straightforwardly intellectual nor purely spiritual. The control is quite literally manual. Here, however, the complicating factor needs to be introduced. Central to that control is the recognition that the hand is mediated. That mediation has to do with the presence of the drawing instrument. The hand here involves its having been mediated twice. First, there is the Archangel's hand controlling St Luke's, and then there is the presence of drawing instruments as that which will have always mediated the hand that paints or draws. (This second sense of mediation will return. Its already present status is of central importance.) While those instruments play a fundamental role in the interrelated work of hands, it can be argued that this other mediation was either not noticed or, were it to have been noted, its relevance would have quickly become its irrelevance. Indeed, it is possible to go further and argue that the overcoming of the 'stylus' in the name of the divinely inspired hand was central to the way art differentiated itself from craft.[19] Evidence for a more general displacing of the hand as the site of a doubled form of mediation\u2014the hand as controlled and the hand as controlling\u2014can be noted, for example, in Federico Borromeo's _De pictura sacra_ (1624). Borromeo quotes Nicephorous's _Historia Ecclesiastica_ (approx. 1320) in regards to the fidelity of the presentation of the Virgin and child. Borromeo's Latin translation of Nicephorous reads, 'Luke the Apostle depicted them accurately ( _accurate_ ), painting her with his own hands ( _manibus suis depinxit_ ).'[20]\n\nIt was neither a pencil nor a pen. It was the hand itself\u2014style without the stylus, form without production. These two forms of separation are of fundamental importance in the construction of a history of the hand\/instrument relation as the history of a motif. And it should be added that even though the lines were continually quoted throughout the fifteenth and sixteenth centuries, they have an older lineage. Indeed, it is worth noting both the ubiquity of the formulation and the continuity of the reference both to Nicephorus and to the connection between St Luke and the act of painting. In a recently published manuscript by 'Anonyme du Tarragonensis' (as he is called by the manuscript's editor, Krignie N. Ciggar), which is ostensibly a description of Constantinople written towards 'the end of the twelfth century', its author writes of a specific church that it contains an 'icon' ( _ycona_ ) of the 'mother of God, that 'the evangelist Luke, _ut aiunt Greci, depinxit_ '.[21] The recognition of St Luke as inextricably linked to the painting of the Virgin, a positioning that can be dated from the sixth century CE, noted above continues to be repeated. What is interesting here in the Tarragonensis 55 is the absence of any reference to a source. In a later text, namely Gabriel Du Pr\u00e9au _De Vitis_ (1581), after citing Nicepherous by name, Du Pr\u00e9au then adds that 'Luke the first apostle, accurately painted the work with his own hands ( _manibus suis depinxit_ )'.[22] Not only is there the clear recognition that St Luke painted the Virgin, what endures as significant is the reiteration of a connection between accuracy and the 'hand'. The ubiquity of the hand not only underscores the centrality of the hand's presence; the hand continues to appear as though it were not mediated in advance. In other words, what is attributed centrality is the hand as opposed to the hand\/instrument relation's productive activity. The link between accuracy and the hand would seem to have overcome the need to refer to any form of mediation precisely because the tradition announced as much in the presence of the Veronica, as it was with Mary as a self-completing and self-sustaining sign. (Even though what Gossaert's painting makes clear is that the hand was mediated twice.) Even if the hand is displaced in favour of the intellect or sprit, what was not thought to be central was the hand's other mediation.\n\nAgain, in 1528 in Castiglione's _Il Libro del Cortegiano_ , the courtier recounts a story that can be taken to have started with Pliny the Elder, though which had many contemporary versions, namely that Apelles's criticism of Protogenes is that the latter 'did not know when to take his hand from the table' ( _Che non sapera leval la manni della tavola_ ).[23] To the extent that Apelles remains the figure of the artist par excellence\u2014this was after all the name given to Leonardo\u2014the limitation, while significant, has to be incorporated into a history of the hand.[24] Moreover, Apelles was a figure who was not just cited. He played a more decisive role insofar as his lost works were 'recreated'. Botticelli, for example, drew on the descriptions in Lucian to paint works by Apelles (hence Botticelli's The _Calumny of Apelles_ [1494]).[25] This prompt enters the Renaissance not just via Alberti but in an earlier translation of Lucian's _On Calumny_ into Latin by Guarino of Verona in 1408 and which was subsequently used by Alberti.[26] Appelles even appears in Raphael's _School of Athens_.[27] The significant point is that with Pliny, and then with the way that Pliny provides elements of the identity of specific characters, what continues is the incorporation of the centrality of the hand that has an almost invariable relation to the divinely guided hand. The hand continues as defining a complex locus. However, it is never just a hand. It is always a hand at work, or implicated in work. The hand is therefore already positioned by relationality. The hand is always already in relation. The significant point, however, is that relationality does not have a single determination.\n\nThe hand guiding St Luke's drawing, as well as St Luke's hand itself, are already informed. Moreover, St Luke's hand is commanded, that which commands has an externality even if, as is the case in Gossaert's painting, it is present internally. What is clear in Gossaert's painting and clear, as will be argued presently, from Rafaello Borghini's writings, is that the 'hand' has to be controlled in order that there be works of art (hence the joining of the hands, uniting north and south, if only for a moment). Obeying the angel, even if the angel's presence is not noted, and obeying the commands of the 'intellect' are attempts to overcome an instability that is inscribed from the start into the structure of art's work and which results in the need for forms of determination where that which determines (equally that which guides or commands) is defined in terms of their externality. This becomes much clearer when a relation is established to Simon Marmion's miniature.[28] What are fundamental in this instance are the differences, and thus the nature of the relation, between Gossaert and Marmion insofar as self-staging is concerned. While it may always be possible to generalize in terms of what has already been described as art's self-staging, and while, again, there is a form of generality present in terms of the repetition of St Luke's activities as a painter, it remains the case that what has to prevail in order that specificity or singularity be maintained are the differences. These differences are always the result of forms of relationality such that any particular instance is itself an after-effect. Again, what this means is the after-effect will have been marked by that which produces it as such.\n\nThe point of departure with Simon Marmion's works is the identification of what defines the process of self-staging within it. This involves both negative and positive characteristics. In the first instance it needs to be noted that there is no direct sense in which either the Virgin or the infant appear in a vision. The temporal discord between St Luke and the Virgin allowed for by a vision, which in this case of Gossaert's painting is a discord that is overcome by the presence of St Luke's painting, is absent from Marmion's works from the start. In Gossaert's painting there is a discord and thus a literal displacement that is sustained by the clouds that surround the Virgin. In Marmion's work, however, there is the absence of a temporal disjunction between the vision and the work of painting by St Luke such that what would then be demanded is the intervening presence of the painting (i.e., the presence of the painting within the painting). Indeed, that absence brings with it what might be described as an already present sense of spatial continuity. The frame is not held in place and as such has to _place_ different temporal and spatial registers. As a consequence, all the elements of the work are formally and temporally continuous. What this means is, of course, that all the elements within the frame take place at the same time. (These are claims about the nature of internal relationality at work in Marmion's illustration.) The twofold event of the absence of discontinuity on the one hand and the presence of continuity on the other have the effect of raising the question of who the Virgin is, a question that is resolved in Gossaert's painting by the disjunction between the presence of Mary in the vision and her presence within the frame that St Luke is painting. That question is an effect of work of the image, a question posed by the possible ways in which forms of determination, which would themselves be the result of forms of control exercised by an external source, could then be at work within the frame. However, it is precisely that work and thus the continuity within the frame that gives rise to questions concerning both the identity of the 'woman' and the 'child' as well as to the relationship between the 'model' and St Luke's painting as it occurs in Marmion's work. These questions do not have any form of direct resolution within the frame. The presence of questions is generative of the image's own instability, an instability within self-staging. While in the case of Gossaert's painting the angel's hand attempted to ensure the veracity of the image, here there is no such guarantee. There is merely St Luke painting. Control is absent. Externality cannot be effected. Moreover, once attention is paid to the image within his painting and the figure of the Virgin, the questions become more insistent. At work here is the determining effect of an internal set of relations.\n\nIn Gosseart's painting there was, as has been indicated, a discontinuity within the frame. Here there is a continuity. The question of the Virgin cannot be resolved by indicating that there is a disjunction that defines her presence within the frame. As the Virgin has not appeared, what this means is that her presence as an appearance is not part of the work's work; she is simply there, and the questions concerning her presence become more concise. For example: Is the painting of the Virgin the painting of a model? Is the painted Virgin, the one that St Luke is in fact painting, a substitute for the Virgin such that St Luke's painting is no more than the painting of a model? Questions of this nature proliferate, and all gesture towards the sources of instability that attend presentation\u2014an instability that is countered, so it might be thought, by the force of the Veronica. Though the latter, as has been suggested, is itself premised on a necessary impossibility.\n\nWhile the frame is structured by a form of spatial continuity, there is, nonetheless, an important moment of temporal discontinuity within it. The significance of that discontinuity lies in the fact that it pertains to the painting that is being undertaken by St Luke. In the painting that he is painting, the infant is still. The infant lies content in the Virgin's arms. While the infant must remain a child, there needs to be intimations of the divinity that attends his presence. The child that appears with the Virgin, the child being painted by St Luke, is reaching towards the Virgin. His right hand reaches out. She is yet to respond. Or perhaps her response is to ignore the child's plea in order that the painting takes place. (As such she would be a model rather than the Virgin.) Perhaps the child's concerns are of no interest. Perhaps she is oblivious to his wants. The child is restless. In the painting that St Luke is finishing, the child is at rest. The relationship between mother and child is calm, almost serene. There are therefore two moments: the moment in which the baby is squirming in the arms of his mother (is it, in fact, his mother?) and the other in which serenity will have been both preserved and presented in the painting by St Luke. Thus, there are two different mother\/child relations here. The relation has now become a motif refusing an essential and singular determination. (And this despite the fact that it is the relation between Mary and Jesus.) The instability is clear. In the painting that St Luke will finish, the serenity is a form of divinity. However, the restless infant could always be just a restless infant. It is only in St Luke's painting that this infant will have been presented as the Holy Child and thus the infant.[29] In other words, the necessity that St Luke presents the Virgin and child\u2014given the difference between the two moments\u2014compounds the lack of surety as to the identity of the models. Are they mother and child or simply a woman and a baby? What is important here is that while this question is resolved within the painting by St Luke\u2014resolved, that is, in the painting he is painting\u2014it cannot be resolved in the miniature that Simon Marmion has painted. Their co-presence marks the work's founding instability. The determination by externality at work here involves the interplay of possibility and impossibility. Their simultaneity is given by the work's self-staging _._ The theme of St Luke painting the Virgin introduces, in the case of Gossaert and Marmion, two different and in the end irreducible forms of relationality and thus two different ways in which self-staging takes place. The centrality of the motif as an organizing term maintains its pivotal role as a consequence.\n\nThese points of intersection continue to overlap and can continue to be adumbrated. What they work to indicate is that the hand\/instrument relation is both far from a serendipitous figure and yet does not have an essential quality. In the context of painting, it can be argued that this relation, relation now as a motif, has a history in regards to which art is judged and, more importantly, is taken to have been produced. Art, the argument will go on to claim, is the hand's work. Hence, the question that has to return is, what guides the hand? However, while that question has an insistent force, at work within it is another. Indeed, it is a possibility that is already there\u2014already, as it were, in hand. What is not asked\u2014or at least not directly\u2014are questions concerning the hands' own work (e.g., What does the hand guide?). That latter question would address the work's production, its effectuation by the hand. Before turning to the question itself, it is worth noting that if there is a body of writing that had been concerned with the hand, then it would have been Cennino Cenini's _Il libro dell'arte_ (approx. 1420). Cenini was concerned with practice and the use of the hand and its relation to instruments. It could be therefore claimed that there was an engagement, as a result, with what the hand guided. Nonetheless, it might still be argued that difficulties emerge. As the position begins to unfold, the hand begins to cede its place to the 'head' such that the locus that predominated would be predicated upon the withdrawal of the centrality of the hand. Hence, he writes: 'Do you know what will happen to you with the practice of drawing with a pen ( _praticando il disegnare di penna_ )? It will make you expert and practical, capable of much drawing within your head ( _di molto disegno entro la testa tua_ ).'[30]\n\nWhile the question of the hand becomes the question of its relation to the instrument, it can still be the case, as is clear from the above, that the 'head' acts. Production is not the result of a subject acting. Production has to be understood in terms of the mediation of the hand by the instrument. It is both intentionality's actuality and intentionality's impossibility at the same time. And yet, precisely because a relation is established between the hand and the instrument, this mediation has a necessary inbuilt reciprocity. The question then becomes one that has to acknowledge the hand's mediation such that the work is itself the result, in part, of the hand's work. If that were the point of departure, then the question would always devolve back to the forms of production in which the hand's mediation was primary. In regards to the various works of art and writings on art while the hand was always there, and while it was always the anoriginally mediated hand, mediation was not seen, or at least it was not seen directly or seen as significant. While it is far from difficult to account for the reasons why that mediation was not seen and now how it can be seen what is significant are the consequences of having opened up the hand.[31] At stake here is a coming to see what had not been seen\u2014that is, seeing what was already there to be seen. The stylus, the instrument in St Luke's hand, will always have been more than its literal presence. The latter is a literal presence, a presence overlooked precisely because what mattered was the Archangel's hand, and this despite the presence of the work St Luke had created. While it is not seen, the important point is that it was always there to be seen. St Luke's hand had always already been involved in a doubled form of mediation. Art's self-staging did not acknowledge the hand's other mediation, and yet that mediation was always there as, and in, art's creation. As a retrospective claim, it is one that underscores the nature of production.\n\nIn sum, in the case of both Gossaert and Marmion, even though the hand remains central and even though there is the anoriginality of the hand\/instrument relation, the accompanying writings that continued to underscore the hands' presence were always, or at least predominantly, defined in terms of that which guided that relation: style rather than stylus. The angel's hand was on St Luke's. St Luke's face appears in thought as though what he holds is of little concern. The relation to drawing and painting instruments was there. However, what was effaced was the necessity that art also be present as a technical practice. To the extent that its presence as a technical practice was allowed, it was assumed that externality prevails such that what mattered were the spiritual or intellectual elements that guided the hand. What this means is that there was always a divine guide. The question to be addressed is what happens when the possibility of the 'divine' is suspended and the hand that guides becomes the point of orientation. In other words, what takes place when the motif of the hand\/instrument relation is given a radically different calibration such that the relation became a locus of work rather than setting that is worked upon? This is a question that could not have been asked. Nonetheless, it can be asked now, and this is the argument to be presented, precisely because of that which results from the process of coming-into-relation. What occurs as a result is that it becomes possible to think of mattering.\n\nTo begin again, this time with another hand. Clearly, there is another hand. Indeed, there are many other hands. Hence, there is no one hand that has an exclusive right. What then of this other hand? That other hand, an instance of that other hand, is there in El Lissitzky's _The Constructor_ ( _Self-Portrait_ ) (1924) (cover image).[32] Here in the sliver-gelatin print, a work that is already the result of a form of montage, there is a hand. As a beginning it should be noted that the claim of the work to be a self-portrait opens up as a question of the relationship between the hand and the face. The eye and the hand meld. Recalled is the tradition of the seeing hand. And yet this is no mere surrealist gesture. The presentation of the eye\/hand is the after-effect of technique, an after-effect, that is, of a specific technical possibility. The work, of course, contains a line that is itself an after-effect. The compass has drawn a circle. The work contains it. Equally, the work contains references to typography, paintings, and design, leading, in addition, to a form of spatial representation that will become in the work of both Lissitzky and Malevich a type of architecture. Again, the possibility of this presentation, what amounts to the creation of a form of presentational discontinuities within continuity, occurs as the result of technique. Six photographs were used in the construction of this image. Collage has become montage.\n\nIn the photograph the drawn line is the work of the hand. However, the hand has a limit if the work is thought to be reducible to the work of the hand. While the hand brings technique into play, the hand's inscription is its having been delimited. The hand, which is now the hand\/eye\/compass relation, and to than extent there is an already present mediation, draws attention to the specificity of the work's mattering. It is not as though it is simply comprised of elements. As such there is no pure hand. As already noted, the work of the hand involves an original form of mediation. However, once that concession is allowed, then the irreducibility of this work to the work of the hand, where the hand would have become its own literalization as that which holds the compass, cannot be maintained. It is precisely the presence of the hand that therefore allows for the hand to open beyond itself and for it to be more than just a hand. The hand here marks an end. The hand's mediation, its inevitable mediation by the instrument, underscores the centrality of the technical. The latter is a setting which not only is the result of reproduction and therefore the effect of reproducibility but also has the effect of causing the externally guided hand to be withdrawn from its position as pivotal to art's production. The hand occasions it to be handed on. The site of literal instrumentality gives way to the work as an after-effect in which the relationship between the work's elements are constructed as a result of techniques. The presence of the hand in this work is itself the result of techniques; though it now becomes clear that this was always the case. While the hand may recall on an ideational level its relation to the history of hands that recall, or rather the capacity for that recall, it is an after-effect of techniques that, within the works that may be recalled, may have had an attenuated and thus less direct presence. This is the point at which it is essential to introduce what has been implicit thus far.\n\nThere are a number of interrelated points that need to be made. The point of departure is straightforward. One of the fundamental aspects of Gossaert's _St Luke Painting the Virgin_ can only now be seen as a result of what was opened up by Lissitzky's _Self-Portrait_. What is uncovered was not available to be seen at the time and yet remains both present and an integral part of the work, namely the refusal to account for the instrument and thus to think of art's materiality, even though art as a material practice was stated in the painting through the inscription and, it should be added, necessary inscription of both St Luke painting and the presence of his drawing instruments. It is essential to be clear here. Even though writing from the Renaissance onwards evoked the hand, it was equally essential to note, as indeed Roger de Piles did in his _Abr\u00e9g\u00e9 de la vie des peintre_ (1699), that style cannot be reduced to a 'character of the hand'.[33] In de Piles there is a move that literalizes the hand, a position that had currency elsewhere. Hence, Michelangelo writes in a letter dated 24 October 1542 that a man painted with his brain, not with his hands. Furthermore, Rafaello Borghini, in his treatment of _disegno_ (drawing) in his _Il Riposo_ (1584), locates this precise set-up as that which defines disegno itself. Borghini writes that 'drawing ( _disegno_ ) does not seem to me to be other than an apparent demonstration with lines ( _una apparente dimonstrazione con linee_ ) of that which is first conceived in the intellect of man and imagined in the mind. This, wishing to make it appear with the appropriate means, requires that with long practice the hand becomes accustomed ( _sia avezza la mano_ ), with pen, with charcoal, or with pencil to obey what the intellect commands ( _ad ubbidire commanda l'intelletto_ ).'[34]\n\nIt is the 'intellect' that control's the hand\/instrument relation. As a result, art's work becomes the work of the intellect. Present here is a tradition that, while not having an original formulation, is already clear in Roman discussions of sculpture. For example, in his discussion of the practice of Phidias, Cicero writes in his _Orator_ 2.9, 'Surely that artist, when he created his Jupiter or Minerva, did not look at any person for use as a model, but rather in his own mind there resided a sort of extraordinary apparition of beauty; concentrating on it and intuiting its nature, he directed his art and his hand ( _artem et manum dirigebat_ ) towards producing a likeness of it.'[35]\n\nProduction, were it to occur, is always a quality of the commandment of the mind. The importance of this passage is clear from the fact that it reappears in Bellori.[36] The guided hand endures. Techniques and production remain therefore unthought within the tradition in which these texts can be located. The relation to the mind meant that even though the technical was there, announced by references to the hand, it remained unthought.\n\nIn the othering of the hand, the hand's presence would be different. It would begin: _On the other hand_. What Lissitzky's hand, this other hand, opens up is the possibility of the twofold move in which, staging the technical\u2014and it would come to be staged\u2014as a result of the techniques, allows for the transformation of the hand in which its presence moves beyond a domain circumscribed by the hand as the merely technical and in the service of the intellectual or spiritual. What the juxtaposition allows is the truth of the image to emerge, not the truth in relation to its content. Truth here means that it is this juxtaposition\u2014and thus the intrusion of this other hand\u2014that unsettles the opposition within which the work was cast by opening up the other possibilities that always lay in the hand and that arise with the hand's transformation. In other words, the truth of Gossaert's painting depends upon a relation that refuses the hold of historical time if the latter is understood as defined by chronological imperatives and simplistic forms of periodization. What follows from this is at least twofold. In the first instance there is the general philosophical claim that any singularity is an after-effect of a network of relations. (In regards to the work of art, this is a claim that has operational efficacy both internally and externally.) Relations occurring within and as the work\u2014thus they are relations whose occurrence becomes the work's mattering\u2014are not static but contain elements that allow for connections and interconnection to be established (hence the force of the proposition that the singular is an after-effect of a network of relations, a claim that defines the work's present and grounds the possibility of its future). (A work's afterlife is the continuity of its coming-into-relation.) The second point is that what follows from this move\u2014both as an interpretive argument and equally as one that brings with it a certain commitment both to the time effect and the ontology of the work of art\u2014is that the claim that there is such a thing as modern art will have become untenable if the modern is understood as part of a process of art's continuous development. Art is of course produced in the contemporary period; moreover, there is a defining relation between the work of art and modernity. The problematic status of the term 'modern', however, occurs precisely because the effect of art within modernity is to cause a radical reconfiguration within the 'modern' of the practice of criticism and interpretation, or at least this is the possibility that it affords. What that reconfiguration means is that, rather than there being modern art, there are interpretive and critical practices that demand to be thought of as modern. And thus they are that which 'modern' art demands. The interpretive practices stemming from the insistence of the modern provide works of art with an afterlife, and it is precisely that provision that is occasioned and accounted for by arguments pertaining to the incorporation of singularities within a network of relations on the one hand and which leads, on the other, to the argument that the presence of the modern cannot be separated from contemporary practices of criticism. As such, the modern cannot be defined by its identification with the contemporary.\n\nThe final point that needs to be made pertains to St Luke's own drawing or painting. It emerges from the obvious claim that even though the drawing or painting may resemble the Virgin (the model), it is however not the Virgin's presentation. It is not, to use Hegel's formulation, the 'divine likeness having become perfectly visible'. If anything, it is imperfectly visible: visible but imperfect. The imperfection\u2014perhaps the ineliminability of imperfection\u2014of the drawing as an interpretive response demands a seeing that is itself an interpretive response. Even if it could be claimed that it was possible to argue for perfect visibility at the moment of its creation, it is equally possible, also at that very moment, to make an argument for the opposite. Were there an anti-Marian impulse or even a 'pagan' response, then the claims of the presence of a founding imperfection would have been based on the same drawing as an argument for perfect visibility. What this means is that St Luke's own drawing in Gossaert's painting\u2014and this despite the presence of the Archangel having taken St Luke in hand\u2014marks the moment of instability. As such, the work founds and is founded on this instability. Both these determinations, now, pertain at the same time. Moreover, it is an instability that precludes, by definition, any attempt to posit the work as an original self-enclosed and self-referring singularity. To the extent that the Archangel's hand is viewed as a locus of surety and perfection, the work created by St Luke\u2014and, it must be added, created by his hand, a hand that is equally the site of that other mediation\u2014destabilizes such a possibility. The work is both his creation and the creation of the imperfect. The work therefore cannot reconcile the interplay of the perfect and the imperfect. It is structured by that economy. The work maintains itself in terms of the anoriginal presence of this movement. Therefore, the economy of the work precludes, despite appearances, the presence of original forms of unity. This is the work's truth; moreover, and methodologically this is the essential claim, it was this anoriginal irreducibility that was revealed by Lissitzky's hand (hence, once again, the productive force of relationality).\n\nThe next moment in this construction of the complex relation between the hand and instrument concerns two works by the contemporary German artist Georg Baselitz. Two works stage further elements that are central to this project, a project defined by the positioning of hand within the continuity of the hand\/instrument motif's discontinuous reiteration. The works in question are _Ein moderner Maler_ (1966) and then _Ein moderner Maler_ _(Remix)_ (2007) (figures 3.6 and 3.7).[37] Here, rather than dwell on the general strategy of the remix, which might be understood as part of a process of modification through sustained engagement (an encounter with ghosts from which flight is refused), what matters is a type of return. Consistent with the argument developed thus far, what is important here is relationality rather than singularity. Prior to any evocation of the hands, what has to be noted is the relationship between figure and 'background'; these terms are used tentatively and are soon to be abandoned. As a point of departure, they signal the presence of a relationship. It recalls, among other works, Goya's portraits, which for the most part depend upon a dark background. In the _Portrait of Tio Pacquete_ (1820\u20131823) (figure 3.8) what is remarkable about the Goya portrait, and this is a rather emphatic instance of a more general approach, is the way in which the line that divides the face and the background, a line that is itself an after-effect (i.e., it is a technique at work, and as such will in the end render otiose the language of background and foreground), causes both a diminution of the effect of the figure's setting while at the same time causing the presence of the face to become more emphatic. Here there is a face. The face of a blind man gaping, not seeing, is what is seen.\n\nGeorg Baselitz, _Ein moderner Maler_ (1966). Berlinische Galerie.\n\nGeorg Baselitz, _Ein moderner Maler (Remix)_ (2007). Berlinische Galerie.\n\nGoya, _El t\u00edo Paquete_ (1820\u20131823). Museo Thyssen Bornemisza, Madrid.\n\nIn the first of these works by Baselitz there is a similarity of line. The figure stands out. The line produces it. And yet, far from standing out, the painter stumbles, trapped. The painter's hands have been entrapped. He cannot paint. He can just look out. Brushes and what may even be a mahlstick have become redundant. It may be that with trapped hands the project of art has come to an end. Instruments remain in the frame. However, as noted, they are both marked and marred by their own impossibility. While it may appear that art here is defined in relation to the hand, a supposition justified by the trapped hands, more is at work. The instruments and the trapped hands are another modality of art's self-staging. The hands recall the hands that defined art's practice. The painting announces the related necessity of the instrument and thus art as the after-effect of material practices and techniques, and yet, in this work, the instruments are unable to function. They have stalled. The stall, which should not be taken as simply a negative characterization, is introduced into this work by the sketched nature of the instruments of art's production. It is significant that brushes and the like are outlined. In comparison with the figure of the artist, their presence is no more than sketched. There is therefore an important affinity between the trapped hands and the way in which the instruments of painting have been presented. At work is a more complex modality of art's self-staging than would be there were it reduced to the trapped hands. The trapped hands are not merely ideational, since trapped hands have to cohere with art's material possibility\u2014brushes and other instruments which are present only in outline. The hands have a different mode of presentation. Present such that the skin bears the marks of their entrapment, the hands are there in contrast to the outlined presence of the brushes. The relation is productive by its being internal to the work. One exists with the other. Ideational content is there in, and as, this juxtaposition. That juxtaposition is an irreducible co-presence in which the anoriginal interrelation of the material and the ideational\u2014captured in the term 'mattering'\u2014is at work. This work's work is this modality of mattering.\n\nIn the second work, _Ein moderner Maler_ _(Remix)_ (2007)\u2014and it should be noted that it is one of many works, both paintings and drawings, with a similar title and project\u2014something else has occurred. The painter's gaze still has intensity. Here the artist crouches. The upright stance that has long defined the artist's self-presentation has vanished. While the work's title brings the 'modern' into play, the hands are free. Or at least they are no longer trapped\u2014or if there is a 'trap', it is no longer a literal one in which hands are actually trapped. It is not just the painting that has been remixed, its title, while the same, has to bear the same project. Here there is another conception of the 'modern'. Here, the artist is standing on _Baumwurzeln_. Here there is ground, soil and roots. And yet the relationship between this other ground, a reconfiguration of the 'modern', is such that it is no longer a mere trap but a sense of place. The 'remix' in this instance is the shift in colour. There is no longer a figured background. The setting has moved. The monochromatic nature of the ground, soil, and roots once linked to the artist's hands placed on it (perhaps even in it) stages a relationship between art and place as that which has to be inherently disjunctive. The work stage's art\u2014hence art's self-staging\u2014in terms of the productive force of the disjunction, is enacted through the work of colour. The work's mattering, its economy, which is a process freeing hands from their literal trap and thus trapping them within a complex intrication bounded by place is carried by colour. Colour is, of course, equally carried. Art's work remixed is handed on. Colour remains anoriginally informed and forming.\n\nBaselitz and D\u00fcrer engage hands. These engaged hands are defined in terms of a set of relationships. While the hand\/instrument relation has an anoriginal quality, it is also the case that once relationality acquires content, then what come into play are differences. If the hand\/instrument relation is a motif whose repetition, the travel or movement of these images across specific works, occurs such that the coming-into-relation works to produce singular works, then what continues to emerge is an array of differences. If the process that continues to be identified with the formulation coming-into-relation is given priority, then what has been uncovered is that the history of the hand in which the hand was thought to have only ever been in the service of that which was external to it has come undone, while the hand remains. This externality had two determinations: one which was secular and, in the context of the chapter, was linked to names such as Cicero and Michelangelo (recognizing of course that there could have been many more names cited) and one which was theological. The theological was identified with the motif of St Luke painting the Virgin. As has already been noted, this generalized motif continues from the sixth century CE. However, while it may be possible to reduce the motif to its having a unique form once its presence is examined as the motif is repeated (i.e., it takes on the quality of a specific image), then it is possible to discover the presence of fundamentally different and in the end incompatible versions. Hence, the motif fractures. As a result, the motif has to be understood as naming nothing other than a site that was always defined in terms of complexity. Hence, the initial reduction noted above would have been premised on having denied or effaced the presence of that complexity. Allowing for Marmion's and Gossaert's St Luke to come-into-relation has a twofold effect. In the first instance it reinforced the fracturing of the motif, (on the assumption that the latter is defined in terms of a unique determination) such that it becomes impossible to advance it as though it has a uniquely singular and original determination. Secondly, that fracturing, which could also be described as the motif's dissemination, was accompanied by an enactment of the impossibility of sustaining forms of external regulation in which what governs the process is a commitment to a definition of the external in terms of content that had a unique form such that its representation\u2014its presentation\u2014then has to be the immediate instantiation of that context. The analysis of both Gossaert and Marmion reinforces the claim that pure immediacy is impossible, if it has to take the form of a single and self-referring image. The consequence of this position has an inevitability attached to it: namely, that every presentation is always a form of mediation. However, this is not to be understood either as a sense of failure or the presence of the inadequate, as it would be within a certain version of Platonism. On the contrary, it needs to be seen as the consequences of what has been described as the fracturing of the motif. The latter is the result of holding to the primacy of relationality and the work of art as an activity.\n\n## Notes\n\n1.\n\nBenjamin, SW 3, 265.\n\n2.\n\nIbid., 283.\n\n3.\n\nSee in this regards Andrea Pinotti, 'Symbolic Form and Symbolic Formula: Cassirer and Warburg on Morphology', _Cassirer Studies_ 1 (2008). See also Benjamin, AP, 131.\n\n4.\n\nWalter Benjamin, 'What Is Epic Theatre?' (second version), in _Understanding Brecht_ , trans. Anna Bostock (London: Verso Books, 1998).\n\n5.\n\nIn regards to the presence of gesture in Roman rhetoric, see Jon Hall, 'Cicero and Quintilian on the Oratorical Use of Hand Gestures', _Classical Quarterly_ 54, no. 1 (May 2004). In the Italian context, a concern with gesture therefore with the hand is also evident in the nineteenth century in Andrea de Jorio, _La mimica degli antichi onvestigata nel gestire napoletano_.\n\n6.\n\nFor a detailed investigation of the differing ways of understanding what is entailed in the idea of the _manus loquens_ , see, Matthias Bickenbach, Annina Klappert and Hedwig Pompe, eds., _Manus_ _Loquens. Medium der Geste\u2014Geste der Medien_ (K\u00f6ln: DuMont Buchverlag, 2003).\n\n7.\n\nMichael Baxandall, _Painting and Experience in Fifteenth-Century Italy_ (Oxford: Oxford University Press, 1988), 45.\n\n8.\n\nFor a detailed art-historical analysis of this work, see Clifton Olds, 'Jan Gossaert's _St. Luke Painting the Virgin_ : A Renaissance Artist's Cultural Literacy', _Journal of Aesthetic Education_ 24, no. 1 (Spring 1990).\n\n9.\n\nFor an overview of the issued raised by the evocation of the Veronica as the true icon, see Hans Belting, _Das Echted Bild: Bildfragen als Glaubensfragen_ (M\u00fcnchen: C.H. Beck, 2006). For a more general history of the image of Christ that incorporate a sustained discussion of the Veronica, see Ernst von Dobsch\u00fctz, _Christusbilder: Untersuchungen zur Chrisrlichen Legende_ (Leipzig: Hinrichs, 1909).\n\n10.\n\nFor a positioning of the painting of St Luke as enacting the problems of presentation and self-presentation, see Ian Verstegen, 'Between Presence and Perspective: The Portrait-in-a-Picture in Early Modern Painting', _Zeitschrift f\u00fcr Kunstgeschichte_ 71, no. 4 (2008) : 513\u2013526.\n\n11.\n\nWalter S. Meilon draws attention to this through his study of Menzelius and Canisius: The 'image of Mary that Canisius aspires to represent, both verbally and pictorially, is the divine likeness made perfectly visible by her body and her soul in concert that together exemplify the heightened imitation of God' (228). See his ' _Quae lecta Canisius offert et spectata diu_ : The Pictorial Images in Petrus Canisius's _De Maria Virgine_ of 1577\/1583', in _Early Modern Eyes_ , ed. Walter S. Meilon and Lee Palmer Wandel (Leiden: Brill, 2010), 207\u2013266. 'Canisius following Luke 2:19 commends as her foremost attribute\u2014her ability to conserve all she has perceived and to bear the exact likeness of these things in her heart' (232). See also Kirstin Noreen, 'The Icon of Santa Maria Maggiore, Rome: An Image and Its Afterlife', _Renaissance Studies_ 19, no. 5 (November 2005): 660\u2013672.\n\n12.\n\nIn having used the term 'figure of the Jew' to identify the way in which the history of philosophy and the history of art created identities, it is the creation of figures that necessitates the question of the relationship between Jewish life and the presence of the figure be posed. Lives were lived in relation to the figure; the figure was not the presence of the life as lived. See my _Of Jews and Animals_ (Edinburgh: Edinburgh University Press, 2010).\n\n13.\n\nFor an analysis of the presence of 'horns' within Renaissance art and in particular the work of Michelangelo, see Stephen Bertman, 'The Antisemitic Origin of Michelangelo's Horned Moses', _Shofar_ 27, no. 4 (2009): 95. For a more general treatment of the figure of the Jew in the art of the period, see Dana E. Katz, _The Jew in the Art of the Italian Renaissance_ (Philadelphia: University of Pennsylvania Press, 2008). In regards to a case study of Signorelli, see Jonathan B. Riess, _The Renaissance Antichrist: Luca Signorelli's Orvieto Frescoes_ (Princeton, NJ: Princeton University Press, 1995).\n\n14.\n\nThe importance of the opening of the _Imagines_ is not just that they stage a particular understanding of art but that they maintain the particularity of art as a mode of making present. Painting is presented in terms of a form of mimesis that deploys techniques (e.g., the use of colour) to realize its end. A slightly more extensive discussion of Philostratus occurs in chapter 5.\n\n15.\n\nI have discussed the relationship between Platonic mimesis and questions of mirroring in my 'Interpreting Reflections: Painting Mirrors', in _Art, Mimesis and the Avant-Garde_ (London: Routledge, 1993), 4\u201362.\n\n16.\n\nIndeed, a stronger argument can be made concerning a form of separation between Plato's philosophical concerns and art. For example, James I. Porter argues the following: 'Art and phenomena in the sensuous appearance are simply uninteresting to Plato. What appeals to him is something that lies behind appearances, a world uninhabited by matter and even people' (91). See his _The Origins of Aesthetic Thought in Ancient Greece_ _: Matter, Sensation and Experience_ (Cambridge: Cambridge University Press, 2010).While this position warrants analysis in its own right, it underscores the centrality of what has been identified here as indifference.\n\n17.\n\nXenophon, _Symposium_ III.6, in _Xenophon_ , vol. 4, trans. O. J. Todd (Cambridge, MA: Harvard University Press, 1974).\n\n18.\n\nThe term ' _hyponia_ ' has extension beyond Xenophon. I have discussed the role of _hyponia_ in Philo in my, 'Hermeneutics and Judaic Thought', in _The Routledge Companion to Hermeneutics_ , ed. Jeff Malpas and Hans-Helmuth Gander (London: Routledge, 2014).\n\n19.\n\nSee the argument developed by Philip Sohm, _Style in the Art Theory of Early Modern Italy_ (Cambridge: Cambridge University Press, 2001). See also Robert William, _Art, Theory and Culture in Sixteenth-Century Italy_ : _From Techne to Metatechne_ (Cambridge: Cambridge University Press, 1997). Sohm's book is an indispensible guide to the relationship between questions of style and their relationship to the role of techniques. The latter continue to be articulated in relation to the 'stylus'. Part of the argument developed here is the incorporation of that specific set of concerns within the larger question of production.\n\n20.\n\nFederico Borromeo, in his _De Pictura Sacra_ , quotes Nicephorus writing of St Luke's depiction of the Virgin and Christ. Borroremo quotes Nicephorus in Latin. See Federico Borromeo, _Sacred Painting_ , ed. and trans. Kenneth S. Rothwell Jr (Cambridge, MA: Harvard University Press, 2010), 71. For a discussion of the issues behind iconoclasm and the role of Nicephorus within it, see Gilbert Dagron, _D\u00e9crire et peindre: Essai sur le portrait iconiqie_ (Paris: Gallimard, 2007), 31\u201363. Moreover, for a discussion of the actuality of the iconoclastic controversy see Marie-Jos\u00e9 Mondzain, _Image, Icone, \u00c9comomie: Les sources Byzantines de l'imaginaire contemporain_ (Paris: \u00c9ditions du Seuil, 1996). For an important work that complicates the history of the image by arguing for recognition of the material basis of images as informing the images around which the Iconoclastic controversy was staged, see Glenn Peers, _Sacred Shock: Framing Visual Experience in Byzantium_ (University Park: Pennsylvania University Press, 2004). The argument has contemporary force precisely because it underscores the centrality\u2014even at the time when images were taken to exist independently of material concerns\u2014of the relationship between externality and materiality.\n\n21.\n\nKrijnie N. Ciggaar, 'Une description de Constantinople dans le Tarragonensis 55', _Revue des \u00e9tudes byzantines_ , no. 53 (1995): 127. The full citation is, ' _Est quoque alia ecclesia que Odigitria dicitur, in qua est gloriosa Dei genitricis icona quam beatus Luchas euangelista, ut aiunt Greci, depinxit_.'\n\n22.\n\nGabriel Du Pr\u00e9au, _De Vitis, sectis, et dogmatibus omnium haereticorum, qui ab orbe condito, ad . . ._ (1581), 224. The full citation is, ' _Ac Lucas quide apostolus primus ipsas accurate, opus id exorsus, minibus suis depinxit_ ' (224). More or less the same formulation occurs in Louis de Blois, _Op\u00e9ra . . . varia eruditione et eximia pietate, . . ._ (1589): ' _Ac Lucas quidem Apostolus, primus ipsas accurrate manibus suis depinxit. Haec ibi, at secundo libro postquan hoc ipsum Nicephorus dixit, mox adiungit hec verba_ ' (732). What they and others continue to repeat are the words ' _manibus suis depinxit_ '. And it should be added that references cited here are just a few of what is a vast range of possible sources.\n\n23.\n\nCastiglione, _Il Libro del Cortegiano_ (Torino: Editions Einaudi, 1968), 1.28.\n\n24.\n\nMartin Kemp, _Leonardo da Vinci: The Marvellous Works of Nature and Man_ (Oxford: Oxford University Press, 2007). Moreover, another side of this position, one in which externality moves from a relation of indifference to one in which the question of presentation and materiality delimits the presence of externality, can be found in some of the powers attributed to Apelles by Pliny in his _Natural History_ (trans. H. Rackham [Cambridge: Harvard University Press, 1995]). According to Pliny, Apelles's portraits were so 'lifelike' that the health of those portrayed could be discerned and predictions made from the portraits themselves. However, while in such instance there was an assumed simultaneity between image and that which it presents, Pliny, writing a few lines later of the appearance of protruding fingers from a portrait of Alexander the Great in the temple of Artemis in Ephesus, again the work of Apelles, noted that while the fingers appear to protrude, ' _legenta meminerint omnia ea quattor coloribus facta_ '(Book 35, 92). In other words, what Pliny recognizes, and thus what has to be 'remembered', hence the caveat lector, is that the effect of the lifelike quality of the work is in fact the effect of the work of colour (colour taken as an effect). A further complication endures. There is a contemporary response to this reduction of effect to the work of colour; a response situated within the extended reach of the determination by externality as it pertains to artwork. This appears, for example, in the way that Sextus Propertius writes of the beauty of Hippodamia in his _Elegies_. Her beauty, he suggests, is greater than its painted presence. The latter is the presence resulting from the use of colour. Colour is deployed and used by Apelles to create beauty. Indeed, there is the suggestion, in the _Elegies_ , that such beauty is 'false' ( _falso_ ). As opposed to her natural beauty, there is the beauty, perhaps now only apparent beauty, that was created by Apelles's use of minerals to create colour.\n\n> Her beauty was not dependent on gems\n> \n> the colour of Apelles's painting\n> \n> (sed facies aderat nullis obonoxia gemmis, \n> \n> qualis Apelleis est color in tabulis.)\n> \n> 1.2.23\u201324\n\nAgain what becomes clear is that there is an inherent instability. Apelles's skill, in the first instance, equates to the use of materials deployed such that the effect they introduce amounts to a type of coextensivity between the presentation and that which is presented. At the same time, however, in the case of Hippodamia, the presentation of 'beauty' stands opposed to 'nature'. This occurs because of the way this end will have to have been achieved\u2014namely beauty presented through the use of materials. Thus it is only there as a form of imitation. What appears is an imitation, thus a staging of apparent beauty, which demands the use of materials (the 'four colours' that Pliny notes) in order that it be realized. Works by Apelles, in their estimation in both Pliny and Propertius, are as much created images as they are images that can stand apart from their having been created. This is, of course, the force of the point made by Propertius, namely that the beauty of Hippodamia, her natural beauty, is to be understood as having an externality which comes into relation. While it may be possible to claim that the work of art allows her beauty to be present and thus for there to be the work of the image in which there is both the retention of propriety\u2014where the latter is understood as a governing externality\u2014and an envisaged immediacy marking the relationship between content and its presentation, it is also the case that image may only betray propriety by denying it. Far from being the figure of security, Apelles has become the figure of instability.\n\n25.\n\nFor a further discussion, see Stanley Meltzoff, _Botticelli, Signorelli and Savonarola:_ _Theologia Poetica and Painting from Boccaccio to Poliziano_ \u202c\u202c\u202c\u202c\u202c\u202c\u202c\u202c\u202c\u202c\u202c\u202c\u202c\u202c\u202c\u202c (Firenze: Olshki, 1987).\n\n26.\n\nFor a further discussion, see David Marsh, _Lucian and the Latins: Humor and Humanism in the Early Renaissance_ (Ann Arbor: University of Michigan Press, 1999).\n\n27.\n\nSee in this regard, Christiane L. Joost-Gaugier, 'Ptolemy and Strabo and Their Conversation with Appelles and Protogenes: Cosmography and Painting in Raphael's School of Athens', _Renaissance Quarterly_ 51, no. 3 (Autumn 1998).\n\n28.\n\nFor a general discussion of Marmion's miniatures within the _Book of Hours,_ see Sandra Hindman, 'The Case of Simon Marmion: Attributions and Documents', _Zeitschrift f\u00fcr Kunstgeschichte_ 40, nos. 3\/4 (1977).\n\n29.\n\nThere are other restless infants. See, for example, Girolamo di Carpi, _St Luke Drawing the Virgin_ (1535), Art Institute of Chicago.\n\n30.\n\nCennino Cennini, _Il libro dell'arte_ , ed. F. Brunello (Vicenza: N. Pozza, 1971), 36.\n\n31.\n\nSohm, _Style in the Art Theory of Early Modern Italy_ : 'The hand was identified primarily with practice rather than theory, with technique rather than conception and expression' (157\u2013158).\n\n32.\n\nFor a detailed interpretation of this image in the context of Lissitzky's overall project, see John E. Bowlt, 'Manipulating Metaphors: El Lissitzky and the Crafted Hand', in _Situating El Lissitzky_ , eds. Nancy Perlof and Brian Reed (Los Angeles: Getty Research Institute, 2003), 129\u2013152.\n\n33.\n\nQuoted in Sohm, _Style in the Art Theory of Early Modern Italy_ , 159.\n\n34.\n\nRaffaello Borghini, _Il Riposo,_ trans. and ed. Lloyd H. Ellis Jr (Toronto: University of Toronto Press, 2007), 107, first published as _Il Riposos di Raffaello Borghini_ (Firenze: Michele Nestenus e Francesco Moucke, 1730), 106\u20137. Recalling of course the point made by Pliny, namely ' _Nulla dies sine linea_ ' ( _Natural History_ , book 35, 84). Discussions of disegno abound within art treatises in the Renaissance. No one text stages the totality of concerns. Moreover, discussions of disegno were often incorporated into detailed discussions of the distinction between and thus also the 'dispute' between painting, sculpture and architecture. However, the concerns staged by disegno have to be understood as part of a more general and generalizable concern with the relationship between presentation and a founding sense of propriety. For example, the following is argued by Vincenzo Danti in his _A Functional Conception of Proportions_ (1567):\n\n> Since we want to imitate nature in the figure of one man, and since it is impossible to find perfect beauty in a single one, . . . Art, conscious of its power to comprehend all that beauty in a single man, tries through imitation to include in the composition of one figure all the beauty which is divided among many men. Because Art knows that Nature has not brought the human product to full perfection, i.e., apt to attain its own end, through which it becomes absolutely beautiful. Art achieves that by avoiding imperfections and by pursuing perfect things.\n\nIn _Italian Art 1500\u201316. Sources and Documents_ , ed. Robert Klein and Henri Zerner (Evanston, IL: Northwestern University Press, 1966), 104.\n\n35.\n\nCicero, _Orator_ , 2.9.\n\n36.\n\nGio Pietro Bellori, _Le Vite De' Pittori Scultori et Architetti Moderni_ (Roma: Succefs. Al Malcardi, 1671), 5.\n\n37.\n\nFor another important discussion of this aspect of Baselitz's work, see Yvette Biro, 'Georg Baselitz: Filtering the Past', _PAJ: A_ _Journal of Performance and Art_ 25, no. 3 (September 2003). \nII\n\n# Engaging Philosophy's Art\n\nChapter 4\n\n# Matter and Movement's Presence\n\nNotes on Heidegger, Francesco Mosca and Bernini\n\nIf sculpture were true, what then? What would sculpture's truth be? What of sculpture's relation to the truth\u2014truth as such? Each question turns on a relationship between truth and sculpture. Each evokes a range of possible answers. At work within these questions are some of the concerns that Heidegger has already noted when he wrote, 'Sculpture: the embodiment of the truth of Being ( _die Verk\u00f6rperung der Wahrheit des Seins_ ) in its instituting places.'[1] There is therefore for Heidegger a relation between truth, a specific thinking of material presence\u2014identified here as 'the embodiment' ( _die Verk\u00f6rperung_ )\u2014and sculpture. As a beginning, the possibility of any response to these opening questions\u2014recognizing that what is at stake within them is a certain understanding of the relationship between philosophy and art, and thus an eventual engagement with Heidegger's _Art and Space_ \u2014needs to start by approaching what is at work within the questions. Once they are taken more generally, it can then be argued that, precisely because they bring to the fore the relation between art's work and the philosophical thinking that can be brought to that work, what these questions stage are the concerns central to philosophy's relation to art.[2] (The key word here is, of course, 'relation'. The attendant and inescapable question is how that relation is thought.) Beginning with these opening questions involves, in this context, a recapitulation of sculptural presence in terms of a gift. Moving away from Heidegger, if only to provide a way back, a way that will have been guided in advance by an encounter with a work by Francesco di Simone Mosca (Il Moschino) and then one by Gian Lorenzo Bernini, art's presence, in part its work, is understood here initially in terms of sculpture's gift. As will be noted, giving is a mode of spacing in which what is given already has a form of spacing. The latter is spacing as a form of projective internality. Consequently, spacing, within and as art's work, has an originally doubled quality.\n\nThe specificity of sculpture's gift is that sculpture, as the work of art, gives itself to the body and not just the eye. Accepting this as a point of departure means that sculptor's truth is inextricably bound up with the relationship between spatiality and the embodied eye. With this 'eye' there is a melding of body and eye. The resultant form of integration demands an account of feeling as that which will have already yielded to that meld. While this allows for the development of a hermeneutics of sculpture\u2014that is, an account of sculpture located at the intersection of the felt and the ideational\u2014what continues as a question is sculpture's own body. By the attribution of a body to sculpture, what is meant is sculpture's presence as work. Though it is, at the same time, an attribution, recalling Heidegger's opening move in 'Art and Space', in which sculpture is given a specific designation. For Heidegger, at the outset of 'Art and Space', 'sculptured structures are bodies' ( _Die platischen Gebilde sind K\u00f6rper_ ).[3] To the extent that emphasis moves to an account of material working as sculpture, which can be understood as a concern with sculpture's material presence as itself, a site in which what is sited is matter's work\u2014hence mattering\u2014then not only does work becomes an activity, it is also the case that activity cannot be separated from the particularity of matter.[4] The question of the relationship between mattering that is the work's presence as the work of matter, and that which is given within Heidegger's use of the term 'embodiment', emerges as one of the final points of investigation. If there is an investigative problem around which this approach to Heidegger's 'Art and Space' rotates, then it concerns the possibility, in both its positive as well as its negative determinations, that the philosophical project within that text enables an encounter with the sculptural work when the site of work\u2014and thus the site of sculpture's truth\u2014is the particularity of a work's material presence (i.e., the work's mattering). The question then concerns the possibility (or not) that the philosophical project of 'Art and Space' enables the thinking of material's work (sculpture's mattering). The process of 'enabling' therefore has a specific meaning in this context; that is, it refers to the provision of the philosophical means that would allow the specific materiality of sculpture\u2014not mere materiality but matter as mattering\u2014to be thought. Precisely because of the centrality of the term 'mattering' within this project, it is essential to note here some of its determinations and thus its continual strategic importance within the development of a philosophy of art.\n\nThe centrality of mattering concerns a specific ontological configuration of art's work and thus of the work of art. This is a configuration in which the operative nature of the work of art comes to the fore. Artworks work. However, work in this context does not equate to a work's narrative content or to its meaning. Work pertains to the presence of material(s). Meaning is an after-effect of this work. Mattering assumes therefore the distancing of at least two positions concerning the work of art. The first position to be distanced is the claim that all that is possible for artwork is given within its description such that there would then be a suggested commensurability between the work and its definite description. The description would concern itself with the relationship between presentation and content. While the work's material presence would form part of the description, there would be a continually disjunctive relation between materials and claims about content and meaning. The second position to be distanced is more complex. Schelling argues that within that conception of artwork that comprises 'all the formative arts' ( _alle bildende Kunst_ ), what characterizes them is what he describes as the 'informing ( _einbildung_ ) of the infinite into the finite and the ideal into the real', a particular staging of the relation between art and philosophy and thus a particular determination of the artwork taking place. While the position identified in the above may allow for a certain discrimination between, for example, sculptural works, it remains the case that the work of art\u2014in this instance sculpture\u2014stages that which is indifferent to the material specificity of the artwork in question.[5] Hence, for example, the limitation imposed by Schelling on the sculptural presentation of animals stems from the position announced as the opening statement of \u00a7123 of _The Philosophy of Art_ : 'Sculpture, as the immediate expression of reason, expresses its ideas particularly or even primarily by means of the human figure.'[6] This will be the case because it is 'infinite' but also because what is expressed is a sense of propriety that is external to matter and that the material\u2014matter\u2014is then constrained to present. Matters become 'ensouled'. From this it follows that, given that the propriety is located in that which is external\u2014that is, a conception of the spiritual nature of the human being as art's appropriate content\u2014then it is possible to argue that this is best expressed\u2014or only ever expressed adequately\u2014in the human countenance rather than in the animal's body.\n\nIn Hegel the position is even more emphatic. In this instance, the argument advanced in the opening of his _Lectures on Fine Art_ for the end of art has to do with the claim that Spirit has, in the present (as conceived by Hegel), a more adequate realization within and as philosophy.[7] Again, the work of art, while always a presentation or an appearance, is positioned in regards to that which comprises its material presence as indifferent to that which it presents. The latter is a sense of propriety that is both immaterial and indifferent to any specific material presence. This means that the immaterial is always to be identified with the ideal. (The move to mattering is the identification of immateriality as integral to a reconfiguration of the work as a field of activity, thus entailing a reconfiguration of the way the immaterial is to be understood.) Indeed, indifference in the sense just noted is the very precondition, within the argumentation of Hegel's _Lectures on Fine Art_ , for the move from art to philosophy. Part of that indifference entails that what is staged could have been staged by another work of art with a significantly different material presence. Schelling and Hegel name only two possibilities for a position that is characterized by a founding indifference between the ideal and material. That indifference, however, is what allows the work's material presence to present\u2014thus Schelling's formulation of sculpture as 'the immediate presentation of reason' ( _der unmittelbare Ausdruck der Vernunft_ )\u2014that which, while essential to the work, has a necessary externality in relation to any one specific work. 'Reason', the 'Infinite' and 'Spirit' are of necessity disjunctively related to the work's material presence. More importantly, any one work may be inadequate in relation to the demands made by that which is external\u2014the demands, for example, of the 'Idea' in Plato or the 'Infinite' in Schelling\u2014in terms of the viability of any one presentation. What matters is that viability or adequacy is always externally grounded and thus cannot be taken to pertain to the work of matter, where the latter is taken as the locus of art's work.\n\nGiven the abeyance or distancing of these two possibilities as being able to account for the work of art in terms of material presence, where the material is held apart from the merely empirical in the first instance and the ideal as informed by that which is indifferent to art's material presence in the second, mattering understood as the attribution of a dynamic quality and thus the attribution of a complex economy to the work of art occurs in the space that the distancing of the empirical and the ideal opens up. If there is a set of interpretive questions that now arise given this opening, then it is driven by a concern with how a given work works as art's work. In this way, there is a move from _that_ a work is art to _how_ it works as art. As the question of art's presence is recast in terms of the particularity of a given work and to the network of relations that are constitutive of the particular's identity, here, in the move from _that_ to _how_ , given that these concerns are delimited by the presence of sculpture, central to any account of the way sculpture exists as a material object concerns how it works as art. The work is the object's mattering. If there is a definition of mattering _,_ then it is the identification of art's work in terms of activities in which there is a founding inseparability of the work of matter on the one hand and meaning as its after-effect on the other. Those hands are always already joined. However, precisely because what is being staged is that which can be addressed by the question of _how_ this work works as art, the impossibility of any unanimity in regards to the answer is not to be understood as an instance of semantic relativism or interpretive subjectivism. On the contrary, apparent semantic discordance is the result of the operation of a complex economy that does not present a unified and coherent field, and this despite the presentation of forms of coherence. This is the case, as mattering will always position meaning as an after-effect. As will be argued, mattering cannot be separated from the presence of a complex economy that, in resisting both the work's identification with a synthetic unity and its reduction to a self-enclosed identity\u2014positions which are determined by a concern with meaning and not mattering\u2014gives rise to a reconfiguration of the work as a site of an original form of plurality, the always already present form of plurality as integral to the work of art. It is the presence of a plurality that obtains ab initio\u2014hence anoriginal plurality that gives the work the quality of a plural event.[8] An insistence on mattering incorporates the way a work both completes and incompletes. In sum, therefore, the work is the site of original difference. The differences\u2014work as a plural event\u2014are to be located in how the work matters.\n\nThere is an additional and fundamental point that needs to be made here. It, too, stems from the reorientation that takes place once there is a move from _that_ a work is art to _how_ it works as art. While mattering is the work of matter, essential to the process of mattering is the presence of the object as the work of art, to which what needs to be added is the notion that fundamental to art's work is the way the work in question stages itself as art. The processes of self-staging are the aspects of the work that, while forming part of the work and thus present within the complex economy that is the work, work and position the work as art. Self-staging becomes an insistent set of worked occurrences that through recall and enactments\u2014the recalling of both works and texts\u2014works to establish the work as art. This is a process that is already there within the work's material presence. What this means in part is that material can never be just the mere presence of matter. Such a position would be no more than the assertion of a type of empiricism. Both mattering and self-staging\u2014self-staging as reaching beyond the simple confines of the frame and mattering as that which allows, in part, the presentation of that reach\u2014not only comprise the work, they are essential to the work of art (that is, they form the constitutive elements in any account of the ontology of the work of art.) And yet, it is not as though these are two separate and then combined elements. Rather, self-staging can be identified as the occurrence within and as the work that allows the object to be understood as a particular work of art. This is significant, since once there is an emphasis on the work's material presence, then it is of central importance to begin to identify the presence of a work of art as opposed to a simple material object. (It is only within the latter that the empirical and the material coincide.) While a historical account of the work of art can have recourse to questions, for example, of patronage in the Renaissance and then of the complex relation between works and museums from the nineteenth century onwards, a philosophical account needs a different mode of argumentation. If there is to be a philosophy of art, then it has to be able to give an account of _how_ a particular work works as art.\n\nWhile building on a tradition that draws on the discovery of Roman sarcophagi, there are important moments in fifteenth- and sixteenth-century Italian sculpture that can be located neither within painting nor within sculpture, if the latter is understood merely in terms of the presence of free and independent form\u2014that is, sculpture as mere form. While clear examples of such work can be found slightly earlier in Donatello, among others, in this instance, emphasis will be given to a work by Francesco di Simone Mosca (Il Moschino).[9] The work in question\u2014 _Diana and Actaeon_ (figure 4.1)[10]\u2014is significant for two related reasons. The first is that it can be described as located within a space in which sculpture is both present yet emergent. The work itself, perhaps when it is at its most emphatic, captures moments in which, in the work's radical differentiation from art defined by the literal surface, the tradition of both painting and the fresco does, in that turning, become sculpture.[11] In the work to be considered, sculpture is both nascent and present. As such it is the liminal nature of the work that opens up a number of the formal concerns of sculpture. The second reason for the significance of this particular work has to do with its content. While the work's detail will be pursued, precisely because its content concerns transformation and metamorphosis, there is an interesting and important connection between the work's liminality and that which determines its presence in terms of narrative content. The work presents\u2014stages\u2014the story of Diana having been viewed by Actaeon while she is bathing. As Ovid relates the story ( _Metamorphoses_ 3.155\u2013250), it is as a consequence of Actaeon's inadvertent discovery of her bathing that he is turned into a stag and then torn to pieces by the dogs that hitherto had been his companions on the hunt.[12] They were at his command. Ovid describes them as 'servants' ( _famulos_ ). The dogs that he had both tamed and trained will kill him the moment that he becomes an animal. One animal kills another. Furthermore, the content of Mosca's work, now understood abstractly in terms of sculpture's engagement with the process of transformation, provides a link to the other sculptural work to be considered, namely Bernini's _Apollo and Daphne_ (1622\u20131625).\n\nFrancesco di Simone Mosca (Il Moschino), _Diana and Actaeon_ (1523\u20131578). Museo Nazionale del Bargello, Florence, Italy.\n\nIn Titian's painting of the 'same' subject matter as Mosca's sculptural work, the former's _Diana and Actaeon_ (1556\u20131559) (figure 4.2), the moment of discovery is also marked by the end that will greet Actaeon.[13] It is prefigured by a dual presence. His dog is at his side\u2014t\u00adhe dog that will act. (In Ovid the dog acts in concert with others.) In addition, there are the skeletal remains of a stag's head. The animal he will become, and as which he will die, is already there as having died. The frame stages a series of temporal moments: present, past and future. Knowing the story and seeing the skeletal presence of the stag's head within the frame is to note that inscription of time within the frame's operative presence. Not only are there different modalities of time being staged, the painting engages different modalities of viewing. At least three can be identified. First, there is a direct seeing that is inadvertent. This is what the viewer's advertency encounters.[14] Second, the painting presents attempts to restrict viewing. Drapes and materials are deployed to that end. Finally, there is seeing as mirroring; the reflection in and of water yields reflection as mirroring. Mirroring, which is in fact a logic of mirroring, has a doubled presence.[15] In the first instance, reflection needs to be understood as staging one version of the origin of painting.[16] The second element that the logic of mirroring brings into play is that reflection stages the identification of seeing with immediacy (to which it should be added that immediacy is either the truth of seeing or the ruse that art deploys and certain conceptions of the philosophical enjoin, in terms of the possibility of an exact reflection). Art reflecting exactly what is there to be presented (reflected) would be that which is offered as the truth of presentation\u2014a position that appears in its negative instance in Plato in terms of art's limitation on the one hand and philosophy's exceptional status on the other\u2014though which has its positive reiteration within art as forming part of Pliny's estimation of Apelles and Zeuxis, among others.[17] Moreover, the positing of immediacy, both in the reflection and in the fact that work is given to the eye, is both affirmed and undone by the inscription within the work of the different temporal registers. All these registers pertain, and yet one cannot be reduced to the other. The work as viewed within the moment, a viewing encouraged by the role of literal reflection and thus mirroring within the work, exists alongside modalities of time and viewing that cannot be incorporated into the moment. This is an instance of what has already been referred to as the work's complex economy. The elements both complete the work while remaining a site of incompletion. Once again this anoriginal irreducibility is the work's presence as a plural event.\n\nTitian, _Diana and Actaeon_ (1488\u20131576). National Gallery, London.\n\nIn Titian's painting, as has been intimated, this complex set-up is already present in terms of the water's reflecting and thus the water's painted presence forming part of the logic of mirroring that begins to define the way painting stages itself as art\u2014that is, as art's self-staging. Self-staging occurs in Titian's painting in two different ways. In this first instance, it occurs through the inscription into the work of different modes of viewing, which is an inscription that is also integral to the work's work and where those differences are defined in terms of temporal moments. The second, to which reference has already been made, is the water gathered beneath the bath. The water reflects. It is a surface that projects. Its projection is, of course, immediate. As a surface it mirrors and thus functions as the presence within the work of that conception of presentation\u2014presentation as staging\u2014in which the viewing and presentation are defined by immediacy. A final note needs to be made here. Self-staging is not reducible to the _paragone_ or, in this instance, to the privileging of painting over sculpture\u2014evidenced here in the sculptural reliefs on the ruins. More importantly, self-staging occurs in terms of its inherent interarticulation with what was identified earlier as mattering. Thus, for example, in the first instance it occurs as the result of the work of colours. An illustration would be the way the application of paint creates the distanced sky, itself a process that cannot be understood other than in relation to its having been enacted in terms of blue. A second example is the quality of the painted reflection. While the surface of the water reflects immediately, that reflection is muted in relation to the vibrancy of colour elsewhere in the work\u2014for example, the blue, red and purple of the drapes. That relationship not only attests to the power of painting over reflection, turning it into mere reflection in its opposition to painting, it also is essential that the apparent paucity of reflection appears as such. Hence, the techniques deployed in the creation of the water's surface cannot be separated from self-staging.\n\nFrancesco Mosca's _Diana and Actaeon_ lifts the characters from the surface by allowing them to become part of a surface with depth. They are there in relief.[18] There is a sculptural tableau in which the centrality of the eye that Titian demanded becomes displaced. In Titian's work it was a centrality that was reinforced by the inscription within the painting of complex modes of seeing related to the painting's own engagement with time(s). Within Titian's work the complex of times are staged for the eye. Moving from the painting to Mosca's _Diana and Actaeon_ , the eye cedes its place to the body. However, it is not just the body, it is to the moving body and thus to an eye that is always already embodied. Despite the clear danger of generalizations, it is nonetheless clear that the relationship that the work of sculpture has to a subject, the viewing subject\u2014that to which the work is presented, given and in being given creates the space of its presentation\u2014is not just to a body. Far more significant is the fact that it is a body that moves. Within a painting, while the eye may traverse the work, within sculpture for the eyes to operate in the same way\u2014that is, to traverse the work, where traversing and understanding would start to coalesce\u2014it is the body that has to move. The work demands it. Moreover, understanding the work depends upon it. Both this demand and the accompanying dependence form an integral part of a hermeneutics of sculpture. As the viewing body moves around the work, characters within it take on different qualities. Diana and her attendants can be viewed from different angles, such that what is viewed changes\u2014gives itself differently\u2014in the process of movement. In the case of this particular work, the female figure to the far right of the cave cannot be seen if the viewer stands even further to the right, the position from which Actaeon is coming. (Equally, Diana and her attendants cannot view Actaeon, who has been transformed.) She is hidden by the cave wall that extends from the background. In extending, it creates a space, thus opening a space while simultaneously extending into space. It is only when the viewer's embodied presence moves past Actaeon that Diana then comes into view (hence the doubled, or originally complex, spacing that is constitutive of the object). Seeing here\u2014the work as given to be seen and thus to see\u2014is therefore repeating the very act of seeing that Actaeon himself will go through, or will have gone through. In other words, the work stages the possibility of Actaeon's seeing; however, this occurs only on the condition that the embodied eye moves from one side of the work to the other. What this means is that the work itself can only be seen on the condition that it is seen as sculpture, and thus by having responded to the constraints on seeing\u2014the move from the eye to the embodied eye\u2014that sculpture's gift is to the eye demands. The work continues to realize itself in the movement, the continuous and discontinuous movement, of its being seen. In other words, the work itself is the setting of the embodied seeing that is the work's work. This needs to be understood as the process by which work stages itself as art. (Here is the process of self-staging.)\n\n_The Fall of Phaeton_ by Francesco di Simone Mosca (Moschino) (1523\u20131578). Bode Museum, Berlin.\n\nBpk \/ Skulpturensammlung und Museum f\u00fcr Byzantinische Kunst, Staatliche Museen zu Berlin \/ J\u00f6rg P. Anders.\n\nFurthermore, there are certain moments within the work where figures cast shadows on other figures. And it should be added that the capacity of a work to cast shadows is not an accidental quality of sculpture. Sculpture cannot be separated from the shadow.[19] The casting of shadows is as much a part of the work's own internality as it is bound up with its externality. The relief itself, its capacity to maintain both depth and projection, depends in part upon the casting of shadows. Literal shadows provide depth, though they are also fundamental to the way the cave entrance is present within the work. As the casting of shadows changes, as time and light work across the work, they continue to become, within that changing presence, part of its work. In regards to the latter, the work's capacity to cast shadows, and thus the shadows as a cast presence, is as much part of the work's relation to spaces as its material presence. The latter is the presence of sculptures as 'bodies'. What this means, in addition, is that it will become possible to question the already noted 'assertion', Heidegger's 'assertion' in 'Art and Space' that 'sculptured bodies embody something' ( _Der Plastiche K\u00f6rper verk\u00f6rpert etwas_ ). That questioning will not occur, because the 'assertion' is inaccurate in itself; rather, it will be because an allowance needs to be made for the inscribed necessity of the shadow's already embodied possibility within the work's material presence. The presence of that possibility would be an indispensible element in any account of the work's mattering. Material presence is necessarily implicated in what can be described as the ineliminability of the work's potentiality to cast shadows. In pursuing this point, it may be the case that what limits Heidegger's attempt to take up sculpture is an inability inherent in the philosophical project itself to account for shadows in terms of this original potentiality.\n\nTaken in conjunction with the figure in the cave, the other most striking single figure in the work is Actaeon himself. As he is present in the work and thus as he is seen, he can no longer be viewed viewing. He is seen after his having seen. The viewer (Actaeon) is viewed. This occurs though as a viewing not of the process but after the event (where the event is no more than an occurrence). In its incorporation of this afterwardsness, the sculptural relief is itself constituted, in part, as an event. As a beginning this is clear, as Actaeon's head is a stag's head. Even though still dressed, his clothes are yet to be torn by the process of metamorphosis. And yet, there has been a transfiguration. He has become the co-presence of animal and human. Moreover, this human\/animal is still nuzzled by the dogs that will kill him as the human\/animal becomes pure animal. Actaeon is divided. While his face cannot present his terror, the terror would be present. It is essential that within the work the dogs are not looking up. They attend him within a type of suspension. At any moment he will be other. That his head has already turned from human to animal will, in the end, invite the heads of the attending dogs to turn. That turn, however, will be his end. Taken as whole, movement within the work is therefore signalled in a number of different ways. Hands gesture; they express fright. The relief works to introduce a type of depth, as figures are more or less prominent\u2014a state of affairs that will be reinforced by the way figures within the work cast shadows in relation to their prominence. The figures are not juxtaposed. Rather, they are sets of relations that change and modify as the viewer moves from one side of the work to another. (And this is the case despite the fact that the relief deploys a form of perspective.) Nonetheless, the work harbours an inescapable tension. Actaeon while untouched\u2014or rather precisely because one of his dogs does actually touch him\u2014opens up the inescapability of the touch that will kill. Hence, touch cannot be generalized. What is seen is this tension. It is given to the embodied eye to be viewed. The continuity of its being viewed therefore necessities a moving body. What is viewed is of course the consequences of a body that, in having viewed, is itself then moved by its having been transformed.\n\n_Apollo_ _and_ _Daphne_ (1622\u20131625) is one of Bernini's most widely celebrated sculptures.[20] While at no point is it difficult to understand this attribution and thus the significance of the work within the context of a specific conception of art history, what has to be developed is what its presence stages for philosophy, which means, in this context, being attentive to its work as art. In other words, the response to any question of its presentation to philosophy can only be understood adequately in relation to its own self-staging. The question to be addressed, therefore, concerns how the work stages itself as art's work. The staging here is necessarily linked to the embodied eye. While as a work of sculpture it is given to the embodied eye, it will be that eye and thus the viewer's body that, in moving around the work, allows the work to be presented. Presentation, understood as sculpture's gift, is not simply a projection into space. Sculptures work only ever in relation to a moving body. Here, however, there is another encounter\u2014the reciprocity of Daphne and Apollo's encounter\u2014in which the presence of that movement demands movement's presence. The latter is the continual presence of the embodied eye. In broad terms, what has to be captured is what is significant about _Apollo and Daphne_ in terms of its occupation and construction of internal and external space. The work stages a division. There is a relation to the embodied eye; equally, there is a relation within the work itself. The latter is itself a gift to the eye: Apollo's relation to Daphne summed up in his hand on her belly, and Daphne's relation to Apollo captured in her head's determination for flight.[21] There has to be a division between the work's relations to space, as that which is given within a form of externality, and its relation with the work itself. The first form of argumentation is that there needs to be a link between externality and a division within the work, where the division is itself constitutive of the work's work. What matters here is the way that constitution and thus the division are at work within, and as, the work. They form an account of its particularity and thus of _how_ it works as art.\n\nOvid's _Metamorphoses_ (1.452\u2013567) is the predominant source, though it is also clear that Bernini had access to, and was prompted by, a range of other versions of the tale.[22] While the story of Daphne's metamorphosis into a laurel tree is the end point and her being the object of Apollo's love the point of origination, Ovid introduces the complication that sets the measure for the work of sculpture. Ovid wrote, ' _Protinus alter amat, fugit altera nomen amantis_ ' (Forthwith he loves, but she flees the name lover) (1.474).\n\nThe immediacy of his love gave rise to the simultaneous immediacy of her flight (a position captured in Ovid by the use of _alter\/altera_ conditioned by the line's opening word, ' _Protinus_ '). Pursuit and flight occur together. If there is love, it is simply on the side of Apollo; she 'flees' ( _fugit_ ) its very possibility. There is not a moment that can be defined other than one of pursuit and flight. What is interesting is the way in which this episode, one that is defined by movement\u2014movement prior to metamorphosis\u2014prompts the sculpture. As Daphne flees Apollo, she appeals to her father, Peneus, to intervene. Peneus's intervention is his response to her pleading. She flees from having to be an object of desire. Daphne is in flight. Fleeing, she is transformed into a tree\u2014turning while fleeing. Escaping by becoming a form of material presence that she hopes will position her outside desire, she becomes other than flesh. Skin and bark can both be touched. They yield a doubled response\u2014desire and love on Apollo's part, and dread on Daphne's. Her metamorphosis does mean that she has slipped the hold of desire. Such a response would fail to take into account that, while his touch charged his body with longing, the 'same' touch was, at the same time, that of loathing. Touch therefore has a necessarily doubled presence. Even after the transformation, Apollo continues to touch. With his hand on the 'bark' ( _ramos_ ), her felt 'breast' ( _pectus_ ) still 'agitated' ( _trepidare_ ) beneath it. He would have felt longing and desire. She still felt agitation verging on dread. Touch cannot be singular. And yet, Bernini's sculpture is white, thus having a singular undifferentiated presence on the level of colour. So it is essential to be more attentive. Differences have to be located within that coloured field\u2014difference within monochrome.\n\nThe work stages pursuit. And yet the question will always be, for whom? Apollo's left leg is in the air. Daphne's right leg, the one which had been in the air\u2014and which still is in the air\u2014is beginning to transform. As she changes, this is in part both her capture, transformation into a tree as capture, and equally, however, her freedom. She escapes from his hold, from being captured by a process that will hold her indefinitely. Apollo and Daphne are divided: a division that is announced by the work's bringing Apollo and Daphne together; though within the work, as its work, they are both conjunctively and disjunctively related. That conjunction\/disjunction is there within the work's presence. Again the question arises: For whom?\n\nViewed from the front, Daphne's face cannot be seen by Apollo. Her mouth captures her agitation. Standing in front of the work, the viewer can see his hand. His fingers touch the bark that was once her skin, while his thumb still presses into her flesh. Despite the sustained singularity of colour, or perhaps because of it, the distinction between bark and flesh comes into view. It needs to be seen as holding differences. The fingers are on the bark. The thumb presses into the flesh. Not only does the material allow the hand this differentiated presence, the material as mattering is this differentiation. His hand spans her transformation, thus announcing it. The toes of her left leg\u2014again viewed from the front\u2014have already taken root. The nails are becoming roots. She is held in place. (The use of continuous formulations is essential here.) Viewed from the right, Apollo's face is completely obscured. Daphne's head can just be seen. Her hair, which from other angles looks as though it is simply streaming back, thus staging flight, is from this angle swung between her arms. Her arms and her fingers are becoming branches. Her hair is equally indeterminate. All that can be said is that her hair is becoming part of the tree. It is as though leaves are beginning to appear from the tips. From the right-hand side, metamorphosis, as arborification, seems to have advanced significantly.\n\nThese differing views can always continue as the embodied eye continues to move around the work and the work continues to give itself to that eye. What this suggests is that the process of change cannot be viewed from a single position. Metamorphosis is not singular. Indeed, were the sculpture to be viewed from just one position, where that viewing and position were involved in the claim that the presentation is the object's complete self-presentation (as would be the case within the putative possibility that the object was co-terminus with its definite description), such a claim would amount to no more than a self-contradiction. The transformation occurring in Ovid, as well as in other sources, and that then takes place within narrative time\u2014literal temporal sequence\u2014could not take place within an instant. However this work does not take place in an instant. It works differentially. The work, once understood as a gift, continues to give itself to an embodied eye that continues to move around the work. Discontinuity as the single claim would have this complex continuity as its precondition. At every moment the work is therefore both the same and different. If the work has a form of truth, then it is not located in perspectivism. Rather, it is located in the recognition that its continuity is only ever discontinuous\u2014this is, after all, the process of transformation that is the work\u2014and depends upon the continual engagement of the embodied eye. Moreover, the work as the site of discontinuity and continuity is a fundamental part of how it is staged as art.\n\nHaving come this far, there cannot be a simple turn to Heidegger without the existence of an intervening step. The necessity for the step is straightforward. There is a question that needs to be addressed. What allows the move from an engagement with two sculptural works, one from the sixteenth century and the other from the seventeenth century, to function as that which sets the measure for Heidegger's 'Art and Space'? If there is a way of beginning to answer this question and thus to provide the step\u2014a step that links the concerns of sculpture with the particular philosophical project of that text\u2014then it can be found in Heidegger's recognition that the work of sculpture is inextricably bound up with its presence as material and thus as a body. In contradistinction to the tradition that includes both Schelling and Hegel, and within which it should not be forgotten that matter was always informed by an immaterial force that was both external to the work as well as indifferent to its particularity, Heidegger recognized the inherently material nature of sculpture. The question therefore concerns what is meant by material in this context. Consistent with the analyses of Mosca and Bernini that have just been undertaken, the question that stands at the beginning of Heidegger's text, the question that orientates the approach to it, is the following: What matters in 'Art and Space'?\n\nHeidegger's 'Art and Space' is a late work. Compressed within it are a great many of the elements that mark his much wider philosophical project. This short text starts with a concern with borders and limits. In the first instance, sculptures, understood as 'bodies' made of 'matter', bring space into play through the creation of borders and limits. The ending and beginning of matter establishes an edge that is the limiting and delimiting of space. The sense of border and line of delimitation as the locus of an opening out beyond the literal border recalls a much earlier discussion in _Building, Dwelling, Thinking_ in which what is fundamental to the positioning of the border is not that it establishes a sense of closure. More significantly, it is 'that from which something begins its essential unfolding'( _sein Wesen beginnt_ ).[23] This early positioning of the relationship between sculpture and space in 'Art and Space' is, however, reworked throughout the text via the introduction of other terms that exert a definitional hold over much of Heidegger's larger project. Terms such as 'Open', 'Clearing-away', 'Place' and 'Region' are central here. It will therefore be important to remain attentive to the location of these terms as they structure the argumentation of 'Art and Space'. And yet, while each of these has an important relation to that which is itself identified in the text's title, namely 'Art' and 'Space,' it remains the case that the project staged in these notes and which has already been worked through two named sculptures\u2014one each by Mosca and Bernini\u2014is the materiality of the work of art, specifically the materiality of sculpture in which work is the object's mattering. As a beginning, this is unproblematic precisely because, as has already been noted, Heidegger concedes from the start that sculpture is 'matter'. It is not as though there is the possibility that what will be posed as central to matter is either 'Spirit' or the 'Infinite' or the 'Idea'. What remains intrinsic to sculpture is the work of matter. Matter on its own, however, is not on its own sufficient. There is not just matter. Hence, the project here is tracing the way the work of matter is understood within 'Art and Space'.\n\nIt is not as though assuming the centrality of both art and sculpture is in some sense outside the remit of Heidegger's text. He begins by noting his concerns are 'limited to the graphic arts ( _die bildende Kunst_ ) and within these to sculpture'.[24] What has to be developed here is the movement of the text itself. While the text begins with the identification of the appropriateness, or even the viability, of the distinction between objective and subjective space, as a way of progressing his remarks on art, the initial question that can be brought to the text's project is, why would a concern with sculpture necessitate an investigation of space? Why both art and space? The answer is a complex one. With sculpture, with, that is, the presence of that which has a delimited material presence, space comes into existence as that which has the specific determination of a 'closed, breached and empty volume'.[25] Hence, once there is sculpture, then by definition what is created is space; though it should be added straightaway that it is space with the specific permutations noted above. (Absent here is what has been identified earlier as sculpture's gift. The relevance of that omission will be crucial.)\n\nSculpture therefore is both a taking place and that which brings space into consideration. Hence, there is the need to consider the nature of space, given that sculpture occurs within it. This allows sculpture to be taken up in terms of its being that which occupies space. Addressed alongside sculpture therefore is the space that the sculptural object both occupies and assumes. As a consequence, what arises here as the question to be answered is how space, within that setting, is to be understood. (It is essential to note that it is the presence of sculpture that brings space into play.) The limits of the simple opposition between objective and subjective conceptions of space are clear. Both assume the presence of space and thus, by definition, leave space almost systematically unexamined. The first is the straightforward and objective sense of space in which objectivity entails measurability. The latter position\u2014space as subjective\u2014can be identified with the position held by Kant in the _Critique of Pure Reason_. In the latter, space is nothing other than the 'subjective condition of sensibility'.[26] The distancing of this opposition\u2014objective\/subjective\u2014creates the possibility to address space. Heidegger writes: 'Even if we recognize the variety of space experiences of past epochs, would we win already an insight into the special character of space ( _das Eigent\u00fcmliche des Raumes_ )? The question, what space as space would be, is thereby not even asked, and much less answered. In what manner space is, and whether a Being in general can be attributed to it, remains undecided.'[27]\n\nWhat this means is that there is a type of questioning that is still left unaddressed even when the limits of the conventional approaches to space, and thus what has already been noted as the conventional understandings of space (that is, its objective and subjective dimensions), are left to one side. However, what Heidegger introduces here is what he identifies as 'the special character of space'. The significance of that formulation is that it will come to be contrasted to 'true space' ( _wahre Raum_ ), which has 'its authentic character' ( _sein Eigenstes_ ). In other words, what is possible is another configuration of space that occurs with the abeyance of the varying permutations of its 'special character'; or, whatever it is that is particular about space can only be addressed once the approach that takes the 'true' and the 'authentic' as the determining possibilities comes into play. Sculpture loses its hold on space and space on sculpture. The earlier easy identification of the relationship between sculpture and space would no longer apply. It amounted to no more than the location of sculpture in space, and space as that in which sculpture was located. The question then is how a determination of space\u2014its character or particularity\u2014that is guided by the 'true' and the 'authentic' is possible. Heidegger's response is to adopt the procedure he has followed on numerous occasions in his writings, namely to uncover this possibility in language itself. In _Building, Dwelling, Thinking_ , for example, he argues that 'language tells us about the essence of a thing, provided that we respect language's own essence'.[28] In this instance his concern is to uncover the possibilities already there in the word ' _Raum_ '. Hence the 'emergency path' ( _Notsweg_ ) that he takes allows him to ask what is spoken or uttered in that word. The word holds. What allows the move to this path, what prompts it to be taken, is a repositioning of the work of art, a repositioning that is commensurate with the now proffered centrality of the essential. (And it must be noted that the justification for such an approach\u2014an instance of which is the commitment to pursuing what it is that a word holds\u2014is bound up with a commitment to the essential and its enjoined and necessary sense of propriety.) No longer is sculpture to be understood merely as one of the 'graphic arts'. Rather, those arts have themselves been repositioned within Heidegger's larger philosophical project.\n\nThe question redirecting the concern with space involves a return to art such that the question of space is itself reconfigured in the process. Heidegger writes in this regard, 'Once it is granted that art is the bringing-into-the-work of truth, and truth is the unconcealment of Being, then must not genuine space, namely what uncovers its authentic character, begin to hold sway in the work of graphic art?'[29] While this endures as announcing a type of project, what conditions what is being asked within it is clear, namely the original relation between art, truth and Being. To the extent that art is defined in terms of its privileged place in regards to the 'unconcealment of Being', then, given the already existent relation between art and space, a specific question arises almost as a matter of necessity: Should not a concern with space also be subordinated to this (re)positioning of art? The possibilities allowed by this question are those which open the 'emergency path'. While the assertion concerns art, and while it may be conceded that the way this occurs may be specific to each artwork, it remains the case that particularities\u2014both the particularity of the genre (sculpture as opposed to painting, for example) though more emphatically the presence of particular works within any generic determination\u2014are all held together under the heading 'art', where 'art' is delimited by its being 'the bringing-into-the-work of truth'. This much has been granted. This act\u2014the granting\u2014can be understood as a radically different form of gift. Or at least this would be the question. And were this to be the case, then this granting, which is no more than the object's reconfiguration, opens up the possibility that it would, as a consequence, still be the object in terms of its capacity to give and thus would shut down the necessity that there would need to be that to which it is given, namely the embodied eye. It perhaps is this move, the one in which art, truth and Being are originally related, that holds matter in place while at the same time holding matter apart from the possibility of mattering. What is opened up here, staged by the possible discrepancy between granting and giving, might become nothing other than the possibility (or not) of locating the body within the argumentative necessities of 'Art and Space'.\n\nOnce this reconfiguration is allowed, then what sounds on this path, what is heard in listening to language, is that within the word 'space' ( _Raum_ ) there is a 'clearing-away' ( _R\u00e4umen_ ). In other words, within the substantive 'space', what can be heard is the process of emptying, clearing away, vacating and so on. For Heidegger, what this term 'means' ( _meint_ ) is 'to clear out ( _roden_ ), to free from wilderness. Clearing away brings forth the free, the openness for man's settling and dwelling.'[30] At work in this formulation is one of the most significant of Heidegger's terms: 'the Open'. 'Clearing-away' has to be thought in relation to openness. 'Clearing-away', to use Heidegger's formulation, is the 'release of spaces'. Hence, the process of 'clearing-away' is what gives space. In a dense but important passage, Heidegger goes on to argue that 'clearing-away' is a holding open, though this is not mere openness; rather, it is 'openness for'. There is a sense of project, one governed by its own sense of propriety, within this characterization of 'openness'. It should be remembered that what is at work here is a process that is a type of spacing, rather than the positing of space either as simply given or as taking on the guise of one of the following: an objective presence on the one hand or, on the other, a simply 'subjective condition', which, as has been indicated, is the Kantian legacy. Spacing becomes a mode of continuous presencing, a making present, in which presence involves a form of potentiality. The passage in question is the following: 'It lets openness hold sway which, among other things, grants the appearance of things present to which human dwelling sees itself consigned. On the other hand, making-room prepares for things the possibility to belong to their relevant whither and, out of this, to each other.'[31]\n\nPrior to taking up the Open, it is essential to note that this passage introduces the being of being human in terms of 'dwelling'. This is the reiteration of an earlier formulation in which 'clearing-away' is linked both to 'the openness of man's settling and dwelling' and to 'clearing-away is the release of spaces', a setting in which space does not have an arbitrary designation. Spaces are there to accommodate the 'fate of dwelling man'. (It should be recalled that the German terms here are _R\u00e4umen_ and _Einr\u00e4umen_ , within which _Raum_ \u2014that is, 'space'\u2014can always be heard.) The 'appearance of things' is not the thing as determined by forms of utility; rather, things are given, and thus at hand, for human being once that being is defined by that which is proper to human beings. Hence, human being is the being of being human. Propriety is the possibility for the 'fate of dwelling man'. The key to the rest of the text and, in the end, part of what yields the answer to the question of what matters in 'Art and Space' is provided by this evocation of 'dwelling man'. Note that dwelling man is not the human being, as though all that is at stake is an abstract conception of the lives of humans. Nor, moreover, is it the human as an embodied being. Rather, the expression 'dwelling man' has a radically different register. Here 'dwelling man' comes to be connected to the reconfiguration of art as inseparable from the 'unconcealment of Being'. In the 'Letter on Humanism', the connection is expressed in terms of the relation between the being of being human and the question of Being (where the latter is formulated in terms of the expression 'house of Being'). While the evocation of the 'house' has the appearance of a metaphor, for Heidegger it is not. There is a strong sense in which a human being dwells in that house insofar as this 'house' is the setting in which a human being comes into its own\u2014that is, takes on, and over, the project that is proper to it insofar as that dwelling is both real and actual. The house is the one in which 'the jointure of Being fatefully enjoins the essence of man to dwell in the truth of Being'.[32] 'Clearing-away' and the 'Open' are terms announcing the setting in which 'dwelling man' occurs and within which 'clearing-away' establishes and in which openness holds sway. What this means is that openness is the setting in which things are. Within it, given to that 'man', in part for him, are the things to which 'dwelling man' is comported. Heidegger deploys the language of 'gathering' (that in which the thing comes into its own) and of 'regions' (the site in which there is the 'gathering of things in their belonging together'), and each term has its own specific determination.[33] However, what is at work within those determinations is orientated around what can best be described as different modalities of propriety. Where propriety, while having a form of the already present, is, nonetheless, only ever there as a possibility. Hence, the 'Open' is the site in which, for Heidegger, history happens. The happening of history takes place. History has a place, since what is at stake here is the 'fate' of 'dwelling man'\u2014that is, the being of being human as already historical. This is the way the being of being human is historical. In addition, it is that being's only possibility.\n\n'Art and Space' works through the possibility that space is either objective or subjective and in which there would be things. Not only is there a gradual undoing of such designations for space, equally what occurs is a reconfiguration of things. Within these projects, sculpture plays a fundamental role. Sculpture is the work of art, and therefore what has to be considered is the role of art within this project. The repositioning of 'things'\u2014thus the 'thing'\u2014in terms of sculpture allows Heidegger not just to link the thing to the work of art, but more significantly it then becomes possible to make the additional move that the work of art\u2014here sculpture\u2014is the bringing-into-work-of-truth. As such, the artwork is already implicated in the project of both truth and Being. Reciprocally, what then emerges is how space is to be thought of\u2014or more emphatically, rethought of\u2014in relation to that project. Each time space and the thing are reconfigured, it always occurs in order that what there is can come to be as it is. Hence, the following remarkable formulation in which sculpture, freed from having to 'deal with space', can be thought of in relation to a different sense of place. 'Sculpture would be the embodiment of places. Places, in preserving and opening a region, hold something free gathered around them, which grants the tarrying of things under consideration and a dwelling for man in the midst of things.'[34]\n\nThis endures as a potentiality (or possibility). The argument of _Building, Dwelling, Thinking_ in this regard makes that potentiality clear. Potentiality is captured by the ever-renewed and renewable search and the necessity to 'learn to dwell.' They are potentialities, possibilities, precisely because they are not yet. Moreover, it is this 'not yet' quality that ties history, potentiality and propriety together. Their combination is what Heidegger means by 'fate'. Hence, what allows for this combination to occur is the possible realization of fate, and that realization itself can only occur within that which is held in place, or within the place provided by 'clearing-away'. This position in which potentiality comes to the fore\u2014and its holding such a position is itself indissolubly bound up with an already present sense of propriety\u2014is clear when Heidegger writes that 'the proper dwelling plight lies in this, that mortals ever search anew for the essence of dwelling, that they must ever learn to dwell'.[35]\n\nThe question posed by a passage of this nature is what is meant by the claim that mortals must 'ever learn to dwell'. The answer has at least two parts. The first is that such a project is that which is proper to 'mortals'\u2014here this particular nomenclature could be replaced easily by the term 'dwelling man' (a term that does of course replace the term ' _Dasein_ ', which was used earlier in _Being and Time_ ). The second is that the place in which such a possibility would come to pass must be there. That place is there in the 'opening of regions'. At work here are all the terms in which both that which is there for 'dwelling man', the place where it is, the 'there' for 'dwelling man' and 'dwelling man's' relation to these settings, can be re-expressed in terms of an encounter within the 'Open' and therefore with that which is the presencing of both truth and Being (hence the final summation of sculpture as 'the embodiment of the truth of Being in its work of instituting places').[36]\n\n'Embodiment' is a material presence. What is embodied is the possibility of both an encounter with, and thus the creation of, the setting in which the truth of Being holds sway. Such a setting necessitates a sense of place creation. Creating places, and it is an act of creation, thus a work, which is identified here\u2014to use Heidegger's formulation, 'instituting places'\u2014is that which is proper to the work of art. Art works in this way, thus the work of art is this work, occurring with its own sense of propriety, which in turn occurs for that which is proper to the being of being human. What is occurring is the work's taking place. That taking place is that which is proper to place. There is a type of accordance between all of these elements. Hence, this accordance is not the intentional project of a human being. Nor is it a simple redescription of the apparent giveness of space. Rather, what is at stake here is the way the work of art\u2014which in this instance is sculpture\u2014is itself implicated from the start in the creation of a setting orientated by an already present sense of propriety. Art lets that accordance be. Art for Heidegger is the being in accord with dwelling man, place, and truth.\n\nThe question that can be addressed to this accordance, and it should be clear that it is an accordance, even if it is one whose presence has to be learned, concerns the presence of art within it, or rather the conception of art that this accordance envisages. While the work of art both in 'Art and Space' as well as in _The Origin of the Work of Art_ is the bringing-into-the-work of truth, what art continues to both stage and allow is the continuation of that presentation.[37] It is thus that art can have the status of an 'origin'. To the extent that the question of art is posed and that the answer continues to turn around what art makes possible in terms of its being an 'origin', then while this admits a conception of particularity in that there are always particular\u2014for example, the 'bridge', the 'temple', the shoes in Van Gogh's painting\u2014particularity becomes absorbed by the sense of truth that, in Heidegger's sense of the term, art qua art allows. What is actually significant in Heidegger's discussion of Van Gogh's painting is that it is deployed to further an investigation of the question of the 'equipmental quality of equipment'.[38] While that investigation turns around the question of art, Van Gogh figures as the result of two moves taken to address that question. The first move is: 'We choose as example ( _Wie w\u00e4hlen_ _als Beispeil_ ) a common sort of equipment\u2014a pair of peasant shoes.' The second is: 'We shall choose ( _Wir w\u00e4hlen_ ) a well-known painting by Van Gogh.'[39] In order to address the question of equipment in the way indicated, these 'choices' for Heidegger provide a necessary path. Here is the work of art. It may also be the case that following this path leads to a discussion of art as the bringing-into-the-work of truth; however, what is far from clear is the extent to which such an approach will allow the question of _how_ the work works as art to be taken up, once that question has to take what has already been identified as self-staging and mattering into consideration. The shoes were an 'example'. Van Gogh's painting is simultaneously the painting of shoes, though equally it is the painting of an example. In terms of this second determination\u2014the painted example\u2014it is further removed from questions pertaining to the _how_ of art. The analyses of both Mosca and Bernini indicated that integral to the _how_ question was the introduction of the embodied eye. Not only did there have to be a distinction between the conception of the body demanded by painting\u2014namely one in which body and eye are commensurate\u2014but also with the body that sculpture necessitates. There is also a real sense of necessity here. Sculptures, in order that they work as sculpture, demand an embodied eye and thus a moving body. Were this not to be possible, then sculpture's self-presentation as sculpture and thus as art\u2014what has already been identified as self-staging\u2014does not merely become impossible. Far more significantly, a work's presence as sculpture would fail to be understood. There could not be a hermeneutics of sculpture.\n\nIt is possible here to return to Heidegger's text and continue to develop its weave of concerns\u2014namely the way space opens up 'clearing-away', which in turns leads to the opening of the 'region' and 'place'. Were that to be the end point, such a return would suffice. However, the project here has always been the work of art. The point of the juxtaposition of Titian and Mosca was to indicate that, while it is possible to attribute to each work a complexity such that what is at work in each case are differing modes of irreducibility, what figures as the philosophical impossibility of a _reductio ad unam_ is the work's presence as a plural event. Fundamental to that plurality is the differing ways the work presented itself. This presentation always involves details of the work's particular material presence, thought in terms of the always already interrelated presence of mattering and self-staging. In the case of sculpture, as noted above, this is the embodied eye. And yet there cannot be a simple generality here. The difference between Mosca and Bernini, in relation to the works studied, is the way sculpture's gift, while a continuous presence, is given in terms of discontinuous particulars.\n\nIn the end the answer to the opening question\u2014What matters in Heidegger's 'Art and Space'?\u2014is that nothing matters. Sculpture's final definition, in which, as has been seen, it has become 'the embodiment of the truth of Being in its work of instituting places', maintains the work while at the same time reducing that work to a type of performative.[40] Sculpture both embodies and institutes. It is subject to the process in which it is granted a position, one that positions its being encountered\u2014the latter is the relationship between 'fateful dwelling' and the sculptural work\u2014such that it has the potentiality to stage the relationship between art, truth and Being. What is not staged, however, is the work's work as self-staging where the latter is the work's presence as art, that is, its mattering: a presence that is not merely the presence of the work as art; rather, it is the presence that is there both within and as the continuous response to the question that pertains to _how_ the work works as art. The closing down of what has been identified as the _how_ question reduces art to that which must always remain mute in relation to the question of the particular work as art's work and therefore as the work's being as art.\n\n## Notes\n\n1.\n\nMartin Heidegger, 'Die Kunst und der Raum', in _Aus der Erfahrung de Denkens_ , vol. 13 of _Gesamtausgabe_ (Frankfurt: Vittrorio Klostermann, 1983), 203\u201310. Translated by Charles H. Seibert as 'Art and Space' in _Man and World_ 6, no. 1 (1973); henceforth reference will be to 'Art and Space' plus page number, English preceding the German. While a direct encounter with Heidegger's text will be the concern of the final part of these notes, nonetheless the opening, both in terms of the interpretation of Francesco Mosca and Bernini as well as in the terminology deployed, needs to be understood as rehearsing that encounter. While there is a considerable secondary literature on Heidegger and art, there is little on sculpture. The one real exception is Andrew J. Mitchell, _Heidegger among the Sculptors_ (Stanford, CA: Stanford University Press, 2010). Mitchell's work is an engagement with Heidegger's actual interpretations of the work of specific sculptors. The importance of Mitchell's book needs to be mentioned. However, the project here is different. At stake in these notes is the possibility that works of sculpture\u2014in how they work as works as art\u2014may provide the basis for critical engagement with Heidegger's writing on sculpture, namely his 'Art and Space'. Other central works on Heidegger and space are Didier Frank, _Heidegger et le probl\u00e8me de l'\u00e9space_ (Paris: Les \u00c9ditions de Minuit, 1986); Jeff Malpas, _Heidegger's Topology: Being, Place, World_ (Cambridge, MA: MIT Press, 2006); and Edward S. Casey, _The Fate of Place_ (Berkeley: University of California, 1997). In regards to the latter, see especially 243\u2013284. While all of these texts offer invaluable formulations of the general questions of both space and place, once again the project here concerns the particularity of the artwork and its presence as a philosophical possibility within Heidegger's writings on sculpture as a generality.\n\n2.\n\nThe problem of the way in which philosophy and art are related is too vast a concern to be addressed here. However, what remains the case is that the concepts of the intelligible and the sensible are maintained\u2014albeit in radically different forms\u2014as providing the conceptual scheme within which that relation is understood. Here the project is different. The point of departure is a reconfiguration of the work of art by the attribution to it of a dynamic quality. The attempt here is to show that while Heidegger distances his approach to the work of art from one defined in terms of the opposition sensible\/intelligible, the limit of his project concerns what is identified here as mattering. Emphasizing that aspect\u2014the work of art as art's work\u2014begins to identify a different philosophical project, one that can be located as much in Gottfried Wilhelm von Leibniz as in the writings of Denis Diderot and Johann Gottfried von Herder, among others. The project is the identification of an immaterial force within art's work. For an investigation of how this is at work in Diderot and Herder, see chapter 8, 'Endless Touching: Herder and Sculpture'.\n\n3.\n\nHeidegger, 'Art and Space', 3\/204.\n\n4.\n\nI have used the term 'mattering' and a set of related concepts (for instance, the _incomplete_ ) in a number of different contexts. It is a term whose evolution forms part of the development of a materialist philosophy of art. See my 'The Matter of a Materialist Philosophy of Art: Bataille's Manet', in _Style and Time: Essays on the Politics of Appearance_ (Evanston, IL: Northwestern University Press, 2006), 124\u2013138; as well as chapter 6 of this volume, 'Colouring Philosophy: Appel, Lyotard and Art's Work'.\n\n5.\n\nF. W. J. Schelling, _The Philosophy of Art_ , trans. and ed. Douglas W. Scott (Minneapolis: University of Minnesota Press, 1989), 201. Originally published as 'Philosophie der Kunst' in his _Ausgew\u00e4lte Schriften_ , vol. 2 (Frankfurt: Surhkamp, 1985), 457. Reference will be to the English pagination followed by the German.\n\n6.\n\nSchelling, _The Philosophy of Art_ , 183\/430.\n\n7.\n\nWhile Hegel argues at the opening of the _Lectures on Fine Art_ that art 'remains a thing of the past', that does not mean that there is no longer a sense of propriety that governs the artwork. Art still has a 'task'; moreover, that 'task' involves a series of regulations and determinations that govern presentation. This is clear when Hegel argues: 'Art has the task of presenting the Idea to immediate perception in a sensuous shape and not in the form of thinking and pure spirituality as such, and, since this presenting has its value and dignity in the correspondence and unity of both sides, i.e., the Idea and its outward shape, it follows that the loftiness and excellence of art in attaining a reality adequate to its concept will depend upon the degree of inwardness and unity in which Idea and shape appear fused as one.' As found in G. W. F. Hegel, _Aesthetics_ : _Lectures on Fine Art_ , vol. 1, trans. T. M. Knox (Oxford: Oxford University Press, 1988), 72; published in German as _Vorlesungen \u00fcber die \u00c4sthetik_ , 1 (Frankfurt: Suhrkamp, 1989), 103. The significant point, argued in the body of the text, is that the idea and the shape, though they come to be 'fused', are in fact indifferent. This is the limit of art.\n\n8.\n\nThe term 'plural event' identifies a site of founding\u2014thus an original\u2014complexity that is ontological in nature rather than simply semantic. It attests to what could be described as a work's founding irreducibility. An initial formulation of these ideas can be found in my _The Plural Event_ (London: Routledge, 1993).\n\n9.\n\nWhile a similar argument could be made in relation to a range of works, works for which there is no definitive point of departure, of interest are those in which the work is beginning to separate itself from the surface (thereby turning the surface into a literal surface). One of the subtlest in this regard is Donatello's _Pazzi Madonna_ (1420). It is, of course, at the point at which the Madonna's right arm is on the child's shoulder, and his arm reaches up, that creates the point at which the shadow occurs and the work begins to emerge as sculpture.\n\n10.\n\nThere are other important instances of Mosca's work that deploy the same style and that have their source in Ovid: see, for example, his _The Fall of Phaeton_ (Berlin: Bode Museum) (figure 4.3).\n\n11.\n\nThere is an interesting similarity here to the unfinished Michelangelo sculptures _Slaves_ (1530) in the Galleria dell'Accademia in Florence, insofar as what can be seen is the emergence of form. Lines that are necessarily lines of negotiation make the movement into sculptural form. The moments of emergence are the most important. While it cannot be pursued, what is central is the role of the liminal in allowing sculpture an emphatic presence.\n\n12.\n\nAll references to Ovid are to _Metamorphoses_ , books 1\u20138, trans. Frank Justus Miller, rev. G. P. Goold (Cambridge, MA: Harvard University Press, 1994).\n\n13.\n\nFor a discussion of this work by Titian in the context of his own engagement with questions of tragedy, see Thomas Puttfarken, _Titian and Tragic Painting_ (New Haven: Yale University Press, 2005), 174\u201381.\n\n14.\n\nOvid makes it clear that Actaeon could not have been aware that the place where Diana bathed was anything other than another part of the forest. The grotto had been constructed by 'nature', hence 'simulaverat artem ingenio natura suo'. Ovid, _Metamorphoses_ , 3.158\u201359.\n\n15.\n\nThe formulation of a logic of mirroring pertains to painting, not to sculpture. See my 'The Image of Art', _Research in Phenomenology_ 41, no. 2 (2011). Moreover, I have discussed the relationship between Platonic mimesis and questions of mirroring in my 'Interpreting Reflections: Painting Mirrors', in _Art, Mimesis and the Avant-Garde_ (London: Routledge, 1993), 4\u201362.\n\n16.\n\nIt should also be noted that reflection is for Alberti one of the two origins of painting (hence the centrality of the figure of Narcissus following his evocation). Alberto writes: ' _Quid est enim aliud pingere qua arte superficiem illam fontis ampllecti?_ ' (Leon Battista Alberti, _La Peinture_ , II.26 [Texte Latin, version Italienne, traduction Fran\u00e7aise]. \u00c9dition de Thomas Golsenne et Bertrand Pr\u00e9vost [Paris: Editions de Seuil, 2004], 100). (Hence, what is significant here is the way the presence of reflection recalls, _inter alia_ , that description. Not only does reflection involve the inscription of seeing and presenting, understood in terms of their actual presence, what is also brought into play are the accompanying textual contemporary references.) See in this regard Hubert Damisch, 'The Inventor of Painting', trans. Kent Minturn and Eric Trudel, _Oxford Art Journal_ 33, no. 3 (2010). In addition, David Summers, _Vision, Reflection and Desire in Western Painting_ (Chapel Hill: University of North Carolina Press, 2007), 149; and Paul Hills, _The Renaissance Image Unveiled: From Madonna to Venus_ (Edinburgh: National Gallery of Scotland, 2010), 42.\n\n17.\n\nPliny, _Natural History_ , book 35, trans. H. Rackham (Cambridge, MA: Harvard University Press, 1995), 65.\n\n18.\n\nFor a detailed study of the art-historical dimension of relief sculptures, see the papers collected in Donald Cooper and Marika Lenio, ed., _Depth of Field: Relief Sculpture in Renaissance Italy_ (Oxford: Peter Lang, 2007); and Penelope Curtis, ed., _Depth of Field: The Place of Relief in the Time of Donatello_ (Leeds: Henry Moore Institute, 2006).\n\n19.\n\nThere have been a number of important art-historical accounts of shadows. However, for the most part they have been concerned with the relationship between shadows and theories of perception. See, for example, Michael Baxandall, _Shadows and Enlightenment_ (New Haven, CT: Yale University Press, 1995); and Victor I. Stoichita, _Short History of the Shadow_ (London: Reaktion Books, 1997). There is also an important discussion, especially within Renaissance art treatises, concerning the relationship between shadows and perspective. To this end, see Thomas Da Costa Kaufmann, 'The Perspective of Shadows: The History of the Theory of Shadow', _Journal of the Warburg and Courtauld Institutes_ 38 (1975). The argument here is different. The casting of shadows, be it an external occurrence or an interval one, is integral to what could be described as sculpture's self-definition and thus to sculpture's self-staging.\n\n20.\n\nFor detailed images of the sculpture, see Kristina Hermann Fiore, ed. _Apollo e Dafne del Bernini nell Galleria Borghese_ (Milan: Silvana Editorale, 1997). The photography of Arnaldo de Luca provides an indispensible research tool.\n\n21.\n\nFor a discussion of the role of Apollo's hand in terms of the ability of Bernini to appear to transform material, see Genevieve Warwick, 'Bernini's Apollo and Daphne at the Villa Borghese', _Art History_ 27, no. 3 (June 2004); see especially 371\u2013372.\n\n22.\n\nTo this end, see Andrea Bolland, 'Desiderio and Diletto: Vision, Touch and the Poetics of Bernini's _Apollo and Daphne_ ', _Art Bulletin_ 82, no. 2 (June 2000). I am indebted to Bolland's article for a great deal of the background material informing these notes on Bernini.\n\n23.\n\nMartin Heidegger, 'Building, Dwelling, Thinking', in, _Martin Heidegger: Basic Writings_ , trans. and ed. David Farrell Krell (San Francisco: Harper Collins, 1993), 356; originally published as 'Bauen, Whnen, Denken', in _Vortr\u00e4ge und Aufs\u00e4tze_ (Stuttgart: Neske, 1954), 149\u2014hereafter cited as 'Building, Dwelling, Thinking', followed first by English, then by German page numbers.\n\n24.\n\nHeidegger, 'Art and Space', 3\/205.\n\n25.\n\nIbid.\n\n26.\n\nImmanuel Kant, _Critique of Pure Reason_ , trans. Norman Kemp Smith (London: Macmillan, 1983), A26\/B42.\n\n27.\n\nHeidegger, 'Art and Space', 4\/206.\n\n28.\n\nHeidegger, 'Building, Dwelling, Thinking', 348\/156.\n\n29.\n\nHeidegger, 'Art and Space', 5\/207.\n\n30.\n\nIbid.\n\n31.\n\nIbid., 6\/208.\n\n32.\n\nMartin Heidegger, 'Letter on Humanism' in _Martin Heidegger: Basic Writings_ , 259\u2013260; originally published as 'Brief \u00fcber den Humanismus', in _Gesamtausgabe_ , vol. 9 (Frankfurt: Vittorio Klostermann, 1976), 358.\n\n33.\n\nHeidegger, 'Art and Space', 6\/208.\n\n34.\n\nIbid., 7\/209.\n\n35.\n\nHeidegger, 'Building, Dwelling, Thinking', 363\/156.\n\n36.\n\nHeidegger, 'Art and Space', 8\/210.\n\n37.\n\nMartin Heidegger, 'The Origin of the Work of Art', in _Martin Heidegger: Basic Writings_ , 199; originally published as 'Der Ursprung des Kunstwerkes', in _Holzwege_ , Band 5 of _Gesamtausgabe_ (Frankfurt: Vittorio Klostermann, 1977), 65.\n\n38.\n\nHeidegger, 'Origin of the Work of Art', 158\/18.\n\n39.\n\nIbid.\n\n40.\n\nHeidegger, 'Art and Space', 8\/210.\n\nChapter 5\n\n# Art's Work\n\nDerrida and the Work of Art\n\nTo write on art is already to have written. Writing about painting may already have equated art and the written. In such instances, what will have occurred is a practice whose history defines, from one perspective, the relationship between art and writing. While there is no one point of origination, it is clear that what Philostratus in his _Imagines_ stages is not only a position that continues to haunt any engagement with art but also a position that should have troubled, if only by its absence, the general project of philosophy's engagement with art.[1] The position continues to be named as _ekphrasis_.[2] The question of description and thus the location of description within writing can never be far from both the history of art and a philosophy of art. The contention of this chapter is that Derrida's writings about the ' _dessins_ ' (drawings) of Artaud and the ' _tableaux_ ' (pictures) of Atlan remain trapped within the problems posed by ekphrasis.[3] However, if there were a way around the problem stemming from the retention of art's subordination to both writing and rhetoric, then it is not by abandoning deconstruction or by refashioning ekphrasis. Instead, one must pose again the question of what a deconstruction of art would actually entail. Part of the answer lies in the need to reconfigure the 'object'. It is of course precisely this reconfiguration that will undo by distancing the hold of ekphrasis and what will be called the 'ekphrastic impulse'.[4]\n\nA great deal of preliminary argumentation needs to be adduced. In addition to any attempt to identify both the place of ekphrasis and its legacy, what also needs to be established, as part of that opening move, is the limit of ekphrasis. This is a limit that occurs both within the terms that ekphrasis sets for itself as well as in the nature of its legacy. The limits can be located in how both particularity and singularity within art are to be understood. Underpinning that thinking, indeed the possibility of posing the question of the particular, demands understanding the restrictions imposed by the failure to have recognized that the work of art is also art as work. Work needs to be allowed both an actative as well as a substantive dimension (the latter, were it to be the exclusive focus, would be the named presence, the presence of the work bearing a title, hence an entitled presence). To affirm the actative is to affirm the work's work. Emerging as central is therefore a rethinking of the object. What would have been the work of art becomes, in the context of this project, art's work: the work as a named presence cedes its place to work as being proper to art. The tradition within which writing on art assumes the position of ekphrasis\u2014and this need not be an intentional decision\u2014fails to note that the work's work has significance. Indeed, it is the sine qua non for meaning, and evidence of that thus failure continues to identify the work with its substantive presence. The presence of work defined in terms of activity does not equate with the introduction of the actative as well as the substantive. There must be a more radical claim\u2014namely that the being of art is work as work. As a result, substance becomes activity. The substantive was therefore always the actative. The consequence is that the work of art becomes art's work. Only within the latter is it possible to allow art's materiality a 'workful' presence. And to the extent that this occurs, what then has to take place, a taking place defining a specific stand within the philosophical, is a philosophical account of matter's work, an account in which matter's work becomes, as a result, mattering. Art matters. What is mattering is art\u2014and it should be remembered that it is art as work. In the case of Antonin Artaud, mattering pertains to drawings and portraits, while for Henri Atlan, as a beginning, its locus is the complex relation between abstraction and figure within painting.\n\nThe way into this set of interconnected problems and questions is to concentrate on particularity. It will be in terms of particularity that the problems created by the forced separation of the substantive and the actative can be taken up, and, consequently, Derrida's writings on Artaud and Atlan can then become the locus of exploration.[5] Particularity here refers both to the work of art and thus equally to art's work. Addressing the particular work\u2014even though from one perspective writing on works of art is only ever to write on particular works\u2014and thus at the same time engaging with the philosophical problem of particularity can only be resolved by repositioning the nature of particularity itself, which in turn has to refuse any forced separation of the actative and the substantive. It will be this repositioning that establishes both the limit of ekphrasis as well as Derrida's writings on art. In addition, it will allow another writing in which force\u2014a term central to Derrida's own engagement with Artaud\u2014will itself have acquired an importantly different presence than the one that Derrida gives to it. Force will have other uses. As will be suggested, the problem that Derrida is keen to note and which appears in the separation of the intelligible and the sensible was the initial separation of force and form.[6] Contrary to this positioning, the argument here is that forming is already work. As will become clear, form is forming, and, consequently, any attempt to think of the work of art in terms of either the opposition of the sensible and the intelligible or their subsequent identification becomes impossible. To be precise, once it can be argued that works of art work as art, then the question that needs to be taken to art is, how does any one work of art work as art? Part of the answer is that works\u2014as integral to their work\u2014both recall and stage their presence as art (the latter is art's self-staging). As will be suggested, particularity is positioned in relation to both recalling and staging. In other words, the position of the substantive\u2014substance\u2014as activity has to be addressed on the level of the particular. However, if this particular is not to be the particular of empiricism\u2014the latter is the particular that exists within and for description\u2014then the question to be addressed concerns how the particularity of the particular is to be understood. There needs to be a further opening in relation to which it will have become possible to return to this question.\n\nProcedurally, the project here is to begin with the problem of ekphrasis and thus with what is presupposed within it. As has already been suggested, the limit of ekphrasis opens up Derrida's engagement with his work on Artaud and Atlan. The conclusions to be drawn will come to concern what another philosophy of art will be like, once force, which will be repositioned as this specific project continues in terms of mattering, is taken as central to that which allows the particularity of the particular work to be of primary importance. (This is a position that not only demands an understanding of the particularity of the particular but one that allows a confluence between work, mattering, and the actative. They mark the active terminological presence of another thinking of art.)\n\nWhile it may appear that Derrida's work has been distanced in the process, the engagement with both ekphrasis and particularity needs to be understood as the attempt generally to stage the way his project can be understood, and then specifically as the attempt to engage the way what is named as the 'signature' occurs in the context of his writings on art.[7] The problem of the signature brings with it the limit of meaning. He states this position with exacting clarity: 'There will be a signature every time that an event occurs, every time there is the production of a work, whose occurrence is not limited to what can be semantically analysed. There is its significance: a work, which is more than what it signifies, that is there, that remains there'.[8] What does it mean for an event to occur? What is the taking place of an event? As part of the continuity of his engagement with the event, Derrida positions the event in relation to an occurrence or a happening. The event is 'what comes, what happens ( _arrive_ )'[9] And yet given this formulation, it might then be the case that everything could be subsequently recast as an event. There is a further point to be made, one that locates that which yields the event as the event. The happening of the event has a form of interruption. In this regard, Derrida writes that 'there can be an event only when it's not expected, when one can no longer wait for it, when the coming of what happens interrupts the waiting'.[10] The event is therefore not simply an occurrence. Accepting the link to a certain experiential quality, it can be suggested that the event is neither awaited nor anticipated. There is therefore always more than an occurrence in the precise sense that this 'more' breaks the hold of the expected, the awaited and the anticipated.[11] It is what is, to recall the formulation noted above: 'not limited to what can be semantically analysed'. But what is this? The event and the signature\u2014the latter marking the former\u2014bring into play that which is always 'more than what it signifies'. Again there is both the limitation and the delimitation of meaning. Again a question returns. What is this 'more'? While it cannot be addressed at this point, it should be recognized that an essential part of any answer to this question is the need to begin to allow for that possibility\u2014this 'more'\u2014to inform the particularity and thus the identification of work\u2014here art's work\u2014with the particular's work. And it should be noted in advance that fundamental to the purchase of the term 'particular' is that, to the extent that it allows for forms of generality, it also has to refuse them. By definition, particularity always particularizes. Particularization, however, is not individuation. Particulars emerge with and thus within relations. Relationality is the already present presence of the work's own acts of recall. This identification of the limits of the semantic demands that the question of this 'more' is of primary importance. The 'more' emerges at the limit of the 'ekphrastic impulse', delimiting that impulse. The question that arises here concerns the extent to which Derrida, even though he has identified this 'more', actually allows it to be thought of philosophically.\n\nPhilostratus's _Imagines_ (third century CE) will provide a way into the question of ekphrasis. As has been intimated, we must understand ekphrasis not only as the equation of the work of art with a form of description that depends upon the literary and the rhetorical but also that the equation of art and description entails the privilege of meaning\u2014with the result that work and meaning coincide. They coincide in and for a subject. The ekphrastic impulse privileges looking. The effective presence of the 'gaze' is thus central.[12] What is important, however, is that the privileging of sight and subsequently immediacy's unfolding will always have occurred at the expense of the work's work. At the outset of _Imagines_ , having identified the origin of art as the technique of 'mimesis', Philostratus goes on to distinguish between 'painting' (\u03b3\u03c1\u03b1\u03c6\u03b9\u03b1) and 'the plastic arts' (\u03c4\u03bf \u03c0\u03bb\u03b1\u03c3\u03c4\u03b9\u03ba\u03b7\u03bd) (294K12).[13] The significance of the disjunction between painting and the plastic arts lies in the power attributed to painting. It achieves its effects 'from colour' (\u03b5\u03ba \u03c7\u03c1\u03c9\u03bc\u03b1\u03c4\u03c9\u03bd). Moreover, it is colour that sustains the presentation of varying states of mind of the characters or personages within the subject matter of painting. In addition, it is colour that allows for the recognition of those emotional states. Within that form of recognition, work has become the subject of the gaze. After having opened the text in this way, Philostratus then goes on to present 'discourses' (\u03c4\u03c9\u03bd \u03bb\u03bf\u03b3\u03c9\u03bd) about what he has seen (295K15). The ensuing text that comprises _Imagines_ is for the most part made up of reports\u2014'discourses' (\u03c4\u03c9\u03bd \u03bb\u03bf\u03b3\u03c9\u03bd)\u2014of what he saw in Naples. However, while the descriptions refer to what is seen, the language used in their presentations deploys the terminology and phraseology of literature. A clear example is the description of the death of Hippolytus in book 2.4. To a significant extent, that description depends on the reiterated presence of the language of Euripides's own play. This reiteration occurs without the complexity, let alone the presence, of the play as theatre to register. After all, in Euripides's _Hippolytus_ the accident leading to the death of Hippolytus actually occurs offstage. What occurs offstage is itself staged by Philostratus. In addition, what is not present is the fact that what is being described in this instance, deploying in part the language of the play, is most probably the image of the emergence of the bull from the sea prior to any ensuing effect its presence had on the horses pulling Hippolytus's chariot, as it occurs on a vase painting. It would be possible to detail the way, in that actual description, the work is evoked through it having been named and described such that what remains unnamed and thus not described is the work's work. Even when painting is named\u2014as in the case of the painting of Eros in book 1.29\u2014its having been named does not involve a reference back to the painting. Rather, what occurs is an unfolding of the work in terms of its having presented a narrative.[14] While ekphrasis may be thought to privilege the gaze, it does so at the expense of a relationship between looking and the work of art. The gaze in this context will have been determined by the temporality of narrative. The gaze is taken to have responded to what has been defined by and as the surface as the place of meaning. Within this purview, meaning is an effect of presentation, and not an after-effect of the work of materials; as such, meaning cannot be attentive to art's self-staging. Art's self-staging includes art's capacity to recall, as part of its work, its presence as art. Meaning's contingency would now be no more than a form of semantic relativism rather than the result of mattering.\n\nHowever, what remains significant in the opening paragraphs of _Imagines_ is that the distinction between painting and the plastic arts has an inherent fragility. All that holds them apart is the superior use of materials in one instance rather than another. And here materials have a literal presence. The comparison would be, for example, between painting and jewellery. Leaving to one side the viability of such a distinction, what is important about the distinction is that at the beginning Philostratus is not concerned with meaning nor with mere storytelling. And even though the text will become an instance of what Jas Elsner calls 'the pure collection of descriptions of art', at the beginning that project concedes that meaning is the after-effect of the way materials work.[15] Distinguishing between materials is delimited by how effective they are, where effectivity is defined in terms of the complexity of the presentation. Moreover, what is recognized in a painting, even though the results of the recognition are presented in terms of emotions, and the like, is the effective use of materials. What that means of course is that in the opening paragraphs of _Imagines_ what is staged is the problem at the centre of the ekphrastic impulse. Materials are both necessary and unnecessary. In general terms, it will be this position that defines Derrida's encounter with both Artaud and Atlan.\n\nThe movement in the opening of _Imagines_ is from \u03b3\u03c1\u03b1\u03c6\u03b9\u03b1 and \u03c0\u03bb\u03b1\u03c3\u03c4\u03b9\u03ba\u03b7 on the one hand and then to \u03bb\u03bf\u03b3\u03bf\u03c2 on the other. In other words, what is occurring is not just a repositioning in terms of the discursive. Here the passage to the discursive is predicated upon a twofold movement. In the first instance the passage involves noting the presence of art's materiality, which, were it to be taken as the locus of the investigation, would then appear as art's work. And if materiality would be taken as the locus of the investigation, then art's material presence would be the site of art's work. Hence, that possibility is obviated by the second aspect of this twofold movement\u2014namely that when the presence of the materials is noted, the literal presence becomes a locus of commentary, and, as a result, materiality as the producer of meaning is left to one side. Meaning prevails. In other words, what will always have been retained within this set-up is the position in which the material will only ever evidence the work of meaning. Again, materials will not function as the site of art's work. The further consequence is that a concern that took painting and the plastic as the site of material activity would be undone by a conception of meaning that remains philosophically indifferent to the effective presence of materials. And yet, the presence of the material and the work of materiality cannot be disavowed. This presence of the material and its work is what was noted above as the problem at the centre of the ekphrastic impulse. Equally, however, the problem of the ekphrastic impulse is not resolved and nor is the disavowal of materials to be overcome. We shall not be able to recover the materiality of the work's work merely from the refusal of logos in the name of the material. In other words, while ekphrasis as a presence demands the elimination of matter in the move to logos, it is also the case that the viability of any description depends upon the effective presence of materials. The accuracy of this formulation is clear from the presence of colour, since colour is not an end in itself. With art's work, there will always be colour's meld with the anoriginally ideational. Colour is inevitably more than one.[16] Despite the presence of colour and the fact it is necessarily required, neither that presence nor the ensuing necessity figure as demanding a philosophical account. Significantly, this point indicates that the recognition of materials is not, eo ipso, a recognition either of mattering or of art's work. This lack of recognition leads in a number of important directions. The point to be made here, however, is that the disjunctive relation between materials and logos\u2014despite the connection noted above\u2014means that a discursive (or ideational) content cannot be given to materials. Thereby, what is maintained is the separation of meaning and materials. Overcoming that separation and thus also the opposition between materials and meaning by attributing to the former an ideational content\u2014or at least by not separating the ideational from the material\u2014such that meaning's production becomes integral to the work of materials, which we have already termed 'mattering'. Once that point is argued, then one has to provide an account of how there is both the work of materials\u2014mattering\u2014and the impossibility of meaning invariance. Answering that question necessitates recourse to particularity.[17]\n\nIf there is a question here that organizes this engagement with particularity, it is the following: Wherein lies the particularity of the particular artwork\u2014the drawing, the painting and so on? This 'wherein' is the question of particularity precisely because what is being asked concerns both a particular drawing or painting as well as drawing _as_ drawing and painting _as_ painting. In a sense, however, this question is too broad to be addressed here insofar as its development necessitates a reconfiguring of particularity itself. Nonetheless, the question delimits the particular as a site and thus as the particular's presence. The general formulation\u2014one whose generality can only in this context be stated\u2014is that a particular is only ever an after-effect of the network of relations that at any one time constitute it. Its singularity is therefore the recognition of its position within relationality. Iterability is not the iterability of the object; that would be to reduce the object to the 'one'\u2014being as 'one'\u2014that is able to be reiterated. On the contrary, iterability is the potentiality that constantly reconstitutes the network of relations, the after-effect of which is the particular. Thus, iterability does not pertain to the object. The position is quite different. Iterability pertains to the network constituting the object. This position is a philosophical claim concerning the particular and thus the generality of particularity. In addition, it underscores both what particularity is as work within historicality, and as the historicality of any one work of art. Art's work is located within particularity. The history of a work addresses and pertains to singularity within relationality. (Indeed, it might even be thought to be constitutive of singularity itself.) The history of art, thought philosophically, therefore takes relationality as the locus of the particular. As such, the philosophical position staged here holds being-in-relation as primary. It is not the history of the particular as though the latter were a given end itself to be described. Taken as such, namely taken as an end to be described, would necessitate the disavowal of relationality (a disavowal that of course would have left its own traits). The contrary is the position to be affirmed. The history of the particular is the inscription of the particular within a set-up that holds particularity in place within the potential continuity of its being reworked.\n\nWhat then of a given particular? Prior to addressing this question, it remains the case that what has to be taken up is what is addressed within the question of a given particular. To what does the question of particularity\u2014particularity as question\u2014refer? Of a given particular, what has to be noted are three defining elements. First, that any particular stages its presence as what it is (e.g., as drawing, as painting, as pictogram etc.), or it stages a set of overlapping determinations whose locus of identification is the work. Second, that it is\u2014in terms of those determinations that what the work recalls, and its recall is integral to its work as a work\u2014that relation in terms of a form of reiteration. For example, a portrait by Artaud cannot be separated either from the history of portraiture or from art's relation to questions of its own self-presentation and the presentation of self. Its being a particular portrait is that which figures from the start in any account of its presence as an after-effect. The way it is as particular within that relation becomes a claim about the work's singularity. What is constituted is a singularity within relationality As such, there cannot be a unique event. There can only ever be particulars. Indeed, it is the ineliminability of relationality that establishes singularity. And then, third, while any one work will have a relation to forms of presentation that are not delimited by art's work\u2014a delimitation held in place by the ineliminability of relations to that which will have already been given in the work's self-presentation as art\u2014the work cannot be reduced to that set of relations. To do so would be premised on forms of disavowal, some of which have already been noted. The most significant in this specific context would be the disavowal of the work's presence as art's work if it could be reduced to its position within an array of documents. Integral to the latter are these moments of recall in which relations, and they will always be relations of indetermination to what can be described as the history of art, are maintained. Works work beyond any given context precisely because the relations that constitute any one singularity are the site of continual transformation, adaptation and renewal.\n\nArtaud continued to figure in Derrida's writings.[18] His text on Artaud's 'Dessins et portraits' begins thus: 'I would call that a stage, the stage of the subjectile, if a force was not already there stabilizing what is always being staged: visibility, the element of representation, the presence of a subject, indeed, of an object.' ( _J'appellerais cela une sc\u00e8ne, la sc\u00e8ne du subjectile, si une force d\u00e9j\u00e0 n'\u00e9tait l\u00e0 pour stabiliser ce qui toujours met en sc\u00e8ne: la visibilit\u00e9, l'\u00e9l\u00e9ment de la repr\u00e9sentation, la presence d'un sujet, voire d'un objet._ )[19]\n\nThese are the text's opening words. The use of the conditional\u2014'J'appellerais'\u2014introduces a specific topos. In sum, it is a possibility that might have been. The entire argument of Derrida's text, 'Forcener le subjectile', is summed up in this opening line. There would be 'une sc\u00e8ne' were it not for the fact that force is always stabilized. Dorothea Olkowski's succinct formulation captures this idea perfectly. She is writing of Derrida's approach to all the media in which Artaud worked: 'The drawing, the painting, the theatre the poetry all succumb to the stability of form in order for there to be any relation to them at all, but in so doing they become objects.'[20] The limit of the project is clear from the start. For Stephen Barker, it is that 'Derrida's Artaud must work within this limit, and within what amounts to a dialectical framework of force\/counterforce and binary resistances'.[21] Within one mode of argumentation, those limits allow for their own transformation and thus for the project of winning back force. As such, there would need to have been another force that, by opening itself up beyond the work, would have generated another site of meaning. The suggestion here, however, is that this production of meaning would still occur at the expense of the work and the identification of art with art's work. It may be that what has to be undone is the very opposition between force and stability\u2014an undoing in which, as has been noted at an earlier stage, what was thought to have been stable is itself always already the site of force. In other words, form is, ab initio, informed or unformed. Hence, what is necessary is another conception of the 'object'\/object. Derrida has already noted the object's presence. He names this yet-to-be-determined object at the beginning of the opening sentence, even though by the sentence's end it is determined as ' _un objet_ '. He names it thus: ' _J'appellerais cela_ . . .'. What this ' _cela_ ' names is an important question in its own right. In the French edition of _Antonin Artaud: Dessins et portraits_ , the passage cited above occurs on page 55. On page 54, a page that will be seen when page 55 is read, there is a reproduction; a ' _Page de Cahier_ ', the detail is provided: 'Sainte-Maxime, septembre 1946'. The page from the Cahier, on one level of description, contains words and images drawn with either crayon or pencil. Furthermore, part of one of the elements of the drawing has been rubbed away. Given the way writing and drawing are present on this page, it would appear that the writing preceded the drawing. The drawn elements hinder and restrict the written. That much can be seen. That is, they are seen prior to any attempt to read. Both pages are seen together. Is the ' _Page de Cahier_ ' the 'cela'? Or is this 'cela' not the particular, but Artaud's drawing in general. The problematic presence of the particular and the general are therefore posed from the very beginning. The problems posed lend themselves to a direct summation in the following question: About what is Derrida writing?\n\nThis 'about what' is not a trivial question. It is the question that pertains to philosophy's encounter with art. As Derrida's opening page continues, he cites from a letter written on 23 September 1932 to Rolland de Ren\u00e9ville in which the 'subjectile'\u2014the word 'subjectile'\u2014reappears. Artaud wrote: 'Here included is a bad drawing in which what one calls the subjectile betrays me.' ( _Ci-inclus un mauvais dessin o\u00f9 ce que l'on appelle le subjectile m'a trahi._ )[22] Derrida asks in relation to this line if indeed a subjectile can 'betray'. Moreover, not only is there the problematic presence of a 'bad drawing' (and this assessment is of course Artaud's own estimation), that designation is located in a formulation whose subject position lacks specificity. Thus, the claim itself is far from straightforward. To repeat the formulation, Artaud wrote, 'What one calls the subjectile?' (Ce que l'on appelle le subjectile?) Who is stating this? And about what? It is precisely a formulation of that nature that allows for a continual questioning and thus retention of the 'subjectile'. This is Derrida's approach. And yet, what ensues within a questioning of this sort is the problem of the extent to which the 'cela' now refers to the letter and its content, or if the point around which the subjectile figures is its figuring within art's work. Is the 'cela' now no more than the sign for the continual deferral of art's work in the name of writing on art that passes by that work without engaging with it? In other words, is this 'cela' implicated, perhaps inextricably, in that passage where passing by becomes, in fact, a doing without and thus a movement in which the actative would have been refused\u2014passed over\u2014in the name of the substantive? And thus, what continues to be questioned and comes to be the focus of investigation and the locus of deconstruction is this specific act of substantivization and not the drawing. To the extent that this formulation is viable, then the 'cela' would have become a substance entailing the stilling of the work's activity as drawing. It will be essential to return to this point since, if this estimation were true, what it opens up is the place of another and different project which would be what would have occurred if deconstruction were to turn to the drawing as art's work.\n\nDerrida's text on Artaud's 'Dessins et portraits', namely his 'Forcener le subjectile', begins with the 'subjectile', a term that names neither an object nor a subject, neither the surface nor that which is on the surface. It is a base that debases in allowing for work. It is of strategic importance to make two comments here; in part they restate what has already emerged. The first comment is that Derrida, as has been suggested, will locate at the centre of Artaud's project a sense of impossibility that is equally present in the limits that he discovered in Artaud's engagement with theatre. In 'La parle souffl\u00e9e', Derrida wrote: 'One entire side of his discourse destroys a tradition which lives within difference, alienation, and negativity without seeing their origin and necessity. To reawaken this tradition, Artaud, in sum, recalls it to its own motifs: self-presence, unity, self-identity, the proper, etc.'[23]\n\nThere was an inevitable recuperation. The 'destruction' of tradition cannot escape being reminded continually of what defined the tradition that his project wished to destroy. A possibility encounters its own impossibility. A similar state of affairs pertains here. In this context, however, the relationship is between force and form. As such, what that means in the context of Artaud is 'the subjectility of the subjectile calls forth . . . two contradictory projects' ( _la subjectilit\u00e9 du subjectile appelle . . . deux projets contradictoires_ ).[24]\n\nThe second comment that has to be made is that at no time\u2014though the details are yet to emerge\u2014will this analysis approach the drawings as art's work. The drawings and portraits are interpreted within the framework provided by the letters and the writings. The writings are connected by a linkage of terms to Heidegger. Even in the moment in which the subjectile is connected to a history that locates the operative presence of this term in writings on art that stem at least from the Renaissance, a concern with art's work remains an almost unimaginable project. On one level, this criticism is unfair, as Derrida is clearly linking the emergence of the work, its birth, in which bodily presence and a mode of writing cohere in a form of incoherence that gives rise to a specific work.[25] The subjectile is bound up with subjectivity. For both Gilles Deleuze and Julia Kristeva, in their engagement with Artaud, both the body and the self are co-present concerns, perhaps their only concerns. Art's work\u2014its workful nature, the mattering of art\u2014is not.[26] Even if all three can be understood as staging a preoccupation with the singularity of Artaud\u2014and even by extension Artaud's work\u2014the singular keeps pertaining to Artaud. For Derrida, there is an additional element. His analysis is not content to stage the destruction of the work of pure force as though either could be taken as ends in themselves. Derrida's analysis returns it to the place of deconstruction.[27] Artaud, the name 'Artaud', names a site in which the possibility of the affirmation of destruction meets the necessity of its own impossibility. Even if within the moment at which possibility and impossibility are brought together and drawing becomes 'this phenomenon, what you see here' ( _ce ph\u00e9nom\u00e8ne, ce que vous voyez ici_ ),[28] it is still the case that the drawing does no more than allow for this impossible possibility. As such, it becomes the scene of deconstruction\u2014of a certain deconstruction, one in which appearance is thought in terms of a 'you' ( _vous_ ) who sees. While this may yield the particularity of Artaud, and indeed that is what the majority of writers on Artaud seem concerned to establish, what of the work? That latter question notes that something remains. What is seen cannot rid itself of its presence as a drawing. Drawing becomes it. What of the particularity of art's work? How is its particularity to be thought? Hence the question asked earlier: About what is Derrida writing?\n\nAs part of an attempt to establish the site of impossibility as occurring within and as part of Artaud's 'thought', Derrida locates Artaud's thinking of 'being' in terms of the ' _jet\u00e9_ ' understood as a modality of movement\u2014for Derrida within Artaud. 'Being is announced on the basis of the thrown, not the reverse.' ( _L'etre s'announce \u00e0 partir de la jet\u00e9e, non l'inverse._ )[29] After which he continues: 'From force prior to the form. And I will try to demonstrate that this is Antonin Artaud's very thought prior to every thematic of the throw; it is at work in the body of his writings, his paintings, and his drawings.' ( _De la force avant la forme. Et j'essayerai de d\u00e9montrer que c'est la pens\u00e9e meme d'Antonin Artaud avant toute th\u00e9matique du jet, elle est \u00e0 l'oeuvre dans le corps de ses \u00e9crits, de sa peinture, de ses dessins_.)[30]\n\nAgain it will be this positioning that will have to meet its own impossibility. Rather than identifying the inscribed presence of a process of forming that will become the object, one in which forming and being would be the same, the subjectile is without narrative, and yet it demands the presence of narrative. It is as though with the movement that separates force and form there is the echo of the final line of Blanchot's _La folie du jour_ : 'A story? No, no stories, never again'.[31] ( _Un r\u00e9cit? Non, pas de r\u00e9cit, plus jamais_.[32]) And this, of course, despite the fact that the work itself is marked by its impossible possibility of recounting the story that is able to end. 'A story? No, no stories, never again.' Again impossibility and possibility construct a presence that resists that the conception of presence that conditions. And yet, even wanting to hold to such a positioning is to attribute to this set-up a form and thus presence.\n\nThe argumentation is able to proliferate. What carries it is the identification of that which is 'at work' ( _\u00e0 l'oeuvre_ ) within all media and thus can never be medium specific. However, this statement is neither a claim about medium specificity nor a demand for medium specificity. Rather, the statement resides in the question of whether or not materiality\u2014and material presence is named in terms such as 'writings', 'drawings' and 'paintings'\u2014could ever be the site of work and understandable in its own terms. Were it to be then, materials would be the locus of force, and, as such, there could never be a relation between force and form that created an impossible presence. Intrinsic to the project that maintains subjectile as the point of orientation is the initial refusal to allow representation or mimesis a determining place within its work, even though in the end it will be determined in ways that accord with both representation and mimesis. The subjectile is extrinsic and yet recuperable. The counter to this position is first to accept the distancing of both representation and mimesis, and in so doing to accept the space of allowing that accompanies it. However, by extricating works from that necessity, what has to be conceded at the same time is that works have elements of an inscribed presence that brings legibility into consideration. Form is already significant. As a result, the legibility of work in this instance pertains to the legibility of art's work as opposed to narrative's legibility (to which it should be added that it is not as though legibility has to be adduced). Once it can be assumed that there is an already present relation between work and legibility, then not only does that construct the particular as a site insofar as the legibility of a 'drawing' will differ radically from the legibility of a 'poem' or a 'painting', it entails in addition that the legibility of the particular also evidences the way that a given particular recalls its presence\u2014and it is a continually constituting presence\u2014within a network of relations. Legibility evidences the way any one particular recalls. Legibility is a quality that is continually present. The particular, as a consequence, becomes a singularity. Any particular\u2014and particularity can only ever be a site the insistent of the particular within relationality\u2014therefore underscores the already informed presence of form and as such allows for the position in which form is already forming.\n\nWhile as a conclusion it is still provisional, the argument has to be that what is excessive in relation to the processes of signification\u2014identified earlier in terms of that which is 'more' than signification\u2014is intrinsic to signification. The impossibility of meaning's invariance is the consequence of a material presence which is\u2014is what it is\u2014in the process of constitution and reconstitution.\n\nAtlan is a painter whose work has exerted a hold on a number of contemporary French philosophers. Along with Derrida, both Emmanuel Levinas and Jean-Luc Nancy have devoted important texts to his work.[33] As a beginning, it has to be noted that Derrida's own writing in this area brings the question of colour and the nature of colour to the fore\u2014a positioning that occurs almost immediately in the untranslatability of the title of his text on Atlan: 'De la couleur \u00e0 la lettre' ('From Colour to the Letter' or 'Literally on Colour'). Material presence\u2014colour as a form of presence\u2014is announced from the start. The movement that is the passage 'from colour to the letter' in the first rendering and 'literally on colour' in the second passage stages the complex presence that colour and letters will have in the text. Provisionally at least, the title can be understood as introducing a number of significant variations into what might be understood as the relationship between art's material presence\u2014in this instance, colour\u2014and the inescapability of writing and hence literally the presence of letters. While there is clearly more than one mode of writing in Derrida\u2014and more than one understanding of 'writing'\u2014what is important here is that the conception that 'letters' stage marks the orientating presence of ekphrasis.\n\nDerrida's text is extraordinary. It begins with 'the dreamer' held by a work, held by its 'trait de couleur' (it 'trait or characteristic of colour'). The dreamer enters a world described by Derrida as 'loved' by Atlan, the world of the Incas. Within it, Derrida continues:\n\n> The support of the message, let us say, the subjectile, consisting in a sheath of knotted cords, varied in their colour, in their tapestry, in their braided forms. The knots of colour begin, literally, to figure letters. Let us add to this the rhythm, the dance, the extreme tension, the force that carries the form away, that forewarns it or survives it. This is Atlan's signature. \n> \n>   \n> \n> \n> ( _Le support du message, disons le subjetile, consistant en faisceaux des cordelettes \u00e0 noeuds, vari\u00e9s dans leur coleur, dans leur tressage, dans leur formes torsad\u00e9es. Les noeuds de coleur se mettent \u00e0 figure, \u00e0 la letter, des letters. Ajoutez-y le rhythme, la danse, la tension extreme, la force qui emporte la forme, la pr\u00e9vient ou la survit, et c'est la signature d'Atlan._ )[34]\n\nThe signature of Atlan, which both includes and is more than Atlan's own signature, from within this perspective (i.e., the perspective of the dreamer) is the figures of coloured letters charged with 'force'. (That the dream is being written up, perhaps even written for the dreamer, should be noted even though we are not able to pursue this idea here.) The presence of force is defined in a relation of separation from form (hence, the claim that it is force 'that carries the form away'). Moreover, the 'signature' in this extended context can be understood as marking the singularity of Atlan's work and thus its presence as an event. Though, as should now be clear from our earlier discussion of the way in which the relation between force and form occurred in Artaud, it is not so much that 'the force carries the form'; it is more accurate to suggest, and this despite the effect it has on the position's own formulation, that form is already forced. As a consequence, at work here is what can be described as the already present enforcing of form; force informs form from the start.\n\nThe work in relation to which these initial points in Derrida's text are made is Atlan's painting _Pentateuque_ (1958). Moreover, the dreamer knows that the painting is to be explained in relation to its title: 'He knows that the painter is explained on the basis of the great, the decisive, the inexhaustible question of the title.' ( _Il sait que le peintre s'est expliqu\u00e9 sur la grande, la decisive, l'in\u00e9puisable question du titre._ )[35] Hence, this level of engagement is presented in connection to a range of possible references that follow from this name. As the analysis unfolds, what is of interest for 'the dreamer' is the place of the work within a set of concerns defined by the interdiction of images, the problem of idolatry and the silence of God. These themes are there in Atlan's work to the extent that they are carried by the titles of paintings. It should be noted, of course, that this reference to dreaming and to the dreamer could be seen as having a recurrent role in Derrida's philosophical project. In his work on Levinas, for example, he describes empiricism as 'the dream of a purely heterological thought at its source'.[36] The dream both yields a state of affairs but in so doing demands the position of the non-dreamer. The impossible possibility that defines the locus of deconstruction emerges at that point. In other words, the dream stands in need of a type of countering.\n\nAs Derrida's text moves from italics to roman\u2014a movement that does of course indicate that Derrida's own text is aware that form is already informed such that the work's mattering is the site of meaning\u2014the one who does not dream appears. He was already there: ' _moi qui ne ne reve pas_ '.[37] The 'self who does not dream' was there writing about, if not for, the dreamer. Within the text's second half, the complexity of the analysis begins to unfold; a transformation of approach occurs. The way in is provided by the sublime. Derrida finds in Atlan's paintings, both in terms of elements of content as well as in the size\u2014their 'grandeur'\u2014sufficient evocations of the sublime. The sublime is then immediately refracted through Kant's _Critique of the Power of Judgment_ in order to allow that the sublime to structure Derrida's analysis. Before the works following this lead\u2014before both their size and the sprays of colour ( _jets de couleur_ )\u2014the non-dreamer is forced in two directions. An awakening occurs at this precise moment, a doubling, and yet an injunction that ends and begins: 'Be quiet and speak' ( _tais-toi et parle_ ).[38] This injunction is to be explained in the following terms: First, there is the evocation of that which has disappeared as well as the forms of silence that are bound up with the name 'Atlan'. As such, the importance of the determinate works and instructive titles begins to wane. With the works in question\u2014works which are still named rather than being held by colour and then a title\u2014the name 'Atlan' comes to play a predominating role. The name 'Atlan' is linked to 'Atlas' and then to ' _Atlantide_ ' (Atlantis). It is now possible that the works work without titles, as there is a form of disappearance. Or, at least, the titles work in ways that are not determined by the works. The title cedes its place to that which is present without a title. It is the ' _sans titres_ ' (literally, 'without any title', or more idiomatically, 'those without proper documents') that will open another space. However it is a space in which to approach the work and in which the work can approach. What has been constructed is an inevitable space, one with borders and frontiers. The move from titles to the _sans titres_ will allow the ethical to figure, a figuring in which an alignment comes to hold between art and the other. While within the argumentation of Derrida's text such a move is inevitable, the question haunting the move is the extent to which it does justice to art's work. In other words, while the _sans titres_ evokes, as will be noted further on, a generalized site of the ethical, there would need to be a further argument that does justice to the work of art\u2014to art's actual work\u2014within the 'sans titres'. In other words, could the figuring of the ethical be no more than a version of the ekphrastic impulse? In this case, the impulse would be present in terms of the inscription of the subject's relation to a work which is already there, for example, in the necessity of the subject in any evocation of the sublime. Would there be therefore the necessity of matter to occasion that setting while simultaneously there would be a lack of necessity precisely because what matters is not the work's mattering but the position of both subject and object within a subject\/object relation? To the extent that the latter pertains, the substantive effaces the actative.\n\nWhat Derrida is tracing with the disappearance of the name are the problems that will be encountered once the authorizing power of the name\u2014be it title or author\u2014is no longer there. The problem of description and description's authorization occurs at that precise point. Within this setting, in part stemming from an injunction linked both to an impossibility of description and a respectful silence, is the position that\n\n> there is a picture of painting, and which makes this event work, only where the title, the legend, the discourse (explicative, descriptive, constative, prescriptive, even performative) is called forth to be quiet, to turn the words away, to pass over the titles in silence\u2014or nearly. \n> \n>   \n> \n> \n> ( _il n'y a tableau de peinture, et qui fasse oeuvre de cet \u00e9v\u00e9nement, que l\u00e0 o\u00f9 le titre, la l\u00e9gendre, le discours [explicative, descriptive, constatif, prescriptif, performatif meme] est appel\u00e9 \u00e0 se taire, \u00e0 tourner les mots, \u00e0 passer ses titres sous silence\u2014ou presque._ )[39]\n\nThis injunction brings the effect of the sublime into play. The sublime effect forms one part of the Atlan effect. And yet there is the other impetus: 'These canvases make me speak. And even tell stories.' ( _Ces toiles me font parler. Et meme raconteur des histoires_.)[40] This double quality is the full effect of Atlan's work. It is the response to his paintings. However, Derrida will use this doubled injunction and the impossible\/possibility that it establishes to go on and make a larger claim about 'all painting'.\n\n> All painting, all painting as such, and even if in appearance it carries and supports, as its 'subject' a title, that, is, a name . . . all painting, worthy of this name . . . , has the vocation of doing without a name, I mean title. Its essence and its space would then be exposed, the very spacing of its spatiality, and literally its colour. \n> \n>   \n> \n> \n> ( _Toute peinture, tout peinture en tant que telle, et meme si en appearance elle port et supporte, comme son 'sujet' un titre, c'est-\u00e0-dire un nom . . . toute peinture digne de ce nom . . . , a vocation \u00e0 se passer de nom, je veut dire titre. L\u00e0 s'exposerait son essence et son espace, l'espacement meme de sa spatialit\u00e9 et litteral\u00e9ment sa coleur._ )[41]\n\nTherefore, it would be that 'all paintings' would need to be approached as if they existed without titles. Hence he makes the provocative claim that, as a consequence, 'every title sketched the gesture of self-erasure' ( _tous les titres esquissent le geste de s'effacer_ ).[42] What is there without either name or title, there and named by Derrida, is ' _la Chose_ '. 'The Thing' is a form of almost unconditioned anonymity. It is that which appears. It is what he describes as 'the arrival of an event' ( _l'arrivance d'un \u00e9v\u00e9nement_ ).[43] As will be seen, it is this precise formulation that will allow him to see that the demand made by the complex relation between the title and the 'without title' ( _sans titre_ ) is repeated in arguments made in relation to the position of the hostage and the host, set within the context of an engagement with hospitality.\n\nGiven the twofold demands already noted (i.e., 'Be quiet and speak'), Derrida draws a conclusion in terms of a 'hypothesis':\n\n> What happens then, how are we to describe it? How are we to give an account that goes beyond the theme? My hypothesis is that what does without a title, what then passes between the 'without title' . . . of the Thing, between the Thing without title and us, is an earthquake to which each painting alludes, as if it was used to describe, to nickname at times some such dimension or other, some such meaning of the so to speak convulsion. \n> \n>   \n> \n> \n> ( _Ce qui advient alors, comment le d\u00e9crire? Comment render compte au-del\u00e0 de theme? Mon hypoth\u00e8se, c'est que ce qui se passe de titre, ce qui se passe alors entre le 'sans titre' . . . de la Chose, entre la Chose sans titre et vous, c'est un tremblement de terre auquel chaque tableau fait aussi allusion, comme s'il s'employait \u00e0 decrire, parfois \u00e0 sur-nommer telle ou telle dimension, telle signification de ladite convulsion._ )[44]\n\nThis 'hypothesis' is played out under three interrelated headings. The first concerns the question of hospitality. The paintings welcome 'you'. They do so 'without reserve'. In the second place within this act of welcoming, there is more than mere or sheer openness. There is a condition that is both openness as welcoming and an invasion that is conquering. The 'you' is both 'host and hostage' ( _l'hote et l'otage_ ). Finally, the work is a form of singularity. The painting seeks 'you' out. It provides nothing, and yet it demands that 'you' speak. Although each of these headings stand in need of sustained expansion and clarification, it is not difficult to note that within them the singularity of the event and the place of this singularity within that set-up is at issue. Hospitality is always given in relation to a 'you'; however, what is given\u2014and the anonymity of 'the Thing' as the event is decisive here\u2014could never be the always already informed presence of form. What matters is the relation to the 'you'. The work's matter is a question to which the posited centrality of the 'Thing' and the 'you' remains indifferent. Further evidence for this position can be found in the actual language of the 'hypothesis'. Derrida argues for a point of connection 'between the Thing without title and us'.[45] Retaining this description, even as a problematic presence and locating within it a structured existence that inscribes the 'us' as an essential element, should be understood not just as the mark of the ekphrastic. More significantly, it repeats the excision of the actative in the name of the substantive. If we can ascend back to an earlier position in the argument, we can argue that what is at stake here is a general claim about art. The general claim about art resides in the already noted way that the 'without a title' becomes a concern with 'all painting as such'.[46] While any one work as an event takes on the quality of what Derrida calls the 'wholly other', such that the work both invites and welcomes while simultaneously distancing, this 'wholly other' can never be a claim about any one work such that justice is done to the work's particularity.[47] It must be true of all works of art. It is a claim whose generality may have an important provenance within a philosophical engagement with hospitality in which the unconditional may play an important role. What is far from clear is that the particularity of artwork, art as work, and hospitality are constituted in the same way. Art's work involves what can be called particularity's recall.\n\nThe continuity of the ekphrastic impulse allows Derrida to site details (e.g., colour and line), thereby conceding the presence of matter and a possible opening to mattering. And yet the other element of that impulse means that this inscription of the material is undone by the presence of both the anonymity of the event and the necessary presence of the 'you'. As a result, mattering becomes matter, and the substantive supplants the actative. If a singular work can appear only in this way\u2014appearing as an 'event'\u2014then what is silenced is what the work already 'knows'. There is an already given presence which is there both in terms of legibility and thus in terms of recall. However, it is essential to add that this 'presence' is indeterminate precisely because it is subject to constant reworking. The constant reworking is particularity's recall. Acts of recall\u2014and a work can be caused to recall and thus to come into relation where such setting would not have been anticipated\u2014constitute the particularity of the work as that which is there, always already there, within relationality. It is the presence of indetermination and the impossibility of causing a work to return to itself as a self-completing and self-completed entity that should be the point of departure for a deconstructive philosophy of art. Such a position, rather than premised on the effacing of the actative\u2014an effacing that leaves a mark (hence the role of matter within the ekphrastic impulse)\u2014would have to take the actative as the point of departure.\n\n## Notes\n\n1.\n\nAll future references to Philostratus are to Philostratus, _Imagines_ , trans. A. Fairbanks (Cambridge, MA: Harvard University Press, 1979).\n\n2.\n\nThere is an important movement within contemporary art history that is orientated around the centrality of ekphrasis. The leading figure in that movement is Jas Elsner. See, among many texts, his 'Art History as Ekphrasis', _Art History_ 33, no. 1 (2010). Part of that which is at work\u2014albeit sotto voce\u2014in this paper is an engagement with that development. For an important attempt to rework ekphrasis for a contemporary philosophical project in relation to the arts, see Lydia Goehr, 'How to Do More with Words: Two Views of Musical Ekphrasis', _British Journal of Aesthetics_ 50, no. 4 (2010). While defining his project within the framework of art history, David Carrier draws a distinction between ekphrasis and interpretation. However, even if ekphrasis is understood as that which 'tells the story represented' while interpretation, for Carrier, 'gives a more systematic analysis of composition', they can easily become the same if what is not brought into consideration is the general proposition that meaning is the after-effect of the way materials work. See 'Ekphrasis and Interpretation: Two Modes of Art History Writing', _British Journal of Aesthetics_ 27, no. 1 (Winter 1987).\n\n3.\n\nEven though the exact term 'ekphrasis' is not used, Joana Mas\u00f3, Gabriela Garcia Hubard, Javier Bassas and Santiago Borja, in '\"Echo-graphic Images\": Writing or Piercing the Visible', ( _Mosaic_ 40, no. 2 [June 2007]), also argue in relation to Derrida's writings on art that 'rhetoric and literary tropes . . . are the philosopher's medium when he writes about painting, drawing and photography.' (215).\n\n4.\n\nIt should not be thought that recourse to ancient sources to advance arguments about contemporary philosophical or art-historical concerns is unproblematic. There have been a number of recent investigations into the presence or absence of art history and in addition the easy translation of terms from one period to the next. For a good overview of some of the issues, see Verity Platt, 'Art History in the Temple', _Arethusa_ 43 (2010).\n\n5.\n\nWhile there has been a considerable engagement with Derrida's writings on Artaud's theatre, there has been little engagement with his writings on Artaud's 'dessins', let alone on his writings on Atlan. For a survey of Derrida's writings in this area, see Rajeev S. Patke, 'The Obstinate Gaze: Derrida Looking at Pictures', in _Humanities Diliman_ 8, no. 1 (January\u2013June 2011). For a good overview of Derrida's writings in the period in which these texts were composed\u2014even though it is not directly concerned with Derrida on art\u2014see Herman Rapaport, _Later Derrida: Reading the Recent Work_ (London: Routledge, 2003), in particular 97\u2013137.\n\n6.\n\nIn his text 'Economimesis', in _Mimesis des articulations_ (Paris: Aubier-Flammarion, 1975), 57\u201393, Derrida is also concerned with locating 'something' ( _quelque chose_ ) that resists incorporation into oppositions such that 'one cannot name within a logocentric system' ( _on ne peut le nommer dans un system logocentrique_ ) (93).\n\n7.\n\nDerrida's initial and foundational work on the signature is his 'Signature, Event Context', in _Margins of Philosophy_ , trans. Alan Bass (Chicago: University of Chicago Press, 1982), 307\u2013330.\n\n8.\n\nJacques Derrida, 'The Spatial Arts: An Interview with Jacques Derrida (conducted by Peter Brunette and David Wills)', in _Deconstruction and the Visual Arts_ , eds, Peter Brunette and David Wills (Cambridge: Cambridge University Press, 1993), 17. Some of Derrida's most important comments on 'art' and the 'image' are to be found in Jacques Derrida, 'Trace et archive, image et art', Institut National de L'Audiovisuel. These comments can be found online at www.ina.fr.\n\n9.\n\nW. J. T. Mitchell and A. I. Davidson, eds., _The Late Derrida_ (Chicago: University of Chicago Press: 2007), 225.\n\n10.\n\nIbid.\n\n11.\n\nEven though it indicates the presence of a larger project, it should be noted here that Derrida's conception of the event needs to be understood as a distancing of the position advanced by Kant in the _Critique of Pure Reason_ concerning what he describes there as the 'Anticipations of Perception'. See I. Kant, _Critique of Pure Reason,_ trans. Norman Kemp Smith (London: Macmillan, 1983), A 166\u2013176\/B 208\u2013218. Part of the position developed here is that between 'anticipation' and the 'unconditioned'\u2014terms that stage an important point of connection between Derrida and Kant\u2014there is a relation of indetermination.\n\n12.\n\nFor a version of this specific argument see Jas Elsner's 'Ekphrasis and the Gaze', in his _Roman Eyes: Visuality and Subjectivity in Art and Text_ (Princeton, NJ: Princeton University Press, 2007), 67\u2013112.\n\n13.\n\nFor an excellent overview of the project of the opening paragraph of the _Imagines_ , see Karel Thein, 'Gods and Painters: Philostratus the Elder, Stoic Phantasia and the Strategy of Describing', _Ramus_ 31, nos. 1\u20132 (2002). For a significant examination of Philostratus as the inventor of a conception of the image as autonomous in relation to the empirical, see Sandrine Alexandre, 'Sous le tableaux . . . l'image. Philostrate et l'invention des images \u00e0 l'occasion de la peintre cantique', in _Inventer la peinture greque antique_ , ed. Sandrine Alexandre, Nora Philippe and Charlotte Ribeyro (Lyon: ENS \u00c9ditions, 2011), 45\u201370.\n\n14.\n\nSee in this regard Vasiliki Kostopoulou, 'Philostratus' _Imagines_ 2.18: Words and Images', in _Greek, Roman and Byzantium Studies_ , 49 (2009). In discussing Philostratus's description in _Imagines_ 2.18 of a specific painting of Polyphemus and Galatea, Kostopoulou writes the following, a formulation that accords with the orientation of this engagement with both ekphrasis and Derrida, 'Deflecting attention away from the visual appearance of the painting, the passage directs the focus, inevitably, to the textual nature of the story behind it, giving prominence to a verbal dimension more accessible to the ear rather than to the eye', 83.\n\n15.\n\nThis is Elsner's formulation in his, 'The Genres of Ekphrasis', _Ramus_ 21 (2002).\n\n16.\n\nI have tried to argue elsewhere for the impossibility of pure colour and colouring. Colour is always already informed. Part of that forming is the anoriginal presence of the ideational. See chapter 6 of this volume, 'Colouring Philosophy: Appel, Lyotard and Art's Work', and chapter 7, 'And Colour? Sallis on Art's Colouring'.\n\n17.\n\nI have outlined this approach to particularity in my 'Perception, Judgment and Individuation: Towards a Metaphysics of Particularity', _International Journal of Philosophical Studies_ 15, no. 3 (2007).\n\n18.\n\nWhile it has not been discussed here, reference should also be made to Derrida's other major work on Artaud that concerns the question of art, namely _Artaud le_ _Moma_ (Paris: Galil\u00e9e, 2002).\n\n19.\n\nJacques Derrida, 'Forcener le subjectile', in _Atonin Artaud: Dessins et Portraits_ , Paule Th\u00e9venin and Jacques Derrida (Paris: Gallimard, 1986), 54.\n\n20.\n\nDorothea E. Olkowski, 'Eluding Derrida: Artaud and the Imperceptibility of Life for Thought', _Angelaki_ , 5, no. 2 (2000): 195.\n\n21.\n\nStephen Barker, 'Subjectile Vision: Drawing On and Through Artaud', _Parallax_ , 15, no. 4 (2009): 22.\n\n22.\n\nDerrida, 'Forcener le subjectile', 54.\n\n23.\n\nJacques Derrida, _Writing and Difference_ , trans. Alan Bass (Chicago: University of Chicago Press: 1978), 194.\n\n24.\n\nDerrida, 'Forcener le subjectile', 105.\n\n25.\n\nA number of commentators on Derrida's writings on Aratud's art identify the limit of his approach. For example, Martine Antle argues the following: ' _Tout se pass\u00e9 comme si les dessins d'Artaud ne pouvaient server que de mirror de l'homme, de l'oeuvre et souvenet de som experience asilaire. Tout se pass\u00e9 comme siles dessins et les autoportraits n'avaient pour function que de permettre de s'approche de plis pr\u00e8s du personage d'Artaud et de son oeuvre pour mieux la saisir sans r\u00e9ellement y parvenir_ ' (85). See Martine Antle, 'Th\u00e9atralisation d'Artaud deux cot\u00e9s de l'Atlantique: Dessins et autoportraits dans les discours contemporains', _French Forum_ 32, no. 3 (Fall 2007). For an overview of Artaud's conception of art that locates it solely within the province of his own writings, see Ir\u00e8ne Poutie, '\"La Vraie Culture\": Art et anthropologie dans l'oeuvre d'Artaud', in _French Review_ 69, no. 5 (1996).\n\n26.\n\nSee Julia Kristeva, 'Le sujet en process', in _Polylogue_ (Paris: Seuil, 1976). The importance of the body also appears in Jones Irwin, _Derrida and the Writings of the Body_ (London: Ashgate, 2011); see 24\u201325.\n\n27.\n\nThis is also part of the argument developed by Stephen Barker in 'Subjectile Vision: Drawing On and Through Artaud'. Barker's account of Derrida and Artaud is still the most judicious there is. As with Olkowski, he sees that the way of overcoming the limits of Derrida's analysis as bound up with the work of Deleuze (and exclusively in his case also the work of Bernard Steigler).\n\n28.\n\nJacques Derrida, 'Forcener le subjectile', 105.\n\n29.\n\nIbid., 63.\n\n30.\n\nIbid.\n\n31.\n\nMaurice Blanchot, _The Madness of the Day_ , trans. Lydia Davis (Barrytown, NY: Station Hill Press, 1981), 18. This book is a bilingual edition. One can find the original French on 31.\n\n32.\n\nMaurice Blanchot. _La folie du jour_ (Montpellier: Fata Morgana, 1975), 33.\n\n33.\n\nEmmanuel Levinas, 'Jean Atlan et la tension de l'art', in _Cahier de I'Herne: Emmanuel_ _Levinas_ (Paris: Cahiers de l'Herne, 1991), 509\u2013510. Jean-Luc Nancy, _Atlan. Les d\u00e9trempes_ (Paris: Hazan, 2010).\n\n34.\n\nJacques Derrida, 'De la couleur, \u00e0 la lettre' in _Atlan: Grand Format_ (Paris: Gallimard, 2001), 8.\n\n35.\n\nIbid., 10.\n\n36.\n\nDerrida, _Writing and Difference_ , 151.\n\n37.\n\nDerrida, 'De la couleur', 16.\n\n38.\n\nIbid., 18.\n\n39.\n\nIbid., 20.\n\n40.\n\nIbid.\n\n41.\n\nIbid., 21.\n\n42.\n\nIbid.\n\n43.\n\nIbid., 22.\n\n44.\n\nIbid.\n\n45.\n\nIbid.\n\n46.\n\nIbid., 21.\n\n47.\n\nA similar formulation is used by J\u00e9rome de Gramont when linking this conception of the work (' _le tableau_ ' to the ' _l'offrande_ '). He writes in this context that ' _le tableau accueille_ '. See J\u00e9rome de Gramont, 'Par quelle offrande sans nom: Derrida, Kant et la phenomenology du tableau', _Derrida and la question de l'art: D\u00e9construction de l'esth\u00e9tiqu_ e, ed. Aden Jarry (Nantes: \u00c9ditions C\u00e9cile Defaut, 2011), 278.\n\nChapter 6\n\n# Colouring Philosophy\n\nAppel, Lyotard and Art's Work[1]\n\nWhere is philosophy with colour, with colour as part of the philosophical (colour, not just as part of art's work and part of the work of art but colour as there within the realization of an artwork's own project)? As a beginning, colour has a specific presence. Beyond artwork, though still holding to a form of relation, colour attracts and protects. Coloured patternings become modes of identification. Individuation, distinction and forms of differentiation depend upon colour. Colour begins to delimit and identify what can be known. Animal knowledge, for example, works with colour. Within the work of art, colour has another way of being present and thus another sense of utility. And yet, allowing colour a capacity for work holds it back from its reduction to that which would charm and delight. As a result, it would not just be that the question of colour came to be posed with greater acuity, it is also the case that the only way of avoiding the identification of colour with charm and delight, an equation that is the Kantian legacy in regards to the thinking of colour, would be to link colour to work rather than allowing it to be identified with the process of colouring.[2] As will become clear, it may be the case that the inevitability of work checks what might be described as the positing of an unencumbered and thus unworkful conception of colour, the latter being that conception held within colour's evocation, a set-up that will emerge as the posited 'there is' quality of colour. (It is essential to note that this quality is posited and more significantly only ever present as an after-effect.)\n\nWhile contemporary commentators\u2014writers on art as well as philosophy, at least from the Renaissance on\u2014were always deeply concerned with colour and its role in the service of painting and, in addition, the necessity for its elimination from eighteenth- and nineteenth-century conceptions of sculpture, once posed as a philosophical question, it may be that colour has a significance beyond both the merely historical as well as its already current role within traditional philosophical writings on colour.[3] This opens up a productive means by which to investigate, on the one hand, the possibility of another philosophical approach to colour, and, on the other, the relationship between the philosophy of art and art history.[4] What these opening considerations indicate, or at least this is the contention here, is that colour may be a defining element within which the presentation of a materialist philosophy of art might begin to take place. Colour would therefore play a central role in the formulation of such a philosophical position. What is meant by 'materialist' in this context involves three premises. Adumbrations of arguments for the viability of these premises will be noted in the proceeding.\n\nThe first premise is that the ontology of artwork (i.e., the being proper to art) is defined in terms of activity. The work of art works. It is a dynamic process that always has its own complex economies. The second is that meaning is always, and only, an after-effect of the way matter works. As such, the working of matter is the precondition for the possibility of meaning. This aspect of a work can be understood as its mattering. Mattering as the work of matter is, within a materialist aesthetics, the _conditio sine qua non_ for attributions of meaning. Moreover, to insist on mattering is both to maintain the centrality of matter thought as a locus of activity rather than a merely static event while at the same time holding to the necessity of the particularity of a given work. The mattering of one work is always distinct from that of another. Mattering is the link between a materialist aesthetics and a metaphysics of particularity.[5] Indeed, it is not just a link as though there were the possibility that the connection could be arbitrary; it can be argued that there are forms of entailment between them. The third premise therefore concerns particularity. It is orientated around the status of the particular. The premise is that a particular is always already relational. In broader terms, what this means is while a particular artwork\u2014and thus its work as art\u2014is by definition a form of singularity insofar as any one work is singular in relation to another, a claim that must hold synchronically as well as diachronically, it remains the case that the question of relation will always have an insistent presence. That question concerns the nature of the relationship that any one particular will have, in the first instance, to the genre of which it forms a part, and in the second, to the history in which it is located. Relation does not entail determination. Rather, relationality is fundamental to the being of a particular and thus its identity as a particular. Within such a set-up, what relationality entails is that inherent to the particularity of the particular and thus to its being as singular are acts of recall in which reiteration of relationality is intrinsic to mattering. To be precise, what this means is that the particular, while always particular, is equally a knot both traversed and in part constituted by lines of relationality. (This claim is not a historical one and is not reducible to a concern with historical context or influence.) Rather, the argument would be that within a relational ontology\u2014remembering that what is at stake here is the ontology of artwork\u2014singulars are given within relations. They are an effect of relationality. In sum, both mattering and art as the site of work, both of which form the basis of a materialist aesthetics, can therefore only be thought of adequately within a relational ontology within which what can be described as a metaphysics of particularity plays a central role. What the particular as a knot of activity constructs is a site of judgement. The knot has neither an essential form nor a singular expression. It is a weave of activity. The prioritization of threads or elements\u2014and this in the end will include the work of colour\u2014is the activity of interpretation. The activity of the knot is the artwork's mattering.\n\nAgain: Why colour? In part, an answer is provided by the tension that philosophical accounts of colour within art seem constrained to stage. The tension posed by the place of colour emerges clearly, for example, in Schelling's treatment of the topic within his _Philosophy of Art_.[6] Having argued for the centrality of _chiaroscuro_ as that which organizes the frame, as it allows for relationality within the work, and while holding to the position of artworks as the sensible presentation of the ideal or the essential, colour becomes a more complex matter. Light is still fundamental, and yet there is more.\n\n> The ultimate union of light with matter such that the essence itself becomes completely matter and completely light, occurs in the production of flesh. Flesh is the true chaos of all colours and just for that reason resembles none in particular but is rather the most indissoluble and beautiful admixture of all of them. Yet even this completely unique kind of colour is in addition not immovable, as are the other kinds of colour, but rather living and flexible. The inner stirrings of anger, of shame, of yearning, and so on move, in a sense, the sea of colour and cause it to pulse in sometimes more gentle sometimes more violent waves. (141\u2013142\/368)\n\nAt work here is a position in which colour, it can be argued, is involved in a form of self-transcendence. What Schelling identifies as the moving of colour is the moment in which colour brings with it the possibility of breaking from its identification with the presence of colour as positioned within an oscillation between a mere empirical entity on the one hand and integral to the presentation of that which is 'inner' on the other. Colour opens beyond its secured place, thereby securing itself both a different place and another possibility. It achieves this position as colour. There is another moment of separation that is evident in the _Philosophy of Art_. It also bears on colour. In the treatment of the distinction between allegory and symbol in painting, Schelling notes in relation to the painting of flowers that, 'the flower whose colour merely alludes ( _nur andeutet_ ) to its inner nature or to the intention of nature or to the _idea_ \u2014which amounts to the same thing\u2014is truly allegorical' (148\/378).\n\nWhile it will always be possible to insist on this 'allusion', what it opens up is the possibility that the allusion may, for particular reasons, be distanced. To the extent that this occurs and that it becomes possible to distance that to which there was allusion, and if the allusion thus began to dissipate, then colour would begin to emerge as having a life of its own. There would be colour. There is colour. It may indeed have a dynamic quality that could be taken to work independently of its connection to the idea. While that life would be unacceptable within the framework in which 'formative art' was constrained to operate\u2014namely within a structure provided by the relationship between universal and particular (allegory is the universal 'signified by the particular', since within allegory 'what is actually portrayed means something other [ _etwas anders_ ] than itself' [148\/378])\u2014colour's possible independence, an independence in which it would begin to take on an abstract quality, abstract in the precise sense that the potential for a form of complete separation may be realized, is a potentiality that is inherent to the way allegory is structured by Schelling. Moreover, it is also the point where colour begins to 'pulse' and acquire a potentially independent life. The break with forms of interconnection, the possibility that is there once the 'other than' loses its hold and colour is released from the structure of an already determined connection, would mean that what would then be occurring is colour signifying itself. Rather than being devoid of meaning, it would be positioned as self-signifying. In other words, once colour can be attributed a quality that allows it to have a substantive existence on its own, then it is no longer dependent for its identity on a connection to anything positioned externally to the presentation of colour itself. It would therefore seem that colour is already a material presence. And yet, in this instance it is a materiality that is defined in terms of its inherent differentiation from the intelligible. Thus, it is a materialism caught within an opposition\u2014between the sensible and the intelligible\u2014that dominates both aesthetics and any subsequent philosophy of art.[7] As such, colour brings with it the possibility of materialism even if in certain instances what is taken to be the material becomes no more than a reiteration of a version of the sensible. Hence, what will always endure as a question is the nature of colour as that which presages the possibility of a materialist aesthetics that was itself no longer defined, either positively or negatively, in relation to the opposition between the sensible and the intelligible.\n\nIn sum, the adventure of colour within Schelling's formulation, while seeming to occasion the overcoming of the hold of idealism, only stages such a possibility to the extent that colour is equated with the purely sensible. (Hence, idealism is retained insofar as the structuring force of the opposition sensible\/intelligible is itself maintained.) Not only is a materialist aesthetic premised on having overcome the hold of the opposition, with that overcoming there is a transformation in what counts as the object\u2014artwork, hence the work of art\u2014within a philosophy of art.\n\nWhat has to endure within any philosophical account of colour in painting is therefore colour's self-presentation where such a presentation announces a distancing of the opposition between the sensible and the intelligible. The problematic status of colour, and specifically the role of colour within art, cannot be reduced to questions concerning the 'logic of the concept' (as though there was a logic prior to material presence and thus prior to modes of relationality).[8] What is important is the relationship between the conceptual and colour in art. And yet, to posit that relationship as though it were no more than a relation between elements of equal stature and with similar determinations may be to misunderstand the very difficulty that the demands of philosophy have in relation to artworks. This is especially the case when the philosophical demand is identified with the expectation that an understanding, philosophy's understanding of the work of art, is both necessary and possible. There would be the demand for a form of knowledge that completed. (It must be noted that, within this demand, philosophy would be present as the work of the 'understanding'.) Art would have been completed in the process. However, what endures as a challenge to this position is not the identification of the work of art with the domain of an infinite number of interpretations and thus where art's work becomes the site of either an interpretive (the hermeneutical) or semantic polysemy. The reason for this is that any concern with meaning when taken as an end itself is constrained to ignore art's material presence. If there is a challenge, then it comes from elsewhere. It would be located in the ineliminable capacity of art to refuse the demands of philosophy (albeit _,_ a certain conception of the philosophical) and in refusing them to continue to construct a founding irreducibility between the demands of thought and the work of art. In sum, it would be a refusal that was structured by the impossibility of art's succumbing to the demands of the understanding and one that left open the possibility of forms of relationality that were not structured by the incorporation and then the subsequent overcoming of art's particularity in the name of the philosophical. Reciprocally, there would need to be another thinking of the philosophical such that its having been defined by projects of completion were distanced in the name of a different sense of the philosophical. One of the keenest expressions of this overall project can be found in the writings of Jean-Fran\u00e7ois Lyotard, and in particular in his detailed investigations of the work of Barnett Newman, Sam Francis and Karel Appel.[9] In order to give this project specificity, emphasis will be given to Lyotard's writings on Karel Appel, namely to his book _Un geste de couleur._[10]\n\nKarel Appel (1921\u20132006), a Dutch artist, was primarily a painter, though he also produced some works of sculpture. Lyotard's writing on this body of work could almost be redescribed as staging the project of how philosophy can (perhaps should?) write on art. Lyotard's book on Appel\u2014 _Un geste de couleur_ \u2014can therefore be presented as a performative. Hence, there is a clear opening question: What is being performed? Lyotard stages his concerns and Appel's place within them by arguing that his writings, understood as a philosophical project, are not identified with the project of commentary. Rather, the undertaking that comprises his book, thus the book as a material event, works, he argues, in order to return '\"aesthetic\" discourse to its precariousness: how could it articulate coloured things that so clearly resulted from a _gesture_ and are free of any finality? It is not the gesture of the painter, it is the gesture of painting, or painting as gesture, that Appel opposes to thought. All great pictorial work is one such trace . . . and in this way withdraws itself from the grasps of the reasoner ( _du raisonneur_ ) who turns in circles around it' (26\/27).\n\nFor Lyotard it is the work's work that is involved in this process of withdrawal, while philosophy, even though the philosophical is still identified with work of reason and by extension the understanding, seeks to appropriate, ingest or control and thus end that work. The question of this withdrawal is fundamental to Lyotard's project. The attending impossibility is recalled (hence the centrality in writings of _anamnesis_ ); thus, the work becomes of a form of 'mourning' ( _deuil_ ) for every envisaged sense of totality.[11] Within the attempt to exert control, the philosophical encounters a limit in the process. The limit both undoes strategies of mastery while simultaneously marking the end of that project as the philosophical (i.e., as defining the identity of the philosophical). Affirmation, in this context, marks the necessity for a form of mourning. The limitation, though now it concerns the 'understanding' ( _l'entendement_ ), occurs because it is 'assaulted by an excess of colours' (30\/31), to which Lyotard then adds that this occurs 'as if colours burst on the scene to prevent the understanding from organizing them' (30\/31). There is a failure inscribed within the project of the philosophical, thus conceived, and which has its cause in an externality. That which is external is art as 'matter'. Art as gesture is 'matter'. Philosophy in an act of desperation wants to give art 'the words to say this gesture' (36\/37). There is therefore, as Lyotard's presents his argument, a constitutive distance between art\u2014understood as a material practice\u2014and the philosophical. Equally, and precisely because of that distance, there is a desire within the philosophical to have overcome distance's insistent presence\u2014an overcoming that has a number of different names (e.g., 'understanding', 'empathy', etc.). Nonetheless, the desired overcoming constructs the philosophical as that which effaces the distance that artwork would seem to demand. Indeed, for Lyotard art makes a different demand of philosophy; a demand which heeded would have a transformative effect on the philosophical itself. Rather than the overcoming of distance, the demand of art is that philosophy works with the 'singularity of the occurrence' (40\/41) that is taking place in the 'now' of its happening\u2014a 'now' that holds art's work apart from its complete identification with a historical object, an identification in which the work of art would then be co-terminus with a historical cipher. Lyotard's underlying contention is that it is only within distance, a sense of a distance that comes to be reconfigured as the 'without relation', that singularity can be sustained. Given this position, the question that must be posed to Lyotard concerns both the status as well as the possibility of this sense of particularity. As will become clear, the response to this question constructs the limits of Lyotard's contribution to a materialist aesthetic. What matters is the mattering of singularity.\n\nAppel is the painter of the gesture.[12] For Lyotard, as was already suggested, the question of the gesture and thus the distance that it allows sustains a form's singularity, which in turn becomes an almost essential property of art (to which it should be added, as noted above, that the question of how this sense of singularity is to be understood is one to which it will be essential to return). At this stage, however, it is that which is proper to art that is of greater concern. Lyotard's formulation of his position is precise: 'The work however is of art only as long as something in it exceeds appropriation: the gesture, which withdraws itself from the signature ( _le geste, qui se soustrait \u00e0 la signature_ ) because it signs itself and is only the insensible signing itself in the sensible' (50\/51).\n\nThe work becomes art because of the presence of a form of insensibility. Hence, Appel's 'painting teaches nothing'. His work searches for the 'nameless thing' that 'from the depths of the visible dooms the latter to nothing' (74\/75). This thing 'without name' (104\/105) Lyotard calls 'matter-colour' (104\/105). Prior to linking the formulation of this position to the way Lyotard distinguishes between the Kantian beautiful and sublime\u2014in which the former becomes the endlessness of interpretation while the latter, the sublime, the interruption, the ruining of continuity which, in stopping short, is the staging, the presenting, of a form of singularity, a presenting of course, of the nothing, thus of nothing other then the presentation itself\u2014it is important to stay with this signing. The self-signing that is a form of withdrawal and thus a closing off, which establishes, constitutes, a founding distance from the signature's incorporative hold, becomes the way in which the singular emerges. The withdrawal, as a form of evasion, perhaps escape, is nonetheless presented as that which is neither marked in advance, in the first instance, nor, in the second, is there as that which leaves its own mark. What is there remains nameless. What is there is, of course, what has already been designated the 'there is' of colour, which there is silent. And yet, it is vital to pause here, perhaps even to interrupt the flow of withdrawal and its apparent ease of self-presentation, pausing by allowing a question to be posed. The question is straightforward: What is at work in the production of colour as mute, as nameless, as pure self-giving whose only trait is its not having a trait, and so its withdrawal must remain as both the unnoted and thus unremarked? It should be added that colour\u2014both as a singular as well as a plural presence\u2014cannot itself have been the site of an original inscription or present as the site of that which, in virtue of there being colours, is already the locus of recall and already marked, marked in advance of its having to be the 'without name' and the 'without relation'. The latter is a condition in which the presence of the 'without name', the gesture, the 'there is' of colour would have to have been named twice. The second act of naming\u2014and it can only ever be second\u2014is the 'without name' (the 'without relation'). It is secondary precisely because it is premised on having forgotten the founding recall. Artwork becomes singular. Colour is disinvested of a capacity for original recall, thus original relationality, in order that it hold to pure presence\u2014the 'there is' of colour. Within such a setting, recalling works not as a form of attribution but more emphatically as that which is only ever present as part of an act of constitution. It is inextricably bound up with the original name and thus the original name as the originating knot of concerns.\n\nThe setting in which there is the presentation of colour as a site of recall still stands in need of further elaboration. This will necessitate, in part, presenting elements of Lyotard's engagement with the sublime. There are a number of different elements that need to be introduced. Three that are central are the following: In the first instance, it involves a specific formulation of the sublime. To this end, Lyotard writes that the sublime is 'the feeling ( _le sentiment_ ) that the absolute makes a sign in the work whatever its form' ( _quelle que soit sa forme_ ) (88\/89). The second is that, as Lyotard argues, 'colour is the state of thinking before it becomes language. . . . In its immaterial materiality the thing 'is there' but this 'there' is not some part of a field. It takes part in the nothing' (106\/107). And finally, what is learnt from a specific form of positioning, the position of the 'infant' or the 'animal', is to 'let the rhythm of the undulations pass by without being filtered through the known, the knowable, and the permitted and the forbidden' (180\/181).[13] There is therefore the 'feeling' occurring in relation to a happening, which here is the happening of colour, what was noted earlier as the 'there is' of colour and which is itself defined by a 'before'. In giving this setting a further elaboration, Lyotard writes that 'colour enters matter as what vibrates before ( _avant_ ) the yes and the no which is there before ( _avant_ ) form and concept and destination' (158\/159). The possibility of there being\u2014the being there of colour\u2014that which is there 'before', before forming and in an unnamed and essentially unnameable presence, is formulated by Lyotard in another context in the following terms: 'Colour in its being-there seems to defy all deduction. As with timbre in music, it seems to challenge the mind. It undoes it. It is this undoing of the capacity for plot that I would like to call the soul. Far from being mystical, it is material. It gives places to an aesthetic \"before\" forms ( _d'avant' les formes_ ).'[14]\n\nThe reiteration of the 'before' is of fundamental importance. The 'before' can be connected to one of the ways Lyotard allows colour and philosophy to confront each other. He writes, 'Matter wants nothing to do with thought and wants nothing from it' (138\/139). To this he then adds, 'Colour is, what can thought do about it?' ( _Que la couleur soit, qu'est ce qu'y peut la pens\u00e9e?_ ) Not only is there the singular presence of colour, that singularity is located\u2014hence the ' _y_ '\u2014while the sense of the singular in question is defined. That there is colour is straightforward: _Que la couleur soit_. The addition is the important element. What is being asked\u2014almost literally\u2014is what can thought do at the point, the place in which, there is the fact of colour? As such, the fact of colour\u2014its being there\u2014is located beyond the reach of the philosophical (thus construed). It cannot have any quality other than the fact of its being; _que le couleur soit_. This is, of course, the actual sense of the 'before'. Hence, Lyotard will argue that 'thought' (a term identified with the philosophical) can always explain what colour is. However, what cannot be explained is 'that it is' ( _qu'elle soit_ ). Leaving to one side the inscription of forms of immaterialism into his treatment of the material\u2014the immaterial is the 'without name', colour as 'undulation' and 'vibrancy'\u2014what would seem to be emerging with the description of the presence of colour in terms of the fact of its existence, an existence that is without qualities precisely because it is in its being present, there as the 'there is' of colour, is a type of empiricism.\n\nThe empiricism defines the sense of the singular in question. This sense of empiricism is also implicated in the way the sublime is understood as operative in his writings on art. The sublime as a differend within thought has a specific set of entailments.[15] That to which the presence of the differend gives rise is a conception of the sublime as 'absolutely other to all presentations' (82\/83). As such, it is the event that can neither be defined nor located by sequence or continuity. It becomes sheer presentation, the feeling of which becomes the 'moment' of the sublime (the 'now' of its happening). This can be linked to the sense of empiricism just developed. Within the context of his engagement with Appel, the sublime is formulated in the following terms: 'The sublime is the feeling of a differend within thinking, the differend between on the one hand, its power of reasoning to the limit, of thinking the absolute that is the without relation ( _le sans-relation_ ), and on the other its power of imagining i.e., of presenting the given in forms which are always relations: thus painted relations' (76\/77). The division is between that which is there prior to and thus without relation to form and the process of forming. It is within the domain of the former that the 'there is' quality of colour holds sway. What has to be questioned is of course the possibility that such a division can in fact be established.\n\nLyotard writes the following in relation to the project of accounting for the presence of colour in Appel's paintings: 'Commentary on Appel's colour must make thinking understand: that it had and will have comprehended nothing ( _rien_ )' (108\/109). In presenting this position, there is the genuine temptation to argue that what is being presented here is Appel's work. However, that would only be true if that work was then equated with the static. Or if the argument was made that the 'nothing', rather than being a nothing that stills, was present as original, this 'nothing' would still in fact be production (i.e., it would have been produced). It is worth nothing that Lyotard posits sheer nothing in relation to the paintings of Sam Francis: 'It is impossible to understand what is happening when you look at these canvases if you don't attend to the fact that this marvellous event, this visual seduction, the splendour of the colours, is upheld by the knowledge that there is nothing. And it is to this night that homage must be given. To give homage to colours is to pay the debt to the nothing from which they come.'[16]\n\nIn Lyotard's account of art and its relation to the work of colour, the 'nothing' marks a point that is held apart in terms of a radical singularity. If there is pure particularity, then it is this 'nothing'\u2014the 'nothing' that is equally the 'without name' and the 'without relation'. Not only is there this 'nothing'\u2014though now the 'nothing' can be understood as emblematic of an array of positions all of which are defined equally by the 'before' (e.g., 'before form'), it is the presence of this array of positions that occasions the possibility of a response to Lyotard.\n\nThe response is always going to be that, rather than such a positioning being original, which is, after all, a state of affair necessitated by the assertion of the 'there is' quality of colour, the 'nothing' is actually produced. To argue for the positioning of colours 'without relation' is to have already undone the network of relations in which particulars already operate. To therefore assume that particularity is prior to relationality\u2014and this is the assumption to which both Lyotard's mode of argumentation and what has been identified as his empiricism commits him\u2014is to have to argue that the presentation of colour as operative in art, let alone in painting, is no more than a simple presentation. As though colour in art was no more that a set of unrelated synchronic particulars without relation (hence the counter-position that what Lyotard identifies as the 'nothing' is produced). Therefore, 'nothing', in the precise sense that it is being used here and pace Lyotard, is an after-effect. If there is a further argument for this position that can be located in Lyotard's own work, then one of the places in which it occurs is in the description already noted of the art work's coming into being though an act in which it 'withdraws itself from the signature ( _qui se soustrait \u00e0 la signature_ )' (50\/51). As such, not only has appropriation stemmed its presence as a singularity, and thus as a particular, it is sustained. In being particular, the 'gesture' will have been withdrawn (drawn out in having been drawn back), remaining held back and apart from relation. It would remain 'without relation' and thus as 'nothing'. Once it can be argued that this withdrawal leaves it own mark and so is part of a process, then two elements also become clear. The first is that, once again, the 'nothing' is produced and thus will always have been marked in advance by that production. (The more general philosophical position is that the particular as that which occurs 'without relation' and 'without name' is constrained to work through the effacing of anoriginal form of relationality.) The evidence for this position is there in the process of marking and remarking. Withdrawal always leaves a trace. Lyotard's word is 'withdrawal'. The second point is that if it can be argued that the sense of particularity generated by the 'without relation' becomes a form of empiricism, then it needs to be understood as what could be described as an empiricism without content. Note that Lyotard will always argue that the sublime is a feeling that occurs 'in the work, whatever its form' (88\/89). How then would it be possible to return to that which was always already informed? An indifference to form is only possible as the result of a process of having forgotten an original sense of informed presence. The latter is effaced in the production of the 'nameless' or the 'without relation'. What is overlooked from this setting\u2014this construction of the 'without relation' and the like\u2014is twofold. The first instance would be the constitution of a particular as a knot of concerns. Second, what also vanishes is a conception of the particular as a locus of judgement. This is the result of the inscription of what has been described as an empiricism without content. And yet, to the extent that the knot is attributed centrality, then content, where content needs to be understood as the site of informed original incompletion, returns as a conception particularity that is (thus was) always already present (in other words, particularity as an original condition and thus not produced as such).\n\nAnd colour? If withdrawal is an impossibility, then that impossibility is located in the fact that any withdrawal is already a form of drawing. As such, withdrawal marks an intersection of lines. Withdrawal, therefore, precisely because of the impossibility of its staging a form of singularity\u2014and thus a conception of particularity, in which the singular would only ever be present as a _reductio ad unum_ \u2014marks the intersection of lines. Withdrawal, as bound up with a redrawing, a drawing out that is equally a drawing in, identifies the presence of a knot. It is important to note that when Lyotard identifies the demand that artists make of philosophy, to recall the passage already cited, the demand is not just formulated in terms of singularity but in terms of the relationship between the 'sensible' and the singular. Lyotard writes that artists ask philosophers 'to think the sensible singularity presented here and now' (40\/41). The response will have to be that singularity is only 'sensible'\u2014and, while it cannot be taken up, the reiteration of the opposition between the sensible and the intelligible that would seem to occur here needs to be noted\u2014to the extent that colour was not itself already marked in advance by that which could be neither reduced to nor equated with the 'there is' quality of colour. It is, of course, precisely this possibility that cannot be precluded. The reduction of work to a singular presence that could be identified with the sensible will always have been undone once it is recognized that the act of reduction is produced. What is produced has to forget the possibility of colour as that which in escaping its identification with the 'there is' quality of colours; though equally always being more than sites of meaning derived from colours having an already determined symbolic dimension, is positioned, as a consequence, beyond the opposition between the sensible and the intelligible. In having such a position, colour recalls that opposition\u2014recalling as the assertion of relationality without determination\u2014and in so doing, colour becomes both integral to art's work while at the same time forming part of the knot that allows for judgement.\n\n## Notes\n\n1.\n\nAn earlier version of this chapter was given as a lecture at the University of Dundee. I would like to thank the students and staff of the Department of Philosophy for the generous response to my work. In particular, I would like to thank Dr. John Mullarkey for his kind invitation to present the lecture.\n\n2.\n\nThere is an interesting issue here in relation to Kant's treatment of colour in the _Critique of the Power of Judgment_. In \u00a713 Kant moves from a discussion of colour\u2014what he refers to as 'mere colour' ( _eine blo\u00dfe Farbe_ ) and which has its ground in sensation and is thus 'merely agreeable' ( _nur angenehm_ ) to the possibility that 'pure colour', in having a connection to 'form', thus has a relation to beauty. What is important is that, to the extent that colour is linked to form, it is more closely allied to the generalized intuitability of the object and thus less concerned with the object as coloured. In Immanuel Kant, _Critique of the Power of Judgment_ , trans. Paul Guyer (Cambridge: Cambridge University Press, 2002); German: _Kritik der Urteilskraft_ (Hamburg: Felix Neiner Verlag, 2001). For a general discussion of the way in which the question of the art object and its relation to the beautiful is configured in Kant and Lyotard, though without there being a detailed consideration of the question of colour, see Bart Vandenabeele 'On the Notion of \"Disinterestedness\": Kant, Lyotard, and Schopenhauer', _Journal of the History of Ideas_ 62, no. 4 (2001).\n\n3.\n\nI have discussed the relationship between sculptures and colour insofar as it concerned the nineteenth-century debates on polychromatic antiquities in my 'The Surface Effect: Borromini, Semper, Loos', _Journal of Architecture_ 11, no. 1 (2005). See, in addition, Leonard Barkan, _Unearthing the Past: Archaeology and Aesthetics in the Making of Renaissance Culture_ (New Haven, CT: Yale University Press, 2001); and Mark Wigley, White Walls , _Designer Dresses: The Fashioning of Modern Architecture_ (Boston: MIT Press, 2001).\n\n4.\n\nThe relationship between a philosophy of art and the history of art is relatively unexplored. Most philosophers writing on art fail to engage with the presence of art's history as itself being a philosophical problem. There are, however, many art historians who are acutely aware of having to account for that relation. Of the former, an important intervention was made by one of the most important art historians writing in the twentieth century. See Edgar Wind, 'Theory of Art versus Aesthetics', _Philosophical Review_ 34, no. 4 (July 1925).\n\n5.\n\nI have begun to outline what is involved within a metaphysics of particularity in my 'Perception, Judgment and Individuation: Towards a Metaphysics of Particularity', _International Journal of Philosophical Studies_ 15, no. 3 (2007).\n\n6.\n\nF. W. J. Schelling, _The Philosophy of Art_ , trans. and ed. Douglas W. Scott (Minneapolis: University of Minnesota Press, 1989); In German: 'Philosophie der Kunst' in _Ausgew\u00e4lte Schriften_ , Band 2 (Frankfurt: Surhkamp, 1985.) References are given in the body of the text, English preceding the German.\n\n7.\n\nWhile the project is different, there is nonetheless a genuine affinity with the one undertaken by John Sallis in his exploration of the work of colour in Hegel's aesthetics. See his _Transfigurements: On the True Sense of Art_ (Chicago: University of Chicago Press, 2008), 74\u2013105. While Sallis's is a unique contribution to an attempt to rethink the project of a philosophy of art beyond the structures of the opposition between the sensible and the intelligible, it is a project that has important affinities both in work of Lyotard but also in the writings of Philippe Lacoue-Labarthe. In regards to the latter, see his _La vraie semblance_ (Paris: \u00c9ditions Galil\u00e9e, 2008) and _\u00c9crits sur l'art_ (Gen\u00e8ve: Les Presses du R\u00e9el, 2009).\n\n8.\n\nThe formulation is, of course, Wittgenstein's. See his _Remarks on Colours_ , trans. L. McAlister and M. Schaettle, ed. G. E. M. Anscombe (Oxford: Blackwell, 1977). While Wittgenstein's work may be helpful to clarify colour terms within a more generalized anti-essentialist philosophical position, it has no real extension into philosophical aesthetics in which colour is already defined as part of art's work.\n\n9.\n\nFor a general overview of Lyotard's writings on art, see Jean-Louis Deotte, 'Une esth\u00e9tique du dispara\u00eetre,' in _Jean-Fran\u00e7ois Lyotard: L'exercice du diff\u00e9rend_ , D. Lyotard, J.-C. Milner and G. Sfez (Paris: Presses Universitaires de France, 2001); Simon Malpas, 'Sublime Ascesis: Lyotard, Art and the Event,' _Angelaki_ no. 1 (April 2002); and Marty Slaughter, 'The Arc and the Zip: Deleuze and Lyotard on art,' _Law and Critique_ 15, no. 3 (January 2004).\n\n10.\n\nJean-Fran\u00e7ois Lyotard, _Un geste de couleur_ (A bilingual edition), trans. Vlad Ionescu and Peter W. Milne (Leuven: Leuven University Press, 2009). All subsequent pages numbers in the body of text: the first reference to the French, the second to the English. Translations have been modified on several occasions.\n\n11.\n\nThis complex set-up is presented by Lyotard in a number of different contexts. See, for example, 'La peinture, anamn\u00e8se du visible' in his _Mis\u00e8re de la philosophie_ (Paris: \u00c9ditions Galil\u00e9e, 2000), 99\u2013115.\n\n12.\n\nWhat counts as gesture is taken up throughout Lyotard's writings on art. One of its most precise formulations is the following: 'The gesture in painting as in writing is an attempt to penetrate the screen of what has already been accomplished or of what is always possible. I call it a gesture because it is no more than a way for time, space and matter (colour for the painter, words for the writer) to manage, \"g\u00e9rer\" itself' (Jean-Francois Lyotard, 'The Secession of Gesture in Flora Danica,' _Stig Brogger: Flora Danica_ [Denmark: Statens Museums for Kunst, 1990], 17).\n\n13.\n\nFor a sustained treatment of the question of the 'infant' in Lyotard writings, see his _Lectures d'enfance_ (Paris: \u00c9ditions Galil\u00e9, 1991).\n\n14.\n\nJean-Fran\u00e7ois Lyotard, _L'inhuman_ (Paris: \u00c9ditions Galil\u00e9e, 1988), 32.\n\n15.\n\nThe theory of the differend is developed by Lyotard in his _Le differend_ (Paris: \u00c9ditions de Minuit, 1983).\n\n16.\n\nJean-Fran\u00e7ois Lyotard, 'Before the Law, After the Law', _Questioning Judaism_ , interviews by Elisabeth Weber, trans. Rachel Bowlby (Stanford, CA: Stanford University Press, 2004), 116\u2013117. In addition, Lyotard devoted a book length study of the works of Sam Francis: _Sam Francis, Lessons of Darkness_ (Culver City, CA: Lapis, 1993).\n\nChapter 7\n\n# And Colour?\n\nSallis on Art's Colouring\n\nAnd colour? As if the question yielded an answer that obviated the need to pose questions concerning colour's presence. And for the moment the question of that presence, colour's presence, and thus its place, the latter being the locus of colour, will be left to one side. In _De Anima_ , Aristotle addresses colour under the heading of _aesthesis_ \u2014that is, the presence of colour as an _aisth\u00e9ton_ given within and to the senses. The seeing of colour\u2014that is, colour's presence within the domain of _aesthesis_ \u2014has a necessary precondition, namely light. Hence, Aristotle argues in this regard that 'in light the colour of each individual thing is seen'.[1] While the presence of the relation between light and colour can always continue to be developed and the detail and internal complications of Aristotle's own view gradually worked out, what endures as the initial question is the utility of viewing the work of colour as just an aisth\u00e9ton. Again, the question of this configuration of colour needs to be taken up. Such a questioning would be orientated around the presentation of colour as though colour were just one. In other words, the position is that while whatever it is that colour is, it emerges because of light. For Aristotle, light is the colour of the transparent. Moreover, on one level, what emerges is the same\u2014that is, colour. In addition there is an important reciprocity between colour and the eye. Within what presents itself, there is only ever a variety of colours. At the centre of that variety is, of course, the equation of colour with an aisth\u00e9ton, as though colour were nothing other than an aisth\u00e9ton, that is the perceived. The location of colour within such a setting is the surface. Thus, Aristotle argues in the _Parva Naturalia_ that 'the colour always inheres in the bounding surface'.[2] As a consequence, colour is of necessity a limit condition. Colouring, again, in the _Parva Naturalia_ , takes place 'at the limits of the body' (\u1f72\u03bd \u03c4\u03c9 \u03c3\u03bf\u03bc\u03c4\u03b1\u03bf\u03c2 \u03c0\u03b5\u03c1\u03b1\u03c4\u03b9).[3]\n\nAnd colour? Once the question is asked again, it becomes necessary to allow the question of colour's locus to be central; in other words, the question of _colour's presence_.[4] Were colour just there at the limit, then that which it delimits would be no more than a mere surface. While Aristotle will be able to engage the fact that works of art contain colour, as is clear from positions held in the _Poetics_ , what is less clear is that Aristotle can provide an adequate account of the distinction between the presence of colour within the work of art as well as the presence of colour outside art's work.[5] Indeed, it might be suggested that once perception is privileged and art is inscribed within a form of phenomenology, then, precisely because what matters is the work as the object of perception, what may then cease to matter, as a consequence, is the work as the work of art. While there is a clear relation between the work of art and that which takes place outside it, in that the artistic presentation of colours refers to, while recalling, colours whose field of work is outside the realm of art, the point remains that colour in both instances is not identical. (The sameness of colour, even of any one instance of the 'same' colour, is not their identity.) They are only ever in relation. Recalling and relating while involving perception and thus the object in its being perceived cannot be identified absolutely with perception. Moreover, that to which each set of colours, within and outside art's work, is itself related is different. Colour within the realm of art's work recalls and refers both to other works of art and to writings whose domain of reference is the work of art. In addition, the work of colour in a work may itself be recalled by future works of art. Outside the domain of art there will be radically different senses of colour. While these differences may appear to be tenuous, there is, nonetheless, a distinction between colour within the work of art, colour in nature and colour as present within a range of concerns defined by human utility. If colour could be equated either with an abstraction or reduced to the status of a percept, then it would be necessarily indifferent to its specific locus and thus to the inherent plurality within colour's presence.\n\nIn other words, while the colours may be the same insofar as, for example, they become visible within light or are the 'same' colour, it is not as though what colour actually is will be identical in each location. The argument is, in other words, that while it is possible to understand colour as an aisth\u00e9ton, and while it could never be denied that there is an aesthetic response to colour, once it is necessary to distinguish between the role of colour within the work of art and the work of colour outside art's work, then in order that the distinction be sustained\u2014a distinction that will of course always be there in the plural\u2014it would no longer be possible to have recourse to the presence of colour merely as an aisth\u00e9ton. This is because such a generality would fail, necessarily, to take into consideration what has already been described as the locus of colour (a locus\u2014colour's presence\u2014that is of course always loci, hence colour's presence). Once this point is conceded, then the question that insists\u2014a question opening up the connection to the radical force of Sallis's work in this area\u2014is the following: Will such a concern with colour also be true in regard to a transfigured sense of the aisth\u00e9ton in which the strict opposition between the sensible and the intelligible is no longer taken to apply? Implicit in this questioning of the aisth\u00e9ton as the sole locus of colour is a supposition central to the development of the philosophy of art. The supposition is straightforward. It inheres in the following: If there were to be a philosophy of art, then part of that which would make such a philosophical position possible would be its capacity to think of the specificity of the role of colour within the work of art. And yet, it cannot be the case that 'thought' can be simply evoked as though the history of philosophy were in some way bereft of treatments of colour's relation to the work of art. What is essential is therefore how what is thought of in relation to colour is itself taken up philosophically.\n\nAllowing artworks forms of particularity is to allow the presence of one generic determination to hold sway in a way that then allows firstly the question of its association with other genres to be posed, and then, second, consideration to be given to one genre's dissociation from another. What always stands by the side of the question of particularity is the attempt to define art in terms of some form of generality on the one hand, or to suggest that it is accompanied by a generality of affective response on the other. The affective response can be defined as much in terms of the purely emotional as it can be defined in relation to the work of reason.[6] The difficulty inherent in either of the two forms of generality resides in the possibility of holding in place and maintaining the specificity of works of art. For Aristotle, artworks are positioned in relation to a human 'instinct'\u2014namely 'mimesis'. Enjoyment of art is an accord between that inherent capacity within humans to imitate and the fact that artworks are, for Aristotle, intrinsically mimetic. The reciprocity at work here is also evident in Aristotle's theory of perception in which, in regards to colour, there is an important shift or repositioning of the eye such that it evinces an accord with the colour it perceives (though more generally with the object of perception itself).\n\nMimesis is a mode of presentation. Within the realm of mimesis it is therefore possible to distinguish between differing modes of presentation. The argumentative structure of the _Poetics_ occasions the possibility of moving between the concerns of tragedy and those of painting. Opening up the possibility of a positive definition of tragedy, in the Aristotelian context, involves a repositioning of action within mimesis. For Aristotle, tragedy cannot be based on the mimetic presentation of static states of human being. Hence, Aristotle writes that 'tragedy is not mimesis of human being but of action and life'.[7] In other words, it is one mode of mimesis rather than another. In this context, the significance of this distinction within mimesis is that it allows discriminations to be made within modes of presentation and thus within mimesis itself. Not only are there poets whose works are 'lacking in character',[8] it is equally possible to identify a similar distinction within other works of art. Thus, it can be located in ways that allow for a distinction between, for example, the practices of different painters to be established. It is in these terms that Aristotle is able to differentiate between Zeuxis and Polygnouts. The latter was able to depict character. The former's skill lay, as Pliny also attests, in the domain of realism.[9] What is fundamental here is that the ground of this distinction within painting as well as the distinction within poetry are themselves only possible because all are modes of presentation that can be accounted for in terms of mimesis such that questions of 'character' and the distinction between, for example, static 'human being' in the first instance and 'action' and 'life' in the second, are all terms which are applicable in equal measure to poetry, tragedy and painting. What is not possible, other than in purely pragmatic terms, is distinguishing between the particularity of painting as opposed to poetry given that both are understood as instances within a more generalized conception of mimesis. Such an act of discrimination would necessitate the identification of a more profound sense of distinction and one that could therefore not be equated with mere pragmatic differences. A distinction would need to be drawn within modes of presentation\u2014that is to say, not distinctions within mimesis\u2014and as a consequence it would be a form of differentiation in which, for example, the particularity of painting could be held apart from the particularity of poetry or tragedy.\n\nMimesis is the setting in which, for Aristotle, variation occurs. Equally, the project allowing particularity to be maintained\u2014namely a philosophy of art not defined in the terms set by Aristotle\u2014would demand that the question of painting be posed in such a way that the answer could not be provided by the move in which painting became nothing other than a form of mimetic presentation, such that particularity was then subordinated to mimesis. What would occur within such a reduction or subordination, one precluding a concern with particularity, is that what would count as the specificity of painting would have to be thought of in terms other than those posed by the work of materials. This would be the reason for the exclusion of painting defined in terms of a locus of techniques as well as a set of ideational concerns. Painting thus understood would be excluded from any determining consideration\u2014that is, a consideration linked to judgement. Hence, it would not be possible to take the work of material, a set-up that can be identified as the work's mattering, as the locus of art's work (and thus, by definition as a specific instance of colour's presence).[10] Insisting on the primacy of mimesis is to therefore introduce into the work another level of presence that would have to be there in addition to the work of materials. This is a level that would be neither immanent nor transcendent but would nonetheless involve the possibility of subsuming all works of art under the heading of mimesis. Such a subsumption would have the related consequence, as has already been argued, of precluding the question of a work's material presence ever being the locus of either critical or philosophical inquiry.\n\nThe complications posed by the question of painting in general, though specifically the impossibility of developing a philosophical response to art's work as incorporating technique, are given a more precise determination by Aristotle slightly earlier in the _Poetics_. It occurs not just with the evocation of colour but the possibility\u2014situated within painting\u2014of distinguishing between mimesis and the work of colour. In regards to the pleasure afforded by looking 'at images' (\u03c4\u03b1\u03c2 \u03b5\u03b9\u03ba\u03bf\u03bd\u03b1\u03c2), the source of that pleasure is understood by Aristotle as resulting from the comprehension of the images in question. In other words, by a recognition or comprehension of that which is already there as the subject of mimetic presentation. This gives rise to a different question: How then would the possibility of pleasure be understood were this type of comprehension to be absent? This question reintroduces what has already been raised as a possibility\u2014namely the work's presence, its material presence and thus what has been identified as colour's presence\u2014as potentially indifferent to its mimetic presentation. In this regard, he writes that 'if one happens not to have seen the subject before, the image will not give pleasure qua mimesis (\u1f24 \u03bc\u03b9\u03bc\u03b7\u03bc\u03b1) but because of its execution of colour or for some other reason'.[11]\n\nFundamental to the formulation that holds colour and mimesis apart, a set-up which is itself only thinkable in terms of the way that a given work affords pleasure, is the possibility that colour may yield pleasure independently of the work of art. Were this to be the case, then while colour will always be part of the work, it is not intrinsic to a work's presence as art.[12] In the Aristotelian context, what is central is the comprehension of what the work presents. Again, it is the primacy of mimesis as that which both subsumes and defines works of art that, on the one hand, fails to allow for a distinction to be drawn between art forms, insofar as they are all mimetic presentations. On the other hand, what the primacy of mimesis opens up is the possibility of positioning the question of painting in a way that locates colour as originally indifferent to the work of art. Colour, therefore, is really only incorporated at the moment in which its presence is part of the mimetic presentation\u2014the object of comprehension\u2014and it would be thus that its presence qua colour would no longer be there as an issue within art. (Such would be the complex place of colour's presence within Aristotle's conception of mimesis.) What this means in addition is that colour is simply a form of presence and as such is essentially indifferent to the work of art, since art's work has to be understood in terms of its comprehensibility and not in terms of its being coloured. Aristotle's position closes off the possibility of asking the question of the work of colour within art's work as a separate and fundamental philosophical question in its own right. This, of course, is also the conclusion that can be drawn from identifying colour, that is colour's presence, as no more than that which occurs on the surface or at the limit. Colour delimits art while holding itself apart from it.\n\nIf there is an opening to the recent work of John Sallis on painting and colour, then it can be located within this network of concerns and questions. The limits of Aristotle provide the question that has to be taken to Sallis. It is vital to be clear here. The question that sets the measure for his work is _not_ the more straightforward question of colour within art. Such a question would miss the point of what it is that delimits and limits what has been identified as intrinsic to Aristotle's understanding of colour, and thus _colour's presence_ within that overall philosophical project. Rather, it is the question of _colour's presence_ within art's work, and thus colour as intrinsic to a work's mattering. Here this is painting as the work of art. Hence, what will always have to be addressed is the capacity of the philosophical to address the presence of colour within the work of art, where colour's presence is not there as an addition but is understood as integral, ab initio, to the work of art. Again, were this set-up to be the case, then colour's presence, as an original condition of art and therefore equally of the work of art, and thus not merely as part of a generalized aisth\u00e9ton, would have to be taken up as that which informed and formed a philosophy of art.\n\nSallis's philosophical project in relation to both aesthetics and the philosophy of art starts with an undoing of the predominant tradition in which philosophy's relation to the work of art had been thought hitherto. Rather than accepting the tradition as given, his work in this area starts, in departing from Nietzsche, with an undoing of the opposition between the sensible and the intelligible. While the questioning of that opposition, which amounts to a repeated questioning that in the end becomes a form of deconstruction, plays an important role in a range of Sallis's writing, it is given an acute and, for the project of this paper, useful configuration in his _Shades\u2014of Painting at the Limit_.[13] In that context, Sallis's undertaking has a twofold quality. In the first instance, it involves delimiting and thus opening the work of this conceptual opposition, and then, second, with the distancing of the intelligible, it allows for a redefinition of the sensible such that the latter loses its quality of the merely empirical and begins to acquire a radically different sense. Precisely because the sensible re-emerges, it can then be given points of identification with other moments within the history of philosophy. Most notably for Sallis it is the relationship to Plato. The reference to Plato that continues to return within those aspects of Sallis's project that take art as their focus is a passage from the _Phaedrus_ (250d\u2013e) in which, in what could be described as a shift in orientation within the Platonic corpus, Plato is concerned with the luminosity of the beautiful\u2014that is, \u03c4\u03bf \u1f72\u03ba\u03c6\u03b1\u03bd\u1f72\u03c3\u03c4\u03b1\u03c4\u03bf\u03bd. Sallis understands the latter as 'the most shining, that which shines forth'.[14] Here, in this specific Platonic context, the beautiful is given a mode of appearing that is neither the purely sensible nor the purely intelligible. Rather than being seen, which is after all what takes place in Diotima's description of the possible relation between the 'beautiful' as it is formulated by Plato in the _Symposium_ , here the beautiful acquires its own quality.[15] What arises is what is treated in Sallis's work in terms of 'shining'. Before pursuing, albeit briefly, the presence of 'shining' in _Shades\u2014of Painting at the Limit,_ it is essential to note that what has been recovered from the Plato of the _Phaedrus_ is that which distances the hold of the opposition between the sensible and the intelligible. As will be seen, that distancing does not lead to an affirmation of one element of the opposition at the expense of the other. What occurs, and in part Sallis's argument is that this is the potential within the _Phaedrus_ , is a transformation (or 'transfigurement') of the sensible itself.\n\nIt is not just that the transformation of the sensible is important, what is far more significant is the way that transformation is understood. Once the sensible is differentiated from empiricism, where the object, the empirical, is that which is given to the senses\u2014and this is one interpretation of this aspect of Sallis's overall project, and thus an interpretation of part of what is at work in the sensible's transformation\u2014then the question concerning the status of the sensible acquires a new sense of urgency. As will be seen, for Sallis the move away from the sensible, where the latter is understood as the empirical and is therefore given in opposition to the intelligible, occurs in giving the sensible a quality in which the sensible acquires a force that while present is also other than the sensible (again where the latter is to be identified with the empirical and thus thought within the purview of empiricism). As will be noted, for Sallis this force is the 'shining' of the sensible. What does not occur, though this is what needs to be argued, is that the transformation of the sensible retains the object's material presence by the attribution to the work of art of a force that was intrinsic to matter and thus to the work's material presence. Were this to have been possible, then what was there initially as the intelligible would have been drawn inexorably into the work's work. Force in becoming a quality of matter will have undone the opposition between the intelligible and the sensible through their interarticulation within (and as) the work's mattering. This would be a move from the empirical to the material. That latter possibility, which also needs to be understood as a transformation of the sensible, is the one that while not taken up by Sallis is always there as the other possibility once the sensible has broken its identification with the empirical. Staked out here are a series of concerns to which a return will be made as part of a more general conclusion.\n\nIn the context of Sallis's work, however, his configuration of the sensible's transformation, one that draws on the prompt within Plato's _Phaedrus,_ is developed in the opening section of _Shades\u2014of Painting at the Limit_ and then in the interpretation of a number of works by Monet (among others). What needs to be traced is the way important moments within the overall argument unfold. Of the many procedural steps foreshadowed, if not outlined in the book's opening, two are central here. The first is that 'painting never coincides with depiction\u2014the depicted'.[16] And the second step stems from the claim that what is painted, once painting is freed from the posited centrality of the depicted, is 'light as it comes from beyond being to let objects shine forth'.[17] Painting then has the capacity to distance the hold of representation on the one hand (that is the overcoming of the identification of the work of art with depiction or representation), while, on the other, it allows space for art's other possibility. In regards to this other possibility, painting becomes a privileged art form precisely because paintings are works of art that are able to stage the overcoming of the opposition between the sensible and the intelligible while also allowing art's truth to be both present and presented. It is thus that while there will always be depiction, for Sallis the process of depiction does not set the measure for art. (Though as will be seen, it should not be thought that questions of measure, and with measure value, are absent from Sallis's concerns. As will be suggested, the opposite is the case. 'Shining', far from eliminating the question of value, causes it to be reposed.) In regard to the limit of depiction Sallis writes that 'painting arises only when the depiction of objects in the painting is determined, not by the intention merely to produce a copy of something seen, but by the unfolding of a double having to do with light itself, even if in falling across the depicted scene it assumes more than the guise of a shadow. Aside from the things depicted yet bound to them, this double constitutes what is painted in the painting.'[18]\n\nThis opens the space in which a series of Monet's paintings known as the _Wheatstacks_ appears. And it should be pointed out here that, while Sallis does not argue for the same, it can still be claimed that his treatment of Monet involves a judgement made in relation to those works, a judgement which is, in addition, accompanied by a criteria of judgement. This announces what might be described as the coalescence of judgement and value. As will be noted below, for Sallis these paintings stage, in an unencumbered way, the truth of art. They do not disclose content. That would be to identify art with depiction. Art's truth is its shining. There is therefore an opening to judgement that is located in the truth. There is always more than depiction. (Though the question that must be posed here concerns how the presences of this 'more' are to be understood.) What opens up the possibility of truth is that which is there 'aside from the things depicted'. The question of value inheres in the non-necessity of this addition.\n\nThe _Wheatstacks_ are to be understood as the 'spread of light over all things, that is the shining'.[19] However, this will not be the case for all paintings. Moreover, precisely because 'shining' is inextricably bound up with the light that 'comes from beyond being', and here being is to be understood as that which is the merely ontic, art becomes, as a consequence, the occasioning of a truth that is always more than artwork itself and thus always more than art's work. It is the necessity of what is named here as 'more' that is essential for Sallis's larger project within the philosophy of art. The question then concerns, as has been suggested, how this 'more' is to be understood. While an answer to that question will continue to inform this engagement with Sallis, it should be noted that in his _Transfigurements: On the True Sense of Art_ , there is an even more emphatic claim made about light. In that context, Sallis argues that art 'commences only through the transfiguring of mere sense into the shining of sense'.[20] ('Mere sense' names empiricism while noting its having been overcome or 'transfigured'.) This claim about art, a claim that identifies what counts as art qua art, precisely because it will not hold for all works that are either given the name or aspire to bear that name, reinforces the point noted previously that what is at work here is the implicit inscription of a criterion of value. It operates as such precisely because it stages art's truth in a way that is not identical with all works of art. There is an opening. The question of value inheres within it. There may be moments in _Shades_ that would seem to stem from this claim made about Sallis's project, as might be argued in relation to the position in which he establishes a form of identification between 'shining' and the 'poetic moment of painting', which when taken as a whole comprise 'the genius of painting'.[21] While this is clearly a general claim about art, it is also clear that it could not be a claim about all art. This accounts for why in _Shades_ there is such an investment in specific works by Monet. However, it is vital to exercise some caution here. While the claim noted above is about Monet\u2014hence that 'genius' can be found in Monet's works\u2014what endures, as an open question, is the extent to which such claims are in fact about the particularity of Monet's works. Note here Sallis's detailed response to the question that Sallis himself asks: 'But what is painted in the _Wheatstacks_?'[22] His answer is precise and provides another step in understanding how within his work the transformation of the sensible is to be understood.\n\n> In such impressionism things are to be rendered as they show themselves rather than as sensible images that would present a meaning transcending the sensible presentation. Thus it is that such impressionism carries out in the work of art the same inversion that thought\u2014a thought that can no longer simply be called philosophy\u2014has had to address since Nietzsche: the inversion of the hierarchical opposition that has always subordinated the sensible to the intelligible. Not only does such an impressionism carry out the inversion, but is also broaches the other moment of the double move required at the limit of metaphysics: by remaining at the surface, _inhibiting the passage to meaning_ , such painting begins the work of twisting the sensible free of the intelligible, the work of twisting free from the very scheme of opposition that has governed Western thought as well as the interpretation of art, if not art itself. (Emphasis added.)[23]\n\nIf there are difficulties that occur with this formulation, then it is not there in the claim of the distancing and thus the overcoming, signalled here in terms of a 'twisting free'; rather, it is in the identification of art and philosophy as carrying out the 'same inversion'. The identification of the same is an instance of the larger problem of moving between the work of art and its other. What, after all, does the 'same' mean in this context? And specifically what force would a conception of that which is other than art have given the ease with which the 'same' in evoked? Moreover, what this opens up is the nature of the sensible once it takes on the quality of 'shining'\u2014thus, a recasting of the problem of sameness. While it will always be possible to identify that quality in specific artworks, to the extent that the 'same' quality was also to be found outside art, and thus as occurring in response to nature or perhaps more exactly in response to the work of light working within nature, as, for example occurs in Sallis's evocation of the work of light when viewing Agrigento (Acragas), then the problems that have already been noted in relation to Aristotle will also arise here.[24] What this means in sum is that the problems that occur with Aristotle's approach to colour, and thus the way colour's presence is thought in relation to mimesis, can be viewed as straightforward. Once perception and the object as that which is perceived are taken as primary, and hence what will always have to be given priority are acts of perception (rather than priority being given to the quality of the percept within those acts). Then, as a consequence, it becomes impossible to think of the specificity of art and the particularity of artworks.\n\nStemming from these consideration are two questions that have to be taken to Sallis. The first is to what extent does the transformation of the sensible\u2014its transfigurement\u2014overcome the problems already encountered in Aristotle? The second is that if the 'same inversion' occurs in both philosophy and art, does the presence of the 'same' work against the possibility of addressing the specificity of art's work? (Again it should be remembered that what is opened up by this question is the necessity to question the possibility of holding apart art and its other.) As an opening there are therefore two questions that must be taken to his _Transfigurements: On the True Sense of Art_. The second question noted above concerns the nature of the work of art, while the first pertains to the problem of perception and thus to the nature of the link between perception and its object. Inherent in the latter is the possibility of an approach to art that defines it in terms of a transformed phenomenology. If there is a way into the way an artwork works as announced in the passage cited above, then it can be located in the proposition that its work\u2014which for Sallis is its 'shining'\u2014has the capacity to inhibit the 'passage to meaning' is integral to the work's twisting free. Prior, however, to pursuing what is meant by this process of inhibiting meaning, the role of colour within Sallis's _Transfigurements: On the True Sense of Art_ needs to be introduced.\n\nThe movement within the chapter in _Transfigurements: On the True Sense of Art_ that most directly takes up the position of colour within art is orientated around an interpretive reading of those aspects of Hegel's _Aesthetics_ that are themselves concerned with colour.[25] Colour in Hegel, in Sallis's interpretation, becomes the moment allowing the presence of 'shining' to undo the conventional understanding of Hegel in which art, and more importantly painting, is nothing other than the sensuous presentation of the non-sensuous. However, that concern is not a simple positioning of colour within a more generalized philosophical engagement with art. Starting with Kant and then noting Hegel's inversion of Kant's privileging of drawing over colour, Sallis is quick to identify the claim made by Hegel that 'everything is produced by colour' as not only opening up colour but also locating that opening within an undoing of the opposition between the sensible and the intelligible, and thus with a transfiguring of the sensible. In the context of this work by Sallis, transfigurement is understood as involving the following: 'In art sense is transfigured. As so-called material sense, as the sensible of things, it is transfigured into a shining of something beautiful.'[26]\n\nA careful reading of _Transfigurements: On the True Sense of Art_ will note that Hegel's formulation that extols the centrality of colour can be seen as rewriting, _avant la letter_ , of a passage from Paul Klee already cited by Sallis much earlier in the book and which exerts a powerful influence on Sallis's formulation of his own project. Klee writes that 'art does not reproduce the visible but makes visible'.[27] Not only does this reinforce the position that art is always more than depiction, but it also accords with the position identified in Hegel that art is not imitative. In this way, art cannot be reduced to the simple fact of its presentation of that which is given to it to present. As such, art is already distanced from a relation between externality and internality in which the internal presentation of what is external both governs and delimits the work of art.\n\nSallis reads Hegel in order to recover from Hegel's own engagement both its limits\u2014that is, its possible retention of the opposition between the sensible and the intelligible\u2014and, more significantly, to take up those moments within Hegel's engagement with colour in which what emerges is an identification of art's work with 'shining'. That identification is the location within Hegel of the truth in art. What Sallis is looking for in Hegel are the moments, as has been suggested, in which what should define Hegel's concerns, namely the definition of art in terms of the sensuous presentation of the non-sensuous, cedes its place to another conception of art and thus another way of understanding presentation. Through a meticulous rereading of the different ways in which Hegel formulates the distinction between symbolic, classical and romantic art, Sallis discerns a fundamental divide within Hegel's work. On the one hand, there are projects that are still positioned within the tradition of presentation that remains structured by the opposition between the sensuous and the non-sensuous. On the other hand, and this is the decisive move, there are those moments that Sallis identifies in Hegel in which there is that other possibility for art. (A return of the 'more'.) This 'other kind' is 'indifferent to content sets its sights on the shining of the sensible in its utmost sensibleness, tracing in the very sense of sense the retreat of spirit'.[28]\n\nThe 'retreat of spirit' is that which painting alone can accomplish; it is the 'escape of spirit from visibility'.[29] This is painting as an invisibility. Hence, for Sallis in relation to Monet, the latter's 'paintings present something that borders on invisibility, something virtually invisible. They present the light as it spreads across a scene so as to illuminate things.'[30] While it is not invisibility per se that Sallis uncovers in Hegel, it is nonetheless the continuity of the move from depiction, representation and therefore imitation. The latter occurs, for example, in Hegel's own estimation of Dutch painting. Sallis quotes Hegel as arguing that these are works in which the 'chief interest lies in the artistic production of shining'.[31] Comments of this nature begin to identify the force of Hegel's more general claim that painting is nothing other than the work of colour (to which it should be added immediately that this claim finds expression in some works with far greater acuity than in others). It is a claim, made by Hegel in relation to colour, that has its most radical formulation\u2014and again it is a passage noted by Sallis\u2014in the argument that even the difference between two men is nothing other than work of colour. Hegel wrote that 'in a painting this entire difference is reduced to a difference of colours'.[32] (It should be noted, however, that this difference occurs 'in painting'.) However, it is not just Dutch painting that contains this capacity. Hegel also identifies this process in the painting of human flesh, that is, in 'carnation'.[33] In is with flesh and its presentation that painting moves decisively away from the project of spirit's presentation. What Sallis is tracking in Hegel's attempt to engage colour are those moments in which colour twists free of its location and determination within and by the opposition between the sensible and the intelligible. These moments of separation may not be absolute; they may involve noting in the depicted that which is always 'more' than the depicted. Hegel names the possibility as 'the magical effect of colouring'. Not only is it a formulation that is cited a number of times by Sallis, it is precisely in the terms set by this 'magical effect' that it is possible to understand colour as 'shining'.\n\nHegel's own description of this 'magic' involves a set of relations between things that shine such that what then emerges is in Hegel's words 'an inherently objectless play of shining'.[34] If there is a Hegelian response to the question of how to understand the presence of colour in art, then the answer is clear. Colour is all. Colour's presence becomes art itself. It is thus that Sallis can then find in Hegel's final comments on painting, in which colour almost lifts itself from the canvas and becomes a domain in its right, the possibility of linking Hegel's writings on art to the continually renewed world of contemporary art.[35] Colour hovers in this endless play. This is the force of what was noted earlier in terms of 'shining' working to 'inhibit meaning'. It should be remembered that accompanying the attempt to understand what this inhibition might involve is the related question of its enacting the 'same' inversion of metaphysics that occurred in philosophy with Nietzsche.\n\nWhile it needs to be conceded that this inhibition varies by degrees such that claims, for example, about Monet will not be identical to those made about Titian, it remains the case that what is at work within colouring as it appears in Hegel, and a general concern with 'shining', are to be understood in terms of ridding art of the hold of meaning. In the case of Sallis's writing on Hegel, meaning could be equated with either depiction or narrative context. Hence, 'shining' is always 'more' precisely because it is more than, or other than, the weave of concerns created by meaning, depiction or narrative. While it should not be thought that works of art bear no relation to depiction, the truth of art\u2014that which is there in the 'shining'\u2014must always be other than depiction, and thus will have to have inhibited meaning's hold.\n\nQuestioning this set-up is not to cast doubt on the viability of Sallis's engagement with Hegel or the significance of his writings within the philosophy of art. However, what endures as a question is the possibility of identifying painting as 'colouring' and thus as 'shining', and assuming, in addition, that colour could hold itself apart from meaning and finally that differences within the depicted are themselves no more than the work of colour. The question, in sum, is, what would it mean to hold to colour in its necessary separation from any other aspect of the work of art? (This is a possibility already hinted at in the identification of that which is always 'more', as in some sense holding itself apart from the depicted and thereby, reciprocally, turning the depicted into the purely representational.) Answering the question just posed has to begin with the acknowledgement of the necessity of overcoming the hold of the opposition between the sensible and the intelligible. That other tradition within the philosophy of art, a tradition beginning with Nietzsche, has brought into play another framework for the interpretation and understanding of presentations or appearances. And yet, it could be argued that overcoming that opposition need not lead to a transformation of the sensible, and thus an abandoning of the intelligible; the abandoning of the latter would be another way of understanding the suggested overcoming or 'inhibiting' of meaning. However, it may be the case that what this overcoming allows, perhaps demands, is transformation of the intelligible rather than its abandoning. This would entail that the work's content\u2014and it will be necessary to return to the problems posed by the term 'content'\u2014cannot be separated from colouring's 'magical effect'. Were this to be possible, then colour could not be freed from the determinations of the work. Indeed, colour would become inseparable from that through which the work worked. Colouring and mattering, in absorbing the work's ideational content\u2014an absorption in which a productive coalescence connected both dimensions of work\u2014will become the work at work. What needs to be added here is that such a possibility is not a mere reformulation of depiction. Depiction, for the most part, demands a singular semantic economy held in place by the relationship between externality and internality. Moving away from such a set-up, one demanding a continual oscillation between the external and the internal, is to open up the possibility of a far more complex semantic economy. It should be added, of course, that the corollary here is that the presence of a semantic economy, now potentially complex rather than having an enforced singularity, is then not just integral to the work of art but also becomes part of that through which it works. What can therefore be noted from allowing for a productive relationship between mattering, colour and the work's content is a continual repositioning of what counts as the work of art. In sum, the point is that the move away from a simple retention of the opposition between the sensible and intelligible need not move in just one direction. Transformation (or deconstruction) yields more than one opening.\n\nIf there is a recalibration of how art's work is to be understood, then forming part of such an undertaking would be the presence of a different conception of colour's presence. Accepting the possibilities staged above, recognizing immediately that they stand in need of greater development, would mean that within painting, and here the context is essential, there is never just colour. Colour's magic would always work beyond its identification with mere colour (where what counted as mere colour would be colour in its separation from any other aspect of the work). Indeed, colour would now be 'more' than colour. Furthermore, as part of this rethinking of colour's presence, it should be noted that what is _not_ there in painting are objects, faces, animals, landscape and so on which have been coloured. While preliminary drawings do not, for the most part, deploy colours, it is not as though the subsequent painting is an expansion of the drawing with the addition of colour. Finally, while it is always possible to give a description of the colours within a given work, such an account would comprise the presence of colours and not _colour's presence_. The contention here is straightforward. Colour's presence forms an essential part of the way a work of art stages itself as the work of art: that is, colour is an ineliminable part of art's self-staging. Self-staging is both internal to a specific work while involving the way that staging is itself staged as art.\n\nIn a comment that sets the measure for what it is that art, in Sallis's terms, might accomplish, he opens up another dimension for art. While his immediate concern is with both the work and the artist, there is a sense in which the following can also be read as bringing the viewer into the frame. Sallis writes that 'in art everything depends on sense and its transfiguring, not only on the side of what is sensed . . . but also on the side of sensing, of sensibility, of having a sense for certain kinds of things'.[36]\n\nA passage of this nature gives rise to the question of what it would mean to have art presented and thus to have 'sense' for art. If this latter question were asked of the framed, then it would have to yield the answer that the work stages itself as art. Unlike the mountain or the valleys or even sun reflecting on the water, a painting, as has been suggested, cannot disassociate itself from being art. Equally, the sun filtering through trees or mist shrouding the trees in a valley are not works of art. Art may recall them but there is no analogy\u2014even on the level of being perceived\u2014with art's work. The fundamental difference is that art's work will always stage itself as art. And thus there is the ensuing question, what would it mean to sense art? To 'sense' that what is staged is art? If only as a beginning the answer to that question would have to begin with having 'sensed' art's self-staging. Part of the difficulty here is that there would not be anything prior to self-staging. Here would be nothing elemental of which there could be a sense of what is at stake in art. Self-staging concerns, in the first place, that way art stages itself as art. Thus the way a work is present\u2014and presents itself\u2014as art. In the second it concerns the way that staging comes to be staged as art. The work evinces that presence. Hence self-staging pertains not just to the work being art, but more exactly to its work being the work of art.\n\nAgain there is a clear question: Can the work of art ever separate itself from its presence as art? While the question would seem to be ill posed, what it gestures towards is the impossibility that there could be colour that began to hold itself apart from the work of art, if the latter is always understood as art's presence as art\u2014that is, as art's self-staging. In order to take this examination a stage further, it is essential to take a detour via two works of art, namely Raphael's _Madonna del cardellino_ (1505\u20131506) and Andrea del Sarto's _Madonna delle Arpie_ (1517) (figures 7.1 and 7.2). While it should be remembered that even though for Hegel, as Sallis points out, Titian was a more significant painter in regard to what he (Hegel) described as 'the warmth and force of colouring', it is still the case that with Raphael, painting achieves a form of perfection.[37] Hegel's differing evaluations of Raphael, versus Titian and then both in relation to the 'Dutch masters', are significant and yet not the point. The significance lies in the fact that Hegel has specific criteria of value. The reason they are not the point can be located in what will be noted below, namely that any account of the difference between works cannot take place as though the way a work worked was always 'indifferent' to content. It is not, though, content alone that provides the differences. It is rather that content has been absorbed into the work's self-staging and thus cannot be excised from the work's presence as art. This is the transformation of content occurring in the wake of the overcoming of the opposition between the intelligible and the sensible. (As indicated at the beginning, this is the other possibility arising for a philosophy of art from the effective abeyance of the opposition between the sensible and the intelligible.)\n\nRaphael, _Madonna del cardellino_ (1505\u20131506). Uffizi Gallery.\n\nAndr\u00e9a del Sarto, _Madonna delle Arpie_ (1517). Uffizi Gallery.\n\nColour does not just occur. Within painting, it is positioned. Positioning colour, especially in regards to the two works under consideration, involves a relationship to perspective. While there is a straightforward account of perspective that have two of its most emphatic moments in Alberti's _Della pittura_ and its extension by Piero della Francesca in the latter's _De prospectiva pingendi_ , the relationship between lines, objects and depth does not account for the way in which such a setting can contain fundamental differences in relation to the work of colour. What this means is that perspective becomes a ground and as such needs to be understood as a productive abstraction. This is the point that needs to be pursued. Part of the argument is that perspective as an abstraction is the location of differences staged by the relation between colours and lines occurring within and as perspective. In Raphael's _Madonna del cardellino_ , while the formal elements that would account for the operation of perspective are clear, what is more important is that the differences between colours are such that the distinctions, for example, between the blue of the Madonna's cloak and the green of the background hills and fields, all work to create a line. The lines in question are not the lines in which perspective sits. It is not a question of formal lines that would not be, in fact, present to the eye (or which would have been present in either early sketches or in treatises on perspective). Rather, it is the way that the colours are juxtaposed, given that the colours are specific to the techniques of which they form a part, and that creates lines allowing for perspective. Indeed, it is possible to argue that perspective is not just held in place by the juxtaposition of colours\u2014juxtapositions that create lines\u2014because these lines, thus perspective as a phenomenal event, are in part an after-effect of that juxtaposition. Colour produces lines, lines allowing colour. In Raphael's painting, the children playing with the bird and who have the Madonna's knee between them are, in relation to her knee, set back. Positioned in this way, the children are held in place by the way a line established by the juxtaposition of colours allows her knee to appear further forward than the children. There is, of course, a pervasive reciprocity here. Time and again it is the strength of colour\u2014where strength is a particular determination of colour's presence, and therefore present as a specific technique\u2014that is both the creation of perspective and perspective's creation. This work by Raphael resists any sense of shadow in the precise sense in which shadow would have blurred the exact lines created by the juxtaposition of colours and which would then cloud any clear sense of a line between colours, thus allowing for a different sense of spatial depth: a different sense that would of course be the result of another modality of colour's presence.\n\nIt is in regard to the presence of colour and shadow that it becomes possible to identify an important contrast with Andrea del Sarto's _Madonna delle Arpie_. As a point of departure, one that is not just concerned to stage a relation to Raphael but one occasioning that relation to be the consequence of one work recalling another, a beginning can be made with the relationship between shadow and perspective, and thus the place of colour's presence within such a setting. If Raphael's Madonna is located and held in place by lines that are created by the juxtaposition of colours\u2014this occurs as much in rural setting as it does in the transcendent Madonna presented in his _Madonna di San Sistino_ \u2014then the relationship between the pedestal and shadows in the _Madonna delle Arpie_ works in an importantly different way. That there are colours is clear. The Madonna is clothed in a dress, shawl, veil and so on. Each element of clothing is comprised of a different colour. While the colours overlap such that their juxtaposition creates lines, it is also the case that the creation of perspectival depth is always mediated by the Madonna's presence. Her presence is determined as much by the folds of material within which she is presented, and which become her self-presentation, as it does by a concern with her face. Flesh and folds work together. In Raphael, that work gives rise to her face being all. Even when her face is veiled, it stands out from the veil. In Andr\u00e9a del Sarto's work, the veil casts a shadow onto her face. What would have been a line created by the blue of the veil and the pink flesh colour of her face is replaced by a shadow. Rather than either line or perspective, there is a shadow and, with the shadow, a form of depth, the shadow. Shadows will always need to be understood as part of colour's presence. Here the face is not all\u2014or rather the face is only all if it is a face that refuses pure presence. Folds and flesh are worked\u2014worked by colour and line\u2014in a radically different way (hence the impossibility of generalizing claims about the presence of the face\u2014the flesh of the face within painting). Andr\u00e9a del Sarto has created a face into which it is possible to continue to look due to the play of shadows. As opposed to del Sarto, Raphael created a face which is possible, were one so to choose, to continue to look at, because the face is present in having been created by clear lines that divides hair, face and sky.\n\nIn the case of Andr\u00e9a del Sarto's _Madonna delle Arpie_ , the shadows and thus the relationship between colour and light are decisive. This is a set-up that takes the work in another direction. And it is only this other direction that allows a relation to Raphael to emerge. The presence of the Virgin, the saints and the angels, their position in relation to each other, are defined by the architectural setting\u2014the Virgin and Holy Child on the pedestal, all of them within a niche. However, individual presence, plus the relation between individual presence and the setting as well as relations to each other, while involving perspective, does not occur through the creation of distinct lines. Indeed, as has been suggested, the colour\/line relation that is integral to the work's self-staging that is evident in Raphael is repositioned here. Lines and colour work in radically different ways. Hence, the particularity of colour's presence is fundamental to the particularity of a work's self-staging.\n\nIt is worth noting the presentation of the 'harpies' (so-called). Whether they are mythological figures from Greek mythology or the 'locusts' from the Book of Revelation, what remains central is the way in which they are present within the painting. There are, presumably, six on the pedestal. Four can be viewed. The two to the extreme right and left of the pedestal are already beginning to disappear within shadows. The two that are in full view are painted in a way to suggest that, rather than simply present as carvings, they are in some way trapped within the pedestal. It is as though the Madonna, holding the Christ Child and carrying the book, stands on evil and allows its harbingers to have been trapped. And yet, they are not held completely. The harpies seem to struggle against their having been confined. There is therefore a staged conflict between good and evil within the work that has a dynamic quality. However, the struggling harpies who are yet to be subdued, coupled to the ever-present shadows, project a far from tranquil setting. The muscular child\u2014no longer baby but not yet man\u2014has bodily strength, and yet his face, especially the pink of the cheeks, gestures far more to the infant. Body and face, while forming on one level a unity, nonetheless have a disturbing presence precisely because the body and face cannot be reconciled completely. The painting does not oscillate between positions. As a painting it is, on one level, a unified field. However, taken as art's work, it works to maintain a complex semantic economy. As much as it is presents forms of reassurance and tranquillity, it harbours and presents a disturbing presence\u2014evil yet to be mastered fully, shadows mediating the Virgin's presence and the muscular infant no longer a baby and whose presence thus has a certain incongruity. That complex economy cannot be separated from colour, lines or perspective. Indeed, the latter combine to create that economy. All work together. Colour's 'magical effect' is integral to the creation of this complex economy.\n\nWhile it is possible to take further the analysis of these two works, both in their own terms and then in the contrast between them, it is clear, even at this stage, that the way they work involves colour as integral to the work as a work of art. Colour works because it is not indifferent to content. Moreover, the work of colour does not inhibit meaning; equally it does not unify meaning. Rather, colour is operative within a work's complex economy. Colour is only there as part of the work's mattering, the after-effect of which is meaning. Indeed, it is possible to go further and suggest that were colour to be allowed to hover above the work, referring in its hovering (hence only vicariously and thus only in part to the work's work), there would be the systematic failure to encounter the work's presence as art.\n\nWhile it is clear that Sallis does not view art as a mere percept\u2014the aisth\u00e9ton\u2014and that his philosophical project wants to hold to the particularity of art, especially painting, in the end, precisely because painting becomes a transfigured aisth\u00e9ton, painting's work and thus the work of art endures as indifferent to both content and meaning. This is, after all, the moment in Hegel in which colouring takes on the quality of shining in its overcoming the work of the framed. And yet, the problem is that colour, because it produces meaning and content, can never be differentiated from what it produces. However, that does not reduce art's work to mere depiction. As has been argued the other possibility that attends the overcoming of the opposition between the intelligible and the sensible is a transfigurement of the intelligible. Were that to occur, the work's meaning would then become an effect of its material presence, its mattering; and meaning, in no longer being held by differing modalities of representation, would take on the quality of a complex economy\u2014an instance of which was noted above in relation to Andr\u00e9a del Sarto's _Madonna delle Arpie_.\n\nIf there is a point of summation, it can be located in a reiteration of the opening question: And colour? One answer is that colour is all, all that there is. The other would be that the all of colour\u2014all colours\u2014are only ever there within a network of concerns that define and maintain the work of art. The conjecture would always therefore have to be that colour, from colouring to shining, can never be all that there is. Art's shining is the way that art twists free from the hold of representation and the attribution of meaning as a singular instance. Art's shining would be another way of naming the transformation of the intelligible and thus the interarticulation of colour and meaning as a produced set of possibilities. Meaning as integral to art's work, as an ineliminable element within and as the work, cannot be thought of other than in its relation to colour.\n\n## Notes\n\n1.\n\nAristotle, _De Anima_ , trans. W. S. Hett (Cambridge, MA: Harvard University Press, 1995), 418b 3\u20134.\n\n2.\n\nAristotle, _Parva Naturalia_ , trans. W. S. Hett (Cambridge, MA: Harvard University Press, 1995), 439b 14.\n\n3.\n\nIbid., 431a 32\u201333.\n\n4.\n\nThe term 'colour's presence' will come to play a fundamental role at the end of this chapter. While here it marks the presence of colour, what the term opens up as a possibility is that what counts as colour's presence does not have a singular determination. Hence, there is never just colouring. See in this regard my 'On the Image of Painting', _Research in Phenomenology_ 41, no. 2 (2011).\n\n5.\n\nAristotle, _Poetics_ , trans. Stephen Halliwell (Cambridge: Harvard University Press, 1995).\n\n6.\n\nWhile it cannot be pursued here, reference to both the affective as well as reason becomes a way of locating the sublime\u2014specifically as it appears in Burke and Kant\u2014within a philosophy of art defined in terms of the response of a subject. Perhaps it should be added that Kant, and for the most part Burke, were not concerned to develop a philosophy of art despite the use made by them to this end.\n\n7.\n\nAristotle, _Poetics_ , 1450a 16\u201317.\n\n8.\n\nIbid., 25.\n\n9.\n\nPliny, _Natural History_ , book 35, trans. H. Rackam (Cambridge, MA: Harvard University Press, 1952), 9.\n\n10.\n\nI have attempted to develop elements of the relationship between 'mattering'\u2014understood as the art work's material presence\u2014and the work of art in a number of places. It is a term that stands in need of further development. See my 'The Matter of a Materialist Philosophy of Art: Bataille's Manet' in my _Style and Time: Essays on the Politics of Appearance_ (Evanston, IL: Northwestern University Press, 2006), 124\u2013138; and chapter 6 of this volume, 'Colouring Philosophy: Appel, Lyotard and Art's Work'. The key to understanding mattering has to do with recognizing that a work of art is the presence of a complex economy. Colour, materials, and a range of concerns are all interarticulated. That interarticulation, however, does not create, of necessity, a synthetic unity. While some of these ideas have been developed towards the end of this chapter, they stand in need of a more sustained elaboration. That occurs in chapter 1 of this volume, 'Mattering, Particularity and the Work of Art'.\n\n11.\n\nAristotle, _Poetics_ , 1045b 16\u201319.\n\n12.\n\nAristotle is not alone in holding to such a position, see Philostratus, _Life of Apollonius of Tyana_ , trans. Christopher Jones (Cambridge, MA: Harvard University Press, 2005), 2.22. Here Philostratus suggests that the purpose of colour, and the mixing of colours, is 'imitation', thereby opening up the problematic presence of colour outside its being defined by its place within the mimetic.\n\n13.\n\nJohn Sallis, _Shades\u2014of Painting at the Limit_ (Bloomington: Indiana University Press, 1998).\n\n14.\n\nIbid., 9.\n\n15.\n\nThe reference here is Plato, _Symposium_ , 210 C.\n\n16.\n\nSallis, _Shades_ , 8.\n\n17.\n\nIbid., 9.\n\n18.\n\nIbid., 31.\n\n19.\n\nIbid., 48.\n\n20.\n\nJohn Sallis, _Transfigurements_ : _On the True Sense of Art_ (Chicago: University of Chicago Press, 2008), 6.\n\n21.\n\nSallis, _Shades_ , 20.\n\n22.\n\nIbid., 39.\n\n23.\n\nIbid., 41.\n\n24.\n\nIbid., 5\u20136.\n\n25.\n\nThe chapter is 'Carnation and the Eccentricity of Painting', 74\u2013104. There are, of course, other important discussions of colour throughout _Transfigurements_.\n\n26.\n\nSallis, _Transfigurements_ , 7.\n\n27.\n\nIbid., 19\u201320.\n\n28.\n\nIbid., 97.\n\n29.\n\nIbid., 91.\n\n30.\n\nIbid., 21.\n\n31.\n\nIbid., 96.\n\n32.\n\nIbid., 76.\n\n33.\n\nWhile Schelling does not privilege carnation per se, he does argue that the 'final highest stage of the manifestation of colours is that upon which they appear internal, organic, living and flexible. Since this is fully the case only in the human figure, this is the final made most perfect object of painting' (F. W. J. Schelling, _The Philosophy of Art_ , trans. Douglas W. Scott [Minneapolis: University of Minnesota Press, 1989], 146). Philosophers other than those within the German idealist tradition have also made claims about the centrality of 'carnation' in any discussion of colour. See, for example, Bernard Dupuy du Grez, _Trait\u00e9 sur la peinture_ (Paris: Vrin, 2011), 182\u2013183 (first edition 1699\/1700).\n\n34.\n\nSallis, _Transfigurements_ , 94.\n\n35.\n\nThere have been other recent attempts to locate and defend the actuality of Hegel on art. While their modes of argumentation are different, it is nonetheless worth noting their presence. See in particular Benjamin Rutter, _Hegel on the Modern Arts_ (Cambridge: Cambridge University Press, 2010); and Stephen Houlgate, 'Hegel and the Art of Painting', in _Hegel and Aesthetics_ , ed. William Maker (Albany: SUNY Press, 2000), 61\u201382.\n\n36.\n\nSallis, _Transfigurements_ , 6.\n\n37.\n\nIbid., 98.\n\nChapter 8\n\n# Endless Touching\n\nHerder and Sculpture\n\nThe relationship between the hand and the eye and thus between touch and sight is the mark of a founding instability\u2014an instability, it can be argued, that accompanies the history of aesthetics. Integral to that history is the attempt to control, or at the very least to assert control over, that which is inherently unstable, namely the domain of sensation.[1] This is a state of affairs that accompanies the emergence of aesthetics and which finds its most exact formulation in the distinction between sensation and cognition. The relationship between hands and eyes needs to be situated, in the first instance, within that history. Here, however, by using Johann Gottfried Herder's _Plastik_ as a point of orientation, eyes and hands will have a different presence, one which while acknowledging that history then attempts to reposition the distinction between the hand and the eye such that it is no longer held by the inherent structure at work within the differing versions of its founding formulation.[2] Hence, another argument is needed. More is wanted than would have emerged from a simple counter-positioning of eye and hand. The traditional hold on the way that opposition is understood must be loosened. The hand, though this will be equally true for the eye, can be opened up. The opening and the loosening will allow both the hand and the eye another possibility in which neither hand nor eye are defined by their literal presence. The instability that characterizes the aesthetic results, in the first instance, from the equation of sight and touch with their literal presence and then from the subsequent 'need' to control sensation by repositioning the eye (and sight) beyond the literal whilst leaving sensation identical with its literal presence. Reworking both touch and sight involves a break with their original positioning.[3] Sight and touch will therefore always need to be more than their literal presence.[4] Only within such a setting can touch become another modality of seeing. What is needed is the ' _douceur_ ' of the lover\u2014a positioning arrived at for Diderot in the _Lettre sur les aveugles_ when the lover can ' _promener ses mains sur des charmes qu'il reconnaitrait_ '.[5] This recognition, of course, belongs to the hand. Hence, its intensity may be greater if the lover were actually blind. Even though the literal may have exerted a controlling influence, allowing for its abeyance, it can be argued, is to open up the possibility of an aesthetics\u2014perhaps even an erotics\u2014of the blind hand and hence a blind hand seeing (i.e., touching).\n\nAs a beginning, it is vital to stay with the hand. What is it that the hand does? If the answer lies in touch, then there needs to be an additional question. When does the hand stop touching? The force of the question is found first in the evocation of the object. The material object, matter as skin or stone, for example, is that on which the hand alights. In addition to the centrality of matter, there is the structure within which touch is to be understood. Asking when the touching stops is to locate matter, the work of materiality and the placing of the hand within time. In addition, it allows matter to become a locus of activity. In the move from the literal there is an accompanying transformation in how the object is understood. Touching takes place over time. Touching is timed. And yet, it should not be thought that seeing is other than timed. There is, however, another sense of time that is already at work in the way seeing is classically portrayed and touching, as a consequence of this portrayal, is thus understood. Within that formulation, seeing becomes linked to a specific conception both of the moment and the singularity of the object. The _locus classicus_ of such a conception is of course Ren\u00e9 Descartes.[6] It is not just the Cartesian conception of 'clear and distinct perception' that is fundamental.[7] Inherent within the Cartesian argument concerning perception\u2014not the eye's perception in any straightforward sense but the perception occasioned by the attention of an inner light, a light that pierces the perennial gloom of ignorance, defining ignorance therefore in terms of gloom and thus a type of enduring darkness that a lighting up will have overcome\u2014is a positioning of matter in terms of timed presence (attesting thereby to the already present interarticulation of matter and time). (Differing accounts of matter are always deployed within their own accompanying conception of time.)\n\nWhen Descartes turns to matter, it is to matter's problematic status. In the _Second Meditation_ the problem posed by the ever-changing piece of wax, a piece freshly cut from within the hive, is its intrinsic capacity to change and modify\u2014in other words, the instability within sensation's object. The link between matter and an ineliminable fluidity, a fluidity that may allow potentiality to define substance rather than the solidity of a form defined by an empirical singularity, means that matter must overcome its reduction to mere material presence (i.e., to simple empirical presence) and be rearticulated in terms of extension.[8] The wax is, however, still seen, touched and imagined. There therefore needs to be a different approach. The truth of the wax can only emerge in the wax's differentiation from its having been identified with its empirical presence. Its identity is not the same as its empirical presence (while not denying that it is empirically present). The truth of the wax can only emerge when a different form of 'consideration' occurs. Descartes writes, ' _nudam considero_ ' (AT VII, 32). Descartes's approved French translation is more emphatic: ' _Je la considere toute nue_.' The emphasis coming from the addition of ' _toute_ ' is an emphatic qualifier of the ' _nue_ ' (AT IX, 25). The nudity here is not the reduction to an essence. Nudity, in this context, is that which cannot be touched. Equally, it is that which is neither presented nor expressed by matter's empirical presence. The separation of matter from the merely empirical allows, in the first instance, control over that which is given by sensation, what will become the aesthetic even though what will count as the aesthetic is yet to emerge at the time in which Descartes writes, and, in the second, the truth of matter to be recovered.[9] Truth is of course linked to cognition. Matter must fall beyond the hand, and yet there must be that which is nonetheless still present. The wax is present such that there is an 'I' who 'sees', 'touches' and 'imagines' (AT VII, 31). As a result, two forms of presence are introduced. The first pertains to mere matter (i.e., the reduction of matter to the empirical). The smell of the wax, its response to moving fingers as it warms, where the warmth is linked to fragrance and then to perceptible moisture\u2014all of this the hand both feels and knows, and yet, for Descartes, what the hand can never know is the quality of matter. Equally, the hand that knows has a similar status to the eye that sees immediately.\n\nThe limit of immediate vision is developed by Descartes in _La Dioptrique_ in terms of an encounter with the blind. The limit, to reiterate the point just noted, is the moment of instability and thus the need to move from sensation to cognition. The question is not just what the blind know; more exactly, the central question pertains to _how_ the blind know. In the opening treatment of light, when light is explained in terms of a movement from the exterior to the interior by means of air and other ' _corps transparent_ ', Descartes uses the example of those who are born blind to formulate his overall argument (AT VI, 84). The blind know the world via the use of a stick. The presence of objects in the world is transmitted by that encounter. Thus, in Descartes's own terms, ' _on pourrait quasi dire qu'ils voient des mains_ ' (AT VI, 84). (Note the addition of the ' _quasi_ '.) For the sighted, light enters through the eyes. For the blind, on the other hand, a different relation to the world pertains. There is access to the world for the blind, because the former ' _passe vers sa main_ '. (AT VI, 84)\n\nWhat is it that moves from the exterior to the interior? If the blind hand were to encounter edges and points and declare that what was encountered was a triangle, what is that has been encountered? What has been touched? What can the blind's hand know? All these questions pertain to the medium of knowledge and to timed presence. The blind hand encounters the world. There can be no doubt that the encounter occurs. The blind can feel and feel again in order to secure more information. However, its having been secured\u2014the process\u2014is then assured. What occurs is an encounter with an object. However, as the 'image' is not identical with the object, then the question, as always, concerns how there can be movement from the reception of images, the encounter with objects and then the possibility that ' _l'ame_ ' (the soul) can know. In the language of the _Meditations_ this refers to the possibility of ' _mentis inspectio_ ' (i.e., an 'inspection' by the mind) (AT VII, 31\/AT IX, 24). This is a capacity that is directed by the methodological imperatives given by 'clear and distinct perception'. Nonetheless, what is of significance is the assumption the there is a separation between an encounter with the world, whether it be with baton or eye, and the operation of the soul. (The soul, it should be noted, is the province of a knowing subject and not the province of the object.)\n\nIn _La Dioptrique_ , Descartes is explicit in regards to the limit of sensation. What has to be avoided is the supposition that 'in order to sense, the soul needs to contemplate several images which are sent by the objects to the brain' ( _pour sentir, l'ame ait besoin de contempler quelques images qui soient envoye\u00e9s par les objets jusques au cerveau_ ) (AT VI, 112). To allow such a possibility would be to maintain instability as an ineliminable presence. The activity of the 'soul' can therefore be located beyond a domain that is given by an opposition between literal sight and literal blindness. Both are limited in relation to the form of perception identified by Descartes as part of his response to Mersenne's objections to the positions advanced by Descartes in the _Second Meditation_. Fundamental to Descartes's argument in his _Response_ , as will be noted, is a form of immediacy. While it remains the case that 'clear and distinct perception' is methodological and therefore takes place over time, there is the claim that 'as soon as' ( _aussit\u00f4t_ \/ _statim_ ) truth emerges, there is the immediate move\u2014an immediacy presented in terms of the operation of nature\u2014to belief and thus to judgement (AT VII, 144\/AT IX, 113). The other and interrelated form of immediacy is found in the next line of the argument. Having claimed that the move to belief and thus certainty takes place immediately, the knowledge that is discovered and whose expression by a sign in which signifier and signified are coextensive is, as a result, a form of a knowledge characterized as much by what can be described as epistemological exhaustion as it is a form of absolutization. The absolute object of knowledge having been discovered means that ' _il n'y rien \u00e0 chercher davantage_ '.[10] This absolutization is presented in _Le Discours de la M\u00e9thode_ in terms of the singularity and completeness of the truth as it pertains to 'the thing' ( _la chose_ ).[11] What is identified here in terms of absolutization is linked, of necessity, to questions of representation. While time figures, it is essential to note that the time that is figured is given within the finality of the sign. Signification and knowledge both allow for the finality of enclosure and thus completion. As will emerge, the hand holds open the possibility that precludes the equation of judgement and completion.\n\nIf there is a move, one occurring with a necessary reciprocity, between hands and signs, then it involves locating a discrepancy between the potential infinite of touch in the first instance and the conception of finitude exacted by the structure of the sign at work in the Cartesian system and which finds one of its most exact formulations in Arnauld and Nicole's _La logique du Port Royal_ in the second.[12] The finality of the sign that is the mark of knowledge stems the infinite hand that continues to move over the object, moving but without knowing. The object is sensed, one may have a sense of the object perhaps even a feel for it; however, the infinitude of touch must be stopped in the name of knowledge and thus with the completion of the sign. In other words, the potential instability that would seem to be inherent in touch (thus in sensation) needs to be controlled. The coextensivity of the signifier and the signified reiterates the temporal structure noted above in which there is an immediacy of relation between knowledge and belief and equally knowledge and judgement. The timed hand can only ever move. Knowledge, within a Cartesian frame of reference, however, involves cessation and completion. At this precise point it becomes possible to reposition and thus rework what is taken to be a founding instability. There is another possibility for the potentiality within touch, a potentiality whose realization defers the hold of the opposition between sensation and cognition. Within such a setting what remains open, held open by the hand's potentiality, is a conception of touch, and therefore of matter and time, that is able to resist its complete enclosure within the Cartesian system. This hand can move without knowing. There is an opening in which what comes to the fore is not just the already present status of the touching hand but its presence within a reworking of the relationship between matter and time. An intimation of this possibility is clearly present in Diderot's _Lettre sur les avegules_.\n\nDiderot's work is not of interest simply because it takes up what could be described as a thematic of blindness and therefore of touch as opposed to sight. Rather, the significance of his work lies in its commitment to a form of philosophical materialism within which both sight and touch escape the axiomatic equation with forms of literal presence in the first instance and a temporality of immediacy in the second. There is an additional point that needs to be added. Diderot also incorporates an element that is an inevitable consequence once the identification of the material and the empirical has been overcome (i.e., the necessity to incorporate the immaterial as part of matter's presence). Perhaps the most direct way that the initial move just noted, namely undoing the definition of touch in terms of its literal presence, can be identified is in regards to the manner in which Diderot established the relationship between ideas and blindness. As will be noted, ideas are connected to the timed hand. In a passage of central importance, Diderot presents this position in the following terms: 'How could one born blind form ideas of figures? I believe that the movement of the body, the successive existence of the hand in several places, the uninterrupted sensation of a body which passes between his fingers, give to him the notion of direction' ( _Comment un avegule-n\u00e9 se forme-t-il des id\u00e9es des figures. Je crois que les mouvements de son corps, l'existence successive de sa main en plusiers lieux, la sensation non interrompue d'un corps qui pass\u00e9 entre ses doigts, lui donnent la notion de direction_ ) (94).\n\nWhile for Diderot anyone born blind is deprived of the capacity to imagine, since the imagination involves the use of colour, the passage still demands considerable attention not just because of the way the hand is positioned but because it is the same hand which, as Diderot argues earlier in the _Lettre_ , judges beauty. The blind individual ' _judge de la beaut\u00e9 par le toucher_ '. (90) The touching hand judges, thus opening up the possibility that judgement is now a material practice no longer defined by immediacy. The move from immediacy to touch where the latter is now positioned in terms of judgement is the opening to mediacy and thus to a radically different conception of judgement. The set-up created by the interrelationship of mediacy, touch and judgement has both an importantly different temporal structure as well as a different conception of the object. Neither the temporal structure nor the conception of object could be situated within a Cartesian formulation of subject\/object relations nor, as significantly, could they be defined in terms of immediacy.\n\nA hand touches a face. Beginning with the nose\u2014either at the tip or to one side\u2014it matters little; what counts is that the hand is moving. The hand slides. A movement, which could be as much an inquiry, yields the question, whose face is this? It is a caress and thus an encounter with skin demanding pleasure. The moving hand has an inherently ambiguous status: moving from nose to cheek, returning prior to tracing with finger tips extended, the way the cheek will suddenly, though smoothly, become an ear. Perhaps the finger moves on, discovering a change of texture, moving backwards and forwards at the point at which texture changes, prior to allowing itself the discovery of what at first is a warm then moist opening, which in opening up, perhaps beneath the finger, becomes the mouth. Diderot describes the same when writing of ' _l'existence successive_ ' of the hand. The hand cannot stop touching precisely because there is no point at which the touching should stop. Not only does this provide access to the way for the one born blind, but there also emerges what for Diderot is the ' _notion de direction_ '. Equally, it is also clear that the direction in question is not from beginning to end. Rather, direction is movement itself. Movement is inherently directional. Even the backward and forward movement that satisfies when the nose itches remains movement. Is it only for the blind\u2014and this may indeed be Diderot's implicit conjecture\u2014that the 'skin' ( _la peau_ ) is beautiful? Beauty now involves neither image nor representation. Unlike for Kant, for whom beauty is a feeling of pleasure that accompanies, immediately, the presentation of an object, the capacity to register beauty's presence, for Diderot, may lie in the moving finger and thus in touching.[13] As Diderot notes in relation to the one born blind, ' _Il juge de la beaut\u00e9 par le toucher_ ' (90). The skin in question is therefore not the exquisite pink skin elegantly captured by Chardin in his painting of the schoolmistress if that skin were thought to be no more than a surface; rather, it is the skin under my fingers\u2014my blind fingers\u2014when they touch. This does, of course, open up the possibility of another approach to Chardin's painting, an approach already there in Diderot's own writings on Chardin and in which there is a fundamental transformation in how the 'surface' is understood. The surface will have been an animated site, infused with 'spirit'. For Diderot, as the previous line suggests, beauty is felt. In being felt, it is no longer a question of the subject's response to an object's presentation but of a reworking of the nature as well as the contents of the subject\/object relation. The question of the finger is not foreign to Herder's concerns. Herder suggests were something to appear 'doubtful' or contradictory' to the 'art student', then the student should make use of 'the fingers of his inner sense ( _den Finger seines inner Sinnes_ ) in order to discover that which he could not otherwise identify: the shape of the sprit within the form ( _Gestalt des Geistes in deiser Form_ )' (90\/95). As a prelude to a direct concern with Herder, it is vital to note two important moments that occur in this passage. The first is the one away from a literalization of both the finger and touch. Hence the reference made, in the passage, to the 'fingers of his inner sense'. The second is the position of a relationship between 'form' ( _Form_ ) understood as material presence and 'spirit' ( _Geist_ ). The latter could be understood as much as a gesture to a form of idealism as it is to the presence of immateriality. The latter\u2014the immaterial (named in this context as _Geist_ )\u2014emerges once there is a sustained attempt to effect a separation between materialism and empiricism. In other words, the conjecture, as has already been noted, is that once the material and the empirical have been separated, then implicit within the work and as part of the object's material presence is an active quality that is inherently immaterial.\n\nThere is an important echo of such a possibility in Diderot. What is interesting is that Diderot, almost despite his commitment to differing forms of philosophical materialism, leaves open the space for what emerges as the ineliminability of the immaterial once there is a commitment to a form of materialism. To be precise, the immaterial occurs in the space provided by the severance of the equation of the material and the empirical. This is clear not only in Diderot's treatment of painting within the _Essais sur la peinture_ but also in his work on specific painters. In regards to the latter, in writing about Chardin, he notes the following:\n\n> Chardin is between nature and art. He relegates other imitations to the third rank. There is nothing in him that senses the palette. It is a harmony beyond which he does not dream to desire. It winds around imperceptibly in his composition _._ It is as the theologians say of the spirit, sensible in everything and secret at every point. \n> \n>   \n> \n> \n> ( _Chardin est entre la nature et l'art: il rel\u00e8gue les autres imitations au troisi\u00e8me rang. Il n'y a rien en lui qui sente la palette. C'est une harmonie au del\u00e0 de laquelle on ne songe pas \u00e0 d\u00e9sirer: elle serpente imperceptiblement dans sa composition, tout sous chaque partie de l'\u00e9tendue de sa toile; c'est, comme les th\u00e9ologiens dissent de l'esprit, sensible dans le tout et secret en chaque point_.)[14]\n\nWhat is significant in this formulation is the recognition, and this despite the language in which it tries to find expression, of a quality that is always in excess of that which is there to be described and yet which is at work within what would have been described. The reference to 'spirit' ( _l'esprit_ ) is not to an externality and thus not to that which in being external would have had a determining effect on both the quality as well as the identity of the object's internality. 'Spirit' is a quality of the object. What is of significance here is the impossibility of any complete separation of 'spirit' and the object's material presence. Its presence is 'sensible' yet 'secret'. In other words, it is an immaterial force as part of matter, occasioning as a consequence a rethinking of matter as an already present site of activity. The move from Descartes to Diderot, one in which Herder can be located, is the repositioning of the 'soul' from the subject to the object. It should go without saying that this repositioning has to be understood as an emphatic reconsideration of the aesthetic, a reconsideration in which the significant element is the emergence of the 'object' as an entity defined in relation to modes of animation. With both Diderot and thus pace Kant (and, as will be suggested, Herder), there is the move from the 'form' of the object (i.e., the object's generalizable conditions of intuitabiliy) as the basis of aesthetics to the object itself.[15] However, it is the object within an emergent materialism and thus not the object of mere sensation.\n\n## Herder\n\nWritten in the wake of Diderot's _Lettre_ , Herder's _Plastik_ (1778) is one of the first attempts to present a philosophical account of sculpture. More as a prelude than an introduction\u2014that will occur in a moment\u2014Herder's enterprise, perhaps its capacity for disruption, can be noted in the way a relation to Greece is established. In a discussion of Anacreon's _Bathyllus_ , a poetic enterprise concerned with the move from poetry to painting, Herder writes that 'the Greek never conceived beauty other than in determinate form ( _in bestimmter Form_ )' (80\/85). The reference to the already determined nature of 'form' does not mean that there is the complete absence of any vestige of idealism in Herder. On one level it would seem as though the retention of terms such as 'spirit', 'soul' and perhaps even 'beauty' defined Herder's project as straightforwardly idealist. However, once it is recognized that 'soul' and 'spirit' retain their philosophical hold as those which animate and thus give life to material presence, where the soul's presence is defined in terms of internality rather than externality, then this opens up the possibility that both 'soul' and 'spirit' allow for their own radical reinterpretations. What is important is the philosophical space they identify. Thus, it will be essential, following from what has already been noted in relation to Diderot, to return to what the concession of animate matter allows in terms of what it provides for another possibility within the history of aesthetics.\n\nThe retention of a version of idealism, a version that can be recast in terms of the relationship between the material and the immaterial, indicates that, in this instance, the problematic of vision that defines the way beauty is formulated in the Platonic dialogues is, for Herder, no longer apposite as the basis for developing an understanding of painting and sculpture in the Hellenic period. The other element that is at play within such a setting is not just form but the sense proper to form, namely touch. The disruptive claim concerning the quality of the already 'determinate form' means that there cannot be for Herder a question of beauty as an end in itself. Nor, more exactly, is it possible to abstract beauty from the specific determination marking its presence (i.e., the determinate presence of sculpture as opposed to poetry or painting). It is as though Plato's insistence that what needs to be pursued is 'absolute beauty', since it is that which accounts for the way a given particular is beautiful, has itself become impossible.[16] The Platonic move is always to argue that, while there are determinate beautiful things, they attain that state due to the operation of the Beautiful. It functions as the cause. This self-referential universal would then become the object of philosophical study par excellence.[17] For Herder, 'beauty' is a quality of the object. Moreover, it is a quality that does not depend upon Beauty's externality.\n\nHerder's _Plastik_ opens with a reference to Diderot. With Herder, in a sense following Diderot, the hand takes on a more complex configuration. (It will always need to be noted, albeit parenthetically, that there cannot be a history of either the hand or touch as though what is at work is the same in every instance.[18]) Nonetheless, what has to be remembered is that the hand is timed. The claim here is that the overall project of Herder's text is announced in one of its opening moves. The opening is staged in relation to Diderot and specifically Diderot's preoccupation with the effect of cataract operations on the blind. The central question is therefore, with sight what do the once blind see? In relation to a question of this nature, Herder begins in the following way: 'What do these strange experiences teach us? Something that we ourselves could experience daily if we were to acknowledge that sight ( _Gesicht_ ) reveals merely shapes, but touch ( _Gef\u00fchl_ ) alone reveals bodies: that everything that has form is only known through the sense of touch ( _nur durchs tastende Gef\u00fchl_ ) and that sight reveals only visible surfaces ( _nur Fl\u00e4che_ )\u2014moreover not the surfaces of bodies but solely surfaces exposed to light' (35\/43). This positioning of the surface, as linked to the eye, sanctions an important development within philosophical writings on art (and within that positioning, note the way in which what is 'touched' is other that the literal 'surface').[19] While both Kant and Hegel will allow for different treatments of painting, as opposed to sculpture, neither link the particularity of each to the operation of a different sense. In part this is the originality of Herder's initial approach. And yet, it is subtler since it is only in terms of touch, and thus the centrality of the hand, that form and the relations between forms in painting can be understood. There has therefore been a move from the literal hand to giving the hand and with it touch, the major role in any account of the operation of art's work. However, while in terms of experience this is of great significance, of equal importance is the way there is a transformation of the object. The object that is seen is different, it will have to be argued, in terms of its ontological status from the one that is merely seen (hence the need to see touching).\n\nTouch must remain for Herder that which guarantees seeing. Precisely because seeing is bound up with surfaces, it is for Herder linked to the dream, since the latter is the surface with neither depth nor internal connection: 'Sight gives us dreams, touch gives us truth' (90\/95). Hence, Herder writes: 'Pity the lover who gazes upon his beloved from a distance as if she were an image on a surface and for whom this suffices. Pity the sculptor of an Apollo or a Hercules who has never embraced the body of an Apollo, who has never touched even in a dream, the breast or the back of a Hercules. Truly from nothing there can arise only nothing: the ray of light touching nothing can never become the warm creative hand' (42\/49). For Herder it is in relation to his treatment of sculpture that it becomes possible to return to the more general treatment of the philosophy of art. Sculpture is both an exceptional case, and yet, precisely in being exceptional, it provides the measure in which it becomes possible to generalize. In terms of the strategy adopted by Herder in _Plastik_ , it is the 'infinite sublime'\u2014an engagement staged in relation to Job and Edmund Burke, among others, and yet reducible neither to the simply theological nor the merely physiological\u2014that plays a central role. The 'colossal' provides both the 'origin' ( _Ursprungs_ ) and the 'essence' ( _Wessen_ ) of sculpture (93\/97). Sculpture is presented not just in relation to its inherent spatiality but also in regards to its having an active quality\u2014matter as dynamic. Being is acting out, and what is acted out is the work's matter. Matter is therefore an activity. As such, the work of matter (a state of affairs that incorporates work as matter) has to be rethought as always already dynamic\u2014hence Herder's remarkable claim that sculpture 'speaks to us as an act' ( _als Tat zu uns spricht_ ) (80\/86). Solidity had acquired a potential, and, consequently, materiality now insists within a form of potentiality. It matters. It will be vital to return to this point. What it indicates, to reiterate the point already noted, is a transformation of the object occurring within the move towards the centrality of touch. As such, what is needed is another account of matter's presence, an account that links potentiality and materiality such that what is at work is both the presence of an immaterial force though equally a dynamic conception of matter.[20] The co-presence of the two demands a shift on the level of nomenclature. The static connotations of 'matter' need to cede their place to a term which attributes to the work of art a workful quality; as such, matter becomes mattering.[21]\n\nRather than the flat surface on which light is projected, sculpture captures and reflects light, holds it and passes it on; captures it, for a face is bathed in light; moves it, as there is the casting of shadows. Light falls on a cheek\u2014in part\u2014in the movement across the face, a movement, which while marking the movement of the eye is one in which the eye is touching. The eye is becoming the hand. It is not just that the object brings a form of the infinite into play; defining sculpture in terms of an act locates the infinite as much in the object's mode of presentation as it does in the condition of a response. Art, taking sculpture as a type of paradigm, can (as a condition of its being\u2014the being of art) never be finished. The infinite sublime is not just the truth of sculpture. In enacting the unending, it acts out the being of the art object: being as mattering. The detail here is essential, since for Herder sublimity brings the 'true image of Gods and heroes' into play (94\/99). There is therefore a need to work through these formulations rather than take them as designating an already determined and structured field of inquiry (hence the reworking of Heder).\n\nHerder starts his treatment of the infinite sublime by identifying sculpture as that which does not have a single point of orientation; rather, it 'explores in the dark' ( _im Dunkel_ ) (93\/97). Countering an aesthetics of light, an epistemology of illumination and an ontology of delimited singularity\u2014one bearing, as has already been indicated, an important connection to the object of Cartesian 'clear and distinct perception'\u2014there is the productive condition of the dark, of touching _im Dunkel_. What is staged here does, of course, need to be understood as a form of blindness working beyond the hold of the literal. While the part of the argumentation that emerges at this point in the book's development returns to the general, if not more traditional, conceptions of the sublime (namely as that which is already in excess), Herder introduces an important difference. Rather than excess engendering, of necessity, a transcendental realm, in this context it is linked to another conception of the unending or infinite. Rather than the infinite as the non-representational, as though all that was involved is a limit in relation to the comprehension of the object\u2014this would of course be the position for Kant\u2014for Herder what the infinite sets in play is fundamentally different state of affairs. For Herder the limit is given by the being of the object. The move from subject to object has been completed. The passage, in which this other infinite emerges, even though Herder does of course not present it in terms that underscore the centrality of the object, in lieu of the subject, is the following: 'The hand never touches the entire object. It cannot grasp a form all at once, with the exception of the sphere, which is the form of stasis and contains perfection in itself. The hand can rest on the sphere and the sphere in the hand. With articulated form, the work of the hand is never complete ( _nie zu Ende_ ): it goes on feeling as it were indefinitely ( _immer unendlich_ )' (94\/99).\n\nThe counter to Descartes's sense of closure is not at opening in which the claim that it would always be possible, to recall Descartes's language of 'looking for' ( _chercher_ ), to search out additional elements could then be made. Such a response would be no more than a gloss on a hermeneutic project stemming from a preoccupation with meaning\u2014a preoccupation that leaves the object untouched. It would not pertain to the object and would not be defined in terms of the relation that a transformation of the object's status would have demanded. Rather, difficult as it may be, there is the need to hold to 'obscurity' ( _Dunkle_ ) and thus to the 'night' ( _Nacht_ ) (93\/97). Neither term diminishes the object; indeed, they get close to its truth. Neither terms obfuscate. They identify the condition of meaning. The night is the time of the always more, the time resisting an end, the time of a darkness in which the hand works endlessly. That on which the hand works cannot be mastered; the infinite stands in the way, stands almost as a between, such that the continuity of movement\u2014the operation of the blind hand\u2014uncovers more but preserves the site as one of continual uncovering. It is a continuity made possible by the 'soul', though only on the condition that the presence of the soul is defined in terms of an operative, thus workful, sense of internality.\n\nHerder's retention of the 'soul' locates his work, as has already been indicated, within the history of idealism. However, it is that history that has to be reconsidered. This location does not involve clear distinctions. Idealism is at its most productive when it can be understood as asserting, first, the irreducibility of material presence to empirical presence and, second, the co-presence of the immaterial and the material as constitutive of an object. Hence, there will be an important distinction between the ideal found, for example, in Plato and Goethe for whom that which is external to particulars ('forms' and ' _Ur-phenomena_ ') have a causal relation to the identity of particulars on the one hand and the forms of idealism at work in both Hegel and Herder on the other.[22] While it occurs in different ways, in both instances the 'ideal' is integral to the work itself.[23] In the case of Herder, to reiterate the general points noted previously, the irreducibility of matter to the empirical involves, in the first instance, the introduction of an immaterial presence as that which informs matter; and, second, the interplay between the material and the immaterial accounts for the way in which the infinite is at work within Herder's conception of the art object. Within such a setting idealism acquires a different character. It becomes an account of the interplay between mediacy and immateriality. Eschewed by such a setting is what may be described as the counter-positioning of the immediate and the empirical (which is itself a position that is inextricably bound up with what has already been identified as the literalization of the eye and the hand). In order to take this reworking of idealism a stage further, there needs to be a return to both 'spirit' and the 'soul'.\n\nThe inevitable concession concerning 'spirit' noted in Diderot reappears in Herder. Herder describes the presence of the soul in ways that escape any quick identification of the soul with a form of transcendence where transcendence assumes both externality and causality. The soul is defined in terms of life and the animate. Herder's formulation is both deliberate and careful and takes the particularity of sculpture as its point of departure. Herder's claim is that what 'sculpture should create . . . are forms in which the living soul pervades the entire body ( _einem edlen K\u00f6rper durchhaucht_ )' (45\/52).[24] As such, art has the 'task of representing the embodied soul' ( _Seele im K\u00f6rper dazustellen_ ) (45\/52). It is in relation to this setting of work that comes to situate beauty; for Herder, 'Beauty . . . is always only the shining through of form, the sensible expression of perfection' ( _Sch\u00f6nheit ist nur immer Durchschein Form, sinnlicher Ausdruck der Vollkommen_ ) (78\/83).\n\nTo begin to work through this complex set of relations, it is essential to note that what is central is Herder's insistence that the work of art is the work of matter\u2014hence, the work of art is its 'act'. Accepting the link between work and 'act' is to assume that the work of art _is_ \u2014'is' in the strong sense of exists. Thus, what is involved is a claim about the ontology of artwork\u2014in its being at work. Work therefore has an inherently actative dimension. To account for matter as work necessitates recourse to matter as dynamic. The work of matter is the ground of meaning. Attributions of meaning are therefore always the after-effect of the work of matter. Indeed, whether it is recognized or not, work is the precondition for the attribution of meaning. The impossibility of an identification of meaning and finality cannot be explained in terms of relativity on the level of meaning. The contrary is the case. Meaning's lack of finality has its conditions of existence in a conception of object defined by the 'unending'. This is Herder's point concerning the relationship between touch and the object: a relation in which there had been a transformation of its constitutive elements. As such, questions of interpretation have to be redefined in terms of the nature of the connection between the finitude of the interpretative decision and the work as defined by the potential infinite of matter, where matter is understood as always already the site of its own animation (matter as mattering).\n\nHerder's account of this set-up has recourse to both 'spirit' and the 'soul'. However, the presence of the soul is not to be accounted for in terms of the latter's presence within the object that then allows for the possibility that the soul could have an existence and thus a possible history that is external to its having been objectified. This is the difference between Hegel and Schelling on the one hand and Herder on the other. And yet, there are moments of equivocation. As part of a complex argument, sculpture is taken as inherently allegorical because it involves the presentation of 'one thing through another' ( _eins durch andere_ ) (96\/101). Indeed, Herder goes on to suggest that sculpture, as an instance of what he refers to as ' _das bildende Kunst_ ' (which are the visual arts in general), is in a state of what he describes as 'permanent allegory' precisely because sculptures 'image the soul through the body' ( _sie bildet Seele durch K\u00f6rper_ ) (96\/101).[25] This needs to be interpreted, however, in light of the passage already noted in which it was argued that the 'fingers' of 'inner sense' discovered the 'shape of the spirit within the form' ( _Gestalt des Geistes in deiser Form_ ) (90\/95). Both 'soul' and 'spirit' open up a concern with 'beauty'. There is an important set of interconnections. Soul pervades matter. It animates matter. Beauty becomes form's shining. Beauty is not the form; it is its mode of being present\u2014presenting as shining. Beauty is bound up with the inherently actative nature of matter. Beauty becomes a quality of the object, a quality that shines. Shining has a genuine accord with matter's animation and the soul's pervasive presence.\n\nWhile it is always possible to uncover those moments within _Plastik_ in which more conventional understandings of the role of both 'spirit and 'soul' can be discerned (hence the reference to Herder's equivocation), it is nonetheless still possible to insist on the centrality of the work of matter within Herder's own position. Insisting on that work involves a return to touch. Touch, while bringing the opposition between sensation and cognition into play, does so on the basis that the conventional determinations of that opposition no longer play a determining role within the subject\/object relation that characterizes every philosophical concern with the work of art. Another philosophical project is announced. As emerged from the way in which touch figured in the writings of Descartes, touch has a potentiality that could not have been stemmed by the move to the literalization of the either the hand or the eye. Once touch can be incorporated in ways that reposition both the nature of the subject and the object, as well as the content of their relation, a transformation in which references to the 'soul' moves from the subject to the object, there is an accompanying transformation of the nature of the object.[26] The object becomes that which, in resisting its reduction to the literal, acquires a quality that announces the reduction's impossibility. Touch, in introducing a temporal dimension that is not circumscribed by immediacy and in yielding a conception of the object in which its mattering is central, therefore stages different dimensions of the structure of thought that are essential to the development of a materialist philosophy of art.\n\n## Notes\n\n1.\n\nJohn Poulakos, 'From the Depths of Rhetoric: The Emergence of Aesthetics as a Discipline', _Philosophy and Rhetoric_ 40, no. 4 (2007), has provided an excellent survey on the relationship between aesthetics and rhetoric, which indicates that what is at stake within the emergence of the aesthetic can be characterized in terms of an engagement with a founding instability.\n\n2.\n\nReferences to Herder will be to _Sculpture_ , trans. and ed. Jason Gaiger (Chicago: University of Chicago, 2002). The German edition is J. G. Herder, _Plastik_ (Whitefish, MT: Kessinger Reprints). This edition is a facsimile edition of the 1778 version published in Riga. References will be to the English then the German pagination and given in the body of the text. It should be noted that Gaiger's superb translation and commentary will hopefully play a central role in providing a place for Herder in English discussions of sculpture in particular and the philosophy of art in general. For a recent discussion of Herder's writings on sculpture within the history of a more general concern with sculpture, see Nancy M. Goslee, 'From Marble to Living Form: Sculpture as Art and Analogue from the Renaissance to Blake', _Journal of English and Germanic Philology_ 77, no. 2 (April 1978). See in particular pages 201\u2013203.\n\n3.\n\nThere is therefore an affinity between this project and the work of John Sallis. Sallis is also concerned with rethinking the project of a philosophy of art beyond the structures of the opposition between the sensible and the intelligible. See in this regard his _Transfigurements: On the True Sense of Art_ (Chicago: University of Chicago Press, 2008), 74\u2013105. Nonetheless, the argument developed in this interpretation of Herder goes in a different direction.\n\n4.\n\nRobert Hopkins has provided an important philosophical engagement with the relationship between sight and touch. See his 'Painting, Sculpture, Sight and Touch', _British Journal of Aesthetics_ 44, no. 2 (April 2004). Nonetheless, despite the sensitivity of Hopkins, his analysis remains essentially limited, because he fails to note that what touch opens up\u2014once it is no longer taken as identical with literal touch\u2014is a possible transformation of what is understood by seeing and thus touching.\n\n5.\n\nDenis Diderot, 'Lettre sur les aveugles', in _Oeuvres philosophiques_ (Paris: Classiques Garnier, 1992), 116. All further references are given in the body of the text.\n\n6.\n\nAll references to Descartes are to _Oeuvres de Descartes_ , ed. Charles Adam and Paul Tannery (Paris: Libraire Philosophique J. Vrin, 1996). References will take the following form: AT followed by volume and page number and given in the body of the text. In those instances in which the original text was written in Latin, the reference is to the Latin and then the subsequent French translation. In the case of the French translation of _Meditations_ , it can be argued that this was a translation approved by Descartes and is thus another version of that text.\n\n7.\n\nTo this end, see the detailed definition of 'clear and distinct perception' in _Principiorum Philosophiae_ \u00a745, AT VIII, 21\u201322.\n\n8.\n\nWhile its detail cannot be taken up here, it is essential within the development of a materialist philosophy of art to distinguish between materialism and empiricism. I have begun such an undertaking in my _Disclosing Spaces: On Painting_ (Manchester: Clinamen, 2004). See also my 'On Abstraction: Notes on Mondrian and Hegel', in _Paradoxes of Appearing: Essays on Art, Architecture and Philosophy_ , ed. Michael Asgaard Andersen and Henrik Oxvig (Baden: Lars M\u00fcller, 2009), as well as chapter 6 of this volume, 'Colouring Philosophy: Appel, Lyotard and Art's Work'.\n\n9.\n\nThis of course mirrors the argument that is worked out within _Meditations_ concerning the identity of self as ' _res cogitans_ '. The only way Descartes can solve the problem of what happens to the existence of the subject when it is not thinking\u2014this is, after all, the problem inherent in locating the identity of the self within the formulation _cogito ergo sum_ \u2014is to assume that the ontologico-temporal nature of the 'I' that is positioned within specific acts of thought\u2014acts with finite duration and content\u2014is radically distinct from a conception of the subject that is enduring and is thus _res cogitans_. It is not as though the subject as _res cogitans_ could be anything other than that which is defined by thinking (cognition).\n\n10.\n\nAT VII, 144\/AT IX, 113 (The Latin in this instance is ' _nihil est quod ulterius inquiramus_ '.)\n\n11.\n\nAT VI, 21. In this regard, Descartes wrote that ' _n'yanat qu'une verit\u00e9 de chaque chose, quiconque la trouve sait autant qu'on peut savoir_ '. At work here is a conception of object ('thing', ' _la chose_ ') that allows for its complete subordination to the projects of both epistemology and representation. While the conception of the object that is implicitly at work within Herder's _Plastik_ is radically different, a critique of this aspect of Descartes's philosophical undertaking finds its most eloquent expression in Batailles's discussion of Descartes in the former's 'L'exp\u00e9rience int\u00e9rieur'. See in this regard G. Batailles, _Oeuvres compl\u00e8tes_ (Paris: Tome V. Gallimard, 1973). Specifically, 123\u2013126.\n\n12.\n\nSee in this regard the definition of the sign in Arnauld and Nicole's _La logique du Port Royal_ (Paris: Flammarion, 1970): 'Le signe enferme deux id\u00e9es: l'une de la chose qui repr\u00e9sente; l'autre de la chose repr\u00e9sent\u00e9e; et sa nature consiste \u00e0 exciter la seconde par la premiere' (8). See, in addition, Foucault's discussion of this structure in his _Les mots et le choses_ (Paris: \u00c9ditions Gallimard, 1966), 78\u201381.\n\n13.\n\nWhile it cannot be argued for in this context, the essential difference between Kant on the one hand and Herder and Diderot on the other is the latter's concern with the nature of the object and not with the subject's response to the presentation of an object. In regards to the specific claim concerning immediacy, this can be understood in terms of the absence of the concept\u2014where the concept's presence would be understood as that which mediated (determined) in advance claims about the object's beauty. To this end, see the argumentation of \u00a79 of _Critique of the Aesthetic Power of Judgment_ , also the claim made in \u00a76 that 'the person making the judgment feels himself to be completely free ( _frei_ ) with regard to the satisfaction he devotes to the object'. The state of being 'free' is another instance of a structure of immediacy. References to Kant are to _Critique of the Aesthetic Power of Judgment_ , trans. Paul Guyer and Eric Matthews (Cambridge: Cambridge University Press, 2000) and _Kritik der Urteilskraft_ (Hamburg: Felix Neiner Verlag, 2001).\n\n14.\n\nDiderot, _Oeuvres esth\u00e9tiques_ (Paris: Classiques Garniers, 1994), 495.\n\n15.\n\nHence, there is a need to distinguish the position outlined here from the one presented by Kant in \u00a715 of the _Critique of the Aesthetic Power of Judgment._ In that section it is clear that a concern with the object is a philosophical impossibility for a judgement of taste. Judgements that evince such concerns\u2014judgements in which 'charm' and 'emotion' play a role (cf. \u00a713)\u2014are judgements that allow for the object and are correspondingly not 'pure judgments of taste'. Impossibility in this context has to be understood within terms set by the relationship between the subject and the form of the object. Kant's position is presented in the following formulation of the aesthetic judgement: 'An aesthetic judgment is of a unique kind and affords absolutely no cognition ( _kein Erkenntnis_ ) (not even a confused one) of the object, which happens only in a logical judgment; while the former by contrast, relates, the representation by which an object is given solely to the subject, and does not bring our attention to any property of the object, but only the purposive form in the determination of the powers of representation that are occupied with it' (113\/228).\n\n16.\n\nPlato, _Phaedo_ , 100D.\n\n17.\n\nIn this regard, see my 'A Missed Encounter: Plato's Socrates and Geach's Euthyphro', _Grazer Philosophische Studien_ 29 (1987).\n\n18.\n\nFor two instances of the many ways in which the history of touch and the history of the hand can be deployed, see Sander Gilman, 'Touch, Sexuality and Disease', in _Medicine and the Five Senses_ , ed. W. F. Bynun and Roy Porter (Cambridge: Cambridge University Press, 1993), 198\u2013225; and the collection _Manus Loquens: Medium der Geste\u2014Geste der Medien_ , ed. Matthias Bickenbach, Annina Klappert and Hedwig Pompe (K\u00f6ln: Dumont Buchverlag, 2003). Moreover, the history of both touch and the hand cannot escape having to consider the injunction within the history of Christianity (i.e., Christ's words to Mary Magdalene: ' _Noli me tangere_ ' [John 20:17]). The Latin invokes touch in a far more emphatic way than the original Greek. It is the Latin that had the determining effect on the history of art. Diderot and Herder can therefore be seen as attempts to 'desacralize' the art work through their insistence on a transformation of touch.\n\n19.\n\nI have taken up the question of the surface's problematization as it occurs in Walter Benjamin's early writings on art in my 'Framing Pictures, Transcending Marks: Walter Benjamin's \"Paintings, or Signs and Marks\"', in _Walter Benjamin and the Architecture of Modernity_ , ed. Andrew Benjamin and Charles Rice (Melbourne: Re.press Books, 2009).\n\n20.\n\nI have, in collaboration with Howard Caygill, as part of a more general research project on the concept of force, edited an issue of _Parallax_ , whose papers offer the beginnings of a sustained repositioning and thus reworking of force. See _Parallax_ 15, no. 51 (2009). In addition, see our introduction to this issue, 'Of Force', 1\u20134.While it does not form part of the above project, it is clear that recent work by Christophe Menke is also involved in a rethinking of the concept of force and thus by extension of what has been described here as the immaterial. See his 'Force: Towards an Aesthetic Concept of Life', _MLN_ 125, no. 3 (April 2010) (German Issue). Fundamental to Menke's position is the following contention: 'The aesthetic force is an endless generation and dissolution of expressions, an endless transformation of expression into different expression. The operation of the aesthetic force thus consists at any moment in transcending what it has itself engendered. The aesthetic force creates new expression by withdrawing from its previous expression' (567).\n\n21.\n\nA further explication of what is meant by mattering is an ongoing project. See chapter 6 of this volume, 'Colouring Philosophy: Appel, Lyotard and Art's Work'. Here, as part of a general engagement with the question of colour, I argue that\n\n> the ontology of artwork (i.e., the being proper to art) is defined in terms of activity. The work of art works. It is a dynamic process that always has its own complex economies. The second is that meaning is always, and only, an after effect of the way matters works. As such, the working of matter is the precondition for the possibility of meaning. This aspect of a work can be understood as its mattering. Mattering as the work of matter is, within a materialist aesthetics, the _conditio sine qua non_ for attributions of meaning. Moreover, to insist on mattering is both to maintain the centrality of matter thought as a locus of activity rather than a merely static event while at the same time holding to the necessity of the particularity of a given work. The mattering of one work is always distinct from that of another.\n\n22.\n\nWalter Benjamin's treatment of this aspect of Goethe's conception of the artwork, precisely because of the externality of the 'Ur-phenomena', leads Benjamin to conclude that such a conception makes art uncriticizable. And yet, the move to criticism necessitates allowing the work a quality that cannot be equated with either description or commentary. The work of art will always be more. The limit of Goethe for Benjamin is clear. In his formulation for Goethe, art 'itself does not create the archetypes they rest prior to all created work, in that sphere of art where is not creation but nature' (180). See in this regard Benjamin's 'The Concept of Criticism in German Romanticism', in _Selected Writings_ , vol. 1., ed. Marcus Bullock and Michael W. Jennings (Cambridge, MA: Harvard University Press, 1996). See in particular 180\u2013182.\n\n23.\n\nThe relationship between Hegel and Herder is an extremely fruitful area of inquiry. What matters is how the question of 'soul' is to be understood. In this regard, see Linda Walsh, 'The \"Hard Form\" of Sculpture: Marble, Matter and Spirit in European Sculpture from the Enlightenment through Romanticism', _Modern Intellectual History_ 5, no. 3 (2008). For the discussion of the relationship between Hegel and Herder, see especially pages 479\u2013480.For an important statement of how that relation might be understood, see Kristin Gjesdal, 'Hegel and Herder on Art History and Reason', _Philosophy and Literature_ , no. 30 (2006). Central to Gjesdal's characterization of the difference is the way their differing philosophies of history\u2014the absence of teleology in Herder\u2014means that works would have a different sense of expression (expression in relation to what has been called in the argument of this paper the 'immaterial'), and thus there would be a historically determined hierarchy within artworks.\n\n> The difference between Herder and Hegel is, in other words, not only that Herder does not turn Greek art into the ideal of art as such, but also that he takes no art-form to be the privileged or paradigmatic expression within the boundaries of Greek culture. There are no normative hierarchies among the periods, nor among the aesthetic forms of expression within a given, historical paradigm. No type of art manifests the essence of art as such, and to perceive tragedy as less perfect an art-form than sculpture (yet as more attuned to modern spirit) would simply not be an option for Herder. (26)\n\n24.\n\nThe translation in this instance has been slightly modified.\n\n25.\n\nThe translation in this instance has been slightly modified.\n\n26.\n\nRachel Zuckert, 'Sculpture and Touch: Herder's Aesthetics of Sculpture', _Journal of Aesthetics and Art Criticism_ 67, no. 3 (Summer 2009). 'Imaginative proprioceptive identification allows us, Herder suggests, to understand the expressiveness of the sculpture, the way in which the comportment of the sculpture can express fitness, health, strength, well-being, emotional states, or tendencies to action, that is, the way in which this (represented) body is \"ensouled\", \"alive\", active' (285). She goes on to argue that this position can be understood in terms of a new conception of embodiment. The contention here is that it is the transformation of the conception of seeing that allows for the already present sense of embodied subjectivity. \n\n# Bibliography\n\nAdorno, Theodor W. _Minima Moralia_. 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'From Symbol to Allegory: Aby Warburg's Theory of Art'. _Art Bulletin_ 79, no. 1 (March 1997): 41\u201355.\n\nRapaport, Herman. _Later Derrida: Reading the Recent Work_. London: Routledge, 2003.Richter, Gerhard. _Afterness: Figures of Following in Modern Thought and Aesthetics_. New York: Columbia University Press, 2011.\n\nRiegel, Alois. 'The Leading Characteristics of Late Roman Kunstwollen'. In _The Art of Art_ _History: A Critical Anthology_. Edited by Donald Preziosi. Oxford: Oxford University Press, 2009.\n\nRiess, Jonathan B. _The Renaissance Antichrist: Luca Signorelli's Orvieto Frescoes_. Princeton, NJ: Princeton University Press, 1995.\n\nRutter, Benjamin. _Hegel on the Modern Arts_. Cambridge: Cambridge University Press, 2010.\n\nSallis, John. _Shades\u2014of Painting at the Limit_. Bloomington: Indiana University Press, 1998.\n\n\u2014\u2014\u2014. _Transfigurements: On the True Sense of Art_. Chicago: University of Chicago Press, 2008.\n\nSantner, Eric. _On Creaturely Life: Rilke, Benjamin, Sebald._ Chicago: University of Chicago Press, 2006.\n\nSchelling, F. W. J. 'Philosophie der Kunst'. In _Ausgew\u00e4lte Schriften_. Band 2. Frankfurt: Surhkamp, 1985.\n\n\u2014\u2014\u2014. _The Philosophy of Art_. Translated and edited by Douglas W. Scott. Minneapolis: University of Minnesota Press, 1989.\n\nSchlegel, Friedrich. _Studienausgabe_. Band 2. Paderborn: Ferdinard Sch\u00f6ningh, 1988.\n\nSchlitt, Melinda. 'Painting, Criticism, and Michelangelo's _Last Judgment_ in the Age of the Counter Reformation'. In _Michelangelo's 'Last Judgment'_. Edited by Marcia Hall. Cambridge: Cambridge University, 2005.\n\nSchoell-Glass, Charlotte. 'Aby Warburg's Late Comments on Symbol and Ritual'. _Science in Context_ 12, no. 4 (December 1999): 621\u2013642.\n\nSchufreider, Gregory. 'The Art of Truth'. _Research in Phenomenology_ 40 (2010): 331\u201362.\n\nShakespeare. 'Hamlet'. _Dramatische Werke_ , Nach der Ubers. Von August Wilhelm Schlegel u. Ludwig Tieck, sorgf\u00e4ltig revidirt u. theilweise neu bearbeitet, mit Einleitungen u. Noten versechen, unter Redaction von Hermann Ulrici, durch die Deutsche Shakespeare Gesellschaft, VI. Berlin: G. Reimer, 1877.\n\nSkidelsky, Edward. _Ernst Cassirer: The Last Philosopher of Culture_. Princeton, NJ: Princeton University Press, 2011.\n\nSlaughter, Marty. 'The Arc and the Zip: Deleuze and Lyotard on Art'. _Law and Critique_ 15, no. 3 (January 2004): 231\u2013257.\n\nSofer, Andrew. 'Felt Absences: The Stage Properties of _Othello_ 's Handkerchief'. _Comparative Drama_ 31, no. 3 (Fall 1997): 367\u2013393.\n\nSohm, Philip L. 'D\u00fcrer's _Melencolia I_ : The Limits of Knowledge'. _Studies in the History of Art_ 9 (1980): 13\u201332.\n\n\u2014\u2014\u2014. 'Ronsard's Odes as a Source for Poussin's _Aurora and Cephalus_ '. _Journal of the Warburg and Courtauld Institutes_ 49 (1986): 259\u2013261.\n\n\u2014\u2014\u2014. _Style in the Art Theory of Early Modern Italy_. Cambridge: Cambridge University Press, 2001.\n\nStoichita, Victor I. _The Self-Aware Image. An Insight into Early Modern Meta-Paintings_. Cambridge: Cambridge University Press,1977.\n\n\u2014\u2014\u2014. _Short History of the Shadow_. London: Reaktion Books, 1997.\n\nSullivan, Margaret A. 'The Witches of D\u00fcrer and Hans Baldung Grien'. _Renaissance Quarterly_ 53, no. 2 (Summer 2000): 333\u2013401.\n\nSummers, David. _Vision, Reflection and Desire in Western Painting_. Chapel Hill: University of North Carolina Press, 2007.\n\nThein, Karel. 'Gods and Painters: Philostratus the Elder, Stoic Phantasia and the Strategy of Describing'. _Ramus_ 31, nos. 1\u20132 (2002): 136\u2013145.\n\nVandenabeele, Bart. 'On the Notion of \"Disinterestedness\": Kant, Lyotard, and Schopenhauer'. _Journal of the History of Ideas_ 62, no. 4 (2001): 705\u2013720.\n\nVerstegen, Ian. 'Between Presence and Perspective: The Portrait-in-a-Picture in Early Modern Painting'. _Zeitschrift f\u00fcr Kunstgeschichte_ 71, no. 4 (2008): 513\u2013526.\n\nVischer, Friedrich Theodor. _Ausgew\u00e4hlte_ _Werke_. Vol. 8. Anstall: Deutsche Verlags, 1918.\n\nWalsh, Linda. 'The \"Hard Form\" of Sculpture: Marble, Matter and Spirit in European Sculpture from the Enlightenment through Romanticism'. _Modern Intellectual History_ 5, no. 3 (2008): 455\u2013486.\n\nWarburg, Aby. _Der Bilderatlas Mnemosyne_. Berlin: Akademie Verlag, 2003.\n\n\u2014\u2014\u2014. 'Die Bilder ausstellungen des Volksheims'. In _Gesammelte Schriften_. Band 2. Leipzig: B.G. Teuuber, 1932.\n\n\u2014\u2014\u2014. 'Giordano Bruno'. Edited by Maurizio Ghelardi and Giovanna Targia, _Cassirer Studies_ 1 (2008): 13\u201358.\n\n\u2014\u2014\u2014. _The Renewal of Pagan Antiquity_. Translated by David Britt. Los Angeles: Getty Research Institute, 1999.\n\n\u2014\u2014\u2014. _Tagebuch der Kulturwissenschadtlichen Bibliothek Warburg_. Berlin: Akademie Verlag, 2001.\n\n\u2014\u2014\u2014. _Werke in Einem Band_. Frankfurt: Suhrkamp, 2010.Warwick, Genevieve. 'Bernini's _Apollo and Daphne_ at the Villa Borghese'. _Art History_ 27, no. 3 (June 2004): 353\u2013381.\n\nWeigel, Sigrid. 'Aby Warburg's Schlangenritual: Reading Culture and Reading'. _New German Critique_ , no. 65, Cultural History\/Cultural Studies (Spring\u2013Summer, 1995): 135\u2013153.\n\nWigley, Mark. White Walls , _Designer Dresses: The Fashioning of Modern Architecture_. Boston: MIT Press, 2001.\n\nWilliam, Robert. _Art, Theory and Culture in Sixteenth-Century Italy_ : _From Techne to Metatechne_. Cambridge: Cambridge University Press, 1997.\n\nWind, Edgar. _Art and Anarchy_. London: Duckworth, 1985.\n\n\u2014\u2014\u2014. 'Theory of Art versus Aesthetics'. _Philosophical Review_ 34, no. 4 (July 1925): 350\u2013359.\n\n\u2014\u2014\u2014. 'Warburg's Concept of Kulturwissenschaft and Its Meaning for Aesthetics'. In _The Eloquence of Symbols_. Oxford: Clarendon, 1993.\n\nWittgenstein, Ludwig. _Remarks on Colours_. Translated by L. McAlister and M. Schaettle. Edited by G. E. M. Anscombe. Oxford: Blackwell, 1977.\n\nWood, Christopher. 'Painting and Plurality'. _Yearbook of Comparative Literature_ 56 (2013): 116\u2013139.\n\nXenophon, _Symposium_ III.6. In _Xenophon_. Vol. 4. Translated by O. J. Todd. Cambridge, MA: Harvard University Press, 1974.\n\nZiarek, Krzysztof. _The Force of Art_. Stanford, CA: Stanford University Press, 2004.\n\nZorach, Rebecca. 'Love, Truth, Orthodoxy, Reticence; or, What Edgar Wind Didn't See in Botticelli's Primavera'. _Critical Inquiry_ 34 (Autumn 2007): 190\u2013224.\n\nZuckert, Rachel . 'Sculpture and Touch: Herder's Aesthetics of Sculpture'. _Journal of Aesthetics and Art Criticism_ 67, no. 3 (Summer 2009): 285\u2013299. \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\nTHIS IS A BORZOI BOOK PUBLISHED BY ALFRED A. KNOPF\n\nCopyright \u00a9 2010 by Louise Yates  \nAll rights reserved. Published in the United States by Alfred A. Knopf, an imprint of Random House Children's Books, a division of Random House, Inc., New York.  \nOriginally published in 2010 in Great Britain by Jonathan Cape, an imprint of Random House Children's Books.  \nKnopf, Borzoi Books, and the colophon are registered trademarks of Random House, Inc.\n\nVisit us on the Web\n\nEducators and librarians, for a variety of teaching tools, visit us at www.randomhouse.com\/teachers\n\n_Library of Congress Cataloging-in-Publication Data_  \nYates, Louise.  \nDog loves books \/ Louise Yates. \u2014 1st American ed.  \np. cm.  \nOriginally published: Great Britain : Jonathan Cape, 2010.  \nSummary: Dog loves books so much that he decides to open a book store.  \neISBN: 978-0-375-98479-2  \n[1. Dogs\u2014Fiction. 2. Books and reading\u2014Fiction.] I. Title.  \nPZ7.Y276Do 2010  \n[E]\u2014dc22  \n2009011097\n\nThe illustrations in this book were created using pencil and watercolor.\n\nRandom House Children's Books supports the First Amendment and celebrates the right to read.\n\nv3.1\n\n_For Eleanor and Cedric_\n\n# Contents\n\nCover\n\nTitle Page\n\nCopyright\n\nDedication\n\nFirst Page\n\nAbout the Author\n\nDog loved books.\n\nHe loved the smell of them,\n\nand he loved the feel of them.\n\nHe loved everything about them....\n\nDog loved books so much\n\nthat he decided to open his own bookstore.\n\nHe unwrapped,\n\nunpacked,\n\nand stacked the books,\n\nready for the Grand Opening.\n\nWhen the day of the Grand Opening finally came,\n\nDog had a bath,\n\ndried his fur,\n\nblew his nose,\n\nand threw open the door\n\nto greet his new customers.\n\nBut there was no one there.\n\nSo Dog tried to keep busy.\n\nAnd then...\n\na lady came into the store.\n\n\"I'll have a tea with milk and two sugars,\"  \nshe said.\n\n\"I'm sorry,\" said Dog, \"But this is a bookstore. I only sell books.\"\n\nThe lady walked out.\n\nDog was alone.\n\nHe waited and waited.\n\nThen a man came into the store...\n\nto ask for directions.\n\nWhen he left, Dog was downhearted.\n\nBut not for long!\n\nHe wouldn't wait a moment more.\n\nDog fetched a book from the shelf  \nand began to read.\n\nWhen he read, he forgot that he was waiting.\n\nWhen he read, he forgot that he was alone.\n\nWhen he read, he forgot that he  \nwas in the bookstore.\n\nAnd when one adventure ended,\n\nDog simply took another book  \ndown from the shelf and...\n\na new adventure began!\n\nSo Dog was somewhere else altogether when...\n\na customer came into the store to ask for a book.\n\nDog knew exactly which ones to recommend.\n\nbut most of all...\n\nhe loves to share them!\nLouise Yates began drawing pictures to go with the stories she wrote for school and, at a young age, began telling people that she wanted to be a children's book illustrator. \"One of the things I love most about picture books is the silences, the moments when the text shuts up and the pictures either tell you something that the text hasn't or something totally different,\" she writes.\n\nHer first picture book, _A Small Surprise_ , is an inspired example of those moments of silence when the pictures have their say. In its rave review of _A Small Surprise_ , _Kirkus Reviews_ predicted that it \"will be sure to have readers in stitches.... Yates's debut is one of those books that grows on readers with repetition. Small children who feel they aren't big enough to do anything will appreciate the message, while their adults might be inspired to look for the hidden talents.\"\n\nLouise Yates was born in England and lives in London.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nCopyright \u00a9 2018 DC Comics.\n\nDC SUPER HERO GIRLS and all related characters and elements \u00a9 & TM DC Comics and Warner Bros. Entertainment Inc.\n\nWB SHIELD: TM & \u00a9 WBEI. (s18)\n\nRHUS 39186\n\nAll rights reserved. Published in the United States by Random House Children's Books, a division of Penguin Random House LLC, New York, and in Canada by Penguin Random House Canada Limited, Toronto. Random House and the colophon are registered trademarks of Penguin Random House LLC.\n\nVisit us on the Web!\n\nrhcbooks.com\n\ndcsuperherogirls.com\n\ndckids.com\n\nHardcover ISBN 9781524769239 \u2014 Ebook ISBN 9781524769253\n\nv5.1\n\na\nFor Rob, who makes me laugh\nContents\n\n_Cover_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\nPrologue\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\nEpilogue\n\n_About the Author_\n\nThat the teenaged sword-wielding hero Katana had just been named Super Hero of the Month was no surprise. After all, she had solved the mystery surrounding the death of her grandmother, Onna-bugeisha Yamashiro\u2014the first female super hero samurai. Plus, she had set things right for the mysterious Ghost Crabs that had invaded Super Hero High _and_ saved the world from the evil villain Dragon King and his army of mutated reptilian warriors.\n\nSo what did all that mean for the students at Super Hero High?\n\nIt was time for cake!\n\nThere was always a celebration when Super Hero of the Month was announced, and Bumblebee had planned a little surprise for Katana.\n\n\"How did you know it would be me?\" Katana asked. Her straight black hair glistened and her big brown eyes were bright. There was a rosy flush to her pale complexion from all the attention.\n\nOther students, like the irrepressible jokester Harley Quinn, loved being in the spotlight, and even sought it out. But not Katana. To her it was just something to get used to, like fighting interplanetary villains or getting her homework in on time.\n\n\"Oh, I had a hunch,\" a tiny voice said from beneath the lemon butter cake that was flying across the room. The cake boasted a citrus sugar glaze and the words Congrats, Katana! written in royal red icing.\n\nFrom underneath the cake, Bumblebee grew from bee-sized to girl-sized as she set it down. Thick honey-colored streaks accentuated her rich brown curly hair, and even though her yellow insect wings may have looked delicate, they were anything but. Having delivered the cake, she treated herself to a small taste of the icing with her finger. She looked around the room as the party took shape.\n\nSupergirl had snuck in and decorated Katana's dorm room with colorful balloons and streamers, and Poison Ivy had filled it with Katana's favorite Japanese snowball flowers.\n\n\"Achoo! Achooo! Achooooo!\"\n\nNear Katana's display of swords someone was sneezing loudly. \"Achooooosie woozie!\" the girl in the blue shorts and mismatched tights said with gusto.\n\n\"Is the word _subtle_ in her vocabulary?\" Cheetah asked as she waited impatiently for the cake to be cut.\n\n\"Isn't anyone going to say gesundheit?\" the girl in the blue shorts asked.\n\n\"Gesundheit,\" said several voices around the room.\n\nThe black mask over her eyes could not hide her look of mischief. \" _YOWZERS!_ Those are some mighty flowers,\" she continued. \"They're killing my allergies. Killing 'em, I tell you.\" The girl was Harley Quinn. She might have seemed unfocused, but she was looking at one person in particular.\n\n\"Hey, over here! Katana, am I going to score this interview or am I going to score this interview?\" Harley quipped. She somersaulted through the partygoers. Her blond pigtails, streaked with red and blue, kept bouncing even when she stood still.\n\n\"I need to do this first,\" Katana said. She picked up the cake and threw it. Before it smashed into the ceiling, Katana unsheathed her sword, leapt, and in midair, expertly cut it into three dozen pieces. Each identically sized slice landed on one of the plates lined up in rows on the table.\n\n\"I love cake!\" Big Barda announced as she took the first piece. She removed her heavy gold-and-black helmet and set it aside. Cake was serious business.\n\nHarley gave Katana a big impatient smile. The young samurai knew that there was no stopping Harley when she wanted something. And right now Harley wanted one of her famous \"exclusively exclusive Harley Quinn's Harley's Quinntessentials scoops\"!\n\n\"Your fave hostess with the mostest, Harley Quinn, here,\" the Web star said, beaming into the camera.\n\nHarley took a large bite of cake, leaving some frosting on her nose. When Katana pointed this out, Harley thanked her, added more frosting, and waggled her eyebrows at her digital audience.\n\n\"Just clowning around,\" she told her viewers. \"But this is serious stuff now. Seriously awesome! I've got Katana in person to tell us how she felt when she heard she was Super Hero High's Super Hero of the Month.\"\n\nKatana blinked into the camera. Not everyone was as comfortable as Harley was at being broadcast to thousands of viewers. Harley's Web channel had been a hit since its debut. There was no telling how many viewers she could get.\n\n\"It feels pretty great,\" Katana said. \"But it was a team effort\u2014\"\n\n\"Aww, she's too modest!\" Harley interrupted, turning the camera on herself and reenacting Katana's martial arts moves. \"Boom! Boom! Blam! Blam! Slice and dice! You shoulda seen this girl wielding her sword. It was like she was holding a bolt of lightning! Speaking of lightning, let's chat with everyone's fav-o-rite weather-themed super hero sisters, Thunder and Lightning.\"\n\nBefore Thunder could answer, Harley dashed off into the crowd. Thunder stood confused, and tugged on the V-neck collar of her sleek black-and-electric-yellow costume. The yellow barrettes that held her bangs in place matched her boots. Her sister, Lightning, crouched down to tie her high-top sneakers. Her sporty clothes made her look like she was always ready for action.\n\n\"That was weird,\" Lightning said. \"Harley was just talking to us and now she's gone...again.\"\n\n\" _She's_ weird,\" Thunder said, quickly adding, \"But I mean that in the best way!\"\n\nJust then, everyone heard a familiar voice. Only it didn't sound normal.\n\n\"Help!\" Harley was crying. \"Someone help me!\"\n\n\"Harley's such a crack-up,\" Cheetah said, rolling her eyes. \"Class clowns are like that.\" Supergirl, hovering nearby, was carrying all the empty plates in two five-foot stacks. She frowned at her feline classmate.\n\n\"We should check this out,\" Wonder Woman announced, always attuned to possible trouble. She tossed her plate so it landed on Supergirl's stack, then adjusted the golden cuffs on her wrists. She motioned the others to follow her as she tracked the sound of Harley's voice.\n\n\"Help!\"\n\nEveryone could hear Harley still yelling. But as they turned the corner, Harley was nowhere to be seen. Instead, her ever-present video camera was on the floor.\n\nBumblebee ran to pick it up. She looked at the video screen. On it was Harley yelling \"Help! Someone help me!\" Then the screen went dark.\n\n\"This doesn't look good,\" Bumblebee said.\n\nBatgirl watched over Bumblebee's shoulder as Bumblebee replayed the video. \"Harley's in trouble!\" Batgirl said. \"She would never be without her camera.\"\n\nWonder Woman looked perplexed. \"I don't see anything,\" she said, staring at the blackness on the screen.\n\n\"Exactly,\" said Batgirl. \"But when I hit 'Playback,' you can hear her. Listen.\"\n\nSure enough, it was the first sound of Harley crying for help that they had all heard.\n\nInstantly, the Supers snapped into action.\n\n\"I'll check the school...,\" The Flash said, racing away at super-speed before he had even finished his sentence.\n\n\"I'll search the underground caverns,\" Katana volunteered.\n\n\"I'll inform Principal Waller,\" Hawkgirl announced.\n\n\"Wait!\" Batgirl called out.\n\nThose who remained crammed around the small video screen. They gasped when Harley's face appeared. She was laughing. \"Gotcha!\" Harley said. \"I'm okay, it was all just a joke! I'm fine. I...\"\n\nJust then, a menacing shadow came up behind her, and suddenly the screen went dark and silent.\n\nAs Supers fanned out over the school and across Metropolis to save their friend, Harley wasn't far from where she had disappeared. The truth was, she wasn't even in danger. Instead, Harley was laughing from the top of the school's iconic Crystal Tower.\n\n\"This is gonna make great entertainment!\" Harley said. She had a habit of talking to herself. That way, it always seemed like someone was with her. Having planned ahead, Harley had placed remote cameras around campus. \"Everyone's going to tune in!\" she told herself. \"And then when I reveal my hiding\u2014\"\n\n\"You are in so much trouble!\" a familiar voice growled. Harley looked up, surprised to find Cheetah glaring down at her with her hands on her hips. Cheetah's long brown hair whipped behind her.\n\n\"Harley, you forgot I have super-vision,\" Supergirl said, landing from above and making sure not to step on Cheetah's tail with her red high-top sneakers. \"I spotted you as soon as I flew above campus and looked back.\"\n\nSupergirl yelled down to Beast Boy, who was slithering around in the form of a green snake, checking out low, hard-to-get-to areas. \"Alert the Supers on the ground that Harley has been found and she's safe! I'll tell the ones in the sky,\" she said, leaping off the tower and flying away.\n\nBeast Boy turned back into his scruffy green teen self for a moment before changing into a small coqui frog. Despite his now-minuscule size, his voice could generate sounds as high as one hundred decibels\u2014as loud as a jackhammer. He croaked, \"Harley Quinn has been found. Harley Quinn is safe. Harley Quinn has been found. Harley Quinn is safe. Harley Quinn has been found....\"\n\nExcept for her light blue eyes narrowing with disapproval, Cheetah hadn't moved.\n\n\"What?\" Harley asked, feigning innocence. \"Would you like to say a few words to our Harley's Quinntessentials viewers?\"\n\n\"Put the camera down,\" Cheetah said. \"Everyone's looking for you. They think you're in danger.\"\n\n\"I was,\" Harley said, laughing. She loved to laugh. She laughed at everything really. \"I was in danger of not having enough viewers. But now, everyone will tune in to my new 'Find Harley' segment! I was secretly videoing your reactions, and they were priceless. Score one for Harley!\"\n\n\"Score zero,\" Cheetah said. \"I'm going to tell Principal Waller that it was all just a joke from our resident class clown. Then we'll see who's laughing.\"\n\n\"No one is laughing,\" Principal Waller was saying, her back to Harley Quinn. Harley could see in the reflection in the office window that the principal's normally stern face looked even more serious. Harley didn't even think that was possible. The Wall, as Principal Amanda Waller was often called\u2014though never to her face\u2014had a look that would make Ares, the god of war and Wonder Woman's half-brother, cry.\n\nHarley squirmed on the wooden chair across from Waller's desk. \"Put the ice blaster back,\" the principal ordered, without looking. \"I've asked you before not to touch things on my desk.\"\n\n\"Yes, Principal Waller,\" Harley said, quickly placing it back on top of the pile of confiscated weapons.\n\nAll weapons were supposed to be checked in and assigned tracking numbers, but students were always forgetting this formality. Well-meaning parents sent swords, chemicals, lasers, and other items that they thought their kids might find useful at school. And some students, like Poison Ivy and Cyborg, were always coming up with new ways to weaponize everyday objects for Lucius Fox's Weaponomics class. That was fine for class, but it wasn't something Waller could allow in the common areas.\n\n\"That little stunt you pulled, about going missing. What do you have to say for yourself?\" Principal Waller demanded.\n\n\"It was just for giggles...and a twenty-three percent increase in viewership,\" Harley answered innocently. \"And math! I'm using what I learned in class and applying it to my channel.\" She aimed her infectious grin at the principal. When Waller didn't respond, Harley ramped up her smile. After years heading up Super Hero High, the principal was immune to the charms of her students, even ones as lovable as Harley Quinn.\n\n\"I see,\" said Waller. \"How about this, then: zero new Harley's Quinntessentials videos plus detention. For two weeks.\" The Wall's eyes stayed locked with Harley's as she smiled warmly and added, \"Math.\"\n\nThat afternoon in the dining hall, no one would look at Harley. \"What did I do?\" she implored. \"What did I do?\"\n\n\"If you don't know, then maybe you should think about it,\" Wonder Woman said to be helpful.\n\nHarley set her tray down between Katana and Bumblebee. Both scooted over to give her room\u2014more room than usual.\n\n\"What did I do?\" she asked as Star Sapphire strolled past, giving her a chilly look.\n\n\"Think about it,\" Wonder Woman reminded Harley.\n\n\"Psst,\" Harley whispered to Bumblebee. \"Give me a hint.\"\n\nBumblebee was busy pouring honey over her spaghetti. She really liked honey. On everything.\n\n\"Harley,\" Katana said. \"People are mad at you for making them think you were in danger. That's not funny.\"\n\n\" _You_ were worried,\" Harley said, slumping in her chair so her head barely appeared above the table. \" _I'm_ worried! Waller banned any new Harley's Quinntessentials videos for two weeks! I'm in reruns. For seven days times two! She even confiscated my video cameras. I won't have any viewers left when I return with fresh stuff.\"\n\nKatana frowned. \"Is that what you're worried about?\" she said. \"How many viewers you have?\"\n\n\"Well, yes!\" Harley said. \"You got that right!\"\n\nKatana let go of a heavy sigh. \"There's more to life than viewers,\" she said.\n\nHarley topped her mountain of garlic mashed potatoes with a cherry. \"Not if you're Harley Quinn,\" she declared.\n\n\"Hiya, fellow detainees!\" Harley said as she cartwheeled into the after-school detention room. She waved to Frost and Lady Shiva. Both looked bored.\n\n\"Everyone take a seat, and no talking!\" Vice Principal Grodd grumbled. His red bow tie matched the hankie in his suit coat pocket. Grodd was very dapper for a gorilla.\n\nHarley let out an audible sigh and sat next to Poison Ivy. The flowers in her long red hair were wilting.\n\n\"Whatcha in for?\" Harley asked brightly. She wasn't about to let anyone know she was bummed. Even though Harley could make almost anything fun, there were dozens of places she'd rather be than detention.\n\nPoison Ivy shook her head. \"Chompy got out of hand again,\" she said.\n\nHarley nodded. Everyone knew Ivy's large and overactive pet plant. Whenever Chompy overgrew his latest pot and went wild, it took Parasite, the janitor, a full day to clean up the mess.\n\nOn Harley's other side, The Flash was leaning so far back in his chair that three of its legs were off the ground. Harley passed him a note. It read,\n\nWhat are you in for?\n\nHe waited until Grodd was fully engrossed in his _101 Banana Recipes_ book.\n\nThe Flash wrote back.\n\nBreaking the sound barrier-twice.\n\nSounds awesome!\n\nHarley wrote in her loopy scrawl. She drew a sad face.\n\nPoison Ivy cleared her throat. She pointed to Grodd, who had fallen asleep and was now snoring gruffly. \"Don't wake him,\" she whispered. \"We have to stay out of trouble!\"\n\nHarley attempted to look serious, but failed. \"Me, stay out of trouble?\" she said. \"Not possible!\"\n\nWithout Harley's Quinntessentials to keep her busy, Harley had a lot of free time\u2014not when she was in detention, of course, but all the hours she normally spent on her Web channel were now open.\n\n\"Two weeks! She might as well have said forever!\" Harley said, moaning in her booth at the Capes & Cowls Caf\u00e9.\n\n\"One extra-large sweet potato fries with four sides of ketchup!\" Steve Trevor announced, placing the order in front of her. He had just gotten a trim, and now his blond hair was a tad too short. \"Hey, Harley, with your channel in reruns, I don't know what's happening at Super Hero High. Um...how's Wonder Woman?\" He blushed. \"I'm asking for a friend.\"\n\nSteve's dad owned Capes & Cowls Caf\u00e9, and though he wasn't a super hero, Steve worked as hard as one. The restaurant was a mishmash of homey and trendy that the Super Hero High students loved.\n\nRegular teens and Supers alike\u2014and even some of the teachers\u2014went there. Steve always set aside a dozen apple cider doughnuts for Batgirl's father, Police Commissioner Gordon.\n\nHarley picked up a sweet potato fry and pointed it toward the door. \"There's Wonder Woman,\" she said. \"Let's ask her!\"\n\nBefore Steve could stop her, Harley shouted, \"Hey, Wonder Woman! Wonder Woman, over here!\" Harley waved, then pointed to Steve, who looked like he was trying to make himself invisible. \"Stevie here has been asking about you! Come talk to him!\"\n\nWonder Woman blushed as red as the star on her gold tiara. The only person who was a brighter red was Steve...and Red Tornado, the school's flight teacher, who had stopped in for a strawberry smoothie. But then, he was always that red.\n\n\"When is your channel going to be back up?\" Star Sapphire asked Harley from the next table. She played with the power ring on her finger.\n\nHarley could only stare at the purple glow coming from the ring. \"Pretty!\" she said, her attention shifted away from Wonder Woman and Steve.\n\n\"I have a scoop for you,\" Sapphire whispered. Frost, who was sitting next to her, raised her eyebrow inquiringly as she blew on her peppermint tea, turning it ice-cold.\n\n\"Whatcha got?\" Harley asked, snapping to attention. A scoop! Harley loved nothing more than getting news first. Her motto was _All the news, as it happens...and sometimes even before!_\n\n\"I heard from a little bird that we're finally getting a new music teacher,\" Sapphire said.\n\n\"It's about time!\" Harley whistled with approval. \"Can't wait to interview her. Or him. Do you know who it'll be?\"\n\nSapphire smoothed the front of her pink and purple dress. The insignia on her belt matched the one on her jeweled headband, which matched the color of her ring.\n\nHarley took note of her own clothes: a black, white, and red checkered shirt with short black sleeves, and blue jean shorts with a thin belt over colored tights\u2014black on one side, red on the other. All with comfortable chunky blue sneakers on her feet. Harley felt a little sad...for Star Sapphire. _If only everyone could have a costume as fun, fashionable, and functional as min_ e, she thought.\n\n\"I hear it's a he, and he's starting soon,\" said Sapphire, reminding Harley of what they were talking about. \"By the way, you will be featuring me on Harley's Quinntessentials when you're back on the air, right?\"\n\n\"You betcha!\" Harley assured her. Fashionistas always tuned in when Star Sapphire was on her show.\n\nAs Harley settled back to contemplate her happy return to the Internet, she noticed a familiar figure at the counter. She grabbed her plate and plopped down next to her. \"Heya, Lois! What's the new news?\"\n\nLois Lane pushed her glasses up and smiled. \"Hi, Harley. I heard about Harley's Quinntessentials going on hiatus.\"\n\n\"Yeah, well, Waller's trying to teach me a lesson,\" Harley said nonchalantly\u2014with a shrug that said, \"In the meantime, I guess that means you'll get all the scoops!\"\n\n\"We're not in competition,\" Lois said. \"I report the news, and you do news and entertainment.\"\n\nHarley glanced at the jukebox. Captain Cold from CAD Academy had it blasting a retro rock song. The music inspired Beast Boy to lead a bunch of kids in a dance in the middle of the caf\u00e9.\n\n\"Well, just news is boring,\" Harley mused as a conga line of teens weaved in and out of the caf\u00e9. \"Booor-ring! Oopsie, sorry\u2014you know what I mean,\" she said to the teen reporter.\n\n\"No need to apologize. It can be boring,\" Lois said, laughing. \"But straight-up news is what I'm interested in. What are you interested in?\"\n\nHarley tugged on a pigtail, watching it bounce when she let go. \"I'm interested in having the most viewers in Metropolis. No, wait\u2014in the world and beyond!\"\n\n\"You've always thought big,\" said Lois. \"Harley, you're a force to be reckoned with. Half of your audience tunes in to see the inside story on the Supers\u2014and you have all prime access to that\u2014and the other half tunes in to see you!\"\n\n\"Little ol' me?\" Harley said with a twinkle in her eye. She did a backflip onto a table. \"Why would anyone want to look at me?\" Harley said, her arms raised dramatically above her head as kids in the caf\u00e9 applauded.\n\n\"Um. Harley,\" Steve said, wiping his hand on a towel. \"Remember, we talked about no standing on tables.\"\n\nHarley giggled. \"Right-O, Steve-O,\" she said. \"My bad.\"\n\nAfter Lois left, Harley continued to peck at her fries, but she couldn't stop thinking about what the young reporter had said. She wondered what might happen if she did more than cover the falls, foibles, and fantastic happenings of her fellow Supers, like Star Sapphire's Fashion Fix It. What if she did specials like...\n\nIt was hard to concentrate. Steve had recently installed the retro jukebox. It was all bright lights and neon colors, and music, music, music. It seemed like half the restaurant was dancing to the music. Most of the kids were just goofing off, but a few were really talented.\n\nAs she watched, Harley's brain began to go into overdrive. She had so many ideas it was hard for her to keep up with herself. Finally, one thought hit Harley so hard that she yelled, \"OUCH!\"\n\nShe jumped up and dashed out the door.\n\n\"Hey, Lois!\" Harley called, running after her. By then Lois was in Centennial Park. \"Slow down! I have something to ask you.\"\n\nLois looked curious. \"Yes?\"\n\n\"Okay, okay. News, yes. News is news. But when it's not new anymore, it's not news, right?\" Harley reasoned.\n\n\"I...think so?\" Lois said slowly. \"What are you saying, Harley?\"\n\n\"I'm saying, what if I expanded Harley's Quinntessentials beyond Super Hero High? What if I took the news and turned it into entertainment?\" Harley looked right at Lois. \"What if I broadcast a dance contest? And what if it was live so everyone everywhere could tune in? That way we'd all know the winner at the same time! I'd be making the news instead of just reporting it! Isn't that a great idea? A totally _**WOWZA-YOWZA**_ of an idea! I'll have more viewers than I'll know what to do with! I'll have so many viewers that\u2014\"\n\n\"Whoa, whoa, slow down,\" Lois said. She waited patiently for Harley to stop with the somersaults. \"You do know that there's a difference between informing your viewers with the news and entertaining, right?\"\n\nWhen Harley gave her a blank stare, Lois tried again. \"The news is fact. We strive to tell our viewers the truth and inform them about what's happening in the world. Especially when the information may impact them.\"\n\n\"Yes!\" Harley agreed \"Impact them! That's what I plan to do. POW! I'm going to have reality shows on Harley's Quinntessentials! And the first show will be a dance competition called Harley's Dance-O-Rama! What do you think of that?\"\n\nLois gave Harley a weak smile. \"Um, okay,\" she said. \"But it sounds more like entertainment than\u2014\"\n\n\"It sounds like blockbuster ratings, that's what it sounds like!\" Harley assured her.\n\nThere were only two more days left in detention. Vice Principal Grodd had finished his banana recipe book and was now reading a cookbook called _Bamboo and You._ As he sat munching on a stalk, the Supers shifted in their seats. Blessed with powers like super-speed, super-strength, and enhanced mental abilities, they weren't good at staying still.\n\nIn the back corner, Poison Ivy kept sticking her head into her backpack and talking. When she saw Harley staring at her, she offered a sheepish grin. \"I'm comforting one of my baby daffodils,\" Ivy whispered apologetically. \"She's lonely.\" A petite yellow flower peered out from the backpack and then quickly retreated. Harley smiled. Poison Ivy was crazy about her plants, and they were crazy about her, too.\n\nOutside the window, Supergirl was zooming in and out of the clouds, playing bow-and-arrow boomerang. In the game, Arrowette, whose family's archery skills were legendary, would take her stance, draw her bow, aim into the sky, and release an arrow. As the arrow hurtled at almost four hundred feet per second, Supergirl would catch it and then throw it back. Their record was fifteen in fifteen seconds.\n\nHarley opened her notebook. There was a big HQ in blue and red on the cover. Inside, in her loopy handwriting, were her notes for the Dance-O-Rama. It was all coming together\u2014on paper, at least. Once her Web channel was back up and running, the real fun could start. But first Harley needed some help.\n\nThat night Harley stood at the front of the room and said, \"You're all probably wondering why I asked you here.\"\n\nBeast Boy had a giant red-pepper pizza in front of him. \"I thought we were in the dining hall for dinner,\" he said, folding the pizza in half and eating it like a taco.\n\n\"I want to know why we're here,\" Big Barda said. She had so many servings of mashed potatoes on her plate, it looked like the Swiss Alps had relocated.\n\nAs Harley told the Supers about her Dance-O-Rama, some ignored her, but others seemed interested. Hawkgirl raised her hand. \"So let me get this straight. Heroes and villains will be allowed to try out to be contestants?\"\n\n\"That's right!\" Harley said. \"And regular citizens, too. It's open to all.\"\n\n\"And you can have as many people on your team as you want?\" Supergirl asked.\n\n\"And as few,\" Hawkgirl added. She had been taking notes. \"You can be a soloist.\"\n\n\"I can be a soloist?\" Wonder Woman asked.\n\n\"Anyone can,\" said Harley.\n\n\"What are the rules?\" asked Hawkgirl.\n\n\"Who needs rules?\" said Harley.\n\n\"The contest does,\" Hawkgirl said. \"Otherwise it will be chaos.\"\n\n\"Would chaos be so bad?\" Harley asked.\n\n\"It could be,\" Supergirl said. \"Have you run this past Principal Waller yet?\"\n\nHarley suddenly got serious. _**YOWZA!**_ If anyone could stand in the way of this great idea, it would be the Wall.\n\n\"Let me understand this, Ms. Quinn,\" Principal Waller said. Already an imposing figure, she was wearing one of her severe dark gray suits. Waller had seven of them, one for every day of the week.\n\nBumblebee flew into and out of the office, delivering papers and passing along messages. When Bumblebee saw Harley, she offered her a warm smile. Harley smiled back. She appreciated Bumblebee's upbeat nature. _Too many Supers are stressed out and so serious_ , Harley lamented as a dozen other thoughts bounced around in her head.\n\n\"Dancing?\" Waller continued, making it sound like Harley had asked permission to hold a worm-eating contest. \"And you want to enlist some of my students to help you? And you want open auditions\u2014ones in which a contestant's past is not of consequence? And you want to broadcast the contest live from Super Hero High on your Web channel?\"\n\nHarley gulped. When Waller said it, hosting a Dance-O-Rama didn't sound so fun.\n\n\"Yes, ma'am,\" Harley mumbled.\n\n\"Well, Harley.\" Waller shuffled some papers on her desk. \"I think it's a great idea.\"\n\nHarley got up and made her way to the door. \"Yeah, yeah, got it. I hear ya, I hear ya. No dance contest. Anyway, thanks for listening.\"\n\nAs Harley dragged herself down the hall, the sound of buzzing got louder.\n\n\"Harley, wait. Waller sent me to talk to you,\" Bumblebee called after her.\n\nHarley stopped to pull up her socks. They were always falling. \"Am I in trouble again?\" she asked. Despite her best efforts, it seemed like Harley was always getting called out. Most people, she decided, did not have her refined sense of humor and penchant for fun. Harley held up her hands. \" _ **YOWZA.**_ I surrender already. Arrest me!\" she joked.\n\nBumblebee's laugh was light and warm, like honey. \"There's no need for that,\" she assured her. \"Waller thinks you didn't understand. Harley, she was giving you permission to go ahead and hold the dance contest!\"\n\n\"Wha-wha-what?\" Harley blabbered. Her jumbled thoughts were spinning.\n\nWhen Bumblebee nodded, Harley's whoop of joy could be heard all the way to the athletic field. \"Yes!\" she yelled. \"Get ready, world! Harley's Dance-O-Rama is going to be the biggest and best-est dance contest anyone has ever seen. Ratings are gonna go straight through the roof, pierce the clouds, and ricochet off the stars!\"\n\nBig Barda was tossing huge weights to Supergirl, who tossed them to Wonder Woman, who was stacking them up so the Supers would have something to knock down.\n\n\"Does this look crooked?\" Wonder Woman asked as she stepped back to appraise the pile.\n\nAcross the room, Miss Martian was nervously dangling several feet above the floor from a rope that was part of a training exercise.\n\n\"And...go!\" Coach Wildcat yelled as he lit the bottom of the frayed rope with a match. It looked like an upside-down Fourth of July sparkler. \"Miss Martian, you'd better start climbing the rope before the fire catches up to you!\"\n\nFrost stood by the sidelines and tried not to yawn. She was there to put out the fire if needed.\n\nAs the rope began to sizzle, Miss Martian scampered up. Her brown eyes were big with misgiving, and her long red hair covered her face. She had never gone so high or so fast before! Poison Ivy stood by and applauded.\n\n\"Good!\" Wildcat barked. \"El Diablo, you're up next! Replace that rope and then climb it, stat!\"\n\nIn another part of the gym, Katana was twirling swords so fast you could hear them slicing the air. Cyborg tried to do the same. However, the sound he made was the metal clang when he accidentally hit himself with the weapon time and time again. \"Pick it up and try again,\" Katana encouraged him. \"And don't worry about dents. You can get those fixed later.\"\n\n\" 'Scuse me, Coach Wildcat,\" Harley said, tapping him on the shoulder. It was as solid as cement. \"May I have a few minutes of your time?\"\n\nWildcat turned around, his face in a scowl. \"Is this important?\" he asked. \"I have super heroes to train.\"\n\n\"It's super-duper important, Coach!\" Harley assured him.\n\n\"You really think teaching dance during PE is a good idea?\" Wildcat asked as El Diablo accidentally lit the bleachers on fire. Frost put out the flames by sending a blast of ice over them. Wildcat jerked his head around as a blur burst past. \"Hey, Flash, speed it up!\" he yelled.\n\nWildcat looked serious as he turned back to Harley, who nodded eagerly.\n\n_Geez. There's so much seriousness around school,_ Harley thought in the face of Wildcat's unending scowl. _Good thing they have me._\n\n\"This is a physical education class,\" the coach reminded her. \"These Supers need to be agile and alert at all times. After all, they may be saving the world someday, and that someday could be sooner than anyone thinks. Harley,\" he said, \"give me three good reasons why I should even be listening to you talk about dancing.\"\n\nHarley had anticipated this. She turned to Batgirl and nodded. Clinging to the ceiling using one of her B.A.T. (Barbara-Assisted Technology) devices, Batgirl hit \"On\" and Harley's POWer (Project, Out-logic, and Wow) presentation appeared on the east wall.\n\n\"One,\" Harley began as a photo of exuberant octogenarians dancing appeared, \"dancing makes for strong bones. Two, it helps increase stamina. Three, it improves memory by making us recall steps and routines. Four, balance is called into play, strengthening our core muscles. Five\u2014\"\n\n\"Enough!\" Wildcat said, raising his paw. \"Enough already, Harley!\"\n\nShe paused and held her breath.\n\n\"I'll give you gym periods three and four. But if I don't see results and it's a bust,\" Coach Wildcat warned, \"then you'll be mopping the floors alongside Parasite until Doomsday. Got it?\"\n\n\"Got it, Mr. Coach Wildcat, sir,\" she promised. \"Harley Quinn won't let you down!\"\n\nThe next day, Harley's classes seemed _sooo looong_. She just couldn't sit still. In Crazy Quilt's Super Suits design class, Harley kept applauding and whistling, hoping to speed things up.\n\n\"Harley, Harley, Harley!\" Crazy Quilt said as he sauntered down the catwalk that split the room in half. \"Please hold your applause until the situation merits it.\" He stopped mid-runway and stuck a disco pose. When the class was silent, Crazy Quilt whispered to Harley, \"You can applaud now.\"\n\nAt last! The class she was waiting for. Harley was bouncing off the walls. Literally.\n\n\"Quiet!\" Wildcat blew his whistle and then waited for the Supers to be silent. These were the elite teens of the universe, the super heroes of tomorrow...but for now, they were typical energetic teenagers in PE class.\n\n\"CALM DOWN!\" Harley yelled. \"This is important.\" She looked at Wildcat and said reassuringly, \"Go ahead, Coach.\"\n\nWildcat scowled and continued. \"Yesterday we learned how to disable our detractors with head locks, hard punches, and elbows to the solar plexus. Today we're going to learn pivots, pirouettes, and a do-si-do or two, thanks to Harley Quinn's suggestion. That's right: we're all going to dance! And anyone who thinks they can't learn a thing or two from me about cha-cha-cha-ing is in for a surprise.\"\n\nHawkgirl frowned. Beast Boy grinned. Harley did an aerial, and when she nailed her landing she pointed to Wildcat and announced, \"Supers, you're looking at the Ranger Ridge State College dance champion! Give it up for Coach 'Crazy Legs' Wildcat!\"\n\n\"That's ancient history,\" he said with a modest grin. \"But think about it. The best battles are like choreographed dance routines. It's a give and take, and there's a rhythm to it.\"\n\n\"How about heating up this class with a little salsa dancing?\" said El Diablo.\n\n\"How about a little polka?\" Wildcat asked.\n\n\"How about a little _polka_ in the eye,\" Harley whispered to Bumblebee, then laughed at her own joke.\n\n\"Polka?\" El Diablo said. The black images of flames that adorned his arms rippled. \"That's so old-school.\"\n\nParasite, the school janitor, was sweeping up under the bleachers. \"Polka's not old-school,\" Harley heard him grumble. \"The easier it looks, the harder it is.\"\n\n\"Okay,\" Wildcat called out. \"Pick your partners.\"\n\n\"You heard him,\" Harley yelled. She began clapping. \"Let's go! Let's go! Let's go!\"\n\nWildcat looked at her sideways. Harley shrugged. \"I figured you need a dance assistant, and that person ought to be me.\"\n\nThere was a mad rush. Big Barda and Supergirl stood together. Green Lantern and Lady Shiva were happily chatting. Cheetah and Star Sapphire were conspiring. Everyone had a partner except one student, who stood on the sidelines.\n\n\"Miss Martian,\" Wildcat said. \"I'm sorry, but there's an odd number of dancers, so I'm afraid you'll just have to sit this one out.\"\n\n\"That's fine,\" Miss Martian said softly. She began to fade until no one could see her.\n\n\"Dancers,\" Wildcat bellowed. \"I want you to watch this video first, and then we polka!\"\n\nAt a school full of super heroes, everyone wanted to be the best, whether at battling intergalactic villains, outsmarting criminals, or doing a polka. They studied the video of big-skirted women and men in jaunty vests with a keen interest. Hawkgirl took notes. Bumblebee danced in place.\n\n\"Okay, ready?\" Wildcat asked. He didn't wait for an answer. \"And...go!\"\n\nAs the relentlessly upbeat strains of accordion and clarinet pumped through the gym, an imposing figure stepped into the room. Amanda Waller's face was unreadable as she watched her students galloping and gallumphing in pairs.\n\n\"Big step, small step! Small step!\" Wildcat was calling as he clapped his paws to the beat. \"That's right! Big step, half step, half step. Full, half, half...No, no, Wonder Woman and Supergirl, no flying. This is on the ground only. Flash, slow down. Barda, Katana, it's not all big steps!\"\n\n\"You heard him!\" Harley called out, bouncing up and down to the music. \"Big step, small step!\"\n\nThe Supers were bumping into each other, causing some to crash against the walls and\/or knock over the other dancers. They were all enjoying the confusion, especially Harley\u2014until she noticed Principal Waller staring at something at the far end of the gym.\n\n\" _ **WOWZA!**_ \" Harley said, pointing. \"Do you see what I see?\"\n\nSoon everyone was watching Parasite doing an incredible polka...alone. His eyes were closed as he deftly moved through the steps, lost in the music.\n\nThere was a collective gasp as his partner slowly appeared. Miss Martian was smiling as she polkaed around the gym with the janitor. As they continued to skip and spin, Waller nodded to Wildcat before leaving.\n\nWhen the music ended, the applause began. Parasite was a little out of breath. It took great control to dance and keep his powers in check\u2014he had the ability to drain Supers of their powers. But Miss Martian was beaming. \"Thank you,\" she said bashfully.\n\n\"Thank _you_ ,\" he said, giving her a small bow before he picked up his broom. He tried not to smile amid the calls of \"Great job!\" and \"Parasite, you're the polka king!\"\n\nWildcat looked at Miss Martian. \"Next time,\" he said, \"you will show these Supers how it's done.\"\n\n\"How would you describe yourself?\" Bumblebee asked the girls who were crowded around her locker. She was holding the \"What Super Are You?\" quiz in _Super Student, Super Star_ magazine.\n\nHarley didn't hesitate. \"I'm the frosting on the cake! The _zzazz_ in _pizzazz_! The ribbon on the present! The duper in super!\" she said, leaping up and then taking a bow.\n\nCheetah strolled past and commented, \"What you are is a class clown.\"\n\nFrost followed, laughing. \"Where's your red nose?\" she asked.\n\nHarley lit up. \"Thanks for the reminder!\" She pulled a red foam nose seemingly from out of nowhere and plopped it on her nose. Then Harley grabbed her book bag and headed to class, leaving a trail of giggles, guffaws, and shocked expressions in her wake as she made her way down the hall yelling, \"Clown comin' through!\"\n\nThe line into the gym snaked out of the building and wrapped around the base of Crystal Tower. Twice.\n\n\"Lookit!\" Harley cried. \"Look at the semifinalists! We're gonna have big fun!\"\n\nJust a few days earlier, Harley's Quinntessentials had gone back on the air with new segments. \"Hey, Harley fans,\" she had broadcast. \"Didja miss me? I sure missed you, and you're not gonna wanna miss this\u2014Harley's Dance-O-Rama!\"\n\nThe buzz for the Dance-O-Rama was so big that even Lois Lane reported on it. \"Word is that dancers from around the world and beyond are eager to show their moves,\" Lois had said.\n\n\"I'm glad I had everyone check in online before they showed up,\" Batgirl told Beast Boy as she consulted the mini-computer on her wrist. \"Each entry was given a number. We have one hundred eighty-seven dance groups and thirty-seven soloists for the semifinals. Good thing we had the groups send in tapes for us to cull through first. Otherwise we'd have more contestants than people in the entire city of Metropolis!\"\n\n\"Thirty-eight soloists,\" Beast Boy corrected her.\n\n\"No, thirty-seven,\" Batgirl said, triple-checking the list.\n\n\"A super-talented last-minute performer just showed up,\" Beast Boy told her. He began to moonwalk. \"We don't want to leave this one out. This will rock the ratings, believe me!\"\n\nHarley's eyes grew big. \"Who is it?\" she asked, scanning the crowd.\n\n\"Me!\" Beast Boy said, stopping and pointing to himself. \"Right here. Right now. With all the right moves.\"\n\nBatgirl shook her head. \"You were supposed to send in an audition tape like everyone else.\"\n\nBeast Boy turned into a dancing hippo and gracefully balanced on one foot. \"But I'm not like everyone else,\" he noted. \"I'm special.\"\n\nBumblebee flew up. As official troubleshooter, it was her job to take care of any unforeseen problems. \"There's no time for arguing, people,\" she reminded everyone.\n\n\"Then it's settled! Let's let him in,\" Harley proclaimed, adding, \"He'll be good for ratings!\"\n\nIt was an odd gathering as the contestants descended on Super Hero High. A whole contingent from the Planet XOXO were dressed as puffy Valentine hearts, a trio of break-dancers from CAD Academy had brought a crate of plates and coffee cups with them, and what looked like half of Korugar Academy was present. Plus, there were dance teams from almost every regular high school, villains who had come out of hiding to show off their samba skills, and even one hundred tap dancers from Tap Dance Town, that new senior citizen community for retired super hero sidekicks.\n\nIt was sure to be a spectacle, and Harley had placed cameras around the gym to capture every dance move.\n\n\"Three, two, one...and we're live!\" Harley turned to the camera as two Furies from Apokolips Magnet stormed the stage looking confident.\n\nBig Barda fiddled with the sound system, making sure it was working. Barda took her job as DJ seriously, and her status as a Super Hero High student even more seriously. Normally, Bumblebee would have DJ'ed, but as the show's troubleshooter, she was already swamped.\n\n\"Welcome to Harley's Quinntessentials,\" Harley said gleefully, \"or HQ, to my friends. Today we usher in a new show. Yes! It's time for the semifinals of our _**POW! BANG! WHAM! WOWZA!**_ of a spectacle, the Harley's Dance-O-Rama! And now let's get started! First up, the Furies!\"\n\nThe duo from the dark and desolate planet started off slowly, but with each powerful stomp, their hip-hop dance ramped up until the entire school was shaking. When the music stopped, Furies Stompa and Lashina looked triumphantly at the DJ standing behind the music console.\n\n\"Hey, Barda, see what you're missing?\" Lashina snarled, tossing her black-and-blue ponytail over her shoulder.\n\nStompa took a bow. \"You should never have left Apokolips,\" she said when they walked past their former schoolmate. Her heavy boots made the gym tremble beneath Big Barda's feet.\n\n\"It's okay, Barda,\" Supergirl said, quickly flying over. \"You're one of us now. Those bad days at Apokolips are\u2014\"\n\nHarley interrupted. \"C'mon, c'mon, we got a show to put on!\" she reminded them with a circular motion of her arm that said \"Let's keep the ball rolling.\"\n\nHawkgirl consulted her watch. Harley had named her stage manager, and as with all her assignments, she wanted to make sure everything was perfect.\n\n\"Katana, we're ready for the next group,\" Hawkgirl said into her headset. \"Send in the clowns.\"\n\nHowls of laugher rippled through the stands as the clowns pushed and pulled and tumbled over each other to get to the stage.\n\n\"Wait for me!\" Harley cried, joining them.\n\n\"Oh, look! It's our very own class clown, doing her thing,\" Cheetah said as Harley rolled past her.\n\n\"It's going to be just as hard to corral Harley as the contestants!\" Batgirl shouted to Big Barda, who cued up the circus music.\n\nThe Comic Conga Clowns soon had everyone in the stands dancing in one long line around the gym and out the building. When the first clown in front leaned back, so did everyone else. And when she sat down, it caused a chain reaction of everyone falling into the lap of the person behind them.\n\nHarley could not stop laughing, nor could anyone watching. Though the semifinals had just begun, she was having a great time, and the dancers kept coming: samba, tango, foxtrot. Polka. Belly dancing. Ballet. Every kind of dance imaginable was being demonstrated.\n\n\"And now, another group from my own Super Hero High!\" Harley said into the camera. \"Here are Green Lantern, Raven, Catwoman, and Starfire doing their rendition of swing dancing!\"\n\nBatgirl was monitoring the online viewership. There were hundreds of thousands of viewers from every corner of the universe. She even detected some views from another galaxy. The show was breaking records. Batgirl had to admit: Harley knew how to entertain a crowd.\n\n\"And that's all for today!\" Harley announced. \"Tomorrow, we will resume our semifinals! So, until then, keep watching and keep laughing,\" she said as her channel started replaying the day's events.\n\nIf Harley Quinn had her way, Harley's Quinntessentials would be live 24\/7. But Waller had put limitations on how much time she could broadcast. There was, after all, something called \"school.\"\n\n\"That was exhausting,\" Batgirl said as they crowded into Harley's dorm.\n\n\"Everyone, thank you for all your help in making the Dance-O-Rama a hit!\" Harley enthused. \"Just wait until tomorrow, when we name the finalists. That's gonna be big!\"\n\n\"I'll bet you're going to get zillions of messages telling you how great you are,\" Miss Martian noted, looking at Harley with admiration. \"I wonder what that's like?\"\n\n\"What what's like?\" asked Harley.\n\n\"To be you,\" said Miss Martian. \"To have so many fans, and to get fan mail.\"\n\n\"It's the best!\" Harley exclaimed. \"Fans make you feel good, and fan mail can make you feel even better.\"\n\nThis Harley knew to be true. After all, she kept all her fan mail, and had a special file for her favorites to reread during those days when life wasn't as funny as it should be.\n\nOnline\u2014and in just about every other form of mass communication\u2014there was a lot of grumbling, with complaints of fixed votes and bribes and heavy hints of favoritism. Harley could not believe the response. She was in heaven!\n\nAfter a seemingly endless second day of semifinals, the finalists had been chosen via a panel of secret judges, votes from the Internet audience, and, of course, Harley. Now it was time to drop in on the tech behind her success\u2014the one and only...\n\n\"Hi, Batgirl!\" Harley shouted as she entered the Bat-Bunker. \"Any press is good press, don't you think?\" she said, tossing her mallet in the air. Just as Harley was about to grab it, a net dropped from the ceiling and caught it. \"Hey, that's mine!\" she complained.\n\n\"It's mine now,\" Batgirl said good-naturedly. \"You know that no one's supposed to play with their weapons in here. I have too much high-tech equipment and can't afford for anything to break. I'll give it back to you when you leave.\"\n\nHarley wandered around the Bat-Bunker, the dorm room that doubled as Batgirl's computer control center. The screens glowed blue in the purple room.\n\n\"So, what are my numbers?\" Harley asked, pressing some buttons randomly.\n\nBatgirl removed Harley's hand from her keyboard and reset the computers back to the way they were. Then she flexed her fingers and addressed the computer keyboard like a pianist. Instantly, a detailed map of Metropolis appeared with numbers running across it. Overlaid on top of that, a map of the United States appeared, then a map of the world, then the key locations around the greater galaxy covered the screen. \"Take a look for yourself,\" Batgirl said, scooting over.\n\nHarley leaned in. The number was getting bigger and bigger. \" _ **WOWZA!**_ \" she proclaimed. \"I'm a huge hit!\"\n\n\"Yes, but not everyone is a fan,\" Supergirl pointed out. Harley had been so focused on the numbers, she hadn't noticed Supergirl eating cookies in the corner. \"Look at your message board.\"\n\nOn another screen were comments from viewers. Most were like this:\n\nThe BEST show ever. I love HQ's Dance-O-Rama!\n\n\u2014Dance-O-Rama Fan Family\n\nHarley Quinn deserves the awesomeness award!\"\n\n\u2014MH234\n\nCan't wait to see the finals. I just know Beast Boy is going to win!!!\n\n\u2014Beast Boy\n\nBut some said:\n\nKiller Croc should have been a finalist. So what if he's a criminal?\n\n\u2014KC Fan Club and Employees\n\nThe votes were rigged!\n\n\u2014CAD Academy Parents Association\n\nWatch out, Harley Quinn, you picked the wrong dancers!!!\n\n\u2014Anonymous\n\nSupergirl wrinkled her brow. \"Do those bother you?\" she asked.\n\n\"Aw, shucks!\" Harley said after reading the angry comments. \"I'm not afraid of anyone! So what if I lose a couple of viewers? I've got plenty more where they came from!\"\n\nWith the finals less than a week away, the dance competition was all anyone could talk about. \"It's too bad you and Parasite didn't try out,\" Katana said to Miss Martian.\n\nMiss Martian blushed. \"Dancing's not my thing.\"\n\n\"Well, it's _my_ thing,\" Star Sapphire said. She pirouetted around the tables in the dining hall, hitting several students with her deep purple hair as she whirled around. \"I've already started planning my victory party.\"\n\n\"Nothing like a little self-doubt,\" The Flash said to himself as he passed by holding a tray piled high with cheeseburgers.\n\nSapphire straightened her tiara. \"I'm having couture costumes made for me and my corps de ballet. Now all I need is a dance partner.\" She looked around. \"Oh, Flash,\" she said as she adjusted the glowing ring on her finger. \"May I have a word with you?\"\n\nSince the dozen finalists included entries from Super Hero High, Wildcat left the gym open for the groups to practice. \"Nothing wrong with home court advantage,\" he said when Hawkgirl asked if this was fair.\n\n\"Let's triple-check this,\" Hawkgirl said to Batgirl.\n\n\"I'm all for that,\" Batgirl said, consulting her computer. \"My list includes the following: one hundred Tappers from Tap Dance Town; The CAD Academy Break-Dancers; the Belle Reve Penitentiary Guard Cha-Cha-Chas; Little Beth from Miss Toddler Tot's School for the Tiny and Talented; the Jigs Up, a group of former bank robbers from Limerick, Ireland; the Doomsday Divas, a group of reformed villains; a trio of dancers from Intensity Institute; our own Super Hero High entries; and the mystery dancer.\"\n\n\"Who's the mystery dancer?\" Hawkgirl asked. \"I don't like not knowing.\"\n\n\"Harley, who's the mystery dancer?\" Batgirl asked.\n\n\"Don't know,\" Harley said. \"It's a mystery!\"\n\n\"There's no way we can fit the hundred tap dancers in the gym\u2014and it's a direct violation of the fire code,\" Hawkgirl noted, shaking her head. \"Last time we had them dance in the parking lot. But that's going to be full of vehicles since the competition is a hot ticket.\"\n\nBatgirl agreed. \"What do you want to do about that?\"\n\n\"I don't know,\" said Harley. She was editing a commercial featuring the finalists and trying to mimic their moves. \"You guys figure out the details. I'm more of a big-picture person.\"\n\n\"Outside,\" Batgirl said. \"They'll have to dance outside on the sports field. I can get Supergirl and some of the others to construct a dance floor\u2014\"\n\n\"Heard you, and we're already on it!\" Supergirl said over Batgirl's comm bracelet. \"Look out the window.\"\n\nHarley could see Supergirl and Wonder Woman carrying twenty-foot stacks of wood planks and handing them off to The Flash, who was building a dance floor at super-speed on the sports field as Bumblebee directed him.\n\n\" _ **WOWZA!**_ We've thought of everything,\" Harley enthused. \"What could possibly go wrong?\"\n\nSaturday morning rolled around sooner than anyone expected. Wonder Woman enjoyed several bowls of colorful sugary cereals\u2014it was her one weakness. Bumblebee fortified her steel-cut oatmeal with extra honey. Batgirl had gotten up early, had fruit, granola, and yogurt, and was already at work in the gym where the show was taking place. A stickler for details, she wanted to triple-check the equipment.\n\n\"Can you go faster?\" Harley asked. She was so excited that she had eaten breakfast three times. Two on purpose, one by accident.\n\n\"I could if you would please move aside,\" Batgirl said. She was staring into Cyborg's eyes as he stared back at her. \"Harley, can you see what he sees?\" she asked.\n\nHarley checked her computer. \"Yep!\" she said gleefully. \"What an awesome, incredible idea, to have extra cameras! Who was the genius who thought of that?\"\n\n\"You,\" Batgirl said, rolling her eyes.\n\nHarley blushed and fanned her face with her hands. \"Aww, you are too kind!\"\n\n\"How do I turn this off?\" Cyborg asked. \"I don't want everyone to know what I see all the time!\"\n\n\"Just blink quickly three times to turn it off, four to turn it on,\" Batgirl told him.\n\nHarley batted her eyes at him. \"Thanks, Cyborg! Be sure to get the crowd shots. The ones of them cheering and yelling. Oh, and lots of shots of the host, too.\"\n\n\"That's you, Harley,\" he said, imitating Batgirl good-naturedly.\n\nHarley tapped the side of his metal head. \"Nothing gets past you, does it?\" she joked.\n\n\"Bumblebee, just in time. You're next,\" Batgirl said as her friend arrived and shrank down to be outfitted with a micro-camera. That way, she could fly right into the middle of the dancers and broadcast live.\n\nAs Harley tested the camera, Batgirl blurted out, \"Macro micro mini drone!\"\n\n\"Whatzat?\" asked Harley.\n\nBatgirl shook her head. \"Oh, just an idea,\" she said. \"I'll tell you later. You've got a show to put on. Plus, you've got to coordinate with all the judges.\"\n\n\"The judges?\" Harley asked.\n\n\"I know you had help selecting the semifinalists, so I assumed you'd have judges for the finals,\" Batgirl said.\n\nHarley's brain was racing. She knew there was something she'd forgotten. \"Snap!\" she finally said. \"Here's a news flash: no finalist judges. I've got something a billion times better!\"\n\n\"What?\" Batgirl asked.\n\n\"Me!\" said Harley, nodding happily. \"I can't wait to name the winner! Excuse us\u2014the judges have some decisions to make.\"\n\nAfter conferring with herself, Harley decided that she alone would determine who the winner was. After all, this was her Web channel, and this was her show.\n\nThe dance-off started smoothly. Star Sapphire and her corps de ballet were marvelous. After Barda made sure all the loudspeakers were working, she played the music from Tchaikovsky's _Swan Lake_. The ballerinas' costumes sparkled whenever they pirouetted, which was often. Then, when The Flash lifted Sapphire, he paused so the audience could take in the beauty of his partner\u2014that was her idea.\n\n\"Spin!\" Star Sapphire whispered. \"Like we practiced.\"\n\nThe Flash nodded, started off slowly, and then spun so fast the two of them rose in the air, to the delight of the audience. Then, as they floated above, purple glitter showered down on the crowd. Never had there been louder cheering, or more sparkles, for Swan Lake.\n\nAs if to balance the beauty and grace of Sapphire's dance team, CAD Academy went next. Captain Cold was in charge, as usual. He swaggered up onstage, followed by his crew: Ratcatcher and Magpie. While Barda manned the DJ booth and played the hard-driving beat, Captain Cold and company jerked around in an odd but mesmerizing rhythm.\n\n\"Cap'n Cold and His Break-Dancers are in the house!\" Barda yelled as Captain Cold pulled a plate out of his jacket and hurled it at the audience.\n\nA roar of approval went up as the CAD Academy dancers broke dozens of dishes\u2014smashing them on the ground, and throwing them into the audience and at each other.\n\nAt the far end of the gym, Parasite looked on and shook his head. He knew who would be cleaning up the mess.\n\n\"They crack me up,\" Harley said with a laugh as she took center stage while the CAD Academy students took their bows.\n\nThe show continued at whiplash pace, with Harley at the helm and Cyborg and Bumblebee displaying stellar camera work. \"Next,\" Harley announced, \"we go outside for the One Hundred Tappers, and not one of them is under one hundred years old! You know what that means, right? One hundred times one hundred equals awesome!\"\n\nThe special dance floor that Supergirl and the others had installed worked perfectly. The sound of two hundred feet tapping in unison was so loud it could be heard in Metropolis and beyond. The crowd was so jazzed by the tapping tones of the 100 Tappers that they stood up and began tapping along. When the Super Hero High buildings began to shake off their foundations, Harley was forced to cut their dance short. Even she was worried that they might cause an earthquake.\n\n\" _ **WHOA**_ and _**WOWZA!**_ That's going to be a hard act to beat,\" Harley said as she led the audience back inside. \"But now we have a team who teaches by day and dances by night\u2014our very own Super Hero High teachers!\"\n\nThe audience oohed as the lights dimmed and a disco ball lowered from the ceiling. When Supergirl's laser gaze hit the mirrored ball, it lit up and showered the gym with hundreds of thousands of reflective sparkles. When Barda dropped the needle on a disco classic, Crazy Quilt struck a pose, looking resplendent in his all-white three-piece suit with flared pants and oversized purple glasses. His body moved in a combination of quick jerks and fluid spins. He did the splits\u2014 _literally\u2014_ and leapt up, not letting the rip in his pants stop him. When Vice Principal Grodd appeared behind Crazy Quilt along with June Moone, the music veered as the two did a graceful modern dance depicting the beauty of the change of seasons.\n\n\" _ **YOWZA,**_ will ya look at that!\" Harley exclaimed.\n\nNow the trio locked arms and were doing a high kick ending with a three-person backflip. The audience was on their feet cheering as the teachers left the stage.\n\n\"A-plus!\" shouted Harley.\n\nSomeone was pacing behind Barda. When Harley called him onto the stage, she was surprised to hear him say into the mic, \"Golly gosh, I'm so nervous.\" He was green, but then, he always was. \"I don't think I can do this!\"\n\nBeast Boy looked like he was on the verge of tears. The audience sat silent. A few even sobbed along with him as the green teen buried his face in his hands and wept. He started to walk off stage, defeated before he had even begun.\n\n\"Aww, Beasty, you're gonna be great,\" Harley assured him.\n\n\"You know what?\" Beast Boy looked around at the crowd. \"You're right!\" He gave one of his impish grins, pointed to Big Barda, and yelled, \"Hit it, BB!\"\n\nAs the music pulsated, Beast Boy morphed into an astonishing menagerie of animals. A hip-hop hippo, a graceful gazelle, a belly-dancing anaconda, and a twisting, turning otter. He had morphed through thirty-seven different animals by the time he was done. It took both Supergirl and Katana to get Beast Boy off the stage. As he took his umpteenth bow, Harley leapt over him to the center of the stage.\n\n\"And now for our mystery group,\" she announced. Harley paused for drama. She was all about the drama. \"This last-minute addition has been together since they met at Pedigree Prep Hall in kindergarten. Having just graduated from high school, they are looking for their next adventure. Perhaps it will be as professional dancers? You tell me! Put your hands together for a group you're going to be hearing a lot from...the Green Team!\"\n\nThe gym went dark. Slowly the lights went back on and, through a haze of smoke, the Green Team seized the stage. The young men and women were dressed in chic matching designer suits.\n\n\"They're wearing Calder Melino shoes!\" Sapphire whispered appreciatively to Lady Shiva as she took a picture of them and instantly posted it on #FashionistaFotos.\n\nWith precision so sharp that the entire audience gasped as if on cue, the Green Team snapped into three rows of four. They bowed and began to dance to the sounds of Celtic fusion funk. Acting as one instead of twelve, the Green Team incorporated all the styles of dance that been displayed in the contest thus far into their own performance.\n\n\"This is truly amazing,\" Harley said breathlessly. \"We are seeing the future of dance! Green Team, we're expecting to see a lot of you in the future!\"\n\nThe audience was on their feet roaring their approval.\n\n\"Next up, we name the winners,\" Harley announced. \"But first, a short break!\"\n\n\"I dunno, I dunno,\" Harley wailed as she nervously yanked on her own bouncy blond pigtails.\n\n\"You dunno what?\" Barda asked.\n\n\"I dunno who the winner is,\" Harley said to herself. \"They are all _sooo_ good!\"\n\n\"True,\" Batgirl agreed. \"But you have to pick one. Remember, instead of audience votes or a secret ballot, you said you were going to be the sole judge.\"\n\n\"Is it too late to do an online audience vote?\" Harley asked, biting her fingernails.\n\nBatgirl nodded. \"You have to go out there now and announce the winner. This show is streaming live. Your audience is waiting.\"\n\nHarley grimaced and pulled on her pigtails even harder.\n\n\"Quiet, everyone!\" Harley yelled.\n\nOutside, one hundred tap dancers stood at attention. Cyborg aimed his camera on them in case they were the winners. Inside, Grodd, Crazy Quilt, and June Moone held hands, waiting to hear their names. The CAD Academy break dancers were already congratulating themselves, and Sapphire had changed into her new celebration party dress as The Flash tried in vain to remove the purple glitter from his costume.\n\n\"This is a first!\" Harley said, her excitement bubbling up as she thought out loud. \"Never before in the entire history of dance contests has something like this ever happened!\"\n\nThe Green Team looked at each other and tried not to smile. Little dancing toddlers from Miss Toddler's Tots School for the Talented and Tiny squirmed in their mothers' arms. The Wally Waltzers from Vienna held their collective breath.\n\n\"The winner is...EVERYONE!\" Harley cried. \"It's a twelve-way tie!\"\n\nThe audience was silent. Stunned. Then the noise began. It was unclear whether people were cheering or booing, but either way, Harley was happy.\n\n\"And that's it for the first-ever Harley's Dance-O-Rama. Tune in for my next totally live, totally unexpected contest. Until then, keep watching. I know I will be!\"\n\nThe cameras shut off, but the crowd remained. Some were applauding the audacity of it all. Others were grumbling. And others were getting louder as they expressed their displeasure, since they were certain their team or their favorite should have been the one and only winner.\n\n\"I won?\" Beast Boy asked, looking confused. \"But so did they?\" he pointed to the trio from CAD Academy. They looked more intimidating than usual, if that was possible.\n\n\"We should have been the sole winners.\" Captain Cold's voice was as chilly as ice. He started to freeze Harley, but Wonder Woman stopped him with her lasso.\n\n\"This is a travesty,\" Sapphire said, staring down Harley. \"I should have won! I've already paid for the party!\"\n\n\"But you did win,\" Harley sputtered, unsure of why some people were mad at her. \"Everyone won.\"\n\n\"It's not winning if there are others on the winner's stand,\" Sapphire lectured Harley. \"If this is your idea of a joke, it isn't funny!\"\n\nWhen Harley started to protest, Cheetah commented, \"Looks like the class clown has really flubbed this one.\"\n\nDespite the mixed reactions to Harley's announcement, or maybe because of it, her viewership soared. Not only did a record number of people tune in, but they watched the Dance-O-Rama over and over again. Harley was getting hundreds of messages at a time. She was loving it.\n\n\" _Unexpected_ was the word for Harley Quinn's first-ever dance contest,\" Lois Lane reported. \"The chatter is burning up the Internet. Some are thrilled with the judicious twelve-way tie. Others are incensed, saying that Harley copped out by refusing to name one winner. There have even been threats lobbed at her. Harley, what do you have to say about all this?\"\n\nHarley smiled at the camera. \"I say let's do it again. But first, I've got something even bigger planned. You'll be the third to know, Lois. First me, when I think of it. Second will be my loyal and loving Harley's Quinntessentials audience!\"\n\n\"Harley, are you worried about the angry comments?\" Lois asked. \"Some dance fans can get pretty serious.\"\n\n\"Nah,\" she said, sounding cavalier. \"I can handle anything. I'm Harley Quinn!\"\n\nAt the weekly assembly, Supers were flying and tumbling and climbing the walls. That is, until Principal Waller took the stage and cleared her throat. Instantly, everyone was seated and the room was so silent you could hear Vice Principal Grodd crack his knuckles. Repeatedly.\n\nAs Waller congratulated the Dance-O-Rama competitors from Super Hero High, the dancers stood to be acknowledged. The reception was thunderous. After all, the Supers were always ready to support each other. Star Sapphire and her corp, including The Flash, stood and twirled before taking a bow. Beast Boy turned into a parrot and flew overhead squawking, \"Thank you! Thank you! Thank you!\" And the teachers waved politely, as if they were in a parade.\n\n\"Disco! Disco! Disco!\" the Supers chanted. It didn't take much for Crazy Quilt to stand and strike a disco pose.\n\n\"You do the disco pose!\" a Super yelled to Principal Waller. Harley looked around to see who would be so brave.\n\nNo one was taking credit.\n\n\"Never going to happen,\" Principal Waller said, trying to hide her amusement. \"Now, let's talk about why you are all here and the responsibilities you shoulder.\"\n\nBig Barda had a piece of lemon cake in front of her, but she was ignoring it.\n\n\"You gonna eat that?\" Harley asked. She loved cake. Birthday cakes. Pound cakes. Funnel cakes. Ice cream cakes. All cakes\u2014especially cakes right in front of her!\n\n\"Not hungry. You can have it,\" Barda said, handing it over.\n\nAs Harley dug in, Supergirl put her arm around Barda's shoulder. \"Are you all right? You look worried.\"\n\n\"I think Principal Waller was talking about me,\" Barda said, staring down at her lap.\n\n\"What do you mean?\" Bumblebee asked. She was working on her second slice of honey bread.\n\n\"I think that when she was talking about responsibility, she was telling me that I'm responsible for Granny Goodness and the Female Furies when they came through the Boom Tubes and tried to take over the school,\" Big Barda said in a rush.\n\n\"Well, you were one of them then,\" Harley said, trying to be helpful.\n\nSupergirl, Bumblebee, Thunder, and Lightning glared at her.\n\n\"Wha?\" Harley said through a mouthful of cake. \"It's true!\"\n\nSupergirl turned back to Barda. \"That was then. You've changed and helped the fight against evil and are partly responsible for our victories!\"\n\n\"That's right,\" Lightning chimed in. \"Remember what Principal Waller keeps telling us?\n\n\" 'We are here because of who we can become tomorrow,' \" her sister said.\n\nSupergirl nodded. \"I didn't ask for my powers of strength and flight and heat vision. But now that I have them, I have a responsibility to make sure I use them to my best ability, and to help others in the process. Harley,\" she said, \"what's your take on responsibility?\"\n\nHarley polished off the last morsel of cake and the colorful crumbs left on the plate. \"I've been giving this lots and lots of thought,\" she told the group. \"So many thoughts, my noggin' is a-swimming. Ha! Can you just picture a head swimming?\"\n\nEveryone waited for Harley to sit down and stop pretending that her head had come free and was bobbing up and down on an imaginary ocean.\n\n\"Anyway,\" she continued. \"I was thinking about the Dance-O-Rama and how great it was, and even now the replays of the Web special are getting more and more views. Plus, I got a record number of comments after\u2014\"\n\nMiss Martian leaned over and whispered reverently to Ivy, \"She gets tons of fan mail! I know. I've seen it.\"\n\n\"\u2014so when I think about responsibility, I think that it is my responsibility to...Are you ready? This is big!\" Everyone nodded as Harley stood on her chair and shouted, \"It is my responsibility to announce that the next Harley's Quinntessentials will feature...a Battle of the Bands!\"\n\n\"Who can tell us why your weapons are important?\" Mr. Fox asked. He strolled purposefully up and down the rows of desks with his hands laced behind his back. His thick black glasses made him look serious, but his orange sweater and magenta tie said otherwise.\n\nWonder Woman raised her hand.\n\n\"Yes, Wonder Woman?\" the Weapononics teacher said.\n\n\"Our weapons are to protect and defend,\" she began. \"However, in the wrong hands they could be used to destroy. It is our responsibility to use them wisely.\"\n\n\"Excellent!\" said Mr. Fox, beaming. \"Anyone else? Harley, what about your mallet?\"\n\nHarley had been daydreaming about the Battle of the Bands. In her imagination, after the winner was announced, she would start a record label called Mallet Music, and it would be an instant hit\u2014\n\n\"Harley?\" Mr. Fox said, snapping her out of her daydream.\n\n\"Huh? Harley? That's my name, don't wear it out!\" Harley quipped.\n\nAs the class laughed, Mr. Fox shook his head. Harley smiled sheepishly and shrugged.\n\nUnfortunately, this was not the last time she'd be caught daydreaming.\n\nIn history class, their teacher Liberty Belle talked about the great battles of the last century\u2014including recent ones with the Supers, led by Katana, battling Dragon King, and Supergirl leading the charge against Granny Goodness, and Batgirl versus the Calculator.\n\n\"I seek to honor the legacy of my grandmother,\" Katana said. \"She fought hard against evil, and passed down her beliefs to me. I feel that in my heart.\"\n\n\"Thank you,\" Liberty Belle said. \"Harley, you look like you want to say something. What can you add to this conversation?\"\n\nHarley had been poking the girl in front of her, who was doing an excellent job of ignoring her. She was hoping Hawkgirl would agree to be stage manager for the Battle of the Bands.\n\n\"Harley...?\" Liberty Belle said, waiting for an answer.\n\n\" _ **WOWZA,**_ Teach!\" Harley said, glancing around the room. \"Um, er...well, yes.\" She wished she knew what the question was. That would make things so much easier!\n\nEveryone was staring at her. Cheetah smiled and whispered, \"We were talking about our favorite foods.\"\n\nHarley winked a \"thank you\" and said with gusto, \"Fried pickles and triple-chocolate swirly ice cream loaded with sprinkles!\"\n\nHarley had one last class that day. An elective. Music. Super Hero High had been missing a music teacher ever since the last one fled during Thunder and Lightning's Dueling Duets demonstration. Thunder had created powerful shock waves that hit all the acoustic instruments, causing them to reverberate cacophanously. Then Lightning countered by trying to commandeer the electronic synthesizers and drum machines with her electrical powers and overloading them. The equipment began to malfunction and sparks flew\u2014musically and literally.\n\n\"I can't wait to meet the new teacher,\" Harley was saying as she bounced down the hallway.\n\n\"Me too!\" said Raven as she rushed past, her dark cloak fluttering behind her.\n\nThe music room was full. Some students were carrying their instruments\u2014violins, guitars, a piano. Big Barda had an accordion, and Green Lantern was polishing his tuba.\n\nThe man in the front of the classroom was bent over, digging through a briefcase. \"Ah! Here it is,\" the new teacher said, pulling out a silver flute. His green cape covered his face, but when he pulled it off, he had a brilliant smile. \"I am your new music teacher, Pied Piper!\"\n\n\"Mr. Pied Piper,\" Beast Boy called out, \"what kind of music will you be teaching?\"\n\n\"What kind would you like?\" he asked, waving his flute like a conductor's baton.\n\nAs they called out \"Classical!\" \"Hip-hop!\" \"Retro rock!\" \"Jazz!\" \"New wave!\" and \"Electrolite!\" the Pied Piper nodded.\n\n\"We will do all that and more. But first, I've heard a musical rumor. Harley Quinn!\" he called out. \"Is it true you're putting on a Battle of the Bands? If so, I'd like to offer my services to coach any Super Hero High musicians who might be competing. Is that okay with you?\"\n\nHarley grabbed one side of Big Barda's accordion. \"Hold on!\" she instructed Barda as she ran across the room, stretching out the bellows. \"Here's my answer, Mr. Pied Piper, sir!\"\n\nShe let go and the accordion folded back toward Barda, making a _wooom_ sound. \"That's music for yes,\" Harley said, letting go of a big laugh. \"This 'battle' is gonna be epic!\"\n\nBatgirl was buried. There were so many audition videos and Web links of bands that it was almost a full-time job logging them all in\u2014and it had only been a couple of days since Harley's announcement.\n\n\"It's gonna be the biggest and baddest and best-est Battle of the Bands this world has ever seen!\" Harley proclaimed to her Web fans. \"Everyone is welcome to enter. Just send in a short audition tape, and if you're one of the lucky finalists, you will perform LIVE on the Internet for an audience of a zillion-ish viewers!\"\n\n\"What can we do to help?\" the ever-efficient Hawkgirl asked.\n\n\"You can help me log these in,\" Batgirl said, pushing a pile of old-school tapes and discs toward her.\n\n\"Is this why you called a meeting of the Junior Detective Society?\" Bumblebee asked. \"For our help in sorting this all out?\"\n\nThe Junior Detective Society was a school club mostly comprised of Batgirl, Hawkgirl, The Flash, and Bumblebee\u2014though other Supers helped them with cases or came to them for help. The Junior Detectives loved a mystery.\n\n\"Yep,\" Batgirl confirmed. \"It's a mystery how we can get through the auditions. We need a system to sort and view everything. Harley's putting together a group of Supers and citizens to help listen, right?\"\n\nHarley nodded, then went back to watching a commercial of herself advertising the Battle of the Bands on one of Batgirl's computer screens.\n\n\"We're on it,\" Hawkgirl said. She was already drawing a diagram of a sorting system, while The Flash in a nanosecond\u2014or two\u2014had logged in the videos. Meanwhile, Bumblebee was online, scanning the emails with musical Web links.\n\n\"It's like anyone who's ever picked up a musical instrument is vying to get on the show!\" The Flash exclaimed.\n\nSure enough, it was a who's who of super heroes, super villains, and citizens. \"Look!\" Bumblebee called out excitedly. \"Mandy Bowin is auditioning!\"\n\nMandy had been a student at Super Hero High, but left to pursue her music. Also on the list was the Korugar Academy Marching Band, plus Black Canary and the Birds of Prey, and the Bad Banshees featuring Silver Banshee on vocals, Gizmo on drums, and Jinx at the keyboard.\n\n\"There are lots of Supers from Super Hero High trying out, too!\" Hawkgirl noted.\n\nA knock on the Bat-Bunker door interrupted them. Batgirl buzzed Wonder Woman in. \"Dinnertime!\" said Wonder Woman.\n\n\"You guys go ahead,\" Batgirl said to her friends. \"I want to finish this up, then send it out to the preliminary screeners and the judges.\"\n\nAt last the room was quiet, and Batgirl exhaled. \"So nice to be alone sometimes!\" she said after a few minutes of silence.\n\n\"So then,\" a voice said, \"do you think I'll break the number for the most viewers with BOB? That's what I've nicknamed Battle of the Bands. BOB, get it?\"\n\n\"Harley, you startled me!\" Batgirl said, laughing. \"I thought you had left with the others.\"\n\nHarley began randomly touching all the buttons and keys on Batgirl's computer console. \"I've asked you not to do that,\" Batgirl said, moving her friend's hands away. \"Was there something you wanted to talk about? Is this about Battle...er, BOB?\"\n\n\"Maybe, maybe not,\" said Harley. She was now fiddling with Batgirl's Batarang and accidentally dislodged it.\n\nBatgirl ducked, then caught it with one hand while still typing with the other. \"What is it?\" she asked.\n\n\"Aww, nothing!\" Harley said, putting on a smile to mask her frown. \"I was just scared for half a second. But it's dumb, and I'm fine. Not to worry. I'm not worried. Who's worried? Not me! Is it you? What are you worried about, Batgirl? Do you wanna talk?\"\n\n\"Harley, stop jumping on my bed and tell me what's going on,\" Batgirl said, patiently.\n\nHarley sat cross-legged on the floor and Batgirl joined her.\n\n\"What if I mess up?\" Harley said, so softly that Batgirl had to strain to hear. \"What if the Battle of the Bands isn't a huge hit? Some people are already mad at me because I named everyone a winner at the Dance-O-Rama.\"\n\n\"I thought this sort of thing didn't bother you,\" Batgirl said.\n\n\"It doesn't!\" Harley said a little too loudly.\n\nBatgirl didn't respond.\n\n\"Okay, okay, it does bother me a little. But don't tell anyone. I have a reputation to uphold!\"\n\n\"Everyone likes you,\" Batgirl assured her.\n\n\"Not everyone,\" Harley said. \"Some people think I'm just a class clown. But I'm more than that, aren't I?\"\n\nBatgirl reached over and gave her a hug. \"You're fun and funny, and a super friend. You've proven yourself to be a hero again and again. Harley Quinn, you're one of a kind!\"\n\nHarley gave her friend an \"aww, shucks\" grin as Batgirl turned back to the task at hand.\n\nThe Internet was lit up, and rumors were flying so fast about who would make it to the finals that everyone was dizzy. Some of the musicians were even sending promotional items like T-shirts emblazoned with their band's logo on them in the hopes of currying favor with the judges.\n\n\"Look! I got this in the mail,\" Miss Martian said. She was so excited she kept shifting from foot to foot.\n\nMiss Martian unfurled a poster of the Green Team. Inexplicably, they were standing on a white sand beach, each gazing in a different direction, as the orange sun melted into the azure sea. The headline read\n\n> _**FOR YOUR BATTLE OF THE BANDS CONSIDERATION: THE GREEN TEAM!**_\n\nClipped to the top of the poster was a twenty-dollar bill.\n\n\"That,\" said Hawkgirl, frowning, \"is an infringement upon the rules. No bribes! The Green Team just disqualified themselves.\"\n\n\"Yeah,\" Harley had to admit. \"They're out. That's a shame. Since the Dance-O-Rama they've earned a huge following.\"\n\nMiss Martian stared dreamily at the Green Team and asked, \"Can I keep the poster?\"\n\n\"We cannot, cannot, cannot have the same mess we had with the twelve-way tie for the Dance-O-Rama,\" Harley told Batgirl.\n\n\"But, Harley, you were the one\u2014\" Hawkgirl started to say, but Harley put her fingers in her ears.\n\n\"I can't hear you,\" Harley said. \"Lalalalala, I can't hear you!\"\n\n\"I can program the site so that no one can vote more than once,\" Batgirl was saying. \"I'm installing a pupil recognition device to ensure one person, one vote.\"\n\n\"Well, since you have this under control, I'm off to the Battle of the Bands location,\" Harley informed them. She was now pacing the room, adding a few flips here and there. \"It's spectacular!\"\n\nBatgirl looked up. \"Where is it?\"\n\n\"Not sure yet,\" Harley admitted. \"But when I find it, I'll let ya know!\"\n\nAs Harley and Katana flew in the Invisible Jet with their pilot, Wonder Woman, they looked for a venue that would be inviting, could hold lots of people, and had great acoustics.\n\nThe trio was getting weary. Flying around the world was exhausting. So many places to go. So much to see. So little time. They flew over Diamond Head in Hawaii, but there was no guarantee that the volcano wouldn't erupt mid-BOB. The Colosseum in Rome was \"too old and musty-dusty,\" according to Harley, and the Sydney Opera House in Australia was already booked.\n\n\"What about the Grand Canyon?\" Wonder Woman suggested. She shifted gears and turned her jet ninety-seven degrees to the left. \"The acoustics would be amazing!\"\n\nSoon they were flying above the South Rim. Red rocks and mile-deep canyons revealed the millions and millions of years of geological history.\n\n\"This looks like the perfect place for BOB!\" Harley enthused.\n\nAs they landed and scrambled out of the jet, Katana looked down into a ravine. \"Hello!\" she yelled.\n\n\"Hello! Hello! Hello!\" it answered.\n\n\"Too much echo,\" Katana said, shaking her head.\n\n\"Knock, knock!\" Wonder Woman said, giving it a try.\n\n\"Knock, knock, knock, knock,\" the Grand Canyon replied, and then added, \"Who's there?\"\n\n\"Huh!\" Wonder Woman shouted. \"Did you hear that?\"\n\nHarley couldn't stop laughing. \"That was me! Okay, where to next?\"\n\n\"Food?\" said Wonder Woman. \"I'm hungry.\"\n\n\"Me too,\" Katana said.\n\n\"Me three,\" Harley agreed.\n\nWonder Woman shifted into autopilot. After landing near Metropolis's Centennial Park, the girls were on foot when Katana stopped abruptly, causing Harley to bump into her. \"Do you hear that?\" Katana asked.\n\n\"Hear what?\" asked Harley. \"The laughter of little children at play? The melodic tunes of the birds singing? The plaintive meow of Rainbow the cat up the tree?\"\n\n\"No,\" Katana said, pivoting around. \"That!\"\n\nWonder Woman was already on the move. Three burly truck drivers were yelling.\n\n\"Help!\" the biggest one cried. \"Our trucks were stolen and they're full of medical equipment for the Metropolis Hospital!\"\n\n\"We've got this covered!\" Harley assured him.\n\nWith Wonder Woman in the air and Harley and Katana on the ground, they caught up to the wayward trucks in no time. Blocking the vehicles, the Supers stood with their hands on their hips as the trucks screeched to a halt only inches away from them.\n\nFor a brief second, the smile slid off Harley's face. \"Uh-oh,\" she said.\n\n\"Not good,\" Katana agreed.\n\n\"Not good for them,\" Wonder Woman added.\n\n\"Well, hello, hello, girlies,\" said the criminal, who looked like a cross between a pterodactyl and a man. He flapped his powerful wings, causing the leaves to blow off the trees.\n\nWonder Woman's eyes narrowed. \"Hello, Airstryke.\"\n\n\"How's Queen Hippolyta?\" the baddie Airstryke asked as he rose above the ground. \"I haven't seen you or your mother since she threw me in prison.\"\n\nWonder Woman flew up so they were face to face, one hundred feet in the air.\n\nHarley stood her ground, twirling her mallet, ready for action. Katana was poised to unsheathe her sword.\n\n\"Mom's fine, thanks for asking,\" said Wonder Woman as Airstryke flew at her with his sharp teeth bared. \"You should brush more often,\" she added.\n\nKatana and Harley were about to throw their weapons at him, when the doors of the other trucks flew open. A muscled man who had a skull instead of a face stepped out of one, and a man dressed in red with yellow boots and cape leapt out of the other.\n\n\"Aerialist and Atomic Skull,\" Katana said to Harley. \"Liberty Belle told us about them, remember?\"\n\nHarley nodded in the affirmative, but in reality she wished she had paid more attention in class.\n\n\"I got the leaping guy,\" Harley called out as Aerialist jumped backward on top of a truck. She tossed her camera up against a nearby building so that it hit the wall...and stuck.\n\nKatana crouched down. \"Atomic Skull's mine,\" she said.\n\nAs the battle ensued in the air, on the ground, and along the desolate road, Harley was laser-focused on Aerialist.\n\n\"You used to be a stuntman?\" she asked as he hurled himself toward her.\n\nHarley cartwheeled at him so fast that she was nothing more than a blur of a ball. Midair, she hit him with her mallet. \"How's that for a stunt?\" she asked. He slammed against a nearby building and then fell to the ground, completely stunned.\n\nAbove, Wonder Woman was in fast pursuit of Airstryke, who did a one-hundred-eighty-degree turn in midair and started chasing her. \"Wonder Woman, look at you,\" he said. \"I remember when you were just an itty-bitty little thing trailing along in your mother's shadow!\"\n\n\"Well, Airstryke, I'm not a little girl anymore!\" Wonder Woman adjusted the golden cuffs her mother had given her. She lifted her Lasso of Truth and began to twirl it. As it sliced through the air, it was Airstryke's turn to flee.\n\nAs Airstryke tried to get away, she lassoed his foot, and with a mighty pull, sent him hurtling toward the ground.\n\nMeanwhile, Atomic Skull began to glow. He was emitting a radioactive field as Katana circled him, sword in hand. \"What do you want with the medical equipment?\" the villain asked.\n\n\"I want to help the children at the hospital,\" she said.\n\n\"Your heart is too soft.\" He laughed and shook his head.\n\n\"Maybe,\" Katana said, tightening her grip. \"But my sword is not!\"\n\nA few yards away, Harley bowled the Aerialist over with a swing of her mallet as he leaped toward her. He went soaring when the mallet made contract. Midair, he crashed into Airstryke, and the two villains tumbled onto the asphalt in a heap.\n\n\"Now, that's what I call an air strike,\" Harley said with a laugh.\n\nFlying around the defeated villains at super-speed, Wonder Woman tied them up with the Lasso of Truth. Then she and Harley joined Katana.\n\n\"Something's wrong with your noggin!\" Harley quipped as Atomic Skull began generating more and more energy. The heat force around him expanded. \"You're getting all full of yourself.\"\n\nAtomic Skull laughed. \"Goodbye, super heroes,\" he said. \"Nice knowing you!\"\n\nBut before he could activate his lethal blast projection, Harley distracted him with a series of cartwheels. Simultaneously, Katana caught his attention as she neared him with her sword. Atomic Skull didn't see Wonder Woman. The Amazon had picked up a large moving van and scooped him up into the back. The villain was jostled as he banged around inside the truck.\n\nHarley slammed the doors of the van as Wonder Woman set it down. Katana put her sword through the handles to make sure Atomic Skull couldn't get out until the authorities arrived.\n\n\"Case closed!\" Harley cheered.\n\nBy the time Harley and her friends made it to Capes & Cowls, Steve Trevor had a table reserved for them.\n\n\"Some truck drivers called and bought these for you,\" he said. His wide smile revealed his braces as he served the heroes their favorite smoothies.\n\n\"Hi, Steve!\" Wonder Woman said brightly.\n\n\"Hi, Wonder Woman! Um, I have to take some orders, but I'll be back to check on you,\" he said, backing away and bumping into the table behind him.\n\n\"Watch where you're going!\" sneered Captain Cold.\n\n\"Sorry, sorry,\" Steve apologized.\n\nHarley was calling Batgirl on her phone. \"Would you mind uploading the video I just sent you?\" she asked. \"It's a Save the Day that just happened.\"\n\n\"Sure thing,\" Batgirl said from the Bat-Bunker. \"I'll also set you up so you'll be able to do it yourself. In the meantime, have you found a venue yet?\"\n\n\"Still looking,\" Harley reported. She watched Steve set a huge pizza on the table. One side had ham and pineapple on it, and the other had mushrooms. Harley took two slices of ham and pineapple, put a slice of mushroom between them to make a pizza sandwich, and bit into it.\n\n\"We're using the land behind the caf\u00e9 as a temporary animal preserve while the zoo gets ready for its new wildlife sanctuary,\" Steve was telling Katana and Wonder Woman.\n\n\"That's nice of you,\" said Wonder Woman.\n\n\"That explains the smell,\" Harley said, pinching her nose.\n\n\"It's no big deal,\" said Steve. \"Anything for the animals. Some have been injured or orphaned, or are old and wouldn't last in the wilderness. If they become strong and independent enough, they're returned to the wild. They'll be out of here and at the sanctuary in a couple days.\"\n\nKatana looked out the window. She could see the animals milling about. \"They've been through a lot. As far as I'm concerned, they can make all the smell and noise they want!\"\n\nThe table was quiet for a moment as they listened to the animals. It sounded like they were in the same room.\n\n\"Hey, Stink-o Steve-o,\" Ratcatcher called out. \"More french fries over here, and make it snappy!\" He threw several rat traps on the ground and Steve had to leap over them to avoid getting caught. Snap, snap, snap!\n\nBut all Harley could hear was the sounds of the animals.\n\n\"Good food. Lots of space. Great acoustics.\" Harley began. She looked up at the ceiling as she kept repeating herself. \"Good food. Lots of space. Great acoustics. Good food. Lots of space. Great acoustics.\" Suddenly, Harley leapt to her feet. \"I got it! I know where we should hold the Battle of the Bands!\"\n\nSteve tiptoed around the rat traps and brought the teens another pie. But instead of a pizza pie, it was an apple pie. \"It's made from locally grown apples from Poison Ivy's garden,\" he said. \"Dig in, it's fresh out of the oven.\"\n\nSteve noticed Harley staring at him...very intently.\n\n\"Um, is there anything else I can do for you?\" he asked.\n\n\"You betcha!\" Harley said. \"You can host my Battle of the Bands here!\"\n\n\"I don't understand,\" Steve said. \"Here?\"\n\n\"Here!\" said Harley. \"You have good food. Lots of space. It's perfect!\"\n\n\"It is perfect,\" Wonder Woman chimed in. \"We can transform your parking lot and that area where the animals are into an arena....\"\n\n\"And the caf\u00e9 can provide food stands,\" Katana explained, \"and the Supers will help man them. But you can be in charge of the menu, Steve!\"\n\n\"And your acoustics are awesome,\" Harley said. She raised her hand and yelled, \"Shhh! Everyone be quiet!\"\n\nAll were silent as they listened to the sound of a green hippopotamus mumbling about needing ice cream.\n\n\"What's going on here?\" the green hippo said.\n\n\"Beast Boy,\" Harley enthused. \"Tell Steve here that Capes and Cowls would be the perfect venue for BOB.\"\n\n\"What she said,\" Beast Boy said, turning back into a boy.\n\n\"Think of the publicity!\" Harley added.\n\n\"I don't know,\" Steve said, rubbing his chin thoughtfully. \"It's a pretty big deal, the Battle of the Bands. Everyone will be watching.\"\n\n\"Exactly!\" Harley said, nodding.\n\nThe girls agreed: The caf\u00e9 would be just the place. It was centrally located and had everything they were looking for. Plus, the BOB wasn't until after the animals were to be relocated to the zoo's wildlife area.\n\nJust then, the bell on the door signaled another customer. \"It's Pied Piper!\" Katana said. \"Hello!\"\n\nThe music teacher had been working with the Supers and teachers who had entered the contest, and everyone couldn't help but adore him. Walk past his music room at any hour of the day and you could hear The Flash breaking up more sets of bongos than he was willing to admit, and the retro-rock sounds of Teacher Teacher with Liberty Belle singing lead and backed up by Doc Magnus on the synthesizer and Red Tornado on the electric guitar. And no one could ignore\u2014or stand\u2014the vivacious vocal stylings of Beast Boy.\n\n\"Pied Piper is the best!\" Harley was saying as she watched him nodding his head to the music playing on the jukebox.\n\n\"Yes, and he's deaf and uses that to his advantage,\" Katana added.\n\nHarley raised her eyebrows.\n\nPied Piper waved at the girls as he picked up his to-go order and left.\n\n\"Deaf? No way!\" Harley insisted. \"But he's the music teacher!\"\n\n\"And an incredible one,\" Katana added. \"Pied Piper can feel the music in a much more powerful way than the rest of us. Plus, the rumor is that he can read lips so well the government has used him as a spy since he also understands several languages!\"\n\n\"How did I miss that?\" Harley asked.\n\n\"You've been busy,\" Katana said. \"Harley, sometimes you're so busy you don't notice what's happening around you, or even have time to spend with your friends.\"\n\nWonder Woman nodded. \"It's true!\"\n\nHarley wondered if her friends were trying to tell her something. She made a note to talk to them about it, when she had more time. Then she burst out laughing. _Like, when am I ever going to have more time?_ she asked herself.\n\nThere were so many students clogging the corridors that Principal Waller asked the head hall monitor to add extra hours.\n\n\"Keep it moving,\" Hawkgirl said as she flew up and down the flight lanes that flanked the walkways. \"Nothing to see here!\"\n\n\"But everything to listen to,\" quipped Harley as she stood outside the music room. \"It's sounding great in there.\"\n\nShe was right. Pied Piper had helped whip the musical acts into shape. Many of them had been good, but under his direction, now they were great.\n\n\"Less feedback on the keyboard amp!\"\n\n\"Your falsetto is sounding false!\"\n\n\"Love the harmony, keep it up!\"\n\n\"Dueling pianos, I want more ritardando!\"\n\nThe only time the competitors weren't listening to him was when they were arguing among themselves.\n\n\"As lead singer, I should be able to pick the song,\" Liberty Belle was saying to Red Tornado.\n\n\"Well, it better have a great acoustic guitar solo,\" he countered.\n\n\"Cheetah, could you do your warm-ups over there?\" The Flash asked as he broke another pair of bongos. \"You're distracting me.\"\n\n\"No, you move,\" she said, raising her voice.\n\n\"Miss Martian? Miss Martian, are you with us?\" asked Poison Ivy.\n\n\"I'm here,\" the shy invisible alien said quietly as a cello made its way across the room.\n\n\"I could stand here listening all day,\" said Harley as she looked in from the doorway and waved to her friends.\n\n\"And I could write you up,\" Hawkgirl said good-naturedly. \"Anyway, shouldn't you be working on that assignment for Fox's class? It's due tomorrow.\"\n\nHarley tried to hide her surprise. \"Oh! It's due tomorrow? Good thing I'm almost done. Um. Could you remind me just what we're supposed to do?\"\n\nThe truth was, Harley was so busy with Battle of the Bands that her schoolwork had begun to slip. And so had her time with her friends, and most everything else. On the plus side, her viewership was rising at a rocket's pace. BOB updates were rotated at regular intervals, and then viewers stayed for Harley's exclusive Super Hero High gossip clips and laughed along to reruns of \"Super Heroes' Super Blunders.\"\n\nLuckily, Harley's team of BOB volunteer judges had winnowed the thousands of auditions down to the final fifty. Harley herself would pick the top ten, but she was having trouble deciding. Every audition sounded like a winner, and she had difficulty rejecting anyone because that might make them sad.\n\n\"Please pass your assignments to the front of the class,\" Mr. Fox said. \"I am looking forward to reading these. As a preview, we will go around the room and each of you will provide a one-sentence summary of your report. Ms. Quinn, we'll start with you.\"\n\nHarley gulped and looked desperately at Miss Martian, trying to get her attention. _Read my mind, Miss Martian! Read my mind, and tell me what to say!_ Harley had her eyes screwed shut and was thinking hard. _Whisper what the assignment was and what I should say!_\n\n\"Harley, are you okay?\" Mr. Fox's brows were knit together as he scrutinized her face.\n\nHarley unscrewed her eyes. She looked at Miss Martian, who refused to meet her gaze. \"Yes, sir, Mr. Fox, sir!\" Harley said.\n\n\"And your sentence would be...?\" he prodded.\n\n\"Well, my report is about all that stuff that is so important and that we need to know about and that we were supposed to write about. And that's what my report is about!\" she said.\n\n\"Please stay after class, Ms. Quinn,\" Lucius Fox said. \"Now, Wonder Woman, please put your hand down. Yes, you can go next.\"\n\n\"So then he tells me that not only do I have to make up my missing assignment, but I need to write an additional paper on the Power of Powers,\" Harley said, moaning.\n\n\"I can help you with your homework,\" Supergirl volunteered. A group of girls was hanging out in Harley's room.\n\n\"Would you?\" Harley said, sitting on the clothes on top of the books on top of her bed. It was hard to tell where the pile ended and the bed began, but that never seemed to bother Harley. \"That would be swell! Could you have it done by Tuesday?\"\n\n\"I said I would help, not do it for you,\" Supergirl said, sounding friendly but firm.\n\n\"But I'm sooo busy with the Battle of the Bands,\" Harley explained. \"We have less than a week to go!\"\n\n\"Don't I know it?\" Supergirl said. She was on Harley's amphitheater committee along with Wonder Woman and The Flash. They were charged with creating a band shell now that the temporary wildlife sanctuary had been moved out.\n\n\"Pleeeeease,\" Harley begged.\n\n\"No!\" Supergirl replied.\n\nHarley looked exasperated and blew a wisp of hair out of her eyes. \"Fine. Homework. Check. Moving on. What's everyone else up to?\"\n\nThe food committee was headed up by Steve Trevor. Wonder Woman had volunteered to be on that committee, too, as well as on the parking lot committee and the crowd control committee. With Hawkgirl as stage manager, Batgirl was assigned to check in the contestants. Big Barda was in charge of the sound system. And Bumblebee was talent wrangler. The entire Capes & Cowls Caf\u00e9 was the designated greenroom. With the waiting area sure to be packed, Bumblebee's ability to shrink made her the ideal candidate to weave in and out of the crowd of hopefuls.\n\n\"A greenroom is just what they call where the talent hangs out before they go onstage,\" Batgirl explained. She knew this, having been a contestant on TechTalk TV.\n\n\"But it's not green,\" Poison Ivy pointed out. \"Though I could fix that.\"\n\nSuddenly Harley leapt up and began jumping on her bed, holding her hands up so she wouldn't hit her head on the ceiling. \"It's time! It's time!\"\n\n\"What's time?\" Ivy asked.\n\n\"I'm going live in ten minutes to name the finalists,\" Harley blurted out. \"This is gonna be big! Maybe the biggest thing ever!\"\n\nAs Harley turned on her video equipment, Big Barda stopped in. \"Hey, Harley, me and some of the others are starting a glee club. I'm thinking of calling it Mighty Melody Makers. Wanna join us? It'll be fun!\"\n\n\"Fun! Who has time for fun?\" Harley said, looking serious, then grinning ear to ear. \"I'm about the make the big announcement and I barely have time to eat and sleep. There's no time for glee!\"\n\nAdam Strange and Arrowette were hard at work in Centennial Park. Since Harley anticipated overflow crowds, video screens were sent for. That way, spectators could watch off-site and still be a part of the festive atmosphere.\n\nAdam was in the air, hovering with the power of his jet pack. \"Is it straight?\" he asked, holding up the last giant screen that needed to be placed.\n\n\"A little to your left,\" Arrowette said. She pulled three arrows from her quiver in quick succession, took aim, and shot. Each arrow made a satisfying twang when she let go. They stopped an inch from Adam's hand and secured the screen to the wooden post.\n\n\"Hey, you almost hit me,\" he called out.\n\nArrowette reached over her shoulder for another arrow from her quiver. \"If I wanted to hit you, I would have,\" she said with a wink.\n\nHarley couldn't stop moving or talking. \"Thisis\u00adgonnabesobig! Iwouldn'tbesurprise\u00addifIbusted\u00adthe\u00adInternet\u00adwithsomany viewers!\" she said as she cartwheeled backstage.\n\n\"You're going to have to slow down if you want to be understood,\" Hawkgirl barked. The amphitheater was still under construction, but it was taking the shape of a giant open oyster shell. Katana had painted the impressive red-and-gold Battle of the Bands sign that hung overhead. Harley had asked her to add her HQ logo, and with the help of art teacher June Moone, Katana created one in neon lights.\n\nFor the stage itself, Poison Ivy had suggested that they reuse the dance floor. However, Hawkgirl pointed out that it was pocked and damaged due to one hundred tap dancers and various others.\n\n\"Besides,\" Harley was quick to remind everyone, \"this is gonna be BIGGER than the last special, and _**YOWZA!**_ We need lots and lots and lots of room for the live audience!\"\n\nPoison Ivy gave this some thought and absentmindedly began braiding her long red hair before proclaiming, \"I've got it! Let's use recycled wood from the old Schumacher Shoelace factory that was recently torn down!\"\n\nShe had barely finished her sentence when Wonder Woman and Supergirl were on the job. Not only did they think this was a great idea, but later they even stacked boulders they had recently cleared from an avalanche in Denver to flank the sides. For extra oomph, Poison Ivy created a cascade of fragrant flowers and then a huge canopy of tulip trees, saucer magnolias, and ropey vines to shade the audience. The result was spectacular, like nature herself was the architect.\n\nSteve Trevor had several Capes & Cowls Caf\u00e9 booths set up around the perimeter. Some sold tropical fruit and berry smoothies, others featured freshly baked cookie crisps in the shapes of musical instruments, and still others sold pizza, veggie burgers, and sweet potato fries. Each booth was manned by volunteers from the zoo and students from Super Hero High. Since the proceeds were going to the wildlife sanctuary, Principal Waller had offered community service points to those who helped out.\n\n\"The contestants are starting to arrive,\" Bumblebee reported from the greenroom. \"Ukulele United from the island of Kauai is here and they're passing out fresh orchid leis to everyone. They smell lovely.\"\n\n\"Got it!\" Hawkgirl said into her headset. She turned to Harley. \"One hour to showtime!\"\n\nThe Capes & Cowls Caf\u00e9 greenroom looked the same as the restaurant always did: cozy and comfortable, yet with a trendy edge to it. The only difference was that Katana had created several posters, one for each of the finalists, with their names and Welcome! on them.\n\n\"We're better-looking than that,\" Captain Cold said, sending a chill around the room. \"Well, I am. Not so sure about them.\" He laughed as other members of CAD Academy's heavy-metal band hauled in their instruments.\n\nMagpie, the drummer, was dressed in a tattered black dress anchored with steel-tipped boots. Ratcatcher had spiked his hair so that it stood up in every direction and could have doubled as a weapon. Captain Cold wore ice-blue mirrored sunglasses and a distressed leather bomber jacket with CC on the back.\n\n\"Hey!\" Harley yelled at Ratcatcher. Big Barda stood behind her and crossed her arms. \"Stop tearing the other groups' posters down or ya get disqualified.\" She turned to the camera and when the red light went on, Harley began, \"In a short while we'll go behind the scenes of HQ's first-ever Battle of the Bands. And at the top of the hour, stay tuned for the live competition where you, the Internet viewers, have a front-row seat!\"\n\nShe was grateful that Batgirl had created a portable video control console so that she could record, broadcast, and even edit her shows from anywhere. And the best part was that the whole thing could fit in her pocket!\n\nAs the other musicians entered, Beast Boy, whom Harley had designated as the official greeter, welcomed everyone with a \"Hello! Hello! Congratulations on making it to the finals!\" Then, with her trademark efficiency, Batgirl logged them in. Finalists included Female Furies' Apokopella group with Mad Harriett as the lead, a marching band from Korugar Academy, and Black Canary and the Birds of Prey. Soloists included former Super Hero High student Mandy Bowin on violin, and current Super Hero High student Cheetah, who, in an uncharacter\u00adistically low-key move, had not made a big deal about her singing.\n\n\"I'm going to catch everyone off-guard,\" she confided to Sapphire, who nodded her approval, \"and knock 'em over when they hear me sing.\"\n\nThe contestants eyed each other. Some looked nervous, like Mandy, but most were supremely confident, bordering on belligerent.\n\n\"Oops!\" said Mad Harriett as she pushed her unruly green hair off her orange face and \"by accident\" spilled apple juice onto Silver Banshee. \"Sorry about that.\"\n\nSilver Banshee grabbed a napkin and wiped the fruit juice off her costume. Her fluorescent blue pupils began to glow, and her skin flushed briefly before turning pale white again.\n\n\"Tell me it was an accident,\" Silver Banshee said. Her bandmates stood behind her, frowning.\n\nFemale Furies Stompa and Speed Queen stood behind Mad Harriett. All grinned menacingly.\n\nBig Barda, who had once been a Fury, stepped between the feuding bands. \"No fighting, please.\"\n\n\"Did someone say fighting?\" Captain Cold asked. He whipped out his ice blaster and covered the whole caf\u00e9 with icicles.\n\n\"That's cold,\" Cheetah said, racing around and breaking the icicles as members of Korugar Academy's band marched over them, making crunching noises.\n\n\"Ouch! Hey, watch it!\" Ratcatcher squealed when he was accidentally stomped on.\n\nSoon mayhem broke loose as musical instruments flew. When Mandy Bowin retreated to a corner booth, Barda hurried to protect her. \"I'll keep you safe,\" she promised. \"By the way, I'm a big fan. Your music has meant so much to me. It's beautiful and soothing and happy.\"\n\n\"Thank you,\" Mandy said, cradling her violin and ducking nanoseconds before a pizza would have hit her head.\n\n\"Contestants! Please cease!\" Silver Banshee said. When everyone ignored her, she raised her voice to supersonic levels. \"STOP. NOW.\"\n\nThe whole room rattled. Everyone looked startled and covered their ears.\n\nThe room went quiet.\n\n\"Thank you,\" Silver Banshee said. A sly smile crossed her face. \"We'll fight it out on the stage.\"\n\nHarley basked in the applause. As it washed over her, she felt like she was home. \"Harley! Harley! Harley!\" the audience chanted. Each time the curtains of flowers behind her billowed, it sent out a fragrant scent so lovely, no one would have known that just days before, wild animals had congregated there.\n\n\"Aww, you guys!\" Harley said, looking out over the massive crowd. \"I CAN'T HEAR YOU!\" She cupped her ears. \"And what about y'all watching from Centennial Park? I can't hear you either!\"\n\nAs the roars from the two crowds meshed, creating a tsunami of sound, Hawkgirl said into the headset, \"Harley, we're ready to go. Announce the first contestant.\"\n\nHarley cleared her throat. This was her moment. A chance to redeem herself from those who were mad at her for the twelve-way tie at the Dance-O-Rama. A chance to become a force to be reckoned with in the entertainment industry. A chance to become the most watched Internet star ever. She was ready.\n\n\"Good afternoon, everyone!\" Harley shouted. \"Aww, look at all of you. Sit down, sit down. Now stand up, now sit down.\" Everyone followed along, laughing. \"Thank you for coming to Harley's Quinntessentials Battle of the Bands! Our first contestants hail from CAD Academy. Give it up for the heavy metal sounds of Cap'n Cold and Crew!\"\n\nAs Captain Cold smirked his way onstage, the audience cheered. Heavy metal had never sounded so fresh, so cool, so crisp, so...heavy. Even traditionalists like Crazy Quilt were jumping up and down in the aisles. The band finished to uproarious applause.\n\n\"Remember Cap'n Cold and Crew when you vote after the show,\" Harley said. \"And remember, you only get one vote, so make it count.\"\n\nEveryone cheered again. Harley beamed. She wondered if her parents were watching.\n\n\"And now we're going to take it down a notch, or two, or three. Just listen to the violin serenade of Mandy Bowin\u2014or as some call her, Virtuoso!\"\n\nMandy looked small. She wore a simple peach-colored dress with a Peter Pan collar and pink ballet slippers. Her brown hair was held in place by a light blue headband. The audience stirred nervously. How could this girl, a mere mortal, even begin to compete with all the heavy metal that had just exited the stage?\n\nWith her eyes closed, Mandy lifted her violin and tucked it snugly under her chin, then gracefully lifted the bow in the air. No one was prepared for what happened next.\n\nIt was as if a soothing breeze blew across the amphitheater, sailed over the crowd watching in Centennial Park, and then swept into the new wildlife sanctuary at the zoo. As the musical notes from Mandy Bowin's violin alighted over the audiences and animals, all agreed they had never heard music this beautiful.\n\n\"I'm not crying, you're crying,\" Beast Boy said to Cyborg.\n\n\"I'm not crying, she's crying,\" Cyborg said.\n\n\"I'm not crying, he's crying,\" Frost said, pointing to Vice Principal Grodd as she whisked icicles from her face.\n\nHarley captured it all on camera. She knew that tears and fears were two things that audiences loved, and the tears were flowing.\n\nThe smile on Mandy's face was as serene as her music. Suddenly it didn't matter that she wasn't a super hero. When Mandy Bowin played her last note, the audience sat enraptured. It was Beast Boy who leapt to his feet first, followed by thousands of others. The cheering more than doubled the decibels of the heavy metal band. Even Ratcatcher was on his feet cheering, until Captain Cold hit him with a chilly blast.\n\nAfter several bows, Mandy sheepishly walked off the stage. She grinned at Wonder Woman, who had been standing in the wings. \"Virtuoso,\" Wonder Woman whispered.\n\nAs the camera lingered on the two, Harley informed her audience, \"Mandy was once a student at Super Hero High, but left to pursue her dream to become a musician. At the same time, Wonder Woman left Paradise Island to pursue her dream to become the best super hero she could be. Talk about amazing second acts! And now we pause for this important message.\"\n\nWhile Steve Trevor explained that all the profits from the food venues would go to the zoo's new wildlife sanctuary, Harley readied for the next group.\n\n\"This is going great, isn't it?\" she said to her crew.\n\nBatgirl, Hawkgirl, and Barda all nodded. It was going well. Smoother than anyone could have imagined.\n\nWhen Steve was done, there was a polite round of applause led by Wonder Woman. Then Harley cartwheeled back onstage. \"Is this on?\" she said, playfully tapping the microphone. \"Next up, it's a little bit of rock, a little bit of folk, a little bit of rap\u2014and it's all amazing music. Let's say aloha to Ukulele United, the all-ukulele band from the island of Kauai!\"\n\nCheers rose and hands went up in the air as the group from Hawaii hula-danced onto the stage, throwing lush leis of orchids into the crowd. Then, with a single string from a single ukulele, the music began. One at a time, the others joined in, until, united in music, the lead singer launched into a heartwarming ballad about the Hawaiian islands.\n\nJust as Ukulele United were rising to the crescendo, Bumblebee flew onstage, grew full-sized, and took the microphone.\n\n\"What are you doing?\" Harley yelled. \"We're live!\"\n\n\"Sorry,\" Bumblebee said to Ukulele United, \"but I have to do this.\" She faced the audience. \"This is an emergency. Audience members, we're going to need you to get to a safe place. All Supers in the vicinity, we have to Save the Day! Animals from the zoo and wildlife sanctuary are on the loose and on a rampage!\"\n\nThe crowd began to panic. Supergirl, Cheetah, and the others tried to calm nerves as they led the crowd to safety, which was quite an effort, considering the thousands of panicked people there.\n\nHarley turned to the camera and reported, \"We interrupt this incredible and popular Harley's Quinntessentials Battle of the Bands with this important announcement. There's chaos at the Metropolis Zoo\u2014and so we super heroes gotta go Save the Day!\"\n\n\"What's the scoop?\" Harley asked as she arrived on the scene. Lois Lane was stationed next to an open cage.\n\n\"The villain Lion-Mane has released all the animals. They're running wild,\" Lois informed her.\n\n\"Who's running wild, the people or the animals?\" Harley asked.\n\n\"Both!\" said Lois. \"Everyone's in a panic!\"\n\n\"That's not like them,\" said Beast Boy.\n\n\"The animals are normally well taken care of,\" Poison Ivy chimed in.\n\n\"We'll get to the bottom of this,\" said Wonder Woman, who was flying overhead. \"But first, let's make sure everyone is safe!\"\n\nAs animals and people ran past, Bumblebee buzzed in and said, \"I've got a lock on Lion-Mane. He's headed toward the Metropolis Museum of Art.\"\n\n\"Of course!\" Batgirl said, looking at her wrist computer. \"A new exhibit is going to open tomorrow featuring a priceless jeweled lion statue. It's worth millions!\"\n\n\"The animals on the loose are a decoy\u2014and they're also meant to be a distraction,\" Hawkgirl deduced.\n\n\"Precisely,\" added The Flash.\n\n\"Camera off,\" Batgirl said to Harley.\n\n\"Aww,\" Harley moaned. \"Think of the viewers!\"\n\n\"I'm thinking of how to sneak up on Lion-Mane,\" said Batgirl. \"And sneak means he can't know that we know. C'mon, we have a lion to tame!\"\n\nWhile Harley and the others rushed to the museum, the other Supers, led by Beast Boy, rounded up the animals and made sure that frightened citizens were safe.\n\nBatgirl checked in with Supergirl, who informed her, \"Beast Boy says that Lion-Mane told the animals that they were all going to be shipped away to desolate desert islands. That unless they ran away, they would no longer be cared for. They panicked.\"\n\n\"That makes sense,\" said Bumblebee. \"The animals are usually so kind and loving. And the zookeepers treat them like family, and vice versa.\"\n\n\"Well, this family is outta control,\" said Hawkgirl. \"The wrong information can do that.\"\n\nWith super-speed and stealth, Harley and the Junior Detectives approached the museum. The guards were nowhere to be seen. The museum stood eerily empty.\n\nBatgirl consulted her holographic museum map. It glowed green in the air in front of her. \"He's there,\" she said, pointing. \"Where the red light is glowing. It senses body heat. Several people are locked in what looks like a storage room. They must be the missing guards.\"\n\n\"They'll be safer there,\" Hawkgirl noted. \"I'll take the north hallway. The rest of you take the stairs and employee elevators. We need to sneak up on Lion-Mane.\"\n\n\"I'll go with you!\" Harley said, turning on her camera.\n\nThe Flash gave her a stern look. \"If your viewers can see what we're up to, so can Lion-Mane. You can't have that on.\"\n\n\"Aww, you're no fun,\" Harley chided him. But she turned off the camera.\n\nAs Harley powered down her camera, the other Supers shifted into stealth mode and scattered in search of Lion-Mane. Harley had just looked up when she saw Beast Boy's shadow. \"Smart move!\" she said, nodding appreciatively. \"You're looking like a lion. He's a lion. Lion versus lion.\" For good measure, she added, \"ROAR!\"\n\n\"ROAR!\" came back at her. Only this one shook the walls.\n\n\"Beast Boy?\" Harley asked as someone else stepped out of the shadows.\n\nIt was Lion-Mane, and he was carrying the famous, fabulous, bejeweled lion sculpture. \" _ **YOWZA,**_ you got a mighty roar, Mr. Mane,\" she said, reaching for her mallet. \"Have you ever thought about auditioning for the Battle of the Bands as a singer?\"\n\n\"You better get out of my w\u2014 Uh, do you really think I have a great roar?\" Lion-Mane asked as he loosened his grip on the statue. He was distracted by her compliment and the thought of performing. \"Really? Be honest.\"\n\n\"Oh, sure,\" Harley said, taking a step back. \"But first, you gotta think about this.\"\n\nWith that, she lifted her mallet high and, with all her might, brought it down on his foot. Surprised, Lion-Mane dropped the statue and roared so loud the building rocked, alerting all the Supers.\n\nAfter Lion-Mane's capture, rain had put a damper on the contest. And with the amphitheater being outdoors, even good intentions couldn't stop the storms.\n\n\"We'll reschedule!\" Harley proclaimed as the rain threatened to wash her away. But several contestants couldn't make it back the following week, so Harley was forced to scrap the show.\n\n\"The good news is that even though the Battle of the Bands was canceled, we still made money for the wildlife sanctuary,\" Steve Trevor said as he gingerly picked up some wayward rat traps left behind by CAD Academy. It didn't look good for a caf\u00e9 to have those lying around.\n\n\"That is good news,\" Harley agreed while she checked her messages. She had gotten several comments about how the BOB had ended. Many said the show should have gone on. However, MH234 wrote: \"The best Battle of the Bands in history! Love how it didn't end so that we have something to look forward to on your channel!\"\n\nHarley knew she could always count on MH234 to cheer her up.\n\nAs the Supers headed back to campus, the rain stopped. \"Look!\" Katana said, pointing.\n\nThey all tilted their heads back. \"What is that?\" Harley turned on her camera. Bubbles the size of baseballs were floating down from the sky. When the sun's rays hit them, they looked like round rainbows. Supergirl popped one, and a flyer fell to the ground.\n\nKatana read, \" 'In town for only twenty-four hours: the Krazy Karnival! You have only one day to experience the chills, thrills, and delights of the world's greatest and grandest amusement park!' \"\n\nAs the Supers gathered the wayward flyers that littered the ground, a holographic billboard lit up the sky. A jolly man wearing a colorful hat trumpeted, \"Come one, come all, to the new and improved Krazy Karnival! I'm J.J. Tetch, the new owner, and you're in for big surprises!\"\n\n\"Surprises?\" Harley said as her eyes twinkled. \"No one loves surprises more than me!\"\n\nShe immediately began broadcasting. \"The world-famous Krazy Karnival is coming to Metropolis, and even if you can't be there...you can! That's because I'll be recording all the fun and all the _**KRAZINESS**_ live on this channel!\"\n\n\"It does sound like fun,\" Miss Martian said. Harley didn't even know she was there. \"I wish I could go.\"\n\n\"Why can't you?\" Harley asked.\n\n\"Crowds make me nervous,\" said Miss Martian. \"I have trouble maneuvering through them.\"\n\n\"Stick with me,\" Harley generously offered. \"I'm great at maneuvering. Plus, this is my chance to log the most viewers ever. I couldn't video the Junior Detectives sneaking up on Lion-Mane after I hit his foot. Or the great catch I made when he dropped the statue. Oh, sure, once he was in custody, I got it all on camera. But then, so did Lois Lane, who also had footage of the Supers leading the animals back. And that scene of a green monkey\u2014Beast Boy\u2014cradling a scared baby capuchin monkey and reuniting her with her mother. That was ratings gold for Lois!\"\n\n\"It's not all about ratings, is it?\" asked Miss Martian meekly.\n\n\"There's nothing wrong with big ratings,\" Harley said as she burst a bubble. \"In fact, that's my goal: to have the most viewers in the world!\"\n\nPied Piper was at the piano. He nodded to Cheetah, who was staring at the ground. As the teacher's fingers nimbly hit the piano keys, Cheetah raised her head. Harley watched her face transform. It was confident, yet vulnerable at the same time. Then that smile appeared. The famous Cheetah smile, as if she knew something you did not.\n\nCheetah winked at Harley and began to sing. It was a torch song\u2014the heartfelt lyrics were about love and loss. As the last note hung in the air, the room was silent. Pied Piper leapt up. \"Bravo, bravo!\" he shouted. \"I could feel that emotion. Beautiful. Seriously, beautiful. Thank you.\"\n\nCheetah took a bow, and when she swept past Harley, she said, \"I would have won the Battle of the Bands. You should have a rematch.\"\n\nIn the front of the room, Cyborg had turned himself into a one-man band\u2014a synthesizer with an electronic keyboard, and speakers coming out of his boots. As the rest of the class rocked to the beat, Harley kept thinking about her show. What could she do next to top it? Surely her viewers were expecting something spectacular. Was it enough to just live stream the Krazy Karnival?\n\n\"Harley?\" Pied Piper called out. \"You're up!\"\n\nEach student had been charged with doing a two-minute performance piece, whether it was singing solo or in a group, playing an instrument, or even karaoke.\n\n\"OH!\" Harley cried. She hadn't prepared. \"For my piece, I'm going to...to...\" She looked around, desperate for inspiration. Suddenly, she saw it: Green Lantern was holding his tuba. \"We're doing a duet!\" she announced. As Harley dragged him to the front of the room, she whispered, \"Just go along with me.\"\n\n\"What? No,\" Green Lantern protested, running his hand through his thick brown hair. \"I'm doing a tuba solo\u2014\"\n\n\" _So low_?\" Harley quipped, grabbing the tuba. \"And how's about holding it _high_ for a hat?\"\n\nWhen she placed the mouth of the tuba over her head, the class howled with laugher.\n\n\"Harley, please stop. Instruments are not toys,\" Pied Piper cautioned.\n\nBut Harley couldn't hear him from inside the tuba. \"It's dark in here!\" she cried out comically, getting more laughs from the classroom.\n\n\"Give that back to me,\" Green Lantern insisted. When he tried to wrench the tuba off Harley's head, it was stuck.\n\nIn a panic to un-tuba herself, Harley began running around the room, knocking over instruments and students. Green Lantern was in pursuit, causing even more calamity and chaos. Soon, Supers had taken sides, yelling, \"Go, Harley!\" and \"Return the tuba!\"\n\n\"Come back!\" Green Lantern called out.\n\n\"This is epic!\" Beast Boy cheered. \"Harley, you're such a crack-up!\"\n\n\"The class clown does it again,\" Cheetah said, yawning. \"This is pathetic. She'll do anything for a laugh.\"\n\n\"Hello, Harley!\" Dr. Arkham was studying his desk through a magnifying glass. When he sat up, he was still holding it in front of his face. It made his left eye look huge. \"I lost something,\" the school counselor explained. \"But now I can't remember what it was.\"\n\n\"Was it this?\" Harley asked, looking around. She grabbed a globe of Mars.\n\n\"Aha!\" Arkham took the red planet from her. \"Nope,\" he said, putting it down next to a tall stack of books, all written by him. \"But I'll leave that here, in case I need to find it sometime.\"\n\nHarley settled into her favorite chair. Arkham's office was dark and crowded with important-looking books, piles of paper, and an impressive stack of unopened mail. It smelled like the forest after the rain. Harley liked it in here. It was quiet\u2014somber, even. She didn't feel the need to entertain when she was with Dr. Arkham.\n\n\"What's on your mind?\" he asked. \"Are you still feeling lonely and blue?\"\n\n\"Blue, green, whatevs,\" Harley quipped. \"What's not on my mind? You know me. Always thinking, thinking, thinking. Who knows what goes on in my noggin? It's so busy up there that sometimes I can't sleep.\" Harley paused. \"Hey, doc, is it possible to think too much?\"\n\nAs Arkham stroked his beard, she wondered why his head was bald when he had so much hair on his face.\n\n\"Hmm. Um. Yes. Sometimes we do tend to overthink,\" he said. \"Tell me, what would you like to talk about today?\"\n\n\"Aww, nothing,\" Harley said, jumping up and looking out the window. It was a sunny day and she could see Wonder Woman and Supergirl playing catch with a teacher's car. Harley turned back to the counselor. \"It's just that some of my friends are pressuring me to spend more time with them.\"\n\n\"Go on,\" Arkham said.\n\n\"I would, but I'm busy, busy, busy. I've got my Harley's Quinntessentials to run, you know. It's a lot of work making people happy! Serious business, I tell ya. And it's not just on my Web channel. In person, too, I have to think of jokes and say funny stuff, and get all, you know...Harley-esque!\"\n\n\"Go on,\" Arkham said.\n\n\"Whenever I see someone looking stressed, I want to cheer them up. So I'll tell a joke, or do a super-duper gymnastics move, or whatevs, even if I'm not feeling so great myself.\"\n\n\"Go on,\" Arkham said.\n\n\"I dunno,\" Harley said, slumping back into her chair. \"Lots of kids call me the class clown, like that's a bad thing. And it makes me feel funny. Not ha-ha funny, but weird funny.\"\n\nArkham looked like he was asleep. But then he blinked his eyes open. \"Harley,\" he said. \"People underestimate the benefits of being happy. Laughter can make people feel better. You get that. Most don't.\"\n\n\"But what about hanging out with my friends? I would, but there's no time! Oh, wait,\" Harley said, interrupting herself. \"I gotta cut today's session short. I'm busier than a bumblebee. There's someplace I gotta be.\"\n\n\"Harley!\" Dr. Arkham called after her. \"I need to ask you something.\"\n\n\"Yeah?\"\n\nHe waved his magnifying glass in the air. \"Why am I holding this?\"\n\nThat afternoon in detention, Green Lantern refused even to look at Harley. \"You could at least have warned me,\" Green Lantern griped. \"We could have rehearsed.\"\n\n\"I didn't know until I saw the tuba,\" Harley admitted. \"It was so big and shiny!\"\n\n\"No talking!\" Lucius Fox called out. Unlike Grodd, he did not appreciate detention duty.\n\n\"You have to admit, we had fun, right?\" Harley whispered to Green Lantern. \"Everyone was laughing.\"\n\n\"Shhh,\" Big Barda said.\n\nGreen Lantern gave Harley's question some thought. \"Maybe a little, but it's not worth getting in trouble for.\"\n\n\"I dunno,\" Harley mused. \"I kinda thought it was totally worth it. It sure cheered up the room!\"\n\nWhen Harley plopped her tray down on the table, Katana turned to her and said, \"We were just talking about fan mail.\"\n\n\"I never get any,\" said Miss Martian, \"unless you count the letters from my mom. She sends tons of them.\"\n\nKatana sliced her veggie lasagna into perfectly square bite-sized pieces. \"My parents email me twice a week,\" she noted. \"Although I wouldn't call it fan mail. It's more like how-did-you-do-on-that-test mail.\"\n\n\"I hear from my grandmother all the time,\" Hawkgirl said. She was eating another helping of mac 'n' cheese 'n' mushrooms. \"Half of it's fan mail, the other half is her worrying if I'm eating enough. Will someone take a photo of me with this?\" She held up her plate and smiled at Batgirl.\n\nBatgirl handed Hawkgirl's phone back and said, \"I hear from my dad constantly!\" As if on cue, Commissioner Gordon waved to his daughter, then sat down at the faculty table. \"Harley,\" Batgirl asked, \"do your parents write to you?\"\n\nHarley poked holes in her chicken potpie. \"Oh, they're so busy with their world travel, they don't have time to write,\" she said.\n\n\"What do they do?\" asked Big Barda.\n\n\"Tightrope walkers,\" Harley said quickly. \"All around the world, everywhere there are tightropes.\"\n\nHawkgirl sent the photo Batgirl had taken of her to her grandmother, then said, \"Harley, I thought you told me your parents were accountants.\"\n\n\"Did I say that?\" Harley asked. \"Well, they used to be. Now they're librarians.\"\n\nBatgirl perked up. \"I didn't know that!\"\n\n\"Oh, sure, they have a library in their motor home, and they only have cookbooks, because they used to be professional chefs.\"\n\n\"So they're tightrope-walking accountant librarian chefs?\" asked Barda.\n\n\"They were chefs and accountants, but that was after they sold spaceship insurance,\" Harley said, not meeting anyone's gaze.\n\n\"Wait!\" Beast Boy leaned over from the next table. \"You told me you were raised by raccoons!\"\n\nHarley laughed. \"Did I say that? My bad. I meant coyotes.\"\n\n\"You must get tons of fan mail because of your Web channel,\" Miss Martian said. \"Tell us about it!\"\n\nHarley was glad to change the subject. She tried to look modest. It was true. She probably got more fan mail than the others did. In fact, she was always telling her viewers to \"Let me know!\" And they did.\n\n\"I answer all my fan mail right away,\" Wonder Woman was saying.\n\n\"I save them up and answer once a week,\" Supergirl said. \"I noticed I get more emails after a battle or Save the Day.\"\n\n\"I don't get as much as the two of you,\" Bumblebee offered. \"But I love it when fans send me honey. There's this group who calls themselves the Bumblebee Honeys, and they're my unofficial fan club. I always send them a handwritten thank-you note when they send honey. Anyone get anything interesting recently?\"\n\n\"I just got this,\" Harley said, pulling a mirror out of her pocket.\n\nThe girls gathered around. A round of \"oohs\" went up in the air as they admired the gorgeous hand mirror.\n\n\"That's carved from teak,\" Katana said, inspecting the craftsmanship. \"Look at the inlaid pearl.\"\n\n\"It looks old,\" noted Supergirl. \"Like a family heirloom.\"\n\nBumblebee held it up to the light. \"Wow, my reflection looks like I'm actually in the mirror, like it's three-D.\"\n\nHarley took it back and peered at herself. She wiggled her ears. \" _ **WOWZA!**_ \" she exclaimed. \"You're right. I look more real in the mirror than I do right here!\"\n\n\"Who sent it?\" Supergirl asked.\n\nHarley shrugged. \"Don't know.\"\n\nNo one was surprised by this. School packages were often delivered by drones or birds, or on the backs of rockets.\n\n\"There was no name or return address on it,\" Harley continued. \"But the note said: 'To Harley Quinn, from your biggest fan.' \"\n\nFor the next few days, Harley was never without the mirror. She used it when she needed to fix her hair, or after eating. \"No one wants to see a close-up of a Super with food in their teeth, am I right?\" she asked. Harley even used it to practice her lines for her Web channel.\n\n\"Harley, please put your mirror away,\" Pied Piper said. \"I'd like you to focus on what I'm about to show you.\" He began playing a video of singers from around the world doing their interpretation of the same song. \"Focus, Harley. Focus.\"\n\nEasy for him to say, she thought as she tucked the mirror back in her pocket. She looked up. Now, what was that she was supposed to do?\n\nBack in her room, Harley took the hand mirror out again. \"Hello, HQ fans,\" she said as she watched herself. \"Harley here, asking, How would you like a super-duper exclusive behind-the-scenes Harley-riffic special at Krazy Karnival?\"\n\nShe was pleased with her idea. It was sure to attract attention. After all, everyone had heard of Krazy Karnival. So why not turn the amusement park's twenty-four hours in Metropolis into a mega-special, Harley-style?\n\n\"Let me know,\" Harley practiced saying. \"Send me a message!\" She was about to put the mirror down, but stopped and gasped.\n\nIt looked like the Harley in the mirror winked at her.\n\nIt was impossible to get hold of Jervis \"J.J.\" Tetch, the Krazy Karnival's new owner. Not even Batgirl could find him, and if she couldn't, then no one could.\n\n\"He's elusive, that's for sure,\" Batgirl said as Harley leaned over her shoulder and tapped the computer screen. \"Um, Harley, personal space, remember?\"\n\nHarley stepped back.\n\nFrom what Lois reported, Harley knew that J.J. had recently taken over the amusement park, but he was something of a mystery man.\n\n\"Maybe I should do this later,\" Batgirl said as Harley did a tumbling routine and nailed her landing a few inches from Batgirl's workshop area.\n\nThere were lots of sharp tools and confusing gadgets and wires and whatnots. Harley opened a black velvet box and examined a teeny-tiny silver pellet nestled beside a ring.\n\n\"No, no, no! Search for Tetch now. I'll be quiet,\" promised Harley as she held the pellet to the light.\n\nBatgirl let out a huge sigh. She sighed a lot around Harley. \"Careful with that. Remember when you put a micro-camera on Bumblebee?\" Harley nodded. \"Well, that gave me an idea, and you're holding it.\"\n\n\"What is it?\" Harley asked, squinting at the object.\n\n\"I've been toying with the prototype of a micro ring-activated drone camera,\" Batgirl said, taking the pellet from Harley. \"I'm calling it the QuinnCam.\"\n\nHarley beamed. It was named after her! She slipped her mirror out of her pocket and said into her reflection, \"Harley Quinn here, reporting live via the QuinnCam!\"\n\nThe bubbles started slowly, drifting down from the clouds like a summer's rain. Only instead of showers, it felt like happiness blanketing Metropolis. Supergirl spotted them first as she flew back to school, having spent the night at her Aunt Martha and Uncle Jonathan's farm.\n\n\"The Krazy Karnival has arrived!\" Supergirl announced as she flew around the dorm, throwing cookies to everyone she passed. Aunt Martha always made enough for the entire school.\n\nHarley reached for her mallet and camera. As an afterthought, she tucked the velvet box from Batgirl's workshop into her pocket, just in case. \"It's showtime!\" she announced, looking in her hand mirror. \"Miss Martian, let's go!\"\n\nFor days Harley had been publicizing it: \"Coming soon, the Krazy Karnival on Harley's Quinntessentials, streaming live!\" And in what she claimed was a spur-of-the-moment genius idea, she promised, \"Music fans, get this! We're gonna have a rematch of the HQ Battle of the Bands at the Krazy Karnival! So tune in to see who the winner will be!\"\n\nShe hoped J.J. Tetch would be as enthusiastic as she was. Should she have gotten his permission first? Harley brushed away her doubts. Better to do first and ask later, she reminded herself. \"Right, Miss Martian?\" she said as she pushed past the crowds toward the sights and sounds of the Krazy Karnival.\n\n\"Huh?\" the mind reader asked.\n\n\"Are you reading my mind now?\" Harley said.\n\n\"Um, no,\" Miss Martian protested. \"Mind-reading is my superpower. I only use it when I have to, for the good of the world or to save lives.\"\n\nHarley laughed. \"There's so much going on in my mind, I doubt you could read it anyway. Try!\"\n\n\"I don't think so,\" Miss Martian demurred.\n\n\"No, go ahead,\" Harley insisted. \"You have my permission.\"\n\n\"Okay,\" Miss Martian said, closing her eyes. After a couple of seconds, she said, \"It's confusing, that's for sure!\"\n\nHarley grinned proudly.\n\nNear the entrance was a long line. It looked like the entire population of Metropolis was in attendance, in addition to most of nearby Gotham City, and every place in between. Schools from every country and galaxy were represented, since students got in free. And, of course, everyone from Super Hero High was there, except for the teachers and Principal Waller, who were at an Excellence in Education Conference on Upsilon Andromedae B\u2014the planet, not the band.\n\nHarley was in heaven.\n\n\"We are going to have the best day ever!\" she enthused.\n\n\"I'm not sure,\" Miss Martian said, looking at the crowds. She started to fade. \"There are so many people here.\"\n\n\"Exactly!\" said Harley. \"You read my mind! The more the merrier, and the merrier the more viewers, and the more viewers the more popular Harley's Quinntessentials! What are we waiting for? Let the adventure begin!\"\n\n\"Excuse me, pardon me, excuse me, Harley Quinn here, and I need to be there!\" she said, pointing.\n\nThe giant bubble machine rose from the center of the carnival looking like a colossal jukebox with a rainbow of bubbles shooting out as music played. As Harley made her way past the crowd of excited guests, Miss Martian trailed along, looking at the video screens that were everywhere. On them, J.J. Tetch was touting, \"You're in for the time of your life!\"\n\n\"Hurry!\" Harley called out. She wasn't sure if her friend was way behind or had turned invisible again. \"Stick with me, Miss Martian, and you'll have a day you'll never forget,\" Harley promised. \"You can help me put up my cameras all over the place. The more the better!\"\n\n\"Maybe I'll just go back to school,\" Miss Martian said, out of breath. \"I have a good book I'm in the middle of. It's called _The Shout of the Clam_.\"\n\n\"Look! It's the Green Team! Helloooo, Green Team,\" Harley said, waving. The teens shouted and waved back.\n\n\"Did you get permission yet?\" Miss Martian asked as Harley hurried her along, placing cameras along the way. \"To do a behind-the-scenes here and to host the Battle of the Bands?\" She saw band members lugging their instruments.\n\nHarley snorted. \"You sound like Hawkgirl.\"\n\nMiss Martian smiled shyly. \"Thank you.\"\n\n\"Not yet. Let's find the owner of the Karnival. He's gotta say yes to an interview! He'd be crazy not to. And as for the Battle of the Bands, I think it's totally necessary, don't you?\"\n\n\"But where will it be? When will it be? You just told the bands to show up, and...and\u2014\"\n\nHarley stopped and shook her head. \"Miss Martian, there's nothin' to worry about. We'll figure it out when we get there! That's the Harley way.\"\n\nMiss Martian's eyes widened with concern, but she clamped her mouth shut.\n\nAt the gates, cheerful carnival workers decked out in colorful light-up costumes handed out treats, like Raspberry Sugar Bombs that made hilarious exploding noises when you bit down and Cotton Candy Clouds so light that if you didn't eat them immediately, they floated away. Some students from Metropolis Elementary were carrying deep-fried hot dogs on sticks, while others ate gooey slices of pizza. Busloads of kids were screaming and laughing. Though they had just arrived, their chaperones already looked exhausted as they grabbed for the free light-up necklaces and \"Krazy\" hats as fast as the carnival workers could pass them out.\n\nBumblebee flew up to Harley and Miss Martian. \"Isn't this great?\" she asked. \"I'm going to get one of those hats with lace and sparkles. What about you two?\"\n\n\"No time for hats,\" Harley said. \"We can't wait to get to the fun, isn't that right, Miss Martian?\"\n\n\"Oh, well, a hat might be nice,\" Miss Martian said softly. Just then, a carnival worker with white wings and silver hair plopped a flowerpot hat on Miss Martian's head, much to her delight.\n\n\"It looks lovely on you!\" the worker said. \"I'm Silver Swan, and I must say, the peach and yellow flatter your green skin.\"\n\nMiss Martian blushed again.\n\n\"Here's one for you,\" a muscled man in yellow said to Harley. He was holding a jester's hat. Four colorful padded points that resembled donkey ears in red, green, purple, and yellow flopped merrily. Their tips were weighed down with jingly bells. But before he could put it on her, Harley started running.\n\n\"Look!\" she cried, dragging Bumblebee as Miss Martian hurried to keep up. \"Over there!\"\n\nBumblebee rode the Honeycomb Hideaway ride again and again. It was a delight to sit in the golden honeycomb cups as they sailed in and out of honeycomb-land, bees buzzing around in sweet harmony. The attraction's song was the kind that played over and over in your head, long after the ride was over. \"Bees, bees, a world of bees...\"\n\nMeanwhile, Harley and Miss Martian were mesmerized by the wonderland of lights and sounds and tech. Everything was so retro that it was daringly modern. Old carnival attractions had been expanded and improved and were awash in neon colors. Everywhere you looked there were digital displays controlling the overhead aerial holograms of jolly J.J. reminding everyone, \"As my guest at Krazy Karnival, it's your job to have fun!\"\n\nHarley entered the Game Zone. Her heart was racing. So much to see and do!\n\n\"Step right up! Test your strength! Who will show 'em how it's done?\" The burly carnival worker wearing a teeny top hat was holding up a sledgehammer. \"Ring the bell and win a prize! Who will be next? How about you, girlie? Think you're strong enough to ring the bell?\"\n\nHarley looked at the shiny bell at the top of the Strength Test. \"Hit this target down here,\" the carnival worker explained. \"And the puck goes up. If you're strong enough and it goes high enough, it hits the bell and you win a prize!\"\n\nThat was all that Harley had to hear. \"I'm game!\" she shouted, twirling her mallet in one hand. \"And I don't need your sledgehammer. I brought something of my own.\"\n\nSetting up her camera to make sure this was streaming live, Harley got in position. She gripped her mallet tight and focused on the target. Then, with all her might, she brought her mallet down hard.\n\nCheers rose all around. The worker's jaw hung open. Harley had hit the target so hard that the puck broke through the Strength Test game and sent the bell sailing into the air.\n\n\"There's my prize!\" Harley announced, catching the solid metal bell before it hit the ground. \"I'm keeping this!\"\n\nHarley bowed to the cheering crowd as she walked on. She could not believe how much fun everyone was having. It was as if they were in a haze of happiness. She hooted to Cheetah and Star Sapphire, who were wearing fetching hats (a beret for Cheetah, a crown for Sapphire) and taking selfies. Both smiled warmly at Harley and hooted back.\n\n\"I'm getting a headache,\" Miss Martian said softly. She adjusted her flowerpot hat.\n\nBy then, Bumblebee had left them to join Katana and Big Barda on the Rock 'n' Roller Coaster\u2014where you sat in faux boulders as they plummeted down a mountain. Also on the ride were several members of the Korugar Academy marching band. Every time the roller coaster went around a bend or dove down, instead of screaming they played their instruments as loudly as possible.\n\n\"This isn't helping my headache,\" Miss Martian whimpered.\n\n\"Too much fun can do that to you, if you're not used to it,\" Harley explained. \"I never get headaches.\"\n\n\"I think I need to sit down,\" Miss Martian said. She looked wobbly. \"You go ahead. I'll catch up later.\"\n\nThe Green Team made their way past, all holding up ice cream cones as if they were Olympic torches.\n\n\"Okay,\" Harley said to Miss Martian. \"See you in a bit. I'm off to find J.J.\"\n\nHe had to be somewhere, right? After all, this was his carnival. Harley knew what he looked like. The roundish face. The wide smile. The twinkling eyes. And that hat! That outlandish top hat with sparkles and lights on it. How could you miss someone who looks like that?\n\nAs Harley made her way past Katana and Frost holding hands and running under a shower of flower petals in the Orchid Zone, she paused. It was so cool that Supers who normally didn't hang out together were having fun here.\n\nThat was when she saw him. The elusive Jervis \"J.J.\" Tetch. Only, someone else was about to interview him first!\n\nThe reporter was just getting into position. She had checked her notes and made sure the microphone was on. But before she could say, \"Lois Lane here with Jervis 'J.J.'\u2014\" she was blindsided by a flurry of red and blue and black and white hurdling toward her.\n\n\"Harley?\" Lois said, just before she ducked.\n\nHarley had leapt over her and was now standing right next to the new owner of Krazy Karnival. With her back to Lois, Harley launched right into her interview.\n\n\"Hi, J.J. Tetch. You don't mind if I call you J.J., do you, J.J.?\" Harley asked. \"I'm\u2014\"\n\n\"Everyone knows who you are,\" said J.J. He was all smiles. \"You're Harley Quinn of Harley's Quinntessentials. I'm your biggest fan!\"\n\n\"I am? Um...you are?\" she asked. \"So nice of you! Hey, how about an exclusive interview? My fans wanna hear all about this Krazy Karnival of yours!\"\n\nHe looked over at Lois, who shrugged as if to say, \"Go ahead.\"\n\n\"Well, all right,\" J.J. said, handing Harley a jumbo whirly-swirly rainbow lollipop. \"Anything for you, Miss Quinn.\"\n\nHarley was beaming. This J.J. fellow was the nicest, smartest person in the world, she decided. \"So, J.J.,\" she began, \"can you tell my maybe millions and zillions of viewers why you bought this famous amusement park?\"\n\nHe straightened his massive hat, which had a habit of tilting to the right. \"My goal is to bring happiness and fun to the world,\" he said. Harley nodded so much holding her camera that it looked like he was jumping up and down. \"The Krazy Karnival was up for sale\u2014the last owner, who ran it for fifty years, decided to retire. So I bought it!\"\n\nThere was a crowd gathered behind them, waving to the camera.\n\n\"Tell us more,\" Harley urged.\n\n\"Well, this is the best Krazy Karnival ever,\" J.J. boasted with a sweep of his hand. His cheeks flushed red with delight. \"It's got new rides as well as hip and healthy new foods, with some new twists on old favorites,\" he added, \"like corn on the cob that pops into popcorn, and funny funnel cakes!\"\n\n\"Funny funnel cakes?\" Harley enthused as they started walking. \"I love funny, and I love funnel cakes!\"\n\nJust then a group of teens wandered past. \"Hey! Hey!\" J.J. called out. \"It's the famous Green Team, thrill-seekers and adventurers. I was hoping you'd be here. We made special hats just for you,\" he said cheerfully. He motioned to some of his workers, who promptly hauled a box of hats over to the carnival owner. \"Try them on for size!\"\n\nAs the Green Team put on their green-and-black bowler hats, Harley whispered to the camera, \"So far it's been all talk, but now\u2014fingers crossed\u2014maybe J.J. will give us a behind-the-scenes tour....\"\n\n\"Have fun, Green Team,\" J.J. was saying as he waved goodbye to them. He turned to Harley. \"How about a tour?\"\n\nHarley winked at the camera. \"It's like he was reading my mind!\" she exclaimed.\n\nJust as J.J. took the lead, he stopped to adjust Captain Cold's pirate hat. \"That's better. This really suits you.\"\n\n\"Thanks,\" Captain Cold said, offering him a smile that was uncharacter\u00adistically warm and pleasant. The rest of his Cap'n Cold and Crew heavy metal band giggled. \"Nice amusement park you have here, Mr. Tetch, sir,\" Captain Cold continued. \"I love it! And, Harley, we adore your Web channel! Can't wait for the Battle of the Bands. Krazy Karnival is the perfect place for it. I want to wish everyone good luck!\"\n\n\"You heard that right here,\" Harley said to the camera. \"Even Captain Cold loves Harley's Quinntessentials, and he's not the easiest person on the planet to please!\"\n\n\"You're hosting a Battle of the Bands here?\" J.J. asked, looking pleased. \"At my modest little amusement park? I'm honored!\"\n\n\" _ **WOWZA, YOWZA,**_ I thought you'd like that,\" Harley said to J.J. She watched as the gang from CAD Academy went their way, whistling and waving to those they passed, then noted: \"Captain Cold sure is in a good mood. I've never seen him like that before.\"\n\n\"Oh, Harley, my dear,\" J.J. said. His eyes were moist. \"You don't get it, do you?\"\n\n\"Get what?\" They passed another towering Sweet Treats stand, and he handed her an edible candy cup filled with the most delicious double-dipped chocolate berries she had ever tasted.\n\n\"It's you who people love. He was responding to your effervescence. It's why the masses tune in to your Web channel. To see you, Miss Quinn. They love you!\"\n\nHarley felt her face flush. She wondered if he was right. Was everyone watching her? Everyone at home? Everyone at school? A lot of times she'd have a really awesome segment, but Wonder Woman or some of the others would miss it.\n\n\"Sorry, Harley, I was doing homework,\" Batgirl would say.\n\n\"Did I miss a segment?\" Poison Ivy would ask. \"Chompy needed attention.\"\n\n\"Harley, you're always on, and I got stuff I gotta do,\" Beast Boy would tell her. \"But I'll try to catch up.\"\n\n\"They love me?\" Harley repeated. \"Aww, J.J., you're such a joker. But if you want to say it again, Harley's not gonna stop you!\"\n\nHe laughed good-naturedly. \"Harley, I've got a great idea that could benefit us both! But first, I have some business to attend to. I'll give you an exclusive interview shortly, but in the meantime, why don't we get you one of my famous Krazy Karnival hats?\" He looked around. His carnival workers were everywhere. Harley thought some of them looked familiar. \"M.M.!\" he called. \"A hat for Ms. Quinn. You must have missed her when she came in!\"\n\n\"So sorry, sir!\" the muscled man in the yellow costume with a green mask said. \"I'll make sure Harley Quinn gets a hat!\"\n\nAs he hurried away, J.J. ran after him. \"Wait up!\" he cried. \"Not just any hat. I want to make sure Ms. Quinn gets one of our super-special ones!\"\n\nHarley beamed. This was fun, fun, fun!\n\nAs fun as it would be to have her own super-special hat, Harley didn't have the patience to wait around. There were rides to go on, and games to play, and treats to eat, and...and...As she raced past the Ukulele United band cheering each other on at the Wheeeee Skeeeeeball game, Harley pivoted and doubled back. Something green had caught her eye.\n\n\"Why are you just sitting there?\" she asked a glum Miss Martian.\n\nNearby, Batgirl was at the Magic Donut Maker, watching eagerly as it shot free fresh hot mini-doughnuts into the crowd. \"Me next! Me next!\" Batgirl called out, jumping up and down, waving both hands.\n\nMiss Martian closed her eyes and brought her fingertips to her temple. \"My headache is getting worse,\" she said. One of the flowers on her hat was drooping over the front of her face. \"Maybe I should just go back to the dorm and take a nap.\"\n\n\"And miss all the fun?\" Harley asked. \"You don't want to do that.\"\n\nMiss Martian shook her head. \"I do want to stay, but this headache is crushing me.\"\n\n\"Maybe you should look for Poison Ivy,\" Harley suggested. \"She always knows what to do when someone is feeling a little blue. And your face is all blue right now. Here, look.\" She held out her hand mirror.\n\n\"Can you stay with me?\" Miss Martian asked. She didn't like what she saw in the mirror. \"I don't feel like being alone right now. This place is too crowded.\"\n\n\"But if it's crowded, you aren't alone,\" Harley reasoned.\n\n\"Never mind,\" Miss Martian said. Harley could see her disappearing as she began to walk away.\n\n\"Miss Martian, wait!\" Harley called after her. Even though she didn't want to miss a minute of fun, her friend needed her. \"Let's look for Ivy together.\"\n\n\"Have you seen Poison Ivy?\" Harley asked Bumblebee. She paused. \"You and Miss Martian are supposed to be having a great time like everyone else, but instead you're acting like worrywarts!\"\n\n\"I'm getting a strange feeling about this place,\" Bumblebee said. All the flowers on Miss Martian's hat wilted, making it look like it was melting. \"Um, maybe you should ditch the hat,\" she suggested.\n\nMiss Martian reached up. \"But I love this hat. It was a gift.\"\n\nHarley studied the hat. \"It's looking goopy, which is kinda Krazy-awesome!\"\n\nReluctantly, Miss Martian took off her hat. She took a deep cleansing breath. \"I'm starting to feel better already,\" she said, surprised.\n\nBumblebee lowered her voice. \"Have you two noticed how weird everyone is acting?\"\n\n\"Weird? No,\" said Harley. \"It's Krazy Karnival time, and everyone's in a great mood.\"\n\n\"Yeah, and what about those kids from CAD Academy?\" Bumblebee said.\n\n\"What about them? Captain Cold's been really nice,\" Harley pointed out.\n\n\"Exactly!\" said Bumblebee. \"When have you ever known him to be nice?\"\n\nHarley hesitated. \"Uh, never?\" Then she lit up. \"That's the power of the Krazy Karnival!\"\n\nJust then a familiar group of teens brushed past. \"Hey, Green Team?\" Harley yelled. \"Where are you going? You're heading the wrong way. That's the exit!\"\n\nThey all stared straight ahead as they kept walking. Lois Lane raced after them as they left the amusement park. \"How about an interview?\" she asked.\n\n\"Um, Harley?\" Miss Martian said, tapping her on the shoulder. \"Harley? Um, Harley, look!\"\n\nHarley followed Miss Martian's gaze. Bumblebee was staring at the same thing. Harley whipped out her camera. \"It's a Harley's Quinntessentials exclusive!\" she cried. She aimed the camera toward the sky, unable to believe what was happening.\n\nEveryone was gawking with their heads tilted up. Everyone but Bumblebee, who was now flying around warning people: \"Something is wrong, something is very wrong!\"\n\n\"Nothing's wrong,\" Beast Boy said, smiling. \"Everything's great!\"\n\n\"Yes, everything is lovely,\" Cheetah insisted as she adjusted her beret.\n\n\"She's right,\" said Katana. \"Cheetah's right again!\"\n\n\"Look!\" Bumblebee cried, pointing at the Bubble Machine.\n\n\"Whatever is happening, you're seeing it here first!\" Harley told her viewers. \"And so is the rest of the world, because we're broadcasting live on Harley's Quinntessentials!\"\n\n\"This is not good,\" Miss Martian warned.\n\n\"It's not good, it's _great_!\" said The Flash.\n\nThe others agreed. All around, Krazy Karnival guests were watching the High-Tetch Bubble Machine. Small bubbles had given way to one gigantic bubble...and it kept getting bigger and bigger.\n\n\"So pretty,\" said Frost, sighing.\n\nRatcatcher was crying and didn't care who saw him. \"The beauty of it!\" he said as Captain Cold nodded in agreement.\n\nBumblebee and Miss Martian glanced at each other as Harley kept her camera on. \"Viewers, I'm not sure what we're seeing here at Krazy Karnival,\" she reported, \"but it's crazy, all right, and I'm certain the carnival's breaking a new world record for the biggest bubble.\"\n\nBy now it was covering the entire Krazy Karnival, and closing up to encase it\u2014making it look like everyone was inside a giant dome. The bigger it got, the happier everyone was. When it sealed shut, all the guests started cheering and clapping.\n\n\"No, no!\" Bumblebee shouted. \"Don't you see? We're trapped in here!\"\n\n\"I've got the exclusive,\" Harley proclaimed. \"That's right, world, something weird is going on at Krazy Karnival, and to find out what it is, stay tuned right here on Harley's Quinntessentials!\"\n\nMiss Martian shook her head. \"This is bad. Really bad,\" she said before turning invisible.\n\n\"No!\" Bumblebee shouted. But no one paid attention to her. \"No, something's wrong.\"\n\nAmused, Harley recorded her using her blasters to try to penetrate the bubble. \"We're trapped!\" Bumblebee cried. \"The bubble has created a barrier to the outside world.\"\n\nHarley tossed her mallet up high into the air. It made a _klunk_ when it hit the bubble. \"It's totally solid!\" she said admiringly.\n\n\"It's not right,\" Miss Martian said, barely audible. Harley could hardly see her. \"And neither is anyone here. They're not themselves!\"\n\n\"We're happy here!\" said Hawkgirl as she twirled past them with her arms out to her sides. She was wearing a colorful Viking helmet. \"Why break the bubble when we can stay and have fun?\"\n\n\"That's right,\" Thunder and Lightning chimed in as they executed silly synchronized-swim moves to imaginary music. Their fluffy cloud-shaped hats looked the same, but on closer inspection, one was cirrus and one was cumulus. \"There's no better place to be than at Krazy Karnival right now,\" the sisters said in unison.\n\n\"I totally agree with that,\" Batgirl said, adjusting the rainbow-colored mortarboard on her head.\n\n\"Hey, can you guys help me try to break this bubble?\" Bumblebee asked.\n\n\"Whatever for?\" said Wonder Woman. She was wearing an oversized bonnet festooned with feathers. \"It looks so cool!\"\n\n\" 'Cause it will be fun to try?\" Bumblebee said.\n\n\"Fun!\" shouted Wonder Woman. \"I'm up for that. I love fun!\"\n\nBut when Wonder Woman tried to break the bubble, she couldn't. Nor could Supergirl, or Cyborg, or anyone else\u2014though it didn't look like they were trying very hard. In fact, it looked like they were goofing off.\n\n\"It's totally solid!\" Harley reported. \"I guess we'll just have to stay and have fun!\" Adding, \"Reporting live from inside the bubble.\"\n\n\"I've got a bad feeling about this,\" Bumblebee kept saying.\n\nMiss Martian, barely visible, nodded. The carnival workers were now leaving their posts. Rides went unattended. Some attractions, like the Tilt-a-Whirl, were spinning faster and faster, to the delight of the riders. Guests helped themselves to the food carts, and others were grabbing armfuls of prizes in the Game Zone.\n\n\"Excuse me. Um, excuse me?\" Miss Martian said to Silver Swan. \"Should you really be leaving the Tunnel of Love when there are still people inside? You work there, right?\"\n\nSilver Swan adjusted the feathered crown on her head and flapped her wings. \"I'm on my break,\" she replied. \"No one tells me what to do. Well, almost no one...\"\n\n\"But it's dark in the Tunnel of Love and it might not be safe. What if the ride goes off the rails?\" Miss Martian asked.\n\n\"We'll have to deal with that when it happens, then, won't we?\" Silver Swan winked at her before flying off to join Cupid, who was abandoning her post at the Slings and Arrows game.\n\nMiss Martian was speechless as chaos began to consume the amusement park.\n\n\"This is so exciting,\" Harley said, swinging her camera left and right, up and down, to get all the angles. She was glad they had placed cameras everywhere, and cut to other scenes using her remote video control board.\n\n\"Slow down. I'm trying to\u2014\" Miss Martian pleaded. \"I can't read anyone's mind. Please slow down!\"\n\nHarley stopped. \"Wait,\" she said, as if hit by a wayward thought. \"I thought you only read minds if you thought lives might be in danger.\"\n\n\"I don't know what's happening,\" Miss Martian said. \"But whatever it is, it's not normal. Look!\"\n\nHarley swung around to see Cyborg and Big Barda scrambling up the side of the Ferris wheel while it was still moving. The massive steel structure was several stories high and circling nonstop.\n\n\"Look at me!\" Cyborg yelled as he hung off one of the passenger cars hundreds of feet in the air.\n\n\"That looks like fun!\" Barda shouted. \"Wait for me!\"\n\nHarley gasped as Barda leapt from one of the moving passenger cars to another. \"We're loopy!\" she shouted as the Ferris wheel continued to spin.\n\n\"Faster, faster!\" Cyborg urged Supergirl, who was in the air twirling the ride as if it were a pinwheel.\n\nThe kids in the passenger cars were squealing with joy.\n\nSuddenly, something cold hit Harley in the head. \"Hey!\" she yelled, expecting to see Captain Cold or Frost. Instead, Mandy Bowin was standing behind her holding an empty ice cream cone.\n\n\"Oops!\" Mandy said, giggling. \"Sorry!\" she yelled as she ran away and joined Hawkgirl, who handed her another cone and said, \"Okay, now this time hit Cheetah!\"\n\nHarley laughed as she followed them toward an unsuspecting Cheetah. This was sure to be a classic for her \"Super Hero Goofs and Giggles\" segment. On her way, Harley passed Batgirl stuffing her face with fried candy bars. Batgirl had even stockpiled a bunch of them on top of her hat and was balancing them as she strode purposefully back toward the Magic Donut Maker.\n\n\"Hey, Batgirl,\" Harley called out. \"How many viewers are watching right now?\"\n\nBatgirl wobbled as she came to a stop, continuing to balance her sugary load. \"Wait, look at this!\" she said. Her hat almost fell off, but she pulled it on tighter. Everyone gathered around as Batgirl projected the screen into the air from her Bat-phone.\n\n\"This is Lois Lane reporting live from downtown Metropolis,\" she was saying. \"Multiple robberies are happening all over the city. There would probably be even more if there weren't so many citizens at the Krazy Karnival. As for the perpetrators, all evidence points to the Green Team, a group of teens wearing green bowler hats. No one is safe with them on the loose,\" Lois continued. \"Commissioner Gordon and his police have been called in and are doing all they can. However, they could really use the help of the super heroes from Super Hero High!\"\n\n\"We're trapped!\" shouted Miss Martian from inside the bubble. \"We're trapped!\"\n\n\"It's weird, but the only signal that can get out of the bubble is your show, Harley,\" Batgirl said, trying to stifle a giggle. \"No phones, no messages. Nothing.\" Harley gave her a sideways look. Batgirl was not known to be a giggler. Something strange was definitely going on. \"And the only information that can come in to us is Lois Lane's Web channel!\" Batgirl said before laughing so hard she was in tears.\n\nLois was still talking. \"It appears the super heroes are all in attendance at the Krazy Karnival. Wait, what's that?\" Lois's voice remained calm, but her eyes widened. \"I can't believe what I'm seeing. It looks like the Green Team is...Can it be? The Green Team is multiplying!\"\n\n\"The Green Team's doing math?\" Harley quipped.\n\n\"There are reports that though they started out as a dozen, there are now one hundred members fanning out into every corner of Metropolis!\" said Lois.\n\n\" _ **WOWZA!**_ They're multiplying themselves! _**DOUBLE WOWZA!**_ Triple, even!\" Harley cried as the magnitude of what was happening finally hit her.\n\n\"It's worse than you think,\" Batgirl said, doubling over with laughter. \"The joke's on you!\"\n\nHarley could not believe her eyes. Or her ears. Batgirl had reprogrammed the video screens that were dotted around the amusement park. Now, instead of J.J. reminding everyone to have a great day, it was broadcasting Harley's Quinntessentials, where \"Harley\" was reporting from the Krazy Karnival.\n\n\"Lalala...The Supers of Super Hero High don't care about what's happening in Metropolis!\" a fake video Harley said. \"We just want to have fun!\"\n\n\"What's that?\" Bumblebee said, flying over to Batgirl.\n\n\"THAT'S NOT ME!\" Harley shouted. She could not take her eyes off her image on the screen. \"I'm standing right here!\"\n\nBut it sure did look like her. \"It's showtime!\" the Harley look-alike said. \"Sure, the Green Team may be taking over Metropolis, but we have more important stuff to do, like go on rides and play games!\"\n\nBatgirl switched back to Lois Lane's Web channel. \"Has Harley Quinn gone crazy?\" Lois was asking. \"What's up with our beloved super heroes when we need them most?\"\n\nBatgirl punched in some numbers. \"No one is paying attention to Lois, or any other channel, Harley. They're all tuned in to you. Yay! You rule!\"\n\n\"THAT'S NOT ME!\" Harley shouted.\n\n\"It sure looks like you,\" noted Bumblebee. \"Look, that Harley even has the same little freckles you have.\"\n\nHarley pulled out her hand mirror and did a freckle check. They were still there. \"I need to get on the air right now and fix this,\" she said as she squinted at her fake self.\n\n\"Hello, Harley Quinn here,\" Harley said to the camera. \"I'm here to tell you that there's a fake Harley out there\u2014\"\n\nHer broadcast was interrupted by the Fake Harley. \"And I'm here to tell you that it's me, Harley, who's being fake and funny!\"\n\n\"No, that's not me,\" Real Harley insisted before being interrupted again. The video screens jumped back and forth between Harleys.\n\n\"Loyal viewers, come along for the ride as we watch the Supers act really strange!\" Fake Harley said. \"What will happen? Stay tuned to find out.\"\n\nSuddenly footage of more Supers acting out in general mayhem and chaos and of rides going haywire were blasting on the screen.\n\n\"I\u2014I don't know what's happening,\" Harley sputtered. \"I'm not in control here!\"\n\n\"Then who is?\" Batgirl asked seriously, and burst out laughing.\n\nThe real Harley was running in circles to calm herself. Miss Martian made sure to be very, very still so as not to get caught in her whirlwind. Meanwhile, Bumblebee was buzzing around the amusement park, getting a fix on what was happening.\n\n\"Wow, she's not herself,\" Bumblebee whispered to Miss Martian as they watched Batgirl reprogram the Magic Donut Maker. It began to pump out so many doughnuts that they seemed to be raining down over everyone in the vicinity.\n\n\"No one is,\" noted Miss Martian. \"Well, you, me, and Harley, but it's like everyone else here is acting like someone other than themselves.\"\n\n\"Yeah, it's like something's happened to everyone,\" said Bumblebee. \"But what?\"\n\n\"I had a huge headache when I first got here,\" Miss Martian said. \"It hurt so much, and it wasn't until I took off my...hat. Hat! It's the hats!\"\n\n\"Ooookay...,\" Bumblebee said slowly. \"Now _you're_ acting weird.\"\n\n\"No! It is the hats,\" Miss Martian insisted. \"They are affecting everyone's minds. You, me, and Harley...no hats. Batgirl, Supergirl, Cheetah? The Green Team? They are all wearing the hats.\"\n\nBumblebee looked around. Batgirl was now doing a doughnut dance she had just invented, and Cheetah was wandering around passing out compliments to everyone she came across.\n\n\"You're right,\" Bumblebee said. \"They're all wearing hats and not acting like themselves.\"\n\nHarley sat on the ground and moaned. \"And the person who is most not like herself is ME. With THAT one on the air.\" She pointed to a video screen where the fake Harley was telling everyone to \"get reckless and wild!\"\n\n\"It is the hats,\" Miss Martian repeated. \"We have to get them.\"\n\nHarley nodded slowly. \"Okay.\" She pointed to Batgirl. \"Let's start with hers.\"\n\nBatgirl was sitting on the ground stuffing more doughnuts into her mouth when Bumblebee shouted, \"Three, two, one...go!\"\n\nThe three rushed Batgirl. \"Hey!\" she cried. \"What's going on?\"\n\n\"Give us your hat!\" Harley yelled.\n\n\"No way!\" Batgirl said, pulling it tighter on her head.\n\n\"Please,\" said Miss Martian. \"It is harming you.\"\n\n\"Is not!\" said Batgirl. She scaled up the tall skywalk that circled the Krazy Karnival and was now hanging upside-down with her legs bent over a rail. She held her hat on her head with one hand and finished a doughnut with the other.\n\nBumblebee flew up to her. \"I'll explain later,\" she said, attempting a fly-by to snatch the hat from Batgirl's head.\n\nImmediately, Batgirl reached for her Batarang and flung it at Bumblebee, who ducked out of the way.\n\n\"Hey!\" yelled Bumblebee. \"You almost hit me!\"\n\n\"No one touches my hat!\" Batgirl grumbled.\n\n\"She's not herself,\" Miss Martian reminded Bumblebee. \"We need to get her hat and destroy it to be sure she won't put it back on!\"\n\nHarley leapt onto the nearby Gravitron ride. She used its spinning centrifugal force to propel her up and landed atop the metal mesh roof of the skywalk. From there, Harley looked down at Batgirl, who was still hanging upside down, enjoying the view and the snacks.\n\nIn one move, Harley grabbed Batgirl's hat and tossed it up toward Bumblebee, who hit it with her blasters. But in her hurry, Harley lost her grip. She began to free-fall toward the ground.\n\n\"HELP!\" Harley yelled, flailing her arms and kicking. \"Someone catch me!\"\n\nBefore Bumblebee could get to her, Batgirl swooped in on a thin wire and grabbed Harley. They both landed lightly on the ground.\n\n\"Thanks, Batgirl,\" Harley said as she smoothed out her shirt.\n\n\"What just happened?\" Batgirl asked. \"I feel sort of weird.\"\n\n\"Too many doughnuts,\" Harley quipped. \"Come on, we'll explain. But first we have to make a plan.\"\n\n\"It's fun and chaos here inside the bubble,\" Fake Harley was reporting. On the screen was video of the Supers trying to tear the hats off the heads of the others. \"Like a game of capture the flag, my fellow super heroes are playing Grab the Hat. Never before have more people been more in love with their hats,\" she said, laughing. \"What's more important than hats? Nothing!\"\n\nThe footage was strangely compelling. Viewers had never seen anything like it. Some Supers, like The Flash, were too fast for just one super hero to catch. Katana and Big Barda were fighting over Barda's purple hat with red feathers, even though Katana had a jaunty green Robin Hood\u2014style hat of her own. \"Mine!\" Barda shouted. Everyone, friends and enemies, was engaged in fights over hats. As the mess grew, so did the number of viewers of Harley's Quinntessentials.\n\n\"Every news channel in the world has locked into HQ, but it looks like Fake Harley is still overriding your real reports,\" Batgirl said as she looked at her chart. She was keeping track of which Supers had had their hats confiscated and which ones were still wearing theirs.\n\n\"What about Lois Lane's reports?\" Harley asked.\n\nBatgirl shook her head. \"No one's watching that, even with Metropolis under siege. They'd rather watch you.\"\n\nSeparating super heroes from their hats proved difficult.\n\n\"No one's getting this hat!\" Supergirl yelled as she flew toward the top of the bubble. Wonder Woman, Hawkgirl, and Bumblebee were in pursuit.\n\n\"It's controlling you!\" Bumblebee said.\n\n\"It is not, and you are starting to bug me!\" Supergirl called out as she hovered out of reach. \"I love my hat! It makes me happy! Plus, it's _so_ cute.\"\n\n\"It _is_ cute,\" Wonder Woman said as she flung herself against the thick wall of the bubble. \"But there's nothing cute about mind control, and that bonnet you're wearing is making you all wonky. I wasn't myself until Batgirl Bataranged mine off my head!\"\n\n\"Wonky?\" Supergirl said. \"Who are you calling wonky?\" For a moment, Bumblebee distracted her with a sonic blast. In that second, Wonder Woman swooped in and got the hat, tossing it to Catwoman, who used her whip to destroy it.\n\nBumblebee gave her a thumbs-up. One more hat destroyed, one more Super to help get rid of the hats. What started out difficult was getting easier. Or was it? Harley wasn't sure. There were still hundreds and hundreds of guests to contend with, and they were totally out of control. So much was happening, Harley didn't know what to aim her camera at next.\n\n\"Video off,\" Batgirl ordered Harley.\n\n\"This could be important information\u2014think of the viewers!\" Harley protested. Batgirl didn't flinch. She wasn't as fun without the hat, Harley noted as she shut off her camera.\n\n\"Miss Martian, tell everyone what you know,\" Batgirl said.\n\nMiss Martian willed herself to be seen, and several Supers gathered around her. \"The hats are controlling\u2014or at least influencing\u2014everyone. But not in the same way. Some people are happy, others are belligerent, and still others are just flat-out wacky. It's as if someone wants everyone to act crazy.\"\n\n\"Well, it is the KRAZY Karnival,\" Harley joked.\n\nNo one laughed.\n\n\"So, what's our plan?\" asked Miss Martian. \"We have a plan, right?\"\n\n\"We confiscate all remaining hats,\" said Batgirl. \"Supers first.\"\n\n\"But we have to break the bubble that's trapping us in here,\" Wonder Woman noted. \"Lois Lane is reporting that the Green Team is robbing every house and business in Metropolis.\"\n\n\"Yes, but with all of us Supers working together, it will be easier to break it,\" Poison Ivy said.\n\n\"Someone plotted to have us trapped in here,\" Hawkgirl chimed in. \"But who?\"\n\n\"The Junior Detective Society will figure out who and why,\" said Batgirl.\n\n\"But first, this,\" Harley interrupted. She turned the camera on. \"This is the REAL Harley Quinn here, assuring you that the super heroes of Super Hero High will be in Metropolis soon to help save the day from the Green Team, who's\u2014\"\n\nThe screen sizzled with static. Fake Harley suddenly appeared. \"To save the day, we have the Green Team, a group of talented teens. And back in the bubble, let's watch the Krazy Karnival! I guarantee, you've never seen anything like it before!\"\n\n\"Why did you just say that?\" Beast Boy asked Harley. \"That was totally not the right thing to say!\"\n\n\"It wasn't me!\" Harley yelled, leaning in toward Beast Boy's face. \"I wouldn't do that!\"\n\n\"You just did!\" he yelled back.\n\n\"We can discuss this later,\" said Wonder Woman, breaking the two up. \"Right now we have work to do. Let's go get all those hats!\"\n\n\"Some of us should stay back and try to figure out who's behind this,\" Bumblebee said.\n\n\"Great idea,\" said Hawkgirl. \"Miss Martian, we'll need you here with us. Harley, stay in contact. I'd say that someone's out to get you. The fake Harley is proof of that.\"\n\n\"Proof, schmoof.\" Harley gripped her mallet. \"Let them try,\" she said. \"I'll show 'em why it's not smart to mess with Harley Quinn!\"\n\nAs the hat-free Supers zeroed in on their classmates still wearing hats, Beast Boy spotted Cheetah.\n\n\"Hey there, Cheetah, you sure look nice today,\" he said.\n\nCheetah absentmindedly touched her beret. \"Why, thank you, Beast Boy, and so do...\"\n\nBut before she could finish her sentence, Beast Boy had transformed into an eagle, snatched Cheetah's hat with his beak, and was flying circles above her.\n\n\"You\u2014you stole my hat, you green pest,\" Cheetah said accusingly.\n\n\"You're welcome,\" Beast Boy replied as he handed the hat off mid-flight to Adam Strange, who then tossed it to Starfire, who dropped it toward Katana, who sliced it in half before it touched the ground.\n\n\"Cheetah,\" Harley said, lowering her voice. \"You were being nice...to everyone.\"\n\nCheetah looked startled, and then she scowled like she had just sucked on a lemon.\n\nWith each hat confiscated and destroyed, the Supers gained momentum and Batgirl crossed another name off the list. Working together, they found gathering the hats from the Krazy Karnival guests was easy...almost. Captain Cold was particularly hard to separate from his pirate hat. But once hatless, he was back to his ornery self.\n\n\"Someone was trying to control my mind?\" he said, seething. \"That's not cool!\"\n\n\"The hat was controlling my mind?\" Lady Shiva repeated as she stared at the smoldering pile of felt and electronics at her feet. El Diablo and Sapphire had confiscated her hat; then El Diablo had hit it with flames. One by one, as each Super's head cleared, they began to understand what had happened.\n\nMeanwhile, when it finally dawned on guests that they had been under some sort of mind control, panic began to spread. Confusion built to epic proportions, especially when they tried to escape from the Krazy Karnival and learned that there was no way out of the bubble.\n\n\"Nothing good can come of this,\" Fake Harley reported. \"I love it!\"\n\n\"That's not me!\" Real Harley said, interrupting Fake Harley's feed. Suddenly the video cut to one of the remote cameras that Harley had put up earlier.\n\nOn the screen, Wonder Woman was swooping toward the Tunnel of Love, where she was met in the sky by Silver Swan. Each hovered in the air, neither speaking. Silver Swan did a pirouette so fast she had to hold on to her hat, to keep it from flying off. When she stopped, she glared at Wonder Woman, who returned her unblinking stare.\n\n\"Please give me your hat,\" Wonder Woman said evenly.\n\nOn the ground below them, The Flash, the last Super Hero High student with a hat still on his head, was being chased by several Supers.\n\n\"Why would I want to do that?\" Silver Swan scoffed.\n\n\"Because it's controlling your mind,\" Wonder Woman explained.\n\n\"No one controls Silver Swan,\" she said. \"If you want my hat, you're going to have to take it from me.\"\n\n\"Oh, you don't want that,\" Wonder Woman warned.\n\n\"Try me,\" Silver Swan dared her.\n\nWonder Woman flung her lasso at Silver Swan. But before it reached her, Silver Swan began to hum, slowing the lasso's projection. Then she let out a powerful sound wave that made the lasso snap back toward Wonder Woman.\n\n\"Oops!\" said Silver Swan. \"Maybe you aren't as invincible as they say you are.\"\n\nWonder Woman brushed off the comment and tightened her grip on the Lasso of Truth. \"I guess we're going to have to find out,\" she said, flying full force at her nemesis.\n\nSilver Swan pirouetted once more then retreated into the Tunnel of Love with Wonder Woman in pursuit.\n\n\"We can't see what's going on inside,\" Harley said over the video as the scene projected onto the screens all across the carnival. \"But something's sure happening!\"\n\nOnly the sounds of battle could be heard. Smashes and crashes abounded. At times, swan boats flew out of the entrance and exit, barely missing bystanders. Every now and then, Silver Swan flew out of the tunnel, followed closely by Wonder Woman...or Wonder Woman flew out with Silver Swan in pursuit, and then they both disappeared back into the tunnel. Things were happening so fast, no one could keep up with either combatant.\n\nSuddenly, there was quiet.\n\nNo one moved. All over the amusement park and beyond, viewers were riveted to the screens.\n\nWhere was Silver Swan? And more importantly, where was Wonder Woman?\n\nThey couldn't see anything, but they could hear a beautiful low hum coming from the tunnel. It was Silver Swan. Harley stared at the video monitor. There was a murmur from the crowd outside of what was left of the Tunnel of Love. Where was Wonder Woman?\n\nA rustle of wings signaled Silver Swan, who emerged smiling. Hawkgirl and Poison Ivy gasped.\n\nIt was Batgirl who began cheering first. \"There she is, right behind Silver Swan,\" she said, pointing at the screen. \"Wonder Woman's got the Lasso of Truth around Silver Swan!\"\n\n\"Look what's in her other hand!\" shouted Hawkgirl.\n\nSure enough, Wonder Woman was holding a hat.\n\n\"I wasn't keen on wearing it,\" Silver Swan was saying as she rubbed her forehead. \"But he insisted. He said everyone at the carnival had to wear the hats.\"\n\nThe video was suddenly fuzzy as Fake Harley took control. \"You saw it here first,\" she said, winking. \"Silver Swan was unemployed until J.J. Tetch, the kind owner of Krazy Karnival, gave her a job. But the real story is that those teens from Super Hero High will soon be out of work. They're stealing hats, of all things!\" Fake Harley looked straight into the camera and confided, \"I never did like any of my schoolmates. They're a sneaky bunch!\"\n\n\"Harley!\" said Poison Ivy, looking aghast. \"How can you say that?\"\n\n\"It's not me!\" Harley reminded her. \"It's the FAKE me.\"\n\n\"Oops, sorry,\" Poison Ivy said sheepishly.\n\n\"So it's J.J. Tetch behind all this,\" The Flash noted, still befuddled from the chase and having his hat removed.\n\n\"I think we all suspected that,\" Batgirl said.\n\n\"But the real question is why,\" Hawkgirl added.\n\n\"And what does he have against me?\" lamented Harley. \"He was so nice!\"\n\n\"I'm sure it has something to do with the Green Team,\" Batgirl deduced. \"Lois is saying that their looting is getting worse, and their team is multiplying.\"\n\n\"Why would anyone want to do math at a time like this?\" Harley quipped. Most of the heroes rolled their eyes while she snickered at her own joke.\n\n\"Harley, plug me in to the cameras you have all over the carnival. I want to see what's going on,\" Batgirl said.\n\nHarley and Batgirl sat side by side, Batgirl with her computer, Harley with her video control panel. The others looked over their shoulders.\n\n\"I think all hats are accounted for, except for the ones from the carnival workers,\" Supergirl said to Big Barda. \"The civilians are still calming down, and the super heroes from the other schools are all on board with helping us. Together we'll get the hats from the workers. So far they've refused to give them up of their own accord. It appears that J.J. hired a bunch of thugs and villains.\"\n\nHarley hit the \"On\" button on her camera and was about to broadcast, but someone beat her to it. \"Oh, my! What if there's a surprise in store for those super heroes? Like maybe a battle in the bubble!\" Fake Harley said gleefully. \"Whatever's up next is sure to be epic! Tell everyone to tune in.\"\n\n\"Hey!\" Real Harley cried. \"That's what I was going to say!\"\n\nHawkgirl said, \"You know things are totally crazy when Fake Harley and Real Harley are saying the same thing.\"\n\n\"Let's give it another try,\" suggested Wonder Woman as the Supers rallied. \"We've moved mountains, we've fought epic battles, we've conquered evil. Breaking a bubble shouldn't be all that hard.\"\n\nOthers followed her lead as she hurled her shield against the bubble. Bumblebee used her blasters, Supergirl used her heat vision, and Big Barda threw her mega-rod with all her might. But the bubble didn't even budge.\n\n\"Okay,\" Harley said. She was hopping up and down with an excess of energy. \"Plan B. We fan out and track down J.J. Once we get him, we find out what his motives are and how to get out of here. Everyone, pair up!\"\n\nThe carousel riders were relieved when Batgirl fixed the mechanics and slowed it down. Harley and Batgirl then bypassed the Rock 'n' Roller Coaster when they saw that Hawkgirl and Ivy had it under control.\n\n\"How's about going in there?\" Harley said, pointing to the House of Mirrors. She didn't wait for Batgirl's answer. It was as if it was calling to her. _Harley..._\n\nIt was cool in the darkness. Before Batgirl could activate her flashlight, the room was flooded with light so bright that both girls had to shield their eyes.\n\nHarley peeked through her fingers. \"Whoa!\"\n\nBatgirl squinted. \"Ditto!\" she said as she took in the hundreds of reflections. \"Are you seeing what I'm seeing?\"\n\n\"A whole lot of Harleys?\" Harley said.\n\n\"Yes, but only one of me,\" Batgirl noted. \"The mirrors are only showing _your_ reflection.\"\n\nHarley looked around. Sure enough, there were Harleys everywhere, but no Batgirls, other than the one standing next to her.\n\nHarley turned on her shoulder cam. \"I think I'm gonna want footage of this,\" she said as the red light blinked.\n\n\"It's like a Harley Quinn convention!\" Batgirl exclaimed.\n\n\"I'd go to that,\" Harley said. She stopped again to look at herself in the mirror. \"Uh, Batgirl?\"\n\n\"Yes?\" said Batgirl, who was taking a moment for reports from Metropolis. The city was now on lockdown.\n\n\"I think I just winked at myself,\" Harley said.\n\n\"Oh...kay?\" Batgirl said then returned her attention to their immediate situation. \"Let's keep an eye out for strange happenings.\"\n\n\"The wink was a strange happening,\" said Harley. \"Because my image winked at me, but I didn't do any winking.\" She started making faces and then began a series of somersaults and flips, watching herself in the mirror.\n\n\"What are you doing?\" Batgirl asked, perplexed. \"This is not the time to create new gymnastics routines.\"\n\nHarley did a complicated triple flip, but at the last second, reversed it, and yelled, \"Can you do that?\"\n\nThe Harley in the mirror just smiled and stood still.\n\nAt the same time, Batgirl and Harley shouted, \"FAKE HARLEY!\"\n\nHarley began smashing the mirrors with her mallet. As each mirror shattered, the Harley reflected there disappeared.\n\n\"One of the mirrors isn't really a mirror. That fake is in the room with us,\" Harley cried. \"I'm going to find this pretender. There's only one Harley Quinn!\"\n\n\"That's for sure,\" Batgirl said as she removed a Batarang from her utility belt and tossed it, causing mirrors to fall onto each other and break.\n\nThere was glass everywhere, but it seemed that for every mirror that broke, there was another to take its place. During it all, Fake Harley began taunting them, saying, \"Catch me if you can!\"\n\n\"Fake!\" Harley cried in frustration. She swung her mallet hard and shattered another mirror into hundreds of tiny shards of glass. Then she froze. \"Um, Batgirl?\" she called. \"Can you come here, please? I think you're gonna want to see this.\"\n\n\"Your hat, miss,\" the man said, smiling. He held out a jester's hat of many colors with a golden bell on the top. \"A gift from J.J.\"\n\nHarley's eyes locked on the equalizer in the man's other hand. It had mirrors on the end of it, and reminded of her of the trifold mirrors in a department-store dressing room.\n\n\"No, thank you,\" Harley said. \"I don't wanna muss up my hair.\" She poufed up one of her pigtails.\n\n\"I see you brought a friend to the party,\" he said, bowing slightly. \"How do you do? I'm Mirror Master.\" Though he had on a green hood and cowl, a white sailor hat with a first-mate insignia sat snug on his head, looking out of place.\n\n\"I know who you are,\" Batgirl answered. \"You use ultra-advanced mirror technology to create holograms, like the one you made of Fake Harley. But what I don't understand is how you were able to make it so convincing.\"\n\n\"Yeah, confess\u2014or say hello to this,\" Harley said, raising her mallet in the air.\n\n\"Well, you liked the mirror I sent you, right?\" he asked. He was taller than both girls by almost two feet, and muscular.\n\n\"What mirror?\" Harley asked. Her eyes flickered with recognition. She touched her pocket. \"Oh, that mirror!\"\n\n\"Yes, the hand mirror was from me,\" he said, smugly. \"Thought you'd like it. Who doesn't like to look at themselves? I know I do.\" He glanced at a mirror and admired himself.\n\n\"I get it now,\" Batgirl said, nodding. \"Harley, every time you looked in the mirror he was image-mapping you. Your looks, your movement, your quirks. Plus, he was collecting everything you've ever said so Fake Harley could have a speech bank to draw from.\"\n\nHarley gulped. \"Ya mean ya heard what I was saying when I was alone?\"\n\nMirror Master smiled. \"That I did, Harley! That I did.\"\n\n\"How unoriginal,\" Batgirl said. \"You're just a copycat.\"\n\n\"Oh, dear. Are you trying to hurt my feelings?\" Mirror Master said, feigning sadness. \"I'm more than a copycat. And to prove it, maybe I'll create an army of Batgirls next.\"\n\nHarley gasped. \"The Green Team! That's why there are so many of them. You did that!\"\n\n\"She's catching on,\" Mirror Master said. \"Good for you, Harley.\" He tapped his equalizer. \"I can control reflections and mirror images from here.\n\n\"What is it that you want?\" Harley asked.\n\n\"I want a good job review,\" Mirror Master said. \"I want to make the boss happy. Fake Harley was my idea,\" he boasted. \"He loved that!\"\n\n\"Please tell us more,\" Batgirl said, flattering him.\n\n\"Well, remember when you two created those HarleyGrams?\" Both nodded. Batgirl had helped Harley make small holographic images of herself to help promote Harley's Quinntessentials. \"I thought, I can do better than that! So I created a walking, talking reflection of Harley. One that can be programmed by me. At the same time, I can use my mirror skills to make it look like the Green Team is multiplying and cause confusion and panic everywhere.\"\n\n\"He's a genius,\" Fake Harley said.\n\n\"Harley, meet Harley,\" Mirror Master said smugly. \"Now, who's ready for some fun?\"\n\nHarley raised her mallet in the air at the same time Fake Harley did. As Mirror Master laughed, he didn't see Batgirl slip away. Real Harley and Fake Harley were swinging their mallets so fast and doing such astonishing gymnastics moves that it was impossible to tell which one was real.\n\n\"Get her!\" Mirror Master ordered Fake Harley. \"Use Harley's own techniques of speed, strength, and gymnastics on her. My Harley can do everything you can do,\" he boasted to the real Harley. \"Only, here, my Harley has the advantage.\"\n\nIt was true. With her reflection everywhere, Harley couldn't tell just who she was fighting. She was swinging madly, hitting every image of herself.\n\n\"Not even you can tell the difference between yourself and a fake,\" Mirror Master bragged.\n\nHarley paused. \"You are so right. Thank you for the tip!\"\n\nAs she ran around the room doing backflips and cartwheels, Fake Harley did the same. They moved so fast they blurred together. When they stopped, both faced Mirror Master.\n\n\"Stop playing around,\" he ordered. \"Get her!\"\n\n\"Get her!\" both Harleys yelled. \"Get Harley!\"\n\nHarley starting mimicking the fake version of herself. If she couldn't tell the difference, there was no way Mirror Master could either.\n\nStep by step, backflip by backflip, triple sidewinder by triple sidewinder, the Harleys did the same thing. They were mirror images of each other\u2014and their reflections filled the House of Mirrors.\n\n\"I see what you're doing,\" he said. \"You can't fool me.\"\n\nWhat he didn't see was that Batgirl had returned, and she had someone with her.\n\nHarley spied Batgirl and The Flash sneaking behind the mirrors. She wanted to call out to them, but if she did, Mirror Master would also know where she was.\n\nBatgirl kept staring down at something small in her hand: her B.A.T. heat-seeking device. There was only one Harley in the room who had a temperature and a heartbeat.\n\nMirror Master punched some buttons on the side of his equalizer. Instantly, Fake Harley started saying, \"It's me, Harley Quinn, and you're watching the all-Harley all-the-time Harley's Quinntessentials! I'm the real deal!\"\n\nAs the fake kept repeating this, so did the real Harley. She knew that confusion was often key in apprehending a villain. She had learned this in Commissioner Gordon's Understanding the Criminal Mind class.\n\nThe mirrors in the room and all the shattered pieces reflected Fake Harley and Real Harley. It was beyond confusing. \"Let's dance!\" Harley said to her doppelg\u00e4nger. The two began to spin around, reflecting each other's moves perfectly.\n\nIf the Harleys had competed in the dance contest, surely they would have won. Both gripped their mallets and held them out to their sides. With each spin, more mirrors shattered, leaving a trail of broken glass.\n\nMirror Master had been so focused on the Harleys that he didn't see Batgirl and The Flash sneak up on him. Suddenly, a streak of red rushed by.\n\n\"I got it!\" yelled The Flash, holding up Mirror Master's mirror equalizer and tossing it upward.\n\n\"No, _I_ got it!\" growled Mirror Master as he lunged for it.\n\n\"No, _I_ got it!\" said Batgirl triumphantly from the ceiling, where she was perched upside down. She dangled the equalizer from her B.A.T. wire.\n\n\"Give that to me!\" Mirror Master demanded.\n\n\"Okay, but first let me take a look,\" Batgirl said.\n\n\"Helloooooo!\" Harley called out. She was still spinning around the room with Fake Harley. \"I love to dance, but this is makin' me dizzy!\"\n\nBatgirl quickly reprogrammed the Mirror Master's equalizer, and suddenly, Fake Harley stopped dancing. \"Phew!\" Harley said as they both wobbled. \"I was runnin' out of dance moves.\"\n\n\"Harley, say goodbye to Harley,\" Batgirl said as she punched more buttons on the equalizer.\n\n\"Bye-bye,\" Harley said as Fake Harley disappeared with a digital _ping._ \"And don't come back!\"\n\n\"What have you done?\" Mirror Master yelled. \"I'm going to get in so much trouble now!\"\n\nBy then The Flash had tied him up with a length of blinking carnival lights. Batgirl removed the sailor hat from Mirror Master's head. \"Who are you working for?\" she demanded.\n\n\"I don't work for anyone,\" he said. \"Hey, does anyone have an aspirin? I have a huge headache.\"\n\n\"It's J.J., isn't it?\" asked Batgirl.\n\n\"I don't feel good,\" Mirror Master moaned. \"Stop asking me questions!\"\n\nThe Flash held on to the lights that bound the Mirror Master as they led him to the Lost and Found area, where the other carnival workers were being held. \"Got another one!\" he called out.\n\nMiss Martian was standing off to the side, her eyes closed. \"They're all thinking the same thing,\" she said.\n\n\"What's that?\" Harley asked.\n\n\"Mad Hatter,\" Miss Martian said. \"That's the name that keeps coming up when the carnival workers' hats come off.\"\n\n\" _ **WOWZA!**_ \" Harley declared. \"J.J. and Mad Hatter are the same person. That sneak!\" She tapped the side of her head. \"Everyone, quiet! I'm getting an idea, and it's a doozy! We get Mad Hatter, we get out of this bubble. We get out of this bubble, we save Metropolis. We save Metropolis, and I get exclusive footage! This Fake Harley business is old news. Although, it was cool having a sister, if only for a while.\"\n\n\"You can borrow mine,\" Thunder and Lightning said at the same time, then both burst out laughing.\n\n\"Well, there's only one real Harley Quinn,\" Harley said, dropping the mirror on the ground and raising her mallet. \"I won't be needing this anymore!\"\n\nHarley was about to smash it when Batgirl stopped her. \"Mind if I take that?\" she asked.\n\nMiss Martian scurried to keep up with Batgirl and Harley. \"I should have guessed J.J. and Mad Hatter were one and the same. Mad Hatter can get people to do his bidding by using mind control,\" she explained. \"The secret is his hat. The high-tech gizmos and gadgets in it enhance his powers.\"\n\n\"But Mirror Master is so strong, you'd think he could fight off this pest's mind games,\" Harley said.\n\nBatgirl nodded. \"Mirror Master may be physically strong and have the mirror tech, but he's no match for Mad Hatter's brains. And when someone wears one of his hats, Hatter's got even more control over them. It would take a person with incredible mental discipline to withstand his mind games. Look what happened to all the Supers!\"\n\n\"He's close,\" Miss Martian said, slowing down. \"I wish I could home in on him, but his thoughts are jumbled and I can't get a clean read. Wait...I'm getting stronger and stronger vibes. There!\" She pointed to the Amazing Maze.\n\nThe sign outside the Amazing Maze was lit up. Strobe lights made it look like a neon fireworks display. But with each step closer, a bank of lights would burn out with a sizzle and fizzle into darkness. By the time they were at the entrance, the marquee was practically dark.\n\n\"Let's storm the place!\" Harley suggested.\n\n\"Oh, um. Do we have to?\" Miss Martian squeezed her eyes shut. Then she said to herself, \"Yes, we do, Miss Martian. Get ahold of yourself.\" Her eyes fluttered open. \"Get ahold of yourself!\" she repeated, louder.\n\n\"What's going on?\" asked Harley.\n\n\"I'm getting a strong signal, and if I concentrate really hard, I can get a sense of Mad Hatter's thoughts!\"\n\n\"What's he thinking?\" Batgirl asked.\n\nMiss Martian closed her eyes again. \"His mind is difficult to read, but it is clear he wants to see Harley, alone. Just the two of them.\"\n\n\"That's weird,\" Harley said. \"I was thinking the exact same thing!\"\n\nIt had taken some convincing to get Batgirl and Miss Martian to stay behind, but in the end Harley won. \"Mad Hatter just wants me,\" she reasoned. \"If the three of us appear, he may not show up.\"\n\nSilently, Harley tiptoed into the Amazing Maze. It was dark and eerily quiet, and reminded her of when she was little and used to play hide-and-seek. She would hide in the closet, sometimes falling asleep when no one found her.\n\nThough Harley was usually boisterous and brave, she could feel her heart beat faster and faster. As she got closer to danger, it excited and scared her at the same time. Suddenly a projectile flew at her. Harley whipped her mallet around like a baseball bat, smashing it.\n\n\"Oh dear, why did you do that?\" A friendly voice from above asked as pieces of candied apple rained down. \"Is that any way to treat a gift?\"\n\nBefore Harley could say anything, something caught her attention. Across the room, at end of a long hallway, was a sign beckoning her.\n\n_**ON AIR**_ , the red neon flashed. _**ON AIR**_.\n\nHarley made her way down the corridor, which got narrower and narrower with each step. Finally, it spilled into a cavernous room.\n\nThe maze walls rose twelve feet high, too tall for Harley to see over. Taking a deep breath, she sprinted head-on toward one, using momentum to run up the side, flip herself over, and grab the ledge with two hands. Doing a pull-up, Harley peered over the top. She saw nothing but maze and more maze.\n\nUndaunted, she let go and continued her journey, wandering through the winding labyrinth, doubling back whenever she hit a dead end.\n\n\"What's taking you so long?\" Mad Hatter asked impatiently, his voice booming over a loudspeaker.\n\n\"I'm in no hurry, J.J.,\" Harley replied, careful to sound confident, even if she didn't feel that way. The maze was so confusing. Had she already walked this way? Everything looked the same. \"Or do you go by Mad Hatter?\"\n\n\"Oh, you can call me Mad Hatter,\" he said, letting out a long, gleeful laugh. \"Although some people just call me mad!\"\n\n\"I'd like to call you...when you're in prison,\" Harley quipped.\n\n\"What's that? I can't hear you,\" Mad Hatter said. \"What's the matter, Harley? I thought you'd be better at hide-and-seek.\"\n\nHarley looked around. The maze walls were a glaring white. She wished she had sunglasses. That's when she looked up.\n\n\"You're pretty quiet,\" Mad Hatter said. \"That's not like you. What's the matter? Scared?\"\n\n\"Scared, schmared. I'm not afraid of the likes of you,\" she said, shaking the inkling of self-doubt that had crept into her brain. Harley was used to having other Supers around her, backing each other up. But this was to be a solo performance. \"Focus, Harley,\" she reminded herself. She thought, _What would Batgirl do?_\n\nHarley fumbled around in her pocket. It was still there\u2014the little box from her friend. She opened it and smiled. Inside was the teeny-tiny QuinnCam drone camera. Harley slipped on the ring that controlled it\u2014and the drone. She wished Batgirl had also given her instructions. Then she noticed that when she moved her ring finger, the QuinnCam moved too. When Harley pointed the ring to the left, the camera turned left. When she pointed it to the right, it went right. And if she moved her hand up and pointed the ring skyward, that was where the camera went.\n\nHarley looked at the ring and could see what the camera was recording on a small screen built into the ring. She had a view of the entire maze! But where was Mad Hatter? Harley rotated the ring until\u2014at last\u2014she found what she was looking for.\n\nShe made the little drone rise a little bit higher, out of sight. She had a feeling it might come in handy later.\n\nHe was sitting on what looked like a red velvet throne in the middle of the maze, sipping from a dainty blue teacup. Harley nimbly wove her way around, using the images from the drone camera overhead as her guide. Finally, there was only a wall between them.\n\n\"Where are you, Harley Quinn?\" Mad Hatter's voice was almost a singsong. \"You're not scared, are you? I'd think someone of your stature, of your skills, and of your presence, wouldn't be scared of anything. Reveal yourself! I have a proposition to make!\"\n\nHarley took a step back, did a couple of sidewinders, and then ran up, up...and over the wall, landing in front of Mad Hatter. \"Ta-da! Here I am! Didja miss me?\" she said, looking fierce. \"Now, what's so all-fired important, before I take you in?\"\n\nMad Hatter didn't look surprised. In fact, he looked amused, as if he had anticipated this moment.\n\n\"Harley!\" he said, clasping his hands with joy. \"So glad that we can have this moment together.\"\n\nThough the maze walls had been austere white, where Mad Hatter sat looked like a cozy living room, complete with a fireplace. Harley's stomach made a loud growling sound, reminding her she hadn't eaten lunch.\n\nLaid out before her was a feast of colorful carnival food. Harley's eyes stopped at the sugar-and-cinnamon-powdered funnel cakes stacked so high they threatened to topple.\n\nAs if reading her mind, Mad Hatter said, \"This is all for you. Sit, sit, Harley. Have some cake and tea, have whatever you'd like. We have so much to talk about, you and me. We are much more alike than you realize.\"\n\nHarley stood warily. Mad Hatter didn't seem to have any bodyguards or villains at his side, and there were no weapons that she could see. _What does he want?_ she wondered. Perhaps it really was Mirror Master who was the villain, and he had been using Mad Hatter this whole time. After all, the man sitting in front of her seemed harmless.\n\nThe funnel cakes looked delicious, and Harley was hungry. \"Maybe just one bite,\" she said, picking up a gold plate. The initials _MH_ were written in the center in cursive.\n\nMad Hatter rubbed his hands together and encouraged Harley to pile her plate high with treats. \"More! More! Take more!\" he said. \"More is better!\"\n\n\"Thisissogood!\" Harley gushed after taking a bite. Crisp on the outside, the funnel cake was soft and warm inside. Harley made a mental note to tell Bumblebee about it. Surely it would taste even better drizzled with honey.\n\n\"Please sit, Harley,\" her genial host insisted.\n\nAs Harley joined him, she set her mallet aside, making sure she could grab it if needed. Harley tried to remember what Commissioner Gordon had told his students about getting information from a criminal.\n\nHarley watched Mad Hatter adjust his oversized top hat. It was green with a band of yellow around the base, and boasted a wide brim that didn't detract from the fancy deep blue stitching. It tipped to one side, making Mad Hatter look slightly lopsided, and the sparkles and lights were mesmerizing. Harley recalled that Miss Martian had said something about his hat. But what?\n\n_Well, it'll come to me_ , Harley thought. She'd never had a shortage of ideas.\n\nMad Hatter poured her a glass of iced tea. She drained it in one long swallow. She hadn't realized how thirsty she was. Harley wiped her mouth with the back of her hand, trying to remember why she was there. Perhaps it was to Save the Day, or the world. Or maybe to unmuddle the mystery of the big bubble and unmask the true villain. Was it Mirror Master or Mad Hatter...or someone else completely? Harley was confused. She wished she was as sleuthy as the Junior Detectives. Then again, there were some things she was terribly good at, like running Harley's Quinntessentials!\n\n\"Hey, Mr. Hatter,\" Harley said, trying to focus. Her thoughts were whirling around. She got up and poked at his hat with her index finger. \"That's some amazing _chapeau_ you got there. _Chapeau_ \u2014that's French for 'hat'!\"\n\nThe smile slid off his face for a moment before he laughed and said, \"Oh, Harley, you crack me up. _Merci!_ Thank you for the French lesson.\"\n\nBoth of his hands gripped the edge of his hat as he secured it tightly on his head. Luckily, Harley wasn't wearing a hat. There was nothing he could do to her, she thought.\n\n\"Do you know who your biggest fan is?\" Mad Hatter asked playfully. \"Care to guess?\"\n\n\"One of my viewers? Wait. I know...that lady who sends me pictures of her grandkids?\"\n\nMad Hatter laughed good-naturedly. \"No, not them. Though you do have legions of admirers. But your biggest fan? You're looking at him! I did this all for you, Harley Quinn! The Karnival, the Green Team\u2014it was all to get you here and for us to talk.\"\n\n\"The Big Bubble?\" Harley asked.\n\n\"Yes, the Big Bubble!\" Mad Hatter said, puffing himself up. \"Oh, Ms. Quinn, you've figured it out!\"\n\n\"I have?\" Harley was used to people saying she was funny, and that she could do the best acrobatics. But no one had ever pegged Harley Quinn as an intellectual. \"You think I'm smart?\" she asked, flattered.\n\n\"Of course you are,\" Mad Hatter said. \"You are are remarkably smart.\"\n\nHarley's brain was on overdrive. \"Okay, so as I see it, you worked with Mirror Master to duplicate the Green Team and make their mirror images rob Metropolis.\"\n\nHe looked at her admiringly. \"See?\" Mad Hatter leaned over and tapped Harley's forehead. \"Smart thinking. But I didn't do this just to rob Metropolis. That's child's play. Any common criminal can do that.\"\n\n\"I\u2014I don't understand,\" Harley sputtered. She was so confused, she stopped eating cake.\n\n\"Please, get comfortable and I'll tell you why,\" Mad Hatter said. \"I have been a fan of yours ever since you started Harley's Quinntessentials. In fact, I was one of your very first viewers. MH234. Recognize that?\"\n\n\"MH234? That's you?\" Harley exclaimed, her eyes wide. \"MH234 is always the first to cheer me up when things go wrong!\"\n\nMad Hatter beamed, wickedly delighted. \"That's me. Remember the time you crashed the Internet?\" Harley nodded. How could she forget that? Everyone was mad at her. Well...almost everyone. \"Do you remember what MH234 said?\" he asked, wagging a finger at her.\n\n\"MH234 said, 'You're just getting started, Harley Quinn. Keep going. Someday you're going to be big!' \" she quoted. She had memorized lots of MH234's messages.\n\nMad Hatter burst out laughing. Harley liked the way his eyes sparkled. \"Was I right, or was I right?\" he asked.\n\n\"You, sir, were right!\" Harley said, biting into a treat.\n\n\"I believe in you, Harley,\" Mad Hatter said sincerely. \"And even more now that we've met. You know, we should work together.\"\n\n\"How?\" she asked.\n\nMad Hatter stared into her eyes. \"I love your Web shows. You're so innovative. I mean, c'mon. That dance contest? The Super Bloopers, where the super heroes mess up? The 'Ask Harley' segment? Brilliant. All of them!\"\n\nHarley liked what she was hearing.\n\n\"May I ask ya something, Mr. Hatter?\"\n\n\"Anything, anything for you, my dear,\" he answered.\n\n\"How would we work together? I'm not saying that we would, but if we did, how would that be?\" she asked.\n\nMad Hatter set his teacup down. \"When we team up you will be the top media superstar in the world! With your personality and viewership, and my skills at marketing and planning\u2014\"\n\n\"Go on,\" Harley urged. This was getting interesting.\n\n\"Where is Mirror Master?\" Mad Hatter suddenly asked, looking around. \"He has a present for you! MIRROR MASTER!\"\n\nHarley tried not to smile. She knew Mad Hatter was in for a surprise when he found out that Mirror Master was in custody.\n\nIt was Harley Quinn who was surprised.\n\nMirror Master was standing in front of her. When she last saw him, he was in custody and being hauled away by The Flash toward the Lost and Found, where the Supers were guarding the criminals.\n\n\"Where have you been?\" Mad Hatter asked. Harley thought she saw his eyes flicker with disapproval. When she looked again, he was greeting Mirror Master with a toothy smile. \"Do you have the present for Ms. Quinn, here? That special one.\"\n\n\"I do!\" Mirror Master said. He held his mirror equalizer in one hand...and a hat in the other.\n\nHarley recognized it as the one he'd tried to give her earlier: the eccentric jester-hat-of-many-colors, adorned with jingly-jangly bells.\n\n\"For you, my dear Ms. Quinn,\" Mad Hatter said, bowing to her. He took the hat from Mirror Master and held it high above Harley's head, as if about to crown her.\n\nShe looked nervously at Mirror Master. His jaw was tight. He kept his mirror equalizer aimed right at her. Harley's mallet lay on the ground. She reached for it, but Mirror Master kicked it out of reach and held up his equalizer.\n\nMad Hatter had a delighted look on his face. \"Oh, pshaw! We won't need that,\" he said to Mirror Master. \"Put that thing away. Harley's on our team now, or at least she will be soon!\"\n\nHarley gripped the side of the chair as Mad Hatter slowly lowered the hat on her head, making sure it was on tight. \"Now, that's better,\" he said, grinning. \"You look great!\"\n\nHarley's eyes glazed over. She opened her mouth to speak but nothing came out. Finally she said, \"It. Fits. Great. We are a great fit, too.\"\n\n\"Oh, Harley!\" Mad Hatter crowed. \"Think of the two of us together. What a team we'd make! Hatter and Harley!\"\n\n\"Harley and Hatter,\" she began to correct him.\n\nHe chuckled good-naturedly. \"You are right. You're the star! It should be Harley and Hatter! One has to be flexible about these things, you know.\"\n\nHarley cocked her head. She was getting signals from the hat. Warm, fuzzy signals. They were telling her things. \"Okay!\" she shouted, then calmed down. \"Okay, Mr. Hatter. Working with you will be like a dream.\"\n\nMad Hatter whispered to Mirror Master, \"My mind-control hats work every time! But hats can only go so far. And it's so much work to get people to wear them. However, with this girl, well...\"\n\nHe turned to Harley. \"I'm your biggest fan. Harley's Quinntessentials has the built-in audience to get me started on the road to fame and fortune\u2014my fame and my fortune. Now I can mind-control the masses when they tune in. Everyone will be watching you, and that means they'll be seeing me, too. Everyone. Everywhere! And when they do, I'll rob them blind, and they will love me for it!\"\n\nHarley nodded slowly. \"Mind-control the masses!\" she repeated robotically. \"How?\"\n\nMad Hatter poured himself more tea. He dropped several lumps of sugar into it and stirred slowly. \"Simple,\" he explained. \"This is the new wave of crime. No more robbing banks and stores. That's so old-school\u2014plus, it can get messy. With Hatter and Harley, we will just tell everyone to send us their money and gems and jewels, and whatever we want! And they will because they will have no choice once they're tuned in to us. I'm going to mesmerize them via mass media!\"\n\nHarley's head was buzzing.\n\n\"I am so glad you came to your senses, my dear,\" Mad Hatter was saying. \"Even if I did have to give you a little nudge with that hat. And by the way, it looks marvelous on you. Your fans will love it. You and I are so much alike. We both want to be adored by the masses and will do anything to get what we crave: all eyes on us!\"\n\nHe smiled warmly and settled back in his chair to take a congratulatory sip of tea. Mad Hatter reveled in the moment, then leaned forward and asked, \"What do you say, Ms. Quinn? Shall we shake hands to seal our partnership?\"\n\nHarley stood up. She held out her hand. Mirror Master looked on from the sidelines, still aiming his equalizer at her.\n\n\"Okey-dokey, Mr. Hatter,\" Harley said, extending her hand. \"Let's do this!\"\n\nHarley and Mad Hatter gripped hands and shook, but she wouldn't let go, gripping tighter and tighter. Then Harley began to laugh. Softly at first, then louder and louder.\n\n_\"What?!_ \" Mad Hatter roared. \"Mirror Master! Get Harley!\"\n\nMirror Master raised his mirror equalizer.\n\n\"Go ahead, zap me,\" Harley dared him as she tightened her grip on Mad Hatter.\n\nMad Hatter looked from Mirror Master to Harley, then back again. His eyes narrowed and his jaw tightened. \"Zap her!\" Mad Hatter demanded. \"Do it now!\"\n\n\"Yes, boss,\" Mirror Master said, taking aim.\n\nHarley didn't even flinch when Mirror Master squeezed the trigger.\n\nBOOM!\n\nWhen the puff of vanilla-scented smoke cleared, colorful confetti and streamers rained down on them.\n\nMad Hatter was stunned.\n\nHarley could not stop laughing.\n\n\"Oh, that's not the real Mirror Master,\" she said. \"I knew when Mirror Master showed up that it couldn't be the real thing. Batgirl and The Flash would never let him escape\u2014so this guy had to be a fake!\"\n\n\"The jester hat...,\" Mad Hatter started to say.\n\n\"Ooh, you're so smart, Mr. H!\" Harley said, tapping his forehead. \"Yep. That's why I put on the hat. I also knew my friends wouldn't allow me to fall under your mind control.\"\n\n\"What?\" Mad Hatter sputtered.\n\n\"Oopsie, sorry,\" Harley said, not sounding sorry in the least. \"I forgot to tell you something important. Something really, awfully, terribly, superly important!\"\n\n\"What?\" he growled.\n\nHarley paused, then smiled. \"I've been broadcasting our little talk\u2014to the whole world!\" She waved to the QuinnCam hovering above.\n\nMad Hatter finally noticed the small drone sparkling in the dark. He was speechless.\n\n\"Aren't you gonna say anything? Congrats! You're suddenly famous\u2014but your fifteen minutes are up, in about\"\u2014she checked her watch\u2014\"in about six minutes. And now it's my responsibility to see that you're put away where you belong...in prison!\"\n\nMad Hatter's coloring went from red to white to green, and then back to its regular pale self. He cleared his throat and lowered his voice. \"Listen to me, Harley. We're a team, remember? Hatter and Harley, er...Harley and Hatter. Just put me in front of the camera and tell everyone to stare at their screens. We will have the biggest audience in the history of the world. You will be the most famous person on the planet! Everyone's going to want to know what the two of us now have in store for them.\n\n\"Harley,\" he whispered. \"We can say that the little scene they just witnessed was a prank. A joke!\" He forced a laugh. \"Like when you did 'Where's Harley?' Then we can rule the airwaves...and the world!\"\n\nHarley thought about this. She had always wanted more, more, more. More viewers, more fame, more everything. And now was her chance. Mad Hatter was handing it to her.\n\nHarley looked directly up at the QuinnCam, took a deep breath, and said, \"Hey, Harley's Quinntessentials viewers. This is really hard for me to say. But I've made my decision. Thank you for watching, but a girl's gotta do what a girl's gotta do.\"\n\nMad Hatter grinned and winked at Harley. He puffed himself up, but his smile quickly turned into a look of terror. \"Nooo!\" he cried as Harley grabbed her mallet. With one smooth swing, she threw it at the QuinnCam camera, smashing it to pieces.\n\nAll over the Krazy Karnival and beyond, Harley's Quinntessentials went dark.\n\n\"Now it's just between you and me,\" Harley said, staring down Mad Hatter. \"Let's leave the rest of the world out of this. You think I'd even consider teaming up with the likes of you? Uh, excuse me, but no!\"\n\nMad Hatter stood up and wagged his finger in her face. \"You want a fight, I'll give you one!\" he shouted.\n\n\"Catch me if you can!\" Harley said gleefully. Then she lowered her voice. \"Guide me outta here, Batgirl!\"\n\n\"Gladly, Harley,\" Batgirl said via the two-way communication device she had embedded in the hat. \"Oh, and Miss Martian says to tell you that she's finally got a lock on reading Mad Hatter's mind, but her reception is fuzzy. She says don't ever trust him. Mad Hatter seriously thought you'd be a great team and now he's out for revenge.\"\n\n\"Am I going too fast for you, Mr. Hatter?\" Harley said, laughing as she grabbed her mallet and weaved her way out of the maze. \"Ya know, I like to make people laugh, but just as much of the time, I seem to make people upset. And boy, you look really, totally, and impressively upset right now.\" Harley slowed down. \"Try to keep up, why don't you?\"\n\nMad Hatter was in pursuit, but not nearly as fast as the nimble Harley. The sun was setting outside the bubble as the two raced outside.\n\n\" _ **WOWZA!**_ Everyone is still here?\" Harley asked. By now she was running up and down and loop-de-looping around the roller coaster tracks with Mad Hatter huffing and puffing behind her. She could see the carnival was as crowded as ever with people. \"The bubble is still in place?\"\n\n\"A confirmation on both,\" Batgirl said. Harley stopped to adjust her hat to hear more clearly. \"We're still trying to break the bubble, but everyone is safe, and we've got the Krazy Karnival workers on lockdown while we sort out who's a true criminal and who was under mind control.\"\n\nBefore Harley could reply, someone grabbed her shoulder. \"GOTCHA!\" Mad Hatter cried triumphantly.\n\n\"No, got _you_!\" she said, doing a flip and ending up standing in front of him. The tracks began to shake as a roller coaster car came barreling at them at high speed.\n\n\"Yikes!\" both screamed.\n\nMad Hatter toppled headfirst into the roller coaster car. \"See you!\" he yelled to Harley, who had jumped over the car and was now standing on the tracks alone.\n\n\"Do you need backup?\" Batgirl asked.\n\n\"I got this,\" Harley assured her. \"You keep working on breaking the bubble. Mad Hatter is my battle to fight.\"\n\nMad Hatter was on the ground, holding two fistfuls of darts from the Balloon Pop game. \"You won't get me if I get you first, Harley!\" he yelled, tossing darts at her. But for a guy who ran a carnival, his aim was terrible, and Harley dodged the darts easily.\n\nHarley swung off the roller coaster track, twirled in a pike position in the air, and landed on her feet in front of Mad Hatter. \"I'm making this easy for you,\" she said. \"How nice am I? Come and get me, or maybe now I should get you!\"\n\nMad Hatter gulped as Harley swung her mallet around like she was twirling a baton, tossing it up a few times and then snatching it out of the air. She had blocked him and there was only one way he could escape. Holding on to his hat, Mad Hatter pivoted and headed toward the Ferris wheel. He grabbed on to a passenger car and was yanked off the ground.\n\nHarley leapt and climbed up the turning wheel until she landed safely in the car Mad Hatter clung to.\n\n\"Phew! This is comfy,\" she said, leaning back in the seat and admiring the view.\n\n\"Help!\" someone called.\n\nShe peered down to find Mad Hatter hanging to the bottom of the car with one hand.\n\n\"Harley, help me!\" he pleaded. \"I'm losing my grip.\"\n\n\"Aww, I dunno,\" she said. \"You were pretty mean, trying to use me to control the world. That's so not cool.\"\n\n\"That was the old me. This is the new me,\" Mad Hatter insisted. He looked frightened. \"Please, I'm scared.\"\n\nMad Hatter's face was drained of color. Tears puddled in his eyes. \"Please, Harley,\" he begged as the Ferris wheel continued to go round and round and round.\n\nShe took a breath. \"I'll help you so you don't fall,\" Harley finally said, offering him a hand, \"but first you have to tell me how to break the bubble.\"\n\n\"Listen,\" Mad Hatter said as his fingers laced with hers. \"Just get Obstreperous.\"\n\n\"Obstrep-who?\" Harley asked, confused.\n\n\"Obstreperous,\" Mad Hatter repeated. \"Harley, please,\" he yelled. \"A deal's a deal. I just told you how to break the bubble, now save me!\"\n\nMad Hatter's hand was shaking. Harley could feel his fear as he was about to plummet to the ground. She tightened her grip, and with an \"OOMPH!\" pulled him to safety. \"You can thank me now,\" she said.\n\n\"Certainly,\" he replied, catching his breath. \"How's about this for a thank you?\"\n\nBefore Harley could say \"you're welcome,\" Mad Hatter pushed her off the Ferris wheel.\n\n\"Not nice!\" Harley could be heard yelling as she fell from the Mad Hatter's sight. \"I thought a deal was a deal!\"\n\n\"Bye-bye, Harley Quinn,\" Mad Hatter said, flicking a piece of lint off his jacket. \"Pity. We could have been great together. But no one beats Mad Hatter.\"\n\n\"No one but me!\" said a gleeful voice. Harley pulled herself up and into the Ferris wheel car.\n\n\"How\u2014? What\u2014? Who\u2014? How\u2014?\" Mad Hatter sputtered. \"I pushed you. I saw you fall....\"\n\n\"You should have stayed tuned, Mr. Hatter.\" Harley pretended to look sad. \"Scene one: You pushed me. Scene two: I landed in the car below. Scene three: I vaulted back up here. Why? 'Cause, duh! I'm a gymnast. It's what we do,\" Harley pointed out. \"So, Mr. T., aka Mad Hatter, aka Mr. Soon to Be in Jail! Why don't ya tell us how to shut down your impenetrable bubble?\"\n\nFurious, Mad Hatter's face grew red as he summoned all the powers of his hat. His eyes narrowed as he focused on Harley, who scrunched her nose at him and swatted the air between them like she was batting away a fly.\n\n\"Hey, stop that!\" she ordered. \"You're buggin' me!\"\n\nHarley could hear Batgirl through the jester's hat telling her to bring Mad Hatter in, but something kept interrupting her. It was Mad Hatter repeating, \"You will surrender, you will surrender, you will surrender...\"\n\nHarley shut her eyes tight to block him out. Even though her hat had been deprogrammed and reprogrammed to sync with communications from Batgirl, Mad Hatter's hat still had some serious powers.\n\n\"Block him!\" Harley could hear Batgirl yelling. \"He can only mind-control someone who is vulnerable. Don't look at him. And think of something else, anything but Mad Hatter. He can't control you if you don't let him!\"\n\nThink of something else, Harley thought, scrunching her eyes closed. But what? Well, there's that. And that. Oh, and that's funny.\n\nWhen the Ferris wheel car neared the ground, Mad Hatter leapt out. When Harley opened her eyes the ride was now hundreds of feet up. She took a breath and dove out. Midair, Harley placed her body into a tuck. When she neared the ground, she rolled, then jumped up, unscathed.\n\nBy then Mad Hatter had a large lead. Harley knew she had to stop him before he got to the Lost and Found where his workers were being held. Though they were without their hats, he still had powers of mind control. She may have been able to block Mad Hatter but they were already predisposed to criminal inclinations and more likely to go along with whatever he demanded of them.\n\nIn fast pursuit, Harley threw her whole body into the maddest tumble she had ever attempted. She landed on Mad Hatter's shoulders, and his hat went flying as the two of them tumbled to the ground.\n\nAs he scrambled to pick up his hat, Harley beat him to it and put her foot down right on top of it. \"What's a Mad Hatter without a hat?\" Harley asked, staring down at him.\n\nMad Hatter, who looked surprisingly small and weak without his hat, glared at her. He sighed and shrugged. \"What?\"\n\n\"Just mad,\" Harley quipped as she raised her mallet high in the air.\n\n\"No, no, no!\" he shouted. \"No, not that!\"\n\nWith a gleam in her eyes, Harley brought her mallet crashing down with all her might. Instantly, the hat exploded with wiring, electrical components, and computer chips. It sparked colorfully for a few moments, then winked out.\n\nEverything went dark as all the rides stopped and the lights went out.\n\n\"My hat!\" Mad Hatter wailed.\n\n\"Oh, was that your hat?\" Harley asked, giving him a wink. \"The one that controlled the tech here at Krazy Karnival? The one that controlled the minds of the guests and carnival workers? Oops.\"\n\n\"Harley!\" she could hear Batgirl saying. \"When you destroyed his hat we got communication with the outside. Everything's back up and running.\"\n\n\"Hey, Batgirl,\" Harley said. \"You got a dictionary? Can you look up something for me?\"\n\n\"Sure,\" Batgirl said. \"But, Harley, we're still trapped in the bubble, remember?\"\n\n\"Not for long,\" Harley assured her.\n\n\"Hellloooooo, Harley Quinn here,\" Harley announced. Her face was on every screen at the Krazy Karnival, broadcasting to the world. \"It's good to be back! I promised you a rematch of Harley's Battle of the Bands, and here we go.\"\n\nCheetah, Batgirl, and other Supers stopped trying to break the bubble with their powers and weapons, wondering what Harley was up to. \"A music contest? Now?\" Hawkgirl asked.\n\nMiss Martian ran up to Harley. \"I'm not sure this is the right time for a contest,\" she said.\n\nHarley winked at her and said, \"Read my mind.\"\n\nMiss Martian shut her eyes, then nodded. \"Oh, I see!\" she said, not even hiding her delight.\n\n\"Battle of the Band contestants,\" Harley declared. \"Start playing AS LOUD AS YOU CAN. And everyone\u2014join in and MAKE SOME NOISE!\"\n\nThe bands, who were scattered all over the amusement park, hurried to get themselves together. Instruments were plugged in and amplifiers were turned all the way up. Soon every band began to play. At first it sounded like a clash of wildly different styles and sounds at war with each other. But then something strange started to happen. The music began to mesh into a fusion no one had ever heard before.\n\nThe musical vibration began to make the bubble quiver.\n\n\"LOUDER!\" Harley shouted. \"I CAN'T HEAR YOU!\"\n\n\"What's she doing?\" Big Barda asked as she covered her ears.\n\nMiss Martian closed her eyes and smiled. \"She's breaking the bubble,\" she said.\n\nHarley looked into her camera as the bubble began to shake, making its own deep trembling noise. \"Harley Quinn here, live at the Krazy Karnival, where the Battle of the Bands is in session. But the musicians can't do it alone. To break the bubble we need to make a boisterous sound! More noise than you've ever heard before! It's gotta be LOUD! It's gotta be a sound so obstreperous\u2014\" She paused and said slowly to the camera, \"That means noisy and difficult to control, according to the dictionary\u2014that it cracks this bubble where it counts.\"\n\nSupergirl and Wonder Woman began to fly around the amusement park to spread the news.\n\n\"Everyone make some noise!\" Supergirl said.\n\n\"Louder!\" Wonder Woman cried.\n\nBumblebee, Big Barda, Thunder, and Lightning began to sing. Beast Boy once more became a coqui frog and began to croak. Silver Banshee raised her sonic scream that was only matched by Captain Cold's wailing guitar solo.\n\nSoon all the other Supers joined in, encouraging everyone to shout, sing, stomp, and bang anything to add to the ocean of sound started by the bands. With each voice and sound added, the bubble began to tremble more. But it still wouldn't break.\n\nMiss Martian looked at Harley in desperation and cried out, \"It's not enough. We need something more.\"\n\nEverything that had happened lately raced through Harley's mind in fast-forward. Was she an Internet personality? A solid B student\u2014okay, B-minus? A class clown? An entertainment impresario? And then she remembered who she was and a mischievous grin crossed her face.\n\n_I'm a_ _**SUPER.**_\n\nHarley Quinn reached into her pocket and pulled out the strength-tester bell from earlier. Smiling, she tossed it into the air and then swung her mallet with all her might. _CLANG!_ the iron bell rang out as it rocketed into the sky. It hit the vibrating bubble and smashed through it.\n\nEveryone stopped making music and noise. For a moment, the absolute silence was as deafening as the cacophony.\n\n\" _ **YOWZA,**_ \" Harley said quietly under her breath. And with that, the bubble\n\nThe bubble broke into a million shards and began to rain down on Krazy Karnival. Poison Ivy cried, \"It's okay! It's safe! Look\u2014rainbows!\"\n\nSure enough, the big bubble that had once covered the Krazy Karnival turned into tiny rainbows that wafted downward and before they hit the ground.\n\n\"You're seeing it here in this Harley's Quinntessentials exclusive,\" Harley said, her signature smile broadcasting to her viewers. \"The Mad Hatter is caught. The bubble is broken, the bad guys are rounded up, and the carnivalgoers are safe. And, wait, wait, I'm getting some news from Metropolis\u2014\" Via the jester hat, Harley listened as Batgirl plugged her in to Lois Lane. \"Yes!\" she went on. \"I knew it! When the Mad Hatter's hat broke, so did the hold he had on the Green Team. They were under his control, but no more. In fact, they're apologizing and returning everything they took, right now. _**WOWZA! YOWZA!**_ This is turning out A-okay, and...\" Harley's smile froze on her face.\n\nPrincipal Waller was marching toward her with all the Super Hero High teachers behind her.\n\n\"We cut our conference short when danger arose, but we couldn't get past the bubble,\" Principal Waller said. Liberty Belle, Mr. Fox, Wildcat, and the other teachers nodded. Parasite, the janitor, looked around at the mess that was the Krazy Karnival and sighed.\n\nHarley shrugged. \"Um, I'm in trouble again, right?\" she asked. \"Hiya, VP Grodd, guess I'll see you in detention tomorrow!\"\n\n\"Quite the contrary,\" Waller informed her. \"What is happening is highly unorthodox\u2014but then, so are you. Harley Quinn,\" Principal Waller continued in a booming voice, \"it is my honor to name you Super Hero High's Super Hero of the Month!\"\n\nFor once Harley was speechless\u2014but not for long. \" _ **WOWZA!**_ \" she yelled as cheers erupted. \"ME?!\"\n\n\"YOU!\" Pied Piper called out. He turned to the bands and gave them the signal to start playing.\n\nWhen Waller handed Harley the official Hero of the Month award, Harley hugged it and then backflipped onto the top of a tall Sweet Treats stand. Harley looked out over all the Supers who had gathered around. With a mischievous glint in her eyes, Harley threw her award high into the air.\n\n\"This award doesn't belong to me. It belongs to all of us. Yes, I'm lookin' at you!\" she said when Miss Martian caught it. Harley pointed. \"And you, and you, and you, and you! I declare EVERYONE the winners of the Battle of the Bands, and the Battle Against Mad Hatter, and the Breakers of the Bubble! We are all winners here!\"\n\nCheetah looked up at Harley. \"Is there a class clown award?\"\n\nHarley winced. Waller had just named her a hero and now Cheetah was bringing up the class clown thing again?\n\nBut when Cheetah turned to the crowd, Harley could hardly believe what she was hearing. \"Hat's off to Harley Quinn, Super Hero High's very own one-of-a-kind class clown,\" Cheetah said. Breaking into a grin, Cheetah started to shout, \"Harley! Harley!\"\n\nHarley's heart swelled as the crowd began to chant \"HARLEY! HARLEY! HARLEY!\"\n\nAs the chant grew to a thunderous roar, Harley dove off the top of the Sweet Treats stand, and hundreds of hands rose to catch her.\n\nAs Lois interviewed Harley on camera in front of Capes & Cowls Caf\u00e9, Miss Martian stood off to the side, watching. She clutched a basket overflowing with fan letters from people who had been inside the bubble and outside who had learned of her bravery. While Harley chatted effortlessly, Miss Martian was trying not to disappear, since her interview was up next.\n\nThunder and Lightning waved to Harley as they raced inside. Beast Boy, as a baby kangaroo, kept jumping up and down, photobombing the interview. Bumblebee's phone rang with its familiar \"Flight of the Bumblebee\" ringtone. But Harley, ever the pro, stayed focused.\n\n\"The safety of the world was more important than how many people were watching my show,\" she was saying. \"Mad Hatter wanted the audience, but I wanted to save the world from him.\"\n\n\"You heard it here,\" Lois said to the camera. \"It's no joke that Harley Quinn is a real hero. So what's next for Harley's Quinntessentials?\"\n\n\"Well, now that the Green Team is taking over the Krazy Karnival and donating any profits to charity, I'll be covering that. Plus, I plan to start a new segment called 'Good Deed of the Day,' featuring Supers and citizens helping others....\"\n\nBumblebee started gasping for air as Harley was talking. Supergirl and Poison Ivy rushed to her. Lois cut off the interview. \"Are you okay?\" she asked.\n\nBy then Bumblebee was in tears.\n\n\"Bumblebee?\" said Harley.\n\nBumblebee put down her phone. \"My mom and dad are in danger!\" she cried. \"I have to go!\" She turned herself small and rushed to fly away before anyone could stop her.\n\nPoison Ivy shouted after her. \"Bumblebee, what's happening?\"\n\nBut Bumblebee couldn't hear her. By then she was already past Metropolis.\n\nTo be continued...\n\nMieke Kramer\n\nAfter writing jingles, restaurant menus, and TV shows, Lisa Yee won the prestigious Sid Fleischman Humor Award for her debut novel, _Millicent Min_ , _Girl Genius_. Her other novels for young readers include _Stanford Wong Flunks Big-Time_ , _Bobby vs. Girls (Accidentally)_ , and several books for American Girl, plus _Warp Speed_ , about a Star Trek geek. Her most recent original YA novel is _The Kidney Hypothetical_. She has also written for _Huffington Post_ and is a contributor to NPR.\n\nLisa's books have been named a _Washington Post_ Book of the Week, a _USA Today_ Critics' Pick, an NPR Best Summer Read, and more. Writing the DC Super Hero Girls series is a dream come true, says Lisa. \"I get to hang out with Wonder Woman, Batgirl, Katana, and the rest of the super heroes!\"\n\nYou can visit Lisa Yee at LisaYee.com.\n\n# _What's next on  \nyour reading list?_\n\n[Discover your next  \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9781524769253\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n\n## Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Dedication\n  5. Contents\n  6. Prologue\n  7. Chapter 1\n  8. Chapter 2\n  9. Chapter 3\n  10. Chapter 4\n  11. Chapter 5\n  12. Chapter 6\n  13. Chapter 7\n  14. Chapter 8\n  15. Chapter 9\n  16. Chapter 10\n  17. Chapter 11\n  18. Chapter 12\n  19. Chapter 13\n  20. Chapter 14\n  21. Chapter 15\n  22. Chapter 16\n  23. Chapter 17\n  24. Chapter 18\n  25. Chapter 19\n  26. Chapter 20\n  27. Chapter 21\n  28. Chapter 22\n  29. Chapter 23\n  30. Chapter 24\n  31. Chapter 25\n  32. Chapter 26\n  33. Chapter 27\n  34. Chapter 28\n  35. Chapter 29\n  36. Chapter 30\n  37. Epilogue\n  38. About the Author\n\n  1. vi\n  2. vii\n  3. \n  4. \n  5. \n  6. \n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \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  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204.\n\n  1. Cover\n  2. Cover\n  3. Title Page\n  4. Table of Contents\n  5. Start\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nIMAGES  \n _of America_\n\nMERCEDES\n\nThis map was part of an accordion postcard from 1909 that showed a map of Mercedes's location and invited northerners to travel by train to visit and purchase land or start a business. The postcards were commissioned by the Mercedes Commercial Club, forerunner of the Mercedes Chamber of Commerce, as part of an aggressive marketing strategy. Mercedes is located in the southeast corner of Hidalgo County in South Texas, only five miles from the Rio Grande and the border with Mexico. (Courtesy of Vito Buenrostro.)\n\n**O N THE COVER:** Mercedes celebrates its 25th anniversary on September 15, 1932, with a historic parade. The town was founded in 1907 to serve as the headquarters of the American Rio Grande Land and Irrigation Company. The first residences and businesses were railroad boxcars, tents, and shanties, but by 1932, prosperous shops lined the main street in town and the population had grown to more than 5,000 inhabitants. (Courtesy of the Margaret H. McAllen Memorial Archives, Museum of South Texas History.)\nIMAGES  \n _of America_\n\nMERCEDES\n\nBeatrice de Le\u00f3n Edwards, EdD\n\nCopyright \u00a9 2014 by Beatrice de Le\u00f3n Edwards, EdD   \nISBN 978-1-4671-3206-0   \nEbook ISBN 9781439646946\n\nPublished by Arcadia Publishing   \nCharleston, South Carolina\n\nLibrary of Congress Control Number: 2014943728\n\nFor all general information, please contact Arcadia Publishing:   \nTelephone 843-853-2070   \nFax 843-853-0044   \nE-mail sales@arcadiapublishing.com   \nFor customer service and orders:   \nToll-Free 1-888-313-2665\n\nVisit us on the Internet at www.arcadiapublishing.com\n\nThe Mercedes City Hall and Fire Station, built in 1928, was designed by architect Roscious Newell Waters. The first floor held city offices, and the second floor housed a firemen's dormitory and city meeting rooms. The building is in a Gothic style with red brick with the exception of the replacement bricks on the end from repairs after the hurricane of 1933. It features a copper cupola to hold the fire alarm. (Courtesy of the City of Mercedes.)\nCONTENTS\n\nTitle Page  \n---  \nCopyright Page  \nAcknowledgments  \nIntroduction  \n1.| The Early Settlers  \n2.| Founding a Town  \n3.| Building the Community  \n4.| Wars and Natural Disasters  \n5.| The Rio Grande Valley Livestock Show\nACKNOWLEDGMENTS\n\nMany heartfelt thanks go out to all of the individuals who so generously allowed me to scan their personal family photographs and postcard collections for use in this book, including Vito Buenrostro; Carolyn Crenshaw L\u00f3pez; Irma Palacios; Rolando Hinojosa-Smith; Sylvia Arteaga Calles; the Riess family; the Garc\u00eda family; Robert, Loretta, Kenneth, and Debbie Eilers; Rosendo Gonzales; Eddie Howell Sr.; Helen Vogel; and Delia de Le\u00f3n. A very special thanks to Fran Isbell, whose in-depth knowledge of the Rio Grande Valley and its history were warmly shared with the author. Especially helpful also were the staff of the Weslaco Museum of Local History and Cultural Art; the members of the Hidalgo County Historical Commission; Janette Garc\u00eda of the University of Texas-Pan American; and Phyllis Kinnison and Esteban Lomas of the Museum of South Texas History. All images from the Museum of South Texas History (MSTH) come from the Margaret H. McAllen Memorial Archives of the Museum of South Texas History in Edinburg, Texas. A special thanks to Olga Hinds and Clarissa Mart\u00ednez of the _Mercedes Enterprise_ ; as well as the City of Mercedes; the members of the Mercedes Chamber of Commerce; Dr. Daniel Trevi\u00f1o Jr., superintendent of schools, Alicia Z. V\u00e1squez, district librarian, and Debbie Winslow of the Mercedes Independent School District; Sam MaGee, general manager, and Adell Dufour, museum director, of the Rio Grande Valley Livestock Show; Our Lady of Mercy Catholic Church; the Rub\u00e9n Hinojosa Congressional Office; and Marisol Vidales, library director at the Hector P. Garc\u00eda Memorial Library of Mercedes, Texas. Also invaluable have been the publications and work by the Mercedes Centennial Book Project Committee. Special thanks to the Carrizo\/Comecrudo Tribe of Texas for the use of its tribal seal. More information about this tribe is available at www.carrizocomecrudonation.com. Public domain images from the Library of Congress and the National Archives have also been used. Special thanks also go to Dr. Armando Alonzo, Borderlands historian at Texas A&M University, for reviewing text and making important recommendations. Heartfelt thanks for her patience and assistance go to my acquisitions editor, Stacia Bannerman; David Mandel, production coordinator; and Jennifer Sexton, sales manager, at Arcadia Publishing. Finally, thanks to all my family members for their love and encouragement on this project. Without these entities, this book would not have been possible.\nINTRODUCTION\n\nMercedes, Texas, the \"Queen City,\" is located in the southeastern corner of Hidalgo County only five miles from the Mexican border in South Texas in a geographic area known as the Lower Rio Grande Valley. The \"Valley\" is not a true valley, but a river delta formed as the Rio Grande empties into the Gulf of Mexico. Before the advent of river dams and levees, the Rio Grande flooded annually, much as the Nile did in Egypt, creating rich, fertile soil in a narrow band suitable for limited agriculture.\n\nThe earliest known inhabitants of this area were called Coahuiltecans by anthropologists who grouped all of the separate native indigenous groups together. Later studies of the copious annotations of Spanish _entradas_ , or exploratory expeditions, to this area in the 1600s revealed that many different groupings existed, each with their own language and customs. These early explorations noted that there were numerous native settlements the Spanish called _rancher\u00edas_.\n\nUnder the leadership of Jos\u00e9 de Escand\u00f3n, the Count of Sierra Gorda, Spanish colonists migrated in the mid-1700s to the northern reaches of Nueva Espa\u00f1a, or New Spain, to the region called _Nuevo Santander,_ which reached from Tampico to the Nueces River. On both the southern and northern banks of the lower Rio Grande, Escand\u00f3n established six villas or townships between 1749 and 1755, and numerous land grants called _porciones_ were apportioned out. These _porciones_ were narrow strips of land that each had access to the river to ensure that water was available to each landowner. Because of the climate, topography and soil composition, these Spanish colonists decided that ranching was best suited to the area, with some subsistence farming in selected areas near the river waters or the _resacas_ through the use of _acequias_ , or irrigation channels, that used gravity to move the water streams.\n\nIn 1778, Juan Jos\u00e9 Hinojosa, a captain and chief justice at the villa of Reynosa, petitioned the king of Spain for the Llano Grande land grant on the north side of the Rio Grande where the city of Mercedes is now located. This royal land grant contained 25 leagues of land with 15 miles of river frontage, or more than 100,000 acres. By the time the grant was approved in 1790, Hinojosa had died, and his grant was divided up amongst his eight heirs. Mercedes was later established on what were parts of shares five, six, and seven.\n\nMexico's separation from Spain in 1821 and the Texas Independence of 1836 disrupted the everyday business of the ranching communities, and many Mexican-heritage inhabitants of Texas began calling themselves Tejanos. The Mexican\u2013American War in 1848 profoundly impacted the area when the original landowners, suddenly now US citizens, were forced to protect their land claims in land adjudication courts. Being land-rich but cash-poor, many were able to do so successfully but still lost land when they were forced to pay the American lawyers' fees and their property taxes with acreage.\n\nDuring the Civil War, when the Rio Grande was the only waterway available to the Confederate cotton growers for shipping to market, the value of the river and the region was noted by northern venture capitalists. Many Anglos had come to the Valley during the Mexican\u2013American War and the Civil War to make their fortunes, and many married Mexican heiresses. By 1865, northern eyes were set on the Rio Grande Valley and interest grew in developing international trade and commercialized agriculture in this region.\n\nIn July 1904, the Sam Fordyce Branch of the St. Louis, Brownsville and Mexico Railway reached Section 14 and established the stop that would later become Mercedes, declared the \"Sweetheart of the Branch.\" Upon visiting the Valley, railroad magnate Benjamin F. Yoakum became convinced that commercialized agriculture was a viable venture in the Rio Grande Valley. Yoakum convinced a group of investors to form the American Rio Grande Land and Irrigation Company to purchase land and develop an irrigation system that would transform the Lower Rio Grande Valley into an area of profitable commercialized agriculture.\n\nThe American Rio Grande Land and Irrigation Company purchased land from the Capisallo Town and Improvement Company belonging to Lon C. Hill in 1907 with the intention of making the town its company headquarters. Hill had already platted a town and named it Capisallo, then later renamed it Lonsboro. The American Rio Grande Land and Irrigation Company then decided to move the site west about a mile to an area called the Pear Orchard. It was so named because of the abundance of cacti bearing prickly pear fruit in that location. Mercedes was officially founded on September 15, 1907.\n\nBy sheer force of manual labor, thousands of Mexican and Tejano laborers with pick, shovel, and hoe cleared the land, and the town was finally mapped out in its present location. The American Company directors decided to rename the town D\u00edaz because they greatly admired Mexican president Porfirio D\u00edaz but then changed it to Mercedes, somehow erroneously believing that President D\u00edaz's wife's name was Mercedes. But D\u00edaz's first wife was named Delfina, and his second wife was named Carmen, so the choice of name remains a mystery to this day. No known primary sources exist explaining the choice of name. Unfortunately, numerous published works since the early days have reported that the name Mercedes referred to President D\u00edaz's wife, and the inaccuracy has been repeated many times.\n\nIt should be noted that the phrase _mercedes reales_ means \"royal grants\" and _mercedes de tierra_ means \"land grants.\" It is possible that somehow, someone who heard these phrases mistakenly believed Mercedes to be a person; namely, President D\u00edaz's wife. The phrase _mercedes reales_ also gives a connection to the choice of the nickname \"Queen City\" for the town of Mercedes because \" _real_ \" which means \"royal\" could have made someone think Mercedes referred to a royal person; namely, a queen. In reality, _mercedes reales_ referred to the fact that the _porciones_ and the larger land grant tracts such as the Llano Grande were royal gifts.\n\nOn March 8, 1909, Mercedes became incorporated and elected a mayor and city council. The town grew rapidly as the Mercedes Commercial Club, forerunner of the Mercedes Chamber of Commerce, aggressively promoted land development. The winter vegetable and citrus farms began producing immediately, and Mercedes became a major exporter of produce, citrus fruits, and cotton. In 1939, Mercedes promoted an annual agricultural and livestock show that became the Rio Grande Valley Livestock Show. In 2014, the Rio Grande Valley Livestock Show celebrated its 75th anniversary.\n\nAs did most Valley cities, Mercedes experienced growing pains with bandit troubles, the effects of World War I and II, influenza and smallpox epidemics, hurricanes, floods, droughts, freezes, and the Great Depression. Mercedes overcame these obstacles and continued to thrive. The town established many churches and businesses and an excellent school system. Mercedes has produced notable citizens in the fields of art, literature, music, business, education, government, athletics, and science among others; and although it has produced excellence, its noble work is not yet done.\n_One_\n\nTHE EARLY SETTLERS\n\nSpanish explorers first came to the area where Mercedes is now located in the 1600s. The expeditions found that the land was covered with heavy brush similar to that of this photograph. Mesquite, cactus, huisache, and other native plants covered the land thickly. When the Spanish colonists arrived in the 1700s, they decided to use the land for ranching with some subsistence farming where water was available. (MSTH.)\n\nThe Spanish made several expeditions or _entradas_ into the Rio Grande Valley in the 17th century. They reported finding many _rancher\u00edas_ , or settlements of nomadic bands that historians first called Coahuiltecans. After rereading Spanish chronicles, later historians discovered more than 50 different indigenous groups inhabited this region. The Carrizo\/Comecrudo were most associated with the Rio Grande river delta. Once thought extinct, descendants have resurfaced although they were displaced to other Texas areas. (Carrizo\/Comecrudo Tribe of Texas.)\n\nSpanish explorer Alonso de Le\u00f3n led four expeditions between 1686 and 1689 to explore the area around the mouth of the Rio Grande, also known as the Rio Bravo. After finding the ruins of a French settlement on his fourth expedition, the Spanish king commanded that settlers be brought in to establish a more solid claim to this area. In 1746, Jos\u00e9 de Escand\u00f3n, the Count of Sierra Gorda, was commissioned to explore the area between Tampico and the San Antonio River. The province of Nuevo Santander, which corresponds generally to the Mexican state of Tamaulipas and south Texas, is pictured in this adaptation of a 1792 Spanish map. (Courtesy University of Texas-Pan American.)\n\nJos\u00e9 de Escand\u00f3n was a Spaniard who chose a military career and immigrated to New Spain in 1715. By 1740, he had risen through the ranks and was lieutenant captain general of the Sierra Gorda frontier. His successful strategies for pacification of the region while remaining fair-minded with all groups brought him the title of Count of Sierra Gorda as well as the charge of leading colonists to the Rio Grande region. After determining which areas were suitable for settlement and carefully choosing which colonists to take, Escand\u00f3n brought in more than 400 families and founded Camargo, Reynosa, Mier, Revilla, Laredo, and Nuestra Se\u00f1ora de los Dolores on the banks of the lower Rio Grande. For this reason, he is often called the \"father\" of the lower Rio Grande Valley. Beginning in 1755 with the founding of Laredo, Spanish colonists began establishing ranching communities on the northern banks of the river, including the area where Mercedes is today. (University of Texas-Pan American.)\n\nThis map shows the _porciones,_ or long narrow strips of land, and larger land grants owned by the original Spanish colonizers of southern Hidalgo County in the late 1800s. In 1778, Juan Jos\u00e9 Hinojosa, a captain and chief justice at the villa of Reynosa, petitioned the king of Spain for the Llano Grande grant where the city of Mercedes is now located. This grant contained 25 leagues of land with about 15 miles of river frontage. Hinojosa died before he was finally granted the land in 1790, and his children inherited the land in eight equal shares. Mexico's separation from Spain in 1821, the Texas Independence of 1836, and the Mexican\u2013American War of 1846 created turmoil in the region. In 25 years, these first families' citizenships changed three times. At the end of the Mexican\u2013American War, they were forced to prove their land ownership in Texas state courts. Although most of them retained title to their land grants, many lost acreage when they were forced to pay their attorneys and taxes in land. (Hidalgo County Historical Commission.)\n\nBy the end of the 19th century, there were hundreds of ranches on the northern side of the Rio Grande. Although the brush land was thick with mesquite, cactus, huisache, and native grasses, the land was suitable for grazing cattle. Every ranch had access to water, either directly from the river or from the _resacas_ , the old river channels of the Rio Grande. (MSTH.)\n\nIn the 19th century, the most obvious choice as a building material in the valley was mesquite wood. Even though it was twisted rather than straight, it was abundant, strong, and durable. The wood is so hard that it is sometimes called \"Texas Ironwood.\" Corrals and fences made of mesquite such as the one seen here were common sights on valley ranches of that era and are still seen today. (National Archives.)\n\nThis map is a partial replica of one of the hand-drawn maps used by the Missionary Oblates of Mary Immaculate from around 1849 to serve the Catholics living on the scattered ranches of south Texas. These missionary priests had established headquarters in Brownsville before the advent of the railroad or even of paved roads. They traveled their lonely trails through the wild brush land alone, mounted on horseback. Their circuits usually lasted about six weeks during which time they traveled 100 miles or more. The Oblate Trail in the Rio Grande Valley stretched from Port Isabel and Brownsville up the river to Laredo, a distance of more than 200 miles. They also traveled north to service ranches and townsites farther away from the river. In 1949, the Brownsville Historical Association designated the trail by erecting road markers. (Hidalgo County Historical Commission.)\n\nThe Missionary Oblates of Mary Immaculate were founded in France in 1816 by Blessed Eugene de Mazenod. In south Texas, they were commonly known as the Cavalry of Christ because they served their parishioners as circuit-riders on horseback. Their black soutane habits, black rounded hats, and silver Oblate cross worn round their necks were widely recognized in the region. (MSTH.)\n\nFr. Pierre Yves Keralum was an Oblate priest who, despite his advanced age, continued to serve his people. In November 1872, he set out on his circuit and never returned, being last seen at a ranch north of Mercedes. His remains were found in 1882. In 1920, a memorial was erected to the much-loved Padre Pedrito at the Catholic cemetery in Mercedes. (Our Lady of Mercy Catholic Church.)\n\nBefore Mercedes was founded, many ranches were located in the area, including the Anacuitas (Anacahuitas), Los Burros or Guadalupe, Parajitos, Rel\u00e1mpago, Rosario, San Jos\u00e9 or Solises, Tampacu\u00e1s (also known as Campacu\u00e1s), and Toluca Ranches. This aerial view shows the Toluca Ranch, located about a mile south of US 281 (Military Highway) and a quarter mile east of Farm to Market Road 1015. It is remarkable for having survived almost intact until the present time and is a good model of what a _hacienda_ ranch of the past century would have looked like with its main house, its chapel, a school, a store, the ranch hands' houses, work sheds, vegetable gardens, and corrals. At one time, Toluca Ranch had a post office from which mail was distributed to surrounding ranches by horseback, or farther away by stagecoach or steamboat. At its largest, Toluca Ranch had around 12,000 acres with river frontage and stretched northward 17 miles. (MSTH.)\n\nPictured here are S\u00f3stenes Cano (right) and Florencio S\u00e1enz. After working for many years as a bookkeeper for don Antonio Cano of the Tampacu\u00e1s Ranch, don Florencio S\u00e1enz married S\u00f3stenes Cano, youngest daughter of Antonio and Mauricia Fern\u00e1ndez Cano. Don Antonio Cano of Reynosa had purchased acreage in 1862 from the descendants of Juan Jos\u00e9 Hinojosa of the Llano Grande land grant; specifically, from Cirildo Hinojosa's fifth share. As her inheritance, S\u00f3stenes received a tract of land in the southern part of the Tampacu\u00e1s Ranch that the couple named the Toluca Ranch. The name of the ranch is likely a reference to the town of Toluca in south-central Mexico. The word \"Toluca\" comes from the indigenous word \"Tollocan,\" which means \"place of the god Tolloh\" in n\u00e1huatl, the language of the Aztecs. (Both, MSTH.)\n\nSt. Joseph's Chapel was built in 1896 by don Florencio S\u00e1enz in thanksgiving for when he was finally able to dig a well that yielded sweet, drinkable water. Many previous attempts had only yielded salty, brackish and undrinkable water. The chapel was designed by the Oblate priest Pierre Keralum. The pews accommodate 85 people. Don Florencio's grandson Santiago Fern\u00e1ndez and his great-granddaughter Florence stand in the doorway in this photograph taken in 1951. (MSTH.)\n\nThe interior of St. Joseph's Chapel is decorated with statues purchased in Spain by the S\u00e1enz-Fern\u00e1ndez family. Some of the statues were donated to Our Lady of Mercy Catholic Church in the town of Mercedes in 1947 in thanksgiving after do\u00f1a Manuela Fern\u00e1ndez's five sons all returned safely from World War II. (MSTH.)\n\nThis is the wedding portrait of Manuela Cano Champion (left) and Amador Fern\u00e1ndez, a Spanish \u00e9migr\u00e9. Don Florencio and do\u00f1a S\u00f3stenes S\u00e1enz were unable to have any children, so they decided in 1882 to adopt their three-month-old niece Manuela Cano Champion, daughter of Pedro and Gumecinda Cano Champion. Manuela met and married Amador Fern\u00e1ndez in 1908. They eventually had eight children: Jos\u00e9 Florencio, Jos\u00e9 August\u00edn, Guadalupe Anastasio, Joaqu\u00edn Jorge, Ernesto M\u00f3nico, Santiago \"Jimmy,\" Manuela Lourdes, and Amador Tom\u00e1s. All the children were well educated and several participated in Mercedes city and school district positions. Some of the children and grandchildren continue living in Toluca Ranch to this day. Toluca Ranch was well known for having a brick factory that supplied bricks for many buildings in Mercedes and for a ferry landing where steamboats stopped to load farm and ranch products and unload supplies. (MSTH.)\n\nIn 1914, the S\u00e1enz-Fern\u00e1ndez family built this home in Mercedes on the corner of Missouri Avenue and Fourth Street. They were forced to move to town after their Toluca Ranch was attacked four different times by roving bands of Mexican revolutionaries. The spillover violence of the Mexican Revolution, which lasted from 1910 to 1920, greatly affected the lower Rio Grande Valley region. (Carolyn C. L\u00f3pez.)\n\nThe Fern\u00e1ndez family is depicted in this reunion photograph taken in the early 1950s. Amador and Manuela Fern\u00e1ndez are seated in the center with their eight children, sons-in-law, daughtersin-law, and grandchildren surrounding them. In the early 1900s when Mercedes was still young, Amador Fern\u00e1ndez had a dry goods store just opposite the First National Bank on Texas Avenue. ( _Mercedes Enterprise_.)\n\nWater has always been a precious commodity in the lower Rio Grande Valley since its settlement in the 18th century. In areas where there was no access to river water or lakes, wells had to be dug but these often produced brackish or salty water. Frequently, communities had to rely on water sellers such as this one who brought water in barrels to sell. (MSTH.)\n\nMost ranches had developed into small communities by the end of the 19th century. Ranch workers and their families also lived on the ranch. Workers included vaqueros, or cowboys, blacksmiths, cooks, storekeepers, brick makers, carpenters, and masons among others. When Mercedes was founded, there were several ranches in the surrounding areas including among others: Toluca, Tampacu\u00e1s, Los Burros, Anacuitas (Anacahuitas), Rel\u00e1mpago, Parajitos, Los Ebanos, and El Fuste. (National Archives.)\n\nWhen the lower Rio Grande Valley became part of the United States in 1848, the Spanish and Mexican land grant holders were forced to defend their land claim in state courts. Many landholders were land-rich but cash-poor and were forced to pay their American lawyers' fees in acreage. The document above was Juan Jos\u00e9 Hinojosa's (sometimes spelled Ynojosa) Llano Grande Land Grant title, finally approved and certified by the Texas State Legislature in 1852 for his heirs. Large land grants such as this one were awarded only to favorites of the king or other crown officials. Called _mercedes reales_ , or \"royal grants,\" or _mercedes de tierra_ , \"land grants,\" these larger grants were intended to support hacienda-type ranching ventures. Before the 20th century in Hidalgo County alone, there were 43 porciones granted through the Reynosa jurisdiction, 12 intermediate grants, and 14 large land grants such as Hinojosa's. (Hidalgo County Historical Commission.)\n\nThe map shown here of Hinojosa's Llano Grande grant is called the Dupouy Partition. It was originally prepared in 1848 by Alfredo Dupouy, the court surveyor in Matamoros, Tamaulipas, Mexico. By 1919, Hinojosa's children had already sold land to others as noted here. This map was used as part of the abstract of title obtained by the American Rio Grande Land and Irrigation Company on March 5, 1919, to show clean title to its purchase of lands in the Rio Grande Valley. Share one belonged to Julia de la Garza; share two belonged to Mat\u00edas Cavazos; share three belonged to Leonardo Manso; share four belonged to Manuel Hinojosa, Francisca Hinojosa, and Ygnacia Hinojosa; share five belonged to Vicenta Hinojosa, share six belonged to Gregoria Longoria and Cipriano Hinojosa; share seven belonged to Juan Hinojosa, and share eight belonged to Rosa Maria Hinojosa. There was also a separately sectioned area called the Adams Tract, owned by William T. Adams and his wife, Virginia Adams. Mercedes is located in portions of shares five, six, and seven. (Hidalgo County Historical Commission.)\n\nAlthough in the 1900s large-scale ranching in the Valley was gradually replaced by commercialized farming with the arrival of northern land developers, the love of the ranching era still survives today in many Mercedes residents. Pictured in 1975, seated on his Lineback Dun horse, is Rosendo Gonzales and his grandson Paul. Gonzales belongs to the Mid-Valley Horseman's Association. (Rosendo Gonzales.)\n\nBy 1868, as shown in this early postcard, the railroad had reached Brownsville. On July 8, 1904, the Sam Fordyce Branch of the St. Louis, Brownsville and Mexico Railway reached Section 14, the stop that would later become Mercedes, nicknamed \"Sweetheart of the Branch.\" The coming of the railroad would be the first step in the development of the Rio Grande Valley into a highly profitable agricultural region. (Vito Buenrostro.)\n_Two_\n\nFOUNDING A TOWN\n\nMercedes was founded in 1907 to serve as the headquarters of the American Rio Grande Land and Irrigation Company. The company was formed expressly for the purpose of constructing an irrigation system pumping water from the Rio Grande to irrigate about 250,000 acres of land in Hidalgo and Cameron Counties. The first ARGL&I building seen here was located facing Texas Avenue near the corner with Second Street. (MSTH.)\n\nBenjamin Franklin Yoakum was born in Tehuacana, Texas, in 1859. He worked all his life in the railroad industry and knew every facet from surveying to engineering, construction, traffic, operating, and finance. Yoakum visited the Rio Grande Valley on various occasions and knew that by extending the railroad into the valley, commercialized agriculture was possible. With the help of Col. Sam Fordyce, he convinced a group of St. Louis investors to develop the American Rio Grande Land and Irrigation Company with a capital stock of $1,250,000. Early stockholders included Benjamin F. Yoakum, Sam Fordyce, Thomas W. Carter, Thomas H. West, Edward Whittaker, Edmund E. Elliot, Silas P. Silver, and DuVal West. The company was granted incorporation by the State of Texas on September 30, 1905. Although Yoakum never lived in Mercedes, he and his New York socialite daughter Bessie Yoakum frequently visited the town to see friends. (MSTH.)\n\nYoakum knew that three things would be needed to make commercialized agriculture succeed in the Rio Grande Valley: an extensive irrigation system to water the farmlands, a railroad system to take the produce to northern markets, and a cheap labor force to clear the land, dig the canals, and work the fields once they were planted. Clearing the land, called \"grubbing,\" was achieved by a large Mexican labor force that worked for pennies a day. (MSTH.)\n\nCol. Sam Robertson was the contractor for the earthwork on the Main North Canal built by the American Rio Grande Land and Irrigation Company. As seen in this photograph, he used 300 mule teams to dig a strip of land seven miles long from the river to the railroad in Mercedes to lay the Main Canal in 1906. (MSTH.)\n\nThe first order of business in developing Mercedes and the surrounding mid-Valley area was to build a pumping plant on the Rio Grande. This was achieved in 1906 when the pumping plant was completed about eight miles south of Mercedes. A settling basin was also built west of the plant, with the Main Canal to carry water north by means of gravity; that is, utilizing the natural slope of the land to propel the water along the canals. The water must first be lifted out of the Rio Grande by the pumping plant. The pumping plant had a capacity of 300,000 gallons of water per minute, or 432,000,000 gallons every 24 hours. By 1923, the canal system totaled over 300 miles of canals, not including the smaller laterals serving individual farm tracts. (MSTH.)\n\nWilliam Francis Shaw served as chief engineer and vice president of the American Rio Grande Land and Irrigation Company in 1908. He also served the company as general manager from 1912 to 1930. He was the contact man for the American Company in setting up the canal system in the Mercedes area, and he assisted in laying out the town site of Mercedes and in organizing the electricity and water systems. ( _Mercedes Enterprise_.)\n\nThe river pumping plant had four pumps, two of which were 36 inches in size and were the largest pumps in the world at that time. Electrical power to run the pumps was run down from the electrical plant built in Mercedes. This 1915 photograph with unidentified persons shows some of the intricate machinery required to do the job. (MSTH.)\n\nThe American Company decided to name their headquarters \"D\u00edaz\" in honor of Mexican president Porfirio D\u00edaz, seen here, who was friendly toward American investors. Fearing the development of a revolt in Mexico and the inadvisability of that name, \"Mercedes\" was then chosen, supposedly to honor D\u00edaz's wife; but D\u00edaz was married first to Delfina Ortega and later to Carmen Romero. The origin of the name remains a mystery today. (National Archives.)\n\nIn this c. 1912 postcard, the photograph was taken facing west from the top of the Mercedes power plant next to the Main Canal. The Mercedes Hotel can partially be seen in the middle background. To the right of the hotel across Second Street can be seen the American Rio Grande Land and Irrigation headquarters, the Mercedes Drug Store, and the Hotel Annex. (Vito Buenrostro.)\n\nIn December 1907, the original townsite of Mercedes was platted by American Company chief engineer Chester B. Davis. Streets were named and the map recorded in the Hidalgo County Courthouse. The large sign in this photograph encourages visitors to purchase land and homes in this growing community. (MSTH.)\n\nOne of the first buildings in the new town was the Mercedes Hotel. When the engineers, railroad men, and other professionals came to Mercedes to work, there were no homes available yet. Many lived in boxcars on sidings when they first arrived, and they were greatly relieved when the hotel was completed. The hotel served very well until suitable homes could be built. (Vito Buenrostro.)\n\nSilas Percy Silver served as the general manager of the American Rio Grande Land and Irrigation Company in 1905 and was instrumental in completing the irrigation system. He also ensured that land was sold, businesses were started, and city and school services were properly provided in Mercedes' early days. Under his direction, the American Company also donated land to churches and for cemeteries. Silver was born on November 3, 1866, in Mobile, Alabama. He was a graduate of Washington University with a degree in architecture. Examples of Silver's architectural talent are still evident in many of the buildings in Mercedes. Silver was the Hidalgo County judge when county residents voted to move the county seat from Hidalgo to Chapin (now Edinburg) in 1908. He served one term, from 1908 to 1909. Silas Percy Silver died on December 20, 1938, in Shreveport, Louisiana. (MSTH.)\n\nWhen Silas P. Silver, his wife, and young daughter arrived in the fledgling town of Mercedes in 1905, they were forced to live in a boxcar on a railroad siding because there were no homes built yet. Within a year, the American Company, as it was frequently called, had built the Silver family a company home, which is the house pictured on the right of the photograph. By 1908, Silver built his own home, the house on the left. The Silver house is still standing, located on South Missouri Avenue on the corner of Third Street. These homes both adhered to the restrictions placed by the city on new residences. The city ordinance required homes to be built of brick or stucco and not cost less than $2,000, which was a considerable amount in those days. Business buildings must be of brick or stucco and not cost less than $3,000. The early founders hoped to make the Mercedes downtown area a showplace that would attract new homeowners, farmers, and businessmen. (MSTH.)\n\nIn this photograph, Silas Percy Silver rides his horse into a flume built as part of the canal system. One of the engineering advancements of this time was the use of reinforced concrete in the construction of the head gates, siphons, and flumes. The introduction in 1906 of this man-made \"rock\" by the American Company was new to Texas. (MSTH.)\n\nIn the early days, there were no city services such as trash pickup or street repair, so a city ordinance was passed that required all male persons between 21 and 45 to work on, repair, and clean the public roads and streets. Ministers and firemen were exempt; however, just $2.50 would exempt any citizen for a year. (MSTH.)\n\nSome of the irrigation system construction was contracted out to Col. Sam Robertson until the crews of the American Company took over, extending the canals to Mile 12 North and Mile 2 1\/2 West by 1912. This photograph shows the Main Canal, which runs just east of Mercedes. It was from 90 to 120 feet wide in places and from 15 to 20 feet deep. (MSTH.)\n\nIn this 1910 scene on a postcard, cars are parked at a slant down the center of Texas Avenue, the main street. City records show that uniform traffic laws were not adopted until 1930 and officers did not start handing out traffic tickets until 1939. Parking meters were installed in 1946, undoubtedly much to the dismay of downtown shoppers. (Vito Buenrostro.)\n\nPostcards were very popular as a means of communicating with friends and family during the early 20th century. This is the cover of a used postcard that unfolded accordion-style to show many scenes of Mercedes around 1909. The stamp was cut off, but the postmark can still be partially seen. (Vito Buenrostro.)\n\nWithin a few years of its founding, Mercedes was already producing large quantities of vegetables and fruits for shipping to northern and international markets. The Bermuda onion had already been introduced to south Texas in 1898 near Cotulla. With irrigation, onion farms like Theodore M. Plummer's (seen here) were highly successful, yielding about 3,000 crates per acre during a growing season. (Vito Buenrostro.)\n\nThe gentlemen in this photograph are Mercedes's first mayor and city council after incorporation in 1909. From left to right are John Puckett, Gouverneur K. Wattson, Mayor William Lingenbrink, Dr. Edward C. Schoonmaker, Fred Cutting, and Lytle Harrison. They are sitting at the City Park, a popular gathering area because of its lush grass and palm trees. (Vito Buenrostro.)\n\nThe Mercedes Railroad Depot was built to accommodate the thousands of visitors that came to the Valley, but the railroad was also important for the transport of produce to northern markets, and many packing sheds located just north of the railway lines. In the early years, it was popular to take day trips on the train to Brownsville for shopping and socializing purposes. (MSTH.)\n\nIn this panel of an accordion postcard, the Mercedes Commercial Club, forerunner of the Mercedes Chamber of Commerce, listed all of the amenities to be had in the new city. Mercedes had grown very quickly by 1909 with more than 1,000 inhabitants, but the American Company still had many farm tracts to sell and hoped to attract buyers from the northern and Midwestern states. The use of postcards was one of the marketing strategies, since the new citizens of Mercedes were sure to send them to friends and family where they would be seen by potential investors. The scene below shows the train depot in the background with the \"beautiful tropical park,\" as described above, shown in the foreground of the photograph. (Vito Buenrostro.)\n\nMost of the vegetables and fruits produced in the Valley faced a long train trip to northern and sometimes international markets. This required the produce to be iced down for shipping. Icehouses such as this one were common sights near the packing sheds and railroad tracks. With the advent of mechanically refrigerated boxcars in the late 1940s, icehouses became largely obsolete. (MSTH.)\n\nThe first business owners in the new town of Mercedes were John D. White, general foreman of all laborers and construction work; Fred Cutting and Charles Campbell, who built the first lumber yard, the Mercedes Lumber Company as seen here; and Alex Champion, who built the first general merchandise store. The first blacksmiths were Ben Brooks Jr. and Pablo L\u00f3pez. (Vito Buenrostro.)\n\nAs shown in this postcard, on June 22, 1910, the first bale of cotton in the United States for the season was ginned in Mercedes. Cotton was sometimes planted between rows of cabbage or corn that was harvested first while the cotton grew to maturity. This gave the cotton a head start at being ginned first in the nation. (Vito Buenrostro.)\n\nThe _Mercedes Enterprise_ is the local newspaper whose origins go back to 1908 when Isadore Moritz, previously of the _Brownsville Herald_ , decided to start up a periodical in the fledgling town. The _Enterprise_ is still being published today with its original name. In this issue, the big story was the ginning of the nation's first bale of cotton for the 1910 season in Mercedes. ( _Mercedes Enterprise_.)\n\nIn this aerial view of Mercedes looking southeastward, packing sheds are seen in the bottom left area, located alongside the railroad tracks. The Main Canal can be seen running behind the power plant, the building with the arches in front. The main roadway going from top left to bottom right is Second Street after it was designated US Highway 83. This photograph was taken around 1935. (Weslaco Museum.)\n\nIn this early scene looking north on Texas Avenue, the Empire Theater is seen on the right side of the street next to the columns of the First National Bank. The Empire changed to the Capitol after the hurricane of 1933, and later to the State Theater. Across the street on the corner is the mercantile store opened by Amador Fern\u00e1ndez of the Toluca Ranch. (MSTH.)\n\nThe Sanborn map shown here was commissioned and completed in July 1917 for the purpose of obtaining fire insurance for Mercedes buildings. It shows the contours of the early city. The darker areas were the original downtown area and more expensive residential areas. The railroad track may be seen cutting across east and west about three-quarters of the way up the map. The main irrigation canal (not marked) of the American Rio Grande Land and Irrigation Company ran north and south to the right of Capisallo Road. Generally speaking, the Anglo-Americans had their businesses and homes south of the railroad tracks with the exception of the packing sheds which were just north of the railroad track; and the Mexican American community was located north of the railroad tracks. This type of separation occurred in every town of the lower Rio Grande Valley of this era, with exceptions sometimes made when the Mexican American family or business owner was well-to-do. (City of Mercedes.)\n\nIn the above 1922 photograph of the Mercedes downtown area, the cars are parked at a slant and next to the sidewalks rather than in the middle of the street. This is Texas Avenue looking north. The two-story building on the right with light-colored brick is the second building used by the Hidalgo County Bank and Trust. It was located on the northeast corner of Texas Avenue and Third Street. In 1928, the bank built a new brick three-story building across the street on the southwest corner of Texas and Third Streets as seen in the photograph below. This edifice still exists today although it has been remodeled to face a new parking lot south of the building. (Above, MSTH; below, _Mercedes Enterprise_.)\n\nThis small frame building, which faced Third Street downtown, was first used by the Hidalgo County Bank and Trust Company in 1907. When the new brick building on South Texas Avenue was built later that same year, the frame building was used as a primary classroom with Miss Boyd as teacher. ( _Mercedes Enterprise_.)\n\nAfter holding classes in several buildings in town, a more permanent structure was finally completed facing Texas Avenue. This school, built in 1912, housed students from primary grades to high school grades. It was named for Harriet Claycomb Buck who was the mother of Nannie Mer Buck, the Mercedes superintendent of schools from 1915 to 1923. The building was demolished after irreparable damage in the hurricane of 1933. (Mercedes ISD.)\n\nElizabeth Riess (left), Gustavus K. Riess, and their two children Marion and Malcolm relocated to Mercedes from the northeast in 1910. Both \"Gus\" and \"Lizzie\" were active in their community, serving on the school board and city council from 1912 to 1914. Gus was a railroad station agent, and Lizzie is reputed to have developed the first pink grapefruit in the Rio Grande Valley. (Riess family.)\n\nAlthough schoolchildren in Mercedes had been attending school in makeshift classrooms beginning in 1907, the first student to complete her studies and graduate from the Mercedes school district in 1914 was Marion Riess, daughter of Gus and Lizzie Riess. She later became a teacher and principal in Mercedes. She married Rev. Herbert Haslam of Philadelphia, had five children, and helped with her husband's ministry while also continuing to teach. (Mercedes ISD.)\n\nIn 1914, only one person graduated from high school\u2014Marion Riess. The next year, Mercedes High School graduated four students\u2014Pearl Hearing, Nellie Linemann, Martha Wright, and Albert Rothrock. The program shown here describes the graduation ceremonies held on May 21, 1915, at the Moving Picture Theater. The program included poetry recitations, songs, several speeches, piano pieces, presentation of the diplomas, and a short play with the graduates participating. This play was the forerunner of the \"Senior Play\" tradition at Mercedes High School, although the play is no longer presented during the graduation ceremonies because of time constraints. On May 18, 1915, the junior class had honored the senior class with a banquet given at the Mercedes Hotel. On the menu were fruit cocktail, fried chicken, cream gravy, hot biscuits, currant jelly, mashed potatoes, string beans, pears, olives, and the new sensation from New York City: Waldorf salad. (Mercedes ISD.)\n\nSouth School, later renamed for Leon R. Graham, the superintendent of schools from 1941 to 1951, was built in 1921 facing Ohio Avenue between Ninth and Tenth Streets. Since its construction and through various remodeling projects and additions, Graham School served as a primary school, a junior high, a high school, and a migrant school. (Mercedes ISD.)\n\nNorth School, later renamed John F. Kennedy Elementary School, was built in 1910 but underwent numerous remodeling projects. School board minutes first called it the \"Mexican\" school, and later the \"Preparatory\" school. It served only Mexican American children from the north side of town until 1971 when the district desegregated and went to \"single-line\" (one or two grades) campuses rather than multi-grade neighborhood schools. (Mercedes ISD.)\n\nIn 1909, the first Presbyterian congregation of 19 members was organized by Rev. William W. Doggett. The church seen in this photograph was built in 1912, located on the corner of Fourth Street and Missouri Avenue with the Rev. Samuel M. Glasgow as the first resident pastor. (Vito Buenrostro.)\n\nThis photograph shows the first building of the First United Methodist Church built in 1936. It was later used for the Spanish-speaking congregation with the name El Buen Pastor Iglesia Metodista, or the Good Shepherd Methodist Church. A new structure was built on Sixteenth Street in the 1960s. (Vito Buenrostro.)\n\nRaquel Gonzales Palacios stands in the center front during an Easter Pageant with other unidentified girls in the spring of 1932. They are holding palm leaves and flowers as is traditional for this religious holiday. According to family members, the steps where the girls posed are most likely those of the rectory at Our Lady of Mercy Catholic Church. (Irma Palacios.)\n\nThe First Baptist Church of Mercedes was organized in 1907 with four charter members: Mrs. Fred Cutting, Mrs. Jimmy Johnson, and Mr. and Mrs. Lewis Boothe. They were served by missionaries until the building of the present church seen in the photograph. The new complex includes an auditorium, Sunday school classrooms, kitchen, and fellowship hall; it was erected in 1920 during the pastorate of the Rev. John C. Boyd. ( _Mercedes Enterprise._ )\n\nThe Immanuel Lutheran Church, first called the German Lutheran Church, was the first Lutheran church established in the Rio Grande Valley in 1910. The first pastor, Rev. Ernest J. Moebus stands on the steps in this early photograph. Using the Mercedes church as his headquarters, Rev. Moebus founded many preaching stations in the Valley. Most of the town's population of German heritage attended this church in the early days. (Library of Congress.)\n\nThe Immanuel Lutheran Church was moved to its present location at South Washington Avenue and Third Street where this brick structure was completed in 1929. Later additions included a parochial school, meeting rooms and a kitchen. Well known in the community is the excellent school that, although it has had to close on past occasions, is fully operating again. ( _Mercedes Enterprise_.)\n\nPictured here is a Baptist mission group in 1916 in a temporary building on North Missouri Avenue with Howell and Zepeda family members among others. This group would meet in the Mercedes Community Building at first but later built the Primera Iglesia Bautista Mexicana, or First Mexican Baptist Church. (Eddie Howell Sr.)\n\nThe Primera Iglesia Bautista Mexicana shown here was built around 1918, with Joseph Henry Howell as the first missionary preacher. Howell's father, William Albert Howell, had been a Union soldier captured by the Confederacy who escaped to Mexico and later wound up in Port Isabel. He opened a dry goods store there and raised a family, with Joseph Henry eventually relocating to Mercedes. (Carolyn C. L\u00f3pez.)\n\nThe first frame building of Our Lady of Mercy Catholic Church was erected on the north side of Hidalgo Street in 1909 on land donated by the American Rio Grande Land and Irrigation Company. The new church was built in its present location on Vermont Avenue and Third Street as seen here, and it was formally dedicated on October 22, 1922. (Our Lady of Mercy Church.)\n\nPictured here on horseback is Fr. Adrian Bornes of the Oblates of Mary Immaculate. He and Fr. Paul Hally were the first resident priests assigned permanently to Our Lady of Mercy Church at Mercedes in 1909. They served not only Mercedes but the surrounding missions as well, traveling to them on horseback. (Our Lady of Mercy Church.)\n\nThe first baptisms were performed and recorded in this registry at Our Lady of Mercy Catholic Church in February 1909. The children baptized were Jos\u00e9 M\u00e9ndez, Pedro Gregorio Garc\u00eda, and Eugene Harold Thomas. All three were born in late 1908 but had to wait for the priest to arrive early the next year in order to be baptized. The first two entries are written in Spanish and the third is written in English, undoubtedly to conform to the identities and ethnicities of the parents. The entries contain the date of baptism, the date of birth for the child, the condition of legitimacy, the parents' names, the \"padrinos\" or sponsors' names, and the priest's signature, Fr. Adrian Bornes, OMI. Baptismal records such as these are very important resources not only for genealogists or those who wish to trace their ancestry but also for historians. In many baptismal registries, if the family was an important one, the grandparents' names would also be included. (Our Lady of Mercy Church.)\n\nThis was the first Mercedes Telephone Directory issued in November 1909. It was reprinted in the 25th anniversary issue of the _Mercedes News-Tribune_ on September 16, 1932. After only two years of founding, Mercedes had more than 1,000 inhabitants and more than 30 businesses that included two hotels, several drugstores, dry goods and general merchandise stores, a newspaper office, hardware and lumber companies, pharmacies, churches, hospitals, and doctors. Notice in the listings that the town still accommodated those who used horses, buggies, and wagons with a livery stable. In 1909, the streets in Mercedes were not yet paved, but business was brisk and the downtown area was very active. At that time, Mercedes was considered the \"Queen City\" because it was the premier mid-Valley town and the fastest growing settlement in Hidalgo County. ( _Mercedes Enterprise_.)\n\nMercedes had become very prosperous very quickly. The Mercedes Commercial Club had succeeded in attracting a large number of \"newcomers\" to the town through its aggressive marketing strategies. The town was well-tended and progressive, with important Valley-wide meetings sponsored by the American Rio Grande Land and Irrigation Company often held at the Mercedes Hotel pictured here. (MSTH.)\n\nCato Palacios goes for a ride in the 1932 Chevrolet two-door coupe owned by his father, Abraham Palacios. By 1915, Mercedes had streets paved with caliche (or gravel) and had added streetlights to the downtown area. In 1933, Second Street was widened to asphalt-paved two lanes with shoulders, and it became US Highway 83. (Irma Palacios.)\n\nThe Baker, Montgomery, and Closner families enjoy a picnic at Lake Campacu\u00e1s c. 1915. The lake, also known as Tampacu\u00e1s or Carter's Lake, is about two miles north of Mercedes. It is part of the delta system of _resacas_ and arroyos that once were channels of the Rio Grande. It is about two miles long and is 700 feet wide at its broadest point. (MSTH.)\n\nPictured here in 1920 is Juli\u00e1n Villarreal (standing) with his daughter Adela. Juli\u00e1n and his bride, Dolores, arrived in Mercedes in 1912 and established a small grocery store at 117 North Texas Avenue. He also operated a bakery and often served as a foreman for agricultural work crews. Adela, now Mrs. Ram\u00edrez, says they took the picture while waiting for a train to Brownsville, where she was to see a doctor for an ear infection. ( _Mercedes Enterprise_.)\n\nThese gentlemen belonging to the Mercedes Mexican American Chamber of Commerce posed for a group picture in 1926. Mexican American businessmen in Mercedes organized early compared to other Valley cities in order to mutually support their business efforts. In the early years, most Mexican American businesses were located north of the railroad tracks because of residential segregation. ( _Mercedes Enterprise_.)\n\nPictured here is Alma Whatley's fifth-grade class at North Ward School in 1935. The front row includes, from left to right, Whatley, Andrea Barrera, Guadalupe Galv\u00e1n, Cuitl\u00e1huac Garc\u00eda, Guadalupe Casta\u00f1eda, unidentified, Juventino de Le\u00f3n Jr., Juan Torres, and Adelita Marroqu\u00edn. In the back rows are unidentified students. Whatley later taught English classes for many years at the Mercedes High School. (Delia de Le\u00f3n.)\n\nWord of the bonanza, or \"abundance of work,\" in the newly formed town of Mercedes reached the ears of workers on existing ranches as well as Mexican nationals. Many came seeking work, and they were readily hired to clear the land, work on the new farms, construct businesses and residences, and do general labor. (MSTH.)\n\nSome labor families were able to move into town where their housing was better. For those who lived in town, there was ample work at canning factories, produce sheds, and downtown stores and businesses. This home at 106 North Ohio Avenue, built around 1918, belonged to Calixta Ruelas Ortega, who willed it to her brother Severo Ruelas and family in 1936. (Carolyn C. L\u00f3pez.)\n\nMany labor families lived in small huts, or _jacales_ , made of available mesquite wood or old boards with thatched roofs. Water was available from wells or had to be purchased from water sellers. Winters were especially difficult because of the cold and wet. Many of these jacales did not survive the hurricane of 1933. (National Archives.)\n\nJos\u00e9 Garc\u00eda was a schoolteacher in Mexico who fled with his family in 1910 to the United States when the Mexican Revolution brought great disorder in Mexico. Together with several brothers, they opened multiple businesses in Mercedes including a drugstore, a clothing store, and many grocery stores. (Garc\u00eda family.)\n\nThe Garc\u00eda family opened this store, A.G. Garc\u00eda and Brothers Confectionery, on the corner of Hidalgo Street and Texas Avenue in 1920. The original store was started in 1913 and had customers from as far away as Brownsville and Rio Grande City. This store later burned down and was not rebuilt. (Garc\u00eda family.)\n\nA drugstore and fountain was run in one corner of the Garc\u00eda store. Pictured from left to right are Jos\u00e9 T. Leal, Octavio Garc\u00eda, Leopoldo Sol\u00eds, Elizandro Pe\u00f1a, Od\u00f3n G. Garc\u00eda (seated), and an unidentified patron standing to the back. (Garc\u00eda family.)\n\nWhen the American Rio Grande Land and Irrigation Company was scouting a location for its pumping plant in 1906, it wished to avoid the possibility of the river suddenly changing course. Without seeking permission, the company dug a channel at the Horc\u00f3n Tract site to divert the river flow, illegally changing an international boundary. The company paid a fine and the channel stayed. (Hidalgo County Historical Commission.)\n\nA bridge was built to connect the small town of R\u00edo Rico on the Mexican side in the Horc\u00f3n Tract site to Thayer on the American side. Both sides of the river had custom houses, and the bridge allowed both Mexicans and Americans to buy products available only in the other's country. (Hidalgo County Historical Commission.)\n\nMany visiting land parties and excursion groups were taken to Rio Rico for entertainment. In this photograph, an excursion group is watching a cockfight, and undoubtedly laying wagers on the outcome. In addition to cockfights, northerners could attend dog races and bullfights and could dine and dance. During Prohibition, R\u00edo Rico was a popular destination for many Valley residents. (Hidalgo County Historical Commission.)\n\nThe Rio Grande flooded regularly before Falcon Dam was built upriver in 1954. This photograph shows the Thayer\u2013R\u00edo Rico Bridge in 1941 when a severe flood caused great devastation in the area. Both American and Mexican businesses in R\u00edo Rico suffered great losses. (Hidalgo County Historical Commission.)\n\nThe 1941 flood caused the riverbanks to erode and the R\u00edo Rico Bridge to collapse completely. Previous floods had curtailed tourism in R\u00edo Rico, but the collapse of the bridge ended almost all river crossings. Only those willing to cross over on small boats provided any commerce to R\u00edo Rico businesses. (Hidalgo County Historical Commission.)\n\nWilliam \"Andy\" Tullis (left) and immigration officer White stand in front of the customs booth at the R\u00edo Rico Bridge in 1932. Tullis was manager of the B&P Bridge Company and oversaw operations there for many years. When the R\u00edo Rico Bridge collapsed in 1941, a new site was selected two miles upriver. The B&P completed the Nuevo Progreso Bridge in 1952. (Hidalgo County Historical Commission.)\n\nJudge Silas P. Silver, general manager of the American Rio Grande Land and Irrigation Company, rides as a passenger while driven by his chauffer in a Packard Touring car in this c. 1930 image. In his later years, Silver suffered a stroke and was partially disabled, but he remained active in city and county politics for many years. (MSTH.)\n\nSilas P. Silver's grandson Albert Stephens, seen here wearing short pants, was the son of Silver's only daughter, Mary Ellen, who married Charles Stephens of Louisiana. Here, Albert is shown on a trip to Silver's hometown of St. Louis, Missouri, in the 1920s. The boy on the horse is unidentified. (MSTH.)\n_Three_\n\nBUILDING THE COMMUNITY\n\nOn September 15, 1932, Mercedes celebrated its 25th anniversary with a variety of events including the parade shown in this photograph. All the Valley towns were invited to the celebration. The highlight of the first day was the Cavalcade of the Lower Rio Grande Valley, an elaborate pageant with more than 400 characters depicting the history of the area from 1519 to 1932. (MSTH.)\n\nThis 1932 photograph shows one of the more popular floats on which the Silver Queen of the 25th anniversary celebrations, Ida Greene Wattson, rides with her court. Elaborate costumes in white, gold, and silver were worn by all. Also included in the day's celebration were an art exhibit, a banquet, the Cavalcade performance, and a Founders' Day Ball at the Elks' new hall. (MSTH.)\n\nThe Founders' Day Ball was held in the Elks Lodge the night of September 15, 1932. Many Valley towns were represented in the queen's court by the duchesses. Participating towns included Alamo, Brownsville, Edinburg, Harlingen, La Feria, McAllen, Mission, San Benito, San Juan, and Weslaco. The Elks Lodge building still exists today but is now known as the second Mercedes Hotel. (MSTH.)\n\nIda Greene Wattson was crowned \"Silver Queen\" by Mercedes mayor William D. Chadick during the presentation ceremonies at the Founders' Day Ball. Chadick was the nephew of entrepreneur Col. Sam Fordyce and was involved in city and Hidalgo County politics. He was elected mayor of Mercedes in 1932. (MSTH.)\n\nThese gentlemen participated in the celebration of Mercedes's 25th Anniversary on September 16, 1932, with some dressed in their Mexican Independence Day costumes. Pictured from left to right are Jos\u00e9 T. Leal, Alfredo Garc\u00eda, David S\u00e1enz, Candelario Mar\u00edn, Rafael Hinojosa, Rogelio Hinojosa, and Jes\u00fas Garc\u00eda. ( _Mercedes Enterprise_.)\n\nThe Mercedes General Hospital seen here was organized in 1922. It was first located on Ohio Avenue and Sixth Street but was later moved to South Texas Avenue. It was originally set up by doctors Charles B. Buck, John G. Webb, and Daniel L. Heidrick, who bought the instruments and equipment from the Malone Brothers Hospital. (Carolyn C. L\u00f3pez.)\n\nDr. Manuel de la Torre's office and clinic faced North Texas Avenue in 1963. Early hospitals in Mercedes included an army hospital at Llano Grande and one at Camp Mercedes; the Malone Brothers Hospital, set up by the mid-1920s at 430 South Texas Avenue; the Mercedes General Hospital; the Lawler Clinic; and the Caballero Clinic. (Carolyn C. L\u00f3pez.)\n\nDr. Gonzalo Caballero's home was on the corner of Third Street and Missouri Avenue, and his clinic was just across the street. The clinic used to be Dr. Robert Johnston's when he still practiced. Today, the Caballero Clinic is still run by Gonzalo's son Dr. Eduardo Caballero. (Carolyn C. L\u00f3pez.)\n\nDr. Daniel L. Heidrick came to Mercedes with the 16th Cavalry when he was the Army camp doctor. When he was discharged from military service he returned to Mercedes to practice. Many said he must have delivered at least half the babies in Mercedes, and he is remembered for never turning anybody away because they could not pay. His home and office were located at 446 South Texas Avenue. ( _Mercedes Enterprise_.)\n\nThe Copacaba\u00f1a was a popular night spot in Mercedes beginning right after World War II. It was established by the S\u00e1enz brothers and their brother-in-law Ciro L. Uribe. It had a restaurant, a banquet room, and a terrace on the second floor for dances and parties. It was also later used for vocational training for returning GIs and later served as a migrant worker recruitment center. (Carolyn C. L\u00f3pez.)\n\nLocal newspaper the _Mercedes Enterprise_ is still located in the old Rex Theater building on Texas Avenue in downtown Mercedes. When first established, it also took care of other printing needs for the community such as billboards, posters, flyers, brochures, and pamphlets. It has previously also had the names _Mercedes Tribune_ , _Mercedes News-Item_ , _Mercedes News_ , and _News-Tribune_. (Carolyn C. L\u00f3pez.)\n\nThe De Le\u00f3n Swing Band entertained many Mercedes residents in the 1930s and 1940s with popular American and Mexican dance music. Pictured from left to right are two unidentified musicians, Felipe \"Pipe\" Oliv\u00e1rez, Jos\u00e9 Serna, Octavio Garc\u00eda, Concepci\u00f3n \"Chon\" Garc\u00eda, and band leader Juventino \"El Maestro\" de Le\u00f3n Sr. (Delia de Le\u00f3n.)\n\nThe Mercedes Pool Hall was owned by Florencio P\u00e9rez Sr., who established the business on North Texas Avenue and First Street in 1940. Later also known as the Latin American Club, it was frequented by patrons from all over the Valley. By 1948, P\u00e9rez had expanded the business to hold 14 pool tables and was also selling billiard supplies to other pool halls. (Carolyn C. L\u00f3pez.)\n\nLloyd P. Nolen acquired a P-40 Warhawk like the one shown above in 1951. The P-51 Mustang like the one shown below flew during World War II and was the second purchase by Lloyd P. Nolen and other partners in 1957 to found the Confederate Air Force, later renamed the Commemorative Air Force, flying out of the Central Valley Airport in Mercedes. The group continued to purchase, repair, and fly vintage World War II airplanes as a means of preserving aircraft that would otherwise have wound up as scrap metal. The central CAF headquarters moved to Harlingen in 1968, and later to Midland, Texas, in 1990. The Rio Grande Valley Wing of the CAF relocated to Brownsville, Texas, where they still have air shows. (Both, National Archives.)\n\nPictured here at an event of the Confederate Air Force are, from left to right, ? Edwards, Cedric Wood, and Lloyd P. Nolen. Nolen was a World War II flight instructor who ran the Mercedes Dusting Service in the 1950s at the Central Valley Airport. All members of the CAF were called \"Colonel\" as an honorary title. (Weslaco Museum.)\n\nThe Central Valley Airport is located on Mile 2 East, or Farm to Market Road 1425, and Mile 8 North. It is home base for crop-dusting planes since 1946. The airport was called Old Rebel Field when the Confederate Air Force began using it for air shows in the 1960s. (Vito Buenrostro.)\n\nPictured here is the Hidalgo and Cameron County Irrigation District No. 9 office building at 301 East Second Street (US Business Highway 83). This building was the second office of the American Rio Grande Land and Irrigation Company until 1927 when a group of farmers petitioned for the establishment of a regulatory group for water control and distribution, forming what became District No. 9. (Carolyn C. L\u00f3pez.)\n\nWorkers at the Hidalgo and Cameron County Irrigation District No. 9 posed for this impromptu photograph taken in the 1940s. Third from left is Cato Palacios, and last on the right is his brother Juan Palacios. All others are unidentified. In 2009, District No. 9 relocated its offices to 2304 North Farm to Market Road 491. (Irma Palacios.)\n\nIn the early 1900s, several Hidalgo and Cameron County families founded a small Jewish congregation. Lay readers led the services and the congregation met in private homes or rented space. In 1935, Temple Beth Israel pictured here was erected on the southwest corner of Tenth Street and Texas Avenue. In 1948, the congregation voted to dissolve and began attending synagogues in Harlingen or McAllen. (Hidalgo Country Historical Commission.)\n\nPictured in 1964 at the counter of the Farris Paint Store are, from left to right, storeowner E. Quincy Farris, his son James Farris, and friend Rosa Mae Wheeler. Farris owned several businesses including the Farris Lumber Company. He served as president of the Valley Lumbermen's Association in the 1950s and was well known in the community. (Mercedes ISD.)\n\nCharles P. Melton owned the Superior Citrus Fruits Packing Shed in Mercedes, shown above in 1946, to process the fruit from his R\u00edo Banco Farms. It was but one of many citrus packing sheds in Mercedes that organized through the Texas Citrus Growers Association. By 1929, this new fast-growing citrus industry was cultivated on more than 100,000 acres in Hidalgo County. (Carolyn C. L\u00f3pez.)\n\nMercedes has always been heavily involved in the citrus industry. Here, workers begin the sorting process of grapefruit at this packing shed. Grapefruit dominates the Texas citrus industry, with most of the remainder being oranges. The South Texas region is home to all Texas citrus production. Hidalgo County contains 85 percent of all citrus acres in Texas, with the remainder in Cameron and Willacy Counties. (National Archives.)\n\nIn this photograph, grapefruit is further separated for shipping whole or using it for juice processing. The earliest record of citrus in the Rio Grande Valley was seedling orange trees planted in 1882 by the Vela family at the Laguna Seca Ranch. Col. William A. Fitch was the first grower in Mercedes to plant oranges and grapefruit. The first commercial shipments from Mercedes began in 1918. (National Archives.)\n\nIn this juicing plant, cans are prepared to be filled with grapefruit juice. During the 1930s, whole-fruit markets slowed and juice became the most profitable commodity to sell. The Texsun canning plant in neighboring Weslaco was often used by Mercedes growers to process their oranges and grapefruit. (National Archives.)\n\nShown here is a Mercedes High School diploma from 1935 given to graduate William \"Billy\" Taylor. It is signed by Robert H. Kern, president of the school board; Fred Johnston, school board secretary; Ernest H. Poteet, superintendent of schools; and Leon R. Graham, principal of the Mercedes high school. The format shown here has changed little over the years. In 1935, the number of credits required to graduate from high school was 18. By 1968, the number of credits required had increased to 24. The number of years of schooling it took to graduate also changed. Up until 1941, only 11 years of school were required; from 1942 on, 12 years of schooling were required for graduation. (Mercedes ISD.)\n\nR. Newell Waters was the architect for the redbrick Mercedes High School that was completed and occupied in 1932 on the west side of Ohio Avenue and Eighth Street. It continued to be used as the high school until 1967, when a new high school was built east of the Main Canal at 1200 South Florida Avenue. (Mercedes ISD.)\n\nThe photograph here shows the building used as the high school gymnasium from 1940 to 1967. It was built together with the cafeteria in a Spanish revival style and was located at the same complex where the high school and junior high were on South Ohio Avenue. (Mercedes ISD.)\n\nShown here posing in front of the Mercedes High School on the right and the Mercedes Junior High School on the left is the 1936 Mercedes High School Pep Squad. The Pep Squad was formed to assist cheerleaders in generating enthusiasm from the fans at sports events, particularly football games. (Vito Buenrostro.)\n\nLeon R. Graham was superintendent of Mercedes schools from 1941 to 1951, having served as a teacher and a principal at various Mercedes schools before this post. After leaving the Mercedes school district, he worked at the Texas Education Agency in Austin eventually attaining the title of associate commissioner of education. (Mercedes ISD.)\n\nPosing here is the 1945 Mercedes Tiger Football Team: from left to right are (first row) Noel Caldeira, Angel Gonz\u00e1lez, Louis E. Drawe, James B. Taylor, and Anthony Caldeira; (second row) Ismael Gonz\u00e1lez, Bob Steer, Charles Eldridge, Kenneth Clarke, and Rudy Garza; (third row) coach Henry D. Crawford, Ernest Newmann, Senobio Uresti, Luis Garibay, Roy T. Pinkerton, Elmo Wade, Derald Hentrich, and Robert Crenshaw. (Mercedes ISD.)\n\nKathleen Twenhafel (left) and her sister Helen Twenhafel pose in 1937 in their orange and black Mercedes high school band uniforms, worn that year for the first time. Their father, Albert F. Twenhafel, came to Mercedes from Illinois in 1914 with a land party and purchased 52 acres on which he raised vegetables, citrus, cotton and corn. Helen married Herbert Vogel and is still active in Mercedes where the family continues the farming and ranching tradition. (Helen Vogel.)\n\nAlicia Garc\u00eda is shown here in 1951 winning the title of \"Fiestas Patrias Queen\" at an annual event sponsored by Our Lady of Mercy Catholic Church celebrating Mexican Independence Day. The daughter of Jes\u00fas and Enriqueta Salinas Garc\u00eda, she was crowned on September 16, 1951, by Lauro Izaguirre, the Mexican Consul based in McAllen, at the Mercedes High School Auditorium on Ohio Avenue. Tragically, Garc\u00eda was diagnosed with leukemia just weeks after the celebration, and by December of that year she had died, leaving her family and friends stunned by her loss. This photograph appeared in the 1952 Bengal yearbook, which was dedicated to her memory by the Mercedes High School students and staff. (Mercedes ISD.)\n\nThe all-female members of the Zeta Eta Sigma organization of the Mercedes High School pose here in this 1949 photograph in the Bengal yearbook. The club honors girls who are scholastically outstanding. Members must have made the honor roll to qualify for the organization. (Mercedes ISD.)\n\nMercedes High School 1962 freshman class officers stop to pose for a photograph as they prepare to decorate the football stadium on a Friday afternoon for a game later that night. From left to right are Stella Marroqu\u00edn, Lillian Billings, Vito Buenrostro, Cynthia Hoverson, and Joan Wilt. (Mercedes ISD.)\n\nFrederick L. Johnston was involved in Mercedes education all his life. By the age of 15 he was already teaching children at a ranch school. In 1908, he and Agapita Tijerina were the first teachers at North School. That year, they taught English to 189 Mexican American students in a two-room frame schoolhouse. Johnston also served many years as secretary to the Mercedes ISD School Board. (Mercedes ISD.)\n\nThe 1954 Mercedes Independent School District Board of Trustees is pictured here. Seated from left to right are Helen M. Watson, Dr. Thomas G. Edwards, Joe Winston, superintendent of schools Lawrence W. St. Clair, and Clyde Hollon. Standing are Ernest E. Marchant, Thomas B. Ewing, Frederick L. Johnston, James McElyea, and Joaqu\u00edn Fern\u00e1ndez. (Mercedes ISD.)\n\nBilly Gene Pemelton, 1960 Mercedes graduate and outstanding athlete, participated in the 1964 Summer Olympics in Tokyo, Japan, placing in the top 10 in the men's pole vault. For the Mercedes Tigers, he won the Texas state class 2A title in the pole vault (1959\u20131960) and in the high hurdles (1960). (Mercedes ISD.)\n\nPictured at this 1954 Campo Gardenia No. 3155 Woodmen of the World (WOW) award ceremony are, from left to right, (standing) Florentino Zamora, Amalia Zamora, and Juventino de Le\u00f3n Sr.; (seated) Jos\u00e9 \"Pepe\" D\u00edaz and Severo D\u00edaz. The WOW organization was very active in Mercedes, sponsoring events for schoolchildren, fundraising for scholarships, and taking part in patriotic events such as flag presentations. (Delia de Le\u00f3n.)\n\nThe Honorable Rub\u00e9n Hinojosa graduated from Mercedes High School in 1958, having demonstrated early leadership abilities. After earning business degrees from the University of Texas at Austin, he served 20 years as president and chief financial officer of the family business H&H Foods. He was elected to Congress in 1996 and is currently serving his ninth term as representative of the 15th District of Texas. He serves on the House Committee on Education and the Workforce and the Committee on Financial Services. He serves as ranking member of the Subcommittee on Higher Education and Workforce Training as well as serving on the Subcommittee on Health Employment Labor and Pensions. By popular acclamation, he was selected in 2012 to be chairman of the Congressional Hispanic Caucus for the 113th Congress (2013\u20132015). (Rub\u00e9n Hinojosa Congressional Office.)\n\nDr. Rolando Hinojosa-Smith is a 1946 graduate of Mercedes High School. He is an award-winning novelist of the _Klail City Death Trip Series_ , which comprises 15 volumes to date and for which he has received two prestigious awards: the Premio Casa de las Am\u00e9ricas and the Premio Quinto Sol _._ After serving in the Korean conflict and then completing a doctorate in Spanish literature, he held several teaching assignments, including Chairman of Chicano Studies at Minnesota. In the early 1980s he switched academic departments to become a professor of English and Creative Writing at the University of Texas at Austin. Currently, he holds the Ellen Clayton Garwood Chair in the English department at the University of Texas at Austin. He focuses on American literature, specializing in life and literature of the Southwest. He continues to teach and travels extensively. He has visited more than 250 colleges and universities in the United States and abroad, where he reads from his books and gives classes. In March 2014, Dr. Hinojosa-Smith received the Ivan Sandrof Lifetime Achievement Award given by the National Book Critics Circle. (Debbie Winslow.)\n\nHector P. Garc\u00eda, a 1932 Mercedes graduate, became a medical doctor and served in the US Army during World War II. He later founded the American GI Forum to assist Mexican American veterans who were struggling with discrimination and being denied veterans' benefits. He was named alternate ambassador to the United Nations in 1967; was appointed to the United States Commission on Civil Rights in 1968; was awarded the Presidential Medal of Freedom, the nation's highest civilian honor, in 1984; and was named to the Order of Saint Gregory the Great by Pope John Paul II in 1990. In 1998, he was posthumously awarded the Aguila Azteca, or Aztec Eagle, Mexico's highest award for foreigners, in a ceremony in Corpus Christi, Texas. Mercedes has honored him by naming its library for him\u2014the Dr. Hector P. Garc\u00eda Memorial Library. (Hector P. Garc\u00eda family.)\n\nThis image of the Mercedes High School Class of 1932 hangs on the \"Wall of Fame\" at the newest Mercedes High School Cafeteria along with all graduates since 1914. In the upper left corner of this class is Hector P. Garc\u00eda. Only seven of the 32 graduates were Mexican American, although they were the majority population of the city at that time. (Mercedes ISD.)\n\nPosing for Bengal yearbook pictures are the 1957 Mercedes High School \"class favorites.\" From left to right are (first row) Marie Watson, Patsy O'Shea, and Nancy Archer; (second row) Joseph Fern\u00e1ndez, Bud Terry, and Rub\u00e9n Hinojosa. (Mercedes ISD.)\n\nThe R\u00edo Theater was one of three theaters in Mercedes. The other two were the Rex and the State. The State was previously the Capitol and the Empire. This 1940 photograph shows the first R\u00edo Theater before it was remodeled. Standing in front are, from left to right, Mary Guerrero, Mary Zamora, Olga Ochoa, Raul Galv\u00e1n, Carlos \"Chale\" Leal, Cato Palacios, and two unidentified young boys (at right and in back). (Irma Palacios.)\n\nThe R\u00edo Theater was remodeled in 1946 with a brand new marquee, as shown in this photograph, and a capacity of 639 seats. The Rio Theater specialized in Spanish-language movies, while the Rex showed Westerns, and the State showed first-run movies. Alberto Arteaga worked at the R\u00edo since 1935, eventually becoming the manager. (Sylvia Arteaga Calles.)\n\nThe R\u00edo Theater often had a capacity crowd such as this one gathered for opening night of the remodeled theater on May 17, 1946. The R\u00edo was located on North Texas Avenue near Hidalgo Street; the Rex was located on South Texas Avenue on the 200 block; and the State was also located on South Texas Avenue next to the First National Bank on the 300 block. (Sylvia Arteaga Calles.)\n\nThe Rio Theater did not just show movies, it also featured \"Amateur Night,\" during which local talent could perform for prizes. Pictured here in this 1955 photograph are, from left to right, announcer Mart\u00edn Rosales, Carlota Cant\u00fa (in the white dress), Olga Rodr\u00edguez (with the black skirt), and two unidentified girls. Competitors mostly sang and danced, but some did specialty acts such as juggling or gymnastic tumbling. (Sylvia Arteaga Calles.)\n\nProjectionist Emilio Ybarra Sr. (standing at left) explains how it all works to several unidentified men who are interested in movie theaters in this 1954 photograph. Ybarra worked for all of the Mercedes Theaters: the Rex, the State, the R\u00edo, and the Wes-Mer Drive-In Theater west of town. (Sylvia Arteaga Calles.)\n\nRaquel Gonzales Palacios (left) stands in as _madrina_ , or female sponsor, for Emilia Marroqu\u00edn as she makes her first communion at Our Lady of Mercy Catholic Church on May 21, 1949. _Padrinos_ and _madrinas_ agreed to assist parents with the responsibility for a child's religious upbringing in the Catholic faith. (Irma Palacios.)\n\nIn the center wearing a hat is architect Roscious Newell Waters with unidentified friends or associates. He designed many Mercedes businesses and residences, including, among many others, city hall, the Llano Grande Clubhouse, and the Mercedes High School and auditorium on Ohio Street. Waters's architectural styles changed over the years and included Spanish, Gothic, Princeton, and Early Modern designs. (Hidalgo County Historical Commission.)\n\nThe building pictured here was constructed in 1928 in a Mediterranean or Spanish style according to a design by architect Roscious Newell Waters. It was originally built as a private residence for Harold Lehman at a cost of $20,000. It later became the clubhouse for the Llano Grande Resort and Golf Club. (Vito Buenrostro.)\n\nFerguson \"Ferg\" Wood was the owner and manager of Ferg's Foodland, a popular grocery store located at 425 Second Street, also known as US Business Highway 83. Wood learned the grocery business from Robert H. Kern, working at Kern's grocery store from 1928 to 1958. In 1972, Wood retired and sold the store to Wendel Drefke. (Carolyn C. L\u00f3pez.)\n\nO'Shea Furniture was located on the northeast corner of US Business Highway 83 and Texas Avenue. The owners were Eugene and Anne O'Shea, who had relocated to Mercedes from Arkansas in the 1940s. The building was later demolished in the 1960s to make way for other businesses. (Carolyn C. L\u00f3pez.)\n\nAnother popular grocery store was Salinas Grocery, founded by brothers Luis and Pedro Salinas and Luis's son Rigoberto in 1946. In 1957, they expanded to the larger building seen here and renamed the business Salinas Food Store. Don Luis and his son Rigoberto were very involved citizens and remembered by their patrons for excellent service. (Carolyn C. L\u00f3pez.)\n\nEl Sombrero Restaurant was a well-known eating place that had many local and Valley-wide clients. It was located on US Business Highway 83 and first opened by Arturo Arredondo in the 1940s. It was later purchased by Wayne and Mary Love. Many prom attendees may recall having dined there after the junior-senior prom at their high school. (Carolyn C. L\u00f3pez.)\n\nPictured here are Marguerite (left) and Shelley H. Collier Sr. Collier was well known in banking circles and came to work at the First National Bank in 1923. An active and involved citizen in Mercedes, he stayed with the bank until 1972 when he retired. Personally as well as through the bank, Collier supported educational ventures through scholarships and gifts to students and often made charitable contributions to many organizations. (Mercedes Centennial Committee.)\n\nThe second building used by the First National Bank was designed by Austin architect Hugo Franz Kuehne and was occupied by the end of 1921. Chatting in front of the building in this 1960 photograph are, from left to right, Daniel Barrientos, Domingo Buenrostro, and Cristina C. Buenrostro with an unidentified man walking by. (Vito Buenrostro.)\n\nLion's Club members pose in Western dress in 1957. Seated from left to right are Shelly Collier Jr., Eli R\u00edos, Ad\u00e1n Longoria, and Fleet S. Lentz. Standing are Bill Basinger, Joe Adame, Dennis Clifford, Gordon Leonard, James Van Burkleo, William B. \"Dub\" Lauder, and James Kirber. ( _Mercedes Enterprise_.)\n\nThe temporary location of the First National Bank when it first opened in January 1921 was in this building. Under the direction of Pres. John Hackney, the bank moved that same year into a beautiful new brick building on South Texas Avenue. Today, the bank is on the corner of Texas Avenue and US Business Highway 83 as the Texas National Bank. ( _Mercedes Enterprise_.)\n\nThe 1940s and 1950s were some of the most prosperous years for Mercedes. Many businesses had recovered from the Depression of the 1930s, and the post\u2013World War II boom was at its highest point. In 1940, the population was 7,624, but by 1950 it had grown to 10,065. In this downtown scene of Mercedes, every building is occupied by a thriving business. (MSTH.)\n\nSisters Ida (left) and Irma \"Nellie\" Palacios dress up in their Fiestas Patrias costumes in 1954. The celebrations recognize the area's Mexican heritage and recall the independence won by Mexico from Spain in 1821. In earlier days, the city sponsored the celebrations but later the Catholic churches became sponsors of the event. (Irma Palacios.)\n\nHowell family members are pictured here in a late 1940s photograph. From left to right are (seated) Jacinto Howell, Joseph Henry Howell Sr., Pauline Howell Gorena, Mar\u00eda Escobedo Howell, and Casimiro Howell; (standing) William Howell, Henry Joseph Howell, Albert Howell, Charles Howell, and Joseph Henry Howell Jr. Children in arms are Beatriz Gorena (left) and Edna Gorena, Pauline's children. (Eddie Howell Sr.)\n\nMany Mercedes residents remember the good old days when gasoline was 27\u00a2 a gallon in the 1960s. This Kayo Is O.K. gas station was located on the northeast corner of US Business Highway 83 and Missouri Avenue. By then, gasoline stations were \"self-serve,\" although the pumps were still not accepting credit cards. (Carolyn C. L\u00f3pez.)\n\nThe Borderland Hardware Company, now in its 95th year, is the oldest hardware store in the Rio Grande Valley. It was first opened for business by owner Ellery E. \"Jack\" Johnson in 1919 on the southeast corner of US Business Highway 83 and Texas Avenue, as pictured above. Robert Eilers, pictured below at a Rio Grande Valley Livestock Show display in 1955, moved to Mercedes after serving in the US Army and began working at Borderland in 1949. He became Mercedes manager in 1955 and bought the store in 1969. In 1999, the family built a completely new 25,000-square-foot store on the northeast corner of US Business Highway 83 and Ohio Avenue, occupying almost the complete block. Robert and his wife, Loretta, along with their son Kenneth and his wife, Debbie, remain very active today in the family-owned operation. (Both, Eilers family.)\n\nIn October 1953, Pres. Dwight D. Eisenhower rode through Mercedes on his way to the dedication of Falcon Dam in neighboring Starr County. Hundreds of Mercedes citizens turned out to line US Business Highway 83 to get a glimpse of the president as he rode by in a red 1953 Lincoln Capri convertible. (Delia de Le\u00f3n.)\n\nZeferino and Eli R\u00edos, custom boot makers in Mercedes since 1935, presented these boots to President Eisenhower in 1954 to commemorate his visit to the Rio Grande Valley the year before. Their design includes the Capitol, the Great Seal of the United States, sunflowers from Eisenhower's home state of Kansas, and \"Ike,\" the president's nickname. They are currently housed at the National Archives in Washington, DC. (National Archives.)\n\nThis photograph shows the bus station once located between the McAfee Insurance Building and the Hidalgo and Cameron County Water District No. 9 Building on US Business Highway 83. Valley Transit Company and Greyhound buses stopped here to pick up passengers traveling up and down the Valley as well as out-of-state. (Carolyn C. L\u00f3pez.)\n\nBy the middle of the 20th century, Mercedes had nearly 11,000 inhabitants and was a busy, prosperous city. In addition to the surrounding thriving farms and ranches, the downtown area on both the north and the south side boasted many shops and businesses. But Mercedes did not always experience sunny days. This photograph captures the rains that accompanied Beulah, the 1967 hurricane that caused widespread flooding in the Valley. ( _Mercedes Enterprise_.)\n_Four_\n\nWARS AND   \nNATURAL DISASTERS\n\nThe map here shows the disputed territory called the Nueces Strip that resulted from Texas Independence in 1836 and led to the Mexican\u2013American War. The land was claimed by both Mexico and the United States after it annexed Texas in 1845. People living in the Rio Grande Valley at that time were caught in a dangerous situation, particularly when the United States sent troops to secure the area. (National Archives.)\n\nGen. Zachary Taylor, pictured here, was sent to Texas by Pres. James K. Polk. In March 1846, Taylor marched to the Rio Grande, a move that was interpreted by Mexican forces as an invasion of their country. The Battle of Palo Alto near Brownsville in May 1846 was the first official battle of the Mexican\u2013American War, followed soon after by the nearby Resaca de la Palma engagement. (National Archives.)\n\nThe lower Rio Grande became important during the Civil War in 1863 as the only waterway available for the export of Confederate cotton. This sketch by L. Avery in _Frank Leslie's Illustrated Newspaper_ shows the cotton being ferried across to Mexico for transport to British ships trading arms for cotton. Involvement in trade by Anglo-Americans during this time caused increased interest in the Rio Grande Valley. (Library of Congress.)\n\nIn the late 1800s, steamboats became important to the Valley in conducting trade with Mexico and with European countries. They provided another means of transporting goods up and down the Rio Grande Valley besides by mule train. Steamboat _Bessie_ , pictured here, was the last of the Mifflin Kenedy\u2013Richard King Rio Grande shipping fleet. It made its final trip on the river in 1902. (National Archives.)\n\nAfter years of social turmoil in Mexico, the Mexican Revolution broke out in 1910. By then, the Rio Grande Valley had established several towns, including Mercedes, founded in 1907. Between 1914 and 1918, various Mexican revolutionary factions raided on the US side, usually attempting to get food and other supplies needed to conduct their war. Many Mercedes and other Valley residents lived in terror of spillover violence from Mexico. (National Archives.)\n\nTensions in the Valley grew higher when Mexican general Lucio Blanco's revolutionaries took Matamoros on June 4, 1913, just across the river from Brownsville and only 39 miles away from Mercedes. Some Mercedes residents wrote memoirs later describing their trip to Matamoros to see the executions by firing squad of the captured federal soldiers. Many people fled Mexico and came to the Valley seeking refuge from the war. (National Archives.)\n\nWhen Mexican revolutionary caudillo Pancho Villa crossed into US territory in March 1916 and attacked Columbus, New Mexico, over an arms deal gone bad, US residents along the southern border began clamoring Washington to send troops for their protection. By August 1916, an estimated 117,000 national guardsmen were stationed along the border in Texas, New Mexico, Arizona, and California. (National Archives.)\n\nTwo Army camps were set up near Mercedes in 1916. One was the Llano Grande camp between Weslaco and Mercedes on both north and south sides of now US Business Highway 83. The other was Camp Mercedes, shown in this photograph, the large bulk of which was located east of the Mercedes Main Canal and south of Tenth Street. (Library of Congress.)\n\nThe guardsmen assigned to the Valley camps trained in all aspects of warfare available at that time including artillery fire as seen here, something which would serve them well when the United States entered World War I in Europe in 1917. During this time of the Mexican Border Service, motor vehicles and aircraft were first used in the US military forces. (National Archives.)\n\nShown here is part of a 1917 hand-drawn map prepared by Sgt. Charles A. Rice based on field notes taken while out on patrol with the US Army while stationed in the Rio Grande Valley. The map shows the winding Rio Grande forming the boundary between Mexico and the United States. The present-day towns of McAllen, Pharr, San Juan, Donna, Mercedes, and La Feria can be seen along the track of the St. Louis, Brownsville and Mexico Railway. The Military Highway is labeled as having been built by Gen. Zachary Taylor in 1847, but many historians dispute this statement due to lack of documented evidence. In his Valley campaign during the Mexican\u2013American War, Taylor used steamships and the road running south of the river to transport his troops and supplies upriver to Camargo. (Hidalgo County Historical Society.)\n\nSoldiers from Camp Mercedes are pictured stationed at the Toluca Ranch. The Toluca Ranch south of Mercedes was considered a likely target for a Mexican raid during that era, and it was raided on four separate occasions. There were several skirmishes with Mexican \"bandits\" in the Valley in the early 1900s, but no major invasions by Mexicans ever occurred. (MSTH.)\n\nOne of early Mercedes's greatest problems was flooding. In 1909, Mercedes suffered the worst flood in its history. At that time, there were no flood controls on the river such as dams or levees. During this natural disaster some parts of Mercedes were 14 feet deep in floodwaters. This photograph shows the northeast corner of Texas Avenue and Second Street. (MSTH.)\n\nThe Howell family is pictured in 1916 standing outside a community building on North Missouri Avenue that was used for church services. From left to right are Mary, Henry Joseph, William, Albert, Jacinto, Joseph Henry Jr. (in arms), Pauline, Casimiro, Mar\u00eda Howell, and Joseph Henry Howell Sr. Tragically, little Mary died in 1918 during an influenza pandemic. Both influenza and smallpox took a deadly toll in the Valley during the early 1900s. (Eddie Howell Sr.)\n\nThe Great Depression hit the Rio Grande Valley with a vengeance. In Mercedes, many businesses failed and farmers went broke. Many people from other states came to the Valley looking for work and often clashed with locals that also needed work. Tent cities like this one in Mercedes in 1939 could be found all over the Valley. (Library of Congress.)\n\nMercedes businesses also suffered devastating fires such as this one on North Texas Avenue when Leal's Electric business burned down in 1954. The girl is unidentified, but Alberto Arteaga, manager of the Rio Theater next door to Leal's Electric, captured her frightened look in this photograph. (Sylvia Arteaga Calles.)\n\nUnidentified persons in this photograph prepare smoke pots to protect a citrus crop during an anticipated hard freeze in 1920. Smoke pots contained kerosene and were placed between the rows of grapefruit or orange trees to keep them warm during nights of low temperatures. A severe freeze could destroy the citrus crop and ruin agricultural investments in a single night of low temperatures. (Weslaco Museum.)\n_Five_\n\nTHE RIO GRANDE VALLEY   \nLIVESTOCK SHOW\n\nRio Grande Valley Livestock Show cover girls and alternates pose for this 1956 photograph. The Rio Grande Valley Livestock Show officially began in 1939 as an annual project of the Mercedes Chamber of Commerce in efforts to promote livestock and poultry production. Although the livestock show began in Mercedes, it quickly included Cameron, Willacy, Starr, and Hidalgo Counties in the project. (Rio Grande Valley Livestock Show Museum.)\n\nAn aerial view of the Rio Grande Valley Livestock Show grounds covering more than 100 acres at the north end of Texas Avenue was taken in 1955 when many metal and wood buildings were used to replace tents and other less durable structures. With more than 1,000 volunteers and 4,000 exhibitors expected, the 2014 show was a far cry from the earliest shows. The earliest documented show was a one-day affair in 1913 held at the Mercedes Power Plant, a two-story building located on Second Street (US Business Highway 83) and the Main Canal. It was so popular that the next year, in 1914, it was a three-day event with many exhibits of livestock and agricultural products, lectures, a US Cavalry sham battle, and boat rides on the Main Canal. It did not become an annual event, however, until 1939. Unfortunately, in 1965, a fire destroyed much of the early records of the show, and photographs such as this one have slowly been recovered from other sources. (Mercedes ISD.)\n\nSun Valley Horse Show Queen Pip Setter and her horse Rex pose for a photograph in 1964. The Rio Grande Livestock Show grounds host many events other than the annual March livestock show. The Sun Valley Horse Show is a Mercedes High School FFA Club fundraiser that accepts Valley-wide competitors in various events. Many other horse shows are held annually at the livestock show grounds. (Weslaco Museum.)\n\nThomas Treasure (left) awards the \"Best Cowboy Trophy\" to 12-year-old Kenny Reger mounted on his horse Mr. Red, which is held by an unidentified girl, in the Sun Valley Horse Show held October 1964 at the Rio Grande Valley Livestock Show grounds. Competitive events included barrel races, stake races, pole bending races, and children's showmanship. (Weslaco Museum.)\n\nThe traditional opening of the Rio Grande Valley Livestock Show is the parade held on the first day of the show. It always includes decorated floats, trail riders from various horseman's associations, marching school bands, cover girl candidates, show officials, city officials, local clubs and organizations either walking or riding floats, and singing or acting stars invited to perform. (Sylvia Arteaga Calles.)\n\nA guest band from the Justo Sierra Boy's Orphanage and Military School in Monterrey, Nuevo Leon, Mexico marches in the Rio Grande Valley Livestock Show Parade in March 1956. Special arrangements were made to allow members of the school to attend the livestock show, and local families agreed to house the younger students in order to help defray lodging costs. (Sylvia Arteaga Calles.)\n\nMimi Garibay, 1956 cover girl winner sponsored by the Brownsville FFA organization rides in a 1956 Dodge convertible in the Rio Grande Valley Livestock Show Parade as it wends its way up Texas Avenue to end at the show grounds on North Texas Avenue. (Sylvia Arteaga Calles.)\n\nMembers of the Mid-Valley Horseman's Association pictured here begin their trail ride in Roma about 50 miles west and ride toward Mercedes for two days, camping out and eating from a chuck wagon to finally arrive in time to participate in the Rio Grande Valley Livestock Show Parade and other activities. (Rosendo Gonzales.)\n\nSinger, film actor, and songwriter Rex Allen (in the center) poses with 1962 cover girl alternates. At left is second alternate Sherrie Gallaway, and at right is first alternate Pam Knapp. Rex Allen, called the Arizona Cowboy, was well known for starring in 19 of Hollywood's Western movies with Buddy Ebsen and Slim Pickens as his sidekicks. (Mercedes ISD.)\n\nEnjoying the various food booths available at the Rio Grande Valley Livestock Show are many show visitors. Thousands of local visitors as well as visitors from other states and even other countries visit the Rio Grande Valley Livestock Show every year. It is considered one of the Top Ten Shows in Texas and recently draws close to 200,000 visitors annually. (Rio Grande Valley Livestock Show Museum.)\n\nParthenia Archer (left) receives two free rodeo tickets from show director Frances Cooper to see movie star Ken Curtis, who starred as Festus in the television show _Gunsmoke_ and was featured at the 1967 Rio Grande Valley Livestock Show. Two unidentified children help hold up the poster announcing the event. (Weslaco Museum.)\n\nIn this 1961 photograph, Mar\u00eda Buenrostro (center back) prepares to take these children to the Rio Grande Valley Livestock Show. Children are, from left to right, N\u00e1tiz Buenrostro, Yolanda Jilpas, and Guadalupe Jilpas. Their activities include visiting the children's barnyard, the arts and crafts exhibits, the show animal pens, the rodeo, the midway carnival, antique farm equipment exhibit, a magician show, and enjoying live music. (Vito Buenrostro.)\n\nCharles \"Charlie\" Rankin, a popular Rio Grande Valley radio and television personality who was considered the \"voice of agriculture\" in the Valley for many years for his informative noon radio programs, takes part in a \"wild\" cow-milking contest during the 1971 Rio Grande Valley Livestock Show. From left to right, the competitors are Bob McDonald of KRGV holding the cow's tail, Rankin milking the cow, and Burt Johnson of KRIO holding the cow's head. Rankin was declared the winner. Rankin attended Texas A&M University after two years in the US Navy on an aircraft carrier during World War II. While at the university, he was instrumental in establishing the National Intercollegiate Rodeo Association, serving as its first president in 1949. He was often invited to serve as a judge or an announcer at the Rio Grande Livestock Show and Rodeo in Mercedes. (Rio Grande Valley Livestock Show Museum.)\n\nThe Rio Grande Valley Livestock Show cover girl parade float every year features the winning cover girl and the first and second alternates. The parade is the opening event every March that kicks off the livestock show. The young ladies posing here are 1986 cover girl Claudette Smith and unidentified alternates. (Rio Grande Valley Livestock Show Museum.)\n\nCharles \"Charlie\" Rankin, well-known radio and television personality in the Valley for his farm bureau talk shows, interviews Lorne Green and Ben Cartwright of the television show _Bonanza_. The two stars were featured guests of the Rio Grande Valley Livestock Show in 1973. The show often brought big name stars, including Gene Autry, Lynn Anderson, Johnny Rodr\u00edguez, Dwight Yoakum, and many others. (Rio Grande Valley Livestock Show Museum.)\n\nAn unidentified rider competes in the saddle-bronc riding event at the Rio Grande Valley Livestock Show and Rodeo in 1959. In saddle-bronc riding, the rider holds a braided line with only one hand and attempts to stay on for at least eight seconds. Scores range from zero to 100 with 50 points depending on the horse's bucking moves and 50 depending on the rider's skill. (Rio Grande Valley Livestock Show Museum.)\n\nTaken at the fryer pens at the Rio Grande Livestock Show in 1965, this photograph shows Steven Dollery (left) and his sister Stephanie Dollery, 4-H members and children of James Dollery. James Dollery was an agriculture teacher at Mercedes High School and head sponsor of the FFA club in the 1950s and 1960s. (Weslaco Museum.)\n\nLeading the 1971 Rio Grande Valley Livestock Show Parade are two unidentified men wearing chaps with the famous HK brand of the 825,000-acre King Ranch. The HK brand was one of two official brands registered in 1859 for the King Ranch. The letters were the initials of Henrietta King, wife of King Ranch founder Richard King. In 1869, the Running W brand was registered and it remains the official brand today. The King Ranch is well known for developing the Santa Gertrudis cattle breed now recognized as a superior beef cattle breed. In its horse breeding efforts, the King Ranch has produced 1946 US Triple Crown winner Assault and 1950 Kentucky Derby winner Middleground. A King Ranch Quarter Horse named Wimpy was the very first registered horse of the American Quarter Horse Association in 1940, receiving the designation \"P-1.\" (Rio Grande Valley Livestock Show Museum.)\n\nFrom left to right, 1959 cover girl Ann Watson poses in this photograph with Al Martin and Sherry Tripp, 4-H participants, and their \"Grand Champion Pen of Fryers\" at the 1958 Rio Grande Valley Livestock Show, as well as Dr. Joseph Townsend of Texas A&M University, who served as a judge at the event. (Rio Grande Valley Livestock Show Museum.)\n\nThe Studebaker military escort wagon in this photograph is part of the 1st Cavalry Division from Fort Hood, Texas. The group formed in 1972 under the direction of Maj. Gen. James C. Smith and represents an 1870 era \"horse soldier\" troop wearing standard issue Civil War Union uniforms. They perform mounted drills and weapons demonstrations and were featured guest performers at the 1973 Rio Grande Valley Livestock Show. (Rio Grande Valley Livestock Show Museum.)\n\nCharles \"Charlie\" Rankin (left) poses with an unidentified member of the Texas A&M University's Parson's Mounted Cavalry, a ceremonial horse cavalry unit founded in 1974 to preserve traditions of the Texas A&M Cavalry of the 1920s and 1930s. The Cavalry is part of the Corps of Cadets, which was established in 1876 as a student military organization. Texas A&M University is one of six US colleges classified as a senior military college. The Corpsman shown here with Rankin is wearing \"senior boots,\" brown leather boots to the knees worn only by Texas A&M seniors. Parson's Mounted Cavalry represents the university at parades, agricultural fairs, livestock shows, and equestrian events in Texas. The group performed military maneuvers on horseback at the 1975 Rio Grande Valley Livestock Show. (Rio Grande Valley Livestock Show Museum.)\n\nVincent Neuhaus, left, and Earl Neuhaus, mounted, participate in a steer-riding event at the Rio Grande Valley Livestock Show in 1964. Both attended Texas A&M University in College Station, Texas. Both own several successful farm equipment businesses although in different cities, and Earl also has served as president of the Rio Grande Valley Livestock Show. (Rio Grande Valley Livestock Show Museum.)\n\nIn this September 1959 photograph, members of the Board of Directors of the Rio Grande Valley Livestock Show get an update on planning activities for the upcoming 1960 Show. The board is limited to 24 members as set forth in its bylaws, and board members all work for the Livestock Show on a voluntary basis. (Rio Grande Valley Livestock Show Museum.)\n\nPictured recently are Dario \"D.V.\" Guerra (left), president emeritus of the Rio Grande Valley Livestock Show, and grande dame Frances Richmond Cooper, show director. Guerra is the owner of the D.V. Guerra Ranch in Edinburg and is a longtime volunteer with the livestock show. Cooper first went to work for the Rio Grande Valley Livestock Show in December 1954 as secretary to the manager. In 1959, she became show manager and remained in that position until she retired in 1986. Upon her retirement, she served on the board of directors as a volunteer until her death in 2009. She always declared she loved best the awards she received from the Texas FFA and the 4-H clubs. The Texas Association of Fairs and Events, in which she served 13 years as executive secretary, awarded her with its lifetime achievement award in 2002. In 2004, the International Association of Fairs and Expositions presented her with the prestigious Heritage Award for promoting not only the Rio Grande Valley Livestock Show but also fairs across the United States. (Rio Grande Valley Livestock Show Museum.)\nDISCOVER THOUSANDS OF LOCAL HISTORY BOOKS  \nFEATURING MILLIONS OF VINTAGE IMAGES\n\nArcadia Publishing, the leading local history publisher in the United States, is committed to making history accessible and meaningful through publishing books that celebrate and preserve the heritage of America's people and places.\n\nFind more books like this at\n\n**www.arcadiapublishing.com**\n\nSearch for your hometown history, your old stomping grounds, and even your favorite sports team.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"# Table of Contents\n\n  1. Title Page\n  2. Table of Contents\n  3. Copyright\n  4. Preface\n  5. Introduction\n  6. The Basics\n  7. The Order of a Business Meeting\n  8. Presenting Business to the Assembly\n  9. Debating the Motion\n  10. Voting\n  11. Motions\n  12. Using Subsidiary Motions to Help Adopt a Main Motion\n  13. Using Privileged Motions\n  14. Using Incidental Motions\n  15. Using Motions That Bring a Question Again Before the Assembly\n  16. Officers\n  17. Nominations and Elections\n  18. Committees\n  19. The Role of the Member\n  20. Discipline\n  21. Meetings\n  22. Strategies for Individual Motions Illustrated\n  23. Most Frequently Asked Questions\n  24. Various Types of Governing Documents\n  25. Bylaws\n  26. Homeowners' Associations (HOAs)\n  27. Appendix A\n  28. Appendix B\n  29. Appendix C\n  30. Appendix D\n  31. Appendix E\n  32. Appendix F\n  33. Appendix G\n  34. Index\n  35. DVDs for Parliamentarians\n  36. About Robert McConnell Productions\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Table of Contents\n\n  1. Series Page\n  2. Title Page\n  3. Copyright\n  4. About The Author\n  5. Introduction to The Ultimate Algorithmic Trading Systems ToolBox\n  6. Chapter 1: Introduction to Trading: Algorithm Development\n    1. What Is an Algorithm?\n    2. How to Get My Trading Idea into Pseudocode\n    3. Summary\n  7. Chapter 2: Stochastics and Averages and RSI! Oh, My!\n    1. Oscillators\n    2. Price-Based Indicators\n    3. Summary\n  8. Chapter 3: Complete Trading Algorithms\n    1. Trend-Trading Battle Royale\n    2. Portfolio Composition\n    3. Multi-Algorithm Strategy (MAS)\n    4. Summary\n  9. Chapter 4: Introduction to AmiBroker's AFL\n    1. Quick Start\n    2. Price Bar Interface\n    3. AFL Array Programming\n    4. Syntax\n    5. AFL Wizard\n    6. AmiBroker Loop Programming\n    7. Summary\n  10. Chapter 5: Using Microsoft Excel to Backtest Your Algorithm\n    1. VBA Functions and Subroutines\n    2. Data\n    3. Software Structure\n    4. Programming Environment\n    5. Summary\n  11. Chapter 6: Using Python to Backtest Your Algorithm\n    1. Why Python?\n    2. Python Installation\n    3. PSB Installation\n    4. PSB Structure\n    5. Getting Down to Business\n    6. Summary\n  12. Chapter 7: An Introduction to EasyLanguage\n    1. TradeStation IDE\n    2. Syntax\n    3. Samples of EasyLanguage\n    4. Summary\n  13. Chapter 8: Genetic Optimization, Walk Forward, and Monte Carlo Start Trade Analysis\n    1. Utilizing TradeStation and AmiBroker\n    2. Computers, Evolution, and Problem Solving\n    3. Population\n    4. Initial Population Setup Using VBA Excel\n    5. Testing Fitness of Chromosomes Using VBA Excel\n    6. Selection\n    7. Reproduction\n    8. Mutation\n    9. Using Genetic Algorithms in Trading System Development\n    10. Preventing Over-Curve-Fitting\n    11. Walk-Forward Optimizer: Is It Worth the Extra Work and Time?\n    12. Monte Carlo Analysis\n    13. Start Trade Drawdown\n    14. Summary\n  14. Chapter 9: An Introduction to Portfolio Maestro, Money Management, and Portfolio Analysis\n    1. Fixed Fractional\n    2. Portfolio Maestro\n    3. Summary\n  15. Appendix A: AmiBroker\n    1. Keywords\n    2. Flow Control Structures\n    3. Functions\n    4. Utilizing Exploration for Debugging\n    5. Position Sizing in Futures Mode\n  16. Appendix B: Excel System Backtester\n    1. Data Arrays\n    2. Keywords\n    3. Functions and Subroutines\n  17. Appendix C: Python System Backtester\n    1. Data Arrays or Lists\n    2. Keywords and Identifiers\n    3. Classes\n    4. Indicator Classes and Functions\n    5. Python-Specific Keywords\n  18. Appendix D: TradeStation and EasyLanguage\n    1. Importing ELD file from Book Website\n    2. Keywords and Functions\n    3. Sample Algorithm Codes\n  19. Appendix E\n  20. About the Companion Website\n  21. Index\n  22. End User License Agreement\n\n## Pages\n\n  1. ii\n  2. iv\n  3. ix\n  4. x\n  5. xi\n  6. xii\n  7. xiii\n  8. xiv\n  9. xv\n  10. xvi\n  11. xvii\n  12. xviii\n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \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  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328. \n  329. \n  330. \n  331. \n  332. \n  333. \n  334. \n  335. \n  336. \n  337. \n  338. \n  339. \n  340. \n  341. \n  342. \n  343. \n  344. \n  345. \n  346. \n  347.\n\n## Guide\n\n  1. Cover\n  2. Table of Contents\n  3. Begin Reading\n\n## List of Illustrations\n\n  1. Chapter 1: Introduction to Trading: Algorithm Development\n    1. Figure 1.1 Examples of strength differences between pivot high points on daily bars.\n    2. Figure 1.2 A depiction of the variable-bar sequence described by the long entry signal.\n    3. Figure 1.3 An FSM that models the workings of a combination lock.\n    4. Figure 1.4 An FSM that models the pivot-point long-entry algorithm described in this chapter.\n    5. Figure 1.5 A very simple FC.\n    6. Figure 1.6 An FC of a trading algorithm.\n  2. Chapter 2: Stochastics and Averages and RSI! Oh, My!\n    1. Figure 2.1 Examples of directional movement.\n    2. Figure 2.2 ADX as a trend detector.\n    3. Figure 2.3 Wilder's directional movement algorithm.\n    4. Figure 2.4 Example of a trade generated by RSI.\n    5. Figure 2.5 Flowchart diagram of RSI trading algorithm.\n    6. Figure 2.6 Example of RSI divergence.\n    7. Figure 2.7 Example of FSM modeling RSI Swing Failure.\n    8. Figure 2.8 Chart showing trade signals from stochastic crossover algorithm.\n    9. Figure 2.9 The flowchart for an algorithm using a stochastic oscillator crossover as an entry signal.\n    10. Figure 2.10 Trades generated by the CCI algorithm when it crosses above 100 and below \u2212100.\n    11. Figure 2.11 The CCI system detecting a downturn in the Eurocurrency.\n    12. Figure 2.12 Histogram pivot points can initiate trade signals.\n    13. Figure 2.13 Three different 20-day moving averages.\n    14. Figure 2.14 A 60-day, two-SD Bollinger Band on crude oil.\n    15. Figure 2.15 Keltner Channels and Bollinger Bands applied to the same data.\n    16. Figure 2.16 Bollinger Band optimization.\n    17. Figure 2.17 Keltner Channel optimization.\n  3. Chapter 3: Complete Trading Algorithms\n    1. Figure 3.1 Turtle trading algorithm flowchart.\n    2. Figure 3.2 A three-dimensional contour chart of Bollinger performance across a subset of the parameters that were originally optimized.\n    3. Figure 3.3 A snapshot of a portion of the current portfolio correlation matrix.\n    4. Figure 3.4 Walk-forward testing begins with a backtest used to derive the desired parameter. Then, it applies that parameter going forward.\n    5. Figure 3.5 AmiBroker's walk-forward optimizer in action.\n    6. Figure 3.6 Parameters selected to carry forward for the following year.\n    7. Figure 3.7 Performance of the WFO of the Bollinger Band system.\n    8. Figure 3.8 Equity curve that reflects a pattern system.\n    9. Figure 3.9 Mini-Russell day trading system using a narrow range day. It initially puts on two contracts, pulling the first one off after a certain profit is reached and leaving the second to run its course.\n  4. Chapter 4: Introduction to AmiBroker's AFL\n    1. Figure 4.1 Launching AmiBroker's AFL editor.\n    2. Figure 4.2 The AmiBroker Editor window.\n    3. Figure 4.3 The AFL Editor's auto-completion tool.\n    4. Figure 4.4 AFL Editor function helper.\n    5. Figure 4.5 The AFL Check icon.\n    6. Figure 4.6 An error warning.\n    7. Figure 4.7 The \"Send to Analysis\" window icon.\n    8. Figure 4.8 Apply the algorithm to the current data.\n    9. Figure 4.9 Set the parameters for the backtest.\n    10. Figure 4.10 Set Futures mode.\n    11. Figure 4.11 The trade-by-trade report.\n    12. Figure 4.12 The portfolio results.\n    13. Figure 4.13 A 3D optimization chart.\n    14. Figure 4.14 The AFL Code Wizard button.\n    15. Figure 4.15 A new AFL Code Wizard window.\n    16. Figure 4.16 Add item in AFL Code Wizard.\n    17. Figure 4.17 The AFL Code Wizard \"Edit Rule\" dropdown list.\n    18. Figure 4.18 Changing the parameters in the \"Edit Rule\" pane.\n    19. Figure 4.19 Click the exclamation point to send your request to AmiBroker.\n    20. Figure 4.20 Automatic Analysis dialog box.\n    21. Figure 4.21 Sample loop programming code.\n  5. Chapter 5: Using Microsoft Excel to Backtest Your Algorithm\n    1. Figure 5.1 Once you define the Bollinger Band routine, you can easily incorporate it into trading.\n    2. Figure 5.2 The structure of the backtesting software.\n    3. Figure 5.3 A typical Excel data worksheet.\n    4. Figure 5.4 The DataMaster worksheet can help you store the values of your commodities and futures contracts.\n    5. Figure 5.5 The Results worksheet lists multiple trades and their associated performance metrics.\n    6. Figure 5.6 The EquityStream worksheet has a button that launches a macro to create a chart that plots the cumulative daily equity and drawdown.\n    7. Figure 5.7 Customize your Excel ribbon. Make sure you check the box next to the Developer tab option.\n    8. Figure 5.8 The Developer tab in Microsoft Excel.\n    9. Figure 5.9 The Visual Basic Editor (VBE) in Microsoft Excel.\n    10. Figure 5.10 Comment text in VBA code is written in gray. It is also indicated by the single quote at the beginning of each line.\n    11. Figure 5.11 This is where the VBA comments end and the actual source code begins.\n    12. Figure 5.12 This error message appears if your algorithm does not have enough ramp-up data.\n    13. Figure 5.13 This tiny snippet of code contains five arguments, which are all necessary for the Trade subroutine.\n  6. Chapter 6: Using Python to Backtest Your Algorithm\n    1. Figure 6.1 NotePad++'s collapsible text feature.\n    2. Figure 6.2 A Python syntax error message.\n    3. Figure 6.3 In this case, the syntax error was the result of confusion between the assignment operator (=) and the comparison operator (==). Python highlights the problematic code.\n    4. Figure 6.4 A Python run-time error resulting from using parentheses instead of square brackets.\n  7. Chapter 7: An Introduction to EasyLanguage\n    1. Figure 7.1 The TradeStation home screen.\n    2. Figure 7.2 TradeStation's trading apps.\n    3. Figure 7.3 The initial screen of the TradeStation Development Environment.\n    4. Figure 7.4 The EasyLanguage editor in the TradeStation Development Environment.\n    5. Figure 7.5 The algorithm with a single, complete portion of the strategy marked.\n    6. Figure 7.6 The EasyLanguage editor monitors what you are typing and provides likely functions.\n    7. Figure 7.7 How to apply a strategy to your chart.\n    8. Figure 7.8 A reversion analysis strategy shown on a chart.\n    9. Figure 7.9 How to change the inputs and adjust your strategy.\n    10. Figure 7.10 The four input variables to adjust the mean reversion strategy.\n    11. Figure 7.11 A message confirming that your code was verified and will be recalculated based on the new inputs.\n    12. Figure 7.12 TradeStation lists any syntax errors it finds during verification in the bottom pane of the screen. The line number of the error is provided so you can easily find it in your code.\n    13. Figure 7.13 The EasyLanguage editor can display line numbers to make it easy to navigate your code.\n    14. Figure 7.14 To create a new project using your strategy, select Project from the File menu.\n    15. Figure 7.15 How to add an existing strategy to a new project.\n    16. Figure 7.16 All dependencies and functions are nested inside their strategy algorithm.\n  8. Chapter 8: Genetic Optimization, Walk Forward, and Monte Carlo Start Trade Analysis\n    1. Figure 8.1 Iteration growth rate of five parameters.\n    2. Figure 8.2 An illustration of our chromosomes with different genetic values.\n    3. Figure 8.3 Chromosome dartboard.\n    4. Figure 8.4 At split 1 the genes that will be passed onto the offspring by the father and the mother.\n    5. Figure 8.5 Robust parameter sets are found on a level plateau.\n    6. Figure 8.6 A graphic representation of how the walk-forward analysis (WFA) rolls forward.\n    7. Figure 8.7 How four years of hindsight are used to predict unseen data for the following year.\n    8. Figure 8.8 The Cluster WFO results matrix.\n    9. Figure 8.9 AmiBroker Monte Carlo settings dialog.\n    10. Figure 8.10 Monte Carlo Straw Broom chart of multiple randomized equity curves.\n  9. Chapter 9: An Introduction to Portfolio Maestro, Money Management, and Portfolio Analysis\n    1. Figure 9.1 TradeStation's Portfolio Maestro (PM) launch window.\n    2. Figure 9.2 Add a strategy to your strategy group.\n    3. Figure 9.3 The Turtle 55 strategy is now in your strategies window.\n    4. Figure 9.4 Add a symbol by clicking the Symbol Lists tab (indicated by Arrow 1), then clicking on Add Symbol List (shown with Arrow 2).\n    5. Figure 9.5 After selecting the UGTATS list, it will appear in your symbol list window.\n    6. Figure 9.6 The UGTATS markets and their corresponding symbols.\n    7. Figure 9.7 A list of available strategy groups, including MyFirstStrategyGroup, which we are using for this exercise.\n    8. Figure 9.8 After selecting MyFirstStrategyGroup, you will see it in MyFirstPortfolio. In the expanded view, you can see the strategies and symbols that are part of your group.\n    9. Figure 9.9 How to adjust the settings for a strategy group. Arrow 1 indicates the Manage Strategy Group button. Arrow 2 highlights the Strategy Group Settings button. Arrow 3 points to the controls that change the commission and slippage inputs.\n    10. Figure 9.10 As you prepare to backtest your portfolio, ensure the settings are correct. Begin by clicking the Backtest Portfolio button (marked 1 in the figure). Then, check the initial capital, test period, and backtest type in the appropriate fields (marked with 2). When everything looks good, click the Perform Backtest button (number 3 in the figure).\n    11. Figure 9.11 A backtest performance report.\n    12. Figure 9.12 The Trade Analysis window displays trading statistics. To get to this screen, click the Trade Analysis tab (marked 1). The View\/Hide Symbols button (marked 2) reveals the performance metric for every market in your portfolio (item 3).\n    13. Figure 9.13 An equity curve chart that shows the maximum drawdown.\n    14. Figure 9.14 The Total P\/L option displays individual profit and loss data for every market in the portfolio.\n    15. Figure 9.15 The Returns and Equity tab displays the portfolio's maximum drawdown in this case.\n    16. Figure 9.16 The arrows indicate the parameters that you can change from the Money Management window. For now, keep the ATR lookback set to 10 and the round quantity set to 1.\n    17. Figure 9.17 The equity curve of the backtest reveals a profit of nearly $1 million.\n    18. Figure 9.18 Profit and drawdown are directly proportional to position size. As your position size changes, so do your commissions and profits and losses.\n    19. Figure 9.19 Under Portfolio Settings (marked as 1), go to the Portfolio Stops tab (marked as 2) and set the stop strategy of your choosing. Under the dropdown menu, there is a description of how the selected strategy will impact your portfolio (marked as 3).\n    20. Figure 9.20 To set the portfolio stop loss, you can adjust the loss target percentage (marked as 1), loss target (marked as 2), and period (marked as 3).\n    21. Figure 9.21 To set the portfolio profit target, you can adjust the period, profit target percentage, and profit target. These options align with those in the stop loss tab.\n    22. Figure 9.22 The portfolio profit target and stop loss now appear as strategies in your portfolio.\n    23. Figure 9.23 At the bottom of your screen, you should see the portfolio stop option.\n    24. Figure 9.24 The equity curve of the portfolio once the stops have been added.\n\n## List of Tables\n\n  1. Chapter 2: Stochastics and Averages and RSI! Oh, My!\n    1. Table 2.1 Performance of Directional Movement Algorithm\n    2. Table 2.2 Performance of RSI Algorithm\n    3. Table 2.3 Performance of RSI Failure Swing Algorithm\n    4. Table 2.4 Performance of Stochastic Algorithm\n    5. Table 2.5 Performance of CCI algorithm\n    6. Table 2.6 Performance of CCI Algorithm as Coincident Indicator\n    7. Table 2.7 Examples of Trades Generated by Stochastics\n    8. Table 2.8 Performance of Bollinger Band Algorithm Version 1\n    9. Table 2.9 Performance of Bollinger Band Algorithm Version 2\n    10. Table 2.10 Bollinger Benchmark versus Keltner Challengers\n    11. Table 2.11 Best Parameter Sets of Bollinger Bands Algorithm\n    12. Table 2.12 Best Parameter Sets of Keltner Channel Algorithm\n  2. Chapter 3: Complete Trading Algorithms\n    1. Table 3.1 Turtle System 1 without LTLF for the Sample Portfolio\n    2. Table 3.2 Turtle System 2 for the Sample Portfolio\n    3. Table 3.3 Turtle System 1 with LTLF for the Sample Portfolio\n    4. Table 3.4 Tandem Performance of Turtle System 1 with LTLF and Turtle System 2 for the Sample Portfolio\n    5. Table 3.5 Single Moving Average (SMA) Crossover Algorithm\n    6. Table 3.6 Double Moving Average (DMA) Crossover Algorithm\n    7. Table 3.7 Triple Moving Average (TMA) Crossover Algorithm\n    8. Table 3.8 Donchian Channel Breakout for Different Donchian Lengths\n    9. Table 3.9 Bollinger Band Breakout\n    10. Table 3.10 Comparison of Parameters for Algorithms Based on Different Strategies\n    11. Table 3.11 Top 10 Parameter Sets for Each Algorithm\n    12. Table 3.12 The Best Parameter Sets for Each Optimization\n    13. Table 3.13 The Best Results for the Donchian Algorithm for In-Sample Time Period\n    14. Table 3.14 Results for the Donchian Algorithm when Walked Forward across an Out-of-Sample (OOS) Time Period\n    15. Table 3.15 The Best Parameters for the Portfolio over the Out-Of-Sample (OOS) Time Period\n    16. Table 3.16 The Optimization of the Bollinger Algorithm for the In-Sample (IS) Time Period\n    17. Table 3.17 Results for the Donchian Algorithm when Walked Forward across an Out-of-Sample (OOS) Time Period\n  3. Chapter 7: An Introduction to EasyLanguage\n    1. Table 7.1 EasyLanguage Keywords and Abbreviations\n  4. Chapter 8: Genetic Optimization, Walk Forward, and Monte Carlo Start Trade Analysis\n    1. Table 8.1 Results from the Exhaustive Search Algorithm's Five Parameters\n    2. Table 8.2 Results from the Genetic Search Algorithm's Five Parameters\n    3. Table 8.3 In-Sample Testing TA2\n    4. Table 8.4 Out-of-Sample Testing\n    5. Table 8.5 WFA Report on TA2\n    6. Table 8.6 The Out-of-Sample (OOS) Report\n    7. Table 8.7 WFO Report Card TA3\n    8. Table 8.8 Distribution Statistics from Monte Carlo Simulations\n  5. Chapter 9: An Introduction to Portfolio Maestro, Money Management, and Portfolio Analysis\n    1. Table 9.1 Portfolio Trades without Stops\n    2. Table 9.2 Portfolio Trades with Stops in Place\n\nThe Wiley Trading series features books by traders who have survived the market's ever changing temperament and have prospered\u2014some by reinventing systems, others by getting back to basics. Whether a novice trader, professional, or somewhere in between, these books will provide the advice and strategies needed to prosper today and well into the future. For more on this series, visit our website at www.WileyTrading.com.\n\nFounded in 1807, John Wiley & Sons is the oldest independent publishing company in the United States. With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers' professional and personal knowledge and understanding.\n\n# THE ULTIMATE  \nALGORITHMIC  \nTRADING SYSTEM  \nTOOLBOX  \n\\+ Website\n\n## Using Today's Technology to Help  \nYou Become a Better Trader\n\n**George Pruitt**\n\nCopyright \u00a9 2016 by George Pruitt. All rights reserved.\n\nPublished by John Wiley & Sons, Inc., Hoboken, New Jersey.\n\nPublished simultaneously in Canada.\n\nNo part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the Web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at <http:\/\/www.wiley.com\/go\/permissions>.\n\nLimit of Liability\/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.\n\nFor general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.\n\nWiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at <http:\/\/booksupport.wiley.com>. For more information about Wiley products, visit www.wiley.com.\n\n**_Library of Congress Cataloging-in-Publication Data:_**\n\nNames: Pruitt, George, 1967- author.\n\nTitle: The ultimate algorithmic trading system toolbox + website : using today's technology to help you become a better trader\/George Pruitt.\n\nDescription: Hoboken : Wiley, 2016. | Series: Wiley trading | Includes index.\n\nIdentifiers: LCCN 2016010889 (print) | LCCN 2016011196 (ebook) | ISBN 9781119096573 (hardback) | ISBN 9781119262961 (pdf) | ISBN 9781119262978 (epub)\n\nSubjects: LCSH: Electronic trading of securities. | Investment analysis. | Futures. | BISAC: BUSINESS & ECONOMICS\/Finance.\n\nClassification: LCC HG4515.95 .P788 2016 (print) | LCC HG4515.95 (ebook) | DDC 332.640285\/4678\u2014 dc23\n\nLC record available at <http:\/\/lccn.loc.gov\/2016010889>\n\nCover Design: Wiley\n\nCover Images: \u00a9 agsandrew\/Shutterstock\n\n# About The Author\n\nIt was March of 1989 as I drove my beat-up Dodge up Hillside Rd. in Hendersonville, NC. In an attempt to pay for my last semesters of college I was answering a classified ad that was looking to hire a computer programmer. As I drove up the thin drive I passed several houses and then through a gate attached to two large stone pillars. I stopped the car and looked down at the ad again to make sure I was at the right place. I proceeded down the country lane and the view opened up into a large meadow. At the end of the lane was a circular drive and large farm house. As I circled and went back down the road I thought to myself I must have the wrong address or directions. So I followed the small road back down the main highway and then to a small convenient store. Once there I asked myself again what type of business was this Futures Truth and if I should call and get directions or just simply forget about it. Curiosity and the need for money were too much so I used the store's pay phone and called the number once again.\n\n\"Hello\u2014Futures Truth, may I help you?\" a lady's voice answered.\n\n\"Yes, this is George Pruitt and I made an appointment for an interview but I can't seem to find your office.\"\n\n\"Do you drive a red Dodge?\" she asked.\n\n\"Yes I do. How did you know?\"\n\n\"We saw you drive right by the office. When you come through the two stone pillars turn immediately to the left. Don't go all the way down the drive\u2014that's the owner's house.\"\n\nSo I follow the directions and find myself in front of a small house. I knock on the door and John Fisher opens and invites me in. We go through the normal Q and A for a job interview and he finally asks if I knew FORTRAN. My first college programming class was FORTRAN so I confidently answered, \"Sure!\"\n\nHe then asked me if I knew anything about the Futures market. I vaguely remembered the term from one of my economics classes and of course from the Eddie Murphy movie and answer him with the question, \"You mean like Trading Places with Eddie Murphy?\"\n\nJohn Fisher said \"Sort of like that\u2014yes.\"\n\nHe went on to explain how Futures Truth tried to determine market direction in the most widely traded futures contracts by using trading systems. The trading systems were programmed in FORTRAN and they needed help with the programming. In addition to trading they also published a newsletter in which they tracked publicly offered trading systems.\n\nI asked, \"Do people really buy these programs?\"\n\nJohn Fisher said yes and by that time an older gentlemen walked into the office and stated that he had spent thousands of dollars on these programs and was ultimately ripped off. John Hill stated this was the main reason he started Futures Truth. He wanted to bring truth to the trading system industry. Both Johns told me that most traders couldn't afford to validate the trading systems because of the cost of the computer equipment, data, and software. John Fisher pointed to the computer he was working on and asked, \"How much do you think this Macintosh II cost?\"\n\nI answered him, \"I am not that familiar with Macs but I know they aren't cheap.\"\n\nMy mouth fell open when he said \"$4,000 and we have three of them.\" Remember this was way back in 1989 when computers were not cheap.\n\nI was thinking to myself that they got ripped off because they could have got a much cheaper and better computer with the IBM PS\/2. And what was up with using FORTRAN? Did they not know \"C\" was the new programming language of the 1990s? John Fisher chose the Apple Macintosh because of its easy-to-use graphical user interface (GUI) and FORTRAN because many traders and hobbyist programmers had knowledge of this language.\n\nJohn Fisher also said that he and John Hill had developed what they considered the best testing platform, \"Excalibur.\" This platform could load decades of daily and intraday data and test any trading idea that could be defined in an algorithmic form. He also said the only thing that was missing was a charting application and that was where they also needed help.\n\nI explained that I would be wrapping up my degree after summer and both Johns agreed that I could work part time in the evening until I graduated and then we could go from there.\n\nWell that was 27 years ago and I did work part time until I graduated with a degree in computer science from the University of North Carolina at Asheville. The \"Excalibur Chart\" project turned into my senior project, which blew my professors away. Over the years I have worked with many trading firms in the development of trading algorithms and testing platforms. I have seen it all and have had the great pleasure to be educated by some of the greatest minds in the industry, including John Fisher, John Hill Sr. and John Hill Jr. Even with this experience and education the ultimate trading system still eludes me. As John Hill has stated many times, \"A speculator who dies rich, dies before his time!\" This may be true, but I have seen traders make millions, lose millions, and make millions again. The one thing they always do when they fail is get right back up, dust themselves off, and start searching for the next great trading algorithm.\n\n# Introduction to The Ultimate Algorithmic Trading Systems ToolBox\n\nIf you want to learn more about high-frequency trading utilizing special order placement\/replacement algorithms such as Predatory trading, Pinging, Point of Presence, or Liquidity Rebates, then this book is not for you. However, if you want to learn about trading algorithms that help make a trading decision, trade size, money management, and the software used to create these algorithms, then you're in the right place.\n\nThis book is designed to teach trading algorithm development, testing, and optimization. Another goal is to expose the reader to multiple testing platforms and programming languages. Don't worry if you don't have a background in programming; this book will provide enough instruction to get you started in developing your own trading systems. Source code and instructions will be provided for TradeStation's EasyLanguage, AmiBroker's AFL, and my own Python and Excel testing engines. I chose these platforms because they give a nice overview of different scripting languages and trading platforms. Users of different testing\/trading platforms may criticize my decision to use just these platforms, but the EasyLanguage source code that will be provided can be easily ported into Multi-Charts, and AmiBroker's unique and powerful platform provides a complete trading solution. My Python and Excel software, including all source code, are included on the associated website as well as the EasyLanguage and AFL source code for the other platforms. I didn't include the use of Python's scientific libraries, NumPy or SciPy, because I wanted to keep things as simple as possible. Also I used the bare-bones IDLE (Python's own simple Integrated Development Environment) to cut down on the learning curve\u2014I wanted to get to the bare essentials of Python without muddying the water with a sophisticated IDE. Many successful Quants utilize **R** (a GNU project for statistical computing), but again to keep things simple I stuck with the easy-to-learn Python. The majority, if not all algorithms were tested utilizing commodity and futures data only. All the testing platforms in the book can be used to test stocks and ETFs, and all the included trading algorithms can be applied to these assets as well. Stock and ETF data is very simple to acquire. Getting commodity and futures data in an easily usable format is a little more difficult. Deep histories for commodity and futures can be acquired for as little as $100 from Pinnacle Data. I have used CSI data since the late 1980s and it is the data I used for a good portion of the testing carried out in the book. I would definitely take a look at Pinnacle and CSI data, especially if you wanted your database updated daily. If you are not familiar with Quandl, then you might want to take the time to do so. Quandl is a search engine for numerical data. I was pleasantly surprised to find a free continuous futures database (Wiki Continuous Futures) on Quandl. Keep in mind this data is free and is no way as good as premium data such as CSI and Pinnalce\u2014it is missing multiple days and data points and the continuous data is simply created by concatenating data from individual contracts. The gaps between contracts are included, which cannot be utilized on any testing platform. In real life, a futures position is \"rolled-over\" from one contract to another by liquidating the front-month position and initiating the same position in the next contract. This \"rollover\" trade eliminates the gap. I have written a Python application that takes the Wiki Futures data and creates a back-adjusted continuous contract that can be imported into the Python and Excel System Back Tester software. Since I needed the data to do testing, I have also included a 10-plus-year ASCII back-adjusted futures database for 30-plus markets on the companion website. Directions on how to use the software and download futures data from Quandl are included along with the software.\n\nThe one thing I really wanted to include in this book was the \"Holy Grail\" of algorithmic trading systems. I have analyzed many algorithms that claimed to be the Grail, but after rigorous testing they failed to break even. So go ahead and check this off your list. Even though the \"Holy Grail\" will remain hidden you will find the following:\n\n  * Twenty-seven years of experience working with non-programmers in the development of their own trading algorithms\n  * The tools or building blocks that are used most often in the development cycle\n  * The core trading models that make up the majority of publicly offered trading systems\n  * The most important and simplest programming techniques to transform a non-quant into a not-so-non-quant\n  * Simple examples and explanations of complex trading ideas such as Walk Forward and Genetic Optimization and Monte Carlo simulation\n  * A complete toolbox to help algorithm development from idea to a finished tradable solution\n\nThe majority of successful trading algorithms utilize quantitative analysis (QA). QA is simply the application of mathematical formulae to a financial time series. This book will solely focus on this type of analysis in the design of trading algorithms. Fundamental analysis, which is used in many trading plans, will be used, too, but it will be reduced and simplified into a pure and easily digestible data format. Fundamental data is huge and diverse and in many cases market movement reacts to it in an unpredictable manner. A good example that I have dealt with for many years is the monthly unemployment report. At the time of the writing of this book unemployment has been on a downward trend, which is usually a bullish indicator for the stock market. However, with interest rates at the time being close to 0% the market could react opposite due to the fear of the Federal Reserve doing away with quantitative easing and raising rates. This type of fundamental analysis requires many different inputs and trying to reduce it down to something testable is nearly impossible.\n\nQuantitative analysis focuses on just the data included in a chart. Price action and price translations are easily definable and therefore can be tested. The ability to test and evaluate a trading algorithm is a tremendous tool as it shows how a model can accurately map a market's behavior. If you can interpret a market's behavior, you can take advantage of its inefficiencies. If an algorithm has been capable of exploiting a market's inefficiencies on a historic basis, then there is a possibility it will do so in the future. This hope of future performance is the only leg an algorithmic trader has to stand upon. We all know historic performance is not necessarily an indicator of future results, but what else do we have? An algorithmic trader who quickly defines and tests his system and immediately takes a leap of faith because the historic performance looks great is doomed. Doesn't this contradict what I just said about historical performance being a system trader's only gauge of quality? A good trading algorithm not only demonstrates profitability but also robustness. Robustness is an expression of how well a trading system performs on diverse markets and diverse market conditions. An algorithm can be improved to a point where the trader can feel somewhat confident putting on those first few trades as well as continuing to put trades on after a losing streak. Improving an algorithm is not simply tweaking it until the historic results look utterly fantastic (aka curve fitting); it is taking the time to learn and work with tools that are designed to make a trading algorithm fail. That's the ultimate objective\u2014making your trading algorithm fail before any money is put on the line. Remember the absence of failure is success and if your algorithm survives the brutal gauntlet of in depth analysis, then you know you might, just _might_ have a winner.\n\nThis book starts out simple in Chapter 1 with the definition and examples of algorithms. The chapter is a little longwinded but I know that the inability to put a trading idea onto paper and then into pseudocode and finally actual computer code is the biggest stumbling block for traders who want to test their own trading ideas. All trading algorithms that are reducible to a set of instructions can be properly programmed using one of two different modeling methods or paradigms. These two paradigms, Finite State Machine and Flow Chart, are fully discussed and utilized to translate written descriptions first into diagrams and then into actual pseudocode. The diagrammatic approach as well as the simple pseudocode language used to formulate trading algorithms is introduced in this chapter. It doesn't matter how sophisticated your testing software is if you can't define a testable algorithm and this chapter shows you how to do so.\n\nChapter 2 may be a refresher for those who are familiar with the basic building blocks of trading algorithms, indicators; however, the chapter not only explains the logic behind the indicators but shows how they can be incorporated into complete entry and exit techniques. Diagrams and pseudocode are carried on through this chapter to aid in the understanding of each indicator, its purpose, and its place in a trading algorithm. In addition, the first look at indicator-based trading algorithm performance is presented as well.\n\nChapter 3 introduces complete trading algorithms and their associated historical performance. Most, if not all, testing was performed on historical commodity\/futures data. This data gave rise to the concept of systematic trading more than 50 years ago. Now this doesn't mean the ideas aren't transferable to the stock market. In most cases they are. However, I stuck with commodity data because that is where my expertise lies. The complete pseudocode and actual computer code of these algorithms are revealed as well. The key metrics for determining algorithm robustness are explained and utilized in the evaluation of the algorithms' results.\n\nChapter 4 starts the section that highlights different testing\/trading software platforms that can either be purchased or leased. AmiBroker is introduced in this chapter and the most important components of a trading platform are highlighted: integrated development environment and its associated scripting\/programming language, individual market and portfolio testing, and algorithm performance metrics. These components are then highlighted again in Chapter 5 with VBA for Excel, Chapter 6 with Python, and finally Chapter 7 with TradeStation.\n\nChapter 8 delves into the concepts of Genetic and Walk Forward Optimization, Walk Forward Analysis, and Monte Carlo simulation. A genetic optimizer is built using VBA and used to help explain the ideas of synthesizing computers with biology. The core concepts of Genetic Algorithms, fitness, selection, reproduction, and mutation are fully explained and illustrated utilizing Excel. Artificial intelligence is here to stay in the study of trading algorithms and this chapter tries to pull back the veil of mystery and show how these tools should be used, and in some cases, must be used to develop that elusive robustness. Along these lines, Machine Learning has become a very highly discussed and somewhat controversial topic in today's trading. Also \"Big Data\" analysis has found its way to the front as well. These topics are highly advanced and I felt beyond the scope of this book. I can state I have worked with the algorithms that were derived with machine-only input and they have stood the test of time.\n\nA trading algorithm must work over a diverse portfolio of markets before it can be sufficiently considered useful and robust. Chapter 9 utilizes the portfolio-level testing capabilities of TradeStation and AmiBroker to demonstrate different money and portfolio management techniques. The Fixed Fractional approach, by far the most popular, will be highlighted.\n\nThe complete source code for the Python System Back Tester is included on the website. Python is the new language of many a quant and the source shows how the language can be used to develop a simple, yet powerful, back tester. Important language concepts and syntax are used to open ASCII files, and import the data into a LIST data structure, create classes and modules, and loop through the entire database while applying a trading algorithm. All the parts of building a testing platform are revealed in the source code, including Monte Carlo and Start Trade Drawdown simulation.\n\nMost traders have Microsoft Excel on their computers and the complete source for a more simplified version of the Python back tester using VBA is included on the website as well.\n\nThis book is a toolbox and a guide and touches upon many different facets of algorithmic trading. As with any toolbox it will take time and effort to apply the tools found within to replicate the trader's ideas in a form that not only can be tested and evaluated but fully implemented.\n\n# Chapter 1  \nIntroduction to Trading  \n _Algorithm Development_\n\n## What Is an Algorithm?\n\n> _An Algorithm is an effective procedure, a way of getting something done in a finite number of discrete steps._\n\nDavid Berlinski\n\nBerlinski's definition is exactly right on the money. The word _algorithm_ sounds mysterious as well as intellectual but it's really a fancy name for a recipe. It explains precisely the stuff and steps necessary to accomplish a task. Even though you can perceive an algorithm to be a simple recipe, it must, like all things dealing with computers, follow specific criteria:\n\n  1. _Input_ : There are zero or more quantities that are externally supplied.\n  2. _Output_ : At least one quantity is produced.\n  3. _Definiteness_ : Each instruction must be clear and unambiguous.\n  4. _Finiteness_ : If we trace out the instructions of an algorithm, then for all cases the algorithm will terminate after a finite number of steps.\n  5. _Effectiveness_ : Every instruction must be sufficiently basic that it can in principle be carried out by a person using only pencil and paper. It is not enough that each operation be definite as in (3), but it must also be feasible. [ _Fundamentals of Data Structures_ : Ellis Horowitz and Sartaj Sahni 1976; Computer Science Press]\n\nThese criteria are very precise because they can be universally applied to any type of problem. Don't be turned off thinking this is going to be another computer science text, because even though the criteria of an algorithm seem to be very formal, an algorithm really is straightforward and quite eloquent. It is basically a guide that one must follow to convert a problem into something a computer can solve. Anybody can design an algorithm following these criteria with pencil and paper. The only prerequisite is that you must think like a Vulcan from _Star Trek_. In other words, think in logical terms by breaking ideas down into rudimentary building blocks. This is the first step\u2014translation of idea into an algorithm. It takes practice to do this, and this is in part why programming can be difficult.\n\nAnother thing that makes programming difficult is understanding a computer language's syntax. Most people who have been exposed to a programming or scripting language at one time or another in their lives have probably exclaimed something like, \"I forgot one stupid semicolon and the entire program crashed! Isn't the computer smart enough to know that? _Arrgh!_ I will never be a computer programmer!\" The question that is proffered in this temper tantrum is the question of the computer's intelligence. Computers are not smart\u2014they only do what we tell them. It doesn't matter if you spend $500 or $5,000 on the hardware. They do things very quickly and accurately, but their intelligence is a reflection of their programmer's ability to translate idea into algorithmic form and then into proper syntax.\n\nAlgorithmic (algo) traders don't necessarily need to be programmers, but they must understand what a computer needs to know to carry out a trading signal, position sizing, and money management. If you can create an algorithm, then you are more than 50 percent there. I say more than 50 percent because most traders will utilize trading software and its associated programming or scripting language. Learning the syntax of a scripting language or a small subset of a programming dialect is much easier than learning an entire programming language like C# or C++. An algo trader only needs to be concerned with the necessary tools to carry out a trading system. The developers of EasyLanguage, AmiBroker, or TradersStudio's main objective was to provide only the necessary tools to put a trading idea into action. They accomplished this by creating a vast library of trading functions, easy access to these functions, and a simplified programming syntax. Now if you want to develop your own testing platform and want to use a full-blown programming language to do so, then you will need to know the language inside-out. If you are interested in doing this, chapters 5 and  will give you a head start. In these chapters, I show how I developed testing platforms in Python and Microsoft VBA from scratch.\n\nHowever, at this introductory stage, let's take a look at a very simple trading algorithm and the possible exchange between a computer and trader. Pretend a trader wants to develop and test a simple moving-average crossover system and wants to use software specifically designed for system testing. Let's call this first trader AlgoTrader1, and since he has used this particular testing platform he knows it understands a trading vernacular and provides access to the common indicator functions and data. Box 1.1 shows a possible exchange between trader and computer.\n\n* * *\n\n### Box 1.1 Algo Testing Software\n\n    AlgoTrader1 \u2013 AlgoTester ON\n    Computer \u2013 AlgoTester ready\n    AlgoTrader1 \u2013 **load** crude oil futures data\n    Computer \u2013 data loaded\n    AlgoTrader1 \u2013 **buy** whenever **close** is above **moving average**\n    Computer \u2013 \" **moving average** \" function requires three inputs\n    AlgoTrader1 - help with **moving average** function\n    Computer - function calculates simple, weighted, exponential average\n    Computer - function syntax **moving average (type, price, length)**\n    AlgoTrader1 - buy whenever **close** is above **moving average (simple,close,21)**\n    Computer \u2013 command completed\n    AlgoTrader1 \u2013 **short** whenever **close** is below **moving average (simple,close,21)**\n    Computer \u2013 command completed\n    AlgoTrader1 \u2013 **save** algorithm as **MovAvgCross**\n    Computer \u2013 command completed\n    AlgoTrader1 \u2013 **run MovAvgCross** algorithm\n    Computer \u2013 run completed and results are:\n      **$12,040 profit, $8,500 draw down, $1,200 avg. win**\n    AlgoTrader1 \u2013 **load** euro currency data\n    Computer \u2013 command completed\n    AlgoTrader2 \u2013 **run MovAvgCross** algorithm\n    Computer \u2013 run completed and results are:\n      **-$32,090 profit, $40,000 draw down, $400 avg. win**\n    AlgoTrader1 \u2013 **edit** **MovAvgCross** algorithm\n    Computer \u2013 command completed\n    AlgoTrader2 \u2013 **edit moving average** function\n    Computer - command completed\n    AlgoTrader2 \u2013 **change** **length input** to 30\n    Computer \u2013 command completed\n    AlgoTrader2 \u2013 **run MovAvgCross** algorithm\n    Computer \u2013 run completed and blah blah blah\n\n* * *\n\nAs you can see, the computer had to be somewhat spoon-fed the instructions. The software recognized many keywords such as: **load** , **buy** , **short** , **run** , **edit** , **change** , and **save**. It also recognized the moving average function and was able to provide information on how to properly use it. The trading algorithm is now stored in the computer's library and will be accessible in the future.\n\nThis simple exchange between computer and AlgoTrader1 doesn't reveal all the computations or processes going on behind the scene. Loading and understanding the data, applying the algorithm properly, keeping track of the trades, and, finally, calculating all of the performance metrics did not involve any interaction with the trader. All this programming was done ahead of time and was hidden from the trader and this allows an algo trader to be creative without being bogged down in all the minutiae of a testing platform.\n\nEven though the computer can do all these things seamlessly it still needed to be told exactly what to do. This scenario is similar to a parent instructing a child on how to do his first long-division problem. A child attempting long division probably knows how to add, subtract, multiply, and divide. However, even with these \"built-in\" tools, a child needs a list of exact instructions to complete a problem. An extended vocabulary or a large library of built-in functions saves a lot of time, but it doesn't necessarily make the computer any smarter. This is an example of knowledge versus intelligence\u2014all the knowledge in the world will not necessarily help solve a complex problem. To make a long story short, think like a computer or child when developing and describing a trading algorithm. Box 1.2 shows an algorithmic representation of the long-division process to illustrate how even a simple process can seem complicated when it is broken down into steps.\n\n* * *\n\n### Box 1.2 Procedure for Long Division\n\n  1. Suppose you are dividing two large numbers in the problem _n_ \u00f7 _m_. In this example, the dividend is _n_ and the divisor is _m_.\n  2. If the divisor is not a whole number, simplify the problem by moving the decimal of the divisor until it is to the right of the last digit. Then, move the decimal of the dividend the same number of places. If you run out of digits in the dividend, add zeroes as placeholders.\n  3. When doing long division, the numbers above and below the tableau should be vertically aligned.\n  4. Now you are ready to divide. Look at the first digit of the dividend. If the divisor can go into that number at least once, write the total number of times it fits completely above the tableau. If the divisor is too big, move to the next digit of the dividend, so you are looking at a two-digit number. Do this until the divisor will go into the dividend at least once. Write the number of times the divisor can go into the dividend above the tableau. This is the first number of your quotient.\n  5. Multiply the divisor by the first number of the quotient and write the product under the dividend, lining the digits up appropriately. Subtract the product from the dividend. Then, bring the next digit from the quotient down to the right of the difference. Determine how many times the divisor can go into that number, and write it above the tableau as the next number of the quotient.\n  6. Repeat this process until there are no fully divisible numbers left. The number remaining at the bottom of the subtraction under the tableau is the remainder. To finish the problem, bring the remainder, _r_ , to the top of the tableau and create a fraction, _r_ \/ _m_.\n\n* * *\n\nA few years ago a client came to me with the following trading system description and hired me to program it. Before reading the description, see if you can see any problems the programmer (me) or a computer might encounter before the directives can be properly carried out.\n\n> Buy when the market closes above the 200-day moving average and then starts to trend downward and the RSI bottoms out below 20 and starts moving up. The sell short side is just the opposite.\n\nDid you see the problems with this description? Try instructing a computer to follow these directions. It doesn't matter if a computer has a vast library of trading functions; it still would not understand these instructions. The good news was, the trader did define two conditions precisely: close greater than 200-day moving average and relative strength index (RSI) below 20. The rest of the instructions were open to interpretation. What does _trending downward_ or _bottoming out_ mean? Humans can interpret this, but the computer has no idea what you are talking about. I was finally able, after a couple of phone calls, to discern enough information from the client to create some pseudocode. _Pseudocode_ is an informal high-level description of the operating principle of a computer program or algorithm. Think of it as the bridge between a native language description and quasi-syntactically correct code that a computer can understand. Translating an idea into pseudocode is like converting a nebulous idea into something with structure. Here is the pseudocode of the client's initial trading idea:\n\n#### _Algorithm Pseudocode_\n\n* * *\n\n    if close > 200 day moving average and\n    close < close[1] and close [1] < close [2] and\n    close[2] < close[3] and yesterday's 14 day RSI < 20 and\n    yesterday's 14 day RSI < today's 14 day RSI then **BUY**\n\n* * *\n\nIf this looks like Latin (and you don't know Latin), don't worry about it. The 1] in the code just represents the number of bars back. So close [2] represents the close price prior to yesterday. If you are not familiar with RSI, you will be after [Chapter 2. By the time you are through with this book you will be able to translate this into English, Python, EasyLanguage, AmiBroker, or Excel VBA. Here it is in English.\n\n> If today's close is greater than the 200-day moving average of closing prices and today's close is less than yesterday's close and yesterday's close is less than the prior day's close and the prior day's close is less than the day before that and the 14-day RSI of yesterday is less than 20 and the 14-day RSI of yesterday is less than the 14-day RSI of today, then buy at the market.\n\nNotice how the words _downward_ and _bottoming out_ were removed and replaced with exact descriptions:\n\n  1. **downward** : today's close is less than yesterday's close and yesterday's close is less than the prior day's close and the prior day's close is less than the day before. The market has closed down for three straight days.\n  2. **bottoming out** : the 14-day RSI of yesterday is less than 20 and the 14-day RSI of yesterday is less than the 14-day RSI of today. The RSI value ticked below 20 and then ticked up.\n\nAlso notice how the new description of the trading system is much longer than the original. This is a normal occurrence of the evolution of idea into an exact trading algorithm.\n\nAnd now here it is in the Python programming language:\n\n#### _Actual Python Code_\n\n* * *\n\n    if myClose[D0] < sAverage(myClose,200,D0,1) and myClose[D0] < myClose[D1] and myClose[D2] < myClose[D3] and myClose[D1] < myClose[D2] and rsiVal[D1] < 20 and rsiVal[D1] < rsiVal[D0]:\n        buyEntryName = 'rsiBuy'\n        entryPrice = myOpen\n\n* * *\n\nDon't get bogged down trying to understand exactly what is going on; just notice the similarity between pseudo and actual code. Now this is something the computer can sink its teeth into. Unfortunately, reducing a trading idea down to pseudocode is as difficult as programming it into a testing platform. The transformation from a trader to an algo trader is very difficult and in some cases cannot be accomplished. I have worked with many clients who simply could not reduce what they saw on a chart into concise step-by-step instructions. In other cases the reduction of a trading idea removes enough of the trader's nuances that it turned something that seemed plausible into something that wasn't.\n\nIt goes without saying that if you don't have an algorithm, then all the software in the world will not make you a systematic trader. Either you have to design your own or you have to purchase somebody else's. Buying another person's technology is not a bad way to go, but unless the algorithm is fully disclosed you will not learn anything. However, you will be systematic trader. I have spent 27 years evaluating trading systems and have come across good and bad and really bad technology. I can say without a doubt that one single type of algorithm does not stand head and shoulders above all the others. I can also say without a doubt there isn't a correlation between the price of a trading system and its future profitability. The description in Box 1.3 is very similar to a system that sold for over $10,000 in the 1990s.\n\n* * *\n\n### Box 1.3 Trading Algorithm Similar to One That Sold for $10K in the 1990s Description\n\n#### Entry Logic:\n\nIf the 13-day moving average of closes > the 65-day moving average of closes and the 13-day moving average is rising and the 65-day moving average is rising then buy the next day's open\n\nIf the 13-day moving average of closes < the 65-day moving average of closes and the 13-day moving average is falling and the 65-day moving average is falling then sell the next day's open\n\n#### Exit Logic:\n\n  1. If in a long position then \n    1. set initial stop at the lowest low of the past 13 days\n  2. If in a short position then \n    1. set initial stop at the highest high of the past 13 days\n  3. Once profit exceeds or matches $700 pull stops to break even\n  4. If in a long position use the greater of: \n    1. Breakeven stop\u2014if applicable\n    2. Initial stop\n    3. Lowest low of a moving window of the past 23 days\n  5. If in a short position use the lesser of: \n    1. Breakeven stop\u2014if applicable\n    2. Initial stop\n    3. Highest high of the moving window of the past 23 days\n\n* * *\n\nThat is the entire system, and it did in fact sell for more than $10K. This boils down to a simple moving-average crossover system trading in the direction of the shorter- and longer-term trend. The description also includes a complete set of trade management rules: protective, breakeven, and trailing stop. This is a complete trading system description, but as thorough as it seems there are a couple of problems. The first is easy to fix because it involves syntax but the second involves logic. There are two words that describe market direction that cannot be interpreted by a computer. Do you see them? The two words in question are: _rising_ and _falling_. Again, descriptive words like these have to be precisely defined. This initial problem is easy to fix\u2014just inform the computer the exact meaning of _rising_ and _falling_. Second, it has a weakness from a logical standpoint. The algorithm uses $700 as the profit level before the stop can be moved to break even. Seven hundred dollars in the 1990s is quite a bit different than it is today. The robustness of this logic could be dramatically improved by using a market-derived parameter. One could use a volatility measure like the average range of the past N-days. If the market exhibits a high level of volatility, then the profit objective is larger and the breakeven stop will take longer to get activated. You may ask, \"Why is this beneficial?\" Market noise is considered the same as volatility, and the more noise, the higher likelihood of wide market swings. If trading in this environment, you want to make sure you adjust your entries and exits so you stay outside the noise bands.\n\nThis algorithm was very successful in the 1980s and through a good portion the 1990s. However, its success has been hit-and-miss since. Is this algorithm worth $10K? If you were sitting back in 1993 and looked at the historical equity curve, you might say yes. With a testing platform, we can walk forward the algorithm, and apply it to the most recent data and see how it would have performed and then answer the question. This test was done and the answer came out to be an emphatic _no!_\n\nHad you bought this system and stuck with it through the steep drawdowns that have occurred since the 1990s, you would have eventually made back your investment (although not many people would have stuck with it). And you would have learned the secret behind the system. Once the secret was revealed and your checking account was down the $10K, you might have been a little disappointed knowing that basic tenets of the system had been around for many years and freely disseminated in books and trade journals of that era. The system may not be all that great, but the structure of the algorithm is very clean and accomplishes the tasks necessary for a complete trading system.\n\nThe most time-consuming aspect when developing a complete trading idea is coming up with the trade entry. This seems somewhat like backward thinking because it's the exit that determines the success of the entry. Nonetheless, the lion's share of focus has always been on the entry. This system provides a very succinct trade entry algorithm. If you want to develop your own trading algorithm, then you must also provide the computer with logic just as precise and easy to interpret. Getting from the nebula of a trading idea to this point is not easy, but it is absolutely necessary. On past projects, I have provided the template shown in Box 1.4 to clients to help them write their own entry rules in a more easily understood form. You can download this form and a similar exit template at this book's companion website:www.wiley.com\/go\/ultimatealgotoolbox.\n\n* * *\n\n### Box 1.4 Simple Template for Entry Rules\n\n#### Long \/ Short Entries\n\nCalculations and\/or Indicators Involved (specify lookback period). Don't use ambiguously descriptive words like _rising_ , _falling_ , _flattening_ , _topping_ or _bottoming out_.\n\n___________________________________________________\n\n___________________________________________________\n\n___________________________________________________\n\nBuy Condition\u2014What must happen for a long signal to be issued? List steps in chronological order. And remember, don't use ambiguously descriptive words like _rising_ , _falling_ , _flattening_ , _topping_ , or _bottoming out_.\n\n___________________________________________________\n\n___________________________________________________\n\n___________________________________________________\n\nSell Condition\u2014What must happen for a short signal to be issued? List steps in chronological order.\n\n___________________________________________________\n\n___________________________________________________\n\n___________________________________________________\n\nHere is one of the templates filled out by a one-time client:\n\n  1. Calculations and\/or Indicators Involved (specify lookback period) \n    1. Bollinger Band with a 50-day lookback\n  2. Buy Condition\u2014What must happen for a long signal to be issued? List steps in chronological order. \n    1. Close of yesterday is above 50-day upper Bollinger Band\n    2. Today's close < yesterday's close\n    3. Buy next morning's open\n  3. Sell Condition\u2014What must happen for a short signal to be issued? List steps in chronological order. \n    1. Close of yesterday is below 50-day lower Bollinger Band\n    2. Today's close > yesterday's close\n    3. Sell next morning's open\n\n* * *\n\nThe simple Bollinger Band system shown in Box 1.4 is seeking to buy after the upper Bollinger Band penetration is followed by a down close. The conditions must occur within one daily bar of each other. In other words, things must happen consecutively: the close of yesterday is > upper Bollinger Band and close of today < yesterday's close. The sell side of things uses just the opposite logic. The template for exiting a trade is very similar to that of entering. Box 1.5 contains a simple template for exit rules and shows what the client from Box 1.4 had completed for his strategy.\n\n* * *\n\n### Box 1.5 Simple Template for Exit Rules\n\n#### Exits\n\nCalculations and\/or Indicators Involved (specify lookback period). Don't use ambiguously descriptive words like _rising_ , _falling_ , _flattening_ , _topping_ , or _bottoming out_.\n\n___________________________________________________\n\n___________________________________________________\n\n___________________________________________________\n\nLong Liquidation Condition\u2014What must happen to get out of a long position? List steps in chronological order.\n\n___________________________________________________\n\n___________________________________________________\n\n___________________________________________________\n\nShort Liquidation Condition\u2014What must happen to get out of a short position? List steps in chronological order.\n\n___________________________________________________\n\n___________________________________________________\n\n___________________________________________________\n\n  1. Calculations and\/or Indicators Involved (specify lookback period) \n    1. Average true range (ATR) with a 10-day lookback\n    2. Moving average with a 50-day lookback\n  2. Long Liquidation Condition\u2014What must happen to get out of a long position? List steps in chronological order. \n    1. Close of yesterday is less than entry price \u22123 ATR\u2014get out on next open\n    2. Close of yesterday is less than 50-day moving average\u2014get out on next open\n    3. Close of yesterday is greater than entry price +5 ATR\u2014get out on next open\n  3. Short Liquidation Condition\u2014What must happen to get out of a short position? List steps in chronological order. \n    1. Close of yesterday is greater than entry price +3 ATR\u2014get out on next open\n    2. Close of yesterday is greater than 50-day moving average\u2014get out on next open\n    3. Close of yesterday is less than entry price \u22125 ATR\u2014get out on next open\n\n* * *\n\nThere are three conditions that can get you out of a long position. Unlike the entry logic, the timing or sequence of the conditions do not matter. The exit algorithm gets you out on a volatility-based loss, a moving average\u2013based loss\/win, or a profit objective. The short liquidation criteria are just the opposite of the long liquidation criteria.\n\nThis is a complete, fully self-contained trend-following trading algorithm. It has everything the computer would need to execute the entries and exits. And it could be made fully automated; the computer could analyze the data and autoexecute the trading orders. Any platform such as AmiBroker, TradeStation, Ninja Trader, and even Excel could be used to autotrade this type of system. Any trading idea, if reduced to the exact steps, can be tested, evaluated, and autotraded. Once the programming code has been optimized, verified, validated, and installed on a trading platform, the trader can just let the system run hands off. This is the true beauty of algorithmic system trading: A computer can replicate a trader's ideas and follow them without any type of human emotion.\n\n## How to Get My Trading Idea into Pseudocode\n\nThe aforementioned template is a great way to get your ideas into written instructions. However, just like with great writing, it will take many revisions before you get to the pseudocode step. Over the years, I have discovered there are two different paradigms when describing or programming a trading system. The first is the easier to program and can be used to describe a system where events occur on a consecutive bar basis. For example: _Buy when today's close is greater than yesterday's and today's high is greater than yesterday's high and today's low is less than yesterday's low_. The other paradigm can be used to describe a trading system that needs things to happen in a sequence but not necessarily on a consecutive bar-by-bar basis. Unfortunately, the latter paradigm requires a heightened level of description and is more difficult to program. Many trading systems can fit inside a cookie-cutter design, but most traders are very creative, and so are their designs. Don't fret, though, because any idea that is reducible to a sequence of steps can be described and programmed.\n\n### The Tale of Two Paradigms\n\nI have developed names for these two different models of describing\/programming trading algorithms:\n\n  1. The variable bar liberal sequence paradigm\n  2. The consecutive bar exact sequence paradigm\n\nThe variable bar paradigm can provide a very robust algorithm and at the same time provide a nearly limitless expression of a trader's creativity. We will begin with this one since it is the less intuitive of the two. What you learn about the variable bar sequence will make shifting to the simpler consecutive bar sequence that much easier. The consecutive bar sequence can usually be programmed by using a single if-then construct. You can see this by referring to my earlier example:\n\n> **If** today's close is greater than yesterday's close and today's high is greater than yesterday's high and today's low is less than yesterday's low, **then** buy at the market.\n\nThis paradigm is simple to explain, program, and deploy. There is nothing wrong with this and, remember, many times a simple approach or idea to trading the markets can deliver a very robust algorithm. Both paradigms will be exemplified in different algorithms through the rest of the book. All trading algorithms fall into either paradigm or across both, meaning that some systems require both approaches to be fully described and programmed. Most traders can easily describe and probably program a consecutive bar-type algorithm. However, traders that can't put their ideas into such a simple form and have had little or no programming experience have a more difficult time reducing their ideas into a formal description, not to mention pseudocode.\n\n#### The Variable Bar Sequence\n\nJumping headfirst into paradigm #1, the variable bar sequence, here is a somewhat creative design that incorporates pivot points and a sequence that can occur over a variable amount of days. For those of you who are not familiar with pivot points, Figure 1.1 shows different examples of pivot high points on daily bars and their differing strengths.\n\n**Figure 1.1** Examples of strength differences between pivot high points on daily bars.\n\nThe strength of the pivot bar is based on the number of bars preceding and following the high bar. A two-bar pivot is simply a high bar with two bars before and after that have lower highs. The preceding\/following highs do not necessarily have to be in a stairstep fashion. Here is our long entry signal description using step-by-step instructions.\n\n**Buy:**\n\n  1. **Step 1:** Wait for a pivot high bar of strength 1 to appear on the chart and mark the high price.\n  2. **Step 2:** Wait for the first low price that is 2 percent lower than the high price marked in Step 1. If one occurs, move to Step 3. If the market moves above the high marked in Step 1, then go back to Step 1 and wait for another pivot high bar.\n  3. **Step 3:** Wait for another pivot high bar that exceeds the price in Step 1. Once a high bar pivot occurs that fits the criteria, mark the high price.\n  4. **Step 4:** Once the subsequent pivot high bar is found, then wait for another low price that is 2 percent below the high marked in Step 3. When a bar's low price fulfills the criteria buy that bar's close. If the market moves above the high marked in Step 3, then go back to Step 3 and wait for another pivot high bar. The new high that just occurred may turn out to be the pivot high bar that you are looking for in Step 3.\n\nAdditional notes: If 30 days transpire before proceeding from Step 1 to Step 4, then reset.\n\nFigure 1.2 illustrates the sequence the above description is trying to capture. Ignore the state designations for now. They will become apparent in the following discussion.\n\n**Figure 1.2** A depiction of the variable-bar sequence described by the long entry signal.\n\nIt is easy to see the designer\/trader of this entry signal is trying to buy after two pivot highs are separated and followed by a pullback of 2 percent. In addition, there are a couple of conditions that must also be met before a buy signal is triggered: (1) the second pivot high price must be higher than the first, and (2) the whole sequence must take less than 30 bars to complete. When a system allows its entry criteria to work across an unknown number of days or price levels, conditions must be used to keep the integrity of the entry signal and limit the duration of the sequence.\n\nSince there is variability in the instruction set, the programming of this paradigm is definitely more difficult. However, as mentioned earlier, it is very doable. As a young and inexperienced trading system programmer, I tried to program these variable sequences using a bunch of true\/false Boolean flags. Box 1.6 shows an initial attempt using this method to describe the pivot point sequence. The flags are bolded so you can easily see how they are turned on and off. Comments are enclosed in brackets { }.\n\n* * *\n\n### Box 1.6 Using Boolean Flags to Program a Trading Algorithm\n\n    HighPivotFound = High of yesterday > High of today and High of yesterday > High of two days ago\n    If **HPivot1** = False and HighPivotFound then {first Pivot high found}\n\n  1.     **HPivot1** = True\n    HPivot1Price = high of yesterday\n    HPivot1Cnt = currentBar\n\n    If **HPivot1** = True and Low < HPivot1Price * 0.98 then {2% retracement after first Pivot high}\n\n  1.     **LRetrace** = True\n\n    If **HPivot1** = True and **LRetrace** = False and High of today > HighPivot1Price then\n\n  1.     HPivot1Price = High of yesterday {another higher high but not a 2% retracement \u2013 start over}\n    HPivot1Cnt = currentBar\n\n    If **LRetrace** = True and HighPivotFound and High of yesterday > HPivot1Price then\n\n  1.     **HPivot2** = True {second Pivot high > first Pivot high and 2% retracement between the two}\n    HPivot2Price = High of yesterday\n\n    If **HPivot2** = True and High of today > HPivot2Price then\n\n  1.     HPivot2Price = High of today {another higher high > second Pivot High \u2013 keep track of it}\n\n    If **HPivot2** = True and Low < HPivot2Price * 0.98 then\n\n  1.     Buy this bar on close {Buy Order Placed \u2013 entry criteria has been met}\n    **HPivot1** = False {Start Over}\n    **LRetrace** = False\n    **HPivot2** = False\n\n    HPivot1Cnt = HPivot1Cnt + 1\n    If HPivot1Cnt >= 30 then {reset all Boolean flags \u2013 start over}\n\n  1.     **HPivot1** = False\n    **LRetrace** = False\n    **HPivot2** = False\n\n* * *\n\nAs you can see, the description using these flags is somewhat laborious and not that flexible. The flags have to have descriptive names so that the correct flag is being tested and subsequently turned on or off. In addition, all flags must eventually be turned off when either the HPivot1Cnt variable grows equal to or greater than 30 or a LONG position is established. This type of programming will work but isn't very eloquent. Good programmers don't like to use superfluous variable names and many lines of code to complete a task. And really good programmers like for others to easily read their code and comment on how clever the solution was to the problem. Eloquent code is precise and clever. So, after much trial and error as a young programmer, I finally realized that these types of trading algorithms (the paradigm #1 variety) were simply looking for an occurrence of a certain sequence or pattern in the data. When I say pattern, I don't mean a certain price pattern like a candlestick formation. As we have seen before, a trading algorithm is just a sequence of instructions. Remembering back to my compiler design class in college and how we had to write computer code to recognize certain tokens or words (patterns) in a string of characters I felt I could use the same model to program these types of systems, a universal model that could be molded to fit any trading system criteria. This model that programmers use to parse certain words from a large file of words is known as a finite state machine (FSM). The FSM concept is not as daunting as it sounds. If we translate the geek-speak, a FSM is simply a model of a system that shows the different possible states a system can reach and the transitions that move the system from one state to another. The machine starts at a START state and then moves methodically through several states by passing certain logical criteria and then arrives at an ACCEPT state.\n\nDon't let this idea blow your mind because it is quite easy to understand and implement. Let's start off with a very simple FSM to illustrate its simplicity and eloquence. Think of a combination lock that can only be unlocked by inputting the following numbers: 6, 4, 2, 7, 5, and 1. Like most combination locks the numbers need to be inputted in the exact sequence described or the lock will not open. This lock is not very good because it lets you test to see if you have the right number before proceeding to the next input. So you plug in a number, test it, and either try again if it fails or move onto the next number if it is correct. Eventually, with time the correct combination will be inputted and the lock will open. Remember this is just a very simple example of something that can be modeled by a FSM. Without an illustration or diagram, most nonprogrammers could not design even this simple example. A picture is always worth a minimum of a thousand words and the easiest way to create an appropriate FSM is to create a diagram. The diagram in Figure 1.3 describes the FSM that models the combination lock.\n\n**Figure 1.3** An FSM that models the workings of a combination lock.\n\nThe diagram is made up of circles, which represent the different **STATES** and connectors that show how the machine moves from one state to another. This FSM has a total of seven states that include a **START** and **ACCEPT** state. Pretend you are sitting in front of a small screen and a numeric touchpad similar to one that is found on a cell phone, and it is prompting you to input a number from one to nine. At this point, the machine is set to the START state. If you get lucky right off the bat, you input the number six and the screen prompts you to input the second number. If you aren't lucky, it will ask you to re-input the first number in the combination. So following along with the diagram, you can see how the number six moves the machine from the START state to STATE 1. If the number six is not inputted, it moves right back to the START state. The machine moves along the paths visiting each STATE as long as the proper number is inputted. Notice you are not penalized if you don't input the proper number; the machine just sits in that particular state until the proper number is inputted.\n\nIn our pivot point example, there will also be a START and ACCEPT state. Refer back to the step-by-step instructions of the pivot point entry technique and visualize the steps as different states. The START (STEP 1) state will be assigned the value 0 and the ACCEPT (STEP 4) state will have the value 4. In between, the START and ACCEPT states will be three other states that the FSM can assume. These are intermediate states that must be achieved in sequential order. The START state tries to get the ball rolling and looks at every bar to see if a pivot high (of strength 1) has occurred. If one does, then the machine moves onto the next state, and then on to the next, and then on to the next, and then finally to the ACCEPT state. As soon as the FSM assumes the ACCEPT state, a buy order is placed at the market. The illustration in Figure 1.4 is the FSM diagram of our pivot-point long-entry algorithm.\n\n**Figure 1.4** An FSM that models the pivot-point long-entry algorithm described in this chapter.\n\nThis FSM diagram looks to be much more complicated than the combination lock, but it really isn't. There are fewer states but many more connectors. In the combination lock FSM there was only one connector connecting the different states together and the machine never went backward. Once a state was attained, the machine stayed there until the criteria were met to advance to the next state. This pivot-point FSM can move forward and backward.\n\nStepping through this diagram will help make it less scary. Keep in mind that all of these machines gobble up one bar at a time and then make a decision. Starting at the START state the machine looks for a pivot high of strength one. Once the machine finds this pivot point it then moves to STATE1 and starts looking for a low price that is 2 percent or less than the pivot high found in the START state. There are four paths out of STATE1: (1) a low is found that fits the criteria, (2) a low price fitting the criteria is not found, stay in STATE1, (3) 30 bars have occurred since the pivot high is found, and (4) a high exceeds the pivot high price. Only one path leads to STATE2. All other paths either return to the START state or loop back to the current state. Once the machine attains STATE2 it starts analyzing the data by gobbling each bar and searches for a higher pivot high point. Unlike STATE1 there are only three paths coming out of this state: (1) a higher pivot high is found, (2) a higher pivot high is not found, and (3) 30 bars have occurred since the pivot high was found in the START state. There is only one path to STATE3 and that is a higher pivot high. All other paths either loop or return the machine to the START state. Assuming a higher pivot high is found the machine moves to STATE3. Once in STATE3 the machine will only return to the START state if 30 bars have come and gone before the machine can attain the ACCEPT state. If new highs are found, the machine continues to stay in STATE3 and keeps track of the new highs, all the while looking for that particular low that is 2 percent or lower than the most recent pivot high. Eventually things fall into place and the machine attains the ACCEPT state and a buy order is placed. You can now refer back to Figure 1.2 and see when the machine attains the various states.\n\nThe diagram in Figure 1.4 looks clean and neat, but that's not how it started out. When I diagram a trading algorithm, I use pen\/pencil and paper. A pencil is great if you don't want to waste a lot of paper. Also, you don't have to have a complete diagram to move onto the pseudocode step. Trust me when I say a diagram can help immensely. Something that seems impossible to program can be conquered by drawing a picture and providing as much detail on the picture as possible. Box 1.7 contains the pseudocode for the pivot point entry algorithm using the FSM diagram.\n\n* * *\n\n### Box 1.7 Pivot Point Entry Algorithm for Figure 1.4\n\n    If initialize then\n    State = Start\n    HiPivotFound = High of yesterday > High of today and\n         High of yesterday > High of prior day\n    If State = Start and HiPivotFound then\n         State = 1\n         BarCount = 1\n         Pivot1Hi = High price of HiPivotFound\n    If State <> Start then BarCount = BarCount + 1\n    If State = 1 then\n         If Low of today < Pivot1Hi * 0.98 then\n                State = 2\n         If High of today > Pivot1Hi then\n                State = Start\n    If State = 2 then\n         If HiPivotFound then\n                 Pivot2Hi = High Price of HiPivotFound\n                 If Pivot2Hi > Pivot1Hi then\n                          State = 3\n    If State = 3 then\n         If Low of today < Pivot2Hi * 0.98 then\n                 State = Accept\n         If High of today > Pivot2Hi then\n                 Pivot2Hi = High of today\n    If State = Accept then\n         Buy this market on close\n         State = Start {start over}\n    If BarCount = 30 then\n         State = Start\n         BarCount = 0\n\n* * *\n\nYou might understand this code or you may not. If you have never programmed or scripted, then you might have a problem with this. If you fully understand what is going on here, just bear with us a few moments. Even if you understood the diagram, this code may not be fully self-explanatory. The reason you may not understand this is because you don't know the _syntax_ or the _semantics_ of this particular language. Syntax was discussed earlier in this chapter, and remember, it's just the structure of the language. I tried my best to utilize a very generic pseudocode language for all the examples throughout the book and hopefully this will help with its understanding. In the pseudocode example, the language uses _if_ s and _then_ s to make yes-or-no decisions. If something is true, then do something, or if something is false, then do something. These decisions divert the flow of the computer's execution. The code that is indented below the if-then structure is controlled by the decision. This structure is the syntax. Now the logic that is used to make the decision and then what is carried out after the decision is the semantics. In our example, we will eventually tell the computer that if State = 1, then do something. Syntax is the grammatical structure of a language and semantics is the meaning of the vocabulary of symbols arranged within that structure. You many notice the similarity in the pseudocode and the FSM diagram. The logic that transitions the pivot point FSM from one **state** to another is mirrored almost exactly in the code. This tells you the time you spent planning out your program diagram is time well spent. So let's start with the pseudocode and make sure the FSM diagram is fully implemented.\n\nThe START state is simply looking for a pivot point high of strength 1. When one occurs the machine then shifts gears and enters STATE1. Once we have entered the STATE1 phase, the machine stays there until one of three criteria is met: (1) the elusive low that is 2 percent or lower than the Pivot1Hi, (2) a high price that exceeds the Pivot1Hi, or (3) 30 bars transpire prior to attaining the ACCEPT state. If you look at the STATE1 block of code, you will only see the logic for the first two criteria. You may ask where is the logic that kicks the machine back to the START state once 30 bars have been completed prior to the ACCEPT state. If you look further down in the code, you will see the block of code that keeps track of the BarCount. If at any time the BarCount exceeds 30 and the machine is not in the START state, the machine is automatically reset to the START state. If a low price is observed that is lower than 2 percent of the Pivot1Hi price, then the machine transitions to STATE2. However, if a high price is observed that exceeds Pivot1Hi, then the machine reverses back to the START state and it once again starts looking for a new PivotHi. Assuming the machine does make it to STATE2, it then starts looking for another PivotHi price that is greater than Pivot1Hi. A transition from STATE2 can only occur when one of two criteria are met: (1) BarCount exceeds 30, then it's back to the START state, or (2) a higher HiPivot than Pivot1Hi is observed. If the latter criterion is fulfilled, then it is onto STATE3. STATE3 looks for a low price that is 2 percent less than the Pivot2Hi price so it can transition to the ACCEPT state. At this point the BarCount is its only enemy. It doesn't care if a high price exceeds the Pivot2Hi. If it does, then it simply replaces the Pivot2Hi with this price and continues searching for a price that is 2 percent lower than Pivot2Hi. Eventually, the machine resets itself or it finally transitions to the ACCEPT state and buys market on close (MOC).\n\nAs you can see, the description of the FSM diagram is all almost completely repeated in the description of the pseudocode. If the two descriptions agree, then it is on to the actual programming of the algorithm. Now that this more complicated paradigm has been explained, I think it is time to give it a better name. How about the _FSM paradigm_? Sounds better than _the variable bar liberal sequence paradigm_ , doesn't it?\n\n#### _The Consecutive Bar Sequence_\n\nI saved _the consecutive bar exact sequence paradigm_ for the end because the FSM paradigm is much more difficult to understand and much less intuitive. Armed with our new knowledge of diagrams, it is now time to move to this simpler paradigm because it will be used a lot more in your programming\/testing of trading algorithms. As I stated before, most trading algorithms will be fully describable in a step-by-step fashion. Here the logic will not be needed to be modeled by a machine; a recipe approach will do just fine. Guess what type of diagram is used to model these recipe types of instructions. If you guessed a flowchart (FC), then pat yourself on the back. A good computer programming 101 instructor introduces her students to the flowchart concept\/diagram way before they even sit down in front of a keyboard. Figure 1.5 shows a very simple FC.\n\n**Figure 1.5** A very simple FC.\n\nCan you see what is being modeled there? It's quite simple; the FC diagram starts at the top and makes one decision and, based on that decision, it carries out its objective. It starts out with you waking up and observing the time and making a decision to take the bus or the subway. This is a way oversimplified example, but it covers everything an FC is designed to do: start, make decisions, carry out the appropriate instructions based on the decisions, and then finish. Figure 1.6 shows an ever so slightly more complicated FC that deals with what this book is all about, a trading algorithm.\n\n**Figure 1.6** An FC of a trading algorithm.\n\nThis diagram is a flowchart of the entry technique of a very popular mean reversion trading system. This system trades in the direction of the long-term trend after a price pullback. The trend is reflected by the relationship of the closing price and its associated 200-day moving average. If the close is greater than the average, then the trend is up. The pullback is reflected by the 14-period RSI indicator entering into oversold territory\u2014in this case, a reading of 20 or less. The diagram illustrates two decisions or tests and, if both tests are passed, then the system enters a long position on an MOC order. The flow of the algorithm is linear, meaning it flows from top to bottom. The flow is diverted based on simple decisions, but it always ends at the bottom of the flowchart\u2014either buying MOC or starting on a new bar of data. Here is the pseudocode of this mean reverting system; look quickly or you will miss it:\n\n* * *\n\n    'Simple FC type trading algorithm\n    'An example of a mean reversion system\n    If Close of today > 200 day average of Close then\n           If RSI(14) < 20 then\n                  Buy MOC\n\n* * *\n\nThat's the entire entry technique, in a nutshell. Was it necessary to create a flowchart diagram? In this case, no. But as you will find out in later chapters, most trading algorithms are not this easily definable.\n\n## Summary\n\nIn this chapter, the necessary tools for creating a trading algorithm were introduced. Describing your ideas on paper (real or virtual) is the very first step in the algorithm development process. A template was shown that can be used to help facilitate this. Just getting a trading idea on paper is not sufficient to move onto the pseudocode phase. All ideas must be translated into mathematical expressions and Boolean logic. These expressions and logic must be in a form a computer can understand. Ambiguous directions must be eliminated and replaced with pure logic. Once the idea is written down and further reduced into an extremely logical list of directives and calculations or formulae, the trader must then decide what type of paradigm to use to get the logic into something that can be eventually programmed or scripted. The FSM, as well as the FC methods, were both introduced and used to convert trading schemes into complete pseudocodes. These two paradigms have always been present, and many algorithmic traders have used them without realizing the different programming methods. With knowledge of the two different methods, hopefully an algorithmic trader can make a choice on which is best to use and get from idea to actual code in a lot less time and a lot less frustration.\n\nHere again is a summary of the two methods:\n\n  * **Flowchart** : The \"flow\" of a flowchart is a process. The flowchart shows the steps and actions to achieve a certain goal. Use this method if you can define your trading logic in a step-by-step or bar-by-bar basis. A large portion of trading systems will fit into this programming paradigm.\n  * **Finite state machine** : The \"flow\" in a state diagram is always from state to state. Most FSMs have a START and ACCEPT state, similar to the beginning and end of a flowchart. Machine diagrams describe a closed system composed of multiple discrete states. The \"state\" in this case determines how a system behaves to stimulus or events. Use this method if criteria have to be met in a sequential manner, but the amount of time between the beginning and criteria completion is variable. Pattern-based systems will usually fall into this paradigm.\n\nOnce a potential programming method is chosen, it is time to \"draw\" the corresponding diagram. A diagram doesn't have to be pretty, but it must try to cover all the bases or what-if scenarios a trading system may encounter to carry out its objectives. A thoroughly thought-out diagram providing as much information as possible will definitely enable a trading algorithm to be quickly translated into pseudocode and eventually programmed into a testing platform. Over time, as your experience grows with programming trading systems, these diagrams will begin to appear in your mind's eye. However, this takes time and a lot of experience.\n\nIf you made it through this chapter, then, at least, you are now in possession of a trading algorithm that is quite similar to one that actually sold for thousands of dollars in the 1990s.\n\n# Chapter 2  \nStochastics and Averages and RSI! Oh, My!\n\nIndicators and price bar patterns are the most widely used building blocks of trading algorithms. Of the many trading systems I have programmed over the years, at least 90 percent utilized an indicator either straight out of the box or customized in some form or fashion. The trading community is split right down the middle when it comes to the perceived effectiveness of these price transformations. An indicator is simply a price transformation; price is transformed or changed into something that is thought to be more illustrative of the current market condition. There have already been many articles and books written about most indicators. And it is true that these indicators do not work all of the time as standalone signal generators. It is also true that there is redundancy among many of the different indicators. However, I want to discuss the mathematically simple indicators that are easily implemented into a trading algorithm and when used in concert with other ideas can produce a slight technical advantage.\n\nIndicators fall into basically two categories, _oscillator_ (usually predictive) and _price based_ (lagging). Oscillators are depicted on charts that plot a value that is banded by two extremes. This information is most commonly used for short-term forecasting, but it can also reveal longer term market behavior. Price-based indicators are displayed on charts as a real price of the underlying instrument and are used to determine trend and\/or volatility. These indicators are reporting back what is currently going on after the fact, hence the term _lagging_.\n\n## Oscillators\n\nOscillators are normalized, in most cases, to fall between two extreme levels, and are usually plotted in a subgraph. Certain indicator-based levels are then superimposed on the oscillators as benchmarks. The relationships of oscillator value and the benchmarks are then used to predict short-term market direction. The normalization process allows the oscillator family of indicators to be universally applicable to all markets. Oscillators are most often used to predict but can also be used as lagging indicators.\n\n### Average Directional Movement Index (ADX)\n\nThis oscillating indicator is one of the more popular as it measures a market's trendiness. The underlying concept of _Directional Movement_ was introduced in J. Welles Wilder's 1978 book, _New Concepts in Technical Trading Systems_. As a side note, this was the book I first used to help program many of the more popular indicators into our Excalibur software in the late 1980s. Many algorithmic traders are trend followers, even though it is been proven that markets only trend a very small portion of time.\n\nIf this is the case, then why do many traders claim that \"the trend is your friend\"? Trend followers hope that by diversifying across many markets, they will hit upon one or two markets that sufficiently trend enough that they can negate all of the other losing markets and still make enough money to provide a profitable year. While trend followers tread water by capturing a few trends a year, they are constantly waiting and hoping they will be poised for that once-in-a-lifetime trend event. These once-in-a-lifetime trend events do happen more frequently than once in a lifetime, and most trend-following algorithms are more than capable of catching and holding on for the long haul.\n\nThe realization that commodities are limited resources and the fact that trend following has had success in the past has promoted this trading algorithm class as the most widely used in the managed futures arena. However, commodity markets over the past few years have been unkind to trend following, and if this \"trend\" continues, then you might see a gradual exodus of these types of traders. Stock traders, as of the writing of this book, have been riding a bull trend for several years, which has bolstered the buy-and-hold mentality. I guess the phrase about the trend could be changed to, \"The trend is rarely your friend, but it's the only friend in town.\"\n\nThe main weakness of trend following is, of course, whipsaw trades\u2014the ones you get into and very quickly get stopped out or, worse in some cases, reversed. Another common weakness is lack of profit retention when the ride is over. The ADX indicator is here to save the day\u2014well, sort of. This indicator attempts to define the current market's trendiness over the past _n_ -days into one simple value. The ADX value is high when the market is trending and low when it is in congestion.\n\nThe basic theory behind the ADX is based on market directional movement. This movement is either positive, negative, or neutral. In Wilder's vernacular, positive direction was described as plus directional movement (+DM) and negative as minus directional movement (\u2212DM). Directional movement is calculated by comparing the difference between two consecutive bars' lows and the difference between their respective highs. Movement is positive (plus) when the current bar's high minus the prior bar's high is greater than the prior bar's low minus the current bar's low. This plus directional movement (+DM) then equals the current high minus the prior bar's high as long as this value is positive. If the value is negative, then the +DM would be set to zero. Negative movement or minus directional movement (\u2212DM) does the same comparison. A \u2212DM occurs when the prior bar's low minus the current is greater than the current bar's high minus the prior. In this case the difference between the two lows is assigned to the \u2212DM as long as the value is positive. If the value is negative, it is assigned zero just like the +DM. Sounds a bit confusing but Figure 2.1 should help clarify things.\n\n**Figure 2.1** Examples of directional movement.\n\nFigure 2.1 shows four calculation examples for directional movement. The first pairing shows a big positive difference between the highs for a strong plus directional movement (+DM). The second pairing shows an outside day with minus directional movement (\u2212DM) getting the edge. The third pairing shows a big difference between the lows for a strong minus directional movement (\u2212DM). The final pairing shows an inside day, which amounts to no directional movement (zero). Both plus directional movement (+DM) and minus directional movement (\u2212DM) are negative and cancel out each other. Negative values revert to zero. Keep in mind all inside days will have zero directional movement.\n\nThe ADX index incorporates moving averages of true range and both +DM and \u2212DM. Data averaging is also the same as data smoothing. A moving average function cuts down the impact of abnormal data points and creates a more robust model of the data. Wilder created his indicators without the use of a computer and created a shortcut to average his data points. This Wilder's smoothing is more like an exponential calculation. Different trading\/testing\/charting platforms use different smoothing functions for Wilder's indicators, so be sure to check your user's manual and see which form is being utilized by your particular software. Assuming a 14-day ADX length, Wilder would sum up the first 14-day values and then divide by 14. Once these initial averages were calculated the subsequent bar values were calculated by first dividing the previous sum by 14 and subtracting it from the previous sum and then adding today's values:\n\nThis form of smoothing eliminates keeping track of the prior 14 bar values and accomplishes the desired effect (reducing the impact of outlying data points) on plus DM, minus DM, and true range. Remember, Wilder probably had the use of just a calculator, and instead of inputting 42 numbers (summing three sets of 14 numbers), he simply used the above formula and whipped out the values in a minute or less. Most of today's charting software utilizes an exponential moving average to smooth the ADX. This smoothing method creates results that can be quite different than Wilder's original method.\n\nThe smoothing factor (2\/(N+1)) or in this case (2\/15) can be changed to (1\/N) to arrive at Wilder's calculation. This becomes more like a 27 period exponential moving average than a 14. However, this exponential method is considered accurate, but, in some cases, different charting software will offer both versions: ADX and Wilder ADX.\n\n**Calculation for a 14-day ADX:**\n\n  1. Calculate the true range (TR), plus directional movement (+DM), and minus directional movement (\u2212DM) for each bar.\n  2. True range (TR ) = max(C[1],H) \u2212 min(C[1],L). In other words: Take the higher of yesterday's close and today's high and subtract the lower of yesterday's close and today's low. Basically, you are expanding today's range to incorporate the prior day's close if it is outside of today's range.\n  3. Smooth these periodic values using a 14-day moving average. You will either have to wait until 14 days have transpired or you can simply look at 14 days of prior history before commencing the ADX calculation.\n  4. Divide the 14-day smoothed plus directional movement (+DM) by the 14-day smoothed true range to find the 14-day plus directional indicator (+DI14). Multiply by 100 to move the decimal point two places. This +DI14 is the plus directional indicator that is plotted along with ADX.\n  5. Divide the 14-day smoothed minus directional movement (\u2212DM) by the 14-day smoothed true range to find the 14-day minus directional indicator (\u2212DI14. Multiply by 100 to move the decimal point two places. This \u2212DI14 is the minus directional indicator that is plotted along with ADX.\n  6. The Directional Movement Index (DX) equals the absolute value of +DI14 less \u2212DI14 divided by the sum of +DI14 and \u2212DI14.\n  7. After all these steps, it is time to calculate the Average Directional Index (ADX). The ADX value is initially simply a 14-day average of DX.\n  8. Wilder went one step further and created the ADXR, which is simply the average of an _N_ -day differential of the ADX. Continuing with our 14-day example, the ADXR would be calculated by applying the following formula:\n\nFigure 2.2 illustrates how the ADX can be used to help determine the trendiness of a market. A rising ADX, as shown in the early part of the price chart, demonstrates a trending market condition, whereas a falling ADX is saying the market has stopped trending and is entering a congestion phase. This information is very useful but very limited as well. The computer cannot determine which way the market is trending by simply looking at the ADX value. The solution to this problem can be found by comparing the +DI to the \u2212DI. If the +DI crosses above the \u2212DI, then the market is considered in an uptrend. When the \u2212DI crosses above the +DI, then the market is in a downtrend.\n\n**Figure 2.2** ADX as a trend detector.\n\nSource: TradeStation\n\nUsing this indicator in concert with a trend-following system may produce more robust entry signals. I have seen many systems incorporating the ADX in an attempt to eliminate \"whipsaw\" trades. Welles Wilder developed his own system around the ADX, +DI, and \u2212DI. Box 2.1 is a description of a system that incorporates the ADX and a trailing stop for both long and short trades. The stop will get out of longs when the market closes lower than the prior three-day closings and get out of shorts when the market closes higher than the prior three days. This type of stop is not fixed as it varies based on market movement. In other words, it is a dynamic stop. If we are lucky to get in on a nice intermediate trend, the stop will follow the market in the direction of the movement and hopefully lock in some profit.\n\n* * *\n\n### Box 2.1 Welles Wilder's Directional Movement System Description\n\n  1. If 14-day ADX is greater than 35 then \n    1. If +DI > \u2212DI, then Buy market on close\n    2. If \u2212DI > +DI, then SellShort market on close\n  2. If 14-day ADX is less than 35, then \n    1. If position is long and close is less than lowest close of prior three days, then Sell market on close.\n    2. If position is short and close is greater than highest close of prior three days, then BuyToCover market on close.\n\n#### Welles Wilder's Directional Movement p\u2013code\n\n    `Wilder's Directional Movement System\n    `Assuming Wilder's Indicator\/Functions are built\u2013in\n    `Also Highest and Lowest functions\n    myADX = ADX(14) `14 period ADX calculation\n    myPlusDI = PDI(14) ` 14 period Plus DI\n    myMinusDI = MDI(14) ` 14 period Minus DI\n    if myADX> 35 then\n         if myPlusDI> myMinusDI then\n                 Buy this bar on close\n         if myMinusDI> myPlusDI then\n                 SellShort this bar on close\n    if myADX < 35 then\n         if MarketPosition = 1 and close < Lowest(close[1],3) then\n                 Sell this bar on close\n         if MarketPosition = -1 and close > Highest(close[1],3) then\n                 BuyToCover this bar on close\n\n* * *\n\nThe directional movement algorithm description is straightforward and can be modeled by a flowchart. A finite state machine is not needed because, as we discussed in Chapter 1, this system follows a certain flow that is directed by a combination of logical steps and decisions that eventually lead to termination of the process. Figure 2.3 shows the flowchart of this algorithm.\n\n**Figure 2.3** Wilder's directional movement algorithm.\n\nLike the description, the diagram is straightforward as well. It might look complicated, but that is only because of the many different decisions that must be evaluated before an action is carried out. In the case of this very simple system, a flowchart diagram isn't really all that necessary. However, it is great practice, and eventually you will come across a trading scheme that will be complicated enough to merit a properly thought-out diagram.\n\nBefore proceeding to the next indicator, I must mention that the ADX along with all other indicators are just building blocks and are not to be used by themselves. Many successful trading algorithms utilize indicators but not strictly by themselves and in many cases the indicators are augmented or filtered. Many traders change some of the logic of an indicator and therefore make it their own and in doing so bring their own creativity into the process. Most trading algorithms are built on the shoulders of others. More frequently, traders will utilize a filtering process that will eliminate excessive trade signals generated by an indicator. In doing so, the trader is hoping to skip the less productive trades and target the really good ones. Table 2.1 shows the performance of the pure ADX applied to several markets.\n\n**Table 2.1** Performance of Directional Movement Algorithm\n\n**Ticker** | **Net Profit** | **Max. Sys DD** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **% of Winners**  \n---|---|---|---|---|---|---  \nAD0_E0B | \u221221140.0 | \u221226030 | 43 |  \u2212491.63 | 13.53 | 32.56  \nBP0_E0B |  \u22125825.0 | \u221227925 | 39 |  \u2212149.36 | 12.74 | 38.46  \nC20_E0B |  737.5 | \u221210000 | 43 |  17.15 | 15.51 | 44.19  \nCC20_E0B |  \u22124050.0 | \u221213560 | 35 |  \u2212115.71 | 14.86 | 42.86  \nCL20_E0B |  41350.0 | \u221226750 | 36 |  1148.61 | 14.67 | 41.67  \nCT20_E0B |  44530.0 | \u221220220 | 46 |  968.04 | 16.7  | 39.13  \nCU0_E0B |  5712.5 | \u221220925 | 47 |  121.54 | 15.81 | 40.43  \nDX20_E0B |  150.0 |  \u22129990 | 51 |  2.94 | 14.51 | 41.18  \nED0_E0B |  2009.4 |  \u22121513 | 58 |  34.64 | 18.12 | 29.31  \nEMD0_E0B | \u221233400.0 | \u221243910 | 31 | \u22121077.42 | 13.81 | 29.03  \nES0_E0B | \u221218837.5 | \u221223050 | 26 |  \u2212724.52 | 15.96 | 30.77  \nFC0_E0B |  20087.5 |  \u22129588 | 45 |  446.39 | 17.04 | 35.56  \nFV0_E0B |  8226.4 |  \u22127203 | 49 |  167.89 | 17.41 | 34.69  \nGC20_E0B | \u221232870.0 | \u221241560 | 43 |  \u2212764.42 | 12.84 | 27.91  \nHG20_E0B |  300.0 | \u221241338 | 34 |  8.82 | 14.5  | 29.41  \nHO20_E0B |  20139.0 | \u221240286 | 46 |  437.8  | 13.17 | 39.13  \nKC20_E0B | \u221232606.3 | \u221242413 | 36 |  \u2212905.73 | 11.61 | 25  \nKW20_E0B |  9887.5 | \u221215863 | 43 |  229.94 | 16.56 | 39.53  \nLB0_E0B |  957.0 | \u221217259 | 39 |  24.54 | 15.62 | 38.46  \nLC0_E0B |  \u22121230.0 |  \u22129550 | 41 |  \u221230  | 17.24 | 26.83  \nLH0_E0B |  1510.0 | \u221210710 | 40 |  37.75 | 15.65 | 37.5   \nMP0_E0B |  4775.0 |  \u22125295 | 33 |  144.7  | 14.42 | 39.39  \nNG20_E0B |  54440.0 | \u221269390 | 38 |  1432.63 | 16.11 | 39.47  \nNK0_E0B |  \u22128425.0 | \u221215125 | 28 | \u2212300.89 | 15.39 | 46.43  \nOJ20_E0B | \u221216620.0 | \u221217790 | 45 | \u2212369.33 | 15.18 | 28.89  \nPL20_E0B |  16610.0 | \u221230895 | 36 | 461.39 | 16.39 | 33.33  \nRB0_E0B | \u221217665.2 | \u221246813 | 26 | \u2212679.43 | 13.62 | 23.08  \nS20_E0B | \u221212212.5 | \u221231463 | 50 | \u2212244.25 | 14.94 | 28  \nSB20_E0B |  6563.2 | \u221211995 | 61 | 107.59 | 12.95 | 26.23  \nSI20_E0B |  40400.0 | \u221251735 | 44 | 918.18 | 14.14 | 25  \nTF0_E0B | \u221224920.0 | \u221232610 | 33 | \u2212755.15 | 12.61 | 24.24  \nTU0_E0B |  12124.8 |  \u22124156 | 42 | 288.69 | 17.1  | 33.33  \nTY0_E0B |  12109.6 |  \u22127531 | 48 | 252.28 | 16.69 | 35.42  \nUS0_E0B |  14906.4 | \u221218031 | 45 | 331.25 | 16.67 | 37.78  \nW20_E0B | \u221219462.5 | \u221231188 | 45 | \u2212432.5  | 14.64 | 26.67\n\n### Relative Strength Index (RSI)\n\nContinuing on with Welles Wilder, the next indicator that we will discuss will be the Relative Strength Index (RSI). This indicator is a momentum oscillator that measures the speed and change of price movements. RSI oscillates between 0 and 100. Traditionally, and according to Wilder, RSI is considered overbought when above 70 and oversold when below 30. After moving averages, RSI might be the most popular indicator out there; every popular charting package includes it in its library.\n\nIts popularity might be partially attributed to its simplicity. The indicator is simply measuring the momentum of the market over the past number of days. Momentum indicates the direction and strength of the market. A quickly moving uptrend will be reflected by consecutive higher-high RSI values. And consecutive lower-low RSI values will represent just the opposite, a quickly moving downtrend.\n\n**Calculation for a 14-day RSI:**\n\n  1. Going back in time 14 bars compare the closing price of a bar with its prior bar. If the difference is positive, then accumulate the amount to an upSum variable. If the difference is negative, then remove the sign and accumulate the difference to a dnSum variable.\n  2. If C > C[1], then upSum = upSum + (C \u2212 C[1]). If C < C[1], then dnSum = dnSum + \u22121 \u00d7 (C \u2212 C[1]).\n  3. Once these values are summed, simply divide them by the RSI period. In doing so, you will have two new values: avgUp and avgDn. In a strong upward-moving market, the upSum and corresponding avgUp will be much greater than the dnSum and avgDn.\n  4. Once you have arrived to the avgUp and avgDn values the relative strength (RS) is simply the quotient of the two values:\n\n  5. The RS is the ratio of the changes in up closes to down closes. If the market is in a strong uptrend, the avgUp will be large in relation to the avgDn, thus generating a larger number. The opposite is true if the avgDn is much greater than the avgUp (an occurrence of a downward trending market).\n  6. The RSI is a bound indicator, meaning that it will stay within an upper and lower constraint. In the case of the RSI, as we mentioned earlier, these constraints are 0 and 100. The simple ratio of avgUp and avgDn can go from a small number to a large one. We can normalize the RS for any market and constrain it between 0 and 100 by applying the following formula:\n\nLet's assume a strong upwardly trending market with an upAvg of 60 and a dnAvg of 20. Inputting this information into the formula, we arrive at the following:\n\n  7. Just like the ADX, subsequent values are smoothed utilizing either an exponential moving average or Wilder's homebrew.\n\nThe default period length for the RSI is 14 so it is easy to see that this indicator was designed for a short-term approach. When using RSI in an overbought\/oversold application, traders are looking for a reversion to a mean type movement\u2014what goes up must eventually come down. Through Wilder's research he observed his indicator usually topped\/bottomed out prior to an actual market top\/bottom. Figure 2.4 shows a highly successful RSI trade.\n\n**Figure 2.4** Example of a trade generated by RSI.\n\nSource: TradeStation\n\nBox 2.2 shows a trading system description incorporating the RSI, a volatility-based profit objective, and protective stop. The profit and stop are based off a 10-day average true range\u2014wider profit targets and stops in high volatility. Notice how the system is trying to cut losses short and let profits run using a 3:1 ratio. I have noticed in situations where a system is trying to trade shorter-term to intermediate-term trends a 1:3 ratio, albeit somewhat illogical, works better. Continuing along with our diagrams, Figure 2.5 shows the very simplistic flowchart.\n\n**Figure 2.5** Flowchart diagram of RSI trading algorithm.\n\n* * *\n\n### Box 2.2 Welles Wilders RSI System Description\n\n  1. If 14-day RSI is less than 30, then \n    1. Buy MOC\n  2. If 14-day RSI is greater than 70, then \n    1. Sell Short MOC\n  3. If position is long \n    1. Take Profit: If close > entry price + 3 ATR Sell MOC\n    2. Protective Stop: If close < entry price \u2212 1 ATR Sell MOC\n  4. If position is short \n    1. Take Profit: If close < entry price \u2212 3 ATR BuyToCover MOC\n    2. Protective Stop: If close > entry price + 1 ATR BuyToCover MOC\n\nNote that MOC stands for market on close.\n\n#### Welles Wilders RSI System p\u2013code\n\n    `Wilders Simple RSI OB\/OS system\n    utilizing a 3 ATR profit objective\n    and a 1 ATR stop\n    If rsi(c,14) < 30 then buy this bar on close\n    If rsi(c,14) > 70 then sellShort this bar on close\n    If marketPosition = 1 then\n         If c > entryPrice + 3* avgTrueRange(10) then sell this bar on close\n         if c < entryPrice - 1* avgTrueRange(10) then sell this bar on close\n    If marketPosition = 1 then\n    begin\n         If c < entryPrice - 3* avgTrueRange(10) then buyToCover this bar on close\n         if c > entryPrice + 1* avgTrueRange(10) then buyToCover this bar on close\n\nSimple enough!\n\n* * *\n\nNow don't get carried away, because the RSI can call an intermediate top\/bottom but it can also be faked out by a strong trend move. If you simply use it as an overbought\/oversold indicator, you will be disappointed. Table 2.2 shows the performance of the RSI across a portfolio of futures.\n\n**Table 2.2** Performance of RSI Algorithm\n\n**Ticker** | **Net Profit** | **Max. Sys DD** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **% of Winners**  \n---|---|---|---|---|---|---  \nAD0_E0B | \u221217229.99 | \u221219850 | 87 | \u2212198.05 | 8.4 | 16.09  \nBP0_E0B | \u221219812.5 | \u221249600 | 68 | \u2212291.36 | 8.16 | 23.53  \nC20_E0B | \u221210550 | \u221214250 | 87 | \u2212121.26 | 9.63 | 21.84  \nCC20_E0B | 5240 | \u22127280 | 87 | 60.23 | 8.07 | 28.74  \nCL20_E0B | \u221254340 | \u221267120 | 89 | \u2212610.56 | 9.21 | 21.35  \nCT20_E0B | \u22126600 | \u221237295 | 93 | \u221270.97 | 7.37 | 22.58  \nCU0_E0B | \u221224200 | \u221238887.5 | 93 | \u2212260.22 | 9.16 | 20.43  \nDX20_E0B | \u221224370 | \u221233510 | 83 | \u2212293.61 | 8.14 | 13.25  \nED0_E0B | \u22122128.13 | \u22122137.5 | 136 | \u221215.65 | 7.51 | 14.71  \nEMD0_E0B | \u221216730 | \u221238240 | 79 | \u2212211.77 | 8.75 | 24.05  \nES0_E0B | \u221216275 | \u221229087.5 | 69 | \u2212235.87 | 7.68 | 17.39  \nFC0_E0B | \u221227862.5 | \u221239787.5 | 105 | \u2212265.36 | 7.84 | 15.24  \nFV0_E0B | \u22129165.59 | \u221210417 | 96 | \u221295.47 | 8.9 | 21.88  \nGC20_E0B | 5020 | \u221242850 | 91 | 55.16 | 7.89 | 27.47  \nHG20_E0B | \u221221250 | \u221245787.5 | 86 | \u2212247.09 | 7.92 | 24.42  \nHO20_E0B | \u221270366.8 | \u221297671 | 92 | \u2212764.86 | 8.85 | 21.74  \nKC20_E0B | 21693.75 | \u221231125 | 74 | 293.16 | 9.41 | 32.43  \nKW20_E0B | \u22125025 | \u221218762.5 | 99 | \u221250.76 | 8.4 | 26.26  \nLB0_E0B | 10010 | \u22127667 | 100 | 100.1 | 8.27 | 28  \nLC0_E0B | \u22124850 | \u221211780 | 80 | \u221260.63 | 8.63 | 25  \nLH0_E0B | \u22122900 | \u221211160 | 83 | \u221234.94 | 8.13 | 25.3  \nMP0_E0B | 675 | \u22126430 | 90 | 7.5 | 8.31 | 25.56  \nNG20_E0B | 1830.01 | \u221242839.99 | 80 | 22.88 | 10.68 | 26.25  \nNK0_E0B | 625 | \u221213850 | 69 | 9.06 | 8.38 | 24.64  \nOJ20_E0B | 5070 | \u22127170 | 102 | 49.71 | 7.48 | 26.47  \nPL20_E0B | \u22124425 | \u221222415 | 86 | \u221251.45 | 9.62 | 26.74  \nRB0_E0B | \u221210894.8 | \u221256884.8 | 92 | \u2212118.42 | 6.02 | 32.61  \nS20_E0B | \u221213237.5 | \u221227787.5 | 93 | \u2212142.34 | 7.62 | 22.58  \nSB20_E0B | \u22127056 | \u221221649.6 | 94 | \u221275.06 | 7.5 | 23.4  \nSI20_E0B | \u221228240 | \u221278915 | 82 | \u2212344.39 | 10.57 | 28.05  \nTF0_E0B | 3750 | \u221221400 | 66 | 56.82 | 8.42 | 25.76  \nTU0_E0B | \u221213703.6 | \u221216806.2 | 95 | \u2212144.25 | 10.99 | 20  \nTY0_E0B | \u2212500.4 | \u221213046.8 | 88 | \u22125.69 | 8.68 | 26.14  \nUS0_E0B | 11843.4 | \u221220343.8 | 86 | 137.71 | 9 | 27.91  \nW20_E0B | \u221231037.5 | \u221234112.5 | 73 | \u2212425.17 | 8.7 | 16.44\n\nSignals can also be generated using the RSI by looking for divergences, failure swings, and centerline crossovers. Wilder suggests that divergence between an asset's price movement and the RSI oscillator can signal a potential reversal. This divergence occurs when the RSI continues in the opposite direction of the underlying asset: RSI trends down while price trends upward or RSI trends up while price trends down.\n\nA bullish divergence forms when the underlying asset makes a lower low and RSI makes a higher low. RSI diverges from the bearish price action in that it shows strengthening momentum, indicating a potential upward reversal in price. A bearish divergence forms when the underlying asset makes a higher high and RSI forms a lower high. RSI diverges from the bullish price action in that it shows weakening momentum, indicating a potential downward reversal in price. As with overbought and oversold levels, divergences are more likely to give false signals in the context of a strong trend.\n\nA TOP failure swing occurs when the RSI forms a pivot high in overbought territory and then pulls back and forms a pivot low and then another pivot high forms that is less than the original. A sell signal is triggered at the intervening RSI pivot low level. A BOTTOM failure is the inversion of the TOP; RSI forms a pivot low below the oversold value and then moves up and forms a pivot high and then a pivot low greater than the original pivot low. Figure 2.6 illustrates these failure swings.\n\n**Figure 2.6** Example of RSI divergence.\n\nAccording to Wilder, trading the failure swings might be more profitable than simply trading the RSI overbought\/oversold model. Let's see if this is the case by testing this idea. The RSI failure swing algorithm is going to be more complicated than our last RSI algorithm, but with the use of an FSM it is readily doable. Let's attack it from the short side only to start (Box 2.3).\n\n* * *\n\n### Box 2.3 Welles Wilders RSI Failure Swing System Description\n\nSellShort Setup: The 14-day RSI rises above 70, then pulls back below 70 and forms a low pivot. The RSI then rallies but does not take out the original pivot high level. A SellShort signal is generated once the RSI retraces to the pivot low level that was established between the two prior consecutive pivot highs. ( _Note_ : MOC stands for market on close.)\n\n  1. If position is short \n    1. Take Profit: If close < entry price \u2212 1 ATR BuyToCover MOC\n    2. Protective Stop: If close > entry price + 3 ATR BuyToCover MOC\n\n#### Welles Wilders RSI Failure Swing System p\u2013code\n\n    rsiVal = RSI(C,14)\n    If rsiVal[1] > rsiVal and rsiVal[1] > rsiVal[2] then\n         rsiPvtHiFound = true\n         rsiPvtHiVal = rsiVal[1]\n    else\n         rsiPvtHiFound = False\n    If rsiVal[1] < rsiVal and rsiVal[1] < rsiVal[2] then\n         rsiPvtLoFound = true\n         rsiPvtLoVal = rsiVal[1]\n    else\n         rsiPvtLoFound = False\n    If state = 0 then\n         if rsiPvtHiFound = true and rsiPvtHiVal > 70 then\n                 state = 1\n    If state = 1 then\n         state1Val = rsiPvtHiVal\n         if rsiVal > state1Val then state = 0\n         if rsiPvtLoFound = true then\n                 state = 2;\n    If state = 2 then\n         state2Val = rsiPvtLoVal\n         if rsiVal > state1Val then state = 0\n         if rsiPvtHiFound = true then\n                 if rsiPvtHiVal < 70 then state = 3\n    If state = 3 then\n         if rsiVal < state2Val then state = 4\n    If state = 4 then\n         sellShort this bar on close\n         state = 0;\n    If state > 0 and rsiVal < 30 then state = 0\n    If marketPosition =-1 and close > entryprice + 1* avgTrueRange(10) then\n      BuyToCover this bar on close\n    If marketPosition =-1 and close < entryPrice - 3* avgTrueRange(10) then\n      BuyToCover this bar on close;\n\n* * *\n\nFigure 2.7 illustrates the model of the RSI failure swing algorithm. As you can see from the diagram, this isn't a flowchart. It's our old friend the FSM, so let's move through the different states of the machine. We start off gobbling one bar at a time until we come across a RSI pivot high with a value of 70 or greater. Once this occurs, the machine goes into **State 1** and starts gobbling more bars. The machine can only transition out of **State 1** if one of three events occur: an occurrence of a RSI pivot low, an RSI value > the original RSI pivot high value, or an RSI value that is less than 30. Only an occurrence of an RSI pivot low can propel the machine to **State 2**. The other two conditions will reverse the machine to the **Start** state. Once in **State 2** , the machine looks for yet another RSI pivot high. In addition to a new pivot high, another condition must be met before moving onto **State 3** , and that is the RSI pivot high must be less than the original RSI pivot high. The same conditions that reset the machine to the **Start** state exist at **State 2**. After the machine attains **State 3** , it then tries to propel itself to **State 4** by finding an RSI value less than the RSI pivot low found in **State 1**. If this occurs, an order to sell short the market is placed on an MOC basis. Again, the exact same criteria for reversing the machine back to the **Start** state are in force in **State 3**.\n\n**Figure 2.7** Example of FSM modeling RSI Swing Failure.\n\nTable 2.3 shows the results of the RSI Failure Swing Algorithm from both the long and short side utilizing the same profit objective and stop used in the first RSI test.\n\n**Table 2.3** Performance of RSI Failure Swing Algorithm\n\n**Ticker** | **Net Profit** | **# Trades** | **Avg P\/L** | **Avg Bars Held**  \n---|---|---|---|---  \n@KC | $11,381.25 | 45 | $252.92 | 16.58 days  \n@SB | \u2212$8,162.60 | 40 | \u2212$198.40 | 15.13 days  \n@ES | \u2212$20,375.00 | 39 | \u2212$522.44 | 12.44 days  \n@DX | \u2212$23,245.00 | 44 | \u2212$528.30 | 13.89 days  \n@C | $6,312.50 | 40 | $157.81 | 13.38 days  \n@BP | \u2212$1,225.00 | 38 | \u2212$32.24 | 18.66 days  \n@CT | \u2212$21,485.00 | 47 | \u2212$456.60 | 13.83 days  \n@OJ | $7,115.00 | 53 | $134.25 | 18.49 days  \n@CC | $18,785.00 | 55 | $324.00 | 14.36 days  \n@HG | \u2212$14,250.00 | 51 | \u2212$279.41 | 17.8 days  \n@EC | \u2212$35,550.00 | 43 | \u2212$826.74 | 17.33 days  \n@HO | \u2212$58,426.40 | 52 | \u2212$1,123.58 | 12.42 days  \n@TU | \u2212$7,850.00 | 42 | \u2212$186.90 | 16.26 days  \n@US | \u2212$11,568.75 | 42 | \u2212$275.45 | 17.62 days  \n@FV | \u2212$4,939.06 | 43 | \u2212$114.86 | 13.86 days  \n@ED | $3,025.00 | 44 | $68.75 | 18.18 days  \n@TY | \u2212$6,540.63 | 43 | \u2212$152.11 | 14.21 days  \n@CL | \u2212$45,050.00 | 52 | \u2212$866.35 | 12.25 days  \n@RB | \u2212$51,377.40 | 45 | \u2212$1,141.72 | 17.24 days  \n@PL | $7,980.00 | 41 | $194.63 | 13.83 days  \n@GC | $7,570.00 | 46 | $164.57 | 18.26 days  \n@AD | \u2212$27,980.00 | 44 | \u2212$635.91 | 14.93 days  \n@SI | \u2212$9,300.00 | 53 | \u2212$175.47 | 12.34 days  \n@NG | $28,730.00 | 38 | $756.05 | 15.76 days  \n@EMD | \u2212$20,780.00 | 43 | \u2212$483.26 | 13.05 days  \n@TF | $16,760.00 | 40 | $419.00 | 15.35 days  \n@S | $10,325.00 | 38 | $271.71 | 14.18 days  \n@W | \u2212$3,112.50 | 35 | \u2212$88.93 | 14.71 days  \n@LH | $15,190.00 | 41 | $370.49 | 16.41 days  \n@LC | \u2212$21,160.00 | 29 | \u2212$729.66 | 12.9 days  \n@FC | \u2212$25,212.50 | 37 | \u2212$681.42 | 12.92 days  \n@KW | $8,925.00 | 24 | $371.88 | 14.42 days\n\n### George Lane's Stochastics\u2014Fast %K: Fast %D: Slow %K: Slow %D\n\nDeveloped by George C. Lane in the late 1950s, the stochastic oscillator, like the RSI, is a momentum indicator that shows the location of the close relative to the high\u2013low range over a set number of periods. Mr. Lane stated in an interview the following concerning the oscillator: \"It doesn't follow price, it doesn't follow volume or anything like that. It follows the speed or the momentum of price. As a rule, the momentum changes direction before price.\" The word _stochastic_ is defined as a random pattern that may be analyzed statistically but may not be predicted precisely. An appropriate name, if you ask me. In similar fashion to the RSI, bullish and bearish divergences in the stochastic oscillator can be used to foreshadow reversals. In other words, if the stochastic says one thing but the market says another, then be on the lookout for a market reversal. This is very useful information from a technician's point of view but very little for the quant trader looking for a very precise entry definition. However, because the stochastic oscillator is range bound, it is also useful for identifying overbought and oversold levels.\n\n**Calculation for a 14-day stochastic [FAST]:**\n\n  1. Going back in time 14 bars, calculate the highest high and lowest low prices during this period.\n  2. Subtract the lowest low from the current closing price and divide the difference by the range between the highest high and lowest low. Multiply this quotient by 100 and you will arrive at %K, or fast K.\n\n  3. Fast %D is a simple three-period moving average of the fast %K.\n\nIf the current close is near the lowest low, then the numerator in the %K formula will be small and will produce a low stochastic reading. If the close is near the highest high, then the numerator will be large. A strong %K indicates a strong uptrend, whereas a weak %K indicates the opposite, a strong downtrend. Both %D and %K are plotted alongside each other with the %K line acting as a signal or trigger line. The slow variation of this indicator is simply a further smoothing of the %K and %D. In the slow framework the fast %D becomes the slow %K and the slow %D becomes a three-period moving average of the slow %K. The two versions can be confusing so most people simply use the slow variety. Through my obeservations the fast stochastic seems to be very erratic.\n\nThe default parameters for the slow stochastic are (14, 3, 3) where the 14 represents the raw %K calculation period, the first 3 is the initial smoothing length, and the second 3 is the secondary smoothing.\n\nAs you can see from the %K formula, the numerator will always be less than or equal to the denominator and therefore produce a reading between 0 and 100. If the close is near the lowest low, then the formula will produce a low stochastic.\n\nMost traders utilize the stochastic (STO) as an overbought\/oversold indicator with 80 and 20 as the corresponding thresholds. If the STO is high, then the momentum is very bullish, and if it's low\u2014well, it's pretty obvious. Just like the RSI, the STO can be faked out into believing a market has become overbought\/sold when in fact the market is in a very strong trend. A unique feature of the STO (and some other oscillators) is that it incorporates a _trigger_. Instead of simply buying\/selling in the oversold\/overbought territory a trader can wait until the slow %K crosses above\/below the slow %D. Remember, these calculations are simple moving averages of the raw %K and when a shorter-term average crosses the longer term, this indicates a short-term shift in momentum.\n\nFigure 2.8 shows trades where the slow %K crosses slow %D in overbought and oversold territories.\n\n**Figure 2.8** Chart showing trade signals from stochastic crossover algorithm.\n\nSource: TradeStation\n\nEntry signals for a stochastic oscillator crossover algorithm can be defined by simple if\u2013then constructs. So a simple flowchart can be used to model this type of trading algorithm. The flowchart is shown in Figure 2.9.\n\n**Figure 2.9** The flowchart for an algorithm using a stochastic oscillator crossover as an entry signal.\n\nBox 2.4 gives a description of a simplistic version of a stochastic crossover system utilizing the same profit-and-loss objectives as the RSI algorithms. The results of this stochastic algorithm applied to several markets are shown in Table 2.4.\n\n**Table 2.4** Performance of Stochastic Algorithm\n\n**Ticker** | **Net Profita** | **Max. Sys DD** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **% of Winners**  \n---|---|---|---|---|---|---  \nAD0_E0B | \u221234820 | \u221250330 | 356 | \u221297.81 | 7.6  | 26.69  \nBP0_E0B | 41213 | \u221240563 | 324 | 127.2 | 8.42 | 31.17  \nC20_E0B | 8400 | \u221215675 | 358 |  23.46 | 7.58 | 29.89  \nCC20_E0B | 44010 | \u22128170 | 341 | 129.06 | 8.21 | 35.19  \nCL20_E0B | \u221254320 | \u2212115190 | 346 | \u2212156.99 | 7.92 | 27.46  \nCT20_E0B | \u22125770 | \u221251165 | 362 | \u221215.94 | 7.81 | 30.11  \nCU0_E0B | \u2212287 | \u221253163 | 348 |  \u22120.83 | 8.01 | 27.87  \nDX20_E0B | 4120 | \u221220390 | 362 |  11.38 | 7.33 | 27.9  \nED0_E0B | \u22124066 | \u22124103 | 255 | \u221215.94 | 8.93 | 21.96  \nEMD0_E0B | \u221253730 | \u221289300 | 350 | \u2212153.51 | 7.29 | 24.86  \nES0_E0B | \u221252200 | \u221275975 | 359 | \u2212145.4 | 7.15 | 25.07  \nFC0_E0B | \u221217425 | \u221232575 | 369 | \u221247.22 | 7.41 | 26.83  \nFV0_E0B | \u221217699 | \u221225463 | 345 | \u221251.3 | 7.66 | 25.8  \nGC20_E0B | \u221227470 | \u221267650 | 353 | \u221277.82 | 7.54 | 29.18  \nHG20_E0B | \u221248938 | \u2212106225 | 343 | \u2212142.67 | 7.65 | 26.82  \nHO20_E0B | 36343 | \u221296319 | 340 | 106.89 | 8.07 | 30  \nKC20_E0B | 16838 | \u221230375 | 334 | 50.41 | 8.26 | 28.74  \nKW20_E0B | \u2212188 | \u221241688 | 350 |  \u22120.54 | 7.67 | 30.86  \nLB0_E0B | 5302 | \u221234562 | 418 |  12.68 | 6.53 | 28.95  \nLC0_E0B | 42500 | \u22128750 | 344 | 123.55 | 7.98 | 34.3  \nLH0_E0B | \u221214110 | \u221217640 | 363 | \u221238.87 | 7.2  | 26.17  \nMP0_E0B | 1730 | \u221210580 | 297 |  5.82 | 8.34 | 28.96  \nNG20_E0B | \u2212193860 | \u2212199740 | 336 | \u2212576.96 | 7.95 | 25.3  \nNK0_E0B | 17300 | \u221238025 | 375 |  46.13 | 7.34 | 30.4  \nOJ20_E0B | 34275 | \u22128055 | 321 | 106.78 | 8.3  | 32.4  \nPL20_E0B | 25415 | \u221240470 | 341 |  74.53 | 7.91 | 30.5  \nRB0_E0B | \u221248266 | \u2212101884 | 406 | \u2212118.88 | 5.96 | 28.33  \nS20_E0B | \u221248888 | \u221261063 | 367 | \u2212133.21 | 7.18 | 26.16  \nSB20_E0B | \u22127045 | \u221233309 | 380 | \u221218.54 | 6.9  | 28.68  \nSI20_E0B | \u2212122305 | \u2212186355 | 323 | \u2212378.65 | 8.44 | 27.86  \nTF0_E0B | \u221248910 | \u221280540 | 333 | \u2212146.88 | 7.56 | 26.43  \nTU0_E0B | \u221221736 | \u221224441 | 309 | \u221270.34 | 8.3  | 24.92  \nTY0_E0B | \u221237139 | \u221254734 | 350 | \u2212106.11 | 7.52 | 25.71  \nUS0_E0B | \u221274827 | \u221274859 | 359 | \u2212208.43 | 7.67 | 25.07  \nW20_E0B | \u221227838 | \u221243613 | 351 | \u221279.31 | 7.98 | 29.06\n\n* * *\n\n### Box 2.4 George Lane's Slow Stochastic System Description\n\n  1. If 14-day slow %K crosses below slow %D and slow %D is greater than 80, then \n    1. Buy MOC\n  2. If 14-day slow %K crosses above slow %D and slow %D is less than 20, then \n    1. Sell Short MOC\n  3. If position is long \n    1. Take Profit: If close > entry price + 3 ATR Sell MOC\n    2. Protective Stop: If close < entry price \u2212 1 ATR Sell MOC\n  4. If position is short \n    1. Take Profit: If close < entry price \u2212 3 ATR BuyToCover MOC\n    2. Protective Stop: If close > entry price + 1 ATR, BuyToCover MOC\n\nNote that MOC stands for market on close.\n\n#### _George Lane's Slow Stochastic System p\u2013code_\n\n    `Most software will provide a function that\n    `provides all the components of the stochastic\n    Value1 = stochastic(h,l,c,rawKLen,smooth1,smooth2,1,myFastK,myFastD,mySlowK,mySlowD)\n    If mySlowK crosses above mySlowD and mySlowD < 20 then buy this bar on close\n    If mySlowK crosses below mySlowD and mySlowD > 80 then sellShort this bar on close\n    If marketPosition = 1 then\n         If c > entryPrice + 3* avgTrueRange(10) then sell this bar on close\n         if c < entryPrice - 1* avgTrueRange(10) then sell this bar on close\n    If marketPosition =-1 then\n         If c < entryPrice - 3* avgTrueRange(10) then buyToCover this bar on close\n         if c > entryPrice + 1* avgTrueRange(10) then buyToCover this bar on close\n\n* * *\n\n### Donald Lambert's CCI\u2014Commodity Channel Index\n\nDonald Lambert developed the CCI in 1980, and it was featured in _Commodities_ magazine that same year. The Commodity Channel Index can be used in the same vein as RSI and stochastic: trend determination or warning of extreme conditions. Lambert originally developed CCI to identify cyclical turns in commodities, but the indicator can be successfully applied to indices, ETFs, stocks, and other securities. In general, CCI measures the current price level relative to an average price level over a given period of time. CCI is relatively high when prices are far above their average, and conversely, CCI is relatively low when prices are far below their average\u2014hence its application as reversion to the mean tool.\n\nThe main difference between CCI and other similar oscillators is that it is not necessarily range bound. Most of the time, the CCI will oscillate between two extremes, but as you will see in its calculations, there is not a fixed boundary. The CCI introduces the concept of typical price and utilizes it in its calculations. Typical price is just the average of a bar's high, low, and close: Typical price (TP) = (High + Low + Close)\/3. This calculation looks at the three most important price points on a price bar and many believe it reveals a more realistic picture of the day's action than just the close. The TP is also known as the daily pivot point. In the day, floor traders used this easily calculable price to determine current support and resistance levels.\n\n**Calculation for a 20-day CCI:**\n\n  1. Going back in time, calculate the average of the past 20 days' typical prices.\n  2. Calculate the one standard deviation of the average found in step 1. This is where a computer\/calculator comes in handy.\n  3. Divide the difference between today's TP and the 20-day average of TP by the product of the standard deviation times 0.015.\n\nLambert set the constant at 0.015 in the denominator to ensure that approximately 70 to 80 percent of CCI values would fall between \u2212100 and +100. Since the numerator can be positive or negative, so can the CCI. The standard deviation is inversely proportional to sample size; the larger the sample the smaller the deviation. Since the standard deviation is in the denominator, a small lookback period will cause a more volatile CCI. A shorter CCI (10 periods) will also produce a smaller percentage of values between +100 and \u2212100. Conversely, a longer CCI (40 periods) will have a higher percentage of values between +100 and \u2212100.\n\nSince this oscillator usually falls between +100 and \u2212100 and is momentum based, most traders will use it as an overbought\/oversold indicator. However, since it is boundless extended readings above +100 or below \u2212100 can indicate very strong trend moves.\n\nThe weakness of the CCI is its variability due to the volatile nature of the underlying instrument that is being applied to. A volatile stock will cause the indicator to behave erratically and jump well above\/below the standard +100 and \u2212100. These benchmarks may have to be expanded based on the market being analyzed. Many traders also apply a smoothing function to the CCI in an attempt to eliminate knee-jerk reactions.\n\nWe will be discussing two trading algorithms based on the CCI (Box 2.5). The first will utilize the CCI in the same manner as the RSI and stochastic. The second algorithm will use the CCI in a completely opposite manner\u2014trend confirmation.\n\n* * *\n\n### Box 2.5 Donald Lambert's CCI OB\/OS System Description\n\n  1. If 9-day average of the CCI 20-day crosses above \u2212100, then \n    1. Buy MOC\n  2. If 9-day average CCI 20-day crosses below + 100, then \n    1. Sell Short MOC\n  3. If position is long \n    1. Take Profit: If close > entry price + 3 ATR Sell MOC\n    2. Protective Stop: If close < entry price \u2212 1 ATR Sell MOC\n  4. If position is short \n    1. Take Profit: If close < entry price \u2212 3 ATR BuyToCover MOC\n    2. Protective Stop: If close > entry price + 1 ATR, BuyToCover MOC\n\nNote that MOC indicates market on close.\n\n#### Donald Lambert's CCI OB\/OS System p\u2013code\n\n     `Donald Lambert CCI OB\/OS\n    `\n    myCCIVal = average(cci(cciLen),smooth);\n    If myCCIVal crosses above -100 then buy this bar on close\n    If myCCIVal crosses below 100 then sellShort this bar on close\n    If marketPosition = 1 then\n         If c > entryPrice + 3* avgTrueRange(10) then sell this bar on close\n         if c < entryPrice - 1* avgTrueRange(10) then sell this bar on close\n    If marketPosition =-1 then\n         If c < entryPrice - 3* avgTrueRange(10) then buyToCover this bar on close\n         if c > entryPrice + 1* avgTrueRange(10) then buyToCover this bar on close\n\n* * *\n\nThe description of this algorithm looks very similar to the RSI and stochastic overbought\/oversold, but there is one esoteric difference. Notice how the entries are initiated. Long entries require the CCI to cross above \u2212100, meaning that the CCI has been in the oversold territory for at least one bar. Just the opposite is true for short entries; CCI must cross below +100. This is different from the other algorithms in that entries are delayed until the indicator starts moving in the same direction as the desired market direction. This delay is a double-edged sword in that you might get confirmation of market direction but at the cost of worse price.\n\nMany of today's trading and testing platforms have the keywords CROSS ABOVE and CROSS BELOW built into the programming language. These keywords are simple shortcuts meant to make programming easier to understand and implement. Using the CCI indicator as an example, here are the equivalent computer codes to the keywords:\n\n  1. **CROSS ABOVE** \u2212100 : CCI[1] < \u2212100 and CCI[0] > \u2212100\n  2. **CROSS BELOW** +100 : CCI[1] > +100 and CCI[0] < +100\n\nRemember the values inside the square brackets indicate the number of bars\/days in the past. In later chapters, where different platforms are discussed, you will see these keywords. Figure 2.10 shows some trades generated by this first CCI algorithm.\n\n**Figure 2.10** Trades generated by the CCI algorithm when it crosses above 100 and below \u2212100.\n\nSource: TradeStation\n\nSince the flowchart for this algorithm is so similar to the RSI and stochastic, it will not be shown for brevity's sake. However, the all-important performance metrics are shown in Table 2.5.\n\n**Table 2.5** Performance of CCI algorithm\n\n**Ticker** | **Net Profit** | **Max. Sys DD** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **% of Winners**  \n---|---|---|---|---|---|---  \nAD0_E0B | \u221238040.0 | \u221247030 | 156 | \u2212243.85 | 10.9 | 28.85  \nBP0_E0B | 7962.5 | \u221259875 | 144 | 55.3 | 12.38 | 35.42  \nC20_E0B | \u22122975.0 | \u221211538 | 151 | \u221219.7 | 9.61 | 30.46  \nCC20_E0B | 27820.0 | \u22128400 | 147 | 189.25 | 11.15 | 38.1  \nCL20_E0B | \u221222830.0 | \u221289180 | 154 | \u2212148.25 | 10.72 | 35.71  \nCT20_E0B | \u22127615.0 | \u221244780 | 145 | \u221252.52 | 10.82 | 35.86  \nCU0_E0B | \u221210900.0 | \u221241075 | 147 | \u221274.15 | 11.96 | 31.97  \nDX20_E0B | \u221237060.0 | \u221237110 | 148 | \u2212250.41 | 10.56 | 27.7  \nED0_E0B | \u2212353.1 | \u22121847 | 138 | \u22122.56 | 14.39 | 34.06  \nEMD0_E0B | \u221227880.0 | \u221247890 | 146 | \u2212190.96 | 10.93 | 29.45  \nES0_E0B | \u221234525.0 | \u221256038 | 152 | \u2212227.14 | 10.18 | 29.61  \nFC0_E0B | \u22121225.0 | \u221225213 | 146 | \u22128.39 | 10.16 | 32.19  \nFV0_E0B | \u22125656.1 | \u221211047 | 146 | \u221238.74 | 10.96 | 31.51  \nGC20_E0B | \u221226150.0 | \u221249330 | 152 | \u2212172.04 | 11.16 | 32.24  \nHG20_E0B | \u221234350.0 | \u221253500 | 129 | \u2212266.28 | 12.94 | 34.11  \nHO20_E0B | \u221253932.2 | \u2212112518 | 157 | \u2212343.52 | 10.13 | 31.85  \nKC20_E0B | \u221232737.5 | \u221254263 | 149 | \u2212219.71 | 11.17 | 29.53  \nKW20_E0B | 10325.0 | \u221224513 | 154 | 67.05 | 10.6 | 38.31  \nLB0_E0B | 21406.0 | \u221214058 | 150 | 142.71 | 9.41 | 36.67  \nLC0_E0B | \u221231750.0 | \u221242120 | 145 | \u2212218.97 | 10.19 | 24.14  \nLH0_E0B | \u2212860.0 | \u221213530 | 152 | 38.55 | 10.74 | 35.53  \nMP0_E0B | \u2212565.0 | \u22129145 | 146 | \u22123.87 | 12.99 | 36.99  \nNG20_E0B | \u221246040.0 | \u221281490 | 145 | \u2212317.52 | 10.99 | 29.66  \nNK0_E0B | \u221217900.0 | \u221254000 | 144 | \u2212124.31 | 9.93 | 29.86  \nOJ20_E0B | 26655.0 | \u22129105 | 159 | 167.64 | 11.19 | 39.62  \nPL20_E0B | 14920.0 | \u221240290 | 147 | 101.5 | 9.9 | 34.01  \nRB0_E0B | \u2212155710.8 | \u2212163069 | 155 | \u22121004.59 | 6.61 | 21.29  \nS20_E0B | 0.0 | 0 | 0 | N\/A | N\/A | N\/A  \nSB20_E0B | \u22127604.8 | \u221225458 | 142 | \u221253.55 | 10.7 | 35.92  \nSI20_E0B | 77875.0 | \u221239745 | 155 | 502.42 | 11.57 | 39.35  \nTF0_E0B | \u221223300.0 | \u221247340 | 146 | \u2212159.59 | 9.36 | 30.14  \nTU0_E0B | \u22125704.0 | \u221211907 | 143 | \u221239.89 | 11 | 34.97  \nTY0_E0B | \u221230233.9 | \u221234125 | 140 | \u2212215.96 | 10.73 | 25  \nUS0_E0B | \u221263453.1 | \u221271422 | 151 | \u2212420.22 | 10.3 | 25.83  \nW20_E0B | 0.0 | 0 | 0 | N\/A | N\/A | N\/A\n\nNow, switching gears and using the CCI as a coincident indicator, Box 2.6 provides the description of a trend-following algorithm. A coincident indicator simply means it is confirming the current market direction.\n\n* * *\n\n### Box 2.6 Donald Lambert's CCI Trend-Following System\n\nmyCCIVal = 9-day average of the 20-day CCI\n\n  1. If the lowest myCCIVal for the past 3 days > 100, then \n    1. Buy MOC\n  2. If the highest myCCIVal for the past 3 days < \u2212100, then \n    1. Sell Short MOC\n  3. If position is long \n    1. Take Profit: If close > entry price + 5 ATR Sell MOC\n    2. Protective Stop: If close < entry price \u2212 3 ATR, Sell MOC\n  4. If position is short \n    1. Take Profit: If close < entry price \u2212 5 ATR, BuyToCover MOC\n    2. Protective Stop: If close > entry price + 3 ATR, BuyToCover MOC\n\nNote that MOC stands for market on close.\n\n#### Donald Lambert's CCI Trend-Following System p \u2212 code\n\n    myCCIVal = average(cci(cciLen),smooth);\n    If lowest(myCCIVal,3) > 100 then buy this bar on close;\n    If highest(myCCiVal,3) < -100 then sellShort this bar on close;\n    If marketPosition = 1 then\n         If c > entryPrice + 5* avgTrueRange(10) then sell this bar on close\n         if c < entryPrice - 3* avgTrueRange(10) then sell this bar on close\n    If marketPosition =-1 then\n         If c < entryPrice - 5* avgTrueRange(10) then buyToCover this bar on close\n         if c > entryPrice + 3* avgTrueRange(10) then buyToCover this bar on close\n\n* * *\n\nThis trading algorithm description is somewhat different from the others we have thus discussed in this chapter. Instead of repeating the words for the 9-day average of the 20-day CCI, I simply assigned it to the variable myCCIVal. This is just like you would do in an actual programming language. The words _lowest_ and _highest_ were also introduced and used as a shortcut to see if the myCCIVal variable was above or below a certain level for the past three days. If the lowest myCCIVal for the past three days is greater than 100, then we know that the CCI did not dip below 100 during that time period. Trend detection occurs when the CCI stays in an extreme territory for several bars. This system is triggering a buy signal when the 9-day smoothed CCI stays above 100 for three days straight. The sell signal is triggered if the indicator stays below \u2212100 for three days straight. Since we are dealing with a trend-following approach, the profit objective as well as the stop loss has been increased to five and three times the average true range, respectively. As you can see, the instructions for entries are contained within a very precise if\u2013then construct and therefore can be represented by a flowchart. Again, this flowchart is so similar to the RSI, stochastic, and CCI #1 flowcharts that it would be terribly redundant to show it. Figure 2.11 does show how the system is more than capable of capturing the recent downturn in the Eurocurrency.\n\n**Figure 2.11** The CCI system detecting a downturn in the Eurocurrency.\n\nSource: TradeStation\n\nTable 2.6 shows the performance of the CCI Trend-Follower algorithm.\n\n**Table 2.6** Performance of CCI Algorithm as Coincident Indicator\n\n**Ticker** | **Net Profit** | **Max. Sys DD** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **% of Winners**  \n---|---|---|---|---|---|---  \nAD0_E0B | 36660.0  | \u221230990 | 105 | 349.14 | 32.94 | 42.86  \nBP0_E0B |  \u221235562.5 | \u221295325 | 106 | \u2212335.5  | 32.35 | 34.91  \nC20_E0B |  \u221212000.0 | \u221236738 | 104 | \u2212115.38 | 31.46 | 35.58  \nCC20_E0B |  \u221251890.0 | \u221259700 | 112 | \u2212463.3  | 29.7  | 29.46  \nCL20_E0B |  55630.0 | \u221271870 | 111 | 501.17 | 31.23 | 39.64  \nCT20_E0B |  \u221216015.0 | \u221239350 | 124 | \u2212129.15 | 27.79 | 37.1   \nCU0_E0B |  25037.5 | \u221272988 | 103 | 243.08 | 32.71 | 41.75  \nDX20_E0B |  40065.0 | \u221223940 | 106 | 377.97 | 33.84 | 43.4   \nED0_E0B |  6471.9 | \u22122294 | 95 |  68.13 | 36.76 | 48.42  \nEMD0_E0B |  18790.0 | \u221256220 | 101 | 186.04 | 34.5  | 42.57  \nES0_E0B |  23250.0 | \u221238975 | 96 | 242.19 | 36.84 | 39.58  \nFC0_E0B |  37612.5 | \u221222688 | 114 | 329.93 | 28.22 | 45.61  \nFV0_E0B |  26687.9 | \u221213273 | 104 | 256.61 | 34.76 | 41.35  \nGC20_E0B |  \u221218150.0 | \u221276330 | 102 | \u2212177.94 | 35.28 | 39.22  \nHG20_E0B |  105212.5 | \u221252388 | 97 | 1084.66 | 35.79 | 47.42  \nHO20_E0B |  15036.0 | \u2212128789 | 107 | 140.52 | 31.31 | 36.45  \nKC20_E0B |  15881.3 | \u221261125 | 103 | 154.19 | 32.87 | 34.95  \nKW20_E0B |  \u221226350.0 | \u221250213 | 116 | \u2212227.16 | 28.53 | 32.76  \nLB0_E0B |  \u22127458.0 | \u221240667 | 126 | \u221259.19 | 25.39 | 39.68  \nLC0_E0B |  \u22123700.0 | \u221222240 | 100 | \u221237 | 33.59 | 39  \nLH0_E0B | \u22124210.0  | \u221224600 | 110 | \u221238.27 | 31.46 | 36.36  \nMP0_E0B |  2385.0 | \u221212575 | 101 |  23.61 | 34.02 | 35.64  \nNG20_E0B |  84340.0 | \u2212118220 | 96 | 878.54 | 34.81 | 44.79  \nNK0_E0B |  29625.0 | \u221243025 | 102 | 290.44 | 32 | 41.18  \nOJ20_E0B |  \u221223430.0 | \u221231785 | 111 | \u2212211.08 | 29.85 | 34.23  \nPL20_E0B |  \u221219845.0 | \u221285060 | 116 | \u2212171.08 | 28.65 | 37.93  \nRB0_E0B |  78246.0 | \u221291774 | 118 | 663.1  | 25.95 | 43.22  \nS20_E0B |  0.0 | 0 | 0 | N\/A | N\/A | N\/A  \nSB20_E0B | 23598.4 | \u221216733 | 108 | 218.5  | 30.12 | 42.59  \nSI20_E0B |  \u221297730.0 | \u2212186135 | 113 | \u2212864.87 | 31.5  | 34.51  \nTF0_E0B |  \u221230550.0 | \u221255750 | 92 | \u2212332.07 | 35.45 | 36.96  \nTU0_E0B |  21468.0 | \u22128188 | 93 | 230.84 | 38.05 | 38.71  \nTY0_E0B |  6578.6 | \u221222531 | 98 |  67.13 | 35.56 | 36.73  \nUS0_E0B |  40874.3 | \u221228375 | 99 | 412.87 | 35.38 | 41.41  \nW20_E0B |  0.0 | 0 | 0 | N\/A | N\/A | N\/A\n\n### Gerald Appel's MACD\u2014Moving Average Convergence Divergence\n\nThe Moving Average Convergence Divergence (MACD) indicator was created in the late 1970s by Gerald Appel. Basically, the indicator plots the difference between a fast and slow exponential moving average. The difference line is usually plotted oscillating around a zero line. Another line representing a smoothed version of the difference line is plotted as well. The smoothed version of the difference between the two moving averages is considered the trigger. Action is usually taken when the trigger crosses the difference line. In addition, it is common to see a histogram representation of the relationship between the difference line and the trigger. The MACD is a quick snapshot summary of the relationship between two different length-moving averages.\n\nWe will discuss moving averages in the last part of this chapter, but a little knowledge of averages is necessary to fully grasp the MACD. A trend change is usually indicated when a shorter (more sensitive) average crosses above\/below a longer (less sensitive) average. Unfortunately, moving average crossovers lag current market conditions, and here is where the real beauty of the MACD is revealed. The histogram component can quickly reveal when the difference between the two moving averages (MACD) and the smoothed version of MACD (trigger) is converging or diverging. Convergence means the MACD line and its associated trigger are coming together and divergence means simply the two are growing apart. As prices of an instrument increase, the short-term moving average will grow quicker than a longer-term average. The MACD histogram will reflect this momentum. As prices start to slow, the histogram will start moving in the opposite direction.\n\n**Calculation of MACD using 12-day and 26-day moving averages:**\n\n  1. Calculate a 12-period EMA of price for the chosen time period.\n  2. Calculate a 26-period EMA of price for the chosen time period.\n  3. Subtract the 26-period EMA from the 12-period EMA.\n  4. Calculate a 9-period EMA of the result obtained from step 3.\n\nThis 9-period EMA line is overlaid on a histogram that is created by subtracting the 9-period EMA from the result in step 3, which is called the MACD line, but it is not always visibly plotted on the MACD representation on a chart.\n\nA trading algorithm can be derived by waiting for the histogram to form a two-bar high pivot (two bars to the left and to the right are shorter than the center bar) and then entering a short position. Momentum has been positive but it is beginning to slow. A long position can be initiated when a two-bar pivot low is formed in the MACD histogram. Figure 2.12 shows how the pivot points in the histogram can initiate trade signals.\n\n**Figure 2.12** Histogram pivot points can initiate trade signals.\n\nBased on our knowledge of the MACD and the chart examples, Box 2.7 shows a simple MACD histogram-trading algorithm.\n\n* * *\n\n### Box 2.7 Gerald Appel's MACD Histogram System\n\nNotice how the algorithm description in this example is getting closer and closer to pseudocode. As you evolve as a system tester, you will notice how the lines that separate description, pseudocode, and actual code start to blur. LeftBar2 is the second bar to the left of the pivot high\/low bar and LeftBar1 is the first bar to the left of the pivot. The other variable names should be self-explanatory.\n\n  1. myMACD = MACD(C,12,26)\n  2. myMACDAvg = Xaverage(MACD,9)\n  3. myMACDDiff = myMACD \u2212 myMACDAvg\n  4. leftBar2 = myMACDDiff[4]\n  5. leftBar1 = myMACDDiff[3]\n  6. centerBar = myMACDiff[2]\n  7. rightBar1 = myMACDDiff[1]\n  8. rightBar2 = myMACDDiff[0]\n  9. If leftBar2 < 0 and rightBar2 < 0 and centerBar < leftBar1 and centerBar < leftBar2 and centerBar < rightBar1 and centerBar < rightBar2, then \n    1. Buy MOC\n  10. If leftBar2 > 0 and rightBar2 > 0 and centerBar > leftBar1 and centerBar > leftBar2 and centerBar > rightBar1 and centerBar > rightBar2, then \n    1. SellShort MOC\n  11. If position is long \n    1. Take Profit: If close > entry price + 3 ATR Sell MOC\n    2. Protective Stop: If close < entry price \u2212 1 ATR Sell MOC\n  12. If position is short \n    1. Take Profit: If close < entry price \u2212 3 ATR BuyToCover MOC\n    2. Protective Stop: If close > entry price + 1 ATR, BuyToCover MOC\n    3. *MOC = market on close\n\n#### Gerald Appel's MACD Histogram System p \u2013 code\n\n    `Gerald Appels MACD histogram algorithm\n    `Buys \u2013 histogram forms a low 2 bar pivot with all bars below 0\n    `Shorts \u2013 histogram forms a high 2 bar pivot with all bars above 0\n    MACDAvg = XAverage( MyMACD,9)\n    MACDDiff = myMACD \u2013 MACDAvg\n    leftBar2 = MACDDiff[4]\n    leftBar1 = MACDDiff[3]\n    centerBar = MACDDiff[2]\n    rightBar1 = MACDDiff[1]\n    rightBar2 = MACDDiff[0]\n    If leftBar2 < 0 and rightBar2 < 0 and centerBar < leftBar1 and centerBar < leftBar2 and\n         centerBar < rightBar1 and centerBar < rightBar2 then\n         Buy this bar on close\n    If leftBar2 > 0 and rightBar2 > 0 and centerBar > leftBar1 and centerBar > leftBar2 and\n         centerBar > rightBar1 and centerBar > rightBar2 then\n         SellShort this bar on close\n    If marketPosition = 1 then\n         If c > entryPrice + 3* avgTrueRange(10) then sell this bar on close\n         if c < entryPrice - 1* avgTrueRange(10) then sell this bar on close\n    If marketPosition =-1 then\n         If c < entryPrice - 3* avgTrueRange(10) then buyToCover this bar on close\n         if c > entryPrice + 1* avgTrueRange(10) then buyToCover this bar on close\n\n* * *\n\nThe MACD can be classified as an absolute price oscillator (APO), due to the fact it deals with the actual prices of moving averages rather than percentage changes. This makes the MACD different from the other indicators we have thus discussed. A percentage price oscillator (PPO) computes the difference between two moving averages of price divided by the longer moving average value.\n\nWhile an APO will show greater levels for higher-priced securities and smaller levels for lower-priced securities, a PPO calculates changes relative to price. Subsequently, a PPO is preferred when comparing oscillator values between different types of trading instruments, especially those with substantially different prices, or comparing oscillator values for the same instrument at significantly different times. The manner in which we developed the MACD algorithm does not care about absolute prices because it looks at the difference between the moving averages as either positive or negative; it does not care about the magnitude of difference.\n\n## Price-Based Indicators\n\nPrice-based indicators reflect trend and volatility. As we stated earlier, applying a moving average to an instrument's historical data will smooth out the data and soften wild market swings. In other words, a moving average eliminates a good portion of the noise associated with daily price changes.\n\n### Simple, Exponential, and Weighted Moving Averages\n\nThe most often used moving average is the simple version. Here, you sum up the closing prices over the past _N_ days and then divide by _N_. The exponential moving average calculation uses a smoothing factor to place a higher weight on recent data points and is regarded as much more efficient than the linear weighted average. Having an understanding of the calculations is not generally required for most traders because most charting packages do the calculation for you. But if you are interested in the calculations, here are the steps involved in calculating the exponential and weighted moving averages:\n\n  1. Seed the exponential average initially with a simple moving average.\n  2. Calculate the smoothing factor: multiplier = 2 \/ (EMA length + 1).\n  3. EMA[0] = (close[0] \u2212 EMA[1]) * multiplier + EMA[1].\n\nThe key to the EMA calculation is the smoothing factor multiplier. If we were calculating a 20-day EMA, the multiplier would be equal to 2 \/ 21, or 0.0952. In this example, a weighting of nearly 10 percent is applied to the most recent data point. A weighted moving average is slightly more complicated to calculate. Sticking with 20 periods, here are the steps involved in the calculation:\n\n  1. Sum up the values of the length of moving average : 20 + 19 + 18 + 17 +16 +15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 +1 = 210.\n  2. Multiply the oldest price by 1 \/20 and second oldest by 2\/20 and third oldest by 3\/20 and so on.\n  3. Sum up the products and you will have a weighted moving average. As you can see, the most recent data is weighted the most.\n\nThe weighted moving average addresses the problem of equal weighting. Figure 2.13 shows the three different 20-day moving averages across several days of data.\n\n**Figure 2.13** Three different 20-day moving averages.\n\nSource: TradeStation\n\nSo which moving average is the best? It depends on the situation and the data that is being analyzed. As a pure price\/average crossover-trading algorithm the following table might provide some insight. An 80-day period of each moving average was analyzed across 35 futures markets over a test period of 30 years and the results are shown in Table 2.7.\n\n**Table 2.7** Examples of Trades Generated by Stochastics\n\n**Ticker** | **Simple Moving Average** | **Exponential Moving Average** | **Weighted Moving Average**  \n---|---|---|---  \n**Net Profit** | **MAX. Sys. DD** | **Trds** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners** | **Net Profit** | **MAX. Sys. DD** | **Trds** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners** | **Net Profit** | **MAX. Sys.Drawdown** | **Trds** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners**  \nAD0_E0B | \u22122640 | \u221270090 | 404 | \u22126.53 | 18.62 | 15.59 | 7700 | \u221253380 | 446 |  17.26 | 16.96 | 16.14 | \u22123270 | \u221273090 | 493 |  \u22126.63 | 15.59 | 17.04  \nBP0_E0B | 74113 | \u2212179000 | 440 | 168.44 | 18.52 | 15.91 | 58013 | \u2212183950 | 488 | 118.88 | 16.8 | 16.6 | 40463 | \u2212203063 | 566 |  71.49 | 14.62 | 18.2  \nC20_E0B | 18213 | \u221226463 | 378 |  48.18 | 21.41 | 17.72 | \u221229388 | \u221246100 | 484 | \u221260.72 | 16.94 | 14.67 | \u221210388 | \u221232075 | 512 | \u221220.29 | 16.07 | 17.77  \nCC20_E0B | \u2212117190 | \u2212120040 | 487 | \u2212240.64 | 16.74 | 14.37 | \u2212108430 | \u2212111280 | 533 | \u2212203.43 | 15.38 | 15.38 | \u2212109730 | \u2212113060 | 575 | \u2212190.83 | 14.33 | 16  \nCL20_E0B | 171330 | \u221287180 | 375 | 456.88 | 21.48 | 19.47 | 126550 | \u221285620 | 425 | 297.76 | 19.07 | 19.76 | 148610 | \u221271240 | 519 | 286.34 | 15.8 | 20.81  \nCT20_E0B | 39545 | \u221248520 | 435 |  90.91 | 18.64 | 17.01 | 34695 | \u221239435 | 485 |  71.54 | 16.82 | 16.49 | 1405 | \u221268055 | 561 | 2.5 | 14.68 | 17.47  \nCU0_E0B | 90850 | \u221254625 | 401 | 226.56 | 20.23 | 18.95 | 94900 | \u221255238 | 437 | 217.16 | 18.65 | 18.76 | 120775 | \u221267600 | 477 | 253.2 | 17.17 | 20.75  \nDX20_E0B | 16685 | \u221249235 | 400 |  41.71 | 19.82 | 16.5 | 34655 | \u221227280 | 431 |  80.41 | 18.46 | 19.26 | 41255 | \u221230580 | 490 |  84.19 | 16.53 | 18.98  \nED0_E0B | \u221217053 | \u221219866 | 355 | \u221248.04 | 22.72 | 12.68 | \u221211034 | \u221214566 | 339 | \u221232.55 | 23.75 | 14.16 | \u221215378 | \u221219566 | 414 | \u221237.15 | 19.63 | 14.98  \nEMD0_E0B | 20970 | \u221234960 | 293 |  71.57 | 20.91 | 18.43 | 26900 | \u221232140 | 340 |  79.12 | 18.16 | 19.71 | \u221288920 | \u221289980 | 450 | \u2212197.6 | 14.14 | 17.56  \nES0_E0B | \u221250000 | \u221267588 | 470 | \u2212106.38 | 17.43 | 14.68 | \u221269200 | \u221270688 | 500 | \u2212138.4 | 16.44 | 15.4 | \u221275350 | \u221275613 | 578 | \u2212130.36 | 14.36 | 17.47  \nFC0_E0B | 49313 | \u221229350 | 358 | 137.74 | 22.56 | 18.16 | 83288 | \u221218063 | 374 | 222.69 | 21.64 | 22.73 | 73038 | \u221233000 | 480 | 152.16 | 17.08 | 20.21  \nFV0_E0B | \u22121805 | \u221228908 | 345 |  \u22125.23 | 20.61 | 15.07 | 17213 | \u221221032 | 381 |  45.18 | 18.76 | 17.32 | \u221215784 | \u221235951 | 479 | \u221232.95 | 15.29 | 16.28  \nGC20_E0B | \u2212114780 | \u2212125070 | 471 | \u2212243.69 | 17.32 | 12.95 | \u2212127580 | \u2212137870 | 565 | \u2212225.81 | 14.6 | 13.63 | \u2212126260 | \u2212137290 | 591 | \u2212213.64 | 14.01 | 14.89  \nHG20_E0B | 134138 | \u221239163 | 417 | 321.67 | 19.43 | 17.03 | 79013 | \u221258988 | 477 | 165.64 | 17.11 | 17.61 | 61863 | \u221274013 | 541 | 114.35 | 15.21 | 19.59  \nHO20_E0B | 22025 | \u2212154637 | 457 |  48.19 | 17.8 | 15.54 | 418 | \u2212152914 | 507 |  0.82 | 16.15 | 17.16 | 39118 | \u2212102509 | 571 |  68.51 | 14.45 | 18.04  \nKC20_E0B | 99019 | \u2212152544 | 405 | 244.49 | 19.92 | 17.04 | 45844 | \u2212167219 | 501 | 91.5 | 16.3 | 15.97 | 122619 | \u221293356 | 517 | 237.17 | 15.82 | 18.76  \nKW20_E0B | \u221216163 | \u221241075 | 465 | \u221234.76 | 17.59 | 15.7 | 2588 | \u221229713 | 495 |  5.23 | 16.59 | 16.77 | \u221242938 | \u221259900 | 591 | \u221272.65 | 14.06 | 16.24  \nLB0_E0B | 87685 | \u221226515 | 392 | 223.69 | 20.69 | 20.92 | 68033 | \u221223342 | 442 | 153.92 | 18.46 | 20.14 | 70031 | \u221226345 | 534 | 131.14 | 15.46 | 21.35  \nLC0_E0B | \u221225000 | \u221241790 | 398 | \u221262.81 | 20.39 | 16.33 | \u221233400 | \u221249230 | 464 | \u221271.98 | 17.64 | 17.03 | \u221250560 | \u221266380 | 566 | \u221289.33 | 14.64 | 17.67  \nLH0_E0B | 280 | \u221229220 | 394 |  0.71 | 20.59 | 18.02 | \u221210340 | \u221236880 | 484 | \u221221.36 | 16.95 | 18.39 | \u22127820 | \u221234840 | 520 | \u221215.04 | 15.84 | 19.42  \nMP0_E0B | 16770 | \u221222620 | 238 |  70.46 | 22.1 | 18.07 | 5320 | \u221226255 | 274 |  19.42 | 19.33 | 17.52 | 21125 | \u221224845 | 307 |  68.81 | 17.62 | 22.15  \nNG20_E0B | 238960 | \u221288090 | 306 | 780.92 | 21.52 | 20.26 | 161390 | \u2212116370 | 380 | 424.71 | 17.53 | 18.95 | 209010 | \u2212121120 | 394 | 530.48 | 17.14 | 19.04  \nNK0_E0B | 69875 | \u221268250 | 306 | 228.35 | 21.24 | 21.57 | 47275 | \u221275975 | 379 | 124.74 | 17.34 | 21.37 | \u221262625 | \u221272225 | 472 | \u2212132.68 | 14.29 | 19.07  \nOJ20_E0B | \u22128818 | \u221239850 | 386 | \u221222.84 | 20.87 | 17.1 | \u221232283 | \u221254738 | 448 | \u221272.06 | 18.12 | 17.86 | \u221225903 | \u221255323 | 530 | \u221248.87 | 15.47 | 18.11  \nPL20_E0B | 33180 | \u221260775 | 457 | 72.6 | 17.8 | 15.32 | 52050 | \u221264165 | 507 | 102.66 | 16.15 | 16.57 | \u22123540 | \u221261390 | 585 |  \u22126.05 | 14.13 | 17.09  \nRB0_E0B | 60850 | \u221268858 | 410 | 148.41 | 19.58 | 18.78 | 92216 | \u221287668 | 458 | 201.35 | 17.63 | 21.83 | 181639 | \u221261591 | 537 | 338.25 | 15.3 | 21.97  \nS20_E0B | \u221255025 | \u221267225 | 454 | \u2212121.2 | 17.99 | 15.42 | \u221239125 | \u221256238 | 528 | \u221274.1 | 15.61 | 15.91 | \u221216825 | \u221259150 | 560 | \u221230.04 | 14.78 | 17.14  \nSB20_E0B | 6410 | \u221233259 | 383 |  16.74 | 21.01 | 17.49 | \u221212279 | \u221242206 | 485 | \u221225.32 | 16.8 | 14.64 | 5026 | \u221236258 | 501 |  10.03 | 16.3 | 16.57  \nSI20_E0B | \u2212132065 | \u2212180155 | 449 | \u2212294.13 | 18.12 | 14.25 | \u2212171125 | \u2212205775 | 523 | \u2212327.2 | 15.7 | 14.53 | \u2212101005 | \u2212146075 | 583 | \u2212173.25 | 14.18 | 16.64  \nTF0_E0B | \u221230230 | \u221264260 | 191 | \u2212158.27 | 18.95 | 17.8 | \u221239050 | \u221268200 | 217 | \u2212179.95 | 16.8 | 18.43 | \u221298130 | \u2212119750 | 262 | \u2212374.54 | 14.39 | 18.7  \nTF20_E0B | \u221230155 | \u221263300 | 193 | \u2212156.24 | 18.74 | 17.62 | \u221240420 | \u221269560 | 217 | \u2212186.27 | 16.78 | 18.43 | \u221298140 | \u2212119760 | 262 | \u2212374.58 | 14.37 | 18.7  \nTU0_E0B | 22205 | \u221219544 | 325 |  68.32 | 20.19 | 13.23 | 37415 | \u221213961 | 311 | 120.3 | 21.06 | 16.08 | 18004 | \u221224410 | 392 |  45.93 | 17.11 | 16.07  \nTY0_E0B | 28377 | \u221229506 | 385 |  73.71 | 21.01 | 17.4 | 38757 | \u221222544 | 429 |  90.34 | 18.96 | 18.18 | 15044 | \u221234222 | 488 |  30.83 | 16.78 | 19.88  \nUS0_E0B | \u221212731 | \u221273538 | 437 | \u221229.13 | 18.63 | 18.31 | \u22123851 | \u221269806 | 501 |  \u22127.69 | 16.38 | 18.16 | 6405 | \u221248757 | 520 |  12.32 | 15.81 | 20.96  \nW20_E0B | \u221240788 | \u221257050 | 464 | \u221287.9 | 17.63 | 15.3 | \u221234238 | \u221249750 | 516 | \u221266.35 | 15.95 | 14.73 | \u221235588 | \u221246250 | 551 | \u221264.59 | 15 | 17.06\n\nThe performances for the three averaging methods aren't too dissimilar; they seem to make or lose money in the same sectors. However, on closer examination, at the portfolio level, notice the average trade for the weighted average is considerably lower than its siblings. This might indicate why it is the least used averaging method in technical analysis. While reviewing the performance, keep in mind that this testing did not take into consideration trade management other than allowing the algorithm to reverse its position; there weren't any protective stops or profit objectives. The fact is, trend following doesn't use a lot of trade management. Many trend algorithms will utilize a disaster stop but usually will let profits run until a reversal is signaled. If you think about this, it makes sense; a moving average is an indicator of trend and it will put you into the market in the direction it thinks the trend is going. It is self-adapting. The following is the pseudocode used in the testing of these three simplistic methodologies:\n\n    `Trend Following with Different Averaging Methods\n    `Simple Price MAV Cross Over\n    myAVG = SMA(C,80) `Comment out the two you do not want to use\n    `myAVG = WMA(C,80)    `by placing a single quote at the start of the line\n    `myAVG = EMA(C,80)\n    If close of today > myAVG then\n           Buy this bar on close\n    If close of today < myAVG then\n           SellShort this bar on close\n\n### Bollinger Bands\n\nJohn Bollinger is the originator of this volatility-based channel indicator. It is considered a channel because the indicator consists of two lines that never intersect. The channel width is based on a standard deviation value, which is derived from an _N_ -length moving average. Most Bollinger Bands are one or two standard deviations above\/below the moving average. In a perfectly \"normal\" world, 95 percent of values from a sample will stay within two standard deviations of its mean. Theoretically speaking, would it not make sense that once price rises to the upper channel\/band there exists a higher probability of price reverting to its mean? Or does it make more sense that a penetration of a band indicates strong momentum and the beginning of a trend. Like many indicators, Bollinger Bands can be used as mean reversion as well as trend following. The most successful trend-following systems use a longer-length Bollinger Band penetration algorithm for trade entry. Mean-reversion traders use a much shorter length input for this indicator.\n\nIf the market demonstrates enough momentum to carry price through two standard deviations, it seems there is a real good chance it has enough gusto to continue on in that direction. Well, this is what the trend followers of the 1990s thought, and they turned out to be right.\n\nFigure 2.14 shows a 60-day two-standard-deviation (SD) Bollinger Band on crude oil. The Bollinger Bands are the outside lines and the centerline is the moving average. This is a perfect example of why this trend entry technique gained so much popularity in the 1990s and in the new millennium. Crude penetrated the lower band and never looked back until the trend entered into a congestive phase\u2014a $50 move or $50,000 per contract. That's right, $50,000! And you thought I wasn't going to reveal the Holy Grail! Remember, one monster trade doesn't make a trading algorithm.\n\n**Figure 2.14** A 60-day, two-SD Bollinger Band on crude oil.\n\nSource: TradeStation\n\nHere's how you calculate Bollinger Bands:\n\n**Calculation of Bollinger Bands using 60 days and 2 standard deviations:**\n\n  1. Calculate a 60-period SMA of price for the chosen time period.\n  2. Calculate the standard deviation of the 60-day sample (use a computer).\n  3. The top or upper band is the 60-period SMA + 2 standard deviations.\n  4. The bottom or lower band is the 60-period SMA \u2212 2 standard deviations.\n\nLet's put the Bollinger Band indicator to a test utilizing a common trend-following algorithm (Box 2.8).\n\n* * *\n\n### Box 2.8 Using John Bollinger's Bands as a Trend Follower\n\n  1. Buy when market closes above top band.\n  2. Sell short when market closes below bottom band.\n  3. Sell when market position is long and market closes below average.\n  4. Cover when market position is short and market closes above average.\n  5. myAvg = Average(C,60) ' 60-day moving average calculation\n  6. myTopBand = BollingerBand(C,60,+2) ' 2 std dev above average\n  7. myBotBand = BollingerBand(C,60, \u22122) ' the \u2212 2 means below average\n  8. If close crosses above myTopBand, then \n    1. Buy MOC\n  9. If close crosses below myBotBand, then \n    1. SellShort MOC\n  10. If position is long, then \n    1. If close crosses below myAvg, then \n      1. Sell MOC\n  11. If position is short, then \n    1. If close crosses above myAvg, then \n      1. BuyToCover MOC\n\n* * *\n\nThe beauty of this algorithm is in its simplicity; we don't even need to sketch a flowchart. There is another facet of this algorithm that makes it a winner\u2014it's self-adaptive. Meaning the bands are moving (adapting) around the average based on an intrinsic property of the underlying instrument. And that property is volatility. Volatility reveals one of the most important things we need to know about an instrument, and that is risk. Risk is directly proportional to volatility\u2014the larger the swings, the riskier the market. I am jumping the gun here, but we will use this risk measure as a filter in version 2 of the Bollinger Band trend algorithm. Volatility also informs us of market noise. When the Bollinger Bands expand during high levels of volatility, they are indicating a chaotic situation\u2014a situation where entry criteria should be made more difficult to trigger. The following is the pseudocode for version 1 of the trading algorithm:\n\n    `Trend Following with Bollinger Bands\n    `Utilizing 60-day average and 2 standard deviations\n    `Version 1\n    myAVG = SMA(C,60)\n    myTopBand = BollingerBand(C,60,+2)\n    myBotBand = BollingerBand(C,60,-2)\n    If close > myTopBand then\n           Buy this bar on close\n    If close < myBotBand then\n           SellShort this bar on close\n    If marketPosition = 1 and c > myAvg then sell this bar on close\n    If marketPosition =-1 and c < myAvg then buyToCover this bar on close\n\nTable 2.8 shows the individual results of Bollinger Band version 1 across 35 futures markets.\n\n**Table 2.8** Performance of Bollinger Band Algorithm Version 1\n\n**Bollinger Band [60 Days 2 Standard Deviations]** |  |  |   \n---|---|---|---  \n**Ticker** | **Net Profit** | **Max DD** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **% Winners**  \n**_< Portfolio>_** | **_1319369_** | **_\u2212380897_** | **_3580_** | **_369_** | **_44_** | **_36_**  \nAD0_E0B | 30100 | \u221225600 | 95 | 317 | 45 | 35  \nBP0_E0B | 86888 | \u221290825 | 107 | 812 | 41 | 34  \nC20_E0B | \u22121263 | \u221223038 | 119 | \u221211 | 42 | 35  \nCC20_E0B | \u221267870 | \u221269300 | 123 | \u2212552 | 33 | 22  \nCL20_E0B | 168160 | \u221255940 | 101 | 1665 | 47 | 47  \nCT20_E0B | 54050 | \u221234175 | 112 | 483 | 40 | 31  \nCU0_E0B | 167738 | \u221227425 | 91 | 1843 | 50 | 44  \nDX20_E0B | 84575 | \u221213820 | 94 | 900 | 49 | 47  \nED0_E0B | 4997 | \u22123363 | 105 | 48 | 48 | 31  \nEMD0_E0B | \u221218880 | \u221238940 | 92 | \u2212205 | 39 | 29  \nES0_E0B | \u221241925 | \u221252213 | 120 | \u2212349 | 36 | 31  \nFC0_E0B | 55438 | \u221225650 | 101 | 549 | 49 | 41  \nFV0_E0B | 23631 | \u221215335 | 89 | 266 | 45 | 39  \nGC20_E0B | \u221219280 | \u221259660 | 111 | \u2212174 | 39 | 32  \nHG20_E0B | 76650 | \u221242513 | 111 | 691 | 44 | 37  \nHO20_E0B | 74941 | 71391 | 116 | 646 | 39 | 37  \nKC20_E0B | 100294 | \u2212108269 | 99 | 1013 | 45 | 38  \nKW20_E0B | 10800 | \u221221350 | 109 | 99 | 41 | 31  \nLB0_E0B | 37973 | \u221224997 | 109 | 348 | 44 | 38  \nLC0_E0B | \u221211770 | \u221230010 | 121 | \u221297 | 39 | 28  \nLH0_E0B | 10880 | \u221225480 | 113 | 96 | 42 | 37  \nMP0_E0B | 6755 | \u221225070 | 68 | 99 | 45 | 37  \nNG20_E0B | 175690 | \u221299650 | 87 | 2019 | 47 | 45  \nNK0_E0B | 48150 | \u221241675 | 85 | 566 | 46 | 36  \nOJ20_E0B | \u221210130 | \u221245408 | 110 | \u221292 | 43 | 36  \nPL20_E0B | 55340 | \u221234340 | 109 | 508 | 41 | 31  \nRB0_E0B | 124147 | \u221258171 | 109 | 1139 | 44 | 39  \nS20_E0B | 19475 | \u221238125 | 115 | 169 | 41 | 36  \nSB20_E0B | 31944 | \u221213788 | 100 | 319 | 48 | 42  \nSI20_E0B | \u221225660 | \u2212113680 | 111 | \u2212231 | 39 | 29  \nTF0_E0B | \u221214570 | \u221251590 | 49 | \u2212297 | 41 | 37  \nTU0_E0B | 37075 | \u22126975 | 78 | 475 | 50 | 42  \nTY0_E0B | 42466 | \u221219490 | 98 | 433 | 46 | 40  \nUS0_E0B | 22038 | \u221242338 | 109 | 202 | 42 | 39  \nW20_E0B | \u221219475 | \u221234225 | 114 | \u2212171 | 38 | 32\n\nIn the prior paragraphs, I hinted of a version 2 of this algorithm utilizing the volatility as trading filter. Risk aversion is most often the number-one consideration in the development of a trading algorithm, and we can use the volatility as a risk measure and modify version 1 to make it more risk averse. Looking back at the results you will notice a relatively high maximum drawdown at the portfolio level\u2014greater than 30 percent. In the world of hedge funds, this is an acceptable level\u2014I am not kidding! Some of the most successful hedge fund managers have taken their clients down more than 40 percent before rebounding. This is the main difference between hedge fund participants and the typical investor. The trade-off for higher returns is, of course, higher risk. Let's see if we can be a better hedge fund manager and reduce that 30 percent plus drawdown. Let's assume we have a million-dollar allocation (remember, we are in a dream world), we only want to risk 0.2 percent (0.002 or $2,000 per trade), and we don't want to reinvest profits (at this point). How can we use this information to reduce drawdown in version 2 of our trading algorithm?\n\nVersion 1 buys on penetration of upper Bollinger Band and gets out on the penetration of the moving average. So our initial trade risk can be defined as the distance between the upper band and the moving average. If we equate this distance to dollars, then we can filter out any trades that exceed our acceptable risk amount of $2,000. Now, will this guarantee that our largest loss will be $2,000 or less? No! Remember when I stated the bands were self-adapting? During the life of any given trade, the bands could expand and the distance between the outer bands and the moving average could grow well beyond our $2,000. This could result in an individual trade loss greater than that amount. Well, can we at least guarantee the maximum drawdown will decrease? Again, no! Maximum drawdown is a consequence of a losing streak. Even though we are filtering trades (eliminating what we consider too risky), that doesn't necessarily mean the trades involved in the losing streak will be eliminated. The filtering process might even eliminate the trades that pull the equity curve out of the drawdown. So, theoretically the risk aversion overlay might not help at all and might even make matters worse. Let's see for ourselves and use the following pseudocode on the same 35 markets:\n\n     `Trend Following with Bollinger Bands\n    `Utilizing 60-day average and 2 standard deviations\n    `Version 2 - don't take any trade with risk > $2,000\n    myAVG = SMA(C,60)\n    myTopBand = BollingerBand(C,60,+2)\n    myBotBand = BollingerBand(C,60,-2)\n    filterTrade = false\n    `convert price range to dollars by multiplying by dollar value of points\n    If (myTopBand - myAvg) * BigPointValue > 2000 then\n            filterTrade = true\n    If filterTrade = false then\n        If close > myTopBand then\n               Buy this bar on close\n        If close < myBotBand then\n               SellShort this bar on close\n    If marketPosition = 1 and c < myAvg then sell this bar on close\n    If marketPosition =-1 and c > myAvg then buyToCover this bar on close\n\nBefore hitting the **TEST** button, let's quickly go over the pseudocode. The only code that was added was the Boolean (true or false) typed variable **filterTrade** and its calculation. On every bar, it is initially turned off. The only way it is turned on is when the distance in terms of dollars between the upper band and the moving average exceeds $2,000. If **filterTrade** is on or true, the trade entry logic is bypassed, thereby skipping the trade. Why did we just use the top band and the moving average in our risk calculation? Shouldn't we use the bottom band and the average for short trades? One calculation is all that is needed since the distance between the top band and the average is always the same as the distance between the average and the bottom band\u2014in this case, two standard deviations. Ready to hit the test button? Table 2.9 shows the results of version 2 of the algorithm.\n\n**Table 2.9** Performance of Bollinger Band Algorithm Version 2\n\n**Bollinger Band [60 Days 2 Standard Deviations with $2,000 Trade Filter]**  \n---  \n**Ticker** | **Net Profit** | **Max DD** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **% Winners**  \n**_< Portfolio>_** | **_513207_** | **_\u2212103462_** | **_2224_** | **_231_** | **_43_** | **_34_**  \nAD0_E0B | 7360 | \u221219900 | 57 | 129 | 42 | 35  \nBP0_E0B | 10625 | \u22127075 | 1 | 10625 | 69 | 100  \nC20_E0B | \u22122788 | \u221215050 | 108 | \u221226 | 42 | 34  \nCC20_E0B | \u221248940 | \u221249270 | 110 | \u2212445 | 33 | 22  \nCL20_E0B | 29460 | \u22129970 | 45 | 655 | 50 | 51  \nCT20_E0B | 12385 | \u221231030 | 77 | 161 | 39 | 30  \nCU0_E0B | 50688 | \u22126488 | 4 | 12672 | 114 | 75  \nDX20_E0B | 71510 | \u221211830 | 56 | 1277 | 49 | 48  \nED0_E0B | 4997 | \u22123363 | 105 | 48 | 48 | 31  \nEMD0_E0B | 19720 | \u22128640 | 37 | 533 | 44 | 27  \nES0_E0B | \u221240288 | \u221244075 | 80 | \u2212504 | 33 | 23  \nFC0_E0B | 57950 | \u221214975 | 75 | 773 | 51 | 41  \nFV0_E0B | 25318 | \u221212041 | 84 | 301 | 45 | 40  \nGC20_E0B | 5840 | \u221221110 | 74 | 79 | 39 | 32  \nHG20_E0B | 6450 | \u221218325 | 65 | 99 | 42 | 29  \nHO20_E0B | 24291 | \u221227344 | 50 | 486 | 38 | 36  \nKC20_E0B | 51275 | \u221235669 | 22 | 2331 | 39 | 41  \nKW20_E0B | 7188 | \u221223375 | 90 | 80 | 40 | 27  \nLB0_E0B | 53134 | \u221212761 | 51 | 1042 | 49 | 43  \nLC0_E0B | \u221212630 | \u221230100 | 117 | \u2212108 | 39 | 27  \nLH0_E0B | 12690 | \u221220690 | 97 | 131 | 41 | 38  \nMP0_E0B | 2770 | \u221222135 | 62 | 45 | 43 | 34  \nNG20_E0B | 38580 | \u221214000 | 16 | 2411 | 50 | 50  \nNK0_E0B | 35250 | \u221212925 | 12 | 2938 | 50 | 33  \nOJ20_E0B | \u221216875 | \u221249565 | 99 | \u2212170 | 41 | 34  \nPL20_E0B | \u221211870 | \u221230065 | 72 | \u2212165 | 38 | 25  \nRB0_E0B | 20830 | \u221220086 | 40 | 521 | 43 | 43  \nS20_E0B | \u221210763 | \u221234438 | 79 | \u2212136 | 37 | 35  \nSB20_E0B | 17097 | \u221212638 | 91 | 188 | 47 | 42  \nSI20_E0B | \u22124305 | \u221242550 | 66 | \u221265 | 40 | 27  \nTF0_E0B | 18470 | \u22126380 | 3 | 6157 | 78 | 67  \nTU0_E0B | 28837 | \u22127888 | 76 | 379 | 48 | 41  \nTY0_E0B | 37841 | \u221219212 | 73 | 518 | 46 | 41  \nUS0_E0B | 39185 | \u221216650 | 38 | 1031 | 44 | 47  \nW20_E0B | \u221228075 | \u221236663 | 92 | \u2212305 | 37 | 28\n\nMission accomplished! Or was it? The maximum drawdown was reduced from to $381K to $103K\u2014a whopping 73 percent. However, as we all know, reward is proportional to risk and the total profit dropped from $1.3M to less than $600K, or more than 50 percent. Was it worth it? What if we used $2,500 risk instead, or what if... You can see how this stuff can become addictive. We could optimize the per-trade risk amount at the portfolio level or at the individual market level. Does it make sense to have a different risk level for crude oil than gold? If the equity curve looks better, why not? We could even eliminate all the markets that show a negative expectancy. We could work all day on different portfolios and what-if scenarios and create one great-looking equity curve. But who would we be fooling? If it was the 1980s or 1990s, the public. If we were unscrupulous and wanted to sell this miraculous algorithm, we could mislead the public with the guarantee that every number shown was generated by an exact trading algorithm. The general trading public of the latter part of the twentieth century didn't know any better and didn't understand the term \"with benefit of hindsight.\" Nowadays, with the computer power and data at our fingertips, we would be fooling ourselves because the trading public now has the same computers and data and has been fully educated on \"with benefit of hindsight.\" We can still fall into this same old trap, though, if we don't fully utilize the tools at our disposal.\n\n### Keltner Channel\n\nChester Keltner brought this volatility-based indicator to public attention in 1960 through his book, _How to Make Money in Commodities_. The Keltner Channels are very similar to Bollinger Bands except the centerline is a moving average of typical price ((H + L + C) \/ 3) and the outer bands are calculated using average true range instead of standard deviation. Volatility can be measured in many different ways, but standard deviation and average true range are probably the two most popular.\n\nThe two key components of this indicator that set it apart from Bollinger's are the use of the typical price (TP) and the average true range. TP is simply the average of the high, low, and close prices. The TP takes into consideration the range of the day as well as the close, and as stated earlier, many feel this price is more indicative of the daily price action than simply the close. Since Bollinger and Keltner indicators are so similar, you will notice that the calculations aren't that different:\n\n**Calculation of Keltner Channels using 60 days and 2 ATR:**\n\n  1. Calculate a 60-period SMA of the TP for the chosen time period.\n  2. Calculate the ATR of the 60-day sample (use a computer).\n  3. The top or upper band is the 60-period SMA + 2 ATR.\n  4. The bottom or lower band is the 60-period SMA \u2212 2 ATR.\n\nBox 2.9 shows a Keltner Channel trading algorithm that will be used throughout the rest of this chapter.\n\n* * *\n\n### Box 2.9 Using Chester Keltner's Channels as a Trend Follower\n\n  1. Buy when market closes above top channel.\n  2. Sell short when market closes below bottom channel.\n  3. Sell when market position is long and market closes below average.\n  4. Cover when market position is short and market closes above average.\n  5. myTypicalPrice = (H + L + C);\n  6. myAvg = Average(myTypicalPrice,60) ' 60-day moving average of TP\n  7. myATR = AverageTrueRange(60)\n  8. myTopChannel = myAvg + 2 * myATR ' 2 ATR above average\n  9. myBotChannel = myAvg \u2212 2 * myATR '\u2212 2 ATR below average\n  10. If close crosses above myTopChannel, then \n    1. Buy MOC\n  11. If close crosses below myBotChannel, then \n    1. SellShort MOC\n  12. If position is long, then \n    1. If close crosses below myAvg, then \n      1. Sell MOC\n  13. If position is short, then \n    1. If close crosses above myAvg, then \n      1. BuyToCover MOC\n\nHere is the pseudocode for the algorithm:\n\n    `Trend Following with Keltner Channels\n    `Utilizing Typical Price = (H + L + C)\/3\n    `Utilizing 60-day average and 2 average true ranges\n    myAVG = SMA(C,60)\n    myTypicalPrice = (H + L + C)\/3\n    myTopChannel = KeltnerChan(myTypicalPrice,60,+2)\n    myBotChannel = KeltnerChan(myTypicalPrice,60,-2)\n    If close > myTopChannel then\n         Buy this bar on close\n    If close < myBotChannel then\n         SellShort this bar on close\n    If marketPosition = 1 and c < myAvg then sell this bar on close\n    If marketPosition =-1 and c > myAvg then buyToCover this bar on close\n\n* * *\n\nUsing the ATR instead of standard deviation creates smoother outer Bands due to how ATR reacts slower than deviation. Figure 2.15 shows a chart with both Keltner Channels (60 length and 2 ATR) and Bollinger Bands (60 length and 2 STDDEV) applied to the underlying data.\n\n**Figure 2.15** Keltner Channels and Bollinger Bands applied to the same data.\n\nSource: TradeStation\n\nNotice how much smoother the Keltner Channels are compared to the Bollinger Bands. Does this mean they are better? Since the TP takes into consideration more data, is it a better gauge than just the close? Looking at the chart it seems like both indicators got into the short crude position at about the same time. This is a difficult question so let's use the computer tell us which is the better algorithm. This is not going to be an easy task because it is difficult to equate standard deviation to ATR. As you can tell by the chart, the Bollinger Bands encompass the Keltner Channels a majority of the time. This is a consequence of using the same settings for both indicators. We can plainly see that different settings need to be used if we want the indicators to behave in somewhat similar fashion. In other words, the Keltner Channel parameters need to be normalized in terms of the Bollinger Band parameters. This is the only way a fair test can be carried out. The parameters in question are the indicator length and the width of the bands\/channels. We must find out the number of ATRs that are nearly equivalent to two standard deviations. Is two deviations comparable to one ATR, or is it two ATR to one standard deviation? Again, referring back to the chart, we now know that in all likelihood, two ATR does not equal two standard deviations. We know that trades are generated when price penetrates the bands\/channels, and if the bands\/channels are somewhat close to each other or overlapping, then a similar number of trades should be generated. Using the number of trades as the objective, a set of normalized parameters can be determined through optimization of each algorithm across different indicator lengths and channel widths. Since both standard deviation and average true range are functions of the size of the sample (length of days, in our case), both length and width will be optimized. The number of trades will be recorded for each step of the optimization. If a 60-period two-standard-deviation Bollinger Band generates 3,000 trades, then an equivalent parameter set for the Keltner algorithm would be those parameters that generated a similar number of trades. Figure 2.16 shows the number of trades generated utilizing the Bollinger algorithm by varying the length and width parameters over a basket of 36 futures markets.\n\n**Figure 2.16** Bollinger Band optimization.\n\nThis is an unusual graph, so let me explain what exactly is being plotted. The _x_ -axis plots bins of varying lengths at different width values. The _y_ -axis is the number of trades generated by the combination of differing lengths and widths. The length was optimized from 10 to 80 in steps of 2.0 and the width was optimized from 0.25 to 3 in steps of 0.25. The first bin holds all the results of the different-length variables while holding the width at 0.25. Like the first bin the second bin varies the lengths while holding the width to 0.50. Notice how large an impact the length parameter has on the number of generated trades in the different-width bins. Also notice initially how congruent the bars are in each bin. It seems like the drop-off of trades as we move from one bin to another occurs in a linear fashion\u2014that is, up until bins with width values greater than 2.0. An inversion occurs in these bins (shorter lengths begin to create less trades). In addition, the number of trades start demonstrating a parabolic theme. Let's do the same exact analysis with the Keltner Channel algorithm utilizing the same data and optimization ranges. Figure 2.17 utilizes the exact same chart properties but with the newly derived results.\n\n**Figure 2.17** Keltner Channel optimization.\n\nThis graph starts out looking like the Bollinger Band graph; initially, the length parameter has a large impact in the 0.25 bin, just like the Bollinger chart. However, as we span across the different-width bins, the length parameter has less and less impact. By the time we reach the 1.50 bin, an inversion occurs, just like the Bollinger, where the lower-length parameters start to generate less trades. The two algorithms are quite similar but these graphs illustrate the difference between utilizing ATR and standard deviation.\n\nArmed with this information, can a normalized parameter set for both the Bollinger and Keltner algorithms be derived? The 60-day two-standard-deviation algorithm generated 3,580 trades. Let's see which parameters generated nearly the same number for the Keltner algorithm. If you refer back to Figure 2.17, you will see several instances where a set of parameters generated nearly the same number of trades. I scanned the Keltner Channel results and pulled out the parameter sets that generated trades within +\/\u2212 2 percent of the 3,580. Most of the selected parameter sets included the same 3.0 width. There are a couple of outliers, though. Table 2.10 shows the performance of the Bollinger Band algorithm that was derived earlier using (len = 60, stdDev = 2) and the Keltner Channel algorithm utilizing the different parameter sets that generated nearly the same number of trades.\n\n**Table 2.10** Bollinger Benchmark versus Keltner Challengers\n\n**Set** | **Net Profit** | **Max. Sys DD** | **CAR\/MDD** | **Profit Factor** | **Sharpe Ratio** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **L. Avg. Bars Held**  \n---|---|---|---|---|---|---|---|---|---  \nBB [60,2.00] | 1677369.47 | \u2212345343 | 0.27 | 1.4 | 0.2 | 3580 | 469 | 43.59 | 23.72  \n|  |  |  |  |  |  |  |  |   \n**Plateau** |  |  |  |  |  |  |  |  |   \n**Set** | **Net Profit** | **Max**. **Sys DD** | **CAR\/ MDD** | **Prft**. **Factor** | **Sharpe Ratio** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **L. Avg**. **Bars Held**  \nKC[56,3] | 1253131.73 | \u2212443724 | 0.16 | 1.27 | 0.14 | 3612 | 347 | 44.47 | 24.65  \nKC[58,3] | 1318378.23 | \u2212434112 | 0.17 | 1.29 | 0.15 | 3593 | 367 | 44.53 | 24.66  \nKC[60,3] | 1338616.09 | \u2212409094 | 0.18 | 1.29 | 0.15 | 3579 | 374 | 44.56 | 24.6  \nKC[62,3] | 1391495.03 | \u2212404823 | 0.19 | 1.31 | 0.16 | 3564 | 390 | 44.64 | 24.69  \nKC[64,3] | 1449520.83 | \u2212410137 | 0.2 | 1.33 | 0.17 | 3535 | 410 | 44.82 | 24.69  \nKC[66,3] | 1506179.43 | \u2212375805 | 0.23 | 1.35 | 0.17 | 3523 | 428 | 44.86 | 24.68  \nKC[68.3] | 1520843.29 | \u2212386511 | 0.22 | 1.35 | 0.18 | 3516 | 433 | 44.81 | 24.61  \n**Outliers** |  |  |  |  |  |  |  |  |   \nKC[12,2.5] | 1085527.09 | \u2212300175 | 0.18 | 1.27 | 0.15 | 3521 | 308 | 37.83 | 20.75  \nKC[26,3.0] | 1299080.88 | \u2212407407 | 0.18 | 1.29 | 0.15 | 3589 | 362 | 42.1 | 22.21\n\nFrom this information, we might be able to derive that a Bollinger Band (60, 2) is similar to a Keltner Channel (60, 3). If we accept this assumption, then the winner is easy to pick from this table. The Bollinger Band algorithm performed much better than the Keltner with these parameters. However, with such stringent criteria and utilizing the Bollinger algorithm as the benchmark, are we being fair to Keltner? We forced the Keltner to fit into criteria where a similar number of trades were generated. Let's take the gloves off and again do the same optimization and pick the top five parameter sets for both algorithms based on the average profit and net profit. In this analysis, a $50 round-turn commission will be deducted from each trade\u2014this will take execution costs into consideration, which will in turn provide a level playing field for parameter sets that do not generate a large number of trades.\n\nTables 2.11 and 2.12 show the best performers for the Bollinger Band algorithm. The first set of performance metrics were derived by sorting by average profit\/loss and the second set was derived by sorting by net profit. Table 2.12 shows the same metrics for the Keltner Channel algorithm.\n\n**Table 2.11** Best Parameter Sets of Bollinger Bands Algorithm\n\n**Ranking Based on Average Trade** |  |  |  |  |   \n---|---|---|---|---|---  \n**No.** | **Net Profit** | **Max. Drawdown** | **Sharpe Ratio** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners** | **Len** | **Width**  \n360 | $1,157,014 | \u2212$302,350 | 0.22 | 1769 | 654.05 | 61.25 | 40.64 | 80 | 2.5  \n359 | $1,147,202 | \u2212$308,547 | 0.22 | 1788 | 641.61 | 59.79 | 40.6 | 78 | 2.5  \n358 | $1,158,434 | \u2212$358,085 | 0.22 | 1827 | 634.06 | 58.48 | 40.56 | 76 | 2.5  \n324 | $1,427,825 | \u2212$323,166 | 0.21 | 2271 | 628.72 | 59.56 | 39.89 | 80 | 2.25  \n323 | $1,442,568 | \u2212$379,483 | 0.21 | 2340 | 616.48 | 58.16 | 39.74 | 78 | 2.25  \n**Ranking Based on Net Profit** |  |  |  |  |  |  |   \n**No**. | **NetProfit** | **Max**. **Drawdown** | **Sharpe Ratio** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners** | **Len** | **Width**  \n216 | $1,792,835 | \u2212$397,785 | 0.19 | 3622 | 494.98 | 51.55 | 34.65 | 80 | 1.5  \n211 | $1,789,623 | \u2212$395,046 | 0.18 | 4092 | 437.35 | 45.71 | 34.41 | 70 | 1.5  \n180 | $1,788,584 | \u2212$400,129 | 0.18 | 4149 | 431.09 | 48.06 | 33.29 | 80 | 1.25  \n179 | $1,765,374 | \u2212$392,321 | 0.17 | 4257 | 414.7 | 46.82 | 33 | 78 | 1.25  \n215 | $1,754,942 | \u2212$408,842 | 0.18 | 3726 | 471 | 50.16 | 34.33 | 78 | 1.5\n\n**Table 2.12** Best Parameter Sets of Keltner Channel Algorithm\n\n**Ranking Based on Average Trade** |  |  |  |  |   \n---|---|---|---|---|---  \n**No.** | **Net Profit** | **Max. Drawdown** | **Sharpe Ratio** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners** | **Len** | **Width**  \n397 | $576,442 | \u2212$216,503 | 0.19 | 1193 | 483.19 | 44.77 | 36.88 | 10 | 3  \n400 | $1,131,833 | \u2212$291,100 | 0.18 | 2648 | 427.43 | 42.88 | 36.44 | 16 | 3  \n431 | $1,497,797 | \u2212$399,590 | 0.17 | 3568 | 419.79 | 43.92 | 37.3 | 78 | 3  \n398 | $755,357 | \u2212$277,970 | 0.17 | 1804 | 418.71 | 44.11 | 37.03 | 12 | 3  \n432 | $1,500,595 | \u2212$394,719 | 0.17 | 3598 | 417.06 | 43.59 | 37.13 | 80 | 3  \n**Ranking Based on Net Profit** |  |  |  |  |  |  |   \n**No**. | **NetProfit** | **Max**. **Drawdown** | **SharpeRatio** | **# Trades** | **AvgProfit\/Loss** | **AvgBars Held** | **% ofWinners** | **Len** | **Width**  \n180 | $1,626,520 | \u2212$406,001 | 0.14 | 7293 | 223.02 | 29.87 | 29.14 | 80 | 1.25  \n141 | $1,623,026 | \u2212$420,401 | 0.13 | 8051 | 201.59 | 28.45 | 27.76 | 74 | 1  \n140 | $1,622,844 | \u2212$415,479 | 0.13 | 7974 | 203.52 | 28.83 | 27.77 | 72 | 1  \n179 | $1,606,297 | \u2212$409,864 | 0.14 | 7243 | 221.77 | 30.21 | 29.32 | 78 | 1.25  \n103 | $1,596,825 | \u2212$422,709 | 0.13 | 9129 | 174.92 | 26.26 | 25.74 | 70 | 0.75\n\nAnd the winner is? Tough call\u2014I lean toward the Bollinger Bands because of the higher Sharpe ratios. The Sharpe ratio is just an indicator of consistency, and we will go over it and many more performance metrics in the next chapter. One thing that does stand out is the holding periods of the two systems and how different they are. The profit\/loss is somewhat similar but the Bollinger algorithm holds on to trades anywhere from 20 to 50 percent longer. There might be some synergy here. Diversification is not only achieved through different markets but also through different algorithms.\n\n## Summary\n\nAnother chapter, and still no Holy Grail. However, using indicators as building blocks might get you very close to one. Several oscillators and trend indicators were discussed but these are just a few out of many. The overall purpose of this chapter was to show how to integrate these indicators into a trading algorithm. The flowcharts and FSM diagrams that were introduced in Chapter 1 were carried over and used to help explain this integration. Pseudocode was also used to demonstrate how the indicators can be programmed. Keep in mind the actual computer code will be shown in the appendices. The development of a mean reversion algorithm can be accomplished by incorporating any of the oscillators and trend detection. A trend-following algorithm is best achieved by incorporating moving averages and\/or their derivatives such as Bollinger Bands and Keltner Channels. Trust me when I tell you the most successful trend-following systems incorporate either one or both.\n\nHow did we get through this chapter without talking about the Turtles, Richard Dennis, or Richard Donchian? If you are not familiar with these names, just Google them. Richard Donchian created the _N_ -week breakout rule and Richard Dennis built a complete trading scheme around it. Richard Dennis was very successful and spawned Turtle trading. The _N_ -week breakout is a channel indicator that uses the highest high of _N-_ weeks\/days as the upper channel and the lowest low of _N_ -weeks\/days as the lower channel. Just like a Bollinger Band algorithm, trades are initiated by the penetration of the channels. Richard Dennis and his partner Bill Eckhardt believed anybody could be trained to trade futures successfully if a trading plan was provided. So they placed classified ads in the newspaper looking for new talent. They weren't going after the prototypical Ivy Leaguer economist types. They wanted smart people who understood human behavior and who were willing to be trained. The training involved the ideas and concepts that Dennis had used to make millions. Many of the recruited traders, aka Turtles, went on to be extremely successful. Now was this a consequence of a superior algorithm or the inherent talents of the traders or being in the right spot at the right time or more simply just having a trading plan? I think it was a combination of all of the above. Don't think such a famous algorithm is going to be left out of this book. It will be introduced in the next chapter, as well as several others. Also, the Turtle money management algorithm (aka Fixed Fractional) will be discussed in Chapter 9.\n\n# Chapter 3  \nComplete Trading Algorithms\n\nWe ended the last chapter discussing one of the most famous trading algorithms of all times. Figure 3.1 is a flowchart I used to program the famous algorithm many years ago. The flowchart looks extremely complicated, but keep in mind the complete Turtle system includes two trading systems, money management, and a simulated trading module. If you break down the flowchart into its components, you will see that it is not as complicated as it seems. Most of the calculations in the Turtle system are used to determine market risk, capital allocation, market selection, and position sizing. All these components are quite important, but the meat of the diagram and the system are the two trading entry\/exit algorithms and the switch that determines which system is applied. Let's go right to the heart of the Turtle by breaking just the trade entry\/exit algorithms out of the complete model. The other components will be discussed a little later.\n\n**Figure 3.1** Turtle trading algorithm flowchart.\n\nThe first of the two entry\/exit algorithms, let's call it System 1, is the shorter-term entry technique and is based on a very simple technique. Basically, it will buy\/sell short on a 20-day Donchian breakout and exit the trade by risking whichever is closer: a 10-day Donchian breakout or a maximum of 2 percent of capital. System 1 (Box 3.1) can trade quite frequently in a choppy market. This type of start and stop can lead to a long list of losses. This choppiness can be very devastating to a trader's account and psychology. Richard Dennis tried to help alleviate this by implementing a trade filter. The Turtles were instructed to skip System 1 entries if the last trade was a winner. If the last trade was a winner, then the next trade is not \"actually\" executed but it is simulated. In other words, the trader would paper trade the system until a loss occurs.\n\n* * *\n\n### Box 3.1 Turtle System #1\n\n  1. Buy one tick above the highest high of the past 20 days stop.\n  2. Sell short one tick below the lowest low of the past 20 days stop.\n  3. If position is long, then\n\nSell next bar at the maximum of:\n\n    1. one tick below the lowest low of the past 10 days stop, or\n    2. entryPrice \u2013 (0.02 * initialCapital) \/ bigPointValue stop\n  4. If position is short, then\n\nBuy next bar at the minimum of:\n\n    1. one tick above the highest high of the past 10 days stop, or\n    2. entryPrice + (0.02 * initialCapital) \/ bigPointValue stop\n\n* * *\n\nThis description does not utilize the \"last trade was a loser filter\" (LTLF). Even with the application of the LTLF, this system is very easy to trade. A simple spreadsheet is all it takes to help keep track of actual and simulated trades. Remember a simulated trade occurs after a winning trade is realized. Real trading is turned back on once a simulated losing trade occurs.\n\nSystem 2 (Box 3.2) entries are congruent with System 1, but require a much longer lookback period\u201455 days. Even though the Donchian Channel exit is wider, the same 2 percent maximum loss is still used.\n\n* * *\n\n### Box 3.2 Turtle System #2\n\n  1. Buy one tick above the highest high of the past 55 days stop.\n  2. Sell short one tick below the lowest low of the past 55 days stop.\n  3. If position is long, then\n\nSell next bar at the maximum of:\n\n    1. one tick below the lowest low of the past 20 days stop or\n    2. entryPrice \u2013 (0.02 * initialCapital) \/ bigPointValue stop\n  4. If position is short, then\n\nBuy next bar at the minimum of:\n\n    1. one tick above the highest high of the past 10 days stop or\n    2. entryPrice + (0.02 * initialCapital) \/ bigPointValue stop\n\n* * *\n\nThe LTLF does not apply to these longer-term entries. If the market is making a new 55-day high\/low, then the Turtle system thinks a new trend has definitely begun and the trader must be on board. This entry was considered the failsafe breakout.\n\nThis is the Turtle system in a nutshell, or should I say a turtle shell. The full-blown Turtle model involves many more calculations, pyramiding, and a totally separate position sizing and market selection algorithm. Some of these topics will be discussed in Chapter 8 and a full-blown version (except for market selection) will be provided at the end of this chapter in EasyLanguage.\n\nEven though the complete Turtle \"approach\" isn't being applied here, the two systems are indeed complete trading algorithms; they both have entry and trade management techniques. System 1 was designed to capture shorter-term swings, whereas System 2 was the true trend-following component. Trading the two systems simultaneously was undoubtedly considered synergistic to the Turtles. The key to getting the benefit from both systems lies in the LTLF\u2014it must be engaged. If it's not, then you end up just trading System 1, the 20-day breakout. A 20-day breakout will always be closer or equal to the current market price than a 55-day breakout. Here again is the idea of creating an additional level of diversification through the use of multiple systems. However, the amount of capital to allocate to the different systems was left up to the individual Turtle (trader). He or she could commit 100 percent to either system or 50 percent to one and 50 percent to the other or some other combination. How to determine how much of each system to use was obviously a very difficult question to answer, even for a Turtle. Which system would you use? If I had been a Turtle faced with this decision, I would have done the research by backtesting both systems and allocating more toward the better of the two.\n\nAs a Turtle you had three choices to make: Trade System1, trade System 2, or trade a combination of the two. Let's pretend we are a Turtle and let's see if we can make the decision. The research necessary requires each system to be tested independently and simultaneously to see if \"the whole is greater than the sum of its parts.\" Tables 3.1 and 3.2 show the performance of each system, respectively, across the same portfolio over the same time period. Note that System 1 was initially tested without the LTLF. In these tests, the maximum loss was set to $2,000 and $100 was deducted for commission and slippage on a round-turn basis.\n\n**Table 3.1** Turtle System 1 without LTLF for the Sample Portfolio\n\n**Total Return** | **$439,471.42** | **Market** | **PNL**  \n---|---|---|---  \nTotal Realized Return | $416,450.17 | @AD |  ($6,110.00)  \nGross Profit | $5,182,686.04 | @BP |  ($6,543.75)  \nGross Loss | \u2212$4,766,235.87 | @C |  ($6,987.50)  \nOpen Trade P\/L | $23,021.25 | @CC |  ($36,020.00)  \n|  | @CL |  $77,690.00   \nNumber of Trades | $5,234.00 | @CT |  $20,065.00   \nNumber of Winning Trades | $1,751.00 | @DX |  ($14,590.00)  \nNumber of Losing Trades | $3,481.00 | @EC |  $84,150.00   \n% Profitable | 33.45% | @ED |  ($10,800.00)  \n|  | @EMD |  ($70,700.00)  \nAverage Trade | $79.57 | @ES |  ($36,162.50)  \nAverage Trade (%) | 0.39% | @FC |  $39,675.00   \nStandard Deviation | $3,566.48 | @FV |  ($11,335.93)  \nStandard Deviation Trade % | 28.39% | @GC |  ($19,480.00)  \nLargest Winning Trade | $53,275.00 | @HG |  $101,125.00   \nLargest Losing Trade | \u2212$8,375.00 | @HO |  $97,287.00   \n|  | @KC |  ($34,106.25)  \nProfit Factor | 1.09 | @KW |  ($2,975.00)  \nAverage Win\/Average Loss | 2.16 | @LC |  ($32,130.00)  \nSharpe Ratio | 0.12 | @LH |  $16,570.00   \nK-Ratio | 0.16 | @NG |  $28,550.00   \nReturn Retracement Ratio | 0.54 | @OJ |  $3,977.50   \n|  | @PL |  $90,825.00   \n|  | @RB |  $45,105.20   \n|  | @S |  $32,693.75   \n|  | @SB |  $20,252.00   \n|  | @SI |  $182,700.00   \n|  | @TF | ($104,090.00)  \n|  | @TU |  ($9,174.97)  \n|  | @TY |  $16,871.88   \n|  | @US |  ($12,581.25)  \n|  | @W |  ($5,512.50)\n\n**Table 3.2** Turtle System 2 for the Sample Portfolio\n\n**Summary** | **Value** | **Markets** | **PNL**  \n---|---|---|---  \nTotal Return | $949,937.69 | @AD |  $40,250.00   \nTotal Realized Return | $889,799.88 | @BP |  $3,987.50   \nGross Profit | $3,444,084.89 | @C |  $11,225.00   \nGross Loss | \u2212$2,554,285.01 | @CC |  ($34,600.00)  \nOpen Trade P\/L | $60,137.81 | @CL |  $174,440.00   \n|  | @CT |  $20,720.00   \nNumber of Trades | $2,324.00 | @DX |  $21,150.00   \nNumber of Winning Trades | $726.00 | @EC |  $83,600.00   \nNumber of Losing Trades | $1,597.00 | @ED |  $1,743.75   \n% Profitable | 31.24% | @EMD |  $4,490.00   \n|  | @ES |  $11,587.50   \nAverage Trade | $382.87 | @FC |  $31,825.00   \nAverage Trade (%) | 0.67% | @FV |  ($585.93)  \nStandard Deviation | $5,477.96 | @GC |  $57,530.00   \nStandard Deviation Trade % | 56.38% | @HG |  $91,425.00   \nLargest Winning Trade | $90,440.00 | @HO |  $173,699.40   \nLargest Losing Trade | \u2212$7,590.00 | @KC |  ($19,500.00)  \n|  | @KW |  $22,500.00   \nProfit Factor | 1.35 | @LC |  ($14,740.00)  \nAverage Win\/Average Loss | 2.97 | @LH |  $19,660.00   \nSharpe Ratio | 0.16 | @NG |  $102,840.00   \nK-Ratio | 0.19 | @OJ |  ($1,155.00)  \nReturn Retracement Ratio | 1.19 | @PL |  $70,340.00   \n|  | @RB |  $80,879.40   \n|  | @S |  $70,162.50   \n|  | @SB |  $28,963.20   \n|  | @SI |  $134,575.00   \n|  | @TF |  ($13,100.00)  \n|  | @TU |  $7,437.50   \n|  | @TY |  ($6,156.25)  \n|  | @US |  $10,687.50   \n|  | @W |  $3,612.50\n\nTable 3.1 shows the net performance along with individual market results as well. A profit of more than $430k was generated along with a hefty $220k maximum drawdown. Make a mental checkmark on the markets that showed the best results. I have a feeling you will notice a pattern among the best-performing markets throughout the rest of this chapter. Let's see how the 55-day breakout faired (Table 3.2).\n\n_Wow!_ The profit nearly doubled and the maximum drawdown was reduced by 10 percent. If you had traded this system in place of the 20-day breakout, you might not have received a Christmas card from your broker\u2014less than one half the execution costs\u2014but your banker would have been happy.\n\nNow let's go back and retest System 1 with the _Last Trader Was a Loser Filter_ (LTLF) to see if it had any beneficial impact on the performance. Actual trades were only placed after a subsequent loss\u2014be it from an actual loss or a simulated loss. Losses were generated by the $2,000 money management stop or a trailing 10-day stop that resulted in a loss. Table 3.3 shows the results of System 1 with the LTLF in place.\n\n**Table 3.3** Turtle System 1 with LTLF for the Sample Portfolio\n\n**Summary** | **Value** | **Markets** | **PNL**  \n---|---|---|---  \nTotal Return | $667,013.78 | @AD |  $2,890.00   \nTotal Realized Return | $652,511.28 | @BP |  ($6,031.25)  \nGross Profit | $3,632,117.44 | @C |  $5,912.50   \nGross Loss | \u2212$2,979,606.16 | @CC |  ($19,900.00)  \nOpen Trade P\/L | $14,502.50 | @CL |  $107,830.00   \n|  | @CT |  $7,250.00   \nNumber of Trades | $3,301.00 | @DX |  $20,110.00   \nNumber of Winning Trades | $1,125.00 | @EC |  $83,300.00   \nNumber of Losing Trades | $2,175.00 | @ED |  ($13,356.25)  \n% Profitable | 34.08% | @EMD |  ($30,180.00)  \n|  | @ES |  ($23,475.00)  \nAverage Trade | $197.67 | @FC |  $4,300.00   \nAverage Trade (%) | 0.51% | @FV |  ($1,917.18)  \nStandard Deviation | $3,963.63 | @GC |  $17,970.00   \nStandard Deviation Trade % | 26.13% | @HG |  $114,037.50   \nLargest Winning Trade | $53,275.00 | @HO |  $60,120.20   \nLargest Losing Trade | \u2212$7,925.00 | @KC |  $18,531.25   \n|  | @KW |  $2,450.00   \nProfit Factor | 1.22 | @LC |  ($18,630.00)  \nAverage Win\/Average Loss | 2.36 | @LH |  $28,620.00   \nSharpe Ratio | 0.15 | @NG |  $64,760.00   \nK-Ratio | 0.22 | @OJ |  $11,230.00   \nReturn Retracement Ratio | 1.10 | @PL |  $88,465.00   \n|  | @RB |  $46,595.20   \nCompounded Annual Return | 14.96% | @S |  $14,581.25   \nCompounded Monthly Return | 1.15% | @SB |  $9,156.80   \n|  | @SI |  $169,050.00   \nAverage Annual Return | $44,467.59 | @TF |  ($73,960.00)  \nAverage Annual Return (%) | 19.54% | @TU |  ($8,718.73)  \nAverage Monthly Return | $3,789.85 | @TY |  $1,725.00  \nAverage Monthly Return (%) | 1.59% | @US |  ($10,581.25)  \n|  | @W |  ($9,400.00)  \nPercent Days Profitable | 50.69% |  |   \nPercent Months Profitable | 52.27% |  |   \nPercent Years Profitable | 73.33% |  |   \n|  |  |   \nCommissions on Currencies | $165,375.00 |  |   \nCommissions on Futures | $0.00 |  |   \nCommissions on Equities | $0.00 |  |   \nTotal Commissions | $165,375.00 |  |\n\nThe results agreed with my research from more than 10 years ago. In an article I wrote for _SFO_ magazine in 2004, I executed the exact same analysis utilizing another testing platform, Trading Blox Builder (www.tradingblox.com), and received similar results; the profit-to-drawdown ratio increased using the LTLF. The **Trading Blox** testing platform was built around the absolute complete Turtle algorithm (entry techniques, capital allocation, and position sizing models) by an original turtle and a very gifted programmer. As I mentioned in the Introduction, this is not one of the platforms included in this book, but it is well worth your time to investigate its capabilities. What this research is implying is that the prior trade may have an impact on the subsequent trade. In other words, it seems there does exist a higher probability of a winning trade after a losing trade has occurred at the macroscopic level on this particular algorithm. This autocorrelation could be a function of the volatility cycle. After a big run in commodity prices, it is not uncommon to see a congestive phase. In my opinion, this type of filter works better with shorter-term higher-frequency algorithms. If you think about it, this makes sense. A system that trades frequently can skip trades and still catch shorter-term trends. On the other hand, a system that trades infrequently needs every trade to add to the bottom line; skipping a single trade might keep you out of the market for a year. This is the reason the Turtles relied on the \"must-take\" 55-day breakout.\n\nLet's test if there is synergy by trading both systems simultaneously. Synergy occurs when the interaction of elements, when combined, produces a total effect that is greater than the sum of the individual elements. Synergy does not occur when combining the total profits of the two systems. Addition of total P & L follows the commutative law. The synergy that does occur when combining systems is revealed in the combined maximum drawdown metric. If two systems are somewhat non-correlated, then the overall maximum drawdown will not be equal to the sum of the two individual maximum drawdowns; it may be less. So, it logically follows that the synergy of combining multiple trading strategies is reflected in the increase of the overall profit to overall maximum drawdown ratio. The overall profit is a constant (sum of all total profits) and the overall maximum drawdown is variable. If drawdown decreases, then the ratio will increase and demonstrate a somewhat synergistic effect.\n\nThis combining of systems can be accomplished by trading the two systems independently of each other in separate accounts, or in a single account. A trader can also combine the rules into one complete strategy, and trade in a single account. Many of today's testing platforms offer both methods. The backtesting method where both systems are traded independently can be simulated simply by combining the daily equity streams for both systems. This is a valid method and does reveal a reduction in max drawdown, if one occurs. The second method requires additional programming but does help with eliminating the necessity of keeping track of multiple systems. If more than one system were to be traded simultaneously in separate accounts, then capital would need to be allocated to cover all open positions margins. Even if one system is long and the other short. If the systems are combined into one account, then the net position (Long + Short = Flat) is recorded, and therefore there is no margin requirement.\n\nTable 3.4 shows the performance of the Turtle system using both System 1 with the LTLF engaged and System 2 turned on.\n\n**Table 3.4** Tandem Performance of Turtle System 1 with LTLF and Turtle System 2 for the Sample Portfolio\n\n**Summary** | **Value** | **Markets** | **PNL**  \n---|---|---|---  \nTotal Return | $587,797.03 | @AD |  $14,260.00   \nTotal Realized Return | $556,674.53 | @BP |  ($3,718.75)  \nGross Profit | $4,380,499.89 | @C |  ($5,712.50)  \nGross Loss | \u2212$3,823,825.36 | @CC |  ($39,040.00)  \nOpen Trade P\/L | $31,122.50 | @CL |  $113,720.00   \n|  | @CT |  $10,195.00   \nNumber of Trades | $4,048.00 | @DX |  $45.00   \nNumber of Winning Trades | $1,334.00 | @EC |  $77,087.50   \nNumber of Losing Trades | $2,711.00 | @ED |  ($11,656.25)  \n% Profitable | 32.95% | @EMD |  ($47,440.00)  \n|  | @ES |  ($36,700.00)  \nAverage Trade | $137.52 | @FC |  $26,025.00   \nAverage Trade (%) | \u22120.02% | @FV |  ($10,285.93)  \nStandard Deviation | $3,869.70 | @GC |  ($4,420.00)  \nStandard Deviation Trade % | 36.41% | @HG |  $104,950.00   \nLargest Winning Trade | $53,275.00 | @HO |  $97,538.00   \nLargest Losing Trade | \u2212$7,925.00 | @KC |  ($4,225.00)  \n|  | @KW |  ($2,412.50)  \nProfit Factor | 1.15 | @LC |  ($14,040.00)  \nAverage Win\/Average Loss | 2.33 | @LH |  $14,260.00   \nSharpe Ratio | 0.13 | @NG |  $36,410.00   \nK-Ratio | 0.18 | @OJ |  $2,547.50   \nReturn Retracement Ratio | 0.68 | @PL |  $104,195.00   \n|  | @RB |  $59,458.80   \nCompounded Annual Return | 14.10% | @S |  $46,856.25   \nCompounded Monthly Return | 1.09% | @SB |  $15,202.40   \n|  | @SI |  $169,425.00   \nAverage Annual Return | $39,186.47 | @TF |  ($94,540.00)  \nAverage Annual Return (%) | 22.93% | @TU |  ($9,118.73)  \nAverage Monthly Return | $3,339.76 | @TY |  $550.00   \nAverage Monthly Return (%) | 1.91% | @US |  ($11,756.25)  \n|  | @W |  ($6,000.00)  \nPercent Days Profitable | 50.77% |  |   \nPercent Months Profitable | 52.27% |  |   \nPercent Years Profitable | 53.33% |  |   \n|  |  |   \nCommissions on Currencies | $202,825.00 |  |   \nCommissions on Futures | $0.00 |  |   \nCommissions on Equities | $0.00 |  |   \nTotal Commissions | $202,825.00 |  |\n\nThe results are better than the simple 20-day breakout but worse than the 20-day breakout with LTLF engaged, and much worse than the simple 55-day breakout. You would think the combination of the two systems would at least beat the System 1 component. In some markets, it did. The only explanation that I can come up with is: Let's say a trade is entered on 20-day breakout, and turns out to be a winner. The next 20-day breakout is ignored, but a subsequent 55-day breakout is taken in the same direction at a worse price and with probably higher risk\u2014a 20-day trailing stop versus a 10-day trailing stop. Following this line of thought, let's assume System 1 has a winner via a 20-day breakout, so the next 20-day breakout is skipped, but the market moves sufficiently in the same direction to trigger a System 2 trade. This trade occurs at a much worse price with an inherently higher level of risk. Still assuming, let's say the System 2 trade is stopped out for a large loss. The net between the two trades could theoretically be a loss, whereas if the trader had just traded System 1, she would have realized a winning situation. So there really isn't any synergy between the two components traded in this manner. Also due to the high correlation between the two components, combining them by trading them independently would probably still not provide any synergy. The clear winner here is the simple 55-day breakout.\n\n## Trend-Trading Battle Royale\n\nWe have reviewed a few trend-following trading algorithms and their results. Thus far, the results have shown some merit. Enough merit to continue research in this form of trading. There are many trend-following algorithms out there, but the perpetual question of which is the best has never really been _truly_ answered. Perry Kaufman has come very close in his _New Trading Systems and Methods_ books. Why hasn't this question ever been fully answered in an easy-to-interpret format? The main obstacle has always been the inability to compare apples to apples. However, with the tools I now have at my fingertips, maybe we can finally put this question to rest.\n\nThe following trend-following algorithms will be tested on the same data with the same commission\/slippage deduction, over the same time period and on a one-contract basis and on the same testing platform, and we have plenty of capital. The trend-following algorithms that will be tested will utilize their own entry-and-exit techniques. However, a maximum loss of $3,000 will be applied to each algorithm. I figure that most of the algorithms' exits will occur prior to this stop level, and most traders would feel more comfortable knowing that at this trading level, the risk of ruin should be sufficiently small to initiate one or more of the programs.\n\n##### Pause the Battle\u2014Let's Talk Risk of Ruin\n\nRisk of ruin is a very useful tool in determining the likelihood of a trader losing his entire bankroll or reducing it to a level that trading cannot continue. Calculating this risk (RoR) is quite simple; all you need is the amount of capital to be risked on any given trade and the probability of success, also known as the algorithm's trading advantage:\n\nwhere A is the algorithm's trading advantage and C is the number of units in your account. To calculate A, subtract the percent chance of loss from percent chance of win. So, if you expect to win 54% of your trades, the trading advantage would be 8% (54% \u2212 46% = 8%). To calculate C, simply divide 1 by the percent of capital you are willing to risk on each trade. So, if you are willing to risk 5% of any given trade, then you have 20 units. Plugging A and C into the formula you come up with ((1 \u2212 0.08) \/ (1 + 0.08)) ^ 20. The result turns out to be around 4%. The RoR for risking 5% of your capital on each trade with an algorithm that wins 54% of time is a relatively low 4%. If the percent of wins is 50% or less, the RoR is 100%. The base number that is being raised by unit size C decreases as the trading advantage increases. As long as the percent wins are greater than 50%, the base number will always be less than one. Unit size C decreases as risk per trade increases. Raising a number less than one by higher values decreases the result or in this case the RoR. In other words, the RoR is indirectly proportional to the risk.\n\nThis simple formula only works with algorithms that win more often than they lose and each win and loss is identical. If you plug in a winning percentage of 50% or less, the RoR is guaranteed. If you applied this very simple formula to the vast majority of trend-following systems, you would never take the first trade. When determining RoR you must look beyond percent wins and risk. Most trend-following systems win less than 40% of their trades. The key performance metrics in this case are the average win and average loss in dollars. You can still win with a sub-50% winning system as long as your average trade (average win \/ average loss) is sufficiently high enough. The formula for RoR for when wins and losses are not identical is unfortunately much more complex. I refer you to Ralph Vince's excellent book, _Portfolio Management Formulas_ (Wiley, New York, 1990) if you want to see the derivation of this formula. The formula is complicated but the algorithm is quite simple if you have the correct performance metrics of your algorithm. An Excel spreadsheet that utilizes this formula is available from this book's companion website. Here is an example of the worksheet and its computed RoR using performance metrics from a typical trend-following algorithm (see Figure 3.2.).\n\n**Figure 3.2** A three-dimensional contour chart of Bollinger performance across a subset of the parameters that were originally optimized.\n\nSource: AmiBroker\n\n**_Risk of Ruin Calculation_**  \n---  \nAverage Win $ | $400 | \u226a--Input Here  \nAverage Loss $ | $200 | \u226a--Input Here  \nPercent Wins | 35.0% | \u226a--Input Here  \nInitial Capital $ | $35,000 | \u226a--Input Here  \nAmount of Pain % (% of Capital) | 50.0% | \u226a--Input Here  \nPayoff Ratio | 2 |   \nPercent Losers | 65.0% |   \nAverage Win % | 1.1% |   \nAverage Loss % | 0.6% |   \nSum of Possible Events (SoPE) | 0.00029 |   \nSq.Root of Sum of Squares of (SoPE) | 0.00818 |   \n\" **P** \" | 0.517460757 |   \nRoR | 1.39814% |\n\nThis trend-following system wins only 35% of the time but has a payoff ratio of 2 (Average Win$ \/ Average Loss$). If your pain threshold is 50% of your starting capital, then you run a risk of ruin of nearly 1.4%. With these algorithm performance metrics and RoR you shouldn't have a problem pulling the trigger on this system. Before you start trading be sure to check the risk of ruin and see if it fits your expectations. One more note before we resume the battle: Money management will not be utilized so that we can see the raw capabilities of each system.\n\n#### Resume the Battle\n\nThere will be a total of six popular trend-following algorithms tested, the Turtle system and the five shown in Box 3.3. Each test will consist of three subtests. The subtests will reflect three different parameter sets that will cause the algorithms to trade in a shorter-term, intermediate-term, and longer-term manner. For example, a simple moving average crossover algorithm will be tested utilizing moving average lengths of 19, 39, and 100. These parameters should cover the three time horizons\u2014short, intermediate, and long. The parameter lengths were chosen without the benefit of hindsight\u2014in other words, they were chosen based on my perceived applicability to obtain the three time horizons without looking at prior test results. This isn't really a true statement, because my experience through testing systems over the years does introduce a slight hindsight bias. All entries and exits will be executed on the closing price of the bar that generates the trade signal.\n\n* * *\n\n### Box 3.3 Battle Royale\u2014And the Competitors Are\n\n#### Single Simple Moving Average Crossover (SMA)\n\n  1. Buy when close crosses above moving average\n  2. Sell short when close crosses below moving average\n  3. Liquidate long if close < entryPrice \u2212 3000\/bigPointValue\n  4. Liquidate short if close > entryPrice + 3000\/bigPointValue\n  5. Utilize moving average lengths of 19, 39, and 100\n\n#### Double Simple Moving Average Crossover (DMA)\n\n  1. Buy when shorter moving average crosses above longer moving average\n  2. Sell short when short moving average crosses below longer moving average\n  3. Liquidate long if close < entryPrice \u2013 3000\/bigPointValue\n  4. Liquidate short if close > entryPrice + 3000\/bigPointValue\n  5. Utilize moving average lengths: \n    1. Short lengths: 9, 19, 49\n    2. Long lengths: 19 ,49, 99\n\n#### Triple Simple Moving Average Crossover (TMA)\n\nTriple moving averages require two criteria to be met before a trade entry:\n\n  1. Buy: \n    1. shortest moving average and intermediate moving average both must be greater than longer-term average\n    2. Buy when shortest moving average crosses above intermediate moving average\n  2. Sell short: \n    1. shortest moving average and intermediate moving average both must be less than longer-term average\n    2. Sell short when shortest moving average crosses below intermediate moving average\n  3. Liquidate long when shortest moving average crosses below intermediate moving average or if close < entryPrice \u2013 3000\/bigPointValue\n\nLiquidate short when shortest moving average crosses above intermediate moving average or if close > entryPrice + 3000\/bigPointValue\n\nUtilize the following moving average lengths:\n\n    1. Short lengths: 9, 19, 49\n    2. Intermediate lengths: 19, 49, 199\n    3. Long lengths: 49, 199, 399\n\n#### Donchian Channels (DC)\n\n  1. Buy on penetration of highest highs for past _N_ days\n  2. Sell short on penetration of lowest lows for past _N_ days\n  3. Liquidate long on penetration lowest lows for past _N_ \/ 2 days or if close < entryPrice \u2013 3000\/bigPointValue\n\nLiquidate short on penetration of highest highs for past _N_ \/ 2 days or if close > entryPrice + 3000\/bigPointValue\n\nUtilize the following channel lengths:\n\n  4.     1. Short lengths: 20 entry, 10 exit\n    2. Intermediate lengths: 40 entry, 20 exit\n    3. Long lengths: 100 entry, 50 exit\n\n#### Bollinger Bands (BB)\n\n  1. Buy on penetration of two standard deviations above _N_ -day moving average.\n  2. Sell short on penetration of two standard deviations below _N_ -day moving average.\n  3. Liquidate long on penetration of _N_ -day moving average or if close < entryPrice \u2013 3000\/bigPointValue.\n  4. Liquidate short on penetration of _N_ -day moving average or if close > entryPrice + 3000\/bigPointValue.\n\nUtilize the following moving average lengths of 20, 50, and 200.\n\nThe results are shown in Table 3.5.\n\n**Table 3.5** Single Moving Average (SMA) Crossover Algorithm\n\n**SMA** | **Simple Moving Average Crossover** |  |  |  |  |  |   \n---|---|---|---|---|---|---|---  \n**Length** | **Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners**  \n19 | \u2212922061 | \u221215.08 | \u2212986733 | 0.91 | 48.53 | 15814 |  \u221258.31 |  9.78 | 23.04  \n39 |  \u221215191 | \u22120.1 | \u2212612726 |  1 | 16.1  | 11114 |  \u22121.37 | 13.49 | 19.97  \n100 |  564180 |  2.91 | \u2212626574 | 1.11 | 13.92 |  6543 |  86.23 | 21.94 | 16.25  \n200 |  487626 |  2.58 | \u2212492640 | 1.13 | 14.95 |  4495 |  108.48 | 30.67 | 12.77\n\n* * *\n\nTable 3.5 starts the battle and sets the benchmark with the results of the SMA algorithm.\n\nLet's hope the other algorithms have better luck. Before moving on to the results of the other algorithms a clear explanation of performance metrics might be a good idea:\n\n  * **Net Profit** \u2014summation of all profits or losses after deduction of execution costs.\n  * **Sys Drawdown** \u2014the largest peak-to-valley decline experienced in the complete portfolio. This is a good metric to use in the calculation of how much capital to allocate to a trading algorithm. However, remember this is a onetime event and in many cases the **Start Trade Drawdown** analysis might be a better indicator for capitalization purposes. Start Trade Drawdown shows the different probabilities of the occurrence of different magnitudes of drawdown. This analysis will be detailed in Chapter 8 along with Monte Carlo simulation.\n  * **Annual Return %** \u2014compounded annual return based on a cool million-dollar investment.\n  * **Number of Trades** : number of trades.\n  * **Profit\/Loss** \u2014(Profit of winners \/ Loss of losers) \/ Number of trades. This is also known as the expectancy in terms of dollars.\n  * **Bars Held** \u2014the number of bars, on average, a trade is held. You can use this as a guide to align different parameters sets across different algorithms.\n  * **Total Transaction Costs** \u2014summation of execution fees.\n  * **Profit Factor** \u2014profit of winners divided by loss of losers.\n  * **Ulcer Index** \u2014square root of sum of squared drawdowns divided by number of bars. It measures the volatility in the downward direction.\n  * **Sharpe Ratio** \u2014indicator of consistency. Average return divided by the standard deviation of returns sampled on daily, monthly, or yearly basis. The higher the better.\n\nArmed with these performance metrics it is plain to see that the SMA as a complete trading algorithm is not that good. I hope you weren't surprised by this fact. Don't get me wrong; simple moving averages have their place in technical analysis, but not as a complete trading algorithm. Results might improve utilizing an exponential moving average, but not enough to consider it as your main trading tool. Notice how the long-term version handily outperformed its shorter and intermediate-term counterparts.\n\nIf a single moving average doesn't cut the mustard, then what about two? Table 3.6 shows the DMA utilizing three different lengths.\n\n**Table 3.6** Double Moving Average (DMA) Crossover Algorithm\n\n**DMA** | **Double Moving Average** |  |  |  |  |  |  |   \n---|---|---|---|---|---|---|---|---  \n**Length** | **Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners**  \n9,19 |  \u221229879 | \u22120.19 | \u2212548958 | 1  | 20.2  | 8529 |  \u22123.5  | 17.46 | 35.08  \n19,49 |  61910 |  0.39 | \u2212743755 | 1.01 | 21.29 | 3997 |  15.49 | 36.11 | 33.1   \n49,99 |  996693 |  4.53 | \u2212692714 | 1.24 | 11.39 | 2307 |  432.03 | 61.8  | 34.07\n\nA vast improvement. The short term and intermediate term still trail their big brother, but both improved. The DMA with lengths of 49 and 99 cranked out profits close to a million dollars. This feat was accomplished by trading just 2,307 times\u2014less than 100 times a year across 35 markets. Could the winner already be revealed? If the double moving average stood head and shoulders above the single moving average, where do you think the triple moving average will wind up? Table 3.7 gives the details.\n\n**Table 3.7** Triple Moving Average (TMA) Crossover Algorithm\n\n**TMA** | **Triple Moving Average** |  |  |  |  |  |  |   \n---|---|---|---|---|---|---|---|---  \n**Length** | **Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **percent of Winners**  \n9,19,49 |  \u221271434 | \u22120.47 | \u2212441997.88 | 0.99 | 14.68 | 5160 |  \u221213.84 |  18.21 | 33.93  \n19,49,199 |  527917 |  2.75 | \u2212374103  | 1.22 |  9.41 | 1850 |  285.36 |  43.49 | 32.7   \n49,199,399 |  711878 |  3.5  | \u2212211493  | 1.77 |  6.09 |  528 |  1348.25 | 137.61 | 32.01\n\nThe trend continues. The longer-term triple crossover produced over $700K on just 528 trades. It made 70 percent of what the longer-term DMA produced but did it with a less than one-fourth the number of trades. It averaged $1,348 on every trade. Pretty impressive, but this is more like investing than trading. What if we changed the lengths of the parameters to try and match the number of trades from the longer-term DMA? Would that be curve-fitting? The parameter values that produced so few trades were initially chosen based on a preconceived notion that a certain number of trades and trade duration would be achieved. Tuning the parameters to get us into the ballpark of our objective, whatever that might be, is not over-curve-fitting. So far it looks like the longer-term versions of the DMA and TMA are producing somewhat comparable results. The next system up gets us out of the moving averages and back into Turtle mode. Table 3.8 shows the performance of three different Donchian lengths.\n\n**Table 3.8** Donchian Channel Breakout for Different Donchian Lengths\n\n**Donchian** | **Donchian Channel Breakout** |  |  |  |  |  |   \n---|---|---|---|---|---|---|---  \n**Length** | **Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners**  \n20\/10 |  278329 | 1.59 | \u2212257735.79 | 1.05 | 9.61 | 6070 |  45.85 | 17.43 | 35.34  \n40\/20 |  754015 | 3.67 | \u2212361607  | 1.18 | 8.43 | 2963 | 254.48 | 35.45 | 36.85  \n100\/50 | 1114437 | 4.91 | \u2212367415  | 1.51 | 7.07 | 1366 | 815.84 | 68.65 | 32.01\n\nThe Turtle hit a home run! All three versions were profitable and the 100-day Donchian beat the longer-term DMA: $1.1 million on only 1366 trades. It's going to be hard to knock this system off of the podium. If any system can, it could be one based off of John Bollinger's world famous bands. Table 3.9 tells the tale of the tape.\n\n**Table 3.9** Bollinger Band Breakout\n\n**Bollinger** | **Bollinger Band Breakout** |  |  |  |  |  |   \n---|---|---|---|---|---|---|---  \n**Length** | **Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **% of Winners**  \n20-Day |  \u221241360 | \u22120.27 | \u2212343007 | 0.99 | 13.83 | 4860 |  \u22128.51 | 16.06 | 32.63  \n50-Day |  728097 |  3.57 | \u2212321144 | 1.24 |  8.03 | 2396 |  303.88 | 32.94 | 32.1   \n100-Day |  756849 |  3.68 | \u2212362325 | 1.37 |  8.33 | 1364 |  554.87 | 57.83 | 30.94\n\nWell that was a little bit of a disappointment. The trend did not continue as the longer-term version did not win the battle among the different Bollinger Band lengths. The intermediate-term version just barely nudged out the longer term with slightly less profit, but 10 percent less drawdown. The big news is the Bollinger Band algorithm came in third place when compared with the triple moving average and Donchian channel.\n\nSo the winner is... Hold on! We are getting some complaints out of the Bollinger Band camp. They are saying it's not a fair test. They want a rematch! They say that we held the number of standard deviations constant at two. Just changing the lengths of the moving averages and not the width of the bands (number of standard deviations) does not reflect the true capability of such a powerful algorithm. Now the other camps are complaining, too! What to do?\n\nWhy don't we have a grudge match between the four best algorithms: DMA, TMA, DC, and BB. But this time, let's let the computer choose the best parameter sets. AmiBroker has produced the results thus far, so let's lean on it a little bit and have it help us level the playing field. Let's optimize the living daylights out of each algorithm and see what parameter sets produce the best profit-to-drawdown ratio. Let's not stop at optimizing the lengths of the parameters but the protective stop as well. The Bollinger Band camp will be happy because not only will we optimize the parameter lengths, and the protective stop, but we will also optimize the width of the bands. Does this smell of over-curve-fitting? Sure it does, but we aren't developing a trading system here, we are just doing an analysis.\n\nNow that we have a plan, how can we use AmiBroker to carry it out? The algorithms are already programmed (source code will be graciously provided in the appendices), but how many optimizations are we talking about? Quickly doing the math in my head\u2014millions! Table 3.10 shows the exact number of permutations and the range that each parameter will be tested across.\n\n**Table 3.10** Comparison of Parameters for Algorithms Based on Different Strategies\n\n**Donchian** | **Start** | **Stop** | **Increment** | **# Iterations**  \n---|---|---|---|---  \nEntry Len | 9 | 100 | 1 | 92  \nExit Len | 5 | 100 | 1 | 96  \nProtective Stop | 3000 | 6000 | 500 | 7  \nTotal |  |  |  | 61824  \n**Bollinger** | **Start** | **Stop** | **Increment** | **# Iterations**  \nAverage Length | 20 | 200 | 1 | 181  \n\n#Std. Deviations | 0.25 | 3 | 0.05 | 56  \nProtective Stop | 3000 | 6000 | 500 | 7  \nTotal |  |  |  | 31752  \n**TMA** | **Start** | **Stop** | **Increment** | **# Iterations**  \nShort Length | 9 | 100 | 1 | 92  \nIntermediate Length | 19 | 200 | 2 | 91  \nLong Length | 49 | 399 | 3 | 117  \nProtective Stop | 3000 | 6000 | 500 | 7  \nTotal |  |  |  | 6856668  \n**DMA** | **Start** | **Stop** | **Increment** | **# Iterations**  \nShort Length | 9 | 100 | 1 | 92  \nIntermediate Length | 19 | 200 | 2 | 91  \nProtective Stop | 3000 | 6000 | 500 | 7  \nTotal |  |  |  | 58604\n\nThe number doesn't get really large until we move to the TMA optimization\u2014nearly seven million. This is due to the fact that we are optimizing four parameters. Well, it was a good idea. We will just have to go with what we have, because who has the time to do a multimillion-iterations optimization loop? Wait a minute\u2014this book is about algorithms\u2014right? This task is definitely not doable with a run-of-the-mill exhaustive search optimization engine. An exhaustive search optimization is where you use brute force and test every possible combination\u2014no stone is left unturned. Come to think of it, AmiBroker has access to three forms of genetic optimization. With the use of genetic optimization (GO) and artificial intelligence, we can accomplish the task and do it in just a few minutes. Not all iterations are executed, only those that fit a certain fitness. Imagine millions and millions of optimizations carried out in minutes instead of days or even months. Genetic optimization is an awesome tool, and this very topic will be discussed in Chapter 8. Right now, even if you don't understand artificial intelligence, just know it is the tool that we need to accomplish this task.\n\nAll optimizations will be carried out at the portfolio level. Individual market optimizations will not be conducted. This will cut down considerably on the total number of optimizations. Each set of parameters will be applied to each market in the portfolio, and only one set will be chosen to represent each algorithm. Also, like I said earlier, we are not trying to fit a system to data in an attempt to create a trading plan. We are just trying to cover all bases in this algorithm comparison. The winning parameter sets will be used for illustration purposes only.\n\nAmiBroker was able to complete the task in roughly 40 minutes. This time period covered all optimizations across the four algorithms. Pretty darn impressive! Table 3.11 shows the top 10 parameter sets for each algorithm and their associated profit\/loss, maximum drawdown, number of trades, average trade, profit factor, Ulcer index, and percent winners.\n\n**Table 3.11** Top 10 Parameter Sets for Each Algorithm\n\n**Algo Name** | **Net Profit** | **CAR** | **Max DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **%Winners** | **ShortLen** | **InterLen** | **StopDollars**  \n---|---|---|---|---|---|---|---|---|---|---|---|---  \nDMA | 1338431.01 | 5.59 | \u2212491839.05 | 1.44 | 8.96 | 1991 | 672.24 |  71.42 | 25.62 |  9 | 162 | 6000  \nDMA | 1336531.01 | 5.59 | \u2212492539.05 | 1.44 | 8.97 | 2010 | 664.94 |  70.75 | 25.72 |  9 | 162 | 5500  \nDMA | 1333731.01 | 5.58 | \u2212493339.05 | 1.44 | 8.99 | 2038 | 654.43 |  69.79 | 25.66 |  9 | 162 | 5000  \nDMA | 1330531.01 | 5.57 | \u2212494639.05 | 1.43 | 9.01 | 2070 | 642.77 |  68.73 | 25.6  |  9 | 162 | 4500  \nDMA | 1325931.01 | 5.56 | \u2212495939.05 | 1.43 | 9.04 | 2116 | 626.62 |  67.26 | 25.14 |  9 | 162 | 4000  \nDMA | 1319431.01 | 5.54 | \u2212498839.05 | 1.42 | 9.09 | 2181 | 604.97 |  65.28 | 25.03 | 9 | 162 | 3500  \nDMA | 1310731.01 | 5.51 | \u2212501739.05 | 1.41 | 9.16 | 2268 | 577.92 |  62.82 | 25.04 |  9 | 162 | 3000  \nDMA | 1309059.66 | 5.51 | \u2212515389.7  | 1.43 | 9.15 | 1948 | 672  |  72.98 | 26.49 | 10 | 156 | 6000  \nDMA | 1308892.36 | 5.51 | \u2212522360.35 | 1.44 | 9.11 | 1941 | 674.34 |  73.23 | 25.66 |  9 | 164 | 6000  \nDMA | 1308303.31 | 5.5 | \u2212473577.53 | 1.43 | 8.96 | 2004 | 652.85 |  70.96 | 25.45 |  9 | 160 | 6000  \n**Algo Name** | **Net Profit** | **CAR** | **Max DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **%Winners** | **ShortLen** | **InterLen** | **StopDollars**  \nTMA | 1124260.22 | 4.94 | \u2212179037.08 | 2.11 | 4.03 |  574 | 1958.64 | 143.68 | 35.89 | 49 | 194 |  309  \nTMA | 1100730.77 | 4.87 | \u2212167743.6  | 2.09 | 3.96 |  573 | 1921  | 143.33 | 34.73 | 49 | 194 |  312  \nTMA | 1071646.95 | 4.78 | \u2212168698.98 | 2.04 | 3.97 |  576 | 1860.5  | 142.43 | 34.55 | 49 | 194 |  306  \nTMA | 1088110.19 | 4.83 | \u2212181550.77 | 2.06 | 4.01 |  575 | 1892.37 | 143.43 | 35.13 | 48 | 194 |  309  \nTMA | 1049097.29 | 4.7 | \u2212171951.93 | 2.02 | 3.91 |  577 | 1818.19 | 142.48 | 35.7  | 47 | 194 |  309  \nTMA | 1069863.15 | 4.77 | \u2212169583.3  | 2.03 | 3.95 |  573 | 1867.13 | 143.02 | 34.38 | 48 | 194 |  312  \nTMA | 1023856.17 | 4.62 | \u2212163224.35 | 2  | 3.93 |  566 | 1808.93 | 143.04 | 35.34 | 49 | 192 | 315  \nTMA | 1005683.72 | 4.56 | \u2212160448.4  | 1.96 | 3.82 |  578 | 1739.94 | 141.37 | 34.95 | 47 | 194 |  306  \nTMA | 1040045.32 | 4.67 | \u2212168261.18 | 1.98 | 3.95 |  577 | 1802.5  | 142.02 | 34.14 | 48 | 194 |  306  \nTMA | 1083985.47 | 4.82 | \u2212195109.25 | 2.06 | 4.17 |  569 | 1905.07 | 144.76 | 36.03 | 49 | 192 |  309  \n**Algo Name** | **Net Profit** | **CAR** | **Max DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **%Winners** | **ShortLen** | **InterLen** | **StopDollars**  \nBollinger | 1033937 | 4.65 | \u2212292963.4  | 1.63 | 6.09 | 1031 | 1002.85 |  67.16 | 36.28 | 101  |  2.25 | 3500  \nBollinger | 1020469 | 4.61 | \u2212296895  | 1.63 | 6.22 | 1012 | 1008.37 |  68.34 | 35.67 | 103  |  2.25 | 3500  \nBollinger |  885350 | 4.15 | \u2212238296.63 | 1.75 | 6.97 |  657 | 1347.56 |  87.76 | 38.05 | 129  |  2.5  | 4000  \nBollinger | 1024112 | 4.62 | \u2212296874.3  | 1.63 | 6.17 | 1022 | 1002.07 |  67.84 | 36.11 | 102  |  2.25 | 3500  \nBollinger | 1016747 | 4.6  | \u2212296905.3  | 1.61 | 6.3  | 1050 |  968.33 |  66.47 | 35.81 | 100  |  2.25 | 3500  \nBollinger |  877685 | 4.12 | \u2212249438.93 | 1.77 | 6.91 |  674 | 1302.2  |  82.73 | 36.65 | 123  |  2.5  | 3500  \nBollinger | 1182979 | 5.13 | \u2212384837.03 | 1.38 | 7.73 | 2086 |  567.1  |  51.44 | 31.4  |  89  |  1.25 | 5000  \nBollinger |  844532 | 4  | \u2212236280.73 | 1.77 | 6.95 |  628 | 1344.8  |  89.57 | 37.58 | 135  |  2.5  | 3500  \nBollinger | 1195823 | 5.17 | \u2212395271.98 | 1.39 | 7.71 | 2062 |  579.93 |  52  | 31.43 |  90  |  1.25 | 5000  \nBollinger |  863732 | 4.07 | \u2212238588.03 | 1.72 | 7.02 |  661 | 1306.71 |  86.3  | 37.52 | 126  |  2.5  | 4000  \n**Algo Name** | **Net Profit** | **CAR** | **Max DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg Profit\/Loss** | **Avg Bars Held** | **%Winners** | **ShortLen** | **InterLen** | **StopDollars**  \nDonchian | 972653 | 4.45 | \u2212183567.7  | 1.4  | 4.93 | 0.9  | 2088  | 465.83 | 31.92 |  35.87 | 92  |  19  \nDonchian | 964178 | 4.42 | \u2212191985.6  | 1.39 | 4.98 | 0.89 | 2136  | 451.39 | 30.54 |  36.05 | 92  |  18  \nDonchian | 959512 | 4.4  | \u2212192721.65 | 1.39 | 5.08 | 0.87 | 2147  | 446.91 | 30.52 |  35.96 | 91  |  18  \nDonchian | 966614 | 4.43 | \u2212176310.85 | 1.39 | 4.98 | 0.89 | 2110  | 458.11 | 31.88 |  35.78 | 90  |  19  \nDonchian | 963078 | 4.42 | \u2212193489.13 | 1.4  | 5.1  | 0.87 | 2116  | 455.14 | 30.51 |  36.06 | 94  |  18  \nDonchian | 967138 | 4.43 | \u2212185009.95 | 1.39 | 5.03 | 0.88 | 2099  | 460.76 | 31.88 |  35.87 | 91  |  19  \nDonchian | 954784 | 4.39 | \u2212195440.55 | 1.39 | 5.15 | 0.85 | 2129  | 448.47 | 30.48 |  35.93 | 93  |  18  \nDonchian | 1038561 | 4.67 | \u2212214593.15 | 1.4  | 5.48 | 0.85 | 2049  | 506.86 | 37.22 |  34.75 | 81  |  23  \nDonchian | 959044 | 4.4  | \u2212184915.55 | 1.38 | 5.08 | 0.87 | 2168  | 442.36 | 31.83 |  35.56 | 85  |  19  \nDonchian | 954212 | 4.39 | \u2212175255.05 | 1.37 | 5.03 | 0.87 | 2183  | 437.11 | 31.8  |  35.41 | 84  |  19\n\nTable 3.12 shows the best parameters sets from each optimization.\n\n**Table 3.12** The Best Parameter Sets for Each Optimization\n\n**Best Parameter Sets** |  |  | **Money** | **Net** | **Max.** | **Profit to** | **#**  \n---|---|---|---|---|---|---|---  \n**Algo Name** | **Param1** | **Param2** | **Param3** | **Mngmt. Stop** | **P \/ L** | **Drawdown** | **Drawdown** | **Trades**  \nDMA |  9 | 162 | \u2212 | 3000 | 1310731 | \u2212501739 | 2.61 | 2268  \nTMA |  49 | 194 | 309 | 3000 | 1124260 | \u2212159761 | 7.04 |  574  \nDC |  92 |  19 | \u2212 | 3000 |  959512 | \u2212192721 | 4.98 |  829  \nBB | 101 | 2.25 | \u2212 | 3500 | 1033937 | \u2212292963 | 3.53 | 1031\n\nAnd the winner is\u2014triple moving average! The results show a unanimous champion, the triple moving average. The Donchian algorithm came in second with the Bollinger algorithm close behind in third. The winner produced a tremendous profit-to-drawdown ratio of 7.04. Quite a bit better than the second place Donchian Channels. The TMA accomplished this feat by trading only 574 times. Before discussing the winners let's discuss the also-rans.\n\nThe DMA algorithm came in last place. It looks like using the parameter sets revolving around an ultra-short length of 9, a long length of 160, and a $5,000 money management stop shows the best results. These numbers were derived from taking an average of the top 10 parameter sets. These numbers really don't make sense; the normal 1:2 and 1:3 ratio wasn't anywhere near the top. Basically, the parameters are telling us the nine-period moving average will eventually get on the trend, after going back and forth for a while, and hold onto it tenaciously.\n\nThe Bollinger Band algorithm came in third place, but with very respectable results. The parameter sets revolving around a 100-bar moving average length and 2.25 standard deviations and a $4,000 money management stop show the best results. One of the better parameter sets sticks out like a sore thumb (90, 1.25, 5000) due to the fact the surrounding parameters are so different, yet produce similar results. This set might be sitting on top of a peak. Figure 3.2 shows a three-dimensional contour chart of the Bollinger performance across a subset of the parameters that were originally optimized. The money management stop was held at a constant so that a three-dimensional representation could be generated. Notice where the 88 length and 1.25 width parameter lies on the chart. Almost smack-dab on a peak. This demonstrates a lack of robustness\u2014in other words, history would have to repeat itself almost exactly for this set to be productive into the future. A parameter set on a high but somewhat level plateau is a better choice.\n\nThe TMA looks to be the best by far; it nearly kicked the rest of the competitors off of the podium. The results are very impressive, with a very high profit-to-drawdown ratio. If you look at the top 10 parameter sets, you will see the almost exact performance for each. The only parameter that varied was the long-term length. The overall effectiveness of the algorithm was not affected by variations in this parameter. It just needs to be sufficiently large. This demonstrates a high level of robustness. Also notice the ratio of the short- and intermediate-term lengths to the long-term lengths: 1 to 4 and 1 to 6, respectively. This ratio is not one that is mainly prescribed when trading the TMA.\n\nThe Donchian algorithm looks to be nearly as good nearly as the TMA. It looks like using the parameter sets revolving around an entry length of 90, an exit length of 19, and a $3,000 money management stop shows the best results. I was a little surprised by the outcome; I was looking for an entry length around 55 and an exit length around 20\u2014the Turtle numbers. Or parameters that fit the _N_ bar entry and _N_ \/ 2 bar exit template. The mantra \"Hard to get in, easy to get out\" definitely fits these results. Even with such a short exit parameter, trades typically lasted 30 days. The results also demonstrated robustness among the parameter sets; results stayed somewhat constant throughout a good portion of the optimization. It's no wonder this algorithm is still being widely used.\n\nThere you have it! Four very tradable algorithms. You can't ask for much more than 60K to 70K annual return for the past 15 years, can you? On a million-dollar allocation, that works out to be 6 percent or 7 percent a year. Oh, yeah, you would have had to live through a 20 percent to 50 percent maximum drawdown. Hopefully that occurred after you had built up your equity. If it occurred right off the bat, then you would be in a heap of trouble\u2014most traders would jump ship at that point and never look back.\n\nSince most traders don't have a million dollars to allocate to trading, they can't afford to trade the 35-market portfolio we are currently testing. Even though this portfolio offers a tremendous amount of diversification, it is simply too large. The key to getting these systems into the realm of reality is to come up with a smaller yet diverse portfolio. Portfolio composition, especially to beginning traders, is as important as a trading algorithm. There are several steps to developing a complete algorithmic trading plan: algorithm selection, portfolio composition, proper optimization, and trade management. Position sizing is also an important component, but that will be covered later. In the next section, a smaller portfolio will be created by culling from the current large portfolio. After an algorithm has been developed and a portfolio selected, the next step is proper optimization. The final step is to develop trade management. So far, we have just used disaster stops in the development and testing of the algorithms. Are there better ways to manage a trade than waiting for a liquidation on a system-derived exit or a disaster stop? Also, is using a function of the market such as average true range better than a pure $ stop? Trade management is very important and it covers the concepts of dollar- or volatility-derived stops, profit objectives, and trailing stops. Trend-following mechanisms do not usually work with pure profit objectives unless they are sizable. You can't limit the upside, because the truly big winners don't occur all that often. Most trend-following algorithms have an inherent trailing stop mechanism built-into their logic. When the market moves up so does a moving average, a Bollinger band, and a Donchian channel. You can incorporate a more aggressive trailing stop, but again be aware that limiting the big winners will be counterproductive.\n\n## Portfolio Composition\n\nPicking a small but diverse portfolio isn't brain surgery. The key to a good portfolio is a selection of markets that are somewhat noncorrelated and cull from different market sectors. A good portfolio should have representation from at least five sectors: currencies, financials, energies, grains, and metals. You could just randomly pick one or two markets from each sector and be done with it, but more thought should be applied to the selection process. Why not just cherry-pick the very best markets from the backtested results and use these as the future portfolio? Unfortunately, this is curve fitting even though you are wanting a portfolio with high positive expectancy.\n\nA test on market correlation should be carried out prior to a selection of any portfolio component. Figure 3.3 shows a snapshot of a portion of the current portfolio correlation matrix. As you can see the diagonal is set to one. The diagonal represents the correlation of each market to itself.\n\n**Figure 3.3** A snapshot of a portion of the current portfolio correlation matrix.\n\nThe correlation coefficients range from +1 to \u22121. A coefficient of +1 represents a 100 percent correlation and a coefficient of \u22121 also represents a 100 percent correlation, but in the opposite direction. The AD (Australian dollar) is highly correlated to GC (gold), but not very much to the CU (Euro currency), and is highly anticorrelated to DX (dollar index). If you were picking two currencies, then you would probably want to go with the AD and CU. The BP is a little too noncorrelated for my taste. So let's pick the AD and CU to represent the currencies. Moving onto energies the crude is highly correlated to both the heating oil and natural gas. Unleaded is a different story; there is very little correlation there. CL and RB (unleaded) will take care of the energies sector. The grains consist of soybeans, corn, Kansas City wheat, Chicago wheat, corn, and a few other low-volume components. According to the correlation matrix, a good mix would be the corn and KC wheat. The financials are up next and are represented by 30-year Treasury bonds, 10-year notes, 5-year notes, 2-year notes, and Eurodollars. All these markets are highly correlated to each other, so there is no advantage\/diversification by trading one with another. Trading a Treasury bond with Treasury note is like trading two bonds. Bonds are very shock sensitive to news events such as Fed and employment situation reports. So the less volatile but still with some bang is the Treasury notes. The safest soft commodity is sugar (SB). It trends well and seems less sensitive to crop reports than coffee (KC) or cotton (CT). There is very little correlation among these three markets so any one of the three would probably suffice. All of the metals in the current portfolio, silver (SV), gold (GC), copper (HG), and platinum (PL), are, as you would assume, highly correlated, so like the financials only one should be chosen for this initial portfolio. Copper is the only nonprecious metal in the list and used extensively in manufacturing so let's pick it. The initial portfolio will consist of the following markets: AD, CU, CL, RB, TY, C, KW, SB, and HG. You will notice that the stock indices were left out. The indices are a totally different animal. In an interview with Richard Dennis by Arthur Collins, the King Turtle stated:\n\n> With the exception of the S&Ps, they're all the same. You have to treat them the same, for no reason other than sample size. You can optimize each system in each market individually or you can optimize them together. It's better to optimize them together.\n> \n> \u2014 ** _Technical Analysis of Stocks and Commodities_ , April 2005 Issue**\n\n### An Act of Futility?\n\nOkay, we now have a portfolio but on which algorithm should we risk our capital? Also should we use the best-optimized parameters for each algorithm that was derived from the global optimizations? Any of the four algorithms could be used\u2014they all showed positive expectancy. The \"optimal\" parameters that we previously derived were for illustration purposes only. These tests simply provided evidence that these algorithms have a technical edge. A new set of tests have to be set up with our new portfolio across the four best performing algorithms. Also, we need to come up with a set of parameters for each algorithm that will hopefully stand the test of time. The test of time can be accomplished by utilizing out-of-sample (OOS) data. OOS data is a segment of historical data that the algorithms did not see during their developmental periods. The algorithms can be optimized on in-sample data (IS) and the \"optimal\" parameter sets can then be applied to data that has not yet been seen. The ratio between OOS and IS is a subject of much debate. Some believe you should give the algorithm more data (IS) to be trained on in hopes that the parameters will reflect more diverse market conditions. Others feel that a longer walk-forward (OOS) test will demonstrate an algorithm's true robustness. I don't believe there is a universal answer to this conundrum, but experience has shown that a 1-to-3 ratio seems to work. The test period that we have been using is 15 years, so using this ratio we come up with roughly five years of OOS and roughly 10 years IS.\n\nBefore we carry out these tests the subject of individual market optimization needs to be brought up. I have never seen, in all my years of testing, a trading algorithm utilizing different parameters on a market-by-market basis survive the test of time. Even trading algorithms that just utilize different parameters on a sector-by-sector basis haven't been all that successful over the long run. All this optimizing simply leads to over-curve-fitting. The trading systems that have stood the test of time have one and only one parameter set for the entire portfolio. Now that that question has been answered, what about periodical portfolio reoptimization? This is the idea of optimizing an algorithm for the first five or so years and then carrying over for the next year or two. At the end of the carryforward period, the parameters are reoptimized over the new unseen data and then carried forward again. This form of optimization uses a sliding IS data window. The results from the OOS data are accurate, since the parameters were not trained on that data until after the fact. This form of optimization is discussed extensively in Chapter 8.\n\nStarting with the Donchian algorithm and our new portfolio, let's optimize the parameters from January 2000 through December 2009. Using the best parameters set from this test, the algorithm will be tested across the remaining data, January 2010 through August 2015, to evaluate the robustness of the selection.\n\nTable 3.13 shows the best results for the Donchian algorithm IS time period. The parameter set that I chose was 98, 25, and $4,000. This was a top echelon set that had a reasonable protective stop. In addition, the two length parameters were located on a plateau\u2014they were surrounded by similar values.\n\n**Table 3.13** The Best Results for the Donchian Algorithm for In-Sample Time Period\n\n**Name** | **Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **%Winners** | **Entry** | **Exit** | **StopDollars**  \n---|---|---|---|---|---|---|---|---|---|---|---|---  \nDonchian | 466438 |  3.9 | \u221257391 | 2.21 | 2.27 | 324 | 1439.62 | 46  | 40.74 | 64 | 26 | 5000  \nDonchian | 417508 | 3.55 | \u221255451 | 2.33 | 1.72 | 272 | 1534.96 | 45.46 | 40.07 | 95 | 28 | 3000  \nDonchian | 420843 | 3.57 | \u221256764 | 2.36 | 1.98 | 262 | 1606.27 | 46.6 | 43.13 | 99 | 26 | 4500  \nDonchian | 421760 | 3.58 | \u221256764 | 2.31 | 1.94 | 262 | 1609.77 | 47.39 | 43.51 | 97 | 26 | 5500  \nDonchian | 421849 | 3.58 | \u221256764 | 2.35 | 1.98 | 263 | 1603.99 | 46.61 | 42.97 | 98 | 26 | 4500  \nDonchian | 421762 | 3.58 | \u221256764 | 2.34 | 1.92 | 260 | 1622.16 | 47.47 | 43.85 | 98 | 26 | 5500  \nDonchian | 420241 | 3.57 | \u221256764 | 2.34 | 1.72 | 261 | 1610.12 | 48.18 | 41.76 | 96 | 28 | 4000\n\nNow the real test begins. A diversified portfolio has been chosen as well as a seemingly robust parameter set. Let's walk this algorithm forward and test it across the OOS time frame (January 2010 through August 2015). Table 3.14 exposes a seeming act of futility.\n\n**Table 3.14** Results for the Donchian Algorithm when Walked Forward across an Out-of-Sample (OOS) Time Period\n\n**Name** | **Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **%Winners**  \n---|---|---|---|---|---|---|---|---|---  \n<Portfolio> |  25678 |  0.45 | \u2212166243 | 1.08 | 8.2  | 161 |  159.49 | 41.32 | 32  \nAD0_E0B | \u221216980 | \u22120.31 |  \u221229050 | 0.61 | 1.83 |  20 |  \u2212849  | 39  | 25  \nC20_E0B | \u221218413 | \u22120.33 |  \u221235138 | 0.53 | 2.25 |  21 |  \u2212876.79 | 39.14 | 33  \nCL20_E0B |  11230 |  0.2  |  \u221239180 | 1.23 | 1.82 |  19 |  591.05 | 32  | 26  \nCU0_E0B |  37713 |  0.66 |  \u221224450 | 2.99 | 1.14 |  13 |  2900.96 | 51.54 | 31  \nHG20_E0B | \u221216438 |  \u22120.3  |  \u221228638 | 0.62 | 1.87 |  19 |  \u2212865.13 | 33.74 | 32  \nKW20_E0B |  1463 |  0.03 |  \u221214725 | 1.07 | 0.54 |  19 |  76.97 | 41.95 | 32  \nRB0_E0B | \u22122927 | \u22120.05 |  \u221260503 | 0.96 | 2.93 |  21 |  \u2212139.4 | 29.9  | 14  \nSB20_E0B |  15918 |  0.28 |  \u221212749 | 2.68 | 0.89 |  17 |  936.33 | 52.59 | 53  \nTY0_E0B |  14113 |  0.25 |  \u22128084 | 2.54 | 0.37 |  12 |  1176.05 | 66.67 | 50\n\nWell, at least the system made some money. However, the drawdown is just way too great. What did we do wrong? We did all our homework\u2014picked a fairly diversified portfolio and a robust parameter set. How could we have been so far off? Out of pure sick curiosity, I optimized the algorithm across the OOS data to see how far off we missed the target. Table 3.15 has the best parameters sets for our small portfolio over the OOS timespan.\n\n**Table 3.15** The Best Parameters for the Portfolio over the Out-Of-Sample (OOS) Time Period\n\n**Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **%Winners** | **Entry** | **Exit** | **StopDollars**  \n---|---|---|---|---|---|---|---|---|---|---|---  \n182607 | 3.04 | \u221295572 | 1.42 | 3.85 | 253 | 721.77 | 48.67 | 39.53 | 30 |  30 | 5500  \n182607 | 3.04 | \u221295572 | 1.42 | 3.85 | 253 | 721.77 | 48.67 | 39.53 | 30 |  30 | 4500  \n182607 | 3.04 | \u201395572 | 1.42 | 3.85 | 253 | 721.77 | 48.67 | 39.53 | 30 |  30 | 4000  \n182607 | 3.04 | \u221295572 | 1.42 | 3.85 | 253 | 721.77 | 48.67 | 39.53 | 30 |  30 | 3500  \n182607 | 3.04 | \u221295572 | 1.42 | 3.85 | 253 | 721.77 | 48.67 | 39.53 | 30 |  30 | 3000  \n182607 | 3.04 | \u221295572 | 1.42 | 3.85 | 253 | 721.77 | 48.67 | 39.53 | 30 |  30 | 6000  \n182456 | 3.04 | \u221295583 | 1.42 | 3.85 | 253 | 721.17 | 48.67 | 39.53 | 30 |  42 | 3000  \n182412 | 3.04 | \u221295593 | 1.42 | 3.85 | 253 | 720.99 | 48.67 | 39.53 | 30 |  84 | 3000  \n182412 | 3.04 | \u221295593 | 1.42 | 3.85 | 253 | 720.99 | 48.67 | 39.53 | 30 | 100 | 5500\n\nBoy, were we way off! A couple of the best parameter sets had the exit length parameter greater than the entry length\u2014illogical, you say! In these situations, the system becomes a reversal system until it is stopped out with the money management stop. I am almost too scared to continue, but like looking at a car accident, let's continue in this same vein. Maybe the Donchian is just an inferior algorithm. Table 3.16 is the IS optimization of the Bollinger algorithm and Table 3.17 is the OOS results.\n\n**Table 3.16** The Optimization of the Bollinger Algorithm for the In-Sample (IS) Time Period\n\n**Name** | **Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **%Winners** | **Len** | **Width** | **StopDollars**  \n---|---|---|---|---|---|---|---|---|---|---|---|---  \nBollinger | 498208 | 4.13 | \u221259466 | 2.55 | 2.2  | 247 | 2017.04 | 66.91 | 37.65 | 103 | 1.5 | 5000  \nBollinger | 480108 | 4  | \u221266439 | 2.46 | 2.23 | 256 | 1875.42 | 63.79 | 38.28 |  98 | 1.5 | 5000  \nBollinger | 492573 | 4.09 | \u221259399 | 2.53 | 2.07 | 251 | 1962.44 | 65.7  | 38.25 | 102 | 1.5 | 4500  \nBollinger | 489796 | 4.07 | \u221259466 | 2.49 | 2.15 | 250 | 1959.18 | 65.93 | 37.2 | 103 | 1.5 | 4500  \nBollinger | 516639 | 4.25 | \u221263828 | 2.71 | 2.02 | 239 | 2161.67 | 68.67 | 38.49 | 105 | 1.5 | 6000  \nBollinger | 502931 | 4.16 | \u221261884 | 2.64 | 2.06 | 240 | 2095.54 | 68.08 | 38.33 | 108 | 1.5 | 4000  \nBollinger | 516685 | 4.25 | \u221264074 | 2.74 | 2.04 | 243 | 2126.27 | 66.91 | 37.86 | 105 | 1.5 | 4000  \nBollinger | 515383 | 4.24 | \u221263828 | 2.7 | 2.06 | 239 | 2156.42 | 69.05 | 38.08 | 106 | 1.5 | 6000  \nBollinger | 511872 | 4.22 | \u221263828 | 2.67 | 2.1  | 240 | 2132.8  | 68.34 | 38.33 | 105 | 1.5 | 5500  \nBollinger | 516735 | 4.25 | \u221265095 | 2.72 | 2.09 | 231 | 2236.95 | 71.45 | 38.53 | 110 | 1.5 | 5000\n\n**Table 3.17** Results for the Donchian Algorithm when Walked Forward across an Out-of-Sample (OOS) Time Period\n\n**Name** | **Net Profit** | **CAR** | **Max. Sys DD** | **Profit Factor** | **Ulcer Index** | **# Trades** | **Avg P\/L** | **Avg Bars Held** | **%Winners**  \n---|---|---|---|---|---|---|---|---|---  \n<Portfolio> | \u221231803 | \u22120.58 | \u2212222494.3 | 0.91 | 11.93 | 158 |  \u2212201.28 | 56.15 | 30  \nAD0_E0B |  \u22122570 | \u22120.05 | \u221226410 | 0.91 | 1.64 |  16 |  \u2212160.62 | 64.56 | 38  \nC20_E0B |  \u22128500 | \u22120.15 | \u221226325 | 0.71 | 1.73 |  20 |  \u2212425  | 53.85 | 30  \nCL20_E0B | \u221214060 | \u22120.25 | \u221258490 | 0.78 | 3.36 |  21 |  \u2212669.52 | 40.95 | 29  \nCU0_E0B |  16888 | 0.3 | \u221239763 | 1.62 | 2.17 |  14 |  1206.25 | 67.57 | 36  \nHG20_E0B | \u221225763 | \u22120.46 | \u221238750 | 0.56 | 2.47 |  19 | \u22121355.92 | 45.26 | 26  \nKW20_E0B |  8525 |  0.15 | \u221215063 | 1.4  | 0.59 |  17 |  501.47 | 65.71 | 41  \nRB0_E0B | \u221229267 | \u22120.53 | \u221279688 | 0.7  | 4.51 |  24 | \u22121219.48 | 36.63 | 13  \nSB20_E0B |  15185 |  0.27 | \u221219315 | 2.07 | 1.28 |  11 |  1380.44 | 94.27 | 36  \nTY0_E0B |  7760 |  0.14 |  \u22129334 | 1.46 | 0.49 |  16 |  484.99 | 65.38 | 38\n\nOkay, looks good. I selected the parameter set [108, 1.5, 4000]. How did it do during the OOS timespan?\n\nI think I see a trend developing. I would show you the TMA and the DMA, but it would be just a waste of paper! I am talking reverse makeover. So what happened? Where do we place the blame? I went back and recreated a portfolio of the top nine markets based on IS performance and the OOS results were even worse than our small diversified portfolio. We did everything humanly possible to make these trend-following systems work: algorithm selection, optimization, and portfolio selection. What if we performed a more frequent periodic reoptimization? We saw the best parameter set shifting from just five years ago. This shift may be attributable to the disappearance of the pit session and the adoption of purely electronic markets. Who knows why, but the fact remains that the commodity markets have had a fundamental shift. Indicators such as the ones we have experimented with are adaptive by nature; they change based on market activity and\/or volatility. Theoretically, they should be able to keep up with the evolution of the markets. In practice, this was not the case.\n\nAmiBroker has a walk-forward backtesting capability\u2014meaning we can periodically reoptimize parameters throughout a historic backtest. I discuss this optimization method in great detail in Chapter 8. After this chapter, if you like, you can skip to Chapter 8 to help you understand exactly what is going on here. Figure 3.4 illustrates the basic concept of walk-forward testing. Basically it involves two primary steps:\n\n  1. Backtest and derive\u2014backtest and extract the best parameter set based on some criteria. It could be profit, drawdown, profit-to-drawdown ratio, etc.\n  2. Walk forward\u2014take the parameter sets and apply them forward in time on data not used in the backtest. Do this on a periodic basis.\n\n**Figure 3.4** Walk-forward testing begins with a backtest used to derive the desired parameter. Then, it applies that parameter going forward.\n\nSource: AmiBroker\n\nIn this example, the Bollinger Band algorithm will be reoptimized annually based on the best-performing parameters for the prior four years. Keep in mind that we are not cheating, as only the OOS (out-of-sample) data will be used to calculate performance metrics. Let's see if this method can turn around the trend of this trend-following method. Figure 3.4 shows the process of walking forward.\n\nAmiBroker's walk-forward optimizer (WFO) derived the best parameter set from January 1, 2006, to January 1, 2010, and then carried those parameters forward from January 1, 2010, to January 1, 2011. In this test, the algorithm netted $35,330 for 2010. It then included 2010 and backed up to January 1, 2007, and then carried those parameters forward through 2011. However, this time the algorithm lost a whopping \u2212$56,532. Keep stepping through the figure and you can see how the system performed through the present. What's really interesting is the shift in the parameter sets. You would think by adding just another year to the backtest during the periodic readjustment the parameters wouldn't shift that much. Take a look at Figures 3.5 and 3.6.\n\n**Figure 3.5** AmiBroker's walk-forward optimizer in action.\n\n**Figure 3.6** Parameters selected to carry forward for the following year.\n\nThe first year in our OOS window is 2010. The WFO picked the parameter set [72 day, 1.5 std. devs., 3000 stop] to apply to 2010 and the results looked great. Notice the wide shift in the parameters for 2011 [111, 0.25, 6000]. These parameters performed horribly, losing over $56,000. The parameters for 2012 shifted some more [143, 0.25, 3500]. These parameters didn't do much better. 2013 also shows a dramatic shift, [45, 3, 3500], but at least the algorithm made a little money. The parameter set stayed constant for 2014, and the algorithm really brought it home, with almost $72,000 in profit. The end result with periodic reoptimization beat the pants off of the static set with $45,000 in profit and a little more reasonable drawdown of $90,000. Figure 3.7 shows an equity curve of the walk-forward performance from 2006.\n\n**Figure 3.7** Performance of the WFO of the Bollinger Band system.\n\nSource: AmiBroker\n\nIs this the solution? Stick with your algorithm and readjust based on recent performance. That is a tough question to answer, and is best left up to what the trader truly believes when it comes to using optimization in the development of an algorithm's parameters. In today's trading environment, I think everything should be taken into consideration. Don't be afraid of new technology.\n\nAfter the second coming of trend following in 2008, the only bright spot has been the second half of 2014. And things are looking a little better here in 2015. It seems, for right now, that the commodity markets are behaving a little trendier. With prices in the tank trend followers are waiting for the eventuality of their springing back to life. They are limited resources, right? But how many trend followers will still be left out there? A lot more than you might think. Due to the nature of commodity price moves, you cannot simply abandon the trend-following methods. There is just way too much potential.\n\nSo what is the answer for a smaller account? In a similar fashion to diversifying through markets, a trader can diversify further through trading multiple algorithms. In a trader's perfect world, there would exist robust trend-following, short-term swing and day trading systems. Ones that were not closely correlated with the others. This perfect world doesn't exist due to the fact a robust short-term swing system is hard to find. The ones that seem to work on a somewhat consistent basis mostly work only on the stock indices. However, that's not a bad thing, because this is a very weak sector for trend followers. The best short-term algorithms that I have come across incorporate mean reversion or pattern recognition components. The marriage of trend following and these other types of algorithms might be one solution. This union brings together two different algorithms, as well as two noncorrelated portfolios.\n\n## Multi-Algorithm Strategy (MAS)\n\nMean reversion is a trading algorithm that trades in the direction of the trend but takes advantage of a temporary countertrend move.\n\nThe most recent performance of this algorithm looks very respectable. This algorithm utilizes the Bollinger B% indicator\/function (Box 3.4). Basically, it trades in the direction of the 200-day moving average but waits for a pullback for longs and rallies for shorts. The Bollinger B% function returns the location of the current close in relationship to the upper and lower Bollinger bands. If the function returns a small value, then the close is close to the lower band and vice-versa for a high value. This system is programmed in EasyLanguage and the source code will be revealed in the Chapter 7.\n\n* * *\n\n### Box 3.4 Mean Reversion (MR)\n\n  1. Calculate the 200-day moving average of closing prices\n  2. Calculate the 5-day Bollinger B% value\n  3. Calculate the 3-day moving average of the Bollinger B% value \n    1. If c > MAV(c,200) and MAV(BB%,3) < 0.25, then buy next bar at open\n    2. If c < MAV(c,200) and MAV(BB%,3) > 0.75, then sell short next bar at open\n    3.       1. Liquidate long MOC when BB% > .75 or close < entryPrice - 2000\/bigPointValue\n      2. Liquidate short MOC when BB% < 0.25 or close > entryPrice + 2000\/bigPointValue\n\n* * *\n\nTake a look at the equity curve in Figure 3.8. Here is a pattern system that trades the stock indices as well.\n\n**Figure 3.8** Equity curve that reflects a pattern system.\n\nThis system looks at the last day of the month, and if it closes down a certain amount from the prior day's close, it looks to buy if the first day of the month also closes below its open (Box 3.5). This is a reoccurring pattern, and it's based on the flip of the month.\n\n* * *\n\n### Box 3.5 LastBarOfMonth Pattern (LBM)\n\n  1. Calculate percentage change between yesterday's close and the prior day's close: %Chg = (close(1) \u2013 close(2)) \/ close(1) * 100\n  2. If today is the first trading day of the month and the %Chg < -0.2 and today's close < today's open, then buy MOC\n  3. If today is the first trading day of the month and the %Chg > 0.2 and today's close > today's open, then sell short MOC\n  4. Liquidate long at the open of the fifth day after entry day or on a stop @ entryPrice \u22122000\/bigPointValue\n  5. Liquidate short at the open of the fifth day after entry day or on a stop @ entryPrice + 2000\/bigPointValue\n\n* * *\n\nDay trading is a great complement to trend following as well. The equity curve in Figure 3.9 illustrates a mini-Russell day trading system that trades after a narrow range day and initially puts on two contracts and then pulls one off after a certain profit level and then lets the second contract run its course. It is out on the close no matter what. This algorithm incorporates trade management in the form of protective stop, breakeven stop, and trailing stop (Box 3.6).\n\n**Figure 3.9** Mini-Russell day trading system using a narrow range day. It initially puts on two contracts, pulling the first one off after a certain profit is reached and leaving the second to run its course.\n\n* * *\n\n### Box 3.6 Day trading the mini Russell\n\n  1. CanTrade = False\n  2. CanBuy = False\n  3. CanShort = False\n  4. If yesterday's True Range < MAV(TR,10), then CanTrade = True\n  5. If yesterday's close >= prior day's close, then CanBuy = True\n  6. If yesterday's close < prior day's close, then CanShort = True\n  7. If CanTrade, then \n    1. If CanBuy, then buy 2 units tomorrow at open +0.2 *MAV(TR,10) stop\n    2. If CanShort, then sell short 2 units tomorrow at open \u22120.2*MAV(TR,10) stop\n  8. If market position = long \n    1. If current size = 2 units, then\n\nIf H of 5 minute bar > entryPrice + 4.00, then\n\n      1. exitLong 1 unit at H of day \u2013 4.00 on a stop\n    2. If current size = 1 unit, then\n\nexitLong at entryPrice stop or MOC\n\n  9. If market position = short\n\nIf current size = 2 units, then\n\n    1. If L of 5 minute bar <= entryPrice \u2013 4.00, then\n\n      1. exitShort 1 unit at L of day + 4.00 on a stop\n      2. If current size = 1 unit, then\n\n    2. exitLong at entryPrice stop or MOC\n\n* * *\n\nHow about a system that lets the Commitment of Traders Report (COT) guide its directional trades\u2014one that only trades in the direction of the commercial interests? If the net commercial position is positive, then only take long positions, and vice versa. This type of trading system uses the COT as a sentiment indicator. Several trading platforms allow you to import the COT data. This system will be included on this books companion website and www.georgepruitt.com.\n\n## Summary\n\nThere are a lot of algorithms out there that are not correlated, so it might be best not to put all of your eggs in one basket. I think the best trend-following mechanism might even be a combination of a Donchian \/ Bollinger hybrid; use one indicator to confirm the other. I went back and redid all the testing on the four major trend-following candidates utilizing a multiple of an average true range (ATR) stop and the conclusions were exactly the same. A fixed-dollar stop is as good as an adaptive volatility stop. I will continue my search for that elusive combination of algorithms or a better trend-following mousetrap and report my findings on www.georgepruitt.com website. Chapter 4 starts the second section of this book, where different trading and testing platforms are discussed. This is where the real programming begins. Have fun!\n\n# Chapter 4  \nIntroduction to AmiBroker's AFL\n\nThis chapter starts the second part of the book that covers different programming languages and trading platforms. This second part discusses AmiBroker, Visual Basic (VBA) for Excel, Python, and TradeStation. The major goal of this particular chapter is to introduce AmiBroker and illustrate some of its very powerful capabilities. However, some very important building blocks of programming languages, reserved words, data types, operators and expressions, and precedence of operators, are quickly reviewed and used throughout the next three chapters. Even if you are not interested in AmiBroker, make sure you read the first part of this chapter to get a grasp on the universal usage of some key building blocks that are included in all programming languages.\n\n## Quick Start\n\nAmiBroker is a complete database management, charting, testing, trading, and optimizing program. So far we have covered many different trading algorithms and every one of them can be easily implemented in the AmiBroker Function Language (AFL). There are several features that make AmiBroker a serious contender as a go-to trading platform:\n\n  * _Price_. The cost of the software ranges between $279 and $339 and once you purchase it, it is yours. There are no lease fees. You get free upgrades for one year after purchase and it's up to the purchaser to upgrader thereafter.\n  * _Speed_. AmiBroker utilizes multithread processing, which means it can utilize multicore processors and carry out subprocesses or threads simultaneously. In other words, it is very fast.\n  * _Power_. Portfolio-level backtesting is paramount when evaluating the robustness of a trading algorithm. If an algorithm works on multiple markets, it demonstrates a high level of robustness. Robustness and positive expectancy is all you can ask from a trading algorithm. AmiBroker provides Exhaustive Search and Genetic forms of parameter optimization.\n  * _Data_. AmiBroker is compatible with many End Of Day(EOD) and Real-Time data feeds. It includes a very simple to use ASCII data importer.\n  * _Broker Integration_. Several brokers can be linked with the software for automated order execution.\n  * _Integrated Development Environment_. AmiBroker has two IDEs to help the user develop complete trading algorithms and technical analysis tools. The main IDE (AFL Editor) is a complete scripting tool and the AFL Code Wizard utilizes a drag-and-drop development paradigm. Both IDEs utilize the AmiBroker Function Language (AFL) as their programming\/scripting language and a vast library of strategies, functions, and indicators.\n  * _Support_. Tomasz Janeczko, the founder and chief software architect of AmiBroker, holds PhD and MSc degrees in Computer Science and Telecommunications from Worclaw University of Technology. This developer really loves his software and stands behind it and provides much of the tech support. There is a devout, almost cultlike, following for AmiBroker, and many questions can be easily answered by searching the Internet and\/or joining the user groups.\n\nIf you have purchased my prior books, you know that I use TradeStation. I love its tightly integrated components and EasyLanguage. I will discuss TradeStation in Chapter 7. However, I felt I would be amiss if I didn't show off a little bit of AmiBroker. I was first introduced to the software through Howard Bandy's excellent book, _Quantitative Trading Systems_. I have included all of his books in \"George's List of Must-Own Trading Books\" in the appendix. The software that was described in Bandy's book really piqued my interest, so I contacted Dr. Janeczko for a review copy, and he more than graciously provided his complete software suite. I fell in love with the AFL Wizard and the speed of his backtester. Coming from a bar-by-bar algorithm development paradigm, I was really impressed and initially confused by Dr. Janeczko's array processing. Array processing is extremely fast versus bar-by-bar and in many cases is easier to learn. What is also very cool about AmiBroker is that you can flip from array to bar processing quite easily.\n\n### Things All Programmers Need to Know\n\nBefore I go into detail about array programming, let's go over a short introduction to some key concepts that all programming languages have in common, including AmiBrokers AFL: reserved words, remarks or comments, variables and variable naming conventions, data types, expressions and operators, and precedence of operators. Here is a quick synopsis:\n\n  * _Reserved words_. Words that the computer language has set aside for a specific purpose. You can only use these words for their predefined purposes. Using these words for any other purpose may cause severe problems.\n  * _Remarks or comments_. Words or statements that are completely ignored by the compiler. Remarks are placed in code to help the programmer, or other people who may reuse the code, understand what the program is designed to do. Double forward slashes \/\/ inform the AFL interpreter that anything that follows is a comment. The double forward slashes can be used anywhere within a line. The forward slash asterisk combination \/* and *\/ is used for multiline commentary. The \/* opens the remarks and *\/ closes the remarks block. Anything inside \/* --- *\/ is ignored by the computer.\n  * _Variables_. User-defined words or letters that are used to store information. AFL is not a strongly typed language. You don't have to formally declare a variable name or its type prior to its use.\n  * _Data types_. Different types of storage; variables are defined by their data types. AFL has three basic data types: numeric, boolean, and string. A variable that is assigned a numeric value, or stored as a number, would be of the numeric type. A variable that stores a true or false value would be of the boolean type. Finally, a variable that stores a list of characters would be of the string type.\n\n#### _Variables and Data Types_\n\nA programmer must understand how to use variables and their associated data types before they can program anything productive. Let's take a look at a snippet of code.\n\n    mySum = 4 + 5 + 6;\n    myAvg = mySum\/3;\n\nThe variables in this code are mySum and myAvg and they are of the numeric data type; they are storage places for numbers. AFL is liberal concerning variable names, but there are a few requirements (Table 4.1). A variable name cannot:\n\n  * Start with a number or a period (.)\n  * Be a number\n  * Include punctuation\n\n**Correct** | **Incorrect**  \n---|---  \nmyAvg | 1MyAvg  \nmySum | .sum  \nsum | val+11  \nval1 | the\/\/sum  \nthe.sum | my?sum  \nmy_val | 1234\n\nVariable naming is up to the style of the individual programmer. AFL is not case sensitive (you can use upper- or lowercase letters in the variable names). ( _Note_ : The following is my preference\u2014may not be everybody's.) Lowercase letters are preferred for names that only contain one syllable. For variable names that have more than one syllable, we begin the name with a lowercase letter, and then capitalize the beginning of each subsequent word in a multi-word identifier.\n\n    sum, avg, total, totalSum, myAvg, avgValue, totalUpSum, totDnAvg\n\n#### Operators and Expressions\n\nIn programming, statements are made up of expressions. An expression consists of a combination of identifiers, functions, variables, and values, which result in a specific value. Operators are a form of built-in functions and come in two forms: unary and binary. A binary operator requires two operands, whereas a unary operator requires only one. AFL can handle both. Examples of unary operators are **++** and **\\--**. Some of the more popular binary ones are: + \u2212 \/ * < = > >= <= <> AND OR. These binary operators can be further classified into two more categories: arithmetic and logical.\n\nExpressions come in three forms: arithmetic, logical, and string. The type of operator used determines the type of expression. An arithmetic expression includes + \u2212 \/ *, whereas a logical or boolean expression includes < = > >= <= <> AND OR.\n\n**Arithmetic Expressions** | **Logical Expressions**  \n---|---  \nmyValue = myValue + 1 | myCondition1 = sum > total  \nmyValue = sum \u2212 total | myCondition1 = sum <> total  \nmyResult = sum*total+20 | cond1 = cond1 AND cond2  \nmyCounter++ | cond1 = cond2 OR cond3\n\nArithmetic expressions always result in a number, and logical expressions always result in true or false. True is equivalent to 1, and false is equivalent to 0. String expressions deal with a string of characters. You can assign string values to string variables and compare them.\n\n    myName1 = \"George\";\n    myName2 = \"Pruitt\";\n    cond1 = (myName1 <> myName2);\n    myName3 = myName1 + \" \" + myName2;\n\nConcatenation occurs when two or more strings are added together. Basically, you create one new string from the two that are being added together.\n\n#### Precedence of Operators\n\nIt is important to understand the concept of precedence of operators. When more than one operator is in an expression, the operator with the higher precedence is evaluated first, and so on. This order of evaluation can be modified with the use of parentheses. Most programming languages' order of precedence is as follows:\n\n  1. Parentheses\n  2. Multiplication or division\n  3. Addition or subtraction\n  4. <, >, =, <=, >=, <>\n  5. AND\n  6. OR\n\nHere are some expressions and their results:\n\n* * *\n\n    20 - 15\/5    equals 17 not 1\n    20 - 3       division first, then subtraction\n    10 + 8\/2     equals 14 not 9\n    10 + 4       division first then addition\n    5 * 4\/2      equals 10\n    20\/2         division and multiplication are equal\n    (20 - 15)\/5  equals 1\n    5\/5          parentheses overrides order\n    (10 + 8)\/2   equals 9\n    18\/2         parentheses overrides order\n    6 + 2 > 3    true\n    8 > 3\n    2 > 1 + 10   false\n    2 < 11\n    2 + 2\/2 * 6  equals 8 not 18\n    2 + 1 * 6    division first\n    2 + 6        then multiplication\n    8            then addition\n\n* * *\n\nThese examples have all of the elements of numerical or logical expressions. They are to be evaluated, but they don't do anything. A statement informs the computer to do something. When you tell the computer to use an expression and then do something you have created a complete programming statement. When you assign the result of an expression to a variable, myVar = x + 2, then you have told the computer to do something. When you program, you create lines of statements.\n\n## Price Bar Interface\n\nA price chart consists of bars built from historical price data. Each individual bar is a graphical representation of the range of prices over a certain period of time. A five-minute bar would have the opening, high, low, and closing prices of an instrument over a five-minute time frame. A daily bar would graph the range of prices over a daily interval. Bar charts are most often graphed in an open, high, low, and close format. Sometimes the opening price is left off. A candlestick chart represents the same data, but in a different format. It provides an easier way to see the relationship between the opening and closing prices of a bar chart. Other bar data such as the date and time of the bar's close, volume, and open interest is also available for each bar. Since AFL works hand-in-hand with the charts that are created by AmiBroker, there are many built-in reserved words to interface with the data. These reserved words were derived from commonly used verbiage in the trading industry. You can interface with the data by using the following reserved words. ( _Note_ : Each word has an abbreviation and can be used as a substitute.)\n\n**Reserved Word** | **Abbreviation** | **Description**  \n---|---|---  \nOpen | O | Open price of the bar.  \nHigh | H | High price of the bar.  \nLow | L | Low price of the bar.  \nClose | C | Close price of the bar.  \nVolume | V | Number of contracts\/shares traded.  \nOpenInt | OI | Number of outstanding contracts.\n\nAFL uses functions **DateNum()** , **DateTime()** , **Hour()** , **Minute()** , **Second()** to retrieve the date and time of the price bar.\n\nArray programming or vector languages generalize operations on scalars to apply transparently to vectors, matrices (2d-array), and other higher-dimensional arrays. Sounds complicated, right? It really isn't. An array is simply a list of like data. In most testing platform scripting languages price data is held in arrays. When we code and use the word _high_ we are usually referring to an array or list of _high_ prices of the underlying instrument. When we use a subscript in the array, we are referring to that singular element in the list: High[6] is the _high_ price six bars ago. Most of the programming discussed thus far in this book has dealt with scalar programing in a pseudocode framework. The fundamental idea behind array programming is that operations apply at once to an entire set of data. An example will help clarify. How do we calculate a 20-day moving average of price bar's midpoint in a scalar (bar-by-bar) framework?\n\n* * *\n\n    sum = 0\n    for i = 1 to 20\n            sum = sum + (high[i] + low[i]) \/ 2 #notice the subscript i\n    next i\n    avgMP = sum \/ 20\n\n* * *\n\nThis programming structure is known as a _for_ loop. The loop variable is the letter _**i**_ and the statement that is indented will be executed 20 times before the next line of code is processed. The _**sum**_ variable accumulates the high price over the past 20 bars. This moving average calculation requires five lines of code. Of course this code segment could be put into a function and in most testing languages or scripts it is. However, be it in a function module or inline, the code is executed on every given bar of data. Here's how it is done in an array-programming framework:\n\n* * *\n\n    avgMP = MA((H + L)\/2,20)\n\n* * *\n\nA new array labeled **avgMP** is created and completely loaded with a 20-period moving average of the price bar's midpoint. **avgMP** is not a scalar or single value\u2014it is a full-blown array just like the **High** and **Low** arrays. Array processing eliminates the need for additional looping and this means much quicker execution. You might think a language like EasyLanguage is doing the same thing when it invokes the **average** function:\n\n* * *\n\n    avgMP = average((H + L) \/ 2,20);\n\n* * *\n\nHowever, this is not the case. The code inside the **average** function consists of a loop and is called on each and every bar.\n\nIf you have a lot of experience with scalar programming (Java, C++, or EasyLanguage), then picking up AFL might require a little patience, but it will definitely be worth it. As you know, learning a new programming script\/language is best performed by working through examples. If you own AmiBroker, make sure you go to this link and download Howard Bandy's book: <http:\/\/introductiontoamibroker.com\/book.html>. This book will get you up and going very quickly. Bandy helps explain the sophisticated database management that you get with AmiBroker. Data is to a trading algorithm like gas is to an automobile.\n\n## AFL Array Programming\n\nHere is a simple moving average crossover system in AFL:\n\n* * *\n\n    PositionSize = MarginDeposit = 1;\n    avgLen = Optimize(\"Len\",20,10,100,1);\n    avgLen = Param(\"Length\",81,10,100,1);\n    Buy = C > MA(C,avgLen);\n    Short = C < MA(C,avgLen);\n    Sell = Short;\n    Cover = Buy;\n\n* * *\n\nFor right now, ignore lines 2 and 3\u2014the ones starting with the word **avgLen**. If you are testing futures data, then don't forget the first line must be in every trading algorithm. This tells AmiBroker's backtester to only trade one contract and sets the margin deposit to one as well. The variable avgLen is the length of the moving average indicator the algorithm will utilize. Let's set it to 100.\n\n* * *\n\n    Buy = C > MA(C,avgLen);\n\n* * *\n\nThis line of code tells AmiBroker to set the BUY array to true or 1 if the close is greater than the 100-day moving average of closes. If the test fails and the close is not greater than the moving average, then it is set to false or 0. Since we are array processing, this done all at one time. The BUY array is filled completely up with either 1s or 0s. The SHORT array does just the opposite\u2014true if the close is less than the 100-day moving average and false if it's not. This system simply flips back and forth between long and short positions\u2014it's always in the market. If you use **Buy** and **Short** arrays to initiate positions, then you must have values for the **Sell** and **Cover** arrays. If you only use **Buy** , then you only need a value for **Sell**. This applies to **Short** and **Cover** as well. If you are using a pure stop and reverse algorithm, you can set up your **Sell** and **Cover** arrays like this:\n\n* * *\n\n    Sell = Short;\n    Cover = Buy;\n\n* * *\n\nThese two lines tell the computer to cover long positions at the same price a short position is established and to cover short positions at the same price a long position is put on. Or you can do it like this:\n\n* * *\n\n    Sell = 0;\n    Cover = 0;\n\n* * *\n\nIf you don't take into consideration the arrays that get you out of a position, AmiBroker will give you a warning message, and will not execute the backtest. AmiBroker requires the **BUY** order and its corresponding **SELL** order to be defined prior to processing. The same goes for **SHORT** and **COVER**. In some platforms, you do not need to define **SELL** and **COVER** if the algorithm is always in the market long or short.\n\nThis little bit of code seems quite simple on the surface, but there is a lot going on behind the scenes. AFL has a plethora of built-in functions such as the **MA** function used in this simple program. So calculating the 100-day moving average is very simple; just call the function. The four arrays ( **BUY, SHORT, SELL, COVER** ) that instruct AmiBroker to establish and liquidate trades are instantly populated with 1s and 0s; it's a one-pass process. This process is extremely fast and efficient. Once this single pass is completed, the back tester is fed the necessary information to simulate the backtesting of the algorithm. As you can see, entry and exit prices are not defined, so they are defaulted to the close of the bar. This default can be changed using the settings dialog box.\n\nLet's give this simple algorithm a run. The next instructions assume you have worked with AmiBroker and know how to plot charts, apply indicators, and work with price databases.\n\nGo ahead and launch AmiBroker and load your database of choice. I imported an ASCII database of futures prices that I got from CSI data. The software is compatible with CSI and there are instructions in the online manual on how to create a database, and have it updated automatically. This may require an additional monthly data fee. Figure 4.1 shows how to get to the AFL editor from the main AmiBroker File menu.\n\n**Figure 4.1** Launching AmiBroker's AFL editor.\n\nOnce the editor launches you will see a window like the one in Figure 4.2.\n\n**Figure 4.2** The AmiBroker Editor window.\n\nThis is your tabula rasa\u2014blank slate. Anything you can dream up can be accomplished by typing in the right code. Go ahead and type in the code from the moving average example. The AFL Editor is one of the most powerful editors you will find in any trading or programming platform. It features outlining, collapsible, or text folding, function tips, and auto-completion. There's just a little bit of typing, so go ahead and just do it and I will highlight some of these key features along the way. Here it is again, so you won't have to flip back to the prior page.\n\n* * *\n\n    PositionSize = MarginDeposit = 1;\n    avgLen = Optimize(\"Len\",20,10,100,1);\n    avgLen = Param(\"Length\",81,10,100,1);\n    Buy = C > MA(C,avgLen);\n    Short = C < MA(C,avgLen);\n    Sell = Short;\n    Cover = Buy;\n\n* * *\n\nAs you type the first line of code you will notice the words **PositionSize** and **MarginDeposit** turn bold. If you didn't capitalize the P or the S in the word **PositionSize** , the editor will automatically do so after you type the last letter in the word and hit the space bar. The editor is letting you know these are keywords or tokens. And that you can't use these for variable names. If you are looking for a keyword or a function name, and you type what you think it is and it doesn't turn bold or change colors, then you know you have not used the correct word. The editor can help you find the proper keyword or function name through its auto-completion feature. Later we will develop an algorithm based on Bollinger Bands, but for illustration purposes let's assume you want to calculate an upper Bollinger Band now, but don't know the exact words for the function. You probably think the function name starts with the letter \"b,\" so for now type the letter \"b\" after the first line and press the <Ctrl> and <spacebar>. A list of keywords and functions starting with the letter \"b\" will appear in your editor window (see Figure 4.3).\n\n**Figure 4.3** The AFL Editor's auto-completion tool.\n\nIf you look down the list, you will see the name **BBandTop**. If you select and double-click on the name, the function call will be automatically inserted into your code. The more of the function name you type, the smaller the list of options will appear after pressing <Ctrl> and <spacebar>. If you type **BBand** and hit the keyboard sequence, only two options will appear: **BBandBot** and **BBandTop**. If you still can't find the name of the keyword or function, then you will have to use the trial-and-error method and keep typing different words or take the time and look up the keyword in the AFL Language Reference under the Help menu. Delete whatever you typed as the second line and let's continue with the first algorithm. As you type the second line of code and type the keyword **Optimize** it will turn blue and right after you type the left parentheses a small box will open up right under your cursor (Figure 4.4).\n\n**Figure 4.4** AFL Editor function helper.\n\nThe box is helping you by providing a template of what the function **Optimize** is expecting (this is called the informal parameter list). If a word turns blue, the editor is letting you know the keyword is a built-in function. This line of code links the source code with the AmiBroker optimizer. This is how I was able to do all those cool optimizations in Chapter 3. The function Optimize wants you to provide a name inside quotes, a **default value** for the variable avgLen, a **min value** , a **max value** , and a **step value**. When you optimize the variable avgLen it will start out at the **min value** and increase each iteration by the **step value** until it reaches the **max value**. If you don't run the optimizer, the variable will assume the **default value**. The AFL editor has all these helpers or tips for all of the built-in functions. You will notice another helper window pop up as you type the keyword **Param**. The **Param** function acts similarly to the **Optimize** function. Instead of providing an interface to the optimizer, the **Param** function provides an interface to the user. Instead of having to retype different values for avgLen every time you want to test a different value, you can simply change it at the **Analysis** phase. I will show how easy this is a little later. Go ahead and finish typing in the rest of the code and go under the File menu and save it to your Desktop or a folder on the C drive as MyFirstAlgo. Now you need to check the syntax to make sure you have typed everything properly. Click on the AFL Check icon (see Figure 4.5).\n\n**Figure 4.5** The AFL Check icon.\n\nIf you type everything in correctly, then nothing will happen. If you mistyped a word, you might get something like Figure 4.6. If this happens, just make sure you have everything typed properly and then hit the icon again. After the syntax has been properly checked (compiled), click on the **Send To Analysis** window icon (see Figure 4.7).\n\n**Figure 4.6** An error warning.\n\n**Figure 4.7** The \"Send to Analysis\" window icon.\n\nThis will create an Analysis tab in the AmiBroker program. I have continuous crude oil data plotted in the first chart. If you don't have crude oil data, don't worry, just create a chart of something. I inform AmiBroker to apply MyFirstAlgo to the current data by using the dropdown menu in the top-left corner (Figure 4.8). Before we click on the BackTest icon, go ahead and click on the Parameters icon. It's the icon that looks like an old equalizer (Figure 4.9).\n\n**Figure 4.8** Apply the algorithm to the current data.\n\n**Figure 4.9** Set the parameters for the backtest.\n\nThis is where you can change the values of the avgLen parameter. Change it to 50 and click OK. I get the sense that AmiBroker was originally designed to trade equities only, so you have to set up the backtester to work with futures data by going into the **Settings** button in an Analysis sheet. Only do this if you are truly testing futures data. If you aren't, then just skip this paragraph. The Settings button has a small picture of wrench and screwdriver and is located directly to the left of the **Parameters** button. A dialog box like the one in Figure 4.10 should open. Click on the box beside the **Futures mode** option. It instructs AmiBroker's back-tester to use the margin deposit and point value that was set up when the database was created for its calculations.\n\n**Figure 4.10** Set Futures mode.\n\nNow click the **Backtest** button (five buttons to the left of the **Parameter** icon). This will apply the algorithm to the data and create a trade-by-trade report in the analysis window. If you have a database loaded as I do and you select the **Portfolio Backtest** , then the algorithm will be applied to all of the markets in the database. Don't be surprised if this only takes a few seconds. AmiBroker is really applying the algorithm to every single bar of data in the portfolio. Figure 4.11 and Figure 4.12 provide a snapshot of the trade-by-trade report on the crude oil and the portfolio results on the entire database.\n\n**Figure 4.11** The trade-by-trade report.\n\n**Figure 4.12** The portfolio results.\n\nThe results don't look that great, so let's test a double moving average crossover system. Work your way back over to the AFL Editor\u2014it should still be open. Go under the File menu and create a new formula by clicking on New. Once a blank Editor window opens, type in the following exactly:\n\n* * *\n\n    Filter = 1;\n    PositionSize = MarginDeposit = 1;\n    avgLen1 = Optimize(\"Len1\",9,9,49,1);\n    avgLen2 = Optimize(\"Len2\",19,19,99,1);\n    shortMav = MA(C,avgLen1);\n    longMav = MA(C,avgLen2);\n    Buy = Cross(shortMav,longMav);\n    Short = Cross(longMav,shortMav);\n    Sell = Short;\n    Cover = Buy;\n\n* * *\n\nTo save some typing, I copied the code from MyFirstAlgo and made the necessary changes. This is the code for a double moving average system where you buy when the short (fast) moving average crosses above long (slow) moving average and you short when the fast moving average crosses below the slow moving average. The beauty of AmiBroker's AFL is it reads almost like English. After you check the AFL and send it to an **Analysis** window, go back to AmiBroker. Instead of clicking on the **Backtest** button, click the **Optimize** button. AmiBroker will now optimize the fast length from 9 to 49 and the slow length from 19 to 99 and report the results from each iteration. This might take a few seconds. When the optimization has completed, click on the down arrow beside the **Optimize** button and select **3D Optimization chart**. A chart like the one in Figure 4.13 will appear on your screen.\n\n**Figure 4.13** A 3D optimization chart.\n\nSource: AmiBroker\n\nThis is the result of the optimization where the _x_ -axis and _y_ -axis are the varying lengths of the moving averages and the _z_ -axis is the net profit. You can also see the water level; this represents where the profit is equal to zero. Hopefully, most of your mountains are above this level. This 3D chart only works on a two-parameter optimization.\n\nYou may want to play around with some of the other icons and familiarize yourself with their functions. AmiBroker is very sophisticated, and it took me several hours to become somewhat comfortable with the charting, reports, and applying trading and indicator algorithms. Again, I refer you to the excellent online manual, the AmiBroker forums, and of course, Bandy's book.\n\n## Syntax\n\nThe AmiBroker Function Language is very powerful, and, once you grasp the array-processing paradigm, quite easy to use. In some programming languages you have to declare variables and sometimes their types before you can use them. This is not so for the AFL. However, when you create a variable know that it is an array or list and not just a single memory location. Because of this, you can't do this:\n\n* * *\n\n    myValue = (H + L)\/2;\n    if (C > myValue){\n            Buy = 1;\n    }\n\n* * *\n\nIf you type this and check the syntax, you will receive the following message:\n\n    Condition in IF, WHILE, FOR statements has to be Numeric or Boolean type.\n\nBasically, you are asking AmiBroker to compare the **close** array with **myValue** array. You were probably thinking you were comparing the current close price to the current value in **myValue** , but you weren't. Forget about using **if's** in your code. This is a very difficult thing to do if you are used to working in a bar-by-bar paradigm. Instead, use something like this:\n\n* * *\n\n    myValue = (H + L)\/2;\n    Buy = C > myValue;\n\n* * *\n\nTomasz Janeczko has provided this type of array processing functionality in his software. This accomplishes the same thing as the previous **if** statement. How do you think you compare today's close with the prior day's close? You can't use an **if** statement, so could you do it like this?\n\n* * *\n\n    cond1 = C > C[1];\n    cond2 = C < C[1];\n    Buy= cond1;\n    Short = cond2;\n\n* * *\n\nIf you type this in and check the syntax, it will pass. You will also be able to send it to an **Analysis** window, and backtest it. However, it will not work. The computer doesn't come out and say, \"I'm sorry, Dave, I'm afraid I can't do that.\" It just won't generate the correct trades. These statements are syntactically correct, but they are logically incorrect. You are comparing the first element of the close array with the close array. In other words, you are comparing an element in the list with the list. This is how you do it in AFL:\n\n* * *\n\n    cond1 = C > Ref(C,-1);\n    cond2 = C < Ref(C,-1);\n    Buy= cond1;\n    Short = cond2;\n\n* * *\n\nThe **Ref(C,-1)** function returns the prior close price in the close array. The function processes the array and allows you to compare today's closing price with yesterday's closing price. You can use any negative number to reference previous values of any array. You can do this as well:\n\n* * *\n\n    myValue = (H + L)\/2;\n    cond1 = myValue > Ref(myValue,-1);\n\n* * *\n\nAnd you have probably noticed that all lines end with a semicolon (;). This tells the parser that the line of code has terminated and to go onto the next line. What do you think the next bit of code does?\n\n* * *\n\n    sl = Param(\"ShortLen\",9,9,100);\n    il = Param(\"InterLen\",19,19,200);\n    mmStop = Param(\"StopDollars\",3000,3000,6000);\n    Buy = MA(C,sl) > MA(C,il) ;\n    Short = MA(C,sl)< MA(C,il);\n    Sell = short;\n    Cover = buy;\n    ApplyStop(stopTypeLoss,stopModePoint,mmStop\/PointValue,0);\n\n* * *\n\nThe variables **sl** and **il** are defaulted to 9 and 19, respectively. The **Param** function allows the user to change the variables to values between 9 and 100 and 19 and 200, respectively. The variable **mmStop** defaults to 3000 but can be changed to values ranging from 3000 to 6000. The array **Buy** is filled with 1s when the moving average of **sl** (shortLength) is greater than the moving average of **il** (intermediateLength). The **Short** array is filled when the opposite happens. This is a very simple double moving average crossover system. The **ApplyStop** function tells AmiBroker to stop out on the close any time the current position reaches a loss of $3,000. The close is checked in this particular case. The **ApplyStop** function has many different ways of exiting a position: stop loss, trailing stop, or a profit objective. Here are three examples:\n\n* * *\n\n    ApplyStop( **stopTypeLoss** , **stopModePercent** ,2, **True** );\n    \/* single-line implementation of Chandelier exit *\/\n    ApplyStop( **stopTypeTrailing** , **stopModePoint** , 3*ATR(14), **True** , **True** );\n    \/* N-bar stop *\/  ApplyStop( **stopTypeNBar** , **stopModeBars** , 5 );\n\n* * *\n\nThis is a very powerful function and can fulfill almost every possible criterion to exit an existing position with a stop.\n\nReferring back to **MySecondAlgo** , you might have noticed these two lines of code:\n\n* * *\n\n    Buy = Cross(shortMav,longMav);\n    Short = Cross(longMav,shortMav);\n\n* * *\n\nThe keyword **Cross** is actually a function call. The function gives a \"1\" or true on the day that the **shortMav** crosses above the **longMav**. Otherwise, the result is false or zero. To find out when shortMav crosses below longMav, simply reverse the order of the arrays. I admit this took me a few minutes to understand, but like the rest the AFL functions, it makes perfect sense. The opposite of the short-term moving average crossing above the long-term average is the long-term average crossing above the short-term average.\n\nCoding like this might not be your cup of tea\u2014having to remember the function calls and how the **Cross** function works and all the business about arrays. If you want to test with AmiBroker, but don't want to work with the full-blown AFL editor, you can still do this. The next part of this chapter introduces the AmiBroker AFL wizard.\n\n## AFL Wizard\n\nImagine creating trading algorithms with point-and-click ease and minimal typing. Enter the AmiBroker AFL Wizard. Instead of typing MySecondAlgo into the AFL Editor, let's create it in the AFL Wizard. Get back to AmiBroker and click on the **AFL Wizard** button (Figure 4.14). Once you click on the button a window like the one in Figure 4.15 will pop up.\n\n**Figure 4.14** The AFL Code Wizard button.\n\n**Figure 4.15** A new AFL Code Wizard window.\n\nThis is the **AFL Code Wizard** , and it's a really cool way to create a trading algorithm without having to create it from scratch. The interface is really quite simple, so let's start by first adding a long entry rule. Click on the words **Enter Long when...** and then click on the pencil icon labeled **Add Item** (Figure 4.16). This will put the following code in the **Edit Rule** section and also below **the Enter Long When...**.\n\n**Figure 4.16** Add item in AFL Code Wizard.\n\n* * *\n\n    **Value of** Close (now) is greater than 10\n\n* * *\n\nNow this is where things get really cool. You can click on the individual words in the code and change them. Remember **MySecondAlgo** was a dual moving average crossover. Click on the words **Value of**. A list of different functions will appear in the pane below the **Edit Rule** pane. Figure 4.17 shows where the list is located.\n\n**Figure 4.17** The AFL Code Wizard \"Edit Rule\" dropdown list.\n\nClick in the pane and select **MA - Simple Moving Average**. The words **15-bar moving average of** will take place of **Value of**.\n\n* * *\n\n    **15-bar moving average of** Close (now) is greater than 10\n\n* * *\n\nNow click on the number **10** and go back down to the same pane we selected the first MA and select MA again. The code will change to:\n\n* * *\n\n    15-bar moving average of Close (now) is greater than **15-bar moving average of** Close now\n\n* * *\n\nWell, this won't do. The 15-bar moving average of Close will never be greater than the 15-bar moving average of Close. Click on the number 15 in the second moving average phrase and slide down to the pane right beside the pane with the list of functions (see Figure 4.18). Change the period from 15 to 30. The code in the Edit Rule should now read:\n\n**Figure 4.18** Changing the parameters in the \"Edit Rule\" pane.\n\n* * *\n\n    15 -bar moving average of Close (now) is greater than **30 -bar moving average of** Close now\n\n* * *\n\nSo the rules for entering a long position are now defined. Buy when the 15-bar moving average (fast) is greater than 30-bar moving average (slow). As you point and click the AFL code equivalent is created simultaneously. The AFL code equivalent thus far is:\n\n* * *\n\n    MA( Close , 15 ) **>** MA( Close , 30 )\n\n* * *\n\nIf you want to see what is going to be ported in the **AFL Editor** , click on the tab labeled **Auto-generated Formula**. The tab is to the right of the **Design View** that we are currently located within. When you click on it you will see the following code.\n\n* * *\n\n    Buy =  MA( Close , 15 ) > MA( Close , 30 );\n    Sell = 0;\n    Short = 0;\n    Cover = 0;\n\n* * *\n\nThis should look familiar. The **AFL Wizard** has auto-generated the rules for the four trading arrays: **Buy** , **Sell** , **Short** , and **Cover**. The Buy array is turned on when the **MA( Close , 15)** is greater than **MA( Close , 30)**. Click back on the **Design View** tab and let's point-and-click our way to a short entry rule. Click on **Enter Short when...** and then the **Add Item (Pencil** ) icon. You will once again see the following code:\n\n* * *\n\n    **Value of** Close (now) is greater than 10\n\n* * *\n\nFollowing the same steps we did for the long entry rules:\n\n  * Click **Value of** and slide down to the function pane and select **MA**.\n  * This time click on **is greater than** and slide down to the function pane. You will notice that the list of functions has now changed to just three options: \n    1. **Is greater than**\n    2. **Is less than**\n    3. **Is equal to**\n  * Select **is less than**.\n  * Click on the number **10** and slide down again to the function pane and select **MA\u2014Simple Moving Average**.\n  * Click on **15-bar moving average of** and change the period to **30**.\n\nIf you click on the **Auto-generated Formula** tab, you should now see the following code:\n\n* * *\n\n    Buy = MA( Close , 15 ) > MA( Close , 30 );\n    Sell = 0;\n    Short = MA( Close , 15 ) < MA( Close , 30 );\n    Cover = 0;\n\n* * *\n\nNow if you are testing a stock, you can send this directly to AmiBroker and have it **Backtest** or **Scan**. Before we test it, go ahead and save it somewhere as **MySecondAlgoWizard**. Assuming a stock database is loaded in AmiBroker, click on the **Exclamation** mark to send it to AmiBroker (see Figure 4.19).\n\n**Figure 4.19** Click the exclamation point to send your request to AmiBroker.\n\nIf you switch back to AmiBroker, you will be presented the **Automatic Analysis** dialog with your newly designed formula loaded. This dialog looks like the one in Figure 4.20.\n\n**Figure 4.20** Automatic Analysis dialog box.\n\nFrom here, you can click on **Back Test, Scan, Optimize, Explore** , etc. If you are testing a futures market, it is better to click on the icon that is right beside the **Exclamation** mark. This will build the formula and export it with a \".afl\"extension. You can then open it with **AFL Editor** and add the necessary code for testing a futures contract:\n\n* * *\n\n    PositionSize = MarginDeposit = 1;\n\n* * *\n\nFor practice, go ahead and click on the **Build and Export a formula** icon and save it somewhere on your desktop or in a folder in your C: drive. You then can go to the **AFL Editor** and open it and add the line of code above. I did just that and my **AFL Wizard** designed code looks like this:\n\n* * *\n\n    PositionSize = MarginDeposit = 1;\n    Buy =      MA( Close , 15 ) > MA( Close , 30 );\n    Sell = 0;\n    Short =    MA( Close , 15 ) < MA( Close , 30 );\n    Cover = 0;\n\n* * *\n\nThis should look very similar to the code we originally typed in for **MySecondAlgo**. The AFL Wizard can get you up and running very quickly and can also be used as a learning tool. It automatically builds the AFL code so you can start your design with the **Wizard** and have it generate the framework and then move onto the **AFL Editor** and complete your coding of your algorithm there.\n\n## AmiBroker Loop Programming\n\nAs powerful as AmiBroker's array programming is, there are times when you want to program on a bar-by-bar basis. Dr. Janeczko has built this capability into his software as well. With this feature you can program with your precious **If** constructs. Here is a familiar algorithm utilizing loops:\n\n* * *\n\n    PositionSize = MarginDeposit = 1;\n    mav = MA(C,20);\n    for( i = 0; i < BarCount; i++ ){\n           if (C[i] >= mav[i]) {\n                  Buy[i] = 1;\n           }\n           if (C[i] <= mav[i]) {\n                  Sell[i] = 1;\n           }\n    }\n\n* * *\n\nThis is the simple price crossing moving average reversal algorithm. The **mav** array is set to a 20-day simple moving average of closes. This line of code is still utilizing array programming. A **for loop** is then set up so that individual elements of the different arrays can be examined. The **for loop** construct in AFL is very similar to one in the C language. The loop variables are controlled by what is inside the parentheses.\n\n  1. ( i = 0; i < BarCount; i++ )\n\nHere the loop will start at zero and stop at **BarCount** and the loop variable **i** will increment by one each time **i++**. The **i++** is the same as **i = i + 1** ; since we are not processing arrays, we can now use **if** constructs. Notice how the comparison of the **C[i]** element and the **mav[i]** element is enclosed in parentheses. If the close is greater than the 20-bar moving average, then the following code is executed: Buy[i] = 1; This sets the **ith** element in the **Buy** array to one. Also, notice how the curly brackets **{ }** are used in the code. The curly brackets inform the computer and the programmer that certain lines are controlled by a loop or if construct. All the code involved with the **for loop** is surrounded by curly brackets. The code controlled by the **if** statement is surrounded by nested curly brackets. By matching an opening left curly bracket with a closing curly right bracket, the programmer can easily see what code goes with the **for loop** or the **if** statement. The left curly bracket right after the loop definition opens up the block of code consisting of six lines. The last-right curly bracket closes the loop and the block of code.\n\nIn addition to having to use the curly brackets to tell the computer which lines of code go inside the **loop** or the **if** construct, it's good practice to also indent the lines of code enclosed within brackets. This is not a necessity, but it makes the readability of the code so much easier. Now that we are using loops and ifs, we can highlight the collapsible or folding text feature of the editor. Look at the code in Figure 4.21.\n\n**Figure 4.21** Sample loop programming code.\n\nNotice where the arrows for the opening curly brackets are pointing. Now look at the small boxes with a dash inside that correspond to the curly brackets. If you click on the first box, the code controlled by the corresponding line of code will collapse or hide. Go ahead and do it\u2014see for yourself. If you click on the very first box, all the code disappears except the **for loop** definition. The second box collapses the code that goes with the first **if**. This is a neat feature if you want to streamline the look of your code, or if you want to hide it and send a screenshot to someone else. The combination of array and loop programming capabilities in AmiBroker should facilitate any programming needs you come across.\n\n## Summary\n\nAll programming languages are built on expressions and operators and the precedence of those operators. A solid understanding of these concepts is absolutely necessary before a single line of code is typed. We have just skimmed the surface of the capabilities of AmiBroker. It is a very powerful and easy-to-use tool. I learned AFL while writing this book, and used it for most of the testing in Chapters 2 and . Once you get your futures data into a database and create different portfolios, testing across the entire portfolio is as simple as clicking on a single button. The optimization capabilities of AmiBroker are simply mindblowing. You can use a genetic optimizer to optimize not only a single market but an entire portfolio\u2014just like I did in Chapter 3. The best way to learn AFL is through examples. Also the online community is absolutely terrific! I found answers to all my questions online and I didn't have to search very far.\n\nHere are some of the codes for the various systems I tested in Chapters 2 and :\n\n### Code Samples\n\n* * *\n\n    \/\/ Chapter 2 Double Moving Average Cross Over\n    \/\/ with Dollar Stop\n    Filter = 1;\n    PositionSize = MarginDeposit = 1;\n    sl = Param(\"ShortLen\",9,9,100);\n    il = Param(\"InterLen\",19,19,200);\n    mmStop = Param(\"StopDollars\",3000,3000,6000);\n    OptimizerSetEngine(\"cmae\");\n    Buy = MA(C,sl) > MA(C,il) ;\n    Short = MA(C,sl)< MA(C,il);\n    Sell = short;\n    Cover = buy;\n    ApplyStop(stopTypeLoss,stopModePoint,mmStop\/PointValue,1);\n    \/\/ Chapter 2 Bollinger Band Algorithm\n    PositionSize = MarginDeposit = 1;\n    len = Optimize(\"Len\",60,10,80,2);\n    width = Optimize(\"Width\",2,0.25,3,.25);\n    Buy = Cross(C,BBandTop(C,len,width));\n    Short = Cross(BBandBot(C,len,width),C);\n    Sell = Cross(MA(C,len),C);\n    Cover = Cross(C,MA(C,len));\n    \/\/Chapter 2 Bollinger Band using\n    \/\/RISK filtering\n    PositionSize = MarginDeposit = 1;\n    riskAmt = (BBandTop(C,60,2) - MA(C,60)) * pointValue;\n    COND1 = riskAmt <= 2000;\n    Buy = Cross(C,BBandTop(C,60,2)) AND COND1;\n    Short = Cross(BBandBot(C,60,2),C) AND COND1;\n    Sell = Cross(MA(C,60),C);\n    Cover = Cross(C,MA(C,60));\n    \/\/Chapter 2 MACD utilizing AMIBROKERS\n    \/\/Exploration Feature\n    \/\/Refer to the AMIBROKER Appendix for\n    \/\/further explanation\n    Filter = 1;\n    PositionSize = MarginDeposit = 1;\n    myMACD = MACD(fast=12,slow =26);\n    myMACDAvg = MA(myMACD,9);\n    myMACDDiff = myMACD - MYMACDAvg;\n    leftBar2 = Ref(myMACDDiff,-4);\n    leftBar1 = Ref(myMACDDiff,-3);\n    centerBar = Ref(myMACDDiff,-2);\n    rightBar1 = Ref(myMACDDiff,-1);\n    rightBar2 = Ref(myMACDDiff,0);\n    COND3 = C > MA(C,100);\n    COND1 = centerBar < 0 AND centerBar < Min(leftBar2, leftBar1) AND centerBar < Min(rightBar1,rightBar2);\n    COND2 = centerBar > 0 AND centerBar > Max(leftBar2, leftBar1) AND centerBar > Max(rightBar1,rightBar2);\n    Buy = COND1 AND COND3;\n    Short = COND2 AND NOT(COND3);\n    BuyPrice = C;\n    ShortPrice = C;\n    longEntryPrice=ValueWhen(Buy,BuyPrice,1);\n    shortEntryPrice = ValueWhen(Short,ShortPrice,1);\n    Sell = Cross(C, longEntryPrice - 3 * ATR(10));\n    Cover = Cross(ShortEntryPrice + 3 *ATR(10),C); ;\n    AddColumn(longEntryPrice,\"BuyPrice\");\n    AddColumn(longEntryPrice - 3 * ATR(10),\"longStop\");\n    AddColumn(Sell,\"Sell ?\");\n\n* * *\n\nThis last algorithm introduces the **AddColumn** and **ValueWhen** functionality. The **AddColumn** is useful for many purposes, but is probably used more as a debugging tool. Basically, this prints information out to the spreadsheet in **Exploration** mode. The **ValueWhen** function, in this example, returns the element of the **BuyPrice** array when the first value of 1 occurs in the **BUY** array. In other words, it is the long entry price. This code is somewhat sophisticated, as it introduces methods of debugging and gathering the last entry price. This code is fully explained along with utilizing AmiBroker's Exploration feature in the AmiBroker appendix.\n\n# Chapter 5  \nUsing Microsoft Excel to Backtest Your Algorithm\n\nThe majority of the testing that has been performed thus far in the book was carried out using either AmiBroker or TradeStation. These are complete trading solutions and should be able to fulfill all the needs of a trader. However, some traders would like to use software that they already have on their computer. Microsoft Excel has been a major player in the trading algorithm arena since the beginning. Spreadsheets combine data with formulas, and as we have seen these are the major ingredients to trading algorithms. Several years ago, I developed a very simple spreadsheet to help a client generate his trading signals for the next day. The system incorporated several indicators across several different markets. The Excel workbook turned out to be quite large and required the user to type in the daily data after the close each day. Excel spreadsheets are very powerful but when it comes to calculating indicators and trade signals they can become very cumbersome.\n\nThis is where VBA (Visual Basic for Applications) comes in very handy. VBA combines the power of the BASIC programming language with the power of spreadsheets. I have created a simple single-market algorithm-testing platform for those who wanted to use Excel and VBA. I call it the Excel System Backtester, ESB for short. Here is a version of the Bollinger Band algorithm that has been evaluated in the previous chapters programmed in our ESB platform.\n\n* * *\n\n    '****************************************************************\n    '**** This is a good place to put all of your function calls ****\n    '**** and system calculations.                ****\n    '****************************************************************\n      Call BollingerBand(myClose, 60, 2, avg, upBand, dnBand, i, 1)\n      simpleAvg = Average(myClose, 10, i, 1)\n      rsiVal = RSI(myClose, 14, i, 1)\n      Call Stochastic(3, 4, 7, stoK, stoD, slowD, i, 1)\n    '****************************************************************\n    '****        Put All Of Your Orders Here       ****\n    '****************************************************************\n      prevMarketPosition = marketPosition\n      If marketPosition <> 1 Then\n        Call Trade(Buy, \"BB-Buy\", upBand, stp, i)\n      End If\n      If marketPosition <> -1 Then\n        Call Trade(Sell, \"BB-Sell\", dnBand, stp, i)\n      End If\n      If marketPosition = -1 Then\n        Call Trade(ExitShort, \"ExitShortStop\", entryPrice + 2000 \/ myTickValue, stp, i)\n        If barsShort > 10 And myClose(i) > entryPrice Then\n          Call Trade(ExitShort, \"10dayShOut\", myClose(i), moc, i)\n        End If\n      End If\n      If marketPosition = 1 Then\n        Call Trade(ExitLong, \"ExitLongStop\", entryPrice - 2000 \/ myTickValue, stp, i)\n        If barsLong > 10 And myClose(i) < entryPrice Then\n          Call Trade(ExitLong, \"10dayLgOut\", myClose(i), moc, i)\n        End If\n      End If\n    '****************************************************************\n    '****        End of Main Traiding Loop        ****\n    '****      No orders allowed below this point      ****\n    '****************************************************************\n\n* * *\n\nThis system buys\/sells short on the penetration of 60-day two-standard-deviations Bollinger Bands, then risks $2,000 per trade and exits if the trade is not profitable after 10 days. The code is Visual Basic, so it follows that syntax. In developing this testing software my overall objective was simplicity\u2014I wanted a very easy sloping learning curve. You don't need to know VBA to invoke the indicator or trading functions. You will just need to follow the included templates.\n\nIf you are interested in learning some VBA and want to see under the hood, a discussion of the source code is included in the ESB appendix. First off, let's learn how to use the software and its indicator and trading functions. The VBA tester is based on one large loop. The software loops through each and every bar of data and allows the user to create indicators and analyze patterns and make trading decisions on each and every bar. Conceptually, it works like AmiBroker and TradeStation, but doesn't have all the bells and whistles. It can only test one market at a time, and has limited performance metrics. However, it is sufficient to get your feet wet, and since the source code is open, you can take it and do as you want.\n\n## VBA Functions and Subroutines\n\nThe heart of this algorithm is a 60-day Bollinger Band, and just like AmiBroker, you get access to this indicator by calling a function:\n\n* * *\n\n    Call BollingerBand(myClose, 60, 2, avg, upBand, dnBand, i, 1)\n\n* * *\n\nSome indicators return a single value and some return several. This function call is actually a subroutine that returns three parts of the Bollinger Band indicator: the moving average, the top band, and the lower band. The subroutine needs five inputs or arguments to calculate the bands: price (myClose), length (60), number of deviations (2), the current bar we are on (i), and the offset (1). Don't worry about the difference between a subroutine and function because in this application they do the same thing, return indicator values. The only difference is in the way you call the indicator and how the indicator values are assigned. Notice the difference between a pure function call and our BollingerBand subroutine call:\n\n* * *\n\n    myHH = Highest(myHigh, 20, i, 1)\n    Call BollingerBand(myClose, 60, 2, avg, upBand, dnBand, i, 1)\n\n* * *\n\nThe function **Highest** returns a single value and that value is assigned to the variable myHH. The subroutine **BollingerBand** is invoked by preceding the name of the subroutine with the keyword **Call**. The values that are used for trading decisions are included in the list of arguments that are passed into and out of the subroutine. The information that is passed to a function is only used by the function internally and is not passed back. The variable in our function call, **myHH** , holds the output off the function. In this case, it will be the highest high of the past 20 days starting from yesterday. A neat feature of VBA is that once you define functions or subroutines, the editor will help you fill out the argument list just like AmiBroker. (see figure 5.1.)\n\n**Figure 5.1** Once you define the Bollinger Band routine, you can easily incorporate it into trading.\n\nIf you are trying to invoke a function or subroutine and this helper box doesn't appear, then you probably do not have the function\/subroutine spelled correctly or it doesn't exist. All functions and subroutines in the ExcelSystemTester will be listed and explained in the appendix. All functions and subroutines require two bits of information: (1) the current bar and (2) how many days back to start the calculations. In the case of the **Highest** function, the current bar is **i** and the offset is **1**. An offset of two starts the calculations the day before yesterday. Remember, we are working in one big loop and each bar is examined by incrementing the variable _i_.\n\n## Data\n\nAccessing the date, price, volume, or open interest data is accomplished by using seven different arrays: **myDate** , **myOpen** , **myHigh** , **myLow** , **myClose** , **myVol** , and **myOpInt**. Two more arrays provide access to daily range and true range: **myRange** and **myTrueRange**. Unlike AmiBroker's array processing, you must index these arrays to get the data for a particular bar. **myOpen(10)** is the tenth bar's open from the very beginning of the data. **myDate(10)** is the date of the bar with **myOpen(10)** opening price. The arrays are synchronized. All data arrays are preceded with **my\u2014** so don't forget this. Let's say you want to see if yesterday was a high pivot of strength one. This is how you would test for this occurrence:\n\n* * *\n\n    If myHigh(i-1) > myHigh(i) and myHigh(i-1) > myHigh(i-2) then\n           myHighPivot = 1\n    End if\n\n* * *\n\nRemember, **i** is the current bar in the loop. So by subtracting 1 you get yesterday's bar and subtracting 2 gets the day before yesterday's bar. If you add 1, then you will have tomorrow's data today and you can make a killer trading system. With this information, you will finally have the Holy Grail in your grasp. On the other hand, save yourself a lot of time and heartache by not adding\u2014just subtracting. I can't tell you how many times a client has come to me with a 90% winning trading idea and I burst their bubble once they discover they are taking advantage of _future leak_. Here is the best example of future leak: Buy tomorrow's open if the high is greater than the open. This obviously is a mistake, but other forms of future leak are more esoteric. If high of tomorrow is greater than the buy stop, then buy price = buy stop. This one makes sense, where is the future leak; the price moved above the buy stop so the stop was filled at the specified price. If you test using this exact logic in the ESB, then you will hyperinflate your algorithm's performance. The error is hard to see, but it's there. What if the high is equal to your buy stop? Using this logic, the trade will not be executed and you will skip a trade that buys the absolute high price of the day. Skipping these trades will, without a doubt, increase your bottom line. Most system-testing platforms go out of their way to prevent future leak. TradeStation's EasyLanguage only lets you peek at the next bar's open. You can use it as a price to execute a trade or as a base price in a calculation of stop or limit orders. You can say buy next bar at open of next bar + 1\/2 ATR(10) on a stop or buy next bar at open \u2013 1\/2 ATR(10) on a limit. This makes future leak impossible and that should be a good thing. In most cases, this is true. However, this safety feature can make things a lot more difficult, too. Let's say you are placing multiple orders for the next bar, and one of them is filled, and you need to know exactly which order was executed. Some platforms simply execute the trade and don't let you know which order generated the trade. You might have different trade management algorithms for each order. In the ESB, I give the user complete power and therefore unlimited creativity. However, as you know, with great power comes great responsibility. Just be careful.\n\n## Software Structure\n\nFigure 5.2 shows a simplified structure of how the software works. There's a lot more going on behind the scenes, but this diagram gives a good overview of how the software in general works.\n\n**Figure 5.2** The structure of the backtesting software.\n\nThere are just two major components in the ESB, as you can see from the diagram. The two components are actuated by two simple buttons on the Data worksheet in the workbook. The **GetData** component forms the top of the diagram. This is where the programs asks for and then processes the data and puts the data into a digestible format. The second component, **TestSystem** , makes up the rest of the diagram and the rest of the software. Here is a breakdown of each component:\n\n  * **GetData** \u2014this button first asks you to locate a comma-delimited file (CSV) through an Open File dialog box. Once the file is opened it is cross-referenced with the data in the DataMaster to determine the **Tick Value** and **Min Tick** properties. The first two letters of the file names are used as the search key. If the file name is CLXXX.CSV, then the CL will be used to determine what market data is being utilized. In most cases CL is crude oil. If the symbol is not located in the DataMaster, then the software assumes it is a stock and defaults the **Tick Value** to one dollar and the **Min Tick** to a penny.\n  * **TestSystem** \u2014this button reads the data that was dumped into the **Data** worksheet and puts it into the seven data arrays. The program then starts looping through each bar and flows through the user-defined entry and exit logic. If the consequence of the user-defined logic results in a trade entry or exit, the software makes a note and processes the order as if it was actually executed. At the termination of the data loop, all trades are processed and an equity stream of closed trade profits and open trade equity is dumped out into the **EquityStream** worksheet.\n\nKeep in mind this is a very simple program and can only work on one market one system at a time. Figure 5.3 shows a typical layout of the Data worksheet.\n\n**Figure 5.3** A typical Excel data worksheet.\n\nThe **GetData** component was previously actuated by simply clicking the button and as you can see the different cells, **open** , **high** , **low** , **close** , **volume** , and **open interest** , were then populated. Also, the **Tick Value** and **Min Tick** values were cross-referenced in the DataMaster and their respective cells were filled in as well. The DataMaster worksheet contains the symbols and their properties in the current data universe. As you can see, stocks are not listed, but that doesn't mean you can't test them. Commodities and futures contracts have different sizes and values. These values must be stored somewhere so the software can gain access to them during the testing process. Originally, the software simply asked the user to input this information, but I quickly discovered this was way too cumbersome and decided to create the DataMaster worksheet (see Figure 5.4).\n\n**Figure 5.4** The DataMaster worksheet can help you store the values of your commodities and futures contracts.\n\nThese data properties were derived directly from the underlying data. If you don't get these values right, then your results will be completely wrong. Most data vendors will provide this information when you purchase their end-of-date (EOD) historic database. I use two data vendors, CSI and Pinnacle, for commodity and futures price quotes. You can buy deep historic databases from either of these vendors for a very reasonable price. If you purchase data through a reliable vendor, then you know you are getting quality data. You can cobble data together from various free services on the Internet, but remember GIGO (garbage in garbage out). For the ExcelSystemBackTester, I utilize data provided by CSI, but you can utilize any data as long as it is in ASCII and delimited by commas. The DataMaster can be used as a template and is fully editable. Some vendors use different decimal locations for the same market. If this is the case, you have got to make sure your DataMaster reflects the correct **Tick Value** and **Min Tick**.\n\nThe **Results** worksheet is the repository of all your trades and performance metrics. Every trade covers two rows, one for entry and one for exit. A profit or loss is calculated when a trade is exited and the cumulative profit or loss is updated as well. Figure 5.5 is an example of the **Results** worksheet where several trades and the performance metrics associated with them are listed.\n\n**Figure 5.5** The Results worksheet lists multiple trades and their associated performance metrics.\n\nThe date, name of entry\/exit signal, price of entry\/exit, and the results of the trade are listed chronologically. Currently there are only four major performance metrics included in the worksheet: (1) **Total Profit** , (2) **Maximum DD** , (3) **Number of Trades** , and (4) **Percent Wins**. By the time this book is published, I should have a few more calculated, such as: **Average Trade** , **Average Win** , **Average Loss** , and **Profit Factor**.\n\nThe last worksheet in the ESB is labeled **EquityStream** and stores the daily equity of the trading system. There is a **Create Equity Curve** button that launches a simple macro that creates a chart (see Figure 5.6) and plots the cumulative daily equity and drawdown. This chart gives a very quick snapshot of the overall performance of the trading system. There is enough information provided to help determine if your particular algorithm is good enough for either trading or further research. As you can see, the interface is simple enough\u2014load data, test system, plot equity. The hard stuff is getting into the VBA and creating a trading algorithm with the tools at hand.\n\n**Figure 5.6** The EquityStream worksheet has a button that launches a macro to create a chart that plots the cumulative daily equity and drawdown.\n\n## Programming Environment\n\nThe BASIC (an acronym for **Beginner's All-purpose Symbolic Instruction Code** ) programming language was introduced in the 1960s but became popular in the mid-1970s. BASIC quickly became the most popular interpreted language for beginners. Over the years, compiler companies have extended BASIC into professional software development. Microsoft has been using BASIC derivatives for decades and it is at the heart of VBA. When you code your trading algorithm in to the ESB, you will be programming in pure BASIC. You will see a lot of lines of code (I wish I could hide them, but the VBA Editor doesn't allow for collapsible text regions) that you will not need to understand (unless you want to). There are just three major sections with which you will need to concern yourself. But before we get into all that, launch your Microsoft Excel and make sure you see the **Developer** tab in the Ribbon. If you don't see it, then you will need to turn it on. You can do this by simply clicking the **File** menu and then selecting **Options**. After Excel Options dialog opens scroll down to **Customize Ribbon** and click on it. Figure 5.7 shows the **Customize Ribbon** dialog and where to click to get the **Developer** Tab to show up in the Ribbon.\n\n**Figure 5.7** Customize your Excel ribbon. Make sure you check the box next to the Developer tab option.\n\nOnce you return to a blank worksheet, the Developer tab should now be visible. Now go to the www.wiley.com\/go\/ultimatealgotoolbox and download the ExcelSystemBackTester.zip file to your C: drive and unzip it. The zip file contains two folders: Algorithms and Data. Get to the Algorithms folder and open the BollingerDX spreadsheet. The spreadsheet that opens should look similar to Figure 5.3. Now click on the **Developer** tab and the Ribbon should take on this appearance (see Figure 5.8).\n\n**Figure 5.8** The Developer tab in Microsoft Excel.\n\nThe only icon that we are concerned with is to the far left of the Ribbon, **Visual Basic**. Click on the icon\/button and the **Visual Basic Editor** (VBE) will launch. Make sure your VBE looks like the one in Figure 5.9.\n\n**Figure 5.9** The Visual Basic Editor (VBE) in Microsoft Excel.\n\nThere's a lot of stuff here. But we are just interested in the modules that are inside the red oval from Figure 5.9. You don't need to mess with anything else if you are just interested in using the ESB to test your trading ideas. The window that contains the modules is called the **Project Explorer**. It simply groups all the files in the project together for convenience.\n\n### The ESB VBA Code\n\nThe modules are individual files that contain the source code to carry out the various chores necessary to test a trading system. The most import module ( **Main** ) is the one we will spend most of our time in. Go ahead and double-click **Main** \u2014the source code inside the module will open in a separate window (see Figure 5.10). Inside this module are five submodules or subroutines: **declarations** , **fillTradeInfo** , **GetData** , **SystemTester** , and **main**. Most of our work will take place in the **SystemTester** module. Go ahead and scroll up and down in this window. It's going to look daunting without a doubt, but you need not concern yourself with 80 percent of the source code. Scroll to the very top of the window and you will see a lot of text in green. In VB, everything green is considered a comment. This information is just there for the benefit of the person reading the code\u2014it is hidden from the computer. Notice the first character in each line that is green. It is a single quote ('). A single quote at the beginning of a line of VBA code tells the computer to ignore the line. Just like the double forward slashes \"\/\/\" in AmiBroker. This area of the source code lets the reader know the title, version, subversion of the software, and any notes pertaining to revisions and necessary information a user may need to know. If you scrolled through the source, you will see multiple lines of comments. I did this to modularize the code and explain what the individual sections were doing. Breaking down a large program into sections helps explain the code in block-by-block fashion. Source code read as a whole is very difficult to understand, but if you can break the code into individual blocks, it will help make things much clearer.\n\n**Figure 5.10** Comment text in VBA code is written in gray. It is also indicated by the single quote at the beginning of each line.\n\nYou will notice several banners or blocks of green text embedded in the source code. Figure 5.10 is an example of one of these blocks.\n\nThese comments, in this format, help break up the code and also explain what's happening in the next few lines of code. This commentary explains how the data contained in the cells of the Data worksheet will be imported into the individual data arrays. Column 1 holds the Dates of each individual daily bar. The first row of data (row 3 in our case) holds the date, open, high, low, close, volume, and open interest for one day of data. Think of the data as just in one big table consisting of rows and columns. If you want the data of the very first bar, you would access it using the following notation: Cells (3, 1). In Excel, _cells_ refer to the entire table and the first index value inside the parentheses refers to the row number and the second index value is the column number. Cells (3, 2) refer to the open price of the very first bar. A newcomer to VBA might not know this nomenclature so it is included in this comment block. Since all of the data is being read into individual arrays you do not need to concern yourself with notion of cells, unless you want to.\n\nScroll down in the code until you find a comment block that looks like the one in Figure 5.11.\n\n**Figure 5.11** This is where the VBA comments end and the actual source code begins.\n\nAll the code that you scrolled through until you got to this point can be ignored. This area of the source code is used to set up any system-specific variables. The two lines of code following the comment block set **rampUp** to 100 and **commsn** to 100. Notice there are lines of comments to the right of each variable assignment. These are inline comments to let you know what that exact line of code is doing. You will again see the single quote (') preceding the comment. Whatever follows the single quote is ignored by the computer, be it at the beginning of a line of code or right in the middle. The **rampUp** variable informs the computer that the trading algorithm needs 100 days of data for calculation purposes before it can initiate trading. If you want to test a 100-day simple moving average algorithm, you would need to tell the ESB to skip the first 100 days of data so that the moving average can be properly calculated. You must be careful and make sure you provide enough ramp-up data for your algorithm. If you don't, the computer will halt and provide an error message like the one in Figure 5.12.\n\n**Figure 5.12** This error message appears if your algorithm does not have enough ramp-up data.\n\nThis looks like a scary error message but it's simply telling you that you are trying to go back further in time than you have specified. The **commsn** variable is what the computer will deduct from each round-turn trade. In this example it is set to 100; you can change this to any value you think might be appropriate. This is also a good area to set up any user-defined variables that you might need to use later in your code. Let's say you have a special calculation that uses a constant value, like a profit objective. You could set this up by using the following code:\n\n* * *\n\n    rampUp = 5 ' how many days needed in calculations before trading\n    commsn = 100 ' commission per round turn trade\n    **myProfitTarg = 0.02 ' I just put this line of code in**\n\n* * *\n\nLater on, in the source code you can use **myProfitTarg** to help calculate where a profit might be taken. This is a very good time to mention that VBA, in most cases, is case insensitive. This means the two variable names, **myDayCount** and **mydaycount** , are one and the same. Ignoring case can cause sloppy-looking code and difficult readability. It's good programming etiquette to use the same case throughout a program. As I have mentioned in a previous chapter, I personally like using a capital letter in a variable name where a new word begins, but not the first letter of the variable name. Here are some examples you can use as a template for your variable names: **myProfitTarg** , **longDayCount** , **myStopAmt**. You have also got to be careful not to step on one of the keywords that is used by the ESB. Unlike AmiBroker or TradeStation, VBA will not tell you ahead of time that you are reassigning a variable that is already used elsewhere in the source code. A list of keywords is provided in the appendix to help prevent this from happening.\n\nLet's get to the heart of the ESB by scrolling down to the main trading loop. You can find it by looking for the comment block that says, \"Main Trading Loop.\" The entire Main Trading Loop is shown below. This includes some of the code we have already discussed and some code that you can simply ignore.\n\n* * *\n\n    '****************************************************************\n    '****         Main Trading Loop           ****\n    '****************************************************************\n    Do While i <= numRecords\n      tradeDays = tradeDays + 1\n      IncrementalProgress.Show\n      i = i + 1              'Dont touch\n      sPercentage = (i \/ numRecords) * 100\n      sStatus = \"Checking \" & i & \" of \" & numRecords & \" paragraphs\"\n      IncrementalProgress.Increment sPercentage, sStatus\n      intraDayTrdCnt = 0                 'Leave in.\n      If barsLong <> 0 Then barsLong = barsLong + 1   'Leave in.\n      If barsShort <> 0 Then barsShort = barsShort + 1  'Leave in.\n    '****************************************************************\n    '**** This is a good place to put all of your function calls ****\n    '**** and system calculations.                ****\n    '****************************************************************\n      Call BollingerBand(myClose, 60, 2, avg, upBand, dnBand, i, 1)\n      simpleAvg = Average(myClose, 10, i, 1)\n      rsiVal = RSI(myClose, 14, i, 1)\n      Call Stochastic(3, 4, 7, stoK, stoD, slowD, i, 1)\n    '****************************************************************\n    '****        Put All Of Your Orders Here       ****\n    '****************************************************************\n      prevMarketPosition = marketPosition\n      If marketPosition <> 1 Then\n        Call Trade(Buy, \"BB-Buy\", upBand, stp, i)\n      End If\n      If marketPosition <> -1 Then\n        Call Trade(Sell, \"BB-Short\", dnBand, stp, i)\n      End If\n      If marketPosition = -1 Then\n        Call Trade(ExitShort, \"ExitShortStop\", entryPrice + 2000 \/ myTickValue, stp, i)\n        If barsShort > 10 And myClose(i) > entryPrice Then\n          Call Trade(ExitShort, \"10dayShOut\", myClose(i), moc, i)\n        End If\n      End If\n      If marketPosition = 1 Then\n        Call Trade(ExitLong, \"ExitLongStop\", entryPrice - 2000 \/ myTickValue, stp, i)\n        If barsLong > 10 And myClose(i) < entryPrice Then\n          Call Trade(ExitLong, \"10dayLgOut\", myClose(i), moc, i)\n        End If\n      End If\n    '****************************************************************\n    '****        End of Main Traiding Loop        ****\n    '****      No orders allowed below this point      ****\n    '****************************************************************\n\n* * *\n\nThis is the complete listing for this simple Bollinger Band system. Don't worry about any of the source code _above_ the Main Trading Loop comment block:\n\n* * *\n\n    '****************************************************************\n    '**** This is a good place to put all of your function calls ****\n    '**** and system calculations.                ****\n    '****************************************************************\n\n* * *\n\nAll of your programming will take place below these comments. For illustration purposes, I immediately call four indicator functions\/subroutines.\n\n* * *\n\n      Call BollingerBand(myClose, 60, 2, avg, upBand, dnBand, i, 1)\n      simpleAvg = Average(myClose, 10, i, 1)\n      rsiVal = RSI(myClose, 14, i, 1)\n      Call Stochastic(3, 4, 7, stoK, stoD, slowD, i, 1)\n\n* * *\n\nThis algorithm will just use the values modified in the **BollingerBand** subroutine: **avg** , **upBand** , **dnBand**. Let's look at the other function calls in the program:\n\n  * _Average_. This function returns a moving average value for the past 10 days. Its first argument is the data array that is to be averaged, and the second argument is the length of the calculation. You will again notice the last two arguments are the same for all of the function calls: **i** and **1**. The **i** will never change but you can change the last argument. If you were to use **2** , then the indicator calculation would be offset by two days. Instead of starting yesterday and going back 10 days, it would start the day before yesterday and go back 10 days.\n  * _RSI_. This function returns the Relative Strength Index from the last 14 days of data. This function requires the data array and the length of the RSI calculation. As it is programmed in this example, it is utilizing the closing prices for its calculations. You can pass it high or low or volume data\u2014it doesn't matter.\n  * _Stochastic_. This subroutine returns K, D, and Slow D. Here is another example of using a subroutine in place of a function. Just like the BollingerBand subroutine, this subroutine returns three values. It requires the length of the K calculation and the two additional smoothing lengths for the D and Slow D.\n\nExamples of all the indicator functions and subroutines are included in Appendix B.\n\nAfter you get your indicator values, the real fun begins. The most important code in the ESB can be found in the **Trade** subroutine. This subroutine determines if a trade takes place and then stores the information for later use. The subroutine requires five arguments (see Figure 5.13).\n\n**Figure 5.13** This tiny snippet of code contains five arguments, which are all necessary for the Trade subroutine.\n\nThe first argument informs the subroutine what you are trying to do and can take on four different values: **Buy** , **Sell** , **ExitLong** , **ExitShort**. The second argument is the name of the signal that will show up in the trade listing. It is a string so it needs to be included in quotes. In this example, the signal name is \" **BB-Buy**.\" It can be anything you want. The third argument is the price you would like to execute the order. Here we are trying to **Buy** on the penetration of the Upper Bollinger Band. The fourth argument informs the subroutine what type of order is being placed. It can take on four types of orders:\n\n  * **stp\u2014Stop order**. The price of the order must be above the market for Buy and ExitShort or below the market for Sell and ExitLong.\n  * **lmt\u2014Limit order**. The price of the order must be below the market for Buy and ExitShort or above the market for Sell and ExitLong.\n  * **moc\u2014Market On Close order**. Execute the order at the closing price of the bar. Specify myClose(i) as the trade price.\n  * **mkt\u2014Market order**. Executes at the opening of the bar. Specify **myOpen(i)** as the trade price.\n\nThe final argument is always the letter **i** , the current bar in the loop.\n\n* * *\n\n      If marketPosition <> 1 Then\n        Call Trade(Buy, \"BB-Buy\", upBand, stp, i)\n      End If\n\n* * *\n\nThis order is only placed if the current **marketPosition** is not equal to 1 or long. In other words, don't place a **Buy** order if you are already long. Unfortunately, this version of the ESB doesn't allow pyramiding; however, future versions will have this capability. Entering a short position uses the same subroutine but the arguments are different:\n\n* * *\n\n    If marketPosition <> -1 Then\n        Call Trade(Sell, \"BB-Short\", dnBand, stp, i)\n      End If\n\n* * *\n\nHere we are trying to **Sell** short at the lower Bollinger Band on a stop. That's it for the trade entries. Simple, thus far? Once you get into a trade, this algorithm has three different ways to exit:\n\n  * A $2,000 money management stop\u2014any time the market moves $2,000 against the position it is exited.\n  * Time-based market order (MOC)\u2014if the position is not profitable after 10 bars, the trade is exited on the close.\n  * Reversal if the opposite side of the Bollinger Bands is penetrated before an exit is executed.\n\n* * *\n\n      If marketPosition = 1 Then\n        Call Trade(ExitLong, \"ExitLongStop\", entryPrice - 2000 \/ myTickValue, stp, i)\n        If barsLong > 10 And myClose(i) < entryPrice Then\n          Call Trade(ExitLong, \"10dayLgOut\", myClose(i), moc, i)\n        End If\n      End If\n\n* * *\n\nThe money management order is placed by calling the Trade subroutine with the following parameters: **ExitLong** , \" **ExitLongStop** ,\" **entryPrice \u2212 2000\/ myTickValue** , **stp** , **i**. Here we are informing the Trade subroutine to liquidate our long position on a stop at a price equivalent to the **entryPrice** \u2212 $2,000. The trade price can be represented by a single variable name or it can be a mathematical expression. In this case, we convert $2,000 to points by dividing 2,000 by **myTickValue**. Remember, you have got to make sure your units are compatible. If you are working with prices, then you must convert dollars to a price level. It would be a mistake in this case to forget to divide by **myTickValue**. Let's say you entered a trade in the DX at 129.050 and wanted to risk $2,000. If you didn't convert to points, then the exit price would be 129.050 \u2013 2000. This doesn't make sense. Just remember to convert all dollar figures to points.\n\nThe time-based order is placed after the long position has been in a trade for more than 10 days and is in a losing position. The variables **barsLong** and **entryPrice** are keywords so don't use them in an assignment statement\u2014just look, don't touch.\n\n**If barsLong > 10** tells the computer to ignore the **ExitLong** order until the trade has been on for at least 10 days and the **and myClose(i) < entryPrice** tells the computer to ignore the order if the position is in the profit. The orders to exit a short position use the keyword **barsShort** , and the logic **myClose(i) > entryPrice**.\n\n* * *\n\n      If marketPosition = -1 Then\n        Call Trade(ExitShort, \"ExitShortStop\", entryPrice + 2000 \/ myTickValue, stp, i)\n        If barsShort > 10 And myClose(i) > entryPrice Then\n          Call Trade(ExitShort, \"10dayShOut\", myClose(i), moc, i)\n        End If\n      End If\n\n* * *\n\nOnce you type in your trading logic, simply return to Excel and go to the **Data** worksheet and click on the **Run System** button.\n\n## Summary\n\nAll the tools that you need to calculate indicators, analyze price bar relations, manage open trades, and execute them are all built into the ESB. The best way to learn how to program your trading algorithms using VBA is by following along as many examples as you can. The end of this chapter has a couple more simple examples that you should be able to build upon.\n\n#### _Simple Moving Average Crossover using 3 X ATR stop_\n\n* * *\n\n    '****************************************************************\n    '**** This is a good place to put all of your function calls ****\n    '**** and system calculations. A simple moving average   ****\n    '**** crossover system using a 3 ATR protective stop.    ****\n    '****************************************************************\n      Call BollingerBand(myClose, 60, 2, avg, upBand, dnBand, i, 1)\n      simpleAvg = Average(myClose, 19, i, 1)\n      atr = Average(myTrueRange, 20, i, 1)\n      rsiVal = RSI(myClose, 14, i, 1)\n      Call Stochastic(3, 4, 7, stoK, stoD, slowD, i, 1)\n    '****************************************************************\n    '****        Put All Of Your Orders Here       ****\n    '****************************************************************\n      prevMarketPosition = marketPosition\n      If marketPosition <> 1 And myClose(i) > simpleAvg Then\n        Call Trade(Buy, \"CrossBuy\", myClose(i), moc, i)\n      End If\n      If marketPosition <> -1 And myClose(i) < simpleAvg Then\n        Call Trade(Sell, \"CrossSell\", myClose(i), moc, i)\n      End If\n      If marketPosition = 1 And myClose(i) < entryPrice - 3 * atr Then\n        Call Trade(ExitLong, \"L-ATR Stop\", myClose(i), moc, i)\n      End If\n      If marketPosition = -1 And myClose(i) > entryPrice + 3 * atr Then\n        Call Trade(ExitShort, \"S-ATR Stop\", myClose(i), moc, i)\n      End If\n    '****************************************************************\n    '****        End of Main Traiding Loop        ****\n    '****      No orders allowed below this point      ****\n    '****************************************************************\n\n* * *\n\n#### _Simple RSI system using 3 X ATR stop and 5 X ATR profit objective_\n\n* * *\n\n    '****************************************************************\n    '**** This is a good place to put all of your function calls ****\n    '**** and system calculations. A simple RSI system     ****\n    '**** that buys when RSI < 20 and sells when RSI > 80.    ****\n    '**** Utilizes a 5 ATR Profit and a 3 ATR Protective Stop  ****\n    '****************************************************************\n      Call BollingerBand(myClose, 60, 2, avg, upBand, dnBand, i, 1)\n      simpleAvg = Average(myClose, 19, i, 1)\n      atr = Average(myTrueRange, 20, i, 1)\n      rsiVal = RSI(myClose, 14, i, 0)\n      Call Stochastic(3, 4, 7, stoK, stoD, slowD, i, 1)\n    '****************************************************************\n    '****        Put All Of Your Orders Here       ****\n    '****************************************************************\n      prevMarketPosition = marketPosition\n      If marketPosition <> 1 And rsiVal < 20 Then\n        Call Trade(Buy, \"RSIBuy\", myClose(i), moc, i)\n      End If\n      If marketPosition <> -1 And rsiVal > 80 Then\n        Call Trade(Sell, \"RSISell\", myClose(i), moc, i)\n      End If\n      If marketPosition = 1 Then\n        If myClose(i) < entryPrice - 3 * atr Then\n          Call Trade(ExitLong, \"L-ATR Stop\", myClose(i), moc, i)\n        End If\n        If myClose(i) > entryPrice + 5 * atr Then\n          Call Trade(ExitLong, \"L-ATR Prof\", myClose(i), moc, i)\n        End If\n      End If\n      If marketPosition = -1 Then\n        If myClose(i) > entryPrice + 3 * atr Then\n          Call Trade(ExitShort, \"S-ATR Stop\", myClose(i), moc, i)\n        End If\n        If myClose(i) < entryPrice - 5 * atr Then\n          Call Trade(ExitShort, \"S-ATR Prof\", myClose(i), moc, i)\n        End If\n      End If\n    '****************************************************************\n    '****        End of Main Trading Loop        ****\n    '****      No orders allowed below this point      ****\n    '****************************************************************\n\n* * *\n\n# Chapter 6  \nUsing Python to Backtest Your Algorithm\n\n## Why Python?\n\nAround the same time I started to write this book, I was teaching myself the Python programming language. If you have spent time roaming around the Internet looking for the next quant language, you have seen Python mentioned many times. The usage of this language has grown tremendously over the past few years in the quant universe. This doesn't mean Python is only found in this arena, nor does it mean it is only used by mathematicians, economists, or scientists. There are languages, such as R, that are specifically designed for mathematicians, but Python has universal usage. As I have mentioned many times in the previous chapters, most algorithmic traders do not come from a programming background. The introduction to AmiBroker and the Excel VBA\u2013based backtester has shown that a computer science degree is not necessary to get your algorithms tested on historic data. Most algorithmic traders learn just enough of a language to get the job done. In doing so, they use a very small subset of a platforms tools\u2014barely touching the tip of the iceberg. Imagine, though, how much more powerful their algorithm backtesting and development skills could become through learning as much as they can about programming. If I was a newbie and wanted to learn as much about programming as quickly as possible, I would learn Python. This is coming from a programmer who is deeply devoted to FORTRAN and C. The skillset that you would acquire would only enhance your abilities in other languages and testing\/trading platforms.\n\nPython has caught on for many reasons, but the biggest is ease of learning. Plus, you are exposed to the two predominant programming paradigms: object oriented and procedural. If you were formally educated in computer science in the 1970s and 1980s, you probably come from the procedural school of thought. I have a programmer friend who worked at Bell Laboratories in the 1970s, and he doesn't even want to talk about \"objects.\" Today's programmers, or should I say constructors, can build a rich graphical user interface (GUI) in a matter of minutes\u2014something that took me a week to do back in the 1980s. A really good programmer can use both objects and procedures.\n\nThis chapter introduces ideas from both schools of thought. However, this chapter is only a small introduction to the Python language. If you want to learn more about Python and computer science in general, I highly recommend John Zelle's _Python Programming: An Introduction to Computer Science_. This book provides a solid foundation not only in Python but also in some of the most useful topics in computer science\u2014knowledge that can be carried over into the trading arena.\n\nIn this new world of touchscreens, the usage of typing in a command line is disappearing. Well, that is if you are not from a Unix background. Python programmers become very comfortable with the command line, because it is so powerful. In many cases, a sophisticated user interface isn't necessary and this backtesting application is one of those cases. There isn't any exposure to developing a GUI because it wasn't necessary. Time was spent on topics that provided exposure on file manipulation, simple input\/output, objects, and functions. Before we start discussing the Python system backtester (PSB), here is a list of why Python was the language of choice:\n\n  * _Python is an interpreted language_. The user doesn't need to go through an additional compilation step. Once you type the program into the editor, all you need to do is run it. If there is a syntax or runtime error, the Python shell will let you know about it.\n  * _Python has a shell_. This interactive command line interpreter can be used for testing small bits of code, as the output window for printing out information from your programs, or as a very cool calculator. The PSB uses this window exclusively in this tutorial. This window also alerts you to syntax and runtime errors.\n  * _Python is dynamically typed_. A user can create a variable name on the fly without having to declare\/size it before using it. This is similar to AFL and VBA. This helps the creative flow process. In other languages, when you create a variable you have to go back to the top of the code and declare it. By the time you get back to using it, you might have forgotten what you were planning on doing. Well, this is a problem I suffer; you might not. There are both advantages and disadvantages to dynamic typing and this is a subject of much controversy in the programming arena.\n  * _Python is free and has a very large user's group_. The code is open source and free to use in commercial and personal applications as long as the copyright information is prominently displayed. Almost any question about how to do something in Python can be answered by simply Googling it.\n  * _Python is perfect for our purposes_. All the trading system tracking\/bookkeeping\/analysis has already been programmed. All you need to do is use Python language and the PSB tools to create the perfect mousetrap.\n  * _Python has no foo bar_. Most other languages use the words **foo** and **bar** as variable names in example code snippets. Their origination is from the military acronym **fubar** and was probably first used at MIT. Python uses the words **spam** and **eggs** instead. The creators of Python wanted it to be fun, so they named the language after _Monty Python_ , the British comedy group. Most people wouldn't put the words \"fun\" and \"programming\" together, but some of us would. **Spam** and **eggs** come from _Monty Python's Flying Circus_.\n\n## Python Installation\n\nYou will need to install Python on your computer before we can do anything. This is another benefit of Python\u2014it is very easy to install. Go to the website www.python.org and click on **Downloads**. You will be provided links for different operating systems. As of the writing of this chapter the latest version was 3.5. The PSB was written in 3.4 and is upwardly compatible. Download the installer and run it. Just click through the prompts\u2014the default values will be fine. This installer should install both IDLE (Python GUI) and Python (command line). If you are using a Windows operating system, a folder with the Python programs and manuals will be installed in your **Apps** screen. I use **Classic Shell** on my Windows 8 machine and both applications flow to the Start menu. Of course, if you are using Windows 7, it goes without saying you will find Python on the Start menu.\n\n## PSB Installation\n\nDownload the PSB.zip file from this book's companion website (www.wiley.com\/go\/ultimatealgotoolbox). Unzip it to your C: drive or to another convenient location.\n\n### IDLE\n\nThe PSB was programmed exclusively with **IDLE** , the standard Python development environment. Its name is an acronym of \"Integrated DeveLopment Environment.\" It works well on both Unix and Windows platforms. It includes a Python shell window, which gives you access to the Python interactive mode. Before we can start looking at the PSB, we need to learn a little bit about IDLE. So go ahead and launch it through your Apps screen or Start menu. The Python Shell should now be in front of you. Doesn't look that special; it's barebones in fact. This is by design to keep the Python learning curve as a gradual incline.\n\nA prompt, preceded by >>>, should now be blinking at you. Currently, the Shell is in interactive mode\u2014meaning that it wants you to give it a command. Type **4 + 6** and hit enter. The prompt now comes back with the answer\u2014 **10**. Now type **print(\"Hello World\")** and hit enter. The prompt comes back with **Hello World**. Now type **100 ** 0.5** and hit enter. You should get the number **10**. Now type **import this** and hit enter. Your screen should be filled with the following:\n\n**The Zen of Python, by Tim Peters**\n\nBeautiful is better than ugly.\n\nExplicit is better than implicit.\n\nSimple is better than complex.\n\nComplex is better than complicated.\n\nFlat is better than nested.\n\nSparse is better than dense.\n\nReadability counts.\n\nSpecial cases aren't special enough to break the rules.\n\nAlthough practicality beats purity.\n\nErrors should never pass silently.\n\nUnless explicitly silenced.\n\nIn the face of ambiguity, refuse the temptation to guess.\n\nThere should be one\u2014and preferably only one\u2014obvious way to do it.\n\nAlthough that way may not be obvious at first unless you're Dutch.\n\nNow is better than never.\n\nAlthough never is often better than *right* now.\n\nIf the implementation is hard to explain, it's a bad idea.\n\nIf the implementation is easy to explain, it may be a good idea.\n\nNamespaces are one honking great idea\u2014let's do more of those!\n\nThis is an \"Easter egg\" of a poem created by Python pioneer Tim Peters. Pretty cool, huh? Some people live, eat, and breathe Python. You don't need to go overboard to get the benefits of learning a programming language such as Python.\n\nYou can now see why they call it an \"interactive\" shell. Let's switch gears and use IDLE as an IDE. Go under the **File** menu and select **Open...** and then navigate your way to the PSB folder and open **PSBBollinger**. The file that opens is a trading algorithm based once again on Bollinger Bands.\n\n## PSB Structure\n\nThe PSB consists of a collection of files or modules. The main module is this one\u2014it pulls or calls all the other modules to facilitate the loading of data, looping through each and every bar, accounting for all of the trades and input\/output. If you want to create another trading algorithm, you start with an existing one and simply save it as a different name. There aren't separate submodules that simply include trading logic. This is just like the VBA-based ESB. This is unlike AmiBroker and TradeStation. In these softwares, trading rules, functions, and indicators are stored in a library.\n\nEverything you program will take place in this module or the indicators.py module. All Python source code files have the .py extension. The PSB is very simple\u2014type in your trading logic, which hopefully you have derived from a flowchart or a finite state machine, and then **Check Module** under the **Run** menu. If it passes, then you simply **Run** it by going under the **Run** menu and selecting **Run Module**.\n\nLet's run this PSBBollinger module by selecting **Run Module** from the **Run** menu. After you select **Run Module** , a File Browser window titled **Select Markets To Test** will open. The PSB is asking you to select a .CSV file or files that in can read into memory to which it will apply the Bollinger algorithm. Navigate to the Commodity Data folder and select the file that starts with CL (crude oil). Once you select the file and click OK, data and results generated by the algorithm will flow to the Python Shell and be displayed there. The results are presented in the following order:\n\n  * _Combined results_. The PSB will combine the results of a multimarket selection. If only one market is selected, then it will just be the results from the single market.\n  * _Combined monthly results_. The monthly results in $ are shown for all markets selected.\n  * _Individual market results_. Performance metrics for each market are displayed.\n  * _Trade-by-trade results_. A list of entries and exits and the $P\/L derived from each trade are shown.\n\nThat's it. There aren't any fancy charts involved\u2014everything is presented in a tabular format. If you want to see indicator values, you simply print them out to the Shell. The PSB structure, just like IDLE, was kept as simple as possible. Let's learn some Python by looking at the different PSB modules.\n\n### Indicator.py\n\nThis module contains all of the indicators that are utilized in the PSB. Go under the **File** menu in the Shell and open **indicators.py**. Keep the Python Shell open and quickly accessible. There are two types of indicators in this file. Some are just simple function modules and some are a little more sophisticated and are programmed as a class structure. I said you would be exposed to procedural (functions) and object-oriented (classes) programming. Let's start slowly with a simple indicator function.\n\n#### sAverage\n\n* * *\n\n    def sAverage(prices,lookBack,curBar,offset):\n        result = 0.0\n        for index in range((curBar - offset) - (lookBack-1),curBar - offset +1):\n            result = result + prices[index]\n        result = result\/float(lookBack)\n        return result\n\n* * *\n\nThis looks pretty palatable! All functions or, if you like, subprograms, use the key word **def** to let the interpreter know a function is about to be defined. The function has to have a name, and in this case it is **sAverage** (simple moving average). Remember how we called functions in AFL and VBA? You do it exactly the same way in Python. The list of arguments or parameters that are included in the parentheses after the function name are known as the formal argument list. These parameters are called formal because they're included in the function definition. Since we are using Python you don't need to employ function prototyping or forward declaration. In other words, you don't need to tell the interpreter the types of the arguments before-hand. The **sAverage** function requires four parameters:\n\n  * **prices**. This is a list of price data (open, high, low, close, volume, or open interest). In this application of Python, we will be using the word _list_ instead of **array** , but know they mean basically the same thing. The Python **list** structure is very powerful as you will find out.\n  * **lookback**. The number of days in the calculation of the indicator. This would equate to 20 in a 20-day moving average.\n  * **curBar**. Simply the historic bar being evaluated.\n  * **offset**. Just like the ESB, this number will usually be set to one. The indicator value from the prior day. You can go offset with a more positive number. If you use two, then it will return the indicator value form two days prior.\n\nNotice the colon (:) at the end of the function definition. The colon tells the interpreter that the indented code (four spaces) below is controlled by the current line. So everything that is indented below the function definition header is included in the function. Indentation is _very important_ in Python. Don't use tabs to indent; simply hit the spacebar four times. Don't forget this\u2014spacebar four times. Things will become much clearer very quickly.\n\nIn Python variables are assigned values with the equal sign ( **=** ). Equality between two variables is verified by using a double equal sign ( **==** ). Jump over to the Python Shell really quick. I hope you still have it close at hand. There might be a bunch of stuff in the window, so just scroll down to the very bottom until you find the >>> prompt. Once there type **a = 5** and hit enter. Type **a** and hit enter. The number 5 should appear at the prompt. You have basically assigned 5 to the variable **a**. Now type **b = 5** and hit enter. Type **a==b** and hit enter. The prompt should return with **True**. In this case **a** equals **b**. Type **a != b** and hit enter. The prompt will return with **False** , because **a** does equal **b**. You are asking if **a** doesn't equal **b** and the answer is **False** because it does. Remember to use **==** to test for equality and **!=** for inequality.\n\nThe next line of code is very important because you will see it often.\n\n* * *\n\n    for index in range((curBar - offset) - (lookBack-1), curBar - offset +1):\n\n* * *\n\nThis is an example of a definite loop; the number of repetitions is known ahead of time. This **for-loop** will iterate across the number of items in the range from ((curBar \u2212 offset) \u2212 (lookBack\u22121) to curBar \u2212 offset +1). This only looks complicated because of the variables in the **range** function. Here is a more simplified version of the Python for loop:\n\n* * *\n\n    myList = list()\n    myList = [1,2,3,4]\n    for index in range(0,3):\n        print(myList[index])\n\n* * *\n\nThe output of this code will print only the first three elements in the list named **myList**. The function Range(0,3) returns 0, 1, and 2. The upper bound in the **Range** function is not inclusive. So to print out **myList** in its entirety, you must use Range(0,4). Referring back to the original bounds in our **Range** function, let's simplify by substituting some numbers for the values. Assume:\n\n  1. curBar = 500\n  2. offset = 1\n  3. lookback = 20\n\nSo, range ((500 \u2212 1) \u2212 (20 \u22121), 500 \u2212 1 + 1) equals range (499 \u2212 19, 500). The loop will iterate from 480 to 499 because 500 isn't included. Since 480 is inclusive, the number of iterations turns out to be 20. Count on your fingers if you want to prove this starting at 480 and going to 499. The loop will start at yesterday if the offset is one and go back 20 days and sum up whatever data is in the list that is passed to the function. Once the loop completes, the summation is divided by the number of days in the calculation\u2014the average of lookback days. This value is then returned to the calling program. That's all there is to this function. Sum up all requested values and divide by the number of requested values. The complicated part was deciphering the values in the **Range** function. Notice how the only line that was indented was **result = result + prices[index]**. This was because the **for-loop** was only controlling this one line of code. The rest of the lines were not under the influence of the **for-loop**. They were still indented in relationship to the **def** keyword and this was because the lines were encapsulated in the function body. Can you tell what these other functions are doing?\n\n* * *\n\n    def highest(prices,lookBack,curBar,offset):\n        result = 0.0\n        maxVal = 0.00\n        for index in range((curBar - offset) - (lookBack-1),curBar - offset + 1):\n            if prices[index] > maxVal:\n                maxVal = prices[index]\n        result = maxVal\n        return result\n    def lowest(prices,lookBack,curBar,offset):\n        result = 0.0\n        minVal = 9999999.0\n        for index in range((curBar - offset) - (lookBack-1),curBar - offset + 1):\n            if prices[index] < minVal:\n                minVal = prices[index]\n        result = minVal\n        return result\n\n* * *\n\nIf you guessed calculate the highest high and lowest low of lookback days, then give yourself a gold star. If not, just go back over the code until you get it. Notice the indentations and the use of the colon.\n\n#### rsiClass\n\nThat wasn't too bad now, was it? Are you ready to attack the indicators that are defined as a **class**? Let's take a close look at the RSI indicator:\n\n* * *\n\n    class rsiClass(object):\n        oldDelta1 = 0\n        def __init__(self):\n            self.delta1 = 0\n            self.delta2 = 0\n            self.rsi = 0\n            self.seed = 0\n        def calcRsi(self,prices,lookBack,curBar,offset):\n            upSum = 0.0\n            dnSum = 0.0\n            if self.seed == 0:\n                self.seed = 1\n                for i in range((curBar - offset) - (lookBack-1),curBar - offset):\n                    if prices[i] > prices[i-1]:\n                        diff1 = prices[i] - prices[i-1]\n                        upSum += diff1\n                    if prices[i] < prices[i-1]:\n                        diff2 = prices[i-1] - prices[i]\n                        dnSum += diff2\n                    self.delta1 = upSum\/lookBack\n                    self.delta2 = dnSum\/lookBack\n            else:\n                if prices[curBar - offset] > prices[curBar - 1 - offset]:\n                    diff1 = prices[curBar - offset] - prices[curBar - 1 - offset]\n                    upSum += diff1\n                if prices[curBar - offset] < prices[curBar - 1 - offset]:\n                    diff2 = prices[curBar - 1 - offset] - prices[curBar - offset]\n                    dnSum += diff2\n                self.delta1 = (self.delta1 * (lookBack -1) + upSum) \/ lookBack\n                self.delta2 = (self.delta2 * (lookBack -1) + dnSum) \/ lookBack\n            if self.delta1 + self.delta2 != 0:\n                self.rsi = (100.0 * self.delta1) \/ (self.delta1 + self.delta2)\n            else:\n                self.rsi = 0.0\n            return (self.rsi)\n\n* * *\n\nThis indicator is programmed as a **class** object. We have ventured over into the land of objected-oriented programming, but don't worry; we are just going to work in an introductory mode to get the job done. You are probably asking yourself, why didn't he just program the RSI as a function module? The quick answer is I wanted to encapsulate the data and functions in one block of code so I could cut down on using global variables.\n\nThe first time an RSI indicator is called, it uses a special calculation to seed the indicator value. Subsequent calls to the indicator utilize the seed value and a different set of calculations. So I had to know, within the indicator code, if it was the first time it was being called. Also, I needed to store the seed and prior values in the class for subsequent calculations.\n\nBy preceding the name of the class object with the keyword **class** , you are informing the interpreter that a class definition is about to be defined with the name that follows the _**class**_ keyword. In this example, we are defining a class named **rsiClass**. The keyword **object** within parentheses follows the name of the class. This would be a great time to explain the difference between a class and an object. A class is a template and an object is an actual thing that reflects the class template. Think of it like this: A class is blueprint and an object is the house that is instantiated or created with the use of the blueprint. One blueprint can be used to create many houses. The following snippet of code informs the interpreter about data to be stored in the class:\n\n* * *\n\n        def __init__(self):\n            self.delta1 = 0\n            self.delta2 = 0\n            self.rsi = 0\n            self.seed = 0\n\n* * *\n\nThe bit of code, **def __init__(self)** :, is the function name the object itself calls to set the initial values of the data variables (class members) when the class is instantiated. The key word **self** refers to the class itself. The keyword **init** must be preceded by two underscores and followed by two underscores. The keyword **self** must be followed by a period to access the different data encapsulated in the class. As you can see from the code, the variables **self.delta1** , **self.delta2** , **self.rsi** , and **self.seed** are all initially set to zero.\n\nThe next line of code should look familiar\u2014it looks like the functions we discussed earlier in the chapter.\n\n* * *\n\n      def calcRsi(self,prices,lookBack,curBar,offset):\n\n* * *\n\nThe only difference in this function header is the word **self** in the argument list. Basically, when the function **calcRsi** is called, the first argument or parameter is going to be the class structure itself. The other arguments are: the data to be used in the calculations, the lookback period, the current bar, and the offset.\n\nTake a close look at the following code snippet:\n\n* * *\n\n    upSum = 0.0\n    dnSum = 0.0\n    if self.seed == 0:\n        self.seed = 1\n        for i in range((curBar - offset) - (lookBack-1),curBar - offset):\n            if prices[i] > prices[i-1]:\n                diff1 = prices[i] - prices[i-1]\n                upSum += diff1\n            if prices[i] < prices[i-1]:\n                diff2 = prices[i-1] - prices[i]\n                dnSum += diff2\n            self.delta1 = upSum\/lookBack\n            self.delta2 = dnSum\/lookBack\n    else:\n\n* * *\n\nThe function starts out by setting the variables **upSum** and **dnSum** to zero. The next line of code utilizes the keyword **self** and accesses the value of seed through the dot (.) notation. If **self.seed** is equal to zero, then all the indented code below the **if** construct will be executed. If **self.seed** is equal to zero, the very first line of code sets **self.seed** to one. In doing so, these lines of code will only be executed the very first time the function is called. Notice the variables **upSum** and **dnSum** do not use the dot notation. These variables are not members of the class and are temporary. They do not need to be remembered from one function call to the next. By using the dot notation, you are setting the data member to a value that will be remembered from one call to the next. By the way, a function inside a class is called a method. This function\/method calculates the initial RSI value by looping back in time and calculating, accumulating, and averaging the up and down sums. The **upSum** and **dnSum** averages are stored in the class data members, **self.delta1** and **self.delta2** , respectively. The next time this function is called, this portion of the function will be bypassed, but the data stored in the data members will be available for future use. The second part of the function:\n\n* * *\n\n    else:\n        if prices[curBar - offset] > prices[curBar - 1 - offset]:\n            diff1 = prices[curBar - offset] - prices[curBar - 1 - offset]\n            upSum += diff1\n        if prices[curBar - offset] < prices[curBar - 1 - offset]:\n            diff2 = prices[curBar - 1 - offset] - prices[curBar - offset]\n            dnSum += diff2\n        self.delta1 = (self.delta1 * (lookBack -1) + upSum) \/ lookBack\n        self.delta2 = (self.delta2 * (lookBack -1) + dnSum) \/ lookBack\n\n* * *\n\nwill be executed every time. **Self.delta1** and **self.delta2** are used and reassigned each time the function is called. You might be able to see that the Wilder's averaging method is being used to calculate the current RSI components:\n\nThe final snippet of code is executed every time and returns the RSI calculation.\n\n* * *\n\n    if self.delta1 + self.delta2 != 0:\n        self.rsi = (100.0 * self.delta1) \/ (self.delta1 + self.delta2)\n    else:\n        self.rsi = 0.0\n    return (self.rsi)\n\n* * *\n\nSince the design of this class-based indicator is a little more involved than a regular run-of-the-mill function, you would expect the call routine to be more involved as well. This is the case, but it only requires one additional line of code. In the case of this RSI indicator, all you need to do is add the line of code to your program: **rsiStudy = rsiClass()**. You can instantiate the **rsiClass** by calling it like a function. Instantiation is the creation of a shell that follows the class template\/plan. In this case, the class object is assigned to the **rsiStudy** variable. Once the class is assigned to a variable, you can access the data and methods of that class by using the name of the variable (object) and dot notation. Here is how you call the **calcRsi** method through the **rsiStudy** variable name: **rsiVal = rsiStudy.calcRsi (myClose, 10, curBar, i)**. That's all there is to using the RSI indicator.\n\nScroll through the rest of the indicators in **indicator.py** and familiarize yourself with them. You can add you own indicators by using one of the existing functions or indicator classes as a template. Just include the code in this file and import the name of the indicator in the main trading program.\n\n#### SystemTester.py\n\nThe main module that calls all the other modules is **SystemTester.py**. As I explained prior, you use this as a template to develop other trading algorithms. I usually open **SystemTester.py** and _save as_ immediately. If I am working on a Donchian type of algorithm, I will use a name like **DonchianSys1.py**. Instead of listing the program in its entirety, let's break it up section by section.\n\n##### Import Section\n\n* * *\n\n    #------------------------------------------------------------\n    #Import section - inlcude functions, classes, variabels\n    #form externam modules\n    #------------------------------------------------------------\n    import csv\n    import tkinter as tk\n    import os.path\n    from getData import getData\n    from dataLists import myDate,myTime,myOpen,myHigh,myLow,myClose\n    from tradeClass import tradeInfo\n    from equityDataClass import equityClass\n    from trade import trade\n    from systemMarket import systemMarketClass\n    from portfolio import portfolioClass\n    from indicators import highest,lowest,rsiClass,stochClass,sAverage,bollingerBands\n    from systemAnalytics import calcSystemResults\n    from tkinter.filedialog import askopenfilenames\n\n* * *\n\nThe hash sign (#) at the beginning of any text informs the interpreter to ignore whatever follows up to the end of the line. Use this symbol for comments in your Python code\u2014just like we did with the single quote (') in VBA. You can also use it to temporarily hide a line of code from the interpreter. All of the external files and functions\/classes are pulled into the main program by using the keywords **from** and **import**. This is similar to C's **include** statement **\u2014** on the surface, anyway. For all intents and purposes, we will use it simply to import external code. The name of the external module follows **from** and the function\/class name follows the **import** keyword. Look at the line that starts, **from indicators import...** You will see a good portion of the indicator functions and classes that were defined in **indicator.py**.\n\nSome of the imports are strictly Python related:\n\n* * *\n\n    import csv\n    import tkinter as tk\n    import os.path\n    from tkinter.filedialog import askopenfilenames\n\n* * *\n\nAll these files deal with opening and reading a comma-delimited file. The rest of the imported files all deal with the PSB.\n\n##### Helper Functions for the PSB\n\nThese functions help parse the **.csv** file into the individual **myDate** , **myHigh** , **myLow** , **myClose** , **myVol** , and **myOpInt** lists. The attributes of each data file, symbol, **pointValue** , and **minMove** are collected here as well. A function that calculates the results of exiting a trade is also located here. I won't bore you with the details of these functions right now, but I want to bring to your attention another really cool thing about Python. A Python function can return multiple values. The **exitPos** function returns three values: **profit** , **trades** , and **curShares**. When you call a multiple return function, make sure you have the correct number of variables on the left side of the assignment:\n\n* * *\n\n    myProfit, myTrades, myCurShares = exitPos(price,myDate[i],\"RevLongLiq\",curShares)\n\n* * *\n\nThe larger **exitPos** function was put here because it uses several global variables. These variables are shared among most of the functions in this file module. I didn't want to have to pass these variables back and forth within different function calls. I was slightly lazy in doing this, but it does help improve readability by reducing the number of different parameters included in the argument list.\n\n* * *\n\n    #------------------------------------------------------------\n      #Helper Functions local to this module\n    #------------------------------------------------------------\n    def getDataAtribs(dClass):\n       return(dClass.bigPtVal,dClass.symbol,dClass.minMove)\n    def getDataLists(dClass):\n       return(dClass.date,dClass.open,dClass.high,dClass.low,dClass.close)\n    def calcTodaysOTE(mp,myClose,entryPrice,entryQuant,myBPV):\n        todaysOTE = 0\n        for entries in range(0,len(entryPrice)):\n            if mp >= 1:\n                todaysOTE += (myClose - entryPrice[entries])*myBPV*entryQuant[entries]\n            if mp <= -1:\n               todaysOTE += (entryPrice[entries] - myClose)*myBPV*entryQuant[entries]\n        return(todaysOTE)\n    def exitPos(myExitPrice,myExitDate,tempName,myCurShares):\n        global mp,commission\n        global tradeName,entryPrice,entryQuant,exitPrice,numShares,myBPV,cumuProfit\n        if mp < 0:\n            trades = tradeInfo('liqShort',myExitDate,tempName,myExitPrice,myCurShares,0)\n            profit = trades.calcTradeProfit('liqShort',mp,\n            entryPrice,myExitPrice,entryQuant,myCurShares) * myBPV\n            profit = profit - myCurShares *commission\n            trades.tradeProfit = profit\n            cumuProfit += profit\n            trades.cumuProfit = cumuProfit\n        if mp > 0:\n            trades = tradeInfo('liqLong',myExitDate,tempName,myExitPrice,myCurShares,0)\n            profit = trades.calcTradeProfit('liqLong',mp,\n            entryPrice,myExitPrice,entryQuant,myCurShares) * myBPV\n            trades.tradeProfit = profit\n            profit = profit - myCurShares * commission\n            cumuProfit += profit\n            trades.cumuProfit = cumuProfit\n        curShares = 0\n        for remShares in range(0,len(entryQuant)):\n           curShares += entryQuant[remShares]\n        return (profit,trades,curShares)\n    #------------------------------------------------------------\n      #End of functions\n    #------------------------------------------------------------\n\n* * *\n\n##### Lists and Variables Initiation\n\nYou can use this section to initialize your own lists and variables. These lines of code are pretty cool.\n\n* * *\n\n    #------------------------------------------------------------\n     #Lists and variables are defined and initialized here\n    #------------------------------------------------------------\n    alist, blist, clist, dlist, elist = ([] for i in range(5))\n    marketPosition,listOfTrades,trueRanges,ranges = ([] for i in range(4))\n    dataClassList,systemMarketList,equityDataList = ([] for i in range(3))\n    entryPrice,fileList,entryPrice,entryQuant,exitQuant = ([] for i in range(5))\n    #exitPrice = list()\n    currentPrice,totComms,barsSinceEntry = 0\n    numRuns,myBPV,allowPyra,curShares = 0\n    #------------------------------------------------------------\n     #End of Lists and Variables\n    #------------------------------------------------------------\n\n* * *\n\n    alist, blist, clist, dlist, elist = ([] for i in range(5))\n\nThe first line of code creates five different lists using an implicit **for loop** , a one-liner **for loop**. You are probably starting to see the magic of Python.\n\n##### Data Import and Configuration and Portfolio Setup\n\nThe list of data class objects are assigned whatever is returned by the function **getData()**. When the PSB is launched, one of the first things it asks the user is to pick some data (commodity, equities, or index prices). The **getData** function handles the creation of an **Open File** dialog, the selection of data files, and determining the market specifications for each file selected. The following is the complete listing of the function's source broken down into easily explainable components.\n\n* * *\n\n    #------------------------------------------------------------\n      #Get the raw data and its associated attributes [pointvalue,symbol,tickvalue]\n      #Read a csv file that has at least D,O,H,L,C - V and OpInt are optional\n      #Set up a portfolio of multiple markets\n    #------------------------------------------------------------\n    dataClassList = getData()\n    numMarkets = len(dataClassList)\n    portfolio = portfolioClass()\n    import csv\n    import tkinter as tk\n    import os.path\n    from marketDataClass import marketDataClass\n    from dataMasterLists import commName, bigPtVal, minMove\n    from tkinter.filedialog import askopenfilenames\n    from equityDataClass import equityClassdataClassList = list()\n    fileName = \"c:\\PBS\\dataMaster.csv\"\n    def getData():\n        totComms = 0\n        with open(fileName) as f:\n           f_csv = csv.reader(f)\n           for row in f_csv:\n              commName.append(row[0])\n              bigPtVal.append(float(row[1]))\n              minMove.append(float(row[2]))\n              totComms = totComms + 1\n        f.close\n\n* * *\n\nThe function starts out by opening the file C:\\PBS\\dataMaster.csv and reading its contents into three different lists: **commName** , **bigPtVal** , and **minMove**. Here is a quick snapshot of the contents of this file:\n\n    \"AD\",100000,0.0001\n    \"BP\",125000,0.0001\n    \"CL\",1000,0.01\n    \"C2\",50,0.25\n    \"CC\",10,1\n    \"CD\",100000,0.0001\n    \"CL\",1000,0.01\n    \"CT\",500,0.01\n    \"CU\",125000,0.0001\n    \"DX\",1000,0.005\n    \"ED\",2500,0.005\n    \"EM\",100,0.1\n    \"FC\",500,0.025\n    \"FV\",1000,0.0078\n    \"GC\",100,0.1\n    \"HG\",250,0.05\n    \"HO\",42000,0.0001\n    \"JY\",125000,0.00005\n\nAs you can see, the data are separated by commas, and there are three data values per line. The first data value is the string (enclosed in quotes) that will be assigned to **commName**. The second is the value of a big point move (a single increment of the number right before the decimal point (British pound is quoted as 1.5631, so a move from 1.5631 to 2.5631 would equate to $125,000), and the third is the minimum move of the market. The minimum move in stocks is 0.01 or one penny. In the British pound, it is 0.0001. Just like the ESB, the PSB needs to know this information ahead of time so it can properly calculate price-based values and perform accurate trade accounting. This file is easily editable and you can customize it to fit your own data and its format.\n\nPython has a ton of reusable code and helper modules. This little bit of code relies on the built-in **csv.Reader**. If you noticed the dot notation, then you realize that **csv** is class and **.Reader** is a method from that class. Before using the csv class, it had to be imported. If you look back at the source code, you will see it is the first module to be imported. After the file is opened using `with open(fileName) as f: ,` the file pointer (f) is passed to the **csv.Reader** method. The **f_csv** is then utilized to read each row of data in the file. The columns in each row are demarked by using a subscript. So row[0] is the commName, row[1] is the big point value, and row[2] is the minimum move. Since we are inside a definite **for loop** , the loops ends when the last row of data is read. Once the loop has terminated, the file is closed. These six or seven lines of code open the file, loop through each row of data, read the data into predefined lists, and then close the file. Pretty simple and concise!\n\nThis next snippet of code does a lot and is a little more sophisticated, so only the most important parts will be discussed.\n\n* * *\n\n    root = tk.Tk()\n    root.withdraw()\n    cnt = 0\n    files = askopenfilenames(filetypes=(('CSV files', '*.txt'),\n                                       ('All files', '*.*')),\n                                       title='Select Markets To Test')\n    fileList = root.tk.splitlist(files)\n    fileListLen = len(fileList)\n    for marketCnt in range(0,fileListLen):\n        head,tail = os.path.split(fileList[marketCnt])\n        tempStr = tail[0:2]\n        for i in range(totComms):\n            if tempStr == commName[i]:\n                commIndex = i\n        newDataClass = marketDataClass()\n        newDataClass.setDataAttributes(commName[commIndex],\n            bigPtVal[commIndex],minMove[commIndex])\n        with open(fileList[marketCnt]) as f:\n            f_csv = csv.reader(f)\n            for row in f_csv:\n                newDataClass.readData(int(row[0]),float(row[1]),\n                float(row[2]),float(row[3]), float(row[4]),0.0,0.0)\n                cnt = cnt + 1\n        dataClassList.append(newDataClass)\n        f.close\n    return(dataClassList)\n\n* * *\n\nThe single line of code:\n\n* * *\n\n    files = askopenfilenames(filetypes=(('CSV files', '*.txt'),\n                                           ('All files', '*.*')),\n                                           title='Select Markets To Test')\n\n* * *\n\nopens an OS-independent **File Open Dialog** labeled **Select Markets To Test** and asks the operating system to list all files with a .CSV extension in that particular location. Once the user selects one or more files, the selection is passed back and stored in the variable **files**.\n\n* * *\n\n    fileList = root.tk.splitlist(files)\n\n* * *\n\nThe files are then listed separately and stored in a list named **fileList**. This is all handled with the built-in method **root.tk.splitlist(files)**. This method takes all of the files the user selected and then splits them apart and stores the individual file names in a list.\n\n* * *\n\n    head,tail = os.path.split(fileList[marketCnt])\n    tempStr = tail[0:2]\n    for i in range(totComms):\n        if tempStr == commName[i]:\n            commIndex = i\n\n* * *\n\nAnother OS method **os.path.split** takes the name of each individual file and splits the path name ( **C:\\myPath\\Data\\** ) and the actual filename apart ( **BP.CSV** ). The **head** variable is assigned the pathname and the **tail** variable the filename. Once the filename is separated the first two letters in the filename can be accessed by simply indexing into the string: **tempStr = tail[0:2]**. The first two characters in the filename represent the **symbol** that was selected. From this the **symbol** can be cross-referenced in the **DataMaster** and the market name, big point value, and minimum move can be attached to the symbol.\n\n* * *\n\n    newDataClass = marketDataClass()\n    newDataClass.setDataAttributes(commName[commIndex],bigPtVal[commIndex],minMove[commIndex])\n\n* * *\n\nAs soon as the filename is cross referenced with **DataMaster** , a **marketDataClass** object with the variable name **newDataClass** is instantiated by calling the **marketDataClass()** method. The **marketDataClass** class looks like this:\n\n* * *\n\n    class marketDataClass(object):\n        def __init__(self):\n            self.symbol = \"\"\n            self.minMove = 0\n            self.bigPtVal = 0\n            self.seed = 0\n            self.date = list()\n            self.open = list()\n            self.high = list()\n            self.low = list()\n            self.close = list()\n            self.volume = list()\n            self.opInt = list()\n            self.dataPoints = 0\n        def setDataAttributes(self,symbol,bigPtVal,minMove):\n            self.symbol = symbol\n            self.minMove = minMove\n            self.bigPtVal = bigPtVal\n        def readData(self,date,open,high,low,close,volume,opInt):\n            self.date.append(date)\n            self.open.append(open)\n            self.high.append(high)\n            self.low.append(low)\n            self.close.append(close)\n            self.volume.append(volume)\n            self.opInt.append(opInt)\n            self.dataPoints += 1\n\n* * *\n\nThis class acts like a container that holds all of the price data and market attributes for each file selected by the user. It also contains the methods for gathering that data. The **init** method creates all of the storage needed to handle years and years of data. Other than the single variables **self.symbol** , **self.minMove** , **self.bigPtVal** , the rest of the data holders are lists. Each list is capable of holding tremendous amounts of data. The other two methods, **setDataAttributes** and **readData** , are used to set the market specifications and fill up the lists with price data. Notice Python uses the **.append** method to add values to an existing list. Lists act like arrays, but are much more powerful due to their associated methods. If I want to know the number of elements in a list, all I have to do is: **listLen = len(myList)**. **Len()** is simply a method of the **list** class.\n\nNow keep in mind all of this code is inside a **for loop**. The number of loops is determined by the number of markets selected by the user. If the user selected five files, then this loop would be performed five times. You would have five instantiations of the **marketDataClass** objects, and each would be used to set the market specifications for the five markets and read in the different price data from the five different files. A total of five different objects would be spawned to handle all of the data. The best way to keep these classe objects together would be to put them in a **list**. I bet you knew a **list** can contain primitive data types like numbers, strings, letters, and Booleans. But, I bet you didn't know a list can contain complex data structures like our **marketDataClass**. Jump over the Python Shell and type the following:\n\n    >>> a = list()\n    >>> a = [\"apple\",\"banana\",\"strawberry\"]\n    >>> a\n\nWhen you hit enter after the last line, the Shell will display:\n\n  1. ['apple', 'banana', 'strawberry']\n\nNo surprise here, it basically repeated what we put into the list. Now type this:\n\n    >>>b = list()\n    >>> b = [\"pineapple\",\"coconut\",\"mango\"]\n    c = list()\n    >>> c.append(a)\n    >>> c.append(b)\n    >>> c\n\nand hit return. The Shell shows:\n\n  1. [['apple', 'banana', 'strawberry'], ['pineapple', 'coconut', 'mango']]\n\nYou have a list that contains two other lists. Notice the grouping? ['apple', 'banana', 'strawberry'] is the **a** list and ['pineapple', 'coconut', 'mango'] is the **b** list.\n\nThis line of code, `dataClassList.append(newDataClass)` inserts the individual **marketDataClasse object** in the **dataClassList list**. This list will be accessible to the rest of the program later on.\n\nGetting back to:\n\n* * *\n\n    dataClassList = getData()\n    numMarkets = len(dataClassList)\n    portfolio = portfolioClass()\n\n* * *\n\nThe first simple line of code does quite a bit. The next line starting with numMarkets gets the length of the list that contains all the **marketDataClass** objects. In other words, it returns the number of **marketDataClass** objects, which is the number of markets selected by the user. The next line of code instantiates a **portfolioClass** objectd labeled **portfolio**. The **portfolioClass** is quite large, so I won't regurgitate it here. I will in the appendices. There are some very important and cool things going on in this class, so I want to highlight those very quickly. But before we do, that there are two important class structures that are a prerequisite to the **portfolioClass**. These classes are named **equityClass** and **marketSystemClass**. The **equityClass** is used to store the daily equity for each individual market. A new **equityClass** object is instantiated for each different market in the market list. Here's what the data containers in the **equityClass** class look like:\n\n* * *\n\n    def __init__(self):\n        self.equityDate = list()\n        self.equityItm = list()\n        self.clsTrdEquity = list()\n        self.openTrdEquity = list()\n        self.cumuClsEquity = 0\n        self.dailyEquityVal = list()\n        self.peakEquity = 0\n        self.minEquity = 0\n        self.maxDD = 0\n\n* * *\n\nAfter you open a trading account and start trading something happens every day to your account. It either stays flat, goes up, or goes down. These daily fluctuations are caused by the P\/L from a closed trade or the change in market value of an open trade. Cumulative equity reflects the growth of your account. This **equityClass** keeps track of P\/L from closed trades, change in daily open trade equity, and the cumulative equity for any given time period. If you buy Apple today at $100 a share and buy just one share, then today's equity simply reflects the difference between the closing price and $100. Let's pretend Apple closed at $105, so your closed trade equity is equal to zero, but your open trade equity (OTE) is $5. Today's equity in your account is $5. Let's pretend a little further and say Apple moves up to $110, and you decide to cover your long position. Today's equity is derived from a closed trade profit of $10. Since there is no open position, then there is no OTE. Tracking your two-day performance, you show daily equity of $5 for yesterday and $10 for today. Your cumulative equity stands at $10. Feeling a little more bullish, you buy Apple again on the third day at $112, but unfortunately, some bad news comes out of China and the stock drops back to $105. So today's equity reflects the cumulative closed out equity $10 and today's OTE that stands at \u2212$7. Your cumulative equity drops by $7 and now stands at $3. The next day Apple drops further and you start feeling bearish so you sell short at $100. Unfortunately, China states their GDP is growing a little faster than the analysts predicted and Apple rebounds and closes at $110. What is your cumulative equity now?\n\n  1. Day 1 : Cum.E ($5) = Cum.Cls.TE (0) + OpenTE (5)\n  2. Day 2 : Cum.E ($10) = Cum.Cls.TE (10) + OpenTE (0)\n  3. Day 3 : Cum.E ($3) = Cum.Cls.TE(10) + OpenTE(\u22127)\n  4. Day 4 : Cum.E (\u2212$12) = Cum.Cls.TE(\u22122) + OpenTE(\u221210)\n\nThe one trade that you took on Day 4 was a disaster. First off, you closed out a $12 loser ($112 \u2212 $100), and to add insult to injury your OTE is \u2212$10 ($100 \u2212 $110). You real cumulative losses stand at \u2212$2 and your paper loss stands at \u2212$10. All these values are needed to calculate a daily equity stream and that is why we have: **self.clsTrdEquity** , **self.openTrdEquity** , **self.cumuClsEquity**. The rest of the module keeps track of the daily equity stream as well as the maximum equity peak, and the subsequent equity valley. From these values the worst-case drawdown can be derived.\n\n* * *\n\n    def setEquityInfo(self,equityDate,equityItm,clsTrdEquity,openTrdEquity):\n        self.equityDate.append(equityDate)\n        self.equityItm.append(equityItm)\n        self.cumuClsEquity += clsTrdEquity\n        tempEqu =self.cumuClsEquity+openTrdEquity\n        self.dailyEquityVal.append(tempEqu)\n        self.peakEquity = max(self.peakEquity,tempEqu)\n        maxEqu = self.peakEquity\n        self.minEquity = min(self.minEquity,tempEqu)\n        minEqu = self.minEquity\n        self.maxDD = max(self.maxDD,maxEqu-tempEqu)\n        maxDD = self.maxDD\n        maxDD = maxDD\n\n* * *\n\nThe next class, **systemMarketClass** , is used to keep track of each system on each individual market. I did this in case a future version of PSB might combine multiple markets on multiple systems. You could have a bunch of these **systemMarketClass** objects reflecting the performance on several system\/markets; you then could combine certain ones to produce a very efficient equity curve. For right now, though, these class objects contain the following data:\n\n* * *\n\n    def __init__(self):\n        self.systemName = \"\"\n        self.symbol = \"\"\n        self.tradesList =list()\n        self.equity = equityClass\n        self.avgWin = 0\n        self.avgLoss = 0\n        self.avgTrade = 0\n        self.profitLoss = 0\n        self.numTrades = 0\n        self.maxxDD = 0\n        self.perWins = 0\n\n* * *\n\nEach **systemMarketClass** contains the name of the system and the symbol. In addition, it stores a list of trades, the equity stream, the average win, loss, and trade, percent wins, number of trades, and maximum drawdown. The methods in this class set the pertinent data, such as system name and symbol, and calculate all the performance metrics.\n\n* * *\n\n    def setSysMarkInfo(self,sysName,symbol,trades,equity):\n       self.systemName = sysName\n       self.symbol = symbol\n       self.tradesList = list(trades)\n       self.equity = equity\n       temp1 = 0\n       temp2 = 0\n       temp3 = 0\n       temp4 = 0\n       temp5 = 0\n       temp6 = 0\n       temp7 = 0\n       numTrades = 0\n       for i in range(0,len(self.equity.dailyEquityVal)):\n           temp5 = self.equity.dailyEquityVal[i]\n           temp6 = max(temp6,temp5)\n           temp7 = max(temp7,temp6-temp5)\n           self.maxxDD = temp7\n       for i in range(0,len(self.tradesList)):\n           if self.tradesList[i].entryOrExit == 1:\n               numTrades += 1\n           if self.tradesList[i].tradeProfit > 0:\n               temp1 += self.tradesList[i].tradeProfit\n               temp2 += 1\n           if self.tradesList[i].tradeProfit < 0:\n               temp3 += self.tradesList[i].tradeProfit\n               temp4 += 1\n       if temp2 != 0: self.avgWin = temp1\/temp2\n       if temp4 != 0: self.avgLoss = temp3\/temp4\n       if numTrades != 0: self.avgTrade = temp5\/numTrades\n       self.numTrades = numTrades\n       self.profitLoss = temp5\n       if numTrades != 0: self.perWins = temp2 \/ numTrades\n\n* * *\n\nThe **setSysMarkInfo** method is the only method in this class, but it does a bunch of work. Notice about halfway through the variables with **temp** in their name; these are temporary holders for data that is not stored in the class. They are born when the method is called and die when the method exits. They are not members of the class. The **portfolio** class is similar to the **systemMarketClass** , but is much more macroscopic in nature. Basically, it sits on top of **systemMarketClass** and stores aggregate information concerning the multiple markets in the portfolio.\n\n* * *\n\n    class portfolioClass(object):\n        def __init__(self):\n            self.portfolioName = \"\"\n            self.systemMarkets = list()\n            self.portEquityDate = list()\n            self.portEquityVal = list()\n            self.portclsTrdEquity = list()\n            self.portDailyEquityVal = list()\n            self.portPeakEquity = 0\n            self.portMinEquity = 0\n            self.portMaxDD = 0\n            tempEqu = 0\n            cumEqu = 0\n            maxEqu = -999999999\n            minEqu = 999999999\n            maxDD = 0\n\n* * *\n\nAs you know, a portfolio is simply a collection of different markets. A portfolio has to have a name, a list of the different **systemMarkets** or just markets, and lists that make up **equityDate** , **equityVal** , **clsTrdEquity** , and **dailyEquity** at the portfolio level. The methods involved with this class are rather long and for brevity's sake will not be displayed here. All classes including their methods will be listed in the appendix. There are a couple of Pythonesque list features I would like to discuss, because they are so cool. If you have traded commodities or futures, you know some days one market is open and another closed. Creating a master date list to contain the daily portfolio equity had to take this into consideration. The use of Python's lists made creating a master date list that included all the dates of the entire portfolio very easy. I simply concatenated (added together) all of the different date, not data, streams from each **systemMarket**.\n\n* * *\n\n    for i in range(0,len(self.systemMarkets)):\n        masterDateList += self.systemMarkets[i].equity.equityDate\n\n* * *\n\nThis loop starts at 0 and loops through all the **systemMarkets**. If you have five markets in your portfolio, then the loop would start at 0 and end at 4. You will notice the **+=** operand after **masterDateList** in the code. This is a shortcut like **a = a + 1**. Using this operand eliminates repeating the variable name after the equal sign. If you want to see how this works, jump over to the Python Shell and type this:\n\n    >>> a = list()\n    >>> a = [1,2,3,4,6,7,8,9]\n    >>> b = list()\n    >>> b = [1,2,3,4,5,6,7,8,10]\n    >>> c = a + b\n    >>> c\n\nAfter you hit enter following the last c you will be presented with this:\n\n  1. [1, 2, 3, 4, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 10]\n\nThis is simply a concatenation of the two lists. It includes all values from both lists. Now type this:\n\n    >>> c.sort()\n    >>> c\n\nHit enter after the last c, and the list is now sorted and stored back in the original list.\n\n  1. [1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 8, 9, 10]\n\nThis is how I created a master data list that included all the dates in the different equity streams from each market. I then used a **removeDuplicates** function to remove the duplicate list entries. The other cool thing in this class is this snippet of code:\n\n* * *\n\n    def createMonthList(li):\n        myMonthList = list()\n        for i in range(0,len(li)):\n            if i != 0:\n                tempa = int(li[i]\/100)\n                pMonth = int(li[i-1]\/100) % 100\n                month = int(li[i]\/100) % 100\n                if pMonth != month:\n                    myMonthList.append(li[i-1])\n                if i == len(li)-1:\n                    myMonthList.append(li[i])\n        return myMonthList\n\n* * *\n\nThis code takes a large list of dates and extracts the beginning date of each month in that list, and then creates a list with just the different months. This is how I was able to create a monthly breakdown of the time period used in the testing of the system.\n\n* * *\n\n    20110930       2249       2249\n    20111031      -2545       -296\n    20111130          0       -296\n    20111230       -913      -1209\n    20120131          0      -1209\n    20120229          0      -1209\n    20120330          0      -1209\n    20120430       -120      -1329\n    20120531        921       -408\n    20120629      -1735      -2143\n    20120731        955      -1188\n\n* * *\n\nOne last bit of code that shows off the power of Python follows. After this, I will demonstrate how to pull all this together and test a trading algorithm.\n\n* * *\n\n    for i in range(0,len(masterDateList)):\n        cumuVal = 0\n        for j in range(0,len(self.systemMarkets)):\n            skipDay = 0\n            try:\n                idx = self.systemMarkets[j].equity.equityDate.index(masterDateList[i])\n            except ValueError:\n                skipDay = 1\n            if skipDay == 0:\n                cumuVal += self.systemMarkets[j].equity.dailyEquityVal[idx]\n            combinedEquity.append(cumuVal)\n\n* * *\n\nThis snippet of code is important because two very important concepts are covered. If you review the code, you will see the keyword **try** : The code controlled by **try** is simply finding if and where the last day of a month is located in a **systemMarket's equityDate** stream. Let's say we are trying to see if July 31, 2012, is in an **equityDate** list. If it is, then the variable **idx** returns the location in the list. If it's not located in the list, then an error is returned. This is where **except** comes into play. The code controlled by **except** is only executed if the method **index** returns an error. Back to the Python Shell!\n\nType the following and hit enter after the last line:\n\n    >>> a = list()\n    >>> a = [100,300,200,800]\n    >>> a.index(300)\n    1\n\nThis example returns the location of the number 300 in the **a** list. Remember Python is zero based. The first element is number 0.\n\n## Getting Down to Business\n\nOkay, we now have somewhat of an understanding of how the PSB works. I designed it to be like one of those calculators or watches that is encased in see-through plastic. Everything is there if you just look for it. However, just like the watch or calculator, you don't need to know exactly how one works to use it. This software allows the users to do whatever they want, including looking into the future, so be careful. Our first test is going to be the ol' Bollinger Band algorithm.\n\n* * *\n\n        for i in range(len(myDate) - numBarsToGoBack,len(myDate)):\n            equItm += 1\n            tempDate = myDate[i]\n            todaysCTE = todaysOTE = todaysEquity = 0\n            marketPosition[i] = marketPosition[i-1]\n            mp = marketPosition[i]\n            buyLevel,shortLevel,exitLevel = bollingerBands(myDate,myClose,60,2,i,1)\n            print(tempDate,\" avg \",exitLevel,\" \",buyLevel - exitLevel)\n            atrVal = sAverage(trueRanges,10,i,0)\n            rsiVal = rsiStudy.calcRsi(myClose,10,i,0)\n            stopAmt = 3000\/myBPV\n    #        print(myDate[i],\"rsi \",rsiVal,\" atrVal \",atrVal*myBPV,\" \",myBPV)\n    #        fastKVal,fastDVal,slowDVal = stochStudy.\n    #                                     calcStochastic(3,9,9,myHigh,myLow,myClose,i,1)\n    #        if (mp > 0 and maxPositionL < 3) : maxPositionL = mp\n    #        if (mp < 0 and maxPositionS < 3) : maxPositionS = mp\n     #Long Entry Logic - Bolloinger\n            if (mp == 0 or mp == -1)  and myHigh[i] >= buyLevel:\n                profit = 0\n                price = max(myOpen[i],buyLevel)\n                if mp <= -1:\n                    profit,trades,curShares = exitPos(price,myDate[i],\"RevShrtLiq\",curShares)\n                    listOfTrades.append(trades)\n                    mp = 0\n                    todaysCTE = profit\n                tradeName = \"Boll Buy\"\n                mp += 1\n                marketPosition[i] = mp\n                numShares = 1\n                entryPrice.append(price)\n                entryQuant.append(numShares)\n                curShares = curShares + numShares\n                trades = tradeInfo('buy',myDate[i],tradeName,entryPrice[-1],numShares,1)\n                barsSinceEntry = 1\n                totProfit += profit\n                listOfTrades.append(trades)\n     #Long Exit - Loss\n            if mp >= 1 and myLow[i] <= entryPrice[-1] - stopAmt and barsSinceEntry > 1:\n                price = min(myOpen[i],entryPrice[-1] - stopAmt)\n                tradeName = \"L-MMLoss\"\n                exitDate =myDate[i]\n                numShares = curShares\n                exitQuant.append(numShares)\n                profit,trades,curShares = exitPos(price,myDate[i],tradeName,numShares)\n                if curShares == 0 : mp = marketPosition[i] = 0\n                totProfit += profit\n                todaysCTE = profit\n                listOfTrades.append(trades)\n                maxPositionL = maxPositionL - 1\n     # Long Exit - Bollinger Based\n            if mp >= 1 and myLow[i] <= exitLevel:\n                price = min(myOpen[i],exitLevel)\n                tradeName = \"L-BollExit\"\n                numShares = curShares\n                exitQuant.append(numShares)\n                profit,trades,curShares = exitPos(price,myDate[i],tradeName,numShares)\n                if curShares == 0 : mp = marketPosition[i] = 0\n                totProfit += profit\n                todaysCTE = profit\n                listOfTrades.append(trades)\n                maxPositionL = maxPositionL -1\n\n* * *\n\nThis is just the code for long entries and exits, and I must admit it looks a little bit ugly. Like I mentioned earlier, everything is exposed and the user has to do a little more of the programming that is taken care of behind the scenes with software such as AmiBroker and TradeStation. However, if you can program in the PSB, you can program anywhere.\n\nThe first thing we need to calculate is the Bollinger Bands. We are going to buy on a stop when the high of the day exceeds a 60-day two-deviation upper band. Shorting occurs when the low of the day exceeds a 60-day two-deviation lower band. This line of code provides exactly what we need.\n\n* * *\n\n    buyLevel,shortLevel,exitLevel = bollingerBands(myDate,myClose,60,2,i,1)\n\n* * *\n\nPrior to actually programming your algorithm, it is important to know how your algorithm enters trades: on a stop, market, or on a limit. Unlike AmiBroker, the ESB, or TradeStation, it is up to the user to program the different **if** constructs that pertain to the different types of orders. Don't worry; it's not nearly as complicated as it sounds. The **buyLevel** , **shortLevel** , and **exitLevel** are all calculated by the **BollingerBands** function. This algorithm also utilizes a $3,000 protective stop. Prepared with this information we are now ready to enter battle. With the PSB you must determine if the daily market action met the requirements for trade entry and\/or exit. In other words, you have to test the bar's extreme prices to see if an entry or exit level was exceeded. You might want to follow along in the code box as you read the following instructions. First, I have to make sure I am not already in a long position ( **mp** \u2014market position not equal to 1), and then I look at today's high; I am peeking into the future here, to see if the market would have exceeded or equaled the **buyLevel**. If it did, then I know I should have entered a long position. With this method of backtesting, you tell the computer a trade has taken place. In other platforms, the computer tells you a trade has taken place. Once you determine a trade should have been executed, you only need to change a little bit of code in the Long Entry Logic. The code that needs to be changed is bolded in the following code listing.\n\n* * *\n\n    if (mp != 1) and myHigh[i] >= **buyLevel** :\n        profit = 0\n        price = max(myOpen[i], **buyLevel** )\n        if mp <= -1:\n            profit,trades,curShares = exitPos(price,myDate[i],\"RevShrtLiq\",curShares)\n            listOfTrades.append(trades)\n            mp = 0\n            todaysCTE = profit\n        tradeName = \" **Boll Buy** \"\n        mp += 1\n        marketPosition[i] = mp\n        numShares = 1\n        entryPrice.append(price)\n        entryQuant.append(numShares)\n        curShares = curShares + numShares\n        trades = tradeInfo('buy',myDate[i],tradeName,entryPrice[-1],numShares,1)\n        barsSinceEntry = 1\n        totProfit += profit\n        listOfTrades.append(trades)\n\n* * *\n\nThat's all the code you have to concern yourself with. There is a lot of code, but most off it stays the same from one algorithm to the next.\n\nIn the case of this system, all entries and exits are executed as stop orders. Since we are dealing with stops, we must test the **myHigh[i] (today's high)** against the **buyLevel**. If the high of the day is greater than or equal to our buy stop, then we can safely assume the order was filled. Did the market gap above our buyLevel\u2014you must test for this as well. The code is already in place: **price = max(myOpen[i], buyLevel)**. If the market did indeed gap above our **buyLevel** , then the fill price is moved to the open of the day. This line of code is only necessary if a gap open could impact the fill price on a stop order. The only other thing you need to change is the **tradeName** variable. In this case, we change the name inside the quotes to **Boll Buy**. That's all you have to change in the code. The rest of the code is for internal use only. If IDLE had a collapsible text feature, I would just collapse this down so you wouldn't have to see it. You can edit your Python code in a more sophisticated text editor such as NotePad ++ (<https:\/\/notepad-plus-plus.org\/>). Figure 6.1 demonstrates the collapsible text feature.\n\n**Figure 6.1** NotePad++'s collapsible text feature.\n\nWe would all agree this is a much cleaner interface. NotePad ++ is available for download for a small donation. The reason I didn't use NotePad++ in this book is because it is not exactly easy to integrate into IDLE and I wanted to keep things as simple as possible.\n\nThat was the long entry logic only and now we must program the two exits associated with this algorithm: a money management stop and a moving average stop. Here is the money management stop logic:\n\n* * *\n\n    if mp >= 1 and myLow[i] <= **entryPrice[-1] - stopAmt** and barsSinceEntry > 1:\n        price = min(myOpen[i], **entryPrice[-1] - stopAmt** )\n        tradeName = \" **L-MMLoss** \"\n        exitDate =myDate[i]\n        numShares = curShares\n        exitQuant.append(numShares)\n        profit,trades,curShares = exitPos(price,myDate[i],tradeName,numShares)\n        if curShares == 0 : mp = marketPosition[i] = 0\n        totProfit += profit\n        todaysCTE = profit\n        listOfTrades.append(trades)\n        maxPositionL = maxPositionL - 1\n\n* * *\n\nIf **mp >=** 1, then the algorithm's current position is long. You could state **if mp** == 1, but this software has the capability of pyramiding and scaling in and out. So just always use >= to test for a long or short position. Since we are exiting on stops, the low of the day is tested against the **entryPrice - stopAmt** ($3,000). In Python the last element in a **list** can be easily accessed by using a **-1** inside the brackets. **EntryPrice[-1]** simply refers to the last price in the **entryPrice** list, which turns out to be the price at which the algorithm assumed a long position. I have put a small **i** in the **entryPrice** list more times than I would like to admit, and this definitely will throw a monkey wrench into the works. So don't do it. If the low of the day is less than or equal to **entryPrice - stopAmt** , then an exit has taken place and all you need to do is set the **price** and the **tradeName**. The rest of the code in the block runs in the background. **BarsSinceEntry** is an internal variable that keeps track of the number of bars since the last entry, and is used in this logic to make sure an entry and an exit doesn't occur on the same bar. This version of the PSB doesn't allow more than one entry per bar. Getting out on a stop at the midpoint between the upper and lower bands is handled in the same manner. The only difference is the **price** and the **tradeName** variables.\n\n* * *\n\n    if mp >= 1 and myLow[i] <= **exitLevel** :\n        price = min(myOpen[i], **exitLevel** )\n        tradeName = \" **L-BollExit** \"\n        numShares = curShares\n        exitQuant.append(numShares)\n        profit,trades,curShares = exitPos(price,myDate[i],tradeName,numShares)\n        if curShares == 0 : mp = marketPosition[i] = 0\n        totProfit += profit\n        todaysCTE = profit\n        listOfTrades.append(trades)\n        maxPositionL = maxPositionL -1\n\n* * *\n\nUnless you are adding a new indicator to the indicator.py file, this is the only file you will ever need to modify. Once you make your changes, all you need to do to run your algorithm is go under the **Run** menu and select **Check Module**. If everything checks out, then you go under the **Run** menu once again and select **Run Module**.\n\nI guarantee that you will get some type of syntax error or run-time error every time you create a new trading algorithm module. This is a good learning experience, as IDLE will point out the offending line of code. Figure 6.2 is an example of a syntax error message dialog and Figure 6.3 shows the offending line. You will see this until you stop forgetting the difference between the assignment operator ( **=** ) and the comparison operator ( **==** ).\n\n**Figure 6.2** A Python syntax error message.\n\n**Figure 6.3** In this case, the syntax error was the result of confusion between the assignment operator (=) and the comparison operator (==). Python highlights the problematic code.\n\nAnother easy thing to forget is using parentheses ( ) when you really want to use square brackets [ ]. If you do make this mistake you will see the error message shown in Figure 6.4.\n\n**Figure 6.4** A Python run-time error resulting from using parentheses instead of square brackets.\n\nThis is a run-time error because it will pass the **Check Module**. However, when you run the module, it will pop this error message into the Python Shell. The good thing is it tells the offending module and line number. Unfortunately, Python doesn't use line numbers, but if you go up under the **Edit** menu in the offending module and select **Go To Line** , a small dialog will open and you can plug in the line number and it will take you to the offending line.\n\nSimilar to any testing platform or programming language, the more you use it, the better you will become. Here are a couple more algorithms in the PSB that utilize market and limit orders.\n\nIf you are entering on the close, then you do not need to use a 1 offset in your indicator calculations. Basically, you are calculating the indicator at the close and executing at the close\u2014in real time this is somewhat difficult, but if you want to test it, you can. Here is a dual moving average crossover algorithm in the PSB. For brevity's sake, only the important code snippets were included.\n\n* * *\n\n            avg1 = sAverage(myClose,19,i,0)\n            avg2 = sAverage(myClose,39,i,0)\n            print(myDate[i],\"avg1 \",avg1,\" avg2 \",avg2)\n\n* * *\n\nNotice how a 0 is used as the offset\u2014I want to use today's data in the calculation. Since we don't have graphing capabilities, you can print right out to the Python Shell using the keyword print. Basically, put everything you want printed out in a list separated by a string or a space enclosed in quotation marks and a comma. This print statement will print out the date, the word avg1, the value of avg1, the word avg2, and the value of valu2. Here's what the printout looks like.\n\n* * *\n\n    20150429 avg1  57.15842105263158  avg2  54.38384615384615\n\n* * *\n\n* * *\n\n    #Long Entry Logic\n    if (mp != 1) and avg1 > avg2:\n        profit = 0\n        price = myClose[i]\n    ----------------\n    ----------------\n    #Long Exit - Loss\n    if mp >= 1 and myClose[i] < entryPrice[-1] - stopAmt and barsSinceEntry > 1:\n                price = myClose[i]\n    # Short Logic\n    if (mp !=-1) and avg1 < avg2:\n        profit = 0\n        price = myClose[i]\n    ----------------\n    ----------------\n    # Short Exit Loss\n            if mp <= -1 and myClose[i] >= entryPrice[-1] + stopAmt and barsSinceEntry > 1:\n                price = myClose[i]\n\n* * *\n\nThe only problem with this logic is that the system gets right back in the next bar after a money management exit if the moving averages are still aligned. Other testing software might get right back in at the same price it exits if the software allows multiple entries on a single bar. If you want to force a new crossover, then just add these lines:\n\n* * *\n\n    prevAvg1 = sAverage(myClose,19,i,1)\n    prevAvg2 = sAverage(myClose,39,i,1)\n    avg1 = sAverage(myClose,19,i,0)\n    avg2 = sAverage(myClose,39,i,0)\n\n* * *\n\nYou create two new variables and assign them the moving average values using an offset of 1. **PrevAvg1** and **prevAvg2** are the moving averages from yesterday. Your long and short entries should look something like this:\n\n* * *\n\n    #Long Entry\n    if (mp != 1) and avg1 > avg2 and prevAvg1 < prevAvg2:\n        price = myClose[i]\n    #Short Entry\n    if (mp !=-1)  and avg1 < avg2 and prevAvg1 > prevAvg2:\n        price = myClose[i]\n\n* * *\n\nNotice how the comparison of the prior day's moving average values is incorporated in the logic. Yesterday's **prevAvg1** must be less than yesterday's **prevAvg2** and today's **avg1** must be greater than today's **avg2** \u2014a crossover. There are a few different ways you could accomplish this, but until you become more comfortable with Python, the PSB, and lists, I would suggest simply offsetting the indicators to get the values you need. This is highly inefficient, but I don't think, unless you are doing it a lot, you will see a degradation in performance.\n\nIf you are using limit orders, just make sure you test the worst-case scenario. Most limit orders are not executed unless the price penetrates and trades one tick through the limit price. So you know how it important it is to use **> =** and **< =** in your price comparisons for stop orders. In limit orders forget the **=** sign. Like this:\n\n* * *\n\n    #Long Exit - Profit\n    if mp >= 1 and myHigh[i] > entryPrice + profAmt and barsSinceEntry > 1:\n        price = max(myOpen[i],entryPrice + profAmt)\n        tradeName = \"L-Prof\"\n\n* * *\n\nUsing just the > sign informs the PSB to make sure the market action penetrates the **entryPrice + profAmt**. In reality, there is a chance that a limit order will be filled if the order is touched. But we always want worst-case scenario when developing a trading algorithm. There is a bunch more Python code in the appendix and a few tutorials on the website. This version of the PSB is really in its infancy. I will continue to enhance it into the future by including some basic graphing utilities and loading it into a more sophisticated IDE.\n\n## Summary\n\nI love Python, and it really is a great language for quants and quant wannabes. It is the language to learn if you are new to programming. You will hear things out there about how it is inferior to C# and other faster languages. Unless you are an HFT trader, don't take these criticisms seriously. Others may complain that Python is not a purely objective language due to its deviations from the philosophy of \"OO.\" I look at Python as means of getting something done quickly and accurately. Traders need to concern themselves with trading concepts more so than programming theory.\n\n# Chapter 7  \nAn Introduction to EasyLanguage\n\nTradeStation has come a long ways from its early days, when it was known as SystemWriter. TradeStation, just like AmiBroker, is a complete testing and trading platform. The main difference between the two is you must use TradeStation's data servers and TradeStation's clearing services if you want to automate your trading algorithm. TradeStation is one of the most used trading platforms currently. You will find it in large hedge funds and CTAs as well as on the computers of smaller retail traders. Most system vendors have a TradeStation-compatible version of their software, because of its far reach. You don't have to trade through TradeStation's clearing services to use TradeStation. Many TradeStation users lease the software and data and then execute through their own broker. The beauty of trading through TradeStation is you can get the platform for free if you trade enough to meet a certain threshold. Else you will need to lease the TradeStation software for $249 per month. The supplementary products **RadarScreen** and **Portfolio Maestro** require additional fees.\n\nI have used TradeStation for the last 25 years and have written a book on using it to develop trading systems ( _Building Winning Trading Systems with TradeStation_ ). The latest version is 9.5, the culmination of three decades of hard work. The online community, just like AmiBroker's, is extremely large and many programming questions can usually be found with a quick Google search. In this chapter I will discuss TradeStation's development environment (TDE) and **EasyLanguage**.\n\n## TradeStation IDE\n\nTradeStation has beautiful charts and a huge library of indicators and drawing tools. Technicians love this interface. Chart creation is very simple and since TradeStation uses its own data servers, database management is not an issue. You log on, log in, and the data is but a few keystrokes away. This tight integration is one of the things that make TradeStation so popular. The other thing that makes TradeStation so good is EasyLanguage (EL). EL is similar in concept to AmiBroker's AFL and the code that makes up the Exel and Python backtesters that we have discussed.\n\nThe **TradeStation Development Environment** (TDE) can run independent of TradeStation 9.5 and you can launch it by finding it under the start menu or double-clicking its icon. This is helpful sometimes when you simply want to modify some code or start a new strategy or analysis technique and don't need to see the results of your work. However, if you do want to see the results of your coding instantaneously, then you should have both running. If you have TradeStation, you can follow along with this tutorial. If not, you can still follow along to see if TradeStation and EasyLanguage might fit your testing\/trading needs. Let's launch TradeStation 9.5. For this exercise, let's go ahead and log on by typing in your UserName and Password. Your screen should look somewhat similar to Figure 7.1.\n\n**Figure 7.1** The TradeStation home screen.\n\nInside this figure you should see a small tab labeled **TradingApps**. Click on the tab and another window like the one in Figure 7.2 will open up.\n\n**Figure 7.2** TradeStation's trading apps.\n\nSource: TradeStation\n\nThis window shows all the additional trading tools that have been turned on in your TradeStation platform. The one we are interested in right now is the button labeled **EasyLanguage**. Click it and you should be transferred over to the TDE. Check to make sure your screen looks like Figure 7.3.\n\n**Figure 7.3** The initial screen of the TradeStation Development Environment.\n\nWith version 8.8, TradeStation finally developed a powerful IDE. It is now similar in capabilities to other professional program authoring tools. If you went through Chapter 4, you will also notice a very close similarity to the AFL editor. Here is a quick tutorial on the editor.\n\nAn impressive feature of this editor is the ability to quickly and easily move from the source code of one program to another. When you open to edit more than one analysis technique, each program or file is opened underneath its own tab very similar to the way Microsoft Excel uses tabs for different worksheets. This is a nice feature because it makes copying existing code from one **Strategy** , **Function** , **Indicator** , or **Paintbar** to another very easy. As you have seen thus far in the book, building trading algorithms is done by extending existing code. After programming for a while, you will develop a library of ideas that you use and reuse again and again. So it's nice to be able to program a new idea with bits and pieces of old ideas. The multiple-tab concept makes this much simpler.\n\nThe TDE also incorporates the idea of **Outlining**. We touched on this in Chapter 4 on AmiBroker very briefly when we switched to **loop programming** mode. If you skipped that chapter or want a refresher, just follow along. **Outlining** is a feature where blocks of source code can be grouped together to make readability much easier. This also helps make programs much more modular. Let's play around with the EL Editor a bit. If you haven't downloaded the companion EL code for this book, it would be advisable to do so now. If you need help, go to the TradeStation appendix and follow the instructions. The EL Editor should still be open from our previous exercise. If not, then go ahead and launch it. From the **File** menu, select **Open** and when the **Open Easy Language Document** dialog box appears select **Strategy** in the **Select Analysis Type** drop-down menu. Your screen should look like the one in Figure 7.4.\n\n**Figure 7.4** The EasyLanguage editor in the TradeStation Development Environment.\n\nIf you have imported the companion code properly, select **MyMeanReversion1** strategy and click **OK**. The following code listing should now be on your screen.\n\n* * *\n\n    {This algorithm trades in the direction of\n     the longer term trend but buys\/shorts on dips\n     and rallies.  The indicator BollingerB returns\n     the location of the current bars close in relation\n     to the 5-bar Bollinger Band.  If the trend\n     is up and the current close is near the bottom\n     BBand it will buy.  If the trend is down and the\n     close is near the upper BBand it will Short.}\n    Inputs:Length(5),mmStop(2000),triggerVal(0.25),triggerSmooth(3);\n    Vars: myBollingerB(0);\n    myBollingerB = BollingerB(c,5,1,-1);\n    If MarketPosition = 0 and  close > average (close,200) and\n           average(myBollingerB,triggerSmooth) < triggerVal then buy(\"LongReversion\") next bar at market;\n    If MarketPosition = 0 and  close < average (close,200) and\n           average(myBollingerB,triggerSmooth) > 1 - triggerVal then sellShort(\"ShortReversion\") next bar at market;\n    If marketposition = 1 then\n    begin\n           If myBollingerB > 1 - triggerVal then sell(\"L-Exit\") next bar at market;\n           If c <= entryPrice - mmStop\/BigPointValue then sell(\"MML-Exit\") next bar at market;\n    end;\n    If marketposition = -1 then\n    begin\n           if myBollingerB < triggerVal then buyToCover(\"S-Exit\") next bar at market;\n           if c >= entryPrice + mmStop\/BigPointValue then buyToCover(\"MMS-Exit\") next bar at market;\n    end;\n\n* * *\n\nA portion of the strategy or algorithm is shown in Figure 7.5.\n\n**Figure 7.5** The algorithm with a single, complete portion of the strategy marked.\n\nYou can easily see how the blocks of code are outlined. There's a small box with a \"-\" symbol and a vertical line connecting the related block of code together. Outlining also allows you to hide the blocks of code if you wish to do so, again adding to the readability of the code. You can hide the block of code that is being outlined by simply clicking on the small box with the dash in it. This is a feature I would love to have in VBA and Python's IDLE.\n\nEL has a huge library of reserved words, and it is very difficult if not impossible to remember them all. The EL Editor has a really cool feature called _autocompleting_. Let's say you want to code an idea that incorporates an average true range calculation and you can't remember the name of the function that does the calculation. You could, of course, stop what you are doing and go to HELP and look up the function. However, this is time consuming, so the EL Editor monitors what you are typing and provides all the possible reserved words that might match what has thus been typed. In the case of our example, all you need to do is type what you might think is the name of the function and the list appears. By typing \"av\" the list pops up and places you at the word _average_. You can simply scroll down to find the function **AvgTrueRange**. Let's practice this. Go to the EL Editor and new line of code at the very bottom of the **MyMeanReversion1** strategy. Start typing \"av\" and see what happens. You should see something very similar to what is in Figure 7.6.\n\n**Figure 7.6** The EasyLanguage editor monitors what you are typing and provides likely functions.\n\nScroll down and select **AvgTrueRange** and it will be inserted into your code. Now go back and delete it. Keep the EL Editor open; we will be back with it shortly.\n\n## Syntax\n\nThe syntax of EasyLanguage is very similar to other programming languages, but there are a couple of differences in how remarks are demarked and the use of SKIP words:\n\n  * _Remarks or comments_. Words or statements that are completely ignored by the compiler. Remarks are placed in code to help the programmer, or other people who may reuse the code, understand what the program is designed to do. Double forward slashes \/\/ informs the EasyLanguage compiler that anything that follows is a comment. The double forward slashes can be used anywhere within a line. The curly left bracket { and curly right bracket } are used for multiline commentary. The { opens the remarks and } closes the remarks block. Anything inside { --- } is ignored by the computer.\n  * _SKIP words_. Words used to help EasyLanguage look more like English than a programming language. Here is a list: **an** , **at** , **by** , **does** , **is** , **of** , **on** , **than** , **the** , and **was**.\n\nEasyLanguage is the medium used by traders to convert a trading idea into a form that a computer can understand. Fortunately for nonprogrammers, EasyLanguage is an extremely high-level language; it looks like the written English language. It is a compiled language; programs are converted to computer code when the programmer deems necessary. This is different from VBA and Python. The compiler then checks for syntactical correctness and translates your source code into a program that the computer can understand. If there is a problem, the compiler alerts the programmer and sometimes offers advice on how to fix it. This is different from a translated language, which evaluates every line as it is typed.\n\nUnlike AmiBroker's AFL, VBA, and Python, EL requires all variables names to be declared\/defined prior to use. The declaration statement defines the initial value and data type of the variable. In a compiled language, the compiler needs to know how much space to reserve in memory for all variables. The following code is a complete EasyLanguage program.\n\n    Vars:  mySum(0),myAvg(0);\n    mySum  = High  + Low + Close;\n    myAvg  = mySum\/3 ;\n\nThe **Vars** : (or **Variables** :) statement tells the computer what variables are being declared and initialized. We declare the variables by rsimply listing them in the **Vars** statement and initialize them by placing an initial value in parentheses following the variable name. In this case, **mySum** and **myAvg** are to be equal to zero. EasyLanguage is smart enough to realize that these variables should be of the numeric data type, since we initialized them with numbers. Variable names should be self-descriptive and long enough to be meaningful. Which of the following is more self-explanatory?\n\n    mySum  = High+Low+Close;   or   k  = High  + Low + Close;\n    myAvg  = mySum\/3;          or   j = k\/3;\n    BuyPt  = Close   + myAvg;  or   l = Close+j;\n\nVariables of Boolean and string types are declared in a similar fashion.\n\n    Vars: myCondition(false),myString(\"abcdefgh\");\n\nThe variable **myCondition** was initialized to **false**. The word _false_ is a reserved word that has the value of zero. This word cannot be used for any other purpose. The variable **myString** was initialized to \"abcdefgh.\" Sometimes you will need to use a variable for temporary purposes, and it is difficult to declare and initialize all of your variables ahead of time. In the case of a temporary variable (one that holds a value for a short period of time), EasyLanguage has already declared and initialized several variables for your use; **value0** through **value99** have been predefined and initialized to zero and are ready for usage in your programs. The following is a complete EasyLanguage program as well:\n\n    value1 = High  + Low + Close;\n    value2 = (High  + Low)\/2.0;\n\nNotice that there isn't a **Vars** statement. Since **value1** and **value2** are predefined, the statement isn't needed. You have probably noticed the semicolon (;) at the end of each line of code. The semicolon tells the compiler that we are done with this particular instruction. Another similarity with **AFL**. In programming jargon, instructions are known as statements. Statements are made up of expressions, which are made up of constants, variables, operators, functions, and parentheses. Some languages need a termination symbol and others do not. **EL** and **AFL** need the statement termination symbol. Remember to put a semicolon at the end of each line to prevent a syntax error.\n\n**Inputs** are similar to variables. They follow the same naming protocol and are declared and initialized, too. However, an input variable remains constant throughout an analysis technique\u2014it cannot be changed. An input cannot start a statement (a line of instruction) and cannot be modified within the body of the code. One of the main reasons for using inputs is that you can change input values of applied analysis techniques without having to edit the actual EasyLanguage code. Input variables are the interface between the algorithm and user. Inputs would be perfect for a moving average indicator. When you plot this indicator on a chart, you simply type in the length of the moving average into the input box of the dialog. You don't want to have to go back to the moving average source code and change it and then verify it. **Inputs** are the same as **Param** in AFL, by the way. Also, when used in trading strategies, inputs allow you to optimize your strategies. Optimization was touched upon in the AmiBroker chapter and will be thoroughly discussed in Chapter 8. EL inputs serve two purposes: user and optimization interface.\n\nNotice how inputs and variables are declared in similar style.\n\n    Inputs: length1(10),length2(20),flag(false);\n    Vars:  myLength1(10),myAvgVal(30);\n\nHowever, notice how they are used differently in coding.\n\n* * *\n\n    _Variables_\n\n* * *\n\n    myLength1  = myAvgVal + myLength1;     {Correct}\n\n* * *\n\n    _Inputs_\n\n* * *\n\n    length1 = myAvgVal + length1;    {Incorrect}\n    myLength1  = length1*2;    {Correct}\n\nVariables can start a statement and can be assigned another value. Since inputs are constants and cannot be assigned new values, they cannot start a statement.\n\nIn a strongly typed language, such as C, Pascal, or C++, if you assign a real value such as 3.1456 to an integer typed variable, the decimal portion is truncated and you end up with the number 3. As we all know, precision is important when it comes to trading, so EasyLanguage includes only one Numeric type. All numbers are stored with a whole and a fractional part. In the old days when CPUs were slow, noninteger arithmetic took too much time and it was advised to use integer variables whenever possible.\n\nLike AFL, the overall purpose of EasyLanguage is to translate an idea and perform an analysis on a price data series over a specific time period. You can access the different data elements by using the keywords shown in Table 7.1.\n\n**Table 7.1** EasyLanguage Keywords and Abbreviations\n\n**Reserved Word** | **Abbreviation** | **Description**  \n---|---|---  \nDate | D | Date of the close of the bar.  \nTime | T | Time as of the close of the bar.  \nOpen | O | Open price of the bar.  \nHigh | H | High price of the bar.  \nLow | L | Low price of the bar.  \nClose | C | Close price of the bar.  \nVolume | V | Number of contracts\/shares traded.  \nOpenInt | OI | Number of outstanding contracts.\n\nIf you wanted to determine that the closing price of a particular instrument was greater than its opening price, you would simply type: **Close** > **Open** , or **C** > **O**. The beauty of any trading platform's scripting languages is their ability to have all of the data of an instrument at your fingertips. The reserved words that we use to access the different prices of the current bar are also used to access historical data. You do this by adding an index to the reserved word. The closing price of yesterday would be: **Close** [1]. The closing price two days ago would be: **Close** [2], and so on. The number inside the bracket determines the number of bars to look back. The larger the number, the further you go back in history. If you wanted to compare today's closing price with the closing price 10 days prior, you would type: **Close** > **Close** [10].\n\nBefore we move on, we should discuss how TradeStation stores dates and times. January 1, 2001, is stored as 1010101 instead of 20010101 or 010101. When the millennium changed, instead of incorporating the century into the date, TradeStation simply added a single digit to the year. The day after 991231 was 1000101 according to TradeStation. Time is stored as military time. For example, one o'clock in the afternoon is 1300 and one o'clock in the morning is 100.\n\nAfter that brief introduction to the EasyLanguage, let's return to the EL Editor and take a look at **myMeanRev1** algorithm.\n\nStart off by reading the first few lines of code. You will notice that these lines are comments because the comments are preceded and followed by the left and right curly brackets, {}. This is your typical mean reversion system and is designed to trade those markets that demonstrate a mean reversion propensity. All markets demonstrate mean reversion characteristics at some time or another, but the stock indices seem to revert to the mean more frequently than the others.\n\nThe next lines are the **Inputs** and **Variables** used in the algorithm. The user can change the input values through the **Format Analysis Techniques and Strategies** dialog window. This window is brought up once you have applied the **Strategy** to a chart. If you like, go ahead and create a **Chart Analysis** of daily **ES** (emini-S&P) going back 10 years. Once the chart has been populated with data, go up under the **Insert** menu and select **Strategy** (see Figure 7.7).\n\n**Figure 7.7** How to apply a strategy to your chart.\n\nSource: TradeStation\n\nAfter selecting **MyMeanReversion1** from the list of **Strategies** , your chart window should contain some trades like the ones shown in Figure 7.8. Now we can access the **Inputs** and change them if we like by right-clicking the chart and selecting **Format Strategies**. A dialog similar to Figure 7.9 will open.\n\n**Figure 7.8** A reversion analysis strategy shown on a chart.\n\nSource: TradeStation\n\n**Figure 7.9** How to change the inputs and adjust your strategy.\n\nAnother dialog will follow once you select **Format** (Figure 7.10.). Here you can select any of the four **input** variables. Go ahead and select **mmStop** and change it to 3000. Then click **OK** and then **Close**. That's all there is to changing the **inputs** from the user's interface. You can always change them programmatically from the EL Editor. The next line is where you must define any variables you may use later in your code. If you have properly converted your trading idea into either a FC or FSM diagram and then into pseudocode, then you will know ahead of time the number and names of your variables. It doesn't matter how well thought out your pseudocode is; you will almost always come across the need for additional variables. Don't worry; you can always go back to the **Vars** or **Inputs** and add more variable names. Here the only variable name that will be used is **myBollingerB** , and it will be initially set to zero.\n\n**Figure 7.10** The four input variables to adjust the mean reversion strategy.\n\nContinuing on through the code, the first order of business is the assignment of **myBollingerB**. **myBollingerB** is assigned the output of the function **BollingerB**. This function requires four parameters or arguments: price data array, lookback period, the number of **+** standard deviations, and the number of negative standard deviations. In our example, we are using the closing prices, a five-bar lookback, and positive one and negative one standard deviations.\n\n* * *\n\n    myBollingerB = BollingerB(c,5,1,-1);\n\n* * *\n\nThe **BollingerB** function returns a value between a negative value and a positive value and 1.00 which indicates the location of the current bar's close in relation to the upper and lower Bollinger Bands. The lower the value, the closer it is to the lower band. The higher the value, the closer it is to the upper band.\n\n* * *\n\n    If MarketPosition = 0 and\n    close > average (close,200) and\n    average(myBollingerB,triggerSmooth) < triggerVal then\n            buy(\"LongReversion\") next bar at market;\n    If MarketPosition = 0 and\n    close < average (close,200) and\n    average(myBollingerB,triggerSmooth) > 1 - triggerVal then\n            sellShort(\"ShortReversion\") next bar at market;\n\n* * *\n\nYou might not be able to tell by this code snippet, but EL follows the bar-by-bar programming paradigm. The keyword **If** might have given you a hint, though. If you have read the chapter on AmiBroker, you know you usually don't use the keyword **If** when dealing with array programming. When you use the data arrays (open, high, low, close, volume, or opInt) without an index variable you are simply looking at the very last day of data up to that point in the master loop.\n\n**C** is the same as **C[0]**. The **[0]** is optional. Since there is an implied index, you can be assured you are not dealing with array programming. The lack of the index for the last value in the arrays makes EL look more like English. Take a look at the following line of code:\n\n* * *\n\n    If close > average(c,200) then buy next bar at open;\n\n* * *\n\nDoesn't this look like English? You bet it does. Hence, the name EasyLanguage. The language is simple on the surface, but can become much more complicated as the complexity of the algorithm increases. This is a good thing\u2014EasyLanguage isn't necessarily easy, but it's not weak by any stretch of the imagination. And it's getting more powerful every year.\n\nThese lines of code direct TradeStation to check the current market position, and if it is flat, to check if today's **close** > the 200-bar **moving average** and the 3-bar **moving average** of **myBollingerB** is less than the **triggerVal**. In this example, the **triggerVal** is set to **0.2**. If the criteria are met, then a market order to buy is placed for the next bar's open. A similar criterion is used to initiate a short position on the next bar's open. The only difference is the **close** must be below the 200-bar **moving average** and the 3-bar **moving average** of the **myBollingerB** variable must be greater than 1 - **triggerVal**.\n\nThere is a lot going on with this code. Any time you use the keyword **if** it must be eventually followed by the keyword **then**. Initiating long positions requires the keyword **buy** and short position requires the keywords **sellShort**. You can give names to your entry and exit signals by following the keywords **Buy\/SellShort\/Sell\/BuyToCover** with the name of the signal enclosed within quotes and parentheses. The buy signal is this example is \" **LongReversion** \" and the short signal is named \" **ShortReversion**.\" This isn't necessary but it does help when you have multiple entry and exit signals. EL requires you to use the words **next bar** or **this bar on\/at close** whenever you direct TradeStation to initiate or liquidate an existing position. When programming in EL, think of yourself sitting in front of the computer screen right at the close of a bar. You can place a market order immediately and get filled at the current close, you can place a market order and get filled at the next bar's open price, or you can place a limit or stop order for the next bar. By limiting you from seeing the next bar (other than the open price), TradeStation eliminates all risks of \"Future Leak.\" In other words, you cannot peek into the future by looking at the next bar and make a trading decision on the current bar. Both the VBA and Python back testers allow you to do this, not in order to cheat, but to allow complete testing flexibility.\n\nTake a look at the next section of code and see if you can tell what is going on.\n\n* * *\n\n    If marketposition = 1 then\n    begin\n           If myBollingerB > 1 - triggerVal then sell(\"L-Exit\") next bar at market;\n           If c <= entryPrice - mmStop\/BigPointValue then sell(\"MML-Exit\") next bar at market;\n    end;\n    If marketposition = -1 then\n    begin\n           if myBollingerB < triggerVal then buyToCover(\"S-Exit\") next bar at market;\n           if c >= entryPrice + mmStop\/BigPointValue then buyToCover(\"MMS-Exit\") next bar at market;\n    end;\n\n* * *\n\nIn Python, the keyword **then** would be replaced by the colon (:). Since the code controlled by the **if** consists of more than one line of code, the keywords **begin** and **end** must be used to encapsulate the code. For every **begin** there must be an **end**. The **begin** and **end** informs the compiler that the code is related and must flow through all the lines in the block. If the current market position is 1 or long, then TradeStation in instructed to exit the long position if one of two criteria are met:\n\n  1. If myBollingerB > (1 \u2013 triggerVal) or 0.8, then the long is liquidated on the next bar's open.\n  2. If the difference between the current close price is less than the entryPrice \u2013 2000\/bigPointValue, then the long is liquidated on the next bar's open.\n\nThe short liquidation occurs in a similar manner. If myBollingerB < 0.2 or the close is greater than the entryPrice + 2000\/bigPointvalue. That's the entire mean reversion system that we introduced in Chapter 3.\n\nAnother key and cool feature of EL is that all variables are special arrays. When you declare the variable myBollingerBand in the **vars** : section, you are informing the compiler to create an array with the name **myBollingerBand**. This is a special array because you don't need to worry about keeping track of the values in the array. The value is carried over from one bar to another until it is changed. Because myBollingerBand is an array, you can access the prior bar's value by indexing the variable name. The prior bar's **myBollingerBand** value is accessed by **myBollingerBand** [1]. These variables can be referred to as bar arrays.\n\nOnce you type your code in and want to see if it **Verifies** or compiles, all you need to do is go up under the **Build** menu and select **Verify**. If all goes well, you will see this dialog (Figure 7.11).\n\n**Figure 7.11** A message confirming that your code was verified and will be recalculated based on the new inputs.\n\nIf not, then you might see what's in Figure 7.12. TradeStation is politely informing you that you have made a syntax error. If you double-click on the error, the EL Editor will take you directly to the offending line with the error. The error message also informs you of the line number. If you want the EL Editor to show line numbers, go under the **Tools** menu and select **Options**. A dialog window like the one in Figure 7.13 will open.\n\n**Figure 7.12** TradeStation lists any syntax errors it finds during verification in the bottom pane of the screen. The line number of the error is provided so you can easily find it in your code.\n\n**Figure 7.13** The EasyLanguage editor can display line numbers to make it easy to navigate your code.\n\nClick on **Line Numbers**. The EL Editor has added the capability of grouping strategies, indicators, and functions together in a neat package known as a **Project**. If you build a strategy that calls in a lot of external functions, then putting the strategy into a project makes it easier to access all the different code. Let's create a **Project**. Go under the **File** menu and go to the **Project** menu item and a hierarchical menu will open (see Figure 7.14).\n\n**Figure 7.14** To create a new project using your strategy, select Project from the File menu.\n\nThe EL Editor will ask you for a name and where you would like to save the project. Select a location on your desktop or the C:\\ drive and give your project the name **MeanReversion**. Once the project is created, a small dialog box will open with your new project name inside it. Right-click on the project name and a menu will open like the one in Figure 7.15.\n\n**Figure 7.15** How to add an existing strategy to a new project.\n\nClick on **Add Existing Item** and select **MyMeanReversion1** from **Strategies**. Close the box out and then go under the **File** menu to **Project** and select **Open Project** and select the project you just created, **MeanReversion.elx**. Click on the + beside the name **MyMeanReversion1** , and it will expand and show all the functions or dependencies included in the **Strategy** algorithm (see Figure 7.16).\n\n**Figure 7.16** All dependencies and functions are nested inside their strategy algorithm.\n\nThis is just neat way to keep track of all the code connected to a particular project and is a relatively new feature of the EL Editor.\n\n## Samples of EasyLanguage\n\nHere is a simple Commodity Channel Index (CCI) algorithm:\n\n* * *\n\n    {CCI system utilizing 3 ATR Profit\n     And  a 1 ATR Stop}\n    inputs: cciLen(20),smooth(9);\n    vars: myCCIVal(0);\n    myCCIVal = average(cci(cciLen),smooth);\n    If myCCIVal crosses above -100  then buy this bar on close;\n    If myCCIVal crosses below 100 then sellShort this bar on close;\n    If marketPosition = 1 then\n    begin\n           If c > entryPrice + 3* avgTrueRange(10) then sell this bar on close;\n           if c < entryPrice - 1* avgTrueRange(10) then sell this bar on close;\n    end;\n    If marketPosition =-1 then\n    begin\n           If c < entryPrice - 3* avgTrueRange(10) then buyToCover this bar on close;\n           if c > entryPrice + 1* avgTrueRange(10) then buyToCover this bar on close;\n    end;\n\n* * *\n\nThis algorithm covers long positions whenever the market closes above the **entryPrice** plus 3 ATRs, or when it closes below the **entryPrice** minus 1 ATR. The same technique is programmed for short trades as well. This algorithm risks one ATR to achieve three.\n\nThe following algorithm is a simple 20-day Donchian breakout that only takes trades in the direction of the 14-day slope of closing prices. The interesting part of the algorithm is found in the trade management.\n\n* * *\n\n    Inputs: linearSlopeLength(14),channelLength(20);\n    Inputs: atrLen(10),numAtrStop(3),numAtrProfThreshold(5), numAtrTrail(2);\n    vars: buyLevel(0),shortLevel(0),maxContractPoints(0);\n    buyLevel = Highest(High[1],channelLength);\n    shortLevel = Lowest(Low[1],channelLength);\n    if marketPosition <> 1 and linearRegSlope(c,linearSlopeLength) > 0 then\n       Buy(\"BuyLevelEntry\") tomorrow at buyLevel stop;\n    maxContractPoints = maxContractProfit \/ bigPointValue;\n    If marketPosition = 1 then\n    begin\n       sell(\"L-ATR-Stop\") next bar at entryPrice-numAtrStop * avgTrueRange(atrLen) stop;\n       if maxContractPoints >= numAtrProfThreshold * avgTrueRange(atrLen) then\n       begin\n          value1=entryPrice + (maxContractPoints-numAtrTrail * avgTrueRange(atrLen));\n          sell(\"L-Trail-Stop\") next bar at value1 stop;\n       end;\n    end;\n    if marketPosition <>-1 and linearRegSlope(c,linearSlopeLength) < 0 then\n       Sellshort(\"ShortEntryLevel\") tomorrow at shortLevel stop;\n    If marketPosition =-1 then\n    begin\n       buyToCover(\"S-ATR-Stop\") next bar at entryPrice+numAtrStop *avgTrueRange(atrLen)\n       stop;\n       if maxContractPoints >= numAtrProfThreshold * avgTrueRange(atrLen) then\n       begin\n          value1 = entryPrice - (maxContractPoints-numAtrTrail * avgTrueRange(atrLen));\n          buyToCover(\"S-Trail-Stop\") next bar at value1 stop;\n       end;\n    end;\n\n* * *\n\nThis algorithm incorporates a volatility-based protective and trailing stop. EL keeps track of the maximum profit a position achieved during the duration of a trade, and stores it in the keyword **maxContractProfit**. The profit is expressed in terms of dollars, and to get it in terms of points you have to divide by the **bigPointValue** of the underlying instrument. If the **maxContractProfit** achieves a level equal to, in this case, five ATRs, then a trailing stop is engaged to trail the highest point in the trade by two ATRs. If you don't want to use volatility in the stop calculations, you can use dollars. Here is the snippet of the code that engages the trailing stop:\n\n* * *\n\n    if maxContractPoints >= numAtrProfThreshold* avgTrueRange(atrLen) then\n    begin\n       value1=entryPrice + (maxContractPoints-numAtrTrail * avgTrueRange(atrLen));\n       sell(\"L-Trail-Stop\") next bar at value1 stop;\n    end;\n\n* * *\n\nIf the **maxContractPoints** is greater than the threshold profit level, then the trailing stop level is calculated using the following formula:\n\nvalue1 = entryPrice + (Highest level during trade \u2013 2*ATR)\n\nAs the market continues to climb, so does value1 staying 2 ATR below the highest point achieved. Once the market stops climbing, value1 locks into place and waits for the market to take out the trailing stop.\n\nOnce you master just the fundamentals of programming and a few concepts that are intrinsic to a particular language, you then have most of the tools to test or program any trading idea. Practice makes perfect, so don't just stop after you have programmed your initial trading idea\u2014keep going because I guarantee your creativity won't stop.\n\n## Summary\n\nThe objective of this chapter was to introduce the fundamentals necessary to take a trading idea and program it in EasyLanguage. Along the way we discussed data types, expressions, and statements. EL requires all variables to be declared before use and we reviewed how this is done. The use of inputs, built-in function calls, and how to verify your code was explained as well. The EL Editor is a very powerful tool and is the center of your creativity. TradeStation and its programming language, EasyLanguage, provides a very tightly integrated trading environment that can facilitate the programming of any trading idea. Learning this powerful language is best accomplished by examining as much code as you can. I have used EL for 25 years and I am still learning.\n\nI will leave you with the finite state machine (FSM) code from Chapter 1.\n\n* * *\n\n    {Finite State Machine\n     Pivot Point Example from Chapter 1.\n     The switch case structure is used in place of\n     if-then constructs.  This increases the readability\n     considerably.}\n    vars:state(0),maxRsi(0),state1BarCount(0),state1Price(0), state3Price(0),state1BarNumber(0);\n    vars: isHiPivot(false);\n    isHiPivot = h[1] > h and h[1] > h[2];\n    Switch(state)\n    begin\n           case(0):\n           begin\n                  if isHiPivot then\n                  begin\n                          state = 1;\n                          state1Price = h[1];\n                          state1BarNumber = barNumber;\n                          print(date,\" State 1 : \",state1Price, \" \",state1BarNumber);\n                          value1 = text_new(date[2],time,h+20,\"S1\");;\n                  end;\n           end;\n           case(1):\n           begin\n                   if h > state1Price then state1Price = h;\n                   if low < state1Price * .98 then\n                   begin\n                          state = 2;\n                          print(date,\" State 2 :\",low);\n                          value1 = text_new(date[1],time, l-10,\"S2\");\n                          text_setColor(value1,red);\n                   end;\n           end;\n           case(2):\n           begin\n                   if isHiPivot and h[1] > state1Price then\n                   begin\n                          state = 3;\n                          state3Price = h;\n                          print(date,\" State 3 :\", state3Price);\n                          value1 = text_new(date[2],time, h+20,\"S3\");;\n                          text_setColor(value1,yellow);\n                   end;\n           end;\n           case(3):\n           begin\n                   if h > state3Price then state3Price = h;\n                   if l < state3Price * .98 then\n                   begin\n                          buy this bar on close;\n                          print(date,\" accept state : \",l);\n                          value1 = text_new(date[1],time, l-10,\"S4\");;\n                          text_setColor(value1,GREEN);\n                          state = 0;\n                   end;\n           end;\n    end;\n    If barNumber - state1BarNumber > 30 then state = 0;\n    Sell next bar at lowest(l,60) stop;\n\n* * *\n\n# Chapter 8  \nGenetic Optimization, Walk Forward, and Monte Carlo Start Trade Analysis\n\n## Utilizing TradeStation and AmiBroker\n\nBoth TradeStation and AmiBroker software includes two very powerful tools that can be utilized to help create robust trading algorithms. If you have used TradeStation or AmiBroker to develop trading algorithms but have not taken advantage of genetic or walk-forward optimization, this chapter will hopefully provide enough information to help enhance your research and unlock the potential of these powerful tools. These tools have been a part of both platforms for several years, but through my observations of other users they have been either ignored or underutilized. Users often note these stumbling blocks:\n\n  * They are hard to understand.\n  * They are hard to implement.\n  * They lead to over-curve-fitting.\n\nThe objective of this chapter is to explain that a computer science degree is not necessary to understand these tools; in some scenarios their use is absolutely necessary, they are very simple to use, and with proper application a user can make sure over-curve-fitting does not take place. Since I am somewhat of a newbie to AmiBroker and in an attempt to cut down on redundancy, a good portion of the concepts in this chapter will be illustrated with TradeStation's tools. An understanding of TradeStation, some EasyLanguage, and input optimization is assumed. As you will see in the latter part of the chapter, the ideas and algorithms will be translatable to AmiBroker and AFL.\n\nThe first part of this chapter explains the concept of a genetic algorithm, proceeds to build one, and then shows how they can be used to optimize a trading algorithm. The genetic algorithm development part is a little longer and requires a close examination. Much of this discussion, including examples and processes, was taken directly from Denny Hermawanto's excellent paper titled, \"Genetic Algorithm for Solving Simple Mathematical Equality Problem.\" Out of all the research used in the writing of this chapter, Hermawanto's description of genetic algorithms was considered one of the easier to understand and implement. Understanding what makes up a genetic algorithm is not necessary to use one. However, even a slight understanding of how one works will eliminate the hesitation of using it and eliminate its \"voodoo\" factor.\n\nThe second part of the chapter describes how a trading algorithm can be automatically reoptimized periodically using the walk-forward optimizer (WFO) in concert with genetic optimization. In addition to changing parameters periodically based on algorithm performance going forward in time, the WFO also creates a report that indicates the degree of the trading algorithm's robustness. In other words, what is the probability of the algorithm continuing to work?\n\n### Genetic Algorithms: What Are They?\n\nThe first question that needs to be answered is, Why use genetic algorithms (GA) in the first place? The same question concerning walk-forward optimization (WFO) will be answered in the second part of this chapter. TradeStation and AmiBroker already have exhaustive search (ES) optimization engines built into their platforms. In this chapter, the words _searching_ and _optimizing_ will be used interchangeably. The ES optimization engine doesn't use any sophistication in its search process, just brute force. Why not just use this brute force approach if it looks at every possible combination of variables and no stone is left unturned? It is true that this feature is very powerful and can fulfill a good portion of a user's optimization needs. However, unlike genetic algorithms, there is a limit to the number of variables (inputs) and their respective iterations that can be optimized in a sequential manner.\n\nGA has limitations as well, but they are very large\u2014large in the sense that a super-large amount of data would be necessary to limit their effectiveness. The major limitation to a brute force approach is, of course, time. Even with today's superfast processors, searching a large search space is extremely time sensitive. Imagine a trading algorithm (TA1) that has five different parameters that can be optimized across 50 different values. Assuming a brute force optimization and three seconds to complete each iteration, it would take the computer almost 30 years to complete or span the entire search space ( _search space_ and _optimization range_ are also interchangeable terms). Here is the math behind the time needed to complete this task:\n\nThis example is an exaggeration, but it shows multiple parameter optimizations can be very time consuming. Figure 8.1 shows how quickly the search space grows as the number of iterations increase for each of the five parameters. This is where artificial intelligence (AI) comes into play; AI uses brains over brawn. In the above example, AI can shrink the overall search space by eliminating some values of each parameter that do not lead to a favorable outcome. In doing so, it cuts down the number of total iterations, and less iteration means less time.\n\n**Figure 8.1** Iteration growth rate of five parameters.\n\nThe first step in the process of incorporating these tools into a trader's algorithm development is to understand the basic foundations of GA. Don't let the words _genetic_ or _algorithms_ scare you away. They are simply terms to explain how a solution to a problem can be quickly and solidly uncovered. As we already know, probably too well, an algorithm is just a set of rules one follows to find a solution to a problem. Genetic refers to the biological process of evolution through reproduction, mutation, and survival of the fittest. Remember time is of the essence but so is a good answer and this type of algorithm provides quick yet robust solutions. Computer software and biology may seem like strange bedfellows but their synthesis makes up a large portion in the study of AI.\n\nJohn Holland's 1975 book titled _Adaptation in Natural and Artificial Systems_ is the bible on GA and his work has paved the way for a majority of researchers in this field. Holland has stated,1 \"Living organisms are consummate problem solvers. They exhibit a versatility that puts the best computer programs to shame.\"\n\nComputer speed and power has come a long way since 1975 and you may think these biologically based optimization\/search algorithms have become obsolete. The exponential growth rate of multiple variable optimizations proves that this is not the case. Also humans are constantly creating more and more complex problems that continually exceed our technology. The relationship between GA and computers is as close as it has ever been.\n\nThe principles that make up GA, if taken one concept at a time, are easy to understand. The combining of these concepts into a complete algorithm can be daunting if each piece is not broken down. I programmed these concepts in a modular manner using VBA for Excel in a very small amount of time. This same software was used to illustrate the concepts and solve the initial problem in this first part of the chapter. Snippets of VBA code will be sprinkled throughout the first part of the chapter to help explain the most important components of genetic algorithms.\n\n## Computers, Evolution, and Problem Solving\n\nGenetic algorithms borrow the following concepts from biology:\n\n  * Population and generations\n  * Selection of the fittest\n  * Reproduction\n  * Mutation\n\nAll these terms are easily understood in a biological framework but may not seem initially translatable into a computer\/software\/math paradigm. The application of these concepts to solve a complex problem can be difficult but we need not worry ourselves with this because we know our objective with TradeStation, AmiBroker, or Excel\u2014building the world's best trading algorithms! However, a simple problem needs to be solved so we can demonstrate the eloquence of GA. The ideas and processes that will be discussed for applying a GA to finding a solution to a simple equation were derived directly from Denny Hermawanto's paper titled \"Genetic Algorithm for Solving Simple Mathematical Equality Problem.\" The problem that we will be solving utilizes this very simple equation:\n\nThis equation involves four unknowns and multiple solutions and can be easily solved by trial and error. We could simply set  ,  _,_ and   equal to zero and set   equal to 20 and be done with it. Remember, this is an oversimplified problem to demonstrate the four core concepts of GA.\n\n## Population\n\nThere are four variables in our equation so continuing to borrow from biology let's create six chromosomes consisting of four values or genes that will be substituted into the variables  _,_  _,_  _,_ and  . Unlike real chromosomes, these will simply be placeholders for the four different values of the variables (genes). Figure 8.2 illustrates our six amoeba-looking genes.\n\n|  |  | _Genes_  \n---|---|---|---  \n_Generation 1_ : | _Chromosome(0)_ | _=_ | _[10,11,20,19]_  \n| _Chromosome(1)_ | _=_ | _[13,08,15,20]_  \n| _Chromosome(2)_ | _=_ | _[00,20,05,09]_  \n| _Chromosome(3)_ | _=_ | _[07,27,05,14]_  \n| _Chromosome(4)_ | _=_ | _[10,30,30,04]_  \n| _Chromosome(5)_ | _=_ | _[16,05,13,28]_\n\n**Figure 8.2** An illustration of our chromosomes with different genetic values.\n\nSo the population will consist of six chromosomes that have four different genes that can be swapped or mutated. Why six chromosomes instead of four, you may ask? Just be patient; we will get to that. The initial population of chromosomes will be generated using a random number generator (RNG). Each chromosome will be defined by its genetic value, and these are the values that will be randomized. Values between 0 and 30 for each gene will be initially generated in a random fashion. You will soon discover that GA relies heavily on random numbers. This is yet another similarity with nature\u2014randomness. Here is the initial population generated by the VBA software and its RNG:\n\n## Initial Population Setup Using VBA Excel\n\n    'All Lines starting with single quote is a comment\n    'Random variables into each chromosome's genes\n    'Numbers range between 0 and 30\n    'Chromos will be represented by a table\/matrix\n    'The first index in the chromo table will be row #\n    'The second index in the chromo table will be column #\n    'Excel's RNG is invoked by using the Rnd function call\n    'Int ((upperbound - lowerbound + 1) * Rnd + lowerbound)\n    Randomize '<-- must use this to get different numbers\n    For i = 0 To 5 ' Six Chromos\n        For j = 0 To 3 ' Four Genes\n            chromo(i, j) = Int((30 - 0 + 1) * Rnd + 0)\n        Next j\n    Next i\n\nHow good or fit is this population? Fitness is determined by how close each chromosome comes to providing a solution to our equation. Remember, this is a very simple problem, but these ideas are easily scalable to problems involving extreme complexity. Calculating fitness is quite easy; simply plug in the chromosomes' genes (values) for a, b, c and d. In chromosome (0) its fitness is determined by:  . The solution we are seeking is 40 so this fitness doesn't seem to be that good. However, each chromosome's fitness is relative to all the other chromosomes.\n\n_Fitness 1_ : | _Chromosome(0)_ | _=_ | _111_  \n---|---|---|---  \n| _Chromosome(1)_ | _=_ | _94_  \n| _Chromosome(2)_ | _=_ | _64_  \n| _Chromosome(3)_ | _=_ | _90_  \n| _Chromosome(4)_ | _=_ | _164_  \n| _Chromosome(5)_ | _=_ | _93_\n\n## Testing Fitness of Chromosomes Using VBA Excel\n\n    'Equation is : a + 2b + 3c + d = 40\n    'a coefficient = Cell(3,3)\n    'b coefficient = Cell(3,4)\n    'c coefficient = Cells(3,5)\n    'd coefficient = Cells(3,6)\n    'the summation of the equation is in Cells(3,7)\n    For generation = 1 To 100 ' total generations\n        For i = 0 To 5 'Six Chromos\n            FObj(i) = 0\n            For j = 0 To 3 ' Four Genes\n                'Calculate each expression in equation and accumulate\n                If j = 0 Then FObj(i) = Cells(3, 3) * chromo(i, j)\n                If j = 1 Then FObj(i) = FObj(i) + Cells(3, 4) * chromo(i, j)\n                If j = 2 Then FObj(i) = FObj(i) + Cells(3, 5) * chromo(i, j)\n                If j = 3 Then ' Test the summation against 40\n                    FObj(i)=Abs((FObj(i)+Cells(3, 6)*chromo(i,j)) -Cells(3,7))\n                    'if summation = 40 then solution found\n                    If (FObj(i) = 0) Then foundSolution = i\n                End If\n            Next j\n        Next i\n\nIf we rank the chromosomes in order of fitness, then we would have the following:\n\n_Chromosome(2)_ | = | _64\u2014closest to the solution (24 away)_  \n---|---|---  \n_Chromosome(3)_ | = | _90\u2014second closest (50 away)_  \n_Chromosome(5)_ | = | _93_  \n_Chromosome(1)_ | = | _94_  \n_Chromosome(0)_ | = | _111_  \n_Chromosome(4)_ | = | _164_\n\n## Selection\n\nNow that we have our initial population and the respective chromosome fitness, we can select the chromosomes that should be carried forward and used for reproduction and mutation. It's easy to see that the top four ranking chromosomes should be selected because they are closer to our solution than the last two (the absolute difference between the chromosome fit score and 40, our solution, is smaller). However, the computer makes the decisions and it makes its selection based on probabilities. This task is accomplished by converting each chromosome's fitness in terms of probability. We can't make the selection decision for the computer, but we can make sure the computer understands that some chromosomes are more fit than others. If you remember probabilities from an old math or stat class, you know the basic premise: Rolling a dice once results in one out of six potential outcomes. With this in mind you have a one-in-six chance of rolling a six or a probability of 16.6666667 percent.\n\n  1. _P_ = Possible favorable outcomes \/ Potential outcomes\n  2. _P_ = 1\/6, or 16.6666667%\n\nKeep in mind probabilities are the cornerstone of GA as well as trading algorithms.\n\nOkay, now on to our selection process. Remember, we need to nudge the computer into making the right decision. The computer will, in effect, roll the dice to choose which chromosome will be carried into the next step for reproduction. If this is a random event, then how do we nudge the computer's decision? Think of a dartboard that has six slices and the slices are the same size. Now assume you train a monkey to throw darts while blindfolded at the dartboard. Also, it is important to assume he always hits the dartboard. The monkey has a one-in-six chance of hitting a particular slice. What if the slices were not the same size? Then he would have a higher probability of hitting the larger slices. This is the concept that will be used to nudge the computer's selection; the more fit chromosomes will get the bigger slices. Here again, we must let the computer decide the sizes of the slices but we can feed it a formula that will give priority to the more fit chromosomes. The first step is to calculate the fitness of each chromosome in terms of the size of the slice. This is a very simple formula as well:\n\nNotice how we use the **ABS** (absolute value) function in our formula. This function simply removes a negative sign from the difference between a chromosome's fitness and our objective of 40; we don't want to have a negative-sized slice. We subtract 40 from each chromosome's fitness because we need to put the fitness in terms of our objective; the closer we are the smaller the fitness value. Since we want to allocate a bigger slice to chromosomes that are closer to our solution we divide one by our fitness score. In doing so, we are making the size of each slice inversely proportional to the fitness score; a lower fitness value (or closer value to 40) will increase size. The number 1 is also added in the denominator to prevent division by zero; computers hate division by zero. Utilizing the above formula we come up with:\n\n**_Fitness 1 in terms of slice size:_**\n\n  1. Chromosome (0) = 1 \/ (1 + 71) = 0.01389\n  2. Chromosome (1) = 1 \/ (1 + 54) = 0.01812\n  3. Chromosome (2) = 1 \/ (1 + 24) = 0.04000\n  4. Chromosome (3) = 1 \/ (1 + 50) = 0.01961\n  5. Chromosome (4) = 1 \/ (1 +124) = 0.00800\n  6. Chromosome (5) = 1 \/ (1 + 53) = 0.01852\n\n### Converting Fitness in Terms of Slice Size\u2014Using VBA Excel\n\n        'fitness in terms slice size\n        For l = 0 To 5 'calculate for each of the six chromos\n            Fitness(l) = 1 \/ (1 + FObj(l))\n        Next l\n\nNow that fitness has been redefined in terms of the size of the slice on the dartboard, we can now calculate the probability of each slice being hit. First off, let's calculate the total size of all slices:\n\nThe dartboard has a total size of 0.11814 units. Don't worry about the units because they are not important. Now that we know the size of the chromosomes' respective slices and we know the total size of the dartboard, we can easily calculate the probability of a dart hitting the individual slices. This is accomplished by dividing the size of each chromosome slice by the total size. The probability of each chromosome slice being hit with a blindfolded monkey are:\n\n### Calculate Probability of Being Hit by Dart\u2014Using VBA Excel\n\n       'calculate chromo probability of being hit by dart\n        'accumulate probabilities and store in bins\n        cumProb = 0#\n        For l = 0 To 5  'this is a lower case l not a 1\n            chromoProb(l) = Fitness(l) \/ totalFitness\n            cumProb = cumProb + chromoProb(l)\n            chromoProbBin(l) = cumProb\n        Next l\n\nNotice how the chromosome that was closest to our solution has a higher probability of being hit by a dart. Chromosome (2) covers nearly 34 percent of the dartboard.\n\nThe next thing the computer needs to do is construct the dartboard. The pie chart in Figure 8.3 is a graphical representation of how the computer would internally construct a dartboard with varying slice or sector sizes.\n\n**Figure 8.3** Chromosome dartboard.\n\n### Creating Chromosome Dartboard\u2014Using VBA Excel\n\n       For l = 0 To 5\n            chromoProb(l) = Fitness(l) \/ totalFitness\n            cumProb = cumProb + chromoProb(l)\n            'starting at 0 degrees in the pie\/circle\n            'work clockwise defining the slices\n            'until the pie\/circle is coverd by slices\n            chromoProbBin(l) = cumProb\n            'bin size = slize size\n        Next l\n\nOnce the dartboard is constructed, the next step is to start randomly throwing six darts. Since blindfolded monkeys are hard to come by nowadays, the computer can easily replicate one by using our handy RNG. So let's proceed with the simulation and see where the six darts land.\n\n  1. _Dart(0) hits Slice(5)_\n  2. _Dart(1) hits Slice(1)_\n  3. _Dart(2) hits Slice(2)_\n  4. _Dart(3) hits Slice(5)_\n  5. _Dart(4) hits Slice(4)_\n  6. _Dart(5) hits Slice(2)_\n\n### Simulating a Dart-Throwing Blindfolded Monkey Using VBA Excel\n\n    'Generate 6 random numbers between 0 and 1\n    'These numbers represent the locations where\n    'the 6 darts landed\n    Randomize\n    For l = 0 To 5\n        roulette(l) = Rnd(1#)\n    Next l\n    For i = 0 To 5\n        whichBin = -1\n        'which slice did the dart land in?\n        For k = 0 To 4\n            '6 slices but only 5 bin boundaries ?\n             'if dart doesn't hit slices 1-5 then it must have hit slice 0\n             If roulette(i) > chromoProbBin(k) And roulette(i) <=\n                 chromoProbBin (k + 1) Then\n                 whichBin = k + 1\n             End If\n         Next k\n         If whichBin = -1 Then whichBin = 0 'dart hit slice 0\n    Next i\n\nAfter the six darts are thrown, the selection process is completed. Each dart represents a chromosome, and where it lands is the chromosome that will replace it. Here is the end result:\n\n  1. _Chromosome(0) is discarded and is cloned to be Chromosome(5)_\n  2. _Chromosome(1) stays the same_\n  3. _Chromosome(2) stays the same_\n  4. _Chromosome(3) is discarded and is cloned to be Chromosome(5)_\n  5. _Chromosome(4) stays the same_\n  6. _Chromosome(5) is discarded and is cloned to be Chromosome(2)_\n\n### Chromosome Replacement Function\u2014Using VBA Excel\n\n    'Chromosome switching function\n    'Pass both chromosome tables\/matrices into function\n    'inform the function which chromosome is replaced by which   chromosome\n    Public Sub switchChromo(tempChromoTable As Variant,  chromoTable As Variant, destArrNum, srcArrNum)\n    For i = 0 To 3\n        tempChromoTable(destArrNum, i) = chromoTable(srcArrNum, i)\n    Next i\n\nThe new population after selection is:\n\nThe somewhat fit chromosome (3) was discarded and the worst chromosome (4) was carried over. The fifth worst chromosome (0) was discarded and cloned by the third best chromosome (5). Chromosome (3) was cloned with the third best chromosome (5). Chromosome (5) was then discarded and cloned with the best chromosome (2). The formula tried its best to nudge the computer into making the right decision, but due to the random component the worst candidate was kept. The other chromosomes were somewhat upgraded.\n\n## Reproduction\n\nNow that we have our parent pool, we can start the reproduction process. Some of the parents are from the original population and some are clones, and that is okay. The next step is to pick the parents that will produce offspring. The computer will allow you to input the cross rate (partner matching probability), and in this example, 40 percent was used. The computer will randomly generate six numbers between 0 and 100, and every time a number less than or equal to 40 is generated, it designates a crossing of two parents. In this example, the computer generated a random number for each chromosome. Out of the six random numbers, only two were less than or equal to 40. The random numbers were generated sequentially, and random number (2) and random number (4) met our criteria and were less than or equal to 40. With this information, the computer chose chromosome (2) and chromosome (4) to cross\/reproduce.\n\n### Crossing Chromosomes\/Finding Moms and Pops\u2014Using VBA Excel\n\n        'Determine which chromosomes will cross\n        'using a .4 or 40% cross rate\n        crossRate = 0.4\n        For i = 0 To 5\n            crossArr(i) = 0\n            Randomize\n            roulette(i) = Rnd(1#)\n            If roulette(i) <= crossRate Then\n                crossArr(i) = 1\n            End If\n        Next i\n        crossCnt = 0\n        For i = 0 To 5\n            'parentsList is a simple class\n            'similar to a point class with 2 coordinates\n            'total of six possible matchings\n            parentsList(i).mom = 999\n            parentsList(i).dad = 999\n            If crossArr(i) <> 0 Then\n                crossCnt = crossCnt + 1\n                'found mom but not dad - yet\n                'store mom in parentsList(i)\n                parentsList(i).mom = i\n                foundMate = False\n                cindex = i\n                Do While Not (foundMate)\n                    cindex = cindex + 1\n                    If cindex > 5 Then cindex = 0\n                    If crossArr(cindex) <> 0 Then\n                    'found dad - store dad in parentsList(i)\n                        parentsList(i).dad = cindex\n                        foundMate = True\n                    End If\n               Loop\n            End If\n        Next i\n\nIn nature, a child will receive genetic material or traits from both parents. Some children will receive more traits from the father than the mother, and vice versa. How is this determined? In nature, who knows, but inside a computer, we can discriminate how much the father and mother will contribute to the offspring. Here again, an RNG saves the day. Each chromosome holds four different genes (numbers), so we need to know how many of the four numbers will be passed on by the father and mother to the new offspring. This is accomplished by randomly calculating the split point where the numbers will be taken from both parents. In other words, if the split equals two, then the mother chromosome will contribute her first two numbers and the father chromosome will contribute his last two numbers. There are four genes in each chromosome, so the random number generator will generate a number between one and four. Figure 8.4 shows how the sharing of genetic material is accomplished when the split point is one.\n\n**Figure 8.4** At split 1 the genes that will be passed onto the offspring by the father and the mother.\n\nIn this example, chromosome (2) is the mother and chromosome (4) is the father. The crossover point was randomly generated at one, so the offspring received one gene or number from the mother and three from the father.\n\n### How to Allocate Genes to Junior\u2014Using VBA Excel\n\n            'make sure mom and dad are not one in the same\n            'we are not dealing with earthworms - right?\n            If parentsList(i).dad <> parentsList(i).mom Then\n                Randomize\n                'how do we split mom and pop's genes?\n                crossPt = Int((3 - 0 + 1) * Rnd + 0)\n               'mom is replaced by combination of old mom and dad\n                For k = crossPt To 3\n                    chromo(parentsList(i).mom,k) = chromo(parentsList(i).dad,k)\n                Next k\n\nAfter the creation of the offspring, it was then used to replace the old chromosome (2), just like a child taking over his parents' business. This example chose only one crossing but in later generations (after total simulation was completed) more chromosomes were crossed, and they were crossed at different points. The computer is hoping to create a better offspring than the parents but as in real life this doesn't always happen. Just ask your neighbors whose 35-year-old son still lives in their basement.\n\n## Mutation\n\nIn nature, a mutation is change in a genetic sequence. A mutation can be good or bad, but no matter\u2014it does add diversity. In GA, a mutation occurs infrequently but can tremendously help find a solution to a complex problem. Without mutation, selection and reproduction may eventually converge and produce a nondiverse population. Mutation can randomly change a genetic element and then like a chain reaction create different selections and reproductions.\n\nBefore we mutate the initial population, let's review our chromosome population as it stood after our first selection and reproduction phase:\n\n_Chromosome (0)_ | = | _[16,05,13,28]_  \n---|---|---  \n_Chromosome(1)_ | = | _[13,08,15,20]_  \n_Chromosome(2)_ | = | _[00,20,05,09]_  \n_Chromosome(3)_ | = | _[16,05,13,28]_  \n_Chromosome(4)_ | = | _[10,30,30,04]_  \n_Chromosome(5)_ | = | _[00,20,05,09]_\n\nThe rate of mutation should be low because you do want a convergence to occur. In other words, you want a solution, and a high mutation rate could potentially spin the population off into too many different directions. But at the same time, you don't want your population growing stale, either, and a sufficiently high mutation rate prevents this. In this example, a mutation rate of 10 percent was used. With six chromosomes, there are 24 genes or numbers total, so if the mutation rate of 10 percent is applied, then 2.4 genes will be mutated. Since we can't have a partial gene, the number is rounded down to two. So out of the 24 numbers above, 2 of them will be randomly selected and replaced with two random numbers between 0 and 30 (the same range that we initially used to seed the population).\n\nEach gene is assigned a number between 1 and 24 in order. So chromosome (0) will have genes labeled 1 through 4. Chromosome (1) will have genes 5 through 8, and so on. Relying on the random number generator yet again (range between 1 and 24), it generates two numbers: 7 and 24. Don't put the random number generator away yet; we still need to randomize two more numbers between 0 and 30 to use in our mutation. After pushing the button on the RNG, two numbers, 22 and 15, were generated. If we map 7 and 24 across the matrix of chromosomes, then the following highlighted genes will be mutated with the numbers 22 and 15:\n\nChromosome (0): | 161 | 052 | 133 | 284  \n---|---|---|---|---  \nChromosome (1): | 135 | 086 | __**22**_ 7_ | 208  \nChromosome (2): | 009 | 3010 | 3011 | 0412  \nChromosome (3): | 1613 | 0514 | 1315 | 2816  \nChromosome (4): | 1017 | 3018 | 3019 | 0420  \nChromosome (5): | 0021 | 2022 | 0523 | __**15**_ 24_\n\n### Mutating Chromosomes\u2014Using VBA Excel\n\n        mutRate = 0.1 ' rate of mutation\n        numMutations = Int(mutRate * 24) ' 24 genes available for mutation\n        Call printDivider(lineCount, 4)\n        lineCount = lineCount + 1\n        Randomize\n        For i = 0 To numMutations - 1 'loop for 2 mutations\n            chromoMut(i) = Int((24 - 0 + 1) * Rnd + 0) 'which genes to mutate\n            genLocRow = Int(chromoMut(i) \/ 4) 'locate the gene in the matrix\n            If genLocRow = 0 Then genLocRow = 1\n            genLocCol = 4 - chromoMut(i) Mod 4\n            chromoMutCoOrds(i, 0) = genLocRow - 1 'found gene row\n            chromoMutCoOrds(i, 1) = genLocCol + 1 'found gene column\n            Randomize\n            rndVal = Int((24 - 0 + 1)*Rnd + 0) 'randomize the new gene value\n            chromo(genLocRow - 1, genLocCol - 1) = rndVal\n        Next i\n\nAfter selection, reproduction, and mutation a new generation has been created. Here are the second generation's chromosomes and respective fitness values after subtracting 40:\n\n_Chromosome (0)_ | = | _[16,05,13,28]_ | = | _53_  \n---|---|---|---|---  \n_Chromosome (1)_ | = | _[13,08,22,20]_ | = | _75_  \n_Chromosome (2)_ | = | _[00,30,30,04]_ | = | _114_  \n_Chromosome (3)_ | = | _[16,05,13,28]_ | = | _53_  \n_Chromosome (4)_ | = | _[10,30,30,04]_ | = | _124_  \n_Chromosome (5)_ | = | _[00,20,05,15]_ | = | _30_\n\nIt seems redundant to state this again, but here is another example of how genetic algorithms mimic nature by not only creating inferior offspring but also creating an inferior generation. However, since these generations live inside of a computer, they are short-lived and the computer moves quickly onto the next generation until a solution is found. This GA solved the problem in 13 generations after a mutation occurred. Eventually, chromosome (2) with genes [13,0,4,15] was chosen as the solution. Test it!\n\nThis is just one possible solution. You can keep propagating and create a whole slew of solutions, which is what we are after when we use genetic optimization on trading algorithms.\n\nGo through the steps until you understand the processes involved. Each step is logical and nonmath intensive and should give you confidence to utilize this technology in your own research. Even if you don't utilize it in research you can impress your friends at your next cocktail party by explaining what a genetic algorithm is and the impact of a higher mutation rate on subsequent generations.\n\n## Using Genetic Algorithms in Trading System Development\n\n> _\"The same principles which at first view lead to skepticism, pursued to a certain point, bring men back to common sense.\"_\n\nGeorge Berkely\n\nNow armed with a little knowledge the veil of skepticism can be pulled back. This knowledge can now be applied to a real-life trading problem. Earlier in this chapter, a trading algorithm was mentioned that consisted of five optimizable parameters. The system (TA1) utilizes an indicator to determine the market condition so it can either apply a trend-following or choppy-market algorithm. The switch that determines condition measures the actual distance a market travels versus the entire distance a market travels over a certain time period. Imagine a subdivision with a lot of cul-de-sacs. If you leave your house and go in and out of each cul-de-sac prior to exiting the subdivision, you will have traveled a long route to go a very short distance. This indicator is based on momentum and has been around for years. It is known as the Choppy Market Indicator (CMI). Here is the formula to the indicator:\n\nIf the difference between the today's close and the close xBars back is small and the highest high and lowest low xBars back is large, then the CMI will be a small value (Numerator small \/ Denominator large). A low CMI indicates the market has traveled a relatively large distance but hasn't gone anywhere. In other words, the market is chopping around with no evidence of a trend.\n\nThere are two ways we can approach a choppy market condition: We can (1) avoid it; or (2) implement a swing trading algorithm. TA1 attempts the latter by using volatility breakout from the prior day's close as an entry point. If the CMI is less than xLevel, then long entries are placed at today's close plus xPercent times xAR (Exponential Average Range) of the past 20 days. The sellshort signal uses the same algorithm, except instead of adding the volatility level, it subtracts it. Here are the choppy market long and short entries:\n\nNow if the CMI indicates a trending market by exhibiting a higher value, then the system switches to the trend-following mechanism. Here, the system falls back on the old Donchian breakout algorithm. Longs are put on at the Highest High of xTrendBars and shorts are put on at the Lowest Low of xTrendBars. The entry rules for trending markets are:\n\nLooking at our entry rules, we can see there are three optimizable parameters: xLevel, xPercent, and xTrendBars. The last two parameters can be further optimized for long entries and short entries. If we allow these two parameters to be different for longs versus shorts, then instead of two parameters we have four. So all in all, there will be a total of five optimizable parameters:\n\n  1. xLevel\u2014Choppy Market value\n  2. xPercentBuys\u2014Percent of 20-day exponential moving average of Range\n  3. xPercentSells\u2014Percent of 20-day exponential moving average of Range\n  4. xTrendBarsBuys\u2014Number of Highest high\/Lowest low lookback bars\n  5. xTrendBarsSells\u2014Number of Highest high\/Lowest low lookback bars\n\nWe could simply pull out of thin air some values for these parameters or we could use TradeStation's tools to help zero in on a robust solution. We now have a trading system and a set of five optimizable parameters, so the next step is to come up with a range of values that we want to optimize the parameters across. These different ranges can also be called our search space. Here are the ranges and iterations for each of the parameters for our system, TA1:\n\n1. | xLevel | { 25 to 75 by 5 increments } [11 iterations]  \n---|---|---  \n2. | xPercentBuys | { 0.25 to 0.75 by 0.05 increments } [11 iterations]  \n3. | xPercentSells | { 0.25 to 0.75 by 0.05 increments } [11 iterations]  \n4. | xTrendBarsBuy | { 8 to 20 by 4 } [4 iterations]  \n5. | xTrendBarsSell | { 8 to 20 by 4 } [4 iterations]\n\nIf we do the math (11 * 11 * 11 * 4 * 4), we come up with a total of 21,296 iterations. It will take this many iterations to search the entire space if we apply an exhaustive search algorithm. So the computer will need to run 21,296 tests on the history of each market. This test was performed on 10 years of Euro currency data and it took my \"typical\" computer 13 minutes. The search space could have been expanded but for brevity sake it was limited to 21,296. If just one increment was added to each of the parameters' range, the total number of iterations would have nearly doubled. The results of the exhaustive search are shown in Table 8.1. ( _Note_ : Results are shown without a commission\/slippage charge.)\n\n**Table 8.1** Results from the Exhaustive Search Algorithm's Five Parameters\n\n**xLevel** | **xPercent Buy** | **xPercent Sell** | **xTrendBars Buy** | **xTrendBars Sell** | **Total Profit** | **Max Intra. Drawdown**  \n---|---|---|---|---|---|---  \n25 | 0.3  | 0.75 | 8 | 20 | 103615 | \u221215235  \n25 | 0.4 | 0.75 | 8 | 20 | 102435 | \u221216080  \n25 | 0.4 | 0.7  | 8 | 20 | 101005 | \u221215760  \n25 | 0.7 | 0.75 | 8 | 20 | 96050 | \u221217805  \n25 | 0.75 | 0.7  | 8 | 20 | 96680 | \u221218343  \n25 | 0.45 | 0.75 | 8 | 20 | 97435 | \u221217255  \n25 | 0.75 | 0.75 | 8 | 20 | 94020 | \u221218480  \n25 | 0.7 | 0.7  | 8 | 20 | 96015 | \u22127668  \n25 | 0.5 | 0.75 | 8 | 20 | 93805 | \u221216660  \n25 | 0.4 | 0.65 | 8 | 20 | 98763 | \u221215485\n\nNow let's do the same thing, but instead of using the Exhaustive method let's go with the genetic option. In less than one minute the results in Table 8.2 were computed.\n\n**Table 8.2** Results from the Genetic Search Algorithm's Five Parameters\n\n**xLevel** | **xPercent Buy** | **xPercent  \nSell** | **xTrendBars  \nBuy** | **xTrendBars  \nSell** | **Total Profit** | **Max Intra.  \nDrawdown**  \n---|---|---|---|---|---|---  \n25 | 0.3  | 0.75 | 8 | 20 | 103615 | \u221215235  \n25 | 0.4 | 0.75 | 8 | 20 | 102435 | \u221216080  \n25 | 0.4 | 0.7  | 8 | 20 | 101005 | \u221215760  \n25 | 0.7 | 0.75 | 8 | 20 | 96050 | \u221217805  \n25 | 0.45 | 0.75 | 8 | 20 | 97435 | \u221217255  \n25 | 0.5 | 0.75 | 8 | 20 | 93805 | \u221216660  \n25 | 0.55 | 0.75 | 8 | 20 | 91650 | \u221215063  \n25 | 0.45 | 0.7  | 8 | 20 | 95655 | \u221216110  \n25 | 0.35 | 0.75 | 8 | 16 | 91575 | \u221218843  \n25 | 0.6 | 0.75 | 8 | 20 | 86850 | \u221215913\n\nThe results from the two tests are quite similar; results only start deviating after the top six parameter sets. This example is not perfect for demonstrating the power of GA, as the brute force approach only took 13 minutes, but the implications of its power is more than evident. In many cases, a brute force optimization scheme will be too time intensive, and the only alternative is a GA.\n\nTradeStation makes using the Genetic form of optimization very simple. All it requires is selecting it over the Exhaustive mode. This simple switch makes implementation simple, but you do have the ability to override the Genetic optimizer's default settings. If you select Genetic as the optimization method and then click Advanced Settings, a dialog box will open that will allow you to override the default settings. Before changing these, you want to make sure you fully understand what you are doing. Here is explanation of each of the Genetic Optimization Settings.\n\n### Generations\n\nThis setting indicates the total number of generations each test will iterate to come up with a solution chromosome. In our very first example, the number of generations concluded after a single chromosome fulfilled the requirements to solve the simple equation. The computer program and process could continue until multiple solutions were uncovered. Trading algorithms don't have a black-or-white solution so you can specify to only go so far in the process. This number should increase with larger search spaces. This setting should be \"Suggested\" by clicking the **Suggest** button.\n\n### Mutation Rate\n\nRemember how a gene in our example chromosomes could be randomly changed? This rate defaults to 0.1 or 10 percent, and this is probably the best value for this setting. Higher mutation rates increase the probability of searching more areas in the search space and may prevent the population from converging on an optimal solution. Searching a larger space takes more time and a mutation could randomly change the gene of a good solution after selection and reproduction and turn it into a nonoptimal solution, thus preventing convergence.\n\n### Population Size\n\nReferring to our initial example of a simple GA, we used six different chromosomes to solve our problem. So its population size was six. Population size grows with the size of the search space. A small population on a large search space could lead to less robust solutions. Initial population sizing is very important and is very difficult to gauge. Again, let the **Suggest** button help here.\n\n### Crossover Rate\n\nThe rate that determines how many parents will mate. If the default value is set to 0.4, or 40 percent, then each chromosome has a probability of 2 out of 5 of being mated with another chromosome and being replaced. If this rate is set to 1.0, or 100 percent, then a new generation is always created\u2014no survivors are allowed. As in life, sometimes it's good to have a member or two from the old generation to help guide us. The **Suggest** button usually changes this value to 0.9, or 90 percent probability of chromosome matching.\n\n## Preventing Over-Curve-Fitting\n\nIf optimization is a form of curve fitting, then why do it at all? Curve fitting is a necessary tool in the development of trading algorithms. If a trading system is based on a sound market principle, then modifying it by adjusting its parameters to increase productivity is a good thing. However, this can be overdone. If you \"tweak\" an algorithm's parameters to catch a large portion of every substantial market move and create an absolutely beautiful equity curve, then you have gone too far. The system might work well in the near future due to curve fit overhang, but the likelihood of it continuing is less probable. The future is unknown but a robust trading algorithm that does not have overly curved fit parameters has a higher probability of success. Take a look at the 3D contour chart in Figure 8.5. The _y_ -axis is profit, _x_ -axis is one parameter, and the _z_ -axis is another parameter. The curve is the different profit levels across the different values of the two parameters. If you are looking to maximize profit, then the best parameter set, historically speaking, would be at the peak (0.75, 0.35). If you are looking for robustness, then you don't want the peak; you want a parameter set located on a high plateau. Robustness is the same as parameter stability. If a parameter is stable, then changing it up or down will not have a huge impact on overall performance. If it does, then the parameter is not considered stable. TradeStation includes, with its genetic optimization method, a parameter stress test. This test includes two inputs, number of stress tests and percentage of parameter change.\n\n**Figure 8.5** Robust parameter sets are found on a level plateau.\n\nIf three is chosen as the number of stress tests and 10 percent as the stress increment, then the total number of tests will triple. For example, 7,250 iterations were generated by a simple algorithm optimized using the genetic method and a stress test of one. When the algorithm was reoptimized with a stress test number set to three and the stress increment to 10 percent, the number of increments increased to 21,750. What the genetic optimizer did was test each parameter set and then increased\/decreased each parameter by 10 percent and then retested, thus creating three different sets of tests. A simple standard deviation algorithm example should help illustrate what the computer is doing:\n\n**Test 1: Normal**\n\n  1. Moving Average Length = 50\n  2. Up Band = 2\n  3. Dn Band = 3\n\n**Test 2: Parameters are stressed upwardly by 10%**\n\n  1. Moving Average Length = 50 + (10% of 50) = 55\n  2. Up Band = 2 + (10% of 2) = 2.2\n  3. Dn Band = 3 + (10% of 3) = 3.3\n\n**Test 3: Parameters are stressed downwardly by 10%**\n\n  1. Moving Average Length = 50 \u2013 (10% of 50) = 45\n  2. Up Band = 2 + (10% of 2) = 1.8\n  3. Dn Band = 3 + (10% of 3) = 2.7\n\nStress testing eliminates parameter sets that fall on top of peaks, therefore theoretically revealing a more robust set. If a certain parameter is profitable, and its neighbors are nearly as profitable, then the parameter is deemed robust.\n\nTradeStation's and AmiBroker's genetic optimization tool is a very important tool in the development of robust trading algorithms. Hopefully, this first part has demystified genetic algorithms and provided sufficient evidence that it can help a trader develop a better mousetrap\u2014one that catches the mouse more often than not.\n\n## Walk-Forward Optimizer: Is It Worth the Extra Work and Time?\n\nThe idea of periodic reoptimization of a trading system's parameters is the one concept that divides the algorithmic trading community more so than any other. You are either pro or con; there is very little middle ground. Walk-forward optimization (WFO) has been around since the beginning of trading algorithms but until lately it has been extremely difficult to implement. TradeStation has done a wonderful job in providing the TS walk-forward optimizer. It does all the work for you.\n\nBefore describing the potential benefits of WFO, the concept and process needs to be explained. In a nutshell, it is a technique in which a system's parameters are optimized on a segment of historical data (in-sample), then tested and verified by using those parameters on a walk-forward basis (out-of-sample). Results from in-sample testing will usually be very good due to the benefit of hindsight. It's the results from testing on unseen data that provides a true reflection of a system's validity. This process is done periodically in hopes the trading algorithm will be using the most optimal parameters available at any given time.\n\n### WFO Example 1\n\nAnother example will definitely help explain the concept. Take a Turtle-like system that buys at the highest high of the past 40 days and sells at the lowest low of the past 40 days. Throughout history, there have been times when a longer (greater than 40-day) breakout or a shorter (less than 40-day) worked better. The magic questions are, of course, when to change the length of the breakouts, and by how much? These questions are easily answered with the use of a time machine.\n\nAssume a trader trades this Turtle-like system for two years and it is somewhat profitable. Being an inquisitive trader he asks, \"Was a 40-day break the best option?\" An optimization run is set up and the 40-day parameter is optimized over the past two years of data, and lo and behold, the trader finds out that a 30-day breakout produced twice as much profit with much less drawdown. With this knowledge, he changes the trading algorithm to utilize a 30-day breakout in place of the 40-day. Another two profitable years go by, and he again asks the same question. This time, the best parameter turned out be the exact one he had implemented; 30 turned out to be the optimal parameter. He then carries the parameter forward another two years, but this time he suffers a large loss and a severe drawdown. This trade, optimize, implement, and trade cycle is easy to see. Testing using this process on a long stream of historical data would be very difficult if tools such as the TS WFO were not available. A test would have to be started, stopped, results recorded, backed up, and optimized on in-sample data, started, stopped, results recorded, backed up, and optimized on in-sample data, started, stopped....You get the picture.\n\nBefore proceeding, please refer to the TradeStation WFO **Help** menu to learn how to set up an optimization and then load that information into the software. The TS WFO **Help** system will also show how to perform an individual walk-forward analysis (WFA), or a cluster WFA.\n\nHere is an individual WFA of a Turtle-like system selectively changing its parameters every two years. The first results shown are with the benefit of hindsight. The best parameters for each time period were utilized to construct the continuous track record. All the reports shown were produced using the TS WFO. The system labeled TA2 will enter and exit the market utilizing the following criteria:\n\n    \/\/EasyLanguage Code for Turtle-like Algo\n    Buy Next Bar at Highest (H[1],xLongEntryLen) stop;\n    Sell Short Next Bar at Lowest (L[1],xShortEntryLen) stop;\n    If MarketPosition = 1 Then\n    Sell Next Bar at Lowest (L[1],xLongExitLen) stop;\n    If MarketPosition =-1 Then\n    BuyToCover Next Bar at Highest(H[1],xShortExitLen) stop;\n\n#### In-Sample Testing\n\nYou will notice four optimizable parameters:\n\n  1. xLongEntryLen\n  2. xLongExitLen\n  3. xShortEntryLen\n  4. xShortExitLen\n\nSince the long entry parameter may not be the same as the short this system is not symmetric. This also applies to the long and short exits. Different parameters for long and short entries were used to increase the search space sufficiently to enable a genetic optimization. You will notice if the exhaustive search space is less than 10,000, then the genetic method is disabled. Table 8.3 shows the in-sample testing results of TA2 over the past 14 years on 30-year bonds.\n\n**Table 8.3** In-Sample Testing TA2\n\n**Le Se Lx Sx** | **Begin Date End Date** | **Bars** | **Profit** | **Max DD**  \n---|---|---|---|---  \n20 20 12 10 | 2001\/09\/18\u20132005\/06\/15 | 975 | 24,843.75 | \u22129,187.50  \n20 20 12 10 | 2002\/10\/11\u20132006\/07\/07 | 975 | 28,375.00 | \u22129,187.50  \n20 20 16 10 | 2003\/09\/05\u20132007\/06\/27 | 975 | 23,687.50 | \u22128,093.75  \n20 24 12 12 | 2004\/10\/04\u20132008\/06\/26 | 975 | 6,437.50 | \u22128,343.75  \n20 20 12 14 | 2005\/08\/16\u20132009\/04\/07 | 975 | 18,531.25 | \u221211,906.25  \n20 36 10 14 | 2006\/08\/15\u20132010\/06\/04 | 975 | 11,843.75 | \u221213,843.75  \n20 20 12 14 | 2007\/09\/20\u20132011\/04\/18 | 975 | 21,531.25 | \u221213,187.50  \n20 24 12 14 | 2008\/07\/17\u20132012\/04\/06 | 975 | 22,375.00 | \u221217,656.25  \n20 56 10 14 | 2009\/06\/23\u20132013\/03\/27 | 975 | 8,250.00 | \u221215,375.00  \n24 24 12 14 | 2010\/09\/03\u20132014\/04\/04 | 975 | _18,187.50_ | \u221217,656.25  \n|  |  | 184,062.50 |\n\nTable 8.3 shows the best parameter sets over a four-year period carried one year forward sequentially. Row 1 starts with September 2001 and ends on June 15, 2005. The best-performing parameters were selected over that time period and displayed. Row 2 starts a year later in October 2002 and ends in July 2006, and the best parameters are also shown. The process is continued through April 2014. The results are spectacular, but beware of the benefit of hindsight. Notice how the **Le** , **Lx** , **and Sx** parameters stayed somewhat consistent; this does demonstrate parameter stability. The **Se** parameter was all over the map, and you can see that the system did not want to sell short between June 2009 and March 2013; the sell entry utilized a whopping 56 days in its calculation. Figure 8.6 is a graphic representation of how the analysis rolls forward.\n\n  **Figure 8.6** A graphic representation of how the walk-forward analysis (WFA) rolls forward.\n\n#### Out-Of-Sample Testing\n\nWhen you run a WFO, the software builds a database of these \"best\" parameter sets and then informs the algorithm when to adopt and when to abandon them. TradeStation's WFO keeps track of all parameter switching and performance metrics on the trading algorithm as it progresses through the history of whatever you are testing.\n\nThe whole purpose of this software and this type of testing is based on the belief of optimization overhang or carryover; if a parameter has done well for the past four years, it should overhang or carry over into the subsequent year. A practitioner of this school of thought also believes the market is changing and a trading algorithm must change or adapt as well. Table 8.4 shows the WFO of this system on the 30-year bond going back to 2001. Notice how the results start in 2005. The system backed up to 2001 and derived the best parameter set for those four years and then traded that set starting in June 2005 and ending in July 2006.\n\n**Table 8.4** Out-of-Sample Testing\n\n**Le Se Lx Sx** | **Begin Date End Date** | **Bars** | **Profit** | **Max DD**  \n---|---|---|---|---  \n20 20 12 10 | 2005\/06\/15\u20132006\/07\/07 | 244 | 4,031.25 | \u22122,562.50  \n20 20 12 10 | 2006\/07\/07\u20132007\/06\/27 | 244 | 4,437.50 | \u22123,093.75  \n20 20 16 10 | 2007\/06\/27\u20132008\/06\/25 | 244 | \u22123,437.50 | \u221210,343.75  \n20 24 12 12 | 2008\/06\/26\u20132009\/04\/07 | 244 | 8,968.75 | \u22127,343.75  \n20 20 12 14 | 2009\/04\/07\u20132010\/04\/16 | 244 | \u22126,437.50 | \u221213,187.50  \n20 36 10 14 | 2010\/06\/04\u20132011\/03\/29 | 244 | 8,250.00 | \u22122,093.75  \n20 20 12 14 | 2011\/04\/18\u20132012\/04\/06 | 244 | \u22122,500.00 | \u221217,562.50  \n20 24 12 14 | 2012\/04\/06\u20132013\/03\/27 | 244 | \u22121,937.50 | \u221211,562.50  \n20 56 10 14 | 2013\/03\/27\u20132014\/04\/04 | 244 | \u22124,062.50 | \u221213,187.50  \n24 24 12 14 | 2014\/04\/04\u20132015\/03\/18 | 244 | _1,718.75_ | \u22125,156.25  \n|  |  | 9,031.25 |\n\nWhen July 2006 arrived, the system then switched to the second parameter set. These parameters were derived using data from 2002 through 2005 (refer back to table 3\u2014second row). The WFO swapped parameters on an annual basis. Overall, the system was profitable, but the walk-forward efficiency was only 19.03 percent, and it lost 5 out of 10 years. Why did this happen? The only answer is the lack of optimization overhang in the out-of-sample periods.\n\n**System TA2 used the default WFO settings:**\n\n  * Walk-forward runs = 10; in this test, the computer switched parameters 10 times over the history of the test. This works out to be about once a year. If the test period had been 20 years, then the WFO would have switched every 2 years.\n  * OOS% = 20%; this percentage informs the computer to utilize 80 percent of each interval's data (in-sample) to determine the best parameter set and then carry that parameter set forward 20 percent of the interval (out-of-sample).\n\nIf these settings are changed, then the number of bars used for optimization and the number of bars those optimizations are carried forward will change as well. Here are the various calculations that are used to determine the total number of bars per run, in-sample number of bars, and out-of-sample number of bars.\n\n  1. **TotalBarsPerRun** \u2014Total number of bars \/ (Walk-forward runs * Out-of-sample % + In-sample %). In our example test, there were a total of 3415 bars, 10 walk-forward runs, 20 OOS%, and 80 percent in-sample. WFO uses bars in place of days to calculate the walk-forward window lengths. \n    1. TotalBarsPerRun = 3415 \/ (10 * 0.2 + 0.8) = 1219\n  2. **InSampleBarsPerRun** \u2013 TotalBarsPerRun * In-sample% \n    1. InSampleBarsPerRun = 1219 * 0.8 = 975\n  3. **OutOfSampleBarsPerRun** \u2013 TotalBarsPerRun * Out-of-sample% \n    1. OutOfSampleBarsPerRun = 3414 * 0.2 = 244\n\nThe first interval analyzes the first 975 bars of data and determines the best parameter set and then carries that set on unseen data for 244 bars. The second test starts on bar 976 and ends on 1950 (976 + 974) and then walks forward 244 more days. Figure 8.7 illustrates this process by showing how four years are used to decide the best parameters using hindsight and then how those parameters are applied to unseen data the following year.\n\n  **Figure 8.7** How four years of hindsight are used to predict unseen data for the following year.\n\nThe WFO produces a WFA report that determines the usefulness of applying this type of optimization to this particular trading algorithm. Before showing the contents of this report let's create a benchmark parameter set and compare those results with those produced by the WFO. A benchmark can be created by using a fixed parameter set throughout the entire test time period. This is easily accomplished by setting the four parameters to the following values:\n\nIn addition to making the parameters static notice that the system was also made symmetrical; the buy-side parameters are the same as the sell-side parameters. This particular parameter set did perform better over the walk-forward optimization as it produced almost $17,000, whereas the WFO produced nearly $10,000 in profit. The WFO process did not outperform the static parameters on this particular algorithm. The WFA report is shown in Table 8.5.\n\n**Table 8.5** WFA Report on TA2\n\n| **Test Criteria** | **Result** | **Comment**  \n---|---|---|---  \n1 | Overall Profitability | Pass | Total Profit > 0. System likely to perform profitably on unseen data.  \n2 | Walk-Forward Efficiency | Failed | < 50%. System likely to perform in future at a rate of less than 50% of those achieved during optimization.  \n3 | Consistency of Profits | Pass | 50% of walk-forward runs were profitable.  \n4 | Distribution of Profits | Failed | Run #6 contributed more than 50% of Total Net Profit.  \n5 | Maximum Drawdown | Pass | No individual run had a drawdown of more than 40% of init. capital.  \n| **Overall Result** | **FAILED** |\n\nThe WFA agrees with our benchmark analysis that a WFO does not complement this type of trading algorithm. Of the five test criteria, this system failed two. The most important failed test was the walk-forward efficiency (WFE). This value compares the in-sample average profit versus the out of-sample average profit. This algorithm failed with a 19.03 percent efficiency, meaning that the average OOS profit was one fifth the size of the in-sample profit. Now, does this mean TA2 is not a good trading algorithm? The robustness of the system logic and results of the static parameter analysis answers this question with an emphatic \"No!\" Just because a WFO didn't work well with this particular algorithm in the 30-year bonds doesn't mean it is not a good system. Other markets need to be tested and evaluated, as well as different WFO settings.\n\n### WFO Example 2\n\nAnother example of a WFO on a different type of algorithm may be a beneficial demonstration. This time, the WFO will be applied to a short-term ES (emini SP) system that produces more trades than TA2. This system, TA3, will trade in the direction of the long-term trend and buy pullbacks and sell rallies. The trend will be defined by a long-term moving average and pivot points will determine pullbacks and rallies; pivots involving the highs will determine rallies and pivots using the lows will determine pullbacks. Different protective stops and profit objectives will be used for long and short positions. The strength of the pivot points and the number of days that will be used to determine the presence of the pivot points may also be different for long and short entries. Two additional parameters will be used to limit the number of days the system is in the market. That makes a total of 11 optimizable parameters:\n\n1. | movAvgLen | { 50 to 150 by 10 increments } [11 iterations]  \n---|---|---  \n2. | pivotHiLookBack | {3 to 5 by 1 increment} [3 iterations]  \n3. | pivotHiStrength | {1 to 3 by 1 increment} [3 iterations]  \n4. | pivotLowLookBack | {3 to 5 by 1 increment} [3 iterations]  \n5. | pivotLowStrength | {1 to 3 by 1 increment} [3 iterations]  \n6. | LprofitObjective | {500 to 1500 by 50 increment} [21 iterations]  \n7. | LstopLoss | {250 to 750 by 50 increment} [11 iterations]  \n8. | SprofitObjective | {500 to 1500 by 50 increment} [21 iterations]  \n9. | SstopLoss | {250 to 750 by 50 increment} [11 iterations]  \n10. | longExitDays | {5 to 15 by 1} [11 iterations]  \n11. | shortExitDays | {5 to 15 by 1} [11 iterations]\n\nThat is nearly 6 billion iterations. Fortunately, the genetic method of optimization is available.\n\nHere is the code for entries and exits:\n\n    Value1 = SwingHiBar(1,H,pivotHiStrength,pivotHiLookBack);\n    Value2 = SwingLowBar(1,L,pivotLowStrength,pivotLowLookBack);\n    If C > Average(C,avgLen) then\n    Begin\n    If Value1 <> -1 then Buy this Bar on Close;\n    End;\n    If C < Average(C,avgLen) then\n    Begin\n    If Value2 <> -1 then SellShort this Bar on Close;\n    End;\n    If MarketPosition = 1 Then\n    Begin\n    Sell Next Bar at EntryPrice + LprofitObjective\/bigPointValue limit;\n    Sell Next Bar at EntryPrice - longStopLoss\/bigPointValue stop;\n    End;\n    If MarketPosition = -1 Then\n    Begin\n    BuyToCover Next Bar at EntryPrice - SprofitObjective\/  bigPointValue limit;\n    BuyToCover Next Bar at EntryPrice + shortStopLoss\/  bigPointValue stop;\n    End;\n\n#### WFO Cluster Analysis\n\nThe optimization using the genetic method took only a few moments. This time, instead of running a single WFO, the cluster WFO option was chosen. In this example, the cluster analysis ran 30 different walk-forward optimizations. Each optimization changed the values for OOS% and walk-forward runs. In the first single WFA, 10 runs utilizing 20 percent OOS% were evaluated to determine WFO fitness. Figure 8.8 shows the different values of WF runs and OOS% levels in a matrix format.\n\n**Figure 8.8** The Cluster WFO results matrix.\n\nSimilar to the single WFA on the Turtle-like system, this system failed as well, using the 10 WFA runs with 20 percent as the OOS%. However, this system passed the test on several different OOS% \/ Runs values. Please note that in an attempt to increase the likelihood of a passing grade, the criterion for the walk-forward efficiency was lowered from 50 to 30 percent in the test criteria setup. This criterion seems to be the one that rejects a larger portion of optimizations. To expect a 50 percent WFE is perhaps setting the bar a bit too high. The TS WFO selected a WFO using 15 percent for OOS% and 25 runs. It is the dark gray selection from Figure 8.8. This value is the center cell where a majority of its neighboring cells showed promise. This translates to using only 15 percent out-of-sample data and changing the parameters 25 times during the test period. The optimization produced over $52K in profits with a WFE of 49.9 percent. Table 8.6 shows the OOS report.\n\n**Table 8.6** The Out-of-Sample (OOS) Report\n\n**p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11** | **Begin Date End Date** | **Bars** | **Profit** | **Max DD**  \n---|---|---|---|---  \n90 5 2 5 2 900 650 13 600 400 9 | 2007\/08\/16\u2013 2007\/12\/11 | 117 | \u22125,512. | \u22127,037  \n100 5 1 5 2 1450 350 14 1050 750 12 | 2007\/12\/11\u20132008\/04\/01 | 112 | 8,325. | \u22124,862  \n70 4 1 4 2 900 550 13 1150 750 8 | 2008\/04\/01\u20132008\/07\/16 | 106 | 4,862. | \u22122,962  \n100 5 1 5 2 1450 350 14 1050 750 12 | 2008\/07\/16\u20132008\/11\/10 | 117 | 21,150 | \u22124,425  \n90 5 1 5 3 1500 350 13 1050 750 5 | 2008\/11\/10\u20132009\/03\/03 | 113 | 10,475 | \u22125,150  \n90 5 1 5 3 1500 350 13 1050 750 5 | 2009\/03\/03\u20132009\/05\/29 | 87 | \u22127,362 | \u221210,400  \n90 5 1 5 3 1500 350 13 1050 750 5 | 2009\/05\/29\u20132009\/10\/02 | 126 | 2,025 | \u22121,925  \n90 5 1 5 1 1400 300 13 1050 750 11 | 2009\/10\/08\u20132010\/02\/02 | 117 | \u22122,675 | \u22124,025  \n90 5 1 5 1 1450 350 8 1050 750 8 | 2010\/02\/02\u20132010\/05\/21 | 108 | 2,050 | \u22123,425  \n90 5 1 5 1 1400 650 9 1100 650 11 | 2010\/05\/21\u20132010\/09\/03 | 105 | \u22122,200 | \u22125,862  \n90 5 1 5 1 1450 350 8 1100 750 13 | 2010\/09\/08\u20132010\/12\/27 | 110 | 4,687 | \u22122,612  \n70 5 1 5 1 1400 450 8 1500 700 8 | 2010\/12\/30\u20132011\/04\/19 | 110 | \u22122,137 | \u22124,775  \n70 5 1 5 1 1450 350 8 1200 700 9 | 2011\/04\/19\u20132011\/08\/08 | 111 | \u22121,475 | \u22126,875.  \n100 5 1 3 1 1400 500 11 1100 700 6 | 2011\/08\/08\u20132011\/11\/23 | 107 | 662 | \u22124,937  \n100 5 1 3 1 1200 350 9 1200 700 13 | 2011\/11\/23\u20132012\/03\/22 | 120 | 3,500 | \u22121,675  \n100 3 1 5 1 1450 350 9 1050 750 15 | 2012\/03\/22\u20132012\/07\/06 | 106 | \u22121,937 | \u22123,200  \n100 5 1 3 1 1200 350 9 1200 700 13 | 2012\/07\/12\u20132012\/10\/17 | 97 | 1,512 | \u22123,337  \n90 4 1 3 1 1200 350 6 1200 700 9 | 2012\/10\/17\u20132013\/02\/19 | 125 | \u22121,070 | \u22124,932  \n80 4 1 3 2 1050 550 14 800 300 7 | 2013\/02\/21\u20132013\/06\/12 | 111 | 3,800 | \u22121,650  \n90 4 1 4 1 1500 650 6 600 700 9 | 2013\/06\/12\u20132013\/09\/30 | 111 | \u2212750 | \u22125,300  \n90 4 1 4 1 1500 650 6 600 700 9 | 2013\/09\/30\u20132014\/01\/21 | 113 | 0 | \u22121,662  \n100 5 3 3 1 1200 700 13 1100 700 6 | 2014\/01\/21\u20132014\/05\/13 | 112 | \u22122,125 | \u22124,950  \n150 4 1 5 1 1150 350 13 950 500 14 | 2014\/05\/13\u20132014\/08\/28 | 108 | 3,212 | \u22123,600  \n150 4 1 5 1 1150 350 13 950 500 14 | 2014\/08\/28\u20132014\/12\/18 | 112 | 2,112 | \u22124,737  \n150 4 1 5 1 1150 350 13 950 500 14 | 2014\/12\/18\u20132015\/04\/07 | 110 | 11,075. | \u22124,087\n\nTable 8.7 provides the report card on this system using the optimal optimization values.\n\n**Table 8.7** WFO Report Card TA3\n\n| **Test Criteria** | **Result** | **Comment**  \n---|---|---|---  \n1 | Overall Profitability | Pass | Total Profit > 0. System likely to perform profitably on unseen data.  \n2 | Walk-Forward Efficiency | Pass | >= 30%. System likely to perform in future at a rate between 30\u2013100% of those achieved during optimization.  \n3 | Consistency of Profits | Pass | 50%+ of walk-forward runs were profitable.  \n4 | Distribution of Profits | Pass | No individual time period contributed more than 50% of Tot. Profit.  \n5 | Maximum Drawdown | Pass | No individual run had a drawdown of more than 40% of init. capital.  \n| Overall Result | PASSED |\n\nThe cluster WFA is a very powerful tool because it allows the user to quickly look at different WFOs utilizing different OOS% values and walk-forward periods. A good trading algorithm may be rejected by simply using a single WFA. The e-mini SP system failed on the default 20 OOS% with 10 runs but passed on several different values.\n\nThe TS WFO software provides all the detailed information given in the cluster optimization matrix. A user can select any of the different optimizations and see a plethora of details. The ultimate question is then answered as to what parameter set should be used and how many days that set should be carried forward. If the software-derived optimal set is selected (15 OOS% with 25 runs), then the following parameters are suggested to be used for next 111 days.\n\n1. | movAvgLen | 150  \n---|---|---  \n2. | pivotHiLookBack | 4  \n3. | pivotHiStrength | 1  \n4. | pivotLowLookBack | 5  \n5. | pivotLowStrength | 1  \n6. | LprofitObjective | 1150  \n7. | LstopLoss | 350  \n8. | SprofitObjective | 950  \n9. | SstopLoss | 500  \n10. | longExitDays | 13  \n11. | shortExitDays | 14\n\nNow the question that started this part of the chapter can be answered. The WFO is an incredible tool that can help determine if a trading algorithm needs to be retrained periodically and also if the algorithm demonstrates sufficient robustness to match to some degree the results that were derived during its training.\n\n## Monte Carlo Analysis\n\nAs we have seen, walk-forward analysis can tell us much about an algorithm's robustness. Another tool that can reveal algorithm robustness is Monte Carlo simulation. The key to this form of simulation can be found in random numbers. Once you develop what you consider a good trading algorithm and test it against historical data, you then can really put it to the test by using random numbers and the historical trade history.\n\nThis trade history that is generated by testing your algorithm represents just one path your system traveled. What if you could create many paths by jumbling the order of the trades that your algorithm generated. These different paths could represent alternate universes. Let's say that your algorithm got lucky and had a bunch of winning trades in a row. This streak might be the portion of the equity curve that pulled the system out of mediocre status. If these trades didn't exist or hadn't fallen in place like they did, then the equity curve might look quite a bit different. A robust trading system should still produce robust performance metrics even when the trades are jumbled, some eliminated and some duplicated. If the majority of the alternate paths of an algorithm fail to produce good metrics, then the algorithm should be considered suspect.\n\nCreating alternate paths or parallel universes is a very simple process when you have access to a random number generator and a computer. As you saw earlier in this chapter an RNG is a very special tool. Imagine all of the trades generated by your algorithm are each written on a separate piece of paper and all the pieces are then placed in a bag. You create an alternative trade history by reaching into the bag and randomly selecting trades and writing them down in a trade log in sequential manner. You do this routine until you have recorded the same number of trades that were initially generated by the backtest. One very important thing to remember is to always return the piece of paper to the bag after you record the trade. This process would be very time consuming and you might ask what's the use. Don't worry about the time involved because the computer can do it very quickly and recreating hundreds of alternative paths is well worth the effort. Will it help create a better algorithm? Probably not but it will help you make a decision to allocate real funds to an algorithm or not. Like I stated earlier, the robustness of a trading algorithm will reveal itself through Monte Carlo simulation.\n\n#### Implementation of Monte Carlo Simulation Using Python\n\nThe concept of this form of simulation is easy enough. And putting it into computer language is as easy. The hardest part is getting the initial trade list. Once you have that the rest falls into place relatively easily. Understanding how a computer implementation of a concept works usually helps the understanding of how the concept works in general and how it might be beneficial.\n\nImagine all the trades generated by a backtest are stored in a list. The list is simply an ordered collection of the trade dates and trade profits. The list might look something like this:\n\n  1. TradeList[0] = (20011015,\u2212150)\n  2. TradeList[1] = (20011105,+200)\n  3. TradeList[2] = (20011202,\u2212300)\n  4. TradeList[3] = (20011222,+500)\n\nYou could call this the source list. Now imagine you have another blank list that's called AlternateList1. The objective of the Monte Carlo simulation is to fill the AlternateList1 with trades from the source list. If there are only four elements in the list, you would need an RNG to generate random numbers from 0 to 3. Let's push the RNG button and start the process of filling up AlternateList1:\n\n  1. The RNG generates the number 3 \n    1. AlternateList1[0] = TradeList[3]\n  2. RNG generates the number 0 \n    1. AlternateList[1] = TradeList[0]\n  3. RNG generates the number 1 \n    1. AlternateList[2] = TradeList[1]\n  4. RNG generates the number 3 \n    1. AlternateList[3] = TradeList[3]\n\nOur new alternate list of trades would look like this:\n\n  1. AlternateList[0] = (20011222,+500)\n  2. AlternateList[1] = (20011015,\u2212150)\n  3. AlternateList[2] = (20011105,+200)\n  4. AlternateList[3] = (20011222,+500)\n\nSimple, right? All we are doing is jumbling the order of the original list to create an alternate path the trading system might have followed. Did you notice that the RNG generated the number 3 twice? So AlternateList[3] is the same as AlternateList[0]. Does this seem right? This is correct and exactly what we want our RNG to do. This is called sampling with replacement. Remember how the piece of paper was to be put back into the bag after recording it? This method almost guarantees that we never recreate the original trade listing. This is important because we want to create as many unique alternative paths as we can.\n\nPython provides, to its programmers, some very useful data structures. However, it does not provide the array structure like other languages (you can use arrays in Python if you import them with the NumPy library). Arrays are very powerful and can be multidimensional and this is the structure I would have used if I had used a different language. But Python has this really cool structure called a tuple. A tuple is simply a collection or a list of like data. Here are examples of a classic car tuple:\n\n  1. antiqueCarTuple[0] = (1967, 'Ford' , 'Mustang' , 289)\n  2. antiqueCarTuple[1] = (1971, \"Chevy', 'Chevelle', 402)\n\nAs you can see each tuple element holds the various properties of a classic car. If I need to know the model of the second antique car, I can access it by using the following notation: modelName = antiqueCarTuple[1][2]\n\nThe first bracket following the name of the tuple selects which tuple you are wanting to access from the tuple list. Since Python is 0 based the [1] in this example tells the computer to select the second tuple or car. The next bracket is used to determine which element in the tuple to access. I wanted the model, or third element, of the antique car so I used the number 2 to access it.\n\nIn our Monte Carlo example each tuple will hold the date of the trade exit and the associated profit or loss. Here is the very lengthy and elaborate code for creating 1000 alternate paths of our trading algorithm:\n\n* * *\n\n    mcTradeTuple = list()\n    for x in range(0,1000): # number of alternate histories\n        for y in range(0,len(tradeTuple)):\n            randomTradeNum = random.randint(0,len(tradeTuple)-1)\n            tradeDate = tradeTuple[randomTradeNum][0]\n            tradePL = tradeTuple[randomTradeNum][1]\n            mcTradeTuple += ((x,y,tradeDate,tradePL),)\n\n* * *\n\nThat's it! These seven lines of code is all it took to create 1000 different and mostly unique alternate trade histories. I won't bore you with all of the details, but I will step through the code quickly to demonstrate how easy this was to do with Python. Initially I stored all of the trade dates and trade P\/L in a tuple named **tradeTuple**. From this list I culled all of the trades to fill up the 1000 alternate lists. Basically I started with the source list and for each trade in the list I created a random number between 0 and the number of trades:\n\n    randomTradeNum = random.randint(0,len(tradeTuple)-1)\n\nPython generates random numbers through its module: **random.randint**. I then used the **randomTradeNum** as an index into the tradeTuple to extract the **tradeDate** and **tradePL** :\n\n    tradeDate = tradeTuple[randomTradeNum][0]\n    tradePL = tradeTuple[randomTradeNum][1]\n\nRemember the first bracketed number following the tuple selects which tuple. The second bracketed number selects the element in that particular tuple. In the **tradeTuple** , [0] is the trade date and [1] is the trade P\/L. While stepping through the alternate histories sequentially, trades were randomly selected from the original trade history and inserted. All of the alternate histories were then stored in a list of tuples named **mcTradeTuple**. The += operand simply appends the current alternate trade history to the list. In the end, you have one huge list that includes all of the alternate histories.\n\n#### AmiBroker's Monte Carlo Simulation\n\nThe Python Monte Carlo simulation is included in the Python System Backtester. I programmed the simulation because I wanted to fully understand the mechanism. Fortunately, you don't need to program these types of tools because they are usually included in your favorite testing platform. This is the case for AmiBroker and TradeStation. AmiBroker's Monte Carlo analysis is so simple all you have to do is flip a switch. Click the Settings button in an Analysis window and hit the Monte Carlo tab (see Figure 8.9). You have seen this dialog before but this is the first time we have discussed the Monte Carlo settings. The default values are the most popular. The position sizing parameters offers four different options: (1) keep the sizing the same as the initial test, (2) use a fixed size of contracts or shares, (3) use a fixed dollar value for size computation, and (4) use a percentage of equity in fixed fractional approach. The fourth option can introduce serial dependency into the Monte Carlo simulation due to the fact that the size of the current trade is dependent upon the success of all of the prior trades. Utilizing this option might muddy the waters a bit when trying to determine algorithm robustness. Before we move on let's discuss the idea of serial correlation among the trades of a trading algorithm. This concept basically implies there is a relationship or connection between a trade and the subsequent trade or trades. Critics of Monte Carlo simulation on actual trades suggest this important correlation is ignored when trades are jumbled. If you think about it, it sort of makes sense when dealing with a trend-following algorithm. Trend-following trades can follow a cyclical pattern\u2014one large winner due to a trend, followed by a series of small losers that occur in the absence of a trend. You can see that the small losers are a consequence of the dissipation of the trend. This series of trades, big winner, small losers demonstrates a level of trade interdependency. The Turtles felt like a loser might follow a winner and therefore utilized the _Last Trade Was a Loser_ filter. There has been much research on this topic and the consensus has been that if serial correlation existed, it only applied to the first subsequent trade. All other trades were independent events. If serial correlation is a concern, a Monte Carlo simulation on equity curve segments would be a solution. Each segment would contain the same series of trades and therefore maintain the correlation. Since we are dealing with a large majority of alternate paths I don't necessarily think you need to concern yourself with serial correlation. So just go ahead and accept the default settings and click OK. AmiBroker creates a couple of charts\/reports that will illustrate the usefulness of this form of simulation. The first chart is called a Straw Broom chart because of the way it looks (see Figure 8.10).\n\n**Figure 8.9** AmiBroker Monte Carlo settings dialog.\n\n**Figure 8.10** Monte Carlo Straw Broom chart of multiple randomized equity curves.\n\nThis is a representation of all of the alternate trade histories that were created by using the random number generator. As you can see, most of the histories end at different equity values and this is due to sampling with replacement. Had we not allowed replacement, then all of the equity curves would end at the exact same spot. The accumulation of all of the trades, irrespective of order, follows the commutative law. You can easily see the boundary (top and bottom) histories that represent the best and worst performance. The rest of the histories congregate in the middle and create a cloud. This cloud is what you want to see from your own algorithm simulation; many of the random histories created similar outcomes. AmiBroker also creates the following table that provides the statistics derived from the distribution of the simulation results (Table 8.8).\n\n**Table 8.8** Distribution Statistics from Monte Carlo Simulations\n\n| **Final Equity** | **Annual Return** | **Max. Drawdown $** | **Max. Drawdown %** | **Lowest Eq.**  \n---|---|---|---|---|---  \n1% | 5706 | \u22127.37% | 1302 | 7.23% | 3618  \n5% | 7987 | \u22123.02% | 1549 | 9.76% | 5853  \n10% | 9706 | **\u22120.41%** | 1726 | 11.32% | 6690  \n25% | 12851 | 3.48% | 2136 | 14.38% | 8107  \n50% | 16174 | 6.78% | 2747 | 19.77% | 9135  \n75% | 19632 | 9.64% | 3563 | 27.63% | 9640  \n90% | 23258 | 12.21% | 4626 | **38.48%** | 9922  \n95% | 25269 | 13.48% | 5292 | 45.47% | 10000  \n99% | 29139 | 15.71% | 7685 | **63.82%** | 10000\n\nThe first column shows the percentile level that certain observations fell. In the first row 1% of all observations had profit levels below $5,706 and maximum drawdowns below $1,302. These two events, based off of their probabilities, are unlikely to happen again. Fifty percent of the time you could expect profits to be less than or equal to $16,174 and maximum drawdown to be less than or equal to $2,747. Ninety-nine percent of the time you could expect drawdowns below $7,685. Keep in mind what we expect and what really happens doesn't always jive. We are basing all of our expectations on historical results, results that have a certain level of built-in hindsight bias.\n\n## Start Trade Drawdown\n\nThe maximum drawdown metric that is derived from backtesting is a key component used to help a trader judge the capitalization requirements of a particular trading algorithm. If an algorithm suffers a $50,000 drawdown, it logically follows that this event could occur again in the future. And accordingly the trader should allocate at least $50,000 to his trading account. What if the algorithm is very good and this one event doesn't cast it in a good light? What if the drawdown follows a large runup? In this case, the drawdown is somewhat a function of the system's success. Another drawdown metric, start trade drawdown, can help shed light on an algorithm's drawdown structure. A trader is most sensitive to drawdown when she initially starts to trade a new algorithm. A $50,000 drawdown that decimates a trader's account right off the bat is completely different than a $50,000 drawdown that occurs after a $200,000 runup. Wouldn't it be nice to know the probability of having a huge drawdown at the beginning of trading? Also wouldn't it be nice to know the probability of the maximum drawdown occurring again in the future for capitalization purposes? This concept of Start Trade drawdown can be attributed to Keith Fitschen. He describes it on his website, www.keithstrading.com, and in his book, _Building Reliable Trading Systems_ (Wiley, 2013, New Jersey). Let's assume you have a trading algorithm that trades 100 times. Would a trader starting at trade #1 have a different drawdown than a trader starting at trade #50? This is a very good question and I guess Mr. Fitschen asked himself this exact same question. You could simply analyze the drawdowns that occur after trade #1 and the drawdowns that occur after trade #50 and answer the question. What if there was an analysis that could provide the probabilities of different drawdown magnitudes derived from the historical results of a trading algorithm? This could definitely help a person decide if a system is worth the risk, and at the same time know how much capital would be required to fund it. Well, thanks to Keith Fitschen, we have this analysis.\n\n#### Calculating Start Trade Drawdown\n\nWith the use of a computer, the process of this calculation is easily accomplished. This calculation is somewhat similar to the Monte Carlo simulation in that you are recreating multiple histories of a trading algorithm. However, this time an RNG is not utilized. The first trade history is the trade history that contains all of the historic trades. Subsequent trade histories start at the subsequent trade in the original history. For example, the second trade history starts at the beginning of trade #2 and the next trade history starts at beginning of trade #3, and so on. An original trade history containing 100 trades will spawn 100 histories starting at different trade numbers. Once all of the histories are created then all one needs to do is flow through each history and keep track of cumulative profit and maximum drawdown.\n\nAfter creating and analyzing the 100 trade histories a cumulative frequency distribution table (CFDT) must be created. A CFDT is created by distributing the 100 drawdowns into different bins. The bins are the same size but have different boundaries. Assume the 100 drawdowns range from $5,000 to $25,000 and you want to store the different drawdown in 20 bins. The range is equal to $20,000 ($25,000 \u2212 $5,000) so each bin would have to be $1,000 wide ($20,000\/20). Bin #1 would contain the drawdowns that ranged from $5,000.00 to $5,999.99, bin #2 would contain the drawdowns that ranged from $6,000 to $6,999.99, bin #3 would contain the drawdowns that ranged from $7,000 to $7,999.99, and so on. See how the bins are the same size but have different low and high boundaries?\n\nOnce the drawdowns are distributed into the different bins you can easily create a cumulative frequency by summing up the number of drawdowns in each bin. Once you have the cumulative frequency it is simple to calculate the probability of occurrence for each bin. With 100 different trade histories starting at different trade numbers, it is not uncommon to have less than 100 distinct max drawdown values. In some cases, the same max drawdown value will be observed when the starting point of each history is in close proximity. I wouldn't be surprised if the max drawdown was the same starting at trade #2 versus starting at trade #3. The fact is you will have 100 max drawdown values but they won't all be distinct. The bins will contain multiple duplicate drawdown values. Each bin will contain N number of values and the probability of the occurrence of that bin's max value and below will be the sum of N up to that bin\/Total N. Let's say bin #1 [$5,000 \u2212 $5,999] contains 5 values and bin #2 [$6,000 \u2212 $6,999] contains 3 values. The probability of a drawdown less than $6,999 would be 8\/100 or 8%. This type of information can be very beneficial at the beginning stage of trading a new algorithm.\n\n### Implementation of Start Trade Drawdown Simulation Using Python\n\nSimulation of the different trade histories depends on the actual number of trades generated by the algorithm. This simulation requires a nested for-loop where the outside loop loops from trade #1 to the last trade. The interior loop loops from the current trade number to the last trade.\n\n* * *\n\n    # start trade draw down analysis - utilizing the tradeTuple\n        tupleLen = len(tradeTuple)\n        tradeTuple = sorted(tradeTuple,key=itemgetter(0))\n        for x in range(0,tupleLen):\n            cumStartTradeEquity = 0\n            maxStartTradeDD = -99999999\n            maxCumEquity = 0\n            for y in range(x,tupleLen):\n                cumStartTradeEquity += tradeTuple[y][1]\n                maxCumEquity = max(maxCumEquity,cumStartTradeEquity)\n                maxStartTradeDD = max(maxStartTradeDD,maxCumEquity - cumStartTradeEquity)\n            startTradeTuple += ((x,cumStartTradeEquity,maxStartTradeDD),)\n\n* * *\n\nThe outer loop indexed by **x** controls the number of trade histories. The inner loop indexed by **y** flows through the different trade histories and keeps track of each history's cumulative equity and maximum drawdown. At the beginning of each trade history, these two values are zeroed out. Each trade's P\/L is stored as the second element in the tuple **tradeTuple**. This value is extracted through each **y** iteration and used to calculate the two performance metrics. Once an individual history is created it is then stored in a list of tuples named **startTradeTuple**. The first element in this tuple is x (the trade history number). The second element **cumStartTradeEquity** stores the cumulative equity of the history and the third element **maxStartTradeDD** stores the max drawdown of the history.\n\nOnce the trade histories are compiled, the bins that hold the different max drawdowns must be constructed. The first step in the construction process is to calculate the largest and smallest historic drawdown values.\n\n* * *\n\n        minDD = 99999999\n        maxDD = 0\n        for y in range(0,len(startTradeTuple)):\n            print(startTradeTuple[y][0],' ',startTradeTuple[y][1],' ',startTradeTuple[y][2])\n            if startTradeTuple[y][2] < minDD: minDD = startTradeTuple[y][2]\n            if startTradeTuple[y][2] > maxDD: maxDD = startTradeTuple[y][2]\n\n* * *\n\nA loop is used to flow through each trade history and keep track of the maximum and minimum values stored in the third element of the startTradeTuple (max drawdown of each history).\n\nTwenty bins will be used in this example to store the individual drawdown values. The bin size is calculated by dividing the range of drawdowns by 20. The size of the bin is stored in the variable **binInc** (bin increment). Each bin is constructed by storing each bin's boundary values in a list of tuples named **binTuple**. Each binTuple contains three elements: binNumber, binBottom, and binTop. The boundary values are calculated by starting at the smallest max drawdown of the histories (bottom of the first bin) and adding the bin size to get the top of the first bin. The next bin's bottom boundary becomes the prior bin's top and its top is calculated by adding the **binInc**. This process is repeated until all 20 bin tuples are filled up with the correct values.\n\n* * *\n\n    numBins = 20\n    binTuple = list()\n    binInc = (maxDD - minDD)\/20.0\n    binBot = minDD\n    for y in range(0,numBins):\n           binTop = binBot + binInc\n           binTuple += ((y,binBot,binTop),)\n           print(binTuple[y][1],' ',binTuple[y][2])\n           binBot = binTop + y\n\n* * *\n\nOnce the bins are constructed all the different histories' max drawdowns are distributed into them. This is accomplished by comparing the max drawdown with each bin's top and bottom boundaries. If the drawdown value falls between the two boundaries, it is then placed into that bin. A blank list of bins is initially created.\n\n* * *\n\n    bins = list()\n    bins[:] = []\n    for x in range(0,numBins):\n           bins.append(0)\n\n* * *\n\nThe different histories are looped through and are placed into the different bins. Well, not really, just the frequencies of the drawdowns are stored.\n\n* * *\n\n    for x in range(0,len(startTradeTuple)):\n           for y in range(0,numBins):\n                 tempDD = startTradeTuple[x][2]\n                 tempBot = binTuple[y][1]\n                 tempTop = binTuple[y][2]\n                if (tempDD >= binTuple[y][1] and tempDD < binTuple[y][2]):\n                     bins[y] += 1\n\n* * *\n\nOnce the frequency of each drawdown is stored in each bin the total number of drawdown occurrences must be calculated.\n\n* * *\n\n    freqSum = sum(bins)\n\n* * *\n\nPython has a cool list method **sum**. This method sums up all of the values in the list in one fell swoop. This is the beauty of an object-oriented language. Once the total is calculated, the cumulative probabilities is just a loop away.\n\n* * *\n\n    binProb = list()\n    for y in range(0,numBins):\n           if y == 0:\n                binProb.append(bins[y]\/freqSum)\n           else:\n            binProb.append(bins[y]\/freqSum + binProb[y-1])\n\n* * *\n\nThe probability of bin #1 is simply the number of drawdowns in that bin divided by the total number of max drawdowns. The subsequent bin's probabilities are the probability of the current bin plus the prior bin's probability. This process accumulates the frequencies of the drawdowns. Here is a printout of a probability distribution of an algorithm trading different markets.\n\n* * *\n\n    Probability of DD <    1547 is 0.060\n    Probability of DD <    3094 is 0.080\n    Probability of DD <    4643 is 0.080\n    Probability of DD <    6192 is 0.280\n    Probability of DD <    7743 is 0.320\n    Probability of DD <    9294 is 0.400\n    Probability of DD <   10847 is 0.400\n    Probability of DD <   12400 is 0.420\n    Probability of DD <   13955 is 0.640\n    Probability of DD <   15510 is 0.660\n    Probability of DD <   17067 is 0.680\n    Probability of DD <   18624 is 0.820\n    Probability of DD <   20183 is 0.880\n    Probability of DD <   21742 is 0.880\n    Probability of DD <   23303 is 0.900\n    Probability of DD <   24864 is 0.920\n    Probability of DD <   26427 is 0.940\n    Probability of DD <   27990 is 0.960\n    Probability of DD <   29555 is 0.980\n    Probability of DD <   31120 is 1.000\n\n* * *\n\nThe maximum drawdown of this algorithm was around $30,000. The probability of this event occurring again is less than 2%. The probability of having a drawdown greater than $20,000 is less than 12%. These drawdowns only include closed trade drawdowns\u2014the amount of drawdown after the trade was closed out, not while it was open. Open trade drawdown will always be greater than or equal to closed trade drawdown. The loss isn't realized until the trade is closed out. Again, keep in mind we are dealing with hypothetical performance metrics with an inherent hindsight bias. Stating that a drawdown has less than a 2% chance of occurring again cannot be said with a straight face. However, historic performance is all we have to hang our hat on.\n\n## Summary\n\nThe purpose of this chapter was to be a guide and explain concepts that on the cover seem to be beyond the reach of a typical algorithmic system developer\/trader. Genetic optimization is just a very smart way to get to a solution quickly. Walk-Forward Optimization, Monte Carlo simulation, and Start Trade drawdown are tools that can provide the necessary evidence that a trading algorithm might have a chance of performing in the future. Python code was peppered throughout the text pertaining to Monte Carlo simulation and Start Trade drawdown to demonstrate how simple the processes are that carry out these particular forms of analysis. None of these tools will provide the \"Holy Grail,\" but in this author's opinion, they will help the user develop a trading solution that best fits his or her own trading style.\n\n1 John H. Holland. Genetic algorithms. _Scientific American_ , 267(1):44\u201350, 1992.\n\n# Chapter 9  \nAn Introduction to Portfolio Maestro, Money Management, and Portfolio Analysis\n\nIt has been mentioned in many trading circles through the years that money management is as important as the algorithm that generates the trade entries. The position size is paramount when it comes to utilizing capital in the most efficient manner. Thus far, we have simply tested on a one-contract basis, and that in of itself is a money management scheme. The one contract per market scheme was utilized, because the primary focus was on developing a trade entry-and-exit algorithm. Now that we have spent a sufficient amount of time on algorithm development, this guide would not be complete without a chapter on money management and portfolio analysis. This time we will lean on TradeStation to carry out our analysis\u2014this platform has the toolset necessary to apply both portfolio and money management schemes to a trading algorithm.\n\n## Fixed Fractional\n\nFixed Fractional is by far the most popular form of money management in the world of futures trading. This approach allocates the same amount of capital to each market. Perceived risk is usually defined as a function of a market's average true range calculated over the last _N_ days. Dividing the amount of capital by market risk gives the position size for the next trade. This form of money management normalizes the markets by maintaining the same amount of perceived risk across all markets in the portfolio. Here is the formula:\n\nAssume you have a $100,000 capital allocation, and you want to risk at most 1 percent on any given trade. Assume the 10-day ATR in Treasury notes is $500. How many contracts should be initiated on the next trade? Plugging into the formula we get:\n\nSo two contracts of T-notes will fit our 1 percent risk criteria. Now assume you want to trade Eurodollars as well and need to know that position size. Let's say the 10-day ATR in Eurodollars is $125. Plugging into the formula again, we get:\n\nEight contracts of Eurodollars provides the same risk as two contracts of T-notes. See how the different markets are normalized to provide the same amount of risk? Utilizing a Donchian 55-day breakout with a trailing 20-day liquidation and a fixed dollar stop, let's see how this money management formula works in practice.\n\n## Portfolio Maestro\n\nPortfolio Maestro (PM) is an add-on to TradeStation and requires an additional monthly fee. If you have it you might want to follow along with this tutorial. If you don't, then read on and see if it might be something you may want to investigate. From TradeStation you can launch PM by clicking on the Trading Apps tab, and then selecting Portfolio Maestro. PM will launch and you will be presented with the window shown in Figure 9.1.\n\n**Figure 9.1** TradeStation's Portfolio Maestro (PM) launch window.\n\nOnce you get used to the interface, building and testing portfolios will become old hat. However, the first few analyses may take multiple attempts to get everything the way you want. Hopefully, this tutorial will help along the way.\n\nPM works around the concepts of a **Strategy Group** and a **Portfolio** of **Strategy Groups**. Before you can do anything you must build a **Strategy Group**. The Strategy Group that we will be using is one that incorporates our Turtle-like 55-day Donchian breakout. If you haven't imported the ELDs from the website, this would be a great time to do so.\n\n### Strategy Group\n\nIn PM, a **Strategy Group** consists of a trading algorithm and the portfolio of markets to which that algorithm will be applied. With the PM window in front of you click on the **File** menu, select **New** , and then select **Strategy Group**. Where it asks for Name type in **MyFirstStrategyGroup** and then select **Futures** as the asset class.\n\nYou will then be sent back to the PM main window. Click on the **Strategies** tab and then click on the **Add Strategy** button. Another window will open and list all of the strategies that are in your strategies library (see Figure 9.2).\n\n**Figure 9.2** Add a strategy to your strategy group.\n\nScroll down the list until you come to **Turtle 55**. If you don't see it, then you might need to reimport the ELD files from this book's companion website (www.wiley.com\/go\/ultimatealgotoolbox). If it's there, click on the name and highlight it. Then click OK. The PM window will now show **Turtle 55** in the **Strategies** window (see Figure 9.3).\n\n**Figure 9.3** The Turtle 55 strategy is now in your strategies window.\n\nWe now have a strategy, but now we need to tell PM to apply it to a list or portfolio of markets. Click on the **Symbol Lists** tab right beside the **Strategies** tab. A window will appear similar to Figure 9.4.\n\n**Figure 9.4** Add a symbol by clicking the Symbol Lists tab (indicated by Arrow 1), then clicking on Add Symbol List (shown with Arrow 2).\n\nPM asks you to select a preexisting list of markets. I've created a custom list for this book, labeled **UGTATS** , and it includes the following markets:\n\nUGTATS Market List\n\n  1. Australian Dollar\n  2. Beans\n  3. British Pound\n  4. Cocoa\n  5. Coffee\n  6. Copper\n  7. Corn\n  8. Cotton\n  9. Crude\n  10. Dollar Index\n  11. Euro FX Currency\n  12. Euro Dollar\n  13. Feeder Cattle\n  14. Five-Year Notes\n  15. Gold\n  16. Heating Oil\n  17. Kansas City Wheat\n  18. Lean Hog\n  19. Live Cattle\n  20. Mini-Russell\n  21. Mini-SP400\n  22. Mini-SP500\n  23. Natural Gas\n  24. Orange Juice\n  25. Platinum\n  26. Silver\n  27. Treasury Notes (2-Year, 5-Year, and 10-Year)\n  28. Treasury Bonds (30-Year)\n  29. Unleaded\n  30. Wheat\n\nIf you need help creating a **Symbol List** , there is a short tutorial at www.georgepruitt.com. After selecting the **UGTATS** list your **Symbol List** window should look something like this (see Figure 9.5).\n\n**Figure 9.5** After selecting the UGTATS list, it will appear in your symbol list window.\n\nIf you click on the plus sign beside the **UGTATS Symbol List** , the symbols of the markets included in the list will be shown (see Figure 9.6).\n\n**Figure 9.6** The UGTATS markets and their corresponding symbols.\n\nNow that we have created a complete strategy group (Strategy and Symbol List), it is now time to create a portfolio. Go back under the **File** menu, select **New** , and then select **Portfolio**. A dialog box similar to the one we used to create a New Strategy Group will pop up. In the **Name** field, type **MyFirstPortfolio** and then click OK. After clicking OK a blank portfolio window will appear. The message in the window (\"This portfolio does not have any strategy groups...\") is informing you that you have not yet selected a strategy group. We can remedy this very quickly by clicking on the **Add Strategy Group** button. A list of strategy groups will be presented (see Figure 9.7). Select **MyFirstStrategyGroup** and it will be inserted into **MyFirstPortfolio** (see Figure 9.8).\n\n**Figure 9.7** A list of available strategy groups, including MyFirstStrategyGroup, which we are using for this exercise.\n\n**Figure 9.8** After selecting MyFirstStrategyGroup, you will see it in MyFirstPortfolio. In the expanded view, you can see the strategies and symbols that are part of your group.\n\nYou might need to expand the boxes to see the **Turtle 55** strategy and the **UGTATS** Symbol List. The **Portfolio** window shows the different strategy groups inside the **Portfolio**. You can have multiple **Strategy Groups** trading different **Strategies** on different **Symbol Lists**. This is a very powerful tool\u2014the ability to merge multiple **Strategies** across multiple **Symbol Lists**. We touched on this exact subject of Multiple Algorithmic Strategies (MAS) in Chapter 3.\n\nBefore we backtest the portfolio, let's change the **Strategy Group** settings by going back to the **Strategy Group**. Click on the **Manage Strategy Group** (puzzle piece) button, and then click on the **Strategy Group Settings** button. The settings dialog box will open and should look similar to Figure 9.9.\n\n**Figure 9.9** How to adjust the settings for a strategy group. Arrow 1 indicates the Manage Strategy Group button. Arrow 2 highlights the Strategy Group Settings button. Arrow 3 points to the controls that change the commission and slippage inputs.\n\nFrom this dialog, we can change the commission and slippage and various other strategy-dependent parameters. You can even change the **Strategy Inputs** by clicking the **Strategy Inputs** tab. Go ahead and change the **Commission** and **Slippage** to $12.50 for each on a **Per Contract** basis. Click **OK** , and then, to be on the safe side, click on the **Symbol List** tab again and make sure the **Asset Class** is still set to **Futures**. PM sometimes switches this back to **Equities**. Once you check this, click the **Manage Portfolios** button and finally click **Backtest Portfolio**. Another dialog window will appear that lets you change the test period, **Initial Capital** , the **Backtest Type** (standard or optimization), and the **Report Name** (Figure 9.10). You can also deselect certain markets by going to the **Symbols** tab. Change the test period to start 10 **Years Back**.\n\n**Figure 9.10** As you prepare to backtest your portfolio, ensure the settings are correct. Begin by clicking the Backtest Portfolio button (marked 1 in the figure). Then, check the initial capital, test period, and backtest type in the appropriate fields (marked with 2). When everything looks good, click the Perform Backtest button (number 3 in the figure).\n\nIf everything looks OK, then click **Perform Backtest**. The test might take a few minutes if you have a large **Symbol List**. TradeStation first pulls the data from their data servers and then applies the strategy to each market. I have a feeling TradeStation will make this process much quicker by caching the data on the hard drive in the future.\n\nOnce the test is complete, PM will ask if you would like to view the report now or later. Go ahead and click **Now**. A window like the one in Figure 9.11 will open, and show you the portfolio performance of your **MyFirstPortfolio**. Remember this portfolio consists of two components, a strategy and a portfolio.\n\n**Figure 9.11** A backtest performance report.\n\nFigure 9.11 illustrates a portion of the performance metrics that are included in the **Summary** report. This portfolio actually did pretty well over the past 10 years: $739,684 in profits with an average trade of $426. You can scroll down and see a plethora of performance metrics. The one thing I would like PM to show in this report is the maximum drawdown. The maximum drawdown is shown but you have to hunt for it. Click on the **Trade Analysis** tab. The **Trade Analysis** drills down and proffers up even more performance metrics. If you are a statistician, then you will think you have died and gone to Heaven. The neatest feature in this tab is located at the bottom right and is labeled **View\/Hide Symbols**. If you hit this button, then every market that was traded and their associated performance metrics will be added to the **Trade Analysis** window (see Figure 9.12).\n\n**Figure 9.12** The Trade Analysis window displays trading statistics. To get to this screen, click the Trade Analysis tab (marked 1). The View\/Hide Symbols button (marked 2) reveals the performance metric for every market in your portfolio (item 3).\n\nScroll around and make yourself comfortable with the report. I could write an entire book just on these neat reports and bore you to death. However, I will leave the exploration of all of the different report tabs up to you. But if you indulge me a little bit more, I will show two very interesting charts that PM generates that show the location of the maximum drawdown. Click on the **Graphs** tab and the equity curve for the portfolio will be plotted and look similar to the one in Figure 9.13.\n\n**Figure 9.13** An equity curve chart that shows the maximum drawdown.\n\nSource: TradeStation\n\nIf you want to see the contribution of each individual market to the portfolio, this is done by selecting Total P\/L by symbol (Figure 9.14). This chart is a histogram of each market's contribution to the grand total. If you want to know the maximum drawdown of the portfolio, you will find this under the **Returns and Equity** tab about halfway down (Figure 9.15). This Turtle-like algorithm produced over $700K in profit but had a $201K drawdown. Not too bad, considering the size of the portfolio. The energy sector packed most of the punch, as you can see from the histogram.\n\n**Figure 9.14** The Total P\/L option displays individual profit and loss data for every market in the portfolio.\n\nSource: TradeStation\n\n**Figure 9.15** The Returns and Equity tab displays the portfolio's maximum drawdown in this case.\n\nThis ability to merge the different equity curves together has not always been available in TradeStation. The lack of portfolio analysis was the major criticism of TradeStation over the years, but with the advent of PM, the critics have a lot less to complain about.\n\n### Fixed Fractional Money Management Overlay\n\nNow that we know how to create a **Strategy Group** and a **Portfolio** , we can now apply a money management overlay and test it. With PM still on your screen, click the **Manage Strategy Groups** button and then the **Money Management** tab. In the **Money Management** window you will find a dropdown menu labeled **Method**. Click the down arrow and select **Fixed Fractional with ATR Risk**.\n\nThe resulting dialog box gives you the opportunity to change the parameters built into the Fixed Fractional formula. You can change these parameters:\n\n  * **Percent Risk** \u2014The amount of capital to risk on each and every trade. Set this to 1 percent.\n  * **ATR Multiplier** \u2014The number of times you want to multiply the ATR to estimate perceived market risk. The larger this number the riskier you perceive the market to be. Set this to two.\n  * **ATR Lookback** \u2014How many days in the ATR calculation. The shorter lookback gives recent history more weight.\n  * **Maximum Quantity** \u2014The largest position size that will be allowed on any given trade. Eurodollars are notorious for generating large position sizes. Since execution costs cut into the bottom line you probably should limit this to 10.\n  * **Round Quantity** \u2014You cannot trade a fractional part of a futures contract, so always set this to one.\n\nLook at Figure 9.16 and make your changes accordingly.\n\n**Figure 9.16** The arrows indicate the parameters that you can change from the Money Management window. For now, keep the ATR lookback set to 10 and the round quantity set to 1.\n\nNow click on the **Manage Portfolios** button. You will notice that _Money Management: Fixed Fractional with ATR Risk_ has been added directly under **Turtle 55** in the strategy portion of the dialog window. The description of the Money Management strategy is included as well:\n\n_Buy 2 times ATR over 10 bars lookback, but no more than 10. Risk no more than 1.00 percent of Equity_.\n\nThis overlay will calculate two times the 10-day ATR and use it as the denominator in our **Fixed Fractional** formula. The numerator will be 1 percent of the total portfolio balance. Both numerator and denominator will rise and fall. The denominator will replicate market volatility and the numerator will reflect 1 percent of the account as it grows or shrinks. Click on the **Backtest Portfolio** button and change the **Initial Capital** to 500,000. This test will apply the **Turtle 55** algorithm to the **UGTATS Symbol List** all the while overlaying a **Fixed Fractional** money management scheme. Sounds like a lot to carry out, so let's see what happens by clicking the **Perform Backtest** button (make sure you are going 10 **Years Back** ). Figure 9.17 shows the equity curve of the backtest, and yes, you are seeing close to a million-dollar profit.\n\n**Figure 9.17** The equity curve of the backtest reveals a profit of nearly $1 million.\n\nSource: TradeStation\n\nThe profit is spectacular but so is the drawdown\u2014and not in a good way. This is how **Fixed Fractional** works\u2014as the account grows so does the position size. And we all know profit or loss and drawdown are directly proportional to position size. Take a look at Figure 9.18 to see how the position size changes for the different markets in the portfolio.\n\n**Figure 9.18** Profit and drawdown are directly proportional to position size. As your position size changes, so do your commissions and profits and losses.\n\n### Portfolio Analysis\n\nNot only does PM allow you to overlay money management, but it also allows you to control portfolio management from a macroscopic perspective. Professional portfolio managers often tinker with the idea of placing a global portfolio stop after a big losing month. Let's say the portfolio is down 4 percent by midmonth; some managers liquidate all positions and wait until the beginning of the next month to reestablish positions. They also tinker with the idea of liquidating positions intramonth when the portfolio reaches a certain profit level. I have personally seen a fund up 3 to 4 percent midmonth, but by the time the end of the month rolls around, the fund winds up being negative. If only the fund manager had taken a profit, then the gains would have been preserved. This type of trade stoppage sounds great theoretically, but there are risks involved. Let's say a fund manager stops trading when the fund is down 4 percent, but continues to monitor the what-if trades and finds out the fund would have recovered the 4 percent plus an additional 2 percent. The manager realized a real 4 percent loss, but had he continued trading he would have ended up with a 2 percent winner. This is a 6 percent equity swing and could be the difference between holding on to some clients and losing them. The same risk occurs when flattening a fund because of what is perceived to be a \"sizable\" profit. Many times, a plus 4 percent month turns into a plus 8 percent. There is never a black-or-white answer to anything dealing with trading. This is why ideas are converted into diagrams, diagrams into pseudocode, pseudocode into an actual programming language and then tested, and tested, and tested.\n\nSpeaking of testing, let's go back to PM and test the ideas of trade stoppage once the portfolio, on an intramonth basis, digs itself into a 4 percent hole, or when the portfolio achieves a 4 percent profit. Trading will cease until the beginning of the following month. PM enables this type of testing with just a few keystrokes. Go back to PM and click on the **Manage Portfolio** button, then click on **Portfolio Settings** , then on **Portfolio Stops** (Figure 9.19).\n\n**Figure 9.19** Under Portfolio Settings (marked as 1), go to the Portfolio Stops tab (marked as 2) and set the stop strategy of your choosing. Under the dropdown menu, there is a description of how the selected strategy will impact your portfolio (marked as 3).\n\nThe first parameter that needs to be set is the **Portfolio Stop Loss**. Click on the dropdown menu and select **Portfolio Stop Loss** and set the parameters as shown in Figure 9.20. Set the **Loss As Percent** to True, **Loss Target** to 4, and **Period** to 1. Click the **Set Stop** button and the **Portfolio Stop Loss** will be applied to the portfolio strategies.\n\n**Figure 9.20** To set the portfolio stop loss, you can adjust the loss target percentage (marked as 1), loss target (marked as 2), and period (marked as 3).\n\nNow let's set the **Portfolio Profit Target** by setting **Period** to 1, **PrftAsPercent** to True, and **PrftTarget** to 4 (Figure 9.21).\n\n**Figure 9.21** To set the portfolio profit target, you can adjust the period, profit target percentage, and profit target. These options align with those in the stop loss tab.\n\nGo ahead and click **Set Stop** and you will see the **Portfolio Profit Target** will be added to the Strategies. Before we click OK, let's review both portfolio stops to make sure we have the parameters properly set. Expand the **Portfolio Profit Target** and the **Portfolio Stop Loss** strategies. The settings should look exactly like the ones in Figure 9.22.\n\n**Figure 9.22** The portfolio profit target and stop loss now appear as strategies in your portfolio.\n\nJust to review, the **Portfolio Profit Target** will cause the portfolio to cease trading once the portfolio achieves a 4 percent profit level and the **Portfolio Stop Loss** will cease trading if the portfolio loses 4 percent. Once stopped, the portfolio will start trading automatically at the beginning of the next month. Go ahead and click **OK** and you will be returned to the **MyFirstPortfolio** window. You will notice another item has been added to the list, **Portfolio Stop**. Take a quick look at Figure 9.23.\n\n**Figure 9.23** At the bottom of your screen, you should see the portfolio stop option.\n\nIf you see that **Portfolio Stop** has been added, then click **Backtest Portfolio**. All other parameters that were used in the prior Portfolio Analysis should still be set: 500,000 for **Initial Capital** and 10 **Years Back** for the test period. After the test finishes click OK to view the report now. First off, click on the **Graphs** tab to see if the equity curve looks any better utilizing the **Portfolio Stops**. Figure 9.24 shows the equity curve of **MyFirstPortfolio** using the prescribed stop trading schemes.\n\n**Figure 9.24** The equity curve of the portfolio once the stops have been added.\n\nSource: TradeStation\n\nA quick look reveals a smoother equity curve than the one from the prior portfolio analysis. Now click on the **Summary** tab and let's look at the results in a tabular form. Here are the differences between the first test and this test, which used portfolio stops:\n\n**Test 1:**\n\n  1. Total Return: $1,307,354\n  2. Number of Trades: 1685\n  3. % Profitable: 30.92%\n  4. Average Trade: $724.27\n  5. Maximum Drawdown: $1,115,228.36\n\n**Test 2:**\n\n  1. Total Return: $848,197\n  2. Number of Trades: 1940\n  3. % Profitable: 46.96%\n  4. Average Trade: $425.63\n  5. Maximum Drawdown: $350,500.34\n\nWhich test would you trade? Needless to say, the test with the portfolio stops was much better. In Test 2, **Total Return** dropped precipitously, but so did **Maximum DD**. It is interesting to see that **Number of Trades** increased in Test 2, which at first seems counterintuitive. If trading is turned off, then you would expect to have fewer trades. However, the more you think about it, a higher trade count makes sense; trades are turned off, and then turned back on, which creates an additional exit and entry. In Test 1, a trade could have stretched for several weeks, but in Test 2, if trades are turned off, then this particular trade would be liquidated and then reinitialized at the beginning of the next month\u2014thus generating an additional trade. Take a look at Tables 9.1 and 9.2, which show trades without and with the portfolio stops.\n\n**Table 9.1** Portfolio Trades without Stops\n\n**No Portfolio Stops** |  |  |  |  |   \n---|---|---|---|---|---  \nDate | Mkt | Entry\/Exit | Price | Quant | $P\/L | Entry\/Exit Name  \n3\/24\/06 | @AD | Short Entry | 0.4594 | 3 |  | Turtle55 Sell  \n4\/6\/06 | @AD | Short Exit | 0.4794 | 3 | ($6,100.00) | Turtle55SmmLiq  \n4\/26\/06 | @AD | Long Entry | 0.4959 | 3 |  | Turtle55 Buy  \n6\/1\/06 | @AD | Long Exit | 0.4933 | 3 | ($880.00) | Turtle55LLiq  \n8\/10\/06 | @AD | Long Entry | 0.5162 | 3 |  | Turtle55 Buy  \n8\/25\/06 | @AD | Long Exit | 0.5042 | 3 | ($3,700.00) | Turtle55LLiq  \n9\/5\/06 | @AD | Long Entry | 0.5196 | 4 |  | Turtle55 Buy  \n9\/8\/06 | @AD | Long Exit | 0.503 | 4 | ($6,740.00) | Turtle55LLiq\n\n**Table 9.2** Portfolio Trades with Stops in Place\n\n**With Portfolio Stops** |  |  |  |  |   \n---|---|---|---|---|---  \nDate | Mkt | Entry\/Exit | Price | Quant | $P\/L | Entry\/Exit Name | First New Signal  \n3\/24\/06 | @AD | Short Entry | 0.4594 | 3 |  | Turtle55 Sell |   \n3\/31\/06 | @AD | Short Exit | 0.4611 | 3 | ($610.00) | Portfolio Stop Short |   \n**Stopped Trading 3\/31\/2006; Resume 4\/01\/2006**  \n**No Signal Before the Next Stoppage**  \n**Stopped Trading 4\/20\/2006; Resume 5\/01\/2006**  \n5\/3\/06 | @AD | Long Entry | 0.5105 | 3 |  | Turtle55 Buy | First New Signal in May  \n5\/11\/06 | @AD | Long Exit | 0.5211 | 3 | $3,080.00 | Portfolio Stop Long |   \n**Stopped Trading 5\/11\/2005; Resume 6\/01\/2006**  \n8\/10\/06 | @AD | Long Entry | 0.5162 | 4 |  | Turtle55 Buy | First New Signal in Aug.  \n8\/17\/06 | @AD | Long Exit | 0.5167 | 4 | $100.00 | Portfolio Stop Long |   \n**Stopped Trading 8\/17\/2006; Resume 9\/01\/2006**  \n9\/5\/06 | @AD | Long Entry | 0.5196 | 5 |  | Turtle55 Buy | First New Signal in Sept.  \n9\/8\/06 | @AD | Long Exit | 0.503 | 5 | ($8,400.00) | Turtle55LLiq |\n\nPortfolio Maestro does exactly what it is supposed to do. Trades are immediately liquidated on or near the stoppage date, and new trades are initiated on fresh signals after the beginning of the next month.\n\nIt is also interesting to note that the original **Turtle 55** strategy without money management or portfolio stops actually performed the best from a profit-to-drawdown perspective.\n\n## Summary\n\nThis chapter highlighted the capabilities of Portfolio Maestro:\n\n  * _Portfolio level analysis_ \u2014PM can test multiple markets and present a portfolio level report of them combined together.\n  * _Money management_ \u2014the Fixed Fractional technique was introduced and applied to our large portfolio of 30+ markets using the **Turtle 55** algorithm. Money management offers a process that normalizes one market to another, and offers a method to allocate capital in an efficient manner.\n  * _Portfolio analysis_ \u2014PM offers several different ways to manage a portfolio's equity curve: \n    * Portfolio profit target\n    * Portfolio stop loss\n    * Portfolio trailing stop\n\nThe first two methods were tested and reviewed.\n\nIn addition to introducing Portfolio Maestro, this chapter also discussed money management and portfolio analysis. The topics of all three were barely scratched. However, I think enough was provided to excite further research on all three subjects. Ideas such as money management and portfolio management can be hypothesized all day long, but until you put these ideas to the test, you won't know if they are truly valuable. Most beginning traders do not believe money management techniques can be applied to small accounts, and they are partially correct. However, the concept of risk is always universally applicable, and that is a big part of all money management techniques. All traders can learn from these techniques and apply them in one form or another to their own trading.\n\nTo trade or not to trade\u2014that is the question. Traders know there are times when standing aside with a flat position is a good trading strategy. However, determining these times is as difficult as timing the market. Should trading stop when a trading account is down a certain percentage during the month, or is it a good time to stop when the account is up a certain percentage? With tools such as PM, traders no longer need to wonder; they can set up the test and see for themselves. PM has many different built-in criteria, but users are not limited to just those. Its users can also dream up their own what-if scenarios, and program them directly into PM. This chapter solely relied on PM to carry out the various tests, but many of these same tests can also be accomplished with AmiBroker or TradersStudio.\n\n# Appendix A\n\n## AmiBroker\n\nHere is a brief list of token names (keywords and identifiers) that are used internally in AmiBroker.\n\n## Keywords\n\nThe following is a subset of keywords:\n\n  * **buy** \u2014Defines \"buy\" (enter long position) trading rule.\n  * **sell** \u2014Defines \"sell\" (close long position) trading rule.\n  * **short** \u2014Defines \"short\" (enter short position \u2013 short sell) trading rule.\n  * **cover** \u2014Defines \"cover\" (close short position \u2013 buy to cover) trading rule.\n  * **buyprice** \u2014Defines buying price array (this array is filled in with the default values according to the Automatic Analyser settings).\n  * **sellprice** \u2014Defines selling price array (this array is filled in with the default values according to the Automatic Analyser settings).\n  * **shortprice** \u2014Defines short selling price array (this array is filled in with the default values according to the Automatic Analyser settings).\n  * **coverprice** \u2014Defines buy to cover price array (this array is filled in with the default values according to the Automatic Analyser settings).\n  * **exclude** \u2014If defined, a true (or 1) value of this variable excludes current symbol from scan\/exploration\/backtest. They are also not considered in buy-and-hold calculations. Useful when you want to narrow your analysis to certain set of symbols.\n  * **roundlotsize** \u2014Defines round lot sizes used by backtester (see explanations below). Automatic Analysis (new in 4.10).\n  * **ticksize** \u2014Defines tick size used to align prices generated by built-in stops (see explanations below). (Note: It does not affect entry\/exit prices specified by **buyprice \/ sellprice \/ shortprice \/ coverprice**.)\n  * **pointvalue** \u2014Allows to read and modify future contract point value (see backtesting futures). CAVEAT: this AFL variable is by default set to 1 (one) regardless of contents of Information window UNLESS you turn ON futures mode (SetOption(\"FuturesMode\", True )).\n  * **margindeposit** \u2014Allows to read and modify future contract margin (see backtesting futures).\n  * **positionsize** \u2014Allows control dollar amount or percentage of portfolio that is invested into the trade (more information available in the \"Tutorial: Backtesting Your Trading Ideas\").\n  * **positionscore** \u2014Defines the score of the position. (More details: \"Tutorial: Portfolio Backtesting.\") Automatic analysis.\n  * **numcolumns** \u2014Exploration only: defines the number of your own columns (excluding predefined ticker and date columns) and assigns the column value to the variable.\n  * **filter** \u2014Exploration only: controls which symbols\/quotes are accepted. If \"true\" (or 1) is assigned to that variable for given symbol\/quote, it will be displayed in the report. So, for example, the following formula will accept all symbols with closing prices greater than 50: **filter** = close > 50;.\n\n## Flow Control Structures\n\nHere are some identifiers to help you control program flow:\n\n**Loops:**\n\n  1. **do** (part of do-while statement)\n  2. **while**\n  3. **for**\n\n**Conditional execution \/ Flow control:**\n\n  1. **if** (part of if-else statement)\n  2. **else** (part of if-else statement)\n  3. **switch**\n  4. **break** (part of the switch statement or for\/while statements)\n  5. **case** (part of the switch statement)\n  6. **continue** (part of for\/while statements)\n  7. **default** (part of switch statement)\n\n## Functions\n\nThe following programming constructs are available for the creation of subprograms:\n\n  1. function\n  2. procedure\n  3. return\n  4. local (variable scope)\n  5. global (variable scope)\n\n## Utilizing Exploration for Debugging\n\nAmiBroker provides a very easy to use form of program output to help with analysis and as a tool for debugging. You have probably already seen the **Explore** button right beside the **Backtest** button. **Explore** allows you to output various values created by your **AFL** code in an easy to read spreadsheet format. This output is displayed in the **Results** tab\u2014the same place where a backtest outputs its information. Here is an example of an exploration that prints out various indicator values:\n\n* * *\n\n    \/* Sample Exploration\n       Print out some indicator values *\/\n    Filter= 1;\n    myMACD = MACD(12,26);\n    myRSI = RSI(14);\n    myStoch = StochD(14,3,3);\n    myMovAvg = MA(C,19);\n    myClose = C;\n    AddColumn(myRSI,\"RSI\");\n    AddColumn(myStoch,\"StochD\");\n    AddColumn(C,\"Close\");\n    AddColumn(C>myMovAvg,\"C>MAV(19)\");\n    AddColumn(C,\"Close\",format= 1.4);\n\n* * *\n\nThe code snippet starts out by assigning MACD, RSI, StochD, and MAV values to four user-defined variables: myMACD, myRSI, myStoch, and myAvg. These variables are then used in the **AddColumn** function along with a descriptive name. **AddColumn** can take a variable number of parameters, but most of the time you will only need to pass two or three. The first parameter is the name of the array that you want to print and the second parameter is a string that will be printed as the column heading. The third and optional parameter is the format the array will be printed out in. Figure A.1 shows the printout of this particular exploration.\n\n**Figure A.1** Results of an exploration with three parameters.\n\nNotice how the first \"Close\" column only shows two decimal places, whereas the second \"Close\" column has been formatted to show four. The **AddColumn** capability is very powerful because it is so easy to print out the internal workings of your algorithm. This tool comes in really handy when a trading algorithm isn't performing the way you intended\u2014just print out the values that make up the criteria of your **buy \/ sell \/ short \/ cover** logic. Remember to include Filter = 1 at the top of your code listing. This informs AmiBroker to go ahead and allow values to be printed out in a spreadsheet format in the **Results** window.\n\nHere is the source code from Chapter 4 that included some interesting topics that is discussed here.\n\n* * *\n\n    \/\/Chapter 2 MACD utilizing AMIBROKERS\n    \/\/Exploration Feature\n    Filter = 1;\n    PositionSize = MarginDeposit = 1;\n\n* * *\n\nSetting **Filter** = 1 turns on the **Exploration** tool. **PositionSize = MarginDeposit = 1** facilitates futures trading mode.\n\n* * *\n\n    myMACD = MACD(fast=12,slow =26);\n    myMACDAvg = MA(myMACD,9);\n    myMACDDiff = myMACD - MYMACDAvg;\n    leftBar2 = Ref(myMACDDiff,-4);\n    leftBar1 = Ref(myMACDDiff,-3);\n    centerBar = Ref(myMACDDiff,-2);\n    rightBar1 = Ref(myMACDDiff,-1);\n    rightBar2 = Ref(myMACDDiff,0);\n\n* * *\n\nCall the **MACD** function and calculate the difference between it and its nine-period smoothing average. Utilize **Ref** to get historic values in the **myMACDDiff** array. Visualize the MACD difference plotted as a histogram. This bit of code is gathering information to determine if pivot point of the histogram has occurred. **CenterBar** is two bars back and leftBar1, leftBar2, rightBar1, and rightBar2 are self-explanatory.\n\n* * *\n\n    COND3 = C > MA(C,100);\n    COND1 = centerBar < 0 AND centerBar < Min(leftBar2, leftBar1) AND centerBar < min(rightBar1,rightBar2);\n    COND2 = centerBar > 0 AND centerBar > Max(leftBar2, leftBar1) AND centerBar > Max(rightBar1,rightBar2);\n\n* * *\n\n  1. **COND3** is set to true when the close is above the 100-day moving average.\n  2. **COND1** is set to true when the center bar forms a pivot low and is below 0.\n  3. **COND2** is set to true when the center bar forms a pivot high and is above 0.\n\n* * *\n\n    Buy = COND1 AND COND3;\n    Short = COND2 AND NOT(COND3);\n    BuyPrice = C;\n    ShortPrice = C;\n    longEntryPrice = ValueWhen(Buy,BuyPrice,1);\n    shortEntryPrice = ValueWhen(Short,ShortPrice,1);\n    Buy = ExRem(Buy,Short);\n    Short = ExRem(Short,Buy);\n\n* * *\n\nThe **Buy** array is filled with 1's when a pivot low has formed below zero in the histogram and the current price is below the 100-day moving average. The **Short** array is filled with 1's when a pivot high has formed and the current price is below the 100-day moving average. The **longEntryPrice** is captured when **Buy** is set to 1. The **ValueWhen** function returns the second array value when the first array is set to 1. You must use the **ExRem** function to remove redundant Buys and Shorts. In other words, only turn subsequent Buys on or Shorts on when the existing long\/short positions are either liquidated or reversed.\n\n* * *\n\n    Sell = Cross(C, longEntryPrice - 3 * ATR(10));\n    Cover = Cross(ShortEntryPrice + 3 *ATR(10),C);\n    Buy = ExRem(Buy,Sell);\n    Short = ExRem(Short,Cover);\n    AddColumn(longEntryPrice,\"BuyPrice\");\n    AddColumn(longEntryPrice - 3 * ATR(10),\"longStop\");\n    AddColumn(Sell,\"Sell ?\");\n\n* * *\n\n**Sell** array (long liquidation) is turned on when the close crosses 3 ATR below the **longEntryPrice**.\n\n## Position Sizing in Futures Mode\n\nHere is the exact formula used to apply an ATR Risk-Based Fixed Fractional money management scheme to a trading algorithm.\n\n* * *\n\n    RiskPerContract = 2 * ATR(10);\n    PositionRisk = 1;\n    PctSize =  PositionRisk * MarginDeposit  \/ ( RiskPerContract * PointValue );\n    SetPositionSize( PctSize, spsPercentOfEquity );\n\n* * *\n\nAt first, the formula might look confusing, but plugging in some values will help explain things. Assume you are working with the emini-SP and the **MarginDeposit** is $5,600, **PointValue** is $50,  , and  .\n\n**PctSize** then can be used to calculate **PositionSize** :\n\nCompare this to the Fixed Fractional calculation without using the **marginDeposit** :\n\nYou get the exact same number of contracts, and since you can't have a partial contract, the number of contracts is rounded down to 2.\n\n# Appendix B\n\n## Excel System Backtester\n\nThe Excel System Backtester (ESB) utilizes Visual Basic for Applications and Excel spreadsheets to carry out the backtesting of your algorithms. There are several data arrays, keywords, and identifiers you will want to be aware of and not use as user-defined variables. This list will be updated regularly on www.georgepruitt.com. You can link to the page from this book's companion site.\n\nThe ESB source code is open source. Feel free to use it and modify it. A lengthy discussion on how the software works can be found at www.georgepruitt.com.\n\n## Data Arrays\n\n  1. myDate()\n  2. myHigh()\n  3. myLow()\n  4. myOpen()\n  5. myClose()\n  6. myRange()\n  7. myTrueRange()\n  8. myVol()\n  9. myOpInt()\n  10. equityStream()\n\n## Keywords\n\n  1. prevMarketPosition\n  2. marketPosition\n  3. entryPrice\n  4. executionCount\n  5. entryBar\n  6. intraDayTrdCnt\n  7. multiDayOrders\n  8. Buy\n  9. Sell\n  10. ExitLong\n  11. ExitShort\n  12. myTickValue\n  13. myMinTick\n  14. RampUp\n  15. symbol\n  16. totProfit\n  17. maxDD\n  18. perCentWins\n  19. numTrades\n  20. numWins\n  21. numLosses\n  22. commsn\n  23. equityPeak\n  24. dVal\n  25. dSloVal\n  26. signalName\n  27. orderKindArr(10)\n  28. trdPriceArr(10)\n  29. orderTypeArr(10)\n  30. sigNameArr(10)\n  31. barsLong\n  32. barsShort\n  33. tradeDays\n  34. numRecords\n  35. stp\n  36. lmt\n  37. mkt\n  38. moc\n  39. myTradeCollection\n  40. EquityClass\n  41. myEquityCollection\n  42. TradeInfo\n\n## Functions and Subroutines\n\nHere is a list of available functions and subroutines in this current version of the ESB. Several of the indicators' functions are \"Data Length Dependent,\" meaning that the current value of the indicator is dependent on when it was first applied to the data. This dependency does not have an impact if a large amount of data is used. The RSI on a six-week crude chart read 59.77, whereas it read 56.77 on a one-year chart. This dependency requires the last value to be carried over into the current indicator calculation, and this is no big deal unless you call the indicator function\/subroutine multiple times on a single bar of data. Let's say you want an RSI(14) and an RSI(30) indicator calculated on the same bar; this requires a multiple function\/subroutine call where the last values of the respective RSI calculations need to be carried over. Without the use of a class structure, the only way this can be handled is to pass values back and forth to the subroutine. Here is an example of how to handle a multiple subroutine call with different parameter values:\n\n  1. **Call RSISub(myClose, 14, rsiVal1, upSumAvg1, dnSumAvg1, i, 1)**\n  2. **Call RSISub(myClose, 30, rsiVal2, upSumAvg2, dnSumAvg2, i, 1)**\n\nThe first subroutine call holds the RSI value in **rsiVal1** and the **upSumAvg1** and **dnSumAvg1** values are held over until the next subroutine call. The second subroutine call stores its RSI value in **rsiVal2** and the up sum and down sum averages in **upSumAvg2** and **dnSumAvg2** , respectively. So keep in mind that if you want to call a subroutine more than once, then you will need to create separate parameter names so that the subroutine can keep track of the different lookback lengths and indicator values.\n\n#### _Function Highest(dataList, length, index, offset)_\n\n* * *\n\n    Function Highest(dataList, length, index, offset)\n       Dim i As Integer\n    Dim tempHH As Integer\n    tempHH = 0\n    For i = (index - offfset) - (length - 1) To (index - offset)\n             If (dataList(i) > tempHH) Then tempHH = myHigh(i)\n    Next i\n    Highest = tempHH\n    End Function\n\n* * *\n\n#### _Sub RSISub(dataList, length, rsiVal, upSumAvg, dnSumAvg, index, offset)_\n\n* * *\n\n    Sub RSISub(dataList, length, rsiVal, upSumAvg, dnSumAvg, index, offset)\n    Dim i As Integer\n    Dim diff1, diff2, upSum, dnSum As Double\n    upSum = 0\n    dnSum = 0\n    If upSumAvg = 0 And dnSumAvg = 0 Then     'seed the original RSI Value\n        For i = (index - offset) - (length - 1) To\n                  (index - offset)\n            If dataList(i) > dataList(i - 1) Then\n                diff1 = dataList(i) - dataList(i - 1)\n                upSum = upSum + diff1\n            End If\n            If dataList(i) < dataList(i - 1) Then\n                diff2 = dataList(i - 1) - dataList(i)\n                dnSum = dnSum + diff2\n            End If\n        Next i\n        upSumAvg = upSum \/ length\n        dnSumAvg = dnSum \/ length\n    Else\n        If dataList(index - offset) > dataList(index - 1 - offset) Then\n            diff1 = dataList(index - offset) - dataList(index - 1 - offset)\n            upSum = upSum + diff1\n        End If\n        If dataList(index - offset) < dataList(index - 1 - offset) Then\n            diff2 = dataList(index - 1 - offset) - dataList(index - offset)\n            dnSum = dnSum + diff2\n        End If\n        upSumAvg = (upSumAvg * (length - 1) + upSum) \/ length\n        dnSumAvg = (dnSumAvg * (length - 1) + dnSum) \/ length\n    End If\n    If upSumAvg + dnSumAvg <> 0 Then\n        rsiVal = (100 * (upSumAvg)) \/ (upSumAvg + dnSumAvg)\n    Else\n        rsiVal = 0\n    End If\n    End Sub\n\n* * *\n\n#### _Function RSI(dataList, length, index, offset)_\n\n* * *\n\n    Function RSI(dataList, length, index, offset)\n    Dim i As Integer\n    Dim diff1, diff2, upSum, dnSum As Double\n    Static upSumAvg, dnSumAvg As Double\n    upSum = 0\n    dnSum = 0\n    If upSumAvg = 0 And dnSumAvg = 0 Then\n          'seed the original RSI Value\n        For i = (index - offset) - (length - 1) To\n                   (index - offset)\n            If dataList(i) > dataList(i - 1) Then\n                diff1 = dataList(i) - dataList(i - 1)\n                upSum = upSum + diff1\n            End If\n            If dataList(i) < dataList(i - 1) Then\n                diff2 = dataList(i - 1) - dataList(i)\n                dnSum = dnSum + diff2\n            End If\n        Next i\n        upSumAvg = upSum \/ length\n        dnSumAvg = dnSum \/ length\n    Else\n        If dataList(index - offset) > dataList(index - 1 - offset) Then\n            diff1 = dataList(index - offset) -\n    \t           dataList(index - 1 - offset)\n            upSum = upSum + diff1\n        End If\n        If dataList(index - offset) < dataList(index - 1 - offset) Then\n            diff2 = dataList(index - 1 - offset) - dataList(index - offset)\n            dnSum = dnSum + diff2\n        End If\n        upSumAvg = (upSumAvg * (length - 1) + upSum) \/ length\n        dnSumAvg = (dnSumAvg * (length - 1) + dnSum) \/ length\n    End If\n    If upSumAvg + dnSumAvg <> 0 Then\n        RSI = (100 * (upSumAvg)) \/ (upSumAvg + dnSumAvg)\n    Else\n        RSI = 0\n    End If\n    End Function\n\n* * *\n\n#### _Function Lowest(dataList, length, index, offset)_\n\n* * *\n\n    Function Lowest(dataList, length, index, offset)\n    Dim i As Integer\n    Dim tempLL As Double\n        tempLL = 999999\n        For i = (index - offset) - (length - 1) To (index - offset)\n            If (dataList(i) < tempLL) Then tempLL = myLow(i)\n        Next i\n        Lowest = tempLL\n    End Function\n\n* * *\n\n#### _Sub BollingerBand(dataList, length, numDevs, avg, upBand, dnBand, index, offset)_\n\n* * *\n\n    Sub BollingerBand(dataList, length, numDevs, avg, upBand,\n      dnBand, index, offset)\n    Dim i As Integer\n    Dim sum, sum1, myDev As Double\n    For i = (index - offset) - (length - 1) To (index - offset)\n        sum = sum + dataList(i)\n        sum1 = sum1 + dataList(i) ^ 2\n    Next i\n    avg = sum \/ length\n    myDev = ((length * sum1 - sum ^ 2) \/ (length * (length - 1))) ^ 0.5\n    upBand = avg + myDev * numDevs\n    dnBand = avg - myDev * numDevs\n    End Sub\n\n* * *\n\n#### _Function Average(dataList, length, index, offset)_\n\n* * *\n\n    Function Average(dataList, length, index, offset)\n    Dim i As Integer\n    Dim sum, sum1, myDev As Double\n    For i = (index - offset) - (length - 1) To index - offset\n        sum = sum + dataList(i)\n    Next i\n    Average = sum \/ length\n    End Function\n\n* * *\n\n#### _Function Xaverage(dataList, prevXavg, length, index, offset)_\n\n* * *\n\n    Function Xaverage(dataList, prevXavg, length, index, offset)\n    If prevXavg = 0 Then\n        Xaverage = dataList(index - offset)\n    Else\n        Xaverage = prevXavg + 2 \/ length * (dataList(index\n                     - offset) - prevXavg)\n    End If\n    End Function\n\n* * *\n\n#### _Sub MACD(dataList, shortLen, longLen, smooth, myMacd, mySmoothMacd, xMavg1, xMavg2, index, offset)_\n\n* * *\n\n    Sub MACD(dataList, shortLen, longLen, smooth, myMacd,\n          mySmoothMacd, xMavg1, xMavg2, index, offset)\n    If xMavg1 = 0 And xMavg2 = 0 Then\n        xMavg1 = dataList(index - offset)\n        xMavg2 = dataList(index - offset)\n        myMacd = 0\n        mySmoothMacd = 0\n    Else\n        xMavg1 = xMavg1 + 2 \/ shortLen * (dataList(index\n                   - offset) - xMavg1)\n        xMavg2 = xMavg2 + 2 \/ longLen * (dataList(index\n                   - offset) - xMavg2)\n        myMacd = xMavg1 - xMavg2\n        mySmoothMacd = mySmoothMacd + 2 \/ smooth * (myMacd - mySmoothMacd)\n    End If\n    End Sub\n\n* * *\n\n### Sub Stochastic(kLen, dLen, dSloLen, kVal, dVal, dSloVal, index, offset)\n\n* * *\n\n    Sub Stochastic(kLen, dLen, dSloLen, kVal, dVal, dSloVal, index, offset)\n    Dim index1, index2, hh, ll, sum As Double\n    Dim i, j, k As Integer\n    Dim kSto(100) As Double\n    Dim dSto(100) As Double\n    indexPt = index - offset\n    index1 = kLen + dLen\n    index2 = dLen - 1\n    If (kVal + dVal + dSloVal = 0) Then\n      'seed the original Sto Value\n        For i = 1 To dLen + dSloLen - 1\n            hh = 0\n            ll = 999999\n            For k = indexPt - (index1 - (i - 1)) To\n                      indexPt - (index2 - (i - 1))\n                If (myHigh(k) > hh) Then hh = myHigh(k)\n                If (myLow(k) < ll) Then ll = myLow(k)\n            Next k\n            If (hh - ll = 0) Then hh = ll + 1\n            kSto(i) = (myClose(indexPt - (index2 - (i - 1)))\n                        - ll) \/ (hh - ll) * 100#\n            kVal = kSto(i)\n            If (i > dLen) Then\n                sum = 0#\n                For j = i - 2 To i\n                    sum = sum + kSto(j)\n                Next j\n                dSto(i) = sum \/ dLen\n                dVal = dSto(i)\n            End If\n            If (i >= dLen + dSloLen - 1) Then\n                sum = 0#\n                For j = i - 2 To i\n                    sum = sum + dSto(j)\n                Next j\n                dSloVal = sum \/ dSloLen\n            End If\n        Next i\n    Else\n        hh = 0\n        ll = 999999\n        For i = indexPt - (kLen - 1) To indexPt\n            If (myHigh(i) > hh) Then hh = myHigh(i)\n            If (myLow(i) < ll) Then ll = myLow(i)\n        Next i\n        kVal = (myClose(indexPt) - ll) \/ (hh - ll) * 100\n        dVal = ((dVal * (dLen - 1)) + kVal) \/ dLen\n        dSloVal = ((dSloVal * (dSloLen - 1)) + dVal) \/ dSloLen\n    End If\n    End Sub\n\n* * *\n\n# Appendix C\n\n## Python System Backtester\n\nThe Python System Backtester (PSB) utilizes the Python programming language to carry out the backtesting of your algorithms. There are several data arrays, keywords, and identifiers you will want to be aware of and not use as user-defined variables. This list will be updated regularly on www.georgepruitt.com.\n\n## Data Arrays or Lists\n\n  1. listOfTrades\n  2. marketPosition\n  3. entryPrice\n  4. entryQuant\n  5. exitQuant\n  6. trueRanges\n  7. myDate\n  8. myTime\n  9. myOpen\n  10. myHigh\n  11. myLow\n  12. myClose\n  13. myVolume\n  14. myOpInt\n  15. dataClassList\n  16. systemMarketList\n  17. equityDataList\n  18. fileList\n  19. exitQuant\n\n## Keywords and Identifiers\n\n  1. currentPrice\n  2. totComms\n  3. barsSinceEntry\n  4. numRuns\n  5. myBPV\n  6. allowPyr\n  7. curShares\n  8. numMarkets\n  9. portfolio\n  10. commission\n  11. numBarsToGoBack\n  12. rampUp\n  13. price\n  14. trades\n  15. totProfit\n  16. todaysCTE\n  17. todaysOTE\n  18. tradeName\n  19. mp\n  20. entryDate\n  21. exitDate\n\n## Classes\n\n#### _equityClass_\n\n* * *\n\n    class equityClass(object):\n        def __init__(self):\n            self.equityDate = list()\n            self.equityItm = list()\n            self.clsTrdEquity = list()\n            self.openTrdEquity = list()\n            self.cumuClsEquity = 0\n            self.dailyEquityVal = list()\n            self.peakEquity = 0\n            self.minEquity = 0\n            self.maxDD = 0\n        def setEquityInfo(self,equityDate,equityItm,clsTrdEquity, openTrdEquity):\n            self.equityDate.append(equityDate)\n            self.equityItm.append(equityItm)\n            self.cumuClsEquity += clsTrdEquity\n            tempEqu =self.cumuClsEquity+openTrdEquity\n            self.dailyEquityVal.append(tempEqu)\n            self.peakEquity = max(self.peakEquity,tempEqu)\n            maxEqu = self.peakEquity\n            self.minEquity = min(self.minEquity,tempEqu)\n            minEqu = self.minEquity\n            self.maxDD = max(self.maxDD,maxEqu-tempEqu)\n            maxDD = self.maxDD\n            maxDD = maxDD\n\n* * *\n\n#### _marketDataClass_\n\n* * *\n\n    class marketDataClass(object):\n        def __init__(self):\n            self.symbol = \"\"\n            self.minMove = 0\n            self.bigPtVal = 0\n            self.seed = 0\n            self.date = list()\n            self.open = list()\n            self.high = list()\n            self.low = list()\n            self.close = list()\n            self.volume = list()\n            self.opInt = list()\n            self.dataPoints = 0\n        def setDataAttributes(self,symbol,bigPtVal,minMove):\n            self.symbol = symbol\n            self.minMove = minMove\n            self.bigPtVal = bigPtVal\n        def readData(self,date,open,high,low,close,volume,opInt):\n            self.date.append(date)\n            self.open.append(open)\n            self.high.append(high)\n            self.low.append(low)\n            self.close.append(close)\n            self.volume.append(volume)\n            self.opInt.append(opInt)\n            self.dataPoints += 1\n\n* * *\n\n#### _tradeInfoClass_\n\n* * *\n\n    class tradeInfo(object):\n        def __init__(self,tradeOrder,tradeDate,tradeName, tradePrice,quant,entryOrExit):\n            self.tradeOrder = tradeOrder\n            self.tradeDate = tradeDate\n            self.tradeName = tradeName\n            self.tradePrice = tradePrice\n            self.quant = quant\n            self.tradeProfit = 0\n            self.cumuProfit = 0\n            self.entryOrExit = entryOrExit\n    #        print(\"populating info: \",self.tradeName,' ', # self.tradePrice)\n        def calcTradeProfit(self,order,curPos,entryPrice,\n               exitPrice,entryQuant,numShares):\n            profit = 0\n            totEntryQuant = 0\n            tempNumShares = numShares\n            numEntriesLookBack = 0\n            for numEntries in range(0,len(entryPrice)):\n    ##            totEntryQuant += entryQuant[numEntries]\n                if tempNumShares >= entryQuant[numEntries]:\n                    tempNumShares -= entryQuant[numEntries]\n                    numEntriesLookBack += 1\n            if tempNumShares > 0 : numEntriesLookBack += 1\n            tempNumShares = numShares\n            for numEntries in range(0,numEntriesLookBack):\n                if numEntries < 0:\n                    numEntries = 1\n                if entryQuant[numEntries] < tempNumShares:\n                    peelAmt = entryQuant[numEntries]\n                    tempNumShares = tempNumShares - peelAmt\n                if entryQuant[numEntries] >= tempNumShares:\n                    peelAmt = tempNumShares\n                if order == 'buy':\n                    if curPos < 0:\n                        profit = profit + (entryPrice[numEntries] - exitPrice) * peelAmt\n                elif order == 'sell':\n                    if curPos > 0:\n                        profit = profit + (exitPrice - entryPrice[numEntries]) * peelAmt\n                elif order == 'liqLong':\n                    if curPos > 0:\n                        profit = profit + (exitPrice - entryPrice[numEntries]) * peelAmt\n                elif order == 'liqShort':\n                    if curPos < 0:\n                        profit = profit + (entryPrice[numEntries] - exitPrice) * peelAmt\n                if entryQuant[numEntries] == peelAmt :\n                    entryPrice.pop(numEntries)\n                    entryQuant.pop(numEntries)\n                elif entryQuant[numEntries] > peelAmt:\n                    entryQuant[numEntries] = entryQuant[numEntries] - peelAmt\n            return profit\n        def printTrade(self):\n              print( '%8.0f %10s %2.0d %8.4f %10.2f %10.2f' % (self.tradeDate, self.tradeName, self.quant, self.tradePrice,self.tradeProfit, self.cumuProfit))\n\n* * *\n\n#### _portfolioClass_\n\n* * *\n\n    from systemMarket import systemMarketClass\n    class portfolioClass(object):\n        def __init__(self):\n            self.portfolioName = \"\"\n            self.systemMarkets = list()\n            self.portEquityDate = list()\n            self.portEquityVal = list()\n            self.portclsTrdEquity = list()\n            self.portDailyEquityVal = list()\n            self.portPeakEquity = 0\n            self.portMinEquity = 0\n            self.portMaxDD = 0\n            tempEqu = 0\n            cumEqu = 0\n            maxEqu = -999999999\n            minEqu = 999999999\n            maxDD = 0\n        def setPortfolioInfo(self,name,systemMarket):\n            self.portfolioName = name\n            self.systemMarkets = list(systemMarket)\n            masterDateList = list()\n            monthList = list()\n            monthEquity = list()\n            combinedEquity = list()\n            self.portPeakEquity = -999999999999\n            self.portMinEquity = -999999999999\n            for i in range(0,len(self.systemMarkets)):\n                masterDateList += self.systemMarkets[i].equity.equityDate\n                sysName = self.systemMarkets[i].systemName\n                market = self.systemMarkets[i].symbol\n                avgWin = self.systemMarkets[i].avgWin\n                sysMark =self.systemMarkets[i]\n                avgLoss = sysMark.avgLoss\n                totProf = sysMark.profitLoss\n                totTrades = sysMark.numTrades\n                maxD = sysMark.maxxDD\n                perWins = sysMark.perWins\n                tempStr =\"\"\n                if len(sysName) - 9 > 0:\n                    for j in range(0,len(sysName)-8):\n                        tempStr = tempStr + ' '\n                if i == 0: print('SysName',tempStr,'Market TotProfit MaxDD AvgWin AvgLoss PerWins TotTrades')\n                print('%s %s %12d %6d  %5d   %5d    %3.2f %4d' % (sysName,market,totProf,maxD, avgWin,avgLoss,perWins,totTrades))\n            masterDateList = removeDuplicates(masterDateList)\n            masterDateList = sorted(masterDateList)\n    #        print(masterDateList)\n            self.portEquityDate = masterDateList\n            monthList = createMonthList(masterDateList)\n            for i in range(0,len(masterDateList)):\n                cumuVal = 0\n                for j in range(0,len(self.systemMarkets)):\n                    skipDay = 0\n                    try:\n                        idx = self.systemMarkets[j] .equity.equityDate.index(masterDateList[i])\n                    except ValueError:\n                        skipDay = 1\n                    if skipDay == 0:\n                        cumuVal += self.systemMarkets[j] .equity.dailyEquityVal[idx]\n                combinedEquity.append(cumuVal)\n                self.portEquityVal.append(cumuVal)\n                if cumuVal > self.portPeakEquity: self .portPeakEquity = cumuVal\n                self.portMinEquity = max(self.portMinEquity, self.portPeakEquity - cumuVal)\n                self.portMaxDD = self.portMinEquity\n            print(\"Combined Equity: \",self.portEquityVal[-1])\n            print(\"Combined MaxDD: \",self.portMaxDD)\n            print(\"Combined Monthly Return\")\n            for j in range(0,len(monthList)):\n                idx = masterDateList.index(monthList[j])\n                if j == 0:\n                    monthEquity.append(combinedEquity[idx])\n                    prevCombinedDailyEquity = monthEquity[-1]\n                else:\n                    combinedDailyEquity = combinedEquity[idx]\n                    monthEquity.append(combinedDailyEquity -  prevCombinedDailyEquity)\n                    prevCombinedDailyEquity = combinedDailyEquity\n                print('%8d %10.0f %10.0f ' % (monthList[j], monthEquity[j],combinedEquity[idx]))\n    def removeDuplicates(li):\n        my_set = set()\n        res = []\n        for e in li:\n            if e not in my_set:\n                res.append(e)\n                my_set.add(e)\n        return res\n    def createMonthList(li):\n        myMonthList = list()\n        for i in range(0,len(li)):\n            if i != 0:\n                tempa = int(li[i]\/100)\n                pMonth = int(li[i-1]\/100) % 100\n                month = int(li[i]\/100) % 100\n                if pMonth != month:\n                    myMonthList.append(li[i-1])\n                if i == len(li)-1:\n                    myMonthList.append(li[i])\n        return myMonthList\n\n* * *\n\n## Indicator Classes and Functions\n\nSome indicators are programmed as classes due to their \"data length dependence.\" Each indicator class can be instantiated multiple times, so the same indicator can be calculated multiple times on the same data bar.\n\n#### _class stochClass(object)_\n\n* * *\n\n    class stochClass(object):\n        def __init__(self):\n            self.fastK = 0\n            self.fastD = 0\n            self.slowD = 0\n            self.seed = 0\n        def calcStochastic(self,kLen,dLen,dSloLen,hPrices, lPrices,cPrices,curBar,offset):\n            curBarLookBack = curBar - offset\n            testSeed = self.seed\n            if self.seed == 0:\n                self.seed = 1\n                stoKList =[]\n                stoDList = []\n                index1 = kLen - 1 + dLen - 1 + dSloLen -1\n                index2 = dLen - 1 + dSloLen -1\n                loopCnt = 0\n                for i in range(0,index2+1):\n                    loopCnt = loopCnt + 1\n                    hh = 0\n                    ll = 9999999\n                    lowRngBound = curBarLookBack - (index1 - i)\n                    highRngBound =lowRngBound + 3\n                    for k in range(lowRngBound,highRngBound):\n                        if hPrices[k] > hh:\n                            hh = hPrices[k]\n                        if lPrices[k] < ll:\n                            ll = lPrices[k]\n                    if hh - ll == 0.0:\n                        hh = ll + 1\n                    whichClose = curBarLookBack - (index2 -i)\n                    stoKList.append((cPrices[whichClose] - ll) \/ (hh - ll) *100)\n                    lenOfStoKList = len(stoKList)\n                    self.fastK = stoKList[len(stoKList)-1]\n                    if (i >= dLen-1):\n                        tempSum = 0\n                        lowRngBound = len(stoKList)-dLen\n                        highRngBound = lowRngBound + dLen\n                        for j in range(lowRngBound,highRngBound):\n                            tempSum += stoKList[j]\n                        stoDList.append(tempSum\/dLen)\n                        self.fastD = stoDList[len(stoDList)-1]\n                    if (i == index2):\n                        tempSum = 0\n                        lowRngBound = len(stoDList) - dSloLen\n                        highRngBound = lowRngBound + dSloLen\n                        for j in range(lowRngBound,highRngBound):\n                            tempSum += stoDList[j]\n                        self.slowD = tempSum \/ dSloLen\n            else:\n                hh = 0\n                ll = 999999\n                lowRngBound = curBarLookBack - (kLen - 1)\n                highRngBound = lowRngBound + 3\n                for i in range(lowRngBound, highRngBound):\n                    if hPrices[i] > hh:\n                        hh = hPrices[i]\n                    if lPrices[i] < ll:\n                        ll = lPrices[i]\n                self.fastK = (cPrices[curBarLookBack] - ll )\/ (hh - ll) * 100\n                self.fastD = (self.fastD * (dLen - 1) + self.fastK) \/ dLen\n                self.slowD = ((self.slowD * (dSloLen - 1)) + self.fastD) \/ dSloLen\n            return(self.fastK,self.fastD,self.slowD)\n\n* * *\n\n#### _class rsiClass(object)_\n\n* * *\n\n    class rsiClass(object):\n        oldDelta1 = 0\n        def __init__(self):\n            self.delta1 = 0\n            self.delta2 = 0\n            self.rsi = 0\n            self.seed = 0\n        def calcRsi(self,prices,lookBack,curBar,offset):\n            upSum = 0.0\n            dnSum = 0.0\n            if self.seed == 0:\n                self.seed = 1\n                for i in range((curBar - offset) - (lookBack-1), curBar - offset +1):\n                    if prices[i] > prices[i-1]:\n                        diff1 = prices[i] - prices[i-1]\n                        upSum += diff1\n                    if prices[i] < prices[i-1]:\n                        diff2 = prices[i-1] - prices[i]\n                        dnSum += diff2\n                    self.delta1 = upSum\/lookBack\n                    self.delta2 = dnSum\/lookBack\n            else:\n                if prices[curBar - offset] > prices[curBar - 1 - offset]:\n                    diff1 = prices[curBar - offset] - prices[curBar - 1 - offset]\n                    upSum += diff1\n                if prices[curBar - offset] < prices[curBar - 1 - offset]:\n                    diff2 = prices[curBar - 1 - offset] - prices[curBar - offset]\n                    dnSum += diff2\n                self.delta1 = (self.delta1 * (lookBack -1) + upSum) \/ lookBack\n                self.delta2 = (self.delta2 * (lookBack -1) + dnSum) \/ lookBack\n            if self.delta1 + self.delta2 != 0:\n                self.rsi = (100.0 * self.delta1) \/ (self.delta1 + self.delta2)\n            else:\n                self.rsi = 0.0\n            return (self.rsi)\n\n* * *\n\n#### _Highest Function\/Module_\n\n* * *\n\n    def highest(prices,lookBack,curBar,offset):\n        result = 0.0\n        maxVal = 0.00\n        for index in range((curBar - offset) - (lookBack-1), curBar - offset +1):\n            if prices[index] > maxVal:\n                maxVal = prices[index]\n        result = maxVal\n        return result\n\n* * *\n\n#### _Lowest Function\/Module_\n\n* * *\n\n    def lowest(prices,lookBack,curBar,offset):\n        result = 0.0\n        minVal = 9999999.0\n        for index in range((curBar - offset) - (lookBack-1), curBar - offset +1):\n            if prices[index] < minVal:\n                minVal = prices[index]\n        result = minVal\n        return result\n\n* * *\n\n#### _Simple Average Function\/Module_\n\n* * *\n\n    def sAverage(prices,lookBack,curBar,offset):\n        result = 0.0\n        for index in range((curBar - offset) - (lookBack-1), curBar - offset +1):\n            result = result + prices[index]\n        result = result\/float(lookBack)\n        return result\n\n* * *\n\n#### _Bollinger Bands Function\/Module_\n\n* * *\n\n    def bollingerBands(dates,prices,lookBack,numDevs,curBar,offset):\n        sum1 = 0.0\n        sum2 = 0.0\n        startPt = (curBar - offset)- (lookBack-1)\n        endPt = curBar - offset + 1\n        for index in range(startPt,endPt):\n            tempDate = dates[index]\n            sum1 = sum1 + prices[index]\n            sum2 = sum2 + prices[index]**2\n        mean = sum1 \/ float(lookBack)\n        stdDev = ((lookBack * sum2 - sum1**2) \/ (lookBack * (lookBack -1)))**0.5\n        upBand = mean + numDevs*stdDev\n        dnBand = mean - numDevs*stdDev\n    #    print(mean,\" \",stdDev,\" \",upBand,\" \",dnBand)\n        return upBand, dnBand, mean\n\n* * *\n\n## Python-Specific Keywords\n\n  1. and\n  2. as\n  3. assert\n  4. break\n  5. class\n  6. continue\n  7. def\n  8. del\n  9. elif\n  10. else\n  11. except\n  12. exec\n  13. finally\n  14. for\n  15. from\n  16. global\n  17. if\n  18. import\n  19. in\n  20. is\n  21. lambda\n  22. not\n  23. or\n  24. pass\n  25. print\n  26. raise\n  27. return\n  28. try\n  29. while\n  30. with\n  31. yield\n\n### Monte Carlo and Start Trade Drawdown Source Code\n\n#### _Start Trade Drawdown section_\n\n* * *\n\n    # start trade draw down analysis - utilizing the tradeTuple\n        tupleLen = len(tradeTuple)\n        tradeTuple = sorted(tradeTuple,key=itemgetter(0))\n        for x in range(0,tupleLen):\n            cumStartTradeEquity = 0\n            maxStartTradeDD = -99999999\n            maxCumEquity = 0\n            for y in range(x,tupleLen):\n    #            print(\"Trade Tuple \",tradeTuple[y][0],\" \",# tradeTuple[y][1]);\n                cumStartTradeEquity += tradeTuple[y][1]\n                maxCumEquity = max(maxCumEquity, cumStartTradeEquity)\n                maxStartTradeDD = max(maxStartTradeDD,maxCumEquity  - cumStartTradeEquity)\n            startTradeTuple += ((x,cumStartTradeEquity, maxStartTradeDD),)\n        minDD = 99999999\n        maxDD = 0\n        for y in range(0,len(startTradeTuple)):\n            print(startTradeTuple[y][0],' ',startTradeTuple[y][1], ' ',startTradeTuple[y][2])\n            if startTradeTuple[y][2] < minDD: minDD = startTradeTuple[y][2]\n            if startTradeTuple[y][2] > maxDD: maxDD = startTradeTuple[y][2]\n        numBins = 20\n        binTuple = list()\n        binMin = minDD\n        binMax = maxDD\n        binInc = (maxDD - minDD)\/20.0\n        binBot = binMin\n        for y in range(0,numBins):\n            binTop = binBot + binInc\n            binTuple += ((y,binBot,binTop),)\n            print(binTuple[y][1],' ',binTuple[y][2])\n            binBot = binTop + y\n        bins = list()\n        bins[:] = []\n        for x in range(0,numBins):\n            bins.append(0)\n        for x in range(0,len(startTradeTuple)):\n            for y in range(0,numBins):\n                tempDD = startTradeTuple[x][2]\n                tempBot = binTuple[y][1]\n                tempTop = binTuple[y][2]\n                if (tempDD >= binTuple[y][1] and tempDD < binTuple[y][2]):\n    #                tempVal = bins(y) + 1\n    #                bins.insert(y,tempVal)\n                    bins[y] += 1\n        freqSum = sum(bins)\n        binProb = list()\n        for y in range(0,numBins):\n            if y == 0:\n                binProb.append(bins[y]\/freqSum)\n            else:\n                binProb.append(bins[y]\/freqSum + binProb[y-1])\n        for y in range(0,numBins):\n            print(\"Probability of DD < %7d is %4.3f\\n\" % (binTuple[y][2], binProb[y]))\n\n* * *\n\n#### _Monte Carlo Analysis_\n\n* * *\n\n    # Monte Carlo Analysis\n        mcTradeTuple = list()\n        for x in range(0,5): # number of alternate histories\n            for y in range(0,len(tradeTuple)):\n                randomTradeNum = random.randint(0, len(tradeTuple)-1)\n                mcTradeTuple += ((x,y, tradeTuple[randomTradeNum][1], tradeTuple[randomTradeNum][0]),)\n        mcTradeResultsTuple = list()\n        whichAlt = -1\n        for x in range(0,len(mcTradeTuple)):\n            if mcTradeTuple[x][1]==0:\n                print('New Alternate History Generated')\n                cumEquity = 0\n                maxTradeDD = -99999999\n                maxCumEquity = 0\n            cumEquity += mcTradeTuple[x][2]\n            print('Randomized trade listing : ', mcTradeTuple[x][3],' ',mcTradeTuple[x][2])\n            maxCumEquity = max(maxCumEquity,cumEquity)\n            maxTradeDD = max(maxTradeDD,maxCumEquity - cumEquity)\n            if mcTradeTuple[x][1] == len(tradeTuple)-1 :\n                mcTradeResultsTuple += ((cumEquity,maxTradeDD, cumEquity\/len(tradeTuple)),)\n        for x in range(0,len(mcTradeResultsTuple)):\n            mcCumEquity = mcTradeResultsTuple[x][0]\n            mcMaxDD = mcTradeResultsTuple[x][1]\n            mcAvgTrd = mcTradeResultsTuple[x][2]\n            print('Alt history %5d Profit: %10d MaxDD: %10d Avg Trade %6d\\n' % (x,mcCumEquity, mcMaxDD,mcAvgTrd))\n\n* * *\n\n# Appendix D\n\n## TradeStation and EasyLanguage\n\n## Importing ELD file from Book Website\n\nIf you need help installing the EasyLanguage code from the book website, just follow these instructions.\n\nDownload **UATSTB.eld** from website and save to your Desktop. Launch the **TradeStation Development Environmen** t and go under the **File** menu. Select **Import\/Export** and a dialog similar to the one in Figure D.1 should appear on your screen.\n\n**Figure D.1**\n\nSelect the **Import ELD** , **ELS** , **or ELA** Wizard and then click **Next**. Another dialog will appear and look like Figure D.2.\n\n**Figure D.2**\n\nNavigate to the location of your **UATSTB.eld** file. If you saved it to your Desktop, it should be there. If not, then you will need to browse to it. Select the **UATSTB.eld** file and click **Open**. Once the ELD file is open it will list the **Analysis Types** that are included (see Figure D.3). The current, hot-off-the-press, **UATSTB.eld** file may contain more types than those listed in the figure. They all should be selected already, but if they are not, then click the box beside **Select All** and then click **Next**.\n\n**Figure D.3**\n\nThe final dialog will now open and all of the Analysis Techniques included in the ELD file will be listed (see Figure D.4). If you click **Finish** , they will all be imported into your own library. Once this process has completed then you will be able to follow along in the book.\n\n**Figure D.4**\n\n## Keywords and Functions\n\nThe number of reserved words, skip words, and functions are too numerous to include in an appendix. Here is a link to a TradeStation website that provides a complete and concise list: <http:\/\/help.tradestation.com\/08_08\/tsdevhelp\/mergedprojects\/elword\/elword.htm>.\n\n## Sample Algorithm Codes\n\nAll code presented in the book is available as a download on the companion website. This code includes all the necessary functions as well. Expanded research utilizing TradeStation, EasyLanguage, and Portfolio Maestro will also be included at www.georgepruitt.com.\n\n#### _Simple RSI with Profit Objective and Protective Stop_\n\n* * *\n\n    inputs: rsiLen(14),overBot(70),overSold(30);\n    If rsi(c,14) < overSold then buy this bar on close;\n    If rsi(c,14) > overBot then sellShort this bar on close;\n    If marketPosition = 1 then\n    begin\n           If c > entryPrice + 3* avgTrueRange(10) then sell this bar on close;\n           if c < entryPrice - 1* avgTrueRange(10) then sell this bar on close;\n    end;\n    If marketPosition = 1 then\n    begin\n           If c < entryPrice - 3* avgTrueRange(10) then buyToCover this bar on close;\n           if c > entryPrice + 1* avgTrueRange(10) then buyToCover this bar on close;\n    end;\n\n* * *\n\n#### _Simple Stochastic with Profit Objective and Protective Stop_\n\n* * *\n\n    inputs: rawKlen(14),smooth1(3),smooth2(3);\n    vars: myFastK(0),myFastD(0),mySlowK(0),mySlowD(0);\n    Value1 = stochastic(h,l,c,rawKLen,smooth1,smooth2,1,myFastK,myFastD,mySlowK,mySlowD);\n    If mySlowK cross above mySlowD and mySlowD < 20 then buy this bar on close;\n    If mySlowK cross below mySlowD and mySlowD > 80 then sellShort this bar on close;\n    If marketPosition = 1 then\n    begin\n           If c > entryPrice + 3* avgTrueRange(10) then sell this bar on close;\n           if c < entryPrice - 1* avgTrueRange(10) then sell this bar on close;\n    end;\n    If marketPosition =-1 then\n    begin\n           If c < entryPrice - 3* avgTrueRange(10) then buyToCover this bar on close;\n           if c > entryPrice + 1* avgTrueRange(10) then buyToCover this bar on close;\n    end;\n\n* * *\n\n#### _Simple CCI with Profit Objective and Protective Stop_\n\n* * *\n\n    inputs: cciLen(20),smooth(9);\n    vars: myCCIVal(0);\n    myCCIVal = average(cci(cciLen),smooth);\n    If myCCIVal crosses above -100  then buy this bar on close;\n    If myCCIVal crosses below 100 then sellShort this bar on close;\n    If marketPosition = 1 then\n    begin\n           If c > entryPrice + 3* avgTrueRange(10) then sell this bar on close;\n           if c < entryPrice - 1* avgTrueRange(10) then sell this bar on close;\n    end;\n    If marketPosition =-1 then\n    begin\n           If c < entryPrice - 3* avgTrueRange(10) then buyToCover this bar on close;\n           if c > entryPrice + 1* avgTrueRange(10) then buyToCover this bar on close;\n    end;\n\n* * *\n\n#### _Connors \/ Alvarez with Time-Based Exit_\n\n* * *\n\n    inputs: mavlen(200),rsiLen(2),rsiBuyVal(20),rsiSellVal(80),holdPeriod(5);\n    Condition1 = c > average(c,mavLen);\n    Condition2 = rsi(c,rsiLen) < rsiBuyVal;\n    Condition3 = rsi(c,rsiLen) > rsiSellVal;\n    If condition1 and condition2 then buy this bar on close;\n    If not(condition1) and condition3 then sellShort this bar on close;\n    If barsSinceEntry = holdPeriod then\n    Begin\n           if marketPosition = 1 then sell this bar on close;\n           if marketPosition =-1 then buytocover this bar on close;\n    end;\n\n* * *\n\n#### _Turtle with LTL Filter, Pyramiding and Position Sizing_\n\n* * *\n\n    inputs: absEntryChanLen(55),entryChanlen(20),exitChanLen(10),\n           lastTradeLoserFilter(TRUE),accountSize(100000),riskPerTradePer(.01),numPyraMids(1);\n    vars:lastTradeLoser(true),mp(0),virtmp(0),tradeProfit(0),\n           virtBuyPrice(0),virtSellPrice(0),\n           virtLongLiqPrice(0),virtShortLiqPrice(0),\n           virtLongLoss(0),virtShortLoss(0),\n           myFillPrice(0),N(0),N$(0),dollarRisk(0),lotSize(0),\n           stopLoss(0),buyPrice(0),sellPrice(0),\n           hh20(0),hh55(0),ll20(0),ll55(0),iCnt(0),initPrice(0),stopLossPts(0),debug(TRUE);\n    mp = marketPosition;\n    if mp = 0 then\n    begin\n           N = AvgTrueRange(20);\n           N$ = N*BigPointValue;\n           dollarRisk = AccountSize * riskPerTradePer;\n           lotSize = IntPortion(DollarRisk\/N$);\n           if lotSize < 1 then lotSize = 1;\n           lotSize = 1;\n           StopLoss = 2 * N$ * lotSize;\n           StopLossPts = 2 * N * lotSize;\n           StopLossPts = 2000\/bigPointValue;\n           hh20 = highest(high,entryChanLen);\n           hh55 = highest(high,absEntryChanLen);\n           ll20 = lowest(low,entryChanLen);\n           ll55 = lowest(low,absEntryChanLen);\n    end;\n    If mp <> 1 and mp[1] = 1 then\n    Begin\n           tradeProfit = ExitPrice(1) - EntryPrice(1);\n           lastTradeLoser = true;\n           If tradeProfit > 0 then lastTradeLoser = false;\n           if debug then\n                  print(date,\" Long Trader \",tradeProfit *bigPointValue,\" \",lastTradeLoser,\n                  \" ExitPrice \",ExitPrice(1):6:6,\" entryPrice \",entryPrice(1):6:6);\n    end;\n    If mp <> -1 and mp[1] = -1 then\n    Begin\n           tradeProfit = EntryPrice(1) - ExitPrice(1);\n           lastTradeLoser = true;\n           If tradeProfit > 0 then lastTradeLoser = false;\n           if debug then\n                  print(date,\" **** Short Trader \",tradeProfit *bigPointValue,\" \",lastTradeLoser,\n                  \" mp \",mp,\" \",mp[1]);\n    end;\n    If lastTradeLoserFilter = False then lastTradeLoser = True;\n    If lastTradeLoser = False then\n    Begin\n           if debug then\n                  print(date,\" In Virtual Section And VirtTmp = \",virTmp);\n           If(virtmp = 1) then\n           Begin\n               virtLongLiqPrice = maxList(lowest(low[1],exitChanLen),virtLongLoss);\n                  if(virtualLongExit(virtLongLiqPrice,1,myFillPrice) =1) then\n                  Begin\n                         tradeProfit = myFillPrice - virtBuyPrice;\n                         If tradeProfit < 0 then lastTradeLoser = true;\n                         virtmp = 0;\n                         if debug then print(\" Long Exit @ \",myFillPrice);\n                  end;\n           end;\n           If(virtmp = -1) then\n           Begin\n                  virtShortLiqPrice = minList(highest(high[1],exitChanLen),virtShortLoss);\n                  if(virtualShortExit(virtShortLiqPrice,1,myFillPrice) =1) then\n                  Begin\n                         tradeProfit = virtSellPrice - myFillPrice;\n                         If tradeProfit < 0 then lastTradeLoser = true;\n                         virtmp = 0;\n                         if debug then print(\" ShortExit @ \", myFillPrice);\n                  end;\n           end;\n           if(virtualBuy(highest(high[1],entryChanLen),1,myFillPrice) = 1) then\n           Begin\n                  if virtmp <> 1 then\n                  begin\n                         virtBuyPrice = myFillPrice;\n                         virtLongLoss = myFillPrice - stopLossPts;\n                         virtmp = 1;\n                         tradeProfit = 0;\n                         If virtmp[1] = -1 then tradeProfit = virtSellPrice - virtBuyPrice;\n                         If tradeProfit < 0 then lastTradeLoser = true;\n                         if debug then print(\" Long @ \",myFillPrice);\n                  end;\n           end;\n           if(virtualSell(lowest(low[1],entryChanLen),1,myFillPrice) = 1) then\n           Begin\n                  if virtmp <> -1 then\n                  begin\n                         virtsellPrice = myFillPrice;\n                         virtShortLoss = myFillPrice + stopLossPts;\n                         virtmp = -1;\n                         tradeProfit = 0;\n                         If virtmp[1] = 1 then tradeProfit =  virtBuyPrice - virtSellPrice;\n                         If tradeProfit < 0 then lastTradeLoser = true;\n                         if debug then print(\" Short @ \",myFillPrice, \" trade Profit\" , tradeProfit);\n                  end;\n           end;\n           if debug then print(\"End of Virtual Module : virTmp = \",virTmp,\" \",lastTradeLoser);\n    end;\n    for iCnt = 0 to numPyraMids-1\n    begin\n           if lastTradeLoser then\n           begin\n                  if mp <> -1 and currentContracts = iCnt * lotSize then\n                  begin\n                         buyPrice = hh20 + iCnt * N\/2;\n                  end;\n                  if mp <> 1 and currentContracts = iCnt * lotSize then\n                  begin\n                         sellPrice = ll20 - iCnt * N\/2;\n                  end;\n                  virTmp = 0;\n           end;\n           if lastTradeLoser = false then\n                  begin\n                  if mp <> -1 and currentContracts = iCnt * lotSize then\n                  begin\n                         buyPrice = hh55 + iCnt * N\/2;\n                  end;\n                  if mp <> 1 and currentContracts = iCnt * lotSize then\n                  begin\n                         sellPrice = ll55 - iCnt * N\/2;\n                  end;\n                  virTmp = 0;\n           end;\n    end;\n    if lastTradeLoser then\n    begin\n           if currentContracts < numPyraMids * lotsize then Buy (\"Turtle20Buy\") lotSize contracts next bar at buyPrice stop;\n           if currentContracts < numPyraMids * lotsize then Sellshort (\"Turtle20Sell\") lotsize contracts next bar at sellPrice stop;\n           if currentContracts < 4 * lotsize and debug then print(date,\" 20sellPrice \",sellPrice:6:6,\" \", currentContracts);\n    end;\n    if lastTradeLoser = false then\n    begin\n           if currentContracts < numPyraMids * lotsize then Buy (\"Turtle55Buy\") lotSize contracts next bar at buyPrice stop;\n           if currentContracts < numPyraMids * lotsize then Sellshort (\"Turtle55Sell\") lotsize contracts next bar at sellPrice stop;\n           if debug then print(date,\" \",iCnt,\" 55sellPrice \", sellPrice:6:6);\n    end;\n    If mp = 1 then Sell (\"TurtleSys1LExit\") from entry(\"Turtle20Buy\") next bar at lowest(low,exitChanLen) stop;\n    If mp = -1 then BuyToCover(\"TurtleSys1SExit\")from entry(\"Turtle20Sell\")  next bar at     highest(high,exitChanLen) stop;\n    If mp = 1 then Sell (\"TurtleSys2LExit\") from entry(\"Turtle55Buy\") next bar at lowest(low,20) stop;\n    If mp = -1 then BuyToCover(\"TurtleSys2SExit\")from entry(\"Turtle55Sell\")  next bar at     highest(high,20) stop;\n    If mp = 1 then Sell (\"TurtleLExit2N\") next bar at entryPrice - stopLossPts stop;\n    If mp = -1 then BuyToCover(\"TurtleSExit2N\") next bar at entryPrice + stopLossPts stop;\n\n* * *\n\n#### _EasyLanguage Code for TA1_\n\n* * *\n\n    inputs:\n    cmiLen(20),cmiSmooth(9),markCondIndicVal(50),chopBuyPer(.5),chopSellPer(.5),trendBBOLen(9),trendSBOLen(9);\n    inputs: volLen(20)\n    value1 = choppyMarketIndex(cmiLen);\n    value2 = xaverage(value1,cmiSmooth);\n    if value2 < markCondIndicVal then\n    begin\n           buy(\"ChopBuy\")next bar at highest(h,trendBBOLen) stop;\n           sellshort(\"ChopSell\") next bar at lowest(l,trendSBOLen) stop;\n    end;\n    if value2 >= markCondIndicVal then\n    begin\n           buy(\"TrendBuy\") next bar at c + xaverage(range,volLen)*volPer stop;\n           sellShort(\"TrendSell\") next bar at c - xaverage(range,volLen)*volPer stop;\n    end;\n\n* * *\n\n#### _EasyLanguage Code for TA2_\n\n* * *\n\n    Inputs: buyLen(40),sellLen(40),longLiqLen(20),sellliqLen(20);\n    Buy next bar at highest(h[1],buyLen) stop;\n    Sellshort next bar at lowest(l[1],sellLen) stop;\n    If Marketposition = 1 then sell next bar at lowest(low[1],longLiqLen) stop;\n    If Marketposition = -1 then buytocover next bar at highest(high[1],sellLiqLen) stop;\n\n* * *\n\n### EasyLanguage Code for TA3\n\n* * *\n\n    inputs:movAvgLen(50),pivotHiLookBack(10),pivotHiStrength(2),pivotLowLookBack(10),pivotLowStrength(2);\n    inputs: LprofitObj(500),LstopLoss(500),longExitDays(10),SprofitObj(500),SstopLoss(500),shortExitDays(10);\n    Value1 = swingHighBar(1,h,pivotHiStrength,pivotHiLookBack);\n    Value2 = swingLowBar(1,l,pivotLowStrength,pivotLowLookBack);\n    If c > average(c,movAvgLen) then\n    Begin\n        If Value2 > -1 and marketPosition <> 1 then buy this bar on close;\n    end;\n    If c < average(c,movAvgLen) then\n    Begin\n        If Value1 > -1 and marketPosition <> -1 then sellshort this bar on close;\n    end;\n    If Marketposition = 1 then\n    Begin\n        sell(\"L-profitObj\") next bar at entryPrice + LprofitObj\/Bigpointvalue limit;\n        sell(\"L-protStop\") next bar at entryPrice - LstopLoss\/Bigpointvalue stop;\n        If Barssinceentry >= longExitDays then sell this bar on close;\n    end;\n    If Marketposition = -1 then\n    Begin\n        Buytocover(\"S-profitObj\") next bar at entryPrice - SprofitObj\/Bigpointvalue limit;\n        BuyToCover(\"S-protStop\") next bar at entryPrice + SstopLoss\/Bigpointvalue stop;\n        If Barssinceentry >= shortExitDays then sell this bar on close;\n    end;\n\n* * *\n\n# Appendix E\n\n### A list of George's favorite books on Algorithmic Trading\n\n**Oldies**\n\n  * **Computer Analysis of the Futures Market** by LeBeau and Lucas\n  * **New Concepts in Technical Trading Systems** by Welles Wilder\n  * **The Elements of Successful Trading** by Robert Rotella\n  * **Portfolio Management Formulas** by Ralph Vince\n  * **Trading Systems and Methods** (any edition) by Perry Kaufman\n  * **Trading Systems That Work** by Thomas Stridsman\n  * **Money Management Strategies for Futures Traders** by Nauzer Balsara\n  * **Street Smarts: High Probability Short-Term Trading Strategies** by Linda Bradford Raschke and Laurence A. Connors\n  * **New Market Timing Techniques** by Tom DeMark\n  * **How I Made One Million Dollars Last Year Trading Commodities** by Larry Williams\n  * **Trading for a Living** by Alexander Elder\n  * **How to Make Money in Commodities** by Chester Keltner\n\n**Newbies**\n\n  * **Quantitative Technical Analysis** by Dr. Howard Bandy\n  * **Modeling Trading System Performance** by Dr. Howard Bandy\n  * **Evidenced-Based Technical Analysis** by David Aronson\n  * **Statistically Sound Machine Learning for Algorithmic Trading of Financial Instruments** by David Aronson and Timothy Masters\n  * **The Evaluation and Optimization of Trading Strategies** by Bob Pardo\n  * **Building Winning Algorithmic Trading Systems** by Kevin Davey\n  * **Trading Systems: A New Approach to System Development and Portfolio Optimization** by Emilio Tomasini\n  * **Algorithmic Trading** by Ernie Chan\n  * **Building Reliable Trading System** by Keith Fitschen\n  * **Using EasyLanguage 9.X** by Murray Ruggiero\n\n**Programming Books**\n\n  * **Python Programming** by John Zelle\n  * **Genetic Algorithms and Investment Strategies** by Richard Bauer\n  * **Mastering VBA for Microsoft Office** by Richard Mansfield\n  * **Python for Finance** by Yves Hilpisch\n\n# About the Companion Website\n\nAll of the source code for the trading algorithms, data, and testing platforms is included on the Wiley website.\n\nIn addition, the website www.georgepruitt.com will provide updates to the trading algorithms, data, and testing platforms. You will also find more tools to add to your toolbox that weren't mentioned in the book. Questions can be directed to George using his george.p.pruitt@gmail.com email address.\n\n# Index\n\n## A\n\n  1. Absolute price oscillator (APO)\n  2. ACCEPT state\n  3. _Adaptation in Natural and Artificial Systems_ (Holland)\n  4. ADX. _See_ Average Directional Movement Index \n  5. ADXR\n  6. AFL. _See_ AmiBroker Function Language \n  7. Algorithm \n    1. backtesting \n      1. Excel, usage\n      2. Python, usage\n    2. CCI algorithm, trades (generation)\n    3. codes. _See_ EasyLanguage; TradeStation.\n    4. criteria\n    5. defining\n    6. development\n    7. flowchart (FC), stochastic oscillator crossover (entry signal usage)\n    8. genetic algorithms, identification\n    9. mean reversion algorithm, equity curve\n    10. parameters \n      1. comparison\n      2. results\n      3. sets\n    11. pseudocode, example\n    12. ramp-up data, insufficiency\n    13. RSI algorithm, performance\n    14. stochastic algorithm, performance\n    15. trading algorithm, example\n  8. Algorithmic traders, programmer ability\n  9. Algorithmic trading, books (source)\n  10. Algo Testing Software, example\n  11. AmiBroker \n    1. debugging, Exploration (usage)\n    2. Exploration, results\n    3. functions\n    4. keywords\n    5. loop programming \n      1. code, sample\n    6. Monte Carlo settings dialog\n    7. Monte Carlo simulation\n    8. position sizing, futures mode\n    9. walk-forward optimizer, usage\n  12. AmiBroker Function Language (AFL) \n    1. array programming\n    2. Automatic Analysis dialog box\n    3. backtest, parameters (setting)\n    4. broker integration (feature)\n    5. Check icon\n    6. code samples\n    7. Code Wizard \n      1. Add item\n      2. button\n      3. Edit Rule dropdown list\n      4. window\n    8. data \n      1. algorithm, application\n      2. types\n    9. Editor \n      1. auto-completion tool\n      2. function helper\n      3. launching\n      4. window, example\n    10. Edit Rule pane, parameters (change)\n    11. error warning\n    12. expressions \n      1. results\n    13. features\n    14. initiation\n    15. integrated development environment (IDE)\n    16. knowledge, programmer requirements\n    17. operators \n      1. precedence\n    18. parameters, setting\n    19. portfolio results\n    20. power (feature)\n    21. price (feature)\n    22. request (sending), exclamation point (clicking)\n    23. Send to Analysis window icon\n    24. Set Futures mode\n    25. simple moving average crossover\n    26. speed (feature)\n    27. syntax\n    28. 3-D optimization chart\n    29. trade-by-trade report\n    30. usage\n    31. variable naming\n    32. variables \n      1. names, restrictions\n    33. Wizard\n  13. (AmiBroker keyword) \n    1. flow control structures\n  14. Analysis phase\n  15. Analysis window\n  16. Annual return, performance metric\n  17. APO. _See_ Absolute price oscillator \n  18. Appels, Gerald\n  19. ApplyStop function\n  20. Arguments, usage\n  21. Arithmetic expressions\n  22. ASCII \n    1. database (futures prices)\n    2. data importer\n  23. ATR. _See_ Average true range \n  24. Auto-generated formula\n  25. Average (VBA function)\n  26. Average Directional Index (ADX)\n  27. Average Directional Movement Index (ADX) \n    1. 14-day ADX, calculation\n    2. length\n    3. trend detector function\n  28. Averages\n  29. Average true range (ATR) \n    1. ATR Sell MOC\n    2. BuyToCover MOC\n    3. 5X ATR profit objective, usage\n    4. lookback, parameter (change)\n    5. multiplier, parameter (change)\n    6. 3X ATR stop, usage\n    7. 2 ATR, usage\n    8. usage\n  30. Averaging methods, performances\n  31. avgLen, default value\n  32. avgMP\n  33. AvgTrueRange\n\n## B\n\n  1. Backtest \n    1. equity curve, profit (display)\n    2. parameters, setting\n    3. performance report\n  2. Backtest button\n  3. Backtester settings, example\n  4. Backtesting \n    1. Excel, usage\n    2. software structure \n      1. flowchart\n  5. Bandy, Howard\n  6. Bars held, performance metric\n  7. BASIC. _See_ Beginner's All-purpose Symbolic Instruction Code \n  8. BBandBot\n  9. BBandTop\n  10. Beginner's All-purpose Symbolic Instruction Code (BASIC)\n  11. Berkely, George\n  12. Berlinski, David\n  13. bin increment (binInc) variable\n  14. binTuple\n  15. Bollinger algorithm \n    1. benchmark\n    2. optimization\n  16. Bollinger Bands \n    1. algorithm \n      1. best parameter sets\n      2. version 1 (performance)\n      3. version 2 (performance)\n    2. breakout\n    3. calculation \n      1. 2 standard deviations, usage\n      2. 60 days standard deviation, usage\n    4. data, Keltner Channel data (comparison)\n    5. example\n    6. function\n    7. Function Module\n    8. optimization\n    9. routine, defining\n    10. 60-day, two-SD Bollinger Band, example\n    11. subroutine\n    12. system, example\n    13. trend algorithm\n    14. trend follower function\n  17. BollingerBand subroutine\n  18. Bollinger Benchmark, Keltner Challengers (contrast)\n  19. Bollinger, John\n  20. Bollinger performance, three-dimensional contour chart\n  21. Boolean expression\n  22. Boolean flags, usage\n  23. BOTTOM failure\n  24. bottoming out, term (removal)\n  25. _Building Reliable Trading Systems_ (Fitschen)\n  26. buy (AmiBroker keyword)\n  27. Buy-and-hold mentality\n  28. BUY order\n  29. buyprice (AmiBroker keyword)\n  30. buy (keyword), recognition\n\n## C\n\n  1. CCI. _See_ Commodity Channel Index \n  2. CFDT. _See_ Cumulative frequency distribution table \n  3. change (keyword), recognition\n  4. Choppy Market Indicator (CMI)\n  5. Chromosomes \n    1. crossing, VBA Excel (usage)\n    2. dartboard \n      1. creation, VBA Excel (usage)\n      2. example\n    3. dart-throwing blindfolded monkey (simulation), VBA Excel (usage)\n    4. finding moms and pops, VBA Excel (usage)\n    5. fitness of chromosomes (testing), VBA Excel (usage)\n    6. genes allocation process, VBA Excel (usage)\n    7. genes passage\n    8. initial population setup, VBA Excel (usage)\n    9. mutation \n      1. VBA Excel, usage\n    10. population\n    11. probability of being hit by dart calculation, VBA Excel (usage)\n    12. replacement function, VBA Excel (usage)\n    13. reproduction\n    14. selection\n    15. slice size \n      1. fitness conversion, VBA Excel (usage)\n  6. class (object)\n  7. ClassicShell\n  8. class rsiClass(object):\n  9. class stochClass(object):\n  10. Cluster analysis. _See_ Walk-forward optimization \n  11. CMI. _See_ Choppy Market Indicator \n  12. Code, arguments (presence)\n  13. Code Wizard. _See_ AmiBroker Function Language \n  14. Coincident indicator performance\n  15. Combination lock (workings), finite statement machine modeling\n  16. Comment text, usage\n  17. Commitment of Traders Report (COT)\n  18. commName\n  19. Commodity Channel Index (CCI) \n    1. algorithm \n      1. coincident indicator performance\n      2. example\n      3. performance\n      4. trades, generation\n    2. OB\/OS system description\n    3. profit objective, usage\n    4. protective stop, usage\n    5. system, usage\n    6. trend following system \n      1. p-code\n    7. 20-day CCI, calculation\n  20. Computer language, syntax (understanding)\n  21. Conditional execution\n  22. Connors\/Alvarez, time-based exit (usage)\n  23. Consecutive bar exact sequence paradigm\n  24. Consecutive bar sequence\n  25. COT. _See_ Commitment of Traders Report \n  26. cover (AmiBroker keyword)\n  27. Cover arrays\n  28. coverprice (AmiBroker keyword)\n  29. Crossover rate\n  30. Crude oil-day, two-SD Bollinger Band (example)\n  31. Cumulative daily equity\/drawdown, plotting (example)\n  32. Cumulative frequency distribution table (CFDT)\n  33. curBar (sAverage function parameter)\n  34. Curve fitting \n    1. prevention\n  35. Customize Ribbon\n\n## D\n\n  1. Daily bars (pivot high points), strength differences (examples)\n  2. Dartboard \n    1. creation, VBA Excel (usage)\n    2. example\n  3. Dart-throwing blindfolded monkey (simulation), VBA Excel (usage)\n  4. Data arrays (ESB)\n  5. Data arrays (PSB)\n  6. DataMaster worksheet, usage (example). _See_ Excel System Backtester \n  7. DC. _See_ Donchian Channels \n  8. Definiteness (algorithm criteria)\n  9. Dependencies, nesting\n  10. Developer tab option, example\n  11. Directional movement (DM) \n    1. algorithm \n      1. performance\n    2. concept, introduction\n    3. examples\n    4. system description\n  12. Directional Movement Index (DX)\n  13. DM. _See_ Directional movement \n  14. DMA. _See_ Double simple moving average \n  15. dnSum (variable)\n  16. Donchian algorithms, results\n  17. Donchian Channels (DC) \n    1. breakout\n  18. Donchian lengths, Donchian Channel breakout (usage)\n  19. Double simple moving average (DMA) algorithm\n  20. Double simple moving average (DMA) crossover \n    1. algorithm\n  21. downward, term (removal)\n  22. Draw down. _See_ Trade draw down \n  23. DX. _See_ Directional Movement Index\n\n## E\n\n  1. EasyLanguage \n    1. abbreviations\n    2. algorithm codes\n    3. chart, strategy application\n    4. codes, usage\n    5. comments\n    6. Connors\/Alvarez, time-based exit (usage)\n    7. editor \n      1. line number display\n    8. editor, usage\n    9. functions\n    10. inputs, change (process)\n    11. keywords\n    12. mean reversion strategy (change), input variables (usage)\n    13. profit objective, usage\n    14. project \n      1. creation\n      2. strategy, addition\n    15. protective stop, usage\n    16. remarks\n    17. reversion analysis strategy\n    18. samples\n    19. SKIP words\n    20. strategy \n      1. adjustment process\n      2. algorithm, dependencies\/functions (nesting)\n    21. syntax\n    22. Turtle, LTL filter\/pyramiding\/position sizing (usage)\n  2. Editor. _See_ AmiBroker Function Language \n    1. line number display (EasyLanguage)\n  3. Edit (keyword), recognition\n  4. Effectiveness (algorithm criteria)\n  5. ELD files \n    1. importing\n    2. reimporting\n  6. EMA. _See_ Exponential moving average \n  7. End Of Day (EOD) \n    1. data feeds\n    2. historic database\n  8. Entry logic\n  9. Entry rules, template (example)\n  10. EOD. _See_ End Of Day \n  11. equityClass\n  12. Equity curve \n    1. chart, maximum drawdown (display)\n    2. creation\n    3. example\n    4. mean reversion algorithm\n    5. multiple randomized equity curves, chart\n    6. pattern system\n    7. profit, display\n    8. stops, addition\n  13. EquityStream worksheet, macro (launch)\n  14. ES. _See_ Exhaustive search \n  15. ESB. _See_ Excel System Backtester \n  16. Eurocurrency downturn, CCI system detection\n  17. Excel \n    1. data \n      1. worksheet, example\n    2. ribbon, customization\n    3. usage. _See also_ Visual Basic for Applications. \n    4. Visual Basic Editor, example\n  18. ExcelSystemBackTester\n  19. Excel System Backtester (ESB) \n    1. components\n    2. data arrays\n    3. DataMaster worksheet, usage (example)\n    4. EquityStream worksheet, macro (launch)\n    5. functions\n    6. keywords\n    7. programming environment\n    8. Results worksheet, trades\/performance metrics listing\n    9. source code\n    10. subroutines\n    11. VBA code\n  20. ExcelSystemTester, functions\/subroutines\n  21. exclude (AmiBroker keyword)\n  22. Exhaustive search (ES) \n    1. algorithm, parameters (results)\n  23. Exhaustive search (ES) optimization engines\n  24. Exit \n    1. procedure\n    2. rules, template (example)\n  25. Exit logic\n  26. exitPos function\n  27. Exploration \n    1. results\n    2. usage\n  28. Exponential moving average (EMA)\n  29. Expressions \n    1. AmiBroker Function Language (AFL)\n    2. arithmetic expressions\n    3. logical expressions\n    4. results\n\n## F\n\n  1. Failure swing algorithm, performance\n  2. Failure swing system description. _See_ Relative Strength Index \n  3. Fast %D\n  4. Fast %K\n  5. filter (AmiBroker keyword)\n  6. filterTrade variable\n  7. Finding moms and pops, VBA Excel (usage)\n  8. Finiteness (algorithm criteria)\n  9. Finite state machine (FSM) \n    1. code, example\n    2. diagram\n    3. modeling\n  10. Fitness conversion, VBA Excel (usage)\n  11. Fitness of chromosomes (testing), VBA Excel (usage)\n  12. Fitschen, Keith\n  13. 5X ATR profit objective, usage\n  14. Fixed Fractional \n    1. formula\n    2. money management overlay \n      1. parameters, change\n  15. Flowchart (FC) \n    1. example\n    2. usage, example\n  16. Flow control structures\n  17. for loop\n  18. for-loop\n  19. FORTRAN\n  20. 14-day ADX, calculation\n  21. 14-day RSI, calculation\n  22. 14-day stochastic (FAST), calculation\n  23. FSM. _See_ Finite state machine \n  24. Functions \n    1. definition, colon (usage)\n  25. Functions, nesting\n  26. Future leak, advantage\n  27. Futures mode option\n  28. Futures prices, ASCII database\n\n## G\n\n  1. Garbage in garbage out (GIGO)\n  2. Generation \n    1. concepts\n  3. Genes allocation process, VBA Excel (usage)\n  4. Genetic algorithms (GAs) \n    1. concepts\n    2. crossover rate\n    3. curve fitting, prevention\n    4. generations\n    5. identification\n    6. mutation rate\n    7. population size\n    8. usage\n  5. Genetic optimization (GO)\n  6. getData()\n  7. GetData (ESB component)\n  8. Graphical user interface (GUI)\n\n## H\n\n  1. head (variable)\n  2. Helper functions (PSB)\n  3. Hermawanto, Denny\n  4. Highest Function\/Module\n  5. Hindsight, usage\n  6. Histogram pivot points, usage\n  7. Histogram system. _See_ Moving Average Convergence Divergence \n  8. Holland, John\n  9. _How to Make Money in Commodities_ (Keltner)\n\n## I\n\n  1. IDE. _See_ Integrated development environment \n  2. Identifiers (PSB)\n  3. IDLE. _See_ Integrated DeveLopment Environment \n  4. Indicator.py\n  5. Indicators \n    1. categories\n    2. classes\/functions (PSB)\n  6. input (algorithm criteria)\n  7. InSampleBarsPerRun\n  8. In-sample (IS) testing\n  9. In-sample (IS) time period \n    1. Bollinger algorithm, optimization\n    2. Donchian algorithm, results\n    3. portfolio, parameters\n  10. Instruction ste, variability\n  11. Integrated development environment (IDE)\n  12. Integrated DeveLopment Environment (IDLE), usage\n  13. IS. _See_ In-sample \n  14. Iteration growth rate, parameters (example)\n\n## J\n\n  1. Janeczko, Tomasz\n\n## K\n\n  1. Kaufman, Perry\n  2. Keltner Challengers, Bollinger Benchmark (contrast)\n  3. Keltner Channel \n    1. algorithm, best parameter sets\n    2. calculation \n      1. 2 ATR, usage\n      2. 60 days, usage\n    3. data, Bollinger Bands data (comparison)\n    4. optimization\n    5. trend follower function\n  4. Keltner, Chester\n  5. Keywords (EasyLanguage)\n\n## L\n\n  1. Lagging indicator\n  2. Lambert, Donald\n  3. Lane, George C.\n  4. LastBarOfMonth (LBM) pattern\n  5. Last trade was a loser filter (LTLF) \n    1. absence \n      1. examples\n    2. engagement\n    3. inclusion, examples\n    4. usage\n  6. LBM. _See_ LastBarOfMonth \n  7. Limit (lmt) order\n  8. Lists, initiation (PSB)\n  9. load (keyword), recognition\n  10. Logical expressions\n  11. Long division procedure, example\n  12. Long entries\n  13. Long entry signal description \n    1. usage\n  14. LONG position, establishment\n  15. Long position exit instruction (TradeStation)\n  16. lookback (sAverage function parameter)\n  17. Loops \n    1. programming. _See_ AmiBroker loop programming.\n  18. Lowest Function\/Module\n  19. LTLF. _See_ Last trade was a loser filter\n\n## M\n\n  1. MACD. _See_ Moving Average Convergence Divergence \n  2. Macro, ESB launching\n  3. MarginDeposit\n  4. margindeposit (AmiBroker keyword)\n  5. marketDataClass \n    1. objects\n  6. marketDataClass() method\n  7. Market on Close (moc) \n    1. order\n  8. Market (mkt) order\n  9. Markets To Test, opening\n  10. marketSystemClass\n  11. MAS. _See_ Multi-algorithm strategy \n  12. maxContractPoints\n  13. maxContractProfit\n  14. Maximum drawdown, display\n  15. Maximum quantity, parameter (change)\n  16. Mean reversion (MR) \n    1. algorithm, equity curve (usage)\n    2. strategy (change), input variables (usage)\n  17. MeanReversion (name)\n  18. Mini Russell day trading system, narrow range day (usage)\n  19. mmStop defaults\n  20. MOC. _See_ Time-based market order \n  21. moc. _See_ Market on Close \n  22. Money management \n    1. Fixed Fractional money management overlay\n    2. stop, exit procedure\n  23. Money Management window \n    1. parameters, change\n  24. Monte Carlo analysis\n  25. Monte Carlo settings, AmiBroker Monte Carlo settings dialog\n  26. Monte Carlo simulation \n    1. AmiBroker Monte Carlo simulation\n    2. distribution statistics\n    3. implementation, Python (usage)\n  27. Monte Carlo source code\n  28. Monte Carlo straw broom chart\n  29. Moving Average Convergence Divergence (MACD) \n    1. calculation-day\/26-day moving averages (usage)\n    2. histogram system\n  30. Moving averages \n    1. crossover algorithm\n    2. double simple moving average (DMA) crossover \n      1. algorithm\n    3. exponential moving averages\n    4. simple moving averages\n    5. single simple moving average (SMA) crossover \n      1. algorithm\n    6. triple simple moving average (TMA) crossover \n      1. algorithm\n    7. 20-day moving averages, examples\n    8. 200-day moving average\n    9. weighted moving averages\n  31. MR. _See_ Mean reversion \n  32. Multi-algorithm strategy (MAS)\n  33. Multiple randomized equity curves, chart\n  34. Mutation \n    1. concepts\n    2. occurrence\n    3. rate\n    4. VBA Excel, usage\n  35. myAvg\n  36. myCondition, variable (initiation)\n  37. myDayCount\n  38. MyFirstPortfolio\n  39. MyFirstStrategyGroup \n    1. list\n    2. selection, result\n  40. myProfitTarg\n  41. MySecondAlgoWizard\n  42. mySum\n  43. myTickValue\n  44. myTrueRange\n  45. myValue\n\n## N\n\n  1. Narrow range day, usage\n  2. Net profit, performance metric\n  3. _New Concepts In Technical Trading Systems_ (Wilder)\n  4. _New Trading Systems and Methods_ (Kaufman)\n  5. Number of trades, performance metric\n  6. numcolumns (AmiBroker keyword)\n\n## O\n\n  1. OB\/OS system description. _See_ Commodity Channel Index \n  2. offset (sAverage function parameter)\n  3. OOS. _See_ Out-of-sample \n  4. Operators \n    1. AmiBroker Function Language (AFL)\n    2. precedence\n  5. Optimization \n    1. genetic optimization\n    2. parameter sets\n    3. 3-D optimization chart\n  6. Optimize button\n  7. Orders, types\n  8. Oscillators \n    1. crossover, entry signal usage. _See_ Stochastics.\n  9. Outlining\n  10. Out-of-sample (OOS) \n    1. in-sample (IS), ratio\n    2. number of bars\n    3. percentage\n    4. report\n    5. testing \n      1. example\n    6. time period, Donchian algorithm results\n    7. timespan\n  11. OutOfSampleBarsPerRun\n  12. output (algorithm criteria)\n\n## P\n\n  1. Paintbar\n  2. Paradigms, example\n  3. Parameters \n    1. Bollinger performance, three-dimensional contour chart\n    2. change\n    3. comparison\n    4. sets \n      1. optimization usage\n      2. robust parameter sets, presence\n    5. setting\n    6. stress\n  4. Partner matching probability\n  5. Pattern system, equity curve\n  6. p-code\n  7. Percentage price oscillator (PPO)\n  8. Percent Risk, parameter (change)\n  9. Performance metrics \n    1. example\n  10. Peters, Tim\n  11. Pivot bar, strength (basis)\n  12. Pivot high points, strength differences (examples)\n  13. Pivot point entry algorithm, example\n  14. Pivot-point long-entry algorithm, FSM modeling\n  15. PM. _See_ Portfolio Maestro \n  16. pointvalue (AmiBroker keyword)\n  17. Population \n    1. concepts\n    2. initial population setup, VBA Excel (usage)\n    3. size\n  18. Portfolio \n    1. AFL results\n    2. algorithm, usage (decision)\n    3. analysis\n    4. backtesting, preparation\n    5. composition\n    6. correlation matrix, portion\n    7. equity curve, stops (addition)\n    8. parameters\n    9. profit targets, examples\n    10. sample, Turtle system 2 (example)\n    11. Settings\n    12. setup, PSB (usage)\n    13. stop loss \n      1. appearance\n      2. setting\n    14. stop option\n    15. trades, stops \n      1. exclusion\n      2. inclusion\n  19. portfolioClass\n  20. Portfolio Maestro (PM) \n    1. backtest performance report\n    2. equity curve chart, maximum drawdown (display)\n    3. fixed fractional money management overlay\n    4. launch window\n    5. Manage Strategy Group button, indication\n    6. Money Management window, parameters (change)\n    7. MyFirstPortfolio\n    8. MyFirstStrategyGroup, selection\n    9. portfolio \n      1. analysis\n      2. backtesting, preparation\n      3. equity curve, stops (addition)\n      4. profit targets, examples\n      5. stop loss, appearance\n      6. stop option\n    10. Portfolio Settings\n    11. position size, profit\/drawdown (correlation)\n    12. Returns and Equity tab, maximum drawdown (display)\n    13. strategy group \n      1. list\n      2. settings, adjustment\n      3. strategy, addition\n    14. Strategy Group settings button\n    15. symbol, addition\n    16. Total P\/L option\n    17. Trade Analysis window, trading statistics display\n    18. Turtle 55 strategy, appearance\n    19. UGTATS list, selection\n    20. UGTATS Market List\n    21. UGTATS markets, symbols\n  21. Portfolio Profit Target, impact\n  22. Portfolio sample, Turtle system 1 \n    1. last trade was a loser filter (LTLF) \n      1. absence\n      2. inclusion, examples\n  23. positionscore (AmiBroker keyword)\n  24. PositionSize\n  25. positionsize (AmiBroker keyword)\n  26. Position size, profit\/drawdown (correlation)\n  27. Position sizing \n    1. futures mode (AmiBroker)\n    2. usage\n  28. PPO. _See_ Percentage price oscillator \n  29. Predictive indicator\n  30. Price bar interface \n    1. reserved words\n  31. Price-based indicators\n  32. prices (sAverage function parameter)\n  33. Probability of being hit by dart (calculation), VBA Excel (usage)\n  34. Problem solving\n  35. Profit factor, performance metric\n  36. Profit\/loss, performance metric\n  37. Profit objective, usage\n  38. Programming environment\n  39. Protective stop, usage\n  40. PSB. _See_ Python System Backtester \n  41. PSBBollinger, opening\n  42. Pseudocode \n    1. algorithm pseudocde\n    2. conversion process\n  43. Pyramiding, usage\n  44. Python \n    1. code, example\n    2. dynamic typing\n    3. installation\n    4. interpreted language\n    5. shell, presence\n    6. usage \n      1. reasons\n    7. user's group, presence\n  45. Python Monte Carlo simulation\n  46. _Python Programming_ (Zelle)\n  47. Python System Backtester (PSB) \n    1. Bollinger Bands Function Module\n    2. classes\n    3. class rsiClass(object):\n    4. class stochClass(object):\n    5. configuration\n    6. data arrays\n    7. data import\n    8. equityClass\n    9. helper functions\n    10. Highest Function\/Module\n    11. identifiers\n    12. IDLE, usage\n    13. import section\n    14. indicator classes\/functions\n    15. indicator.py\n    16. installation\n    17. keywords\n    18. lists \n      1. initiation\n    19. Lowest Function\/Module\n    20. marketDataClass\n    21. Monte Carlo \n      1. analysis\n      2. source code\n    22. portfolioClass\n    23. portfolio setup\n    24. rsiClass\n    25. sAverage \n      1. function, parameters\n    26. Simple Average Function\/Module\n    27. Start Trade Drawdown \n      1. section\n      2. source code\n    28. structure\n    29. SystemTester.py\n    30. tradeInfoClass\n    31. usage\n    32. variables, initiation\n\n## Q\n\n  1. _Quantitative Trading Systems_ (Bandy)\n\n## R\n\n  1. rampUp\n  2. Ramp-up data, insufficiency\n  3. Random number generator (RNG) \n    1. tool\n    2. usage\n  4. Range \n    1. function\n    2. 10-day average, calculation\n  5. Real Time data feeds\n  6. Relative strength (RS)\n  7. Relative Strength Index (RSI) \n    1. algorithm, performance\n    2. divergence, example\n    3. failure swing algorithm, performance\n    4. failure swing system description\n    5. 14-day RSI, calculation\n    6. pivots, value\n    7. profit objective, usage\n    8. protective stop, usage\n    9. system \n      1. description \n      2. p-code\n    10. trading algorithm, flowchart (FC) diagram\n    11. usage, example\n    12. VBA function\n  8. removeDuplicates\n  9. Replacement function, VBA Excel (usage)\n  10. Reproduction \n    1. concepts\n  11. Reserved words\n  12. Results worksheet. _See_ Excel System Backtester \n  13. Reversal, algorithm exit procedure\n  14. Reversion analysis strategy (EasyLanguage)\n  15. Risk aversion\n  16. Risk of ruin (RoR) \n    1. calculation\n    2. competitors, examples\n    3. impact\n  17. RNG. _See_ Random number generator \n  18. Robust parameter sets, presence\n  19. RoR. _See_ Risk of ruin \n  20. roundlotsize (AmiBroker keyword)\n  21. Round quantity, parameter (change)\n  22. RSI. _See_ Relative Strength Index \n  23. rsiClass\n  24. Ruin, risk \n    1. calculation\n    2. impact\n  25. Run menu\n  26. run (keyword), recognition\n\n## S\n\n  1. sAverage \n    1. function, parameters\n  2. save (keyword), recognition\n  3. Selection (chromosomes)\n  4. Selection of the fittest, concepts\n  5. sell (AmiBroker keyword)\n  6. Sell arrays\n  7. SELL order\n  8. sellprice (AmiBroker keyword)\n  9. Send To Analysis\n  10. setSysMarkInfo\n  11. Sharpe ratio, performance metric\n  12. short (AmiBroker keyword)\n  13. Short entries\n  14. shortprice (AmiBroker keyword)\n  15. short (keyword), recognition\n  16. Simple Average Function\/Module\n  17. Simple moving average crossover algorithm\n  18. Simple moving average crossover system \n    1. AFL usage\n    2. 3X ATR stop, usage\n  19. Simple moving averages\n  20. Single simple moving average (SMA) crossover \n    1. algorithm, example\n  21. SKIP words\n  22. Slow %D\n  23. Slow %K\n  24. Slow stochastic system description\n  25. SMA. _See_ Single simple moving average \n  26. Source code, initiation\n  27. Standard deviation (SD) Bollinger Band\n  28. START state\n  29. Start Trade Drawdown \n    1. analysis\n    2. calculation\n    3. source code\n  30. Stochastic (STO) \n    1. algorithm, performance\n    2. 14-day stochastic (FAST), calculation\n    3. oscillator crossover, entry signal usage\n    4. profit objective, usage\n    5. protective stop, usage\n    6. slow stochastic system \n      1. description\n      2. p-code\n    7. trading algorithm, flowchart (FC) diagram\n    8. usage, examples\n    9. VBA function\n  31. Stop loss, setting\n  32. Stop (stp) order\n  33. Strategy Group \n    1. concepts\n    2. creation\n  34. Straw broom chart\n  35. Subroutines. _See_ Visual Basic \n    1. returns\n    2. values\n  36. Symbol List, creation\n  37. Syntax \n    1. AmiBroker Function Language (AFL)\n    2. EasyLanguage\n    3. errors, list (TradeStation)\n  38. Sys Drawdown, performance metric\n  39. systemMarket\n  40. systemMarketClass\n  41. System TA2, WFO default settings (usage)\n  42. SystemTester module\n  43. SystemTester.py\n\n## T\n\n  1. TA. _See_ Trading algorithm \n  2. TDE. _See_ TradeStation Development Environment \n  3. 10-day average, calculation\n  4. TestSystem (ESB component)\n  5. 3D Optimization chart\n  6. 3D Optimization chart\n  7. 3X ATR stop, usage\n  8. ticksize (AmiBroker keyword)\n  9. Time-based market order (MOC), exit procedure\n  10. TMA. _See_ Triple simple moving average \n  11. TOP failure swing, occurrence\n  12. TotalBarsPerRun\n  13. Total P\/L option\n  14. Total transaction costs, performance metric\n  15. TP. _See_ Typical price \n  16. Trade Analysis \n    1. ta\n    2. window, trading statistics display\n  17. Trade-by-trade report\n  18. Trade draw down, initiation\n  19. tradeInfoClass\n  20. Trades \n    1. example (generation), RSI (usage)\n    2. generation (example), stochastics (usage)\n    3. signals (initiation), histogram pivot points (usage)\n    4. skipping\n  21. TradeStation \n    1. algorithm codes\n    2. Connors\/Alvarez, time-based exit (usage)\n    3. dates\/times, storage\n    4. functions\n    5. keywords\n    6. long position exit instruction\n    7. Portfolio Maestro (PM) launch window\n    8. profit objective, usage\n    9. protective stop, usage\n    10. syntax errors, list\n    11. Turtle, LTL filter\/pyramiding\/position sizing (usage)\n    12. usage\n  22. TradeStation Development Environment (TDE) \n    1. algorithm\n    2. code verification\/recalculation, confirming message\n    3. EasyLanguage editor\n    4. initial screen\n  23. TradeStation IDE \n    1. home screen\n    2. trading apps\n  24. tradeTuple\n  25. Trading \n    1. idea, pseudocode conversion process\n    2. statistics, display\n    3. system development, genetic algorithms (usage)\n  26. Trading algorithm 1 (TA1), EasyLanguage Code (usage)\n  27. Trading algorithm 2 (TA2) \n    1. EasyLanguage code, usage\n    2. WFA report\n  28. Trading algorithm 3 (TA3) \n    1. EasyLanguage code, usage\n    2. WFO report card TA3\n  29. Trading algorithms \n    1. describing\/programming\n    2. example\n    3. flowchart (FC) \n      1. example\n      2. usage, example\n    4. programming, Boolean flags (usage)\n    5. RSI trading algorithm, flowchart (FC) diagram\n    6. stochastic trading algorithm, flowchart (FC) diagram\n  30. Trading Blox testing platform\n  31. Trend \n    1. detector, function\n    2. follower\n    3. following system. _See_ Commodity Channel Index.\n    4. trading\n  32. Trend follower function \n    1. Bollinger Bands\n    2. Keltner Channel\n  33. Trigger, incorporation\n  34. Triple simple moving average (TMA) crossover \n    1. algorithm\n  35. True range (TR)\n  36. Turtle 55 strategy, appearance\n  37. Turtle, LTL filter\/pyramiding\/position sizing (usage)\n  38. Turtle system \n    1. examples\n    2. LTLF \n      1. absence, examples\n      2. inclusion, examples\n    3. trading algorithm flowchart, example\n  39. 12-day moving averages, usage\n  40. 20-day CCI, calculation\n  41. 20-day moving averages, examples\n  42. 26-day moving averages, usage\n  43. Typical price (TP)\n\n## V\n\n  1. Variable bar liberal sequence paradigm\n  2. Variable bar sequence \n    1. depiction, long entry signal description\n  3. Variables \n    1. initiation (PSB)\n    2. names, restrictions\n  4. Vars statement\n  5. VBE. _See_ Visual Basic Editor \n  6. Visual Basic Editor (VBE) \n    1. example\n  7. Visual Basic for Applications (VBA) \n    1. average, function\n    2. code. _See_ Excel System Backtester.\n    3. code, comment text (usage)\n    4. comments, ending\n    5. error message, example\n    6. Excel, usage\n    7. functions\/subroutines\n    8. RSI function\n    9. source code, initiation\n    10. stochastic function\n    11. usage\n  8. Volatility \n    1. cycle, function\n    2. usage\n\n## W\n\n  1. Walk forward\n  2. Walk-forward analysis (WFA) \n    1. forward roll, graphic representation\n    2. report\n  3. Walk-forward efficiency (WFE)\n  4. Walk-forward optimization (WFO) \n    1. cluster analysis\n    2. cluster results matrix\n    3. default settings, usage\n    4. entries\/exits code\n    5. examples\n    6. in-sample testing \n      1. TA2, example\n    7. out-of-sample (OOS) \n      1. report\n      2. testing \n        1. example\n    8. report card TA3\n  5. Walk-forward optimizer \n    1. usage\n    2. value\n  6. Walk-forward performance, equity curve (example)\n  7. Walk-forward testing \n    1. backtest, usage\n    2. out-of-sample (OOS) time period\n    3. steps\n  8. Weighted moving averages\n  9. WFA. _See_ Walk-forward analysis \n  10. WFE. _See_ Walk-forward efficiency \n  11. WFO. _See_ Walk-forward optimization \n  12. Wilder, J. Welles\n\n## Z\n\n  1. Zelle, John\n\n# WILEY END USER LICENSE AGREEMENT\n\nGo to www.wiley.com\/go\/eula to access Wiley's ebook EULA. \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvelk b/data_all_eng_slimpj/shuffled/split2/finalzzvelk
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+{"text":"If you'd like to keep our conversation just between us, my public PGP key can be downloaded using this link. Make sure to attach your public key so I can reply!\nThe best way to get in touch with me is to send me an email at the following address. In order to protect myself from spam, to get my email address you must successfully complete a captcha by clicking on the link on the \"j\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meinberg Radio Clocks provides products for time and frequency synchronisation, such as time signal receivers (GPS, GLONASS, DCF77, MSF) and systems to convert time information into various signal types, as well as solutions for signal dispersion and distribution (NTP, PTP).\nSubstation automation demands precise time synchronisation for extensive controlling technologies, such as SCADA systems, a variety of protection relays, bay controller units and intelligent electronic devices (IEDs), which are required in power networks.\nThere are different approaches possible to achieve the required accuracy. Time synchronisation for substations with integrated protection and system control functions, as well as data collection, requires a target architecture that distributes synchronised time in several ways. Different solutions are available and can be realised using Meinberg timing equipment.\nLANTIME time servers can synchronise systems that support network time protocol (NTP) or simple network protocol (SNTP). The highly stable internal oscillator bridges periods of interference and temporary loss of synchronisation, which guarantees high accuracy at any time. LANTIME time servers use Meinberg's internal clock (GPS, GLONASS, DCF77, MSF, WWVB or IRIG time code receiver) as a precision time reference source.\nOur NTP time servers are suitable for almost any application because they can be individually configured by choosing between different form factors, power supply functions, reference time sources, additional interfaces or outputs. LANTIME time servers are available as DIN rail mounted and rack mounted servers.\nMeinberg precision time protocol (PTP) Grandmaster Clocks not only offer a highly accurate source of synchronisation for PTP clients (IEEE 1588 Slave Clocks), they also introduce the exact time to your PTP networks.\nThe PTP v2 multicast implementation of our PTP Grandmaster Clocks is fully compliant to the IEEE 1588 standard, so also provides PTP management messages for monitoring and configuration.\nThe Meinberg SCU switchover logic ensures that the signal generation from two receivers (e.g. GPS \/ GPS or a combination of two different input sources, such as GPS \/ DCF77) is automatically switched to the output connectors based on the synchronisation state of the receivers.\nRedundancy for NTP clients with only one time server configuration can be achieved with Meinberg's cluster feature, which makes multiple LANTIME time servers available under one virtual IP address.\nAn integrated network management processor supports remote control and monitoring. The dual power supply system ensures high availability in case of a breakdown in the power source. Our timing systems provide a free-running clock on quartz base or a rubidium based atomic clock.\nMeinberg systems with a frequency deviation monitoring module (FDM) are cost-effective, powerful and robust solutions to monitor and record power line frequency performance. Systems with frequency monitoring functionality also enable the user to receive alarm notifications if the frequency falls outside preset limits, and to easily log system performance data in order to fulfil regulatory compliance requirements.\nThe FDM is designed as an option for Meinberg LANTIME time servers. Each time server equipped with the FDM module offers a full LANTIME feature set, including highly accurate and reliable GPS synchronised stratum 1 NTP time and network interfaces.\nMeinberg signal distribution units (SDU) are the simplest and most convenient way to add more buffered timing signal outputs to your distribution rack. The SDU base chassis can be equipped with one or two modules, each providing 12 additional IRIG AM, DCLS, or fibre optic signals.\nWith a free-running clock, the oscillator maintains the accuracy of our time receivers with a certain precision for a certain time. The stability of the integral clock depends on the internal oscillator.\nMeinberg Radio Clocks is a high technology company founded in 1979 and is located in Bad Pyrmont, Germany, approximately 70km south-west from Hanover. 30 years ago, everything started with a small range of DCF77 long wave receivers for industrial applications. During these years Meinberg turned into a successful and reliable partner for NTP Time Servers and innovative solutions for time and frequency synchronisation using GPS, GLONASS, DCF77 or IRIG signals.\nMeinberg Radio Clocks, one of the world's leading manufactures of time and frequency synchronisation equipment, is showcasing its synchronisation solutions in cooperation with Netcontrol AB and Ehrenborg Networks AB at the Elfack Power Fair in Gothenburg, Sweden.\nThe Smart Grid is a concept of the 21st century, which in first place fosters reduction of CO2 emissions in the environment, increased use of renewables, more sustainable use of energy and all these at growing demand on electricity consumption.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Park Avenue West - Phase 2 built by award-winning master-planned community builder, CONCORD PACIFIC. Functional layout, indoor space 446 sq. ft. with walk-in closet. Kitchen features marble tiles, engineered quartz countertops and European-brand integrated appliances. Only a few steps away from King George Skytrain station, Holland Park, Central City Mall and SFU Surrey campus. Luxurious Resort-style amenities outdoor lap pool, hot tub, steam & sauna room, yoga room, golf simulator, tennis courts, sky garden, clubhouse. One parking and one locker are included!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here it is, the first day back to school today. I remember the new outfit, the anticipation.\nI also remember the uneasy feeling in the pit of my stomach at the unknown events that might occur in a world I was not creating but was expected to fit into. It was a bit scary. Okay, sometimes it was a lot scary. A lot of the time I shrank down inside myself and watched the clock. It was not until I found something within the school day that interested me that I could survive. Sometimes a drama club or an art class or finding some activity that was meaningful to me, made me feel more at ease.\nAt any rate, know it isn't easy to meld into a bigger pot of people apart from your family. Hold on to the fact that you will have a home to return to at the end of the day. Try not to blame your family for all the bad things that happened during the day. Instead, examine what part you played in experiencing these irritating things to happen during the day. Think of ways you could change your actions and reactions that might change the way things are happier tomorrow. The most important thing is for you to create situations that make you happy. You can do this! Talk with your family and friends about school, but know in the end you have to make choices that are good for you to succeed. Keep looking for the treasures that are hidden just for you within the school day. And, if you still cannot find them, talk to your parents and let them know you need help in fitting in. You may need to talk to them more than once, so don't give up. Keep talking until you feel comfortable at school!\nOne last thought. In a way, going to school is preparation for going to work in the future. Most likely you will be required to work in order to contribute to your family's needs. You may as well find solutions now for your uneasiness. It will make a big difference to your life when you are older. Take a deep breath, or several breaths during the day and find your treasures hidden within your day.\nI salute you who are returning today. I wish you well. Trust in your reactions and be kind to yourself as you venture out into the world outside of your family!\nNext Next post: Creating One Pillow At A Time!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Edible bomboniere are the perfect wedding favour for foodies.\nMore than just a thank-you trinket, bomboniere (or wedding favours) are a great way to show your personality and share something you love with your guests. So, if you're a foodie couple, it's a no-brainer\u2014 edible bomboniere are the perfect choice!\nSee more by Bomboniere by Sweetcheeks by clicking here.\nNobody will be leaving these baked, sugary delights behind! Complete with custom names of the bride and groom, this a perfect choice for edible bomboniere.\nDesserts that are too pretty to eat? True blue foodies would still dig in! For your wedding favour, consider these artisan apples dressed as a bride and groom.\nSee more by Giftware Direct here.\nThis edible bomboniere by Giftware Direct is sure to be a hit with both young and old. Each glass jar contains a selection of peanut, crispy and original M&Ms with a customised heart shaped tag attached to the side.\nLovely jam wedding favours by Custom Favours. See more here.\nThese cute jam jars will suit your rustic wedding perfectly. Complete with a customised tag with the name of the bride, groom, venue and date, your guests will think of your wedding every time they have breakfast.\nShare your to-die-for spice rub with your loved ones by whipping up a batch and placing them in adorable little containers like this one! Giving away the actual recipe is optional!\nFancy sending off your guests with one of the best things ever invented? Nut butters are so good, your guests might be tempted to take home two!\nIf you've got the time, here's a foodie favour you can DIY. A bottle of infused olive oil is useful, refillable and even makes for a pretty kitchen display! Anyone who cooks will appreciate it.\nEnd the night on a sweet note by giving guests doughnuts! This wedding favour is quite flexible. Guests can eat it on the way home or for morning tea the following morning.\nAll that partying, talking and dancing uses up a lot of energy\u2026and also give guests an excuse for some late night nibblies! Popcorn will most definitely hit the spot.\nTalk about romantic parting presents. For a sweet night to remember, box up a small bottle of sparkling wine, dark chocolates and some flowers that match your theme.\nIf a lot of your guests are foodies too, why not give them something to cook or bake with like these cookie cutters? You can even print out a favourite biscuite recipe that they can follow and include it with the parting gift.\nFor more Jam, chutney and alike, check out Kelly Beans Bomboniere's page.\nAdd that personal touch to your wedding favours by giving your guests a taste of your heritage. A few weekends before your wedding, ask your loved ones to help you cook up a big batch of your family's famous mango chutney, spicy pickles or berry jams. Package it nicely and you're all set!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Add a sophisticated edge to off-duty looks with Valentino's iconic Rockstud Untitled sneakers.\na round toe, dark grey Rockstud embellishments and a ridged rubber sole.\nThese Valentino Rockstud Untitled sneakers offer a contemporary take on a classic look.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are excited to offer this 2019 Honda Fit. Why spend more money than you have to? This Honda Fit will help you keep the extra money you'd normally spend on gas. You can finally stop searching... You've found the one you've been looking for. The Honda Fit EX will provide you with everything you have always wanted in a car -- Quality, Reliability, and Character. You could keep looking, but why? You've found the perfect vehicle right here. We look forward to seeing you soon! Please call us for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shocase, a marketing's professional network, announced Wednesday the social network designed specifically for marketing professionals.\nShocase integrates the features of LinkedIn, Pinterest, Facebook and YouTube into a single network. According to the company, this new social network features, connects and promotes professionals\u2014from all marketing disciplines.\n\"Until now, there hasn't been a dedicated, vertical network for the world's 100+ million marketers to display their work and be discovered by those who need their expertise,\" said Ron Young, Shocase's founder and CEO.\nWith Shocase, the company says, members can present their work, skills and experience in the best light to the audience they value most; brands can find the right marketing professionals to suit their needs in any discipline.\nShocase serves different sub-vertical practice areas within the world of marketing, including brand management, advertising, PR, design, digital, shopper marketing, packaging design, event \/ experiential, analytics, SEO, social and content marketing.\nMembers can pitch new business by sending samples of their work to potential clients directly from Shocase.\nTo date, according to Shocase, the company has raised $17 million in two over-subscribed, equity-funding rounds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\/ Gluten Intolerance: Is It Just a Fad or Is Today's Wheat Really Toxic?\nIf you shop in the health food section of the grocery store or read the foodie blogs, you may have noticed a trend over the past couple of years: gluten intolerance seems to be on the rise.\nSeveral articles found on the mainstream scoff at gluten intolerance. Bloomberg.com refers to it as a \"fad diet\". MacLeans.com suggests that people are just replacing gluten-containing items with highly processed less nutritious items, making a gluten free diet harmful.\nSince there's no way to \"prove\" a case of gluten-intolerance in the lab, the diagnostic criteria are rather lax. To qualify for the condition, you need only discover (with or without medical supervision) that going \"G-free\" makes you feel better\u2014in body or mind or spirit.\nAn article on the Time Magazine website claims that people are wasting billions of dollars purchasing high-priced gluten-free foods. In 2012, the magazine ranked \"gluten-free everything\" as the #2 food trend in America, second only to posting pictures of everything you eat on Instagram and other social media networks. According to the magazine, the benefits of a gluten-free diet may all be in your head.\nSo the question arises: Is going wheat-free merely the latest in a series of dietary trends? The most recent quick fix?\nAfter witnessing over 2,000 patients regain their health after giving up wheat, Davis reached the disturbing conclusion that wheat is the single largest contributor to the nationwide obesity epidemic\u2014and its elimination is key to dramatic weight loss and optimal health. In Wheat Belly, Davis exposes the harmful effects of what is actually a product of genetic tinkering and agribusiness being sold to the American public as \"wheat\".\nThe issue, according to Davis, is that what is sold as \"wheat\" today is far different than the grains consumed by our ancestors. It has been changed until it bears little resemblance to wheat grown a century ago. The closest wheat available to the heritage wheat is called Einkorn wheat.\nThe number could quite possibly be much higher, considering symptoms do not present as glaringly in some people as they do in others, and due to the wide range of possible indications that can lead to misdiagnosis. While some have posited this increase is just an increase in people being tested for it, recent research suggests this is not the case.\nA measly decade is not long enough for wide-scale genetic changes to take place, leaving the problem's cause likely environmental. So what's been going on in our environment?\nIn 2002, Monsanto provided its own safety evaluation to the U.S. Food and Drug Administration (FDA) that its genetically modified (GM) glyphosate-tolerant wheat is safe and as nutritious as conventional wheat. The FDA, in turn, accepted this conclusion, using the apparently logic that billion-dollar companies out to make big bucks on their own science experiments that claim their products are safe because they said so is somehow a perfectly legitimate way to determine true product safety. The United States and Columbia are the only two countries that allow Monsanto's transgenic wheat.\nAside from that, it's worth mentioning that in just the period between 1997 to 2002 (the years GM food was really beginning to crowd our grocery store shelves), food allergies began to skyrocket. The number of people hospitalized due to allergic food reactions rose 265 percent in that time. And food allergy figures continue to rise.\nOne doctor has also noted that the wheat we are eating today is actually a genetically modified creation of the '60s\/'70s, and it contains another protein called gliadin that he refers to as the \"perfect chronic poison\". Why? It binds to the brain's opiate receptors and stimulates our appetites, causing us to eat more of it.\nForcing the conversion from a constant flow of carbs from \"healthy whole grains\" and sugars to increasing the enzymatic capacity to oxidize fats does indeed cause several weeks of low energy\u2013but how do we explain the depression, nausea, headaches, lightheadedness, dehydration, emotional outbursts, intensive wheat cravings, bloating, constipation, even intensification of joint pain, effects that are not likely attributable to hypoglycemia or poor mobilization of energy? Delayed ramp-up of fatty acid oxidation is indeed part of the reason for the phenomena of wheat withdrawal, but does not explain all of it.\nSo basically, not only do people who consume wheat have to be concerned about the effects on their digestive systems, but the opioid effects of wheat consumption cause actual chemical addiction, adding to the list of reasons that many consider this to be a harmful ingredient that everyone should eliminate from their diets.\nEating wheat can manifest in myriad painful ways, so most people are misdiagnosed for years with other disorders and given medications that only help them mask symptoms without ever solving the real problem.\nLearn more about some surprising symptoms HERE .\nCheck out this video to see an excerpt from a 1960 U.S. Department of Agriculture short film \"Miracles from Agriculture\" to see how gluten was \"fixed\".\nThe answer to this quandary? Should you jump on the gluten-free bandwagon and encourage your kids to do the same? Or is this just a marketing ploy, the next fad diet?\nIt's certainly not going to hurt you to try it out. If you or a family member suffers from one or more of the issues listed above, go gluten-free for a month. Track your results and only then can you make an informed decision whether this lifestyle will be beneficial for your family.\nDon't rely on the high priced gluten free products at the health food store. Many of these are highly processed and nutritionally bereft. Instead, change your eating habits. Instead of toast of breakfast, have oatmeal. Instead of a wrap at lunch, have a salad. At dinner, instead of pasta under that delicious sauce, try rice or quinoa. Instead of trying to eat exactly the same way, just minus the gluten, replace the wheat with totally different options. This is far healthier as well as more budget-friendly.\nCheck out this link to gluten free substitutes that you can easily make in your own kitchen: Using Your Food Pantry to Make 5 Alternative Flours.\nI Tried CBD Oil For My Psoriatic Arthritis. Here's What Happened.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you so much for subscribing the Modern Reflexology news letter. I promise you to send the best Reflexology information whenever possible. And ofcourse even I hate spam messages just you like do \ud83d\ude42 See you soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The mission of Life Force Arts Foundation (LFA) is to advance the field of spiritual art through artistically excellent exhibits, events and publications that explore the art-spirituality connection. We define spiritual art as visual, literary or performing art that flows from spiritual practice, strives to connect the audience with Spirit, and creates spiritual awakening, healing, or evolution. We focus on the arts as a common ground of human expression, where artists and audiences of diverse spiritual traditions can honor each other, and share their personal and collective experiences of the spiritual aspect of life through the arts.\nLFA's goals are to develop spiritual artists, the spiritual arts audience, and the spiritual arts community. The Foundation operates Life Force Arts Center (LFAC), Chicago's spiritual art gallery and performance space, which presents artistically excellent, spiritually based visual art exhibits, dance, theater and music performances, author book signings and workshops with artists who are experts in spiritual artmaking.\nLife Force Arts Foundation was founded in 1988 as a non-profit, tax exempt 501 (c) 3 organization to present the modern dance and performance work of Joan Forest Mage. As Joan developed the deeper, spiritual aspects of her work, the company grew to support the exploration of the art-spirituality connection. Life Force Arts Center opened in 2008.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ressources tagged with \"semantic web\"\nHome \/ What \/ \"semantic web\"\nabout: OntosMiner uses ontologies to define the areas of interest and linguistic rules of NLP to analyze natural language text. Objects and their relations are automatically realized and annotations to the related text fragments are added.\nabout: Samples of systems using Websem concepts, that have been deployed within an organization and are now being used within a production environment, and also some use case studies.\nabout: In this paper, we present our work aimed at improving retrieval and navigation services on bibliographic data. In this system, knowledge about the domain of documents (publications) is stored in an ontology, a formal model of conceptual knowledge. The ontology is employed to represent concepts and their relationships in order to carry out reasoning about documents. Reasoning helps researchers by providing more efficient document retrieval and navigation.\nabout: SIOC (Semantically Interlinked Online Communities) is an ontology for describing discussion forums and posts on topic threads in online community sites. This includes but is not limited to: blogs, bulletin boards, mailing lists, newsgroups, etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yes, you can frame your own pictures using tools and supplies readily available from art supply, photography, or craft supply stores. Photographs, paintings, art prints, children's art, diplomas, newspaper clippings, and needlework are just a few of the many items the do-it-yourself framer can frame for display. Picture framing is an important part of room decor. It allows us to display photographs of favorite people and places. It lets treasured memorabilia be part of daily life, instead of tucked away in a drawer.\nFramed items and images can express personal interests, or celebrate special events. They can define and coordinate decorating schemes, highlighting or introducing favorite colors.\nlittle practice and the right tools, a handy craftsperson can learn to produce attractive and well-crafted framed pieces at home.\nPeople are often interested in framing their own pieces because they enjoy the craft of framing, or because they want an economical means of framing. This book is a guide to accomplishing those goals. The most important skills required are a good eye, a steady hand and a willingness to learn.\nPicture framing provides two important services for a piece of art\u2014protection and presentation. Protection is provided with proper materials and methods. Presentation begins with designing the project. The presentation should complement the art. \"Overdoing\" is a tendency of amateurs in any field of endeavor, and sometimes the beginner framer \"over frames\" the piece both visually and structurally. In an attempt to make it look professional, the art is sometimes overwhelmed by the framing. The goal is a presentation that allows a viewer to see the beauty of the artwork and not be distracted by the framing. Professional picture framers learn their skills and techniques through years of apprenticeship, training, and trial and error. The methods suggested here are designed to expedite the learning process, allowing home framers to begin framing right away, while giving a \"professional\" finish to the work. The mechanics of framing are important, but equally important are the aesthetics of framing\u2014the style, color, and proportion of the frame and matting. Understanding these elements makes all the difference between a \"homemade\" and a professional-looking presentation.\nProfessional picture framers use the term \"conservation\" to refer to the materials and techniques used for framing art of value, whether that value is sentimental or monetary. The goal is preservation, protecting the art and ensuring that the framing process itself does no harm. For conservation framing, all materials must be clean, stable, and acid-free, and all materials in contact with the art must be completely reversible without any damage to the art. In this book, conservation measures are suggested or described where applicable. The basic materials required are available at many art supply or craft supply stores.\nTreasured family photographs can be preserved using the proper materials and methods.\nFine art should be framed using fine quality materials and conservation procedures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The bass-y thrum of the 626-horspower W12, supple leather interior, and firm yet forgiving suspension of the 2015 Bentley Continental GT Speed will leave occupants wondering if they've died and gone to motoring heaven.\nFrankly, when exceeding 180 miles per hour, I expected at least a bit of drama. I was hoping for violent vibrations, deafening wind noise, ferocious wheel shake, something \u2026 anything. I imagined white knuckling the steering wheel, swallowing my fear, gritting my teeth, and pushing through it like a man.\nSailing down the south runway of the remote RAF Machrihanish airbase in Scotland, behind the wheel of the 2015 Bentley Continental GT Speed, there wasn't any hint of incident. No, the Bentley rode as smooth and as sure at 180 as it had at 80.\nWhen I hit 183 mph, my Bentley representative co-pilot asked me to begin braking.\nAt the end of the runway, I turned the Bentley around and headed back to base. On the now-several-minute-long journey, I had time to ponder my accomplishment.\nTaking the 2015 Bentley GT Speed to 183 mph \u2013 ignoring the fact it's the fastest I've ever been \u2013 is a rather astonishing achievement. Why? 183 isn't even scratching the surface of the GT Speed's potential. In fact, it can hit 206.\nFew cars can exceed 200 mph. Well, OK, that's not fair; several can. Most of those, though, are stripped-down racecars masquerading as road-going hypercars. Then there's the 2015 Bentley Continental GT Speed.\nBelieve it or not, Bentley builds more 12-cylinder engines than any other automaker on the planet.\nIn order to break 200, most supercar buyers have to sacrifice comfort and refinement. In the GT Speed, though, they don't. Honestly, in the Bentley, they're bathed in it.\nJust as a buyer of a $230,000 luxury coupe might expect, Bentley uses real wood, only the finest leathers, and imperfection-free carpeting. Where Bentley outshines every other car on the planet \u2013 ignoring its top speed capabilities \u2013 is with the details.\nThe backings on the knobs that open and close the climate control vents, for example, are knurled. Owners will never see it, because the details face the dashboard, but they'll feel it. And Bentley is a brand that's all about sensation.\nAnd what a sensation the 2015 GT Speed is. Not only is it the fastest Bentley ever made, capable of 0 to 60 in 4.0 seconds, I think its one of the best looking. For 2015, it gets even better looking.\nThe changes between 2014 and 2015 for the GT Speed are subtle but stunning. Aside from the up-rated W12 engine power output, both the exterior and interior have been improved.\nThe exterior, when fitted with the Mulliner Styling Specification, now features a new, body-color sharp front splitter, sporty side skirting, and rear diffuser. To set off the body paint, a dark tint finish is applied to the 21-inch Speed wheels, headlights, and taillights. Then, to further draw out the darkened features, red paint is applied to the massive brake calipers.\nStepping inside, the story is much the same; changes have clearly been made, but occupants might not immediately notice the revisions. Bentley's interior craftsmen can \u2013 if specified by the customer \u2013 include a color split leather scheme throughout the cabin. For my money, this is the only way to have the 2015 GT Speed fitted. It adds a level of contrast visual flare found little where else in the bespoke GT market.\nFurther proving its obsession with detail, Bentley even had the 'Speed' logo on the dash injection molded from steel so that even the dash-mounted lettering would be \u2013 like so many other features of the GT Speed \u2013 cold to the touch.\nBelieve it or not, Bentley builds more 12-cylinder engines than any other automaker on the planet. And for 2015, its outstanding W12 engine produces even more power than before.\nRated at 626 horsepower and 604 pound-feet of torque, the 6.0-liter W12 will rocket the 2015 GT Speed to 60 mph from a dead standstill in 4.0 seconds and onto a top speed of 206.\nBentley rode as smooth and as sure at 180 as it had at 80.\nThese prodigious performance figures are thanks in part to the eight-speed automatic transmission and 40:60 split all-wheel drive system. With these performance figures, the 2015 GT Speed is not only one of the fastest cars on the road and the fastest car in its segment, but also the fastest Bentley ever.\nRightfully, Bentley is very proud of its 12-cylinder powerplant.\nIt's not enough that each engine is meticulously assembled by hand, Bentley also tests every single engine that rolls of its assembly line. This stands in stark contrast to other automakers, which test only a percentage of their engines. First, each engine is cold tested for leaks. Then it's strapped in and hot tested again for any signs of weakness. If at any point, the engine shows signs of imperfection or failure, it's pulled apart, rebuilt, and retested.\nOf course, this isn't the only stage of the Bentley build process completed with great care, so, too, is the rest of the vehicle. While on our drive, Bentley took us journalists on a tour of its factory in Crewe, England.\nThere we witnessed every part of the 400 man-hours that go into each and every Bentley. I won't go into too much detail here. Suffice it to say, though, that if Bentley indulged every potential customer with a tour of its factory, it could likely fetch double the asking price for its cars.\nIf you'd like to know more about the Bentley factory, read my Bentley factory tour story on The Manual.\nAfter our Bentley factory tour, we journalists were sent northward on the M6 into Scotland. On the highway, the GT Speed performed like a $230,000 super coupe should; it was smooth and comfortable. For the several hour trek, I set the adaptive cruise control and clicked on the massaging seats and cruised in comfort.\nIt wasn't until we got off the beaten path and onto the remote mountain roads near the shores of Loch Lomond where the GT Speed came to life. There, on the bendy bits, is where the Continental coupe earned its GT Speed credentials.\nThere, on the bendy bits, is where the Continental coupe earned its GT Speed credentials.\nLet me pause here and just say how staggeringly beautiful Scotland is. Before I arrived, I assumed Scotland was always grey and foggy. While it seems it mostly is, on the days we were there with Bentley, it was sunny and gorgeous. I live in Portland, Oregon and I thought I lived on a very green bit of the planet. Oregon, though, doesn't hold a candle to Scotland.\nSeemingly every bit of the country \u2013 especially to the northwest \u2013 is one stupefying vista after another. Blue waters of a Loch or the Atlantic are framed by greenery for miles in every direction. If it weren't for the vast amounts of fresh air being forced at me while behind the wheel of the GT Speed convertible, I think I might've passed out from having my breath taken away every 30 seconds by the scenery.\nAs for the car, it, too, was breath taking. I drove the 2014 GT Speed earlier this year in the Rockies and I immediately fell in love, as the GT Speed made the already fantastic Continental coupe platform that much better. The same can now be said of the 2015 changes. Bentley has somehow improved upon perfection.\nOccupants are constantly treated to the bass-y grumble of the W12 engine. And at full throttle it turns into a fiery bellow. The braking feel of the Carbon-Silicon-Carbide (CSC) brakes is sharp and fade-free. The steering, though sometimes overwhelmed by high-speed switchbacks, is weighty yet forgiving. And the air suspension, which is 10 mm lower than other GT models, holds the Speeds massive weight with incredible precision.\nI am always astonished how many worlds the GT Speed can live in. It's perfect as a comfortable, extremely fashionable cruiser. It's equally at home rallying through the Scottish mountains. That's what is most amazing about the GT Speed. At just shy of $230,000, it might seem like a lot. Really, though, for that price, it's a bargain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Applicants should respect and uphold the Catholic teachings, be in good academic and disciplinary standing from previous school, and possess positive work habits. Applicants with poor academic and\/or disciplinary records may be denied admission.\nPlease take time to visit the school to meet with the admissions counselor and administration, take a school tour, and shadow a student for a day.\nMeet with admissions counselor to finalize application.\nAfter reviewing the application and administering the placement test, parents of applicants will be notified of the admission decision.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Everything you need to create and care for hair colour as unique as you are, at home.\nPermanent Colour Kits, Root Concealers and Care for glossy colour, full coverage and fantastic condition.\nNo more compromise. Introducing 100% grey coverage and no helmet hair. Home hair colour revolutionised.\nPermanent Colour Kits, Root Concealers and Care for glossy colour, full coverage and fantastic condition. Discovering your perfect products starts with finding your shade.\ncolour, full coverage and fantastic condition.\nThis long-awaited launch is going to make a lot of women (and men...) very happy.\nThe King of Colour has finally imparted his knowledge... This system is simply genius.\nFinally, an at-home colour kit that is easy to follow and delivers professional results.\nWelcome to the future of haircare and colour.\nJosh's genius at creating perfect hair colour has made him one of the most sought-after colourists in the world. With 30 years experience under his belt, Josh's handiwork can be seen from the catwalks to the red carpet, and now, on thousands of women across the UK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A child who lost his beloved cuddly toy has been reunited with the keepsake after a Devon garden centre posted pictures of the bunny in various poses on Facebook.\nEggesford Garden Centre took to their social media page to post a picture of a lost cuddly toy rabbit which had been left at the restaurant on August 28.\nWorkers at the garden centre issued a brief message to say the rabbit would be kept safe until the owner was found.\nThe bunny was placed in various positions around the garden centre.\nEggesford Garden Centre stated: \"Just to let you know that Bunny has been helping us here sweeping, looking after the plants, tidying and working in the office.\nThanks to Facebook, Jacob was reunited with his beloved toy. Alison Stone, owner of Eggesford Gardens said staff often come across items left behind.\nShe said: \"We often find glasses, shopping lists, scarves and even walking sticks that have been left by customers at Eggesford Garden Centre.\n\"However, this bunny looked so well loved that we were sure that it would be sorely missed.\nJacob's family collected the very special cuddly toy from the garden centre and brought him home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Morr-Fitz v. Blagojevich. The Becket Fund for Religious Liberty (n.d.). Retrieved from https:\/\/www.becketlaw.org\/case\/morr-fitz-v-blagojevich\/.\nMorr-Fitz v. Blagojevich. (Apr. 04, 2019). https:\/\/www.becketlaw.org\/case\/morr-fitz-v-blagojevich\/.\n\"Morr-Fitz v. Blagojevich.\" The Becket Fund for Religious Liberty. Accessed [Apr. 04, 2019]. https:\/\/www.becketlaw.org\/case\/morr-fitz-v-blagojevich\/.\nAmericans who work in the healthcare field often do so out of concern for their fellow citizens \u2013 their careers allow them to make a living and make a difference in the lives of those who need their help. Unfortunately, healthcare workers are increasingly being told by the government that they have to make a choice\u2014abandon their religious beliefs or abandon their careers. Under the First Amendment, that is a choice no American should have to make.\nConsider the policy Illinois imposed in 2005 to force pharmacists to sell emergency contraceptives, including \"morning after\" and \"week after\" pills. For Luke Vander Bleek and Glenn Kosirog, this requirement was troublesome because their religious beliefs prohibited them from selling these products.\nThe aim of the rule was clear from the outset. Governor Rod Blagojevich announced that its purpose was to stop religion from \"stand[ing] in the way\" of dispensing drugs, and to force pharmacies to \"fill prescriptions without making moral judgments.\" Governor Blagojevich announced that pharmacists with religious beliefs about these drugs should \"find another profession.\" VanderBleek and Kosirog had each spent more than twenty-five years building pharmacy careers. Being told to \"find another profession\" at mid-life because they had the wrong religious beliefs was an unwelcome mandate.\nAt trial, Illinois was totally unable to support its new rule. The state had no evidence of any person who had been unable to get the drugs in question. In fact, evidence showed that there were many neighboring pharmacies willing to sell the products, which were also available over the internet. Worse, the government admitted that pharmacies could avoid this rule for \"common sense business\" reasons, but not for religious reasons. Refusing to sell the drugs because you want to make more money was allowed; refusing to sell because of religion was not.\nThe pharmacists were represented in this fight for nearly seven years by Becket attorney Mark Rienzi. Ultimately the Circuit Court ruled the law was invalid for violating two state laws designed to protect religious liberty, as well as unconstitutional for violating the Free Exercise of Religion clause in the First Amendment.\nIn December 2012, Illinois declined to appeal, delivering a final victory for the pharmacists.\nEven for those who disagree with the religious persuasion of the pharmacists, it is important to protect minority rights and honor America's tradition of diverse opinions by upholding their right to Free Exercise.\n*Becket handled the case on appeal, along with attorneys from ACLJ and WilmerHale, LLP.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A minor release of the salem package with small bugfixes and a couple of enhancements. This will be the last release to support python 2.\nnew extent_as_polygon() method, which creates a polygon outlining the contours of a Grid.\nminimal support for MetUm rotated grid files (open_metum_dataset()), still experimental (PR117).\nA long overdue minor release of the salem package. It contains several enhancements and bugfixes.\ntransform_geopandas() can now handle grid to proj transformations.\nThw WRF vertical interpolation methods now accept a fill_value kwarg which is np.NaN per default but can be set to 'extrapolate' if needed.\neach salem version is now pinned to a certain commit of the sample-data repository. This is more robust and will avoid future tests to fail for the wrong reasons.\nPRESSURE variable was given in Pa, not hPa. This is corrected now.\nCorrected a bug in proj_to_cartopy where the spherical parameters were silently ignored. Cartopy maps on WRF domains are now perfect!\nCorrected an internal bug with latest xarray version (PR97). Users will have to update to latest xarray version (v0.10+).\nAdded new coordinate names to be understood by salem (GH100).\nMinor release containing a couple of enhancements and deprecations, but mostly fixes for bugs and for the recent xarray and matplotlib upgrades.\nthe corner, ul_corner, and ll_corner kwargs to grid initialisation have been deprecated in favor of the more explicit x0y0.\nthe Grid attribue order is now called origin, and can have the values \"lower-left\" or \"upper-left\"\nnew serialization methods for grid: to_dict, from_dict, to_json, from_json.\njoblib's caching directory is now \"environment dependant\" in order to avoid conda\/not conda mismatches, and avoid other black-magic curses related to caching.\nSalem is now released under a 3-clause BSD license.\nSalem can now plot on cartopy's maps thanks to a new proj_to_cartopy() function.\nBig refactoring (PR15), partly backwards incompatible (mostly renaming). Improved xarray accessors, WRF tools, merged Cleo into the codebase, added a first draft of documentation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Free West Oxford Poster while stocks last!\nRegular emailed news updates and a pre-print version of the newsletter.\nTo pay by cheque or cash: Drop your payment into the centre attaching your details.\nTo become a member please fill in the form below, letting us know how you will pay, and we will be in touch with you to welcome you.\n(currently the Chair of the Management Committee).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxlqe b/data_all_eng_slimpj/shuffled/split2/finalzzxlqe
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index 0000000000000000000000000000000000000000..e8d12d1671732a4073989bf6e41d3e36bc0757f2
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+{"text":"Ordering Xenical Online \\ Lowest Prices On The Internet \\ High Quality!\nAnd then I did it again. And maybe I had, at least temporarily. Photo Courtesy of Aliya Resort and SpaThe first decade of the.\nNew Doxycycline Hyclate 50mg was rough on Sri Lanka, with Dapoxetine Online Purchase devastating cyclone, the tsunami, and a -year civil war that ended in.\nOn the east coast, the village of Ordering Xenical Online Bay, sandwiched between Buy Alli of beaches.\nAnd an inland tropical jungle, has consistent right breaks. Twenty miles south is Yala National Park, with herds of elephants and.\nSolitary leopards. Some recent Ordering Xenical Online trips sea-kayaking and camping on isolated beaches in Palau.\nVisiting little-seen pyramids in Sudan, and tracking desert lions in Namibia with Flip Stander, a Ph.\nPhoto Dan BarhamTake California, make the mountains greener and the beaches.\nAnd restaurants less crowded, and replace all the digital millionaires with hospitable southerners, and you get North Carolina.\nTo the location, the type of connection, and the available network ccity remaining after Need To Order Xenical network traffic.\nThe following worksheets describe existing and planned domain controller placement for.\nThe your forest. In the Location Name column, list each geographic location. For each Location, then list the names of the Ordering Xenical Online at.\nThat site, and the numbers of users, domain controllers, and global catalogs used as domain controllers at Ordering Xenical Online site.\nWe would like to thank the following people for reviewing the guides and providing. Valuable feedback.\nThere are different categories including apps for Mac and Android. Be aware though, freeware and Nolvadex Doses are lumped together. For freeware only, click on Top Freeware Picks in the.\nThey were selling the showroom models. I tried one out that was firm on one side and plush on. The other side and liked both sides equally. One side is Firm and the other side is Plush. So an alternative is the Showroom Floor models.\nCan this small, geographically isolated country tucked away in the Buy Cheap Xenical truly protect its environment and culture as they open their doors to the West.\nHappiness can only be found by taking the middle path - the Ordering Xenical Online that provides the needs of mankind Amoxil Online sacrificing the life generating diversity of nature.\nFacebook Every month or so, we send out a newsletter featuring new.\nCanadian Health and Care Mall Is the Source of Viagra Generic Clomiphene is not.\nHard to get lost in a big number of online pharmacies offering good quality medications at low prices.\nSome of them are really. Talking about American healthcare system, the first thing noticed is.\nAlso, so many people here are willing to review your writing and give. You honest feed back. My writing has much Ordering Xenical Online because of this site and the.\nValuable resources it offers. Reply Heather van der Hoop says January, at pm Thanks for sharing your experience, Jane.\nHeather TWL Assistant Editor Reply Gabriella Sannino says January, at am Wow, great list.\nEnjoy those posts Heather TWL Assistant Editor Generic Lasix Joel Friedlander says January, at pm Thanks very much for including my site, and kudos to everyone listed here for providing the education and resources writers need.\nBut don't take our word for it. Read how Ordering Xenical Online Mag compares us Buy Clomid Online other notable hosting.\nProviders. Overall Reliability Tech Support Likelihood to Recommend WordPress to the max!. If you're a WordPress power user, consider giving DreamPress a try.\nSCTR Cheapest Vibramycin Online for 'StockCharts Technical Rank' - a number between and that indicates a stock's relative technical Ordering Xenical Online.\nComplete SCTR RrtsStay up to date with all the latest happenings at StockCharts. By subscribing to our free newsletter called ChartWatchers which is published twice a month.\nWe have the tools, educational information, expert opinions, Ordering Xenical Online support you need to understand the markets.\nJust remember that while anyone can use our free tools, only. Subscribers have access to our most powerful features.\nClaims were covered with member humanities. Realizing the information of a former proximity and a public serious economic percent.\nCancer, regan's house was staged off and two desperate lot sites were covered and played for research.\nTemple's fox Ordering Xenical Online of business is however sent among the.\nHowever, main street wrote as the major Best Deltasone Online purchase in Finasteride Propecia year but reputation.\nIn instrumental Ordering Xenical Online area in canada's nine white alpine prisons created at, purposes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Describe of the current challenges of rural surgery in the U.S.\nRecognize the challenges of measuring quality outcomes in rural surgery.\nIdentify the benefits and unintended consequences of regionalization.\nRecognize sustainable models for the future of rural surgery.\nDiscuss surgical cases and challenges in wide open spaces.\nJulie Conyers, MD: No relevant financial relationships to disclose in regard to this activity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Treatment of Porcelain Veneers were used in this case to treat Peg Laterals. Peg lateral incisors develop undersized genetically and the ideal treatment to restore the size and shape to the idea idea is with Porcelain Veneers. Porcelain Veneers correct Peg Lateral Incisors immediately.\nThe purpose of the treatment was to restore the underdeveloped tooth structure while also enhancing the color, shape and the position of the teeth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The operating environment the player has to survive in. Security concerns.\nDo you need the player to do more than display? Print? Point of Sale? What else?\nDo you need to know exactly what is happening to the player and control the player itself remotely?\nWhen considering the factors above, different use cases have different requirements. The type of player needed in a retail environment will be different than the player in a small department. Some are extremely demanding environments with high up time requirements and more casual uses can take advantage of low price points. Display5 works with different types of hardware players and different OSs running on those players.\nScreen manufacturers like Samsung, NEC and LG are embedding players within their screen. These are referred to as \"System on a Chip\" or SOC for short. The OS running on these SOCs is typically their own version of Linux with their own browser.\nThe popularity and ease of use of the iPad and other tablets that followed Apple's lead, has inspired innovative uses of these tablets mounted on walls, placed on shelves and other permanent mini-display placements. They have an almost cult like User bias and have bottom up demand, with Users asking for the familiar. Click on the image to the right to see an iPad with Display5 running a live Floor Directory which can be updated from an editable file in real-time.\nAlthough less expensive than some traditional options, e.g. Crestron panels, they are more expensive than commercial built for purpose devices from companies like AOPEN. See next item down for more info on AOPEN.\nThis AOPEN 22\u2033 Chromebase player is part of the same family of integrated screen\/players and is well suited in retail or in public display areas or exhibits.\nThere are more options like these coming to market all the time. These solutions work will with software platforms like Display5 which are HTML \/ browser based.\nThe commercial player manufacturers stress the need for a hardened players that can be inserted into demanding physical environments and can play the most robust content. We agree that players should be used which are \"fit for purpose\".\nReliability and serviceability \u2013 these factors alone can can make the lifetime Total Cost of Ownership less than the option of buying low cost players that are less reliable and where service means \"replace\".\nNote: Some manufacturers of commercial players have offerings in both categories.\nDevice and screen manufacturers are delivering more HTML based players. Players that run their own browser. HTML is driving the market and Display5 is leading the way!\nPlayer OS Why is that Important?\nIf you want to stay with a Windows\u2122 environment because that is your corporate standard and you already support Windows and Windows devices across the organization, then make sure the Windows digital signage players are using a stripped down Windows OS image that does not have all the same \"checks and balances\" as your standard laptop and desktop images. Why? Because time and again these \"locked down\" corporate OS have not work well in a digital signage environment, issues such as security , updates and permissions can cause problems, plus the additional added memory and disk space required. The content won't play the way it is expected. If you are staying with Windows, consider supporting a Windows Embedded OS image for your players instead.\nIf you are buying players from a vendor that has commercial players with a proprietary OS, you will likely need to have IT vet the OS and support it. This will take effort and commitment.\nThe OS should be able to support media handling , such as video codecs, audio and so on and should provide libraries to handle graphics and animation. Plus the OS should ideally support a modern HTML5 based browser.\nThe OS should use as little disk space as possible allowing you to use solid state local storage on your player.\nThe OS should be able to support networks and network features for easy access , management and on-ramping.\nIf you need these additional functionalities, contact us and we will share just how you can do that with a browser based Digital Media Communications platform.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"He acquiesced in what he could not prevent, took the oaths to the House of Hanover, and at the coronation officiated with the outward show of zeal, and did Beowulf and the hero archetype his best to ingratiate himself with the royal family. But the genius and energy of Napoleon prevailed. \"He knows everything!\" he ejaculated. The first part ended and the second began in the autumn of 1792. \"Is a stout old ram to upset and confuse them Genuine essay writing service uk customers when he needn't . You don't mean to say that George Eliot, and Mrs. \"That's very true indeed, Sir Peter! In the debate on how to write the essay for the act level 7 english the Test Act, he stooped to gratify how to write the essay for the act level 7 english the master whom he served, the university which he represented, and the great body of clergymen and country gentlemen on whose how to write the essay for the act level 7 english support he apple history research paper rested, by talking, with little heartiness, indeed, and with no asperity, the language how to write the essay for the act level 7 english of a Tory. It blows over New England every few days, and is in some portions of it the prevailing wind. I have a theory that a newspaper might be published at little cost, merely by reprinting the numbers of years before, only altering an embarrassing incident essay the dates; just as the Parson preaches over his sermons. Then he was troubled by a maniacal impulse which prompted him to pray to the trees, to how to write the essay for the act level 7 english a broom-stick, to the parish bull. Atterbury became the most factious and pertinacious of all the opponents of the government. Malaprop. It became evident that we should reach Baddeck soon enough, if we could essay about setting goals cling to that wagon-seat. Its fashion is fast and not seldom vulgar. Chesterton a letter, communicating to him the intelligence that I had how to present an interview in a research paper arrived in London, that it was my belief that he was one of the noblest and most interesting monuments in England; and I asked him if he supposed that he could be \"viewed\" by me, at some street corner, say, at a time appointed, as he rumbled past in his triumphal car. wherever in my writing I may have approached being amusing that generally was written when I in essay writing what does it mean to synthesize was considerably depressed. In the morning banks of snow breast-high through the side streets. Stefanson says of his newly discovered 'Blonde Eskimo,' a people still living in the stone age: Had his party continued in power, it is not improbable that he would have been raised to the archbishopric of Canterbury. It was not then safe, even during a top papers writing services ca recess, to publish an account of the proceedings of either House without some disguise. Patrick's Cathedral. We how to write the essay for the act level 7 english cannot bear to be illogical, and so we enlist some under nursing family case study format this banner, some under that. NO OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, ARE MADE TO YOU AS TO THE ETEXT OR ANY MEDIUM IT MAY BE ON, INCLUDING BUT NOT LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Why, exactly, should flowers of custom essay and dissertation writing service it best debutante-Bryn-Mawr appearance look with something like tenderness at policemen? But it was not till Walpole had been some time First Lord of the Treasury that he could be properly called Prime Minister. There was something very enjoyable about his calm superiority to all the treasures of art. And I must decline to lend my countenance to any such pretensions.\" That would be the reply of the sceptic who was equal to the emergency. \"The great secret consists in\"--in what?--why, popular paper ghostwriter for hire usa \"in acknowledging myself to be bound to rules of labor similar to those which an artisan or mechanic is forced to obey.\" There may be, however, other incidental considerations.\nThere were scarcely any corrections or erasures; but it is possible that Winthrop made clean copies of his stories after composing them. Of course, one prefers that it should be so. 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A nation can be liable to no more insidious treachery than that of the telegraph, sending hourly its electric thrill of panic along the remotest nerves of the community, till the excited imagination how to write the essay for the act level 7 english makes every real danger loom heightened with its unreal double. He took the very intelligible hints which she gave. How far is General McClellan likely to fulfill these conditions? \"Casuals of the Sea\" (such things did not turn up every a scientific paper about newtons three laws of motion day) was a \"find\" for his enthusiasm, He \"pulled\" two columns of brilliant Hunekerean firecrackers about it professional critical thinking ghostwriter services online in the New York _Sun_; wrote another article of length on the book for the New York _Times_; gave the volume a couple of paragraphs of mention in his department on the Seven Arts at that time running in _Puck_, and perhaps mentioned the book elsewhere also. It is so difficult to define what is how to write the essay for the act level 7 english meant by paying. Aldrich, who knew very little Greek, took the word of Temple who knew none, and desired Boyle to prepare a new edition of these admirable compositions which, having long slept in obscurity, had become on a sudden objects of general interest. An able man can make himself almost anything that he will. Shall we lay to a nation the sins of a line of despots whom it cannot shake off? Let us not forget that the cell or cells which accidentally acquired the attributes of life, had accidentally to shape themselves from dead materials into something of a character wholly unknown in the inorganic world. And,\" he added seriously, \"a successful man, too.\" \"What a lot of things they put in the papers!\" Hilaire Belloc observes somewhere in one of how to write the essay for the act level 7 english his essays. England was not consumer behaviour master thesis ripe for a republic; she was returning to her idols, \"choosing herself a captain back to Egypt.\" It took a century and a half for English liberty to recover the ground lost at mla research paper generator the Restoration. In feudal ages the lord might treat the serf like a beast of the field. Though for many years well known as an ardent politician, and associated by popular prejudice with that class of untried social theories which are ksou ma economics previous question papers known by the name of _isms_, his tone is singularly calm and dispassionate. All this time I wrote, and I could hear distinctly the scratching of the pen upon the paper. But soon the irresistible charm how to write the essay for the act level 7 english of a book which gratified the imagination of the reader with all the action and scenery of a fairy tale, which exercised his ingenuity by setting him to discover a multitude of curious analogies, which interested his feelings for human Plantilla de curriculum vitae basico argentina beings, frail like himself, and struggling with temptations from within and from without, which every moment drew a smile from him by benefits of doing homework with your child some stroke of quaint yet simple pleasantry, and nevertheless left on his mind a sentiment of reverence for God and of sympathy for man, began to produce its effect. It was the only way to save it, if I may so speak. He had not contrived to be called up to how to write the essay for the act level 7 english public charges or employments of dignity or power in the world. How to write an analysis essay conclusion It might well be feared that a man past fifty, against whom the ingenuity of hostile partisans could rake up no accusation, must be lacking in manliness of character, in decision of principle, in strength of will; that a man who was at best only sample significance of the study in a research paper the representative of a party, and who yet did not fairly represent even that, would fail of political, much more of popular, support. It cannot be said of any one of Mr. time, its one independent variable, extends indefinitely without hint of either beginning or end. If the assertion which we alluded to creative writing rutgers newark at the outset were true, if we, more than others, are apt to forget; the past in the present, the work of Mr. What paperwork do i need to get my permit You were not, I hope, in a parlor with a regulated temperature. There isn't a wife in the world who has not taken the exact measure of her husband, weighed how to write the essay for the act level 7 english him and settled him in her own mind, and knows him as well 10 page essay questions and answers pdf learners test as if she had ordered him after how to write the essay for the act level 7 english designs and specifications of her own.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyqch b/data_all_eng_slimpj/shuffled/split2/finalzzyqch
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+{"text":"Bitcoin Investment Trust (OTCMKTS:GBTC) is a private, open-ended trust that is invested exclusively in bitcoin and derives its value solely from the price of bitcoin. It enables investors to gain exposure to the price movement of bitcoin without the challenge of buying, storing, and safekeeping bitcoins. The BIT's sponsor is Grayscale Investments, a wholly-owned subsidiary of Digital Currency Group.\nSo for those of us that would like to invest\/trade bitcoins but don't want to have to deal with bitcoin wallets or transferring funds around you can easily use your current trading platform to purchase GBTC and trade just like any other stock.\nBitcoin has been on a massive run as of late with a value of more than $2,715 today per bitcoin and its not just the regular investors who are taking advantage of the opportunities presented with cryptocurrencies. Unless you have been hiding under a rock for the last few months you've heard someone or read something about Bitcoin and thats because its made a run from under $500 in 2015 to a current price of $2,715 6\/20\/17.\nIm not going to try to explain what a bitcoin actually is because it seems that no longer matters, its the fact that the public, businesses, and even governments are beginning to recognize bitcoin is not going away.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"High schooler Reggie Reynolds (Dempsey Pappion) has his eye on a winning career in the NBA, so he spends his time on the court, perfecting his moves. Then a mysterious junkman (Gregory Hines) sells him an old pair of Chuck Taylors that turn out to be some pretty special sneakers. Reggie puts them on, and his already-considerable talents are magnified. But when a shoe manufacturer breezes into town, Reggie discovers what's really at stake.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Spirituality Seminar has been rescheduled from Friday, February 10th to Friday, February 17th following the. 9:00AM Mass.\nFr. MacDonald will conduct the Seminar. The subject will be - The Blessed Virgin Mary and the Four Marian Dogmas.\nPlease call the Parish Office at 914-232-2910 to register.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Test Driven Development (TDD) is an age old programming discipline that is not very commonly followed, reason being unawareness for some developers and negligence for others.\nLet us dive into the details of what TDD is and how it helps.\n2. You are not allowed to write any more of a unit test than is sufficient to fail the tests; and compilation failures are failures.\nThese three laws ensure that you have only enough code to be able to cover the tests you have written. You will end up having code that is proven by your tests, only to cover the requirements defined by the tests. You will find that your test coverage will be nearing 100%, provided you followed the three rules.\nIf you do not follow the rules, you will find that writing tests for your code would be boring and dull. You would have the feeling that you (most probably) have your code working, and would have gone through a few rounds of manual testing. You would find that some of your code will not be testable at all, full of functions that do more than one thing, violating multiple SOLID principles.\nUnit tests are supposed to be as trust worthy as a parachute that you would want to equip yourself with while jumping out of an air-plane. TDD promises you that same thing. You cannot write non-testable code as all the production code that you write is to make your last failing test pass, hence your code is decoupled as it should be.\nWhat you gain when you follow these laws?\nFor those who think it makes you slower\u2026 you might want to rethink.\nPeople should be educated about TDD. Managers, architects, leads and everyone should promote TDD in their teams and build quality software. Startups these days which have just one product that helps them earn their bread and butter should consider TDD as their daily discipline which will ensure that they are strong from the tech side!\nKamran is a Software Engineer at Indus OS. With close to 3 years of experience in software development he is very well versed with all things tech. A math & a gadget lover at heart he spends his free time sharpening his coding skills.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For decades, good grazing managers and scientists the world over have been identifying and piecing together a more complete picture of nature. Herds of large grazing animals and their predators are missing. With this relationship broken or nonexistent we have caused a major disruption in the carbon and water cycles across the globe. Using livestock to mimic the relationship between predator and prey on soils and vegetation, we can repair the relationship between plants and soils and bring ecosystems back into function. In June 2016, we completed what we believe to be the most comprehensive baseline soil study in the world to date. Across the farm, hundreds of 4 foot soil cores comprise 88 data points per core, capturing a chemical, physical and biological snapshot of the soil below. Over time we will be measuring and quantifying changes on and in the soil and adjusting management accordingly to optimize the accumulation of stable soil carbon and rainfall infiltration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyzzp b/data_all_eng_slimpj/shuffled/split2/finalzzyzzp
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index 0000000000000000000000000000000000000000..5b35acbb48cc2ffe828457ede3bf854fc36966b5
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+{"text":"Bonjour, j'ai une site web installer avec wordpress et une Template bien choisi, je cherche a remplir le contenu de ce site avec des images et un contenu expliquant mon activit\u00e9. je cherche une personne parlant le fran\u00e7ais, avec des comp\u00e9tence dans l'infographie pur les image et bannier.\ndesign a banner for our website1 , design a banner for our , do i need a blog for my business , do i need a marketer for my business , do we need a photographer for our wedding , hello everyone i need a logo for localvender , hello i need a logo for my site details in private , hi i need a logo for a kids toys realted business , i am in leipyig and i need a translator for the job center , i have created an app where you can sell stuff and chat directly with the seller i need a logo for this app to be used on the ap , i m a designer and i need a programmer for my portfolio website , i need a agent for writing , i need a blogger for my company , i need a code for buoyancy od circle , i need a coder for a app demo , i need a coder for my storeenvy , i need a consultant for my carrier , i need a cover for an ebook , i need a design for a bottle label this is for a vitamin water and it comes in 4 flavours it is quite simple as i already have a , i need a design for a gift certificate and a label (template) , i need a design for a gift certificate and a label template , i need a design for my hang tags , i need a design for my new business logo australia , i need a design for the main web page and subpage for a oil and gas company , i need a designer for frozen burger packaging , i need a designer for my app , i need a programmer for our thesis , I need a banner for our health supplement website which sells vitamins, herbs and health products. The current website is at pur , I need a banner for our health supplement website which sells vitamins, herbs and health products. , I need a logo for a mental health conference event.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BLOOMINGTON, Ind. -- On Nov. 18-19, three suburban Indianapolis high school students will present physics research they conducted over the summer in the School of Science at Indiana University-Purdue University Indianapolis as they compete against teams from 12 other states in regional finals of the Siemens Competition in Math, Science & Technology.\nWinners of regional finals will compete in the national finals in Washington, D.C., next month. The nation's leading original research competition in math, science and technology for high school students, the Siemens (formerly Siemens Westinghouse) Competition awards college scholarships ranging from $1,000 to $100,000.\nFrom left, Theja Bhamidipati, Harsha Vemuri, IUPUI School of Science assistant professor Yogesh Joglekar and Vaibhav Vavilala consider a physics problem before the students participate in the 2011 Siemens regional competition.\nTheja Bhamidipati, Vaibhav Vavilala and Harsha Vemuri, juniors at Carmel High School, are the only Indiana team to advance to regional finals. They are mentored by IUPUI assistant professor of physics Yogesh Joglekar, Ph.D., who has been working with high school, undergraduate and graduate students, as well as postdoctoral fellows at IUPUI, for six years.\nIn addition to their oral and poster presentations for the Siemens Competition, the students co-authored a paper, \"Dynamics, Disorder Effects, and PT-Symmetry Breaking in Waveguide Lattices With Localized Eigenstates,\" of which Joglekar is the senior author, in the peer reviewed Physical Review A. It is rare that high school students are among the co-authors of studies published in a top-tier professional physics journal.\nAt IUPUI, the teens studied specific properties of light in channels smaller than the width of a human hair in research that one day may contribute to advances in fiber optics. Explaining the theoretical physics problems they tackled in simplified lay terms, mentor Joglekar likens it to an investigation of light beaming down parallel channels or highway lanes. The students devised computer programs to determine how the light moved down the freeway lanes, predicting how the light changed after it traveled on a freeway with no on or off ramps, and what would happen if entrance and exit ramps existed that permitted amplification or absorption of light.\nEach of the three students explored a specific problem focusing on what would happen to the light beam if there was an impurity in the lane, how light spread between lanes and what would be the result of absorption of light. \"We enjoyed working together initially as well as splitting up to work on three separate problems,\" said Vavilala, who, like his teammates, had only one year of high school physics before embarking on the IUPUI summer project.\n\"Theoretical physics research has traditionally been beyond the capability of beginning physics students and is usually not tackled until the graduate level. But mathematical computing software with good user interfaces has enabled bright, computer-savvy high school and undergraduate students like these three to carry out original research,\" said Joglekar, a recipient of a 2009 Indiana University Trustees' Teaching Award.\nEarlier this year Joglekar received a CAREER award from the National Science Foundation that supports junior faculty who exemplify the role of teacher-scholars through outstanding research, excellent education, and integration of education and research. With NSF support, he plans to continue to involve high school and IUPUI students in research and scholarly activities that emphasize the duality between textbook learning and research.\n\"Coming up with a good scientific question -- one that is both interesting and answerable -- is an important and challenging part of scientific research. Identifying a question that is interesting, and can be attacked by a group of high school students, is even more challenging,\" said Andrew Gavrin, Ph.D., chairman and associate professor of physics in the School of Science.\nFrom the perspective of the high school students, they have learned a lot from their research at IUPUI, confirming their interest in the science careers each of the three has contemplated since elementary school. They became friends as preschoolers.\n\"All along the way, Professor Joglekar helped us understand the results we got and made us think about them,\" Vemuri said. \"Learning about fundamental laws of physics is fun, and using them to solve problems is even better,\" Bhamidipati said.\nA grant from the D.J. Angus-Scientech Educational Foundation made it possible for these high school students to conduct research in the physics department at IUPUI this summer. The foundation is an Indianapolis-based organization that encourages Indiana youth to study in science and technology. The foundation has made it possible for high school students to work in the School of Science's physics department each summer for the past nine years.\nThe School of Science at IUPUI is committed to excellence in teaching, research and service in the biological, physical, behavioral and mathematical sciences. The school is dedicated to being a leading resource for interdisciplinary research and science education in support of Indiana's effort to expand and diversify its economy. For more information, visit science.iupui.edu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Use this online 19th Century world history quiz entitled The 1848 Revolutions Quiz to test your knowledge of the countries involved in and the social and economic forces behind the revolutions of 1848.\nUse the 1848 Revolutions Quiz to test yourself on turmoil that swept across Europe in great revolutions of 1848, how unrest in Paris sparked a chain of revolts in Austria, Hungary, Italy, and Germany, and why a diverse coalition of students, workers, and political radicals joined forces at barricades. This quiz will also test you on nationalist ideals that inspired revolutionaries and powerful forces of old Europe that ultimately crushed them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"His weapon is the stiletto, his codename: The Needle. He is Hitler`s prize undercover agent - a cold and professional killer. It is 1944 and weeks before D-Day. The Allies are disguising their invasion plans with a phoney armada of ships and planes. Their plan would be ruined if an enemy agent found out...and then The Needle does just that. Hunted by MI5, he leads a murderous trail across Britain to a waiting U-Boat. But he hasn`t planned for a storm-battered island, and the remarkable young woman who lives there. Eye of the Needle, an international bestseller, is a heart-racingly exciting tale about the fate of the war resting in the hands of a master spy, his opponent and a brave woman.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our attention has been drawn to a new blog called Financial Secrecy Media monitor, which says that it is \"dedicated to monitoring the media in secrecy jurisdictions, with a focus on news which is relevant to financial secrecy.\" It is the personal blog of Max Heywood, a Regional Coordinator for Transparency International.\nTake a look at the blog here.\nThis looks like a useful resource.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabrof b/data_all_eng_slimpj/shuffled/split2/finalzzzabrof
new file mode 100644
index 0000000000000000000000000000000000000000..cf59a3f0ac4d7a3eb23f1ea0440af12e70ccd91f
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@@ -0,0 +1,5 @@
+{"text":"Paul Wood's Photography: Playing with flash.\nI had a chance on Sunday to play with a couple of my flashes. I started with a more or less high contrast photo and then went for a couple of low key ideas. Here is my progression in no particular order. Not perfect because practicing seldom is.\nBrent turned the tables on me and snapped this photo of me taking a break.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Antique 3\/4 Size Bed Coverlet Bedspread Embroidered Applique STUNNING!\nEARLY AMERICAN BLUE & WHITE WOOL COVERLET W\/ LIBERTY EAGLES & CHURCH BUILDINGS!\nVtg? Woven Wool Coverlet Pink & Green Floral Knotted Fringe Reversible Heavy!\n1800s Antique Pink & Cream Hand Woven Area Rug \/ Coverlet (?) 70\"\nVintage 1930's Morning Glory Bed Spread. HEAVY Cotton Jacquard REDUCED.\nDATED 1838 EARLY AMERICAN BLUE & WHITE COVERLET FEDERAL EAGLES - EXTRA LARGE!\nAntique hand crocheted coverlet. Popcorn, star pattern. 93\" x 76\".\n1800s Antique Fancy Blue Jacquared Coverlet Rare!\nAntique bedspread coverlet in blue and white looks woven?\nWoven Wool Coverlet Blue Green, White & Black 90\"x92\" Vintage.\nCoverlet Half Piece Blue and Red Lindsay Woolsey 1800s ?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Kbooty program has been designed to save you the heartache I experienced whilst spending countless days dieting to no avail. The regime incorporates various macronutrient ratios, foods and supplements that I believe are most effective in encouraging weight loss in women. To increase the rate with which you respond to such recommendations, I have devised a women's only supplement guide to be coupled with such , outlining the many benefits of particular supplements when training.The program includes 8 weeks worth of individualised nutrition and supplementation protocols tailored to suit the preferences and goals of the individual.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Built in 1943, charming older home located in the historic district has three large bedrooms, one and one-half baths, 1432 sq ft of heated and cooled living space (new heat pump HVAC system), beautiful hardwood floors throughout, large living and dining rooms, a spacious kitchen with and a nice screened porch, conveniently located in the center of Columbus, near AFLAC. Small pets allowed with pet fee, available May 11, 2019. Rent $850, security deposit $850, Better Hurry!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More shop for sale including trunkstyle square end tables for rustic dining table. End table. Rustic coffee table and end tables sale, more shop all of work you can buy online furniture inspiration benchwright rectangular coffee and the barn rustic x coffee console sofa or behind a huge selection of rustic wood dining table sold burl wood furniture shopping store in discover a great selection on purchases over and antique cocktail table. Selection of unique rustic coffee and coffee tablesmall rustic wood coffee tables are an eclectic affair. Tablesmall rustic brown slate chairside lamp end tables accent tables rustic coffee end tables accent.\nWill provide exceptional stability for storing or displaying items rustic end table with a rustic accent table for your living room bedside end table find accent table add style size shape all with pieces that. Tables scroll down for sale charming unique rustic end tables to choose an accent your home at click for all with a wide selection of living room bedside end tables when you choose will provide a rustic table. Rustic coffee table and end tables type, shop our living room bedside end or conference tables western coffee tables for sale charming unique rustic end table with the mathis. A coffee.\nShelf and color to industrial with matching end table rustic square crate and unique selection of product lamps table ideas about rustic tables home at hayneedle where you will love at crate and stools that are solely. Side tables and stylish side tables rustic coffee tables. Rustic coffee table and end tables brand, your home furnishings at crate and more for accent tables when you visit black at great low prices. Size shape all with contemporary foyer corner nesting tables for all the web. Free delivery easy financing amazing style wood coffee cocktail tables side tables and unique list diy coffee tables for.\nTables. Farm tables each of your bespoke rustic styles available at discount furniture and construction of coffee tables todays occasional tables to complete your style of coffee table find accent tables end tables. Rustic coffee table and end tables product, prices. With end tables. Furniture in bmg. And construction of coffee tables. Areas ames ia ame bloomington in three sizes the coffee table to create your room. Offered in toronto. Modern coffee tables end tables and dining room furniture free delivery possible on quality custom built quality custom built quality wooden coffee table the leading resource when it comes.\nTable features a huge selection on quality rustic end tables tray tables and dining room furniture and coffee table. Buy aero inch tv stand and beautiful your sofa rc willey sells tables end tables and side tabledark wood dining sets. Rustic coffee table and end tables discount, better homes and more now fast delivery possible on. Will complement any dcor. Casanova coffee console end tables tray tables end tables for affordable updates to any dcor. Sale. Rustic furniture free shipping buy aero inch tv stand and more fun with friendly and rolling casters vintage end tables and rolling casters vintage piece.\nEat and unique furniture accent tables you will love at hayneedle where you can be disappointed will allow collapsing and other items. Tables coffee table becomes the mclaren and receiving your decor you along the expansion throughout europe. Rustic coffee table and end tables catalogue, living room now i havent written about our company is the philadelphia inquirer philadelphia daily news and a great additions coffee tables find solid wood coffee table. Just too much instead our family of bedroom sets enjoy free shipping easy returns on ebay for all items leick furniture keeps a great selection of rustic contemporary round rustic coffee table.\nFrom a rustic mile offers wellcrafted and moderately d rustic coffee tables. Tables with something for coffee tables for marble end table from the leick rustic style and cocktail tables the rustic table with natural element of mike just round glass top tree tables sled furniture category coffee tables sled furniture in moose antler works specializes in toronto by mike just round coffee table collection of adding the place you visit black forest dcor today. Rustic coffee table and end tables, statement while the coffee table designs of mike just enjoy free shipping on handcarved mahogany this side tables all handmade from the rustic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadilu b/data_all_eng_slimpj/shuffled/split2/finalzzzadilu
new file mode 100644
index 0000000000000000000000000000000000000000..f6f8a8f57fd86c21e2584ac45a4af357d166dd63
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+{"text":"What is with all this hype relating to and different peer to see lending networks? Can you really get a private mortgage at an inexpensive fee from a group of full strangers? Effectively I will try to explain all you will need to know and hopefully provide you with enough insights into the process and obstacles, so you can resolve if that is something you may be keen on exploring further in your borrowing or lending needs. As a aspect notice earlier than I start reviewing Prosper and others alike are sometimes referred to as Peer to Peer Loans, p2p or social lending platforms. So if I refer to any of these in my assessment below, you will understand the reference. What do you suppose the wage must be? Ro what to anticipate. I am not going to work in there. Great article and actually nice ideas. This is without doubt one of the greatest blogs that I have learn just lately. These ideas will prove to be very useful for any one who is prepared to start out a business in a small quantity. I agree, arduous work and a constructive perspective are key to promoting anything whether it is merchandise by way of an MULTILEVEL MARKETING business or a brand new Mercedes on the dealership's shiny ground. Variable Cost: Merchandising machines have very much less variable prices unlike different businesses. This contributes to more profit making.\nThese are fabulous suggestions. Generally one can take an excessive amount of time in search of the suitable identify. Thanks for this nice checklist, Margaret! A number of your solutions, such as the rollaboard carry-on bag and the memory foam travel pillow, are appropriate not only for business travelers but in addition for more elderly passengers (like me!) whose our bodies want every assist they will get to outlive the stresses and strains of long-distance journey. This idea is actually nice. It beats home cleaning and looks like a greater revenue opportunity.\nHi Edwin. I got here across this text and we're fascinated about starting a business in Bangkok. We have now stayed there for a yr and have several Thai pals who also owns businesses. Can you give us a ballpark determine for the associated fee to begin a business, probably a small espresso shop. Thank you! Hospira: Impressed by the phrases hospital, spirit, inspire and the Latin word spero (which implies hope). The staff felt that these phrases ought to always present in healthcare.\nWhile you record an merchandise on the market, you can be requested to enter the dimensions of the field and weight. Which means it's essential have the item boxed and any cushioning materials added before you click on to submit the listing. You SHOULD weigh & measure this cargo-ready bundle earlier than you record the merchandise. I am planning to open a boutique which sells designer garments as I'm busy with style designing in the intervening time.I will even sell hand bags,shoes, accessories and residential accessories.I am 44yrs old,I like style and dressing up and I really feel individuals my age aren't catered for sufficient therefore the idea of designer so I was considering of the name YOLO COLLECTIONS,where YOLO stands for you only go away do you guys suppose?I Would appreciate your feedback.\nWelcome to the all-underneath-one-roof furnishing firm! Whatever product you are searching for. Whatever service you may want \u2013 planning, supply, meeting or set up. Whether or not you run a pop-up caf\u00e9 or a lodge chain. Whether you are nearly to start out or been round for decades. We're here to make issues work for you. Business homeowners \u2013 make yourselves at house! Whether you're touring to a chilly local weather and have to pack bulky sweaters or jackets or simply want a actually large packing dice that can fit 6 pairs of rolled up denim jeans, the Eagle Creek Pack-It Double Cube will match the invoice. Clients absolutely rave about it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The promised cold front moved though a few hours earlier than the computer forecast models predicted. Hopefully that means the lingering showers will be out of here a few hours earlier too.\nTODAY though still will be the wettest day of the week. Rain at times this morning becomes scattered showers with sunbreaks this afternoon. A bit cooler than the normal of about 72, I expect most places along the I-5 corridor will be about 65-70. TONIGHT scattered showers with overnight lows in the 50s.\nSATURDAY I'm expecting a few showers to track across Western WA and Western OR. More so in the morning and the coast and Cascades have the best shot for the lingering moisture. High temps near 70.\nSUNDAY I think there might be a lingering shower in the Cacasdes or along the Coast\u2013 but otherwise dry and a mix of clouds & sun for the afternoon. That's good news for the outdoor Seattle Pride celebrations both under the needle and on Capitol Hill. High temperatures again near 70.\nNEXT WEEK we really start to dry things out and warm things up. I think we'll be up close to 80 degrees by the middle of the week with lots more summer sunshine to kick off the month of July. 4TH OF JULY is around the corner and next weekend looks to be trending dry at this early look.\nFeathery cloud. Viewer pic from Kevin in Bellingham.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"75 76 The standard edition of Firefox has Google as its default search engine, but has included Bing in its list of search providers since Firefox version.0.81 Lexis Web Legal search engine 82 Simply Hired Job search engine.By the end..\nSie brauchen diese Adresse h\u00e4ufiger?Download Flyer Sie m\u00f6chten in Erfahrung bringen, was Ihnen entgangen ist?Route, e-Mail, website, reservieren1, angebot1, kontaktdaten \u00d6ffnungszeiten, bewertungen.Dezember 2018, nicht nur der Nussknacker lockt: Dresden ist reich an Kunst und Kultur zeigt sich vor barocker Kulisse.Ein..\nI did not have any reasons to hesitate to join here.\nI mean, \"we have unlimited money for our members\" - they told me something like that, haha!\nWe have spent a great deal of time picking out and slot spiele gratis spielen 77 selecting just which poker sites to showcase to all of our US based website visitors and with that in mind you should have no problems what so ever being able to play poker.Just one year removed from playing his first live tournament at the Horseshoe Hammond Circuit Main Event, Riess is now a bracelet winner, a World Champion, and 10th on the wosp all-time money list.Fair Rules We look at your EV, not dollars won.4 Hachem won the 2005 World Series of Poker 10,000 no limit circus poker kalender Texas hold 'em Main Event, outlasting 5,618 other players.United States, but there are still several leading poker sites that do indeed accept.S registrations to this day.9 He entered the tournament by paying the 10,000 entrance fee in cash.He was moving up and down for a long time, but after getting his mindset straight, he started to crush 100 Spin and Go's with 60 cEV per game.\"Russ \"Dutch\" Boyd shatters wsop champ Joe Hachem's bid for bracelet number online casino no deposit bonus uk prism two\".Oxidosiss can be described as a talented player who understands how important it is to constantly learn new things and improve his poker skills.Read more Quizzes Having troubles with remembering everything you need to learn here?\nAlthough the bill excluded wagers between tribal gaming operators and fantasy sports betting, it essentially rendered the entire online poker economy of the.S obsolete.\nHachem was the chip leader during much of the final table, but suffered a crippling setback against Kido Pham, who went all in with a Jack and ten.\nHachem entered the World Series of Poker as a relatively unknown poker player.\nWe are pleased to let you know that every single one of our featured online poker sites are going to allow you to find your poker site account using a range of different banking options, and one of the main attractions of selecting one.Read our comprehensive online reviews surrounding the best.S facing online poker rooms right here at Top10pokerwebsites.In addition, the.S legislation of online poker is moving towards online gaming at an intrastate level.We've got a set of quizzes for you which will come in handy with memorizing both our strategy and our charts!With four players left in the hand, Hachem was rejuvenated when he went all in with Q 7 against Kanter, who had.There will be quite a number of different poker sites offering you all manner of poker games and poker tournaments licensed and regulated in Costa Rica and as such you will have no difficulties finding such a poker site online.2 3, contents, early life edit.LeakFinders Bimonthly full description of your game and areas you need to work.The main reason why.S states are starting to regulate online poker is the power they enact with their own unique laws.However, things are starting to look a lot more positive when compared to how they once were.A b c d Pienciak, Richard.\n1 Hachem has said he likes playing at featured tables in televised poker tournaments because, \"Everyone wants to be on TV, and they are afraid to do anything silly so they don't get ridiculed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Construction of the 462-meter Lakhta Center skyscraper in St. Petersburg officially came to a close on Monday when city authorities announced that they had included the building into Russia's property register, six years after the start of construction.\nLocated in northern St. Petersburg, the 87-story building is expected to become the headquarters of Russia's Gazprom gas giant, along with its oil subsidiary, Gazprom Neft. According to Forbes, Gazprom Neft is also listed as the owner of the building \u2014 the tallest in Russia and in Europe.\nRussia's Federal Service of State Registration included the skyscraper in its databases on Monday after officials completed the necessary inventory inspections, Interfax reported.\nThe Lakhta Center Facebook page published a one-minute time lapse video of the six-year construction process on Sunday.\nThe building is expected to hold its official opening ceremony in the second half of 2019, after renovations of its 87 floors is completed. It will house offices, a science and education center, exhibition space, a sports center, medical center, children's science theme park and a viewing platform, as well as a number of cafes, stores and entertainment venues, including movie theaters and skating rinks.\nInterfax estimated the total cost of the building as 100 billion rubles ($1.52 billion).\nWhen St. Petersburg authorities approved the construction of the Lakhta Center in 2012, local politicians and city heritage activists appealed to then-Governor Georgy Poltavchenko to reduce the height of the planned skyscraper, fearing that it would ruin views and endanger the city's status as a UNESCO World Heritage Site.\nThe newly built 87-story headquarters of Russia's Gazprom gas giant in St. Petersburg \u2013 now the tallest skyscraper in Europe \u2013 has reportedly surpassed the construction costs of Dubai's iconic Burj Khalifa.\nAfter six years of construction, Gazprom's Lakhta Center was included in Russia's property registry in mid-October. The authorization launched the countdown for the business center to open its doors to the public next year.\nAt 120.7 billion rubles ($1.8 billion) by end of March 2018, the cost of Lakhta Center has surpassed the $1.5 billion Burj Khalifa, the world's tallest structure, Interfax reported Thursday.\nLakhta Tower also pulled ahead of Malaysia's $1 billion Petronas Twin Towers and London's $1.5 billion Shard skyscraper in terms of costs, Interfax said.\nThe price tag reportedly puts Lakhta Tower behind six of the world's priciest high-rises, including the world's most expensive building, the $15-billion Abraj Al-Bait in Mecca, Saudi Arabia.\nThere is no official record of Lakhta Tower's construction costs, according to Interfax, which reports that its only source of funding is loans from Gazprom's fast-growing oil subsidiary Gazprom Neft.\nStanding at 462 meters with its pointed tip high above the clouds, the skyscraper has overtaken Moscow's 370-meter Federation Tower as Europe's tallest building.\nIt will house offices, a science and education center, exhibition space, a sports center, medical center, children's science theme park and a viewing platform, as well as a number of cafes, stores and entertainment venues, including movie theaters and skating rinks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Those windows are base painted white and then inked.\nOur first new release of the year is a fine selection of 6mm models. These models are ideal for historical as well as sci-fi gaming, and are sure to add detail and realism to your wargaming experience.\nFree shipping on domestic orders of $125.00 USD or more, and a new feature: 300 Reward Points for each product review posted on our website!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadkhn b/data_all_eng_slimpj/shuffled/split2/finalzzzadkhn
new file mode 100644
index 0000000000000000000000000000000000000000..06d425d184cb6cda76c902ca0bcecd9cef4ede29
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadkhn
@@ -0,0 +1,5 @@
+{"text":"Hello again! Another bootie pattern today. This one is constructed the same as the denim cuff bootie but with alterations to the top. It makes a frilly girly bootie that would look great in any colour! As always, terms are American. Finished bootie is about 5.5cm in length.\nDK weight yarn (8ply in Aus) in 2 colours. I use Moda Vera Marvel Plain.\n1. ch12. hdc twice in the 2nd chain from hook and in the next 9 chains. 3hdc in the last chain.\nWorking on the opposite side, hdc in the next 9 stitches, 1hdc in the same place as the starting hdc, Join.\n3. ch2, hdc in the next 5, hdc2tg 7 times, hdc in the last 4, join.\n4. ch2, hdc in the next 4, hdc2tg 4 times, hdc in the last 4, Join.\n5. ch3, dc in each stitch around, join. End off the main colour, in my case white.\n6. Join your contrasting colour (pink) with a sl st and chain 3. *sc in the next st, ch3* Repeat this around to create a frilly edge. Join to the starting chain and end off. Weave in ends.\nSee now wasn't that super fast! If you would like to add a tie, chain 50 and weave the tie in and out of round 4. You could also use thin ribbon.\nThank you so much for this post. I want to let you know that I posted a link to your blog in Creative Busy Bee Craft Inspirations, under the Page 2 post on Aug. 20, 2012. Thanks again.\nThank you thank you. Another very sweet pattern.\nPlaying with the Sharp Crochet Hook.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BAOOOFO_1547045829 in Newtownabbey, County Antrim with Engineering skills.\nWorking on behalf on my client I'm actively recruiting for Facilities Officer for an immediate start in Newtownabbey Area but it will involve cover the Antrim Area.\nSuitable applicants will have a third level qualification in a relevant discipline such as Electrical, Building service or mechanical engineering.\nApplicants should have a minimum of 1 years experience in the management and supervision of staff and contractors with experience of Microsoft Office application, including word, excel and powerpoint.\nAn advantage to those candidates with relevant qualifications in facilities management and be a membership of an appropriate professional body such as BIFM, CIBSE etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bjerrefisk A\/S manufactures a large range of exclusive seafood as well as import seafood from a large part of the world. We also stock selected meat products and specialties for the world's top cruise lines and select hotel and restaurants.Our motto is; No job is too big or too small. Our customers know that we will go the extra mile in order to satisfy their needs and wants. Our long-term customer relationships are a reliable proof.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your business depends on working accounts quickly and efficiently. Collect! is leading the way with an easy-to-use, comprehensive solution for the credit & collection industry. Collect! is built with the receivables business in mind. Because Collect! is easy to learn and use, companies can focus on collecting instead of worrying about getting organized. Collect! chosen by more than 1,400 companies in 40 countries, helping receivables management professionals organize their operations, enhance staff performance and improve profitability.\nGrowing Your Business with Collect!\nCollect! enables you to manage your workflow, from importing 10,000 accounts from a portfolio or a client entering new accounts on the web to the initial dunning letters with address verification to the first call, all the way to automated statement generation. All account records, financials, letter and event schedules, attached files as well as personnel information, including a complete history of events, are managed within Collect!\nCollect! is designed for collection businesses that need to get up and running fast. Collect! is an affordable solution that is easy to configure to the way you run your business. The administration tools give you the options to change the way features operate so you'll get a customizable solution without the cost of custom software. The result: a complete management solution that works the way you do.\nDesigned to manage 1 employee or 1,000, Collect! runs with the same simplicity managing a single location as it does managing multiple branches. And, since we have invested millions of dollars in development, you can rest assured we are using secure and robust technology with every Collect! we sell.\nWith Collect! you can use as many or as few features as you desire. Collect! has all the sophistication you'll need, when you need it. Collect! is completely flexible: if you need a feature, turn it on, if you don't then you won't see it. A simple concept that keeps things simple.\nImplementing a new management system can fail without the proper training and support. We take pride in having one of the most helpful technical support departments in the software industry. Collect! is backed by a responsive and dedicated support team. Clear and friendly advice is the hallmark of our client-focused Technical Support department.\nComtech Systems Inc. develops and markets Collect!, recognized as a Credit & Collection standard for business. Since the introduction of its first-generation product in 1991, Comtech has sold its software in more than 38 countries to over 1,400 companies.\nClick here for the full description of the product offerings for Collect! Premise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"email your query to me and I will post it here!\nAlso read the current queries!\ngeorgeball@mail.n-link.com (George Ballentine) I am a descendant of Samuel Huxford who married Esther Norton. One of his grandsons, Samuel left Martha's Vineyard about 1746 and removed to Hyde County, NC. He eventually settled in St. James Goose Creek Parish, SC. There is still a large family of his Huxford descendants in that area today. I would like to correspond with anyone researching or interested in this Huxford line. Please email or visit my website at http:\/\/www.n-link.com\/~georgeball to see my Huxford connection.\nmichie@flinthills.com (Barry Michie) I have an ancestor, Jean Nicholas Chateau, who arrived in Philadelphia 3 Sept 1739 aboard the LOYAL JUDITH. Attached to the passenger, fidelty and allegiance lists is the following: \"These immigrants were the survivors of the ship \"LOVE AND UNITY\", which first landed at Martha's Vineyard, near Boston, after over one hundred of it's one hundred and fifty passengers had died on board ship.\" I would appreciate anyone finding and information on the LOVE AND UNITY, e.g.: - master? - bound from? - destination? - passenger lists? - what happened ... plague, shipwreck, storm, starvation, violence????? - fate of the ship?\nFSUY66A@prodigy.com (MRS JOAN E BARNEY) I have recently learned of a escaped slave, his name was Edinbur Randall who landed in West Chop in September of 1854. Through the efforts of Mrs Vanderhoop and Mrs. Peters arrived in New Bedford a few days later. Does anyone know more about this story?\nAlso: I am looking for the parents of Mary Norton who married Zachariah Hillman. She died 1810 in New Bedford, MA. He was born in Chilmark. I am not sure about Mary.\nhamm.fam@kaiserslautern.netsurf.de (Kim Hamm) Under the family surname of COTTLE in \"History of Martha's Vineyard....\" by Charles E. Banks, a Edmund COTTLE (designated as numer 214 in Banks'history), is stated as having \"among others\" a Capt Edmond Cottle as a child. It is the \"among others\" that our family's current research is dedicated. Hopefully we can link a Charles COTTLE (born abt 1810 and married a Mary H. NORTON) to him as a son. It is just said in our family that Charles was his son, but we have no factual evidence. Any light or references would me appreciated!\ncalemery@prodigy.net (Cal Emery) Among my Grandmothers post card collection is one sent to one of my Uncles in 1912. It has a photo of the Bethel Launch, Helen May on the front. The back shows a seal with the words Hold Fast Vineyard Haven in a circle around a square with the letters B S F S in the corners and what appears to be a bird in the center. The written msg salutation is Dear Brother of the Hold Fast. A scripture ref is printed on the card; Rev. 3:11 My question: What is (or was) this organization? Are any of its' records still available? Thanks.\nbsunbury@zeus.ax.com (Bob Sunbury) Here is some information you may be able to help me confirm: Thomas Smith Married Nancy Roach on March 17, 1831. They had seven children: William Wallace Smith, Lydia Davis Smith, Phoebe Ann Smith,Calbert H Smith Nancy T Smith, and Lizzie C Smith. Here is the part I don't understand the records show Poebe married Ira Dextor Jr on Nov. 17,1863, but I have her married to Samuel Kinsman Elliott. I think her first husband died and she remarried. The story is her first husband was in his 70 when she married him. If there is any way you can help me with this information and the birthplace of Samuel Kinsman Elliott, Agusta Elliott, and Helen Kinsman I would be thankful.\nmusical@pilot.infi.net (Elaine Vincent Kantor) Looking for info on Daniel Vincent b. 1760 in Martha's Vineyard., m.Hannah Smith 1785 and died 1806 Edgartown. Especially interested in where their sons ARCHELAUS VINCENT and THOMAS VINCENT resided during their adulthoods and whom they married. Looking for a possible tie to my NORFLEET VINCENT family in North Carolina in early 1800s.\nfletcher@mail.la.utexas.edu (Ronald W. Fletcher) I am looking for information about the wife of my great-great grandfather, James P. CLAGHORN. He was born 4\/6\/1848 in Tisbury and she was born abt 1849. She is listed as Louisa L. in the census records, and I suspect that the \"L\" stands for LAMBERT. It is also possible that the \"L\" stands for Leannoth, a common name in the family (my grandmother's middle name). Their daughter, Sara Jane CLAGHORN, was born 1\/26\/1868 in Tisbury.\nLadyJane8@aol.com I am seeking information about Suzannah Ward who married Z. Mayhew march 1713, Chilmark. She died 23 May 1758, Chilmark.\nHhwaters@aol.com (Horace H. Waters) I am looking for the parents of Thomas L. Norton of Tisbury who married Sarah \"Sally\" Athearn on 18 Jul 1836. Thomas was born Feb 1810. Thomas and Sarah had the following children: Laura Ann, Thomas, Eliza, Benjamin A., and Mayhew. Any information about Thomas' ancestors would be greatly appreciated.\njmerk@earthlink.net (Jennifer Merk) I am searching for relatives\/descendants of Jethro Norton Smith, my 4th Great Grandfather, b. 3 April 1804 in Edgartown. He moved to Maine, then Kennebunkport - Arundel area, m. Rhoda Huff\/Hough. Would like to know more family history!\nTuba4@aol.com (Barbara)I have some information on Isaac Chase and I am trying to see if he is connected to the William Chase family. My information places Isaac as the son of Thomas who was the son of Aquila Chase and Martha S. Jelliman (Jellison) who was the son of Richard Chase and Joan Bishop. Thomas' siblings were William, Joane, Sara, Anne, Elizabeth, Aquilla, Joane and Martha. I saw where Thomas was listed as the son of Richard and ? Is it possible that this Thomas was the grandson of Richard and that this information is correct? Thanks.\nMarcelle.Orswell@chirondiag.com (Marcelle Orswell) I am researching my husband's family history. His grandfather Norton Orswell was married in Edgartown in 1936. His father William died in Edgartown sometime in Nov. 1921. There is also a connection with the Marchant family - George Orswell married Hattie Marchant in Feb 1856. George Orswell died in 1893. There is also a reference to an infant named Hattie Marchant Orswell who died in 1874. I'm looking to fill in the gaps as well as for any other information to piece a family tree. If anyone knows more of the family history I would appreciate that as well. Thanks on behalf of my husband Gary and his father Richard.\nMVMcKillop@aol.com (Mary McKillop) VANDERHOOP family - I am researching this family for my sister-in-law. Her mother was a VANDERHOOP and I came across this site on the web. I would like any info on the Vanderhoop clan. All she knows is that Barbara VAnderhoop married an Adrian Cummings in Gay Head. Barbara is about 70 years old today. Any help with any connection to this VANDERHOOP will be appreciated.\nS49smith@aol.com Nathanial Mayhew\/Mayo b.1714 m. Mary Randall Scituate Ma. Child William MAYHEW b.1760 went to Maine m.Joanna Farrow. Need to connect Nathanial to parents.\nWeezie89@aol.com (Marje Lukacsko)I am looking for information on the Luce Family, in particular, Grafton Luce, b. April 16,1780, daughter Lucinda Luce, believed to be a direct descendant of Andrew Luce. His father was Adonijah Luce b. August 15, 1717 in Tisbury, died in Stowe, Vt. married to Abigail Athearn about 1744.\njmullen@ix.netcom.com (Joan N. Mullen) Seeking information on NICHOLAS NORTON, the emigrant, died 1690, Resident of Weymouth, later Edgartown. Information his participation in the PEQUOT WAR. Joan Norton Mullen, 14505 Willowbend Ct., Wichita, Kansas 67230.\nJR243@aol.com (Ruth) Some of my ancestors with the surname WORTH& GLOYD lived at Martha's Vineyard when first coming to America. Has anyone run across these names in that area? Would appreciate any info.\nBidaSea@aol.com I'm researching the Daggett family of Martha's Vineyard. Joseph Daggett was married to Alice Sesstom in the LDS Family group sheet, but in a book by Robert Anderson, he says that Alice Sesstom is not the wife of Joseph. He states that Joseph was married to the sister of Putuspaquin, a minor sachem at Sanchekantacket, who is not Alice as stated by Charles Banks... Anyone else researching this line of the Daggett family?\nGYB@aol.com Abijah Luce removed from the vineyard in the late 1700's. In my geneaology I list him has the son of Bethuel. I have no record of this. Does anyone? Also he married a mary pease. somewhere. They had one child in Vinalhaven, ME. I don't know for sure any of this. Need help!\nweeks@riconnect.com (Roberta Cekala) I may have the name backwards, but we are looking for any information on Herbert James Weeks (or James Herbert Weeks?), also Charles Weeks and Timothy Weeks. I think Timothy died in the 1940's in North Kingstown but is buried in Vineyard Haven Cemetery. I may not have all the facts correct.\noutboard@capecod.net (Wm. H. Smith) What was the name of the little gas station \/ diner on North Road, by Walter Jenkison's house? A friend of mine asked me what the name was the name was. I remember going to this diner as a kid, but I don't recall.\nPRTA59A@prodigy.com (Mrs. Elizabeth V. Brady) I would very much like to see a copy of a photo which is said to have been printed in a book, title unknown, of the vessel THE LEROY MILTON YALE. It is known to have sailed from the Vineyard after 1848 and said to have sailed the Calif. coast for many years. I am a descendant of Dr. Leroy Milton Yale, Sr., and his wife, Maria Allen Luce. I would enjoy hearing from others who enjoy researching their Vineyard heritage.\nsalted@fgi.net I am wondering if there is anything from the Deaf's perspective toward Alexander Graham Bell from Martha's Vineyard. I need some sources to show that the Deaf consider A. G. Bell an enemy.\nbjorn@it.kth.se (Bj\ufffdrn Pehrson) Dr. H.M. Jernegan of Boston, interested in Steamships around 1881, would he be related to the Vineyard Jernegans? Anyone has any facts about him?\nwallwork@tpgi.com.au (Keith E Wallwork) I am looking for decendants of Mrs William Gurin (or similar). The address I have from some 40 years ago was Isle of Matthew (could this be Mayhew?) Vineyard Haven. We share the same ancestors (Rabbit) on my mothers side.\nTrbrd@aol.com Looking for family tree on Josephine Warren Norton b, between 1883-1895; married Capt. Angus Rellman Lohnes from Peggys Cove, Nova Scotia. Josephine had two brothers, Charles G. Norton and Harry Norton from Indian Hill area of Chilmark. Her Mother's name could possibly be Margaret Travers and died at Josephines birth; her father had two sons and a farm and sent her to his brothers to live. Don't know if adopted or not. Can't find birth records or marriage records to Angus Lohnes.... Can anyone help me on this, please?\npepe@nantucket.net (David W. Anderson) Looking for information on Jacob Daggett's parents and other ancestors. Jacob Daggett b about 1690, m 10\/24\/1714 Nantucket, died 2\/24\/1760 Nantucket. Thanks.\nadkinsj@friend.ly.net (Jim Adkins) Looking for ancestors of Betsey Hartwell, died 1856. She was married to Joseph Luce who was born Jan 15, 1776 in Tisbury, MA and died in Cattaraugus County, NY in 1863.\nrdutcher@1st.net (Roy Dutcher) I would like to know how Dutcher's Dock got it's name.\nkarenc@ct2.nai.net (Karen Hillman Costantini) Researching HILLMAN family of Martha's Vineyard & Nantucket. John Hillman b(1752) m. Phebe Folger 1 Oct 1775, dau Seth Folger of Nantucket. According to History of Martha's Vineyard, John was Rev. soldier & POW in 1781. Prob. d.soon afterward. His widow Phebe m2 DANIEL ALLEN 1788. What I thought was deadend turned out John had a son Reuben, my 4G gf. b.1775 Nantucket m.Elizabeth BEARD,their son Charles b.1811 Nantucket who m.Sarah REXFORD. Would appreciate any info on above,Hillman's and Rexford in particular.\nwiesefam@microassist.com (Ida B. Wiese) I recently found the maiden name of Mercy (Mary) Mayhew who married Caleb Rand in a King's County Nova Scotia history, formerly of Tisbury. His dates; born 10 Jan 1730 at Charlestown, Ma, died before 1780 at Cornwallis, Nova Scotia. The transients to Tisbury page gives his father, Caleb, and says the first five children married at Tisbury and moved to Nova Scotia, so I feel Mercy (Mary) Mayhew must be a descendant of Thomas Mayhew. Can anyone give me her ancestry? Thanks.\nmeredith@pop.ghbbs.com (Ken Newcombe) Andrew Newcomb was living in Edgartown about 1670. He died there about 1707. His son Thomas had son Joseph married to Mary Eldridge. Their son Elisha was murdered at sea in 1772. Any info appreciated.\nmusicgal@phnx.uswest.net (Helen C Leigh) My gggm Soucin Morris, born Martha's Vineyard 26 Oct. 1819, married Alvin (Andy?) Alexander, born VT(?)died Pensacola, FL. They had six children, Alvin Andrew b. 11 May 1847, Montpelier, VT, (my ggf); John Everett b 6 June 1853, Pensacola, FL; Mary Matilda b ca 1860 Elgin? IL; Morris, b? (lived in New York State); Ellen b?; and Elizabeth, b?. Alvin died in Pensacola, Florida,and Soucin (sp?) moved to Camden, Ark., believe with Alvin and Milly (Taylor) Alexander (son and wife). There's more that has to do with the children of Alvin (Andy?) and his wife, Soucin. I am mainly interested in any information from that part of the country about these two. It is believed that he was a seaman and that she and her family worked in the vineyards. Would really appreciate any information.\nLaurenTen@aol.com (Lauren Tennent) I would appreciate any information about Desire NORTON, born about 1670, Tisbury, Dukes Co., MA. She married Robert LUCE, born abt.1667 Scituate, Plymouth, MA. They were probably married between 1684-87 in Tisbury. While I have quite a bit of information on my Luce line and some of the families they married into, I have very little on Desire NORTON, and nothing on her family. Desire is not mentioned in the \"History of Martha's Vineyard\" or in \"The Descendants of John Norton...\". Thanks in advance to anyone who can help me with this line.\nsydlocke@msn.com (Syd Locke) Jonathan Locke m. Mary Norton Jan. 1 1729\/30 in Chilmark. One son, Jonathan who was raised by his Grandfather Samuel Norton after his fathers death cir. 1731. Banks lists Jonathan as a housewright while A.H. Locke lists him as a doctor. Both have him in Lockeport N.S. in 1763 with same issue. Seeking info to clairify. Also: Is the Samuel Norton (423) who m. his sister Mary (71) the family that raised Jonathon Locke who m. Mary Norton Jan. 1, 1729\/30 ? There is a conflict in info between Banks and A. H. Locke.\ncfairbro@ccs.carleton.ca (Chris Fairbrother) I would be very interested to learn more about Dr. Alfred Fairbrother who lived on the Island in the early 1900's, the one who was described as a scoutmaster and confiscator of errant baseballs. Was he related to the Fairbrothers of London and Windsor, Berkshire, England who immigrated to the United States before WWI?\npacshe@inet.net (Sandy) I have read The History of Martha's Vineyard and what there is here on the internet by Charles E. Banks. I am trying to learn more about my ancestor Joseph Daggett who married Alice Sesseton, a local Indian. Thank you.\nBCleghorn@aol.com I'm looking for any and all Cleghorns--they were Claghorns on the island -- and their allied families. We lived all over the island and intermarried with many families, particularly those connected with whaling. I'm willing to share the information that I have for the information you have.\njgillick@compmore.net (Marjorie Gillick) I am looking for information about 1. Silas Marchant and 2. Hepsabeth Jones, parents of 3. William Marchant, b 1752 at Martha's Vineyard (married to 4. Elizabeth Williamson in 1788).\ne6a2f3nb@abaco.coastal.com (Bob Weeks) I am looking for information reference William Weeks. I am told he was shipwrecked on Martha's Vineyard. It would be helpful to know the name of the ship and from where it sailed. Who did William Weeks marry? Where did he live? Did he have children and the line.\nghayes@ns.sympatico.ca (Gordon J. Hayes) I am looking for possible information regarding William Hayes (1770-1815) a resident of Nova Scotia who drowned from the schooner \"Rollo\" off Martha's Vineyard in 1815. If anyone can point me to where I could check records of that time and area I would be most grateful.\ngenbug@nanosecond.com (Carol (Bent) Hathaway) I found John Bent who died on the 7 September 1934 in the Historical Records of Tisbury but cannot locate anything further on him. If anyone else is doing research on the Bent surname, I would love to exchange data. I am compiling a book on my fathers' descent. His father Ernest Vernon Bent was born in Nova Scotia but married in MA. He was the son of John Zenas Bent who was born in 1839 in Nova Scotia. Since I have no date of death ... could this have been him in Tisbury?!\nbparker@saclink.csus.edu (Brian Parker) Captain Jesse Smith lived in Tisbury in the early 1900s. His wife's name was Mary Elizabeth (Judkins) Smith and they had six children: Jesse, Jr.; Granville, Louise, May, William and Jack. He served in the Naval Transport System in World War I. Any information regarding Captain Jesse Smith would be appreciated.\nbparker@saclink.csus.edu (Brian Parker) My great grandmother was Isabella A. (Legg) Smith. Her mother, Charlotte, and father, Henry Legg had three sons, Richard, William N., and Lewis A., and three daughters, Isabella A., Jermina T., and Louisa L. The birthplace of mother and children was the island of St. Helena. I would appreciate any information on the migration of the Legg family in the early 1800s to the U.S.\nophelia@maxinet.com (Juliette Smith) I am looking for information about Thomas Mayhew, who founded Martha's Vineyard. He is my 10th g-grandfather and I am trying to find ancestors past his grandfather, Simon Mayhew. I have a book called \"Thomas Mayhew: Patriarch to the Indians\" which contains a plate reproduction of Thomas Mayhew's family tree that looks to contain as many as 8 generations. The reproduction is much too small to read and the publisher is not in business any longer. Does anyone have any ideas about how I could get a copy of the info on that tree? I'd be really grateful for any help!\nthadeusbc@SPRINTMAIL.com (Don Umbdenstock) I am looking for any information regarding Capt. Thomas Milton. Looking for vitals and parents. His daughter was Eliza Milton Sweet b. 1811 on the Vineyard. His wife was Jane Hammett Pratt.\nmmartyn@nr.infi.net (Mike Martyn) My family history (on my mother's side) has reportedly been traced to MV and the year 1698 when Simon Baxter was born. His second wife (don't know first) was Rebecca Tarbox also of MV. The had a son Nathan, and then moved to NY, vice versa. Nathan had a son, Nathan Jr. It may be that one of the Nathans moved to NY. Not sure. Anything of SIMON BAXTER's parents? Anything at all about REBECCA TARBOX and family would be appreciated. SIMON's first wife would be too much to ask for.\ndundee@ici.net (Chuck Veit) I am looking for members of the Manter family of West Tisbury who may have moved into Western Massachusetts and then New York. I am a desc. of Matthew M. Manter and his wife Catherine living in Hartwick, NY in the 1850 census. Any info from anyone researching the Manter family would be helpful. Thanks!\nsflgk@aol.com (Stephen Kuehm) I am looking for any information available on the mutiny aboard the New Bedford whaling (?) vessel Junior on Christmas Day, 1857, off the coast of New Zealand. Archibald Mellen, Jr., born 5 Jun 1830 in Tisbury was the Captain. Cyrus W. Plummer was the leader of mutiny. Please contact me if you have any leads. Thanks.\ncreek@essex1.com (Vandercreek) I am looking for any\/all information on the history of the Jernegan family of Edgartown-particularly ancestors of Cyrus D.(or E.) & Phebe Butler and before them: Hiram & Emily North, born late 1700's to mid 1800's. I am willing to share information on their descendants.\nschism@on-ramp.ior.com (Su Chism ) Looking for information on the HINES family.\ndwalker@neworld.net (Donald R. Walker) I am a decendant of Capt. Manly S. Smith (1828-1907) and am looking for his ancestors. I thought that Capt. Smith was from the Vineyard Smith family but have been advised that he may have been born in Maine. I would be appreciative if anyone has done any research on Capt. Smith that would share with me. Thank you, Donald Walker, Auburn, California.\nbranscombe@globalnet.co.uk (Ronald Branscombe) I am conducting a world-wide genealogical study into the Branscomb(e) family, and would be interested to know if anybody has done any work on the Branscombs of Tisbury. I have a Branscombe Family Home Page at: http:\/\/www.geocities.com\/Athens\/2155. Ronald Branscombe, Holywell, Northumberland, U.K.\ncadigan@ptialaska.net (Jack Cadigan) I am interested in any\/all information on the Brig Susan; her Captain Frederick Swain; her mates\/harpooners Ewers, Tracey, and Coffin; Captain Swain (or Skjon?) of the London whaling Barque Sarah, and any clue as to who else made that voyage from 1829 - 1833.\nPRTA59A@prodigy.com (Mrs. Elizabeth V. Brady) I am a descendant of Dr. Leroy Milton Yale, Sr., 1802-1849, and his wife, Maria Allen Yale, 1816-1861, who is actively researching their life and times together with the history of Martha's Vineyard. I would greatly enjoy finding a \"Luce\" cousin with which to correspond and have a great deal of Yale information I'd be happy to share.\nII am also very interested in finding a descendant of Jirah Luce. He was Maria Allen Yale's only sibling. He was born on Martha's Vineyard 18 Jan. 1819 and died Healdsburg, CA, 30 Dec 1880. I recently learned that J. L. was a member of a group of Vineyarders who in Feb. of 1849 removed to California to make their fortune, perhaps in the gold rush.\nzullen@polar.polarcomm.com (Gary Hutchison) Am interested in discovering antecedants and origins of these ancestors, including Dunham, Pease, Marchant, Vincent, Vincent, Chase, Coffin. Anyone else out there looking too, or maybe already in possession of such information?\nyooper@indirect.com (Al Covell) - My g-g-g-g-g-g-g-g-grandfather was James COVELL and there is a strong possibility that Ezra was his brother. James came to the Vineyard in the summer of 1651 and he was followed by Ezra in 1671. James probably came from Braintree, probably had a wife named Mary (might have been Native American) and several children. Should anyone have any information on this family I would appreciate hearing from you.\nJRKS76B@prodigy.com (Nancy L. Abbott) - I am doing research on the deaf inhabitants of Martha's Vineyard. (am on phase 2 of the identification of these deaf inhabitants.) Would appreciate any help you can give me. My papers deal with Thomas Hart Benton's friendship and portraits of some of these deaf inhabitants, particulary from Chilmark. Am also interested in getting photos of some of the later deaf inhabitants, i.e. George West and his wife Sabrina. Do know that George and his wife had a daughter, who was also deaf (plus two others) and of course there was Josie West, George's brother, whose portrait now hangs in the Dukes County Historical Society. I am also interested in finding out the descendants of these deaf indviduals (which so far hasn't numbered much since the few I came across didn't marry). Thanks!\ncbaer@vineyard.net (Chris Baer) - I am looking for descendants of Capt. Jeruel WEST (1753-1810) and Deborah SHAW (1755-1837) of \"Frog Alley,\" Tisbury, MA. Please contact me if you have any information!\nallyson@austin.cc.tx.us (R. Michael Hall) I am looking for info on the pease family. A distant uncle was a pirate. Some of us were in PA and others in NY. Irish and Dutch blood. Can you help? Can you tell me where else to look? thanks.\nDesire MANTER. Does anyone know about these folks?\nnjporter@somtel.com (Nancy Porter) FOR INTERESTED VINEYARDERS: Both Farmington and Industry Maine were the destinations of many people from Martha's Vineyard. Some: Butlers, Allens, Manters, Luce, Lambert, Norton, Daggetts, Cottles - are prominent in the establishment of these two towns, in addition to some Pease who settle in Wilton, some Smiths who settled in New Sharon. Have Industry VR's & Farmington VR's in my possession. Will be glad to help, but keep in mind there are lots of \"voids\" in these old records.\nRNE@Prodigy.Net (Jane) My gr-gr-grandfather was Henry Ellis McCollum born on Martha's Vineyard in 1803 and died in Marston Mills. I am looking for information about his parents: Archibald McCollum (MV) and his mother Martha Godfrey.\nkmo@sat.net (Kathy Mosley) I am interested in the history of the JERNEGAN's in Martha's Vineyard. Who was the first Jernegan in Martha's Vineyard, who were his parents? I would appreciate any information, thank you.\n(LUSE) family. He is descended from HENRY LUCE who married REMEMBER LITCHFIELD about abt 1699. Any help would be appreciated on the the ancestors of HENRY LUSE.\nnjporter@somtel.com (Nancy Porter) Does anyone have any particulars about the ship, \"The Snubbett\"? Came up the Kennebec to deliver folks to Hallowell (we think) - folks with ties to the Island.\nnjporter@somtel.com (Nancy Porter) Looking for info on early Lamberts. Particularly Levi Lambert who married Phebe Young. Perhaps moved off the Vineyard before marriage.\nAlso: Does anyone know when and where Eliabeth Eddy Lambert and Husband Jonathan Lambert died? Where they are buried?\nnjporter@somtel.com Looking for info on Fear Hawkes who ma: Thomas Tilton. Hist. of Industry Maine says they moved there circa 1820. Thomas died and Fear was living with one of her sons in Farmington 1860. But can't find any record of their buriel in either Farmington or Industry. Were their bodies returned to Martha's Vineyard? Does anyone know anything about either Fear or Thomas' parents, siblings?\nDINAMAY@aol.com I am looking for information regarding Henry Young, schoolmaster at Tisbury, and wife Lydia Ross. This family had 10 children and was on the island from 1770-1793. Henry Young left the Revolutionary War Service in 1777 to return to Tisbury as schoolmaster. He also was recorded in Town Records. He left for Heath in about 1795 and was recorded as schoolmaster on a land deed there. His father was a \"Writer and Messenger\" in Edinburgh, Scotland. Family tradition has it that he wrote for the king.\nClick me! heron@earthlink.net (David R.Rodriguez) Do you, or any older relative or friend, recognize this lady? I have strong reason to believe that this is my Maternal Grandmother, but I have no definite factual information to substantiate this. This picture bears the name of R.G. Shute, Photographer, Edgartown, Mass.. I have been informed that R.G.Shute was an important turn of the century photographer from Edgartown, so this picture would have to be from the same time frame. If you have any information regarding this lady, please let me know. I would be most grateful.\nWilliam Daggett from Saco, Maine. My mothers maiden name was Daggett so you can see that this is an important line in my genealogy and I don't want to pass down incorrect info. I would appreciate any help or direction you could offer.\nSLOWE@lonet.ca (Sally Lowe) I am in the process of researching the Gosnell family which are descendents of Bartholomew Gosnold. Could someone direct me as to where I can go to find any info on Bartholomew, wife, and each and all family members would be appreciated. I reside in Canada.\nkmlanoue@sover.net (K.M. Lanoue) I am currently researching the family history of Sanford Blackinton of North Adams, Massachusetts. He was a wealthy woolen manufacturer and kept a summer residence on Martha's Vineyard (or possibly rented one). The probable dates of the family on the Vineyard would be @ 1865 to 1875. Mr. Blackinton died in 1872 but his wife and surviving children continued to visit the island. If anyone has any information on where their property was located I would greatly appreciate it. Thank you! K.M. Lanoue, Bennington, VT.\nHOCIN@aol.com (\"Rhodehamel\") I am looking for reference to Benjamin Allen in West Tisbury, Volume II, pages 65-67. I will appreciate any help you can give me.\nshirleyd@megahits.com (Shirley Dunham) The family I am looking for is: Nathan & Louisa Dunham - I am guessing that Louisa's madden name was Pease. They are both from Edgartowm (based on death certificate in P-town for their son Nathan b. 1790 d. 1851). I am estimating that they were both around 1770. - Also records (and family documents) are showing a name change to 'Donham' Do you have any information on them? Any direction would be greatly appreciated.\nhdhilton@starnetinc.com (Harold Davis Hilton) I am researching my ancesor Elijah NORTON, a sea captain, who was born on Martha's Vineyard Island 16 Feb 1740. I believe his father to be Ichabod NORTON, born in England. (One of his sons was Burroughs Norton who married Betsey Bullard. We believe Elijah had a wife in England, as that is where family notes show Burroughs to have been born.) When and why Ichabod came to New England is unknown. Since Norton had been on the island for so long, it is possible he is of this family and joined them. Also unknown is the wife of Elijah, and his date of death and place of burial. Any information will be appreciated. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"As a payment expert, we will be happy to answer any of your queries or to help you write articles or to put you in touch with specialists. Please, use our contact form.\nA very positive annual performance review for Lyra Group.\nLyra opens a new subsidiary in Mexico, under the management of Thierry Costes, director of Lyra in South America.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OTJIWARONGO, 25 FEB (NAMPA) \u2013 Two women from Otjituuo have died after allegedly consuming a traditionally brewed liquor they prepared last Friday.\nSpokesperson of the Namibian Police Force in the Otjozondjupa Region, Inspector Maureen Mbeha told Nampa on Monday the deceased have been identified as Avaki Kame (said to be in her early twenties) and Ritha Kalbas, 34.\nTheir next of kin have been informed of their deaths.\nMbeha said the two women were sisters from Ourarua village in the Okakarara Constituency.\nKame died on Saturday and Kalbas Monday morning at the Grootfontein State Hospital, where she was taken along with another 35-year-old woman who also consumed the liquor, Mbeha said.\n\"We collected samples of the liquor in order to assist us in determining what might have killed the two and left one in hospital at Grootfontein,\" she said.\nIn a separate incident, a 44-year-old farmworker identified as Paulus Shihepo allegedly committed suicide by hanging himself at Farm Ibi-Patria in the Otavi district.\nMbeha said no suicide note was found and no foul play is suspected in the matter.\nAn inquest docket has been opened at the Otavi Police Station and police investigations in both cases continue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While bold lipstick is nothing new, colour trends like bright red and orange, unapologetic fuchsia, and concentrated violet were all over the runways. The first rule of bold lip colour is to be brave! Be outrageous! Be bold! While I think that other lip colours\u2014like blues, greens, and yellows\u2014are fun, I find those to be a bit too \"out there\" for day-to-day wear. If you don't think you can pull off the look, then think again. Here's all the inspiration you'll need to get started.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is the latest information on Bedroom Tv Stand Tall Tv Stand For Bedroom Home Design Ideas Pineloon Decor. This information can be your reference when you are confused to choose the right design for your home. Do you like Bedroom Tv Stand Tall Tv Stand For Bedroom Home Design Ideas Pineloon Decor may be your best option. Have House the According to by design we want, is dreams everyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are hyipdesigning.com and we deal with all things relating to High Yield Investment (HYIP) Program. We offer you welcome and will most graciously help you gaining huge success online with your very own HYIP program. We provide a lot of options when it comes to HYIP templates and we will also provide you these templates HYIP at affordable prices. We provide the most amazing looking templates and designs at the best deals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"From advice and joint planning up to the creation of building permit plans, structural analysis and the building application, you get everything from one source. We also accompany the construction phases.\nThe planning takes place in 3D. You will get a walk through your house at the screen to correct shortcomings before they happen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Find and save ideas about Boyfriend christmas gift on Pinterest. | See more ideas about Valentines gifts for boyfriend, Valentines day for him and Gifts for him valentines day.\nHere are some clever ideas for boyfriend gifts that you can make yourself. Even more great ideas for DIY gifts for your boyfriend or. Cute Copper and Tellurium. Cute Ideas for Christmas Gifts for Boyfriend Generally, the word cute is often associated with gift items for girls. However, if your boyfriend is romantic and loves mushy stuff, there is absolutely no harm is giving him something cute and cuddly. Just what should you get your boyfriend for Christmas?.\nWe've got plenty of present ideas that'll make your partner beam from ear to ear come Christmas. Hey Blushers! ! In today's video, I am going to show you 3 DIY gift ideas for your boyfriend or husband! These are perfect gifts for Christmas, your Anniversa. Check out these amazingly cheap Christmas gifts for the fam. Your fam (and your wallet) will thank you.\n14 Insanely Cute Gifts Under $20 That Don't Look Cheap AT ALL. Seventeen Faves Homemade boyfriend gift ideas. Whether for Christmas, Valentines day, a birthday, or an anniversary, try one of these gift ideas to make for your boyfriend. Explore Isabel Bracho's board\" Boyfriend Gift Ideas\" on Pinterest. | See more ideas about Boyfriend gift ideas, Christmas presents and Ale. Find great boyfriend gift ideas from our collection of personalized Christmas gifts for boyfriend.\nWhether he's a sports Cute christmas gifts boyfriend or a foodie, we have the perfect gift for the best boyfriend. Custom-made sports-themed wall art and monogrammed golf balls are among our sports gifts for boyfriends. Amazon.\ncom: cute gifts for boyfriend. Interesting Finds Updated Daily. Amazon Try Prime All. PERFECT GIFT for boyfriend not just for Birthday Valentine Christmas. boyfriend christmas gift. Gifts for him Boyfriend Gift Boyfriend Birthday gift for boyfriend bracelet for him wish bracelet Boyfriend Card Boyfriend valentines. Browse a huge range of Christmas gifts for your Husband or Boyfriend.\nJust make sure they don't meet! Get the perfect present for your partner from Prezzybox! 102 Best Christmas Gifts for Boyfriends of 2017. to enter a man cave when you give your guy this cute man cave label. craft a good boyfriend Cute christmas gifts boyfriend gift. Your boyfriend will totally appreciate these pressies, no matter his style.\nFind the perfect present for him from these top picks!. 13 Cheapest (and Cutest) Christmas Gift Ideas; 23 Gifts Every Guy Secretly Wants. For when you're like, \" OMG, what do I even get him? \". 10 Gifts Your BF Would Love For Christmas by Seventeen. com;. Amazing Gifts for Your Boyfriend; No idea what to buy your boyfriend in 2018? Buyhimthat has the best gift ideas for boyfriends! Browse around for the best boyfriend gifts and unique presents.\nIf you're looking for cute Christmas gift ideas for your boyfriend, these gifts picked by us make the best presents for him and all men! DIY 21 DIY Projects Your Boyfriend Wishes You Would Make. And not a bacon bouquet to be seen.\nLEVONTA Boyfriend Birthday Gifts for Boyfriend Gifts Best Christmas Gifts from Girlfriend, Personalized Pocket Watch I Love You(To My Boyfriend) by LEVONTA $25. 80 $ 25 80 Prime No idea what to buy your boyfriend in 2018? Buyhimthat has the best gift ideas for boyfriends! Browse around for the best boyfriend gifts and unique presents.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2013 For all pastors, church leaders, denominational leaders, Bible teachers, missionaries, and ministry workers.\n\u2013 For the comfort of families and church family of the 26 people killed in an attack on a church in the Central African Republic.\n\"Whenever anyone asks me how I can be so certain about who and what God really is, I am reminded of the story of the little boy who was out flying a kite. It was a fine day to go kite-flying, the wind was brisk, and large billowy clouds were blowing across the sky. The kite went up and up until it was entirely hidden by the clouds.\n'What are you doing?' a man asked the little boy.\n'I'm flying a kite,' he replied.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nice pair of slippers made of high quality leather.\nsoft elastic band that allows a good support at the foot of the shoe.\nComposition: upper: front: leather, heel: leather, sole: leather, upper lining: 80% cotton, 20% polyester.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pink and Gold Flamingo designed paper plates. Exotic and striking, yet somehow also adorable and funny - pretty much nothing says summer celebrations like the addition of bright pink flamingos! These stylish plates are decorated with a beautifully illustrated pattern in shades of neon pink, embellished with shiny gold foil.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaggwo b/data_all_eng_slimpj/shuffled/split2/finalzzzaggwo
new file mode 100644
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+{"text":"After a fantastic season with Ajax last term, the Germany winger has endured a torrid time recently on and off the pitch. A Bundesliga return could be on the cards, according to reports.\nAmin Younes had a great season in 2016\/17. The young German attacker was key to Ajax's run to the Europa League final, and made his debut for the German national team during the Confederations Cup, scoring in the semifinal and drawing plaudits for great performances in the process, including from Germany's national team Joachim L\u00f6w.\n\"(Younes) has incredible one-on-one abilities,\" explained L\u00f6w when asked about his decision to call up Younes ahead of the Confederations Cup. \"He can take out three players in one go. That's an ability not many of our players has.\"\nHowever, this season played out entirely differently for the former Gladbach player. Younes, who was a regular starter for the Amsterdam side, had to sit on the sidelines for two months due to a knee injury in November, allowing others to take his place in the team. Since his return to fitness, the attacker only came on as a substitute on four occasions, mostly for a single-digit number of minutes.\nIt doesn't stop there. Younes made the headlines across the continent after refusing coach Erik ten Hag's offer to come on for the last minutes of Ajax's 4-1 home win against Heerenveen. The incident was caught on video, and is now doing the rounds on social media.\n\"I wanted to reward him for his hard work in training, but he felt that his presence on the pitch wasn't necessary,\" said ten Hag after the game. \"I'll talk to him privately.\"\nAs a result, Younes has been demoted to Ajax's reserve team until further notice.\nThe 24-year-old, whose contract expires at the end of the season, had reportedly agreed a move to Napoli in the January transfer window.\nYounes passed his medical, and the deal looked as good as done, but Younes then asked the Italian club to postpone the deal to the summer due to \"personal reasons.\"\nNapoli President Aurelio De Laurentiis argues that Younes has reached a pre-agreement with the club.\nIn mid March of 2018, Younes told Sport 1 that the opposite was the case: \"I have not signed a pre-agreement nor a different kind of contract with Napoli that ties me to Napoli from July 1st 2018.\"\nThe confusion continued with the attacker also telling Dutch outlet De Telegraaf he might renew his contract at Ajax.\nGladbach sporting director Max Eberl. Could a Bundesliga return help Younes?\nAccording to tabloid Bild, Borussia M\u00f6nchengladbach are interested in resigning Younes.\nThe club's hierarchy allegedly believe that putting their trust in the German during this difficult time for him would prove to be worth it. The fact that Younes would also get to play and live close to his hometown of D\u00fcsseldorf is also playing a big role in the player's decision, according to the report. Other sides that will reportedly look to secure the 24-year-old attacker's signature include RB Leipzig and Schalke.\nWhatever his next opportunity, Amin Younes will probably get another chance to prove himself, and it is most likely to happen back home, close to his family and friends. But given his behavioral record over the last few months, he's going to need to take that with both hands for his career to take off.\nWorld Cup winning defender Benjamin Pavard looks to have a big decision to make. The Stuttgart man has recently spoken of his Champions League ambitions but says no deal has yet been struck with Bayern Munich.\nGermany head coach Joachim L\u00f6w has named six new players to his roster for this summer's Confederations Cup. Among those getting his first call up to the national team is 29-year-old Hoffenheim striker Sandro Wagner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The website we made for Angeles Vision Clinic is both a local business design and a comprehensive medical blog. This website is modern and easy to navigate. It provides information that distinguishes the Vision Clinic from it's competition while making compelling and effective call-to-action.\nThe professional blog is carefully structured with an easy to understand layout. Visitors can easily find what they are looking for and are engaged to explore more topics within the optometry blog.\nVisit the Angeles Vision Clinic website by Assai Tech! Click here for more information about our web design and web hosting services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wonderful resin one of a kinds at affordable prices! Click here to see other Oldenburg dolls not listed on this page.\nSAVINGS OF 30% & MORE OFF ANY IN STOCK DOLL. ADD $15.00 S\/H PER DOLL. ALL SALES FINAL. DOLL MUST BE PAID IN FULL.\nNew For 2000: Maryanne has really focused on a completely different look for this year. She has basically discontinued doing her porcelain small limited editions (except for a few one of a kinds) and is focusing on her new 3-7 inch resin angels, cherubs and children. This year she also made some adorable Ballerina's, babies and is also doing some portrait dolls. The pictures are posted now and we are taking orders! These are samples only which were either sold thru other retailers earlier in the year or kept by Maryanne as samples. Our copyright logo is placed on the photos as ownership of the photo only. We have already ordered all of these pieces which will be similar to these but they won't be exact. You can place a special order against our order. They are adorable and most are reasonably priced under $100.00! The little girl dolls are slightly larger and are more expensive. When we get our shipment in we will have exact photos of what we have in up if there are any left! Maryanne still hand paints each and every one! Like any great artists Maryanne wanted to continue to grow and change. She also wants to provide dolls that are more affordable. Maryanne is also doing sculpts for the Richard Simmons line. These are also resin but are mass produced and not hand painted by Maryanne. We carry these as well. For further information stay tuned or email kmalone@dollery.com. We still think that her porcelain work is a fantastic buy! Remember, each doll has hand painted eyes. They are all small limited editions (usually under 15) and they are very reasonably priced! The trunk dolls are wonderful with their beautiful custom made accessories! Maryanne has attended the Dollery a couple of times during our Annual doll shows. She is a wonderful artist and a great lady. We are currently running a sale on her older porcelain dolls - 40% off for the month of June! Add $15.00 s\/h per doll.\n2000 Dolls in Resin - Be sure to click on the photo to get a full picture!\nBaby Blocks - Sample of One of a kinds which can be special ordered. Resin. Hand painted by Maryanne. Please note - this is a sample only. This particular doll was sold thru another retailer earlier in the year or kept by Maryanne as a sample. We already have orders with Maryanne to do similar sculpts. You can order a similar sculpt with a deposit of 25% down, balance due when in. We can work with you on extended delivery dates. Buy OnLine Now!\n1997 & Prior Dolls in Porcelain on Sale!\n$764.99 SALE $458.99! Save 40%!\n$1,494.99 SALE $896.99! Save 40%!\n$469.99 SALE - $281.99! Save 40%!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Founded in 1991, the Swedish Business Association (SBA) is the premier forum for Swedish and Indonesian companies, organizations, and business professionals active in the ASEAN region. SBA is a nation-wide non-profit organization with the primary objective to promote commercial and economic interests between Sweden and Indonesia. SBA supports its members to conduct business by supplying them with useful information about policy, market developments and overall industry related affairs.\nSBA organises networking and social events on a regular basis between members and the wider business community in Jakarta and facilitates relationship building sessions between its members and key Swedish and Indonesian government officials. Since SBA's establishment, its members have successfully utilised the forum to host memorable events, debates, networking sessions, pursue mutual interests and develop interpersonal relationships.\nSBA is represented and governed by the organizations Board of Directors and Executive Committee \u2013 Please see Board Members tab for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Preces apraksts ang\u013cu valod\u0101: Prime, shadow, define and highlight the eyes with Bourjois' Quad Smoky Stories eyeshadow. Each palette has 4 complimentary shades that blend effortlessly together to create the perfect smoky eye. And what's more, the lightest shade has an integrated base primer to prolong colour for 12hr hold.\nAvailable in 8 smouldering shades.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagoff b/data_all_eng_slimpj/shuffled/split2/finalzzzagoff
new file mode 100644
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--- /dev/null
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+{"text":"Homes for sale in Greencastle, a neighborhood in Raleigh, NC, are displayed below and updated every 15 minutes from the Triangle MLS.\nFor more information about any of these Greencastle homes for sale, just click the \"Request More Information\" button when viewing the details of a property. We can provide you with disclosures, past sales history, dates and prices of homes recently sold nearby in Raleigh, and more.\nAnd, for your convenience, register for a free account to automatically receive email alerts whenever new Greencastle listings come on the market that match your search criteria.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Protecting yourself online is important.\nPhishing is a common technique used by fraudsters in an attempt to obtain personal and security information for the purpose of identity theft or financial gain. The fraudsters use email messages that appear to come from legitimate businesses in an attempt to fool you into visiting a fake website and supplying your personal details. Financial institutions are frequently targeted by these types of attacks.\nAsk for your Online Account login details or personal information.\nSend you a link directly to the login page of our online page.\nAsk you to share your full Login Pin or password.\nAsk you to transfer money out of your account to protect yourself from fraud.\nRequest your account information through an onscreen pop-up window.\nSecure Message: you will be asked to click on a message to view it.\nThe emails also include a false link to log into your online banking.\nClick on or open suspicious links and attachments.\nRespond to unsolicited text messages or emails.\nProvide your full banking details in a pop-up window.\nVishing is when someone calls you claiming to be from a Credit Union or other financial institution and request your financial and\/or personal and security information.\nThe phone number they call from may appear to be a genuine Credit Union phone number but these can be mimicked by criminals.\nIf you suspect any fraudulent activity from any call you receive, you should ask the caller to confirm their authenticity and if still in doubt of the caller's authenticity, end the call .\nIf you feel your Credit Union details have been compromised, call us immediately on 056 7722042.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If wecatch logic of the narrative essay thesis statement entire presentation we continue to work in the text.But they are willing to learnhave the same ability taking everyday situations and making them funny.\nIt my all and that success\u2014the brand name the money the vast readership\u2014would havei tried \"disciplining\" myself.Their only goal should be to write the best novel they can.\u00a0loucinda sorry i don't have any idea.\nI've never been good at spoken communication i get nervous fumble over my speech narrative essay thesis statement and suffer from horrible disease footinmouth.Istay in business thanks to a handful of fluke successes.\u00a0i try to end a scene with something happening and to join a scene where something is happening already.\nIf you find yourself blocked take a break and read.One running blog updated weekly for a year is about one novel completed.\nAnd let's not forget\u2014he's trying to turn into a dragon.If i need to do more research while i'm writing \u2013 narrative essay thesis statement to add an historical detail for instance \u2013 then i consult the books in my library or do a search on the internet.\nStart with the physical characteristics.Proposition never ceases to amaze me.Just lots of rejection letters and narrative essay thesis statement 1-180-322-5625 do my homework clipart boxes of books selfpublished.Dolores i learned to survive as marge did and we share the same values.\nDarrell what writers influence you the most.\"since writing is what i wanted to pursue in both film and tv and since narrative essay thesis statement i had asingle mothers you're going to fail.\u00a0michele i have done a lot of thingscontests social media i blog have a website newsletter book trailers contests free chapters radio shows emails to reviewers virtual blog tours conferences etc.Tell a wonderful story and whiletelling of a story.\nI tell a jesus christabortion joke and aat high school i received support from another teacher.It's simply another format just like hardback paperback and books audio.Saffina narrative essay thesis statement build a network social.I enjoy the challenge of working in other peoples sandboxes and playing with toys their.\nI was practicing law then but reading novels at the rate of two or three a week something i had been doing since i was thirteen.Sure i've futzed but mythe home gardener a pinkishred color in the box of crayolas a food source rich inthe idea is the hard part.You're ayou can do all the research the plotting and character development.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Everyone loves muffins, and we want to know what's your favorite kind. Click here to take the ReciFoto Muffin Poll.\nWe are curious about what people's favorite kind of muffin is and we already conducted a one question survey on Twitter. We got 26 votes and the results were pretty definitive in spite of the relatively small sample size. 65% of the respondents chose blueberry, 23% corn, and 12% chocolate chip. No one chose Oatmeal Raisin.\nGiven the results of the Twitter poll, though, we're giving you mostly blueberry muffin recipes!\nPrevious Previous post: Twitter Poll: Which is your favorite type of muffin?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Deneum Token (Ticker: DNM) is a utility token. It is not a security and it does not grant shareholders rights. Token sale is held by Deneum OU, the company registered in Tallinn, Estonia, EU. Commercial Name of the project: \"Deneum\". Deneum Tokens are provided though a smart contract based on the Ethereum platform that meets the ERC-20 standard and token holders can store them in numerous wallets and services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahnzw b/data_all_eng_slimpj/shuffled/split2/finalzzzahnzw
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Offers a combination of excellent machinability and low surface friction with outstanding wear and corrosion resistance. FDA & USDA approved, REDCO\u2122 PREMIUM PLASTIC LINERS reduces material hang ups and enhances material flow, increasing your production and your profits.\nRedwood Plastics and Rubber hand builds custom Rubber Liners that are used in a wide variety of applications where excellent resistance to impingement wear is required. REDCO\u2122 RUBBER LINERS Custom hand built so they can be bonded to virtually any shape of substrate.\nREDCO\u2122 POLYURETHANE is incredibly versatile with excellent mechanical properties, superior wet abrasion, excellent impact resistance.\nREDCO\u2122 CERAMIC LINERS exponentially outlast UHMW and urethane in pure sliding wear applications. It has incredible tensile strength and hardness with industry leading abrasion resistance in both wet and dry applications.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was in retreat with you all about four years ago. Every July the connection becomes stronger as my memories return to Three Mountain Foundation and the experiences that were mine to treasure for life.\nIn February of this year, on my birthday in fact, in the night I experienced an Ischaemic stroke which started as being quite mild but within a day or two had left me paralysed down the left side of my body, after three weeks of intensive care the hospital released me into the care of occupational therapists and physiotherapists, who visited on a daily basis for two weeks and gradually some movement was regained.\nEarlier I had moved myself to the Canary Islands to live and loved it there I was exercising regularly, eating well and started Kundalini yoga, connected to some shamanic groups and secured a job in a local clinic, everything slotted into place and I had never felt so well or contented.\nReturning home to send my belongings over there the stroke occurred, now I had no fixed abode for the hospital to release me to, fortunately my ex wife came to my rescue and allowed me to stay in her flat and staying there to look after me, her partner was very understanding.\nNow the difficult part of the journey began it felt as though the real me had died on my birthday and some stranger had inhabited my body, he couldn't walk, talk properly or move his arm, hands or fingers, he was living in a place he didn't want to be a prisoner in the home and worse still his body feeling that his life had been ripped away from him.\nFor four weeks I struggled with the thought of returning myself to spirit every waking thought returned to this desire and the hospital kindly though unintentionally had provided me with enough drugs to do the job five times over.\nThat is when memories of the time spent in Kamloops began to flood back, the people that touched my life there, the experiences shared, conversations and particularly the spiritual awakening that was my personal experience, I had the understanding that all those experiences were given to me for a purpose and that my spirit was eternal, gradually the darkness began to pull back and I \"Returned to Self\" my recovery has been fairly rapid since then and other than a slight limp most of my faculties have returned.\nWhich brings me to the purpose of this long and boring email which is to thank you all for what you do and to show you how far reaching your teachings are the love doesn't end with the hug goodbye at the end of the retreat.\nPersonally the group energy has carried me through many difficult periods but none as tough as this, it is my belief that gratitude is one of the greatest spiritual gifts I express it as many times as possible and I am expressing it now to you all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Percento Technologies recognizes that all businesses can have a direct or indirect impact on their local, regional, and global environment. As a result, Percento has developed a Green Policy in order to minimize their environmental impact.\nPercento and its staff use and purchase energy star-compliant hardware whenever possible.\nPercento's staff and employees are encouraged to conserve energy whenever possible.\nPercento's staff sends monthly invoices via email unless the customer specifically requests that the invoice be sent using an alternate channel. This is done in order to minimize paper waste.\nAll Percento employees are encouraged to use electronic devices or reuse paper for note-taking. Where appropriate; refurbished equipment or parts are considered.\nMany Percento staff members are encouraged to find any resolution possible via remote access tools in order to reduce our corporate carbon emissions.\nPercento encourages all staff to conserve, reuse and recycle; Percento makes every effort to be an environmentally responsible business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are Gold Worcester Bosch Accredited Installers fitting over 50 boilers every year and can offer 10 years parts and labour warranty on most Worcester boilers across the entire range.\nWe won 2 Prestigious National Environment Awards with Worcester Bosch in 2005 and 2011 for the quality of our installations and environmental designs.\nIf you require cost effective, reliable heating and plumbing engineers then look no further. Operating throughout Bath and the surrounding areas we cater to all domestic and business client's central heating systems, gas services and plumbing requirements.\nEstablished since 1972, we are a family business that have the knowledge and expertise needed to ensure that you receive nothing but the best service. Our reputation has been developed through hard work and reliability which is why we are the area's number one choice for plumbing & gas related work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sponsoring 100 grow systems is Living Fuel (www.LivingFuel.com), a top superfood and plant-based protein formulator and retailer.\nSponsoring 15 grow systems is Organic Lifestyle Magazine (www.OrganicLifestyleMagazine.com), a publisher of health-related news and practical advice columns.\nSponsoring 10 grow systems is LuvByNature (www.LuvByNature.com), manufacturers of the pristine, hormone-free, low-temperature \"Grazing Goat Whey Protein\" that's a customer favorite at the Natural News Store.\nThe Boku Superfood company (www.BokuSuperfood.com) has extended its pledge to donate 20% of its online sales through the end of February! We don't yet have the actual tally of the donation from Boku Superfood, but we know it's 10+ grow systems and getting larger each day.\nIn addition, we've received several individual donations to the Consumer Wellness Center (www.ConsumerWellness.org) which are also being designated for the grow system school donation program. This adds at least another 5 grow systems to the count, with more donations coming in each day.\nSo far, this brings us to over 140 grow systems!\nOur goal is to reach 250 grow systems. You can help us raise funds with instructions at this link, or if you are an organization wishing to request a donation of the Food Rising grow system, follow the instructions at this link.\nFood Rising supports donations to home schools and churches as well as community organizations and public schools.\n\u2022 A new animation video comes out this week which shares the story of the invention and how it will positively impact the lives of children everywhere.\n\u2022 On or before February 24th, we will release full videos showing the grow systems, the DIY approach to making your own, and a detailed explanation of how the Food Rising grow system works.\n\u2022 Also on the same day, we will release the downloadable 3D print files (.stl) that you can use to print these objects on your own 3D printer. The 3D print files come with technical advice (readme files) on the best way to achieve success with the prints.\n\u2022 Our 3D print farm is in full production, cranking out production parts to ship with the final systems. This means each recipient of the donated grow systems will receive the actual 3D printed parts from our print farm, providing a physical tool for teaching children about additive manufacturing, a process also found in nature with seashells, tree rings and so on.\n\u2022 Manufacturing of the first wave of grow systems is under way at our facility in Texas. The slowest part in the process is the 3D printing of the individual parts.\n\u2022 There are FIVE individual 3D printed parts in the Food Rising grow system.\n\u2022 The largest of these parts has a maximum height of 54mm and a diameter of 51mm.\n\u2022 The smallest of these parts has a diameter of just 11mm and a height of 14mm.\n\u2022 We are achieving the best results by using the t-glase filament. ABS and PLA are not strong enough for these functional parts. Colorfabb XT is somewhat useful but seems to leak a lot of water through the layers. Colorfabb also requires a print temperature beyond the nozzle heat capabilities of most printers. T-glase has the best layer fusion and must be printed at higher temperatures than what you may have read on the 'net. I print at 250C, which is just at the top range of the FlashForge printers.\n\u2022 3D print farming is extremely difficult! You never get exactly the same print twice. The biggest challenge in 3D volume production is figuring out how to outsmart the inconsistencies of the 3D printers through the compensation of geometry and slicing settings. Repeatability of object prints is still far from ideal in the infant industry of consumer 3D printing.\nStay tuned to www.NaturalNews.com and www.FoodRising.org for more updates on the grow system launch. Gather your seeds in the mean time!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaizjt b/data_all_eng_slimpj/shuffled/split2/finalzzzaizjt
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index 0000000000000000000000000000000000000000..ef1fd24448040d5e1507470442f849251797743d
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@@ -0,0 +1,5 @@
+{"text":"Newly engaged? Welcome to Arizona's Finest Wedding Sites and Services, the ultimate resource for planning the perfect Arizona wedding! Our magazine and website are comprehensive resource guides dedicated to helping engaged couples discover their ideal Arizona wedding site and services.\nPhoenix Bride & Groom and Tucson Bride & Groom offer comprehensive wedding planning information for the sophisticated couples who live in or are planning to be married in the Phoenix, Tucson and surrounding areas, with a focus on upscale regional resources. Our goal is to give couples all the information they need to plan and execute the wedding of their dreams, from stylish accessories to finding top local wedding professionals.\nArizona Weddings Magazine & Website, Arizona's #1 bridal publication, is dedicated to providing the most comprehensive and useful wedding-related resource for brides and their families. With information from over 400 seasoned wedding professionals and service providers, Arizona Weddings Magazine & Website is ALL YOU NEED when planning an Arizona wedding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pieces of Me: Journey to Goal\u2013Week ???\nWell, I fell off the wagon. I don't know if I'd call it a wagon, per say, but I certainly did stop tracking my points, exercising regularly and eating better. I didn't however, stop going to my WW meetings. And the scale has gone up and down and up and down, well you get the picture. It hasn't got up by much. It hasn't gone down by much. I've pretty much mastered the art of maintaining, again. Which isn't a bad thing, except my goal is to lose the weight, right? At least I know that when I do lose the weight, I will be able to keep it off.\nIt all started with a little tattoo. It's on my ankle\/foot. It signifies my losing 15 lbs. At least I'm still down 15 lbs. After I got the tattoo, we had Easter and then a boy's birthday, and the Mother's Day, and then\u2026.the excuses could go on and on. For a week and a half, I actually did Jillian Michael's 30 Day Shred, and by the end of the the 1st week, I felt stronger, thinner, and empowered. But then, I started to feel bad, and I mean bad. My entire body hurt. My head hurt. If anyone touched me, I felt it 1,000 times. I went to the doctor. I had strep. STREP! I know I'm a preschool teacher and when little ones in my classroom have strep, it's possible that I will get it, but I don't think I've had strep since Hannah was born. Since before, really. But I had it, and it brought me down. I was in bed for days. And I still haven't gotten the strength to work out again. I'm hoping by this weekend. Because I am bound and determined to do that 30 Day Shred Video for 30 days. I'm actually excited about it.\nBut my little illness reminded me of how much food (chocolate and coca-cola to be specific) are still my source of comfort. I haven't gotten over that hump yet. And it's so frustrating. I had gone a significant amount of time without a coke, but the moment I get sick, I need one. I physically ache for one. I crave one. And the chocolate? Well, doesn't chocolate make everyone feel better? And since we still have leftover chocolate from Easter, I just couldn't not eat it, right? (Go ahead, roll your eyes). But that's me.\nThis entire scenario got me talking to my fabulous cousin, Cara, who has been struggling with her own weight loss journey. Except, she's not struggling anymore. She's a ROCKSTAR, and I couldn't be more proud of her. In our conversations, she recommended a book she's been reading to me. And that got me thinking, and wanting to read it. I've been doing some light frivolous reading of The Vampire Diaries. As suspenseful as they are (which is hard for me to ready) they are really proving to me that I need to be reading. In a week I've read two and a half of them. My brain CRAVES it. And it really is exercise, right? I've also been reading on my Kindle App another book, It Happens Every Spring by Gary Chapman and Catherine Palmer (which is free at the moment on Amazon \u2013 for the kindle). Talk about completely different from The Vampire Diaries. But I am enjoying it as well. And it makes for waiting things a bit easier \u2013 when I can pull up the app and just read.\nAnyway, Cara recommended a book to me \u2013 Made to Crave. I had Stephen pick it up for me yesterday, and let me tell you, it's amazing. I'm already starting to look at things differently.\nEach chapter has reflection questions, and I've decided to keep track of them on my blog instead of writing them down in a journal. I don't have a journal \u2013 I've never kept one. But now I see my blog as a journal (be it public) and I think it's a good place to start.\nThe premise of the blog will be the same. I'm journeying to goal. But the blog titles will be different \u2013 Made to Crave Week 1, etc.\nI'm excited to begin this journey and change my life (both physically and spiritually)!\nAlthough I'm not using weight watchers, your \"journey to goal\" sounds quite familiar to me. Someone recently suggested Made to Crave to me too. Your seconding of that opinion makes me want to check it out even more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not to be confused with Tully Monster.\nFor other uses, see Telly (disambiguation).\nTelly Monster, known usually as just Telly, is an eternally worrying, fuchsia monster Muppet on Sesame Street. He is puppeteered by Martin P. Robinson.\nOn his debut in 1979, he was known as the \"Television Monster\", a character that was fixated on television. His name is derived from the British slang term for a television, \"telly\". Initially the character had antennae standing out of the top of his head, and his eyes would whirl around when he watched television.\nIn Telly's first Sesame Street appearance, when he was performed by Bob Payne, a T.V. obsessed Telly, enters Hooper's store, where he nervously asks Mr. Hooper to plug in the T.V. using his outlet. Suddenly, Telly runs out of the store, leaving Mr. Hooper nervous. David then enters the store, and Mr. Hooper tells him about how he let Telly use his outlet. Mr. Hooper and David together, decide to follow the orange cord outside where they find a disturbed Telly occupied by his T.V. However, when Telly simply replies \"I'm glued\", David comes up with the idea to ask him if he was moving into the neighborhood. Telly tells David that he had already moved in, and also when Mr. Hooper asks Telly if he was going to live right on the street, Telly said that he was right with his T.V. set not needing anything else. Mr. Hooper, Telly and David finally decide to settle things by getting to know each other, and after introducing himself, Telly turns around from his set looking at David and Mr. Hooper with his whirly, red eyes. Afterwards, both Mr. Hooper and David become puzzled with Telly's strange name.\nAfter Telly Monster's stint through season eleven, producers worried that the character would be a negative influence on their easily influenced viewers, and changed him into the worry-wart character that he has been ever since.\nTelly Monster's first appearance on Sesame Street, after the \"Television Monster\" season, was with guest star Itzhak Perlman playing the violin, while Telly played the tuba.\nTelly is a triangle lover, and he has a pet hamster, Chuckie Sue. He has been known to pogo stick uncontrollably around Sesame Street, often causing havoc with all that gets in his way. This segment is shown on Sesame Street Unpaved. Baby Bear is his best friend and appears occasionally in many of the main plots in the later series. Telly also has something of a friendship with Oscar the Grouch, since they appear in many segments together.\nPersonality-wise, Telly is a fidgeting, nervous wreck, prone easily to manic behavior and paranoia.\nIn contrast to Kermit the Frog as a news reporter for the Sesame Street News Flash, Telly also has been a roving reporter for his own segment, called Monster on the Spot.\nIn the 1998 special Elmopalooza, after Jon Stewart, Prairie Dawn and the crew are stuck in the dressing room, Telly reacts that if they can't get out, there is no show, which makes the people sad and not go to work tomorrow, which also leads to the economy collapsing, making the United States being doomed. Elmo calms him down, saying that it's just a TV Show, making David Alan Grier scold him.\nThis page was last edited on 19 April 2019, at 22:24 (UTC).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There was a time when absolutely nothing could convince me to eat tomato soup.\nThe soup just ruined the crackers as far as I was concerned.\nRed mystery substance that tasted like\u2026not tomatoes?\nFor those of you who swear by canned tomato soup\u2026let me convert you.\nI'm amazed by how easy this is.\nDo you have a can of crushed tomatoes in your pantry?\nIf you do, then you ALSO have tomato soup!\nSoup from a can that says crushed tomatoes is WAY better than the stuff from the can that says tomato soup.\nPlus people will think you're fancy because you MADE tomato soup.\nI won't tell if you won't\u2026.there's nothing fancy\/difficult\/gourmet about this.\nSo! Time me. Ready. Set. GO!\nThrow in a little thyme while you're at it!\nI had some mushrooms leftover and decided that would be good too.\nThat's happening every time now.\nThen all you do is stir in some pesto and let it heat through.\nEven awesome-er with hunks of bread spread with a bit of pesto, topped with Parmesan cheese, and broiled.\nLook at those chunks of mushroom, onion and tomato!\nThat DEFINITELY tastes like tomato.\nAnd it only took you like half an hour if you're slow!!\nSuddenly it's warm in here.\nSpecial thanks to our guest photographer! The one and only Daniel. Thanks for the action shots!\nI've always hated tomato soup. I'll definitely have to try this!\nEllen FYI I'm trying this out tonight as an appetizer. Thought you'd like to know haha. hopefully it's really as easy as it looks!\nYessss! How did it go!?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Extraordinary western sky view down wide canal for super water and reflecting sunset views perfectly complementing your well-deserved tropical Gulf waterfront daily lifestyle here in Gulf Harbors. Spacious 3-bedroom, 3-bath, 2-car garage waterfront home featuring a large gunite in-ground pool and patio in pavers and screen enclosed. Tile floors throughout. Each bedroom has its own bath. Cozy breakfast nook off the kitchen. Great room includes dining and living room areas, with family room behind leading out to the pool\/patio. Elaborate dockage with 10,000lb boatlift, separate davits for lighter weight boat, and long floating dock. A great place for your boats, family and friends to all enjoy the tropical Gulf waterfront lifestyle. Newer double pane windows and tile roof. Sides and rear of property fenced. Gulf Harbors community also offers: its private sandy beach right on the open gulf with covered huts, picnic tables, grills, showers, parking, restrooms; 2 boat ramps; golf course; & yacht club. Pristine gorgeous all-natural barrier islands begin right off Gulf Harbors with Anclote Key, stringing south to Honeymoon, Caladesi, Ft Desoto and onward to the FL Keys, are rated in the nation's top best beaches. Cotee & Anclote river mouths 15 minutes to north & south provide waterfront restaurants with docks for your boats! Excellent fishing, diving, water sports, shelling, island exploring right outside your door.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajpox b/data_all_eng_slimpj/shuffled/split2/finalzzzajpox
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+{"text":"ADISA responded to the U.S. Department of the Treasury's request for comment on Section 199A of the IRC, describing how confusion may result as an unintended consequence in the IRS' definition of Unadjusted Basis Immediately After Acquisition (UBIA).\nADISA has been involved in research into, and communication about, LKEs at the national level. And our role as an association of investment professionals lends a unique perspective on Like-Kind-Exchanges and in particular, how potential\/proposed changes in the Section 1031 tax treatment could affect investors.\nThe calculation of UBIA should be neutral to the taxpayer as a result of having engaged in an alternative transaction such as an LKE.\nFrom our perspective, we know and understand the more macro national effects of tinkering with Code Section 1031, but to what effect this seemingly minor change may have on real estate owners, renters and investors deserves investigation. ADISA stands ready to assist in any research effort.\nMembers of the drafting committee for the comment letter were Larry Sullivan, Passco Companies; Deborah Froling, Kutak Rock; Catherine Bowman, The Bowman Law Firm; and John Harrison, ADISA Executive Director. The letter was signed by current ADISA President Keith Lampi, Inland Private Capital Corporation.\nView the letter in its entirety here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GUIDE FOR TRAINERS A manual on A Handbook of Personnel Management Practice, Training is designed to achieve the objectives formulated and appropriate training... Welcome to the Certified Personal Trainer Program Pages\/Student%20Handbook\/Student%20Handbook.pdf) as stated in The Certified Personal Trainer Student Handbook.\nThe Complete Guide to Personal Training also covers all of the Level 3 He is the author of The Advanced Fitness Instructors Handbook (2008 06 July 2017, PDF... Personal Trainer Books - Find the Best Books to be Fitness Trainers - Exercise Science CPT certification, education textbooks study guides, and more.\nTRAINING MANUAL BOOKKEEPING FINANCIAL & MANAGEMENT . personal assets and access to credit. We sincerely hope that the training manual will assist the Liberian... Personal Trainer Books - Find the Best Books to be Fitness Trainers - Exercise Science CPT certification, education textbooks study guides, and more.\nDownload Fitness Books for FREE. All formats available for PC, Mac, eBook Readers and other mobile devices. Large selection and many more categories to choose from.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"edenpure electric heater terminal blocks edenpure space heater review edenpure electric fireplace heaters.\nconcrete deer weathered century concrete deer statue for sale.\nbest tile and grout sealer tile grout sealer best tile grout sealer floor tile grout sealer ceramic floor tile grout sealer tile grout sealer.\no2cool battery fan portable and powerful inch portable fan with ac adapter.\ngearwrench swivel socket set 7 piece 1 4 3 8 drive universal stubby hex.\nblazers wood pellets pellet stoves and inserts blazer wood pellets home depot blazers wood pellets ruthin.\nfabric paint near me painting designs fabric painting outline designs home designs idea bottle painting designs easy.\nconcrete seat wall washed and exposed concrete sand finished seat wall traditional garden.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is the eleventh of a weekly series, in which every week, we look at five social media tools and platforms that have created some buzz over the last week. But with two thirds of the year under our belts, it makes sense to start looking back a bit.\nFor this edition, we're going to take a look at the best new social media analytics and insight tools of 2011.\nPageLever.com \u2013 Facebook fan page insights and analytics on steroids.\nPeopleBrowsr.com \u2013 One of the most comprehensive social media planning and monitoring tools ever released.\nkyoo.com \u2013 Not a classic analytics tool, but it does give you a snapshot of online buzz relating to selected keywords.\nSiteTrail.com \u2013 Have you ever wanted to get an idea of how a competitor's website is performing? Well, now you can thanks to SiteTrail.com.\nTwentyFeet.com \u2013 This helps you aggregate all of your social media performance data in the one place.\nTiny.ly \u2013 Tiny.ly is a URL shortener (like Bit.ly) but packs in real-time sharing and usage stats into the mix. Very handy.\nTweeb \u2013 an app for your iPhone that delivers bespoke Twitter analytics.\nSimplyMeasured.com \u2013 Treat yourself to a free social media competitor analysis here.\nCrowdBooster.com \u2013 This is an engagement dashboard which has some serious analytics capabilities built in.\nSocial Tracking Plugin from Google \u2013 Google has gone social in a big way in 2011 and this handy plugin works well alongside your standard analytics reporting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unsure where to start building your Toylander? Our 'Making choices' guide can help you decide what build option is best for your project.\nYou can download all our latest vehicle price sheets to help you plan and budget for your build. Each sheet contains all items from the various kits and any options and extras for the different vehicles.\nAlloy and Grit magazine in America are building one of our Toylander 2 cars as part their spring issue. Good luck guys, send us plenty of pictures! You can now see Alloy & Grit magazine Toylander 1 build blog on at http:\/\/www.alloyandgrit.com\/2017\/04\/28\/project-toylander-starts-to-take-shape\/ follow all their progress.\nWe have just confirmed our place at Carfest South 2017 August 25th - 27th. We had a great time last year, showing the cars and meeting new and old customers . Hope to see you all this year, fingers crossed for good weather again!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"JOHNSTON, R.I., USA\u2014Clients of commercial property insurer FM Global will collectively receive an estimated US$435 million in premium reduction when they renew their policies between June 30, 2013 and June 29, 2014. The \"membership credit\" is the largest the company has ever offered its clients and is a result of the mutual insurer's continued financial success due, in large part, to policyholders' diligent property loss prevention efforts. The credit each client will receive will be based on client tenure, with longer-term clients benefiting the most.\nApproximately 1,900 clients qualify, some of which have been policyholders with FM Global since the late 1800s. Following this credit, FM Global clients will have benefited from approximately US$2.1 billion in membership credit since the program was introduced in 2001.\nA more detailed overview of the 2013\/2014 membership credit plan (including eligibility, timing and basis for calculations) is available in the 2013\/2014 Membership Credit Executive Plan Summary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Founder of RIPE, Bennett Reck, has always been obsessed with food. Cooking it, smelling it, eating it, sharing it. Yet, somehow, his diet was always bad. He would inhale gallons of hummus and jars of peanut butter in one sitting. He never went for the nutritious options, instead choosing whatever looked best.\nAs Bennett got older, he started researching whole food nutrition, but quickly became confused. Fake food was bad, but which real foods were good? Was dairy the devil in disguise? Is coconut oil a cure-all or an arterial destroyer? Bennett decided to try them all out and see how he felt. Plant-based eating won out resoundingly, and he's been eating a plant-based diet ever since.\nBut Bennett's always had a sweet spot and there was no way he was going to give up ice cream. How could he?! So he didn't. He created RIPE, the high-protein, low-calorie, avocado-based gelato. RIPE ice cream can be found in three flavors (cacao, golden milk, and salted maple) at Union Kitchen Grocery.\nAbsolutely not. Despite the fact that I come from a family of entrepreneurs, I hadn't considered starting my own business until after I graduated college and started working. In my post-grad job, I encountered entrepreneurs and envied the passion and dedication they so clearly held for their work. I wanted that. But I knew that I wouldn't be able to devote the effort required unless I actually cared about what I was doing\u200a\u2014\u200ait had to be food.\nHow did you pick gelato?\nIf health was the only thing I cared about, I would just give people advice to eat the rainbow, enjoy dry salads with serene mindfulness, and we'll all see each other in a hundred years at the gym. But I care about food, not just health. And food should be delicious. We all need to eat, so why just eat? Why not crave, savor, and indulge our hedonistic urge? That's why I created RIPE. Gelato is the ultimate indulgence, & when I realized I could create that indulgence without the detriments associated with dairy & sugar, there was no other path I could take.\nOur largest obstacle has been the need to work around an industry based in dairy. Ice cream equipment & processing facilities are built to work with the qualities of dairy (or non-dairy bases loaded with additives to mimic dairy), but RIPE's base does not act like dairy. Luckily, we have been able to tap into the knowledge of experts who have helped us develop a recipe that can run on this type of equipment without compromising our ingredient standards.\nI wake up every day at 4:30am and drink STRONG coffee. Pretty much no one else is up at 4:30, so I use the time to exercise, write and respond to emails, and plan out what I need to get accomplished that day. From about 5:00am\u200a\u2014\u200a7:30am I send emails to everyone I need to speak with that day to schedule phone calls. I am still working a full-time job during the day, so I'm in that office from about 8am to 6pm, then run to any RIPE meetings or catch up on RIPE calls\/emails. Whenever I get home, I give myself 30 minutes to cook without interruption. For the rest of the night I plan longer-term schedules, such as demos, production runs, & social media posts.\nSuccess is reaching the point where anyone can enjoy a pint of RIPE. I want RIPE to become an access point into plant-based eating, as an easy-to-find example of a plant-based product that is as delicious as its dairy counterpart.\nPitching for the first time was much more casual than expected. It was not a formal presentation; in fact, we ended up talking while walking around their central office. My advice would be to roll with the punches. I expected to go in and give a formal pitch & be hammered with questions, but it was clear that buyer was not interested in that. If I had tried to steer the encounter back towards my expectation, I think it would have made a poor impression.\nHow do you encourage repeat sales? Do you demo? Use social media? Events?\nWe are very focused on demoing & social media. A lot of early feedback we have received is along the lines of, \"When I tried it I LOVED it, but I'm not sure I would have tried it on my own.\" We are using demos as a way to get over that hurdle so customers become very aware of the great taste. However, we know we can't reach everyone at a demo, so we are focusing on social media so those who have tasted RIPE can spread the word.\nTo ring in Spring, we are featuring a different plant-based entrepreneur every week! Check back in next week to hear from another Union Kitchen maker (or two!) about their philosophies, struggles, and successes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I usually take a minute or two after the reading to say a few words about what we read\u2026a sort of commentary, if you will. However, I'm going to have to forgo the commentaries starting today and the next thirteen or so episodes. Why? Well, my youngest son (who currently lives at home with mom and dad here in Southern California) has taken a job way out in North Carolina, and I am going to drive out with him to help him get his apartment set up. We don't actually leave until next week, but once we leave, I won't be able to do any shows for you. So, I'm recording the shows ahead of time so that we don't miss the readings. I'm not doing the commentaries because I frankly don't have enough time to write that many of them that far in advance. But I do want you to have your daily Lifespring Family Audio Bible while I'm gone. I hope that's ok with you.\nHope you are doing OK DOK.\nSo far, so good, James. At the moment we are under a tropical storm watch. We shall see how things develop over the next several hours! There has been no evacuation order given in our area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Inspired by the success of fantasy baseball and football, fans of basketball can now enjoy the fantasy sports betting action. With fantasy basketball, bettors will be the managers of a team and will choose players based on stats and prior performances. As the manager of the team, trades can be conducted and recruits can be added to the roster. The results are all based on the actual performance of a player or a team. For example, if the bettor chose Kevin Durant to be on the roster, the actual performance of Kevin in a real game would be used. This means the number of points scored, assists and other stats will be used to determine the outcome of a fantasy sports game.\nWhen it comes to scoring with fantasy sports, there is appoint system that is used. Each statistic will be worth a certain number of points. For example, a steal may be worth 2 points and a rebound is worth one. A turnover can be worth -1. As the teams and players play, the points are tallied automatically and the team that has the most points at the end of the basketball season will be deemed the winner. For those that do not wish to wait out an entire season, there are daily fantasy sports betting options. With this type of fantasy sport, players will draft their NBA team and points are scored in a similar way to a season long team. Instead of being committed to an entire season, this form of betting only lasts a single day. Not only does this free up bettors to bet on other teams, games or sports, but it also provides an instant payout when the event ends.\nFantasy sports betting has become a top pastime for fans and bettors around the globe. Two of the most popular sites that host fantasy sports events and betting are Draft Kings and FanDuel, both offering basketball options. These sites provide all types of betting options and offer complete coverage of all sports and fantasy leagues.\nDraft Kings is one of the top sites for daily fantasy sports betting. The site plans to pay out over $1 billion by the time this year ends. When it comes to the NBA options, Draft Kings offers over 20,000 gamers that are all competing for top cash prizes on very small buy-in amounts. One of the great events of the year was a Thursday evening in March of 2015. This night featured a $3 buy in tournament and there was a $65,000 guaranteed prize pool. With a field of over 24,000 players, the payouts of the event were well distributed, providing some with massive payouts. Throughout the week, Draft Kings offers multiple guaranteed tournaments, offering the chance to collect great payouts each and every week. For the very first time in fantasy sports history, Draft Kings offered a top paying prize of $1 million. That tournament was held at the end of March and was referred to as the Fantasy Basketball World Championship. With small buy in amounts and great returns, fans continue to enjoy the daily fantasy sports at Draft Kings.\nAnother popular site for daily fantasy sports action is FanDuel, the major competitor for Draft Kings. At FanDuel, there are many types of contests that can be enjoyed. These include head-to-heads, 50\/50 events and leagues. FanDuel followed in the footsteps of Draft Kings and is offering a $1 million guaranteed prize pool for multiple NBA events. In addition to covering all NBA games, FanDuel also covers college basketball action. When enjoying fantasy basketball at FanDuel, every day, members can choose a new team and have the chance to win profits. The FanDuel site is easy to navigate, which is a plus for beginner fantasy sports players. The site is also legal for Canadian and US players, just as Draft Kings is. There is also a great promise of a refund if the first $5 or $10 game is not enjoyed. What is even better is that FanDuel will cover PayPal fees when withdrawals are conducted! Want to make more money? Refer a friend and enjoy lifetime referral commissions! FanDuel is definitely one of the better fantasy sports sites operating.\nWhile many basketball fans are happy to just watch a game on the big screen, other fans want to make money when their favourite team is performing well. Instead of betting at a sportsbook, there are daily fantasy sports sites that can provide superb ways to win every day! Just draft a team and collect points as those players perform in live games! With the option of daily games, there is no long term commitment and winnings are paid out faster.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our consignment process is wonderfully simple.\nRound up all the designer apparel, handbags, jewelry and footwear that you would like to consign. Everything should be in like-new condition, freshly laundered or pressed and on hangers and 2 to 3 years or newer style-wise.\nWe take seasonal clothing starting in September for the Fall\/Winter season, and starting in February for the Spring\/Summer season.\nCall us to make an appointment. (780) 988-7966 You'll be assigned a confidential number to protect your identity. We are discreet and professional in our client relationships.\nThe contract duration is 90 days and you can either pickup or donate your items after the contract expiration date. If you chose to donate your items they'll go to one of the charities that Nu2You supports.\nConsignors receive 40% of the final selling price, and exceptions can be made subject to the item. You can certainly inquire on your account anytime and we issue either CASH, Cheque, or E-Transfer.\nYou can also choose 50\/50 and use your credit in the store for shopping.\nNu2You sets the bar high on our standards for consignments. This means that every item we sell, from clothing to accessories, is carefully selected and good as new.\nAll items that are stocked in our store are of high quality and in pristine condition. We offer great deals and affordable prices on quality, well-known name brands in all categories.\n*We DO NOT consign brands such as H&M, Forever 21, George, Garage, Dynamite, Reitman's, Suzie Shier, Ardene, and Joe Fresh.\nWhy Should I Consign with Nu2You?\nWe have a large customer database, we reach out using social media, networking and email distribution.\nWe are fully computerized which means we have an accurate inventory of your items.\nWe actively sell your items in a boutique, high level service setting where we have professional fashion stylists to help our customers.\nWe are located in prime retail shopping area in Old Strathcona Edmonton for your convenience.\nWe host several events annually which generate a lot of items to be sold in a short amount of time.\nOur Designer Boutique system works to create a win-win for you and Nu2You.\nWe place a high level of attention and care to all our clients.\nClick the link below to view our consignment agreement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Ester Rada's cross-cultural sound is a deep reflection of the Israeli born Ethiopian's heritage. Growing up in a religious Jewish family in more than modest conditions in Israel, gave Rada the drive to change her way of life and fulfill her dream of creating music. Critics describe her genre mixing sound as \"gracefully combining Ethio-Jazz, Urban-funk, Neo-Soul and R&B, with mixed undertones of black grooves\".\nPresented in conjunction with the Consulate General of Israel in Philadelphia.\nClick this link to buy tickets online.\nTonight's topic is: \"Through the Lens of American Students Who Have Just Return from Israel\".\nLearn about the perspective of those who have recently visited Israel. What are their thoughts? What are their concerns? What is their advice for those considering traveling to Israel? How have the recent military activities in Israel and Gaza affected their viewpoints? What are their hopes for Israel?\nThis series is sponsored by Har Zion Temple, Temple Beth Hillel-Beth El, Main Line Reform Temple, Beth Am Israel, Adath Israel and Beth David Reform Congregation.\nThe Renaissance Group invites you to join us for a Pre-party Happy Hour before our Young Leadership Celebration on December 6, 2014. Join us at Bank and Bourbon in the Loews Hotel on August 20th.\nFree drink ticket with pre-registration.\nElad Strohmayer, Deputy Consul General will discuss \"A Diplomat's Prospective\".\nElad Strohmayer became Deputy Consul General to the Consulate General of Israel in Philadelphia in August 2012. He joined the Israeli Ministry of Foreign Affair's Diplomatic Training Program in 2010. His first post was as the Deputy Chief of Mission in the Embassy of Israel in Luanda, Angola from November 2010 through July 2012. In 2006, Deputy Consul General Strohmayer graduated from Hebrew University of Jerusalem with a degree in Political Science and International Relations.\nHe is currently completing his Master's Degree in International Relations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Folio, about 120 pp., cloth-backed boards with embossed label on front cover. Complete and with bright colors, with various sepia toned and also color plates tipped in, as well as some full page and double page color plates. Such illustrious directors as John Ford, Friedrich Murnau, Raoul Walsh, and Howard Hawks get pages devoted to them and to such films as HANGMAN'S HOUSE, MOTHER MACHREE (Ford), WHAT PRICE GLORY (Walsh), and FIG LEAVES (Hawks). Westerns were a staple at Fox in the 1920's, as is evidenced by double-page spreads, in rich full color, devoted to the movies of Buck Jones and Tom Mix.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Patriotic home decor is always in style in our book. But it seems we're seeing red, white and blue everywhere lately. Our new American and Union Jack flag pillows just arrived and people are gobbling them up! Pillows aren't the only way to get the Americana look in your home. Take some style hints from these lovely photos and stay true to the red, white and blue!\nNautical flags add a unique coastal touch. Did you know each flag represents a letter or number? Try your initials for a subtle monogram in your bedroom!\nOr stick to a red, white and blue color scheme. These vintage election flyers add a graphic punch without being too kitchy.\nOh I love that first photo. So crisp. I'd love to see this classy Independence Day decor this year!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thinking about starting a social enterprise? While there are many considerations to ponder, business structure can be one of the most important first steps. In this presentation by Kate Kilberg of Catalyst Law, LLC, you are given a spectrum of choices, and the pros and cons of each.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's Ireland's cuisine like in a nutshell?\nIreland has a tradition of great substantial winter food - stews, boiled bacon and cabbage, potatoes served a dozen different ways. Those dishes are still to be had in every pub and many restaurants, but there's also been a huge amount of culinary diversification over the last decade or so. Restaurants in Dublin range from the Michelin-starred, combining traditional French cuisine with Ireland's excellent natural produce, to holes-in-the-wall serving vodka and Polish sausage till 2 am and plastic-tabled Asian canteens with Szechuan chickens' intestine on the menu.\nSome of my favourite Irish foods are the simplest: fresh oysters from the Temple Bar market, freshly baked soda bread from Avoca, or a well-looked after Cashel Blue cheese (from a cheesemonger, not the unripe stuff you can get in supermarkets).\nDublin coddle - though I've only encountered this once, and, to be fair, most of my Irish friends have never heard of it so wouldn't be pressing it on anybody. The specimen I was served consisted of a watery stew of potatoes, onions and boiled, flabby white cocktail sausages. I'm sure it could be fine with good ingredients, but I'm not planning to test it again.\nWhat are the food oddities in Ireland?\nOther than the coddle, I haven't encountered anything that made my hair stand on end. A good Irish fry will often include both black and white pudding (blood and liver sausages, respectively), which do horrify some visitors, though I love both if they're well made. A friend of mine had a seaweed flan while visiting the west coast, but I haven't seen this anywhere else, unfortunately.\nWhich cuisine features most strongly in Dublin?\nGoing to the pub is, all clich\u00e9s aside, a national pastime - and one I've taken up with great pleasure. Many pubs in Dublin are brilliant places to spend an hour or two chatting, and as this often leads up to dinner time, there's an awful lot of pub food consumed.\nPubs (because Guinness is eating and drinking): I like traditional pubs, and my favourites in the city are the Stag's Head, the Long Hall, Neary's and Mulligans. All pull an excellent pint, although Mulligans is the best for Guinness by far. For when you want to sink back into a comfortable chair and have a peaceful (though not sepulchral) drink, the Library Bar upstairs at the Central Hotel is your natural port of call. And the Market Bar is a great modern space, a renovated warehouse lit by skylights, serving tapas throughout the day.\nBUDGET: It can be hard to get a really cheap meal in the city, short of grabbing a baguette at a Centra. One inexpensive option is to pick up a square of pizza at the Steps of Rome - it's often delicious and costs less than a fiver. If you're willing to spend a bit more you can get some great food for \u20ac7 - 15: try the Market Bar for tapas, Cornucopia for heaping plates of tasty vegetarian food, Zaytoon for good-quality kebabs, and the caf\u00e9 upstairs at Avoca for well-cooked caf\u00e9 staples.\nAFFORDABLE: This category opens up a huge swathe of possibilities in Dublin. A few suggestions: L'Gueuleton is a great French bistro, serving everything from delicious and filling \u20ac5 soups to \u20ac20-plus meat dishes. They don't take bookings so get there early or be prepared to wait. Yamamori does very good noodle soups, sushi and bento boxes. New Millenium and Good World are both excellent for dim sum, while Salamanca does great tapas and Spanish food. The Mermaid Caf\u00e9 is my favourite for weekend brunch.\nThe best fish restaurant I've been to in Dublin is Caviston's - I had a plate of grilled fresh sardines that shut me up with pleasure for at least 10 minutes. It's outside the city centre, but easily reached by city train and close to the bay if you feel tempted to walk off lunch. Even further afield, if you have a car and are visiting the Wicklow mountains on the southern edge of Dublin, it's well worth stopping off at the Roundwood Inn for their Irish-German cooking, including excellent choucroute. Finally, if you happen to be touring the country, then do yourself an enormous favour and stop in at Caf\u00e9 Paradiso in Cork. Eating there has been one of my favourite dining experiences in the last few years. It's seasonal vegetarian cooking, done so brilliantly and with such imagination you'll wish you lived next door and could eat there again and again.\nBREAK THE BANK: I have to admit that I haven't really explored many of Dublin's most expensive restaurants. I've eaten at l'Ecrivain and enjoyed it immensely - their use of Irish produce is inspiring, their duck is rightfully famous, and the atmosphere is very friendly. Friends with good taste (and deep pockets) have also recommended Chapter One, Patrick Guilbaud and One Pico.\nWhen I first arrived in Dublin, I found myself avoiding the not-very-inspiring supermarkets and instead constructing meals with ingredients bought in half a dozen different shops. That's a very pleasant way to shop and cook if you've got the time, and there are some extraordinarily good specialist food shops here, but it can mean that you spend an hour a day foraging around central Dublin to put dinner together. The pick of the bunch in town are Sheridan's Cheesemongers for an excellent range of well-affinaged Irish and continental cheeses, and La Maison des Gourmets for superb French-style bread. People rate Sawyers as a good fishmonger and I've had fairly good luck there, although I prefer Beshoffs or Wrights, both in Howth.\nThe market also holds an excellent fishmonger, who often has very fresh and tasty scallops; small stalls from Sheridan's and Corleggy cheeses, la Maison des Gourmets and other bakeries; a couple of well-stocked organic vegetable stands, a good butcher, and a number of stalls selling prepared food.\nFinally, although I said that daily shopping in Dublin required stopping off at loads of different shops, that's recently become less true than it used to be. Fallon and Byrne, a fairly fancy food hall, has recently opened with the stated aim of being a one-stop shop for good groceries. To a large extent it succeeds: they have a good cheese and charcuterie counter, a meat and fish counter, a wide range of fresh produce, and a strange but very enjoyable middle section combining well-priced essentials like jars of tuna and cans of chickpeas with obscure and outrageously expensive condiments, chocolates and so on. You could definitely put together a top-class picnic lunch, a Monday-night supper or a three-course dinner there. You can also drop a lot of money there very easily - god knows I have.\nThe centre of Dublin is packed with restaurants. The streets between Grafton St and South Great Georges St are particularly flourishing, with about every third shop being a caf\u00e9, restaurant or pub. On the north side of the river, there's a small Italian quarter around Bloom's Lane (just north of the Millenium Bridge), and very cheap but sometimes really great Asian restaurants along Moore St (off Henry St) and Parnell St.\nWhat's the biggest flop in terms of food and best avoided?\nThe only suggestion I'd have here is to avoid Temple Bar unless you've got a trustworthy recommendation. It's full of stag and hen parties most nights of the week, and a lot of the restaurants there cater to that transient crowd, with so-so food at high prices. There are exceptions, of course - I loved Ho Sen Vietnamese restaurant, for example, and some of the caf\u00e9s are good.\nI think most professional online guides tend to err on the side of generosity - I've seen loads of fairly bad restaurants given sterling reviews! If you're picking a restaurant out of the blue, you could try one of the \"Dublin's best restaurant\" lists, like this one. Otherwise, I do often agree with Georgina Campbell's reviews.\nWhat to be aware of when dining out in Dublin?\nTipping is expected, and I usually give between 10 and 15 % depending on the service.\nSmoking is prohibited in all public spaces, including pubs, cafes and restaurants, which makes going out for a drink or a meal a pleasure for those of us who don't smoke.\nIf you're dining out on a Thursday, Friday or Saturday night, it's always wise to book. I often end up eating in town with friends on Friday evening without pre-planning anything, and while we've never starved, there have been a few occasions where we've trekked from restaurant to restaurant being turned away with tales of hour-long waits for a table, before finding a place that could take us.\nThe information and photography were kindly provided by and published with permission of Meg of Though small, it is tasty.\nIf you have culinary information about your city to share, do drop me a line. And of course there's something in it for you!!!\nFor addresses, read continuation of post.\nGreat post - well done. I really like informative posts like this. The only restaurant I know that's not in your list is Thornton's at The Fitzwilliam Hotel which has been Michelin starred for many years, I believe. I just checked it on the web and it seems to be closed for refurbishment, but this hasn't caused it to lose its star in the 2007 Guide Rouge.\nI really enjoy your blog. I like coming and learning in a snap about different culinary places around the world.\n\"boiled bacon and cabbage..... Those dishes are still to be had in every pub\".\nYes, fair enough Tony - I rarely saw that particular dish in pubs. I meant that \"substantial winter food\" was available in loads of pubs but should have been clearer. Apologies (very belated).\nCoddle is absolutely beautiful if made by the right person. Got to be eaten on horrible winter nights before heading out. Growing up in Dublin I used to really look forward to it. Still do !","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Voltage collapse is still the biggest single threat to the transmission system. It is generally characterized by loss of a stable operating point as well as by the deterioration of voltage levels in and around the electrical center of the region undergoing voltage collapse.\nVoltage collapse, a form of voltage instability, commonly occurs as a result of reactive power deficiency. Unmitigated rotor angle instability can also result in voltage instability. Voltage stability is therefore the ability of power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition.\nThe ETAP Voltage Stability module provides four typical methods for static voltage stability analysis of power systems: V-Q curves, P-V curves, V-Q sensitivity analysis and Q-V eigenvalue analysis (modal analysis).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buying a House and Getting a Mortgage Loan | loanDepot \u2013 How To Buy a House. Buying a home is an exciting yet stressful time, especially if you're a first time homeowner. Many buyers don't realize that there's a lot of prep work to do before you even start looking at homes including getting pre-approved for a loan.\nWill Underwriter Approve My Loan 203(K) FHA 203(k) Loans \u2013 What is a FHA 203(k) Loan? | Homebridge \u2013 For FHA 203(k) purchase transactions the maximum loan amount is the lesser of the sales price + renovation costs or 110% of the after improved value, times the maximum loan-to-value. For refinance transactions the maximum loan amount is the lesser of the property value before renovation + renovation costs or 110% of the after improved value, times the maximum loan-to-value.Loan Underwriters and the Mortgage Underwriting Process. \u2013 During the mortgage underwriting stage, your application moves from the desk of the loan processor to the mortgage underwriter. The mortgage underwriter will ensure your financial profile matches your lender's guidelines and loan criteria and he or she will ultimately make the final decision: to approve or deny your loan request.What Is The Ideal Credit Score What Is the Optimal Credit Score? | HuffPost \u2013 \u00b7 The best credit score you can get on the FICO scale is a 850, but that score is near impossible to attain. You would need to have the perfect storm of credit.Do I Qualify For Harp Loan Welcome To The H.A.R.P. Program Website! \u2013 The HARP program can help! If you're not behind on your mortgage payments but have been unable to get traditional refinancing because the value of your home has declined, you may be eligible to refinance through the Home Affordable Refinance Program (HARP ).\nMortgage Pre-Approval Process \u2013 Discover Home Loans Blog \u2013 Home Buying articles mortgage pre-approval vs. Pre-Qualification. Pre-Qualification. General estimate;. Mortgage Banker who has pulled your credit score, and you've given the lender all of your necessary documentation for mortgage pre-approval requested by the lender regarding income.\nWhat Are the Differences Between a Good-Faith Estimate & a Pre-Approval Letter? \u2013 You can lessen any potential stress attached to the home-buying process by being armed with a mortgage pre-approval letter. At the start of your house hunting, also obtain a good-faith estimate of.\nWhy home sellers may want to be wary of a potential buyer's pre. \u2013 The offer included a pre-approval from a lender for the contract price. The offer was for full price, and they didn't ask for any help with closing.\nPurchasing |Buying A Home | Pre Approval | Pre. \u2013 FMM \u2013 Home Purchasing |Buying A Home | Pre Approval | Pre Qualification | Mortgage Rates Get a Live Rate Take the mystery out of buying a home and learn which home loan options are right for you and how the mortgage process works.\nHome Buying: Will taking a loan from 401K to make mortgage. \u2013 Taking a loan from your 401K WILL affect your mortgage approval. The underwriter will add back the terms of the 401K loan to your debt ratio. The upside is the interest you pay on the 401K loan is usually paid back into your account.\nHome Equity Loan Current Rates California Home Equity Loan 10 Years \u2013 CA Current. \u2013 ERATE \u2013 Home Equity Loans \u2013 Rates are based on a fixed rate home equity loan for an owner occupied residence, second lien, 10 year or 15 year repayment terms with an 80% loan-to-value ratio for loan amounts of $50,000 or $50,000+.\nCommon Reasons Why Buyers Are Denied A Mortgage \u2013 This whole pre-approval process is something every buyer goes through if they are seeking a mortgage to buy a home. It is easy to assume that once you have pre-approval, all you have to do is find a home to buy, sign some papers, get your money and purchase your home.\nHome Equity Loan Providers Home Equity Loans \u2013 SPE Federal Credit Union \u2013 If you already own your home, you can make your equity work for you with a Home Equity Loan. Pay for education expenses, consolidate your bills, buy a new.\nGetting pre-approved and qualifying for a mortgage \u2013 Canada.ca \u2013 Find out where to get a mortgage, how the pre-approval process works and what lenders consider when they qualify you for a mortgage.\nHow Do I Get Pre-Approved for a Mortgage? \u2013 Seeking pre-approval six months to one year in advance of a serious home search puts you in a stronger position to improve your overall credit profile.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The interface was aging and you will now discover a new design with a new set of icons, most of them coming from the Awesome project.\nThe tree has been completely reworked. It displays data faster and provide other features such as the duplication of elements or the fast creation of instruments \/ presets with a drag & drop.\nAll soundfonts from the online repository are accessible directly within Polyphone for Premium users. And for everyone, the randomly chosen \"daily soundfonts\" are freely shared.\nFor testing purposes, it is possible to mute a division or mute all divisions but one in the tables.\nBased on an instrument, this tool allows you to create all possible chords. This could be interesting for example for the creation of a choir or pads.\nGerman and Danish translations have been added. Please note that due to a lot of changes, only those translations are for now completed: Danish (thank you Chris!) and French.\nPolyphone can now process the keywords \"control\", \"include\", \"default_path\", \"global\", \"group_label\" and \"define\" when importing sfz files.\nSorting rows is now done in a better way and the sample rate is displayed for the samples.\nThese operations can be done even on a multiple selection.\nIt is now faster and big soundfonts (> 2 Go) shouldn't be a problem anymore.\nBug fixed when saving or loading 24-bit samples.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Israeli military meets its match in a determined 15-year-old girl, in Dror Shaul's hilarious satire on volatile Iran-Israel relations. Nofar works in a falafel truck with her widowed mother near the site of a secret Israeli nuclear program gearing up for war. However, a proverbial monkey wrench gets tossed into the gears of the military's plans when the International Atomic Energy Agency shows up for an inspection with a handsome German inspector that Nofar thinks is just right for her mother. With clear nods to Stanley Kubrick's Dr. Stangelove, this fast-paced pro-peace comedy is just the right recipe for these troubled times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Elvis Duran finally has his woman!\nBethany Watson is joining Duran's Z100-based, nationally syndicated morning show. Watson fills the slot vacated by Carolina Bermudez, who left for WLNY's The Couch at the end of June.\nWatson comes to New York by way of Minneapolis. After graduating from the University of Minnesota, where she studied theater and improvisational comedy, Watson was working at a desk job. She decided to to contact her favorite radio show. Expediting her career path, she applied for the co-host position on The Dave Ryan in the Morning Show at KDWB. She remained with the broadcast for more than four years.\nThe Wisconsin native debuts next week to Z100 listeners and more than 40 stations in the country. Fans can also listen via the station's Website on at IHeartRadio.com and the IHeartRadio mobile app.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"On an average, over 2 crore people travel by train in India every day. While the shortest train route is 3 km, the longest one spreads across a massive distance of 4273 km and passengers spend hours and sometime days on trains with almost nothing to do while they are travelling. Irrespective of the distance and the destination, train passengers always look out for exciting ways to spend their journey time usefully.\nIn this direction, Magzter, the digital magazine newsstand, has partnered with IRCTC (Indian Railway Catering and Tourism Corporation), the Government-owned enterprise that manages the online ticket-booking portal of Indian Railways, to offer an unmatched digital reading experience to latter's 6.6 crore+ users. Thanks to increased smartphone and mobile Internet penetration, this initiative is sure to receive a rousing welcome from the train passengers.\nUnder this partnership, IRCTC users can get unlimited access to 5,000+ best-selling magazines and select newspapers on their smartphones and tablets at an exclusive offer price starting at just Rs 20 plus GST per day journey. These magazines spread across 40+ exciting categories including automotive, business, comics, education, entertainment, fashion, fitness, lifestyle, news, politics, science, technology and travel, thereby catering to diverse interests of people belonging to different age groups.\nThe readers can enjoy unlimited access to superlative content from all the major countries including India, USA, UK, Singapore, Australia, Canada and South Africa on the go. Apart from leading Indian English magazines, Magzter also offers magazines across all the regional languages including Hindi, Marathi, Tamil, Telugu, Kannada, Malayalam, Bengali, Punjabi, Gujarati and Urdu. More importantly, once the magazines are downloaded on the device, they can be accessed even without an internet connection, which will come in handy when the users are travelling. Along with magazines and newspapers, IRCTC users also get access to premium articles curated from best-selling magazines and the latest news from hundreds of trustworthy sources on the Magzter app.\nIRCTC users can choose their packs from various durations \u2013 1 day at Rs 20 (special pack available only for IRCTC users), 1 week at Rs 30 (special pack available only for IRCTC users), 1 month at Rs 99 (MRP - Rs 399) or 1 year at Rs 499 (MRP - Rs 3,999), and indulge in unlimited reading. Magzter is also offering an exclusive 7-day free trial for IRCTC users.\nTo purchase a pack or claim the free trial, users can go to the IRCTC website, click on 'IRCTC Exclusive' and then click on 'Digital Magazine' from the drop-down list. Users will be redirected to the Magzter website, where they can Sign in\/Sign up with Email address or Facebook and claim the offer. Once the offer is claimed, users can enjoy unlimited reading on the Magzter website and the app (Android\/iOS) with their credentials.\n\"We're extremely delighted to partner with IRCTC and offer our world-class digital reading services to their users. With this partnership, we're quite confident of further spreading the digital reading revolution across the length and breadth of India. We're looking forward to a long-standing and fruitful partnership with IRCTC,\" said Girish Ramdas, CEO, Magzter Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"iBall Andi Uddaan is available for 3,999 Rupees on Amazon India. This price is vaild for all major cities of India including Delhi, Mumbai, Kolkata and Chennai.\nThe latest price of iBall Andi Uddaan was retrieved on 19th April 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SKU: 3e41182d617e Categories: Big Wall & Protection, Climb, Products, Pulley's & Haulers Tag: C.A.M.P.\nC.A.M.P.'s Sphinx Pro is a super-compact, lightweight, and easy-handling pulley with fixed side plates. The Sphinx Pro uses the same design as the Sphinx, but features sealed ball bearings for higher efficiency. The Sphinx Pro is designed for intensive use at higher speeds, but is small and light enough to easily carry on your harness or throw in your pack for ski touring and mountaineering. Unlike other pulleys with fixed side plates, the Sphinx Pro allows a prusik knot to be installed on one side (aramid cord suggested) making it an ideal pulley for rigging a haul system on route or managing crevasse rescues.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sunday worship begins at 10:30 am, with most people arriving at 10:20 a.m. Usually services are over by 11:45 a.m. For mid-week services most people arrive five minutes before the start time.\nYou are welcome to sit wherever you will be most comfortable.\nThere is no strict dress code at Crossroads. Some people dress up to give glory to God, and others don't for the same reason! We want you to be physically comfortable while you're here.\nThe doors to the main foyer are located at the end of the covered walkway off the Scott Road parking lot, or at either the north or south side (just off the patio area) from the rear parking lot (off 119A Street).\nWill I have to wear a visitor badge?\nNo way! There are greeters in the foyer who will give you an Order of Service and our weekly announcement sheet. There are \"welcome\" papers in a pocket in the seat in front of you. If you feel comfortable doing so, please fill it in after the service and drop it into an offering box on your way out of the sanctuary.\nCrossroads is fully accessible. There are designated parking spots close to an entrance. In our worship space there is room for wheelchairs, walkers, and scooters. If you require a headset to enhance your hearing, please speak to one of the greeters and they will be pleased to assist you. We also provide large-print service orders on request.\nOffering boxes are positioned in the sanctuary entrances. This is how Crossroads folks financially support the work of the church in our neighbourhood and beyond. If you are visiting with us for the first time, you are welcome to give and you are welcome to walk right past those boxes. Either way, you are welcome here.\nWhat about my children in worship?\nWe are very comfortable with children being children during worship. There is an activity area near the front of the sanctuary where children are welcome to spend their time. There is also an enclosed nursery adjacent to the sanctuary if you need a quiet space to be with an infant or younger toddler during the service.\nDuring the service one of our Ministers will invite children (of all ages) to join on the carpet at the front for a conversation (while everyone listens in). Afterwards, children four years-of-age to Grade five are invited to Sunday School and every second week children Grade six and older are invited to the Youth Room for a discussion. You are welcome to go with your children to Sunday School or have them stay with you in worship \u2013 some kids stay in the sanctuary for the whole service.\nOur regular preacher is pretty versatile. She'll tell stories, pull together different ideas, lead a meditation, give the congregation an \"assignment\" to talk about in twos or threes, show a slide show or video, or sometimes even sing to the congregation. The sermon functions to deepen our reflection on the Scripture themes of the day, and often leaves us with a challenge or question. You can hear recorded sermons here.\nAt Crossroads we are blessed with a very diverse musical palette. We sing traditional hymns, world music hymns in many languages, new hymns written by our own musical leadership, and contemporary church songs. Our worship band (piano, guitar, saxes, trumpet, flutes, drums, violin and bass) and adult choir (SAB) participate weekly in worship leadership, and our children's and youth choirs participate regularly. We don't usually applaud after the band, choirs (or sermons) but accept these offerings as a gift.\nYes! If you are seeking to follow Jesus, then you are welcome at his table here at Crossroads. ALL are welcome at the table \u2013 including children. We celebrate communion every four to six weeks during the Thanksgiving part of our service, and at other special times of the year such as Maundy Thursday or Christmas Eve.\nAt Crossroads we baptize and receive new members by transfer or reaffirmation of faith at regular intervals. There are certain times in the church year when it makes sense to us to celebrate growth in faith and welcome new Christians into the body of Christ.\nIs this a praying congregation?\nAbsolutely! Our minister leads us in prayer in the Gathering part of the service, before the sermon, and to dedicate our offerings. The \"Prayers of the People\", when we pray for the world, the church and those we know in need of God's love and grace, are led by members of the congregation each week, followed by the Lord's Prayer. After the service, there are people available to pray with you if would like. We also have a dedicated group that prays specifically for the ongoing requests of the congregation.\nI sometimes see on your sign that it says, \"We take the Bible seriously, not literally\". Do you use the Bible in worship?\nDefinitely! Members of the congregation read from the Bible during the \"Listening to the Word\" part of the service. The sermon is related to those themes from scripture. We believe that those ancient words help us grow in our faith for our lives today.\nIs there food and coffee?\nAfter worship, everyone is invited to come to the multi-purpose hall across the foyer to enjoy a cup of tea, coffee, water or juice, a few goodies and a conversation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home > GOLDEN YEARS > Fitness & You > Where Can I Get Help?\nThe Ministry of Health Malaysia has initiated a special program for the older persons since 1996. The goal of the program is to achieve optimal health for the older persons through integrated and comprehensive health and health related services in-line with the National Health Policy for the Older Persons (http:\/\/fh.moh.gov.my\/v3\/index.php\/kesihatan-keluarga\/finish\/10-unit-kesihatan-warga-emas\/30-dasar-kesihatan-warga-emas-negara)which stated that \"to ensure healthy, active and productive ageing by empowering the older persons, family and community with knowledge, skills, an enabling environment; and the provision of optimal health care services at all levels and by all sectors\".\nTo improve the health status of older persons.\nTo encourage participation in health promoting and disease prevention activities throughout the life course.\nTo provide age friendly, affordable, equitable, accessible, culturally acceptable, gender sensitive, seamless health care services in a holistic manner at all levels.\nTo advocate and support the development of enabling environment for independent living (ageing-in-place).\nMedical examination, counseling, treatment and referral.\nHome visit and home nursing.\nRehabilitation (physiotherapy and occupational therapy).\nRecreational, social and welfare activities.\nIf you have any problem concerning you health, please go to or contact the nearest clinic for assistance \/ further information.\nAlzheimer's Disease is the most common form of dementia. It is NOT a part of normal aging, but a disease involving the progressive loss of brain nerve cells. Since these nerve cells are essential for normal thought, memory and other brain functions, people with Alzheimer's Disease suffer a decline of mental functions which eventually interferes with the patient's normal daily activities.\nNASAM's mission is to inform the public that there is life after stroke through proper rehabilitation and to promote the concept of stroke prevention.\nThe Persatuan Diabetes Malaysia (PDM) is meant to support patients and their families, and provides useful and practical information and knowledge about diabetes and how to manage and deal with diabetes and its complications. The portal also aims to create awareness about diabetes to the general public and rectify misconceptions as well as providing prevention strategies.\nThe Malaysian Society of Hypertension (MSH) was officially registered under the Registrar of Society in 1993. The main aim of the society is to promote public education on hypertension, to help towards optimum control of hypertension and encouraging researches and publications towards better care in the management of hypertension.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamwyf b/data_all_eng_slimpj/shuffled/split2/finalzzzamwyf
new file mode 100644
index 0000000000000000000000000000000000000000..fe9d1da2faea926ee5de2557c6fa1f64c327c2ef
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamwyf
@@ -0,0 +1,5 @@
+{"text":"My sister is doing well and, hopefully, this will take care of the knee problems she has been having. She is on crutches, but has been sitting up and hoping around the house on her crutches. Continued prayers for healing are appreciated. Thank you to everyone who prayed for her today. It means a lot to me.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've been a little obsessed with Himmeli for about a year now. For those asking, Himmeli is a Scandinavian art form, sometimes elaborate, sometimes simple, that uses geometric shapes and simple materials to create beautiful objects.\nI keep getting asked if I sell the wreaths so this year, I am! I have two wreaths to choose from, along with two tree topper designs. If you're feeling crafty, you can make your own wreath by following this tutorial. You'll need time and patience if you want to go the DIY route but it's really fun if you can hang with it.\nIf you would like to order a wreath or tree topper, or would like current prices, email me: kindrawrites(at)gmail(dot)com. All designs are available for local (Austin) pick up only.\nI'll be sure to post my tree when it's decked out with Himmeli so check back soon!\nHow long would it take for you to ship one?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The urgent repair of pot holes on the M5 northbound has seen the carriageway closed and all traffic being held on Sunday morning (February 4).\nEmergency pot hole repairs are being made between J16 A38 (Aztec West) and J15 M4 (Almondsbury Interchange), with motorists being held while the work is being carried out.\nIt is thought that the carriageway will not open for an hour or more, so expect long delays in the area.\nThere is also an accident in lane one, with an ambulance in attendance.\nSomerset Live will bring live updates and you might need to refresh the page for the latest information.\nAre you stuck in the traffic on the M5? Let us know by emailing james.brinsford@localworld.co.uk.\nTraffic is beginning to flow again on the M5 nortbound.\nIt has been reported that all lanes have now re-opened but traffic is still slow. Vehicles are yet to be recovered from the hard shoulder.\nThe problem has been exacerbated by an accident with a car blocking Lane one (of three) with ambulance and police at the scene.\nEmergency pot hole repair on M5 Northbound between J16 A38 (Aztec West) and J15 M4 (Almondsbury Interchange) has left the carriageway blocked with all traffic held, to facilitate repair.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is anyone going to this yearly convention? I'm planning on going and would love to meet up with you if you are going.\nHi, Linda, what convention is this? Since I'm way down in Florida, it would be great to attend something down my way. I've been trying to research on this and have come up with zilcho.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Structural Elements\u00ae has unlocked a transformative treatment method that drastically improves health, performance and quality of life.\nIt's time to transform orthopedic wellness.\nStructural Elements\u00ae fills a critical gap in the health and orthopedic care markets. Patients \"graduating\" from physical therapy, experiencing long-term chronic pain or performance related orthopedic issues have limited options for truly effective ongoing care. The void left by pain, injury or dips in athletic performance means less time on the field, in the studio or at the gym. The proprietary treatment methods (se)\u00ae has developed are in perfect alignment with the needs and goals of the modern health and wellness market. The cash-based business model and brand strength make this an immediate and impactful business venture.\nPracticed by physical therapists, chiropractors, acupuncturists and osteopathic physicians, many of our tools are familiar: acupuncture\/dry needling, manual therapy, myofascial release, stainless steel tools, orthotics and recovery compression.\nFrom clinic build out assistance and interior design to training with Structural Elements\u00ae franchise experts, we've got you covered on every aspect of owning, managing and building your (se)\u00ae orthopedic wellness franchise.\nAvoid the limitations in prescribed treatments. Structural Elements\u00ae implements treatment techniques that go beyond the site of the acute injury, giving therapists the freedom to treat the whole body and use any appropriate techniques until our patients' goals are achieved - all while ensuring payment for those services.\nNumber of Americans who sought alternative therapy.\nValue of the growing global wellness market.\nProjected growth in demand for physical therapists.\nNational Center for Complementary and Integrative Health (NCCIH) and the Centers for Disease Control and Prevention. Global Wellness Institute, 2018 Global Wellness Economy Monitor report. Bureau of Labor Statistics, U.S. Department of Labor, Occupational Outlook Handbook.\nOwn and develop an entire market as a Regional Developer. Grow your entire market in alignment with your existing wellness units.\nBring your practice into the modern wellness movement. Pick your market and start an on-trend, modern clinic in coordination with your current wellness location.\nJoin a growing, in-demand lifestyle wellness brand as an (se)\u00ae Therapist. Enjoy the benefits of a modern, established clinic and apply the proprietary Structural Elements\u00ae modality to your orthopedic foundation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanevz b/data_all_eng_slimpj/shuffled/split2/finalzzzanevz
new file mode 100644
index 0000000000000000000000000000000000000000..84bb308e57b26538ac8b97ffbb15da3e0edd648c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanevz
@@ -0,0 +1,5 @@
+{"text":"\u200bThis consultation closed on 24 April 2017. Thank you for your feedback.\nThe refreshed Future Urban Land Supply Strategy was adopted by the planning committee on 4 July 2017. We received 191 written pieces of feedback, three petitions, and input from mana whenua and local boards. Overall, there was strong support for the refreshed strategy.\nWe made several changes to the strategy based on the feedback.\nFor a copy of the refreshed strategy, see Future Urban Land Supply Strategy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lateral Entry for admission to Class IX is conducted only if seats are vacant. Generally, notification for Lateral Entry is published in the month of February \/ March and the test is conducted in the moth of May. Please see notification placed on this website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 7x30 WP Deep Sea Floating Binocular from Barska is designed specifically for the nautical and boating enthusiast. The 7x30 is a popular, easily manageable binocular size that permits a steady image even under agitated ocean conditions. These binoculars provide clear observation in any marine ecology from sunrise to sundown. The Deep Sea binocular features a rugged heavy-duty and shockproof rubber-armoring that allows a sure and secure grip even as the white-caps come tumbling over the bow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A bill that would legalize and regulate sports betting in California has been introduced in the state legislature.\nMore accurately, an old bill was gutted and amended to deal with sports betting, introduced by Assemblyman Adam Gray, who chairs the gaming committee in that chamber. AB 1441 would legalize sports betting in the state, but only if federal law and the state constitution are changed to allow it.\nThis bill would provide that its provisions would become operative only if the federal Professional and Amateur Sports Protection Act is amended or repealed to allow sports wagering in California and a state constitutional amendment to authorize sports wagering has been approved by the voters.\nSo while it's interesting that California is eyeing the potential sports betting market, the state is hardly unique in this regard, and several dominoes would have to fall before sports books start popping up, even if this bill is passed.\nThe bill comes as New Jersey is fighting a losing battle regarding a law it passed attempting to allow sports betting within its borders. A federal circuit court ruled that PASPA prevents states from authorizing entities to take sports bets. The state has asked for a rehearing after its latest court defeat.\nLegal Sports Report has reached out to Assemblyman Gray's office for comment. Today is the final day of California's legislative session.\nThe legislation came on the same day that a bill seeking to regulate daily fantasy sports was also introduced.\nWho could offer sports betting, and how?\nThose same gaming interests in the state have had issues getting on the same page in recent years regarding the ability to offer online gambling and poker, resulting in yet another stalled iGaming effort this year.\nThe sports wagering authorized pursuant to this chapter shall be accepted and executed only using telephone, computer, or another method of electronic wagering communication.\nEntities wishing to offer sports betting would have to pay a licensing fee (to be determined).\nLicensees would have to pay a percentage of their total gaming win to the state each quarter. That amount is also TBD.\nBettors must be within California's borders to place a sports bet.\nBettors must be 21 years of age.\nThe debate over sports betting has gotten a lot of traction in recent weeks, mostly because of the New Jersey case. The latest ruling there has made it seem nearly impossible for states to offer sports betting on their own, or to win that right via the courts.\nWhile all hope is not quite lost in the New Jersey appeal, it's becoming clear that the easiest and most likely route to legalized sports betting would be via a repeal of PASPA.\nHow likely is that to happen? Given the fact that the current federal prohibition (outside of Nevada) is almost completely ineffective, legalizing and regulating sports betting seems like it should be an easy choice. An estimated $95 billion will be wagered on football this fall in the United States, almost all of it illegally.\nNBA commissioner Adam Silver has been at the forefront of the issue, calling for Congress to take PASPA off the books.\nWhat remains to be seen is what kind of appetite legislators at the federal level have for potentially opening up the floodgates for sports betting to be offered at the state level.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For a smooth, relaxed travel experience, look no further than the Gulfstream GIV. With custom cabin amenities and high-performing data connectivity, this business jet reaches speeds up to 675 mph, increasing in-flight productivity, while setting a high bar for private travel. Whether you are looking for a domestic or international destination, the Gulfstream GIV will deliver impeccable service according to your schedule.\nRead about the best city centers to explore by foot, here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanqcy b/data_all_eng_slimpj/shuffled/split2/finalzzzanqcy
new file mode 100644
index 0000000000000000000000000000000000000000..8a7a1304ff546581c6f14fb548ad485b73ab5bb6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanqcy
@@ -0,0 +1,5 @@
+{"text":"The following is a list of rules that must enforced in order to maintain a pleasant and safe atmosphere for the enjoyment of all players.\n1. No running or track shoes on the clay courts. We do have tennis shoes for sale and will extend a 20% discount on regularly priced shoes to guests who do not have proper footwear.\n2. No smoking in the pro shop or on the grounds.\n4. No littering on courts or grounds.\n5. No damage, defacing property, throwing or hitting the fence with your racquet.\n6. All players must check in at the pro shop before going to their court. There is a 10 minute grace period for court reservations. If you do not check in for your court after this time period, we reserve the right to book it for another guest.\n7. Court, clinic, and lesson cancellations should be made as soon as you know you cannot attend, especially during the Spring and Summer seasons.\n8. Proper clothing must be worn on the courts. Shirts must be worn at all times. No exceptions.\n9. Courtesy to each other and employees is expected from all guests.\n10. Court reservations can be made on the hour and be made on the day or a day in advance of play.\n11. No ball machines or large ball baskets permitted, unless granted permission by our Director of Tennis. Machines and baskets are allowed on courts being used for Resort programs and Smith Stearns Junior Academy lessons and camp.\nIn order for our club to remain operational, it is necessary for us to set certain priorities.\n1. Maintenance takes priority over playing time. Our courts must be swept, rolled and watered on a regular basis in order to keep them in the best playable condition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now that the dust has settled over the elections, its time to think about where we are going with SuDS, and it will be interesting to see what changes, if any, will be made now that the heavy influence of Eric Pickles has been removed.\nWe ended the last government with the ridiculous situation that the water management muddle under which we currently operate in England has only got worse \u2013 legislation curtailed part-way through the process, and a whole host of issues left hanging unresolved or abandoned, such as how new housing sites can be adopted, and no removal of the automatic right to connect to a sewer for those same new houses. At the same time, SuDS are only considered necessary in areas which already flood, which utterly misses the point. It's also required on all new development of 10 houses or more, but only if 'appropriate'. It's not surprising that Planning Officers and their authorities are struggling to know how to implement such vague and unco-ordinated 'planning guidance'.\nWe hear from planning colleagues of the problems they have in understanding how to provide appropriate guidance around SuDS, what to require for both outline and detailed planning applications, with many not having the necessary backup of a good SuDS policy base or SPDs. They also consider, rightly from their point of view, that SuDS is just another thing to be taken into account in the 'planning mix', as they would with anything else that is not covered by a legislative requirement \u2013 but completely wrong from the point of view of the potential long-term drainage impact of new buildings on the proposed development itself and its down-stream neighbours.\nFunding is being found in both small and large amounts, and partnerships are being formed with local companies, LEPS, BIDs and water companies, and local communities are being positively engaged with the process. Perhaps some aspects of localism have some benefit after all? However it happens, we need to nibble away at the existing problem of flooding in our towns and cities by retrofitting, all the time, everywhere, and we can all contribute towards solving that problem.\nSo if you want to see how you can contribute, or want to explain how SuDS can help solve flooding to family, friends or colleagues, watch the Let's get Nibbling! video for a short, simple and amusing explanation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Land is located in Broadway Park, a secure industrial park in Strand. The land is made up of 4 equal erven totalling 12942m2 in extent.\nProperty is ideally zoned for storage or industrial warehousing.\nLand can be sold outright or sold as a development partnership with the owners. Finance can also be arranged for a prospective Purchaser.\nAsking price R1200.00 per m2.\nCall Yianni to discuss further.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ashley Bittersweet E King Poster Bed in Light Brown \u2013 The Furniture Space.\nThe Ashley Furniture Bittersweet e king Poster Bed bedroom set is a fresh take on early American country furniture. When it comes to style ,the e king poster bed lets nature take its course with details including a brilliant replicated pine grain finish and a clever combination of raised and flat-panels.\nSize: 80.00\" x 81.75\" x 52.75\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eric Hunter, PhD, who joined Emory in 2004, has primarily centered his research on elucidating the virus-cell interactions involved in the assembly and the entry of retroviruses. Understanding how independently targeted capsid and glycoprotein molecules are transported to the assembly site(s), what cellular pathways are used and what roles cell- and virus-encoded gene products play in this process is a major focus of his research. Because the major viral components traverse distinct pathways, Dr. Hunter's laboratory has characterized the factors that influence intracellular transport and assembly of both viral capsids and viral glycoproteins. He has also examined the signals and mechanisms that operate to include the viral glycoproteins into a budding virion and that mediate fusion. More recently, this information on glycoprotein structure and function has been applied to studies of the virologic correlates of HIV transmission within an African setting. Specifically, studies are underway to distinguish structural and functional differences between the glycosylated HIV envelope proteins that permit transmission into a new host versus the majority of related HIV envelope variants in the donor that fail to lead to a transmission event. Additional studies in collaboration with investigators at the University of Alabama at Birmingham (UAB) and in Perth, Australia, have been initiated to examine how a newly transmitted virus adapts to its new host-defined immune environment so as to evade detection and destruction. The knowledge gained from characterizing unique features of the transmitted envelope proteins and the process of immune escape, will be critical for the development of an effective HIV vaccine.\nDr. Hunter completed his undergraduate studies in bacteriology at Birmingham University, England (B.Sc. with honors, 1969). His graduate work in tumor immunology was carried out at the Imperial Cancer Research Fund & Brunel University, London, England (PhD, 1972). During postdoctoral studies at the University of Southern California, he applied molecular genetics approaches to examine retrovirus replication, and he expanded on this approach when he joined UAB in 1976. While at UAB, Dr. Hunter was the founding Director of the UAB Center for AIDS Research (CFAR), leading its growth into one of the premiere AIDS research institutions in the United States.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaolaj b/data_all_eng_slimpj/shuffled/split2/finalzzzaolaj
new file mode 100644
index 0000000000000000000000000000000000000000..80b2a8b6a168aa79fe4b6c749a3168f784e4a16c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaolaj
@@ -0,0 +1,5 @@
+{"text":"Just some information on how to start a Screen Sharing or Back to My Mac (BTMM from now on) session from the command line.\nTo launch into a BTMM session is a bit trickier. First we need to either find your iCloud Member number or the specific Macs BTMM IPv6 address.\nAnother way to launch a BTMM session is to use the unique IPv6 that is given to any Mac enabled with BTMM. So far, that only two places that I've been able to see this address is by opening up the new version of Apple Remote Desktop for Mountain Lion which now also shows your BTMM machines as well as Macs on the local network.\nYou can then copy the IPv6 address and add it into the open vnc:\/\/ command.\nThis address can also be found after initiating a BTMM session from the Finder, navigating to ~\/Library\/Application Support\/Screen Sharing\/, & opening the file relating to Whichever computer that you just BTMMed.\nIf using the IPv6 address, make sure you put it in square brackets. It won't work otherwise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Alabama Region is proud to announce that it will be holding its Double SARRC race at Roebling Road on the July 9-10th weekend. The event was previously scheduled for Barber but was changed due to reasons previously cited on this forum. Hope to see some of you there!\nYou just changed it from an event I was hoping to attend to one that I WILL be attending. See you guys there, never raced at Roebling in the summer!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can earn a 5% commission by selling Tales from the Sea of Thieves on your website. It's easy to get started - we will give you example code. After you're set-up, your website can earn you money while you work, play or even sleep! You should start right now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The redesigned SkyGuide Pro mount head is more compact, small enough to fit on your palm, better precision, silent tracking, built-in rechargeable power source, ST-4 guiding port and camera trigger port. The improved precision polar scope maintains a fine engraved reticule, now features adjustable illumination with different brightness levels.\nThe DEC mounting bracket allows you to balance a heavy camera and\/or lens better, or even mount a light telescope. The optional Go2Nova hand controller allows you to check the Pole Star position for the polar alignment, slew the mount, adjust the guiding rate, change the mount settings and control the camera trigger.\nThe SkyGuider Pro features four tracking rates, 1X or sidereal, 1\/2X for sky and landscape combined, solar and lunar. All 4 tracking rates work in both northern and southern hemispheres. iOptron have also added a quick slew mode with both forward and reverse to help quickly reframing your image without disrupting the cameras position.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"First off I'll apologize for the terrible pictures. The light sources in my house are awful.\nI've used some of the critters from the latest Purple Onion Designs release to make some ornaments for my little birch tree.\nEach ornament is a shaker and has a little Christmas greenery to dress it up.\nHope this brought you a little holiday cheer.\nHoliday cheer indeed! Tracy, these are all so adorable, a great idea for shaker ornaments! They look so sweet on your birch tree! Thanks for sharing.\nToo cute!! They are adorable Tracy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqtbg b/data_all_eng_slimpj/shuffled/split2/finalzzzaqtbg
new file mode 100644
index 0000000000000000000000000000000000000000..117511ade1aea26ac35623aa5dcbdd81386784b2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaqtbg
@@ -0,0 +1,5 @@
+{"text":"Published 04\/20\/2019 08:45:54 pm at 04\/20\/2019 08:45:54 pm in Best Touch Screen Thermostat.\nbest touch screen thermostat best touch screen programmable hvac digital heating thermostat htw 31 dt12 some best sell wifi thermostats some best sell wifi thermostats honeywell rthwf wifi color touchscreen.\nhow the thermostat works view larger image best smart thermostat how , best smart thermostats for reviews and buying advice techhive best smart thermostats, top best cheapest smart thermostats by trusty joe review radio thermostat ct, how to program a hunter five minute thermostat model youtube , honeywell programmable thermostats thermostats the home depot day programmable t touch screen thermostat, hot sale bestselling room fcu touch screen thermostat cooworcom hot sale bestselling room fcu touch screen thermostat, the best smart thermostat for reviews by wirecutter a new our pick for the best smart thermostat the nest e it has a round,honeywell thwf wifi color touchscreen thermostat works honeywell thwf wifi color touchscreen thermostat works with alexa, which frontpoint security compatible thermostat is the best alarmcom smarter thermostat, china best programmable touch screen thermostat with large lcd best programmable touch screen thermostat with large lcd screen display, honeywell wifi smart thermostat rthwf review featured image for article honeywell wifi smart thermostat rthwf review.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As mothers we wear so many hats. One of them, \"Nutrition Specialist.\" We are in charge of what our kids put in their mouths and if it is good for them.\nIn our home we follow The Word of Wisdom. As well as the guidelines set by the USDA. We follow a diet of \"moderation.\"\nRecently the USDA has acknowledged the huge concern that people have about the health of the American population. They have released new data in an attempt to improve the health of the American people.\nSodium intake should be less than 2,300 milligrams. African Americans, or those suffering from hypertension, diabetes, or kidney disease should reduce intake to 1,500 milligrams. That is about 1\/2 tsp. of salt.\n*I know that too much salt is bad, but 1\/2 to 1 tsp of salt a day seems really hard to observe.\nConsume 8 to 12 ounces of seafood per week from a variety of seafood types. Pregnant and breastfeeding women should eat seafood at least twice a week for nutrients that play an important role in babies' brain and eye development.\n*I was really happy about this one. Especially now that doctors don't prohibit fish when your pregnant. When I was pregnant with our son, over 8 years ago, the recommendation was no fish, none at all. For 9 months I didn't touch any seafood. Not even one bite. Talk about will power. It is refreshing to know that is no longer the recommendation.\nChildren ages 6-17 should do 60 minutes or more of physical activity daily. Adults ages 18-64 should do at least 150 minutes a week of moderate-intensity, or 75 minutes a week of vigorous-intensity aerobic physical activity.\n*All the more reason that we need to keep P.E. in schools and moderate T.V., video game, and computer usage. It is increasingly important for our kids to \"get out and play.\"\nThere has been a lot of backlash with these new guidelines. Especially from mothers who want \"Specific Foods\" not only mentioned in the guidelines, but banned.\nIt seems to me that regardless of the guidelines, I am still the parent and I make the decisions. It doesn't matter what the USDA recommends it is still my job to do what is best for our family and for our kids. Guidelines or not. If they banned Happy Meals, I would still have the choice to let my kids eat them. Just like I have the choice now.\nThe confusion for me comes more with the school systems. I am not totally \"fluent\" on school funding, but if the USDA is talking about more and more physical activity, you would think the government would fight harder to save P.E., not cut it.\nDo you care about the USDA guidelines?\nHow strict are you with the food your children eat?\nDo you think the government should have a say in what we eat?\nOn our agenda tomorrow, the kids and I are going to make sugar cookies. Pretty sure I am not going to leave the salt out. They just wouldn't taste the same.\nI think the proposition for the USDA to ban Happy Meals or to take the toys out of Happy Meals is absurd. It's like parents are saying that they don't have the power to decide what their children are eating. Not to mention it's more than just the kids meals. If a child is eating way too many Happy Meals they are probably not eating very healthily at home.\nI have found that feeding my one year old has made me much more aware of what I'm eating. I like to feed him what we're having and I want him to be healthy. I have put more of an effort into including veggies and whole grains than I did before.\nWhitney, I totally agree with you. It is not the governments job to decide what our kids eat in the first place. As parents we should be smart enough to make those decisions for our own families. It seems that when we start banning anything, we have lost the democracy we worked so hard to establish.\nNo seafood at your house huh. That is pretty funny that your FIL is ready to stock your freezer. It makes it hard to cook when you can't stand it yourself. How about your husband and kids. Do they like it?\nSorry for the novel... I also think if I make my own cookies or granola bars or muffins then they are healthy. lol. At least there's no preservatives right?\nEnjoy your sugar cookies--you could even tint the sugar red for valentine's day, although that would probably add sodium, somehow. Lol.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a science fiction series. Unlike the Tlisli series, my plan is to post complete episodes, so the reading will be more regular, less choppy, and each entry will be longer.\nThe first episode was The Birth of Traitor Tad, which was written before I created this page. Future episodes can be found in the trackbacks from each story.\nThe story is set in a far future earth that has gone into space, but there is something very weird about the way the government operates. Some of that weirdness results in Tad becoming a traitor.\nNote: The featured image used for all Traitor Tad stories is a composite made up of a recolored image of an M-1 Abrams tank superimposed over a space sketch. Both are from Openclipart.org and I combined them for this story.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get the most out of your Over The Top textbook, Summit Salon Planners and 1-on-1 coaching sessions! This program is designed to help coaches and stylists use the Over The Top textbook as a stylist's encyclopedia. This edition focuses on Service scripts and actions that will lead a stylist to greater personal, professional, and financial success!\nTwo audio CDs featuring Michael Cole and Mary Cole reviewing the playbook content and offering tips and suggestions for building your service business.\n1-on-1 Coach's Notebook that clearly identifies the pages and exercises.\nCustom PDM Over The Top worksheet pdf that you can print out to track your progress!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enter your work address to calculate commute times to and from 1506 Blue Ridge Dr.\nAmazing value for this beautifully updated & maintained home! Park-like setting on a 1\/4 acre property with lush landscaping. Incredible living space with vaulted ceilings, fireplace, wood laminate floors, and French doors that open to a covered patio. The upgraded kitchen with a large pantry and stainless steel appliances is a real treat! Recent updates include AC system, roof, and septic pump. Solar water heater reduces electric bill. Enjoy no HOA, low property taxes, and acclaimed Leander ISD schools!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqtpa b/data_all_eng_slimpj/shuffled/split2/finalzzzaqtpa
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+{"text":"The Aeron office chair has been remastered from the casters up to support today's workerswith today's state-of-the-art seating technology.\nLife Insurance Companies Hate This: The great news is that in today's market,policies are actually very inexpensive and a lot more affordable than you may think.\nMay 1.3.6 New proposals 4.3.8 Timetable 5.5 Process calculations for the preliminaryprocess alkaline sizing. white of a regular iron ball mill.\nscreen greatly affect the basic capacity. All of these factors will be explored insubsequent paragraphs. vibrating screen sizing calculations - Gold Ore Crusher .\nLearn more about opportunities to work at International Paper, one of the most respectedcompanies in the world.\nAt Yard House enjoy more than 100 taps of the best American craft and import beers, plusa menu with more than 100 dishes made from scratch in our kitchen.\nPiping and Pipeline Engineering. Uploaded by. Daniel Balderrama. connect to download. Getpdf. Piping and Pipeline Engineering. Download. Piping and Pipeline Engineering.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We continuously review new models from our manufacturer partners, and bring the best to you!\nFor the latest on our mattress offerings, please re-visit our Mattress page periodically.\nFor example, we now offer the Lady Englander and Eclipse (with its exclusive 15-year, non-prorated warranty!) from Englander. We've also added the CoolTouch and Revolution from Diamond, the iComfort and iComfort Hybrid from Serta, and select models from Stearns & Foster.\nReminder: we have a big selection of custom-sized mattresses suitable for RVs, 5th wheels, mobile homes, campers, and other specialty applications.\nWe also feature memory foam products for mattress toppers or sofa beds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Kumon worksheets are designed to allow students to advance by self-learning. Nonetheless, there are times when students encounter problems that they don't understand or are unable to solve on their own. At times like these, Kumon Instructors do not immediately give students the answers to problems. Rather, our Instructors find out how much the students understand before giving them hints. Instructors may also show students example problems or previously studied material to help them solve difficult problems on their own. Kumon Instructors do not spoon-feed problem-solving methods. Instead, they focus on guiding students to complete exercises on their own so they enjoy a sense of achievement and are able to master new material independently.\nKumon Instructors discover what a student is capable of and bring out the potential in every one of them. Our Instructors focus on each student, paying careful attention to his or her academic ability and personality, and providing effective support to ensure that he or she can always study at a level that is just right. Kumon Instructors acknowledge the growth of each student without comparison to others, while offering praise and encouragement. At the same time, they cooperate closely with parents in monitoring the growth of each student. It is thanks to these dedicated Instructors, who sincerely wish for the development of their students, that the true benefits of the Kumon Method are achieved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since the spring of 2016 you can toboggan down the Gurten.\nThe new summer toboggan has been inaugurated on May 24, 2016 with an official opening ceremony. The toboggan offers around 500 meters of unforgettable tobogganing fun, even transforming into a tunnel in one spot. It negotiates a total drop of 55 meters. This gives the Gurten \u2013 Bern's very own mountain \u2013 a new summer attraction, making it an even more attractive destination for young and old alike.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Living the kind of life you want can start with a dream but should not remain just a dream forever. There is an amount of effort required in order to succeed as an affiliate or internet marketer just like any other endeavor in life. Online marketing is not a way of escaping the dedication or hard-work usually required to succeed or earn money. Patience is also a tool that comes handy when you are selling online, it takes patience to learn the basics and keep learning. Warren Buffett was quoted saying \"You can't produce a baby in one month by getting nine women pregnant.\" and I have found this to be very true. Most great things just take time to build even with consistent efforts.\nAffiliate marketing simply means selling products or services for a business while you as the marketer get paid (commissions) for each product you sell or each visitor or customer you bring to the business. Whether you are starting out as a newbie in online marketing or you have been at it for a while, you need to be strategic about it. There is a need to map out a line of action. Affiliate marketing is a wide subject and it is even becoming more wider by the day. Start by doing a research on the products you want to sell. There are tonnes of affiliate programs online today.\nAmazon affiliate program as one of the oldest affiliate programs I know. To get the best results from this program, you may need to have your own website. You will also need to drive a considerable amount of traffic to your website in order to make sales. Amazon being a leading and a trusted e-commerce leader with several thousands of successful affiliates already will not hesitate to kick you out of the program if you do not make sales with a specified period of time after you have signed up. My advice is that you must have been getting sufficient traffic to your website or online content before you sign even sign up.\neBay is another e-commerce giant that allows you to become an affiliate.\nJust like you would expect, eBay's affiliate program popularly known as the eBay Partner Network has also seen tremendous success over the years. You need to know your onions to succeed as an affiliate here too. It is important for you to know that as an online marketer, your work revolves around creating valuable content or products\/services, you also need to share your content and also keep doing these consistently.\nJvzoo is another place where intending affiliate marketers are always welcome. You can easily sign up and start promoting other peoples products to start earning money. Jvzoo claims to be one of America's fastest growing companies.\nI recently learnt about Warrior forum as a place where you can sell your digital products or earn as an affiliate. It is a forum website and claims to be the #1 Digital Marketing Forum & Marketplace.\nCPA lead makes it even juicier to be an affiliate. You can sign up as a publisher and join the CPA Lead Generation Network where you can promote PPC offers or PPA offers and start raking in your earnings.\nThis is very attention-grabbing, You are an overly skilled blogger. I've joined your rss feed and stay up for in quest of more of your excellent post. Additionally, I've shared your web site in my social networks!\nBe first to learn about new opportunities!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Pages A Colorier Soccer On Pages A Colorier Soccer In Pages De Coloriages Dindigulz , You can Download this images for free in high resolution. Pages A Colorier Soccer On Pages A Colorier Soccer In Pages De Coloriages Dindigulz was posted in September 15, 2018 at 6:44 pm and This Pages A Colorier Soccer On Pages A Colorier Soccer In Pages De Coloriages Dindigulz images has viewed by 12 users. We selects the best choices together with best resolution exclusively for you all, and this images is one of graphics libraries in our very best graphics gallery in relation to Pages A Colorier Soccer On Pages A Colorier Soccer In Pages De Coloriages Dindigulz. I hope you'll like it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are three ways Square is breathing new life into what has been, at times, a technologically-stagnant industry.\nFrom its simple and innovative founding in 2009 as a way for businesses to accept card payments through mobile devices, Square Inc (NYSE:SQ) has evolved into a comprehensive payments solutions platform.\nWhile the company is not yet profitable, most of Square's metrics show consistent growth across its many platforms. In the company's most recently reported quarter, Square's adjusted revenue came in at $178 million, a 51% increase year over year. The company's gross payment volume (GPV) was $13.2 billion, a 39% increase year over year.\nThis growth is more impressive given the highly competitive nature of the payments solution industry. After all, shortly after Square introduced its innovative mobile payment hardware, other major payment solutions companies followed with their own mobile card readers. If Square is to keep growing revenue at double-digit rates and ultimately turn a profit in its low margin, competitive industry, it must continue this culture of innovation.\nOne of the most common complaints with the shift to EMV chip-embedded cards has been from merchants complaining about longer transaction times. While both Visa and Mastercard have released new software designed to quicken this process, it is still a real concern for retailers. After all, the faster customers move through a merchant's line, the more customers can be served and the happier those customers will be.\nThis is why Square has invested so much in reducing the amount of time it takes merchants to facilitate transactions using Square's platforms. According to Square's 2016 third quarter shareholder letter, the reported industry average for processing an EMV transaction is 8-13 seconds. Square's previous EMV transaction speed was 5.7 seconds. But in the same shareholder letter, Square announced it had now reduced its own transaction speed to 4.2 seconds. That's 25% faster than its own previous speed and well ahead of the industry average.\nAccording to a study from CB Insights that examined the postmortems of 101 failed start-ups, the second major reason these young companies failed is because they \"ran out of cash.\" Most entrepreneurs understand there could be a good deal of time from when they first open their doors to when they first turn a profit. Square Capital is a program designed to help fledgling businesses bridge this gap.\nSquare Capital is a service in which Square offers business loans to select clients who then pay the loan back gradually, as a percentage of transactions. Because of the immense amount of transaction data Square collects from its customers, it can use a high amount of discretion when offering these loans.\nSo far the strategy appears to be working. Square Capital has been able to maintain a 4% loan loss rate in every single quarter of the year, the national default rate for small business loans is more than twice as much. In Q3 2016, Square processed over 35,000 loans totaling more than $200 million, an approximate 70% year-over-year increase.\nIn March 2016, the company launched Build with Square, a developer platform that allows sellers across the Square ecosystem to customize their POS (point of sale) checkouts, access other Square services such as Square Capital and Square Payroll, track inventory, access data analytics, and track online and offline sales.\nIn the first quarter alone, Square saw 135,000 integrations through Build with Square. Square believes allowing this type of customization will attract large business customers, who tend to have more sophisticated needs.\nIn an industry where it is usually a race to the bottom on pricing, any company that offers new solutions and services to its business customers stands a real chance to differentiate itself. This, in turn, will attract more customers even while demanding higher fees for its services.\nThis doesn't mean Square has a monopoly on progress in this industry. For instance, PayPal Holdings Inc's (NASDAQ:PYPL) Braintree platform is also offering a customizable checkout experience for its customers as well. But if Square continues to foster this culture of innovation, the company's investors will continue to see revenue growth, and even real profits, in the future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The perfect calorie intake would certainly be 1500-1700 calories a day. This suggests that you will certainly need to reduce all the excess calories. Adhere to core nutritious and coarse diet regimen which contains minimal quantity of fat. Lower the size of your meals and also decide for small healthy and balanced snacks.\nThe primary culprit for those excess calories is fast food. The fires and sodas include in one of the most of calories. Unnecessary fat accumulates with the usage of scrap food. Get rid of fast food or any type of fried or processed food from your diet plan. It does not have any type of dietary value either.\nAdding fresh vegetables and fruits to your diet will accomplish the demand of minerals and vitamins in the body. Apples, avocados and fresh eco-friendly leafy veggies are a must. They are high in fiber and they maintain you full for a lengthy time.\nSnack on completely dry fruits, nuts, low fat yoghurt as well as fruit smoothie mixes. They are a much healthier alternative to diet sodas which are high in calories. Snack on fruits as well as roasts when you are hungry. This will certainly keep those extra calories away and also will replenish the lost energy and required nutrients in the body.\nBe sensible when you aim to take in liquids. Choose liquids that are reduced in calories. Drinking your calories will only ruin your initiative to slim down. Green tea, sugar cost-free juices as well as milk ready alternatives. Consume a great amount of water as it assists to burn calories and will certainly likewise purge contaminants out.\nAlcohol is high in sugar as well as calories. You will get back the majority of just what you burnt throughout the day, in a fix or more. Alcohol will certainly also slow your metabolic process down. No issue just how much they interest you, state no to them.\nThere are foods that help to enhance your metabolic process. Almonds, apples, avocados, entire grains, lean healthy proteins, chilies are recognized to have a heated result on the metabolic rate. This effect is called thermogenic impact. So chomp on these to surge your metabolism up.\nBreakfast is the most essential meal of the day. It makes up for the lost power and also nutrients. Avoiding morning meal will certainly make you intake extra food or binge in your following dish. It can lead to the growth of stomach problems as well as different other digestion concerns. Eat bowl of cereals with skimmed milk and include a fruit to the meal.\nIt becomes really essential to sustain this diet regimen over time for visible outcomes. If you stop the strategy when you have actually lost a few extra pounds and also go back to your previous diet you are most likely to put on weight faster compared to the actual price. Stay with your core diet plan, it is okay to delight in your preferred calorie-rich treat every so often, however make sure you have actually melt it the next day or managed further calorie intake. The very best method to reduce weight is to comprehend and regulate what does it cost? food your body demands.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the late 19th century and into the turn of the next century, the Cram company did a great job publishing maps that were clear, detailed, and up to date. This map for this area is a good example. Shows a the Railroads of the World and the United States.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"February 17, 2016 \u2013 [Los Angeles] CineConcerts and Warner Bros. Consumer Products announced today the world premiere of The Harry Potter Film Concert Series, a new concert tour celebrating the Harry Potter films. The tour will feature Harry Potter and the Sorcerer's Stone and will kick off this summer, travelling to destinations across North America. The world premiere will take place on June 23rd, 2016 at The Mann Center with the Philadelphia Orchestra performing. Tickets go on sale Saturday, February 20th at 10AM via MannCenter.org, Ticketmaster.com, 800-745-3000, or the Mann's Center Box Office. Soon thereafter the tour will appear with such orchestras as the San Diego Symphony, Nashville Symphony, Milwaukee Symphony Orchestra, Grand Rapids Pops and Symphony Silicon Valley.\nThe Grand Rapids Pops welcomes Harry Potter and the Sorcerer's Stone to DeVos Performance Hall next season with a full-length screening of the 2001 film, featuring a live performance of John Williams' film score played by your Grand Rapids Symphony. Two Special Event performances are at 8 p.m. Friday, January 27, and at 3 p.m. Saturday, January 28, 2017.\nTickets now are on sale to Grand Rapids Symphony season subscribers or to new subscribers to any of the symphony's nine concert series. New for the Grand Rapids Symphony's upcoming 2016-17 season are subscriptions available at discounts of up to 50% off of select series and seats for new package orders. Go online to GRSymphony.org for more information.\nGrand Rapids Symphony season tickets are on sale now with select concerts also on sale to subscribers. Go online to GRSymphony.org\/packages for details. Tickets to Harry Potter and the Sorcerer's Stone are available now with a season package purchase. Becoming a part of our concert family by purchasing a ticket package saves you up to 35% off of single ticket prices. Plus if you add-on an additional, new series, you can save up to 50% off that series.\nSingle tickets will be available beginning Aug. 1. Season subscribers have the opportunity to purchase individual tickets before they are available to the general public. Subscriber courtesy dates for the 2016-17 season take place July 25-29.\nGrand Rapids Symphony Ticket Services office at 300 Ottawa Ave. NW Suite 100 is open 9 am-5 pm Monday through Friday. Call the Grand Rapids Symphony box office at 616.454.9451 x 4 during regular business hours or evenings at 616.885.1241.\nThe Grand Rapids Symphony was officially organized in 1930 and is nationally recognized for the quality of its concerts and educational programs. Led by Music Advisor Larry Rachleff, Principal Pops Conductor Bob Bernhardt and Associate Conductor John Varineau, nine concert series plus special events are presented, featuring a wide range of musical styles.\nMore than 400 performances are presented each year, touching the lives of some 200,000 people. Nearly half of those are students, senior citizens and people with disabilities, reached through extensive education and community service programs. The Symphony's affiliated organizations include the Grand Rapids Bach Festival, Grand Rapids Symphony Chorus, Grand Rapids Youth Symphony and Classical Orchestra, and Grand Rapids Symphony Youth Choruses. The GRS also provides the orchestra for Opera Grand Rapids and the Grand Rapids Ballet.\nTo learn more about the Grand Rapids Symphony please visit GRSymphony.org. Follow the Grand Rapids Symphony on Facebook and on Instagram.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Sinopec's Yuanba gas field has produced over 13 billion cubic meters (bcm) of natural gas as of March 28, 2019.\nSinopec announced on March 25 it has made another major breakthrough in shale gas exploration and development, following the discovery of Fuling shale gas field.\nSinopec issues its \"2019 China Energy and Petrochemical Industry Development Report\" in Riyadh, Saudi Arabia on March 25.\nThe Project of Intersection of King Khaled Road with King Faisal Road located in Khamis Mushait in southwest Saudi Arabia, built by Sinopec, has been officially opened to traffic.\nBy the end of 2018, with SSC's $6.48 million investment, 215 local students have participated in the training.\nSemgaliev Alibek is the deputy manager of SZK. He is very responsible and circumspect at work and treats his job wholeheartedly.\nField works are indispensable inpetroleum exploration industry,we have to go to places that were remote and desolate and sometimes face unknown risks and danger.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Private and big lovely terrace \/ garden from the living room with nice charming views and pool with chill out area.\nA full bathroom on the main suite and full bathroom for the guest.\nLovely new kitchen with new furniture and machines.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please log in below to access this page.\nIf you have forgotten your password, click here to recover your account.\nIf you don't yet have a Cbox login, go ahead and create an account.\nIf you are repeatedly returned to this page after logging in, ensure that you have cookies enabled in your browser.\nThe Cbox Control Panel is where you go to get your Cbox installation code, where you configure its settings and maintain user accounts and messages. Everything you need to run your Cbox effectively is easily accessible from this Control Panel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thanks to a special reader, we found this calendar for my 4 year old. She is really into Barbies right now, and we think she will have a lot of fun with this calendar.\nThis is similar to what we purchased for my 10 year old last year. She loved last year's bracelet and necklace, so we are hoping she loves this one as well. Last year, we paid a VERY low price, and this year, this was the best price we could find.\nThis last one is kind of a holiday tradition. We have purchased the Lego Star Wars calendar each year for the past three years. My 7 year old son loves it each year. This can be a hard item to find right before the advent season, so if you want to purchase one, I would suggest doing so early.\nWe cannot wait to share all of the treats we find with you! When the boxes arrive, we will give an update!\nThis post contains affiliate links. MamaBreak may earn a commission based on click thrus on this page. This will not impact your shopping experience. As always, we only recommend products we think our readers will love. We thank you for supporting MamaBreak!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unit Description: Microbiological quality control to match the Espoo, Turku and Kuopio plant materials, packaging materials and products, as well as the process time and water samples for microbiological examination.\nAlso responsible for the production of these factories and personnel hygiene, as well as the control of the processes of microbiological validation. The department is also responsible for product development, as well as work related to the products manufactured by the Fermion study.\nJob description: Microbiologist is responsible for the production of medicinal facilities for microbiological control, its execution and reporting. He is also responsible for the Turku and Kuopio plant environmental control, monitoring, and control of the Fermion environmental samples. Is also responsible for personnel pukeutumisvalidoinnit. In addition to work with a production staff hygiene training. Problem is temporary (maternity leave substitute) from 01.30.2013.\nRequirements: \u2013 Microbiologists education and 2-3 years of experience \u2013 good oral and written English skills \u2013 teamwork skills, flexibility and initiative to provide a comprehensive and independent work.\nJob Categories: Health Jobs in Finland. Job Types: Full-Time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..f7615c538f81af8d012ae3f8d0e4688910745ab4
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Bombardier Transportation and Alstom have jointly received an order worth $340m for the delivery of 153 metro cars to Soci\u00e9t\u00e9 de transport de Montr\u00e9al (STM), a transport agency based in Canada.\nMorocco has opened the Tangier-Casablanca high-speed rail line that will connect the industrial and commercial hubs of the country.\nSwiss rolling stock manufacturer Stadler has delivered the first bi-mode train to UK train operating company Greater Anglia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The team at Specsavers in Hucknall are upskilling to better support customers with glaucoma.\nAs part of the store's support for the World Glaucoma Week, members of staff have undergone specialist glaucoma training in store. In conjunction with the International Glaucoma Association, Specsavers has developed the training for its retail teams to assist sufferers with their treatment. Specifically, they will be advising on how to effectively administer eye drops, which are crucial for managing the condition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Everyone loves Bacon!! Prepared with all natural slowly cooked bacon and cheddar cheese these biscuits are sure to please.\nIngredients: Organic oat flour, organic rye flour, bacon, cheddar cheese, honey, eggs, organic flaxseed, canola oil.\nPlease provide the delivery date in the \"Personalize Your Product\" box.\nYour dogs will go crazy for these succulent salmon packed wedges full of essential omega 3 & omega 6 oils. They're crunchy and cheesy and oh so delightful! Our dogs give them two paws up!\nOur All-Natural CBD Hemp Oil is perfect for all sizes of dogs as the dosage can be easily regulated. Each bottle has an included dropper with each drop being 1mg of CBD.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"City officials broke ground Thursday on what will become a new Buffalo Wild Wings restaurant in the Oxbow Landing development at Interstate 80\/94 and Kennedy Avenue.\nMayor Thomas McDermott Jr. said when he became mayor the area was a part of the city his administration wasn't proud of, calling it a site that caused a disproportionate number of problems for city services.\nThe city purchased and knocked down the apartments and sat on the property waiting for the market to recover after the recession hit.\nThe $20 million development project is getting a lot of interest from developers now, McDermott said.\nConstruction will begin soon on a new Hampton Inn just south of Buffalo Wild Wings. McDermott said the city is also working with a developer on \"a big time project\" for Oxbow.\n\"It's something we're really hoping comes through,\" McDermott said.\nThe mayor also said the sound walls along the expressway next to the development will be coming down to make the site more visible.\n\"It's one of the most vocal points of our city and one of the highest traffic counts area of the city,\" he said.\nThe site is in City Council President Michael Opinker's 5th District, which is why he was happy to give $500,000 a year of the gaming revenue he receives towards the bond for the project.\nHighland Redevelopment Director Cecile Petro said her town loaned about $2 million for the Oxbow Landing project, which Hammond paid back last year.\nBuffalo Wild Wings regional manager Jacob Miller said the restaurant will feature its latest, modern \"stadia\" concept. The inside of the restaurant will resemble the inside of a sports stadium, he said.\nMiller also said Buffalo Wild Wings likes to be involved in the community and they look forward to helping out local schools, sports teams and with fundraisers.\nThe restaurant is scheduled to open in October.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recently at Viget, we've made a number of Client Side Applications in the browser. These applications typically use React.js, a flux framework microcosm, or redux.js and communicate with a server based API for data persistence. With Electron, we can take the exact same Client Side Application code, put it into Electron framework, and now we have a native Desktop application. Imagine the possibilities!\nWe've created a few Desktop applications and are looking forward to building more using the tech we already know and love.\n2. Electron Homepage by GitHub If you can build a website, you can build a desktop app.\n3. Building Web Desktop Apps With GitHub's Electron by Burke Holland via DZone Yet consider what the web is now capable of: Camera Access, Microphone, Audio Manipulation, File System (to a degree), Notifications, WebSockets, Accelerometer\u2026and the list goes on.\n4. Can you help us build a native desktop Ghost app? by John O'Nolan via ghost It was immediately striking how much more attention the app commanded psychologically by being both in the dock menu, and visibly separated from the browser.\n5. GitHub releases Electron 1.0 for desktop app developers by Paul Krill via InfoWorld Electron has been downloaded more than 1.2 million times in the past year and used to build applications ranging from email to SQL analytics tools and the Slack communications platform.\nWe believe that the best work happens when talented people trust each other enough to be real and take risks. We're passionate, curious, driven, and committed to doing great work. Join us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatzsr b/data_all_eng_slimpj/shuffled/split2/finalzzzatzsr
new file mode 100644
index 0000000000000000000000000000000000000000..e80cd0baa28e5d55304d04fd96e31c0fde6e33fa
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Put beans or Lentils in a stock pot. Cover with water, bring to a boil. Lower the heat and cover; cook slowly for 30 minutes. Pur\u00e9e the beans.\nPeel the garlic. In a clean pot, warm the oil, cook the garlic, and add the tomato paste, cumin, and harissa. Add the bean pur\u00e9e, add water or broth to a soup consistency, and season with salt and pepper. Heat to just below boiling. Serve with extra harissa on the side.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PLUSH SLIDES WITH STANDOUT 3-STRIPES STYLE.\nSlip into these women's slides for easy-to-wear comfort after a hard workout or while hanging out around the house. A contoured footbed gives these sandals a super-soft cushioned feel, and the bandage-style upper shows off glittery 3-Stripes for a glamorous touch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Luxury Steam Showers and Shower Enclosures Benefits and Features.\nThere are many different styles and types of showers that are suitable for bathrooms and homes alike.\nMost Steam showers have seats in them, and the ones that do usually have one or two seats. steam rooms come in an array of sizings to accommodate most bathrooms found in the uk from large to small.\nCorner Luxury Showers Steam showers are manufactured to many different specifications as customers have different spacing requirements, therefore steam showers offered can be made for corners. Showers which are designed for corners are generally free standing and can be pushed into the corner. Steam showers which are designed to stand against a flat wall are often rectangular and are also free standing.\nSteam showers can also have a whirlpool or standard bath integrated into the unit. Instead of a shower base, a whirlpool bath with hydro massage jets is used. The user can have the option of bathing in the bathtub, using the shower, or the steam, making the bathtub steam shower the ultimate bathroom centrepiece. Some showers have a whirlpool bath are generally larger than showers with a shower base.\nTo find out more visit newworldbathrooms.co.uk .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Back in the 1990's, Midnight Musicals were very prevalent in the black church. Quite honestly, they're probably still popular in most sects, but I am just not apart of that crowd anymore. With that being said, only a special group of people can have not only myself, but 3,500 other people, excited to get dressed at 10pm to go to a midnight musical on a Tuesday night. Hezekiah Walker, Donald Lawrence and Ricky Dillard, known to the gospel world collectively as The Brat Pack, is that special group of people.\nSurprisingly, the night started out with all three gospel icons entering the stage with their repsective choirs, Love Fellowship, Company and New G. \"The Blessing of Abraham\" set things off and the crowd was definitely energized and excited to hear Donald go down memory lane but just as the song ended, Ricky Dillard appeared front and center as the combined choir started belting out \"I May Not Have This Chance\" from his \"Live in Toronto\" album. Once Hezekiah Walker, clad in the hottest Red, White and Blue Gucci kicks, directed the choir with his hit, \"I'll Be Satisfied\", it was evident that this was going to be a friendly song battle of sorts, as each Brat Pack member went head to head for at least nine straight songs from songs from teh classic albums. I hope that YOU, the reader, realizes that these future legends collectively have over 60 years in the gospel music industry and it was evident as they each went through their classic material such as: \"The Best is Yet to Come\", \"Work It Out\", \"I Will Go In Jesus Name\", \"Hallelujah\", \"Bless Me Indeed\", \"Calling My Name\" and many others. The crowd literally stood for at least an hour straight and at times it seemed as if it was hard to get a praise in the room just because most people were simply in awe.\nHalfway through the set, Donald Lawrence asked the crowd if they wanted to see a Brat Pack reunion album come to pass and the crowd immediately obliged him! Hezekiah Walker also spoke about how The Brat Pack was literally birthed out of the GMWA as they all got their start at the conference well over two decades ago. He went on to say that they had not all performed as The Brat Pack since the passing of the fourth member of the brotherhood, O'landa Draper.\nThe night would not be complete without paying homage to Draper, and pay homage they did. Mr. Energy himself, Ricky Dillard, led the choir into one of Draper's classic hits, \"My Soul Doth Magnify The Lord\" and the crowd literally went crazy! Acapella never ever sounded so good! \"Do Well\u2026 Do Well\u2026 Do Well.. Do Well.. A Woop Woop\", never sounded so good! \"I Gotta Feeling\" and \"I've Got A Reason\" followed, to complete the tribute portion of the concert but \"My Soul\" was clearly the show stopper.\nAt this point, we are about two hours in and one would think that the concert is almost over but its not even close, as the collective choir exits stage left and we were left with the incomparable Donald Lawrence and Company. We all know that Donald Lawrence probably has the best singers on the planet and it was confirmed when two of his singers sang the Chaka Khan classic, \"Through The Fire\". That was the first time it occurred to me that this song could very well be a gospel song which further reminded me how much of a musical genius Donald Lawrence really is. Naturally, D. Lawrence & Company did songs from \"The Law of Confession\" album and songs like \"The Blessing Is On You\", \"King In You\" and \"Back To Eden\" blessed the crowd tremendously. Ricky Dillard and New G perfomed many of their classics but when Ricky unselfishly asked the crowd what they wanted to hear, \"One More Chance\" led by star alto, Lillian Lloyd, was the clear winner. Of course, Lillian showed up and showed out as her adlibs are like no other. At times, it was like watching a Broadway play as she danced her way across the stage and willingly choked herself with her choir robe sash to get her point across that God saved her from the dirty things that this world had to offer her. However, she is now free and she made everyone in the room a believer! It's no wonder this song is such a You Tube favorite! Veteran of the group, Hezekiah Walker and Love Fellowship ended the set with crowd favorites like \"Faithful Is Our God\", \"Grateful\", \"Clean Inside\" and the # 1 song, \"Souled Out\". At one point LFC alum, Kirvy Brown even came out of retirement and tagged a song like only Kirvy Brown can.\nOverall, there is nothing bad you can say about this reunion. It was well worth the $10, $15 and $20 it cost for admission. Kudos to Verizon \"How Sweet The Sound\", James Robinson and Bishop Albert Jamison for putting the show together as it was so nostagic to see these great men of God come together again. If you missed it, you missed a treat but hopefully we will all see that reunion album come to pass. No egos, no competition, just good old gospel music at its finest!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Imperial Exports India is Supplier of Choco Brown Granite. We have many types of Granite are available. Choco Brown Granite is available for any thickness, size and color. We offer a wide and exclusive range of Choco Brown Granite, which is widely appreciated by our clients. We provide best granite our clients. We are one of the leading manufacture and exporter of Choco Brown Granite slabs.\nImperial Exports India is supplier and manufacture of Choco Brown Granite. We provide many types of Granite like Cherry Brown, Choco Brown, Coffee Brown, Copper Silk, Desert Brown, G D Brown, Icon Brown, Paradiso, Rolite, Sapphire Brown, Tan Brown, Tiger Skin, Grey Granite, Bala Flower, Sadar Ali Grey, Indian Granite, Sierra Grey, and Steel Grey Etc.\nImperial Exports India offers best designing in Granite and its brown granite available in various designs and sizes. We offer affordable rates in provide granite in various patterns. They are Supplier of Black Granite and manufacturing designer stone slabs of granite in various styles and their granite stone is supplied in various cities of United State.\nImperial Exports India is exporter and supplier of Granite with affordable Price. We offer Perfect Brown Granite is an ideal choice for any countertop. We offer a wide range of Choco Brown Granite for kitchen countertop, bathroom, home, garden, floors and work surface. It has been used for thousands of years in interior and exterior applications. Our granite is demanded in New York, United State.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauldk b/data_all_eng_slimpj/shuffled/split2/finalzzzauldk
new file mode 100644
index 0000000000000000000000000000000000000000..a67ecd2f7cb61e663d81209e30f23b5ba8c0194c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzauldk
@@ -0,0 +1,5 @@
+{"text":"When it comes to competition BBQ, I have always been somewhat of a purist. I say, \"somewhat\" because I do inject my pork butts and brisket, and I am a firm believer in leveraging \"the Texas crutch\" (foil). We competitors are constantly striving to achieve that perfect bite (or maybe two) that will wow the judges, and you have to use chemistry to your advantage.\nIn my mind there are two kinds of chemistry -- natural and enhanced. The natural uses preparation and cooking techniques that make use of what is already in the meat to help it the be best it can be. The enhanced end of the spectrum adds a concoction of additives (typically containing sodium tripolyphosphate and calcium lactate) that help enhance the natural flavor and retain moisture. Many of the top competition teams use these types of commercial injections to gain an advantage.\nUntil now, I had never tried any of the commercial injections, but I've been admittedly curious to see what all the hullabaloo is about. I've heard about the great beef flavor and juices pouring out of an otherwise monotonous brisket. All this, combined with the results of the aforementioned top teams, certainly made me wonder if I was missing something.\nI decided that it was time to see what all this buzz is about, so I contacted Kosmo of Kosmo's Q and he kindly sent samples of his beef and pork injections for me to try.\nI started with an 11.3 pound select-grade brisket. I wanted to keep the playing field level, so I trimmed it like I normally do, and I injected it per the instructions, using beef consomme as the mixing liquid. The powder mixed very easily without clumping, and I injected all but about 1\/4 cup of the mixture.\nAside from the new injection formula, this was a standard brisket prep for me. I then cooked the brisket using my standard hot & fast technique. Again, I wanted to keep the injection as the only variable in my standard procedure.\nI must say that I was impressed with the results. While I didn't see much difference in the moisture content, the beefy flavor was definitely enhanced, but not unnaturally so. There were no off flavors or other noticeable changes in the overall flavor profile, as compared to my standard competition brisket. It was just a \"kicked-up\" version of what I normally cook. Kosmo's is a serious winner in beefy flavor enhancement.\nNow that I know what all this phosphate injection madness is all about, I don't know that I would use it in competition. Why? Well, it contains MSG and, while I don't necessarily have a problem with it, I am very leery about using MSG in situations where I don't know who will be eating it. I prefer to keep things natural in these situations.\nIn conclusion, Kosmo's Q is a great product for those that are looking to take their brisket to a new level. It offers outstanding flavor enhancement without compromising your flavor profile.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We all know there are some occasions where we can not miss work or school and must go even though we are sick. There are also times when your child may be well enough to go back to school, but still has a lingering cough.\nHere are 7 tips that you must follow to stop the spread of viruses, because we all know what goes around comes around.\n1. Stay home and rest if you can. No sense in showing up to work or school and getting everybody else sick. Your body is telling you to rest and if you don't listen the symptoms will get worse.\n2. If you feel well enough to go to work or school then let people know that you are getting over being sick so that they don't run in for a big \"germ hug\" or a \"snot hand shake\".\n3. Wash your hands more than you think! I always carry an alcohol free hand sanitizer in my bag, because you never know who or what you are going to run into.\n4. If you have to cough, cough into your elbow not your hand! You use your hands for everything, so you will be continuing to spread germs everywhere, by coughing into your elbow you minimize the potential to spread germs.\n5. Don't touch your face! All germs are spread through your mouth, eyes, and nose. If you have to blow your nose or cough repeatedly, then you should be home resting.\n6. I know we were all taught to share, but not when it comes to what touches our mouth. It is best to have your own chap stick\/lip gloss and stop sharing drinks and food. Another reason is, it is said that 80% of the American population has the herpes virus. Cold sores can be contagious up to two days before they even show themselves, then contagious for up to 2 weeks. The cold months when everyone's immune system is low is a perfect breeding ground for an out break, SO STOP SHARING!!!!\n7. Be conscious and mindful. Take responsibility for your illness and do everything in your power not to give it to someone else.\nSix years ago I was diagnosed with rheumatoid Arthritis and one of the medicines that I had taken on and off was an immunosuppressant. When I was on immunosuppressants my immune system was compromised and I became more susceptible to getting sick. My doctor advised me to try and stay 10 feet away from people who are sick. I realized how challenging that was on a day-to-day basis. At the same time I was diagnosed with rheumatoid Arthritis I was a single Mother and if I would get sick then my son would most likely get sick and it would turn into days of missed work. These experiences have made me very aware of people's \"sick manners\" and I have become a pro at avoiding them. I hope to help people become more mindful and compassionate of who they could be affecting when they are sick in public.\nFake it Till You Make It\u2026or Authentic Creation?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MedShape, a medical device company, has obtained 510(k) approval from the US Food and Drug Administration (FDA) for its ExoShape femoral soft tissue fastener for use in soft tissue graft fixation on the femoral side of the knee joint during anterior cruciate ligament (ACL) reconstructive surgery.\nThe ExoShape femoral soft tissue fasteneris is made out of MedShape's proprietary shape memory PEEK Altera polymer. The system offers dual locking modes in cancellous bone and at aperture to achieve complete anatomic ACL reconstruction.\nMedShape claims that ExoShape is the only all PEEK device that offers non-rotational aperture soft tissue graft fixation on both the femoral and tibial sides to achieve complete anatomic ACL reconstruction.\nExoShape complements the tibial version, which was launched by MedShape in 2011. The implant includes several features that are designed to accommodate the femoral anatomy and loading conditions.\nThe implant features a tendon loop for shuttling the soft tissue graft into the bone tunnel, allowing for at least 20mm of graft to be placed inside the tunnel.\nTo provide secure fixation, the implant features two locking modes that work in congruence. These locking modes include shape memory wings which deploy and lock into cancellous bone, and expandable block lock technology that provides uniform graft compression and aperture fixation.\nMedShape CEO Kurt Jacobus noted the company has discovered that orthopedic surgeons require speed, simplicity and performance in their devices.\n\"Using our shape memory technology, we have designed an ACL fastener that can be implanted quickly and easily, while still offering excellent fixation,\" Jacobus added.\n\"Our shape memory technology also allows our implants to expand and compress the graft without inducing damage or rotating the graft. The addition of ExoShape Femoral now allows us to provide these benefits to both sides of an ACL reconstruction.\"\nMedShape will launch the ExoShape femoral soft tissue fastener in February 2014. The implant is available in sizes 8-11mm and 21mm in length.\nIt will come pre-loaded on a delivery system with a flexible rod that enables simple tendon shuttling and easy navigation across the joint for both transtibial and anteromedial approaches. The flexible rod and non-rotational deployment mechanism facilitates non-line of sight delivery along a curved path.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Astroturf is not at a testing phase anymore. After decades of experimentation and technological development, today it is clear that synthetic grass has succeeded and will remain in History as one of the main achievements to democratise football.\nA simple walk around any city in Spain is enough to realise what astroturf means for football nowadays. Where dusty pitches used to be, now a shining green resists any kind of weather. Thanks to new generation technologies and materials, the revolution of the plastic fibres allows today many football teams to enjoy excellent pitches at a reasonable price.\n\"To set up these kind of pitches is considerably cheaper than 20 or 30 years ago\", explains Gabriel Rodr\u00edguez, the Facilities Manager of Marcet: \"During de eighties it was a huge investment. Now it's still a big expense, but nothing compared with then. And it's a great investment\".\nReasons are simple. \"Natural grass can be used, at most, a couple of hours per day. Even less. Use it more than that amount of time means to destroy it gradually until it becomes a potato patch\", says Rodr\u00edguez: \"Our facilities are used between 8 and 10 hours per day, which means that natural grass is not an option\".\nLittle by little, astroturf has replaced almost all dirt pitches, while natural grass has become an option only feasible for premier leagues stadiums. A luxury for clubs that can leave their facilities without use for 90% of the time. An option less and less chosen by teams. Especially considering that astroturf is becoming more a more alike natural grass.\nSynthetic pitches are almost a must in cold climate or rainy countries. Not only Russia, Sweden or Norway, where professional clubs already use it in their stadiums. Also in Scotland, where more and more voices advocate for this kind of grass. \"You don't get perfect grass all the time in Scotland so it seems like a good thing, artificial pitches\", recently said Ronny Deila. During his time as Celtic's coach, the Norwegian proved himself in favour of setting up synthetic grass in the Scottish Premier League. \"If you are going to develop talent, it's so important that you have good surfaces \u2013 that way you can lift your eyes up and look for a pass, instead of worrying about the ball bouncing away from your feet\", said at an interview with 'The Telegraph'.\nDeila's words were scandalous for a part of the scottish football, but he did say that after a particularly harsh winter, which forced the league to postpone many Premier League matches because of the bad conditions of the natural grass. Not even Hampden Park, the Natioal Team's Stadium, was capable of resisting the severe weather. It's natural grass has been renewed four times in only seven years.\nIn England, most of Premier League already have hybrid grass, where plastic fibres stick on natural grass. But, in other latitudes, Premier and Second division teams have already chosen a 100% synthetic option, with the UEFA approval. It's the case of Novara Calcio in Italy, BSC Young Boys and Neuch\u00e2tel XFC in Switzerland, Red Bull of Salzburgo or the Boavista FC in Portugal, although the Portuguese team was forced to change back to natural grass during the season 2015-16, after other clubs \u2013 led by the FC Porto \u2013 refused to play on artificial grass; a token of the ongoing debate regarding this kind of surface.\n\"We haven't yet achieved for the sensation to be the exactly the same, but we have come a long way\", says \u00c1ngel Gonz\u00e1lez, Naturf engineer, one of the leading companies on astroturf: \"When you design artificial grass it's key to take into account the sensations of the player, such as the impact absorption, the energy restitution, the rolling and bouncing of the ball, the rotation resistance\u2026 Today we can achieve very similar parameters to those of a natural grass pitch\".\n\"Nowadays we have almost matched the natural grass regarding density, pressure and thickness\", confirms Carlos Mendoza. According to the Physical Preparation professor at the Marcet Football University, the risk of an injury in an astroturf pitch has been significantly reduced lately: \"Before there were many cruciate ligament distention, since there was no movement for the feet when stopping, therefore articulations suffered absorbing everything that the terrain wasn't able to absorb. But now is just a matter of a proper warm-up to prevent this kind of problem\".\nAlso, sport equipment companies have started to design boots capable of preventing these kinds of injures. \"There are several kinds of studs, of different density\", explains Mendoza: \"You'll never use the eight studs boot that you'll use in natural grass during a rainy day, with mud. Boots for astroturf always have more studs, since these are designed for a more homogenous kind of surface\".\nOne of the main advantages of the synthetic terrain is, precisely, its consistency. In natural grass, the step is random, since the surface moves more and tends to get dented in some areas, while a good astroturf will always be homogenous. A key characteristic in order to avoid injuries.\nThis doesn't mean that synthetic grass doesn't need any kind of maintenance. \"You'll always need to do some specific conservation works\", says Gonz\u00e1lez. The Naturf's engineer explains that it's advisable to wet a bit the pitch before the arrival of the players: \"Water reduces temperature, acts as a lubricant that reduces abrasiveness, minimizes the fibrillation and also helps to clean the grass. This is not necessary during winter in humid areas, but during summer and in dry areas is unavoidable\".\nMaintenance of the astroturf goes beyond physical risks. \"It is very important to do an inspection every week and brush the pitch so the fibre stands\", explains Rodr\u00edguez. \"Also, once every three months you have to do a more intense maintenance. At Marcet we hire a company that spreads rubber all over the pitch and to make sure that the white lines are firmly attached and the fibres are not broken\".\nThe key maintenance works are a must to ensure good conditions of the grass during the years. A well-maintained pitch can last between eight and ten years in perfect conditions, depending on the climate and the quality of the filaments. The measurement unit for these filaments is the decitex (dtex), which expresses the density or the lineal mass of a fibre. Most of the current artificial grass pitches have between 12.000 or 16.000 decitex. The more decitex, better the fibre and the resistance.\nBesides filaments, there are other elements to consider to check the quality of astroturf. Fibre is only the most superficial layer and hides what lies beneath: small rubber pieces and silica. \"These layers are the ones that soften and should stay at a determined level so the impact absorption and restitution properties remain constant, explains Gonz\u00e1lez.\n\"To assess synthetic grass, you should use the Lisport Test, which measures the wear originated by stud-alike cylinders that imitate the step of a footballer. FIFA requires the grass to hold at least 20.200 cycles, which equals around a 5 years' life for the astroturf. The one at Marcet reaches 100.000 cycles\" says the engineer.\nQuality determines the price, which is between 26\u20ac and 50\u20ac per square metre, according to Rodr\u00edguez: \"To that, you must add the maintenance costs, which are higher if you outsource it. It might cost you an extra 4.000\u20ac or 5.000\u20ac every three months\". It's a considerable amount, but a guaranteed amortization. And a must for those pitches that will be used more than two hours per day, which is the majority\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many believe Apple introduced the world to payments without cash or card. Long before Apple Pay came into being, Paypal, Google Wallet and few other channels were already being used. However, the adoption rate for Google Wallet and other similar options was dismal, with only a few enthusiasts using it. While Apple may not have the frontrunner advantage, it definitely is revolutionary, as Apple has claimed.\nApple already has most of the leading banks as partners, not to mention the three big credit card issuers. Operating in two regions, UK and US, it has an enviable list of participating banks, with Barclays and HSBC promising to support Apple Pay in the future. However, other companies are lining up to dethrone Apple from its newfound success, with the Korean competitor announcing Samsung Pay and MCX's (Merchant Customer Exchange) CurentC ready to launch next month.\nWith Apple entering the untapped market, banks are facing increased competition in the sphere, threatening traditional banking in more ways than one. Banks can no longer ignore the onslaught of technology in banking but will they rise to the challenge? Will Apple, Google or Facebook be your next bank? Is this the end of cash and card payments? Is one touch mobile payment the future of transactions? Let's find out.\nApple wants its users to get rid of their physical wallet, hoping to bring those functionalities to the phone. Apple Pay allows users to make payments, store boarding passes, tickets, coupons, and credit\/debit card information. Users can then make purchases at the touch of a button, with the fingerprint scanner giving it the much needed security for such an operation.\nApple Pay makes use of NFC (Near Field Communication) to make transaction (s) seamless. It also uses the tokenization technology, wherein a one-time code is generated for each transaction. Tokenization works by substituting a random string of characters for the actual card details. As Tim Cook said during the unveiling of the service, sensitive card details will not be stored on phone and the merchant cashier would not see a user's name or card data, thereby securing data. The thumbprint biometric authentication adds another layer of security, giving it an edge over other traditional transactions. Also, in case of a breach, hackers will be confronted with useless scrambled data instead of actual card information.\nAccording to AlixPartners Study, 63% of 18-25 year old's surveyed are already using mobile payments, up from 56% just six months earlier. The key to get this target audience on board is a seamless user experience. The ease in the process means quick transactions and most importantly more data. The additional data will include user contact information, mobile location, merchant, data, purchase history, in addition to legacy credit transaction information. The key question is are banks ready for this Big Data? There needs to be well thought out and solid mechanism to collect, manage, visualize and monetize this information which will be coming in large volume, different variety and at high velocity.\nWhat's in it for banks\/Financial Institutions?\nThe introduction of PayPal, Apple Pay, GoogleWallet, Android Pay, Samsung Pay and other competitors entering financial services domain have left banks in a fix. Most see this evolution and revolution as a threat to banks, but also realize that it could play an optimal role in the migration from plastic cards to mobile commerce. Some even think Apple Could Become Your Next Bank, but that is definitely not the case for now, without dismissing the possibility of it happening someday.\nBig Data technologies have enabled harnessing these technologies in an effectively. Hexanika prides itself for leveraging latest data management technologies to develop solutions and assists banks in first understanding how new technologies can be adopted. Please feel free to contact us.\nSocial Media Junkie and Writer. As Content Specialist at Hexanika, he enjoys sharing insights about various aspects of technology, Big Data\/Hadoop, Banking and Regulatory Issues along with other topics related to the company's field of interest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaunwv b/data_all_eng_slimpj/shuffled/split2/finalzzzaunwv
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index 0000000000000000000000000000000000000000..678753c7c880079f9670113f94b3cb137849451c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaunwv
@@ -0,0 +1,5 @@
+{"text":"Blogging is going to go on hiatus for the month of July as my husband and I prepare to move to a new place. This means my time is mostly going to be focused on packing, cleaning, and generally trying not to die. I hope to come back with some stronger content, more reviews, personal posts. I also have a trip to Japan coming up as well, so you'll definitely get some of that on the blog during the month of September.\nThis entry was posted in Uncategorized and tagged hiatus, personal post, Sam on July 1, 2018 by samcmar.\nHi everyone! I realize I've been fairly quiet on this blog lately. A lot of it has to do with my personal life and how busy I've been. I've been working on purging parts of my life in different ways, and I'm trying to work on dealing with my grief by doing different things that make me happy. Today, I went on an adventure to a beautiful waterfront park near where I live and my best friend and I took my dog for a long hike, seeing beautiful landscapes. I thought I'd share some photos from that wonderful experience.\nDuring our walk we saw a \"Ghost Bicycle.\" I thought this was gorgeous looking, so I snapped this photo. I do wonder what the real story behind this is though.\nDifferent shots along the park path of Lake Ontario on the East side.\nDakota having some fun swimming in Lake Ontario. She looks SO HAPPY. She is best dog after all.\nSome important words of wisdom.\nAnd we had an amazing walk. Dakota was pooped and have a huge nap (except when she thought something was happening in the car, then she'd shoot up to look). This was such a calm and great experience for me. We walked nearly 10,000 steps and it was just so good to be next to the lake, staring out at the water and realizing how important these little moments truly are.\nThis entry was posted in Uncategorized and tagged dogs, personal post, Sam on August 2, 2017 by samcmar.\nIf you can't tell from this picture, my reading mojo came back full force. It's been over a month since my mother passed away and reading fell a bit on the wayside. As I was telling Molly, I just haven't had the motivation for awhile to read, and now all of a sudden I just started flying through books again. I loved the majority of the books that I read above and I'm hoping June is equally a stronger month.\nFor June I want to do something different. My husband and I are planning to move in the fall and obviously my TBR shelf of shame is too huge to take with me. My goal between now and the fall is to read my own books and see how many of them I can knock out. This is where for the next few months I going to get you all to help me out. I asked on Twitter for people to come up with challenges for me.\nA book that is has purple on the cover.\nSomething by a Canadian author.\nA book with a dog in it or as the main character.\nRead something that has blue on the cover.\nAt the end of the month I will let you all know if I accomplished these challenges, as well as share all the physical books I read. I am hoping to share an updated picture of the Shelf of Shame to give you a sense of how much has been read. We'll see how this goes, and hopefully by September my TBR will look more manageable.\nThis entry was posted in Uncategorized and tagged personal post, Sam, tbr on June 4, 2016 by samcmar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1 Mystery of the Swimming Gorilla, The McMahon, P. J.\n2 Case of the Smiling Shark, The McMahon, P. J.\n3 Mystery of the Morphing Hockey Stick, The McMahon, P. J.\n4 Case of the Psychic Hamster, The McMahon, P. J.\n5 Mystery of the Disappearing Dinosaurs, The McMahon, P. J.\n6 Case of the Singing SeaDragons, The McMahon, P. J.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I now have a new room.\nIt's been a bit mad recently - I went to London, had a day of activities at Royal Holloway, left for Berlin the next day, spent a week there (which I will write a separate post on), then spent a week in Gottingen (also in Germany), returned to Edinburgh from Bremen, had one night there, packed my university bag, went to Leeds to see my sister (she has just moved there for her Masters) then to London the next day, new room, and first day on my new job the day after.\nI signed up to a bunch of new societies and met lots of lovely new people.\nTomorrow I will sleep in, do some vintage shopping, have a work phone call and meet my team from the Global Leadership Programme to sort out some more plans for the exhibition in November.\nMy university course officially starts back on Friday, and I am a little apprehensive about the expectations for the dissertation, but excited to get back into things.\nHopefully I will develop my disposable camera this week - which has some pictures from Germany on there - I will share some on here.\nI am still in the seemingly impossible process of sorting out all of my computer files, which are a mess - and somewhat resemble my brain right now.\nOnce I have those things sorted, and have made my room a little more pretty - I could post a uni room tour as well as all the other posts I have been meaning to write. I think I might write a Royal Holloway specific post, and maybe something on media careers etc.\nLet me know what you think and what you are particularly nosey about (ha).\nWill be posting music that I like at the moment \/ other favourites in the next days - before I get properly back into uni mode.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Protocol Education are seeking to appoint a Level 3 qualified Nursery Nurse to join our successful supply cover team, to be working in local Children's Centres and Nurseries in Sheffield.\nDesired Nursery Nurses will specialise in Foundation Stage settings. You will be warm and friendly with a desire to really make a difference during this crucial stage in a child's development. As a dedicated practitioner you will be committed to ensuring that children leave the setting to enter the first years in school with all the key social skills required.\nAt Protocol Education we take our responsibilities to you seriously, our aim is to ensure you are well supported and rewarded for your efforts, even if it is just to say 'thank you' for a job well done.\nIf you are a qualified Nursery Nurse who is able to enthuse, motivate and engage children and would like the challenge of working in a variety of settings, then apply now for immediate consideration.\nPlease note: ad-hoc positions like this do often turn into long-term and permanent roles for regular staff.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After tasting the most amazing apple pie ever in Bruges I had to try my hand at making one just as good. I did a bit of research and found that there is a difference between English apple pie and Dutch apple pie.. for one thing they call it Appeltaart. It's a lot sweeter than English apple pie and it is absolutely amazing. There is no need to sweeten it up with custard, just a side of ice cream to give it that lovely hot cold contrast.\n-Grab a bowl and add the flour, butter, sugar, cinnamon and 1 and a half of the eggs. Stick your hands in and mix the ingredients together needing it in to a dough adding a little more flour as and when you need to.\n- Once you have a nicely even and smooth dough take a quarter of it away and set aside. Flour your (clean) surface and start rolling out your dough. My tin is a spring form 22.5cm tin but I think this mixture will fit up to 24cm.. this seems to be a more popular size of tin? If you haven't got a spring form tin butter the sides so the pie will slide out a little easier.\n- Preheat your oven to 170 degrees (150 if fan assisted) and start prepping your filing.\n- Peel, core and slice your apples. In a bowl add the sugar, sultanas, cinnamon, lemon juice, and ground almonds and mix together. Then add in your apples and cover your apple slices in the mixture I prefer to do this with my hands so I don't break up all my thinly slices apple pieces trying to do it with a spoon.\n- Layer your apples and sultanas in the tin. You might have a bit of liquid left over drizzle this over your filing but don't add too much as you don't want your pie to be soggy. Then sprinkle over some more sugar (optional) cinnamon, and your almond flakes.\n- Grab your left over dough and cut it in to thin strips. This is the first time I had ever done a lattice and I think it will need a lot of practice to make it look perfect but I think for a first attempt mine came out rather well. I started doing all the strips going across first and then once I had made sure they fit I peeled them back and started threading through the other side. I made a pathetic attempt to search youtube for the perfect way to do this but found it no help.\n- Once you are satisfied with your lattice... whisk the remaining half an egg and brush it on the top of your pie, use it to stick and fold down the edges to seal your pie.\n- Pop it in the oven for about an hour, you may need to turn it to make it even all over, once it's golden brown all over on the top take it out and leave it to cool.\n- Once those agonising 30minutes are up slice up you pie and serve with creamy ice cream. Dust with lots of icing sugar to give it that extra Dutch touch.\nThe result was perfect, better than I could of hoped for. Absolute heaven in my mouth. The sweetness of the dough combined with the tart apples, moist on the inside crunchy on the outside, warm apple pie with cold ice cream- my ideal dessert. I can't wait to eat some more of it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauwed b/data_all_eng_slimpj/shuffled/split2/finalzzzauwed
new file mode 100644
index 0000000000000000000000000000000000000000..62d743207daff126f5dc471f1e5de67807abbcef
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"This will be our second year playing the FairviewCon event in our home city of San Gabriel. The event is n Thursday 8\/2\/18 on Fairview Ave. between North and South Circle Dr. from 6:30-10:00pm. We go on at 7:30pm and will be playing two sets! There will be vendors, food trucks, art, comics, raffles and more! The event is free, open to the public, and all ages. The Pasadena Star news caught up with Frankie last year to interview him about the event and growing up in the area. Read the article here.\nSurf Guitar 101 Convention 2015 Information!\nPasadena Star News reviews The Volcanics!\nDouble Crown Records caught up with Frankie just before the release of their latest album \"The Lonely One\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I love the addition of the wood! Yay, go ambitious hubby!\nwow, those posts made a huge difference, and painting them white looks so fresh!\nGood luck with the rest of your porch.\nThat looks great! Kudos to you and your husband.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No meeting facilities were found in Cape Vogel, Papua New Guinea. Your search was widened to include meeting facilities within 300 miles of Cape Vogel, Papua New Guinea. To find more area meeting facilities, click the Zoom link shown above.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An approaching World Trade Organization deadline makes higher prices likelier for European goods made with parts and materials imported from China.\nOn Dec. 11, the European Union could grant China \"market economy status,\" making it harder under WTO rules for the EU to protect its industries from what it deems as unfair trade practices by Beijing.\nOr, the EU could draft new trade rules that do away with the distinction between market and nonmarket economies that the bloc has used in the past.\nThe odds are against China getting the market title by the deadline, said Edwin Vermulst, a partner at Brussels-based law firm VVGB Advocaten.\n\"There is simply too much pressure from southern EU states and industrial sectors with a protectionist agenda to maintain, or even increase, the high level of duties,\" Mr. Vermulst said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On dining table portable tablet aimed at the lowest prices in home movable height adjustable laptop tables a computer desks might not be your skills as good computer desk shop with an internal battery. Prices shop for dining table stock photos illustrations and tablets in computer table pc or laptop table clocks a virtual column is a similar design elsewhere i would try my hand at best prices and made with a device class of the times tables computer table with an integrated installing the multiplication games let you prepare for computer table products about of other royaltyfree stock.\nDesigns for the best office and dual monitor computer table and your work style choose traditional modern study table designs for each table designs furniture table wood yaheetech modern simple design home monarch specialties inc hollow core l shaped puter designs furniture and are plenty of computer graphics designs a drafting table designs. Measurement units metric footprint. Computer table designs, desks with drawers is about study tables are office computer room and save ideas about computer desk. Home office desks wood tables in wooden computer table collections subscribe my channel for home office computer furniture info inside internet cafe computer.\nPopular visitor terms Table for computer, computer table designs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavhav b/data_all_eng_slimpj/shuffled/split2/finalzzzavhav
new file mode 100644
index 0000000000000000000000000000000000000000..eebcdbe8d46fb282ccde0b485f044f58cef7997e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavhav
@@ -0,0 +1,5 @@
+{"text":"Do I have to keep sightseeing when I travel?\nI think Japanese people make full their schedule whole day long when going to travel.\nEspecially when they participate in sightseeing tours.\nI am now in South Africa at a tourist destination called Sea Point beside the Table Mount and Lion's Head.\nI stayed at a hotel which is located 2 minutes away from the beach.\nIt is a dormitory room for 8 people, but the internet is fast, the room is comfortably and wide.\nThe hotel has own market, so you can finish your meal without going out.\nWhy do I have to go out when I'm in such an environment?\nSo I have not done anything yet for sightseeing.\nI was just walking around the city.\nThere are two major reasons why I came to Cape Town.\nOne is to meet friend who worked together by ship.\nI could see her 2 days ago.\nI did not feel the flow of such time and we enjoyed chatting with drinking alcohol.\nAnother purpose is to climb Table Mountain and take pictures. You can't find a mountain like table mountain even if you look around the world.\nI think I will do it in next week.\nWhen you search in internet, Cape Town has many sights.\nI am planning to go to Cape of Good Hope on the southernmost tip of Africa and to go to Boulder Beach where there are plenty of wild penguins.\nI do not get motivated.\nI often stay at home all day when I don't get motivated.\nSince I came to sightseeing spot all the way, I did not go sightseeing, and I'm writing a blog and using wifi at the hotel.\nI think that it would be nice to travel way in such a form if you have plenty of time.\nIt seems like traveling to live?\nWe went to the shopping mall in the middle of sightseeing area by walking along the beach together. About 3 hours round trip.\nDeparting in the evening, We came back to the hotel around 9pm.\nIt was okay even if I walked after sun went down.\nIt is said that you should not walk alone by a local friend.\nCape Town has a very beautiful sunset.\nThere were a lot of runners in the evening, some people were training with equipment.\nApparently there are many Germans in Cape Town.\nA German who was walking around the city with me, said that I hear German languages \u200b\u200ba lot.\nI heard that there are quite many German who study PhD in Cape Town.\nCape Town is a very beautiful city.\nEspecially if you are looking at sunset or night view.\nThe city spreads out under the Table Mountain and the Lion's Head which can be said as the symbol of the city.\nIt seems to be chosen to live by European retired people because of the good weather.\nI do nothing in Cape Town and I am having a relaxing day.\nIt is not too hot, it's not cold, with a pleasant ocean breeze blowing, while watching the beautiful sunset in the evening.\nIf it is so comfortable, I think that I do not have to do anything here.\nI think that I will spend days without doing anything for next 2 days.\nI think this is not bad way of a journey, What do you think?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When choosing remote controlled fans you should always consider the room area. Always pick the appropriate blade size of your Hunter Pacific ceiling fans. If you do have a larger room area it is better to get two or more fans in order to provide better cooling ability for your room.\nNowadays, consumers are much wiser in terms of choosing high-quality products that are economical and at the same time durable. Using Martec Ceiling Fans is more cost efficient and low maintenance as compared to using air conditioners.\nHunter Pacific Ceiling fans are also an ideal fixture for your kitchen. Aside from its cooling effects, it helps stir away hot air especially when cooking and baking.\nHunter Ceiling Fans are best installed in the bedroom. On a sweltering summer night, you can open a window and turn the fan on. The ceiling fan will immediately circulate the cool night breeze throughout your room and make your slumber refreshing and comfortable.\nUse ceiling fans with lights. Having a light fixture with your fan helps you save on electricity cost. The Hunter Pacific ceiling fans offer a variety of electric fans with built in halogen lights. Take your pick on colourful and well-designed remote controlled fans that can add charm in your rooms.\nOur sturdy Hunter Pacific ceiling fans provide you with a reversing switch for both summer and winter operations. During summer, move the switch so fan blades rotate counterclockwise. This will help circulate a breeze inside your living room or office. Switch your ceiling fan during winter so that the blades spin clockwise. This well then push warm air back into you room.\nFor more information on Hunter Pacific ceiling fans and other products, please feel free to email us or give us a call at 07 3421 3322. Our friendly and knowledgeable online staff will only be too happy to assist you with any enquiries you may have.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It always seems like I'm rushing to get ready in the morning! Is there anything that you find that helps you manage your time better or anything that helps you get your hair done more quickly? Also, how long does it take your hair to dry and any tips on getting it to dry more quickly?\nTry wearing it in braids at night (2-10 depending on how long your hair is). Then, in the morning, spritz the braids with a leave in (if you need to), unbraid them, and go!\nI take one shower at Night and I conditioner wash my hair, moisturize it with something, then while it's still pretty wet I put gel in it then put it up in a semi tight super high ponytail towards the top of my head so I won't sleep on it. (aka pineapple it) that way I can shower in the morning without getting my hair wet then put my makeup on and be done. I always get pretty good to decent looking hair this way. Hope it helps u some.\nThe night before I have early classes I wash my hair at night, apply my all my product, throw it up in a pineapple and cover it with a satin bonnet. In the morning I just have to take my hair down and spritz it with my homemade refresher spray. Hope this helps you!\ni usually wash my hair every few days, because i have NO time in the morning to do that. So at night, I wash my hair and mousse it to control my after-shower poofiness (but for me at least a little poofiness is inevitable lol) and then i braid it or put it in a loose ponytail. When i wake up i rinse my hair down under the bath tub faucet or the sink and get it really wet like i would in the shower, but i don't wash it. And then i style it and as long as my hair doesn't decide to be stupid, it usually takes me about 10 to 15 minutes. 10 being a ponytail and 15 leaving it down. I get up at 6:50, my hair's done by seven, and then i leave for school at 7:20.\nFirst I take showers during the night. I co wash with Garnier fructis naturals conditioner (don't use this! It has silicones!!!! I just have to use it up before I can get a better one) then I scrunch out some of the water so my hair is not dripping wet while I change. Then I add some of my conditioner Detangle my hair and then add herbal essences set me up gel to my hair. Then I let my hair dry for about ten minutes then i plop with a long sleeved t shirt in the morning I do touch ups with some gel ad then I am ready to go!\nUsually I take a shower at night since I have to wake up super early to wash my hair in the morning. But overnight I'll put my hair in a pineapple and put a satin sleep cap over it, and then I just take my pineapple out in the morning and that's it.\nOn the days that i dont shower in the morning i take one the night before and then i put it in french braids with leave in conditioner and when i wake up all i have to do is take out the braids and put in a some mousse \uf48b hope that helped!\nI wake up at 530, and hope in the shower. while in the shower i will wet my hair in cold water and comb out the knots. After i apply Garnier's sleek&Shine leave in conditioner while hair is soaking wet, making sure every strand of hair is applied with this from roots to ends. After i put it in a bun so i can keep the conditioner locked into the hair. If i'm taking my time i finish by 605 maximum. I get dressed and then put my hair back down and apply ion miracle leave-in sold at sally's. I tie it up wash my face apply some face lotion and put it back down and apply a tad bit if needed in certain parts, and by then its 630 ish and i have to leave the house at 640 for my bus stop.\nI follow the same method! Sometimes I keep my hair twisted\/braided throughout the next day to make it completely dry, then unbraid\/untwist the next morning after.\n2.) Put rinse-out conditioner in hair and detangle ends. Leave it in for 5-10 minutes as I finish shower.\nMy hair typically takes a while to completely dry but after an hour or so it's pretty dry. If I go to bed with it totally wet it will still be wet in the morning.\nAt night I like to let it air dry for half an hour to an hour before I put it up in a pineapple and sleep. Then when I wake up I have great curly hair and all I have to do is apply some leave in conditioner and anti-frizz cream. I think if you do your shower the night before, you won't have to spend so much time in the morning on your hair. If I ever shower in the morning, I just let it air dry then apply leave-in conditioner and it looks fine but I like the way it looks with my night routine and that way I'm not rushed in the morning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spotlight metal tripod floor lamp the green head, spotlight floor lamp. 65 inch black 3 light tree lamp spotlight floor lamp. Finding 10 best spotlight floor lamps warisan lighting.\nFloor lamps: awesome spotlight floor lamp spotlight floor, industrial chrome spotlight floor lamp modernism. Floor lamp spotlight. Spotlight floor lamp modern furniture brickell collection.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1) *The Negative Thinkers* These are the set of people who are always online to give negative comments to any business opportunity, they served as destiny stopper, their negative comment in the group usually denied people to achieve any given opportunity that could have changed their life situation.\n2) *Doubters* These are the set of people, who always think every online business is *SCAM* due to their previous experience. But even if you introduced something genuinely to them, their internal doubtful habit will deny them to take action on a good platform that can change their story!\n5). *The Excuse givers:* They see the opportunity but their interest is killed by fear, excuses and analysis paralysis. Not risk takers, always mistrusting and prefer to keep money. They like to see cash in their pockets than in an investment - No fluidity.\n*6). The Believers:* Believe in the possibilities of an opportunity but take No action, or maybe just decide to begin for some reasons without passion and conviction. Eventually they give up along the way.\n7). *Action Takers:* - They just don't believe, they put work to their belief. They have come to trust that which they believed. They learn more while earning and earn more with focus and strategy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawjbx b/data_all_eng_slimpj/shuffled/split2/finalzzzawjbx
new file mode 100644
index 0000000000000000000000000000000000000000..c0c5a39e976ffedbc9dfa90659d04eb49bd8bffb
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"While Rick enjoyed farming and carpentry work, driving semi was by far his favorite job, which became a passion he passed on to his son. His favorite pastimes included fishing with his grandson and cutting wood with the boys. Rick had a knack for making conversation with anyone and everyone. You could often find him at Kwik Star getting coffee and socializing with the guys. He was always willing to help anyone he could.\nRick is survived by his wife, Janette; his daughter, Jessica (Brandon) Wilder; his son, Mike Snitker (Paige and Oaklynn Bolsinger); and Rick's (a.k.a. Poppy's) pride and joy was his only grandson, Collin Wilder. He is also survived by three brothers, Bruce (Connie) Snitker, Craig (Kathy) Snitker and Clark (Norma) Snitker; two sisters-in-law: Judy Benzing and Carolynn Brown; two brothers-in-law, Jerry (Celia) Brown and Joel (Jeanne) Brown; as well as many nieces and nephews.\nHe was preceded in death by his parents, Joann Johnson and Nathan Snitker; his father- and mother-in-law, John and Ella Brown; and two brothers-in-law, Ronald Benzing and Jack Brown.\nLeonard-Grau Funeral Home and Cremation Service of Monona assisted the family with arrangements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Art historian A insists the meaning of this work of art can be found in room one. Upon rushing through the passageway to room one, however, we could not find any meaning.\nArt historian B's books all conclude with the meaning of this work of art being found in room two. Upon rushing through the passageway to room two, we could not find any meaning.\nArt historian C argues in lectures to university students, that the meaning of this work of art is divided between rooms three and four. Upon rushing through the passageway to room three then four, however, we could not find any meaning.\nReturning to the passageway, convinced this work of art has no meaning, we little realised we were standing right within it. For later the meaning was found to be here, in this passageway, between the imaginary space of fiction and the real space of a blue cube \u2013 a passageway through which we have been rushing all along.\nABC art: blue cube is available to collect.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hendrix came from outer space, I am convinced of it. The first Experience album came out in 1967 which was an amazing year for music as we all know, Cream, The Beatles, The Stones, The Who, etc. But Jimi was something else, he sounded like no one, he looked like no one, and he scared the hell out of every one. In his short life he changed music and guitar forever. 40 years after his death his presence is as strong as ever. He is well known for his pyrotechnics but his ballad playing is just as amazing. On this lick I am borrowing the chord progressing from \"May This Be Love\" from one of my favorite Hendrix style licks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wall panels with hidden screw consist from two rollformed plates and polyurethane filling. They are the most commonly used for cladding of commercial and industrial buildings.\nThey are manufactured in three rollforming variants: microprofiles, linear and flat rollforming with PUR and PIR filling.\nStandard panel width is 100 cm, but they can be also 60 cm wide, which is especially attractive when external flat profile with wood decor is chosen.\nFacade panels can be 50, 60, 80, 100, and 120 mm thick.\nWith selection of suitable colors or type of rollforming, extraordinary esthetic effects can be achieved implementing the designed architectural solution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After New Orleans, we hopped on a plane and flew to San Diego! Adam and I had about three days to ourselves in between Death to Stock stops which would explain the lack of photos. I had been shooting so much prior that I mostly left my camera at our hotel.\nIf we weren't at the beach or In N' Out\u2026we were hanging out with Vanessa and Aaron! I've known Vanessa for years through blogging and it was so amazing to hang out in real life!\nWe could definitely see ourselves moving to San Diego someday.\nHi Allison! I used to follow your old blog when I had my old blog, Glass and Sable. I'm back on the online space and I'm loving this new space of yours, too. I recently moved out to SoCal and I'm glad you had a warm welcome out here. Hope you're doing well! P.S. just signed up for Death to Stock and it's amazing!\nAh I love hearing that you used to read along. I'm jealous you're in SoCal while we head into the winter months here. Hope you're loving DTTSP!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxdmb b/data_all_eng_slimpj/shuffled/split2/finalzzzaxdmb
new file mode 100644
index 0000000000000000000000000000000000000000..87afdc46a86d8b1a52ee6d27a182a62cfc23a09f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxdmb
@@ -0,0 +1,5 @@
+{"text":"Whether you're a foodie, a beauty junkie or a gin lover, variety is the spice of life and we are always striving to discover new and exciting products. We love subscription boxes (we may be slightly biased) and having a suprise delivery of all your favourite things arrive at your doorstep every month, is the best feeling!\nSubscription boxes are on the rise so whatever your passion, chances are there is a box out there for you! These are our favourite subscription boxes out there.\nBeer 52 sends out 8 or 10 delicious craft beers, every 28 days. The Beer 52 team curates a themed selection of craft beers every month covering a range of different styles.\nBloom & Wild is redefining the flower delivery experience, making the giving and receiving of beautiful fresh flowers the joy it is meant to be. The Bloom and Wild team source flowers seasonally, responsibly and carefully to be sent to a frequency of your choice.\nReceive seasonal, artisan cheeses from the UK and the Continent as either quarterly, bi-monthly or monthly. You can also buy a one-off month or receive a cheese parcel from The Cheese Society for three months.\nHello Fresh take the guess work out of your evening meal by sending you chef-curated recipes and ingredients you enjoy cooking. You have the choice of when to receive your ingredients and how many people to cater for. The boxes are nutritious and can also be tailored for both vegetarians and meat eaters alike!\nWith all the thousands of beauty brands out there, it can be tricky to find the right product for you. Birch Box will send you a surprise beauty box every month, packed full of new and interesting products. The handy member login area means you can tailor the deliveries to cater for your like and skin types.\nWe couldn't make this list without including our own wonderful box now could we! Gin lovers around the country receive an amazing full sized bottle of craft gin on a monthly, bi-monthly or quarterly basis. There are delicious gourmet snacks, tasty and interesting mixers and a club magazine packed full of ginny info, interviews, recipes and fun facts. Join today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Success Stories - Ben from Pathways to the Future on Vimeo.\nSuccess Stories - Elizabeth from Pathways to the Future on Vimeo.\nSuccess Stories - Adam from Pathways to the Future on Vimeo.\nThe annual Ability Works Awards honor outstanding candidates from each WVDRS district. Learn more about a few of the thousands of people who achieve vocational rehabilitation each year in West Virginia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With the sameweaning and extubation criteria where to buy ampicillin for betta fish extubation waspossible sooner in the CTGI group when com-pared to controls ( p < 0.05) (Fig. Shown on the bottom is the Hr associated with a continuous scorederived from Nt-proBNP, C-reactive protein, and troponin i.\nShe is dressed in scrubs, seated in the department supervisor's of? ce. Tuberculosis surveillance and monitoring in Europe 2013. Reconfiguration interferes with the bowel's ability tocoordinate a contraction thereby minimizing high-pressure peaks. Patients who undergo novel procedures, whether as a \"one-off\" or part of atrial, are particularly prone to the therapeutic misconception, where a research sub-ject does not appreciate the distinction between the research and clinical paradigmand attributes therapeutic intent to a research procedure (Lidz and Appelbaum2002). When compared withsurvivors, non- survivors had lower TC-O2 andhigher TC-CO2 values. Ifthe use of CTGI is indicated, it must be managedpreferentially with a security device, presented inFig. Toronto MayorMel Lastman went on record as saying, \"Let me be clear; it's safe to live inToronto, and it's safe to visit Toronto\" (\"City of Toronto Disputes WorldHealth Organization Travel Advisory\" 2003). However, in that study, the cutoff was set at14 days prior to data collection. However, in a minority of patientswith focal MND of the arms, respiratory involvement mayensue.\nAnti-inflammatory Actions of Adjunctive Tetracyclines andOther Agents in Periodontitis and Associated Comorbidities. Hence where to buy ampicillin for betta fish neither Scr nor BUN is considered a sensitiveindicator of renal function. Other important predisposing conditions are abortionor pregnancy [77, 86]. Thus where to buy ampicillin for betta fish factors(e.g., insulin and growth factors) that stimulate Cys (cystine) uptake by cells generallyincrease intracellular-GSH concentrations.\nThetreatment must be \"reasonable and necessary\" to receive reimbursement. Yet where to buy ampicillin for betta fish thebenefit of adding rifampin remains unknown without a study comparing the combinationversus monotherapy.\nEffectiveness of three different doses of carvedilol for exertional angina.Carvedilol-Angina Study Group. In situa-tional phobias where to buy ampicillin for betta fish propranolol may be added as andwhen required. Hierarchical agglomerative clusteringgroups together the data in a stepwise fashion matching oneexpression profile with the next where to buy ampicillin for betta fish one at a time, until all theprofiles are matched and form clusters, as shown in Figure 4.3.K-means clustering starts out by randomly clustering theexpression profiles into groups or bins and then performingthe comparisons between the bins, by selecting like profilesto stay and putting different profiles into other bins, until allthe profiles are matched and reorganized into bins or clustersthat show similarity. Used stairs, marking time, with B hand rails, 6 steps up and 6 steps down x2. MBP-Gwas important to describing gut dynamics and the sustainedpresence of free MBP in the blood (a result of enterohepaticrecirculation of MBP-G followed by conversion to free MBPin the GI). However,mesenteric nodules are seen with similar frequency in perito-neal tuberculosis and peritoneal carcinomatosis. They returned on a similar westward flight threeweeks later.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WONDERREEL is online entertainment tailored for big kids, their passions and their tastes.\nWe will offer a collection of the finest children's film and video from all over the world, including award-winning adventure, comedy, animation, drama and documentary\u2013variety unprecedented in US children's media.\nWe'll use big-data technology to find the shows in our collection that will appeal to each individual child, then we program a personal channel for every child.\nKids will be able to take charge of their viewing with controls designed to make it easy to discover new favorites in our collection, then enjoy them with friends.\nAnd with easy, intuitive tools, parents will be able supervise their children's viewing\u2013at whatever level makes sense to your family.\nThat's why we call WONDERREEL entertainment as smart as your kids.\n\"Planet Wonderreel\" is a website where grownups who know and love kids can learn about WONDERREEL\u2014and also lend their support.\nWONDERREEL is unlike any other project in the history of children's media. We're making new uses of digital technology. We're searching the world for terrific shows that otherwise might never be seen in the US. We don't even care if they're \"hits\"\u2013as long as there are kids who find them fun, engaging, even rewarding. Every show will be available in its original language, in English, and before long, in Spanish too.\nAll this makes WONDERREEL an expensive project to launch. We're not going to show ads to children. We're not taking government money either. WONDERREEL will be supported by the families it serves, through monthly subscriptions. So it's important to demonstrate to our backers that moms, dads, grandparents and everybody who loves kids believe in WONDERREEL and want their kids to have this rich entertainment alternative.\nBut Planet Wonderreel is more than that. It's a place where the people who make WONDERREEL can stay in touch with the people who entrust their kids to us during screen time. We are dedicated to serving children and their families. We want to hear from you\u2014your ideas, your concerns and, well, sure, your praise too!\nTEAM WONDERREEL is an online workshop where kids will have their say in designing WONDERREEL. It runs from September through December 2016.\nWhere does WONDERREEL get shows?\nWONDERREEL is working with public broadcasters, producers and filmmakers on every continent to curate our collection. Most of our shows are beloved by children in the countries where they were made, whether broadcast on TV or shown in theaters. Many have been selected for major children's film and television festivals. Quite a few are festival prize-winners.\nWe're not looking for super-mega-hits or fad programs, but the best of the best\u2013whether they're archival classics, innovative experiments or films and series that have proven themselves with kids in the lands and languages they come from.\nAll over the world, from Chile to China, Pakistan to Poland, independent filmmakers and public broadcasters make amazing, hilarious, thrilling, enlightening films and series. They shoot adventures, mysteries, drama, animation, documentaries\u2013even news reported by kids. These producers take children as seriously as HBO or Discovery take grownups. They don't talk down to kids. They honor their smarts.\nWONDERREEL is designed to make shows like that available to big kids in the US and beyond.\nIf by \"interactive\" you mean, \"Are there video games on WONDERREEL?\", the answer is no. We believe big kids deserve a place where they can experience the power of story and meet characters much like themselves. So do our creative partners. That's the heart of the film and video on WONDERREEL.\nDo kids have control over their WONDERREEL experience?\nIs the platform responsive to the interests and needs of children and families?\nCan kids use WONDERREEL to share great experiences with friends and family, near and far?\nAnd does WONDERREEL, as it grows, plan to introduce areas where kids can express their own creativity?\nWe hope not. We've designed WONDERREEL as a sustainable business that returns value to our investors. It's a new form of business, however: a \"public benefit corporation.\" That means that we're equally committed, by charter and by law, to our public service mission: bringing rich, diverse cultural resources to kids 6-11.\nCan I see WONDERREEL now?\nWhat's wrong with broadcast and cable\/satellite TV for kids?\nAbsolutely nothing! They do a great job with the technology at their disposal.\nBut broadcast and cable\/satellite are hampered by the simple fact that they can only show one program at a time. So their programs need to have mass appeal. They need hits.\nThe Internet changes that. A service can offer thousands of shows at a time to hundreds of thousands of kids\u2013anytime, anywhere. We can provide infinitely more choice than broadcast or cable\/satellite. Even better, we can bring just the right shows to every child.\nThat's all possible online. And WONDERREEL intends to take full advantage of that opportunity.\nIsn't this what PBS is for?\nWe love PBS. But PBS stations face the same constraints as any other broadcaster. They can only show one program at a time.\nPBS Kids needs to make the most of a few hours before kids head off to school every morning, then a few more between the time they come home and prime time. With a few brilliant exceptions, they've chosen to use that time to serve preschoolers.\nWONDERREEL will pick up where PBS Kids has to stop\u2013with quality, diversity and choice in full-length video for school-age kids, 24\/7, across every digital platform.\nIsn't there enough video for kids online?\nWell, there's lots of video for kids online\u2013that's for sure. But one of our colleagues calls it cooked-spaghetti programming: throw stuff against the wall and see if it sticks.\nYouTube provides kids delicious little video snacks. Netflix and Amazon invite them to binge on shows they've already seen on TV and at the movies (plus a few originals, to be fair).\nadd your name and the names of organizations you represent to our list of endorsers.\nLike WONDERREEL on Facebook, follow us on Twitter and Pinterest, and tell all your friends about us.\nadvise us by taking our pricing survey.\nshare your thoughts in our suggestion box.\ncheck out our ideas for volunteering.\nAnd please return often, because we'll introduce new opportunities to help as Planet Wonderreel unfolds.\nI have a different question.\nWe're eager to answer. Please drop your question in our suggestion box and we'll get right back to you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apex Design is a downtown Denver-based consulting firm specializing in transportation planning, traffic engineering and operations, and technology for surface transportation. We are well-respected by our clients and teaming partners. Apex promotes a culture of professional growth through training, leadership development and encouraging involvement in professional organizations. Our team maintains an attitude of mutual support and collaboration. We share best practices and lessons learned across projects, and foster great relationships through quarterly team building and social activities.\nThis position is an excellent opportunity for a motivated and highly-skilled roadway design professional who wishes to be part of our expanding team. The successful candidate will play an important role in the design of numerous major roadway system improvement projects throughout Colorado for state and local public agencies, and federal public-land roadways.\nDesign and analysis of freeways, interstate highways, interchanges, urban and rural roadways, and shared-use trails.\nPrepare studies, reports, plans, specifications, and cost estimates for a wide variety of transportation projects.\nAssist with design of multidisciplinary traffic and Intelligent Transportation System (ITS) projects.\nSupport project managers with client relations, stakeholder management, and sub consultant coordination.\nCoordinate and work with a team of project managers, design engineers, technicians, and support staff to successfully complete a variety of projects.\nTake ownership of projects from start to finish.\nSupport in the pursuit of projects through qualifications, scope, schedule and fee development.\nBachelor's Degree in civil engineering from an accredited college or university.\n5+ years of experience in the design of freeways, interstate highways, interchanges, urban and rural roadways, and shared use trails.\nKnowledge and experience with MicroStation, InRoads and\/or OpenRoads\/Geopak, AutoCAD Civil 3D, and other applicable software common to the field of transportation engineering.\nExperience with Colorado DOT (or other state DOT) plan preparation procedures and standards.\nExperience with Federal Lands Highway plan preparation procedures and standards is a plus.\nHydraulics and traffic experience is a plus.\nSelf-motivated, detail-oriented professional with strong organizational skills.\nA desire to grow personally and professionally with each project while working in a team environment to create designs and drawings supporting our clients' overall project goals.\nColorado Professional Engineer License or ability to transfer a license from another state.\n*All benefits are subject to change at the discretion of Apex Design, PC.\nApex Design is a Denver-based consulting firm specializing in transportation planning, traffic engineering and operations, and technology for surface transportation. Apex is well-respected by our clients and teaming partners, and our staff takes pride in being able to show friends and family how we are making a difference with our projects.\nWe promote a culture of professional growth through training, leadership development and encouraging involvement in professional organizations. Our team maintains an attitude of mutual support and collaboration. We share best practices and lessons learned across projects, and foster great relationships through quarterly team building and social activities.\nPut simply\u2026we are proud of what we do and who we work with!\nApex is currently leading the design or implementation oversight for the technology elements of many major projects in Colorado. We work closely with the Colorado Department of Transportation (CDOT), E-470 Public Highway Authority, High Performance Transportation Enterprise (HPTE), Regional Transportation District (RTD), Panasonic, City and County of Denver, and other public agencies. With those relationships in mind, we are looking for a motivated and talented Senior Intelligent Transportation Systems (ITS) engineer that is ready to join our team and engage in projects that are making lives better.\nWe have opportunities for project management, design and delivery on a variety of projects, including fiber optic communications, Ethernet networks, connected vehicle solutions, tolling systems, ramp metering, software development oversight, traffic signals, transit and roadway ITS, systems integration and testing, and more. If you have good communications skills, enjoy a healthy team environment, and have 8+ years of experience with ITS, systems engineering, connected vehicles, tolling systems or transit control systems, you may be the perfect fit for this position!\nWe offer a competitive compensation package, and the ability to grow both professionally and personally within the engineering community. To apply, submit a cover letter and resume to careers@apexdesignpc.com attention to Melissa Rosas, PE, President.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Greetings, Brainiacs! This week we discuss a little bit of current events to start off our data download. We talk some exciting Superhero news (including CASTING RUMORS!) , Star Wars Universe trailers and some shocking and exciting news about who will be producing the next trilogy! As always, we will be giving some insight into what we've been watching and listening to this week, and we will each discuss our current obsessions to get you obsessed!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The sun sets everyday. From the day we are born till the day we die the sun rises and sets. An yet it is only in our sunset years that we start to appreciate the real sunsets and rises. For we know that they are getting limited in number for us to see.\nIt is hard to be positive 24\/7. In fact i think it is impossible. With the amount of negativity in the world and the ease we seem to have access to it, via mms and online, it is going to het to you no matter how strong you are and how positive you think. But you know what? That is ok\u2026.we are human. To be human is to have feelings and we must feel all of these feelings sometimes to remind us that we are human.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The past 24 months have brought Virtual Reality and Augmented Reality to the public's attention. Virtual Reality has already been in use in Surgical Robotics and has been implemented by several medical device companies and is in use in many operating rooms.\nClinicians in behavioral health have developed treatment protocols for depression, and others. Some are using it for diagnostics as well as treatment. The field is ripe for study. PubMed.com is a valuable resource for my readers.\nOur source for information in today's post is Dr. Walter Greenleaf (Stanford University). sponsored by Crowdcast a live streaming application.\nVirtual Reality shares the limelight now with another superstar, Artificial Intelligence. Undoubtedly the two will join forces, merge and become Virtual Intelligence or some other eponym as an eye catcher for engagement. At the least our lexicon is changing rapidly.\nAs a research scientist, Dr. Greenleaf's focus has been on age-related changes in cognition, mood, and behavior. His early research was on age-related changes in the neuroendocrine system and the effects on human behavior. He served as the Director of the Mind Division, Stanford Center on Longevity, where his focus was on age-related changes in cognition. He is currently a Distinguished Visiting Scholar at Stanford University's MediaX Program, a Visiting Scholar at Stanford University's Virtual Human Interaction Lab, the Director of Technology Strategyat the University of Colorado National Mental Health Innovation Center, and a member of the Board of Directors for Brainstorm: The Stanford Laboratory for Brain Health Innovation and Entrepreneurship.\nThe $69 billion merger between CVS and Aetna will close imminently after New York signed off on the deal Monday.\nThe deal is now expected to close Wednesday. Aetna and CVS say that the merger will improve health-care outcomes and reduce costs immediately.\nThey have plans to turn CVS's 10,000 pharmacies and clinics into community-based sites of care with nurses and other health professionals available to give diagnoses or do lab work.\nThe merger also means that there will no longer be any independent pharmacy benefit managers in the U.S.\nThe deal was cleared by the Department of Justice in October, and New York was the last state regulatory approval that the companies needed.\nAs part of New York's approval, CVS and Aetna agreed to some concessions, including enhanced consumer and health insurance rate protections, privacy controls, cybersecurity compliance, and a $40 million commitment to support health insurance education and enrollment.\nCVS to complete Aetna merger after clearing final hurdle | TheHill: The $69 billion merger between CVS and Aetna will close imminently after New York signed off on the deal Monday.\nthe NRA has criticized doctors for making guns a public health issue.\nAn emergency room physician was mortally wounded along with two other persons.\nThose on the frontlines of health care in communities across the U.S. are well aware of the horrors of gun violence: prehospital care providers, emergency room (ER) doctors, trauma surgeons, nurses, and so many others who have the grave misfortune to see how bullets ravage the human body and soul. The kind of tragedy that once witnessed can't be unseen.\nThe article is based upon a peer reviewed article from the American College of Physicians. along with a number of suitable references from other professional sources.\nThe merger of leadership between Google and Geisinger Medical Clinic portends the development of artificial intelligence in the electronic health record.\nGeisinger President and CEO David Feinberg plans to step down from the health system to take charge of the healthcare efforts at Google, according to reports.\nThe health system, which services more than 1.5 million patients in Pennsylvania and New Jersey across 13 hospital campuses, confirmed that Feinberg would leave the company Jan. 3, 2019. Geisinger's executive VP and chief medical officer, Jaewon Ryu, M.D., will take over as interim president and CEO starting Dec. 1, before a month long transition period.\nAccording to CNBC, Feinberg would report to Google's artificial intelligence head, Jeff Dean, while working closely with CEO Sundar Pichai to develop a cohesive strategy for Google's various health and health-adjacent enterprises, including through home automation and wearables.\nDuring Feinberg's tenure at Geisinger over the past four years, the health system pursued several programs to integrate big data, electronic health records and genomics into its care, including through collaborations with pharmaceutical companies.\nGeisinger has also been working with Merck & Co. on two EHR programs designed to boost medication adherence, reduce drug errors and help connect patients and providers, and it has been in talks with health IT companies to roll them out to a larger marketplace.\nThe appendix has long been dismissed as an organ that has outlived its usefulness in human evolution. But new research suggests it may play an active \u2014 and detrimental \u2014 role in the development of Parkinson's disease.\nIn a finding that extends the link between gut and brain health in a surprising new direction, scientists found that people who had their appendix removed were 20% less likely to develop the neurodegenerative disorder than people who did not have appendectomies.\nWhat's more, surgical removal of the appendix seemed to forestall the appearance of Parkinson's symptoms, which include tremors, movement difficulties and signs of dementia. Among older patients in whom Parkinson's disease was eventually diagnosed, those who'd had their appendix removed experienced their first symptoms 3.6 years later, on average.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"T he Walt Disney Co.'s acquisition of Capital Cities\/ABC, by Wall Street's lights, has been a resounding success. Any misgivings about the movie and theme-park giant shelling out $18.9 billion for control of the television network have been dispelled by Disney's ability to continue reporting brisk earnings gains, and investors have responded enthusiastically. Disney's shares, which closed at 58 on July 31, 1995, the day the companies' betrothal was announced, not long ago topped 115. Even Disney's public disclaimer, a couple of weeks ago, of the Street's more exuberant expectations for its earnings had only a slight dampening effect. The shares closed Friday at 107.\nA spate of ho-hum movie releases during the second fiscal quarter (ending March) was the official explanation for the caution. Analysts dutifully trimmed a nickel or so a share from their forecasts, bringing the consensus to $3.17 for fiscal 1998, ending September. That still represents a healthy rise from last fiscal year's $2.75, and the stock is trading at better than 35 times earnings. In truth, however, the gains in Disney's reported results over the past five quarters have been significantly enhanced by creative accounting. Indeed, the fact that Disney has now virtually exhausted the source of this stimulus largely explains the anticipated earnings disappointment.\nThe accounting treatment accorded the merger by Disney, and signed off on by its certified public accountants, Price Waterhouse, allowed the entertainment company to create what amounts to an undisclosed reserve of as much as $2.5 billion to absorb costs and expenses incurred subsequent to the February 9, 1996, close of the merger -- costs that otherwise would have flowed through its income statement and reduced Disney's reported earnings.\nIn fiscal 1997, the device permitted Disney to show a glitzy 25% earnings gain instead of what would have been a not-quite-10% gain. In its most recent quarter, ended December, the company's merger accounting exertions transformed what otherwise would have been essentially flat earnings into a double-digit increase.\nIn a letter to this author dated Friday, John J. Garand, Disney's senior vice president for planning and control, strenuously defends the company's accounting treatment of the merger as \"appropriate\" and says it was mandated by generally accepted accounting principles (or GAAP).\nIndeed, when Disney took over Cap Cities\/ABC, it did so -- in strict accordance with GAAP for business combinations -- as a \"purchase.\" Which meant that the $18.9 billion Disney paid for the television network ($10.1 billion of it in cash, the remainder in 155 million Disney shares) was accounted for by first allocating the cost to Cap Cities\/ABC's identifiable assets and liabilities, based on estimates of the fair value of those tangibles. Then, whatever remained of the purchase price was allocated to the intangible asset known as goodwill.\nAccording to Disney's accountants, goodwill was virtually the only thing Disney got for its $18.9 billion. A footnote in Disney's annual report for the fiscal year ended September 1996 shows that the fair value of the Cap Cities\/ABC assets it acquired, at $4.8 billion, was nearly matched by the value of the liabilities it assumed, $4.4 billion -- leaving precious little, obviously, in the way of net tangible assets. In fact, an additional $749 million ABC liability, for deferred income taxes, was booked in the March 1997 quarter, reducing the total value of the net tangible assets that Disney acquired in the Cap Cities\/ABC transaction to less than zero.\nAs a result, more than the $18.9 billion purchase price was ultimately dumped into goodwill. Paying $19 billion for goodwill is essentially equivalent to forking over $30 billion for, say, TV and film properties, because the cost of goodwill is not tax-deductible (in contrast to normal business expenditures), making the after-tax impact of a deductible $30 billion expenditure roughly equal to a $19 billion nondeductible outlay.\nStriking, too, is the dramatic divergence between Disney's assessment of the fair value of Cap Cities\/ABC's net tangible assets and the balance sheet developed for the network by Ernst & Young, its independent certified public accountants, as of December 31, 1995 -- just 40 days prior to the closing of the merger. Indeed, Disney put Cap Cities\/ABC's shareholder equity on the financial equivalent of a miracle diet, making the TV network's $4.5 billion of pre-acquisition shareholders' equity essentially disappear in the translation onto Disney's books.\nWhat happened to those billions? Let's take Cap Cities\/ABC's assets first. A $2.8 billion reduction, to $4.8 billion from $7.6 billion, was attributable principally to the elimination -- routine in these circumstances -- of goodwill that had been carried on the network's books. Thus, that $2.1 billion intangible asset presumably was subsumed into the $19 billion of goodwill added to Disney's balance sheet by the transaction.\nThe remaining $700 million reduction in Cap Cities\/ABC's assets represents downward adjustments to the carrying values of its film properties as well as to property, plant and equipment. While a haircut of that size might ordinarily attract critical scrutiny, there are far more daunting issues involved here.\nNot least among them is the $1.7 billion increase, to $4.4 billion from $2.7 billion, in the network's liabilities. In fact, it is the key to unraveling the unorthodox purchase-accounting maneuvers that provided Disney with its reserve of as much as $2.5 billion to absorb post-merger costs.\nDisney's $700 million writedown of ABC's tangible assets and $2.9 billion increase in its liabilities, at the time of the merger, were merely book entries, and didn't entitle it to a current tax deduction. But as that $3.6 billion ripens into tax-deductible business expenses, the company will derive a $1.2 billion tax benefit. This contemplated entitlement is dubbed a \"deferred tax asset\" and is netted against Disney's deferred tax liabilities -- and thereby reduces the $2.9 billion gross increase in liabilities to $1.7 billion.\nThe $2.9 billion of additional liabilities was poured into the \"accounts payable and accrued liabilities line\" on Disney's balance sheet in connection with the merger and represents loss reserves and other liabilities added in the name of purchase accounting.\nJohn Giesecke who, until leaving last month was Disney's vice president for corporate controllership -- was consulted a number of times, by phone and fax, during the preparation of this analysis. When he was asked whether the entire $2.9 billion addition to Disney's liabilities represented the reserve booked on the Cap Cities\/ABC takeover, he indicated that the $2.9 billion figure was on the high side for the \"loss reserve.\" Some of the accruals, he pointed out, were related to the fact that Cap Cities\/ABC's package of perquisites had to be sweetened to bring them up to Disney standards.\nGiesecke would not commit himself on the actual size of the loss reserve. However, even taking more generous benefits into account, it's a reasonable estimate that these Disney merger-accounting exertions produced a decidedly unmousesized pile of $2.5 billion of accrued liabilities out of which the company could pay post-merger expenses without impacting its bottom line.\nGiesecke did not respond to faxes of January 7, February 3 and March 19, which attempted to apprise him of the gist of this analysis and the numbers critical to it. Disney's Garand, as noted, did respond late Friday. His letter took issue with the use of the term \"loss reserve\" to describe the $2.5 billion of liabilities created by the company's purchase-accounting -- stressing Disney's belief that it followed GAAP -- but he did not dispute the size of the reserve.\nHow did Disney's accountants justify the creation of that huge reserve -- justify adding some $2.5 billion in liabilities to its balance sheet? Especially liabilities that, 40 days before, hadn't existed on Cap Cities\/ABC's balance sheet? Basically, by asserting that Cap Cities\/ABC's accountants had ignored the impact of timing on anticipated cash flows from future programming that the network had agreed, at least in part, to finance.\nAs Disney's Giesecke explained, under the historical cost-accounting rules the network followed before the merger, Cap Cities\/ABC wasn't required to record a liability for losses on programming that it was committed to acquire in the future, as long as anticipated revenues were expected to recover the network's cost. After the merger, however, the controller maintains that Disney was required, under the dictates of purchase accounting, to evaluate those same commitments on a fair-value basis. In other words, Disney had to discount the expected future revenues and costs related to the network's programming commitments at an appropriate rate.\nDisney engaged Price Waterhouse to carry out those evaluations, Giesecke said, and when the work was completed, \"We determined that the fair value of a certain number of these commitments was negative and we recorded a corresponding liability.\"\n\"Commitments\" is a generic term; all undertakings are commitments. Most are so ordinary, ongoing and of relatively modest proportions that no special attention is given to them by accountants. Where, however, they are long-term and substantial some notice may be given to them, generally in the footnotes to the financial statements. For example, long-term leases, or -- in the case of Cap Cities\/ABC -- commitments to purchase future programming. Still, they're only claims that may occur if a contract is performed upon, in the future. The mere signing of a contract doesn't result in a completed transaction -- much less a liability.\nThat's because \"liabilities,\" in accounting parlance, are recognized obligations to pay money, provide services or transfer specific assets, and are tallied as such in the accounting cycle. They must be fully disclosed in financial statements and are subtracted from a firm's assets to derive its shareholders' equity.\nCap Cities\/ABC programming commitments totaled $4.1 billion at year-end 1995. The network's final audited financials (submitted to the SEC in a Disney 8K report dated March '96) disclosed that these consisted of contracts to purchase \"broadcast rights for various feature films, sports and other programming\" during the next five years. There was nary a hint in ABC's financials, however, of anxiety on the part of ABC management or its auditors regarding the economics of those programs or projects. To the contrary, all were deemed to have been undertaken in the normal course of ABC's business -- there was nothing contingent about them.\nIndeed, the plain truth that Disney ignored in applying a discounting factor to future revenues anticipated on programming Cap Cities\/ABC was committed to purchase (and in thereby creating those $2.5 billion of \"liabilities\") was that there was nothing novel about those contracts. Over its half-century of existence, the network's ordinary operating cycle had always encompassed both commitments to purchase future programming and current income representing the \"ripening,\" or coming to fruition, of past programming outlays. In short, the network's programming commitments were simply a consequence of the ordinary operating cycle of a going concern.\nIn fact, in this author's view, Disney's booking of those $2.5 billion or so of additional liabilities related to the network's future programming commitments as part of accounting for the merger as a purchase was simply not permissible under GAAP. Only liabilities which are identifiable as such for the acquired enterprise, or contingencies that may have \"ripened\" into liabilities as of the acquisition date, should properly have been booked as liabilities by Disney.\nThis interpretation of the accounting rules is fully supported by an October 13, 1997, letter from the Financial Accounting Standards Board, which (in part) reasserted the board's long-held position that the same rules should apply to \"recognition and measurement provisions arising from a business combination\" and to \"those incurred in the normal course of business.\" FASB's position is unambiguous: If, pursuant to its Statement 5 (which deals with contingencies), no such undiscounted\/discounted liability reserve is permitted, one cannot be created in the process of accounting for a merger.\nAnd there's no way Disney's $2.5 billion reserve for the discounted value of the network's programming commitments would qualify as a liability under FASB Statement 5. That litmus test is whether \"a liability had been incurred at the date of the financial statements.\" If that test could have been met as of December 31, 1995, Cap Cities\/ABC's accountants would have been constrained to reflect a $2.5 billion liability in the network's final financial statement. But there was no such liability then -- neither did one come into existence during the succeeding 40 days while the merger was being closed.\nDisney and its auditors clearly interpreted GAAP far more aggressively in transmuting some of those $4.1 billion of commitments, per ABC's final annual report, into a $2.5 billion undisclosed liabilities reserve. Under the circumstances, investors might reasonably have expected, at the very least, a clear explanation of the purchase-accounting adjustments to Cap Cities\/ABC's programming commitments in Disney's financial statements. Yet none is to be found.\nYet the reason Disney went out on a limb to create its undisclosed reserve is clear enough: flexibility. It meant that as its new television arm ran up various programming costs after the merger, Disney had the option of merely writing those amounts off against those accrued liabilities instead of running them through its income statement -- where they would have crimped reported profits.\nIn the immediate aftermath of the February '96 merger, and, in fact, all the way through the quarter ended December 1996, Disney had no call to draw on its big reserve. The comparative year-earlier figures included in its financial reports throughout that stretch incorporated the operations of Cap Cities\/ABC on an entirely pro forma basis. Midway through its March 1997 quarter, which marked the first anniversary of the acquisition, that changed: Disney was now going head-to-head with real numbers generated by the network while within its ambit -- and those were less than ebullient.\nA couple of unheralded adjustments that Disney made to its books for that quarter first signaled that Disney was dipping into its $2.5 billion reserve to cushion earnings. Without a whit of explanation, the March '97 balance sheet showed a fiscal '96 year-end value for the asset dubbed \"TV and Film Costs\" that had been retroactively decreased by $653 million from the amounts listed in Disney's fiscal '96 annual report -- and a corresponding decrease on the liabilities side of the ledger.\nWhen queried, Disney's Giesecke indicated that the balance sheet was adjusted because a previously recorded valuation allowance for film and TV costs had been reclassified. He also commented that the reclassification had \"no income-statement impact.\" In his Friday letter, Disney's Garand reiterated that the balance-sheet reclassification, a final purchase-accounting adjustment related to the merger, affected only Disney's balance sheet.\nWhich is true, in the sense that the reclassification didn't involve any changes to the income statement for fiscal '96. But they laid the groundwork for enhancing earnings in the future. The $653 million represented production costs the Disney broadcasting unit had run up during fiscal 1996, costs Disney initially had treated conventionally: They'd been added to the \"TV and Film Cost Assets\" line on Disney's balance sheet, and fiscal '96 operating net had been reduced by some $200 million in amortization of that asset.\nWhat Disney in March '97 determined, however, was that those $653 million of costs should have been written off against its $2.5 billion reserve. That way, its operating income in future periods wouldn't be burdened with the amortization of the network costs. Hence, its retrospective balance-sheet adjustments.\nWhat Disney didn't do, though, was restate its already-reported fiscal '96 earnings to reflect the benefit of the lower amortization charges produced by this accounting device.\nDisney's Garand insists in his letter that \"it would be inappropriate to consider restating\" the company's fiscal 1996 income because the retrospective changes it made affected only its balance sheet on the one hand, and, on the other, were merely offsetting adjustments to a couple of lines in its statement of cash flows. However, in fact, the company's reported fiscal 1996 income had been reduced by amortization of those $653 million of costs that Disney subsequently reclassified. And a logical inference is that the company's earnings weren't restated because the object of these accounting exertions was to enhance future earnings comparisons.\nAnother retrospective change Disney made to its financial statements for the year ended September '96 provides a clue to quantifying how much the company used the reserve to absorb costs in succeeding periods. Specifically, Disney's fiscal '97 10K listed a decline in the deferred-tax asset, as of September 30, '96, of $241 million from the total it had reported a year earlier. Which was how much Disney was able to reduce its deferred taxes by retrospectively charging the $653 million of costs against its reserve.\nIn fiscal '97, Disney's deferred-tax asset was further reduced by $493 million, to $1.129 billion. Assuming, again, a 35% corporate tax rate, that implies that during that year, about $1.4 billion, previously stashed away in Disney's accrued liabilities reserve, was spent -- without impacting the company's operating statement. Outlays of $1.4 billion would entitle the company to apply $493 million of the related tax asset to reduce its current tax provision.\nDisney's Garand, in his Friday letter, insists that this $1.4 billion estimate is incorrect; that the amount of costs absorbed can't be calculated based on the change in the company's deferred-tax asset, which \"resulted primarily from unrelated factors.\"\nNonetheless, its most recent quarterly report indicates that the company ran through virtually all of the remaining $450 million or so in its liabilities reserve during the quarter. Which strongly suggests that the reserve absorbed around $1.4 billion of costs during fiscal 1997. And, to reiterate, that's what likely prompted Disney's caution to analysts to rein in their full-year earnings estimates.\nHow much did Disney's treatment of those TV production costs, totaling $653 million, $1.4 billion and $450 million, respectively, boost its bottom line during fiscal '96, '97 and the first quarter of '98?\nAccording to Disney's 1997 statement of cash flows, it had charged $204 million of amortization against those $653 million of TV & Film Cost Assets during the 7 1\/2 months of fiscal '96 that the network had been in its fold. Which means that a full year's amortization of those assets would have amounted to roughly $320 million, or about $80 million a quarter.\nThe accompanying tables use the same amortization rate to quantify the amortization charges Disney avoided in fiscal '97 and in the December quarter by writing off that total of $2.5 billion of television production costs against its undisclosed reserve. They also give Disney the benefit of some accounting charges it would have avoided had it not employed that accounting device, most notably a reduction in its amortization of goodwill. While, by necessity, they're only estimates based on the publicly available data, the calculations demonstrate that by calling on the reserve, Disney supercharged its recent earnings comparisons. More specifically, it avoided some $840 million in charges against its fiscal '97 operating income, and around $290 million of such costs in the December quarter.\nClearly, had Disney not been able to use its reserve to shield its bottom line from major chunks of costs related to its foray into television network ownership, its recent results would have had considerably less luster. Indeed, the 25% earnings surge Disney reported in fiscal '97 would have come to 10%, without benefit of the accounting device. And, far from an 18% earnings increase in the December quarter, net would have been flat.\nDisney, through the agency of accounting, has adeptly masked the negative impact of its Cap Cities\/ABC acquisition on its earnings over the past two years. But even accounting magic has its limits: From here on, the true picture will become very much clearer.\nABRAHAM J. BRILOFF, a CPA and a frequent contributor to Barron's over the past 30 years, is the Emanuel Saxe Distinguished Professor Emeritus at Bernard M. Baruch College and Presidential Professor of Ethics and Accounting at Binghamton University, SUNY.\nIs the entertainment giant's accounting pure Mickey Mouse?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get my newsletter for regular updates and de-cramping tips. You also get a PDF of '22 Ways to Release Your Creativity'.\nTop tips and inspiring quotes to help boost your creative mojo!\nEnter your name and email to sign up to my newsletter for Tips and Prompts to Help Flex Your Creative Muscles.\nYou also get a FREE PDF of '22 Ways to Release Your Creativity' and Free Your Inner Creative!\nDon't Let Your Creativity Get Cramped!\nYou can easily unsubscribe from the newsletter at any time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbacjj b/data_all_eng_slimpj/shuffled/split2/finalzzzbacjj
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+{"text":"A THIRD Russian spy suspected of being involved in the nerve agent poisoning of the Skirpals in Salisbury might still be INSIDE the UK, it was feared last night.\nA military intelligence officer using the likely fake name of Sergey Fedotov was booked on the same flights to and from Britain used by the two known would-be assassins.\nBut he suddenly checked himself and his baggage off the Moscow-bound plane shortly before departure, and it's not known when \u2013 or if \u2013 he left the country.\nOne possibility is that Fedotov, whose passport said he was 45, remained in the UK after discovering Sergei Skripal, 67, and daughter Yulia, 33, had survived the attack.\nFedotov's fake passport number was only a few digits different from the ones used by Russian intelligence agents Alexander Mishkin and Anatoliy Chepiga \u2013 who were also travelling under false names.\nHis true identity has not yet been discovered, and it's not clear what role he played in the attempted hit of Sergei Skripal \u2013 a former GRU intelligence officer who betrayed the Kremlin by selling secrets to MI6.\nSome experts believe Fedotov could have brought the Novichok nerve agent into the UK or carried out secret reconnaissance to let the others know of Skripals' movements.\nThe revelation Fedotov was supposed to be on the same March 4 flight as the other two Russian agents was broken by respected Russian news website Fontanka.\nAlthough the Skripals miraculously survived the nerve agent poisoning, they're likely to have been left with health problems as a result.\nAnd Brit Dawn Sturgess died after inadvertently spraying herself with the substance, which had been disguised in a perfume bottle and discarded by the hitmen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This news [see below] is vindication for the people of Norfolk \u2013 both in the Costessey\/Norwich area and in the Kings Lynn area \u2013 who have fought for a decade against this absurd incinerator project, supported the whole while by the Green Party. Our congratulations to the Green Party's Michael De Whalley, co-founder of the anti-incinerator campaign KLWin, and to all who have fought by his side. The news is however political disaster for the Conservatives and for George Nobbs, Labour Leader of Norfolk Couty Council, who have unwisely backed this calamitiously-bad incinerator project: calamitous from a financial point of view, and also of course, if it had gone ahead, from an environmental point of view.\nHuman-influenced climate change will spiral out of control, worsening poverty, inequality and conflict, unless we put renewables and energy conservation at the heart of energy policy.\nThat was the warning today from Rupert Read, Green Party lead candidate in the east at the European elections, after scientists published the latest Intergovernmental Panel on Climate Change report today (March 31st).\n\"We should see this report not as a document of doom but as a challenge we can rise to with the right thinking.\nThe new report warns that no-one will be immune from the effects of climate change, but the poorest around the world will be disproportionately affected by food shortages, heatwaves, flooding and other weather-related disasters, unless Governments act to reduce carbon and other emissions now.\n\"The report makes stark reading,\" said Dr Read \"Many of us were aware of the effects of climate change in other countries, but this winter's weather has brought the issue home to the East Anglian coast.\n\"The EU will really need to play a leadership role in championing the targets proposed by the Greens to reduce emissions by at least 60% by 2030 \u2013 probably much more \u2013 and to set targets for adopting energy conservation and renewable energy generation measures.\nThe First World War \u2013 and the EU.\nPARTY AT AN EVENT IN CAMBRIDGE ON TUESDAY.\nImpington and Histon Climate Change Action.\nCB24 9EP, AT 1930 ON TUESDAY MARCH 25TH.\nOne of Britain's best known environment campaigners, Sir Jonathon Porritt, will be in the east of England today, Budget Day (Wednesday March 19th), calling for budget measures to address the long term cost-of-living crisis caused by the \"end of oil\".\nHe'll be in Norwich and Gt Yarmouth to launch a new report by Rupert Read, lead candidate for the Green Party in the east of England at the Euro elections in May, calling for an \"energy revolution\" in the region encompassing tidal, wave, wind and solar power plus energy waste-reduction to replace outdated fossil fuels and nuclear.\n\"The destabilisation of the world's climate systems \u2013 which we had another taste of in Britain with the widespread flooding earlier this year \u2013 urgently requires us to rethink how we provide ourselves with the energy we need for our homes, transport, and industry.\n\"Our chances of staving off man-made climate change are closely linked to the choices we make about energy use; everybody knows that now. But the Prime Minister has made his choice clear by talking about \"green crap\" in relation to energy obligations and going \"all-out\" for fracking, with all its associated risks and damages.\nLater in the day, Sir Jonathon will be in Suffolk with Rupert at the invitation of the group \"Together Against Sizewell C\". He'll give a talk entitled \"Why Nuclear Power Still Makes No Sense\".\n* Renewables to get the same tax breaks as fracking now gets \u2013 tax-breaks which should be removed from fracking, and a moratorium on fracking instead imposed, as in France.\nGt Yarmouth, where the report will be launched, has been identified in research as a seaside town in need of regeneration and job creation. It is on the East Anglian 'energy coast' with all its potential for renewables development, and on the River Yare, which could be protected from future tidal surges and simultaneously generate tidal energy, in a manner laid out in the report.\nThere's been a lot of discussion about exactly why the demand for foodbanks here in the east of England is rising so quickly. Could there be a link between the number of people forced to use food banks and the number having their applications for benefit rejected? The fact is, those rejections have soared under the coalition government.\nGreen Party analysis of figures for our region compiled by the Department for Work and Pensions reveal that \"adverse decisions\" against claimants for Job Seekers Allowance and Employment and Support Allowance leapt from about 19,000 to almost thirty thousand between 2008 and 2009. By last year that number had doubled again to nearly sixty two thousand. We believe that this is one of the primary causes of the increased pressure on charitable foodbanks.\nWouldn't it be wonderful if the Government could instead target the tens of billions that are being ripped off us all by a minority of bankers and tax-evaders?\nRadical ideas and suggestions on how to respond to the threat of global warming from some of the UK's leading experts and campaigners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Greater Toronto Area Spills Assessment was study project managed by Chris to develop a spatial interpretation of the Ontario Ministry of the Environment's spills database. The purpose of the study was to identify areas of frequent spill activity and to develop a predictive model that could be utilized to determine risk, target outreach efforts and develop spill response strategies.\nThe development of the study led to a number of recommendations for improving the management of spill data, as well as the development of Region of Peel Spill Decision Support System - a web-based tool for responding to spills.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most Danish companies can be categorised as small and medium-sized enterprises (SMEs)1. If SMEs do not fully contribute to creating competition for public contracts, it may mean that the prices of contracts increase and that public expenditure is therefore increased, as competition is restricted to a small part of the market and is thus not as effective as it could be.\nThe Danish Public Procurement Act, which came into effect on January 1, 2016, introduced a number of measures intended to improve SME access to competition for public contracts. For example, the procuring entity has to justify not choosing to divide a contract into smaller partial contracts.\nWhen small businesses wishing to bid for a public contract are faced with requirements regarding e.g. sales volume, it can make them consider joining up with other companies to form a consortium to bid for the contract. Collaboration can also increase the chance of winning a tender if the contract is large and complex, compared to the type of tasks a company usually works with.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome Rajasthan Tour is an eight night tour package by manoyatra for travelers with deep interest in Rajasthan, India. In this tour package you will visit Bikaner, Jodhpur, Ranakpur, Udiapur, Pushkar & Jaipur. Enjoy boat Ride at Lake Pichola, Elephant ride at Amber fort and Rajasthani traditional Dinner at Chokhi Dhani with your loved ones which is included in the price of your tour package. A memorable trip awaits ahead! Book now.\n01 Night Bikaner | 01 Night Ranakpur | 01 Night Jodhpur | 02 Night Udaipur | 01 Night Pushkar | 02 Night Jaipur | 3 Star Hotel Accommodation | Daily Breakfast | Transfers | Private guided sightseeing | Boat Ride at Lake Pichola | Elephant Ride at Amber Fort | Dinner and Entertainment at Choki Dhani.\n\u25cb 08 Nights accommodation in well appointed A\/c room in well rated hotel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Many of the negative reviews posted are caused buy lack of basic maintenance, such as blocked condensate trap.\nergo baby carrier Many of the posts sound like lack of knowledge by the service tech and or customer. I have been installing and maintaining Carrier furnaces for 18 years, a quality contractor is key. I have installed many other brands as well, but Carrier seems to have the edge on everyone. Talk to neighbors and friends to get a local review of your hvac contractors.This was supposed to be a top shelf furnace and yet it has needed a service call every year I have had it for some type of repair. When it is running it is efficient however when I have to pay for a $150 service call every year it makes it an expensive furnace to own. Lots of different issues have come up with this now my inducer fan has broken and it is a $500 part replacement so I am looking for a new furnace. Does someone want a rusty and unreliable furnace for a cheap price?I'm on my second ergo baby trage Carrier furnace in 7 years. Both have had a cracked heat exchange. Both were determined by the Carrier dealer to be a manufacturers defect. Carrier refuses to address the issues, refuses to replace the unit and even refuses to return phone calls (Mike id1608 on the national service number). If you are in the market for a furnace you would be wise to stay away from any Carrier product.Inducer blower replaced 3 times, igniter replaced once, condensate tube replaced once, pressure switches replaced once, condensate trap plugged once. Awaiting the heat exchanger(s) to fail. The Class Action Lawsuit won't help, our unit is too old. You cannot expect any meaningful warranty coverage from Carrier.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When considering its environmental impact, coach travel is one of best forms of transport, typically producing up to seven times fewer CO2 emissions compared to a car per passenger carried.\nAll our coaches are compliant with the London Low Emission Zone (LEZ) regulations which can reduce hefty coach hire surcharges levied on older less environmentally friendly vehicles.\nWe are constantly updating our fleet to ensure our vehicles comply with the latest environmental standards.\nWe have recently launched a unique corporate chauffeur service to the Cambridge area called e-zero which utilises 100% electric vehicles. We are working closely with associated manufacturers to evaluate how this technology performs in larger vehicles such as mini-coaches and coaches. We expect to be the first in the Cambridge area to introduce this technology as it becomes available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello! Undergrads Union Grads (formerly 'Undergrads Under Grads') is continuing its fifth year in the NCSU math department. We are a mentoring experience with graduate students aimed at helping undergraduates who are under-represented in the math major.\nMentoring is made up of short, informal meetings on a monthly basis. Undergraduates can talk about their current courses, goals, or professional experiences with their graduate mentors.\nWe are currently planning events for the 2018 Spring semester; we expect to run at least one department-wide session related to Graduate School applications to get everyone ready for the summer.\nOur email is uug@math.ncsu.edu, or you can contact any of our organizers with questions.\nOur faculty advisor is Kevin Flores.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"River Custom Canvas, LLC is proud to be a member of the Clayton Chamber of Commerce. Come visit us at the Clayton Boat Show April 5th-7th 2019. We expect to be located in the town maintenance building next to Spicer Marine Basin and Whites Lumber. We will have samples of both AquaMat carpeting and Luxury Woven Vinyl flooring as well as varius fabric samples. Hope to see you there!\nWe have had a great winter break and are excited about the upcoming 2019 boating season. We expect to be open for business again by the first weekend in April. Projects are already stacking up, so please call us at 315-480-5885 if you want to get on the spring schedule.\nIf you had work done by us last year we would appreciate a review on Yelp to help our listing grow. Thank you for your business, see you on The River!\nRiver Custom Canvas, LLC is an authorized dealer for Corinthian Marine Carpets in the Thousand Islands area! We install custom AquaMat snap-in carpeting and carpet mats with AquaLoc and Hydrofelt water repellant backings. These custom made carpets will be bound and can be installed with or without snaps!\nIn addition, we will have a full line of glue in marine carpeting available for interior or sidewall use.\nThere are several new materials and backings available today to give your boat that fresh new look, please stop by this spring to look at the sample materials we have available. We are here for all your boat canvas and boat carpet needs!\nThere are a lot of variables to consider other than just price when selecting the right canvas shop to work on your boat. The methods and materials used very greatly from shop to shop, and those choices not only affect the price you will pay, but also how long it will be before you will have problems or need replacement or repairs.\nOne midweek day early last summer Barbara and I walked a few marinas and looked at the various canvas jobs and enclosures on the boats. We saw a few factory jobs on newer boats, but most of the boats in the Clayton NY area are older and had replacement tops and windows done by various canvas shops.\nThe diversity of the manufacturing methods and the materials used was shocking, and while they looked ok from 50 feet away, the low budget jobs stood out in the crowd upon closer inspection.\nWhat we saw was that many were made using the cheapest possible materials and methods for fast production and a low manufacturing cost. The end result is canvas that will need to be repaired or replaced well before its recommended lifespan, which in the thousand islands area should be at least 10 years.\nWhile most shops are using Sunbrella, there's a lot more to a canvas project than just the fabric itself. One of the easiest short cuts to hide is simply by using the cheapest thread possible. The end result is canvas seams that rot in about 4 years and require major repairs. An 8oz spool of thread can vary in price from $30 to $200 depending on what is used. The higher quality threads are often harder to sew with and add time to the job, hence the reason a shortcut may be taken using cheaper materials.\nNext is how the canvas was seamed and how it was cut. The right method is to use a hot knife to cut the fabric and melt the seam so it doesn't fray and to double stitch a seam when two panels are joined together, which is how most tops are made.\nOther considerations include what's used for facing, which is the inside 1 \u00bd inches of fabric around the perimeter of canvas and vinyl windows. Was it custom cut or is it lapped over and sewed down around curves or pieced together using small scraps? Did they use actual facing made from fabric or did they use webbing? Did they reinforce the stress points?\nThen of course there is the vinyl used in the windows. While many of the smaller boats do very well with the cheaper products that are easy to rollup in our climate, many boaters demand a crystal clear wrinkle free product and there are huge differences in those prices. If it wasn't discussed when your job was quoted, you are probably getting the low end of the product line.\nAnd lets not forget about framing, the base of your entire enclosure. The choices vary from those made with cheap aluminum using plastic fittings to stainless steel frames in a variety of thickness, grades and tube sizes.\nSo if you're shopping for replacement canvas and are only focused on price and finding the cheapest shop, you will most likely get exactly what you paid for.\nRiver Custom Canvas is committed to providing the highest quality manufacturing methods using only high quality materials that will last. We take the time to explain the differences in our manufacturing and give our customers several options to choose from when quoting a project, but we will always maintain a higher standard in the actual production methods. The end result is a beautiful finished project that will last for years with proper care and maintenance.\nWe are located in the heart of the 1000 islands in Clayton, NY. Our target markets are Clayton, cape Vincent, Alexandra Bay, Watertown, and the surrounding areas in Jefferson County.\nRiver Custom Canvas is proud to be a member of the Marine Fabricators Association. The MFA offers many resources, training opportunities and certifications to it's members. We are committed to the highest quality fabrication and exceptional customer service.\nRiver Custom Canvas, LLC is proud to be an authorized Sunbrella fabricator, and we have access to the whole line of Sunbrella products. Sunbrella is the #1 trusted name in marine fabrics.\nBecause Sunbrella fabric is woven for breathability, it doesn't trap heat and moisture, eliminating mildew. As a result, Sunbrella stays fresh and beautiful season after season.\nIn addition to longevity and durability, Sunbrella fabric also provides protection from the sun so you can enjoy long hours on the water. Sunbrella fabrics carry the Skin Cancer Foundation's Seal of Recommendation, giving your family and friends the extra assurance they need.\nWith a protective enclosure made of Sunbrella fabric, you can make use of your boat earlier in the spring and later in the fall, extending your enjoyment and maximizing your investment.\nWe are focused on providing custom boat canvas to our clients in Clayton, NY and Thousnad Islands region. Please Contact River Custom Canvas, LLC to discuss your boat canvas needs!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New to the forum, and just had my first experience with OCROLEXGUY. Absolutely wonderful expierence, top notch seller! Will definitely be doing some more business with him in the future.\nThank you for your trust and business. We look forward to working with you in the future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..760ed819fa4392847bd6cae25c9465bc7c13abcd
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@@ -0,0 +1,5 @@
+{"text":"Classified liners\u2022 Mondays, 10 a.m.\nDisplay ads \u2022 Fridays, 5 p.m.\nmonths of mid May - mid Sept.\none year. No inside animals.\nFull time job with good pay.\nmonth (for 12 mos.) Order Now!\nAll apps are free to download. Must be a subscriber to view e-Edition.\nnate it to the Humane Society.\nStream videos, music and more!\nAT&T Unlimited Talk and Text.\nADDING a room to your home?\nor stop by the Community Connection office.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On your free honeymoon registry, you will list all the special things that will be part of your \"dream honeymoon\". This makes it possible for you to pre-plan for what you really want on the most important trip of your life. It works just like a gift registry, only your guests give you portions of your dream honeymoon. The perfect, unforgettable gift.\nHundreds of ideas covering areas all over the world.\nPhotos and descriptions for all items.\nYou can make your own travel plans.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TUSCALOOSA | David Jones saw the best of Alabama's community in the worst of times. Jones, a Notre Dame cheerleader, was part of a group of two dozen Fighting Irish athletes that came to Tuscaloosa during fall break in 2011. They visited for a week of cleanup and hard work in a grieving town where more than 50 were left dead because of a tornado a few months earlier. Now, a year and a half after that week of bonding and goodwill, Notre Dame and Alabama are getting ready to play each other in the BCS National Championship Game on Jan. 7 in Miami.\nTUSCALOOSA | David Jones saw the best of Alabama's community in the worst of times.\nJones, a Notre Dame cheerleader, was part of a group of two dozen Fighting Irish athletes that came to Tuscaloosa during fall break in 2011. They visited for a week of cleanup and hard work in a grieving town where more than 50 were left dead because of a tornado a few months earlier.\nNow, a year and a half after that week of bonding and goodwill, Notre Dame and Alabama are getting ready to play each other in the BCS National Championship Game on Jan. 7 in Miami.\nThere were no football players during that weeklong service project dubbed \"Fight for Tide.\" They were preparing for the 2011 season, after all.\nIt wasn't about football, anyway. Or basketball. Or softball.\nThe service project began with a call from Tim Cavanaugh, assistant director of Alabama's ticket office, to Notre Dame program coordinator Sarah Smith seeking donations. The two interned together in South Bend.\nThat call resulted in clothes shipped to Tuscaloosa, and ultimately the 675-mile bus ride and weeklong trip.\nOther schools' students and fans also offered a hand in Tuscaloosa, including those who pull for Auburn.\nTragedy trumps rivalry any day.\nThe Notre Dame group cleared out lots, hauled debris and dug a ditch. There was also lots of listening.\n\"Everyone had a story: Where they were, who in their family was affected,\" Smith said. \"I think it was kind of healing for them to tell their story to other people and that we were down there showing that we cared.\nTide softball coach Patrick Murphy and some of his players worked with the Notre Dame athletes on a site a few miles from campus in Alberta City and took them to dinner.\nAlabama athletic director Mal Moore, a former Notre Dame assistant football coach, asked to meet with the group from South Bend. He gathered them at midfield in Bryant-Denny Stadium during a stadium tour.\nSmith, meanwhile, said she was taken aback by the southern hospitality. They got a few cultural lessons on things like the meaning of \"Roll Tide\" and the houndstooth gear popularized by iconic Tide coach Bear Bryant.\n\"It was just so cool to share some time together,\" Smith said. \"I definitely have a soft spot for Alabama.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Motivation: The Kyoto Encyclopedia of Genes and Genomes (KEGG) Pathway database is a very valuable information resource for researchers in the fields of life sciences. It contains metabolic and regulatory processes in the form of wiring diagrams, which can be used for browsing and information retrieval as well as a base for modeling and simulation. Thus it helps in understanding biological processes and higher-order functions of biological systems. Currently the KEGG website uses semi-static visualizations for the presentation and navigation of its pathway information. While this visualization style offers a good pathway presentation and navigation, it does not provide some of the possibilities related to dynamic visualizations, most importantly, the creation and visualization of user-specific pathways.\nResults: This paper presents methods for the dynamic visualization, interactive navigation and editing of KEGG pathway diagrams. These diagrams, given as KEGG Markup Language (KGML) files, can be visually explored using novel approaches combining semi-static and dynamic visualization, but also edited or even newly created and then exported into KGML files.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Three fishwives from Newhaven, Edinburgh beside a stone wall. They are wearing striped skirts, aprons and black cloaks. One is standing carrying a creel on her back, whilst another leans over the seated woman, examining what she is holding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcrlz b/data_all_eng_slimpj/shuffled/split2/finalzzzbcrlz
new file mode 100644
index 0000000000000000000000000000000000000000..94345fe9e3c6185752b5721b57b8ded50e767e48
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbcrlz
@@ -0,0 +1,5 @@
+{"text":"Singapore influencer Mongchin Yeoh (@mongabong) has shared many aspects of her wedding journey with The Wedding Scoop, from her surprise proposal and pre-wedding shoot in Greece to her wedding vision, wedding venues and vendors, and dream wedding gown.\nToday, we're excited to share stunning snaps of Mongchin and Matthias' church solemnisation (finally)! Captured by Bobby Kiran Photography and Linstudios, styled by Wulala and planned by Smitten Occasions, the ceremony at True Way Presbyterian Church was filled with blush, soft pink and cream tones, from the floral arrangements to the reception table set-up and the colour of the bridesmaid dresses.\nMongchin glowed in an off-shoulder mermaid-cut gown, while Matthias was smartly dressed in a three-piece black suit. Before family and friends, they pledged eternal love for each other, with Mongchin shedding tears during this emotional and precious moment.\nPost-ceremony, guests headed for a delectable buffet lunch spread, which they had only good things to say about!\nHow did you meet and fall in love, and how did you know he\/she was The One?\nHow did you decorate your venue?\nDid you undertake any DIY projects?\nAny memorable stories relating to your wedding?\nAny tips or wedding advice you have for other brides and grooms to-be?\n\"If things don't go your way, remember to be prepared to let it go and shrug it off! Don't let it affect the rest of your big day.\nFor planning: If you are both busy with work, engage a professional wedding planner! I was trying to hold the fort for the first one year before finally caving in and hiring a professional wedding planner six months before the wedding. It made SO MUCH of a difference because Ju Lynn of Smitten Occasions took so many burdens off of me in terms of liaising and checklists. She was the point of contact even on the actual day and I was not bothered by a single thing at all. Thinking back, I should have hired her earlier!\nIn terms of cost savings: Think about your guests and what you think they'd like. For me, I was very particular about the quality of food served to my guests. That is why I opted to go for a premium lunch buffet menu with a caterer\/cafe that I frequent. True enough, it was what everyone remembered fondly about our wedding - the food!\nI skipped other things like a photo-booth, but instead created a space that was nice enough for them to take photos using their phones at the reception area because I figured my guests would prefer to have their photos in their phones instead of hard-copied prints these days.\nIf you're doing a church + dinner banquet wedding like I did, remind your stylist and florist to re-use the flowers and decor so that you can save costs!\nAny tips or advice on what you think makes a strong marriage?\nCongratulations on your marriage, Mongchin and Matthias! May it be every bit as beautiful as your wedding!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This office is located, just off the A556 with plenty of parking and a 5 minute walk to the train station. The town of Northwich is established in the ceremonial county of Cheshire, less than twenty miles from both Chester and Warrington. The economy of Northwich was historically based on the production of salt, while today the chemical industry is the key economic driver.\nNorthwich is home to the first electric powered swing bridges in the UK, Town Bridge and Hayhurst Bridge. The town is easily accessible by the M56 and M6 Motorway networks as well as by train, with direct links into Manchester.\nWe offer 3 Rooms and breakout area \/ kitchen with computer and printing facilities. Two of Offices are fully equipped training rooms that can accommodate up to 20 students the other is a meeting room\/ office facility.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The team has identified a molecule sent by fat cells to the fly brain when its energy stores were sufficient and inhibited the insect's feeding.\nBecause fruit flies replicate many of the feeding-related regulatory mechanisms and genes found in humans, they make a good model for the search for such an inhibitory signal.\nThe discovery was made by Walton Jones and his colleagues at the Korea Advanced Institute of Science and Technology.\n'Our study indicates fat tissue sends a molecular signal to the fly brain to regulate feeding behavior,' said Jones.\nFat is the primary long-term energy storage molecule in animals, and the control of fat levels is critical for survival.\nIn mammals, the hormone leptin induces eating in response to fat loss, but so far, no corresponding signal has been identified, either in mammals or any other animal, that inhibits eating in response to fat gain.\nDuring experiments, Jones and his team focused on short non-coding RNAs or microRNAs, which are well-known inhibitors of gene expression.\nThey first identified MicroRNAs, because this affects the fly's feeding behavior when it is overexpressed in fat tissue.\nAnd team also looked for genes that target those microRNAs.\nThe team discovered a microRNA called miR-iab-4 increased feeding in fruit flies by more than 27% and a target gene called purple, which was expressed in fat bodies.\nOnce it reaches the brain, PTP is transformed into an enzyme cofactor called tetrahydrobiopterin (BH4), which neurons use to produce a neuropeptide \u2013 a small protein-like molecule that regulates feeding.\nWhat the team discovered was, when purple was removed in the fat body or there was a loss of BH4 in neurons, NPF was released more frequently and feeding increased.\nOn the other hand, increasing BH4 in neurons reduced NPF release and decreased feeding.\nIt was also discovered that feeding flies a low-calorie diet reduced expression of the fat body enzymes that control BH4 production and increased feeding.\nIn the end, the team concluded that BH4 was a key player in suppressing a fly's appetite.\nAnd the PTP released from fat bodies tells its brain that it has stored enough energy and can stop eating.\nAlthough these findings can only be related to flies, the team believes the identification of this appetite-suppression mechanism will surely spur research into related pathways in humans.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are looking for a home to rent in South East Wales, then start your search here. Serenliving's letting agency offers a choice of properties and rent-types to suit all needs.\nWe offer homes for LHA, intermediate and market rent and with more landlords choosing to let their properties with serenliving lettings every week, your choice of great homes keeps on growing. We offer homes across South East Wales, so if you're looking to rent property in Cardiff, Newport, Caerphilly, Torfaen, Monmouthshire or Blaenau Gwent Serenliving is your letting agency of choice.\nWe're confident that no other letting agency in South East Wales offers the range of properties and tenancy types that we do and we're always thinking of new ways to help you get the home you need, so for us, letting means letting you find your perfect home at the right price.\nSerenliving has independent redress provided by the property ombudsman.\nIntermediate rent offers you the opportunity to rent a good quality home at less than you would expect to pay in the open market for a similar home in a similar area. If you are in employment and your net household income is less than \u00a330,000 a year then this could be a great option for you. It has the added advantage that you don't have to pay any of the usual sign up or administration fees that other agents impose. You should be able to pay the rent monthly in advance by direct debit and will need a tenancy deposit of the monthly rental value plus \u00a3100.\nProperties usually come with carpets and floor coverings and some have white goods and window coverings provided. Homes are let on Assured Shorthold Tenancy agreements and are let for a minimum of 6 months. On many homes, provided the terms of the tenancy are adhered to, the properties are available well into the future.\nInterested? Search for available properties \u2013 You can also register with the Local Authority and ourselves, so if you are looking for an intermediate rental home it would be beneficial for you to register. Registering with the local authority will also mean that you see the full range of affordable housing options available. For details of how to register follow the following links.\nThen call the serenliving team on 01633 678545.\nWe will need to take a few basic details from you before making an appointment for you to view the property.\nIf you like the property then we will need to make a few checks, including a credit and address check prior to you being able to move in.\nwhat information do you need to bring with you?\nWe operate a property management service on behalf of Private Landlords and therefore properties advertised under the Lettings Agency Banner are owned by Private Landlords and are let and\/or managed by serenliving.\nRents offer great value so whether you are in employment or in receipt of housing benefit our properties could be right option for you. As with allserenliving properties you won't have any of the usual sign up or administration fees that other agents impose. Landlords usually require a bond of 1 months rent and a months rent in advance. See our tenancy deposit section if you need help gaining a tenancy bond.\nHomes are let on Assured Shorthold Tenancy agreements and are let for a minimum of 6 months. On many homes, provided the terms of the tenancy are adhered to, Landlords are happy for you to carry on renting well into the future.\nInterested? Available properties are shown on our homes for rent page. If you are looking for a home within Torfaen these are first advertised via theTorfaen Home Seeker so if you are looking for a home in Torfaen it would pay to register. Click on the link for details.\nfound a home that you like?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Four Just Men was Edgar Wallace's first bestseller. The four men are young, handsome, immensely wealthy vigilantes (including a European prince) who kill criminals who are beyond the reach of the law in the name of justice. Wallace wrote two sequels and the book was adapted as a silent film in 1921, a film in 1939 and as a British television series in 1959.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdhsk b/data_all_eng_slimpj/shuffled/split2/finalzzzbdhsk
new file mode 100644
index 0000000000000000000000000000000000000000..0220676fcac2f3447f78159363ebb83224bbc94b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdhsk
@@ -0,0 +1,5 @@
+{"text":"> started, but wait 18-24 months to complete.\n>>>> the AWT\/Swing\/Java2D discussion Mario mentions.\n>>> too, assuming (or course) that the Plum Hall test was valid.\n>> with Plum Hall for contributing to GCC.\n>> invalidated a TCK test and got it added to the exception lists.\n>> rewriting the tests so we have a free replacement.\n>>>> and not rely on the JCK.\n>> them what the details of a particular method or class really are.\n>> I think having a free TCK should be one of the goals for JDK7 at least.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Plug your iPhone and USB memory stick into the USB slots on your computer, and boot up the software. Then from the upper left corner of its startup interface, click the drop-down arrow to change to your iPhone for getting iPhone tunes onto USB drive.... Manually Transfer All Songs from iPod to USB Stick (Win) This way is only available for iPod, but not iPhone and iPad. If you want, you can do things the old-fashioned way and manually transfer your music from iPod to your flash drive.\nConnect the USB flash drive to the computer and wait for windows to initialize it. 3. Connect the iPhone to a USB port on the computer using its data sync cable. Wait a few seconds for Windows to... Plug your iPhone and USB memory stick into the USB slots on your computer, and boot up the software. Then from the upper left corner of its startup interface, click the drop-down arrow to change to your iPhone for getting iPhone tunes onto USB drive.\nI want to know how I can download apps from Android market using my PC and transfer it to my phone. I don't have .apk manager. I tried to download Astro file manager, but it is a .exe file, wherein .apk is not recognized.\nIf you don't have an Android phone, you can download apps manually to your Fire TV from thwe b. Go back through the Settings menu until you get to the Fire TV home page. Then, using either the Search tool on the far left or the Alexa voice search button on your remote, search for \"Downloader.\" Highlight the result below to get to the app page on the Amazon Appstore.\nPlug your iPhone and USB memory stick into the USB slots on your computer, and boot up the software. Then from the upper left corner of its startup interface, click the drop-down arrow to change to your iPhone for getting iPhone tunes onto USB drive.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Moved by the plight of Tamils displaced during the three-decade-long civil war, Sri Lankan President Maithripala Sirisena has promised to set up a special task force to resettle them within six months.\n\"I will set up a special Presidential Task Force to resettle these people in their original places within 6 months. I will bring the military, police and the state agencies together to achieve this,\" Sirisena said addressing a Christmas event in Jaffna yesterday. \"I went inside their homes and myself saw the sorry conditions they live in,\" the President said.\nSirisena also criticised those who accuse him and Prime Minister Ranil Wickremesinghe of compromising national security by easing the rigid security conditions imposed in the north. \"I invite those who shout from Colombo about national security to come and inspect these living conditions. I will provide all facilities for them to visit the north if they want to,\" he said.\n\"The national security is not under threat\u2026 the national security is a question in the lives of these people who have been displaced 25 years ago,\" he said. He said although his predecessor Mahinda Rajapaksa's government had been able to finish the war and bring in peace, true reconciliation has not been achieved.\n\"The war was won, peace has returned but the root causes of the war still remain,\" he said.\nSirisena since January has brought in some relief measures for the Tamil minority in response to their long felt demands. Sirisena's government released civilian lands occupied by the military for over 30 years, lifted travel restrictions to the north and appointed civilian governors by removing the ex-military governors in the two provinces.\nHe charged that the Rajapaksa followers in his own party were trying to topple the government by using alleged national security challenges as an issue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Honorable Jean Michel NINTCHEU and Joshua OSIH have been detained in Douala by law enforcement officials and taken to an unknown destination, the camerounweb editorial team reports from a source on the scene.\nThe city of Douala is beset by a contingent of bunkers for several hours this Wednesday, January 23, 2019.\nAs a reminder, the Honorable Jean Michel NINTCHEU organized a march in three boroughs of the country's economic capital to express his frustration at the bad policy that led to the withdrawal of CAN 2019 in Cameroon.\nA crisis meeting at the headquarters of the SDF Central region following the arrest of these two parliamentarians of the SDF.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finally, a revelation occurred to me. There were reinforcements getting ready to attack me and my mountain. To stop their plan I would first have to destroy the town and maybe the reinforcements wouldn't continue coming. Then a puzzling thought popped into my head, Why would the goblins team with dwarves? I thought goblins were evil and would only ever team up with the wargs? I pushed that thought out of my head and flew out of my mountain like a bullet.\nNow in the end, the score board has not changed \u2013 I'm waiting for the audiences vote\u2026 which is you! If you are lucky enough to do student feedback for me, please please please say who you think should've won, and why!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdudk b/data_all_eng_slimpj/shuffled/split2/finalzzzbdudk
new file mode 100644
index 0000000000000000000000000000000000000000..43dbe2295582c455f13d4d737d2eb7b3f5b8e9d0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdudk
@@ -0,0 +1,5 @@
+{"text":"Modern, 4 bedroom, spacious mid-terraced family house located in the popular Manor Park area of Sheffield with excellent amenities and transport links nearby.\n3 double bedrooms and 1 very large attic master bedroom with en suite.\nLarge modern upstairs bathroom with bath and shower.\nDouble glazed and fully central heated.\nLarge lawn garden to the rear with a deck area and additional outhouse\/utility room.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"i. Adem; bir erkek ad\u0131. Adams apple bak. apple. not to know one from Adam tan\u0131yamamak the old Adam insanlar\u0131n g\u00fcnah i\u015flemeye olan tabii e\u011filimi.\ni. \u00c2dem Baba, \u00c2dem. Adam's apple anat. \u00e2demelmas\u0131.\n5. employee; servant; retainer; helper.\nprov. husband, man. Adam\u0131m! colloq. My friend!\nadam adama defans sports man-to-man defense.\nadam adama savunma sports man-to-man defense.\nadam almamak (for streets) to be very crowded with people, be teeming with people.\nadam azman\u0131 enormously large person.\nadam ba\u015f\u0131na apiece, each, for each person.\nadam be\u011fenmemek to be overcritical of people.\nadam boyu the height of a man.\nadam\u0131n\u0131 bulmak \/\u0131n\/ to find the right person to do (a job).\nadama \u00e7evirmek \/\u0131\/ to put (something) in good repair, put (something) in good shape.\nadam de\u011filim! colloq. I'll be damned.\nadama d\u00f6nmek\/benzemek to look presentable; to look like somebody; to look like something.\nadam\u0131na d\u00fc\u015fmek (for a job) to come the way of someone who is really suited to do it.\n1. to be the making of (someone); to mature, make a man\/a woman of.\n2. to put (something) in good repair, rejuvenate.\n3. to set (a place, an organization) to rights, put (a place, an organization) on its feet; to make (a place, an organization) into something, make (a place, an organization) thrive.\nadam evlad\u0131 a person of good family and upbringing.\n1. properly, suitably, in the right way.\nadam\u0131na g\u00f6re (adapting one's approach) to suit the individual.\nadam i\u00e7ine \u00e7\u0131kmak\/kar\u0131\u015fmak to mix with people.\nadam istemek (for a project, for someone) to require a good man, need a person who's really worth his salt.\nadam kald\u0131rmak to kidnap someone, abduct someone.\nadam k\u0131tl\u0131\u011f\u0131\/yoklu\u011fu shortage of qualified and capable people.\n1. to know how to use someone for one's own benefit.\n2. to know how to get someone to work, know how to get work out of someone, know how to work someone.\n1. to grow up and become a responsible member of society.\n2. (for something in disrepair) to be given a new lease on life, be put in good repair.\n1. to be very good at, be highly skilled in (a job).\n2. to be the right person for (a job).\n3. to be (someone's) man, be one of (someone's) men, be in the employ of; to be a retainer of.\n4. to be someone whom one can rely on, be someone whom one can trust.\nadam \u00f6ld\u00fcrmek to murder someone, commit murder.\nadam sa\u011fmak to trick a man out of his money, milk people.\nadam\/insan sarraf\u0131 a good judge of people.\nadam se\u00e7mek to show favor, play favorites, not to give a fair chance to everyone.\nadam sen de! colloq. Don't worry.\/Take it easy!\/Never mind.\nadam s\u0131ras\u0131na ge\u00e7mek\/girmek to win a place of respect and responsibility.\nadam vurmak to commit murder.\nadam yerine koymamak \/\u0131\/ to think (someone) to be beneath consideration, consider (someone) to be of no importance.\nThe name given in the Bible to the first man, the progenitor of the human race.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The congestion of traffic around Tudor, from East Northern Lights to Lake Otis Parkway, has had the city of Anchorage proposing a solution for years. However, due to differing priorities as Anchorage grows, the U-Med District Northern Access Project still lacks the funding it needs to proceed.\nThe project was proposed to alleviate the traffic by extending the road.\n\"The project is an extension of Elmore through Bragaw, connecting another north-south corridor between Providence Drive and Northern Lights Boulevard,\" Sean Baski, project manager from the Department of Transportation, said.\nSince 2003, the project was already in thought and analyzed to determine the best course to take in order to create a practical alleviation.\nThe confidence for the project is strong, but with no money, it still won't be able to move forward.\n\"The project was a state-funded project; money was provided to the municipality of Anchorage, and then the municipality of Anchorage then passed those funds over to the Department of Transportation, and then we were developing the project, sometime in 2015-2016. The city administration removed their support from the project and since it was their project, the Department of Transportation then archived the project. Now it's archived and no longer being advanced,\" Baski said.\nDespite this, there is a clear demand for such an extension in the road.\n\"I think the project has a clear purpose and need, and benefits the U-Med district and solves problems that are real in the U-Med area,\" Osgood said.\n\"The University of Alaska Anchorage, APU, the different medical organizations in the area all supported the project, so there's definitely support for the project from the business side and then also from area residents, but there's definitely some area residents that did not support it,\" Baski said.\nThough the project may have reached its end, Osgood believes the project is important enough to be considered once again in the future.\n\"I do believe that the project will be back at some point because there is a clear need for it. And if the U-Med district will continue to grow, they will need construction of a larger infrastructure,\" Osgood said.\nPlans like the U-Med District Northern Access Project are proposed in order to create a safer traffic pattern. The goal is to decrease the congestion in traffic, and as the area grows, the goal becomes more relevant. With the U-Med road project at its final end, this goal is still nowhere near completion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FETID ZOMBIE \"Decade of Death\" compilation CD\/Book, celebrating 10 years of underground death metal, is now available from Toge No Jizo (Japan). The CD features select tracks from the FETID ZOMBIE discography as well as a 184-page A5-size book featuring unpublished sketches and other artwork related to the band. For customers outside of Asia, copies will be available from the FETID ZOMBIE Bandcamp website and Riddickart webstore this winter.\nTHRASH ATTACK Magazine (Germany) Issue 11 cover art. The illustration is based on the characters from the old CIRITH UNGOL (Michael Whelan) and ASHBURY (Ernie Polo) covers. The magazine features interviews with both bands as well as others like ETERNAL CHAMPION, RIPPER, etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u00c1lvaro Enrigue is the author of five novels and two story collections. He is a recipient of the Joaqu\u00edn Mortiz Prize and the the New York Public Library's Cullman Fellowship. His last novel, Sudden Death, was awarded the Premio Herralde and published in 14 languages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbeubm b/data_all_eng_slimpj/shuffled/split2/finalzzzbeubm
new file mode 100644
index 0000000000000000000000000000000000000000..f00799746c93c28d0909ac793efccae3c79d3e38
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbeubm
@@ -0,0 +1,5 @@
+{"text":"i miss christine. not this weekend, but the next one, i get to see\u2026 - ...but why is the rum gone?\ni hate my internship. im getting used to the people though. almost all of them seem pretty ok. i take my breaks with brian and sometimes anna, and other people come and go, but im getting to know pretty much all of them. i'm sure anna thinks i'm quiet, and maybe slightly snobby. i don't really care. she may be hot, but she's very straight (engaged to be married sometime in october), and i like holly more. and i like christine even more. the rest of them are nice people, and they seem to like me allright. i guess thats the part that counts the most, for me. i hate sitting at a desk, doing nothing but thinking-type work, but as long as everything's ok with me and the people there, i'm ok. my happiness depends way too much on other people.\ni miss christine. holly makes me miss christine even more. i know i was never happy with the christine situation, but at least i never ignored her for 2 weeks straight... and there i was almost not hiding it. that's what i miss the most. here i am so deep in the closet that i can't even see the light from the room. there i was in the closet, but i was almost to the edge. i wasn't jumping on every chance to deny it, like i do here. if the whole jim situation happened here, it would have been his age from square one. none of this 'no, it's not his age... no, it's nothing about him... about me? what? gotta run...' that's loser, almost ready to tell the truth, but not quite. and this is loser deep in lies, and happy that way. big difference. am i rambling? sorry. i miss christine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For all the train geeks, National Rail Museum is one of the museums in Delhi you can't afford to miss. Railways in India was first developed during British' rule for easy transportation of goods. This move was followed by adding passenger carriage. The first passenger journey was complete in the year 1853 between Bombay and Thane. Since then Indian Railway has seen many changes and advancements and the National Rail Museum is the narration of the journey of Indian Railway from 1853 to today. Part of every Delhi Darshan day sightseeing tour, the museum located in Chanakyapuri, National Rail Museum is a haven for every history and heritage lover.\nOpened on February 1, 1977, National Rail Museum has recently gained its reputation to be a major tourist attraction in Delhi with train enthusiasts coming to witness the glorious journey. A remarkable showcase of Indian Railways' heritage, the museum exhibits some of the oldest steam engines and carriages. The heritage passenger carriages in comparison to modern bogey, stand miles apart. A kid-friendly museum, the museum will excite the kids as well as adults. The museum has some of the rarest exhibitions like the Fairy Queen, the oldest surviving steam engine in the world. There are also many other tourist attractions inside the museum like the 3D simulator, Rail restaurant, and the gift shop. 3D simulator allows you to witness the journey of some of the heritage trains of India and be a passenger on it. Apart from the outdoor display, the museum also boasts of an interior museum which has various modern and retired signalling system. Visitors are also treated with a train ride inside the Museum's park. You can choose from various heritage train rides ranging from the likes of BPSMT Steam Ride and toy train. A nice place to get photographed in a vintage train, the National Rail Museum is sure to be your favourite museum in Delhi.\nWitness the entire journey of Indian railway, from its maiden voyage to modern advancements.\nOutdoor gallery featuring miniature operational railway routes and exhibiting heritage trains and engines.\nNearest Metro Station is Dhaula Kuan(2 km). You can either take Airport Express Line or Pink Line Metro to reach Dhaula Kuan. You can take auto from the metro station.You can also take a bus from Central Secretariat Metro Station to reach National Rail Museum.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Business Cloud Essentials is a cloud-based ERP software which delivers a single, fully-integrated platform for businesses to use financials, CRM, stock control, supplier and logistics management, payroll and dashboard reporting. The platform is delivered as a Software-as-a-Service (SaaS) so it's easily accessible through any web browser and the pricing model allows businesses to scale their usage as required.\nOur ERP software provides business owners and managers with improved efficiencies and streamlined processes through an integrated solution. In today's fast paced business environment, we know how important it is for businesses to be able to work quickly and maximise productivity. Using Business Cloud Essentials, we help SMEs re imagine the way they work, helping their business scale and grow.\nAccess all business information with one login for Business Cloud Essentials, uniting all data from across the organisation in the software.\nReduce manual processing of information and maximise productivity by using one, central system for all business activity.\nGet a better understanding of your business activity and performance with real-time reporting using Business Cloud Essentials.\nWith ERP software, organisations can benefit from instant access to their data. It's no longer necessary to be in the office to run your business.\nTurn on period management to give greater control over your financial transactions.\nAuto recieve goods on GRNs to increase efficiency when booking goods in.\nRecord GL budgets to enable you to track performance against budget.\nMRP PO consolidation to simplify your supplier ordering process.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sydney Tea Festival - Tea Revolution!\nWe had the most amazing day at the Sydney Tea Festival and it was so wonderful to see all your gorgeous happy faces on the day. Things got busy fast with tea lovers pouring in from all around Sydney (5,500 of you!) thirsty for speciality drops of delicious tea.\nThe craziness hit around lunch time and didn't stop! All was an exciting blur of our swishing infusers pouring endless cups of our organic infusions, tea lovers chattering and slurping away in tea heaven to the backdrop pitter patter of soft rain outside the Carriageworks warehouse.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you buy a subscription for oneself or as a gift, it will automatically be renewed for the exact same term length at the renewal rate then obtainable to clients. If you happen to be in average shape, hunting to get into fantastic shape then this magazine has lots of articles on overall health, exercising and diet program that will aid you. Only in people with high APA levels are well being concerns probably to materialize.\nThe organization mentioned that for the duration of the time amongst getting the Net order and cancellation in January, 5 problems of Current Wellness have been sent to you. A quantity of tests have confirmed this protection which includes human trials conducted by Serge Renaud of the French National Institute of Health and Medical Research. Not only did the overall health magazine covers about overall health , it also give some lesson of a straightforward 1st aids that we must do ahead of we could attain the physicians. Vitamin E has been known for decades to be lifesaving against heart of the well being meals neighborhood thinks of vitamin E as tocopherols.\nAuthors, particularly new ones, can fret about the possibility of receiving poor testimonials for their self published books. Women's Overall health magazine gives a well rounded viewpoint for ladies searching for news and tips in the regions of fitness, diet program, beauty, style, and relationships.\nIf you have a friend who is a overall health freak, why don't you subscribe to 2 or 3 well being magazines, as a present for him. Yes, nails can be a mirror to your deteriorating health and a timely evaluation can save you just before it's as well late. And right now, with well being care transformation, Oregon is further along in population overall health than most states. For the duration of Christmas and Christmas present giving, it is true, we hardly think of well being. Very useful ideas for living a heathly life style, along with great physical exercise tips and recipes as effectively. Our thanks go to the National Wellness Federation and Scott Ideas for his shattering report on the current CODEX deliberations in Germany.\nWhen we believe in terms of total toxic load, we can recognize why optimal functioning of the liver's detoxication systems is basic to our overall health and wellness. Overall health magazines also give them great suggestions in creating muscle tissues, and in escalating their protein intake. It is so relaxing for me i commence falling asleep, folks ask how so it appears so stressing, not for me. i want they would bring the magazine back. Apart from news and associated health answer and difficulties comes from the overall health magazine are from properly identified doctors and scientist. Even though it is been half a dozen years because Magic Crochet Magazine exited the industry there are nonetheless a lot of people who are interested in this crochet magazine. Winners of this year's Wavemaker Awards will be notified by June 1 and will be invited to celebrate their achievement at an awards showcase in London on July six.\nThe parent kit is completely free of charge and will be mailed to a residence or downloaded from the internet site. For a profitable sale, a consumer need to be ready, willing and capable\u2026 yes, all 3! The men and women that are saying great factors about sea salt, are attempting to sell you a specific brand of it. They do not care about your wellness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhgmi b/data_all_eng_slimpj/shuffled/split2/finalzzzbhgmi
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+{"text":"oh no, not another Vee coolant drip thread!\nRight down to the appearance of the coolant drip.\nAll the hoses going to the thermostat housing are fairly new because I replaced them in the last year or so, except the small one going downward in the photo (described as \"Hose, Water Bypass\" in the parts diagram). I don't think that's the source of the leak though.\nMight this point to a need for a new thermostat? Since that housing has no gasket or seal aside from the thermostat itself, I can't think of what else it could be allowing the leak.\nAlso, I'm really sick and tired of taking stuff out of the way to replace cooling system hoses. Partly because I just did it a few weeks ago to replace the short hoses that go to the cylinder heads from the thermostat housing.\nWould it be so wrong to paint a bit of exhaust gasket maker on that joint? Might stop the drips until I feel like doing the job \"properly,\" at least.\nLast edited by DesertBike; 12-25-2018 at 03:15 PM.\nIf you just did work on the cooling system that is probably where your problem lies. A clamp may need a tighten or be moved an bit. Even new hoses can be defective. First thing is to locate the leak. Then if gasket maker will stanch the drip for a while go for it. At least you are still riding. Up here in the great white north most bikes are snuggled in their beds.\nI too have had drips on both my 1000's. I found that moving the coolant reservoir plug up about3\/8\" to leave the overflow hose to vent solved the problem on both. Worth a try.\nI think the coolant reservoir plug thing was the key. Moved it up so the overflow tube was uncovered and got the bike good and warmed up. No more drips so far.\nSeems like the coolant system gets too pressurized, and springs a leak at the thermostat if you have the reservoir plug in all the way. You'd think the owner's manual or at least the shop manual would mention this.\nI have a leak also. What exactly is this \"coolant reservoir plug\"?\nFWIW, I still haven't fully resolved this issue. It seems totally random as to when it does or doesn't happen. A few weeks ago I got a few drops of coolant on my left boot, and could see that coolant had been leaking out of the thermostat, dripping on to the cylinder head and then down the left side of the engine.\nFound a little bit of coolant in my skid plate when I took it off to clean some stuff last weekend, too. First time I'd had the plate off in months, so it was recent. It is an Enduro Guardian plate so there is one corner where coolant can pool to an extent. Not more than several drops' worth but still Not Good.\nBut over the last few weeks, I've been unable to actually observe the coolant dripping out of the thermostat housing.\nI finally had enough and ordered a replacement thermostat. Don't know how else to permanently fix this. The thermostat itself is the only rubber component in the area that's leaking. Haven't installed it yet though.\nLast edited by DesertBike; 04-10-2019 at 10:11 AM.\nMine is also completely random. Not sure if I should worry about it? Yesterday took it out for about a 30 mile ride, got home, no leaks. Again, mine appears to be coming out the overflow tube, about where my right foot would be if at a stop and foot was on the ground.\nOut the overflow tube is no biggie. Don't worry about it. That's why the tube is there. In my experience, temperature swings while parked can cause a drip.\nIf it becomes excessive, then you probably need to overhaul the waterpump.\nFWIW: Mine did that for a while after rebuilding the waterpump and then stopped. I haven't seen a drip in several years.\nTo the OP: Is it possible the new hoses you installed are not oriented properly? They have a bend to them and different diameters at each end. It is also possible the O-rings on the unions are not seated correctly. Or, as you suspect, the thermostat could be your source. I haven't dug into the thermostat, but did the hoses and unions last year.\nThat was going to be my suggestion, but as you say - sometimes the overflow will drip a little for no obvious reason. It's only if it's all the time that you need to think about a water pump rebuild, IMO.\nTo the OP: Is it possible the new hoses you installed are not oriented properly?\nNo. Suzi did us a solid here - it's almost impossible to install them wrong.\nThey have a bend to them and different diameters at each end. It is also possible the O-rings on the unions are not seated correctly. Or, as you suspect, the thermostat could be your source. I haven't dug into the thermostat, but did the hoses and unions last year.\nO-rings (#9 in the Radiator Hose parts fiche) are original AFAIK. I have new ones, but decided not to attempt replacement when I put on new connector hoses. Too much work, and the leak wasn't coming from there anyway. Well enough alone and all that.\nHaven't had any more leaks from those hoses or their fittings, so I suppose that was a good call.\nReason I think it's the thermostat that needs R&R is that the only coolant drips I can see are coming from there.\nI'm replacing the thermostat today, but it's tough going. Even with the fuel tank removed, there is almost no room to work. Removing the airbox will allow a smidge more room to swing a (stubby) wrench. I really don't want to remove the throttle bodies.\nO-rings (#9 in the Radiator Hose parts fiche) are original AFAIK.\nThere's your source, I'm betting.\nApparently they are infamous. I didn't know about it either until mine went.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Furniture \u2014 This furniture, is one of the 14 designs published under the gallery: Home Interior Remodeling Ideas. In the next page, we have the home interior remodeling with well best finish remodeling small interior decorating ideas simple [ ... ]. While in\tthe previous page, we have the home interior remodeling photo of exemplary home interior remodeling inspiring good interior remodeling popular [ ... ].","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The America Ferrera starrer will be back with another round of 22 episodes.\nNBC is keeping Superstore on its schedule.\nThe network has handed out an early season four renewal to the America Ferrera starrer. Season four will consist of another full 22 episodes, in line with the current third cycle of the Universal Television comedy.\n\"Superstore is one of our signature NBC comedies, and we're so pleased to be bringing it back for the 2018-19 season,\" outgoing NBC Entertainment president Jennifer Salke said Wednesday in making the announcement.\nIn its third season, Superstore has grown 6 percent (1.7) year-over-year in the all-important adults 18-49 demographic and 3 percent (5.2 million) among total viewers. Factoring in multiplatform viewing, the series grows to a 2.9 in the demo with 35 days of delayed viewing.\nSuperstore joins a 2018-19 NBC roster that includes previously renewed comedies The Good Place and Will & Grace, as well as drama This Is Us. NBC also has straight-to-series drama The Gilded Age, from the creator of Downton Abbey, which is expected to debut in 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Click on play button to listen high quality(320 Kbps) mp3 of \"Okmalumkoolkat Gqi\u2764\". This audio song duration is 1:48 minutes. This song is sing by Okmalumkoolkat Gqi\u2764. You can download Okmalumkoolkat Gqi\u2764 high quality audio by clicking on \"Download Mp3\" button.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fees are based on record searches of one subject in one jurisdiction and a seven or ten-year scope.\nInternational strategies are available; fees are based on country\/region.\nLevels I \u2013 V: seven years, one jurisdiction.\nLevel VI: 10 years, unless a particular state or local law limits the reporting period to seven years.\nLevels V-VI: eight business days.\n25-percent surcharge of total fee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
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+{"text":"Hue Medical College was established on 28th, March 1957. Its training scale has been increased by years, especially after the reunification. Since 2007, Hue Medical College was renamed as Hue University of Medicine and Pharmacy (Hue UMP).\nIn general, Hue UMP is running under the direction of Ministry of Education and Training (MOET) and Hue University. In particular, the University follows the direction of Ministry of Health (MOH) to develop its health specialty. According to the revised Law of Education issued in July 2015, Hue UMP, like other training centers across Vietnam, is becoming independent in conducting its training curriculum. This significant change gives promising opportunities of curriculum revision and improvement to the University. Within 60-year development, Hue UMP has trained about 20.000 doctors, pharmacists, bachelors and more than 8.000 graduate students.\nIn addition to training, Hue UMP also focuses on doing research and applying new medical technology in healthcare. The University has carried out several successful scientific researches at both national and international level. The number of international publication has increased over the years.\nMore importantly, thanks to the strengthened international relations and cooperation, the University has been supported and funded by 30 international projects to build up its human resource capacity as well as infrastructure and facilities.\nThe vision and strategic plan of the University toward the year 2030 is to become the national leading university; strongly focus on developing high quality training program and joint training program; improve the capacity of doing scientific research and applying technology; enhance the quality of medical services; completely conduct the model of University- University Hospital.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"*Cast in bright silver with faceted lucite stones in white and beautiful floral designs throughout. These packages of beads are variety packages and will contain exactly as shown in the image. The beads contain clear and clear AB cz stones that go with the floral motif so beautifully and elegantly!!\nThey are approx. 15 mm height, 33 mm wide and have approx. 11 mm between the loops.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This entry was posted in Africa, Mozambique, South Africa, Uncategorized and tagged Africa, Colorado Traveling Ducks, elephants, giraffe, Kruger National Park, kudu, lion, monkey, rhinos, safari, South Africa, zebra. Bookmark the permalink.\nHow fantastic to see these wonderful animals close up.\nWe loved seeing so many. However, this was my last safari on this trip. Thanks for reading and commenting.\nWas new to me also. Thanks for reading and commenting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eric & Monique are a husband and wife singer-songwriter duo who consider themselves to be a mixture of both musicians and missionaries. In this conversation we explore how they met, some stories from their time on the road and more.\nIf you are not acquainted with their music, check out the video below!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Those of us who serve on the board of directors, on the staff, or volunteer at The Family Center want to extend a great-big, enthusiastic thank you to the citizens of this generous community who helped make the recent Salvation Army Red Kettle Campaign a big success.\nI especially want to thank the merchants who allow us to ring the bells in front of their establishments and the nice folks at General Motors and the UAW for collecting at the plant.\nSpace will not allow me to list and thank all those who rang bells for us, regardless of cold, rain, and wind. Many friends of The Family Center ring every year multiple times, no matter the conditions. Finally, I want to thank everyone in this county who put something in one of the red buckets, or maybe just came by and smiled or said, \"Merry Christmas!\" Because of your generosity, than $38,000 will be used to assist those in need.\nWhile I have your attention, I want to also thank the many groups and kind individuals, including you, James Bennett, and The Daily Herald, who together make our annual Turkey Giveaway and our Santa's Workshop events so successful.\nBesides the Salvation Army Red Kettle Campaign, The Family Center receives donations from area churches and individual supporters throughout the year.\nOur thrift store, Common Threads (at the corner of Alpine Drive and Hatcher Lane), has generated much-needed funds to expand our offerings in the food pantry and provide more help with rent and power bills. We are grateful for the growth and success of this endeavor and thank Gina Russell and her great group of volunteers and all you who shop there and\/or donate clothing or other things for us to sell. Volunteers can even work extra hours to earn store discounts or assistance with a utility bill.\nWe also have a couple of fun-filled evenings to raise funds, and we thank all those who purchased tickets for the \"Reverse Raffle\" and the \"Taste of the Town\" events last year.\nUsing these funds in 2018, we were able to help more than 2,000 families with rent or utility bills. Also, more than 4,600 families, representing nearly 10,000 individuals, used the food pantry.\nThis community also donated almost 200,000 pounds of food last year, valued at over $300,000.\nTo find out more about the work of The Family Center and Common Threads, or to get information about becoming a volunteer, please call (931) 388-3840.\nDirector Dawn Taylor also publishes a quarterly newsletter and frequent updates. You can call and give your e-mail address to receive these. You may also follow The Family Center on Facebook. We are extremely appreciative of all your support, whether it is by contributions, volunteer hours, or kind thoughts and prayers for our efforts to help our fellow citizens.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbikaw b/data_all_eng_slimpj/shuffled/split2/finalzzzbikaw
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+{"text":"Persimmons are by far my favorite fruit. They are orange colored, very sweet and contain lots fiber. They are also rich in vitamin C, beta carotene, calcium, folate, iron, magnesium and potassium.\nThey are believed to reduce the risk of atherosclerosis, improve lipid function and even combat hangovers!\nI eat them by cutting them horizontally in half and scooping the flesh out with a spoon. You can also eat them with the skin, just make sure they are ripe.\nAn article published by the Journal of Skin Pharmacology and Physiology concluded that an encapsulated fruit and vegetable concentrate, helps skin be healthier and therefore age less.\nYou may be wondering where to get this concentrate from, no need to look for any. Eating fresh fruits and vegetables and preparing your own juices everyday will probably achieve the same results (I haven't tested this theory, but I am pretty sure!). Always include lots of greens and a variety of different colored fruits and vegetables.\nEat healthy, be happy and as proven recently, beautiful and young as well!\nFennel (Hinojo in spanish) is a vegetable packed with disease preventing antioxidants. It can be eaten raw or cooked (I like it better raw) and it has the texture of celery but with a sweet anise flavor.\nIt is very soothing to the digestive system and it is also diuretic. It contains powerful phytochemicals that strengthen blood vessels, improve circulation and reduce heart disease. Fennel is rich in vitamins B, C, E and K, beta-carotene, folate, calcium, iron, potassium, zinc and fiber.\nIn a bowl toss fennel with 1 tablespoon olive oil and salt. In another bowl mix the rest of the ingredients together, except for the pomegranate seeds. Divide the greens on to 4 plates, top with fennel and pomegranate seeds, enjoy.\nHigh Fructose Corn Syrup (HFCS) is extremely common in processed foods like soft drinks, fruit juices, cereals, chips, breads, soups, dressings, sauces, yogurt, cakes, pet food and many more.\nHFCS is composed of glucose and fructose and it is produced from corn. It's a very cheap sweetener so big companies love to use it in their products.\nIt has been associated to overweight, diabetes, cardiovascular disease and some studies conclude that appetite control is suppressed as well. This means that the hormone telling your brain it's full is not working! So you'll eat more and not feel satisfied.\nOh! And by the way it's the fastest of the sugars to be converted into fat.\nPlease read the labels on every packaged or processed product you buy, this fattening chemical is everywhere. Be careful, remember the only way to be save is to cook everything form scratch. Eat healthy, be happy!\nWe are starting a new week and recently a new year, so why not start a new diet?\nTry eliminating the following foods from your diet during this week, I can bet you will feel so great you'll want to keep eating this way.\nIf there are 2 books I could never ever live without, they would have to be these two by Terry Walters. Not only does she give you the healthiest recipes, but she explains step by step how to do the basics, from cooking grains, to choosing the best fruits and vegetables!\nHer books are divided by seasons and the ingredients she uses are in their most natural, unprocessed state. All the recipes are wholesome and nutritious and by the way, did I mention delicious!\nSo, if you are looking into healthier eating, losing weight or even just a cooking book, these are the ones to buy!\nThe Macrobiotic diet is more than just a diet, it can be considered a way of life.\nThis is a very balanced and nutritious diet. It keeps you healthy (studies say if might prevent cancer and other degenerative diseases) and keeps you at your ideal weight.\nMacrobiotics include lot's of grains (50%) and vegetables (30%) which have to be well chewed. Nuts, legumes, beans and miso soup (15%) must also be included. It can be similar to being a vegetarian, but in this case fish is permitted in small quantities (5%).\nFoods are consumed in their most natural state. Refined and processed foods, dairy, sugar, alcohol, coffee, meats, eggs and poultry are off the list.\nTo eat more like a macrobiotic try incorporating all kinds of different grains to your diet: brown rice, barley, buckwheat and sea vegetables like Kombu, Wakame, Noru and Hijiki.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The human mind grows every day. This is a given. It is the way that it grows that is the focus of this particular article because there are many ways in which a mind can grow and not improve. There are also many ways that a mind can grow and actually deteriorate. Intellectual growth must be in areas that are conducive to the further production of biologically and psychologically beneficial outcomes that aid the intellect of the individual and all of those that surround this individual.\nWhen the mind is fed disinformation, then it will tend to grow in ways that are not conducive to beneficial outcomes. However, if the mind perceives this information as being possibly deceptive, then it will grow in a beneficial manner. Therefore, if the mind perceives disinformation as being legitimate, then it will deteriorate. The mind must often learn to sort the wheat from the chaff in order to more effectively process data in a correct way. In order to perform this task, then it is necessary that the mind possess some rudimentary skills of logical analysis and constantly develop and exercise these skills so that they may grow and be able to handle more and more types of both misinformation and disinformation.\nWhen the logical faculties of the intellect are more thoroughly trained and can absorb information in a more correct fashion, correct intellectual growth will occur. When this type of growth occurs, the mind of any individual that has had its intellectual faculties hindered by incorrect information masquerading as truth will suddenly experience a renaissance of new development. Ironically, many intellects that are never properly challenged with the deceitful nature of misinformation and disinformation will not be as likely to be able to learn at greater rates of development until they are properly challenged and learn to overcome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"#1761807). They remain in universe.\nthings to detect any packaging regressions. I found none.\nany issues before release if possible.\ndefault, so from a distribution perpsective I believe this will be safe.\nNevertheless, all testing is valuable and appreciated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rabbit body diagram Diagram credit the ibicenco rabbit or conill pag232s deivissa. Its precise origins are unknown but its a dutch style rabbit with a white collar muzzle and limbs and a darker body that can be After a period of growth the hair remains in the follicle in a resting stage to eventually be sloughed from the body. Figure 83. Diagram of hair in skin the both found in rabbit hairs. Figure 93 In science i found that men were more likely than women to scamper down niche mathematical rabbit holes. There are exceptions to see human nature in terms of free body diagrams flowcharts and.\nRabbit body diagram Conspirators hardly hold on to one lie like a scatter diagram they splash the space with many lies i dont know if any of us has seen a rabbit escaping danger hop hop hop so these We talked about the point when the liquid would penetrate the skull and touch the brain the body temperature fans went down a theory rabbit hole why cersei would abandon her son whether or not Cumulative data is shown in a bar diagram fig. 4k. Dpds expressing apol1g1 and apol1g2 or an anti cd2ap antibody rabbit polyclonal sc 25272 santa cruz. Ip fractions were probed for different.\nRabbit body diagram A few days ago i wrote a post about the basins of the moon a result of a trip down a rabbit hole of book research thats why geologic time is usually diagramed in tall columnar diagrams like Every cell in your body after all carries about two or three meters of dna while a 1.5 millisecond drop of 0.56 na indicated a fold exactly in the middle. The diagram at right makes it clearer The connectome is in essence the wiring diagram of the brain. Its a physical map of all much of that spaghetti tangle of neurons deals with phenomena specific to the body the brains in..\nModern pieces run the gamut from the highly abstract like john emanuels pastel blue torso which reduces the bodys structure to a series of and neddi hellers blue rabbit a creepy Childish though it may seem in these vehicles theres actually a diagram of a turtle and a rabbit on the throttle the custom boat only needs 6 inches of water to run but the manmade body of.\nIt's possible to get or download caterpillar-wiring diagram from several websites. If you take a close look at the diagram you will observe the circuit includes the battery, relay, temperature sensor, wire, and a control, normally the engine control module. With an extensive collection of electronic symbols and components, it's been used among the most completed, easy and useful wiring diagram drawing program. Rabbit Body Diagram. The wiring diagram on the opposite hand is particularly beneficial to an outside electrician. Sometimes wiring diagram may also refer to the architectural wiring program. The simplest approach to read a home wiring diagram is to begin at the source, or the major power supply. Basically, the home wiring diagram is simply utilized to reveal the DIYer where the wires are.\nIn a parallel circuit, each unit is directly linked to the power supply, so each system gets the exact voltage. There are 3 basic sorts of standard light switches. The circuit needs to be checked with a volt tester whatsoever points. Rabbit Body Diagram. Each circuit displays a distinctive voltage condition. You are able to easily step up the voltage to the necessary level utilizing an inexpensive buck-boost transformer and steer clear of such issues. The voltage is the sum of electrical power produced by the battery. Be sure that the new fuse isn't blown, and carries the very same amperage.\nThe control box may have over three terminals. After you have the correct size box and have fed the cable to it, you're almost prepared to permit the wiring begin. Then there's also a fuse box that's for the body controls that is situated under the dash. Rabbit Body Diagram. You will find that every circuit has to have a load and every load has to have a power side and a ground side. Make certain that the transformer nameplate power is enough to supply the load that you're connecting.\nCat body diagram without t rabbit body diagram rabbit body language rabbit body chart rabbit body view robot body diagram rabbit body types rabbit body diagram fear.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Being a great cook day in and day out means making the best of what you have on hand. A great cook is a versatile cook. Of course, it helps if you have a versatile recipe and a few hints along the way. Tomato and Beef Soup is a versatile recipe.\nUse whatever kind of small pasta you have in the cupboard. Small because it is easier to eat in soup. I always have barley on hand \u2013 because I like it and use it often \u2014 but not always split peas and lentils because I buy them only for specific recipes and don't replace them immediately. Increase the amount of barley or pasta to make up for the missing ingredient. Diced tomatoes make a nicer looking, neater to eat soup but you can use whatever kind of tomatoes you have on hand. If you've been cooking for a while, these things may seem common sense. But if you are just starting out in the kitchen or \"expanding\" your horizons, these little tips can be helpful. And if you are working, raising children and a million other things, it's easy to be so tired you can't even think. #mommybrain is real!\nYou can make Tomato and Beef Soup from scratch in just over an hour. If you brown one pound packages of ground beef and freeze them when you come home from the grocery, it will save some hassle on a busy evening after work. Just measure the beef broth and dry ingredients, thaw a package of browned ground beef and dump it, the tomatoes, tomato paste and pasta into the broth mixture after 45 minutes of simmering and simmer another 15 minutes.\nYou can make and freeze several batches for cooking later. Place peas, barley, lentils, dried minced onion, Italian seasoning, rice, garlic powder and pepper in a quart zipper bag. Measure pasta into snack size zipper bag and add to the same bag with the other dry ingredients. Brown ground beef and place in gallon zipper bag. Add the quart bag with the dry ingredients and label before freezing. I have a very small freezer so I'd rather mark cans of tomato paste, diced tomatoes and beef broth with the name of soup and set aside in special spot in the pantry. But if you have freezer room, go ahead and add tomatoes, tomato paste and even the beef broth to the ground beef in the gallon zipper bag and tightly seal. Place in a second gallon bag and add the quart bag into the second bag to make one neat package. Label with name and cooking directions. Set in the refrigerator the night before so it will be thawed and ready to cook when you get home from work the next day.\nFilling and, oh so good, Beef and Tomato Soup is easy to make from scratch. OR you can make ahead and freeze, for quick cooking.\nPour beef broth into large pan. Add split peas, barley, lentils, rice and seasonings. Bring to boil, cover and then lower heat to simmer for 45 minutes.\nBrown ground beef, drain fat if needed. Stir tomatoes and tomato paste into ground beef. Cover and keep warm over low heat while broth finishes simmering.\nStir beef and tomato mixture into broth mixture. Add pasta. Cover and continue to simmer for fifteen minutes or until pasta, lentils, peas and barley are tender.\nPlace peas, barley, lentils, dried minced onion, Italian seasoning,rice, garlic powder and pepper in a quart zipper bag. Measure pasta into snack size zipper bag and add to the same bag with the other ingredients.\nBrown ground beef and place in gallon zipper bag. Add the quart bag with the dry ingredients and label before freezing.\nMark cans of tomato paste, diced tomatoes and beef broth with name of soup and set aside in pantry. OR add them to the ground beef in the gallon zipper bag and tightly seal, then place in a second gallon bag and add the quart bag into the second bag.\nI'm so glad you stopped by today! Be sure to follow A Pinch of Joy so you don't miss a thing! If you found this helpful or inspiring, please share! Your support of A Pinch of Joy is appreciated!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbinxc b/data_all_eng_slimpj/shuffled/split2/finalzzzbinxc
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+{"text":"Weeks - if not months - after we reported that a new P!nk and Eminem collaboration was waiting to be released, she has finally confirmed its existence.\n'Revenge' will finally be out this Friday when the 'What About Us' singer's new album Beautiful Trauma is released, although she says it would never have happened if it wasn't for some good old red wine.\n\"And I wrote him an email and I said I love you, I've always loved you,\" she revealed on Lorraine. \"I've loved you since you gave me your autograph at the 2001 MTV Music Video Awards.\"\nPerhaps we'll see some of this wine-infused recording session (and love letter-writing) in On The Record: Beautiful Trauma, the behind-the-album documentary premiering on Apple Music this week.\nWe cannot WAIT to hear this one!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What are the closest stations to Chung C\u01b0 Ho\u00e0ng Anh Gia Lai 1?\nL\u00ea V\u0103n L\u01b0\u01a1ng is 157 meters away, 3 min walk.\nLotte Mark is 247 meters away, 4 min walk.\nSi\u00eau Th\u1ecb Lotte is 360 meters away, 5 min walk.\nWhich Bus lines stop near Chung C\u01b0 Ho\u00e0ng Anh Gia Lai 1?\nThese Bus lines stop near Chung C\u01b0 Ho\u00e0ng Anh Gia Lai 1: 140, 72.\ndata-title=\"Chung C\u01b0 Ho\u00e0ng Anh Gia Lai 1\"\nTraveling to Chung C\u01b0 Ho\u00e0ng Anh Gia Lai 1 in Ho Chi Minh has never been so easy. Use Moovit to get detailed step by step directions as you travel from your current location or from a top attraction or any major public transit station.\nView detailed routes on a map, see bus and train schedules, arrival times, and service alerts so you know exactly how to get to Chung C\u01b0 Ho\u00e0ng Anh Gia Lai 1.\nWhen traveling to Chung C\u01b0 Ho\u00e0ng Anh Gia Lai 1 use Moovit's Live Directions with Get Off Notifications to know exactly where and how far to walk, how long to wait for your line , and how many stops are left. Moovit will alert you when it's time to get off \u2014 no need to constantly re-check whether yours is the next stop.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Direct Chill casting for slabs is a new development for producing high-quality billets for slabs; Kalpatech manufactures premium quality steel billets of copper. These D.C. Caster For Slabs & Billets Of Copper manufactured by Kalpatech has international standards. The range of D.C. Caster For Slabs & Billets Of Copper has features like higher strength. The D.C. Caster For Slabs & Billets Of Copper have excellent pressure tightness and excellent conduction towards heat and electricity.\nTextile Reeds (Stainless Steel) are used for looms weaving wire mesh, papermaking mesh. Kalpatech has a lot of experience in making these stainless steel reeds. Kalpatech uses advanced equipment in making these stainless steel reeds. Kalpatech also offers perfect inspection measurement for stainless steel reed making.\nThese stainless reeds are being made by in house stainless steel tapes which are at par with Laminage De Precision so they are durable. And these stainless steel reeds are made under experienced supervision.\nFor the first time in India, Kalpatech Engineers backed up by over two decades of experience in the field of copper with Vertical Continuous Casting Plant for wider copper and copper alloy strips. Vertical Continuous Casting Plant for copper is the best process to achieve casting for wider copper and copper alloy strips.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The agenda is here and the minutes of the 2015 AGM are here. Nominations or proposed items to be discussed at the AGM can be posted in the 2016 AGM forum to provide opportunity for others to view and, if appropriate, comment on them. Alternatively, please email them to secretary at halcrowpensioners dot org dot uk. An updated agenda will be issued before the meeting. The AGM is the opportunity for the members to provide their views on the direction of HPA.\nThe HPA committee is always keen to welcome new members. If you would like to help or would like to improve your understanding of pensions-related issues then please contact the HPA secretary who can also provide additional information about what the committee does.\nFor those who haven't already paid, the HPA treasurer looks forward to receiving your 2016 subscription. It remains at \u00a310. Payment by bank transfer is preferred because there are no transaction charges although you can also pay by credit card through our PayPal portal. Details of the payment methods are here. Access to the 2016 AGM will be restricted to those who have paid their 2016 subscription. If you have paid your subscription then please check that you have access to the forum. If you haven't registered then please do so or if you have forgotten your forum registration details then please email the webmaster.\nHPS isn't the only pension scheme with funding problems. One which has been making the news recently is the BHS staff pension scheme. If the FT website cooperates then you can read several recent articles about this scheme. Alternatively, there is a report here. The BA pension scheme is also going to provide some income for the lawyers. Some interesting history is also revealed by this report of a parliamentary debate.\nHPA members occasionally put pen to paper and their works deserve a mention. First in line is our chairman's hypothesis about building the pyramids. A very recent publication is Alan Cooke's Our Man in Phuket. The introdutory blurb fails to mention the 34 years of Halcrow employment. Also worthy of a mention is this book by a member's daughter which might well be of greater interest to wives because it reflects on the challenges of a spouse on a foreign posting.\nOne of the motions passed at the AGM was to collect location data for members in order to identify which MPs might be best targeted in the event that parliamentary support is needed (the Pensions Regulator may be more responsive to MPs than it is to pension scheme members). A field has been added in the newsletter subscriber database. All newsletter recipients are therefore requested to check and update their data using the link to their personal preferences at the bottom of the email newsletter. If you have any difficulties please contact the webmaster.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You are here: Products 20 Slot Bamboo Universal Knife Block Without Knives. Knife Storage Organizer and Holder by Shenzhen Knives.\nHome \/ Home & Kitchen,Kitchen & Dining,Cutlery & Knife Accessories,Knife Blocks & Storage,Knife Blocks \/ 20 Slot Bamboo Universal Knife Block Without Knives. Knife Storage Organizer and Holder by Shenzhen Knives.\nKeep all your favorite knives within arm's reach! No more fumbling around searching for the right knife. Unlike most knife blocks, our beautiful bamboo storage block stores up to 19 knives, with extra slots to store kitchen scissors and a sharpening rod. We heard your feedback and complaints, and we have improved our knife slots design compared to existing competitor knife blocks: We designed the knife slots for easier handling and made them larger to better accommodate those larger knives. We made all the knife slots wider, allowing you to fit a larger variety of blade sizes; so you don't have to worry about not being able to fit your specialty knives, or having to search for the correct slot every time. The Shenzhen Knives 20 slot bamboo knife block makes storing knives hassle-free, so you can spend less time fumbling and more time cooking!\nWe will sell a big amount of 20 Slot Bamboo Universal Knife Block Without Knives. Knife Storage Organizer and Holder by Shenzhen Knives. in our BG Store LLC, so that you may buy the most suitable choice for you.\nWhen you don't desire to purchase the brand-new 20 Slot Bamboo Universal Knife Block Without Knives. Knife Storage Organizer and Holder by Shenzhen Knives. by simply big price, you can request product via our retail outlet.\nIf you want to arrangement 20 Slot Bamboo Universal Knife Block Without Knives. Knife Storage Organizer and Holder by Shenzhen Knives. in America and other international locations and get these products of top quality and reasonably low price, you can create the good alternative by visiting each of our shop. For obtaining to search for retail store, because precisely what you may need being a customer is straightforward to find from the catalog of goods on the website involving BG Store LLC. We offer the top 20 Slot Bamboo Universal Knife Block Without Knives. Knife Storage Organizer and Holder by Shenzhen Knives. plus the best in order to the locals all over the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjmgi b/data_all_eng_slimpj/shuffled/split2/finalzzzbjmgi
new file mode 100644
index 0000000000000000000000000000000000000000..041269cc71e7a393e218249236db3dd23b215a37
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjmgi
@@ -0,0 +1,5 @@
+{"text":"Set your folder permissions and specify access for multiple users, ensuring that you and your team are the ones collaborating the files you want. Features RAID capability, get reliable data sharing even if a single hard drive fails, so that you can keep on working!\nBe efficient with your work.\nEnable remote access, so you can edit your photos, videos, and documents where ever you go in this world. Built-in SD card reader, conveniently allows you to copy your multimedia files from the SD card to the network storage.\nDirectly edit your photos and video files on the network storage device. Powered with high end specifications, our network storage device deliver fast read and write speed as if you're working directly from your computer system.\nStay mobile with your work and share it among your clients.\nZyxel incorporates advanced P2P technologies to enable you to quickly browse your work files from your network storage device where ever you are.\nFlexibility to get your devices connected together.\nWhether you choose to utilize SAMBA or WebDAV to share your files, the Zyxel network storage device provides to flexibility for you to choose on how you connect with all your colleagues.\nDeliver your finished artworks with its link aggregation for the faster data transfer speed.\nGet your digital media online up to 30% faster than traditional single gigabit Ethernet connection, by utilizing the network storage's dual Ethernet ports.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NEWMARKET \u2014 The closing of Newmarket's nearly 20-year-old Head Start Program in June \u2014 caused by the federal sequestration \u2014 has forced the parents of more than a dozen children to either drive to a nearby community for the program or to just stay home.\n\"They sent us a letter in May and said that we're closing,\" said Jim Hilton, Newmarket's recreation director.\nThe Newmarket Head Start Program was run out of the Newmarket Recreation Department's building, which leased the site to Head Start for approximately $12,000 a year. The program was about to complete its 18th year when the decision was made to kill it in May.\nAcross the county, thousands of parents are finding themselves in a situation similar to the situation in Newmarket.\nAccording to the national Head Start website, funding was cut nationally by about 5 percent for the nearly 50-year-old program. And multiple Head Start locations have been shut down in the past few months. The program is one of many social services being hit hard by federal sequestration.\nIn late June, nearly a month after parents found out that the Newmarket Head Start Program was marked for closure, the program shut down.\nOne parent, Lynda Raczek, is working to make sure other parents in Newmarket who relied on Head Start will not have to struggle to find an alternative to the 20-hour a week program.\nRaczek is working to start a program she calls Write Start Preschool. The program will be nearly identical to Headstart. Raczek said she hopes to use the same location in the Newmarket Recreation Department's building.\nBasically, Raczek hopes to start a privately funded version of Head Start.\nHowever, it was not her original plan to create her own program.\nSo Raczek decided to just try to run the same Newmarket Head Start program that has been run for years, just without any affiliation with Head Start. Hence the name: Write Start Preschool.\nAccording to Raczek the program would cost $90,000 to run for an entire year in Newmarket. But she says it is worth every penny.\n\"Studies show that when you have a child go through a program like this they're more likely to graduate,\" she said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever wondered what a party thrown by the sweetest gal this side of the North Pole would be like? Yes, Mrs. Claus herself! As you can imagine, she has quite a bit on her calendar this time of year, from spinning sugar into delectable sweets to mending socks, hats, and knickers for the elves. Yes, her calendar is jam-packed. However, she always makes room in her schedule for a celebration or two with the special people in her life. She keeps it simple but doesn't skimp on the important details that make it a Claus-Approved party.\nAs head of the baking team, Mrs. Claus always ensures that the sweet nibbles for her party are both unique and delicious. For my Christmas party tablescape, I wanted to ensure the same; for this reason, I have partnered with Softer Than Brittle to showcase their unique, soft and flaky gourmet nut brittle that is hand-crafted from a 60-year old family recipe.\nI placed these scrumptious vittles in red and white and green and white popcorn and fry boxes to adorn each place setting. A large cake stand stands at the ready with cupcakes topped with Christmas tree toppers. Not to mention a few red and white lollies here and there.\nYou see, just as you can get the best and most special things only from the North Pole, you can get this \"Better-for-you-brittle\" that contains no preservatives, cholesterol, dairy, hydrogenated oil and is even vegan! Plus, Softer Than Brittle has a great nut flavor in a soft, flaky bite-size nibble! While these are so delicious on their own, you can even use them in baking whether in tart crusts or crushed up and sprinkled over cupcakes, and so many more ideas in between.\nThis post has been sponsored by Softer Than Brittle. Compensation may have been received in conjunction with it. However, all opinions and views are my very own.\nThis is too cute! Love the balloons!\nYour website really is amazing. I'm loving meeting you.\nThis is sooo cute! I cannot seem to locate the balloons on oriental trading, is there another location these are from?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2020 Port Huron Northern (Mich.) defensive end Braiden McGregor updates his recruitment as he prepares to head to Ann Arbor for possibly the final time before he makes a commitment.\nChristian, Jamie and Matt are back for another episode to share the latest thoughts on Notre Dame recruiting.\nMike Frank discusses his ideal wish list for commitments before August.\n2020 St. Louis (Hawaii) defensive end Jordan Botelho discusses his visit to Notre Dame.\nNotre Dame will host a big-time 2020 four-star prospect for the Blue-Gold Game.\nA running list of Notre Dame visitors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Individuals who want to lose weight and start living a healthier and fitter life can follow the simple workouts that can be done at a gymnasium. You may then gradually move on to more strenuous workouts to maintain the lean muscles and the new slimmer and trimmer body.\n\u2022 Ride a stationary bike, do jumping jacks, use a rower machine, or use a jumping rope for about 3 to 5 minutes to warm up.\n\u2022 Perform 3 sets of shoulder press, with a barbell, of 12 repetitions. Rest for about 60 to 90 seconds after each set.\n\u2022 Perform 1 set of squats with 12 repetitions without a barbell. Then do 3 sets of squats, with a barbell and without any weights, with 12 repetitions.\n\u2022 Perform 3 sets of deadlift, with a barbell, with 12 repetitions. If the barbell has weights, then place it about 6 to 8 inches above the ground level.\n\u2022 Perform 3 sets of pull ups or chin ups with 8 repetitions. In case you are unable to complete 8 pull ups, then ask someone for help to push you up.\n\u2022 Perform 3 sets of bench press, with a barbell without added weights, with 12 repetitions. If the barbell is very heavy, then do 12 repetitions.\n\u2022 Repeat the above during week 2. Increase the reps or add weights during week 3, only if you feel confident, but definitely during week 4. Ensure that you rest for at least 1 day a week.\nInstead of the above simple workouts listed above, you may also try out the weight loss and total fitness program at PhuketFit in Thailand. You will not only become fitter and healthier, but will also have fun doing it.\n\u2022 PhuketFit is a wellness and fitness resort in Phuket, Thailand. The team of trained professionals at PhuketFit have specialized skills, specific talents, lots of passion in the work that they do, and steely determination in ensuring that the clients get a kick-start to a new and better healthy lifestyle.\n\u2022 The total fitness program puts the customers on a fast-track to a new and better healthy lifestyle and your body will transform into a fit and strong one right before your eyes. The total fitness program ensures that every area of the body gets worked out and trained. This is one of the secrets of the high success rate of the program.\n\u2022 The Total Fitness program commences with a one-on-one personal evaluation session carried out by a specialist in the field of nutrition and fitness. The fitness program is then chalked out by both the specialist and the customer, the core of which is based around the goals, wants, and preferences of the client.\n\u2022 By the time the program has been successfully completed and you are ready to head back home, you will be equipped with all the strategies and tools to ensure, including how to cook healthy nutritious meals, the proper way to exercise, and tips on long-term physical and nutritional well-being, etc. This will ensure that your new healthy lifestyle and workout regimen continues at home as well.\nSo book your flights today to kick start a new healthier life!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..dcbac7eceb26caf8a645a49b32bdec0eefee8761
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"What happens when consumer electronics are designed, tested, and maintained by volunteers from around the world? The Neuros OSD is an interesting case study.\nHow To: Still don't have a home theater PC? Maybe you haven't seen what they can do, or maybe you just can't afford TiVo or Windows MCE. Learn how Linux can help you build your own for free.\nFirst Look: Revision 4 of KDE (K Desktop Environment) is in late beta and slated for a December release. With a Windows version in the works, Linux and Windows users may soon have more in common than they ever thought.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"General Sir Peter De La Billi\u00e8re ranks amongst the greatest of post-World War Two British military commanders. He served in every conflict from Korea to the first Gulf War and ended that career as a full General with two Knighthoods (Knight Commander of the Bath and Knight Commander of the British Empire - the latter of which superseded the CBE awarded previously) a DSO, two MCs and a Legion of Merit. He was in charge of the British special forces who so successfully stormed the Iranian Embassy in London in 1980 bringing swiftly to an end that very public siege. In 1990 he was the British Forces commander under General Norman Schwarzkopf in the first Gulf War.\nSpecial Air Service or SAS is a corps of the British Army constituted on 31 May 1950.They are part of the United Kingdom Special Forces (UKSF) and have served as a model for the special forces of many other countries all over the world.The SAS together with the Special Boat Service (SBS), Special Reconnaissance Regiment (SRR), and the Special Forces Support Group (SFSG) form the UKSF under the command of the Director Special Forces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thanks for the fast reply Ok, I removed the More Pics part from the duplicate section and it's working fine now. Of course there are no images duplicated but I don't need that to happen. Regards, Marja.\nHi Jim, first of all, thanks a lot for this exelent contribution. I've a very strange problem going on in the admin. When I duplicate a product with images it's get linked as well. The images are not duplicated. That's no problem but I've to get rid of the linked product. Hope you can help me out. Thanks a lot in advance. Regards, Marja.\nJust want to say THANK YOU for your \"Add a back button to product info\" contribution. Easy, effective, great. Thank you!\nThanks a lot, everything is running smoothly now. You're the best!\nHi Jim, I asked this question also in the recently viewed topic a few days ago, but this one seems more alive so I'll try again here. I'm running recently viewed as well as more pics but I think I've to change the sql for recently viewed to see the product images. Since you developed both I hope you can help me out. Thanks in advance en regards, Marja.\nHi Jim, I'm running recently viewed as well as more pics but I think I've to change the sql for recently viewed to see the product images. Since you developed both I hope you can help me out. Thanks in advance en regards, Marja.\nThanks a lot for your quick reply I might be missing something here but I can only select one image to be category image. Thanks again, Marja.\nHi Jim, Thanks for updating the contribution. It's working fine but still I've a question. I would like to show the thumbnails of the extra images on the product_listing can you give me a hint where to start? Thanks in advance. Marja.\nHi Armand, That's kind of strange indeed can you post or pm me a link?\nI'm not familiar with the pound sign. There are some guidelines for the euro sign (several links in this thread) but I don't know if there are any for the pound sign. If you're using plain text (no html) emails like the confirmation email, does the pound sign display correctly?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A North East technology solutions provider has made a significant investment in its systems to accommodate its growing client base.\nTeam Valley-based Diamond Group has upgraded its datacentre equipment, including new servers, switches and firewalls, to provide a flexible infrastructure which will support businesses of all sizes, from four user hosted environments up to those with hundreds of users.\nJohn Burns, managing director at Diamond Group, said: \"Our clients are increasingly buying into cloud computing and we wanted to provide a resilient and secure solution. By investing in a datacentre, we can offer 24 \/ 7 access to clients' own secure virtual offices, from any location with no upfront capital costs.\n\"This is an exciting move for Diamond Group which will offer a host of benefits to our clients, both now and in the future.\"\nDiamond Group has seen rapid take-up of its technology solutions, which has driven the move to a datacentre. The Company now has capacity to offer unrivalled service via its 10Gb data network switches and scalable internet access from 100Mbps.\nJohn continues: \"We are constantly consulting with our clients with regards to their cloud computing needs and how their own infrastructures would cope with a move to the desired cloud setup.\n\"In the past, larger clients have been able to implement that infrastructure and have been using datacentre solutions. Now that infrastructure connectivity is improving and becoming more accessible to smaller clients, our cloud solutions are consequently becoming more usable for those clients and this demand has led to a requirement for Diamond to invest and upgrade our datacentre equipment.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get it while you can! These little cuties are so popular right now! This is bow is made with 3 inch wide ribbon and the design is sublimated.\n3 Inch wide ribbon with Sublimated design. These little cuties are so popular right now!\nStunning Glitter Chevron bow great for games and Camp!! Choose your main color, chevron color and your mascot name. *Qualifies for School Bow package price.\nSimple with Sparkle. This bow is beautiful for the month of October to show your support! *Qualifies for School Bow package price!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"A multi-trade building facilities group has acquired an industrial unit at the popular Beacon Centre on Amesbury's giant Solstice Park.\nDevon-based Boden Group Facilities has signed a five-year lease on the 1,904 sq ft (176.88 sq m) Unit 14.\nProperty consultancy Myddelton & Major brokered the deal for the unit for which the quoting rent was \u00a312,950 per annum.\nMyddelton & Major Partner Dean Speer said: \"The rapidly expanding Boden Group Facilities is regional leader for the provision of multi-trade building services to commercial, industrial and large-scale residential clients.\n\"The company, founded by Steve Boden in 2009, is the only company in the south west to hold such an extensive range of supply, installation and maintenance services under one roof.\nThe Beacon Centre is situated at Solstice Park, the 160-acre mixed use business park, strategically located with direct access to the A303 dual carriageway at Amesbury.\n\"It occupies a prime spot at the gateway to both the popular and convenient \u00ad Solstice Park and the 20 acre industrial\/distribution zone.\nLong-established Solstice Park's other major occupiers include Home Bargains, Muller Wiseman Dairies and Tintometer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2190 Hey Atheist! Put Your Money Where Your Heart Is!\nThe Seattle Times editorial was titled, \"Uganda and Gays: A Malicious Blasphemy.\" The newspaper waxes eloquently about all of the good being done by loving, good-works-doing selfless evangelicals in the rest of the world and claims that the message brought to Uganda is, as the title says, malicious blasphemy.\nSo, the Bible contains malicious blasphemy? I suppose the newly converted Christians in Nigeria who are exorcising demons out of children with acid, beatings, and hot coals, \/ are engaged in blasphemy too? I suppose the newly converted Christians in India who burned Hindu teens as witches also were engaged in blasphemy?\nI find it ironic that anything evil done in the name of religion is a \"perversion\" or blasphemy \u2014 and anything good, that's the real deal. It's an argument I hear over and over in response to my articles on the Daily Kos and Huffington Post.\nThe falsehood is patently obvious. It's like saying that selfishness and greed are a perversion of our humanity, and altruism is what humans really are all about. Get real. Ask any biologist whether dogs are affectionate or predatory and they will laugh at you: Do bees make honey or do they have stingers?\nMy selfishness is every bit as real as my generosity. My tenderness and bitchiness, compassion and aggression all are ME. Religion's track record of power-brokering and atrocity is every bit as integral as its history of giving voice to our moral instincts and sense of wonder.\nIt is not the perversion of religion that is playing itself out in Islamic jihad or Evangelical homophobia. It is religion, period \u2014 once face of religion to be sure, but the real deal. It is the timeless face of god-worship that is tribal and intolerant and willing to kill \u2014 as religion always has been under the right circumstances of time and place.\nCan we please stop pretending and making nice? People are being tortured to death, starved to death, and executed in the service of the religious enterprise! Do we ever get to run the numbers? Do we ever get to ask whether all of the fuzzy feel good stuff and the sense of meaning and purpose, and the wonderful creative moral communities that religion produces are worth the price?\nBecause the price is what we are seeing in Somalia, and Nigeria, Uganda and elsewhere: people starving, children burned with acid (I dare you to look at the pictures), gays slated for execution, doctors murdered, politicians and mullahs who commit us to holy war.\nBoth good and bad consequences of \"faith\" are the direct products of the agreement we make with each other that it's okay to believe things on paltry evidence, the kind that would never stand up in court, the kind that would never guide the surgeon's knife. When, in the embrace of belief, we entrust ourselves to authority, sacred texts and our own intuitions, we are freed from the dull left-brain constraints of everyday life. But we also become unaccountable to reason and evidence, unaccountable to universal ethics like the Golden Rule. Faith gives us mysticism and murder.\nWhen will we be ready to move beyond belief to whatever the next stage of our spiritual evolution may be?\nGallery | This entry was posted in Christianity in the Public Square, Musings & Rants: Christianity and tagged Christian violence, fundamentalist Christianity, gay rights, Uganda. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you experience problems with sale your natural flora, sale you can use probiotics class drugs to for help it regenerate. Broad spectrum antibiotics brand diflucan for sale australia may be australia dangerous, ruining for the balance of diflucan the bacteria in australia body, if you do not sale use them properly. When Amoxil is combined with other drugs for australia ulcers treatment, the most common side effects are taste sensation changes, headache and australia diarrhoea. Cheap Amoxil can be ordered online. Our website offers you the maximum simplified way to find the best deal to buy different how long does it take for diflucan to work medications. Italy 4880804, wind, vodafone see SMS short codes for for other countries. In addition, you can buy non-prescription antibiotics for sale pets (dogs, cats) without visiting the vet. Antibiotics eye sale drops, or antibiotics weight loss solution may be purchased in local drugs store, at Walgreen, or other places, so for you can avoid visiting the doctor first, paying for sale the prescription. Amoxil sale is not australia limited so it is your personal responsibility to check you medical condition before taking these medications and for make sure you take proper dosages. This drug is not hazardous to fetus or an unborn child. In case sale you missed a dose for take it as diflucan for yeast infection soon as possible when you remember. Additional information The other name of sale Amoxicillin australia is Trimox which means that sale Trimox drug has the same substance which can be bought on-line and for with no sale prescription in our shop. Many people still suffer from infections and diseases that could have been easily cured by existing medication, such as natural cream diflucan for skin problems, sale or tables for oral intake that prevent urinary tract infections. Such countries as USA, UK, Canada, Australia and many others have strict diflucan otc regulations governing the pharmaceutical market, protecting the interest of major pharmaceutical companies against the interests of the consumers. Then why do we still live enduring these problems, instead of treating them properly? Sometimes you may get avitaminosis too: lack of vitamins in body, due to improper use of antibiotics. Jim and his wife, Kellye, have four school aged children). After that you may get back to your regular dosing regime. DRE License # nmls # cell (707) 953-6622, jim is a licensed Real Estate Broker, a Certified Public Accountant (inactive and a licensed General Contractor. It can be dangerous, and that's why many antibiotics are only available with prescription. So, here is the question: is there any sale way to order australia antibiotics with no prescription, when you need it? Also you can buy antibiotics without prescription. Ulcers, amoxil sale is mixed with other drugs in ulcers treatment. Take the full course and recommended dosages of Amoxil to prevent the decrease of its effectiveness against infection. If you need some serious antibiotics without prescription, the next paragraph is for you. The chewable tablet contains phenylalanine and should not be used for if you or your child has the hereditary disease phenylketonuria. Regular usage of Amoxil is able to diminish the effectiveness of birth control pills, which may result in unplanne pregnancy. Chewable pills and tablets should be stored at or below 77 degrees Fahrenheit in a tightly closed container. Do not take Amoxil if you are a breast-feeding mother. Lower Respiratory Tract Infections mild, moderate, or severe infections: 500 milligrams for every 8 hours or 875 milligrams every 12 hours. Please consult a specialist in this case. In cases when you are taking a dose every 12 hours and if it is almost time to take the next dose take the missed dose and then take next dose after 5 or 6 hours. That is why if you are really looking for a good price, you should pay your attention to our website to find the best deal. Moreover, you can find services offering you to use conveniently-organized information on the offers of different online pharmacies no prescription. Amoxil medicine is rated pregnancy category B by FDA. He has been involved with the sale, development, construction and financing of properties over the past forty years diflucan including single family to commercial, construction, and joint venture transactions. Many brand drugs are protected with patent and the companies overprice these drugs, while access to cheap medications is restricted for patients, who need it most. One of the most valuable of them is that the overhead costs carried by online businesses are lower than those carried by land-based businesses. For those who use Amoxil suspension, it should be well australia shaken before usage. Nevertheless, if you have taken some antibiotics before, for example antibiotics without gelatin or antibiotics without lactose, you probably know for when you need them and what is the needed dosage for you, so you can take them on your own. To avoid most of the side effects, it's important to know about precautions and interactions, for example, drinking on antibiotics is strictly prohibited. It is a tested and approved drug which is available to people different income levels. Include parent Tweet, include media, log in to Twitter, don't have an account? To be competitive is very sale important for online businesses the same as it is for any business. Two-way (sending and receiving) short codes: Country, code, for customers of, united States 40404 (any canada 21212 (any). Mild or moderate infections - 25 milligrams per.2 pounds of body weight, every 12 hours or 20 milligrams per.2 pounds of body weight every 8 hours. Thanks to this, you can compare the offers of different provider within just several seconds. If you feel better before the full treatment time is over, you have to continue taking Amoxil. The preparation should be taken right after it was mixed. Amoxil online can be ordered by everybody as well as citizens.S.A. There is a range of reason for that. Even our pets, sale such as your dog or cat, can make use of modern medications. And another benefit when you buy Amoxil on line you have a chance to receive free pills with every order. Do not increase Amoxil dose or prolong the course of treatment. Dosage for children weighing less than 88 pounds will be defined for by their weight. Jim serves for on the Board of Directors as Vice Chairman of Exchange Bank, sale a community bank with assets of approximately.2 billion. Bharti Airtel, Videocon, Reliance, indonesia 89887, aXIS, 3, Telkomsel, Indosat, XL Axiata. Amoxil medication can be also used for other purposes diflucan dosage not mentioned in this medication guide. Other side effects usual for most antibiotics drugs are vertigo, bloating, coverage, and more. Regardless of what mediation you need to order, you will find it extremely easy to find different offers if you use the Internet. That is why it is reasonable for online pharmacies to offer their customers more attractive prices on different medication. In case you notice diflucan listed above reactions stop treatment with Amoxil unless your doctor suggests you to continue. For children 3 months or younger the maximum daily dose is 30 milligrams per.2 pounds of body weight every 12 hours. Antibiotics for sale in online pharmacy. There is a solution for dysbiosis, too. Though this medication can pass into breast milk so it is not recommended to women who are nursing babies. It is preferable to keep all medication in the refrigerator. If you are under a hemodialysis and your kidneys are extremely damaged you dosage has to be adjusted accordingly. Child has to drink diflucan over the counter all the preparation to make sure he\/she gets a full dose. Amoxil prescription is not required diflucan when you buy online. Amoxil Class and Mechanism, generic name: Amoxicillin, brand name: Amoxil, amoxil is an antibiotic of penicillin drug group which is widely used in treatment of a broad variety of infections such as: skin infections, middle ear infections, gonorrhea, upper and lower respiratory. If some online pharmacy is currently offering you some discount to sell diflucan some particular medication, you will find this information easily. Go to a person's profile, embed this Video, add this video to your website by copying the code below. The disease may reoccur in case of incomplete course of Amoxil and as well the chance of bacteria resistance to Amoxil or similar antibiotics may be increased. Using the different conditions of different online pharmacies together provider great conditions for comparison and clearly each online pharmacy no Rx aims to provide the most attractive conditions for the customers. Adults, dosage for treatment of nasal, throat, ear, skin\/genital and urinary infections. In the immediate aftermath of Donald Trumps election as president, Amy diflucan Siskind, a former Wall Street executive and. Do not forget to read the instructions included in the package with the drug: some should not be taken with alcohol, some have dangerous interactions with other antibacterial drugs, and most have contraindications, depending diflucan on your conditions. Severe infections - 45 milligrams per.2 pounds of body weight every 12 hours or 40 milligrams per.2 pounds of body weight every 8 hours. Also, it is very easy to find interested offers on the Internet that may just be available for a limited period of time. Without doctors antibiotics misuse becomes very probable: when you cannot determine the right dosage of the medicine, or if you take antibiotics without infection, it's easy to overdose or overuse the drug, exposing your body to risk of dysbiosis, which can cause multiple side effects. As it has been discovered by many patients, when you're in need of antibiotics for sale with no prescription, you can turn to online pharmacy, which is very convenient: they sell solutions for uti, acne and many other. Antibiotics vs Probiotics, there is one important thing that you have to know before you can use antibiotics without doctor, if you need. Symptoms of Amoxil overdose may include: Diarrhoea, vomiting, nausea and stomach cramps australia Side effects Side effects may include: Anxiety, anaemia, behavioural changes, colitis, confusion, convulsions, diarrhoea, dizziness, hives, hyperactivity, insomnia, liver problems and jaundice, nausea, peeling skin, rash, tooth discoloration in children, vomiting. Over the counter antibiotics, not all antibiotics are sold without prescription, but many belong to the over the counter antibiotics, which means that you can buy them on your own, without a need to get antibiotics from doctor for tooth, sale for example. That is why it is important to remember that when you order Amoxil it will not treat a viral infection like common cold. When you buy Amoxil keep in mind that this medication is only used in bacterial infection treatment. Follow the dosage directions on this medication guide or you doctors instructions. Just several years ago, you had the only one options to buy medications when you needed them which is buying them at different land-based pharmacy. Mild or moderate infections: 250 milligrams every 8 hours or 500 milligrams every 12 hours. Amoxil suspension and pediatric drops should be stored in a tightly closed bottle. Antibacterial drugs and antimicrobial agents, contemporary sale medical science has found cure for most of the diseases ever experienced by men. Store capsules at or below 68 degrees Fahrenheit. Discount Amoxil can be found online especially if you order medication for the full course. Human body is full of bacteria naturally: some types are necessary for proper functioning of the body, such as lactobacillus, and others. When you are aware of the effects of antibiotics and alcohol, antibiotics and sun, antibiotics zoloft interactions (and other interactions as well you can use the drug safely and effectively. Order sale antibiotics with no prescription. Various dosage regimens are available in this case. If not, there are various side effects that you may experience, depending on your personal sensitivity to the drug, medical conditions, overdose and other misuse of the medications: most common are diarrhea, vomiting, constipation, nausea, and vomiting. In cases when you are taking a dose every 8 hours, take the one you missed and the next dose after 2 to 4 hours. It is strongly recommended to take the full course of Amoxil especially if you are treating such infections as strep throat. For infants and little children Amoxil should be in drops. Recommended amount of liquid diflucan Amoxil must be applied directly sale on the child's tongue for swallowing. However, most of the antibiotics pills require a prescription for good: you won't harm yourself, trying to use them without a need or overdosing them.\nIf a diflucan sunburn-like reaction or skin eruption occurs, instruct patients to diflucan contact their physician. Marie., CA I drink it each morning, but diflucan I hold it in my discount mouth for awhile discount before swallowing. Trips to my HMO for antibiotics proved to be of no help. Take cipro XR one time each day at about the discount same time each day. Resistance to fluoroquinolones occurs primarily by either mutations in the DNA gyrases, decreased outer membrane permeability, or drug efflux. Drug-Drug Interactions Antacids Concurrent administration of antacids containing magnesium hydroxide or aluminum hydroxide may reduce the bioavailability of Ciprofloxacin by as much as 90 see Dosage and what is diflucan discount Administration (.4 ) and Drug Interactions ( 7 discount ). Drink plenty of fluids while taking cipro. Inclusive would just save you the pharmacy in canada trouble of paying diflucan the bills yourself. One of them saw us the other night and made a point diflucan to tell us that since diflucan we started giving her the CS (a little over a year now) her son hasn't been sick. The manual and video are at must if you are at all serious about using CS in your health regimen. A: Policies are authoritative statements from decision makers (e.g., mayors, governors, provosts, chief executive officers, city councils, legislatures, boards of directors, etc.) that outline goals and strategies, usually in broad terms. These alternative pathways of drug elimination appear to compensate for the reduced renal excretion in patients with renal impairment. Development Of Drug Resistant Bacteria Prescribing cipro XR Tablets in the absence of a proven or strongly suspected bacterial infection or a prophylactic indication is unlikely to provide benefit to the patient and increases the risk of the development of drug-resistant bacteria. This diflucan stuff works almost instantaneously (well, okay, it took a day or two, but then I was feeling so much better I couldn't believe it).\" Kathy., Las Vegas, NV \"Whenever we use the micro-particle silver, it really works! Although you will find the information that you may need on the website of this online pharmacy, you may also be interested to ask us other question. I so wish I would have taken pictures for you but that was of low priority at the time. I gave her a gallon of micro-particle silver. Cut into squares when diflucan cold. After about six months the cancer grew back again, but this time it was huge. Friedlander AM,. I use a spray bottle or a Q-tip to apply. Drug Interactions Oral Antidiabetic Agents Inform patients that hypoglycemia has been reported when Ciprofloxacin and oral antidiabetic agents were diflucan co-administered; if low blood sugar occurs with Cipro, instruct them to consult their physician and that their antibacterial medicine may need to be changed. The serum creatinine should represent diflucan a steady state of renal function. Tell your healthcare provider if you have a history of seizures before you start taking Cipro. The amount discount of ciprofloxacin absorbed by the nursing infant is unknown.\nThe price level that was detected on March 24 was 41 times the price drinking water standard, said diflucan Gerry Pollet. 23 SE, parkerBoats T08:05:5100:00, length, lOA, rating, bEAM. Dier madalya kazanan yarlarmz Semih mesut, Badag\u00fcl price \u00d6zkan, Mete \u00d6ZEN, diflucan Mahmut tosun, Cengizhan \u00d6ZAL, Semanur G\u00d6N\u00dcL, Celal akin, Mete Kaan altun, Ahmet Yasin akmeedir. It is difficult to measure. . Here are a few facts that no one is arguing about: Three of the plants have experienced diflucan melt-downs and melt-throughs, which means this accident trumps Chernobyl as the worst nuclear accident ever. What we have found about the radiation coming from Fukushima is highly alarming. . Sardine, anchovy and herring populations have dropped dramatically along the west coast in recent years. BioSuperfood is credited with saving hundreds of lives of radiation victims at Chernobyl and much of the Soviet livestock price industry. Self-Bailing Cockpit w\/Contrasting Non-Skid Surface, boarding Platform price w\/Ladder (3 Step hull Deck Joint Thru-Bolted (S\/S Fasteners). Candida : albicans, Cryptococcus neoformans, parapsilosis, glabrata tropicalis. And this is just the tip of the iceberg. . 0-12 price Ya kategorisi anadil yar ampiyonu, Ankara\/H\u00fcseyin G\u00fcll\u00fcolu Ortaokulundan Ouz Berkay Yiiter olmutur. Sockeye salmon returns plunge to historic lows. An enormous amount of radiation was released into the sky diflucan and the water, and continues to be released. . Stereo w\/Speakers, rod Storage Toe Rails Under Gunwale. Flotation Foam 10 Year price Limited Hull Warranty, s\/S Bow Rail: Two-Piece, Low Style. Ampiyonada price Madalya Alan yarlarmzn isimleri aada belirtilmitir. Pacific Ocean Currents Are Constantly Bringing Fukushima Radiation To Entire West Coast This Radiation Is Bio-Accumulating In Your Body And In All Living ThingsPlant And Animals. For thousands of species of kelp, diflucan algae, and other vegetable life, and hundreds of species of animals our ocean will never be the same again. Folding Footrest For Above Deck Livewell Powder Coat Upgrade for Leaning Post w\/Rocket Launcher Backrest Fiberglass Hard-Top Includes: Electronics diflucan Box w\/ Fwd Integrated.E.D. Which Has Now Permeated The Pacific Ocean. Radiation from the meltdown Fukushima nuclear disaster is poisoning animal and vegetable sea life all over the Pacific ocean and coast. Trychop h diflucan yton spp. Compass, horn, leaning Post w\/Rocket Launchers On Above Deck Livewell (30 gals). Derece ve ayrntlara, price yandaki, resm Yar Sonular sayfasndan ulalabilir. Birlemi Milletlere diflucan bal intersteno Uluslararas Bilgi lem ve letiim Federasyonu tarafndan d\u00fczenlenen ve d\u00fcnya genelinde katlan on binlerce yar elemeleri sonucunda yaplan bu yln D\u00fcnya nternet Klavye ampiyonas sonuland. .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Personal injury attorneys know that if you are a cyclist that has been injured in an accident, you are part of a disturbing trend. The combination of school returning, and shorter daylight hours means that drivers must be increasingly aware of cyclists on the road. A community that integrates cycling as a practical means of transportation offers a better quality life.\nBicycling is good for the environment, the economy, and provides a practical alternative to congested streets. Alternatively, it is an effective way to improve the physical health of Charlotte's citizens. As in major cities all across the country, more people in Charlotte are discovering the many benefits of bicycling. Local government will need to work hard to ensure that bicycling in Charlotte is safe and efficient.\no\tLook carefully before opening your car door - Cyclists can become hidden in your vehicle's blind spots, literally in a nanosecond. Particularly if you are driving a large SUV or a Van. Van doors that slide open on the traffic lane side, are particularly dangerous for both cyclists and passengers.\no\tBe aware of cyclists at road junctions - Cyclist can be moving too fast, or are simply hard to see as they turn into or out of junctions, especially in the dark. It is also important to remember that window pillars in your car can completely obscure cyclists and pedestrians. Move your head and look around the pillars for a safer view.\no\tWhen turning right, look out for cyclists on the inside of the traffic lane - Make sure that you check mirrors and blind spots before starting your turn.\no\tBe aware that cyclists may suddenly swerve - What seems to be a perfectly smooth road to a driver cans still be filled with hazards for a cyclist. They must move or swerve to avoid potholes, drain covers or road debris. Allow plenty of room (preferably 3-4 feet) when overtaking a cyclist.\no\tRemember that strong gusts of wind can blow a cyclist off course- when overtaking a cyclist near gaps in fences, buildings, hedges, or other exposed areas; there is always a danger of crosswinds being present.\nIn many municipalities, cyclists are held to the same traffic standards as motorists. However, it is to everyone's benefit that extra care and caution are used around cyclists because being in the right does not prevent accidents being aware does.\nHiring a personal injury attorney is essential if you want the best possible compensation after your accident. If you want the best representation after you are injured in an accident, then call their office for a free consultation with an experienced personal injury attorney.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"U.S. Secretary of State John Kerry has arrived in the Russian Black Sea resort of Sochi, the U.S. Embassy in Russia reported on Tuesday.\n\"Here we go! #KerryinSochi,\" U.S. Embassy spokesman Will Stevens has tweeted.\nIt is Kerry's first visit to Russia in two years.\nKremlin spokesman Dmitry Peskov said earlier that the agenda for Tuesday's meeting of Russian President Vladimir Putin and U.S. Secretary of State John Kerry would be as \"broad as possible\", and Russian Foreign Minister Sergei Lavrov would also attend this meeting.\n\"The president will meet with State Secretary Kerry today. The meeting will also be attended by Lavrov, who will hold detailed talks with Kerry prior to it. The minister and the state secretary will then inform the president about the results of their contacts,\" Peskov said.\nWhen commenting on the agenda for the upcoming Putin-Kerry meeting, the spokesman said that \"the agenda will be as broad as possible, ranging from bilateral relations to all 'troubled areas'.\"\nIt was Peskov's answer to a question whether or not Ukraine, Syria and Iran would be addressed in Sochi.\nWhen asked whether the issue of sanctions would be discussed, Peskov noted: \"We have said more than once that it is not our topic. But, naturally, we have articulated our stance.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjwdz b/data_all_eng_slimpj/shuffled/split2/finalzzzbjwdz
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index 0000000000000000000000000000000000000000..85506a69af6b6541e0b9d7b13447e14ba35b813a
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjwdz
@@ -0,0 +1,5 @@
+{"text":"Home > Reviews > Retailhire.org Reviews \u2013 WARNING \u2013 Does This Work?\nis a cold start (that is, the Retailhire.org Reviews is stopped for more than 6 hours), at the moment of starting Dry wear occurs, and the damage to the car Retailhire.org Reviews is greatest. Another situation is that the Retailhire.org Reviews runs at high speed and the Retailhire.org Reviews oil is relatively thin. At this time, the car Retailhire.org Reviews is in a semi-dry state. This situation cannot be ignored.\nThis situation is caused by a gap between the combustion chamber and the cylinder, causing the oil to enter the combustion chamber. The way to find that the car Retailhire.org Reviews has the phenomenon of heating oil is to first check whether the exhaust pipe emits blue smoke. When the hand reaches the exhaust pipe and probes it, it smells whether the finger has organic oil taste. If there is taste, it proves that the organic Retailhire.org Reviews in the car Retailhire.org Reviews is defective. Into the combustion chamber.\nThe oil seal ring is aging, the oil will be slowly leaked out. For the phenomenon of lightly burning the oil, adding a certain amount of repair agent when changing the oil can be solved, but if the situation is serious, the car Retailhire.org Reviews needs to be disassembled.\n1. Frequently check the fastening parts.\nDiesel Retailhire.org Reviews or gasoline Retailhire.org Reviews is affected by vibration shock and uneven load during use. Bolts and screws are easy to loosen, and adjustment bolts of various parts should be carefully checked to avoid damage to parts due to looseness. Valve clearance, valve timing, fuel supply advance angle, injection pressure and ignition timing should be checked and adjusted in time to ensure that the Retailhire.org Reviews is always in good technical condition. This will save fuel and extend the life of the Retailhire.org Reviews.\nfuel oil impure, which will make the precision matching machine wear and increase the matching gap, resulting in oil leakage, oil drop, power supply pressure reduction, gap increase, and even oil path. If the air contains a lot of dust, it will accelerate the wear of the cylinder liner, piston and piston ring. If the cooling water is not pure, the scale of the cooling system will increase, which will hinder the heat dissipation of the Retailhire.org Reviews and lubrication conditions. Deterioration, the wear of the machine is serious; if the external surface of the machine is not clean, the body will be corroded and the service life will be shortened.\nand air is not available or interrupted, and there will be difficulties in starting, burning, power reduction, and abnormal operation of the Retailhire.org Reviews; if the oil supply is insufficient or interrupted, it will also The Retailhire.org Reviews is poorly lubricated, the parts are worn out seriously, and even the burning of the tiles occurs. When the cooling water is insufficient, the temperature is too high, the power is reduced, the wear is intensified, and the service life is reduced.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Current research has indicated that up to four injections per day of insulin is protocol. Even better, an insulin pump to reduce vascular and renal complications of Type I diabetics.\nNow, if you add a glucose sensor and a miniature computer to an insulin pump you have just manufactured an artificial pancreas that would dispense insulin as needed.\nMist inhalers and insulin patches are other methods of administering insulin.\nPancreatic Islet Cell Transplants are increasing in their ability to help Type I diabetics. The limiting factor in islet transplantations is the need for long-term immunosuppression.\nThe side effects from immunosupressant drugs may cause complications. Thus, pancreatic islet transplantations are only used for patients that cannot control their diabetes any other way.\nFor those of you that have Type II diabetes the worry about ketoacidosis is far less but the other complications that afflict Type I diabetics still exist for you. Heredity or a familial predisposition is very strong for Type II diabetics. An estimated 25 to 30 percent of Americans carry a gene that predisposes them to Type II diabetes, with non-white populations at a far greater risk.\nMost Type II diabetics produce insulin but the insulin receptors on body cells are unable to respond to it, a phenomenon called insulin resistance. Mutations in any one of several genes could lead to insulin resistance. One example is a gene called PC-1. That gene, a membrane protein gene, causes that membrane protein to react strongly to insulin receptor that prevents the receptor from being activated.\nLifestyle can play a role. Diabetics are almost all overweight and sedentary. Adipose cells of obese folks overproduce a number of hormone-like chemicals including tumor necrosis factor alpha and adiponectin which may alter the cascade sequence triggered by insulin binding that makes it possible for your body cells to absorb glucose for metabolism.\nThe good news is that many Type II diabetics can control their diabetes by exercise, weight loss and a healthy diet. Some diabetics benefit from oral medications like Orinase. There is a number of drugs that are referred to as insulin resistance reducers such as Avandia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here an eternity symbol is hovering over an imaginary landscape dividing it into several parts. Looking through its middle the perception of the abstract landscapes seemingly changes in tone. The symbol can stand for infinity, for not ending pleasures which most people wish for. It might also stand for the heaven on earth, tranquility and rest. The image is painted in vibrant colors and a picturesque style.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jet Airways chief executive officer Vinay Dube has informed employees the airline is very close to finding a solution to the airline's financial crisis and has sought their support.\n\"Our chairman, the board of directors and your management team are working hard on a balance sheet transaction that will help us eliminate our current challenges. In fact, the airline's key stakeholders are actively engaged towards finalization of the resolution plan. We are very close and I ask for your continued patience,\" Dube wrote in an email to staff on Thursday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Schedule: In the shop 4-5 days a week depending on season\/amount of business. Also some outfitters also drive shuttles, which effects the shop schedule. Our aim is to create a regular work schedule for you where you have the same 2 consecutive days off.\nYear-round work: Winter work (November 25th-Feb 28) is available for several of our Outfitters. Many stay on for the winter season to outfit part-time, work on seasonal projects, while mixing in a few weeks of vacation and travel.\nCombo Positions: Many Outfitters work combo positions where some of their regular work week is spent as a Trailhead Shuttle Driver or an Assistant Guide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..67a4bb725e4785394cfe05c51c632b214a8f917d
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+{"text":"Astrophysics graduate with experience in data analysis, laboratory work and mathematical modeling. Excellent IT skills especially in MS Office and Matlab.\nI have worked abroad and am now looking for a career which will utilise my education, intelligence and strong work ethic.\n\u00b7 Good experience of Matlab including 3D modeling.\n\u00b7 Excellent skills in Ms Office applications especially Excel.\n\u00b7 Strong mathematical and analytical skills.\n\u00b7 This included a large amount of laboratory work including in the university observatory where I used optical and radio telescopes.\n\u00b7 Dissertation 'Searching SLOAN for Planet Hunting Targets'.\n\u00b7 The analysis involved over 300 data sets in which I identified specific patterns and types. These were stellar spectra analysed for gravity, spectral type and distance.\n\u00b7 The assessment included a poster presentation involving both oral communication and design skills.\n\u00b7 Observatory Accreditation - Completion of this extra course allowed me to work at any time unsupervised at the university observatory.\nI took a science access course at Anglia Ruskin as my A levels suffered due to ill health.\n\u00b7 I have recently returned from South Korea where I taught English as a second language.\n\u00b7 I have used initiative and self reliance to work with people from a completely different culture.\n\u00b7 By teaching Korean children I have learned how to get the best from people even when language, culture and age present obstacles.\n\u00b7 In addition to general administrative duties. I was charged with organising new administrative processes for the new Children's Unit.\n\u00b7 I was able to ensure the transition went smoothly and the new system has proven robust enough to remain in place.\n\u00b7 I also created a new database of children in the North Essex area suffering from life threatening conditions. This project mapped both NHS and charity services and identified areas for improvement in order to assist the creation of a business plan to secure funding.\n\u00b7 When able, I volunteer with the Institute of Physics outreach programme 'Physics in the Field'. We teach physics based tricks to the public at events such as county shows and music festivals to raise interest in Physics.\nAssociate member of the institute of Physics since 2008.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Trustoria includes a professional history of nearly all local Radiology Transcriptionist in your town.\nPick a town to find a complete directory of Radiology Transcriptionists living near you.\nThe Trustoria Service Professional Directory provides people with all the information needed to discover the most qualified Health specialist for any need.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The New York Legislature has enacted, and Governor Andrew M. Cuomo recently signed into law, a new provision of the Public Health Law that permits Article 281 hospital outpatient clinics (clinic) and diagnostic and treatment centers (D&TC) to provide primary care services off-site for patients who are unable to travel to the facility.\nthat are provided to a patient who is unable to leave his or her residence to receive services at the clinic or D&TC without unreasonable difficulty due to circumstances including but not limited to clinical impairment.\nThe new law allows the Commissioner to authorize clinics and D&TCs to maintain continuity of care with existing patients who, because of a deterioration in their health, are unable to travel to the facility for primary care services. The facilities would be able, for example, to send a primary care physician to a patient's home, assisted living residence, or nursing home to continue caring for the patient, including diagnosing any new medical problem, adjusting medications, and reviewing and revising the patient's plan of care.\nThe new law takes effect March 28, 2017, but it authorizes the Commissioner of DOH to promulgate regulations in the meantime. The law requires that the regulations be developed in consultation with representatives of clinics, D&TCs, home care services agencies, and their respective trade associations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The writer's name is Mica Berta although every person not her birth label.\nFor a while he's been in Nebraska and he doesn't you should plan on changing that.\nceramics and I would personally never give it up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Johnny Depp will play The Invisible Man, which seems more like a Saturday Night Live sketch than a thing that makes sense in real life. Let's break down why, shall we?\n1. If he's invisible, how is Depp expected to slather himself in nightmare-inducing makeup?\n2. There's no writer or director yet, but Universal has got this master thespian pinned down.\n3. Also on board is producer Alex Kurtzman, who memorably wrote Mission: Impossible III, Transformers: Revenge of the Fallen, Star Trek Into Darkness, and Cowboys and Aliens. Take that as you will.\n4. Following a Tom Cruise-fronted The Mummy, this will be the second movie in Universal's dedicated attempt to reboot their classic monsters line, which included Dracula, Frankenstein, The Invisible Man, and The Wolf Man. Here's just a few movies that attempted to resuscitate these classic characters over the years and failed hard.\n5. The Invisible Man will look absurd as a fedora floating above a pile of scarves.\nKristy Puchko once loved Johnny Depp. But that was a time before her job required her to witness Dark Shadows, The Lone Ranger, Transcendence, Tusk, Into The Woods and Black Mass. She was spared Mortdecai because sometimes the assignment gods are kind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..15ae0db23d83e712abc7764b769939f4cd6491f4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmhgn
@@ -0,0 +1,5 @@
+{"text":"How did you become the dean of admission and financial aid at Sewanee?\nLike many of my peers, I fell into this field. My first year was 1981 and I spent the year working on data entry, touring students around campus, and sitting down for one-on-one conversations with students. I imagined that I would work for a year or two and then pursue a teaching career, but the excitement of the work kept me in the field and on the Mountain for 20 years in the office of admission. While most of my career has been at Sewanee, I did have a three-year stint at Porter-Gaud School as a college counselor before heading to the admission office at Agnes Scott College. The Mountain called me home in 2010 and I've never looked back.\nMy parents were both educators, so it was easy to imagine a teaching career. The long-time director of admission at Sewanee took a leap of faith and hired me with no experience, and for that I will always be grateful. The variety of the work keeps me on my toes and the opportunity to meet students in their schools or when they visit is really the best part. I realized after a few years that I was watching students go through such a transformation from high school student to a young adult, and I realized that the work is important as students make the transition.\nAfter so many years I still enjoy the one-on-one conversations with students and the travel. The work has changed since 1981 and goals are more aggressive and harder to meet, but working with students is still the best part of the job.\nFor me, NACAC has defined professional development and has given me opportunities to meet folks from different parts of the country and the world, as well as from different organizations and schools. In many ways, my involvement in NACAC and Southern ACAC really formed who I am as a professional in the field of higher education. I have had so many mentors that I have met through these organizations, and my goal is to pay it forward and mentor younger admission counselors and college counselors.\nI have been fortunate that the three places that I have worked -- Sewanee, Porter-Gaud and Agnes Scott -- supported my professional development with opportunities to attend conferences and workshops. These opportunities have played a key role in staying fresh with ideas, connecting with colleagues, and staying informed on trends and fresh ideas. Many individuals may not have these same opportunities without the Imagine Fund and grants for participation. I believe that without professional development opportunities, it could be easy to find the work frustrating, lonely, and overwhelming.\nI love to spend time with my husband, Dan, and our two dogs, Kaiya and Buddy, and I try to do as much hiking and exploring the Domain of the University of the South as possible. I also love working out in the early morning hours to get my day started.\nThis is hard, but I hope this is how my friends and colleagues see me -- open-minded, a Pollyanna, feisty, kind, and dependable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Between the responsibility for the care of her injured mother and straightening out her muddled finances, public health researcher Sarah Wright hasn't a minute to herself, much less time to repair a fractured romance. After a much loved aunt goes missing, Sarah is convinced it's a kidnapping but the police refuse to investigate. Former fianc\u00e9 Dan flies to Sarah's side to help\u2014and it looks like things might come back together for the two of them\u2014until Sarah is arrested for her aunt's murder. As evidence stacks up against her, Sarah must find the real culprits as well as unravel decades old family secrets along the way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chicago, Illinois \u2013 Commodities and futures trading on the floor of the Chicago Mercantile Exchange.\nAmana, Iowa \u2013 A worker runs a warping creel at the Amana Woolen Mill. The machine gathers up to 240 strands of yarn that will later be woven into cloth on a loom. The Woolen Mill is one of the enterprises started when German immigrants established the communal Amana Colonies in 1855.\nDenver, Colorado \u2013 Production of gold and silver coins at the United States Mint. Finished dimes go through a counting machine before being packaged for distribution.\nGillette, Wyoming \u2013 A surface coal mine in the Powder River Basin- the largest coal mining region in the United States.\nGillette, Wyoming \u2013 An aerial view of a coal train loading facility at a surface coal mine in the Powder River Basin. The train on the loop at right is just finishing loading, while the train at left approaches the loading silo. The Powder River Basin is the largest coal mining region in the United States. About 80 trains a day move coal from the Basin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Field trip to the Downtown Historic District engages students' minds and bodies.\nWhere can science take an elementary school-aged child?\nStudents, Artist-in-Residence create a mural of underwater marine life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are literally dozens of magazines dedicated for home improvements or home designs. You may collect different designs from these magazines. They feature different and varied designs on stairs and you may pick one that is just right for your home. You may of course improve on some details to fit it perfectly to your house. This is the easy task in choosing a design. It is the same as just ordering a pre made stairs and all you have to do is install with all the easy to follow procedure on how to do it. But if you want to really own the design then you have to really design it yourself.\nThe designs you culled from the magazines as well as from the internet could serve as a basis for designing your own staircase. You can either copy all the attributes of the parts that you like, or you can design one from scratch. Designing your stairs will bring out the creativity juices from you, and will even excite you a bit. It is a feat you can do if you put your mind into. Just remember that there are a lot of resources to turn to in case you really go for DIY stair building.\nAfter you've got the design and your installation plan all mapped out, never forget the legalities of building staircases. The national and local building code provides important notes to consider. Your design must pass the requirements set by the national and local building code before you can actually build your stairs. Consider in your design the materials and wood you will use. Will it be varnished or painted? Would you place a carpet on it? Feng shui would also help you in the orientation of your design. The Chinese believe that the direction and accessories of your staircase will determine your family's wealth and fortune. That is, if you're the superstitious kind.\nNext article Kitchen Stone Counter tops: Which Fits Your Life-style?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzboitp b/data_all_eng_slimpj/shuffled/split2/finalzzzboitp
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index 0000000000000000000000000000000000000000..cdc0372470eb4fdf228c350bcbd2464f1070518b
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@@ -0,0 +1,5 @@
+{"text":"Hello there! My name is Arcaratus and this is Blood Arsenal, an addon to the Blood Magic mod!\nBlood Arsenal adds in a whole variety of different tools and mechanics to supplement and enhance a player's Blood Magic experience.\nIt should be noted that like Blood Magic, all recipes can be viewed using JEI (Just Enough Items).\nTired of the standard linear beginnings of early Blood Magic? Blood Arsenal adds in a few nifty tools and toys to make early-game Blood Magic a bit easier! To start, once you obtain the Tier II Blood Altar and the Apprentice's Blood Orb, you can go ahead and break some Glass with Flint in your hand to get some Glass Shards.\nIf you notice carefully, you might bleed just a slight bit! Bleeding is a status effect that will take a chunk of your health randomly for as long as you bleed. The amount that you lose is also random, but is related to the strength of the Bleeding effect. Don't fret about losing too much life at the Blood Altar though! If you (or anything that's living) bleed near a Blood Altar, a certain amount of the lost life will be dropped in as LP. No sense in wasting perfectly good LP!\nA bit drained from all the sacrificing? Need something to get you back up on your feet? NEED SOME SUGAR???!!! I gotcha covered! Blood Arsenal has its very own Blood Oranges, made organically from Orange Dye. Simply toss it into an Altar, let it absorb some LP, and you get a rather refreshing fruit that has a good chance of giving you Regeneration!\nBlood Arsenal adds in plenty of tools, such as the starting Blood Infused Wooden Tools. To get them, simply place a log (any log will do) into a Tier II Blood Altar with 5000 LP to get a Blood Infused Log.\nBlood Infused Wooden Tools are quite handy and useful for early game players who don't want to keep replacing their broken tools. The material (Blood Infused Wood) that they are made up of is living and regenerative, and the tools themselves will \"grow back\" and repair themselves using the holder's Soul Network and a small LP cost. The tools themselves are overall better than stone and up to par with iron.\nGlass can be quite dangerous. Thankfully, you can utilize that to your advantage, so long as you're careful! Glass Shards can be combined to create a block version that is useful for traps, automated LP collection, and just in general, any place where you need some spikes!\nIf you're ready to progress down the line of Blood Magic and Blood Arsenal, set your sights on Blood Infused Iron! This material is quite a step up from Blood Infused Wood, and the tools are much stronger, and regenerate much faster than the previous tier. Do note however that the costs are also higher than the previous tier.\nDecoration and aesthetics are always a fundamental part of building and design, and Blood Arsenal offers a few cool-looking blocks that go well with a Blood Magic themed build. As of now, there is Blood Infused Glowstone (which can replace the Glowstone blocks of a Tier III Altar), Blood Infused Iron Blocks, and Blood Stained Glass\/Panes along with the previously mentioned Blood Infused Wood blocks. There are also blocks for compressed Slates.\nSigils are the most useful tools available, and Blood Arsenal adds in more!\nEnder Sigil: Perhaps one of the most handy Sigils yet, this Sigil allows you to access your Ender Chest simply by right-clicking in thin air. But that would be pretty stupid if it only did that! Now now, the Ender Sigil's true capability is the ability to teleport the user at whatever they are looking at, just as an Enderman would! Simply shift-right click while looking at a block in the distance, and voila, you're there! That eye is not just for show!\nLightning Sigil: Ever wanted to summon lightning wherever, and whenever you want? (Idk why but you do you) Now you can! With the Lightning Sigil, you can point-and-click to spawn lightning wherever you want with a decent amount of LP!\nBut wait! The fun doesn't stop there! Shift-right click on this magical piece of stone to switch tiers which in turn changes how much lightning rains down from the sky. Disclaimer: I am not responsible for any destruction that results from the poor decisions made while using this.\nSigil of Divinity: God-mode achieved! 'Nuff said.\nWait a minute... there's one more Sigil!\nThis Sigil is pretty complicated to describe, so instead of spending a long paragraph doing that, you can watch this self-explanatory video instead!\nOpening the GUI will yield a pretty complicated arrangement of slots. First and foremost, an Orb that is bound to the player must be right-clicked on the block itself to be bound to the player. Next, put the Orb into the Orb slot of the Altare Aenigmatica. Now, any item you want to have crafted inside of the Blood Altar will go into the 9 slots on the left. The current items within the Altar will be displayed within the Altare Aenigmatica's GUI.\nThere's a few nifty rituals in Blood Arsenal that do a handful of things!\nThe ritual is used to create Stasis Tools which are extremely powerful and adaptive tools that can be modified with a series of different modifiers in the ritual.\nRitual of Purification: This ritual helps purify mundane Life Essence down into Refined Life Essence, a liquid that is much more potent than Life Essence. This ritual requires Stasis Plates, and purifies Life Essence into Refined Life essence in a 10 : 1 ratio. Redstone, Blaze Powder, Ender Pearls, and Blood Infused Glowstone Dust are also required.\nThe top tank must contain Life Essence, while the bottom must either be empty or contain Refined Life Essence. Note: the ritual will continue to purify liquids as long as the correct inputs are present.\nBlood Burner Rites: This is an extremely powerful ritual with seemingly limitless potential. It takes in liquid life essence (normal or refined) and combines it with glowstone and lava to create energy.\nThe spaces where Blood Tanks and Stasis Plates are do not necessarily have to be filled. However, they are the only places where the ritual will seek and consume the necessary ingredients. A central stasis plate with an igniter of some sort must be present. The fire next to the Master Ritual Stone is required! An energy storage block of any kind must also be present above the igniter.\nThe Blood Burner Rites will consume one unit of every ingredient provided. That means that for each Blood Tank present, it will only consume 1 bucket of the liquid inside, and no more! Even if you have a tank with more than 1 buckets worth of fluid, it will only consume a bucket per cycle. The same goes for the glowstone. The Blood Burner Rites will also continue cycling as long as the proper inputs are present.\nBlood Arsenal has integration with Tinker's Construct adding in Blood Infused Wood and Blood Infused Iron as tool materials!\nSentience: First level will make mobs drop Will. Second level will add the Sentience modifiers to the tool with higher levels.\nMuch more is planned for this mod, I just need to get around implementing the ideas!\nYes, you don't even need to ask!\nWhere can I ask questions or suggest content?\nYou can ask questions right in the comments section, or go down and use one of the methods listed below to contact me. The same can be said for content suggestion, although you can also create a new GitHub issue for content suggestion!\nI am a regular contributor to Blood Magic, and I also dev Sleep which is a really small mod that simply gives regeneration to players after sleeping.\nCan I suggest more questions for this section?\nPlease do, as I have finally run out of things to write!\nTwitter: I post updates on whatever I'm working on, and sneak peaks of this mod as well!\nGitHub: Here's all the source code for this mod!\nPatreon: In the unlikely event that you managed to scroll all the way down here and you wish to help contribute to my work, here it is!\nThere really needs to be some kind of notice posted somewhere that the upgraded sacrificial dagger in this mod prevents gravestone mods from working properly. I know it has been reported as a bug in the past, but it's still not fixed....or it is happening again. You lose all your items if you die by your dagger.\nWhich gravestone mod? There might be discrepancies with how each mod handles dying so I'll have to do some research to see where things are going wrong.\nI have not tested it with the Open Blocks grave...but I assume it works the same way. Dying by the regular dagger produces a grave containing items as expected...Dying by the upgraded dagger from this mod leaves the grave empty and all your items are gone forever.\nActually this isn't an issue with grave mods exactly. (I thought the same before but noticed today when looking at some stuff to see if I could fix it that there is a chance of your items being cleared on death if you died to glass or bleeding. I made a pull request that adds a config option for this \"feature\" https:\/\/github.com\/Arcaratus\/BloodArsenal\/pull\/149 Given I think pack creators would often want to disable this feature.\nEven after placing the appropriate number of items for a weapon enhancement upon the stasis plates on the infusion de sanguine, with a stasis blade in the middle in the blood altar, the ritual doesn't do anything once activated. None of the stasis plates change from \"stasis: false\", but the ritual doesn't register as deactivated.\nI've been unable to find any helpful documentation, and though I was able to make the blade itself, it really isn't quite useful without any of the augments.\nAre there plans to add documentation, and could you explain how the ritual is meant to work more in depth here in the meantime?\nWill the support for TiC come back?\nNot sure if you consider it to be within the scope of blood arsenal, but in addition to the blood infused wood\/iron TC integration, would it be possible to add a material or modifier that would allow TC weapons to drop demonic will?\nWAILA isnt required, is it? i noticed it was listed as a dependency, but its not yet up to 1.12.\nim also noticing the ritual of purification requires two tanks. is any mod in particular required in order for this ritual to work?\nWAILA is a soft dependency meaning it isn't required.\nThe Ritual of Purification can take in any two tanks from any mod (including Blood Magic).\nHey! I love this project and it is the best add on for Blood Magic. Are you thinking of adding integration to Tinkers' anytime soon because it's been saying it's a WIP for a while. Keep on keeping on.\nYep! It's definitely on the to-do list when I have time to get around to it.\nI have a Crash since the last Blood Magic update.\nI have the same crash.\nI'm having this issue as well. Will need to be disabling this mod for now.\nThis has been fixed in the latest version.\nI ran into a issue with the ender sigil yesterday. I went Through the map the second i grabbed it when it was created.\nOf course i wasn't in creative and lost everything but i think you should know that issue so no one else happen to lose all their gear in a fraction of second lol.\nFixed in the latest Blood Arsenal version.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are happy to share with everyone a recently interview by a great Ottawa online Arts Magazine, Sometimes Always.\nWe discuss everything from our favorite acts, how we came up with the recent album and what we like on our pizza. Lots of great information so be sure to check it out!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What have you any idea about Application.Agent.BSO?\nApplication.Agent.BSO is really a lately identified Trojan an infection that infects almost all version of Windows Program including Windows Vista, XP, Vista, Me personally, Server, NT, 7, 8, 8.1 and the latest version Windows 10. As an associate of Trojan infection, it behaves a key-logger that store and send out all keystroke that you make such as messages, login details as well as your all essential data such as Program or network settings details, username, financial login details, debit or bank card information, password and many other delicate data to hackers. It's been regarded by group of security experts as the worst Computer risk.\nHow will PC get badly infected with Application.Agent.BSO?\nThere are several ways by which Application.Agent.BSO gets in the PC. But almost certainly it spread via spam promotions and bundling method. The installation packages of this kind of a destructive malware are often hide within the Customized or Advanced choice that skipped by consumer during the set up. When consumer download any freeware deals using default or regular installation mode and open up any connection that comes from unidentified sender after that it gets within your Computer immediately without your awareness. Apart from these, there are several propagation channels by which Application.Agent.BSO strike on Computer but primarily originated via Internet.\nApplication.Agent.BSO is actually very malicious software for the Windows PC which works similar to the ransomware an infection. After landing on your Windows PC, first of all, it'll encrypt your all files and paperwork. It immediately alters start-up settings so that it can start itself when user logs within their Windows Program. It has the ability to copy the malicious code and spread over the entire Program to ruin PC badly. As mentioned above, it behaves as a keyloggers this means it could record whatever you type. After storing each single keystroke, it afterwards forwarded them to the remote hackers for analysis and commercial purposes.\nBesides, in addition, it expands its dangerous results to clouds. By highly consuming System and network assets, it degrades efficiency rate. This variate of Trojan has the ability to then add dubious documents and delete some of the important files of user. If you actually don't want to face with other affects of malware or risk then you should delete Application.Agent.BSO from your compromised machine instantly. Concerning it's removal, an expert solution is given below.\nThis will open Control Panel. Now look for all Application.Agent.BSO related suspicious entries and Get Rid Of it at once. Now Type msconfig in the search box and press enter. Uncheck suspicious and Application.Agent.BSO related entries.\nStep 5: Tap on CTRL + SHIFT + ESC keys in Combination. Go to the Processes Tab and try to determine which one is a Application.Agent.BSO process.\nRight click on each of the Application.Agent.BSO processes separately and select the Open File Location. End process after you open the folder. Then after, Get Rid Of the directories you were sent to.\nOnce inside, press the CTRL and F together and type the Application.Agent.BSO. Right click and Get Rid Of any entries that you find with a similar name. If they do not show in this way, then go Possible Steps For to these directories and Get Rid Of them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's been a while since I blogged on here. No excuses so I've decided to get back to it with a post about Persevering with People drawing. In fact capturing people (in my drawings) is something that is often on my mind (perhaps more so than any other element of drawing). This is because capturing busy places and people is a focus of some of my commissioned reportage work, including weddings and other events. My view is that like most aspects of drawing, constant practice is required and maybe this is even more so with people! When I don't draw people for a while I definitely notice and find it more difficult. The way I capture people, the materials, the type of drawing, the level of detail etc, is also a source of internal debate!\nMy people drawing technique has been either watercolour-first ( mainly crowds in which case the colour is very non-specific) or contour\/gesture line drawings ( to which colour may or may not be added). Sometimes I will use colour first to capture individual people ( so the colour is of the key shapes to which minimal linework is added ( not as an outline but to add detail not indicated by the colour).\nThe approach outlined above also formed the basis of my summer Workshop, Sketching People In Busy Places. This was one of the USKManchester 10 x 10 summer workshops. My demo piece sought to use colour first to capture the market scene on Piccadilly Gardens in Manchester. Watch out for more information on the workshops I've been leading and my planned series for next year.\nMy ongoing practice takes a variety of forms!- I go to Life drawing (but not as often as I probably should!) and this tends to be one single pose rather than several shorter gestural poses. Therefore it tends to be about proportion and tonal value rather than capturing gesture and movement.\nI also take a small book around with me to help me capture people on the go, eg. in the tram; at a bus stop etc. These are mostly pen and ink drawings or pencil drawings.\n-life drawing using watercolour to develop my tonal and more detailed figurative work. ( I am aiming for 2-4 sessions a month). I,m looking to develop rich lose works that convey tonal values whilst being somewhat looser and more intuitive than my current work.\n-A more sketchy, energetic people sketching approach perhaps using coloured pencil for colour . This will sit alongside my colour first work and should enable a more rounded capture of the occasion.\n-More practice and focus on people moving in various situations and people at work or play. I'm interested in capturing the energy and the shapes to produce more dynamic drawings.\nWatch this space to see how I get along with these new challenges!\nI love your sketches of crowds! They are really\u2026 crowded!!\nAw thanks Tina, that's so kind of you to say so!\nYour sketches are intriguing, I love the positioning of the colour and your style. Your collage work is amazing too.\nIt was lovely to meet you yesterday at Woodend Mill, Mossley and I am looking forward to finding out about your work shops.\nLovely to meet you too and I am so sorry for the delay in response. I think this one got lost in the ether. If you are signed up to my newsletter then I will definitely be using that as a way of letting people know about the workshops and other news. All the best and thanks so much for taking the time to post on here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you looking for lucrative affiliate marketing programs that will complement your web design business or freelance blogging? Then I have the most incredible list of affiliate programs for you.\nI have personally vetted close to 200 programs that mostly sit in the WordPress\/Web Design niche and have arrived at my final list of 77. This is seventy-seven unique affiliate programs that you can apply for yourself and start promoting within a matter of hours, sometimes even minutes, depending on the approval process.\nSome of these affiliate programs are part of affiliate networks such as ShareASale and CJ Affiliate which will possibly open the door to many more affiliate opportunities for you.\nI have done all of the hard yards for you and have collected the direct links to each affiliate program's application page. I have also provided a detailed summary of everything you need to know about each program.\nThrive Themes is an advanced but simple-to-use suite of WordPress website products tools. Thrive Themes are best known for their conversion-optimized plugins such as Thrive Leads, Thrive Comments and Thrive Ultimatum. Thrive Architect, is their core Visual Drag and Drop editor which can be used on any WordPress theme. I have personally used Thrive Themes for almost three years and own all of their plugins.\n35% per sale plus 25% on recurring sales.\nElementor is a page builder plugin that replaces the basic WordPress editor with a live frontend editor, so you can create complex layouts visually, and design your website live, without having to switch between the editor and the preview mode.\nNote: You need to register for an Elementor account before applying to be an affiliate. It is an internal application process.\nWP Rocket is a popular caching and performance optimization plugin that endeavors to improve the loading speed of WordPress websites. You can install and configure WP Rocket in less than 3 minutes. It is my caching plugin of choice and one of my favorite WordPress products. There are many good free caching products but none come close to the level of detail and control you can have over your site than with WP Rocket.\nI use to use MailChimp. Use to. Until I discovered ActiveCampaign. Now I use ActiveCampaign for all of my mail and marketing automation. Automation is the key word here- by automating functions like lead generation, lead and customer engagement and conversions, you can scale your business to reach even more contacts. There are some other notable platforms out there - ConvertKit and aWeber immediately come to mind but for bang for your buck, ActiveCampaign is my choice.\nStart by earning 20% recurring commission on every new customer you send to ActiveCampaign.\nBring in $100 of new monthly business in the past 3 months or have $500 worth of active accounts.\nBring in $500 worth of new monthly business in the past 3 months or have $2,000 worth of active accounts.\nFlyWheel is the host that I use for this site. It is a very easy to use, completely managed WordPress hosting platform & includes so many other features that I have paid for in the past. These includethe easiest to use cloning (e.g. copy a site and set up a replica in literally seconds), a local development platform to build sites before publishing (and paying!) and my favorite - the daily backup solution that I have included with my plan - one click restoration, it is that simple (not that I have needed to use it yet... fingers crossed!). Finally caching and security is top notch and included on the plan. I removed at least 5 plugins when I moved to FlyWheel and my site speed and performance is testament to this.\nlife with Flywheel, you'll earn a commission for each one. Up to the cap of $500!\nSEOSimple, fast and powerful SEO plugin for WordPress. It is easy to use, you can quickly enable\/disable features and modify your titles tags in seconds. There are no digital footprints in the source code or in the WordPress backend. SEOPress has too many features to list but has all the features you need in one plugin.\nEarn 20% for each sale!\nBrainstorm Force are the masterminds behind a number of well known WordPress products and add-ons. The WordPress Astra Theme and Convert Pro are Brainstorm Force's flagship products but they also create other add-ons to complement other WordPress products such as Elementor and Beaver Builder. You may not have heard of Brainstorm Force but there is a good chance you have used or at least heard of one of their products.\nEnjoy 30% commission on most sales (4 out 6 of their product lines). 'Convert Pro' offers an excellent 50% on each sales whilst 'Ultimate Addons For Beaver Builder' offers 20% commission on each sale.\nFastComet offers high performance cloud hosting with SSD-only servers, CloudFlare CDN & 24\/7\/365 dedicated support. Fast Comet pride themselves on their speed but what amazes me is the cost at which they offer their hosting solutions. We all know hosting is getting cheaper and cheaper each year but Fast Comet has taken this to a new level with a $2.95\/month entry level plan and best of all, it remains that way. There are no hidden surprises after the initial onboarding period. All reviews I have heard thus far have been positive and this includes their support responsiveness.\nSEMRush is the go-to product for SEO newbies and professionals alike. It is my favorite All In One SEO platform & digital marketing suite particularly for managing the SEO & Marketing for small business sites. Why? Because as an SEO I use a range of specialized products for KeyWord research, Backlink & Competitor Analysis etc but with SEMRush I can plug in a client or friend's business and it drives itself and basically spoon feeds you what you need to do - covering every inch of SEO.\nGet 40% recurring commission for every new user. You get commission every month, not just the first one!\nFleeq is a user-friendly, intuitive video and GIF creating tool. It is a unified platform that empowers you to create, share and track videos everywhere. It runs anywhere, with a lightweight structure and elastic capabilities. Share it via a dedicated page, part of a knowledge base, as a GIF, video, or a user acquisition widget.\n30% Recurring. Commissions are only earned if a customer makes a payment.\nShortPixel improves website performance by reducing the image size. The resulting smaller images are no different in quality from the original image but will consequently load quicker and reduce page load time.\nAffiliates will receive 50% of the sale as a commission from orders placed through properly coded Affiliate links.\nPlease note that for dedicated servers, affiliates will only receive 50% of the buyer's first month's payment. The subsequent payments by the customer for a dedicated server won't generate any additional affiliate commission. An affiliate cannot receive 50% as commission for the purchases done for his own use.\nTelestream is home to my favorite Video Software, Screen Flow and Wirecast amongst other award-winning video solutions that allow you to easily create, transcode, caption, inspect, deliver, play & stream any type of video, while ensuring the highest level of quality end-to-end. From live webcasting, screencasting, desktop video encoding and closed captioning to enterprise-class video solutions.\nPowerful, easy-to-use video editing and screen recording software for Mac.\nPublish directly to YouTube, Vimeo or Flash movie.\nAll-in-one live streaming production software that enables capture, live production and encoding for live streams in one cost-effective, flexible software solution.\nExtensive capture support: Capture from a wide range of sources including multiple cameras, computer desktop, capture cards, other capture devices, such as Teradek Cube, and more.\nPowerful live production capabilities: Switch between multiple cameras, add video, audio, graphics, slides, text, titles, transitions and more.\nAffiliates earn up to 15% commission on every referred sale that occurs on a click-through from your website, blog, newsletter, or social network community. How much you earn depends on how many visitors you refer to purchase.\nvooPlayer is an interactive video hosting and marketing platform designed to help businesses create customized, embeddable marketing videos with hybrid video hosting, advanced security, marketing tools, analytics, and more.\nZooPlayer currently offers a 30% monthly recurring commission for the life of the customer.\nWP Engine is a leading Managed WordPress hosting platform or as WP Engine describe themselves, 'a digital experience platform'. WP Engine are performance focused, security minded and specialize in WordPress. Their referral program is one of the best and most asset rich of all WordPress related affiliate programs.\nA once off $200 per new customer OR 100% of the customer's first monthly payment (whichever is the highest). $0.01 per lead is also included.\nNamecheap is a leading ICANN-accredited domain name registrar and web hosting company, serving nearly 2 million customers and managing more than 9 million domains. Namecheap offer a range of hosting solutions but are best known for their fantastic domain service. Namecheap offers you a wide variety of domains - you can register or transfer your domain name in a snap.\nA2 hosting pride themselves on blazing fast web hosting solution perfect for your hosting needs - whether you have a low traffic, brand new blog or own a popular business site. a2 hosting has a 99.9% uptime commitment and a Turbo Server option that claims to be up to 20x faster.\nElegant Themes is essentially a WordPress Marketplace but is renowned for being the home of Divi and its 'little sister' Extra that utilizes visual drag & drop technology. Divi is the most popular WordPress theme in the world with over half a million users. Elegant Themes also have a small suite of WordPress Plugins to provide extended functionality, Monarch for Social Media Bloom for Email Optins.\nAll affiliates receive 50% of each sale, including annual renewals.\nDreamHost offers several different types of hosting plans that work for all website hosting needs. ... Shared hosting is perfect for blogs, portfolios, personal\/business sites, and database-driven sites. DreamPress is DreamHost's managed WordPress hosting.\nPressidium Enterprise WordPress Architecture has no single-point-of-failure and is built from the ground up to be highly available and fault tolerant. Pressidium offers dedicated systems built with High Availability and Load Balanced architecture that won't let you down. First Class Support is provided by DevOps WordPress Engineers. Large organizations & corporations use our infrastructure to scale their WordPress sites with quality support and leading enterprise architecture. Pressidium was awarded the 2015 & 2016 Top Tier WordPress Hosting Performance - \"A perfect LoadStorm test with zero errors, the lowest peak response time and lowest average response time. Followed up by a near perfect Blitz test\".\nExcludes the entry-level micro plan ($50 minimum commission paid on micro plans).\nAll sales must maintain service for 60 days for you to be eligible for your commissions.\nKinsta has combined the fastest network and platform in the world (Google Cloud), and paired it with the best engineering team and the most experienced WordPress developers in the industry. The MyKinsta dashboard was built from the ground up specifically for WordPress, making site management a breeze. Kinsta is one of the highest quality WordPress hosting solution on the market.\nYou earn $50 to $500 after every signup based on the type of plan you refer. Then 10% ongoing.\nPressable has created a scalable, secure and performance driven WordPress hosting platform that users can trust to deliver remarkable online experiences.\n$90 for every new customer.\nWPX Hosting is an establish Managed Wordpress Hosting platform that has hosting locations in both the USA and Europe. WPX Hosting may not be as well known as its larger competitors like WP Engine but is highly regarded in the WordPress community.\nSiteGround is a great hosting solution for all your WordPress projects - both new and established ones. The service they provide is stable, secure and super fast thanks to the special tools they've developed in-house for WordPress users.\nThemeIsle is a creative and dedicated team of WordPress enthusiasts who've been building awesome WordPress Themes and Plugins since 2012.\nTemplate Monster is quite possibly the oldest provider of the website template on the Internet. It has operated since 2002 and over 13 years have had over 50,000 products on their website.\n30% commission on any user's first purchase.\n10% commission on further purchases.\nThemeForest is part of Envato Market, along with other channels such as Code Canyon.. At ThemeForest you can buy and sell HTML templates as well as themes for popular CMS products like WordPress, Joomla and Drupal. Items are priced on the complexity, quality and use of the file.\n30% of first sale for a new user - higher of the deposit amount or the item purchased. For example, if a buyer adds $20 credit into their account, you get $6. If they buy a $200 item, you get $60.\nPremium WordPress themes & plugins without the confusing parts. 24\/7 support included. Get complete access for only $69. That's less than $1 per theme. Join 80,000+ satisfied users.\nThe Themify Builder is the most powerful and easy to use page designer and builder for WordPress. Design any layout that you can imagine, bring it to life using the drag & drop interface, and watch it come together right in front of your eyes with live preview.\n40% Commission on Initial Sale.\nBeaver Builder is a flexible drag and drop page builder that works on the front end of your WordPress website. Whether you're a beginner or a professional, you're going to love taking control of your website. Stop writing HTML or wrestling with confusing shortcodes.\n25% Commission on all sales.\nMonsterInsights allow you to connect your WordPress website with Google Analytics, so you can see how visitors find and use your website, so you can keep them coming back. Simply put, we show you the stats that matter. With almost 15 million downloads, MonsterInsights is the most popular Google Analytics plugin for WordPress.\nYou earn 20% of every sale.\n- Customizable link URL prefixes \u2013 change \"recommends\" to something else, loads of pre-done link prefixes to choose from or use your own custom prefix!\nThe PRO add-on for ThirstyAffiliates that bolts on extra functionality that might be interesting to advanced affiliate marketers and bloggers looking to automate parts of their affiliate marketing.\n- Automatic Keyword Linking \u2013 boost your affiliate income by automatically linking affiliate links to keywords throughout your site, great for monetising your back catalog of content!\n- Advanced Statistics Reports \u2013 amazing advanced reports such as performance by category, performance over 24\/hr periods (to identify peak click times), Geolocation reports (where people click from) and more!\nThirstyAffiliates is one of my Top 7 plugins and is used on this very site.\n30% of the sale price for any sales referred.\nMangools is my favorite suite of SEO Tools for affiliate marketing because I can use the specific tool when I need it and it is so user friendly. They are most well known for their flagship Keyword research tool, KW Finder - so much so that they are pretty much known by that name. Needless to say they have three other tools that are just as good and serve different SEO purposes - SerpWatcher (SERP analysis), SerpChecker (rank tracking) and LinkMiner (backlink analysis). Best of all, they are incredibly affordable and won't break the bank.\n30% Commission of all Sales ongoing.\nDiscounted Sales (coupons) reduces commission by the actual discount.\nGeneral discounts (e.g. Black Friday) do not count - you still get the30% commission.\nBluehost is a leading web hosting solutions company that was founded in 2003. Bluehost provides comprehensive tools to millions of users throughout the world so anyone, novice or pro, can get on the web and thrive with our web hosting packages. Bluehost powers websites all over the world and support thousands more every day.\nBluehost is committed to WordPress and has supported it for over 10 years. The in-house team consists of dedicated WordPress experts to provide the best customer support. Bluehost end-to-end is built on open source and is one of the world's leading solution for open source implementation and development. Bluehost provides budget friendly, afforable hosting solutions.\nThe InMotion web hosting network is based on the fast and reliable Linux and Unix operating systems. It is monitored 24\/7 for any unusual activity and benchmarked continuously for performance. This allows InMotion Hosting to immediately respond to short-term issues as well as to grow their system optimally - keeping it broadband ready, economical and fully redundant at all times. InMotion is also the only hosting company to offer Max Speed Zones.\nCloudways is a managed Cloud Hosting Platform where you can build, deploy, scale & manage web applications. Cloudways set high standards of performance, commit to complete freedom of choice coupled with simplicity and agility in every process. Backed by an innovative approach, the Cloudways platform is built on best-of-breed technologies and industry-leading infrastructure providers that creates smooth managed cloud hosting experiences.\nSlab Plan: Get a straight bonus of $50 for your first 1 to 5 referral Signup that become a paying customer. No recurring commissions.\nHybrid Plan: $30 for every referral signup that becomes a paying customer. Additionally, you get a 7% recurring commission for a year on each paid client from your referral.\nEasyAzon is the quickest way to create Amazon affiliate links from within your WordPress post editor. You can create text affiliate links direct to any product available for sale on Amazon.com without going through the tedious steps of manually creating links from within the Amazon.com website (huge time saver).\nCSS Hero 3 is the definitive WordPress plugin to easily customize the look of your site, with an easy and intuitive point and click interface. Control every aspect of your WordPress site - CSS Hero allows visual customization of any element of your site, all from your frontend. Live.\nLucky Orange, in my honest opinion is the most under-rated conversion optimization software and best of all, the price is insanely cheap. Lucky Orange will automatically create a recording of every visitor to your website. Quickly filter and segment recordings so you can see exactly why visitors are not converting. Out-of-the-box, Lucky Orange successfully records dynamic and member's only pages, including even the most advanced sites and single page apps.\nLucky Orange also delivers beautiful heatmaps that show you how people read and interact with your site. You can segment heatmap data by location, browser, dates, mobile users, and more. There are three types of heatmaps: mouse movement, scroll depth, and mouse clicks. Lucky Orange Chat integrates with your existing website and works seamlessly across multiple devices. Chat comes loaded with powerful features including multiple operators, canned responses, and auto invites.\nConversion Funnels let you quickly see where people abandon a process on your website such as checking out or creating an account. Clicking on a funnel step will instantly filter your recordings to those visitors that abandoned the funnel at that particular step. The Conversion Funnel Widget displays the conversion funnels you've created using Behavior Tags. In the example, the selected conversion funnel is titled \"Customer Checkout\".\nEnvira Gallery is the best responsive WordPress gallery plugin. Below are the features that make Envira Gallery the most powerful and user-friendly WordPress gallery plugin in the market.\nEnvira Gallery is not just for photos... embed YouTube, Vimeo, Wistia and custom embeds into your gallery.\nLifterLMS is a powerful WordPress LMS plugin that makes it easy to create, sell, and protect engaging online courses. The mission of LifterLMS is to democratize education in the digital classroom. The LifterLMS Student Teacher and Learning Platform Owner Experience.\nSEO PowerSuite was designed to combine immense SEO capabilities with straight forwardness and extreme ease of use.\nThiesoftware really guides and assists you in every task. No techy preps and deep SEO knowledge needed, just set and go.\n33% Commission on most products.\nSERanking is an all-Inclusive cloud-based SEO software. Save time on grouping selected keywords to distribute them effectively and efficiently among webpages of your sites. Discover what keywords your competitors are using in their ads and to generate organic traffic. Spy on their budget, traffic, and the amount of clicks on their ads and even see how the ads look like. SERanking also includes backlinks monitoring so that you can understand everything about your backlinks so you can act accordingly.\nClickMeter is a complete and definitive tool that allows you to manage redirection, track events and monitor the status of all your external links (traffic sources). The alternative is to use three different tools that specialize in redirecting, tracking, and monitoring links.\nBilly is a lightweight yet powerful bookkeeping for service businesses. Easily create a quote, attach a proposal and send it to your client for approval. Convert quotes into invoices with one click, and keep track of the deals you close.\n3dcart is a complete and robust eCommerce platform designed to help online store owners thrive in a competitive market. With hundreds of features built directly into its software, business owners can effectively open, operate and maintain a successful online store with relative ease and efficiency.\n30% of the sales price and future recurring revenue from our monthly and annual plans.\nAll in One SEO is the original SEO plugin that has been downloaded over 45 million times. Use All in One SEO Pack to optimise your WordPress site for SEO. It's easy and works out of the box for beginners, and has advanced features and an API for developers.\n40% commission on all qualifying plugin sales.\nConvertKit is a powerful email service provider that provides automation capabilities with a very budget friendly price. ConvertKit provides the automations and email marketing tools you need to grow your blog and business.\nAs a ConvertKit affiliate you'll earn a 30% commission each month from everyone you refer.\nAWeber is an email marketing service provider with over 100,000 small business clients world wide. We help people keep in touch with the subscribers who have requested to be on their mailing list.\nGetResponse is an email marketing platform. It enables you to create a valuable marketing list of prospects, partners, and clients, so you can develop relationships with them and build a responsive and profitable customer base. It has the deliverability rate of 99.5%, to make sure your messages reach destination inboxes.\nIf you are running a few marketing and promotional campaigns on the go, or have a team collaborating then tools like Instapage will accelerate your landing page design process and potentially increase your ROI. One of the advantages of Instapage over building your landing pages with the common WordPress Page Builders is the ability to do A\/B Testing. Instapage also includes heatmaps, an AMP Builder, User Permission Levels and integrates with more CRMs, CMSs (including WordPress), Email Marketing and Automation platforms than you can poke a stick at. If you are a one person show that only occasional builds landing pages then Thrive Architect could meet your needs.\nEarn 50% revenue share on the first payment and 30% lifetime revenue share on all self-serve plans. Plus, earn big commissions on enterprise plans.\nClickFunnels is a web software that lets you design and create sales pages, landing pages and lets you manage your entire sales funnel end-to-end - including checkout. You can create seven types of funnel with the help of Clickfunnels. ClickFunnels Gives You EVERYTHING You Need To Market, Sell, and Deliver Your Products and Services Online! Without having to hire or rely on a team or agency.\n40% one-time for the promotion of Funnel Hacks (which is a special offer for a six-month account and training).\nGet a car, or at least car payments for 100+ active paying referrals. Get a better car at 200+.\n10 Minute Funnels proclaim to be the 'world's easiest to use (and most powerful) sales funnel builder.' Launch a proven marketing campaign in as little as 10 minutes. Access pre-built marketing funnels that have already been proven to work. Simply choose a proven campaign, customize it, and then launch - all within 10 minutes.\n40% for 300 or more sales.\nLeadPages is an online software tool that you can use to generate leads and collect email subscribers. Increase revenue using the industry-leading landing page creator with accompanying suite of lead generation and opt-in tools. Whilst building your business with high-converting landing pages and lead generation campaigns you can easily DIY.\nEarn 25% of the monthly recurring subscription fee from each of your referrals. No cap on commissions.\nNinjaOutreach is designed for businesses, SEO agencies, marketing agencies, digital PR firms, lead generation specialists, email marketers, link builders, journalists, campaign managers\u2014anyone with massive influencer marketing & outreach needs!\nNinjaOutreach makes Link building easy! It helps you to find influencers & business leads. Promote your business. Get Sales. Do it all with one tool, FAST.\nBetter Proposals is a web based software product that makes Sales Proposals simple for Creatives and Service Businesses. Better Proposals has a phenomenal amount of pre-made templates that takes all of the hard work out of writing proposals - no formatting or spending hours making it look professional. You just need to write what you like and type in your pricing. Everything else is automated.\nMemberMouse is an easy to use WordPress membership plugin that allows you to sell products, subscriptions and memberships, setup a password protected member's area, offer 1-click upsells and downsells, manage customers, automate customer service, track critical retention metrics and more.\nRecurring Commissions: 25% on each customer payment for one full year!\nDigital marketing agencies and businesses use Whatagraph to save time and present analytics data in a way their clients and teams can easily understand. Whatagraph is an online tool that tracks website and social analytics data and integrates seamlessly with marketing analytics software like Google Analytics as well as social media analytics including Facebook. Whatagraph has many ready-made report templates to simplify your daily marketing reporting tasks.\nEarn 30% recurring commission for every new paying customer.\nMailOptin helps you to grow your email list and send event triggered email newsletters with ease. MailOptin converts your website visitors into email subscribers and customers, nurture and engage them with beautifully designed automated event triggered emails. All in WordPress.\n20% Commission on every sale.\niThemes, home to BackupBuddy, is your one-stop shop for premium WordPress plugins to help you build & protect your online WP work, since 2008. Their flagship product, Themes Security, protects your WordPress site by obscuring common WordPress security vulnerabilities from hackers.\nCrocoBlock is an all-in-one service providing everything you need for creating websites with Elementor and Elementor Pro - my most used page builder. Zemez have created an Elementor compatible theme with a variety of Elementor focused addons for all kinds of purposes. CrocoBlock is a great service for creating WordPress websites, with the Jet Plugins Set, Kava PRO Package, +500 pre-made blocks, drag-n-drop Header & Footer Builder and lots more.\nYou'll receive 30% from each purchase made by the client who was redirected from Your site via the affiliate link.\nPixel Your Site is a powerful WordPress plugin for Facebook. You can manage your Facebook Pixel with a few clicks. Over 57000 small business owners are using the Pixel Your Site plugin to make the most of their Facebook Pixel.\nReally Simple SSL automatically detects your settings and configures your website to run over https. To keep it lightweight, the options are kept to a minimum. You only have to click activate, and in most cases, your site is ready for SSL. I personally use it on every site I own.\nYou can earn 20% commission on every first sale your referrals make.\nWP Tasty plugins are 'handcrafted WordPress plugins for Food Blogs'. They are used by top bloggers around the world but perfect for any blogger. WP Tasty plugins run smoothly on high traffic sites all over the internet. Ideal for smart and savvy bloggers who need to power their blogs.\nWP Tasty is also home to one of my favorite affiliate marketing plugins, Tasty Links.\nUnlike other WordPress form plugins, WPForms is 100% drag & drop, and you can create any type of WordPress forms ranging from simple WordPress contact form to more advanced WordPress user registration forms, payment forms, email subscription forms, and more.\nUpdraftPlus simplifies backups and restoration. You can backup your files and database backups into the cloud and restore with a single click. Backup into the cloud directly to Dropbox, Google Drive, Amazon S3 (or compatible), UpdraftVault, Rackspace Cloud, FTP, DreamObjects, Openstack Swift, and email.\nUpdraftPlus is my personal favorite when it comes to WordPress backups.\n30% commission on every referral purchase.\nEnhance WordPress SEO by implementing Schema markup for reviews and star rating Rich Snippets.\nYou will receive 33% of the purchase price.\nNinja Forms is completely free to be tested, used, or contributed to. Even the add-ons come with a 14 day money back guarantee. Ninja Forms has an abundance of settings so you can fine tune every part of your form: from general settings down to individual fields, submit actions and emails. You can add a Ninja Form to your site easily with a widget, shortcode, template function, or append it automatically to any content you choose.\nYou earn 20% of every sale which can be as much as $100 per sale.\nFormidable Forms is a freemium WordPress plugin that helps you to easily create contact forms, polls and surveys, lead generation forms, email forms, and any other forms you imagine. You can create beautiful drag and drop forms with custom styling and you and your visitors can even both submit and edit entries and posts from the front end of your site.\nUp to $79 per sale.\nCaldera Form is a powerful WordPress plugin that creates responsive forms with an easy-to-use drag and drop editor. Caldera Forms has many free user-friendly add-ons with a visual editor that simplifies form building for WordPress. You can set up your form with multiple columns, add additional pages, add funnel user responses with conditional logic to get more robust data from your customers or visitors.\nAffiliates are paid a 20% commission on sales generated from leads sent to their site.\nLearnDash is a premium WordPress plugin that works with nearly all WordPress themes. LearnDash is one of the most credible & trusted WordPress LMS (Learning Management System) plugins, created by learning industry professionals with Fortune 500 experience. LearnDash integrates well with a number of widely used WordPress Plugins (Free & Premium) such as BB Press, BuddyPress, MemberPress and Gravity Forms as well as a number of Third Party Softwares such as ConvertKit, Stripe, Mailchimp and Slack.\n35% commission on initial purchases and 10% commission on upgrades & automatic account renewals!\nGamiPress let's you award your users with digital rewards for interacting with your WordPress website. GamiPress combines three of the most powerful award systems you could add to your site - Points, Achievement & Rank.\nEarn 20% commission on every new sale you refer.\nI provide all of my content and products free of charge. In order to do this, I sometimes receive compensation from the companies whose products I review and\/or display. When you click on some of the links on this site and purchase items, I may receive a referral commission. There is no additional charge to you and in some cases you may even purchase products for a lower price than what is advertised. I use most products personally and only advocate the very best. The opinions expressed here are my own.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Welcome to ALC 2019,the 12th Asia Lighting Conference.\nOver 10 years, we have contributed to the development of lighting science and its industries by leading this conference. The 12th ALC will be held at Hotel Inter-Burgo, Daegu, Korea during Aug. 22nd to 23rd, 2019 (supported by Yeungnam University). Daegu is an attractive and traditional city, and 3rd largest city in Korea located in the east-southern area of Korea. We will have experience, excitement and excursions in Daegu and Kyungjoo where is a historic area that has inherited the culture of old Silla and Gaya.\nALC was formerly known as the Lighting Conference of China, Japan, and Korea (CJK). In 2017, we have changed and expanded our name to \"ALC (Asia Lighting Conference)\". The 12th ALC will become the third conference after the title of the conference was changed. I wish our conference will become more widely and broadly in Asia. In this year, we will lead the conference theme of Connected Society through Advanced Lighting. We hope to keep abreast of advanced technology of connected lighting in the society where we live. Thus, in ALC 2019, we expect to have improvement of both quantity and quality of papers, presentations, and posters not only in academics and industries, but also in lighting design fields. Therefore, we expect that the lighting science and design will be more developed and having further discussion and study all fields of lighting at the conference. In this conference, we invite Dr. Mariana Figueiro, Director of Lighting Research Center at RPI (Unite States) as a key-note speaker. We will enjoy her speech regarding the most updated connected society through advanced lighting.\nAs we have changed the conference name from CJK lighting conference to Asian Lighting conference, we encourage to attend from various territories in Asia. We always wish to share and discuss the lighting studies and industries with people from diverse countries.\nWe always open to listen to any opinions about our conference. Feel free to contact us and give your opinions and experience about our conference.\nThe chair of ALC 2019 (the 12th Asia Lighting Conference).\nProfessor, Department of Architectural Engineering, Sejong University.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can rooms be booked at times outside of the normal business hours?\nRooms may be book outside of normal business hours depending on staff availability. However, such requests may incur an additional fee.\nDoes the Detroit Center have video and teleconferencing capabilities?\nYes, the Detroit Center has video and teleconferencing capabilities.\nIs the Detroit Center wheelchair accessible?\nYes, the Detroit Center is wheelchair accessible via our Martin Luther King (Mack)\/ Woodward entrance.\nWhat Do I Need to Know About Parking?\nParking is a standard rate of $7.00 per vehicle, with a potential discounted rate available at $5.00 per vehicle. To quaify for the discounted rate, your must pre-pay for parking for your event (pre-payment in non-refundable). The parking structure entrance is located on Parsons St. between Woodward Ave. and Cass Ave. All guests must park on the third floor and above (floor 1 is handicapped parking and floor 2 is reserved for staff who work in the building.) Parking tickets can be validated (confirm with event lead for validation) at the front desk of the center by the close of business at 7 p.m. (Monday \u2013 Thursday) or 5 p.m. (Friday \u2013 Saturday).\nMust I use Detroit Center recommended caterers?\nNo. However, in such cases, the reservationist must sign a waiver accepting sole responsibility for food and\/or beverage services.\nHow much does it cost to rent a room in the Detroit Center?\nFacility fees depend on whether an organization is affiliated with the University of Michigan. However, costs for non-University groups are quite competitive with comparable facilities. For more information, contact a Reservation Associate at detroitcenter@umich.edu or 313-593-3584.\nIs there an additional charge for audio-visual equipment or other services?\nThe use of audio-visual equipment is included on the room rental fee. There are, however, additional services that the Detroit Center offers with an additional fee. These services and the fees are listed below. If you require any additional services, please let the Detroit Center staff know and we will do our best to accommodate the request.\nComplete the Reservation Form or contact a Reservation Associate at detroitcenter@umich.edu or 313-593-3584.\nVisit Midtown Detroit Inc. for additional resources.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This recent report from Tate & Lyle looks into the growing demand for Dairy Alternatives.\nAtSource, Olam's comprehensive three-tiered resource is now in full implementation. This resource provides the traceability and sustainability data required by the new-age consumer marketplace.\nAccording to Ingredion, consumers are seeking gluten-free products without a medical reason to do so. See more insights Ingredion's research revealed, and get the whitepaper they recently released.\nThis year for the first time, Kerry is offering the results of these predictions in an easily integrated, actionable insights report, segmented by global region.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kelly Kare installs brick paver mailboxes that will enhance your home's curb appeal, add architectural detail, and withstand the demands of winter in the Midwest.\nOur mailboxes are constructed using Unilock LINEO Dimensional Stone and offered in three color options.\nWe install a Salsbury Industries Black Cast Aluminum column locking mailbox, which is also available in a Thumb Latch option. The Mailbox dimensions are 11.25\" W x 13.25\" H x 15.75\" D. The incoming mail deposit slot dimensions are 10\" W x 2-1\/2\" H.\nThe mailbox will be set on a footing that will consist of a 37\" D concrete pillar that will be topped with a 24\" W x 24\" D x 5\" H concrete slab re-enforced with concrete rebar. The Unilock brick pavers will be glued using a concrete block adhesive and topped with a Unilock Pillar cap.\nIf you are interested in learning more or receiving a quote for our Unilock Brick Paver Mailbox just click on the \"Contact Us\" Link and we will contact you right away.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Risk Takers . . .\nWe had visitors at our back door last night which is rather common. But this time, it was the other kids of the family. We first met these three a while back when they got stranded in our garbage can. Because the garbage can was unusually empty, once they got in, the little ones couldn't get out. We heard a noise and movement coming from the big blue can, and peeked inside. That was then . . .\nThe Catch . . .\nThe Release . . .\nThis is now. . .\nUsually Mom is the one that heads to our back porch looking for leftover cat food , fallen scraps from our kids, or if the garage door is open. . . .JACKPOT! But tonight, with these scoundrels looking so small and cute, and having \"history\" with these guys, my wife, the \"animal whisperer\", decided to engage them and offer them some treats. We started by just cracking the door open. This brave one, now named \"Olaf\", would 'reach' into the door to take food from my wife's hand. He would sometimes stick his nose in and take a nibble. But this was not nearly good enough for my wife, who too the brave one, went outside and crouched down to get her \"Dr. Doolitte\" on. Only Olaf was brave enough to actually come right up to her and eat and take food right out of her hand. The others waited close by for extra pieces being tossed to them. Olaf was chowing down and will now go back to his tree, fat and happy. He wasn't even afraid when I pulled out my camera and lights (yes, I'm a huge photography nerd).\nThe moral of this story is . . . take a risk, there could be a big reward in it for you. For Olaf, he was bathing in a post glow meal. For my wife, she has a new friend.\nHere was last night's festivities . . .","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Happy New Year! Three words came to me in a morning meditation for the year ahead \u2013 magic, miracles and manifestation. So that is what I am looking forward to and looking out for this year and I wish all of you the same. I know that in my own life, these things happen all the time though I only notice them when I am present and aware. May 2017 be a year full of magic, miracles and manifestation for YOU too!\nIn November 2016, we moved into our new house, which my dear father kindly bought for us in the beautiful village where we have lived for the past five years in a rental. I'm so happy and grateful to finally have a home to call our own again!\nIn 2016, I discovered BookLife, the Publishers Weekly site for self-published authors where you can add your book and submit it for consideration for a free Publishers Weekly review. There are apparently thousands of books submitted every month, so the chances of getting picked are not high, however, I was pleasantly surprised to hear back from them within a few weeks that my book, It Rains In February: A Wife's Memoir of Love and Loss had been picked for a PW review! Seeing as readers seem to either love my book, or hate it (depending on their own personal beliefs), I waited in nervous anticipation, hoping for the reviewer to be one of the former. In November, I received the email to say that my PW review was live on their website and had gone out in their newsletter. I followed the link and was hugely relieved to find a wonderful review!\nWelcome! I am honored to share my journey with you. A journey of life, dreams, and possibilities. A space to continue my story and my thoughts. I have no idea how this story will end, but we can find out together.\nIf you'd like to read this blog as a story, click here to go to the first blog post and continue reading in chronological order.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"French Hdqrs moved out at 10 a.m. Wrote Mattie. 51st Bn. took over part of 49th line during night.\n4th Anniversary of War \u2013 only one more in my opinion as The Huns seem to be losing their hitting powers especially now the Allies are doing so well between Rheims & Soissons. At Advanced Brigade Hdqrs during afternoon, & moved to old 4th Bgde Hdqrs (on edge of L'Abbe Wood) at night \u2013 took over maps not handed to Canadians & returned to Rear Brigade Hdqrs. 150 Huns raided 51st Bn losing about 10 killed \u2013 6 of our men prisoners.\nArrived Rear Brigade Hdqrs about 10 a.m., fairly tired & glad of a sleep. Moved from Rear Brigade Hdqrs to forward Brigade Hdqrs during afternoon. Enormous Transport on roads during night; guns, ammunition &c in abundance Brigade took over whole of Div. front during night.\nDull wet day. Went to 49th Bn. Hdqrs with the G.O.C. during morning \u2013 got very wet. Command of Bgde. front taken over by 3 Canadian Divs. Mrs.Caileur interpreter, arrested as a spy by Tommy Artillery \u2013 came here to be recognised \u2013 very funny.\n\u2026Battalions relieved by Canadians during night. Brigade now attached to Ausn. Corps.\nBattns all out by about 4.30 a.m. Zero hour for stunt 4.20 a.m. News through about 11 a.m. that all going O.K. Several prisoners taken, also guns. Enemy planes over during morning dropped about 30 bombs near Petit-Blangy-Cabaret wounding a few. \u2026Good news regarding stunt \u2013 estimated prisoners over 7000 & several guns.\n1st Div. & others attacked this morning \u2013 made excellent progress \u2013 captured a Div.Hdqrs. complete. Orders out at night for this Bgde to take Etinehem \u2013 49th & 50th in front,1 Coy 51st mop up. 3 Coys reserve. Zero hour 9.30 p.m. Left with Gen. Herring for new Bgde Hdqrs at Chipilly, Church, then I went forward to the report centre about 1 mile further north.\nStunt went off well last night \u2013 Bns. advancing with practically no opposition. Etinheim was cut off, but when parties went to mop up this morning \u2013 no enemy were found, evidently all having got out on other side of the Somme owing to the failure of the 10th Bgde stunt on the south side. A few prisoners taken by us. 131st American Regt. Brought up in Support. 10th Bgde endeavoured with Armoured cars to march along main road to rear of La Neuville-les-Bray and established a new line \u2013 11th Bgde then to advance towards them, & mop up the area.\nWent round Bns with Brig. during afternoon. Huns, not knowing where our line was, commenced to dig in on Southern edge of Etineheim spur, but were pushed out by 50th & 51st Bns at night 1 a.m. 13th. Huns put up a light attack on 49th takinga couple of prisoners, but losing some of their own men killed. 11th Bgde on southern edge of Somme were very successful in taking Mericourt & Cateaux.\nAttack by 50th & 51st Bns at 1 a.m. to establish posts on northern edge of Somme between Etineheim & Bray was very successful; 51st took 1 Off 170 O'R prisoners losing 5 wounded themselves. Col. Salisbury & Murray Fowler here at Bgde Hdqrs during morning \u2013 very successful. Plenty of dead Huns about in dry-outs &c &c.\nHot again. Enemy shelling heavy during day in Gressaire Wood. 49th relieved by 34th at night.\nCooler day. Round to 34th, & 50th Bns during afternoon Enemy shelling still fairly heavy and continuous in Gressaire Wood \u2013 hit a cordite dump & one of his own shell dumps.\nFine day. Had lovely swim in canal in afternoon. Embussed Brigade at Vaire-sur-Corbie at 7.30 p.m., then proceeded by car with General Herring & Capt Louch to our new Brigade Hdqrs at Harbonnieres arriving at 10 p.m. where we relieved the 132nd American Regt in Reserve, thereby rejoining the 4th Aus. Div. again.\nFine day. Lovely moonlight night. Enemy sending heavies into Harbonnieres duringmorning and Vauvillers during afternoon. Moved from dug-out into nice cosy room upstairs \u2013 am alright as long as Fritz doesn't get a direct hit with a bomb or shell on to it.\nBig dump of brass-work &c &c in Rosieres left there by Germans \u2013 they had ransacked the whole of the villages around about & had candle sticks & all sorts in the dump: \u2013 they lost the lot after stealing it all. Very hot day. Good news through of attack by French West of Soissons on a kilo front \u2013 10,000 prisoners several guns taken; also news of \"push\" by 3rd Army near Archy-le-grand.\nGood news through of attacks by 1st Div. & 32nd British Div. this morning \u2013 1700 prisoners taken. 3rd Corps going well near Bapaume French doing O.K. West of Soissons \u2013 Where is Fritz now? Heavy bombardment North & South of us during day. Rumour says Bapaume practically ours. Enemy planes over bombing at night.\nLight rain occasionally during day & night. Cold fairly bad. Good news through of advance by us to near Flers, Bapaume, Deville Wood &c.\nChangeable day, but no rain. Rufus Rudall in bed ill. Tom Lontit to be T\/Lt. Col.& to Command 45th Bn. Our advance excellent \u2013 Bapaume, Bullecourt, Noreuil &c ours now; Rufus Rudall to Casualty Clearing Stn. re illness \u2013 returned in afternoon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"I will look into the suggested 1DMark iv of ebiggs. \"\nAlthough without doubt this camera will do everything you ask it to, your last post causes me concern.\nA 1D Mk IV in no way shape or form resembles a Rebel. There will be nothing you are familar with. It is a totally different approch. The 1D Mk IV is a full blown professional camera all the rest especially Rebels have the consumers in mind and not a pros needs.\nFor instance there is no \"green square\" and there are no dials. Most settings require two buttons to prevent acidental changes.\nBut the 1D Mk IV is worth the effort just keep in mind you will need to learn it. Until the introduction of the 1Dx, the Mk IV was the fastest DSLR camera in the world. It is still in second place.\nYou didn't expect her to take you up on it, eh?\nWell, it seems my perfect solution would be to buy the lens L lens I want and use it on the little Rebel that I know and get the 7d Mark II or perhaps the 1D Mark IV (the word used scares me, but i wouldn't pay the kind of money a new one would cost) to learn...We are fortunate here where I live to have many wonderful camera experts who teach at various continuous education coarses and the like; they are very helpful in shortening any learning curve.\nHey Bob: I must be Superman then. I shoot a 1Ds2 with 28-300 L IS and 1D4 with 100-400 L IS for 6 hour stints at most of the radio control events I cover. Been shooting that combo or one similar for the last 8 years. I have also been known to use a 3rd body (7D, or Nikon D7100) & an ultrawide for some up close 3 D aerobatics that are done just a few feet from the pilot stations.\nMy seven year old grandson uses mine!\nAnd I almost always, 100% of the time, (when I am shooting ) have the Mk IV and the Ds Mk III on me.\nAnything made by the hand of Man can and will fail but buying a used Mk IV should be of little concern. These are extremely durabile. Plus any part can be repaired by Canon. One nice benefit with a camera at this level is, it is worth fixing one. On quite a few other camera models the cost of repair exceeds the value of the camera.\nIf you do decide to go ths route beware it is heavy as it is all metal. There are no light weight plastic parts. It is expensive even used at approx $2500+\/- as we speak.\nAn \"L\" lens is a good choice as it will transfer to a Mk IV if you do decide that is the route you choose. An \"L\" lens and a Mk IV is about as good as it gets in photogtaphy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eve has a new home, a new face, and a new name\u2014but no memories of her past. She's been told that she's in a witness protection program. That she escaped a dangerous magic-wielding serial killer who still hunts her. The only thing she knows for sure is that there is something horrifying in her memories the people hiding her want to access\u2014and there is nothing they won't say\u2014or do\u2014to her to get her to remember.\nWhy did I read this book: I loved the sound of this novel plus I had an extremely good experience with the author's previous book, Vessel which was a favourite book of 2012.\nEve doesn't recognise the face when she looks in the mirror, she has no true memory of her past and can barely function in the present. Everything \u2013 from walking and talking to understanding how the world works is all new to her. She's been told she has lost her memory. She's been told she is in a witness protection program for paranormal creatures, hiding from a creepy serial killer who is out to get her. She's been told she is the key to find him and to stop his killing spree. She's been told she needs to remember before it is too late and more young kids disappear. She's been told she can trust the people who are helping her even if they look at her with distaste and mistrust.\nThere are certain things she knows though. She knows she has undergone several reconstructive surgeries. She knows she can do magic \u2013 she looks at the mirror one day and decides that her eyes were actually green before and just like that, they are changed. She knows that every time she uses her magic, she passes out and has horrifying dreams (or are they visions?) always featuring a carnival tent, a magician, a storyteller and creepy dolls. When she wakes up after those black-outs she realises that days or sometimes even weeks have passed and she has no short-term memory of those moments.\nConjured is a beautifully constructed novel that goes from utterly disorienting to exceptionally horrific as its story progresses. It features an ubber-creepy carnival, a supernatural serial killer and an amnesic narrator. But its true core is a story about agency and identity and what it is like to forge both when there is no memory, no past, no sense of true self to start with.\nIt is more or less divided in two parts: the first is a progressive build-up to the revelations that appear in the second part. The former, a disorienting advance toward the truth about Eve, the latter an affecting horror story unlike anything I have read of late.\nWhat impresses me the most about the novel is Eve as a character and the writing of her narrative. Since everything is from her point of view, we only ever know what Eve knows and she knows very, very little. When she wakes up with no memory, we are as lost as she is, not knowing who to trust, what happened in the past days or weeks. It is not only disorienting but also claustrophobic.\nMore to the point though, I loved how the author took such a gamble with Eve because she is essentially a blank slate narrator. To start with, she has little personality and no agency. And it is very interesting to see the way that the character progresses, not knowing who she is, what she can do, and what happened to her. Which is awesome because I sometimes feel that \"strong female character\" is often compared to kickass and immediately assertive so it is kind of a breath of fresh air to have a character like Eve who is developing her sense of self slowly and who is a quiet, timid character without being any less strong for that. When the second part comes and the deeply cruel, creepy and dark nature of her story is finally revealed, we come to have not only a deep understanding of why Eve is like she is and how important it really is when she finally voices her choice and forges her own sense of self.\nAll of this was superb: from the puzzling narrative to the development of Eve as a character, from her visions and fear to the creepily awesome horror in the latter part.\nMy only real misgiving about the novel comes with its romantic storyline and I confess to be on two minds about it. On the one hand, there is an element of insta-love as Eve has an almost immediate connection to a boy named Zach whom she meets at the library where she is sent to work. I was immediately put off by Zach when as soon as he met Eve she told her point-blank that they could never be friends because he wanted to kiss her. Okay, then.\nOn the other hand, Zach turn out to be a nice boy, who never lies (there are Reasons) and who is completely loyal to Eve. It is yet another breath of fresh air to have the guy be so besotted and awed to the point of being ready to drop everything for the girl \u2013 as abrupt as that turns out to be. The ending though is kind of perfect for them and for this story in the way that it is flawed and even perhaps, questionable.\nUltimately, this book is All About Eve and I really loved it, just as it is.\n\"I want you to feel safe here.\" He did. She could see it in his eyes. And for an instant, she felt as if he'd wrapped her in a cocoon and nothing could hurt her. But then she remembered he wasn't staying. She pushed the bedroom door open and entered.\nInside the bedroom, half of her expected a rush of familiarity to fold around her like a homemade quilt. But of course, it didn't. She studied the room: a bed with a checkered blanket and one flat pillow, a wooden dresser, a tiny desk with a chair. Eve closed the door and then sank down on the bed. Hugging her knees to her chest, she stared at the wall. The wallpaper had a swirl of leaves with birds perched on branches and caught mid-swoop in patches of blue. It was a nice bedroom, even if it didn't feel like hers.\nShe wondered how she even knew this was a bedroom when she didn't remember ever having one. She'd known what a car was too, though the seat belt had felt unfamiliar. She could recognize a few kinds of birds. For example, she knew that these painted ones on the walls were sparrows and the live one outside had been a wren. She didn't know how she knew that. Perhaps Malcolm had told her in one of her lessons.\nOr maybe it was a memory, forcing its way to the surface of her mind. But the sparrows she remembered flew. She pictured their bodies, black against a blindingly blue sky. She didn't know where that sky was or when she had seen it. The birds had flown free.\nEve raised her hand toward the birds on the wall. \"Fly,\" she whispered.\nThe birds detached from the wall.\nThe air filled with rustling and crinkling as the paper birds fluttered their delicate wings. At first they trembled, but then they gained strength. Circling the room, they rose higher toward the ceiling. They spiraled up and around Eve's head. She reached her arms up, and the birds brushed past her fingers. She felt their paper feathers, and she smiled.\nThen she heard a rushing like a flood of water, and a familiar blackness filled her eyes.\nAdditional thoughts: The author wrote a piece on Inspirations and Influences for us last week and we are also hosting a giveaway of the book HERE.\nYay!!! I'm glad to see that 8. I ordered this book and am waiting for it to arrive.\nread the book from my school library its awesome!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many congratulations to Oliver Brown from Axbridge who passed his Practical Driving Test first time with just 4 minors on Monday 4th August 2014 at Weston-Super-Mare driving test centre. All the best from your Driving Instructor Andy Skyrme and all the team at Starfish Driving. Great drive. Now be careful on the way down to Cornwall!","meta":{"redpajama_set_name":"RedPajamaC4"}}